Modeling two-nucleon knock-out in neutrino-nucleus scattering

Modeling two-nucleon knock-out in neutrino-nucleus scattering

Kajetan Niewczas

Kajetan Niewczas 2p2h Ghent 06.11.2020 1 / 44

Neutrino oscillation experiments

P2f(νµ→ ν^µ) = 1−sin²(2θ) sin²

∆m²L 4E_ν



0.00 0.02 0.04 0.06 0.08 0.10 0.12

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

(expected)

normed νµ fux

E_ν [GeV]

E_ν^rec= 2(Mn− E^B)E_µ− (EB²− 2MⁿEB+m²_µ) 2[M_n− E^B−E_µ+|~k^µ| cos θ^µ]

Kajetan Niewczas 2p2h Ghent 06.11.2020 2 / 44

Detected rate of ν_α events

R_να ∼ Φνµ (E_ν)× Pνµ→να({Θ},E_ν)× σνα (E_ν)× det.

Eventrate Incomingflux Oscillationprobability Cross section Efficiency

Knowledge of neutrino-nucleuscross sections:

→ allows to reconstructneutrino energy from the detectedfinal states,

→ is the crucial uncertainty inoscillation analyses,

but...

→ is an advanced computational problem,

→ current precision is not exceeding 20%,

→ constraints from ND are not enough.

K. Abe et al., Phys.Rev.Lett. 121 (2018) 171802 (edited)

Kajetan Niewczas 2p2h Ghent 06.11.2020 3 / 44

Nuclear response

⌫_µ0.8 < cos ✓_µ< 0.9

Elastic

N^⇤ DIS

Coherent

GR

QE

N^⇤ DIS

2N

Nucleon

Nucleus

Nucleonresponse Nucleusresponse !

T. Van Cuyck

Kajetan Niewczas 2p2h Ghent 06.11.2020 4 / 44

Dimensionality of the problem

k_µ

k_ν

(ω,q)

p_n p_p

any binary scattering with on-shell particles

4 four-vectors =16 variables

- 4 : on-shell relations - 4 : 4-mom. conservation - 3 : nucleon rest frame - 2 : neutrino alongˆz 3 independent variables

→ we can fix incoming energy (E^ν)

→ the cross section is rotationally invariant (φ^µ)

→ the final formula is 1-dimensional, e.g. dσ/dq²

Kajetan Niewczas 2p2h Ghent 06.11.2020 5 / 44

Dimensionality of the problem

k_µ

k_ν

(ω,q)

p_n p_p p_i

scatterings including an off-shell target

3 independent variables

+ 3 : nucleus rest frame + 1 : off-shell nucleon

7 independent variables

+ 3 : every on-shell particle

→ we can fix incoming energy (E^ν)

→ the cross section is rotationally invariant (φ^µ)

→ the final formula is at least 5-dimensional

Kajetan Niewczas 2p2h Ghent 06.11.2020 6 / 44

ComputingνA cross section

Monte Carlo generator

→ ^{generateevents}

→ ^coverwhole phase space

→ ^{useful but}approximated e.g.NuWro

◊^ËvCCWCC+ vCLWCL+ vLLWLL+ vTWT + vT TWT T+ vT CWT C

+vT LWT L+ h(vT^ÕWT^Õ+ vT C^ÕWT C^Õ+ vT L^ÕWT L^Õ)^È. (4.23)

d /d✏e0d⌦e0dTad⌦ad⌦b(10 ³⁶cm²/MeV²)

MEC pn + pp (initial pairs) SRC pn + pp (initial pairs)

90 180

270 360

✓a( ) 0

90 180 270 360

✓b( )

90 180

270 360

✓a( ) 0

90 180 270 360

✓b( ) 0 0.2 0.4 0.6 0.8 1

90 180

270 360

✓a( ) 0

90 180 270 360

✓b( ) 0 5 10 15 20 25 30 35 40

Figure 4.4: The ¹²C(e, e^ÕNaNb) cross section (Na = p, Nb = p^Õ, n) at ‘e = 1200 MeV,

‘e^Õ = 900 MeV, ◊e^Õ= 16^¶and T_p= 50 MeV for in-plane kinematics. Left with SRCs, right with MECs, the bottom plot shows the (◊a, ◊b) regions with P12<300 MeV/c.

Figure 4.5: The¹²C(‹µ, µ^≠N_aN_b) cross section (Na= p, Nb= p^Õ, n) at ‘‹µ= 750 MeV,

‘µ= 550 MeV, ◊µ= 15^¶and Tp= 50 MeV for in-plane kinematics. Left with SRCs, right with MECs, the bottom plot shows the (◊a, ◊b) regions with P12<300 MeV/c.

60

◊^ËvCCWCC+ vCLWCL+ vLLWLL+ vTWT+ vT TWT T+ vT CWT C

+vT LWT L+ h(vT^ÕWT^Õ+ vT C^ÕWT C^Õ+ vT L^ÕWT L^Õ)^È. (4.23)

d /d✏e0d⌦e0dTad⌦ad⌦b(10 ³⁶cm²/MeV²)

MEC pn + pp (initial pairs) SRC pn + pp (initial pairs)

90 180

270 360

✓a( ) 0

90 180 270 360

✓b( )

90 180

270 360

✓a( ) 0

90 180 270 360

✓b( ) 0 0.2 0.4 0.6 0.8 1

90 180

270 360

✓a( ) 0

90 180 270 360

✓b( ) 0 5 10 15 20 25 30 35 40

Figure 4.4: The¹²C(e, e^ÕNaNb) cross section (Na= p, Nb= p^Õ, n) at ‘e = 1200 MeV,

‘e^Õ= 900 MeV, ◊e^Õ= 16^¶and Tp= 50 MeV for in-plane kinematics. Left with SRCs, right with MECs, the bottom plot shows the (◊a, ◊b) regions with P12<300 MeV/c.

Figure 4.5: The¹²C(‹µ, µ^≠NaNb) cross section (Na= p, Nb= p^Õ, n) at ‘‹µ= 750 MeV,

‘µ= 550 MeV, ◊µ= 15^¶and Tp= 50 MeV for in-plane kinematics. Left with SRCs, right with MECs, the bottom plot shows the (◊a, ◊b) regions with P12<300 MeV/c.

60

Detailed calculation

→ ^computecross sections

→ fixed kinematics

→ precise butexpensive e.g.Ghent group

Kajetan Niewczas 2p2h Ghent 06.11.2020 7 / 44

Contents

• History of 2p2h modeling

• Theoretical formalism of the Ghent group

◦ ^Kinematics

◦ Nucleon wave functions

◦ Short-range correlations

◦ Meson-exchange currents

• Experimental prospects

T. Van Cuyck, N. Jachowicz, R. González-Jiménez et al., Phys.Rev.C 95 (2017) 054611 T. Van Cuyck, N. Jachowicz, R. González-Jiménez et al., Phys.Rev.C 94 (2016) 024611

Kajetan Niewczas 2p2h Ghent 06.11.2020 8 / 44

The MiniBooNE puzzle

An attempt to make apure CCQE measurement...

Graphic from S. Dolan 8

What do we actually measure?

Many modes contribute to any

measurement Integrated over broad ω region Difficult to tune theory models!

S. Dolan

Kajetan Niewczas 2p2h Ghent 06.11.2020 9 / 44

The MiniBooNE puzzle

An attempt to make apure CCQE measurement...

→suffered from hugemodel dependencies

L. Alvarez-Ruso, Nucl.Phys.B Proc.Suppl. 229-232 (2012) 167-173 (Neutrino 2010)

Kajetan Niewczas 2p2h Ghent 06.11.2020 10 / 44

The theoretical framework:

language of response functions

Kajetan Niewczas 2p2h Ghent 06.11.2020 11 / 44

Cross section formula

CCνA scattering

e(‘e, ke)

A(EA, pA) e^Õ(‘eÕ, keÕ)

“^ú(Ê, q)

ig_µ‹

Q2

B(EB, pB)

ieJlep^µ(q) ≠ ieJnucl^‹ (q)

(a) eA scattering.

‹(‘‹, k‹)

A(EA, pA) µ(‘µ, kµ)

W^±(Ê, q)

igµ‹

M_W²

B(EB, pB)

≠ i_2Ô2^q Jlep^µ(q) ≠ i_2Ô2^q cos ◊cJnucl^‹ (q)

(b) CC ‹A scattering.

Figure 2.2: Feynman diagrams for lepton-nucleon scattering processes.

The lepton and hadron currents are defined as

Jµ^lep(q) © u(kf, sf)J^‚µ^lepu(ki, si) = u(kf, sf) “µ(1 + h“⁵) u(ki, si), (2.16) Jµ^nuc(q) © È f|J^‚_µ^nuc(q) | iÍ, (2.17) where we introduced the Dirac spinors u(ki, si) and u(kf, sf) for the incoming and scattered lepton. The | iÍ and | fÍ refer to the initial and final nuclear states and the operatorsJ^‚_µ^lepandJ^‚_µ^nuc(q) are the lepton and nuclear current operators in momentum space. The structure of the nuclear current operator will be discussed throughout this work.

The lepton current operator is known exactly from field theory. In the lepton current, h = 0 for (unpolarized) electron scattering and h equals ≠(+) for neutrino (anti- neutrino) interactions, reflecting the V ≠ A structure of the weak interaction. By convention, the factor 1/2 of the spin-projection operator (1 + h“⁵)/2 is absorbed in the weak coupling constant.

For the nuclear current, we rely on the rules corresponding to the Feynman rules but for nuclear physics [7–9], as bound states cannot be described in field theory.

These nuclear current matrix elements are the building blocks of the nuclear responses and will be at the center of our modeling efforts. They contain all the dynamical information of the electromagnetic or electroweak interaction.

23

EM eA scattering

e(‘e, ke)

A(EA, p_A) e^Õ(‘e^Õ, ke^Õ)

“^ú(Ê, q)

igµ‹

Q²

B(EB, pB)

ieJlep^µ(q) ≠ ieJnucl^‹ (q)

(a) eA scattering.

‹(‘‹, k‹)

A(EA, pA) µ(‘µ, kµ)

W^±(Ê, q)

igµ‹

M_W²

B(EB, p_B)

≠ i_2Ô2^q Jlep^µ(q) ≠ i_2Ô2^q cos ◊cJnucl^‹ (q)

(b) CC ‹A scattering.

Figure 2.2: Feynman diagrams for lepton-nucleon scattering processes.

The lepton and hadron currents are defined as

Jµ^lep(q) © u(kf, sf)J^‚µ^lepu(ki, si) = u(kf, sf) “µ(1 + h“⁵) u(ki, si), (2.16)

Jµ^nuc(q) © È f|J^‚_µ^nuc(q) | iÍ, (2.17)

where we introduced the Dirac spinors u(ki, si) and u(kf, sf) for the incoming and scattered lepton. The | iÍ and | fÍ refer to the initial and final nuclear states and the operatorsJ^‚µ^lepandJ^‚µ^nuc(q) are the lepton and nuclear current operators in momentum space. The structure of the nuclear current operator will be discussed throughout this work.

The lepton current operator is known exactly from field theory. In the lepton current, h = 0 for (unpolarized) electron scattering and h equals ≠(+) for neutrino (anti- neutrino) interactions, reflecting the V ≠ A structure of the weak interaction. By convention, the factor 1/2 of the spin-projection operator (1 + h“⁵)/2 is absorbed in the weak coupling constant.

For the nuclear current, we rely on the rules corresponding to the Feynman rules but for nuclear physics [7–9], as bound states cannot be described in field theory.

These nuclear current matrix elements are the building blocks of the nuclear responses and will be at the center of our modeling efforts. They contain all the dynamical information of the electromagnetic or electroweak interaction.

Currents:

Jµ^lep(q) ≡ ¯^u(k_f,s_f)ˆJ_µ^lepu(k_i,s_i) = ¯u(k_f,s_f)γ_µ(1+hγ⁵)u(k_i,s_i) Jµ^nuc(q) ≡ hΨf| ˆ^Jµ^nuc|Ψii

where h=0 for (unpolarized) electrons, and h=−(+)for (anti)neutrinos

Kajetan Niewczas 2p2h Ghent 06.11.2020 12 / 44

Cross section formula

CCνA scattering

e(‘e, ke)

A(EA, pA) e^Õ(‘eÕ, keÕ)

“^ú(Ê, q)

igµ‹

Q2

B(EB, pB)

ieJlep^µ(q) ≠ ieJnucl^‹ (q)

(a) eA scattering.

‹(‘‹, k‹)

A(EA, pA) µ(‘µ, kµ)

W^±(Ê, q)

igµ‹

M_W²

B(EB, pB)

≠ i_2Ô2^q Jlep^µ(q) ≠ i_2Ô2^q cos ◊cJnucl^‹ (q)

(b) CC ‹A scattering.

Figure 2.2: Feynman diagrams for lepton-nucleon scattering processes.

The lepton and hadron currents are defined as

Jµ^lep(q) © u(kf, sf)J^‚µ^lepu(ki, si) = u(kf, sf) “µ(1 + h“⁵) u(ki, si), (2.16) Jµ^nuc(q) © È f|J^‚_µ^nuc(q) | iÍ, (2.17) where we introduced the Dirac spinors u(ki, si) and u(kf, sf) for the incoming and scattered lepton. The | iÍ and | fÍ refer to the initial and final nuclear states and the operatorsJ^‚_µ^lepandJ^‚_µ^nuc(q) are the lepton and nuclear current operators in momentum space. The structure of the nuclear current operator will be discussed throughout this work.

The lepton current operator is known exactly from field theory. In the lepton current, h = 0 for (unpolarized) electron scattering and h equals ≠(+) for neutrino (anti- neutrino) interactions, reflecting the V ≠ A structure of the weak interaction. By convention, the factor 1/2 of the spin-projection operator (1 + h“⁵)/2 is absorbed in the weak coupling constant.

For the nuclear current, we rely on the rules corresponding to the Feynman rules but for nuclear physics [7–9], as bound states cannot be described in field theory.

These nuclear current matrix elements are the building blocks of the nuclear responses and will be at the center of our modeling efforts. They contain all the dynamical information of the electromagnetic or electroweak interaction.

23

EM eA scattering

e(‘e, ke)

A(EA, p_A) e^Õ(‘e^Õ, ke^Õ)

“^ú(Ê, q)

igµ‹

Q²

B(EB, pB)

ieJlep^µ(q) ≠ ieJnucl^‹ (q)

(a) eA scattering.

‹(‘‹, k‹)

A(EA, pA) µ(‘µ, kµ)

W^±(Ê, q)

igµ‹

M_W²

B(EB, p_B)

≠ i_2Ô2^q Jlep^µ(q) ≠ i_2Ô2^q cos ◊cJnucl^‹ (q)

(b) CC ‹A scattering.

Figure 2.2: Feynman diagrams for lepton-nucleon scattering processes.

The lepton and hadron currents are defined as

Jµ^lep(q) © u(kf, sf)J^‚µ^lepu(ki, si) = u(kf, sf) “µ(1 + h“⁵) u(ki, si), (2.16)

Jµ^nuc(q) © È f|J^‚_µ^nuc(q) | iÍ, (2.17)

where we introduced the Dirac spinors u(ki, si) and u(kf, sf) for the incoming and scattered lepton. The | iÍ and | fÍ refer to the initial and final nuclear states and the operatorsJ^‚µ^lepandJ^‚µ^nuc(q) are the lepton and nuclear current operators in momentum space. The structure of the nuclear current operator will be discussed throughout this work.

The lepton current operator is known exactly from field theory. In the lepton current, h = 0 for (unpolarized) electron scattering and h equals ≠(+) for neutrino (anti- neutrino) interactions, reflecting the V ≠ A structure of the weak interaction. By convention, the factor 1/2 of the spin-projection operator (1 + h“⁵)/2 is absorbed in the weak coupling constant.

For the nuclear current, we rely on the rules corresponding to the Feynman rules but for nuclear physics [7–9], as bound states cannot be described in field theory.

These nuclear current matrix elements are the building blocks of the nuclear responses and will be at the center of our modeling efforts. They contain all the dynamical information of the electromagnetic or electroweak interaction.

23

Matrix elements:

M^Wfi = −ⁱ√^GF

2cosθcJν^lep(q)Jnuc^ν (q) M^γfi = −^ie

2

Q²Jν^lep(q)Jnuc^ν (q)

Kajetan Niewczas 2p2h Ghent 06.11.2020 13 / 44

Cross section formula

CCνA scattering

e(‘e, ke)

A(EA, p_A) e^Õ(‘eÕ, keÕ)

“^ú(Ê, q)

igµ‹

Q²

B(EB, p_B)

ieJlep^µ(q) ≠ ieJnucl^‹ (q)

(a) eA scattering.

‹(‘‹, k‹)

A(EA, pA) µ(‘µ, kµ)

W^±(Ê, q)

ig_µ‹

M_W²

B(EB, pB)

≠ i_2Ô2^q Jlep^µ(q) ≠ i_2Ô2^q cos ◊cJnucl^‹ (q)

(b) CC ‹A scattering.

Figure 2.2: Feynman diagrams for lepton-nucleon scattering processes.

The lepton and hadron currents are defined as

Jµ^lep(q) © u(kf, sf)J^‚_µ^lepu(ki, si) = u(kf, sf) “µ(1 + h“⁵) u(ki, si), (2.16) Jµ^nuc(q) © È f|J^‚_µ^nuc(q) | iÍ, (2.17) where we introduced the Dirac spinors u(ki, si) and u(kf, sf) for the incoming and scattered lepton. The | iÍ and | ^fÍ refer to the initial and final nuclear states and the operatorsJ^‚_µ^lepandJ^‚_µ^nuc(q) are the lepton and nuclear current operators in momentum space. The structure of the nuclear current operator will be discussed throughout this work.

The lepton current operator is known exactly from field theory. In the lepton current, h = 0 for (unpolarized) electron scattering and h equals ≠(+) for neutrino (anti- neutrino) interactions, reflecting the V ≠ A structure of the weak interaction. By convention, the factor 1/2 of the spin-projection operator (1 + h“⁵)/2 is absorbed in the weak coupling constant.

For the nuclear current, we rely on the rules corresponding to the Feynman rules but for nuclear physics [7–9], as bound states cannot be described in field theory.

These nuclear current matrix elements are the building blocks of the nuclear responses and will be at the center of our modeling efforts. They contain all the dynamical information of the electromagnetic or electroweak interaction.

23

EM eA scattering

e(‘e, ke)

A(EA, pA) e^Õ(‘e^Õ, ke^Õ)

“^ú(Ê, q)

igµ‹

Q²

B(EB, pB)

ieJlep^µ(q) ≠ ieJnucl^‹ (q)

(a) eA scattering.

‹(‘‹, k‹)

A(EA, pA) µ(‘µ, kµ)

W^±(Ê, q)

igµ‹

M_W²

B(EB, pB)

≠ i_2Ô2^q Jlep^µ(q) ≠ i_2Ô2^q cos ◊cJnucl^‹ (q)

(b) CC ‹A scattering.

Figure 2.2: Feynman diagrams for lepton-nucleon scattering processes.

The lepton and hadron currents are defined as

Jµ^lep(q) © u(k^f, sf)J^‚µ^lepu(ki, si) = u(kf, sf) “µ(1 + h“⁵) u(ki, si), (2.16)

Jµ^nuc(q) © È f|J^‚µ^nuc(q) | iÍ, (2.17)

where we introduced the Dirac spinors u(ki, si) and u(kf, sf) for the incoming and scattered lepton. The | iÍ and | fÍ refer to the initial and final nuclear states and the operatorsJ^‚µ^lepandJ^‚µ^nuc(q) are the lepton and nuclear current operators in momentum space. The structure of the nuclear current operator will be discussed throughout this work.

The lepton current operator is known exactly from field theory. In the lepton current, h = 0 for (unpolarized) electron scattering and h equals ≠(+) for neutrino (anti- neutrino) interactions, reflecting the V ≠ A structure of the weak interaction. By convention, the factor 1/2 of the spin-projection operator (1 + h“⁵)/2 is absorbed in the weak coupling constant.

For the nuclear current, we rely on the rules corresponding to the Feynman rules but for nuclear physics [7–9], as bound states cannot be described in field theory.

These nuclear current matrix elements are the building blocks of the nuclear responses and will be at the center of our modeling efforts. They contain all the dynamical information of the electromagnetic or electroweak interaction.

The cross section is propotional to the square:

X

if

M^Wfi

² = ^G

2 F

2 cos²θcL_µνH^µν X

if

M^γ_fi² = ^e

4

4Q²L_µνH^µν

Kajetan Niewczas 2p2h Ghent 06.11.2020 14 / 44

Cross section formula

CCνA scattering

e(‘e, ke)

A(EA, p_A) e^Õ(‘eÕ, keÕ)

“^ú(Ê, q)

ig_µ‹

Q²

B(EB, pB)

ieJlep^µ(q) ≠ ieJnucl^‹ (q)

(a) eA scattering.

‹(‘‹, k‹)

A(EA, pA) µ(‘µ, kµ)

W^±(Ê, q)

igµ‹

M2 W

B(EB, pB)

≠ i_2Ô2^q Jlep^µ(q) ≠ i_2Ô2^q cos ◊cJnucl^‹ (q)

(b) CC ‹A scattering.

Figure 2.2: Feynman diagrams for lepton-nucleon scattering processes.

The lepton and hadron currents are defined as

Jµ^lep(q) © u(k^f, sf)J^‚_µ^lepu(ki, si) = u(kf, sf) “µ(1 + h“⁵) u(ki, si), (2.16) Jµ^nuc(q) © È f|J^‚_µ^nuc(q) | iÍ, (2.17) where we introduced the Dirac spinors u(ki, si) and u(kf, sf) for the incoming and scattered lepton. The | iÍ and | fÍ refer to the initial and final nuclear states and the operatorsJ^‚_µ^lepandJ^‚_µ^nuc(q) are the lepton and nuclear current operators in momentum space. The structure of the nuclear current operator will be discussed throughout this work.

The lepton current operator is known exactly from field theory. In the lepton current, h = 0 for (unpolarized) electron scattering and h equals ≠(+) for neutrino (anti- neutrino) interactions, reflecting the V ≠ A structure of the weak interaction. By convention, the factor 1/2 of the spin-projection operator (1 + h“⁵)/2 is absorbed in the weak coupling constant.

For the nuclear current, we rely on the rules corresponding to the Feynman rules but for nuclear physics [7–9], as bound states cannot be described in field theory.

These nuclear current matrix elements are the building blocks of the nuclear responses and will be at the center of our modeling efforts. They contain all the dynamical information of the electromagnetic or electroweak interaction.

23

EM eA scattering

e(‘e, ke)

A(EA, pA) e^Õ(‘eÕ, keÕ)

“^ú(Ê, q)

igµ‹

Q²

B(EB, pB)

ieJlep^µ(q) ≠ ieJnucl^‹ (q)

(a) eA scattering.

‹(‘‹, k‹)

A(EA, pA) µ(‘µ, kµ)

W^±(Ê, q)

igµ‹

M_W²

B(EB, p_B)

≠ i_2Ô2^q Jlep^µ(q) ≠ i_2Ô2^q cos ◊cJnucl^‹ (q)

(b) CC ‹A scattering.

Figure 2.2: Feynman diagrams for lepton-nucleon scattering processes.

The lepton and hadron currents are defined as

Jµ^lep(q) © u(kf, sf)J^‚µ^lepu(ki, si) = u(kf, sf) “µ(1 + h“⁵) u(ki, si), (2.16)

Jµ^nuc(q) © È f|J^‚µ^nuc(q) | iÍ, (2.17)

where we introduced the Dirac spinors u(ki, si) and u(kf, sf) for the incoming and scattered lepton. The | iÍ and | fÍ refer to the initial and final nuclear states and the operatorsJ^‚µ^lepandJ^‚µ^nuc(q) are the lepton and nuclear current operators in momentum space. The structure of the nuclear current operator will be discussed throughout this work.

The lepton current operator is known exactly from field theory. In the lepton current, h = 0 for (unpolarized) electron scattering and h equals ≠(+) for neutrino (anti- neutrino) interactions, reflecting the V ≠ A structure of the weak interaction. By convention, the factor 1/2 of the spin-projection operator (1 + h“⁵)/2 is absorbed in the weak coupling constant.

For the nuclear current, we rely on the rules corresponding to the Feynman rules but for nuclear physics [7–9], as bound states cannot be described in field theory.

These nuclear current matrix elements are the building blocks of the nuclear responses and will be at the center of our modeling efforts. They contain all the dynamical information of the electromagnetic or electroweak interaction.

23

Leptonic tensor:

L_µν ∝

k_i,µk_{f ,ν}+k_{f ,ν}k_i,µ+g_µνm_im_f −^gµνk_i·^kf −^ih_µναβk_i^αk_f^β the axial term (−^ih_µναβk_i^αk_f^β) drops down for electrons (h=0)

Kajetan Niewczas 2p2h Ghent 06.11.2020 15 / 44