In this note we consider effects of a Two-Photon Exchange (TPE) in inclusive
DIS and semi-inclusive DIS (SIDIS). In particular, transverse single spin
asymmetries are generated in inclusive DIS if more than one photon is exchanged
between the lepton and the hadron. We briefly summarize the TPE in DIS in the
parton model and extend our approach to SIDIS, where a new leading twist
$\sin(2\phi)$ contribution to the longitudinal beam spin asymmetry shows up.
Possible TPE effects for the Sivers and the Collins asymmetries in SIDIS are
power-suppressed.Comment: 4 pages, 1 figure, Presented at the 18th International Symposium on
  Spin Physics (SPIN 2008), Oct. 6-11, 2008, Charlottesville, VA, US

A. Metz

Donal B. Day

Donald G. Crabb

M. Schlegel

Matt Poelker

Simonetta Liuti

Xiaochao Zheng

Yelena Prok

English

arXiv

Crossref

Two-Photon Exchange in (Semi-)Inclusive DIS

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Two-Photon Exchange in (Semi-)Inclusive DIS
M. Schlegel∗ and A. Metz†
∗Theory Center, Jefferson Lab, Newport News, VA 23606, USA
†Department of Physics, Temple University, Philadelphia, PA 19122, USA
Abstract. In this note1 we consider effects of a Two-Photon Exchange (TPE) in inclusive DIS
and semi-inclusive DIS (SIDIS). In particular, transversesingle spin asymmetries are generated in
inclusive DIS if more than one photon is exchanged between thlepton and the hadron. We briefly
summarize the TPE in DIS in the parton model and extend our appro ch to SIDIS, where a new
leading twist sin(2φ) contribution to the longitudinal beam spin asymmetry showsup. Possible
TPE effects for the Sivers and the Collins asymmetries in SIDI are power-suppressed.
Keywords: Two Photon Exchange, Inclusive DIS, SIDIS
PACS: 13.60.Hb
TWO-PHOTON EXCHANGE IN INCLUSIVE DIS
It was already argued in the 60s that, due to time-reversal inv riance, single spin asym-
metries (SSAs) are forbidden in inclusive deep-inelastic lepton scattering off nucleons if
one considers only a One-Photon Exchange between the leptonand the nucleon (Christ-
Lee theorem [1]). Early experiments confirmed this statement and found transverse sin-
gle target spin asymmetries in DIS being consistent with zero at the percent level [2, 3].
Measurements of this observable have been repeated very recently at HERMES with
higher precision (of the order 10−3), and the effect is still compatible with zero [4]. An
ongoing Hall A experiment at Jefferson Lab even plans to measur the target SSA in
DIS with an accuracy of the order 10−4 [5].
If one considers DIS beyond the One-Photon Exchange approximation, transverse
SSAs may well exist [1, 6, 7]. TPE can generate a nontrivial phase in the amplitude
of the process — a necessary condition for a SSA — and thereforpr vide a nonzero
single spin correlation of the formεµνρσ SµPν lρ l′σ (cf. Fig. 1). The transverse SSAs
generated by TPE were calculated in the naive parton model inRef. [6] by assuming
that both photons couple to the same quark (Fig. 1). The observables were considered
in the Bjorken limit where the momentum transfer to the nucleon,q2 = (l− l′)2 ≡−Q2,
is large, and power corrections in 1/Q were neglected. In the following we will briefly
discuss this calculation.
Both the transverse target and lepton beam SSA receive theirleading contributions
in α — the electromagnetic fine structure constant — from an interfer nce of One-
Photon Exchange DIS amplitudes and Two-Photon Exchange amplitudes (Fig. 1). In the
1 Authored by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE-AC05-06OR23177.
The U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or
reproduce this manuscript for U.S. Government purposes.
l, S l
′
P, S
Ph
l, S l
′
P, S
FIGURE 1. Two-Photon Exchange in the parton model. Merely the coupling of the two photons to the
same quark is considered. Also the complex conjugate diagrams (not shown here) have to be taken into
account.Left Panel: TPE in inclusive DIS.Right Panel: TPE in semi-inclusive DIS.
parton model for DIS one considers the hard electromagneticinteraction of a quasi-free
parton with the lepton and treats the other partons as spectators (impulse approximation).
We assume that, for the particular observables we are interested in here, the dominant
contribution (in the sense of a twist expansion) of TPE is picked out if one works with
a single active parton. This approximation is represented in Fig. 1. The two-photon box
diagram is the only QED diagram in the parton model which carries an imaginary part.
Other radiative corrections of the same order inα such as bremsstrahlung effects are
purely real and hence cannot give rise to transverse SSAs.
The result for the transverse lepton beam SSA, calculated from the left graph in Fig. 1
(and its complex conjugate counterpart), reads [6]
ATU (xB,y,φs) = α
ml
2Q
|~ST |sin(φs)
y2
√
1− y
1− y+ 12y2
∑q e3q f1(xB)
∑q e2q f1(xB)
, (1)
wherexB = Q2/(2P ·q) andy = (P ·q)/(P · l) denote the common DIS variables,eq the
quark charge,|~ST | the transverse polarization vector of the lepton, andφs the angle of
this vector with respect to the lepton plane. The ordinary unpolarized parton distribution
of a quark flavorq is represented byf q1 . We point out that divergent terms, which appear
at intermediate steps of the calculation and can be regulated by a photon mass, cancel
in the final result (1). Since the asymmetry (1) is not only proportional toα ≃ 1/137
but also to the lepton massml one can expect rather small effects. In fact, by assuming
u-quark dominance in a proton target and an electron beam onereadily estimates this
asymmetry to be of the order 10−6. Asymmetries roughly two-hundred times larger can
be generated by a muon beam. One might also speculate about enhanc d results from
effects beyond the naive parton model. In Refs. [8, 9, 10, 11](double) logarithms of the
type log(Q2/m2l ) were advocated in connection with the transverse beam SSA inelastic
lepton-nucleon scattering. Such logarithmic terms might also increase the beam SSA in
DIS considered here. However, further work is required in order to decide whether and
how precisely such effects show up for the DIS case.
We now turn to the parton model result for the transverse targe SSA in DIS. The con-
tribution of the left diagram in Fig. 1 (and its complex conjugate counterpart) reads [6]
AUT (xB,y,φs) = α
xBM
2Q
y(1− y)√1− y
1− y+ 12y2
|~ST |sin(φs)
(
ln
Q2
λ 2
+finite
)∑q e3qg
q
T (xB)
∑q e2q f
q
1 (xB)
, (2)
whereM denotes the nucleon mass, andgT a higher twist parton distribution. Note
that a small photon massλ has been introduced in order to get a finite result. The
divergence forλ → 0 signals a violation of electromagnetic gauge invariance in the
parton model (for the particular observableAUT ). It was suggested that gauge invariance
might be restored by considering contributions beyond the naive parton model such
as multiparton correlations [6]. Since the SSA (2) is a twist-3 observable one indeed
expects contributions from quark-gluon correlations. Whether the inclusion of such
terms provides a finite result remains to be seen. Another candid te for the restoration of
electromagnetic gauge invariance are multiquark correlations, i.e. configurations where
the two photons couple to two different quarks.
Focusing on the finite contributions in (2) one expects the asymmetry to be at most of
the order 10−3 (for x ≃ 0.1). (A very rough estimate ofAUT , where one just considers
the suppression due toα, gives an effect of the order 10−2.) However, because of the
divergence, a reliable numerical result can hardly be extracted from Eq. (2). Additional
contributions beyond the naive parton model not only are supposed to cancel the diver-
gence but also to yield finite terms which have to be taken intoaccount as well. It is
also worthwhile to mention that a quark-mass effect toAUT — a particular contribution
beyond the parton model — was studied in detail in Ref. [7]. Such an effect is propor-
tional to the quark transversity distribution and leads, according to [7], to an asymmetry
of the order 10−4. In addition, we note that a diquark spectator model calculation for the
neutron also predictsAUT ≃ 10−4 [12].
TWO-PHOTON EXCHANGE IN SEMI-INCLUSIVE DIS
It is straightforward to extend the naive parton model calcul tion of Ref. [6] to SIDIS in
the kinematical regime where the transverse momentumPh⊥ of the produced hadron
is much smaller than the momentum transferred to the nucleon, Ph⊥ ≪ Q. For this
kinematics, observables can be described in the parton model in t rms of transverse
momentum dependent (TMD) parton distributions and fragmentation functions. Like in
inclusive DIS, TPE in general also generates SSAs in SIDIS. In the following we discuss
results for the SSAs calculated in the naive parton model. The contribution of the right
graph in Fig. 1 reads for a longitudinally polarized lepton beam and a produced pion,
Asin(2φ)LU = α
y
(
1+ 2−y1−y lny
)
1− y+ 12y2
sin(2φ)
∑q e3qC
[
2(~h·~kT )(~h·~pT )−~kT ·~pT
2Mmπ
h⊥q1 H
⊥q
1
]
∑q e2qC
[
f q1 D
q
1
] . (3)
The angleφ represents the angle between the lepton and the production plane (in
the conventions of Ref. [13]), whileD1 denotes the unpolarized TMD fragmentation
function,h⊥1 the Boer-Mulders function [14], andH
⊥
1 the Collins function [15]. We also
use the unit-vector~h ≡ −~Ph⊥/|~Ph⊥|. The symbolC [. . .] denotes a convolution in the
transverse momentum space (w being some kinematical prefactor),
C [w f D] =
∫
d2kT d
2pT δ (2)(~kT −~pT −~Ph⊥/zh)w(~kT ,~pT ) f q(xB,~k2T )Dq(zh,~p2T ). (4)
Both TMD functions in Eq. (3) are so-called (naive) time-reversal odd entities for
which initial/final state interactions play an important role. We assume that the interplay
between initial/final state interactions and TPE does not spil the QCD-factorization in
terms of TMD correlation functions.
Like Eq. (1), the SSA in (3) is free of divergences. Furthermore, it is a leading twist
sin(2φ) modulation of the beam spin asymmetry that is absent in a model-in pendent
decomposition of the SIDIS cross section into structure functio s assuming One-Photon
Exchange [16, 17, 13]. This means that the asymmetry (3) is generated by TPE only,
and possible sin(2φ) modulations from QCD radiative corrections in the One-Photon
Exchange approximation are ruled out. A similar effect has also been discussed in [18]
for TPE in electro-excitation of a∆ resonance. Since Eq. (3) is proportional toα
and since one expects the Boer-Mulders effect in unpolarized SIDIS from a cos(2φ)
modulation to be of the order of a few percent (see, e.g., the model calculation [19]) one
can roughly estimate the asymmetry in (3) to be of the order 10−4−10−3. At this point
one might again speculate about possible logarithmic enhancements as discussed above
for the transverse beam SSA in inclusive DIS.
We also calculated the target asymmetriesAUL andAUT , where we again encounter
divergences like in the case of the transverse target SSA in DIS. The comments on this
feature which we made above apply here, too. Interestingly,in the case of the target
SSAs, TPE provides only twist-3 contributions to azimuthalmodulations like sinφ for
AUL, as well as sinφs and sin(2φ − φs) for AUT . Such twist-3 modulations are already
present in the One-Photon Exchange approximation [13]. In particular, we find that for
AUT TPE corrections to the modulations sin(φ −φs) and sin(φ +φs) which are typically
analyzed in terms of the (leading twist) Sivers and Collins effect, respectively, appear at
most at the level of twist-4.
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