Off-shell renormalization in the presence of dimension 6 derivative operators. III. Operator mixing and $\beta$ functions

We evaluate the one-loop $\beta$ functions of all dimension 6 parity-preserving operators in the Abelian Higgs-Kibble model. No on-shell restrictions are imposed; and the (generalized) non-polynomial field redefinitions arising at one-loop order are fully taken into account. The operator mixing matrix is also computed, and its cancellation patterns explained as a consequence of the functional identities of the theory and power-counting conditions.


I. INTRODUCTION
New physics beyond the Standard Model (SM) can be characterized in a model independent and systematic fashion within the Effective Field Theories (EFTs) framework, in which the (renormalizable) tree-level SM action is supplemented with the terms (k ≥ 5) where O [k] i are k-dimensional operators whose dimension dictates the suppression of the corresponding coefficients c [k] i in terms of powers of a high-energy scale Λ. The resulting Standard Model Effective Field Theory (SMEFT) action is not renormalizable in the usual (power counting) sense; it is, nevertheless, renormalizable in the modern sense [1], as all the divergences can be cancelled through the renormalization of the (infinite) number of terms in the bare action while respecting the symmetries of the theory.
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) mixing between 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 suitable choice of the basis in the space of local operators [3][4][5][6]; additionally, alien operators have been shown to be cohomologically trivial and therefore have vanishing on-shell correlators [3] (for a review see also [7]). This fact is at the basis of recent computations in the literature [8][9][10][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 this purpose one needs to extend the approach adopted in the power-counting renormalizable case [13][14][15] to EFTs; in particular, one must work out a procedure to fix the generalized field redefinitions (GFRs) that do arise in these models. Here 'generalized' means that, at variance with the power-counting renormalizable case, these redefinitions are not linear in the quantum fields (in fact, not even polynomial already at one-loop order, as we will show). 2 The matching of the couplings order by order in the loop expansion, once the GFRs' effects are taken into account, is the next technical step required to match the model with its UV completions 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 overlapping divergences with local counter-terms.
To attain these goals, in [16] it has been developed a general theory for the recursive subtraction of off-shell UV divergences order by order in the loop expansion applicable to EFTs displaying a spontaneously broken symmetry phase. This is achieved by solving the Slavnov-Taylor (ST) identity to all-orders, which allows in turn to disentangle the gaugeinvariant contributions to the off-shell one-particle irreducible (1PI) amplitudes from those associated 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 rise to an infinite number of one-loop divergent diagrams, maximally violates power counting.
In particular, the complete renormalization of all the radiatively generated dimension 6 operators has been carried out together with the determination of the full g-dependence of the β-function coefficients.
Before moving on to consider the full dimension 6 SMEFT [18], there is just one aspect that has been left out in the study of its Abelian sibling: namely, the analysis of the full off-shell renormalization when all inequivalent parity-preserving dimension 6 operators (classified according to [12]) are added to the power counting renormalizable action. And this constitutes precisely the subject of the present paper.
From the point of view of the EFT renormalization programme of [1], what we achieve here is to fully evaluate all the terms appearing in the renormalized action S at one loop (in the relevant sector of dimension ≤ 6), expressed as At zero antifields, ∆ 1 collects one-loop gauge-invariant counterterms. The renormalized action has the same form as the original bare action S 0 ; in particular, it can be expanded on a basis of gauge-invariant operators (in the zero antifield sector). However, these counterterms are not enough to renormalize the theory: one must also take into account the effects of GFRs, that are implemented according to a canonical transformation with respect to the Batalin-Vilkovisky (BV) bracket associated with the gauge symmetry of the model [1]. The transformed bare action S ′ 0 takes then the form where F 1 is the one loop term in the loop expansion F (t) = tF 1 + · · · of the generator of the canonical transformation responsible for the field-antifield redefinition: Being canonical, this transformation pre- j ) = 0, and is obtained by solving the differential equationṠ 0 (t) = (F (t), S 0 (t)) with the boundary condition S 0 (0) = S 0 , see [1]. Such canonical transformation generalizes the usual linear wave function renormalizations of the power-counting renormalizable cases. It plays a crucial and ubiquitous 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 to make 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's formulations 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 is explicitly carried out in Sect. VI. Finally, in Sect. VIII we describe the one-loop mixing matrix in the original theory and compare our results with the literature. Conclusions are presented in Sect. IX. A number of technical issues are discussed in a set of Appendices presented at the end of the paper: functional identities of the X-theory and the propagators in 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.

II. NOTATIONS AND CONVENTIONS
In the X-formalism approach of [19], the tree-level vertex functional takes the form In the expression above, the first line represents the action of the Abelian HK model in the Xformalism, where the usual scalar field φ ≡ 1 with v the vacuum expectation value (vev) is supplemented with a singlet field X 2 , that provides a gaugeinvariant parametrization of the physical scalar mode. Notice also that we defined φ 0 = σ+v with σ having a zero vev. The field X 1 plays instead the role of a Lagrange multiplier: when going on-shell with this field one recovers the constraint 1 X 2 ∼ 1 v (φ † φ − v 2 /2), which once inserted back into the first line of Eq. (2.1), cancels the m 2 -term leaving the usual Higgs quartic potential with coefficient ∼ M 2 /2v 2 . Hence, Green's functions in the target theory 2 have to be m 2 -independent, a fact that provides a very strong check of the computations, due to the ubiquitous presence of m 2 both in Feynman amplitudes as well as invariants.
The X 1,2 -system comes together with a constraint BRST symmetry, ensuring that the number of physical degrees of freedom in the scalar sector remains unchanged in the Xformalism 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: 1 Going on-shell with X 1 yields the condition so that the most general solution is + η, η being a scalar field of mass m. However, in perturbation theory the correlators of the mode η with any gauge-invariant operators vanish [16], so that one can safely set η = 0. 2 We define as 'target' theory the original theory defined in terms of conventional fields.

5
The associated ghost and antighost fields c,c are free. The constraint BRST differential s anticommutes with the (usual) gauge group BRST symmetry of the classical action after the gauge-fixing introduced in the fifth line of Eq. (2.1): Here ω (ω) is the U(1) ghost (antighost); the latter field is paired into a BRST doublet with the Lagrange multiplier field b, enforcing the usual R ξ gauge-fixing condition The third line of Eq. (2.1) contains the dimension 6 parity preserving subset of the gaugeinvariant operators described in [12], modulo for the fact that we use the zero expectation We thus see that the classical powercounting renormalizable action is supplemented in the X-formalism by the X 2 -dependent Notice that the operator O [6] 3 is special in the sense that it does not give rise in the X-theory to new interaction vertices: rather it modifies the propagator of the X 2 -field by rescaling the p 2 -term [21] (the full set of propagators of the model is summarized in Appendix C).
Notice also that in comparison with the conventions of [16,17] we have rescaled the higher dimensional coupling constants by a factor v/Λ in order to obtain, when mapping back to the target theory, the standard 1/Λ 2 pre-factor for dimension 6 operators.
To maintain a detailed comparison with [1], we provide in the following some technical details.
The relevant BV bracket is the one associated with the gauge symmetry, the constraint BRST symmetry invariance being exhausted in the X 1 -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 and thus there is no need to introduce the gauge antifield A * µ for controlling quantum corrections 4 (although algebraically one is allowed to). Also, in the R ξ -gauge that we employ, there is no need to introduce the antifieldω * , coupled to the Nakanishi-Lautrup field b = sω: in fact, see Appendix B, the b-equation (B1) and the antighost equation (B2) imply that at the quantum level there is no dependence on the field b and moreover that the antighost dependence can be reabsorbed by the antifield redefinition (B4). On the other hand, in the formulation of [1], where one introduces bothω * and A * µ , the antighost-dependent sector of the action is recovered from the antifield couplings d 4 x (A * µ sA µ + χ * sχ) via a canonical transformation with fermionic generator F = d 4 x F ξω (that incidentally exactly yields the antifield redefinition in Eq. (B4)). Thus, the dimension ≤ 6 sector of S 0 is At one loop order further operators will be radiatively generated starting from Γ (0) . Those operators can be however expressed in the target theory as gauge invariant polynomials in the field φ, its (symmetrized) covariant derivatives, the field strength and its ordinary derivatives. This set of variables is particularly suited in order to obtain the coefficients of 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 independent operators is carried out in some detail in Appendix E. 4 This latter fact can be easily understood since the coupling does not generate any interaction vertex involving A * µ , due to the aforementioned linearity of the BRST transformation of A µ in the quantum fields.
Returning to Eq. (2.1), we notice that the terms in the third line of Eq.(2.1) respect both BRST symmetries and thus they do not violate either the X 1 -equation (A6) or the ST identity (A1). Finally, in the fourth row we have added the external sources T 1 , R, U required to define the X 2 -equation at the quantum level in the presence of additional non power-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 in the X-theory for all the operators giving rise to contributions to dimension 6 operators in the target theory.
In what follows subscripts denote functional differentiation with respect to fields and external sources. Thus, amplitudes will be denoted as, e.g., Γ (1) χχ , meaning . 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 cancellations in gauge invariant operators and in particular, as we will see, the crucial role of the GFRs in ensuring the gauge independence of ostensibly gauge invariant quantities.
Consider now the UV divergent contributions to one-loop amplitudes. They form a local functional (in the sense of formal power series) denoted by Γ (1) . Since Γ (1) belongs to the kernel of the linearized ST operator S 0 defined in Eq. (A3), i.e., the nilpotency of S 0 ensures that Γ (1) is the sum of a gauge-invariant functional I (1) and a cohomologically trivial contribution S 0 (Y (1) ): with GFRs described by the cohomologically trivial term S 0 (Y (1) Ultimately, we are interested in the UV divergences of dimension 6 gauge invariant operators in the target theory. To identify the invariants in the X-theory contributing to these operators the mapping function from the X-to the target theory is needed. As explained in [16,17] this amounts to solving the X 1,2 -equations in the X-theory via the replacements in Eq. (A8) and then going on-shell with X 1,2 . At the one loop level it is sufficient to impose the classical equations of motions for X 1,2 . The X 1 -equation gives whereas the classical X 2 -equation of motion yields (at zero external sources) By inserting Eqs. (3.5) and (3.6) into the solutions of the X 1,2 -equations (A9a) we obtain the explicit form of the mapping for the HK model:

IV. DIMENSION SIX OPERATORS COEFFICIENTS
For computing the UV coefficients of dimension 6 gauge-invariant operators in the target theory, we need to consider, see Appendix D: 1. Operators which only depend on the external sources and contribute to dimension 6 operators in the target theory again due to the mapping in Eq. Clearly, all the associated UV coefficients λ i , θ i and ϑ i will be ξ-independent. In order to fix them, we need to evaluate a certain number of Feynman amplitudes and derive the projections of these operators on the relevant 1-PI Green's functions. However, and as already 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 from GFRs. Indeed, the latter prove essential in order to ensure gauge independence of the UV coefficients 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 is to work out the GFRs controlled by S 0 (Y (1) ). One needs to take them into account appropriately, otherwise the renormalization of gauge invariant operators is affected by spurious contributions arising from the incorrect subtraction of UV divergences to be removed by GFRs. In particular GFRs play a crucial role in ensuring the gauge independence of the UV coefficients of gauge invariant operators, as we will explicitly show.
In the Algebraic Renormalization approach we adopt, GFRs can be written in terms of two classes of invariants as with P and Q some local functionals 5 depending on the fields (collectively denoted by Φ) and the external sources (collectively denoted by ζ) and S 0 the linearized ST operator in Eq. (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 5 We remind the reader that in EFTs field redefinitions are, in general, non-linear in the quantized fields. the case where P and Q are constant. Since one has that the P -invariant is fixed in this case by the amplitude Γ ωχ * . Similarly, if P depends on the fields and the gauge invariant sourcesc * , R, T 1 , U, the P -invariant can be fixed by looking at antifield-dependent 1-PI amplitudes. Indeed, since the antighost equation (B2) entails that the dependence on the antighost at loops higher than one only happens via the combination χ * in Eq. (B4), we do not need to consider antighost amplitudes and antifield-dependent ones are sufficient.
The Q-invariant is trickier. Let us first notice that it does not project on χ * , σ * antifielddependent monomials: To understand the Q-invariant role in the renormalization of the theory, we remark that it depends only on the combination φ 0 ; therefore it is useful to rewrite the counting operator in terms of φ, φ † , i.e., Next, observe that we are only interested in the case when the right-hand side (r.h.s.) is evaluated at X 1,2 = 0 6 ; an explicit computation shows that the r.h.s. is indeed gaugeinvariant (remember that we need to use the antifield χ * , as a consequence of the antighost equation): On the other hand, the gauge-invariant operators of the renormalizable Abelian HK model This provides a consistent definition of the independent gauge invariant operators generalizing the corresponding set of independent physical parameters discussed in the powercounting renormalizable case. 7 Observe that as announced the m 2 -dependence has disappeared.
We also notice that in the Landau gauge (ξ = 0) ghosts are free and the theory enjoys an exact global invariance with α a constant parameter. As a consequence of this rigid U(1) invariance the only allowed cohomologically trivial invariants in the Landau gauge are those of the Q-type; P -invariants do not arise. We will verify this property in the explicit computations that follow. On the other 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 than the fields, as is well known in the literature [24].
We now list the monomials in the expansion of P, Q contributing to the projections needed to fix the coefficients of the dimension 6 operators in Eqs. (D1), (D2j) and (D3). Using the notation we obtain The different coefficients can be then evaluated by projection onto the relevant Feynman amplitudes; their values are then i.e., the polynomial Q is gauge-invariant, as expected; moreover, as anticipated, all ρ's coefficients 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 linear order in the higher dimensional couplings. For that purpose we need to apply the mapping in 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 image of the source T 1 under the mapping is proportional to g 1 and hence from the T 1 sector we receive contributions at the linearized level only from amplitudes linear in T 1 , whose coefficients need to be evaluated at z = 0. By taking these observations into account, one easily sees that the GFRs in the target theory at linear order in the g i 's and z couplings take the following form: 14 From Eq.(5.15) we see that the GFRs are non-multiplicative already at one loop and in the 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 can proceed to project on the one-loop amplitudes required to determine the coefficients of the invariants (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 and Feynman gauge and found to coincide as required.

A. Pure external sources invariants
The non zero ϑ i coefficients are

C. Gauge invariants depending only on the fields
The non zero λ i coefficients are

VII. MAPPING TO THE TARGET THEORY
The UV coefficients in the target theoryλ i can be obtained by: applying the mapping in Eq. (3.7) to the invariants in Eqs. (D1) and (D2j); combining the projections with the operators in (D3); and, finally, using the results (6.1), (6.2) and (6.3). Notice that for these coefficients all m 2 -dependent contributions must cancel out; we have checked this explicitly.
The coefficients so obtained represents the complete one-loop renormalizations of the corresponding operators; in particular, no linearized approximation in the higher dimensional couplings g i 's has been made so far. However, as the resulting general expressions are rather lengthy, we report below the non zero coefficientsλ i at linear order in the g i couplings: We hasten to emphasize that GFRs do contribute also at the linearized level, as has been discussed in detail in Section V A. Failure to take their contributions into account would lead to an erroneous determination of the coefficients in Eq.(7.1).
The g i 's, z contributions to the β functions can then be easily determined from Eq. (7.1), leading to:

VIII. ONE-LOOP MIXING MATRICES
We are now in a position to compare our results with those in the literature [25]. By inspecting Eq.(7.1) we obtain the mixing matrix represented in Table I. We find agreement with the results of [25] with the exception of the mixing of φ 4 D 2 operators with F 2 φ 2 . More specifically, a closer inspection of Eq. (7.1) shows that the operator respects the mixing pattern derived in [25], whereas the operator does not since it mixes with There is an elegant cohomological interpretation of this result. One can find S 0 -invariant combinations of gauge invariant operators that do not depend on the antifields, in very much the same way as in Eq. (5.14). Notice that these invariants depend on σ, χ only via φ and 20 they are generated by Z 2 (now to be understood in the target theory). In particular one which is gauge-invariant. Thus any invariant of the form This means that there is the freedom to replace the invariant I 6 with the linear combination of I 2 and I 3 in Eq. (E16) up to a cohomologically trivial S 0 -invariant. This transformation induces the following shift on the space of theλ's parameters: In order to study the one-loop amplitudes dependence on the g i 's and z beyond the single higher-dimensional operator insertion approximation commonly used in the literature, we have 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 underlying amplitudes decomposition made transparent by the X-formalism. As explained above, the λ's are a linear combination of the λ's coefficients multiplying gauge invariant operators which are independent from external sources of the X-theory, and of the coefficients ϑ, θ's associated with invariants involving external sources insertions (the UV behaviour of which is more constrained than that of the fields). In particular, we find for the relevant operators in Table II : The ϑ 1 -terms can be neglected: they can only induce a z-dependence and thus do not contribute to the cancellations in Table II. Hence, the problem is reduced to the determination of the g i 's dependence of the λ's coefficients in the X-theory. One immediately sees that these coefficients cannot depend on g 3 since this is a trilinear vertex in X 2 that does not contribute to the 1-PI amplitudes of the starting theory at one loop. Thus, the last row of Table II must hold, as the only possible dependence on g 3 at one loop arises from the mapping to the target theory in Eq.(3.7) and therefore governed by external amplitudes involvingc * and/or R external sources, which do not enter in Eq. (8.7).
The remaining three forbidden dependences just seem to be an accidental consequence of the one-loop Feynman diagrams; as a result, cancellation patterns do not seem to lend themselves to an easy generalization to higher orders.

IX. CONCLUSIONS
In the present paper we have completed the investigation of the one-loop off-shell renormalization of the Abelian Higgs-Kibble model supplemented at tree-level with all dimension 6 parity preserving on-shell inequivalent gauge-invariant operators. This was the last step towards the analysis of the SU(2)×U(1) case.
We have shown that the X-theory formalism provides an effective way to work out the relevant GFRs, which in turn are found to have an ubiquitous effect on the one-loop UV coefficients of dimension 6 operators. In fact, since the GFRs are non linear and even non polynomial in the fields, it is advantageous to employ cohomological tools in order to disentangle the UV coefficients of the gauge-invariant operators from the spurious (and gauge-dependent) contributions associated with GFRs.
We have provided a full one-loop computation going beyond the customary linearized approximation in the higher dimensional couplings. All coefficients have been evaluated Application of the method presented to the SMEFT is currently under investigation.
Appendix A: Functional Identities in the X-theory

ST identities
The ST identity (also known as the master equation in the BV approach) associated to the gauge group BRST symmetry reads or, at order n in the loop expansion, where S 0 is the linearized ST operator: S 0 maps the antifields σ * , χ * into the equations of motion of the fields σ, χ, while it acts on the fields as the BRST operator s. Notice that, as explained before, we do not introduce an antifield for the gauge field A µ since in the Abelian case treated here the gauge BRST transformation is linear.
The ST identity for the constraint BRST symmetry is Notice that this equation stays the same irrespectively of the presence of higher-dimensional gauge invariant operators added to the power-counting renormalizable action.
The X 2 -equation is in turn given by 3. Solving the X 1,2 -equations At order n, n ≥ 1 in the loop expansion the X 1,2 -equations reduce to By using the chain rule for functional differentiation it is straightforward to see that Eqs. (A8) entail that Γ (n) only depends on the combinations: c * =c * + 1 v ( + m 2 )(X 1 + X 2 );