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PHYSICS LETTERS

26 August 1982

TEMPERATURE DEPENDENCE OF COUPLING CONSTANTS K. BABU JOSEPH, V.C. KURIAKOSE and M. SABIR High Energy Physics Group, Department o f Physics, University o f Cochin, Cochin 682022, India Received 27 January 1982 Revised manuscript received 5 April 1982

We study the temperature-dependence of coupling constants at the one-loop level for massive ~o4 theory and massive scalar electrodynamics (SED). It is found that the scalar coupling constant h for m 2 > 0 decreases with temperature leading to a phase transition to a non-interacting phase. In a model with m 2 < 0, h increases as In 7". The gauge coupling constant of SED increases uniformly with temperature.

Finite temperature effects in quantum field theory are currently of considerable importance in particle physics and cosmology. The dependence of the coupling constant on temperature has wide applications. For instance, it has been realised that if the temperature dependence of the coupling constant is included in the calculations the amount o f supercooling associated with the first order phase transitions in grand unified theories m a y be drastically reduced [ 1 ]. Witten has defined a temperature-dependent coupling constant [2] which has been employed in various formulations [ 1,3 ]. This note contains an attempt to calculate the temperature behaviour of the coupling constant at the one4oop level. We have chosen massive qo4 theory and scalar electrodynamics (SED) as models. The form o f the temperature-dependence for the ~o4 theory is also investigated using the renormalisation group. qO4 theory. The lagrangian under consideration is £ = -~(3u~p)2 -- ~m2~0 2 -- (X/4!)~0 4 ,

X > 0.

(1)

The coupling constant X appearing in this lagrangian is unrenormalised and the renormalised coupling constant can be obtained b y the vertex renormalisation procedure [4]. The lowest order graph which is of order of X (fig. 1) is given b y the single vertex: 1'(1) = -iX(27r) 4 84(k 1 + k 2 - k 3 - k4).

(2)

There will be three graphs (fig. 2) o f order of k 2, which give identical contributions and hence it is sufficient to consider any one o f them, yielding

k,

k~

Fig. 1. Bare vertex in ~o4 theory. 120

Fig. 2. Vertex correction in ~o4 theory at one-loop level. 0 0 3 1 - 9 1 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 O 1982 North-Holland

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Fig. 3. Lowest order vertex correction in SED.

x2fd4ks

d4k6 (270 4 64(kl + k 2 - k 5 - k6) (270 4 64(k 5 + k 6 - k 3 - k4)

1.(2)='-2-

(2rr)~(27r)4

(k?~tn2;

(3)

( k 2 _ m 2)

In the imaginary time formalism the Feynman rules at finite temperature correspond to their zero temperature counterparts with the following replacements.

f d4k _+i (2rr) 4

d3k

/3 ~fi2~) 3

with k 0 = 09, 09n =

2rrni//3(bosons),/3 is the

inverse of temperature, and

(2rr)4 64 (k 1 + k 2 + k3 +'" ") -+ (2Tr)3 (/3/i) 6 (COn1 + t°n2 +C°n3 + '" ) 6 3 (k 1 + k2 + k3 +''')" Thus eq. (3) becomes i.(2 ) _ X2 1 ~ f ~ d 3 k (270 3 fl--6 63(/t:1 + k 2 _ k 3 _ k 4 ) 1 2 --i/3 n a (2rr)J i (CO/'/1+con2--con3--con4 ) (O92_ k 2 _ m2)2 "

(4)

Expressing 1.~ as 1./3= p(1) + p(2) where P~ = -i~tfl(2rr) 4 64(k 1 + k 2 - k 3 - k4) we can find that the temperaturedependent coupling constant in the one-loop approximation is given by X~=X

3X21 g-~t - d3k

1

(091-

T j

(s)

2'

where E 2 = k 2 + m 2 and we have set the external momenta to be zero. The summation is done first using the identity

n=l (n 2 + a 2)

2a 2

+~-coth(za),

(6)

which yields d3k ( 1 Xt3 = X - [email protected](-2~)3 4E 3

+

1 2E 3 [exp (/3Ek)

--1 ] +

13exp (/3E/c) ) 2E~eex~LTk ) 7-112

"

(7)

The temperature-independent term cancels with the renormalisation counter term at zero temperature, so that X~ = x + 4 7 r 2 ~ k 2 d k -

d(

1

dk 2 (k 2 + m2) 1/2 {exp[/3(k 2 + m2) 1/2] - 1)

)

(8)

Putting x 2 =/32k2,/3m = a, we have 121

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PHYSICS LETTERS

J

(

X~ = X + 33"~2 x 2 dx 8 1 4rr2 0 8x 2 (x2+a2)l/2{exp[(x2+a2)l/2]

26 August 1982

)

(9)

- 1} '

which can be evaluated in the high temperature limit giving X~ = X - ( 3 X 2 / 8 ~ 2 ) [ a T / 2 m

(10)

+ ~ In ( m / 4 ~ T ) ] .

In the absence of spontaneous symmetry breakdown (SSB) the scalar coupling constant is seen to decrease with temperature. If we neglect the In T term at high temperatures then a critical temperature Ta may be def'med corresponding to the vanishing of the coupling constant Ta = 16rrm/3X - (m/~) in (m/47r).

(11)

Ta signals the emergence of a non-interacting phase for the field system. It is known that at zero temperature the model can possess asymptotic freedom only for a negative value of X [5]. The present result must be contrasted with this because it holds for the physically interesting case of positive X. If the onset of "no-interaction" is a genuine phase transition, then it should also work in the reverse. It is not difficult to glean a few illustrations from physics where forces get weakened with rise of temperature. The rupturing of chemical bonds and disappearance of the phonon field picture in solids under thermal agitation are examples that bring out this mechanism at least in a qualitative manner. However, for a clear and specific quantitative comparison regarding the behaviour of scalar coupling constants the anharmonic vibration problem has to be studied in the context of a self coupled phonon field at f'mite temperature. We encounter an imaginary mass in eq. (1) for a theory with SSB. However the imaginary terms may be separated, and it is hoped that they will disappear when higher order effects are taken into account [6]. The resulting expression for X~ reads X~ = X + (3X2/16rr 2) In (47rT).

(12)

It is seen that with SSB X increases uniformly with temperature. This provides a justification for Witten's well known recipe [2]. The present result may be applied to the Ginzburg-Landau model for superconductivity which predicts a variation of the penetration depth 6 with the quartic coupling constant X : ~ ~ ~ [7]. At high temperatures, since X increases the penetration depth 8 increases and as a result, superconductivity is inevitably lost.

Renormalisation group approach. The above calculations can be checked using the renormalisation group [8]. Following Coleman and Weinberg [9] the renormalisation group equations are written (TS/ST +/3 8/8X + (1 + 3,m)m 8/8m + 7~oc 8/8~Oc)V = 0, (TSIOT + (38lax + (1 + 7m)m alSm + 7~ c 8/8~% + 2 7 ) z

(13) =

0,

(14)

where V is the effective potential at finite temperature and Z is the scale of the field. The temperature-dependent effective potential at the one-loop level evaluated in the high temperature approximation, is given by [6] V = ~ m 2tp2 + (X/4 !)~0c 4 + (M4/647r2)ln (TZ/47r2m 2) + 1 M 2 T 2 _ M 3 T[ 12%

(15)

where M 2 = m 2 + 1 X~o2. It is convenient to use V(4) = 84 V/~¢ 4 in place of V. Introducing a scale factor t, t =

In(Tim), eqs. (13) and (14) can be rewritten (-SlOt + ~ 818X + ~0 c 818~0c + 4~)V (4) = 0,

(16)

(--818t + ~8/8X + ~0 c 8/8~0 c + 2 ~ Z = 0,

(17)

where/] = ~/Tm and ~ = 8/7m. Using the zero-loop values for V(4) and Z, viz. X and 1 respectively, we can fmd

= ½(8/St)Z(O,X), 122

~3= (8/St)V(4)(O,X) - 4~X.

(18)

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PHYSICS LETTERS

In the one-loop approximation, the values of V (4) and Z are given by V(4)(t, X) = X~ = X - (3~2/8rr2)OrT/Zm - ½ In (T/m)),

(19)

Z(t, ?,) = 1.

(20)

Hence ~ = 0,

(21)

t7 =-(3x2/8rr2)(½rre t - {t).

If we assume a functional dependence of the form (22)

= dX~/dt, we then fred

?v~ =

X

(23)

1 - (3X/16~2)ln (r/m) + (3X/16u) (T/m) It is remarkable that perturbation and renormalisation group calculation predict the same type of thermal behaviour for the coupling constant. Massive scalar electrodynamics. We will extend the foregoing calculations to the case of gauge coupling constant in SED described by the lagrangian £ = (3 u + i e A u ) ~ , ( b u _ i e A u ) ~ _ m2(~o,~o) _ X(~o,~p)2 _ 1 F

F u.

(24)

The lowest order vertex renormalisation correction is of order e 4 and comes from fig. 3 which yields 4 Fd4k3 d4k4 d4q (2n) 484(k l _ k 3 _ q ) F (2) = 2e g u u J ( ~ 4 (27r)4 (2~) 4 (kl + k3) (k2 + k4) X (2rr) 4 64(k 2 - k 4 - q)(2rr) 4 84(k 3 + p l - P 2 - k4) (k 2 _ m 2 ) q 2 ( k 2 _ m2 ) •

(25)

As before, the vertex renormalisation at finite temperature can be written

r(2)=2e4guv--~nf~(27r)3~800i

( n, +°°7l 2 --°9I'/3 --0o~4 )'63(k'm+ P l

1 2--P2 )-- k2)2 (oo2_ " k2

(26)

Defining ['~ as Pt3 = p(1)+ p(2), where p(1) = 2ie2guv(2rr)4 64(k I +Pl - P2 - k2) and P~ = 2ie2guv(27r)4 64(k 1 + Pl - P2 - k 2 ) w e fred e~ = e 2 + ~ - ~ / " d3k 1 /J n a(2rr)3 (co2 _ k 2 _ m2) 2"

(27)

At high temperatures this simplifies to e 2 = e 2 + (e4/47r2)[TrT/2m + ~1 In (rn/47rT)].

(28)

We have presented two models wherein the scalar and gauge coupling constants are temperature-dependent, SSB is the critical factor that determines the nature of the temperature variation. In the early universe when there 123

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was thermal equilibrium the temperature must have varied with time approximately as T ~ (t) -1/2 [10]. This implies that the coupling constants could vary with time under conditions o f thermal equilibrium, thus realising Dirac's hypothesis o f time variation of constants of nature. The thermal behaviour of non-abelian gauge coupling constants is currently under investigation. One o f us (V.C.K.) is thankful to the University Grants Commission, New Delhi, for the award o f a Teacher Fellowship.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

124

M. Sher, Phys. Rev. D24 (1981) 1699. E. witten, Nucl. Phys. B177 (1981) 477. L.F. Abbott, Nucl. Phys. B185 (1981) 233. C. Nash, Relativistic quantum fields (Academic Press, New York, 1978). R.A. Brandt, Ng Wing-chiu and Y. Wai-Bong, Phys. Rev. D19 (1979) 503. L. Dolan and R. Jackiw, Phys. Rev. D9 (1974) 3320. D.R. TiUey and J. Tilley, Superfluidity and superconductivity (Van Nostrand, Reinhold, 1974). M.B. Kislinger and P.D. Morley, Phys. Rev. D13 (1976) 2771. S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888. A.D. Dolgov and Ya.B. Zeldovich, Rev. Mod. Phys. 53 (1981) 1.