Light nuclei and electroweak probes

Tae-Sun ParkSungkyunkwan University (SKKU)

in collaboration withYoung-Ho Song & Rimantas Lazauskas

FB19@Bonn, Germany, Aug.31~Sep.05, 2009

Contents• Motivations (with brief intro. to HBChPT)

• Part I: Magnetic dipole (M1) currents up to N3LO and n+d →3He+γ

• Part II: Gamow-Teller(GT) currents up to N3LO and applica-tions

• Part III: M1 currents and n+3He → 4He+γ

Motivation: M1 & GT are …• important in astro-nuclear physics

– M1: n+p→d+γ, n+d →3He+γ, n+3He → 4He+γ (hen), …– GT: p+p→d+e++ν (pp), p+3He→4He+e++ν (hep), …

• providing dynamical information not embodied in Hamilton-ian– 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 )

Heavy-baryon Chiral Perturbation Theory

1. Degrees of freedom: pions & nucleons up to L

(, , , ) + high energy part => appears as local operators of p’s and N’s.

• 2. Expansion parameter = Q/Lc

Q : typical momentum scale and/or mp,

Lc : mN ~ 4p fp ~ 1 GeV .

L = L0 + L1 + L2 + with L ~ (Q/Lc) 3. Weinberg’s power counting rule for irreducible diagrams.

Current operators in HBChPT J ~ (Q/Lc)

with = 2 (nB-1) + 2 L + i i, i = di + ni/2 + ei - 1

※Additional suppression: ~ (1, Q), 5 ~ (Q, 1)

※ GT up to N3LO : 1B + 2B(1π-E )+ 2B(CTs)※ M1 up to N3LO: 1B + 2B(1π-E )+ 2B(CTs) +2B(2π-E)

V0, Ai(GT) Vi (M1), A0

1B 1=LO Q = LO

2B: 1π-E (i = 0) Q ∙ Q = NLO

2B: 1π-E (i = 1) Q^3=N3LO

2B:CTs with LECs Q^3=N3LO Q ∙ Q^3=N3LO

2B: loop (2π-E, 1πC) Q^4 Q ∙ Q^3=N3LO

3B-tree Q^4 Q ∙ Q^4

Comments on matrix elements <f |J |i>

• Wave functions– Available potentials: phenomenological, EFT-based– Short-range behavior : highly model-dependent !

• Electro-weak current operators J

– At N3LO, 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

• <f |JCT |i> = C0(L) <f |( )ij L(rij) |i> – For a given w.f. and L, determine C0(L) so as to reproduce the experi-

mental values of a selected set of observables that are sensitive on C0

• 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 …

• C0(L) : L-dependent (to compensate the difference in short-range)• Net amplitude <f |(J1B +J1π +… +JCT |i> : L-independent • will be proven numerically by checking L-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…)

Part I:

M1 properties of

A=2 and A=3 systems (n+d →3He+γ)

M1 currents up to N3LO

J= J1B + J1p + (J1pC + J2p + JCT) = LO + NLO + N3LO

• g4S and g4V appear in – (2H), (3H), (3He), …– Cross sections/spin observables of npd, ndt, …

• We can fix g4S and g4V by – A=2 sector: (2H) and (npd)– A=3 sector: (3H) and (3He)– …

Details of (n+d →3He+γ) calculation

• nd and Rc (photon polarization of nd capture) calculated• g4s and g4v : fixed by (3H) and (3He)• Various potential models are considered

– Av18 (+ U9)– EFT (N3LO) NN potentials of Idaho group– EFT (N3LO) NN potentials of Bonn-Bochum group with diff. L

• {L, L’}= {450,500}, {450, 700}, {600, 700} MeV = (E1, E4, E5)– INOY (can describe BE’s of 3H and 3He w/o TNIs )

• Wave functions: Faddeev equation

nd and Rc

L-depdendence (model: INOY)inputs: (3H) and (3He)

Λ[MeV]

μ(d) σ(np)[mb]

σ (nd)[mb]

- Rc

500 0.8584 330.9 0.501 0.466

600 0.8584 330.7 0.497 0.465

700 0.8585 330.5 0.495 0.465

800 0.8583 330.5 0.495 0.465

900 0.8583 330.4 0.496 0.465

Exp. 0.8574 332.6(6) 0.508(15) 0.420(30)

※Pionless EFT: nd= 0.503(3) mb, -Rc= 0.412(3)Sadeghi, Bayegan, Griesshammer, nucl-th/0610029

Sadeghi, PRC75(‘07) 044002

Model-dependence

Model μ_d σ_np[mb]

σ_nd[mb]

-Rc2a_nd[fm]

BE(H3)[MeV]

BE(He3)[MeV]

Av18 0.858 331.9 0.680(3) 0.435 1.266 7.623 6.925

Av18+U9 0.860 330.6 0.478(3) 0.458 0.598 8.483 7.753

INOY 0.859 330.6 0.498(3) 0.465 0.551 8.483 7.720

I-N3LO 0.857 330.4 0.626(2) 0.441 1.101 7.852 7.159

E1-N3LO 0.858 328.7 0.688(4) 0.438 1.263 7.636 6.904

E4-N3LO 0.859 331.0 0.609(4) 0.448 1.024 7.930 7.210

E5-N3LO 0.855 330.9 0.879(8) 0.411 1.781 7.079 6.403

Exp. 0.8574 332.6(7) 0.508(15) 0.420(30) 0.65(4) 8.482 7.718

M1(J=1/2) ≈ 2(B3), M1(J=3/2) ≈ 4(B3)

A trial for improvement

• Replace B3 by B3exp

mn=n(B3) + mn

=> mn’=n(B3

exp) + mn = n(B3

exp) + mn - n(B3)

• mn’

– m2’ = -21.89±0.24, m4

’ = 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

Another way to correct the long-range part

• Adjust parameters of 3N potential to have correct scatter-ing 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

• A2p = -29.3 KeV -36.955 KeV

– BE(3H) and 2and are OK, but BE(3He) is over-bound by 40 KeV.

– Introduce charge-dependent term in U9, and all the B.E.s and scattering lengths can be reproduced• A2p -36.575 KeV for 3He

Model-dependence (improved)Model μ_d σ_np

[mb]σ_nd[mb]

-Rc2a_nd[fm]

BE(H3)[MeV]

BE(He3)[MeV]

<m_n’> 0.491(8) 0.463(3)Av18+U9* 0.862(1) 330.9(3) 0.477(3) 0.457(2) 0.623 8.482 7.718

I-N3LO+U9* 0.859(1) 330.2(4) 0.479(4) 0.468(2) 0.634 8.482 7.737Exp. 0.8574 332.6(7) 0.508(15) 0.420(30) 0.65(4) 8.482 7.718

• L-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

ji

FijijB

i N

iiiiiiiAB

AAA

mpppgA

LON

LONLO234OPE

2

22

2

1

OPE 4F

p

Part II:

Gamow-Teller channel (pp and hep)

There is no soft-OPE (which is NLO) contributions

))((ˆ)(ˆ2 2124

jijijjiiN

AFij dd

fmgA

p

Thanks to Pauli principle and the fact that the contact terms are effec-tive only for L=0 states, only one combination is relevant:

61ˆ

32ˆ

31ˆ2ˆˆ

4321 ccddd R

The same combination enters into pp, hep, tritium-b decay (TBD), -d capture, d scattering, … .

We use the experimental value of TBD to fix ,then all the others can be predicted !

Rd̂

Results(pp)

L (MeV) 1B 2B 500 1.00 4.85 0.076 0.035

0.041 600 1.78 4.85 0.097 0.031

0.042 800 3.90 4.85 0.129 0.022

0.042

Rd̂

Rd̂

Rd̂

Rd̂

with -term, L-dependence has gone !!!

the astro S-factor (at threshold) Spp= 3.94 (1 0.15 % 0.10 %) 10-25 MeV-barn

Rd̂

Results(pp)

0

0.020.04

0.060.08

0.1

0.120.14

2B (w/o CT) 2B

500600800

Results(hep)Reduced matrix element with respect to L (MeV)

- 1

-0.50

0.51

1.5

22.5

1B 2B(w/o CT) 2B 1B+2B

500600800

Part III:

The hen process (n+3He → 4He+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

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, < f |J1B|i> ≈0 .

– |f=4He> : 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– < f |J|i> 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.

Details of (n+3He → 4He+γ) calculation

• Various potential models are considered– Av18– I-N3LO: NN potentials of Idaho group– INOY (can describe BE’s of 3H and 3He w/o TNIs )– Av18 + UIX– I-N3LO+UIX* (UIX parameter is adjusted to have B(3H) exactly)

• Wave functions: Faddeev-Yakubovski equations in coordinate space

• M1 currents up to N3LO• g4s and g4v : fixed by (3H) and (3He)

• hen cross section: model-dependent (but correlated with ERPs)

Model n3He [b] B (4He) [MeV]

B (3H) [MeV]

B (3He) [MeV]

3an3He [fm] rHe4 [fm] PD(4He) [%]

Av18 80(12.2) 24.23 7.623 6.925 3.43-0.0082i 1.516 13.8

I-N3LO 57.3(7.9) 25.36 7.852 7.159 3.56-0.007i 1.520 9.30

I-N3LO+UIX* 44.4(6.7) 28.12 8.482 7.737 3.44-0.0055i 1.475 10.9

Av18+UIX 49.4(8.5) 28.47 8.483 7.753 3.23-0.0054i 1.431 16.0

INOY 34.4(4.5) 29.08 8.482 7.720 3.26-0.0058i 1.377 5.95

Exp. 55(3); 54(6)

28.3 8.482 7.718 3.28(5)-0.001(2)i

1.475(6)

• How hen cross section is correlated to ERPs ? n3He ∝ ζ≡ [q (anHe3/rHe4)2]-2.75 or n3He ∝ q-5 PD

2/3

Model n3He

[b]n3He [b]ⅹζexp/ζ

Av18 80(12.2) 54.8(8.4)

I-N3LO 57.3(7.9) 54.5(7.5)

I-N3LO+UIX* 44.4(6.7) 56.4(8.5)

Av18+UIX 49.4(8.5) 54.8(9.4)

INOY 34.4(4.5) 53.9(7.5)

Exp. 55(3); 54(6)

55(3); 54(6)

n3H

e(b)

• Cutoff-dependence: (Re Mhen) vs Λ[MeV]

500 600 700 800 900

-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20

1B 1B+2B (w/o CT)total

Convergence ?

Effective ME

• Observation: If we omit something (say, <J2p>), values of g4s and g4v should also be changed to have the correct (3H) and (3He) without <J2p>.

• “effective <J2p>” ≡ changes in the net ME if we omit <J2p>

naïve <J> effective <J>

1B 0.1518 0.1518

2B: 1π (NLO) -0.1657 0.0749

2B: 1πC(N3LO) -0.1465 -0.0093

2B: 2π (N3LO) -0.0445 -0.0010

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 N3LO. N4LO calculations will be interesting to

check the convergence of the calculations.