Quantum fluctuations of the electroweak sphaleron: Erratum and addendum

Abstract

We correct an error in our treatment of the tadpole contribution to the fluctuation determinant of the sphaleron, and also a minor mistake in a previous estimate. Thereby the overall agreement between the two existing exact computations and their consistency with the estimate is improved considerably. PACS number(s): 11.15.Kc, 02.60.JhWe have presented recently [l] a n exact computation of the fluctuation determinant of the electroweak sphaleron. The results disagreed substantially with those of an earlier evaluation of this quantity [2]. Moreover, neither of the two exact computations agreed with analytical estimates [3,4] that are expected to be good a t small Higgs boson masses, essentially approximations in which higher gradient terms are neglected, a fact that has been repeatedly criticized (see, e.g., 151).While we thought that possibly the gradient-type expansions were to blame for this discrepancy, the failure of compatibility between approximations and exact results can be traced back to our treatment of the tadpole contributions. Indeed we removed all tadpole contributions to the Higgs field completely, in a misinterpretation of the renormalization and rescaling prescription of Refs. [2,4]; however, a finite piece has to be restituted. In order to understand this point, which has considerable numerical consequences, we review shortly these contributions (see [6], especially Appendix C). This cannot be done consistently within the three-dimensional asymptotic theory since the approximation of dismissing all but the lowest Matsubara frequency is not justified for these divergent contributions.The tadpole contribution including all Matsubara frequencies readsThe first term is the T = 0 contribution which goes into the Higgs boson mass renormalization. The second term can be expanded at high temperature as T J x2dX 1 1 7r Jx2 + m2/T2 exp (Jx2 + m2/T2) -1 up to terms of order 1nT or lower. The term quadratic in T can be absorbed [2, 41 into the T dependence of the vacuum expectation value of the Higgs field. The linear term is part of the well-known TG3 term of the effective potential and without this contribution the latter is incomplete (see, e.g., the discussion of this termHere mi are the masses circulating in the loop, ci are their couplings to the Higgs field, and Ho(x) is the Higgs profile. The coefficients ci can be identified as coefficients of the terms proportional to Ht -1 in the diagonal elements of the potential given in Appendix A of [I]; they are given below. The momentum integral including the factor T/2 can be rewritten [6] 'Electronic address: baa~keOhet.~hysik. uni-dortmund.de FIG. 1. The fluctuation determinant. We plot the logarithm of the fluctuation determinant n as a function of the ratio X/g2. Our corrected results are given as triangles, those of Ref. [2] as squares. The solid line is the estimate based on the effective potential.

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