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Дата изменения: Tue Aug 15 14:57:18 2000
Дата индексирования: Mon Oct 1 22:52:51 2012
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The problem

Every time when we are trying to create a model containing a massive vector particle we meet a problem caused by bad asymptotics of its propagator

$$\frac{i}{ (2\pi)^4}\, \frac{g_{\mu\nu} -k_\mu k_\nu/m^2}{m^2-k^2}\;.$$ (1)

The $( g_{\mu\nu} - k_\mu k_\nu/m^2 )$ factor gives a projection on physical degrees of freedom in the polarization space. This term appears because a 4-component vector field is used to describe a particle with 3 degrees of freedom. The $k_\mu k_\nu/m^2$ term leads to a fast growth of amplitudes at high energies, what breaks unitarity and is not compatible with the renormalizability of theory.

The problem mentioned above is solved in the framework of gauge field theories where the gauge symmetry is responsible for mutual cancellation of rapidly growing contributions of separate diagrams [Bjorken&Drell]. Let our model of particle interaction be based on a gauge theory. Then on a step of numerical evaluation we expect a cancellation of contributions which come from various Feynman diagrams. Consequently, finite precision numerical calculations may lead to wrong results. Accompanying problem is a cumbersome expression for each diagram as a result of appearance of mutually canceling terms.

At the same time there is a freedom in formulation of Feynman rules for gauge theories caused by an ambiguity of gauge fixing terms [Bjorken&Drell]. These terms modify the quadratic part of the Lagrangian and consequently may improve the vector particle propagator. Indeed, in the case of t'Hooft-Feynman gauge the propagator of vector particle takes the form

$$\frac{i}{ (2\pi)^4} \frac{g_{\mu\nu}}{m^2 - k^2}\;.$$ (2)

that provides a formulation of the theory where the problem of vector particle propagator is solved explicitly.

A price for this solution is an appearance of three additional unphysical particles in the model. They are a couple of Faddeev-Popov ghosts and one Goldstone ghost. All of them have scalar type propagators with the same mass $m$. Opposite to (1) the propagator (2) does not vanish when it projected onto the temporal polarization state

$$e^0=k/m\;\;,$$ (3)

that also leads to the appearance of additional unphysical state. The main principles of gauge invariance guarantee [Bjorken&Drell] that an expression for the amplitude should be the same for any gauge if only physical incoming and outgoing states are considered.

Generally the t'Hooft-Feynman gauge solves the problem of cancellations. But while calculating processes with incoming or outgoing massive vector particles, we meet a similar problem. Indeed, we need to multiply the diagram contributions by polarization vectors. The polarization vectors $(e^1,e^2,e^3)$ constitute an orthonormal basis in the sub-space orthogonal to a momentum $k$. Due to the relation

$$e^1_{\mu} e^1_{\nu} + e^2_{\mu} e^2_{\nu} + e^3_{\mu} e^3_{\nu} = k_\mu k_\nu/m^2 -g_{\mu\nu}$$ (4)

at least one of the polarization vectors has large (of order of $k$) components for any choice of polarization basis. Let vector $k$ have the components

$$k=(\sqrt {m^2 + p^2}, 0,0,p)\;.$$ (5)

Then the polarization vectors can be chosen as

$$e^1=(0,1,0,0)\;;$$ (6)

$$e^2=(0,0,1,0)\;;$$ (7)

$$e^3=(p/m,0,0,\sqrt {1 + p^2/m^2)}\;.$$ (8)

The first two vectors correspond to transversal polarizations and the third one corresponds to a longitudinal one. We see that the longitudinal vector has large ( of order of $k$) components. It may imply a fast increase of cross-sections of processes with the longitudinal polarizations at high energies or an appearance of cancellations between various diagrams. Indeed the second case is realized and, hence, we have got a problem with diagram cancellations. When evaluating squared diagrams we get cumbersome expressions again as a result of substitution of the projector (4).