Electroweak reactions in “a few-body system”

Collaborators:BD Carlsson, D Odell, L Platter, A Ekstrom,

T Papenbrock, C Forssen, G Rupak and S Bacca

Bijaya Acharyaacharya@uni-mainz.de

July 14, 2018

Outline

1 Chiral effective field theory with external currents

2 The proton-proton fusion reaction

3 “Luscher’s Formula” for capture matrix elements

4 P-wave contribution to the pp fusion reaction

5 Muon capture by deuteron

6 Electromagnetic response functions of the deuteron at largefour-momentum transfer

7 Outlook and summary

χEFT with external currents

degrees of freedom: pions and nucleons (and, sometimes, also thedelta) plus external (electroweak, dark . . .) probes.

NN forces given by pion exchanges and contact interactions.

+(Q/Λ)0 :

(Q/Λ)2 : + + + +

+ + + +(Q/Λ)3 : cD

c1, c3, c4

πN processes and nuclear currents that couple to external probesderived from same theory — many LECs are common!

cDc3, c4

c1, c2, c3, c4

The NNLOsim Interactions [Carlsson et al. (2016)]

Weinberg counting, Gaussian regulators with cutoff ΛEFT ∼ 500 MeV.

nonperturbative resummation of all NN interactions up to (Q/Λ)3 inthe Schrodinger equation.

simultaneous fitting of LECs to πN and NN scattering data, EB and〈rc2〉 of 2,3H and 3He, Q(2H), as well as EA

1 (3H), with bothexperimental and theoretical uncertainties.

42 interactions — 7 values for ΛEFT in the range (450, 600) MeV and6 different truncations of the NN scattering energy in input data —have same long-distance properties.

covariance matrix of the LECs known for each of the 42 interactions— correlations between all LECs taken into account.

errors in LECs propagate to calculated observables.

p + p → d + e+ + νe

Picture credit: A. Ray

Cross section parametrized as S-factor, S(E ) = σ(E ) E eπ√

mp/E α .

“Gamow window” at E < 10 keV. Maclaurin’s series useful,S(E ) = S(0) + E S ′(0) + . . .

lab experiments practically impossible — rigorous analysis oftheoretical uncertainty important.

Semi-leptonic weak cross sections and decay rates

σ, Γ ∼∑spins

∫phase space

|〈f |HW |i〉|2

Below GeV scale, weak interaction Hamiltonian can be considered azero-range coupling between leptonic and nucleonic currents,

HW =G√

2

∫d3x [jα(x)Jα(x) + h.c.] .

Need matrix elements of nucleonic current operator Jα betweennucleonic initial and final states.

Other calculations

“realistic” approaches: educated guesses for potentials and currents[Bethe and Critchfield (1938), Bahcall and May (1969),Kamionkowsky and Bahcall (1994), Schiavilla et al. (1998)]

hybrid approach: current operator derived in EFT sandwichedbetween phenomenological wavefunctions. [Park et al. (1998), Park et al. (2003)]

Effective Field Theories

EFT(π) [Kong and Ravndal (2001), Butler and Chen (2001),Ando et al. (2008), Chen et al. (2013)]χEFT [Marcucci et al. (2013)]

Lattice

EFT [Rupak and Ravi (2015)]QCD [Savage et al. (2017)]

Our error budget

Theory uncertainty ∼ 0.7% from:

Numerical solution of Schrodinger equation ∼ 0.05%

Polynomial fit to S(E ) ∼ 0.005%

Statistical uncertainties in LECs ∼ 0.4%

Energy range of the input NN scattering data . 0.2%

Cutoff dependence [O(α2) corrections not shown here]:

460 480 500 520 540 560 580 600

ΛEFT (MeV)

4.02

4.04

4.06

4.08

4.10

4.12

S(0

)(1

0−23

MeV

fm2)

3.75

3.8

3.85

3.9

3.95

4

4.05

4.1

4.15

4.2

4.25

S(0

) [

10

-23 M

eV

fm

2]

RecommendedPionless EFTChiral EFT (Pisa-JLab)

Chiral EFT (Ours)

Lattice QCD

The recommended value [Adelberger et al. (2011)] was obtained from“range of values of published calculations”.

The error in the Pisa-JLab calculation [Marcucci et al. (2013)] wasestimated by using two different values for ΛEFT, 500 and 600 MeV.

Pisa-JLab calculation is the only one that includes p-wave pp state.They find a ∼ 0.5% contribution at threshold that grows with E .

Pisa-JLab calculation used a basis of Laguerre polynomials to representtheir wave functions,

φ am (β; r) ∝ e−

12βrL a

m (βr), (1)

which makes extension to A > 2 systems more convenient.

5 10 15 20 25 r

-1.0

-0.5

0.5

1.0

r ϕ01/2 (1; r)

r ϕ11/2 (1; r)

r ϕ21/2 (1; r)

r ϕ31/2 (1; r)

Truncating m at some integer n effectively imposes an UV and an IRcutoff. Dirichlet boundary at r = L, which scales as

L = (4n + 2a + 6)β−1 . (2)

For the leading (Gamow-Teller) matrix element at E = 50 keV :

matrix element (%)

binding energy (ppm)

36 38 40 420.01

0.05

0.10

0.50

1

L [fm]

Rela

tive

Err

or

Finite basis is widely used to solve nuclear many-body problems.

UV truncation error easy to avoid for nuclear interactions but hard tomodel [Konig et al. (2014)].

For IR corrections, equivalent of Luscher’s formula already exists forbound state properties [More et al. (2012)].

We derive similar analytic formula for finite-volume correction incapture/fusion reactions.

“Luscher’s Formula” for capture matrix elements

Iλ(p; η; L) =

∫ L

0dr u(L)(r) rλ up(r) ,

u(L)(r)→A∞e−γ∞r

[1− e−2γ∞(L−r)

]+ (1− δl0)O( 1

γ∞r ) +O(e−γ∞[2L+r ])

up(r)→ sin[pr − ηp log(2pr) + σl + δl − πl

2

]+O( 1

pr )

for weak capture, the LO matrix element has λ = 0

more generally, λ = 0, 1, . . ., and is related to the multipolarity of thetransition

At γ∞L 1, λ, η,

∆Iλ(p; ηp; L) =2A∞γ∞γ2∞ + p2

Lλ e−γ∞L sin

(δl + σl −

πl

2+ pL− ηp log 2pL

).

“Luscher’s Formula” for capture matrix elements

numerical

analytic

35 40 45 50 55

0.0000

0.0005

0.0010

0.0015

0.0020

L [fm]

Δℐ

0(k

;η

;L)[f

m1/2]

numerical

analytic

35 40 45 50 55

-0.0002

-0.0001

0.0000

0.0001

0.0002

L [fm]

Δℐ

0(k

;η

;L)[f

m1/2]

Figure: The IR correction to the radial matrix element of the LO (Gamow-Teller)operator for pp fusion at E = 50 keV (left) and 1 MeV (right). The numerical resultswere calculated using Harmonic oscillator bases of varying dimensionality.

“Luscher’s Formula” for capture matrix elements

depends on the type of basis functions used only through L.

contains no fit parameters at A = 2.

should work for A > 2 nuclei. However, one might have to use ourformula as an extrapolant with γ∞, A∞ and δl fit to numerical dataat several L values.

gets better at larger L and higher energies, i.e. smaller ηp.

needs improvement/extension for application to the rp-process regime.

P-wave contribution to pp fusion in pionless EFT

Anticipated uncertainty ∼ larger ofQ

mπ.

1

20and

γ

mπ∼ 1

3

= +

−itC(E;p′,p) −iTC(E;p′,p)

= +

L(LO)W = −

GV√2

(l0+N†τ−N

+ gAl+ · N†στ−N)

only contributes for nonzero recoil momentum(k). k/γ . 0.01

L(NLO)W = −

GV√2

[− gAl

0+N†σ ·

i←→∇

2mτ−N

− l+ ·(N†

i←→∇

2mτ−N

)]Q/m . 0.01

Anticipated size of the p-wave contribution to S-factor ∼ O(10−4).

Results so far ...

|M(NLO) +M(rec)|2 =16

(Gv√

2

)2 (EνEe + pν · pe)

(3|Trec|2 + 2gA

2TNLO · T∗NLO

)+ 6 (pν · TNLO) (pe · T∗NLO) + 3 (EνEe − pν · pe)TNLO · T∗NLO

− (6 + 4g2A) (Eνpe + Eepν) · TNLOT∗rec + 2g2

A (3EνEe − pν · pe) |Trec|2,

where

Trec = −√

8πγ e iσ1e−πηp/2 (pν + pe) · p |Γ(2 + iηp)|1

(γ2 + p2)2e2ηp arctan p/γ , (3)

and

TNLO =√

8πγ e iσ1e−πηp/2 p

m|Γ(2 + iηp)|

1

p2 + p2η2p

[1 +

p2 + 2pηpγ − γ2

γ2 + p2e2ηp arctan p/γ

].

The MuSun experiment at PSI

The MuSun experiment aims at measuring µ− + d → νµ + n + n capturerate with 1.5% precision.

µ−

νµ

dn

n

e+

νe

p

d

p

first precise measurement of weak reaction on NN system.

simplest possible test of semileptonic nuclear calculations.

input data to constrain pp fusion rate without contaminating it withNNN physics.

χEFT calculation of muon capture by deuteron

Higher-energy process than pp fusion — pion-pole diagrams need tobe added:

Combining with the recent Roy-Steiner extraction of ci′s

[Hoferichter et al. (2015, 2016), Siemens et al. (2017)], with remainingrelevant LECs fitted NN data, allows us to fix cD from 1S0 capturerate.

Upto O(Q3), predicts NN electroweak currents and pion-range part ofNNN interaction, cD .

Important test of q-dependence of nuclear matrix elements and of thesingle-nucleon axial form factor.

Muon capture: Results

NNLOsim : ΓD(1S0) = 252.4+2.5

−2.1 s−1 .

NNLORS1 : ΓD

(1S0) = 252.8± 4.6 s−1 .Additional uncertainty from nucleon axial radius 2 : ∆ΓD

(1S0) ∼ 3.9 s−1 .Including higher partial waves : ΓD = 397.8 s−1 .

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0cD

230

240

250

260

270

280

Γ1S

0

D[s−

1]

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5

dR

3.95

4.00

4.05

4.10

4.15

4.20

S(0

)[1

0−23

MeV

fm2]

0.60

0.65

0.70

0.75

0.80

〈EA 1〉

4.00 4.02 4.04 4.06 4.08 4.10 4.12

Spp(0) [10−23 MeV fm2]

235

240

245

250

255

260

265

270

Γ1S

0

D[s−

1]

triton β-decay

1With NN contacts fitted to Tlab < 200 MeV phaseshifts from Granada PWA, and2H binding and radius.

2From variation of r 2A in the 1-σ interval (0.24,0.68) fm2.

The electromagetic response functions

The longitudinal response function

RL(q, ω) =∑f

|〈ψf |ρ(q)|ψi 〉|2 δ(ω + md − Ef ) (4)

and the transverse response function

RT (q, ω) =∑f

|〈ψf |jT (q)|ψi 〉|2 δ(ω + md − Ef ) (5)

are related to the cross section in the one-photon exchange limit by

dσ

dΩdω= σMott

([qµqµq2

]2

RL − [qµqµ2q2

− tan2(θ/2)]RT

). (6)

50 100 150 ω [MeV]

0

0.005

0.01

0.015

0.02

RL(q

=3

00

MeV

,ω)

[ M

eV-1

] EM responses of the deuteron in the impulse approximation

100 150 200 250 ω [MeV]

0

0.001

0.002

0.003

0.004

0.005

0.006

RL(q

=5

00

MeV

,ω)

[MeV

-1]

jmax

= 2

jmax

= 4

jmax

= 6

jmax

= 10

jmax

= 15

50 100 150 ω [MeV]

0

0.005

0.01

RT(q

=3

00

MeV

,ω)

[ M

eV-1

]

100 150 200 250 ω [MeV]

0

0.002

0.004

0.006

0.008

0.01

0.012

RT(q

=5

00

MeV

,ω)

[ M

eV-1

]

25 50 75 100 125 150 ω [MeV]

0

0.005

0.01

0.015

0.02

RL(q

=3

00

MeV

,ω)

[ M

eV-1

] EM responses of the deuteron in the impulse approximation

100 150 200 250 300 ω [MeV]

0

0.005

0.01

RL(q

=5

00

MeV

,ω)

[MeV

-1]

This work (w/ Idaho N3LO)

Expt. (BLAC,1988)

AV18 (Shen et al.)

25 50 75 100 125 150 ω [MeV]

0

0.005

0.01

0.015

RT(q

=3

00

MeV

,ω)

[ M

eV-1

]

100 150 200 250 300 ω [MeV]

0

0.005

0.01

0.015

RT(q

=5

00

MeV

,ω)

[ M

eV-1

]

Next up: νd and ν-nucleus scattering

Picture credit: Bacca and Pastore

ν experiments use event generators that need nuclear physics inputfor ν-nucleus cross section.

The ab initio approach, with chiral EFT interactions, can provideimportant benchmark for models that go into these generators.

Inclusive ν-induced breakup of 2H.

Extend to ν-induced one/two-nucleon knockout from 4He, 12C, 16O,. . ., 40Ar using nuclear many-body methods.

Summary

1 Calculated pp fusion cross section. Result agrees with another χEFTcalculation once their result is corrected for basis truncation error.

2 Derived analytic expressions for p-wave contributions to pp fusionreaction in pionless EFT. Numbers due soon!

3 Performed uncertainty analysis of the 1S0 µd capture rate andobtained correlation with the pp fusion S-factor.

4 Upcoming experimental µ-d capture rate value, combined withRoy-Steiner determination of ci

′s, fixes electroweak currents andpion-exchange part of NNN force completely from πN and NN sectors.

5 Better determination of nucleon axial form factor is required for moreprecise calculations of finite-qµ nuclear weak processes.

6 Will calculate ν-nucleus cross sections starting from ν-d .

References

Bethe and Critchfield (1938)

Phys. Rev. 54 (1938) 248

Bahcall and May (1969)

Astroph. J. 155 (1969) 501

Kamionkowsky and Bahcall (1994)

Astroph. J. 420 (1994) 884

Schiavilla et al. (1998)

Phys. Rev. C 58 (1998) 1263

Park et al. (1998)

Astroph. J. 507 (1998) 443

Park et al. (2003)

Phys. Rev. C 67 (2003) 055206

Marcucci et al. (2013)

Phys. Rev. Lett. 110 (2013) 192503

Kong and Ravndal (2001)

Phys. Rev. C 64 (2001) 044002

References

Butler and Chen (2001)

Phys. Lett. B 520 (2001) 87

Ando et al. (2008)

Phys. Lett. B 668 (2008) 187

Chen et al. (2013)

Phys. Lett. B 720 (2013) 385

Rupak and Ravi (2015)

Phys. Lett. B 741 (2015) 301

Savage et al. (2017)

Phys. Rev. Lett. 119 (2017) 062002

Adelberger et al. (2011)

Rev. Mod. Phys. 83 (2011) 195

Carlsson et al. (2016)

Phys. Rev. X 6 (2016) 011019

Konig et al. (2014)

Phys. Rev. C 90, (2014) 064007

More et al. (2012)

Phys. Rev. C 87 (2012) 044326

References

Hoferichter et al. (2015, 2016)

Phys. Rev. Lett. 115 (2015) 192301

Phys. Rept. 625 (2016) 1

Siemens et al. (2015, 2016)

Phys. Lett. B 770 (2017) 27

Marcucci et al. (2013)

Phys. Rev. Lett. 110 (2013) 192503

Carlsson et al. (2016)

Phys. Rev. X 6 (2016) 011019