USC(NT)-93-6

Submitted to Internat’l J. Mod. Phys. E

Neutrino-Nucleus Reactions

Kuniharu
Kubodera^{*}^{*}*Work supported in part by
National Science Foundation Grant No. PHYS-9006844.

Department of Physics and Astronomy

University of South Carolina

Columbia, SC 29208, U.S.A.

and

Satoshi Nozawa

Department of Physics

Queen’s University

Kingston,Canada, K4L 3N6

ABSTRACT

The current status is reviewed of theoretical treatments of neutrino-nucleus reactions that are relevant to the detection of astrophysical neutrinos. Various nuclear physics aspects involved in the evaluation of the neutrino-nucleus reaction cross sections are critically surveyed.

Chapter 1 Introduction

Although many topics can come under the title: “Neutrino-Nucleus Reactions”, we shall be concerned here with neutrino-nucleus reactions that pertain to the terrestrial observation of astrophysical neutrinos. The neutrinos play fundamental roles in various astrophysical phenomena. The wealth of these phenomena is eruditely described in, e.g., Bahcall [2] and Fukugita [3]. Obviously, terrestrial experiments to detect these neutrinos are highly valuable sources of astrophysical information, and observational neutrino astrophysics is an extremely rapidly expanding research field [4]. It is useful to distinguish three different energy regimes of the astrophysical neutrinos. (i) The low energy region (), which includes the solar neutrinos and the lower energy part of the supernova neutrinos. (ii) The medium-energy region () exemplified by the higher energy region of supernova neutrinos. (iii) The high-energy region () represented by the solar flare neutrinos and atmospheric neutrinos, whose energy can reach as high as GeV. Neutrino-nucleus reactions offer a great variety of ways to detect these astrophysical neutrinos. Generally, the following types of neutrino-nucleus reactions are possible:

(1) | |||||

(2) | |||||

(3) | |||||

(4) |

In the charged-current (CC) reactions, eqs. (1),(2), the script is limited to the electron family unless the incident energy is high enough to produce a muon or a tau-lepton, whereas the neutral-current (NC) reactions, eqs.(3),(4), can always occur for and . The final nuclear state can in general be either bound or unbound (if the latter is allowed energetically), and all these will contribute to inclusive events in which only the final leptons are monitored. On the other hand, in a particular type of experiment such as a radiochemical experiment, only the contribution of particle-bound final states will be registered.

To set the stage, we first give a brief survey of the solar neutrino problem. The Sun generates its energy primarily by changing protons into -particles. The actual reaction chain through which takes place involves many intermediate nuclear reactions, some of which are weak-interaction processes emitting neutrinos. The standard solar model (SSM) [2, 5] gives a definite prediction of the neutrino flux coming from each of these neutrino-emitting reactions. The first observation of the solar neutrinos was achieved by Davis and his collaborators [6] through a radiochemical counting of Ar produced in the reaction . According to the SSM, the Cl detector of Davis et al’s Homestake facility should register SNU (1 SNU events per target atom per second), but the observed value is only SNU [6]-[10]. This clear deficit of the neutrino flux is the solar neutrino problem, which is one of the most fundamental issues in today’s astrophysics. An independent confirmation of the deficit of the solar neutrino flux has been obtained by the Kamiokande experiment [11, 12, 13], in which the neutrino flux is monitored through the leptonic reaction occurring in the large water Cerenkov counter. The neutrino flux observed by the Kamiokande [13] is times the SSM prediction, corroborating the existence of the solar neutrino problem. One thing to be noted here is that the neutrino detection thresholds of the Homestake and Kamiokande facilities are rather high. The Kamiokande with 7.5 MeV is sensitive only to the neutrinos coming from the reaction

Bahcall and Bethe [17] have recently carried out a systematic study of solar-model explanations. They studied a collection of 1000 precise solar models in which each input parameter (the principal nuclear reaction rates, the solar composition, the solar age, and the radiative opacity) for each model was drawn randomly from a normal distribution with the mean value and standard deviation appropriate to that variable. None of these 1000 solar models was compatible with the results of the chlorine or the Kamiokande experiments. Even if the solar models were artificially modified to reproduce a B neutrino flux in agreement with the Kamiokande experiment, none of these fudged models agreed with the chlorine observation. Furthermore, the GALLEX and SAGE experiments were found to disagree with the prediction of the solar models by and , respectively. Thus, the totality of the existing solar neutrino data seems to strongly indicate that the solar neutrino problem has at least part of its origin in the hitherto unknown properties of the neutrinos.

The particle-physics explanations invoke, in one form or another, a mechanism to convert electron neutrinos into other particles (neutrino oscillation) so that not all of the original ’s produced in the core of the Sun would reach the terrestrial detectors (for a review, see e.g. [18]). The most promising particle-physics solution so far proposed is the Mikheyev-Smirnov-Wolfenstein (MSW) effect [19, 20], which can be summarized as follows. The three neutrino states, , and , are assumed to have different masses, with the electron neutrino the lightest. In the solar core with high electron densities, the neutrinos acquire effective masses through the neutrino-electron interactions. The neutral-current interaction simply gives a universal shift of the neutrino masses, but the charged-current interaction, which in the absence of muons and -leptons is operative only for , increases the selectively so that, for sufficiently high electron densities, the effective electron neutrino mass can become heavier than those of the other neutrinos. If the Hamiltonian involves a piece that causes lepton-flavor mixing, then in a critical region where , or , it is possible for to transform into the other neutrino species via resonant transitions. This phenomenon is called the MSW effect [19, 20]. The parameters characterizing the MSW mechanism are the neutrino mass differences in free space and the strengths of flavor mixings. By adjusting these parameters one can vary the region of the electron neutrino spectrum that is changed into the other neutrinos through the resonant transitions. The contemporary literature abounds in articles that deal with the determination of the parameters characterizing the MSW model or its variants. At present, the bare neutrino mass difference of about to eV with appropriate corresponding mixing strengths is compatible with the existing data. However, this does not constitute unambiguous evidence that the MSW mechanism is operative in the Sun; the direct confirmation of the MSW effect needs an additional set of experiments. Neutrino-nucleus reactions are of great importance in this context, since in some favorable cases they allow us to identify various types of neutrino reactions individually. Of particular significance is the detection of neutral-current (NC) reactions, eq.(3); since the NC reaction rate is the same regardless of neutrino species, a measurement of the NC reactions gives the bolometric flux of all neutrino species [21, 22, 23]. Therefore, by combining information on the NC reactions with that on the CC reactions, one can obtain definitive information on neutrino oscillation, or more specifically, on the MSW effect. In the planned Sudbury Neutrino Observatory (SNO) [24, 25], which uses a 1,000-tonne, heavy-water Cerenkov counter, one aims at the separate registration of the CC and NC reactions on the deuteron with the detection threshold as low as 5 MeV. The data obtained at the SNO will provide unambiguous information on the neutrino flux for individual flavors and consequently on the neutrino oscillation. There are other nuclear targets which are important in this context, and we will discuss them later in the text.

The neutrino-nucleus reactions are not the only place where neutral-current interactions appear. The leptonic reactions:

(5) | |||||

(6) |

where or , do involve the neutral-current contributions. More specifically, for , the reactions can occur through both the charged and neutral currents, whereas for and only the neutral current will participate. Therefore, water Cerenkov counters and liquid scintillation counters in which these leptonic reactions provide main signals do count both neutral-current (NC) and charge-current (CC) events. It is to be noted, however, that the NC and CC events cannot be registered separately by monitoring the leptonic reactions, eqs.(5),(6). This situation makes the use of nuclear targets particularly attractive.

We now turn to astrophysical neutrinos other than the solar neutrinos. The large water Cerenkov counters at the Kamiokande and IMB allowed the first observation of the supernova neutrinos coming from SN1987A. For this spectacular success it was crucial that these facilities had sufficiently high time resolutions to clearly define supernova-related neutrino events. Since all the existing nuclear-target detectors use the radio-chemical counting method, their time resolution is not sufficient to allow the monitoring of supernova neutrinos. In the near future, however, counter experiments involving nuclear targets will become possible. The above-mentioned Sudbury Neutrino Observatory (SNO) [24, 25] is a notable example. As will be discussed in more detail later, the SNO’s ability to detect the NC reactions with high time-resolution can be important in the context of the supernova neutrinos as well. We will also discuss some other nuclear targets which can be very useful in connection with supernova neutrino detection.

In considering the high-energy astrophysical neutrinos [group (iii) in the above-described classification], it is important to realize that, as the incident neutrino energy increases, the nuclear reactions become progressively more important as compared to the leptonic processes. This implies that even the water Cerenkov counter and liquid scintillation counter, which for low-energy neutrinos primarily detect the electrons from the leptonic reactions, will start detecting the nuclear reactions [26]. As will be discussed in the following, this offers yet another interesting use of these facilities. A notable example is the water Cerenkov counter measurement of the ratio of the muonic neutrino flux to the electronic neutrino flux in the atmospheric neutrinos.

One of the fundamental prerequisites for nuclear targets
envisaged as neutrino detectors is that
reliable estimates of the relevant reaction cross sections
be available.
This presents an interesting and challenging task
to nuclear physicists.
^{†}^{†}†The cross sections of the leptonic processes,
eqs.(5),(6),
by contrast are accurately known
from the standard electroweak theory.
There is in fact one particular type of
neutrino-nucleus reaction for which the cross section
can be determined mode-independently;
for the coherent scattering
,
the cross section is determined
by the number of nucleons in .
The coherent process plays an important role
in dark matter search [27]
as well as in certain types of
solar neutrino experiments [28, 29, 30],
but we will not discuss the coherent scattering here.
For a review of this topic, see [2].
The great variety of relevant nuclear targets
and the wide energy range of the astrophysical neutrinos
to be considered necessitate the use of many different approaches
that can be found in the nuclear physics arsenal.
In this review,
we survey the current status of theoretical attempts
at obtaining reliable estimates of the neutrino-nucleus reactions
of astrophysical interest.
To highlight the basic aspects of
various approaches so far used and their possible limitations,
we shall organize our topics
according to the calculational methods involved.
We describe in chapter 2 the direct microscopic calculation
which is relevant to the lightest nuclei.
In chapter 3 we discuss a model-independent determination
of the cross sections
based on the elementary-particle treatment.
Chapter 4 deals with the empirical
effective operator method.
The use of (p,n) reactions
as calibrators of the Gamow-Teller matrix elements
is surveyed in chapter 5.
Chapters 6 and 7 are concerned with high-energy
neutrino-nucleus reactions.
The Fermi gas model and its possible refinement
are discussed in chapter 6,
while the treatment of semi-inclusive
reaction is addressed in chapter 7.
Additional remarks are made in chapter 8.

Chapter 2 “Ab initio” calculations based on realistic wave functions

In the so-called realistic description of nuclei one works with the nuclear Hamiltonian

(7) |

where is the mass number, is the nucleon kinetic energy, and is the realistic nucleon-nucleon potential. Although is called the realistic Hamiltonian due to the fact that reproduces the observed properties of the two-nucleon systems reasonably well, is an effective Hamiltonian of highly operational nature. To arrive at starting from the fundamental QCD picture is far from trivial. First, the dynamical degrees of freedom of quarks and gluons need to be translated into the effective degrees of freedom of hadrons. Second, the Hilbert space of hadrons must be truncated down to those of non-relativistic nucleons interacting via potentials. Despite these fundamental problems, it is generally believed that solutions for the Schrödinger equation with the Hamiltonian give reasonably reliable nuclear wave functions, and it is usually considered as an ab initio approach to calculate matrix elements of nuclear electromagnetic or weak-interaction processes with the use of these wave functions along with the standard non-relativistic reduction of the single-nucleon responses to the relevant currents [31, 32]. Two major problems encountered in this ab initio approach are as follows. First, it is in general extremely difficult to obtain an exact solution of the -body Schrödinger equation , and hence one is usually forced to work with rather drastically truncated model wave functions ; shell-model wave functions are the most commonly used and most successful example. Now, if the matrix element of a nucleon observable is calculated using model wave functions and , one expects . This difference is called core-polarization effect. Another point is that eliminating all hadronic degrees of freedom but those of the nucleons is an approximation and, even if one can calculate exactly, it may still differ from the corresponding true matrix element. One usually accounts for this difference by adding to many-body operators , which represent the effects of the eliminated hadronic degrees of freedom. These additional terms are called the exchange-current operators.

Now, to calculate the core-polarization and exchange-current effects in complex nuclei is one of the most challenging problems in nuclear physics and, although many important achievements have so far been made in this domain, the problem is far from solved. However, nuclei with small mass numbers () offer important exceptions. For these lightest nuclei one can obtain sufficiently “realistic” wave functions either by directly solving the Schrödinger equation or through variational calculations, eliminating thereby the core-polarization problem. For these systems, therefore, one can carry out ab initio calculations of the transition matrix elements of electromagnetic and weak-interaction processes, provided a reasonably reliable method to incorporate the exchange-current effects exists.

2.1. reactions

The A=2 systems are the simplest nuclei, for which one can obtain reasonably reliable wave functions corresponding to the realistic Hamiltonian eq.(7). This allows us to estimate with high accuracy the cross sections of the neutrino-deuteron reactions. These cross sections are important for deriving useful astrophysical information from the experimental data that will soon become available at the Sudbury Neutrino Observatory (SNO) [24, 25]. The neutrino-deuteron reactions that are relevant to the SNO are:

(8) | |||||

(9) | |||||

(10) | |||||

(11) |

As mentioned earlier, one of the great advantages of the SNO is its capability to detect both the neutral- and charged-current reactions and record them separately, and this is crucially important [23] for proving that the MSW effect [19] is indeed a solution for the solar neutrino problem [2].

The earlier calculations [33, 34, 35] were mainly meant for low-energy solar neutrinos and therefore included only s-wave for the final N-N relative motion. Extended calculations that take account of higher partial waves and, correspondingly, the contributions of the forbidden-type transitions were carried out by Ying, Haxton and Henley [36] (to be referred to as YHH1) and by Tatara, Kohyama and Kubodera [37] (to be referred to as TKK) up to 170 MeV. Unfortunately, the cross sections obtained in these two detailed calculations show alarmingly large discrepancies for higher energies; the YHH1 results become significantly larger than those of TKK around MeV and the cross section ratios reach towards MeV. To clarify the origin(s) of these discrepancies, Doi and Kubodera [38] (to be referred to as DK1) carried out a systematic comparison of the formalisms used in YHH1 and TKK. The numerical results of the independent calculation in DK1 [38] suggested possible numerical errors in YHH1, and this suggestion was confirmed by the authors of YHH1. The results of a revised estimation by Ying, Haxton and Henley [39] (to be referred to as YHH2) are in essential agreement with those of TKK and DK1.

Beyond the 20 % level precision, however, there still exist discrepancies among the results of TKK, DK1 and YHH2. The cross sections in TKK begin to be systematically smaller than the corresponding numbers in DK1 and YHH2 as surpasses MeV, and the differences for some channels reach 20 % towards 170 MeV. It has been demonstrated in DK1 that this discrepancy arises because TKK dropped the term quadratic in , the space component of the vector current. At low energies, the contribution of is known to be much less important than that of the time-component . TKK, assuming that this behavior would persist for higher energies, ignored the quadratic term that appears in squaring the transition matrix element to obtain the cross section. According to DK1, however, the contribution of is comparable to or even larger than that of for MeV, and ignoring leads to an underestimation of the cross sections by up to 20 %. Meanwhile, of DK1 becomes progressively larger than the corresponding results in TKK as decreases from 7 MeV; e.g., at MeV, . (At this low energy the contribution of ignored in TKK is completely negligible.) A recent study by Doi and Kubodera [40] (to be referred to as DK2) indicates that this behavior is caused by the approximate Q-value expression used for this channel in DK1. As far as comparison with YHH2 is concerned, we remark that the exchange-currents effects are altogether ignored in YHH2. This is estimated [37] to lead to a general underestimation of the cross sections by up to 5 %. Thus each of the published works, TKK [37], DK1 [38] and YHH2 [39], has some point(s) to be improved upon.

Kohyama and Kubodera [41] have extended the calculation of Tatara et al. [37] by including the contributions of the terms. The results of Kohyama and Kubodera [41] and those of DK2 [40] have been found to agree with each other within 2 %. Since these two calculations are independent, both in the calculational methods and in choosing various input parameters, it is reassuring that their results agree to this degree. (According to TKK, different choices of the nuclear potential from among realistic nucleon-nucleon interactions affects the cross sections by up to 1 %.) We tabulate in table 1 in Appendix the cross sections for the reactions eqs.(8)-(11) obtained by Kohyama and Kubodera [41]; 5 % uncertainties are attached to the cross sections. Table 1 represents the best available estimates of the neutrino-deuterium cross sections up to MeV.

One generally expects that the ratios and are much less sensitive to the nuclear models than the absolute cross sections themselves. Indeed, the and obtained by Kohyama and Kubodera [41] agree with the corresponding values given in DK2 [40] within 0.2 %. Therefore, it seems reasonably safe to assign 1 % uncertainties to the calculated values of and .

The average cross section for the electronic neutrinos from the decay calculated using the results of [41] is: . This is consistent with the observed value [42].

Apart from testing the MSW effect for the solar neutrinos, Bahcall et al. [35] considered the merit of the SNO in determining the masses of heavy-flavor neutrinos through arrival-time-delay measurements of supernova neutrinos. Some other possible uses of the SNO for solar and supernova neutrinos were discussed by Balantekin and Loreti [43]. The SNO can also be used to study neutrinos of extremely high energies such as solar-flare and atmospheric neutrinos. The above described “ab initio” calculation probably can be extended up to several hundred MeV, but its reliability will diminish rather significantly for these high energies. At yet higher energies, the results are expected to approach those of the high-energy approximation to be discussed in chapter 6.

2.2. Neutrino-production reaction on the lightest nuclei

Although our review is primarily concerned with neutrino-nucleus reactions, this is a good place to discuss the latest development in the theoretical treatment of neutrino-production reactions on the lightest nuclei. As mentioned earlier, the pp reaction

(12) |

is the primary solar thermonuclear reaction and produces a dominant part of the solar neutrino flux [2]. The “hep” reaction

(13) |

contributes only a tiny portion of the solar neutrino flux [2], but the maximum neutrino energy of the hep process ( MeV) is the largest of all the solar neutrinos. Carlson et al. [44] have carried out a detailed ab initio calculation of the cross sections for these reactions, using as a nucleon-nucleon interaction the Argonne potential. To reduce the uncertainty in the axial exchange current operator, its matrix element has been adjusted so as to reproduce the measured Gamow-Teller matrix element for tritium -decay. For the pp reaction, the calculated astrophysical factor and its derivative at zero energy are: MeV b, and MeV b; these are close to the values quoted in [5]. Carlson et al. find that exchange-current contribution enhances the capture rate by 1.5 %. This is considerably smaller than the enhancement found in earlier works [45, 46]. In the absence of Coulomb distortion the transition matrix element of the pp reaction is just the hermitian conjugate of that of the reaction. For the thermal pp reaction, however, the Coulomb repulsion between the two protons is extremely important and responsible for the reduction of exchange-current contribution as compared with the typical values found for reaction [37].

As for the HeHe reaction rate, Carlson et al. [44] obtained MeV b, which is smaller by a factor of than the values obtained earlier by Werntz and Brennan [47] and by Tegner and Bargholtz [48]. This change is caused primarily by the exchange current effect, which cancels the impulse approximation contribution. This is one of the most dramatic examples of the exchange-current effect.

Chapter 3 Elementary-particle treatment (EPT)

The elementary-particle treatment (EPT) was first introduced by Kim and Primakoff [49] and by Fujii and Yamaguchi [50]. In EPT, instead of describing nuclei in terms of nucleons or other constituents, one treats nuclei as “elementary” particles with given quantum numbers. A transition matrix element for a given process is parametrized in terms of the nuclear form factors solely based on the transformation properties of the relevant current and nuclear states [49, 51, 52]. Insofar as all of these nuclear form factors can be deduced empirically from experimental data, one can make a totally model-independent prediction on every observable for the given transition. In fact, this requirement is an extremely stringent one, and so far the EPT calculation has been carried out completely only for the triad of B -C -N.

This particular =12 case, however, deserves special attention. One reason is that the C nucleus, which is abundantly contained in ordinary liquid scintillators, can be a very important nuclear target in neutrino astrophysics. Although the threshold energies of the neutrino reactions on C are too high for solar neutrino detection, they are low enough for observing energetic neutrinos originating from stellar collapses. The neutrino energies up to MeV are expected to be of relevance for stellar collapse neutrinos. Another reason is provided by the intensive programs of beam-dump neutrino experiments at Los Alamos [53, 54, 55, 56] and the Rutherford Laboratory [77, 58]. Some of these experiments directly aim at observing the neutrino -C reactions. In other particle-physics experiments [53] main interest is in the scattering but here also, to eliminate the dominant background due to the neutrino -C reaction, one needs to know its cross sections with reasonable accuracy.

The processes of relevance here are the superallowed neutral-current (NC) reactions

(14) | |||||

(15) |

and the superallowed charged-current (CC) reactions

(16) | |||||

(17) |

The final nuclear states of the above reactions form a triad of states. The existence of these superallowed transitions is favorable for simultaneous monitoring of the NC- and CC-reactions. (The elastic scattering leading to is unobservable in normal circumstances.) Fukugita, Kohyama and Kubodera [59] showed that EPT allows the completely model-independent determination of the cross sections for the reactions eqs.(14)-(17) for 100 MeV. A similar treatment was presented by Mintz and Pourkaviani [60, 61]. Although the use of EPT is at present limited to the transitions listed above, this is not an obstacle for monitoring these exclusive processes, as discussed in [59]. Here we summarize the calculation of Fukugita et al. [59].

The effective Hamiltonian responsible for the NC reactions is given by

(18) |

where is the Fermi constant, , and , with being the Weinberg angle. (The isoscalar neutral current, which cannot cause nuclear excitations, has been dropped.) The effective Hamiltonian for the CC reactions is given by

(19) |

where is the Cabibbo angle; , and ; . The most general form of the matrix elements of the vector and axial-vector currents for the NC reactions is

(20) |

(21) |

where , , is the polarization vector of the spin-1 nucleus, and is the average of the initial and final nuclear masses. All information on nuclear dynamics is contained in the nuclear form factors, , , , and . Here the nuclear form factors are classified in the “cartesian” representation. An alternative formalism is described in [51, 63]. For the CC reactions the Wigner-Eckart theorem in isospin space leads to

(22) |

(23) |

Now, the contribution of the term to the observables is proportional to and therefore can be dropped in the present energy regime ( MeV) for which muon-production can be ignored. For the remaining nuclear form factors one can first consider their values at , and examine their -dependence later. Define , , and . can be determined from the width of the -decay , and the CVC. From MeV [64], we obtain . and can be determined from the decay and the decay . The -value of the -decay is written as

(24) |

with the maximum -particle energy. The measurement of the -ray angular distribution from aligned B and N parent nuclei gives the angular correlation quantities , which are related to as [65, 66, 67].

(25) |

The experimental values [68, 69], and , give . Then the contribution of the term in eq.(24) is for MeV. The importance of this term relative to 1 being smaller than typical isospin-symmetry breaking effects within nuclei (), the term in eq.(24) can be ignored. Experimentally [64], for , and for . The difference between and gives a measure of the isospin symmetry breaking quoted above (insofar as there is no second-class currents). Using s, which is the average of and with the error corresponding to their difference, we obtain .

As grows, one must consider the -dependence of the nuclear form factors. The total cross section for the NC reactions including the -dependence is given by

(26) | |||||

(27) | |||||

(28) | |||||

where the upper (lower) sign refers to the neutrino (anti-neutrino) reaction; with being the neutrino scattering angle; with MeV; with . The cross sections for the CC reactions are given by

(29) |

where the upper (lower) sign refers to the neutrino (anti-neutrino) reaction. Here is as defined previously except the replacements , and ; denotes the Fermi function for the Coulomb correction. , with MeV ( MeV) for (). The above expression has been obtained by assuming that the -dependences of and are the same as that of . From an EPT analysis of the process, Nozawa et al. [70] showed that holds within a 10 % accuracy at least up to . For no direct experimental information is available but, since its contribution is minor, its precise form has little significance. Now, can be determined model-independently from obtained from inelastic electron scattering: . According to [71],

(30) |

where fm, and .

Using the EPT summarized above, Fukugita et al. [59] were able to determine the cross sections for eqs. (14)-(17) up to MeV within accuracies; these uncertainties reflect the ambiguities in the input data and isospin symmetry breaking in nuclei. Similar results were obtained by Mintz and Pourkaviani [60, 61]. As in the A=2 case discussed earlier, the ratio of the NC cross section to the CC cross section is expected to be less affected by the nuclear physics ambiguities. When we vary the nuclear form factors within 10 % (which are typical errors given in ref. [59]), is found to change by less than 2 %.

We now remark on another semi-empirical method developed by Walecka and Donnelly to estimate the matrix elements of weak-interaction transitions in light nuclei. In the Walecka-Donnelly (W-D) method [72, 73, 22], one assumes that the weak-interaction transition operator can be approximated by a sum of single-particle multipole operators that arise in the impulse approximation and that the electomagnetic data can be used to place constraints on the nuclear matrix elements of these operators. Because of these basic assumptions it should be obvious that, contrary to the often-made claim, the W-D method is not a model-independent approach. Apart from the fundamental assumption pertaining to the impulse approximation, an important question is to what extent one can identify the effective matrix element of a single-nucleon operator appearing in the weak-interaction process with that of the that appears in the electromagnetic process. Chemtob and Rho [76] emphasized that the exchange-currents for the electromagnetic interaction can be drastically different from those of the weak processes. This point has been reemphasized in [44]. The EPT is free from these problems and hence, from a formal point of view, definitely a preferable approach. Having said this, we must quickly add that EPT in its rigorous form can be used at present only for the very limited case of A=12. By contrast, the W-D method, owing to the additional assumptions made on the possible form of the transition operators, can be applied to many cases where the lack of necessary experimental information renders EPT at moment powerless. Thus, in practice, the W-D method is a very useful approach insofar as some independent information is available to check its reliability for given individual cases. The use of experimental data to constrain the parameters involved in the W-D method is in general expected to reduce nuclear-model dependence significantly, but the remaining possible uncertainties should be kept in mind.

Donnelly and Peccei [22] used the W-D method to estimate the cross sections for the NC reactions eqs. (14),(15). While a good agreement is seen between Donnelly and Peccei’s results and those of Fukugita et al. [59] at lower incident energies, disagreement becomes apparent for MeV, and it amounts to 60 % at MeV. Furthermore, while Fukugita et al. report an appreciable difference between and at higher energies due to the interference term , this behavior is not reproduced in [22]. These discrepancies seem to signal the breakdown of some assumptions involved in the W-D method [22] as applied to higher incident neutrino energies. On the other hand, for low neutrino energies, the EPT results justify the W-D method [72], in which the leading-order impulse-approximation transition matrix element in the weak interaction process is deduced from the observed M1 strength. For the estimate of Fukugita et al. [59] is significantly smaller than the published result of the W-D method [74]; specifically, for the cross section averaged over the spectrum due to the -decay, the EPT gives , which is about 25 % lower than that of Donnelly [74]. However, a more recent result of the W-D method [75] agrees with the EPT result.

The KArlsruhe-Rutherford Medium Energy Neutrino (KARMEN) experiment has been intensively pursued at the spallation facility of the Rutherford Laboratory [57, 58, 77]. The pulsed beam dump neutrino source provides monoenergetic with MeV from decay at rest as well as and with energies up to 52.8 MeV coming from the subsequent decay. A 56 t liquid scintillation calorimeter [77] is used as massive live target of C. The NC reaction eq.(16) has been observed [78] by monitoring the emission back to the ground state ( MeV, B.R = 94 %). Experimentally [78], represents the average over the spectrum coming from the decay. This agrees well with theoretical estimates 59, 61, 63]. The reaction can be identified by taking the delayed coincidence of the recoil and the from the N -decay. The KARMEN measurement [79, 80] gives , which is consistent with the earlier Los Alamos result [54]: . These experimental values are in good agreement with recent theoretical estimates given in [59, 75, 60]. [ , where

EPT calculations on the -C reactions for MeV were carried out by several authors. In particular, Mintz and Pourkaviani [62] extended the EPT calculation up to GeV. The reliability of such extension, however, is not well established since the calculation involves the nuclear form factors at very high , which are not known empirically except . There have also been attempts to apply EPT to nuclear systems other than the C target [63] or to processes [60]. Here again, the problem is that there is not enough data to determine all relevant nuclear form factors empirically. The lack of experimental information necessitates the introduction of a number of additional theoretical assumptions, which are usually motivated by the impulse approximation results. This ad hoc procedure largely nullifies the predictive power of the original EPT.

Koetke et al. [55], using the in-flight pion decay neutrino source at LAMPF, measured the cross sections for and at an average neutrino energy of 202 MeV. The cross section for the latter exclusive process is . This is considerably higher than obtained in the W-D method [81, 82, 83] and an “EPT” estimate of [60]. Here we have used the quotation mark since, as discussed above, a completely model-independent EPT analysis is not feasible in this case. According to Mintz and Pourkaviani [60], one can increase the predicted value sufficiently by enhancing the pseudoscalar form factor Although it is generally believed that the exchange-current effect quenches rather than enhances, there may be other competing nuclear effects. (For recent attempts to determine the effective inside nuclei using the radiative -capture, see ref.[84] for experiment and ref.[85] for theory.) The inclusive reaction will be discussed later.

Chapter 4 Empirical effective operator method (EEOM)

Going back to the “realistic” nuclear Hamiltonian eq. (7), we recall that, even if we forgo the basic problem of deriving from the fundamental QCD and decide to be content with the operational usefulness of , it is in general prohibitively difficult to determine true eigenstates ’s of and calculate the nuclear matrix elements of an observable using ’s. This is true even when we limit ourselves to the nucleon-only regime, ignoring the exchange-current problem. One therefore usually splits the space subtended by ’s into two parts, a relatively manageable model space and the remainder , and calculate observables using wave functions ’s that belong to . The phenomenological success of the shell model suggests choosing as the space of the lowest shell-model configurations; if necessary, slightly excited configurations can be included in . Once the space of ’s is truncated to that of ’s, and must undergo corresponding transformations, and so that and should hold. The nucleon-nucleon interactions that feature in are called the effective interactions, a somewhat historical terminology since from today’s viewpoint the original itself is a highly “effective” entity. Similarly, is called the effective operator. The difference between and represents the core-polarization effect. The formal framework to obtain the effective operators by incorporating effects of the eliminated space does exist (for review, see [86, 87]), but applying this formalism to actual complex nuclei encounters many practical difficulties. Exceptionally, in the simplest cases of closed-shell nuclei plus or minus one nucleon, explicit calculations can be carried out to lowest orders of perturbation, as exemplified by the extensive work of Tower and Khanna [93, 94] and the University of Tokyo group (Arima, Shimizu, Hyuga et al) [95]. These authors calculated the magnetic moments, M1 transition strengths , and the Gamow-Teller strengths (GT), taking into account up to second-order core polarization effects. Furthermore, the exchange-currents effects were carefully estimated and included in the final expressions of their effective operators. The results of these highly elaborate calculations are thoroughly documented in [94, 95].

Parallel to these theoretical attempts to derive the effective operators for the special cases, a useful semi-empirical approach which can cover a wider variety of nuclei has been developed by Wilkinson, Brown, Warburton and Wildenthal [88, 89, 90, 91, 92]. We shall refer to this approach as the empirical effective operator method (EEOM). As will be explained below, EEOM finds a useful application in estimating the cross sections of low-energy neutrino-nucleus reactions for nuclei up to the -shell. We outline EEOM, taking the Gamow-Teller (GT) operator as an example [92].

Consider a GT transition and suppose we are given reasonably realistic nuclear wavefunctions for and . By “realistic” we usually mean the best available shell-model wavefunctions, which, as today’s standard demands, are eigenvectors of a full matrix of covering all possible configurations within a given major shell. The shell-model GT matrix element is given by

(31) | |||||

where the summation over and goes over all the single-particle orbits contained in the model space , and the one-body-transition density is obtained from shell-model wavefunctions for an operator with rank in ordinary (isospin) space of () via

(32) |

As stated, even if the model space is reasonably realistic, there are two significant sources of corrections to , one from core polarization and the other from the exchange-current effect. If is large enough, the core polarization effect arises from rather highly excited states, and consequently its dependence on and will be weak. The same should hold for the exchange-current effect, which comes from the eliminated hadronic degrees freedom. In EEMO, therefore, one simulates these effects by replacing in eq. (31) with an effective matrix element

(33) |

while keeping unchanged. This replacement will give an effective GT strength . To determine the effective coupling constants, , and for a given major shell, one takes a sufficiently large number of GT decays that belong to this major shell and whose values, or equivalently (GT)’s, are known experimentally. One then adjusts , and in such a manner that the (GT)’s calculated with would optimally reproduce the chosen set of experimental (GT)’s. Once , and are determined, one can predict all other (GT)’s involving the same major shell.

The EEMO has proven to be highly successful in correlating a great majority of the observed magnetic moments, M1 strengths, and Gamow-Teller strengths over a wide range in the periodic table [90, 91, 92]. Furthermore, the effective coupling constants determined in EEOM show reasonable agreement with the corresponding quantities obtained theoretically for the closed-shell plus/minus one nucleon systems [94, 95].

Two p-shell nuclei, B and C, have recently been considered as useful targets to simultaneously monitor the NC and CC reactions for the solar neutrinos. In fact, Raghavan, Pakvasa and Brown’s proposal [96] to use the B target is already being intensively pursued in the BOREX and BOREXINO projects. We describe here how the EEOM (or a closely related approach) was used to estimate the cross sections on these targets.

4.1. C target

The possible advantage of a C-enriched scintillation counter as a solar-neutrino detector was suggested by Arafune et al. [97] and further studied in [99]. These authors emphasized that the replacement of C in a scintillator with C not only makes the detector much more sensitive to CC reactions, but also allows efficient detection of the NC reactions, and that even the natural abundance of C (1 %) may be large enough to detect NC reactions in a large volume scintillator. The relevant reactions are the CC reaction

(34) |

and the NC reaction

(35) |

For the solar neutrino energy region. only the following final states need to be considered [98]. For the CC reactions, , , , and in N, with MeV for the ground state transition. For the NC reactions, , , and in C.

The total cross section for the CC reaction is given in impulse approximation by

(37) | |||||

where . The Fermi matrix element contributes only to a transition between isomultiplet members. One often uses the reduced transition strengths (F) and (GT) defined as

(38) | |||||

(39) |

The NC cross section is given by

(40) |

For the CC, if the value of decay is known, is given model-independently in terms of :

(41) |

This applies to . From for , we obtain model-independently . Here stands for the cross section averaged over the B neutrino spectrum. (Due to the threshold energy MeV, only the neutrinos from B decay are relevant here.) As is a super-allowed transition, one can expect a strong feeding of N(gnd). For the transitions to the excited states in N, for which no experimental values are available, one must rely on theoretical