Analytical evaluation of the anomalous fermion-number nonconservation at high temperatures in the (1 + 1)-dimensional Abelian Higgs model

Abstract

The Abelian Higgs model with a fermionic current nonconserved due to an anomaly is considered in 1 + 1 dimensions. The one-loop expression for the rate of the fermionic-number nonconservation at high temperatures is obtained analytically for arbitrary values of the Lagrangian parameters.The non-Abelian nature of the standard electroweak theory remains a subject of intense interest. The existence of the 9 vacuum in this SU(2) x U ( 1 ) gauge theory leads to the nonconservation of leptonic and baryonic numbers which is, however, negligibly small at zero temperature. Nevertheless, as was pointed out in Ref. 2 in matter at high temperatures which took place in the early Universe anomalous nonconservation of fermionic numbers is not suppressed. The relevant considerations are usually performed in the Ao=O gauge. There is a static energy barrier (SEB) between the classical vacua with different values of the Chern-Simons n~m b e r .~ At high temperatures the system has enough energy to pass through the SEB (Ref. 4) via classical thermodynamical fluctuations. At temperatures smaller than the height E, of the SEB the probability l-of the transitions over the barrier is small and may be evaluated in the semiclassical approximation l-=Aexp( -E,/T). Here the preexponential factor A is important.' Exact analytical evaluation of the ~r e e x~o n e n t i a l factor A in 3+ 1 dimensions is a serious So the semiclassical calculations in various tov models are v a l~a b l e .~-'~ In Ref. 10 the r q version of , -the Abelian Higgs model in 1 + 1 dimensions was shown to reproduce many essential features of the real case. It was solved analytically in the limit g '/h >> 1 (where g is gauge and h is scalar self-coupling constants) for integer values of the ratio g / m . In this paper we give an analytical solution for arbitrary values of the coupling constants g and h.The theory under consideration is defined by the Lagrangian of the form where @, W, and A, are scalar, spinor, and vector gauge fields, respectively. The particle spectrum contains vector and Higgs bosons with masses rn; "g 'c ' and rnh =2hc 2.The gauge-invariant fermionic current J, =-1Jy,ty is not conserved due to an anomaly:Nonconservation of the fermionic number is associated with the fluctuations of gauge fields which in the AoPO gauge change the value of the Chern-Simons number. The theory has a 9-vacuum structure. The classical vacua with different values of the Chern-Simons number are separated by SEB, the minimum height of which E, is nonzero. A statistical system built in the vicinity of one such vacua is slightly unstable with respect to penetration through the SEB. The decay rate of such a state coincides with the rate of anomalous fermionic-number nonconservation in hot plasma. l 2 In the one-loop approximation it is related to the imaginary part of thk free energy y:5,12 the coefficient K is to be defined later. The relation (3) is useful because there is a regular representation for the free energy in terms of the Matsubara functional integral. The functio...

Electroweak baryon number non-conservation in the early Universe and in high-energy collisions

“…³ÍÑÓÑÔÕß ÄÞÔÑÍÑÕÇÏÒÇÓÂÕÖÓÐÞØ Ô×ÂÎÇÓÑÐÐÞØ ÒÇÓÇØÑ-AEÑÄ Ä àÕÑÌ ÕÇÑÓËË ÃÞΠÄÞÚËÔÎÇÐÂ Ä [76,106]: [116]. ±ÓÇAEÒÑÎÑÉËÏ, ÚÕÑ ÍÑÎËÚÇÔÕÄÑ ÒÖÊÞÓÇÌ, ÑÃÓÂÊÖáÜËØÔâ Ä ÇAEËÐËÙÇ ÑÃÝÇÏÂ Ä ÇAEËÐËÙÖ ÄÓÇÏÇÐË, ÓÂÄÐÑ T,  ÔÍÑÓÑÔÕß ÔÕÇÐÑÍ ÒÖÊÞÓÇÌ…”

One-loop fermionic corrections to the instanton transition in the two dimensional chiral Higgs model

“…This is checked in Fig. 2 9 The accuracy can be checked by calculating the instanton number (13) or the action of the instanton for m 2 H =m 2 W 1 that is known to be v 2 [15]. The results of the numerical integration agree to 13 decimals with the action in the latter case and at least 7 for the instanton number in any case (see Fig.…”

“…2 The determinants in the high temperature sphaleron transition in 1 1 dimensions were computed in Refs. [6,9]. L ÿ 1 4 F F i j @ ÿ i e 2 5 A j ÿ V 1 2 jD j 2 if j j 1 5 2 j ÿ if j j 1 ÿ 5 2 j : (1) The charges of the left-and right-handed fermions differ by a sign, e L ÿe R e 2 , and the symmetry breaking potential is chosen to be V 4 jj 2 ÿ v 2 2 .…”

“…This model has been studied as a toy model for the fermionic number nonconservation in the electroweak theory in a number of papers; see, e.g. [5][6][7][8][9][10]17].…”

One-loop corrections to the instanton transition in the Abelian Higgs model: Gel’fand-Yaglom and Green’s function methods

“…In the context of the baryon number violation the high temperature sphaleron transition in this model has been studied [22,23] for which exact classical solutions and an exact expression of the sphaleron determinant [24,25] are known, thus providing a complete one-loop semiclassical transition rate which can be studied numerically on the lattice, e.g. by measuring the fluctuations of the Chern-Simons number.…”