###### Abstract

We show that initial-state interactions contribute to the distribution in unpolarized Drell-Yan lepton pair production and , without suppression. The asymmetry is expressed as a product of chiral-odd distributions , where the quark-transversity function is the transverse momentum dependent, light-cone momentum distribution of transversely polarized quarks in an unpolarized proton. We compute this (naive) -odd and chiral-odd distribution function and the resulting asymmetry explicitly in a quark-scalar diquark model for the proton with initial-state gluon interaction. In this model the function equals the -odd (chiral-even) Sivers effect function . This suggests that the single-spin asymmetries in the SIDIS and the Drell-Yan process are closely related to the asymmetry of the unpolarized Drell-Yan process, since all can arise from the same underlying mechanism. This provides new insight regarding the role of quark and gluon orbital angular momentum as well as that of initial- and final-state gluon exchange interactions in hard QCD processes.

SLAC-PUB-9561

hep-ph/0211110

November 2002

Initial-State Interactions in the Unpolarized Drell-Yan Process
^{*}^{*}*Work partially supported
by the Department of Energy, contract DE–AC03–76SF00515, and by the
LG Yonam Foundation.

Daniël Boer, Stanley J. Brodsky, and Dae Sung Hwang

Department of Physics and Astronomy, Vrije Universiteit, De Boelelaan 1081

NL-1081 HV Amsterdam, The Netherlands

e-mail:

Stanford Linear Accelerator Center

Stanford University, Stanford, California 94309, USA

e-mail:

Department of Physics, Sejong University, Seoul 143–747, Korea

e-mail:

PACS numbers: 12.38.-t, 12.38.Bx, 13.88.+e, 13.85.Qk

## 1 Introduction

Single-spin asymmetries in hadronic reactions have been among the most challenging phenomena to understand from basic principles in QCD. Several such asymmetries have been observed in experiment, and a number of theoretical mechanisms have been suggested [1, 2, 3, 4, 5, 6]. Recently, a new way of producing single-spin asymmetries in semi-inclusive deep inelastic scattering (SIDIS) and the Drell-Yan process has been put forward [7, 8]. It was shown that the exchange of a gluon, viewed as initial- or final-state interactions, could produce the necessary phase leading to a single transverse spin asymmetry. The main new feature is that despite the presence of an additional gluon, this asymmetry occurs without suppression by a large energy scale appearing in the process under consideration. It has been recognized since then [9], that this mechanism can be viewed as the so-called Sivers effect [1, 10], which was thought to be forbidden by time-reversal invariance [4]. Apart from generating Sivers effect asymmetries, the mechanism offers new insight regarding the role of orbital angular momentum of quarks in a hadron and their spin-orbit couplings; in fact, the same matrix elements enter the anomalous magnetic moment of the proton [7]. The new mechanism for single target-spin asymmetries in SIDIS necessarily requires non-collinear quarks and gluons, and in the Sivers asymmetry the quarks carry no polarization on average. As such it is very different from mechanisms involving transversity (often denoted by or ), which correlates the spin of the transversely polarized hadron with the transverse polarization of its quarks.

In further contrast, the exchange of a gluon can also lead to transversity of quarks inside an unpolarized hadron. This chiral-odd partner of the Sivers effect has been discussed in Refs. [6, 11], and in this paper we will show explicitly how initial-state interactions generate this effect. Goldstein and Gamberg reported recently that is proportional to in the quark-scalar diquark model [12]. We confirm this and find that these two distribution functions are in fact equal in this model. Although this property is not expected to be satisfied in general, nevertheless, one may expect these functions to be comparable in magnitude, since both functions can be generated by the same mechanism. We investigate the consequences of the present model result for the unpolarized Drell-Yan process. We obtain an expression for the asymmetry in the lepton pair angular distribution. Here is the angle between the lepton plane and the plane of the incident hadrons in the lepton pair center of mass. This asymmetry was measured a long time ago [13, 14] and was found to be large. Several theoretical explanations (some of which will be briefly discussed below) have been put forward, but we will show that a natural explanation can come from initial-state interactions which are unsuppressed by the invariant mass of the lepton pair.

## 2 The unpolarized Drell-Yan process

The unpolarized Drell-Yan process cross section has been measured in pion-nucleon scattering: , with deuterium or tungsten and a -beam with energy of 140, 194, 286 GeV [13] and 252 GeV [14]. Conventionally the differential cross section is written as

(1) |

These angular dependencies^{†}^{†}†We neglect and
dependencies, since these are of higher order in
[15, 16] and are expected to be
small. can all
be generated by perturbative QCD corrections, where for instance
initial quarks radiate off high energy gluons into
the final state. Such a perturbative QCD calculation at
next-to-leading order leads to at very small transverse momentum of the lepton pair.
More generally, the Lam-Tung relation [17]
is expected to hold at order and the relation is hardly modified by
next-to-leading order () perturbative QCD corrections
[18]. However, this relation is not satisfied by the experimental
data [13, 14].
The Drell-Yan data shows remarkably large values of ,
reaching values of about 30% at transverse momenta of the lepton
pair between 2 and 3 GeV (for
and extracted in the Collins-Soper frame [19]
to be discussed below). These large values of are not compatible
with as also seen in the data.

A number of explanations have been put forward, such as a higher twist effect [20, 21], following the ideas of Berger and Brodsky [22]. In Ref. [20] the higher twist effect is modeled using an asymptotic pion distribution amplitude, and it appears to fall short in explaining the large values of .

In Ref. [18] factorization-breaking correlations between the incoming quarks are assumed and modeled in order to account for the large dependence. Here the correlations are both in the transverse momentum and the spin of the quarks. In Ref. [6] this idea was applied in a factorized approach [23] involving the chiral-odd partner of the Sivers effect, which is the transverse momentum dependent distribution function called . From this point of view, the large azimuthal dependence can arise at leading order, i.e. it is unsuppressed, from a product of two such distribution functions. It offers a natural explanation for the large azimuthal dependence, but at the same time also for the small dependence, since chiral-odd functions can only occur in pairs. The function is a quark helicity-flip matrix element and must therefore occur accompanied by another helicity flip. In the unpolarized Drell-Yan process this can only be a product of two functions. Since this implies a change by two units of angular momentum, it does not contribute to a asymmetry. In the present paper we will discuss this scenario in terms of initial-state interactions, which can generate a nonzero function .

We would also like to point out the experimental observation that the dependence as observed by the NA10 collaboration does not seem to show a strong dependence on , i.e. there was no significant difference between the deuterium and tungsten targets. Hence, it is unlikely that the asymmetry originates from nuclear effects, and we shall assume it to be associated purely with hadronic effects. We refer to Ref. [24] for investigations of nuclear enhancements.

We compute the function and the resulting asymmetry explicitly in a quark-scalar diquark model for the proton with an initial-state gluon interaction. In this model equals the -odd (chiral-even) Sivers effect function . Hence, assuming the asymmetry of the unpolarized Drell-Yan process does arise from nonzero, large , this asymmetry is expected to be closely related to the single-spin asymmetries in the SIDIS and the Drell-Yan process, since each of these effects can arise from the same underlying mechanism.

The Tevatron and RHIC should both be able to investigate azimuthal asymmetries such as the dependence. Since polarized proton beams are available, RHIC will be able to measure single-spin asymmetries as well. Unfortunately, one might expect that the dependence in (measurable at RHIC) is smaller than for the process , since in the former process there are no valence antiquarks present. In this sense, the cleanest extraction of would be from .

## 3 Cross section calculation

In this section we will assume nonzero and discuss the calculation of the leading order unpolarized Drell-Yan cross section (given in Ref. [6] with slightly different notation)

(2) | |||||

This is expressed in the so-called Collins-Soper frame [19], for which one chooses the following set of normalized vectors (for details see e.g. [25]):

(3) | |||||

(4) | |||||

(5) |

where , are the momenta of the two incoming hadrons and is the four momentum of the virtual photon or, equivalently, of the lepton pair. This can be related to standard Sudakov decompositions of these momenta

(6) | |||||

(7) | |||||

(8) |

with , via the identification of the light-like vectors

(9) | |||||

(10) |

The azimuthal angles lie inside the plane orthogonal to and . In particular, = , where gives the orientation of , the perpendicular part of the lepton momentum ; is the angle between (the direction of ) and . In the cross sections we also encounter the following functions of , which in the lepton center of mass frame equals , where is the angle of the momentum of the outgoing lepton with respect to (cf. Fig. 1):

(11) | |||||

(12) |

Furthermore, we use the convolution notation

(13) |

where are lightcone momentum fractions and is the flavor index.

In order to obtain the cross section expression one contracts the lepton tensor with the hadron tensor [6, 23]

(14) |

where . The correlation function is parameterized in terms of the transverse momentum dependent quark distribution functions [11]

and similarly for .

We end this section by giving the resulting expression for [6]

(16) |

## 4 Asymmetry calculation

The above cross section in terms of and can be represented by the diagram in Fig. 3.

Insertion of the parameterization of and will yield the asymmetry, among many other terms. However, in the lowest order quark-scalar diquark model the diagram Fig. 3 will not lead to nonzero in , and consequently, also not to a nonzero asymmetry. To generate such an asymmetry we will include initial-state interactions corresponding to diagrams such as those depicted in Fig. 3. Following the reasoning of Refs. [9, 26], this should be equivalent to Fig. 3 with an effective (and ) with nonzero function. Here we do not intend to give a full demonstration of this in the Drell-Yan process; a generalized factorization theorem which includes transverse momentum dependent functions and initial/final-state interactions remains to be proven [27]. Instead we present how to arrive at an effective from initial/final-state interactions and use this effective in Fig. 3. Also, for simplicity we will perform the explicit calculation in QED. Our analysis can be generalized to the corresponding calculation in QCD. The final-state interaction from gluon exchange has the strength , where are the photon couplings to the quark and diquark.

The diagram in Fig. 3 coincides with Fig. 6(a) of Ref. [28] used for the evaluation of a twist-4 contribution () to the unpolarized Drell-Yan cross section. The differences compared to Ref. [28] are that in the present case there is nonzero transverse momentum of the partons, and the assumption that the matrix elements are nonvanishing in case the gluon has vanishing light-cone momentum fraction (but nonzero transverse momentum). This results in an unsuppressed asymmetry which is a function of the transverse momentum of the lepton pair with respect to the initial hadrons. If this transverse momentum is integrated over, then the unsuppresed asymmetry will average to zero and the diagrams will only contribute at order as in Ref. [28].

First we will calculate the matrix to lowest order (called ) in the quark-scalar diquark model which was used in Ref. [7]. (Although the model is based on a point-like coupling of a scalar diquark to elementary fermions, it can be softened to simulate a hadronic bound state by differentiating the wavefunction formally with respect to a parameter such as the proton mass.) As indicated earlier, no nonzero and will arise from . Next we will include an additional gluon exchange to model the initial/final-state interactions (relevant for timelike/spacelike processes) to calculate and do obtain nonzero values for and . Our results agree with those recently obtained in the same model by Goldstein and Gamberg [12]. We can then obtain an expression for the asymmetry from Eq. (16) and perform a numerical estimation of the asymmetry.

### 4.1 matrix in the lowest order ()

As indicated in Fig. 4 the initial proton has its momentum given by , and the final diquark . We use the convention , .

We will first calculate the matrix to lowest order () in the quark-scalar diquark model used in Ref. [7]. By calculation of Fig. 4 one readily obtains

with a constant . The normalization is fixed by the condition

(18) |

In Eq. (4.1) we used the relation

(19) | |||||

This model is similar to the so-called spectator model (see e.g. Ref. [29]), where in addition a vector diquark is included and the coupling constant is treated as a form factor (in order to guarantee convergence). Of course, this can be assumed in the present model calculation as well and will be discussed in Section 4.4. Assuming real form factors, the functions and are strictly zero in the spectator model.

#### 4.1.1 Calculation of

For the calculation of the denominator of the asymmetry one needs to know the function , which can be obtained from given in Eq. (3):

(20) |

We now take and for the numerator spinor contraction, we calculate

(21) | |||||

Then, from Eqs. (4.1), (20) and (21), we arrive at

(22) | |||||

where we define , and

(23) |

Since we consider the proton state with mass as a bound state composed of a quark with mass and a diquark with mass , the function as given in Eq. (23) is always nonzero and positive.The integral in Eq. (18) with given in Eq. (22) can for instance be regulated by assuming a cutoff in the invariant mass: , and the value of is adjusted to satisfy the normalization condition Eq. (18) [30].

### 4.2 matrix with final-state interaction ()

In order to obtain the matrix with final-state interaction (called ), from which one can trivially obtain the one with initial-state interaction, we calculate the diagram given in Fig. 5(b). This is equal to the diagram calculated by Ji and Yuan [31] to obtain nonzero , starting from the formal gauge invariant definition of this transverse momentum dependent distribution function [9, 26]. In Fig. 5(b) we attached the virtual photon line to the later end of the eikonal line in order to emphasize that the final-state interaction effect has become an ingredient of the distribution functions of the target proton. In reality, the whole eikonal line should be considered to be at the same point.

Defining through Fig. 5(b) (in the Feynman gauge), we have

where we used The derivation of the starting formula of Eq. (4.2) is given in the Appendix. This underlies the step from Fig. 5(a) to Fig. 5(b) and hence the step from Fig. 3 to Fig. 3.

For in Eq. (4.2), we consider only the contribution from the imaginary part of , that is, the contribution from . There is no contribution from the real part of , since the hermitian conjugate term cancels it. Then, we have

When we perform the integration, we have

#### 4.2.1 Calculation of

We now apply this to and for the numerator spinor contraction we calculate

(29) | |||||

where we used and