Parton Distributions in Light-Cone Gauge:
Where Are the Final-State Interactions?
1
Feng Yuan and Xiangdong Ji
Department of Physics, University of Maryland, College Park, Maryland 20742
Abstract. We show that the final-state interaction effects in the single target spin asymmetry
discovered by Brodsky et al. can be reproduced by either a standard light-cone gauge definition
of the parton distributions with a prescription of the light-cone singularities consistent with the
definition with a gauge link involving the gauge potential at the spatial infinity.
INTRODUCTION
Recently, Brodsky and collaborators have re-examined the significance of the parton
distributions measurable in deep-inelastic and other high-energy scattering. They found
that the final state interactions (FSI) between the struck quark and target spectators
yield distinct physical effects such as shadowing and single-spin asymmetry [1, 2].
These effects are of course contained in the light-cone gauge-link explicitly present
in the definition of the parton distributions in the non-singular gauges, in which the
gauge potential vanishes at the spacetime infinity. In the light-cone gauge, however, the
gauge-link vanishes by choice, and the parton distributions in the conventional definition
become parton densities which are entirely determined by the ground state light-cone
wave functions.
In this paper, we argue that the standard definition of the parton distributions in
the light-cone gauge requires a unique prescription for the light-cone singularities -the
one that is constrained to reproduce the light-cone gauge link in the covariant gauge.
Alternatively, if one demands the initial state wave function be real, then the usual
light-cone gauge link in the definition of the parton distributions is incomplete. It ought
to be supplemented with an additional contribution. In the non-singular gauges, this
new eikonal factor does not contribute. But in the gauges such as the light-cone gauge
where the gauge potential does not vanish asymptotically, the additional gauge link
is responsible for the final state interactions. We use the example of the single spin
asymmetry discussed in Ref. [2] to show that the FSI physics is faithfully reproduced in
this approach.
1
Supported by the US Department of Energy DE-FG02-93ER-40762.
CP675, Spin 2002: 15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
Sources and Polarimeters, edited by Y. I. Makdisi, A. U. Luccio, and W. W. MacKay
© 2003 American Institute of Physics 0-7354-0136-5/03/$20.00
484
SIVERS FUNCTION IN COVARIANT GAUGE
The transverse-momentum parton distribution in Covariant gauge is defined,
f x k d ξ d 2 ξ iξ k ξ k e
2π 3
Pψ ξ ξL†ξ ∞ ξ γ L0∞ 0ψ 0P 1
2
(1)
where the path-ordered light-cone gauge link[3, 4, 5, 6]
Lξ ∞ ξ P exp
ig
∞
ξ
A ξ ξ d ξ (2)
In hard scattering, the gauge link L∞ 0 arises from the final state interactions between
the struck quark and the gluon field in the target spectators[3, 6].
In the non-singular gauges, the above definition yields the correct gauge-invariant
parton distributions. As an example, let us first calculate the asymmetrical part of the
transverse momentum distribution in a nucleon due to its transverse polarization, the socalled Sivers function [7]. Since we are only interested in the matter of principle, we use
the simple model introduced in Ref. [2]to study the polarization asymmetry discussed
there.
Expanding Eq. (1) to the first order in g and dropping the leading term which does not
yield any asymmetry in the transverse momentum distribution, we have
x k f1T
d ξ d 2 ξ iξ k ξ k Pψ ξ ξ n
e
2π 3
∞
n ie1 Aξ 0d ξ γ ψ 0P hc 1
2∑
n
(3)
0
where e1 is the charge of the struck quark and n represents the intermediate di-quark
follows Ref. [8] except the kinematic factor is included here. At
states. The notation f 1T
one-loop order, we have contributions from Fig. 1,
x k f1T
ig2 e1 e2
42π 3 Λk2 d4q
1
U PS k mγ k q mU PS 2
2π 4
q iε
21q x iε q k q2 1 m2 iε P k q12 λ 2 iε
hc
(4)
where qµ is the gluon momentum. M, m and λ are the masses of the nucleon, quark and
diquark, respectively. U PS is the on-shell spinor for the nucleon with momentum P
2 denotes
and polarization S. Λk
Λk k x1 x
2
2
2
2
M mx 1λ x
485
2
(5)
q
k+q
k
P
P
FIGURE 1. One-loop contribution to the spin-dependent transverse momentum distribution in the
nucleon
The final result for the Sivers function is
x k f1T
2
g2 e1 e2 1 xm xM αβ γ
1 Λk
ε
k
P
S
ln
γ
α β k2 Λ0
2
2π 4
4Λk
(6)
With this Sivers function, the SSA found by Brodsky et al.[2] is reproduced, and so this
calculation demonstrates that the standard definition of the parton distribution in the nonsingular gauge does take into account properly the effects of the final-state interactions
[4].
SIVERS FUNCTION IN LIGHT-CONE GAUGE
In the light-cone gauge A 0, however, the light-cone gauge link L vanishes. Where are
the final state interactions? To find the answer, we consider all contributions to f x k at one-loop order in both Feynman and light-cone gauges[5]. In the light-cone gauge,
the gluon propagator has singularity at q n 0 and requires a regularization[6]. The
parton distribution in Eq. (1) is valid for the light-cone gauge only when the following
light-cone gauge propagator is used[5]
Dµν q i gµν qµ nν qν nµ q2
q n iε
(7)
where the direction of qµ is toward the struck quark in its initial state. In other words,
one now does not really has a freedom to choose the regularization for the light-cone
singularity, contrary to the popular belief.
In general, if the gauge potential does not vanish at large ξ , it has a non-vanishing
contribution to a gauge link at ξ ∞. Therefore, the definition of the parton distribution
in Eq. (1) is no longer gauge invariant because the two light-cone links generated by
ψ 0 and ψ ξ ξ are not connected at ξ ∞. If one makes a gauge transformation
U ξ which does not vanish at ξ ∞, an SU(3) matrix U † ξ ∞ ξ U ξ ∞ 0
pops up in the distribution after the transformation. Therefore, Eq. (1) must be modified
to a form that is invariant under a singular gauge transformation. Motivated by the above
consideration, we modify the eikonal phase in Eq. (1) to,
1 ∞
L0 ∞ 0 ∆L P exp ig
d ξ A ξ ∞ ξ (8)
0
486
where the path in the transverse direction is largely arbitrary.
Consider the parton distribution in Eq. (1) with the gauge link ∆L,
x k f1T
d ξ d 2 ξ iξ k ξ k Pψ ξ ξ n
e
2π 3
∞
n ie1 d ξ A∞ ξ γ ψ 0P hc 1
2∑
n
ξ
(9)
Going to the momentum space, we have
x k f1T
ig2 e1 e2
42π 3Λk d 4q
U PS k mγ k q mU PS
2π 4
iq ∞
e q k21q2 x m2qiε P k q12 λ 2 iε q2 1 iε
hc(10)
where 1q comes from the n q q term in the light-cone propagator for the gluon.
Using
eiq L
lim iπδ q (11)
L∞ q
which is true in the sense of principal-valued distribution, we recover the result in
Eq. (6).
A consistency check follows when replacing q by q iε in Eq. (11). The exponential factor becomes expε ∞ 0. Therefore, if Eq. (7) is used from the light-cone
gauge propagator, the new gauge link does not contribute. However, the parton distributions defined with the extra gauge link free one from choosing a specific prescription for
1q [6]. Any prescriptions in fact will yield the same result.
SUMMARY AND DISCUSSIONS
To summarize, we have shown that the final state interactions can be taken into account
in the light-cone gauge by either a gauge propagator chosen according to the physics of
light-cone gauge link in the usual parton distribution, or an extra gauge link at ξ ∞
in the parton distribution.
In the Drell-Yan porcess, the gauge link in the parton distributions arises from the
initial state interactions rather than from the final state. Correspondingly, the gauge
link in Eq.(1) for the parton distributions will end up with ξ ∞, and the extra
gauge link in light-cone gauge in Eq. (8) will take integral at ξ ∞ as well. As a
consequence, the Sivers function for DY process will have overall sign difference from
the DIS process[2, 4, 6], and the naive universality of the parton distributions will not be
valid any more.
Several interested transverse momentum dependent parton distribution functions are
sensitive to the quark orbital angular momentum of the proton, which is very important to understand the proton spin sum rule[9]. There are many observables which are
487
potentially sensitive to, although they do not directly measure the orbital angular momentum itself. For example, the Pauli form factor F2 Q2 of the proton, the generalized
parton distributions, higher-twist structure functions, and the P -dependent parton distributions. All of these observables have been recently correlated in the framework of
light-cone wave functions for three-quark Fock state of the proton[10].
ACKNOWLEDGMENTS
The authors thank A. Belitsky for collaboration, and S. Brodsky for a number of useful
discussions.
REFERENCES
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