Cosmological consequences of a hierarchical supersymmetric model

Volume 112B, number 6

PHYSICS LETTERS

27 May 1982

COSMOLOGICAL CONSEQUENCES OF A HIERARCHICAL SUPERSYMMETRIC MODEL

So-Young PI Lyman Laboratory of Physics. Harvard University, Cambridge, MA 02138, USA and Research Laboratories of Mechanics, University of New Hampshire, Durham, Nil 03824. USA Received 4 December 1981

Unusual finite-temperature behavior of Witten's supersymmetricmodel is discussed. A high-temperaturephase persists to the weak-interaction scale. Initial monopole density can be strongly suppressed and production of heavy particles is inhibited. Difficultieswith baryon generation are suggested.

Recently Witten has proposed an interesting possible solution [1] to the mass hierarchy problem in grand unified theories [2], based on a spontaneously broken supersymmetric SU(5) model. He assumed that the small scale (weak-interaction scale) is fundamental, the large one (the unification scale) being generated dynamically. In Witten's model supersymmetry breaking is obtained by a variant of the O'Raifeartaigh method [3]. The idea that the large mass scale is induced by the small fundamental scale is attractive; not only does it give a possible solution to hierarchy problems, but also it has interesting cosmological consequences. In our paper, we shall discuss the possible phase transitions, monopole production are baryon generation in the early universe as is implied by this model. The discussion is qualitative; detailed analysis will be described in a later publication. First we introduce the model. These are two com.plex Higgs fields A~ and Y}, in the adjoint representation of SU(5), and one singlet Higgs field X. The Higgs field in the fundamental representation is ignored for simplicity. The scalar potential is given by [4]

t r A Y + 2gXA[ 2

3,

3).

(2)

Y is parallel to A at the minimum and is given by Y = (g/X)X diag(2, 2, 2,

3, - 3 ) .

(3)

However, X is undetermined at the tree level. The broken gauge symmetry is SU(3) × SU(2) X U(1). The one-loop effective potential must be calculated in order to determine X. The one-loop effective potential is given by [5] F Vl(~b) = ~(--1)£ M/4(q~)lnMi2(q~)//~2, (4) i 647r~ where the sum runs over all helicity states,Mi(q~ ) is the field dependent mass of the ith such state, F = 1 for fermions, F = 0 for bosons,/1 is a renormalization mass. Witten has shown [ 1] that for large X, X >>M, the effective potential evaluated at the minimum given by eqs. (2) and (3) has the following form, which indudes the lowest-order term. g2

29X2 50e 2

VI(X) = 3092 +X2 1 -~g2 +X2/30

+ X2(trA2A*2 _ X t r A 2 trA .2)

+e 2 tr(i[A,A*] +i[Y, y * ] ) 2 ,

A = [gM/(30g 2 +X2) 1/2] diag(2, 2, 2,

M4g2X2 (

Vo(A, Y , X ) = g 2 l t r A 2 M212

+tr[X(AY + YA)-~X~

M is the only mass scale in the theory, and it characterizes supersymmetry breaking. To minimize the energy, one must have

× I n {Sl2/i.t2)+O(lX[ 2)

807r2

forX>>M.

(5)

(1) If 29X2 - 50e 2 < 0, X increases without limit until

0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland

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perturbation theory breaks down. However, Witten argues that asymptotic freedom will force the gauge coupling, e 2, which really depends on X, to vanish with increasing X. Hence the effective coupling 29X2 50 e2 changes sign at some large value X 0. Then a stable minimum will be produced near XO, which can be interpreted as the unification mass scale ( ~ 1014 GeV). Therefore, X 0 is independent of the fundamental scale M of the theory and is determined by the renormalization group equation. SU(5) is strongly broken by the vacuum expectation value of Y according to eq. (3). The interesting early-universe phase transitions, which we study in this paper, are a consequence of a peculiar feature of this model: the direction of symmetry breaking is entirely determined by the A field, which has small mass characteristics. Therefore, as we shall demonstrate, the temperature of the symmetrychanging phase transition is also of the order of the small mass, since it is governed by the field A. For our analysis we shall need the finite-temperature effective potential [6,7]. It is given by, up to one-loop corrections,

27 May 1982

i +

2

-

+ O(Mi4(~b),M/6(¢)/T2) ]

(for fermions).

(8b)

For T~Mi(4)), the "temperature dependence in fiT(+) disappears as exp [-Mi(¢)/T ] and the zero-temperature effective potentials V0(q~) + V0(¢) dominate. For the regions between T ~ M i ( ¢ ) and T >>M i (+) the temperature corrections have not been reliably computed. In this model the finite-temperature potential has the following form: for T>>Mi(A,X , Y), we keep only the first two terms in eqs. (8a) and (Sb). Veff(T,A,X, Y)= Vo(A,X , Y) - ~orrZN(T) T 4 + aAT2 trAA* + o y T 2 tr YY* + o x T 2 X X * . (9a) Vo(A,X, Y) is as in eq. (I);N(T) =Nb(T ) + Nf(T)" g7 where N b and Nf are the total number of distinct helicity states for bosons and fermions with mass ~ T; oX, ay ando X are functions ofe 2, ~,2,g2, of order (e2, ~,2,g2) and are positive in this model. For T ~ M i ( A , X , Y)

Veff(T, q~) = V0(q~) + V;(q)),

(6a)

V1T(¢) = V7(¢) + eT(¢),

(6b)

Veff(T,A,X, Y)= Vo(A,X, Y) + vO(A,X, Y). (9b)

where q~denotes scalar fieldsin the theory. V0(¢) is the classical potential; V°(+) is the one-loop zero-temperature contribution of eq. (4); FT(O) is the oneloop finite-temperature potential and is given by [7]

For regions between Mi(A ,X, Y) > T and Mi(A ,X, Y) T, we shall simply extrapolate the two limiting forms (9a) and (9b). The above finite-temperature effective potential shows that essentially, nothing happens until the temperature becomes the fundamental mass scale M of the theory. Let us discuss this in some detail: at very high temperature T >>X 0, where X 0 is the large mass scale (>~ 1014 GeV), an absolute minimum exists atA = X = Y = 0. The universe is in the SU(5) phase. As the temperature decreases, the A field may undergo a second-order phase transition. The highest possible critical temperature is

ao

27r2 "

0

X ln{1 -- (--1)F exp {-- [x 2

+Mi2(¢)/r2]}}.

(7)

Here M i (q)) and F are defined as in eq. (4). At high temperature, T >>M i (¢), F~r(¢) ma~;,be expanded as

[61

TA = (2/aA)ll2gM.

fflT(~b)= ~ [--~o~riT 4 i

+ AMi2(~)T2 - A.-1M3(+)T -

A~r-2Mia(~)lnMiZ(g>)lr2

+ O(M4(¢),Mi6(¢)/T2)] 442

(for bosons),

(8a)

(10)

This phase transition will actually occur at TA if the minimum of X stays at X = 0 until this temperature. Therefore, the minimum of A will be at Amin = 0 for the temperatures higher than TA . In the following we argue that no phase transitions occur for T ~ TA but they do occur only for T <~TA .

(1) T>~ TA . First, let us consider the behaviour of the X and Y fields for the temperatures T >-. TA where Ami n = 0. At very high temperatures an absolute minimum exists at X = Y = 0. This minimum will be present until the temperature becomes <~ O(M). However, as the temperature decreases well below the mass scale X 0 (>~ 1014 GeV), a non-zero minimum may develop at large X and Y. For the regions where X, Y >> T, T is much less than the field dependent mass M i and as shown in eq. (9b) the behavior of X and Y is dominated by the zero-temperature effective potential. When Amin = 0, unlike the zero-temperature situation, the classical potential does not determine the direction of Y nor the magnitudes of X and Y. These are fixed by the one-loop corrections evaluated atAmin = 0 and arbitrary Y and X. A general calculation of this quantity is very lengthy. Let us study the one-loop correction when the Y field points in the SU(3) X SU(2) X U(1) direction. We calculate vO(A, X, Y) for A = 0, Y = y diag(2, 2, 2, - 3 , - 3 ) with y and X arbitrary. We took A = 0 because that is the equilibrium value for temperatures T >~TA , as shown above. We find that all the e 2 dependence cancels. This tells us that, for T>~ TA, the oneloop potential does not develop a minimum as a consequence of the renormalization group running coupling e 2. The same is true for the SU(4) X U(1) direction. This means that the large vacuum expectation value X 0 at T = 0 is induced by the non-zero vacuum expectation value o f A . Therefore, X = Y = 0 is the only minimum as long as Amin = 0 and A will have an independent secondorder phase transition at TA = (2/oA )l/2gM ~--0(3//). (2) T <. TA . When T ~ TA , A will have a nonzero minimum of the form in eq. (2), and the gauge symmetry will be softly broken to SU(3) X SU(2) X U(1). For X, Y >> T, the effective potential will have the form eq. (5) asAmi n becomes non-zero. A minimum will develop near X 0 due to the effective potential in eq. (5). ForX, Y < T, Ymin is parallel toA and is given by

Ymin = [2XgO2Xmin/(°y T2 + 2~'202)] X diag(2, 2, 2 , - 3 ,

T¢M

27May 1982

PHYSICS LETTERS

Volume 112B, number 6

3)

(g/X) Xmin diag(2, 2, 2, - 3 , - 3 ) ,

(11)

where v 2 ~ g2M2/(3092 + X2). Xmi n is determined by the effective potential, Veff(X, T)at small X. For T ~ TA and 0 <~X ~ O(1}/), the one-loop finite-temperature and the one-loop zero-temperature effective potentials ate of same order [~ O(M4)]. Therefore, we keep subdominant terms in eq. (8a), (8b) and combine with contributions from eqs. (1) and (4) to get the following potential: M492~k2- ~Tr2N(T)T 4 + g1 o x T 2 IXI 2 Veff(X, T ) = 30g2 + 32 -

+ ~i (--1~2 M4(X) ln T2/t.z2

(12a)

• 64rr ___.__+ M4g2X 2 T~0 30g 2 + X2

+ V°(X).

(12b)

The term - ~rr-IM/3(X) T in eq. (8a) is neglected because it can be imaginary and therefore unreliable [6]. Eqs. (12a) and (12b) describe two different behaviors of X depending on the sign of d 2 vO(x)/dX 2 IX=0. The sign depends in general on the range of parameters of the potential. The magnitude of d 2 V~(X)/ dX21x=0 is ~M2x(g 4, X4, e4). We shall briefly describe the two possible cases, Let us take d 2 V°(X)/dX 2 IX=0 > 0. In this case d2 Veff (X, T)/dX2 Ix=0 is always positive. When the minhnum X = 0 become metastable at T < O(gM), there will be a first-order phase transition. This is a purely quantum-mechanical process in which a finite region of space tunnels through the barrier in the potential to form a bubble of the new phase [8]. The phase transition occurs at the temperature TX when the bubble nucleation rate becomes larger than the fourth power of the expansion rate of the universe [9]. The bubble nucleation depends on the values of the theory's parameters and therefore so does Tj(. After the tunneling is completed at X~ [Veff(X = 0) ? t = Veff(X0) ] X will increase continuously from X 0 to

x0. Alternatively, we take the curvature d 2 V°/ dX21x=0 < 0. Then dZVeff(X, T)/dX2Ix=o can be negative and the transition from the minimum X = 0 to X 0 can occur without the tunneling process. The critical temperature TX at which the minimum X = 0 becomes unstable is determined by eq. (12a) and Ty < O(gM, eM, XM). In both cases, as X increases, the SU(3) X SU(2) 443

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× U(1) symmetry breaking will become strong and the masses of the particles which depend on X become heavy. However, we notice that the vacuum-energy difference V(X = O) - V(X = X0) is only of O(M 4) and the phase transition temperature is also of O(M). A puzzle presents itself: from where can the energy come to make the heavy particles whose masses eventually should exceed 1014 GeV? One possibility is that the particles that should become heavy decay as X increases; so in fact no heavy particles are formed. Of course, this decay cannot occur for magnetic monopoles. Presumably they will slow down the increase of X. Now we examine baryon-number production and monopole production. Baryon-number production. Let us consider the conventional out-of-equilibrium decay scenario for baryon-number generation through the decay of heavy bosons, with constant massM b [10]. The decay rate P is given by F = (~]//b where a = e2/41r. Baryon number will arise only when the universe is out of equilibrium; this requires that the ratio K, defined by K =- I/HI T=Mb where H is the universe's expansion rate, be less than or equal to 1. (Computer calculation [ 11] indicates that baryons can be produced even for K slightly greater than 1 .) However, in Witten's model, K~-- 1014 a, whenM b is assumed to be a constant, O(M), and the universe is in thermal equilibrium so that net baryon number is not produced. The fact that M b increases does not alter the conclusion. To establish this, we calculated ~/b/nb, due to the increase o f M b for the colored Higgs boson decay, where n b is the equilibrium density distribution of the heavy bosons. If/'~b/nb were greater than F = ~Mb(t), then thermal equilibrium cannot be maintained and baryon asymmetry could arise. However, a rough estimate gives the opposite conclusion; we find [ ' > ;lb/n b at T ~ M b ( t ). (b ) Monopole production. When SU(5) breaks down to SU(3) × SU(2) × U(1) at T = TA light monopoles will be produced prolifically due to large fluctuations in the A field. These monopoles annihilate each other strongly when they are closely together. However, as the universe expands, the annihilation becomes negligible. Preskill's [12] estimate for the monopole density to entropy ratio, at temperatures well below the critical temperature, is given by

444

nm/T3 ~ 1 0 ' 6 Mm/M p,

27 May 1982 (13)

where M m is the mass of the monopoles. In standard SU(5) theory, where M m ~ 1016 GeV, Preskill's calculation tells us that the monopole annihilation can reduce nm/T3 only to O(10-10). However, the standard scenario of helium synthesis requires rim~T3 ~< O(10 -19) f o r m m ~ 1016 GeV. The bound on the present value of nm/T3 imposed by the observed Hubble constant and deceleration parameter is ~10 -24 f o r m m ~ 1016 GeV. This is the well-known problem of initial monopole production in the standard SU(5) theory ,1 Eq. (17) tells us that monopole density is proportional to the mass of monopoles. In the scenario we are considering, the monopole mass is O(M/a) for the temperature T >~Tx , and it increases with temperature for T<~ Tx . A rough estimate shows that if Tx ~< 10 -3 TA , the annihilation can reduce the monopole density to nm/n~, ~ O(10 - 2 ° ) whenM is taken to be the weak-interaction scale. If these monopoles become heavy as X increases to X0, their mass density will be within the bound at the time of helium synthesis. We summarize our qualitative analysis of a SU(5) supersymmetric model in which a small scale is fundamental, the large one being generated dynamically: (a) SU(5) stays unbroken until the temperature scale becomes the fundamental scale. We conjecture that this is a model-independent result. [When SU(5) stays unbroken until the weak-interaction scale, the gauge coupling will become strong and there may be a dynamical symmetry breaking of SU(5). This possibility has not been considered here.] (b) Since SU(5) symmetry breaking occurs at the temperature o f the weak-interaction scale, M, and the energy difference between the SU(5) phase and the SU(3) X SU(2) X U(1) phase is of O(M4), it may not be possible to produce superheavy particles. (c) Baryon production poses difficulties for this model. Since the expansion rate of the universe is very small at the time of phase transition, it is difficult for the universe to be out of thermal equilibrium with respect tobaryon generating reactions. (d) Suppression of initial monopoles is possible. Detailed calculations to supplement the qualitative 4:1 Some papers on the suppression of initial monopole production in grand unified theories are mentioned in ref. [13]

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analysis presented here are in progress. During the preparation o f this manuscript, I learned that similar work is being done by I. Affleck and P. Ginsparg. I thank L. Girardello, A. Guth, R. Jackiw, J. Preskill and E. Witten for useful discussions. This research is supported b y the Department o f Energy under Grant No. ER-78-5-02-4999. I thank the Physics Department o f Harvard University for their hospitality. N o t e d added. Ginsparg [14] concludes in agreement with m y result, that the symmetry of the theory stays unbroken until the temperature reaches small mass scale M of the theory. However, he also alleges the existence o f another minimum at large X for T >~ T A . I find no evidence of such a second minimum in my calculation, and therefore disagree about this point which is irrelevant to the principal result. References

[1] E. Witten, Trieste preprint (1981). [2] S. Weinberg, Phys. Rev, D13 (1976) 974; D19 (1978) 1277; E. Gildner and S. Weinberg, Phys. Rev. D13 (1976) 3333.

27 May 1982

[3] L. O'Raifeartaigh, Nucl. Phys. B96 (1975) 331. [4] E. Witten, Princeton preprint (1981), to be published in Nuclear Physics B. [5] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 788; E. Witten, talk Harvard University (1981). [6] L. Dolan and R. Jackiw; Phys. Rev. D9 (1974) 3320. [71 S. Weinberg, Phys. Rev. D9 (1974) 3357. [8] S. Coleman, Phys. Rev. D15 (1977) 2929. [9] A.H. Guth and E. Weinberg, Phys. Rev, D23 (1981) 876. [10] Yu. Ignatiev, N.V. Frasnikov, ViA. Kuzmin and A.N. Tavkhelidze, Phys. Lett. 76B (1978) 436; M. Yoshimura, Phys. Rev. Lett. 41 (1978) 281;42 (1979) 146(E); S. Dimopoulos and L. Susskind, Phys. Rev. D18 (1978) 4500; B. Toussaint, S.B. Trieman, F. Wilczek and A. Zee, Phys. Rev. D19 (1979) 1036; J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Phys. Lett. 80B (1979) 360; S. Weinberg, Phys. Rev. Lett. 42 (1979) 850. [11] J.N. Fry, K.A. Olive and M.S. Turner, Phys. Rev. D22 (1980) 2953. [12] J. Preski/1, Phys. Rev. Lett. 43 (1979) 1365. [13] A. Guth and H. Tye, Phys. Rev. Lett. 44 (1980) 631; P. Langacker and S.Y. Pi, Phys. Rev. Lett. 45 (1980) 1; G. Lazarides and Q. Shafi, Phys. Lett. 94B (1980) 149. [14] P. Ginsparg, Phys. Lett. l12B (1982) 45.

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