Abstract
For gauge theory, the matrix element for any physical process is independent of the gauge used. However, since this is a formal statement, it does not guarantee this gauge independence in every case. An example is given here where, for a physical process in the standard model, the matrix elements calculated with two different gauge – the Rξ gauge and the unitary gauge – are explicitly verified to be different. This is accomplished by subtracting one matrix element from the other. This non-zero difference turns out to have a subtle origin. Two simple operators are found not to commute with each other: in one gauge these two operations are carried out in one order, while in the other gauge these same two operations are carried out in the opposite order. Because of this result, a series of question are raised such that the answers to these question may lead to a deeper understanding of the Yang–Mills non-Abelian gauge theory in general and the standard model in particular.
Highlights
Since the standard model of Glashow, Weinberg, and Salam [1] is a Yang–Mills non-Abelian gauge theory [2], the matrix element in the standard model for any physical process is expected to be independent of the gauge chosen
The corresponding calculation in the unitary gauge was carried out much later by Gastmans, Wu, and Wu [10]. This result with the unitary gauge was later verified by Christova and Todorov using an unsubtracted dispassion relation [11]. It has been puzzling why these two results, one with the Rξ gauge and the other with the unitary gauge, do not agree with other; the main result Eq (95) of the present paper shows, through a detailed analysis, why they should not agree. (g) It is interesting, and important, to delve into the underlying reason in more detail why, as found here and contrary to expectation, the Rξ gauge and the unitary gauge do not give the same answer for this Higgs decay H → γ γ through one W loop
What we have shown here is that the underlying reason for the disagreement between the Rξ gauge and the unitary gauge is very subtle for the present case
Summary
Since the standard model of Glashow, Weinberg, and Salam [1] is a Yang–Mills non-Abelian gauge theory [2], the matrix element in the standard model for any physical process is expected to be independent of the gauge chosen It is the purpose of the present paper to investigate the validity of this gauge independence, by studying a specific example through explicit calculation. As seen from (1) and (2), the W propagator depends on the choice of the gauge, the contribution from the W loop can potentially depend on the gauge used With these consideration, the physical process to be studied in this paper is the Higgs decay (3) through one W loop. The physical process to be studied in this paper is the Higgs decay (3) through one W loop This matrix element is to be calculated both in the Rξ gauge and the unitary gauge, and the results are to be compared. A number of important question are to be raised later in Section 8, and hopefully the answer to some of these questions will teach us how this calculation, and a number of other calculations in the standard model, should be carried out
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