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

2. Lake Louise Winter Institute 2005, February 25 Neutron beta-decay: probing C,P,T-invariance snsn p pepe sese n e p Observable T-violating correlations: R (T,P): s n [p e × s e ] D (T):s n [p p × p e ] V (T,P):s n [p p × s e ] L (T):p p [p e × s e ] d  ~  (1 + b m/E + A (s n p e )/p + G (s e p e )/E + N (s n s e ) + Q (s e p e )(s n p e )/E(E + m) + R (s e [s n × p e ])/E ) Energy scales: m = 0.511 MeV,  M = 1.2933 MeV, M p = 938.27 MeV

3. Lake Louise Winter Institute 2005, February 25 Neutron beta-decay law General Hamiltonian of the neutron beta-decay: H = (  p  n )(C S  e  + C S ’  e    ) + (  p    n )(C V  e    + C V ’  e      ) + 1/2(  p    n )(C T  e    + C T ’  e      ) - (  p      n )(C A  e      + C A ’  e    ) + (  p    n )(C P  e    + C P ’  e  ) + H.C. Standard Model: C S = C S ’ = C T = C T ’ = C P = C P ’= 0 C V = -C V ’ = -G F /√2 C A = -C A ’ = g A G F / √2 g A ≈ 1.26 due to QCD corrections (V-A) law g W u d u d u d R≠0 may indicate Scalar and Tensor interactions

4. Lake Louise Winter Institute 2005, February 25 Measurements of R-type correlations DecayCorrelationResult ×10 3 Location 19 Ne→ 19 Fe, s Ne [p e ×s e ]-79 ± 53Princeton  0 →  -,p s  [p p ×s p ]-100 ± 70BNL  0 →  -,p s  [p p ×s p ]-94 ± 60CERN  + →e +, e,  s  [p e ×s e ]7 ± 23SIN 8 Li→ 8 Be,e -, e s Li [p e ×s e ]1.6 ± 2.2PSI n→p,e, e s n [p e ×s e ]? ± 5PSI (2005) S = Im[(C S + C S ’)/C A ] T = Im[(C T + C T ’)/C A ] Experimental constraints on S and T (1  bands are shown): R  = 2 Im[ 2(C T C A ’*+ C T ’C A *) + (C S C A ’*+ C S ’C A *- C V C T ’*- C V ’C T *)] - 2  m/p e Re[ 2(C T C T ’*- C A C A ’*) + (C S C T ’*+ C S ’C T *- C V C A ’*- C V ’C A *)]

5. Lake Louise Winter Institute 2005, February 25 Theoretical predictions in SM n p e 0-th order: R   = 0 1-st order: R   = -2G F 2  (g A 2 - g A )m/p e  = G F 2 (1 + 3g A 2 ) R (1) ~ 8.3×10 -4 m/p e The origin of this result and the factor (g A 2 - g A ): J = (J 0,J z,J +1,J -1 ) - lepton current, proton at rest, nucleons - plane waves d  ~ |‹p|H|n›| 2 = |‹p|H|n›| 2 V + |‹p|H|n›| 2 A + |‹p|H|n›| 2 VA After integrating over neutrino directions: |‹p|H|n›| 2 V = 2g 2 |J 0 | 2 = 2g 2 (   e  e ) |‹p|H|n›| 2 A = 2g 2 g A 2 (|J z | 2 + 2|J +1 | 2 ) = 2g 2 g A 2 ((  + e  e )+ 2(  + e  z )  e )) |‹p|H|n›| 2 VA = -2g 2 g A (iJ 0 J z * + c.c.) = -4g 2 g A  + e  z  e ) Coulomb-distorted wavefunction (exact potential solutions at R→0):  + e  z  e = F(Z,E)( - v z /c + p z (  e p) + (1 -  2 ) 1/2 /E [p×[  e ×p]] z -  /E [  e ×p] z )  + e  e = F(Z,E)(1 – (  e v)/c) – no contribution to R 

6. Lake Louise Winter Institute 2005, February 25 Types of further corrections R ≈ 8.3×10 -4 m/p e, R  (1) = -2G F 2  (g A 2 - g A ) m/p e R  (2) = R  (1) (1 +  R  (kinematic) +  R  (radiative) +  R  (finite size) )  /  = 2.3×10 -3 – further radiative corrections m/M p = 5×10 -4 – proton recoil effects  = 7.29×10 -3 – corrections to lepton wavefunctions p R N ~ 10 -3 – higher angular momenta emissions (for non-point-like nucleons) n p or p S z = 1/2 S z = ±1/2 ((L e + S e ) + (L + S )) z = J = 0,1 L e, - not constrained 1) Higher L (Dirac quantum number  ) suppressed by centrifugal effect 2) n→p transition only favors certain  -matrix combinations (v p << c) “Allowed approximation”

7. Lake Louise Winter Institute 2005, February 25 Finite nucleon size effects “normal approximation”: leading orders in  v N /c,  R N / e nuclear structure:  -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-  prediction: g A = 1.09 ‹p|  + (i J Y Jm )*|n› = (C.-G.C.) ‹Y J › ‹p|  + (i J   Y Jm )*|n› = (C.-G.C.) ‹   Y J › ‹p|  + (i L  T L Jm ) † |n› = (C.-G.C.) ‹  T L J › ‹p|  + (i L  T L Jm ) † |n› = (C.-G.C.) ‹  T L J › (C.-G.C.) = ‹½ (M’) J(m) | ½ (½)› lepton wavefunctions: : 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 p e, MeV/c  R  (finite size) d  = dE d  × 2 g 2 (  M - E) 2   =±1/2   = ±1/2 × |   =±1, ±2,…  J = 0,1 e -i     A* J   J -1 ‹1/2(  )1/2(  )|J(  +  )› ‹I’(M’)J(  +  )|I(M)›| 2 Expansion in terms of nuclear momenta (E.Konopinski):

9. Lake Louise Winter Institute 2005, February 25 Proton recoil effects Including higher powers of m/M n,  M/M n, p e /M n, we obtain:  R  (kinem) = - (E 2 (5 + 11g A ) +  M 2 (2 + 8g A ) –  ME (7 + 13g A ) – 6g A m 2 ) / (6g A (  M - E) M n ) p e, MeV/c  R  (kinematic)

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

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

12. Lake Louise Winter Institute 2005, February 25 Summary and conclusions Theoretical uncertainties to R-correlation in the process n→p,e -, 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