Conserved vector current in QCD-like theories and the gradient flow

We present analytical results for the Euclidean 2-point correlator of the flavor-singlet vector current evolved by the gradient flow at next-to-leading order ($O(g^2)$) in perturbatively massless QCD-like theories. We show that the evolved 2-point correlator requires multiplicative renormalization, in contrast to the nonevolved case, and confirm, in agreement with other results in the literature, that such renormalization ought to be identified with a universal renormalization of the evolved elementary fermion field in all evolved fermion-bilinear currents, whereas the gauge coupling renormalizes as usual. We explicitly derive the asymptotic solution of the Callan-Symanzik equation for the connected 2-point correlators of these evolved currents in the limit of small gradient-flow time $\sqrt{t}$, at fixed separation $|x-y|$. Incidentally, this computation determines the leading coefficient of the operator-product expansion (OPE) in the small $t$ limit for the evolved currents in terms of their local nonevolved counterpart. Our computation also implies that, in the evolved case, conservation of the vector current, hence transversality of the corresponding 2-point correlator, is no longer related to the nonrenormalization, in contrast to the nonevolved case. Indeed, for small flow time the evolved vector current is conserved up to $O(t)$ softly violating effects, despite its $t$-dependent nonvanishing anomalous dimension.


Introduction and main results
The gradient flow for classical Yang-Mills theory was first formulated in [1,2], see also [3] for a nice review of the subject. The gradient flow equation is a specific map of the elementary gauge field A µ (x) to its gradient-flow evolved (smeared) version B µ (t, x), for a given initial condition at t = 0. The parameter √ t ≥ 0, with the dimension of a time, or equivalently a length, can be interpreted as the gradient-flow time, or equivalently the smearing radius.
The last decade has seen a revived interest in the gradient flow, whose properties and uses are being further investigated in the context of quantum Yang-Mills theory coupled to matter fields, more specifically, QCD-like theories formulated in the continuum or on a Euclidean lattice. In [4] the authors provided a lattice formulation of the Yang-Mills gradient flow to study large-N properties of Yang-Mills theory.
Later on, the one-loop renormalization of the gradient-flow evolved Yang-Mills Lagrangian density in the Wilsonian normalization 1 , i.e., g 2 B a µν B a µν with B a µν the canonically normalized evolved gauge-field strength, was derived in [6]. A systematic analysis of the all-order renormalization properties of gradient-flow evolved elementary gauge fields, and implications for the renormalization of pure-glue local composite operators was then provided in [7], see also [8]. The renormalization of gradient-flow evolved elementary fermion fields was first investigated in [9].
Interestingly, gradient-flow type equations -that can be seen as generalized diffusion equations -are employed in many other contexts, ranging from physics to engineering, often to smear microscopic effects in mechanical systems. In quantum field theory, we note that stochastic quantization -see, e.g., [10] for a reviewinvolves analogous techniques, though fundamentally different in purpose.
In fact, the gradient-flow equation for gauge fields used here coincides with the Langevin equation for the stochastic quantization of a Yang-Mills theory with the noise term removed. Yet, the gradient flow in the present paper only acts on the operators in the correlators and it never involves the Lagrangian that occurs in the definition of the vacuum expectation value.
In this work we further explore the properties of gradient-flow evolved composite operators in the fermion sector of QCD-like theories. In particular, we will compute to O(g 2 ) the Euclidean 2-point correlator Π V µν (t, x − y) = J V µ (t, x)J V ν (t, y) of the flavor-singlet vector current J V µ (t, x) =χ(t, x)γ µ χ(t, x) evolved to a gradient-flow time √ t. Our main result is derived in Sec. 5.3: x − y, µ, g(µ)) = Z 2 Jt (g(µ), ε) Π V µν (t, x − y, ε, g) where Π V µν is the bare correlator in dimensional regularization, with g the bare coupling, and Π V µν,0 (t, x − y) is the leading order evolved correlator in Eq. (5.5): Equation (1.1) shows that the evolved 2-point correlator requires a multiplicative renormalization. Hence, the evolved vector current acquires a t-dependent anomalous dimension, in contrast to the nonevolved case, where it has no anomalous dimension. The result in Eq. (1.1) is consistent with results for 1-point correlators of evolved fermion bilinears [9,11,12] 2 . Importantly, it confirms that such renormalization ought to be identified with a universal renormalization of the evolved elementary fermion field [9], whereas the gauge coupling renormalizes as usual.
Our result is thus consistent with studies so far [7,9], which suggest that the only renormalization of evolved fermionic composite operators is the one induced by the renormalization of the evolved elementary fermion fields and the gauge coupling, independently of their tensor structure. Equation (1.1) thus yields the leading O(g 2 ) contribution to the anomalous dimension of all evolved fermion-bilinear currents.
Moreover, we will make contact with the nonevolved case by deriving the leading contribution to the OPE of the evolved fermion-bilinear currents from the corresponding connected 2-point correlators in the limit of small gradient-flow time √ t, at fixed separation |x − y|. The universal UV asymptotics of the leading coefficient c(t) in the OPE [7,9]: of a generic renormalized evolved fermion-bilinear current J R (t, x) -with J(t, x) and J(x) multiplicatively renormalizable -as the renormalization-group invariant coupling g( √ t) → 0 reads: 2 The gradient flow in QCD-like theories We consider perturbatively massless QCD-like theories with gauge group SU (N ) and N f flavors of Dirac fermions in the representation R. We work in Euclidean metric throughout this paper, with Hermitian gamma matrices γ µ , and we employ anti-hermitian generators for The gradient-flow equation for the gauge field reads [6]: in the Wilsonian normalization of the gauge field. The gauge field B µ (t, x) is the solution of Eq.
T a is the bare gauge field and T a , a = 1, . . . N 2 −1 are the generators of SU (N ). The dot in Eq. (2.1) stands for the derivative with respect to t and D µ = ∂ µ +[B µ , · ]. The flow time, √ t, acts as the smearing radius for the gauge field B µ (t, x). The parameter α 0 can be seen as a gauge fixing parameter in Eq. (2.1), and we will be working with α 0 = 1, along with the Feynman gauge chosen in the Lagrangian.
The generalization of Eq. (2.1) to fermion fields can be formulated as in [9], and reads:χ and here it is implicit that the T a in the B µ field are in the representation R of the fermions.

Solutions of the gradient-flow equations
We start by considering the well-known integral form of Eq. (2.1) that is solved iteratively for the gauge field: and: The scalar kernel can be conveniently rewritten in operator notation as: , it is a nonlocal Gaussian regulator of a Dirac delta distribution -often referred to in this context as a smearing -whose Fourier transform (FT) is an entire analytic function of order two. By means of the "exponential-of-Laplacian" notation in Eq. (2.9), Eq. (2.6) reads: Analogously, the integral form of Eq. (2.3) that is solved iteratively for the fermion fields reads: where the quantities ∆ and ← − ∆ contain the evolved gauge field and read: By means of Eq. (2.9), Eq. (2.11) also reads: where the exponential-of-Laplacian acts on the expression inside brackets {. . .}.

Free propagators
We briefly review the expressions for the free propagators, both nonevolved and evolved, in coordinate space together with their limiting behaviors as t → 0 at fixed nonzero separation (x − y), or (x − y) → 0 at fixed positive t.
The nonevolved gauge field propagator in Feynman gauge is: where D(x − y) is the scalar propagator in d Euclidean dimensions: The nonevolved free fermion propagator is: with SU (N ) indices (which we will mostly keep implicit) i, j = 1, ..., d(R), where d(R) is the dimension of the representation R, and: We also recall the relation between the fermion and scalar propagators: The above formulas are readily generalized to the evolved case, where the free propagators do not receive contributions from the second flow-time integral terms in Eqs. (2.6) and (2.11). Hence: and: where we have introduced a convenient notation: the bar over the coordinate in the last line of Eqs. (2.21) and (2.22) represents the exponential-of-Laplacian, and the subscript the associated flow time. In this notation the scalar kernel in Eq. (2.9) The evolved gauge and fermion propagators now satisfy, respectively: where the evolved scalar propagator is: (2.25) and the evolved fermion propagator is: with γ(a, z) the lower incomplete gamma function. The gradient-flow evolution thus amounts to replacing the gamma functions in Eqs. (2.15) and (2.18) with their lower incomplete counterpart, whose integral representation is: Note that the evolved fermion propagator in Eq. (2.30) actually vanishes in this limit due to its Lorentz structure 5 , differently from the scalar propagator in Eq. (2.29). Viceversa, Eqs. (2.25) and (2.26) recover the nonevolved result in the limit t → 0 at fixed nonzero separation, since γ(a, (x − y) 2 /(8t)) → Γ(a).
The result of the combined limits t → 0 and (x − y) → 0 thus depends on the order in which the two limits are taken in the following sense. For the scalar propagator one always produces a singularity, which is in x-space -the one of the nonevolved case -or in t-space, i.e., in the flow coordinate, when taking first t → 0, or (x − y) → 0, respectively. The fermion propagator, instead, vanishes when taking first (x − y) → 0, whereas it recovers the original x-space singularity when taking first t → 0. This makes clear that we need to consider the latter limit, i.e., t → 0 at nonzero separations in order to make contact with the nonevolved correlators of the original quantum field theory.

The vector current evolved by the gradient flow
We introduce the evolved vector current: written in terms of the evolved fermion fields χ andχ.
For the perturbative calculation of the 1-and 2-point correlators in Secs. 3 and 5, respectively, we go from the Wilsonian to the canonical normalization by rescaling the bare gauge field everywhere A µ (x) → gA µ (x), with g the bare gauge coupling. After rescaling, we conveniently rewrite all fields as expansions in powers of g: and for later use we write explicitly the expressions for χ n ,χ n and B µ,n for n ≤ 2: All fields in the flow integral in Eq. (2.33) are functions of (s, x), derivatives are always with respect to x and the exponential-of-Laplacian acts on the expressions inside brackets {. . .}. Analogously, we expand the evolved vector current in Eq. (2.31) in powers of the coupling: where the currents J V µ,n (t, x) for n = 0, 1, 2 read: with χ n andχ n , for n = 0, 1, 2 in Eq. (2.33).
3 The 1-point correlator ∂ µ J V µ (t, x) evolved by the gradient flow and (D µ = ∂ µ + B µ ): we recognize that O + is the divergence of the vector current O + = ∂ µ J V µ , whereas O − is the operator that enters the fermion equation of motion. The 1-point correlator O − (t, x) has been studied to O(g 2 ) in [11], where, differently from its nonevolved counterpart, it was found to be nonzero and to renormalize with a new counterterm induced by the gradient flow.
In this section we consider the 1-point correlator O + (t, x) = ∂ µ J V µ (t, x) . Its nonevolved version vanishes (before and after subtraction of divergences) independently of whether J V µ is conserved or not. In fact, (∂ µ J V µ (x)) R = ∂ µ J V µ R (x) = 0 holds for the simple reason that the momentum at the vertex for J V µ vanishes, but, in the vector case, also because J V µ R (x) itself vanishes due to Lorentz invariance and the fact that J V µ is odd under charge conjugation 6 .
For the evolved correlator, the same chain of identities: holds provided one can write the evolved current in terms of its Fourier transform as J V µ (t, x) = p e ipxJ V µ (t, p). This is true for the bare current, with χ andχ in Eq. (2.11), and it cannot be spoiled by renormalization, analogously to the nonevolved case. Again, the last equality in Eq. (3.4) is also implied by Lorentz invariance and the fact that J V µ (t, x) is odd under charge conjugation. We verify Eq. (3.4) diagrammatically up to O(g 2 ) to illustrate the coordinate space approach in the context of evolved correlators. The diagrams that contribute to ∂ µ J V µ (t, x) to order g 2 are shown in Fig. 1 7 , following the notation explained in App. A.

Leading order, O(g 0 )
The leading order contribution is given by diagram D01 in Fig. 1 and reads: The naming of the diagrams corresponds to the one in [11].
where d(R) is the dimension of the fermion representation R, and . . . 0 stands for the connected contribution to the path-integral average over the Euclidean free-theory measure. In the third line we used Eq. (2.19), which implies that the correlator vanishes at leading order. We work with dimensional regularization in d = 4 − 2ε dimensions.
3.2 Next-to-leading order, O(g 2 ) The next-to-leading order contribution is given by diagrams D02 through D06 in Fig. 1. Diagram D02 contains the insertion of two vertices from the QCD action -QCD vertices in short -, and is the only diagram present at O(g 2 ) in the nonevolved case, where it vanishes. Diagram D03 contains the insertion of one QCD vertex, while D04-D06 do not contain QCD vertices. We show that each one of these five contributions vanishes separately. The contribution from D02 reads: where we employed γ ρ γ α γ ρ = (2 − d)γ α , the relation: and the symmetry relations The contribution from D03 reads: where we employed Eq. (2.19). The final contribution is given by: By Eq. (2.35), one obtains: where in the last line we used where we employed the evolved fields in Eq. (2.33). In the third equality of Eq. (3.11), the first term comes from D06 and it vanishes because S λ (y − x) + S λ (x − y) = 0, the second term is from D04 and vanishes for the same reason, the last term is also from D04 and vanishes since δ ab δ ij T a , T b ij = tr [T a , T a ] = 0. Hence, we have shown diagrammatically how the 1-point correlator of the divergence of the evolved vector current vanishes to O(g 2 ), as anticipated in Eq. (3.4) and in full analogy with the nonevolved case.

The 2-point vector correlator in massless QCD-like theories
We introduce the massless bare nonevolved 2-point vector correlator in Euclidean coordinate space: where . . . defines the path-integral average in the Euclidean theory and J V µ =ψγ µ ψ is the bare flavor singlet vector current. For nonzero separations, contact terms do not occur in Eq. (4.1) and the correlator is multiplicatively renormalizable.

Solution of the Callan-Symanzik equation
For later comparison with the gradient-flow evolved case, we recall the renormalization-group equation and its solution for the connected 2-point correlator of a general gauge invariant and multiplicatively renormalizable current of canonical energy dimension D. For simplicity, we consider the scalar case Π = J(x)J(y) conn with, for example, J =ψψ. For correlators of pure-glue operators and further details of this construction see [14,15].
Multiplicative renormalization in coordinate space at nonzero separation implies: where Π R and Π are the renormalized and bare correlators, respectively, Z J is the multiplicative renormalization factor, Λ is the ultraviolet cutoff in some regularization, µ is the renormalization scale and g is the Yang-Mills running coupling. Based on dimensional considerations, we can rewrite Π R in the massless theory in terms of the dimensionless 2-point correlator Π R as 9 : Then the Callan-Symanzik equation, i.e., the statement that the bare correlator is µ independent: translates into the following equation for Π R : where γ J is the anomalous dimension of J, and β(g) is the QCD beta function: Since Π R depends on (x − y) only through the dimensionless product (|x − y|µ), Eq. (4.5) also reads: The structure of Eq. (4.7) now implies that the dimensionful correlator in Eq. (4.3) factorizes as follows: where F is a dimensionless renormalization-group invariant (RGI) function of the RGI running coupling g(|x − y|) -therefore not determined by Eq. (4.7) -and Z 2 J is determined by integrating Eq. (4.7) between a reference scale µ −1 and |x − y|: where we introduced the perturbative expansions: with β 0 and β 1 the universal, i.e., renormalization-scheme independent one-and twoloop coefficients of the QCD beta function, respectively. Importantly, the second line of Eq. (4.9) determines the universal UV asymptotics of Z 2 J as g(|x − y|) → 0: Furthermore, the third and fourth line of Eq. (4.9) contain the perturbative expansion of Z 2 J by means of the perturbative expansion of g 2 (z) [14]: (4.12) with z = |x − y|, in terms of g 2 (µ) to order g 6 (µ), and valid for scales µ −1 and z close to zero and O(1) logarithms log(zµ). For a proof of Eq. (4.12) see [15].
For massless fermions, the validity of the above analysis extends with appropriate account of the Lorentz structure to the connected 2-point correlators of all flavor singlet and non-singlet fermion bilinear currents -scalar, pseudoscalar, vector, axial and tensor 10 .
In the case of the 2-point vector correlator in Eq. (4.1), the well known all-order result Z J = 1, hence γ J = 0 through Eq. (4.6), is a consequence of the conservation of the flavor singlet vector current. It entails the renormalization-group invariance of the correlator Π V R,µν = Π V µν and that of the vector current J V R,µ = J V µ . An important observation is that, in order to fulfill these properties and at difference with the scalar case, the Lorentz structure of the vector correlator starts to change at O(g 4 (µ)) in perturbation theory. This was explicitly shown in [13]. Therefore, in this case we replace the all-order solution in Eq. (4.8) with: 13) in terms of dimensionless coefficients A n and B n . The leading (n = 0) and next-toleading (n = 1) orders have A n = B n . We now proceed to review the perturbative expression for the correlator Π V R,µν up to next-to-leading order O(g 2 ). The calculation up to O(g 8 ) for N = 3 and fermions in the fundamental representation can be found in [13].
As also for the evolved case in Sec. 5, we present our results in the most general case, i.e., for N colors and N f Dirac flavors in a representation R. With antihermitian SU (N ) generators T a in the representation R one has: with T (R), C 2 (R) and d(R) the index, Casimir and dimension of the representation R, respectively, and d(G) = N 2 − 1 is the dimension of the adjoint representation G of su(N ).
Our results will be expressed in terms of the dimension d(R) and Casimir C 2 (R) of the fermion representation R. We recall that in the fundamental representation, The leading order result, from the left diagram in Fig. 2, is: The next-to-leading order contributions are associated to the last two diagrams in Fig. 2. The UV divergences of both diagrams exactly cancel each other 11 , so that the next-to-leading order result is renormalization-group invariant. Indeed, we have verified that the results reported in [13] and ancillary files 12 for the vector correlator in coordinate space with N = 3 and fermions in the fundamental, up to and including O(g 6 (µ)), can be rewritten in terms of g(|x − y|) in Eq. (4.12) only. For comparison with the evolved case, our generalized result to order g 2 (µ) reads: It manifestly satisfies the transversality condition: to order g 2 (µ). Equation (4.17) is equivalent to the statement that the corresponding current is conserved and not renormalized. At order g 4 (µ) and higher, transversality forces the Lorentz structure of the vector correlator to change. We can understand it as follows. The 2-point correlator of the conserved vector current does not renormalize, thus only the RGI coupling g(|x − y|) induces a coordinate dependence beyond the one of the leading order correlator. This dependence starts at order g 4 (µ), as implied by Eq. (4.12). Hence, 11 One can verify the exact cancellation of the short-distance divergences in coordinate space following a known method nicely explained in [17], chapter 11. 12 http://www-ttp.particle.uni-karlsruhe.de/Progdata/ttp10/ttp10-42/ the Lorentz structure of the vector correlator must change at order g 4 (µ) in order to still guarantee transversality. This applies iteratively to higher orders. For the 2-point correlators of possibly higher spin nonconserved currents, where operator mixing may or may not occur, a change of the Lorentz structure at higher orders is in general allowed. Interestingly, the 2-point correlator of the multiplicatively renormalizable nonconserved tensor currentψσ µν ψ does not change its Lorentz structure to the orders computed in [13], i.e., to O(g 6 ).

The 2-point vector correlator evolved by the gradient flow
We now consider the Euclidean 2-point correlator of flavor singlet vector currents evolved to the same flow time √ t. Analogously to Eq. (2.34), we write the bare correlator as follows: The leading order contribution reads: with J V µ,0 in Eq. (2.35). Π V µν,0 is derived in Sec. 5.1. The next-to-leading order contribution is derived in Sec. 5.2 and can be conveniently divided into four terms: with the currents given in Eq. (2.35). The first two classes of diagrams, I and II, contain respectively the insertion of two QCD vertices and one QCD vertex. Our aim is now to establish the renormalization properties of the evolved Π V µν , and how it differs from the nonevolved case where it does not renormalize. Therefore, when deriving the next-to-leading order contribution in Sec. 5.2, we will concentrate on the divergent parts only. The diagrammatic notation for Secs. 5.1 and 5.2 is explained in App. A. The leading order contribution is given by: 2. This has the following implications.
Thus, the spacetime derivative of the small t expansion in Eq. (5.6) yields: showing that the small t evolution induces an exponentially soft violation of transversality. We recall that in the nonevolved case the transversality condition is respected up to contact terms. In fact, in the evolved case, the spacetime derivative of Eq. (5.5) 13 γ(n + 1, z) = n!(1 − e −z e n (z)), with e n (z) = n k=0 z k /k!, for n = 0, 1, 2, . . ..

reads:
showing that the violation of transversality at a generic fixed t is proportional to the derivative of a smeared Dirac delta distribution, i.e., a contact term smeared by the gradient flow. For each equality in Eq.  .7) and (2.9) for the scalar kernel K 2t . We may also consider the alternative limit of vanishing separation (x − y) → 0 at fixed flow time. In this case, the expansion in Eq. (2.28) inserted in Eq. (5.5) yields for the 2-point correlator: showing that it vanishes at zero spacetime separation, i.e., the short-distance singularity has been removed by the gradient flow.
This exercise makes manifest the noncommutativity of the two limits t → 0 and (x − y) → 0 at the level of 2-point correlators 14 .

Next-to-leading order, O(g 2 )
We treat separately the four contributions in Eq. (5.4).

Type I contribution
The first contribution in Eq. (5.4) is associated to the diagrams in Fig. 4 and it is given by: 14 Of course, the evolved correlator vanishes asymptotically for spacetime separations much larger than the smearing radius √ t, hence, transversality is trivially recovered asymptotically in this limit. and: Type I diagrams are the direct generalization of the two diagrams that contribute at O(g 2 ) to the nonevolved correlator, see Sec. 4. In the latter case, the UV divergence of I.1 cancels the one of I.2 rendering the correlator finite. In particular, the divergence, i.e., the non integrable short-distance singularity of the self-energy contribution I.1 arises at z 1 = z 2 -the coordinates of the internal vertices in Fig. 4 -, while the divergence of I.2 is at z 1 = z 2 = x and z 1 = z 2 = y.
In the evolved case, we note that the propagators that contribute to the divergence of I.1 are not modified by the flow. Hence, I.1 is UV divergent and it generates the same divergence as in the nonevolved case. In I.2, the two fermion propagators that potentially contribute to the divergence are now modified by the flow, thus their short-distance behavior is altered as explained in Sec. 2.2. This is enough to render I.2 finite, in contrast to the nonevolved case. In App. C we rewrite all contributions in terms of the momentum space representations of the (evolved) propagators and (evolved) Dirac delta's. Then, the fact that I.2 is finite is again manifest due to the exponential flow factors e −tk 2 , with t the external flow time and k the internal loop momentum associated to the propagators that generate the UV divergence in the nonevolved case.

Type II contribution
The second contribution in Eq. (5.4) is associated to the diagrams in Fig. 5 and it is given by: and: Diagram II.1 is divergent, where the short-distance singularity arises when the black and white blobs in Fig. 5 coalesce. Diagram II.2 is instead finite, due to the presence of evolved propagators. Again, this is also manifest in the momentum integrals in App. C, for the same reasons as I.2 in type I contribution.

Type III contribution
The third contribution in Eq. (5.4) is associated to the diagrams in Fig. 6, and we employed: when inserting J V µ,2 of Eq. (2.35) in the correlator. Diagrams III.1 and III.2 come from the termχ 2 γ µ χ 0 (χ 0 γ µ χ 2 ) in J V µ,2 . In fact,χ 2 (χ 2 ) in Eq. (2.33) has three terms, yet, the contribution coming from B µ,2 vanishes. Diagram III.3 comes from the term χ 1 γ µ χ 1 in J V µ,2 . It is convenient to rewrite the total contribution from III.1 and III.2 as follows, see App. B for the derivation: where (II.1) is the type II contribution in Eq. (5.12) and: Finally, III.3 is given by: The contribution III.1 is UV divergent and we further derive it in Sec. 5.3. The remaining two contributions III.2 and III.3 are both finite, though this is less straightforward to see in coordinate space expressions due to nested flow integrals and exponential-of-Laplacian actions. The finiteness of III.2 is further established in App. D using integration by parts in the momentum expression of App. C. Finally, the finiteness of III.3 is manifest in its momentum expression in App. C, analogously to I.2 and II.2.

Type IV contribution
The last contribution in Eq. (5.4) is associated to the diagrams in Fig. 7 and it is given by: and: Both IV.1 and IV.2 are finite. This is established by considering the momentum space expressions in App. C, analogously to I.2, II.2 and III.3; the k integrals for both contributions contain the factor e −tk 2 , thereby excluding the possibility of developing a divergence.

Total UV divergence at O(g 2 )
We have established in Sec. 5 We have noticed in Sec. 5.2.1 that the contribution I.1 in Eq. (5.10) is UV divergent due to the one-loop fermion self-energy insertion, which is not modified by the flow. Appendix E shows the final result for I.1 in d dimensions -where the gradient-flow modifications occur outside the self-energy insertion -and its εexpansion. The latter yields: where we employed tr(T a T a ) = −C 2 (R)d(R), Π V µν,0 (t, x − y) is the leading order contribution in Eq. (5.5) and the dots stand for finite contributions that do not enter the renormalization of the correlator.
The last divergent contribution is III.1 in Eq. (5.16). Its calculation is straightfoward upon noticing that the integral in z in Eq. (5.16) yields: 15 All contributions are IR finite. This is important, otherwise spurious UV divergences would be produced in dimensional regularization.
where we employed the propagator in Eq. (2.25) and the expansion of the γ function in Eq. (2.28), see App. E for the ε-expansion. Thus we obtain: is the leading order contribution in Eq. (5.5) and the dots stand for finite contributions that do not enter the renormalization of the correlator.
We conclude that the bare 2-point vector correlator evolved by the gradient flow is no longer UV finite at the next-to-leading order in perturbation theory, i.e., O(g 2 ).
The total UV divergence at O(g 2 ), from the sum of Eqs. (5.21) and (5.23) and after multiplying Π V µν,2 (t, x − y) by the bare coupling g 2 reads: Thus the evolved correlator acquires a renormalization not present in the nonevolved case.

Including the gradient-flow renormalization factor, Z χ
The presence of a UV divergence and thus of a renormalization in the 2-point correlator of the evolved vector current may initially come as a surprise. However, this calculation with the result in Eq. (5.24) provides an explicit verification of the fact that the renormalization factor Z χ introduced in [9] has indeed a universal nature, arising as a new renormalization of the fermion fields χ(t, x) andχ(t, x) evolved by the gradient flow. In fact, by introducing the renormalized evolved fermion fields in the representation R: with renormalization factor [9]: where g(µ) is the renormalized coupling, and employing the renormalization factor Z Jt = Z χ for the evolved current, we obtain the renormalized 2-point vector correlator at O(g 2 (µ)): R,µ = Z Jt J V µ =χ R γ µ χ R and the QCD renormalized coupling g(µ), which is related to the bare coupling g as follows: Since the 1/ε poles of the dimensionally regularized expression in Eq. (5.27) exactly cancel, we conclude that the renormalization of the evolved elementary fermion fields and that of the coupling is the only one required for the evolved 2-point vector correlator at O(g 2 ) in perturbation theory. The arguments presented in [7][8][9] further suggest that this property extends to all orders in perturbation theory. Moreover, the results in [7][8][9] imply that all evolved fermion-bilinear currents acquire the same renormalization factor Z Jt , and thus the same anomalous dimension. The latter can be obtained from its definition in Eq. (4.6), with Z J replaced by Z Jt . In M S-like schemes it reads: 16 We refer to a Lorentz structure of the type δ µν /2 − a(x − y) µ (x − y) ν /(x − y) 2 with a = 1.

OPE of the evolved currents from their 2-point correlators
In full analogy with the analysis in Sec. 4 for the connected 2-point correlator of a generic nonevolved gauge-invariant and multiplicatively renormalizable current, the Callan-Symanzik equation for the connected 2-point correlator of an evolved gaugeinvariant current of canonical dimension D, and that renormalizes as J R (t, x) = Z Jt J(t, x) reads: x − y, µ, g(µ)) = 0 (5.31) with γ Jt in Eq. (5.30). Equation (5.31) is implied by the renormalization-group invariance (µ independence) of the evolved bare correlator Π, related to Π R as: analogously to the nonevolved case in Eq. (4.2), in a regularization with UV cutoff Λ. However, differently from the nonevolved case, the Callan-Symanzik Eq. (5.31) poses a two-scale problem, with scales √ t and |x − y|. We can solve the equation in the limit in which one of the two scales dominates in the UV. Specifically, we are interested in the limit t → 0 at fixed |x − y|, hence √ t |x − y|, to establish an asymptotic relation between the evolved and nonevolved correlators.
The validity of the OPE for composite operators evolved by the gradient flow in the small t limit has been verified in [7,9]. Thus, for a generic multiplicatively renormalizable evolved fermion-bilinear current in the small t limit we write: where . . . is its vacuum expectation value, J R (x) is the renormalized nonevolved local current and c(t) is the leading coefficient of the OPE in the small t limit. We can determine c(t) from the small t expansion of the corresponding evolved 2-point correlator Π R . Such small t expansion reads: where in the right-hand side the dimensionless coefficient C = c 2 is the square of the OPE coefficient c(t) in Eq. (5.33), and Π R is the renormalized nonevolved correlator. The latter is given by Eq. The factorization of the dependence on √ t and |x − y| occurs in the first term of the right-hand side in Eq. (5.34), which is of O(t 0 ) times logarithms of √ tµ 18 . The term ∆Π R refers to contributions of O(t) times logarithms of √ tµ and |x − y|µ. These higher order contributions have in general a nonfactorizable dependence on the two scales √ t and |x − y|. Therefore, for multiplicatively renormalizable J R (t, x) and J R (x) the renormalization-group equation for C is now implied by Eq. (5.32) for the evolved correlator and Eq. (4.2) for the nonevolved one, which yield: Hence, C has an anomalous dimension given by the difference of the anomalous dimensions of the evolved and the nonevolved Π R . The solution can be written as: where F is a dimensionless RGI function of the RGI running coupling g( √ t), with F → 1 as g( √ t) → 0 implied by perturbation theory, and: Equation (5.37) coincides with Eq. (4.9) for the nonevolved case once γ J has been replaced with γ Jt − γ J and g(|x − y|) with g( √ t). Then, by means of the latter substitutions, the UV asymptotic expression for Z 2 in Eq. (5.37) is given by the second line of Eq. (4.9).
With C given by Eq. (5.36), the small t expansion of the evolved correlator thus reads: showing that the evolved correlator has the same Lorentz structure of the nonevolved one at leading order in the small t expansion, i.e., up to O(t) contributions. The leading universal UV asymptotics of Eq. (5.38) as g( √ t) → 0 then reads: and the leading universal UV asymptotics of the first OPE coefficient: for J R (t, x) in Eq. (5.33) thus follows: In the case of the evolved vector current one has γ (0) We have also noticed that in the nonevolved vector case, Eq. (4.8) is replaced with Eq. (4.13), which takes into account a change of the Lorentz structure at higher orders in perturbation theory.
Accordingly, by employing Eq. (4.13) the small t expansion for the evolved flavor singlet vector correlator has the explicit form: with A n = B n for n = 0, 1, and we employed the coefficient C from Eqs. (5.36) and (5.37) with γ J = 0. Equation (5.42) shows explicitly that the leading contribution to the small t expansion of the evolved vector correlator in Eq. (5.42) inherits its Lorentz structure from the nonevolved correlator order by order in g 2 (|x − y|). Finally, we note that the leading n = 0 term in Eq. (5.42) reproduces the explicit O(g 2 (µ)) result in Eq. (5.27) by means of the perturbative expansion: Jt log (tµ 2 ) + . . . All the other finite contributions manifestly vanish in the same limit and only contribute at higher orders to the OPE.
In the opposite limit of vanishing separation at fixed t > 0, all of the above finite contributions are expected to vanish as a consequence of the regulating effect of the gradient-flow smearing.
Finally, we mention that "evolved contact terms" resulting from the smearing of nonevolved contact terms, may be expected to contribute to evolved 2-point correlators. In the vector case, dimensional analysis tells us that nonevolved contact terms are of the type δ µν ∆δ (4) (x − y) and ∂ µ ∂ ν δ (4) (x − y). Hence, they may lead to evolved contact terms of the type δ µν ∆K t (x − y) and ∂ µ ∂ ν K t (x − y), respectively. These terms no longer vanish at nonzero separation at fixed t > 0, and they can contribute starting at O(t) to the OPE in Eq. (5.42).
6 Current conservation and renormalization 6.1 Nonevolved case: conservation implies nonrenormalization We briefly review a simple argument for how conservation of a nonevolved gaugeinvariant local current implies its nonrenormalization. The conservation of the local and gauge-invariant vector current J V µ (x) is the consequence of an exact nonanomalous global U (1) symmetry and it is encoded in the corresponding Ward identity once the theory is quantized.
The conservation of J V µ (x) then implies that the associated gauge-invariant and dimensionless charge Q is also conserved. It follows that Q cannot depend on any unphysical scale. Hence, it cannot acquire an anomalous dimension and the nonrenormalization of the vector current, J V µ (x) = J V R,µ (x), thus follows. The nonrenormalization and conservation of J V µ (x) in turn imply the transversality of the corresponding 2-point correlator.

Evolved case: conservation does not imply nonrenormalization
The situation is fundamentally different for the evolved vector current J V µ (t, x). We have seen that the latter acquires an anomalous dimension γ Jt , which enters the leading term of the small t expansion of the evolved 2-point vector correlator as shown in Eq. (5.42). Yet, the same leading term fulfils transversality and current conservation, despite the presence of an anomalous dimension. Indeed, specifically: x − y, µ, g(µ)) = = C( √ tµ, g(µ)) ∂ x µ Π V R,µν (x − y, µ, g(µ)) + ∂ x µ ∆Π R,µν (t, x − y, µ, g(µ)) = ∂ x µ ∆Π R,µν (t, x − y, µ, g(µ)) (6.1) and the first term in the right-hand side of the first equality has an anomalous dimension, but vanishes because the nonevolved correlator is transversal, i.e., ∂ x µ Π V R,µν (x − y, µ, g(µ)) = 0. It is then clear that a nonzero anomalous dimension is allowed because the evolved current depends on the additional (unphysical) gradient-flow scale √ t, which parametrizes its nonlocality.
On the other hand, the violation of transversality and the nonconservation of the evolved vector current do occur through the second term ∂ x µ ∆Π R in the righthand side of Eq. (6.1). Hence, they are a soft-breaking effect of O(t) induced by the smearing action of the gradient flow that vanishes as t → 0. This agrees with the explicit results at O(g 0 ) in Eqs. (5.7) and (5.8) and the classical leading-order consideration in App. F.
As a side note, one could also relate the nonconservation of the evolved vector current to the lack of a corresponding exact symmetry in a d + 1-dimensional theory that includes the flow direction, along the lines of [7,9,18].

Conclusions
We have studied the renormalization properties of the gradient-flow evolved flavor singlet 2-point vector correlator in perturbatively massless QCD-like theories, and showed that, in contrast to the nonevolved case, the correlator is renormalized and the evolved vector current acquires an anomalous dimension: Our result confirms that this anomalous dimension is induced by the renormalization of the evolved elementary fermion field first derived in [9], and thus applies to all evolved fermion-bilinear currents. Our result is also in agreement with results in the literature for 1-point correlators of evolved fermion bilinears [9,11,12]. The Callan-Symanzik equation for the connected 2-point correlators of generic multiplicatively renormalizable evolved fermion-bilinear currents now involves two scales, i.e., the flow time √ t and the separation |x − y|. We made connection with the nonevolved case by deriving the solution of the Callan-Symanzik equation in the limit of small gradient-flow time √ t, at fixed separation |x − y|. Incidentally, the leading order contribution to this expansion also determines the leading OPE coefficient for the corresponding evolved current in the small t limit.
We have also discussed how, interestingly, conservation of the evolved vector current and transversality of the corresponding 2-point correlator no longer imply nonrenormalization, at difference with the nonevolved case. In particular, the leading contribution to the small t expansion of the evolved 2-point vector correlatorwhich is O(t 0 ) times any power of logarithms -fulfils transversality and the current is conserved despite the presence of the anomalous dimension γ Jt . This is due to the presence of the additional gradient-flow scale √ t, so that renormalization logarithms at O(t 0 ) only depend on the product √ tµ. Violation of transversality and nonconservation do occur at O(t) in the OPE as a soft breaking effect induced by the nonlocality of the evolved current.
Note added: We are aware 19 that another group has been independently computing the gradient-flow evolved 2-point vector correlator at next-to-leading order with the same result.

A Diagrammatic notation
In this appendix we introduce the diagrammatic notation employed in Fig. 1 and Figs. 3 -7 for the gradient-flow evolved 1-point and 2-point correlators. The adopted notation follows [11].

Propagators
The Feynman rule in Euclidean coordinate space for the gradient-flow evolved fermion propagator reads: where S(x t −ȳ s ) is given in Eq. (2.26), and we use the same line as for the nonevolved fermion propagator -the latter is obtained for t = s = 0. Analogously, for the evolved gluon propagator one has: with D(x t −ȳ s ) in Eq. (2.25).

Flow-time integrals
The flow-time integrals and associated kernels that are present in the second term (interaction part) of the solution to the flow equation in Eq. (2.6) for the gluon and Eq. (2.11) for the fermion are represented by a double line that always ends in a white blob representing a gradient-flow interaction vertex, i.e., a vertex induced by the flow. Explicitly, for the fermionic case in Eq. (2.11): x,t,i y,s,j where we only highlighted the flow-integral in the second term of Eq. (2.11) associated with the double line, and the dashes stand for a combination of lines emanating from the vertex. The structure of gradient-flow vertices is further explained around Eq. (A.5). The second term in the gauge field solution in Eq. (2.6), analogously represented by double gluon lines, does not occur in this work.

Vertices
The QCD vertex that enters our calculations is represented by a filled blob. The Feynman rule in Euclidean coordinate space and in the case of gradient-flow evolved fields at the vertex reads: with obvious replacements in the case of nonevolved fields. We now discuss the structure of gradient-flow vertices that appear in this work, i.e., of the type in Eq. (A.3). Specifically, for the fermion field χ(t, x) they represent the second term in Eq. (2.11), which can be written as an expansion in powers of g starting at O(g). Thus at O(g n ) this vertex corresponds to χ n (t, x) defined in Eq. (2.32), with n ≥ 1. The explicit expressions for χ 1,2 are in Eq. (2.33), and we reproduce them here together with the corresponding gradient-flow vertices.
The lowest order gradient-flow vertex in this work is given by: and at the next-to-leading order one has: In this appendix we derive Eq. (5.15), which reads: We start with: associated to the diagrams (III.1) and (III.2) in Fig. 6. After performing the Wick contractions, we obtain: We then employ the relation: to rewrite the second term in Eq. (B.3) as: The last term in Eq. (B.5) exactly cancels (II.1) in Eq. (5.12), after noting that:
(E. 16) where here the finite terms include the finite parts from the ε-expansions in Eqs. (E. 12) and (E.13), while the dots stand for O(g 4 ) contributions and additional O(g 2 ) finite terms.
We add that at t > 0 the short-distance limit of the O(g 2 ) contributions computed here is nonsingular and vanishes. This is shown using: which then combines with the vanishing limit of Π V µν,0 (t, x).