The radiative neutron capture on 3 He ( 3 He+n→ 4 He+  ) in effective field theory Young-Ho Song Seoul National University in collaboration with T.-S. Park, K. Kubodera, D.-P. Min, M. Rho

1.Motivation and Reviews 2. Formalism 3. Numerical Results 4. Conclusion

Motivation The radiative neutron capture on 3 He ( 3 He+n→ 4 He+  ) at threshold is closely related to the hep process( 3 He+p→ 4 He+e+ e ) They shows much similarity and both can be calculated in a same low energy effective theory scheme.

3 He+p→ 4 He+e+ e No experiment Large theoretical uncertainty

hep theory Old schematic wave function (‘52-’91) S-factor :4~630 ( MeV-b) Modern wave function (‘91-’01) S-factor( MeV-b) ’91 (Carlson et al.)1.3 ’92 (Schiavilla et al.)  S 0 = 2.3(“standard value”) ’01 (Marcucci et al) 9.64 Effective field theory with hybrid method(MEEFT) TSP et al., PRC67(’03)055206, nucl-th/ S (theory)=(8.6  1.3) MeV-b How can we test our prediction ? Can we test our hep prediction by applying the same method to the hen process ?

3 He+n→ 4 He+  Experiment:  (exp)= (55 ±3)  b, (54 ± 6)  b Theory : –Old schematic wave function  b : (’81) Towner & Kanna  b : (’91) Wervelman –Modern wave function (112, 140)  b : (’90: VMC) Carlson et al ( 86, 112)  b : (’92: VMC) Schiavilla et al a( 3 He - n)= (3.50, 3.25) fm Accurate recent exp: a( 3 He - n)= 3.278(53) fm

hen and hep Hen matrix elements by Schiavilla et al (’92: VMC) 100 M.E. (fm 3/2 ) IA MI Leading One-body contribution is highly suppressed. Pseudo-orthogonality between initial and final wave function → meson exchange current plays essential role. 2.Cancellation between one-body contribution and meson exchange current contribution → amplify uncertainty → We need accurate 4-body wave function and exchange current operator

Formalism Heavy baryon chiral perturbation theory –Gives a systematic way to obtain current operators up to N 3 LO –Gives a systematic treatment of the short range physics –However, it is yet hard to calculate 4-body system Standard nuclear physics approach –Gives a very accurate phenomenological potential –Reliable 4-body wave function can be obtained –However, there is no systematic way of treating short range physics. There is no way of systematic error estimation

EFT Operator EFT W.F. Hybrid Method SNPA W.F. Short range physics  isoscalar and isovector M1 in n + p  D +    -d capture rate   -d scattering  Etc. ≈ C 0   (r)

hybrid method + renormalization procedure for the short ranged contributions By using the hybrid method, obtain the wave function and the matrix elements. And then, fix the value of C 0 at given cutoff so as to reproduce other known experimental data. The value of C 0 is model-dependent, which cancels out the model-dependence of  f  r  i  so as to have model- independent  f | O short |  i , which is the renormalization condition. Once we fix the value of C 0, we can predict other processes which depends on the C 0. MEEFT strategy for M=  f  i 

MEEFT Strategy for M(Hen)=  f  i    VMC wave functions with Av14 + Urbana VIII   Up to N 3 LO in heavy-baryon chiral-perturbation theory (HBChPT) Weinberg’s power counting rule for irreducible diagrams.

the two-body currents in momentum space are valid only up to a certain cutoff  This implies that when we go to coordinate space, the currents should be appropriately regulated. This is the key point in our approach. Cutoff defines the energy momentum scale of EFT that divides low energy degrees of freedom and high energy d.o.f.

Up to N 3 LO, no three-body (and higher-body) current operators appear. To control the short-range physics consistently,  we apply the same (Gaussian) regulator  for all the A=2,3 and 4 systems, with  eV Only two constants to be fixed

How to fix g 4s and g 4v ? They are not fixed by the symmetry. In principle, they can be fixed by solving QCD at low-energy, but... Magnetic moments of 3 H and 3 He also depend on g 4s and g 4v. For each , we fix the values of g 4s and g 4v by imposing the condition that the experimental values of  ( 3 H) and  ( 3 He) should be reproduced.

 (MeV) g 4s g 4v Numerical Result

Results( M 2B /M 1B ) of the hen process

Summary  (theory)= (49 ±7)  -b, which is consistent with the exp., (55 ±3)  -b, (54 ± 6)  -b. MEEFT works well for the hen process, providing the 1 st satisfactory theoretical prediction for the hen cross section. Possible extensions (future works) –Different wavefunctions Wave functions with V low-k –Going to higher order –Investigating other few-body (electro-weak) processes