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

Home , Sphaleron

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

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

L = F F +(D )(D ) V () ; (1)

F = @ A @ A ;

D =(@ ig A );

2 2

V()=(jj ) ;

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

V sp ontaneously breaks the U(1) symmetry, yielding a gauge b oson of mass M = gv

2v . and a scalar Higgs particle of mass M =

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

L>L ; L = : (2)

1 1

This critical value of L is to b e contrasted with the one obtained in the framework of

the non-gauge mo del: L = 2 (ref. [13]).

In the A = 0 gauge, when the residual gauge symmety under space-dep endent

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

(x)= )( x) ; exp(

A =0 ;

(3)

A = :

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( 1) ; (4)

dz 2

whose p erio dic solutions are given in terms of the Jacobi elliptic function sn(z ):

(z )=kb(k)sn(b(k )z; k) ; b (k)= : (5)

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

q = + n if m is o dd ; q = n if m is even : (6b)

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

x ; (7) z =

the energy of the sphaleron takes the form

2K(k)

3 2 2 2

E = 2v E (k ) ; E (k )= [( ) +( 1) ]dz : (8)

E (k ) is nite and monotonically increasing from k =0 (L=L )tok=1 (L =1):

E (0) = ; E (1) = : (9)

4 2

For L mL new branches of solutions app ear, which can b e considered successions

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

exp( )i v(z )+ (z ;z )+i (z ;z ) ; =

1 1 0 1 2 0 1

A = a (z ;z ) ;

(10)

0 0 0 1

A = + a (z ;z ) :

1 1 0 1

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)

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

= ! : (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

=1 ; (16)

ful lled by the classical solution = tanh(z ), obtained in the limit L !1.

Eq. (13) is the central equation in ref. [13]. Recently,itwas revisited in ref. [14] in

the light of the Lame equation and its lowest solutions were presented in terms of the

Lame p olynomials. (For completeness we give some of these p olynomials in App endix

A.) Therefore, we refrain from presenting more details ab out the analytical solutions of

eq. (13). Just note that for the sphaleron solution (i.e. m = 1) due to the anti-p erio dic

conditions for the function , the rst relevant eigenvalue for the gauge mo del is a zero

mo de given (in the notations of App endix A) by

(cd)0

0 2 2

N =2 ; =cdE ; ! = b (k )h 2 0 :

This zero mo de is asso ciated with the translational invariance of the mo del.

Equation (14) is also a Lame equation. For = N (N +1);Ninteger, 2N + 1 of the

eigenvectors are given by the Lame p olynomials (given in App endix A for N =0;1;2;3).

This sector, describing the a uctuations, contains only p ositive mo des.

Wenow turn to the coupled system (15). First note that the equations ab out the

m-sphaleron can b e easily solved in the limit L = mL , since then k = 0 and = 0,

and the system decouples into two harmonic equations.

For even values of m and L = mL the eigenvalues organize as follows

2j 2j

2 2

; ! (m; j )=2 ; (17a) ! (m; j )= 2+2

m m

j b eing an integer which lab els the states. For o dd values of m one needs to takeinto

account the anti-p erio dic b oundary conditions satis ed by the uctuations. Here it is

convenient to double the length of the circle, but to retain only those mo des that are

compatible with the p erio dicity conditions. These are given by the subset

2j +1 2j

2 2

! (m; j )=2+2 ; ! (m; j )=2 ; (17b)

m m

of the set (17a).

Nowwe apply this strategy to the system (15), cho osing m = 2. The eigenmo des

of the sphaleron then app ear automatically by selecting only the subset (17b). When 5

= N (N + 1), and N a non-negativeinteger, the rst few (lowest) mo des of eq. (15)

can b e obtained analytically by expanding each comp onent as a linear combination of

the Lame p olynomials of degree less or equal to N +1. (Only p olynomials with the

same parityenter in the combinations.) In the case L =2L the eigenvalues have the

following order

2 2

2 (1) ; 0 (3) ; 2 (2) ; 6 (2) ; 8 (2) ; :::: ; 2j 2 (2) ; 2j (2) ;

with multiplicities indicated in parenthesis. The ab ove degeneracies are lifted for k 6=0.

In the following we denote the branches app earing in the ab ove list by

a ; b; c; d ; e; f ; g; h ; i; j ; ::: ;

resp ectively. Also, we de ne

a = F; =G : (18)

1 2

The eigenvalues ! and the eigenfunctions F and G, that are obtained from the Lame

p olynomials, then read for N =0;1 and 2

N=0

2 2

! = b ; F =0; G=dn;

2 2 2

! =k b ; F =0; G=cn;

(19)

! =0; F =1; G =0;

! =0; F =0; G =;

N=1

2 0 2

! = 2; F =; G= 1;

0 2 2 2

; G= ; =2kb ; F = + !

2 2 2

! =k b ; F = cn; G = ;

(20)

2 2 2 0

! =2kb ; F = + ; G =;

2 2

! =b ; F = dn; G = k cn;

2 0 2

! =2; F =; G= ;

f 6

N=2

2 2

! = b (1 2 1+3k );

2 2 2

! = k b (1 k +3);

2 2

2 4

! =42b 1k +k ;

2 2 2

! =k b (1 + k +3);

! =2;

(21)

2 2 2

! =b (1+4k );

2 2

! = b (1 + 2 1+3k );

! =6;

2 2

2 4

! = 4+2b 1k +k ;

2 2 2

! = b (4 + k );

where we presented only the eigenvalues. The corresp onding eigenfunctions F and G

for N = 2 are given in App endix B. When p ossible, we expressed the functions F and

0 2

G in terms of (see eq. (5)) and of = kb cndn. The elliptic functions (sn; cn; dn)

have arguments b(k )z ;k. In general there are 2(2N + 1) mo des that are p olynomials

in sn; cn; dn [16].

All mo des constructed in this way are normal mo des ab out the 2-sphaleron solution.

2 2

The branches lab elled ! and ! corresp ond to negative mo des of eq. (15). For the

sphaleron solution only the negative mo des ! should b e retained and the p ositive

mo des lab elled c; d; i; j .

We did not succeed in obtaining analytical solutions for the mo de equations of the

sphaleron (and the 2-sphaleron) for arbitrary values of . Numerical analysis, however

indicates that the eigenvalues fall smo othly b etween the analytically obtained values for

= N (N +1) ;N integer. For small k this is con rmed by a p erturbative calculation 7

of the mo des (i.e. for L L ). The results for the rst few eigenvalues are

2 2 2 3

! = 2+k (2 )+O(k );

2 2 3

! = 2 2k + k ( 1) + O (k );

2 2 2 3

! = k + O (k );

2 2 3

! =2 2k + k ( 1) + O (k );

(22)

3 2 2

3) + O (k ); ! =2+k (

2 2 3

! =2+k ( 3) + O (k );

2 2 2 3

! ! = 6+5k ( 2) + O (k );

g h

2 2 2 2 3

! ! =8+k ( 12) + O (k ):

i j

These expressions are valid for any > 0. Note, that the degeneracy of the branches

g and h and of i and j is lifted only by terms of order k .

When one investigates the system (15) in the background of higher multi-sphalerons

(m> 2) all the mo des discussed ab ove are still present but additional branches of

eigenvalues app ear in b etween. For instance, in the limit L =4L , the sp ectrum for the

4-sphaleron reads

3 1 5 9

2 (1) ; (2) ; 0 (3) ; (2) ; 2 (2) ; (2) ; (2) ; 6 (2) ; ::: : (23)

2 2 2 2

Here we nd analytical solutions also for non-integer values of N .Following the

same lines as previously,we nd in the case N =1=2 (i.e. =3=4) four mo des

analytically which read

p p

3 2

2 2

! = ; F = dnz cnz (dnz cnz ); G = dnz cnz (dnz k cnz ) ;

2 k

p p

2 2

dnz cnz (dnz k cnz ); G =2k dnz cnz (dnz cnz ) : ; F = ! =

(24)

CONCLUSIONS

The classical solutions of the 1+1-Ab elian Higgs mo del on a circle of length L form

a sequence of m solutions, when L mL , where L is the critical length at which the

1 1

rst non-trivial sphaleron solution app ears. They share this remarkable feature with

the sphaleron solutions of the theory on the circle. Dep endent up on the value of the 8

Higgs mass, a similar sequence of sphaleron solutions is present in the Weinb erg-Salam

theory [17,18].

For the theory the analysis of the small uctuations ab out the sphaleron solu-

tions leads to a Lame equation [14]. Thus the lowest solutions of the eigenvalue problem

are known analytically and the eigenvectors are just the Lame p olynomials.

For the Ab elian Higgs mo del the analysis of the small uctuations ab out the sphale-

ron solutions is more involved. Here one obtains a system of four Lame-like equations, of

which only two decouple in general. Furthermore, the equations dep end on an additional

parameter, , the ratio of the Higgs and vector meson masses. Nevertheless, for certain

values of this parameter, namely for = N (N + 1), with N integer, the lowest solutions

can b e constructed analytically in terms of the Lame p olynomials.

For the two decoupled equations the eigenvectors are simply the Lame p olyno-

mials of degree 2 and N . For the two coupled equations, the eigenvectors are linear

combinations of the Lame p olynomials, which for = N (N + 1) consist of the Lame

p olynomials of degree N 1. There are 2(2N + 1) such eigenvectors [16]. (The case

N = 0 is sp ecial, since here all equations decouple.) It is intriguing, that analytical

solutions of this typ e can b e found, since the construction involves 4N + 6 parameters,

solving 4N + 14 algebraic equations. 9

APPENDIX A

In this App endix we present some details ab out the Lame equation and the rst

few Lame p olynomials. When k is xed (0 k<1) the Lame equation

d E

2 2

+[hN(N +1)k sn (z; k)]E =0 (A1)

admits an in nite but discrete sp ectrum. (The equation is considered on a p erio d of

sn(z; k)). For integer values of N the rst few eigenvectors of eq. (A1) are the Lame

p olynomials. There are 2N + 1 such p olynomials, which are given b elow for N =0;1;2;3.

02 2

The notations are those of ref. [15] with k =1k .

N =0:

(u)0

uE (z )=1; h =0 : (A2)

N =1:

(s)0

0 2

sE (z )=snz; h =1+k ;

(A3)

cE (z )=cnz; h =1 ;

1 1

(d)0

0 2

dE (z )=dnz; h = k :

N =2:

(sc)0

0 2

scE (z )=snzcnz; h =4+k ;

(sd)0

0 2

sdE (z )=snzdnz; h = 1+4k ;

(A4)

(cd)0

0 2

cdE (z )=cnzdnz; h =1+k ;

2 02

1k k 1+k

(u)0;1

0;1

2 2

2 02

; h = 2(1 + k ) 2 1 k k : uE (z )=sn z

2 2

N =3:

(scd)0

2 0

= 4+4k ; (z )=snzcnz dnz; h scdE

2 4

2(1 + k ) 4 7k +4k

0;1

sE (z )=sn z snz;

(s)0;1

2 4

= 5(1 + k ) 2 h 4 7k +4k ;

2 02

2+k 4k k

0;1

(A5)

cE (z )=cnz sn z ;

2 02

h = 5+2k 2 4k k ;

2 4

1+2k 1k +4k

0;1

; (z )=dnz sn z dE

(d)0;1

2 4

h = 2+5k 2 1k +4k :

3 10

When N is half an o dd integer there are also analytical solutions of the Lame

equation. The eigenvectors then are algebraic functions of snz ,cnzand dnz . The

solutions corresp onding to N =1=2 and N =3=2 are

N =1=2:

1+k

dnz +cnz; h = : (A6) E =

1=2 1=2

N =3=2:

2 4

dnz +cnz dnz +(k 1 1k +k )cnz ; E =

3=2

(A7)

2 4

h = (1 + k ) 1 k + k :

3=2

Unlike the Lame p olynomials, these solutions have a p erio d 8K (k ). As a consequence

other mo des (with the same eigenvalue h) can b e obtained by shifting z in (A6), (A7)

by2K(k), which is equivalenttochanging the sign in frontofcnz each time it app ears.

APPENDIX B

Here we give the eigenfunctions F and G of eq. (15) which can b e expressed in

terms of the Lame p olynomials for =6.

2 2 2

dn ! 3b

2 2 2 0

a) F =6 cn+(! 3b )cn; G = (2 cn ) dn( );

k 6k

2 2 2 2 0 2 2 2 2

b) F =6 dn + (! 3b k )dn; G = (2 dn k cn) cn(! 3b k ) ;

b b

0 3

c) F = 2 ; G = ( );

d) F; G see branchb with ! ! ! ;

b d

e) F = ; G= ;

2 2 2

f ) F =2k cn kb cn; G = dn;

g ) F; G see brancha with ! ! ! ;

a g

; h) F = ; G=

i) F; G see branchc with ! ! ! ;

c i

2 2 2 2

j ) F =2 dn k b dn; G = k cn: 11

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