Background Field Method and Generalized Field Redefinitions in Effective Field Theories
BBackground Field Method and Generalized Field Redefinitionsin Effective Field Theories.
A. Quadri ∗ INFN, Sezione di Milano, via Celoria 16, I-20133 Milano, Italy (Dated: February 21, 2021)
Abstract
We show that in a spontaneously broken effective gauge field theory, quantized in a general back-ground R ξ -gauge, also the background fields undergo a non-linear (albeit background-gauge invari-ant) field redefinition induced by radiative corrections. This redefinition proves to be crucial inorder to renormalize the coupling constants of gauge-invariant operators in a gauge-independentway. The classical background-quantum splitting is also in general non-linearly deformed (in anon gauge-invariant way) by radiative corrections. Remarkably, such deformations vanish in theLandau gauge, to all orders in the loop expansion. PACS numbers: 11.10.Gh, 12.60.-i, 12.60.Fr ∗ [email protected] a r X i v : . [ h e p - t h ] F e b . INTRODUCTION In the absence of direct resonance signals of new physics beyond the Standard Model(BSM) at the LHC, indirect experimental searches of BSM physics have become increasinglypopular in recent years, e.g. lepton flavour universality violations [1–4] or searches for non-resonant Higgs boson pair production (for a recent review see [5]).In this context the SM Effective Field Theory (SMEFT) [6–8] provides a consistent the-oretical tool in order to describe the energy regime up to some higher energy scale Λ. Theadvantage of the SMEFT is that it takes into account the constraints arising from the invari-ance under the SU(3) × SU L (2) × U Y (1) gauge group in a model independent way, withoutthe need to know the precise form of its ultraviolet (UV) completion.In this approach the SM Lagrangian is supplemented by higher dimensional gauge-invariant operators suppressed by powers of Λ. Renormalizability by power-counting is thenlost and new UV divergences arise order by order in the loop expansion. As in any effectivegauge theory, they must be subtracted by a combination of generalized (i.e. non-linear andin general not even polynomial [9]) field redefinitions and the renormalization of the couplingconstants, associated with gauge-invariant operators of increasing dimensions [10, 11].That such a program can indeed be completed in a recursive way by adding local counter-terms while preserving the relevant symmetries of the theory is a key result established manyyears ago [11] in the setting of the Batalin-Vilkovisky (BV) formalism (for a review see e.g.[12]).The BV formalism can be seen as a generalization of the BRST quantization proce-dure [13–15] that applies also to non power-counting renormalizable models. The Slavnov-Taylor (ST) identity [16, 17], encoding at the quantum level the BRST invariance of theclassical gauge-fixed action, is translated into the BV master equation.From a physical point of view the BV master equation ensures, as well as the ST identity,physical unitarity of the theory, i.e. the cancellation of unphysical ghosts in the intermediatestates [13, 18–20].Due to the huge number of operators arising in effective field theories it is natural toapply the background field method (BFM) [21–34] technique in order to simplify the task ofcomputing the radiative corrections. The BFM is particularly advantageous since it allowsto retain (background) gauge invariance to all orders in perturbation theory. The resulting2 IG. 1. Maximally UV divergent one-loop amplitudes generated by the vertex ( φ∂φ ) background Ward identity is linear in the quantum fields, unlike the ST identity, and henceis easier to study. Use of the BFM has been recently advocated in the context of the(geometric) SMEFT in Refs [35–37].In power-counting renormalizable theories both the background and the quantum fieldsrenormalize linearly. Linearity of the renormalization of the background fields together withbackground gauge invariance yields powerful relations between counter-terms that are oneof the main virtues of the BFM [38].The situation is significantly more involved in effective gauge theories. For instance atypical derivative-dependent dim.6 interaction ∼ ( φ∂φ ) gives rise already at one loop to aninfinite number of UV-divergent amplitudes, generated by configurations with two powersof the internal loop momentum from the derivative-dependent interaction at each vertex.They are compensated by two inverse powers from each propagator (see Figure 1), so thatthe UV degree of divergence of these Feynman amplitudes is always 4, irrespectively of thenumber of the external φ -legs.The task of evaluating the required counter-terms in spontaneously broken effective gaugefield theories is simplified in the so-called X -formalism [39–41] by the use of a gauge-invariantfield coordinate for the physical scalar mode, namely X ∼ v (cid:16) φ † φ − v (cid:17) , where φ is theusual Higgs doublet and v its vacuum expectation value (v.e.v.).Let us consider e.g. the two-derivatives vertices ( φ∂φ ) arising from the gauge-invariantinteraction (cid:16) φ † φ − v (cid:17) ( D µ φ ) † D µ φ , D µ being the covariant derivative. This operator is rep-resented in the X -formalism by ∼ X ( D µ φ ) † D µ φ [9]. Since the X -amplitudes are uniquelyfixed by the functional identities of the theory [9], one needs to consider only graphs with3nternal X -lines, so that at least one derivative acts on the external φ -legs, thus reduc-ing the UV degree of divergence of the amplitudes that need to be evaluated. There isonly a finite number of UV divergent amplitudes of this type, so one can renormalize thissector of the theory by a finite number of independent (i.e. not fixed by the symmetries)local counter-terms, while diagrams in Figure 1 are automatically taken into account by analgebraic resummation induced by the functional identities of the model [39–41].In effective gauge theories quantum fields undergo generalized non-linear field redefini-tions (GFRs). The X -formalism provides an effective way to separate the renormalizationof the gauge coupling constants from the physically spurious contributions controlled by theGFRs [39–41]. GFRs play a crucial role in carrying out the correct recursive off-shell renor-malization of the one-particle-irreducible (1-PI) amplitudes, since only once the appropriateGFRs have been implemented, the renormalization of the coupling constants turns out tobe gauge-independent [9, 42, 43].When effective gauge theories are quantized in the BFM, the question arises of whetheralso the background fields undergo a non-linear redefinitions, and if such a redefinition isbackground gauge-invariant.This is a non trivial issue that can be studied by combining the X -formalism with theAlgebraic Renormalization approach to the BFM [31–34]. Compatibilty between the STidentity and the background Ward identity is obtained by extending the BRST differential s to the background fields, collectively denoted by (cid:98) Φ, and by pairing them with anticommutingvariables Ω (cid:98) Φ , so that s (cid:98) Φ = Ω (cid:98) Φ , s Ω (cid:98) Φ = 0 . (1.1)The corresponding extended ST identity uniquely fixes (in a background gauge-invariantway) the dependence of the vertex functional on the background fields (cid:98) Φ.In the present paper we extend the X -formalism to the BFM and study the renormaliza-tion of the Abelian Higgs-Kibble model supplemented by dim.6 operators, as a playgroundtowards the renormalization of the SMEFT in the BFM approach.We find that:1. the tree-level background-quantum splittingΦ = (cid:98) Φ + Q Φ
4s in general deformed in a non-linear (and gauge-dependent) way, unlike in the power-counting renormalizable case where only multiplicative Z -factors arise both for back-ground and quantum fields;2. a noticeable exception is the Landau gauge, where no such deformation of the tree-levelbackground-quantum splitting happens, to all orders in the loop expansion;3. as a consequence of the radiative corrections to the background-quantum splitting andof the GFRs, background fields also undergo a non-linear redefinition;4. the redefinition of the background fields is background gauge-invariant. This resultfollows from non-trivial cancellations between the non gauge-invariant contributionsto the background-quantum splitting and the non gauge-invariant terms in the GFRs;5. the background and quantum field redefinitions are crucial in order to properly renor-malize the coupling constants in a gauge-independent way.The paper is organized as follows. In Sect. II we set up our notations, present theclassical action of the Abelian Higgs-Kibble model with dim.6 operators in the X -formalismand introduce the BFM tree-level vertex functional, together with the background gauge-fixing. In Sect. III we study the compatibility between the background Ward identity andthe mapping from the X -formalism to the standard φ -representation (target theory). Weprove that 1-PI amplitudes in the target theory are background gauge-invariant if those inthe X -theory are. In Sect. IV we study the local solutions to the background Ward identitythat are relevant for the classification of the UV divergences of the theory. In Sect. V wesolve the ST identity in order to fix the dependence on the background fields. We findthat the tree-level background-quantum splitting is non-trivially deformed at the quantumlevel. The gauge dependence of such corrections is studied in the Feynman and in theLandau gauge. In Sect. VI we obtain the generalized field and background redefinitionsfor the X -theory by combining the effect of the deformation of the background-quantumsplitting and of the GFRs. As a non-trivial check, we show that at zero quantum fields Q Φ = 0 subtle cancellations happen that make the vertex functional invariant w.r.t. thebackground transformation of the background fields only, in agreement with the backgroundWard identity. Finally in Sect. VII we provide the explicit form of the GFRs both for5ackground and quantum fields in the ordinary φ -formalism by applying the mapping fromthe X - to the target theory. Conclusions are presented in Sect. VIII.Appendices contain the discussions of some aspects of the Algebraic Renormalization ofthe theory. In Appendix A we enumerate the functional symmetries of the model. Ap-pendix B is devoted to the parameterization of background gauge field redefinitions. Therenormalization of the tadpole and its gauge dependence are studied in Appendix C. II. BFM TREE-LEVEL VERTEX FUNCTIONAL
We start from the tree-level vertex functional of the Abelian Higgs-Kibble model supple-mented by dim.6 gauge-invariant operators in the so-called X -formalism of [43]:Γ (0) = (cid:90) d x (cid:104) − 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 ) (cid:105) . (2.1)The field content of the model includes the Abelian gauge field A µ , the usual scalar field φ ≡ √ φ + iχ ) = 1 √ σ + v + iχ ) , φ = σ + v , with v denoting the v.e.v., and a singlet field X describing in a gauge invariant way thephysical scalar mode of mass M . Indeed, if one goes on-shell in Eq.(2.1) with the auxiliaryfield X , that plays the role of a Lagrange multiplier, we obtain the constraint( (cid:3) + m ) (cid:16) φ † φ − v − vX (cid:17) = 0 , so that the field X must fulfill the condition X = v (cid:16) φ † φ − v (cid:17) + η, η being a scalar fieldof mass m . However, it can be proven that in perturbation theory the correlators of themode η with any gauge-invariant operators vanish [9], so that one can safely set η = 0 andby going on-shell perform in Eq. (2.1) the substitution X ∼ v ( φ † φ − v / . m -term cancels out and one gets back the usual Higgs quartic potential with coefficient ∼ M / v plus the set of dim.6 parity-preserving operators arising from the third line ofEq.(2.1) (we use the same notations as in [42]): (cid:79) [6]1 = (cid:90) d x F µν (cid:16) φ † φ − v (cid:17) ∼ (cid:90) d x vX F µν , (2.2a) (cid:79) [6]2 = (cid:90) d x (cid:16) φ † φ − v (cid:17) ∼ (cid:90) d x v X , (2.2b) (cid:79) [6]3 = (cid:90) d x (cid:16) φ † φ − v (cid:17) (cid:3) (cid:16) φ † φ − v (cid:17) ∼ (cid:90) d x v X (cid:3) X , (2.2c) (cid:79) [6]4 = (cid:90) d x (cid:16) φ † φ − v (cid:17) ( D µ φ ) † D µ φ ∼ (cid:90) d x vX ( D µ φ ) † D µ φ. (2.2d)We notice that the parameter m must disappear in the correlators of the gauge-invariantoperators of the target theory (i.e. the one obtained by going on-shell with the X , -fields),as can be checked explicitly at the one loop order [9, 41–43].In Eq.(2.1) ¯ ω, ω are the Faddeev-Popov antighost and ghost fields, while b is theNakanishi-Lautrup field enforcing the gauge-fixing condition (cid:70) ξ = 0with (cid:70) ξ ≡ ∂A + ξevχ , (2.3) ξ being the gauge parameter.The tree-level vertex functional (2.1) is invariant both under the usual gauge BRSTsymmetry sA µ = ∂ µ ω ; sφ = ieωφ ; sσ = − eωχ ; sχ = eω ( σ + v ); sω = 0 ; s ¯ c = b ; sb = 0 ; sX = sX = sc = s ¯ c = 0 , (2.4)and a constraint BRST symmetry (cid:115) X = vc ; (cid:115) c = 0; (cid:115) ¯ c = φ † φ − v − vX , (2.5)while all other fields are (cid:115) -invariant. The latter symmetry ensures that the number ofphysical degrees of freedom in the scalar sector remains unchanged in the X -formalism withrespect to the standard formulation relying only on the field φ [39, 40]. c, ¯ c are the ghostand antighost fields of the constraint BRST symmetry. They are free.7he two BRST differentials s, (cid:115) anticommute.Several external sources need to be introduced in the vertex functional (2.1) in order toformulate at the quantum level the symmetries of the theory, as a consequence of the non-linearity in the quantized fields of the operators involved: the antifields [12] σ ∗ , χ ∗ , i.e. , theexternal sources coupled to the relevant BRST transformations that are non-linear in thequantized fields, the antifield ¯ c ∗ coupled to the constraint BRST variation of ¯ c in Eq.(2.5),and the sources T , U and R , coupled to the gauge-invariant operators in the fourth lineof Eq.(2.1). The latter sources are needed in order to define at the quantum level the X , -equations of motion, as summarized in Appendix A.The main virtue of this approach is that several relations among 1-PI Green’s functionsof the effective field theory, that are hidden in the standard formulation, becomes manifestas they are encoded in the X , -equation and the related system of external sources. Inparticular the X -formalism is suited in order to evaluate the GFRs and disentangle thegauge-invariant renormalization of the coupling constants [9, 42, 43].In order to formulate the theory in the background field method we introduce the back-ground gauge field (cid:98) A µ and the background scalar (cid:98) φ ≡ √ ( (cid:98) σ + v + i (cid:98) χ ). They transform asthe corresponding fields under a background gauge transformation of parameter α , namely δA µ = ∂ µ α , δ (cid:98) A µ = ∂ µ α , δφ = ieαφ , δ (cid:98) φ = ieα (cid:98) φ . (2.6)If the gauge-fixing functional (cid:70) ξ in Eq.(2.3) is replaced by (cid:98) (cid:70) ξ = ∂ µ ( A µ − (cid:98) A µ ) + ξe ( (cid:98) φ χ − (cid:98) χφ ) , (2.7)the tree-level vertex functional Γ (0) becomes background gauge invariant provided that: i)all other fields and external sources are required to be δ -invariant, with the exception of theantifields σ ∗ , χ ∗ ; ii) σ ∗ , χ ∗ are gathered in a complex antifield φ ∗ ≡ √ ( σ ∗ + iχ ∗ ) transformingas a scalar in the fundamental representation: δφ ∗ = ieαφ ∗ . (2.8)In order to ensure the compatibility of the background gauge invariance with the ST identity,one also needs to introduce for each background field (cid:98) Φ an anti-commuting variable Ω Φ pairing with the background field into a BRST doublet [31, 33, 34]: s (cid:98) A µ = Ω µ , s (cid:98) σ = Ω ˆ σ , s (cid:98) χ = Ω ˆ χ , s Ω µ = s Ω ˆ σ = Ω ˆ χ = 0 . (2.9)8his procedure uniquely fixes the dependence of the vertex functional on the backgroundfields in the sector at zero ghost number, since in this sector the background-dependentpart of the vertex functional can be recovered by a canonical transformation that respectsthe ST identity (when the latter is equivalently rewritten as the Batalin-Vilkovisky masterequation) [44, 45]. Being a canonical transformation, the physical content of the theory isnot modified by the introduction of the background fields [32–34, 44, 45].The tree-level vertex functional Γ (0) in the presence of the background fields thus acquiresan Ω-dependence generated by the gauge-fixing term:Γ (0)g . f . = (cid:90) d x s (cid:104) ¯ ω (cid:16) ξ b − (cid:98) (cid:70) ξ (cid:17)(cid:105) = (cid:90) d x (cid:104) ξb − b (cid:16) ∂A + ξevχ (cid:17) + ¯ ω (cid:16) (cid:3) ω + ξe v ( σ + v ) ω (cid:17) + b (cid:16) ∂ (cid:98) A − ξe (cid:98) σχ + ξe (cid:98) χ ( σ + v ) (cid:17) + ¯ ωξe (cid:16)(cid:98) σ ( σ + v ) + (cid:98) χχ (cid:17) ω − ¯ ω∂ µ Ω µ + ξe ¯ ω Ω (cid:98) σ χ − ¯ ωξe Ω (cid:98) χ ( σ + v ) (cid:105) . (2.10)The last two lines in the above equation contain the additional terms proportional to thebackground fields and their BRST partners. At (cid:98) A µ = (cid:98) σ = (cid:98) χ = 0 as well as Ω (cid:98) σ = Ω (cid:98) χ = Ω µ =0 we recover the gauge-fixing and ghost terms in Eq.(2.1).The background tree-level vertex functional is then obtained by the replacementΓ (0) → Γ (0) + (cid:90) d x (cid:104) b (cid:16) ∂ (cid:98) A − ξe (cid:98) σχ + ξe (cid:98) χ ( σ + v ) (cid:17) + ¯ ωξe (cid:16)(cid:98) σ ( σ + v ) + (cid:98) χχ (cid:17) ω − ¯ ω∂ µ Ω µ + ξe ¯ ω Ω (cid:98) σ χ − ξe ¯ ω Ω (cid:98) χ ( σ + v ) (cid:105) . (2.11)The ghost number is assigned as follows. A µ , σ, χ, X , X , b have ghost number zero. c, ω and the background BRST partners Ω µ , Ω (cid:98) σ , Ω (cid:98) χ have ghost number one. ¯ c ∗ has ghost numberzero. The antifields σ ∗ , χ ∗ have ghost number -1.Since the theory is non-anomalous, the full vertex functional Γ is invariant under theST identity, the background Ward identity and the X , -equations, as summarized in Ap-pendix A.Γ can be expanded in the loop parameter as follows:Γ = ∞ (cid:88) n =0 (cid:126) n Γ ( n ) . (2.12)9erturbation theory is carried out order by order in the loop expansion by recursively im-posing the functional identities of the model while subtracting the UV divergences by meansof suitable local (in the sense of formal power series) counter-terms. III. BACKGROUND WARD IDENTITY FOR THE TARGET THEORY
Eventually we are interested in the 1-PI Green’s functions in the standard φ -formalism,that are obtained from the vertex functional of the X -theory by going on shell w.r.t. thefields X , [9, 41–43].The procedure amounts to carry out the replacements in Eq.(A7a) and then substitute X , with the solution of their equations of motion, order by order in the loop expansion.Let us denote by (cid:101) Γ the vertex functional of the target theory (i.e. the generating func-tional of the 1-PI Green’s functions in the φ -formalism). Since the functional differentialoperators for the X , -equations in Eqs.(A4) and (A5) and for the background Ward identityin Eq.(A13) commute: [ (cid:66) X , (cid:87) ] = [ (cid:66) X , (cid:87) ] = 0 , (3.1)we conclude that (cid:101) Γ is also background gauge-invariant.It is instructive to check this result at one loop order in the sector of operators up todimension 6, for which the explicit form of the mapping has been worked out in [9, 42, 43].At one loop we need to solve the tree-level equations of motion for X , [42]. The X -equationof motion yields X = 1 v (cid:16) φ † φ − v (cid:17) , (3.2)while the classical X -equation of motion gives (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 . (3.3)By inserting Eqs. (3.2) and (3.3) into the replacements in Eq.(A7a) we obtain the explicitform of the mapping at one loop:¯ (cid:99) ∗ → − ( 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) , (3.4a) (cid:84) → g Λ (cid:16) φ † φ − v (cid:17) ; (cid:85) → g Λ (cid:16) φ † φ − v (cid:17) ; (cid:82) → g v (cid:16) φ † φ − v (cid:17) . (3.4b)It can be seen by direct inspection that the r.h.s. of the above Equations only containgauge-invariant operators. Since Γ (1) is background gauge invariant and the replacementsin Eqs.(3.4a) and (3.4b) transform background gauge-invariant sources into backgroundgauge-invariants combinations in the target theory, we conclude that (cid:101) Γ (1) is automaticallybackground gauge-invariant. IV. LOCAL SOLUTIONS TO THE BACKGROUND WARD IDENTITY
Let us denote by Γ ( n ) the UV-divergent part of the n -th order vertex functional. Providedthat the UV divergences have been subtracted up to order n − ( n ) is a local functional (in the sense of formal power series) inthe fields, the external sources and their derivatives.If the regularization scheme is symmetric (as it happens e.g. for dimensional regulariza-tion), UV divergences must also fulfill the same background Ward identity in Eq.(A13): (cid:87) (Γ ( n ) ) = 0 . (4.1)Since the n -th order UV divergences are local, we need to solve Eq.(4.1) in the space oflocal functionals. Moroever by Eq.(A12) we can use the redefined antifield χ ∗(cid:48) and then set¯ ω = b = 0 (since the n -th order vertex functional n ≥ b and the onlydependence on the antighost ω is via χ ∗(cid:48) , as can be seen from Eq.(A11)).An efficient way to obtain the most general solution to Eq.(4.1) in this functional spaceis to carry out the following change of variables: A µ → Q µ = A µ − (cid:98) A µ , σ → ˜ φ − v ≡ v ( (cid:98) φ φ + (cid:98) χχ ) − v , χ → ˜ χ ≡ v ( (cid:98) φ χ − φ (cid:98) χ ) ,σ ∗ → ˜ σ ∗ ≡ v ( (cid:98) φ σ ∗ + (cid:98) χχ ∗ ) , χ ∗(cid:48) → (cid:102) χ ∗(cid:48) ≡ v ( (cid:98) φ χ ∗(cid:48) − σ ∗ (cid:98) χ ) . (4.2)It is easy to see that Q µ , ˜ φ , ˜ χ, ˜ σ ∗ , (cid:102) χ ∗(cid:48) are gauge-invariant. Moreover they reduce to theoriginal fields and antifields at zero backgrounds.11ccordingly, the most general solution Γ ( n ) [ A µ , σ, χ ; (cid:98) A µ , (cid:98) σ, (cid:98) χ ; σ ∗ , χ ∗ ] to Eq.(4.1) can bewritten as follows:Γ ( n ) = Γ ( n ) [ Q µ , ˜ φ − v, ˜ χ ; 0 , ,
0; ˜ σ ∗ , ˜ χ ∗ ] + (cid:80) ( n ) ( (cid:98) A µ , (cid:98) σ, (cid:98) χ ) . (4.3)The first term in the r.h.s. of Eq.(4.3) is obtained by replacing the quantum fields ( A µ , σ, χ )and their antifields with their background gauge-invariant counterparts in Eq.(4.2). Thesecond term (cid:80) ( n ) is the most general solution in the kernel of the operator (cid:87) , namely agauge invariant formal power series built out from the background fields scalar (cid:98) φ and itsbackground covariant derivatives and from the background field strength (cid:98) F µν = ∂ µ (cid:98) A ν − ∂ ν (cid:98) A µ and its ordinary derivatives, that vanishes at zero background fields.The background Ward identity is unable to fix the ambiguities encoded by (cid:80) ( n ) ( (cid:98) A µ , (cid:98) σ, (cid:98) χ ).One thus needs to make recourse to the extended ST identity in order to select out of thegeneral solution to the background Ward identity in Eq.(4.3) the unique vertex functional(in the sector with zero ghost number) depending on the background fields and compatiblewith the ST identity itself.A remark is in order here. A different basis is often used in BFM calculations, namelythe quantum fields are defined as Q µ ≡ A µ − (cid:98) A µ , q σ ≡ σ − (cid:98) σ, q χ ≡ χ − (cid:98) χ , i.e. the variablesover which one integrates in the path integral. We collectively denote these fields by Q Φ .The background Ward identity in the Q Φ -variables reads (cid:87) (Γ) = − eq χ δ Γ δq σ + eq σ δ Γ δq χ − ∂ µ δ Γ δ (cid:98) A µ − e (cid:98) χ δ Γ δ (cid:98) σ + e ( (cid:98) σ + v ) δ Γ δ (cid:98) χ − eχ ∗ δ Γ δ (cid:98) σ ∗ + eσ ∗ δ Γ δχ ∗ = 0 . (4.4)We notice that at Q Φ = 0 Eq.(4.4) states that the vertex functional at zero quantum fieldsis background-gauge invariant: (cid:87) ( Γ | Q Φ =0 ) = (cid:110) − ∂ µ δδ (cid:98) A µ − e (cid:98) χ δδ (cid:98) σ + e ( (cid:98) σ + v ) δδ (cid:98) χ − eχ ∗ δδ (cid:98) σ ∗ + eσ ∗ δδχ ∗ (cid:111) Γ | Q Φ =0 = 0 . (4.5)At Q Φ = 0 the redefined fields in Eq.(4.2) reduce to gauge-invariant combinations, namely Q µ | Q Φ =0 = 0 , ˜ φ (cid:12)(cid:12)(cid:12) Q Φ =0 = 1 v ( (cid:98) φ + (cid:98) χ ) , ˜ χ | Q Φ =0 = 0 . (4.6)As a consequence of Eq.(4.6), at zero quantum fields Γ ( n ) in Eq.(4.3) reduces to a backgroundgauge-invariant functional, in agreement with Eq.(4.5).12 . BACKGROUND-QUANTUM SPLITTING In the physical sector at zero ghost number the dependence on the background fields isuniquely fixed by the ST identity in Eq.(A1) once the 1-PI Green’s functions of the quantizedfields and the correlators involving the sources Ω (cid:98) Φ are known.By taking a derivative w.r.t Ω µ , Ω (cid:98) σ , Ω (cid:98) χ and then setting c = ω = Ω µ = Ω (cid:98) σ = Ω (cid:98) χ = b = 0we get Γ (cid:48) (cid:98) σ = − (cid:90) d x (cid:104) Γ (cid:48) Ω (cid:98) σ σ ∗ Γ (cid:48) σ + Γ (cid:48) Ω (cid:98) σ χ ∗ Γ (cid:48) χ (cid:105) , Γ (cid:48) (cid:98) χ = − (cid:90) d x (cid:104) Γ (cid:48) Ω (cid:98) χ σ ∗ Γ (cid:48) σ + Γ (cid:48) Ω (cid:98) χ χ ∗ Γ (cid:48) χ (cid:105) . (5.1)In the above equation we have denoted by a prime the functionals evaluated at c = ω = Ω µ =Ω (cid:98) σ = Ω (cid:98) χ = b = 0. Moreover in order to simplify the notations we denote by a subscript thefunctional differentiation w.r.t the field or external source, e.g. Γ χ = δ Γ δχ . When the momentaof the fields and external sources are displayed as arguments of the corresponding amplitudes,we understand that the functional derivatives of the vertex functional are evaluated at zerofields and external sources, i.e. we refer to the specific 1-PI amplitudes. For instance thetwo-point 1-PI function with one σ and one background (cid:98) σ legs will be denoted by Γ (1) (cid:98) σ ( − p ) σ ( p ) . If one would use q σ , q χ with the corresponding antifields q ∗ σ , q ∗ χ , an extra dependence onΩ (cid:98) σ , Ω (cid:98) χ would arise, since sq σ = sσ − s (cid:98) σ = − eωχ − Ω (cid:98) σ , sq χ = sχ − s (cid:98) χ = eω ( σ + v ) − Ω (cid:98) χ , (5.2)so that Γ (0) q ∗ σ Ω (cid:98) σ and Γ (0) q ∗ χ Ω (cid:98) χ would not vanish, thus introducing additional terms in the r.h.s.of Eq.(5.1). For this reason we prefer to use the ( A µ , σ, χ )-basis in solving the extended STidentity for the background dependence.Let us project Eq.(5.1) at first order in the loop expansion. Since we use the basis( A µ , σ, χ ), there is no tree-level 1-PI amplitude involving the antifields σ ∗ , χ ∗ together withthe background ghosts Ω (cid:98) σ , Ω (cid:98) χ . Hence we findΓ (1) (cid:48) (cid:98) σ = − (cid:90) d x (cid:104) Γ (1) (cid:48) Ω (cid:98) σ σ ∗ Γ (0) (cid:48) σ + Γ (1) (cid:48) Ω (cid:98) σ χ ∗ Γ (0) (cid:48) χ (cid:105) , Γ (1) (cid:48) (cid:98) χ = − (cid:90) d x (cid:104) Γ (1) (cid:48) Ω (cid:98) χ σ ∗ Γ (0) (cid:48) σ + Γ (1) (cid:48) Ω (cid:98) χ χ ∗ Γ (0) (cid:48) χ (cid:105) . (5.3)We are interested in the background dependence of the UV divergences of the theory, thatare local. Hence we can solve Eqs.(5.3) in the space of local functionals. Moreover at b = 013here is no dependence of the tree-level vertex functional on the background, again as aconsequence of the use of the basis ( A µ , σ, χ ).Thus in order to recover the full dependence on the background fields, once one knows theamplitudes at zero background, we just need to expand the kernels Γ (1) (cid:48) Ω (cid:98) σ σ ∗ , Γ (1) (cid:48) Ω (cid:98) σ χ ∗ , Γ (1) (cid:48) Ω (cid:98) χ σ ∗ , Γ (1) (cid:48) Ω (cid:98) χ χ ∗ in powers of the fields, antifields and the backgrounds and then solve the functional differ-ential equations Eq.(5.3) by integrating over ˆ σ, ˆ χ .Since the kernels are gauge-dependent, we proceed to a separate discussion for the Feyn-man and the Landau gauge. A. Feynman gauge
In the Feynman gauge ξ = 1 the kernels are non-vanishing. We notice that by power-counting they contain at most logarithmic divergences, so we can drop derivative-dependentterms in their local expansion around zero momentum and write (we omit the coefficientsvanishing by parity):Γ (1) (cid:48) Ω (cid:98) σ σ ∗ = (cid:90) d x (cid:104) γ Ω (cid:98) σ σ ∗ + γ Ω (cid:98) σ σ ∗ σ σ + γ Ω (cid:98) σ σ ∗ (cid:98) σ (cid:98) σ + 12 γ Ω (cid:98) σ σ ∗ σσ σ + 12 γ Ω (cid:98) σ σ ∗ χχ χ + γ Ω (cid:98) σ σ ∗ σ (cid:98) σ σ (cid:98) σ + γ Ω (cid:98) σ σ ∗ χ (cid:98) χ χ (cid:98) χ + 12 γ Ω (cid:98) σ σ ∗ (cid:98) σ (cid:98) σ (cid:98) σ + 12 γ Ω (cid:98) σ σ ∗ (cid:98) χ (cid:98) χ (cid:98) χ + γ Ω (cid:98) σ σ ∗ T T + . . . (cid:105) . (5.4)The dots stand for terms with more than two fields σ, χ and their backgrounds as well asadditional powers of the external sources. We truncate the expansion to the order requiredfor the comparison with the explicit results of [42].More specifically the coefficients can be obtained by evaluating the UV divergent part ofthe 1-PI Green’s functions involving insertions of Ω (cid:98) σ , σ ∗ and the other fields and externalsources, so for instance γ Ω (cid:98) σ σ ∗ = Γ (1)Ω (cid:98) σ ( − p ) σ ∗ ( p ) (cid:12)(cid:12)(cid:12) p =0 , γ Ω (cid:98) σ σ ∗ σ = Γ (1)Ω (cid:98) σ ( − p − p σ ∗ ( p ) σ ( p ) (cid:12)(cid:12)(cid:12) p = p =0 , (5.5)and so on. A similar expansion holds for the other kernels.By explicit computation we find the following results (to the accuracy required to renor-malize dim.6 operators [42])Γ (1) (cid:48) Ω (cid:98) σ σ ∗ = (cid:90) d x M A π v (cid:15) (cid:104) − z z χ v − T + . . . (cid:105) , (1) (cid:48) Ω (cid:98) σ χ ∗ = (cid:90) d x M A π v (cid:15) χv (cid:104) z z − z ( z − z ) σv + . . . (cid:105) , Γ (1) (cid:48) Ω (cid:98) χ σ ∗ = (cid:90) d x M A π v (cid:15) χv (cid:104) z z − z ( z − z ) σv + . . . (cid:105) , Γ (1) (cid:48) Ω (cid:98) χ χ ∗ = (cid:90) d x M A π v (cid:15) (cid:104) z z − z (1 + z ) σv + z (3 z − z ) σ v + z (1 + z ) χ v − z ) T + . . . (cid:105) . (5.6)We notice that in this specific case there is no dependence of the kernels in Eq.(5.6) on thebackground fields, so the integration of Eq.(5.3) is trivial and yields a linear dependence onthe background fields themselves:Γ (1) (cid:48) = − (cid:90) d x (cid:104)(cid:16)(cid:98) σ Γ (1) (cid:48) Ω (cid:98) σ σ ∗ + (cid:98) χ Γ (1) (cid:48) Ω (cid:98) χ σ ∗ (cid:17) Γ (0) (cid:48) σ + (cid:16)(cid:98) σ Γ (1) (cid:48) Ω (cid:98) σ χ ∗ + (cid:98) χ Γ (1) (cid:48) Ω (cid:98) χ χ ∗ (cid:17) Γ (0) (cid:48) χ (cid:105) + Γ (1) (cid:48) (cid:12)(cid:12)(cid:12) (cid:98) σ = (cid:98) χ =0 . (5.7)The last term in Eq.(5.7) denotes the UV divergent part of the vertex functional at zerobackground fields. It has been evaluated in [42] for the relevant sector of operators up todimension 6.Eq.(5.7) is of particular significance. It states that the background-quantum splitting isnon-trivially modified at the quantum level according to the following redefinitions: σ → (cid:98) σ + q σ − (cid:98) σ Γ (1) (cid:48) Ω (cid:98) σ σ ∗ − (cid:98) χ Γ (1) (cid:48) Ω (cid:98) χ σ ∗ ,χ → (cid:98) χ + q χ − (cid:98) σ Γ (1) (cid:48) Ω (cid:98) σ χ ∗ − (cid:98) χ Γ (1) (cid:48) Ω (cid:98) χ χ ∗ . (5.8)Once applied to the tree-level vertex functional Γ (0) , such redefinitions generate the linearterms in the background fields in the r.h.s. of Eq.(5.7).We emphasize that the kernels Γ (1) (cid:48) Ω Φ Φ depend on the fields and external sources in acomplicated way, so that Eq.(5.8) is a highly non-linear redefinition w.r.t. the quantumfields.In the limit z → (1) (cid:48) Ω (cid:98) σ σ ∗ = γ Ω (cid:98) σ σ ∗ , Γ (1) (cid:48) Ω (cid:98) χ σ ∗ = 0 , Γ (1) (cid:48) Ω (cid:98) σ χ ∗ = 0 , Γ (1) (cid:48) Ω (cid:98) χ χ ∗ = γ Ω (cid:98) χ χ ∗ (cid:12)(cid:12) z =0 = 0 , so in this limit Eq.(5.8) implies that the background-quantum splitting is modified linearly,as expected by power-counting renormalizability of the theory at z = 0.15 . Landau gauge At variance with the Feynman gauge, in the Landau gauge ξ = 0 all the four kernels areidentically zero since there are no interaction vertices involving the background ghosts Ω’sAs a consequence the classical background-quantum splitting Φ = (cid:98) Φ + Q Φ does notreceive any radiative corrections. This holds true to all orders in the loop expansion. Thusthe dependence on the background fields only originates from the (undeformed) background-quantum splitting. VI. GENERALIZED FIELD REDEFINITIONS IN THE BFM
We are now in a position to study the generalized field redefinitions (GFRs) arising atone loop order in the presence of the background. Together with the renormalization ofthe coupling constants they allow to recursively remove the UV divergences of the theorytogether with the coupling constants renormalization.In particular the background generalized field redefinitions (BGFRs) can be obtained ina straightforward way by changing the variables from the ( A µ , σ, χ )-basis to the Q Φ -basis inEq.(5.7) and then setting the quantum fields to zero.Let us start from the second term in the r.h.s. of Eq.(5.7), that is present both inthe Feynman and Landau gauge. According to the general results of [11] and the explicitcomputations of [42, 43], the functional Γ (1) (cid:48) (cid:12)(cid:12)(cid:12) (cid:98) σ = (cid:98) χ =0 decomposes into the sum over a set ofintegrated local gauge invariant operators (cid:73) j with gauge-independent coefficients c j andthe functional Y (1) , responsible for the generalized field redefinitions of the quantum fields:Γ (1) (cid:48) (cid:12)(cid:12)(cid:12) (cid:98) σ = (cid:98) χ =0 = (cid:88) j c j (cid:73) j + Y (1) . (6.1)Both the coefficients c ’s and the functional Y (1) have been evaluated in [42, 43] for operatorsup to dimension 6.To this approximation the (cid:83) -exact functional Y (1) can be written as Y (1) = (cid:83) (cid:90) d x (cid:104)(cid:16) ρ + ρ σ + ρ σ + ρ χ + ρ T T (cid:17) (cid:90) The coefficients c j run over three classes of invariants in the classification of [42, 43]: gauge-invariantoperators only depending on the fields; gauge-invariant operators only depending on the external sources;gauge-invariant mixed operators depending both on the external sources and the fields. (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) (cid:90) (cid:105) , (6.2)where we use the notation (cid:90) ≡ ( σ ∗ σ + χ ∗ χ ); (cid:90) ≡ ( σ ∗ ( σ + v ) + χ ∗ χ ) . (6.3)The coefficients ρ ’s and ˜ ρ ’s are gauge-dependent and have been explicitly evaluated in [42].In the Feynman gauge the operators arising in the functional Y (1) are not gauge invariant,while in the Landau gauge they are.In order to prove this result let us notice that the combination (cid:90) in Eq.(6.3) is indeedgauge invariant, since (we can drop b -dependent terms by Eq.(A9)): (cid:90) = φ δ Γ (0) δφ + φ † δ Γ (0) δφ † = − 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 + ieφ ∗ φω − ie ( φ † ) ∗ φ † ω , (6.4)where we have introduced the notation φ ∗ = 1 √ σ ∗ − iχ ∗ (cid:48) ) . (6.5)Then by using the values of the coefficients computed in [42] one obtains Y (1) (cid:12)(cid:12)(cid:12) ξ =0 = (cid:83) (cid:90) d x M A π v (cid:15) (cid:104) − 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) (cid:105) (cid:90) = (cid:90) d x M A π v (cid:15) (cid:104) − 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) (cid:105) × (cid:104) − 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 (cid:105) + . . . (6.6)where the dots stand for additional terms of dimension ≥ X , -dependent terms (thatare recovered by the replacement in Eq.(A6)) and antifield-dependent terms that we do notneed to consider.The r.h.s. of Eq.(6.6) is gauge-invariant by inspection, as anticipated.17e also notice that in Landau gauge there is a combined field renormalization for σ + v ,as a consequence of the rigid global U(1) invariance holding true in this gauge [46].In the Feynman gauge instead the functional Y (1) reads Y (1) (cid:12)(cid:12)(cid:12) ξ =1 = (cid:83) (cid:90) d x M A π v (cid:15) (cid:110)(cid:104)
11 + z − z (1 + z ) σv + z (3 z − M A (1 + z ) v σ v − z (1 + z ) χ v (cid:105) (cid:90) + z z ) χ v (cid:90) (cid:111) = (cid:90) d x M A π v (cid:15) (cid:110) zv z ) χ v Γ (0) σ ++ (cid:104)
11 + z − z (1 + z ) σv + z (3 z − M A (1 + z ) v σ v + z ( z − z ) χ v (cid:105) ( σ Γ (0) σ + χ Γ (0) χ ) (cid:111) + . . . (6.7)where again the dots stand for additional terms not contributing to the renormalizationof physical operators with dimension ≤ X , -dependent contributions. The r.h.s ofEq.(6.7) is not gauge-invariant, as can be directly seen.We now collect all the factors contributing to the classical equations of motion for σ, χ in Eq.(5.7). There are two types of contributions: • one is associated with the deformation of the background-quantum splitting at oneloop order (the first term between square brackets in the r.h.s. of Eq.(5.7)); • the second is induced by the GFRs of the quantum fields (described by the functional Y (1) ). It is convenient to parameterize Y (1) as Y (1) = (cid:90) d x (cid:16) F (1) σ Γ (0) σ + F (1) χ Γ (0) χ (cid:17) + . . . , (6.8)where the coefficients of the classical equations of motion F (1) σ , F (1) χ are gauge-dependent functionals depending on the fields and the external sources and the dotsstand for antifield-dependent contributions that do not matter for the present discus-sion.In order to make the connection with the usual BFM formalism, we eventually switch to the( Q µ , q σ , q χ )-basis. By looking at the coefficients of Γ (0) σ , Γ (0) χ in Eqs.(5.7) and (6.8) we derivethe full renormalization of the fields, encoded in the following equations σ R = (cid:98) σ + q σ − (cid:98) σ Γ (1) (cid:48) Ω (cid:98) σ σ ∗ − (cid:98) χ Γ (1) (cid:48) Ω (cid:98) χ σ ∗ + F (1) σ , R = (cid:98) χ + q χ → (cid:98) χ + q χ − (cid:98) σ Γ (1) (cid:48) Ω (cid:98) σ χ ∗ − (cid:98) χ Γ (1) (cid:48) Ω (cid:98) χ χ ∗ + F (1) χ . (6.9)The BGFRs are obtained from the r.h.s. of the above equation after setting Q Φ = 0 (orequivalently A µ = (cid:98) A µ , σ = (cid:98) σ, χ = (cid:98) χ ).It turns out that such BGFRs are background-gauge invariant (although the kernels andthe F -contributions in Eq.(6.9) are not separately background gauge-invariant).At variance with the power-counting renormalizable case, the BGFRs are non-linear. A. Background gauge invariance at Q Φ = 0 Let us now check that at zero quantum fields one recovers a background gauge-invariantvertex functional in agreement with the background Ward identity Eq.(4.5).In the Landau case this is obvious by inspection since in that gauge Y (1) (cid:12)(cid:12)(cid:12) ξ =0 is separatelygauge invariant while the kernels Γ (1)Ω (cid:98) ΦΦ ∗ are vanishing, so the whole r.h.s. of Eq.(6.1) is gauge-invariant and there are no contributions from the quantum deformation of the background-quantum splitting.On the other hand, in the Feynman gauge the functional Y (1) (cid:12)(cid:12)(cid:12) ξ =1 is not backgroundgauge invariant at Q Φ = 0. Background gauge invariance is only recovered for the sum (5.7)once the contribution from the kernels is taken into account.In fact, as shown in Appendix B, once one sets to zero the quantized fields the UVdivergent part of the 1-PI vertex functional in the Feynman gauge reduces toΓ (1) (cid:48) ξ =1 (cid:12)(cid:12)(cid:12) Q Φ =0 = (cid:88) j c j (cid:73) j − (cid:90) d x M A π v (cid:15) (cid:110) z z (cid:16) (cid:98) φ † (cid:98) φ − v (cid:17) − v z (3 z − z ) (cid:16) (cid:98) φ † (cid:98) φ − v (cid:17) (cid:111) (cid:90) | Q Φ =0 . (6.10)Again by using Eq.(6.4) we see that the above expression is gauge-invariant, as expected.A comment is in order here. By comparing Eq.(6.10) with Eq.(6.6) we see that thecoefficients of (cid:98) φ † (cid:98) φ − v / (cid:98) φ † (cid:98) φ − v / coincide, while the constant term is vanishingin Feynman gauge.The difference can be traced back to the gauge dependence of the tadpole renormal-ization, as discussed in Appendix C, and offers an interesting example of a more generalissue. While the functional Γ (1) (cid:48) ξ =1 (cid:12)(cid:12)(cid:12) Q Φ =0 is background gauge-invariant, in agreement with thebackground Ward identity Eq.(4.5), this does not mean that the coefficients of the local19auge-invariant operators in Γ (1) (cid:48) ξ =1 (cid:12)(cid:12)(cid:12) Q Φ =0 are also gauge-independent. It turns out that sucha gauge independence only holds modulo the equations of motion of the theory, i.e. (froma cohomological point of view) only modulo (cid:83) -exact terms that are accounted for by theBGFRs. VII. GFRS IN THE TARGET THEORY
The final form of the background-quantum splitting in the target theory can be eventuallyread off from Eq.(6.9) by applying the mapping in Eqs.(3.4). The BGFRs in the target theoryare recovered by setting afterwards Q Φ = 0.Several comments are in order. First of all the coefficient F (1) σ at zero background andzero quantum fields represents the renormalization of the v.e.v. Since in the Landau gauge F (1) is proportional to σ + v , as a consequence of the fact that only the invariant (cid:90) entersin Eq.(6.6), we conclude that no independent renormalization of the v.e.v. is present in theLandau gauge. This is a well-known result is power-counting renormalizable theories [46]that extend to the EFT case, being a consequence of the rigid global U(1) symmetry holdingtrue in this gauge.In the approximation of Eqs.(5.6) (linear in the source T ) the σ, χ redefinitions in thepresence of the backgrounds (cid:98) σ, (cid:98) χ read : σ R = (cid:98) σ + q σ − M A (1 − δ ξ ;0 )8 π v (cid:15) (cid:110)(cid:104) − z z χ v − g v Λ (cid:16) σ + vσ + 12 χ (cid:17)(cid:105)(cid:98) σ + (cid:104) z z − z ( z − z ) σv (cid:105) χv (cid:98) χ (cid:111)(cid:12)(cid:12)(cid:12)(cid:12) χ = (cid:98) χ + q χ σ = (cid:98) σ + q σ + M A π v (cid:15) (cid:110) δ ξ ;0 + (cid:104)(cid:16) − z z − g v Λ (cid:17) δ ξ ;0 + 2 δ ξ ;1 z (cid:105) σv − (cid:104) z (1 + z ) + g v Λ (cid:16) δ ξ ;0 + δ ξ ;1 (1 + z ) (cid:17)(cid:105) σ v + (cid:104) z (1 + z ) ( δ ξ ;1 − δ ξ ;0 ) − g v Λ δ ξ ;0 (cid:105) χ v + (cid:104) z (1 + z ) (cid:16) z − δ ξ ;0 + 2(3 z − δ ξ ;1 (cid:17) − g v Λ (cid:16) δ ξ ;0 + δ ξ ;1 (1 + z ) (cid:17)(cid:105) σ v + (cid:104) − z z (cid:16) − z z δ ξ ;0 + δ ξ ;1 (cid:17) − g (cid:16) δ ξ ;0 + δ ξ ;1 (1 + z ) (cid:17)(cid:105) χ σv + z ( − δ ξ ;1 (1 + z ) (cid:16) z + ( − δ ξ ;0 (cid:17) χ σ v (cid:105)(cid:111)(cid:12)(cid:12)(cid:12)(cid:12) χ = (cid:98) χ + q χ σ = (cid:98) σ + q σ + . . . , R = (cid:98) χ + q χ − M A (1 − δ ξ ;0 )8 π v (cid:15) (cid:110)(cid:104) z z − z ( z − z ) σv (cid:105) χv (cid:98) σ + (cid:104)
11 + z − z (1 + z ) σv + z (3 z − z ) σ v + z (1 + z ) χ v − z ) g v Λ (cid:16) σ + vσ + 12 χ (cid:17)(cid:105)(cid:98) χ (cid:111)(cid:12)(cid:12)(cid:12)(cid:12) χ = (cid:98) χ + q χ σ = (cid:98) σ + q σ + M A π v (cid:15) (cid:110) δ ξ ;0 + 21 + z δ ξ ;1 − (cid:104) z (1 + z ) δ ξ ;0 + 4 zδ ξ ;1 (1 + z ) + 2 g v Λ (cid:16) δ ξ ;0 + δ ξ ;1 (1 + z ) (cid:17)(cid:105) σv + (cid:104) z (1 + z ) (cid:16) ( z − δ ξ ;0 + (3 z − δ ξ ;1 (cid:17) − g v Λ (cid:16) δ ξ ;0 + δ ξ ;1 (1 + z ) (cid:17)(cid:105) σ v − (cid:104) z (1 + z ) (cid:16) (1 + z ) δ ξ ;0 + (1 − z ) δ ξ ;1 (cid:17) + g v Λ (cid:16) δ ξ ;0 + δ ξ ;1 (1 + z ) (cid:17)(cid:105) χ v + ( − δ ξ ;1 z (1 + z ) (cid:104) z + ( − δ ξ ;0 (cid:105) σχ v (cid:111) χv (cid:12)(cid:12)(cid:12)(cid:12) χ = (cid:98) χ + q χ σ = (cid:98) σ + q σ + . . . (7.1)The dots stand for terms cubic in σ or of dimension ≥ q χ = q σ = 0 in Eq.(7.1). As already noticed, in thislimit the vertex functional becomes background gauge-invariant w.r.t. the variation of thebackground fields only. Accordingly the BGFRs are generated by a multiplicative redefini-tion of the background fields by a gauge-invariant polynomial that can be immediately readoff from Eq.(B1) : (cid:98) σ R (cid:98) χ R = (cid:104) a + a T + a (cid:16) (cid:98) φ † (cid:98) φ − v (cid:17) + a (cid:16) (cid:98) φ † (cid:98) φ − v (cid:17) + . . . (cid:105) (cid:98) σ + v (cid:98) χ (7.2)with the coefficients a ’s given by Eq.(B4). Notice that these coefficients are in generalgauge-dependent. VIII. CONCLUSIONS
In the present paper we have investigated the renormalization of the quantum and back-ground fields in a spontaneously broken gauge effective field theory. We have shown thatin this class of models, where power-counting renormalizability is lost, both the backgroundand the quantum field renormalize in a non-linear way.One must take into account the contributions from the radiative deformation of the21lassical background-quantum splitting as well as the effect of the non-linear GFRs of thequantum fields.At zero quantum fields Q Φ = 0 one recovers background gauge invariance of the vertexfunctional w.r.t the transformation of the background fields. This property is reflected onthe background gauge-invariance of the background-dependent counter-terms.However, despite such background gauge invariance at zero quantum fields, the coeffi-cients of the background invariants that are proportional to the equations of motion are ingeneral gauge-dependent.Consequently the correct renormalization of gauge-invariant operators requires to takeinto account the effect of the BGFRs already at one loop order.For higher order computations, the much more complicated background and quantumgeneralized field redefinitions must be carried out in order to achieve the symmetric sub-traction of the theory under consideration.The tools and results in the present paper pave the way to further applications to non-Abelian effective gauge theories and in particular to the SMEFT. ACKNOWLEDGMENTS
Useful discussions with D.Anselmi and D.Binosi are gratefully acknowledged.
Appendix A: Symmetries of the theory
Several functional identities hold for Γ: • the Slavnov-Taylor (ST) identity associated with the gauge BRST symmetryThe ST identity for the vertex functional Γ generated by the gauge BRST differential s reads (cid:83) (Γ) = (cid:90) d x (cid:104) ∂ µ ω δ Γ δA µ + δ Γ δσ ∗ δ Γ δσ + δ Γ δχ ∗ δ Γ δχ + b δ Γ δ ¯ ω + Ω µ δ Γ δ (cid:98) A µ + Ω (cid:98) σ δ Γ δ (cid:98) σ + Ω (cid:98) χ δ Γ δ (cid:98) χ (cid:105) = 0 . (A1) • the constraint ST identity 22he ST identity associated with the BRST differential (cid:115) is (cid:83) C (Γ) = (cid:90) d x (cid:104) vc δ Γ δX + δ Γ δ ¯ c ∗ δ Γ δ ¯ c (cid:105) = (cid:90) d x (cid:104) vc δ Γ δX − ( (cid:3) + m ) c δ Γ δ ¯ c ∗ (cid:105) = 0 , (A2)where in the second term of the above equation we have used the fact that the fields c, ¯ c are free: δ Γ δ ¯ c = − ( (cid:3) + m ) c ; δ Γ δc = ( (cid:3) + m ) c . (A3) • the X , -equationsSince c is a free field, the constraint ST identity Eq. (A2) reduces to the X -equationof motion (cid:66) X (Γ) ≡ δ Γ δX − v ( (cid:3) + m ) δ Γ δ ¯ c ∗ = 0 . (A4)The X -equation is in turn given by (cid:66) X (Γ) ≡ δ Γ δX − 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 − (cid:104) (1 + z ) (cid:3) + M (cid:105) X − v ¯ c ∗ . (A5)Both Eqs.(A4) and (A5) are unaltered by the presence of the background fields. Atorder n , n ≥ X , -equations read δ Γ ( n ) δX = 1 v ( (cid:3) + m ) δ Γ ( n ) δ ¯ c ∗ , (A6a) 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 . (A6b)By using the chain rule for functional differentiation we see that by Eqs. (A6) Γ ( n ) canonly depend on the combinations:¯ (cid:99) ∗ = ¯ c ∗ + 1 v ( (cid:3) + m )( X + X ); (cid:84) = T + g v Λ X , (cid:85) = U + g v Λ X ; (cid:82) = R + g v X . (A7a)Notice that the combinations in the r.h.s. of Eq.(A7a) are background gauge invariant.23 the b -equation δ Γ δb = ξb − ∂ µ ( A µ − (cid:98) A µ ) − ξe [( (cid:98) σ + v ) χ − (cid:98) χ ( σ + v )] . (A8)By projecting Eq.(A8) at order n ≥ b -dependence is confined at tree level: δ Γ ( n ) δb = 0 , n ≥ . (A9)Hence in studying higher order 1-PI Green’s functions one can safely set b = 0. • the antighost equation δ Γ δ ¯ ω = (cid:3) ω + ξev δ Γ δχ ∗ − ∂ µ Ω µ + ξe Ω (cid:98) σ χ − ξe Ω (cid:98) χ ( σ + v ) . (A10)At order n ≥ δ Γ ( n ) δ ¯ ω = ξev δ Γ ( n ) δχ ∗ . (A11)Eq.(A11) entails that at order n ≥ ω only happens via thecombination χ ∗(cid:48) = χ ∗ + ξev ¯ ω . (A12) • the background Ward identity (cid:87) (Γ) = − ∂ µ δ Γ δA µ − eχ δ Γ δσ + e ( σ + v ) δ Γ δχ − ∂ µ δ Γ δ (cid:98) A µ − e (cid:98) χ δ Γ δ (cid:98) σ + e ( (cid:98) σ + v ) δ Γ δ (cid:98) χ − eχ ∗ δ Γ δ (cid:98) σ ∗ + eσ ∗ δ Γ δχ ∗ = 0 . (A13) Appendix B: Parameterization of Background Generalized Field Redefinitions
We parameterize the terms proportional to the classical equations of motion for σ, χ inEq.(5.7) at Q Φ = 0 (i.e. (cid:98) A µ = A µ , (cid:98) σ = σ, (cid:98) χ = χ ) as follows (cid:90) d x (cid:110) a + a T + a (cid:16) (cid:98) φ † (cid:98) φ − v (cid:17) + a (cid:16) (cid:98) φ † (cid:98) φ − v (cid:17) + . . . (cid:111) (cid:90) | Q Φ =0 = (cid:90) d x (cid:104)(cid:16) r + r T T + r σ (cid:98) σ + r σ (cid:98) σ + r χ (cid:98) χ + . . . (cid:17) Γ (0) σ (cid:12)(cid:12) Q Φ =0 (cid:16) r χ (cid:98) χ + r χT (cid:98) χT + r χσ (cid:98) χ (cid:98) σ + r χσ (cid:98) χ (cid:98) σ + r χ (cid:98) χ + . . . (cid:17) Γ (0) χ (cid:12)(cid:12) Q Φ =0 (cid:105) , (B1)where the dots denote terms at least quadratic in T , cubic in σ or of dimension ≥ r ’s are known since they are linear combinations of the ˜ ρ ’s in Eq.(6.2)and the γ ’s in the kernel expansion Eq.(5.4), namely r = v ˜ ρ , r σ = ρ + ˜ ρ + v ˜ ρ − γ Ω (cid:98) σ σ ∗ ,r T = v ˜ ρ , r σ = ρ + ˜ ρ + v ˜ ρ − γ Ω (cid:98) σ σ ∗ σ − γ Ω (cid:98) σ σ ∗ (cid:98) σ , r χ = v ˜ ρ − γ Ω (cid:98) χ σ ∗ χ − γ Ω (cid:98) χ σ ∗ (cid:98) χ ,r χ = ρ + ˜ ρ − γ Ω (cid:98) χ χ ∗ , r χT = ρ T + ˜ ρ T − γ Ω (cid:98) χ χ ∗ T ,r χσ = ρ + ˜ ρ − γ Ω (cid:98) σ χ ∗ χ − γ Ω (cid:98) σ χ ∗ (cid:98) χ − γ Ω (cid:98) χ χ ∗ σ − γ Ω (cid:98) χ χ ∗ (cid:98) σ ,r χσ = ρ + ˜ ρ − γ Ω (cid:98) χ χ ∗ σσ − γ Ω (cid:98) χ χ ∗ (cid:98) σ (cid:98) σ − γ Ω (cid:98) χ χ ∗ (cid:98) σσ − γ Ω (cid:98) σ χ ∗ σχ − γ Ω (cid:98) σ χ ∗ (cid:98) σχ − γ Ω (cid:98) σ χ ∗ σ (cid:98) χ − γ Ω (cid:98) σ χ ∗ (cid:98) σ (cid:98) χ ,r χ = ρ + ˜ ρ − γ Ω (cid:98) χ χ ∗ χ − γ Ω (cid:98) χ χ ∗ (cid:98) χ − γ Ω (cid:98) χ χ ∗ (cid:98) χχ . (B2)The coefficients a ’s can be expressed as linear combinations of the r ’s. The linear system isover-constrained, so we obtain some consistency conditions that have to be fulfilled.We find a = r χ , a = r χT , va = r χσ , a v a = r χσ , (B3a) r = va , r σ = v a + a , r σ = 32 va + v a , r χ = v a , r χ = a , r T = va . (B3b)Eqs.(B3a) fix the coefficients a ’s, while Eqs.(B3b) are the consistency conditions that mustbe fulfilled. One finds for the a ’s: a = M A (1 − δ ξ ;1 )16 π v (cid:15) , a = − M A (1 − δ ξ ;1 )8 π v (cid:15) ,a = − π v zM A z (cid:15) , a = 116 π v M A z (3 z − z ) (cid:15) . (B4)It is then easy to check that they obey Eqs.(B3b). Appendix C: Tadpole renormalization
The coefficient c in Eq.(B1) is related to the renormalization of the tadpole Γ (1) σ (0) .25e begin by studying the background taadpole. By taking a derivative of the ST identityEq.(A1) w.r.t. Ω (cid:98) σ and then setting all the fields and external sources to zero we obtainΓ (1) (cid:98) σ (0) = 0 , (C1)i.e. the background tadpole vanishes (in the ( A µ , σ, χ )-basis) in any gauge as a consequenceof the ST identity.The UV-divergent part of the σ -tadpole Γ (1) σ (0) can be read off from Eq.(6.1):Γ (1) (cid:48) (cid:12)(cid:12)(cid:12) (cid:98) σ = (cid:98) χ =0 = (cid:88) j c j (cid:73) j + Y (1) ⊃ λ (cid:90) d x (cid:16) φ † φ − v (cid:17) + c (cid:90) d x (cid:104) ( σ + v )Γ (0) σ + χ Γ (0) χ (cid:105) . (C2)By taking a derivative of Eq.(C2) w.r.t σ and then setting fields and external sources to zerowe obtain Γ (1) σ (0) = vλ − m vc . (C3)The coefficient λ reads λ = 116 π v z ) (cid:110) (1 + z )[ M + M A (1 + z ) ] m + 2[ M + 3 M A (1 + z ) ] (cid:111) (cid:15) , (C4)where M A = ev is the mass of the vector meson A µ .In Feynman gauge c vanishes so that λ = 1 v Γ (1) σ (0) (cid:12)(cid:12)(cid:12)(cid:12) ξ =1 . (C5)Eq.(C3) then implies in the Landau gauge: c | ξ =0 = 1 m v (cid:16) Γ (1) σ (0) (cid:12)(cid:12)(cid:12) ξ =1 − Γ (1) σ (0) (cid:12)(cid:12)(cid:12) ξ =0 (cid:17) . (C6)This is a consistency relation satisfied by the coefficient c in Eq.(C6) that can be easilyverified by explicit computation. [1] R. Aaij et al. (LHCb), Phys. Rev. Lett. , 151601 (2014), arXiv:1406.6482 [hep-ex].[2] R. Aaij et al. (LHCb), JHEP , 104 (2016), arXiv:1512.04442 [hep-ex].[3] A. M. Sirunyan et al. (CMS), Phys. Lett. B , 517 (2018), arXiv:1710.02846 [hep-ex].
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