aa r X i v : . [ h e p - ph ] O c t Spin asymmetry at large x F and k ⊥ Matti J¨arvinen Department of Physical Sciences and Helsinki Institute of Physics, POB 64, FIN-00014University of Helsinki, FinlandE-mail: [email protected]
Abstract.
We suggest that the large single spin asymmetries observed at high momentumfractions x F and transverse momenta k ⊥ of the pion in p ↑ p → π ( x F , k ⊥ ) + X arise from thecoherence of the soft interactions with the hard parton scattering process. Such coherence can bemaintained if x F → k ⊥ → ∞ , while k ⊥ (1 − x F ) ∼ Λ QCD stays fixed. Analogous coherenceeffects have been seen experimentally in the Drell-Yan process at high x F . We find that the p ↑ p → πX production amplitudes have large dynamic phases and that helicity flip contributionsare unsuppressed in this limit, giving rise to potentially large single spin asymmetries.
1. Introduction and motivation
Single pin asymmetry (SSA) is the dependence of a cross section on a single measured spin. Thesize of a transverse SSA is characterized by the analyzing power A N = dσ ↑ − dσ ↓ dσ ↑ + dσ ↓ ∝ Im[ M → M ∗← ] (1)where l ( ↔ ) refer to the transverse spin (helicity) of the polarized particle. Sizeable A N ’s havebeen observed in p ↑ p → π ( x F , k ⊥ ) + X [1, 2] and in pp → Λ ↑ ( x F , k ⊥ ) + X [3]. At the highestmeasured longitudinal momentum fractions x F ≃ . p ↑ p → π ± X at the E704experiment [1] ( √ s = 20 GeV) the asymmetry rises to | A N | ∼ .
4, and increases for transversemomenta above k ⊥ = 0 . ep ↑ → e + X ) [4]. Recently the asymmetry has been seen to persist at muchlarger center of mass energy √ s = 200 GeV in π production at STAR [2]. The asymmetryincreases with k ⊥ up to k ⊥ ≃ . x F = 0 . . . . . A N requires a helicity flip and a large (dynamical) phasebetween the two helicity amplitudes. Due to these requirements A N vanishes the standardleading twist collinear QCD factorization [5] but can be described by using generalized schemes.The E704 and STAR data have been fitted using transverse momentum dependent factorization[6] and including twist-three effects [7]. However, while these approaches are able to reproducethe x F dependence of A N , they predict that A N ∝ Λ QCD /k ⊥ in apparent conflict with the data.Notice that largest asymmetries have been observed at very large x F ≃ . p ↑ p → π ( x F , k ⊥ ) + X . In order to produce such a pion within thestandard leading twist QCD factorization one quark must carry a momentum fraction x & . z & . Present address: Institut for Fysik og Kemi, Syddansk Universitet, Campusvej 55, 5230 Odense M, Denmark equirements imply a very small production cross section. In fact, QCD at leading twist wasfound to underestimate the E704 pion production cross section by an order of magnitude at high x F [8] which casts doubt on the applicability of factorization based approaches on SSAs in thiskinematic region. On the other hand, the cross section measured at higher √ s and lower x F bySTAR is consistent with leading twist QCD.
2. Coherence at large x F As noted above present approaches fail to produce the total cross section at large x F and the k ⊥ dependence of A N in p ↑ p → π ( x F , k ⊥ ) + X . These shortcomings motivate us to suggest thatthe asymmetry is a large x F coherence effect [9]. Such effects in unpolarized Drell-Yan havebeen studied previously [10]. The angular distribution of the muon pair in πN → µµ ( x F ) + X provides a clear signature of the coherence effects which set in at high x F . When the intrinsichardness of the contributing pion Fock states becomes comparable to the virtuality Q of thephoton the angular distribution of the muons, which is 1 + cos θ at leading twist, turns intosin θ . This phenomenon was subsequently observed in the Drell-Yan data [11] where the changeof angular distribution occurs at x F ≃ . Q ≃
20 GeV .In general, the increase of coherence effects at large x F can be understood as follows (see [12]).The lifetime τ of a Fock state inside a rapidly moving proton is the inverse of the (light-front)energy difference ∆ E between the Fock state and the proton P + ∆ E = M − X i k i ⊥ + m i x i ; X i x i = 1 (2)where x i ( > k i ⊥ , and m i are the momentum fraction, the transverse momentum, and the massof parton i , respectively, P + is the proton light-front momentum, and M is the proton mass.When one quark carries a large x i ∼ x F → x j ∼ − x F →
0. Hencethe lifetime of the state τ ∼ / ∆ E ∼ (1 − x F ) P + / Λ QCD becomes short. The incoherence of suchstate with a hard quark requires (1 − x F ) P + / Λ QCD ≫ τ hard ∼ P + /k ⊥ or k ⊥ (1 − x F ) ≫ Λ QCD .When x F grows large enough, i.e. , k ⊥ (1 − x F ) ∼ Λ QCD , (3)the hard scattering becomes coherent with the soft physics.
3. Coherence effects in p ↑ p → πX Sizeable coherence effects were observed in Drell-Yan for x F ≃ . Q ≃ − k ⊥ ∼ p ↑ p → πX in a similar manner as in Drell-Yan above. We takethe scale ∼ k ⊥ of perturbative QCD to be large, k ⊥ → ∞ , but we keep k ⊥ (1 − x F ) fixed (insteadof x F ) so that (3) holds. In this limit single quark factorization fails and several partons fromthe same parent hadron contribute coherently to the process.A leading contribution to the process in the limit of fixed k ⊥ (1 − x F ) is shown in figure 1.A short lived Fock state with one fast quark ( x ∼
1) is created via gluon exchange. The fastquark scatters with the target obtaining a large transverse momentum k ⊥ . The quark finallypicks up a slow antiquark and the pion is formed through a gluon exchange which equalizes themomentum fractions. The interactions within the slow quark system (indicated by the dashedcircle in figure 1) are soft with transverse momentum scale ∼ Λ QCD . The condition (3) impliesthat all parts of the diagram are fully coherent.Recall that we need a helicity flip and a large, helicity-dependent phase to create a sizeableasymmetry. As a helicity flip is suppressed in the hard interactions, the flip must occur in . p π ± ± + − ±∓− + + x ∼ k ⊥ soft Figure 1.
Our mechanism forthe SSA in p ↑ p → πX . Seetext for explanation.the soft subprocesses. This is modeled in figure 1 by helicity changing one gluon exchange.The helicity flip vertex is indicated by a dot and ± are the helicities of the quarks in the twointerfering amplitudes. A dynamical phase is obtained from the hard subprocess as indicatedby the vertical dashed cut.Using some further simplifications the two helicity amplitudes of figure 1 can be estimated [9].A sizeable helicity dependent phase indeed arises when the coherence condition (3) is satisfied.This is a proof of principle that large A N is possible in the kinematic limit of fixed k ⊥ (1 − x F ).
4. Conclusion
We demonstrated that large x F coherence effects may explain the large asymmetries of p ↑ p → π ( x F , k ⊥ ) + X . In the limit of large k ⊥ with k ⊥ (1 − x F ) fixed the overall coherence of thescattering process is maintained which leads to large dynamical phases and possibly to largesingle spin asymmetries. Our mechanism may be able to reproduce the experimental result that A N increases with k ⊥ even for k ⊥ & Λ QCD . Note that simply assuming that the maximum of A N as a function of k ⊥ is set by the coherence requirement (3) shifts the maximum from thestandard expectation k ⊥ , max ∼ Λ QCD to a larger value.
Acknowledgments
This work was supported in part by the Academy of Finland through grant 102046, byGRASPANP, the Finnish Graduate School in Particle and Nuclear Physics, and by the MagnusEhrnrooth foundation.
References [1] Adams D L et al. (E704) 1991
Phys. Lett.
B261 et al. (STAR) 2004
Phys. Rev. Lett. Preprint hep-ex/0310058 )Surrow B (STAR) 2007
AIP Conf. Proc.
Preprint arXiv:0705.3483[hep-ex] )[3] Bunce G et al.
Phys. Rev. Lett. et al. Phys. Rev.
D40 et al. (HERMES) 2005
Phys. Rev. Lett. Preprint hep-ex/0408013 )[5] Kane G L, Pumplin J and Repko W 1978
Phys. Rev. Lett. Phys. Lett.
B362
Preprint hep-ph/9503290 )D’Alesio U and Murgia F 2004
Phys. Rev.
D70
Preprint hep-ph/0408092 )[7] Qiu J W and Sterman G 1999
Phys. Rev.
D59
Preprint hep-ph/9806356 )Kouvaris C, Qiu J W, Vogelsang W and Yuan F 2006
Phys. Rev.
D74
Preprint hep-ph/0609238 )[8] Bourrely C and Soffer J 2004
Eur. Phys. J.
C36
Preprint hep-ph/0311110 )[9] Hoyer P and J¨arvinen M 2007
JHEP
039 (
Preprint hep-ph/0611293 )[10] Berger E L and Brodsky S J 1979
Phys. Rev. Lett. et al. Phys. Rev. Lett. et al. (NA10) 1986 Z. Phys.
C31
Nucl. Phys.