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Higher Order Electroweak
Corrections for Parity Violating
Analog of GDH Sum Rule
Krzysztof Kurek
Andrzej Sołtan Institute for Nuclear Studies, Wasaw
In collaboration with Leszek Łukaszuk
P.v.analog of GDH sum rule
Dispersion relations and derivation of the p.v.s.r.:
 Leszek Łukaszuk, Nucl.Phys.A 709 (2002) 289-298
Applications: proton/deuteron target:
 Krzysztof Kurek & Leszek Łukaszuk,
Phys.Rev.C 70 (2004) 065204
 Krzysztof Kurek, Proceedings of X Workshop on
High Energy Spin Physics, NATO ARW DUBNA-SPIN-03,
editors: A.V.Efremov and O.V.Teryaev, Dubna 2004, p.109.
Outlook
Revival of interest in parity violating Compton
scattering.

Dispersion relations and low energy behaviour.
Sum rules for p.v. spin polarizabilities .

P.v. analog of GDH sum rule.
_______________________________________________



Derivative of the GDH sum rule.
(Pascalutsa,Holstein,Vanderhaeghen, 2004)
P.v. analog of GDH sum rule for elementary targets :
- lowest order in EW perturbative theory;
- derivative of the p.v.analog of GDH sum rule
(one-loop EW corrections ).
Introduction
+ photo-production
 Polarized
photon
asymmetry
in

New experiments based on intense polarized beams
near
the threshold
canopportunity
be a good candidate
to part
of photons
give the
to test a weak
of photon-hadron
interactions
(parity
measure
p.v. pion-nucleon
couling
h1; ifviolating,
large, p.v.)
dominates low energy nucleon weak interactions
to knowledge
large range.of p.v. couplings in nucleon-meson
dueThe
 Similar
is expected for
the low
energy Compton
(nucleon-nucleon)
forces
is important
for
scattering.
understanding
the non-leptonic, weak hadronic
1
 h has been measured in nuclear and atomic
interactions (p.v. couplings are poorly
known).
18
systems; the disagreement between F and
133Cs experiments is seen.
We are looking for model independent relations
(sum rules) involving parity violating reactions
Asymptotic states in SM and the
Compton amplitudes
Collision theory and SM:
 Asymptotic states – stable particles (photons,
electrons and at least one neutrino, 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, FadeevKulish, Frohlich, Buchholz et al.. 1991)
Asymptotic states in SM and the
Compton amplitudes


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
Dispersion relations and low energy
behaviour
Let’s consider Compton amplitude:
For Re() >0 we get the physical Compton amplitude;
For Re() <0 the limiting amplitude can be obtained applying complex
conjugation :
Dispersion relations and low energy
behaviour
Coherent amplitudes (related to cross section):
crossing
Here T inv. is
not demanded
Normalization (Optical theorem):
Dispersion relations and low energy
behaviour
Causality, crossing, unitarity  dispersion relation for amplitude f
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.Khriplovich et al..,
Sov.Phys.JETP 82(1996) 616
Dispersion relations and low energy
behaviour
Unpolarized target
Superconvergence hypothesis
and p.v. analog of GDH sum rule
Subtraction point is taken at  =0 and - due to LET –
no arbitrary constants appear in the dispersion
formulae for fh(-)
Assuming superconvergence:
fh(-) () → 0 with →

¯¯¯¯¯¯¯
Parity violating analog of GDH sum rule
GDH (p.c.) sum rule and p.v. analog
of GDH sum rule
For ½ spin target the above formula is equivalent to:
allowing T-violation
Nucl.Phys.B 11(1969)2777
Anomalous magnetic moment
Electric dipole moment
Lowest order SM, see also:
S.Brodsky,I.Schmidt, 1995
(2+ 2)
GDH sum rule (p.c.)
GDH sum
and p.v.(1965)
analog
of
S.B. rule
Gerasimov,Yad.Fiz.2
598
S.D. Drell, A.C. Hearn,
GDH sum
rule
Phys.Rev.Lett. 16 (1966) 908
The formulae from previous slide are equivalent to
pair of sum rules in the form:
Let us emphasise that only if the p.v. sum rule
is true the formula
become
equivalent and identical with those from Almond.
In such a case the photon momentum direction can be ignored
and the p.c. sum rule reduce to the standard form of GDH sum
rule used in literature.
Two questions:


Asymptotic high energy behavior of the
cross sections – to guarantie the sum rule
integral converge
Higher order EW corrections
High energy contribution
Small? Numerically yes but not clear if
integral converge;
- Parton model: contribution from sea/gluon ~1/x;

→ ∞ means x → 0 !
- summation over: x → worse 1/xα ,Ermolaev talk
energy2 ≈ s
- saturation model: Log(s)?
Some indications but I don’t know definite answer.

Also true for „standard” p.c. GDH sum rule
Higher order Electroweak corrections
GDH (I*) sum rule in QED
Non-trivial elementary target example:
* C.K.Iddings,Phys.Rev.B 138 (1965)446
To obtain anomalous magnetic
moment of electron/muon
there are– two
possibilities:
Anomalous magnetic moment of electron
J. Schwinger
- One loop direct calculations
(Phys.Rev.73 (1948)4161): = /2 - GDH integral = 0 up tp 2
a’ la Schwinger
First non-zero contribution : 3
- Calculate α3 cross sections and
integrate to GDH s.r.
 + e →  + e ( 3 virtual corrections)
 + e → e + e + e ‾( pair production)
 + e → e +  +  (double photon Compton )
D.A. Dicus, R.Vega Phys.Lett.B 501 (2002)44
Derivative of the GDH sum rule
(V.Pascalutsa, B.R.Holstein,M.Vanderhaeghen, 2004)
Introduce „classical” value of a.m.m,
Then the total a.m.m = κ0 + δκ,
δκ – quantum (loop) corrections
Note that in the theory of explicit Pauli term GDH s.r. is not valid,
since there now exists a tree-level contribution to the Compton
amplitude which cannot be reproduced by a dispersion relations using
The degrees of freedom included in the theory
(photons and spin ½ fermions in case of QED)
High energy degree of freedom –
Integrated out of the theory
„New” sum rule
Derivative of the GDH sum rule
(V.Pascalutsa, B.R.Holstein,M.Vanderhaeghen, 2004)
Taking the limit to the theory with vanishing classical a.m.m
(κ0 = 0, δκ → κ) we back to GDH sum rule but we obtain a new
sum rule by taking the first derivative with respect to κ0 of both
lhs and rhs of the above equation.
Valid for non-perturbative as well as for any given order in
perturbation theory.
Derivative of the GDH sum rule
(V.Pascalutsa, B.R.Holstein,M.Vanderhaeghen, 2004)
To the lowest order it reduces to:
Linear relation: tree cross sections
enough, no renornalization, etc.!!
Gives 0
Reproduce imiediately Schwinger result:
P.v. analog of GDH sum rule
The photon scattering off elementary
lepton targets – tree level
e → Z0e
 → We
e → W
e → Z0e
 → We (multiplied by 0.1)
e →  W (multiplied by 5)
P.v. sum rule satisfied for every process separately,
also separately for left- and right- hand side 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).
EW corrections – the crosscheck of
the method (trick)
 The procedure:
calculate tree–level cross sections for all
of contributing processes but with modified couplings:
explicit Pauli term according PHV.
 This is equivalent to one-loop EW corrections to GDH sum rule
(EW corrections to anomalous magnetic moment) and one-loop
corrections to p.v. analog of GDH s.r.
____________________________________________________
 To crosscheck of the method: we consider also another
modified coupling – „axial” Pauli term – should be responsible
for T violated amplitude which appears in SM on the level of
two-loop. It means that our „modified” tree-level cross sections
should be zero themselves, not after integration in s.r.
Results


Anomalous magnetic moment (GDH sum rule): agree
with known EW corrections (e.g. Phys.Rev.D5
(1972)2396, Czarnecki et al.., a.m.m of muon).
The results of GDH integral for W and Z0 bosons
separately are not coincidate with a.m.m. contribution
from these bosons calculated in one-loop direct method;
The contribution from both integral added together
reproduce correct result
Test for T-violated calculation passed: the cross sections
with modified „axial” Pauli couplings gives zero on the
level of cross sections (no electric dipole moment on
one-loop level as should be)
The method seems to be working in the case of EW theory

Results cont.
Integral of p.v.analog of GDH sum rule
is zero for Z boson but differs from 0 for W!
 Signal that one-loop EW corrections can violate p.v.
analog of GDH sum rule.
 Question of aplicability of the method
(PHV trick not working?), modified LET?
subtraction point not at zero?
need confirmation (one-loop direct calculations of
the cross sections)
New cotribution
New physics ?
