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Multinucleon excitations in neutrino–nucleus scattering: connecting different microscopic models for the correlations

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Abstract

The problem of nucleon–nucleon correlations and meson exchange currents has been vividly debated in connection with the neutrino–nucleus cross sections. In this work we focus on nucleon–nucleon correlations by discussing a formal correspondence between the approaches based on independent particles and the ab initio approaches involving correlated wave functions. We use a general technique based on unitary transformation mapping the Fermion operators relative to bare nucleons into quasi-particle operators relative to dressed nucleons. We derive formulas for spectral functions, response functions, momentum distribution, separation energy, general enough to be applied with any kind of effective nucleon–nucleon interaction. We establish the relation between the non-energy-weighted sum rule and the Fermi sea depopulation. With our tools we evaluate whether approaches based on effective interactions are compatible with the expected amount of correlations coming from ab initio calculations. For this purpose we use as a test the Fermi sea depopulation and the value of the kinetic energy per nucleon.

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Notes

  1. We remind that in the case of Lyon, Valencia and Ghent groups long-range nuclear correlations are also taken into account, via the Random Phase Approximation (RPA).

  2. The \( \sigma \tau \) operators are replaced by the usual 1/2 to 3/2 transition operators ST in the case of coupling to the \(\varDelta \).

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Authors and Affiliations

  1. Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, 69622, Villeurbanne, France

    G. Chanfray & M. Ericson

  2. Theory Department, CERN, 12111, Geneva, Switzerland

    M. Ericson

  3. IPSA-DRII, 63 boulevard de Brandebourg, 94200, Ivry-sur-Seine, France

    M. Martini

  4. Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Sorbonne Université, Université Paris Diderot, CNRS/IN2P3, Paris, France

    M. Martini

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  1. G. Chanfray
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Appendices

The one-body Green’s function and the mass operator

In this appendix we remind some textbook formulas (see for example Ref. [91]) of the many-body theory.

The one-body Green’s function is defined as :

$$\begin{aligned} G_{k',k}(t)=\left\langle 0\left| -i\,\mathcal{{T}}\left( a_{k'}(t) , a^\dagger _{k}(0) \right) \right| 0 \right\rangle =\int \frac{\mathrm{d}\varOmega }{2\pi } e^{-i\varOmega t} G_{k',k}(\varOmega ). \end{aligned}$$
(61)

Its Lehmann representation can be written in a dispersive form as :

$$\begin{aligned} G_{k',k}(\varOmega ) =\int \mathrm{d}E\left( \frac{S^p_{k',k}(E)}{\varOmega -E+i\eta }+\frac{S^h_{k,k'}(E)}{\varOmega -E-i\eta }\right) \end{aligned}$$
(62)

and is entirely known once the hole and particle spectral functions, defined as

$$\begin{aligned} S^h_{k,k'}(E)&=\sum _n \langle 0|a^\dagger _k|n \rangle \langle n|a_{k'}|0 \rangle \, \delta \left( E+E_n^{A-1}-E_0^A\right) \nonumber \\ S^p_{k',k}(E)&=\sum _n \langle 0|a_{k'}|n\rangle \langle n|a^\dagger _k|0 \rangle \, \delta \left( E-E_n^{A+1}+E_0^A\right) , \end{aligned}$$
(63)

are known.

The single particle propagator in nuclear matter is obtained as the diagonal component one-body Green’s function:

$$\begin{aligned} G(E,\mathbf{k})= & {} \int \mathrm{d}t\, e^{iEt}\left\langle 0\left| -iT\left( a_\mathbf{k}(t), a^\dagger _\mathbf{k}(0)\right) \right| 0\right\rangle \nonumber \\= & {} \left( E-(k^2/2M)-M(E,\mathbf{k})\right) ^{-1}, \end{aligned}$$
(64)

where \(M(E,\mathbf{k})\) is the mass operator.

The real and imaginary part of the nuclear matter single particle propagator can be expressed in terms of the spectral functions as:

$$\begin{aligned} Re\, G(E,\mathbf{k})= & {} \int \mathrm{d}E'\,\frac{S^p(E',\mathbf{k})+S^h(E',\mathbf{k})}{E-E'} \end{aligned}$$
(65)
$$\begin{aligned} Im\, G(E,\mathbf{k})= & {} -\pi \left( S^p(E,\mathbf{k})-S^h(E,\mathbf{k})\right) . \end{aligned}$$
(66)

The imaginary part of the mass operator \(M(E,\mathbf{k})\) (i.e. the optical potential), assuming that it is always smaller than the real part (see Ref. [62]) , is related to the single particle propagator and to the spectral functions by the following relations :

$$\begin{aligned} -\frac{1}{\pi } Im\, G(E,\mathbf{k})\simeq -\frac{1}{\pi }\frac{Im\, M(E,\mathbf{k})}{(E-\epsilon _k)^2} =S^p(E,\mathbf{k})-S^h(E,\mathbf{k}), \end{aligned}$$
(67)

where \(\epsilon _k\) is the Hartree-Fock single particle energy. Inserting in Eq. (67) the expression of the spectral functions given by Eqs. (25), (24), we obtain the expression of Eq. (26) for the imaginary part of the mass operator.

Response function in the factorization approximation

In this appendix we explicitly derive the expression of the response function in the factorization approximation. The response function, already defined in Eq. (3), can be explicitly written as:

$$\begin{aligned} R(\omega ,\mathbf{q})= & {} \sum _{n}\,\left| \left\langle n \left| \sum _{j=1}^A\,\mathcal{{O}}(j)\, e^{i\mathbf{q}\cdot \mathbf{x}_j} \right| 0 \right\rangle \right| ^2\,\delta (\omega -E_n + E_0)\nonumber \\= & {} \sum _{n}\sum _{k'_1, k'_2,k_1,k_2} \langle k'_1|\mathcal{{O}}^\dagger e^{-i\mathbf{q}\cdot \mathbf{x}}|k'_2 \rangle \nonumber \\&\times \langle k_2|\mathcal{{O}} e^{i\mathbf{q}\cdot \mathbf{x}}|k_1\rangle \langle 0|a^\dagger _{k'_1}a_{k'_2}|n\rangle \langle n|a^\dagger _{k_2} a_{k_1}|0 \rangle \nonumber \\\equiv & {} \left( -\frac{1}{\pi }\right) Im \varPi (\omega ,\mathbf{q}). \end{aligned}$$
(68)

The response function is the imaginary part of the polarization propagator \(\varPi (\omega ,\mathbf{q})\), which can be expressed as:

$$\begin{aligned} \varPi (\omega ,\mathbf{q})= & {} \int \mathrm{d}t\, e^{i\omega t}\,\sum _{k'_1, k'_2,k_1,k_2} \langle k'_1|\mathcal{{O}}^\dagger e^{-i\mathbf{q}\cdot \mathbf{x}}|k'_2\rangle \langle k_2|\mathcal{{O}} e^{i\mathbf{q}\cdot \mathbf{x}}|k_1 \rangle \,\nonumber \\&\times \left\langle 0\left| -i\,\mathcal{{T}}\left( a^\dagger _{k'_1}(t)a_{k'_2}(t) , a^\dagger _{k_2}(0) a_{k_1}(0)\right) \right| 0 \right\rangle \nonumber \\\approx & {} \int i\mathrm{d}t\, e^{i\omega t}\,\sum _{k'_1, k'_2,k_1,k_2}\langle k'_1|\mathcal{{O}}^\dagger e^{-i\mathbf{q}\cdot \mathbf{x}}|k'_2\rangle \langle k_2|\mathcal{{O}} e^{i\mathbf{q}\cdot \mathbf{x}}|k_1 \rangle \nonumber \\&\times \langle 0|-i\,\mathcal{{T}}\left( a^\dagger _{k'_1}(t) , a_{k_1}(0)\right) |0 \rangle \nonumber \\&\times \langle 0|-i\,\mathcal{{T}}\left( a_{k'_2}(t) , a^\dagger _{k_2}(0) \right) |0 \rangle \nonumber \\= & {} \int i\mathrm{d}t\, e^{i\omega t}\,\sum _{k'_1, k'_2,k_1,k_2} \langle k'_1|\mathcal{{O}}^\dagger e^{-i\mathbf{q}\cdot \mathbf{x}}|k'_2 \rangle \nonumber \\&\langle k_2|\mathcal{{O}} e^{i\mathbf{q}\cdot \mathbf{x}}|k_1 \rangle (-1) G_{k_1,k'_1}(-t) G_{k'_2,k_2}(t),\nonumber \\ \end{aligned}$$
(69)

where we have applied the factorization scheme which consists to approximate the two-body Green’s function as a product of two one-body Green’s functions. Using the Lehmann representation, given in Eq. (62), for the one-body Green’s function, one obtains for the polarization propagator:

$$\begin{aligned}&\varPi (\omega ,\mathbf{q}) =\sum _{k'_1, k'_2,k_1,k_2} \langle k'_1|\mathcal{{O}}^\dagger e^{-i\mathbf{q}\cdot \mathbf{x}}|k'_2 \rangle \langle k_2|\mathcal{{O}} e^{i\mathbf{q}\cdot \mathbf{x}}|k_1 \rangle \nonumber \\&\quad \times (-i)\int \frac{d\varOmega _1}{2\pi }\frac{d\varOmega _2}{2\pi } \,2\pi \,\delta (\omega -\varOmega _2+\varOmega _1)\, G_{k_1,k'_1}(\varOmega _1) G_{k'_2,k_2}(\varOmega _2)\nonumber \\&=\sum _{k'_1, k'_2,k_1,k_2} \langle k'_1|\mathcal{{O}}^\dagger e^{-i\mathbf{q}\cdot \mathbf{x}}|k'_2\rangle \langle k_2|\mathcal{{O}} e^{i\mathbf{q}\cdot \mathbf{x}}|k_1\rangle \nonumber \\&\quad \times \int \frac{dE_1}{2\pi }\frac{dE_2}{2\pi }\left( \frac{S^h_{k_1,k'_1}(E_1)S^p_{k'_2,k_2}(E_2)}{\omega -E_2+E_1+ i\eta }\nonumber \right. \\&\quad \left. -\frac{S^p_{k_1,k'_1}(E_1)S^h_{k'_2,k_2}(E_2)}{\omega -E_2+E_1- i\eta }\right) . \end{aligned}$$
(70)

The response function is obtained by taking the imaginary part (for positive \(\omega \)) of Eq. (70), leading to:

$$\begin{aligned}&R(\omega ,\mathbf{q})=\sum _{k'_1, k'_2,k_1,k_2} \langle k'_1|\mathcal{{O}}^\dagger e^{-i\mathbf{q}\cdot \mathbf{x}}|k'_2\rangle \langle k_2|\mathcal{{O}} e^{i\mathbf{q}\cdot \mathbf{x}}|k_1 \rangle \nonumber \\&\quad \times \int _{-\infty }^{\varepsilon _F} dE_1 \int _{\varepsilon _F}^{\infty } dE_2\, S^h_{k_1,k'_1}(E_1)\,S^p_{k'_2,k_2}(E_2)\,\delta (\omega -E_2+E_1). \end{aligned}$$
(71)

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Chanfray, G., Ericson, M. & Martini, M. Multinucleon excitations in neutrino–nucleus scattering: connecting different microscopic models for the correlations. Eur. Phys. J. Spec. Top. 230, 4357–4372 (2021). https://doi.org/10.1140/epjs/s11734-021-00291-x

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