Feynman rules for the Standard Model Effective Field Theory in R ξ -gauges

Transcription

1 Publised for SISSA by Springer Received: May 3, 017 Accepted: June 10, 017 Publised: June 7, 017 Feynman rules for te Standard Model Effective Field Teory in R ξ -gauges A. Dedes, a W. Materkowska, b M. Paraskevas, a J. Rosiek b and K. Suxo a a Department of Pysics, Division of Teoretical Pysics, University of Ioannina, Ioannina GR-45110, Greece b Faculty of Pysics, University of Warsaw, Pasteura 5, Warsaw, Poland adedes@cc.uoi.gr, weronika.materkowska@fuw.edu.pl, mparask@grads.uoi.gr, janusz.rosiek@fuw.edu.pl, csoutzio@cc.uoi.gr Abstract: We assume tat New Pysics effects are parametrized witin te Standard Model Effective Field Teory SMEFT written in a complete basis of gauge invariant operators up to dimension 6, commonly referred to as Warsaw basis. We discuss all steps necessary to obtain a consistent transition to te spontaneously broken teory and several oter important aspects, including te BRST-invariance of te SMEFT action for linear R ξ -gauges. Te final teory is expressed in a basis caracterized by SM-like propagators for all pysical and unpysical fields. Te effect of te non-renormalizable operators appears explicitly in triple or iger multiplicity vertices. In tis mass basis we derive te complete set of Feynman rules, witout resorting to any simplifying assumptions suc as baryon-, lepton-number or CP conservation. As it turns out, for most SMEFT vertices te expressions are reasonably sort, wit a noticeable exception of tose involving 4, 5 and 6 gluons. We ave also supplemented our set of Feynman rules, given in an appendix ere, wit a publicly available Matematica code working wit te FeynRules package and producing output wic can be integrated wit oter symbolic algebra or numerical codes for automatic SMEFT amplitude calculations. Keywords: Effective Field Teories, Beyond Standard Model ArXiv eprint: Open Access, c Te Autors. Article funded by SCOAP 3. ttps://doi.org/ /

2 Contents 1 Introduction and motivation Notation and conventions for te SMEFT Lagrangian 4 3 Mass eigenstates basis in SMEFT Higgs mecanism 5 3. Te gauge sector Gauge-Goldstone mixing Fermion sector 10 4 Corrections to te SM couplings 10 5 Gauge fixing and FP-gosts in R ξ -gauges 14 6 Feynman rules and Matematica implementation 17 7 Conclusions 18 A SMEFT Feynman rules 19 A.1 Propagators in te R ξ -gauges 1 A. Lepton-gauge vertices A.3 Lepton-Higgs-gauge vertices 3 A.4 Quark-gauge vertices 35 A.5 Quark-Higgs-gauge vertices 37 A.6 Quark-gluon vertices 5 A.7 Higgs-gauge vertices 55 A.8 Gauge-gauge vertices 84 A.9 Higgs-gluon vertices 90 A.10 Gluon-gluon vertices 9 A.11 Four-fermion vertices 93 A.1 Lepton and baryon number violating vertices 97 A.13 Gost vertices 99 B SMEFT Matematica package for FeynRules 10 1

3 1 Introduction and motivation After te discovery of te Higgs boson, te picture of te Standard Model SM [1 3] being a spontaneously broken gauge teory at an Electroweak Scale EW wit v 46GeV as been teoretically establised and experimentally confirmed to a significant accuracy. Neverteless, new pysics beyond te SM may be idden in te experimental errors of measurements tat are becoming increasingly accurate at te LHC. Suc penomena can be parametrized in terms of te so-called SM Effective Field Teory SMEFT [4 6], 1 were, assuming Λ to be te typical energy scale of te SM extension, te observable effects are suppressed by powers of te expansion parameter v/λ. Te SM s weak response to a more fundamental teory effective or not living at Λ may be due to te fact tat suc a scale is far above te EW scale i.e., Λ v, or because non-renormalizable, UV-dependent couplings are, someow, small. Besides te verification of te SM gauge group and content, a renewed interest in te SMEFT arises from te fairly recent completion of all gauge invariant, independent, mass dimension-6 operators, first conducted in a study by Bucmüller and Wyler [10] in 1985 and lately amended by te Warsaw university group [11] in 010. We sall refer to tis set of operators as te Warsaw basis. In tis basis tere are 59+1 baryon-number conserving and 4 baryon-number violating operators. If pysics beyond te SM lies not too far from te EW scale, so tat is invisible, but also not too close to te EW scale, so tat te effective field teory description EFT does not fail, ten SMEFT observables sould encode possible deviations from te SM to order v/λ no matter wat te fundamental UV teory is. A serious attempt in calculating suc observables sould start by first writing down te Feynman rules for propagators and vertices for pysical fields, after spontaneous symmetry breaking SSB of te effective teory, in a way tat consistently renders te teory renormalizable in te modern sense - ere of absorbing infinities into a finite number of counterterms up to order v/λ. One major criterion for tis to be realized is tat te gauge boson propagators vanis for momenta p as p so tat te teory satisfies usual power counting rules for renormalizability, as in te SM for example. In 1971, t Hooft [13] and B. Lee [14] sowed tat tis can be realized in a linear gauge wic a year later extended to a larger class of renormalizable gauges by Fujikawa, Lee, Sanda [15], and Yao [16]. Tis class of renormalizable gauges, called R ξ -gauges, can be parametrized by one or more arbitrary constants, collectively written as ξ. In addition to te smoot beavior of te propagators, R ξ -gauges allow for eliminating unwanted mixed terms between pysical gauge bosons and unpysical Goldstone scalar fields in spontaneously broken gauge teories. To te best of our knowledge, quantization of SMEFT in linear R ξ -gauges does not exist in te literature tus far. Wat complicates te picture of quantization in R ξ -gauges, or as a matter of fact in every oter class of gauges, is twofold: a field redefinitions and reparametrizations and b mixed field strengt operators. A careful treatment of te 1 For reviews see refs. [7 9]. In counting, we include te lepton-number d = 5 violating operator [1] but do not count ermitian conjugated operators and suppress fermion flavor dependence.

4 former to retain gauge invariance is necessary [17] wile properly rotating away but not completely eliminating from vertices te latter, results in SM-like propagators for pysical and unpysical fields. More specifically, in tis paper we consider SSB of te Warsaw basis teory and present a full set of Feynman rules in R ξ -gauge in a mass basis, wit te following features: No restriction is made for te structure of flavor violating terms and for CP-, leptonor baryon-number conservation, SMEFT is quantized in R ξ -gauges written wit four different arbitrary gauge parameters, ξ γ,ξ Z,ξ W,ξ G for better cross cecks of pysical amplitudes. Gauge fixing and gost part of te Lagrangian is cosen to be SM-like and preserve Becci, Rouet, Stora [18], and Tyutin [19] BRST invariance. All bilinear terms in te Lagrangian ave canonical form, bot for pysical and unpysical Goldstone and gost fields; all propagators are diagonal and SM-like. Feynman rules for interactions are expressed in terms of pysical SM fields and canonical Goldstone and gost fields. We are aware tat in te literature tere are many calculations done already witin SMEFT, including several articles wit loop calculations usually performed in unitary or non-linear gauges, see for example ref. [9] and references terein. However, we tink tat a full set of Feynman rules written and coded in te symbolic computer program in te R ξ -gauges, including in addition te most general structure of te flavor violating terms, is someting tat can largely simplify furter suc analyses. Especially, aving suc collection is useful because te number of primary vertices in SMEFT in R ξ -gauges is uge: 383 witout counting te ermitian conjugates surprisingly, for most SMEFT vertices te Feynman rules are reasonably sort, wit an exception of self-interactions of 4, 5 and 6 gluons. An explicit diagrammatic representation for all interaction vertices will minimize possible mistakes tat arise from missing terms or even entire diagrams in amplitude calculations. Furtermore, implementation of tem as a model file to te FeynRules package[0] produces an output ready to be furter used in symbolic or numeric programs for amplitude calculations. Te procedure we followed in deriving te SMEFT Feynman rules consists of te following steps: 3 1. witin te Warsaw basis, given for reference in section, we perform te SSB mecanism and furter field and coupling rescalings wit constant parameters wic ave no effect on te S-matrix elements up to OΛ 3 corrections. Tey make all bilinear terms of gauge, Higgs and fermion fields canonical [section 3],. we discuss oblique corrections to te SM vertices, coming from te constant field and coupling redefinitions wen moving from weak to mass basis [section 4], 3 Steps 1 and ave been discussed in numerous earlier papers e.g., ref. [1], but we include tem ere for completeness and consistency. 3

5 field l j Lp = ν Lp e Lp e Rp fermions q αj Lp = u α Lp d α Lp scalars u α Rp d α Rp ϕ j ϕ + = ϕ 0 ypercarge Y Table 1. Te SM matter content in te gauge basis. Isospin, colour and generation indices are indicated wit j = 1,, α = and p = 1...3, respectively. 3. we introduce suitable R ξ -gauge fixing and gost terms in te Lagrangian, in a way tat renders also te gost propagators diagonal. Te new terms eliminate te unwanted gauge-goldstone mixing and establis BRST invariance. Tus, in te mass basis of SMEFT all quadratic terms of pysical SM particles and unpysical Goldstone bosons and gosts become SM-like [section 5], 4. we evaluate Feynman rules for all sectors of te teory in R ξ -gauges. [appendix A]. Ten, in section 6 and in appendix B we describe te features of te SMEFT model file for FeynRules package and a set of programs generating automatically relevant Feynman rules, bot in Matematica and Latex/axodraw format. We conclude in section 7. Notation and conventions for te SMEFT Lagrangian Trougout tis article we use te notation and conventions of ref. [11]. However, in order to distinguis between te fields and parameters of te initial, gauge basis and te final, mass basis, we use primed notation for fermion fields and teir Wilson coefficients in te former, reserving te unprimed symbols for te pysical mass eigenstates basis, were flavor space rotations ave been performed. In addition, and not to clutter te notation furter as compared to ref. [11], we absorb te teory cut-off scale Λ in te definitions of Wilson coefficients, rescaling tem appropriately as C 5 X /Λ C5 X, C6 X /Λ C 6 X. For completeness and reference, in tables and 3 we list all, gauge independent, dimension-6 operators of te Warsaw basis derived in ref. [11]. Te only dimension- 5 operator, te lepton-number violating operator [1], reads Q νν = ε jk ε mn ϕ j ϕ m l k Lp T Cl n Lr ϕ l Lp T C ϕ l Lr,.1 were C is te carge conjugation matrix in notation of ref. [11]. Ten te full gauge invariant Lagrangian, up to OΛ 3, takes te form L = L 4 SM +Cνν Q 5 νν + X C X Q 6 X + f C f Q 6 f,. were Q 6 X denotes dimension-6 operators tat do not involve fermion fields, i.e., operators entitled as X 3,ϕ 6,ϕ 4 D,X ϕ columns of table, wile Q 6 f denotes operators tat 4

6 contain fermion fields among oter fields i.e., all oter operators in tables and 3. Te renormalizable part of te Lagrangian is we suppress generation indices ere, L 4 SM = 1 4 GA µνg Aµν 1 4 WI µνw Iµν 1 4 B µνb µν +D µ ϕ D µ ϕ+m ϕ ϕ 1 λϕ ϕ + i l L /Dl L +ē R /De R +q L /Dq L +ū R /Du R + d R /Dd R l LΓ e e Rϕ+ q LΓ u u R ϕ+ q LΓ d d Lϕ..3 As compared to ref. [11] we sligtly cange te notation for te gauge group generators wile keeping all oter conventions identical. Te covariant derivative ten reads, D µ = µ +ig B µ Y +igw I µt I +ig s G A µt A,.4 were te weak ypercarge Y assigned to te fields is given in table 1. In fundamental representation, te generators for SU read T I = τ I / wit τ I I=1,,3 being te Pauli matrices and for SU3 read T A = λ A / wit λ A A=1,...,8 being te Gell-Mann matrices. Te field strengt tensors are: G A µν = µ G A ν ν G A µ g s f ABC G B µg C ν,.5 W I µν = µ W I ν ν W I µ gǫ IJK W J µw K ν,.6 B µν = µ B ν ν B µ..7 Finally, we consider te SMEFT accurate up to OΛ 3 corrections and terefore all relations obtained witin it are accurate up to tis level of approximation. We will implicitly make use of tis property in our derivations witout making any furter notice. 3 Mass eigenstates basis in SMEFT As usual, in order to identify pysical and unpysical degrees of freedom in te presence of SSB, one needs to diagonalize te resulting mass matrices for all fields. However, in SMEFT tere is an extra intermediate step involving field rescalings, since SSB also affects te canonical normalization of te kinetic terms. In te following sections we discuss tis procedure step by step. 3.1 Higgs mecanism Te relevant operator terms contributing to te Higgs potential are L H = D µ ϕ D µ ϕ+m ϕ ϕ λ ϕ ϕ +C ϕ ϕ ϕ 3 + C ϕ ϕ ϕ ϕ ϕ + C ϕd ϕ D µ ϕ ϕ D µ ϕ. 3.1 Minimization of te potential results in a corrected vacuum expectation value vev, wic reads [1], m v = λ + 3m3 λ 5/ Cϕ. 3. 5

7 X 3 ϕ 6 and ϕ 4 D ψ ϕ 3 Q G Q G Q W Q W f ABC G Aν µ G Bρ ν G Cµ ρ Q ϕ ϕ ϕ 3 Q eϕ ϕ ϕ l pe rϕ fabc GAν µ G Bρ ν G Cµ ρ Q ϕ ϕ ϕ ϕ ϕ Q uϕ ϕ ϕ q pu r ϕ ε IJK W Iν µ W Jρ ν W Kµ ρ Q ϕd ϕ D µ ϕ ϕ D µ ϕ Q dϕ ϕ ϕ q pd rϕ ε IJK WIν µ Wν Jρ Wρ Kµ X ϕ ψ Xϕ ψ ϕ D Q ϕg ϕ ϕg A µνg Aµν Q ew l pσ µν e rτ I ϕw I µν Q 1 ϕl ϕ i D µ ϕ l pγ µ l r Q ϕ G ϕ ϕ G A µνg Aµν Q eb l pσ µν e rϕb µν Q 3 ϕl ϕ i D I µ ϕ l pτ I γ µ l r Q ϕw ϕ ϕw I µνw Iµν Q ug q pσ µν T A u r ϕg A µν Q ϕe ϕ i D µ ϕē pγ µ e r Q ϕ W ϕ ϕ W I µνw Iµν Q uw q pσ µν u rτ I ϕw I µν Q 1 ϕq ϕ i D µ ϕ q pγ µ q r Q ϕb ϕ ϕb µν B µν Q ub q pσ µν u r ϕb µν Q 3 ϕq ϕ i D I µ ϕ q pτ I γ µ q r Q ϕ B ϕ ϕ B µν B µν Q dg q pσ µν T A d rϕg A µν Q ϕu ϕ i D µ ϕū pγ µ u r Q ϕwb ϕ τ I ϕw I µνb µν Q dw q pσ µν d rτ I ϕw I µν Q ϕd ϕ i D µ ϕ d pγ µ d r Q ϕ WB ϕ τ I ϕ W I µνb µν Q db q pσ µν d rϕb µν Q ϕud i ϕ D µ ϕū pγ µ d r Table. Dimension-6 operators oter tan te four-fermion ones from ref. [11]. For brevity we suppress fermion ciral indices L, R. Notice tat in all our expressions and Feynman rules tat follow we use only tis vev. As usual, we next expand te Higgs doublet field around te vacuum, Φ + ϕ = v +H +iφ 0 Te Lagrangian bilinear terms of te scalar fields are ten given by, L Bilinear H = CϕD v C ϕ v µ H By rescaling te fields as wit te constant factors 1 m 3 4 λv v4 C ϕ H 1+ 1 CϕD v µ Φ 0 + µ Φ µ Φ = Z H, = Z Φ 0, G ± Φ ±, 3.5 Z CϕD v C ϕ v, 3.6 Z CϕD v, 3.7 6

8 LL LL RR RR LL RR Q ll l pγ µl r l sγ µ l t Q ee ē pγ µe rē sγ µ e t Q le l pγ µl rē sγ µ e t Q 1 qq q pγ µq r q sγ µ q t Q uu ū pγ µu rū sγ µ u t Q lu l pγ µl rū sγ µ u t Q 3 qq q pγ µτ I q r q sγ µ τ I q t Q dd d pγ µd r d sγ µ d t Q ld l pγ µl r d sγ µ d t Q 1 lq l pγ µl r q sγ µ q t Q eu ē pγ µe rū sγ µ u t Q qe q pγ µq rē sγ µ e t Q 3 lq l pγ µτ I l r q sγ µ τ I q t Q ed ē pγ µe r d sγ µ d t Q 1 qu q pγ µq rū sγ µ u t Q 1 quqd Q 8 quqd Q 1 ud ū pγ µu r d sγ µ d t Q 8 qu q pγ µt A q rū sγ µ T A u t Q 8 ud ū pγ µt A u r d sγ µ T A d t Q 1 qd q pγ µq r d sγ µ d t Q 8 qd q pγ µt A q r d sγ µ T A d t LR RL and LR LR B-violating [ ][ ] Q ledq l j p e r d sq j t Q duq ε αβγ ε jk d α p T Cu β r q γj s T Cl k t [ ][ ] q j p u rε jk q k s d t Q qqu ε αβγ ε jk q αj p T Cq βk r u γ s T Ce t [ ][ q j p T A u rε jk q k s T A d t Q qqq ε αβγ ε jnε km q αj p T Cq βk r q γm s T Cl n [ ][ ] Q 1 lequ l p j e rε jk q s k u t Q duu ε αβγ d α p T Cu β r u γ s T Ce t Q 3 lequ l j p σ µνe rε jk q k s σ µν u t Table 3. Four-fermion operators from ref. [11]. For brevity we suppress fermion ciral indices L,R. one obtains te pysical Higgs field and Goldstone fields,g ± wit canonically normalized kinetic terms. Te tree-level squared mass of te normalized Higgs field now reads, [ M = m 1 m 3C ϕ λ 4λC ϕ +λc ϕd ] = λv 3C ϕ λc ϕ + λ CϕD v t ] 3. Te gauge sector Te Lagrangian terms wic are relevant for gauge boson propagators read, L EW = 1 4 WI µνw Iµν 1 4 B µνb µν +D µ ϕ D µ ϕ +C ϕw ϕ ϕwµνw I Iµν +C ϕb ϕ ϕb µν B µν +C ϕwb ϕ τ I ϕwµνb I µν +C ϕd ϕ D µ ϕ ϕ D µ ϕ, 3.9 L QCD = 1 4 GA µνg Aµν +C ϕg ϕ ϕg A µνg Aµν, 3.10 were τ I are te Pauli matrices. Oter, potentially relevant operators of te teory, containing B µν, W I µν and G A µν influence only CP-violating vertices. Teir bilinear terms are total derivatives and do not affect propagators. Terefore, we neglect tem in our discussion ere. 7

9 To simplify te above expressions, it is convenient to introduce barred fields and couplings, suc as W I µ Z g W I µ, ḡ Z 1 g g, B µ Z g B µ, ḡ Z 1 g g, Ḡ A µ Z gs G A µ, ḡ s Z 1 g s g s, were for our constant, field and coupling rescalings, we coose Z g 1C ϕw v, Z g 1C ϕb v, 3.11 Z gs 1C ϕg v. We note tat suc transformations do not violate gauge invariance. Tey preserve te form of te covariant derivative wic now reads, D µ = D µ = µ +iḡ B µ Y +iḡ W I µt I +iḡ s Ḡ A µt A, 3.1 wile te field strengt tensors rescale te same way as teir respective fields. Te particular coice of eq renders te kinetic terms for te electroweak fields canonical, wit an exception of te mixed Q ϕwb operator in eq Furtermore, te last redefinition of eq is sufficient to define massless pysical, canonically normalized gluon fields, as g A µ ḠA µ In terms of barred electroweak gauge bosons, Bµ and W µ, te bilinear part of te Lagrangian reads, L Bilinear EW = 1 4 W 1 µν W 1µν + W µν W µν 1 4 +ḡ v 8 W µ 1 W 1µ + W µ W µ + v W3 µ ḡ ḡḡ 8 Z B µ ḡḡ ḡ W3 µν B µν W3µ B µ 1 ǫ ǫ 1 W3µν B µν, 3.14 were we ave defined, ǫ C ϕwb v From eq one identifies immediately te pysical carged gauge bosons W ± µ, as W ± µ = 1 W 1 µ i W µ, 3.16 wit te mass M W = 1 ḡv

10 Te neutral gauge boson mass basis is obtained troug te congruent matrix transformation [], producing simultaneously canonical kinetic terms and diagonal masses. It reads, W3 µ B µ = X Zµ A µ, 3.18 wit te matrix X taking te form, X = 1 ǫ cos θ sin θ ǫ 1 sin θ cos θ Straigtforward calculation leads to a mixing angle [1, 3] tan θ = ḡ ḡ + ǫ 1 ḡ ḡ, 3.0 wereas for gauge boson masses we obtain M Z = 1 ḡ +ḡ v 1+ ǫḡḡ ḡ Z +ḡ, M A = One can easily verify tat te poton remains massless from te vanising determinant of te mass matrix in eq Note also tat te X transformation affects te trace of tis matrix, tus producing te ǫ-dependence for M Z. 3.3 Gauge-Goldstone mixing Te operators relevant for Goldstone bosons kinetic terms give also rise to Goldstone-gauge boson mixing. Tey read, L H D µ ϕ D µ ϕ + C ϕd ϕ Dµ ϕ ϕ Dµ ϕ, 3. wic, in te presence of SSB, generate te unwanted terms L GEW = i ḡv W µ 1 µ Φ + µ Φ + ḡv W µ µ Φ + + µ Φ ḡv Z W3 µ µ Φ 0 + ḡ v Z Bµ µ Φ After expressing L GEW in terms of te pysical gauge bosons and Goldstone bosons, one arrives to te familiar expression, L GEW = im W W + µ µ G W µ µ M Z Z µ µ. 3.4 Tus, in mass basis all Wilson coefficients in te bilinear gauge-goldstone mixing ave been absorbed in te definitions of fields and masses. As we discuss in section 5, suc a property essentially allows to adopt te standard R ξ -gauge fixing also for SMEFT loop calculations. 9

11 3.4 Fermion sector Te operators relevant to fermion masses are L f = i l L / Dl L +ē R / De R + q L / Dq L +ū R / Du R + d R / Dd R l LΓ e e Rϕ+ q LΓ u u R ϕ+ q LΓ d d Rϕ+H.c. [ ] + ϕ ϕ l LC eϕ e Rϕ+ϕ ϕ q LC uϕ u R ϕ+ϕ ϕ q LC dϕ d Rϕ+H.c. [ ] + C νν ϕ l L T C ϕ l L+H.c., 3.5 were Γ e,u,d and C eϕ,c uϕ,c dϕ are general complex 3 3 matrices, C νν is a symmetric complex 3 3 matrix and primed fields denote te fields in te interaction gauge basis group and generation indices are suppressed. Te fermion kinetic terms remain unaffected by SSB, wile te mass terms read L mass = 1 ν T L CM ν ν L ē LM ee R ū LM uu R d LM d d R +H.c., 3.6 wit te 3 3 mass matrices equal to M ν = v C νν, M e = v Γ e C eϕ v M u = v Γ u C uϕ v, M d = v Γ d C dϕ v,. 3.7 To diagonalize lepton and quark masses we rotate te fermion fields by te unitary matrices, ψ X = U ψx ψ X, 3.8 wit ψ = ν,e,u,d, X = L,R and te unprimed symbols denoting te mass eigenstates fields. Ten, te singular value decomposition for carged fermion mass matrices results in U e L M eu er = M e = diagm e,m µ,m τ, U u L M uu ur = M u = diagm u,m c,m t, 3.9 U d L M d U d R = M d = diagm d,m s,m b, wile te diagonal neutrino mass matrix is obtained troug U T ν L M ν U νl = M ν = diagm ν1,m ν,m ν3, 3.30 wit all fermion masses now being real and non-negative. 4 Corrections to te SM couplings Corrections to te interactions described by te dimension-4 SM Lagrangian can come eiter as genuine new vertices generated by iger order operators, or from te dimension- 4 vertices modified by te sifts in te fields and parameters necessary to express tem 10

12 in te mass eigenstates basis. In tis section we discuss te second class of oblique corrections. In terms of pysical gauge bosons, te electroweak part of te covariant derivative its QCD part parametrized in terms of ḡ s -coupling is uncanged compared to te SM, reads D EW µ = µ +i ḡ T + W + µ +T W µ +iḡx 11 T 3 +ḡ X 1 YZ µ +iḡx 1 T 3 +ḡ X YA µ. 4.1 Te pattern of electroweak symmetry breaking results in a conserved electric carge, identified troug te standard relation Q = T 3 +Y. Te electromagnetic gauge invariance of te broken teory manifests troug te corrected electroweak unification condition, ē = ḡ X = ḡ X 1, 4. wic couples te poton only to te electric carge wile keeping it massless. Using eq. 4. and te property det X = 1 one can always express te covariant derivative in te familiar form, D EW µ = µ +i ḡ T + W + µ +T W µ +iḡ Z T 3 sin θ QZµ +iēqa µ, 4.3 were te modified couplings now read, ḡḡ ē = ḡ 1 ǫḡḡ +ḡ ḡ, +ḡ ḡ Z = ḡ +ḡ 1+ ǫḡḡ ḡ +ḡ. 4.4 In summary, after redefinitions of fields and couplings in mass basis, corrections to gauge interactions originating from te sift in te gauge and Higgs sector parameters depend only on two additional Wilson coefficients: C ϕwb, responsible for te mixing of electroweak gauge boson kinetic terms, and, C ϕd appearing troug te pysical Z 0 -boson mass see eq Furtermore, te C ϕd operator breaks te custodial invariance as tis is described by te anomalous value of te ρ parameter, ρ = J C.C J N.C. = ḡ M Z ḡ Z M W = 1+ 1 CϕD v. 4.5 Asitiswellknown, tisisstronglyconstrainedbyprecisionewexperiments, attelevelof 0.1% [4]. Consequently, sizable oblique corrections in te gauge sector could potentially arise only from te gauge boson kinetic mixing ǫ defined in eq Anoter set of oblique corrections originates in te flavor sector of SMEFT after diagonalization of te fermion mass matrices [see section 3.4]. In te SM, only te products U u L U dl and U e L U νl appear in te carged quark and lepton current couplings after flavor rotations: tey are identified as te CKM [5] and PMNS [6, 7] mixing matrices, 11

13 respectively. However, in SMEFT te fermion-fermion-w ± couplings contain additional contributions from operators, witout affecting te fermion bilinear terms of te model. Te relevant part of te Lagrangian as te form: { L c.c. = ḡ [ } W µ + ū p γ µ U u L 1+v C v ϕq3 U dl ]pr P L + U u R C ϕud U dr d r pr ḡ W + µ ν p γ µ [ U e L 1+v C ϕl3 U νl ] pr P L e r +H.c. 4.6 As a result, one can identify te pysical, altoug not unitary any more, mixing matrices for quark and leptons, troug: K CKM K U u L 1+v C ϕq3 U dl, 4.7 U PMNS U U e L 1+v C ϕl3 U νl. 4.8 In wat follows we also redefine te Wilson coefficients of te operators involving fermionic currents, by absorbing into tem te fermion flavor rotations from gauge to te mass basis. In tis way we are able to express te mass basis Lagrangian entirely in terms of te unprimed fields, Wilson-coefficients K-, and U-mixing matrices. In some cases te redefinitions are not unique, as in te operators involving left fermion SU doublets one can adsorb into te Wilson coefficient eiter te rotation matrix of te lower or upper constituent of te doublet. We coose it always to be te lower field e L or d L rotation, as in tis way flavor violating K or U matrices appear explicitly in less experimentally constrained u-quark or neutrino couplings see also discussion in ref. [8]. Our redefinitions are collected in table 4. Finally, Higgs boson interactions wit fermions are affected by te transition to te pysical mass eigenstates bot universally, due to te cange of Higgs-boson normalization in eq. 3.6, and in a flavor dependent way, due to te modified relation in eq. 3.7 between fermion masses and te Yukawa couplings. Te Higgs-fermion-fermion interaction Lagrangian in mass basis is, L ψψ = ē ū d [ Me v [ Mu v [ Md v 1 14 CϕD v +C ϕ v C eϕ v ] e+h.c CϕD v +C ϕ v C uϕ v ] u+h.c CϕD v +C ϕ v C dϕ v ] d+h.c., 4.9 wit te diagonal fermion mass matrices above, defined in eq Note tat te dimension-5 operator in eq..1, induces also a Higgs-neutrino-neutrino vertex but tis is igly suppressed since it is proportional to neutrino masses. 1

14 C eϕ = U e L C eϕ U er C ll f1 f f 3 f 4 = U el g f U el g4 f 4 U el g 1 f 1 U el g 3 f 3 C ll g1 g g 3 g 4 C dϕ = U d L C dϕ U dr C ee f1 f f 3 f 4 = U er g f U er g4 f 4 U er g 1 f 1 U er g 3 f 3 C ee g1 g g 3 g 4 C uϕ = U u L C uϕ U ur C le f1 f f 3 f 4 = U el g f U er g4 f 4 U el g 1 f 1 U er g 3 f 3 C le g1 g g 3 g 4 C ew = U e L C ew U er C qq1 f1 f f 3 f 4 = U dl g f U dl g4 f 4 U dl g 1 f 1 U dl g 3 f 3 C qq1 g1 g g 3 g 4 C eb = U e L C eb U er C qq3 f1 f f 3 f 4 = U dl g f U dl g4 f 4 U dl g 1 f 1 U dl g 3 f 3 C qq3 g1 g g 3 g 4 C dg = U d L C dg U dr C dd f1 f f 3 f 4 = U dr g f U dr g4 f 4 U dr g 1 f 1 U dr g 3 f 3 C dd g1 g g 3 g 4 C dw = U d L C dw U dr C uu f1 f f 3 f 4 = U ur g f U ur g4 f 4 U ur g 1 f 1 U ur g 3 f 3 C uu g1 g g 3 g 4 C db = U d L C db U dr C ud1 f1 f f 3 f 4 = U ur g f U dr g4 f 4 U ur g 1 f 1 U dr g 3 f 3 C ud1 g1 g g 3 g 4 C ug = U u L C ug U ur C ud8 f1 f f 3 f 4 = U ur g f U dr g4 f 4 U ur g 1 f 1 U dr g 3 f 3 C ud8 g1 g g 3 g 4 C uw = U u L C uw U ur C qu1 f1 f f 3 f 4 = U dl g f U ur g4 f 4 U dl g 1 f 1 U ur g 3 f 3 C qu1 g1 g g 3 g 4 C ub = U u L C ub U ur C qu8 f1 f f 3 f 4 = U dl g f U ur g4 f 4 U dl g 1 f 1 U ur g 3 f 3 C qu8 g1 g g 3 g 4 C ϕl1 = U e L C ϕl1 U el C qd1 f1 f f 3 f 4 = U dl g f U dr g4 f 4 U dl g 1 f 1 U dr g 3 f 3 C qd1 g1 g g 3 g 4 C ϕl3 = U e L C ϕl3 U el C qd8 f1 f f 3 f 4 = U dl g f U dr g4 f 4 U dl g 1 f 1 U dr g 3 f 3 C qd8 g1 g g 3 g 4 C ϕe = U e R C ϕe U er C quqd1 f1 f f 3 f 4 = U ur g f U dr g4 f 4 U dl g 1 f 1 U dl g 3 f 3 C quqd1 g1 g g 3 g 4 C ϕq1 = U d L C ϕq1 U dl C quqd8 f1 f f 3 f 4 = U ur g f U dr g4 f 4 U dl g 1 f 1 U dl g 3 f 3 C quqd8 g1 g g 3 g 4 C ϕq3 = U d L C ϕq3 U dl C lq1 f1 f f 3 f 4 = U el g f U dl g4 f 4 U el g 1 f 1 U dl g 3 f 3 C lq1 g1 g g 3 g 4 C ϕd = U d R C ϕd U dr C lq3 f1 f f 3 f 4 = U el g f U dl g4 f 4 U el g 1 f 1 U dl g 3 f 3 C lq3 g1 g g 3 g 4 C ϕu = U u R C ϕu U ur C ld f1 f f 3 f 4 = U el g f U dr g4 f 4 U el g 1 f 1 U dr g 3 f 3 C ld g1 g g 3 g 4 C ϕud = U u R C ϕud U dr C lu f1 f f 3 f 4 = U el g f U ur g4 f 4 U el g 1 f 1 U ur g 3 f 3 C lu g1 g g 3 g 4 C νν = Uν L C νν U νl C qe f1 f f 3 f 4 = U dl g f U er g4 f 4 U dl g 1 f 1 U er g 3 f 3 C qe g1 g g 3 g 4 C ed f1 f f 3 f 4 = U er g f U dr g4 f 4 U er g 1 f 1 U dr g 3 f 3 C ed g1 g g 3 g 4 C eu f1 f f 3 f 4 = U er g f U ur g4 f 4 U er g 1 f 1 U ur g 3 f 3 C eu g1 g g 3 g 4 C ledq f1 f f 3 f 4 = U er g f U dl g4 f 4 U el g 1 f 1 U dr g 3 f 3 C ledq g1 g g 3 g 4 C lequ1 f1 f f 3 f 4 = U er g f U ur g4 f 4 U el g 1 f 1 U dl g 3 f 3 C lequ1 g1 g g 3 g 4 C lequ3 f1 f f 3 f 4 = U er g f U ur g4 f 4 U el g 1 f 1 U dl g 3 f 3 C lequ3 g1 g g 3 g 4 C duq f1 f f 3 f 4 = U ur g f U el g4 f 4 U dr g1 f 1 U dl g3 f 3 C duq g1 g g 3 g 4 C qqu f1 f f 3 f 4 = U dl g f ]U er g4 f 4 U dl g1 f 1 U ur g3 f 3 C qqu g1 g g 3 g 4 C qqq f1 f f 3 f 4 = U dl g f U el g4 f 4 U dl g1 f 1 U dl g3 f 3 C qqq g1 g g 3 g 4 C duu f1 f f 3 f 4 = U ur g f U er g4 f 4 U dr g1 f 1 U ur g3 f 3 C duu g1 g g 3 g 4 Table 4. Definitions of te Wilson coefficients multiplying te fermionic currents in te mass basis. We suppress te flavor indices for te two-fermion operators as te contraction is non-ambiguous ere. For te four-fermion vertices we assume summation over repeating indices. 13

15 5 Gauge fixing and FP-gosts in R ξ -gauges Compared to te SM, te procedure of gauge fixing in SMEFT involves additional features. A consistent and convenient, for practical purposes, coice of gauge fixing conditions and gost sector sould fulfil te following requirements: Cancel te unwanted Goldstone-gauge boson bilinear mixing, as in SM. Lead to SM-like propagators in terms of te effective mass basis parameters and fields. Preserve te BRST invariance of te full Lagrangian in te presence of gauge fixing and gost terms. Let us notice tat te gauge basis Lagrangian in terms of barred couplings and fields, as obtained troug eq. 3.11, keeps te same form up to rescaling factors. For te dimension-4 terms it reads, L 4 SM = 1 4 Z g s Ḡ A µνḡaµν 1 4 Z g W µν I W Iµν 1 4 Z g Bµν Bµν + D µ ϕ D µ ϕ+m ϕ ϕ 1 λϕ ϕ + i l L / Dl L +ē R / De R + q L / Dq L +ū R / Du R + d R / Dd R l LΓ e e Rϕ+ q LΓ u u R ϕ+ q LΓ d d Rϕ, 5.1 wile all iger dimensional operators remain unaffected at te considered order. Eac term in te barred Lagrangian is still manifestly SU3 SU U1 invariant, despite te presence of Z-factors. Terefore, we may equivalently use tis Lagrangian to gauge fix te teory. Our coice for te gauge fixing term in te electroweak sector reads L GF = 1 F ˆξ 1 F, 5. wit te gauge fixing functionals F i defined troug F = F 1 F F 3 F 0 µ W1µ Wµ µ = µ W3µ µ Bµ vˆξ iḡ Φ+ Φ ḡ Φ+ +Φ ḡzg Φ 0 0 ḡ ZG Φ and a 4 4 symmetric matrix ˆξ introduced as ξ W 0 ξ W ˆξ = ξz 0 X ξ A X,

16 wit X being te mixing matrix of te neutral electroweak gauge bosons in eq Wit suc a coice in gauge basis, te transformations wic diagonalize and rescale te electroweak gauge and Goldstone bosons also bring te gauge fixing term in a familiar form. After substituting te mass basis fields into eq. 5.3, we arrive at te expression L GF = 1 ξ W µ W + µ +iξ W M W ν W ν iξ W M W G 1 ξ Z µ Z µ +ξ Z M Z 1 ξ A µ A µ, 5.5 wic looks identical to te SM one in te standard linear R ξ -gauges and as all terms required to eliminate te unwanted Goldstone-gauge mixing of eq. 3.4, troug a total derivative. As previously mentioned in section 3.3, suc a standard coice for R ξ -gauges is possible since, in mass basis, all Wilson coefficients of te unwanted terms become absorbed in masses and fields. Te gauge fixing conditions violate gauge invariance and we need to introduce a gost term in te Lagrangian to compensate and restore te more general BRST invariance. A convenient and consistent coice for a gost term takes te form L FP = N Ê ˆM F N, 5.6 were te gauge basis gost, anti-gost fields are defined as N i = N 1,N,N 3,N 0, Ni = N 1, N, N 3, N 0, respectively and we ave also introduced te symmetric 4 4 matrix, Ê = X 1 X Te gauge fixing functionals F i cosen in eq. 5.3 are linear in te fields and terefore testandard Faddeev-Popov FP treatment wit determinants applies. 4 Teexplicitform of ˆMF can be always obtained by performing an infinitesimal gauge transformation on F i. However, since we also wis to demonstrate te BRST invariance of te SMEFT action we follow instead an equivalent derivation of ˆMF wit te elp of te BRST-operator, s. It reads, ˆM ij F Nj = sf i, 5.8 were lowercase Latin indices run in te electroweak space {i, j}=1,,3,0. Despite te presence of constant mixing matrices in te gauge fixing functionals, te s-operator transforms te fields included in F i, in a way identical to SM, as sϕ = iḡ Yϕ N 0 iḡt I ϕ N I, sϕ = +iḡ ϕ YN 0 +iḡϕ T I N I, s B µ = µ N 0, s W µ I = µ N I ḡǫ IJK WJ µ N K In te FP-treatment, it is clear tat te matrix Ê factors out from te determinant as detê ˆM F = detêdet ˆM F, affecting te pat integral wit an irrelevant constant factor. 15

17 Ten, ˆMF reads explicitly, 0 W µ 3 W µ 0 ˆM F N = N +ḡ µ W µ 3 0 W µ 1 0 W N 5.10 µ W µ H +v Φ 0 Φ+ +Φ ḡ Φ + +Φ ḡ + vḡˆξ Φ 0 H +v i Φ+ Φ iḡ Φ + Φ ḡ 4 ZG 0 Φ+ +Φ izg 0 Φ+ Φ ZG N H +v 0 ḡ ḡ Z G H +v 0 ḡ ḡ Z G H +v ḡ Z 0 ḡ G H +v 0 ḡ ḡ Z Φ+ +Φ iḡ ḡ Z Φ+ Φ Once again, te cosen form of eq. 5.6 wit te presence of te matrix Ê, makes te transformation wic diagonalizes te gauge bosons kinetic terms and masses to diagonalize also gost bilinear terms. By defining gost and anti-gost fields in mass basis troug te relations 1 N 1 in = η ±, N 3 η Z = X, N 0 η A 1 N 1 ±i N = η ±, 5.11 N3 η Z = X, 5.1 N 0 η A all occurrences of te X matrix in bilinear gost terms become absorbed, leaving tem in a canonical form wit squared masses ξ W MW, ξ ZMZ and zero for te corresponding poton gost. Again, te gost propagators are SM-like see appendix A.1. Neverteless, corrections from iger dimensional operators appear explicitly in gost vertices as it was also mentioned in ref. [9]. Te BRST invariance of te SMEFT action not including te gauge fixing and gost sector, follows immediately from its gauge invariance. In order to establis BRST for te gauge fixing and gost sector, as well, we consider, sn 0 = 0, sn I = ḡ ǫijk N J N K, 5.13 s N i = F j ˆξ 1 Ê 1 ji Using eq. 5.8 and eq. 5.14, te property ˆξ 1 = ˆξ 1 and te relation s ˆM F N = 0, wic is associated wit te nilpotency of BRST, one obtains sl GF = 1 F s i ˆξ 1 ij F j = F i ˆξ 1 ij sf j i = s N Êij ˆMjk F Nk = s Ni ij Ê ˆMjk F Nk = sl FP Hence, te full Lagrangian now remains invariant under BRST-symmetry transformations. 16

18 As easily noticed, te BRST transformation on all gauge basis fields, besides antigosts, is identical to SM. Terefore, for tis set of fields it is nilpotent. Te gauge fixing functionals F i, altoug modified by te presence of new constant mixing matrices, are still linear functions of te same fields as in SM i.e., gauge and Goldstone bosons. Tus, te BRST transformation for tem is also nilpotent, satisfying s F i = sm ij F Nj = 0, wic can be always verified explicitly. Finally, we note tat te presence of constant matrices in te transformation for anti-gosts is in practice irrelevant. Tis is because one can always introduce auxiliary fields [30, 31] in a suitable manner witout eventually affecting te gauge fixing and gost terms. Te coice L GF = B ÊF + 1 B ʈξÊB, 5.16 is equivalent to eq. 5. wen te equations of motion are taken for te auxiliary fields B i. Canging only te transformation for anti-gosts, into s N i = B i and introducing te new one sb i = 0 for te auxiliary fields, one can verify tat te action remains BRST-invariant. Moreover, te BRST transformation on all fields is now nilpotent, tat is s = In te QCD-sector, an analogous discussion of te R ξ -gauges is far more trivial. In terms of barred fields and couplings, te gauge fixing and gost terms read wit simply, L GF +L FP = 1 ξ G F A F A + η A GM AB F η B G, 5.18 F A = µ g Aµ, M AB F η B = η A ḡ s µ f ABC g Bµ η C G Feynman rules and Matematica implementation In appendix A we ave collected te Feynman rules for SMEFT propagators and interaction vertices in te R ξ -gauges. Most of te vertices are reasonably compact and for many processes tey can be readily used even for manual calculations. We did not display explicitly only te five and six gluon self-interactions as tey are, after symmetrizing in all Lorentz and color indices, very long and it is unlikely tat tey can be used in any calculations witout te use of computer symbolic algebra programs. Apart from te printed version, we ave developed a publicly available Matematica code calculating te same set of Feynman rules, suc tat its output can be directly fed to oter symbolic or numerical packages for ig energy pysics calculations. Our code works witin te FeynRules package [0] and is constructed as a model file for FeynRules supplied wit set of auxiliary programs performing te field redefinitions described earlier in te paper. In addition, tese programs perform some extra simplifications, on top of te ones done by FeynRules, like Fierz transformations in four-fermion interactions assuring 17

19 tat all terms in a given vertex are always added wit te same ordering of fermion indices, wenever possible. Similarly, tere as been made several simplifications in te gluon vertices based on Jacobi identity. Our package contains also routines generating automatically Latex output for SMEFT Feynman rules. If necessary, users can run te SMEFT-code to obtain a subset of vertices for cosen Wilson coefficients, relevant just to teir analysis. Te SMEFT package for FeynRules, wit instructions for te user, can be downloaded from ttp:// In appendix B we describe ow to install and run our package. Very recently it as appeared in te literature a Matematica program, called DsixTools [3] tat calculates te Renormalization Group Equation RGE running of Wilson coefficients for te operators listed in tables and 3. Tis code is complementary to our SMEFT code wen calculating renormalized amplitudes at leading and, up to modifications, next to leading order in perturbation teory. 7 Conclusions It is a central problem in particle pysics today to categorize and parametrize New Pysics effects tat are expected to arise by new effective operators at some scale Λ. In tis article we analyzed te structure of Standard Model Effective Field Teory SMEFT including non-renormalizable operators up to dimension 6. For te first time in literature we derived te complete set of Feynman rules for tis teory quantized in linear R ξ -gauges. More precisely, we started from te well known Warsaw basis of ref. [11], were te complete set of independent gauge invariant d 6 operators is given, and identified te mass eigenstate fields after Spontaneous Symmetry Breaking SSB. In acieving tat goal, we performed constant and gauge invariant field and coupling redefinitions in suc a way tat all pysical and unpysical fields possess canonical kinetic terms. Furtermore, we constructed gauge fixing functionals wic in mass basis ave a form of te linear R ξ - gauges used routinely in te SM loop-calculations. A general set of different gauge fixing parameters for eac gauge field as been introduced, for completeness and for additional cross-cecks of te teory. In order to restore te broken gauge symmetry after adding te gauge fixing terms, a set of Faddeev-Popov gosts as been introduced. Te gost Lagrangian as been cosen suc tat te gost propagators again ave te SM-like structure, wile te effect of iger dimensional operators appears explicitly only in teir interaction vertices. We also proved tat our SMEFT action preserves BRST invariance and provide te reader wit pertinent transformations in section 5. In summary, after establising all steps described above, te bilinear part of SMEFT Lagrangian and all, pysical and unpysical, field propagators expressed in terms of pysical masses ave exactly te same structure as in te SM altoug certain relations of masses and couplings, suc as te ρ-parameter for example, are modified by te new operators. Te effect of new d = 5 and 6 operators appears explicitly only in triple and iger multiplicity vertices, eiter as modifications of te SM ones or as genuine new interactions beyond te SM. 18

20 Witin te mass basis considered ere, we constructed te complete set of Feynman rules in te linear R ξ -gauges, not resorting to any restriction suc as CP- or baryonlepton-number conservation. Te Feynman Rules for te total 383 vertices not counting te ermitian conjugate ones, wic are about four times more tan te SM vertices, are given in appendix A. All Feynman rules were derived using te FeynRules code and a set of auxiliary programs created by te autors to perform field redefinitions, various simplifications and an automatic translation to Latex/axodraw format. All propagators and vertices are bot listed explicitly in te appendix A and provided as a publicly available Matematica package, tat can be downloaded from ttp:// Te reader can consult appendix B for programming and installation details. On te practical side, we believe tat our SMEFT collection of Feynman rules sould significantly facilitate future penomenological analyses, saving time in deriving from scratc often lengty expressions in a complicated teory. In addition, our Feynman rules elp to avoid possible mistakes and omissions of diagrams, wic could easily appen wen taking into account only some parts of te full Lagrangian, as tis is done in many studies so far. Furtermore, te publicly available SMEFT model file for FeynRules package tat accompanies tis article, can be directly used as an input file to oter ig energy pysics computational computer programs, again streamlining te calculation of future SMEFT pysical predictions. Acknowledgments AD would like to tank Apostolos Pilaftsis for illuminating discussions about loop calculations in effective field teory. Te work of JR and WM is supported in part by te National Science Centre, Poland, under researc grants DEC-015/19/B/ST/0848, DEC- 015/18/M/ST/00054 and DEC-014/15/B/ST/0157. JR would also like to tank University of Ioannina and CERN for ospitality during is visits. AD and KS would like to tank University of Warsaw for ospitality. KS acknowledges full financial support from Greek State Scolarsips Foundation I.K.Y. Tis researc was implemented troug a scolarsip by te Greek State Scolarsips Foundation I.K.Y wic was financed by te action BACKING OF POSTDOCTORAL RESEARCHERS troug te Operational Program Development of workforce, Education and Lifelong Learning, wit priority axes 6, 8, 9 and as been co-financed by te European Social Fund ESF and Greek national funds. A SMEFT Feynman rules In tis appendix we list te complete set of Feynman rules for SMEFT in te pysical mass eigenstates field basis and in R ξ -gauges. In our notation for interaction vertices, indices and momentum for eac particle external leg carry a common number label. External indices appear explicitly in te diagrams but momenta are suppressed for a better visual result. Te convention for number labels is displayed below for te four possible topologies of SMEFT. Momenta are always considered incoming. 19

21 p p 1 p 1 p p 1 p p 3 p 3 p 1 p 5 5 p p 4 4 p 3 p 3 5 p 1 p 4 p 5 p 6 6 In addition to te notation defined in te main paper, we use te following symbols: Index type Flavor generation Spinor Color in triplet representation quarks Color in adjoint representation gluons Symbols f i,g i s i m i a i,b i Lorentz µ i,ν i,α i,β i,... Finally, η µν denotes te Minkowski metric tensor wit signature +,,,. An important remark sould be made about Lorentz indices contraction. After all te teoretical work, discussed in sections 4 and 5, deriving expressions for te Feynman rules is straigtforward but tedious. In order to save time and minimize te possibility of misprints, Feynman rules were generated fully automatically by a specialized Matematica code directly producing Latex output. However, it was difficult to implement in suc a Matematica to Latex translator te proper positioning of Lorentz indices, suc tat upper and lower repeating indices are contracted. Tus, in te expressions of tis appendix one sould assume tat repeating Lorentz indices are always contracted in a covariant way, even if tey are not subscript-superscript pairs. Altoug te final output for SMEFT Feynman Rules is automatized we ave made an effort to furter simplify vertices manually wenever possible. For example, a great deal of simplification appens in 4, 5, and 6-point gauge boson vertices. 4 0

22 A.1 Propagators in te R ξ -gauges W ± µ ν Z 0 µ ν [ i k MW η µν k µ k ν ] 1ξ W k ξ W MW [ i k MZ η µν k µ k ν ] 1ξ Z k ξ Z MZ A 0 µ ν g µ,a ν,b η ±, η ± η Z, η Z η A, η A η G, η G a b i [η µν k 1ξ A kµ k ν k iδ ab k ] [η µν 1ξ G kµ k ν ] i k ξ W MW i k ξ Z MZ i k iδ ab k i k G ± f g 1 g k ξ Z M Z i k ξ W M W i k M iδ g 1g /k m f Note tat f above stands for any fermion in te teory, f = ν,l,u,d. Apart from Kronecker delta in flavor indices δ g 1g, quark propagators sould be multiplied by δ m 1m in color indices too. 1

23 A. Lepton-gauge vertices ν f 1 i ḡ +ḡ δ f1 f γ µ 3 P L iḡḡ v ḡ +ḡ δ f 1 f C ϕwb γ µ 3 P L ν f 1 e f 1 e f 1 e f e f Z 0 µ 3 A 0 µ 3 Z 0 µ iv ḡ +ḡ U g f U g 1 f 1 C ϕl1 g 1 g γ µ 3 P L 1 iv ḡ +ḡ U g f U g 1 f 1 C ϕl3 g 1 g γ µ 3 P L + iḡḡ ḡ +ḡ δ f 1 f γ µ 3 iḡ ḡ v ḡ +ḡ 3/ δ f 1 f C ϕwb γ µ 3 ḡ v + ḡ +ḡ pν 3 Cf ew f 1 σ µ 3ν P L +Cf ew 1 f σ µ 3ν ḡv ḡ +ḡ pν 3 Cf eb f 1 σ µ 3ν P L +Cf eb 1 f σ µ 3ν i ḡ +ḡ δ f 1 f ḡ ḡ γ µ 3 P L +ḡ γ µ 3 iḡḡ v + ḡ +ḡ δ 3/ f 1 f C ḡ ϕwb ḡ γ µ 3 P L ḡ γ µ 3 ḡv + ḡ +ḡ pν 3 Cf ew f 1 σ µ 3ν P L +Cf ew 1 f σ µ 3ν ḡ v + ḡ +ḡ pν 3 Cf eb f 1 σ µ 3ν P L +Cf eb 1 f σ µ 3ν + 1 iv ḡ +ḡ C ϕl1 f 1 f γ µ 3 P L + 1 iv ḡ +ḡ C ϕl3 f 1 f γ µ 3 P L + 1 iv ḡ +ḡ C ϕe f 1 f γ µ 3 e f iḡ Uf f 1 γ µ 3 P L vp ν 3Ug 1 f 1 Cg ew 1 f σ µ 3ν ν f 1 W + µ 3 e f e f 1 W + µ 3 + ḡv σ µ 3µ 4 P LCf ew f 1 +Cf ew 1 f σ µ 3µ 4 W µ 4

24 ν f e f 1 A 0 µ 3 + ḡḡ v ḡ +ḡ U g 1 f σ µ 3µ 4 P LC ew g 1 f 1 e f 1 e f 1 W µ 4 ν f Z 0 µ 4 W µ 3 A.3 Lepton-Higgs-gauge vertices e f ḡ v ḡ +ḡ U g 1 f σ µ 3µ 4 P LC ew g 1 f v δ f 1 f m lf1 γ 5 v 4 δ f 1 f C ϕd m lf1 γ 5 v/p 3 P LC ϕl1 f 1 f v/p 3 P LC ϕl3 f 1 f v/p 3 C ϕe f 1 f e f e f 1 i v δ f 1 f m lf1 ivδ f1 f C ϕ m lf1 + iv 4 δ f 1 f C ϕd m lf1 + iv PLC eϕ f f 1 +C eϕ f 1 f e f ν f 1 i v m l f Uf f 1 +i vug 1 f 1 C ϕl3 g 1 f /p 3 P L +m lf ν f ν f 1 vu gf/p 3 P LUg 1 f 1 Cg ϕl1 1 g +vu g f /p 3 P LUg 1 f 1 Cg ϕl3 1 g 3

25 e f e f 1 + iv PLC eϕ f f 1 +C eϕ f 1 f +i i /p 3 P L /p 4 P L C ϕl3 f 1 f +i C ϕl1 f 1 f /p 3 P L /p 4 P L /p 3 /p 4 C ϕe f 1 f e f 1 e f 1 G e f e f + iv PLC eϕ f f 1 +C eϕ f 1 f + v PLC eϕ f f 1 C eϕ /p 3 P L /p 4 P L C ϕl3 f 1 f C ϕl1 f 1 f f 1 f /p 3 P L /p 4 P L /p 3 /p 4 C ϕe f 1 f e f e f 1 + 3iv PLC eϕ f f 1 +C eϕ f 1 f e f ν f 1 +ivug 1 f 1 C eϕ g 1 f +i /p 3 P L /p 4 P L Ug 1 f 1 C ϕl3 g 1 f 4

26 ν f ν f 1 +iu g f /p 3 P L /p 4 P L U g 1 f 1 C ϕl1 g 1 g +iu g f /p 3 P L /p 4 P L U g 1 f 1 C ϕl3 g 1 g ν f 1 ν f 1 G ν f ν f Zµ 0 4 U g f /p 3 P L /p 4 P L U g 1 f 1 C ϕl1 g 1 g +U g f /p 3 P L /p 4 P L U g 1 f 1 C ϕl3 g 1 g +iv ḡ +ḡ U g f U g 1 f 1 C ϕl1 g 1 g γ µ 4 P L iv ḡ +ḡ U g f U g 1 f 1 C ϕl3 g 1 g γ µ 4 P L e f ḡ + ḡ +ḡ pν 3 C ew f f 1 σ µ 3ν P L +C ew f 1 f σ µ 3ν e f 1 A 0 µ 3 ḡ ḡ +ḡ pν 3 Cf eb f 1 σ µ 3ν P L +Cf eb 1 f σ µ 3ν e f ḡ + ḡ +ḡ pν 4 C ew f f 1 σ µ 4ν P L +C ew f 1 f σ µ 4ν e f 1 ḡ + ḡ +ḡ pν 4 C eb f f 1 σ µ 4ν P L +C eb f 1 f σ µ 4ν Z 0 µ 4 +iv ḡ +ḡ C ϕl1 f 1 f γ µ 4 P L +iv ḡ +ḡ C ϕl3 f 1 f γ µ 4 P L +iv ḡ +ḡ C ϕe f 1 f γ µ 4 5

27 ν f ν f 1 G iḡvu g f U g 1 f 1 C ϕl1 g 1 g γ µ 4 P L e f 1 ν f 1 W µ + 4 e f W µ + 4 e f G p ν 4C ew f f 1 σ µ 4ν P L iḡvc ϕl1 f 1 f γ µ 4 P L iḡvc ϕe f 1 f γ µ 4 /p 3 P L /p 4 P L Ug 1 f 1 C ϕl3 g 1 f ν f ḡ ḡ 3U g1 +ḡ pν f σ µ 3ν P LC ew g 1 f 1 e f 1 A 0 µ 3 ḡ ḡ +ḡ pν 3U g1 f σ µ 3ν P LC eb g 1 f 1 i ḡḡ v ḡ +ḡ U g 1 f C ϕl3 f 1 g 1 γ µ 3 P L G ν f e f 1 G ḡ ḡ 4U g1 +ḡ pν f σ µ 4ν P LC ew + g 1 f 1 ḡ ḡ +ḡ pν 4U g1 f σ µ 4ν P LC eb g 1 f 1 + i ḡ v ḡ +ḡ U g 1 f C ϕl3 f 1 g 1 γ µ 4 P L Z 0 µ 4 6

28 e f ν f 1 p ν 4U g 1 f 1 C ew g 1 f σ µ 4ν i ḡvu g 1 f 1 C ϕl3 g 1 f γ µ 4 P L e f 1 e f 1 W µ + 4 ν f Wµ 4 e f A 0 µ 3 +ip ν 4U g1 f σ µ 4ν P LC ew g 1 f 1 i ḡ ḡ +ḡ pν 3 Cf ew f 1 σ µ 3ν P L Cf ew 1 f σ µ 3ν + i ḡ ḡ +ḡ pν 3 Cf eb f 1 σ µ 3ν P L Cf eb 1 f σ µ 3ν e f e f 1 i ḡ ḡ +ḡ pν 4 Cf ew f 1 σ µ 4ν P L Cf ew 1 f σ µ 4ν i ḡ ḡ +ḡ pν 4 Cf eb f 1 σ µ 4ν P L Cf eb 1 f σ µ 4ν Z 0 µ 4 e f e f PLC eϕ f f 1 C eϕ f 1 f G 7

29 e f e f 1 G + i PLC eϕ f f 1 +C eϕ f 1 f e f e f 1 e f 1 e f + 3 PLC eϕ f f 1 C eϕ f 1 f + i PLC eϕ f f 1 +C eϕ f 1 f e f e f PLC eϕ f f 1 C eϕ f 1 f e f e f 1 + 3i PLC eϕ f f 1 +C eϕ f 1 f 8

30 e f ν f 1 +iu g 1 f 1 C eϕ g 1 f G e f ν f 1 ν f 1 e f +iu g 1 f 1 C eϕ g 1 f +iu g 1 f 1 C eϕ g 1 f ν f A 0 µ 3 ν f 1 ḡ iḡḡ +ḡ U g f Ug 1 f 1 Cg ϕl1 1 g γ µ 3 P L ḡ iḡḡ +ḡ U g f Ug 1 f 1 Cg ϕl3 1 g γ µ 3 P L G ν f ν f 1 G + i ḡ ḡ ḡ +ḡ U g f Ug 1 f 1 Cg ϕl1 1 g P L + i ḡ ḡ ḡ +ḡ U g f Ug 1 f 1 Cg ϕl3 1 g P L Z 0 µ 5 9

31 e f A 0 µ 3 e f 1 ḡ iḡḡ +ḡ Cϕl1 f 1 f γ µ 3 P L+ ḡ iḡḡ +ḡ Cϕl3 f 1 f γ µ 3 P L ḡ iḡḡ +ḡ Cϕe f 1 f γ µ 3 G e f e f 1 G Zµ 0 5 ν f ν f 1 + i ḡ ḡ ḡ Cϕl1 +ḡ f 1 f P L i ḡ ḡ + ḡ Cϕe +ḡ f 1 f i ḡ ḡ ḡ Cϕl3 +ḡ f 1 f P L +i ḡ +ḡ U g f U g 1 f 1 C ϕl1 g 1 g P L i ḡ +ḡ U g f U g 1 f 1 C ϕl3 g 1 g P L Z 0 µ 5 ν f ν f 1 +i ḡ +ḡ U g f U g 1 f 1 C ϕl1 g 1 g P L i ḡ +ḡ U g f U g 1 f 1 C ϕl3 g 1 g P L Z 0 µ 5 e f e f 1 +i ḡ +ḡ C ϕl1 f 1 f P L+i ḡ +ḡ C ϕl3 f 1 f P L+i ḡ +ḡ C ϕe f 1 f Z 0 µ 5 30

32 e f e f 1 +i ḡ +ḡ C ϕl1 f 1 f P L+i ḡ +ḡ C ϕl3 f 1 f P L+i ḡ +ḡ C ϕe f 1 f Z 0 µ 5 ν f ν f 1 G W µ + 5 ν f ν f 1 G +ḡu g f U g 1 f 1 C ϕl1 g 1 g P L iḡu g f U g 1 f 1 C ϕl1 g 1 g P L W + µ 5 e f e f 1 G +ḡc ϕl1 f 1 f P L +ḡc ϕe f 1 f W + µ 5 e f G e f 1 iḡc ϕl1 f 1 f P L iḡc ϕe f 1 f W + µ 5 31

33 ν f A 0 µ 3 e f 1 ḡḡ + ḡ +ḡ U g 1 f C ϕl3 f 1 g 1 γ µ 3 P L G ν f A 0 µ 3 e f 1 G ν f e f 1 G Zµ 0 5 i ḡḡ ḡ +ḡ U g 1 f C ϕl3 f 1 g 1 γ µ 3 P L ḡ ḡ +ḡ U g 1 f C ϕl3 f 1 g 1 P L ν f G e f 1 + i ḡ ḡ +ḡ U g 1 f C ϕl3 f 1 g 1 P L Z 0 µ 5 e f ν f 1 G i ḡu g 1 f 1 C ϕl3 g 1 f P L W + µ 5 3

34 e f ν f 1 i ḡu g 1 f 1 C ϕl3 g 1 f P L e f ν f 1 ν f e f 1 W µ + 5 W µ + 5 G W µ + 4 i ḡu g 1 f 1 C ϕl3 g 1 f P L ḡu g1 f σ µ 4µ 5 P LC ew g 1 f 1 W µ 5 e f e f 1 W + µ 4 i ḡ σ µ 4µ 5 P LCf ew f 1 Cf ew 1 f σ µ 4µ 5 W µ 5 e f e f 1 W + µ 4 + ḡ σ µ 4µ 5 P LCf ew f 1 +Cf ew 1 f σ µ 4µ 5 W µ 5 33

35 ν f A 0 µ 3 e f 1 iḡḡ ḡ +ḡ U g 1 f σ µ 3µ 5 P LC ew g 1 f 1 W µ 5 ν f e f 1 Wµ 4 Zµ 0 5 ν f A 0 µ 3 e f 1 Wµ 5 + iḡ ḡ +ḡ U g 1 f σ µ 4µ 5 P LC ew g 1 f 1 + ḡḡ ḡ +ḡ U g 1 f σ µ 3µ 5 P LC ew g 1 f 1 ν f e f 1 W µ 4 ḡ ḡ +ḡ U g 1 f σ µ 4µ 5 P LC ew g 1 f 1 Z 0 µ 5 e f A 0 µ 3 e f 1 G ḡḡ ḡ +ḡ σµ 3µ 5 P LC ew f f 1 W + µ 5 34

36 e f G e f 1 W + µ 4 + ḡ ḡ +ḡ σµ 4µ 5 P LC ew f f 1 Z 0 µ 5 A.4 Quark-gauge vertices u f u f A 0 µ 3 Z 0 µ 3 iḡḡ 3 ḡ +ḡ δ f 1 f γ µ 3 + iḡ ḡ v 3 ḡ +ḡ δ 3/ f 1 f C ϕwb γ µ 3 ḡ v ḡ +ḡ pν 3 Cf uw f 1 σ µ 3ν P L +Cf uw 1 f σ µ 3ν ḡv ḡ +ḡ pν 3 Cf ub f 1 σ µ 3ν P L +Cf ub 1 f σ µ 3ν i + 6 ḡ +ḡ δ f 1 f ḡ 3ḡ γ µ 3 P L +4ḡ γ µ 3 iḡḡ v 6 ḡ +ḡ δ 3/ f 1 f C 3ḡ ϕwb ḡ γ µ 3 P L 4ḡ γ µ 3 ḡv ḡ +ḡ pν 3 Cf uw f 1 σ µ 3ν P L +Cf uw 1 f σ µ 3ν ḡ v + ḡ +ḡ pν 3 Cf ub f 1 σ µ 3ν P L +Cf ub 1 f σ µ 3ν + 1 iv ḡ +ḡ K f1 g K f g 1 C ϕq1 g g 1 γ µ 3 P L 1 iv ḡ +ḡ K f1 g K f g 1 C ϕq3 g g 1 γ µ 3 P L + 1 iv ḡ +ḡ C ϕu f 1 f γ µ 3 d f 1 d f A 0 µ 3 iḡḡ + 3 ḡ +ḡ δ f 1 f γ µ 3 iḡ ḡ v 3 ḡ +ḡ δ 3/ f 1 f C ϕwb γ µ 3 ḡ v + ḡ +ḡ pν 3 Cf dw f 1 σ µ 3ν P L +Cf dw 1 f σ µ 3ν ḡv ḡ +ḡ pν 3 Cf db f 1 σ µ 3ν P L +Cf db 1 f σ µ 3ν 35