Download A path to lepton-nucleus reactions from first principles

Canada’s National Laboratory for Particle and Nuclear Physics
Laboratoire national canadien pour la recherche en physique nucléaire
et en physique des particules
A path to lepton-nucleus
reactions from first principles
The Atomic Nucleus
Sonia Bacca | Theory Department | TRIUMF
In collaboration with:
Nir Barnea, Gaute Hagen, Oscar Javier Hernandez, Mirko Miorelli,
Giuseppina Orlandini, Thomas Papenbrock, Tianrui Xu, et al.
Electroweak Reactions
TRIUMF Double Beta Decay Workshop, May 11-13, 2016
Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada
Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada
Friday, 13 May, 16
Nuclear Chart
From First Principles
• Describe the nucleus as a system of interacting protons and neutrons and solve the
(non-relativistic) many-body problem without approximations or only with controllable
approximations.
Aim:
1) Develop a strong predictive theory in the framework of light nuclei
where one can test nuclear theory ingredients
2) Then extend it towards heavier systems and/or to observables where
experiments are hard
• To provide useful numbers for low-energy nuclear physics:
- Double beta decay processes
- interaction of neutrinos with nucleonic matter
Lowenergy
• To provide interpretations for particle physics experiments:
- neutrino-interactions with nuclei
Higher-energy
Sonia Bacca
Friday, 13 May, 16
2
Ab-initio Theory Tools
Nuclear Currents
Nuclear Forces
s1
r2
Traditional Nuclear Physics
s2
AV18+UIX, ..., J2
...
µ
J =
sA
r1
m
rA
H|
i
= Ei |
N
+
µ
JN N
m
+
N
+ ...
N
N
exchange currents
i
H = T + VN N + V3N + ...
Effective Field Theory
N2LO, N3LO ...
subnuclear d.o.f.
µ consistent with
J
r·J =
High precision two-nucleon potentials:
well constraint on NN phase shifts
Three nucleon forces:
less known, constraint on A>2 observables
V
i[V, ⇢]
Cross
section
D
E2
ˆ
/
f |Jµ | 0
Review Paper:
Electromagnetic Reactions on Light Nuclei
S. Bacca and S. Pastore
J. Phys. G: Nucl. Part. Phys. 41 123002 (2014).
Sonia Bacca
Friday, 13 May, 16
µ
JN
3
The Continuum Problem
R(q, !) =
/ R(q, !)
f
ˆ
f |Jµ (q)| 0
E
2
(Ef
E0
!)
“
“
ground
state
Z
D
X
bound
excited state
Counts
Cross
Section
400
2-body break-up
3-body break-up
...
Excitation Energy
A-body break-up
continuum
300
giant
resonances
200
narrow
resonances
100
0
0
5
10
20
25
ω [MeV]
Excitation
Energy
15
30
35
Sonia Bacca
Friday, 13 May, 16
40
4
Lorentz Integral Transform
Efros et al., Nucl.Part.Phys. 34 (2007) R459
Response in the continuum
R(⇤) =
⇧
⌅ ⇥
ˆµ ⇥0
⇥f JÔ
f
⇤
2
R(⇥)
d⇥
(⇥
)2 +
L( , ) =
(H
E0
(Ef
2
E0
⇤)
= ˜| ˜⇥
ˆµ | ⇥0 ⇥
˜ = JÔ
+ i ) | ⇥⇥
|˜
• Due to imaginary part the ˜solution is unique
• Since the r.h.s. is finite, then | has bound state asymptotic behaviour
You can use any good bound state method!
L( , )
inversion
R(!)
with the exact final state interaction
Sonia Bacca
Friday, 13 May, 16
e.g. Hyperspherical Harmonics,
No Core Shell Model,
Coupled Cluster Theory
5
Medium-mass nuclei
Develop new many-body methods that can extend the frontiers to heavier and neutron nuclei
• CC is optimal for closed shell nuclei (±1, ±2)
Coupled Cluster Theory
Uses particle coordinates
CC future aims
T
|⇥(⌥
r
,
⌥
r
,
...,
⌥
r
)
=
e
| 0(⌥r1 , ⌥r2 , ..., ⌥rA )
1
2
A
0
reference SD with
any sp states
T =
CC theory now
T(A)
T1 =
cluster expansion
tai a†a ai
ia
1 X ab † †
T2 =
tij aa ab aj ai
4
...
ij,ab
T1
T2
T3
CCSD
CCSDT
Sonia Bacca
Friday, 13 May, 16
48Ca
from first principles
Hagen et al., Nature Physics 12, 186-190 (2016)
3.6
Rp [fm]
3.5
3.4
3.3
3.2
3.1
0.15
0.18
0.21
3.3
Rskin [fm]
3.4
3.5
3.6
Rn [fm]
2.0
2.4
↵D [fm3]
Ab initio with three nucleon forces from chiral EFT
Density Functional Theory
7
Friday, 13 May, 16
LIT with Coupled Cluster Theory
New theoretical method aimed at extending ab-initio calculations towards medium-mass
S.B. et al., PRL 111, 122502 (2013)
z ⇤ )| ˜ i = J µ |
(H
with
z = E0 +
L( , ) =
D
˜|˜
0i
+i
E
(H̄
H̄ = e
T
HeT
¯ =e
⇥
T
⇥eT
L( , ) =
with
Formulation based on the solution of an
¯
z ⇤ )| ˜ R (z ⇤ )i = ⇥|
D
E
˜ L| ˜ R =
h
1 n
¯ † | ˜ R (z ⇤ )i
= h0̄L |⇥
2⇡
| ˜ R (z ⇤ )i = R̂(z ⇤ )|
Equation of Motion with source
Present implementation in the CCSD scheme
0i
| ˜ R (z)i
io
0i
No approximations done so far!
T = T1 + T2
R̂ = R̂0 + R̂1 + R̂2
Sonia Bacca
Friday, 13 May, 16
8
LIT with Coupled Cluster Theory
New theoretical method aimed at extending ab-initio calculations towards medium mass
Validation for 4He
Comparison of CCSD with exact hyperspherical harmonics (EIHH) with NN forces at N3LO
EIHH
EIHH
6
CCSD
-2
-3
L [fm MeV 10 ]
8
4
2
Γ=10 MeV
0.5
0
-20
0
20
40
60
σ [MeV]
80
100
The comparison with exact theory is very good!
120
R(ω) [mb/MeV]
2
He
EIHH
EIHH
CCSD
0.4
0.3
0.2
0.1
0
Sonia Bacca
Friday, 13 May, 16
4
20
40
60
80
100
ω [MeV]
9
Extension to heavier nuclei
With Mirko Miorelli, PhD student at UBC
PRC 90, 064619 (2014)
100
40
Ca
Ahrens et al.
σγ(ω) [mb]
80
LIT-CCSD
NN (N3LO)
60
40
20
0
0
20
40
60
80
100
ω[MeV]
Sonia Bacca
Friday, 13 May, 16
10
Neutrino-nucleus cross section
Neutrino long baseline experiments (T2K, Miniboon, LBNE, etc.) require theoretical
input to simulate the interaction of neutrinos with the detector material (12C, 16O, 40Ar, ...)
So far very simple models (RMF) are used and ab-initio calculations with reliable
error estimates are sought for
0
e, ⌫
, Z 0, W ±
A
0
e , ⌫ or `
Neutrino scattering ⬄ electron scattering
(!, q)
X
• Quasi Elastic part can be
studied with the LIT method
and coupled-cluster theory
• Need to test theory on
electron scattering data first
probe
[MeV]
Ultimate goal: addressing inclusive neutrino -16O interactions for T2K
Sonia Bacca
Friday, 13 May, 16
11
Inelastic e-Scattering
A(e,e’)X
Virtual Photon
µ
Pf
µ
k’
µ
µ
µ
q = k − k’
qµ = (ω , q)
µ
k
( , q)
can vary independently
µ
P0
Inclusive cross section A(e,e’)X
⇤
2
4
2
⇥
d ⇥
Q
Q
2
= ⇥M
R
(⇤,
q)
+
+
tan
L
4
d d⇤
q
2q2
2
2
Q
=
with
and
M
qµ2 = q2
2
and
scattering angle
Mott cross section
Sonia Bacca
Friday, 13 May, 16
⌅
RT (⇤, q)
12
Inelastic e-Scattering
A(e,e’)X
Virtual Photon
µ
Pf
µ
k’
µ
µ
µ
q = k − k’
qµ = (ω , q)
µ
k
( , q)
can vary independently
µ
P0
Inclusive cross section A(e,e’)X
2
⇤
4
⇥
2
d ⇥
Q
Q
2
= ⇥M
R
(⇤,
q)
+
+
tan
L
4
d d⇤
q
2q2
2
RL (⇤, q) =
RT (⇥, q) =
⌅
⇤
f
⌅
⇤
f
Friday, 13 May, 16
|⇥
|⇥
f | ⇥(q) |
f | JT (q) |
2
0 ⇤|
Ef
2
0 ⇤|
Sonia Bacca
⌅
RT (⇤, q)
E0
q
⇤+
2M
E0
q
⇥+
2M
2
2
Ef
⇥
⇥
13
4He(e,e’)X
Inelastic e-Scattering
Calculation of RL ( , q) with the LIT and hyperspherical harmonics
Medium-q kinematics
S.B. et al., PRL 102, 162501 (2009)
20
Bates
Saclay
-1
q=400 MeV/c
-3
10
RL [10 MeV ]
8
6
4
Bates
Saclay
5
-1
q=300 MeV/c
-3
-3
-1
RL [10 MeV ]
RL [10 MeV ]
Bates
Saclay
15
6
10
4
q=500 MeV/c
3
2
5
2
0
40
60
80
100 120
ω [MeV]
140
160
180
0
1
60
80 100 120 140 160 180 200 220 240
ω [MeV]
0
50
100
150
200
250
ω [MeV]
300
350
e’
AV18+UIX:
PWIA
e
γ
4
He
*
p
3
Full FSI:
H
Strong effect of FSI which can be accounted rigorously with the LIT
New project: compare ab-initio methods to account for FSI with spectral function approach
(with Noemi Rocco)
Sonia Bacca
Friday, 13 May, 16
14
Coulomb Sum Rule with CC
With Tianrui Xu, B.Sci at UBC
1
SL (q) =
Z
RL (⇤, q) =
Z
1
!th
⌅
⇤
f
|⇥
d!
RL (!, q)
Total inelastic strength
p2
GE (Q2 )
f | ⇥(q) |
2
0 ⇤|
q
⇤+
2M
2
Ef
E0
⇥
MEC are not
important for
q<2 fm-1
Expand charge operator in multipoles and calculate it for different q
We will use a two-body force derived from chiral EFT (Entem and Machleidt N3LO)
Sonia Bacca
Friday, 13 May, 16
15
Coulomb Sum Rule in
4He
Convergence in terms of multipoles
0.6
Isovector CSR
0.5
0.4
0.3
0.2
0
+1
+2
+3
+4
+5
+6
+7
+8
4
He
Benchmarking with HH
1.0
0.1
100
200
300
q [MeV/c]
400
500
Isovector CSR
0.0
0
0.8
4
He
0.6
0.4
0.2
0.0
0
Sonia Bacca
Friday, 13 May, 16
CCSD, Nmax=16
HH
100
200
300
q [MeV/c]
400
500
16
Coulomb Sum Rule in
4He
Convergence in terms of multipoles
0.6
Isovector CSR
0.5
0.4
0.3
0.2
0
+1
+2
+3
+4
+5
+6
+7
+8
4
He
T.Xu et al., EPJ Web of Conferences 113, 04016 (2016)
Comparison with experiment
1.0
0.1
0.8
100
200
300
q [MeV/c]
400
He
500
SL(q)
0.0
0
4
0.6
0.4
r
P
im
l
e
LIT-CCSD
Buki et. al, w tail
World Data, w tail
0.2
0.0
0
Sonia Bacca
Friday, 13 May, 16
y
r
a
in
100
200
300
q [MeV/c]
400
500
17
Coulomb Sum Rule in
16O
Preliminary
16
O
Next address transverse and weak response in 16O with
two-body
currents
Sonia Bacca
Friday, 13 May, 16
18
Two-body Currents
See talks by S.Pastore, A.Schwenk
Two-body currents (or MEC) play a crucial role in double beta decay and lepton-scattering
Example of the Transverse sum rule
Lovato et al., PRL 111, 092501 (2013)
2.0 (b)
12
C
1- and 2-body current
ST (q)
1.5
1.0
1-body current
AV18+IL7
EC
/ AV18+IL7
3
0.5
0.0
0
IA
j / AV18+IL7
IA+MEC
j
/ AV18+IL7
1
2
-1
3
q (fm )
Sonia Bacca
Friday, 13 May, 16
19
Two-body Currents
With Oscar Javier Hernandez, PhD student at UBC
At NLO you get about 70% of the
total effect of 2-body currents
Experiment
NLO current
NLO
LO (1-body)
500
600
Total 2-body currents
one pion exchange diagrams
Seagull
700
Pion-in-Flight
Adapted from L. Girlanda, et al.
EPJ Web of Conferences 3, 01004 (2010)
Sonia Bacca
Friday, 13 May, 16
20
Two-body Currents
With Oscar Javier Hernandez, PhD student at UBC
• Performed the multipole expansion of the NLO current at finite q
need for neutrino scattering
• Check theZ low-momentum case of magnetic dipole for the deuterium susceptibility
M
2↵
RM (!)
=
d!
3
!
Deuteron
Our work
Friar*
Our work
Our work
Friar*
Potential
AV18
AV18
AV18
AV18
AV18
Current
LO
LO
NLO
NLO+∆
NLO+∆
Susceptibility [fm3]
0.0678
0.0678
0.0756
0.0776
0.0774
*J. Friar and G.L. Payne, Phys. Rev. C 56 619 (1997)
Sonia Bacca
Friday, 13 May, 16
21
Two-body Currents
With Oscar Javier Hernandez, PhD student at UBC
• Performed the multipole expansion of the NLO current at finite q
need for neutrino scattering
• Check theZ low-momentum case of magnetic dipole for the deuterium susceptibility
M
2
=
3
RM (!)
d!
!
Deuteron
Our work
Friar*
Our work
Our work
Friar*
Potential
AV18
AV18
AV18
AV18
AV18
Current
LO
LO
NLO
NLO+∆
NLO+∆
*J. Friar and G.L. Payne, Phys. Rev. C 56 619 (1997)
Susceptibility [fm3]
0.0678
0.0678
11% enhancement
0.0756
14% enhancement
0.0776
0.0774
}
}
• To do: apply to heavier nuclei using coupled-cluster theory
Sonia Bacca
Friday, 13 May, 16
22
Magnetic Sum Rule
SM =
Z
d! RM (!)
Validation for 4He
Comparison of coupled cluster with exact hyperspherical harmonics (EIHH) with NN forces at N3LO
1-body operator Preliminary
Method
in µN2
EIHH
1.69
CCSD
1.62
CCSD+(T)
1.74
CCSD+(T)
1.74
Sonia Bacca
Friday, 13 May, 16
23
Outlook
LIT
+
w.f.
from CC
+
Jµ
some of the technology developed for this program can be useful to
• Possibly
double-beta decay program:
➡ Gamow-Teller strength functions
➡ Role of subnuclear degrees of freedom in quenching factors
Thank you for your attention!
Sonia Bacca
Friday, 13 May, 16
24