Light-cone distribution amplitudes of vector meson in large momentum effective theory

We investigate the leading twist light-cone distribution amplitudes (LCDAs) of vector meson in the framework of large momentum effective theory. We derive the matching equation for the LCDAs and quasi distribution amplitudes. The matching coefficients are determined to one loop accuracy, both in the ultraviolet cut-off and dimensional regularization schemes. This calculation provides the possibility of studying the full $x$ behavior of LCDAs and extracting LCDAs of vector mesons from lattice simulations.

of evaluating PDFs and LCDAs from the first principle of QCD [24,[36][37][38][39]. Thus LaMET provides one more approach of accessing LCDAs of vector mesons by lattice simulation. Before the lattice evaluation is performed, it is necessary to determine the matching coefficient between the LCDAs and quasi-DAs, in QCD perturbation theory.
The present paper is devoted to the perturbative matching between the quasi and light-cone distribution amplitudes of vector mesons in LaMET. We will study the twist-2 LCDAs of vector meson and the corresponding quasi-DAs. The main aim of this work is to derive the matching equation for quasi and light cone distribution amplitudes. To do this, we will calculate the one loop corrections to both the quasi and light-cone distribution amplitudes, then work out the matching coefficients to one loop accuracy. This work will provide the possibility of extracting LCDAs of vector meson from future lattice simulations.
The rest of this paper is organized as follows. In Sec. II, we present the definitions of twist-2 LCDAs for the transversely and longitudinally polarized states, and their corresponding quasi-DAs. In Sec. III, we calculate the oneloop corrections to the LCDAs and quasi-DAs, in the UV cut-off scheme. In Sec. IV, the LaMET matching equation will be derived. We summarize in Sec. V. The results under dimensional regularization and matching coefficients with a finite UV cut-off Λ will be arranged in the Appendices.

II. DEFINITIONS OF LIGHT-CONE AND QUASI DISTRIBUTION AMPLITUDES
Before introducing the quasi-DAs, we first revisit the LCDAs. We adopt the light-cone coordinate system to discuss the LCDAs. In light-cone coordinate system, any four-vector a can be expressed as a µ = (a + , a − , a ⊥ ) = ((a 0 + a 3 )/ √ 2, (a 0 − a 3 )/ √ 2, a 1 , a 2 ). The two unit light-cone vectors are denoted as n µ = (0, 1, 0 ⊥ ) and l µ = (1, 0, 0 ⊥ ). The inner product of four vector a and b then reads In QCD, the LCDAs are defined by the matrix elements of non-local gauge invariant quark bilinear operators, in which the two fermion fields are separated in the n direction. At the leading twist, there are two LCDAs φ ⊥ V and φ V corresponding to the transversely (denoted by "⊥") and longitudinally (denoted by " ") polarized states of the vector meson. We first introduce the non-local operators in coordinate space where Γ = γ + γ α ⊥ for transversely polarized vector meson, and Γ = γ + for longitudinally polarized vector meson. W (ξ − , 0) is the Wilson line with the end points (0, ξ − , 0 ⊥ ) and (0, 0, 0 ⊥ ). In LCDAs the Wilson line is light-like where P denotes that the exponential is path ordered. We also need the Fourier transformation of these operators, which are denoted by x ≡ k + /P + is the longitudinal momentum fraction with k + be the momentum of quark. Then, the LCDAs of the transversely and longitudinally polarized vector meson are defined by the matrix elements of O Γ V (x), in which O Γ V (x) is sandwiched between the meson and vacuum states where f ⊥ V and f V are the decay constants of the vector meson V , P and ǫ * are the momentum and polarization vector of meson V , respectively. The decay constants are defined by the local operators then, the LCDAs can be expressed as the ratio of the non-local and local matrix elements In LaMET, one can define the quasi-DAs similarly. It is more convenient to discuss quasi observables in the original Cartesian coordinate system. The unit vector of the z direction is denoted by n µ z = (0, 0, 0, 1). The inner product of n z and an arbitrary vector a gives n z · a = a z = −a z . To define the quasi-DAs, we introduce two non-local bilinear operators, in which the fermion fields are separated on the z direction and their Fourier transformation where Γ = γ z γ α ⊥ and γ z for the transverse and longitudinal components, respectively. Here the Wilson line W is along the z direction, with zn µ z = (0, 0, 0, z) and the origin of coordinates (0, 0, 0, 0) as its end points. Then, the quasi-DAs of the transverse and longitudinal components of a vector meson are defined by the matrix elements of the operators as We note that although the light-cone and quasi operators are different, the decay constants in Eqs. (4) and (9) should be the same. The reason is that either the quasi or light-cone operator is the µ = + or µ = z component of the operatorψγ µ γ α ⊥ ψ orψγ µ ψ. The Lorentz index is only carried by the polarization vector ǫ * , while the decay constants are scalar quantities, therefore they are independent of the Lorentz indices. The decay constants are related to the matrix elements of local operators by The quasi-DA can be expressed as the ratio of the non-local and the local matrix elements as One can immediately find that the LCDAs and quasi-DAs are normalized to 1, i.e., from the definitions.

III. ONE LOOP RESULTS
To examine the factorization and determine the matching coefficients at one loop level, we first replace the meson state V, P, ǫ * | with its lowest Fock state Q(x 0 P )Q((1−x 0 )P )|. P is the total momentum of the quark and anti-quark, x 0 P and (1 − x 0 )P are the momenta of the Q andQ, respectively, with 0 < x 0 < 1. Then the matrix elements with the Fock state as their final state can be calculated in perturbation theory. Direct calculation at tree level leads to We will perform our calculation under Feynman gauge. The Feynman diagrams at one loop level are presented by Fig. 1. The distribution amplitudes of the Fock state is calculable in perturbation theory, thus can be expanded in series of α s . Up to one loop level, we have On the other hand, the matrix element of O Γ V (x), up to one loop level, can be expressed by Here (1) denotes the one loop correction to the matrix element in which the self energy of quark has been excluded. The contributions from quark's self energy are involved in δZ is the one loop correction of quark's self energy. Meanwhile, the local matrix element is also corrected at one loop where δZ is the one loop vertex correction of the local operator. Since O Γ V and O Γ V are the µ = + and µ = z components of operatorψγ µ γ α ⊥ ψ orψγ µ ψ, respectively, δZ should be same for light-cone and quasi local operators. From Eqs. (5) and (10), one can get that From Eqs. (15) and (16), we immediately have By comparing Eq. (18) and Eq. (14), one can identify that From Eq. (18), one can indicates that since quark's self energy cancels between the non-local and local matrix elements, it will not contribute to the distribution amplitudes. Therefore we have no need to consider Fig. 1 (e) and (f). The general discussions of one loop correction above are also applicable to the LCDAs. In the following calculations, we will introduce a small gluon mass m g to regularize the collinear divergence. For the UV divergence, we will employ two schemes: one is adding an UV cut-off Λ on the transverse momentum, another one is the dimensional regularization (DR). In this section we devote to the cut-off scheme, the DR results will be arranged in Appendix A.
To express the one loop results of quasi and light cone distribution amplitudes, we introduce the generalized plus distribution "+", which are defined by where T (x) is an arbitrary smooth test function. The generalized plus function regularizes the pole of divergent integral at x = x 0 .

A. Transverse distribution amplitudes
We now list the results of distribution amplitudes for the transversely polarized vector meson diagram by diagram. In Fig. 1(a), the internal gluon is not connected to the Wilson line. For this diagram, we have for both LCDA and quasi-DA. In Fig. 1(b)(c), one end of the internal gluon is attached to the Wilson line, thus there is an eikonal propagator, which is proportional to 1/(x − x 0 ). The contributions from Fig. 1 for LCDA, and for quasi-DA. For Fig. 1 for LCDA, and for quasi-DA. Fig. 1(d) is the one loop correction to Wilson line's self energy, which is proportional to n 2 , n is the direction vector of Wilson line. This contribution vanishes for LCDA since n 2 = 0, but do not vanish for quasi-DA. Then the results read and Note that for quasi-DA, this diagram contributes a linear divergence. Perturbative calculation on quasi-PDFs also shows the existence of the power-like UV divergence [17,18,22]. The power divergences have to be subtracted properly. A renormalization scheme has been proposed to subtract the linear divergence based on the auxiliary field formalism [19,21,40,41]; another approach is to replace the straight Wilson line with the non-dipolar Wilson lines [42]. It has been known that the power divergence in Wilson line's self energy can be canceled by introducing a "mass counter term" of the Wilson line [43]. Since the source of the linear divergence is the Wilson line's self energy, the improved quasi-PDFs and DAs are proposed by adding such a mass counter term, to subtract the linear divergence [24,40]. In the same spirit, one can also define the improved quasi-DAs of vector meson. To do this, we replace the operator in Eq. (9) by the "improved" operator where δm is the mass counter term of the Wilson line. It has been shown that δm can be extracted by using the static quark potential non-perturbatively [44]. Perturbative calculation shows that the contribution from δm cancels the linearly divergent term in Eq. (28). Therefore, one can get the result for improved quasi-DAs just by subtracting the linearly divergent term.
In the above results, the LCDAs are only non-zero in the physical regions 0 < x < x 0 and x 0 < x < 1, while the quasi-DAs have non-zero support in all of the four regions x < 0, 0 < x < x 0 , x 0 < x < 1 and x > 1. However, the collinear divergence only exists in the physical regions 0 < x < x 0 and x 0 < x < 1. One can also notice that the LCDAs and quasi-DAs are symmetric under variable substitution x ↔ 1 − x, x 0 ↔ 1 − x 0 .

B. Longitudinal distribution amplitudes
The one loop results of distribution amplitudes for longitudinally polarized vector meson are listed below diagram by diagram.
For Fig. 1(a), we have for LCDA, and for quasi-DA. Note that according to Eq. (20), we have subtract the vertex correction of the local operator which can be expressed as an integral of the non-local matrix element. Therefore the contributions above have been reformed to the generalized plus distribution. For Fig. 1(b), the results are for LCDA, and for quasi-DA. Similarly, for Fig. 1(c), we have for LCDA, and for quasi-DA. Fig. 1(d) receives contribution from Wilson line's self energy, which is proportional to n 2 , n is the direction vector of Wilson line. This contribution vanishes for LCDAs since n 2 = 0, but does not vanish for quasi-DAs. The results reads for LCDA, and is the result of quasi-DA. Similar to the transverse quasi-DA, this diagram also contributes a linear divergence to the longitudinal quasi-DA. As we have discussed in the last subsection, the linear divergence can be cured by introducing a mass counter term of Wilson line. The improved quasi-DAs have already been defined by Eq. (29). The one loop results under the improved definition can be got by subtracting the linearly divergent term in Eq. (37). At last, since that all of the results above are represented by the generalized plus distribution, thus they are zero under the integration, which is the normalization condition given by Eq. (12).

IV. THE MATCHING EQUATION
In this section, we present the matching equation connecting the LCDAs and quasi-DAs. In LaMET, if the factorization holds, the quasi-DA φ Γ V can be factorized as where y is constrained by 0 < y < 1. Here Z Γ is the perturbatively calculable function, hence can be expanded in the series of α s as By recalling the tree level result in Eq. (13), one can find that the one loop correction to the matching coefficient can be attributed to the difference between LCDA and quasi-DA at one loop level, and for Z (1) , the result reads In other regions, Z ⊥ and Z (1) are zero. One can notice that Z Γ (x, y, P z , Λ) = Z Γ (1 − x, 1 − y, P z , Λ). We should note that the plus distribution here is to subtract the singularities located at x = y, which is a little different from the one defined in Eq. (21). One can immediately find that the collinear divergence, which is regularized by m g , canceled out between LCDAs and quasi-DAs, thus the matching coefficients are free of IR divergence. Thus we have proved the LaMET factorization for DAs of vector meson at one loop level. There are also UV divergence which are regularized by the cut-off Λ. As we have discussed in Sec. III, the linear divergence will be subtracted by introducing δm, the mass counter term of Wilson line. Therefore, the matching coefficients of LCDAs and the improved quasi-DAs are the same to Eqs. (41) and (42) except the linearly divergent terms, hence the improved matching coefficients have only the logarithm UV divergence. The relation between improved matching coefficients and Eqs. (41)(42) is given by In Sec. III, we have taken the Λ → ∞ limit, the O(P z /Λ) contributions have been neglected. However, at present it is difficult to take too large value of P z in lattice simulations, in fact, Λ and xP z are of the same order in a practical calculation on the lattice. Therefore, it is valuable to consider the finite Λ corrections to the matching coefficients. The matching coefficients with a finite cut-off have been derived for the quark PDF [17,38] and LCDA of pion [24]. In Appendix B we will list the one loop matching coefficients of vector meson's distribution amplitudes with a finite cut-off Λ.
Since LCDAs do not depend on P z , one can take derivative with ln P z on both sides of the factorization formula Eq. (38), and derive the evolution equation of quasi-DAs with P z where V Γ (x, y) = d ln Z Γ (x, y, P z , Λ)/d ln P z is the evolution kernel, and the superscript "imp." denotes that the quasi-DAs are under the improved definition. With the Z Γ calculated in the above, we arrive at where θ(x) is the Heaviside step function. These functions are the Brodsky-Lepage kernels. It indicates that the evolution of quasi-DAs with P z shares the same behavior with the scale evolution of LCDAs, which are dominated by the Efremov-Radyushkin-Brodsky-Lepage (ERBL) equation [1,[45][46][47]. This evolution equation can be used to resum the large logarithm of P z which appears in the perturbative calculations. The P z evolution behavior for quasi-PDFs have already been reported, see, e.g., Refs. [10,18]. Since the P z evolution equation of quasi-DAs is equivalent to the ERBL equation of LCDAs, one can expect that when P z → ∞, the quasi-DAs converge to the same asymptotic form with LCDAs. Therefore, it seems that the asymptotic form is the UV fixed point for both LCDAs and quasi-DAs.

V. SUMMARY
In the framework of large momentum effective theory, we have performed one loop calculation on the leading twist light-cone distribution amplitudes as well as the quasi distribution amplitudes of the vector meson. The distribution amplitudes of both transversely and longitudinally polarized meson have been discussed. Based on the perturbative calculation under UV cut-off and DR schemes, we have examined the LaMET factorization and found that the collinear divergence cancels between light-cone and quasi distribution amplitudes. The matching coefficients have been determined at one loop accuracy. We also get the meson momentum evolution equation for quasi distribution amplitudes, and find that the evolution kernels are identical with the Brodsky-Lepage kernels of light-cone distribution amplitudes. The results of the present work will be useful to extract light-cone distribution amplitudes of vector mesons from the future lattice simulations.
For practical simulation on the lattice, the renormalization of quasi-DAs is necessary. In the present work the calculation is performed in a naive cut-off scheme and the renormalization is absent. Furthermore, the one loop calculation is not on the discrete but the continuum quasi-PDFs. Therefore, a calculation based on lattice perturbation theory, is necessary to fill the gap. Another approach is to renormalize the quasi-DAs in a nonperturbative renormalization scheme, such as the RI/MOM scheme, which has been employed to renormalize quasi-PDFs on the lattice. These issues will be discussed in the future works.

Transverse distribution amplitudes
We list here our results of distribution amplitudes for transversely polarized vector meson under dimensional regularization. For Fig. 1(a), we have For Fig. 1(b), we have For Fig. 1(c), we have Fig. 1(d) is the self energy of wilson line. For a Wilson line along the light-cone direction, the self energy is zero. For a space like Wilson line, the self energy is linearly divergent. However, in DR scheme, one can assign a finite value to the linearly divergent self energy with analytical continuation. Thus we have, In the results of LCDAs, we have performed the MS subtraction. For the quasi-DAs, the results of all the one loop diagrams are finite. However, one can notice that when x → ±∞, the quasi-DA behaves as ∝ 1/x, which is logarithmically divergent. One can take the convolution of the quasi-DA and an arbitrary test funtion T (x), e.g., T (x) = 1. The integral is zero since the quasi-DAs is of type [f (x)] + , but it is due to the cancelation of two logarithmically divergent integrals. Thus a renormalization is needed to make the integrals converge. One calculation on the quasi-PDF based on RI/MOM scheme has been performed in Ref. [35]. The renormalization on quasi-DAs will be discussed in a forthcoming work.

Longitudinal distribution amplitudes
Now we list our results for the distribution amplitudes of longitudinally polarized vector meson under dimensional regularization.
For Fig. 1(a), we have According to Eq. (18), there is a contribution from the vertex correction of the local operator, which can be expressed as a integral of φ Γ V (x) or φ Γ V (x). Note that we have added the contribution from δZ (1) V δ(x − x 0 ) here, so the contributions above have been reformed to the generalized plus distribution.