Ⅱ-4. Hadron Physics

RIKEN Accel. Prog. Rep. 51 (2018)

Coulomb-nuclear interference eﬀects on forward π 0 production inpolarized-proton–nucleus collisions
G. Mitsuka∗1

M = eiϕ MC + MH ,(1)

where ϕ is the Coulomb phase, MC is the Coulomb
scattering amplitude, and MH is the hadronic scattering amplitude. The Coulomb scattering amplitude via
a one-photon exchange is expressed asoﬀ-shell nucleon propagator, FπN ∗ N and FIP N N ∗ are
the form factors at each vertex, ANN
is the IP N NIPelastic scattering amplitude, s = (pa + pb )2 , and s1 =(p1 + k)2 .







Fig. 1. Diagram of the amplitudes driven by the photon(γ ∗ ) or Pomeron (IP ) exchange in proton-nucleus collisions.

The diﬀerential cross section for single π 0 production
is expressed with the Källén function λ(x, y, z) ≡ x2 +y 2 + z 2 − 2xy − 2yz − 2zx as∫
λ(s1 , m2p , m2π )1/2
π 2 |M |2dσ=
ds1 .
dq 28λ(s, m2p , m2p )(2π)5s1(4)Figure 2 shows the cross section in pAu collisions at
sNN = 200 GeV. The dominant amplitude transits
from Coulomb to hadronic at q 2 ∼ 0.02 GeV2 . This
indicates that single spin asymmetries for forward π 0 s,
most likely produced below 0.1 GeV2 , are signiﬁcantly
modiﬁed by the interference between the Coulomb andhadronic interactions.
The estimation of single-spin asymmetries and extension of the presented framework to forward neutrons will be a topic of future investigation.
√dσ/dq2 (mb/(GeV/c)2)

It was reported from the PHENIX experiment atBNL-RHIC that the transverse single spin asymmetry, denoted as AN , for forward neutrons measured intransversely polarized-proton–nucleus (pA) collisions√at sNN = 200 GeV is far diﬀerent from that in√proton–proton (pp) collisions at s = 200 GeV.1)
I presented in Refs. 2–3) that ultra-peripheral pAcollisions (UPCs, also known as Primakoﬀ eﬀects) contribute to the measured AN modestly in pAl collisions
and signiﬁcantly in pAu collisions, and that UPCs together with hadronic interactions successfully explain
the PHENIX results. In UPCs, virtual photons (γ ∗ )emitted from the relativistic nucleus interact with thepolarized protons and then produce the neutrons andother particles.
In the Monte Carlo simulations discussed inRefs. 2–3), electromagnetic eﬀects (UPCs) and hadronic
eﬀects are taken into account independently. However, the interference between these two eﬀects, called
the Coulomb-nuclear interference (CNI) eﬀects, would
have nonzero amplitudes in the very small momentumtransfer region. In this report, I present the implementation of the CNI eﬀects for forward π 0 s in polarizedproton–nucleus collisions. Forward π 0 production is
described by a simpler mechanism than that for forward neutrons. Natural units h̄ = c = 1 are usedthroughout.
The scattering amplitude M for single pion production in the CNI eﬀects is given by

103

102

CNICoulomb

2

MC = Ze(pb + p2 )µµ

F (q )⟨p1 , k|J µ |pa ⟩,q2
∗

(2)

0

where ⟨p1 , k|J |pa ⟩ is the γ + p → p + π transitioncurrent4) and F (q 2 ) is the form factor. The kinematic
variables are deﬁned in Fig. 1. The hadronic scattering
amplitude via a one-Pomeron exchange5) is obtainedby replacing a virtual photon with a Pomeron in Fig. 1:2MH =FA gπNN FπN ∗ N (p21f ) FIPNN ∗ (p21f ) ANN
IP (s, q )/2s(3)× ū(p1 )iγ5 SN (s1 )γ µ u(pa )ū(p2 )γµ u(pb ) ,where nuclear eﬀects are taken into account in FA ,gπNN is the pion-nucleon coupling constant, SN is the∗1

Hadronic
0RIKEN Nishina Center- 108 -

0.02

0.04

0.06

0.08
0.1
q2 ((GeV/c)2)

Fig. 2. Diﬀerential cross sections for Coulomb (blue dotted), hadronic (red dashed), and CNI eﬀects (blacksolid).

References
1) C. Aidala et al., (PHENIX Collaboration), Phys. Rev.
Lett. 120, 022001 (2018).
2) G. Mitsuka, Phys. Rev. C 95, 044908 (2017).
3) G. Mitsuka, RIKEN Accel. Prog. Rep. 50, 15 (2017).
4) D. Drechsel, S. S. Kamalov, L. Tiator, Eur. Phys. J. A
34, 64 (2007).
5) P. Lebiedowicz, A. Szczurek, Phys. Rev. D 87, 074037(2013).