Singularities, boundary conditions and gauge link in the light cone gauge

# Singularities, boundary conditions and gauge link in the light cone gauge

## Abstract

In this work, we first review the issues on the singularities and the boundary conditions in light cone gauge and how to regularize them properly. Then we will further review how these singularities and the boundary conditions can result in the gauge link at the infinity in the light cone direction in the Drell-Yan process. Except for reviewing, we also have verified that the gauge link at the light cone infinity has no dependence on the path not only for the Abelian field but also for non-Abelian gauge field.

## I introduction

The light cone gauge was widely used as an approach to remove the redundant freedom in quantum gauge theories. The Yang-Mills theories were studied on the quantization in light cone gauge by several authors Bassetto:1984dq (); Leibbrandt:1983pj (). In perturbative QCD, the collinear factorization theorems of hard processes can be proved more conveniently and simply in light cone gauge than in other gauges Amati:1978by (); Ellis:1978ty (); Efremov:1978cu (); Libby:1978qf (); Libby:1978bx (). Actually only in light cone gauge, the parton distribution functions defined in QCD hold the probability interpretation in the naive parton modelFey71 (). However in light cone gauge, when we calculate the Feynman diagrams with the gauge propagator in the perturbative theory, we have to deal with the light cone singularity ,

 Dμν(q)=1q2+iϵ(gμν−nμqν+nνqμq+). (1)

There have been a variety of prescriptions suggested to handle such singularities Bassetto:1983bz (); Lepage:1980fj (); Leibbrandt:1987qv (); Slavnov:1987yh (); Kovchegov:1997pc () from a practical point of view. Afterwards, it was clarified Bassetto:1981gp () that the gauge potential cannot be arbitrarily set to vanish at the infinity in the light cone gauge, the spurious singularities are physically related to the boundary conditions that one can impose on the potentials at the infinity. Different pole structures for regularization mean different boundary conditions. It should be emphasized that the above conclusion does not restrict to the light cone gauge, it holds for any axial gauges.

The non-trivial boundary conditions at the infinity in the light cone gauge also clarifies another puzzle in the transverse momentum dependent structure functions of nucleons. In the covariant gauge, in which the gauge potential vanishes at the space-time infinity, the transverse-momentum parton distribution can be given by operator matrix elements Collins:1981uw (); Collins:1989bt (); Collins:2003fm ()

 q(x,→k⊥) = 12∫dy−2πd2y⊥(2π)2e−ixp+y−+ik⊥⋅y⊥ (2) ×⟨P|¯ψ(y−,y⊥)n/L†[∞,y⊥;y−,y⊥] ×L[∞,0⊥;0,0⊥]ψ(0,0⊥)|P⟩,

where

 L[∞,y⊥;y−,y⊥]≡Pexp(−ig∫∞y−dξ−A+(ξ−,y⊥)), (3)

In this paper, we will devote ourselves to reviewing the above works and putting them together with the emphasis on mathematical rigor. However, through the reviewing, we will try to discuss them in a different way or point of view, which can be also regarded as the complementary to the previous work. Except for reviewing, we also have verified that the gauge link at the light cone infinity has no dependence on the path not only for the Abelian field but also for non-Abelian gauge field, which has not been discussed in the previous works.

We will organize the paper as follows: in Sec. II, we present some definitions and notations which will be used in our paper. In Sec. III, we discuss how the singularity can arise in the light cone gauge, how different singularities correspond to different boundary conditions and how we regularize them properly. In Sec. IV, we derive the transverse gauge link or more general gauge link in light cone gauge in Drell-Yan process. In Sec. V, we verify that the gauge link at the light cone infinity has no dependence on the path for non-Abelian gauge field. In Sec. VI, we give a brief summary.

## Ii definitions and notations

In our work, we will choose the light cone coordinate system by introducing two lightlike vectors and and two transverse spacelike vectors and

 nμ = 1√2(1,0,0,1)≡[0,1,0⊥], (4) ¯nμ = 1√2(1,0,0,−1)≡[1,0,0⊥] (5) nμ⊥1 = (0,0,1,0)≡[0,0,1,0], (6) nμ⊥2 = (0,0,0,1)≡[0,0,0,1] (7)

where we have used square brackets ‘[ ]’ to denote the components in the light-cone coordinate, compared with the usual Cartesian coordinate denoted by the parentheses ‘( )’. In such coordinate system, we can write any vector as or , where .

Since we will consider the non-Abelian gauge field all through our paper, we will use the usual compact notations for the non-Abelian field potential and strength, respectively,

 Aμ≡Aaμta,  Fμν≡Faμνta, (8)

where is the fundamental representation of the generators of the gauge symmetry group.

For the sake of conciseness, we would like to introduce some further notations. We will decompose any momentum vector and the gauge potential vector , as the following

 kμ = ~kμ+xpμ,  Aμ=~Aμ+A+¯nμ (9)

where , , and . Meanwhile, for any coordinate vector , we will make the following decomposition,

 yμ = ˙yμ+y−nμ (10)

where . With these notations, it is very easy to show . In the light cone gauge , the gauge vector . When no confusion could arise, we will write as for simplicity.

## Iii singularities and boundary conditions in the light cone gauge

In this section, we will review how the singularities appear in the light cone gauge, how they are related to the boundary conditions of the gauge potential at the light cone infinity and how we can regularize them in a proper way consistent with the boundary conditions. Although this section is mainly based on the literature Bassetto:1981gp () and Belitsky:2002sm (), there are also a few differences from them. For example, we will discuss the non-Abelian gauge field from the beginning to the end, while in the original works, only the Abelian gauge field was emphasized. Besides we will make the maximal gauge fixing from the point of view of linear differential equation.

With the light cone gauge condition , let us consider the non-Abelian counterpart of Maxwell equations,

 DμFμν=−jν (11)

where . We can rewrite the above equations in another form

 ∂μ∂μAν−∂ν∂μAμ=−Jν (12)

where we have defined . Contracting both sides of Eq.(12) with and taking the light cone gauge condition into account yields

 nν∂ν∂μ~Aμ=nνJν. (13)

Integrating the above equation gives rise to

 ~∂μ~Aμ(+∞,~x)−~∂μ~Aμ(−∞,~x)=∫+∞−∞dx−nνJν (14)

Since , in general, need not be true, we can not arbitrarily choose both and at the same time. One of these boundary conditions can be arbitrarily chosen while the other one must be subjected to satisfy the constraint (14). This is just why we can not choose the boundary conditions arbitrarily in light cone gauge. In fact, this conclusion holds for any axial gauges. From the Fourier transforms of Eq.(12),

 k2Aν−kν(k⋅A)=−J (15)

and together with the light cone gauge condition, it is easy to obtain the formal solutions

 ~Aμ=∫d4keik⋅xk2(−~Jμ+~kμk+J+) (16)

where and are the Fourier transforms of and ,respectively. It is obvious that there is an extra singularity at in the solution (16). If we assume that the currents are regular at , it is easy to verify that the different pole prescriptions correspond to different boundary conditions. In our paper, we will consider three different boundary conditions,

 Advanced : ~A(+∞,˙y)=0 Retarded : ~A(−∞,˙y)=0 Antisymmetric : ~A(−∞,˙y)+~A(∞,˙y)=0 (17)

which correspond to three different pole structures, respectively,

 1k+−iϵ,     1k++iϵ,    12(1k++iϵ+1k+−iϵ). (18)

where the last prescription is just the conventional principal value regularization. In the next section, we will deal with the Fourier transform of the gauge potential,

 ~Aμ(k+,˙y)≡∫∞−∞dy−eik+y−~Aμ(y−,˙y) (19)

In order to pick out the contribution of the gauge potential at the infinity, we need a mathematical trick by manipulating this integration by parts,

 ∫∞−∞dy−eik+y−~Aμ(y−,˙y) (20) = ik+∫∞−∞dy−eik+y−∂+~Aμ(y−,˙y)

where . We will see that once we choose the prescriptions (18) according to the boundary conditions (III), we will obtain the gauge link at the light cone infinity.

We have seen that we cannot choose the boundary conditions arbitrarily, now we will discuss how to fix the gauge freedom as maximally as possible. These have been also discussed in the appendix in Ref.Belitsky:2002sm (), we will take them into account from the point of view of differential equations. Under a general gauge transformation, the gauge potential transforms as,

 Aμ→S−1AμS+igS−1∂μS (21)

In order to eliminate the light cone component , we obtain the gauge transformation by solving the equation,

 nμ∂μS=ignμAμS (22)

This equation is an ordinary linear differential equation, whose solution is well known,

 S = P{exp[ig∫x+x+0nμAμ(ξ,x−,x⊥1,x⊥2)dξ]} (23) ×~S(x−,x⊥1,x⊥2)

where is an arbitrary unitary matrix which does not depend on . This freedom allows us to set one of the residual three components of zero on the three dimensional hyperplane at . Without loss of generality, we can set by solving the following equation

 ¯nμ∂μ~S=ig¯nμAμ~S. (24)

The solution is given by

 ~S = P{exp[ig∫x−x−0dξ¯nμAμ(x+0,ξ,x⊥1,x⊥2)]} (25) ×S⊥(x⊥1,x⊥2).

There is still an arbitrary unitary matrix which depends only on . We can use this freedom to further set one of the residual transverse components of the gauge potential zero, e,g, =0, at the two dimensional hyperplane (, ) by solving

 nμ⊥1∂μS⊥=ignμ⊥1AμS⊥(x⊥1,x⊥2) (26)

The solution is given by

 S⊥ = P{exp[ig∫x⊥1x0⊥1dξnμ⊥1Aμ(x+0,x−0,ξ,x⊥2)]} (27) ×S1⊥(x⊥2).

We can continue to set the only left transverse components at the straight line [, , ] by solving

 nμ⊥2∂μS1⊥=ignμ⊥2AμS1⊥(x⊥2) (28)

The solution is given by

 S1⊥ = P{exp[ig∫x⊥1x0⊥1dξnμ⊥2Aμ(x+0,x−0,x0⊥2,ξ)]} (29) ×S2⊥

With only a trivial global gauge transformation left, we have maximally fixed our gauge freedom. Although we cannot choose the boundary conditions of the gauge potential arbitrarily in the light cone gauge, the constraint that the field strengths should vanish at the infinity requires that the gauge potential must be a pure gauge,

 Aμ=1igω−1∂μω (30)

where with . We can expand the above pure gauge as

 Aμ = 1igω−1∂μω (31) = ∂μϕ+i2![∂μϕ,ϕ]+i23![[∂μϕ,ϕ],ϕ]+⋅⋅⋅

## Iv Gauge link in the light cone gauge in Drell-Yan process

In this section, we will review how the singularities and boundary conditions in the light cone gauge can result in the gauge link at the light cone infinity. Since the detailed derivation of transverse gauge link had been made for semi-inclusive deeply inelastic scattering Belitsky:2002sm (); Gao:2010sd (), we will discuss the Drell-Yan process in details in order to avoid total repeating. For simplicity, we will set the target to be a nucleon and the projectile be just an antiquark.

The tree scattering amplitude of Drell-Yan process corresponding to Fig. 1 reads

 Mμ0=¯u(q−k)γμ⟨X|ψ(0)|P⟩, (32)

where denotes the momentum of initial quark from the proton with the momentum , and and are the momenta of the anti-quark and virtual photon, respectively.

The one-gluon amplitude in the light cone gauge corresponding to Fig. 2 reads,

 Mμ1 = ∫d4k1(2π)4d4y1 ei(k−k1)⋅y1¯u(q−k)γρ1q/−k/1(q−k1)2+iϵ (33) ×⟨X|~Aρ1(y1)γμψ(0)|P⟩.

In order to obtain the leading twist contribution, we only need the pole contribution in the quark propagator,

 ^Mμ1 = ∫d3~k1(2π)4d3˙y1p+dx12πdy−1 ei(~k−~k1)⋅˙y1+i(x−x1)p+y− (34) ×¯u(q−^k)γρ1q/−^k/12p⋅(^k1−q)1(x1−^x1−iϵ) ×⟨X|~Aρ1(y1)γμψ(0)|P⟩

where with an extra denotes that only the pole contribution is kept and with , which is determined by the on-shell condition . In Eq.(34), we have separated the integral over and from the others in order to finish integrating them out first. Now we need to choose a specific boundary condition for the gauge potential at the infinity. Let us start with the advanced boundary condition . Using Eq. (20) for the advanced boundary condition, we have

 ^Mμ1 = ∫d3~k1(2π)4∫d3˙y1 ∫dx12π∫dy−1 (35) ×ei(~k−~k1)⋅˙y1ei(x−x1)p+y−¯u(q−k)γρ1 ×q/−^k/12p⋅(^k1−q)1(x1−^x1−iϵ)i(x−x1−iϵ) ×⟨X|∂+~Aρ1(y1)γμψ(0)|P⟩

Finish integrating over and :

 ∫dx12πdy−1ei(x−x1)p+y−1(x1−^x1−iϵ) (36) ×i(x−x1−iϵ)∂+~Aρ1(y1) = −∫dy−1(θ(−y−)ei(x−^x1)p+y−+θ(y−)) ×1x−^x1∂+~Aρ1(y1) = 1x−^x1~Aρ1(−∞,˙y1)+higher twist.

where only the leading term in the Tailor expansion of the phase factor is kept, because the other terms are proportional to , which will contribute at higher twist level.

However if we choose the retarded boundary conditions, we can have

 ^Mμ1 = ∫d3~k1(2π)4∫d3˙y1 ∫dx12π∫dy−1 (37) ×ei(~k−~k1)⋅˙y1ei(x−x1)p+y−¯u(q−k)γρ1 q/−^k/12p⋅(^k1−q)1(x1−^x1−iϵ)i(x−x1+iϵ) ×⟨X|∂+~Aρ1(y1)γμψ(0)|P⟩.

Integrating out and first yields

 ∫dx12π∫dy−1ei(x−x1)p+y−1(x1−^x1−iϵ) (38) ×i(x−x1+iϵ)∂+~Aρ1(y1) = −∫dy−1(θ(−y−)ei(x−^x1)p+y−−θ(−y−)) ×1x−^x1∂+~Aρ1(y1) = higher twist.

We can see the retarded boundary condition does not result in leading twist contribution in the Drell-Yan process. If we choose the antisymmetric boundary condition, which corresponds to the principal value regularization, we obtain

 ∫dx12π∫dy−1ei(x−x1)p+y−1(x1−^x1+iϵ) (39) ×PVi(x−x1)∂+~Aρ1(y1) = ∫dy−112(2θ(y−)ei(x−^x1)p+y−−θ(y−)+θ(−y−)) ×1x−^x1∂+~Aρ1(y1) = 1x−^x1~Aρ1(+∞,˙y1)+higher twist,

where PV denotes principal value. In the above derivation, we notice that the presence of the pinched poles are necessary to pick up the gauge potential at the light cone infinity. Actually these pinched poles have selected the so-called Glauber modes of the gauge fieldIdilbi:2008vm (). Although there is no leading twist contribution in the retarded boundary condition, it was shown in Belitsky:2002sm (), that all the final state interactions have been encoded into the initial state light cone wave functions. In principal value regularization, the final state scattering effects appear only through the gauge link , while in advanced regularization, it appear through both the gauge link and initial light cone wave functions. In the following, we will only concentrate on the advanced boundary condition. Only keep leading twist contribution and inserting Eq. (36) into Eq. (35), we have

 ^M1 = ∫d3~k1(2π)4d3˙y1 ei(~k−~k1)⋅˙y1¯u(q−k)γρ1q/−^k/12p⋅(^k1−q) (40) ×1x−^x1⟨X|~Aρ1(−∞,˙y1)ψ(0)|P⟩.

Using Eq. (31), only keeping the first Abelian term and performing the integration by parts over , we obtain

 ^M1 = ∫d3~k1(2π)4d3˙y1 ei(~k−~k1)⋅˙y1¯u(q−k)(~k/−~k/1) (41) ×q/−^k/12p⋅(^k1−q)−ix−^x1⟨X|ϕ(−∞,˙y1)ψ(0)|P⟩.

We can calculate these Dirac algebras and finally obtain,

 ^M1 = ¯u(q−k)⟨X|iϕ(−∞,0)ψ(0)|P⟩. (42)

where we have dropped all the higher twist contributions. Now let us further consider the two-gluon exchange scattering amplitude plotted in Fig. 3,

 Mμ2 = ∫d4k2(2π)4d4k1(2π)4d4y2d4y1 ei(k−k2)⋅y2+i(k2−k1)⋅y1 (43) ×¯u(q−k)γρ2q/−k/2(q−k2)2+iϵγρ1q/−k/1(q−k1)2+iϵ ×⟨X|~Aρ2(y2)~Aρ1(y1)γμψ(0)|P⟩.

Analogously to the case of , we will only keep the pole contribution,

 ^Mμ2 = ∫d3~k2(2π)3d3~k1(2π)3d3˙y2d3˙y1p+dx22πp+dx12πdy−2dy−1 (44) ×ei(~k−~k2)⋅˙y2+i(~k2−~k1)⋅˙y1+i(x−x2)p+y−2+i(x2−x1)p+y−1 ×¯u(q−k)γρ2q/−^k/22p⋅(^k2−q)γρ1q/−^k/12p⋅(^k1−q) ×1(x2−^x2−iϵ)1(x1−^x1−iϵ) ×⟨X|~Aρ2(y2)~Aρ1(y1)γμψ(0)|P⟩.

With the regularization (20) and (18), we can integrate out and first,

 ^Mμ2 = ∫d3~k2(2π)3d3~k1(2π)3d3˙y2d3˙y1p+dx12πdy−1 (45) ×ei(~k−~k2)⋅˙y2+i(~k2−~k1)⋅˙y1+i(x−x1)p+y−1 ×¯u(q−k)γρ2q/−^k/22p⋅(^k2−q)γρ1q/−^k/12p⋅(^k1−q) ×1(x−^x2−iϵ)1(x1−^x1−iϵ) ×⟨X|~Aρ2(−∞,˙y2)~Aρ1(y1)γμψ(0)|P⟩.

Further integrating out and gives rise to

 ^Mμ2 = ∫d3~k2(2π)3d3~k1(2π)3d3˙y2d3˙y1 ei(~k−~k2)⋅˙y2+i(~k2−~k1)⋅˙y1 (46) ×¯u(q−k)γρ2q/−^k/22p⋅(^k2−q)γρ1q/−^k/12p⋅(^k1−q) ×1(x−^x2−iϵ)1(x−^x1−iϵ) ×⟨X|~Aρ2(−∞,˙y2)~Aρ1(−∞,˙y1)γμψ(0)|P⟩

Using Eq. (31), only keeping the first Abelian term, we have

 ^Mμ2 = ∫d3~k2(2π)3d3~k1(2π)3d3˙y2d3˙y1 ei(~k−~k2)⋅˙y2+i(~k2−~k1)⋅˙y1 (47) ×¯u(q−k)γρ2q/−^k/22p⋅(^k2−q)γρ1q/−^k/12p⋅(^k1−q) ×1(x−^x2−iϵ)1(x−^x1−iϵ) ×⟨X|~∂ρ2ϕ(−∞,˙y2)~∂ρ1ϕ(−∞,˙y1)γμψ(0)|P⟩.

Using the integration by parts, we can integrate out and and obtain

 ^Mμ2 = ∫d3