Two-photon exchange in electron deuteron scattering

0.12 MB·

by  A. P. Kobushkin

TWO-PHOTON EXCHANGE IN ELECTRON DEUTERON SCATTERING
XX International Baldin Seminar on High Energy Problems
RELATIVISTIC NUCLEAR PHYSICS & QUANTUM CHROMODYNAMICS
Dubna, October 4 — 9, 2010
A.P. Kobushkin1† and Ya.D. Krivenko-Emetov2
1 (1) Bogolyubov Institute for Theoretical Physics
1 Metrologicheskaya Street 14B, 03680 Kiev, Ukraine
0
(2) Institute for Nuclear Research, Prospekt Nauki 47, 03680 Kiev, Ukraine
2
E-mail: [email protected]
n †
a
J

0 Abstract
1
It is shown that the amplitude of elastic ed scattering beyond Born approximation

] contains six generalized form factors, but only three linearly independent combina-
h
t tions of them (generalized charge, quadrupole and magnetic formfactors) contribute
-
l to the reaction cross section in the second order perturbation theory. We examine
c
u two-photon exchange and find that it includes two types of diagrams, when two
n
virtual photons interact with the same nucleon and when the photons interact with
[
different nucleons.We discuss contribution of the two-photon exchange in reaction
1
observables, generalized and structure functions and tensor polarization of the
v
A B
7 deuteron.
6
8
1
.
1
1 Introduction
0
1
1
A study of electromagnetic structure of the deuteron, the simplest nucleon system, provides
:
v
with important information about nucleon-nucleon interaction. Because the deuteron is a
i
X
spin-1 system its electromagnetic current characterized by three form factors, charge G ,
C
r
a quadrupole GQ and magnetic GM form factors. Due to smallness of the fine structure con-
stant α the form factors are usually extracted from experimentally measurable observables in
the framework of Born approximation (one-photon exchange, OPE). Nevertheless theoretical
calculations [1, 2, 3, 4] show that effects beyond OPE may significantly change results of such
procedure.
In what follows we calculate amplitude of two-photon exchange (TPE) of the elastic ed
scattering, one of the mostly important effects beyond OPE, and estimate TPE contribution
in observables of the process.
2 Observables beyond one-photon exchange
From P and T invariance it follows that elastic scattering amplitude of a spin-1 particle
2
(electron) on a spin-1 particle (deuteron) is determined by 12 invariant amplitudes. Putting
the electron mass to zero reduces the number of the invariant amplitude (form factors) to 6
and the spin structure of the amplitude in the Breit frame may be specified by the following
1
parametrization:
cos θ η h h
G11 2 − 2G10 G1,−1
Tλ′λ;h =  η2G1−0h G0p0cos θ2 − η2G1h0 . (1)
−h η −h cos θ
pG1,−1 2G10 G1p1 2
 
Here Tλ′λ;h is reduced amplitude, which is conpnected with the usual amplitude by
16πα
= EeEdTλ′λ,h, (2)
M Q2
λ and λ′ are spin projections of the deuteron and h is sign of electron helicity; E and E are
e d
electron and deuteron energies and θ is the scattering angle in the Breit system; η = Q2/4m2;
d
h = f +hsin θf , h = f +hsin θf . (3)
G10 1 2 2 G1,−1 3 2 4
The form factors , , f , ..., f are complex functions of the two independent kinematical
11 00 1 4
G G
variables, for example Q2 and θ.
In Ref. [4] instead of the form factors , , f , ..., f the following their linear combi-
11 00 1 4
G G
nations were introduced
= 2η , = + 4η ,
G11 GC − 3 GQ G00 GC 3 GQ
f = +g sin2 θ, f = g , (4)
1 GM 1 2 2 GM − 1
f = g , f = g .
3 2 4 3
We call (Q2,θ) and (Q2,θ) the generalized electric, quadrupole and magnetic from
Q M
G G
factors.
Bystandardcalculationsonederivesthedifferentialcrosssectionandcomponentsoftensor
polarization of the deuteron
dσ
=σ ,
M
dΩ S
η [8( e ∗ + η 2)+ 2(1+2tg2θ)]
t = − ℜ GCGQ 3|GQ| |GM| 2 ,
20
3√2
S (5)
√η
t = 2cos θη e ∗ G e sin2 θg +g 2sin2 θ cos θη eg G ,
21 √3 − 2 ℜ GMGQ − Mℜ 2 3 2 − 2 2 ℜ 1 Q
S
cos2(cid:2)θη 2 4sin2 θηG eg (cid:0)+4cos θ(G (cid:1) 2ηG ) eg (cid:3)
t =− 2 |GM| − 2 Mℜ 1 2 C − 3 Q ℜ 2.
22 2√3cos2 θ
2S
In Eqs. (5) σ is the Mott cross section,
M
= + tg2 1θ ,
S A B 2 LAB
(Q2,θ) = (Q2,θ) 2 + 8η2 (Q2,θ) 2 + 2η (Q2,θ) 2, (6)
A |GC(cid:0) (cid:1)| 9 |GQ | 3 |GM |
(Q2,θ) = 4(1+η)η (Q2,θ) 2
B 3 |GM |
and ∗ = G G +δ ∗ + ∗ δ , (K,L) = C,Q,M.
GKGL K L GKGL GK GL
The advantage of using the form factors , and is that the expression for the cross
C Q M
G G G
section and t have the same form as in OPE approximation. Nevertheless the Rosenbluth
20
separation of the structure functions (Q2,θ) and (Q2,θ) can no longer be done because
A B
they depend on two variables.
2
e e′ e e′
p p′ n n′
d d′ d d′
n p
e e′ e e′
N(1) N′(2) N(1) N′(2)
N(2) N′(1) N(2) N′(1)
d d′ d d′
Figure 1: Two-photon exchange diagrams. The top diagrams correspond to the amplitudes
I and I, the bottom diagrams to the amplitudes II and II.
Mp Mn MP MX
3 Calculation of the two-photon exchange
In our calculation of TPE we consider two types of diagrams, where the virtual photons
interact directly with the nucleons
= I + II. (7)
2
M M M
One of them, I = I + I, corresponds to diagrams, where both photons interact with
M Mp Mn
the same nucleon (Fig. 1, top). The other type, II = II + II, corresponds to the
M MP MX
diagrams, where the photons interact with different nucleons (Fig. 1, bottom).
Important input in calculation of I is TPE amplitude for a nucleon N. It has the
M
following structure [5]
4πα
= u¯′γ u p~′ σ′ Hµ p~ σ , (8)
M2γN Q2 h µ h N N N
D (cid:12) (cid:12) E
(cid:12) (cid:12)
where Hµ is the “effective hadron current” (cid:12)b (cid:12)
N
q Pµ
b Hµ = ∆FNγµ ∆FN[γµ,γν] ν +FNK γν . (9)
N 1 − 2 4m 3 ν m2
N N
In Eqs. (8) and (9) pb and pe′ are theenucleon momentae, σ and σ′ are the nucleon spin
N N
projections, p~ σ and p~′ σ′ are the nucleon spinors, K = (k + k′)/2, P = (p + p′ )/2;
N N N N
| i | i
∆FN and ∆FN may be called corrections to the Dirac and Pauli form factors of nucleon N
1 2
and FN is a new form factor. All the quantities ∆FN, ∆FN and FN are of order α. They
3 1 2 3
areecomplex feunctions of two kinematical variables, e.g. Q2 and ν = 4PK.
Ceonsidering the deuteron structure nonrelativisteically oene gets ethe following expressions
3
◦
θ = 180
10−5
)
4
/
2
Q
(
2d
G
/
B 10−6
10−7
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
2 2
Q (GeV/c)
Figure 2: Results of calculations for at θ = 180◦ (the curves are explained in the text).
LAB
B
G (t) = (1+t/0.71)−2 is dipole form factor. Experimental data are from [7].
d
for appropriate TPE form factors for the elastic ed scattering [4]
3√2 1
δ I = 2δ S I0 (Q)+I0 (Q) , δ I = δ S I2 (Q) I2 (Q) ,
GC GE 00 22 GQ η GE 20 − 2√2 22
(cid:20) (cid:21)
M (cid:2) (cid:3)
δ I = 3δ S I0 (Q)+I2 (Q) +2δ S I0 (Q) 1I0 (Q)+ 1I2 (Q)+ 1I2 (Q) ,
GM m 2 GE 22 22 GM 00 − 2 22 2 20 2 22
(cid:26) (cid:20) q (cid:21)(cid:27)
E (cid:2) (cid:3)
gI = ǫ e , gI = gI = 0,
1 − mF3 2 3
where = 2MFS I0 (Q) 1I0 (Q)+ 1I2 (Q)+ 1I2 (Q) , generalized nucleon electric
F3 m 3 00 − 2 22 2 20 2 22
and magnetic form fahctors are defined byq(see Ref [6]) i
e
ν ǫν
δ N = ∆FN τ∆FN + FN, δ N = ∆FN +∆FN + FN,
GE 1 − 2 4m2 3 GM 1 2 4m2 3
N N
uτ ≈(r)4ηis,νth≈e rmadNiaEledaenuedteǫriosnthweacveoemfumnocntiloynuesfoerdoproblaitraizlamtioomneepnatruammeℓteearn,dIℓLδ′ℓ(QS)==e1(δ0∞pdr+jLδ 21nQ),r uℓ′(r)uℓ(r),
ℓ GE 2R GE (cid:0)GE (cid:1)
etc.
The amplitude II was calculated within hard-photon approximation, i.e. assuming that
M
each intermediate photon carries about half of the transferred momentum ∆ ∆ q.
1 ∼ 2 ∼ 2
Omitting tedious calculations (see Ref. [4]), we obtain
κ
δ II = κ G 1ηG , δ II = G ,
GC EE − 3 MM GQ −2 MM
2(cid:0)κG (cid:1)κG cos2 θ (10)
δ II = EM , gII = EM 2, gII = gII = κηcos θG .
GM 1+sin2 θ 1 1+sin2 θ 2 3 2 MM
2 2
4
10−3
◦
θ = 120
)
4
/ 10−4 A
2
Q
(
2d
G
/
)
B
d
n
a 10−5
A
(
B
10−6
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
2 2
Q (GeV/c)
Figure 3: and at θ = 120◦. Full curve for ONE+TPE (TPE is calculated with
LAB
A B
CD-Bonn deuteron wave function), dot-dashed for ONE approximation.
Here
G = Gp(1Q2)Gn(1Q2), G = Gp (1Q2)Gn (1Q2),
EE E 4 E 4 MM M 4 M 4 (11)
G = 1 Gp(1Q2)Gn (1Q2)+Gp (1Q2)Gn(1Q2)
EM 2 E 4 M 4 M 4 E 4
and (cid:2) (cid:3)
128αE 1 d3pd3p′U (p)U (p′)
e 0 0
κ = , where = . (12)
− Q4 C C (2π)3 4E E (p p′)
Z 1+ e (p +p′) 8cos θ e x − x +i0
Qm z z − 2 Q2
d
In Eq. (12) U (p) is S-component of the deuteron wave function in the momentum representa-
0
tion. To evaluate the integral above one can use the integral representation for the propagator
1
= i ∞dτei(α+i0)τ and reduce to a one-dimensional integral
α+i0 − 0 C
R
−1
∞ dy Q2
= if eifyu2(y), where f = Q2 4E 4cos2 θ + . (13)
C − Z0 y2 0 " es 2 m2d#
4 Numerical calculations and conclusions
In Figure 2 we compare results of our calculations for at θ = 180◦ with ONE analysis
LAB
B
of Ref. [10] (dot-dashed). For TPE calculations we have used the deuteron wave functions
for CD-Bonn and Paris potentials. ONE+TPE with TPE calculated with CD-Bonn deuteron
[8] wave function and Paris [9] deuteron wave function are given by full and dashed curves,
respectively. One sees that at Q2 > 2 GeV2 TPE contribution becomes more than 10% in ,
B
while in it is not significant, see Figure 3.
A
5
0.2
0.15
0.1
0.05
2 0
2
t
-0.05
-0.1
-0.15
-0.2
0 0.5 1 1.5 2 2.5
2 2
Q (GeV)
Figure 4: t at θ = 70◦. Dashed line is for OPE-approximation, full and dashed-dot lines
22 LAB
are for OPE+TPE with TPE calculated with CD-Bonn and Paris deuteron wave functions,
respectively. Data are from [12] and [11] (circles and boxes, respectively).
TPE effect in t was found significant (Figure 4), but in t and t its contribution is
22 20 21
minor (<1%).
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(2010).
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