1. 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

2. 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 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.

3. Outlook  Revival of interest in parity violating Compton scattering.  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. _______________________________________________  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; - lowest order in EW perturbative theory; - derivative of the p.v.analog of GDH sum rule - derivative of the p.v.analog of GDH sum rule (one-loop EW corrections ). (one-loop EW corrections ).

4.  New experiments based on intense polarized beams of photons give the opportunity to test a weak part of photon-hadron interactions (parity violating, p.v.)  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  + 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 h  1 ; if large, measure p.v. pion-nucleon couling h  1 ; 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 scattering. scattering.  h  1 has been measured in nuclear and atomic systems; the disagreement between 18 F and systems; the disagreement between 18 F and 133 Cs experiments is seen. 133 Cs experiments is seen. We are looking for model independent relations (sum rules) involving parity violating reactions Introduction

5. 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) 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) 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) 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)

6. 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. 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)): 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 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  G F order contribution; massive Z 0 and W  or H bosons) + any order in strong interactions + any order in strong interactions

7. 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 :

8. Dispersion relations and low energy behaviour Coherent amplitudes (related to cross section): Normalization (Optical theorem): crossing Here T inv. is not demanded

9. Dispersion relations and low energy behaviour Causality, crossing, unitarity  dispersion relation for amplitude f

10. Dispersion relations and low energy behaviour Low Energy Theorem (LET) for any spin of target: A.Pais, Nuovo Cimento A53 (1968)433 I.B.Khriplovich et al.., Sov.Phys.JETP 82(1996) 616 P, K

11. Dispersion relations and low energy behaviour Unpolarized target

12. Superconvergence hypothesis and p.v. analog of GDH sum rule ¯¯¯¯¯¯¯ Assuming superconvergence: f h (-)  (  )  → 0 with  →  Subtraction point is taken at  =0 and - due to LET – no arbitrary constants appear in the dispersion formulae for f h (-)  Parity violating analog of GDH sum rule

13. For ½ spin target the above formula is equivalent to: GDH (p.c.) sum rule and p.v. analog 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

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

15. Two questions:  Asymptotic high energy behavior of the cross sections – to guarantie the sum rule integral converge  Higher order EW corrections

16. High energy contribution SSSSmall? Numerically yes but not clear if integral converge; - Parton model: contribution from sea/gluon ~1/x; energy2 ≈ s → ∞ means x → 0 ! - summation over: x → worse 1/x α,Ermolaev talk - saturation model: Log(s)? Some indications but I don’t know definite answer. AAAAlso true for „standard” p.c. GDH sum rule

17. Higher order Electroweak corrections

18. 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 sum rule in QED GDH (I*) sum rule in QED To obtain anomalous magnetic moment of electron/muon there are two possibilities: - One loop direct calculations a’ la Schwinger - Calculate α 3 cross sections and integrate to GDH s.r.

19. 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) 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 = κ 0 + δκ, δκ – quantum (loop) corrections High energy degree of freedom – Integrated out of the theory „New” sum rule

20. 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 (κ 0 = 0, δκ → κ) we back to GDH sum rule but we obtain a new sum rule by taking the first derivative with respect to κ 0 of both lhs and rhs of the above equation. Valid for non-perturbative as well as for any given order in perturbation theory.

21. 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.!! Reproduce imiediately Schwinger result: Gives 0 Nice trick

22. P.v. analog of GDH sum rule The photon scattering off elementary lepton targets – tree level  e → Z 0 e  → We  e → W  e → Z 0 e  → 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).

23. 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.

24. 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). 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 Z 0 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

25. Results cont.  Integral of p.v.analog of GDH sum rule is zero for Z boson but differs from 0 for W! 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? (PHV trick not working?), modified LET? subtraction point not at zero? subtraction point not at zero? need confirmation (one-loop direct calculations of the cross sections) need confirmation (one-loop direct calculations of the cross sections) New cotribution New physics ?