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

Lake Louise Winter Institute 2005, February 25

Neutron beta-decay: probing C,P,T-invariance

sn

pp

pe

se

n

e

p

Observable T-violating correlations:

R (T,P): sn [pe × se]

D (T): sn [pp × pe]

V (T,P): sn [pp × se]

L (T): pp [pe × se]

d ~(1 + b m/E + A (snpe)/p + G (sepe)/E + N (snse)

+ Q (sepe)(snpe)/E(E + m) + R (se[sn × pe])/E )

Energy scales: m = 0.511 MeV, M = 1.2933 MeV, Mp = 938.27 MeV

Lake Louise Winter Institute 2005, February 25

Neutron beta-decay law

General Hamiltonian of the neutron beta-decay:H = (pn)(CSe + CS’e) + (pn)(CVe + CV’e) + 1/2(pn)(CTe + CT’e) - (pn)(CAe + CA’e) + (pn)(CPe + CP’e) + H.C.

Standard Model: CS = CS’ = CT = CT’ = CP = CP’= 0CV = -CV’ = -GF/√2CA = -CA’ = gA GF/√2

gA ≈ 1.26 due to QCD corrections

(V-A) law

g

W

u dud

ud

R≠0 may indicate Scalar and Tensor interactions

Lake Louise Winter Institute 2005, February 25

Measurements of R-type correlations

Decay Correlation Result ×103

Location

19Ne→19Fe, sNe[pe×se] -79 ± 53 Princeton

0→-,p s[pp×sp] -100 ± 70 BNL

0→-,p s[pp×sp] -94 ± 60 CERN

+→e+,e, s[pe×se] 7 ± 23 SIN

8Li→8Be,e-,e sLi[pe×se] 1.6 ± 2.2 PSI

n→p,e,e sn[pe×se] ? ± 5 PSI (2005) S = Im[(CS + CS’)/CA]

T = Im[(CT + CT’)/CA]

Experimental constraints on S and T (1 bands are shown):

R = 2 Im[ 2(CTCA’*+ CT’CA*) + (CSCA’*+ CS’CA*- CVCT’*- CV’CT*)] - 2 m/pe Re[ 2(CTCT’*- CACA’*) + (CSCT’*+ CS’CT*- CVCA’*- CV’CA*)]

Lake Louise Winter Institute 2005, February 25

Theoretical predictions in SMn

p

e0-th order: R= 01-st order:

R = -2GF2(gA

2 - gA)m/pe

= GF2(1 + 3gA

2)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 wavesd ~ |‹p|H|n›|2 = |‹p|H|n›|2

V + |‹p|H|n›|2A + |‹p|H|n›|2

VA

After integrating over neutrino directions:|‹p|H|n›|2

V = 2g2|J0|2 = 2g2 (ee)

|‹p|H|n›|2A = 2g2gA

2(|Jz|2 + 2|J+1|2) = 2g2gA2((+

ee)+ 2(+ez)e))

|‹p|H|n›|2VA = -2g2gA(iJ0Jz* + c.c.) = -4g2gA+

eze)

Coulomb-distorted wavefunction (exact potential solutions at R→0): +

eze = F(Z,E)( - vz/c + pz(e p) + (1 - 2)1/2/E [p×[e×p]]z - /E [e×p]z )+

ee = F(Z,E)(1 – (e v)/c) – no contribution to R

Lake Louise Winter Institute 2005, February 25

Types of further corrections

R ≈ 8.3×10-4 m/pe , R(1) = -2GF2 (gA

2 - gA) m/pe

R(2) = R(1)(1 + R(kinematic) + R(radiative) + R(finite size))

• /= 2.3×10-3 – further radiative corrections• m/Mp = 5×10-4 – proton recoil effects• = 7.29×10-3 – corrections to lepton wavefunctions • pRN ~ 10-3 – higher angular momenta emissions

(for non-point-like nucleons)

n p or p

Sz = 1/2 Sz = ±1/2

((Le + Se) + (L + S))z= J = 0,1

Le, - not constrained

1) Higher L (Dirac quantum number ) suppressed by centrifugal effect

2) n→p transition only favors certain -matrix combinations (vp << c)

“Allowed approximation”

Lake Louise Winter Institute 2005, February 25

Finite nucleon size effects

“normal approximation”: leading orders in vN/c, RN/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: gA = 1.09

‹p|+(iJ YJm)*|n› = (C.-G.C.) ‹YJ›

‹p|+(iJ YJm)*|n› = (C.-G.C.) ‹YJ›

‹p|+(iL TLJm)†|n› = (C.-G.C.) ‹TL

J›

‹p|+(iL TLJm)†|n› = (C.-G.C.) ‹TL

J›

(C.-G.C.) = ‹½ (M’) J(m) | ½ (½)›

lepton wavefunctions:: free Bessel functionse: numerical solutions for spherically-symmetrical potential matching inside and outside p

Lake Louise Winter Institute 2005, February 25

Finite nucleon size effects

pe, MeV/c

R(finite size)

d = dE d× 2 g2 (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):

Lake Louise Winter Institute 2005, February 25

Proton recoil effects

Including higher powers of m/Mn, M/Mn, pe/Mn, we obtain:

R(kinem) = - (E2(5 + 11gA) + M2(2 + 8gA) – ME (7 + 13gA) – 6gAm2) / (6gA (M - E) Mn)

pe, MeV/c

R(kinematic)

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)

np

e

R(radiative)

pe, MeV/c

(Yokoo, Suzuki, Morita; Vogel, Werner):

Lake Louise Winter Institute 2005, February 25

All Coulomb corrections

pe, MeV/c

Depending on the experimental setup, more calculations are needed to establish the theoretical uncertainty to R-correlation.

pe, MeV/c

R(Coulomb)

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