INTERNATIONAL DOCTORAL
SCHOOL OF THE USC
Luigi
Bella on e
PhD Thesis
P ecision physics a High
Luminosi y LHC and u u e
collide s
San iago de Compos ela, 2023
Doc o al P og amme in Nuclea and Pa icles Physics
DOCTORAL THESIS
PRECISION PHYSICS AT HIGH
LUMINOSITY LHC AND FUTURE
COLLIDERS
Luigi Bella on e
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!submi !my! hesis,! ollowing! he!app op ia e!p ocedu e! o! he!Regula9ons!and!decla e! ha :!!
1) The! hesis!co e s! he! esul s!o ! he!elabo a9on!o !my!wo k.!
2) I ! his!is! he!case,! he! hesis! e e s! o! he!collabo a9ons!wi hin! his!wo k.!
3) I!confi m! ha ! he! hesis!does!no !incu !in!any!kind!o !plagia ism! om!o he !au ho s!no !wo ks!
submiDed!by!me!in!o de ! o!ob ain!o he !9 les.!
4) The! hesis!is! he!final! e sion!submiDed! o !i s!de ence!and!bo h! he!p in ed! e sion!and! he!
elec onic! e sion!p o ide! he!same!con en .!
And!I!ag ee! o!submi ! he!Documen a y!Commi men !o !supe ision!in! he!e en ! ha ! he!o iginal!
e sion!is!no !deposi ed!a ! he!School.!!
In!San3ago+de+Compos ela,+25 h+Augus +2023.+
Elec onic+signa u e+
M /
Ms.
Luigi+Bella on e
Thesis!
Ti le:
P ecision+physics+a +High+Luminosi y+LHC+and+ u u e+collide s
SUPERVISOR/TUTOR AUTHORISATION
M /Ms
Pie Paolo Gia dino
As:
Supe iso
Thesis i le:
P ecision physics a High Luminosi y LHC and u u e collide s
STATES:
Tha his hesis co esponds o he wo k ca ied ou by M /Ms Luigi Bella on e, unde my
supe ision/ u o ing and he eby au ho ise i s p esen a ion, aking in o accoun ha i mee s all he
ele an equi emen s s a ed in he Doc o al S udies Regula ions o he USC, and as i s di ec o / u o i
does no incu in he causes o abs en ion es ablished in he 40/2015 Law.
In San iago de Compos ela, 24 Augus 2023
Elec onic signa u e
SUPERVISOR/TUTOR AUTHORISATION
M /Ms
Nés o A mes o Pé ez
As:
Tu o
Thesis i le:
P ecision physics a High Luminosi y LHC and u u e collide s
STATES:
Tha his hesis co esponds o he wo k ca ied ou by M /Ms Luigi Bella on e, unde my
supe ision/ u o ing and he eby au ho ise i s p esen a ion, aking in o accoun ha i mee s all he
ele an equi emen s s a ed in he Doc o al S udies Regula ions o he USC, and as i s di ec o / u o i
does no incu in he causes o abs en ion es ablished in he 40/2015 Law.
In San iago de Compos ela, 28 Augus 2023
Elec onic signa u e
Resumo
O Modelo Es ´anda (SM) da ´ısica de pa ´ıculas ´e o ma co e´o ico ac ual que desc ibe,
a ni el undamen al, as in e acci´ons en e pa ´ıculas elemen ais. A cons uci´on de coli-
sionado es de al a ene x´ıa o La ge Had on Collide (LHC) no CERN, o maio acele ado
de pa ´ıculas cons u´ıdo a a o de ago a, pe mi iu que os cien ´ı icos p obasen os seus
modelos a escalas de ene x´ıa cada ez maio es. Nos ´ul imos anos, o LHC cen ou cada
ez m´ais a s´ua a enci´on nas medidas de p ecisi´on dos p ocesos do SM, xa que xogan
un papel c ucial na explo aci´on da no a ´ısica. A ´en ase na ´ısica de p ecisi´on am´en
mo i ou o desen ol emen o da ac ualizaci´on de al a luminosidade (HL) do LHC e a p o-
pos a de no as m´aquinas de al a p ecisi´on, como o FCC-ee. Nes e con ex o, o es o zo
e´o ico debe iguala o aballo expe imen al, po que as medidas p ecisas equi en dunha
p ecisi´on igual nas p edici´ons do SM. Es e a gumen o am´en se aplica ´as eo ´ıas m´ais
al´a do Modelo Es ´anda (BSM) e ´ol ese especialmen e ele an e no caso das Teo ´ıas
de Campo E e i as (EFTs), onde as co ecci´ons de o de supe io poden e un impac o
signi ica i o nos l´ımi es da no a ´ısica. O en oque p incipal des e ese ´e o c´alculo p e-
ciso dos p ocesos no SM e no ma co da Teo ´ıa E e i a do Modelo Es ´anda (SMEFT),
inclu´ındo gg-HH, p oduci´on de D ell Yan qq-ll e obse ables elec o ebles. Finalmen e,
o obxec i o ´e ace p edici´ons p ecisas que poidan se p obadas nos colisionado es do u-
u o, como o HL LHC e o FCC-ee. O es udo des a ´a ea ´e moi impo an e e p opo ciona
in o maci´on c ucial sob e a es u u a da no a ´ısica m´ais al´a do SM.
11
12 Resumo
Resumen
El Modelo Es ´anda (SM) de la ´ısica de pa ´ıculas es el ma co e´o ico ac ualmen e
es ablecido que desc ibe, a ni el undamen al, las in e acciones en e las pa ´ıculas el-
emen ales. La cons ucci´on de colisionado es de al a ene g´ıa como el La ge Had on
Collide (LHC) en el CERN, el acele ado de pa ´ıculas m´as g ande jam´as cons uido,
ha pe mi ido a los cien ´ı icos p oba sus modelos a escalas de ene g´ıa cada ez m´as al as.
En los ´ul imos a˜nos, el LHC ha cen ado cada ez m´as su a enci´on en las mediciones
p ecisas de los p ocesos del SM, ya que desempe˜nan un papel c ucial en la explo aci´on de
la nue a ´ısica. El ´en asis en la ´ısica de p ecisi´on ambi´en ha mo i ado el desa ollo de
la ac ualizaci´on de al a luminosidad (HL) del LHC y la p opues a de nue as m´aquinas
de al a p ecisi´on, como el FCC-ee. En es e con ex o, el es ue zo e´o ico debe iguala el
abajo expe imen al, po que las mediciones p ecisas equie en una p ecisi´on igual en
las p edicciones del SM. Es e a gumen o ambi´en se aplica a las eo ´ıas m´as all´a del
Modelo Es ´anda (BSM) y se uel e especialmen e ele an e en el caso de las Teo ´ıas
de Campo E ec i as (EFT), donde las co ecciones de o den supe io pueden ene un
impac o signi ica i o en los l´ımi es de la nue a ´ısica. El en oque p incipal de es e esis
es el c´alculo p eciso de los p ocesos en el SM y en el ma co de la Teo ´ıa E ec i a del
Modelo Es ´anda (SMEFT), incluyendo gg-HH, p oducci´on de D ell Yan qq-ll y obse -
ables elec od´ebiles. Finalmen e, el obje i o es hace p edicciones p ecisas que puedan
se p obadas en u u os colisionado es, como el HL LHC y el FCC-ee. El es udio de
es a ´a ea es muy impo an e y p opo ciona in o maci´on c ucial sob e la es uc u a de la
nue a ´ısica m´as all´a del SM.
13
14 Resumen
Summa y
The S anda d Model (SM) o pa icle physics is he cu en ly es ablished heo e ical
amewo k which desc ibes, a a undamen al le el, in e ac ions among elemen a y pa -
icles. The cons uc ion o high-ene gy collide s such as he La ge Had on Collide
(LHC) a CERN, he la ges pa icle accele a o e e buil , allowed scien is s o es
hei models a inc easingly high ene gy scales. In ecen yea s, he LHC has inc eas-
ingly ocused i s a en ion on he p ecise measu emen s o SM p ocesses, as hey play
a c ucial ole in he explo a ion o new physics. The emphasis on p ecision physics
has also mo i a ed he de elopmen o he high luminosi y (HL) LHC upg ade, and
he p oposal o new high p ecision machines such as he FCC-ee. In his con ex , he
heo e ical e o has o ma ch he expe imen al wo k, because p ecise measu emen s
equi e an equal p ecision o he SM p edic ions. This a gumen applies also o heo ies
Beyond he S anda d Model (BSM), and i becomes pa icula ly ele an in he case o
E ec i e Field Theo ies (EFTs), whe e highe -o de co ec ions can signi ican ly impac
he bounds on new physics. The main ocus o his hesis is he p ecision calcula ion
o p ocesses in he SM and S anda d Model E ec i e Field Theo y (SMEFT) ame-
wo k, including gg →HH, D ell Yan p oduc ion qq →ll, and elec oweak obse ables.
Finally, he ul ima e goal is o make accu a e p edic ions ha can be es ed on u u e
collide s, such as HL LHC and Fcc-ee. The s udy o his a ea is e y impo an , and i
gi es c ucial in o ma ion on he s uc u e o new physics beyond he SM.
15
16 Summa y
Objec i es and Me hodology
The main goal o his hesis is o es physical phenomena a LHC and u u e collide s
by p o iding accu a e calcula ions o sensi i e obse ables.
His o ically, p ecise calcula ions o obse ables ha e played a c ucial ole in ou sea ch
o unde s and he Na u e. Indeed, hey allowed us o igo ously es he heo ies o -
mula ed o comp ehend he uni e se. Fo his eason, in his hesis we calcula ed he
pe u ba i e co ec ions o SM and SMEFT p ocesses ha a e phenomenologically el-
e an o physics a he LHC and u u e collide s. These include:
•The NLO co ec ion o he p oduc ion o a pai o Hbosons and HZ bosons
om gluon usion. The calcula ion o hese p ocesses is ex emely impo an o
he expe imen al measu emen o he Higgs p ope ies and i used o measu e he
Higgs wid h. In pa icula , he double higgs p oduc ion is signi ican ly ele an
because o he iple Higgs coupling in ol ed in he calcula ion.
•The calcula ion o EWPOs a he nex - o-leading o de (NLO) QCD and elec-
oweak expansions o he SMEFT wi h an a bi a y la o s uc u e o he
e mion ope a o s. Nume ical NLO SMEFT i s o EWPOs a e expec ed o ha e
a s ong dependence on he assumed la o s uc u es and we will check his using
a ious popula assump ions o la o symme ies.
•D ell Yan p oduc ion in he SM EFT a NLO; his p ocess is e y impo an o
de ec new physics signals and p ecise heo e ical p edic ions in he e ec i e ield
heo y a e c ucial. The e ec s o ou e mions ope a o s on his ampli ude is a
new challenge and i will gi e us essen ial knowledge on he s uc u e o he BSM
physics, in pa icula on he la ou s uc u e.
All he equi ed he calcula ions a e pe o med analy ically. Indeed while he e a e
pu ely nume ical me hods ha could be used, hyb id nume ical/analy ical echniques
a e gene ally mo e lexible and easie o adap o b oade uses, such as EFTs, han he
pu ely nume ical ones. Fu he mo e, hey a e compu a ionally less in ensi e, making
hem be e sui ed o build e icien Mon e Ca lo gene a o s. Fo hese easons analy ical
calcula ions a e mo e con enien o ou aims.
17
18 Objec i es and Me hodology
Me hodology
As explained, he main objec i e o his hesis is he s udy o p ecision physics in he
SM and i s ex ensions. This in ol e he applica ion o ools o he p ecise calcula ion
o p ocesses ele an o he phenomenology o he LHC and u u e collide s.
To achie e ou goals we use public and p i a e codes implemen ed in he so wa e Ma he-
ma ica. The ele an ampli udes o he sca e ing o he decay in analysis a e gene a ed
wi h FeynA s [1] and we use Feyncalc, Li eRed and Fi e [2,3,4] o educe he am-
pli udes in o a combina ion o Mas e In eg als (MI). Feynman in eg als a e calcula ed
h ough s anda d me hods when possible and h ough he ene gy expansions echniques
when necessa y. A one loop le el we use he Passa ino-Vel man unc ions as basis o
he MI and he analy ical e alua ions is done h ough FeynHelpe s [5]. We use he Pad´e
App oximan s o imp o e he con e gence o ene gy expansions o allow us o ex end
he alidi y o hese echniques o egions o he phase space whe e hese app oaches
should o he wise ail. This me hod is applied o physical p ocesses such as gg →HZ
and gg →HH in o de o imp o e he accu acy o he esul s ob ained in [6,7] and
allowing us o ma ch wi h he esul s o [8]. In his way we a e able o make a p edic ion
a e e y poin o he pa ame e space wi h high p ecision.
The p oblems a ising by he p esence o γ5and se e al Di ac nma ices a e ea ed wi h
K eime ’s egula iza ion scheme [9], a powe ul ool ha p ese es he chi al symme y
o he S anda d Model in all ba e pe u ba i e calcula ions. All he SMEFT calcula ions
a e done in he Wa saw basis wi h he FeynA s model ile ob ained om [10]. Finally,
we use plo s and ba diag ams o p esen ou phenomenological esul s.
Chap e 1
In oduc ion
The in es iga ion o he na u al phenomena has been one o he d i ing o ces behind
scien i ic ad ancemen . Rega ding pa icle physics, his pu sui has led o he de elop-
men o he S anda d Model (SM), an ou s anding success ul amewo k ha desc ibes
and classi ies he elemen a y pa icles and hei in e ac ions. This culmina ed wi h he
disco e y o he Higgs boson in 2012 [11,12], ma king a u he essen ial s ep owa ds
a sa is ac o y unde s anding o he mechanism unde lying he elec oweak-symme y
b eaking (EWSB). Ne e heless, e en wi h i s iumphs, he SM lea es nume ous heo-
e ical and expe imen al ques ions unanswe ed. Examples o he la e include neu ino
oscilla ions and da k ma e . F om a heo e ical poin o iew, we can men ion he
hie a chy o he e mion masses, he mys e y o he o igin o he Higgs po en ial and he
inclusion o g a i y (beyond he classical le el ) in he SM amewo k.
In his con ex , pa icle collide s ha e played a c ucial ole in shaping ou unde s anding
o he undamen al laws o he uni e se. The cons uc ion o high-ene gy collide s such
as Te a on, LEP and he La ge Had on Collide (LHC) a CERN, he la ges pa icle
accele a o e e buil , allowed scien is s o es hei models a inc easingly high en-
e gy scales d i ing disco e ies and con i ming heo ies. A e he disco e y o he Higgs
boson, none o he measu ed obse ables indica ed clea ly he ene gy scale o physics
beyond he S anda d Model (BSM). Indeed, only e en s p edic ed by he SM ha e been
seemingly de ec ed a pa icle collide s.
In ecen yea s he p ecise measu emen o SM p ocesses has aken a p ominen ole
as a undamen al ins umen in he sea ch o new physics. Anomalies in p ecision mea-
su emen s, like de ia ions om expec ed alues, can be an indi ec indica ion o new
phenomena. Indeed, h ough o -shell and loop e ec s, we can obse e he in luence o
new pa icles wi hou di ec ly p oducing hem.
In gene al, he phenomena we a e looking o ha e e y low p obabili y o occu ing,
making he de ec ion p ocess demanding. This aspec is pa icula ly ele an in p eci-
sion physics, since he measu emen o small a ia ions om he expec ed p edic ions
equi es la ge amoun o da a. One o he cen al mo i a ions behind he planned high
luminosi y (HL) LHC upg ade [13] is he de elopmen o he capabili ies o he LHC
19
20 CHAPTER 1. INTRODUCTION
as a p ecision machine. Inc easing luminosi y acili a es he gene a ion o mo e da a,
which no only enables a mo e comp ehensi e examina ion o es ablished mechanisms
bu , as al eady men ioned, opens also he doo o he obse a ion o unexpec ed new
phenomena.
Fo ins ance, he upcoming HL LHC is p ojec ed o gene a e a minimum o 15 mil-
lion Higgs bosons annually, a signi ican inc ease compa ed o he abou h ee million
collec ed du ing he LHC ope a ion in 2017 [14]. This enhancemen will allow us o pu
mo e s ingen cons ain s on many SM pa ame e s. In his con ex , gluon usion, which
is al eady he mos ele an p oduc ion mechanism [15,16] o Higgs physics, will be
e en mo e signi ican . I plays an impo an ole in double-Higgs p oduc ion, gg →HH,
and i is also ele an in he p oduc ion o a Higgs boson in associa ion wi h a Zboson,
gg →ZH. These wo sca e ings a e e y impo an o Higgs physics. The i s one
ecei es con ibu ions om he Higgs ilinea in e ac ion, one o ew SM couplings s ill
no measu ed. The second one is one o main channel o ZH and i allows o s udy
Higgs decays o bo om qua ks, a icky p ocess o obse e in had onic collide s due o
QCD backg ound. Fo his eason, in his hesis we s udied he nex - o-leading-o de
(NLO) i ual con ibu ions o hese p ocesses.
Indeed, he expe imen al e o needs o be complemen ed by an equi alen le el o
heo e ical wo k. This is essen ial because, in o de o he measu emen s o hold im-
po ance, ou SM p edic ions ha e o ma ch he p ecision o expe imen al esul s.
This easoning also applies o heo ies beyond he SM, and i is specially impo an
when conside ing E ec i e Field Theo ies (EFT). Indeed, wi hou elying on a spe-
ci ic BSM model, an ex ension o he SM wi h highe -dimensional ope a o s p o ides
an impo an way o desc ibe indi ec BSM phenomena. One o he mos commonly
used me hod o s udy new physics e ec s is he S anda d Model E ec i e Field Theo y
(SMEFT), which man ains he same ield composi ion and symme ies as he SM. In
his con ex , in es iga ing he p ope ies o massi e elec oweak gauge bosons (W and
Z bosons) has signi ican po en ial o indi ec sea ches o BSM physics. We can di ide
in wo classes he p ocesses ha ha e been s udied o explo e his di ec ion. The i s
one includes he mul i-boson p ocesses a high-ene gy collide s; he second one in ol es
he e mion sca e ing p ocesses media ed by s- o -channel W/Z bosons, also known
as he Elec oweak P ecision Obse ables (EWPOs). In his hesis we ocus on he la e .
In he las decades, EWPOs ha e o e ed clea indica ions ha he SM accu a ely de-
sc ibes physics a he EW scale, since hey a e sensi i e o modi ica ions o he gauge-
boson– e mion couplings and o he gauge-bosons masses. In ecen yea s he analysis
o hese measu emen s has been ex ended also beyond he he SM amewo k. Indeed,
wi hin he SMEFT app oach, he compa ison o EWPOs wi h heo e ical p edic ions can
pu signi ican bounds on he coe icien s associa ed wi h new physics. In his con ex ,
a leading o de (LO) in SMEFT nume ous s udies ha e been pe o med, ex ac ing
limi s on he coe icien s o dimension-6 ope a o s om global i s on EWPOs and o he
2.2. NON-ABELIAN HIGGS MECHANISM: THE ELECTROWEAK INTERACTIONS27
The in e ac ions in ol ing Zand Wwill also in ol e he masses o hese pa icles, which
espec he ela ion
mW=mZcos θW.(2.24)
This means ha all he e ec s o Wand Zexchange p ocess, a leas a ee-le el, can
be w i en in e ms o mZ,eand θW.
I is also use ul o de ine he pa ame e :
ρ≡m2
W
m2
Zcos2θW
= 1.(2.25)
The equa ion (2.25) is expe imen ally con i med a he pe mille le el [38]. As we
will see below, i is ela ed o he so-called cus odial symme y o he SM. Wha e e
mechanism one in oduces o s udy beyond he S anda d Model (BSM) heo ies, is highly
cons ained o ep oduce his esul .
2.2.1 SM e mion couplings o he gauge bosons
Using he co a ian de i a i e in equa ion (2.22) we can de e mine he couplings o he
ec o bosons o he SM e mions, once he quan um numbe o e mions ields a e se .
Taking in accoun he ac ha (expe imen ally) he W boson couple only o le -handed
qua ks and lep ons, we assign he le -handed SM e mions o double s o SU(2)Wwhile
making he igh -handed single unde his g oup. Once he τ3 alue has been speci ied
o each e mions, he alue Ycan be se using he elec ic cha ge ope a o . The igh -
handed e mions ha e τ3= 0, so he hype cha ge is equal o Q. In pa icula , he
igh -handed neu ino is comple ely uncha ged, unde he SM gauge g oup. Ins ead, o
he le -handed ield we ha e
lL=νe
e−L
, qL=u
dL
(2.26)
and assigning Y=−1
2 o he lep on double and Y= +1
6 o he qua k double , we e-
p oduce he co ec elec ic cha ge. No ice ha , he le - and he igh - handed e mions
li e in di e en undamen al ep esen a ions o he SM gauge g oup, so an use ul way
o look a hem is o hink o hese componen s as dis inc pa icles.
The kine ic Lag angian ollows di ec ly om he quan um numbe s assignmen and is
L=i¯
lL/
DlL+i¯qL/
DqL+i¯eR/
DeR+i¯uR/
DuR+i¯
dR/
DdR(2.27)
whe e he co a ian de i a i e is gi en by equa ion (2.22). We omi ed he igh -handed
neu ino in he kine ic e m since would ha e ze o coupling bo h o SU(2) and U(1).
Expanding he co a ian de i a i e, he Lag angian 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
whe e
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)
whe e in he EM cu en is w i en in he Di ac no a ion since he le and he igh
handed couplings a e equal.
No ice ha he JWcu en s, mix he up and he down componen s o each le double .
2.3 Mass e ms in he SM
In w i ing he Higgs ield couplings o he e mions, we will i s analyze he qua k
couplings o he Higgs ield ϕand hen we will discuss o lep on case.
The six le -handed qua ks a e a anged in o h ee double o SU(2)w, as ollows
qL=ui
diL
=u
dL
,c
sL
,
bL(2.34)
ha ing hype cha ge Y= +1
6. On he o he hand, he six igh -handed qua ks a e single s
unde SU(2)w: he up- ype qua ks ca y hype cha ge Y=2
3, while he down- ype ones
ha e Y=−1
3:
ui
R={uR, cR, R}, di
R={dR, sR, bR}(2.35)
Wi hou assuming any la ou conse a ion (o any o he addi ional symme y), he
mos gene al, eno malizable, gauge in a ian e m ha in ol es he ields 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)
whe e λij
uand λij
da e gene ic 3 ×3 complex ma ices. We can simpli y he o m o his
Lag angian by making a chi al ans o ma ion in la ou space. To ind he app op ia e
ans o ma ions we no ice ha by squa ing λuwe ob ain wo He mi ian ma ices λuλ†
u
and λ†
uλu. This can be diagonalized by de ining he uni a y ma ices Uuand Wu
λuλ†
u=UuD2
uU†
uλ†
uλu=WuD2
uW†
u(2.37)
2.3. MASS TERMS IN THE SM 29
whe e D2
uis a diagonal ma ix wi h posi i e eigen alues. Then we can w i e
λu=UuDuW†
u(2.38)
whe e Duis he diagonal ma ix whose diagonal elemen s a e he posi i e squa e- oo s
o he eigen alues o D2
u. By simila conside a ions we can w i e λdas
λd=UdDdW†
d(2.39)
No ice ha he uni a y ma ices Uu,d and Wu,d can s ill be mul iplied on he igh by
he same phase ma ix:
eiα10 0
0eiα20
0 0 eiα3
(2.40)
wi hou modi ying he ma ices λu,d.
We a e now eady o make he change o a iables wi h he ollowing ans o ma ions
ui
R=Wij
uui
Rdi
R=Wij
ddi
R.(2.41)
Since he igh -handed qua ks ui
Rand di
Rdo no couple o each o he h ough he kine ic
e ms, he ma ices Wuand Wddisappea om he heo y.
Again, we pe o m he change o a iables:
ui
L=Uij
uui
Rdi
L=Uij
ddi
L.(2.42)
These ans o ma ions elimina e Uuand Ud om he e ms in (2.37) ha in ol e he
lowe componen o he ield ϕ. In uni a y gauge only hese e ms su i e. I we now
de ine
mu=
√2Dumd=
√2Dd,(2.43)
in uni a y gauge, he equa ion (2.37) becomes:
Lm=−mij
u¯ui
Luj
R( 1 + h
)−mij
d¯
di
Ldj
R( 1 + h
) + h.c (2.44)
The ans o ma ions ca ied ou in equa ion (2.42) ha e some implica ions in he kine ic
Lag angian, because uLand dLa e mixed by he weak in e ac ions.
The only ele an e ms a e he couplings o he le -handed qua ks wi h he gauge
bosons W±:
¯uLγµdL→¯uLγµU†
uUddL.(2.45)
This means ha he cha ge-changing weak in e ac ions mix he h ee gene a ions o ui
L
qua ks wi h h ee gene a ions o di
Lqua ks, h ough a uni a y ma ix:
VCKM =U†
uUd(2.46)
whe e VCKM is known as he Cabibbo-Kobayashi-Maskawa (CKM) mixing ma ix.
In gene al, a uni a y 3 ×3 ma ix can be pa ame ized using six imagina y phases and
30 CHAPTER 2. STANDARD MODEL
h ee o a ion angles. Ne e heless, in ou case we can s ill emo e2 i e o his phases
by pe o ming a change o a iables on he qua k ields:
ui
L,R →eiαiui
L,R di
L,R →eiβidi
L,R (2.47)
The inal o m o VCKM will con ain h ee o a ion angle and one imagina y phase. The
la e is esponsible o he CP iola ion o weak in e ac ion.
Finally, we discuss he Higgs coupling wi h he lep ons. Like o he qua ks sec o we
ha e 3 gene a ions o lep ons ha co espond o h ee le -handed double s
lL=νi
eiL
=νe
dL
,νµ
µL
,ντ
τL(2.48)
each one wi h hype cha ge Y=−1
2, while he e a e only h ee igh -handed single
lep ons and hey ha e hype cha ge Y=−1 ( he cha ged lep ons)
ei
R={eR, µR, τR}.(2.49)
We now w i e he Yukawa couplings in ol ing he abo e ields and he ield ϕas
Lm=−λij
e¯
li
Lϕei
R+h.c. (2.50)
Neglec ing he igh -handed neu ino we ha e only one coupling. Following he same
s eps we did o he qua k sec o , we can w i e
λe=UeDeW†
e(2.51)
and make he change o a iables
ei
L→Uij
eej
Lνi
L→Uij
eνj
Lei
R→Wij
eej
R(2.52)
In uni a y gauge he Lag angian (2.50) eads
Lm=−mij
e¯ei
Lej
R( 1 + h
) + h.c (2.53)
whe e we ha e de ined he mass ma ix
me=
√2De(2.54)
No ice ha we ha e elimina ed all he uni a y ma ices om he heo y. In ac ,
since ei
Land νi
L ans o m in he same way, he couplings wi h he gauge bosons W±a e
in a ian unde his ans o ma ion. The esul ing lep on heo y does no iola e CP
symme y and u he p ese es he lep on numbe pe each gene a ion. This p ope y
is called lep on uni e sali y.
2This ac is ela ed o equa ion (2.40).
2.3. MASS TERMS IN THE SM 31
CKM ma ix and Wol es ein pa ame iza ion
The CKM ma ix is a uni a y 3 ×3 ma ix de ined on he qua k gene a ion space. I
can be pa ame e ized by h ee mixing angles and he CP- iola ing KM phase. O he
many possible con en ions, a s anda d choice has become [39]:
VCKM =
c12c13 s12c13 s13e−iδ
−s12c23 −c12s23s13eiδ c12c23 −s12s23s13eiδ s23c13
s12s23 −c12c23s13eiδ −c12s23 −s12c23s13eiδ c23c13
(2.55)
whe e sij = sin θij,cij = cos θij and δis he phase esponsible o he CP- iola ing
phenomena. No ice ha θij can be chosen o lie in he i s quad an , so sij,cij ≥0.
Since, i is known expe imen ally ha
s13 ≪s23 ≪s12 ≪1 (2.56)
i is con enien o in oduce ano he pa ame iza ion o exhibi his hie a chy; we de ine
he ou Wol ens ein pa ame e s [40]:
λ=s12
Aλ2=s23
Aλ3(ρ−iη) = s13e−iδ
hen, o λ3o de we he CKM ma ix eads:
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 iola ion can be de e mined by measu ing ρ−iη.
2.3.1 Cus odial symme y
Suppose o u n o he gauge couplings in equa ion (2.12). The ield ϕcan be ea anged
in a ma ix ields Φ as
Φ = (iσ2ϕ∗, ϕ) = ϕ∗
0ϕ+
−ϕ∗
+ϕ0(2.58)
and we can ew i e he Lag angian (2.12):
Lh=1
2T [(∂µΦ)†∂µΦ] + µ
2T [Φ†Φ] −λ
2T [(Φ†Φ)2] (2.59)
This Lag angian is in a ian unde a global symme y SU(2)L⊗SU(2)R:
Φ→LΦR†(2.60)
32 CHAPTER 2. STANDARD MODEL
wi h L∈SU(2)Land R∈SU(2)R.
The ield Φ, because o he po en ial o m, acqui es a non anishing VEV:
⟨Φ⟩ ∼ 0
0 (2.61)
This la e is s ill in a ian unde he subg oup SU(2)V⊂SU(2)L⊗SU(2)R, which
co esponds o he ans o ma ions:
Φ→VΦV†(2.62)
The global symme y is spon aneously b oken, wi h a b eaking pa e n SU(2)L⊗SU(2)R→
SU(2)Vand we ha e again ha h ee Golds one bosons eme ge om he heo y. Now,
suppose o gauge he SU(2)Lg oup. The Lag angian is s ill in a ian unde he global
SU(2)L⊗SU(2)Rg oup and he Golds one bosons, by he Higgs mechanism, can be
”ea en” o gi e mass o he gauge ields. In his case he physical eigens a es would co -
espond o linea combina ion o he SU(2)Lgauge ields, and he masses o he ec o
bosons would be exac ly degene a e:
m2
W=m2
Z=g2
4 2(2.63)
whe e gis he gauge coupling o SU(2)L.
In he SM, his scena io occu s when all he Yukawa couplings a e iden ical and
g′→0, esul ing in he Wand Zbecoming exac ly degene a e. The la e app oxima e
symme y is called cus odial symme y and i p o ec s he pa ame e ρ om quan um
co ec ions, gi ing he exac ela ion ρ= 1 a ee-le el. Fu he mo e, assuming ha
he hype cha ge co esponds o he hi d gene a o o SU(2)R,T3
R, i also explains why
only mZ ecei es a con ibu ion om he hype cha ge coupling g′. Finally, we no ice
ha he op qua k mass has an impo an ole on he explici b eaking o he cus odial
symme y. Indeed, being he hea ies elemen a y pa icle in he SM, he qua k op
in oduces signi ican quan um co ec ions o he ρpa ame e , causing a de ia ion (a
ew pe cen [41,42]) om he LO alue ρ= 1.
2.4 Beyond ee le el
The Lag angian we jus buil in oduces a numbe o ee pa ame e s ha a e di ec ly
ela ed o he physical obse ables. Indeed, a he Leading O de (LO) in he pe u ba-
ion heo y, he calcula ions in ol e only he simples Feynman diag ams. The esul s
a e ypically ini e quan i ies and can be di ec ly in e p e ed as physical obse ables.
Howe e , when highe -o de co ec ions a e aken in o accoun , we ha e mo e compli-
ca ed Feynman diag ams in ol ed and loops o i ual pa icles come in o play. The
loops in Feynman diag ams in oduce in eg als o e momen a o he i ual pa icles
2.4. BEYOND TREE LEVEL 33
and, when e alua ed, can lead o di e gen e ms. An example o hese in eg als is:
λZd4k
(2π)4
1
k2+m2∼ ∞,(2.64)
whe e λis a dimensionless coupling. The p esence o di e gen in eg als is an indica ion
ha he heo y equi es u he ea men .
2.4.1 Regula iza ion
In QFT, a ixed o de in he pe u ba ion heo y, we can expec a ini e numbe o
in eg als (di e gen o no ). Indeed, physical quan i ies con ains sums, p oduc s and
con olu ions o hem. Howe e , as men ioned in he p e ious sec ion, some o he
in eg als may be di e gen , bu his is no necessa ily a an unsol able p oblem. Indeed,
i a single in eg al does no con e ge he eason may be ha we isola ed he in eg al
om he es o he heo y. This may happen, o example, when we a e conside ing non
obse able pa ame e s, ha a e indeed no physical. In his case he di e gence is no a
p oblem, bu i is jus a ma e o e-pa ame iza ion. A heo y whe e all he di e gences
may be consis en ly emo ed in a ini e numbe o s eps is called eno malizable3.
As i s s ep owa ds he eno maliza ion o a heo y, we will b ie ly desc ibes how o
deal wi h in eg als in he o m (2.64). The idea is o in oduce an a i icial p esc ip ion
o make ou in eg al ini e; an example is he in oduc ion o a cu -o Λ:
λZ|k|≤Λ
d4k
(2π)4
1
k2+m2∼Λ2.(2.65)
The pa ame e in oduced is called egula o and he p esc ip ion o ende ing di e -
gen diag ams ini e is called egula iza ion. O cou se in he limi Λ → ∞ he in eg al
is s ill di e gen , bu his ick allow us o pos pone his ope a ion un il we ha e calcu-
la ed quan i ies ha a e physically ele an . Indeed he p esence o a di e gence in a
single in eg al is no an issue i , a e a e-pa ame iza ion, he physical quan i ies admi
he emo al o he egula o , which in his scheme co esponds o he limi Λ → ∞.
Se e al egula iza ion schemes we e p oposed in he li e a u e, bu he one ha es ab-
lished i sel o be he a ou i e s a egy in heo e ical pa icle physics is he Dimensional
Regula iza ion. This me hod p ese es Lo en z and gauge in a iance and has he use ul
cha ac e is ics o being applicable also on in a ed di e gences. The idea o dimensional
egula iza ion is e y simple: we ex end he 4-dimensional domain o de ini ion o he
analy ical unc ion in o D= 4 −2ϵdimensions:
λµ4−DZdDk
(2π)D
1
k2+m2∼2
4−D∼1
ϵ.(2.66)
3In a se ies o pape s o 1971-1972, G. ’Hoo and M. Vel man p o ed ha he Elec oweak Theo y
is eno malizable [43].
34 CHAPTER 2. STANDARD MODEL
No ice ha in Ddimensions, he coupling λhas non–ze o mass dimension 4 −D, so we
eplaced λ→λµ4−D. The new λis dimensionless and µis an a bi a y ene gy scale. As
we can see, he e ϵplays he ole o he egula o , he limi o conside is ϵ→0 and he
di e gence mani es i sel as a pole in ϵ.
2.4.2 Di ac Algeb a
In gene al, o consis en ly e alua e he in eg and in in eg als o he o m 2.66, we will
need a gene alisa ion o he me ic and o he Di ac algeb a o Ddimensions. Some o
he p esc ibed con inua ions o he Lo en z and Di ac algeb a a e e y na u al:
gµνgµν =D,
{γµ, γν}= 2gµν 1,
γµγµ=1
2gµν{γµ, γν}
=gµνgµν =D. (2.67)
Howe e o he p esc ip ions a e a mo e sub le and equi e a mo e ca e ul explana ion.
Indeed, since Dimensional egula iza ion was in oduced, i was e iden ha dealing
wi h chi al couplings was a complica ed issue. The p oblem eme ges because he usual
4-dimensional γ5, de ined as
γ5=−iγ0γ1γ2γ3,(2.68)
does no ha e a canonical ex ension in o D dimension. Indeed [43], we canno expec o
p ese e he an icommu a ion ela ion
{γµ, γ5}= 0 (2.69)
and, a he same ime, handle he aces o he Di ac ma ices wi h he usual ma he-
ma ical ules. In a nu shell, as poin ed ou by D. K ame [44] and ’ Hoo -Vel man [43,
45], one has o decide be ween keeping he cyclic p ope y o he ace (HVBM scheme)
o keeping he an icommu a ion ela ion 2.69 o he γ5(K ame scheme). Clea ly, he
scheme used has a ele an impac on he he Feynman ules and on he Di ac aces. In
he HVBM scheme, he an icommu a ion ule 2.69 is abandoned and he γ5de ini ion
is ex ended o a −2ϵ-dimensional subspace:
{γµ, γ5}= 0 4 dimensions
[γµ, γ5] = 0 −2ϵdimensions.(2.70)
HVBM scheme p o ides a na u al ea men o handle Di ac aces. Howe e , i has he
signi ican d awback o iola e he he chi al symme y o he SM (hence o Wa d Iden i y
2.4. BEYOND TREE LEVEL 35
) in all pe u ba i e calcula ions in ol ing chi al couplings. The solu ion p oposed by
B ei enlohne -Maison (BM in he HVBM scheme) in ol es he es o a ion o he chi al
symme y as pa o he eno maliza ion p ocess. The idea is o in oduce addi ional
ini e coun e e ms alongside he usual di e gen ones. I is impo an o no e ha his
es o a ion o chi al symme y is no limi ed o in e ac ions explici ly in ol ing γ5bu is
expec ed o all in e ac ions in he SM. Addi ionally, hese ini e coun e e ms need o be
calcula ed beyond O(ϵ0) o consis en ly es o e chi al symme y a highe pe u ba i e
o de s. Thus, he eno maliza ion p ocedu e can be e y demanding.
Fo his eason, we will use he K ame scheme o deal wi h aces o Di ac ma ices.
Indeed, K ame scheme does no need he in oduc ion o any coun e e m, simpli ying
subs an ially he calcula ion and, as we will see below, his p ocedu e does no b eak
he chi al symme y o he SM. Ne e heless, i is impo an o men ion ha K ame
scheme has no been comple ely accep ed and emb aced as a p e e able choice o HVBM
and he e a e s ill doub s abou i s use beyond one loop [46,47]. Howe e , in his hesis
we will use he K ame scheme only o one loop calcula ions.
2.4.3 K ame scheme
The ules o handling Di ac ma ices in K ame scheme a e[44]:
•An icommu a ion ules:
{γµ, γν}= 2gµν 1,
{γµ, γ5}= 0.(2.71)
•Fo he aces o Di ac ma ices wi hou γ5:
T [γµ1γµ2. . . γµ2n−1] = 0,
T [γµ1γµ2. . . γµ2n] = 4 X
σ
(−1)sgn(σ)gµi1µj1gµi2µj2. . . gµinµjn.
(2.72)
•Fo he aces o Di ac ma ices in ol ing γ5we ha e:
T [γµ1γµ2. . . γµ2n−1γ5] = 0,
T [γµ1γµ2. . . γµ2nγ5] = 4iX
σ
(−1)sgn(σ)ϵµin+1 µin+2 µjn+1 µjn+1 gµi1µj1. . . gµin+2 µin+2 ,
(2.73)
wi h
1 = i1<··· < in+ 2, ik< jk,
whe e ϵµ1µ2µ3µ4is he 4-dimensional Le i-Ci i a enso .
36 CHAPTER 2. STANDARD MODEL
•I is o bidden o use cyclic p ope y o he ace when an odd numbe o γ5is
in ol ed.
•I he e is mo e han one diag am con ibu ing o a gi en p ocess, all he aces
mus be ead s a ing a he same poin 4.
•In he case whe e an anomalous axial cu en is in ol ed, he ace o he anoma-
lous g aph mus be ead s a ing om an axial ec o e ex, in o de o ul ill he
usual con en ion o conse ed ec o cu en s. In he case o mul iple axial ec o
e ices a symme ic choice o he eading p esc ip ion mus be used.
No ice ha when one is compu ing Di ac aces wi h a mos ou Di ac gammas and
γ5, he K ame scheme is equi alen o he Nai e Dimensional Regula iza ion (NDR).
Mo e de ails abou he di e ences and he analogies be ween he schemes discussed a e
explained in [46].
2.5 Reno maliza ion
We know b ie ly desc ibe he eno maliza ion p ocedu e, ollowing [48]. A p ac ical way
o ca y ou he eno maliza ion is he so-called coun e e m app oach. The basic idea
behind he coun e e m app oach is o in oduce addi ional e ms, known as coun e -
e ms, in o he Lag angian o he heo y. Fi s s ep, we ha e o eplace he pa ame e s
o he Lag angian wi h hei ba e quan i ies (in he ollowing deno ed wi h a ”0” sub-
sc ip ), meaning ha hey canno be di ec ly in e p e ed as physical quan i ies. Then,
he ba e quan i ies a e sepa a ed in o wo componen s: a ini e eno malized pa and
a di e gen coun e e m o eno maliza ion cons an . E e y ba e pa ame e K0is spli
as:
K0=ZKK= (1 + δK)K.(2.74)
Then inse ing his ans o ma ions in o he classical Lag angian we ha e
L0=L+δL(2.75)
whe e Lis he same as he o iginal Lag angian bu wi h ba e quan i ies eplaced by
eno malized ones and δLis he coun e e m Lag angian. This la e g oups all he
po en ial UV e ms and leads o an addi ional se o Feynman ules. A e imposing he
eno maliza ion condi ions on he eno malized quan i ies, we can ob ain hei nume ical
alues using expe imen al inpu s. The choice o he inpu obse ables and he eno mal-
iza ion condi ions de e mine he eno maliza ion scheme. The complexi y o he heo y
oge he wi h a bi a iness in he choice o he ‘bes ’ eno maliza ion scheme can make
i di icul o compa e and o ollow di e en app oaches. In he li e a u e a e p esen
many di e en schemes [49]; depending on he physical p ocess we a e conside ing, i
may be con enien o use a eno maliza ion scheme a he han ano he .
4In he D ell Yan sca e ing, he e mion chains o he LO will p o ide us a na u al eading poin .
3.2. METHOD 43
he Dynkin index. Fo he double higgs p oduc ion we ha e:
Aµν
HH =Aµν
1F1+Aµν
2F2(3.10)
whe e F1and F2a e he o m ac o s associa ed o he spin-0 and spin-2 p ojec o s,
espec i ely. Fo he HZ p oduc ion we ha e:
Aµνρ
HZ =
7
X
i
Pµνρ
iAiHZ (3.11)
whe e Pia e a basis o o hono mal p ojec o s and AiHZ a e he associa ed scala o m
ac o s. In bo h he p ocesses, hey depend only on scala quan i ies, namely he op
qua k mass m , he masses o he ex e nal pa icles and he pa onic Mandels am a i-
ables. Addi ionally, aking all momen a o be incoming, we de ine he pa onic Mandel-
s am a iables as
ˆs= (p1+p2)2,ˆ
= (p1+p3)2,ˆu= (p2+p3)2,(3.12)
and he ans e se momen um pTo he inal-s a e pa icles can be w i en as
p2
T=ˆ
ˆu−m2
3m2
4
ˆs.(3.13)
As sugges ed in e s. [29,56], i he ampli ude o he p ocess is w i en in e ms o
(an i)symme ic o m ac o s wi h espec o he exchange ˆ
↔ˆu, hen i is su icien o
discuss only he o wa d con ibu ion o he c oss sec ion. The e o e, in he ollowing
we will always assume ha |ˆ
| ≤ |ˆu|and ha
ˆ
=−1
2ˆs−m2
3−m2
4−qλ(ˆs, m2
3, m2
4)−4ˆs p2
T,(3.14)
whe e λ(a, b, c) = a2+b2+c2−2ab −2ac −2bc is he K¨all´en unc ion.
3.2 Me hod
In gene al, he deg ee o di icul y in he e alua ion o loop diag ams g ows wi h he
numbe o ene gy scales p esen in he diag am. In he case o single-Higgs p oduc ion
he ele an diag ams ea u e a iangula opology and, consequen ly, depend upon
only wo scales, namely he Higgs mass, mH, and he op mass2,m . In his case, he
unc ional dependence o he esul upon he op mass can be exp essed in e ms o one
single a iable, m2
H/m2
. Due o his simpli ied one-scale si ua ion, exac analy ic esul s
o he NLO co ec ions a e a ailable since many yea s [57,58,59,60].
In he case o p ocesses wi h wo pa icles in he inal s a e he si ua ion is mo e com-
plica ed. Indeed hese p ocesses ecei e con ibu ions no only om iangle diag ams,
2All he qua ks bu he op a e assumed o be massless.
44 CHAPTER 3. HIGGS PRODUCTION FROM GLUON FUSION
ha can be calcula ed adap ing he exac analy ic esul s ob ained o single-Higgs p o-
duc ion, bu also om box- opology diag ams. In pai p oduc ion, gg →HH, he
box diag ams depend upon ou scales, namely ˆs, ˆ
, m , mH, whe e ˆs, ˆ
, and ˆua e he
Mandels am a iables which sa is y he condi ion
ˆs+ˆ
+ ˆu= 2 m2
H.(3.15)
Conce ning associa ed p oduc ion, gg →ZH, a i h ene gy scale is p esen , i.e. he
mass o he Z ec o boson, mZ.
Exac analy ic esul s o wo-loop box diag ams wi h se e al ene gy scales canno
be de i ed wi h he p esen compu a ional echnology. Ins ead, usually wo di e en
s a egies a e ollowed in o de o e alua e he wo-loop box con ibu ion in Higgs p o-
duc ion ia gluon usion. a) A ully nume ical exac e alua ion [61,62,63,64,65].
b) An app oxima e analy ic e alua ion ha akes ad an age o hie a chies among he
a ious ene gy scales p esen in he diag ams, in o de o educe he numbe o scales
in he p oblem. Thus, i s alidi y is es ic ed o speci ic egions o he phase space.
The me hod used is based on he expansion o he diag ams in e ms o a ios o small
ene gy scales s. la ge ene gy scales, in o de o ob ain a esul ha e ains an exac
dependence upon he la ge ene gy scales. Conce ning he small ones, in o de o simpli y
u he he e alua ion, expansions in e m o a ios be ween small ene gy scales is o en
used.
The o me s a egy, al hough accu a e, is e y demanding om a compu a ional
poin o iew, equi ing a high deg ee o op imiza ion in o de o ob ain a esul in a
easonable, al hough usually qui e long, compu e ime. Fu he mo e, his app oach is
no e y lexible wi h espec o he modi ica ion o he inpu pa ame e s.
S a egy b) p o ides accu a e esul s alid in speci ic egions o he phase space
wi hou equi ing hea y compu a ional wo k, i.e. in a sho compu e ime. Examples
o his app oach o e alua ing he wo-loop box con ibu ion a e:
i) The in ini e- op-mass limi [66,67] e ined by he inclusion o powe s in he la ge
op-mass expansion (LME) [68,69,70,71]. He e, m is assumed o be he la ge
ene gy scale while ˆs, ˆ
, mH, and in associa ed p oduc ion also mZ, a e conside ed
o be he small ones. Thus, he alidi y o his app oach is es ic ed o phase-
space egions whe e ˆs/(4m2
)≤1. The ad an age o his app oxima ion a e he
a he simple esul s, which can be exp essed in e ms o a ional unc ions and
loga i hms o he kind log(m2
/ˆs).
ii) The e alua ion ia an expansion in he ans e se momen un, pT, o he inal-s a e
pa icles [29,56]. He e, ˆsand m a e assumed o be he la ge ene gy scale while
mH, mZand pT, ha can be aded o ˆ
, a e conside ed o be he small ones. The
alidi y o his app oach is es ic ed o phase-space egions whe e |ˆ
|/(4m2
)≲1.
The analy ical complexi y o his app oach is highe han in i), as gene alised poly-
loga i hms and wo ellip ic in eg als occu in he inal esul s. The e alua ion o
he la e can be easily pe o med using he esul s o Re . [72].
3.2. METHOD 45
iii) The e alua ion ia a high-ene gy (HE) expansion [30,73,74]. He e ˆs, ˆ
a e as-
sumed o be he la ge ene gy scale while m , mHand mZ, wi h m ≫mH, mZ,
a e conside ed o be he small ones. The alidi y o his app oach is es ic ed o
phase-space egions whe e |ˆ
|/(4m2
)≳1. The HE expansion leads o analy ical
esul s ha can be exp essed in e ms o ha monic polyloga i hms.
i ) The e alua ion ia an expansion in e ms o small ex e nal masses [75,76,77].
He e ˆs, ˆ
, m a e assumed o be he la ge ene gy scale while mHand mZa e
conside ed o be he small ones. This app oach basically co e s he en i e phase
space o he conside ed p ocesses. Howe e , since he educ ion o scales in his
app oach is minimal, one ends up wi h he e alua ion o Mas e In eg als (MIs)
ha a e much mo e complica ed han hose appea ing in he i)–iii) cases. As a
consequence he e alua ion o he box con ibu ion in any poin o he phase space
equi es a longe compu e ime han in he app oaches i)–iii).
As an al e na i e app oach, e s. [78,79] p oposed o econs uc he ull esul om
i s LME e sion, supplemen ed by he non-analy ic pa o he diag ams nea he op
h eshold, ia a con o mal mapping and Pad´e app oximan s.
In [80] we p opose an al e na i e way o de i e he ull op-mass dependence in Higgs
p oduc ion ia gluon usion, based on he me ging o he pTexpansion in ii) wi h he HE
expansion in iii) ha indi idually a e alid in complemen a y egions o he phase space.
Since he nume ical e alua ions o he wo expansions a e qui e as om a compu a ional
poin o iew, ou p oposal allows a as e alua ion o he i ual co ec ions o Higgs
p oduc ion ia gluon usion ha is accu a e in he en i e phase space.
The key poin o ou analysis is o ex end he ixed-o de esul s bo h in he pT
expansion [29,56] and in he HE expansion [73,74] up o o beyond hei bo de o
alidi y, i.e. ˆ
≃4m2
, in o de o me ge he wo analy ic app oxima ions. This is done by
cons uc ing a [1/1] Pad´e app oximan o he pT- esul and a [6/6] Pad´e app oximan
o he HE- esul . We poin ou ha he ex ension o he HE expansion ia Pad´e
app oximan s has been al eady conside ed in e s. [73,74].
3.2.1 Pade app oximan
In he o wa d egime, he alidi y o bo h he pTand HE expansions is limi ed by he
condi ion
|ˆ
| ≃ 4m2
,(3.16)
i.e. o any ixed alue o ˆs, he pTexpansion p o ides eliable esul s when |ˆ
|≲4m2
while he HE expansion is accu a e o |ˆ
|≳4m2
, i he ixed ˆs > 4m2
. Howe e ,
we ind ha in he icini y o he poin |ˆ
|= 4m2
he ixed-o de esul s in he pT
expansion and in he HE expansion a e bo h di e gen (see ig. 3.2). As a consequence,
a s aigh o wa d combina ion o he pT-expanded and he HE-expanded esul s canno
allow o an accu a e desc ip ion o he abo e egion, and his ac p e en s a ull co e age
46 CHAPTER 3. HIGGS PRODUCTION FROM GLUON FUSION
o he phase space. We poin ou ha his si ua ion does no change subs an ially when
highe o de s in bo h he expansions a e compu ed.
Al e na i ely, he con e gence o he expanded esul s can be imp o ed by conside -
ing he espec i e Pad´e app oximan s. Indeed, s a ing om a gi en Taylo expansion
o an exac unc ion (x) a ound x= 0 up o he i s e ms
(x)≃
−1
X
k=0
ckxk,(3.17)
i is possible o cons uc he associa ed Pad´e app oximan , de ined as
[m/n](x) = p0+p1x+···+pmxm
1 + q1x+. . . qnxn,(3.18)
p o ided ha m+n+ 1 = . Speci ically, by Taylo -expanding he .h.s. o eq. (3.18),
he {pi, qj}coe icien s o he Pad´e app oximan can be w i en in e ms o he ckones
known om eq. (3.17), by sol ing a sys em o linea equa ions. Usually, [m/n] Pad´e
app oximan s such ha m=ngi e he bes imp o emen in he con e gence o he
o iginal Taylo expansion, and we conside only hese combina ions in ou s udy. In
he pT-expanded esul s, a NLO, only he i s h ee e ms in eq. (3.17) a e known and
he e o e we a e limi ed o cons uc a [1/1] Pad´e app oximan (we will e e o his as
he pT-Pad´e). Ins ead he a ailabili y o many e ms in he HE-expansion esul s allows
o conside se e al [n/n] app oximan s (de ined as HE-Pad´e).
When calcula ing he pT-Pad´e, ca e is o be aken in he ea men o he expansion
pa ame e s. As discussed in e s.[29,56], no only he pTbu also he masses o he
ex e nal pa icles a e unde s ood as small pa ame e s. Since hese a e all ea ed on
he same oo ing wi h espec o he la ge scales se by ˆsand m , we can w i e he
gene al exp ession o a pT-expanded o m ac o Fin he ampli ude in e ms o a
scaling pa ame e 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)
whe e m3is in e p e ed as mHand mZ o gg →HH and gg →ZH, espec i ely,
and ∆m= (m2
4−m2
3)/2 is included only o he ZH case (see e . [56]). S a ing om
eq. (3.19) we can hen ob ain he co esponding [1/1] Pad´e app oximan wi h espec o
he limi x→0
[1/1](x) = p0+p1x
1 + q1x,(3.20)
wi h
p0=c0p1=c1−c0c2
c1
q1=−c2
c1
,
and subsequen ly se x= 1 in eq. (3.20).
We wan o cla i y a possible sou ce o ambigui y conce ning he limi o alidi y o
he pTexpansion. Indeed, while in he p e ious wo ks we sugges ed ha his expansion
3.2. METHOD 47
is alid o p2
T≲4m2
, as he compa ison a LO be ween he pT-expanded and exac
esul seems o indica e, in his pape we ollow a mo e conse a i e app oach and we
conside as limi o alidi y o he pTexpansion |ˆ
|≲4m2
. Addi ionally, we checked
ha he same complemen a i y o he pTand HE expansions can be obse ed when
choosing p2
T= 4m2
as limi o alidi y.
We now discuss he p ocedu e adop ed o cons uc he Pad´e app oximan s om
he HE expansion. Following he p esc ip ion o e . [81] (see also [82,83]), we ini ially
a ange he a ious o de s F(i)o he HE expansion o a gi en o m ac o as ollows
F(x) = F(0) +
L
X
l=1 F(2l−1)m(2l−1)
+F(2l)m(2l)
xl=
L
X
l=0
dlxl,(3.21)
whe e o de s ela ed o odd powe s o m a e g ouped wi h he o de s ela ed o he nex
e en powe . Then, we cons uc [n/n] app oximan s in xwi h 2n=L om eq. (3.21)
using he analy ic exp essions a ailable in [84,85], and se ing x= 1. We ema k ha
ou Pad´e app oximan s a e ob ained in a ully symbolic way, whe eas in e s. [81,82] all
he kinema ical quan i ies a e ixed o he espec i e nume ical alues be o e he Pad´es
a e cons uc ed in x. Fu he mo e, in compa ison o e s.[81,74], we only s udied [n/n]
HE-Pad´es up o n= 6. In hose e e ences Pad´es wi h n > 6 we e also conside ed in
o de o ex apola e he esul s in he egion |ˆ
|<4m2
, o a ixed ˆs, and cha ac e ize he
ela i e unce ain ies o di e en [m/n] Pad´es. In ou case, because he egion |ˆ
|<4m2
,
is mo e accu a ely desc ibed by he esul s o he pTexpansion, we ind ha a [6/6] HE-
Pad´e is mo e han enough o pe o m he me ging wi h he pT- esul and, a he same
ime, o desc ibe accu a ely he high-ene gy egion.
The pT-Pad´e and he HE-Pad´e ex end he ange o alidi y o each expansion beyond
i s limi . As discussed in he nex sec ion, he pTand he HE Pad´es a e accu a e
enough o b idge he gap a ound he phase-space egion |ˆ
| ≃ 4m2
. Then, an accu a e
app oxima ion o he exac esul o any phase-space poin (ˆs, ˆ
) can be ob ained by
choosing as swi ching poin be ween he Pad´e-imp o ed expansions any poin in he
egion |ˆ
| ∼ 4m2
. Fo simplici y we choose o use he pT-Pad´e when |ˆ
|<4m2
and he
HE-Pad´e when |ˆ
| ≥ 4m2
, o any ixed alue o ˆs. We ecall ha in ou discussion
we jus conside he o wa d egion |ˆ
| ≤ |ˆu|, while he esul in he complemen a y
phase-space egion is ob ained using he symme y o ou o m ac o s unde ˆ
↔ˆu.
No icing ha , when |ˆ
| ≤ |ˆu|, he maximum absolu e alue o ˆ
as a unc ion o ˆsis
gi en by |ˆ
|max = 1/2(ˆs−m2
3−m2
4) ou choice co esponds o using he pT-Pad´e up o
he pa onic ene gy ˆsc= 8m2
+m2
3+m2
4. In his ene gy egion (√ˆsc≃500 GeV o
gg →HH and gg →ZH) a he LHC mo e han 2/3 o he had onic c oss sec ion is
concen a ed.
48 CHAPTER 3. HIGGS PRODUCTION FROM GLUON FUSION
3.3 The pTand HE expansions s he exac esul s
a LO
In his sec ion we assess he eliabili y o ou me ging p ocedu e by s udying how well
he combina ion o he pT-Pad´e and he HE-Pad´e can ep oduce he exac LO esul s
o HH and ZH p oduc ion ia gluon usion. Fo he sake o simplici y, we discuss in
de ail only he gg →HH p ocess, bu we e i ied ha simila conclusions can be d awn
o gg →ZH. We ecall ha he ampli ude o gg →HH can be exp essed as
Aµν =Gµ
√2
αS(µR)
2πδabTFˆs[Aµν
1F1+Aµν
2F2],(3.22)
and ha bo h iangle and box diag ams con ibu e o F1
F1=F△
3m2
H
ˆs−m2
H
+F□,(3.23)
whe eas he F2 o m ac o ecei es con ibu ion only om boxes. Ou goal is o imp o e
he e alua ion o he box con ibu ions, he e o e we ocus on he discussion o F□and
F2. The LO esul s o hese o m ac o s, deno ed as FLO
□and FLO
2, a e shown in
ig. 3.2, o ixed alues o he pa onic cen e -o -mass ene gy.
Only la ge alues o ˆsa e shown in ig. 3.2 because o small ˆs alues he pT-expanded
esul s a e e y accu a e [29]. The pT-expanded and HE-expanded esul s a e ep esen ed
by he blue and pu ple solid lines, espec i ely, and hey de ia e om he exac esul ,
shown as a solid black line, a |ˆ
|/4m2
≃1, as an icipa ed in he p e ious sec ion. The
ligh blue dashed line s ands o he [1/1] pT-Pad´e, while he pink dashed line ep esen s
he [6/6] HE-Pad´e. One can see ha he Pad´e esul s show an imp o ed con e gence
wi h espec o he ixed-o de expansions. The bo om pa o he plo s in ig. 3.2 shows
he a io o he expanded and Pad´e esul s o he exac one. Indeed, ig. 3.2(a,b) shows
ha in he case o FLO
□ o |ˆ
|/4m2
= 1 he di e ences o he Pad´e esul s wi h espec
o he exac p edic ion a e negligible. Fo he F2 o m ac o , whose con ibu ion o
he c oss-sec ion is much smalle han he one o he F1 o m ac o , he di e ence is
always below 5%, see ig. 3.2(c,d). We no ice ha , when compa ing he accu acies o
he Pad´e app oximan s, la ge disc epancies can be a ibu ed o he pT-Pad´e. Indeed,
being he la e a [1/1] Pad´e, i is expec ed o be a less e ined app oxima ion han
he [6/6] HE-Pad´e. S ill, in he case o FLO
□ he di e ences be ween he wo Pad´e nea
|ˆ
|/4m2
= 1 a e negligible. We also no ice ha , as ˆsinc eases, la ge alues o |ˆ
|a e
allowed by he kinema ics, and he ela i e impo ance o he HE expansion inc eases.
The imp o emen in con e gence p o ided by he Pad´e app oximan s is such ha
he me ging o he wo esul s discussed in he p e ious sec ion can ep oduce he exac
p edic ion wi h good accu acy o e e y alue o ˆ
, o any ˆs. While we e ain om
showing mo e examples he e, we no e ha we s udied he beha iou o all he box
con ibu ions o gg →HH and gg →ZH a se e al alues o ˆs. We explici ly checked
ha , among he a ious possibili es, a [6/6] HE-Pad´e is mo e han enough o an accu a e
me ging. Fu he mo e, we obse ed ha he alue |ˆ
|= 4m2
is a good choice as a me ging
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
a io o ull
−ˆ
/(4 m2
)
|FLO
|
ull
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
a io o ull
−ˆ
/(4 m2
)
|FLO
|
ull
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
a io o ull
−ˆ
/(4 m2
)
|FLO
2|
ull
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
a io o ull
−ˆ
/(4 m2
)
|FLO
2|
ull
PTexp
PTexp [1/1]
HE
HE [6/6]
(d)
Figu e 3.2: Modulus o he box o m ac o s con ibu ing o gg →HH a LO, o a
ixed alue o (a,c) √ˆs= 0.9 TeV and (b,d) √ˆs= 2 TeV. In he uppe pa o each plo ,
he exac p edic ion (solid black line) is shown oge he wi h he pTand HE expansions
(solid blue and pu ple lines, espec i ely) and wi h he [1/1] pTand [6/6] HE Pad´e
app oximan s (dashed ligh blue and pink lines, espec i ely). The bo om pa o each
plo shows he a io o he abo e esul s o he exac p edic ion.
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
a io o ull
MHH [GeV]
ˆσ(0) [ b]
ull
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
a io o ull
MZH [GeV]
ˆσ(0) [ b]
ull
small pTand HE expansion
(b)
Figu e 3.3: Pa onic c oss sec ion a LO o (a) gg →HH and (b) gg →ZH. The
uppe pa o each plo shows he exac p edic ion (solid line) oge he wi h he me ging
o he pTand HE Pad´e app oximan s (dashed line). The bo om pa o each plo shows
he a io o he me ged esul o he exac p edic ion.
poin o he pTand HE Pad´e app oximan s. The high le el o accu acy o ou me ging
me hod can be obse ed in ig. 3.3, whe e he pa onic c oss sec ion a LO is shown o
gg →HH and gg →ZH. One can see ha de ia ions o he combina ion o he pT-
and HE-Pad´e wi h espec o he exac p edic ion ne e exceed 1%.
3.4 Me ging he pTand HE expansions a NLO
In he p e ious sec ion we showed ha he me ging o he pT- and HE-Pad´e can ac-
cu a ely ep oduce he exac LO p edici on. In his sec ion we p esen he me ging
o he NLO pT-expanded and HE-expanded esul s imp o ed by he espec i e Pad´e
app oximan s.
In ig. 3.4 he NLO con ibu ions o F□and F2a e shown3. The ela i e beha iou
o he a ious app oxima ions is analogous o wha we obse ed a LO. Fo low alues
o |ˆ
| he ixed-o de and Pad´e-imp o ed pT-expanded esul s ag ee well. Inc easing he
alue o |ˆ
|up o he me ging egion, |ˆ
| ∼ 4m2
, he pT-Pad´e becomes close o he Pad´e-
imp o ed HE expansion. Fo alues abo e |ˆ
|= 4m2
he pTand HE Pad´e app oximan s
show small de ia ions as expec ed. The NLO s udy shows he same quali a i e beha iou
as he LO one. This makes us con iden ha he p oposed me ging p ocedu e wo ks
well also a NLO.
3FNLO
□and FNLO
2a e he o m ac o s as de ined in eq. (3.22) bu do no con ain he double iangle
diag ams ha can be exp essed in e ms o p oduc s o one-loop in eg als and as such a e compu ed
analy ically in exac op-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
/(4 m2
)
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
/(4 m2
)
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
/(4 m2
)
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
/(4 m2
)
PTexp
PTexp [1/1]
HE
HE [6/6]
(d)
Figu e 3.4: Modulus o he box o m ac o s con ibu ing o gg →HH a NLO, o a
ixed alue o (a,c) √ˆs= 0.9 TeV and (b,d) √ˆs= 2 TeV. The pTand HE expansions a e
shown as solid blue and pu ple lines, espec i ely, while he [1,1] pT- and [6,6] HE-Pad´e
a e shown as dashed ligh blue and pink lines, espec i ely.
52 CHAPTER 3. HIGGS PRODUCTION FROM GLUON FUSION
MHH [GeV] ˆ
[GeV2]VPade
in Vg id
in
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: Compa ison o a ious nume ical alues o V in aken om he g id o e . [86]
wi h ou Pad´e cons uc ion.
We now compa e ou e alua ion o he i ual co ec ions o he di-Higgs p oduc-
ion p ocess4wi h he nume ical esul p o ided as a g id in e . [86]. This e e ence
summa izes he wo k o e . [81], whe e he nume ical calcula ion in exac op-mass de-
pendence o e . [62] was supplemen ed by he esul in he HE expansion o e . [73].
The compa ison is done on he quan i y
∆ˆσ i =Zˆ
+
ˆ
−
αs
32π2
1
ˆs2V indˆ
, (3.24)
whe e he ini e pa o he i ual co ec ions V in is de ined as in e . [78]. The esul s
a e shown in ig. 3.5. We no e ha V in depends on he choice o he IR sub ac ion,
his is why in he lowe panel o ig. 3.5 he di e ence be ween he expanded esul s
and he nume ical g id o e . [81] is shown (di ided by he Bo n esul ), a quan i y
ha is independen o he IR sub ac ion e m. The g id o e . [86] shows e y good
ag eemen wi h ou esul s a e e y in a ian mass, excep o he i s ew bins a low
MHH. The eason is a la ge unce ain y o he nume ical g id on he low MHH bins, ha
a e desc ibed by only a ew poin s in he nume ical g id due o hei small con ibu ion
o he o al c oss sec ion. Fo mode a e and la ge MHH we obse e di e ences below 1%
in he a io be ween ou esul s and he ones o e . [81]. We con i m hese indings by
compa ing ou esul s o V in wi h he alues gi en in he g id [86] o a ious poin s
a ixed (MHH,ˆ
). We ind a good ag eemen , as can be in e ed om able 3.1 o some
ep esen a i e alues. We no ice ha in he egion whe e bo h expansions pe o m less
well we see di e ences o a ew pe cen , al hough he la e will be educed in ∆ˆσ i due
o he in eg a ion o e ˆ
.
Finally, we show ha ou me ging app oach is lexible wi h espec o he modi ica-
ion o he inpu pa ame e s by compu ing he i ual co ec ions o a ious eno mal-
isa ion schemes o he op qua k mass. I was no ed in e s. [63,64] ha he di-Higgs
p oduc ion p ocess su e s om a la ge unce ain y associa ed o he eno malisa ion
4We no e ha o ZH p oduc ion no public code including he esul s o he ull compu a ion [65]
is cu en ly a ailable. Hence we e ain om making any compa isons o ZH p oduc ion.
4.2. HEFT 59
o ope a o s a e gene a ed and only a ini e numbe o coun e e ms is necessa y. This
p ocedu e is e ec i e bu elies on ce ain assump ions o be meaning ul. To ensu e he
alidi y o his app oach, one mus assume ha con ibu ions om highe -dimensional
ope a o s and double inse ions o ope a o s a e much smalle han single inse ions. In
his con ex , we can ake he SM as an example. Le ’s assume ha he cu -o o he
SM is Λ = 1 TeV
dim-6 ope a o : Cdim−6
2
Λ2∼Cdim−6
16
wo dim-6 ope a o s: C2
dim−6
2
Λ4∼C2
dim−6
162
dim-8 ope a o : Cdim−8
4
Λ4∼Cdim−8
162(4.1)
whe e Cdim is he Wilson coe icien o he ope a o and is he Higgs e ha we
used o make he con ibu ion adimensional1. By applying his escaling, we ob ain he
condi ions Cdim−6≪16 and Cdim−8≪Cdim−6. These condi ions appea easonable,
bu i is impo an o no e ha he Wilson coe icien s a e no couplings hemsel es.
The e o e, hey can ha e la ge alues wi hou necessa ily indica ing he need o non-
pe u ba i e new physics.
4.2 HEFT
As men ioned in he In oduc ion, he calcula ions pe o med in he hesis a e ca ied
ou in he SM and i s mos common EFT ex ension, he SMEFT amewo k. Howe e ,
we no ice ha he e is ano he common s a egy o build an EFT ex ension o he SM,
called Higgs E ec i e Field Theo y (HEFT). In his app oach, unlike SMEFT whe e
he Higgs is he usual SM double , he Higgs ield is conside ed o be a single unde
GSM =SU(2)W×U(1)Y, while he Golds one bosons, esul ing om he symme y
b eaking p ocess, a ange hemsel es in a non-linea ep esen a ion o GSM :
U= exp(2iϕjTj
).(4.2)
Al hough we will no del e in o his me hod in he hesis, we hink i is ele an o
men ion al e na i e amewo ks o he one employed in his wo k. O e all, he main
dis inc ions be ween HEFT and SMEFT lie in he ene gy scale, deg ees o eedom and
he scope o applicabili y o desc ibing BSM physics.
In gene al HEFT includes SMEFT, hus all NP signals can be de ec ed wi hin his
amewo k. Howe e , in he limi Λ ≫ , HEFT is equi alen o SMEFT [91], bu he
1 is he scale o he SM in e ac ions, so i is logical o use his pa ame e .
60 CHAPTER 4. STANDARD MODEL EFFECTIVE FIELD THEORY
calcula ions a e mo e complex and complica ed. The e o e, assuming Λ ∼TeV, we can
use SMEFT ins ead o HEFT o simpli y he calcula ions, while s ill cap u ing he ull
ange o BSM physics e ec s.
4.3 SMEFT
The concep o he S anda d Model E ec i e Field Theo y (SMEFT) has been de eloped
and s udied by many esea che s in he ield o pa icle physics. I is p obably no
possible o a ibu e i s in en ion o a single indi idual o g oup. Indeed his app oach
has been buil upon he ounda ions o EFT and he unde s anding o he SM. Va ious
physicis s ha e con ibu ed o he de elopmen and explo a ion o SMEFT, including
[92,93,94].
The SMEFT Lag angian is cons uc ed as an expansion in highe dimension ope a o s,
wi h he SU(3) ×SU(2) ×U(1) gauge symme y unb oken,
L=LSM + Σ∞
k=5Σn
a=1
Ck
a
Λk−4Ok
a.(4.3)
whe e he ope a o s o mass-dimension k,Ok
a, a e cons uc ed om SM ields and all o
he e ec s o he beyond he SM (BSM) physics eside in he coe icien unc ions, Ck
a.
The ocus o mos o SMEFT esea ch has been on dimension-6 ope a o s and mo e
ecen ly on dimension-8. This is due o he ac ha hese ope a o s p o ide he dom-
inan con ibu ions and i is commonly assumed ha any new physics would ini ially
mani es i sel h ough a dimension-6 ope a o . Indeed, a dimension 5, only one ope -
a o a ises and i in oduces lep on numbe iola ion and yields Majo ana masses o
neu inos. The s ingen cons ain s on his ype o iola ion needs ei he e y high
ene gy scales o e y small coe icien s o i he expe imen al limi s. See Appendix A.1
o a u he discussion.
4.4 Wa saw basis
A dimension 6 in he SMEFT amewo k, as discussed in [95], he e is a possibili y o
edundancy among he ope a o s. In his con ex , edundancy e e s o he p esence
o ope a o s ha can be ela ed o each o he h ough equa ions o mo ion (EOM),
in eg a ion by pa s (IBP), ield ede ini ions o unde lying symme ies. Indeed, ce ain
combina ions o ope a o s may lead o he same physical e ec s o ha e equi alen con-
ibu ions o obse ables. As a esul , including all possible ope a o s in he analysis
may lead o o e coun ing o edundan desc ip ions o he same physics. To add ess his
issue, i is common o apply echniques such as he IBP o he EOM o build an op-
e a o basis. These me hods help o sys ema ically iden i y he independen s uc u es
and cons uc a minimal se o ope a o s o ca ch he ele an BSM physics. Indeed, by
emo ing edundan ope a o s, we can ob ain a mo e e icien and concise desc ip ion
4.4. WARSAW BASIS 61
o he heo y, acili a ing he calcula ions and imp o ing he comp ehensibili y o he
esul s. Las ly, we no ice ha he iden i ica ion o edundan ope a o s equi es me icu-
lous conside a ion o he symme ies, ields con en , and in e ac ions o he heo y. Fo
his eason, he choice o ope a o basis plays a c ucial ole in de e mining he app o-
p ia e se o independen ope a o s in SMEFT.
The mos used ope a o basis in he SMEFT amewo k is he Wa saw basis, in oduced
by [96]. This basis choice has many ad an ages:
•Reno maliza ion g oup in es iga ion.
The Wa saw basis has been al eady s udied in he con ex o he eno maliza ion
g oup and he associa ed RGEs ha e been comple ely calcula ed [97,98,99] a one loop
o de . This allows o pe o m calcula ions a NLO in his basis wi hou he need o a
change o basis which would also in ol e a SM ield ede ini ion.
•FeynRules implemen ed in Ma hema ica.
The comple e se o Feynman ules o he Wa saw basis ha e been de i ed in Rξgauges
[100] and he inal heo y is exp essed in a basis cha ac e ized by SM-like p opaga o s
o all physical and unphysical ields. Fu he mo e, in [101], he au ho s made publicly
a ailable a Ma hema ica code wo king wi h he FeynRules package, allowing au oma ic
SMEFT ampli ude calcula ions.
•Compa ison o esul s.
The Wa saw basis is he cu en ly mos commonly used basis in SMEFT amewo k.
Using he same basis in physics is use ul o acili a e communica ion, p omo ing consis-
ency and a oiding con usion be ween esea che s.
4.4.1 Comple e se o dimension-6 ope a o
The EFT we a e building has o ollow only wo cons ain s: he ope a o s mus espec
he gauge and he Lo en z symme ies. Addi ional equi emen s, such as CP conse -
a ion and/o he absence o ope a o s ha iola e ba yon symme y, can be imposed.
Howe e , i should be no ed ha by imposing hese condi ions, one is al eady making
speci ic assump ions abou he unde lying BSM physics. In he nex sec ions, ollowing
[96], we pe o m he classi ica ion o all possible dimension-6 ope a o s using he same
no a ion in oduced in 2.2.
4.4.2 Dimension 5 ope a o s
Fo ou analysis, i is con enien o ini ially conside only he le -handed e mions
ψ∈l, ec, q, uc, dcas undamen al ields, using he cha ge conjuga es o he SU(2)W-
single e mions. Wi hin his con en ion, he e a e h ee possibili ies o he e mionic
62 CHAPTER 4. STANDARD MODEL EFFECTIVE FIELD THEORY
cu en s 2:¯
ψ1γµψ2,¯
ψT
1Cψ2,¯
ψ1Cσµνψ2. By conside ing bosonic objec s wi h he app o-
p ia e numbe o Lo en z indices and emembe ing ha Xµν3is an isymme ic, comple e
se s o building blocks o o his class o ope a o s can be easily de e mined o each
e mionic cu en . Fo dimension 5 we ha e:
¯
ψ1γµψ2: (ϕD)
¯
ψT
1Cψ2: (ϕ2, D2)
¯
ψ1Cσµνψ2: (X, D2) (4.4)
whe e ϕis he Higgs ield and Dis he SM co a ian de i a i e. Recalling he hype -
cha ges assigned in 2.2, we see ha he cu en s in ol ing Ccan ne e ha e hype cha ge
0 while o he ec o cu en s i can ne e be equal o ±1
2. This means ha he classes
(X, D2, ϕD) can be disca ded. Thus he only class we mus conside is ψ2ϕ2. The hype -
cha ge o he Higgs p oduc , i i is no 0, can be ±1. The only e mionic cu en ha
can cancel i is he one buil ou o wo lep on double s. The e o e he only possibili y
is:
ϵjkϵmnϕjϕm(lk
p)TCln
≡(e
ϕ†lp)TC(e
ϕ†l )T.(4.5)
This ope a o s iola es lep on numbe conse a ion and, a e ele oweak symme y
b eaking, i gene a es neu ino masses and lep on mixings.
Bosonic Ope a o s
Fo pu ely bosonic ope a o s, due o cons ain s om he SU(2)Wgauge g oup, i is
necessa y o ha e an e en numbe o Higgs ields ϕand an e en numbe o co a ian
de i a i es D so ha all Lo en z indices mus be con ac ed. Thus, he only po en ial
ield con en s o dimension-6 bosonic ope a o s a e X3, X2ϕ2, X2D2, Xϕ4, XD4, Xϕ2D2,
ϕ6, ϕ4D2and ϕ2D4. I is possible o demons a e ha all hese ope a o s can be educed
by he EOM (o symme ies conside a ions) o ope a o s con aining e mions o o classes
X3, X2ϕ2, ϕ6and ϕ4D2. The class Xϕ4canno appea because he e is no any o he
objec ha can be used o con ac he Lo en z indices. XD4can be mo ed in X2D2
because all possible con ac ions (including hose wi h ϵµνρσ ) lead o he appea ance
o a leas one in a ian de i a i e commu a o [Dµ, Dν]∼Xµν, which places i in he
X2D2class. Fo ou classi ica ion, since we a e in e es ed o O(1
Λ2), he EOMs can be
de i ed di ec ly om he SM Lag angian. We ha e:
(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 o Lo en z indices
3We ecall ha Xµν e e s o he gauge ield enso and can be equal o WI
µν, GA
µν o Bµν.
4.4. WARSAW BASIS 63
Fu he mo e, we de ine
ϕ†i
↔
Dµϕ≡ϕ†(Dµϕ)−i(Dµϕ)†ϕand ϕ†i
↔
Da
µϕ≡ϕ†τaDµϕ−i(Dµϕ)†τaϕ.
In his wo k, we ha e o ganized he ope a o classes in such a way ha hose wi h ewe
co a ian de i a i es a e conside ed ”lowe classes”. Following [96], we will see ha i
is possible o educe ope a o s om highe o lowe classes. I classes ha e an equal
numbe o de i a i es, he o de ing is de e mined by he numbe o X enso s, wi h
lowe classes ha ing ewe X enso s. We begin wi h he class o ope a o s ha can be
educed o lowe class:
-ϕ2D4
He e, we can conside only ope a o s whe e all he de i a i es ac on a single ϕ
ield, because o he possibili ies a e equi alen up o o al de i a i es. Con ibu-
ions in ol ing ϵµνρσ can be dis ega ded, because hey esul in e ms in he o m
[Dµ, Dν]∼Xµν, which co esponds o lowe classes. Simila ly, he o de ing o he
co a ian de i a i es ac ing on ϕcan be chosen a bi a ily. We can ake ad an age
o his eedom o use DµDµϕas he ”building blocks” o he ope a o s conside ed.
This allows us o mo e his class o lowe classes h ough he EOM 4.6.
-ϕ2XD2
In his class, we conside Xbeing po en ially dual and we igno e ϵµνρσ. Indices o X
need o be con ac ed wi h bo h de i a i es, hus we ha e o conside h ee cases:
(i) Each de i a i e ac s on a di e en ϕ ield. We can elimina e his possibili y
h ough in eg a ion by pa s, neglec ing o al de i a i es. (ii) Bo h de i a i es ac
on a single objec . This esul s again in [Dµ, Dν]∼Xµν leading o he ϕ2X2class.
(iii) One de i a i e ac s on Xand he o he on ϕ. By EOM we can mo e o lowe
classes ϕ4D2and ψ2ϕD.
-X2D2
Fo his class, simila ly o he ϕ2D4, we can ocus ou a en ion only on ope a o s
whe e all he de i a i es ac on a single enso . The case whe e bo h he de i a i es
a e con ac ed wi h ϵµνρσ o wi h a single enso is excluded because [Dµ, Dν]∼
Xµν. The e o e, we allow he enso X o be dual and igno e ϵµνρσ o he wise.
In he case whe e each de i a i e is con ac ed wi h a di e en enso , we use
[Dµ, Dν]∼Xµν o change he o de hei o de and h ough EOM we we can mo e
i o lowe class. The only possibili y is ha he de i a i es a e con ac ed wi h
hemsel es e
XµνDρDρXµν. Howe e , using he Bianchi iden i y
e
XµνDρDρXµν =−e
Xµν(DρDµXνρ +DρDνXρµ) (4.7)
and again using [Dµ, Dν]∼Xµν and EOM we can educe his case o he lowe
class.
We can now e iew he ope a o s appea ing in able 4.1:
64 CHAPTER 4. STANDARD MODEL EFFECTIVE FIELD THEORY
•X3
He e we ha e he i s class o ope a o s appea ing in he Wa saw basis. We deno e
X, Y, Z he possible di e en enso s ha may appea in his class. We allow hem
o be dual, hus we o ge abou ϵµνρσ o he wise. The only possible non anishing
and independen con ac ion o Lo en z indices eads Xν
µYρ
νZµ
ρ. Indeed, all he
h ee enso s mus be di e en because XαµXβνZµνgαβ is ze o by he an isimme y
o Z. Fu he mo e, nei he o he wo enso s can be dual, as Xν
µe
Xρ
ν=−1
4δρ
µXαβ e
Xαβ
is symme ic while Zis an isymme ic. The only possibili y is ha o ge a gauge
single om h ee di e en enso s, we ha e o use he s uc u e cons an s ABC
and ϵIJK .
•X2ϕ2
The Higgs combina ions can be single s o iple s o SU(2)W. Hype cha ge condi-
ions u he cons ain he o ms o hese p oduc s o be w i en as ϕ†ϕand ϕ†τIϕ.
In o al we ha e eigh possible ope a o s, as lis ed in able 4.1.
•ϕ6
Hype cha ge cons ain s imply ha exac ly h ee o he Higgs ields mus be com-
plex conjuga e. As in he p e ious case, we can conside enso p oduc s o single s
and iple s o SU(2)W. The combina ion o h ee iple s can only esul in an
o e all single i i is ully an isymme ic. Howe e , since all he iple s a e iden-
ical, his combina ion e alua es o ze o. Two iple s and one single is a possible
combina ion (ϕ†τIϕ)(ϕ†τIϕ)(ϕ†ϕ), bu due o he equa ion
τI
jkτI
mn = 2δjnδmk −δjkδmn (4.8)
i esul s equals o (ϕ†ϕ)3. Hence, we ha e only one independen ope a o in his
class.
•ϕ4D2
As in he o he cases, he hype cha ge cons ains wo Higgs ields o complex
conjuga ed. Conside ing ha he wo de i a i es mus be con ac ed, hey ha e
o ac on wo di e en ϕ ields, o he EOM leads he ope a o o lowe classes. I
he de i a i es ac on wo conjuga ed o wo unconjuga ed ields, we can elimina e
hose possibili ies by IBP. On he o he hand, i one de i a i e ac s on a conjuga ed
ield and he o he on an unconjuga ed ield, ou SU(2)W enso p oduc con ains
ou dis inc undamen al ep esen a ions, indica ing he p esence o exac ly wo
independen single s.
4.4. WARSAW BASIS 65
X3ϕ6and ϕ4D2ψ2ϕ3
QG ABCGAν
µGBρ
νGCµ
ρQϕ(ϕ†ϕ)3Qeϕ (ϕ†ϕ)(¯
l′
pe′
ϕ)
Qe
G ABC e
GAν
µGBρ
νGCµ
ρQϕ□(ϕ†ϕ)□(ϕ†ϕ)Quϕ (ϕ†ϕ)(¯q′
pu′
e
ϕ)
QWεIJK WIν
µWJρ
νWKµ
ρQϕD ϕ†Dµϕ∗ϕ†DµϕQdϕ (ϕ†ϕ)(¯q′
pd′
ϕ)
Q
WεIJK
WIν
µWJρ
νWKµ
ρ
X2ϕ2ψ2Xϕ ψ2ϕ2D
QϕG ϕ†ϕ GA
µνGAµν QeW (¯
l′
pσµνe′
)τIϕWI
µν Q(1)
ϕl (ϕ†i
↔
Dµϕ)(¯
l′
pγµl′
)
Qϕ
e
Gϕ†ϕe
GA
µνGAµν QeB (¯
l′
pσµνe′
)ϕBµν Q(3)
ϕl (ϕ†i
↔
Da
µϕ)(¯
l′
pτIγµl′
)
QϕW ϕ†ϕ WI
µνWIµν QuG (¯q′
pσµνTAu′
)e
ϕ GA
µν Qϕe (ϕ†i
↔
Dµϕ)(¯e′
pγµe′
)
Qϕ
Wϕ†ϕ
WI
µνWIµν QuW (¯q′
pσµνu′
)τIe
ϕ WI
µν Q(1)
ϕq (ϕ†i
↔
Dµϕ)(¯q′
pγµq′
)
QϕB ϕ†ϕ BµνBµν QuB (¯q′
pσµνu′
)e
ϕ Bµν Q(3)
ϕq (ϕ†i
↔
Da
µϕ)(¯q′
pτIγµq′
)
Qϕ
e
Bϕ†ϕe
BµνBµν QdG (¯q′
pσµνTAd′
)ϕ GA
µν Qϕu (ϕ†i
↔
Dµϕ)(¯u′
pγµu′
)
QϕWB ϕ†τIϕ WI
µνBµν QdW (¯q′
pσµνd′
)τIϕ WI
µν Qϕd (ϕ†i
↔
Dµϕ)( ¯
d′
pγµd′
)
Qϕ
WB ϕ†τIϕ
WI
µνBµν QdB (¯q′
pσµνd′
)ϕ Bµν Qϕud i(e
ϕ†Dµϕ)(¯u′
pγµd′
)
Table 4.1: Dimension-6 ope a o s o he han he ou - e mion ones ( om [96]). Fo
b e i y we supp ess e mion chi al indices L, R.
4.4.3 Single- e mionic-cu en ope a o classi ica ion
Fo his class o ope a o s i is con enien o use he same no a ion in oduced in 4.4.2.
Again, by simila conside a ions, we ha e h ee possible e mionic cu en s. They a e:
¯
ψ1γµψ2: (XD, ϕ2D, D3)
¯
ψT
1Cψ2: (ϕ3, ϕD2)
¯
ψ1Cσµνψ2: (Xϕ, ϕD2) (4.9)
Fo scala and enso e mionic cu en s, we no ice ha he nume o higgs ields is
always odd. As a esul , hese cu en s mus o m isospin double s. Using he s anda d
no a ion wi h igh -handed single s, hese cu en s a e ep esen ed by ¯
ψ1ψ2and ¯
ψ1σµνψ2.
Simila ly, ec o cu en s combine wi h an e en numbe o Higgs ields, hus hey can
only o m isospin single s o iple s. Wi h his conside a ion, no ec o cu en s wi h
C en e in o ou conside a ions, e en i he isospin single s a e aken as igh -handed.
The e o e , we can go back o he s anda d no a ion in he ollowing discussion. As o
he bosonic case, we will use he EOM 4.6 and as well as he classical EOM o e mions
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
Fu he mo e, we will need o ecall he iden i ies:
γµγν=gµν −iσµν, γµγνγρ=gµνγρ+gνργµ−gρµγν−ϵµνρσγργ5.(4.11)
As in he p e ious sec ion, we s a wi h he ope a o s ha can be educed o lowe
classes:
-ψ2D3
He e, h ee co a ian de i a i es a e con ac ed wi h he ec o ial cu en ¯
ψγµψ.
As discussed o he classes ϕ2D4and X2D2, we can elimina e de i a i es ac ing on
¯
ψ”by pa s” and choose he o de ing o he de i a i es ac ing on ψ. By choosing
he o de ing as ¯
ψDµDµ/
Dψ, we can educe by EOM o lowe classes.
-ψ2ϕD2
In his class, we conside scala and enso ial cu en s. De i a i es ac ing on ¯
ψ
can be emo ed ”by pa s”. The cases ¯
ψσµνψDµDνϕand ϕ¯
ψσµνDµDνψcan be
mo ed o lowe classes ecalling ha [Dµ, Dν]∼Xµν. We s ill ha e 4 possible
combina ions:
¯
ψψDµDµϕ, ϕ ¯
ψDµDµψ, (Dµϕ)¯
ψσµνDνψ, (Dµϕ)¯
ψDµψ.
The cases ¯
ψψDµDµϕand ϕ¯
ψDµDµψ∼ϕ¯
ψ/
D/
Dψ can be educed o lowe classes
by EOM; (Dµϕ)¯
ψσµνDνψcan be mo ed o he las case ecalling ha σµν =
−i(gµν −γµγnu). Conside ing ha :
2(Dµϕ)¯
ψDµψ= (Dµϕ)¯
ψγµ/
Dψ + (Dµϕ)¯
ψ/
Dγµψ
=−¯
ψ←
/
DγµψDµϕ−¯
ψγµγνψDµDνϕ+. . . (4.12)
whe e he ” . . . ” s ands o a o al de i a i e and lowe classes. Finally, using he
EOM and ecalling he ela ion 4.11, we can mo e i o lowe classes.
-ψ2XD
Simila ly o o he cases, we assume ha Xcan be dual and neglec ϵµνρσ o he wise.
Since we a e dealing wi h a ec o ial cu en , ecalling ha Xis an isymme ic,
he de i a i e mus be con ac ed wi h X. I i ac s on X, we can mo e i o lowe
classes ( o he usual enso ) h ough EOM o we can use he Bianchi iden i y
Dρe
Xρµ = 0 ( o he dual enso ). The only emaining exp ession o be conside ed
is Xµν ¯
ψγµDνψ. I is educed o lowe classes as ollows:
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 he las equa ion he i s wo e ms a e educed o lowe classes h ough EOM.
Fo he las e m we ha e:
DρXµν ¯
ψγργµγνψ= 2 ¯
ψγµψDρXµν −iϵρµνσ ¯
ψγσγ5ψDρXµν.(4.14)
In bo h he cases (Xdual o no ), using EOM and Bianchi iden i y, we can mo e
his e ms o lowe classes.
We con inue ou e iew discussing he o he ope a o s appea ing in able 4.1:
•ψ2ϕ3
The e mion cu en mus ake he o m o an isospin double and colo single ,
like he Yukawa in he SM. The numbe o conjuga ed and un-conjuga ed scala
ields in ϕ3is ixed o each o he e mion cu en s due o hype cha ge cons ain s.
As a esul , he only possibili ies o his class a e he Yukawa e ms mul iplied by
ϕ†ϕ.
•ψ2Xϕ
In his case we ha e o conside only enso ial cu en s. Again, o espec hy-
pe cha ge cons ain s, he e mion cu en mus be analogous o he SM Yukawa
couplings. Fo each enso ield Xµν, he e is only one possible con ac ion wi h
each e mionic cu en . The dual enso s ˜
Xwould no yield any new esul s, due
o he iden i ies:
ϵαβµνσµν = 2iσαβγ5and γ5ψL,R =∓ψL,R.(4.15)
•ψ2ϕ2D
I he de i a i e ac s on any o he e mion ields, i s con ac ion wi h he ec o
cu en leads o equa ions o mo ion (EOMs) and b ings us o he p e iously
discussed lowe class ψ2ϕ3. The e o e, we can jus conside he case whe e he
de i a i e ac on he scala ields. The Higgs ields combined can gi e isospin
single s o iple s and colo single s. The e mion cu en s mus obey o he
same selec ion ules, allowing o p ecisely he cu en s lis ed in able 4.1, wi h
he excep ion o he uγµdcu en , which equi es He mi ian conjuga ion. The
numbe o conjuga ed and unconjuga ed Higgs ields is de e mined by hype cha ge
condi ions. We begin by emo ing de i a i es by using in eg a ion by pa s on one
o he scala s. We hen o m isospin single s o iple s om p oduc s o ϕ1and
Dµϕ2acco ding o he s uc u e o he co esponding e mion cu en s. A his
poin , we ob ain ope a o s ha di e om hose in able 4.1 only by he p esence
o Dins ead o ↔
D. The poin is ha he ope a o wi hou ↔
D, a e no He mi ian
and we ha e o e i y i he h.c. a e independen om hem o no . No ice ha
his issue does no occu o all he o he ope a o s (also o Qϕud), since hey a e
He mi ian by cons uc ion. Fo he emaining se en ope a o s, conside ing he
symme ic combina ion we ha e:
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 mo e hem lowe classes by EOM. The e iew on single- e mion-cu en
ope a o s is comple ed.
4.4.4 Fou - e mion ope a o classi ica ion
Fou - e mion ope a o s a e nume ous bu s aigh o wa d o classi y. Fi s , simila o
he p e ious sec ion, we conside only le -handed e mions ψ∈l, ec, q, uc, dc. Besides
i ial ou comes in ol ing p oduc s o wo ze o-hype cha ge cu en s (indica ed in 4.2
as (¯
LL)(¯
LL),(¯
RR)( ¯
RR),(¯
LL)( ¯
RR),), we ha e only a ew o he possible ield con en s:
(¯
l¯ecdcq),(qucqdc),(lecquc),(qqql),(dcucucec),(qq¯uc¯ec),(ql¯uc¯
dc) (4.17)
and he associa ed he mi ian conjuga e. The i s h ee cases gene a e B-Conse ing
ope a o s while he o he ou gene a e B- iola ing ones. We s a by conside ing he
exp essions wi h wo ψand wo ¯
ψ. In his case, by spin conside a ions, we ha e only
one single pai ing; looking a he SU(2)Wquan um numbe s, we see ha he e a e wo
double s and wo single s ields, meaning ha we can ha e only one single . Finally,
including also he colou indices, we ha e
¯
3⊗3 = 8 ⊕1 B-conse ing
3⊗3⊗3 = 10 ⊕8⊕8⊕1 B- iola ing (4.18)
hus we ha e only one ope a o o each o h ee conside ed cases. In Table 4.2, hey a e
indica ed as Qledq, Qduq, Qqqu, using he s anda d igh -hand no a ion. In he o he ou
cases in 4.17, we ha e ou le handed e mions. Again, by applying spin cons ain s
and Fie z iden i ies, we can elimina e edundan cases. Finally, aking in o accoun
isospin and colo index con ac ions, we inalize he cons uc ion o he ou - e mion
ope a o s. I can be eadily con i med ha he lis ed ope a o s o m a comple e basis
o ou - e mion ope a o s, as discussed in [96].
4.5 SMEFT couplings
In SMEFT, simila ly o wha happens in he SM, he weak gauge bosons and he e mions
acqui e mass due o he SSB mechanism. Howe e , unlike in he SM, o iden i y he
physical deg ees o eedom one needs o pe o m an ex a in e media e s ep in ol ing
ield escalings, since SSB also a ec s he canonical no maliza ion o he kine ic e ms.
This applies also o he SM e mions whe e, be o e he usual diagonaliza ion o he
Yukawa couplings, we need o ede ine he SM ields and couplings. This p ocess is
accu a ely desc ibed in [100] and i in ol es se e al s eps ha we will no epo en i ely
in his hesis, bu we will jus show he main esul s.
5.2. NOTATION 75
s uc u e o e mion ope a o s, among o he s. Unde s anding he unce ain ies inhe en
in hese assump ions is c ucial o in e p e ing SMEFT i s[115].
The assump ions abou he la o s uc u e in oduce signi ican model dependence
in o he SMEFT p edic ions. I is s aigh o wa d o implemen a gene al la o s uc-
u e o he ee le el p edic ions o obse ables[116,117,118,119,120,121,122] and
he one-loop MS eno maliza ion o he dimension-6 coe icien unc ions is known o
an a bi a y la o s uc u e[97,98,99]. The one-loop nex - o-leading o de (NLO) elec-
oweak p edic ions o physical obse ables, howe e , ypically in ol e a la ge numbe
o 4- e mion and 2- e mion ope a o s wi h po en ially complica ed la o s uc u es. In
addi ion, he e mion ope a o s in oduce new sub le ies in he eno maliza ion p oce-
du e. Exis ing calcula ions o he one-loop elec oweak co ec ions o EWPO, Higgs, and
di-boson da a do no include he mos gene al la o s uc u e o he 2- qua k and 4-
qua k ope a o s in he loops[123,124,125,126,127,128,129]. He e, we p esen a gene -
aliza ion o p e ious NLO SMEFT calcula ion o EWPOs [130,131,132] which included
4- e mion ope a o s, bu no 2- e mion ope a o s, o allow o an a bi a y la o s uc-
u e. The co ec ions o he EWPOs om 4- e mion ope a o s in he U(3)5symme ic
case a e in [128]. The ole o la o assump ions in i s o op and bo om qua k da a
has been ex ensi ely examined in he li e a u e and hose analyses a e complemen a y
o ha p esen ed he e[133,134,135,132].
We se he CKM ma ix o be diagonal, which implies ha only ope a o s con aining
pai s o iden ical la o e mions con ibu e. We u he wo k o linea o de in he
SMEFT coe icien s and se all masses o he han he op qua k o be 0. We wan o
ema k ha hese wo condi ions cons ain he ligh e mion Yukawa couplings o be
ze o, wi hou addi ional assump ions, since he SMEFT ope a o s ha would induce
a modi ica ion o he SM Yukawas do no in e e e wi h he SM ampli udes a linea
o de .
5.2 No a ion
Re aining only he dimension-6 ope a o s, we calcula e obse ables, Ob, o one-loop as
an expansion in 1
Λ2and keep only he linea e ms since he SMEFT is eno malizable
o de by o de in powe s o 1
Λ2,
Ob=Ob,SM + Σn
a=1
Ca
Λ2βab,(5.2)
whe e βab is p ocess dependen and depends on he kinema ic in a ian s and he inpu
pa ame e s, and Ob,SM is he SM p edic ion. We use he Feynman ules om Re . [100],
wi h gene al la o s uc u es o he 2- and 4- e mion ope a o s, al hough we assume
ha he CKM ma ix is diagonal, which has implica ions o he e mion s uc u es o
he ope a o s ha con ibu e o ou calcula ion, as we will see. Fu he mo e, we assume
ha he SMEFT does no in oduce new sou ces o CP iola ion, ha is we assume he
coe icien s o all he ope a o s o be eal.
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 ope a o s con ibu ing o he Zand Wpole obse ables o his
s udy a ee le el. i, j, k, l = 1,2,3 a e gene a ion indices.
Ou goal is he dimension-6 SMEFT calcula ion o NLO QCD and NLO elec oweak
co ec ions o elec oweak p ecision obse ables (EWPOs) wi h a bi a y la o s uc-
u es o he ope a o s in ol ing e mions. The echnical de ails a e in he nex sec ion
and in he ollowing sub-sec ions we discuss he e ec s o la o on he p edic ions.
5.2.1 EWPOs
In comple e gene ali y, he Wa saw basis con ains 2499 ba yon numbe conse ing
dimension-6 ope a o s. Much o he p oli e a ion o ope a o s is associa ed wi h he
la o s uc u e. O cou se, mos o he la o s uc u es will no con ibu e o a gi en
obse able and we begin by conside ing EWPOs a ee le el. The Zand Wboson pole
obse ables ha we conside a e,
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)
No e ha we do no use he e ec i e mixing angle in ou i s, since i is de i ed om
he asymme ies.
The ope a o s ha con ibu e o he EWPOs a LO a e comp ised o wo bosonic
ope a o s wi h no la o s uc u e (OϕW B and OϕD) and 8 e mionic ope a o s, o which
he e a e 7 ope a o s wi h 2 e mionic indices ( he 2- e mion ope a o s), and 1 ope a o
wi h 4 e mionic indices ( he 4- e mion ope a o ). The explici o ms o he ope a o s
ha appea a LO a e epo ed in Table 5.1. These ope a o s change he couplings o
he Zand Wbosons o e mions, and explici p edic ions o he measu ed quan i ies
o Eq. 5.3 a e gi en in Appendix A o Re . [124]. The only 4- e mion ope a o ha
is ele an o EWPOs a LO is Oll[ijkl] which has he symme y Oll[ijkl] = Oll[klij]
and only Oll[2112] = Oll[1221] con ibu es o he obse ables o Eq.5.3. The indices
(i, j, k, l = 1,2,3) e e o he e mion gene a ion.) Dimension-6 ope a o s in ol ing he
elec on and he muon gi e con ibu ions o he decay o he µ, changing he ela ion
be ween he e , , and he Fe mi cons an Gµ,
Gµ≡1
√2 2−1
√2Λ2Cll[1221] + 1
√2Λ2C(3)
ϕl [11] + C(3)
ϕl [22].(5.4)
A NLO, he EWPOs o Eq. 5.3 ecei e con ibu ions om 22 addi ional ope a o s
(which a e de ined in Re . [124]), which we classi y acco ding o he numbe o e mions:
5.2. NOTATION 77
•4 bosonic ope a o s:
OϕB,OϕW ,O□,OW.(5.5)
•2 2- e mion ope a o s:
OuB[ij],OuW [ij].(5.6)
No ice ha only OuB[33] and OuW [33] con ibu e o he EWPOs a NLO i all
e mions excep he op a e massless.
•16 4- e mion ope a o s:
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)
Fi e o he NLO-gene a ed 4- e mion ope a o s ha e a la o symme y,
Oee[ijkl],O(3)
qq [ijkl],O(1)
qq [ijkl],Ouu[ijkl],Odd[ijkl]≡OY[ijkl] = OY[klij].(5.8)
I is con enien o ca ego ize he ope a o s acco ding o hei dependence on he
la o since only speci ic s uc u es con ibu e o he EWPOs a NLO:
•A) 2- e mion ope a o s: 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 conside he CKM ma ix o be diagonal which has
he consequence ha he coe icien s o he ope a o s in Class A ha e diagonal
la o s uc u es:
CX[ij] = E(i)
Xδij, i, j = 1,2,3,(5.9)
esul ing in 3 independen coe icien s o each ope a o . No ice ha he ope a o
Oϕu[33] i s con ibu es o he EWPOs a NLO. Fu he mo e, as no ed abo e,
OuB[ij] and OuW [ij] en e in ou calcula ions only wi h coe icien s CuB[33] and
CuW [33] espec i ely. Toge he he e a e 23 independen coe icien s in Class A.
•B) 4- e mion ope a o s in ol ing only iden ical e mion ep esen a ions: OY[ijkl]≡
Oee[ijkl], O(3)
qq [ijkl], O(1)
qq [ijkl], Ouu[ijkl], Odd[ijkl], Oll[ijkl]. Since hey s em om
he combina ion o wo e mion cu en s belonging o iden ical ep esen a ions once
we equi e ha he CKM ma ix is he uni ma ix, he e a e only wo ways la o
is allowed o ” low” h ough hese ope a o s: OY[iijj] and OY[ijji]. Fu he mo e,
hese ope a o s a e subjec o he la o symme y in Eq. 5.8, esul ing in he
coe icien s ha ing he la o s uc u e,
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
esul ing in 9 independen coe icien s o each ope a o . Howe e , he la o s uc-
u e o Oee[ijkl] is u he cons ained by he Fie z iden i y (eiγµej)(ekγµel) =
(eiγµel)(ekγµej), which imposes he equali y A(ij)
ee =B(ij)
ee , educing he numbe
o independen coe icien s o Oee[ijkl] o 6. The only coe icien o Class B ha
does no con ibu e o he EWPOs a NLO is Cuu[3333]. In o al, we ha e 50
independen coe icien s in Class B con ibu ing.
•C) 4- e mion ope a o s wi h 2 di e en e mion ep esen a ions: 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]. Fo hese ope a o s, ou choice o a diagonal CKM ma ix
equi es ha he la o mus low only in one way: OY[iijj]. The e o e, he coe -
icien s o hese ope a o s ha e he la o s uc u e:
CZ[iijj] = D(ij)
Z, i, j = 1,2,3,(5.11)
wi h no u he es ic ions, co esponding o 9 independen coe icien s o each
ope a o , o a o al o 99 independen coe icien s in Class C ha con ibu e o
he obse ables o Eq. 5.3 a NLO.
In he mos gene al la o case, we see ha EWPOs compu ed o NLO ecei e con i-
bu ions om 178 independen coe icien s: 6 om bosonic ope a o s, 23 om 2- e mion
ope a o s, and 149 om 4- e mion ope a o s. We nex conside di e en la o assump-
ions o he e mionic ope a o s in o de o educe he numbe o ope a o s ha need o
be conside ed. The la o assump ions we conside a e U(3)5, minimal la o iola ion
(MFV), U(2)5, hi d gene a ion cen ic, hi d gene a ion phobic, hi d gene a ion phobic
+U(2)5, and a la o less s uc u e. We will discuss each o hese assump ions in de ail
in he ollowing sub-sec ions.
5.3 Scena ios
5.3.1 Fla o Assump ions: U(3)5
In he absence o Yukawa couplings, he SM e mions ha e a global U(3)5symme y,
G3≡U(3)q×U(3)l×U(3)u×U(3)d×U(3)e.(5.12)
The in oduc ion o Yukawa in e ac ions in he SM Lag angian p ese es a global hype -
cha ge symme y U(1)Y, a ba yonic symme y U(1)B, and h ee lep onic symme ies,
U(1)eU(1)µand U(1)τ.
Ou i s app oxima ion when imposing la o symme ies on SMEFT p edic ions
is o assume ha he dimension-6 SMEFT coe icien s ha e he G3symme y o he
(Yukawa-less) SM. This symme y p e en s he gene a ion o he ope a o s OuW [33] and
OuB[33] since hey ca y a le - igh e mionic cu en .
Unde G3, he o he ope a o s in Class A espec he o m
Class A : CX[ii] = EX, i = 1,2,3,(5.13)
5.3. SCENARIOS 79
All he 2- e mion ope a o s ha con ibu e o he EWPOs au oma ically ha e he s uc-
u e o Eq. 5.13, and he e a e 7 eal coe icien s in his class, consis en wi h he coun ing
o Re . [119].
Rega ding he ope a o s o Class B, bo h con ac ions o e mion indices
( iγµ i)( jγµ j),( iγµ j)( jγµ i),(5.14)
a e in a ian unde U(3)5, so he coe icien s o Class B educe o
Class B : CY[iiii] = AY+BY, CY[iijj] = AY, CY[ijji] = BY,
i=j&i, j = 1,2,3.(5.15)
Remembe ing ha Bee =Aee, he e a e 11 independen coe icien s in Class B.
Finally, he ope a o s in Class C ha e he s uc u e
Class C : CZ[iijj] = DZi, j = 1,2,3,(5.16)
o a o al o 11 independen coe icien s also in Class C. In o al, i we impose a G3
symme y on he SMEFT Lag angian, we a e le wi h 29 independen coe icien s con-
ibu ing o he EWPOs a one loop.1
5.3.2 Fla o Assump ions: MFV
The Yukawa couplings b eak he U(3)5global symme y in he SM. I we assume ha
his is he only sou ce o b eaking o he G3symme y, we a e led o he MFV sce-
na io2[136]. The SM Yukawa couplings a e conside ed as U(3)5auxilia y ields wi h he
ans o ma ions,
Yu∼(3,1,3,1,1)
Yd∼(3,1,1,3,1)
Ye∼(1,3,1,1,3) .(5.17)
I is always possible o choose a basis such ha Yu, Yd, and Yea e diagonal ma ices.
We emind he eade ha we ha e assumed ha he CKM ma ix is diagonal and he
only non-ze o e mion mass is assumed o be he op qua k mass, M . The coe icien s o
he ope a o s con aining a leas 2 op qua ks a e modi ied, while he o he coe icien s
e ain he G3s uc u e. Ou implemen a ion o he MFV scena io wi h a massless b
qua k is equi alen o a global U(2)q×U(3)l×U(2)u×U(3)d×U(3)esymme y.
In Class A, he only ope a o s wi h a la o s uc u e ha depends on he hi d
gene a ion e mions a e 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 no e ha he (RR)(RR) ope a o , O(8)
ud , does no con ibu e o EWPOs a one-loop, and so ou
coun ing o he U(1)5ope a o s ag ees wi h Re . [118].
2Since we assume ha he CKM ma ix is diagonal, he e is no la o iola ion in ou implemen a ion
o he MFV scena io.
80 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES
Since o OuW and OuB only he CuW [33] and CuB[33] en e in ou calcula ions, a o al o
12 independen coe icien s con ibu e o he EWPOs a NLO, in ag eemen wi h Table
9 o [119].
The ope a o s in Class B in ol ing cha ge 2
3- qua ks all ha e 4 uR ields o 4 qL
ields. Re aining he con ibu ions up o O(M2
2) , he coe icien s o Ouu,O(3)
qq and O(1)
qq
a e subjec o he ollowing ela ions in he MFV scena io,
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 educes he numbe o independen coe icien s o 4 o each ope a o . The e-
maining ope a o s in Class B sa is y he U(3)5 ela ions o Eq. 5.15, hus esul ing in
17 independen coe icien s in Class B.
In Class C, he ope a o s Olu,Oqe,O(1)
qd ,Oeu,O(1)
ud ,O(1)
lq O(3)
lq ha e 2 op qua ks and
hey sa is y he ela ions:
Class C : CZ[ααii] = DZ, CZ[33ii] = D(3)
Z, α = 1,2, i = 1,2,3
and hence we ha e 2 independen coe icien s o each ope a o .
The only ope a o in Class C wi h 2 cha ge- 2
3qua ks con ibu ing o EWPOs is O(1)
qu
which sa is ies he ela ions,
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 he e a e 4 independen la o s uc u es o C(1)
qu . The emaining ope a o s
in Class C, Oed,Old and Ole, sa is y he U(3)5 ela ions and con ibu e 1 independen
ope a o each. O e all, we ha e 21 independen coe icien s in Class C. Excluding he
ope a o s O(8)
qd ,O(8)
qu ,and O(8)
ud which do no con ibu e o EWPOs a NLO, ou coun ing
is in ag eemen wi h Table 2 o [120] and Table 9 o [119].
5.3.3 Fla o Assump ions: U(2)5
He e we conside a scena io whe e he new physics dis inguishes be ween he 3 d gene -
a ion and he i s 2 gene a ions. The i s 2 gene a ions a e assumed o ha e he global
symme y[118],
G2≡U(2)q×U(2)l×U(2)u×U(2)d×U(2)e.(5.18)
Since we conside only he op qua k mass o be non-ze o, he U(2)5symme y is un-
b oken.
Using he classi ica ion o ope a o s gi en in Sec. 5.3.1, he G2symme y equi es
ha ope a o s o Class A ha e he o m:
Class A : CX[αα] = EX,
CX[33] = E(3)
X, α = 1,2.(5.19)
5.3. SCENARIOS 81
The e a e 16 independen coe icien s in his class.
The Class B ope a o s sa is y
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 ha each ope a o has 5 independen coe icien s. As be o e, Oee has an ex a con-
s ain coming om he Fie z iden i ies, which educes he numbe o independen co-
e icien s o 3 o his ope a o . The e o e, since Ouu[3333] does no con ibu e, Class B
has 27 coe icien s con ibu ing o EWPOs a NLO in o al.
The Class C ope a o s, OC, con ain 2 di e en e mion ep esen a ions and he U(2)5
symme y 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 he e a e 4 independen coe icien s co esponding o each ope a o s uc u e, o a
o al o 44 ope a o s.
5.3.4 Fla o Assump ions: Thi d Gene a ion Cen ic
In his scena io, only ope a o s in ol ing he 3 d gene a ion e mions a e non-ze o. An
example o such a scena io migh be a Z′boson ha only in e ac s wi h he 3 d gene -
a ion. Ope a o s o Class A in his scena io ha e he o m:
Class A : CX[αα] = 0,
CX[33] = E(3)
X, α = 1,2.(5.21)
The e a e 9 eal coe icien s in his class.
The Class B and C ope a o s sa is y
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.
Fo a o al o 5 coe icien s in Class B and 11 coe icien s in Class C.
5.3.5 Fla o Assump ions: Thi d Gene a ion Phobic
In his scena io, we assume ha he new physics only couples o he 1s 2 gene a ions.
An example o such a model can be ound in [137]. We assume no u he symme y,
bu no e ha he ope a o s con ibu ing o EWPOs do no ha e an a bi a y la o
s uc u e, as desc ibed a he beginning o his sec ion.
82 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES
Ope a o U(3)5MFV U(2)53 dgen speci ic 3 dgen phobic 3 dgen phobic + U(2)5Fla o less
Class A7 12 16 9 14 7 9
Class B11 17 27 5 23 11 6
Class C11 21 44 11 44 11 11
To al 29 50 87 25 81 29 26
Table 5.2: Numbe o independen ope a o s con ibu ing o NLO p edic ions o EW-
POs in a ious la o scena ios.
The la o s uc u e o he ope a o s o Class A is
Class A : CX[αα] = E(α)
X.
CX[33] = 0, α = 1,2,(5.22)
o a o al o 14 independen coe icien s.
Fo Class B we ha e
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,
co esponding o 4 independen coe icien s o each ope a o (3 o Oee), o a o al o
23 coe icien s in Class B.
Finally o Class C he a ailable s uc u es a e
Class C : CZ[ααββ] = D(αβ)
Z
CZ[αα33] = CZ[33αα] = CZ[3333] = 0, α, β = 1,2,(5.23)
which again gi es 4 independen coe icien s o each ope a o , o a o al o 44 coe icien s
in Class C.
5.3.6 Fla o Assump ions: Thi d Gene a ion Phobic + U(2)5
As a speci ic case o he p e ious example, we conside a scena io whe e, a e assuming
ha new physics couples only o he 1s 2 gene a ions, we u he assume he exis ence
o a U(2)5symme y.
In his case, he la o s uc u e o he ope a o s o Class A is
Class A : CX[αα] = EX,
CX[33] = 0, α = 1,2,(5.24)
o a o al o 7 independen coe icien s.
5.4. NLO CALCULATIONS 83
Fo Class B we ha e
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)
co esponding o 2 independen coe icien s o each ope a o (1 o Oee), o a o al o
11 coe icien s in Class B.
Finally o Class C he a ailable s uc u es a e
Class C : CZ[ααββ] = DZ
CZ[αα33] = CZ[33αα] = CZ[3333] = 0, α, β = 1,2,(5.26)
which again gi es 1 independen coe icien o each ope a o , o a o al o 11 coe icien s
in Class C.
5.3.7 Fla o Assump ions: Fla o less Scena io
This is he scena io employed in ou p e ious calcula ions[130], whe e la o plays no
ole and we simply d op all la o indices. In his scena io, we assume ha he ope a o s
ha e no la o s uc u e, ha is we make he eplacemen s,
CX,Y,Z[. . . ] = AX,Y,Z .(5.27)
The e o e in his case he e a e 26 coe icien s ha con ibu e.
We summa ize he esul s o ou discussion o la o scena ios in Table 5.2. The
numbe o ope a o s con ibu ing o EWPOs a NLO a ies d ama ically depending on
he la o s uc u e and is signi ican ly smalle han he 178 independen coe icien s
ha con ibu e wi h no assump ions abou la o . We will see ha hese choices ha e
la ge e ec s on he nume ical esul s o i s o EWPOs.
5.4 NLO calcula ions
This sec ion desc ibes ou calcula ional p ocedu e o he NLO co ec ions o he elec-
oweak p ecision obse ables. Some o he esul s can be ob ained om ou p e ious
calcula ion[130] by gene alizing he la o s uc u e o each obse able. Howe e , includ-
ing a gene al la o s uc u e a NLO equi es a new calcula ion o many o he la o
s uc u es and obse ables. The e o e, we ha e pe o med he calcula ion in all gene -
ali y and in his sec ion we desc ibe he de ails o he calcula ion and he assump ions
ha a e implici in he nume ical esul s o he nex sec ion.
A ee le el, he in e ac ions a e jus he Z(and W) decays o 2 e mions shown in
Fig. 5.1 and he e ec o including he dimension-6 SMEFT con ibu ions is o change
84 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES
Z
(a) (b)
γ
(c)
Figu e 5.1: Z decays o e mions. The black ci cle ep esen s he ee le el SMEFT
in e ac ion.
he ec o boson couplings o e mions. The ee le el SMEFT couplings o e mions o
he Zand Wa e gi en in e ms o ou inpu pa ame e s (α, MZ, Gµ),
L≡2MZq√2GµZµgZq
L+δgZq
Lqγµq+ZµgZu
R+δgZu
RuRγµuR
+ZµgZd
R+δgZd
RdRγµdR+ZµgZl
L+δgZl
Llγµl
+ZµgZe
R+δgZe
ReRγµeR+ZµδgZν
RνRγµνR
+ 2MW[α, MZ, Gµ]sGµ
√2Wµ(1 + δgWq
L)uLγµdL+δgW q
RuRγµdR
+Wµ(1 + δgWl
L)νLγµeL+δgW ν
RνRγµeR+h.c.,
(5.28)
whe e a ee le el,
gZ
R=−s2
WQ and gZ
L=T
3−s2
WQ
s2
W≡1−M2
W[α, MZ, Gµ]
M2
Z
,(5.29)
T
3=±1
2, and M2
W[α, MZ, Gµ] is he mass o he Wboson w i en in e ms o he
(α, MZ, Gµ) inpu pa ame e s as de ined la e in Sec. 5.4.4. Analy ic exp essions o he
shi s in he couplings in e ms o he Wa saw basis coe icien s can be ound in [138,
139].
We use he SMEFT Feynman ules o Re . [100] as implemen ed in FeynRules [101]
om which a FeynA s [140] model ile is ob ained. In his way, we a e able o gene a e
he ele an one-loop and eal emission ampli udes, which include all he QCD and EW
5.5. RESULTS 91
Measu emen Expe imen ”Bes ” heo y
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( om ALR had 0.15138 ±0.00216 0.1469 ±0.0004 [156,157]
Ae( om ALR lep) 0.1544 ±0.0060 0.1469 ±0.0004 [156,157]
Ae( om Bhabba pol) 0.1498 ±0.0049 0.1469 ±0.0004 [156,157]
Aµ0.142 ±0.015 0.1469 ±0.0004 [156,157]
Aτ( om 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 Wo ld A e 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 o he wise ci ed, expe imen al esul s a e aken om Table 10.5 o
he Pa icle Da a G oup[153]. The heo y esul s include he ull se o 2-loop con i-
bu ions o he Zpole obse ables, along wi h highe o de co ec ions when known.
The heo y p edic ions a e compu ed using he o mulae in he indica ed e e ences and
ou inpu pa ame e s, and he heo y e o s include he pa ame ic unce ain ies on M
and Mh[156], along wi h he es ima ed heo y unce ain ies desc ibed in he espec i e
pape s.
92 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES
Fo example, we can s udy he χ2 o he coe icien C(1)
qq in he MFV scena io. The
χ2 unc ion is gi en by:
χ2
MFVC(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 he abo e p ocedu e, o ma ginalize C(1)
qq [1,1,2,2] and C(1)
qq [3,3,3,3], we ha e
∂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)
Se ing he ma ginalized coe icien s o hei minimum we ind:
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, ha now depends only on C(1)
qq [1,1,3,3] and C(1)
qq [1,3,3,1], is used o
ca y ou he equi ed analysis o he MFV scena io.
We begin by conside ing limi s on he 2- e mion ope a o s (Class A) ha con ibu e
o he Zand Wboson obse ables lis ed in Eq. 5.3. Table 5.4 compa es he LO and
NLO esul s o he SMEFT coe icien s in he U(3)5, MFV, and 3 d gene a ion cen ic
scena ios. We no e ha he U(3)5and la o less scena ios a e iden ical o he coe icien s
o he 2- e mion ope a o s. The limi s on la o s uc u es ha a e no lis ed in he able
can be de i ed using he esul s o Sec ion 5.2, al hough o cla i y we lis limi s on some
o he non-independen coe icien s. The di e ences be ween he LO and NLO i s a e
in gene al qui e small. The single pa ame e limi s a e compa ed g aphically in Fig. 5.4
whe e we see di e ences up o ac o s o 2 be ween he a ious la o assump ions.
The con ibu ions o op qua k loops o he 2- e mion ope a o s ha e been s udied in
[164]. Table 3 o his e e ence p esen s he 95% CL single pa ame e limi s 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
3 d gene a ion cen ic LO
3 d gene a ion cen ic NLO
95 % CL limi s on 2- e mion ope a o s om EWPOs
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C(3)
q[33]
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C(1)
q[33]
<la exi sha1_base64="d4KcH i GGmCQRyLh0nRUuz3 I=">AAAB9XicbVDLSgNBEOyN xh UY9eBoPgKewaUY+BXDxGMA/Y GF2MpsMmZ1d5qGEJ /hxYMiX 0Xb/6Nk2QPmljQUFR1090Vppwp7b TmF WNzq7hd2 nd2z8oHx61VWIkoS2S8ER2Q6woZ4K2NNOcdlNJcRxy2gnHjZn eaRSsUTc60lKgxgPBYsYwdpKD41+1k HDJmpX6sF/XLF bpzoFXi5aQCOZ 98ld kBATU6EJx0 5np qIMNSM8Lp NQziqaYjPGQ+pYKHFMVZPO p+jMKgMUJdKW0Giu/p7IcKzUJA5 Z4z1SC17M/E/zzc6ugkyJlKjqSCLRZHhSCdoFgEaMEmJ5hNLMJHM3o ICE M A2qZEPwll9eJe2LqndV9e4uK3U3j6MIJ3AK5+DBNdThFp QAgISnuEV3pwn58V5dz4W QUnnzmGP3A+ wChuJHk</la exi >
Cu[33]
<la exi sha1_base64="9+yleScHiIiO IwGC e62F h164=">AAAB9XicbVBNS8NAEJ3U 1q/oh69LBbBU0msqMdCLx4 2A9IY9lsNu3SzSbsbpQS+j+8eFDEq//Fm//GbZuD j4YeLw3w8y8IOVMac 5 kp 6xubW+X ys7u3 6B XjUUUkmCW2ThCeyF2BFORO0 Znm JdKiuOA024wbs787iOViiXiXk9S6sd4KFjECNZGemgO8n46Yiice W6P7C Ts2ZA60S yBVKNAa2F/9MCFZTIUmHC luU6q/RxLzQin00o/UzTFZIyH1DNU4JgqP59 PUVnRglRlEhTQqO5+nsix7FSkzgwnTHWI7XszcT/PC/T0Y2 M5FmmgqyWBRlHOkEzSJAIZOUaD4xBBPJzK2IjLDERJugKiYEd/nlVdK5qLlXN us pwijjKcAKncA4uXEMDbqEFbSAg4Rle4c16sl6sd+ j0Vqyiplj+AP 8weHn5HT</la exi >
Cd[33]
<la exi sha1_base64="Smjd2QdNG5/PR8OYZB0I6KkB3+c=">AAAB9XicbVDLSgNBEOyN xh UY9eBoPgKewaUY+BXDxGMA/Y GF20psMmX0wM6uEJ /hxYMiX 0Xb/6Nk2QPmljQUFR1093lJ4I bd VmF WNzq7hd2 nd2z8oHx61VZxKhi0Wi1h2 apQ8AhbmmuB3UQiDX2BHX/cmPmdR5SKx9G9niTohXQY8YAzqo300Ohn WTECU7dWs3 ly 21Z6D BInJxXI0eyX 3qDmKUhRpoJqpT 2In2Mio1ZwKnpV6qMKFsTI oGh REJWXza+ekjOjDEgQS1ORJnP190RGQ6UmoW86Q6pHa mbi 95bqqDGy/jUZJqjNhiUZAKomMyi4AMuESmxcQQyiQ3 xI2opIybYIqmRCc5ZdXS ui6lxVnb LS 3O4yjCCZzCOThwDXW4hSa0gIGEZ3iFN+ Je He Y9Fa8HKZ47hD6zPH4kokdQ=</la exi >
Ce[33]
<la exi sha1_base64="GF4 HK+8nKYxxq71+iuxjW5V A8=">AAAB/XicbVDLSsNAFL2p 1p 8bFzM1iEuimJFXVZ6MZlB uANobJdNIOnTyYmQg1BH/FjQ F3Po 7 wbp20W2n gwuGce7n3Hi/mTC L+jYKK6 6x FzdLW9s7un l/0JZRIgh kYhHou hSTkLaUsxxWk3FhQHHqcdb9yY+p0HKiSLwjs1iakT4GHI Eaw0pJ HjXc B+PGOLZ Vqxz7Je ea4Z mqWjOgZWLnpAw5mq751R9EJAloqAjHU ZsK1ZOioVihNOs1E8kjTEZ4yH aR igEonnV2 oVO DJA CV2hQjP190SKAykngac7A6xGc Gbi 95 UT5107KwjhRNCTzRX7CkY QNAo0YIISxSeaYCKY hWRERaYKB1YSYdgL768TN nV uya 9elO WHkcRjuEEKmDDFdThBp QAgKP8Ay 8GY8GS/Gu/Exby0Y+cwh/IHx+QPYnJQg</la exi >
C(1)
l[33]
<la exi sha1_base64="cwB5gUzscm6h/ms6HyQQgP Fi+c=">AAAB/XicbVDLSsNAFL2p 1p 8bFzM1iEuimJFXVZ6MZlB uANobJdNIOnTyYmQg1BH/FjQ F3Po 7 wbp20W2n gwuGce7n3Hi/mTC L+jYKK6 6x FzdLW9s7un l/0JZRIgh kYhHou hSTkLaUsxxWk3FhQHHqcdb9yY+p0HKiSLwjs1iakT4GHI Eaw0pJ HjXc B+PGOLZ VqpnWW9Ws1xzbJV WZAy8TOSRlyNF3zqz+ISBLQUBGOpezZVqycFA FCKdZqZ9IGmMyxkPa0zTEAZVOO s+Q6daGSA/E pChWbq74kUB1JOAk93BliN5KI3F /zeony 52UhXGiaEjmi/yEIxWhaRRowAQlik80wUQw SsiIywwUTqwkg7BXnx5mbTPq/Zl1b69KNe PI4iHMMJVMCGK6jDDTShBQQe4Rle4c14Ml6Md+Nj3low8plD+APj8w bsJQi</la exi >
C(3)
l[33]
Figu e 5.4: Limi s on coe icien s o 2- e mion ope a o s in he U(3)5(LO in black, NLO
in ed), MFV (LO in g een, NLO in blue) and 3 d gene a ion cen ic (LO in magen a,
NLO in cyan) scena ios wi h a single non-ze o ope a o and ma ginalizing o e he o he
independen la o s uc u es o each ope a o .
NLO:Solid;LO:Dashed
MFV
U(3)5
3 dGenCen
-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
3 dGenPh
3 dGenPh+U(2)5
Fla Less
-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
Figu e 5.5: 95% CL limi s on C(1)
ϕq [ij] unde la o assump ions desc ibed in he ex .
Resul s a LO a e d awn wi h dashed lines, esul s a NLO a e d awn wi h solid lines.
On he le we p esen C(1)
ϕq [11] s. C(1)
ϕq [33] in he U(3)5(black), MFV (blue) and
3 d gene a ion cen ic (magen a) scena ios. In hese scena ios C(1)
ϕq [22] = C(1)
ϕq [11]. On
he igh we p esen C(1)
ϕq [11] s. C(1)
ϕq [22] in he 3 d gene a ion phobic (g een), 3 d
gene a ion phobic + U(2)5(o ange) and la o less ( iole ) scena ios. In he i s wo
scena ios C(1)
ϕq [33] = 0, while in he la o less scena io C(1)
ϕq [33] = C(1)
ϕq [22] = C(1)
ϕq [11]. All
o he coe icien s a e se o 0.
CuB[33], Cϕu[33] and C(−)
ϕq [33] ≡1
2(C(1)
ϕq [33] −C(3)
ϕq [33]). Ou esul s ag ee wi h hese o
∼10 −20%, which is consis en wi h he use o sligh ly di e en se s o da a as inpu .
94 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES
In Fig. 5.5 and in Figs. A.1-A.6 in he Appendix A.3, we show on he le hand side
he limi s on he 2- e mion ope a o s in he U(3)5, 3 d gene a ion speci ic, and MFV
scena ios . On he igh hand side o hese plo s we show he 3 d gene a ion phobic
and 3 d gene a ion phobic + U(2)5scena ios, whe e he new physics only couples o he
i s and second gene a ions, and he la o less scena io. I is appa en ha he limi s
one quo es on hese ope a o s is highly dependen on he assumed la o scena io. I is
in e es ing o no e in Fig. 5.5, he la ge di e ences in he shapes o he LO and NLO i s
in he MFV and 3 d gene a ion phobic scena ios and ha he limi s on he 3 d gene a ion
phobic scena io a e conside ably weake han in he o he scena ios.
Tables 5.5 and 5.6 show he 95% CL limi s on coe icien s in he la o schemes
discussed in Sec. 5.2 o he Class B and Class C ope a o s. In gene al, he e is a s ong
dependence on he la o scena io. This dependence is much la ge han o he 2- e mion
ope a o coe icien s and he la o less scena io gi es much mo e s ingen bounds o
many coe icien unc ions han is he case in he o he la o scena ios.
The mos p ecise limi s a e on Cll and a e shown in Fig. 5.6 o se e al la o
scena ios. I is clea ha using he la o less scena io gi es limi s ha a e ac o s o
O(10−100) mo e p ecise han in he MFV o U(3)5scena ios o Cll. This is unde s ood
om Eq. 5.4 whe e we see ha he only la o s uc u e ha con ibu es o Gµis
Cll[1221] and we obse e ha he limi s on Cll in he U(2)5scena io a e he weakes .
In Fig. 5.7, we show he s ong co ela ion be ween Cll[1221] and Cll[3333] in he MFV
scena io and be ween Cll[1221] and Cll[1122] in he U(2)5and 3 d gene a ion phobic
scena ios. We no e ha he e a e only 2 independen Cll coe icien s.
Ma ginalized single pa ame e limi s on he Class B ope a o coe icien s C(1)
qq and
C(3)
qq a e shown in Fig. 5.8. The U(3)5and MFV esul s o hese ope a o s a e wi hin
a ac o o 2 o each o he , while he 3 d gene a ion speci ic and 3 d gene a ion phobic
scena ios a e weakly cons ained. Sample co ela ions be ween di e en la o s uc u es
a e shown in Fig. 5.9, demons a ing he sensi i i y o he la o assump ions o hese
i s.
The 4- e mion Class C ope a o s can be s udied indi idually, and ma ginalizing o e
he independen la o s uc u es gi es he limi s in Fig. 5.10. The limi s ob ained in he
la o less se up in his case also simila o hose o he MFV and U(3)5 la o s uc u es.
The 3 d gene a ion speci ic and 3 d gene a ion phobic scena ios gi e weak bounds o
hese ope a o s. In addi ion, he e a e la ge co ela ions be ween he di e en la o
s uc u es, as illus a ed in Fig. 5.11.
5.6 Fu u e pe spec i es
As discussed in his chap e , he e i ica ion o consis ency wi hin he SM o EWPO
is an impo an me hod, along wi h di ec sea ches o new pa icles, o in es iga e he
po en ial exis ence o BSM physics. Indeed, explo ing he p ope ies o massi e elec-
oweak gauge bosons (W and Z bosons) holds g ea p omise o indi ec BSM sea ches,
as he in e ac ions in ol ing pho ons and gluons a e subjec o s ingen cons ain s
imposed by he unb oken gauge symme ies. In his con ex , he achie able p ecision o
5.6. FUTURE PERSPECTIVES 95
0.01 0.1 1 10 100 1000
|d|
U(3)5
MFV
Fla o less
U(2)5
3 d gene a ion phobic + U(2)5
95% CL limi s on C
ll ange om EWPOs
(Coe icien s a e no all independen )
<la exi sha1_base64="+HZEkH41yE5gHP0HH jwe92wLkg=">AAAB9HicbVBNSwMxEJ2 X7V+VT16CRbBU9lYUY+FXjxWsB+wXUo2zbah2eyaZA l6e/w4kER /4Yb/4b03YP2 pg4PHeDDPzgkRwbVz32ylsbG5 7xR3S3 7B4dH5eOT o5TRVmLxiJW3YBoJ hkLcONYN1EMRIFgnWCcWPudyZMaR7LRzNNmB+RoeQhp8RYyW/0M4HEzMO1G b75YpbdRdA6wTnpAI5m 3yV28Q0zRi0lBB Pawmxg/I8pwK is1Es1SwgdkyHzLJUkY PFk P0IVVBiiMlS1p0EL9PZGRSO pFNjOiJiRX Xm4n+el5 wzs+4TFLDJF0uClOBTIzmCaABV4waMbWEUMX YiOiCLU2JxKNgS8+ I6aV9V8U0VP1xX6m4eRxHO4BwuAcM 1OEem ACCk/wDK/w5kycF+ d+Vi2Fpx85hT+wPn8AS5LkQI=</la exi >
Cll[1331]
<la exi sha1_base64="U/SLJLnwh91D EKE+WY1CnlyeUw=">AAAB9HicbVBNSwMxEJ2 X7V+VT16CRbBU9kUUY+FXjxWsB+wXUo2zbah2ew2yRbK0 /hxYMiX 0x3 w3pu0e PXBwOO9GWbmBYng2 ju 1PY2 7Z3S ulw4Oj45PyqdnbR2ni IWjUWsugHRTHDJWoYbwbqJYiQKBOsE48bC70yZ0jyWT2aWMD8iQ8lDTomxk /oZwKJuYd Nez3yxW36i6BNgnOSQVyNP l 94gpmnEpKGCaO1hNzF+RpThVLB5qZdqlhA6JkPmWSpJxLS LY+eoyu DFAYK1 SoKX6eyIjkdazKLCdETEj e4 xP88LzXh Z9xmaSGSbpaFKYCmRg EkAD hg1YmYJoY bWxEdEUWosTmVbAh4/eVN0q5V8W0VP95U6m4eRxEu4BKuAcMd1OEBm ACChN4hld4c6bOi/Pu KxaC04+cw5/4Hz+ACs+kQA=</la exi >
Cll[1221]
<la exi sha1_base64="j9eGoWCxNe0ReUnWDgPg L qgCM=">AAAB9HicbVBNS8NAEJ3U 1q/qh69LBbBU0lE1GOhF48V7Ae0oWy2m3bpZhN3J4US+ju8eFDEqz/Gm//GbZuD j4YeLw3w8y8IJHCoO +O4WNza3 neJuaW//4PCo HzSMnGqGW+yWMa6E1DDpVC8iQIl7ySa0yiQ B2M63O/PeHaiFg94jTh kSHSoSCUbSSX+9nkshZ17Pw++WKW3UXIO Ey0kFcjT65a/eIGZpxBUySY3pem6C kY1Cib5 NRLDU8oG9Mh71qqaMSNny2OnpELqwxIGG bCslC/T2R0ciYaRTYzojiyKx6c/E/ 5 ieOdnQiUpcsWWi8JUEozJPAEyEJozlFNLKNPC3k YiG K0OZUsiF4qy+ k9ZV1bupeg/XlZqbx1GEMziHS/DgFmpwDw1oAoMneIZXeHMmzo z7nwsWw OPnMK +B8/gAoMZD+</la exi >
Cll[1111]
<la exi sha1_base64="I1Fj3G5zIiVQZoQNzOi1/nYFmN0=">AAAB9HicbVBNSwMxEJ2 X7V+VT16CRbBU9kUUY+FXjxWsB+wXUo2zbah2ew2yRbK0 /hxYMiX 0x3 w3pu0e PXBwOO9GWbmBYng2 ju 1PY2 7Z3S ulw4Oj45PyqdnbR2ni IWjUWsugHRTHDJWoYbwbqJYiQKBOsE48bC70yZ0jyWT2aWMD8iQ8lDTomxk /oZwKJuYdx eb3yxW36i6BNgnOSQVyNP l 94gpmnEpKGCaO1hNzF+RpThVLB5qZdqlhA6JkPmWSpJxLS LY+eoyu DFAYK1 SoKX6eyIjkdazKLCdETEj e4 xP88LzXh Z9xmaSGSbpaFKYCmRg EkAD hg1YmYJoY bWxEdEUWosTmVbAh4/eVN0q5V8W0VP95U6m4eRxEu4BKuAcMd1OEBm ACChN4hld4c6bOi/Pu KxaC04+cw5/4Hz+ACs8kQA=</la exi >
Cll[1122]
<la exi sha1_base64=" SzNTx+IwjdeJMiJ4pINUekgLkQ=">AAAB9HicbVBNSwMxEJ2 X7V+VT16CRbBU9lYUY+FXjxWsB+wXUo2zbah2eyaZA l6e/w4kER /4Yb/4b03YP2 pg4PHeDDPzgkRwbVz32ylsbG5 7xR3S3 7B4dH5eOT o5TRVmLxiJW3YBoJ hkLcONYN1EMRIFgnWCcWPudyZMaR7LRzNNmB+RoeQhp8RYyW/0M4HEzMO4V P75YpbdRdA6wTnpAI5m 3yV28Q0zRi0lBB Pawmxg/I8pwK is1Es1SwgdkyHzLJUkY PFk P0IVVBiiMlS1p0EL9PZGRSO pFNjOiJiRX Xm4n+el5 wzs+4TFLDJF0uClOBTIzmCaABV4waMbWEUMX YiOiCLU2JxKNgS8+ I6aV9V8U0VP1xX6m4eRxHO4BwuAcM 1OEem ACCk/wDK/w5kycF+ d+Vi2Fpx85hT+wPn8AS5HkQI=</la exi >
Cll[1133]
<la exi sha1_base64="XMojbGTgcEK3BJkem4G V/k/I40=">AAAB9HicbVBNS8NAEJ3U 1q/qh69LBbBU0lU1GOhF48VbC20oWy2m3bpZhN3J4US+ju8eFDEqz/Gm//GbZuD j4YeLw3w8y8IJHCoO +O4W19Y3N eJ2aWd3b/+g HjUMnGqGW+yWMa6HVDDpVC8iQIlbyea0yiQ/DEY1W +45h I2L1gJOE+xEdKBEKR FK 2XSSKnnUsL 1euuFV3D JK JxUIEejV/7q9mOWRlwhk9SYjucm6GdUo2CST0 d1PCEshEd8I6likbc+Nn86Ck5s0q hLG2pZDM1d8TGY2MmUSB7YwoDs2yNxP/8zoph d+JlSSIldssShMJcGYzBIg aE5QzmxhDI 7K2EDammDG1OJRuC /zyKmldVL3 qnd/Vam5eRxFOIFTOAcPbqAGd9CAJjB4gmd4hTdn7Lw4787Ho Xg5DPH8A O5w80ZZEG</la exi >
Cll[3333]
Figu e 5.6: Limi s on coe icien s o Cll[ijkl] in a ious la o scena ios, whe e i, j, k, l =
1,2,3. Only Cll is aken o be non-ze o in his igu e, and he independen la o s uc-
u es no shown a e ma ginalized o e . Exac numbe s a e gi en in Table 5.5.
MFV
3 dGenCen
Fla Less
-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
3 dGenPh
3 dGenPh+U(2)5
Fla Less
-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
Figu e 5.7: 95% CL limi s on Cll in a ious la o scena ios wi h all o he coe icien s
aken o be 0. On he le we p esen Cll[3333] s. Cll[1221] in he MFV (blue), 3 d
gene a ion cen ic (black) and la o less (magen a) scena ios. On he igh we p esen
Cll[1122] s. Cll[1221] in he U(2)5(cyan), 3 d gene a ion phobic (g een), 3 d gene a ion
phobic + U(2)5(o ange) and la o less (magen a) scena ios. All he independen la o
s uc u es no p esen in he plo s a e ma ginalized o e .
a ious u u e expe imen s has undamen al impo ance. Indeed, ac o s such as he s a-
is ical size o he da a sample and he in e play be ween expe imen al and heo e ical
sys ema ic unce ain ies play a c ucial ole. In he epo [21], an es ima ion o p ojec ed
s a is ical and sys ema ic unce ain ies is p oposed o a ious u u e linea collide s. In
his sec ion, using he da a in Table 3 [21], we show he impac o p ojec ed u u e
collide s e o s (ILC-GigaZ and Fcc-ee) on he ange o he bounds SMEFT coe icien s,
while also compa ing hem wi h he cu en alues. We s ess ha he e alua ions o
sys ema ic e o s a e ough es ima es in e ms o hei magni ude. Fo a mo e accu a e
assessmen , i would be necessa y ins umen a ion de ails and o he ools ha a e cu -
96 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES
0.01 1 100 10000 1e+06
|d|
U(3)5
MFV
Fla o less
3 d gene a ion phobic + U(2)5
3 d gene a ion speci ic
95% CL limi s on C
qq
(1,3) anges om EWPOs
(Coe icien s a e no all independen )
<la exi sha1_base64="MMF0m ZnjHSwghGE9s2RAun5/0w=">AAAB+3icbVDLSsNAFJ3UV62 WJduBo QNyVTRF0WunFZwT4gjWEynbZDJ5N0ZiKWkF9x40IR /6IO//GaZuF h64cDjnXu69J4g5U9px q3CxubW9k5x 7S3 3B4ZB+XOypKJKF E FI9gKsKGeC jXTnPZiSXEYcNoNJs25332kU FI3O ZTL0QjwQbMoK1kXy73PTT6TR7SK oInORge bFa mLADXCcpJBeRo+ ZX xCRJKRCE46VcpETay/FUjPCaVbqJ4 GmEzwiLqGChxS5aWL2zN4bpQBHEbSlNBwo 6eSHGo1CwMTGeI9Vi enPxP89N9PDGS5mIE00FWS4aJhzqCM6DgAMmKdF8ZggmkplbIRljiYk2cZVMCGj15XXSqd QVQ3dXVYa9TyOIjgFZ6AKELgGDXALWqANCHgCz+AV FmZ9WK9Wx/L1oKVz5yAP7A+ wAgoJMl</la exi >
C(1)
qq [1111]
<la exi sha1_base64="4Epg7OBjjgk9iHjUwU62iFQW3yE=">AAAB+3icbVDLSsNAFJ3UV62 WJdugkWom5IJoi4L3bisYB+QxjCZT qhk0k6MxFLyK+4caGIW3/EnX/j M1CWw9cOJxzL/ eEySMSmXb30ZpY3N e6e8W9nbPzg8Mo+ XRmnApMOjlks+gGShFFOOooqR qJICgKGOkFk9bc7z0SIWnM79UsIV6ERpyGFCOlJd+s xsOs0 sjq8yF0IHc zzZ dsBew1gksSA0UaP m12AY4zQiXGGGpHShnSg Q0JRzEheGaSSJAhP0Ii4mnIUEelli9 z61w QyuMhS6u IX6eyJDkZSzKNCdEVJjue Nx 88N1XhjZdRnqSKcLxcFKbMU E1D8IaUkGwYjNNEBZU32 hMRIIKx1XRYcAV19eJ12nAa8a8O6y1nSKOM gFJyBOoDgGjTBLWiDDsDgCTyDV/Bm5MaL8W58LF LRjFzA 7A+PwBI6uTJw==</la exi >
C
(1)
qq
[1122]
<la exi sha1_base64="VleThwgLmY00I DQ8uO 9WM/MHA=">AAAB+3icbVDLSsNAFJ3UV62 WJdugkWom5IJoi4L3bisYB+QxjCZT qhk0k6MxFLyK+4caGIW3/EnX/j M1CWw9cOJxzL/ eEySMSmXb30ZpY3N e6e8W9nbPzg8Mo+ XRmnApMOjlks+gGShFFOOooqR qJICgKGOkFk9bc7z0SIWnM79UsIV6ERpyGFCOlJd+s xsOs0 sjq8yF3oONDzzZ dsBew1gksSA0UaP m12AY4zQiXGGGpHShnSg Q0JRzEheGaSSJAhP0Ii4mnIUEelli9 z61w QyuMhS6u IX6eyJDkZSzKNCdEVJjue Nx 88N1XhjZdRnqSKcLxcFKbMU E1D8IaUkGwYjNNEBZU32 hMRIIKx1XRYcAV19eJ12nAa8a8O6y1nSKOM gFJyBOoDgGjTBLWiDDsDgCTyDV/Bm5MaL8W58LF LRjFzA 7A+PwBI62TJw==</la exi >
C(1)
qq [1221]
<la exi sha1_base64="Mhyn105IX q3n0 oWEeS 36YMgk=">AAAB+3icbVDLTsJAFJ3iC/FVcelmIjHBDemAUZckbFxiIo+k1GY6TGHC9MHM1Eia/oobFx j1h9x5984QBcKnuQmJ+ cm3 8WLOpLKsb6Owsbm1 VPcLe3 Hxwemc l owSQWiHRDwS Q9LyllIO4opT uxoDjwOO15k9bc7z1SIVkU3q ZTJ0Aj0LmM4KVllyz3HLT6TR7SK oI MRajQc16xYNWsBuE5QTiogR9s1 wbDiCQBDRXhWEobWbFyUiwUI5xmpUEiaYzJBI+o WmIAyqddHF7Bs+1MoR+JHSFCi7U3xMpDqScBZ7uDLAay1V L 7n2Ynyb5yUhXGiaEiWi/yEQxXBeRBwyAQlis80wUQw SskYywwUTqukg4B b68T 1G qqob LS Oex1EEp+AMVAEC16AJbkEbdAABT+AZ II3IzNejH jY9laMPKZE/AHxucPJ aTKQ==</la exi >
C(1)
qq [1133]
<la exi sha1_base64="1LUyw5lYm43oVnG43MBmB8SB Q=">AAAB+3icbVBNS8NAEJ34We X Ec wSLUS0mqqMdCLx4 2A9IY9hs +3SzSbd3Ygl5K948aCIV/+IN/+N2zYHbX0w8Hh hpl5QcyoVLb9bay b2xubRd2i 7+weH5lGpLaNEYNLCEY EN0CSMMpJS1HFSDcWBIUBI51g3Jj5nUciJI34 Z GxA RkNMBxUhpyTdLDT+dTLKH OKcZ+6FhuebZb qz2G EicnZcjR9M2 Xj/CSUi4wgxJ6Tp2 LwUCUUxI1mxl0gSIzxGQ+JqylFIpJ Ob8+sM630 UEkdHFlzdX EykKpZyGge4MkR JZW8m/ue5iR ceCnlcaIIx4 Fg4RZK JmQVh9KghWbKoJwoLqWy08QgJhpeMq6hCc5ZdXSb Wda6qz 1luV7L4yjACZxCBRy4hj cQhNagOEJnuEV3ozMeDHejY9F65qRzxzDHxi PyzUky0=</la exi >
C(1)
qq [3333]
<la exi sha1_base64="aM9 3I CNq9zC/Mj1pR0kjyhpes=">AAAB+3icbVBNS8NAEJ34We X Ec wSLUS0mqqMdCLx4 2A9IY9hs +3SzSbd3Ygl5K948aCIV/+IN/+N2zYHbX0w8Hh hpl5QcyoVLb9bay b2xubRd2i 7+weH5lGpLaNEYNLCEY EN0CSMMpJS1HFSDcWBIUBI51g3Jj5nUciJI34 Z GxA RkNMBxUhpyTdLDT+dTLKH HJxn mOhuebZb qz2G EicnZcjR9M2 Xj/CSUi4wgxJ6Tp2 LwUCUUxI1mxl0gSIzxGQ+JqylFIpJ Ob8+sM630 UEkdHFlzdX EykKpZyGge4MkR JZW8m/ue5iR ceCnlcaIIx4 Fg4RZK JmQVh9KghWbKoJwoLqWy08QgJhpeMq6hCc5ZdXSb Wda6qz 1luV7L4yjACZxCBRy4hj cQhNagOEJnuEV3ozMeDHejY9F65qRzxzDHxi PyO4kyc=</la exi >
C(3)
qq [1111]
<la exi sha1_base64="6emjwJ9B5x/MpIJMG/IGNasRJNc=">AAAB+3icbVDLSsNAFJ34 PUV69JNsAh1UzJR1GWhG5cV7APSGCbTST 0MklnJmIJ+RU3LhRx64+482+c llo64ELh3Pu5d57goRRqWz721hb39jc2i7 lH 39g8OzaNKR8apwKSNYxaLXoAkYZST qKKkV4iCIoCR BuDnzu49ESB zezVNiBehIachxUhpyTc TT+bTPKH HZxn sQOo7nm1W7bs9h RJYkCoo0PLN /4gxmlEuMIMSelCO1FehoSimJG83E8lSRAeoyFxNeUoI LL5 n1plWBlYYC11cWXP190SGIimnUaA7I6RGc mbi 95bq CGy+jPEkV4XixKEyZpWJ FoQ1oIJgxaaaICyo XCIyQQVjqusg4BL +8SjpOHV7V4d1l eEUcZTACTgFNQDBNWiAW9ACbYDBE3gG +DNyI0X4934WLSuGcXMM gD4/MHJsOTKQ==</la exi >
C(3)
qq [1122]
<la exi sha1_base64="u+H+gQ+D7zO/7usF+ 4lMNay Ug=">AAAB+3icbVDLSsNAFJ34 PUV69JNsAh1UzJR1GWhG5cV7APSGCbTST 0MklnJmIJ+RU3LhRx64+482+c llo64ELh3Pu5d57goRRqWz721hb39jc2i7 lH 39g8OzaNKR8apwKSNYxaLXoAkYZST qKKkV4iCIoCR BuDnzu49ESB zezVNiBehIachxUhpyTc TT+bTPKH HZxn QcaDnm1W7bs9h RJYkCoo0PLN /4gxmlEuMIMSelCO1FehoSimJG83E8lSRAeoyFxNeUoI LL5 n1plWBlYYC11cWXP190SGIimnUaA7I6RGc mbi 95bq CGy+jPEkV4XixKEyZpWJ FoQ1oIJgxaaaICyo XCIyQQVjqusg4BL +8SjpOHV7V4d1l eEUcZTACTgFNQDBNWiAW9ACbYDBE3gG +DNyI0X4934WLSuGcXMM gD4/MHJsWTKQ==</la exi >
C(3)
qq [1221]
<la exi sha1_base64="Y6NzwbaA4MLZzQ jgU6g dP STA=">AAAB+3icbVDLTsJAFJ3iC/FVcemmkZjghnTAqEsSNi4xkUdSajMdpjBhOi0zUyNp+i uXGiMW3/EnX/jAF0oeJKbnJxzb+69x48Zlcq2 43CxubW9k5x 7S3 3B4ZB6XuzJKBCYdHLFI9H0kCaOcdBRVjPRjQVDoM9LzJ62533skQ KI36 ZTNwQjTgNKEZKS55ZbnnpdJo9pNXGReZA2Gi4nlmxa/YC1jqBOamAHG3P/BoMI5yEhC MkJQO GPlpkgoihnJSoNEkhjhCRoRR1OOQiLddHF7Zp1 ZWgFkdDFlbVQ 0+kKJRyF q6M0RqLFe9u i 5yQquHFTyuNEEY6Xi4KEWSqy5kFYQyoIVmymCcKC6ls PEYCYaXjKukQ4O L66Rb 8G G y7 DT eRxFcA OQBVAcA2a4Ba0QQdg8ASewS 4MzLjxXg3Ppa BSO OQF/YHz+ACnOkys=</la exi >
C(3)
qq [1133]
<la exi sha1_base64="egWFez6 MDxwzg7GScHAPqIpP20=">AAAB+3icbVDLTsJAFJ3iC/FVcemmkZjghnTAqEsSNi4xkUdSajMdpjBhOi0zUyNp+i uXGiMW3/EnX/jAF0oeJKbnJxzb+69x48Zlcq2 43CxubW9k5x 7S3 3B4ZB6XuzJKBCYdHLFI9H0kCaOcdBRVjPRjQVDoM9LzJ62533skQ KI36 ZTNwQjTgNKEZKS55ZbnnpdJo9pNXGRebARgO6nlmxa/YC1jqBOamAHG3P/BoMI5yEhC MkJQO GPlpkgoihnJSoNEkhjhCRoRR1OOQiLddHF7Zp1 ZWgFkdDFlbVQ 0+kKJRyF q6M0RqLFe9u i 5yQquHFTyuNEEY6Xi4KEWSqy5kFYQyoIVmymCcKC6ls PEYCYaXjKukQ4O L66Rb 8G G y7 DT eRxFcA OQBVAcA2a4Ba0QQdg8ASewS 4MzLjxXg3Ppa BSO OQF/YHz+ACnSkys=</la exi >
C(3)
qq [1331]
<la exi sha1_base64="Pbo06+93/R ELd7y+qHLV1uDmHU=">AAAB+3icbVDLTsJAFJ3iC/FVcelmIjHBDWnBqEsSNi4xkUdSajMdpjBhOi0zUyNp+i uXGiMW3/EnX/jAF0oeJKbnJxzb+69x48Zlcqy o3CxubW9k5x 7S3 3B4ZB6XuzJKBCYdHLFI9H0kCaOcdBRVjPRjQVDoM9LzJ62533skQ KI36 ZTNwQjTgNKEZKS55ZbnnpdJo9pNXGReY0NFzP Fg1awG4TuycVECO md+DYYRTkLCFWZISse2YuWmSCiKGclKg0SSGOEJGhFHU45CI 10cXsGz7UyhEEkdHEFF+ iRSFUs5CX3eGSI3lqjcX//OcRAU3bkp5nCjC8XJRkDCoIjgPAg6pIFixmSYIC6p hXiMBMJKx1XSIdi L6+Tb 1mX9Xsu8 Ks57HUQSn4AxUgQ2uQRPcgjboAAyewDN4BW9GZ wY78bHs Vg5DMn4A+Mzx8 7JM </la exi >
C(3)
qq [3333]
<la exi sha1_base64="WV9Jdui5339VUWjU6KulSgVe+9M=">AAAB+3icbVDLTsJAFJ3iC/FVcelmIjHBDemAUZckbFxiIo+k1GY6TGHC9MHM1Eia/oobFx j1h9x5984QBcKnuQmJ+ cm3 8WLOpLKsb6Owsbm1 VPcLe3 Hxwemc l owSQWiHRDwS Q9LyllIO4opT uxoDjwOO15k9bc7z1SIVkU3q ZTJ0Aj0LmM4KVllyz3HLT6TR7SK oI NRo4Ec16xYNWsBuE5QTiogR9s1 wbDiCQBDRXhWEobWbFyUiwUI5xmpUEiaYzJBI+o WmIAyqddHF7Bs+1MoR+JHSFCi7U3xMpDqScBZ7uDLAay1V L 7n2Ynyb5yUhXGiaEiWi/yEQxXBeRBwyAQlis80wUQw SskYywwUTqukg4B b68T 1G qqob LS Oex1EEp+AMVAEC16AJbkEbdAABT+AZ II3IzNejH jY9laMPKZE/AHxucPJ qTKQ==</la exi >
C(1)
qq [1331]
Figu e 5.8: Limi s on sample o coe icien s o Class B 4- e mion ope a o s in a ious
la o scena ios. δis he ange o he 95% con idence le el limi s. Only a single ope a o
is aken o be non-ze o and he la o s uc u es no shown a e ma ginalized o e . Exac
numbe s a e gi en in Table 5.5.
MFV
U(3)5
Fla Less
-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
Fla Less
-6-4-20246
-20
-10
0
10
20
Cqq
(3)[1,3,3,1]
Cqq
(3)[1,1,3,3]
95%CL
Figu e 5.9: 95% CL limi s on C(1)
qq (le ) and C(3)
qq ( igh ), in a ious la o scena ios wi h
all o he coe icien s aken o be 0. On he le we p esen C(1)
qq [1331] s. C(1)
qq [1133]
in he MFV (g een), 3 d gene a ion cen ic (o ange) and la o less ( iole ) scena ios.
On he igh we p esen C(3)
qq [1331] s. C(3)
qq [1133] in he MFV (g een), 3 d gene a ion
cen ic (o ange) and la o less ( iole ) scena ios. All he independen la o s uc u es
no p esen in he plo s a e ma ginalized o e .
en ly una ailable. Fu he mo e, i should be no ed ha wi hin his app oach, we lack
p ojec ed expe imen al alues, p e en ing us om pe o ming a i o he obse ables.
Howe e , we a e no in e es ed in he ac ual possible anges o he coe icien s because
ou goal is o make an es ima ion o he size o he bounds based on u u e collide s
p ojec ions.
5.6. FUTURE PERSPECTIVES 97
0.001 0.01 0.1 1 10 100
|d|
U(3)5
MFV
Fla o less
3 d gene a ion phobic + U(2)5
3 d gene a ion speci ic
U(2)5
95% CL limi s on C
lq
(1,3) anges om EWPOs
(Coe icien s a e no all independen )
<la exi sha1_base64="EGmi/KLejW 6mSA4QQ LPCudMh4=">AAAB+3icbVDLSsNAFJ34 PUV69LNYBHqpmSKqM CNy4 2AekMUymk3boZBJnJmIJ+RU3LhRx64+482+c llo64ELh3Pu5d57goQzpR3n21pb39jc2i7 lH 39g8O7aNKV8WpJLRDYh7L oAV5UzQjmaa034iKY4CTn BpDXze49UKhaLOz1NqB hkWAhI1gbybc LT/jD/l9VkPnuY Qo+H5d WpO3PAVYIKUgUF2 79NRjGJI2o0IRjpVzkJN LsNSMcJqXB6miCSYTPKKuoQJHVHnZ/PYcnhllCMNYmhIaz X ExmOlJpGgemMsB6 ZW8m/ue5qQ6 YyJJNVUkMWiMOVQx3AWBBwySYnmU0MwkczcCskYS0y0ia sQkDLL6+SbqOOLu o9qLabBRxlMAJOAU1gMAVaIIb0AYdQMATeAa 4M3K R 3 pY K5Zxcwx+AP 8wcb0ZMi</la exi >
C(1)
lq [1122]
<la exi sha1_base64=" SWaawF60YEGIjFRYd1GNgG4EwU=">AAAB+3icbVBNS8NAEJ3U 1q/Yj16CRahXkpSRT0We FYwX5AGsNmu22XbjZxdyOWkL/ixYMiX 0j3 w3b sc PXBwOO9GWbmBTGjU n2 1FYW9/Y3Cpul3Z29/YPzMNyR0aJwKSNIxaJXoAkYZST qKKkV4sCAoDR BpDnzu49ESB xOzWNiReiEadDipHSkm+Wm37KH L7 OqcZe65huebFb mz2G EicnFcjR8s2 /iDCSUi4wgxJ6Tp2 LwUCUUxI1mpn0gSIzxBI+JqylFIpJ Ob8+sU60M GEkdHFlzdX EykKpZyGge4MkR LZW8m/ue5iRpeeynlcaIIx4 Fw4RZK JmQVgDKghWbKoJwoLqWy08RgJhpeMq6RCc5ZdXSadecy5 zu1FpVHP4yjCMZxAFRy4ggbcQA agOEJnuEV3ozMeDHejY9Fa8HIZ47gD4zPHyT6kyg=</la exi >
C(1)
lq [3333]
<la exi sha1_base64="Y 0k q6 VVbsqF/ohaB 1MelcQ=">AAAB+3icbVDLSsNAFJ3UV62 WJdugkWom5KJoi4L3bisYB+QxjCZT qhk0mcmYgl5F cuFDE T/iz 9x2mah QcuHM65l3 CRJGpbL b6O0 6xuVXe uzs7u0 mI V oxTgUkHxywW/QBJwignHUUVI/1EEBQFjPSCSW m9x6JkDTmd2qaEC9CI05DipHSkm9WW37GH L7 H5+l sQOo7nmzW7Yc9h RJYkBoo0PbN 8EwxmlEuMIMSelCO1FehoSimJG8MkglSRCeoBFxNeUoI LL5 n1qlWhlYYC11cWXP190SGIimnUaA7I6TGc mbi 95bq Cay+jPEkV4XixKEyZpWJ FoQ1pIJgxaaaICyo XCYyQQVjquig4BL +8S pOA1424O1F ekUcZTBMTgBdQDBFWiCG9AGHYDBE3gG +DNyI0X4934WLSWjGLmCPyB8 kDHumTJA==</la exi >
C(3)
lq [1122]
<la exi sha1_base64="YCY29qHJnWw/OLHao uP0iUZD2Q=">AAAB+3icbVDLSsNAFJ34 PUV69LNYBHqpmRaUZeFblxWsA9IY5hMp+3QySTOTMQS8i uXCji1h9x5984bbPQ1gMXDu cy733BDFnSj O 7W2 G5 V3YKe7u7R8c2keljooSSWibRDyS QA ypmgbc00p71YUhwGnHaDSXPmdx+pVCwSd3oaUy/EI8GGjGB JN8uN 2UP2T3aaV+n kI1eueb5edqjMHXCUoJ2WQo+XbX/1BRJKQCk04VspFTqy9FE NCKdZsZ8oGmMywSPqGipwSJWXzm/P4JlRBnAYSVNCw7n6eyLFoVLTMDCdIdZj ezNxP88N9HDay9lIk40FWSxaJhwqCM4CwIOmKRE86khmEhmboVkjCUm2sRVNCGg5ZdXSadWRZdVdH Rb TyOA gBJyCCkDgCjTADWiBNiDgCTyDV/BmZdaL9W59LF X HzmGPyB9 kDI STJg==</la exi >
C(3)
lq [1133]
<la exi sha1_base64=" M20xy4N 0m3wnTq 2cAeC2q1g=">AAAB+3icbVDLSsNAFJ34 PUV69LNYBHqpmRaUZeFblxWsA9IY5hMp+3QySTOTMQS8i uXCji1h9x5984bbPQ1gMXDu cy733BDFnSj O 7W2 G5 V3YKe7u7R8c2keljooSSWibRDyS QA ypmgbc00p71YUhwGnHaDSXPmdx+pVCwSd3oaUy/EI8GGjGB JN8uN 2UP2T3aaV+n n1OkKeb5edqjMHXCUoJ2WQo+XbX/1BRJKQCk04VspFTqy9FE NCKdZsZ8oGmMywSPqGipwSJWXzm/P4JlRBnAYSVNCw7n6eyLFoVLTMDCdIdZj ezNxP88N9HDay9lIk40FWSxaJhwqCM4CwIOmKRE86khmEhmboVkjCUm2sRVNCGg5ZdXSadWRZdVdH Rb TyOA gBJyCCkDgCjTADWiBNiDgCTyDV/BmZdaL9W59LF X HzmGPyB9 kDI yTJg==</la exi >
C(3)
lq [3311]
<la exi sha1_base64="KaI9XgnOoVpYi1YmPZ7u5KhY4ow=">AAAB+3icbVDLSsNAFL3xWes 1qWbwSLUTUlaUZeFblxWsA9IY5hMp+3QycOZiVhC sWNC0Xc+iPu/BunbRbaeuDC4Zx7u ceP+ZMKs 6N bWNza3 gs7xd29/YND86jUkVEiCG2TiEei52NJOQ pWzHFaS8WFAc+p11/0pz53UcqJI COzWNqR gUciGjGClJc8sNb2UP2T3aaV+njl1Ddczy1bVmgO EjsnZcjR8sy /iAiSUBDRTiW0 G WLkpFooRT NiP5E0xmSCR9TRNMQBlW46 z1DZ1oZoGEkdIUKzdX EykOpJwG u4MsB LZW8m/uc5iRpeuykL40TRkCwWDROOVIRmQaABE5QoP UEE8H0 YiMscBE6biKOgR7+eVV0qlV7cuq X Rb TyOApwAqdQARuuoAE30II2EHiCZ3iFNyMzXox342PRumbkM8 wB8bnDygSkyo=</la exi >
C(3)
lq [3333]
Figu e 5.10: Limi s on sample o coe icien s o Class C 4- e mion ope a o s in a ious
la o scena ios, wi h he independen la o s uc u es o each ope a o ma ginalized
o e . Exac numbe s a e gi en in 5.6.
MFV
3 dGenCen
3 dGenPh+U(2)5
Fla Less
-1012
-5
0
5
10
15
Clq
(1)[3,3,3,3]
Clq
(1)[1,1,2,2]
95%CL
U(2)5
MFV
3 dGenCen
3 dGenPh+U(2)5
Fla Less
-8-6-4-2024
-2
-1
0
1
2
Clq
(3)[3,3,3,3]
Clq
(3)[1,1,2,2]
95%CL
Figu e 5.11: 95% CL limi s on C(1)
lq (le ) and C(3)
lq ( igh ), in a ious la o scena ios wi h
all o he coe icien s aken o be 0. On he le we p esen C(1)
lq [3333] s. C(1)
lq [1122] in
he MFV (g een), 3 d gene a ion cen ic (black), 3 d gene a ion phobic + U(2)5(o ange)
and la o less ( iole ) scena ios. On he igh we p esen C(3)
lq [3333] s. C(3)
lq [1122] in
he U(2)5(blue), MFV (g een), 3 d gene a ion cen ic (black), 3 d gene a ion phobic +
U(2)5(o ange) and la o less ( iole ) scena ios.. All he independen la o s uc u es
no p esen in he plo s a e ma ginalized o e .
Me hod
Simila ly o he app oach used in he p e ious sec ion, we e alua e only one non ze o
ope a o a ime and hen we explo e di e en la o scena ios. Due o he p e ious
assump ions, we conside only single pa ame e limi s, ma ginalizing o e he o he
independen la o s uc u es. The ma ginaliza ion is he same desc ibed in 5.5. As
al eady men ion, since we do no ha e p ojec ed expe imen al alues, we se hem o
hei ”bes ” heo e ical SM alue, indica ed in Table 5.3. In his way, he χ2will no
depend on he cen al alue o he obse ables bu only on he p ojec ed unce ain ies.
Finally, we do no expec ha he heo e ical unce ain ies will emain he same a
98 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES
Ope a o LO U(3)5NLO U(3)5LO MFV NLO MFV 3 dgen cen ic 3 dgen cen ic
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 anges on 2- e mion ope a o s wi h a ying la o assump-
ions desc ibed in he ex . O he la o s uc u e o a gi en ope a o a e ma ginalized
o e , and di e en coe icien s se o 0. The en ies labelled wi h a ”-” co espond o
ope a o s ha do no con ibu e, while hose labelled wi h a ”*” a e no independen .
The la o s uc u es gi en no in he able can be always de i ed om hose in he
able using he ela ions de ailed in he ex .
momen he collide s will un. I is logical o hypo hesize ha hey will dec ease hus,
in his sec ion, we assume ha hey will be one hal o he cu en ones [38]. Al hough
simplis ic, his assump ion seems easonable when conside ing he ime u u e collide s
will ope a e a ull capaci y (T > 20 yea s). In Table 5.7 a e indica ed he da a used o
his analysis including ILC-GigaZ and Fcc-ee expec ed expe imen al unce ain ies and
expec ed heo e ical unce ain ies.
5.6.1 Resul s
We begin o show he esul s s a ing wi h Class A. We de ine he expec ed upg ade as
ηFcc/ILC(Ci) = δcu en (Ci)
δFcc/ILC(Ci)(5.54)
whe e δcu en (Ci) and δF cc/ILC(Ci) a e espec i ely he cu en and he p ojec ed size o
he bounds o a ce ain coe icien Ci. Table 5.8 shows he ob ained he 95% CL lim-
i s on some o he coe icien s o di e en la o scena io and di e en u u e collide s.
In his class he cu en size o he bounds is O(C)≲1 o all he coe icien s and in
all he scena ios. Conside ing u u e collide s bounds we obse e a signi ican imp o e-
men o all he ope a o s. The minimum enhancemen is obse ed o he coe icien
C(1)
ϕl [11] in he 3 d GenPh scena io, when compa ed o he ILC-GigaZ p ojec ed bounds
ηILC(C(1)
ϕl [11]) = 3.075. The maximum, again in he 3 d GenPh scena io, is o Cϕu[22]
wi h η cc(Cϕu[22]) = 63.008.
5.6. FUTURE PERSPECTIVES 99
Ope a o U(3)5MFV U(2)53 dgen 3 dgen 3 dgen Fla o less
speci ic 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 esul s o 95% CL limi s on Class B ope a o s when he coe icien s
o all o he ope a o s a e se o 0 and he di e en la o s uc u es o a gi en ope a o
a e ma ginalized o e . The limi s ma ked wi h a ”*” a e ob ained using he ela ions
gi en in he ex and do no ep esen independen coe icien s. We label wi h an ”x”
he cases whe e no limi can be de i ed.
Finally we no ice ha o some he coe icien s, depending on he scena io, he p o-
jec ed upg ade is almos he same o Fcc-ee and ILC-GigaZ. Fo example, in Fig. 5.12
we show he esul s o Cϕu[33]. Fo his coe icien , we do no see signi ican di e ences
be ween he wo u u e collide p ojec ions.
Table 5.9 p esen s some o he 95% CL limi s on 4- e mion coe icien s o mul iple
scena ios. As discussed in he p e ious sec ion, la o has a d ama ic e ec on he
100 CHAPTER 5. ELECTROWEAK PRECISION OBSERVABLES
Ope a o U(3)5MFV U(2)53 dgen speci ic 3 dgen phobic +U(2)5Fla o less
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 esul s o 95% CL limi s on Class C ope a o s when he coe icien s
o all o he ope a o s a e se o 0 and he di e en la o s uc u es o a gi en ope a o
a e ma ginalized o e . The limi s ma ked wi h a ”*” a e ob ained using he ela ions
gi en in he ex and do no ep esen independen coe icien s. We label wi h an ”x”
he cases whe e no limi can be de i ed.
la o bounds. In pa icula , depending on he scena ios, we ha e seen ha some o he
coe icien s in Class B & C may ha e a ange o alues O(δ(Ci)) ≫10. I is in e es ing o
check i a coe icien ha is no signi ican ly cons ained wi h he cu en expe imen al
p ecision, can p o ide mo e use ul in o ma ion in u u e collide s.
In Fig 5.13, we show he esul s o C(1)
qq [1122] and C(3)
qq [1331]. Fo C(1)
qq [1122], we
see ha bo h he scena ios would no cons ain he coe icien su icien ly, in o de
o ha e signi ican bounds. Indeed, al hough we obse e ηF cc = 33.50 in he MFV
6.3. LO DY CALCULATIONS 107
Figu e 6.1: LO Feynman diag ams o DY p ocess. Realized wi h [185]
.
A∼ASM + Σi
C6
i
Λ2A6
i,LO + Σj
D6
j
16π2Λ2A6
j,NLO .(6.2)
He e ASM , A6
i,LO, and A6
j,NLO ep esen he SM, dimension-6 ee le el, and dimension-6
1−loop con ibu ions, espec i ely. We de ine he linea SMEFT esul as:
|A|2
lin ≡ | ASM |2+2Re(ΣiA∗
SM
C6
i
Λ2A6
i,LO)
+2Re(ΣiA∗
SM
D6
i
16π2Λ2A6
j,NLO) (6.3)
whe e we no ice ha Eq. 6.3 is no posi i e de ini e. Fo ou pu poses, we de ine he
quad a ic SMEFT esul as:
|A|2
quad ≡ | ASM + Σi
C6
i
Λ2A6
i,LO + Σj
D6
j
16π2Λ2A6
j,NLO |2.(6.4)
In SMEFT phenomenology s udies and global i s, he quan i y de ined in Eq. 6.4 is
commonly used. Ne e heless, i is impo an o no e ha we do no include wo ypes
o O(1
Λ4) e ms in Eq. 6.4: he in e e ence o he dimension-8 ope a o s wi h he SM
esul [184], and he double inse ions o he dimension-6 ope a o s in he ampli ude. In
he ollowing we will neglec hese e ms, because hey a e beyond he scope o cu en
NLO EW SMEFT calcula ions.
6.3 LO DY calcula ions
In his sec ion we show he LO esul s o q(p1)q(p2)→l+(p3)l−(p4). Fig 6.1 shows
he wo possible opologies p esen a LO. In he ollowing, we conside up qua k ini ial
s a es, he down qua k case is gi en in Appendix A.1. In e ms o helici y ampli udes
we ha e:
MXY =u(p2)γµPXu(p1)·u(p3)γµPYu(p4),(6.5)
108 CHAPTER 6. DRELL YAN PROCESS
wi h PL,R =1∓γ5
2. A ee le el we ha e
ALO = ΣXY GXY MXY (6.6)
wi h
GXY =GSM
XY +δGXY (6.7)
whe e he δ e m ep esen s he SMEFT con ibu ions.
The e o e, o he up qua k ini ial s a es we ha e
GSM
LR =−2(s−4m2
W)(m2
W−m2
Z)
3s 2(s−m2
Z)GSM
LL =2sm2
W+ 8m4
W+sm2
Z−8m2
Wm2
Z
3s2 2−3s 2m2
Z
GSM
RL =−4(s−2m2
W)(m2
W−m2
Z)
3s 2(s−m2
Z)GSM
RR =8(s−m2
W)(−m2
W+m2
Z)
3s 2(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)
3s 2(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
6s2 2−6s 2m2
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)
3s 2(s−m2
Z)(−Cll[1221] −Cll[2112] + 2C(1)
ϕl[11] + 2C(3)
ϕl[22])),
(6.9)
whe e s= (p1+p2)2.
The SMEFT coe icien s con ibu ing a his o de a e
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 o e helici y ampli udes and a e aging o e he spin and colo we ha e
|ALO(s, )|2≡1
12ΣXY |GXY |2|MXY |2
≡1
12 |ALO |2.(6.11)
The pa onic c oss sec ions, a e aged o e spin and colo a e gi en by:
dˆσ
d =1
48πs2(|GLL |2+|GRR |2)u2+ (|GLR |2+|GRL |2) 2
ˆσLO =1
16πs2Z0
−s
d |ALO(s, )|2,(6.12)
whe e = (p1−p3)2and u= (p1−p4)2. The nume ical esul s o he ee le el
4− e mion ope a o s has been ex ensi ely s udied in he li e a u e [186,163,187,188,
189]. Acco ding o [190], he D ell Yan p ocess can gi e use ul in o ma ion no only
om he high-ene gy egime. Indeed, due o i s la ge c oss sec ion, i allows us o pu
signi ican cons ain s on he dimension-6 SMEFT coe icien s e en a ela i ely low
ene gies, p o iding a comp ehensi e unde s anding o he SMEFT con ibu ions o he
neu al DY p ocess. Indeed, e en a LHC ene gies, he one-loop EW con ibu ions o
Zdecays in he SMEFT can each O(10 −20)%. In p e ious wo k [191], whe e a se o
SMEFT con ibu ions o neu al D ell-Yan p ocess we e compu ed, i has been shown
ha HL-LHC will ha e signi ican sensi i i y o hese e ec s.
6.4 SMEFT NLO DY calcula ions:
Vi ual con ibu ions
In his sec ion, we p o ide a de ailed accoun o he calcula ion o i ual NLO co ec ions
wi hin he SMEFT o he D ell-Yan p ocess. The no a ion ollows ha o p e ious wo ks
110 CHAPTER 6. DRELL YAN PROCESS
a) b)
c) d) e)
Figu e 6.2: Samples o Feynman diag ams con ibu ing a NLO o DY p ocess. Realized
wi h [185]. a) Reno maliza ion coun e e ms, b) Ve ex,
c) Box, d) 4- e mion e ex, e) 4- e mion box.
[123,130,166,124]. The ele an ope a o s o he 1−loop ampli udes include hose
om Eq. 6.10 as well as addi ional ones lis ed 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)
whe e, as al eady discussed in Chap e 5, we no ice ha only ope a o s con aining pai s
o iden ical la o e mions can con ibu e. In his wo k, we will s a he explo a ion o
he e ec s in DY p oduc ion o 4- e mion ope a o s ha i s a ise a 1-loop o de as well
as he co ec ions o he 4- e mion ope a o s appea ing a LO. The i ual co ec ions
o D ell-Yan encoun e bo h UV and IR di e gences, which we handle by wo king in
d= 4−2ϵdimensions using he dimensional egula iza ion in oduced in Chap e 2. We
schema ize he con ibu ions o he i ual NLO co ec ions in o box, e ex, 4- e mion
e ex iangle, 4- e mion box iangle and eno maliza ion coun e e m con ibu ions,
as shown in Fig. 6.2. We no ice ha ou compu a ion includes only NLO co ec ions
ha in e e e wi h he LO ampli udes. Unlike [165], due o he 4 e mion ope a o s
inclusion, we see ha also opologies ha do no appea a he SM le el a e in ol ed in
he calcula ion.
We use FeynRules [192] ou ines o con e RξFeynman ules [100] o he SMEFT
in o a FeynA s [1] model ile, hen using FeynCalc [2] we compu e ampli udes and we
educe 1-loop in eg als o Passa ino-Vel man in eg als. We do no ix he gauge in he
SMEFT Lag angian and we e i y a he end he cancella ion o ξ e ms, in o de o
ha e a u he check on ou compu a ion. Excluding he eno maliza ion coun e e ms,
we de e mine he one loop con ibu ions by con ac ing NLO ampli udes wi h MXY ,
6.4. SMEFT NLO DY CALCULATIONS: VIRTUAL CONTRIBUTIONS 111
ac ing as p ojec o s in he massless e mion limi . To be consis en in he SMEFT
expansion, ou app oach equi es o conside Feynman diag ams wi h a mos a single
SMEFT ope a o inse ion. This ensu es eno malizabili y by a oiding di e gences ha
canno be coun e ed wi hou in oducing highe -o de ope a o s a leading o de . As
al eady discussed in Sec ion 2.4.1, he ea men o γ5in D dimension equi es pa icula
ca e. Indeed, he complex Di ac aces o he SMEFT aise di icul ies in he calcula ion
p ocess. To e alua e he Di ac aces, ou choice be ween he NDR o K eime scheme
depends on he speci ic class o Feynman diag ams illus a ed in Figu e 6.2.
Reno maliza ion coun e e ms
The p opaga o con ibu ions a e de i ed di ec ly om he SMEFT gauge boson and
e mion 2-poin unc ions [151,152]. In he eno maliza ion o he LO ampli ude, we
employed a mixed OS/MS scheme in oduced in Chap e 2. SM pa ame e s a e eno -
malized in he OS scheme, while SMEFT ope a o coe icien s a e ea ed as MS objec s.
We use he {Gµ, MZ, MW}scheme o inpu pa ame e s, wi h co ec ions o gauge boson
masses de ined as
M2
V=M2
0,V −ΠV V (M2
V),(6.14)
whe e V=Z, W, he 0 indica es he ba e quan i ies and ΠV V (M2
V) a e he 2-poin
unc ions o Re s. [151,152] compu ed on-shell. As in Chap e 5, he ela ion o Eq. 6.1
is modi ied a 1-loop,
Gµ+1
2√2Λ2Cll[2112] + Cll[1221]−√2
2Λ2C(3)
ϕl [11] + C(3)
ϕl [22]≡1
√2 0
(1 + ∆ )
(6.15)
whe e 0is he squa e o he minimum o he po en ial a ee le el and he analy ic
exp ession o ∆ in he SMEFT is gi en in Re . [123]. The e ec i e ield heo y coe -
icien s o he dimension-6 ope a o s a e ea ed as MS quan i ies, de ined a he scale
o he measu emen , i.e. he EW scale. The poles o he one-loop coe icien s Cia e
ex ac ed om Re s. [97,98,99],
Ci(µR) = C0,i −1
2ˆϵ
1
16π2γijCj,(6.16)
whe e µis he eno maliza ion scale, γij a e he one-loop anomalous dimensions,
µR
dCi
dµR
=1
16π2γijCj,(6.17)
and ˆϵ−1≡ϵ−1−γE+ log(4π).
Ve ex
Ve ex con ibu ions in ol e 1PI e ex ampli udes ha a e ex ac ed by con ac ing
ampli udes wi h MXY . We no ice ha he p esence o SMEFT ope a o s does no aise
112 CHAPTER 6. DRELL YAN PROCESS
he issue o he D-dimensional e alua ion o he Di ac ace, because a mos 4 Di ac
ma ices a e in ol ed. Thus, in his case we can use he NDR scheme when γ5is in ol ed
in he calcula ion.
Box
In his case we conside ed ”SM ype” box diag ams, meaning ha in his subsec ion
we s udied box opologies ha appea al eady a he SM le el. As in he p e ious case
we ex ac he box con ibu ions by con ac ing ampli udes wi h MXY , using hem as
”p ojec o s”. In his case, we ha e e mion chains ha in ol e up o 6 Di ac ma ices
and he e o e we use he K eime scheme using a consis en eading poin . As discussed
in Sec ion 2.4.3, he eading poin p esc ip ion e ec i ely speci ies wi h which Di ac
ma ix o s a he non-cyclic ace in he K eime scheme. Ou choice is o begin
eading he Di ac aces om he p ojec o inse ion.
4- e mion e ex iangle
The diag ams discussed in his subsec ion a e he NLO co ec ions o he 4- e mion
ope a o s appea ing a LO; in pa icula we conside ed he case whe e a gauge boson
is ”connec ing” 2 lep ons o 2 qua ks. The e mion chains appea ing in hese diag ams
ha e he some o m as hose in he e ex 6.4 and, a e p ojec ing h ough MXY , we
can use he NDR scheme o e alua e he Di ac aces.
4- e mion box iangle
Finally, we discuss NLO 4 e mion eynman diag ams wi h a gauge boson connec ing
di e en ype o e mions, e.g qua k and lep on. Acco ding o he K eime scheme
when we ha e mo e han 5 Di ac ma ices and we a e dealing wi h se e al axial cu en
we should employ a symme ic eading poin . Howe e , e en i we ha e se e al γ5
in ol ed in he calcula ions we a e handling open e mion chains, no closed e mion
loops. This means we can educe he e mion chains using he Di ac algeb a be o e we
p ojec h ough MXY . A e his ope a ion, he chains appea ing in hese diag ams will
ha e he same o m o 6.4. By he same easoning, we use he same eading poin o he
box diag ams, s a ing o ead ou Di ac aces om he p ojec o inse ion. Finally, we
no ice ha no employing a eading poin would no gi e us he co ec esul s o hese
diag ams, as expec ed. Indeed, as explici ly e i ied, his choice would lead o gauge
and chi al iola ing e ms.
6.5. SMEFT NLO DY CALCULATIONS: REAL CONTRIBUTIONS 113
Figu e 6.3: Real con ibu ions Feynman diag ams o DY p ocess. Realized wi h [185].
6.5 SMEFT NLO DY calcula ions:
Real Con ibu ions
The NLO esul equi es he eal con ibu ions om bo h pho on and gluon emission
qq →l+l−γ, qq →l+l−g . (6.18)
In Fig 6.3 we show a sample o Feynman diag ams con ibu ing a his o de . The
IR singula i ies a e egula ed using phase space slicing wi h dimensional egula iza ion
D= 4 −2ϵ, as in Re s. [193,194,195,196]. A e egula ing he IR singula i ies by
including he collinea and so limi s o he 2 →3 con ibu ions, ϵcan be se o 0.
As discussed in [197,198], he so limi s o he 2 →3 sca e ing p ocesses ha e a
uni e sal o m ha is he same o bo h he SM and he SMEFT. The so con ibu ion
is cha ac e ized by pho on o gluon ene gies wi h he condi ion Eγ, Eg<∆E, whe e
∆E ep esen s a chosen in ini esimally small cu o . The so pa onic c oss sec ion is
de ined in e ms o he lowes o de SMEFT c oss sec ion o Eq. 6.11,
dˆσso = = 1
16πs2Z0
−s
d |ALO(s, )|2αδEW
so (s, ) + αSCFδQCD
so (s).(6.19)
I is impo an o poin ou ha αis ede ined wi hin he con ex o he SMEFT. A
his o de , he ac o s in luencing αa ise om he Zmass de ini ion and he SU(2)/U(1)
mixing
α=√2Gµm2
W(m2
Z−m2
W)
πm2
Z−1
Λ2
m2
W
2m2
Zπm2
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 no ice ha no adjus men s a e in oduced due o ac o s like CϕW .
114 CHAPTER 6. DRELL YAN PROCESS
The so unc ions a e
δEW
so =Q2
q q(s) + Q2
l l(s) + 2QqQlh(s, )
δQCD
so = q(s)
(s) =
log µ2
4∆E2
πϵ +
2 log2µ2
4∆E2−π2
4π+1
πϵ2
h(s, ) =
1
2log2µ2
4∆E2−Li2s
+ 1+ log µ2
4∆E2log −s
−π2
4
π+
log µ2
4∆E2+ log −s
πϵ +1
πϵ2.
(6.21)
Fo so gluons, ake Ql= 0 and αQ2
q→αsCF. Adding he i ual one loop con ibu ions
and Eq. 6.21, he ϵ2dependences cancel, lea ing jus he ϵsingula e ms.
Conside pho ons emi ed om he ini ial qua k wi hin an angle δθ. The ini ial s a e
collinea con ibu ions a e abso bed in o he de ini ion o he PDFs [193]. The e o e, in
he MS scheme, we ha e
ˆσq
PDF =Q2
q
16πs2Z0
−s
d |A(s, )LO |2α δPDF (s)
δPDF (s) = 1
π
1
2log µ2
µ2
!4 log 2∆E
√s+ 3+
2 log 2∆E
√s+3
2
ϵ
,(6.22)
whe e µFis he ac o iza ion scale. The QCD con ibu ion is ound wi h he eplacemen
αQ2
q→αsCF.
When he pho ons a e emi ed om he inal s a e lep on,
ˆσl
coll =Q2
l
16πs2Z0
−s
d |A(s, )LO |2αδcl(s)
(6.23)
wi h δcl(s) gi en by
δcl(s) = 1
π1
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, pu ing hese e ms oge he we ha e:
ˆσa=1
16πs2
1
12 Z0
−s
d |ALO(s, ) + δANLO(s, )|2,(6.25)
whe e
6.6. PHENOMENOLOGY 115
200 400 600 800
Mll [GeV]
0
2
4
6
8
10
[(dσ (EFT) / dσ(SM)-1] (%)
Fla o less
U(2)5, Tqq
3[1221]=0
U(3)5, Tqq
3[1221]=0
√S=13.5 TeV, D ell-Yan
Fla o Scena ios, Tqq
3[1122]=1
C(3)
qq [1221]
C(3)
qq [1221]
C(3)
qq [1122]
Figu e 6.4: O(3)
qq in di e en la o assump ions, √S= 13.5 TeV
δANLO(s, ) = δA i (s, ) + 1
2ALO(s, )αδEW
so (s, ) + αsCFδQCD
so (s)
+(αQ2
q+αsCF)δPDF (s) + αδcl(s) (6.26)
and δA i (s, ) is he one loop eno malized ampli ude calcula ed in Sec . 6.4 and
as always ALO and αa e SMEFT quan i ies. We no e ha Eq. 6.26 is a ini e objec
and we a e ee o apply i a linea o quad a ic o de in he SMEFT ollowing Eqs.
6.3 and 6.4. Indeed, ˆσahas no dependence on ϵ, hus we can ake he limi ϵ→0.
6.6 Phenomenology
The analy ical esul s ob ained in his chap e a e going o be implemen ed in Mon eca lo
p og ams, in pa icula in POWHEG code, in o de o include also con ibu ions coming
om PDF ac o iza ion and ha d non-collinea con ibu ions om he 2 →3 p ocess.
Being an ongoing p ojec , we can no p esen a de ailed phenomenological analysis a
his s age. Howe e , we can p o ide a p elimina y s udy o he phenomenology in ol ing
4- e mion ope a o s i s appea ing a NLO. Fo he la o scena ios, we use he same
no a ion in oduced in Sec ion 5.3.
In Fig 6.4, we show ou esul s o O(3)
qq in di e en la o assump ions o √s=
13.5 TeV. Speci ically, we a e plo ing he pe cen age di e ence be ween he SMEFT
and SM di e en ial c oss sec ions as a unc ion o dilep on 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] (%)
Fla o less
U(2)5, Tqq
3[1221]=0
U(3)5, Tqq
3[1221]=0
D ell-Yan
Fla o Scena ios, Tqq
3[1122]=.1
Solid (do ed), ECM=13.5 (100) TeV
C(3)
qq [1221]
C(3)
qq [1221]
D ell Yan
C(3)
qq [1122]=0.1
Figu e 6.5: O(3)
qq di e en cen e -o -mass ene gies.
1 and se ing C(3)
qq [1221] = 0 we no ice ha in all he la o scena ios, he SMEFT
con ibu ion g ows wi h he dilep on ene gy, wi h U(3)5 eaching 9% while he la o less
and U(2)2 emaining below 4%.
In Fig 6.5, we compa e di e en cen e -o -mass ene gies: √s= 13.5 TeV, which
is he ene gy a which he LHC uns, and √s= 100 TeV, he expec ed ene gy o
he u u e collide Fcc-ee. We se C(3)
qq [1122] = 0.1 and C(3)
qq [1221] = 0. Due o he
linea dependence o he SMEFT coe icien on he c oss sec ion, his esul s in a simple
escaling o dσEFT. We see ha inc easing he ene gy scale doesn’ signi ican ly a ec
he con ibu ions in any o he la o scena ios. Howe e , we do obse e a sys ema ic
educ ion o a ew pe cen be ween he ene gy scales.
In Fig 6.6, we show he co ela ion be ween he coe icien s C(3)
qq [1122] and C(3)
qq [1221]
in he U(2)5scena io, assuming
dσEFT
dσSM −1>2% o 10%, Mll >500 GeV.(6.27)
Acco ding o [199], he cu en DY ela i e unce ain y o muon dilep on mass Mll >500
GeV is ≳19% . Unde hese condi ions, we can de ec BSM e en s in his channel only
i hey de ia e mo e han 19% om he SM. I is easonable o assume ha he p ecision
will inc ease a u u e collide s and ha his upg ade will allow us o measu e possible
de ia ions wi h highe sensi i i y. Fo his eason, in his analysis we conside ed wo
scena ios, a ”conse a i e” one a 10% and a mo e ”op imis ic” one a 2% .
We no ice ha , due o he linea app oxima ion, we can no se bounds on he 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
6s 2(s−m2
Z)
(−Cll[1221] −Cll[2112] + 2C(1)
ϕl[11] + 2C(3)
ϕl[22]))).
(A.2)
124 APPENDIX A.
A.3 EWPOs
O he plo s o 2- e mion ope a o s
In his subsec ion, we show a collec ion o plo s in ol ing he 2- e mion ope a o s, as
desc ibed in Chap e 5.
NLO:Solid;LO:Dashed
MFV
U(3)5
3 dGenCen
-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
3 dGenPh
3 dGenPh+U(2)5
Fla Less
-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
Figu e A.1: 95% CL limi s on C(3)
ϕq [ij] unde la o assump ions desc ibed in he ex .
Resul s a LO a e d awn wi h dashed lines, esul s a NLO a e d awn wi h solid lines.
On he le we p esen C(3)
ϕq [11] s. C(3)
ϕq [33] in he U(3)5(black), MFV (blue) and
3 d gene a ion cen ic (magen a) scena ios. In hese scena ios C(3)
ϕq [22] = C(3)
ϕq [11]. On
he igh we p esen C(3)
ϕq [11] s. C(3)
ϕq [22] in he 3 d gene a ion phobic (g een), 3 d
gene a ion phobic + U(2)5(o ange) and la o less ( iole ) scena ios. In he i s wo
scena ios C(3)
ϕq [33] = 0, while in he la o less scena io C(3)
ϕq [33] = C(3)
ϕq [22] = C(3)
ϕq [11]. All
o he coe icien s a e se o 0.
A.3. EWPOS 125
NLO:Solid;LO:Dashed
MFV
U(3)5
3 dGenCen
-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
3 dGenPh
3 dGenPh+U(2)5
Fla Less
-0.2 0.0 0.2 0.4
-0.2
0.0
0.2
0.4
Cϕu[1,1]
Cϕu[2,2]
95%CL
Figu e A.2: 95% CL limi s on Cϕu[ij] unde la o assump ions desc ibed in he ex .
Resul s a LO a e d awn wi h dashed lines, esul s a NLO a e d awn wi h solid lines.
On he le we p esen Cϕu[11] s. Cϕu[33] in he U(3)5(black), MFV (blue) and 3 d
gene a ion cen ic (magen a) scena ios. In hese scena ios Cϕu[22] = Cϕu[11], also no ice
ha o he MFV scena io only he NLO esul s can be ob ained. On he igh we
p esen Cϕu[11] s. Cϕu[22] in he 3 d gene a ion phobic (g een), 3 d gene a ion phobic
+U(2)5(o ange) and la o less ( iole ) scena ios. In he i s wo scena ios Cϕu[33] = 0,
while in he la o less scena io Cϕu[33] = Cϕu[22] = Cϕu[11]. All o he coe icien s a e
se o 0.
NLO:Solid;LO:Dashed
MFV
3 dGenCen
-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
3 dGenPh
3 dGenPh+U(2)5
Fla Less
-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
Figu e A.3: 95% CL limi s on Cϕd[ij] unde la o assump ions desc ibed in he ex .
Resul s a LO a e d awn wi h dashed lines, esul s a NLO a e d awn wi h solid lines.
On he le we p esen Cϕd[11] s. Cϕd[33] in he MFV (blue) and 3 d gene a ion cen ic
(magen a) scena ios. In hese scena ios Cϕd[22] = Cϕd[11]. On he igh we p esen
Cϕd[11] s. Cϕd[22] in he 3 d gene a ion phobic (g een), 3 d gene a ion phobic + U(2)5
(o ange) and la o less ( iole ) scena ios. In he i s wo scena ios Cϕd[33] = 0, while
in he la o less scena io Cϕd[33] = Cϕd[22] = Cϕd[11]. All o he coe icien s a e se o 0.
126 APPENDIX A.
NLO:Solid;LO:Dashed
MFV
3 dGenCen
-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
3 dGenPh
3 dGenPh+U(2)5
Fla Less
-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
Figu e A.4: 95% CL limi s on Cϕe[ij] unde la o assump ions desc ibed in he ex .
Resul s a LO a e d awn wi h dashed lines, esul s a NLO a e d awn wi h solid lines.
On he le we p esen Cϕe[11] s. Cϕe[33] in he MFV (blue) and 3 d gene a ion cen ic
(magen a) scena ios. In hese scena ios Cϕe[22] = Cϕe[11]. On he igh we p esen
Cϕe[11] s. Cϕe[22] in he 3 d gene a ion phobic (g een), 3 d gene a ion phobic + U(2)5
(o ange) and la o less ( iole ) scena ios. In he i s wo scena ios Cϕe[33] = 0, while in
he la o less scena io Cϕe[33] = Cϕe[22] = Cϕe[11]. All o he coe icien s a e se o 0.
NLO:Solid;LO:Dashed
MFV
3 dGenCen
-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
3 dGenPh
3 dGenPh+U(2)5
Fla Less
-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
Figu e A.5: 95% CL limi s on C(1)
ϕl [ij] unde la o assump ions desc ibed in he ex .
Resul s a LO a e d awn wi h dashed lines, esul s a NLO a e d awn wi h solid lines.
On he le we p esen C(1)
ϕl [11] s. C(1)
ϕl [33] in he MFV (blue) and 3 d gene a ion
cen ic (magen a) scena ios. In hese scena ios C(1)
ϕl [22] = C(1)
ϕl [11]. On he igh we
p esen C(1)
ϕl [11] s. C(1)
ϕl [22] in he 3 d gene a ion phobic (g een), 3 d gene a ion phobic
+U(2)5(o ange) and la o less ( iole ) scena ios. In he i s wo scena ios C(1)
ϕl [33] = 0,
while in he la o less scena io C(1)
ϕl [33] = C(1)
ϕl [22] = C(1)
ϕl [11]. All o he coe icien s a e
se o 0.
A.3. EWPOS 127
NLO:Solid;LO:Dashed
MFV
3 dGenCen
-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
3 dGenPh
3 dGenPh+U(2)5
Fla Less
-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
Figu e A.6: 95% CL limi s on C(3)
ϕl [ij] unde la o assump ions desc ibed in he ex .
Resul s a LO a e d awn wi h dashed lines, esul s a NLO a e d awn wi h solid lines.
On he le we p esen C(3)
ϕl [11] s. C(3)
ϕl [33] in he MFV (blue) and 3 d gene a ion
cen ic (magen a) scena ios. In hese scena ios C(3)
ϕl [22] = C(3)
ϕl [11]. On he igh we
p esen C(3)
ϕl [11] s. C(3)
ϕl [22] in he 3 d gene a ion phobic (g een), 3 d gene a ion phobic
+U(2)5(o ange) and la o less ( iole ) scena ios. In he i s wo scena ios C(3)
ϕl [33] = 0,
while in he la o less scena io C(3)
ϕl [33] = C(3)
ϕl [22] = C(3)
ϕl [11]. All o he coe icien s a e
se o 0.
128 APPENDIX A.
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Lis o publica ions
In he ollowing pages, we show all he ele an in o ma ion abou he publica ions used
in he hesis. We ha e ep oduced pa o hei con en in Chap e s 3and 5.
Gluon Fusion P oduc ion a NLO:
Me ging he T ans e se Momen um and he High-Ene gy Expansions
Au ho s
Luigi Bella on ea, Giuseppe Deg assib, Pie Paolo Gia dinoa,
Ramona G ¨obe c, Ma co Vi ib
(a) Ins i u o Galego de F´ısica de Al as Ene x´ıas, Uni e sidade de San iago de Compos ela, 15782 San iago de Compos ela, Galicia-Spain
(b) Dipa imen o di Ma ema ica e Fisica, Uni e si `a di Roma T e and INFN, sezione di Roma T e, I-00146 Rome, I aly
(c) Dipa imen o di Fisica e As onomia ’G. Galilei’, Uni e si `a di Pado a and INFN, sezione di Pado a, I-35131 Pado a, I aly
PhD S uden Con ibu ion
Collabo a ion in he calcula ions p esen ed in he pape , as well as in he discussions and w i ing.
Used in Chap e : 3
Jou nal and A icle In o ma ion
Jou nal name: Jou nal o High Ene gy Physics
Publishe : Sp inge Na u e
ISSN: 1029-8479
Yea o publica ion: 2022
DOI: 10.1007/JHEP07(2022)069
Impac ac o in 2022: 5.4
The a icle is dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License (CC-BY 4.0), which pe mi s
any use, dis ibu ion and ep oduc ion in any medium, p o ided he o iginal au ho (s) and sou ce a e c edi ed.
145
146 Lis o publica ions
The Impo ance o Fla o in SMEFT Elec oweak P ecision Fi s
Au ho s
Luigi Bella on ea, Sally Dawsonb, Pie Paolo Gia dinoa,
(a) Ins i u o Galego de F´ısica de Al as Ene x´ıas, Uni e sidade de San iago de Compos ela, 15782 San iago de Compos ela, Galicia-Spain
(b) High Ene gy Theo y G oup, Depa men o Physics, B ookha en Na ional Labo a o y, Up on, NY 11973, USA
PhD S uden Con ibu ion
Collabo a ion in he calcula ions p esen ed in he pape , as well as in he discussions and w i ing.
Used in Chap e : 5
Jou nal and A icle In o ma ion
Jou nal name: Jou nal o High Ene gy Physics
Publishe : Sp inge Na u e
ISSN: 1029-8479
Yea o publica ion: 2023
DOI: 10.1007/JHEP05(2023)208
Impac ac o in 2022: 5.4
The a icle is dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License (CC-BY 4.0), which pe mi s
any use, dis ibu ion and ep oduc ion in any medium, p o ided he o iginal au ho (s) and sou ce a e c edi ed.
Pe missions o con en euse
In Fig 3.1 we show plo s om [15,16]. The ep oduc ion o hese con en s is allowed
unde he C ea i e Commons A ibu ion License (CC-BY 4.0), which pe mi s any use,
dis ibu ion and ep oduc ion in any medium, p o ided he o iginal au ho (s) and sou ce
a e c edi ed.
All he o he igu es p esen ed in his hesis a e o iginal wo ks o he au ho o hey
ep oduce con en s om he pape s men ioned in A.3. As al eady s a ed, he use hese
igu es is allowed unde he C ea i e Commons A ibu ion License (CC-BY 4.0).
147
148 Pe missions o con en euse
