Pis'ma v ZhETF, vol. 101, iss. 10, pp. 733-738
© 2015 May 25
A study of neutrino decay in magnetic field with the "worldline instanton" approach
P. Satunin1')
Institute for Nuclear Research of the RAS, 117312 Moscow, Russia Submitted 1 April 2015
We study the process of neutrino decay to electron and IF-boson in the external magnetic field using the semiclassical "worldline instanton" approach. Being interested only in the leading exponential factor, we make calculations in a toy model, treating all particles as scalars. This calculation determines the effective threshold energy of the reaction as a function of the magnetic field. Possible astrophysical applications are discussed. It is emphasized that the method is general and is applicable to a decay of an arbitrary neutral particle into charged ones in the external electromagnetic field.
DOI: 10.7868/S0370274X1510001X
Introduction. Recent detection of very high energy (up to 1015 eV) neutrinos in the IceCube experiment [1, 2] can open a new branch of astrophysics - very high energy neutrino astronomy. Although the angular resolution of neutrino detectors is rather low, it will significantly improve in the near future, and may reach the level sufficient for identification of neutrino sources.
One class of potential neutrino sources are pulsars and magnetars. These objects generically have a super-strong magnetic field around them in a radius of several kilometres. This raise a question: Can neutrinos escape the region of strong magnetic field if they are produced inside it? Neutrino dispersion relation in the external magnetic field is modified [3], so its decay, forbidden in the absence of the field, becomes allowed. The main decay channels are v —> ve+e~ and v —> e~W+.
These processes have been studied in many works [4-10] (see also the book [11]). The dependence of their widths on the magnetic field and neutrino energy exhibits the following common feature. At small fields or energies they are exponentially suppressed while when the energy or magnetic field exceed certain values the suppression disappears. In other words, the above reactions proceeds effectively only above a certain threshold energy, which depends on the value of the magnetic field.
The reaction
(f)
being of the first order in the weak coupling constant, gives the leading contribution to the neutrino decay width once the energy exceeds the threshold.
^e-mail: satunin@ms2.inr.ac.ru
It was analysed for subcritical magnetic fields2' in [4-6] and for supercritical fields in [7]. The reaction (1) reduces the neutrino mean free path to the values shorter than the astrophysically relevant distances just after it leaves the regime of exponential suppression [5]. This will produce a cutoff in the spectrum of neutrino sources if the latter possess strong magnetic fields in the region of neutrino emission.
We study the process (1) using the semiclassical "worldline instanton" method. This method is technically much simpler than the standard approach [4-7], based on the exact expressions for electron and VF-boson wave functions (or propagators) in the magnetic field, and provides an independent verification of the results existing in the literature.
Worldline path integral approach [12] is a powerful tool to study non-perturbative phenomena in quantum field theory, such as particle production in a classical external field. The well-known example is the Schwinger effect [13] - creation of electron-positron pairs from vacuum in a constant electric field. Affleck et al. showed
[14] that the rate of the process can be expressed as the quantum mechanical partition function of an auxiliary system describing periodic motion of a charged particle in the external field, analytically continued to Euclidean time domain. The corresponding path integral can be evaluated in the saddle point approximation. The method was generalized to pair production in time-dependent and space-dependent electric fields
[15], including the case of pair production induced by
2'The critical, or Schwinger, magnetic field is obtained as Hcr = rn?eJe ~ 4 • 1013 G, where me and e are the electron mass and charge.
a photon in the initial state [16]. In [17] this approach was used to study decay of photon to e+e~ pair in the magnetic field.
It was shown [14, 15], that the exponential part of the rate of such Schwinger-like processes does not depend on the spin of charged particles (spin dependence appears only in the pre-exponential factor). So, for simplicity we will consider in our work all particles participating in the process as scalars.
The width of neutrino decay. We consider neutrino decay to an electron and M/+-boson in the external magnetic field. We are interested only in the main exponential behaviour of the result, which should be independent of particle spins. Instead of the electroweak theory for simplicity we consider a toy model with scalar particles. The Lagrangian of the model is
+ D^DfiX ~ mwX*X + + + gi4>*X + h-c. (2)
Here £ is a real massless scalar field, representing "neutrino", 4> and x are scalar "electron" and "VF-boson"; me and mw denote their masses. Interaction term includes constant g of a mass dimension. Fields 4> and x interact with a gauge field A^, which has the standard kinetic term. Covariant derivative T>M is defined as usual, D^ = - ieAM) 4>.
In terms of our toy model we consider the process of a £ particle (neutrino) decay to a pair of 4> particle and x antiparticle (scalar electron and VF-boson, respectively). Neutrino with four-momentum kM = (w, k) propagates orthogonally to the uniform magnetic field H. We choose the coordinate system where the magnetic field is directed along the x-axis, neutrino momentum -along the y-axis, so k = (0, w,0). The reaction is kine-matically allowed if uj > mw + me, we study it well above the threshold, uj mw + me. Let us mention that all subsequent formulas are valid if the electron is replaced by muon or tau-lepton.
Following the optical theorem, the width of £ can be obtained from the imaginary part of its self-energy:
T = ¿ImS(fc), (3)
where £(&) is the Fourier transform of the correlator:
s(y-z) = {x,\y)4>{y)4>,-{z)x{z))+{4>,-{y)x{y)x,-{z)4>{z)).
(4)
The first term in Eq. (4) corresponds to creation of a <f> particle and x antiparticle; the second term - to creation of a x particle and 4> antiparticle. We will concentrate
only on the first term for two reasons. First, in the model (2) the exponential parts of both terms are equal, so for simplicity we can restrict to one of them. Second, in the realistic case of the Standard Model neutrino self-energy does not contain an analogy of the second term due to the lepton charge conservation.
The two-point Green function can be represented as (see [18]):
POO
{4>*(z)4>(y))= / dT1{y\e-T(Dl+m')\z) = Jo
= r-^-e-^L r{Tl)=Z" DXe-f«^QMdr (5)
Jo NjXiA DXG '(5)
where
. o
Xtl
£qm = - (6)
i2
and N = Dxe- fo ~iLdT is a normalization
factor. Summation over repeated Greek indices with Euclidean signature is understood. The notation denotes derivative over r. The formula similar to (5) is valid for the (x*(y)x(z)) correlator. Substituting both correlators into the self-energy (4) we introduce integrals over xM(0) and xM(Ti) and corresponding delta-functions, fixing the boundary conditions at the points r = 0, r = T\. Thus, we obtain a single path integral with periodical boundary conditions:
- z) <x
p.b.c
x ¿^MTi) - y^e-m2^-m2^-foTl+T2 £QMdr (7)
The notation p.b.c means periodical boundary conditions xm(t) = xm(t + Ti + T2). Setting + =0 without loss of generality and making the Fourier transformation of (7), we obtain3':
TcxJmJ^^-J^^ J Dx,5^[x,(0)+xM(T1)]x
p.b.c
x e-miT1-m2wT2-J^1 + T2 LQMdr+ik^x^T^-x^0)] _ ^
Here we introduce the euclidean neutrino momentum kE = (iuj, k). We break the integral in the exponent of (8) into two parts: the first part is defined at the segment r G [0, Ti], in the second part r e pi, Ti + T2]. For convenience we make a change of variables: in the first
3'ln calculations it is convenient to consider delta functions in the integral representation.
A study of neutrino decay in magnetic field with the uworldline instanton" approach
735
integral we set r' = tj^tt, in the second - r' = -ijrT. The (minus) exponent of (8) becomes
f1'2 i2 f1'2 5[xM] = m2Ti + / 7^rdT ~ ie / A^k^dr + Jo i Jo
r 1
+ ni^To +
f 1 ^»2
——(It — ie
'1/2
8 T2
I 1/2
A^x ¡¿drr
(9)
Thus, the paths describing electron and TF-boson correspond to r e [0,1/2] and [1/2, f] respectively. The expression (9) has the form of a sum of two Euclidean actions of two relativistic particles with different masses in the external electromagnetic field. Two sources of opposite signs located at the proper times 0 and 1/2, are added into the action. The strength of the sources depends on the neutrino momentum.
We expect that the integrals on the r.h.s. of (8) can be evaluated in the saddle point approximation. The saddle point equations for xM(r) give the classical trajectories (in general complex), which should be substituted into the action (9). If the action (9) on the solution is parametrically large, the width of the process is suppressed by the exponent of the action (with the minus sign). For further calculations it is convenient to choose the gauge AM = —^F^x^. Varying the action (9) over we obtain the equations for different regions of parameter r (we denote the solutions in the two regions
xi1] and x
(2) \
and
„(2) _
= -A0(r-i
„(2)
„(2)
-A
—¿A(2)sh
402 [ r - -
(15)
(2)
ch
402 T-
- Ch02
Here, for simplicity, instead of T.we use dimension-less parameters 0i = TieH. Other parameters are determined in the following way:
An =
4 w 6it
oj sh0j W eff 811(0,+02)'
eHOi + P' "yv ■ ^ ^' (16)
here j = 2 if i = 1 and vice versa. This solution is shown in Fig. f. Substituting solution (f4), (f5) into (9), we
IX'-)
\ 3me 1 2o)eH > \ \ \ i \ *
m„
eH
\ \ i
—i—
3 m
2 /
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