# Inclusive neutrino scattering off deuteron from threshold to GeV energies

###### Abstract

Background: Neutrino-nucleus quasi-elastic scattering is crucial to interpret the neutrino oscillation results in long baseline neutrino experiments. There are rather large uncertainties in the cross section, due to insufficient knowledge on the role of two-body weak currents. Purpose: Determine the role of two-body weak currents in neutrino-deuteron quasi-elastic scattering up to GeV energies. Methods: Calculate cross sections for inclusive neutrino scattering off deuteron induced by neutral and charge-changing weak currents, from threshold up to GeV energies, using the Argonne potential and consistent nuclear electroweak currents with one- and two-body terms. Results: Two-body contributions are found to be small, and increase the cross sections obtained with one-body currents by less than 10% over the whole range of energies. Total cross sections obtained by describing the final two-nucleon states with plane waves differ negligibly, for neutrino energies MeV, from those in which interaction effects in these states are fully accounted for. The sensitivity of the calculated cross sections to different models for the two-nucleon potential and/or two-body terms in the weak current is found to be weak. Comparing cross sections to those obtained in a naive model in which the deuteron is taken to consist of a free proton and neutron at rest, nuclear structure effects are illustrated to be non-negligible. Conclusion: Contributions of two-body currents in neutrino-deuteron quasi-elastic scattering up to GeV are found to be smaller than 10%. Finally, it should be stressed that the results reported in this work do not include pion production channels.

###### pacs:

25.10.+s, 25.30.Pt## I Introduction

In last few years, inclusive neutrino scattering from nuclear targets has become a hot topic. Interest has been spurred by the anomaly observed in recent neutrino quasi-elastic scattering data on C Aguilar08 (); Butkevich10 (), i.e. the excess, at relatively low energy, of measured cross section relative to theoretical calculations. Analyses based on these calculations have led to speculations that our present understanding of the nuclear response to charge-changing weak probes may be incomplete Benhar10 (), and, in particular, that the momentum transfer dependence of the axial form factor of the nucleon, specifically the cutoff value of its dipole parameterization Juszczak10 (), may be quite different from that obtained from analyses of pion electro-production data Amaldi79 () and measurements of the reaction in the deuteron at quasi-elastic kinematics Baker81 (); Kitagaki83 () and of and elastic scattering Ahrens87 () ( GeV versus GeV). However, it should be emphasized that the calculations on which these analyses are based use rather crude models of nuclear structure—Fermi gas or local density approximations of the nuclear matter spectral function—as well as simplistic treatments of the reaction mechanism, and should therefore be viewed with skepticism.

In this paper, we calculate cross sections for inclusive neutrino scattering off deuteron in a wide energy range, from threshold up to 1 GeV. The motivations for undertaking such a work are twofold. The first is to provide a benchmark for studies of electro-weak inclusive response in light nuclei we intend to carry out in the near future. The second motivation has to do with plans Garvey12 (), still under development, to determine the neutrino flux in accelerator-based experiments from measurements of inclusive cross sections on the deuteron. In particular, in charged-current neutrino capture on deuteron, the final states can be measured, in principle, very well. Clearly, accurate predictions for these cross sections are crucial for a reliable determination of the flux.

A number of studies of neutrino-deuteron scattering at low and intermediate energies ( MeV) were carried out in the past decades, see Ref. Kubodera94 () for a review of work done up to the mid 1990’s. These efforts culminated in the Nakamura et al.’s 2001 and 2002 calculations of the cross sections for neutrino disintegration of the deuteron induced by neutral and charge-changing weak currents. These calculations were based on bound- and scattering-state wave functions obtained from last-generation realistic potentials, and used a realistic model for the nuclear weak current, including one- and two-body terms. The vector part of this current was shown to provide an excellent description of the radiative capture cross section for neutron energies up to 100 MeV Nakamura01 (), while the axial part was constrained to reproduce the Gamow-Teller matrix element in tritium -decay Nakamura02 (). The Nakamura et al. studies have played an important role in the analysis and interpretation of the Sudbury Neutrino Observatory (SNO) experiments Ahmad02 (), which have established solar neutrino oscillations and the validity of the standard model for the generation of energy and neutrinos in the sun Bahcall04 ().

In the present work, we use the same theoretical framework as the authors of Refs. Nakamura01 (); Nakamura02 (), but include refinements in the modeling of the weak current—which however, as shown in Sec. V, will turn out to have a minor impact on the predicted cross sections—and extend the range of neutrino energies up to 1 GeV. While the theoretical approach is essentially the same, nevertheless the way in which the calculations are carried out in practice is rather different from that used in those earlier papers, which relied on a multipole expansion of the weak transition operators, and evaluated the cross section by summing over a relatively large number of final two-nucleon channels states. In contrast, we evaluate, by direct numerical integrations, the matrix elements of the weak current between the deuteron and the two-nucleon scattering states labeled by the relative momentum (and in given pair spin and isospin channels), thus avoiding cumbersome multipole expansions. Differential cross sections are then obtained by integrating over (and summing over the discrete quantum numbers) appropriate combinations of these matrix elements, i.e. by calculating the weak response functions. The techniques developed here for the deuteron should prove valuable when we will attempt the Green’s function Monte Carlo calculation of these response functions (or rather, their Laplace transforms Carlson92 ()) in nuclei.

This paper is organized as follows. In Sec. II and Appendix A we present the neutrino and antineutrino differential cross sections expressed in terms of response functions, while in Sec. III we provide a succinct description of the neutral and charge-changing weak-current model. In Sec. IV we outline the methods used to obtain the two-nucleon bound and continuum states, and discuss the numerical evaluation of the response functions. A variety of results for the neutral current processes H() and H(), and charge-changing processes H() and H() are presented in Sec. V, including the sensitivity of the calculated cross sections to (i) interaction effects in the final states, (ii) different short-range behaviors of the two-body axial weak currents, and (iii) different potential models to describe the two-nucleon bound and continuum states. In order to illustrate the effects of nuclear structure, we compare these cross sections to those obtained in a naive model in which the deuteron is taken to consist of a free proton and neutron (the free nucleon cross sections are listed for reference in Appendix B). Concluding remarks and an outlook are given in Sec. VI.

## Ii Inclusive neutrino scattering off deuteron

The differential cross section for neutrino () and antineutrino () inclusive scattering off deuteron, specifically the processes

(1) |

induced by neutral weak currents (NC), and the processes

(2) |

induced by charge-changing weak currents (CC), can be expressed as

(3) | |||||

where = for the NC processes and = for the CC processes, and the () sign in the last term is relative to the () initiated reactions. Following Ref. Nakamura02 (), we adopt the value GeV as obtained from an analysis of super-allowed -decays Towner99 ()—this value includes radiative corrections—while is taken as 0.97425 from PDG (). The initial neutrino four-momentum is , the final lepton four momentum is , and the lepton scattering angle is denoted by . We have also defined the lepton energy and momentum transfers as and , respectively, and the squared four-momentum transfer as . The Fermi function with accounts for the Coulomb distortion of the final lepton wave function in the the charge-raising reaction,

(4) |

it is set to one otherwise. Here , is the gamma function, is the deuteron radius ( fm), and is the fine structure constant. Radiative corrections for the CC and NC processes due to bremsstrahlung and virtual photon- and -exchanges have been evaluated by the authors of Refs. Towner98 (); Kurylov02 () at the low energies ( MeV) relevant for the SNO experiment, which measured the neutrino flux from the B decay in the sun. These corrections are neglected in the present work, since its focus is on scattering of neutrinos with energies larger than 100 MeV. We are not (or not yet, at least) concerned with providing cross section calculations with % accuracy in this regime. Lastly, the nuclear response functions are defined as

(5) | |||||

(6) | |||||

(7) | |||||

(8) | |||||

(9) |

where and represent, respectively, the initial deuteron state in spin projection and the final two-nucleon state of energy , and is the deuteron rest mass. The three-momentum transfer is taken along the -axis (i.e., the spin-quantization axis), and is the time component (for ) or space component (for ) of the NC or CC.

The expression above for the CC cross section is valid in the limit , in which the lepton rest mass is neglected. At small incident neutrino energy, this approximation is not correct. Inclusion of the lepton rest mass leads to changes in the kinematical factors multiplying the various response functions. The resulting cross section is given in Appendix A.

## Iii Neutral and charge-changing weak currents

We denote the neutral and charge-changing weak currents as and , respectively. The former is given by

(10) |

where is the Weinberg angle ( PDG ()), and denote, respectively, the isoscalar and isovector pieces of the electromagnetic current, and denotes the isovector piece of the axial current (the on the isovector terms indicates that they transform as the -component of an isovector under rotations in isospin space). Isoscalar contributions to associated with strange quarks are ignored, since experiments at Bates Spayde00 () and JLab Ahmed12 () have found them to be very small.

The charge-changing weak current is written as the sum of polar- and axial-vector components

(11) |

The conserved-vector-current (CVC) constraint relates the polar-vector components of the charge-changing weak current to the isovector component of the electromagnetic current via

(12) |

where are isospin operators. We now turn to a discussion of the one- and two-body contributions to the NC and CC.

### iii.1 One-body terms

The isoscalar components of the one-body electromagnetic current are given by

(13) | |||||

(14) | |||||

(15) |

and the corresponding isovector components of are obtained by the replacements

(16) |

where and are the isoscalar/isovector combinations of the proton and neutron electric () and magnetic () form factors, and are the position and momentum operators of nucleon , and are its Pauli spin and isospin matrices, and is the nucleon mass (0.9389 GeV). Note that we have decomposed and into transverse () and longitudinal () components to the momentum transfer , and have used current conservation to relate the latter to the isoscalar and isovector charge operators and . The isovector components of the axial weak neutral current are given by

(17) | |||||

(18) | |||||

where is the nucleon axial form factor, and denotes the anticommutator. The operators above include terms of order in the non-relativistic expansion of the single-nucleon covariant currents. These have been neglected in the study of Ref. Nakamura02 (). The proton and neutron electromagnetic and nucleon axial form factors are parametrized as

(19) | |||||

(20) | |||||

(21) |

from which the isoscalar and isovector combinations are obtained as . The proton and neutron magnetic moments are and in units of nuclear magnetons (n.m.), and the nucleon axial-vector coupling constant is taken to be PDG (). The values for the cutoff masses and used in this work are 0.833 GeV and 1 GeV, respectively. The former is from fits to elastic electron scattering data off the proton and deuteron Hyde04 (), while the latter is from an analysis of pion electroproduction Amaldi79 () and neutrino scattering Baker81 (); Kitagaki83 (); Ahrens87 () data. Uncertainties in the dependence of the axial form factor, in particular the value of , could significantly impact predictions for the neutrino cross sections under consideration. As mentioned earlier, recent analyses of neutrino quasi-elastic scattering data on nuclear targets Juszczak10 () quote considerably larger values for , in the range (1.20–1.35) GeV.

The polar-vector () and axial-vector () components of the charge-changing weak current are obtained, respectively, from and by replacing

(22) |

However, in the case of , in addition to the terms entering Eqs. (17)–(18), we also retain the induced pseudoscalar contribution, given by

(23) |

where the induced pseudoscalar form factor is parametrized as

(24) |

This form factor is not well known Gorringe04 (). The parameterization above is consistent with values extracted Czarnecki07 (); Marcucci12 () from precise measurements of muon-capture rates on hydrogen Andreev07 () and He Ackerbauer98 (), as well as with the most recent theoretical predictions based on chiral perturbation theory Bernard94 (). This contribution vanishes in NC-induced neutrino reactions.

### iii.2 Two-body terms

Two-body terms in the neutral and charge-changing weak currents have been discussed in considerable detail in Refs. Carlson98 (); Marcucci00 (); Marcucci05 () (and references therein). We list the terms included in the present study—i.e., the subset of those derived in the above references expected to give the dominant two-body contributions to the processes of interest here—in the following two sub-sections for clarity of presentation and future reference in Sec. V. Unless stated otherwise, they are given in momentum space, and configuration-space expressions follow from

(25) |

where and , and are the initial and final momenta of nucleon , and

(26) |

These configuration-space operators are used in the calculations reported below.

#### iii.2.1 Two-body vector terms

The two-body isovector current operator consists of pseudoscalar- and vector-meson (referred to as -like and -like) exchange, and -excitation terms,

(27) |

The -like and -like exchange currents read:

(28) | |||||

(29) | |||||

where

(30) |

and

(31) | |||||

(32) | |||||

(33) |

Here , , are the isospin-dependent central, spin-spin, and tensor components of the two-nucleon interaction (the AV18 in the present study), and are spherical Bessel functions. The factor in the expression for ensures that its volume integral vanishes. In a one-boson-exchange (OBE) model, in which the isospin-dependent central, spin-spin, and tensor interactions are due to - and -meson exchange, the functions , , and simply reduce to

(34) | |||||

(35) | |||||

(36) |

where and are the meson masses, , and and are the pseudovector , and vector and tensor coupling constants, and the hadronic form factors are parameterized as

(37) |

While the AV18 interaction is not a OBE model, nevertheless the effective propagators , , and projected out of its , , and components are quite similar to those listed above with cutoff masses in the range (1.0–1.5) GeV. We note that the -like and -like currents with the , , and defined in Eq. (30) satisfy by construction the current conservation relation with the AV18 , , and interaction components (for a discussion of the issue of current conservation in relation to the momentum-dependent terms of the AV18, see Ref. Marcucci05 ()).

The isovector -excitation current is written in configuration space as (for a derivation based on a perturbative treatment of -isobar degrees of freedom in nuclear wave functions, see Ref. Carlson98 ())

(38) |

where and are spin- and isospin-transition operators converting a nucleon into a isobar and satisfying the identity

(39) |

is the to transition potential,

(40) |

is the tensor operator obtained by replacing with , the regularized spin-spin and tensor radial functions and are defined as

(41) | |||||

(42) |

and , is the coupling constant ( from the width of the ), and the parameter in the short-range cutoff function is taken as (from the AV18). Finally, the electromagnetic transition form factor is parameterized as

(43) |

where the transition magnetic moment is 3 n.m., as obtained in an analysis of data in the -resonance region Carlson86 (). This analysis also gives =0.84 GeV and =1.2 GeV.

The two-body isoscalar current operator considered in the present study only includes the contribution associated with the transition mechanism,

(44) |

where

(45) |

The combination of coupling constants is taken as 1.37, and the cutoff masses and as 0.75 GeV and 1.25 GeV, respectively, from a study of the deuteron magnetic form factor Carlson91 (). The dependence of the electromagnetic transition form factor is modeled by using vector-meson dominance,

(46) |

where is the -meson mass.

The two-body isovector and isoscalar electromagnetic charge operators and consist of terms associated with -like and -like exchanges

(47) |

where

(48) | |||||

(49) |

and

(50) | |||||

(51) |

with and as defined in Eqs. (30). The nucleon electromagnetic Dirac and Pauli form factors and are obtained from

(52) | |||||

(53) |

with .

The polar-vector components of the charge-changing weak current are obtained from via CVC, which implies the replacements

(54) |

(55) |

in Eq. (38), and the replacement (22) in Eqs. (48)–(49). Only the transverse components (perpendicular to ) of the vector part of the NC and CC are explicitly included in the calculations to follow. Their longitudinal components have already been effectively accounted for by the replacement in Eq. (15) (and the similar one for the isovector terms). Lastly, we note that in the study of Ref. Nakamura02 () the -meson exchange and transition contributions to the two-body vector current, and - and -exchange contributions to the two-body vector charge have been neglected. Furthermore, the -exchange and excitation currents are regularized by introducing a monopole form factor ( fm), which naturally leads to a different short-range behavior of these currents than obtained here.

#### iii.2.2 Two-body axial terms

Set I | Set II | |
---|---|---|

1-b (NR) | +0.9213 | +0.9213 |

1-b (RC) | –0.0085 | –0.0085 |

2-b () | +0.0078 | +0.0123 |

2-b () | –0.0042 | –0.0055 |

2-b () | +0.0123 | +0.0196 |

2-b () | +0.0263 | +0.0159 |

0.614 | 0.371 |

The axial parts of the neutral and charge-changing weak current operators consist of contributions associated with - and -meson exchanges, the axial transition mechanism, and a excitation term

(56) |

where the isospin component is either for NC or for CC. The - and -meson exchange and transition axial currents read, respectively,

(57) |

(58) | |||||

(59) |

while the excitation axial current is obtained from Carlson98 ()