Gradient flow exact renormalization group

The gradient flow bears a close resemblance to the coarse graining, the guiding principle of the renormalization group (RG). In the case of scalar field theory, a precise connection has been made between the gradient flow and the RG flow of the Wilson action in the exact renormalization group (ERG) formalism. By imitating the structure of this connection, we propose an ERG differential equation that preserves manifest gauge invariance in Yang--Mills theory. Our construction in continuum theory can be extended to lattice gauge theory.


Introduction
The gradient flow [1][2][3][4][5][6] is a continuous deformation of a gauge field configuration A a µ (x) along a fictitious time t ≥ 0. It is given by a gauge-covariant diffusion equation is the field strength of the flowed or diffused field B a µ (t, x), 1 and is the covariant derivative with respect to B a µ (t, x). The gradient flow bears a close resemblance to the coarse graining along renormalization group (RG) flows [7]. This aspect of the gradient flow has been investigated from various perspectives [6,[8][9][10][11][12][13][14][15][16][17]. In this paper we further our understanding of how the gradient flows are related to the RG flows by using the exact renormalization group (ERG) formalism (for reviews of ERG, see for instance Refs. [18][19][20]).
In scalar field theory, the analogue of Eq. (1.1) would be [21] ∂ t ϕ(t, x) = ∂ µ ∂ µ ϕ(t, x), ϕ(t = 0, x) = φ(x). (1.4) It is actually possible to make a precise connection between the gradient flow and the flow of a Wilson action under ERG [15] (see also Ref. [17]). In D dimensional Euclidean space, the ERG differential equation for the Wilson  where K and k are cutoff functions satisfying K(p) = 1 for |p| → 0, 0 for |p| → ∞, , k(p) and ∆(p) ≡ −2p 2 dK(p) dp 2 . (1.8) The origin of the anomalous dimension η τ in the above has been elucidated in Ref. [23]. Particularly for K(p) = e −p 2 , it has been shown [15] that the correlation functions of the 1 f abc is the structure constant defined from the anti-hermitian generator T a of the gauge group by [T a , T b ] = f abc T c . 2 Throughout this paper, we use abbreviations, p ≡ d D p (2π) D , δ(p) ≡ (2π) D δ (D) (p). (1.5) preserves gauge invariance. In Sect. 3.4, we solve the ERG equation in the lowest approximation, i.e., in the lowest order in a parameter λ (3.10). This parameter turns out to provide a convenient expansion parameter analogous to the conventional gauge coupling. In Sect. 4, we generalize the construction of the Wilson action in Sect. 3.1 to lattice gauge theory. We conclude the paper in Sect. 5. There is a short appendix to Sect. 3 about the normalization of the gauge field.
In this paper, we only present the basic idea and basic equations for our formulation of Yang-Mills theory; we defer possible applications for future studies.

Scalar field theory
As pointed out in Ref. [25], the change of a Wilson action S τ under a change of the cutoff scale in Eq. (1.6) can be formulated as an equality of modified correlation functions. In terms of dimensionless variables, Eq. (38) of Ref. [25] with t → 0, ∆t → τ , and e ∆tγ → Z The anomalous dimension in Eq. (1.6) and the wave function renormalization factor Z τ are related by Here, the modified correlation functions are defined by [25] 3) where the ordinary correlation functions are denoted with single brackets: (2.4) In terms of ordinary correlation functions, Eq. (2.1) reads Now, let us choose the Gaussian as the cutoff function K. We then have where is the diffused scalar field in Eq. (1.4) given in momentum space. In terms of functional integrals, this reads Using field variables in coordinate space we get δ δφ(p) = d D x e ipx δ δφ(x) and δ δϕ(t,p) = d D x e ipx δ δϕ(t,x) . Hence, we can rewrite Eq. (2.9) as (2.14) The first equality is obvious. In the second equality, we have made the replacement, , which is justified in front of the delta function. Then, we have interchanged δ δφ(x ′ ) and φ(x ′ ) neglecting an infinite constant δ δφ(x ′ ) φ(x ′ ) = δ (D) (x = 0) because this contributes only to the constant term in S τ [φ]. Finally, using the relation , (2.15) we obtain an ERG equation Here, the derivative with respect to x ′ does not act on x ′ in δ δφ(x ′ ) . Switching back to momentum space, we get where, as in Eq. (2.8), we identify the flow time t and the scale parameter τ by The field B ′b ν (t, x ′ e τ ) in the delta function is diffused from the integration variable A ′ by the flow equation Note that we have added a "gauge fixing term" with the parameter α 0 [3,4] to the original flow equation (1.1); this term suppresses the gauge degrees of freedom along the diffusion and guarantees the finiteness of gauge non-invariant correlation functions of the diffused gauge field in perturbation theory [4]. This somewhat peculiar addition is due to our tacit assumption of perturbation theory in this section. In fact, we exclude this term in lattice gauge theory discussed in the next section. In transcribing Eq. (2.13) to gauge theory, we have set Z τ = 1 because the diffused field does not receive wave function renormalization [4]; we will see that this choice is consistent with an effective presence of a cutoff in the Wilson action. We have also adopted k(p) = p 2 which yields Under a change of the scale parameter τ , Eq. (3.1) preserves the partition function: (3.4) The first equality follows from the vanishing of a total derivative [dA] δ δA a µ (x) F[A] = 0 for any well-behaved functional F[A]; for the second equality, we have used Eq. (3.1). The invariance of the partition function, expected of a Wilson action, remains formal unless the functional integral in the most right-hand side of Eq. (3.4) is regularized. In perturbation theory, at least, we can give a gauge invariant meaning to the last integral by dimensional regularization. With the lattice transcription of Eq. (3.1) in the next section, the invariance of the partition function can be given a rigorous meaning.
Another important relation that follows immediately from Eq.
This is analogous to Eq. (2.7) in scalar field theory. As for the right-hand side, note that the flow equation (3.3) can be written as an integral equation [3,4]: is the integration kernel of a linear diffusion, and Using Eq. (3.6), we can express δB δA , necessary on the right-hand side of Eq. (3.5), as a power series in B. The right-hand side of Eq. (3.5) is then given by correlation functions of the diffused field B.
We now suppose that the "bare" action S τ =0 [A] contains a gauge coupling g 0 . Setting 6 g 0 = µ ǫ Z g (ǫ)g, where µ is an arbitrary mass scale and D = 4 − 2ǫ, we take ǫ → 0 for a continuum limit. By a general theorem [4] the right-hand of Eq. (3.5) has a finite limit. Hence, the correlation functions with respect to S τ [A] on the left-hand side of Eq. (3.5) are finite in the continuum limit. This suggests that our definition of the Wilson action (3.1) implements effectively an ultraviolet cutoff for the Wilson action. 7 6 Here, Z g (ǫ) = 1 − g 2 (4π) 2 β0 2ǫ + O(g 4 ) and β 0 = 11 3 C A , where C A is the Casimir of the adjoint representation, f abc f bcd = C A δ ab . 7 In a lattice transcription of Eq. (3.1) in the next section, the presence of an ultraviolet cutoff in the Wilson action is obvious.

Gauge invariance
We next show that S τ [A] defined by Eq. (3.1) is invariant under any infinitesimal gauge transformation of the scaled gauge potential The τ dependent factor λ acts like a coupling constant: An infinitesimal gauge transformation on A is but the corresponding gauge transformation on A is modified by λ as , we first note that the first factor in Eq.
is invariant under the transformation (3.12) because the functional derivative transforms in the adjoint representation under Eq. (3.12): (3.14) We next examine the argument of the delta function in Eq. (3.1). Under the transformation (3.12), we find (we write x ′ as x for simplicity) In the third line above, we can replace e −τ (D−2)/2 A c ν (x) by B ′c ν (t, xe τ ) since ω is infinitesimal, and the two are equal when ω = 0. The last line implies that the gauge transformation (3.12) on the external variable A induces a gauge transformation on B ′b ν (t, xe τ ) with the gauge function −ω b (xe τ ): In the functional integral (3.1), the integration variable A ′ and the diffused gauge field B ′ are related by the flow equation (3.3). We wish to show that there is a gauge transformation on A ′ that gives the gauge transformed B ′ , given by Eq. (3.16), as the solution of the diffusion equation (3.3). To show this, let us consider an infinitesimal gauge transformation on the diffused field B that depends on the flow time s (we save t for t = e 2τ − 1): This changes the flow equation (3.3) to If we choose ξ as the solution to the linear diffusion equation, where the gauge potential A a µ (x) under the tilde ( ) is replaced by the rescaled potential, Eq. (3.9).
Using a relation analogous to Eq. (2.15) (with δ (D) (x) replacing D(x)): (3.25) Here, the gauge potential A a µ (x) is replaced by the combination (3.24) if it appears under the hat. This is our ERG equation for Yang-Mills theory.
Note that without the hat, Eq. (3.25) would involve only the first order differentials of S τ , and our ERG equation would be merely a change of variables. It is the differential operator in the hat (3.24), whose origin is the exponentiated second order differentials in Eq. (3.22), that introduces higher-order differentials in Eq. (3.25).
Once the ERG equation (3.25) has been obtained, we may forget the original construction (3.1) and the gradient flow behind it. Under the ERG flow, the gauge invariance is preserved in the sense explained in Sect. 3.2.
For completeness, we give a little more explicit form of the ERG equation (3.25): In deriving this, we have interchanged the order of A c µ (x); this is justified because f abc is anti-symmetric in b ↔ c. To write a differential equation for S τ , we multiply e −Sτ from the left of Eq. (3.26) and write covariant derivatives explicitly to obtain Differentiating e Sτ further, we obtain a non-linear ERG equation that involves up to quartic differentials of S τ : . (3.28)

Approximate solution to O(λ 0 )
From Eq. (3.28), we see that the parameter λ, whose original definition is Eq. (3.10), provides a convenient expansion parameter which organizes terms in the ERG equation. We expand the Wilson action in powers of λ as (3.29) where w n = O(λ 0 ). By substituting this into the right-hand side of Eq. (3.28), we obtain terms of the form ∞ n=2 λ n−2 1 n! d D x 1 · · · d D x n W a1···an n,µ1···µn (x 1 , . . . , x n )A a1 µ1 (x 1 ) · · · A an µn (x n ). (3.30) Therefore, the expansion of the Wilson action in the form (3.29) is consistent with the ERG equation (3.28).
In this paper, we study only the lowest order O(λ 0 ) terms in some detail, 9 postponing the higher-order calculations for future studies. We thus set Equation (3.28) then gives ∂ ∂τ In deriving this, we have neglected δ (D) (x = 0) assuming dimensional regularization. Imposing the translational and rotational invariance and global gauge invariance, we can write where T (p) and L(p) are functions of p 2 . Equation (3.32) then gives The general solution is given by where C(p) and D(p) are arbitrary functions of p 2 . Locality demands that C(p) and D(p) can be expanded in powers of p 2 at p = 0: Unitary demands C 0 > 0 and D 0 > 0. 9 This is the only term for the abelian gauge theory.
14 As τ → +∞, the action S τ [A] approaches an infrared fixed point S * [A], corresponding to constants C 0 and D 0 : (3.37) Since C 0 > 0 and D 0 > 0 are arbitrary, their variations give marginal operators: (3.38) It can be seen that these correspond to the change of normalization of the gauge field A (see Appendix). 10 Infinitesimal C n and D n , on the other hand, give where n = 1, 2, . . . , which correspond to irrelevant operators at the fixed point. If we make a particular choice C 0 = 1 and D 0 = ∞ in Eq. (3.36), the fixed point action becomes transverse: (3.40) and the marginal operator at the fixed point is given by It is important to pursue the above analysis to higher orders in λ to see how the ordinary beta function arises in our formalism.

Lattice gauge theory
In the previous section, we have constructed a gauge invariant Wilson action and its associated ERG equation for a generic Yang-Mills theory in continuum R 4 . We now tailor the construction for lattice gauge theory. For simplicity, we consider an infinite volume lattice Z 4 . The discrete coordinates on Z 4 render our ERG transformation discrete. This discreteness is introduced through "block-spins." Let us pick a fixed "block-spin" factor b from one of the integers, 2, 3, . . . We then define a "block-spin" link variable by where U (x, µ) is a conventional link variable on the Z 4 lattice; here,μ denotes the unit vector in the µ direction. This U (x, µ) is regarded as a link variable on the coarse lattice bZ 4 scaled by the factor b.
We then divide the range of the scale factor τ , originally continuous in 0 ≤ τ < ∞, into the contiguous intervals n∆τ < τ ≤ (n + 1)∆τ, n = 0, 1, 2, . . . , The nth interval corresponds to the scaling of x by a factor between b n and b n+1 . Multiplying a lattice coordinate x ∈ Z 4 by e ∆τ = b gives the coordinate bx on the coarse lattice bZ 4 . Now, we consider a continuous change of the Wilson action within one of the intervals in Eq. (4.2). A natural extension of Eq. (3.1) for the interval τ = (n∆τ, (n + 1)∆τ ] would be the discrete transformation from S n to S n+1 , given by 11 This needs a fair amount of explanation, which we give below. First, ∂ a x,µ is a link differential operator defined by (see also Appendix A of Ref. [3]) where T a denotes a (anti-hermitian) generator of the gauge group. The exponentiated link differential operator in Eq.
where ∂ x,µ ≡ T a ∂ a x,µ . The initial value at τ = 0 is given by the "block-spin" link variable (4.1) constructed from the integration variable U ′ defined on Z 4 : It is the value of W τ at τ = ∆τ that appears in the delta function. A possible choice of S w [W ] is the plaquette action, where the sum runs over the plaquettes p belonging to the coarse lattice bZ 4 , and W (p) is the product of the "block-spin" link variables around p. Note that the lattice flow equation (4.6) is written in terms of the scale factor τ rather than the flow time t = b 2n e 2τ − 1. We have used ∂ ∂t = b −2n e −2τ ∂ 2∂τ and absorbed the factor b 2n e 2τ into the right-hand side; this prescription 11 Note that the formula (3.1) can be used to relate the Wilson actions between two non-zero τ s.
is natural because we have rescaled the lattice coordinates by the factor b 2n e 2τ compared with n = 0. Thanks to this prescription, the ERG transformation (4.4) from S n to S n+1 does not depend on n explicitly. We obtain the lattice Wilson  Hence, the partition function is preserved just as in Eq. (3.4).
As for the gauge invariance, we first note that a gauge transformation is given by If ω is infinitesimal, the link differential operator transforms in the adjoint representation, (4.12) where the link differential operator acts on U g on the left-hand side, but it acts on U of U g on the right. This shows that (∂ a x,µ ∂ a x,µ F[U ]) U →U g = ∂ a x,µ ∂ a x,µ F[U g ], and in Eq. (4.4) the gauge transformation on U and the first exponentiated link differential operator commute.
The gauge transformation (4.11) acts on the delta function in Eq. (4.4) as (we set x ′ → x for simplicity) This shows that the gauge transformation (4.11) on U induces an inverse gauge transformation W g −1 ∆τ on W ′ ∆τ defined on the coarse lattice bZ 4 . Now, if W ′ τ is the solution of the lattice flow equation (4.6) with the initial condition U ′ , given by Eq. (4.7), then W ′g −1 τ is the solution with the initial condition U ′g −1 as long as g does not depend on τ ; this follows from the property (4.12). Hence, the gauge transformation g on U induces the inverse gauge transformation g −1 on the initial condition U ′ . To obtain this transformation on bZ 4 , we can introduce the following gauge transformation on Z 4 : otherwise. (4.14) This gauge transformation commutes with the second exponentiated link differential operator in Eq. (4.4) and, as long as S n [U ′ ] is gauge invariant, the resulting Wilson action S n+1 [U ] is also gauge invariant. This completes our argument for the gauge invariance of the lattice ERG transformation. The structure of our Wilson action defined recursively by Eq. (4.4) resembles the "lattice effective action" that has been advocated and studied in Refs. [8,9]. Our definition is different in two crucial aspects, however: Eq. (4.4) has exponentiated link differential operators, and the lattice points are rescaled in each step of the ERG transformation. As we have emphasized in the previous section, these two are essential ingredients for obtaining an ERG differential equation that is non-linear in the Wilson action and entails scale transformation of space.
Finally, let us derive an ERG differential equation in lattice gauge theory that follows from the definition (4.4) of the Wilson action. For this, we define S n+1 (τ )[U ] by We have introduced a diffusion factor τ so that As τ → 0+, S n+1 (τ ) reduces essentially to S n , written for the block-spin link variables U defined by Eq. (4.7): The dependence of S n+1 (τ ) on the diffusion factor τ is given by the differential equation, = exp x,µ,a For the first equality above, we have used the lattice flow equation (4.6) in evaluating , which follows from the definition of the link differential operator (4.5). It is understood that the operator ∂ ′c y,σ acts on W ′ τ . For the second equality, we have rewritten ∂ ′c y,σ as the derivative on U , ∂ ′c y,σ → −∂ c y,σ ; this identity holds because the link differential operator acts on the delta function as d ν)). This link differential operator on U can be put outside to act on the integral over U ′ . Then, we can replace ∂ c y,σ S w [W ′ τ ] by ∂ c y,σ S w [U ] thanks to the delta function. Therefore, from Eq. (4.15), we get an ERG differential equation By integrating this from τ = 0+ to τ = ∆τ , we restore the finite change of the Wilson action in Eq. (4.4). Thus, our ERG transformation in lattice gauge theory consists of the rescaling of lattice points by Eq. (4.17) and the diffusion from τ = 0+ to τ = ∆τ by Eq. (4.19). See Eq. (4.16). As we have shown, this transformation preserves the partition function and manifest gauge invariance of the Wilson action. It is important to note that neither Eq. (4.17) nor Eq. (4.19) depends explicitly on n. This implies a possibility of finding a fixed point solution, S n+1 = S n . The technique in Ref. [2] appears helpful to study such questions.

Conclusion
Imitating the structure of the Wilson action in scalar field theory, expressed by the field diffused by the flow equation, we have constructed a manifestly gauge-invariant Wilson action and its associated ERG differential equation in Yang-Mills theory. The construction, extended to lattice gauge theory, provides a non-perturbative gauge invariant Wilson action of Yang-Mills theory. We have presented only the basic idea and basic relations in this paper; we expect many future applications including analytic or numerical searches for non-trivial RG fixed points in gauge theory. We can also expect extensions in various directions, such as inclusion of matter fields and search for a reparametrization invariant ERG formulation of quantum gravity. It should be also interesting to clarify a possible relation to the other gauge invariant ERG formulations of gauge theory [30][31][32][33][34].
A. Normalization of the gauge field In Sect. 3, we have normalized the gauge field A a µ (x) so that the rescaled field A a µ (x) ≡ λA a µ (x), defined by Eq. (3.10), has the ordinary gauge transformation (3.11). In fact this is not the only choice of normalization. We can change the normalization of A a µ (x) arbitrarily so that the rescaled field is given by Let S z,τ [A] be the Wilson action of this field. We should then obtain This implies [25] e Sz,τ [A] = exp For where ǫ is infinitesimal, we obtain Hence, S z,τ satisfies the same ERG equation (3.25) as S τ except with the addition of on the right-hand side. We can interpret − dz(τ ) dτ as the anomalous dimension of the gauge field.
The marginal operator O 0 (p), Eq. (3.41), that we have found at the end of Sect. 3 is in fact the operator N ; we find We believe that the right choice of the anomalous dimension is necessary to obtain a fixed point of the ERG transformation.