Two-vertex loop diagrams in the two-dimensional subspace induced by a superstrong magnetic field: polarization operator
- PDF / 520,676 Bytes
- 7 Pages / 612 x 792 pts (letter) Page_size
- 86 Downloads / 175 Views
EMENTARY PARTICLES AND FIELDS Theory
Two-Vertex Loop Diagrams in the Two-Dimensional Subspace Induced by a Superstrong Magnetic Field: Polarization Operator and Photon and Axion Decay to Neutrinos V. V. Skobelev* Moscow State Industrial University, Avtozavodskaya ul. 16, Moscow, 115280 Russia Received April 26, 2010; in final form, October 11, 2010
Abstract—The absence of divergences and singularities in effectively two-dimensional field theory induced by a superstrong magnetic field is demonstrated for the example where loop diagrams involving two vector and two pseudovector vertices are calculated. The form of effective low-energy Lagrangians for (γνν) and (aνν) interactions in a superstrong magnetic field is presented. The role of photon and axion decays to neutrinos in the early universe is discussed. DOI: 10.1134/S1063778811040144
1. INTRODUCTION A calculation of matrix elements associated with diagrams involving electron loops in a strong magnetic field whose induction satisfies the condition B B0 = m2 /e = 4.41 × 1013 G (where m is the electron mass and e is an elementary charge) has some special features stemming from the fact that the respective mathematical formalism becomes effectively two-dimensional [1]. Among other things, we will see that divergences typical of the fourdimensional case disappear along with some special features of the theory, such as the Adler chiral anomaly. The objective of the present study is to demonstrate this for the example of two-vertex loop diagrams describing photon and axion decays to neutrinos (see Figs. 1 and 2, respectively). Preliminarily, the methods used here will be illustrated within QED for the example of calculating the photon polarization operator P , which was previously found by the present author in [2] (the respective result with allowance for temperature effects was obtained in [3]) within the two-dimensional covariant formalism being developed, the divergence being removed under the assumption of a gauge-invariant structure of the polarization tensor. Specifically, we have kμ kν , (1) Pμν (k) = P (k) gμν − 2 k k2 = k02 − k32 , *
where the 3 axis is aligned with the field direction, as in the previous studies of the present author. Here, it will be shown that the divergence is canceled without recourse to regularization procedures and that a gauge-invariant structure is obtained automatically. We start from the definition of the polarization tensor in the form d4 xei(kx)
Pμν (k) = 4παi
(2)
× tr [γμ G(x)γν G(−x)] , α = e2 = 1/137, where G(x) is the coordinate-difference-dependent part of the Green’s function in a superstrong mag iγ (x1 + y1 )× netic field [1] (the phase factor exp 2 (x2 − y2 ) in the two-vertex loop is canceled). Substituting the expression for it into Eq. (2) and going over to two-dimensional γ matrices and contractions [1], we obtain 2 k⊥ 2iαγ (3) Pμν (k) = 2 exp − 2γ (2π) γ
μ, ν = 0, 3,
e
ν
e
ν–
Fig. 1. Photon decay to neutrinos in a superstrong field.
E-mail: [email protected]
624
×
d2 ptr γμ
d2 p = dp0 dp3 ,
TWO
Data Loading...
