1. Light nuclei and electroweak probes Tae-Sun Park Sungkyunkwan University (SKKU) in collaboration with Young-Ho Song & Rimantas Lazauskas FB19@Bonn, Germany, Aug.31~Sep.05, 2009

2. Contents Motivations (with brief intro. to HBChPT) Part I: Magnetic dipole (M1) currents up to N 3 LO and n+d → 3 He+γ Part II: Gamow-Teller(GT) currents up to N 3 LO and applications Part III: M1 currents and n+ 3 He → 4 He+γ

3. Motivation: M1 & GT are … important in astro-nuclear physics –M1: n+p→d+γ, n+d → 3 He+γ, n+ 3 He → 4 He+γ (hen), … –GT: p+p→d+e + +ν (pp), p+ 3 He→ 4 He+e + +ν (hep), … providing dynamical information not embodied in Hamiltonian –Electric channel (E1, E2) : Siegert’s theorem: (∂∙J = 0) ⇔ Hamiltonian –For M1 : ∂∙J = 0 by construction, gauge-inv says little here –For GT : ∂∙J ≠ 0. a good playing ground of EFTs (ideal to explore short-range physics )

4. Heavy-baryon Chiral Perturbation Theory 1. Degrees of freedom: pions & nucleons up to  ( , , ,  ) + high energy part => appears as local operators of  ’s and N’s. 2. Expansion parameter = Q/   Q : typical momentum scale and/or m ,   : m N ~  f  ~ 1 GeV. L = L 0 + L 1 + L 2 +  with L ~ (Q/   ) 3. Weinberg’s power counting rule for irreducible diagrams.

5. Current operators in HBChPT J  ~ (Q/   ) with = 2 (n B -1 ) + 2 L +  i i, i = d i + n i /2 + e i - 1 ※ Additional suppression:   ~ (1, Q),    5 ~ (Q, 1) ※ GT up to N 3 LO : 1B + 2B(1π-E )+ 2B(CTs) ※ M1 up to N 3 LO: 1B + 2B(1π-E )+ 2B(CTs) +2B(2π-E) V 0, A i (GT)V i (M1 ), A 0 1B1=LOQ = LO 2B: 1π-E ( i = 0) Q ∙ Q = NLO 2B: 1π-E ( i = 1) Q^3=N 3 LO 2B:CTs with LECsQ^3=N 3 LOQ ∙ Q^3=N 3 LO 2B: loop (2π-E, 1πC)Q^4Q ∙ Q^3=N 3 LO 3B-treeQ^4Q ∙ Q^4

6. Comments on matrix elements Wave functions –Available potentials: phenomenological, EFT-based –Short-range behavior : highly model-dependent ! Electro-weak current operators J –At N 3 LO, there appear NN contact-terms (CTs) in GT/M1 –CTs represent the contributions of short-ranged and high-energy that are integrated-out –How to fix the coefficients of them (LECs) ? Solve QCD => Oh no ! Determine from other experiments => usual practice of EFTs, i.e., renormalization procedure

7. = C 0 (  ) –For a given w.f. and , determine C 0 (  ) so as to reproduce the experimental values of a selected set of observables that are sensitive on C 0 Model-dependence in short-range region: –In EFTs, differences in short-range physics is into the coefficients (LECs) of the local operators. –We expect that … C 0 (  ) :  -dependent (to compensate the difference in short-range) Net amplitude :  -independent will be proven numerically by checking  -dependence Model-dependence in long-range region: –Long-range part of ME: governed by the effective-range parameters (ERPs) such as binding energy, scattering length, effective range etc –In two-nucleon sector, practically no problem ∵ most potentials are very good in 2N sector –In A ≥ 3, things are not quite trivial (we will see soon…)

8. Part I: M1 properties of A=2 and A=3 systems (n+d → 3 He+γ)

9. M1 currents up to N 3 LO J= J 1B + J 1  + (J 1  C + J 2  + J CT ) = LO + NLO + N 3 LO

10. g 4S and g 4V appear in –  ( 2 H),  ( 3 H),  ( 3 He), … –Cross sections/spin observables of np  d , nd  t , … We can fix g 4S and g 4V by –A=2 sector:  ( 2 H) and  (np  d  ) –A=3 sector:  ( 3 H) and  ( 3 He) –…

11. Details of (n+d → 3 He+γ) calculation  nd and R c (photon polarization of nd capture) calculated g 4s and g 4v : fixed by  ( 3 H) and  ( 3 He) Various potential models are considered –Av18 (+ U9) –EFT (N 3 LO) NN potentials of Idaho group –EFT (N 3 LO) NN potentials of Bonn-Bochum group with diff.  { ,  ’}= {450,500}, {450, 700}, {600, 700} MeV = (E1, E4, E5) –INOY (can describe BE’s of 3 H and 3 He w/o TNIs ) Wave functions: Faddeev equation

12.  nd and R c

13.  -depdendence (model: INOY) inputs:  ( 3 H) and  ( 3 He) Λ [MeV] μ (d) σ (np) [mb] σ (nd) [mb] - Rc 5000.8584330.90.5010.466 6000.8584330.70.4970.465 7000.8585330.50.4950.465 8000.8583330.50.4950.465 9000.8583330.40.4960.465 Exp.0.8574332.6(6)0.508(15)0.420(30) ※ Pionless EFT:  nd = 0.503(3) mb, -R c = 0.412(3) Sadeghi, Bayegan, Griesshammer, nucl-th/0610029 Sadeghi, PRC75(‘07) 044002

14. Model-dependence Model μ _d σ _np [mb] σ _nd [mb] -R c 2 a_nd [fm] BE(H3) [MeV] BE(He3) [MeV] Av180.858331.90.680(3)0.4351.2667.6236.925 Av18+U90.860330.60.478(3)0.4580.5988.4837.753 INOY0.859330.60.498(3)0.4650.5518.4837.720 I-N3LO0.857330.40.626(2)0.4411.1017.8527.159 E1-N3LO0.858328.70.688(4)0.4381.2637.6366.904 E4-N3LO0.859331.00.609(4)0.4481.0247.9307.210 E5-N3LO0.855330.90.879(8)0.4111.7817.0796.403 Exp.0.8574332.6(7)0.508(15)0.420(30)0.65(4)8.4827.718

15. M1(J=1/2) ≈  2 (B 3 ), M1(J=3/2) ≈  4 (B 3 )

16. A trial for improvement Replace B 3 by B 3 exp m n =  n (B 3 ) +  m n => m n ’ =  n (B 3 exp ) +  m n =  n (B 3 exp ) + m n -  n (B 3 ) m n ’ – m 2 ’ = -21.89±0.24, m 4 ’ = 12.24±0.05 –  nd ’ =0.491±0.008 mb, Rc ’ =0.463±0.003 –Cf) data: 0.508±0.015 mb & 0.420±0.030

17. Another way to correct the long-range part Adjust parameters of 3N potential to have correct scattering lengths and B.E.s –Range parameter (a cutoff parameter) and the coefficients of U9 potential is adjusted C = 2.1 fm -2  3.1 fm -2 A 2  = -29.3 KeV  -36.955 KeV –BE( 3 H) and 2 a nd are OK, but BE( 3 He) is over-bound by 40 KeV. –Introduce charge-dependent term in U9, and all the B.E.s and scattering lengths can be reproduced A 2   -36.575 KeV for 3 He

18. Model-dependence (improved) Model μ _d σ _np [mb] σ _nd [mb] -R c 2 a_nd [fm] BE(H3) [MeV] BE(He3) [MeV] 0.491(8)0.463(3) Av18+U9*0.862(1)330.9(3)0.477(3)0.457(2)0.6238.4827.718 I-N3LO+U9*0.859(1)330.2(4)0.479(4)0.468(2)0.6348.4827.737 Exp.0.8574332.6(7)0.508(15)0.420(30)0.65(4)8.4827.718  -independence is found to be very small, short-range physics is under control. Model-dependence is tricky –Naively large model-dependence –But strongly correlated to the triton binding energy –Model-independent results could be obtained either by making use of the correlation curve or by adjusting the parameters of 3N potential to have correct ERPs

19. OPE 4F  Part II: Gamow-Teller channel (pp and hep) There is no soft-OPE (which is NLO) contributions

20.  hanks to Pauli principle and the fact that the contact terms are effective only for L=0 states, only one combination is relevant:  he same combination enters into pp, hep, tritium-  decay (TBD),  -d capture,  d scattering, …. We use the experimental value of TBD to fix, then all the others can be predicted !

21. Results(pp)  (MeV)  500 1.004.85  600 1.784.85  800 3.904.85  with -term,  -dependence has gone !!! the astro S-factor (at threshold) S pp = 3.94 (1  0.15 %  0.10 %) 10 -25 MeV-barn 

22. Results(pp)

23. Results(hep) Reduced matrix element with respect to  (MeV)

24. Part III: The hen process ( n+ 3 He → 4 He+γ ) σ (exp)= (55 ±3) μb, (54 ± 6) μb 2-14 μb : (1981) Towner & Kanna 50 μb : (1991) Wervelman (112, 140) μb : (1990) Carlson et al ( 86, 112) μb : (1992) Schiavilla et al (49.4±8.5, 44.4±6.7) μb : this work

25. hen(M1) & hep(GT) are hard to evaluate Leading 1B contribution (IA) is highly suppressed due to pseudo-orthogonality, : –The major part of |  f > and |  i > belong to different symmetry group, cannot be connected by r- independent operators, ≈0. –|  f = 4 He> : most symmetric (S4) state, all N’s are in 1S state –|  i > : next-to-most symmetric state (S31) ; 1-nucleon should be in higher state due to Pauli principle – is sensitive to MECs and minor components of the wfs. Requires realistic A=4 wfs for both bound and scattering states !! Accurate evaluation of MECs needed – Short-range contributions plays crucial role !! Coincidental cancellation between 1B and MEC occurs.

26. Details of ( n+ 3 He → 4 He+γ ) calculation Various potential models are considered –Av18 –I-N3LO: NN potentials of Idaho group –INOY (can describe BE’s of 3 H and 3 He w/o TNIs ) –Av18 + UIX –I-N3LO+UIX* (UIX parameter is adjusted to have B( 3 H) exactly) Wave functions: Faddeev-Yakubovski equations in coordinate space M1 currents up to N 3 LO g 4s and g 4v : fixed by  ( 3 H) and  ( 3 He)

27. hen cross section: model-dependent (but correlated with ERPs) Model  n3He [  b] B ( 4 He) [MeV] B ( 3 H) [MeV] B ( 3 He) [MeV] 3 a n3He [fm]r He4 [fm]P D ( 4 He) [%] Av1880(12.2)24.237.6236.9253.43-0.0082i1.51613.8 I-N3LO57.3(7.9)25.367.8527.1593.56-0.007i1.5209.30 I-N3LO+UIX*44.4(6.7)28.128.4827.7373.44-0.0055i1.47510.9 Av18+UIX49.4(8.5)28.478.4837.7533.23-0.0054i1.43116.0 INOY34.4(4.5)29.088.4827.7203.26-0.0058i1.3775.95 Exp.55(3); 54(6) 28.38.4827.7183.28(5)- 0.001(2)i 1.475(6)

28. How hen cross section is correlated to ERPs ?  n3He ∝ ζ≡ [q (a nHe3 /r He4 ) 2 ] -2.75 or  n3He ∝ q -5 P D 2/3 Model  n3He [  b] ⅹ ζ exp /ζ Av1880(12.2)54.8(8.4) I-N3LO57.3(7.9)54.5(7.5) I-N3LO+UIX*44.4(6.7)56.4(8.5) Av18+UIX49.4(8.5)54.8(9.4) INOY34.4(4.5)53.9(7.5) Exp.55(3); 54(6)  n3He (  b)

29. Cutoff-dependence: (Re M hen ) vs Λ[MeV]

30. Convergence ?

31. Effective ME Observation: If we omit something (say, ), values of g 4s and g 4v should also be changed to have the correct  ( 3 H) and  ( 3 He) without. “effective ” ≡ changes in the net ME if we omit naïve effective 1B 0.1518 2B: 1π (NLO)-0.1657 0.0749 2B: 1πC(N 3 LO)-0.1465-0.0093 2B: 2π (N 3 LO)-0.0445-0.0010

32. Discussions For all the cases, short-range physics is under control. Long-range part is tricky : To have correct results, one needs –either the potential that produces all the relevant effective range parameters (ERPs) correctly –or the correlation of the results w.r.t. ERPs –Once the correct ERPs have been used, all the results are in good agreement with the data So far up-to N 3 LO. N 4 LO calculations will be interesting to check the convergence of the calculations.