Precision physics at High Luminosity LHC and future colliders

INTERNATIONAL DOCTORAL SCHOOL OF THE USC Luigi Bellafronte PhD Thesis Precision physics at High Luminosity LHC and future colliders Santiago de Compostela, 2023 Doctoral Programme in Nuclear and Particles Physics DOCTORAL THESIS PRECISION PHYSICS AT HIGH LUMINOSITY LHC AND FUTURE COLLIDERS Luigi Bellafronte INTERNATIONAL PHD SCHOOL OF THE UNIVERSITY OF SANTIAGO DE COMPOSTELA PHD PROGRAMME IN NUCLEAR AND PARTICLES PHYSICS SANTIAGO DE COMPOSTELA 2023 !DECLARATION+BY+THE+THESIS+AUTHOR+ I!submit!my!thesis,!following!the!appropriate!procedure!to!the!Regula9ons!and!declare!that:!! 1) The!thesis!covers!the!results!of!the!elabora9on!of!my!work.! 2) If!this!is!the!case,!the!thesis!refers!to!the!collabora9ons!within!this!work.! 3) I!confirm!that!the!thesis!does!not!incur!in!any!kind!of!plagiarism!from!other!authors!nor!works! submiDed!by!me!in!order!to!obtain!other!9tles.! 4) The!thesis!is!the!final!version!submiDed!for!its!defence!and!both!the!printed!version!and!the! electronic!version!provide!the!same!content.! And!I!agree!to!submit!the!Documentary!Commitment!of!supervision!in!the!event!that!the!original! version!is!not!deposited!at!the!School.!! In!San3ago+de+Compostela,+25th+August+2023.+ Electronic+signature+ Mr/ Ms. Luigi+Bellafronte Thesis! Title: Precision+physics+at+High+Luminosity+LHC+and+future+colliders SUPERVISOR/TUTOR AUTHORISATION Mr/Ms Pier Paolo Giardino As: Supervisor Thesis title: Precision physics at High Luminosity LHC and future colliders STATES: That this thesis corresponds to the work carried out by Mr/Ms Luigi Bellafronte, under my supervision/tutoring and hereby authorise its presentation, taking into account that it meets all the relevant requirements stated in the Doctoral Studies Regulations of the USC, and as its director/ tutor it does not incur in the causes of abstention established in the 40/2015 Law. In Santiago de Compostela, 24 August 2023 Electronic signature SUPERVISOR/TUTOR AUTHORISATION Mr/Ms Néstor Armesto Pérez As: Tutor Thesis title: Precision physics at High Luminosity LHC and future colliders STATES: That this thesis corresponds to the work carried out by Mr/Ms Luigi Bellafronte, under my supervision/tutoring and hereby authorise its presentation, taking into account that it meets all the relevant requirements stated in the Doctoral Studies Regulations of the USC, and as its director/ tutor it does not incur in the causes of abstention established in the 40/2015 Law. In Santiago de Compostela, 28 August 2023 Electronic signature Resumo O Modelo Est´andar (SM) da f´ısica de part´ıculas ´e o marco te´orico actual que describe, a nivel fundamental, as interacci´ons entre part´ıculas elementais. A construci´on de colisionadores de alta enerx´ıa o Large Hadron Collider(LHC) no CERN, o maior acelerador de part´ıculas constru´ıdo ata o de agora, permitiu que os cient´ıficos probasen os seus modelos a escalas de enerx´ıa cada vez maiores. Nos ´ultimos anos, o LHC centrou cada vez m´ais a s´ua atenci´on nas medidas de precisi´on dos procesos do SM, xa que xogan un papel crucial na exploraci´on da nova f´ısica. A ´enfase na f´ısica de precisi´on tam´en motivou o desenvolvemento da actualizaci´on de alta luminosidade (HL) do LHC e a proposta de novas m´aquinas de alta precisi´on, como o FCC-ee. Neste contexto, o esforzo te´orico debe igualar o traballo experimental, porque as medidas precisas requiren dunha precisi´on igual nas predici´ons do SM. Este argumento tam´en se aplica ´as teor´ıas m´ais al´a do Modelo Est´andar (BSM) e v´olvese especialmente relevante no caso das Teor´ıas de Campo Efetivas (EFTs), onde as correcci´ons de orde superior poden ter un impacto significativo nos l´ımites da nova f´ısica. O enfoque principal deste tese ´e o c´alculo preciso dos procesos no SM e no marco da Teor´ıa Efetiva do Modelo Est´andar (SMEFT), inclu´ındo gg-HH, produci´on de Drell Yan qq-ll e observables electrofebles. Finalmente, o obxectivo ´e facer predici´ons precisas que poidan ser probadas nos colisionadores do futuro, como o HL LHC e o FCC-ee. O estudo desta ´area ´e moi importante e proporciona informaci´on crucial sobre a estrutura da nova f´ısica m´ais al´a do SM. 11 12 Resumo Resumen El Modelo Est´andar (SM) de la f´ısica de part´ıculas es el marco te´orico actualmente establecido que describe, a nivel fundamental, las interacciones entre las part´ıculas elementales. La construcci´on de colisionadores de alta energ´ıa como el Large Hadron Collider (LHC) en el CERN, el acelerador de part´ıculas m´as grande jam´as construido, ha permitido a los cient´ıficos probar sus modelos a escalas de energ´ıa cada vez m´as altas. En los ´ultimos a˜nos, el LHC ha centrado cada vez m´as su atenci´on en las mediciones precisas de los procesos del SM, ya que desempe˜nan un papel crucial en la exploraci´on de la nueva f´ısica. El ´enfasis en la f´ısica de precisi´on tambi´en ha motivado el desarrollo de la actualizaci´on de alta luminosidad (HL) del LHC y la propuesta de nuevas m´aquinas de alta precisi´on, como el FCC-ee. En este contexto, el esfuerzo te´orico debe igualar el trabajo experimental, porque las mediciones precisas requieren una precisi´on igual en las predicciones del SM. Este argumento tambi´en se aplica a las teor´ıas m´as all´a del Modelo Est´andar (BSM) y se vuelve especialmente relevante en el caso de las Teor´ıas de Campo Efectivas (EFT), donde las correcciones de orden superior pueden tener un impacto significativo en los l´ımites de la nueva f´ısica. El enfoque principal de este tesis es el c´alculo preciso de los procesos en el SM y en el marco de la Teor´ıa Efectiva del Modelo Est´andar (SMEFT), incluyendo gg-HH, producci´on de Drell Yan qq-ll y observables electrod´ebiles. Finalmente, el objetivo es hacer predicciones precisas que puedan ser probadas en futuros colisionadores, como el HL LHC y el FCC-ee. El estudio de esta ´area es muy importante y proporciona informaci´on crucial sobre la estructura de la nueva f´ısica m´as all´a del SM. 13 14 Resumen Summary The Standard Model (SM) of particle physics is the currently established theoretical framework which describes, at a fundamental level, interactions among elementary particles. The construction of high-energy colliders such as the Large Hadron Collider (LHC) at CERN, the largest particle accelerator ever built, allowed scientists to test their models at increasingly high energy scales. In recent years, the LHC has increasingly focused its attention on the precise measurements of SM processes, as they play a crucial role in the exploration of new physics. The emphasis on precision physics has also motivated the development of the high luminosity (HL) LHC upgrade, and the proposal of new high precision machines such as the FCC-ee. In this context, the theoretical effort has to match the experimental work, because precise measurements require an equal precision of the SM predictions. This argument applies also to theories Beyond the Standard Model (BSM), and it becomes particularly relevant in the case of Effective Field Theories (EFTs), where higher-order corrections can significantly impact the bounds on new physics. The main focus of this thesis is the precision calculation of processes in the SM and Standard Model Effective Field Theory (SMEFT) framework, including gg →HH, Drell Yan production qq →ll, and electroweak observables. Finally, the ultimate goal is to make accurate predictions that can be tested on future colliders, such as HL LHC and Fcc-ee. The study of this area is very important, and it gives crucial information on the structure of new physics beyond the SM. 15 16 Summary Objectives and Methodology The main goal of this thesis is to test physical phenomena at LHC and future colliders by providing accurate calculations of sensitive observables. Historically, precise calculations of observables have played a crucial role in our search to understand the Nature. Indeed, they allowed us to rigorously test the theories formulated to comprehend the universe. For this reason, in this thesis we calculated the perturbative corrections to SM and SMEFT processes that are phenomenologically relevant for physics at the LHC and future colliders. These include: •The NLO correction to the production of a pair of Hbosons and HZ bosons from gluon fusion. The calculation of these processes is extremely important for the experimental measurement of the Higgs properties and it used to measure the Higgs width. In particular, the double higgs production is significantly relevant because of the triple Higgs coupling involved in the calculation. •The calculation of EWPOs at the next-to-leading order (NLO) QCD and electroweak expansions of the SMEFT with an arbitrary flavor structure for the fermion operators. Numerical NLO SMEFT fits to EWPOs are expected to have a strong dependence on the assumed flavor structures and we will check this using various popular assumptions for flavor symmetries. •Drell Yan production in the SM EFT at NLO; this process is very important to detect new physics signals and precise theoretical predictions in the effective field theory are crucial. The effects of four fermions operators on this amplitude is a new challenge and it will give us essential knowledge on the structure of the BSM physics, in particular on the flavour structure. All the required the calculations are performed analytically. Indeed while there are purely numerical methods that could be used, hybrid numerical/analytical techniques are generally more flexible and easier to adapt to broader uses, such as EFTs, than the purely numerical ones. Furthermore, they are computationally less intensive, making them better suited to build efficient Monte Carlo generators. For these reasons analytical calculations are more convenient to our aims. 17 18 Objectives and Methodology Methodology As explained, the main objective of this thesis is the study of precision physics in the SM and its extensions. This involve the application of tools for the precise calculation of processes relevant for the phenomenology of the LHC and future colliders. To achieve our goals we use public and private codes implemented in the software Mathematica. The relevant amplitudes for the scattering or the decay in analysis are generated with FeynArts [1] and we use Feyncalc, LiteRed and Fire [2,3,4] to reduce the amplitudes into a combination of Master Integrals (MI). Feynman integrals are calculated through standard methods when possible and through the energy expansions techniques when necessary. At one loop level we use the Passarino-Veltman functions as basis for the MI and the analytical evaluations is done through FeynHelpers [5]. We use the Pad´e Approximants to improve the convergence of energy expansions to allow us to extend the validity of these techniques to regions of the phase space where these approaches should otherwise fail. This method is applied to physical processes such as gg →HZ and gg →HH in order to improve the accuracy of the results obtained in [6,7] and allowing us to match with the results of [8]. In this way we are able to make a prediction at every point of the parameter space with high precision. The problems arising by the presence of γ5and several Dirac nmatrices are treated with Kreimer’s regularization scheme [9], a powerful tool that preserves the chiral symmetry of the Standard Model in all bare perturbative calculations. All the SMEFT calculations are done in the Warsaw basis with the FeynArts model file obtained from [10]. Finally, we use plots and bar diagrams to present our phenomenological results. Chapter 1 Introduction The investigation of the natural phenomena has been one of the driving forces behind scientific advancement. Regarding particle physics, this pursuit has led to the development of the Standard Model (SM), an outstanding successful framework that describes and classifies the elementary particles and their interactions. This culminated with the discovery of the Higgs boson in 2012 [11,12], marking a further essential step towards a satisfactory understanding of the mechanism underlying the electroweak-symmetry breaking (EWSB). Nevertheless, even with its triumphs, the SM leaves numerous theoretical and experimental questions unanswered. Examples of the latter include neutrino oscillations and dark matter. From a theoretical point of view, we can mention the hierarchy of the fermion masses, the mystery of the origin of the Higgs potential and the inclusion of gravity (beyond the classical level ) in the SM framework. In this context, particle colliders have played a crucial role in shaping our understanding of the fundamental laws of the universe. The construction of high-energy colliders such as Tevatron, LEP and the Large Hadron Collider (LHC) at CERN, the largest particle accelerator ever built, allowed scientists to test their models at increasingly high energy scales driving discoveries and confirming theories. After the discovery of the Higgs boson, none of the measured observables indicated clearly the energy scale of physics beyond the Standard Model (BSM). Indeed, only events predicted by the SM have been seemingly detected at particle colliders. In recent years the precise measurement of SM processes has taken a prominent role as a fundamental instrument in the search of new physics. Anomalies in precision measurements, like deviations from expected values, can be an indirect indication of new phenomena. Indeed, through off-shell and loop effects, we can observe the influence of new particles without directly producing them. In general, the phenomena we are looking for have very low probability of occurring, making the detection process demanding. This aspect is particularly relevant in precision physics, since the measurement of small variations from the expected predictions requires large amount of data. One of the central motivations behind the planned high luminosity (HL) LHC upgrade [13] is the development of the capabilities of the LHC 19 20 CHAPTER 1. INTRODUCTION as a precision machine. Increasing luminosity facilitates the generation of more data, which not only enables a more comprehensive examination of established mechanisms but, as already mentioned, opens also the door to the observation of unexpected new phenomena. For instance, the upcoming HL LHC is projected to generate a minimum of 15 million Higgs bosons annually, a significant increase compared to the about three million collected during the LHC operation in 2017 [14]. This enhancement will allow us to put more stringent constraints on many SM parameters. In this context, gluon fusion, which is already the most relevant production mechanism [15,16] for Higgs physics, will be even more significant. It plays an important role in double-Higgs production, gg →HH, and it is also relevant in the production of a Higgs boson in association with a Zboson, gg →ZH. These two scatterings are very important for Higgs physics. The first one receives contributions from the Higgs trilinear interaction, one of few SM couplings still not measured. The second one is one of main channel for ZH and it allows to study Higgs decays to bottom quarks, a tricky process to observe in hadronic colliders due to QCD background. For this reason, in this thesis we studied the next-to-leading-order (NLO) virtual contributions for these processes. Indeed, the experimental effort needs to be complemented by an equivalent level of theoretical work. This is essential because, in order for the measurements to hold importance, our SM predictions have to match the precision of experimental results. This reasoning also applies to theories beyond the SM, and it is specially important when considering Effective Field Theories (EFT). Indeed, without relying on a specific BSM model, an extension of the SM with higher-dimensional operators provides an important way to describe indirect BSM phenomena. One of the most commonly used method to study new physics effects is the Standard Model Effective Field Theory (SMEFT), which mantains the same field composition and symmetries as the SM. In this context, investigating the properties of massive electroweak gauge bosons (W and Z bosons) has significant potential for indirect searches of BSM physics. We can divide in two classes the processes that have been studied to explore this direction. The first one includes the multi-boson processes at high-energy colliders; the second one involves the fermion scattering processes mediated by s- or t-channel W/Z bosons, also known as the Electroweak Precision Observables (EWPOs). In this thesis we focus on the latter. In the last decades, EWPOs have offered clear indications that the SM accurately describes physics at the EW scale, since they are sensitive to modifications of the gaugeboson–fermion couplings and of the gauge-bosons masses. In recent years the analysis of these measurements has been extended also beyond the the SM framework. Indeed, within the SMEFT approach, the comparison of EWPOs with theoretical predictions can put significant bounds on the coefficients associated with new physics. In this context, at leading order (LO) in SMEFT numerous studies have been performed, extracting limits on the coefficients of dimension-6 operators from global fits on EWPOs and other 2.2. NON-ABELIAN HIGGS MECHANISM: THE ELECTROWEAK INTERACTIONS27 The interactions involving Zand Wwill also involve the masses of these particles, which respect the relation mW=mZcos θW.(2.24) This means that all the effects of Wand Zexchange process, at least at tree-level, can be written in terms of mZ,eand θW. It is also useful to define the parameter: ρ≡m2 W m2 Zcos2θW = 1.(2.25) The equation (2.25) is experimentally confirmed at the per mille level [38]. As we will see below, it is related to the so-called custodial symmetry of the SM. Whatever mechanism one introduces to study beyond the Standard Model (BSM) theories, is highly constrained to reproduce this result. 2.2.1 SM fermion couplings to the gauge bosons Using the covariant derivative in equation (2.22) we can determine the couplings of the vector bosons to the SM fermions, once the quantum number of fermions fields are set. Taking in account the fact that (experimentally) the W boson couple only to left-handed quarks and leptons, we assign the left-handed SM fermions to doublets of SU(2)Wwhile making the right-handed singlet under this group. Once the τ3value has been specified for each fermions, the value Ycan be set using the electric charge operator. The righthanded fermions have τ3= 0, so the hypercharge is equal to Q. In particular, the right-handed neutrino is completely uncharged, under the SM gauge group. Instead, for the left-handed field we have lL=νe e−L , qL=u dL (2.26) and assigning Y=−1 2to the lepton doublet and Y= +1 6to the quark doublet, we reproduce the correct electric charge. Notice that, the left- and the righthanded fermions live in different fundamental representations of the SM gauge group, so an useful way to look at them is to think of these components as distinct particles. The kinetic Lagrangian follows directly from the quantum numbers assignment and is L=i¯ lL/ DlL+i¯qL/ DqL+i¯eR/ DeR+i¯uR/ DuR+i¯ dR/ DdR(2.27) where the covariant derivative is given by equation (2.22). We omitted the right-handed neutrino in the kinetic term since would have zero coupling both to SU(2) and U(1). Expanding the covariant derivative, the Lagrangian becomes L=i¯ lL/ ∂lL+i¯qL/ ∂qL+i¯eR/ DeR+i¯uR/ ∂uR+i¯ dR/ ∂dR+ (2.28) +g(W+ µJµ+ W+W− µJµ− W+ZµJµ Z) + eAµJµ EM (2.29) 28 CHAPTER 2. STANDARD MODEL where Jµ+ W=1 √2(¯νLγµeL+ ¯uLγµdL) (2.30) Jµ− W=1 √2(¯eLγµνL+¯ dLγµuL) (2.31) Jµ Z=1 cos θW¯νLγµ(1 2)νL + ¯eLγµ(−1 2+ sin2θW)eL+ ¯eRγµ(sin2θW)eR + ¯uLγµ(1 2−2 3sin2θW)uL+ ¯uRγµ(−2 3sin2θW)uR +¯ dLγµ(−1 2+1 3sin2θW)dL+¯ dRγµ(1 3sin2θW)dR(2.32) Jµ EM = ¯eγµ(−1)e+ ¯uγµ(+2 3)u+¯ dγµ(−1 3)d(2.33) where in the EM current is written in the Dirac notation since the left and the right handed couplings are equal. Notice that the JWcurrents, mix the up and the down components of each left doublet. 2.3 Mass terms in the SM In writing the Higgs field couplings to the fermions, we will first analyze the quark couplings to the Higgs field ϕand then we will discuss to lepton case. The six left-handed quarks are arranged into three doublet of SU(2)w, as follows qL=ui diL =u dL ,c sL ,t bL(2.34) having hypercharge Y= +1 6. On the other hand, the six right-handed quarks are singlets under SU(2)w: the up-type quarks carry hypercharge Y=2 3, while the down-type ones have Y=−1 3: ui R={uR, cR, tR}, di R={dR, sR, bR}(2.35) Without assuming any flavour conservation (or any other additional symmetry), the most general, renormalizable, gauge invariant term that involves the fields Qi L, ui R, di R and ϕis: Lm=−λij d¯qi Lϕdi R−λij u¯qi L(iσ2ϕ∗)ui R+h.c (2.36) where λij uand λij dare generic 3 ×3 complex matrices. We can simplify the form of this Lagrangian by making a chiral transformation in flavour space. To find the appropriate transformations we notice that by squaring λuwe obtain two Hermitian matrices λuλ† u and λ† uλu. This can be diagonalized by defining the unitary matrices Uuand Wu λuλ† u=UuD2 uU† uλ† uλu=WuD2 uW† u(2.37) 2.3. MASS TERMS IN THE SM 29 where D2 uis a diagonal matrix with positive eigenvalues. Then we can write λu=UuDuW† u(2.38) where Duis the diagonal matrix whose diagonal elements are the positive square-roots of the eigenvalues of D2 u. By similar considerations we can write λdas λd=UdDdW† d(2.39) Notice that the unitary matrices Uu,d and Wu,d can still be multiplied on the right by the same phase matrix:   eiα10 0 0eiα20 0 0 eiα3 (2.40) without modifying the matrices λu,d. We are now ready to make the change of variables with the following transformations ui R=Wij uui Rdi R=Wij ddi R.(2.41) Since the right-handed quarks ui Rand di Rdo not couple to each other through the kinetic terms, the matrices Wuand Wddisappear from the theory. Again, we perform the change of variables: ui L=Uij uui Rdi L=Uij ddi L.(2.42) These transformations eliminate Uuand Udfrom the terms in (2.37) that involve the lower component of the field ϕ. In unitary gauge only these terms survive. If we now define mu=v √2Dumd=v √2Dd,(2.43) in unitary gauge, the equation (2.37) becomes: Lm=−mij u¯ui Luj R( 1 + h v)−mij d¯ di Ldj R( 1 + h v) + h.c (2.44) The transformations carried out in equation (2.42) have some implications in the kinetic Lagrangian, because uLand dLare mixed by the weak interactions. The only relevant terms are the couplings of the left-handed quarks with the gauge bosons W±: ¯uLγµdL→¯uLγµU† uUddL.(2.45) This means that the charge-changing weak interactions mix the three generations of ui L quarks with three generations of di Lquarks, through a unitary matrix: VCKM =U† uUd(2.46) where VCKM is known as the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix. In general, a unitary 3 ×3 matrix can be parametrized using six imaginary phases and 30 CHAPTER 2. STANDARD MODEL three rotation angles. Nevertheless, in our case we can still remove2five of this phases by performing a change of variables on the quark fields: ui L,R →eiαiui L,R di L,R →eiβidi L,R (2.47) The final form of VCKM will contain three rotation angle and one imaginary phase. The latter is responsible of the CP violation of weak interaction. Finally, we discuss the Higgs coupling with the leptons. Like for the quarks sector we have 3 generations of leptons that correspond to three left-handed doublets lL=νi eiL =νe dL ,νµ µL ,ντ τL(2.48) each one with hypercharge Y=−1 2, while there are only three right-handed singlet leptons and they have hypercharge Y=−1 (the charged leptons) ei R={eR, µR, τR}.(2.49) We now write the Yukawa couplings involving the above fields and the field ϕas Lm=−λij e¯ li Lϕei R+h.c. (2.50) Neglecting the right-handed neutrino we have only one coupling. Following the same steps we did for the quark sector, we can write λe=UeDeW† e(2.51) and make the change of variables ei L→Uij eej Lνi L→Uij eνj Lei R→Wij eej R(2.52) In unitary gauge the Lagrangian (2.50) reads Lm=−mij e¯ei Lej R( 1 + h v) + h.c (2.53) where we have defined the mass matrix me=v √2De(2.54) Notice that we have eliminated all the unitary matrices from the theory. In fact, since ei Land νi Ltransform in the same way, the couplings with the gauge bosons W±are invariant under this transformation. The resulting lepton theory does not violate CP symmetry and further preserves the lepton number per each generation. This property is called lepton universality. 2This fact is related to equation (2.40). 2.3. MASS TERMS IN THE SM 31 CKM matrix and Wolfestein parametrization The CKM matrix is a unitary 3 ×3 matrix defined on the quark generation space. It can be parameterized by three mixing angles and the CP-violating KM phase. Of the many possible conventions, a standard choice has become [39]: VCKM =  c12c13 s12c13 s13e−iδ −s12c23 −c12s23s13eiδ c12c23 −s12s23s13eiδ s23c13 s12s23 −c12c23s13eiδ −c12s23 −s12c23s13eiδ c23c13  (2.55) where sij = sin θij,cij = cos θij and δis the phase responsible of the CP-violating phenomena. Notice that θij can be chosen to lie in the first quadrant , so sij,cij ≥0. Since, it is known experimentally that s13 ≪s23 ≪s12 ≪1 (2.56) it is convenient to introduce another parametrization to exhibit this hierarchy; we define the four Wolfenstein parameters [40]: λ=s12 Aλ2=s23 Aλ3(ρ−iη) = s13e−iδ then, to λ3order we the CKM matrix reads: VCKM =  1 0 0 0 1 0 0 0 1 + −λ2 2λ Aλ3(ρ−iη) −λ−λ2 2Aλ2 Aλ3(1 −ρ−iη)−Aλ20 +O(λ4) (2.57) and CP violation can be determined by measuring ρ−iη. 2.3.1 Custodial symmetry Suppose to turn off the gauge couplings in equation (2.12). The field ϕcan be rearranged in a matrix fields Φ as Φ = (iσ2ϕ∗, ϕ) = ϕ∗ 0ϕ+ −ϕ∗ +ϕ0(2.58) and we can rewrite the Lagrangian (2.12): Lh=1 2Tr[(∂µΦ)†∂µΦ] + µ 2Tr[Φ†Φ] −λ 2Tr[(Φ†Φ)2] (2.59) This Lagrangian is invariant under a global symmetry SU(2)L⊗SU(2)R: Φ→LΦR†(2.60) 32 CHAPTER 2. STANDARD MODEL with L∈SU(2)Land R∈SU(2)R. The field Φ, because of the potential form, acquires a non vanishing VEV: ⟨Φ⟩ ∼ v0 0v(2.61) This latter is still invariant under the subgroup SU(2)V⊂SU(2)L⊗SU(2)R, which corresponds to the transformations: Φ→VΦV†(2.62) The global symmetry is spontaneously broken, with a breaking pattern SU(2)L⊗SU(2)R→ SU(2)Vand we have again that three Goldstone bosons emerge from the theory. Now, suppose to gauge the SU(2)Lgroup. The Lagrangian is still invariant under the global SU(2)L⊗SU(2)Rgroup and the Goldstone bosons, by the Higgs mechanism, can be ”eaten” to give mass to the gauge fields. In this case the physical eigenstates would correspond to linear combination of the SU(2)Lgauge fields, and the masses of the vector bosons would be exactly degenerate: m2 W=m2 Z=g2 4v2(2.63) where gis the gauge coupling of SU(2)L. In the SM, this scenario occurs when all the Yukawa couplings are identical and g′→0, resulting in the Wand Zbecoming exactly degenerate. The latter approximate symmetry is called custodial symmetry and it protects the parameter ρfrom quantum corrections, giving the exact relation ρ= 1 at tree-level. Furthermore, assuming that the hypercharge corresponds to the third generator of SU(2)R,T3 R, it also explains why only mZreceives a contribution from the hypercharge coupling g′. Finally, we notice that the top quark mass has an important role on the explicit breaking of the custodial symmetry. Indeed, being the heaviest elementary particle in the SM, the quark top introduces significant quantum corrections to the ρparameter, causing a deviation (a few percent [41,42]) from the LO value ρ= 1. 2.4 Beyond tree level The Lagrangian we just built introduces a number of free parameters that are directly related to the physical observables. Indeed, at the Leading Order (LO) in the perturbation theory, the calculations involve only the simplest Feynman diagrams. The results are typically finite quantities and can be directly interpreted as physical observables. However, when higher-order corrections are taken into account, we have more complicated Feynman diagrams involved and loops of virtual particles come into play. The loops in Feynman diagrams introduce integrals over momenta of the virtual particles 2.4. BEYOND TREE LEVEL 33 and, when evaluated, can lead to divergent terms. An example of these integrals is: λZd4k (2π)4 1 k2+m2∼ ∞,(2.64) where λis a dimensionless coupling. The presence of divergent integrals is an indication that the theory requires further treatment. 2.4.1 Regularization In QFT, at fixed order in the perturbation theory, we can expect a finite number of integrals (divergent or not). Indeed, physical quantities contains sums, products and convolutions of them. However, as mentioned in the previous section, some of the integrals may be divergent, but this is not necessarily a an unsolvable problem. Indeed, if a single integral does not converge the reason may be that we isolated the integral from the rest of the theory. This may happen, for example, when we are considering non observable parameters, that are indeed not physical. In this case the divergence is not a problem, but it is just a matter of re-parametrization. A theory where all the divergences may be consistently removed in a finite number of steps is called renormalizable3. As first step towards the renormalization of a theory, we will briefly describes how to deal with integrals in the form (2.64). The idea is to introduce an artificial prescription to make our integral finite; an example is the introduction of a cut-off Λ: λZ|k|≤Λ d4k (2π)4 1 k2+m2∼Λ2.(2.65) The parameter introduced is called regulator and the prescription for rendering divergent diagrams finite is called regularization. Of course in the limit Λ → ∞ the integral is still divergent, but this trick allow us to postpone this operation until we have calculated quantities that are physically relevant. Indeed the presence of a divergence in a single integral is not an issue if, after a re-parametrization, the physical quantities admit the removal of the regulator, which in this scheme corresponds to the limit Λ → ∞. Several regularization schemes were proposed in the literature, but the one that established itself to be the favourite strategy in theoretical particle physics is the Dimensional Regularization. This method preserves Lorentz and gauge invariance and has the useful characteristics of being applicable also on infrared divergences. The idea of dimensional regularization is very simple: we extend the 4-dimensional domain of definition of the analytical function into D= 4 −2ϵdimensions: λµ4−DZdDk (2π)D 1 k2+m2∼2 4−D∼1 ϵ.(2.66) 3In a series of papers of 1971-1972, G. t’Hooft and M. Veltman proved that the Electroweak Theory is renormalizable [43]. 34 CHAPTER 2. STANDARD MODEL Notice that in Ddimensions, the coupling λhas non–zero mass dimension 4 −D, so we replaced λ→λµ4−D. The new λis dimensionless and µis an arbitrary energy scale. As we can see, here ϵplays the role of the regulator, the limit to consider is ϵ→0 and the divergence manifest itself as a pole in ϵ. 2.4.2 Dirac Algebra In general, to consistently evaluate the integrand in integrals of the form 2.66, we will need a generalisation of the metric and of the Dirac algebra to Ddimensions. Some of the prescribed continuations of the Lorentz and Dirac algebra are very natural: gµνgµν =D, {γµ, γν}= 2gµν 1, γµγµ=1 2gµν{γµ, γν} =gµνgµν =D. (2.67) However other prescriptions are far more subtle and require a more careful explanation. Indeed, since Dimensional regularization was introduced, it was evident that dealing with chiral couplings was a complicated issue. The problem emerges because the usual 4-dimensional γ5, defined as γ5=−iγ0γ1γ2γ3,(2.68) does not have a canonical extension into D dimension. Indeed [43], we cannot expect to preserve the anticommutation relation {γµ, γ5}= 0 (2.69) and, at the same time, handle the traces of the Dirac matrices with the usual mathematical rules. In a nutshell, as pointed out by D. Kramer [44] and ’t Hooft-Veltman [43, 45], one has to decide between keeping the cyclic property of the trace (HVBM scheme) or keeping the anticommutation relation 2.69 of the γ5(Kramer scheme). Clearly, the scheme used has a relevant impact on the the Feynman rules and on the Dirac traces. In the HVBM scheme, the anticommutation rule 2.69 is abandoned and the γ5definition is extended to a −2ϵ-dimensional subspace: {γµ, γ5}= 0 4 dimensions [γµ, γ5] = 0 −2ϵdimensions.(2.70) HVBM scheme provides a natural treatment to handle Dirac traces. However, it has the significant drawback of violate the the chiral symmetry of the SM (hence of Ward Identity 2.4. BEYOND TREE LEVEL 35 ) in all perturbative calculations involving chiral couplings. The solution proposed by Breitenlohner-Maison (BM in the HVBM scheme) involves the restoration of the chiral symmetry as part of the renormalization process. The idea is to introduce additional finite counterterms alongside the usual divergent ones. It is important to note that this restoration of chiral symmetry is not limited to interactions explicitly involving γ5but is expected for all interactions in the SM. Additionally, these finite counterterms need to be calculated beyond O(ϵ0) to consistently restore chiral symmetry at higher perturbative orders. Thus, the renormalization procedure can be very demanding. For this reason, we will use the Kramer scheme to deal with traces of Dirac matrices. Indeed, Kramer scheme does not need the introduction of any counterterm, simplifying substantially the calculation and, as we will see below, this procedure does not break the chiral symmetry of the SM. Nevertheless, it is important to mention that Kramer scheme has not been completely accepted and embraced as a preferable choice to HVBM and there are still doubts about its use beyond one loop [46,47]. However, in this thesis we will use the Kramer scheme only for one loop calculations. 2.4.3 Kramer scheme The rules for handling Dirac matrices in Kramer scheme are[44]: •Anticommutation rules: {γµ, γν}= 2gµν 1, {γµ, γ5}= 0.(2.71) •For the traces of Dirac matrices without γ5: Tr[γµ1γµ2. . . γµ2n−1] = 0, Tr[γµ1γµ2. . . γµ2n] = 4 X σ (−1)sgn(σ)gµi1µj1gµi2µj2. . . gµinµjn. (2.72) •For the traces of Dirac matrices involving γ5we have: Tr[γµ1γµ2. . . γµ2n−1γ5] = 0, Tr[γµ1γµ2. . . γµ2nγ5] = 4iX σ (−1)sgn(σ)ϵµin+1 µin+2 µjn+1 µjn+1 gµi1µj1. . . gµin+2 µin+2 , (2.73) with 1 = i1<··· < in+ 2, ik< jk, where ϵµ1µ2µ3µ4is the 4-dimensional Levi-Civita tensor. 36 CHAPTER 2. STANDARD MODEL •It is forbidden to use cyclic property of the trace when an odd number of γ5is involved. •If there is more than one diagram contributing to a given process, all the traces must be read starting at the same point4. •In the case where an anomalous axial current is involved, the trace of the anomalous graph must be read starting from an axial vector vertex, in order to fulfill the usual convention of conserved vector currents. In the case of multiple axial vector vertices a symmetric choice of the reading prescription must be used. Notice that when one is computing Dirac traces with at most four Dirac gammas and γ5, the Kramer scheme is equivalent to the Naive Dimensional Regularization (NDR). More details about the differences and the analogies between the schemes discussed are explained in [46]. 2.5 Renormalization We know briefly describe the renormalization procedure, following [48]. A practical way to carry out the renormalization is the so-called counterterm approach. The basic idea behind the counterterm approach is to introduce additional terms, known as counterterms, into the Lagrangian of the theory. First step, we have to replace the parameters of the Lagrangian with their bare quantities (in the following denoted with a ”0” subscript), meaning that they cannot be directly interpreted as physical quantities. Then, the bare quantities are separated into two components: a finite renormalized part and a divergent counterterm or renormalization constant. Every bare parameter K0is split as: K0=ZKK= (1 + δK)K.(2.74) Then inserting this transformations into the classical Lagrangian we have L0=L+δL(2.75) where Lis the same as the original Lagrangian but with bare quantities replaced by renormalized ones and δLis the counterterm Lagrangian. This latter groups all the potential UV terms and leads to an additional set of Feynman rules. After imposing the renormalization conditions on the renormalized quantities, we can obtain their numerical values using experimental inputs. The choice of the input observables and the renormalization conditions determine the renormalization scheme. The complexity of the theory together with arbitrariness in the choice of the ‘best’ renormalization scheme can make it difficult to compare and to follow different approaches. In the literature are present many different schemes [49]; depending on the physical process we are considering, it may be convenient to use a renormalization scheme rather than another. 4In the Drell Yan scattering, the fermion chains of the LO will provide us a natural reading point. 3.2. METHOD 43 the Dynkin index. For the double higgs production we have: Aµν HH =Aµν 1F1+Aµν 2F2(3.10) where F1and F2are the form factors associated to the spin-0 and spin-2 projectors, respectively. For the HZ production we have: Aµνρ HZ = 7 X i Pµνρ iAiHZ (3.11) where Piare a basis of orthonormal projectors and AiHZ are the associated scalar form factors. In both the processes, they depend only on scalar quantities, namely the top quark mass mt, the masses of the external particles and the partonic Mandelstam variables. Additionally, taking all momenta to be incoming, we define the partonic Mandelstam variables as ˆs= (p1+p2)2,ˆ t= (p1+p3)2,ˆu= (p2+p3)2,(3.12) and the transverse momentum pTof the final-state particles can be written as p2 T=ˆ tˆu−m2 3m2 4 ˆs.(3.13) As suggested in refs. [29,56], if the amplitude of the process is written in terms of (anti)symmetric form factors with respect to the exchange ˆ t↔ˆu, then it is sufficient to discuss only the forward contribution to the cross section. Therefore, in the following we will always assume that |ˆ t| ≤ |ˆu|and that ˆ t=−1 2ˆs−m2 3−m2 4−qλ(ˆs, m2 3, m2 4)−4ˆs p2 T,(3.14) where λ(a, b, c) = a2+b2+c2−2ab −2ac −2bc is the K¨all´en function. 3.2 Method In general, the degree of difficulty in the evaluation of loop diagrams grows with the number of energy scales present in the diagram. In the case of single-Higgs production the relevant diagrams feature a triangular topology and, consequently, depend upon only two scales, namely the Higgs mass, mH, and the top mass2,mt. In this case, the functional dependence of the result upon the top mass can be expressed in terms of one single variable, m2 H/m2 t. Due to this simplified one-scale situation, exact analytic results for the NLO corrections are available since many years [57,58,59,60]. In the case of processes with two particles in the final state the situation is more complicated. Indeed these processes receive contributions not only from triangle diagrams, 2All the quarks but the top are assumed to be massless. 44 CHAPTER 3. HIGGS PRODUCTION FROM GLUON FUSION that can be calculated adapting the exact analytic results obtained for single-Higgs production, but also from box-topology diagrams. In pair production, gg →HH, the box diagrams depend upon four scales, namely ˆs, ˆ t, mt, mH, where ˆs, ˆ t, and ˆuare the Mandelstam variables which satisfy the condition ˆs+ˆ t+ ˆu= 2 m2 H.(3.15) Concerning associated production, gg →ZH, a fifth energy scale is present, i.e. the mass of the Zvector boson, mZ. Exact analytic results for two-loop box diagrams with several energy scales cannot be derived with the present computational technology. Instead, usually two different strategies are followed in order to evaluate the two-loop box contribution in Higgs production via gluon fusion. a) A fully numerical exact evaluation [61,62,63,64,65]. b) An approximate analytic evaluation that takes advantage of hierarchies among the various energy scales present in the diagrams, in order to reduce the number of scales in the problem. Thus, its validity is restricted to specific regions of the phase space. The method used is based on the expansion of the diagrams in terms of ratios of small energy scales vs. large energy scales, in order to obtain a result that retains an exact dependence upon the large energy scales. Concerning the small ones, in order to simplify further the evaluation, expansions in term of ratios between small energy scales is often used. The former strategy, although accurate, is very demanding from a computational point of view, requiring a high degree of optimization in order to obtain a result in a reasonable, although usually quite long, computer time. Furthermore, this approach is not very flexible with respect to the modification of the input parameters. Strategy b) provides accurate results valid in specific regions of the phase space without requiring heavy computational work, i.e. in a short computer time. Examples of this approach of evaluating the two-loop box contribution are: i) The infinite-top-mass limit [66,67] refined by the inclusion of powers in the large top-mass expansion (LME) [68,69,70,71]. Here, mtis assumed to be the large energy scale while ˆs, ˆ t, mH, and in associated production also mZ, are considered to be the small ones. Thus, the validity of this approach is restricted to phasespace regions where ˆs/(4m2 t)≤1. The advantage of this approximation are the rather simple results, which can be expressed in terms of rational functions and logarithms of the kind log(m2 t/ˆs). ii) The evaluation via an expansion in the transverse momentun, pT, of the final-state particles [29,56]. Here, ˆsand mtare assumed to be the large energy scale while mH, mZand pT, that can be traded for ˆ t, are considered to be the small ones. The validity of this approach is restricted to phase-space regions where |ˆ t|/(4m2 t)≲1. The analytical complexity of this approach is higher than in i), as generalised polylogarithms and two elliptic integrals occur in the final results. The evaluation of the latter can be easily performed using the results of Ref. [72]. 3.2. METHOD 45 iii) The evaluation via a high-energy (HE) expansion [30,73,74]. Here ˆs, ˆ tare assumed to be the large energy scale while mt, mHand mZ, with mt≫mH, mZ, are considered to be the small ones. The validity of this approach is restricted to phase-space regions where |ˆ t|/(4m2 t)≳1. The HE expansion leads to analytical results that can be expressed in terms of harmonic polylogarithms. iv) The evaluation via an expansion in terms of small external masses [75,76,77]. Here ˆs, ˆ t, mtare assumed to be the large energy scale while mHand mZare considered to be the small ones. This approach basically covers the entire phase space of the considered processes. However, since the reduction of scales in this approach is minimal, one ends up with the evaluation of Master Integrals (MIs) that are much more complicated than those appearing in the i)–iii) cases. As a consequence the evaluation of the box contribution in any point of the phase space requires a longer computer time than in the approaches i)–iii). As an alternative approach, refs. [78,79] proposed to reconstruct the full result from its LME version, supplemented by the non-analytic part of the diagrams near the top threshold, via a conformal mapping and Pad´e approximants. In [80] we propose an alternative way to derive the full top-mass dependence in Higgs production via gluon fusion, based on the merging of the pTexpansion in ii) with the HE expansion in iii) that individually are valid in complementary regions of the phase space. Since the numerical evaluations of the two expansions are quite fast from a computational point of view, our proposal allows a fast evaluation of the virtual corrections to Higgs production via gluon fusion that is accurate in the entire phase space. The key point of our analysis is to extend the fixed-order results both in the pT expansion [29,56] and in the HE expansion [73,74] up to or beyond their border of validity, i.e. ˆ t≃4m2 t, in order to merge the two analytic approximations. This is done by constructing a [1/1] Pad´e approximant for the pT-result and a [6/6] Pad´e approximant for the HE-result. We point out that the extension of the HE expansion via Pad´e approximants has been already considered in refs. [73,74]. 3.2.1 Pade approximant In the forward regime, the validity of both the pTand HE expansions is limited by the condition |ˆ t| ≃ 4m2 t,(3.16) i.e. for any fixed value of ˆs, the pTexpansion provides reliable results when |ˆ t|≲4m2 t while the HE expansion is accurate for |ˆ t|≳4m2 t, if the fixed ˆs > 4m2 t. However, we find that in the vicinity of the point |ˆ t|= 4m2 tthe fixed-order results in the pT expansion and in the HE expansion are both divergent (see fig. 3.2). As a consequence, a straightforward combination of the pT-expanded and the HE-expanded results cannot allow for an accurate description of the above region, and this fact prevents a full coverage 46 CHAPTER 3. HIGGS PRODUCTION FROM GLUON FUSION of the phase space. We point out that this situation does not change substantially when higher orders in both the expansions are computed. Alternatively, the convergence of the expanded results can be improved by considering the respective Pad´e approximants. Indeed, starting from a given Taylor expansion of an exact function f(x) around x= 0 up to the first rterms f(x)≃ r−1 X k=0 ckxk,(3.17) it is possible to construct the associated Pad´e approximant, defined as [m/n](x) = p0+p1x+···+pmxm 1 + q1x+. . . qnxn,(3.18) provided that m+n+ 1 = r. Specifically, by Taylor-expanding the r.h.s. of eq. (3.18), the {pi, qj}coefficients of the Pad´e approximant can be written in terms of the ckones known from eq. (3.17), by solving a system of linear equations. Usually, [m/n] Pad´e approximants such that m=ngive the best improvement in the convergence of the original Taylor expansion, and we consider only these combinations in our study. In the pT-expanded results, at NLO, only the first three terms in eq. (3.17) are known and therefore we are limited to construct a [1/1] Pad´e approximant (we will refer to this as the pT-Pad´e). Instead the availability of many terms in the HE-expansion results allows to consider several [n/n] approximants (defined as HE-Pad´e). When calculating the pT-Pad´e, care is to be taken in the treatment of the expansion parameters. As discussed in refs.[29,56], not only the pTbut also the masses of the external particles are understood as small parameters. Since these are all treated on the same footing with respect to the large scales set by ˆsand mt, we can write the general expression for a pT-expanded form factor Fin the amplitude in terms of a scaling parameter x F(x) = 2 X N=0 xNX i+j+k=N cijk (p2 T)i(m2 3)j(∆m)k≡ 2 X N=0 xNcN(3.19) where m3is interpreted as mHand mZfor gg →HH and gg →ZH, respectively, and ∆m= (m2 4−m2 3)/2 is included only for the ZH case (see ref. [56]). Starting from eq. (3.19) we can then obtain the corresponding [1/1] Pad´e approximant with respect to the limit x→0 [1/1](x) = p0+p1x 1 + q1x,(3.20) with p0=c0p1=c1−c0c2 c1 q1=−c2 c1 , and subsequently set x= 1 in eq. (3.20). We want to clarify a possible source of ambiguity concerning the limit of validity of the pTexpansion. Indeed, while in the previous works we suggested that this expansion 3.2. METHOD 47 is valid for p2 T≲4m2 t, as the comparison at LO between the pT-expanded and exact result seems to indicate, in this paper we follow a more conservative approach and we consider as limit of validity for the pTexpansion |ˆ t|≲4m2 t. Additionally, we checked that the same complementarity for the pTand HE expansions can be observed when choosing p2 T= 4m2 tas limit of validity. We now discuss the procedure adopted to construct the Pad´e approximants from the HE expansion. Following the prescription of ref. [81] (see also [82,83]), we initially arrange the various orders F(i)of the HE expansion for a given form factor as follows F(x) = F(0) + L X l=1 F(2l−1)m(2l−1) t+F(2l)m(2l) txl= L X l=0 dlxl,(3.21) where orders related to odd powers of mtare grouped with the orders related to the next even power. Then, we construct [n/n] approximants in xwith 2n=Lfrom eq. (3.21) using the analytic expressions available in [84,85], and setting x= 1. We remark that our Pad´e approximants are obtained in a fully symbolic way, whereas in refs. [81,82] all the kinematical quantities are fixed to the respective numerical values before the Pad´es are constructed in x. Furthermore, in comparison to refs.[81,74], we only studied [n/n] HE-Pad´es up to n= 6. In those references Pad´es with n > 6 were also considered in order to extrapolate the results in the region |ˆ t|<4m2 t, for a fixed ˆs, and characterize the relative uncertainties of different [m/n] Pad´es. In our case, because the region |ˆ t|<4m2 t, is more accurately described by the results of the pTexpansion, we find that a [6/6] HE- Pad´e is more than enough to perform the merging with the pT-result and, at the same time, to describe accurately the high-energy region. The pT-Pad´e and the HE-Pad´e extend the range of validity of each expansion beyond its limit. As discussed in the next section, the pTand the HE Pad´es are accurate enough to bridge the gap around the phase-space region |ˆ t| ≃ 4m2 t. Then, an accurate approximation of the exact result for any phase-space point (ˆs, ˆ t) can be obtained by choosing as switching point between the Pad´e-improved expansions any point in the region |ˆ t| ∼ 4m2 t. For simplicity we choose to use the pT-Pad´e when |ˆ t|<4m2 tand the HE-Pad´e when |ˆ t| ≥ 4m2 t, for any fixed value of ˆs. We recall that in our discussion we just consider the forward region |ˆ t| ≤ |ˆu|, while the result in the complementary phase-space region is obtained using the symmetry of our form factors under ˆ t↔ˆu. Noticing that, when |ˆ t| ≤ |ˆu|, the maximum absolute value of ˆ tas a function of ˆsis given by |ˆ t|max = 1/2(ˆs−m2 3−m2 4) our choice corresponds to using the pT-Pad´e up to the partonic energy ˆsc= 8m2 t+m2 3+m2 4. In this energy region (√ˆsc≃500 GeV for gg →HH and gg →ZH) at the LHC more than 2/3 of the hadronic cross section is concentrated. 48 CHAPTER 3. HIGGS PRODUCTION FROM GLUON FUSION 3.3 The pTand HE expansions vs the exact results at LO In this section we assess the reliability of our merging procedure by studying how well the combination of the pT-Pad´e and the HE-Pad´e can reproduce the exact LO results for HH and ZH production via gluon fusion. For the sake of simplicity, we discuss in detail only the gg →HH process, but we verified that similar conclusions can be drawn for gg →ZH. We recall that the amplitude for gg →HH can be expressed as Aµν =Gµ √2 αS(µR) 2πδabTFˆs[Aµν 1F1+Aµν 2F2],(3.22) and that both triangle and box diagrams contribute to F1 F1=F△ 3m2 H ˆs−m2 H +F□,(3.23) whereas the F2form factor receives contribution only from boxes. Our goal is to improve the evaluation of the box contributions, therefore we focus on the discussion of F□and F2. The LO results for these form factors, denoted as FLO □and FLO 2, are shown in fig. 3.2, for fixed values of the partonic center-of-mass energy. Only large values of ˆsare shown in fig. 3.2 because for small ˆsvalues the pT-expanded results are very accurate [29]. The pT-expanded and HE-expanded results are represented by the blue and purple solid lines, respectively, and they deviate from the exact result, shown as a solid black line, at |ˆ t|/4m2 t≃1, as anticipated in the previous section. The light blue dashed line stands for the [1/1] pT-Pad´e, while the pink dashed line represents the [6/6] HE-Pad´e. One can see that the Pad´e results show an improved convergence with respect to the fixed-order expansions. The bottom part of the plots in fig. 3.2 shows the ratio of the expanded and Pad´e results to the exact one. Indeed, fig. 3.2(a,b) shows that in the case of FLO □for |ˆ t|/4m2 t= 1 the differences of the Pad´e results with respect to the exact prediction are negligible. For the F2form factor, whose contribution to the cross-section is much smaller than the one of the F1form factor, the difference is always below 5%, see fig. 3.2(c,d). We notice that, when comparing the accuracies of the Pad´e approximants, larger discrepancies can be attributed to the pT-Pad´e. Indeed, being the latter a [1/1] Pad´e, it is expected to be a less refined approximation than the [6/6] HE-Pad´e. Still, in the case of FLO □the differences between the two Pad´e near |ˆ t|/4m2 t= 1 are negligible. We also notice that, as ˆsincreases, larger values of |ˆ t|are allowed by the kinematics, and the relative importance of the HE expansion increases. The improvement in convergence provided by the Pad´e approximants is such that the merging of the two results discussed in the previous section can reproduce the exact prediction with good accuracy for every value of ˆ t, for any ˆs. While we refrain from showing more examples here, we note that we studied the behaviour of all the box contributions to gg →HH and gg →ZH at several values of ˆs. We explicitly checked that, among the various possibilites, a [6/6] HE-Pad´e is more than enough for an accurate merging. Furthermore, we observed that the value |ˆ t|= 4m2 tis a good choice as a merging 3.3. THE pTAND HE EXPANSIONS VS THE EXACT RESULTS AT LO 49 0.9 1.0 1.1 0 0.5 1 1.5 2 2.5 3 0.3 0.4 0.5 0.6 0.7 √ˆs=0.9 TeV ratio to full −ˆ t/(4 m2 t) |FLO | full PTexp PTexp [1/1] HE HE [6/6] (a) 0.9 1.0 1.1 0 1 2 3 4 5 0 0.1 0.2 √ˆs=2.0 TeV ratio to full −ˆ t/(4 m2 t) |FLO | full PTexp PTexp [1/1] HE HE [6/6] (b) 0.9 1.0 1.1 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 √ˆs=0.9 TeV ratio to full −ˆ t/(4 m2 t) |FLO 2| full PTexp PTexp [1/1] HE HE [6/6] (c) 0.9 1.0 1.1 0 1 2 3 4 5 0 0.1 0.2 √ˆs=2.0 TeV ratio to full −ˆ t/(4 m2 t) |FLO 2| full PTexp PTexp [1/1] HE HE [6/6] (d) Figure 3.2: Modulus of the box form factors contributing to gg →HH at LO, for a fixed value of (a,c) √ˆs= 0.9 TeV and (b,d) √ˆs= 2 TeV. In the upper part of each plot, the exact prediction (solid black line) is shown together with the pTand HE expansions (solid blue and purple lines, respectively) and with the [1/1] pTand [6/6] HE Pad´e approximants (dashed light blue and pink lines, respectively). The bottom part of each plot shows the ratio of the above results to the exact prediction. 50 CHAPTER 3. HIGGS PRODUCTION FROM GLUON FUSION 0.95 1.0 1.05 400 600 800 1000 1200 1400 1600 1800 2000 0.0 0.5 gg →HH ratio to full MHH [GeV] ˆσ(0) [fb] full small pTand HE expansion (a) 0.95 1.0 1.05 400 600 800 1000 1200 1400 1600 1800 2000 0.0 0.5 1.0 1.5 2.0 gg →ZH ratio to full MZH [GeV] ˆσ(0) [fb] full small pTand HE expansion (b) Figure 3.3: Partonic cross section at LO for (a) gg →HH and (b) gg →ZH. The upper part of each plot shows the exact prediction (solid line) together with the merging of the pTand HE Pad´e approximants (dashed line). The bottom part of each plot shows the ratio of the merged result to the exact prediction. point for the pTand HE Pad´e approximants. The high level of accuracy of our merging method can be observed in fig. 3.3, where the partonic cross section at LO is shown for gg →HH and gg →ZH. One can see that deviations of the combination of the pT- and HE-Pad´e with respect to the exact prediction never exceed 1%. 3.4 Merging the pTand HE expansions at NLO In the previous section we showed that the merging of the pT- and HE-Pad´e can accurately reproduce the exact LO prediciton. In this section we present the merging of the NLO pT-expanded and HE-expanded results improved by the respective Pad´e approximants. In fig. 3.4 the NLO contributions to F□and F2are shown3. The relative behaviour of the various approximations is analogous to what we observed at LO. For low values of |ˆ t|the fixed-order and Pad´e-improved pT-expanded results agree well. Increasing the value of |ˆ t|up to the merging region, |ˆ t| ∼ 4m2 t, the pT-Pad´e becomes close to the Pad´eimproved HE expansion. For values above |ˆ t|= 4m2 tthe pTand HE Pad´e approximants show small deviations as expected. The NLO study shows the same qualitative behaviour as the LO one. This makes us confident that the proposed merging procedure works well also at NLO. 3FNLO □and FNLO 2are the form factors as defined in eq. (3.22) but do not contain the double triangle diagrams that can be expressed in terms of products of one-loop integrals and as such are computed analytically in exact top-mass dependence [70]. 3.4. MERGING THE pTAND HE EXPANSIONS AT NLO 51 1.6 1.8 2 2.2 2.4 0 0.5 1 1.5 2 2.5 3 √ˆs=0.9 TeV |FNLO | −ˆ t/(4 m2 t) PTexp PTexp [1/1] HE HE [6/6] (a) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0 1 2 3 4 5 6 7 √ˆs=2.0 TeV |FNLO | −ˆ t/(4 m2 t) PTexp PTexp [1/1] HE HE [6/6] (b) 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 √ˆs=0.9 TeV |FNLO 2| −ˆ t/(4 m2 t) PTexp PTexp [1/1] HE HE [6/6] (c) 0.2 0.4 0.6 0.8 1 1.2 1.4 0 1 2 3 4 5 6 7 √ˆs=2.0 TeV |FNLO 2| −ˆ t/(4 m2 t) PTexp PTexp [1/1] HE HE [6/6] (d) Figure 3.4: Modulus of the box form factors contributing to gg →HH at NLO, for a fixed value of (a,c) √ˆs= 0.9 TeV and (b,d) √ˆs= 2 TeV. The pTand HE expansions are shown as solid blue and purple lines, respectively, while the [1,1] pT- and [6,6] HE-Pad´e are shown as dashed light blue and pink lines, respectively. 52 CHAPTER 3. HIGGS PRODUCTION FROM GLUON FUSION MHH [GeV] ˆ t[GeV2]VPade fin Vgrid fin 280.9 −7.783 ·1039.548 ·10−69.410 ·10−6 411.4 −6.627 ·1044.520 ·10−44.510 ·10−4 586.96 −6.925 ·1044.930 ·10−44.943 ·10−4 716.55 −1.816 ·1054.430 ·10−44.298 ·10−4 1048.93 −2.133 ·1052.952 ·10−43.104 ·10−4 1855.32 −1.678 ·1062.497 ·10−42.498 ·10−4 Table 3.1: Comparison of various numerical values of Vfin taken from the grid of ref. [86] with our Pad´e construction. We now compare our evaluation of the virtual corrections for the di-Higgs production process4with the numerical result provided as a grid in ref. [86]. This reference summarizes the work of ref. [81], where the numerical calculation in exact top-mass dependence of ref. [62] was supplemented by the result in the HE expansion of ref. [73]. The comparison is done on the quantity ∆ˆσvirt =Zˆ t+ ˆ t− αs 32π2 1 ˆs2Vfindˆ t, (3.24) where the finite part of the virtual corrections Vfin is defined as in ref. [78]. The results are shown in fig. 3.5. We note that Vfin depends on the choice of the IR subtraction, this is why in the lower panel of fig. 3.5 the difference between the expanded results and the numerical grid of ref. [81] is shown (divided by the Born result), a quantity that is independent of the IR subtraction term. The grid of ref. [86] shows very good agreement with our results at every invariant mass, except for the first few bins at low MHH. The reason is a large uncertainty of the numerical grid on the low MHH bins, that are described by only a few points in the numerical grid due to their small contribution to the total cross section. For moderate and large MHH we observe differences below 1% in the ratio between our results and the ones of ref. [81]. We confirm these findings by comparing our results for Vfin with the values given in the grid [86] for various points at fixed (MHH,ˆ t). We find a good agreement, as can be inferred from table 3.1 for some representative values. We notice that in the region where both expansions perform less well we see differences of a few percent, although the latter will be reduced in ∆ˆσvirt due to the integration over ˆ t. Finally, we show that our merging approach is flexible with respect to the modification of the input parameters by computing the virtual corrections for various renormalisation schemes of the top quark mass. It was noted in refs. [63,64] that the di-Higgs production process suffers from a large uncertainty associated to the renormalisation 4We note that for ZH production no public code including the results of the full computation [65] is currently available. Hence we refrain from making any comparisons for ZH production. 4.2. HEFT 59 of operators are generated and only a finite number of counterterms is necessary. This procedure is effective but relies on certain assumptions to be meaningful. To ensure the validity of this approach, one must assume that contributions from higher-dimensional operators and double insertions of operators are much smaller than single insertions. In this context, we can take the SM as an example. Let’s assume that the cut-off for the SM is Λ = 1 TeV dim-6 operator: Cdim−6 v2 Λ2∼Cdim−6 16 two dim-6 operators: C2 dim−6 v2 Λ4∼C2 dim−6 162 dim-8 operator: Cdim−8 v4 Λ4∼Cdim−8 162(4.1) where Cdim is the Wilson coefficient of the operator and vis the Higgs vev that we used to make the contribution adimensional1. By applying this rescaling, we obtain the conditions Cdim−6≪16 and Cdim−8≪Cdim−6. These conditions appear reasonable, but it is important to note that the Wilson coefficients are not couplings themselves. Therefore, they can have large values without necessarily indicating the need for nonperturbative new physics. 4.2 HEFT As mentioned in the Introduction, the calculations performed in the thesis are carried out in the SM and its most common EFT extension, the SMEFT framework. However, we notice that there is another common strategy to build an EFT extension to the SM, called Higgs Effective Field Theory (HEFT). In this approach, unlike SMEFT where the Higgs is the usual SM doublet, the Higgs field is considered to be a singlet under GSM =SU(2)W×U(1)Y, while the Goldstone bosons, resulting from the symmetry breaking process, arrange themselves in a non-linear representation of GSM : U= exp(2iϕjTj v).(4.2) Although we will not delve into this method in the thesis, we think it is relevant to mention alternative frameworks to the one employed in this work. Overall, the main distinctions between HEFT and SMEFT lie in the energy scale, degrees of freedom and the scope of applicability for describing BSM physics. In general HEFT includes SMEFT, thus all NP signals can be detected within this framework. However, in the limit Λ ≫v, HEFT is equivalent to SMEFT [91], but the 1vis the scale of the SM interactions, so it is logical to use this parameter. 60 CHAPTER 4. STANDARD MODEL EFFECTIVE FIELD THEORY calculations are more complex and complicated. Therefore, assuming Λ ∼TeV, we can use SMEFT instead of HEFT to simplify the calculations, while still capturing the full range of BSM physics effects. 4.3 SMEFT The concept of the Standard Model Effective Field Theory (SMEFT) has been developed and studied by many researchers in the field of particle physics. It is probably not possible to attribute its invention to a single individual or group. Indeed this approach has been built upon the foundations of EFT and the understanding of the SM. Various physicists have contributed to the development and exploration of SMEFT, including [92,93,94]. The SMEFT Lagrangian is constructed as an expansion in higher dimension operators, with the SU(3) ×SU(2) ×U(1) gauge symmetry unbroken, L=LSM + Σ∞ k=5Σn a=1 Ck a Λk−4Ok a.(4.3) where the operators of mass-dimension k,Ok a, are constructed from SM fields and all of the effects of the beyond the SM (BSM) physics reside in the coefficient functions, Ck a. The focus of most of SMEFT research has been on dimension-6 operators and more recently on dimension-8. This is due to the fact that these operators provide the dominant contributions and it is commonly assumed that any new physics would initially manifest itself through a dimension-6 operator. Indeed, at dimension 5, only one operator arises and it introduces lepton number violation and yields Majorana masses for neutrinos. The stringent constraints on this type of violation needs either very high energy scales or very small coefficients to fit the experimental limits. See Appendix A.1 for a further discussion. 4.4 Warsaw basis At dimension 6 in the SMEFT framework, as discussed in [95], there is a possibility of redundancy among the operators. In this context, redundancy refers to the presence of operators that can be related to each other through equations of motion (EOM), integration by parts (IBP), field redefinitions or underlying symmetries. Indeed, certain combinations of operators may lead to the same physical effects or have equivalent contributions to observables. As a result, including all possible operators in the analysis may lead to overcounting or redundant descriptions of the same physics. To address this issue, it is common to apply techniques such as the IBP or the EOM to build an operator basis. These methods help to systematically identify the independent structures and construct a minimal set of operators to catch the relevant BSM physics. Indeed, by removing redundant operators, we can obtain a more efficient and concise description 4.4. WARSAW BASIS 61 of the theory, facilitating the calculations and improving the comprehensibility of the results. Lastly, we notice that the identification of redundant operators requires meticulous consideration of the symmetries, fields content, and interactions of the theory. For this reason, the choice of operator basis plays a crucial role in determining the appropriate set of independent operators in SMEFT. The most used operator basis in the SMEFT framework is the Warsaw basis, introduced by [96]. This basis choice has many advantages: •Renormalization group investigation. The Warsaw basis has been already studied in the context of the renormalization group and the associated RGEs have been completely calculated [97,98,99] at one loop order. This allows to perform calculations at NLO in this basis without the need for a change of basis which would also involve a SM field redefinition. •FeynRules implemented in Mathematica. The complete set of Feynman rules for the Warsaw basis have been derived in Rξgauges [100] and the final theory is expressed in a basis characterized by SM-like propagators for all physical and unphysical fields. Furthermore, in [101], the authors made publicly available a Mathematica code working with the FeynRules package, allowing automatic SMEFT amplitude calculations. •Comparison of results. The Warsaw basis is the currently most commonly used basis in SMEFT framework. Using the same basis in physics is useful to facilitate communication, promoting consistency and avoiding confusion between researchers. 4.4.1 Complete set of dimension-6 operator The EFT we are building has to follow only two constraints: the operators must respect the gauge and the Lorentz symmetries. Additional requirements, such as CP conservation and/or the absence of operators that violate baryon symmetry, can be imposed. However, it should be noted that by imposing these conditions, one is already making specific assumptions about the underlying BSM physics. In the next sections, following [96], we perform the classification of all possible dimension-6 operators using the same notation introduced in 2.2. 4.4.2 Dimension 5 operators For our analysis, it is convenient to initially consider only the left-handed fermions ψ∈l, ec, q, uc, dcas fundamental fields, using the charge conjugates of the SU(2)W- singlet fermions. Within this convention, there are three possibilities for the fermionic 62 CHAPTER 4. STANDARD MODEL EFFECTIVE FIELD THEORY currents 2:¯ ψ1γµψ2,¯ ψT 1Cψ2,¯ ψ1Cσµνψ2. By considering bosonic objects with the appropriate number of Lorentz indices and remembering that Xµν3is antisymmetric, complete sets of building blocks for for this class of operators can be easily determined for each fermionic current. For dimension 5 we have: ¯ ψ1γµψ2: (ϕD) ¯ ψT 1Cψ2: (ϕ2, D2) ¯ ψ1Cσµνψ2: (X, D2) (4.4) where ϕis the Higgs field and Dis the SM covariant derivative. Recalling the hypercharges assigned in 2.2, we see that the currents involving Ccan never have hypercharge 0 while for the vector currents it can never be equal to ±1 2. This means that the classes (X, D2, ϕD) can be discarded. Thus the only class we must consider is ψ2ϕ2. The hypercharge of the Higgs product, if it is not 0, can be ±1. The only fermionic current that can cancel it is the one built out of two lepton doublets. Therefore the only possibility is: ϵjkϵmnϕjϕm(lk p)TCln r≡(e ϕ†lp)TC(e ϕ†lr)T.(4.5) This operators violates lepton number conservation and, after eletroweak symmetry breaking, it generates neutrino masses and lepton mixings. Bosonic Operators For purely bosonic operators, due to constraints from the SU(2)Wgauge group, it is necessary to have an even number of Higgs fields ϕand an even number of covariant derivatives D so that all Lorentz indices must be contracted. Thus, the only potential field contents for dimension-6 bosonic operators are X3, X2ϕ2, X2D2, Xϕ4, XD4, Xϕ2D2, ϕ6, ϕ4D2and ϕ2D4. It is possible to demonstrate that all these operators can be reduced by the EOM (or symmetries considerations) to operators containing fermions or to classes X3, X2ϕ2, ϕ6and ϕ4D2. The class Xϕ4cannot appear because there is not any other object that can be used to contract the Lorentz indices. XD4can be moved in X2D2 because all possible contractions (including those with ϵµνρσ ) lead to the appearance of at least one invariant derivative commutator [Dµ, Dν]∼Xµν, which places it in the X2D2class. For our classification, since we are interested to O(1 Λ2), the EOMs can be derived directly from the SM Lagrangian. We have: (DµDµϕ)j=m2ϕj−λ(ϕ†ϕ)ϕj−¯eλ† elj+ϵjk ¯qkλuu−¯ dλ† dqj, (DρGρµ)A=gs(¯qγµTAq+ ¯uγµTAu+¯ dγµTAd), (DρWρµ)I=g 2(ϕ†i↔ DI µϕ+¯ lγµτIl+ ¯qγµτIq), (DρBρµ) = g′Yϕϕ†i↔ Dµϕ+g′X ψ∈{l,e,q,u,d} Yψ¯ ψγµψ. (4.6) 2up to Lorentz indices 3We recall that Xµν refers to the gauge field tensor and can be equal to WI µν, GA µν or Bµν. 4.4. WARSAW BASIS 63 Furthermore, we define ϕ†i ↔ Dµϕ≡ϕ†(Dµϕ)−i(Dµϕ)†ϕand ϕ†i ↔ Da µϕ≡ϕ†τaDµϕ−i(Dµϕ)†τaϕ. In this work, we have organized the operator classes in such a way that those with fewer covariant derivatives are considered ”lower classes”. Following [96], we will see that it is possible to reduce operators from higher to lower classes. If classes have an equal number of derivatives, the ordering is determined by the number of Xtensors, with lower classes having fewer Xtensors. We begin with the class of operators that can be reduced to lower class: -ϕ2D4 Here, we can consider only operators where all the derivatives act on a single ϕ field, because other possibilities are equivalent up to total derivatives. Contributions involving ϵµνρσ can be disregarded, because they result in terms in the form [Dµ, Dν]∼Xµν, which corresponds to lower classes. Similarly, the ordering of the covariant derivatives acting on ϕcan be chosen arbitrarily. We can take advantage of this freedom to use DµDµϕas the ”building blocks” of the operators considered. This allows us to move this class to lower classes through the EOM 4.6. -ϕ2XD2 In this class, we consider Xbeing potentially dual and we ignore ϵµνρσ. Indices of X need to be contracted with both derivatives, thus we have to consider three cases: (i) Each derivative acts on a different ϕfield. We can eliminate this possibility through integration by parts, neglecting total derivatives. (ii) Both derivatives act on a single object. This results again in [Dµ, Dν]∼Xµν leading to the ϕ2X2class. (iii) One derivative acts on Xand the other on ϕ. By EOM we can move to lower classes ϕ4D2and ψ2ϕD. -X2D2 For this class, similarly to the ϕ2D4, we can focus our attention only on operators where all the derivatives act on a single tensor. The case where both the derivatives are contracted with ϵµνρσ or with a single tensor is excluded because [Dµ, Dν]∼ Xµν. Therefore, we allow the tensor Xto be dual and ignore ϵµνρσ otherwise. In the case where each derivative is contracted with a different tensor, we use [Dµ, Dν]∼Xµν to change the order their order and through EOM we we can move it to lower class. The only possibility is that the derivatives are contracted with themselves e XµνDρDρXµν. However, using the Bianchi identity e XµνDρDρXµν =−e Xµν(DρDµXνρ +DρDνXρµ) (4.7) and again using [Dµ, Dν]∼Xµν and EOM we can reduce this case to the lower class. We can now review the operators appearing in table 4.1: 64 CHAPTER 4. STANDARD MODEL EFFECTIVE FIELD THEORY •X3 Here we have the first class of operators appearing in the Warsaw basis. We denote X, Y, Z the possible different tensors that may appear in this class. We allow them to be dual, thus we forget about ϵµνρσ otherwise. The only possible non vanishing and independent contraction of Lorentz indices reads Xν µYρ νZµ ρ. Indeed, all the three tensors must be different because XαµXβνZµνgαβ is zero by the antisimmetry of Z. Furthermore, neither of the two tensors can be dual, as Xν µe Xρ ν=−1 4δρ µXαβ e Xαβ is symmetric while Zis antisymmetric. The only possibility is that to get a gauge singlet from three different tensors, we have to use the structure constants fABC and ϵIJK . •X2ϕ2 The Higgs combinations can be singlets or triplets of SU(2)W. Hypercharge conditions further constrain the forms of these products to be written as ϕ†ϕand ϕ†τIϕ. In total we have eight possible operators, as listed in table 4.1. •ϕ6 Hypercharge constraints imply that exactly three of the Higgs fields must be complex conjugate. As in the previous case, we can consider tensor products of singlets and triplets of SU(2)W. The combination of three triplets can only result in an overall singlet if it is fully antisymmetric. However, since all the triplets are identical, this combination evaluates to zero. Two triplets and one singlet is a possible combination (ϕ†τIϕ)(ϕ†τIϕ)(ϕ†ϕ), but due to the equation τI jkτI mn = 2δjnδmk −δjkδmn (4.8) it results equals to (ϕ†ϕ)3. Hence, we have only one independent operator in this class. •ϕ4D2 As in the other cases, the hypercharge constrains two Higgs fields to complex conjugated. Considering that the two derivatives must be contracted, they have to act on two different ϕfields, or the EOM leads the operator to lower classes. If the derivatives act on two conjugated or two unconjugated fields, we can eliminate those possibilities by IBP. On the other hand, if one derivative acts on a conjugated field and the other on an unconjugated field, our SU(2)Wtensor product contains four distinct fundamental representations, indicating the presence of exactly two independent singlets. 4.4. WARSAW BASIS 65 X3ϕ6and ϕ4D2ψ2ϕ3 QGfABCGAν µGBρ νGCµ ρQϕ(ϕ†ϕ)3Qeϕ (ϕ†ϕ)(¯ l′ pe′ rϕ) Qe GfABC e GAν µGBρ νGCµ ρQϕ□(ϕ†ϕ)□(ϕ†ϕ)Quϕ (ϕ†ϕ)(¯q′ pu′ re ϕ) QWεIJK WIν µWJρ νWKµ ρQϕD ϕ†Dµϕ∗ϕ†DµϕQdϕ (ϕ†ϕ)(¯q′ pd′ rϕ) Qf WεIJK f WIν µWJρ νWKµ ρ X2ϕ2ψ2Xϕ ψ2ϕ2D QϕG ϕ†ϕ GA µνGAµν QeW (¯ l′ pσµνe′ r)τIϕWI µν Q(1) ϕl (ϕ†i ↔ Dµϕ)(¯ l′ pγµl′ r) Qϕ e Gϕ†ϕe GA µνGAµν QeB (¯ l′ pσµνe′ r)ϕBµν Q(3) ϕl (ϕ†i ↔ Da µϕ)(¯ l′ pτIγµl′ r) QϕW ϕ†ϕ WI µνWIµν QuG (¯q′ pσµνTAu′ r)e ϕ GA µν Qϕe (ϕ†i ↔ Dµϕ)(¯e′ pγµe′ r) Qϕ f Wϕ†ϕf WI µνWIµν QuW (¯q′ pσµνu′ r)τIe ϕ WI µν Q(1) ϕq (ϕ†i ↔ Dµϕ)(¯q′ pγµq′ r) QϕB ϕ†ϕ BµνBµν QuB (¯q′ pσµνu′ r)e ϕ Bµν Q(3) ϕq (ϕ†i ↔ Da µϕ)(¯q′ pτIγµq′ r) Qϕ e Bϕ†ϕe BµνBµν QdG (¯q′ pσµνTAd′ r)ϕ GA µν Qϕu (ϕ†i ↔ Dµϕ)(¯u′ pγµu′ r) QϕWB ϕ†τIϕ WI µνBµν QdW (¯q′ pσµνd′ r)τIϕ WI µν Qϕd (ϕ†i ↔ Dµϕ)( ¯ d′ pγµd′ r) Qϕ f WB ϕ†τIϕf WI µνBµν QdB (¯q′ pσµνd′ r)ϕ Bµν Qϕud i(e ϕ†Dµϕ)(¯u′ pγµd′ r) Table 4.1: Dimension-6 operators other than the four-fermion ones (from [96]). For brevity we suppress fermion chiral indices L, R. 4.4.3 Single-fermionic-current operator classification For this class of operators it is convenient to use the same notation introduced in 4.4.2. Again, by similar considerations, we have three possible fermionic currents. They are: ¯ ψ1γµψ2: (XD, ϕ2D, D3) ¯ ψT 1Cψ2: (ϕ3, ϕD2) ¯ ψ1Cσµνψ2: (Xϕ, ϕD2) (4.9) For scalar and tensor fermionic currents, we notice that the numer of higgs fields is always odd. As a result, these currents must form isospin doublets. Using the standard notation with right-handed singlets, these currents are represented by ¯ ψ1ψ2and ¯ ψ1σµνψ2. Similarly, vector currents combine with an even number of Higgs fields, thus they can only form isospin singlets or triplets. With this consideration, no vector currents with C enter into our considerations, even if the isospin singlets are taken as right-handed. Therefore , we can go back to the standard notation in the following discussion. As for the bosonic case, we will use the EOM 4.6 and as well as the classical EOM for fermions i/ Dl =λeeϕ, i / De =λeϕ†l, i / Dq =λuue ϕ+λddϕ, i / Du =λ† ue ϕ†q, i / Dd =λ† dϕ†q. (4.10) 66 CHAPTER 4. STANDARD MODEL EFFECTIVE FIELD THEORY Furthermore, we will need to recall the identities: γµγν=gµν −iσµν, γµγνγρ=gµνγρ+gνργµ−gρµγν−ϵµνρσγργ5.(4.11) As in the previous section, we start with the operators that can be reduced to lower classes: -ψ2D3 Here, three covariant derivatives are contracted with the vectorial current ¯ ψγµψ. As discussed for the classes ϕ2D4and X2D2, we can eliminate derivatives acting on ¯ ψ”by parts” and choose the ordering of the derivatives acting on ψ. By choosing the ordering as ¯ ψDµDµ/ Dψ, we can reduce by EOM to lower classes. -ψ2ϕD2 In this class, we consider scalar and tensorial currents. Derivatives acting on ¯ ψ can be removed ”by parts”. The cases ¯ ψσµνψDµDνϕand ϕ¯ ψσµνDµDνψcan be moved to lower classes recalling that [Dµ, Dν]∼Xµν. We still have 4 possible combinations: ¯ ψψDµDµϕ, ϕ ¯ ψDµDµψ, (Dµϕ)¯ ψσµνDνψ, (Dµϕ)¯ ψDµψ. The cases ¯ ψψDµDµϕand ϕ¯ ψDµDµψ∼ϕ¯ ψ/ D/ Dψ can be reduced to lower classes by EOM; (Dµϕ)¯ ψσµνDνψcan be moved to the last case recalling that σµν = −i(gµν −γµγnu). Considering that: 2(Dµϕ)¯ ψDµψ= (Dµϕ)¯ ψγµ/ Dψ + (Dµϕ)¯ ψ/ Dγµψ =−¯ ψ← / DγµψDµϕ−¯ ψγµγνψDµDνϕ+. . . (4.12) where the ” . . . ” stands for a total derivative and lower classes. Finally, using the EOM and recalling the relation 4.11, we can move it to lower classes. -ψ2XD Similarly to other cases, we assume that Xcan be dual and neglect ϵµνρσ otherwise. Since we are dealing with a vectorial current, recalling that Xis antisymmetric, the derivative must be contracted with X. If it acts on X, we can move it to lower classes (for the usual tensor) through EOM or we can use the Bianchi identity Dρe Xρµ = 0 (for the dual tensor). The only remaining expression to be considered is Xµν ¯ ψγµDνψ. It is reduced to lower classes as follows: Xµν ¯ ψγµDνψ=1 2Xµν ¯ ψ(γµγν/ D+γµ/ Dγν)ψ=1 2Xµν ¯ ψ(γµγν/ D−/ Dγµγν)ψ+Xµν ¯ ψγµDνψ =⇒1 4Xµν ¯ ψ(γµγν/ D−/ Dγµγν)ψ=1 4Xµν ¯ ψγµγν/ Dψ +1 4Xµν ¯ ψ← / Dγµγνψ+ +1 4DρXµν ¯ ψγργµγνψ+. . . (4.13) 4.4. WARSAW BASIS 67 In the last equation the first two terms are reduced to lower classes through EOM. For the last term we have: DρXµν ¯ ψγργµγνψ= 2 ¯ ψγµψDρXµν −iϵρµνσ ¯ ψγσγ5ψDρXµν.(4.14) In both the cases (Xdual or not), using EOM and Bianchi identity, we can move this terms to lower classes. We continue our review discussing the other operators appearing in table 4.1: •ψ2ϕ3 The fermion current must take the form of an isospin doublet and color singlet, like the Yukawa in the SM. The number of conjugated and un-conjugated scalar fields in ϕ3is fixed for each of the fermion currents due to hypercharge constraints. As a result, the only possibilities for this class are the Yukawa terms multiplied by ϕ†ϕ. •ψ2Xϕ In this case we have to consider only tensorial currents. Again, to respect hypercharge constraints, the fermion current must be analogous to the SM Yukawa couplings. For each tensor field Xµν, there is only one possible contraction with each fermionic current. The dual tensors ˜ Xwould not yield any new results, due to the identities: ϵαβµνσµν = 2iσαβγ5and γ5ψL,R =∓ψL,R.(4.15) •ψ2ϕ2D If the derivative acts on any of the fermion fields, its contraction with the vector current leads to equations of motion (EOMs) and brings us to the previously discussed lower class ψ2ϕ3. Therefore, we can just consider the case where the derivative act on the scalar fields. The Higgs fields combined can give isospin singlets or triplets and color singlets. The fermion currents must obey to the same selection rules, allowing for precisely the currents listed in table 4.1, with the exception of the uγµdcurrent, which requires Hermitian conjugation. The number of conjugated and unconjugated Higgs fields is determined by hypercharge conditions. We begin by removing derivatives by using integration by parts on one of the scalars. We then form isospin singlets or triplets from products of ϕ1and Dµϕ2according to the structure of the corresponding fermion currents. At this point, we obtain operators that differ from those in table 4.1 only by the presence of Dinstead of ↔ D. The point is that the operator without ↔ D, are not Hermitian and we have to verify if the h.c. are independent from them or not. Notice that this issue does not occur for all the other operators (also for Qϕud), since they are Hermitian by construction. For the remaining seven operators, considering the symmetric combination we have: ϕ†(Dµ+← Dµ)ϕ¯ ψγµψ=∂µ(ϕ†ϕ)ϕ¯ ψγµψ= (ϕ†ϕ)ϕ¯ ψ(/ D+← / D)ψ(4.16) 68 CHAPTER 4. STANDARD MODEL EFFECTIVE FIELD THEORY and we can move them lower classes by EOM. The review on single-fermion-current operators is completed. 4.4.4 Four-fermion operator classification Four-fermion operators are numerous but straightforward to classify. First, similar to the previous section, we consider only left-handed fermions ψ∈l, ec, q, uc, dc. Besides trivial outcomes involving products of two zero-hypercharge currents (indicated in 4.2 as (¯ LL)(¯ LL),(¯ RR)( ¯ RR),(¯ LL)( ¯ RR),), we have only a few other possible field contents: (¯ l¯ecdcq),(qucqdc),(lecquc),(qqql),(dcucucec),(qq¯uc¯ec),(ql¯uc¯ dc) (4.17) and the associated hermitian conjugate. The first three cases generate B-Conserving operators while the other four generate B-violating ones. We start by considering the expressions with two ψand two ¯ ψ. In this case, by spin considerations, we have only one single pairing; looking at the SU(2)Wquantum numbers, we see that there are two doublets and two singlets fields, meaning that we can have only one singlet. Finally, including also the colour indices, we have ¯ 3⊗3 = 8 ⊕1 B-conserving 3⊗3⊗3 = 10 ⊕8⊕8⊕1 B-violating (4.18) thus we have only one operator for each of three considered cases. In Table 4.2, they are indicated as Qledq, Qduq, Qqqu, using the standard right-hand notation. In the other four cases in 4.17, we have four left handed fermions. Again, by applying spin constraints and Fierz identities, we can eliminate redundant cases. Finally, taking into account isospin and color index contractions, we finalize the construction of the four-fermion operators. It can be readily confirmed that the listed operators form a complete basis for four-fermion operators, as discussed in [96]. 4.5 SMEFT couplings In SMEFT, similarly to what happens in the SM, the weak gauge bosons and the fermions acquire mass due to the SSB mechanism. However, unlike in the SM, to identify the physical degrees of freedom one needs to perform an extra intermediate step involving field rescalings, since SSB also affects the canonical normalization of the kinetic terms. This applies also to the SM fermions where, before the usual diagonalization of the Yukawa couplings, we need to redefine the SM fields and couplings. This process is accurately described in [100] and it involves several steps that we will not report entirely in this thesis, but we will just show the main results. 5.2. NOTATION 75 structure of fermion operators, among others. Understanding the uncertainties inherent in these assumptions is crucial for interpreting SMEFT fits[115]. The assumptions about the flavor structure introduce significant model dependence into the SMEFT predictions. It is straightforward to implement a general flavor structure for the tree level predictions for observables[116,117,118,119,120,121,122] and the one-loop MS renormalization of the dimension-6 coefficient functions is known for an arbitrary flavor structure[97,98,99]. The one-loop next-to-leading order (NLO) electroweak predictions for physical observables, however, typically involve a large number of 4-fermion and 2-fermion operators with potentially complicated flavor structures. In addition, the fermion operators introduce new subtleties in the renormalization procedure. Existing calculations of the one-loop electroweak corrections to EWPO, Higgs, and di-boson data do not include the most general flavor structure for the 2- quark and 4- quark operators in the loops[123,124,125,126,127,128,129]. Here, we present a generalization of previous NLO SMEFT calculation of EWPOs [130,131,132] which included 4-fermion operators, but not 2-fermion operators, to allow for an arbitrary flavor structure. The corrections to the EWPOs from 4-fermion operators in the U(3)5symmetric case are in [128]. The role of flavor assumptions in fits to top and bottom quark data has been extensively examined in the literature and those analyses are complementary to that presented here[133,134,135,132]. We set the CKM matrix to be diagonal, which implies that only operators containing pairs of identical flavor fermions contribute. We further work to linear order in the SMEFT coefficients and set all masses other than the top quark to be 0. We want to remark that these two conditions constrain the light fermion Yukawa couplings to be zero, without additional assumptions, since the SMEFT operators that would induce a modification to the SM Yukawas do not interfere with the SM amplitudes at linear order. 5.2 Notation Retaining only the dimension-6 operators, we calculate observables, Ob, to one-loop as an expansion in 1 Λ2and keep only the linear terms since the SMEFT is renormalizable order by order in powers of 1 Λ2, Ob=Ob,SM + Σn a=1 Ca Λ2βab,(5.2) where βab is process dependent and depends on the kinematic invariants and the input parameters, and Ob,SM is the SM prediction. We use the Feynman rules from Ref. [100], with general flavor structures for the 2- and 4-fermion operators, although we assume that the CKM matrix is diagonal, which has implications for the fermion structures of the operators that contribute to our calculation, as we will see. Furthermore, we assume that the SMEFT does not introduce new sources of CP violation, that is we assume the coefficients of all the operators to be real. 76 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES Oll[ijkl](¯ liγµlj)(¯ lkγµll)OϕWB(ϕ†τaϕ)Wa µνBµν OϕD ϕ†Dµϕ∗ϕ†Dµϕ Oϕe[ij] (ϕ†i ↔ Dµϕ)(eRiγµeRj)Oϕu[ij] (ϕ†i ↔ Dµϕ)(uRiγµuRj)Oϕd[ij] (ϕ†i ↔ Dµϕ)(dRiγµdRj) O(3) ϕq [ij] (ϕ†i ↔ Da µϕ)(¯qiτaγµqj)O(1) ϕq [ij] (ϕ†i ↔ Dµϕ)(¯qiτaγµqj)O(3) ϕl [ij] (ϕ†i ↔ Da µϕ)(¯ liτaγµlj) O(1) ϕl [ij] (ϕ†i ↔ Dµϕ)(¯ liτaγµlj) Table 5.1: Dimension-6 operators contributing to the Zand Wpole observables of this study at tree level. i, j, k, l = 1,2,3 are generation indices. Our goal is the dimension-6 SMEFT calculation of NLO QCD and NLO electroweak corrections to electroweak precision observables (EWPOs) with arbitrary flavor structures for the operators involving fermions. The technical details are in the next section and in the following sub-sections we discuss the effects of flavor on the predictions. 5.2.1 EWPOs In complete generality, the Warsaw basis contains 2499 baryon number conserving dimension-6 operators. Much of the proliferation of operators is associated with the flavor structure. Of course, most of the flavor structures will not contribute to a given observable and we begin by considering EWPOs at tree level. The Zand Wboson pole observables that we consider are, MW,ΓW,ΓZ, σh, Re, Rµ, Rτ, Rs, Rc, Rb, Ae, Aµ, Aτ, As, Ac, Ab, Ae,FB, Aµ,F B, Aτ,F B, AF B,s, AF B,c, AF B,b .(5.3) Note that we do not use the effective mixing angle in our fits, since it is derived from the asymmetries. The operators that contribute to the EWPOs at LO are comprised of two bosonic operators with no flavor structure (OϕW B and OϕD) and 8 fermionic operators, of which there are 7 operators with 2 fermionic indices (the 2-fermion operators), and 1 operator with 4 fermionic indices (the 4-fermion operator). The explicit forms of the operators that appear at LO are reported in Table 5.1. These operators change the couplings of the Zand Wbosons to fermions, and explicit predictions for the measured quantities of Eq. 5.3 are given in Appendix A of Ref. [124]. The only 4-fermion operator that is relevant for EWPOs at LO is Oll[ijkl] which has the symmetry Oll[ijkl] = Oll[klij] and only Oll[2112] = Oll[1221] contributes to the observables of Eq.5.3. The indices (i, j, k, l = 1,2,3) refer to the fermion generation.) Dimension-6 operators involving the electron and the muon give contributions to the decay of the µ, changing the relation between the vev, v, and the Fermi constant Gµ, Gµ≡1 √2v2−1 √2Λ2Cll[1221] + 1 √2Λ2C(3) ϕl [11] + C(3) ϕl [22].(5.4) At NLO, the EWPOs of Eq. 5.3 receive contributions from 22 additional operators (which are defined in Ref. [124]), which we classify according to the number of fermions: 5.2. NOTATION 77 •4 bosonic operators: OϕB,OϕW ,O□,OW.(5.5) •2 2-fermion operators: OuB[ij],OuW [ij].(5.6) Notice that only OuB[33] and OuW [33] contribute to the EWPOs at NLO if all fermions except the top are massless. •16 4-fermion operators: Oed[ijkl],Oee[ijkl],Oeu[ijkl],Olu[ijkl],Old[ijkl],Ole[ijkl], O(1) lq [ijkl],O(3) lq [ijkl]Oqe[ijkl],O(1) qd [ijkl],O(3) qq [ijkl],O(1) qq [ijkl], O(1) qu [ijkl],O(1) ud [ijkl],Ouu[ijkl],Odd[ijkl].(5.7) Five of the NLO-generated 4-fermion operators have a flavor symmetry, Oee[ijkl],O(3) qq [ijkl],O(1) qq [ijkl],Ouu[ijkl],Odd[ijkl]≡OY[ijkl] = OY[klij].(5.8) It is convenient to categorize the operators according to their dependence on the flavor since only specific structures contribute to the EWPOs at NLO: •A) 2-fermion operators: OX[ij]≡Oϕe[ij],Oϕu[ij],Oϕd[ij],O(3) ϕq [ij],O(1) ϕq [ij],O(3) ϕl [ij], O(1) ϕl [ij],OuB[ij],OuW [ij]. We consider the CKM matrix to be diagonal which has the consequence that the coefficients of the operators in Class A have diagonal flavor structures: CX[ij] = E(i) Xδij, i, j = 1,2,3,(5.9) resulting in 3 independent coefficients for each operator. Notice that the operator Oϕu[33] first contributes to the EWPOs at NLO. Furthermore, as noted above, OuB[ij] and OuW [ij] enter in our calculations only with coefficients CuB[33] and CuW [33] respectively. Together there are 23 independent coefficients in Class A. •B) 4-fermion operators involving only identical fermion representations: OY[ijkl]≡ Oee[ijkl], O(3) qq [ijkl], O(1) qq [ijkl], Ouu[ijkl], Odd[ijkl], Oll[ijkl]. Since they stem from the combination of two fermion currents belonging to identical representations once we require that the CKM matrix is the unit matrix, there are only two ways flavor is allowed to ”flow” through these operators: OY[iijj] and OY[ijji]. Furthermore, these operators are subject to the flavor symmetry in Eq. 5.8, resulting in the coefficients having the flavor structure, CY[iiii] = F(i) Y, CY[iijj] = A(ij) Y, CY[ijji] = B(ij) Y, A(ji) Y=A(ij) Y, B(ji) Y=B(ij) Y, i =j&i, j = 1,2,3,(5.10) 78 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES resulting in 9 independent coefficients for each operator. However, the flavor structure of Oee[ijkl] is further constrained by the Fiertz identity (eiγµej)(ekγµel) = (eiγµel)(ekγµej), which imposes the equality A(ij) ee =B(ij) ee , reducing the number of independent coefficients for Oee[ijkl] to 6. The only coefficient of Class B that does not contribute to the EWPOs at NLO is Cuu[3333]. In total, we have 50 independent coefficients in Class B contributing. •C) 4-fermion operators with 2 different fermion representations: OZ[ijkl]≡Oed[ijkl], Oeu[ijkl], Olu[ijkl], Old[ijkl], Ole[ijkl], O(1) lq [ijkl], O(3) lq [ijkl], Oqe[ijkl], O(1) qd [ijkl], O(1) qu [ijkl], O(1) ud [ijkl]. For these operators, our choice of a diagonal CKM matrix requires that the flavor must flow only in one way: OY[iijj]. Therefore, the coefficients of these operators have the flavor structure: CZ[iijj] = D(ij) Z, i, j = 1,2,3,(5.11) with no further restrictions, corresponding to 9 independent coefficients for each operator, for a total of 99 independent coefficients in Class C that contribute to the observables of Eq. 5.3 at NLO. In the most general flavor case, we see that EWPOs computed to NLO receive contributions from 178 independent coefficients: 6 from bosonic operators, 23 from 2-fermion operators, and 149 from 4-fermion operators. We next consider different flavor assumptions for the fermionic operators in order to reduce the number of operators that need to be considered. The flavor assumptions we consider are U(3)5, minimal flavor violation (MFV), U(2)5, third generation centric, third generation phobic, third generation phobic +U(2)5, and a flavorless structure. We will discuss each of these assumptions in detail in the following sub-sections. 5.3 Scenarios 5.3.1 Flavor Assumptions: U(3)5 In the absence of Yukawa couplings, the SM fermions have a global U(3)5symmetry, G3≡U(3)q×U(3)l×U(3)u×U(3)d×U(3)e.(5.12) The introduction of Yukawa interactions in the SM Lagrangian preserves a global hypercharge symmetry U(1)Y, a baryonic symmetry U(1)B, and three leptonic symmetries, U(1)eU(1)µand U(1)τ. Our first approximation when imposing flavor symmetries on SMEFT predictions is to assume that the dimension-6 SMEFT coefficients have the G3symmetry of the (Yukawa-less) SM. This symmetry prevents the generation of the operators OuW [33] and OuB[33] since they carry a left-right fermionic current. Under G3, the other operators in Class A respect the form Class A : CX[ii] = EX, i = 1,2,3,(5.13) 5.3. SCENARIOS 79 All the 2-fermion operators that contribute to the EWPOs automatically have the structure of Eq. 5.13, and there are 7 real coefficients in this class, consistent with the counting of Ref. [119]. Regarding the operators of Class B, both contractions of fermion indices (fiγµfi)(fjγµfj),(fiγµfj)(fjγµfi),(5.14) are invariant under U(3)5, so the coefficients of Class B reduce to Class B : CY[iiii] = AY+BY, CY[iijj] = AY, CY[ijji] = BY, i=j&i, j = 1,2,3.(5.15) Remembering that Bee =Aee, there are 11 independent coefficients in Class B. Finally, the operators in Class C have the structure Class C : CZ[iijj] = DZi, j = 1,2,3,(5.16) for a total of 11 independent coefficients also in Class C. In total, if we impose a G3 symmetry on the SMEFT Lagrangian, we are left with 29 independent coefficients contributing to the EWPOs at one loop.1 5.3.2 Flavor Assumptions: MFV The Yukawa couplings break the U(3)5global symmetry in the SM. If we assume that this is the only source of breaking of the G3symmetry, we are led to the MFV scenario2[136]. The SM Yukawa couplings are considered as U(3)5auxiliary fields with the transformations, Yu∼(3,1,3,1,1) Yd∼(3,1,1,3,1) Ye∼(1,3,1,1,3) .(5.17) It is always possible to choose a basis such that Yu, Yd, and Yeare diagonal matrices. We remind the reader that we have assumed that the CKM matrix is diagonal and the only non-zero fermion mass is assumed to be the top quark mass, Mt. The coefficients of the operators containing at least 2 top quarks are modified, while the other coefficients retain the G3structure. Our implementation of the MFV scenario with a massless b quark is equivalent to a global U(2)q×U(3)l×U(2)u×U(3)d×U(3)esymmetry. In Class A, the only operators with a flavor structure that depends on the third generation fermions are Oϕu,O(3) ϕq ,O(1) ϕq ,OuW and OuB. Class A : CX[αα] = EX, CX[33] = E(3) X,OX≡Oϕu ,O(3) ϕq ,O(1) ϕq , α = 1,2 C˜ X[ii] = E˜ X,O˜ X≡Oϕe,Oϕd,O(3) ϕl ,O(1) ϕl , i = 1,2,3. 1We note that the (RR)(RR) operator, O(8) ud , does not contribute to EWPOs at one-loop, and so our counting of the U(1)5operators agrees with Ref. [118]. 2Since we assume that the CKM matrix is diagonal, there is no flavor violation in our implementation of the MFV scenario. 80 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES Since for OuW and OuB only the CuW [33] and CuB[33] enter in our calculations, a total of 12 independent coefficients contribute to the EWPOs at NLO, in agreement with Table 9 of [119]. The operators in Class B involving charge 2 3- quarks all have 4 uRfields or 4 qL fields. Retaining the contributions up to O(M2 t v2) , the coefficients of Ouu,O(3) qq and O(1) qq are subject to the following relations in the MFV scenario, Class B : CY[1122] = CY[2211] = AY, CY[1221] = CY[2112] = BY CY[33αα] = CY[αα33] = A(3) Y, CY[3αα3] = CY[α33α] = B(3) Y CY[αααα] = AY+BY, CY[3333] = 2(A(3) Y+B(3) Y)−AY−BY, α = 1,2 which reduces the number of independent coefficients to 4 for each operator. The remaining operators in Class B satisfy the U(3)5relations of Eq. 5.15, thus resulting in 17 independent coefficients in Class B. In Class C, the operators Olu,Oqe,O(1) qd ,Oeu,O(1) ud ,O(1) lq O(3) lq have 2 top quarks and they satisfy the relations: Class C : CZ[ααii] = DZ, CZ[33ii] = D(3) Z, α = 1,2, i = 1,2,3 and hence we have 2 independent coefficients for each operator. The only operator in Class C with 2 charge- 2 3quarks contributing to EWPOs is O(1) qu which satisfies the relations, Class C : C(1) qu [ααββ] = Dqu(1) , C(1) qu [3333] = D(33) qu(1) C(1) qu [33αα] = D(3) qu(1) , C(1) qu [αα33] = D(¯ 3) qu(1) , α, β = 1,2, and hence there are 4 independent flavor structures for C(1) qu . The remaining operators in Class C, Oed,Old and Ole, satisfy the U(3)5relations and contribute 1 independent operator each. Overall, we have 21 independent coefficients in Class C. Excluding the operators O(8) qd ,O(8) qu ,and O(8) ud which do not contribute to EWPOs at NLO, our counting is in agreement with Table 2 of [120] and Table 9 of [119]. 5.3.3 Flavor Assumptions: U(2)5 Here we consider a scenario where the new physics distinguishes between the 3rd generation and the first 2 generations. The first 2 generations are assumed to have the global symmetry[118], G2≡U(2)q×U(2)l×U(2)u×U(2)d×U(2)e.(5.18) Since we consider only the top quark mass to be non-zero, the U(2)5symmetry is unbroken. Using the classification of operators given in Sec. 5.3.1, the G2symmetry requires that operators of Class A have the form: Class A : CX[αα] = EX, CX[33] = E(3) X, α = 1,2.(5.19) 5.3. SCENARIOS 81 There are 16 independent coefficients in this class. The Class B operators satisfy Class B : CY[ααββ] = AY, CY[αββα] = BY, CY[αααα] = AY+BY, CY[αα33] = CY[33αα] = A(3) Y, CY[3αα3] = CY[α33α] = B(3) Y, CY[3333] = F(3) Y, α =β, α, β = 1,2. so that each operator has 5 independent coefficients. As before, Oee has an extra constraint coming from the Fierz identities, which reduces the number of independent coefficients to 3 for this operator. Therefore, since Ouu[3333] does not contribute, Class B has 27 coefficients contributing to EWPOs at NLO in total. The Class C operators, OC, contain 2 different fermion representations and the U(2)5 symmetry implies, Class C : CZ[ααββ] = DZ, CZ[3333] = D(33) Z CZ[33αα] = D(3) Z, CZ[αα33] = D(¯ 3) Z, α, β = 1,2.(5.20) and there are 4 independent coefficients corresponding to each operator structure, for a total of 44 operators. 5.3.4 Flavor Assumptions: Third Generation Centric In this scenario, only operators involving the 3rd generation fermions are non-zero. An example of such a scenario might be a Z′boson that only interacts with the 3rd generation. Operators of Class A in this scenario have the form: Class A : CX[αα] = 0, CX[33] = E(3) X, α = 1,2.(5.21) There are 9 real coefficients in this class. The Class B and C operators satisfy Class B & C : CY,Z[ααββ] = CY[αββα] = 0, CY,Z[αα33] = CY,Z[33αα] = CY[3αα3] = CY[α33α] = 0, CY[3333] = F(3) Y, CZ[3333] = D(3) Z, α, β = 1,2. For a total of 5 coefficients in Class B and 11 coefficients in Class C. 5.3.5 Flavor Assumptions: Third Generation Phobic In this scenario, we assume that the new physics only couples to the 1st 2 generations. An example of such a model can be found in [137]. We assume no further symmetry, but note that the operators contributing to EWPOs do not have an arbitrary flavor structure, as described at the beginning of this section. 82 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES Operator U(3)5MFV U(2)53rdgen specific 3rdgen phobic 3rdgen phobic + U(2)5Flavorless Class A7 12 16 9 14 7 9 Class B11 17 27 5 23 11 6 Class C11 21 44 11 44 11 11 Total 29 50 87 25 81 29 26 Table 5.2: Number of independent operators contributing to NLO predictions for EW- POs in various flavor scenarios. The flavor structure of the operators of Class A is Class A : CX[αα] = E(α) X. CX[33] = 0, α = 1,2,(5.22) for a total of 14 independent coefficients. For Class B we have Class B : CY[αααα] = F(α) Y, CY[ααββ] = A(αβ) Y, CY[αββα] = B(αβ) Y, CY[αα33] = CY[33αα] = CY[3αα3] = CY[α33α] = CY[3333] = 0, α=β, α, β = 1,2, corresponding to 4 independent coefficients for each operator (3 for Oee), for a total of 23 coefficients in Class B. Finally for Class C the available structures are Class C : CZ[ααββ] = D(αβ) Z CZ[αα33] = CZ[33αα] = CZ[3333] = 0, α, β = 1,2,(5.23) which again gives 4 independent coefficients for each operator, for a total of 44 coefficients in Class C. 5.3.6 Flavor Assumptions: Third Generation Phobic + U(2)5 As a specific case of the previous example, we consider a scenario where, after assuming that new physics couples only to the 1st 2 generations, we further assume the existence of a U(2)5symmetry. In this case, the flavor structure of the operators of Class A is Class A : CX[αα] = EX, CX[33] = 0, α = 1,2,(5.24) for a total of 7 independent coefficients. 5.4. NLO CALCULATIONS 83 For Class B we have Class B : CY[ααββ] = AY, CY[αββα] = BY, CY[αααα] = AY+BY, CY[αα33] = CY[33αα] = CY[3αα3] = CY[α33α] = CY[3333] = 0, α=β, α, β = 1,2,(5.25) corresponding to 2 independent coefficients for each operator (1 for Oee), for a total of 11 coefficients in Class B. Finally for Class C the available structures are Class C : CZ[ααββ] = DZ CZ[αα33] = CZ[33αα] = CZ[3333] = 0, α, β = 1,2,(5.26) which again gives 1 independent coefficient for each operator, for a total of 11 coefficients in Class C. 5.3.7 Flavor Assumptions: Flavorless Scenario This is the scenario employed in our previous calculations[130], where flavor plays no role and we simply drop all flavor indices. In this scenario, we assume that the operators have no flavor structure, that is we make the replacements, CX,Y,Z[. . . ] = AX,Y,Z .(5.27) Therefore in this case there are 26 coefficients that contribute. We summarize the results of our discussion of flavor scenarios in Table 5.2. The number of operators contributing to EWPOs at NLO varies dramatically depending on the flavor structure and is significantly smaller than the 178 independent coefficients that contribute with no assumptions about flavor. We will see that these choices have large effects on the numerical results of fits to EWPOs. 5.4 NLO calculations This section describes our calculational procedure for the NLO corrections to the electroweak precision observables. Some of the results can be obtained from our previous calculation[130] by generalizing the flavor structure of each observable. However, including a general flavor structure at NLO requires a new calculation for many of the flavor structures and observables. Therefore, we have performed the calculation in all generality and in this section we describe the details of the calculation and the assumptions that are implicit in the numerical results of the next section. At tree level, the interactions are just the Z(and W) decays to 2 fermions shown in Fig. 5.1 and the effect of including the dimension-6 SMEFT contributions is to change 84 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES Z (a) (b) γ (c) Figure 5.1: Z decays to fermions. The black circle represents the tree level SMEFT interaction. the vector boson couplings to fermions. The tree level SMEFT couplings of fermions to the Zand Ware given in terms of our input parameters (α, MZ, Gµ), L≡2MZq√2GµZµgZq L+δgZq Lqγµq+ZµgZu R+δgZu RuRγµuR +ZµgZd R+δgZd RdRγµdR+ZµgZl L+δgZl Llγµl +ZµgZe R+δgZe ReRγµeR+ZµδgZν RνRγµνR + 2MW[α, MZ, Gµ]sGµ √2Wµ(1 + δgWq L)uLγµdL+δgW q RuRγµdR +Wµ(1 + δgWl L)νLγµeL+δgW ν RνRγµeR+h.c., (5.28) where at tree level, gZf R=−s2 WQfand gZf L=Tf 3−s2 WQf s2 W≡1−M2 W[α, MZ, Gµ] M2 Z ,(5.29) Tf 3=±1 2, and M2 W[α, MZ, Gµ] is the mass of the Wboson written in terms of the (α, MZ, Gµ) input parameters as defined later in Sec. 5.4.4. Analytic expressions for the shifts in the couplings in terms of the Warsaw basis coefficients can be found in [138, 139]. We use the SMEFT Feynman rules of Ref. [100] as implemented in FeynRules [101] from which a FeynArts [140] model file is obtained. In this way, we are able to generate the relevant one-loop and real emission amplitudes, which include all the QCD and EW 5.5. RESULTS 91 MeasurementExperiment ”Best” theory ΓZ(GeV) 2.4955 ±0.0023 2.4943 ±0.0006 [154,155,156] Re20.804 ±0.05 20.732 ±0.009 [154,155,156] Rµ20.784 ±0.034 20.732 ±0.009 [154,155,156] Rτ20.764 ±0.045 20.779 ±0.009 [154,155,156] Rb0.21629 ±0.00066 0.2159 ±0.0001[154,155,156] Rc0.1721 ±0.0030 0.1722 ±0.00005[154,155,156] σh41.481 ±0.033 41.492 ±0.008[154,155,156] Ae(from ALR had 0.15138 ±0.00216 0.1469 ±0.0004 [156,157] Ae(from ALR lep) 0.1544 ±0.0060 0.1469 ±0.0004 [156,157] Ae(from Bhabba pol) 0.1498 ±0.0049 0.1469 ±0.0004 [156,157] Aµ0.142 ±0.015 0.1469 ±0.0004 [156,157] Aτ(from SLD) 0.136 ±0.015 0.1469 ±0.0004 [156,157] Aτ(τpol) 0.1439 ±0.0043 0.1469 ±0.0004 [156,157] Ac0.670 ±0.027 0.66773 ±0.0002[156,157] Ab0.923 ±0.020 0.92694 ±0.00006[156,157,158] As0.895 ±0.091 0.93563 ±0.00004[156,157] Ae,FB0.0145 ±0.0025 0.0162 ±0.0001 [156,157] Aµ,FB0.0169 ±0.0013 0.0162 ±0.0001 [156,157] Aτ,FB 0.0188 ±0.0017 0.0162 ±0.0001 [156,157] Ab,FB0.0996 ±0.0016 0.1021 ±0.0003 [156,157,158] Ac,FB0.0707 ±0.0035 0.0736 ±0.0003 [156,157] As,FB0.0976 ±0.0114 0.10308 ±0.0003 [156,157] MW(GeV) PDG World Ave 80.377 ±0.012 80.357 ±0.006[159,160] ΓW(GeV) 2.085 ±0.042 2.0903 ±0.0003[161] Table 5.3: Unless otherwise cited, experimental results are taken from Table 10.5 of the Particle Data Group[153]. The theory results include the full set of 2-loop contributions for the Zpole observables, along with higher order corrections when known. The theory predictions are computed using the formulae in the indicated references and our input parameters, and the theory errors include the parametric uncertainties on Mt and Mh[156], along with the estimated theory uncertainties described in the respective papers. 92 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES For example, we can study the χ2for the coefficient C(1) qq in the MFV scenario. The χ2function is given by: χ2 MFVC(1) qq = 21.0184 −0.183188 C(1) qq [1,1,2,2] + 0.00528431 C(1) qq [1,1,2,2]2 −0.193834 C(1) qq [1,1,3,3] + 0.00725802 C(1) qq [1,1,2,2]C(1) qq [1,1,3,3] + 0.120819 C(1) qq [1,1,3,3]2+ 5.33703 C(1) qq [1,3,3,1] −0.2973 C(1) qq [1,1,2,2]C(1) qq [1,3,3,1] −0.271204 C(1) qq [1,1,3,3]C(1) qq [1,3,3,1] + 4.19859 C(1) qq [1,3,3,1]2−5.92823 C(1) qq [3,3,3,3] + 0.163898 C(1) qq [1,1,2,2]C(1) qq [3,3,3,3] + 0.0186148 C(1) qq [1,1,3,3]C(1) qq [3,3,3,3] −4.83774 C(1) qq [1,3,3,1]C(1) qq [3,3,3,3] + 3.85299 C(1) qq [3,3,3,3]2. (5.51) Following the above procedure, to marginalize C(1) qq [1,1,2,2] and C(1) qq [3,3,3,3], we have            ∂χ2 MFV ∂C(1) qq [1,1,2,2] = 0 ∂χ2 MFV ∂C(1) qq [3,3,3,3] = 0 =⇒ C(1) qq [1,1,2,2] →8.062020 −0.9688560 C(1) qq [1,1,3,3] +27.44810 C(1) qq [1,3,3,1] C(1) qq [3,3,3,3] →0.597832 + 0.0181909 C(1) qq [1,1,3,3] +0.0439991 C(1) qq [1,3,3,1]. (5.52) Setting the marginalized coefficients to their minimum we find: χ2 MFV(C(1) qq [1,1,3,3], C(1) qq [1,3,3,1]) = 18.5079 −0.124191C(1) qq [1,1,3,3] + 0.117473C(1) qq [1,1,3,3]2+ 0.048042C(1) qq [1,3,3,1] −0.0711661 C(1) qq [1,1,3,3]C(1) qq [1,3,3,1] + 0.012012C(1) qq [1,3,3,1]2. (5.53) The new χ2, that now depends only on C(1) qq [1,1,3,3] and C(1) qq [1,3,3,1], is used to carry out the required analysis for the MFV scenario. We begin by considering limits on the 2-fermion operators (Class A) that contribute to the Zand Wboson observables listed in Eq. 5.3. Table 5.4 compares the LO and NLO results for the SMEFT coefficients in the U(3)5, MFV, and 3rd generation centric scenarios. We note that the U(3)5and flavorless scenarios are identical for the coefficients of the 2-fermion operators. The limits on flavor structures that are not listed in the table can be derived using the results of Section 5.2, although for clarity we list limits on some of the non-independent coefficients. The differences between the LO and NLO fits are in general quite small. The single parameter limits are compared graphically in Fig. 5.4 where we see differences up to factors of 2 between the various flavor assumptions. The contributions of top quark loops to the 2-fermion operators have been studied in [164]. Table 3 of this reference presents the 95% CL single parameter limits on CuW [33], 5.5. RESULTS 93 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 U(3)5 LO U(3)5 NLO MFV LO MFV NLO 3rd generation centric LO 3rd generation centric NLO 95 % CL limits on 2-fermion operators from EWPOs <latexit sha1_base64="cxMhC6DRuO5n6tWO0B5jV7L/gZY=">AAAB/XicbVDLSsNAFL3xWesrPnZuBotQNyWxoi4L3bisYB+QxjCZTtuhk4czE6GG4K+4caGIW//DnX/jtM1CWw9cOJxzL/fe48ecSWVZ38bS8srq2npho7i5tb2za+7tt2SUCEKbJOKR6PhYUs5C2lRMcdqJBcWBz2nbH9UnfvuBCsmi8FaNY+oGeBCyPiNYackzD+te2o2HDN1nd2m5epo51arrmSWrYk2BFomdkxLkaHjmV7cXkSSgoSIcS+nYVqzcFAvFCKdZsZtIGmMywgPqaBrigEo3nV6foROt9FA/ErpChabq74kUB1KOA193BlgN5bw3Ef/znET1r9yUhXGiaEhmi/oJRypCkyhQjwlKFB9rgolg+lZEhlhgonRgRR2CPf/yImmdVeyLin1zXqpZeRwFOIJjKIMNl1CDa2hAEwg8wjO8wpvxZLwY78bHrHXJyGcO4A+Mzx/je5Qn</latexit> C(3) q[33] <latexit sha1_base64="UAxqonphRPshoBvI/WzVO6/Snuw=">AAAB/XicbVDLSsNAFL3xWesrPnZuBotQNyWxoi4L3bisYB+QxjCZTtuhk4czE6GG4K+4caGIW//DnX/jtM1CWw9cOJxzL/fe48ecSWVZ38bS8srq2npho7i5tb2za+7tt2SUCEKbJOKR6PhYUs5C2lRMcdqJBcWBz2nbH9UnfvuBCsmi8FaNY+oGeBCyPiNYackzD+te2o2HDN1nd2nZPs2catX1zJJVsaZAi8TOSQlyNDzzq9uLSBLQUBGOpXRsK1ZuioVihNOs2E0kjTEZ4QF1NA1xQKWbTq/P0IlWeqgfCV2hQlP190SKAynHga87A6yGct6biP95TqL6V27KwjhRNCSzRf2EIxWhSRSoxwQlio81wUQwfSsiQywwUTqwog7Bnn95kbTOKvZFxb45L9WsPI4CHMExlMGGS6jBNTSgCQQe4Rle4c14Ml6Md+Nj1rpk5DMH8AfG5w/gZ5Ql</latexit> C(1) q[33] <latexit sha1_base64="d4KcHfivGGmCQRyLh0nRUuz3vfI=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKewaUY+BXDxGMA/YrGF2MpsMmZ1d5qGEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1090Vppwp7brfTmFtfWNzq7hd2tnd2z8oHx61VWIkoS2S8ER2Q6woZ4K2NNOcdlNJcRxy2gnHjZnfeaRSsUTc60lKgxgPBYsYwdpKD41+1ktHDJmpX6sF/XLFrbpzoFXi5aQCOZr98ldvkBATU6EJx0r5npvqIMNSM8LptNQziqaYjPGQ+pYKHFMVZPOrp+jMKgMUJdKW0Giu/p7IcKzUJA5tZ4z1SC17M/E/zzc6ugkyJlKjqSCLRZHhSCdoFgEaMEmJ5hNLMJHM3orICEtMtA2qZEPwll9eJe2LqndV9e4uK3U3j6MIJ3AK5+DBNdThFprQAgISnuEV3pwn58V5dz4WrQUnnzmGP3A+fwChuJHk</latexit> Cu[33] <latexit sha1_base64="9+yleScHiIiOfIwGCfe62Ffh164=">AAAB9XicbVBNS8NAEJ3Ur1q/oh69LBbBU0msqMdCLx4r2A9IY9lsNu3SzSbsbpQS+j+8eFDEq//Fm//GbZuDtj4YeLw3w8y8IOVMacf5tkpr6xubW+Xtys7u3v6BfXjUUUkmCW2ThCeyF2BFORO0rZnmtJdKiuOA024wbs787iOViiXiXk9S6sd4KFjECNZGemgO8n46YiicevW6P7CrTs2ZA60StyBVKNAa2F/9MCFZTIUmHCvluU6q/RxLzQin00o/UzTFZIyH1DNU4JgqP59fPUVnRglRlEhTQqO5+nsix7FSkzgwnTHWI7XszcT/PC/T0Y2fM5FmmgqyWBRlHOkEzSJAIZOUaD4xBBPJzK2IjLDERJugKiYEd/nlVdK5qLlXNffustpwijjKcAKncA4uXEMDbqEFbSAg4Rle4c16sl6sd+tj0Vqyiplj+APr8weHn5HT</latexit> Cd[33] <latexit sha1_base64="Smjd2QdNG5/PR8OYZB0I6KkB3+c=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKewaUY+BXDxGMA/YrGF20psMmX0wM6uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ4IrbdvfVmFtfWNzq7hd2tnd2z8oHx61VZxKhi0Wi1h2fapQ8AhbmmuB3UQiDX2BHX/cmPmdR5SKx9G9niTohXQY8YAzqo300OhnvWTECU7dWs3rlyt21Z6DrBInJxXI0eyXv3qDmKUhRpoJqpTr2In2Mio1ZwKnpV6qMKFsTIfoGhrREJWXza+ekjOjDEgQS1ORJnP190RGQ6UmoW86Q6pHatmbif95bqqDGy/jUZJqjNhiUZAKomMyi4AMuESmxcQQyiQ3txI2opIybYIqmRCc5ZdXSfui6lxVnbvLSt3O4yjCCZzCOThwDXW4hSa0gIGEZ3iFN+vJerHerY9Fa8HKZ47hD6zPH4kokdQ=</latexit> Ce[33] <latexit sha1_base64="GF4rHK+8nKYxxq71+iuxjW5VtA8=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1iEuimJFXVZ6MZlBfuANobJdNIOnTyYmQg1BH/FjQtF3Pof7vwbp20W2nrgwuGce7n3Hi/mTCrL+jYKK6tr6xvFzdLW9s7unrl/0JZRIghtkYhHouthSTkLaUsxxWk3FhQHHqcdb9yY+p0HKiSLwjs1iakT4GHIfEaw0pJrHjXctB+PGOLZfVqxz7Jerea4ZtmqWjOgZWLnpAw5mq751R9EJAloqAjHUvZsK1ZOioVihNOs1E8kjTEZ4yHtaRrigEonnV2foVOtDJAfCV2hQjP190SKAykngac7A6xGctGbiv95vUT5107KwjhRNCTzRX7CkYrQNAo0YIISxSeaYCKYvhWRERaYKB1YSYdgL768TNrnVfuyat9elOtWHkcRjuEEKmDDFdThBprQAgKP8Ayv8GY8GS/Gu/Exby0Y+cwh/IHx+QPYnJQg</latexit> C(1) l[33] <latexit sha1_base64="cwB5gUzscm6h/ms6HyQQgPrFi+c=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1iEuimJFXVZ6MZlBfuANobJdNIOnTyYmQg1BH/FjQtF3Pof7vwbp20W2nrgwuGce7n3Hi/mTCrL+jYKK6tr6xvFzdLW9s7unrl/0JZRIghtkYhHouthSTkLaUsxxWk3FhQHHqcdb9yY+p0HKiSLwjs1iakT4GHIfEaw0pJrHjXctB+PGOLZfVqpnWW9Ws1xzbJVtWZAy8TOSRlyNF3zqz+ISBLQUBGOpezZVqycFAvFCKdZqZ9IGmMyxkPa0zTEAZVOOrs+Q6daGSA/ErpChWbq74kUB1JOAk93BliN5KI3Ff/zeonyr52UhXGiaEjmi/yEIxWhaRRowAQlik80wUQwfSsiIywwUTqwkg7BXnx5mbTPq/Zl1b69KNetPI4iHMMJVMCGK6jDDTShBQQe4Rle4c14Ml6Md+Nj3low8plD+APj8wfbsJQi</latexit> C(3) l[33] Figure 5.4: Limits on coefficients of 2-fermion operators in the U(3)5(LO in black, NLO in red), MFV (LO in green, NLO in blue) and 3rd generation centric (LO in magenta, NLO in cyan) scenarios with a single non-zero operator and marginalizing over the other independent flavor structures of each operator. NLO:Solid;LO:Dashed MFV U(3)5 3rdGenCen -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 -0.04 -0.02 0.00 0.02 0.04 0.06 Cϕq (1)[1,1] Cϕq (1)[3,3] 95%CL NLO:Solid;LO:Dashed 3rdGenPh 3rdGenPh+U(2)5 FlavLess -0.05 0.00 0.05 0.10 0.15 0.20 -0.1 0.0 0.1 0.2 0.3 Cϕq (1)[1,1] Cϕq (1)[2,2] 95%CL Figure 5.5: 95% CL limits on C(1) ϕq [ij] under flavor assumptions described in the text. Results at LO are drawn with dashed lines, results at NLO are drawn with solid lines. On the left we present C(1) ϕq [11] vs. C(1) ϕq [33] in the U(3)5(black), MFV (blue) and 3rd generation centric (magenta) scenarios. In these scenarios C(1) ϕq [22] = C(1) ϕq [11]. On the right we present C(1) ϕq [11] vs. C(1) ϕq [22] in the 3rd generation phobic (green), 3rd generation phobic + U(2)5(orange) and flavorless (violet) scenarios. In the first two scenarios C(1) ϕq [33] = 0, while in the flavorless scenario C(1) ϕq [33] = C(1) ϕq [22] = C(1) ϕq [11]. All other coefficients are set to 0. CuB[33], Cϕu[33] and C(−) ϕq [33] ≡1 2(C(1) ϕq [33] −C(3) ϕq [33]). Our results agree with these to ∼10 −20%, which is consistent with the use of slightly different sets of data as input. 94 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES In Fig. 5.5 and in Figs. A.1-A.6 in the Appendix A.3, we show on the left hand side the limits on the 2-fermion operators in the U(3)5, 3rd generation specific, and MFV scenarios . On the right hand side of these plots we show the 3rd generation phobic and 3rd generation phobic + U(2)5scenarios, where the new physics only couples to the first and second generations, and the flavorless scenario. It is apparent that the limits one quotes on these operators is highly dependent on the assumed flavor scenario. It is interesting to note in Fig. 5.5, the large differences in the shapes of the LO and NLO fits in the MFV and 3rd generation phobic scenarios and that the limits on the 3rd generation phobic scenario are considerably weaker than in the other scenarios. Tables 5.5 and 5.6 show the 95% CL limits on coefficients in the flavor schemes discussed in Sec. 5.2 for the Class B and Class C operators. In general, there is a strong dependence on the flavor scenario. This dependence is much larger than for the 2-fermion operator coefficients and the flavorless scenario gives much more stringent bounds for many coefficient functions than is the case in the other flavor scenarios. The most precise limits are on Cll and are shown in Fig. 5.6 for several flavor scenarios. It is clear that using the flavorless scenario gives limits that are factors of O(10−100) more precise than in the MFV or U(3)5scenarios for Cll. This is understood from Eq. 5.4 where we see that the only flavor structure that contributes to Gµis Cll[1221] and we observe that the limits on Cll in the U(2)5scenario are the weakest. In Fig. 5.7, we show the strong correlation between Cll[1221] and Cll[3333] in the MFV scenario and between Cll[1221] and Cll[1122] in the U(2)5and 3rd generation phobic scenarios. We note that there are only 2 independent Cll coefficients. Marginalized single parameter limits on the Class B operator coefficients C(1) qq and C(3) qq are shown in Fig. 5.8. The U(3)5and MFV results for these operators are within a factor of 2 of each other, while the 3rd generation specific and 3rd generation phobic scenarios are weakly constrained. Sample correlations between different flavor structures are shown in Fig. 5.9, demonstrating the sensitivity to the flavor assumptions of these fits. The 4-fermion Class C operators can be studied individually, and marginalizing over the independent flavor structures gives the limits in Fig. 5.10. The limits obtained in the flavorless setup in this case also similar to those of the MFV and U(3)5flavor structures. The 3rd generation specific and 3rd generation phobic scenarios give weak bounds for these operators. In addition, there are large correlations between the different flavor structures, as illustrated in Fig. 5.11. 5.6 Future perspectives As discussed in this chapter, the verification for consistency within the SM of EWPO is an important method, along with direct searches for new particles, to investigate the potential existence of BSM physics. Indeed, exploring the properties of massive electroweak gauge bosons (W and Z bosons) holds great promise for indirect BSM searches, as the interactions involving photons and gluons are subject to stringent constraints imposed by the unbroken gauge symmetries. In this context, the achievable precision of 5.6. FUTURE PERSPECTIVES 95 0.01 0.1 1 10 100 1000 |d| U(3)5 MFV Flavorless U(2)5 3rd generation phobic + U(2)5 95% CL limits on C ll range from EWPOs (Coefficients are not all independent) <latexit sha1_base64="+HZEkH41yE5gHP0HHtjwe92wLkg=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9lYUY+FXjxWsB+wXUo2zbah2eyaZAtl6e/w4kERr/4Yb/4b03YP2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TRVmLxiJW3YBoJrhkLcONYN1EMRIFgnWCcWPudyZMaR7LRzNNmB+RoeQhp8RYyW/0M4HEzMO1Gvb75YpbdRdA6wTnpAI5mv3yV28Q0zRi0lBBtPawmxg/I8pwKtis1Es1SwgdkyHzLJUkYtrPFkfP0IVVBiiMlS1p0EL9PZGRSOtpFNjOiJiRXvXm4n+el5rwzs+4TFLDJF0uClOBTIzmCaABV4waMbWEUMXtrYiOiCLU2JxKNgS8+vI6aV9V8U0VP1xX6m4eRxHO4BwuAcMt1OEemtACCk/wDK/w5kycF+fd+Vi2Fpx85hT+wPn8AS5LkQI=</latexit> Cll[1331] <latexit sha1_base64="U/SLJLnwh91DrEKE+WY1CnlyeUw=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9kUUY+FXjxWsB+wXUo2zbah2ew2yRbK0t/hxYMiXv0x3vw3pu0etPXBwOO9GWbmBYng2rjut1PY2t7Z3Svulw4Oj45PyqdnbR2nirIWjUWsugHRTHDJWoYbwbqJYiQKBOsE48bC70yZ0jyWT2aWMD8iQ8lDTomxkt/oZwKJuYdrNez3yxW36i6BNgnOSQVyNPvlr94gpmnEpKGCaO1hNzF+RpThVLB5qZdqlhA6JkPmWSpJxLSfLY+eoyurDFAYK1vSoKX6eyIjkdazKLCdETEjve4txP88LzXhvZ9xmaSGSbpaFKYCmRgtEkADrhg1YmYJoYrbWxEdEUWosTmVbAh4/eVN0q5V8W0VP95U6m4eRxEu4BKuAcMd1OEBmtACChN4hld4c6bOi/PufKxaC04+cw5/4Hz+ACs+kQA=</latexit> Cll[1221] <latexit sha1_base64="j9eGoWCxNe0ReUnWDgPgfLtqgCM=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GOhF48V7Ae0oWy2m3bpZhN3J4US+ju8eFDEqz/Gm//GbZuDtj4YeLw3w8y8IJHCoOt+O4WNza3tneJuaW//4PCofHzSMnGqGW+yWMa6E1DDpVC8iQIl7ySa0yiQvB2M63O/PeHaiFg94jThfkSHSoSCUbSSX+9nkshZ17Pw++WKW3UXIOvEy0kFcjT65a/eIGZpxBUySY3pem6CfkY1Cib5rNRLDU8oG9Mh71qqaMSNny2OnpELqwxIGGtbCslC/T2R0ciYaRTYzojiyKx6c/E/r5tieOdnQiUpcsWWi8JUEozJPAEyEJozlFNLKNPC3krYiGrK0OZUsiF4qy+vk9ZV1bupeg/XlZqbx1GEMziHS/DgFmpwDw1oAoMneIZXeHMmzovz7nwsWwtOPnMKf+B8/gAoMZD+</latexit> Cll[1111] <latexit sha1_base64="I1Fj3G5zIiVQZoQNzOi1/nYFmN0=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9kUUY+FXjxWsB+wXUo2zbah2ew2yRbK0t/hxYMiXv0x3vw3pu0etPXBwOO9GWbmBYng2rjut1PY2t7Z3Svulw4Oj45PyqdnbR2nirIWjUWsugHRTHDJWoYbwbqJYiQKBOsE48bC70yZ0jyWT2aWMD8iQ8lDTomxkt/oZwKJuYdxreb3yxW36i6BNgnOSQVyNPvlr94gpmnEpKGCaO1hNzF+RpThVLB5qZdqlhA6JkPmWSpJxLSfLY+eoyurDFAYK1vSoKX6eyIjkdazKLCdETEjve4txP88LzXhvZ9xmaSGSbpaFKYCmRgtEkADrhg1YmYJoYrbWxEdEUWosTmVbAh4/eVN0q5V8W0VP95U6m4eRxEu4BKuAcMd1OEBmtACChN4hld4c6bOi/PufKxaC04+cw5/4Hz+ACs8kQA=</latexit> Cll[1122] <latexit sha1_base64="vSzNTx+IwjdeJMiJ4pINUekgLkQ=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9lYUY+FXjxWsB+wXUo2zbah2eyaZAtl6e/w4kERr/4Yb/4b03YP2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TRVmLxiJW3YBoJrhkLcONYN1EMRIFgnWCcWPudyZMaR7LRzNNmB+RoeQhp8RYyW/0M4HEzMO4VvP75YpbdRdA6wTnpAI5mv3yV28Q0zRi0lBBtPawmxg/I8pwKtis1Es1SwgdkyHzLJUkYtrPFkfP0IVVBiiMlS1p0EL9PZGRSOtpFNjOiJiRXvXm4n+el5rwzs+4TFLDJF0uClOBTIzmCaABV4waMbWEUMXtrYiOiCLU2JxKNgS8+vI6aV9V8U0VP1xX6m4eRxHO4BwuAcMt1OEemtACCk/wDK/w5kycF+fd+Vi2Fpx85hT+wPn8AS5HkQI=</latexit> Cll[1133] <latexit sha1_base64="XMojbGTgcEK3BJkem4GtV/k/I40=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU1GOhF48VbC20oWy2m3bpZhN3J4US+ju8eFDEqz/Gm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHjUMnGqGW+yWMa6HVDDpVC8iQIlbyea0yiQ/DEY1Wf+45hrI2L1gJOE+xEdKBEKRtFKfr2XSSKnnUsLv1euuFV3DrJKvJxUIEejV/7q9mOWRlwhk9SYjucm6GdUo2CST0vd1PCEshEd8I6likbc+Nn86Ck5s0qfhLG2pZDM1d8TGY2MmUSB7YwoDs2yNxP/8zophrd+JlSSIldssShMJcGYzBIgfaE5QzmxhDIt7K2EDammDG1OJRuCt/zyKmldVL3rqnd/Vam5eRxFOIFTOAcPbqAGd9CAJjB4gmd4hTdn7Lw4787HorXg5DPH8AfO5w80ZZEG</latexit> Cll[3333] Figure 5.6: Limits on coefficients of Cll[ijkl] in various flavor scenarios, where i, j, k, l = 1,2,3. Only Cll is taken to be non-zero in this figure, and the independent flavor structures not shown are marginalized over. Exact numbers are given in Table 5.5. MFV 3rdGenCen FlavLess -60 -40 -20 020 40 -0.05 0.00 0.05 Cll[3,3,3,3] Cll[1,2,2,1] 95%CL U(2)5 3rdGenPh 3rdGenPh+U(2)5 FlavLess -20 -10 010 20 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 Cll [ 1,1,2,2 ] Cll[1,2,2,1] 95%CL Figure 5.7: 95% CL limits on Cll in various flavor scenarios with all other coefficients taken to be 0. On the left we present Cll[3333] vs. Cll[1221] in the MFV (blue), 3rd generation centric (black) and flavorless (magenta) scenarios. On the right we present Cll[1122] vs. Cll[1221] in the U(2)5(cyan), 3rd generation phobic (green), 3rd generation phobic + U(2)5(orange) and flavorless (magenta) scenarios. All the independent flavor structures not present in the plots are marginalized over. various future experiments has fundamental importance. Indeed, factors such as the statistical size of the data sample and the interplay between experimental and theoretical systematic uncertainties play a crucial role. In the report [21], an estimation of projected statistical and systematic uncertainties is proposed for various future linear colliders. In this section, using the data in Table 3 [21], we show the impact of projected future colliders errors (ILC-GigaZ and Fcc-ee) on the range of the bounds SMEFT coefficients, while also comparing them with the current values. We stress that the evaluations of systematic errors are rough estimates in terms of their magnitude. For a more accurate assessment, it would be necessary instrumentation details and other tools that are cur- 96 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES 0.01 1 100 10000 1e+06 |d| U(3)5 MFV Flavorless 3rd generation phobic + U(2)5 3rd generation specific 95% CL limits on C qq (1,3) ranges from EWPOs (Coefficients are not all independent) <latexit sha1_base64="MMF0mtZnjHSwghGE9s2RAun5/0w=">AAAB+3icbVDLSsNAFJ3UV62vWJduBotQNyVTRF0WunFZwT4gjWEynbZDJ5N0ZiKWkF9x40IRt/6IO//GaZuFth64cDjnXu69J4g5U9pxvq3CxubW9k5xt7S3f3B4ZB+XOypKJKFtEvFI9gKsKGeCtjXTnPZiSXEYcNoNJs25332kUrFI3OtZTL0QjwQbMoK1kXy73PTT6TR7SKvoInORgefbFafmLADXCcpJBeRo+fZXfxCRJKRCE46VcpETay/FUjPCaVbqJ4rGmEzwiLqGChxS5aWL2zN4bpQBHEbSlNBwof6eSHGo1CwMTGeI9VitenPxP89N9PDGS5mIE00FWS4aJhzqCM6DgAMmKdF8ZggmkplbIRljiYk2cZVMCGj15XXSqdfQVQ3dXVYa9TyOIjgFZ6AKELgGDXALWqANCHgCz+AVvFmZ9WK9Wx/L1oKVz5yAP7A+fwAgoJMl</latexit> C(1) qq [1111] <latexit sha1_base64="4Epg7OBjjgk9iHjUwU62iFQW3yE=">AAAB+3icbVDLSsNAFJ3UV62vWJdugkWom5IJoi4L3bisYB+QxjCZTtqhk0k6MxFLyK+4caGIW3/EnX/jtM1CWw9cOJxzL/feEySMSmXb30ZpY3Nre6e8W9nbPzg8Mo+rXRmnApMOjlks+gGShFFOOooqRvqJICgKGOkFk9bc7z0SIWnM79UsIV6ERpyGFCOlJd+stvxsOs0fsjq8yF0IHcfzzZrdsBew1gksSA0UaPvm12AY4zQiXGGGpHShnSgvQ0JRzEheGaSSJAhP0Ii4mnIUEelli9tz61wrQyuMhS6urIX6eyJDkZSzKNCdEVJjuerNxf88N1XhjZdRnqSKcLxcFKbMUrE1D8IaUkGwYjNNEBZU32rhMRIIKx1XRYcAV19eJ12nAa8a8O6y1nSKOMrgFJyBOoDgGjTBLWiDDsDgCTyDV/Bm5MaL8W58LFtLRjFzAv7A+PwBI6uTJw==</latexit> C (1) qq [1122] <latexit sha1_base64="VleThwgLmY00IfDQ8uOr9WM/MHA=">AAAB+3icbVDLSsNAFJ3UV62vWJdugkWom5IJoi4L3bisYB+QxjCZTtqhk0k6MxFLyK+4caGIW3/EnX/jtM1CWw9cOJxzL/feEySMSmXb30ZpY3Nre6e8W9nbPzg8Mo+rXRmnApMOjlks+gGShFFOOooqRvqJICgKGOkFk9bc7z0SIWnM79UsIV6ERpyGFCOlJd+stvxsOs0fsjq8yF3oONDzzZrdsBew1gksSA0UaPvm12AY4zQiXGGGpHShnSgvQ0JRzEheGaSSJAhP0Ii4mnIUEelli9tz61wrQyuMhS6urIX6eyJDkZSzKNCdEVJjuerNxf88N1XhjZdRnqSKcLxcFKbMUrE1D8IaUkGwYjNNEBZU32rhMRIIKx1XRYcAV19eJ12nAa8a8O6y1nSKOMrgFJyBOoDgGjTBLWiDDsDgCTyDV/Bm5MaL8W58LFtLRjFzAv7A+PwBI62TJw==</latexit> C(1) qq [1221] <latexit sha1_base64="Mhyn105IXtq3n0voWEeSt36YMgk=">AAAB+3icbVDLTsJAFJ3iC/FVcelmIjHBDemAUZckbFxiIo+k1GY6TGHC9MHM1Eia/oobFxrj1h9x5984QBcKnuQmJ+fcm3vv8WLOpLKsb6Owsbm1vVPcLe3tHxwemcflrowSQWiHRDwSfQ9LyllIO4opTvuxoDjwOO15k9bc7z1SIVkU3qtZTJ0Aj0LmM4KVllyz3HLT6TR7SKvoIrMRajQc16xYNWsBuE5QTiogR9s1vwbDiCQBDRXhWEobWbFyUiwUI5xmpUEiaYzJBI+orWmIAyqddHF7Bs+1MoR+JHSFCi7U3xMpDqScBZ7uDLAay1VvLv7n2Ynyb5yUhXGiaEiWi/yEQxXBeRBwyAQlis80wUQwfSskYywwUTqukg4Brb68Trr1GrqqobvLSrOex1EEp+AMVAEC16AJbkEbdAABT+AZvII3IzNejHfjY9laMPKZE/AHxucPJraTKQ==</latexit> C(1) qq [1133] <latexit sha1_base64="1LUyw5lYm43oVnG43MBmB8SBftQ=">AAAB+3icbVBNS8NAEJ34WetXrEcvwSLUS0mqqMdCLx4r2A9IY9hst+3SzSbd3Ygl5K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5QcyoVLb9baytb2xubRd2irt7+weH5lGpLaNEYNLCEYtEN0CSMMpJS1HFSDcWBIUBI51g3Jj5nUciJI34vZrGxAvRkNMBxUhpyTdLDT+dTLKHtOKcZ+6FhuebZbtqz2GtEicnZcjR9M2vXj/CSUi4wgxJ6Tp2rLwUCUUxI1mxl0gSIzxGQ+JqylFIpJfOb8+sM630rUEkdHFlzdXfEykKpZyGge4MkRrJZW8m/ue5iRrceCnlcaIIx4tFg4RZKrJmQVh9KghWbKoJwoLqWy08QgJhpeMq6hCc5ZdXSbtWda6qzt1luV7L4yjACZxCBRy4hjrcQhNagOEJnuEV3ozMeDHejY9F65qRzxzDHxifPyzUky0=</latexit> C(1) qq [3333] <latexit sha1_base64="aM9r3ItCNq9zC/Mj1pR0kjyhpes=">AAAB+3icbVBNS8NAEJ34WetXrEcvwSLUS0mqqMdCLx4r2A9IY9hst+3SzSbd3Ygl5K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5QcyoVLb9baytb2xubRd2irt7+weH5lGpLaNEYNLCEYtEN0CSMMpJS1HFSDcWBIUBI51g3Jj5nUciJI34vZrGxAvRkNMBxUhpyTdLDT+dTLKHtHJxnrmOhuebZbtqz2GtEicnZcjR9M2vXj/CSUi4wgxJ6Tp2rLwUCUUxI1mxl0gSIzxGQ+JqylFIpJfOb8+sM630rUEkdHFlzdXfEykKpZyGge4MkRrJZW8m/ue5iRrceCnlcaIIx4tFg4RZKrJmQVh9KghWbKoJwoLqWy08QgJhpeMq6hCc5ZdXSbtWda6qzt1luV7L4yjACZxCBRy4hjrcQhNagOEJnuEV3ozMeDHejY9F65qRzxzDHxifPyO4kyc=</latexit> C(3) qq [1111] <latexit sha1_base64="6emjwJ9B5x/MpIJMG/IGNasRJNc=">AAAB+3icbVDLSsNAFJ34rPUV69JNsAh1UzJR1GWhG5cV7APSGCbTSTt0MklnJmIJ+RU3LhRx64+482+ctllo64ELh3Pu5d57goRRqWz721hb39jc2i7tlHf39g8OzaNKR8apwKSNYxaLXoAkYZSTtqKKkV4iCIoCRrrBuDnzu49ESBrzezVNiBehIachxUhpyTcrTT+bTPKHrHZxnrsQOo7nm1W7bs9hrRJYkCoo0PLNr/4gxmlEuMIMSelCO1FehoSimJG83E8lSRAeoyFxNeUoItLL5rfn1plWBlYYC11cWXP190SGIimnUaA7I6RGctmbif95bqrCGy+jPEkV4XixKEyZpWJrFoQ1oIJgxaaaICyovtXCIyQQVjqusg4BLr+8SjpOHV7V4d1lteEUcZTACTgFNQDBNWiAW9ACbYDBE3gGr+DNyI0X4934WLSuGcXMMfgD4/MHJsOTKQ==</latexit> C(3) qq [1122] <latexit sha1_base64="u+H+gQ+D7zO/7usF+f4lMNayrUg=">AAAB+3icbVDLSsNAFJ34rPUV69JNsAh1UzJR1GWhG5cV7APSGCbTSTt0MklnJmIJ+RU3LhRx64+482+ctllo64ELh3Pu5d57goRRqWz721hb39jc2i7tlHf39g8OzaNKR8apwKSNYxaLXoAkYZSTtqKKkV4iCIoCRrrBuDnzu49ESBrzezVNiBehIachxUhpyTcrTT+bTPKHrHZxnrvQcaDnm1W7bs9hrRJYkCoo0PLNr/4gxmlEuMIMSelCO1FehoSimJG83E8lSRAeoyFxNeUoItLL5rfn1plWBlYYC11cWXP190SGIimnUaA7I6RGctmbif95bqrCGy+jPEkV4XixKEyZpWJrFoQ1oIJgxaaaICyovtXCIyQQVjqusg4BLr+8SjpOHV7V4d1lteEUcZTACTgFNQDBNWiAW9ACbYDBE3gGr+DNyI0X4934WLSuGcXMMfgD4/MHJsWTKQ==</latexit> C(3) qq [1221] <latexit sha1_base64="Y6NzwbaA4MLZzQtjgU6gfdPrSTA=">AAAB+3icbVDLTsJAFJ3iC/FVcemmkZjghnTAqEsSNi4xkUdSajMdpjBhOi0zUyNp+ituXGiMW3/EnX/jAF0oeJKbnJxzb+69x48Zlcq2v43CxubW9k5xt7S3f3B4ZB6XuzJKBCYdHLFI9H0kCaOcdBRVjPRjQVDoM9LzJ62533skQtKI36tZTNwQjTgNKEZKS55ZbnnpdJo9pNXGReZA2Gi4nlmxa/YC1jqBOamAHG3P/BoMI5yEhCvMkJQOtGPlpkgoihnJSoNEkhjhCRoRR1OOQiLddHF7Zp1rZWgFkdDFlbVQf0+kKJRyFvq6M0RqLFe9ufif5yQquHFTyuNEEY6Xi4KEWSqy5kFYQyoIVmymCcKC6lstPEYCYaXjKukQ4OrL66Rbr8GrGry7rDTreRxFcArOQBVAcA2a4Ba0QQdg8ASewSt4MzLjxXg3PpatBSOfOQF/YHz+ACnOkys=</latexit> C(3) qq [1133] <latexit sha1_base64="egWFez6fMDxwzg7GScHAPqIpP20=">AAAB+3icbVDLTsJAFJ3iC/FVcemmkZjghnTAqEsSNi4xkUdSajMdpjBhOi0zUyNp+ituXGiMW3/EnX/jAF0oeJKbnJxzb+69x48Zlcq2v43CxubW9k5xt7S3f3B4ZB6XuzJKBCYdHLFI9H0kCaOcdBRVjPRjQVDoM9LzJ62533skQtKI36tZTNwQjTgNKEZKS55ZbnnpdJo9pNXGRebARgO6nlmxa/YC1jqBOamAHG3P/BoMI5yEhCvMkJQOtGPlpkgoihnJSoNEkhjhCRoRR1OOQiLddHF7Zp1rZWgFkdDFlbVQf0+kKJRyFvq6M0RqLFe9ufif5yQquHFTyuNEEY6Xi4KEWSqy5kFYQyoIVmymCcKC6lstPEYCYaXjKukQ4OrL66Rbr8GrGry7rDTreRxFcArOQBVAcA2a4Ba0QQdg8ASewSt4MzLjxXg3PpatBSOfOQF/YHz+ACnSkys=</latexit> C(3) qq [1331] <latexit sha1_base64="Pbo06+93/RrELd7y+qHLV1uDmHU=">AAAB+3icbVDLTsJAFJ3iC/FVcelmIjHBDWnBqEsSNi4xkUdSajMdpjBhOi0zUyNp+ituXGiMW3/EnX/jAF0oeJKbnJxzb+69x48Zlcqyvo3CxubW9k5xt7S3f3B4ZB6XuzJKBCYdHLFI9H0kCaOcdBRVjPRjQVDoM9LzJ62533skQtKI36tZTNwQjTgNKEZKS55ZbnnpdJo9pNXGReY0NFzPrFg1awG4TuycVECOtmd+DYYRTkLCFWZISse2YuWmSCiKGclKg0SSGOEJGhFHU45CIt10cXsGz7UyhEEkdHEFF+rviRSFUs5CX3eGSI3lqjcX//OcRAU3bkp5nCjC8XJRkDCoIjgPAg6pIFixmSYIC6pvhXiMBMJKx1XSIdirL6+Tbr1mX9Xsu8tKs57HUQSn4AxUgQ2uQRPcgjboAAyewDN4BW9GZrwY78bHsrVg5DMn4A+Mzx8v7JMv</latexit> C(3) qq [3333] <latexit sha1_base64="WV9Jdui5339VUWjU6KulSgVe+9M=">AAAB+3icbVDLTsJAFJ3iC/FVcelmIjHBDemAUZckbFxiIo+k1GY6TGHC9MHM1Eia/oobFxrj1h9x5984QBcKnuQmJ+fcm3vv8WLOpLKsb6Owsbm1vVPcLe3tHxwemcflrowSQWiHRDwSfQ9LyllIO4opTvuxoDjwOO15k9bc7z1SIVkU3qtZTJ0Aj0LmM4KVllyz3HLT6TR7SKvoIrNRo4Ec16xYNWsBuE5QTiogR9s1vwbDiCQBDRXhWEobWbFyUiwUI5xmpUEiaYzJBI+orWmIAyqddHF7Bs+1MoR+JHSFCi7U3xMpDqScBZ7uDLAay1VvLv7n2Ynyb5yUhXGiaEiWi/yEQxXBeRBwyAQlis80wUQwfSskYywwUTqukg4Brb68Trr1GrqqobvLSrOex1EEp+AMVAEC16AJbkEbdAABT+AZvII3IzNejHfjY9laMPKZE/AHxucPJrqTKQ==</latexit> C(1) qq [1331] Figure 5.8: Limits on sample of coefficients of Class B 4-fermion operators in various flavor scenarios. δis the range of the 95% confidence level limits. Only a single operator is taken to be non-zero and the flavor structures not shown are marginalized over. Exact numbers are given in Table 5.5. MFV U(3)5 FlavLess -60 -40 -20 020 40 60 -20 -10 0 10 20 Cqq (1)[1,3,3,1] Cqq (1)[1,1,3,3] 95%CL MFV U(3)5 FlavLess -6-4-20246 -20 -10 0 10 20 Cqq (3)[1,3,3,1] Cqq (3)[1,1,3,3] 95%CL Figure 5.9: 95% CL limits on C(1) qq (left) and C(3) qq (right), in various flavor scenarios with all other coefficients taken to be 0. On the left we present C(1) qq [1331] vs. C(1) qq [1133] in the MFV (green), 3rd generation centric (orange) and flavorless (violet) scenarios. On the right we present C(3) qq [1331] vs. C(3) qq [1133] in the MFV (green), 3rd generation centric (orange) and flavorless (violet) scenarios. All the independent flavor structures not present in the plots are marginalized over. rently unavailable. Furthermore, it should be noted that within this approach, we lack projected experimental values, preventing us from performing a fit of the observables. However, we are not interested in the actual possible ranges of the coefficients because our goal is to make an estimation of the size of the bounds based on future colliders projections. 5.6. FUTURE PERSPECTIVES 97 0.001 0.01 0.1 1 10 100 |d| U(3)5 MFV Flavorless 3rd generation phobic + U(2)5 3rd generation specific U(2)5 95% CL limits on C lq (1,3) ranges from EWPOs (Coefficients are not all independent) <latexit sha1_base64="EGmi/KLejWr6mSA4QQfLPCudMh4=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBHqpmSKqMtCNy4r2AekMUymk3boZBJnJmIJ+RU3LhRx64+482+ctllo64ELh3Pu5d57goQzpR3n21pb39jc2i7tlHf39g8O7aNKV8WpJLRDYh7LfoAV5UzQjmaa034iKY4CTnvBpDXze49UKhaLOz1NqBfhkWAhI1gbybcrLT/jD/l9VkPnuYtQo+H5dtWpO3PAVYIKUgUF2r79NRjGJI2o0IRjpVzkJNrLsNSMcJqXB6miCSYTPKKuoQJHVHnZ/PYcnhllCMNYmhIaztXfExmOlJpGgemMsB6rZW8m/ue5qQ6vvYyJJNVUkMWiMOVQx3AWBBwySYnmU0MwkczcCskYS0y0iatsQkDLL6+SbqOOLuvo9qLabBRxlMAJOAU1gMAVaIIb0AYdQMATeAav4M3KrRfr3fpYtK5Zxcwx+APr8wcb0ZMi</latexit> C(1) lq [1122] <latexit sha1_base64="fSWaawF60YEGIjFRYd1GNgG4EwU=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16CRahXkpSRT0WevFYwX5AGsNmu22XbjZxdyOWkL/ixYMiXv0j3vw3btsctPXBwOO9GWbmBTGjUtn2t1FYW9/Y3Cpul3Z29/YPzMNyR0aJwKSNIxaJXoAkYZSTtqKKkV4sCAoDRrrBpDnzu49ESBrxOzWNiReiEadDipHSkm+Wm37KHrL7tOqcZe65huebFbtmz2GtEicnFcjR8s2v/iDCSUi4wgxJ6Tp2rLwUCUUxI1mpn0gSIzxBI+JqylFIpJfOb8+sU60MrGEkdHFlzdXfEykKpZyGge4MkRrLZW8m/ue5iRpeeynlcaIIx4tFw4RZKrJmQVgDKghWbKoJwoLqWy08RgJhpeMq6RCc5ZdXSadecy5rzu1FpVHP4yjCMZxAFRy4ggbcQAvagOEJnuEV3ozMeDHejY9Fa8HIZ47gD4zPHyT6kyg=</latexit> C(1) lq [3333] <latexit sha1_base64="Yt0kfq6ttVVbsqF/ohaBt1MelcQ=">AAAB+3icbVDLSsNAFJ3UV62vWJdugkWom5KJoi4L3bisYB+QxjCZTtqhk0mcmYgl5FfcuFDErT/izr9x2mahrQcuHM65l3vvCRJGpbLtb6O0tr6xuVXeruzs7u0fmIfVroxTgUkHxywW/QBJwignHUUVI/1EEBQFjPSCSWvm9x6JkDTmd2qaEC9CI05DipHSkm9WW37GHvL7rH5+lrsQOo7nmzW7Yc9hrRJYkBoo0PbNr8EwxmlEuMIMSelCO1FehoSimJG8MkglSRCeoBFxNeUoItLL5rfn1qlWhlYYC11cWXP190SGIimnUaA7I6TGctmbif95bqrCay+jPEkV4XixKEyZpWJrFoQ1pIJgxaaaICyovtXCYyQQVjquig4BLr+8SrpOA1424O1FrekUcZTBMTgBdQDBFWiCG9AGHYDBE3gGr+DNyI0X4934WLSWjGLmCPyB8fkDHumTJA==</latexit> C(3) lq [1122] <latexit sha1_base64="YCY29qHJnWw/OLHaofuP0iUZD2Q=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBHqpmRaUZeFblxWsA9IY5hMp+3QySTOTMQS8ituXCji1h9x5984bbPQ1gMXDufcy733BDFnSjvOt7W2vrG5tV3YKe7u7R8c2keljooSSWibRDySvQArypmgbc00p71YUhwGnHaDSXPmdx+pVCwSd3oaUy/EI8GGjGBtJN8uNf2UP2T3aaV+nrkI1eueb5edqjMHXCUoJ2WQo+XbX/1BRJKQCk04VspFTqy9FEvNCKdZsZ8oGmMywSPqGipwSJWXzm/P4JlRBnAYSVNCw7n6eyLFoVLTMDCdIdZjtezNxP88N9HDay9lIk40FWSxaJhwqCM4CwIOmKRE86khmEhmboVkjCUm2sRVNCGg5ZdXSadWRZdVdHtRbtTyOArgBJyCCkDgCjTADWiBNiDgCTyDV/BmZdaL9W59LFrXrHzmGPyB9fkDIfSTJg==</latexit> C(3) lq [1133] <latexit sha1_base64="tM20xy4Nt0m3wnTqrt2cAeC2q1g=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBHqpmRaUZeFblxWsA9IY5hMp+3QySTOTMQS8ituXCji1h9x5984bbPQ1gMXDufcy733BDFnSjvOt7W2vrG5tV3YKe7u7R8c2keljooSSWibRDySvQArypmgbc00p71YUhwGnHaDSXPmdx+pVCwSd3oaUy/EI8GGjGBtJN8uNf2UP2T3aaV+nrn1OkKeb5edqjMHXCUoJ2WQo+XbX/1BRJKQCk04VspFTqy9FEvNCKdZsZ8oGmMywSPqGipwSJWXzm/P4JlRBnAYSVNCw7n6eyLFoVLTMDCdIdZjtezNxP88N9HDay9lIk40FWSxaJhwqCM4CwIOmKRE86khmEhmboVkjCUm2sRVNCGg5ZdXSadWRZdVdHtRbtTyOArgBJyCCkDgCjTADWiBNiDgCTyDV/BmZdaL9W59LFrXrHzmGPyB9fkDIfyTJg==</latexit> C(3) lq [3311] <latexit sha1_base64="KaI9XgnOoVpYi1YmPZ7u5KhY4ow=">AAAB+3icbVDLSsNAFL3xWesr1qWbwSLUTUlaUZeFblxWsA9IY5hMp+3QycOZiVhCfsWNC0Xc+iPu/BunbRbaeuDC4Zx7ufceP+ZMKsv6NtbWNza3tgs7xd29/YND86jUkVEiCG2TiEei52NJOQtpWzHFaS8WFAc+p11/0pz53UcqJIvCOzWNqRvgUciGjGClJc8sNb2UP2T3aaV+njl1Ddczy1bVmgOtEjsnZcjR8syv/iAiSUBDRTiW0rGtWLkpFooRTrNiP5E0xmSCR9TRNMQBlW46vz1DZ1oZoGEkdIUKzdXfEykOpJwGvu4MsBrLZW8m/uc5iRpeuykL40TRkCwWDROOVIRmQaABE5QoPtUEE8H0rYiMscBE6biKOgR7+eVV0qlV7cuqfXtRbtTyOApwAqdQARuuoAE30II2EHiCZ3iFNyMzXox342PRumbkM8fwB8bnDygSkyo=</latexit> C(3) lq [3333] Figure 5.10: Limits on sample of coefficients of Class C 4-fermion operators in various flavor scenarios, with the independent flavor structures of each operator marginalized over. Exact numbers are given in 5.6. MFV 3rdGenCen 3rdGenPh+U(2)5 FlavLess -1012 -5 0 5 10 15 Clq (1)[3,3,3,3] Clq (1)[1,1,2,2] 95%CL U(2)5 MFV 3rdGenCen 3rdGenPh+U(2)5 FlavLess -8-6-4-2024 -2 -1 0 1 2 Clq (3)[3,3,3,3] Clq (3)[1,1,2,2] 95%CL Figure 5.11: 95% CL limits on C(1) lq (left) and C(3) lq (right), in various flavor scenarios with all other coefficients taken to be 0. On the left we present C(1) lq [3333] vs. C(1) lq [1122] in the MFV (green), 3rd generation centric (black), 3rd generation phobic + U(2)5(orange) and flavorless (violet) scenarios. On the right we present C(3) lq [3333] vs. C(3) lq [1122] in the U(2)5(blue), MFV (green), 3rd generation centric (black), 3rd generation phobic + U(2)5(orange) and flavorless (violet) scenarios.. All the independent flavor structures not present in the plots are marginalized over. Method Similarly to the approach used in the previous section, we evaluate only one non zero operator at time and then we explore different flavor scenarios. Due to the previous assumptions, we consider only single parameter limits, marginalizing over the other independent flavor structures. The marginalization is the same described in 5.5. As already mention, since we do not have projected experimental values, we set them to their ”best” theoretical SM value, indicated in Table 5.3. In this way, the χ2will not depend on the central value of the observables but only on the projected uncertainties. Finally, we do not expect that the theoretical uncertainties will remain the same at 98 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES Operator LO U(3)5NLO U(3)5LO MFV NLO MFV 3rdgen centric 3rdgen centric LO NLO C(3) ϕq [33] [-0.0029,0.020] [-0.0024,0.020] [-0.019,0.042] [-0.019,0.069] [-0.011,0.045] [-0.0062,0.069] C(3) ϕq [11] [-0.0029,0.020]* [-0.0024,0.020]* [-0.0093,0.024] [-0.012,0.020] 0 0 C(1) ϕq [33] [-0.0070,0.060] [-0.011,0.053] [-0.0067,0.060] [-0.027,0.055] [-0.011,0.045] [-0.016,0.059] C(1) ϕq [11] [-0.0070,0.060]* [-0.011,0.053]* [-0.032,0.10] [-0.071,0.18] 0 0 Cϕu[33] [-0.021,0.13]* [-0.0037,0.11] - [-0.042,0.27] - [-0.020,0.29] Cϕu[11] [-0.021,0.13] [-0.0037,0.11]* [-0.021,0.13] [-0.031,0.10] 0 0 Cϕd[33] [-0.21,-0.0024] [-0.18,0.0036] [-0.21,-0.0024] [-0.18,0.0036] [-0.28,0.012] [-0.37,0.00083] Cϕe[33] [-0.0033,0.019] [-0.0036,0.018] [-0.0033,0.019] [-0.0036,0.018] [-0.040,0.028] [-0.041,0.022] C(1) ϕl [33] [-0.011,0.0050] [-0.012,0.0049] [-0.011,0.0050] [-0.012,0.0049] [-0.026,0.033] [-0.029,0.029] C(3) ϕl [33] [-0.015,-0.00041] [-0.013,-0.00029] [-0.015,-0.00041] [-0.013,-0.00029] [-0.020,0.033] [-0.022,0.031] CuW [33] 0 0 - [-0.90,0.15] - [-0.90,0.15] CuB[33] 0 0 - [-0.61,0.061] - [-0.61,0.061] Table 5.4: 95% CL allowed ranges on 2-fermion operators with varying flavor assumptions described in the text. Other flavor structure of a given operator are marginalized over, and different coefficients set to 0. The entries labelled with a ”-” correspond to operators that do not contribute, while those labelled with a ”*” are not independent. The flavor structures given not in the table can be always derived from those in the table using the relations detailed in the text. moment the colliders will run. It is logical to hypothesize that they will decrease thus, in this section, we assume that they will be one half of the current ones [38]. Although simplistic, this assumption seems reasonable when considering the time future colliders will operate at full capacity (T > 20 years). In Table 5.7 are indicated the data used for this analysis including ILC-GigaZ and Fcc-ee expected experimental uncertainties and expected theoretical uncertainties. 5.6.1 Results We begin to show the results starting with Class A. We define the expected upgrade as ηFcc/ILC(Ci) = δcurrent(Ci) δFcc/ILC(Ci)(5.54) where δcurrent(Ci) and δF cc/ILC(Ci) are respectively the current and the projected size of the bounds of a certain coefficient Ci. Table 5.8 shows the obtained the 95% CL limits on some of the coefficients for different flavor scenario and different future colliders. In this class the current size of the bounds is O(C)≲1 for all the coefficients and in all the scenarios. Considering future colliders bounds we observe a significant improvement for all the operators. The minimum enhancement is observed for the coefficient C(1) ϕl [11] in the 3rd GenPh scenario, when compared to the ILC-GigaZ projected bounds ηILC(C(1) ϕl [11]) = 3.075. The maximum, again in the 3rd GenPh scenario, is for Cϕu[22] with ηfcc(Cϕu[22]) = 63.008. 5.6. FUTURE PERSPECTIVES 99 Operator U(3)5MFV U(2)53rdgen 3rdgen 3rdgen Flavorless specific phobic phobic+U(2)5 C(3) qq[1133] [-0.80,0.81] [-17.6,15.4] x 0 0 0 [-0.48,0.09] C(3) qq[1331] [-4.38,6.42] [-5.43,6.00] x 0 0 0 [-0.48,0.09]* C(3) qq[1122] [-0.80,0.81]* [-31.2,45.4] x 0 x [-303,375] [-0.48,0.09]* C(3) qq[1221] [-4.38,6.42]* [-44.8,26.3] x 0 x [-836,672] [-0.48,0.09]* C(3) qq[3333] [-5.15,7.21]* [-89.4,90.5]* x [-2.94,20.4] 0 0 [-0.48,0.09]* C(3) qq[1111] [-5.15,7.21]* [-60.9,56.6]* x 0 x [-460,369]* [-0.48,0.09]* C(1) qq[1133] [-0.20,1.84] [-18.6,17.1] x 0 0 0 [-0.89,1.52] C(1) qq[1331] [-1.39,0.94] [-60.0,51.6] x 0 0 0 [-0.89,1.52]* C(1) qq[1122] [-0.20,1.84]* [-1623,1408] x 0 x [-189,169] [-0.89,1.52]* C(1) qq[1221] [-1.39,0.94]* [-1276,1470] x 0 x [-109,106] [-0.89,1.52]* C(1) qq[3333] [-0.66,1.86]* [-2.62,3.42]* x [-0.26,1.84] 0 0 [-0.89,1.52]* C(1) qq[1111] [-0.66,1.86]* [-153,133]* x 0 x [-85.0,62.3]* [-0.89,1.52]* Cll[1133] [-5.65,2.52] [-5.65,2.52] [-46.1,22.2] 0 0 0 [-.0029,.021] Cll[1331] [-0.076,0.047] [-0.076,0.047] [-38.2,65.4] 0 0 0 [-.0029,.021]* Cll[1122] [-5.65,2.52]* [-5.65,2.52]* [-18.4,14.2] 0 [-11.4,17.4] [-9.3,5.3] [-.0029,.021]* Cll[1221] [-0.076,0.047]* [-0.076,0.047]* [-0.25,0.22] 0 [-0.16,0.27] [-0.13,0.088] [-.0029,.021]* Cll[3333] [-5.72,2.56]* [-5.72,2.56]* [-44.0,191] [-55.2,41.6] 0 0 [-.0029,.021]* Cll[2222] [-5.72,2.56]* [-5.72,2.56]* [-18.6,14.4]* 0 [-90.2,27.0] [-9.4,5.4]* [-.0029,.021]* Cll[1111] [-5.72,2.56]* [-5.72,2.56]* [-18.6,14.4]* 0 [-26.7,26.3] [-9.4,5.4]* [-.0029,.021]* Cee[1133] [-0.80,4.07] [-0.80,4.07] x 0 0 0 [-1.07,5.4] Cee[1122] [-0.80,4.07]* [-0.80,4.07]* x 0 x [-0.67,6.02] [-1.07,5.4]* Cee[3333] [-1.60,8.13]* [-1.60,8.13]* x [-36.7,19.7] 0 0 [-1.07,5.4]* Cee[1111] [-0.80,4.07]* [-0.80,4.07]* x 0 x [-1.34,12.0] [-1.07,5.4]* Cuu[1133] x x x x x x [-1.67,0.30] Cdd[3333] x x x [-428,5.54] x x [-68.4,2.38] Table 5.5: NLO results for 95% CL limits on Class B operators when the coefficients of all other operators are set to 0 and the different flavor structures of a given operator are marginalized over. The limits marked with a ”*” are obtained using the relations given in the text and do not represent independent coefficients. We label with an ”x” the cases where no limit can be derived. Finally we notice that for some the coefficients, depending on the scenario, the projected upgrade is almost the same for Fcc-ee and ILC-GigaZ. For example, in Fig. 5.12 we show the results for Cϕu[33]. For this coefficient, we do not see significant differences between the two future collider projections. Table 5.9 presents some of the 95% CL limits on 4-fermion coefficients for multiple scenarios. As discussed in the previous section, flavor has a dramatic effect on the 100 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES Operator U(3)5MFV U(2)53rdgen specific 3rdgen phobic +U(2)5Flavorless C(3) lq[1133] [0.026,0.90] [0.03,0.90] [-0.0078,1.45] 0 0 [0.026,0.90] C(3) lq[3311] [0.026,0.90]* [-0.66,1.13] [-9.62,22.2] 0 0 [0.026,0.90]* C(3) lq[1122] [0.026,0.90]* [0.03,0.90]* [-1.43,1.78] 0 [-0.98,2.02] [0.026,0.90]* C(3) lq[3333] [0.026,0.90]* [-0.66,1.13]* [-7.43,3.69] [-1.33,0.88] 0 [0.026,0.90]* C(1) lq[1122] [-0.66,0.27] [-0.73,0.27] x 0 [-5.20,15.2] [-0.66,0.27] C(1) lq[3333] [-0.66,0.27]* [-1.50,1.79] x [-1.50,1.62] 0 [-0.66,0.27]* Clu[1122] [-0.20,0.49] [-0.20,0.55] x 0 [-2.6,7.6] [-0.20,0.49] Clu[3333] [-0.20,0.49]* [-1.33,1.13] x [-1.35,1.27] 0 [-0.20,0.49]* Cqe[1122] [-0.20,1.06] [-66.2,142] x 0 [-25.7,3.34] [-0.20,1.06] Cqe[3333] [-0.20,1.06]* [-2.38,6.35] x [-2.26,1.25] 0 [-0.20,1.06]* Ced[1122] [-3.86,15.3] [-3.86,15.3] x 0 [-3.29,25.8] [-3.86,15.3] Ced[3333] [-3.86,15.3]* [-3.86,15.3]* x [-117,42.8] 0 [-3.86,15.3]* Cld[1122] [-9.11,3.26] [-9.11,3.26] x 0 [-15.2,5.21] [-9.11,3.26] Cld[3333] [-9.11,3.26]* [-9.11,3.26]* x [-72.9,59.3] 0 [-9.11,3.26]* Cle[1122] [-7.09,8.13] [-7.09,8.13] x 0 [-10.5,14.5] [-7.09,8.13] Cle[3333] [-7.09,8.13]* [-7.09,8.13]* x [-92.8,55.5] 0 [-7.09,8.13]* Ceu[1122] [-0.81,0.16] [-0.88,0.11] x 0 [-12.9,1.66] [-0.81,0.16] Ceu[3333] [-0.81,0.16]* [-1.07,1.73] x [-1.05,1.95] 0 [-0.81,0.16]* C(1) ud [1122] [-0.30,7.68] [-154,90.2] x 0 [-13.9,75.7] [-0.30,7.68] C(1) ud [3333] [-0.30,7.68]* [-9.3,27.0] x [-0.24,18.4] 0 [-0.30,7.68]* C(1) qd[1122] [-13.3,0.47] [-202,128] x 0 [-29.4,139] [-13.3,0.47] C(1) qd[3333] [-13.3,0.47]* [-27.2,8.27] x [-24.4,0.11] 0 [-13.3,0.47]* C(1) qu[1122] [-2.82,0.54] x x 0 [-71.0,14.1] [-2.82,0.54] C(1) qu[3333] [-2.82,0.54]* x x [-3.07,0.44] 0 [-2.82,0.54]* Table 5.6: NLO results for 95% CL limits on Class C operators when the coefficients of all other operators are set to 0 and the different flavor structures of a given operator are marginalized over. The limits marked with a ”*” are obtained using the relations given in the text and do not represent independent coefficients. We label with an ”x” the cases where no limit can be derived. flavor bounds. In particular, depending on the scenarios, we have seen that some of the coefficients in Class B & C may have a range of values O(δ(Ci)) ≫10. It is interesting to check if a coefficient that is not significantly constrained with the current experimental precision, can provide more useful information in future colliders. In Fig 5.13, we show the results for C(1) qq [1122] and C(3) qq [1331]. For C(1) qq [1122], we see that both the scenarios would not constrain the coefficient sufficiently, in order to have significant bounds. Indeed, although we observe ηF cc = 33.50 in the MFV 6.3. LO DY CALCULATIONS 107 Figure 6.1: LO Feynman diagrams for DY process. Realized with [185] . A∼ASM + Σi C6 i Λ2A6 i,LO + Σj D6 j 16π2Λ2A6 j,NLO .(6.2) Here ASM , A6 i,LO, and A6 j,NLO represent the SM, dimension-6 tree level, and dimension-6 1−loop contributions, respectively. We define the linear SMEFT result as: |A|2 lin ≡ | ASM |2+2Re(ΣiA∗ SM C6 i Λ2A6 i,LO) +2Re(ΣiA∗ SM D6 i 16π2Λ2A6 j,NLO) (6.3) where we notice that Eq. 6.3 is not positive definite. For our purposes, we define the quadratic SMEFT result as: |A|2 quad ≡ | ASM + Σi C6 i Λ2A6 i,LO + Σj D6 j 16π2Λ2A6 j,NLO |2.(6.4) In SMEFT phenomenology studies and global fits, the quantity defined in Eq. 6.4 is commonly used. Nevertheless, it is important to note that we do not include two types of O(1 Λ4) terms in Eq. 6.4: the interference of the dimension-8 operators with the SM result [184], and the double insertions of the dimension-6 operators in the amplitude. In the following we will neglect these terms, because they are beyond the scope of current NLO EW SMEFT calculations. 6.3 LO DY calculations In this section we show the LO results for q(p1)q(p2)→l+(p3)l−(p4). Fig 6.1 shows the two possible topologies present at LO. In the following, we consider up quark initial states, the down quark case is given in Appendix A.1. In terms of helicity amplitudes we have: MXY =u(p2)γµPXu(p1)·u(p3)γµPYu(p4),(6.5) 108 CHAPTER 6. DRELL YAN PROCESS with PL,R =1∓γ5 2. At tree level we have ALO = ΣXY GXY MXY (6.6) with GXY =GSM XY +δGXY (6.7) where the δterm represents the SMEFT contributions. Therefore, for the up quark initial states we have GSM LR =−2(s−4m2 W)(m2 W−m2 Z) 3sv2(s−m2 Z)GSM LL =2sm2 W+ 8m4 W+sm2 Z−8m2 Wm2 Z 3s2v2−3sv2m2 Z GSM RL =−4(s−2m2 W)(m2 W−m2 Z) 3sv2(s−m2 Z)GSM RR =8(s−m2 W)(−m2 W+m2 Z) 3sv2(s−m2 Z)(6.8) and δGLR =1 Λ2(CϕD(4m4 W−sm2 Z) 3s2−3sm2 Z +2CϕWBm2 W(−5s+ 8m2 W)q−1 + m2 Z m2 W 3s(s−m2 Z)−Cqe[1122] +(4m2 W−m2 Z)Cϕe[22] 3s−3m2 Z +2(−m2 W+m2 Z)C(1) ϕq[11] s−m2 Z +2(m2 W−m2 Z)C(3) ϕq[11] s−m2 Z −(s−4m2 W) (m2 W−m2 Z) 3s(s−m2 Z)−Cll[1221] −Cll[2112] + 2C(1) ϕl[11] + 2C(3) ϕl[22]), δGRL =1 Λ2(CϕD(4m4 W−2sm2 Z) 3s2−3sm2 Z +4CϕWBm2 W(−3s+ 4m2 W)q−1 + m2 Z m2 W 3s(s−m2 Z)−Clu[2211] +4(m2 W−m2 Z)C(1) ϕl[22] 3(s−m2 Z)+4(m2 W−m2 Z)C(3) ϕl[22] 3(s−m2 Z)+(−2m2 W+m2 Z)Cϕu[11] s−m2 Z −2(s−2m2 W)(m2 W−m2 Z) 3sv2(s−m2 Z)−Cll[1221] −Cll[2112] + 2C(1) ϕl[11] + 2C(3) ϕl[22]), δGLL =1 Λ2(CϕD(8m4 W−sm2 Z) 6s2−6sm2 Z +2CϕWBm2 W(−3s+ 8m2 W)q−1 + m2 Z m2 W 3s(s−m2 Z)−C(1) lq [2211] +C(3) lq [2211] + (4m2 W−m2 Z)C(1) ϕl[22] 3s−3m2 Z +(4m2 W−m2 Z)C(3) ϕl[22] 3s−3m2 Z +(−2m2 W+m2 Z)C(1) ϕq[11] s−m2 Z +(2m2 W−m2 Z)C(3) ϕq[11] s−m2 Z +2sm2 W+ 8m4 W+sm2 Z−8m2 Wm2 Z 6s2v2−6sv2m2 Z −Cll[1221] −Cll[2112] + 2C(1) ϕl[11] + 2C(3) ϕl[22]), 6.4. SMEFT NLO DY CALCULATIONS: VIRTUAL CONTRIBUTIONS 109 δGRR =1 Λ2(CϕD(4m4 W−4sm2 Z) 3s2−3sm2 Z +16CϕWBm2 W(−s+m2 W)q−1 + m2 Z m2 W 3s(s−m2 Z)−Ceu[2211] +4(m2 W−m2 Z)Cϕe[22] 3(s−m2 Z)+2(−m2 W+m2 Z)Cϕu[11] s−m2 Z +4(s−m2 W)(−m2 W+m2 Z) 3sv2(s−m2 Z)(−Cll[1221] −Cll[2112] + 2C(1) ϕl[11] + 2C(3) ϕl[22])), (6.9) where s= (p1+p2)2. The SMEFT coefficients contributing at this order are CϕWB, CϕD, C(3) ϕl [11], C(3) ϕl, [22], C(1) ϕl [22], Cϕe,[22], C(3) ϕq,[11], C(1) ϕq [11], Cϕu,[11], Cll,[1221], Cll[2112], C(3) lq [2211], C(1) lq, [2211], Cqe[1122], Clu[2211], Ceu[2211].(6.10) Summing over helicity amplitudes and averaging over the spin and color we have |ALO(s, t)|2≡1 12ΣXY |GXY |2|MXY |2 ≡1 12 |ALO |2.(6.11) The partonic cross sections, averaged over spin and color are given by: dˆσ dt =1 48πs2(|GLL |2+|GRR |2)u2+ (|GLR |2+|GRL |2)t2 ˆσLO =1 16πs2Z0 −s dt |ALO(s, t)|2,(6.12) where t= (p1−p3)2and u= (p1−p4)2. The numerical results of the tree level 4−fermion operators has been extensively studied in the literature [186,163,187,188, 189]. According to [190], the Drell Yan process can give useful information not only from the high-energy regime. Indeed, due to its large cross section, it allows us to put significant constraints on the dimension-6 SMEFT coefficients even at relatively low energies, providing a comprehensive understanding of the SMEFT contributions to the neutral DY process. Indeed, even at LHC energies, the one-loop EW contributions to Zdecays in the SMEFT can reach O(10 −20)%. In previous work [191], where a set of SMEFT contributions to neutral Drell-Yan process were computed, it has been shown that HL-LHC will have significant sensitivity to these effects. 6.4 SMEFT NLO DY calculations: Virtual contributions In this section, we provide a detailed account of the calculation of virtual NLO corrections within the SMEFT for the Drell-Yan process. The notation follows that of previous works 110 CHAPTER 6. DRELL YAN PROCESS a) b) c) d) e) Figure 6.2: Samples of Feynman diagrams contributing at NLO for DY process. Realized with [185]. a) Renormalization counterterms, b) Vertex, c) Box, d) 4-fermion vertex, e) 4-fermion box. [123,130,166,124]. The relevant operators for the 1−loop amplitudes include those from Eq. 6.10 as well as additional ones listed in Eq. 6.13 CW, Cϕ□, CϕW , CϕB, CuW [33], CuB[33], C(3) ϕl [ii], C(1) ϕl [ii], Cϕe[ii], C(3) ϕq [ii], C(1) ϕq [ii], Cϕu[ii], Cϕd[ii], Cee[ijkl], Cll[ijkl], C(1) qq [ijkl], C(3) qq [ijkl], Cuu[ijkl], Ced[iijj], C(1) ud [iijj], C(1) qu [iijj], Cqe[iijj], C(1) qd [iijj], Clu[iijj], Clu[ijkl], C(3) lq [iijj], C(1) lq [iijj], Cle[iijj], Cld[iijj], Ceu[iijj],(6.13) where, as already discussed in Chapter 5, we notice that only operators containing pairs of identical flavor fermions can contribute. In this work, we will start the exploration of the effects in DY production of 4-fermion operators that first arise at 1-loop order as well as the corrections to the 4-fermion operators appearing at LO. The virtual corrections to Drell-Yan encounter both UV and IR divergences, which we handle by working in d= 4−2ϵdimensions using the dimensional regularization introduced in Chapter 2. We schematize the contributions to the virtual NLO corrections into box, vertex, 4-fermion vertex triangle, 4-fermion box triangle and renormalization counterterm contributions, as shown in Fig. 6.2. We notice that our computation includes only NLO corrections that interfere with the LO amplitudes. Unlike [165], due to the 4 fermion operators inclusion, we see that also topologies that do not appear at the SM level are involved in the calculation. We use FeynRules [192] routines to convert RξFeynman rules [100] for the SMEFT into a FeynArts [1] model file, then using FeynCalc [2] we compute amplitudes and we reduce 1-loop integrals to Passarino-Veltman integrals. We do not fix the gauge in the SMEFT Lagrangian and we verify at the end the cancellation of ξterms, in order to have a further check on our computation. Excluding the renormalization counterterms, we determine the one loop contributions by contracting NLO amplitudes with MXY , 6.4. SMEFT NLO DY CALCULATIONS: VIRTUAL CONTRIBUTIONS 111 acting as projectors in the massless fermion limit. To be consistent in the SMEFT expansion, our approach requires to consider Feynman diagrams with at most a single SMEFT operator insertion. This ensures renormalizability by avoiding divergences that cannot be countered without introducing higher-order operators at leading order. As already discussed in Section 2.4.1, the treatment of γ5in D dimension requires particular care. Indeed, the complex Dirac traces of the SMEFT raise difficulties in the calculation process. To evaluate the Dirac traces, our choice between the NDR or Kreimer scheme depends on the specific class of Feynman diagrams illustrated in Figure 6.2. Renormalization counterterms The propagator contributions are derived directly from the SMEFT gauge boson and fermion 2-point functions [151,152]. In the renormalization of the LO amplitude, we employed a mixed OS/MS scheme introduced in Chapter 2. SM parameters are renormalized in the OS scheme, while SMEFT operator coefficients are treated as MS objects. We use the {Gµ, MZ, MW}scheme for input parameters, with corrections to gauge boson masses defined as M2 V=M2 0,V −ΠV V (M2 V),(6.14) where V=Z, W, the 0 indicates the bare quantities and ΠV V (M2 V) are the 2-point functions of Refs. [151,152] computed on-shell. As in Chapter 5, the relation of Eq. 6.1 is modified at 1-loop, Gµ+1 2√2Λ2Cll[2112] + Cll[1221]−√2 2Λ2C(3) ϕl [11] + C(3) ϕl [22]≡1 √2v0 (1 + ∆r) (6.15) where v0is the square of the minimum of the potential at tree level and the analytic expression for ∆rin the SMEFT is given in Ref. [123]. The effective field theory coefficients of the dimension-6 operators are treated as MS quantities, defined at the scale of the measurement, i.e. the EW scale. The poles of the one-loop coefficients Ciare extracted from Refs. [97,98,99], Ci(µR) = C0,i −1 2ˆϵ 1 16π2γijCj,(6.16) where µis the renormalization scale, γij are the one-loop anomalous dimensions, µR dCi dµR =1 16π2γijCj,(6.17) and ˆϵ−1≡ϵ−1−γE+ log(4π). Vertex Vertex contributions involve 1PI vertex amplitudes that are extracted by contracting amplitudes with MXY . We notice that the presence of SMEFT operators does not raise 112 CHAPTER 6. DRELL YAN PROCESS the issue of the D-dimensional evaluation of the Dirac trace, because at most 4 Dirac matrices are involved. Thus, in this case we can use the NDR scheme when γ5is involved in the calculation. Box In this case we considered ”SM type” box diagrams, meaning that in this subsection we studied box topologies that appear already at the SM level. As in the previous case we extract the box contributions by contracting amplitudes with MXY , using them as ”projectors”. In this case, we have fermion chains that involve up to 6 Dirac matrices and therefore we use the Kreimer scheme using a consistent reading point. As discussed in Section 2.4.3, the reading point prescription effectively specifies with which Dirac matrix to start the non-cyclic trace in the Kreimer scheme. Our choice is to begin reading the Dirac traces from the projector insertion. 4-fermion vertex triangle The diagrams discussed in this subsection are the NLO corrections to the 4-fermion operators appearing at LO; in particular we considered the case where a gauge boson is ”connecting” 2 leptons or 2 quarks. The fermion chains appearing in these diagrams have the some form as those in the vertex 6.4 and, after projecting through MXY , we can use the NDR scheme to evaluate the Dirac traces. 4-fermion box triangle Finally, we discuss NLO 4 fermion feynman diagrams with a gauge boson connecting different type of fermions, e.g quark and lepton. According to the Kreimer scheme when we have more than 5 Dirac matrices and we are dealing with several axial current we should employ a symmetric reading point. However, even if we have several γ5 involved in the calculations we are handling open fermion chains, not closed fermion loops. This means we can reduce the fermion chains using the Dirac algebra before we project through MXY . After this operation, the chains appearing in these diagrams will have the same form of 6.4. By the same reasoning, we use the same reading point of the box diagrams, starting to read our Dirac traces from the projector insertion. Finally, we notice that not employing a reading point would not give us the correct results for these diagrams, as expected. Indeed, as explicitly verified, this choice would lead to gauge and chiral violating terms. 6.5. SMEFT NLO DY CALCULATIONS: REAL CONTRIBUTIONS 113 Figure 6.3: Real contributions Feynman diagrams for DY process. Realized with [185]. 6.5 SMEFT NLO DY calculations: Real Contributions The NLO result requires the real contributions from both photon and gluon emission qq →l+l−γ, qq →l+l−g . (6.18) In Fig 6.3 we show a sample of Feynman diagrams contributing at this order. The IR singularities are regulated using phase space slicing with dimensional regularization D= 4 −2ϵ, as in Refs. [193,194,195,196]. After regulating the IR singularities by including the collinear and soft limits of the 2 →3 contributions, ϵcan be set to 0. As discussed in [197,198], the soft limits of the 2 →3 scattering processes have a universal form that is the same for both the SM and the SMEFT. The soft contribution is characterized by photon or gluon energies with the condition Eγ, Eg<∆E, where ∆Erepresents a chosen infinitesimally small cutoff. The soft partonic cross section is defined in terms of the lowest order SMEFT cross section of Eq. 6.11, dˆσsoft = = 1 16πs2Z0 −s dt |ALO(s, t)|2αδEW soft(s, t) + αSCFδQCD soft (s).(6.19) It is important to point out that αis redefined within the context of the SMEFT. At this order, the factors influencing αarise from the Zmass definition and the SU(2)/U(1) mixing α=√2Gµm2 W(m2 Z−m2 W) πm2 Z−1 Λ2 m2 W 2m2 Zπm2 WCϕD + 4qm2 W(m2 Z−m2 W)CϕWB + (m2 Z−m2 W)2(C(3) ϕl [11] + C(3) ϕl [22]) −Cll[1221] −Cll[2112].(6.20) We notice that no adjustments are introduced due to factors like CϕW . 114 CHAPTER 6. DRELL YAN PROCESS The soft functions are δEW soft =Q2 qfq(s) + Q2 lfl(s) + 2QqQlh(s, t) δQCD soft =fq(s) ff(s) = log µ2 4∆E2 πϵ + 2 log2µ2 4∆E2−π2 4π+1 πϵ2 h(s, t) = 1 2log2µ2 4∆E2−Li2s t+ 1+ log µ2 4∆E2log −s t−π2 4 π+ log µ2 4∆E2+ log −s t πϵ +1 πϵ2. (6.21) For soft gluons, take Ql= 0 and αQ2 q→αsCF. Adding the virtual one loop contributions and Eq. 6.21, the ϵ2dependences cancel, leaving just the ϵsingular terms. Consider photons emitted from the initial quark within an angle δθ. The initial state collinear contributions are absorbed into the definition of the PDFs [193]. Therefore, in the MS scheme, we have ˆσq PDF =Q2 q 16πs2Z0 −s dt |A(s, t)LO |2α δPDF (s) δPDF (s) = 1 π 1 2log µ2 µ2 f!4 log 2∆E √s+ 3+ 2 log 2∆E √s+3 2 ϵ ,(6.22) where µFis the factorization scale. The QCD contribution is found with the replacement αQ2 q→αsCF. When the photons are emitted from the final state lepton, ˆσl coll =Q2 l 16πs2Z0 −s dt |A(s, t)LO |2αδcl(s) (6.23) with δcl(s) given by δcl(s) = 1 π1 6(−6 log(2∆E √s)(log(δθ)−2 log(2µ) + log(s)) + 3(4 log(2∆E √s) + 3) log( 2µ √s√δθ ) −9 2(log(δθ)−2 log(2µ) + log(s)) −12 log2(2∆E √s)−4π2+ 39) + 2 log(2∆E √s) + 3 2 ϵ (6.24) Finally, putting these terms together we have: ˆσa=1 16πs2 1 12 Z0 −s dt |ALO(s, t) + δANLO(s, t)|2,(6.25) where 6.6. PHENOMENOLOGY 115 200 400 600 800 Mll [GeV] 0 2 4 6 8 10 [(dσ (EFT) / dσ(SM)-1] (%) Flavorless U(2)5, Tqq 3[1221]=0 U(3)5, Tqq 3[1221]=0 √S=13.5 TeV, Drell-Yan Flavor Scenarios, Tqq 3[1122]=1 C(3) qq [1221] C(3) qq [1221] C(3) qq [1122] Figure 6.4: O(3) qq in different flavor assumptions, √S= 13.5 TeV δANLO(s, t) = δAvirt(s, t) + 1 2ALO(s, t)αδEW soft(s, t) + αsCFδQCD soft (s) +(αQ2 q+αsCF)δPDF (s) + αδcl(s) (6.26) and δAvirt(s, t) is the one loop renormalized amplitude calculated in Sect. 6.4 and as always ALO and αare SMEFT quantities. We note that Eq. 6.26 is a finite object and we are free to apply it at linear or quadratic order in the SMEFT following Eqs. 6.3 and 6.4. Indeed, ˆσahas no dependence on ϵ, thus we can take the limit ϵ→0. 6.6 Phenomenology The analytical results obtained in this chapter are going to be implemented in Montecarlo programs, in particular in POWHEG code, in order to include also contributions coming from PDF factorization and hard non-collinear contributions from the 2 →3 process. Being an ongoing project, we can not present a detailed phenomenological analysis at this stage. However, we can provide a preliminary study of the phenomenology involving 4-fermion operators first appearing at NLO. For the flavor scenarios, we use the same notation introduced in Section 5.3. In Fig 6.4, we show our results for O(3) qq in different flavor assumptions for √s= 13.5 TeV. Specifically, we are plotting the percentage difference between the SMEFT and SM differential cross sections as a function of dilepton mass. Assuming C(3) qq [1122] = 116 CHAPTER 6. DRELL YAN PROCESS 100 200 300 400 500 600 700 800 Mll [GeV] 0 0.2 0.4 0.6 0.8 [(dσ (EFT) / dσ(SM)-1] (%) Flavorless U(2)5, Tqq 3[1221]=0 U(3)5, Tqq 3[1221]=0 Drell-Yan Flavor Scenarios, Tqq 3[1122]=.1 Solid (dotted), ECM=13.5 (100) TeV C(3) qq [1221] C(3) qq [1221] Drell Yan C(3) qq [1122]=0.1 Figure 6.5: O(3) qq different center-of-mass energies. 1 and setting C(3) qq [1221] = 0 we notice that in all the flavor scenarios, the SMEFT contribution grows with the dilepton energy, with U(3)5reaching 9% while the flavorless and U(2)2remaining below 4%. In Fig 6.5, we compare different center-of-mass energies: √s= 13.5 TeV, which is the energy at which the LHC runs, and √s= 100 TeV, the expected energy for the future collider Fcc-ee. We set C(3) qq [1122] = 0.1 and C(3) qq [1221] = 0. Due to the linear dependence of the SMEFT coefficient on the cross section, this results in a simple rescaling of dσEFT. We see that increasing the energy scale doesn’t significantly affect the contributions in any of the flavor scenarios. However, we do observe a systematic reduction of a few percent between the energy scales. In Fig 6.6, we show the correlation between the coefficients C(3) qq [1122] and C(3) qq [1221] in the U(2)5scenario, assuming dσEFT dσSM −1>2% or 10%, Mll >500 GeV.(6.27) According to [199], the current DY relative uncertainty for muon dilepton mass Mll >500 GeV is ≳19% . Under these conditions, we can detect BSM events in this channel only if they deviate more than 19% from the SM. It is reasonable to assume that the precision will increase at future colliders and that this upgrade will allow us to measure possible deviations with higher sensitivity. For this reason, in this analysis we considered two scenarios, a ”conservative” one at 10% and a more ”optimistic” one at 2% . We notice that, due to the linear approximation, we can not set bounds on the SMEFT A.2. DRELL YAN 123 δGLL =1 Λ2(−8CϕWBpm6 W(−m2 W+m2 Z) 3s2−3sm2 Z−CϕD(4m4 W+sm2 Z) 6s2−6sm2 Z−C(1) lq [2211] −C(3) lq [2211] −(2m2 W+m2 Z)C(1) ϕl[22] 3s−3m2 Z−(2m2 W+m2 Z)C(3) ϕl[22] 3s−3m2 Z +(−2m2 W+m2 Z)C(1) ϕq[11] s−m2 Z +(−2m2 W+m2 Z)C(3) ϕq[11] s−m2 Z +−4m2 W(s+m2 W)+(s+ 4m2 W)m2 Z 6sv2(s−m2 Z) (−Cll[1221] −Cll[2112] + 2C(1) ϕl[11] + 2C(3) ϕl[22]))). (A.2) 124 APPENDIX A. A.3 EWPOs Other plots for 2-fermion operators In this subsection, we show a collection of plots involving the 2-fermion operators, as described in Chapter 5. NLO:Solid;LO:Dashed MFV U(3)5 3rdGenCen -0.01 0.00 0.01 0.02 -0.02 0.00 0.02 0.04 0.06 0.08 Cϕq (3)[1,1] Cϕq (3)[3,3] 95%CL NLO:Solid;LO:Dashed 3rdGenPh 3rdGenPh+U(2)5 FlavLess -0.04 -0.02 0.00 0.02 0.04 -0.05 0.00 0.05 0.10 Cϕq (3)[1,1] Cϕq (3)[2,2] 95%CL Figure A.1: 95% CL limits on C(3) ϕq [ij] under flavor assumptions described in the text. Results at LO are drawn with dashed lines, results at NLO are drawn with solid lines. On the left we present C(3) ϕq [11] vs. C(3) ϕq [33] in the U(3)5(black), MFV (blue) and 3rd generation centric (magenta) scenarios. In these scenarios C(3) ϕq [22] = C(3) ϕq [11]. On the right we present C(3) ϕq [11] vs. C(3) ϕq [22] in the 3rd generation phobic (green), 3rd generation phobic + U(2)5(orange) and flavorless (violet) scenarios. In the first two scenarios C(3) ϕq [33] = 0, while in the flavorless scenario C(3) ϕq [33] = C(3) ϕq [22] = C(3) ϕq [11]. All other coefficients are set to 0. A.3. EWPOS 125 NLO:Solid;LO:Dashed MFV U(3)5 3rdGenCen -0.05 0.00 0.05 0.10 -0.1 0.0 0.1 0.2 0.3 Cϕu[1,1] Cϕu[3,3] 95%CL NLO:Solid;LO:Dashed 3rdGenPh 3rdGenPh+U(2)5 FlavLess -0.2 0.0 0.2 0.4 -0.2 0.0 0.2 0.4 Cϕu[1,1] Cϕu[2,2] 95%CL Figure A.2: 95% CL limits on Cϕu[ij] under flavor assumptions described in the text. Results at LO are drawn with dashed lines, results at NLO are drawn with solid lines. On the left we present Cϕu[11] vs. Cϕu[33] in the U(3)5(black), MFV (blue) and 3rd generation centric (magenta) scenarios. In these scenarios Cϕu[22] = Cϕu[11], also notice that for the MFV scenario only the NLO results can be obtained. On the right we present Cϕu[11] vs. Cϕu[22] in the 3rd generation phobic (green), 3rd generation phobic +U(2)5(orange) and flavorless (violet) scenarios. In the first two scenarios Cϕu[33] = 0, while in the flavorless scenario Cϕu[33] = Cϕu[22] = Cϕu[11]. All other coefficients are set to 0. NLO:Solid;LO:Dashed MFV 3rdGenCen -0.20 -0.15 -0.10 -0.05 0.00 -0.4 -0.3 -0.2 -0.1 0.0 Cϕd[1,1] Cϕd[3,3] 95%CL NLO:Solid;LO:Dashed 3rdGenPh 3rdGenPh+U(2)5 FlavLess -0.3 -0.2 -0.1 0.0 0.1 -0.6 -0.4 -0.2 0.0 0.2 Cϕd[1,1] Cϕd[2,2] 95%CL Figure A.3: 95% CL limits on Cϕd[ij] under flavor assumptions described in the text. Results at LO are drawn with dashed lines, results at NLO are drawn with solid lines. On the left we present Cϕd[11] vs. Cϕd[33] in the MFV (blue) and 3rd generation centric (magenta) scenarios. In these scenarios Cϕd[22] = Cϕd[11]. On the right we present Cϕd[11] vs. Cϕd[22] in the 3rd generation phobic (green), 3rd generation phobic + U(2)5 (orange) and flavorless (violet) scenarios. In the first two scenarios Cϕd[33] = 0, while in the flavorless scenario Cϕd[33] = Cϕd[22] = Cϕd[11]. All other coefficients are set to 0. 126 APPENDIX A. NLO:Solid;LO:Dashed MFV 3rdGenCen -0.005 0.000 0.005 0.010 0.015 0.020 -0.04 -0.02 0.00 0.02 Cϕe[1,1] Cϕe[3,3] 95%CL NLO:Solid;LO:Dashed 3rdGenPh 3rdGenPh+U(2)5 FlavLess -0.005 0.000 0.005 0.010 0.015 0.020 0.00 0.02 0.04 0.06 Cϕe[1,1] Cϕe[2,2] 95%CL Figure A.4: 95% CL limits on Cϕe[ij] under flavor assumptions described in the text. Results at LO are drawn with dashed lines, results at NLO are drawn with solid lines. On the left we present Cϕe[11] vs. Cϕe[33] in the MFV (blue) and 3rd generation centric (magenta) scenarios. In these scenarios Cϕe[22] = Cϕe[11]. On the right we present Cϕe[11] vs. Cϕe[22] in the 3rd generation phobic (green), 3rd generation phobic + U(2)5 (orange) and flavorless (violet) scenarios. In the first two scenarios Cϕe[33] = 0, while in the flavorless scenario Cϕe[33] = Cϕe[22] = Cϕe[11]. All other coefficients are set to 0. NLO:Solid;LO:Dashed MFV 3rdGenCen -0.010 -0.005 0.000 0.005 -0.02 0.00 0.02 0.04 Cϕl (1)[1,1] Cϕl (1)[3,3] 95%CL NLO:Solid;LO:Dashed 3rdGenPh 3rdGenPh+U(2)5 FlavLess -0.015 -0.010 -0.005 0.000 0.005 0.010 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 Cϕl (1)[1,1] Cϕl (1)[2,2] 95%CL Figure A.5: 95% CL limits on C(1) ϕl [ij] under flavor assumptions described in the text. Results at LO are drawn with dashed lines, results at NLO are drawn with solid lines. On the left we present C(1) ϕl [11] vs. C(1) ϕl [33] in the MFV (blue) and 3rd generation centric (magenta) scenarios. In these scenarios C(1) ϕl [22] = C(1) ϕl [11]. On the right we present C(1) ϕl [11] vs. C(1) ϕl [22] in the 3rd generation phobic (green), 3rd generation phobic +U(2)5(orange) and flavorless (violet) scenarios. In the first two scenarios C(1) ϕl [33] = 0, while in the flavorless scenario C(1) ϕl [33] = C(1) ϕl [22] = C(1) ϕl [11]. All other coefficients are set to 0. A.3. EWPOS 127 NLO:Solid;LO:Dashed MFV 3rdGenCen -0.015 -0.010 -0.005 0.000 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 Cϕl (3)[1,1] Cϕl (3)[3,3] 95%CL NLO:Solid;LO:Dashed 3rdGenPh 3rdGenPh+U(2)5 FlavLess -0.020 -0.015 -0.010 -0.005 0.000 0.005 0.010 -0.025 -0.020 -0.015 -0.010 -0.005 0.000 0.005 Cϕl (3)[1,1] Cϕl (3)[2,2] 95%CL Figure A.6: 95% CL limits on C(3) ϕl [ij] under flavor assumptions described in the text. Results at LO are drawn with dashed lines, results at NLO are drawn with solid lines. On the left we present C(3) ϕl [11] vs. C(3) ϕl [33] in the MFV (blue) and 3rd generation centric (magenta) scenarios. In these scenarios C(3) ϕl [22] = C(3) ϕl [11]. 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Gluon Fusion Production at NLO: Merging the Transverse Momentum and the High-Energy Expansions Authors Luigi Bellafrontea, Giuseppe Degrassib, Pier Paolo Giardinoa, Ramona Gr¨oberc, Marco Vittib (a) Instituto Galego de F´ısica de Altas Enerx´ıas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia-Spain (b) Dipartimento di Matematica e Fisica, Universit`a di Roma Tre and INFN, sezione di Roma Tre, I-00146 Rome, Italy (c) Dipartimento di Fisica e Astronomia ’G. Galilei’, Universit`a di Padova and INFN, sezione di Padova, I-35131 Padova, Italy PhD Student Contribution Collaboration in the calculations presented in the paper, as well as in the discussions and writing. Used in Chapter : 3 Journal and Article Information Journal name: Journal of High Energy Physics Publisher: Springer Nature ISSN: 1029-8479 Year of publication: 2022 DOI: 10.1007/JHEP07(2022)069 Impact factor in 2022: 5.4 The article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 145 146 List of publications The Importance of Flavor in SMEFT Electroweak Precision Fits Authors Luigi Bellafrontea, Sally Dawsonb, Pier Paolo Giardinoa, (a) Instituto Galego de F´ısica de Altas Enerx´ıas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia-Spain (b) High Energy Theory Group, Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA PhD Student Contribution Collaboration in the calculations presented in the paper, as well as in the discussions and writing. Used in Chapter : 5 Journal and Article Information Journal name: Journal of High Energy Physics Publisher: Springer Nature ISSN: 1029-8479 Year of publication: 2023 DOI: 10.1007/JHEP05(2023)208 Impact factor in 2022: 5.4 The article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Permissions of content reuse In Fig 3.1 we show plots from [15,16]. The reproduction of these contents is allowed under the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. All the other figures presented in this thesis are original works of the author or they reproduce contents from the papers mentioned in A.3. As already stated, the use these figures is allowed under the Creative Commons Attribution License (CC-BY 4.0). 147 148 Permissions of content reuse