Background field method in the gradient flow

The Yang--Mills gradient flow and its extension to the fermion field provide a very general method to obtain renormalized observables in gauge theory. The method is applicable also with non-perturbative regularization such as lattice. The gradient flow thus offers useful probes to study non-perturbative dynamics of gauge theory. In this work, aiming at possible simplification in perturbative calculations associated with the gradient flow, a modification of the gauge-fixed version of the flow equation, which preserves gauge covariance under the background gauge transformation, is proposed. This formulation allows for example a very quick one-loop calculation of the small flow time expansion of a composite operator that is relevant to the construction of a lattice energy--momentum tensor. Some details of the calculation, which have not been given elsewhere, are presented.


Introduction
The Yang-Mills gradient flow [1,2] and its extension to the fermion field [3] provide a very general method to obtain renormalized observables in gauge theory. The method is applicable also with non-perturbative regularization such as lattice. The gradient flow thus offers useful probes to study non-perturbative dynamics of gauge theory. See Ref. [4] for a recent review.
As noted in Ref. [1], for perturbative consideration of the gradient flow, it is useful to introduce a "gauge fixing term" to the flow equation that breaks gauge covariance. This gauge fixing term gives rise to a Gaussian damping factor also for gauge degrees of freedom which ensures a good convergence property of momentum integrals; this then facilitates perturbative consideration of the gradient flow such as the proof of its renormalizability [2,3].
In this work, aiming at possible simplification in perturbative calculations associated with the gradient flow, a modification of (the gauge-fixed version of) the flow equation of Ref. [1] is proposed. Our proposal is where t > 0 is the flow time which parametrizes the flow of the gauge field B µ (t, x); D ν and G ν µ stand for the covariant derivative and field strength of the flowed gauge field, respectively. The term being proportional to α 0 is the "gauge fixing term" mentioned above. To define this term, we decompose the gauge fields into the background and quantum parts as . Then in Eq. (1.1), is the covariant derivatives with respect to the background fieldB µ (t, x). The idea is that, as the conventional background field method [5,6,7,8,9], the "gauge fixing term" is designed so that covariance under the background gauge transformation is preserved; the quantum fields transform as the adjoint representation under the background gauge transformation. In addition to Eq. (1.1), we postulate that the background field obeys its own flow equation: whereĜ ν µ (t, x) is the field strength of the background fieldB µ (t, x).
Since the present study is already published in Ref. [10], in these proceedings, I will reproduce some materials which were not explicitly given in Ref. [10]. In particular, since the most interesting result obtained in Ref. [10] is a one-loop calculation of the small flow time expansion of the composite operator G a µρ (t, x)G a νρ (t, x), which is relevant to the construction of a lattice energymomentum tensor via the gradient flow [11,12] (see also Ref. [13]), we will present some details of the calculation 1 which were omitted in Ref. [10]; we do not treat the fermion flow in the present article. In this way, I hope that the present article becomes complementary to Ref. [10]. Our notational convention is identical to that of Ref. [10]; in particular, generators of the gauge group are normalized as tr(T a T b ) = (−1/2)δ ab .

Small flow time expansion relevant to the energy-momentum tensor
The initial condition in the flow equation (1.1), A µ (x), is subject of the functional integral with the Boltzmann weight, specified by the Yang-Mills and background-gauge-fixing actions: here D ≡ 4 − 2ε is the spacetime dimension. The Faddeev-Popov ghost action corresponding to this background gauge fixing is given by From these, the tree-level propagators in the presence of the background field are given by, for the Feynman gauge λ 0 = 1, is the background covariant derivative in the adjoint representation; (D 2 ) ab =D ac µD cb µ and In what follows, as Ref. [10], we assume that the background fieldÂ µ (x) obeys the Yang-Mills equation of motion:D νFν µ (x) = 0. Eq. (1.3) then implies that the background gauge field does not flow,B(t, x) =Â(x). This assumption considerably simplifies all the expressions and, in particular, the tree-level propagator of the flowed quantum field in the presence of the background field, for the "Feynman gauge" α 0 = 1, 2 is given by [ Now, for the construction of a lattice energy-momentum tensor in Refs. [11,12], one has to find the coefficients ζ 11 (t) and ζ 12 (t) in the small flow time expansion [2] of the form, For our background-quantum decomposition, where we have usedB(t, x) =Â(x). A similar expansion holds also for F a µρ F a νρ . Thus, in the tree-level, in which the quantum fields are treated as zero, we have and from these, in the tree level. For the one-loop calculation, as noted in Refs. [15], it is convenient to consider the correlation function   (2.13) To study the t → 0 behavior of this, we set δ (x − y) = d D p (2π) D e ipx e −ipy , and moves the plain wave e ipx to the most left-hand side under the limit lim y→x as the Fujikawa method [16,17]. For this, we noteD µ e ipx = e ipx (ip µ +D µ ), (2.14) and rescale the integration variable as (2.16) and this momentum integration identically vanishes for any D. This shows that one-loop tadpole diagrams do not contribute to the O(t 0 ) terms in Eq. (2.7).
Next, we consider one-loop diagrams which arise from the contraction of quantum fields in the last line of Eq. (2.8) by propagators (2.3) and (2.6). A procedure being similar to that led to Eq. (2.15) yields Eq. (3.16) of Ref. [10]; it is where P µα,νδ ,β γ ≡ δ µα δ νδ δ β γ − δ µα δ νγ δ β δ − δ µβ δ νδ δ αγ + δ µβ δ νγ δ αδ . (2.18) We note that, in the present background-gauge-covariant formulation, this compact expression (2.17) contains all the information equivalent to the tedious diagrammatic expansion computed in Ref. [11]. We first compute the second term of Eq. (2.17): The expansion of this expression for t → 0 is easy because, as noted in Ref. [10], only terms symmetric under µ ↔ ν contribute by definition. From Eqs. (A1) and (A2) of Ref. [10], we immediately see that the expansion yields The last term in Eq. (2.17) gives rise to the same contribution. The computation of the first term of Eq. (2.17) is somewhat complicated. We first consider the expression without the factor P µα,νδ ,β γ ; (2.21) We then use Eqs. (A1)-(A3) of Ref. [10]. We can neglect O(ξ −D/2 ) terms in the integrand, because these terms give rise to O(t −1 ) terms in Eq. (2.17) for D → 4 which must be absent from gauge invariance. After the expansion, we carry out the momentum integration. Then since where ε(t) −1 ≡ 1/ε + ln(8πt). These coefficients are fundamental for the construction of a lattice energy-momentum tensor in Refs. [11,12]. The present simple calculational scheme revealed that there were errors in the original diagrammatic calculation in Ref. [11] (the diagrams in which the mistakes were made have been identified; see Ref. [18]). See also the errata for Refs. [11,12,13].

Conclusion
In the present work, we introduced a background-gauge-covariant gauge fixing in the gradient flow equation. At least in the one-loop order, this formulation allows a very efficient calculational scheme for the small flow time expansion as we illustrated for the composite operator that is relevant to the construction of a lattice energy-momentum tensor. Because of its efficiency, further applications, including two-loop computation of the small flow time expansion, are expected.
I would like to thank Kazuo Fujikawa, Kenji Hieda, and Hiroki Makino for enjoyable discussions. The work of H. S. is supported in part by Grant-in-Aid for Scientific Research 23540330.