Electroweak processes with two- and three- nucleon systemscrex.fmf.uni-lj.si/eep17/Golak_Bled_2017_07_04.pdf · Rev. Lett. 115, 122301 (2015)) LENPIC ... Pieter Maris, Hugh Potter,

Electroweak processes

with two- and three- nucleon

systems

J.Golak,

R.Skibiński, H. Witała, K.Topolnicki,

E.Epelbaum, H. Kamada, A. Nogga. L.E. MarcucciJAGIELLONIANUNIVERSITY

International Workshop on (e,e’p) Processes (EEP17)

Bled, 4 July, 2017

Outline

Introduction: elements of our formalism

Electromagnetic reactions with 2N and 3N systems

Muon capture on 2H, 3He and 3H

Selected reactions with (anti)neutrinos

Conclusions and outlook

EEP17, Bled, 4 July 2017

A very efficient momentum space framework to deal with nucleon-nucleon

scattering, nucleon-deuteron scattering and many electroweak processes has

been constructed and tested:

Phys. Rept. 274, 107 (1996); Phys. Rept. 415, 89 (2005);

Eur. Phys. J. A24, 31 (2005)

Limitations: nonrelativistic character and lack of the Coulomb force in the 3N

continuum

Calculations performed with semi-phenomenological 2N and 3N potentials:

Bonn B, AV18, Urbana IX, older chiral potentials from the Bonn/Bochum

group and recently with the improved chiral potentials from E. Epelbaum et al.

Introduction


Formalism


initial

lepton

final

lepton

initial

nuclear

system

final

nuclear

system

intermediate

boson

nuclear

current

operator

if jN *2 NNT fi Lαβ

Formalism (cont.)


Lαβ known analytically

if mimf jN from ab initio calculations in

momentum space

Dynamical ingredients (1): 2N and 3N Hamiltonians

4

)3(

4

)2(

4

)1(

4321

3

0

4321

3

0123121323

3

03

12

2

02

V

N

NN

N

N

N

VVVVVVH

VVVVHVVVVHH

VHH

used to generate nuclear bound and scattering states

contain 2N and 3N potentials

12212 jjjj N

Formalism (cont.)


Dynamical ingredients (2): nuclear EM and weak single-nucleon, 2N

and 3N current operators

)3(

)3(

123123

)2(

)2(

123132

)1(

)1(

123231

)3(

123

)2(

123

)1(

123123132231

1231323123213

123

jjj

j

N

jjjjjjjjj

jjjjjjjjj

jjjjjjjj

describe interactions of the electroweak probe with nuclear system

deuteron state with Ed < 0

Formalism (reactions with 2H)


dddN EH 2

dd jN '

dN

N

oadN jGtpjN 2

2

0122

)( 1

elastic channel

break-up

channel

12

2

0121212 ViEGtVt N

0,2

0)()(

2

m

pEEH N

free 2N propagator

Lippmann-Schwinger equation

N0

N

HiElimEG

2

0

2

0

1)(

f

N0

fHiE

ilim

3

)(

formal definition including

the channel state

Formalism (reactions with 3He and 3H)

two-body or three-body

break-up channel with final

scattering states


NjN 3' elastic or quasielastic channel with

initial and final bound states

bN EH3

3N bound state with Eb < 0

generated by the Faddeev equation

iNf jN 3

)(

(1) 3N force decomposed as

V4(i) is symmetric under the exchange of nucleons j and k, i≠j≠k≠i

(3) 2N off-shell t-matrix generated via LSE:

(2) free 3N propagator

)3(

4

)2(

4

)1(

44VVVV

Operators in 3N space:

1

3

0111 tGVVt N

(4) permutation operator: 23132312 PPPPP



N0

N

HiElimEG

3

0

3

0

1)(

UPGtGVPPGt

jPGtGVPGtU

NNN

i

NNN

3

01

3

0

)1(

4

3

01

3

01

3

0

)1(

4

3

01

112

1

)1(1112

1


),,( .. QEjUU mc

3N internal

energy

magnitude of the three

momentum transfer

Auxiliary equation for


UPGtUP

jPGtjPN

UPjPN

N

NN

i

N

NiNN

NdiNdNd

3

0133

3

01333

)1(1)1(1

)1(1


Quadratures

to obtain nuclear matrix elements for arbitrary exclusive kinematics !

Semi-exclusive and inclusive observables are calculated by suitable

integrations over the phase space domains.


and kk

span the x-z plane !

Θ* and Φ* define

the direction

of the initial 3He (3H) spin

Example: electron scattering

zkkQkk

kky ˆ|| ,ˆ


phase space factor

(kinematical factor) Mott cross section

General formula for the (exclusive) cross section

Only three assumptions:

(1) one-photon approximation

(2) nuclear current conservation

(3) final electron helicity is not measured

h the only dependence on

the initial electron helicity !

0ˆ||0 NQ

NNQN z

zQ


Electron scattering (cont.)

Kinematical factors vi



,2

11 yx NiNN

yx NiNN 2

11

Dynamical quantities (response functions) Ri



Unpolarized inclusive electron scattering



inc

TT

inc

LLMott

inc RvRv

Using the completeness relation for final nuclear states with E>0,

response functions Rinc can be calculated without explicit integrations

over the full two-body and three-body phase space !

i

fi

m

iNi

ifif

mm

f

inc

AB

AHEB

BAEEdfR

)(

)(

3

*)()(

,

A, B are the j0, j±1 components of the current operator



,112

3

2

1

3

01

*3

01

*

i

i

m

A

N

iB

N

i

m

AiBi

inc

AB

UPGBUPGAi

BAi

R

C

N

C UPG 13

0

C

NNN

iC

N

C

UPGVGtPGt

jGtU

11

)1(1

3

0

)1(

4

3

01

3

01

3

01

and

where



The two methods for inclusive response functions provide an excellent test

of numerics.

They have been applied to electron scattering, photodisintegration

reactions, muon capture, pion absorption, radiative pion capture, CC and

NC neutrino induced reactions.

The case without 3N force is particularly interesting, since

A

iAA UjU )1(

Input (not only !) from ChEFT

General strategy in the few-nucleon systems:

• use (consistent) dynamical ingredients (2N and 3N potentials,

electroweak current operators)

• solve the dynamical equations (Schrödinger equation, Lippmann-

Schwinger equation, Faddeev equations)

• give predictions for nuclear structure and reaction observables

• confront results of theoretical calculations with experimental data to

improve your input


Calculations employing the AV18 and Urbana IX potentials together with

related EM and weak current operators are still very important →

benchmark opportunities

Input from ChEFTWhile (new_data || better_theory) {

prepare best possible dynamical ingredients (2N and 3N

potentials, electroweak current operators);

solve the dynamical equations (Schrödinger equation,

Lippmann-Schwinger equation, Faddeev equations);

give predictions for nuclear structure and reaction observables;

confront results of theoretical calculations with experimental

data to improve your input;

}


Which chiral results should be used ?

Idaho group ? (R. Machleidt, F. Sammarruca,

Physica Scripta 91, 083007 (2016); D.R. Entem et al., arXiv:1703.05454 [nucl-th])

Bochum/Bonn group ? (E. Epelbaum et al., Eur. Phys. J. A51, 53 (2015); Phys.

Rev. Lett. 115, 122301 (2015))

LENPIC (Low Energy Nuclear Physics International Collaboration)

established to coordinate few-nucleon and many-nucleon calculations

http://www.lenpic.org

``to understand nuclear structure and reactions with chiral forces’’


Chiral NN potentials with non-local regularization fail to some extent in

the description of Nd scattering observables

Improved chiral NN potentials from E. Epelbaum et al. reveal very

welcome properties in 2N and 3N systems

Work on consistent 3N potentials up to N3LO is being finalized

Input from ChEFT (cont.)

LENPIC

Sven Binder, Kai Hebeler,

Joachim Langhammer, Robert Roth

Andreas Nogga

Pieter Maris, Hugh Potter, James Vary

Evgeny Epelbaum, Hermann Krebs

Hiroyuki Kamada

Richard J. Furnstahl,

Jacek Golak, Roman Skibiński,

Kacper Topolnicki, Henryk Witała

Ulf-G.Meißner

Veronique Bernard

Angelo Calci



Work on EM and weak current operators consistent with the improved

chiral forces not yet finished

Leading two-pion-exchange current operator

S. Kölling et al., Phys. Rev. C80, 045502 (2009)

Corrections to one-pion exchange and short-range contributions

S. Kölling et al., Phys. Rev. C84, 054008 (2011)

Nuclear axial current operators to fourth order

H. Krebs et al., Annals of Physics 378, 317 (2017)


Leading two-pion-exchange current operatorPhys. Rev. C80, 045502 (2009) Nonvanishing contributions

come from

2

3

1

3

10

2

9

2

8

2

7

2

6

2

5

2

4

2

3

2

10

1

9

1

8

1

7

1

6

1

5

1

4

1

3

1

,,,,,,,,,

,,,,,,,,

ffffffffff

ffffffff

Nonvanishing contributions come from

SSSSS

SSSSSS

fffff

ffffff

3

3

2

3

8

2

7

2

1

2

8

1

7

1

6

1

5

1

4

1

1

1

,,,,

,,,,,,



Spin operators

Isospin operators

…



Older chiral forces at N2LO with leading one-pion exchange and two-

pion exchange currents

np),(H2

SNC

SNC+1π

SNC+1π+2πAV18

[deg]..mc

p

Older chiral forces: EM reactions

np),(H2

D. Rozpędzik et al.


pdHe ),(3

Eγ= 12 MeV Eγ= 20.5 MeV Eγ= 50 MeV


Similar behaviour in 3N system !


dpeeHe )',(3

MeV/c 50 MeV, 20

MeV 80',30 MeV,100

Q

EE o

e

MeV/c 120 MeV, 30

MeV 70',88 MeV,100

Q

EE o

e


two-body break-up of 3He

very small sensitivity of unpolarized xs to current operator !


pnpeeHe )',(3

MeV 50 MeV, 20

MeV 80',30 MeV,100

Q

EE o

e

MeV 120 MeV, 30

MeV 70',88 MeV,100

Q

EE o

e


three-body break-up of 3He


pnpeeHe )',(3

MeV50 MeV,20

MeV80',30 MeV,100

Q

EE o

e

MeV120 MeV,30

MeV70',88 MeV,100

Q

EE o

e



Improved chiral forces: EM reactions

What can be done without explicit EM current operators ?

Use Siegert theorem to implicitly include many-nucleon contributions:

Replace a part of a electric multipole by a Coulomb multipole calculated from

the single nucleon charge density.

Calculate magnetic multipoles from the single nucleon current operator

Pretty simple in momentum space !


The total deuteron photodisintegration cross section


np),(H2


Improved chiral forces: EM reactionsnp),(H2

Eγ= 30 MeV

Eγ= 100 MeV

R=0.9 fm

various orders

truncation

errors

N4LO

various R-values



Eγ= 60.8 MeV

Eγ= 19.8 MeV

pn),(H2

R=0.9 fm

various orders

truncation

errors

N4LO

various R-values


OLD

R=1 fm

N4LO

The c.m. neutron-deuteron capture

cross section at Enlab = 9 MeV


H3 dn



MeV9

H

n

3

E

dn

MeV29

He

p

3

E

dp

MeV95

He

p

3

E

dp

R=0.9 fm

various orders

truncation

errors

N4LO

various R-values


Muon capture from the lowest K-shell

of the muonic atom studied with the single

nucleon weak current operator

≈ 10 % error in predictions

2

'

)'()()(

22

1

'

3

100

mZE

mm

mmm

emZ

rr

Z

Z

rmZ

K

negligible for Z=1,2 when

compared to the muon or

nucleon mass

reduced mass

Muon capture on 2H and 3He

p

n

Phys. Rev. C 90, 024001 (2014)


Methods developed for electromagnetic reactions can be easily

applied to following reactions

nnn

nnpe

nde

e

nnd

H

H

H

HH

3

3

3

33

Muon capture on 2H and 3He


quadruplet

states

doublet

states

Hyperfine structure in deuteron

Muon capture on 2H


F=1/2 PW F=1/2 full F=3/2 PW F=3/2 full

SNC 351.8 382.3 9.8 11.4

SNC+MEC 356.9 391.0 10.3 12.1

Doublet (F=1/2) and quadruplet (F=3/2) capture rates in s-1

calculated with the AV18 NN potential (neutron mass is used)

agrees with results of the Pisa group: L.E.

Marcucci et al., Phys. Rev. C83, 014002 (2011)

Muon capture on 2H

μ-+d → νμ+n+n


Chiral

order

R=0.8 fm R=0.9 fm R=1 fm R=1.1fm R=1.2 fm Γmax - Γmin

LO 396.0 397.4 398.4 398.9 399.2 3.3

NLO 384.2 385.8 387.2 388.6 389.8 5.7

N2LO 385.0 386.1 387.2 388.3 389.3 4.3

N3LO 386.8 386.4 385.2 384.3 383.2 3.6

N4LO 385.5 386.1 386.3 385.6 384.6 1.7

AV18 382.3

Doublet capture rates (F=½) in s-1 calculated with the improved chiral

potentials and the single nucleon current operator with RC

very weak dependence on the regulator parameter R

μ-+d → νμ+n+n

Muon capture on 2H


Results from the MuSun experiment will be very important !

http://muon.npl.washington.edu/exp/MuSun/

our present

prediction without

2N currents

Muon capture on 2H

expected

error size

in MuSun


Chiral

order


LO 1610 1618 1610 1594 1572 46

NLO 1330 1357 1381 1405 1427 97

N2LO 1337 1356 1376 1395 1415 78

N3LO 1314 1304 1289 1278 1266 48

N4LO 1296 1307 1308 1299 1285 23

AV18 1353


Total capture rates in s-1 calculated with the improved chiral potentials and

the single nucleon current operator with RC

μ-+3He → νμ+3H

Muon capture on 3He: triton channel


No relativity in the kinematics !

nde

H3 nnpe

H3

kinematically allowed regions

Muon capture on 3He: breakup channels


← best predictions !


Total capture rates


best

predictions !

Muon capture on 3H: 3-body breakup channel



R=0.9 fm

various orders

truncation

errors

N4LO

various R-values


Chiral

order


LO 262 282 312 350 392 130

NLO 536 525 515 504 492 44

N2LO 547 539 529 518 507 40

N3LO 584 586 592 596 603 19

N4LO 590 584 583 587 595 12

AV18 604




μ-+3He → νμ+n+d

Muon capture on 3He: two-body break-up


Chiral

order


LO 95 99 105 113 120 26

NLO 159 157 154 151 148 11

N2LO 161 159 157 154 151 10

N3LO 169 169 171 172 175 6

N4LO 170 169 169 170 173 4

AV18 169




μ-+3He → νμ+p+n+n

Muon capture on 3He: three-body break-up


Reactions with (anti)neutrinos


We follow papers by:

S. Nakamura et al., Phys. Rev. C 63, 034617 (2001)

Doron Gazit and Nir Barnea, Phys. Rev. C 70, 048801 (2004)

E. O'Connor et al., Phys. Rev. C 75, 055803 (2007)

Doron Gazit and Nir Barnea, Phys. Rev. Lett. 98, 192501 (2007)

G. Shen et al., Phys. Rev. C 86, 035503 (2012)

A. Baroni and R. Schiavilla, arXiv: 1704.02002 [nucl-th]

Reactions with (anti)neutrinos


ppldl

CC

CF NG

skusku2

cos),(1)','( 5

EMN

q

eskusku

2

2

),()','(

CC

CF NG

skvskv2

cos)','(1),( 5

nnldl

Prescription is simple: If you start with , , replacenpede

Reactions with (anti)neutrinos (cont.)


npd ll

NC

F NG

skusku2

),(1)','( 5

EMN

q

eskusku

2

2

),()','(

NC

F NG

skvskv2

)','(1),( 5

npd ll

And for the reactions with the neutral current, construct the

corresponding nuclear current operator and replace



Quasielastic antineutrino scattering on 3He

results based on AV18

with the single nucleon

current including

relativistic corrections

HeHee

33

HHe 33

d

ddqel

tot

2

sin2



Elastic (anti)neutrino scattering on 3He and 3H

results based on AV18

with the single nucleon

current including

relativistic corrections

.33 etcHeHe ee

d

ddel

tot

2

sin2

Conclusions and outlook

• Very robust momentum space framework to deal with many electroweak

processes has been constructed and tested (limitations)

• New input: improved chiral 2N and 3N potentials (even 4N potentials) from the

Bochum/Bonn group are available

• Substantial improvement in description of many observables

• LENPIC (Low Energy Nuclear Physics International Collaboration) to coordinate

few-nucleon and many-nucleon calculations

• Consistent electroweak current operators are still under construction

• New chiral potentials from the Idaho group and related current are also on the

market

• New precise data are badly needed - MESA (Mainz Energy-Recovering

Superconducting Accelerator) will be of great importance !


Expected MESA parameters

E= 150 MeV

E’ > 20 MeV

ϴe > 10 deg

ideal to study few-nucleon

dynamics within the

nonrelativistic framework

with the input from ChEFT !E’ [MeV]

ϴe

[de

g]

Conclusions and outlook (cont.)


magnitude of three-momentum

transfer vs. internal energy of

3N system


But before MESA starts

• Energy ranges and phase-space regions best suited to study the nuclear

current operator and three-nucleon force effects should be identified for

considered reaction channels

• Achievable accuracy of theoretical predictions for various observables

should be estimated

• Work on consistent chiral potentials and EM current operators has to be

finalized


Various observables in deuteron electrodisintegration (polarization might be

crucial !)

Two-body break-up of 3He

1. unpolarized proton angular distributions (for a wide range of angles)

2. 3He analyzing power

3. Spin-dependent helicity asymmetries

Three-body break-up of 3He

1. Semi-exclusive cross sections (proton and neutron) at various emission

angles with respect to the momentum transfer

2. 3He analyzing power

3. Spin-dependent helicity asymmetries

What should be measured ?



Thank

you !


Documents

Electroweak processes with two- and three- nucleon systemscrex.fmf.uni-lj.si/eep17/Golak_Bled_2017_07_04.pdf · Rev. Lett. 115, 122301 (2015)) LENPIC ... Pieter Maris, Hugh Potter,