DVCS on spinless nuclear targets in impulse approximation

13 Pages·0.18 MB·English

by  V. Guzey

RUB-TP2-19/03
DVCS on spinless nuclear targets in impulse
approximation
V. GUZEY
Institut fu¨r Theoretische Physik II, Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany
M. STRIKMAN
Department of Physics, the Pennsylvania State University, State College, PA 16802, USA
3
0 Abstract
0
2 Within the impulse approximation, we derive expressions for the amplitude of deeply virtual

n Comptonscatteringonspinlessnucleiintermsofthegeneralizedpartondistributionsofthenucleon.
a
As an application, nuclear effects in the beam-charge and single-spin asymmetries are discussed.
J

4
2
1 Introduction

1
v
6 Deeply Virtual Compton Scattering (DVCS) attracts a large amount of interest, both exper-
1 imentally and theoretically. The initial results on DVCS have been reported by HERMES
2
1 [1], ZEUS [2] and H1 [3] experiments at DESY and the CLAS experiment [4] at TJNAF;
0 the recent progress in the theoretical understanding of DVCS is summarized in the review
3
articles [5],[6],[7].
0
/ DVCS on a given target accesses new non-perturbative objects – Generalized Parton
h
p Distributions (GPDs) of the target. These distributions contain encoded information on the
-
p quark-gluon structure of the target.
e To date most of the experimental and theoretical work dedicated to DVCS concern with
h
: nucleon and pion targets. However, there are already first data on DVCS on nuclear (Deu-
v
terium, Neon) targets taken at HERMES [8]. On the theoretical side, nuclear GPDs may
i
X
allow one to better understand the nature of nuclear forces [9].
r
a Previously, the impulse approximation approach to nuclear effects in coherent DVCS on
Deuterium and other spin-1 nuclei was considered in Refs. [10]. The aim of this paper is to
apply the impulse approximation to DVCS on spin-0 nuclei and, in particular, to obtain a
qualitative estimate of the Fermi motion effect on the beam-charge asymmetry for DVCS a
spinless nucleus of Neon that was recently measured at HERMES [8].
In order to make clear the principal steps of the derivation, we first consider the calcula-
tion of the electromagnetic form factor of a spinless nucleus.
2 Electromagnetic form factor of a spinless nucleus
The electromagnetic (charge) form factor of a spinless nucleus, Fe.m.(t), is defined using the
A
matrix element of the operator of the electromagnetic current, Jµ(0), between the states of
the initial and final nucleus with momenta P and P′ = P +q (t q2 < 0):
A A A ≡
P +q Jµ(0) P = (2P +q)µFe.m.(t). (1)
h A | | Ai A A
1
The matrix element in Eq. (1) can be evaluated using the impulse approximation which is
based on the intuitive picture that the interaction with the nucleus is a two-step process:
First, the nucleus is decomposed into non-relativistically moving nucleons; then, these nu-
cleons interact independently with the probe. As a result, the matrix elements between the
nuclear states can be expressed in terms of the nucleon matrix elements.
The consistent derivation of the impulse approximation and discussion of its limitations
was given in Ref. [11]. In what follows, we shall present a less rigorous treatment of the
impulse approximation. It is convenient to start from the covariant expression for the matrix
element in Eq. (1) (see Ref. [12])
d4p d4p
P +q Jµ(0) P = 1 ... A Γ¯ (p +q,p ,...,p )
h A | | Ai X Z i(2π)4 i(2π)4 A 1 2 A
nucleons
1 1 ( 1)A−1
− Γˆµ (q) − − Γ (p ,p ,...,p )
×(cid:16)pˆ +qˆ m(cid:17) e.m. (cid:16)pˆ m(cid:17)(pˆ m)...(pˆ m) A 1 2 A
1 1 2 A
− − − −
i(2π)4δ4(P p ... p ), (2)
A 1 A
× − − −
¯
with the incoherent sum of electromagnetic interactions with each nucleon, Γ and
nucleons A
Γ thPe covariant vertex functions describing the transition of A nucleons to the nucleus, and
A
Γˆµ the electromagnetic vertex of the nucleon,
e.m.
iσµνq
Γˆµ (q) = γµF (t)+ νF (t), (3)
e.m. 1 2m 2
where F and F are the nucleon elastic form factors; m is the nucleon mass.
1 2
The covariant vertices Γ¯ and Γ in Eq. (2) can be related to non-relativistic nuclear
A A
wave function of the target. This non-relativistic reduction of Eq. (2) can be carried out as
follows.
Firstly, there are A four-dimensional integrations in Eq. (2). One integral (over p ) is
1
taken using the energy-momentum delta-function. Other A integrals over energies of the
spectator nucleons are taken in the approximation that only non-relativistic nucleons with
positive energy are present in the nuclear wave function. This enables one to take the residue
over the nucleon mass pole, for instance,
dp0 2π
2 = i . (4)
Z (p )2 m2 +iǫ − 2m
2
−
This procedure puts all spectator nucleons on their mass shell. Moreover, because of the
energy-momentum conservation at each nuclear vertex, the interacting nucleon is also on the
mass shell (ignoring small nuclear binding).
Secondly, since all nucleons are on the mass shell, the Dirac spinors in the propagators
of the bound nucleons can be replaced with those of the free ones:
pˆ+m = u(i)(p)u¯(i)(p). (5)
X
i
Thirdly, the covariant nuclear vertex functions Γ¯ and Γ are expressed in terms of the
A A
corresponding non-relativistic nuclear wave function, Φ , for instance
A
A−1
u¯(β)(p )u¯(i2)(p )...u¯(iA)(p )Γ = (2π)32m 2 (p2 m2)Φ (p ,β;p ,i ;...;p ,i ). (6)
1 2 A A (cid:16) (cid:17) 1 − A 1 2 2 A A
2
Note that the nuclear wave function is normalized to the number of nucleons A. With
Eqs. (4)-(6) in mind, the expression for the matrix element in Eq. (2) simplifies
P +q Jµ(0) P = d3p Φ∗(p +q,α)Φ (p ,β)u¯(α)(p +q)Γˆµ (q)u(β)(p ). (7)
h A | | Ai X Z iX A 1 A 1 1 e.m. 1
nucleons α,β
Here d3p denotes A 1 3-dimensional integrals, d3p = d3p ...d3p . Also we keep only
i i 2 A
−
the momentum and spin indices of the struck nucleon in the argument of the nuclear wave
function; summation over the polarization of the spectator nucleons is assumed.
For a spinless nucleus, only the terms with α = β survive in the product Φ∗(p +
A 1
q,α)Φ (p ,β). Introducing the nuclear wave function averaged over polarization of its nu-
A 1
cleons, Φ (p ),
A 1
1
Φ (p ,β) = Φ (p ), (8)
A 1 A 1
√2
we arrive at
1
P +q Jµ(0) P = d3p Φ∗(p +q)Φ (p ) u¯(α)(p +q)Γˆµ (q)u(α)(p )
h A | | Ai 2 nuXcleonsZ i A 1 A 1 Xα 1 e.m. 1
1
= d3p Φ∗(p +q)Φ (p )Tr Γˆµ (q)Pˆ(p ,q) , (9)
2 X Z i A 1 A 1 (cid:16) e.m. 1 (cid:17)
nucleons
where Pˆ(p ,q) is some sort of ”off-forward propagator”, Pˆ(p ,q) = u(α)(p )u¯(α)(p +q).
1 1 α 1 1
This propagator can be decomposed in the basis of Dirac matrices wPith the result
Pˆ(p ,q) = pˆ +qˆ/2+m+pˆ qˆ/(2m). (10)
1 1 1
Using the definition of Γˆµ in terms of the form factors F (t) and F (t) and then taking the
e.m. 1 2
trace of Dirac matrices in Eq. (9), we obtain
P +q Jµ(0) P = (2P +q)µFe.m.(t) = d3p Φ∗(p +q)Φ (p )(2p +q)µG (t),
h A | | Ai A A X Z i A 1 A 1 1 E
nucleons
(11)
wherethenucleonchargeformfactorisintroduced, G (t) = F (t)+t/(4m2)F (t). Asaresult
E 1 2
of the non-relativistic reduction, Eq. (11) is intrinsically reference frame dependent and it
is good only with accuracy (p2/m2). For a more consistent treatment, one would need to
O 1
use the light-cone (infinite-momentum frame) description of the nuclear wave function, see
reviews in [11, 13].
Inthenon-relativistic fort = 0we canusewithaccuracy (p2/m2 )theµ = 0component
O 1 N
of Eq. (11) to fix normalization of the wave function
m ZGp(0)
Fe.m.(0) = d3p Φ (p ) 2G (0) = E d3p Φ∗(p ) 2 = Z. (12)
A MA nuXcleonsZ i| A 1 | E A Z i| A 1 |
Here we introduced the proton (Gp) and neutron (Gn) charge form factors with the proper-
E E
tiesGp(0) = 1andGn(0) = 1. Wealsousedp0 msince theaccuracyofthenon-relativistic
E E 1 ≈
convolution approximation is (p2/m2 ).
O 1 N
3
3 DVCS on a spinless nuclear target
The amplitude for deeply virtual Compton scattering (DVCS) on any hadronic target is a
function of 5 variables: ξ, Q2, t, φ and φ . The parameter ξ is an analog of the Bjorken
⊥
variable x ,
Bj
Q2 2x
Bj
2ξ = = , (13)
2p¯q 2 x + ∆2x
− Bj Q2 Bj
where p¯= (p+p′)/2 = p+∆/2 with p and p′ the momenta of the hadron in the initial and
final states, ∆ = p′ p and ∆2 = t. The angle φ is the angle between the lepton and hadron
−
scattering planes; the angle φ is associated with the target polarization and, hence, it does
⊥
not appear in the present discussion: The amplitudes considered below are averaged over
φ (see Ref. [5] for the detailed discussion of kinematics of DVCS). In the following we shall
⊥
keep an explicit dependence on the variables ξ and t only.
We are interested in sufficiently large x , x 0.05, where nuclear modifications of the
Bj Bj
≥
parton densities are small and where average longitudinal distances in the DVCS process are
comparable or smaller than the average internucleon distances. In this case, the use of the
impulse approximation is well justified. In this approximation, for a spinless nuclear target,
the DVCS amplitude, Tµν, can be read off immediately from Eq. (9)
A
1
Tµν(ξ,t) = d3p Φ∗(p +∆)Φ (p )Tr Tˆµν(ξ′,t)Pˆ(p ,q) , (14)
A 2 X Z i A 1 A 1 (cid:16) 1 (cid:17)
nucleons
where Tˆµν is the amputated DVCS amplitude (without the external spinor lines) for the free
nucleon [6],
1 1
Tˆµν(ξ′,t) = ( gµν) dxC+(x,ξ′) H(x,ξ′,t)nˆ+E(x,ξ′,t)/(2m )iσµν∆ +... . (15)
2 − ⊥Z−1 (cid:16) N ν(cid:17)
Here, C+(x,ξ′) = 1/(x ξ′ +iǫ) +1/(x+ξ′ iǫ); H and E are the GPDs of the nucleon;
− −
the ellipses denote the terms vanishing after taking the trace; Pˆ(p ,q) is the ”off-forward”
1
propagator defined in Eq. (10). The variable ξ′ is defined with respect to the interacting
nucleon,
Q2 2x′ 2(x /α)
2ξ′ = = Bj = Bj , (16)
(2Ap +∆)q 2 x′ + ∆2x′ 2 (x /α)+ ∆2(x /α)
1 − Bj Q2 Bj − Bj Q2 Bj
where we introduced the Bjorken variable defined with respect to the struck nucleon, x′ =
Bj
Q2/(2Ap q),andconnecteditwithx byintroducingthefactorα = x /x′ = A(p q)/(P q).
1 Bj Bj Bj 1 A
Combining Eqs. (13) and (16) gives
ξ′ = ξ 2−xBj + ∆Q22xBj . (17)
α × 2 (x /α)+ ∆2(x /α)
− Bj Q2 Bj
The deviation of α from unity and, thus, ξ′ from ξ, is a measure of the importance of the
Fermi motion of the bound interacting nucleon. Since α = 1+ ( p /m), it is legitimate to
1
O | |
study the effects of α = 1 within the impulse approximation.
6
4
Substituting Eq.(15)intoEq.(14),takingthetraceandusing theon-mass-shellcondition
2(p ∆)+∆2 = 0, we obtain the following result for Tµν
1 · A
1 1
Tµν(ξ,t) = d3p Φ∗(p +∆)Φ (p ) ( gµν) dxC+(x,ξ′)
A X Z i A 1 A 1 h2 − ⊥Z−1
nucleons
∆2
H(x,ξ′,t)+E(x,ξ′,t) 2(p n)+(∆ n) . (18)
1
× (cid:16) 4m(cid:17)(cid:16) · · (cid:17)i
The light-like four-vector n is completely defined by the vectors P¯ and q [6]
A
q +2ξP¯
A
n = , (19)
Q2/(4ξ)+M¯2ξ
A
with M¯2 = M2 ∆2/4. Then,
A A −
p q +2ξp P¯ Q2/(2x)(α/A)+2ξP p ξ∆2/2
1 1 A A 1
p n = = − (20)
1 Q2/(4ξ)+M¯2ξ Q2/(4ξ)+M¯2ξ
A A
and
∆n = 2ξ. (21)
−
Thus, the final expression for the amplitude of DVCS on a spin-0 nucleus is given by the
convolution formula, Eq. (18), with ξ′, p n and ∆n given by Eqs. (17), (20) and (21).
1
On the other hand, the DVCS amplitude on a spinless nuclear target can be, at leading
twist, expressed intermsofa singleGPD,H , (theLorentzstructure oftheDVCS amplitude
A
is the same for all spinless hadrons, and the amplitude for the pion target can be found, for
example, in [14])
1
Tµν(ξ,t) = (gµν) dxC+(x,ξ)H (x,ξ,t). (22)
A − ⊥Z A
−1
Equating Eqs. (18) and(22), we find the connection between nuclear and nucleon generalized
parton distributions in the form
1 1 1
dxC+(x,ξ)H (x,ξ,t) = d3p Φ∗(p +∆)Φ (p ) dxC+(x,ξ′)
Z A 2 Z i A 1 A 1 Z
−1 −1
∆2 ∆2
Z(Hp(x,ξ′,t)+Ep(x,ξ′,t) )+(A Z)(Hn(x,ξ′,t)+En(x,ξ′,t) )
×(cid:16) 4m − 4m (cid:17)
2(p n)+(∆ n) , (23)
1
×(cid:16) · · (cid:17)
where the superscripts ”p” and ”n” denote the proton and neutron GPDs.
Equation (23) has both real and imaginary parts. They can be separated with the result
1 1 1 1
dx P( )+P( ) H (x,ξ,t) = d3p Φ∗(p +∆)Φ (p )
Z−1 (cid:16) x ξ x+ξ (cid:17) A 2 Z i A 1 A 1
−
1 1 1 ∆2
dx P( )+P( ) Z(Hp(x,ξ′,t)+Ep(x,ξ′,t) )
×Z−1 (cid:16) x ξ′) x+ξ′) (cid:17)(cid:16) 4m
−
∆2
+(A Z)(Hn(x,ξ′,t)+En(x,ξ′,t) ) 2(p n)+(∆ n) (24)
1
− 4m (cid:17)(cid:16) · · (cid:17)
5
and
1
H (ξ,ξ,t) H ( ξ,ξ,t) = d3p Φ∗(p +∆)Φ (p )
A − A − 2 Z i A 1 A 1
∆2
Z(Hp(ξ′,ξ′,t) Hp( ξ′,ξ′,t)+(Ep(ξ′,ξ′,t) Ep( ξ′,ξ′,t)) )
×(cid:16) − − − − 4m
∆2
+(A Z)(Hn(ξ′,ξ′,t) Hn( ξ′,ξ′,t)+(En(ξ′,ξ′,t) En( ξ′,ξ′,t)) )
− − − − − 4m (cid:17)
2(p n)+(∆ n) , (25)
1
×(cid:16) · · (cid:17)
where P(1/x) denotes the principal value integral.
4 Beam-charge and single-spin asymmetries
Generalizedpartondistributionfunctionsaregreatinnumber anddependonmanyvariables.
Thus, one studies special observables (asymmetries) involving GPDs aiming to study only
certainaspectsofGPDsandonlyafewatatime. Inthissectionweconsider thebeam-charge
and single-spin asymmetries for DVCS on a spinless nuclear target.
The beam-charge asymmetry, A , is measured by scattering unpolarized leptons of op-
C
posite charges (positrons and electrons) on unpolarized hadronic targets. The asymmetry
is then defined as the function of the angle φ, the angle between the leptonic and hadronic
scattering planes (the dependence on Q2, ξ and t is implicit),
N+(φ) N−(φ)
A (φ) = − , (26)
C
N+(φ)+N−(φ)
where N+ (N−) is proportional to the scattering cross section for the incoming positron
(electron). It is important to note that the DVCS amplitude, , interferes with the
DVCS
T
purelyelectromagneticBethe-Heitleramplitude, ,sothattheDVCSasymmetriesdepend
BH
T
and, in some kinematics, are dominated by interference of both amplitudes, . For instance,
I
the asymmetry A probes the real part of the interference between the DVCS and Bethe-
C
Heitler amplitudes.
The expressions for the DVCS, Bethe-Heitler and interference contributions to the scat-
tering cross section on the pion target (since the pion can be replaced by any spin-0 hadronic
target, the expressions for the asymmetries are also valid for spinless nuclei) are derived in
[14]. The beam-charge asymmetry can be written as
(λ = 1)+ (λ = 1)
A (φ) = I I − , (27)
C
2 2 + (λ = 1)+ (λ = 1)+ (λ = 1)+ (λ = 1)
BH DVCS DVCS
|T | I I − T T −
where λ the helicity of the incoming (massless) lepton and
Fe.m.(t) ∆2 2 Q2
(λ) = cI + Km(cI cos(mφ)+λsI sin(mφ))+ K3cI cos(3φ) ,
I −x2Bjy3∆2P1P2(cid:16)Q2 0 mX=1 m m M2 3 (cid:17)
(Fe.m.(t))2 2
2 = cBHKmcos(mφ),
|TBH| −x2Bjy2(1+ǫ2)∆2P1P2 mX=0 m
6
1
(λ) 2 = cDVCS +K(cDVCScos(φ)+λsDVCS sin(φ))
|TDVCS | x y2Q2(cid:16) 0 1 1
Bj
Q2
+ K2cDVCS cos(2φ) . (28)
M2 2 (cid:17)
Here and are dimensionless lepton propagators (divided by Q2); K and ǫ2 and kine-
1 2
P P
maticsfactors; thecoefficients c ands aregivenbyEqs. (31),(32),(33)and(11)ofRef.[14].
i i
Note that while the result of Ref. [14] includes also the twist-three terms, we consider only
the leading, twist-two, contribution.
In general, exact Eqs. (24) and (25) enable one to evaluate A in the most general case.
C
However, the main goal of this paper is to write down simple and yet reliable expressions,
where the effects associated with the nuclear target are presented in a transparent form. To
thisend,letusconsiderthefollowingkinematicsforDVCSonanucleartarget: ∆2 = t Q2,
≪
Q2 equals a few GeV2 and x 0.1. These conditions correspond to the kinematics of the
Bj
≥
HERMES experiment at DESY. In this kinematics, the Bethe-Heitler process dominates
the cross section and, moreover, we can neglect the terms proportional ǫ2 and ∆2/Q2 and
keep only the leading terms in powers of K (K √∆). The simplified expression for the
∝
beam-charge asymmetry reads
KcI
A (φ) = cos(φ) 1
C ycBHFe.m.(t)
0 A
= cos(φ)K8(2−2y +y2)xBj R−11dx(cid:16)P(x−1ξ)+P(x+1ξ)(cid:17)HA(x,ξ,t) , (29)
− ycBH Fe.m.(t)
0 A
where 1 dx P( 1 )+P( 1 ) H (x,ξ,t) is given by Eq. (24) and the nuclear charge form
factor FR−e1.m.(t(cid:16)) caxn−ξbe paraxm+eξtr(cid:17)izeAd phenomenologically using the experimental data on the
A
nuclear charge radius or, alternatively, can be evaluated using Eq. (11). In order to get a
first estimate of the influence of nuclear medium on A , in the limit t = 0 let us evaluate the
C
asymmetry ignoring the Fermi motion effects so that α = 1. Neglecting the (x ) effects
Bj
O
in the definition of ξ in Eq. (13) and the (ξ2) effects in the definition of p n in Eq. (20),
1
O
one has p n = α/A = 1/A. Then the nominator of the last term in Eq. (29) (see Eq. (24))
1
becomes
1 1 1
dx P( )+P( ) H (x,ξ,0)
Z−1 (cid:16) x ξ x+ξ (cid:17) A
−
1 1 1
= dx P( )+P( ) ZHp(x,ξ,0)+(A Z)Hn(x,ξ,0) , (30)
Z−1 (cid:16) x ξ x+ξ (cid:17)(cid:16) − (cid:17)
−
where in the last step we have used that ξ′ = ξ in the considered approximation.
The denominator of the last term in Eq. (29) is Fe.m.(0) = Z. In order to quantify
A
the resulting nuclear effects, one can introduce the ratio of the beam-charge asymmetries
measured with the nuclear and the proton targets, A /Aproton. Using Eqs. (29) and (30),
C C
the ratio can be presented in the following form
A (φ) 1 dx P( 1 )+P( 1 ) Hp(x,ξ,0)+(A/Z 1)Hn(x,ξ,0)
C = R−1 (cid:16) x−ξ x+ξ (cid:17)(cid:16) − (cid:17) . (31)
Aproton(φ) 1 dx P( 1 )+P( 1 ) Hp(x,ξ,0)
C R−1 (cid:16) x−ξ x+ξ (cid:17)
7
The immediate consequence of Eq. (31) is that the ratio A /Aproton is greater than unity,
C C
i.e. the beam-charge asymmetry for the nuclear target is larger than the corresponding
asymmetry for the proton.
Thesecondkindofasymmetry, thesinglespinasymmetry, A ,ismeasured byscattering
LU
longitudinally polarized leptons of opposite helicities on an unpolarized hadronic target. The
resulting asymmetry is defined as
Nλ=1(φ) Nλ=−1(φ)
A (φ) = − , (32)
LU Nλ=1(φ)+Nλ=−1(φ)
where Nλ=1 (Nλ=−1) is proportional to the scattering cross section for the incoming positron
with positive (negative helicity). This asymmetry probes the imaginary part of the interfer-
ence between the DVCS and Bethe-Heitler amplitudes. In the notations of Eq. (27), A
LU
can be written in the form
(λ = 1) (λ = 1)
A (φ) = I −I − . (33)
LU
2 2 + (λ = 1) (λ = 1)+ (λ = 1) (λ = 1)
BH DVCS DVCS
|T | I −I − T −T −
In the approximation that the Bethe-Heitler process dominates the cross section, a sim-
plified expression for the single-spin asymmetry can be presented
KsI
A (φ) = sin(φ) 1
LU ycBHFe.m.(t)
0 A
K8y(2 y)x H (ξ,ξ,t) H ( ξ,ξ,t)
Bj A A
= sin(φ) − − − , (34)
ycBH Fe.m.(t)
0 A
where H (ξ,ξ,t) H ( ξ,ξ,t) is given by Eq. (25). In order to obtain a rough estimate of
A A
− −
the influence of nuclear effects on A , the latter can be evaluated in the limit t = 0 and
LU
ξ′ = ξ using Eq. (25). Then one immediately obtains for the ratio of the nuclear to the
proton asymmetries, A /Aproton,
LU LU
A (φ) Hp(ξ,ξ,0) Hp( ξ,ξ,0)+(A/Z 1) Hn(ξ,ξ,0) Hn( ξ,ξ,0)
LU = − − − (cid:16) − − (cid:17) . (35)
Aprotonφ) Hp(ξ,ξ,0) Hp( ξ,ξ,0)
LU − −
Again, like in the case of A /Aproton, one concludes that the single-spin asymmetry for the
C C
nuclear target is larger than that for the proton.
Similarly to the case of the matrix element of the electromagnetic current considered in
Sec. 2, Eqs. (24) and (25) are reference frame dependent: The three-vector ∆ entering the
argument of the nuclear wave function depends on the reference frame. This is an intrinsic
problem of the impulse approximation. In what follows we shall use the laboratory reference
frameandchooseq = (q0,0,0, qz ). Then thevariableα = 1+pz/mandfromthecondition
−| | 1
(q ∆)2 = 0, we obtain that ∆z x m.
bj
− ≈ −
Having fixed the kinematics, we can discuss two important effects not included in the
approximate expressions of Eqs. (29) and (34). Firstly, our exact impulse approximation
expressions, Eqs. (24) and (25), describe non-static bound nucleons so that, on average,
α > 1 (since ∆z < 0, the nuclear wave function favors positive pz), and, hence, ξ′ < ξ.
1
8
This means that the GPDs of the bound nucleons, H and E, are probed at smaller values of
variableξ which leadstoanadditionalenhancement oftheratiosA /Aproton andA /Aproton
C C LU LU
on the top of the ”combinatoric” enhancement by the term (A/Z 1)Hn.
−
Secondly, ourfinalexpressionsfortheasymmetries, Eqs.(29)and(34),whichareobtained
in the approximation of a small momentum transfer ∆ and dominance of the Bethe-Heitler
process, assume that DVCS on a nuclear target is coherent, i.e. the target remains intact.
However, as soon as t = 0, both coherent and incoherent (nucleus breaks up) contributions
6
enter the total cross section so that the expressions for A and A should be modified
C LU
accordingly. In order to achieve this, we generalize the expression for the sum of the incohent
and coherent contributions to the cross section of pion-nucleus production of two jets [15] to
the case of DVCS on nuclei 1.
J T
J T
A A A A
a) b)
Figure 1: The schematic representation of the interference between the Bethe-Heitler (J)
and DVCS (T) amplitudes on nuclei: There are Z attachments of both J and T to the same
proton (a) and Z(A 1) attachments of J to the proton and T to a different nucleon (b).
−
The modified asymmetries become
K8(2 2y +y2)x
Bj
A (φ) = cos(φ) −
C − ycBH
0
1 dx P( 1 )+P( 1 ) ZHp(x,ξ,t)+Z(A 1)Fe.m.(t′)H (x,ξ,t′)
R−1 (cid:16) x−ξ x+ξ (cid:17)(cid:16) − A A (cid:17) (36)
× ZF (t)+Z(Z 1)(Fe.m.(t′))2
(cid:16) 1 − A (cid:17)
and
K8y(2 y)x
Bj
A (φ) = sin(φ) −
LU ycBH
0
1
ThemaindifferencefromRef.[15]isthatinthepresentcaseonedealswithinterferenceoftwoamplitudes:
One is coupled to the protons only and another is coupled to all nucleons with approximately the same
strength.
9
Z(Hp(ξ,ξ,t) Hp( ξ,ξ,t))+Z(A 1)Fe.m.(t′)(H (ξ,ξ,t′) H ( ξ,ξ,t′))
(cid:16) − − − A A − A − (cid:17)(37)
× ZF (t)+Z(Z 1)(Fe.m.(t′))2
(cid:16) 1 − A (cid:17)
where t′ = tA/(A 1). The schematic representation of theoriginof the combinatoric factors
−
Z and Z(A 1) is given in Fig. 1. The first terms in the nominator and denominator of
−
Eqs. (36) and (37) describe the contribution coming from the attachment in the “in” and
“out” states to the same nucleon (it gives the dominant incoherent term at large t) that,
at small t (neglecting the neutron contribution suppressed by the smallness of the electro-
magnetic form factors and also neglecting the contribution of the GPDs E), is proportional
to the number of protons, Z, times the GPD H of the free proton. This contribution has
a slow t-dependence governed by the proton elastic form factor F (t). The contribution
1
given by the second term in the nominator and denominator of Eqs. (36) and (37) is due to
the attachment to two different nucleons. It is mostly coherent and it has a much steeper
t-dependence dictated essentially by the nuclear charge form factor Fe.m.(t).
A
The main point of considering Eqs. (36) and (37) is the following. If the experimental
equipment does not allow to extract the purely coherent DVCS so that Eqs. (29) and (34)
can be used, the experimental asymmetries present a sum of the coherent and incoherent
contributions and are given by Eqs. (36) and (37). While A /Aproton and A /Aproton are
C C LU LU
significantly larger than unity for coherent nuclear DVCS (the ratios of the asymmetries are
close to the factor of two in the considered kinematics), A /Aproton = A /Aproton = 1 for
C C LU LU
the incoherent part. Thus, the inclusion of the incoherent contribution decreases the ratio
of the asymmetries.
In order to illustrate these points, we consider DVCS on spinless nuclei of Neon (A = 20
and Z = 10) and Krypton (spinless isotope with A = 76 and Z = 36). Figure 2 presents
the ratio of the nuclear to proton asymmetries, A /Aproton and A /Aproton, as a function
C C LU LU
of t at x = 0.1 and Q2 = 2.5 GeV2 and fixed cos(φ). This choice of x and Q2 roughly
corresponds to the kinematics of the HERMES DVCS experiment with nuclei [8]. The solid
curves is the full result including both incoherent and coherent contributions; the dashed
curves include onlythecoherent partofthecrosssection. Thefollowing two featuresofFig.2
are of interest. Firstly, the ratio of the asymmetries for the coherent contribution (dashed
curves) is significantly larger than unity. This is expected from the approximate expressions
of Eqs. (31) and (35). As discussed above, an additional t-dependent enhancement arises
because the bound nucleon GPDs, which enter the complete expressions of Eqs. (24) and
(25), are probed at smaller values of ξ than for the free proton.
Secondly, theinclusion oftheincoherent contributionsignificantly reduces theratioofthe
asymmetries (solid curves), especially at larger values of t , where the coherent contribution
| |
is suppressed by the smallness of the nuclear form factor. From the practical point of view,
this means that if the experimental resolution in t is poor, the asymmetries extracted from
the experiment are obtained from the t-averaged data. This means for Eqs. (36) and (37)
that first one integrates separately the nominator and denominator over t and only then
the ratio is taken. Integrating the nominators and denominators of A , A , Aproton and
C LU C
Aproton over t from t = x m /(1 x + x m2 /Q2) to t = 0.1 GeV for Neon
LU | | | |min Bj N − Bj Bj N | |max
(the maximal t measured by HERMES [8]) and to t = 0.05 GeV for Krypton (the
max
| | | |
coherent contribution for Krypton occupies a smaller t-range), and then taking the ratio of
10