The continuum linear dilaton

Vol. 52 (2021) Acta Physica Polonica B No 6–7 THE CONTINUUM LINEAR DILATON∗ ∗∗ Eugenio Megías Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional Universidad de Granada Avenida de Fuente Nueva s/n, 18071 Granada, Spain Mariano Quirós Institut de Física d’Altes Energies (IFAE) The Barcelona Institute of Science and Technology (BIST) Campus UAB, 08193 Bellaterra (Barcelona), Spain (Received April 29, 2021; accepted May 13, 2021) Continuum spectra can be a way out to alleviate the tension generated by the elusiveness of narrow resonances of new physics in direct experimental searches. Motivated by the latter, we consider the linear dilaton model with a continuum spectrum of KK modes. It is provided by a critical exponential bulk potential for the scalar field stabilizing the distance, between the UV boundary at y= 0 and a naked (good) singularity at y=ys, in proper coordinates, which corresponds in conformal coordinates to zs→ ∞. The cutoff Msin this theory is an intermediate scale Ms≃10−5MPl and the warped factor solves the hierarchy between Msand the TeV, while the hierarchy between MPl and Mshas to be solved by a (Little) String Theory with coupling gs≃10−5. The Standard Model is localized on a 4D IR brane. The graviton and radion Green’s and spectral functions have a continuum of states with a TeV mass gap, and isolated poles consisting of the 4D graviton and the light radion/dilaton. We construct the effective field theory below the mass gap where the continua of KK modes are integrated out, generating a set of dimension eight operators which contribute to low-energy electroweak precision observables, and high-energy violation of unitarity in vector boson scattering processes. The radion mass depends on the stabilizing UV brane potential and its wave function is localized toward the IR which enhances its coupling with the SM fields. DOI:10.5506/APhysPolB.52.711 ∗Funded by SCOAP3under Creative Commons License, CC-BY 4.0. ∗∗ Contribution to the special volume in memoriam of Prof. Martinus Veltman. (711) 712 E. Megías, M. Quirós 1. Introduction No clear deviation has been found, so far, at present (LHC, ... ) and past (Tevatron, LEP/SLC, .. . ) particle physics experiments, from the predictions of the Standard Model (SM) of electroweak and strong interactions. However, given a number of observational facts which cannot be coped with by the SM (dark matter and dark energy, the baryon asymmetry of the universe, . . . ), and some theoretical drawbacks as, among others, its sensitivity to the ultraviolet (UV) scale (a.k.a. hierarchy problem), it is generally believed that the SM is an effective theory and that some UV completion, with beyond the SM (BSM) physics, is needed. This fact has motivated a plethora of BSM models aiming to UV completing the SM, thus solving some of the above issues. In fact, Martinus Veltman was one of the pioneers to recognize the hierarchy problem and the need for a UV completion of the SM, in a seminal paper published in 1981 by Acta Physica Polonica B [1]. One of the most successful BSM models was proposed in 1999 by Lisa Randall and Raman Sundrum [2], where the hierarchy between the fourdimensional (4D) Planck scale MPl and the TeV scale is provided by a warped fifth dimension, in a five-dimensional (5D) space with a non-factorizable metric and two branes, the UV brane and the infrared (IR) brane. This theory predicts the existence of a discrete spectrum made out of towers of Kaluza– Klein (KK) states, with masses in the TeV range, associated with the SM fields, as e.g. the graviton. However, the elusiveness of narrow resonances in direct searches [3,4] led people to imagine different solutions to the hierarchy problem (leaving aside the possibility of superheavy KK modes [5,6]), either with broad resonances [7] or even a continuum of resonances heavier than a mass gap [8,9], evading direct searches and thus detectable only by indirect measurements. In theories with a warped extra dimension, the brane distance is stabilized by means of a bulk scalar field ¯ φwith brane potentials (the Goldberger– Wise mechanism [10]), and different theories are classified by the behavior of the bulk scalar field in the limit of ¯ φ→ ∞ [11]. In fact, in the absence of the IR boundary, it is found in Ref. [11] that there is a critical behavior for the spectrum to be a continuum with a mass gap, when the bulk potential behaves as V(¯ φ)∝e2¯ φin the limit of ¯ φ→ ∞. If the bulk potential goes faster than e2¯ φ, there is a discrete spectrum, and if it goes slower, there is a continuum spectrum without any mass gap. For this critical behavior of the potential, the stabilizing field ¯ φbehaves linearly in conformally flat coordinates znear the asymptotic limit. This linear behavior was obtained as the 5D effective theory of a class of type II strings, known as the Little String Theory (LST) [12], in the decoupling limit of very small string coupling. The Continuum Linear Dilaton 713 In this paper, we will then study 5D warped theories with a bulk potential which is V(¯ φ)∝e2¯ φ, for all values of the field ¯ φ, thus providing a warped realization of the so-called linear dilaton models. We have found two classes of theories, depending on the sign of the metric slope with respect to the conformal coordinate. (i) For the case of negative slope [13–15], gravity decouples in the limit of z→ ∞ so that an IR brane is compelling. The size of the bulk cutoff is the TeV so that the SM fields should be localized on the UV brane and the whole hierarchy problem has to be solved by the LST, with a string scale at the TeV and a coupling as tiny as ∼10−15. In the presence of the brane, the graviton and radion KK modes are discrete with a mass gap and separated by ∼30 GeV. (ii) For the case of positive slope, in the limit of z→ ∞, gravity is correctly described for a cutoff at an intermediate scale ∼10−5MPl, so that the string scale has to be fixed by the LST at that intermediate scale and the string coupling is small ∼10−5, but larger than in the case of negative slope. The SM has to be localized on the IR brane, which is not a boundary of the space, and the spectrum for the graviton and radion is a continuum with a gap related to the metric slope. The distance between the UV boundary and the IR brane is stabilized by a Goldberger–Wise mechanism. The discrete spectrum of Kaluza–Klein gravitons in Randall–Sundrum theories was considered in Refs. [16–21]. In this paper, we have studied the linear dilaton theory, with positive slope metric in conformally flat coordinates and continuum graviton and radion spectra. The contents of the paper are as follows. In Sec. 2, the gravitational background is analyzed in detail, with the discussion of linear dilaton models with different sign slopes, and their connection with the Little String Theory. The Green’s functions and spectral functions for the graviton are studied in Sec. 3. The graviton Green’s functions contain a massless isolated pole, corresponding to the 4D graviton and a continuum of KK modes with a TeV mass gap. The coupling of the graviton continuum with the SM fields is studied in Sec. 4, where a class of dimension eight operators is obtained after integrating out the continuum of KK modes in the effective theory. The radion Green’s and spectral functions are studied in Sec. 5. The radion Green’s function has a continuum of resonances and an isolated pole in the first Riemann sheet corresponding to a mass, below the continuum mass gap, which depends on the UV stabilizing brane potential. Integrating out the continuum leaves an effective theory with dimension eight operators similar to the case of the graviton KK modes. Our conclusions are drawn in Sec. 6. 714 E. Megías, M. Quirós 2. The gravitational background We consider a slice of 5D space-time between a brane at the value of y=y0= 0 in proper coordinates, the UV boundary brane, and a (possible) admissible singularity [22] placed at y=ys, a value which has to be determined dynamically. In addition, we will introduce an IR brane, at y=y1< ys, responsible for electroweak breaking, where we will assume the SM sector to be localized. The 5D action of the model, with metric defined by ds2=gMN dxMdxN≡e−2A(y)ηµνdxµdxν−dy2,(1) including the stabilizing bulk scalar φ(x, y), with mass dimension 3/2, reads as S=Zd5xp|det gMN |−1 2κ2R+1 2gMN (∂Mφ)(∂Nφ)−V(φ) −X αZ Bα d4xq|det ¯gµν|λα(φ)−1 κ2Z B0 d4xq|det ¯gµν|K0,(2) where κ2= 1/(2M3 5),M5being the 5D Planck scale, V(φ)and λα(φ)are the bulk and brane potentials of the scalar field φ, and the index α= 0, 1 refer to the UV and IR branes, respectively. The IR brane is responsible for the generation of the IR scale ∼TeV and contains the brane Higgs potential which spontaneously breaks the electroweak symmetry, thus solving the hierarchy problem between M5and the TeV scale for the considered value of A(y1), as we will see. In Eq. (2), the 4D induced metric is ¯gµν = e−2A(y)ηµν, where the Minkowski metric is given by ηµν = diag(1,−1,−1,−1). The last term in Eq. (2) is the usual Gibbons–Hawking–York boundary term [23,24], where K0is the extrinsic UV curvature. In terms of the metric of Eq. (1), the extrinsic curvature term reads as [25]K0=−4A0(y0). Note that the extrinsic curvature at the singularity is canceled by the action of the determinant. The equations of motion (EoM) read then as A00 =κ2 3φ02+κ2 3X α λα(φ)δ(y−yα),(3) A02=−κ2 6V(φ) + κ2 12φ02,(4) φ00 −4A0φ0=V0(φ) + X α λ0 α(φ)δ(y−yα),(5) where the prime symbol (0)will hereafter stand for the derivative of a function with respect to its argument. The EoM in the bulk can also be written The Continuum Linear Dilaton 715 in terms of the superpotential W(φ)as [26] φ0=1 2 ∂W ∂φ , A0=κ2 6W , (6) and V(φ) = 1 8∂W ∂φ 2 −κ2 6W2(φ).(7) The brane potential terms in the EoM are responsible for boundary and jumping conditions for the fields in the branes. In particular, by integrating the equations in a neighborhood of each brane, and using Eq. (6), we get on the UV boundary W(φ(y0)) = λ0(φ(y0)) , W0(φ(y0)) = λ0 0(φ(y0)) ,(8) where the Z2orbifold conditions have been used. On the other hand, on the IR brane, we have to impose continuity conditions for W(φ)and W0(φ),i.e. ∆W(φ(y1)) = 0 ,∆W0(φ(y1)) = 0 ,(9) where ∆Xis the jump when crossing the brane. Simple brane potentials satisfying boundary (8) and jumping (9) conditions, and fixing dynamically the values of φat the branes, i.e. vα≡φ(yα), are given by λ0(φ) = W(φ) + 1 2γ0(φ−v0)2, λ1(φ) = 1 2γ1(φ−v1)2.(10) Integrating the gravitational 5D Lagrangian by parts in the bulk, and after using the background EoM, one can see that there are contributions to the potential localized on the boundaries as ∓e−4A(yα)W[27], where the ∓sign corresponds to the boundaries yα= (y0, ys). While this contribution vanishes on the singularity at y=ys, it gives a contribution to the UV boundary such that the effective UV brane potential is U0(φ) = λ0(φ)−W(φ) = 1 2γ0(φ−v0)2,(11) which is dynamically minimized for φ=v0. Moreover, on the IR brane, there is not such boundary contribution and there the effective potential is U1(φ) = λ1(φ)which is minimized for the value of φ=v1. For convenience, we will define the dimensionless field ¯ φ≡κφ/√3. The properties of the 5D theory depend on the superpotential behavior in the limit ¯ φ→ ∞ [11]. In particular, when the asymptotic superpotential behavior is exponential eν¯ φ, for ν < 1, the spectrum is continuous without mass gap, for ν > 1there is a mass gap and a discrete spectrum, and for the critical value νc= 1, the spectrum is continuous with a mass gap. 716 E. Megías, M. Quirós We will hereby consider the critical case νcwhere the superpotential and bulk potential are W(¯ φ) = 6k κ2e¯ φ, V (¯ φ) = −9k2 2κ2e2¯ φ,(12) where k.M5is the parameter which determines the 5D curvature. The model defined by superpotential (12) has a singularity at a finite value of the proper coordinate y=ys, as in soft wall models. It leads to a gapped continuum spectrum, and the hierarchy problem is, more conventionally, solved in the same way as in RS theories, with fundamental scales M5and k, and a derived TeV scale after warping. The solution for the background in proper coordinates is1 ¯ φ(y) = −log[k(ys−y)] , A(y) = −log(1 −y/ys).(13) We will also consider, as in general soft wall models, two branes, at y= 0 (the UV boundary) and y=y1(the IR or Higgs brane) where we assumed the Standard Model, and in particular the Higgs, to be located, such that the values of the IR brane location y1and the singularity yswill be dynamically determined by the brane potentials λα(φ), fixing the field ¯ φat the values ¯v0≡κv0/√3and ¯v1≡κv1/√3in the UV and IR branes, respectively, such that kys= e−¯v0, ky1= e−¯v0−e−¯v1.(14) Notice that the first expression demands that kys>0. Moreover, the solution of the hierarchy problem between M5and the TeV scale is achieved for a given value of the warp factor at the IR brane A(y1)≡A1, which imposes the relation ¯v1−¯v0=A1.(15) As we will see in Secs. 3and 5, the squared mass gap of the continuum graviton and radion spectrum is then given by m2 g=9 4ρ2,(16) with ρ=±1/ys.(17) Using that dz=±eA(y)dy, where ±corresponds to the sign of ρ, the relation between conformally flat and proper coordinates in the model of Eq. (12) turns out to be ρ·(z−z0) = −log(1 −y/ys),(18) 1The solution of the EoM, Eq. (6), leads in fact to A(y) = ¯ φ(y) + c, where cis a constant that can be fixed by choosing A(0) = 0 so that c=−¯ φ(0) = −¯v0. The Continuum Linear Dilaton 717 and then the background in these coordinates is given by ¯ φ(z) = A(z) + ¯v0, A(z) = ρ·(z−z0),(19) where we fix z0≡1/k. Therefore, in conformally flat coordinates, the dilaton is linear and the corresponding model is dubbed as linear dilaton model (LDM). According to Eq. (17), the parameter ρcan have both signs and, accordingly, two classes of theories are implemented. 2.1. The ρ < 0 case: the discrete LDM Let us now consider the solution with a negative sign of the parameter ρ in (17) [13,14]: ρ=−1/ys. Now we will fix the values of the scalar field in the branes as ¯v0≃0and ¯v1≃ −|A1|, and then the relation between ρand kis given by k=|ρ|e−¯v0≃ |ρ|.(20) In this case, we will consider as the fundamental interval z0< z < z1<∞, with two branes located on them. The interval [z0, z1]is then mapped into the interval [−|y1|,0] in proper coordinates and the metric and background field profiles are given by A(z) = −|ρ|(z−z0), A(y) = −log(1 + |ρy|),(21) ¯ φ(z) = A(z) + ¯v0,¯ φ(y) = A(y) + ¯v0,(22) so that A(y)<0,A1≡A(y1) = ¯v1−¯v0<0. Then |ρy1|= e|A1|−1≃e|A1|,|ρ|(z1−z0) = |A1|,(23) and the location of the IR brane is dynamically fixed by ¯v0and ¯v1. The relationship with the 4D Planck scale is here given by κ2M2 Pl = z1 Z 0 e−3Adz=1 3|ρ|e3|A1|−1⇒MPl ≃2M3 5 3|ρ|1/2 e3|A1|/2. (24) The hierarchy problem is solved at the string level, as M5∼ |ρ| ∼ TeV, the warped factor in this theory is required to be |A1| ≃ 23, which leads to a value of |y1| ≃ 10−7cm, small as compared to that of the ADD model, but greater (by a factor e|A1|)than the one for the ρ > 0case. The spectrum of this theory is discrete with the lightest KK mode mass being O(ρ). Let us notice that this theory does not admit a continuum spectrum, as sending z1→ ∞ leads to MPl → ∞, which means that gravity 718 E. Megías, M. Quirós is decoupled. As the aim of this paper is considering continuum spectra for linear dilaton models, this class of models with ρ < 0will not be considered here2. In this theory, the hierarchy problem has to be entirely solved by the string theory which should set the cutoff scale in the 5D theory at the TeV. Therefore, here the SM is located in the UV brane. Notice that in this setup, M5≃ρ≃TeV are fundamental scales, while the 4D Planck scale is a derived scale. 2.2. The ρ > 0 case: the continuum LDM In this paper, we will consider the ρ > 0case in (17), and the IR brane location in conformally flat coordinates z1is dynamically determined, as well as the value of y1, by the IR fixing of the dilaton at the value ¯v1. We will here fix ¯v1= 0 and ¯v0=−A1<0. Then, one finds from Eq. (14) that kys= eA1and k(ys−y1)=1. The hierarchy problem is then solved by fixing A1such that ρ=O(TeV), as given by Eq. (17) for k.M5with ρ=ke−A1.(25) The value of M5is determined by the relation of M5and kwith the 4D Planck scale MPl given by κ2M2 Pl = ys Z 0 e−2Ady⇒M5=3 2ρM2 Pl1/3 ,(26) which yields, for ρ=O(TeV),M5≃1013 GeV ≃10−5MPl, and correspondingly a warp factor A1≃23. In conformal coordinates, the location of the branes in units of ρare at ρz0= e−A1and ρz1=A1+ e−A1≃A1, while the singularity is located at infinity, zs→ ∞. The length of the fundamental region is y1≃ys≃10−17 cm, much larger than the corresponding one in the RS model ∼10−31 cm, but still much smaller than that in ADD theories [28]∼10−2cm (for the case of two extra dimensions) where only gravity propagates in the bulk. In this case, we can consider two kinds of fundamental intervals for our theory: — When the fundamental interval is [0, y1],i.e. [z0, z1]in conformal coordinates, the theory spectrum is discrete, the first mode mass being O(ρ). — When the fundamental interval is [0, ys],i.e. [z0,∞)in conformal coordinates (as we will consider in this paper), this theory predicts a continuum spectrum with an O(ρ)gap: the continuum linear dilaton model (CLDM). 2Detailed phenomenological studies of this model have been done in Refs. [13–15]. The Continuum Linear Dilaton 719 Let us notice that in this theory, the warp factor solves the hierarchy problem between the intermediate scale M5and the TeV scale. This means that the UV completion of this theory should be a string theory with the string mass at the intermediate scale, Ms≃M5, thus solving the hierarchy problem between the Planck scale and M5. This class of models does not support gauge bosons propagating in the bulk of the extra dimension, so that we will consider the whole SM localized at the IR brane. Notice that in this setup, as in RS models, M5is a fundamental scale, while the TeV scale and the 4D Planck mass are derived from the theory warp factor. 2.3. Connection with Little String Theory As we have seen in the previous section, the hierarchy problem between the 4D Planck scale, MPl, and the 5D Planck scale, M5≃1013 GeV, has to be solved by a string theory with a string scale at the intermediate value Ms≃M5. As the relation between Msand MPl in string theory is given by M2 Pl =1 g2 s M8 sV6,(27) where gsis the string coupling and V6the volume of the compactified dimensions, imposing MsMPl requires, either V6`6 s, with `s= 1/Ms, or gs1. The second possibility, i.e. V6∼l6 sand gs∼Ms/MPl 1can be realized in type II string theories, where the size of the gauge coupling is unrelated to gs, but fixed by the geometry (radii) of the compact dimensions where gauge interactions propagate. In the limit of gs→0, a type II string theory, dubbed Little String Theory3, was constructed where non-Abelian gauge interactions are localized on a stack of (Neveu–Schwarz) NS5-branes, a 6D space with 1+3 flat dimensions and two extra longitudinal dimensions compactified on a torus T2with size ∼`2 s 4. In the gravity decoupling limit gs→0, the NS5-branes give rise to the 6D LST, which is strongly coupled and seems to have no Lagrangian description. By holography, one can relate the 6D strongly coupled theory in the absence of gravity to a 7D theory with gravity weakly coupled with a linear dilaton. Upon compactification of the two extra dimensions in T2, we obtain a 5D theory with weakly interacting gravity and a linear dilaton, as that studied in the previous section. Here, there is a fundamental difference between the two previous theories with ρ < 0and ρ > 0. — In the case of ρ < 0, by making the extra dimension infinite, i.e. z1→ ∞, one gets from Eq. (24) that MPl → ∞ and so gravity is 3For a review, see Ref. [12]. 4The four extra transverse dimensions are compactified in a manifold, and we will assume all compact dimensions have a size ∼`s. 726 E. Megías, M. Quirós 3.2. Spectral functions In this section, we find it convenient to work in a basis with flat extra dimensional coordinate y,i.e. with wave function ¯ hµν(x, y)as ¯ hµν(x, y)=e−A(y)hµν(x, y),(60) with a corresponding Green’s function ¯ Ghy, y0= e−A(y)Ghy, y0e−A(y0).(61) For time-like momenta, p2>0, all Green’s functions are complex for values of p>mg= 3ρ/2, which is not associated with a particle threshold decay, an intrinsic property of e.g. unparticle theories [32]. In this way, we can define the corresponding spectral functions as ¯ρhy, y0;s=−1 πIm ¯ Ghy, y0;s+i, s ≡p2.(62) In Fig. 3, we show ¯ρh(y0, y0;p),¯ρh(y0, y1;p)and ¯ρh(y1, y1;p)as functions of p/ρ where the prefactors, defined as F00 =ρ , F01 =ρρ k1/2,F11 =ρρ k(63) make them scale-invariant [9]. By using the identity lim →0+ 1 x+i =P1 x−iπδ(x),(64) 0 2 4 6 8 10 0.00 0.02 0.04 0.06 0.08 0.10 0.12 pΡ FΡ00× ΡhHy0,y0L 0 1 2 3 4 5 6 -0.2 -0.1 0.0 0.1 0.2 pΡ FΡ01× ΡhHy0,y1L 1.4 1.5 1.6 1.7 1.8 1.9 2.0 0 1 2 3 4 5 6 pΡ FΡ11× ΡhHy1,y1L Fig. 3. Scale-invariant spectral functions F00 ·¯ρh(y0, y0;p)(left panel), F01 · ¯ρh(y0, y1;p)(middle panel) and F11 ·¯ρh(y1, y1;p)(right panel) as a function of p/ρ, for a continuum graviton. We have used A1= 23 in all panels and assume time-like momenta p2>0. The Continuum Linear Dilaton 727 one can see that the small pbehavior of the Green’s functions provided in Sec. 3.1 implies the existence of a Dirac delta behavior in the spectral functions at p= 0 ¯ρhy, y0;s= 3ρe−A(y)−A(y0)δ(s) + ··· .(65) This delta function appears in all the Green’s functions of Fig. 3. Notice that, although the spectral functions ¯ρh(y0, y0)and ¯ρh(y1, y1)are positive definite, the spectral function propagating from the UV to the IR brane ¯ρh(y0, y1)is not. This fact just challenges the physical interpretation of the spectral function in a 4D quantum field theory, which is positive definite by its probabilistic interpretation. To understand the positivity of the spectral function in our theory, we have to consider, from the 4D point of view, the spectral function ¯ρh(y, y0;s) as the matrix element (y, y0)of an operator ˆρh,i.e. (ˆρh)y0 y≡¯ρhy, y0;s(66) acting on the infinite-dimensional space parametrized by the coordinate y. The matrix action on a vector vy≡v(y)is thus represented by the integral, e.g. Py0(ˆρ)y0 yvy0≡Rdy0¯ρh(y, y0;s)v(y0). In parallel with the definition of the operator ˆρh, one can define, from the Green’s functions ¯ Gh(y, y0), the operator ˆ Ghsuch that ˆρh=−1 πIm ˆ Gh,where Im ˆ Gh=1 2iˆ Gh−ˆ G† h.(67) The elements of ˆρhthen form an infinite-dimensional matrix whose positivity properties will be analyzed now. When taking into account the property of Eq. (48), the matrix ˆρhturns out to have a factorizable form, i.e. one finds the following explicit expressions for the spectral function (ˆρh)y0 y= ˆρy·ˆρy0,(68) where, for p2≥m2 g, ˆρy=s2 3πρR (1 + R2) 1 (1 −¯y)1/2Im(1 + iR) (1 −¯y)3 2iR,(69) with R(p) = p(4/9) ·p2/ρ2−1.(70) 728 E. Megías, M. Quirós Given the factorization property, Eq. (68), it turns out that the operator ˆρhis positive semidefinite: all its eigenvalues are zero except one λ(p)(i.e. det ˆρh= 0), which is given by the trace of the matrix, i.e. λ(p) = tr ˆρh=1 ρ 1 Z 0 ¯ρh(¯y, ¯y;s)d¯y . (71) Using Eq. (68), one can see that there is a divergence at the value ¯y= 1, so that the expression for λneeds to be regularized. We will do it by introducing the cutoff ¯in the integral (71) which now will extend from 0to 1−¯, so that the integral will be dominated by its value at 1−¯, giving a term ∝(−log ¯)which will be the leading one. As we will see, this divergence will cancel out when computing physical observables, so it will not require any renormalization procedure. It turns out that λ(p)is computed as λ(p) = δp2+−log ¯ 2πρ λun(s) + O¯0, λun(s) = s−m2 g−1/2, (72) where δ(p2)is the contribution to the spectral function of the graviton zero mode6and λun(s)the contribution to the spectral function from the continuum, or unparticle contribution with a mass gap mgand dimension dun = 3/2[33]. Therefore, in the diagonal basis, the matrix ˆρd hhas all elements null except one, which can be chosen to be the element with y=y0=ys, which is equal to λ,i.e. ˆρd hy0 y=λ(p)δy,ysδys,y0,(73) and the (infinite) orthogonal rotation Oy0 yfrom ˆρh→ˆρd hhas, in particular, elements Oys y=ˆρy √λ.(74) Now, the contribution of ˆρhto a physical process where φ(y)is the profile along the extra dimension of the initial state in the tensor Tµν(x, y)with a coupling for fields ¯ hµν(x, y)given by L5D =−1 √2M3/2 5 eA(y)¯ hµν(x, y)Tµν(x, y),(75) 6Its correct normalization comes from the prefactor in Eq. (65) as 3ρRys 0e−2A(y)= 1. The Continuum Linear Dilaton 729 and ψ(y0)the profile of the final state in Tρσ(x, y0), is given by the element tr φTeA·ˆρh·eAψ= tr φTeAOT·ˆρd h·OeAψ =X y,y0 φyeAyOTy ysˆρd hys ys Oys y0eAy0ψy0=X y,y0 φyeAyˆρy √λ·λ·ˆρy0eAy0ψy0 √λ = ys Z 0 dydy0φ(y)eA(y)¯ρhy, y0eA(y0)ψy0= ys Z 0 dydy0φ(y)ρhy, y0ψy0, (76) where we see that the divergence in the calculation of the eigenvalue λcancels out, while the last equality is written in terms of the spectral density ρhin the basis hµν(x, y). In particular, if the initial-state function is located at the brane y=yα, and the final state is localized at the brane y=yβ, then φ(y)∝δ(y−yα),ψ(y0)∝δ(y0−yβ), and the result of Eq. (76) is given by tr φTeA·ˆρh·eAψ= eA(yα)+A(yβ)¯ρh(yα, yβ;p) = ρh(yα, yβ).(77) The functions ¯ρh(yα, yβ;p)are those plotted in Fig. 3. 4. Coupling of the graviton with SM matter fields We are assuming fields located in the brane y=yα. Then, the usual form of the interaction Lagrangian in the 4D effective theory is given by the Lagrangian L5D =−1 √2M3/2 5 Tµν(x, y)hµν(x, y)δ(y−yα),(78) where Tµν(x, yα)is the energy-momentum tensor of the matter fields localized at yα7. In particular, for the SM fields living in the IR brane at y=y1, the energy-momentum tensor is given by Tµν = 2DµH†DνH+i¯ ψγµDνψ−FµρFνρ −ηµνLSM ,(79) where Dµis the SM covariant derivative, ψcorresponds to all SM leftand right-handed fermions and Fµν is the field strength of different gauge fields 7We are assuming here the simplified case where matter lives in some brane, as e.g. the SM which is living in the IR brane, or perhaps some dark sector which could live in the UV brane. For matter (SM singlets) propagating in the extra dimension, one should replace the interaction term in Eq. (78) by Rdy Tµν (x, y)hµν (x, y). 730 E. Megías, M. Quirós Fµν =Wa µν, Bµν. The term proportional to ηµν does not contribute to the different vertices as the tensor hµν is traceless. Notice that in the broken phase, when hHi=v/√2(0,1)T, the massive gauge bosons have contributions to Tµν proportional to m2 VVµVν. The graviton zero mode wave function, h0 µν(x, y) = h0(y)h0 µν, where h0(y) = √3ρis canonically normalized as Rys 0dye−2Ah2 0= 1, couples with the energy-momentum tensor at the IR brane as −1 MPl Tµν(x, y1)h0 µν(x).(80) We will now consider the coupling with matter of the continuum of KK modes, with Green’s function Gh(y, y0). The effective field theory (EFT) for matter localized at the brane yα, for momenta pρ, provides the dimension eight operator Oh(x, yα)with Wilson coefficient c(yα)as LEFT(yα) = c(yα)Oh(x, yα), Oh(x, yα) = Tµν(x, yα)Dµν,ρσTρσ(x, yα) = Tµ νTν µ−1 3Tµ µ2,(81) where the Wilson coefficients here have mass dimension −4. In particular, for the SM which is localized in the IR brane, c(y1) = −1 6 1 ρ4.(82) Thus, gravitational interactions are suppressed by the TeV scale ρ, reflecting the fact that the continuum of KK modes is localized toward the IR. On the contrary, if there is some extra matter localized in the UV brane, SM singlets, the corresponding Wilson coefficient would be c(y0) = −1 9 1 ρ2M2 Pl .(83) In fact, if we define the effective coupling geff(yα)as |c(yα)| ≡ g2 eff(yα)1 ρ2,(84) we can see that geff(y0)≃1/MPl, while geff(y1)≃1/ρ. 4.1. Low-energy constraints The effective Lagrangian in Eq. (81) does give rise, in particular, to the dimension eight operators, in the notation of Refs. [34,35] LEFT ⊃ 2 X i=0 fSi ρ4OSi,(85) The Continuum Linear Dilaton 731 where OS0=DµH†DνHDµH†DνH,OS1=DµH†DµHDνH†DνH, OS2= (DµH†DνH)(DνH†DµH), fS0=−1 3, fS1=2 9, fS2=−1 3.(86) The contributions of these effective operators to the observables S, T, U have been computed in Ref. [34] as8 αT =−15 16π2(mW/ρ)4fS0+fS2+2 5fS11 + c2 Ws2 W c2 W log(ρ/mW), (87) where αis the fine structure constant, and sW(cW)the sine (cosine) of the electroweak mixing angle θW, while S=U= 0. Using now the values in Eq. (86), we get αT ≃13 24π2(mW/ρ)41 + c2 Ws2 W c2 W log(ρ/mW),(88) which provides a very mild bound on the value of ρas αT .4×10−5 (3×10−6) for ρ&500 GeV (1 TeV). The small value of the Tparameter comes mainly because this effect stems from a dimension eight operator, and thus is suppressed by the fourth power of 1/ρ. 4.2. High-energy constraints The effective Lagrangian in Eq. (81) does also give rise to a number of dimension eight operators, which contribute to an anomalous quartic gauge coupling (aQGC) as LEFT ⊃X j fTj ρ4OTj+X k fMk ρ4OMk,(89) where, using the notation of Refs. [34,35], we have OT0= (WµνWµν)WαβWαβ,OT2= (WµαWνα)WµβWνβ, OT5= (WµνWµν)BαβBαβ,OT7= (WµαWνα)BµβBνβ, OT8= (BµνBµν)BαβBαβ,OT9= (BµαBνα)BµβBνβ,(90) 8We thank Prof. O.J.P. Éboli for a private communication on this result. 732 E. Megías, M. Quirós and OM0= (WµνWµν)DαH†DαH,OM1= (WµαWνα)DµH†DνH, OM2= (BµνBµν)DαH†DαH,OM3= (BµαBνα)DµH†DνH, (91) with Wilson coefficients fT0=1 18 , fT2=−1 6, fT5=1 9, fT7=−1 3, fT8=1 18 , fT9=−1 6, fM0=−2 9, fM1=2 3, fM2=−2 9, fM3=2 3.(92) The LHC constraints on the above operators are obtained from the CMS experiment [36–38]. The strongest constraints are over the operators OT0 and OT2which translate into the 95%.L. lower bound mg&1.3TeV. Projections in FCC-hh, at √s= 100 TeV and integrated luminosities up to 30 ab−1, have been made on the anomalous WWγγ couplings [39] which, for leptonic decay channels of the Ws in the final state, yield future bounds reaching values as mg&7TeV. Of course, the presence of aQGC induces violation of unitarity, e.g. in longitudinal gauge boson scattering processes involving four-vector particles, as the corresponding scattering amplitudes grow with ˆs2, where √ˆsis the center-of-mass energy, since the SM cancellation fails. This issue has been generally considered for the operators OSi,OTjand OMkin Refs. [35,40]. The unitarity violation indicates a failure of the EFT to describe the corresponding processes at such large values of √ˆs. In particular, using the general results in Ref. [35], the unitarity constraints imply an upper bound as √ˆs.2mgfor the validity of the EFT. 5. The radion The radion field F(x, y)is defined as the scalar perturbation of the metric ds2= e−2A(y)−2F(x,y)ηµνdxµdxν−[1 + G(x, y)]2dy2, φ(x, y) = φ(y) + ψ(x, y),(93) with F(x, y) = F(y)R(x). When considering an appropriate gauge choice, the EoM for the y-dependent part become [41] p2e2AA00(y)−1−2F(y) + d dye2AA00(y)−1∂ye−2AF(y)= 0 , F0−2A0F=φ0ψ , G = 2F . (94) The Continuum Linear Dilaton 733 After rescaling the field by F(z) = e3A(z)/2φ0(z)˜ F(z), one can cast the EoM in a Schrödinger-like form as −˜ F00(z) + VF(z)˜ F(z) = p2˜ F(z),(95) where the potential is given by VF(z) = 9 4A02(z) + 5 2A00(z)−A0(z)φ00(z) φ0(z)−φ000(z) φ0(z)+ 2 φ00(z) φ0(z)2 .(96) This potential turns out to be equal to the constant value VF(z) = m2 g, where mg= 3ρ/2is the mass gap for the radion. This value of the mass gap equals that of the graviton in previous sections. 5.1. The radion Green’s functions After making the field redefinition F(x, y)→κF(x, y), as for the case of the graviton, the EoM for the radion Green’s function GF(y, y0;p)[41,42] is the same as the one for the graviton, cf. Eq. (38). After fixing the value of y0, we can divide the yspace into the following domains: 0≤y≤y0and y0≤y≤ys. Then, the general solution is GF(y, y0;p) = (CI 1·(ys−y)3 2∆− F+CI 2·(ys−y)3 2∆+ Fy < y0< ys CII 1·(ys−y)3 2∆− F+CII 2·(ys−y)3 2∆+ Fy0< y < ys , (97) where we have defined ∆± F=∆± hand δF=δh,cf. Eq. (40). The Green’s function is subject to boundary and matching conditions in the UV and IR branes, as well as for y=y0. These read G0 F0, y0= 1 3κ2W(φ(y)) −2p2e2A(y) U00 0(φ(y))!GF(y)y=0 ,(98) ∆GFy0, y0= 0 ,∆G0 Fy0, y0= e4A(y0), ∆GFy1, y0= 0 ,∆G0 Fy1, y0= 0 , where the localized effective potential in the UV brane U0(φ)is defined by Eq. (11), and its second derivative turns out to be U00 0(φ(0)) = γ0−2ρ, which in the following we will denote by U00 0. In addition, we should impose regularity in the IR, i.e. we consider CII 1= 0. After implementing the boundary and matching conditions in the general solution, one finds GFy, y0=1 3ρ 1 δF (1 −¯y↑)3 2∆+ F ×−(1 −¯y↓)3 2∆− F+1 + 3U00 0 2ρ δF Φ(p)(1 −¯y↓)3 2∆+ F,(99) 734 E. Megías, M. Quirós where Φ(p) = p2 ρ2−U00 0 4ρ(1 + 3δF).(100) The analytical expressions of the brane-to-brane Green’s functions are GF(y0, y0;p) = U00 0 2ρ2 1 Φ(p),(101) GF(y0, y1;p) = U00 0 2ρ2 1 Φ(p)e−3A1 2∆+ F,(102) GF(y1, y1;p) = 1 3ρ 1 δF e3A1−1 + 1 + 3U00 0 2ρ δF Φ(p)e−3A1δF,(103) and their low-momentum behaviors are G−1 F(y0, y0,1)≃ pρ−2ρ+Op2,(104) G−1 F(y1, y1)≃ pρ−6ρ4 2k3+ρ3+Op2.(105) In the following, we will denote the zero momentum limits of the brane-tobrane Green’s functions as Gαβ F≡limp→0GF(yα, yβ;p). We plot in Fig. 4the result for the Green’s functions GF(y0, y0),GF(y0, y1) and GF(y1, y1), normalized to their zero momentum limits, as functions of p/ρ, for time-like momenta p2>0. For space-like momenta p2<0, the Green’s functions are purely real. We plot in Fig. 5the Green’s functions as functions of |p|/ρ, in the latter case. 0 2 4 6 8 10 0.2 0.5 1.0 2.0 5.0 pΡ ÈGFHy0,y0LGF00 È 0 2 4 6 8 10 0 1 2 3 4 5 pΡ e-3A12×ÈGFHy0,y1LGF01 È 1.0 1.5 2.0 2.5 3.0 0.5 1.0 5.0 10.0 50.0 100.0 pΡ ÈGFHy1,y1LGF11 È Fig. 4. Plots of |GF(y0, y0;p)/G00 F|(left panel), |GF(y0, y1;p)/G01 F|(middle panel), and |GF(y1, y1;p)/G11 F|(right panel) as functions of p/ρ. We have used A1= 23 and U00 0=kin all panels, and assume time-like momenta p2>0. The Continuum Linear Dilaton 735 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 ÈpÈΡ GFHy0,y0LGF00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 ÈpÈΡ GFHy0,y1LGF01 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 ÈpÈΡ GFHy1,y1LGF11 Fig. 5. Plots of GF(y0, y0;|p|)/G00 F(left panel), GF(y0, y1;|p|)/G01 F(middle panel), and GF(y1, y1;|p|)/G11 F(right panel) as functions of |p|/ρ. We have used A1= 23 and U00 0=kin all panels, and assume space-like momenta p2<0. Notice that (unlike the graviton case) the Green’s functions do not have an isolated massless mode as their behavior in the limit p→0, as shown in Eqs. (104)–(105), yields a constant value and not an isolated singularity. This point is in agreement with previous studies on the subject in Ref. [11]. The function Φ(p)given by Eq. (100) has a single zero, either in the first or second Riemann sheet. Let us write the equation Φ(mF) = 0, with mFthe mass of the radion, in the form of U00 0 ρ=4m2 F/ρ2 1+3δF with δF=±q1−(4/9) ·m2 F/ρ2,(106) where the +(−)corresponds to the first(second) Riemann sheet. We display in the left panel of Fig. 6the parametric dependence of U00 0/ρ with mFas given by Eq. (106). One can see that U00 0≥0demands that mF≤mg when considering the 1st Riemann sheet, while √2ρ<mF≤mgin the 2nd Riemann sheet, so that the mass is below the mass gap, except for U00 0/ρ = 9 where it has the same value. 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