TEMPERATURE EFFECTS ON VORTEX AND MONOPOLE IN QUANTUM FIELD THEORY

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TEMPERATURE EFFECTS ON VORTEX AND MONOPOLE IN QUANTUM FIELD THEORY

R. Manka, G. Vitiello

To cite this version:

R. Manka, G. Vitiello. TEMPERATURE EFFECTS ON VORTEX AND MONOPOLE IN QUANTUM FIELD THEORY. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-117-C3-123.

�10.1051/jphyscol:1989319�. �jpa-00229460�

JOURNAL DE PHYSIQUE

Colloque C3, supplément au n°3, Tome 50, Mars 1989 C3-H7

TEMPERATURE EFFECTS ON VORTEX AND MONOPOLE IN QUANTUM FIELD THEORY

R. MANKA and G. VITIELLO*

Department of Theoretical Physics, Silesian University, PL-40007

Katowice, Poland

*Dipartimento di Fisica dell'Universita, 1-84100 Salerno, Italy and e Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Italy

Résumé : Nous étudions les effets de la température sur la condensation de bosons en champ quantique dans le cas de systèmes avec "kinks", vortex et monopoles.

Abstract - Me study the effects of temperature on boson condensation in Quantum Field Theory in the case of systems with kinks, vortices and monopoles.

1 - INTRODUCTION

The vortex solution in the scalar electrodynamics / l / (the abelian Higgs model in 4-dimensions/2/) is the first model of string which is a subject of much study in recent time in the perspective of constructing free from infinities theories for elementary particles. The vortex solution also explains us many condensed matter phenomena as the vortex in super- conductivity/3/ or the filamentary structures in biological systems/4/.

On the other hand, the non-abelian gauge theories lead to monopole solutions whose existence is crucial in the understanding of many problems in elementary particles and of the evolution of the early universe/5/. In the present paper we focus our attention on the effects of non-zero temperature on topologically non trivial soliton solutions as vortex and monopole. Me consider also the kink solution in one-dimensional models /6/.

We treat such solutions as extended objects created by non-homogeneous quantum boson condensation in the vacuum state of the theory and use the variational method to study the temperature effects. We find that a critical temperature T= may exist at which a phase transition occurs and the soliton solution vanishes. We study symmetry restoration at T= and point out some phenomena as thermal contributions to mass terms.

2 - S0LIT0NS SOLUTIONS AT NON-ZERO TEMPERATURE

The simplest case to treat is the kink solution in one-dimensional model

/&/. We consider the quantum real scalar field ;|> (x>, x=x1,t, with

lagrangian

(1)

The field equation for the quantum field i, is

(2)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989319

C3-118 JOURNAL DE PHYSIQUE

*it.,

a ⁼ - p*. -k~vz ^{3, ; doe;}

^{d .}

In the variatronal approach at non-zero temperature T we consider the Boso- liubov inequality /8/

where F is the free energy F=-KTlnTr(exp(-i;H)), H o and Fo are the Hamiltonian and the free energy for the trial system, respectively.

denotes statistical averages: < W 0 = (1/Z) Tr (exp (-CHo) A ) with Z=TR (exp(-RHO? )

.

Our aim /9/ is to separate in the interacting system an effective excitation against an effective vacuum background. This vacuum background can be regarded as the quasi-classical temperature dependent field. The trial lagrangian and field equation are

and the connection between the fields and p is postulated as

with expectation value in the temperature dependent ground state given by

The free energy F* Iv(x,:')) is obtained as

where D is space-time dimensions number (D=2 in the present case) and

Eq.(8) is the generalization of the Eiinzburg-Landau functional and appears to be the classical energy functional for the classical field v(x,(') /9.10/. We use then the variational equations >6/33.t:= 0 and

a F , / $ / = o -

The equation for the "condensate function" v(x,;:) is then obtained

When v is constant in space-time (homogeneous condensation) we have

which, by assuming v#O, gives

Eq. ( 12) shows that at T=T, such that =

3A < eZ& ~ Z L

We have thus symmetry restoration at T, (see eq.(7)).The static kink solution to eq. (10) is

which goes to zero at I;,: the kink thus disappears at the critical temperature T., For D=4, in the thin-wall approximation, Linde / l l / has considered in relation with cosmological problems the spherical symmetry solution related to the kink. Our results show that a phase transition can occur at T.,

We consider now the vortex and the monopole.

a) The vortex string equations.

We consider a complex scalar field ,I. ( x ) , x=x+, t, i=1,2,3, and a vector gauge field Clli (x) with lagrangian

The lagrangian (15) is invariant under the U(1) local gauge transformations

/

Ay(*l-+ ~,(x/ = A _/"

+ & ^XI

and.under global U(1) which is assumed to be spontaneously broken

JOURNAL DE PHYSIQUE

The field

,

.

It is useful to introduce the dynamical maps /7,12/ in terms of asymptotic fields, say in-f ields

b i n and Us" are the ghost field and Proca field respectively /7,12,16/.

The U(1) transformations are induced by the asymptotic field translations (dynamical rearrangement of symmetry)

The transformations (21) control the boson condensation. In the case of homogeneous boson condensation (i.e. v=v(b)) we find then

where MZ(i?) denotes the mass of the gauge vector field. We thus see that a critical temperature T, may exist at which v(R=)=O. Moreover, M Z ( f l ) gets contributions from thermal effects (see the term e'<:p2:>,- in eq.(22)) and this term does not go to zero even at T,. This is a manifestation of the fact that the Hilbert space for the physical states at T, is unitarily non-equivalent t o the state space at T=O. In the case of non-homogeneous boson condensation we have

which by writing vtx,R) as v(x,fl)=v(d)f ( x ) and going to cylindrical coor- dinates, lead to the vortex string equations%

where we used the choice 1r tr, o , z ) =-n,-l/e, tr. a. z ) =., ~.+~Cl(r) /er. We observe that as T->O eq5. (24) reduce to the equations for the vortex at T=O 1 while the term <:,a":>, in the second of the eq. (24) does not go to zero even at T=T,. A detailed discussion of eqs. (24) and their numerical solutions are in ref. /12/.

b) The monopole.

- -

We consider the real scalar triplet (x) and the gauge vector field Cl (x) x=(x*.t) = 1,2,3, with S0(3) invariant lagrangian /17/ /4

The S O ( 3 ) global invariance is assumed to be spontaneously broken

< o p / e ~ ~ / o p ) ) ⁼

The field j:# (x) is the Higgs field and 4.1 (x) and ~ l . , ~ ( x ) are the

C3-122 JOURNAL DE PHYSIQUE

Goldston: fields. The dynamical map for .!,,(XI in terms of asymptotic

fi 4 A

fields i , p and Ak.: is obtained as /13/.

For C I l i we use

r

Here

$1=3i3*~

4

By using the 't Hooft-Polyakov ansatz /13/

we obtain the field equations for the fields K(r) and A(r)

which reduce to the well-known monopole equations for T 3 0 . For the homogeneous condensation case we have

and a critical temperature T, may exist at which symmetry is restored. We

also have for the mass of the gauge vector field

2

-,

^"

bit; =je JY)

1-il3) - 3-ezZ <ArbAl?

4-

which, a s in the vortex case, exhibit thermal contributions which do not vanish at T,. See also eqs. (31) which as T->T, get still contributions from statistical averages although symmetry is restored.

3

-

We have shown that effects of non-zero temperature may affect in a drastic way the kink. the vortex and the monopole shape up to a critical temperature T, at which these extended objects disappear due to thermalization. At T,, the symmetry of the system is restored; however, some contributions to mass terms of the gauge field are still present denoting the unitarily inequivalence of the state space at T=T, t o the one at T=O. In our study we used the framework of quantum boson condensation and the variational method technique. For an approach in terms of thermo- field dynamics /7/ see refs./l2/ and /13/.

REFERENCES

/1/ H. B. Nielsen and P. Olesen, Nucl. Phys. @SJ (1973) 45.

H.J. de Vega and F.A. Sckaposnik, Phys. Rev. D14 (1976) 1100.

/2/ P. Higgs, Phys. Rev.

145

T.W.B. Kibble, Phys. Rev.

155

F. Englert and R. Brout, Phys. Rev. Lett. 1=( (1964) 321.

/3/ V.L. Ginzburg and L.D. Landau, Zh. Eksp. Theor. Fiz. 20 (1950) 1064.

A.A. Abrikosov, Sov. Phvs. JEPT 5 (1957) 1174.

/4/ E. Del Giudice, S. Doglia, M. Milani and G. Vitiello, Nucl. Phys. B- F S 17 (1986) 185.

/5/ D.A. Kirzhinits, JEPT Lett.

15

A.H. Guth and E. J. Weinberg, Nucl. Phys 8212 (1983) 321.

A.D. Linde, Nucl. Phys. (1983) 421.

/6/ P.K. Bullough and P.J. Caudrey, Solitons (Springer, Berlin, 1980).

/7/ H. Umezawa, H. Matsumoto and M. Tachiki, Thermo-field Dynamics and Condensed States (North-Holland, Amsterdam, 1982).

/8/ R.P. Feynman, Statistical Mechanics (W.A. Benjamin, Reading, Mass.

1972).

/9/ R. Manka, J. Kuczynski and G. Vitiello, Nucl. Phys. (1986) 533.

/10/ E. Del Giudice, M. Milani, R. Manka and G. Viticllo, Phys. Letters B

- (1988) 661.

/11/ A.D. Linde, Nucl. Phys. & (1983) 421.

/12/ R. Manka and G. Vitiello, Topological solitons and temperature effect.^ in gauge field theories. I. The vortex string, Salerno

-

Katowice Preprint 1988.

/13/ R. Manka and G. Vitiello, Topological solitons and temperature ef- fects in gauge field tehories. 11. The monopole, in preparation.

/14/ C. Bechi, A. Rouet and R. Stora, Commun. Math. Phys.

42

Ann. Phys. (NY) (1976) 287.

/IS/ T. Kugo and I. Ojima, Progr. Theor. Phys. Suppl., 6& (1979) 1.

/16/ H. Matsumoto, N.J.Papastamatiou, H. Umezawa and G. Vitiello, Nucl.

Phys. I33 (1975) 61.

/17/ G. 't Hooft, Nucl. Phys. B79 (1974) 276.

A.M. Polyakov, JEPT Lett. 2 0 (1974) 194.