Strong coupling phase transitions in supersymmetric grand unified models

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22 August 1985

STRONG COUPLING PHASE TRANSITIONS IN SUPERSYMMETRIC GRAND UNIFIED MODELS David B. R E I S S School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA Received 11 February 1985

The determination of the temperature at which a grand unified model becomes strongly coupled should be based upon a physical quantity such as the screening length rather than the ad hoc condition that the opening becomes 0(1). I use a recent calculation of this screening length (the inverse electric mass) to discuss some aspects of strong coupling behavior in the cosmology of supersymmetric grand unified models. Significant effects may occur in a variety of cases. An interesting possibilit is that there may be a pair of confining and deconfining phase transitions at a temperature as low as the supersymmetry breaking scale (O(TeV)). I present illustrative examples for these effects.

In investigations of the cosmological consequences of particular grand unified models based on a gauge group G we are often interested in determining the scale at which G (or a subgroup of G) becomes strongly coupled. For example, in certain supersymmetric models at high temperatures the system is in the phase characterized by the gauge group G [say, SU(5)] ; and, as the universe cools, the system may be found in a metastable comfiguration with gauge group G, but with a very small tunneling probability into a stable configuration. This overwhelmingly small probability might persist until very low temperatures (of the order o f a TeV, or perhaps considerably lower [1,2] ) if the system never became strongly coupled. The phase transition into the SU(3)c × SU(2)L X U(1)y phase would occur far too late to be consistent with cosmology. However, since the gauge coupling grows with decreasing temperature (and since generally there is a "Landau pole" at f'mite temperature), there will occur a temperature at which the system becomes strongly coupled. If we choose the value of the temperature at which a = g2/41r ~ 1, this would happen at a temperature o f order 108-1012 GeV (for example in minimal supersymmetric models [3], among a host of other models). The occurrence of this strongly coupled regime can cause the system to make the transition out of the metastable phase into the phase 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

characterized by the lower free energy [ 1 , 2 , 4 - 6 ] . This effect has also been used to attempt to solve the problem o f breaking the degeneracy of the vacua in supersymmetric models [4], in the primordial monopole problem [2,6,7] and in a variety of other contexts. To estimate where the system becomes strongly coupled by searching for the scale at which ~ ~, 1 is rough at best [8]. It seems to make more physical sense to use an expression for the screening length (as a function of temperature) in this determination [8,9]. For very special cases, of which pure SU(3)c is the best example, it turns out that the temperature at which the confining phase transition occurs (the temperature at which the electric screening length becomes infinite is lower than this generally, allowing for the possibility of a metastable supercooled plasma as the temperature goes from higher to lower values [9] ) is very close to that at which a ~ 1. This is born out by a recent calculation of the gluon's electric mass, Mel (the inverse screening length) with the inclusion of the so-called "plasmon" term which arises from the summation of an infinite set of infrared divergent finite-temperature graphs [9]. Such an expression for the electric mass has recently been calculated for an arbitrary grand unified model [9]. In this calculation of Me1 restricted to an arbitrary N = 1 305

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supersymmetric grand unified model it was found that the temperature at which such a system becomes strongly coupled generally does not occur near the position a ~ 1. In fact, such models may have the strongly coupled phase occur at a temperature for which a ~ 1/20, which, for many of these models, is a value that is never attained below the Planck temperature. The reason for singling out supersymmetric grand unified models is the presence of Yukawa and quartic scalar couplings with gauge strength. In the absence of such large couplings the change in where the system becomes strongly coupled (as compared to that obtained from the condition a ~ 1) is not as drastic as in the supersymmetric case (where, also, the coupling itself is generally larger as compared to that in conventional grand unified models at the same temperature). For a supersymmetric grand unified model with n R chiral multiplets falling into the irreducible representation R of the gauge group G, it was found that

[9] 2

2

+

t3/2

/

-

31r1/2 a3/2 (T(adj) +

2_+ n R T ( R ) )

1 ~ [nRd(R)/d(adj) ] [M2(R)/T 2] 3/2,

'/r

(1)

where Tr(TaT b) = T(R)6 ab for the irreducible rep2 = 27raC2(R)T 2 is the high temresentation R, Msc perature mass of the scalar bosons in the chiral multiplet, C2(R ) = d(adj) T(R)/d(R) is the eigenvalue of the quadratic Casimir operator acting on the representation R, and "adj" stands for the adjoint representation. In this expression we have assumed that all Yukawa and quartic scalar couplings not coming from gauge terms are small relative to the gauge couplings (expressions for the more general case are given in ref. [9] ). The last two terms in eq. (1) which are O(a3/2) are the "plasmon" contributions. There are two points of view that one may take with regard to the expressions calculated in ref. [9] for M2r In the conclusions to that paper they were referred to as the "strong" and "weak" points of view. The "strong" view is that the expression for M21 calculated in the weak coupling expansion to O(a3/2) actually approximately reproduces the true behavior of M2eleven near places where M21 = 0 (i.e. where the 306

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electric screening length becomes infinite). If that is true, then setting M21 ~ 0 corresponds to looking for the temperature near which a con fining-deconfining phase transition is expected to occur. The "weak" view is that the expression for M21 to O(a 3/2) is much better to use to determine where the weak couplingexpansion breaks down by setting M21 ~ 0 (that is, by noting where the O(~ 3/2) term becomes comparable to the O(a) term) than the naive condition a ~ 1. From this point of view - and using the breakdown of the weak coupling expansion as a hallmark of where strong coupling behavior sets in - this paper is an exploration of what new physical possibilities can arise (such as the possibility of multiple SU(3)c phase transitions). One further comment: The question of what the correct expansion parameter is for perturbation theory (or the weak coupling expansion) is an old one. It certainly is not a in the situations that we are discussing here. This can be easily gleaned from a quick glance at table 1 and from the numerical results discussed in what follows. Hence the contentions that the condition a ~ 1 is inappropriate for the determination of where the weak coupling expansion breaks down, and that a much better estimate can be obtained by considering where M21 ~ 0 when computed to O(a3/2). To some extent, a determination of which of the "strong" or "weak" points of view is more likely to be correct would require a calculation o f M 2, to O(a2), although it might require a truely non-perturbative analysis. To determine the temperature at which the strongly interacting phase is expected to have set in we use the condition M2~ ~ 0 and the renormalization group expression for the running coupling to lowest order (specialized to a supersymmetric model), a(T) = 27r[27rot-1 (M)

+(3T(adj)- ~ n R T ( R ) ) l n ( T / M ) ] -1 .

(2)

(where T < M and there are no mass thresholds between T and the reference scale M). The expressions (1) and (2) are valid for any gauge group G (in a supersymmetric model) for which the temperature at which we are examining the system is either large compared to any bare scalar masses that may be in the lagrangian or small compared to them

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Table 1

M2el/T2 as a function of`* for a variety of choices of chiral representation content for SU(5) and SU(3) computed from eq. (1). A1SO shown is the value `*crit for which MZel/T2 = 0 in each case. M2el/T2

Gauge group

Chiral representations

acrit

[SU(5)]

pure supersymmetric SU(5)

1/3.6

3 la - 59a 3/2

24

1/12.6

63`* - 224cx3/2

3(5 + 10) 3(5 + 10) + 24

1/12.9 1/19.9

69a - 248`*3/2 101`* - 451`*3/2

3(5 + 10) + (5 + 5) 3(5 + 10) + (5 + 5) + 24

1/14.2 1/21.1

75`* - 283`*3/2 107`* - 491`*3/2

3(5 + 10) + 2(5 + 5) 3(5 + 10) + 2(5 + 5 ) + 24

1/15.1 1/22.2

82`* - 319 `*3/2 113`* - 532~ 3/2

3(5 + 10) + (5 + 5) + (10 + 1"-0) 3(3 + 10) + (5 + 5) + (10 + 1"-0)+ 24

1/18.1 1/24.4

94`* - 400`*3/2 126`* - 622`*3/2

3(5 + 10) + 2(5 + 5) + (10 + 1"-'0) 3(3 + 10) + 2(5 + 5) + (10 + 1"-0)+ 24

1/18.9 1/25.5

101a - 439`*3/2 132`* - 666`*3/2

3(3 + 10) + 2(5 + 5) + 2(10 + 1--0) 3(3 + 10) + 2(5 + 5) + 2(10 + 1-"0)+ 24

1/22.7 1/28.6

119`* - 567`*3/2 151`* - 808`*3/2

6(3 + ]) 6(3 + ]) + 8 6(3 + 3) + 2(3 + 3) + 8

1/9.8 1/14.1 1/15.9

57`* - 178`*3/2 75`* - 282`*3/2 88`* - 351`*3/2

[SU(3)]

[in which latter case those scalar multiplets do n o t c o n t r i b u t e to the sums in eqs. (1) and ( 2 ) ] . The expression is o n l y approximate outside o f these ranges

[9]. To illustrate some o f the u n u s u a l consequences of determining where strong coupling behavior occurs b y using the expressions (1) and (2) we analyze the class o f SU(5) models discussed b y K o u n n a s , Lahanas and N a n o p o u l o s [10]. In these models there are six mass scales: AQC D < M W < M 3 , 8 < M T < M X < M p . Three o f these are adjustable: M 3 8 , M T , a n d M X. M3, 8 represents a scale ( ~ 1 0 4 - i'06 GeV) at which a light color octet occurs [as well as an SU(2)L triplet] a n d M T represents a scale ( ~ 1 0 1 0 - 1 0 1 2 GeV) at which heavier color triplets occur - those which are responsible for the generation o f a b a r y o n asymmetry. There is also a d y n a m i c a l l y generated scale in these models: the scale at which SU(5) becomes strongly interacting and hence [ 1 , 2 , 4 - 6 ] precipitates an SU(5) breaking transition. This critical temperature is computed in ref. [10] from a ~- 1 to be T c ~ 109 ~ 1010 GeV. We will see presently t h a t this last result is n o t correct w h e n calculated via consideration of the electric screening length. The list of chiral SU(5) multiplets is as follows [10] :

three families in the representation ( ] + 10)F and Higgs' in the representations 2(5 + 5)H + (10 + i'0)H + 24 H. There is also the possible inclusion o f (50 + 50)H and other representations which we will assume to have masses of O(Mp) and which we will therefore n o t include in this discussion;however, note that the inclusion o f more representations w o u l d only strengthen m a n y of the conclusions. To analyze this model at temperatures intermediate b e t w e e n M w a n d M p we will need the SU(3)c × SU(2)L × U ( 1 ) y representations that have masses of order M w , M3, 8 and M T. T h e y are as follows:
3(3, 2, - 1 ) + 3(3, 1, - 2 ) + 3(3, 1 , 4 )

from 3(5 + 10)F , MW:

M3, 8" MT:

2(1, 1, 6) + 2(1, 1, - 6 )

from (10 + 1-'0)H,

+ 2(1, 2, --3) + 2 ( 1 , 2 , 3)

from 2(5 + 5)H'

(8, 1, 1) + (1, 3, 1) 2(3, 1, --2) + 2(3, 1 , 2 )

from 24H, from 2(5 + 5)H"

First we will consider the system in the SU(5) 307

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phase at temperatures a b o v e M X. In ref. [10] the value of c~5(Mx) is found to be a 5 ( M x ) = 1/7.3 (for M3,8 = 105 GeV and M T = 1011 GeV). From table 1 we see that the critical value of ot5 for a model with representation content 3(5 + 10)F + 2(5 + 5)H + (10 + 10) H + 24 H is acrit --- 1/25.5. Thus, as advertized earlier, we find that in the SU(5) phase the system is strongly coupled (and possibly confining) already at T ~ M x . Let us continue to use this model as an example. If we c o m p u t e M 2el for SU(3)c f o r M T "~ T "~M X [i.e., SU(3)c with the chiral representations 6(3 + 3) + 2(3 + 3) + 8] we see from table 1 that the critical value for astrong is 1/15.9 for this representation content and hence, since 1/12.4 < astrong < 1/7.3 in this temperature range [10], SU(3)c is strongly interacting for these temperatures. The same calculation, performed for M3, 8 < T < M T [where the SU(3)c chiral representation content is 6(3 + 3) + 8], yields a critical value (from table 1) for astrong of 1/14.1 whereas 1/12.8 < O~stron~ < 1/12.4 in this temperature interval [10]. Thus, for this temperature range, we again have the result that SU(3)c could be strongly interacting. Between M w and M3, 8 the chirat representation content is 6(3 + 3) (just quark superfields) which gives a critical value for O~strong of 1/9.8. In this range, however [10], 1/12.8 < Otstrong < 1/10, thus SU(3)c is weakly coupled at least for much of this range. We have the unusual behavior of, as we go from low temperatures to high temperatures, SU(3)c going from being strongly interacting to weakly interacting to strongly interacting. This final strongly interacting SU(3)c then matches up with a strongly 'interacting SU(5) at the scale M X . Thus far we have used this model to illustrate two points. First, that it is often the case that a supersymmetric grand unified model may be in the confining phase at the unification temperature. Second, it is possible that there are other unexpected confinem e n t - d e c o n f i n e m e n t phase transitions depending on the presence of mass thresholds. With regard to the first point we must be a bit more careful (although we will come to the same conclusion in this model). In calculating the critical value of a near and above M x we have assumed that all of the representations 3(5 + 10)F + 2(5 + 5)H + (10 + I-'0)H + 24 H contribute to M~e1. Let us assume that this is true. However, let us assume that the bare (zero 308

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temperature) masses of 2(5 + 5)H + (10 + I"O)H + 24 H are all such that these representations do not contribute to Me21at temperatures below M x [in the SU(5) phase]. In that case the critical value of ~ is no longer the value acrit = 1/25.5 that it was above M x. Rather, it now has the value Ctcrit = 1/19.9. This does not affect the conclusions above since this value is still much smaller than the value of t~ one would expect at this temperature from a naive consideration of the renormalization group. However, this effect is worth noting since many (minimal) supersymmetric grand unified models have couplings at Unification that are closer to 1/20 (this, as a consequence of there being fewer multiplets contributing to the renormalization group equations at intermediate scales). For such models we might have a circumstance as follows: As the temperature lowers from the unification temperature where the system is in the confining phase, degrees of freedom turn off leading to a deconfining transition. Then, as the temperature continues to decrease, the growing coupling causes the system to become confining again. It is conceivable that such a system may still supercool in the phase characterized by the grand unified group G in spite of being confining above M x if it has not had sufficient time to effect the gauge phase transition before the degrees of freedom turn off. In this case it is the lower temperature confining-deconfining phase transition that determines at what temperature the gauge phase transition occurs. Even if the model is not confining above the unification temperature (as determined from a calculation of Me21) we may still get significantly different results than through the condition tx ~ 1. To illustrate this let us consider the minimal supersymmetric model [3] based upon the representation content 3(5 + 10)F + (5 + 5)H + 24H" With two light Higgs doublets the value of ct5 at unificaiton is 1/23.5 (with the choice Ctstrong(Mw) = 1.7.1 used in ref. [3] ), whereas the critical value for a 5 from table 1 is 1/21.1 ; thus, the system is probably unconfined above M x ~ 1017 GeV. However, below MX, if the relevant chiral multiplets are 3(5 + 10), the critical value ofct 5 is 1/ 12.9. From eq. (2), cz5 attains this value at a value of the temperature o f M x / T ~ 10 3 as opposed to the value 6 X 107 obtained using ~5 ~ 1 [4]. Thus, even in this minimal context we cannot obtain the strongly interacting scale ( ~ 1010 GeV) often needed for cosmolog-

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ical model building purposes. (Note further that, as mentioned earlier, that the critical temperature for the confining phase transition will generally be higher than that at which M21 = 0; hence, the actual value of Mx/T will be lower than that calculated above.) We can note one other unusual possibility for SU(3)c. It may be that SU(3)c becomes contiming at a temperature corresponding to the supersymmetry breaking scale (a few TeV) and then becomes deconfining again at a temperature somewhat higher than this. This is because, in order to calculate M21 below the supersymmetry breaking scale, we must only include the three families of quarks: neither their superpartners nor the gauginos contribute. Such a calculation gives a critical value for a s of 1/3.6 (see the appendix of ref. [9] for the general expression that applies to this case); whereas, with the inclusion of the superpartners and the gauginos (which both become important above the supersymmetry breaking scale) the critical value is 1/9.8. Since a reasonable value for as(Mw) is in the range 1/7 to 1/10, we see that such a phase transition is possible ; but, as some of these possible values for as(Mw) are quite close to the critical value and since the supersymmetry breaking scale is unknown up to a factor of ten or so, we cannot state this conclusion with confidence. However, it is intriguing that there might be new SU(3)c phase transitions at such a low temperature. Note also that the splitting between particles and their superpartners would possibly lead to "time structure" in the behavior of the system near other mass thresholds. Thus, for example, in the SU(5) model of ref. [10] analyzed above, there can possibly be other phase transitions near M3, 8 or M T. The details depend upon the mass spectrum of particles and their superpartners. The occurrence of a plasmon term in the high temperature expression for a boson's mass is not restricted to the electric mass of the gauge vector. Such a term also arises for the mass of a Higgs scalar. (The simplest case of massless ~b 4 theory gives a high temperature mass m2(T) = XT 2 to O(X) and a mass m2(T) = [X - (3/¢r)X 3/2 ] T 2 with the inclusion of the infrared plasmon effects [9] .) This leads to two speculations. There is a theorem [11] that to lowest order inverse gauge symmetry breaking (the possibility that the symmetry decreases with increasing temperature [12] ) may not occur in a supersymmetric grand uni-

22 August 1985

fled model [as long as there are no explicit U(1) factors in the ultimate gauge group]. This result depends on the positive semidefiniteness of the non-zero temperature scalar mass matrix. The inclusion of the" plasmon term can alter this: Note that this contribution often carries a sign relative to the lowest order result. Another speculation is that gauge symmetry breaking in a grand unified model may be driven (or catalyzed) by the plasmon contribution in circumstances where the relevant scalar's masses would otherwise not attain negative values (again because of the occurrence of the relative sign); and, in fact, it may be due (in the supersymmetric case) to the gauge contributions [O(a3/2)] to the temperature mass of the scalars. (In early versions o f N = 1 locally supersymmetric models, a large t-quark Yukawa coupling (or such a coupling for a fourth generation) was used to cause the scalar's mass squared (which had positive mass squared at high energy scales and so would otherwise not have been able to induce symmetry breaking) to run to negative values [13] ). As a timal point, it would be interesting to see what effect multiple SU(3)c phase transitions might have on the properties of the axion: in particular the axion "energy crisis" [14]. These issues are currently under investigation. I am happy to thank Joe Kapusta and Serge Rudaz for conversations, constructive criticisms and for critical comments on the manuscript. I wish to thank the Lewes Center for Physics, where this work was started, for its hospitality. I also thank Julius Reiss, who wanted to see his name in print. This work was supported in part by the Department of Energy contract DE-AC02-83ER40105.

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[9] J. Kapusta, D.B. Reiss and S. Rudaz, University of Minnesota preprint DOE/ER/4105-513 (1985). [10] C. Kounnas, A.B. Lahanas, D.V. Nanopoulos and M. Quiros, Nucl. Phys. B236 (1984) 438; Phys. Lett. 132B (1983) 95. [11] H.E. Haber, Phys. Rev. D26 (1982) 1317. [12] S. Weinberg, Phys. Rev. D9 (1974) 3357. [ 13 ] L. Atvarez-Gaum6, J. Polchinski and M. Wise, Nucl. Phys. B221 (1983) 495 ;

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J. Ellis, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. 121B (1983) 123; J. Ellis, J.S. Hagelin, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. 125B (1983) 275; L. Ib~nez and C. Lopez, Phys. Lett. 126B (1983) 54; Nucl. Phys. B233 (1984) 545. [14] L. Abbott and P. Sikivie, Phys. Lett. 120B (1983) 133; J. Preskill, M. Wise and F. Wilczek, Phys. Lett. 120B (1983) 127; M. Dine and W. Fischler, Phys. Lett. 120B (1983) 137.