Deep Electroproduction of Photons and Mesons on the Deuteron

0.56 MB·English

by  F. Cano

EPJ manuscript No.
(will be inserted by the editor)
Deep Electroproduction of Photons and Mesons on the Deuteron
F. Cano1 and B. Pire2
1 DAPNIA/SPhN,CEA–Saclay, F-91191 Gif-sur-YvetteCedex, France.
2 CPhT, E´cole Polytechnique,F-91128 Palaiseau Cedex, France; UMR C7644 of CNRS.
July 17, 2003
0
1 Abstract. We study deeply virtual Compton scattering and deep exclusive meson electroproduction on
0 a deuteron target. We model the Generalized Quark Distributions in the deuteron by using the impulse
2 approximation for the lowest Fock-space state on the light-cone. We study the properties of the resulting
GPDs, and verify that sum rules violations are quite small in the impulse approximation. Numerical
n
predictions are given for the unpolarized cross sections and polarization asymmetries for the kinematical
a
J regimesrelevantforJLabexperimentsandforHERMESatHERA.Weconcludethatthesignalofcoherent
scattering on the deuteron is comparable to the one on the proton at least for low momentum transfer,
1
providing support to the feasibility of the experiments. The short distance structure of the deuteron may
1
thusbe scrutinized in the nearfuture.

3
v PACS. 2 4.85.+p, 12.38.Bx, 25.30.-e
1
3
2 1 Introduction known nuclear system and represents the most appropri-
7
ate starting point to investigate hard exclusive processes
0
Thestudyofhardexclusiveprocesses,suchasdeeplyVir- off nuclei[9]. On the other hand, these processes could of-
3
tual Compton Scattering (DVCS) feranewsourceofinformationaboutthepartonicdegrees
0
of freedom in nuclei, complementary to the existing one
/
h eA e′γA′ (1) from deep inelastic scattering. Experimentally, deuteron
→
p
targets are quite common and as a matter of fact, DVCS
- and deep exclusive meson electroproduction (DEMP)
p experiments are being planned or carried out at facilities
e eA eMA (2) likeCEBAFatJLabandHERMESatHERA,wheresome
′ ′
h → datahavealreadybeenreleased[10].Oneshouldofcourse
v: where A is a hadron(usually a nucleon,here a deuteron), distinguish the case where the deuteron serves merely as
i andMameson(usuallyaρoraπ)orapairofmesons(of a source of slightly bound protons and neutrons from the
X
relativelysmallinvariantmass)inthekinematicaldomain case where the deuteron acts as a single hadron. In the
r ofalargemomentumtransferQ2 betweentheleptonsbut formercase,the scatteringis incoherentandthe deuteron
a
a small momentum transfer (t) between the hadrons, has will break up during the reaction. In the latter case, to
been recently demonstrated to open the possibility of ob- whichwedevoteourstudy,thedeuteronstaysintactafter
tainingaquitecompletepictureofthehadronicstructure. the scattering. The fact that this occurs in a non negli-
The informationwhich can be accessedthrough these ex- gible fraction of events is not evident to everybody, since
perimentsisencodedbythe GeneralizedPartonDistribu- it is usual, but uncorrect, to mix the concepts of hard
tions, GPDs[1,2] (for recent reviews see[3]), which give in and destructive reactions. Indeed, as estimates given be-
particularinformationonthetransverselocationofquarks low will show, the very nature of deep electroproduction
in the hadrons[4]. Recent measurements of the azimuthal in the forward region is that the target is not violently
dependence of the beam spin asymmetry in DVCS [5,6] shattered by the hard probe. The fragile nature of the
haveprovidedexperimentalevidencetosupportthevalid- deuteronthusdoesnotpreventitfromstayingintact.This
ityoftheformalismofGPDsandtheunderlyingQCDfac- pictureshouldofcoursebe experimentallytestedthrough
torization of short-distance and long-distance dominated the comparison of rates for coherent and incoherent elec-
subprocesses. troproduction. The need for a deuteron recoil detector is
The theoretical arguments used in deriving factoriza- primordial in this respect.
tion theorems in QCD for the nucleon[7] target case can
be applied to the deuteroncase as well,and therefore one The paper is organized as follows: In section 2 we
can develop the formalism of GPDs for the deuteron[8]. remind the reader of the formalism of Generalized Par-
Fromthetheoreticalviewpoint,itisthesimplestandbest ton Distributions for spin-1 targets in general and the
2 F. Cano, B. Pire: Deep Electroproduction of Photons and Mesons on theDeuteron
γ∗ γ
deuteron in particular. In section 3 we explain in detail
the construction of the impulse approximation to evalu-
ate the helicity amplitudes and in section 4 we derive the
deuteron GPDs from the helicity amplitudes and study
the properties and implications with a numerical model. N(p1 ) ~ ~ N(p 1’)
Insection5wegivetheusefulformulaeforcalculatingthe H, H, E, E
DVCS cross section. In section 6 we show our numerical χ χ’
estimates for the usual observablesin the DVCS case and
D(P) D(P’)
comment on the feasibility of experiments. In section 7
we examine the electroproduction of mesons. Througout N(p )
2
the paper we will limit ourselves to the quark sector of
the GPDs, which is a good approximation provided the Fig. 1. Estimatingtheheliticyamplitudesfortheγ∗D→γD
intheimpulseapproximation.Thefinalresultisaconvolution
Bjorken variable x is not too small. This variable is de-
Bj model between the deuteron wave function and the GPDs for
fined as usual as xBj = 2QP2q, i.e., it is given in lab frame thenucleon (upperblob).
by x = Q2 , where M is· the deuteron mass and ν the
Bj 2Mν
virtualphotonenergy.Gluoneffectswillbeneededforun- 1 1
derstanding higher energies experiments. Previous short dxH (x,ξ,t) = dxH˜ (x,ξ,t) = 0. (6)
5 4
reports on our results have been presented at recent con- Z−1 Z−1
ferences [11]
Takingthe forwardlimit ofthe matrix elements defin-
ing GPDs leads to the relations [8] between GPDs and
2 GPDs in the deuteron: definitions and basic parton densities in the deuteron (with obvious notations)
as :
properties
q1(x)+q 1(x)+q0(x)
Aparametrizationofthenon-perturbativematrixelements H (x,0,0)= − ,
1
which determine the amplitudes in DVCS and DEMP on 3
a spin-one target were given in terms of nine GPDs for H (x,0,0)= q0(x) q1(x)+q−1(x),
the quark sector[8]: 5 − 2
Vλ′λ = dκeixκ2P¯.n P′,λ′ ψ¯( κn)γ.nψ(κn) P,λ H˜1(x,0,0)= q↑1(x)−q↑−1(x) (7)
2π h | − | i for x > 0. The corresponding relations for x < 0 involve
Z
= ǫ βV(i)ǫαH (x,ξ,t), (3) the antiquark distributions at x, with an overall minus
′∗ βα i −
sign in the expressions for H and H .
i=1,5 1 5
X
In all this paper we will restrict to the quark contri-
Aλ′λ = dκeixκ2P¯.n P′,λ′ ψ¯( κn)γ.nγ5ψ(κn) P,λ bution. Gluon contributions are expected to be small at
2π h | − | i
Z medium energies but should be included in a more com-
= ǫ′∗βA(βiα) ǫαH˜i(x,ξ,t), (4) plete description of the process. Therefore we will limit
ourselves to values of x not smaller than 0.1.
i=1,4 Bj
X
where P,λ represents a deuteron state of momentum P
and po|lariziation λ, P¯ = (P +P )/2, and nµ is a light-
′
like vector with n+ = 0,n = 0. Due to the spin-one 3 Helicity Amplitudes in the Impulse
character of the target, the⊥re are more GPD’s than in Approximation
the nucleon case, but at the same time the set of po-
larization observables which in principle could be mea- The impulse approximationis the zeroth order to explain
sured is also richer. Not much is known about these non- the photon-nucleus interaction, but it is the first model
perturbativeobjectswhichencodethewayquarksarecon- onehas to analyzesince the bulk ofthe physicsis already
fined in deuterons, except a limited set of sum rules and contained in it.
somelimiting casevalues.Sumrules[8]relatethese GPDs In the previous section we mentioned that the rele-
to the usual deuteron form factors : vant quantities are the deuteron GPDs . For the sake of
1 simplicity we will model the matrix elements Vλ′λ (3)and
dxH (x,ξ,t) =G (t) (i=1,2,3),
i i Aλ′λ (4)andrecoverthe GPDsjustbyusingthe relations
Z−1 giveninthe appendix.Furthermore,sincewe aregoingto
1
dxH˜ (x,ξ,t) =G˜ (t) (i=1,2), (5) limit ourselves to the quark content of the deuteron and
i i
it is an isoscalar target, we will denote:
Z−1
or lead to a null average :
1 1 Vλu′λ =Vλd′λ ≡Vλq′λ (8)
dxH (x,ξ,t) = dxH˜ (x,ξ,t) = 0,
Z−1 4 Z−1 3 and a similar relation holds for Aλ′λ.
F. Cano, B. Pire: Deep Electroproduction of Photons and Mesons on theDeuteron 3
Let us denote by Pµ (P µ) the momentum of the in- Therefore,wehavethattherelevantkinematicalquan-
′
coming (outgoing) deuteron and λ (λ) its polarization tities for the nucleons that make up the initial deuteron
′
state,thatsometimeswewilldenoteby0,+or .Toper- are (α α ) :
1
− ≡
formthisanalysiswewillchooseasymmetricframewhere
the averagemomentum P¯µ =(Pµ+P µ)/2 has no trans-
′
versecomponents.We willalsoneedalight-likevectornµ p+ = α(1+ξ)P¯+
1
to define, together with P¯µ, the light-cone and satisfying p+ = (1 α)(1+ξ)P¯+ (17)
P¯ n=1. To be more concrete, it is convenientto choose 2 −
a f·rame where P¯µ moves fast to the right. p1 +p2 = ∆⊥
Themomentumtransfer∆µ =P µ Pµ hasalongitu- ⊥ ⊥ − 2
′
−
dinal and a transverse component. The skewness controls and for the final deuteron we have(α α ) :
the fraction of momentum transferedin the ’+’ direction: ′ ≡ ′1
ξ =−(P +∆P·n′)·n =−2∆P¯++ (9) pp′2′1++ ==(α1′(−1−α′)ξ()1P¯−+ξ)P¯+ (18)
Withtheseconsiderations,thefour-vectorscorrespond- ∆
ing to each deuteron are 1: p′1⊥+p′2⊥ = 2⊥
We can now use the decomposition of the deuteron
M2+∆ 2/2 ∆ states in terms of nucleon states and the wave function
Pµ =((1+ξ)P¯+, 2P¯+(1+⊥ξ) ,− 2⊥) (10) defined in Appendix A1, Eq. (71), to get :
M2+∆ 2/2 ∆
P′µ =((1−ξ)P¯+, 2P¯+(1 ⊥ξ) , 2⊥) (11) 2 1+ξ 1
The invariant momentum transf−er is written as : Vλq′λ = (16π3)Z dαdα′dp1⊥dp′1⊥s1−ξ√αα′
α(1+ξ) 2ξ
t=∆2 =−4ξ2M12+ξ2∆⊥2 (12) · δΘ2((αp(′11⊥+−ξp)1⊥−x ∆⊥ξ))δΘ(α(α′−(1+ξ)1−2ξξ−) ) (19)
− · −| |− −
momTehnetupmosittrivanitsyfeorft∆0 f2⊥orimapfilxieesdtξh:at there is a minimal · λ′1X,λ1,λ2χ∗λ′(α′,k′⊥,λ′1,λ2)χλ(α,k⊥,λ1,λ2)
4ξ2M2 1 dκeiκx p ,λ ψ¯ ( κn)γ nψ (κn)p ,λ
t0 =−1 ξ2 (13) · 2Z 2π h ′1 ′1| q −2 · q 2 | 1 1i
−
and at the same time, for a given t, there is un upper In the equation above, the variables of the spectator
bound on the allowed values for ξ : nucleonhavebeeneliminatedjustbyusingthenormaliza-
tion properties of the one-particle states. The arguments
ξ2 −t (14) of the deuteron wave function (see appendix A1 for de-
≤ 4M2 t tails), k and k , are the transverse momentum of the
′
− active n⊥ucleon in⊥a frame where P = 0 and P = 0
The GPDs depend in addition of one more variable ′
respectively. Their relationship with⊥the transvers⊥e mo-
x which is defined as the fraction of average momentum
mentum in the symmetric frame is :
carried by the partons in the ’+’ direction :
k¯ n
∆
x= P¯··n (15) k⊥ ≡k1⊥ =p1⊥ +α 2⊥ (20)
with k¯µ = (kµ +kµ)/2. Therefore, the longitudinal mo- ∆
mentum of the init′ial parton is (x+ξ)P¯+, whereas the k′⊥ ≡k′1⊥ =p′1⊥ −α′ 2⊥ (21)
final one has (x ξ)P¯+, deliveringa longitudinaltransfer
∆+ = 2ξP¯+ to−the deuteron. The Heaviside functions in Eq. (20) ensure the posi-
− tivityofthe’plus’momentumcarriedbythenucleonsand
Now letus turnourattentionto the kinematics atthe
putalowerboundontheintegrationoverα.Thefirstone
nucleonlevelandlet us define the fractionoflongitudinal
standsforprocesseswhere x >ξ whereasthe secondone
momentum carried by each nucleon in the deuteron as :
| |
acts when we are on the ERBL region where x <ξ.
| |
αi = Pp+i+ ; α′i = Pp′i++ (16) locaTlhqeuraerfkoroep,ewreateonrdbeutpweweinthonmea-ntruicxleeolnemsteanttess,owf ahicnhonis-
′ parameterized in terms of nucleon GPDs. To take advan-
1 we use the following notation for the four-vectors tage of usual parameterizations found in the literature, it
(v+,v−,v⊥),with v± = √12(v0±v3) is convenient to keep on working on a symmetric frame.
4 F. Cano, B. Pire: Deep Electroproduction of Photons and Mesons on theDeuteron
Hwiotwheinvetrh,esidnecuetneruocnle,otnhsecsayrmrymseotmriceftrraamnsevfeorrsethmeodmeuentetruomn + √t20m−tηλ1EIS(xN,ξN,t)δλ′1,−λ1 (28)
(cid:21)
is not the symmetric frame for the active nucleon.
and a similar expression for Aq
Byperformingatransverseboostitispossibletoeval- λ′λ
uate the matrix element in (20) in a symmetric frame,
which has the property: 2 1+ξ 1
p˜′1⊥ +p˜1⊥ =0 (22) Aqλ′λ = (16π3)Zαmin dαdα′dk⊥dk⊥′s1−ξ√αα′
1 α α(1+ξ) 2ξ
where we have marked with a˜ the quantities in this δ2(k′ k − ∆ )δ(α′ − )
boosted frame. Since it is a transverse boost, it does not · ⊥− ⊥−(cid:18)1−ξ(cid:19) ⊥ − 1−ξ
tcvohercatnwogrietnhtµhneios’tn+roa’tncscovhmearnpsgoeencdeonemtistphooenfre,tnhstiens,cveie.ceitt.oni˜rsµsa.=Tlighnheµt.-lilgikhet-vleikce- · λ′1X,λ1,λ2χ∗λ′(α′,k′⊥,λ′1,λ2)χλ(α,k⊥,λ1,λ2)
In that frame, the parameterization of the nucleon 2λ ( 1 ξ2 H˜IS(x ,ξ ,t)
GPDs is made with variables that refer to the initial and · 1 − N N N
(cid:20) q
final nucleon, i.e., we define : ξ2
− 1Nξ2 E˜IS(xN,ξN,t))δλ′1λ1
− N
xN = ¯p˜k¯˜·n˜n˜ = α(1+xξ) ξ (23) + 2pλ1ξN√t20m−tηλ1E˜IS(xN,ξN,t)δλ′1,−λ1 (29)
1· − (cid:21)
∆˜ n˜ ξ The factor 2 in front of the formulae above stands
ξ = · = (24)
N −2¯p˜ n˜ α(1+ξ) ξ for the number of nucleons, so that the isoscalar nucleon
1· − GPDs (HIS, EIS, ...) is the isoscalarcombination within
Due to the lower bound on the values of α, we have one single nucleon :
ξ ξ. Moreover, it can be checked that x = xN, which
N ≥ ξ ξN
isconsistentwiththefactthatweareprobingqq¯distribu- 1
HIS(x ,ξ ,t)= Hu(x ,ξ ,t)+Hd(x ,ξ ,t) .
tionamplitudesinthedeuterononlythroughthenucleon. N N 2 N N N N
In other words, when we enter the ERBL region in the (30)
(cid:2) (cid:3)
deuteron(i.e.when x =ξ), we doso atthe nucleonlevel ThephasethatgoeswiththenucleonhelicityflipGPDs
| |
(i.e. when x =ξ ). is given by :
N N
| |
The transverse momentum of the nucleon that inter-
acts with the photon is, after the boost : 2λ∆˜x i∆˜y
η = − (31)
λ ∆˜
| ⊥|
∆˜ For the sake of clarity we have omitted the Heaviside
p˜1 = ⊥ (25) function in the integrals above but recall that there is a
⊥ − 2
lower bound on the value of α, which is
∆˜
p˜′1⊥ = 2⊥ (26) 2ξ x +ξ
α =max ,| | (32)
∆˜ = (1+ξN)∆ +2ξNp1⊥ (27) min (cid:26)1+ξ 1+ξ (cid:27)
⊥ ⊥
Now we can use the parameterization of the nucleon
4 Deuterons GPDs
matrix element given in the appendix, and after some al-
gebra and changes in the integration variables we reach
Oncewehaveobtainedthehelicityamplitudesitisstraight-
the final result for V :
λ‘λ forward to get from them the deuteron GPDs, just from
thedefinitionsgiveninEq.(3,4).Todosoinasimpleway,
one can use the light-cone polarization vectors given in
2 1+ξ 1
Vλq′λ = (16π3)Zαmin dαdα′dk⊥dk′⊥s1−ξ√αα′ [p8e]n.dTihxe. analytical expressions are summarized in the ap-
1 α α(1+ξ) 2ξ Atthispointonemayarguethatdefinitionsofdeuteron
δ2(k′ k − ∆ )δ(α′ − ) GPDs werenotactuallyneccessaryto reachthe results of
· ⊥− ⊥− 1 ξ ⊥ − 1 ξ
(cid:18) − (cid:19) − the preceding section and that one can derive the cross
· λ′1X,λ1,λ2χ∗λ′(α′,k′⊥,λ′1,λ2)χλ(α,k⊥,λ1,λ2) sBeucttiiotnsshaonudldobbeseermvapbhlaess,izdeidretchtalytftrhoemgeEnqusi.n(e2o8b)jaencdts(t2h9a)t.
parametrizethehadronicstructurearetheGPDs,therest
· ( 1−ξN2 HIS(xN,ξN,t) beingjustkinematics.TheGPDshavewelldefinedproper-
(cid:20) q tiesinsomelimitsandtheiranalysiscouldhelpusintest-
ξ2 ing the soundness of a model, as a complementary check
− 1Nξ2 EIS(xN,ξN,t))δλ′1λ1 to the comparison with experimental data
− N
p
F. Cano, B. Pire: Deep Electroproduction of Photons and Mesons on theDeuteron 5
D D’ D D’ D D’
(a) (b) (c)
Fig. 2. Deuteron generalized parton distributions in the im-
pulse (a and b) and beyond (c) the impulse approximation
4.1 Deuteron wave function
We need a specific model for the spatial deuteron wave Fig. 3. (a) Longitudinal momentum distribution of the nu-
function. As can be seen in the appendix, in the lower cleon within the deuteron. (b) Gap between the fractions of
longitudinal momentum carried by the active nucleon before
Fock-spaceapproximationonecanlinkthelight-conewave
and after the interaction as a function of ξ(x ) and α
functiontotheusualinstant-formwavefunctionthrougha Bj
identificationof variables.We havechosena parametriza-
tionofthespatialwavefunctiongivenbytheParisPoten-
of the ratio of the binding energy divided by the nucleon
tial [12] which has a S-wave supplemented with a D-wave
mass.
component with a probability of 5.8 %. We do not ex-
In the impulse approximation, the active nucleon af-
pect a strong dependence on the chosen parametrization
ter the interaction with the photon carries a fraction of
for the deuteron wave function. Most of them are identi-
longitudinal momentum which is given by
cal in the low-momentum region since they are strongly
constrained by the well known form factors. Differences x
Bj
between parametrizationsare significantonly in the large α′ =α− 1 x (1−α) . (36)
Bj
momentum region. Since we are going to limit ourselves −
to the low-momentum transfer region, we will not be es- In Fig. 3.b we plot the difference α α′ as a function
−
pecially sensitive to the tail of the wave function. of α and for several values of the skewness. We see that
Nonetheless, before going through the details of the forxBj >0.1thisdifferenceislargerthanthewidthofthe
results,letusdiscusssomefeaturesofthedeuteronGPDs momentum distribution, and therefore, we will inevitably
that may be expected from quite general grounds. The have a too fast or too slow nucleon (in the longitudinal
skewnessparameterξ determines the momentum transfer direction). In this case the central region of momentum,
in the longitudinal direction: where a maximal contribution is expected, is missed and
thenthe crosssectionswilldecreaseveryfastwithx .In
Bj
∆+ (P + P+)= 2ξP¯+ , (33) other words, there is an increasing difficulty in forming a
′
≡ − − coherent final state as the longitudinal momentum trans-
and in the generalized Bjorken limit this is entirely fixed fer, i.e. x increases. In that case other coherent mecha-
Bj
by the kinematics of the virtual photon (ξ xBj/2). In nisms,whichcouldinvolvehigherFock-spacecomponents,
≈
the impulse approximation, this momentum transfer has will presumably become dominant. Not much is known
to be provided by the active nucleon, and after that, the about these states, but it should be emphasized that the
finalstateofthisactivenucleonstillhastofitintothefinal suppression of the diagram of Fig. 1 occurs at x as low
Bj
deuteron. Since the deuteron is a loosely bound system, as 0.2, so that there is room to check the importance of
one cannot have a very asymmetrical sharing of longitu- the contribution of these ’exotic’ states.
dinal momentum between the nucleons and one may thus The choice of the nucleon GPDs deserves a more de-
guess that the formation of the coherent final state will tailed discussion.
be strongly suppressed in the impulse approximation for
large skewness.
To be more quantitative let us define the longitudinal
4.2 Modelling nucleon GPDs
momentumdistributionofthenucleoninthedeuteronas:
LetusfirstconsiderthehelicityconservingnucleonGPDs.
dk dβ Following [13] we have taken a factorized ansatz for the t
nλ(α)= (16⊥π)3 |χλ(β,k⊥,λ1,λ2)|2δ(α−β) , dependence of the nucleon GPDs:
λX1,λ2Z
(34)
which is normalized according to Hu(x ,ξ ,t) =hu(x ,ξ )1Fu(t) (37)
N N N N 2 1
dαnλ(α)=1 . (35) Hd(xN,ξN,t) =hd(xN,ξN)F1d(t) (38)
Z H˜q(x ,ξ ,t) =h˜q(x ,ξ )F˜q(t) (39)
N N N N
In Fig. 3.a we show n (α) evaluated with the wave
0
functionfromtheParispotential[12].Thisdistributionis and neglected the strange quark contributions Hs. The
strongly peaked at α = 0.5 and its width is of the order flavour decomposition of the proton and neutron Dirac
6 F. Cano, B. Pire: Deep Electroproduction of Photons and Mesons on theDeuteron
form factor, for which we have taken the usual dipole pa- region.A particulartype of contribution in the ERBL re-
rameterizations [14], gives: gionis the Polyakov-WeissD-term[19],whichwewillnot
include in our analysis.
Fu(t) =2Fp+Fn (40)
1 1 1
Fd(t) =2Fn+Fp (41) 4.3 Results
1 1 1
For the axial form factor we have taken F˜q(t) = (1 Withtheingredientsmentionedbeforeweplotinfigures4
− and 5 the corresponding generalized quark distributions,
t/M2) 2 with M = 1.06 GeV [15]. For hq and h˜q we
A − A which is flavorblind for the deuteron case. The support
follow the ansatz based on double distributions:
of these functions is 1 < x < 1 but we have plotted
− −
only the central region. In addition, due to the assump-
1 1 x′ tion made when modelling H˜ for the nucleon (the non-
hq(x ,ξ )= dx − dy δ(x x ξ y )q(x) contribution of the polarized sea) we have that H˜ (x
N N Z0 ′Z−1+x′ ′" N − ′− N ′ ′ ξ,ξ,t) vanishes. i ≤
The rapid falloff of the GPDs with x reflect the fact
δ(x +x ξ y )q¯(x) π(x,y ), (42) that the impulse approximation, i.e, the single nucleon
N ′ N ′ ′ ′ ′
− − # contribution cannot account for very large longitudinal
1 1 x′ momentum.
˜hq(x ,ξ )= dx − dy δ(x x ξ y ) Noticealsothehugedifferencesinthescalesofthevar-
N N ′ ′ N ′ N ′
Z0 Z−1+x′ − − ious GPDs : for the vector sector, H3 dominates over the
·∆qV(x′)π(x′,y′), (43) others, whereas H4 or H5 are very small. The respective
3 (1 x)2 y2 sizesmayberelatedtothevaluesofthedifferentdeuteron
′ ′
π(x′,y′)= − − . (44) formfactorsfortheGPDsthathaveasum-ruleconnection
4 (1 x)3
− ′ to them (see Eq.5).
The form factors that we have used in the current
where we have only considered the polarizationof the va-
(G ,G ,G ) are related with the usual charge monopole,
lence quarks. To avoid numerical problems with the inte- 1 2 3
G , magnetic dipole, G and charge quadrupole, G in
grals in the low x region we have followed the procedure C M Q
the following way :
explained in [8]. Throughout this work we have taken the
parameterization MRST 2001 NLO [16] for the unpolar-
izedpartondistributionsandtheparameterizationLSS01 2
[17] for the polarized ones. G1(t)= GC(t) ηGQ(t)
− 3
Concerning the helicity flip nucleon GPDs, Eq and
G (t)= G (t) (45)
E˜q, we can safely neglect the latter since we deal with 2 M
2
an isoscalar target. The former is suppressed in the Vλ′λ (1+η)G3(t)= GM(t) GC(t)+ 1+ η GQ(t)
amplitudes,Eq.(28)bykinematicalfactors.However,one − 3
(cid:18) (cid:19)
might think that there could be physicalsituations which
could be sensitive to this GPD: in the amplitudes where whith η = 4−Mt2. With the flavour decomposition of the
λ =λthetransitionwithHIS isdoneatthecostofusing formfactorsfor the deuteron, we have: Gu =Gd Gq =
th′e6 D-waveofthedeuteron,i.e.,bymakinguseofangular 3G . The dominant form factor is Gq duei mainily≡toithe
i 3
momentum. The term with EIS could flip the helicity at sizeofG (t)(see[20]),andifweconsidertheformfactors
Q
thenucleonlevelandtherefore,the(rathersmall)D-wave as a normalization condition for the GPDs, it is natural
admixture in the deuteron is not necessary. that H dominates over the other GPDs.
3
Unfortunately, most of the observables are dominated Notice, however, that the fact that a GPD is large
by amplitudes with λ = λ. We have checked that effects doesnotmeannecessarilythatitplaysamajorroleinthe
′
duetoEIS arenegligibleandwehaveonlyshownhowtiny observables: it has to be multiplied by the corresponding
they are for the sum rules, just for illustrative purposes. kinematical coefficients.
Furthermore, when modelling Eq following the steps ex- It is worthwhile mentioning that we have plotted the
plainedin[13],onerealizesthattheisoscalarcombination GPDs at a particular value of t,i.e. we cannotset t=0
is suppressed. Recall that Eq is normalized to the Pauli to study the ξ and x behaviou−r. The reason is that, even
formfactor,thatintheforwardlimitgivesjusttheanoma- ifweassumeafactorizedformforthetdependence inthe
lous magnetic moment, very small for the isoscalar case. nucleon, in the deuteron we cannot isolate this t depen-
Let us stress that the available models of GPDs are dence. In fact, there are two sources of t dependence in
fraught with uncertainties, in particular in the ERBL re- H and H˜ : first the explicit t dependence in the nucleon
i i
gion.There,GPDsdescribetheemissionofaqq¯pairfrom GPDs and, the most important one, the transverse mo-
the target,andanansatzonlyusingthe informationfrom mentum in the deuteron wave function. Then, we cannot
usual parton densities should be used with care. Notice circumvent the kinematical relationship between ξ and t,
also that, while for x>ξ GPDs are bounded from above Eq. (12) and for a non-vanishing ξ we have inevitably a
[18], no analogous constraints are known in the ERBL non-vanishing t.
F. Cano, B. Pire: Deep Electroproduction of Photons and Mesons on theDeuteron 7
4.4 Sum Rules : tests and discussions
In the impulse approximation we have retained only the
lowest Fock-space state of the deuteron (Fig. 2, a and b).
As we see, the qq¯ components which are tested in the
region x < ξ, are considered only within the nucleon
| |
itself (Fig. 2.b).
We have also neglected the NN¯ components in the
deuteronwavefunction,whichcouldgiverisetodiagrams
like the one in Fig. 2.c. When one evaluates the elastic
deuteron form factor [21,22] this is an exact approxima-
tion since one can always choose a frame where the mo-
mentum transfer is purely transverse and in that case,
∆+ = 0 and no pairs can be created from or annihilated
into a photon.
IntheDVCSandDEMPcasesthereisanon-vanishing
momentumtransferinthelongitudinaldirection,controled
by the skewness parameter ξ. Therefore, there are neces-
sarily diagrams where the final photon goes out from the
anhilation of, for example, a NN¯ pair or a qq¯(see figure
2.c). One has to include these higher Fock-space compo-
nents to recover Lorentz invariance.
Lorentz invariance is the physicalreasonwhy the sum
rules (5), obtained by integrating the GPDs over x, be-
come ξ-independent. When we perform that integration
we have matrix elements of local operators (form factors)
thatcannotdependonthe artifactsofthe kinematics,i.e.
oftheseparationbetweentransverseandlongitudinalmo-
mentum transfer (see [23] for a more detailed discussion
Fig. 4. Generalized Quark Distributions for the deuteron at on this point).
Q2=2 GeV2, ξ=0.1 and t=−0.25 GeV2. We can make use of this relationship between the ξ-
independence of the sum rules and the contribution of
higherFock-spacestates inthe deuteronto checkhow ac-
curate the impulse approximation is. We have plotted in
figure 6 the quantities:
1
I (ξ)= dxHq(x,ξ,t) (46)
i i
Z−1
For a fixed t, the functions I (ξ) would be constant if
i
the impulse approximation was exact : straight lines in
thefigureshowthe’theoretical’valuesofthesumruleac-
cording to the experimental parameterizationof the form
factors. Any residual ξ dependence of I (ξ) is a measure
i
of the importance of the higher Fock-space states that
we have not included in our description. Looking at this
figure and to the corresponding one for the axial case, we
canseethatthisdependenceisfairlymild,whichindicates
that, in the kinematical regime that we are interested in,
the deuteron is essentially a two-nucleon state2. As ξ in-
creases the impulse approximation would become a too
rough approximation with respect to Lorentz invariance.
One should distinguish between the variation in ξ of
the quantities I (ξ) and the particular values they take,
i
Fig. 5. Generalized polarized Quark Distributions for the which are sensitive also to the details of the employed
deuteron at Q2 = 2 GeV2, ξ = 0.1 and t = −0.25 GeV2.
H˜1 (upper-left), H˜2 (upper-right), H˜3 (lower-left), H˜4 (lower- 2 We checked that the inclusion of the nucleon GPD EIS
right), does not introduce any improvement at all: its contribution
vanishes exactly at ξ =0, and is always small at other values
of ξ.
8 F. Cano, B. Pire: Deep Electroproduction of Photons and Mesons on theDeuteron
model.Infigure6weseethatthepointsobtainedwithour
calculations are quite close to the experimental parame-
terization. Obviously, the models works better at smaller
ξ, for the reasons exposed above. In fact, in figure 7, we
showthe t-dependence ofthe sumrulesatξ =0,where it
isclearlyseenthatresultsagreesquitewellwiththeexper-
imentalparameterization,whenavailable,orwiththeval-
ues imposed by time reversal or Lorentz invariance. This
comes as no surprise since it is well known that the light-
cone deuteron wave function is able to give the deuteron
form factors at the momentum transfer we are working
(see[24]forarecentreview).Inthecontextofourdiscus-
sion, at ξ = 0 the pair creation or annihilation with the
photon vanishes in the light-cone formalism.
One final remark concerns also the subtleties of the
light-cone formalism: nucleons are on-shell, i.e., they ver-
ify, with the notation employed in the previous section,
that:
m2+p2
i
p−i = 2p+ ⊥ (47)
i
But, they are offlight-cone energyshell, andas a con-
sequence, if P+ =p+1 +p+2 and P =p1⊥ +p2⊥ (as it is
⊥
the case),one has that P− 6=p−1 +p−2. Therefore,one has
that the momentum transfer t defined from the deuteron
variables does not coincide with the one defined from the
variables of the active nucleon. Moreover,the upper limit
over ξ2 is not t . If nevertheless one enforces ξ2 to
N 4m−2 t N
havethis upper lim−it, this leads only to a tiny shift in the
value ofα in the integrals.Fromthe practicalpointof
min
view, these differences are too tiny to be seen in the nu-
mericalcalculation,unlessonegoestoverylargevaluesof
Fig. 6. Sum rules for the vector GPDs at Q2 =2 GeV2 and
t. This just reflects the fact that the off-shellness effects
− t = −0.5 GeV2. Solid lines are the expected theoretical re-
in the light-cone energy are of the order of the binding
sultandpointsaretheresultsobtainedwithourmodel.Filled
energy over the longitudinal momentum [25]: points:thenucleonGPDEIS isnotincluded;Emptypoints:
EIS included.
V
P−−(p−1 +p−2)∝ P+ (48)
and in our case this is of the order of the binding energy where the three form factors have been measured in the
of the deuteron over the center of mass energy, i.e, very low and medium momentum transfer ranges[20]. The lep-
small. tonic tensor Lµν is given by
5 DVCS Amplitudes and Cross Sections 1 1
Lµν =u¯(k′,h′) γµ γν +γν γµ u(k,h)
(k + q ) (k q )
Therearetwoprocessesthatcontributeto thedeeply vir- (cid:20) 6 ′ 6 ′ 6 −6 ′ (cid:21)
(51)
tualComptonscatteringamplitudeofEq.1underconsid-
The Bethe-Heitler process is thus completely known in
eration. The first one is the Bethe-Heitler process where
terms of already measured form factors.
the outgoing photon is produced from the lepton line. Its
The second process where the photon is emitted from
amplitude (for either electrons or positrons) is given by
thehadronicpartismoreinterestingintermsofthestudy
e3 of the hadronic structure. It is called virtual Compton
TBH =− t ǫ∗µ(q′,λ′)jν(0)Lµν (49) scattering since it can be decomposed in a γ∗A → γB
process. Its amplitude T is written as
VCS
whereeistheprotoncharge.Thedeuteroncurrentisgiven
by: e3
TVCS =±Q2 Ω(h,λ)MH′λ′,Hλ (52)
jµ =− G1(t)(ǫ′∗·ǫ)2P¯µ+G2(t) (ǫ′∗·2P¯)ǫµ+(ǫ·2P¯)ǫ′µ∗ Xλ
− G3(t)(ǫ′∗·2P¯)(ǫ·2P¯)MP¯µ2(cid:2) (50(cid:3)) wpohseirterotnhseaunpdpetrhseigfnunisctfioornelΩectcroomnsesanfrdomthethloewderecoonmepfoor-
F. Cano, B. Pire: Deep Electroproduction of Photons and Mesons on theDeuteron 9
Fig. 8. Sumrulesfortheaxial-vectorGPDsatQ2 =2GeV2
and t=−0.5 GeV2.
Fig. 7. Sum rules for the vector GPDs at Q2 = 2 GeV2 1 1 1
dx + e2Vq
and fixed ξ = 0 as a function of t. Solid lines represent the x ξ+iη x+ξ iη q λ′λ
expected theoretical values whereas points are the results of Z−1 (cid:18) − − (cid:19)Xq
our evaluation. +iǫµναβp˜ n˜ (56)
α β
1 1 1
dx e2Aq
x ξ+iη − x+ξ iη q λ′λ
sition of the leptonic current in terms of the polarization Z−1 (cid:18) − − (cid:19)Xq
vector of the virtual photon with the convention ǫ =+1.
0123
For completeness, let us first write down the formula
for the cross section of the Bethe Heitler process on the
Q
u¯(k′,h)γνu(k,h)= Ω˜(h,λ)ǫν(q,λ) (53) deuteron
√1 ǫ
Xλ − (4πα )3
λ T 2 = em [ A(t)+ B(t)] (57)
Ω˜(h,λ)= δλ0√2ǫ (√1+ǫ+2hλ√1 ǫ)e−iλφ | BH| t2 KA KB
− √2 −
(cid:20) (cid:21) X
whereA,Baretheelasticstructurefunctionsofthedeuteron,
Sometimes we will also use: which are well known in the low momentum region, and
the kinematical coefficients are:
Q
Ω(h,λ) Ω˜(h,λ) (54)
≡ √1 ǫ
− 2M2
The photon-deuteron helicity amplitudes are defined B = [(2(k q′)+t)2+(2(k q′)+Q2)2]
K −(k q )(k q ) · ·
as: · ′ ′· ′
4t
= + (M2+s+Q2 2s )
A B kp
MH′λ′,Hλ =ǫ∗µ(q′,λ′)ǫν(q,λ)Hµν (55) K −K (k′·q′) −
t
and + (2M2+Q2 2s )2 (Q2 2t)2
kp
2(k q )(k q ) − − −
· ′ ′· ′
Hµν =(gµν p˜µn˜ν n˜µp˜ν) +4(Q2+s s (cid:8))2+4t(t+s s ) (58)
kp kp
− − − −
(cid:9)
10 F. Cano, B. Pire: Deep Electroproduction of Photons and Mesons on theDeuteron
with s =(k+p)2.
kp
The VCS amplitude gives a contribution to the cross
sectionwhichmaybedecomposedintermsofitsazimuthal
dependence as
1(4πα )3
|TVCS|2 = 3Q2(1emǫ) 2|MH′1,H1|2+2|MH′1,H−1|2
− H,H′
X X (cid:0)
+4ǫMH′1,H0 2
| |
+4 ǫ(1+ǫ) cosφRe[MH′1,H0MH∗′1,H−1−MH′1,H0MH∗′1,H1]
− 4pǫcos(2φ)Re[MH′1,H−1MH∗′1,H1] . (59)
(cid:1)
The interference between the two processes leads to
a contribution to the DVCS cross section which may be
Fig. 9. Unpolarized Differential Cross section for DVCS for
written as
typical kinematics at JLab (left panel) and HERMES (right
2(4πα )3 panel).Dashed-dottedline:BHonly;dashedline:DVCSonly;
em
(TVCSTB∗H+TV∗CSTBH)=∓3Qt√1 ǫ full line: BH + DVCS + Interference.
X −
2hRe[ǫ (q ,λ =+1)j LµνΩ˜(h,λ)]
· ∗µ ′ ′ ν We present in Fig. 9 the unpolarized cross section for
H,H′,λ,h
X low (left panel) and medium (right panel) energy reac-
· Re[MH′1,Hλ] (60) tions.TheBethe-HeitlerandVCScontributionsareshown
as well as their interference. The relative importance of
where the upper sign stands for electrons and the lower
thesecontributionsdepend muchonthe productionangle
one for positrons.
of the final photon, as can be read from the figure. To
The study of the initial electron helicity dependence
discuss the feasibility of the experiment, a comparison to
may be expressed through the following weighted contri-
the proton target case is welcome. This is shown on Fig
butions to the cross section
10 for medium energy reactions.
Coherent deep VCS is certainly not a negligible effect
2h TBH 2 =0 (61) at small values of t and we can expect that this process
| |
should soon become observable so that some knowledge
X
of the deuteron GPDs will become accessible. More than
testingthevalidityoftheimpulseapproximation,thegoal
4(4πα )3 ǫ
2hT 2 = em (62) of such an experiment is to observe some definite devia-
| VCS| 3 Q2 1 ǫ
r − tionfrom the impulse approximationpredictions,thereby
X
H,H′ sinφIm[MH′1,H0MH∗′1,H1−MH′1,H0MH∗′1,H−1] idneduitceartoinng. Tsoomsecrnuotnin-tizreivisaulchshoerfftecdtiss,taintciescionntetreensttionfgthtoe
X turn to some more specific observables, such as spin and
charge asymmetries.
2(4πα )3 The beam spin asymmetry is defined as
em
2h(T T +T T )=
VCS B∗H V∗CS BH ∓3Qt√1 ǫ
· X 2hIm[ǫ∗µ(q′,λ′ =+1)jνLµνΩ˜(h,λ)−] ALU(φ)= ddσσ↑↑((φφ))−+ddσσ↓↓((φφ)) (64)
H,H′,λ,h
X where φ is the angle between the lepton and hadron scat-
· Im[MH′1,Hλ] (63) tering planes. The numerator is proportional to the in-
terference between the Bethe-Heitler and the VCS ampli-
where the upper sign stands for electrons and the lower
tudes.
one for positrons.
Ourpredictionscalculatedwithourmodelizeddeuteron
GPD’s are shown on Fig. 11 for JLab and Hermes ener-
gies.The signof the asymmetry is reversedfor a positron
6 Numerical Results for DVCS
beam.Suchasizableasymetryshouldbequiteeasilymea-
sured.Itwillconstituteacrucialtestofthevalidityofany
Ourmodelenablesusnowtoestimatethe crosssectionof model.
coherentdeeplyvirtualComptonscatteringonthedeuteron. It has been shown[2] that, asymptotically, the beam
We are particularly interested by the forthcoming exper- spin asymmetry exhibits a sin(φ) azimuthal dependence.
iments at JLab and Hermes at DESY, and we thus shall We have performeda Fourier decomposition of the asym-
present results for the kinematics of these experimental metry A obtained for the deuteron and, indeed, we
LU
set ups. havecheckedthedominanceofthesin(φ)component,even