Post on 24-Feb-2016
description
Bethe-Salpeter approach and lepton-, hadron-deuteron scattering
V.V.BurovS.G.Bondarenko, E.P.Rogochaya, M.V.Rzjanin, G.I Smirnov– JINR, Dubna
A.A.Goy, V.N.Dostovalov, K.Yu.Kazakov, A.V.Molochkov, D.Shulga, S.E.Suskov – FESU, Vladivostok, Russia
S.M.Dorkin- Dubna Univ., A.V.Shebeko – Kharkov, M.Beyer – RU, Rostock, W.-Y Pauchy Hwang – NTU Taipei, Taiwan,
N.Hamamoto, A.Hosaka, Y.Manabe, H.Toki –RCNP, Osaka, Japan
1. Introduction2. Basic Definitions.3. Separable Interactions 4. Summary
Introduction Study of static and dynamic electromagnetic properties of light
nuclei enables us to understand more deeply a nature of strong interactions and, in particular, the nucleon - nucleon interaction.
Urgency of such researches is connected to a large amount of experimental data, and also with planned new experiments, which will allow to move in region of the large transfer momenta in elastic, inelastic, and deep-inelastic lepton - nucleus reactions.
At such energies an assumptions of nucleus as a nucleon system is not well justified. For this reason the problems to study in intermediate energy region the nonnucleonic degrees of freedom (Δ-isobars, quarks etc.) and Mesonic Exchange Currents (MEC) are widely discussed.
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Introduction However, in spite of the significant progress being
achieved in this way, the relativistic effects (which a priori are very important at large transfer momenta) are needed to be included.
Other actively discussed problem is the extraction of the information about the structure bound nucleons from experiments with light nuclei . Such tasks require to take into account relativistic kinematics of reaction and dynamics of NN interaction. For this reason const-ruction of covariant approach and detailed analysis of relativistic effects in electromagnetic reactions with light nuclei are very important and interesting.
Bethe - Salpeter approach give a possibility to take into account relativistic effects in a consistent way.
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Bethe –Salpeter Formalism Let us define full two particle Green Function:
Bethe –Salpeter Equation for G:
where
, ; , 1 2 1 2 1 2 1 2G , ; , 0 0 ,x x y y T x x y y
, ; ,, ; , 1 2 1 2 1 2 1 2
4
, ; , 1 2 1 21
, ; , 1 2 3 4 , ; , 3
0
4
0
1 2
, ; , , ; ,
, ; ,
, ; , , ; ,
G
G
G
G ,
kk
K
x x y y x x y y
i dw x x w w
w w w w w w y y
1 2, ; , 1 2 1 2 1 2 2
01, ;G , ,F Fx x y y S x x S y y
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Let us make Fourier transformation of :
where is total, are relative 4-momentum
The expressions for and are similar.
0, ,K G G
1 2 1 2 1 2 1 2
1 2 1 21 2 1 2
, ; , ; ,
exp ,2 2
p p P dx dx dy dy x x y y
x x y yiP ip x x ip y y
V K
p pP
1 2 1
1 2 2
2/ 2 2
P q q q P pp q q q P p
G
1q
2q
1q
2q
p p
0G20/09/2012
Full Green function for two particle system is:
The full one particle Green function is:
1 2 4;
1 2; ;4
, ;2 2
, ; , ; .2 2
G
G2
F F
F F
P Pp p P S p S p p p
P P dkiS p S p p k P k p PV
1 1 .
0FS pp pm i
We will use propagators without mass operator: FS p S p
Mass operator
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T-matrix
0GV TGLet us introduce T-matrix:
; ; ;4
1 2;
, ; , ; , ;2
, ; .2 2
dkp p P p p P i p k P
P PS k
T
T
V
S k
V
k p P
The BSE for T-matrix is:
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Thus a bound state corresponds to a pole in aT-matrix at (M is the mass of the bound
state):
2 2P M
2 2; ;, ; , ,, ,
;p p P R p pP p pP M
TP
P
; , ;R p p P is regular at 2 2P M function,
,P p is a vertex function.
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Vertex function of BSE Let us write the vertex function of BSE:
One can express it by the BS Amplitude:
1 21 2 1 2
1 2
, exp2
0 ,
x xP p dx dx iP ip x x
T x x D P
denotes a state of the deuteron with total momenta
,D P .P
1 2, , .2 2P PP p S p S p P p
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We can obtain the vertex function of BSE using that T- matrix for bound state has pole at :2 2P M
;4
1 2
, , ;2
, .2 2
dkP p i p k P
P PS k S k k
V
P
BSE for vertex function , .P p
,P p ,P k
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The normalization condition follows from: , 0 , 2D P J D P iP
2 2
1 242 , , .
2 22 P M
dk P PP i P k S k S k P kP
BS equation for Amplitude:
1 2
;4, , ; , .2 2 2P P dkP p iS p S p p k P PV k
1. E.E. Salpeter and H.A.Bethe, Phys.Rev. C84(1951) 12322. S. Mandelstam, Proc.Roy.Soc. 233A (1955) 248.3. S.Bondarenko et.al, Prog.Part.Nucl.Phys. 48(2002)449;4. S.Bondarenko et.al, NP, A832(2010)233; NP, A848
(2010) 75; NP, B219-220c (2011) 216; FBS, 49 (2011) 121; PLB, 705(2011)264; JETP Letters, 94(2011)800.
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Solution of BS Equation Separable Kernel of Interaction
– BSE for T-matrix after partial expansion can be written as:
– Separable anzats:
20 0 0 0 02
0 0 0 0 0
, , , ; , , , ;2
, , , ; , ; , , , ; .
ip p p p s p p p p s dq q d q
p p q q s
T
TS q q s q q p
V
p sV
0 0 0 01
, , , ; , , ,N
ij i j ij jiij
V p p p p s g p p g p p
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Then for T – matrix we can write:
00 01
0, ,; .,, ,N
ij i jij
s g p pp p pp s gpT p
Substitution V , T in BSE for T–matrix we can find : ij s 1 1 ,ijij ijs H s
where the can be written as: ijH s
2 0 02
0 0 , ; .2
, ,ij i ji dq q d q S q q s g q q gH s q q
Then radial part of BSA has following form:
00 0, ; ,, ,iij jg cpp p s spp S p
N
i,j=1
where coefficients satisfy the equation: jc s
, 1
0.N
i ik kj jk j
c s H s c s
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NN-scattering Let us consider NN-scattering in -channel ( -
notation). In this case nucleons are on mass shell:
and T-matrix can be parameterized as:
Here are phase shifts of waves, - is mixing parameter. For low energy NN – scattering we can express
phase shift through scattering length a, effective radius of interaction :
3 31 1S D 2 1S
JL
* 20 0 0, / 4 / 2,labp p p p p s m m
= = E
20
* 2
cos 2 1 sin 22 .sin 2 cos 2 1
S DS
S D D
iis
i i
e i eiTp s i e e
S D
2 3* * *01cot .2Srp s p O p
a
3 31 1S D
0r
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Covariant Graz-II kernel of interaction As a starting point we will use for channels: 3 3
1 1S D
221 0
1 0 222 20 11
220
2 0 222 20 12
2 22 20 2 0
3 0 22 22 2 2 20 21 0 22
1 0 2 0 3 0
1, ,
, ,
1, ,
, , , 0
S
S
D
D D S
p pg p p
p p
p pg p p
p p
p p p pg p p
p p p p
g p p g p p g p p
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Solution of BSE The solution of BSE with separable potential can be written as:
Properties of the deuteron and low energy NN-scattering .
31
31
2 3
1 1
3
1
0 0
0 03 3
, ,
, ,
;
;
Sij ij
i
j
S
jD
D
j
j
g p p g p p
g p
c
p g p
s
c s p
%
NR 4 2.225 0.2499 0.8565 0.0241 1.786 5.419 RIA 4.82 2.225 0.2812 0.8522 0.0274 1.78 5.420 Exp. 2.224
60.286 0.8574 0.0263 1.759 5.424
Dp
31S
D2D MeVDQ
2fm / 2e m fm fmD /D S 0r a
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Vertex function 31
0 4 , .S
g p p p p
± 1± 0.6
± 0.2 0 0.2 0.4 0.6 0.8 1p4 (G eV )
00.2
0.40.6
0.81
p (G eV )
0
0.2
0.4
0.6
0.8
1
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Vertex function 31
0 4 , .D
g p p p p
± 2± 1
01
2p 4 ( G e V )
0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2
p ( G e V )
0
0 . 0 5
0 . 1
0 . 1 5
0 . 2
0 . 2 5
0 . 3
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19
Relativistic description
220
22 pp pp
Ps
pPpQ
22
1|)(|
p
pgYamaguchi
011)( 222
022 ip
gmassofcenter
p
ppp
2222
11)(
pg
massofcenter
QpQ
0220 ipg p p
Qg No poles
422220 )(
1)(
p
g p pp
Y. Avishai, T. Mizutani, Nucl. Phys. A 338 (1980) 377-412
K. Schwarz, J. Frohlich, H.F.K. Zingl, L. Streit, Acta Phys. Austr. 53 (1981) 191-202
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At 0:1) Poles do not cross the counter2) Good limit
ip Esp 2/)2,1(0
222)4,3(0 ip p
222)6,5(0 ip p
R.E. Cutkosky, P.V. Landshoff, D.I. Olive, J.C. Polkinghorne, Nucl. Phys. B 12 (1969) 281-30020/09/2012
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Form factors of the separable kernel
41
221
220
220][
1 )()(
p
pg
pp
pP
The uncoupled channels
242
222
220
2202
3220][
2 ))(()()(
)(
p
pppg c
ppp
pP
3P0,1P1,3P1:
1S0:
41
221
220
2201][
1 )()()(
ppg c
ppppS
242
222
220
22202
220][
2 ))(())(()(
p
ppg c
pppppS
43
223
220
220][
3 )()()(
p
pgp
ppS
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The coupled channel 3S1-3D1
41
221
220
2201][
1 )()()(
ppg c
ppppS
242
222
220
22202
220][
2 ))(())(()(
p
ppg c
pppppS
)))(()(())(()( 4
3222
3222
0431
2231
220
22203
220][
3
pp
ppg c
ppppppD
44
224
220
220][
4 )()()(
p
pgp
ppD
0)()()()( ][2
][1
][4
][3 pppp DDSS gggg
)()(
)()()(][
2244233222111
][1144133122111
13
pcccc
pccccpS
S
g
ggS
)()(
)()()(][
4444343242141
][3344333232131
13
pcccc
pccccpD
D
g
ggD
The vertex functions of the deuteron
220
2
20
04 ))02((|)]|,()[2(||
)2(2 kEMkMEddk
Mp 2
ikgkki
kd
ldk
dl
The normalization
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1P1+,3P1
+
consts ijij )(
1S0+,3P
0+
ijij )()( 0 sss
3S1+-
3D1+:
20
)(ms
s
ijij
0|)(|det 1
dij Ms
The calculation scheme
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Р-waves:
n
i 1
2exp2exp2 ))(())()(( iii sss
1S0+: 2exp2exp
1
2exp2exp2 )()())(())()(( aaasss iii
n
i
3S1+-3D1
+:
2exp2exp2expexp
1
2exp2exp2exp2exp2
)()())(())()((
))(())()(())(())()((
aaasss
ssssss
iii
iDiDiDiSiSiS
n
i
The Minimization procedure
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as (fm) r0s (fm)
MY3 -23.750 2.70
MYQ3 -23.754 2.78
Experiment -23.748(10) 2.75(5)
pd(%) at(fm) r0t(fm) Ed(MeV)
MY4 6 5.417 1.75 2.2246
MYQ4 6 5.417 1.75 2.2246
CD-Bonn 4.85 5.4196 1.751 2.224575
Graz II 4.82 5.42 1.78 2.225
Experiment - 5.424(4) 1.759(5) 2.224644(46)
1S0+
:
3S1+-
3D1+:
O. Dumbrajs et al., Nucl Phys. B 216 (1983) 277
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Inelasticity!
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SAID (http://gwdac.phys.gwu.edu)CD-Bonn: R. Machleidt, Phys. Rev. C 63 (2001)
024001SP07: R.A. Arndt et al., Phys. Rev. C 76 (2007)
025209
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0.0 0.5 1.0 1.5 2.0 2.5 3.0
0204060
T L ab (G eV )
-60-40-20
020 M Y 2
S P 07M Y I2
(1 P 1+ )(d
eg)
(1 P 1+ )(d
eg)
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0.0 0.5 1.0 1.5 2.0 2.5 3.00
20406080
T L ab (G eV )
-60-40-20
0 M Y 2S P 07M Y I2
(3 P 0+ )(d
eg)
(3 P 0+ )(d
eg)
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0 1 2 3
0.0
0.2
0.4
0.6
0.8(
MYI
2 )2-(
exp )2
T L ab (G eV )
B re it-W ig n er R es o n an c e fit
T L a b ,1=0.872 G eV , M *1=2.27 G eV , 1=0.199 G eV ;T L a b ,2=1.595 G eV , M *2=2.55 G eV , 2=1.335 G eV ;F u l l fit
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0.0 0.5 1.0 1.5 2.0 2.5 3.00
20406080
T L ab (G eV )
-80-60-40-20
0M Y 2S P 07M Y I2
(3 P 1+ )(d
eg)
(3 P 1+ )(d
eg)
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Graz II: L. Mathelitsch, W. Plessas, M. Schweiger, Phys. Rev. C 26 (1982) 65
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0.0 0.5 1.0 1.5 2.0 2.5 3.0
0204060
T L ab (G eV )
-80
-40
0
40 M Y 6S P 07M Y I6
(1 S 0+ )(d
eg)
(1 S 0+ )(d
eg)
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0 1 2 30
20406080
T L ab (G eV )
-120-80-40
04080
120
M Y 6S P 07M Y I6-n ewS P 00,F G A
(3 S 1+ )(d
eg)
(3 S 1+ )(d
eg)
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0 1 2 3-80-60-40-20
0
T L ab (G eV )
-80-60-40-20
02040
M Y 6S P 07M Y I6-n ewS P 00,F G A(3 D 1+ )
(deg
)(3 D 1+ )
(deg
)
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Deuteron “Wave function”
0.0 0.5 1.0 1.5 2.0 2.5 3.010 -3
10 -2
10 -1
10 0
10 1
10 2
| S|(G
eV-3
/2)
p (G eV /c )
M Y Q 6 M Y 6 G raz II (N R ) G raz II P ar is
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Deuteron “Wave function”
0.0 0.5 1.0 1.5 2.0 2.5 3.010 -3
10 -2
10 -1
10 0
| D|(G
eV-3
/2)
p (G eV /c )
M Y Q 6 M Y 6 G raz II (N R ) G raz II P ar is
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Summary BS approach
– is full covariant descriptions of two body system;– allows to build the multirank covariant separable
potential MYN and MYIN of the neutron-proton interaction for coupled and uncoupled partial-waves states with the total angular momentum J=0,1,2 till the kinetic energy 3GeV.• The description of the phases and inelasticity parameter with
MYN and MYIN is very good.• Deuteron MYN wave functions are very close to
nonrelativistic one at small momenta less then 0.7 GeV/c.• Dibaryon resonances are proposed.
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Summary BS approach:
– can give very reasonable explanation structure functions, form factors and tensor polarization of deuteron in elastic eD-scattering;
– gives in one iteration approximation pair mesonic currents– gives foundations of light cone dynamics approaches;– gives good instrument to study polarization phenomena in
elastic, inelastic, deep-inelastic lepton deuteron scattering;– is a powerful tool for investigation of the reactions with the
deuteron (as well as reactions with the few-body systems).
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Plans To investigate the influence of the complex part of the
interaction kernel (namely, influence of the inelasticity parameter) to the exclusive cross section and polarization characteristics of the deuteron electrodisintegration for several kinematic conditions (Sacle, JLab).
To calculate the observables in the photo- and hadron-deuteron reactions.
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Evolution of Nucleon Structure in Nuclei Let us consider Deep Inelastic Scattering (DIS) leptons from
nuclei:
Cross section can be written as:
Lepton tensor has form:
Hadron tensor we write as:
l A l X
2
4 .,,L k Wdq
k P p
1, .2
s s s s
ss
L k k u k u k u k u k
4 41, 2 .2 n
n
W P p P j n n j P P q p
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Structure functions in DIS Hadron tensor can be related to amplitude for forward Compton
scattering T-matrix by means of the unitary relation:
Using Gauge invariance condition:
we can write q0 is the photon energy):
1, Im , .2
W P q T P q
, 0,q W P q
222
1 2 2 2 2
,, ,
W qq q P q P qW P q W q g P q P qq M q q
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In Bjorken limit
we can write the hadron tensor in following form:
Here are scale invariant structure functions (SF).
2 2
21 1
22 2
2
, ,
, ,
, ,
2
q Q
MW q F x
W q F x
qxM
1 22 2 2
1,q q P q P qW P q g F x P q P q F xq P q q q
1 2,F x F x
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Basic Approximations There are three basic groups of models for
explanation of the EMC effect by taking into account:– Nucleon separation energy, relativistic fermi-motion, NN-
correlations– Non-nucleon degrees of freedom;– The quark confinement radius changes.
Basic Approximations:– The one boson approximation in the bound state equation;– Treatment of the DIS amplitude as an incoherent sum of
amplitudes on individual constituents;– Representation of the hadron tensor of the bound nucleon in
the same form as for free nucleon
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The first assumptions allows us to use BSE. The second assumptions allows to treat the squared amplitude of
DIS on the nucleus as the sum of the squared amplitudes for scattering on individual constituents.
The available experimental data for DIS on nuclei is mainly in the region x>10-3 and Q2>1GeV2, and shows that the ratio FA
2/FD2
is independent of Q2. In the calculations we shall restrict ourselves to the Bjorken limit, where the first and second approximations are well justified.
The third assumptions which allows the hadron tensor of a virtual nucleon to be represented through SF of free nucleon. But this representation is valid when the nontrivial differences between scattering on free and bound nucleon are small. There are three such differences or so called of shell effects: – Impossibility of using the condition of gauge invariance for the bound
nucleon; – The contribution of antinucleon degrees of freedom;– The unsynchronous of bound nucleons.
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Nuclear Compton Amplitude Nuclear Compton amplitude can be written as:
where , relative time is:
Bethe-Salpeter vertex function is:
4, , 0 , ,A iqxT P q i d xe A P T j x j A P , ,2 1, 0 , , , ,A A
P PnA P T j x j A P dZdZ Z G Z x Z Z
1, 1,n nZ z z dZ dz dz ;
, 2 ,, , .A AP n PndZdZ S Z G Z Z Z
0 0.1
1 n
i j ij
z zn
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The kernel of the integral BS equation is:
The kernel is the anzats of the theory:– One bozon exchange kernel:
– Separable form of the kernel:
– BS vertex in the momentum space is:
1 12 2, , , .n nnG Z Z S Z Z G Z Z
1 12
,
, , ;n m i m j i ji j m
G Z Z z z z z
2,
, .n ij i ji j
G Z Z g Z g Z
14 41 , 1, , .
n
j jj
i k xA
n P nnS P K P K dx dx e x x
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The BS Amplitude of Compton scattering for deuteron
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Structure functions of deuteron Using Mandelstam technique and neglecting
terms of order 1/Q2 and (MD – 2E)2 we arrive to expression for structure function of deuteron:
0 /2
3 23
3 232 2
2
0
2
2
2
2
3
22
2
22,
2
4
,
.,D
D
D
D
D ND N
NN
N
D
D
D
D D
ND
D
NN
E MD
k
M EM E
M
F M k
M k
x
E
M
E kd k mM ME
xM
F x
dF
E kM
x
M E
d
k
x
F x k
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Normalization conditions The momentum sum rule is:
The baryon sum rule is:
0 0
3 2
3 3
2
2
0
2 ,22
,2
22
2
4
n
D D
D
DD
D
D
k
D
k
Dd k m E E
M MEM E
M EM k
MM Mk
EM k
0 0
3 2
3 220
2 2 1.,2
,4 22 nDD k k
DD
D
DMd k mM kE
M Ek
M M Ek M
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Nonrelativistic limit Let us expand the enegry of the bound nucleon in power of p2/m2 :
where T=2E-2m is nucleon kinetic energy and =M-2m is the binding energy, and analog of nonrelativistic function is:
The normalization condition has the form:
22
33
2 22
3 1 ,22
N
NN
ND ND N
k dF xF x F x
dxd k T xk k
m m
0 / 2
2
22 2
3.
4 2,
DD
E MD kD
k MME
kM E
m
3
32 1.
2k kd
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EMC – effect (1983)Ratio structure functions Fe/D
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1.4
1.3
1.2
1.1
1.0
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European Muon CollaborationData for the ratio of iron and deuterium structure functions are from the EMC [Aubert J.J. e.al. PL, 123B, 275(1983)]
( □ ) and from the SLAC [Arnold R.G. at al., PRL, 52,727(1984)] (•) experiments. Theory [Akulinichev et al.
Preprint INR P-0382(1984)]: the values V= -50Mev, PF= 270 Mev/c have been used in numerical calculations
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Wrong!!!
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In Quasipotential (QP) approach with synchronous nucleons:
33
32 2
2 21 .
22
NDD N
D ND
D
Tkd k xm
dFk
xF x F x
dxm
Fermi motion
NRlimit
QP
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56
2 2/Fe DF x F x 4
2 2/He DF x F x
EMC - effect
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Ratio of SF’s in BS approach
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Nuclear effects for the ratio of SF’s.Universal description for all A.
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Summary BS approach:
– is full covariant descriptions of two body system;– allows to describe the properties of deuteron with separable
potential;– can give very reasonable explanation structure functions,
form factors and tensor polarization of deuteron in elastic eD-scattering;
– gives in one iteration approximation pair mesonic currents– gives foundations of light cone dynamics approaches;– gives good instrument to study polarization phenomena in
elastic, inelastic, deep-inelastic lepton deuteron scattering;
EMIN-2012 7220/09/2012
Summary BS approach:
– allows by the model-independently the SF of light nuclei to be calculated in terms of SF of nuclear fragments and three-dimensional momentum distribution;
– gives the good explanation of the behavior for SF’s ratios of the light nuclei to the SF of the free nucleon;
– indicates that the modification of the nucleon structure of lightest nuclei is a manifestation of unsynchronous behavior of bound nucleon;
– gives new understanding fundamental properties of nucleon, mainly its time deformation in relativistic bound system.