Higher Order Electroweak Higher Order Electroweak Corrections for Parity ViolatingCorrections for Parity ViolatingAnalog of GDH Sum RuleAnalog of GDH Sum Rule

Krzysztof KurekAndrzej Sołtan Institute for Nuclear Studies, Wasaw

In collaboration with Leszek Łukaszuk

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

P.v.analog of GDH sum ruleP.v.analog of GDH sum rule

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.

OutlookOutlook

Revival of interest in Revival of interest in parity violating Compton parity violating Compton scattering.scattering.

Dispersion relations and low energy behaviour.Dispersion relations and low energy behaviour. Sum rules for p.v. spin polarizabilities .Sum rules for p.v. spin polarizabilities . P.v. analog of GDH sum rule.P.v. analog of GDH sum rule.______________________________________________________________________________________________

Derivative of the GDH sum rule. Derivative of the GDH sum rule. (Pascalutsa,Holstein,Vanderhaeghen, 2004)(Pascalutsa,Holstein,Vanderhaeghen, 2004)

P.v. analog of GDH sum rule for elementary targets :P.v. analog of GDH sum rule for elementary targets : - lowest order in EW perturbative theory;- lowest order in EW perturbative theory; - - derivative of the p.v.analog of GDH sum rulederivative of the p.v.analog of GDH sum rule (one-loop EW corrections ).(one-loop EW corrections ).

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

The knowledge of p.v. couplings in nucleon-The knowledge of p.v. couplings in nucleon-meson 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; if large, ; if large, dominates low energy nucleon weak interactions dominates low energy nucleon weak interactions due to large range.due to large range. 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. We are looking for model independent relations We are looking for model independent relations (sum rules) involving parity violating reactions(sum rules) involving parity violating reactions

IntroductionIntroduction

Asymptotic states in SM and Asymptotic states in SM and the Compton amplitudesthe Compton amplitudes

Collision theory and SM:Collision theory and SM: Asymptotic states – stable particlesAsymptotic states – stable particles (photons, (photons,

electrons and electrons and at least one at least one neuneuttrino, proton rino, proton and stable atomic ions)and stable 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 Asymptotic states in SM and the Compton amplitudesthe Compton amplitudes

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 be will 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 Compton amplitude:

For Re() >0 we get the physical Compton amplitude;For Re() <0 the limiting amplitude can be obtained applying complexconjugation :

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

Coherent amplitudes (related to cross section):

Normalization (Optical theorem):

crossing

Here T inv. is not demanded

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

Causality, crossing, unitarity 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

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

Unpolarized target

SSuperconvergence hypothesisuperconvergence hypothesisand pand p.v. analog of GDH sum .v. analog of GDH sum rulerule

¯¯¯¯¯¯¯

Assuming superconvergence:fh

(-) () → 0 with →

Subtraction point is taken at =0 and - due to LET –no arbitrary constants appear in the dispersion formulae for fh

(-)

Parity violating analog of GDH sum rule

For ½ spin target the above formula is equivalent to:

GDH (p.c.) sum rule and p.v. GDH (p.c.) sum rule and p.v. analog of GDH sum ruleanalog of GDH sum rule

(2+ 2)

Anomalous magnetic moment

Electric dipole moment

allowing T-violation

Lowest order SM, see also: S.Brodsky,I.Schmidt, 1995

Nucl.Phys.B 11(1969)2777

GDH sum rule and p.v. analog GDH sum rule and p.v. analog of GDH sum ruleof GDH sum rule

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 ignoredand the p.c. sum rule reduce to the standard form of GDH sumrule used in literature.

GDH sum rule (p.c.)S.B. Gerasimov,Yad.Fiz.2 (1965) 598S.D. Drell, A.C. Hearn, Phys.Rev.Lett. 16 (1966) 908

Two questions:Two questions:

Asymptotic high energy behavior of Asymptotic high energy behavior of the cross sections – to guarantie the the cross sections – to guarantie the sum rule integral convergesum rule integral converge

Higher order EW correctionsHigher order EW corrections

High energy contributionHigh energy contribution

Small? Numerically yes but not clear ifSmall? Numerically yes but not clear if integral converge;integral converge; - Parton model: contribution from sea/gluon ~1/x; - Parton model: contribution from sea/gluon ~1/x;

energyenergy22 ≈ ≈ s s → ∞ means x ∞ means x → 0 !

- summation over: x → worse 1/xα ,Ermolaev talk,Ermolaev talk - saturation model: Log(s)?- saturation model: Log(s)?Some indications but I don’t know definite answer.Some indications but I don’t know definite answer.

Also true for „standard” p.c. GDH sum ruleAlso true for „standard” p.c. GDH sum rule

Higher order Electroweak Higher order Electroweak correctionscorrections

Non-trivial elementary target example:Non-trivial elementary target example:

D.A. Dicus, R.Vega Phys.Lett.B 501 (2002)44

Anomalous magnetic moment of electron – J. Schwinger

(Phys.Rev.73 (1948)4161): = /2 - GDH integral = 0 up tp 2

First non-zero contribution : 3

+ e → + e ( 3 virtual corrections) + e → e + e + e ( pair production) + e → e + + (double photon Compton )

‾

* C.K.Iddings,Phys.Rev.B 138 (1965)446

GDH GDH (I*) sum rule in QED sum rule in QED

To obtain anomalous magnetic moment of electron/muonthere are two possibilities:- One loop direct calculations a’ la Schwinger- Calculate α3 cross sections and integrate to GDH s.r.

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 usingThe degrees of freedom included in the theory (photons and spin ½ fermions in case of QED)

Derivative of the GDH sum ruleDerivative of the GDH sum rule(V.Pascalutsa, B.R.Holstein,M.Vanderhaeghen, (V.Pascalutsa, B.R.Holstein,M.Vanderhaeghen, 2004)2004)

Introduce „classical” value of a.m.m,Then the total a.m.m = κ0 + δκ, δκ – quantum (loop) corrections

High energy degree of freedom – Integrated out of the theory

„New” sum rule

Derivative of the GDH sum ruleDerivative of the GDH sum rule(V.Pascalutsa, B.R.Holstein,M.Vanderhaeghen, (V.Pascalutsa, B.R.Holstein,M.Vanderhaeghen, 2004)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 newsum rule by taking the first derivative with respect to κ0 of bothlhs 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 ruleDerivative of the GDH sum rule(V.Pascalutsa, B.R.Holstein,M.Vanderhaeghen, (V.Pascalutsa, B.R.Holstein,M.Vanderhaeghen, 2004)2004)

To the lowest order it reduces to:

Linear relation: tree cross sectionsenough, no renornalization, etc.!!

Reproduce imiediately Schwinger result:

Gives 0

Nice tr

ick

P.v. analog of GDH sum ruleP.v. analog of GDH sum ruleThe photon scattering off The photon scattering off elementary lepton targets – tree elementary lepton targets – tree levellevel

e → Z0e → Wee → 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 EW corrections – the crosscheck of the method crosscheck of the method (trick)(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.

ResultsResults

Anomalous magnetic moment (GDH sum rule): agree Anomalous magnetic moment (GDH sum rule): agree with known EW corrections (e.g. Phys.Rev.D5 with known EW corrections (e.g. Phys.Rev.D5

(1972)2396, Czarnecki et al.., a.m.m of muon). (1972)2396, Czarnecki et al.., a.m.m of muon).

The results of GDH integral for W and ZThe results of GDH integral for W and Z0 0 bosons bosons separately are not coincidate with a.m.m. separately are not coincidate with a.m.m. contribution from these bosons calculated in one-contribution from these bosons calculated in one-loop direct method; The contribution from both loop direct method; The contribution from both integral added together reproduce correct resultintegral added together reproduce correct result

Test for T-violated calculation passed: the cross Test for T-violated calculation passed: the cross sections with modified „axial” Pauli couplings gives sections with modified „axial” Pauli couplings gives zero on the level of cross sections (no electric dipole zero on the level of cross sections (no electric dipole moment on one-loop level as should be)moment on one-loop level as should be)

The method seems to be working in the case of EW The method seems to be working in the case of EW theorytheory

Results cont.Results cont.

Integral of p.v.analog of GDH sum ruleIntegral of p.v.analog of GDH sum rule is zero for Z boson is zero for Z boson but differs from 0 for W!but differs from 0 for W! Signal that one-loop EW corrections can Signal that one-loop EW corrections can

violate p.v. analog of GDH sum rule.violate p.v. analog of GDH sum rule. Question of aplicability of the methodQuestion of aplicability of the method (PHV trick not working?), modified LET?(PHV trick not working?), modified LET? subtraction point not at zero? subtraction point not at zero? need confirmation (one-loop direct need confirmation (one-loop direct

calculations of the cross sections)calculations of the cross sections)New cotribution New cotribution New physics New physics ??