Parity Violating analogue of GDH sum rule

1
Parity Violating analogue of GDH sum rule
Leszek Lukaszuk, Nucl.Phys.A 709 (2002)
289-298) Krzysztof Kurek Leszek Lukaszuk,
Phys.Rev.C 70(2004)065204
Frascati, 11 February, 2005
2
Motivation

• The knowledge of p.v. couplings in nucleon-meson
  
• (nucleon-nucleon) forces is important for
• understanding the non-leptonic, weak hadronic
• interactions (p.v. couplings are poorly known).
• Polarized photon asymmetry in ?
  photo-production
• near the threshold can be a good candidate to
• measure p.v. pion-nucleon couling h?1.
• Similar is expected for the low energy Compton
• scattering.
• h?1 has been measured in nuclear and atomic
• systems the disagreement between 18F and
• 133Cs experiments is seen.

• The rising interest in GDH sum rule and its Q2
  generalizations has started with the new
  generation of precise spin experiments.
• New experiments based on intense polarized beams
  of photons give also the opportunity to test a
  weak part of photon-hadron interactions (parity
  violating, p.v.)

3
Asymptotic states in SM and the limitations of
considerations concerning the Compton amplitudes

• Collision theory and SM
• Asymptotic states stable particles (photons,
  electrons and neurinos, proton and stable atomic
  ions)
• Existence of unstable particles source of
  concern in Quantum Field Theory (Veltman, 1963,
  Beenakker et al..,2000)
• Each stable particle should correspond to an
  irreducible Poincaré unitary representation
  problem with charged particles, QED infrared
  radiation ? well established procedure exists in
  perturbative calculus only. (Bloch-Nordsic,
  Fadeev-Kulish, Frohlich, Buchholz et al.. 1991)

4
Asymptotic states in SM and the limitations of
considerations concerning the Compton amplitudes

• Forward amplitudes no radiation
• Strong interactions no asymptotic states of
  quarks and gluons in QCD (confinement). Physical
  states are composite hadrons.
• R.Oehme (Int. J. Mod. Phys. A 10 (1995))
• The analytic properties of physical
  amplitudes are the same as those obtained on the
  basis of an effective theory involving only the
  composite, physical fields
• The considerations concerning Compton amplitudes
  will be
• limited to the order ? in p.c. part and to the
  order ?2 in the
• p.v. part ( they are infrared safe and at low
  energies are
• ?GF order contribution massive Z0 and W? or H
  bosons)
• any order in strong interactions

5
Dispersion relations and low energy behaviour
Lets consider forward Compton amplitude

For Re(?) gt0 we get the physical Compton
amplitude For Re(?) lt0 the limiting amplitude
can be obtained applying complex conjugation and
exploiting invariance with respect to rotation
6
Dispersion relations and low energy behaviour
Coherent amplitudes (related to cross section)
crossing
Normalization (Optics theorem)
We shall not use P, C, T invariance
7
Dispersion relations and low energy behaviour
Analyticity, crossing, unitariry ? dispersion
relation for amplitude f
8
Dispersion relations and low energy behaviour
Low Energy Theorem (LET) for any spin of target
P, K
A.Pais, Nuovo Cimento A53 (1968)433 I.B.Khriplovic
h et al.., Sov.Phys.JETP 82(1996) 616
9
Sum rules for p.v. spin polarizabilities and
superconvergence hypothesisP.v. analogue of GDH
sum rule
Subtraction point is taken at ? 0 and - due to
LET we get the dispersion formula for fh(-)?
Unpolarized target
Assuming superconvergence fh(-)? (?)
? 0 with ???

?

Parity violating analogue of GDH sum rule
10
GDH (p.c.) sum rule and p.v. analogue of GDH sum
rule
For ½ spin target the above formula is equivalent
to


Nucl.Phys.B 11(1969)2777
Anomalous magnetic moment Electric dipole moment
(?2 ?2)
11
The photon scattering off elementary lepton
targets
?e ? Z0e (solid line) ?? ? We (dotted -
multiplied by 0.1) ?e ? ? W (dashed - multiplied
by 5)
?e ? Z0e ?? ? We ?e ? ?W
P.v. sum rule satisfied for every process
separately, also separately for left- and
right- handed electron target.

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).
12
Proton target
13
GDH measurement and the saturation
experimental point of view
14
Saturation hypothesis for p.v. sum rule
Lets consider sum rule in the form
And define the F quantity
15
Saturation hypothesis for p.v. sum rule

• 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.

16
The pion photoproduction models for ?N ? p? with
weak interactions efects taken into account

• HB?PT
• (J-W,Chen, X.Ji, Phys.Rev.Lett.86 (2001)4239
• 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)
• Effective lagrangian approach with one particle
  exchange domination and with vertices structure
  taken into account.
• (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)
• Both approaches give similar results close to
  threshold.
• In our paper (KK, LL, Phys.Rev.C) the effective
  lagrangian
• approach has been used.

17
Contribution to the p.v. ?0 and ? production
amplitude according to Hwang-Henley pole
model
Additional contribution for charged
pion a) and b) nucleon pole, c) - ? pole
a) , b) - nucleon pole ,c) , d) , e) , f) - ?
pole, g), h) vector meson poles
18
The effective Lagrangians characterizing the
couplings among the hadrons (Hwang-Henley)
i 1,2,3 and
0
19
Parity violating couplings in Hwang-Henley model

• ?NN (h?1, h?2, h?3) izoscalar, izovector,
  izotensor
• ?NN (h?0, h?1) izoscalar and izovector
• ?NN h?1
• ?N? - f? , taken 1 (in units 10-7)
• ??N µ, (free parameter (-15,15), in units
  10-7)
• ??? - hE , (free parameter (-17,17), in units
  10-7)

8 models have been considered (B. Desplanques,
Phys.Rep. 297,(1998)1). The values of p.v.
couplings (in models) are based on the
caclulations of the quark- quark weak
interactions with strong interactions
corrections, symetry and exprimental data
(hyperons decays) taken into account.
20
Parity violating coupling constants
The p.v. meson-nucleon coupling constants are
calculated from the flavour-conserving part of
weak interactions
p.v. Hamiltonian
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. )
21
Parity violating coupling constants
K1 - absence of strong int. corr.
?
Ann.Phys.124(80)449
?
Factorization approximation
SU(6)W
Nucl.Phys.A335(80)147
N.Kaiser,U.G.Meissner, Nucl.Phys.A
489(88)671, 499(89)699,510(90)759
based on chiral model
-7
22
SU(6)W
Balachandram, Phys.Rev. 153 (1967)
1553 S.Pakwasa, S.P.Rosen, Phys.Rev. 147
(1966)1166

• SU(6)W subgroup of SU(12), all transformations
  which leave untouched ?0 and ?3
• Decomposition SU(3)XSU(2)W
• SU(2)W weak isospin
• Generators i?k ?5 (SU(2)W)
• SU(6)W symmetry related to fixed direction
  useful in description of two-body decays

Factorization matrix element factorizes into two
parts Matrix element of current between vacuum
and meson and Matrix element of another currents
between nucleons
23
The cross sections and asymmetries according to
Hwang-Henley pole model
Cross sections and asymmetries (or polarized
cross sections) given by sum of the products of
formfactors and relevant couplings
The unpolarized cross section for pion
photoproduction - good agreement with data.
Having couplings calculated for 8 considered
models and the formfactors taken from
Hwang-Henley pole model the differences of the
polarized cross sections are calculated. The
saturation hypothesis with saturation energy ?sat
0.55 GeV is assumed and free parameters hE
and ? are selected to satisfy condition F
(?sat) lt 0.1 .
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Results
25
Results non-saturated models

• Models 2 and 3 do not satisfy the quick
  saturation hypothesis for any hE and ?
• additional structure should be seen above 0.55
  GeV to satisfy sum rule
• If saturation energy shifted to 1 GeV then ?100
  pb is expected for ?? in energy of photon between
  0.55-1 GeV quite large.
• This might indicate that it is desirable to look
  for p.v. effects in this region
• Remaining considered models satisfy hypothesis
  additional measurements of asymmetries can help
  to distinguish between different models

?
26
The asymmetries for different saturated models.
Model 4
Model 5
(A in 10-7 units, E? in GeV)
27
Results saturated models

• Combining the measurements of ?0 and ?
  asymmetries together would allow to select
  models or group of models.
• Lets define
• A0sat , Asat , A0th , Ath are ?0 and ?
  asymmetries for saturation and threshold energy
  region, respectively.
• Then
• Asat gt0 selects models 1 and 8 in addition
• A0th gt 0 (and/or A0sat lt 0) ? 1
• A0th lt 0 (and/or A0sat gt 0) ? 8

28
Results saturated models

• Asat lt-610-7 (large) selects 4 and 5 in
  addition
• A0th ? -210-7 ? 5
• A0th ? 0 ? 4
• -610-7 lt Asat lt0 selects 1,4,6,7,8 in
  addition
• A0th lt 0( ? -110-7) ? 7
• A0th ? 0 ? 1,4,6,8 - then combinnig with
  Ath and A0sat
• Athgt1 and A0th lt0 select (4 and 6)
  and (1 and 8)

29
Experimental 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 Spin
• Physiscs 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.

30
Experimental feasibility
Taking 60 ?A current at 12 GeV electron beam and
1mm Au plate target we calculate the photon
bremsstrahlung spectrum as follows
For energy range from 0.137 GeV (threshold) to
0.55 GeV (saturation) it reads 1.9109
events/sec. 0.137 0.3 GeV ? 7108
events/sec 0.4 0.55 GeV ?
2.7108 events/sec
Spectrum of photons ? 1/? - bremsstrahlung sum
rule type. For 1cm long liquid hydrogen target
the number of events /sec. is
108 -109 events/sec seems to be large but the
same rate 109is expected in LHC and the
relevant detection techniques are
feasible (E.Longo, Nucl.Inst. and Meth.A 486
(2002)7)
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Experimental feasibility

• 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 6103 - 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 41013 61013 and 1.51012
  4.51012 events,
• respectively. Corresponding beam time
• 6104 8.5104 and 6103 - 1.7104 sec.

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

32
Concluding remarks

• The sum rule has been checked within lowest order
  of the electroweak theory for the photon-induced
  processes with elementary lepton targets. It
  would be interesting to check this sum rule in
  higher perturbative orders.
• In analogy with observed feature of GDH sum rule
  on proton the quick saturation hypothesis has
  been formulated.
• 8 models with different sets of p.v. couplings
  have been analyzed in the frame of effective
  lagrangian and pole model approach

33
Concluding remarks

• Models with the largest p.v.pion couplings h?1 do
  not saturate below 0.55 GeV and the contribution
  from higher energies cross sections are needed
• It is argued that the measurements of the ?0 and
  ? asymmetries at the threshold and close to
  saturation point allow to distinguish between
  saturated models (p.v. couplings)
• The verification of our predictions seems to be
  experimentally feasible with the beam time of the
  order of 105 sec. in the near future experimental
  facilities (JLab)

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35
SU(6)W
Balachandram, Phys.Rev. 153 (1967)
1553 S.Pakwasa, S.P.Rosen, Phys.Rev. 147
(1966)1166

• SU(6)W subgroup of SU(12), all transformations
  which leave untouched ?0 and ?3
• Decomposition SU(3)XSU(2)W
• SU(2)W weak isospin
• Generators i?k ?5 (SU(2)W)
• SU(6)W symmetry related to fixed direction
  useful in description of two-body decays

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