Products of Current Operators in the Exact Renormalization Group Formalism

Given a Wilson action invariant under global chiral transformations, we can construct current composite operators in terms of the Wilson action. The short distance singularities in the multiple products of the current operators are taken care of by the exact renormalization group. The Ward-Takahashi identity is compatible with the finite momentum cutoff of the Wilson action. The exact renormalization group and the Ward-Takahashi identity together determine the products. As a concrete example, we study the Gaussian fixed-point Wilson action of the chiral fermions to construct the products of current operators.


I. INTRODUCTION
It is a principle of quantum field theory that the invariance of a theory under a continuous transformation implies the conservation of a current. When a theory is expressed by a Wilson action with a finite momentum cutoff, the principle holds for the Wilson action. In [1] an energy-momentum tensor was constructed from the invariance of the Wilson action under translations and rotations. In this paper we would like to consider the Wilson action of chiral fermions with global flavor symmetry to construct multiple products of the conserved current operator.
To build the Wilson action, we use the exact renormalization group (ERG) formalism.
(See for example [2][3][4][5] and references therein.) The Wilson action satisfies a well defined differential equation under the continuous change of scale. We adopt a convention that each time we integrate more of the high-momentum fluctuations, we introduce a change of scale to restore the same cutoff function. The continuum limit corresponds to a trajectory parametrized by a logarithmic scale parameter t so that a fixed point is reached in the limit t → −∞.
The Wilson action of a theory in the continuum limit has all the short distance physics incorporated into the vertices of the action. The full theory is obtained by further integration of the fields with momenta below the cutoff. The Wilson action is determined by the ERG differential equation whose solution is parametrized by the relevant variables of the theory.
Composite operators can be considered as infinitesimal changes of the Wilson action, and they also obey well defined differential equations under the change of logarithmic scale. The general properties of products of composite operators have been discussed in [6]. We follow and extend the discussions there by considering the multiple products of current operators.
ERG is good at handling the short distance singularities via ERG differential equations. Well defined ERG differential equations admit only the solutions consistent with locality, i.e., the vertices of the action and composite operators must be analytic at zero momenta. This is the guiding principle we follow throughout the paper.
Though we consider only chiral fermion fields as dynamical fields, our discussion of current operators is easy to modify in the presence of other dynamical fields, for example in the case of QCD with massless quarks.
Our subject obviously overlaps with the construction of chiral gauge theories using the ERG formalism. (See, for example, [7] and references therein.) For example, the derivation of chiral anomaly using the ERG formalism was done in the context of gauge theory. (See, for example, Sec. 9 of [4] and [8].) The multiple products of current operators require a much lighter formalism.
The paper is organized as follows. In Sec. II, we introduce a current operator for a generic Wilson action of chiral fermions under the assumption of global continuous symmetry. In Sec. III, we introduce multiple products of current operators and derive the ERG equations satisfied by them. By coupling the current with an external gauge field, we construct a composite operator in terms of which we can consider all the products of current operators at once. In Sec. IV, we consider the Ward-Takahashi (WT) identity satisfied by the products of currents. Our discussion is not completely closed, and we need to introduce the identity as a working hypothesis. The ERG differential equation and the WT identity are mutually consistent, and they together characterize the products of current operators. In Sec. V we discuss the changes to the ERG equation and the WT identity caused by the shortdistance singularities of the operator products. In Sec. VI we consider the products of current operators for the free theory. Though this section is all about 1-loop diagrams, the example elucidates the general formalism given in the preceding sections.
Please note that we use the following condensed notation for momentum integrals:

II. CURRENT COMPOSITE OPERATORS
We consider a theory of chiral fermion fields ψ,ψ satisfying where The theory is determined by its Wilson action with a fixed UV cutoff. The cutoff is given in terms of a smooth momentum cutoff function K(p), such as e −p 2 , that is 1 at p = 0 and vanishes as p → ∞. We parametrize the Wilson action by a logarithmic scale parameter t and demand that it obey the ERG differential equation where γ t is an anomalous dimension of the chiral fermion field, the trace is for both spinor and flavor indices, and the minus in front of the trace is due to the Fermi statistics.
We assume that the Wilson action S t describes a continuum limit; as we take t → −∞, we obtain a UV fixed-point: All the physics beyond the fixed cutoff scale of 1 has been incorporated into the action. By integrating the fluctuations of momenta less than 1, we get full correlation functions of the fields.
We define the correlation functions by where is the high-momentum propagator. The correction involving the cutoff function is a technicality typical in the ERG formalism. Thanks to the correction, though, the correlation functions satisfy the simple scaling relation operator O t (p) is a functional whose correlation functions are defined by where the exponentiated differential operator does not act on O t (p). We define O t (p) so that its correlation functions satisfy the scaling relation For simplicity we have taken −y, the scale dimension of O t (p), independent of t. For (11) to be valid, O t (p) must satisfy the ERG differential equation where D t , acting on functionals, is defined by The simplest example of a composite operator is Though they are composite operators, they have the same correlation functions as the elementary fields ψ(p),ψ(−p): We are now ready to discuss symmetry. We assume that the correlation functions have global symmetry: where U is an arbitrary unitary matrix that acts on the flavor indices of ψ andψ. (U may be a U(N) matrix if we have N flavors.) For infinitesimal transformations we obtain where T a are hermitian matrices normalized by and satisfying the commutation relation (We will omit the summation symbol for the repeated indices c from now on.) To express (17) as an operator equation, we introduce an equation-of-motion composite operator by where Ψ,Ψ are defined by (14). E a is a total derivative of the exponentiated Wilson action, and it has correlation functions − ψ(p 1 ) · · · T a ψ(p + p i ) · · · ψ(p n )ψ(−q 1 ) · · ·ψ(−q n ) t + ψ(p 1 ) · · · ψ(p n )ψ(−q 1 ) · · ·ψ(p − q i )T a · · ·ψ(−q n ) t .
The symmetry (17) is equivalent to In fact this is equivalent to what we usually consider as the invariance of the action In Appendix A we show that this is equivalent to (23) .
Since E a (p) is a local operator, it must be proportional to the momentum: where the current J a µ (p) must be a local composite operator. Unless there is a local operator j a µ (p) orthogonal to p µ p µ j a µ (p) = 0 , (25) defines the current J a µ (p) unambiguously. Since E a (p) has scale dimension 0, J a µ (p) must have scale dimension −1. In coordinate space J a µ (x) = p e ipx J a µ (p) has scale dimension D − 1.
As an example, let us consider the Gaussian fixed-point theory for which We find This implies

III. PRODUCTS OF CURRENT OPERATORS
We wish to define multiple products of currents. The product of two currents is defined where P is a local counterterm necessary to make the product a composite operator; the bare product J a µ (p)J b ν (q) is not a composite operator in the sense introduced in the previous section. P also takes care of the short-distance singularity occurring when the two currents come close together. For the product to be a composite operator of scale dimension −2, it must satisfy where D t is given by (13). This implies Similarly, we define the product of three currents as and so on for the higher order products. We note that P a 1 ···an µ 1 ···µn (p 1 , · · · , p n ) gives the shortdistance singularity due to all the n currents coming together simultaneously, and it is proportional to the delta function in momentum space unless there is a composite operator of scale dimension −n or less available. (That means is given by The ERG equations for the higher order counterterms are given similarly.
To consider all the local products of current operators simultaneously, we introduce a classical gauge field coupled to the current: so that its exponential is a composite operator. We assign the scale dimension −D + 1 to the source field A a µ so that e Wt[A] becomes a composite operator of scale dimension 0, satisfying the ERG equation where D t is defined by (13).

IV. COMMUTATION RELATION -WARD-TAKAHASHI IDENTITY
We now wish to consider the "commutation relation" of two currents. The quotation mark is put because it needs to be explained. Our commutation relation is an operator which amounts to the Ward-Takahashi (WT) identity We wish to explain the above and its generalization to higher order products in this section.
We define an equation-of-motion composite operator by where are the composite operators satisfying Hence, we obtain This gives the second term of the right-hand side of (41). Let (41) then amounts to This equality is plausible but not obvious, and it needs an explanation. We will check this later explicitly for the Gaussian theory. Here we satisfy ourselves by checking the consistency of (49) with Bose symmetry of the current operator, which requires the product be symmetric under the interchange. The product may depend on which divergence we calculate first. Calculating p µ J a µ (p) first, (48) gives Hence, for consistency, we must find To compute the right-hand side, we consider correlation functions: Hence, we obtain Then, the consistency condition (52) gives which is indeed satisfied by (49).
We have thus checked at least that (40) is consistent with the Bose symmetry of the current. We adopt (40) and its generalization to higher orders as our working hypothesis: For the correlation functions, this gives The WT identity (56) we just introduced is compactly expressed in terms of the composite Expanding this in powers of the external source A, we can easily check the equivalence to (56). Multiplying an infinitesimal ǫ a (−p) and integrating over p, we can rewrite this as where is an infinitesimal gauge transformation.

V. CORRECTIONS TO THE ERG EQUATION AND THE WT IDENTITY
We have identified two important properties of e Wt [A] . One is the ERG differential equation (39), and the other is the gauge invariance (59). Both may receive corrections due to short distance singularities. Since the nature of singularities depends on the space dimension D, we specify D = 4 in the following discussion.
We first consider possible corrections to the ERG equation. The product of n current operators has scale dimension −n, and it can mix with operators of the same scale dimension.
As for the mixing with the delta function δ( i p i ), we only need to consider which mix with the delta function δ( i p i ) with appropriate powers (quadratic, linear, none) of momenta. This gives a new ERG differential equation: where f is a linear combination of the products of two A's with two derivatives, three A's with one derivative, and four A's with no derivative. Consistency with (59) gives the gauge invariance of f . Hence, we obtain where In fact the gauge invariance (59) itself may also get corrected as where F a (p; A) is a polynomial of A with scale dimension −4. This is the familiar chiral anomaly. [9,10] For (64) to be consistent with (61), F a (p; A) must be independent of t, i.e., the anomaly must be scale independent.
The algebraic structure of the anomaly is well known. [11] For completeness, let us derive it using the ERG formalism. By definition of δ ǫ , we must obtain where Using (64) twice, we obtain where we have used (54). Hence, (65) gives the desired algebraic constraint A well-known nontrivial solution to this is given by [12] p The construction of e W [A] is guided by two equations. One is the ERG differential equation where b is a constant, and D is defined by .
The other is the WT identity with anomaly where A is a constant. Both b and A are determined as we construct P a 1 ···an µ 1 ···µn (p 1 , · · · , p n ) from n = 2 to higher n. b is determined by locality. Locality implies the analyticity of P's at zero momenta. We must choose b appropriately to guarantee that (71) admits a solution satisfying locality. Similarly, the coefficient A of the chiral anomaly is determined by locality. The solution to (71) admits a couple of free parameters consistent with locality.
We tune them to satisfy (73) as much as possible. What is left is the anomaly.
In the following we sketch the calculation of d α 1 ···αn for n = 2, 3, 4. The case n = 2 is sufficient to determine the coefficient b, but we need the case n = 3 to determine A. We A. Product of Two n = 2 d ab αβ (p 1 , p 2 ) = δ ab d αβ (p, −p) satisfies the ERG equation and the Ward identity The analyticity of d αβ (p, −p) at p = 0 demands that the rhs (82) be free of quadratic terms in p. (If there were, we would obtain a nonlocal p 2 ln p dependence.) To expand the integral on the rhs of (82) in powers of p, we use where ǫ 1234 = 1, and where h ′ (q) ≡ d dq 2 h(q), etc. We obtain where The integrand is a total derivative, and the value of the integral is independent of the choice of the cutoff function K(p).
the general solution of (82) is given by where A, B are free parameters. The subtractions make the integrand of order e 2t as t → −∞, and the integral is convergent.
We can fix A using the WT identity (83). First note To determine A we compute the rhs of (83): Consistency with (90) demands and The first equation must hold since the WT identity is an operator equation consistent with ERG. We verify it explicitly in SubSec. 5 of Appendix B. There, in SubSec. 4, we also B is left arbitrary.
Let us stop here to examine the asymptotic behavior of d αβ (p, −p) for large p. In principle we could obtain the asymptotic behavior using the solution (89). Instead, it is easier to go back to (82) and (83), which give Hence, we obtain the asymptotic behavior determined by the constant b. Using this we can construct the continuum limit as [13] This satisfies Since D αβ depends on the constant B, we can rewrite this as D αβ (p, −p) is also transverse: The two-point function of the current is now obtained as which is transverse, and satisfies the scaling relation B. Product of Three n = 3 d αβγ (p 1 , p 2 , p 3 ) satisfies the ERG equation where b is given by (88), and the WT identity where A is to be determined.
Analyticity of d αβγ at p i = 0 requires the absence of terms linear in p i from the rhs of (104). (p i would imply nonlocal p i ln p j .) Let us check it. Expanding the integral in momenta, we obtain the linear terms as The integrand is a total derivative, and we obtain The general solution is given by where c αβγδ are arbitrary constants, not determined by (104). We note that the integrand behaves as e t as t → −∞, and the integral is convergent. The particular form of the linear terms is required by the cyclic symmetry: The most general form of c αβγδ is given by where s, t, u are constants. 1 We now wish to show that we can choose s, t, u, and A so that (105) is valid. Since (105) is consistent with (104), we only need to check the terms quadratic in momenta. Using we obtain where A is given by (94).
We next consider the small momentum behavior of the integral on the rhs of (105): where the first integral, whose integrand is a total derivative, can be calculated as (See SubSec. 2 of Appendix B.) Hence, we obtain the rhs of (105) as We next compute the small momentum behavior of the lhs of (105). From we obtain Matching this with (115), we obtain which determine the low momentum behavior We also obtain the coefficient of the anomaly as 2 Let us stop here to examine the asymptotic behavior of d αβγ (p 1 , p 2 , p 3 ) for large momenta.
Instead of taking the asymptotic limit of (108), we go back to (104) and (105). For large momenta, (104) gives and (105) gives (121) gives the dominant asymptotic behavior which is proportional to the coefficient b. Hence, we can construct the continuum limit as This satisfies the scaling relation and the WT identity: The continuum limit of the connected three-point function defined by satisfies the scaling relation and the WT identity C. Product of Four n = 4 d αβγδ (p 1 , p 2 , p 3 , p 4 ) must satisfy the ERG equation where b is given by (88), and the WT identity where A is given by (120).
We would like to check two things. As for (130), we would like to check the vanishing of the rhs at zero momenta. (A constant would imply nonlocal ln p.) As for (131), we would like to check its validity at the first order in momenta.
The rhs of (130) gives (rhs) The integrand is a total derivative, and we obtain (See SubSec. 1 of Appendix B.) Hence, the rhs vanishes at zero momenta as desired.
We now wish to check (131) to first order in momenta. (130) determines only the momentum dependence of d αβγδ , but its value at p i = 0 is left undetermined. The most general form, consistent with cyclic symmetry, is where s 4 , t 4 are constants so that To compare this with the rhs, we first compute where we have used (119), and 1 2 (s − t) is given by (118b). We next compute Hence, the rhs of (131) is The last term vanishes because We have thus checked the validity of (131).
Finally we examine the asymptotic behavior of d αβγδ (p 1 , p 2 , p 3 , p 4 ) for large momenta.
(130) and (131) give The first equation gives the asymptotic behavior Hence, a continuum limit is obtained as ≡ lim t→+∞ d αβγδ (p 1 e t , p 2 e t , p 3 e t , p 4 e t ) − bt (δ αβ δ γδ + δ βγ δ αδ − 2δ αγ δ βδ ) , which satisfies the scaling relation and the WT identity Hence, the connected four-point function defined by satisfies the scaling relation and the WT identity

D. Recapitulation
Let us recapitulate the results of this section by writing down equations for e W [A] , a composite operator of scale dimension 0. The ERG differential equation is given by The WT identity is given by W [A] is determined uniquely by the above two equations up to a constant multiple of the gauge invariant If we define we can rewrite the ERG equation as is the 1-loop beta function.

VII. CONCLUSIONS
In this paper we have discussed the multiple products of current operators using the exact renormalization group (ERG) formalism. The multiple products are characterized by two mutually consistent equations: one is the ERG differential equation and the other is the Ward-Takahashi (WT) identity. We have argued that these two equations suffer changes due to the short-distance singularities of the products, and the revised equations are given by (61) for ERG and (64) for the WT identity. In Sec. VI we have calculated the multiple products explicitly by solving these equations for the Gaussian fixed-point. The guiding principle in these calculations is the locality of the operators. Since the momenta below the cutoff have not been integrated, the coefficient functions for the products of the current are analytic at zero momenta.
There are some future directions we can consider. We may consider a theory such as QCD with fields other than the chiral fermions. Or we may consider a more nontrivial fixed-point Wilson action. We also think it interesting to study the multiple products of other composite operators such as the energy-momentum tensor.