Light-nuclei spectra and electroweak response: a …...Light-nuclei spectra and electroweak...
Transcript of Light-nuclei spectra and electroweak response: a …...Light-nuclei spectra and electroweak...
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
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
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
Spectra of few-nucleon systems from LQCDBeane et al. (2013)
NPLQCD calculations (mπ = 806 MeV)
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
2N potential from LQCD and nuclear spectraAoki et al. (2012); McIlroy et al. (2017)
LQCD calculation of 2N potential by HAL collaboration
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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 + · · ·
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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)
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
Many of the developers of the basic model . . .
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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)
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
np (T = 0 and 1) and pp phase shifts
Ia-Ib: Elab = 125 MeV IIa-IIb: Elab = 200 MeV
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
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)
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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•
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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 . . .
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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)
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
Outlook
Slide by Lonardoni
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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)
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
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
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
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:
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
Outlook
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)
Basic model
Chiral 2Ninteractions
Chiral 3Ninteractions
EWKinteractions
EWK QEresponse
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*