Ward identities for the standard model effective field theory

We derive Ward identities for the standard model effective field theory using the background-field method. The resulting symmetry constraints on the standard model effective field theory are basis independent, and constrain the perturbative and power-counting expansions. A geometric description of the field connections, and real representations for the SU ð 2 Þ L × U ð 1 Þ Y generators, underlies the derivation.


I. INTRODUCTION
The standard model (SM) is an incomplete description of observed phenomena in nature. However, explicit evidence of new long-distance propagating states is lacking. Consequently, the SM is usefully thought of as an effective field theory (EFT) for measurements and data analysis, with characteristic energies proximate to the electroweak scale ( ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2hH † H p i ≡v T ), such as those made at the LHC or lower energies.
The standard model effective field theory (SMEFT) is based on assuming that physics beyond the SM is present at scales Λ >v T . The SMEFT also assumes that there are no light hidden states in the spectrum with couplings to the SM; and a SUð2Þ L scalar doublet (H) with hypercharge y h ¼ 1=2 is present in the EFT.
A power-counting expansion in the ratio of scales v T =Λ < 1 defines the SMEFT Lagrangian as The higher-dimensional operators Q ðdÞ i are labeled with a mass dimension d superscript, and multiply unknown, dimensionless Wilson coefficients C ðdÞ i . The sum over i, after nonredundant operators are removed with field redefinitions of the SM fields, runs over the operators in a particular operator basis. In this paper we use the Warsaw basis [1]. However, the main results are formulated in a basis independent manner and constrain relationships between Lagragian parameters due to the linear realization of SUð2Þ L × Uð1Þ Y in the SMEFT.
The SMEFT is a powerful practical tool, but it is also a well-defined field theory. Many formal field-theory issues also have a new representation in the SMEFT. This can lead to interesting subtleties, particularly when developing SMEFT analyses beyond leading order. When calculating beyond leading order in the loop (ℏ) expansion, renormalization is required. The counterterms for the SMEFT at dimension five [2,3], and six [4][5][6][7] are known and preserve the SUð3Þ × SUð2Þ × Uð1Þ symmetry of the SM. Such unbroken (but nonmanifest in some cases) symmetries are also represented in the naive Ward-Takahashi identities [8,9] when the background-field method (BFM) [10][11][12][13][14][15] is used to gauge fix the theory. In Ref. [16] it was shown how to gauge fix the SMEFT in the BFM in R ξ gauges, and we use this gauge-fixing procedure in this work.
The BFM splits the fields in the theory into quantum and classical background fields (F → F þF), with the latter denoted with a hat superscript. By performing a gaugefixing procedure that preserves the background-field gauge invariance, while breaking explicitly the quantum-field gauge invariance, the Ward identities [8] are present in a "naive manner"-i.e., the identities are related to those that would be directly inferred from the classical Lagrangian. This approach is advantageous, as otherwise the gaugefixing term, and ghost term, of the theory can make symmetry constraints nonmanifest in intermediate steps of calculations.
The BFM gauge-fixing procedure in the SMEFT relies on a geometric description of the field connections, and real representations for the SUð2Þ L × Uð1Þ Y generators. Using this formulation of the SMEFT allows a simple Ward-Takahashi identity to be derived, which constrains the n-point vertex functions. The purpose of this paper is to report this result and derivation. 1

II. PATH INTEGRAL FORMULATION
The BFM generating functional of the SMEFT is given by The integration over d 4 x is implicit in L GF . The generating functional is integrated over the quantum-field configurations via DF, with F field coordinates describing all longdistance propagating states. J stands for the dependence on the sources that only couple to the quantum fields [18]. The background fields also effectively act as sources of the quantum fields. S is the action, initially classical, and augmented with a renormalization prescription to define loop corrections. The scalar Higgs doublet is decomposed into field coordinates ϕ 1;2;3;4 , defined with the normalization The scalar kinetic term is defined with a field space metric introduced as where ðD μ ϕÞ I ¼ ð∂ μ δ I J − 1 2 W A;μγI A;J Þϕ J , with real generators (γ) and structure constants (ε A BC ) defined in the appendix. The corresponding kinetic term for the SUð2Þ L × Uð1Þ Y spin-one fields is where A; B; C; … run over f1; 2; 3; 4g, (as do I, J) and Extending this definition to include the gluons is straightforward.
A quantum-field gauge transformation involving these fields is indicated with a Δ, with an infinitesimal quantum gauge parameter Δα A . Explicitly, the transformations are The BFM gauge-fixing term of the quantum fields W A is [16] The introduction of field space metrics in the kinetic terms reflects the geometry of the field space due to the powercounting expansion. These metrics are the core conceptual difference of the relation between Lagrangian parameters, compared to the SM, in the Ward identities we derive. The field spaces defined by these metrics are curved; see Refs. [19][20][21]. The background-field gauge fixing relies on the basis independent transformation properties of g AB and h IJ , 2 and the fields, under background-field gauge transformations (δF) with infinitesimal local gauge parameters δα A ðxÞ given by where we have left the form of the transformation of the fermion fields implicit. Here i, j are flavor indicies. The background-field gauge invariance of the generating functional, i.e., is established by using these gauge transformations in conjunction with the linear change of variables on the quantum fields. The generating functional of connected Green's functions is given by where J ¼ fJ A μ ; J I ϕ ; J f ; Jfg. As usual the effective action is the Legendre transform 1 Modified Ward identities in the SMEFT have been discussed in an on-shell scheme in Ref. [17]. 2 The explicit forms of g AB and h IJ are basis dependent. The forms of the corrections for the Warsaw basis at L ð6Þ are given in Ref. [16].
Here our notation is chosen to match Ref. [22]. S-matrix elements are constructed via [22][23][24] The last term in Eq. (11) is a gauge-fixing term for the background fields, formally independent from Eq. (6), and introduced to define the propagators of the background fields.
Finally, we define a generating functional of connected Green's functions W c ½Ĵ as a further Legendre transform [24], withF ¼ fW A ; ϕ I g and

III. WEAK EIGENSTATE WARD IDENTITIES
The BFM Ward identities follow from the invariance of Γ½F; 0 under background-field gauge transformations, In position space, the identities are For some n-point function Ward identities, the background fields are set to their vacuum expectation values (vevs). This is defined through the minimum of the classical action S, where the scalar potential is a function of H † H, which we denote as hi. For example, the scalar vev defined in this manner is through ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2hH † H p i ≡v T and explicitly hϕ J i with an entry set to the numerical value of the vev does not transform viaγ I A;J . A direct relation follows between the tadpoles (i.e., the one-point functions δΓ=δφ I ) and, hφ J i, given by Requiring a Lorentz-invariant vacuum sets the tadpoles for the gauge fields to 0. Thus, for the scalars γ B hϕ J i ≠ 0 and the unbroken combination ðγ 3 þγ 4 Þhϕ J i¼0 corresponds to Uð1Þ em . Equation (17) with B ¼ 3, 4 does not given linearly independent constraints. This leads to the requirement of a further renormalization condition to define the tadpole δΓ=δφ 4 to vanish. The Ward identities for the two-point functions are The three-point Ward identities are

IV. MASS-EIGENSTATE WARD IDENTITIES
The mass-eigenstate SM Ward identities in the BFM are summarized in Ref. [15]. The transformation of the gauge fields, gauge parameters, and scalar fields into mass eigenstates in the SMEFT iŝ withÂ C ¼ ðŴ þ ;Ŵ − ;Ẑ;ÂÞ,Φ L ¼ fΦ þ ;Φ − ;χ;Ĥ 0 g. This follows directly from the formalism in Ref. [16] (see also Ref. [25]). The matrices U, V are unitary, with ffiffi ffi g p AB ffiffi ffi The square root metrics are understood to be matrix square roots and the entries are hi of the field space metrics entries. The combinations ffiffi ffi g p U and ffiffiffi h p V perform the mass-eigenstate rotation for the vector and scalar fields, and bring the corresponding kinetic term to canonical form, including higher-dimensional-operator corrections. We define the mass-eigenstate transformation matrices to avoid a proliferation of index contractions. The structure constants and generators, transformed to those corresponding to the mass eigenstates, are defined as The background-field gauge transformations in the mass eigenstate are The Ward identities are then expressed compactly as In this manner, the "naive" form of the Ward identities is maintained. The BFM Ward identities in the SMEFT take the same form as those in the SM up to terms involving the tadpoles. This is the case once a consistent redefinition of couplings, masses, and fields is made.

V. TWO-POINT FUNCTION WARD IDENTITIES
The Ward identities for the two-point functions take the form

VI. PHOTON IDENTITIES
The Ward identities for the two-point functions involving the photon are given by Using the convention of Ref. [15] for the decomposition of the vertex function an overall normalization factors out of the photon two-point Ward identities compared to the SM, and The latter result follows from analyticity at k 2 ¼ 0.

VII. W AE ;Z IDENTITIES
Directly, one finds the identities terms. By definition, the vev is defined as ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2hH † H p i ≡v T . The substitution of the vev leading to theẐ boson mass in the SMEFT (M Z ) absorbs a factor in the scalar mass-eigenstate transformation matrix as ffiffiffiffiffiffiffiffiffiffiffiffiffiffi If a scheme is chosen so that δΓ=δφ 4 vanishes, then transformation to the mass-eigenstate basis of the one-point vector δΓ=δφ i is still vanishing in each equation above. One way to tackle tadpole corrections is to use the Fleischer-Jegerlehner (FJ) tadpole scheme; for discussion see Refs. [26,27].

VIII. A;Z IDENTITIES
The mapping of the SM Ward identites for Γ AZ in the BFM given in Ref. [15] to the SMEFT is As an alternative derivation, the mapping between the mass eigenstate (Z; A) fields in the SM and the SMEFT (Z, A) reported in Ref. [28] directly follows from Eq. (27). Input parameter scheme dependence drops out when considering the two-point function Γ AZ in the SM mapped to the SMEFT and a different overall normalization factors out.

IX. CONCLUSIONS
We have derived Ward identities for the SMEFT, constraining both the perturbative and power-counting expansions. The results presented already provide a clarifying explanation to some aspects of the structure of the SMEFT that has been determined at tree level. The utility of these results is expected to become clear as studies of the SMEFT advance to include subleading corrections.