weak reactions and chiral eft in few body systems

Weak Reactions and Chiral EFT in Few Body Systems

M. Viviani

INFN, Sezione di Pisa &Department of Physics, University of Pisa

Pisa (Italy)

Inelastic Reactions in Light Nuclei

October 6-10, 2013IIAS, Jerusalem (Israel)

M. Viviani (INFN-Pisa) Weak Reactions and Chiral EFT Jerusalem, October 10, 2013 1 / 40

Outline

1 Introduction

2 Wave functions

3 Muon Capture

4 pp fusion

5 PV processesp − p longitudinal symmetry~n − 3He → p − 3H longitudinal asymmetry

6 Conclusion & Outlook


Weak reactions of interest (A ≤ 4)

Astrophysical interest:

p + p → 2H+ e+ + νe

p + 3He → 4He+ e+ + νe

neutrino scattering

νx + d → νx + p + n

νe + d → e− + p + p

νx +4He → νx + X

Beta decay & Muon catpures:3H → 3He+ e− + νe

µ− + d → n + n + νµ

µ− + 3He → 3H+ νµ (70%)

µ− + 3He → n + d + νµ (20%)

µ− + 3He → n + n + p + νµ (10%)

Tfi = 〈Ψf |J lµ(Jh)µ†|Ψi 〉Calculation of |Ψi,f 〉: H|Ψi,f 〉 = E |Ψi,f 〉Construction of the operator

Jhµ =∑

i=1,A J(1)µ (i)+

∑

i<j J(2)µ (i , j)+ · · ·

Consistency between H and Jµ?

Accuracy? How to improve the

predictions/estimates?

→ Effective Field Theory (EFT) + χPT


χEFT

Advantagesmore contact with QCD

systematic and controlled expansion of nuclear potential/transition operators

[Weinberg, 1990], [Ordonez & Van Kolck, (1992)], [Meissner (1992)], [Park, Rho &Kubodera (1995)], [Bernard et al., 1995], [Epelbaum et al., 2003], . . .

Used to study a large variety of nuclear processes

Degrees of freedom at energy Q > Λ integrated out

L useful for processes of energy Q ≪ Λ

Λ ≈ 500 MeV – 700 MeV (problem with the ∆)

→ organize the expansion in powers of Q/Λ (possible since chiral symmetry imposes

derivative couplings)

“pionfull” χEFT: degrees of freedom: pions and nucleons (Λ ∼ 500 MeV)

“pionless” χEFT: degrees of freedom: nucleons (Λ ≤∼ mπ)


Low energy constants (LEC’s)

Include all possible Lagrangian terms contributing at a given Qν

Each term is multiplied by a “coupling constant” → the LEC

They encode our ignorance of what happens for Q > Λ

There are (for a given ν) a finite number of LEC’s

Calculated using QCD work in progress

Fitted to some experimental data or extracted from some model

The LEC’s depend on Λ – the observables should not depend on it

→ the same LEC’s enter the nuclear potentials & currents

U = e i~τ ·~π/fπ u =√U uµ = i [u†(∂µ − irµ)u − u(∂µ − ilµ)u

†]

rµ, lµ external right- and left-handed currents L = Lππ + LπN + LNN + LNNπ + · · ·

LπN = N(

iγµDµ −mN +gA

2γµγ5uµ

)

N

︸︷︷︸

O(Q)

+∑

i=1,7

ciNOiN

︸︷︷︸

O(Q2)

+ · · · [Fettes et al ., (1998)]

LNNπ = −d1Nγµγ5uµNNN +

id2

4Nσµν [uµ, τa]NNγνγ

5τaN + · · · [Van Kolck, (1994)]


Weak interaction: coupling with the W± at low energy → current-current interaction

we can assume rµ = 0 and lµ = −4GV√2J lµτ

+

J lµ =∑

l=e,µ,τ

ψlγµ(1− γ5)ψνl

take the terms linear in J lµ and (J lµ)† & expand in powers of the pion fields

LI = −GV√2J lµ

[

N(γµ − gAγµγ5)τ+N +

√2fπ∂µπ

+ + · · ·]

+ h.c.

Unified description of different processes

Example: LI = · · ·+ c3Ψuµ(uµ)†Ψ+ · · · where uµ ≈ − 1fπ∂µ~π · ~τ + 4GV√

2τ+Jℓµ


Derivation of the potential and current

Define operators V and K = J lµ(Jh)µ† acting only on the nucleonic d.o.f

NN Potential1 Derive T -matrix of the process using

standard TOPT

2 NR expansion the obtained amplitudes

3 T = T (0) + T (1) + T (2) + · · ·T (n) ∼ Qn

4 Define V = V (0) + V (1) + V (2) + · · ·such that

TV = V + VG0TV G0 =1

Ei − H0 + iǫ

5 TV = T up to a given order n0(V (n)G0V

(m) ∼ Qn+m+1 )

V (0) = T (0)

V (1) = T (1) − V (0)G0V(0)

. . .

Current1 Derive T -matrix of the process using

standard TOPT

2 T = T (−3) + T (−2) + T (−1) + · · ·T (n) ∼ Qn (e.g. pp fusion)

3 DefineK = K (−3) + K (−2) + K (−1) + · · · suchthat T = 〈Ψf |K |Ψi 〉 up to a given order

4 Ψi , Ψf fully interacting nuclear statesΨi = φi + G0VΨi

5 φi,f asymptotic (free) states

〈Ψf |K |Ψi 〉 = 〈φf |(I+VG0+VG0VG0+. . . )K

×(I+G0V+G0VG0V+. . . )|φi 〉K (−3) = T (−3)

K (−2) = T (−2)−V (0)G0K(−3)−K (−3)G0V

(0)

. . .


Example: OPE

TOPE=V1V2

E1 − E ′1 − ωk

δk,p1−p′1+

V1V2

E2 − E ′2 − ωk

δk,p′1−p1

ωk =√

m2π + k2 Vi =

gA

2fπu(p′i , s

′i )γ

µγ5ikµτi,a√

2ωku(pi , si ) ≈

gA

2fπ

iσi · k√2ωk

on shell (in the C.M.) E ′i = Ei → we recover the expression of the OPEP

The third diagram is of order Q → cancelled by −V (0)G0V(0) → no V (1)

1

∆E − ωk≈ − 1

ωk

[

1 +∆E

ωk+

(∆E

ωk

)2

+ · · ·]

(∆E/ωk )2 is of order Q2 – ambiguity in the definition ov V (2)

V (2) = V(0)OPE

f (E ′1 − E1,E

′2 − E2)

ω2k

f (0, 0) = 0

V (2) enters in the expression of V (n>3) andK (n>0)

All the expression are equivalent – related bya unitary transformation


NN & 3N forces

NN force: N3LO “N3LO” model – [Entem & Machleidt, 2003]

3N force: N2LO “N2LO” model – [Navratil, 2007], [Marcucci et al., 2012]

In progress: 3N force at N3LO & N4LO [Krebs et al., 2012]


Weak transition operators

K =GF√2

[

Vµ − Aµ]

J lµ

Vµ = (ρV , JV )

Aµ = (ρA, JA)

ρV and JV : related via CVC to the EM current derived at N4LO [Park et al., 1996],[Koelling et al., 2009-2011], [Pastore, MV, et al., 2009,2011]

ρA and JA: N3LO [Park et al., 2003] (N4LO in progress)


In the following: ρV and JV via CVC from [Song, Park & Lazauskas (2009)] (N4LO)ρA and JA from [Park, MV, et al. (2003)] (N3LO)

LEC’s to be fixed: g4S and g4V in the vector current ⇒ fit to the A = 3 magnetic moments

+ 1 LEC dR entering the axial current [Gazit’s talk]

note [Koelling et al., 2009-2011], [Pastore, MV, et al., 2009,2011]:In the vector current there are 5 new LEC’s [see Epelbaum’s talk]


In the following: ρV and JV via CVC from [Song, Park & Lazauskas (2009)] (N4LO)ρA and JA from [Park, MV, et al. (2003)] (N3LO)

LEC’s to be fixed: g4S and g4V in the vector current ⇒ fit to the A = 3 magnetic moments

+ 1 LEC dR entering the axial current [Gazit’s talk]

note [Koelling et al., 2009-2011], [Pastore, MV, et al., 2009,2011]:In the vector current there are 5 new LEC’s [see Epelbaum’s talk]

D Ec c d

R

ijkV j (A)

dR =MN

ΛχgAcD +

1

3MN(c3 + 2c4) +

1

6

[Gardestig and Phillips, PRL 96, 232301 (2006)][Gazit et al., PRL 103, 102502 (2009)]

fit cD (dR) to GTexp and cE to B(A = 3) (using the N3LO/N2LO model)⇒ cD ; cEMAX and cD ; cEMIN

cD ; cE g4S g4VΛ=500 MeV −0.20;−0.208 0.207± 0.007 0.765± 0.004Λ=600 MeV −0.32;−0.857 0.146± 0.008 0.585± 0.004


Wave functions

Nuclear wave functions

Non-local potentials with a strong state dependence + very complicate 3N force

Methods for A ≥ 3: Faddeev-Yakubovsky Equations, GFMC, Variational methods(Gaussians, NCSM, HH)

HH method [Kievsky, MV, et al.,J. Phys. G 35, 063101 (2008)]

EIHH method [Barnea et al., (2000)], [Bacca et al., (2012)]

H =∑

i

p2i2M

+∑

i<j

V (i , j) +∑

i<j<k

W (i , j , k) + . . .

Search for accurate solution of HΨ = EΨ

Expansion of Ψ on the basis of Hyperspherical Harmonics functions

A high-precision variational approach to three- and four-nucleon bound and zero-energyscattering states, A. Kievsky, S. Rosati, M. Viviani, L.E. Marcucci, and L. Girlanda J.Phys. G, 35, 063101 (2008)


Comparison of 4N calculations [PRC 84, 054010 (2011)]

HH = expansion on a complete basis + Kohn variational principleFY & AGS = solution of the Faddeev-Yakubovsky or AGS equations

p − 3He elastic scattering

N3LO potential

AGS= Deltuva & FonsecaFY= Lazauskas & Carbonell

0 60 1200

100

200

300

400

500

dσ/d

Ω [m

b/sr

]

Famularo 1954Fisher 2006AGS

2.25 MeV

0 60 1200

0,2

0,4

Ay0

Fisher 2006George 2001

0 60 120θc.m. [deg]

0

0,1

0,2

A0y

Daniels 2010

0 60 120

Mcdonald 1964Fisher 2006

4.05 MeV

0 60 120

Fisher 2006

0 60 120θc.m. [deg]

Daniels 2010

0 60 120 180

Mcdonald 1964

5.54 MeV

0 60 120 180

Alley 1993

0 60 120 180θc.m. [deg]

Alley 1993Daniels 2010





N3LO potential


0 60 1200

100

200

300

400

500

dσ/d

Ω [m

b/sr

]

Famularo 1954Fisher 2006AGSHH

2.25 MeV

0 60 1200

0,2

0,4

Ay0


0 60 120θc.m. [deg]

0

0,1

0,2

A0y

Daniels 2010

0 60 120


4.05 MeV

0 60 120

Fisher 2006

0 60 120θc.m. [deg]

Daniels 2010

0 60 120 180

Mcdonald 1964

5.54 MeV

0 60 120 180

Alley 1993

0 60 120 180θc.m. [deg]






N3LO potential


0 60 1200

100

200

300

400

500

dσ/d

Ω [m

b/sr

]

Famularo 1954Fisher 2006AGSHHFY

2.25 MeV

0 60 1200

0,2

0,4

Ay0


0 60 120θc.m. [deg]

0

0,1

0,2

A0y

Daniels 2010

0 60 120


4.05 MeV

0 60 120

Fisher 2006

0 60 120θc.m. [deg]

Daniels 2010

0 60 120 180

Mcdonald 1964

5.54 MeV

0 60 120 180

Alley 1993

0 60 120 180θc.m. [deg]



Study of the 3N force

p − 3He elastic scattering Ep = 5.54 MeV

0

100

200

300

400

dσ/dΩ [mb/sr]

0

0.1

0.2

0.3

0.4

0.5 Ay

-0.1

0

0.1

0.2A

0y

0 30 60 90 120 150 180θ

c.m. [deg]

-0.1

0

0.1

Ayy

0 30 60 90 120 150 180θ

c.m. [deg]

-0.1

0

0.1

Axx

0 30 60 90 120 150 180θ

c.m. [deg]

-0.2

-0.1

0

0.1A

xz

cyan band= NN force only blue band= NN+3N forcesdata: [Fisher et al., PRC 74, 034001 (2006)] [Alley & Knutson, PRC 48, 1890 (1993)]

[Daniels et al., PRC 82, 034002 (2010)]


Muon captures in light nuclei (A ≤ 3)

µ− + d → n + n + νµ

Two hyperfine states: f = 1/2 and 3/2

Dominant capture from f = 1/2 → ΓD

1960 1965 1970 1975 1980 1985 1990year

250

300

350

400

450

500

550

ΓD [

s-1]

Wang et al. (1965)Bertin et al. (1973)Bardin et al. (1986)Cargnelli et al. (1986)

New measurement in progress: MuSun(PSI, Switzerland)

µ− + 3He → 3

H+ νµ

Hyperfine states: (f , fz ) = (1, ±1, 0)and (0, 0)

dΓ

d(cos θ)=1

2Γ0[1 + AvPv cos θ

+AtPt(3 cos2 θ − 1

2) + A∆P∆

]

Pv = P1,1 − P1,−1

Pt = P1,1 + P1,−1 − 2P1,0

P∆ = 1− 4P0,0

Γ0= total capture rate, Av,t,∆= angularcorrelation parameters

Ackerbauer et al., (1998):

Γ0=1496(4) s−1

Souder et al., (1998): Av=0.63 ± 0.09(stat.)+0.11

−0.14 (syst.)


Results: ΓD(µ− + d)

NN potential: N3LO [Entem & Machleidt, (2003), (2012)]

χEFT 1S03P0

3P13P2

1D23F2 Total

IA – Λ = 500 MeV 238.8 21.1 44.0 72.4 4.4 0.9 381.7IA – Λ = 600 MeV 238.7 20.9 43.8 72.0 4.4 0.9 380.8

FULL – Λ = 500 MeV 254(1) 20.5 46.8 72.1 4.4 0.9 399(1)FULL – Λ = 600 MeV 255(1) 20.3 46.6 71.6 4.4 0.9 399(1)

ΓD =399(3) s−1 (conservative assumption)

Theoretical “error”: from the uncertainties in cD , cE , g4S , g4V

Calculation performed assuming GPS = GχPTPS = 7.99± 0.20

[Marcucci et al., PRC 83, 014002 (2011), PRL 108, 052502 (2012)]


Comparison with data and previous calculations

1960 1965 1970 1975 1980 1985 1990year

350

360

370

380

390

400

410

420

430

440

450

ΓD [s-1

]

Ricci et al. (2010) [SNPA - NJI]Marcucci et al. (2011) [SNPA+χEFT*]Wang et al. (1965)Bertin et al. (1973)Bardin et al. (1986)Cargnelli et al. (1986)Marcucci et al. (2012) [χEFT]Adam et al. (2012) [χEFT]Ando et al. (2002) [χEFT* - AV18]


Results: Γ0(µ− + 3

He)

χEFT(N3LO/N2LO) Γ0IA – Λ = 500 MeV 1362IA – Λ = 600 MeV 1360

FULL – Λ = 500 MeV 1488(9)FULL – Λ = 600 MeV 1499(9)

[Marcucci et al., PRL 108, 052502 (2012)]

Γ0 =1494(13) s−1

vs. Γ0(exp)=1496(4) s−1

[Gazit, PLB 666, 472 (2008)] χEFT* – AV18/UIX → 1499(16) s−1

If GPS is left free ⇒ GPS =8.2 ±0.7 vs. GχPTPS =7.99±0.20

MUCAP experiment (µ− − p capture ) GχexptPS =8.06±0.48±0.28

[arXiv:1210.6545]


pp fusion

σ(E) =1

(2π)3G2V

vm5

e f (E)∑

M

|〈d ,M|A−|pp〉|2 S(E) = S(0) + S ′(0)E +1

2S ′′(0)E2 + . . .

Goal: < 1% accuracy

Dominant contribution from the 1S0wave

P-wave contribution: ∼ 1%

Two-body contribution: ∼ 1%

pp wave functionEM interaction:VC1 + VC2 + VDF + VVP + · · ·VVP ∼ exp(−2me r): sizeable effect atlow energies

Necessity to solve the Schroedingerequation up to r > 1, 000 fm

1% effect

SNPA: [Schiavilla et al., 1998], “Hybrid” χEFT: [Park et al., 2003]S(0) = 3.94(1± 0.0015± 0.0010± ǫ) (only 1S0 wave)

errors from uncertainties in gA, fit of the tritium β-decay, etc; ǫ “systematic error” (?)

See also the review paper: [E. G. Adelberger et al., Rev. Mod. Phys. 83, 195 (2011)[arXiv:1004.2318] ]


Effect of MEC at N3LO

c3,4 = Mc3,4

d1,2 =Mf 2πgA

d1,2

j(2)dR

=gA

Mf 2πdR(~τ1 × ~τ2)−(σ1 × σ2)

j(2)c3,a =

2

3

gAm2π

Mf 2πc3

σ1τ1− + σ2τ2−ω2

j(2)c4,a = −2

3

gAm2π

Mf 2π

(

c4+1

4

)(~τ1 × ~τ2)−(σ1 × σ2)

ω2k

j(2)c3,b

= −2gA

Mf 2πc3

(

τ1−(σ1 · k) k−1

3k2

σ1

ω2

)

j(2)c4,b

=gA

Mf 2π

(

c4+1

4

)(σ1 · k) k× σ2−1

3k2

σ1 × σ2

ω2

×(~τ1 × ~τ2)−

j(2)OPE = − gA

2Mf 2π(~τ1 × ~τ2)−

ik · (σ1 − σ2) Q

ω2

Astrophysical factor at Ec.m. = 2.5 keV(10−25MeV · b)

dR = d1 + 2d2 +1

3c3 +

2

3c4 +

1

6

N3LO5001b 4.0357dR 3.8860c3, a 3.8558c4, a 3.9633c3, b 4.0061c4, b 4.0825OPE 4.0876


Results with the EFT

Effect of the EM interactions VC2 + VDF + VVP + · · ·1S0 contribution only

S(0)× 1023 [MeV fm2] S ′(0)/S(0) [MeV−1] S ′′(0)/S(0) [MeV−2]N3LO+VC1 4.03 11.53 226M3LO+VEM 4.00 11.42 239

S(0) [×10−25 MeV b]

1S03P0

3P13P2

IA Λ=500 MeV 3.961(2)FULL Λ=500 MeV 4.008(5) 4.011(5) 4.020(5) 4.030(5)

Λ=600 MeV 4.008(5) 4.010(5) 4.019(5) 4.029(5)

Summary: S(0) = (4.030± 0.006)× 10−25 MeV bTheoretical uncertainty reflects our knowledge of the LECs (gA, g4S , dR , . . . )

SNPA & “hybrid”-EFT: (4.01± 0.01)× 10−25 MeV b [Adelberger et al., (2011)]/πEFT: (3.99± 0.14)× 10−25 MeV b [Chen et al., (2012)] [DeLeon et al., this workshop]


p + p → d + e+ + νe astrophysical factor

Effect of two-body currents

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1E [MeV]

2

4

6

8

10

12

14S(

E)

[ x

10-2

3 MeV

fm

2 ]

IAFULL

Calculation with N3LO - only the 1S0 wave


Parity violation

“Standard model” weak interaction between quarks at low energies

Lweak = GF√2

(

Jµ†W JWµ + Jµ†Z JZµ

)

JµW = “charged” current JµZ “neutral” current

JµW = 2 cos θcqLγµτ−qL q =

(ud

)

sw = sin θW cw = cos θW

JµZ =1√2cW

−2

3s2W

(

qRγµqR + qLγ

µqL

)

+(2− 2s2W

)qLγ

µτzqL − 2s2W qRγµτzqR

,

under chiral transf. qL → LqL and qR → RqR

AL,R = qL,RγµqL,R Ba

L,R = qL,RγµτaqL,R

PV terms:

isoscalar terms ∼ ~BL · ~BL − ~BR · ~BR

isovector terms ∼ (AR + AL)(B3L − B3

R)

isotensor terms ∼ Iab(BaLB

bL − Ba

RBbR)

→ PV interaction at the nuclear level constructed to violatechiral symmetry in the same way [Kaplan & Savage, 1992]

AL,R → AL,R

BL → qLγµL†τaLqL

BR →qRγ

µR†τaRqR

Iab =

−1 0 00 −1 00 0 +2


PV Lagrangian

P- and C-odd terms up to one four-gradient [Kaplan & Savage, 1992]LEC’c ∼ GF f

2π ≈ 10−7

LPV ,0∆T=1 = −h1πfπ

2√2NX 3

−N ∼ − h1π√2N(~π × ~τ)zN + O(π2)

LPV ,1∆T=0 = −h0VNuµγ

µN

LPV ,1∆T=1 = +

h1V2

NγµNTr(uµX3+)−

h1A2Nγµγ5NTr(uµX

3−)

LPV ,1∆T=2 = h2V I

abN(X aRuµX

bR + X a

LuµXbL )γ

µN

−h2A2I abN(X a

RuµXbR − X a

LuµXbL )γ

µγ5N

X aL = uτ au†

X aR = u†τ au

X a± = X a

L ± X aR

XAL →

(huL†)τ a(Lu†h†),etc

N → hN

h ≡ h(L,R, π)non-linear matrix

+ 5 contact terms of order Q [Girlanda, 2008]corresponding to the 5 Danilov parameters [Danilov] → /πEFT


PV potential at order O(Q)

k = p′1 − p1 = −(p′2 − p2) Q = (p′1 + p1 − p′2 − p2)/2

− gAh1π

2√2fπ

i(~τ1 × ~τ2)z(σ1+σ2)·k

ω2k

(∼ Q−1)

+term1

m2N

(∼ Q1)

C1

4πfπm2π

(σ1 + σ2) ·Q+C2

4πfπm2π

i(σ1 × σ2) · k+ . . . (∼ Q1)

+gAh

1π

2√2fπ

m2π

(4πfπ)2i(~τ1 × ~τ2)z (σ1 + σ2) · kL(k) (∼ Q1)


∼∫

d3qf (q2)q · σ → 0

− ∼∫

d3qf (q2)q · σ → 0

−

+

+gAh

1π

2√2fπ

g2Am2π

(4πfπ)2

(

4(τ z1 + τ z2 )z (σ1 + σ2) · kL(k)

+i(~τ1 × ~τ2)z (σ1 + σ2) · k[

H(k)− 3L(k)])

(∼ Q1)


s =√

k2 + 4m2π

L(k) =1

2

s

klog

(s + k

s − k

)

H(k) =s2 − k2

s2L(k)

Fourier transforms

vπ(r) =1

2π2

∫ ∞

0dk k2j0(kr)

fΛ(k)

k2 +m2π

L(r) =1

2π2

∫ ∞

0dk k2j0(kr)fΛ(k)L(k)

H(r) =1

2π2

∫ ∞

0dk k2j0(kr)fΛ(k)H(k)

Z(r) =1

2π2

∫ ∞

0dk k2j0(kr)fΛ(k)

Cutoff function

fΛ(k) = exp(−k4/Λ4)

Form chosen to not modify the behaviour at small k


0 1 2 3 4 5 6 7r [fm]

0,0001

0,001

0,01

0,1

YukawianOPE - EFT

0 1 2 3 4 5 6 7r [fm]

1e-05

0,0001

0,001

0,01

0,1

1

OPEL(r)H(r)Z(r)


Further terms. . .

PV LπN of order Q2

All possible indipendent terms P- and C -odd

∆I = 0 ∆I = 1 ∆I = 2

Nγµγ5DνN〈uµuν〉 NN〈χ+X 3

−〉 IabN[[X aR , u

µ], [X bR , u

ν ]]σµνN − R ↔ L

NF−µνσ

µνN NX 3−N〈χ+〉 IabN[X a

R [uµ, uν ]X b

RσµνN − R ↔ L

Ni [χ−,X 3+]N IabNN〈X a

RuµX b

Ruν〉 − R ↔ L

Other 11 terms Other 5 terms

Only the two red terms contribute to the NN PV potentialThey give simply a renormalization of h1π


Summary

Already derived by: [Zhu et al., 2005] – [Desplanques et al., 2008](some differences)

Vπ = VOPE−LO + VOPE−NNLO + VTPE−triangle + VTPE−box ∼ h1π

V CT =C1

Λχm2π

Q · (σ1 − σ2) +C2

Λχm2π

ik · (σ1 × σ2)

+C3

Λχm2π

(τ1 × τ2)z ik · (σ1 + σ2) +C4

Λχm2π

(τ1 + τ2)zQ · (σ1 − σ2)

+C5

Λχm2π

Iij (τ1)i (τ2)

jQ · (σ1 − σ2)


p − p longitudinal symmetry

Experiments

Bonn [Eversheim et al., 1991] Az (13.6MeV) = (−0.97± 0.20)× 10−7

PSI [Kistryn et al., 1987] Az (45MeV) = (−1.53± 0.21)× 10−7

TRIUMF [Berdozet al., 2003] Az (221MeV) = (+0.84± 0.34)× 10−7

Appz (θ,E) =

σ+1/2(θ,E)− σ−1/2(θ,E)

σ+1/2(θ,E) + σ−1/2(θ,E)Appz (E) =

∫

θ1≤θ≤θ2dr App

z (θ, k)σ(θ,E)∫

θ1≤θ≤θ2dr σ(θ,E)

Isospin matrix elements

〈T = 1,Tz = +1|(τ1 × τ2)z |T = 1,Tz = +1〉 = 0

〈T = 1,Tz = +1|(τ1 + τ2)z |T = 1,Tz = +1〉 = 2

〈T = 1,Tz = +1|Iab(τ1)a(τ2)b|T = 1,Tz = +1〉 = 2

Appz (E) = a0(E)h1π + a1(E)C ′

1 + a2(E)C2 C ′1 = C1 + 2C4 + 2C5


Coefficients a0(E ), a1(E ), a2(E )

E [MeV] a0 a1 a2Λ = 500 MeV

13.6 0.2532 −0.0850 −0.220145 0.5634 −0.1823 −0.4493221 −0.2443 −0.0846 +0.1770

Λ = 600 MeV13.6 +0.2384 −0.1024 −0.211145 +0.5491 −0.2088 −0.4450221 −0.1971 −0.0415 0.1563

Note: at low energies the observable is dominated by the transition 1S0 ↔ 3P0, and we canshow that ai (E) ∼

√E


Fit of the LEC’s

Reasonable range for h1π = 0÷ 12.0× 10−7, “best value” h1π = 4.56× 10−7

[Desplanques, Donogue, & Holstein, 1980]Lattice QCD h1π ≈ 1.0× 10−7 Wasem, 2012

Λ [MeV] C ′1 × 107 C2 × 107 C ′

1 × 107 C2 × 107 C ′1 × 107 C2 × 107

h1π = 4.56× 10−7 h1π = 1.0× 10−7 h1π = 12.0× 10−7

500 −2.16989 10.00329 −1.66058 5.33275 −3.14844 18.97703600 −2.80108 10.37958 −2.67065 5.92568 −3.05166 18.93707

0 50 100 150 200 250 300E

p [MeV]

-2

-1.5

-1

-0.5

0

0.5

1

Azpp

[X

10-7

]

N3LO500N3LO600


By taking into account the experimental uncertainties . . .a0(E)h1π + a1(E)C ′

1 + a2(E)C2 = Appz (E)±∆A

ppz (E)

-10 -5 0 5 10C

1

0

5

10

15

20

C2

E=221 MeVE=13.6 MeVE=45 MeV

hπ

1=1 x 10

-7

For E = 13.6 and 45 MeV almost coincident bandsA’BCD’=region of overlap


By taking into account the experimental uncertainties . . .a0(E)h1π + a1(E)C ′

1 + a2(E)C2 = Appz (E)±∆A

ppz (E)

-10 -5 0 5 10C

1

0

5

10

15

20

C2

E=221 MeVE=13.6 MeVE=45 MeV

hπ

1=1 x 10

-7

A

B

C

D

A’ B’

C’

D’

For E = 13.6 and 45 MeV almost coincident bandsA’BCD’=region of overlap


Longitudinal asymmetry in ~n + 3He → p + 3

H @ ORNL

~n − p → d + γ NPDGAMMA experiment (analyzing data): sensitive to h1π~n − 3He → p − 3H (under costruction) using part of the NPDGAMMA apparatus

VPV =gAh

1π

2√2fπ

O(1)0 +

C1

4πfπO

(0)1 +

C2

4πfπO

(0)2 +

C3

4πfπO

(1)3 +

C4

4πfπO

(1)4 +

C5

4πfπO

(2)5

0+ and 0− scattering dominated by T = 0 resonances

EFT:

π: ∼ h1π(τi × τj )z isovector ∆T = 1

contact terms 1 and 2: ∆T = 0 (C1 ≈ −1, C2 ≈ 10)

contact terms 3 and 4: ∆T = 1 (C3 ≈ 0,C4 ≈ −1)

contact term 5 : ∆T = 2 (C5 ≈ +1)

→ dominant contribution from contact term 2 (?)


Contributing waves

initial state (n − 3He) q ≈ 0: 1S0,3S1

final state (p − 3H) q = 0.165 fm−1:

J = 0: 1S0,3P0

J = 1: 3S1 −3D1,

1P1 −3P1

Neglecting 3D1, we have to compute the matrix elements T JLS,L′S′ ≡ T ex

LS,L′S′

PC PV1S0 → 1S0 T 0

00,001S0 → 3P0 T 1

00,113S1 → 3S1 T 0

01,013S1 → 1P1 T 1

01,103S1 → 3P1 T 1

01,11

PC T-matrix elements + ΨJπLS : using the KVP+HH method, starting from a NN+3N

interaction model (neglecting the PV potential)

PV T-matrix elements: T J0J,1S = 〈T ΨJ−

1S′|VPV |ΨJ+

0J 〉 (Monte Carlo code by R. Schiavilla)


Results for the PV EFT potential Preliminary

az = h1πa0 + C1a1 + C2a2 + C3a3 + C4a4 + C5a5

Inter. a0 a1 a2 a3 a4 a5N3LO/N2LO −0.153 +0.027 0.021 −0.111 −0.023 −0.002

h1π C1 C2 C3 C4 C5

Param. (units 10−7) 4.56 −1 10 0 −1 +1

product (units 10−7) −0.70 −0.027 +0.21 0.0 +0.023 −0.002

az −0.50× 10−7

h1π = 0: az = +0.20× 10−7, h1π = 12× 10−7: az = −1.4× 10−7

Important contribution also from the long-range OPE term

If h1π can be measured by NPDGAMMA, the measure of az useful to extract C2


Conclusion & Outlook

Weak reactions in light nuclei:

New theoretical study of muon capture in χEFT

Γ0(µ− + 3He): nice agreement theory vs. experiment

ΓD(µ− + d):

some discrepancies among different theoretical works

MuSun: more accurate experimental results → important test of χEFT

New refined calculation of the pp fusion up to 100 keV

Study of PV processes: p − p and p − 3He asymmetries

In the future:

µ− + 3He → n + d + νµ

µ− + 3He → n + n + p + νµ

p + 3He → 4He+ e+ + νe

other PV processes ~n + p → d + γ, ~n + X spin rotation, . . .

in progress: jA, ρA derived at N4LO [Epelbaum et al.] , [Schiavilla et al.]


Collaborators

L. Girlanda - INFN & Salento University, Lecce (Italy)

R. Schiavilla - Jefferson Lab. & ODU, Norfolk (VA, USA)

S. Pastore - USC, Columbia (SC, USA)

M. Piarulli, A. Barone & F. Spadoni (PhD students) - ODU, Norfolk (VA, USA)

A. Kievsky & L. E. Marcucci - INFN & Pisa University, Pisa (Italy)

D. Cartisano (graduate student) Pisa University, Pisa (Italy)


Thank you for your attention!


weak reactions and chiral eft in few body systems

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