light-nuclei spectra and electroweak response: a …...light-nuclei spectra and electroweak...

Basic model

Chiral 2Ninteractions


EWKinteractions

EWK QEresponse

Outlook

Light-nuclei spectra and electroweakresponse: a status report

R. Schiavilla

Theory Center, Jefferson Lab, Newport News, VA 23606, USAPhysics Department, Old Dominion University, Norfolk, VA 23529, USA

October 20, 2017

Basic model



EWKinteractions

EWK QEresponse

Outlook

Spectra of few-nucleon systems from LQCDBeane et al. (2013)

NPLQCD calculations (mπ = 806 MeV)

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2N potential from LQCD and nuclear spectraAoki et al. (2012); McIlroy et al. (2017)

LQCD calculation of 2N potential by HAL collaboration

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Outline

The basic model of nuclear theoryChiral 2N potentialsChiral 3N potentials and light-nuclei spectraNuclear electroweak transitionsNuclear electroweak response in quasi-elastic regimeOutlook

Basic model



EWKinteractions

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The basic model

Effective potentials:

H =

A∑i=1

p2i

2mi+

A∑i<j=1

vij︸︷︷︸th+exp

+

A∑i<j<k=1

th+exp︷︸︸︷Vijk + · · ·

Assumptions:Quarks in nuclei are in color singlet states close tothose of N ’s (and low-lying excitations: ∆’s, . . . )Series of potentials converges rapidlyDominant terms in vij and Vijk are due to π exchange

leading πN coupling =gA

2 fπτa σ ·∇φa(r)

Effective electroweak currents:

jEW =

A∑i=1

ji +

A∑i<j=1

jij +

A∑i<j<k=1

jijk + · · ·

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Yukawa potential in classical mechanics

What is the connection between meson-exchangeinteractions and their representation in terms of vij?A simple model: a classical scalar field φ(r, t)interacting with static particles:

L =1

2

[φ2 − |∇φ|2 − µ2 φ2

]− g φ

A∑i=1

δ (r− ri)

Lowest-energy configuration occurs in the static limitφ(r, t)→ φ(r) (Poisson-like equation of electrostatics)

∇2 φ− µ2 φ = g

A∑i=1

δ (r− ri)

Energy of the field (up to self energies) in this limit

Eφ =1

2

∫drφ(r) g

A∑i=1

δ(r− ri) = − g2

4π

A∑i<j=1

e−µrij

rij

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Yukawa potential in quantum mechanics

Scalar-field Hamiltonian is

H =∑k

ωk a†kak︸︷︷︸

H0

+ g

A∑i=1

∑k

1√2ωk V

(ak eik·ri + a†k e−ik·ri

)︸︷︷︸

H′

Set of shifted harmonic oscillators; exact eigenenergiesof field given by

Enk =∑k

ωk nk−g2A∑

i<j=1

14π

e−µrijrij︷︸︸︷∫

dk

(2π)3

eik·(ri−rj)

ω2k︸︷︷︸

energy shift

In CM and QM the scalar field energy in the presenceof static particles can be replaced by a sum of vY (rij)

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Scattering between slow-moving particles

In potential theory to leading order

T Y 1fi =

∫dr e−ip

′·r vY (r) eip·r = − g2

ω2q︸︷︷︸

vY (q)

(q = p− p′)

In meson-exchange theory (to leading order)

TM1fi =

∑I

final state︷︸︸︷〈p′,−p′; 0|H ′|I〉〈I|H ′

initial state︷︸︸︷|p,−p; 0〉

Ei − EI= − g

2

ω2q

= +p −p

p’ −p’

p −p

p’ −p’

p −p

p’ −p−qq

Basic model



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Beyond leading order

(a3)(a2)(a1) (a4)

(b2)(b1) (b3) (b4)

(c2) (c4)(c1) (c3)

(Y2)

p’

p

−p’

−p

q

q1

2p−q q1 −p1

TM2fi = T Y 2

fi + correction

Obtain from TM2fi − T Y 2

fi correction term v(2)(q,

p′+p︷︸︸︷Q )

such that vY (q) + v(2)(q,Q) reproduces TM2fi

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Many of the developers of the basic model . . .

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χEFT formulation of the basic model

χEFT is a low-energy approximation of QCDLagrangians describing the interactions of π, N , . . . areexpanded in powers of Q/Λχ (Λχ ∼ 1 GeV)Their construction has been codified in a number ofpapers 1

L = L(1)πN + L(2)

πN + L(3)πN + . . .

+L(2)ππ + L(4)

ππ + . . .

L(n) also include contact(NN

)(NN

)-type interactions

parametrized by low-energy constants (LECs)Initial impetus to the development of χEFT for nuclei inthe early nineties 2

1Gasser and Leutwyler (1984); Gasser, Sainio, and Svarc (1988); Bernard et al. (1992); Fettes et al. (2000)

2Weinberg (1990)–(1992)

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Chiral 2N potentials with ∆’sPiarulli et al. (2015); Piarulli et al. (2016)

Two-nucleon potential: v = vEM + vLR + vSR

EM component vEM including corrections up to α2

Chiral OPE and TPE component vLR with ∆’s

Short-range contact component vSR up to order Q4

parametrized by (2+7+11) IC and (2+4) IB LECsvSR functional form taken as CRS (r) ∝ e−(r/RS)2

withRS=0.8 (0.7) fm for a (b) models

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np (T = 0 and 1) and pp phase shifts

Ia-Ib: Elab = 125 MeV IIa-IIb: Elab = 200 MeV

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Deuteron wave functions

0 1 2 3 4 5r(fm)

0

0.1

0.2

0.3

0.4

0.5

fm-1

/2

NV2-IaNV2-IbNV2-IIaNV2-IIbAV18

s-wave

d-wave

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Dominant features of 2N potential

v in T,S=0,1 (d-like) v in T,S=1,0 (quasi-bound)

0 1 2 3 4r(fm)

-200

0

200

400

600

MeV

Chiral NV2-IaAV18

300 MeV

550 MeV

0 1 2 3 4r(fm)

-100

0

100

MeV

Chiral NV2-IaAV18

Strong state-dependent correlations

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Impact on pair momentum distributionsSchiavilla et al. (2007); Wiringa et al. (2014); 12C(e, e′NN) JLab exp: Subedi et al. (2008)

0 1 2 3 4 510-2

100

102

104

10612C

0 1 2 3 4 510-2

100

102

104

10610B

0 1 2 3 4 510-2

100

102

104

1068Be

0 1 2 3 4 510-2

100

102

104

1066Li

0 1 2 3 4 510-2

100

102

104

106

q (fm-1)

12(q

,Q=0

) (fm

3 )

4He

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Ab initio methods

Hyperspherical harmonics (HH) expansions for A= 3and 4 bound and continuum states

|ψV 〉 =∑µ

cµ |φµ〉︸︷︷︸HH basis

and cµ from EV =〈ψV |H|ψV 〉〈ψV |ψV 〉

Quantum Monte Carlo for A > 4 bound states

Figure by Lonardoni

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Outlook

GFMC for A ≤ 12RMP by Carlson et al. (2015)

Propagation in imaginary time

E0 = limτ→∞

〈ψV |H e−τ H |ψV 〉〈ψV |e−τ H |ψV 〉

Exponential growth with A (in 12C st-states ∼ 4× 106)

ψV =∑s≤2A

∑t≤2A

φst(r1, . . . , rA)χst(1, . . . , A)

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Failures of basic model with 2N potentials only

spectra of light nucleiIntermediate-energy Nd and low-energy nα scatteringnuclear matter E0(ρ) and neutron star masses

8 9 10 11 12 13 14 15 16R (km)

0

0.5

1

1.5

2

2.5

3

M (

MO• )

Causality: R>2.9 (G

M/c2 )

ρ central=2ρ 0

ρ central=3ρ 0

35.1

33.7

32

Esym

= 30.5 MeV (NN)

1.4 MO•

1.97(4) MO•

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Chiral 3N potentials with ∆’s

3N potential up to N2LO:

Short-range contact terms depend on LECs cD and cEcD and cE fixed by fitting Eexp

0 (3H) = –8.482 MeV and nddoublet scattering length aexp

nd = (0.645± 0.010) fm

w/o 3N with 3N

Model cD cE E0(3H) E0(3He) E0(4He) 2and E0(3He) E0(4He)

Ia 3.666 –1.638 –7.825 –7.083 –25.15 1.085 –7.728 –28.31Ib –2.061 –0.982 –7.606 –6.878 –23.99 1.284 –7.730 –28.31IIa 1.278 –1.029 –7.956 –7.206 –25.80 0.993 –7.723 –28.17IIb –4.480 –0.412 –7.874 –7.126 –25.31 1.073 –7.720 –28.17

Better strategy: fix cD and cE by reproducing Eexp0 (3H)

and the GTexp matrix element in 3H β-decay . . .

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Outlook

Spectra of light nucleiPiarulli et al. (2017)

100 100

90 90

80 80

70 70

60 60

50 50

40 40

30 30

20 20

Ene

rgy

[MeV

]

4He 6He 6Li7Li

8He

8Li

8Be

9Li

9Be

10He

10Be 10B

11B

12C

0+ 0+2+

1+3+2+1+

3/21/27/25/25/2 0+

2+

2+1+3+

0+2+

4+

2+1+3+

3/21/25/2

3/25/21/2

0+

0+2+2+

3+1+

3/2

0+

GFMC calculations

AV18+IL7EXPNV2+3-Ia

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Outlook

Nuclear axial currents at one loopBaroni et al. (2016) & (2017); Pastore et al. (2017)

A single unknown LEC in j5 fixed in 3H β-decay (onlyN2LO and N3LO terms shown below)

GT m.e.’s in A= 6–10 nuclei (AV18/IL7 potential withχEFT axial current)

(3+,0)

(1+,0)

(0+,1)

(1+,0)

10B

10C

98.54(14)%< 0.08 %

(0+,1)

E ~ 0.72 MeV

E ~ 2.15 MeV

1 1.1 1.2

Ratio to EXPT

10C

10B

7Be

7Li(gs)

6He

6Li

3H

3He

7Be

7Li(ex)

gfmc 1bgfmc 1b+2b(N4LO)Chou et al. 1993 - Shell Model - 1b

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µ− capture on 2H and 3HeMarcucci et al. (2012)

χEFT predictions with conservative error estimates:

Γ(2H) = (399± 3) sec−1 Γ(3He) = (1494± 21) sec−1

Errors due primarily to:experimental error on GTEXP (0.5%)uncertainties in EW radiative corrections 1 (0.4%)cutoff dependence

Using ΓEXP(3He) = (1496± 4) sec−1, one extracts

GPS(q20 = −0.95m2

µ) = 8.2± 0.7

versus GEXPPS (q2

0 = −0.88m2µ) = 8.06± 0.55 2 and a χPT

prediction of 7.99± 0.20 3

Upcoming measurement of Γ(2H) by the MuSuncollaboration at PSI with a projected 1% error . . .

1These corrections increase rate by 3%, see Czarnecki et al. (2007); 2From a measurement of ΓEXP(1H),

Andreev et al. (2013); 3Bernard et al. (1994), Kaiser (2003)

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CC and NC ν-A scattering

Large program in accelerator ν physics (MicroBooNE,NOνA, T2K, Minerνa, DUNE, . . . )

rate ∝∫dE Φα(E)P (να → νβ;E)σβ(E,E ′)

Determination of oscillation parameters dependscrucially on our understanding of

ν flux Φα(E)ν-A cross section σβ(E,E ′)DUNE νµ flux MiniBooNE

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GFMC calculation of EM response in 12CCarlson and Schiavilla (1992); Lovato et al. (2013–2016)∫ ∞

0dω e−τω Rαβ(q, ω)=〈i | j†α(q) e−τ(H−Ei) jβ(q) | i〉

Inversion back to Rαβ(q, ω) by maximum entropymethods

longitudinal transverse

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0 50 100 150 200 250 300 350 400

𝑅00

/𝐶2 u,

u

𝜔[MeV]

|𝐪| = 570 MeV/c

GFMC 𝑂1u+2uGFMC 𝑂1uWorld data

0.00

0.02

0.04

0.06

0.08

0 50 100 150 200 250 300 350 400

𝑅uu

/𝐶2 u,

u

𝜔[MeV]

|𝐪| = 570 MeV/cGFMC 𝑂1u+2uGFMC 𝑂1uWorld data

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

𝑆 u(𝑞

)

𝑞 [fm−1]

GFMC 𝑂1u+2uExp-TRExpExp-TFF

← Coulomb sum

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Outlook

NC responses and cross sections in 12CLovato et al. in preparation

Inclusive ν/ν (−/+) cross section given in terms of fiveresponse functions

dσ

dε′ldΩl∝[v00R00 + vzz Rzz − v0z R0z +

dominant︷︸︸︷vxxRxx ∓ vxy Rxy

]

0

2

4

6

8

10

12

0 100 200 300 400 500

𝑅𝑥𝑥

[GeV

−1 ]

𝜔[MeV]

VNC 1𝑏VNC 12𝑏ANC 1𝑏ANC 12𝑏NC 1𝑏NC 12𝑏

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Outlook

Slide by Lonardoni

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Subleading 3N potentials

Different strategies for constraining the (10) LECs insubleading contact 3N potential:

Nd scattering observables at low energiesSpectra of light- and medium-weight nuclei andproperties of nuclear/neutron matterpd scattering at 3 MeV

AFDMC for A = 16 and 40

Piarulli et al. (2017) Lynn et al. (2016); Lonardoni et al. (2017)

Basic model



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Nuclear weak interactions: low energy regime

Shell model in agreement with exp if geffA ' 0.7 gA

Understanding “quenching” of gA in nuclear β decaysRelevant for neutrinoless 2β-decay since rate ∝ g4

A

Martinez-Pinedo et al. (1996)0ν–2β amplitude

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Nuclear matrix element in 0ν–2β decayEngel and Menendez (2017)

Ca Se Xe

Test case1: 8He(0+; 2)→8Be(0+; 2)

standard axial∝ τ+i τ+j σi · σj

1Mereghetti et al. (2017)

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The HH/QMC team

The ANL/JLAB/LANL/Pisa quadri-axis

A. Baroni1 (USC)J. Carlson (LANL)S. Gandolfi (LANL)L. Girlanda (U-Salento)A. Kievsky (INFN-Pisa)D. Lonardoni (LANL)A. Lovato (ANL)

L.E. Marcucci2 (U-Pisa)S. Pastore3 (LANL)M. Piarulli4 (ANL)S.C. Pieper (ANL)R. Schiavilla (ODU/JLab)M. Viviani5 (INFN-Pisa)R.B. Wiringa (ANL)

Computational resources from ANL LCRC, LANL OpenSupercomputing, and NERSC

Theory Center support for August 2017 collaborationmeeting is gratefully acknowledged

1ODU Ph.D. 2017; 2ODU Ph.D. 2000; 3ODU Ph.D. 2010; 4ODU Ph.D. 2015; 5CEBAF theory postdoc 1994

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Outlook

EM operators up to one loopPastore et al. (2009) & (2011); Kölling et al. (2009) & (2011); Piarulli et al. (2013)

Contributions to jγ up to order eQ:

Five unknown LEC’s in jγ at eQLO for ργ at eQ−3 and no OPE corrections at eQ−1

No unknown LEC’s in ργ up to eQ

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Outlook

The LEC’s characterizing jγ at one loop

d’s could be determined by (γ, π) data on the nucleon

dV1 , dV2 fixed by assuming ∆ dominance, dS1 , cS , and cV

fixed by fitting A = 2 and 3 EM observablesThree-body currents at N3LO vanish:

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3He/3H magnetic f.f.’sPiarulli et al. (2013)

Two-body (isovector) contributions play crucial role

Magenta shaded area: LEC cV fixed by σγnpRed shaded area: LEC cV fixed by µV (3He/3H)

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Outlook

Magnetic moments in A ≤ 10 nucleiPastore et al. (2013)

GFMC calculations use AV18/IL7 (rather than chiral)potentials with χEFT EM currentsPredictions for A > 3; about 40% of µ(9C) due tocorrections beyond LO

-3

-2

-1

0

1

2

3

4

µ (µ

N)

EXPT

GFMC(IA)GFMC(FULL)

n

p

2H

3H

3He

6Li

6Li*

7Li

7Be

8Li 8B

9Li

9Be

9B

9C

10B

10B*

light-nuclei spectra and electroweak response: a …...light-nuclei spectra and electroweak...

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