Transverse spin asymmetries for W-production in proton-proton collisions

We study parity-even and parity-odd polarization observables for the process $p \, p \to l^{\pm} \, X$, where the lepton comes from the decay of a $W$-boson. By using the collinear twist-3 factorization approach, we consider the case when one proton is transversely polarized, while the other is either unpolarized or longitudinally polarized. These observables give access to two particular quark-gluon-quark correlation functions, which have a direct relation to transverse momentum dependent parton distributions. We present numerical estimates for RHIC kinematics. Measuring, for instance, the parity-even transverse single spin correlation would provide a crucial test of our current understanding of single spin asymmetries in the framework of QCD.


Introduction
It has long been recognized that production of W -bosons in hadronic collisions can provide new insights into the partonic structure of hadrons, with polarization observables being of particular interest. In this context the parity-odd longitudinal single spin asymmetry (SSA) in proton-proton scattering plays a very important role, both for leptonic as well as hadronic final states (see [1][2][3][4][5][6][7][8][9][10][11][12] and references therein). A major aim of looking into this observable is to get new and complementary information on the quark helicity distributions inside the proton.
In the meantime, also a few studies for W -production with transversely polarized protons are available [13][14][15][16]. These papers mainly focus on a particular parity-even transverse single spin effect in p p → W ± X (with a subsequent decay of the W ± into a lepton pair) that is related to the transverse momentum dependent Sivers function f ⊥ 1T [17] in the polarized proton. Such an observable could, in principle, be measured at the Relativistic Heavy Ion Collider (RHIC) in Brookhaven. In order to have clean access to transverse momentum dependent parton distributions (TMDs) like the Sivers function, one has to reconstruct the W -boson in the experiment. However, what one measures is p p → l ± X, and the detectors at RHIC do not allow to fully determine the momentum of the W .
The kinematics for inclusive production of a single lepton in proton-proton collisions coincides with the one for inclusive production of a jet or a hadron, for which mostly collinear factorization is used in the literature. In this Letter, we compute transverse spin observables for p p → l ± X in the collinear twist-3 formalism at the level of Born diagrams. The machinery of collinear twist-3 factorization was pioneered already in the early 1980's [18,19], and in the meantime frequently applied to transverse spin effects in hard semi-inclusive reactions (see [20][21][22][23] and references therein).
If one of the protons in p p → l ± X is transversely polarized, and the other is either unpolarized or longitudinally polarized, one can identify two parity-even and two parity-odd spin observables. We will discuss below that, in the collinear twist-3 approach, these four observables contain two specific twist-3 quark-gluon-quark correlation functions. One is the so-called ETQS (Efremov-Teryaev-Qiu-Sterman) matrix element [18,20,21], which is related to a particular moment of the transverse momentum dependent Sivers function as shown in [24,25]. The second is related to the TMD g 1T [23,26], where we use the TMD-notation of Refs. [27][28][29][30].
In addition to the analytical results, we provide numerical estimates for typical RHIC kinematics ( √ s = 500 GeV). All the observables are peaked around l T ≈ M W /2, with l T representing the transverse momentum of the lepton and M W the W -boson mass. In each case we predict clearly measurable effects. For the parity-even transverse SSA A e T U our numerical results are very close to those obtained in Ref. [15] on the basis of factorization in terms of transverse momentum dependent parton correlators.
Before presenting our results we emphasize that measuring A e T U would provide a crucial test of our present understanding of transverse SSAs in QCD. In particular, this means that such a measurement would test the same physics -the gluon exchange between the remnants of the hadrons and the active partons -which underlies the famous process-dependence of the Sivers function and of related timereversal odd parton distributions [31]. In other words, experimental results for A e T U in p p → l ± X, even if analyzed in terms of collinear parton correlators, would check a crucial ingredient of TMDfactorization [32][33][34]. Such a check, in essence, can be considered to be as fundamental as measuring the sign of the Sivers asymmetry in the Drell-Yan process.

Analytical results
We start by fixing the kinematical variables for the process p p → l ± X, and assign 4-momenta to the particles according to p(P a ) + p(P b ) → l ± (l) + X .
By means of these momenta we specify a coordinate system throughê z =P a = −P b ,ê x =l T (with l T representing the transverse momentum of the jet), andê y =ê z ×ê x . Mandelstam variables are defined by while on the partonic level one haŝ where k a and k b denote the momentum of the active quark/antiquark in the protons; see also Fig. 1(a). The momentum fraction x a characterizes the (large) plus-momentum of the quark/antiquark in the proton moving alongê z through k + a = x a P + a . 1 Likewise, one has In the second step in (4) we express x a , for a given √ s, through l T = | l T | and the pseudo-rapidity η = − ln tan(ϑ/2) of the lepton, since transverse momenta and (pseudo-)rapidities are commonly used to describe the kinematics of a final state particle in proton-proton collisions. Next, we turn to the polarization observables for p p → l ± X, which we compute in the collinear factorization framework. As already mentioned, we focus on the situation when one proton is transversely polarized, while the other is either unpolarized or longitudinally polarized. One finds the 1 For a generic 4-vector v, we define light-cone coordinates according : contribution from quark-gluon-quark correlation. This diagram, together with its Hermitian conjugate which is not displayed, needs to be taken into account when computing twist-3 observables.
following expression for the cross section: 2 with In Eq. (5), ϑ w is the weak mixing angle, V ab is a CKM matrix element, M is the proton mass, M W is the W -mass and Γ W its decay width. We also use ε ij T ≡ ε −+ij with ε 0123 = 1. The transverse spin vector of the proton moving alongê z is denoted by S aT , whereas λ b represents the helicity of the second proton. The lower limit of the x b -integration is given by x min b = −t/(s + u). One can project out the four spin-dependent components of the cross section in (5), in order, through In these formulas, '↑ x/y ' ('↓ x/y ') denotes transverse polarization alongê x/y (−ê x/y ) for the proton moving in theê z -direction, whereas '+' and '−' represent the helicities of the second proton. The dots in Eq. (5) indicate longitudinal single spin and double spin observables, as well as four possible correlations for double transverse polarization. In collinear factorization, the latter are at least twist-4 effects in the Standard Model. Note that double transverse polarization observables for W -production were also discussed in connection with potential physics beyond the Standard Model (see [35,36] and references therein).
We computed the (twist-2) unpolarized cross section in the first line of (5) on the basis of diagram (a) in Fig. 1 by applying the collinear approximation to the momenta k a and k b of the active partons. The result contains the ordinary unpolarized quark distribution f a 1 for a quark flavor a. The hard scattering coefficients H ab andH ab in Eq. (5), expressed through the partonic Mandelstam variables in (3), read In Eq. (10), one has to replaceû byt for ab =ūd,ūs, ud, us.
The four cross sections in (6)-(9) represent twist-3 observables. Calculational details for such observables in collinear factorization can be found in various papers; see, e.g., Refs. [21][22][23][37][38][39]. We merely mention that one has to expand the hard scattering contributions around vanishing transverse parton momenta. While for twist-2 effects only the leading term of that expansion matters, in the case of twist-3 the second term is also relevant. In addition, the contribution from quark-gluon-quark correlations, as displayed in diagram (b) in Fig. 1, needs to be taken into consideration. The sum of all the terms can be written in a color gauge invariant form, which provides a consistency check of the calculation.
The quark-gluon-quark correlator showing up in σ e T U and σ o T L is the aforementioned ETQS matrix element T a F (x, x) [18,20,21]. The peculiar feature of this object is the vanishing gluon momentumthat's why it is also called "soft gluon pole matrix element". If the gluon momentum becomes soft one can hit the pole of a quark propagator in the hard part of the process, providing an imaginary part (nontrivial phase) which, quite generally, can lead to single spin effects [18,20,21]. Note also that in our lowest order calculation no so-called soft fermion pole contribution (see [40] and references therein) emerges. For σ o T U and σ e T L another quark-gluon-quark matrix element -denoted asg a ; see, in particular, Refs. [23,26,41] -appears, together with the familiar twist-3 quark-quark correlator g a T (and, in the case of σ o T L , together with the quark helicity distribution g a 1 ). We use the common definitions for f 1 , g 1 , and g T . The quark-gluon-quark correlators T F andg are specified according to 3 with F µν representing the gluon field strength tensor, and D µ = ∂ µ − igA µ the covariant derivative. Equations (11) and (12) hold in the light-cone gauge A + = 0, while in a general gauge Wilson lines need to be inserted between the field operators.
It is important that T F andg are related to moments of TMDs. To be explicit, one has [23][24][25][26] where we use the conventions of Refs. [27][28][29][30] for the TMDs f ⊥ 1T and g 1T . In Eq. (13) we take into account that the Sivers function f ⊥ 1T depends on the process in which it is probed [31,42]. In order to make numerical estimates we will exploit the relations in (13), (14).
Finally, note that, due to the pure vector-axialvector coupling of the W -boson, no chiral-odd parton correlator shows up in any of the four spin correlations in (5), which makes those observables rather clean. The situation is different if one considers single lepton production from the decay of a virtual photon or of a Z-boson.

Numerical results
Now we move on to discuss numerical results for the polarization observables by limiting ourselves to the transverse single spin effects. This means, we consider the two spin asymmetries A e T U and A o T U , with σ e T U and σ o T U from Eq. (6) and (7), respectively, and σ U U denoting the unpolarized cross section. Note that the definition of A e T U corresponds to the one of the transverse SSA A N , which has been extensively studied in one-particle inclusive production for hadron-hadron collisions; see [43][44][45] for recent experimental results from RHIC.
To compute σ U U we use the unpolarized parton densities from the CTEQ6-parameterization [46]. For the ETQS matrix element we use the relation (13) between T F and the Sivers function, and take f ⊥ 1T from the recent fit provided in Ref. [47] on the basis of data from semi-inclusive DIS. (For experimental studies of the Sivers effect we refer to [48,49], while extractions of the Sivers function from data can be found in [47,[50][51][52][53][54].) In the case of A o T U one needs input for g T andg. For g T we resort to the frequently used Wandzura-Wilczek approximation [55] (see [56] for a recent study of the quality of this approximation) whereas forg we use (14) and a Wandzura-Wilczek-type approximation for the particular k T -moment of g 1T in (14) [57], leading tog We mention that (17) and a corresponding relation between chiral-odd parton distributions were used in [58,59] in order to estimate certain spin asymmetries in semi-inclusive DIS. The comparison to data discussed in [59] looks promising, though more experimental information is needed for a thorough test of approximate relations like the one in (17). Measuring the SSA A o T U could provide such a test. The helicity distributions g a 1 in (16) and (17) are taken from the DSSV-parameterization [60]. The transverse momentum of the lepton l T serves as the scale for the parton distributions.
The numerical estimates are for typical RHIC kinematics, i.e., √ s = 500 GeV. We present the asymmetries either as function of η for fixed l T or vice versa.  Figure 2: A e T U for p p → l ± X as a function of η (left) and l T (right) for √ s = 500 GeV. The solid line is for l − -production, and the dashed line is for l + -production.

=0.5 η
We start by discussing the parity-even asymmetry A e T U . As shown in the right plot in Fig. 2, this observable is peaked around l T ≈ M W /2 -a feature that does not depend on the value of η. To be more precise, the peak is at l T = 41 GeV, i.e., slightly above M W /2. The peak in the polarized cross section σ e T U gets enhanced in the asymmetry, because the unpolarized cross section drops rather fast when going beyond l T = M W /2. (As a side-remark we point out that the asymmetry in the peak region is completely dominated by the third term in the 2nd line in (5) containing the factor K(ŝ).) Nevertheless, in this kinematical region we expect A e T U to be measurable. As discussed in the introduction, in this context it is important to recall that information on the sign of the asymmetry is already sufficient for a crucial test of our current understanding of transverse SSAs.
In particular in the peak region, the asymmetry is larger for l − -production (W − -production) than for l + -production, which is partly due to the rather large Sivers function for d-quarks obtained in the fit of Ref. [47]. The l − -asymmetry and l + -asymmetry come with opposite sign because the Sivers function for u-quarks and d-quarks have an opposite sign. Note also that both asymmetries change sign as function of l T . Therefore, whether the sign of the asymmetry can be measured unambiguously may critically depend on the l T -resolution in the experiment.
As the η-dependence of A e T U in left plot in Fig. 2 shows, the asymmetry is maximal in the positive η range, when a large-x parton from the polarized proton participates in the hard scattering. Obviously, by integrating over a suitable η-range one may optimize between magnitude of the asymmetry on the one hand and the size of the statistical error bars on the other. Moreover, it is worthwhile to mention that the contributions from the antiquark Sivers functions are not negligible in the backward region. (Here we refer to a corresponding discussion on the Sivers asymmetry in the Drell-Yan process for proton-proton collisions in [61], where the strong sensitivity to the Sivers function for antiquarks was already pointed out.) It is also interesting that for both l + -production and l − -production the overall magnitude of A e T U is very similar to the predictions presented in Ref. [15], where TMD-factorization was used.
Let us now turn to the parity-odd transverse SSA A o T U , which is displayed in Fig. 3. Again, this asymmetry has a pronounced peak at l T = 41 GeV, and it is largest for l + -production (up to about 8 %). As outlined above, our prediction for A o T U is based on the Wandzura-Wilczek-type approximation leading to (17), which probably represents the most uncertain part of our calculation. Nevertheless, the asymmetry should be within experimental reach. Like in the case of the parity-even SSA, also A o T U is almost entirely determined by the K(ŝ)-term in the 3rd line in (5). This implies that, due to the relation (14), it gives rather clean access to the TMD g 1T , which so far is experimentally unconstrained. Therefore, in any case, a measurement of A o T U would provide very interesting new information.

Summary
We have studied transverse spin asymmetries for the process p p → l ± X, where the lepton is produced in the decay of a W -boson. If one of the protons is transversely polarized, and the other is either unpolarized or longitudinally polarized, there exist two parity-even and two parity-odd spin asymmetries. We computed these asymmetries in collinear twist-3 factorization at the level of Born diagrams. Moreover, for the two transverse single spin asymmetries A e T U and A o T U -defined through Eq. (15) and (6), (7) -we made numerical estimates for typical kinematics at RHIC ( √ s = 500 GeV). In the following we summarize our main results: • The analytical results for all four spin-dependent cross sections are given by two particular quark-gluon-quark correlators, which have a direct relation to transverse momentum dependent parton distributions: the Sivers function f ⊥ 1T and the TMD g 1T ; see Eqs. (13), (14). Measuring these observables could therefore provide new information on the structure of the proton that goes beyond the collinear parton model.
• The parity-even SSA A e T U is largest for l − -production (up to about 8 %), and it is peaked for transverse momenta l T of the lepton slightly above M W /2. (Actually, all the asymmetries studied in this Letter are significant only in a relatively narrow region around l T ≈ M W /2.) Measuring the sign of this asymmetry can, in essence, provide an as crucial test as measuring the sign of the Sivers asymmetry in Drell-Yan would do: it can test our present understanding of the underlying dynamics of transverse SSAs and at the same time check an important ingredient of TMD-factorization, namely the influence of the Wilson-line which is generated by the interaction between the active partons and the remnants of the protons. (For related work we refer to [15,31,42,50,[62][63][64].) • To the best of our knowledge the parity-odd SSA A o T U was never before explored in the literature. We find A o T U to be largest for l + -production (also up to about 8 %, like A e T U for l − -production). This observable is directly related to (a moment of) the TMD g 1T , for which at this time no experimental information exists.
In general, we believe that W -physics for polarized proton-proton collisions is very promising not only in the case of longitudinally polarized protons, but has also a considerable discovery potential for transverse polarization.