Dominant Contribution in Pion Production Single-Spin Asymmetries
aa r X i v : . [ h e p - ph ] F e b DOMINANT CONTRIBUTION IN PION PRODUCTIONSINGLE-SPIN ASYMMETRIES Philip G. Ratcliffe , † , Marco Ramilli (1) Dipartimento di Fisica e Matematica, Universit`a degli Studi dell’Insubria (2)
Istituto Nazionale di Fisica Nucleare, Sezione di Milano † E-mail: philip.ratcliff[email protected]
Abstract
Working with a completely collinear twist-3 factorized cross-section formula, weidentify two largely dominant partonic sub-processes, which contribute to the single-spin asymmetries in semi-inclusive pion production, in the region of large p T andmedium–large x F . During the past years, different models have been developed in an attempt to explain themechanism behind the single-spin asymmetries observed experimentally in high-energyhadronic interactions. The approach based on the study of the hadronic cross-sectioncontribution given by the twist-3 components in the operator product expansion of partonmatrix elements turns out to be particularly interesting: taking into account such termsprovides a consistent model. However, at the same time the complexity of the calculationalframework unfortunately increases, since twist-3 contribution are characterized by thepresence of an additional gauge-field term, which in turn implies an extra gluon in thesub-processes, see for example [1, 2].Restricting our analysis therefore to a particular class of processes (pion productionin proton–proton collisions), our principal aim is to identify which, if any, among allpossible partonic sub-processes provide the dominant contributions to the asymmetryand to understand the origin of the suppression of the other terms. We can thus list a setof criteria (which we call “selection rules” ) summarizing these mechanisms. To simplifyour analysis, we shall extract a totally collinear cross-section formula, in the axial gaugeand in the limit of x F →
1, valid for large p T . We shall now go into detail, first by providing an expression for the twist-3 contributionto the cross-section through the study of the pole behavior of the Bjorken variables, andthen by analyzing the causes of the suppression of many other sub-processes. Talk presented at the XII Advanced Research Workshop on High Energy Spin Physics (DSPIN–07),Dubna, 3–7 Sept. 2007. .1 The poles Working in axial gauge, thus setting A + = 0, allows us to write the twist-3 contributionto the cross-section in the following way: dσ ( τ =3) ≃ Tr n Φ αA ( x , x )S β ( x , x ) o g ⊥ αβ , (1)where Φ αA ( x , x ) is the multi-parton matrix element and the index α is completely trans-verse, due to gauge choice. Moreover, in the axial gauge, the relation between Φ αA ( x , x )and Φ αF ( x , x ) assumes a very simple form (see [3], Eq. 7.3.30):( x − x )Φ αA ( x , x ) = − i Φ αF ( x , x ) , (2)demonstrating that if Φ αF ( x , x ) is different from zero for x = x , then Φ αA ( x , x ) musthave a pole.The analysis of the hard part is also crucial for the pole structure; there are twodifferent possibilities for the extra gluon, generated at twist-3, to interact significantly:with the on-shell fragmenting parton (the so-called final-state interactions, FSI) and withthe on-shell parton coming from the unpolarized nucleon (initial-state interactions, ISI);the important feature of these interactions is the presence of an extra internal propagator,whose Dirac structure has the form · · · 6 k P · k ) k α − ( x − x ) γ α Px − x − iε ! · · · , (3)where k µ is the four-momentum of the on-shell parton and P µ is the four-momentum ofthe polarized hadron.By also taking into account the pole behavior originating in the multi-parton matrixelement, it is possible to separate the trace over the Dirac indices into two traces, eachone with a different pole structure: the first, known as the single-pole contribution, wherethe ( x − x ) term in the numerator cancels the pole contribution of the matrix element,and the other, called the double-pole contribution, where no such cancelation occurs. Inorder to maintain the cross-section a real quantity, we are forced to take the imaginarypart of these poles, remembering thatIm x − x ± iε ) ! = ∓ iπδ ( x − x ) , (4)Im x − x ± iε ) ! = ∓ iπδ ′ ( x − x ) . (5)Using these relations and integrating the derivative of the delta function by parts, weobtain the following expression for the twist-3 contribution to the cross-section: dσ ( τ =3) = Z dx dx ′ dzz ε P h S ⊥ T ( dG F ( x, x ) dx H DP ( x, x ′ , z )+ G F ( x, x ) H SP ( x, x ′ , z ) ) f ( x ′ ) D ( z ) , (6)where we have omitted the color factors and the sum over flavor indices; ε µνT is the an-tisymmetric tensor in the transverse directions, G F ( x, x ) is the multi-parton distributionfunction evaluated at the pole (owing to the delta functions), f ( x ′ ) is the unpolarizedquark density and H represents the hard-scattering partonic cross-sections, with DP and SP standing respectively for double pole and single pole.2 .2 “Selection Rules” Given such an expression for the cross-section at twist three, we list here the set ofprinciples we have adopted to identify the possibly dominant contributions:- first, we expect DP contributions to be much more relevant than SP ones, owingto the presence of the derivative of the multiparton density function, which endowsthe asymmetry with a behavior in x roughly as A N ∼ − x ) (for x F approachingunity, the Bjorken x of the incoming parton also approaches unity), thus enhancingthe contribution of such terms for growing x F ;- for x F → | T | ≪ | U | ≪ | S | , we expect the t -channel diagrams to be dominant;for the same reason, remembering the power suppression of the hard parts given inEq. 3, we expect FSI to give a greater contribution than ISI;- we neglected the contributions given by polarized gluons and by sea quarks sincethese may reasonably be expected to be small.In order to test our model and the selection rules described above, we have evaluatedthe single-spin asymmetries for the reaction p ↑ p → π + X for the STAR kinematical range( √ S = 200 GeV and 1.3 GeV/ c < P hT < . c , see for example [4]). Restricting ouranalysis to the contribution given only by the t -channel diagram involved in the process,in Fig. 1a we present a comparison between the data points and the resulting predictiongiven by our model; we note that there is good agreement with data for values of x F greater than 0 . − . (a) (b) Figure 1. (a)
The theoretical curve represents the prediction for the SSA in π productionevaluated at P hT = 2 . (b) Here we plot the samecurve as in Fig. 1a, compared to the FSI DP term in a quark–gluon (here labeled qg ) sub-processand the FSI DP in a quark–quark subprocess. In Fig. 1b we also plot the total asymmetry, but together with the contribution givenby the two major sub-processes we have identified, i.e. the t -channel FSI DP terms.Comparing these curves, we can see how the two sub-processes mentioned provide almostentirely the value of the asymmetry in the kinematical range of x F > .
4; for lower valuesof this variable, we expect all the neglected contribution to become more important.3
Conclusions
To summarize then:- we have obtained an expression providing predictions for the single-spin asymme-tries for pion production consistent with data, in a completely collinear framework,without appealing to any collinear expansion;- using such an expression and a simple set of criteria, we have also been able toidentify two largely dominant subprocesses, which are almost entirely responsiblefor the asymmetries in the x F → References [1] A.V. Efremov and O.V. Teryaev, Phys. Lett. B , 383 (1985).[2] J.W. Qiu and G. Sterman, Phys. Rev. D , 014004 (1998).[3] V. Barone, A. Drago and P.G. Ratcliffe, Phys. Rept.359