Parity Violating analogue of Parity Violating analogue of GDH sum ruleGDH sum rule

Frascati, 11 February, 2005

Leszek Łukaszuk, Nucl.Phys.A 709 (2002) 289-298)Krzysztof Kurek & Leszek Łukaszuk, Phys.Rev.C 70(2004)065204

The knowledge of p.v. couplings in nucleon-meson The knowledge of p.v. couplings in nucleon-meson (nucleon-nucleon) forces is important for (nucleon-nucleon) forces is important for understanding the non-leptonic, weak hadronic understanding the non-leptonic, weak hadronic interactions (p.v. couplings are poorly known).interactions (p.v. couplings are poorly known). Polarized photon asymmetry in Polarized photon asymmetry in ++ photo-production photo-production near the threshold can be a good candidate to near the threshold can be a good candidate to measure p.v. pion-nucleon couling hmeasure p.v. pion-nucleon couling h

11. . Similar is expected for the low energy Compton Similar is expected for the low energy Compton scattering.scattering. hh

11 has been measured in nuclear and atomic has been measured in nuclear and atomic systems; the disagreement between systems; the disagreement between 1818F and F and 133133Cs experiments is seen.Cs experiments is seen.

MotivationMotivation

The rising interest in GDH sum rule and The rising interest in GDH sum rule and its Q2 generalizations has started with its Q2 generalizations has started with the new generation of precise spin the new generation of precise spin experiments.experiments.

New experiments based on intense New experiments based on intense polarized beams of photons give also the polarized beams of photons give also the opportunity to test a weak part of opportunity to test a weak part of photon-hadron interactions (parity photon-hadron interactions (parity violating, p.v.)violating, p.v.)

Asymptotic states in SM and the Asymptotic states in SM and the limitations of considerations limitations of considerations concerning the Compton concerning the Compton amplitudesamplitudesCollision theory and SM:Collision theory and SM: Asymptotic states – stable particlesAsymptotic states – stable particles (photons, (photons,

electrons and neurinos, proton and stable electrons and neurinos, proton and stable atomic ions)atomic ions)

Existence of unstable particles – source of Existence of unstable particles – source of concern in Quantum Field Theoryconcern in Quantum Field Theory (Veltman, (Veltman, 1963, Beenakker et al..,2000)1963, Beenakker et al..,2000)

Each stable particle should correspond to an Each stable particle should correspond to an irreducible Poincarirreducible Poincaréé unitary representation unitary representation – – problem with charged particles, problem with charged particles, QED infrared QED infrared radiationradiation → well established procedure exists in → well established procedure exists in perturbative calculus perturbative calculus onlyonly. (Bloch-Nordsic, Fadeev-. (Bloch-Nordsic, Fadeev-Kulish, Frohlich, Buchholz et al.. 1991)Kulish, Frohlich, Buchholz et al.. 1991)

Asymptotic states in SM and the Asymptotic states in SM and the limitations of considerations limitations of considerations concerning the Compton concerning the Compton amplitudesamplitudes Forward amplitudes – no radiationForward amplitudes – no radiation Strong interactions: no asymptotic states of quarks and Strong interactions: no asymptotic states of quarks and

gluons in QCD (confinement). Physical states are gluons in QCD (confinement). Physical states are composite hadrons.composite hadrons.

R.Oehme R.Oehme (Int. J. Mod. Phys. A 10 (1995)):(Int. J. Mod. Phys. A 10 (1995)):

„ „The analytic properties of physical amplitudes are the The analytic properties of physical amplitudes are the same as those obtained on the basis of an effective same as those obtained on the basis of an effective theory involving only the composite, physical fields”theory involving only the composite, physical fields”

The considerations concerning Compton amplitudes The considerations concerning Compton amplitudes will will be be

limited to the order limited to the order in p.c. part in p.c. part and to the and to the order order 22 in the in the p.v. partp.v. part ( they are infrared safe and at low energies are ( they are infrared safe and at low energies are GGFF order contribution; massive Z order contribution; massive Z00 and W and W or H bosons)or H bosons) + any order in strong interactions+ any order in strong interactions

Dispersion relations and low Dispersion relations and low energy behaviour energy behaviour

Let’s consider forward Compton amplitude:

For Re() >0 we get the physical Compton amplitude;For Re() <0 the limiting amplitude can be obtained applying complexconjugation and exploiting invariance with respect to rotation :

Dispersion relations and low Dispersion relations and low energy behaviour energy behaviour

Coherent amplitudes (related to cross section):

Normalization (Optics theorem):

crossing

We shall not use P, C, T invariance

Dispersion relations and low Dispersion relations and low energy behaviour energy behaviour

Analyticity, crossing, unitariry dispersion relation for amplitude f

Dispersion relations and low Dispersion relations and low energy behaviour energy behaviour

Low Energy Theorem (LET) for any spin of target:

A.Pais, Nuovo Cimento A53 (1968)433I.B.Khriplovich et al.., Sov.Phys.JETP 82(1996) 616

P, K

Sum rules for p.v. spin Sum rules for p.v. spin polarizabilities and polarizabilities and superconvergence hypothesissuperconvergence hypothesisP.v. analogue of GDH sum ruleP.v. analogue of GDH sum rule

¯¯¯¯¯¯¯

Assuming superconvergence:fh

(-) () → 0 with →

Subtraction point is taken at =0 and - due to LET –we get the dispersion formula for fh

(-)

Unpolarized target

Parity violating analogue of GDH sum rule

GDH (p.c.) sum rule and p.v. GDH (p.c.) sum rule and p.v. analogue of GDH sum ruleanalogue of GDH sum rule

For ½ spin target the above formula is equivalent to:

Nucl.Phys.B 11(1969)2777

(2+ 2)

Anomalous magnetic moment

Electric dipole moment

The photon scattering off The photon scattering off elementary lepton targetselementary lepton targets

e → Z0e → Wee → W

e → Z0e (solid line) → We (dotted - multiplied by 0.1)e → W (dashed - multiplied by 5)

First time calculations done (for W boson) by Altarelli, Cabibo, Maiami , Phys.lett.B 40 (1972) 415. Also discussed by S. Brodsky and I. Schmidt , Phys.Lett. B 351 (1995) 344. (for details see also: A. Abbasabadi,W.W.Repko hep-ph/0107166v1 (2001), D. Seckel, Phys.Rev.Lett.80 (1998) 900).

P.v. sum rule satisfied for every process separately, also separately for left- and right- handed electron target.

Proton targetProton target

GDH measurement and the GDH measurement and the saturation: saturation: experimental „point of view”experimental „point of view”

Saturation hypothesis for p.v. sum Saturation hypothesis for p.v. sum rulerule

Let’s consider sum rule in the form:

And define the F quantity:

Requirement that F() does not exceed prescribed small value at = sat determines saturation energy.

The usefulness of such definition of saturation is based on the assumption that there is no large contribution to the sum rule integral from photons with energy higher than sat .

For the GDH on proton – according to experimental data sat and F(sat ) can be estimated as follows:

sat 0.5-0.6 GeV and F(sat ) 0.1 (10%), respectively.

Saturation hypothesis for p.v. sum Saturation hypothesis for p.v. sum rulerule

The pion photoproduction models for The pion photoproduction models for γγN → pN → p with weak interactions efects with weak interactions efects taken into accounttaken into account

HBHBχχPTPT (J-W,Chen, X.Ji, Phys.Rev.Lett.86 (2001)4239; (J-W,Chen, X.Ji, Phys.Rev.Lett.86 (2001)4239; P.F.Bedaque, M.J.Savage,Phys.Rev.C 62 (2001)018501;P.F.Bedaque, M.J.Savage,Phys.Rev.C 62 (2001)018501; J-W.Chen,T.D.Cohen,C.W.Kao, Phys.Rev.C 64 (2001)055206)J-W.Chen,T.D.Cohen,C.W.Kao, Phys.Rev.C 64 (2001)055206)

Effective lagrangian approach with one particle Effective lagrangian approach with one particle exchange domination and with vertices structure exchange domination and with vertices structure taken into account. taken into account.

(W-Y.P.Hwang, E.M.Henley, Nucl.Phys.A 356 (1981)365, (W-Y.P.Hwang, E.M.Henley, Nucl.Phys.A 356 (1981)365, S-P.Li, E.M.Henley, W-Y.P.Hwang, Ann.Phys. 143 (1982)372)S-P.Li, E.M.Henley, W-Y.P.Hwang, Ann.Phys. 143 (1982)372)

Both approaches give similar results close to threshold.Both approaches give similar results close to threshold.In our paper (KK, LŁ, Phys.Rev.C) the effective the effective

lagrangian lagrangian approach has been used.approach has been used.

Contribution to the p.v. Contribution to the p.v. 0 and +

production amplitude according to Hwang-Henley pole model

a) , b) - nucleon pole ,c) , d) , e) , f) - pole,g), h) – vector meson poles

Additional contribution for

charged pion:a) and b) – nucleon

pole,c) - + pole

The effective Lagrangians The effective Lagrangians characterizing the couplings among characterizing the couplings among the hadrons (Hwang-Henley)the hadrons (Hwang-Henley)

i = 1,2,3 and:

0

Parity violating couplings in Parity violating couplings in Hwang-Henley modelHwang-Henley model

ρρNNNN – (h– (hρρ11, h, hρρ

22, h, hρρ33)) ; izoscalar, izovector, ; izoscalar, izovector,

izotensorizotensor ωωNNNN – – (h(hωω

00, h, hωω11)) ; izoscalar and izovector ; izoscalar and izovector

NNNN – – hh11

NN - - ff , taken 1 (in units 10 , taken 1 (in units 10-7-7)) γγNN – – μμ**, („free” parameter: (-15,15), in units 10, („free” parameter: (-15,15), in units 10-7-7)) γργρ - - h hEE , („free” parameter: (-17,17), in units 10 , („free” parameter: (-17,17), in units 10-7-7))

8 models have been considered 8 models have been considered (B. Desplanques, Phys.Rep. 297,(1998)1)(B. Desplanques, Phys.Rep. 297,(1998)1)..The values of p.v. couplings (in models) are based on the The values of p.v. couplings (in models) are based on the caclulations of the quark- quark weak interactions with strong caclulations of the quark- quark weak interactions with strong interactions corrections, symetry and exprimental data interactions corrections, symetry and exprimental data (hyperon’s decays) taken into account.(hyperon’s decays) taken into account.

Parity violating coupling Parity violating coupling constantsconstants

p.v. Hamiltonian

The p.v. meson-nucleon coupling constants are calculated from the flavour-conserving part of weak interactions :

and strong interactions effects from QCD should be accounted for. (K label in table presented on next slide, more details in: B. Desplanques, Phys. Rep. 297 (1998)1. )

Parity violating coupling Parity violating coupling constantsconstants

-7

Ann.Phys.124(80)449

Nucl.Phys.A335(80)147

N.Kaiser,U.G.Meissner,Nucl.Phys.A 489(88)671,499(89)699,510(90)759

Factorization approximation

SU(6)W

based on chiral model

K=1 - absence of

strong int. corr.

The cross sections and asymmetriesThe cross sections and asymmetries according to Hwang-Henley pole model

The unpolarized cross section for pion photoproduction - good agreement with data.

Cross sections and asymmetries(or polarized cross sections) givenby sum of the products of formfactors and relevant couplings

Having couplings calculated for 8 considered models and the formfactors taken from Hwang-Henley pole model the differences of the polarized crosssections are calculated. The saturation hypothesis with saturation energysat = 0.55 GeV is assumed and „free” parameters hE and * are selected to satisfy condition F (sat) < 0.1 .

ResultsResults

Models 2 and 3 Models 2 and 3 do not satisfy the quick do not satisfy the quick saturation hypothesissaturation hypothesis for any h for any hEE and and * *

additional structure should be seen above additional structure should be seen above

0.55 GeV to satisfy sum rule;0.55 GeV to satisfy sum rule; If saturation energy shifted to 1 GeV then If saturation energy shifted to 1 GeV then

100 pb is expected for 100 pb is expected for in energy of photon in energy of photon between 0.55-1 GeVbetween 0.55-1 GeV – quite large. – quite large.

This might indicate that it is desirable to look This might indicate that it is desirable to look for p.v. effects in this regionfor p.v. effects in this region

Remaining considered models satisfy Remaining considered models satisfy hypothesis; additional measurements of hypothesis; additional measurements of asymmetries can help to distinguish between asymmetries can help to distinguish between different modelsdifferent models

Results: „non-saturated” Results: „non-saturated” modelsmodels

The asymmetries for different The asymmetries for different „saturated” models.„saturated” models.

Model 4 Model 5

(A in 10(A in 10-7-7 units, E units, E in GeV) in GeV)

Results: „saturated” modelsResults: „saturated” models

Combining the measurements of Combining the measurements of 00 and and + +

asymmetries together would allow to select asymmetries together would allow to select models or group of models.models or group of models.

Let’s define:Let’s define:

AA00satsat , A , A++

sat sat , A, A00thth , A , A++

thth are are 00 and and ++ asymmetries for saturation and threshold asymmetries for saturation and threshold energy region, respectively.energy region, respectively.

Then: Then:

AA++satsat >0 >0 selects models selects models 1 1 andand 8 8; in addition; in addition

AA00thth > 0 (and/or A > 0 (and/or A00

satsat < 0) < 0) → → 11

AA00thth < 0 (and/or A < 0 (and/or A00

satsat > 0) > 0) → → 88

Results: „saturated” modelsResults: „saturated” models

AA++satsat <-6*10 <-6*10-7 -7 (large)(large) selects selects 44 and and 5; 5; in addition in addition

AA00thth -2*10 -2*10-7 -7 → → 55

AA00thth 0 0 → → 4 4

-6*10-6*10-7 -7 < A< A++satsat <0 <0 selects selects 1,4,6,7,8; 1,4,6,7,8; in additionin addition

AA00thth < 0( < 0( -1*10 -1*10-7-7) ) → → 77

AA00thth 0 0 → → 1,4,6,8 - then combinnig with 1,4,6,8 - then combinnig with AA++

thth and A and A00satsat::

AA++thth>1 and A>1 and A00

thth <0 select <0 select (4 (4 and and 6) 6) and and (1 (1 andand 8) 8)

Experimental feasibilityExperimental feasibility

The intensity and polarization of the electron beam at JLab allow to produce an intense, circularly polarized beams of photons from the bremsstrahlung process.

Ch.Sinclair et al.. Letter of intent 00-002, JLab.

B. Wojtsekhowski, W.T.H. van Oers, (DGNP collaboration),PHY01-05,JLab, AIP Conference proceedings SPIN 2000, 14 –th International SpinPhysiscs Symposium, Osaka, Japan, October 16-21, 2000; published June 2001, ISBN 0-7354-3.

The 12 GeV upgrade of CEBAF, White Paper prepared for the NSAC Long Range Planning Exercise, 2000, L.S. Cardman et al..,editors,Kees de Jager, PHY02-51, JLab.

For energy range from 0.137 GeV (threshold) to 0.55 GeV (saturation) it reads 1.9*109 events/sec.; 0.137 – 0.3 GeV → 7*108 events/sec 0.4 – 0.55 GeV → 2.7*108 events/sec

108 -109 events/sec seems to be large but the same rate 109is expected in LHC and the relevantdetection techniques are feasible(E.Longo, Nucl.Inst. and Meth.A 486 (2002)7)

Experimental feasibilityExperimental feasibility

Taking 60 A current at 12 GeV electron beam and 1mm Au plate target we calculate the photon bremsstrahlung spectrum as follows:

Spectrum of photons 1/ - „bremsstrahlung” sum rule type.For 1cm long liquid hydrogen target the number of events /sec. is

To verify quick saturation hypothesis: sum rule ntegral should be measured up to 0.55 GeV and: if the results comes 40 -110 pb – the hypothesis is not satisfied - in this case one needs 1013 – 1014 events which correspond to 6*103 - 6* 104 sec. of beam time; much smaller results would indicate the possibility of quick saturation. example: model 5: low energy contribution (up to 0.3 GeV) is positive: 20-28 pb,saturation region (0.4-0.55 GeV) is negative: (-10)–(-14) pb,It demands 4*1013 – 6*1013 and 1.5*1012 – 4.5*1012 events, respectively. Corresponding beam time: 6*104 – 8.5*104 and 6*103 - 1.7*104 sec.

Experimental feasibilityExperimental feasibility

To overcome statistics the large number of events is needed (signal higher than fluctuation of total production):

Concluding remarksConcluding remarks

The sum rule has been checked within lowest The sum rule has been checked within lowest order of the electroweak theory for the order of the electroweak theory for the photon-induced processes with elementary photon-induced processes with elementary lepton targets.lepton targets. It would be interesting to It would be interesting to check this sum rule in higher perturbative check this sum rule in higher perturbative orders.orders.

In analogy with observed feature of GDH sum In analogy with observed feature of GDH sum rule on proton the quick saturation rule on proton the quick saturation hypothesis has been formulated.hypothesis has been formulated.

8 models with different sets of p.v. couplings 8 models with different sets of p.v. couplings have been analyzed in the frame of effective have been analyzed in the frame of effective lagrangian and pole model approachlagrangian and pole model approach

Concluding remarksConcluding remarks

Models with the largest p.v.pion couplings hModels with the largest p.v.pion couplings h1 1

do not saturate below 0.55 GeV and the do not saturate below 0.55 GeV and the contribution from higher energies cross contribution from higher energies cross sections are needed sections are needed

It is argued that the It is argued that the measurements of the measurements of the 00 and and + + asymmetries asymmetries atat the threshold and close the threshold and close to saturation point allow to distinguish to saturation point allow to distinguish between „saturated” models (p.v. couplings)between „saturated” models (p.v. couplings)

The verification of our predictions seems to be The verification of our predictions seems to be experimentally feasible with the beam time of experimentally feasible with the beam time of the order of 10the order of 1055 sec. in the near future sec. in the near future experimental facilities (JLab)experimental facilities (JLab)

SU(6)SU(6)WW

SU(6)SU(6)WW – subgroup of SU(12), all – subgroup of SU(12), all transformations which leave untouched transformations which leave untouched 0 0 and and 33

Decomposition: SU(3)XSU(2)Decomposition: SU(3)XSU(2)WW

SU(2)SU(2)WW – weak isospin – weak isospin

Generators: iGenerators: ikk 5 5 (SU(2)(SU(2)WW))

SU(6)SU(6)WW – symmetry related to fixed direction; – symmetry related to fixed direction; useful in description of two-body decaysuseful in description of two-body decays

Bałachandram, Phys.Rev. 153 (1967) 1553S.Pakwasa, S.P.Rosen, Phys.Rev. 147 (1966)1166

Factorization: matrix element factorizes into two parts:Matrix element of current between vacuum and meson andMatrix element of another currents between nucleons