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
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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
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Formalism
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initial
lepton
final
lepton
initial
nuclear
system
final
nuclear
system
intermediate
boson
nuclear
current
operator
if jN *2 NNT fi Lαβ
Formalism (cont.)
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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.)
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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)
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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
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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
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Formalism (reactions with 3He and 3H)
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
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),,( .. QEjUU mc
3N internal
energy
magnitude of the three
momentum transfer
Auxiliary equation for
Formalism (reactions with 3He and 3H)
UPGtUP
jPGtjPN
UPjPN
N
NN
i
N
NiNN
NdiNdNd
3
0133
3
01333
)1(1)1(1
)1(1
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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.
Formalism (reactions with 3He and 3H)
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
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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
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Electron scattering (cont.)
Kinematical factors vi
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Electron scattering (cont.)
,2
11 yx NiNN
yx NiNN 2
11
Dynamical quantities (response functions) Ri
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Electron scattering (cont.)
Unpolarized inclusive electron scattering
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Electron scattering (cont.)
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
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Electron scattering (cont.)
,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
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Electron scattering (cont.)
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
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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;
}
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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’’
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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
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Input from ChEFT (cont.)
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)
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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
,,,,
,,,,,,
Input from ChEFT (cont.)
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Spin operators
Isospin operators
…
Input from ChEFT (cont.)
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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.
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pdHe ),(3
Eγ= 12 MeV Eγ= 20.5 MeV Eγ= 50 MeV
Older chiral forces: EM reactions
Similar behaviour in 3N system !
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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
Older chiral forces: EM reactions
two-body break-up of 3He
very small sensitivity of unpolarized xs to current operator !
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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
Older chiral forces: EM reactions
three-body break-up of 3He
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pnpeeHe )',(3
MeV50 MeV,20
MeV80',30 MeV,100
Q
EE o
e
MeV120 MeV,30
MeV70',88 MeV,100
Q
EE o
e
Older chiral forces: EM reactions
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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 !
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The total deuteron photodisintegration cross section
Improved chiral forces: EM reactions
np),(H2
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Improved chiral forces: EM reactionsnp),(H2
Eγ= 30 MeV
Eγ= 100 MeV
R=0.9 fm
various orders
truncation
errors
N4LO
various R-values
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Improved chiral forces: EM reactions
Eγ= 60.8 MeV
Eγ= 19.8 MeV
pn),(H2
R=0.9 fm
various orders
truncation
errors
N4LO
various R-values
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OLD
R=1 fm
N4LO
The c.m. neutron-deuteron capture
cross section at Enlab = 9 MeV
Improved chiral forces: EM reactions
H3 dn
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Improved chiral forces: EM reactions
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
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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)
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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
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quadruplet
states
doublet
states
Hyperfine structure in deuteron
Muon capture on 2H
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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
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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
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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
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Chiral
order
R=0.8 fm R=0.9 fm R=1 fm R=1.1fm R=1.2 fm Γmax - Γmin
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
very weak dependence on the regulator parameter R
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
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No relativity in the kinematics !
nde
H3 nnpe
H3
kinematically allowed regions
Muon capture on 3He: breakup channels
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← best predictions !
Muon capture on 3He: breakup channels
Total capture rates
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best
predictions !
Muon capture on 3H: 3-body breakup channel
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Muon capture on 3He: breakup channels
R=0.9 fm
various orders
truncation
errors
N4LO
various R-values
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Chiral
order
R=0.8 fm R=0.9 fm R=1 fm R=1.1fm R=1.2 fm Γmax - Γmin
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
very weak dependence on the regulator parameter R
Total capture rates in s-1 calculated with the improved chiral potentials and
the single nucleon current operator with RC
μ-+3He → νμ+n+d
Muon capture on 3He: two-body break-up
EEP17, Bled, 4 July 2017
Chiral
order
R=0.8 fm R=0.9 fm R=1 fm R=1.1fm R=1.2 fm Γmax - Γmin
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
very weak dependence on the regulator parameter R
Total capture rates in s-1 calculated with the improved chiral potentials and
the single nucleon current operator with RC
μ-+3He → νμ+p+n+n
Muon capture on 3He: three-body break-up
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Reactions with (anti)neutrinos
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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
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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.)
EEP17, Bled, 4 July 2017
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
Reactions with (anti)neutrinos (cont.)
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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
Reactions with (anti)neutrinos (cont.)
EEP17, Bled, 4 July 2017
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 !
EEP17, Bled, 4 July 2017
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.)
EEP17, Bled, 4 July 2017
magnitude of three-momentum
transfer vs. internal energy of
3N system
Conclusions and outlook (cont.)
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
EEP17, Bled, 4 July 2017
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 ?
Conclusions and outlook (cont.)
EEP17, Bled, 4 July 2017
Thank
you !
EEP17, Bled, 4 July 2017