PHYSICAL REVIEW C
VOLUME 58, NUMBER 2
AUGUST 1998
Low-energy Compton scattering of polarized photons on polarized nucleons
D. Babusci,1 G. Giordano,1 A. I. L’vov,2 G. Matone,1 and A. M. Nathan3
1
INFN-Laboratori Nazionali di Frascati, Frascati, Italy
2
P.N. Lebedev Physical Institute, Moscow, Russia
3
Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green Street, Urbana, Illinois 61801
~Received 13 March 1998!
The general structure of the cross section of g N scattering with polarized photon and/or nucleon in initial
and/or final state is systematically described and exposed through invariant amplitudes. A low-energy expansion of the cross section up to and including the order O( v 4 ) is given that involves ten structure parameters of
the nucleon ~dipole, quadrupole, dispersion, and spin polarizabilities!. Their physical meaning is discussed in
detail. Using fixed-t dispersion relations, predictions for these parameters are obtained and compared with
results of chiral perturbation theory. It is emphasized that Compton scattering experiments at large angles can
fix the most uncertain of these structure parameters. Predictions for the cross section and double-polarization
asymmetries are given and the convergence of the expansion is investigated. The feasibility of the experimental
determination of some of the structure parameters is discussed. @S0556-2813~98!02407-8#
PACS number~s!: 25.20.Dc, 13.60.Fz, 11.80.Cr, 13.88.1e
I. INTRODUCTION
Compton scattering on the proton at low and intermediate
energies has thus far been studied mainly with unpolarized
photons. Many recent data are available on the unpolarized
differential cross section both in the region below pion
threshold @1–3# and in the delta region @4–8#. They have led
to a determination of the dipole electric and magnetic polarizabilities of the proton, have given useful constraints on
pion photoproduction amplitudes near the delta peak, and
have provided sensitive tests for different models of Compton scattering, such as those based on resonance saturation
@9–12#, chiral perturbation theory @13,14#, and dispersion relations @15–19#.
With the advent of new experimental tools such as highly
polarized photon beams, polarized targets, and recoil polarimetry @20#, it becomes possible to study the very rich spin
structure of Compton scattering. In particular, many additional structure parameters of the nucleon, such as the spin
@21–24# and quadrupole @25# polarizabilities, could be measured in such new-generation experiments and used for testing hadron models at low energies. The first attempt to determine the ‘‘backward’’ spin polarizability from
unpolarized experiments has recently been reported @26#.
Therefore, it is timely to give a detailed description of the
appropriate polarization observables and their relationship to
the low-energy parameters that might be measured in such
experiments. That is the main purpose of the present report.
This paper is organized as follows. In Sec. II, we introduce the invariant Compton scattering amplitudes. In Sec. III
we develop a general structure for the g N scattering cross
section with polarized photons and/or polarized nucleons in
the initial and/or final state. In Sec. IV we do a low-energy
expansion of the invariant amplitudes and develop formulas
for the low-energy expansion of the cross section and spin
observables. In the process, we introduce and discuss the
physical meaning of the parameters ~polarizabilities! which
are required to describe the cross section up to and including
the order O( v 4 ). In Sec. V we give theoretical predictions
0556-2813/98/58~2!/1013~29!/$15.00
PRC 58
for all these polarizabilities by using fixed-t dispersion relations and compare them with available predictions of chiral
perturbation theory. We then investigate the range of validity
of our low-energy expansion and show that it is generally
valid below the pion threshold. Finally, we investigate quantitatively the dependence of different observables on the polarizabilities and recommend particularly sensitive experiments to perform. Some of the details related to the
definitions and physical meaning of the polarizabilities are
contained in the appendixes.
II. INVARIANT AMPLITUDES
The amplitude T f i for Compton scattering on the nucleon,
g ~ k ! N ~ p ! → g 8~ k 8 ! N 8~ p 8 ! ,
~2.1!
^ f u S21 u i & 5i ~ 2 p ! 4 d 4 ~ k1p2k 8 2 p 8 ! T f i .
~2.2!
is defined by
Constrained by P and T invariance, it can be expressed in
terms of six invariant amplitudes T i as @27,28,15,17,18#
H
T f i 5ū 8 ~ p 8 ! e 8 * m 2
2
N mN n
1i
N2
P m8 P n8
P 82
~ T 1 1 g •KT 2 !
~ T 3 1 g •KT 4 ! 1i
P m8 N n 1 P 8n N m
P 82K 2
P m8 N n 2 P 8n N m
J
P 82K 2
g 5 g •KT 6 e n u ~ p ! ,
g 5T 5
~2.3!
where e and e 8 are the photon polarization vectors, u and u 8
are the bi-spinors of the nucleons (ūu52m, m is the nucleon
mass!, and g 5 5( 01 10 ). The orthogonal four-vectors P 8 , K, N,
and Q are defined as
1013
© 1998 The American Physical Society
1014
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
P m8 5 P m 2K m
P•K
K2
,
1
P5 ~ p1p 8 ! ,
2
N m 5 e mabg P 8 a Q b K g ,
1
K5 ~ k 8 1k ! ,
2
T 2 52A HL
4 ,
T 3 5A HL
2 ,
A 55
T 6 52A HL
6 .
s5 ~ k1 p ! ,
u5 ~ k2p 8 ! ,
2
t5 ~ k2k 8 !
~2.6!
H
12ms8 * •s vv 8
FS
FS
12
A 65
~2.7!
1
~ T 2T 4 ! ,
4n 2
t2
m
4m 2
~ m 4 2su ! 54 n 2 1t2
2
t522 vv 8 ~ 12z ! ,
.
~2.8!
h 52 vv 8 ~ 11z ! ,
~2.9!
v 85 v 1
F
v
t
5 v 11 ~ 12z !
2m
m
G
21
.
~2.10!
We will reserve the symbols v , v 8 , and z for these lab-frame
variables. Note that in the lab frame, N m 5(0,N), where N
5(m/2)k8 3k is orthogonal to the reaction plane.
In terms of the A i , the Compton scattering amplitude T f i
in the lab frame assumes the following form:
1
A 2 5 @ 2T 5 1 n ~ T 2 1T 4 !# ,
t
1
2me8 * •e vv 8
N~ t !
t
~ T 2T 4 ! ,
4n 2
where v 5k 0 , v 8 5k 08 are the photon energies, z5cos u is
the photon scattering angle, and
are the usual Mandelstam variables and s1u1t52m .
These functions have no kinematical singularities but they
are subject to kinematical constraints arising from the vanishing of the denominators in the decomposition ~2.3! in
cases of forward or backward scattering. Therefore, it is useful to define the following linear combinations @18,19,29#:
T f i5
2mT 6 2
1
1
n5 ~ v1v8!,
2
2
1
A 1 5 @ T 1 1T 3 1 n ~ T 2 1T 4 !# ,
t
1
h
G
G
t
~ T 2T 4 ! ,
4n 2
The amplitudes A i ( n ,t) are even functions of n , they have no
kinematical singularities or constraints, and they have dimension m 23 .
In the lab system ~the nucleon N at rest! the kinematic
invariants n , t and h read
~2.5!
2
T 1 2T 3 2
1
~ T 1T 4 ! ,
4n 2
h5
The amplitudes T i are functions of the two variables
n 5(s2u)/4m and t, where
2
F
F
with
T 5 52A HL
3 ,
T 4 52A HL
5 ,
h
A 45
1
1
Q5 ~ p2p 8 ! 5 ~ k 8 2k ! ,
2
2
~2.4!
where the antisymmetric tensor e mabg is fixed by the condition e 012351.
Unfortunately, there is no accepted standard for the definitions of the invariant amplitudes in nucleon Compton scattering. The definitions we use here may differ from those
used in other works. We follow the conventions of Refs.
@17–19#, which are related to the so-called Hearn-Leader
amplitudes A HL
used in Refs. @15,28# by
i
T 1 5A HL
1 ,
1
A 35
PRC 58
12
t
4m 2
t
4m 2
D
D
~ 2A 1 2A 3 ! 2
~ A 1 2A 3 ! 1
n2
m2
n2
m2
A 5 2A 6
A 5 2A 6
G
G
22i s•e8 * 3e nvv 8 ~ A 5 1A 6 ! 12i s•s8 * 3s nvv 8 ~ A 5 2A 6 !
F S D
F S D
F S D
1i s•k̂ s8 * •e v 2 v 8 A 2 1 12
v8
n
A 4 1 A 5 1A 6
m
m
2i s•k̂8 e8 * •s vv 8 2 A 2 1 11
v
n
A 4 2 A 5 1A 6
m
m
2i s•k̂ e8 * •s v 2 v 8 2A 2 1 12
F
G
G
v8
n
A 4 2 A 5 1A 6
m
m
S D
1i s•k̂8 s8 * •e vv 8 2 2A 2 1 11
v
n
A 1 A 1A 6
m 4 m 5
G
GJ
,
~2.11!
PRC 58
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
where N(t)5(12t/4m 2 ) 1/2 and the two magnetic vectors s,
s8 are defined as
s5k̂3e,
s8 5k̂8 3e8 .
~2.12!
or c.m. frame. We will use the prime to distinguish the
Stokes parameters for initial or final photon, j i and j i8 .
Note that the above-defined Stokes parameters transform
under parity as
P
We will consider the general structure of the cross section
and double-polarization observables in four related reactions:
W → g 8N 8,
gW N
~3.1a!
W → gW 8 N 8 ,
gN
~3.1b!
W 8,
gW N→ g 8 N
~3.1c!
W 8.
g N→ gW 8 N
~3.1d!
We start by introducing the polarization variables.
Photon polarization properties are conveniently described
in terms of the Stokes parameters j i (i51,2,3) @30# which
define the photon polarization matrix density as follows @31#:
S
1
1 11 j 3
^ e a e b* & 5 ~ 11 s• j! ab 5
2
2 j 1 1i j 2
j 1 2i j 2
12 j 3
D
.
ab
~3.3!
and j 2 are Lorentz invariant. They give the degree of linear
and circular polarization, respectively. Moreover, j 2 561
corresponds to the right ~left! helicity state, provided the xyz
frame is right handed. The total degree of photon polarization is given by j 5 Aj 21 1 j 22 1 j 23 <1. The values j 1 and j 3
separately are frame dependent, although they are still invariant with respect to boosts or rotations in the x g z g -plane.
They define the angle w that the electric field makes with the
x g z g -plane:
cos 2w 5
j3
,
jl
sin 2w 5
j1
.
jl
~3.4!
To fix the azimuthal freedom in j 1 and j 3 , we first choose a
frame in which all the momenta k, p, k8 and p8 are coplanar.
This choice is not too restrictive and includes both the lab
and c.m. frames. In such a frame and for any polarized photon, either g or g 8 , we take the y g axis in Eq. ~3.2! to lie
along the direction of k̂3k̂8 . Then the appropriate z g axis is
given by k̂ or k̂8 , and the x g axis is directed along (k̂3k̂8 )
3k̂ or (k̂3k̂8 )3k̂8 , respectively. The angle w in Eq. ~3.4!
gives the angle between the electric field and the reaction
plane. Thus defined, the Stokes parameters do not depend on
further specification of the frame and are the same in the lab
P
j 2→ 2 j 2 ,
j 3→ j 3 ,
~3.5!
under time inversion (k̂→2k̂ e→2e* ) as
T
T
j 1→ 2 j 1 ,
T
j 2→ j 2 ,
j 3→ j 3 ,
~3.6!
and under crossing (k̂→k̂8 e→e8 * ) as
cross
j 1 → j 18 ,
cross
cross
j 2 → 2 j 28 ,
j 3 → j 38 .
~3.7!
The nucleon polarization density matrix is specified by a
polarization four-vector S which is orthogonal to the nucleon
four-momentum p @31#:
1
2
^ u ~ p ! ū ~ p ! & 5 ~ g •p1m !~ 11 g 5 g •S ! .
~3.8!
Introducing also the polarization three-vector z in the
nucleon rest frame, one can relate z and S through the boost
transformation,
~3.2!
Here the photon polarization vector e m is taken in the radiation gauge e•k̂50 and a , b 51,2 denote either of two orthogonal directions x g ,y g which are in turn orthogonal to the
photon momentum direction z g 5k̂. Such a definition of j i is
manifestly frame dependent; nevertheless, the quantities
j l 5 Aj 21 1 j 23
P
j 1→ 2 j 1 ,
III. CROSS SECTIONS AND ASYMMETRIES
A. General structure of the cross section
1015
S5 z1
S0
p,
p 0 1m
S 05
p•S p• z
,
5
p0
m
~3.9!
where A2S 2 5 u zu <1 gives the degree of nucleon polarization. We apply the notations S, z, and S 8 , z8 for the initial
and final nucleons, respectively.
Note that the vectors z, z8 are frame dependent and undergo Wigner rotation around the y axis when a boost in the
reaction plane is applied. Nevertheless, z is the same in the
lab and c.m. frames, although that is not the case for z8 .
In both the lab and c.m. frames, the differential cross
section of the reactions ~3.1! reads
ds
5F 2 u T f i u 2
dV
with
F5
5
1 v8
,
8pm v
1
8 p As
,
lab frame,
c.m. frame.
~3.10!
Here the square of the Lorentz-invariant matrix element T f i ,
appropriately averaged and summed over polarizations, has
the same generic form in all four cases ~3.1!,
3
u T f iu 25
( @ W 0i 1K•SW 1i 1Q•SW 2i 1N•SW 3i # j i ,
i50
~3.11!
where we set j 0 51, and for the moment we disregard optional primes distinguishing polarization variables for initial
and final particles. Since S and N are axial vectors and j 1 , j 2
have odd P parity, some of the invariant functions W i j must
vanish:
1016
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
W 015W 025W 105W 135W 205W 235W 315W 3250.
~3.12!
W 8 ! u 2 5W 001W 03j 38 1N•S 8 ~ W 301W 1
u T f i ~ g N→ gW 8 N
33j 38 !
8 1W 1
8!
1K•S 8 ~ 2W 1
11j 1
12j 2
In terms of the remaining W i j , the generic expression ~3.11!
gets the following specific form for the individual cases
given in Eq. ~3.1!:
W → g 8 N 8 ! u 2 5W 001W 03j 3 1N•S ~ W 301W 1
u T f i ~ gW N
33j 3 !
1
1K•S ~ W 1
11j 1 1W 12j 2 !
1
1Q•S ~ W 1
21j 1 1W 22j 2 ! ,
~3.13a!
W → gW 8 N 8 ! u 2 5W 001W 03j 83 1N•S ~ W 301W 2
u T f i~ g N
33j 8
3!
1
1Q•S 8 ~ W 1
21j 8
1 2W 22j 8
2 !.
W 00~ n ,t ! 5W 00~ 2 n ,t ! ,
W 30~ n ,t ! 52W 30~ 2 n ,t ! ,
W 03~ n ,t ! 5W 03~ 2 n ,t ! ,
1
W2
33~ n ,t ! 52W 33~ 2 n ,t ! ,
~3.13b!
~3.14!
W 8 ! u 2 5W 001W 03j 3 1N•S 8 ~ W 301W 2
u T f i ~ gW N→ g 8 N
33j 3 !
2
1K•S 8 ~ W 2
11j 1 1W 12j 2 !
2
1Q•S 8 ~ W 2
21j 1 1W 22j 2 ! ,
~3.13c!
~3.13d!
Note that the same functions W i j determine the cross section
in Eqs. ~3.13a! and ~3.13d! @as well as in Eqs. ~3.13b! and
~3.13c!# which are related through T invariance; the negative
signs in Eqs. ~3.13b!, ~3.13d! are easily found from Eq. ~3.6!.
The relationship between the cases ~3.13a! and ~3.13b! is
determined by the crossing symmetry of the amplitude T f i
and Eq. ~3.7!:
2
1K•S ~ 2W 2
11j 8
1 1W 12j 8
2!
8 2W 2
8!,
1Q•S ~ W 2
21j 1
22j 2
PRC 58
1
W2
11~ n ,t ! 5W 11~ 2 n ,t ! ,
1
W2
12~ n ,t ! 5W 12~ 2 n ,t ! ,
1
W2
21~ n ,t ! 5W 21~ 2 n ,t ! ,
1
W2
22~ n ,t ! 5W 22~ 2 n ,t ! .
In terms of the invariant amplitudes T i or A i , the functions W i j read ~cf. Ref. @32#!:
1
1
W 005 ~ 4m 2 2t !~ u T 1 u 2 1 u T 3 u 2 ! 2 ~ s2m 2 !~ u2m 2 !~ u T 2 u 2 1 u T 4 u 2 ! 1m ~ s2u ! Re ~ T 1 T *
2 1T 3 T *
4!
2
2
2t u T 5 u 2 1 ~ m 4 2su ! u T 6 u 2
1
1
1
5 ~ 4m 2 2t !~ t 2 u A 1 u 2 1 h 2 u A 3 u 2 ! 2 ~ t 3 u A 2 u 2 2 h 3 u A 4 u 2 ! 2 n 2 t ~ t18 n 2 ! u A 5 u 2 1 h ~ t 2 12m 2 h ! u A 6 u 2
4
4
2
H
J
1 2
2
1 Re 2 n 2 t 2 ~ A 1 1A 2 ! A *
5 1 h ~ 4m A 3 1tA 4 ! A *
6 ,
2
~3.15a!
1
1
W 035 ~ 4m 2 2t !~ u T 1 u 2 2 u T 3 u 2 ! 2 ~ s2m 2 !~ u2m 2 !~ u T 2 u 2 2 u T 4 u 2 ! 1m ~ s2u ! Re ~ T 1 T *
2 2T 3 T *
4!
2
2
5
ht
2
Re $ @~ 4m 2 2t ! A 1 14 n 2 A 5 # A 3* 14m 2 A 1 A 6* % ,
~3.15b!
W 30524 Im ~ T 1 T *
2 1T 3 T *
4!
528 n Im ~ tA 1 A 5* 1 h A 3 A 6* ! ,
~3.15c!
W6
3354 Im @ 2T 1 T *
2 1T 3 T *
4 62T 5 T *
6#
H
5 Im 28 n @@ tA 1 2 ~ t14 n 2 ! A 5 # A *
6 1 h A 3A *
5 #6
J
2
~ tA 2 24 n 2 A 5 !~ h A *
4 1tA *
6! ,
m
~3.15d!
2
W6
115 Im @~ 4m 2t !~ T 1 1T 3 ! T *
6 1m ~ s2u !~ T 2 1T 4 ! T *
6 7t ~ T 2 2T 4 ! T *
5#
5 Im
H
J
t
2
@~ 4m 2 2t ! A 1 14 n 2 A 5 #~ h A *
4 1tA *
6 ! 62 n t ~ tA 2 24 n A 5 ! A *
6 ,
2m
~3.15e!
PRC 58
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
1017
2
W6
125Re @~ 4m 2t !~ 2T 1 1T 3 ! T *
6 1m ~ s2u !~ 2T 2 1T 4 ! T *
6 6t ~ T 2 1T 4 ! T *
5#
H
5Re 2
h
2m
J
2
@~ 4m 2 2t ! A 3 14m 2 A 6 #~ h A *
4 1tA *
6 ! 62 n t ~ tA 2 24 n A 5 ! A *
5 ,
~3.15f!
W6
215Im@ 4m ~ T 1 2T 3 ! T *
5 1 ~ s2u !~ T 2 2T 4 ! T *
5 7mt ~ T 2 1T 4 ! T *
6 6 ~ s2u !~ T 1 1T 3 ! T *
6#
2
2
52Im$ 2m ~ tA 2 24 n 2 A 5 !@ h A *
3 1 ~ t14 n ! A *
6 # 6 n @ tA 1 2 ~ t14 n ! A 5 #~ h A *
4 1tA *
6 !%,
~3.15g!
W6
225Re@ 24m ~ T 1 1T 3 ! T *
5 2 ~ s2u !~ T 2 1T 4 ! T *
5 6mt ~ T 2 2T 4 ! T *
6 7 ~ s2u !~ T 1 2T 3 ! T *
6#
52Re$ 2mt ~ tA 2 24 n 2 A 5 ! A *
1 7 n h A 3~ h A *
4 1tA *
6 !%.
Below the pion photoproduction threshold, the amplitudes
T i , A i are real, and therefore only the six structures, W 00 ,
6
W 03 , W 6
12 , and W 22 , are different from zero.
B. Asymmetries
The thirteen invariant quantities W i j standing in Eq.
~3.13! intervene directly in the definition of polarization observables which can be measured experimentally. Not all of
them are independent, since with six complex invariant amplitudes there are only 11 independent observables. In the
following we shall classify them according to the number of
polarization degrees of freedom involved in the process. We
shall define some n-index asymmetries S i , S a , and S i a ,
where i5(1,2,3) or (1 8 ,28 ,38 ) refers to the photon Stokes
parameters j i or j 8i , and a 5(x N ,y N ,z N ) or (x N8 ,y N8 ,z N8 ) refers to the right-handed axes along which the nucleon spin z
or z8 may be aligned. At this point we have to relate the
products such as K•S, Q•S, and N•S in Eq. ~3.13! to the
spin vector z ~or z8 ), and eventually fix a frame. We choose
the lab frame for practical reasons; for any other frame only
K,Q
the coefficients C x,z,x 8 ,z 8 introduced below have to be recalculated. We will indicate all necessary changes to be done
for the c.m. frame.
Directing the y N and y N8 axes along k3k8 , we choose the
z N axis along the photon beam direction k̂. Note that since
zc.m.5 zlab , all the asymmetries obtained in this way in the
lab frame for the reactions ~3.1a!,~3.1b! with the polarized
nucleon N are the same in the c.m. frame. In the case of the
polarized final nucleon, we choose the z N8 axis to lie along
the nucleon recoil momentum p8 . Such an axis depends on
whether we use the lab or c.m. frame so that we get different,
though related, asymmetries.
We can further simplify the notation. The axes
(x g ,y g ,z g ) and (x N ,y N ,z N ) are identical and we will denote
them in the following by simply (x,y,z). For (x N8 ,y N8 ,z N8 ) we
will use (x 8 ,y 8 ,z 8 ). Note that, despite the fact that the y and
y 8 axes are the same, we will keep the prime when using y 8
because it identifies which nucleon (N or N 8 ) we are referring to. All these axes are shown in Fig. 1.
The set of observables is defined as follows.
Polarization-independent observable (n50), or simply
the unpolarized cross section,
d s̄
5F 2 W 00 .
dV
~3.16!
~3.15h!
Single-polarization observables (n51), of which there
are two for each of the reactions ~3.1!:
~i! The beam asymmetry for photons which are linearly
polarized either parallel or perpendicular to the scattering
plane ( j 3 561) and unpolarized nucleons ~target and recoil!. The same quantity gives the degree of linear polarization ( j 83 ) of the photon scattered from unpolarized nucleons:
S 3 5S 3 8 5
S
s i 2 s'
i
'
s 1s
D
5
z 5 z 8 50
S
s i 8 2 s' 8
i8
'8
s 1s
D
5
z 5 z 8 50
W 03
W 00
~3.17!
~the primed polarizations i 8 or ' 8 refer to the final photon
state!. This asymmetry is often designated as S.
~ii! The target asymmetry or recoil polarization for unpolarized photons, whereby either N or N 8 is polarized ( z y or
z 8y 561) along the 6y56y 8 directions:
S y 5S y 8 5
5C Ny
S
s y 2 s 2y
s y 1 s 2y
W 30
.
W 00
D
5
j 5 j 8 50
S
s y 8 2 s 2y 8
s y 8 1 s 2y 8
D
j 5 j 8 50
~3.18!
Here the coefficient
FIG. 1. Axes to project out polarizations in the lab frame. The
various y axes (y g , y N , y g8 , and y N8 ) all point out of the plane of
the figure.
1018
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
C Ny 5
m
m
vv 8 sin u 5 A2 h t
2
4
~3.19!
~as given through both lab and invariant quantities! determines the scalar product N•S5C Ny z y . It is just the negativey component of N m . The asymmetry S y 5S y 8 is often designated as P, the recoil nucleon polarization.
Double-polarization observables (n52), of which there
are five for each of the reactions ~3.1!:
~i! The beam-target asymmetry for incoming linearly polarized photons, either parallel or perpendicular to the scattering plane ( j 3 561), and target nucleon polarized perpendicular to the scattering plane. The same quantity can be
measured as the correlation of polarizations of the outgoing
particles g 8 and N 8 :
S 3y 5S 3 8 y 8 5
5
~ s i 2 s' ! y 2 ~ s i 2 s' ! 2y
'
i
~ s 1 s ! y 1 ~ s 1 s ! 2y
~ s i 8 2 s' 8 ! y 8 2 ~ s i 8 2 s' 8 ! 2y 8
~ s i 8 1 s' 8 ! y 8 1 ~ s i 8 1 s' 8 ! 2y 8
5C Ny
W1
33
W 00
.
~3.20!
Another independent quantity is the correlation of the beam
polarization and the polarization of recoil nucleon, or the
correlation of the target polarization with the final photon
polarization:
S 3y 8 5S 3 8 y 5
5
S 28z5
~ s i 2 s' ! y 8 2 ~ s i 2 s' ! 2y 8
'8
i8
s R1 s
W 001N•SW 30
s Rx 2 s Lx
Q 1
C Kx W 1
121C x W 22
s Rx 1 s x
W 00
S 2z 5
5
L
s Rz 2 s Lz
s Rz 1 s Lz
5
,
Q 1
C Kz W 1
121C z W 22
W 00
S 2z 8 5
R
L
R
L
R
L
R
L
s x82 s x8
s x81 s x8
s z82 s z8
s z81 s z8
~3.24!
.
Q
Q•S5C Q
x z x 1C z z z . ~3.25!
S 28z85
2
C x8W 2
121C x 8 W 22
K
5
W 00
2
C z8W 2
121C z 8 W 22
5
Q
W 00
,
1
C x8W 1
122C x 8 W 22
K
5
s x 88 1 s x 88
L
s z 88 2 s z 88
Q
W 00
1
C z8W 1
122C z 8 W 22
K
L
5
s z 88 1 s z 88
R
,
~3.27!
L
R
Q
K
s x 88 2 s x 88
R
S 28x85
L
,
Q
W 00
.
K,Q
They are given by the coefficients C x 8 ,z 8 appearing in the
expansions
Q•S 8 5C x 8 z 8x 8 1C z 8 z z88 ,
~3.28!
K•S 8 5C x 8 z 8x 8 1C z 8 z z88 ,
K
K
Q
Q
which in the lab frame read
K
~3.23!
C x 8 52
Ah
2N ~ t !
Q
and the scattered photon-target asymmetries
W 00
Numerically, C K,Q
x,z are the same in the lab and c.m. frames.
Another four asymmetries, which are depend linearly on
those in Eqs. ~3.23! and ~3.24!, describe the correlations of
the recoil polarization with the polarization of the photons g
or g 8 :
R
~3.22!
Such a quantity survives when the nucleon spin lies in the
reaction plane. Considering the cases with the target spin z
aligned in 6x or 6z directions, we can introduce the beamtarget asymmetries
S 2x 5
s Rz 8 1 s Lz 8
Q 2
C Kz W 2
122C z W 22
,
1
t ~ s1m 2 !
CQ
5
v
2
v
cos
u
52
.
!
~
8
z
2
4m ~ s2m 2 !
S 2x 8 5
~ii! The asymmetries with circular photon polarizations
( j 2 or j 82 561). The general expression for the asymmetry
with circularly polarized photons follows from Eq. ~3.13!.
For example, for the reaction ~3.1a!
.
5
W 00
1
m A2 h t
,
C Kx 5C Q
x 52 v 8 sin u 52
2
2 ~ s2m 2 !
'8
1
K•SW 1
121Q•SW 22
s Rz 8 2 s Lz 8
Q 2
C Kx W 2
122C x W 22
In terms of lab or invariant quantities, they read
~3.21!
5
L
s Rx 8 1 s Lx 8
K•S5C Kx z x 1C Kz z z ,
W2
~ s 2 s ! y 2 ~ s 2 s ! 2y
33
N
5C
.
y
i8
i8
'8
'8
W
00
~ s 1 s ! y 1 ~ s 1 s ! 2y
s R2 s L
5
Here the coefficients C K,Q
x,z determine the scalar products
~ s i 1 s' ! y 8 1 ~ s i 1 s' ! 2y 8
i8
s Rx 8 2 s Lx 8
t ~ s1m 2 !
1
s2m 2
2
C Kz 52 ~ v 1 v 8 cos u ! 52
,
2
2m
4m ~ s2m 2 !
~3.26!
'
i
S 28x5
PRC 58
,
C z85
Q
C x 8 50,
A2t
2
N~ t !
K
C z85
~ lab! ,
n A2t
,
2mN ~ t !
~3.29!
PRC 58
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
where N(t)5(12t/4m 2 ) 1/2. Since the z 8 axis in the c.m.
frame is different from that in the lab frame, the correspondK,Q
ing C x 8 ,z 8 are different too. Except for the signs, they coincide with C K,Q
x,y in Eq. ~3.26!:
K
Q
C x 8 52C x 8 5C Kx 5C Q
x ,
Q
C z 8 5C Q
z
K
C z 8 52C Kz ,
s p /41 s 2 p /4
5
1
K•SW 1
111Q•SW 21
W 001N•SW 30
S 1x 5
S 1z 5
p /4
s px /41 s 2
x
5
p /4
s zp /42 s 2
z
~3.31!
.
Q 1
C Kx W 1
111C x W 21
W 00
Q 1
C Kz W 1
111C z W 21
5
2 p /4
s zp /41 s z
W 00
,
~3.32!
S 18x5
S 18z5
p /48
s px /48 2 s 2
x
p /48
s px /48 1 s 2
x
p /48
s zp /48 2 s 2
z
p /48
s zp /48 1 s 2
z
5
5
Q 2
2C Kx W 2
111C x W 21
W 00
Q 2
2C Kz W 2
111C z W 21
W 00
,
,
and a similar linearly dependent set
S 1x 8 5
S 1z 8 5
S 18x85
S 18z85
p /4
2 p /4
p /4
2 p /4 5
p /4
2 p /4
p /4
2 p /4 5
s x8 2 s x8
2
C x8W 2
111C x 8 W 21
s x8 1 s x8
W 00
K
Q
s z8 2 s z8
2
C z8W 2
111C z 8 W 21
s z8 1 s z8
W 00
p /4
2 p /4
s x8 82 s x8 8
p /4
2 p /4
s x8 81 s x8 8
p /4
2 p /4
s z8 82 s z8 8
p /4
2 p /4
s z8 81 s z8 8
K
,
,
K
Q
1
2C x 8 W 1
111C x 8 W 21
W 00
1
2C z 8 W 1
111C z 8 W 21
K
5
w
s wi 5 s 2
2i ,
2 s̄
5S 2x z x 1S 2z z z ,
s i 2 s'
2 s̄
~3.35!
5S 3 1S 3y z y ,
one can write the differential cross section in the following
compact form:
d s d s̄
5
$ 11S y z y 1F• j% ,
dV dV
~3.36!
which, being supplied with appropriated primes, is valid for
any combination of ingoing or outgoing polarized particles.
Thus, with unpolarized and linearly polarized initial or
final photons, one can access two observables: d s̄ /dV and
S 3 , respectively. When the nucleon ~target or recoil! polarization is added, four more asymmetries appear, S y and S 1x ,
S 1z , S 3y ~and their primed companions!, all of which vanish
below the pion threshold. When circularly polarized photons
are used with polarized nucleons, two more asymmetries appear, S 2x and S 2z , both of which survive below the pion
threshold.
It is important to emphasize that the formulas such as
~3.10! and ~3.36!, being used for polarized particle~s! in the
final state, give the total yield of the particles and their polarization density matrix. To get a partial cross section of a
particle produced with specific final polarization~s!, one must
use an average instead of a sum over polarizations. That is
equivalent to inserting the statistical factor 1/g into these
formulas, where g51,2,2,4 for the cases ~3.1a!–~3.1d!, respectively.
IV. LOW-ENERGY EXPANSIONS
,
Q
W 00
.
One can get another representation ~with x→2x, etc.! for
the above asymmetries by using the following relations
among the cross sections that have opposite in-plane nucleon
~target or recoil! polarizations:
s Ri 5 s L2i ,
s R2 s L
5S 1x z x 1S 1z z z ,
Q
~3.33!
5
F 25
2 s̄
F 35
With the same considerations discussed for Eq. ~3.23!, one
can define the asymmetries
p /4
s px /42 s 2
x
s p /42 s 2 p /4
~3.30!
~iii! The asymmetries with photons linearly polarized at
w 56 p /4 with respect to the scattering plane ( j 1 561). For
example, in the case ~3.1a!,
s p /42 s 2 p /4
and similarly for the primed cross sections ~with the polarized final photon!. These relations are due to parity conservation.
Introducing the generic quantities
F 15
~ c.m.! .
1019
~ i56x,6z,6x 8 ,6z 8 !
~3.34!
A. Invariant amplitudes and polarizabilities
A very general method for obtaining low-energy expansions of physical amplitudes for different reactions consists
in introducing invariant amplitudes free from kinematical
singularities and constraints and expanding them in a power
series @33,34#. Following this method, we first decompose
the invariant amplitudes A i into Born and non-Born contributions,
A i ~ n ,t ! 5A Bi ~ n ,t ! 1A NB
i ~ n ,t !
~ i51, . . . 6 ! .
~4.1!
The Born contribution is associated with pole diagrams involving a single nucleon exchanged in s- or u-channels and
g NN vertices taken in the on-shell regime,
G m ~ p 8 ,p ! 5q g m 2
k
@ g , g • ~ p 8 2 p !# .
4m m
~4.2!
1020
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
It is completely specified by the mass m, the electric charge
eq, and the anomalous magnetic moment e k /2m of the
nucleon:
PRC 58
1
T Thomson52r 0 q 2 e8 * •e .
8pm fi
~4.9!
~4.3!
The non-Born contribution to T f i is determined by the
structure constants introduced in the expansion ~4.6!. Its
spin-independent part starts with a vv 8 term,
where e is the elementary charge, e 2 .4 p /137, and q51,0
for the proton and neutron, respectively. The pole residues r i
read
1
T NB,nospin5 vv 8 ~ a E e8 * •e 1 b M s8 * •s ! 1O~ v 2 v 8 2 ! .
8pm fi
~4.10!
A Bi ~ n ,t ! 5
me 2 r i
~ s2m 2 !~ u2m 2 !
r 1 522q 2 1r 3
t
4m
2
4m vv 8
r 2 52q k 12q 2 1r 3
,
r 3 5r 5 5 k 2 12q k ,
52
e 2r i
r 45 k 2,
,
t
4m
2
Here the constants a E and b M ,
,
4 p a E 52a 3 2a 6 2a 1 ,
r 6 522q 2 2r 3 . ~4.4!
The Born contribution to T f i possesses all the symmetries of
the total amplitude T f i , including gauge invariance, and
takes all singularities of T f i at low energies.
The non-Born parts A NB
of the invariant amplitudes,
i
which contain all the structure-dependent information, are
regular functions of n 2 and t and can be expanded as a power
series in n 2 and t. Since
t522 vv 8 ~ 12z ! ,
n 2 5 vv 8 1
t2
16m 2
,
~4.5!
such an expansion can be recast as a power series in the
cross-even parameter vv 8 :
A NB
i ~ n ,t ! 5a i 1 vv 8 ~ a i, n 22 ~ 12z ! a i,t ! 1•••.
Here the low-energy constants
a i 5A NB
i ~ 0,0 ! ,
a i,t 5
a i, n 5
S D
] A NB
i
]t
S D
] A NB
i
]n 2
~4.6!
H
5r 0 e8 * •e 2q 2 1
, ...
1r 0 s8 * •s
vv 8
4m 2
4m 2
1
H ~eff2 ! ,nospin52 4 p ~ a E E2 1 b M H2 !
2
~4.7!
4 p b n 52a 3,n 2a 6,n 1a 1,n 1
@ k 2 12q k 2q 2 ~ 12z !#
@ q 2 2z ~ k 1q ! 2 # .
J
~4.8!
Here r 0 5e 2 /4p m. The leading term in Eq. ~4.8! reproduces
the Thomson limit,
~4.12!
of the nucleon with external electric and magnetic fields,
leading to the amplitude ~4.10!. Due to its interference with
the Thomson amplitude ~4.9!, the contribution of the polarizabilities, Eq. ~4.10!, results in a O( v 2 ) effect in the differential cross section d s̄ /dV in the case of the proton (q
51) and in a O( v 4 ) effect in the case of the neutron (q
50).
The O( v 4 ) terms in Eq. ~4.10! are also easily read out
from Eqs. ~2.11! and ~4.6!. They are determined by the dipole polarizabilities and the following combinations of n and t-derivative constants a i, n and a i,t of the amplitudes A 1 ,
A 3 and A 6 :
4 p a n 52a 3,n 2a 6,n 2a 1,n 2
n 5t50
n 5t50
vv 8
are identified as the dipole electric and magnetic polarizabilities of the nucleon, just in accordance with an O( v 2 ) effective dipole interaction
,
parametrize the structure of the nucleon as seen in its twophoton interactions.
The expansion ~4.6! directly leads to a corresponding
low-energy expansion of the total amplitude T f i in the lab
frame. We first consider the spin-independent part of Eq.
~2.11!. The spin-independent part of the Born term T Bf i follows from Eq. ~2.11!:
N ~ t ! B,nospin
T
8pm fi
4 p b M 52a 3 2a 6 1a 1 ,
~4.11!
a5
m2
a5
m2
,
,
~4.13!
4 p a t 52a 3,t 2a 6,t 2a 1,t 1
4 p b t 52a 3,t 2a 6,t 1a 1,t 1
a3
4m 2
a3
4m 2
,
.
These polarizability-like quantities are constant coefficients
of energy- and angle-dependent corrections to the dipole interaction ~4.12! that enter to next order in photon energy.
The recoil correction ;1/m 2 in Eqs. ~4.13! is explained in
Appendix C.
We now introduce linear combinations of the parameters
~4.13! which have a more direct physical meaning. If we
consider the partial-wave structure of the amplitude T f i ~see
Appendix A!, we can relate the t-derivative constants in Eq.
~4.13! to the quadrupole polarizabilites of the nucleon @25#:
PRC 58
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
a E2 512a t ,
b M 2 512b t .
~4.14!
A nonrelativistic example is given in Appendix B. The quantities
a E n 5 a n 22 a t 1 b t ,
b M n 5 b n 22 b t 1 a t , ~4.15!
which we call ‘‘dispersion polarizabilities,’’ describe the v
dependence of the dynamic dipole polarizabilities. In terms
of the parameters a E2 , b M 2 , a E n , and b E n , the corresponding effective interaction of O( v 4 ) has the form
where g i are structure parameters ~often called the ‘‘spin
polarizabilities’’! which are linear combinations of the constants a i ~see Appendix A!. In the following we consider a
linear combination of these parameters:
g E1 52 g 1 2 g 3 5
g M 15 g 45
1
~ a 2a 4 12a 5 1a 2 ! ,
8pm 6
1
~ a 2a 4 22a 5 2a 2 ! ,
8pm 6
~4.20!
1
H ~eff4 ! ,nospin52 4 p ~ a E n Ė2 1 b M n Ḣ2 !
2
2
1
4 p ~ a E2 E 2i j 1 b M 2 H 2i j ! ,
12
g E2 5 g 2 1 g 4 52
~4.16!
g M 2 5 g 3 52
where the dots mean a time derivative and
1
E i j 5 ~ ¹ i E j 1¹ j E i ! ,
2
1
H i j 5 ~ ¹ i H j 1¹ j H i !
2
~4.17!
are the quadrupole strengths of the electric and magnetic
fields.
We next consider the spin-dependent part of Eq. ~2.11!.
The spin-dependent part of T f i starts with O( v ) terms that
come from the Born contribution. To leading order one has
@35#
1
n
T B,,spin
52ir 0
~ q 2 s•e8 * 3e 1 ~ k 1q ! 2 s•s8 * 3s !
fi
8pm
2m
k 1q
1ir 0 q
~ v 8 s•k̂8 s8 * •e 2 vs•k̂ e8 * •s !
2m
1O~ v 2 ! .
~4.18!
The omitted higher-order terms can be read out from Eqs.
~2.11! and ~4.3!.
As shown in Eq. ~2.11!, the non-Born part of T spin
f i starts
with O( v 3 ) terms. They are determined by the four constants a 2 , a 4 , a 5 , and a 6 . Due to their interference with the
spin-dependent O( v ) terms of Eq. ~4.18!, they give rise to a
O( v 4 ) correction to the unpolarized cross section. In the
case of a polarized proton and a circularly polarized photon,
these terms appear at the order O( v 3 ).
The O( v 3 ) terms in T NB,spin
were considered in several
fi
papers @21–23#. In the most recent ~and best known! work
@23# they are parametrized as
1
T NB,spin5i v 3 g 1 s•e8 * 3e
8pm fi
1i v 3 g 2 ~ s•k̂8 3k̂ e8 * •e 2 s•e8 * 3e k̂8 •k̂ !
1i v 3 g 3 ~ s•s e8 * •k̂ 2 s•s8 e•k̂8 !
1i v 3 g 4 ~ s•e8 * 3k̂ e•k̂8 2 s•e3k̂8 e8 * •k̂
22 s•e8 * 3e k̂8 •k̂ ! 1O~ v 4 ! ,
1021
~4.19!
1
~ a 1a 6 1a 2 ! ,
8pm 4
1
~ a 1a 6 2a 2 ! ,
8pm 4
which have a more transparent physical meaning, as explained in Appendixes A and B. The parameters g E1 and
g M 1 describe the spin dependence of the dipole electric and
magnetic photon scattering, E1→E1 and M 1→M 1,
whereas g E2 and g M 2 describe the dipole-quadrupole amplitudes M 1→E2 and E1→M 2, respectively. The amplitude
~4.19! implies an effective spin-dependent interaction of order O( v 3 )
1
H ~eff3 ! ,spin52 4 p ~ g E1 s•E3Ė1 g M 1 s•H3Ḣ
2
22 g E2 E i j s i H j 12 g M 2 H i j s i E j ! .
~4.21!
To summarize, we have obtained a low-energy expansion
of the amplitudes for nucleon Compton scattering. To
O( v 4 ), the cross section and polarization observables are
determined by 10 polarizability parameters: ~i! Two dipole
polarizabilities: a E and b M , ~ii! two dispersion corrections
to the dipole polarizabilities: a E n and b M n , ~iii! two quadrupole polarizabilities: a E2 and b M 2 , and ~iv! four spin polarizabilities: g E1 , g M 1 , g E2 , g M 2 . These polarizabilities
have a simple physical interpretation in terms of the interaction of the nucleon with an external electromagnetic field.
Equivalently, these ten parameters are linear combinations of
the low-energy constants, Eq. ~4.7!, representing the zeroenergy limit of the six invariant amplitudes A i plus two combinations of both the n and t derivatives of these amplitudes.
To illustrate the interplay among all these polarizabilities,
we consider the two limiting cases of forward and backward
scattering.
~i! The amplitude for forward scattering,
S
1
v2
2e8 * •e ~ A 3 1A 6 !
@ T f i # u 50 5
8pm
2p
1
D
v
i s•e8 * 3e A 4 ,
m
has the following low-energy decomposition:
~4.22!
1022
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
has the following low-energy decomposition:
1
5e8 * •e @ 2r 0 q 2 1 v 2 ~ a E 1 b M !
@T #
8 p m f i u 50
1 v ~ a n 1 b n ! 1O~ v !#
4
1
@T #
8 p m f i u5p
6
S
1i vs•e8 * 3e 2
D
5N ~ t ! e8 * •e $ 2r 0 q 2 1 vv 8 ~ a E 2 b M !
r 0k 2
1 v 2 g 1O~ v 4 ! ,
2m
1 v 2 v 8 2 ~ a n 2 b n 24 a t 14 b t ! 1O~ v 3 v 8 3 ! %
~4.23!
1i Avv 8 s•e8 * 3e
where
g 52 g E1 2 g M 1 2 g E2 2 g M 2 5
a4
2pm
1
~ a 1a 6,n ! .
2 p 3,n
~4.25!
1i
5
~8pm!
2
H
q 4 ~ 11z 2 ! 1
vv 8
4m 2
~8pm!
r 20
2m
6
1
~8pm!
W 6,B
22 5
2
r 20
2m
7
D
~4.29!
A decomposition analogous to Eq. ~4.1! can be applied
also to the invariant functions W i j :
W i j 5W Bi j 1W NB
ij .
~4.30!
The Born contribution is given by
J
~4.31a!
~8pm!
W 6,B
12 5
2
S
1
a5
4a 1,t 2a 1,n 2 2 .
2p
m
@ 4q 3 ~ q12 k !~ 12z ! 2 12q 2 ~ 9210z1z 2 ! k 2 14q ~ 322z2z 2 ! k 3 1 ~ 32z 2 ! k 4 # ,
1
1
~4.28!
B. Cross sections
A5
n
t
s•e8 * 3e A 2 1 12
A5
m
4m 2
r 20
B
W 005
2
a 2 1a 5
2pm
a n 2 b n 24 a t 14 b t 5 a E n 2 b M n 2 121 a E2 1 121 b M 2
~4.26!
1
~4.27!
is the ‘‘backward-angle spin polarizability,’’ and
H S D
S
D
F S D GJ
4m 2
J
g p 52 g E1 1 g M 1 1 g E2 2 g M 2 52
1
vv 8
t
52
e8 * •e 12
@T #
8 p m f i u5p
2pN~ t !
4m 2
t
r0
~ k 2 14q k 12q 2 !
2m
where
~ii! The amplitude for backward scattering,
3 A 12
H
1 vv 8 g p 1O~ v 2 v 8 2 ! ,
~4.24!
is the ‘‘forward-angle spin polarizability,’’ and
a n 1 b n 5 a E n 1 b M n 1 121 a E2 1 121 b M 2 52
PRC 58
W B0352
2
H F
r 20
2
F
~ 12z 2 ! q 4 1
~ 11z ! q 2 2
vv 8
4m 2
vv 8
4m
2
G
~ k 2 12q k ! 2 ,
G
~4.31b!
~ 12z !~ k 2 12q k ! @ k 2 1q ~ 12z !~ q1 k !#
J
n
~ 12z !~ k 2 12q k !@ k 2 12q k 1q ~ 12z !~ q1 k !# ,
2m
H F
~ 12z ! q 2 1
vv 8
4m
2
G
~4.31c!
~ 12z !~ k 2 12q k ! @ k 2 12q k 1q ~ 12z !~ q1 k !#
J
n
~ 11z !~ k 2 12q k !@ k 2 1q ~ 12z !~ q1 k !# .
2m
~4.31d!
The non-Born contributions are more conveniently written by separating the terms of different order in vv 8 . In particular, for
W 00 and W 03 we have
PRC 58
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
~2!
~4! 2
2
3
3
W NB
0k 5U 0k vv 8 1U 0k v v 8 1O~ v v 8 !
1023
~4.32!
~ k50,3! ,
where
1
~ 8pm !2
1
~8pm!
2
U ~002 ! 52r 0 q 2 @~ 11z 2 ! a E 12z b M # ,
~4.33a!
U ~032 ! 5r 0 q 2 ~ 12z 2 ! a E ,
~4.33b!
1
1
r0
U ~004 ! 5 ~ 11z 2 !~ a 2E 1 b 2M ! 12z a E b M 1
~ 12z !@~ 11z !~ k 2 12q k !~ a E 1z b M ! 12q 2 „2z a E 1 ~ 11z 2 ! b M …#
2
4m 2
~8pm!
2
F
2r 0 q 2 ~ 11z 2 ! a E n 12z b M n 1
G
z3
3z 2 21
r0
P~ z !,
a E2 1
bM2 1
6
12
2m
H
F
~4.33c!
G
J
1
r0
z
1
r0
R .
U ~034 ! 5 ~ 12z 2 ! 2 ~ a 2E 2 b 2M ! 2
~ k 2 12q k !~ a E 1z b M ! 1r 0 q 2 a E n 1 a E2 2 b M 2 1
2
2
6
12
2m
4m
~8pm!
1
2
Here P(z) and R are polynomials in z of the third and zero
order, respectively, which are determined by the spin polarizabilities as follows:
P ~ z ! 5 @ q 2 ~ 112z23z 2 ! 22q k ~ 12z ! 2 12 k 2 z # g E1
1 @~ q 2 12q k !~ 322z2z 2 ! 1 k 2 ~ 32z 2 !# g M 1
1 @ 2q 2 ~ 123z 2 12z 3 ! 22q k ~ 11z23z 2 1z 3 !
and
R5q 2 ~ g E1 2 g M 2 ! 2 ~ k 1q ! 2 ~ g M 1 2 g E2 ! .
1 @ 2q 2 ~ 12z ! 2 14q k ~ z2z 2 ! 12 k 2 z # g M 2 ,
~4.34!
1
~8pm!
where
U ~122 ! 52
r0
~ 11z !@ k 2 1q ~ 12z !~ k 1q !#~ a E 1 b M ! 1r 0 q 2 ~ 11z !@ g E1 1 g M 1 1z ~ g E2 1 g M 2 !# ,
2m
2
U ~222 ! 52
r0
~ 12z !@ k 2 12q k 1q ~ 12z !~ k 1q !#~ a E 2 b M ! 2r 0 q 2 ~ 12z !@ g E1 2 g M 1 1z ~ g E2 2 g M 2 !# ,
2m
~4.37b!
2
U ~123 ! 52
r0
~ 12z ! $ @ 2 k 2 14q k 1q ~ 12z !~ k 1q !#~ g E1 2 g M 1 !
2m
1
~8pm!
1 @~ k 2 12q k !~ 11z ! 1q ~ 12z !~ k 1q !#~ g E2 2 g M 2 ! % ,
1
~8pm!
~ k51,2! ,
~4.36!
2
1
~8pm!
~4.35!
For the invariants W 12 and W 22 we can stop the expansion at
O( v 3 ), since they appear in the squared amplitude multiplied by terms of O( v ). Therefore we have
W 6,NB
5U ~k22 ! vv 8 6U ~k23 ! nvv 8 1O~ v 2 v 8 2 !
k2
1 k 2 ~ 3z 2 21 !# g E2
~4.33d!
2
U ~223 ! 52
H
~4.37a!
~4.37c!
a E1 b M
r0
1 ~ k 1q !@ 2 k 1q ~ 12z !#~ g E1 1 g M 1 !
~ 11z ! @ k 2 1q ~ 12z !~ k 1q !#
2m
m
J
2 @~ k 1q ! 2 ~ 12z ! 2q k ~ 11z !#~ g E2 1 g M 2 ! .
~4.37d!
1024
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
F G S DH S
In order to illustrate the interplay between polarizabilities
and the cross sections ~in the lab frame!, we consider below
the case of polarized photon beam and polarized target at u
50, p /2, and p .
~i! Forward scattering:
F G
d s̄
dV
u 50
5 @ r 0 q 2 2 v 2 ~ a E 1 b M !# 2 1r 20
v
d s̄
dV
u5p
1r 20
2
4m 2
k4
1r 0
~4.38!
u 50
52r 20 q 2
F
v 2
v3
k 1r 0 k 2 ~ a E 1 b M !
m
m
12r 0 q 2 v 3 g 1O~ v 5 ! ,
d s̄
dV
5
u 5 p /2
S DH F
v8
v
2
r 20
2
q 41
F G
d s̄
S3
dV
J
v 2v 82 2
~ k 14 k q12q 2 ! g p 1O~ v 6 !
m
G
S DH
2
r0 2
~ k 14 k q12q 2 !
m
1O~ v 5 ! ,
2
2
r 20
2
q 41
vv 8
4m
G
5
u 5 p /2
2
G
F
D
~4.41!
G
vv 8
4m
~ k 2 12 k q ! 1r 0 q 2
2
v 2v 82
2m 2
bM
J
2
r 0v
G
F
~ k 2 12 k q ! 2 1r 0 vv 8 a E q 2 2
vv 8
~4.42!
G
1
~ k 2 12 k q ! 2 v 2 v 8 2 ~ a 2E 2 b 2M !
2
4m
2
J
D
1
r0 2 2 2
b
1
v v 8 @ q ~ g E1 2 g M 2 ! 2 ~ k 1q ! 2 ~ g M 1 2 g E2 !# 1O~ v 6 ! , ~4.43!
12 M 2
2m
S D H F
v8
v
J
where g p is the backward-angle spin polarizability ~4.28!.
~iii! At 90°:
S
S DH F
S
d s̄
dV
~ k 2 14 k q12q 2 ! 2
r0 2 2
v v 8 @ 2 ~ 2 k q2q 2 ! g E1 1 ~ k 1q ! 2 ~ 3 g M 1 2 g E2 ! 2q 2 g M 2 # 1O~ v 6 ! ,
2m
v8
5
v
u 5 p /2
S 2z
4m 2
~ 3 ~ k 1q ! 4 24 k q 3 1q 4 ! 2r 0 vv 8 a E q 2 2
2
1r 0 q 2 v 2 v 8 2 a E n 2
F
vv 8
v8
5n
v
u5p
~4.39!
vv 8
4m
d s̄
S 2z
dV
1
1
1 v 2 v 8 2 ~ a 2E 1 b 2M ! 2r 0 q 2 v 2 v 8 2 a E n 2 b M 2
2
12
1
D
3 @ r 0 q 2 2 vv 8 ~ a E 2 b M !# 12r 0 q 2 vv 8 g p
where g is the forward-angle spin polarizability ~4.24!.
~ii! Backward scattering:
F G
m2
~4.40!
and
G
vv 8
3 ~ a E n 2 b M n 2 121 a E2 1 121 b M 2 !
and
d s̄
dV
q 2 11
1 v 2 v 8 2 ~ a E 2 b M ! 2 22r 0 q 2 v 2 v 8 2
v4 2
2r 0
k g 1O~ v 6 !
m
S 2z
2
3 @ r 20 q 2 22r 0 vv 8 ~ a E 2 b M !#
22r 0 q 2 v 4 ~ a E n 1 b M n 1 121 a E2 1 121 b M 2 !
F
v8
v
5
PRC 58
r0
v8 2
k q 32
~ k 12 k q !~ k 1q ! 2
2m
4m
G
1
vv 8
n
vv 8 ~ k 2 1 k q1q 2 !~ a E 1 b M ! 2q 2 vv 8 g E1
@~ k 1q ! 2 b M 2 k q a E # 2
2
2m
4m
1
n
vv 8 @ k q ~ g E1 2 g M 2 ! 2 ~ 2 k 2 14 k q1q 2 ! g M 1 1 ~ k 1q ! 2 g E2 # 1O~ v 5 ! ,
2m
J
~4.44!
PRC 58
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
F
S 2x
d s̄
dV
G
5
u 5 p /2
S D H F
v8
v
2
r 0v 8 2
v 2
r0 2
q ~ k 1q ! 2 1
k ~ k q12q 2 !
2m
4m
vv 8
n
vv 8 ~ k 2 1 k q1q 2 !~ a E 1 b M ! 2q 2 vv 8 g M 1
@~ k 1q ! 2 a E 2 k q b M # 1
2m
4m 2
2
n
vv 8 @ k q ~ g M 1 2 g E2 ! 2 ~ 2 k 2 14 k q1q 2 ! g E1 1 ~ k 1q ! 2 g M 2 # 1O~ v 5 ! .
2m
J
0
0
NB
p
p
A NB
2 ~ n ,t ! 5A 2 ~ t ! 2A 2 ~ 0 ! 1Ā 2 ~ n ,t ! ,
~4.46!
2 0
g p NN F p 0 gg
0
t3 ,
A p2 ~ t ! 5 T p5 ~ t ! 5
2
t
t2m 0
~4.47!
where
p
with t 3 51 or 21 for the proton and neutron, respectively,
and where
3
m p0
64p
2
~4.48!
F p 0 gg
is the p 0 two-photon decay width. The relative sign of the
p NN coupling g p NN and the p 0 gg coupling F p 0 gg is negative. In Eq. ~4.46!, Ā NB
2 is a smoother function of n ,t than
and
is
better
approximated
by a polynomial in vv 8
A NB
2
~which is just the constant a 2 to the order we consider!. In
terms of the spin polarizabilities, this means that the following substitutions have to be done in all the expansions:
g E1 → g E1 1X 2a f ~ t ! ,
g M 1 → g M 1 2X 2a f ~ t ! ,
g E2 → g E2 2X 2a f ~ t ! ,
g M 2 → g M 2 1X 2a f ~ t ! ,
~4.49!
where
0
A p2 ~ 0 !
,
X 2a 5
8pm
G
1
For the low-energy expansion of the amplitudes and structure functions, two further remarks must be considered. First,
as is evident from Eq. ~2.3!, the t-channel p 0 exchange contributes only to T 5 ~and thus only to A 2 ). Owing to the small
pion mass m p 0 , this diagram is a rapidly varying function of
t and therefore it is profitable to keep its expression unexpanded. This can be achieved with the following replacement
in the non-Born part of the amplitude A 2 :
G p 0 → gg 5
1025
f ~ t !5
t
2
m p 0 2t
.
~4.50!
Generally, after the separation of the p 0 -exchange contribution, all the low-energy expansions discussed in this section
become valid provided vv 8 /m p2 is a small parameter. The
radius of convergence of the vv 8 series is, up to a small
correction of order O(m p /m), equal to
u v u <m p
~4.51!
~4.45!
and is determined by the pion photo-production threshold
where the amplitudes have a singularity and acquire an
imaginary part. Another close singularity is due to the
t-channel exchange of two pions, which gives the same restriction ~4.51!.
The second remark is that the low-energy expansions of
the structure functions W i j , when used within the radius of
convergence, give an accurate account of the dependence of
the cross sections on the dipole polarizabilities a E , b M , and
the spin polarizabilities g E1 , g M 1 , g E2 , g M 2 . In the expres6
sions for W 6
12 and W 22 , the spin polarizabilities enter to leading order O( vv 8 ) as an interference with the Thomson amplitude ~4.9!. The subleading terms of order O( nvv 8 )
appear due to interference with the amplitude ~4.18!. Numerically the latter terms are enhanced by the anomalous
magnetic moment @for example, k 2 14q k 12q 2 512.4 in
Eq. ~4.27! for the proton# and are as important as the leading
terms. On the other hand, in the expansions for W i j , the
leading order contribution of the quadrupole and dispersion
polarizabilities a E2 , b M 2 , a E n , b M n are already of order
O( v 2 v 8 2 ), which is the highest order included in the expansion. Therefore the subleading terms which describe the interference of these polarizabilities with the amplitude ~4.18!
are not included, although they may be comparable in magnitude to the leading terms. For this reason, it is generally
more accurate to use low-energy expansions for the invariant
amplitudes A i ( n ,t), including all ten polarizabilities that appear to the order considered, and then to calculate the structure functions W i j through Eqs. ~3.15!. This more accurate
procedure will be used when we discuss the low-energy observables in Sec. V.
In principle, all the polarizabilities we have defined can be
determined from experiment. For example, in the proton case
once the dipole polarizabilities a E and b M have been determined @from low-energy experiments that are sensitive only
to terms of order O( v 2 )#, the angular behavior of the coefficients W 12 and W 22 ~from the measurements of S 2x and
S 2z ) enables a determination of all the spin polarizabilities
g E1 , g M 1 , g E2 , g M 2 . Then the angular dependence of the
unpolarized cross section allows one to determine the remaining polarizabilities a E2 , b M 2 , a E n , b M n . In the next
section, we investigate the feasibility of this approach.
V. DISCUSSION
In this section we summarize some theoretical predictions
for the polarizabilities, examine the question of convergence
of the low-energy expansion, and discuss the sensitivity of
various observables to these polarizabilities.
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
1026
A. Theoretical predictions for polarizabilities
We now consider various theoretical predictions for the
polarizabilities in order to establish the accuracy that should
be achieved in experiments to make them useful for resolving theoretical ambiguities. We restrict ourselves to estimates based on chiral perturbation theory and dispersion relations, since these have proved to be successful in
applications to strong and electromagnetic interactions of
hadrons at low energies.
In its standard form, chiral perturbation theory gives
leading-order ~LO! predictions for the polarizabilities entirely in terms of the pion mass m p , the axial coupling constant of the nucleon, g A 51.257360.0028, and the pion decay constant f p 592.460.3 MeV. The nucleon mass m
which describes recoil corrections does not enter into the LO
predictions but appears in the next-to-leading order together
with some additional parameters. The LO-contributions
dominate in the chiral limit m p →0 and are expected to provide semiquantitative estimates in the real world. All results
quoted here were obtained by Bernard et al., as summarized
in Ref. @13#. Using explicit formulas from that work, we
have
a ~ELO! 510X 1 ,
3
!
a ~ELO
n 5 X3 ,
4
LO!
a ~E2
57X 3 ,
LO!
g ~E1
525X 2 1X 2a ,
LO!
g ~E2
5X 2 2X 2a ,
b ~MLO! 5X 1 ,
7
!
b ~MLO
n 5 X3 ,
6
!
b ~MLO
2 523X 3 ,
~5.1!
!
g ~MLO
1 52X 2 2X 2a ,
!
g ~MLO
2 5X 2 1X 2a .
scribe the p 0 exchange ~4.47! with couplings fixed by the
chiral anomaly ~Wess-Zumino-Witten! and by the
Goldberger-Treiman relation:
F p 0 gg 52
2 m 2ND
D
,
Here the quantities X i are proportional to ith powers of the
inverse pion mass:
4 p g ~MD1! 5
E
X 15
51.2331024 fm3 ,
24m p
X 25
X 2a 5
E2
12p m p2
E 2t 3
2
p g Am p0
X 35
E2
20m p3
12p f p
2
,
g p NN 5g A
m
.
fp
~5.3!
4 p b ~MDn! 5
m 2ND
D
2
,
2 m 2ND
,
D3
D!
4 p g ~E2
5
D!
4 p a ~E2
5
2 m ND Q ND
2D
Q 2ND
2D
,
~5.4!
~and nothing for the other polarizabilities, as discussed in
detail in Appendix D!. Here m ND and Q ND are the magnetic
and quadrupole transition moments, respectively. Depending
on how the M 1-coupling is extracted from the experimental
radiative width of the D ~with relativistic or nonrelativistic
phase space!, the D-pole contribution to b M was evaluated in
@14,36# as
51.1131024 fm4 ,
5611.331024 fm4 ,
52.9631024 fm5 ,
e 2N c
where N c 53. Accordingly, we use the neutral pion mass
m p 0 5134.97 MeV to evaluate X 2a . 1
Calculation to the next order~s! involves new parameters
~counterterms! that are usually estimated via an approximation that takes into account the nearest resonances and/or
loops with strange particles. Such a procedure may lead to
rather different results for polarizabilities, depending on further details ~for example, compare Ref. @13# and Ref. @14#;
see also below!. Among the nucleon resonances, the
D(1232) is special because it is separated from the nucleon
by a relatively small energy gap D5m D(1232) 2m.2m p and
has a large coupling to the p N channel. Therefore this resonance is particularly important for low-energy phenomena
such as the polarizabilities. In Refs. @14,36#, the D(1232)
was considered as a partner of the nucleon in the chiral expansion, which was reformulated in terms of a generic small
energy scale e 5O(m p ,D). In the following we invoke only
one component of such an approach, the D(1232)-pole contribution. We ignore the contribution of pion loops with an
intermediate D(1232). These terms are relatively small for
the spin polarizabilities, although large for a E and b M . We
take into account both M 1 and E2 couplings of the D. Although the quadrupole contribution is of higher order in e ,
numerically it is not negligible for the spin polarizability
g E2 . For the D-pole contribution we have
4 p b ~MD ! 5
2
PRC 58
b ~MD ! 512.0 @ 14#
~5.2!
where E5eg A A2/(8 p f p ) is the threshold photoproduction
amplitude for charged pions in the chiral limit. Since the
values X 1 , X 2 , X 3 are determined by pion loops with charged
pions, we use the charged pion mass m p 5139.57 MeV for
obtaining numbers for these X’s. The terms with X 2a de-
or
7.2 @ 36#
~5.5!
1
In principle, the difference between the masses of p 1 and p 0
runs beyond the accuracy of leading-order predictions. Its consistent treatment has to involve radiative corrections of O(e 2 ). However, we assume that the use of the experimental masses of the
pions in the present context still makes sense because this is expected to reduce the size of the counterterm contributions that come
from the full treatment.
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
PRC 58
1027
TABLE I. Polarizabilities of the nucleon given by chiral perturbation theory to leading order in m p ~the
columns labeled p 0 and ‘‘loop’’ @13,24#!. Also the D-pole contribution is given with the larger strength of
Eq. ~5.5! ~see discussion in text!. Other predictions are based on dispersion relations @19,42#. Separately
given are the p 0 -exchange contribution (A as
2 ) and the contribution of excitations. The set HDT @42# uses pion
photoproduction multipoles of Hanstein et al. @43#. The set SAID is the result of the present work based on the
solution SP97K @41#. In the columns p 0 and A as
2 , the proton or neutron case corresponds to t 3 51 or 21,
respectively.
CHPT
~leading order!
p0
A as
2
loop
aE
bM
(1024 fm3 )
a En
bMn
a E2
bM2
(1024 fm5 )
g E1
gM1
g E2
gM2
g
gp
(1024 fm4 )
D pole
12.3
1.2
2.2
3.5
20.7
28.9
11.3 t 3
211.3 t 3
211.3 t 3
11.3 t 3
245.3t 3
25.5
21.1
1.1
1.1
4.4
4.4
Dispersion relations
excitations1A as
1
proton
neutron
HDT
SAID
HDT
SAID
11.9a,c
1.9a,c
12.0
13.3b,c
1.8b,c
23.8
9.1
27.5
222.4
5.3
0.2
4.0
0.75
24.8
4.8
11.2 t 3
211.2 t 3
211.2t 3
11.2 t 3
245.0t 3
24.5
3.4
2.3
20.6
20.6
10.8
23.4
2.7
1.9
0.3
21.5
7.8d
22.4
9.2
27.2
223.5
25.5
3.4
2.6
20.6
20.2
12.1
25.6
3.8
2.9
20.7
20.4
13.0
Experimental values for the proton are a E 512.161, b M 52.171 @3#.
Experimental values for the neutron are a E 512.662.5 @44# and 065 @45#. See also the criticism of the
former result in Ref. @46#.
c
a E 2 b M 510.0 and 11.5 is used as input for the proton and neutron, respectively. This guarantees that
A as
1 (0) is the same for the proton and neutron.
d
Experimental value reported for the non-p 0 part of g p for the proton is 17.363.4 @26#.
a
b
~units are 1024 fm3 ).2 With the generally accepted phenomenological magnitude of m ND ~see, for example, @37#!, b (D)
M
.(12213) @29,38#. In the Skyrme model @39# or large-N c
QCD @40#, the transition magnetic moment is related to the
isovector magnetic moment of the nucleon as m ND 5 A2 m VN
53.33(e/2m), so that b (D)
M 512.0. All these contributions are
summarized in Table I, where the larger magnitude of Eq.
~5.5! has been assumed. With the fixed ratio Q ND / m ND 5
20.25 fm ~see Appendix D!, the use of the other value
would reduce all the D-pole contributions by 40%.
Another way to calculate the polarizabilities is provided
by dispersion relations for the amplitudes A i ( n ,t). In the
following we give results of the fixed-t dispersion relations
which have the form @19#
Re A i ~ n ,t ! 5A Bi ~ n ,t ! 1
1A as
i ~ t !.
2
2
P
p
E
n max
n thr
Im A i ~ n 8 ,t !
n 8d n 8
n 822 n 2
~5.6!
This difference also illustrates how estimates of counterterms
may depend on assumptions.
Here the dispersion integral is taken from the pion photoproduction threshold to some large energy ~actually, it is 1.5
GeV!. The remaining so-called asymptotic contribution is t
dependent but only weakly energy dependent. From unitarity, the imaginary part of the amplitudes A i and the integral
in Eq. ~5.6! can be calculated by using experimental information on single- and double-pion photoproduction which is
available at energies below n max . In the present context we
use the latest version of the single-pion photoproduction amplitudes provided by the partial wave analysis of the VPI
group @41# ~the code SAID, solution SP97K!. Also, doublepion photoproduction is taken into account in the framework
of a model which includes both resonance mechanisms and
nonresonant production of the p D and r N states. Details of
this procedure are given in Ref. @19#.
The quantities A as
i (t) take into account contributions of
high energies into the dispersion relations. According to
Regge phenomenology, only the amplitudes A 1 and A 2 can
have a nonvanishing part at high energies and thus have a
large asymptotic contribution. For the other amplitudes
(A 3,4,5,6), the dispersion integrals provide a very good estimate of the corresponding non-Born parts. Thus, we can reliably find through Eq. ~5.6! the constants a 3,4,5,6 and a 3,t ,
a 6,t , a 3,n , a 6,n . Moreover, the constant a 1,n can be found as
well, since it is not affected by the asymptotic contribution.
1028
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
Uncertainties appear only for the constants a 1 , a 2 , and a 1,t ,
which get a sizable contribution from the asymptotic part of
the amplitude.
Following the arguments of Ref. @19#, we relate the
asymptotic contributions A as
1,2(t) to the t-channel exchange of
the s and p 0 mesons, respectively. The couplings of the p 0
are well known, and we include this contribution by virtue of
Eq. ~4.47!, in which we use experimental values for couplings and introduce a monopole form factor with L p
5700 MeV. The experimental magnitude of the product
g p NN F p 0 gg 520.33 GeV21 is .3% higher than that given
by the anomaly equation ~5.3!. However, the form factor
reduces the p 0 strength by (m p /L p ) 2 .4% and makes the
resulting contribution at t50 very close to that given by the
anomaly ~see Table I!. Nevertheless, beyond the p 0 , the amplitude A 2 can have contributions from other heavier exchanges and thus have an additional piece that weakly depends on t.3 Therefore the fixed-t dispersion relation for
A 2 ( n ,t) should not be considered as a reliable source of
information on a 2 .
The amplitude A 1 ( n ,t) gets a large asymptotic contribution from s exchange, which contributes to the constant a 1
and therefore to the difference of the dipole polarizabilities
a E 2 b M . Since this difference is experimentally known for
the proton to be a E 2 b M 5(1062)31024 fm3 @3#, as determined by low-energy Compton scattering data, the constant
a 1 is known too. However, the quantity a 1,t is not fixed by
these data and cannot be unambiguously predicted from the
dispersion relations ~5.6!. Using data on g p scattering at
higher energies ~above 400 MeV!, the t dependence of the
2
asymptotic contribution A as
1 (t) at 2t;0.5 GeV was estimated in @19#. It corresponds to a monopole form factor with
the cutoff parameter M s .600 MeV ~‘‘mass’’ of the s ).
However, the t dependence of A as
1 (t) at smaller t might be
somewhat steeper as suggested from independent estimates.
Therefore, the fixed-t dispersion relation for A 1 ( n ,t) should
not be considered as a reliable source of information on a 1,t .
With these precautions, we put into Table I the results of
saturating the dispersion integrals by known photoproduction
amplitudes and by the asymptotic contributions discussed
above. We assume that A as
1 is the same for the proton and
neutron. Depending on which photoproduction input is used,
the integrals are slightly different. We give both our results,
which use the SAID solution SP97K as an input and a model
for double-pion photoproduction, and the results by Drechsel
et al. @42#, which use photoproduction amplitudes by Hanstein et al. ~HDT! @43# ~and ignore the double-pion channel!.
The main difference between these two evaluations comes
A recent analysis @26# of unpolarized Compton scattering data
suggests the existence of such an additional contribution, since the
spin polarizability g p found there deviates considerably from that
predicted by the dispersion integral ~5.6! for the amplitude A 2 and
by the p 0 asymptotic contribution ~see Table I!. Even after allowing
for an uncertainty of ;62 in the integral contribution given in
Table I, as suggested by a sizable integrand at energies above 1
GeV where calculations of Im A i are very model dependent, such a
strong deviation is difficult to understand theoretically and needs to
be confirmed by additional experimental measurements.
3
PRC 58
TABLE II. Combinations of polarizabilities of the nucleon
which do not depend on the asymptotic contributions A as
1,2(t). Notation is the same as in Table I.
CHPT ~LO!
D pole
a E1 b M a
(1024 fm3 )
13.6
12.0
a En1 b M n
a E2 1 b M 2
a E n 1 41 a E2
b M n 1 41 b M 2
(1024 fm5 )
5.7
11.8
7.4
1.2
g E1 1 g M 1
g E2 1 g M 2
g E1 1 g E2
g M 11 g M 2
(1024 fm4 )
26.7
2.2
24.4
0
DR ~SAID!
proton
neutron
13.8
15.1
5.3
0.2
0.05
5.3
5.3
5.1
3.1
3.5
6.8
3.7
4.4
3.4
4.0
0.75
0.75
4.0
20.7
2.2
21.5
3.0
21.8
2.2
22.6
3.0
Experimental value for the proton is a E 1 b M 515.262.6 @3#. A
recent evaluation of this quantity in terms of the Baldin sum rule
yields 13.6960.14 ~stat! for the proton and 14.4060.66 ~stat! for
the neutron @48#.
a
from near-threshold energies where the SAID and HDT multipoles E 01 are rather different ~see Ref. @42# for details!.
For some other ~generally similar! dispersion estimates, see
Refs. @22,47,48#. Available experimental data are also given
in the footnotes to Table I, including measurements of a E
and b M for the proton ~see Ref. @3#, and references therein!,
measurements of a E for the neutron @44,45# ~see also @46#
for critical discussion!, and a recent estimate of g p for the
proton @26#. It is apparent from Table I that it is necessary to
measure the spin polarizabilities to an accuracy ;131024
fm4 in order to obtain constraints valuable for the theory.
Similarly to Ref. @42#, we can isolate quantities that can
be reliably predicted by the dispersion relations ~5.6! ~up to
small uncertainties coming from the photoproduction amplitudes!, namely those that do not depend on a 1 , a 2 , and a 1,t
~see Table II!. In the absence of precise data on doublepolarized Compton scattering at low energies, these predictions can be used to diminish the number of unknown parameters during fits and to help in determination of
unconstrained combinations. These latter combinations are
a E 2 b M ~which has already been measured!, g p , and a E2
2 b M 2 . Therefore, the most interesting and informative experiments on Compton scattering are those that are sensitive
to the parameters g p and a E2 2 b M 2 , which mainly contribute to the backward scattering amplitude. Below we discuss
which observables are the best suitable to this aim. The
above remarks apply principally to the proton. The case of
the neutron can considered in a similar manner. However,
since a free neutron target is not available, neutron polarizabilities are studied through elastic or inelastic nuclear
Compton scattering, thereby introducing additional uncertainites due to Fermi motion, meson exchange currents,
final-state interactions, etc. ~see, e.g., Refs. @49,50#!.
B. Convergence of the low-energy expansion
In order to investigate the convergence of the low-energy
expansion, we use the same fixed-t dispersion relations de-
PRC 58
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
1029
FIG. 2. Calculated cross sections and asymmetries for the proton. A comparison is given between a full dispersion calculation ~full
curves! and the low-energy expansion ~dashed curves! in which the amplitudes are expanded to O( v 2 v 8 2 ). The upper panels are the
unpolarized cross section at 0° ~left! and 180° ~right!. The lower two panels are the unpolarized cross section ~left! and S 3 ~right! at 90°.
FIG. 3. Calculated asymmetries for the proton. A comparison between a full dispersion calculation ~full curves! and the low-energy
expansion ~dashed curves! in which the amplitudes are expanded to O( v 2 v 8 2 ). The upper panels are S 2x ~left! and S 2z ~right! at 90°. The
lower two panels are S 2z at 180° ~left! and 0° ~right!.
1030
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
PRC 58
FIG. 4. Calculated cross sections and asymmetries for the neutron. A comparison is given between a full dispersion calculation ~full
curves! and the low-energy expansion ~dashed curves! in which the amplitudes are expanded to O( v 2 v 8 2 ). The upper panels are the
unpolarized cross section at 0° ~left! and 180° ~right!. The lower two panels are the unpolarized cross section ~left! and S 3 ~right! at 90°.
scribed in the preceding section to make an exact prediction
for the invariant amplitudes at any energy. From these the
W i j can be calculated exactly and predictions for the various
observables can be made. These can be compared with predictions calculated by truncating the A i to O( v 2 v 8 2 ), as
described in Sec. IV. Such a comparison is shown in Figs. 2
and 3 for the proton and Figs. 4 and 5 for the neutron. For
the proton, the expansion works very well for energies up to
the pion threshold. For the neutron, it appears to work less
well due to the fact that the Born contributions are considerably smaller than for the proton. The full calculation deviates
rapidly from the expansion once the pion threshold is
crossed.
C. Sensitivity of observables to polarizabilities
We now address the question of the measurement of the
polarizabilities we have defined, with the principal focus being on the spin polarizabilities. To the order of our expansion, the double polarization observables are not sensitive to
the quadrupole and dispersion polarizabilities, whereas the
unpolarized cross section and S 3 are sensitive to all of the
polarizabilities. Therefore it seems reasonable to use double
polarization measurements to constrain the g i , then unpolarized measurements to measure the remaining polarizabilities.
At extreme forward and backward angles, the observable S 2z
is sensitive to the g i in the combinations that give g @Eq.
~4.24!# and g p @Eq. ~4.28!#, respectively. Two other linear
combinations can be obtained from measurements of S 2x
and S 2z at 90°. Indeed, the formulas given at the end of Sec.
IV suggest that these measurements are primarily sensitive to
g M 1 and g E1 , respectively, at least to the lowest order. In
Figs. 6 and 7 we show these observables as a function of
energy, as calculated using our low-energy expansion with
polarizabilities fixed by the SAID dispersion relation values
given in Table I. In order to evaluate the sensitivity of the
observable to the polarizability, we have adjusted various
quantities by 431024 fm4 relative to the value in Table I.
This amount is comparable to the pion loop contribution to
each of the polarizabilities but is somewhat larger than the
typical discrepancy among the competing theories.
We conclude that in the energy regime below pion threshold where the low-energy expansion is valid, it will be very
difficult to measure the spin polarizabilities to an accuracy
that can discriminate among the theories, at least at the extreme forward and backward angles. In particular, the backward spin asymmetry S 2z is almost completely insensitive to
theoretically motivated changes to g p , whereas the forward
spin asymmetry is only moderately sensitive to changes in g .
Somewhat more sensitive are the asymmetries at 90°, which
might provide some useful constraints on g E1 and g M 1 . Most
promising is S 2x for the neutron, which is remarkably sensitive to changes in g M 1 .
At higher energy, the spin asymmetries are more sensitive
to the spin polarizabilities, but of course the low-energy expansion is no longer valid. Dispersion theory provides a convenient formalism for interpreting Compton scattering data
beyond the low-energy approximation, but only for those
polarizabilties not already constrained by the same disper-
PRC 58
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
1031
FIG. 5. Calculated asymmetries for the neutron. A comparison between a full dispersion calculation ~full curves! and the low-energy
expansion ~dashed curves! in which the amplitudes are expanded to O( v 2 v 8 2 ). The upper panels are S 2x ~left! and S 2z ~right! at 90°. The
lower two panels are S 2z at 180° ~left! and 0° ~right!.
FIG. 6. Plots for the proton indicating the sensitivity of the double polarization observables to the spin polarizabilities as a function of
energy. The solid curves use the SAID dispersion values for all of the spin polarizabilities ~Table I!. The upper panels show S 2x ~left! and S 2z
~right! at 90°. The long and short dashed curves increase by 431024 fm4 the values of g E1 and g M 1 , respectively. The lower panels show
S 2z at 180° ~left! and 0° ~right!. The dashed curves decrease by 431024 fm4 the values of g ~right! or g p ~left!.
1032
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
PRC 58
FIG. 7. Plots for the neutron indicating the sensitivity of the double polarization observables to the spin polarizabilities as a function of
energy. The solid curves use the SAID dispersion values for all of the spin polarizabilities ~Table I!. The upper panels show S 2x ~left! and S 2z
~right! at 90°. The long and short dashed curves increase by 431024 fm4 the values of g E1 and g M 1 , respectively. The lower panels show
S 2z at 180° ~left! and 0° ~right!. The dashed curves decrease by 431024 fm4 the values of g ~right! or g p ~left!.
sion relations. As discussed above, g p and a E2 2 b M 2 are not
well constrained by the dispersion relations due to potentially unknown asymptotic contributions to a 2 and a 1,t , respectively. We therefore investigate whether Compton scattering in the so-called dip region between the D(1232) and
higher resonances might usefully constrain these two parameters. In Figs. 8 and 9 we present calculations of the unpolarized cross section and the spin observables S 3 , S 2x , and
S 2z in the energy range 200–500 MeV. In these calculations,
the parameter a 1,t was adjusted by changing the s mass in
as
2
the asymptotic contribution A as
1 (t)5A 1 (0)/(12t/M s ),
as
where A 1 (0) was already fixed by experimental data on
a E 2 b M . The parameter a 2 was adjusted by adding to the
asymptotic contribution from the p 0 exchange a contribution
of heavier exchanges, i.e., by using in the dispersion relap0
tions the ansatz A as
2 (t)5„A 2 (t)1C…F(t), where C was an
adjusted constant and F(t) was a monopole form factor with
the cutoff parameter L p '700 MeV. In the unpolarized
cross section for the proton, a change of g p from 237 to
241 is indistinguishable from a change in the s mass from
500 to 700 MeV ~which changes a E2 2 b M 2 from 49 to 38!.
However, these possibilities are easily distinguished with
S 2z , so that a combination of unpolarized and polarized
measurements in this energy range offers the possibility of
placing strong constraints on both g p and a E2 2 b M 2 . Of
course, any practical determination of the polarizabilities
from Compton scattering data, especially at energies above
the D peak, has to take into account uncertainties in the photopion multipoles used to evaluate the dispersion integrals.
At the moment, these uncertainties are not negligible.4
VI. SUMMARY
The general structure of the Compton scattering amplitude from the nucleon with polarized photons and/or polarized nucleons in the initial and/or final state has been developed. A low-energy expansion of the amplitude to O( v 4 )
has been given in terms of ten polarizabilities: two dipole
polarizabilities a E and b M , two dispersion corrections to the
dipole polarizabilities a E n and b M n , two quadrupole polarizabilities a E2 and b M 2 , and four spin polarizabilities g E1 ,
g M 1 , g E2 , and g M 2 . The physical significance of the parameters has been discussed, and the relationship between these
and the cross section and spin observables below the pion
threshold has been established. We have also presented theoretical predictions of these parameters based both on fixedt dispersion relations and chiral perturbation theory. We have
established that the range of validity of our expansion extends to the pion threshold. We have shown that low-energy
experiments will have to be very precise to resolve the theoretical ambiguities in the polarizabilities. However, we have
suggested that measurements at higher energy might help fix
the most theoretically uncertain of them, particulary the
For example, see Refs. @8,26# for arguments against the latest
solution SP97K of Ref. @41# regarding the strength of the M 11
multipole and Refs. @42,43# for possible problems with the E 01
multipole.
4
PRC 58
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
1033
FIG. 8. Plots for the proton indicating the sensitivity of S 2x ~bottom left!, S 2z ~bottom right!, S 3 ~top right!, and the unpolarized cross
section ~top left! to changes in the the backward spin polarizability g p and the parameter a 1,t at u Lab5135° as a function of energy. The solid
curve uses a 1,t corresponding to M s 5500 MeV and g p 523731024 fm4 ; the long dashes correspond to 500 MeV and 241; and the short
dashes correspond to 700 MeV and 237.
FIG. 9. Plots for the neutron indicating the sensitivity of S 2x ~bottom left!, S 2z ~bottom right!, S 3 ~top right!, and the unpolarized cross
section ~top left! to changes in the the backward spin polarizability g p and the parameter a 1,t at u lab5135° as a function of energy. The solid
curve uses a 1,t corresponding to M s 5500 MeV and g p 55831024 fm4 ; the long dashes correspond to 500 MeV and 62; and the short
dashes correspond to 700 MeV and 58.
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
1034
backward spin polarizabilitiy g p and the difference of quadrupole polarizabilities a E2 2 b M 2 .
1
5 r 1 ~ R 1 1R 2 ! 1 r 3 ~ R 3 1R 4 12R 5 12R 6 ! ,
@T #
8 p W f i u 50
ACKNOWLEDGMENTS
1
5 r 1 ~ R 1 2R 2 ! 1 r 3 ~ R 3 2R 4 22R 5 12R 6 ! .
@T #
8 p W f i u5p
~A3!
A.L. appreciates the hospitality of the University of Illinois at Urbana-Champaign and the Institut fur Kernphysik at
Mainz where a part of this work was carried out. This work
was supported in part by the National Science Foundation
under Grant No. 94-20787.
Some authors @28,23,13,36# utilize a different spin basis, and
the following identities provide links with the notation of
those works:
APPENDIX A: MULTIPOLE CONTENT
OF POLARIZABILITIES
e8 * •k̂ e•k̂8 5x r 1 2 r 2 ,
1. Center-of-mass amplitudes
where x5k̂8 •k̂ and where we have used e•k̂5e8 * •k̂8 50.5
In particular, the amplitudes A i from @36# ~we denote them
here by A H
i ) read
~A1!
where W5 As and the spin basis r i reads
r 1 5e8 * •e ,
r 2 5s8 * •s ,
AH
1 5c ~ R 1 1xR 2 ! ,
r 3 5i s•e8 * 3e ,
AH
2 52cR 2 ,
AH
3 5c ~ R 3 1xR 4 12xR 5 12R 6 ! ,
r 4 5i s•s8 * 3s ,
AH
4 5cR 4 ,
r 5 5i ~ s•k̂ s8 * •e 2 s•k̂8 e8 * •s ! ,
r 6 5i ~ s•k̂8 s8 * •e 2 s•k̂ e8 * •s ! .
D
HS
HS D
H S
H S
D
H S
H S D
R 1 5C c 1 2A 1 2
W2
m
W2
m
2
2
A3 2
A3 1
m
J
D
D
W2
m
2
W
2
m
2
A3 2
A3 1
W2
m
W2
J
n
W
c A 2 c A ,
m 2 5 m 3 6
R 4 5C ~ W2m ! 2 ~ 12x ! A 1 1 ~ 11x !
R 6 5C ~ W2m ! 2 A 1 2
AH
6 52cR 6 , ~A5!
n
W
c 2A 52 c 3A 6 ,
m
m
R 3 5C ~ W2m ! 2 ~ x21 ! A 1 1 ~ 11x !
R 5 5C ~ W2m ! 2 2A 1 2
AH
5 52c ~ R 4 1R 5 ! ,
where c54 p W/m.
The c.m. amplitudes R i are related to the invariant amplitudes ~2.7! by
~A2!
In the particular cases of forward or backward scattering,
R 2 5C c 1 A 1 2
~A4!
i ~ s•s e8 * •k̂ 2 s•s8 e•k̂8 ! 52 r 3 2 r 6 ,
6
( r iR i~ v , u * ! ,
i51
i s•k̂8 3k̂ e8 * •e 5x r 3 1 r 4 2 r 5 ,
i ~ s•e8 * 3k̂ e•k̂8 2 s•e3k̂8 e8 * •k̂ ! 52x r 3 2 r 5 ,
In the c.m. frame, the amplitude T f i of nucleon Compton
scattering can be represented by six functions R i of the energy v and the c.m. angle u * as @51,53#
T f i 58 p W
PRC 58
~A6!
n
W
c 3A 52 c 2A 6 ,
m
m
S
A 1 ~ W 2 2m 2 ! A 2 1
2 3
S
J
J
n
W
c A 2 c A ,
m 3 5 m 2 6
A 1 ~ W 2 2m 2 ! 2A 2 1
2 3
W3
m
W3
m
D
S
A 12 ~ W2m ! 2 n A 5 1
3 4
D
S
A 12 ~ W2m ! n A 5 1
3 4
W2
A
m 6
W2
A
m 6
DJ
DJ
,
.
A few other relations can be obtained from Eq. ~A4! by doing a dual transformation, e→s5k̂3e, s→2e5k̂3s ~and the same for primed
vectors! which is just a p /2 rotation of the polarizations. Under such a transformation, r 1 ↔ r 2 , r 3 ↔ r 4 , and r 5 ↔ r 6 .
5
PRC 58
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
Here x5cos u* and
C5
~ s2m 2 ! 2
64p s 2
1035
In the important case of low energies ~or very heavy
nucleon!, one has
,
c 1 54mW1 ~ W2m ! 2 ~ 12x ! ,
R 1 52c ~ A 3 1A 6 1A 1 ! ,
R 2 52c ~ A 3 1A 6 2A 1 ! ,
c 2 54W ~ W2m ! 2 ~ W2m ! ~ 12x ! ,
2
c 3 54W 2 2 ~ W2m ! 2 ~ 12x ! .
~A7!
Note also that the invariants n , t, h are
s2m 2 1t/2
,
n5
2m
h5
R 15
R 25
R 35
2m 2
~ x11 ! .
R 4 52c 8 ~ A 6 2A 5 ! ,
2R 5 5c 8 ~ A 4 1A 6 1A 2 ! ,
~ s2m 2 ! 2
t5
~ x21 ! ,
2s
~ s2m 2 ! 2
R 3 52c 8 ~ A 6 1A 5 ! ,
~A8!
~A9!
2R 6 5c 8 ~ A 4 1A 6 2A 2 !
up to higher orders in v /m. Here c5 v 2 /4p and c 8
5 v 3 /4p m.
A multipole expansion of the amplitudes R i has the form
@52–54#
l2
l2
9 ! 2 @~ l11 ! f l1
( $ @~ l11 ! f l1
EE 1l f EE #~ l P 8
l 1 P l21
M M 1l f M M # P 9
l %,
l>1
l2
9 ! 2 @~ l11 ! f l1
( $ @~ l11 ! f l1M M 1l f l2M M #~ l P l8 1 P l21
EE 1l f EE # P l9 % ,
l>1
l2
l2
l1
9 2l 2 P 8l ! 2 @ f l1
9 22 f l1
( $ @ f l1
EE 2 f EE #~ P l21
MM2 f MM#P9
l 12 f EM P l11
MEP9
l %,
l>1
~A10!
R 45
R 55
R 65
l2
l1
9 2l 2 P l8 ! 2 @ f l1
9 22 f l1
( $ @ f l1M M 2 f l2M M #~ P l21
EE 2 f EE # P l9 12 f M E P l11
EM P l9 % ,
l>1
l2
l2
l1
- ! 2 @ f l1
- # 22 f l1
9 12 P ( $ @ f l1
EE 2 f EE #~ l P 9
l 1 P l21
MM2 f MM#Pl 1 f EM @~ 3l11 ! P 9
l 12 P l21
M E @~ l11 ! P l11
l #%,
l>1
l2
l1
- ! 2 @ f l1
- # 22 f l1
9 12 P ( $ @ f l1M M 2 f l2M M #~ l P 9l 1 P l21
EE 2 f EE # P l 1 f M E @~ 3l11 ! P 9
l 12 P l21
EM @~ l11 ! P l11
l #%.
l>1
Here P l 5 P l (x) are Legendre polynomials of x5cos u*. The
l6
multipole amplitudes f TT 8 with T,T 8 5E,M correspond to
transitions Tl→T 8 l 8 and the superscript indicates the angular momentum l of the initial photon and the total angular
momentum j5l6 21 . Due to T invariance,
~ l11 ! 2
f l1
,
EM 5 f M E
~ l11 ! 2
f l1
.
M E 5 f EM
~A11!
Keeping only dipole-dipole and dipole-quadrupole transitions in these formulas, we obtain
12
R 1 52 f 11
EE 1 f EE ,
11
11
R 3 5 f 12
EE 2 f EE 16 f EM ,
R 5 526 f 11
ME ,
12
R 2 52 f 11
MM1 f MM ,
11
11
R 4 5 f 12
M M 2 f M M 16 f M E ,
~A12!
R 6 526 f 11
EM
~plus higher multipoles, which introduce an angular dependence to the amplitudes R i ).
2. Polarizabilities to order O„ v 3 …
Low-energy expansions of the amplitudes R i are obtained
from Eqs. ~A6! and ~4.6! @22#. Leading nonvanishing terms
of R i are given by the Born term, in accordance with the
low-energy theorem by Gell-Mann–Goldberger–Low:
S
R B1 5r 0 q 2 211 ~ 11x !
R B2 52r 0 q 2
R B3 52r 0 q 2
D
v
1O~ v 2 ! ,
m
v
1O~ v 2 ! ,
m
v
1O~ v 2 ! ,
2m
R B4 52r 0 ~ k 1q ! 2
v
1O~ v 2 ! ,
2m
~A13!
1036
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
R B5 5O~ v 2 ! ,
R B6 5r 0 q ~ k 1q !
v
1O~ v 2 ! ,
2m
which describe a spin dependence of the dipole transitions
E1→E1 and M 1→M 1, and to the quantities
where r 0 5e 2 /4p m and eq is the electric charge of the
nucleon. Structure-dependent ~i.e., non-Born! contributions
to R i start with the terms
2
3
R NB
1 5 v a E 1O~ v ! ,
2
3
R NB
2 5 v b M 1O~ v ! ,
3
4
R NB
3 5 v ~ g 1 12 g 3 ! 1O~ v ! ,
3
4
R NB
5 52 v ~ g 2 1 g 4 ! 1O~ v ! ,
1
~ a 1a 6 1a 1 ! ,
4p 3
3
4
R NB
4 5 v g 2 1O~ v ! ,
~A14!
3
4
R NB
6 52 v g 3 1O~ v ! ,
g 15
b M 52
1
~ a 2a 5 2a 2 ! ,
4pm 4
1
g 35
~ a 2a 4 2a 6 ! ,
8pm 2
g 25
1
~ a 1a 6 2a 1 ! ,
4p 3
1
~ a 2a 6 ! ,
4pm 5
1
NB
~ a 1a 6 1a 2 ! .6 v 23 ~ f 11
ME! ,
8pm 4
1
NB
~ a 1a 6 2a 2 ! .6 v 23 ~ f 11
EM ! ,
8pm 4
~A18!
which describe transitions to quadrupole states, M 1→E2
and E1→M 2. In terms of these quantitities, the structuredependent parts of the amplitudes R 3 to R 6 read
3
4
R NB
3 5 v ~ 2 g E1 1 g M 2 ! 1O~ v ! ,
3
4
R NB
4 5 v ~ 2 g M 1 1 g E2 ! 1O~ v ! ,
~A19!
3
4
R NB
5 52 v g E2 1O~ v ! ,
~A15!
1
g 45
~ a 2a 2 2a 4 22a 5 ! .
8pm 6
Here the constants g i are those that appeared in Eq. ~4.19!.
The physical meaning of these polarizabilities can be understood by using the multipole expansion ~A12!. Comparing with Eq. ~4.10!, we identify the polarizabilities as leading
terms in the structure-dependent multipoles:
3
4
R NB
6 52 v g M 2 1O~ v ! .
The quantities g E1 , g M 1 , g E2 , and g M 2 are related to the spin
polarizabilities d , k , d 1 and k 1 of Levchuk and Moroz @22#
~they call them gyrations! by
d 52 g E1 1 g M 2 ,
d 1 52 g E2 ,
g 35 g M 2 ,
12 NB
v 2 b M . ~ 2 f 11
MM1 f MM! ,
k 52 g M 1 1 g E2 ,
k 1 52 g M 2 ,
~A20!
to the spin polarizabilities g i of Ragusa @23# by
g 1 52 g E1 2 g M 2 ,
12 NB
v 2 a E . ~ 2 f 11
EE 1 f EE ! ,
g 2 52 g M 1 1 g E2 ,
g 45 g M 1 ,
~A21!
to the spin polarizabilities a 1 , b 1 , a 2 , b 2 of Babusci et al.
@24# by
11
11 NB
v 3 g 1 . ~ f 12
EE 2 f EE 26 f EM ! ,
~A16!
11
11 NB
v 3 g 2 . ~ f 12
M M 2 f M M 16 f M E ! ,
NB
v 3 g 3 . ~ 6 f 11
EM ! ,
12 NB
v 3 g 4 . ~ f 11
MM2 f MM! .
It is seen that some of these g i describe mixed effects of
spin-dependent dipole scattering and dipole-quadrupole transitions. A more transparent physical meaning can be ascribed
to the quantities
g E1 52 g 1 2 g 3 5
g E2 5 g 2 1 g 4 52
g M 2 5 g 3 52
where coefficients are directly read out from Eq. ~A9!:
a E 52
PRC 58
1
~ a 2a 4 12a 5 1a 2 !
8pm 6
12 NB
. v 23 ~ f 11
EE 2 f EE ! ,
1
12 NB
g M 15 g 45
~ a 2a 4 22a 5 2a 2 ! . v 23 ~ f 11
MM2 f MM! ,
8pm 6
~A17!
a 1 54 g M 2 ,
b 1 524 g E2 ,
a 2 522 g E1 22 g M 2 ,
b 2 52 g M 1 12 g E2 ,
~A22!
and to the forward- and backward-angle spin polarizabilities
by
g 52 g E1 2 g M 1 2 g E2 2 g M 2 ,
g p 52 g E1 1 g M 1 1 g E2 2 g M 2 .
~A23!
The parameter d in @24# is d 52 g p .
3. Quadrupole and dispersion polarizabilities
Effects described by the constants a n , a t , b n , and b t
correspond to the following contributions of order O( v 4 ) to
the spin-independent amplitudes R 1 and R 2 :
d R ~14 ! 5 v 4 @ a n 1 ~ 2x22 ! a t # ,
d R ~24 ! 5 v 4 @ b n 1 ~ 2x22 ! b t #
~A24!
@these are not all terms of order O( v 4 ) in R 1,2 , as is discussed in Appendix C#. The constants in Eq. ~A24! are re-
PRC 58
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
lated to O( v 4 ) terms in dipole-dipole and quadrupolequadrupole transitions. Introducing weighted sums over
projections of the total angular momentum j,
l2
f El 5 ~ l11 ! f l1
EE 1l f EE ,
l2
f M l 5 ~ l11 ! f l1
M M 1l f M M ,
~A25!
1. Quadrupole polarizability a E2
Let us consider a charged particle in the bound s-wave
state u 0 & affected by an external electric potential A 0 (r). Expanding the interaction eA 0 (r) in powers of r, we get the
quadrupole interaction with the external field,
1
1
V5 er i r j ¹ i ¹ j A 0 ~ 0 ! 52 Q i j E i j .
2
6
we have
R 1 5 f E1 12x f E2 2 f M 2 ,
R 2 5 f M 1 12x f M 2 2 f E2 .
~A26!
Comparing with Eq. ~A24!, we conclude that the constants
a t and b t are proportional to the electric and magnetic quadrupole polarizabilities of the nucleon @25#,
a E2 512a t .12v 24 ~ f E2 ! NB,
b E2 512b t .12v 24 ~ f M 2 ! NB.
~A27!
The normalization coefficient here is explained in Appendix
B. It is chosen to conform to the definitions used in atomic
physics where, for example, dynamic electric polarizabilities
of the hydrogen read @25,55#
4 p a El ~ v ! 5e
2
(
nÞ0
S
1
E n 2E 0 2 v
D
The factor 4 p arises because we use units in which e
.4 p /137.
The combinations
2
a E n 5 a n 22 a t 1 b t ,
~A29!
are identified as O( v 4 ) terms in the dipole amplitudes f E1
and f M 1 ; that is, dispersion effects in the dynamic dipole
polarizabilities
a E1 ~ v ! 5 a E 1 v 2 a E n 1•••,
b M 1 ~ v ! 5 b M 1 v 2 b M n 1•••.
~A30!
For the hydrogen atom,
(
nÞ0
u ~ D z ! n0 u 2
,
E n 2E 0
4 p a E n 52
u ~ D z ! n0 u 2
(
nÞ0 ~ E 2E
D5er.
n
Here Q i j 5e(3r i r j 2r 2 d i j ) is the quadrupole moment of the
system and
1
E i j 52¹ i ¹ j A 0 5 ~ ¹ i E j 1¹ j E i ! ,
2
E i i 50,
~B2!
is the quadrupole strength of the field. The energy shift of the
particle caused by the quadrupole interaction ~B1! is given
by second-order perturbation theory,
DE52
1
1
~ Q i j E i j ! 0n ~ Q pq E pq ! n0
52 XE i j E i j .
36nÞ0
E n 2E 0
36
~B3!
(
Here n numerates excited states u n & and their energies E n .
The quantity X is defined as a coefficient in the expression
1
~A28!
b M n 5 b n 22 b t 1 a t
~B1!
@~ Q i j ! 0n ~ Q pq ! n0 1H.c.#
(
nÞ0 E n 2E 0
1
1
u @ r l P l ~ cos u !# n0 u 2 .
E n 2E 0 1 v
4 p a E 52
1037
0!
3
,
~A31!
APPENDIX B: NORMALIZATION
OF THE POLARIZABILITIES
In this appendix we explain the normalizations of the polarizabilities and effective interactions, which are defined in
Sec. IV, by using a simple nonrelativistic model. We discuss
the quadrupole polarizability a E2 and the spin polarizabilities g E2 and g M 1 .
S
D
2
5X d i p d jq 1 d iq d j p 2 d i j d pq .
3
~B4!
The right-hand side ~RHS! of Eq. ~B4! is the most general
tensor T i j pq which has vanishing traces T i.i pq 5T i j p. p 50 and
is symmetric under i↔ j, p↔q, or i j↔pq.
Taking i, j,p,q5z, we relate X to the quadrupole polarizability a E2 :
X512p a E2 ,
4 p a E2 [
u ~ Q zz ! n0 u 2
1
.
2 nÞ0 E n 2E 0
(
~B5!
Finally, the energy shift ~B3! takes the form of an effective
quadrupole potential
nospin
H E2
52
eff
1
4 p a E2 E i j E i j .
12
~B6!
2. Quadrupole spin polarizability g E2
Now let us consider a particle moving around a heavy
nucleus. We assume that both the particle and the nucleus
have spin and that the total spin of the system in the ground
state u 0 & is 1/2. In the presence of both an electric quadrupole and magnetic dipole interaction,
1
V52 Q i j E i j 2M i H i ,
6
~B7!
where M i is the magnetic moment operator, the corresponding energy shift of the system, ( nÞ0 V 0n V n0 /(E 0 2E n ), has a
mixed E2-M 1 term,
1038
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
4. Compton scattering amplitude
1
1
@~ Q i j E i j ! 0n ~ M k H k ! n0 1H.c.#
6 nÞ0 E n 2E 0
(
DE52
1
52 Y E i j ~ s i H j 1 s j H i ! .
6
~B8!
Here Y is defined as
1
@~ Q i j ! 0n ~ M k ! n0 1H.c.#
(
nÞ0 E n 2E 0
S
D
2
5Y s i d jk 1 s j d ik 2 s k d i j .
3
~B9!
1
1
4 p g E2 s z [2
@~ Q zz ! 0n ~ M z ! n0 1H.c.# ,
4 nÞ0 E n 2E 0
~B10!
(
where the last equation explicitly shows the normalization
and physical meaning of g E2 . Such a polarizability can exist
if there are tensor forces inside the system. Finally, the energy shift ~B3! takes the form of an effective potential
1
H E2,spin
5 4 p g E2 E i j ~ s i H j 1 s j H i ! 54 p g E2 E i j s i H j .
eff
2
~B11!
3. Dipole spin polarizability g M1
Now, let us assume that the above spin-1/2 system scatters a photon through a magnetic dipole interaction 2M
•H(t). The corresponding Compton scattering amplitude
through excited intermediate states reads
D
1 M1
~ M j ! 0n ~ M i ! n0 ~ M i ! 0n ~ M j ! n0
T f i 5 v 2 s 8j * s i
1
.
2m
E n 2E 0 2 v
E n 2E 0 1 v
nÞ0
~B12!
~B13!
where the parameter g M 1 is defined as a coefficient in the
equation
1
(
nÞ0 ~ E 2E
n
0!
2
@~ M j ! 0n ~ M i ! n0 2H.c.# 54 p g M 1 i e i jk s k .
~B14!
The scattering amplitude ~B13! can be associated with an
effective spin-dependent interaction
1
M 1,spin
H eff
52 4 p g M 1 s•H3Ḣ.
2
where H i j 5 12 (¹ i H j 1¹ j H i ) and H i i 50, the Compton scattering amplitude reads
1
1
2 ! ,no spin
T ~f E2,M
5 v 4 a E2 ~ 2z r 1 2 r 2 !
i
8pm
12
1 4
v b E2 ~ 2z r 2 2 r 1 ! ,
12
~B17!
where r i are given in Eq. ~A2!. The spin-dependent dipolequadrupole interaction
H ~effE2,M 2 ! ,spin54 p ~ g E2 E i j s i H j 2 g M 2 H i j s i E j ! ~B18!
results in the Compton scattering amplitude
1
T ~ E2,M 2 ! ,spin5 v 3 g E2 ~ r 4 2 r 5 ! 1 v 3 g M 2 ~ r 3 2 r 6 ! .
8pm fi
~B19!
The spin-dependent dipole interaction
1
H ~effE1,M 1 ! ,spin52 4 p e i jk s k ~ g E1 E i Ė j 1 g M 1 H i Ḣ j !
2
~B20!
gives the Compton scattering amplitude
1
T ~ E1,M 1 ! ,spin52 v 3 ~ g E1 r 3 1 g M 1 r 4 ! .
8pm fi
~B21!
APPENDIX C: QUADRUPOLE POLARIZABILITIES
AND RELATIVISTIC CORRECTIONS
TO THE DIPOLE INTERACTION
Its spin dependent part at low energies is
1
T M 1,spin52i g M 1 v 3 s•s8 * 3s ,
8pm fi
1
4 p ~ a E2 E i j E i j 1 b M 2 H i j H i j ! ,
12
~B16!
1
Y 5212p g E2 ,
S
Now we give a summary of interactions and scattering
amplitudes based on the above normalizations. Note that the
corresponding electric and magnetic effective interactions
are related through the duality transformation, E i →H i , H i
→2E i .
With the effective quadrupole interaction
H ~effE2,M 2 ! ,no spin52
The RHS of Eq. ~B9! is the most general tensor T i jk which
i
has vanishing trace T i.k
50 and is symmetric under i↔ j.
Taking i, j,k5z, we relate Y to the quadrupole spin polarizability g E2 :
(
PRC 58
~B15!
The polarizabilities of the nucleon can only be given an
exact meaning through definition. The simplest definition of
the multipole polarizabilities a El and b M l is that they are the
appropriately normalized coefficients of the v 2l terms in the
partial-wave amplitudes of Compton scattering, ( f El ) NB and
( f M l ) NB @25,56#. However, we do not follow this approach
for the quadrupole polarizabilities because it leads to some
unwanted features when relativistic effects are taken into account. Considering O( v 4 ) terms in the amplitudes, we want
to exclude contributions that arise merely as relativistic recoil corrections to the dipole polarizabilities.
We would like to associate with the polarizabilities a El
and b M l those nucleon-structure effects in the amplitude T f i
that are even functions of the photon energy or momentum
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
PRC 58
and do not depend on the nucleon spin. However, both the
energy and the spin depend on the reference frame. If the
frame is changed, the energy undergoes a Lorentz transformation and the Pauli spinors of the nucleon undergo a
Wigner rotation. If the amplitude T f i associated with the polarizabilities a El , b M l is chosen to be spin independent in
the c.m. frame, it would be spin dependent in other frames,
including the lab and Breit frames. Moreover, since the electric and magnetic fields are not invariant under Lorentz transformations, the splitting of structure effects into electric and
magnetic parts, ; a El and ; b M l , may also depend on the
frame. Giving a relativistically sound definition, we have to
be cautious when choosing a frame and using correspondence with notions of classical physics.
The c.m. frame is not good in this respect. The c.m. amplitudes R i do not possess all the symmetries that the amplitude T f i itself has. The crossing transformation,
Since the nucleon spin in the lab or Breit frame is the same,6
the amplitude in the lab frame will be spin independent also.
Nevertheless, we do not wish to directly relate individual
spin-independent terms in Eq. ~2.11! of different orders in
vv 8 with appropriate polarizabilities. That is partly because
the amplitude ~2.11! is not symmetric with respect to the
initial and final nucleon. The factors vv 8 e8 * •e and
vv 8 s8 * •s in Eq. ~2.11! represent the electric and magnetic
fields taken in the rest frame of the initial nucleon, whereas a
sound definition should use the fields in the frame in which
the nucleon is at rest, at least on average. Some of the
O( v 2 v 8 2 ) terms in Eq. ~2.11! are actually the result of a
Lorentz transformation of O( v 2 ) terms in the Breit frame.
For the above reasons, we choose as a definition of the
Compton scattering amplitude related with the dipole polarizabilities the expression
~a ,bM !
E
T f i,B
e,k↔e 8 * ,2k 8 ,
~C1!
brings the total momentum of the g N system, k1 p, out of
rest, so that the amplitudes R i are neither odd nor even functions of the energy. Therefore, an effective covariant interaction ~i.e., an effective Lagrangian!, which describes the
polarizabilities and possesses the symmetries of the total amplitude T f i , would result in c.m. amplitudes R i that contain
terms of mixed order in v , both even and odd. It would be
difficult to rely on individual terms in R i when identifying
the polarizabilities, except for terms of lowest order.
The amplitudes in the lab frame are also not good, because of the lack of symmetry between the initial and final
nucleon. In particular, the PT transformation,
e,k↔e 8 ,k 8 ,
s,p↔2 s,p8 ,
pB8 52QB ,
H
spin
m
2
T no
f i,B 5ū 8 ~ p 8 ! e 8 *
2
~C2!
5
~C4!
P m8 P 8n
P 82
S
T 11
m 2n T 2
m 2 2t/4
DJ
H F S D
G
F S D
N mN n
N2
S
T 31
m 2n T 4
m 2 2t/4
D
e nu
2m n 2
t
e8 * •e 2A 1 2 12
A3
N~ t !
4m 2
2
n 2A 5
m 2 2t/4
2A 6
1s8 * •s A 1 2 12
t
4m 2
A 31
n 2A 5
m 2 2t/4
2A 6
GJ
,
~C5!
~C3!
respectively. In such a frame, both T invariance and crossing
symmetry are fulfilled in the simplest way and, importantly,
the nucleon is at rest on average, (p1p8 ) B50. That is why,
in the course of an analysis of elastic eN scattering, the Breit
frame rather than the c.m. frame is used to relate the amplitude of the reaction eN→eN with physically meaningful
structure functions of the nucleon, the electromagnetic form
factors G E and G M . For some deeper motivation in favor of
the Breit frame and its relation with the language of wave
packets, see Ref. @57#.
Therefore in constructing our definitions, we choose to
postulate that the polarizability interaction and the related
Compton scattering amplitude are spin independent in the
Breit frame. It will be spin dependent in the c.m. frame.
[4 pv 2Bū 8 u ~ eB8 * •eBa E 1sB8 * •sBb M ! ,
where both the energy v B and all polarizations are taken in
the Breit frame. Neither O( v 4 ) terms nor recoil corrections
;t/m 2 are explicitly included here. The factor ū 8 u
5 A4m 2 2t52mN(t) is spin independent in the Breit frame
and serves only for a covariant normalization.7 Note also that
v B5 n /N(t). Since the spin-independent part of the Compton scattering amplitude in the Breit frame reads
applied to the g N system, brings the initial nucleon out of
rest. That is why Eq. ~2.11! contains both even and odd
powers of the photon energies.
The best choice is provided by the Breit frame, in which
the nucleon before and after photon scattering has the momentum
1
pB5 ~ k8 2k! B5QB ,
2
1039
the invariant amplitudes A i corresponding to Eq. ~C4! are
6
That is because the Wigner angles for the nucleon-spin rotation
between the lab and Breit frames vanish for both the initial and final
nucleon. The Wigner angle, uW }V3v, depends on the velocity v of
the nucleon itself and the relative velocity V of the frames. In the
case of the transformation between the lab and Breit frames, uW
50 for the initial nucleon N, because it is at rest, v5p50. Also,
uW 50 for the final nucleon N 8 , because V}p8 1p5p8 is parallel to
the nucleon velocity v}p8 . The similar Wigner angles for photons
are generally not zero, so that the photon polarizations are different
in the lab and Breit frames.
7
Cf. the definition of the electromagnetic form factors of the
nucleon @57#.
1040
~aE ,bM !
A1
522 p ~ a E 2 b M ! ,
~aE ,bM !
A3
52
2 p~ a E1 b M !
,
12t/4m 2
~C6!
(a ,b )
and other A i E M are zero. Using Eq. ~2.11!, we find the
corresponding scattering amplitude in the lab frame:
~a ,bM !
Tfi E
5
H
8 p m vv 8
e8 * •e a E 1s8 * •s b M
N~ t !
2
t
4m 2
J
~ e8 * •e 2s8 * •s !~ a E 2 b M ! , ~C7!
where a recoil correction ;t/m appears as a result of the
no-recoil ansatz in the Breit frame, Eq. ~C4!.
With the above definition of the contribution of the dipole
polarizabilities, we write the remaining terms of the nonspin
Born amplitude T NB,no
as
fi
~a ,bM !
spin
T NB,no
2T f i E
fi
58 p m vv 8 $ e8 * •e ~ n 2 a n 1t a t !
1s8 * •s ~ n 2 b n 1t b t ! % 1O~ v 6 ! .
~C8!
They are given by the parameters a n , a t , b n , and b t in Eq.
~4.13!, which determine quadrupole and dispersion polarizabilities, as discussed in Appendix A.
APPENDIX D: POLE CONTRIBUTION OF THE D„1232…
TO POLARIZABILITIES
To calculate the contribution of the D-isobar excitation
into the polarizabilities, we write an effective g ND interaction in the form similar to Eq. ~B7!:
@10#
@11#
@12#
@13#
@14#
@15#
@16#
1
H eff52M•H2 Q i j ¹ i E j .
6
F. J. Federspiel et al. Phys. Rev. Lett. 67, 1511 ~1991!.
A. Zieger et al., Phys. Lett. B 278, 34 ~1992!.
B. E. MacGibbon et al., Phys. Rev. C 52, 2097 ~1995!.
E. Hallin et al., Phys. Rev. C 48, 1497 ~1993!.
G. Blanpied et al., Phys. Rev. Lett. 76, 1023 ~1996!.
C. Molinari et al., Phys. Lett. B 371, 181 ~1996!.
J. Peise et al., Phys. Lett. B 384, 37 ~1996!.
G. Blanpied et al., Phys. Rev. Lett. 79, 4337 ~1997!.
T. Ishii et al., Nucl. Phys. B165, 189 ~1980!; Y. Wada et al.,
ibid. B247, 313 ~1984!.
S. Capstick and B. D. Keister, Phys. Rev. D 46, 84 ~1992!.
V. Pascalutsa and O. Scholten, Nucl. Phys. A591, 658 ~1995!.
O. Scholten and A. Yu. Korchin, V. Pascalutsa, and D. Van
Neck, Phys. Lett. B 384, 13 ~1996!.
V. Bernard, N. Kaiser, and U.-G. Meissner, Int. J. Mod. Phys.
E 4, 193 ~1995!.
T. R. Hemmert, B. R. Holstein, and J. Kambor, Phys. Rev. D
55, 5598 ~1997!.
W. Pfeil, H. Rollnik, and S. Stankowski, Nucl. Phys. B73, 166
~1974!.
I. Guiaşu, C. Pomponiu, and E. E. Radescu, Ann. Phys. ~N.Y.!
114, 296 ~1978!.
~D1!
Here M and Q i j are the magnetic dipole and electric quadrupole transition operators and are characterized by the matrix elements
^ D,6 21 u M z u N,6 21 & 5 m ND ,
2
@1#
@2#
@3#
@4#
@5#
@6#
@7#
@8#
@9#
PRC 58
BABUSCI, GIORDANO, L’VOV, MATONE, AND NATHAN
^ D,6 21 u Q zz u N,6 21 & 56Q ND .
~D2!
Since the interaction ~D1! involves M 1 and E2 transitions
into the j53/2 state, it contributes to the multipoles f 11
MM ,
11
,
f
and
therefore
to
the
polarizabilities
b
,
g
,
a
f 22
M
M
1
E2 ,
EE
ME
g E2 .
Using these matrix elements and Eq. ~B4! from Appendix
B, we find 4 p a E2 5Q 2ND /(2D). From Eq. ~B9! we get
4 p g E2 52 m ND Q ND /(2D). Using the same matrix elements,
the Wigner-Eckart theorem, and Eq. ~B14!, we find 4 p g M 1
5 m 2ND /D 2 . Finally, the equations 4 p b M 52 m 2ND /D and
4 p b M n 52 m 2ND /D 3 are magnetic analogs of Eq. ~A31!.
In terms of the ratio R5E 11 /M 11 of the resonance multipoles of pion photoproduction taken at the resonance energy (E g ,lab5340 MeV!,
Q ND 12
5 R.20.25 fm,
m ND k
~D3!
where k is the photon energy of the decay D→ g N in the rest
frame of the D and R.22.75% ~we take an average of
22.5% @58# and 23.0% @8#!.
@17# D. M. Akhmedov and L. V. Filkov, Sov. J. Nucl. Phys. 33,
1083 ~1981!.
@18# A. I. L’vov, Sov. J. Nucl. Phys. 34, 597 ~1981!.
@19# A. I. L’vov, V. A. Petrunkin, and M. Schumcher, Phys. Rev. C
55, 359 ~1997!.
@20# See for example, LSC Collaboration, D. Babusci et al., BNL61005, 1994; GRAAL Collaboration, J. P. Bocquet et al.,
Nucl. Phys. A622, 124c ~1997!.
@21# K. Y. Lin, Nuovo Cimento A 2, 695 ~1971!.
@22# M. I. Levchuk and L. G. Moroz, Proc. Acad. Sci. Belarus 1, 49
~1985! @in Russian#.
@23# S. Ragusa, Phys. Rev. D 47, 3757 ~1993!.
@24# D. Babusci, G. Giordano, and G. Matone, Phys. Rev. C 55,
R1645 ~1997!.
@25# I. Guiaşu and E. E. Radescu, Ann. Phys. ~N.Y.! 120, 145
~1979!; 122, 436 ~1979!.
@26# J. Tonnison, A. M. Sandorfi, S. Hoblit, and A. M. Nathan,
Phys. Rev. Lett. 80, 4382 ~1998!.
@27# L. I. Lapidus and C. K. Chao, Zh. Éksp. Teor. Fiz. 41, 294
~1961! @Sov. Phys. JETP 14, 210 ~1961!#.
@28# A. C. Hearn and E. Leader, Phys. Rev. 126, 789 ~1962!.
PRC 58
LOW-ENERGY COMPTON SCATTERING OF POLARIZED . . .
@29# V. A. Petrun’kin, Sov. J. Part. Nucl. 12, 278 ~1981!.
@30# J. D. Jackson, Classical Electrodynamics, 2nd ed. ~Wiley, New
York, 1975!, p. 277.
@31# V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics ~Pergamon, New York, 1982!.
@32# G. V. Frolov, Zh. Éksp. Teor. Fiz. 39, 1829 ~1960! @Sov. Phys.
JETP 12, 1277 ~1961!#; M. G. Ryskin and G. V. Frolov, Yad.
Fiz. 13, 1270 ~1971! @Sov. J. Nucl. Phys. 13, 731 ~1971!#.
@33# H. D. I. Abarbanel and M. L. Goldberger, Phys. Rev. 165,
1594 ~1968!.
@34# W. A. Bardin and W.-K. Tung, Phys. Rev. 173, 1423 ~1968!.
@35# M. Gell-Mann and M. Goldberger, Phys. Rev. 96, 1433
~1954!; F. Low, Phys. Rev. 96, 1428 ~1954!.
@36# T. R. Hemmert, B. R. Holstein, J. Kambor, and G. Knochlein,
Phys. Rev. D 57, 5746 ~1998!.
@37# T. E. O. Ericson and W. Weise, Pions and Nuclei ~Clarendon
Press, Oxford, 1988!.
@38# N. C. Mukhopadhyay, A. M. Nathan, and L. Zhang, Phys. Rev.
D 47, R7 ~1994!.
@39# G. Adkins, C. Nappi, and E. Witten, Nucl. Phys. B228, 552
~1983!.
@40# E. Jenkins and A. V. Manohar, Phys. Lett. B 335, 452 ~1994!.
@41# R. A. Arndt, I. I. Strokovsky, and R. L. Workman, Phys. Rev.
C 53, 430 ~1996!; solution SP97K; Phys. Rev. C 56, 577
~1997!.
@42# D. Drechsel, G. Krein, and O. Hanstein, nucl-th/9710029.
@43# O. Hanstein, D. Drechsel, and L. Tiator, Phys. Lett. B 385, 45
1041
~1996!; 399, 13 ~1996!; Nucl. Phys. A632, 561 ~1998!.
@44# J. Schmiedmayer, P. Riehs, J. A. Harvey, and N. W. Hill,
Phys. Rev. Lett. 66, 1015 ~1991!.
@45# L. Koester et al., Phys. Rev. C 51, 3363 ~1995!.
@46# T. L. Enik et al., Yad. Fiz. 60, 648 ~1997! @Phys. At. Nucl. 60,
567 ~1997!#.
@47# A. M. Sandorfi, C. S. Whisnant, and M. Khandaker, Phys. Rev.
D 50, R6681 ~1994!.
@48# D. Babusci, G. Giordano, and G. Malone, Phys. Rev. C 57,
291 ~1998!.
@49# K. W. Rose et al., Nucl. Phys. A514, 621 ~1990!.
@50# M. I. Levchuk, A. I. L’vov, and V. A. Petrun’kin, Few-Body
Syst. 16, 101 ~1994!; M. I. Levchuk and A. I. L’vov, FewBody Syst., Suppl. 9, 439 ~1995!.
@51# L. I. Lapidus and C. K. Chao, Zh. Éksp. Teor. Fiz. 37, 1714
~1959! @Sov. Phys. JETP 10, 1213 ~1960!#.
@52# V. I. Ritus, Zh. Éksp. Teor. Fiz. 32, 1536 ~1957! @Sov. Phys.
JETP 5, 1249 ~1957!#.
@53# A. P. Contogouris, Nuovo Cimento 25, 104 ~1962!.
@54# Y. Nagashima, Prog. Theor. Phys. 33, 828 ~1965!.
@55# C. K. Au, J. Phys. B 11, 2781 ~1978!.
@56# P. A. M. Guichon, G. Q. Liu, and A. W. Thomas, Nucl. Phys.
A591, 606 ~1995!.
@57# F. J. Ernst, R. G. Sachs, and K. C. Wali, Phys. Rev. 119, 1105
~1960!.
@58# R. Beck et al., Phys. Rev. Lett. 78, 606 ~1997!.