Salpeter approach and lepton hadrondeuteron scattering VVBurov SGBondarenko EPRogochaya MVRzjanin GI Smirnov JINR Dubna AAGoy VNDostovalov KYuKazakov ID: 393129
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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, RussiaS.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
IntroductionBasic Definitions.Separable Interactions Summary
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IntroductionStudy 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. 220/09/2012EMIN-2012Slide3
IntroductionHowever, 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.320/09/2012EMIN-2012Slide4
Bethe –Salpeter FormalismLet us define full two particle Green Function:Bethe –Salpeter Equation for G:where
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Let us make Fourier transformation of :where is total, are relative 4-momentum
The expressions for and are similar.
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Full Green function for two particle system is:The full one particle Green function is:
We will use propagators without mass operator:Mass operator6
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T-matrix
Let us introduce T-matrix:The BSE for T-matrix is:720/09/2012EMIN-2012Slide8
Thus a bound state corresponds to a pole in aT-matrix at (M is the mass of the bound state):
is regular at function,
is a vertex function.8
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Vertex function of BSELet us write the vertex function of BSE:One can express it by the BS Amplitude:
denotes a state of the deuteron with total momenta 920/09/2012EMIN-2012Slide10
We can obtain the vertex function of BSE using that T- matrix for bound state has pole at :
BSE for vertex function
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The normalization condition follows from:
BS equation for Amplitude:E.E. Salpeter and H.A.Bethe, Phys.Rev. C84(1951) 1232S. Mandelstam, Proc.Roy.Soc. 233A (1955) 248.
S.Bondarenko et.al, Prog.Part.Nucl.Phys. 48(2002)449;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.
11
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Solution of BS EquationSeparable Kernel of InteractionBSE for T-matrix after partial expansion can be written as:
Separable anzats:1220/09/2012EMIN-2012Slide13
Then for T – matrix we can write:
Substitution V , T in BSE for T–matrix we can find : where the can be written as:
Then radial part of BSA has following form:
where coefficients satisfy the equation:
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NN-scatteringLet 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 :
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Covariant Graz-II kernel of interactionAs a starting point we will use for channels:
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Solution of BSEThe solution of BSE with separable potential can be written as: Properties of the deuteron and low energy NN-scattering .
%
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.
2.2246
0.286
0.8574
0.0263
1.759
5.424
16
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Vertex function
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Vertex function
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Relativistic description
Yamaguchi
No poles
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
19
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At 0:Poles do not cross the counter
2) Good limitR.E. Cutkosky, P.V. Landshoff
, D.I. Olive, J.C. Polkinghorne, Nucl. Phys. B 12 (1969) 281-3002020/09/2012
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Form factors of the separable kernel
The uncoupled channels3P0,1P1,3P1:1S0:
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The coupled channel3S
1-3D1
The vertex functions of the deuteron
The normalization
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1P1+,3P1
+1S0+,3P0+
3
S
1
+
-
3
D
1
+
:
The calculation scheme
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Р-waves:
1S0+:3S1+-3D1+:
The Minimization procedure24
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as (fm)r0s (fm)MY3-23.7502.70
MYQ3-23.7542.78Experiment-23.748(10)2.75(5)pd(%)at(fm)r0t(fm)Ed(MeV)MY465.4171.752.2246MYQ465.4171.752.2246CD-Bonn4.855.41961.7512.224575Graz II4.82
5.421.782.225Experiment-5.424(4)1.759(5)2.224644(46)
1
S
0
+
:
3
S
1
+
-
3
D
1
+
:
O.
Dumbrajs
et al.,
Nucl
Phys. B 216 (1983) 277
25
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Inelasticity!
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SAID (http://gwdac.phys.gwu.edu)CD-Bonn: R. Machleidt, Phys. Rev. C 63 (2001) 024001
SP07: R.A. Arndt et al., Phys. Rev. C 76 (2007) 0252092920/09/2012EMIN-2012Slide30
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3120/09/2012EMIN-2012Slide32
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Graz II: L. Mathelitsch, W. Plessas, M. Schweiger, Phys. Rev. C 26 (1982) 65
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Deuteron “Wave function”4820/09/2012EMIN-2012Slide49
Deuteron “Wave function”4920/09/2012EMIN-2012Slide50
SummaryBS approachis 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.5020/09/2012EMIN-2012Slide51
SummaryBS 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 currentsgives 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).5120/09/2012EMIN-2012Slide52
PlansTo 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.5220/09/2012EMIN-2012Slide53
20/09/2012EMIN-201253Evolution of Nucleon Structure in NucleiLet us consider Deep Inelastic Scattering (DIS) leptons from nuclei: Cross section can be written as:
Lepton tensor has form:Hadron tensor we write as:Slide54
20/09/2012EMIN-201254Structure functions in DISHadron 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 (n = q0 is the photon energy):Slide55
20/09/2012EMIN-201255In Bjorken limitwe can write the hadron tensor in following form:
Here are scale invariant structure functions (SF).Slide56
20/09/2012EMIN-201256Basic ApproximationsThere are three basic groups of models for explanation of the EMC effect by taking into account:
Nucleon separation energy, relativistic fermi-motion, NN-correlationsNon-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 nucleonSlide57
20/09/2012EMIN-201257The 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 FA2/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.Slide58
20/09/2012EMIN-201258Nuclear Compton AmplitudeNuclear Compton amplitude can be written as:
where , relative time is:Bethe-Salpeter vertex function is:Slide59
20/09/2012EMIN-201259The 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:Slide60
20/09/2012EMIN-201260The BS Amplitude of Compton scattering for deuteronSlide61
20/09/2012EMIN-201261Structure functions of deuteronUsing Mandelstam technique and neglecting terms of order 1/Q2 and
(MD – 2E)2 we arrive to expression for structure function of deuteron:Slide62
20/09/2012EMIN-201262Normalization conditionsThe momentum sum rule is:The baryon sum rule is:Slide63
20/09/2012EMIN-201263Nonrelativistic limitLet us expand the enegry of the bound nucleon in power of p2/m2
:where T=2E-2m is nucleon kinetic energy and e=M-2m is the binding energy, and analog of nonrelativistic function is: The normalization condition has the form:Slide64
EMC – effect (1983)Ratio structure functions Fe/D20/09/2012EMIN-201264
1.41.31.21.11.0Slide65
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 calculations20/09/2012EMIN-201265Wrong!!!Slide66
20/09/2012EMIN-201266In Quasipotential (QP) approach with synchronous nucleons:
Fermi motionNRlimitQPSlide67
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EMC - effectSlide68
20/09/2012EMIN-201268Ratio of SF’s in BS approachSlide69
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20/09/2012EMIN-201270Nuclear effects for the ratio of SF’s.Universal description for all A.Slide71
20/09/2012EMIN-201271SummaryBS 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 currentsgives foundations of light cone dynamics approaches;gives good instrument to study polarization phenomena in elastic, inelastic, deep-inelastic lepton deuteron scattering;Slide72
20/09/2012EMIN-201272SummaryBS 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.