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Sphalerons and Normal Modes in the 1+1-Dimensional Abelian Higgs

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Sphalerons and Normal Modes in the 1+1-Dimensional Abelian Higgs CERN-TH.6598/92 Sphalerons and Normal Mo des in the 1+1-dimensional Ab elian Higgs Mo del on the Circle Yves Brihaye CERN, Geneva, Switzerland Stefan Giller, Piotr Kosinski Department of Theoretical Physics, UniversityofLodz Pomorska 149/153, 90236 Lo dz, Poland Jutta Kunz Instituut vo or Theoretische Fysica, Rijksuniversiteit te Utrecht NL-3508 TA Utrecht, The Netherlands and FB Physik, Universitat Oldenburg, Postfach 2503 D-2900 Oldenburg, Germany Abstract The Ab elian Higgs mo del is considered on the circle. The p erio dic sphaleron solutions are constructed explicitly. The equations for the normal mo des ab out these solutions resemble Lame equations. For sp ecial values of the Higgs eld mass a number of modes are obtained analytically, including in particular the negative mo des. CERN-TH.6598/92 July 1992 INTRODUCTION In the standard mo del baryon and lepton numb er are not strictly conserved [1]. In fact, the non-conservation of baryon and lepton numb er is a non-p erturbative phe- nomenon. Baryon numb er violating pro cesses involving instantons may b e relevantat very high energies [2,3], while sphaleron [4] mediated pro cesses are relevant at high temp eratures in the early universe [5,6]. In particular, the baryon asymmetry of the universe may p ossibly b e obtained at the electroweak phase transition. The sphaleron of the Weinb erg-Salam theory is an unstable solution of the classical eld equations, describing the top of the energy barrier b etween top ologically inequiva- lentvacua. It is known only approximately via the numerical solution of a set of coupled non-linear di erential equations [4,7]. Likewise, the normal mo des of oscillation ab out the sphaleron are known only numerically. They enter into the baryon decay rate, along with the classical energy of the sphaleron. Attempts to evaluate the baryon decay rate at nite temp erature are confronted with many technical problems, aggravated by the lack of a closed form for the classical solution and for the nomal mo des [8]. This is a motivation to study simpler mo dels. In this resp ect the Ab elian Higgs mo del in 1 + 1 dimensions is a particularly useful mo del [9-12]. Manton and Samols [13] p erformed a very interesting study of sphalerons on the 4 circle for the simple mo del. Considering the space variable on the circle they constructed the sphalerons analytically, treating the radius of the circle as a parameter. Only recently was it realized [14] that the mo de equation investigated in ref. [13] is in fact a Lame equation, whose lowest solutions can b e expressed in terms of Lame p olynomials and thus b e given analytically,too. In this pap er, we consider the Ab elian Higgs mo del on the circle, and we analyze the classical solutions and the normal mo des in the same spirit as Manton and Samols [13]. The p erio dic sphaleron solutions are given analytically in terms of the Jacobi function sn(z ). The equations for the normal mo des are technically more complicated than those of refs. [13,14], b ecause one channel of the mo de equations consists of a system of coupled, but Lame-like, Schrodinger equations which dep end on two external parameters: the radius of the circle and the ratio of the masses of the gauge and Higgs b osons. In spite of these complications wehave obtained analytical solutions for sp ecial values of the mass ratio. The solutions are then given in terms of the Jacobi functions sn(z ), cn(z ) and dn(z ). 1 CLASSICAL SOLUTIONS We start from the Lagrangian density in 1 + 1-dimensions 1 L = F F +(D )(D ) V () ; (1) 4 F = @ A @ A ; D =(@ ig A ); 2 v 2 2 V()=(jj ) ; 2 where, along with refs. [9,13,14], we assume that the space co ordinate takes value on a ~ ~ circle of length L.For later convenience we de ne L M L=2. The Higgs p otential H V sp ontaneously breaks the U(1) symmetry, yielding a gauge b oson of mass M = gv W p 2v . and a scalar Higgs particle of mass M = H Wenow consider the classical equations of motion of the Lagrangian (1). Besides the vacua, a trivial static nite energy solution exists for all values of L. This constant sphaleron solution corresp onds to the maximum of the Higgs p otential. Non-trivial solutions of the equations of motion exist for suciently large values of L, in fact for p L>L ; L = : (2) 1 1 2 This critical value of L is to b e contrasted with the one obtained in the framework of 1 p the non-gauge mo del: L = 2 (ref. [13]). 1 In the A = 0 gauge, when the residual gauge symmety under space-dep endent 0 transformations is exploited to eliminate the p erio dic part in the phase of the Higgs eld, the non-trivial static solutions read v M 2i xq H p (x)= )( x) ; exp( ~ 2 2 L A =0 ; (3) 0 2q A = : 1 ~ gL Here x represents the space co ordinate and q represents the Chern-Simons charge of the solution. As shown b elow, p erio dicity requires 2q to b e an integer. The function (z ) ob eys the well-known kink equation 2 d 2 = 2( 1) ; (4) 2 dz 2 whose p erio dic solutions are given in terms of the Jacobi elliptic function sn(z ): 2 2 (z )=kb(k)sn(b(k )z; k) ; b (k)= : (5) 2 1+k Here k; (0 k 1), is a parameter which xes the p erio d of the function sn(z ). In fact, the p erio d of the function sn(z; k) in its rst argumentis4K(k), where K (k ) denotes the complete elliptic function of the rst kind. ~ In order for , as given in eq. (3), to b e a p erio dic function on the interval [0; L], the length of the circle L and the parameter k should b e related as follows 2mK (k ) ; (6a) L = b(k ) and q should b e 1 q = + n if m is o dd ; q = n if m is even : (6b) 2 It is understo o d that m and n are integers and m> 0. The solution obtained for m = 1 has the lowest energy, and is therefore referred to as the sphaleron of the mo del. (We show b elow that it has a single direction of instability.) De ning the dimensionless variable z as M H x ; (7) z = 2 the energy of the sphaleron takes the form Z 2K(k) p d 3 2 2 2 E = 2v E (k ) ; E (k )= [( ) +( 1) ]dz : (8) sp dz 0 E (k ) is nite and monotonically increasing from k =0 (L=L )tok=1 (L =1): 1 4 p E (0) = ; E (1) = : (9) 3 4 2 For L mL new branches of solutions app ear, which can b e considered successions 1 of m sphalerons. Therefore we will refer to them as m-sphalerons in the following. Their energies branch o from the energy branch of the constant solution = 0 like in Fig. 1 of ref. [13]. 3 NORMAL MODES To study the normal mo des of oscillation ab out the solutions (3), we consider general uctuations around these con gurations, which are considered as a classical background. We parametrize the elds using the notation of ref. [12]: 2iqx 1 p exp( )i v(z )+ (z ;z )+i (z ;z ) ; = 1 1 0 1 2 0 1 ~ 2 L A = a (z ;z ) ; (10) 0 0 0 1 2q A = + a (z ;z ) : 1 1 0 1 ~ gL The functions a ;a are p erio dic (in z ) on [0, L], while the functions ; are p erio dic 0 1 1 1 2 (resp. anti-p erio dic) if 2q is even (resp. o dd), which is due to the phase factor in front of in eq. (10). Let us now consider the Euclidean version of the action given ab ove (eq. 1), which is suitable for nite temp erature. We x the gauge degrees of freedom bycho osing the background gauge condition @a 2M @a i 0 H 1 G(a; )= ( ) ; = + : (11) @z @z 2 M 0 1 W The gauge xed action density then reads 1 1 2 2 2 S = (F ) + jD j + V ()+ G : (12) gf 4 2 Expanding the action (12) in p owers of the uctuations, the normal mo des ab out the classical solutions ob ey the following set of Schrodinger equations (obtained from the quadratic form in the uctuations) 2 2 (@ @ +6 2) = ! ; (13) 1 1 1 1 2 2 2 (@ @ + )a = ! a ; (14) 1 1 0 0 2 2 0 @ @ + 2 a a 1 1 1 1 2 = ! : (15) 0 2 2 2 @ @ +( + 2) 2 1 1 2 2 The equations for the uctuations a and decouple, while the equations for a 0 1 1 and remain coupled in general. However in the case L = L (i.e. k =0,=0)and 2 1 in the case = 0 the system (15) decouples obviously. It can also b e decoupled bya 4 constant transformation in the case L = 1 (i.e. k=1). This can b e seen by using the identity d 2 =1 ; (16) dz ful lled by the classical solution = tanh(z ), obtained in the limit L !1. 1 Eq. (13) is the central equation in ref.
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