Baryon asymmetry and low energy parity restoration

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PHYSICS LETTERS

21 January 1982

BARYON ASYMMETRY AND LOW ENERGY PARITY RESTORATION A. MASIERO Max-Planck-Institut far Physik und Astrophysik, Munich, Fed. Rep. Germany and G. SENJANOVIC Max-Planck-Institut flir Physik und Astrophysik, Munich, Fed. Rep. Germany and Brookhaven National Laboratory 1, Upton, LL N Y 11973, USA Received 25 March 1981

We examine the question of baryon number generation in the early universe in theories with low intermediate mass scales It is shown that the recently proposed O(10) grand unified theory with parity restoration at E ~. few hundred GeV allows for the creation of matter-antimatter asymmetry in accord with cosmological observations. This evades a recent general argument for incompatibility between weakly broken left-right symmetry and substantial baryon generation.

1. In the last couple of years, the problem of dynamical baryon number generation has received a great deal of attention [ 1 ]. It was shown that the presence of baryon number violating interactions in grand unified theories leads naturally to the creation of m a t t e r antimatter asymmetry in the very early universe. The conventional picture of grand unification [2] assumes the desert in energies between M w and the unification energy M X ~- 1014 GeV, which corresponds to the masses of superheavy gauge bosons which mediate proton decay. In such theories the CP violating decays of these superheavy (Higgs) bosons were presumably responsible for the development of B asymmetry, when they ran out of equilibrium at T ~ M X. It was at first believed that CP violation had to be hard (i.e. intrinsically present in the basic lagrangian) in order to remain at T~. 1014 GeV so as to lead to nonvanishing asymmetry. It was realized [3], however, that if symmetry does not get restored at high temperature (a consistent possibility in gauge theories), soft CP violation, which results from spontaneous symmetry breaking of the initially CP conserving theory, can do the job as well. That fact will be important in our analysis given 1 Precent address. 0 031-9163/82/0000-0000/$ 02.75 © 1982 North-Holland

below. It turns out [4] also, that baryon asymmetry could be generated not only in grand unified theories, but also in partially unified models with intermediate mass scales. Actually, the question of intermediate mass scales is interesting in itself. It was shown recently [5] that simple grand unified theories do allow low intermediate mass scales, not far f r o m M w. For example, one has a consistent O(10) theory with parity restoration at very low energies, E ~ ( 2 - 3 ) M w. It is then important to know whether such models can account for the explanation of matter-antimatter asymmetry and the prediction n B/n 7 ~ 1 0 - 9 - 1 0 -11 as dictated by observation. The problem has become recently more acute as Kuzmin and Shaposhnikov [6] have stressed the light connection between left-right symmetry and baryon asymmetry generation, concluding that this latter requires right-handed weak gauge bosons extremely massive in comparison with the left-handed ones. In this letter we present a simple and natural, though somewhat unconventional mechanism for baryon generation in theories where the breaking of left-right symmetry appears at such an extremely low energy scale. Before we describe it in detail, we shall first discuss the model and point out the difficulties associated 191

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with the naive picture of high temperature behaviour of gauge theories and baryon number creation. 2. We offer now a telegraphic summary of basic features of the O(10) grand unified model [7] with low energy parity restoration (for details, see ref. [5]). O(10) is the minimal realistic left-right symmetric grand unified theory. It actually contains the SU(2)L ® SU(2)R ® SU(4)c group of Pati and Salam [8]. Since the adjoint representation of O(10) is a 45-dimensional two-index antisymmetric representation, there are forty-five gauge bosons: twenty-one of them associated w.ith SU(2)L ® SU(2) R ® SO(4) c and twenty-four superheavy gauge bosons which are color triplets and come in two SU(2)L doublets (yi), (x i). The fermions are placedin IJg dimensional spinorial representations, in left-right symmetric manner:

(u, di

e

'

dC

e+ L '

(1)

where the superscript c stands for charge conjugate fields. Since we are interested in left-right symmetry at low energies [8,9], we imagine the following chain of symmetry breaking: O(10) M ~ SU(2)L ® SU(2)R ® U(1)B-L ® SU(3)c M~ SU(2)L ® U(1),y ® SU(3)c M ~ U(1)m ® SU(3)c "

(2) A gauge theory is never completely specified without its Higgs sector. One that furnishes the desired symmetry breaking is the following one: (a) 45 dimensional representation dPij = --dPji (i, j = 1..... 10). If we write alP45 =Lijcbii where Lij =-Lji are the generators of the group, it is easy to see that (~) = (B - L)oxbreaks O(10) ~ SU(2)L ® SU(2)R ® U(1)B_ L ® SU(3) c. Therefore,M X ~ g o X. (b) 126 dimensional five-index antisymmetric representation q5126 --= c~i/klm, which contains an SU(5) Singlet and SU(2)R triplet field AR, with (A R) = (~126) = VR . At this stage left-right symmetry is broken and M R ~ g V R. (c) Since 16 X 16 = 10 + 120 + 126, the light Higgs sector consists in principle of al three above representations. We will come to it below. Before we proceed, we would like to mention the 192

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basic predictions of the above scheme. By the virtue of A R being a triplet, the right-handed neutrino gets [10] a large Majorana mas~, of order o f M R. In turn, v L gets a tiny Majorana mass, inversely proportional to M R [10] and whose precise value depends on the choice of the possible light Higgs sector and the value o f M R. Now, the phenomenological analysis [5] of charged and neutral current processes tied up with unification constraints [5] suggest either M R ~ 109 GeV or M R ~ {2-3) M w. The second possibility is rather interesting, since it provides a testable alternative to conventional models with a desert above M w. In what follows we shall stick to it. Its characteristics are [5] : (i) Unification scaleMw ~ 1018 GeV, which makes the proton effectively stable (rp ~ 1042 yr). (ii) Substantial lepton number violation through neutrino-less double decay 07 ~ 10 - 4 - 1 0 -5). (iii) Nonvanishing Majorana masses for left-handed neutrinos. If we choose the light Higgs qbL = ~10, a 10 dimensional representation, we get * 1 m v = m2u/mvR

(3)

which is true generation by generation and where u stands for the associated T3L = 1/2 quark. For mvR ~ M R ~ ( 2 - 3 ) M w, one gets my, > 0.5 MeV mvr > 250 MeV, which are the experimental bounds. We would like to point out that if only ~10 is present, then the quark-lepton mass relations become at low energies [12]

mb.~3mr,

ms~3m~,

m d ~ 3 m 1,

(4)

which are not good for m d and m s. Therefore, a light Higgs is probably a combination of 10, 120 and 126 which we will simply denote by qbL. We will use the notation: = ~, 1018 GeV, (qb126> = (~R > = M R ~ ( 2 - 3 ) M w and (~L) = M W. 3. We offer now a discussion on the possible origin of baryon asymmetry in this model. As is well known, the necessary ingredients to have a calculated baryon number are [ 1] : microscopic baryon number violation; departure from equilibrium; and CP violation at high temperature. Since baryon number violation is intrinsically present in this theory at all temperatures, we have only to discuss CP violation. In what follows, we *I This is as in the original work of Gell-Mann et al. and Yanagida [ 11 ].

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take CP violation to be soft, i.e. due to spontaneous symmetry breaking [13]. Let us be a bit more precise. We said above that qbL contains 126 Higgs, among other fields. It is possible to arrange for the solution which minimizes the potential that (~L(126)) be complex [14], thus leading to CP violation at T = 0. The question becomes whether the theory remains CP noninvariant at high T. Namely, by analogy with ferromagnets and as explicitly confirmed [ 15 ] in the single Higgs model, the symmetry is expected to be restored above the critical temperature T c ~ M w (or so), which would then rule out soft CP violation as a viable mechanism for the baryon production. Fortunately, the above is not true in general. In the models with at least two Higgs multiplets, the symmetry is not necessarily restored [3]. We illustrate it on a simple toy model with two scalar fields q51 and qb2. One can show then, from a high temperature analysis of the potential, that there is a solution which looks like T < Tc ~(~bl):

(qbl) @ 0 :/= (~b2),

T>Tc:

(qbl) = 0,

(5) (~b2)= c T ,

where c is a dimensionless number (usually small ~.g). Symmetry may remain broken at high T (it should be kept in mind that one vacuum expectation value always has to vanish). Therefore, in our case we have an option: at T >>Mw,MR either (alpR) =/=0 or (qb L) 0 *2. In any case, CP invariance is broken since either of the vacuum expectation values can be complex. We now give our scenario for the development of matter-antimatter asymmetry in the early universe. As in the conventional scheme [ 1 ], it wilt be :the decays of Higgs bosons which would produce the baryons. A comment is needed regarding their and the X gauge bosons' masses. Since VX ~ 1018 GeV, we expect M X ~ 1017-1018 GeV. Now, since the masses of exotic Higgs bosons (leptoquarks) are also proportional to VX, they will clearly be superheavy, with the precise values of m H depending of ~ 4 couplings. The constraint on m H is provided by the requirement that their decays be out of equilibrium for T .2 Strictly speaking, it can be shown that one of the vacuum expactation values in the theory has to vanish at high T. In our case, we are going to require at least two ( q~)'s ~ 0 at high T; for example (q~R) and, say, one of the (~L)'S.

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~ m H. For T ~ m H : FH(T ~ m H) ~ h2(mH)m H ,

H(T~mH) ~ m 2 /Mp '

(6)

where PH is heavy Higgs decay rate; H is the Hubble constant: H ~ T2/Mp and h(mH) is the value of the Yukawa couplings at T ~ m H. Therefore, PH(T mH) ~ H ( T ~ mH) implies

m H ~ h2(mH)M P .

(7)

For h(mH) ~ 1 0 - 2 , we get m H ~ 1015 GeV. In the rest of this paper, we will assume m H ~ 1015 GeV. Before tracing the early history ,of the universe to see how the B asymmetry developed, we wish to stress a difference from the other, "conventional", approaches. In our case, as we have seen, (q5R) and some (d#L) *2 are taken to be nonvanishing at all temperatures, their values being ~.gT at high T. Therefore, the right-handed neutrinos vR (and some ordinary fermions) will be massive at high T. It should be clear that our situation differs completely from the recently analyzed role of superheavy fermions (in particular, VR) in baryon production [16]. Indeed, in our scheme the left-right symmetry gets broken only at a very low energy scale, so that, at T ~ 0, rnvR is of the order of about one hundred GeV or so. However, due to the symmetry non-restoration at high T, v R gets a mass proportional to Tin the early stages of the universe we are discussing now. As we shall see, these massive fermions at high T are going to play an important role in our scenario for B production. Let us start examining the gauge boson (X) decays. As PX ~ m2°tx/(m2x + T2) 1/~, with a x ~ 10 - 2 , we get P x < H a t T~-.m X ~-. 1018 GeV. At T ~ r n x , I"X eventually catches up with H and so X bosons start decaying, producing a nonvanishing baryon number. Since in our picture we want to get a calculable nB/n. r from the superheavy Higgs boson (H) decays, it is important to establish if some B violating interaction was in equilibrium at T ~ mH, thus wiping out the B asymmetry due to X decays. Fortunately, this is the case; indeed, the B violating direct and inverse decays of the massive fermions happen at a rate I"F (see fig. 1):

VF ~ (1927r3)-lh4T5/m4,

(8)

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~

j

J

I

I

tH I

Fig. 1. The decay of massive fermions (m F <~ T), denoted F, into lighter ones through the exchange of superheavy Higgs bosons.

so that using h ~ 10-1 (say for VR) for T ~ 1016 GeV, when X decays are over, I"F ~ H , thus assuring that no B asymmetry could develop. Thus for T ~ mH, we can safely consider n B ~ 0. As T drops below mH, the inverse Higgs decays are becoming more and more suppressed by the Boltzmann factor exp(-mH/T), but the decay rate gets faster than the expansion rate of the universe. At say, T = ~ m H 1014 GeV, we get: I'H(1014 GeV)~h2(1014 GeV)m H ~ 1011 GeV,

(9)

H(1014 GeV)-~ 109 GeV.

Let us now come to an estimate of the ~B generated in H decays. We must consider the interference of the diagrams given in fig. 2a and fig. 2b. The complex mass insertions in the Higgs propagator of fig. 2b ensure CP non-conservation and they come about due to the symmetry breaking at high T: ('~i) ~ g ei°'iT. Clearly, in order to get an imaginary part we need i = 1,2 at least, and this explains why we have required at least two VEV's to be non-vanishing at all T (see footnote 2). We stress again that it does not matter whether both these two VEV's belong to the Higgs sector providing mass to the ordinary fermions or if one of them corresponds to (~R). For reasons we are going to explain later, it is safer to take them belonging to different Higgs representations of SO(10), say 10 and 126. This mixing in the propagator induces a suppression factor

a)

b)

Fig. 2. The baryon number violating decays of superheavy Higgs bosons: (a) the tree level amplitude and (b) one-loop amplitude. The nonvanishing ~d~ results from the interference of these two graphs.

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in ZkB of order ~ ( ~ ) 2 / m 2 ~ ~T2/m 2 ~ 10 -3. For the rest, our calculation of zkB does not differ from the standard results [17] s~that: nB/n ~ ~ 10-14--10-10, which is in accord with observation: (nB/n.r)ob s 10-11--10 -9 . Our model shows that grand unified theories with weakly broken left-right symmetry can account for the baryon asymmetry of the universe. Actually, a claim has been made recently [6] that the above cannot be true, as the breaking of parity has to be almost as strong as superstrong symmetry breaking in order to have a consistent baryon production. The crucial point in ref. [6] is that in a left-right symmetric theory the net amount of z2xBis proportional to the amount of left-right asymmetry, i.e. zkB~ IVL - VR I/MH , where

V L ~MWL/g,

VR ~ M W R / g .

Now, in a "conventional" scheme where left-right symmetry is restored at high T, it is then clear that one must require VR to be not too inferior to M H if enough n B has to be produced. However, in a situation of symmetry non-restoration, even though IVL - VR[T~.O ~ M H , this does not prevent I VL - VRI "~ ~ T, at high T. and then IVL - VRI/M H ~ X / a " 10-1, when the relevant H decays take place. Clearly, also in our case, care must be taken not to have VL ~ VR at high T, a situation which could arise if the two tadpoles in fig. 2b corresponded to Higgs bosons belonging to the same representation. The safest solution is thus to choose cI,1 and q~2 in two different representations. In this case,
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(ii) the washing of any initial baryon number is established through (massive) fermionic decays and mverse decays at T ~ 1016 GeV; (iii) nB/n. v '-~ 10-14--10 -10 is produced through the decays of superheavy Higgs bosons (m H ~ 1015 GeV) at I ' ~ 1014-1015 GeV; (iv) finally, the necessary l e f t - r i g h t asymmetry is kept strong at high T, again due to symmetry nonrestoration. Before closing, we would like to comment on the question of the possible domain structure [18] of the universe in our model. Namely, CP violation, as we said, is soft, i.e. due to spontaneous symmetry breaking. If eventually symmetry is restored, then we would expect the existence of domains of matter and antimatter, by analogy with ferromagnets. Now, it is not clear whether symmetry gets gets restored, since near T ~-Mp q u a n t u m gravitational effects are non.negligible and not known. It is important to realize, however, that symmetry may never be restored, in which case there would be no domains. We wish to acknowledge useful discussions with Rabi Mohapatra and Roberto Peccei. One of us (G.S.) thanks R. Peccei and the rest of the Max-Planck theory group for their warm hospitality. The work of G.S. was in part supported by the US Department of Energy.

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[4] A. Masiero and R.N. Mohapatra, Max-Planck preprint MPI-PAE/PTh 46•80. [5] T. Rizzo and G. Senjanovi6, BNL preprints (1981). [6] V.A. Kuzmin and M.E. Shaposnikov, Phys. Lett. 92B (1980) 115. [7] H. Georgi, in: Particles and fields (ALP,New York, 1975); H. Fritzsch and P. Minkowski, Ann. Phys. 93 (1975) 193; for further references, see P. Langacker, SLAC preprint (1981). [8] J.C. Pati and A. Salam, Phys. Rev. D10 (1974) 235. [9] R.N. Mohapatra and J.C. Pati, Phys. Rev. DI1 (1974) 566, 2558; G. Senjanovi6 and R.N. Mohapatra, Phys. Rev. D12 (1975) 1502. [10] R.N. Mohapatra and G. Senjanovi6, Phys. Rev. Lett. 44 (1980) 912; Phys. Rex,. D23 (1981) 165. [11] M. Gell-Mann,P. Ramond and R. Slansky, unpublished, reported by P. Ramond at the First Workshop on Grand unification (1980); T. Yanagida, KEK lectures (1980). [12] A. Buras, J. Ellis, M.K. GaiUardand D.V. Nanopoulos, Nucl. Phys. B135 (1978) 66. [13] T.D. Lee, Phys. Rev. D8 (1973) 1226; Phys. Rep. C9 (1974) 148; for a review, see G. Senjanovi6 invited talk at XX Intern. Conf. on High energy physics (Madison, 1980), to be published in the proceedings. [14] J.A. Harvey, P. Ramond and D.B. Reiss, Phys. Lett. 92B (1980) 909. [15] D.A. Kirzhnitz and A.D. Linde, Phys. Lett. 42B (1972) 471; S. Weinberg, Phys. Rev. D9 (1974) 3537; L. Dolan and R. Jackiw, Phys. Rev. D9 (1974) 2904; C. Bernard, Phys. Rev. D9 (1974) 3312. [16] T. Yanagida and M. Yoshimttra, Phys. Rev. Lett. 45 (1980) 71; J.A. Harvey, E.W. Kolb, D.B. Reiss and S. Wolfram, Caltech preprint (1980); R. Barbieri, D.V. Nanopoulos and A. Masiero, Phys. Lett. 98B (1981) 191; F.R. Klinkhamer, G. Branco, J.P. Derendinger, P. Hut and A. Masiero, Astron. Astrophys., to be published. [17] J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Phys. Lett. 80B (1979) 360; D.V. Nanopoulos and S. Weinberg, Phys. Rev. D20 (1980 2480; S. Barr, G. Segr6 and A. Weldon, Phys. Rev. D20 (1979) 2494; A. Yildiz and P. Cox, Phys. Rev. D2i (1980) 906; for further references see P. Langacker, SLAC preprint (1981). [18] Ya. B. Zeldovich, I. Ya. Kobzarev and L.B. Okun, Soy. Phys. JETP 40 (1975) 1; R.W. Brown and F. Stecker, Phys. Rev. Lett. 43 (1979) 315; G. Senjanovi6 and F. Stecker, Phys. Lett. 96B (1980) 285 ; K. Sato, Nordita preprint (1980). 195