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Higher Order Electroweak Corrections for Parity ViolatingAnalog 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 • Polarized photon asymmetry in + photo-production near the threshold can be a good candidate to measure p.v. pion-nucleon couling h1; if large, dominates low energy nucleon weak interactions due to large range. • Similar is expected for the low energy Compton scattering. • h1 has been measured in nuclear and atomic systems; the disagreement between 18F and 133Cs experiments is seen. We are looking for model independent relations (sum rules) involving parity violating reactions • 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 understanding the non-leptonic, weak hadronic interactions (p.v. couplings are poorly known).
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, Fadeev-Kulish, 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 Wor 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 hypothesisand 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 and p.v. analog of GDH sum rule 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 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 thestandard 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; energy2 ≈ s →∞ means x → 0 ! - summation over: x → worse 1/xα,Ermolaev talk - saturation model: Log(s)? Some indications but I don’t know definite answer. • Also true for „standard” p.c. GDH sum rule
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: • One loop direct calculations • a’ la Schwinger • Calculate α3 cross sections and • integrate to GDH s.r. 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 ) ‾ 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: Nice trick Linear relation: tree cross sections enough, no renornalization, etc.!! Gives 0 Reproduce imiediately Schwinger result:
P.v. analog of GDH sum ruleThe photon scattering off elementary lepton targets – tree level e → Z0e → We (multiplied by 0.1) e → W (multiplied by 5) e → Z0e → We e → W P.v. sum rule satisfiedfor 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 ?