A formalism for weak interactions in nuclei

A formalism for weak interactions in nuclei

A. Samana^I; F. Krmpoti^I; A. Mariano^II

^IInstituto de Física, Universidade de São Paulo C. P. 66318, 05315-970 São Paulo, SP, Brazil

^IIDepartamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, La Plata, 1900, Argentina

ABSTRACT

The neutrino-nucleus cross-section and the muon capture rate are discussed within a simple formalism which facilitates the nuclear structure calculations. The corresponding formulae only depend on four types of nuclear matrix elements, which are currently used in the nuclear beta decay. We have also considered the non-locality effects arising from the velocity-dependent terms in the hadronic current. We show that for both observables in 12C the higher order relativistic corrections are of the order of ~ 4% only, and therefore do not play a significant role.

The semileptonic weak interactions with nuclei include a rich variety of processes, such as the neutrino (antineutrino) scattering, charged lepton capture, e^± decays, etc, and we have at our disposal the results of more than a half-century of beautiful experimental and theoretical work. At the time being their study is mainly aimed to inquire on possible exotic properties of the neutrino associated with its oscillations and massiveness by means of exclusive and inclusive scattering processes on nuclei, which are often used as neutrino detectors. An example is given by the recent experiments performed by both the LSND and the KARMEN Collaborations, looking for n_m ® n_e and _m ® _e oscillations with neutrinos produced by accelerators [1, 2]. Thus the knowledge of reactions induced by neutrinos on nuclei becomes a crucial step for the interpretation of experiments on neutrinos, and a reliable prediction of these cross sections is a challenging problem from the nuclear structure point of view.

The weak interaction formalism most frequently used in the literature is that of Walecka [3], where, in close analogy with the electromagnetic transitions, the nuclear operators are classified into Coulomb (), longitudinal (), transverse electric (^el) and transverse magnetic (^mag). We carry out a different multipole expansion of the V – A hadronic current and express all observables in terms of the vector (V) and axial vector (A) nuclear matrix elements.

The weak Hamiltonian is expressed in the form [4]

where

is the hadronic current operator, and

is the plane waves approximation for the matrix element of the leptonic current; G = (3.04545 ± 0.00006)×10^–12 is the Fermi coupling constant (in natural units),

is the momentum transfer (P_i and P_f are momenta of the initial and final nucleon (nucleus), M is the nucleon mass, is the mass of the charged lepton, and g_V, g_A, g_M and g_P are, respectively, the vector, axial-vector, weak-magnetism and pseudoscalar effective dimensionless coupling constants. Their numerical values are

The finite nucleon size (FNS) effect is incorporated via the dipole form factor with a cutoff L = 850 MeV, i.e., as:

To use (1) with the non-relativistic nuclear wave functions, the Foldy-Wouthuysen transformation has to be performed on the hadronic current (2). When the velocity dependent terms are included this yields¹ :

where the operator p º –iÑ acts on the nuclear wave functions, and the following short notation has been introduced:

For the neutrino-nucleus reaction k = – q_n, with º {, } and q_n º {E_n,q_n}, and the corresponding cross section from the initial state |J_iñ to the final state |J_fñ reads

where F(Z + 1,) is the Fermi function, q º · and

with y_i(r) º ár|J_iM_iñ and y_f(r) º á r|J_fM_fñ being the nuclear wave functions.

The transition amplitude can be cast in the form:

where the lepton traces L₄, L₄₀ and L_{m = –1, 0, 1} are defined in Ref.[5] , and the nuclear matrix elements

contain the operators :

Similarly, for the capture rate one gets [5]

where f_1S is the muonic bound state wave function evaluated at the origin, and

where the effective charges are

and the coupling constants are now

As an application, we have evaluate the contribution of the non-locality effects, arising from the velocity-dependent terms in the hadronic current, in weak processes with neutrinos and muons within the triad {¹²B,¹²C,¹²N}. The numerical calculations were performed within the particle number projection charge-exchange RPA (PQRPA) [6], by employing the same configuration space (nl = (1s,1p,1d,2s,1f,2p)) and the same residual force (V = –4p (v_sP_s + v_tP_t)d(r)) as in Ref.[5]. We also use the same parameterization as in this work for the coupling strengths within the particle-particle (pp) and particle-hole (ph) channels: = = 23.92 MeV-fm³, and = /0.6 = 39.86 MeV-fm³.

In the Table I are given the results for inclusive folded flux-averaged neutrino scattering cross sections

with = e^–,m^–, and where (E_n) is the normalized neutrino flux.

We have presented a somewhat new formalism for the neutrino-nucleus cross-section and the muon capture rate, based on a multipole expansion of the V – A hadronic current in terms of the transition operators (13), which are currently used in the beta decay. The resulting formulae are more simpler than those developed by Walecka [3] and as such facilitate the nuclear structure calculations. We have also considered the non-locality effects arising from the velocity-dependent terms in the hadronic current. We show that for both observables in ¹²C the higher order relativistic corrections are of the order of ~ 4% only, and therefore do not play a very significant role.

References

[1] LSND Collaboration, C. Athanassopulus et al. , Phys. Rev. C58, 2489 (1998).

[2] KARMEN Collaboration, K. Eitel et al. , Nucl. Phys. B, Proc.Suppl. 77, 212 (1999).

[3] J.D. Walecka, Muon Physics, ed. V.M. Hughes and C.S Wu (Academis, New York, 1975).

[4] C. Barbero, F. Krmpotic, and D. Tadic, Nucl. Phys. A628, 170 (1998); C. Barbero, F. Krmpotic, A. Mariano and D. Tadic, Nucl. Phys. A650, 485 (1999).

[5] F. Krmpotic, A. Mariano and A. Samana, Phys.Lett. B541, 298 (2002).

[6] F. Krmpotic, A. Mariano, T.T.S. Kuo, and K. Nakayama, Phys. Lett. B319, 393 (1993).

Received on 03 October, 2003

1

There is a misprint in eq.(5) of Ref.[5].

Publication Dates

• Publication in this collection
  26 Oct 2004
• Date of issue
  Sept 2004

History

• Received
  03 Oct 2003