Off-shell renormalization in the presence of dimension 6 derivative operators. III. Operator mixing and β functions
aa r X i v : . [ h e p - ph ] J a n Off-shell renormalizationin the presence of dimension 6 derivative operators.III. Operator mixing and β functions D. Binosi ∗ and A. Quadri † European Centre for Theoretical Studies in Nuclear Physics andRelated Areas (ECT*) and Fondazione Bruno Kessler, Villa Tambosi,Strada delle Tabarelle 286, I-38123 Villazzano (TN), Italy INFN, Sezione di Milano, via Celoria 16, I-20133 Milano, Italy (Dated: January 21, 2020)
Abstract
We evaluate the one-loop β functions of all dimension 6 parity-preserving operators in the AbelianHiggs-Kibble model. No on-shell restrictions are imposed; and the (generalized) non-polynomialfield redefinitions arising at one-loop order are fully taken into account. The operator mixingmatrix is also computed, and its cancellation patterns explained as a consequence of the functionalidentities of the theory and power-counting conditions. PACS numbers: 11.10.Gh, 12.60.-i, 12.60.Fr ∗ [email protected] † [email protected] . INTRODUCTION New physics beyond the Standard Model (SM) can be characterized in a model indepen-dent and systematic fashion within the Effective Field Theories (EFTs) framework, in whichthe (renormalizable) tree-level SM action is supplemented with the terms ( k ≥ S [ k ]0 = Z d x X i c [ k ] i O [ k ] i , (1.1)where O [ k ] i are k -dimensional operators whose dimension dictates the suppression of thecorresponding coefficients c [ k ] i in terms of powers of a high-energy scale Λ. The resultingStandard Model Effective Field Theory (SMEFT) action S ≡ Z d x L SM | {z } P k =2 S [ k ]0 + X k ≥ S [ k ]0 , (1.2)is not renormalizable in the usual (power counting) sense; it is, nevertheless, renormalizablein the modern sense [1], as all the divergences can be cancelled through the renormalizationof the (infinite) number of terms in the bare action while respecting the symmetries of thetheory.When addressing operator mixing in such theories on-shell calculations are sufficient.Indeed while it has been known since a long time that there is ultraviolet (UV) mixingbetween gauge invariant and gauge variant (unphysical) operators (also known as ‘alien’operators [2]), it has also been shown that such mixing can be made to vanish by a suitablechoice of the basis in the space of local operators [3–6]; additionally, alien operators have beenshown to be cohomologically trivial and therefore have vanishing on-shell correlators [3] (fora review see also [7]). This fact is at the basis of recent computations in the literature [8–11]as it implies that for certain purposes, e.g. , when evaluating anomalous dimensions and/or S -matrix elements, one can consider only on-shell inequivalent operators [12].A separate issue, however, is the evaluation of the β -functions of the theory. For thispurpose one needs to extend the approach adopted in the power-counting renormalizablecase [13–15] to EFTs; in particular, one must work out a procedure to fix the generalizedfield redefinitions (GFRs) that do arise in these models. Here ‘generalized’ means that, atvariance with the power-counting renormalizable case, these redefinitions are not linear inthe quantum fields (in fact, not even polynomial already at one-loop order, as we will show).2he matching of the couplings order by order in the loop expansion, once the GFRs’ effectsare taken into account, is the next technical step required to match the model with its UVcompletions while respecting the locality of the low energy theory also at higher loop orders,since it allows to unequivocally fix the correct counter-terms needed to subtract overlappingdivergences with local counter-terms.To attain these goals, in [16] it has been developed a general theory for the recursivesubtraction of off-shell UV divergences order by order in the loop expansion applicable toEFTs displaying a spontaneously broken symmetry phase. This is achieved by solving theSlavnov-Taylor (ST) identity to all-orders, which allows in turn to disentangle the gauge-invariant contributions to the off-shell one-particle irreducible (1PI) amplitudes from thoseassociated with the gauge fixing and field redefinitions, which, in a general EFT, can be(and indeed are) non polynomial (and cannot obviously be accessed staying on-shell). Next,in [17] this algebraic technique has been applied to study the Abelian Higgs-Kibble (HK)model in the presence of the dimension 6 operator ( g/ Λ) φ † φ ( D µ φ ) † D µ φ , which, giving riseto an infinite number of one-loop divergent diagrams, maximally violates power counting.In particular, the complete renormalization of all the radiatively generated dimension 6operators has been carried out together with the determination of the full g -dependence ofthe β -function coefficients.Before moving on to consider the full dimension 6 SMEFT [18], there is just one aspectthat has been left out in the study of its Abelian sibling: namely, the analysis of thefull off-shell renormalization when all inequivalent parity-preserving dimension 6 operators(classified according to [12]) are added to the power counting renormalizable action. Andthis constitutes precisely the subject of the present paper.From the point of view of the EFT renormalization programme of [1], what we achievehere is to fully evaluate all the terms appearing in the renormalized action S at one loop (inthe relevant sector of dimension ≤ S = S + ~ ∆ + · · · . (1.3)At zero antifields, ∆ collects one-loop gauge-invariant counterterms. The renormalizedaction has the same form as the original bare action S ; in particular, it can be expanded on abasis of gauge-invariant operators (in the zero antifield sector). However, these countertermsare not enough to renormalize the theory: one must also take into account the effects of3FRs, that are implemented according to a canonical transformation with respect to theBatalin-Vilkovisky (BV) bracket associated with the gauge symmetry of the model [1]. Thetransformed bare action S ′ takes then the form S ′ = S + ~ [∆ + ( F , S )] + · · · , (1.4)where F is the one loop term in the loop expansion F ( t ) = ~ tF + · · · of the generator ofthe canonical transformation responsible for the field-antifield redefinition: Φ → Φ ′ (Φ , Φ ∗ ),Φ ∗ → Φ ∗ ′ (Φ , Φ ∗ ) on S ′ [Φ ′ , Φ ′ ∗ ] = S [Φ , Φ ∗ ]. Being canonical, this transformation pre-serves the fundamental BV brackets (Φ ′ i , Φ ′ ∗ j ) = δ ij , (Φ ′ i , Φ ′ j ) = (Φ ′ ∗ i , Φ ′ ∗ j ) = 0 , and isobtained by solving the differential equation ˙ S ( t ) = ( F ( t ) , S ( t )) with the boundary con-dition S (0) = S , see [1]. Such canonical transformation generalizes the usual linear wavefunction renormalizations of the power-counting renormalizable cases. It plays a crucial andubiquitous role in the SMEFT renormalization program, as we will show.The paper is organized as follows. In Sect. II we set up our notation and, in order tomake the work self-contained we briefly review the most salient features of the X -formalism.Then, in Sects. III and IV the parameterization of the one-loop UV divergences both in the X - and the target (original) theory is presented and the mapping between the two theory’sformulations derived. GFRs are studied and their form explicitly obtained in Sect. V,whereas the renormalization of dimension 6 gauge invariant operators in the X -theory isexplicitly carried out in Sect. VI. Finally, in Sect. VIII we describe the one-loop mixingmatrix in the original theory and compare our results with the literature. Conclusions arepresented in Sect. IX. A number of technical issues are discussed in a set of Appendicespresented at the end of the paper: functional identities of the X -theory and the propagatorsin Appendices A, B and C; the list of gauge invariant operators in Appendix D; and, finally,the on-shell operator reduction relations in Appendix E.4 I. NOTATIONS AND CONVENTIONS
In the X -formalism approach of [19], the tree-level vertex functional takes the formΓ (0) = Z d x h − F µν F µν + ( D µ φ ) † ( D µ φ ) − M − m X − m v (cid:16) φ † φ − v (cid:17) − ¯ c ( (cid:3) + m ) c + 1 v ( X + X )( (cid:3) + m ) (cid:16) φ † φ − v − vX (cid:17) + z ∂ µ X ∂ µ X + g v Λ X ( D µ φ ) † ( D µ φ ) + g v Λ X F µν + g v X + T ( D µ φ ) † ( D µ φ ) + U F µν + RX + ξb − b (cid:16) ∂A + ξevχ (cid:17) + ¯ ω (cid:16) (cid:3) ω + ξe v ( σ + v ) ω (cid:17) + ¯ c ∗ (cid:16) φ † φ − v − vX (cid:17) + σ ∗ ( − eωχ ) + χ ∗ eω ( σ + v ) i . (2.1)In the expression above, the first line represents the action of the Abelian HK model in the X -formalism, where the usual scalar field φ ≡ √ ( φ + iχ ) = √ ( σ + v + iχ ) with v the vacuumexpectation value (vev) is supplemented with a singlet field X , that provides a gauge-invariant parametrization of the physical scalar mode. Notice also that we defined φ = σ + v with σ having a zero vev. The field X plays instead the role of a Lagrange multiplier: whengoing on-shell with this field one recovers the constraint X ∼ v ( φ † φ − v / m -term leaving the usual Higgsquartic potential with coefficient ∼ M / v . Hence, Green’s functions in the target theory have to be m -independent, a fact that provides a very strong check of the computations,due to the ubiquitous presence of m both in Feynman amplitudes as well as invariants.The X , -system comes together with a constraint BRST symmetry, ensuring that thenumber of physical degrees of freedom in the scalar sector remains unchanged in the X -formalism with respect to the standard formulation relying only on the field φ [20, 21].More precisely, the vertex functional (2.1) is invariant under the following BRST symmetry: s X = vc ; s φ = s X = s c = 0; s ¯ c = φ † φ − v − vX . (2.2) Going on-shell with X yields the condition( (cid:3) + m ) (cid:16) φ † φ − v − vX (cid:17) = 0 , so that the most general solution is X = v (cid:16) φ † φ − v (cid:17) + η, η being a scalar field of mass m . However, inperturbation theory the correlators of the mode η with any gauge-invariant operators vanish [16], so thatone can safely set η = 0. We define as ‘target’ theory the original theory defined in terms of conventional fields. c, ¯ c are free. The constraint BRST differential s anticommutes with the (usual) gauge group BRST symmetry of the classical action after thegauge-fixing introduced in the fifth line of Eq. (2.1): sA µ = ∂ µ ω ; sω = 0; s ¯ ω = b ; sb = 0; sφ = ieωφ. (2.3)Here ω (¯ ω ) is the U(1) ghost (antighost); the latter field is paired into a BRST doublet withthe Lagrange multiplier field b , enforcing the usual R ξ gauge-fixing condition F ξ = ∂A + ξevχ, (2.4)with ξ the gauge fixing parameter.The two BRST symmetries can both be lifted to the corresponding ST identities at thequantum level, provided one introduces a suitable set of so-called antifields, i.e. , externalsources coupled to the relevant BRST transformations that are non-linear in the quantizedfields. The antifield couplings are displayed in the last line of Eq. (2.1); the ST identitiesare instead summarized in Appendix A.The third line of Eq. (2.1) contains the dimension 6 parity preserving subset of the gauge-invariant operators described in [12], modulo for the fact that we use the zero expectationvalue combination φ † φ − v ∼ vX instead of φ † φ . We thus see that the classical power-counting renormalizable action is supplemented in the X -formalism by the X -dependentoperators O [6]1 = Z d x F µν (cid:16) φ † φ − v (cid:17) ∼ Z d x vX F µν , (2.5a) O [6]2 = Z d x (cid:16) φ † φ − v (cid:17) ∼ Z d x v X , (2.5b) O [6]3 = Z d x (cid:16) φ † φ − v (cid:17) (cid:3) (cid:16) φ † φ − v (cid:17) ∼ Z d x v X (cid:3) X , (2.5c) O [6]4 = Z d x (cid:16) φ † φ − v (cid:17) ( D µ φ ) † D µ φ ∼ Z d x vX ( D µ φ ) † D µ φ. (2.5d)Notice that the operator O [6]3 is special in the sense that it does not give rise in the X -theoryto new interaction vertices: rather it modifies the propagator of the X -field by rescalingthe p -term [21] (the full set of propagators of the model is summarized in Appendix C). In the spirit of [12] we drop operators that are on-shell equivalent, i.e. , that differ by terms vanishing oncethe classical equations of motion are imposed. v/ Λ in order to obtain, when mapping back tothe target theory, the standard 1 / Λ pre-factor for dimension 6 operators.To maintain a detailed comparison with [1], we provide in the following some technicaldetails.The relevant BV bracket is the one associated with the gauge symmetry, the constraintBRST symmetry invariance being exhausted in the X -equation, as shown in Appendix A,see Eqs. (A4) and (A6). Next, as the gauge group is Abelian: there is no ghost antifield,since sω = 0; the BRST transformation of the gauge field is linear in the quantized fields andthus there is no need to introduce the gauge antifield A ∗ µ for controlling quantum corrections (although algebraically one is allowed to). Also, in the R ξ -gauge that we employ, there isno need to introduce the antifield ¯ ω ∗ , coupled to the Nakanishi-Lautrup field b = s ¯ ω : infact, see Appendix B, the b -equation (B1) and the antighost equation (B2) imply that atthe quantum level there is no dependence on the field b and moreover that the antighostdependence can be reabsorbed by the antifield redefinition (B4). On the other hand, in theformulation of [1], where one introduces both ¯ ω ∗ and A ∗ µ , the antighost-dependent sector ofthe action is recovered from the antifield couplings R d x ( A ∗ µ sA µ + χ ∗ sχ ) via a canonicaltransformation with fermionic generator F = R d x F ξ ¯ ω (that incidentally exactly yields theantifield redefinition in Eq. (B4)). Thus, the dimension ≤ S is X k =1 S [ k ]0 (cid:12)(cid:12)(cid:12) A ∗ µ =¯ ω ∗ =0 ≡ Γ (0) (cid:12)(cid:12) b =¯ ω =0 . (2.6)At one loop order further operators will be radiatively generated starting from Γ (0) . Thoseoperators can be however expressed in the target theory as gauge invariant polynomialsin the field φ , its (symmetrized) covariant derivatives, the field strength and its ordinaryderivatives. This set of variables is particularly suited in order to obtain the coefficientsof the one loop invariants controlling the UV divergences of the theory [7]. Additionally,some of these operators will be on-shell equivalent; the reduction to on-shell independentoperators is carried out in some detail in Appendix E. This latter fact can be easily understood since the coupling Z d x A ∗ µ sA µ = Z d x A ∗ µ ∂ µ ω does not generate any interaction vertex involving A ∗ µ , due to the aforementioned linearity of the BRSTtransformation of A µ in the quantum fields. X -equation (A6) or theST identity (A1). Finally, in the fourth row we have added the external sources T , R, U required to define the X -equation at the quantum level in the presence of additional nonpower-counting renormalizable interactions, see Eq.(A7). III. ONE-LOOP UV DIVERGENCES
In this section we will work out the parameterization of the one-loop UV divergences inthe X -theory for all the operators giving rise to contributions to dimension 6 operators inthe target theory.In what follows subscripts denote functional differentiation with respect to fields andexternal sources. Thus, amplitudes will be denoted as, e.g. , Γ (1) χχ , meaningΓ (1) χχ ≡ δ Γ (1) δχ ( − p ) δχ ( p ) (cid:12)(cid:12)(cid:12)(cid:12) p =0 . (3.1)A bar denotes the UV divergent part of the corresponding amplitude in the Laurent ex-pansion around ǫ = 4 − D , with D the space-time dimension. Dimensional regulariza-tion is always implied, with amplitudes evaluated by means of the packages FeynArts and
FormCalc [22, 23]. As already remarked, all amplitudes will be evaluated in the Feynman( ξ = 1) and Landau ( ξ = 0) gauge; this will allow to explicitly check the gauge cancellationsin gauge invariant operators and in particular, as we will see, the crucial role of the GFRsin ensuring the gauge independence of ostensibly gauge invariant quantities.Consider now the UV divergent contributions to one-loop amplitudes. They form a localfunctional (in the sense of formal power series) denoted by Γ (1) . Since Γ (1) belongs to thekernel of the linearized ST operator S defined in Eq. (A3), i.e. , S (Γ (1) ) = 0 , (3.2)the nilpotency of S ensures that Γ (1) is the sum of a gauge-invariant functional I (1) and acohomologically trivial contribution S ( Y (1) ):Γ (1) = I (1)gi + S ( Y (1) ) , (3.3)8ith GFRs described by the cohomologically trivial term S ( Y (1) ). Eq. (3.3) bears in fact aclose resemblance with Eq. (1.4), as, for the model at hand, we find the identifications∆ = − I (1)gi (cid:12)(cid:12)(cid:12) b =¯ ω =0 ; ( F , S ) = − S ( Y (1) ) . (3.4)Ultimately, we are interested in the UV divergences of dimension 6 gauge invariant oper-ators in the target theory. To identify the invariants in the X -theory contributing to theseoperators the mapping function from the X - to the target theory is needed. As explainedin [16, 17] this amounts to solving the X , -equations in the X -theory via the replacementsin Eq. (A8) and then going on-shell with X , . At the one loop level it is sufficient to imposethe classical equations of motions for X , . The X -equation gives X = 1 v (cid:16) φ † φ − v (cid:17) , (3.5)whereas the classical X -equation of motion yields (at zero external sources)( (cid:3) + m )( X + X ) = − ( M − m ) X − z (cid:3) X + g v Λ ( D µ φ ) † D µ φ + g v Λ F µν + g v X . (3.6)By inserting Eqs. (3.5) and (3.6) into the solutions of the X , -equations (A9a) we obtainthe explicit form of the mapping for the HK model:¯ c ∗ → − ( M − m ) v (cid:16) φ † φ − v (cid:17) − zv (cid:3) (cid:16) φ † φ − v (cid:17) + g Λ ( D µ φ ) † D µ φ + g Λ F µν + g (cid:16) φ † φ − v (cid:17) , (3.7a) T → g Λ (cid:16) φ † φ − v (cid:17) ; U → g Λ (cid:16) φ † φ − v (cid:17) ; R → g v (cid:16) φ † φ − v (cid:17) . (3.7b) IV. DIMENSION SIX OPERATORS COEFFICIENTS
For computing the UV coefficients of dimension 6 gauge-invariant operators in the targettheory, we need to consider, see Appendix D:1. Operators which only depend on the external sources and contribute to dimension 6operators in the target theory again due to the mapping in Eq. (3.7). They are listedin Eq. (D1), and their UV coefficients denoted by ϑ i ’s;9. Mixed field-external sources gauge-invariant operators contributing to dimension 6operators in the target theory under the mapping in Eq. (3.7); these are listed inEq. (D2j), and their UV coefficients by θ i ’s;3. Dimension 6 field-dependent gauge-invariant operators that do not involve externalsources; these are listed in Eq. (D3) and their UV coefficients denoted by λ i ’s.Clearly, all the associated UV coefficients λ i , θ i and ϑ i will be ξ -independent. In orderto fix them, we need to evaluate a certain number of Feynman amplitudes and derive theprojections of these operators on the relevant 1-PI Green’s functions. However, and asalready noticed, UV divergences of the latter cannot be parameterized in terms of the λ i ’s, θ i ’s and ϑ i ’s coefficients alone, since one needs to take into account contributions fromGFRs. Indeed, the latter prove essential in order to ensure gauge independence of the UVcoefficients of gauge invariant operators, as we will soon explicitly show. V. GENERALIZED FIELD REDEFINITIONS
The first and most difficult step for carrying out the off-shell renormalization program isto work out the GFRs controlled by S ( Y (1) ). One needs to take them into account appro-priately, otherwise the renormalization of gauge invariant operators is affected by spuriouscontributions arising from the incorrect subtraction of UV divergences to be removed byGFRs. In particular GFRs play a crucial role in ensuring the gauge independence of the UVcoefficients of gauge invariant operators, as we will explicitly show.In the Algebraic Renormalization approach we adopt, GFRs can be written in terms oftwo classes of invariants as S Z d x h P (Φ; ζ )( σ ∗ σ + χ ∗ χ ) + Q (Φ; ζ )( σ ∗ ( σ + v ) + χ ∗ χ ) i , (5.1)with P and Q some local functionals depending on the fields (collectively denoted by Φ)and the external sources (collectively denoted by ζ ) and S the linearized ST operator inEq. (A3). For convenience, we refer to these terms as P - and Q -invariants.In order to get a better insight on the parameterization in Eq. (5.1) let us first consider We remind the reader that in EFTs field redefinitions are, in general, non-linear in the quantized fields. P and Q are constant. Since one has that S Z d x ( σ ∗ σ + χ ∗ χ ) = Z d x h σ δ Γ (0) δσ + χ δ Γ (0) δχ + σ ∗ eχω − χ ∗ e ( v + σ ) ω i ⊃ − Z d x evχ ∗ ω, (5.2)the P -invariant is fixed in this case by the amplitude Γ (1) ωχ ∗ . Similarly, if P depends on thefields and the gauge invariant sources ¯ c ∗ , R, T , U , the P -invariant can be fixed by looking atantifield-dependent 1-PI amplitudes. Indeed, since the antighost equation (B2) entails thatthe dependence on the antighost at loops higher than one only happens via the combination e χ ∗ in Eq. (B4), we do not need to consider antighost amplitudes and antifield-dependentones are sufficient.The Q -invariant is trickier. Let us first notice that it does not project on χ ∗ , σ ∗ antifield-dependent monomials: S Z d x ( σ ∗ ( σ + v ) + χ ∗ χ ) = Z d x h ( σ + v ) δ Γ (0) δσ + χ δ Γ (0) δχ + σ ∗ eχω − χ ∗ e ( v + σ ) ω i ⊃ Z d x v ¯ c ∗ − Z d x vm σ. (5.3)However, Eq. (5.3) clearly shows that it yields a contribution to ¯ c ∗ (and the σ -tadpole).To understand the Q -invariant role in the renormalization of the theory, we remark thatit depends only on the combination φ ; therefore it is useful to rewrite the counting operatorin terms of φ, φ † , i.e. , S Z d x ( σ ∗ ( σ + v ) + χ ∗ χ ) = S Z d x (cid:16) φ δ Γ (0) δφ + φ † δ Γ (0) δφ † (cid:17) . (5.4)Next, observe that we are only interested in the case when the right-hand side (r.h.s.) isevaluated at X , = 0 ; an explicit computation shows that the r.h.s. is indeed gauge-invariant (remember that we need to use the antifield e χ ∗ , as a consequence of the antighostequation): S Z d x (cid:16) φ δ Γ (0) δφ + φ † δ Γ (0) δφ † (cid:17) = Z d x h − φ † D φ − m v (cid:16) φ † φ − v (cid:17) φ † φ − ∂ µ T ( φ † D µ φ + h . c . ) − T ( φ † D φ + h . c . ) + 2¯ c ∗ φ † φ i . (5.5) X , -amplitudes being fixed in a purely algebraic way by Eq. (A8) σ ∗ , χ ∗ has disappeared; as a consequence thisinvariant contains a combination of gauge-invariant operators that vanish on-shell. Letus now consider what happens in the power-counting renormalizable case ( T = 0 and z = g i = 0). Imposing the mapping in Eq. (3.7) on the r.h.s. of Eq. (5.5) we obtain : S Z d x (cid:16) φ δ Γ (0) δφ + φ † δ Γ (0) δφ † (cid:17) = − Z d x h φ † D φ + 2 M v (cid:16) φ † φ − v (cid:17) − M (cid:16) φ † φ − v (cid:17)i . (5.6)On the other hand, the gauge-invariant operators of the renormalizable Abelian HK modelwith dimension ≤ Z d x F µν ; Z d x ( D µ φ ) † D µ φ ; Z d x (cid:16) φ † φ − v (cid:17) ; Z d x (cid:16) φ † φ − v (cid:17) , (5.7)whereas the number of physical parameters is 3, which are usually chosen to be: the gaugecoupling e associated with the coefficient of the field strength squared; the mass of the vectormeson M A , which is related to the renormalization of the vev via the tadpole invariant; and,finally, the mass of the physical scalar M , which appears with the quartic potential invariant.The scalar kinetic covariant term is related instead to the wave function renormalization ofthe two-point Higgs field and as such cannot have physical effects. If we denote by Z / the coefficient of the corresponding invariant (5.6), the combination in the r.h.s. of thatequation is exactly the one related to the wave function renormalization φ → (1 + Z / ) φ .Motivated by these remarks, we choose to express all Q -invariants in the X -theory of theform Z d x Q (Φ; ζ ) φ † D φ, (5.8)with Q (Φ; ζ ) gauge-invariant, as a linear combination of gauge invariant operators and co-homologically trivial invariants of the form S Z d x Q (Φ; ζ )( σ ∗ ( σ + v ) + χ ∗ χ ) . (5.9)This provides a consistent definition of the independent gauge invariant operators gener-alizing the corresponding set of independent physical parameters discussed in the power-counting renormalizable case. Observe that as announced the m -dependence has disappeared.
12e also notice that in the Landau gauge ( ξ = 0) ghosts are free and the theory enjoysan exact global invariance δφ = ieαφ ; δφ † = − ieαφ † (5.10)with α a constant parameter. As a consequence of this rigid U (1) invariance the only allowedcohomologically trivial invariants in the Landau gauge are those of the Q -type; P -invariantsdo not arise. We will verify this property in the explicit computations that follow. On theother hand, notice that in a general gauge, Q need not be gauge-invariant and both P and Q -type invariants are required, due to the fact that the vev renormalizes differently thanthe fields, as is well known in the literature [24].We now list the monomials in the expansion of P, Q contributing to the projections neededto fix the coefficients of the dimension 6 operators in Eqs. (D1), (D2j) and (D3). Using thenotation Z ≡ ( σ ∗ σ + χ ∗ χ ); Z ≡ ( σ ∗ ( σ + v ) + χ ∗ χ ) , (5.11)we obtain Y (1) = S Z d x h(cid:16) ρ + ρ σ + ρ σ + ρ χ + ρ T T (cid:17) Z + (cid:16) ˜ ρ + ˜ ρ σ + ˜ ρ σ + ˜ ρ χ + ˜ ρ σχ + ˜ ρ T T + ˜ ρ T T T + ˜ ρ T T σ + ˜ ρ T T χ (cid:17) Z i . (5.12)The different coefficients can be then evaluated by projection onto the relevant Feynmanamplitudes; their values are then ρ = (1 − δ ξ ;0 )8 π v M A z ǫ ; ρ = − (1 − δ ξ ;0 )4 π v zM A (1 + z ) , (5.13a) ρ = (1 − δ ξ ;0 )8 π v z (3 z − M A (1 + z ) ǫ ; ρ = − (1 − δ ξ ;0 )8 π v zM A (1 + z ) ǫ , (5.13b) ρ T = − (1 − δ ξ ;0 )8 π v M A (1 + z ) ǫ ; ˜ ρ = (1 − δ ξ ;1 )16 π v M A , (5.13c)˜ ρ = − (1 − δ ξ ;1 )8 π v zM A z ; ˜ ρ = (1 − δ ξ ;1 )8 π v z ( z − M A (1 + z ) , (5.13d)˜ ρ = ( − δ ξ ;0 π v zM A z ; ˜ ρ = − ( − δ ξ ;0 z [3 z + ( − δ ξ ;0 ]16 π v (1 + z ) M A ǫ , (5.13e)˜ ρ T = − (1 − δ ξ ;1 )8 π v M A ǫ ; ˜ ρ T T = (1 − δ ξ ;1 )8 π v M A ǫ , (5.13f)˜ ρ T = (1 − δ ξ ;0 )8 π v z (2 + z ) M A (1 + z ) ǫ ; ˜ ρ T = 0 . (5.13g)13otice that in Landau gauge Y (1) reduces to Y (1) (cid:12)(cid:12)(cid:12) ξ =0 = S Z d x M A π v ǫ h − T + 4 T − v z z (cid:16) φ † φ − v (cid:17) + 2 v z (3 z − z ) (cid:16) φ † φ − v (cid:17) i Z , (5.14) i.e. , the polynomial Q is gauge-invariant, as expected; moreover, as anticipated, all ρ ’scoefficients vanish in this gauge. A. GFRs in the target theory
It is instructive to obtain the explicit form of the GFRs in the target theory at linearorder in the higher dimensional couplings. For that purpose we need to apply the mappingin Eq. (3.7) to Y (1) retaining only the terms linear in the g i ’s and z .We remark that the coefficients in Eq. (5.13) only depend on z . Moreover, the imageof the source T under the mapping is proportional to g and hence from the T sectorwe receive contributions at the linearized level only from amplitudes linear in T , whosecoefficients need to be evaluated at z = 0. By taking these observations into account, oneeasily sees that the GFRs in the target theory at linear order in the g i ’s and z couplingstake the following form: φ ′ χ ′ = ( M A π v " − z )(1 − δ ξ ;0 ) + (1 − δ ξ ;1 ) − h (1 − δ ξ ;0 ) (cid:16) g v Λ + 2 zv (cid:17) + (1 − δ ξ ;1 ) (cid:16) g v Λ + zv (cid:17)i σ − (cid:16) zv + g Λ (cid:17) σ − (cid:16) zv + g Λ (cid:17) χ − zv σχ + · · · ǫ σχ + ( M A π v " − δ ξ ;1 − − δ ξ ;1 ) (cid:16) zv + g v Λ (cid:17) σ − zv σχ − (1 − δ ξ ;1 ) (cid:16) zv + g Λ (cid:17) σ − h ( − δ ξ ;0 zv + (1 − δ ξ ;1 ) g Λ i χ + · · · ǫ v , (5.15)where the dots denote higher dimensional contributions that are not relevant in the oneloop renormalization of the dimension 6 operators under consideration. Notice also thatthe contribution proportional to the constant spinor ( v, T is associated with the Q -typeinvariants. 14rom Eq.(5.15) we see that the GFRs are non-multiplicative already at one loop and inthe linearized approximation. VI. RENORMALIZATION OF GAUGE INVARIANT OPERATORS
Once the cohomologically trivial sector has been fixed as in Eq.(5.12) and (5.13) we canproceed to project on the one-loop amplitudes required to determine the coefficients of theinvariants (D1), (D2j) and (D3). As the methodology is illustrated in detail in Ref. [17],we report here only the results, which have been explicitly evaluated in both Landau andFeynman gauge and found to coincide as required.
A. Pure external sources invariants
The non zero ϑ i coefficients are ϑ = − π M + (1 + z ) M A (1 + z ) ǫ ; ϑ = 116 π − M + 3(1 + z ) M A (1 + z ) ǫ , (6.1a) ϑ = 3 M A π ǫ ; ϑ = − π M (1 + z ) ǫ , (6.1b) ϑ = 116 π z + z (1 + z ) ǫ ; ϑ = 316 π M + (1 + z ) M A (1 + z ) ǫ , (6.1c) ϑ = 9 M A π ǫ ; ϑ = 14 π (1 + z ) ǫ , (6.1d) ϑ = 332 π M + (1 + z ) M A (1 + z ) ǫ ; ϑ = 3 M A π ǫ , (6.1e) ϑ = 18 π h M A + M (1 + z ) i ǫ ; ϑ = 18 π (1 + z ) ǫ , (6.1f) ϑ = 3 M A π ǫ ; ϑ = 14 π M A (1 + z ) ǫ , (6.1g) ϑ = 132 π z + z (1 + z ) ǫ ; ϑ = 3 M A π ǫ , (6.1h) ϑ = 116 π (1 + z ) ǫ ; ϑ = − π M A (1 + z ) ǫ , (6.1i) ϑ = 144 M A π ǫ ; ϑ = − π z + 3 z + z (1 + z ) ǫ , (6.1j) ϑ = − π M + 2(1 + z ) M A (1 + z ) ǫ ; ϑ = − π z ) ǫ , (6.1k)15 = 3 M A π ǫ ; ϑ = − M π (1 + z ) ǫ , (6.1l) ϑ = 18 M A π ǫ ; ϑ = − π z ) ǫ . (6.1m) B. Mixed field-external sources invariants
The non zero θ i coefficients are θ = − π v z ) h − z ) M + 2(1 + z ) M A + (2 + 4 z + 3 z + z ) m i ǫ , (6.2a) θ = 18 π v z ) n ( z − M + 6(1 + z ) M A − (1 + z ) h M + (1 + z ) M A i m o ǫ , (6.2b) θ = 3 M A π v ǫ , (6.2c) θ = − π v z ) h (1 + z ) m + 4 M i ǫ , (6.2d) θ = − π v z ) h − z (1 + z ) + 4(1 + z ) g v Λ + (2 + z ) g v Λ i ǫ , (6.2e) θ = − π v z ) n z + 3 z + z ) m − (1 + z ) h − z + − (cid:16) g v Λ (cid:17) z + 3 g v Λ i M A + h g v Λ (3 + z ) i M , (6.2f) θ = − M A π v z ) h − z + g v Λ (cid:16) g v + 4Λ (cid:17)i ǫ , (6.2g) θ = − g v π Λ z ) ǫ , (6.2h) θ = − g π Λ z ) ǫ , (6.2i) θ = − π v z ) h − z ) M A + (cid:16) g v Λ (cid:17) M + (2 + 4 z + 3 z + z ) m i ǫ , (6.2j) θ = 3 M A π v ǫ , (6.2k) θ = − g π Λ z ) ǫ , (6.2l) θ = 18 π v z (1 + z ) h (1 + z ) (5 + z ) M A + 4(2 − z ) M + 4(1 + z ) m i ǫ , (6.2m)16 = − π v z ) n (1 + z ) (2 + 3 z + 3 z + z ) m + 4(1 + z ) m h (1 − z ) M + (1 + z ) M A i + 4 h − z ) M A + (1 − z + z ) M io ǫ , (6.2n) θ = 6 M A π v ǫ , (6.2o) θ = 12 π v z ) h z − M + ( z − m i ǫ , (6.2p) θ = g v π Λ z ) ǫ , (6.2q) θ = − π v Λ z ) n − g v M + (1 + z ) h (2 + z ) g v + 4 g v z (2 g v + Λ )+ 4 z (8 g v + 4 g v Λ − (1 + z )Λ i M A o ǫ , (6.2r) θ = − g v π Λ z ) h M + 2(1 + z ) M A i ǫ , (6.2s) θ = g v π Λ z ) ǫ , (6.2t) θ = − π v z (1 + z ) ǫ , (6.2u) θ = 18 π v z ) h − z ) M + 6(1 + z ) M A + (1 + z )(3 M + 2(1 + z ) M A ) m i ǫ , (6.2v) θ = 36 M A π v ǫ , (6.2w) θ = 1 π v z ) ǫ , (6.2x) θ = 18 π v z ) h (2 − z ) M + 2(1 + z ) M A + (2 + 5 z + 6 z + 4 z + z ) m i ǫ , (6.2y) θ = 14 π v (1 − z )(1 + z ) ǫ , (6.2z) θ = 6 M A π v ǫ , (6.2aa) θ = 14 π v z ) h − z ) M + (1 + z ) m i ǫ . (6.2bb)17 . Gauge invariants depending only on the fields The non zero λ i coefficients are λ = 116 π v z ) n (1 + z ) h M + (1 + z ) M A i m + 2 h M + 3(1 + z ) M A io ǫ , (6.3a) λ = 132 π v z ) n − z ) M + 4 m M A (1 + z ) + 12(1 + z ) M A + 4 m M (1 − z ) + (1 + z ) (2 + 2 z + z ) m o ǫ , (6.3b) λ = − π v z (1 + z ) n − z ) M + 2(1 + z ) m + (1 + z ) h − z ) M + (1 + z ) (5 + z ) M A i m o ǫ , (6.3c) λ = − π v z ) n (1 + z ) h
16 + 4 z + 3 g v Λ + 12 g v Λ i M A + g v Λ (cid:16) − g v Λ (cid:17) M o ǫ , (6.3d) λ = g v π Λ z ) 1 ǫ , (6.3e) λ = 164 π v z ) nh z + 4(1 − z ) g v Λ + (1 + z ) g v Λ i M + (1 + z ) (cid:16) z − g v Λ − g v Λ (cid:17) M A + 4(1 + z ) g v Λ m o ǫ , (6.3f) λ = 132 π v z ) nh z − z ) g v Λ + (5 + z ) g v Λ i M (6.3g)+ 3(1 + z ) h z (3 + z ) − z ) g v Λ − (5 + 3 z ) g v Λ i M A + (1 + z ) h − z (1 + z ) + 4(1 + z ) g v Λ + (2 + z ) g v Λ i m o ǫ , (6.3h) λ = − π v z ) n g v Λ M + (1 + z ) h (cid:16) g v Λ + g v Λ (cid:17) + 2 g v Λ + 24 g g v Λ + g v Λ i M A o ǫ , (6.3i) λ = − g v π Λ z ) 1 ǫ , (6.3j) λ = 1128 π v Λ z ) n − g v Λ M + (1 + z ) h g v Λ − g v Λ − z ) g g v Λ − z ) (cid:16) g v Λ + g (cid:17)i M A − z ) g v Λ m o ǫ . (6.3k)18 II. MAPPING TO THE TARGET THEORY
The UV coefficients in the target theory ˜ λ i can be obtained by: applying the mappingin Eq. (3.7) to the invariants in Eqs. (D1) and (D2j); combining the projections with theoperators in (D3); and, finally, using the results (6.1), (6.2) and (6.3). Notice that for thesecoefficients all m -dependent contributions must cancel out; we have checked this explicitly.The coefficients so obtained represents the complete one-loop renormalizations of thecorresponding operators; in particular, no linearized approximation in the higher dimensionalcouplings g i ’s has been made so far. However, as the resulting general expressions are ratherlengthy, we report below the non zero coefficients ˜ λ i at linear order in the g i couplings:˜ λ ∼ − π v h zM A + ( M − M A ) g v Λ − M A g v Λ + M g v Λ i ǫ , (7.1a)˜ λ ∼ − π v h(cid:16) M M A + 42 M (cid:17) z + 4 (cid:16) M + M M A − M A (cid:17) g v Λ − g v Λ M A + (cid:16) M + M A (cid:17) g v Λ i ǫ , (7.1b)˜ λ ∼ − π v h zM (18 M + 5 M A ) + 2(5 M + 2 M M A − M A ) g v Λ − M A g v Λ + (8 M + M A ) g v Λ i ǫ , (7.1c)˜ λ ∼ π v h zM A − (3 M + 7 M A ) g v Λ i ǫ , (7.1d)˜ λ ∼ π v h z (5 M + 3 M A ) − M A g v Λ + 12 M A g v Λ i ǫ , (7.1e)˜ λ ∼ π v (4 M + 11 M A ) (cid:16) z − g v Λ (cid:17) ǫ , (7.1f)˜ λ ∼ − π v h g v Λ ( M + 5 M A ) + M A (cid:16) − z + g v Λ (cid:17)i ǫ , (7.1g)˜ λ ∼ − π v g v Λ (2 M + M A ) 1 ǫ . (7.1h)We hasten to emphasize that GFRs do contribute also at the linearized level, as has beendiscussed in detail in Section V A. Failure to take their contributions into account wouldlead to an erroneous determination of the coefficients in Eq.(7.1).The g i ’s, z contributions to the β functions β i = (4 π ) dd log µ ˜ λ i (7.2)can then be easily determined from Eq. (7.1), leading to: β ⊃ − v h zM A + ( M − M A ) g v Λ − M A g v Λ + M g v Λ i , (7.3a)19 ⊃ − v h(cid:16) M M A + 42 M (cid:17) z + 4 (cid:16) M + M M A − M A (cid:17) g v Λ − g v Λ M A + (cid:16) M + M A (cid:17) g v Λ i , (7.3b) β ⊃ − v h zM (18 M + 5 M A ) + 2(5 M + 2 M M A − M A ) g v Λ − M A g v Λ + (8 M + M A ) g v Λ i , (7.3c) β ⊃ v h zM A − (3 M + 7 M A ) g v Λ i , (7.3d) β ∼ v h z (5 M + 3 M A ) − M A g v Λ + 12 M A g v Λ i , (7.3e) β ⊃ v (4 M + 11 M A ) (cid:16) z − g v Λ (cid:17) , (7.3f) β ⊃ − v h g v Λ ( M + 5 M A ) + M A (cid:16) − z + g v Λ (cid:17)i , (7.3g) β ∼ − v g v Λ (2 M + M A ) . (7.3h) VIII. ONE-LOOP MIXING MATRICES
We are now in a position to compare our results with those in the literature [25]. Byinspecting Eq.(7.1) we obtain the mixing matrix represented in Table I. We find agreementwith the results of [25] with the exception of the mixing of φ D operators with F φ . Morespecifically, a closer inspection of Eq. (7.1) shows that the operator I = Z d x (cid:16) φ † φ − v (cid:17) ( D µ φ ) † D µ φ, (8.1)respects the mixing pattern derived in [25], whereas the operator I = Z d x (cid:16) φ † φ − v (cid:17) ( φ † D φ + h . c . ) , (8.2)does not since it mixes with I = Z d x F µν (cid:16) φ † φ − v (cid:17) . (8.3)There is an elegant cohomological interpretation of this result. One can find S -invariantcombinations of gauge invariant operators that do not depend on the antifields, in very muchthe same way as in Eq. (5.14). Notice that these invariants depend on σ, χ only via φ and20 φ φ D φ F φ φ D × φ TABLE I. One-loop operator mixing matrix in the Abelian HK model. Shaded entries denote a van-ishing coefficient. The × indicates an entry that should vanish according to the non-renormalizationtheorem of [25] but that does not given the coefficients in Eq. (7.1). they are generated by Z (now to be understood in the target theory). In particular onefinds S Z d x Z = Z d x (cid:16) φ δSδφ + φ † δSδφ † (cid:17) = Z d x h − ( D φ ) † φ − φ † D φ − M v (cid:16) φ † φ − v (cid:17) φ † φ i , (8.4)which is gauge-invariant. Thus any invariant of the form S Z d x Q ( φ, φ † , A µ ) Z , (8.5)is gauge invariant if Q is a gauge-invariant polynomial. Being cohomologically trivial, theabove family of invariants can be added order by order in the loop expansion without chang-ing the physical observables of the theory. Intuitively the simultaneous variation of the co-efficients of the operators entering in the invariants (8.5) cannot affect the physics since thevariation is proportional to the equations of motion.This is an example of the aforementioned fact that the mixing between gauge-invariantand alien operators (which are cohomologically trivial with respect to the linearized SToperator) can be made to vanish by a suitable basis choice in the space of local operators [2–6]. This means that there is the freedom to replace the invariant I with the linear com-bination of I and I in Eq. (E16) up to a cohomologically trivial S -invariant. Thistransformation induces the following shift on the space of the ˜ λ ’s parameters: e λ → e λ − M e λ ; e λ → e λ − M v e λ . (8.6)For this new basis then, the non-renormalization theorem of [25] hold true.21 λ e λ e λ e λ e λ e λ e λ e λ e λ e λ zg g g TABLE II. Dependence of the e λ i ’s on the higher dimensional coupling constants. Shaded entriesdenote that the dependence of the e λ i parameter on the corresponding coupling constant vanishes. In order to study the one-loop amplitudes dependence on the g i ’s and z beyond the singlehigher-dimensional operator insertion approximation commonly used in the literature, wehave reported in Table II the dependence of the (shifted) ˜ λ ’s coefficients on the g i ’s and z ,based on the full one loop computation carried out in the present paper.The vanishing entries in Table II can be partially understood in terms of the underlyingamplitudes decomposition made transparent by the X -formalism. As explained above, the e λ ’s are a linear combination of the λ ’s coefficients multiplying gauge invariant operatorswhich are independent from external sources of the X -theory, and of the coefficients ϑ, θ ’sassociated with invariants involving external sources insertions (the UV behaviour of whichis more constrained than that of the fields). In particular, we find for the relevant operatorsin Table II : e λ = λ + g ϑ Λ ; e λ = λ ; e λ = λ + g ϑ Λ ; e λ = λ . (8.7)The ϑ -terms can be neglected: they can only induce a z -dependence and thus do notcontribute to the cancellations in Table II. Hence, the problem is reduced to the determi-nation of the g i ’s dependence of the λ ’s coefficients in the X -theory. One immediately seesthat these coefficients cannot depend on g since this is a trilinear vertex in X that doesnot contribute to the 1-PI amplitudes of the starting theory at one loop. Thus, the lastrow of Table II must hold, as the only possible dependence on g at one loop arises fromthe mapping to the target theory in Eq.(3.7) and therefore governed by external amplitudesinvolving ¯ c ∗ and/or R external sources, which do not enter in Eq. (8.7).The remaining three forbidden dependences just seem to be an accidental consequenceof the one-loop Feynman diagrams; as a result, cancellation patterns do not seem to lendthemselves to an easy generalization to higher orders.22 X. CONCLUSIONS
In the present paper we have completed the investigation of the one-loop off-shell renor-malization of the Abelian Higgs-Kibble model supplemented at tree-level with all dimension6 parity preserving on-shell inequivalent gauge-invariant operators. This was the last steptowards the analysis of the SU(2) × U(1) case.We have shown that the X -theory formalism provides an effective way to work out therelevant GFRs, which in turn are found to have an ubiquitous effect on the one-loop UVcoefficients of dimension 6 operators. In fact, since the GFRs are non linear and evennon polynomial in the fields, it is advantageous to employ cohomological tools in orderto disentangle the UV coefficients of the gauge-invariant operators from the spurious (andgauge-dependent) contributions associated with GFRs.We have provided a full one-loop computation going beyond the customary linearizedapproximation in the higher dimensional couplings. All coefficients have been evaluatedboth in Feynman and in Landau gauge and the gauge independence of the UV coefficientsof the gauge invariant operators explicitly checked. As expected, it does not hold unless theeffects of GFRs are properly accounted for.We find that the pattern of operator mixing cancellations studied in the previous literatureonly holds off-shell if an appropriate choice of the on-shell equivalent operators is made. Thiscan be traced back to the freedom of adding cohomologically trivial combinations of gauge-invariant operators at one loop order, thus selecting a particular basis of gauge-invarianton-shell inequivalent operators.Application of the method presented to the SMEFT is currently under investigation.23 ppendix A: Functional Identities in the X -theory1. ST identities The ST identity (also known as the master equation in the BV approach) associated tothe gauge group BRST symmetry reads S (Γ) = Z d x h ∂ µ ω δ Γ δA µ + δ Γ δσ ∗ δ Γ δσ + δ Γ δχ ∗ δ Γ δχ + b δ Γ δ ¯ ω i = 0 , (A1)or, at order n in the loop expansion, S (Γ) ( n ) = S (Γ ( n ) ) + n − X j =1 (cid:16) δ Γ ( j ) δσ ∗ δ Γ ( n − j ) δσ + δ Γ ( j ) δχ ∗ δ Γ ( n − j ) δχ (cid:17) = 0 , (A2)where S is the linearized ST operator: S (Γ ( n ) ) = Z d x h ∂ µ ω δ Γ ( n ) δA µ + eω ( σ + v ) δ Γ ( n ) δχ − eωχ δ Γ ( n ) δσ + b δ Γ ( n ) δ ¯ ω + δ Γ (0) δσ δ Γ ( n ) δσ ∗ + δ Γ (0) δχ δ Γ ( n ) δχ ∗ i = s Γ ( n ) + Z d x h δ Γ (0) δσ δ Γ ( n ) δσ ∗ + δ Γ (0) δχ δ Γ ( n ) δχ ∗ i . (A3) S maps the antifields σ ∗ , χ ∗ into the equations of motion of the fields σ, χ , while it acts onthe fields as the BRST operator s . Notice that, as explained before, we do not introducean antifield for the gauge field A µ since in the Abelian case treated here the gauge BRSTtransformation is linear.The ST identity for the constraint BRST symmetry is S C (Γ) ≡ Z d x h vc δ Γ δX + δ Γ δ ¯ c ∗ δ Γ δ ¯ c i = Z d x h vc δ Γ δX − ( (cid:3) + m ) c δ Γ δ ¯ c ∗ i = 0 , (A4)where in the latter equality we have used the fact that both the ghost c and the antighost¯ c are free: δ Γ δ ¯ c = − ( (cid:3) + m ) c ; δ Γ δc = ( (cid:3) + m )¯ c. (A5) X , -equations By using Eq. (A5) one sees that Eq. (A4) reduces to the X -equation of motion δ Γ δX = 1 v ( (cid:3) + m ) δ Γ δ ¯ c ∗ . (A6)24otice that this equation stays the same irrespectively of the presence of higher-dimensionalgauge invariant operators added to the power-counting renormalizable action.The X -equation is in turn given by δ Γ δX = 1 v ( (cid:3) + m ) δ Γ δ ¯ c ∗ + g v Λ δ Γ δT + g v Λ δ Γ δU + g v δ Γ δR − ( (cid:3) + m ) X − h (1 + z ) (cid:3) + M i X − v ¯ c ∗ . (A7)
3. Solving the X , -equations At order n , n ≥ X , -equations reduce to δ Γ ( n ) δX = 1 v ( (cid:3) + m ) δ Γ ( n ) δ ¯ c ∗ , (A8a) δ Γ ( n ) δX = 1 v ( (cid:3) + m ) δ Γ ( n ) δ ¯ c ∗ + g v Λ δ Γ ( n ) δT + g v Λ δ Γ ( n ) δU + g v δ Γ ( n ) δR . (A8b)By using the chain rule for functional differentiation it is straightforward to see thatEqs. (A8) entail that Γ ( n ) only depends on the combinations:¯ c ∗ = ¯ c ∗ + 1 v ( (cid:3) + m )( X + X ); T = T + g v Λ X , U = U + g v Λ X ; R = R + g v X . (A9a) Appendix B: The b - and the gauge ghost equation The set of the functional identities holding in the X -formulation of the Abelian HK modelis completed by: • The b -equation: δ Γ δb = ξb − ∂A − ξevχ ; (B1) • The antighost equation: δ Γ δ ¯ ω = (cid:3) ω + ξev δ Γ δχ ∗ . (B2)25t orders n ≥ b - and the antighost equations imply δ Γ ( n ) δb = 0; δ Γ ( n ) δ ¯ ω = ξev δ Γ ( n ) δχ ∗ , (B3)so that at higher orders the vertex functional does not depend on the Nakanishi-Lautrupfield b and the dependence on the antighost is only via the combination e χ ∗ ≡ χ ∗ + ξev ¯ ω. (B4) Appendix C: Propagators1. The X − σ sector Diagonalization of the quadratic part of the action in this sector is achieved by setting σ = σ ′ + X + X . Then one has∆ σ ′ σ ′ = ip − m ; ∆ X X = − ip − m ; ∆ X X = i (1 + z ) p − M . (C1)Several comments are in order here. At g , g , g = 0 no higher dimensional interactionsvertices are present. However, the model is still non power-counting renormalizabile, sincethe derivative interaction of the X , -system ∼ ( X + X ) (cid:3) ( φ † φ ) violates power-countingrenormalizability as a consequence of the fact that the combination X ≡ X + X has apropagator falling down as 1 /p for large p at z = 0, as can be seen from Eq. (C1):∆ XX = ∆ X X + ∆ X X ∼ − iz z p . (C2)On the other hand at z = 0 ∆ XX goes as 1 /p for large momenta and this compensates thetwo momenta from the Xφ † φ interaction vertex, giving rise to a power-counting renormal-izable model (at zero g i ’s) [21].
2. The gauge and ghost sector
The diagonalization in the gauge sector is obtained by redefining the Nakanishi-Lautrupmultiplier field b ′ = b − ξ ∂A − evχ. (C3)26hen, the A µ -propagator is∆ µν = − i (cid:16) p − M A T µν + 1 ξ p − M A (cid:17) ; M A = ev, (C4)whereas the the Nakanishi-Lautrup, pseudo-Goldstone and ghost propagators are∆ b ′ b ′ = i ξ ; ∆ χχ = ip − ξM A ; ∆ ¯ ωω = ip − ξM A . (C5)As usual, ξ = 0 corresponds to the Landau gauge, whereas ξ = 1 is the Feynman gauge.Finally, the ghost associated to the constraint BRST symmetry is free:∆ ¯ cc = − ip − m . (C6)27 ppendix D: List of Gauge-invariant Operators1. Pure external sources invariants ϑ Z d x ¯ c ∗ ; ϑ Z d x T ; ϑ Z d x U ; ϑ Z d x R, (D1a) ϑ Z d x (¯ c ∗ ) ; ϑ Z d x T ; ϑ Z d x U ; ϑ Z d x R , (D1b) ϑ Z d x ¯ c ∗ (cid:3) ¯ c ∗ ; ϑ Z d x T (cid:3) T ; ϑ Z d x U (cid:3) U ; ϑ Z d x R (cid:3) R, (D1c) ϑ Z d x ¯ c ∗ T ; ϑ Z d x ¯ c ∗ U ; ϑ Z d x ¯ c ∗ R ; ϑ Z d x T U, (D1d) ϑ Z d x T R ; ϑ Z d x U R ; ϑ Z d x ¯ c ∗ (cid:3) T ; ϑ Z d x ¯ c ∗ (cid:3) U, (D1e) ϑ Z d x ¯ c ∗ (cid:3) R ; ϑ Z d x T (cid:3) U ; ϑ Z d x T (cid:3) R ; ϑ Z d x U (cid:3) R, (D1f) ϑ Z d x (¯ c ∗ ) ; ϑ Z d x T ; ϑ Z d x U ; ϑ Z d x R , (D1g) ϑ Z d x (¯ c ∗ ) T ; ϑ Z d x (¯ c ∗ ) U ; ϑ Z d x (¯ c ∗ ) R ; ϑ Z d x ¯ c ∗ T , (D1h) ϑ Z d x ¯ c ∗ U ; ϑ Z d x ¯ c ∗ R ; ϑ Z d x ¯ c ∗ T U ; ϑ Z d x ¯ c ∗ T R, (D1i) ϑ Z d x ¯ c ∗ U R ; ϑ Z d x T U ; ϑ Z d x T R ; ϑ Z d x T U , (D1j) ϑ Z d x T R ; ϑ Z d x T U R ; ϑ Z d x U R ; ϑ Z d x U R . (D1k)
2. Mixed field-external sources invariants θ Z d x ¯ c ∗ (cid:16) φ † φ − v (cid:17) ; θ Z d x T (cid:16) φ † φ − v (cid:17) ; θ Z d x U (cid:16) φ † φ − v (cid:17) , (D2a) θ Z d x R (cid:16) φ † φ − v (cid:17) ; θ Z d x ¯ c ∗ ( D µ φ ) † D µ φ ; θ Z d x T ( D µ φ ) † D µ φ, (D2b) θ Z d x U ( D µ φ ) † D µ φ ; θ Z d x R ( D µ φ ) † D µ φ ; θ Z d x ¯ c ∗ (cid:16) φ † D φ + h . c . (cid:17) , (D2c)28 Z d x T (cid:16) φ † D φ + h.c. (cid:17) ; θ Z d x U (cid:16) φ † D φ + h . c . (cid:17) ; θ Z d x R (cid:16) φ † D φ + h . c . (cid:17) , (D2d) θ Z d x ¯ c ∗ (cid:16) φ † φ − v (cid:17) ; θ Z d x T (cid:16) φ † φ − v (cid:17) ; θ Z d x U (cid:16) φ † φ − v (cid:17) , (D2e) θ Z d x R (cid:16) φ † φ − v (cid:17) ; θ Z d x ¯ c ∗ F µν ; θ Z d x T F µν , (D2f) θ Z d x U F µν ; θ Z d x RF µν ; θ Z d x (¯ c ∗ ) (cid:16) φ † φ − v (cid:17) , (D2g) θ Z d x T (cid:16) φ † φ − v (cid:17) ; θ Z d x U (cid:16) φ † φ − v (cid:17) ; θ Z d x R (cid:16) φ † φ − v (cid:17) , (D2h) θ Z d x ¯ c ∗ T (cid:16) φ † φ − v (cid:17) ; θ Z d x ¯ c ∗ U (cid:16) φ † φ − v (cid:17) ; θ Z d x ¯ c ∗ R (cid:16) φ † φ − v (cid:17) , (D2i) θ Z d x T U (cid:16) φ † φ − v (cid:17) ; θ Z d x T R (cid:16) φ † φ − v (cid:17) ; θ Z d x U R (cid:16) φ † φ − v (cid:17) . (D2j)
3. Gauge invariants depending only on the fields λ Z d x (cid:16) φ † φ − v (cid:17) ; λ Z d x (cid:16) φ † φ − v (cid:17) , (D3a) λ Z d x (cid:16) φ † φ − v (cid:17) ; λ Z d x ( D µ φ ) † D µ φ, (D3b) λ Z d x (cid:16) φ † D ( µνµν ) φ + h . c . (cid:17) ; λ Z d x (cid:16) φ † φ − v (cid:17) ( φ † D φ + h . c . ) , (D3c) λ Z d x (cid:16) φ † φ − v (cid:17) ( D µ φ ) † D µ φ ; λ Z d x F µν , (D3d) λ Z d x ∂ ρ F ρµ ∂ σ F σµ ; λ Z d x F µν (cid:16) φ † φ − v (cid:17) , (D3e)where D ( µνµν ) denotes complete symmetrization over µ, ν : D ( µνµν ) φ ≡ [( D ) + D µ D ν D µ D ν + D µ D D µ ] φ. (D4)Notice that in the text we have denoted by I j the invariant with coefficient λ j .29 ppendix E: On-shell Reduction of dim.6 Field-Dependent Gauge Invariant Oper-ators We consider in this Appendix the on-shell reduction of dimension 6 operators in the targettheory. The relevant classical gauge-invariant action S is obtained from the first four linesof Eq.(2.1) by going on-shell with X , .The corresponding equations of motion for the gauge field and the scalar φ are δSδA µ = ∂ ρ F ρµ + i h φ † D µ φ − ( D µ φ ) † φ i , (E1a) δSδφ = − ( D φ ) † − M v (cid:16) φ † φ − v (cid:17) φ † , (E1b) δSδφ † = − ( D φ ) − M v (cid:16) φ † φ − v (cid:17) φ. (E1c)Since we will be interested only in the one-loop corrections that are linear in the g i ’s and z we can limit ourselves to the leading order equations of motion in Eq. (E1); also we recallhere the identity [ D µ , D ν ] = − iF µν . (E2)The on-shell independent dimension 6 operators can be chosen to be I , I and I .Notice that the operator in the tree-level vertex functional Z d x (cid:16) φ † φ − v (cid:17) (cid:3) (cid:16) φ † φ − v (cid:17) = Z d x (cid:16) φ † φ − v (cid:17)h ( D φ ) † φ + φ † ( D φ ) + 2( D µ φ ) † D µ φ i , (E3)can be represented in terms of invariants in the contractible pairs basis as in the r.h.s. ofthe above equation. Therefore we just need to reduce all I i ’s invariants in terms of I , I and I by using the equations of motion (E1).Let us start from I . This operator contains three terms, namely: Z d x φ † D φ ; Z d x φ † D µ D D µ φ ; Z d x φ † D µ D ν D µ D ν φ. (E4)Then one finds that: • Integration by parts gives: Z d x φ † ( D ) φ = Z d x ( D φ ) † D φ ∼ Z d x M v (cid:16) φ † φ − v (cid:17) φ † φ (E5)30here the equations of motion for φ, φ † have been used in the last line. Hence weobtain Z d x φ † ( D ) φ ∼ Z d x n M v (cid:16) φ † φ − v (cid:17) + M v (cid:16) φ † φ − v (cid:17) o = M v I + M v I . (E6) • The second term can be rewritten as follows Z d x φ † D µ D D µ φ = Z d x φ † D µ D ρ D µ D ρ φ + Z d x φ † D µ D ρ [ D ρ , D µ ] φ = Z d x φ † D µ D ρ D µ D ρ φ − i Z d x ( D ρ D µ φ ) † F ρµ φ, (E7)where in the last line we have used Eq. (E2) and integrated by parts. Now − i Z d x ( D ρ D µ φ ) † F ρµ φ = − i Z d x ([ D ρ , D µ ] φ ) † F ρµ φ = − Z d x F ρµ φ † φ, again by using Eq. (E2). Eventually we arrive at the result Z d x φ † D µ D D µ φ = Z d x φ † D µ D ρ D µ D ρ φ − Z d x F ρµ (cid:16) φ † φ − v (cid:17) − v Z d x F ρµ = Z d x φ † D µ D ρ D µ D ρ φ − v I − I . (E8) • We are finally left with the decomposition of the last term in Eq. (E4). One has Z d x φ † D µ D ρ D µ D ρ φ = Z d x h φ † D φ + φ † D µ [ D ρ , D µ ] D ρ φ i = Z d x h ( D φ ) † D φ − iF µρ ( D µ φ ) † D ρ φ i , (E9)where we have used Eq. (E2) and integrated by parts. It is convenient to split the lastterm in the above equation as follows i Z d x F µρ ( D µ φ ) † D ρ φ = i Z d x F µρ n ( D µ φ ) † D ρ φ + ( D µ φ ) † D ρ φ } = Z d x n − i ∂ ρ F ρµ h φ † D µ φ − ( D µ φ ) † φ i − i F µρ h φ † [ D µ , D ρ ] φ + ([ D ρ , D µ ] φ ) † φ io = Z d x n − i ∂ ρ F ρµ h φ † D µ φ − ( D µ φ ) † φ i − F µρ φ † φ o . (E10)31y using the A µ -equation of motion (E1) the first term in the last line of the aboveequation becomes − i Z d x ∂ ρ F ρµ h φ † D µ φ − ( D µ φ ) † φ i ∼− Z d x (cid:16) φ † D µ φ − ( D µ φ ) † φ (cid:17)(cid:16) φ † D µ φ − ( D µ φ ) † φ (cid:17) = Z d x n φ † φ ( D µ φ ) † D µ φ − h φ † D µ φ φ † D µ φ + h . c . io . (E11)Integrating by parts the last term in the last line of the above equation one finds − Z d x h φ † D µ φ φ † D µ φ + h . c . i = Z d x n φ † φ ( D µ φ ) † D µ φ + 12 φ † φ h φ † D φ + ( D φ ) † φ io , (E12)and thus − i Z d x ∂ ρ F ρµ h φ † D µ φ − ( D µ φ ) † φ i ∼− Z d x (cid:16) φ † D µ φ − ( D µ φ ) † φ (cid:17)(cid:16) φ † D µ φ − ( D µ φ ) † φ (cid:17) = Z d x n φ † φ ( D µ φ ) † D µ φ + 12 φ † φ h φ † D φ + ( D φ ) † φ io . (E13)Putting everything together we find Z d x φ † D µ D ρ D µ D ρ φ ∼ = Z d x n ( D φ ) † D φ − φ † φ ( D µ φ ) † D µ φ − φ † φ h φ † D φ + ( D φ ) † φ i + 12 F µρ φ † φ o ∼ Z d x n M v (cid:16) M v (cid:17)(cid:16) φ † φ − v (cid:17) − (cid:16) φ † φ − v (cid:17) ( D µ φ ) † D µ φ + 12 F µρ (cid:16) φ † φ − v (cid:17) − v D µ φ ) † D µ φ + M (cid:16) M v (cid:17)(cid:16) φ † φ − v (cid:17) + 14 M v (cid:16) φ † φ − v (cid:17) + v F µρ o = 14 M v I + M (cid:16) M v (cid:17) I + M v (cid:16) M v (cid:17) I − v I − I + v I + 12 I . 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