Single and double polarization observables in timelike Compton scattering off proton

1.1 MB·English

by  Marie Boër

Timelike Compton scattering off the proton and Generalized Parton Distributions.
M. Bo¨er1, M. Guidal1 and M. Vanderhaeghen2
1Institut de Physique Nucl´eaire d’Orsay, CNRS-IN2P3, Universit´e Paris-Sud, F-91406 Orsay, France
2Institut fu¨r Kernphysik and PRISMA Cluster of Excellence,
Johannes Gutenberg Universit¨at, D-55099 Mainz, Germany.
Westudytheexclusivephotoproductionofaleptonpairofftheprotonwiththeaimofstudyingthe
proton quark structure via the Generalized Parton Distributions (GPD) formalism. After deriving
theamplitudesoftheprocessescontributingtotheγP →P(cid:48)e+e−,theTimelikeComptonScattering
and the Bethe-Heitler process, we calculate all unpolarized, single- and double- beam-target spin
observables in the valence region in terms of GPDs.
PACSnumbers:
5
1 I. INTRODUCTION details of the formalism and of their properties.
0
2
Thescatteringoflightonmatter,whichcangenerically

n becalledComptonscattering,isapowerfultooltoinves-
a e- (k) e- (k')
tigate its inner structure. Nowadays, understanding the
J
structure of hadrons in terms of quark and gluon (par-
1
tons) degrees of freedom, i.e. the basic constituents of g * (q) g (q')

] matterknowntothisday, isthesubjectofanintensere-
h search effort. Only these past fifteen years or so, thanks hard
p to the emergence of high intensity, high energy (multi- soft x+x x-x
-
p GeV) and high duty-cycle lepton accelerators, Compton
e scattering at the partonic level starts to be investigated
h experimentally in an efficient way. P (p) P' (p')
[
A particular case of Compton scattering at the par-

1 tonic level is the Deeply Virtual Compton Scattering
v (DVCS) process on the proton P, i.e. γ∗P →γP(cid:48) where t
0 the initial virtual photon γ∗ is radiated from an incom-
7
ing lepton beam (see Fig. 1-top). It is of particular in-
2
0 terest as the amplitude of the process allows to access e- (k)
0 some essential operators of Quantum Chromo-Dynamics
. (QCD,thefundamentaltheorygoverningtheinteractions e+ (k')
1
0 of quarks and gluons). Indeed, at sufficiently large vir- g (q) g * (q')
tuality of the initial photon (Q2 =(k−k(cid:48))2), the DVCS
5
hard
1 amplitude can be factorized into an elementary “hard”
: (perturbative) scattering process γ∗q →γq (where q is a soft x+x x-x
v
quark of the proton), exactly calculable from Quantum
i
X Electro-Dynamics as well as perturbative QCD, and a
P (p) P' (p')
r “soft” (non-perturbative) QCD bilocal matrix elements.
a
The Fourier transforms into momentum space of these
QCD matrix elements are the so-called Generalized Par-
t
ton Distributions (GPDs). In DVCS on the nucleon,
at QCD leading-twist order there are four quark helic-
ity conserving GPDs (H, E, H˜ and E˜) which can be
Figure 1: Top: QCD leading-twist DVCS diagram; Bottom:
accessed and which correspond to the four independent
QCD leading-twist TCS diagram (the crossed diagrams of
helicity-spintransitionsbetweentheinitialquark-proton both processes are not shown). The dashed line illustrates
system and the final one. The GPDs contain a wealth the factorization between
of information on the partonic structure of the proton:
the longitudinal momentum and transverse space distri-
butionsofthequarksandgluons,thecorrelationbetween DVCSiscurrentlywidelyinvestigated, experimentally
these momentum and space distributions, sensitivity to aswellastheoretically. Ref.[4]compilesallexistingdata
the quark-antiquark content in the proton, the quark or- in the valence region and presents some first informa-
bital momentum contribution to the proton spin, etc... tions on the partonic structure of the proton that can
Wereferthe readerto thereviews[1–4] onGPDs forthe beextracted, withinsomeapproximations, fromthefirst
2
DVCS data through the GPD formalism. One example we show the impact of next-to-leading-twist corrections
of a pioneering result is the first quantitative evaluation on the cross sections and on the asymmetries. We will
of the increase of the transverse size of the proton as conclude in the eighth section.
smaller longitudinal momentum fractions of partons are
probed.
However, the GPD information is difficult to extract: II. FORMALISM
DVCS cross sections are small (of the order of pb), there
isthecompetingBethe-Heitler(BH)processwhichleads We are studying the process:
tothesamefinalstateePγ butwherethefinalphotonis
emittedfromtheincomingorscatteredleptonandwhich γ(q)P(p)→ P(cid:48)(p(cid:48))e−(k)e+(k(cid:48)) (1)
is therefore not related to GPDs but nevertheless inter-
feres. Furthermore, there is the need of measuring a se- in a GPD framework, i.e. when the final photons virtu-
riesofspin(beamand/ortarget)observablestoconstrain ality Q(cid:48)2 = (k+k(cid:48))2 is sufficiently large and the proton
the different GPDs, etc... It would therefore be useful to momentum transfer t = (p−p(cid:48))2 is sufficiently small so
investigateifsupplementaryand/orcomplementarycon- thatthefactorizationillustratedinFig.1-bottomcanbe
straintsonGPDscouldbeobtainedfromprocessesother applied. From DVCS, Deep Inelastic Scattering (DIS)
than DVCS. In this spirit, we investigate in this arti- and Drell-Yan analysis and experiments, it is believed
cle the related process of Timelike Compton Scattering that Q(cid:48)2 > 2 GeV2 and −t < 1 GeV2 (or −t < 30%)
Q(cid:48)2
(TCS) which corresponds to the exclusive photoproduc- should define such a reasonable phase space. Regarding
tion of a lepton pair on the proton: γP →P(cid:48)γ∗ (cid:44)→e+e− Q(cid:48)2,oneshouldalsoavoidregionswhereonecanhavethe
and which is displayedin Fig. 1-bottom. Like for DVCS, production of vector mesons, decaying into e+e− pairs
at sufficiently large virtuality of the final virtual photon (for instance, the broad ρ(cid:48)(1700)). Finally, one should
(Q(cid:48)2 =(k+k(cid:48))2), it is predicted that the process factor- considerthesquaredcenter-of-massenergyoftheincom-
izes and is sensitive to GPDs, the same ones accessed in ing photon and target proton s=(q+p)2 (cid:38) 4 GeV2, in
DVCS. order to minimize possible contributions from the Dalitz
The TCS process was originally investigated in terms decay of proton resonances.
of GPDs about ten years ago in Ref. [5]. In this pio-
neeringwork, analyticalformulasintermsofGPDswere
derived for the unpolarized and the circularly polarized A. Kinematics
beamcrosssectionsoftheprocessγP →P(cid:48)e+e−, i.e. on
aprotontarget. Veryrecently,asecondarticlecontinued
We will use the notation of Ji [7] for GPDs, i.e. GPDs
theinvestigationbystudyingthelinearlypolarizedbeam
dependonthethreevariablesx,ξ andtwherethequark
cross section [6].
longitudinal momentum fractions x and ξ are defined
However, in order to obtain simple analytical expres-
w.r.t. the average proton momentum P and proton mo-
sions, a few approximations were used in the calculation
mentum transfer ∆, respectively. We therefore define:
oftheTCSamplitude(forinstancemasscorrectionterms
oftheorderofm2/Q(cid:48)2 wheremisthemassoftheproton 1
P = (p+p(cid:48)), (2)
were neglected). In the present work, we waive some of
2
these approximations and present calculations of differ- ∆=(p(cid:48)−p)=(q−q(cid:48)), (3)
ent observables. In addition, besides unpolarized cross
sections,westudyallsingleanddoublebeam-targetspin
and we also introduce the average photon momentum
observables. We focus in this article on a proton target.
This article is organized as follows: in the next sec- 1
q¯= (q+q(cid:48)). (4)
tion, we present the general theoretical formalism of the 2
TCS process, in particular the expression of the QCD
GPDs are matrix elements of QCD operators which are
leading-twistamplitudeintermsofGPDs,andoftheac-
defined at equal light-cone time. It is therefore conve-
companying BH process. We discuss some experimental
nient to use a frame where the q¯ and P momenta are
considerations in the third section. In the fourth sec-
collinear along the z-axis and in opposite directions. We
tion,wepresentournumericalresultsfortheunpolarized
cross section of the γP → P(cid:48)e+e− process and we com- define the lightlike vectors along the positive and nega-
tive z directions as:
pare them to the previous work of Ref. [5]. In the fifth
section, wepresentourresultsforsinglespinobservables √
p˜µ =P+/ 2(1,0,0,1), (5)
(beamandtarget)andwecompareourworkforthebeam
√
polarization observables to the results of Ref. [5]. In the nµ =1/P+·1/ 2(1,0,0,−1), (6)
sixth section, we present our results for the double-spin
beam-targetobservables. Foreachcase,wewillshowthe and we define the light-cone components a± by a± ≡
√
dependence of the observables on different GPDs and its (a0±a3)/ 2. Wehavep˜2 =n2 =0andp˜·n=1. Inthis
sensitivityondifferentkinematics. Intheseventhsection light-cone frame, introducing the + components of the
3
∆ and q¯ four-vectors, ξ˜and ξ˜(cid:48) respectively, the various
four-vectors involved can be decomposed as:
D
m¯2 q + D q - 2
Pµ =p˜µ+ 2 qn¯2µ, (7) (-~x ' -2 ~x ) (qx +- ~xk' ) (-~x ' + ~x )
q¯µ =−ξ˜(cid:48)p˜µ− nµ, (8)
2ξ˜(cid:48)
D D
∆µ =−2ξ˜p˜µ+ξ˜m¯2nµ+∆µ, (9) k - k +
⊥ 2 2
~ ~
with m¯2 = m2 − ∆2 and m is the proton mass. We (x + x ) (x - x )
4
relate the final photon virtuality to the average photon
momentum:
q¯2 =Q(cid:48)2/2−∆2/4. (10)
We have to relate the light-cone momentum fractions D D
ξ˜and ξ˜(cid:48) to the kinematical variables which are experi- q + 2 q - 2
~ ~ ~ ~
mentallyaccessible. Todoso, weintroducethevariables (-x ' - x ) (-x ' + x )
ξ and ξ(cid:48):
q¯2 −Q(cid:48)2+∆2/2 k - q
ξ(cid:48) =− = , (11) D ~ D
2P.q¯ 2(s−m2)+∆2−Q(cid:48)2 k - (x + x ') k +
2 ~2
~
ξ =−∆.q¯ = Q(cid:48)2 . (12) (x + x ) (x - x )
2P.q¯ 2(s−m2)+∆2−Q(cid:48)2
The light-cone momentum fractions ξ˜ and ξ˜(cid:48) are re-
lated to the kinematical variables ξ and ξ(cid:48) by:
Figure 2: Leading order diagrams for the process γq → γq.
1+ξ˜(cid:48)2m¯2/q¯2 We indicate between parenthesis the + components of the
ξ˜=ξ· , (13) four-vectors.
1−ξ˜(cid:48)2m¯2/q¯2
2
ξ˜(cid:48) =ξ(cid:48)· . (14)
(cid:112)
1+ 1−4ξ(cid:48)2m¯2/q¯2 with, in the Bjorken limit where ξ =−ξ˜(cid:48),
In the asymptotic limit, where mass and ∆ terms are
neglected relatively to Q(cid:48)2, we have: HµTνCS (18)
1
Q(cid:48)2 1 (cid:90) (cid:18) 1 1 (cid:19)
ξ˜=ξ =−ξ˜(cid:48) =−ξ(cid:48) = . (15) = (−g ) dx +
2s−Q(cid:48)2 2 µν ⊥ x−ξ−i(cid:15) x+ξ+i(cid:15)
−1
(cid:18) (cid:19)
∆
. H(x,ξ,t)u¯(p(cid:48))n/u(p)+E(x,ξ,t)u¯(p(cid:48))iσαβn β u(p)
B. Timelike Compton amplitude α2m
1
ThetwodiagramsofFig.2havetobecalculated. The i (cid:90) (cid:18) 1 1 (cid:19)
− ((cid:15) ) dx −
leading order amplitude reads: 2 νµ ⊥ x−ξ−i(cid:15) x+ξ+i(cid:15)
−1
1
TTCS ∝ γν (k/+/q¯)γµ (cid:18) ∆.n (cid:19)
H (k+q¯)2+i(cid:15) . H˜(x,ξ,t)u¯(p(cid:48))n/γ u(p)+E˜(x,ξ,t)u¯(p(cid:48))γ u(p) ,
5 5 2m
1
+ γµ(k/−/q¯)γν
(k−q¯)2+i(cid:15) where we used the metric:
(cid:20)
1 1 1
∝ n/(p˜µnν +p˜νnµ−gµν).( + ) (−gµν)⊥ =−gµν +p˜µnν +p˜νnµ , (19)
2 x−ξ˜(cid:48)+i(cid:15) x+ξ˜(cid:48)−i(cid:15)
((cid:15) ) =(cid:15) nαp˜β.
(cid:21) νµ ⊥ νµαβ
1 1
−n/γ i(cid:15)µνκλn p˜ ( − ) . (16)
5 λ κ x−ξ˜(cid:48)+i(cid:15) x+ξ˜(cid:48)−i(cid:15) The GPDs entering Eq. 18 are proton GPDs, i.e. they
read, in terms of quark flavors:
ThefullTCSamplitude,correspondingtothediagram
of Fig. 1-bottom (plus the associated crossed diagram) H (x,ξ,t) = 4Hu/p + 1Hd/p + 1Hs/p. (20)
reads then TCS 9 9 9
e3 In this work, we will take the GPD parametrizations
TTCS =− u¯(k)γν v(k(cid:48))(cid:15)µ(q)HTCS, (17)
q(cid:48)2 µν from the VGG model [1, 8–10], which are summa-
4
rized in Ref. [4] and based on the Radyushkin double-
distribution ansatz for the (x,ξ)-dependence [11–13] and e-(k)
on a Reggeized ansatz for the t-distribution [1, 10]. At g (q)
a couple of instances, in order to estimate the model de- e+(k')
pendence of our calculations, we will use the factorized
ansatz for the t-dependence of the H GPD [8]. Also, we
will occasionally study the sensitivity of observables to g * (q')
the so-called D-term [14], which is included in the VGG
model and whose parametrization can be found as well
in Refs. [1, 8–10]. P (p) P' (p')
C. Gauge invariance
The TCS amplitude is not exactly gauge invariant,
only in the Bjorken limit. To restore gauge invariance,
twist-3 corrections of order ∆⊥/Q are needed. As a first g (q) e+(k')
step in adressing this issue and estimating its effect, we
e-(k)
propose to add a correction term to the twist-2 vector
part tensor as follows:
Hµν =Hµν − Pµ ·(∆ ) ·Hκν (21) g * (q')
LO 2P ·q¯ ⊥ κ LO
Pν
+2P ·q¯·(∆⊥)λ·HLµOλ P (p) P' (p')
PµPν
− ·(∆ ) ·(∆ ) ·Hκλ ,
4(P ·q¯)2 ⊥ κ ⊥ λ LO
where Hµν stands for the tensor of Eq. 18. This is a
LO
generalizationoftheprescriptionproposedinRefs.[9,15]
for DVCS (we also refer the reader to Refs. [16–18] for Figure 3: The Bethe-Heitler diagrams.
further discussions on the issue of gauge invariance in
the DVCS amplitude). One can readily check that Hµν
respects gauge invariance both w.r.t. initial and final
photons, i.e. q Hµν =0, and q(cid:48)Hµν =0. The impact of
µ ν
adding this correction to the observables is shown at the can be considered to be known with good accuracy.
end of this paper, in the section VII. In this work, we take the parametrizations issued from
Refs. [19, 20].
D. The Bethe-Heitler amplitude
The TCS process is accompanied by the BH process,
involvingthetwodiagramswhicharepresentedinFig.3.
E. Cross section
Their amplitude reads:
e3
TBH =− u¯(p(cid:48))Γν u(p)(cid:15)µ(q) (22) At fixed beam energy Eγ or longitudinal momentum
∆2
transfer ξ, there are four independent kinematical vari-
(cid:32) (cid:33)
k/−/q /q−k/(cid:48) ables for the process γ(q)P(p) → P(cid:48)(p(cid:48))e−(k)e+(k(cid:48)). A
u¯(k) γ γ +γ γ v(k(cid:48)),(23)
µ(k−q)2 ν ν(q−k(cid:48))2 µ naturalchoiceistotake: Q(cid:48)2 andt=∆2 thatwealready
defined,andthetwoanglesθ andφoftheelectroninthe
with the virtual photon-proton electromagnetic vertex γ∗ center-of-mass (with the z(cid:48)-axis along the direction
matrix of the γ∗ in the γ∗ −P(cid:48) center of mass). We illustrate
in Fig. 4 the different variables involved. In addition, we
Γν =γν F (t)+ iσνρ∆ρ F (t). (24) displaythepolarizationangleΨbetweenthepolarization
1 2m 2 vector of the incoming photon and the scattering plane
in the γ∗−p C.M..
The BH amplitude depends on the proton Dirac and
Pauli form factors F (t) and F (t) which, at small t The 4-differential unpolarized cross section is then ex-
1 2
5
y

III. EXPERIMENTAL CONSIDERATIONS
x

z
  TheonlydatafortheprocessγP →P(cid:48)e−e+ whichare
e-­‐
 (k)
  ⤴Φ
in the phase space of concern for our study, have been
⤴Ψ
collected and analyzed a few years ago by the CLAS col-
γ
 (q)
  laboration using the ≈ 6 GeV electron beam of Jeffer-
γ*
 (q’)
  son Lab. The data were actually obtained in “quasi-
photoproduction” mode. This means that the scattered
N’
 (p’)
  e+
 (k’)

t
  electron from the beam is not detected in CLAS and
γ
 N

 CM
  N
 (p)
  is considered to be in almost the same direction as the
beamline. This results in very low Q2 electroproduc-
tion, i.e. “quasi-photoproduction”. This pilot analysis
can be found in Ref. [21]. The ≈ 6 GeV beam energy,
combinedwiththe≈1034cm−2s−1 luminosity,allowedto
reachmaximumQ(cid:48)2 valuesof3GeV2,whichcorresponds
to an invariant mass of the e+e− system M of ≈ 1.8
e+e−
GeV. This is close to the mass of several vector mesons
decaying into e+e−, in particular the broad ρ(cid:48)(1700). In
ordertohaveaTCSinterpretationascleanaspossible,it
isofcourseadvisabletoavoidsuchresonanceswhichcon-
tribute to the process γP →P(cid:48)e−e+. The data analysis
of Ref. [21] is therefore difficult to interpret in terms of
GPDs but it nevertheless demonstrates that the process
γP →P(cid:48)e−e+ can be measured at JLab.
⤴Φ
The forthcoming JLab energy upgrade to 11 GeV al-
e-­‐
 (k)

lowstoexploreaQ(cid:48)2 regionbetween4and9GeV2 which
x
  correspondstoaM regionbetween2and3GeV,i.e.
y
  N’
 (p’)
  θ
  a vector meson resoen+aen−ce-free region between the ρ(cid:48) and
z'
  the J/Ψ. In the case of CLAS12, the upgraded CLAS
detector associated to the JLab energy increase, there is
e-­‐
 e+
 CM
  e+
 (k’)
  also a luminosity gain of a factor 10. These upgrades
have led to the first dedicated TCS accepted proposal at
JLAB [22]. Itwilluseasimilar“quasi-photoproduction”
technique as used in the pioneer 6 GeV analysis. In ad-
dition, there will be the improvement of the detection
Figure4: Toppanel: schemeofthereactionplaneγP →γ∗P(cid:48) of the low Q2 scattered electron via a dedicated tagging
(forTCS)intheγP CMframeandillustrationofsomeofthe equipement, supplementing CLAS12. This allows, be-
kinematicvariables. Redarrowsshowthepolarizationvectors sides the measurement of unpolarized cross sections, to
of the photon beam and proton target. obtainlinearlypolarizedphotonsobservables,bymeasur-
Bottom panel: decay angles in the e−e+ CM frame. φ and θ ing the azimuthal dependence of the scattered electron.
are respectively the azimuthal and polar angles between the
Finally, with a polarized electron beam, which is avail-
e− direction and the virtual photon direction in the γ∗−P(cid:48)
ableatJLab,onecanaccesscircularlypolarizedphotons
CM frame.
observables. Itisthereforeexpectedthatinthenextfew
years, numerous γP → P(cid:48)e−e+ data which can lend to
GPD interpretation will be available.
pressed as: We show in Fig. 5 the kinematical domain which can
be accessed with the upgraded JLab. We display in blue
d4σ the (ξ, Q(cid:48)2) phase space accessible for TCS with an 11
(γP →P(cid:48)e+e−) (25)
dQ(cid:48)2dtdΩ GeV electron beam, assuming that the real photon is
1 1 1 provided by bremsstrahlung of the electron and that its
= |TBH +TTCS |2, energy is in E ∈[5,11] GeV. We have applied two cuts:
(2π)464(2mE )2 γ
γ Q(cid:48)2 ∈[4,9] GeV2 and −t∈[0,1] GeV2. The motivations
where|TBH+TTCS |2 isaveragedoverthetargetproton arerespectivelytostayintheregionfreeofvectormesons
resonances and minimize higher twist corrections to the
helicities and beam polarizations and summed over the
TCS formalism, which grow with t . We overlap in red
final proton helicities. Q(cid:48)2
inthissamefigurethe(ξ,Q2)phasespaceaccessiblewith
an 11 GeV beam for DVCS. We have applied the cuts:
−t ∈ [0,1] GeV2, for the same reason as for TCS, and
6
s>4 GeV2 in order to stay above the baryon resonance the intermediate θ values. For instance, at θ=90◦, the
region. calculation shows only two small “bumps” at φ=0◦ and
One notes the large intersection between the DVCS φ=180◦.
andtheTCSphasespaces. Measurementsofobservables TheseparticularshapesareduetotheBHprocessand
sensitive to GPDs in the common (ξ,Q2) region by both its singularities. Indeed, in the BH diagrams of Fig. 3,
processes should bring strong constraints on the extrac- when the electron (positron) is emitted in the direction
tion of GPDs and tests of factorization and universality. of the initial photon, i.e. θ=0◦ (θ=180◦), the propa-
gator of the positron (electron) becomes singular and
creates a peak in the φ distribution at φ=180◦ (φ=0◦).
Intuitively, θ=0◦ (θ=180◦) forces all particles to be in
the same plane, i.e. φ=180◦ (φ=0◦). The kinematics
θ=0◦, i.e. the electron is in the direction of the photon
beam,correspondstoφ=180◦ becausethevirtualphoton
is emitted by the positron, not the electron (see Fig. 6).
WedisplayalsoinFig.6thecontributionofTCSalone.
In this calculation, we have used only the GPD H. The
inclusion of the other GPDs barely changes the curves.
In contrast to the BH, the TCS is almost flat in φ for
all θ values. It is clear that the process γP → P(cid:48)e+e−
is largely dominated by the BH. There is never less than
an order of magnitude between BH and TCS.
In Fig. 6, we show the curves BH+TCS for a series of
θ angles as well as the BH alone for θ=45◦ and θ=90◦.
Only at θ=90◦, where one is far from the two BH sin-
gularities, we have a visible difference between the two
curves and therefore a sensitivity to TCS. It is of the or-
Figure5: Kinematicaldomainaccessibleasafunctionofξand derof30%atφ=180◦. Asonegetsclosertooneofthetwo
Q2 for DVCS (red plain surface) and in ξ and Q(cid:48)2 for TCS BHsingularities(θ=45◦ forinstance),thetwocurvesBH
(blue dotted surface) with an 11 GeV electron beam. For andBH+TCSareessentiallyindistinguishableandthere
DVCS, the cuts −t∈[0,1] GeV2 and s>4 GeV2 have been is no sensitivity to TCS.
applied and for TCS, the cuts Eγ ∈ [5,11] GeV, −t ∈ [0,1] Finally,weshowinFig.6,ourcalculationsofBH+TCS
GeV2 and Q(cid:48)2 ∈[4,9] GeV2 have been applied.
(and of BH alone) for θ integrated over the range
[π/4,3π/4]. In order to maximize count rates, from an
experimental point of view, it is interesting to integrate
We now present our results for the calculations of the over θ. We still have a sensitivity to TCS, however it is
unpolarized cross sections, single spin asymmetries and of the order of 5%, i.e. lesser than at fixed θ=90◦: the
double-spinasymmetries, respectivelyinsections IV, V integration over θ dilutes the sensitivity to TCS.
and VI. In these sections, all calculations will be done We display in the top panel of Fig. 7 the Q(cid:48)2-
in the Bjorken limit and at leading order (we refer the dependence,atξ =0.2and−t=0.4GeV2,ofdσ/dQ(cid:48)2dt
readertoRefs.[23–25]forworksonNext-To-Leadingor-
for the BH and TCS processes. The 2-fold cross section
der corrections in α to the TCS amplitude). In section
s has been integrated over the decay angles: φ ∈ [0,2π]
VII,wewillstudytheeffectofkeepingtheexactkinemat- and θ ∈[π,3π]. We also display in this figure the results
ics presented in section IIA and of the gauge-invariance 4 4
of the analytical formulaes of Ref. [5]. For TCS, we have
restoration prescription described in section IIC.
of course used the same GPD parameterization for both
calculations (only H in this case).
In order to better appreciate the comparison, we dis-
IV. UNPOLARIZED CROSS SECTION
playinthebottompanelofFig.7theratiosofourcalcu-
lationstothoseofRef.[5]. Theagreementisexcellentfor
Wediscussinthissectiontheunpolarizedcrosssection the BH process, as it should, since there was no approx-
of the γP → P(cid:48)e+e− process, which therefore includes imation done in the derivation of the analytical formula
the BH and TCS processes. Fig. 6 shows our calculation forthisprocessinRef.[5]. ForTCS,weshowtwocurves
of the φ-dependence of the 4-fold differential cross sec- fortheratio. Indeed,inRef.[5],intheanalyticalformula
tion dσBH at ξ =0.2, −t=0.4 GeV2, Q(cid:48)2 =7 for the TCS process, the nucleon mass was neglected in
dQ(cid:48)2dtdφd(cosθ)
GeV2 and for different θ values. The φ-shape is strongly the phase space factor, i.e. the cross section is propor-
dependent on the θ value. As θ tends to 0◦, the φ dis- tional to 1 rather than 1 (this is not the case for
s2 s−m22
tribution peaks towards φ=180◦ and as θ tends to 180◦, the BH process where the phase space factor is exact
the φ-distribution peaks towards φ=0◦ (or 360◦). There in Ref. [5]). In the bottom panel of Fig. 7, we plot the
isasmoothtransitionbetweenthesetwosingularitiesfor TCScrosssectionofRef.[5]with(bluedottedcurve)and
7
the TCS analytical formula of Ref. [5].
4)
-V
Ge 10-2
b.
n
dt ( BH: this work
2Q’ 10-3 BH: Berger et al
d
σ/ TCS: this work
d
TCS: Berger et al
10-4
4 5 6 7 8 9
4) 10-2 Q’2 (GeV2)
-
V
e
G
.
b
n
( 10-3
Ω
d al
dt er et 1.02
2 g 1
Q’ er
σ / d 10-4 work / B 00..9968 BTCHS
d s 0.94
hi TCS (+mass)
t
0.92
10-5
0.9
0 50 100 150 200 250 300 350
4 5 6 7 8 9
φ (deg.) Q’2 (GeV2)
Figure 6: Unpolarized cross section dσ for the
dQ(cid:48)2dtdφd(cosθ)
process γP → P(cid:48)e+e− (for BH + TCS and TCS alone) as Figure 7: Top panel: the two-fold differential cross section
a function of φ, at ξ=0.2, −t = 0.4 GeV2, Q(cid:48)2 = 7 GeV2 dσ/dQ(cid:48)2dtforBH(highestcurves)andTCS(lowestcurves)as
for different fixed θ values: 10◦,45◦, 90◦, 135◦, 170◦ and for a function of Q(cid:48)2 for ξ =0.2 and −t=0.4 GeV2. The calcu-
θ integrated over [π/4,3π/4]. For two values of θ (45◦ and lations have been integrated over φ∈[0,2π] and θ ∈[π,3π].
4 4
90◦) and for θ integrated over [π/4,3π/4], we also show the Thesolidcurvesareourcalculationswhilethedashedcurves
calculation of the BH alone. are from Ref. [5]. Bottom panel: ratio of our calculations to
those of Ref. [5]. For the TCS process, the blue dotted curve
contains the nucleon mass term in the phase space factor in
thecalculationofRef.[5]whilethereddashedcurvedoesnot.
without(reddashedcurve)thenucleonmassterminthe
phase space factor. In this way, one can distinguish the
differencesbetweenthetwocalculationsoriginatingfrom To end this section concerning the unpolarized cross
the trivial phase space factor from those, more subtle, section, we show in Fig. 8 the influence of the D-term
coming from the TCS amplitude. It is seen in this figure on the three-fold differential cross section dσ/dQ(cid:48)2dtdφ,
thatatthelowestQ(cid:48)2 valuesandatthepresentlyconsid- calculated for ξ = 0.2, −t = 0.4 GeV2 and with θ in-
eredkinematics, thedifferenceinthecrosssectioncalcu- tegrated over [π,3π]. It modifies the amplitude of the
4 4
lationsdependingontheprescriptiontakenforthephase cross section at φ=0◦ and φ=180◦ by about 10%.
space factor can reach more than 10%. In both cases, it
isalsoseenthatthedifferencebetweentheBergeretal.’s
calculations and ours diminishes as Q(cid:48)2 increases, as ex-
pected since terms of the order of m2 were dropped in
Q(cid:48)2
8
dependence of A for ξ=0.2, −t = 0.2 GeV2, Q(cid:48)2 = 7
(cid:96)U
GeV2 for θ = 45◦, θ = 90◦ and θ integrated over
[π/4,3π/4] and φ = 0◦ and φ = 10◦. The approximate
shapeoftheasymmetryisacos(2Ψ)whichisreminiscent
-4)V 1.6 of the modulation which is predicted for the so-called
Ge 1.5 H Σ asymmetry in single meson photoproduction. This
b. 1.4 cos(2Ψ) modulation appears explicitely in the analytical
p H+D-term
φ ( 1.3 expressions of Ref. [6].
d
dt 1.2 We note that the BH alone produces an asymmetry.
2dQ’ 1.1 It is actually the dominant contribution. The TCS pro-
σ/ ducesonlyvariationsoftheamplitudeatthepercentlevel
d 1
around the BH, making this observable not very favor-
0.9
abletostudyTCSandGPDs. Theamplitudeandphase
0.8 of the asymmetry depend strongly on the decay angles:
0 50 100 150 200 250 300 350 it is the strongest as θ approaches 90◦ and the phase in-
φ (deg.) creases as φ increases. This phase shift due to φ is also
apparent in Ref. [6].
In Fig. 9, we have used only the contribution of the H
GPD for TCS. In Fig. 10, we show the t-dependence of
Figure 8: The three-fold differential cross section
the A asymmetry for BH alone, BH+TCS (with only
dσ/dQ(cid:48)2dtdφ as a function of φ with (red dashed curve) and (cid:96)U
without(blacksolidcurve)theD-term,calculatedatξ=0.2, H) and BH+TCS (with H+H˜). Calculations have been
−t=0.4 GeV2 and with θ integrated over [π,3π]. done for ξ=0.2, Q(cid:48)2 = 7 GeV2, φ = 0◦, Ψ = 180◦ and
4 4
for θ = 45◦, θ = 90◦ and θ integrated over [π/4,3π/4].
Depending on t, we notice some sensitivity of the A
(cid:96)U
observable to the GPDs H and H˜.
V. SINGLE SPIN ASYMMETRIES
In Fig. 11, we show the peculiar φ-dependence of A
(cid:96)U
We now turn to the single spin asymmetries: beam a(stmΨal=l)0in◦flautenoucer sotfatnhdeaDrd-tkeirn) (AΨmLUem.00..46 aBBtHHi+,c θT=sC9.S0,o W,θ φ==90e0oo, aφ=l0soo show the
or target. Photons beams can be polarized linearly or
0.2 BH, θ=45o, φ=0o
circularly. Target polarization vectors can be oriented 0
along the x−, y− or z−axis in the γP → γ∗P(cid:48) plane -0.2 BH+TCS, θ=45o, φ=0o
(lasteee sFpigin. 4a)s.ymFomreptroileasriaznatdi,onfoollbosweirnvgabnloest,atwioenwsiullsecdalcfour- ) (AΨLU 00..46 BBHH+, θT=C9S0,o ,θ φ==900oo, φ=0o --00..64 BBHH+, 4T5CoS<θ, <4153o<5θo<, φ1=305oo, φ=0o
Dj.VTChSe,fiwrestwiinlldetaxgitrheefemrsAtoij,thi.ee.pwoliatrhiztawtioonintdyicpeesoifatnhde 0.02 BBHH,+ θT=C4S5,o ,θ φ==405oo, φ=0o 0BH50, θ=10900o1,5 0φ=21000o250 30Ψ0 (de35g0.)
-0.2
beam: U for an unpolarized beam, (cid:12) for a circularly po- )Ψ 0.6-0.4 BH, 45o<θ<135o, φ=0o
larized beam and (cid:96) for a linearly polarized beam. The (U -0.6 BH+TCS, 45o<θ<135o, φ=0o
L 0 50 100 150 200 250 300 350
second index j refers to the polarization of the target A 0.4 BH, θ=90o, φ=10o Ψ (deg.)
andcantakethevaluesU, x, y orz, withobviousmean-
0.2
ings. In this section, dedicated to single-polarization ob-
servables, we will therefore consider successively the five 0
independentasymmetriesA ,A ,A ,A andA .
(cid:96)U (cid:12)U Ux Uy Uz
-0.2
-0.4
A. Linearly polarized photons
-0.6
WeintroducetheangleΨbetweenthepolarizationvec-
0 50 100 150 200 250 300 350
tor of the photon and the plane spanned by the photon
Ψ (deg.)
beam and the (e+e−) system, which contains the x-axis
(see Fig. 4). Then, we define:
Figure 9: A asymmetry as a function of Ψ for BH and for
(cid:96)U
A (Ψ)= σx(Ψ)−σy(Ψ), (26) BH+TCS (only GPD H contribution) at ξ=0.2, −t = 0.4
(cid:96)U σ (Ψ)+σ (Ψ) GeV2, Q(cid:48)2 =7 GeV2 and for different sets of θ and φ angles.
x y
where σ (σ ) stands for the 4-fold differential cross sec-
x y
tions dσ with a photon beam polarized
dQ(cid:48)2dtdφd(cosθ)dΨ
in the x-(y-)direction. We display in Fig. 9 the Ψ-
9
ALU 0 θ=90o : BH
-0.1 BH+TCS, H
-0.2 BH+TCS, H+H~ ALU 0.4
-0.3 θ=45o : BH 0.3
-0.4 BH+TCS, H 0.2
-0.5 ~
-0.6 BH+TCS, H+H 0.1
-0.7 45o<θ<135o : BH 0
0.2BBHH0++.3TTCCSS0.4,, HH+0.H5~ 0.6 |t| 0(.(7GeV2) -0.1
-0.2
BH
-0.3
H
-0.4
H+Dterm
0 50 100 150 200 250 300 350
LU 0 φ (deg.)
A
-0.1
-0.2
-0.3 Figure 11: A(cid:96)U asymmetry as a function of φ for ξ = 0.2,
Q(cid:48)2 =7GeV2,−t=0.4GeV2 andθ∈[π/4,3π/4]. Theblack
-0.4
solid curve corresponds to BH only, the red dashed curve to
-0.5 BH+TCS with only the GPD H contribution and the blue
-0.6 dotted curve to BH+TCS with the GPD H and the D-term
contributions.
-0.7
0.2 0.3 0.4 0.5 0.6 0.7
|t| ((GeV2)
sees that the integration over θ does not strongly reduce
the signal. Such integration allows to maximize count
rates.
Figure10: A asymmetryasafunctionoftforξ=0.2,Q(cid:48)2 = In Fig. 12 (top panel), we also compare our results
(cid:96)U
7 GeV2, Ψ = 0◦, φ = 0◦ and θ = 45◦, θ = 90◦ and θ ∈ to those of Ref. [5]. In all cases, our calculations pro-
[π/4,3π/4]. duce amplitudes a few percents larger. This might be
attributed to some mass correction terms to the TCS
amplitude which are present in our calculation and not
in Ref. [5].
B. Circularly polarized photons Fig. 12 (bottom panel) shows the A asymmetry
(cid:12)U
for θ integrated over [45◦,135◦] using different GPDs
We define: parametrizations for TCS.
InFig.13,weshowforξ =0.2,Q(cid:48)2 =7GeV2,φ=90◦
σ+−σ−
A = , (27) and θ integrated over [45◦,135◦], the t-dependence of
(cid:12)U σ++σ−
A and its sensitivity to different GPDs. We notice
(cid:12)U
where σ± stand for the 4-fold differential cross sections that the magnitude of A(cid:12)U increases with |t| and that
dσ for the two photon spin states, right and there is a sensitivity of this observable to all four GPDs,
dQ(cid:48)2dtdφd(cosθ)
especially at large | t |. We also display in this fig-
left polarized.
ure our calculation with the factorized ansatz for the
We display in Fig. 12 (top panel) our results for A
(cid:12)U
as a function of φ at Q(cid:48)2 = 7 GeV2, ξ = 0.2, −t = 0.4 t-dependence of the H GPD in order to illustrate the
GeV2 for θ=45◦, 90◦ and θ integrated over [45◦,135◦]. model-dependence of our results.
The TCS is calculated here with only the H GPD. In all
kinematics, we obtain a sin(φ) shape with a significant
C. Polarized targets
amplitude,upto≈25%. WeobservethattheBHdoesn’t
produce any asymmetry. Any signal therefore reflects a
contribution from TCS. This is due to the fact that, as We define:
was shown in Ref. [5], this observable is sensitive to the σ+−σ−
imaginary part of the amplitude and that the BH am- AUi = σ++σ−, (28)
plitude is purely real. The amplitude of the asymmetry
dependsonθ. Itismaximalfor θ=90◦ whereBHismin- where σ± stands for the four-fold differential cross sec-
imalanditdecreasesasθ tendstoθ=0◦ (or180◦). Since tions dσ for the two target spin orientations
dQ(cid:48)2dtdφd(cosθ)
the BH does not produce on its own an asymmetry, one + and − along the axis i=x,y or z.
10
U
A
0.2
U
0
A
0.1 -0.05 BH only
H (reggeized)
~
-0.1 H+H
0 H+E
BH only ∀ θ -0.15 H+E~
this work: θ ∈ (450,1350) H (factorized)
-0.1 this work: θ = 900 -0.2
this work: θ = 450
-0.2 Berger: θ ∈ (450,1350) -0.25
Berger: θ = 900
Berger: θ = 450 -0.3
0 50 100 150 200 250 300 350 0.2 0.3 0.4 0.5 0.6 0.7
φ (deg.) |t| ((GeV2)
Figure 13: The A asymmetry as a function of t for
(cid:12)U
BH+TCS at ξ = 0.2, Q(cid:48)2 = 7 GeV2, φ = 90◦ and θ inte-
gratedover[45◦,135◦]. TCSiscalculatedwithdifferentGPD
U
A contributions.
0.2
0.1
In this figure, TCS is calculated with different GPDs.
0 Depending on the value of t, the three asymmetries are
BH only
sensitivetotheGPDsH,H˜ andEinvariousproportions.
-0.1 H
~ We also display in this figure our calculation with the
H+H
-0.2 H+E factorized ansatz for the t-dependence of the H GPD in
~ order to illustrate the model-dependence of our results.
H+E
0 50 100 150 200 250 300 350
φ (deg.)
Figure 12: Top panel: The A asymmetry as a function
(cid:12)U
of φ for θ = 45◦, θ = 90◦ and θ ∈ [45◦,135◦]. The TCS
is calculated here with only the H GPD. We compare our
calculations to those of Ref. [5]. Bottom panel: The A
(cid:12)U
forθ∈[45◦,135◦]usingdifferentsGPDsparametrizationsfor
TCS. The calculations are done for Q(cid:48)2 = 7 GeV2, ξ = 0.2,
−t=0.4 GeV2.
WeshowinFig.14ourresultsfortheφ-dependenceof
A , A and A for Q(cid:48)2 =7 GeV2, ξ =0.2, −t=0.4
Ux Uy Uz
GeV2 for θ integrated over [π, 3π]. Like for A , it is
4 4 (cid:12)U
advantageous to integrate over θ, in order to maximize
count rates, since the signal is barely reduced. The TCS
is calculated with different GPD contributions. We ob-
serve sinφ or cosφ shapes with amplitudes between 10
and 15%. Like for A , the BH doesn’t produce any
(cid:12)U
asymmetryandanynon-zeroasymmetrydirectlyreflects
the strength of GPDs.
WeshowinFig.15thet-dependenceofA , A and
Ux Uy
A atφ=90◦,0◦and90◦respectively,forthekinematics
Uz
ξ = 0.2, Q(cid:48)2 = 7 GeV2 and θ integrated over [π, 3π].
4 4