Transverse single spin asymmetries in photon production

Transverse single-spin asymmetries (SSA) in inclusive reactions are now considered to be directly related to the transverse momentum ${\bf k}_{T}$ of the fundamental partons involved in the process. We find that the ideal probe to extract information on the gluon Sivers function is the transverse SSA of prompt photon production $p p^{\uparrow} \to \gamma X$, at large $p_T$. The following related processes, $p p^{\uparrow} \to \gamma + jet + X$, $p p^{\uparrow} \to \gamma^* + X \to \mu^+ \mu^- + X$ and $\bar{p} p^{\uparrow} \to \gamma + X$ are also briefly discussed.

At present there is a wealth of experimental observations of single spin asymmetries (SSA) in many different processes. Large SSA have been measured in pp ↑ → πX, where one proton is transversely polarized, and in which the produced pion prefers to come out, either to the right or to the left of the plane formed by the beam direction and the proton polarization vector, depending on its charge. This effect was first observed at FNAL more then ten years ago, in experiments done by the E704 Collaboration [1], at center-of-mass (c.m.) energy √ s ∼ 20 GeV. It occurs also at √ s = 200 GeV, as observed recently for π 0 production by the STAR Collaboration [2], in the first spin run at BNL-RHIC. Although the data appear to have very little energy dependence, a careful study of the unpolarized cross section leads to conclude that the SSA, in these two energy regimes, may have two different dynamical origins [3]. Several SSA have been also measured in hyperon (and antihyperon) inclusive production pN → Y ↑ X, at various energies [4], but a suitable detailed interpretation of these rich polarization data is still missing. Moreover, recently an azimuthal asymmetry has been also observed in semi-inclusive deep-inelastic scattering (SIDIS) l p ↑ → l πX, for targets polarized transversely (A U T ) and longitudinally (A U L ) relative to the direction of the unpolarized incoming lepton beam direction [5,6].
Although these SSA are not yet fully understood, they are expected to give valuable information on the orbital angular momenta of quarks and gluons inside the hadron. Furthermore, they provide us with an understanding of QCD at the amplitude level, which comes from the fact that the SSA is proportional to the interference of a spin flip and a non spin flip amplitude, out of phases. Therefore in perturbation theory such an interference effect, which requires an imaginary part, is generated at the one loop level. The interference is between wave functions with angular momenta J z = ±1/2 and hence contains information on the partons orbital angular momenta [7]. Moreover, the required matrix element measures the spin-orbit correlation S · L within the target hadrons wavefunction, the same matrix element which produces the anomalous magnetic moment of the proton, the Pauli form factor, and the generalized parton distribution E which is measured in deeply virtual Compton scattering.
In practice, essentially two mechanisms have been proposed in order to explain the SSA. The first one is to generalize the parton distribution functions by considering distributions that depend on the transverse momenta k T of these partons, and the second is to take into account higher twist operators [8]. Recently it was shown that there is a direct relation between these two approaches, so in fact they are expected to produce very similar effects. In the case of the k T dependent distribution functions, the SSA can be produced either by quark distributions, which is called the Sivers effect [9], proposed long time ago, or by quark fragmentation functions, which is called the Collins effect [10]. For some time it was thought that the Sivers function vanished, but this was shown not to be the case in an explicit simple model calculation [7].
In general both the Sivers and the Collins effects will be present in a specific reaction, although there are some cases in which only one of them contributes. For example, the Collins effect is the only mechanism that can lead to asymmetries A U T and A U L , defined above. On the other hand, it does not appear in some electroweak interaction processes, where there is only the Sivers effect. In this paper we will concentrate on the Sivers function, whose existence was proved by considering final state interactions in a diquark model [7,11]. The diquark model can only predict the Sivers function for the valence quarks, and it is also of interest to calculate it for sea quarks or for gluons. In fact, the gluon Sivers function was mentioned for the first time in Ref. [12], and only recently it was also considered in jet correlations [13] and in D meson production [14] in p ↑ p collisions. Just as the quark Sivers function is related to the hadrons anomalous magnetic moment, the gluon Sivers function is connected with the gluons contribution to the same anomalous magnetic moment, a quantity which in general is difficult to obtain.
The direct photon production in pp collisions can provide a clear test of shortdistance dynamics as predicted by perturbative QCD, because the photon originates in the hard scattering subprocess and does not fragment, which immediately means that the Collins effect is not present. This process is very sensitive to the gluon structure function, since it is dominated by the quark-gluon Compton subprocess in a large photon transverse momentum range. Prompt-photon production, pp(pp) → γX, has been a useful tool for the determination of the unpolarized gluon density and it is considered one of the most reliable reactions for extracting information on the polar-ization of the gluon in the nucleon [15]. Some years ago, the E704 Collaboration [16] at FNAL measured single spin asymmetries for direct photon production in pp collisions at 200 GeV/c. Although the single spin asymmetry for the direct-photon production was found consistent with zero, within the experimental uncertainty, there is nowadays a real possibility to increase the precision of the measurement. In this letter, we show how to relate the transverse SSA to the gluon Sivers function.
There are only two hard scattering processes for the direct photon production in high p T collisions. One is the lowest-order Compton subprocess, qg → γq and the other one is the lowest-order annihilation subprocess, qq → γg. However, since the first subprocess is dominant in pp → γX collisions, the unpolarized cross section for producing a photon of transverse momentum p T and rapidity y can be written approximately as where q i (x, k T ) [ G(x, k T ) ] is the quark [gluon] distribution function with specified k T . A priori k T , the magnitude of k T , is expected to be small compared to √ s, where s is the center of mass energy of the reaction pp → γX. Therefore in order to simplify our discussion, we will use the following expressions which are valid only in the collinear approximation. The subprocess cross section is and by replacingû byt, one obtains the other internal cross section occurring in Eq. (1). Here α is the fine structure constant, α s is the strong coupling constant, e q denotes the quark charge andŝ,t,û stand for the Mandelstam variables for the parton subprocess According to the general definition of the k T -dependent parton distributions f (x, k T ) (f = q, G) inside a transversely polarized proton, where spin-up is labeled with ↑ and down with ↓, it is clear that whereas for the Sivers functions [9] we have Here S p denotes the transverse polarization of the proton of three-momentum p andp is a unit vector in the direction of p. The correlation proposed by Sivers corresponds to a time-reversal odd triple vector product. Now we can define the SSA as where d∆ N σ = dσ ↑ − dσ ↓ , whereas dσ = dσ ↑ + dσ ↓ and we have A priori the k T -dependence of all these parton distributions is unknown, but as an approximation one can assume a simple factorized form for the distribution functions and take for example, as in Ref. [12], where λ(k T ) is flavor independent, and a similar expression for the corresponding Sivers functions In such a situation ¶ , it is clear that the SSA will also factorize and then it reads where p T is the transverse momentum of the photon produced at the c.m. energy √ s, and H(p T ) is a function of p T , the magnitude of p T . We also recall the well known relation between y and x F , namely x F = 2 sinhy(p T / √ s). To achieve that it is necessary to determine in Eqs. (1) and (8), the range of integration over x a and to study the relative magnitude of x a and x b . As an example, using Eq. (2) with √ s = 200 GeV and p T = 20 GeV, the results for x min versus x F are shown in Fig. 1(a) and we find that x min ≈ x F in the region x F > 0.3. On the other hand, x b versus x a is shown in Fig. 1(b) and we see that when x a is integrated over ¶ The simplifying assumptions used above for the kinematics in the collinear approximation (see Eq. (2)), is justified by taking Gaussian expressions for λ(k T ) and η(f T ).
the range [x min , 1], the main contribution comes from the low x b values. Therefore, when we look at the large x F region, where x a is large but x b is small, the asymmetry can be approximately expressed as  [19]. We hope this will be a good motivation to undertake the measurement of the SSA, but we know that the extraction of the gluon Sivers function, even if it turns out to be large, will not be straightforward. Among the various effects which might dilute the SSA, it is important to mention the effects of QCD gluon resummation [20,21] and Sudakov effects have been shown to lead to significant suppression of the SSA considered in Ref. [13].
Other similar processes are pp ↑ → γ + jet + X, muon pair production pp ↑ → γ * + X → µ + µ − + X and pp ↑ → γ + X. The first reaction is certainly very interesting also, because by detecting simultaneously the photon and the jet, one has both rapidities to consider and Eq. (12) becomes simpler, with no integrations. For muon pair production, the outgoing photon is monitored by its conversion to muon pairs and this process is more difficult to study experimentally. Finally, in the case of pp ↑ → γ +X, the quark annihilation process qq → γg dominates, which makes it unpractical.
Therefore, the ideal probe to extract the gluon Sivers function is the transverse single spin asymmetry of prompt photon production at high p T , and RHIC is obviously very suitable to realize this important measurement with good precision.