Grand Bang

Volume 99B, number 5

PHYSICS LETTERS

5 March t981

GRAND BANG A.D. LINDE Lebedev Ph)'sical Institute, Moscow, USSR Received 25 November 1980

It is shown that symmetry breaking between strong and electroweak interactions in grand unified theories typically is a strongly first order phase transition. In many theories this phase transition proceeds as a ~eat explosion all over the universe at t ~ 10 .35 s after the Big Bang.

Recently, there has been nmch interest in symmetry breaking phase transitions [1 3], which occur with the decrease of the temperature in grand unified theories (GUTs) [4]. There are two main reasons for such an interest in the phase transitions in GUTs. Firstly, it is the desire to have a more detailed scenario of the cosmological baryon production [5], and secondly, it is the need to solve the primordial monopole problem in GUTs [6] (see in this cotmection ref. [7]). In most papers discussing these problems it has been assumed that phase transitions in GUTs typically are second order or weakly first order, see e.g. refs. [8,9]. The same statement is contained also in a recent paper [101, in which a detailed discussion of phase transitions in GUTs was given. However, the criterion used in ref. [10] to distinguish between strongly and weakly first order phase transitions is not quite adequate (see below). In the present paper it will be shown that phase transitions with symmetry breaking between strong and electroweak interactions, which occur in GUTs at temperatures T ~ 1014-1015 GeV, practically always are strongly first order [11 ]. These phase transitions proceed from a supercooled symmetric phase as an explosion with a large release of thermal energy. In this sense phase transitions in GUTs differ greatly from the phase transition with symmetry breaking in the Weinberg-Salam model, which is typically (for most natural relations between the coupling constants) a second order or a weakly first order phase transition [1-3].

Let us first consider the phase transition with symmetry breaking S U ( 5 ) ~ SU(3)c X SU(2)L X U(I )y in the minimal SU(5) model [4]. The effective potential in this model at temperature T = 0 at the classical level is given by [12]: V((I)) =

1 2 tr q)2 + aa 1 (tr ~2)2 + 7b i tr ~4 , 5~t

(1)

where t-l) is the Higgs field in the adjoint representation. Let us denote 7 = ½(15a + 7b). l f b > 0, 7 > 0, then the symmetric state with vanishing classical field (I~ is unstable, and SU(5) breaks spontaneously to SU(3)c × SU(2)L × U(1)y by the appearance of the field ~I, = v diag(1, 1 , 1 , - 5 ,3- 5 ) "3

(2)

where at T = 0 v ( r = o) = v o = u / v ~ - .

(3)

It is known that at sufficiently high temperatures T > Tq the symmetric phase v = 0 becomes stable, since the curvature of the temperature-dependent effective potential V((!b, T) at v = 0 becomes positive [1-3]: /~2(T) =/32 - / 3 T 2,

(4)

front which it follows that T 2 =/22//3. Cl

(5)

For the minimal SU(5) model it can be easily shown [8,10] that the coefficient/3 in (4), (5) is given by

0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company

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/3 = ~o(75g 2 + 130a + 9 4 b ) ,

(6)

where g2 is the effective gauge coupling constant, which is of the order 0.3 at T ~ Tc~ in this model. At T >> Tci the only stable phase is the phase u = 0. With a decrease of temperature the asymnretric phase v(T) 4=0 may appear at some temperature Tc2 > Tel, and this phase becomes energetically favourable at some other temperature T e, Tc2 > T c > Tcl [2,3], see fig. 1. However, the first order phase transition from the phase v = 0 to v(T) 4:0 is a t u n n e l i n g process [13]. Estimates based on the theory of such processes in spontaneously b r o k e n gauge theories at a finite temperature [13] indicate that f o r a , b ~ 1 the tunneling probability is very small and the phase transition proceeds from a strongly supercooled state at T ~ Tca, whereas for sufficiently large a, b the phase transition occurs somewhere near T c and the supercooling is comparatively small. Let us first consider the case when the phase transition to the phase v(T) 4:0 occurs near T = T c . If v e = u(T = Tel ) = 0, or Vc "~ V0 = v(T = 0) = U/N/~-, the phase transition is called second order or weakly first order. If, however, the j u m p of the field u at T ~ Tcx is large, so that v c ~ v 0, the phase transition is strongly first order. To calculate v c let us write an equation for v at T 4=0 [ 1 - 3 , 10]: dV(qb, T)/do = 1@v ( _ U 2 + 3'02) + F f T ) : 0.

(7)

The t e m p e r a t u r e - d e p e n d e n t term I ' ( T ) is a sum of c o n t r i b u t i o n s of all particles, interacting with the field

,I,

[lOl:

5 March 1981 8

11

F(T) = v[i~=l (3" + Sb)F(T, mi) + ~ (7 + lOb)F(T, mi) i=9

(8) 24

+ 33"F(T, mo) + ~

i=13

3"F(T mi) + ~ g2F(T, mx)1 .

Here mi, m o are masses o f various Higgs mesons, m x is the mass of the vector mesons X and Y, and 1

F(T, m) -

27r2 X ?

o

---

(9) -

k2dk

(k2 + mZ~a:Cexpi(~5 -+ m2)li21T] - 1 }

At m / T ~ 1 the value o f F ( T , m) is given by [1]:

F(T, m)

= ~ T 2 [1 -

(3/Tr)(m/T)].

(10)

If we Were calculating in the renormalizable covariant gauges, in which m i = 0 for the Goldstone particles i = 13, ..., 24, we would o b t a i n a large c o n t r i b u t i o n of these particles to V(T) even at T ~ mi, where ms. are masses of physical particles in the theory [10]. This would be obviously a gauge artifact, and therefore we prefer to calculate V(T) in the C o u l o m b gauge, in which the masses m i for i = 13, ..., 24 are equal to rex, as it should be in all physical gauges , 1 . In this case from (7), (8) and (10) it follows that v[/-t2 -- /3T2

3`02 + (Tv/3OTr)Q(g 2, 7, b)] = 0 ,

(11)

where

Q(g2,7, b) = 7 7 ( 1 0 b ) 1/2 + 4 ( 1 0 b ) 3/2

(12)

+ 3X/~ 3'3/2 + lSx/~-Tg + l~---"~SN/-2g3• F r o m eq. (11) it follows that the phase o 4= 0 exists only at T < Tcz, where

vo Vc

T22 = T21 (1 - O2/36007r2/33")-1 . o

.

TC I

TC TC2

Fig. 1. Temperature dependence of the symmetry breaking parameter v(T) in the minimal SU(5) model. The symmetry breaking phase transition occurs somewhere between Tc and Tel. It is seen that (Tc2 - Tcl)/Tcl ~ 1, which is in agreement with ref. [10], but nevertheless vc ~ vo, e.g. the phase transition actually is strongly first order. 392

(13)

Now from (12), (13) it follows that e.g. at a ~ b ~ g 2 = 0.3 (To 2 -

Tc,)/To, < 1,

(14)

,1 For a discussion of the gauge invariance problem of o (T) and of the preferable role of the Coulomb gauge see ref.[2].

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i.e. the hysteresis loop is small, which is in agreement with the conclusion, obtained in ref. [10]. However, a large size of the hysteresis loop is not a necessary criterion for a phase transition to be strongly first order, contrary to what was assumed in ref. [10]. Indeed, we shall see that the discontinuity of the phase transition, i.e. the jump of the field v at the critical temperature Tcl, is very large, vc ~ v0, and therefore the phase transition actually is strongly first order. To show this, let us compute uc = u(T = Tcl ). At T = Tel the two first terms in eq. (11) are cancelled, and from (3), (11) it follows that Vc = ( O / 3 O ~ v ~ ) v o

.

(is)

For a = b -- g2 = 0.3 eq. (15) implies that uc ~ 0.75v 0 ,

(16)

i.e. the phase transition is strongly first order. The reason why this conclusion is quite consistent with the inequality (14) is obvious from fig. 1. One should note that eq. (16) is to be slightly improved, since at uc ~ 0.75u 0 some of the masses m i, m0, m x are comparable with Tcl, and therefore eq. (10) is not quite exact. In that case one should calculate F(T, rn) numerically for different m and solve numerically eq. (11) for vc. The result, however, only slightly differs from that obtained above: vc --~ 0.6v 0 .

(16')

Some other corrections of a few percent may appear if one wishes to calculate not v(T), but a more physically relevant quantity: (Z-1/2(T))u(T), where Z ( T ) is the residue of the pole of the Green function of the Higgs field H 0 [2]. With all these corrections taken into account it can be shown that symmetry breaking in

the minimal SU(5) model is strongly firsz order for any relations between a, b and g2 (for g2 ~ 0.3) * z Unfortunately we cannot claim that our results are absolutely reliable. Indeed, higher order corrections to the value of Tcl might be large due to infrared divergences, which appear in higher orders of perturbation theory in the thermodynamics of massless YangMills fields in the symmetric phase v = 0 [3,14]. This is a general difficulty of the theory of phase transitions in gauge theories [31, and in this sense the usual statement [1,2] that the phase transition in the Weinberg For footnote see next colomn.

5 March 1981

Salam model typically is second order or weakly first order also is not quite reliable. However, one may argue that due to high-tenlperature corrections the Yang-Mills fields in the phase u = 0 acquire some mass ~g2 T, and after this mass generation the higher order corrections do not modify the lowest order expressions for mi(T), V(~, T). Tc and Tc. [3,14]. l Another problem is that, as we have already mentioned, the phase transition actually occurs not at T = T q , but somewhere b e t w e e n Tcl and T c. From fig. 1 it is obvious that this also will not alter our statement that the phase transition in the SU(5) model is strongly first order. Now let us try to understand what the reason is for the great difference between the types of phase transitions in the SU(5) model and in the Weinberg-Salam theory? The answer to this question is connected with the fact that very many particles contribute to F(T) in GUTs. Therefore the critical temperature Tcl diminishes and becomes of the same order as (or even less than) the particle masses, which appear in the asymmetric phase at T ~ To,. For example, in the SU(5) model considered above rcl ~ 0.8/a for a ~ b ~ g 2 = 0.3, and H just gives a typical mass scale in the asymmetric phase (for uc ~ u0). But in this case the value of F(T) in the asymmerrie phase becomes strongly suppressed [due to suppression o f F ( T , m) at m ~> T, see eqs. (9), (10)], and consequently the decrease o f u c = u(T = Tca ) as compared with u0 = u(T = 0) becomes small. As was shown above, this effect is already important in the minimal SU(5) model. With an increase of the number of elementary particles the critical temper,2 A very interesting situation occurs at a, b ~(375/128~2)g 4, and, consequently, if

mo> @(g/27r)mx ~ 0.2rex, where m0 is the mass of the ttiggs meson H0. ~Ihisis a rather severe constraint on m0, analogous to the constraint on the Higgs meson mass in the Weinberg-Salam model obtained in ref. [13]. The above-mentioned constraints on y and m0 are slightly stronger than those obtained in ref. [10]. 393

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ature Tcl decreases and the phase transition becomes more and more strongly first order. For example in the minimal SO(10) model the coefficient ~3in eq. (4) becomes [9] /3 = ~0(75g2 + 280a + 112b).

(6')

In this case at a ~ b ~ g2 ~ 0.3 the value of Tct becomes of order 0.6~, which is less than in the SU(5) model, and consequently all the above-mentioned effects become more pronounced. Now let us take into account that with a decrease of T the value of I'(T) in the asymmetric phase decreases exponentially [see eq. (9)]. It is clear therefore, that in GUTs based on the groups E 6, E7, SU(8) etc., in which there are thous a n d s o f elementary particles, phase transitions always (or almost always) are strongly first order. As we have already emphasized, first order phase transitions with symmetry breaking proceed from a supercooled symmetric phase at some temperature between T c and Tcl. During this transition all the energy of the symmetric vacuum u = 0 rapidly transforms into thermal energy. Moreover, during the phase transition X, Y mesons and superheavy Higgs mesons acquire masses of the same order or greater than Tc, and after their decay, which starts just after the phase transition, their energy also is transformed into thermal energy of light particles. The total energy released after a phase transition in GUTs typically is of the same order as the total thermal energy of matter at T ~ T c, and in some special cases (e.g. at a, b ,~g2 in the minimal SU(5) model) it is even much greater. This means that in the course of the expansion of the universe there was an explosion all over the universe, and the temperature of the universe abruptly increased, The strength of this explosion depends on the degree of supercooling at the critical temperature. According to grand unified theories, this explosion occurred at T ~ 1014-1015 GeV, at t ~ 10 .35 s after the Big Bang, and for obvious reasons we call this explosion the Grand Bang. Another reason to give such a name to the symmetry breaking phase transition in GUTs is its importance for the further evolution of the universe and even for our own existence. Indeed, as we have already mentioned, the decay processes of X, Y mesons and superheavy Higgs mesons, which probably have led to the excess of matter over antimatter in the universe [5], have started just after the Grand Bang. 394

5 March 1981

In the minimal SU(5) model for b > 0, 3' > (375/ 1287r2)g 4 * 2, and without cubic terms in V(d~) there should be at least one Grand Bang , 3 . In more complicated theories, based on the groups E6, ET, E8, SU(8) etc. there should be a series of Grand Bangs before symmetry breaks down to SU(3)c × SU(2)L X U(1)y. We would like to emphasize that the existence of several Grand Bangs in the very early universe seems to be an almost inavoidable implication of GUTs. On the other hand, the existence of great explosions, shaking the whole universe and changing abruptly its temperature, the rate of its expansion and the properties of elementary particles, is rather surprising and unexpected. Some years ago there was no reason to believe that there were any catastrophic events in the early universe after the Big Bang. One may say that the young Universe was believed to behave properly as a quite respectable Lady. Now, after the investigation of cosmological consequences of GUTs, one may feel even much greater respect to our Universe: It appears that this old Lady has a violent past, which only now becomes partially known to us, and a further study of her biography seems intriguing and interesting. N o t e added. After this paper has been written, 1 received a preprint "Strongly first order phase transitions in GUTs" by Daniel [15]. In this preprint an attempt has been made to verify our statement, contained in ref. [11] and in the preprint version of the present paper, where it was pointed out that phase transitions in GUTs are strongly first order. The results of the perturbation theory calculations by Daniel practically coincide with ours (except for some small differences, connected with his choice of the covariant gauge, in which m i = 0 for i = 13 .... ,24, see the discussion of this point in our paper). However, he still argued by use of the e-expansion that for some particular relations between the coupling constants, e.g. for a ~ b ~ 0.05, g2 ~ 0.3, the phase transition is weakly first order, whereas from our results it follows that in

+3 A recent detailed investigation of this problem by Kuzmin, Shaposhnikov and Tkachev (to be published) shows that for certain relations between a, b and g2 a two-step phase transition SU(5) -* SU(4) × U(1) -~ SU(3) × SU(2) × U(I) may take place in this model, this phase transition also being strongly first order.

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this case uc ~ 0.Tu 0 and the phase transition is strongly first order. In this c o n n e c t i o n we w o u l d like to n o t e that the e-expansion usually is inapplicable for the investigation o f strongly first order phase transitions, and a particular form of this expansion used by Daniel diverges in the physical limit e -+ I.

References

[6]

[7] [8]

/1] D.A. Kirzhnits, JETP Lett. 15 (1972) 529; D.A. Kirzhnits and A.D. Linde, Phys. Lett. 42B (1972) 471 ; S. Weinberg, Phys. Rev. D9 (1974) 3357; L. Doland and R. Jackiw, Phys. Rev. D9 (1974) 3320: D.A. Kirzhnits and A.D. Linde, Zh. Eksp. Teor. Fiz. 67 (1974) 1 263 [JETP 40 (1975) 628]. [2] D.A. Kirzhnits and A.D. Linde, Ann. Phys. (NY) 101 (1976) 195. [3] A.D. Linde, Rep. Progr. Phys. 42 (1979) 389. [4] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [5] A.D. Sakharov, Pis'ma Zh. Eksp. Teor. Fiz. 5 (1967)32; V.A. Kuzmin, Pis'ma Zh. Eksp. Teor. Fiz. 13 (1970) 335; M. Yoshimura, Phys. Rev. Lett. 41 (1978) 281 ;

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[15]

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S. Dimopoulos and L. Susskind, Phys. Rev. DI 8 (1978) 4500; D. Toussaint, S. Treiman, F. Wilczek and A. Zee, Phys. Rev. D19 (1979) 1036; S. Weinberg, Phys. Rev. Let~. 42 (1979) 859. Ya.B. Zeldovich and M.Yu. Khlopov, Phys. Lett. 79B (1978) 329; J.P. Preskill, Phys. Rev. Lett. 43 (1979) 1365. A.D. Linde, Pis'ma Zh. Eksp. Teor. Fiz. 32 (1980) 535; A.D. Linde, Phys. Lett. 96B (1980) 293. A.II. Guth and S.-H. Tye, Phys, Rev. Lett, 44 (1980) 631. G. Lazarides, M. Magg and Q. Shaft, CERN preprint Ref. TH.2856 (1 980). M. Daniel and C.E. Vayonakis, CERN preprint Ref. TH. 2860 (1980). A.D. Linde, Bielefeld Univ. preprint BI-TP-80/20 (1980). A. Buras, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B135 (1978) 66. A.D. Linde, Phys. Lett. 70B (1977) 306; 92B (1980) 119; Phys. Lett. B, to be published. A.D. Linde, Phys. Lett. 96B (1980) 289; D.J. Gross, R.D. Pisarski and L.G. Yaffe, Princeton Univ. preprint (1980). M. Daniel, Phys. Lett. 98B (1981) 371.

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