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Coulomb corrections to R-correlation in the polarized neutron decay Alexey Pak

University of Alberta, 2005. Lake Louise Winter Institute 2005, February 25. Coulomb corrections to R-correlation in the polarized neutron decay Alexey Pak. In collaboration with A. Czarnecki. Lake Louise Winter Institute 2005, February 25. Neutron beta-decay: probing C,P,T-invariance.

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Coulomb corrections to R-correlation in the polarized neutron decay Alexey Pak

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  1. University of Alberta, 2005 Lake Louise Winter Institute 2005, February 25 Coulomb corrections to R-correlation in the polarized neutron decayAlexey Pak In collaboration with A. Czarnecki

  2. Lake Louise Winter Institute 2005, February 25 Neutron beta-decay: probing C,P,T-invariance Energy scales: m = 0.511 MeV, DM = 1.2933 MeV, Mp = 938.27 MeV n Observable T-violating correlations: sn se R (T,P): sn [pe ×se] D (T): sn [pp × pe] V (T,P): sn [pp × se] L (T): pp [pe × se] e pe n pp p dG ~ x(1 + b m/E + A(snpe)/p + G(sepe)/E + N(snse) + Q(sepe)(snpe)/E(E + m) + R(se[sn × pe])/E )

  3. Lake Louise Winter Institute 2005, February 25 Neutron beta-decay law General Hamiltonian of the neutron beta-decay: H = (ypyn)(CSyeyn + CS’yeg5yn) + (ypgmyn)(CVyegmyn + CV’yegmg5yn) + 1/2(ypsmnyn)(CTyesmnyn + CT’yesmng5yn) - (ypgmg5yn)(CAyegmg5yn + CA’yegmyn) + (ypg5yn)(CPyeg5yn + CP’yeyn) + H.C. Standard Model: CS = CS’ = CT = CT’ = CP = CP’= 0 CV = -CV’ = -GF/√2 CA = -CA’ = gA GF/√2 gA ≈ 1.26 due to QCD corrections R≠0 may indicate Scalar and Tensor interactions (V-A) law d d u u g d u W

  4. Lake Louise Winter Institute 2005, February 25 Measurements of R-type correlations Experimental constraints on S and T (1s bands are shown): S = Im[(CS + CS’)/CA] T = Im[(CT + CT’)/CA] Rx = 2 Im[ 2(CTCA’*+ CT’CA*) + (CSCA’*+ CS’CA*- CVCT’*- CV’CT*)] - 2 a m/pe Re[ 2(CTCT’*- CACA’*) + (CSCT’*+ CS’CT*- CVCA’*- CV’CA*)]

  5. Lake Louise Winter Institute 2005, February 25 p Theoretical predictions in SM n e 0-th order: Rx(0) = 0 1-st order: Rx(1) = -2GF2a(gA2 - gA)m/pe x = GF2(1 + 3gA2) R(1) ~ 8.3×10-4 m/pe The origin of this result and the factor (gA2 - gA): J = (J0,Jz,J+1,J-1) - lepton current, proton at rest, nucleons - plane waves dG ~ |‹p|H|n›|2 = |‹p|H|n›|2V + |‹p|H|n›|2A + |‹p|H|n›|2VA After integrating over neutrino directions: |‹p|H|n›|2V = 2g2|J0|2 = 2g2 (F+eFe) |‹p|H|n›|2A = 2g2gA2(|Jz|2 + 2|J+1|2) = 2g2gA2((F+eFe)+ 2(F+e(1+sz)Fe)) |‹p|H|n›|2VA = -2g2gA(iJ0Jz* + c.c.) = -4g2gA(F+eszFe) Coulomb-distorted wavefunction (exact potential solutions at R→0): F+eszFe = F(Z,E)( - vz/c + pz(se p) + (1 - a2)1/2/E [p×[se×p]]z- a/E [se×p]z) F+eFe = F(Z,E)(1 – (se v)/c) – no contribution to R g

  6. Lake Louise Winter Institute 2005, February 25 Types of further corrections • R ≈ 8.3×10-4 m/pe , Rx(1) = -2GF2 a (gA2 - gA) m/pe • Rx(2) = Rx(1)(1 + DRx(kinematic) + DRx(radiative) + DRx(finite size)) • a/p = 2.3×10-3 – further radiative corrections • m/Mp = 5×10-4 – proton recoil effects • a = 7.29×10-3 – corrections to lepton wavefunctions • pnRN ~ 10-3 – higher angular momenta emissions (for non-point-like nucleons) ((Le + Se) + (Ln + Sn))z= J = 0,1 Le,n - not constrained 1) Higher L (Dirac quantum number k) suppressed by centrifugal effect 2) n→p transition only favors certain g-matrix combinations (vp << c) n p p or Sz = ±1/2 Sz = 1/2 “Allowed approximation”

  7. Lake Louise Winter Institute 2005, February 25 Finite nucleon size effects “normal approximation”: leading orders in avN/c, aRN/le nuclear structure: b-moments - calculated in MIT bag model • MIT bag model: • non-interacting m=0 quarks • constant pressure on the spherical bag boundary • lowest levels identified • as N-D • prediction: gA = 1.09 ‹p|t+(iJ YJm)*|n› = (C.-G.C.) ‹YJ› ‹p|t+(iJg5YJm)*|n› = (C.-G.C.) ‹g5YJ› ‹p|t+(iLsTLJm)†|n› = (C.-G.C.) ‹sTLJ› ‹p|t+(iLaTLJm)†|n› = (C.-G.C.) ‹aTLJ› (C.-G.C.) = ‹½(M’) J(m) | ½ (½)› lepton wavefunctions: n: free Bessel functions e: numerical solutions for spherically-symmetrical potential matching inside and outside p

  8. Lake Louise Winter Institute 2005, February 25 Finite nucleon size effects Expansion in terms of nuclear momenta (E.Konopinski): dG = dE dW × 2 g2 (DM - E)2Sm=±1/2 Sm=±1/2 × |Sk=±1, ±2,… SJ = 0,1 e-id(k)c+km A*JkrJ-1 ‹1/2(m)1/2(m)|J(m + m)› ‹I’(M’)J(m + m)|I(M)›|2 DRx(finite size) pe, MeV/c

  9. Lake Louise Winter Institute 2005, February 25 Proton recoil effects Including higher powers of m/Mn,DM/Mn, pe/Mn, we obtain: DRx(kinem) = - (E2(5 + 11gA) + DM2(2 + 8gA) – DME (7 + 13gA) – 6gAm2) / (6gA (DM - E) Mn) DRx(kinematic) pe, MeV/c

  10. Lake Louise Winter Institute 2005, February 25 Radiative corrections Following diagrams are considered with the Coulomb-distorted electron wavefunction (ultraviolet divergence cut at L = 81 GeV) (Yokoo, Suzuki, Morita; Vogel, Werner): p g n DRx(radiative) e n pe, MeV/c

  11. Lake Louise Winter Institute 2005, February 25 All Coulomb corrections DRx(Coulomb) pe, MeV/c pe, MeV/c Depending on the experimental setup, more calculations are needed to establish the theoretical uncertainty to R-correlation.

  12. Lake Louise Winter Institute 2005, February 25 Summary and conclusions • Theoretical uncertainties to R-correlation in the process • n→p,e-,n have been analyzed, including: • proton recoil • radiative corrections • finite nucleon size effects • Current and the next generation experiments will not hit • the SM background Thank you for your attention

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