Quasi-parton distribution functions: two-dimensional scalar and spinor QCD

We construct the quasi-parton distributions of mesons for two-dimensional QCD with either scalar or spinor quarks using the $1/N_c$ expansion. We show that in the infinite momentum limit, the parton distribution function is recovered in both leading and sub-leading order in $1/N_c$.


I. INTRODUCTION
Light cone distribution amplitudes are central to the description of hard exclusive processes with large momentum transfer. They account for the non-perturbative quark and gluon content of a hadron in the infinite momentum frame. Using factorization, hard cross sections can be split into soft partonic distributions convoluted with perturbativly calculable processes. The partonic distributions are inherently non-perturbative. They are currently estimated using experiments, lattice simulations or models.
Recently one of us [1] has suggested that the light cone hadronic wavefunctions can be recovered from Euclidean correlators in hadronic states using instead quasi-parton distribution functions through pertinent renormalization in the infinite momentum limit. Preliminary lattice simulations have proven very promising [2,3]. The purpose of this letter is to explore this construct in two-dimensional scalar and spinor QCD in the non-perturbative 1/N c expansion.
Two-dimensional scalar QCD has a smooth large N c limit with a confining spectrum [4][5][6]. In this model the current correlators exhibits many features of fourdimensional QCD in contrast to two-dimensional spinor QCD [7]. In the deep inelastic regime the results exhibit expected scaling laws, and are overall in support of the Feynman partonic picture and the light cone expansion. In this paper, these two models will be used interchangeably to test the concept of the quasi-distributions in a non-perturbative context, as they differ by a minor change in the algebra of the pertinent bosonic operators. Specifically, we construct the quasi-parton distributions for both scalar and spinor QCD in leading and subleading order in 1/N c and show that they merge with the expected light cone distributions in the infinite momentum limit without additional renormalization. Our leading conclusion for two-dimensional spinor QCD is in agree- * Electronic address: xji@physics.umd.edu; Electronic address: yizhuang.liu@stonybrook.edu; Electronic address: ismail.zahed@stonybrook.edu ment with a recent study [8].
The organization of the paper is as follows: in section II we discuss a canonical quantization of two-dimensional scalar QCD in the axial gauge. We make explicit the Hamiltonian of the model in leading order in 1/N c using bosonized fields. Some renormalization issues are also discussed. In section III we explicit the wavefunction for scalar QCD in the light cone limit. In section IV we construct the quasi-parton distribution function in leading order in 1/N c , and show that it reduces to the light cone wavefunction in the infinite momentum limit. We also discuss the leading correction in 1/P . In section V, we show how to generalize the bosonization scheme algebraically for both scalar and spinor QCD, and use it for a systematic organization of the operators in 1/N c . This scheme is used in section VI, VII to correct the light cone parton distribution and quasi-distribution in spinor two-dimensional QCD through standard perturbation theory. We show that the subleading corrections to the quasi-parton distribution function merges with the parton distribution function in the infinite momentum limit without renormalization. Our conclusions are in section VIII. In the Appendix we summarized some elements of two-dimensional spinor QCD pertinent for our canonical analysis both in light-cone and axial gauge.

II. QUANTIZATION OF SCALAR QCD IN AXIAL GAUGE
We first discuss the general structure of the Hamiltonian in two dimensions for scalar SU(N) QCD in the axial gauge A 1 = 0. The same discussion for two-dimensional spinor QCD in both the light-cone and axial gauge is summarized briefly in the Appendix. The starting Lagrangian is In terms of the canonical momenta π † = Π φ = (D 0 φ) † and π = Π φ † = D 0 φ, the corresponding Hamiltonian reads The equation of motion for A 0 is a constraint equation that can be solved in terms of φ, π , to yield the canonical Hamiltonian To proceed, we will use a free-like representation for the field and its conjugate However, instead of the free dispersion law E k = √ k 2 + m 2 , we will use an arbitary E(k) that will be fixed self-consistently below in the planar approximation, with E k → |k| asymptotically.
A. Hamiltonian to order 1/ √ Nc The Hamiltonian (3) is quartic in a k , a † k . We now choose to bosonize it, by re-writing it in terms of the quadratic operators In leading order of 1/ √ N c , Using (5-6) and the identity a (T a ) ij (T a ) kl = δ il δ kj − 1 N δ ij δ kl , the Hamiltonian (3) now reads to order 1/ √ N c as Here λ = g 2 N c is the standard t ′ Hooft coupling. We have made use use of the notation M (k) = M (k, −k), N (k) = N (k, k), and For a consistent expansion in 1/N c , we can eliminate the √ N c term in (7) by setting Π − (k) = 0. The result is a gap equation for E(k) The leading order Hamiltonian simplifies to The integral in the gap equation (9) and subsequently the Hamiltonian contains a divergence and requires regu-larization. For that we regularize 1 (k+k1) 2 using the standard principal value (PV) prescription It is readily seen that Π − is finite but Π + diverges as with Π r finite. We have checked that for physical states (on mass shell) the ǫ-contributions cancel out (see below). The solution to the gap equation (9) that asymptotes E k → |k| still suffers from a logarithmic divergence even after the PV prescription, namely This is actually related to the mass divergence for the scalar one-loop self energy, and renormalizes the scalar mass From here on, we will refer to Π + as the renormalized momentum operator, and m as the renormalized mass, and omit the r-label for convenience. With this in mind, the renormalized gap equation (9) now reads

III. WAVE-FUNCTION AND LIGHT CONE LIMIT
To construct the light cone wave-function of the scalar quarks, we define and use them to re-write (10) in the form The bi-local operator M (p, q) can be decomposed in modes where the first contribution refers to the light cone wavefunction describing a pair of scalar quarks moving forward in the light front, while the second contribution refers to a pair moving backward in the front form. The pair is characterized by a relative momentum p and a center of mass momentum P . Here m n , m † n are canonical bosonic annihilation and creation operators. The equation of motion follows by commutation We can check that the ǫ-dependent divergences noted in the momentum operator cancel out. Indeed, using (12) the LHS in (19) both of which cancel out. This checks the consistency of the renormalization procedure for scalar QCD. No such renormalization is needed for spinor QCD.
In the large momentum limit P the equation simplifies. For that we set p = xP , k = yP , and take P → ∞ on both sides of (19). In this limit the backward wavefunction vanishes φ − → 0. Since and the equation of motion (19) involves only the forward wavefunction in the form where we have defined φ + n (xP, P ) = φ n (x), and PV refers to the principal value of the integral. (22) was obtained initially in [5] using different arguments.

IV. QUASI-PARTON DISTRIBUTION FUNCTION
The light cone distribution for scalar quarks is just |φ n (x)| 2 in leading order in 1/N c . We now show that to the same order, the light cone distribution function and the quasi-distribution function as defined in [1] are in agreement without further normalization. For that, we define the quasi-distribution functioñ where |P refers to the meson state. In the axial gauge, the Wilson line W [z, 0] = 1. Using the mode decomposition (4) and the relations (6) we obtain for the quasidistributioñ For P → ∞, we have E n = P and xP = E(xP ) and all φ − vanish. The quasi-parton distribution function reduces identically to the parton distribution function |φ n (x)| 2 . For finite P , (24) shows that the backward moving pair in φ − contributes. To assess this quantitatively, we now expand in 1 P the contributions φ ± in (24). For that, we go back to (19) and expand in 1 P , namely The coefficients β 1 is fixed through a straightforward Taylor expansion of Π + , while β 2 is fixed by the gap equation. Their explicit form is not needed for the general arguments to follow. With this in mind, the leading correction to φ − is and the subleading correction for φ + = φ(x) + 1 Here we have defined In general , this equation is solved in the same Hilbert space that defines K 0 − H 0 , if we note that K 0 − H 0 is hermitian in the space defined with the measure φ † φ where the set of φ n forms a complete basis set. The formal solution to (28) is The 1 P expansion now clearly shows that the the rate at which the quasi-distribution (24) approaches the asymptotic light-cone distribution |φ n (x)| 2 is smooth for all x = 0, 1. It is singular for x = 0, 1 through the contribution of the backward moving pair φ − in (26). So the large P limit should be taken before the x → 0, 1 limits at the edges.

V. ALGEBRAIC STRUCTURE
The algebraic framework we have developed allows us to go beyond the leading order in 1/N c , and therefore check the proposal in [1] beyond the leading order we have so far established. For that, we note that the bilocal operators (5) obey a closed algebra with N † 12 = N 21 . The sign assignment for the bosonization of scalar QCD is s = +1 as all underlying operators are bosonic.
A solution to this algebraically closed set can be found by organizing the bi-local operator in 1/N c , where M 0 satisfies the commutation relation in the infinit N c limit. In terms of (31-32) the solution to (30) can be found by inspection in leading and next to leading order It is important to note that the expantion of the N 's starts at the second order! From now on to avoid cluttering, we omit the 0 for the large N c asymptotic operator. When the operators in (33) are inserted back into the Hamiltonian, we obtain a complete expression for the first three terms of the 1/N c expanded Hamiltoinian in terms of the large N c asymptotic operators that define the Hilbert space. Specifically, to order 1 Thus, up to order 1/N 2 c we encounter six M interactions, but up to oder 1/N c √ N c we are still dealing with more tractable quartic and qubic terms. Our algebraic treatment differs notablly from the one presented in [9] in that in ours the algebra is corrected which is required for a consistent expansion. The resulting effective hadronic Hamiltonian is different.

VI. CORRECTION TO THE PDF IN SPINOR QCD
In so far our discussion has concentrated on twodimensional scalar QCD where we have established that the quasi-parton distribution function reduces to the parton distribution function in leading order in 1/N c . We have checked that this is also the case for two-dimensional spinor QCD, in agreement with a recent study [8]. In the Appendix we have briefly summarized the key changes from scalar to spinor in the light cone and axial gauge.
Since in the spinor version, the underlying fields are fermionic and not bosonic, the algebraic structure (30) differs from scalar to spinor QCD only in the sign switch s = +1 → −1, with exactly the same bosonized Hamiltonian (34). Also, to avoid unecessary long formula we will only discuss the 1/N c corrections to the parton distribution function in two-dimensional spinor instead of scalar QCD. The arguments for both models are similar, but the formula for scalar QCD are laboriously long as we have checked, with exactly the same conclusion.
Using the definitions for spinor QCD in the Appendix, we use for the bi-local mesonic operator M in the light cone gauge the decomposition which satisfies (30) with s = −1. To order 1/N c , the Hamiltonian for two-dimensional spinor QCD is the same as in (34), which after inserting (35) yields the first two leading contributions to the interaction of the form The quartic contribution in (36) is only shown schematically. It is of order 1/N c , and apparently relevant for the 1/N c correction to the parton distribution function. However, by simple inspection it gives zero contribution when acting on a free and leading meson contribution to the state, i.e.
It will be dropped. Therefore the leading correction to the parton distribution function is given by Here |P 1 is the first order perturbation of the meson state m † i (P ) |0 , which by standard perturbation theory reads Inserting (39) into (38) and carrying out the contractions yield as a correction to the leading parton distribution function and

VII. CORRECTION TO THE QUASI-PDF IN SPINOR QCD
In this section we derive the 1/N c correction to the quasi-parton distribution function for two-dimensional spinor QCD and show that it is in agreement with the 1/N c correction to the parton distribution we just established in the large momentum limit. For that, we switch to the description of two-dimensional spinor QCD in the axial gauge using the changes in the Appendix.
In the axial gauge, the Hamiltonian is writtent in terms of m n (P ) and φ ± . The structure of the Hamiltonian is still of the form (34). We now note that the contributions to the first order shift of the state |P 1 of the form m † m † m † always carries φ − . In the large momentum limit these terms drop out as we have shown earlier, so they will be ignored. The only surviving terms in the Hamiltonian at large momentum are also of the form m † m † m + c.c.
With the above in mind and to be more specific, the parts of the Hamiltonian (34) that will contribute to the quasi-parton distribution function in leading order in perturbation theory are of the form The ensuing shifts caused by (43) on the mesonic state to first order in 1 Nc are respectively of the form with and the coefficients f ijk and f ijkl are where we have set The last contribution f − ijk involves at least one φ − and therefore drops out in the large momentum limit, so it will not be quoted.
All contributions of the form f ijkl involve at least one φ − and also drop out in the large momentum limit. More specifically, in the large momentum limit, we set P i = P → +∞, and we change our variables to P 1 = xP , P 2 = yP , and P 3 = zP , then any term which contains φ − (x 1 P, x 2 P ) vanishes in this limit, an example is the f 1234 term.
The parton fractions are constrained kinematically. For instance, the energy denominator 1 E xP + E yP + E z − E p implies 0 < x, y, z < 1 in leading order in 1/P , otherwise the contribution is subleading. In this case, the only term in H 1 which contains only φ + (first contribution in (34)) will reduce to the light cone gauge term if one identifies the creation operators in both cases using More specifically, the first order correction to the quasiparton distribution function is proportional to with |P corrected to first order. There are two type of contributions in (49) as we now discuss. First, the m † m term. For this only the |i 1 in the shift of the state contributes, and the specific contribution with only φ + is 2 sin θ(xP ) 2 In the large momentum limit, we have p k = yP , and p l = (1 − y)P as discussed above. The first term is nonzero if 0 < x < y, and the second term is always zero for 0 < x < 1 since (x + y) > y. Thus by shifting y → y + x with 0 < y < 1 − x, and taking care of factors of P , this contribution matches the correction to the parton distribution function in the light cone gauge (40). Second, the mm + m † m † term comes with at least one φ − , and is always zero in the large P limit as discussed above. It follows, that the order 1/N c contribution to the quasi-parton distribution matches the parton distribution in the large momentum limit without renormalization in two-dimensional spinor QCD. We have explicitly checked that the same holds for two-dimensional scalar QCD.

VIII. CONCLUSIONS
Using a bosonized form of two-dimensional scalar and spinor QCD, we have analyzed the quasi-parton distribution of a meson state. In the infinite momentum limit, the quasi-distribution matches the parton distribution on the light cone both in leading and sub-leading order without further renormalization, but the limit is subtle at the parton fractions x = 0, 1. This provides a nonperturbative check on the proposal put forth by one of us [1] for extracting the QCD light cone partonic distributions from their quasi-distribution counterparts using pertinent equal-time Euclidean correlators through suitable matching at large momentum.