Radiative neutron [beta]-decay in effective field theory

Journal of Research of the National Institute of Standards and Technology, July-August, 2005 by Susan Gardner, Veronique Bernard, Ulf-G. Meissner, Chi Zhang

We consider radiative [beta]-decay of the neutron in heavy baryon chiral perturbation theory. Nucleon-structure effects not encoded in the weak coupling constants [g.sub.A] and [g.sub.V] are determined at next-to-leading order in the chiral expansion, and enter at the O(0.5%)-level, making a sensitive test of the Dirac structure of the weak currents possible.

Key words: neutron [beta]-decay; radiative corrections.

1. Framework

Experimental studies of [beta]-decay at low energies have played a crucial role in the rise of the Standard Model (SM) [1]. In recent years, continuing, precision studies of neutron [beta]-decay have been performed, to better both the determination of the neutron lifetime and of the correlation coefficients. To realize a SM test to a precision of [approximately equal to]1% or better requires the application of radiative corrections [2]. One component of such, the "outer" radiative correction, is captured by electromagnetic interactions with the charged, final-state particles, in the limit in which their structure is neglected. In this, neutron radiative [beta]-decay enters, and we consider it explicitly. We do so in part (i) to study the hadron matrix elements in O(1/M), as the same matrix elements, albeit at different momentum transfers, enter in muon radiative capture [3], and (ii) to test the Dirac structure of the weak current, through the determination of the circular polarization of the associated photon [4, 5]. Here we report on our recent work--please see Ref. [6] for all details.

In neutron radiative [beta]-decay, bremsstrahlung from either charged particle can occur, and radiation can be emitted from the effective weak vertex. In the pioneering work of Ref. [4] only the bremsstrahlung terms are computed--this suffices only if all O(1/M) terms are neglected. Here we describe a systematic analysis of neutron radiative [beta]-decay in the framework of heavy baryon chiral perturbation theory (HBCHPT) [7, 8, 9] and in the small scale expansion (SSE) [10], including all terms in O(1/M), i.e., at next-to-leading order (NLO) in the small parameter [epsilon] [6]. We note that [epsilon] collects all the small external momenta and quark (pion) masses, relative to the heavy baryon mass M, which appear when HBCHPT is utilized; in case of the SSE, such is supplemented by the [DELTA](1232)-nucleon mass splitting, relative to M, as well. These systematic approaches allow us to calculate the recoil-order corrections in a controlled way.

We consider n(p) [right arrow] p(p') [e.sup.-]([l.sub.c]) [bar.v.sub.e]([l.sub.v]) [gamma](k), where p, p', [l.sub.e], [l.sub.v], and k denote the four-momentum of the neutron, proton, electron, anti-neutrino, and photon, respectively--we denote the photon energy by [omega]. At low energies, the matrix element for radiative neutron [beta]-decay decomposes into two pieces,

M(n [right arrow] p[e.sup.-][bar.v.sub.e][gamma]) = i[[g.sup.[alpha][beta]]/[M.sub.W.sup.2]][<[bar.v.sub.e][e.sup.-]|[J.sub.[alpha].sup.-]|0> <[bar.v.sub.e][e.sup.-][gamma]|[J.sub.[alpha].sup.-]|0>], (1)

in terms of the leptonic weak current ([J.sup.-]), as well as the hadronic vector (V) and axial.vector (A) currents. Note that [[epsilon].sub.[mu]] is the photon polarization vector and [M.sub.W] is the W-boson mass. The first term includes bremsstrahlung from the proton, as well as radiation from the effective weak vertex, whereas the second term describes bremsstrahlung from the electron. We now turn to the leptonic and hadronic matrix elements which appear. The leptonic current matrix elements follow from QED, in concert with the V-A structure of the weak current. The latter, cum Lorentz and translational invariance [11], also fixes ; the form factors which appear therein can be determined from experiment. To compute the remaining matrix elements, , we employ HBCHPT. Thus the heavy baryon is treated non-relativistically, and its interactions are organized in powers of [epsilon]. We work in O(1/M) throughout, so that our matrix elements include photon emission from the weak vertex as well. For consistency we also treat in the non-relativistic limit, expanding to O(1/[M.sup.2]) throughout. We note that the pertinent two- and four-point functions can be taken directly from Ref. [3], after relabeling the momenta and such [6]. Working in the Coulomb gauge [epsilon]* * v = 0 for the photon and making use of the transversality condition [epsilon]* * k = 0, we find is of O(1/M), so that only electron bremsstrahlung makes an O(1) contribution to radiative neutron [beta]-decay.

2. Results

We now present our results [6]. We show the photon energy spectrum d[GAMMA]/d[omega] in Fig. 1, and for the total branching ratio, which depends on the range chosen for [omega], we find,

[omega] [member of] [0.005 MeV, 0.035 MeV], Br : 2.59 * [10.sup.-3],

[omega] [member of] [0.035 MeV, 0.100 MeV], Br : 1.11 * [10.sup.-3],

[omega] [member of] [0.100 MeV, [[omega].sup.max] = 0.782 MeV], Br : 0.72 * [10.sup.-3], (2)

The branching ratio determined for [omega] [member of] [0.035 MeV, 0.100 MeV] can be compared directly with the experimental limit of Br < 6.9 * [10.sup.-3] (90% CL) [12], with which it is compatible. In Fig. 1 we superimpose the numerical results we find with those using the leading order form of [[summation].sub.spins]|M|[.sup.2]. The two curves can scarcely be distinguished; indeed, the recoil-order corrections to the matrix elements are no larger than O(0.5%). The SSE contribution is itself of O(0.1%). In contrast, the recoil-order corrections to the A and a correlations in neutron [beta]-decay are of O(1-2%) [13]; apparently, the appearance of an additional particle in the final state makes the recoil-order corrections smaller still.