Baryogenesis in a model where CP is broken spontaneously

Volume 145B, number 3,4

PHYSICS LETTERS

20 September 1984

BARYOGENESIS IN A MODEL WHERE CP IS BROKEN SPONTANEOUSLY

Gino SEGRI~ a Department of Physics, Universityof Pennsylvania, Philadelphia,PA 19104, USA Received 17 May 1984

We examine a model recently proposed by A. Nelson in which CP is broken spontaneously at the GUT scale. We show that in this model a baryon asymmetry of order 10-2 0 is generated, where 0 is the argument of the determinant of the quark mass matrix.

In the standard SU(3)c × SU(2)L × U(1) model, the strong interactions contain a T, or equivalently CP, violating parameter

£o = (Og2/327r2) F ' ~

(1)

By a chiral U(1) rotation, the complex phase of the quark mass matrix,MQ may be rotated into 0 or, equivalently, we say there is an overall strong CP violating parameter = 0 + arg d e t M Q .

(2)

Experimental limits on the electric dipole moment of the neutron [1] *x say 0-~ 10 -9 [2]. The strong CP problem is to explain why this parameter is so small. We believe generally that the smallness of a dimensionless parameter e is an indication that the symmetry of the theory increases as we let e ~ 0 [3]. This does not happen in conventional models with CP violation so it is not natural to have 0 be small. Several solutions have been proposed for this strong CP problem, none of them entirely satisfactory. Setting the up quark mass, m u ~ 0 is ruled out by experiment [4]. The interesting approach, first proposed by Peccei and Quinn [5] has a variety of problems of its own. The latest version, the so-called invisible axion models [6,7], are not ruled out, but they can hardly 1 Supported in part by the US Department of Energy under Contract No. EY-76-C-02-3071. *1 Ref. [1] contains a review of experiments and a list of references. 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

be called economical as they require new symmetries, new particles, new scales [8] and fine tunings. Another line of enquiry has been to impose CP as a discrete symmetry of the lagrangian, broken softly (or spontaneously) by the vacuum expectation values (VEVs) of scalar fields. At tree level we then have = 0; since CP is broken spontaneously we do not expect the 0 obtained by going beyond tree approximation to be infinite. This class of models *2 however have a hard time getting 0 to be less than 10 -9. Basically this can only be accomplished by having additional symmetries and several Higgs multiplets. This kind of construction has problems of its own, e.g. neutral Higgs boson mediated flavor changing neutral currents [10]. Recently Ann Nelson [11 ] has proposed an interest. ing variation of models where CP is a discrete symmetry. In her model the low energy world is exactly the same as the minimal SU(5) model of Georgi and Glashow [12]. In addition there is an SO(3) flavor symmetry and some new superheavy flavor singlet fermions. There are Higgs bosons with non-zero VEVs which connect the heavy fermions to the flavor triplet light fermions. An additional global U(1) is required in her original model. Labelling states by their SU(5)gaug e X SO(3)family X U(1)globa I quantum numbers, Nelson's model [11] has the following Yukawa couplings

.2 Representative samples are given in ref. [9]. 231

Volume 145B, number 3,4

PHYSICS LETTERS

.Cyu k = ~ku [10,3,01 [10,3,01(5,1,0) + Xd [10,3,01 [5,3,01(5,1,0) + hi[10,3,0] [lO,1,-1](ri,3,1) +&tg,3,0l [5,1,-1](ri,3,1) + m 1 [10,1,1] [10,1,-1] + m 215,1,1] [5,1,-1]

+ h.c.

(3)

The fields represented by square bracket quantum numbers are fermions and those by round brackets are scalar bosons (r/denotes representations such as 1 or 24 which break SU(5) at the GUT scale). All coupling constants Xu d, hi,fi, rnl 2 are real by CP invariance. The (ri,3,1~ break SU(5), SO(3) and the global U(1) all at the same time, and further, by having complex VEVs, break CP spontaneously as well. They cause the ordinary light family triplet fermions to mix with the superheavy fermions so CP is introduced into the low energy sector. This now still contains only three families of fermions, albeit a mixture of the SO(3) triplet and singlet fermions. Similarly, there is only one light Higgs doublet which couples to fermions. Moreover, although the model clearly contains CP violating phases in the mass matrix, arg detMQ is in fact zero at tree level. Since 0 vanishes because CP is broken spontaneously, we have that is zero in this model at tree level. At the one-loop level there are contributions to g, but they are suppressed by factors of heavy fermion mass (Mi) divided by the scale of GUT breaking ~ )

~ (gigi/32rr2) (MtM]/# 2) ln(MtM]/i.t2),

(4)

where gi is an appropriate fermion coupling constant. Having ~ 10 -9 [2] requires that the heavy fermions acquire masses which are of order 10 -3 or smaller than the GUT breaking scale/.t. This can be achieved if the ml,2,in (3) are such that ml.2//~ <~ 10 -3 and fi, hi "~g, the gauge coupling constant. These tunings are natural in the sense that setting one of these parameters to zero increases the symmetry of the theory. At the two-loop level in this model, there are contributions to 0 not suppressed by fermion masses. They give 0 ~ 10 -11 At this point, one may well ask if this is really progress. We have after all introduced new particles, new symmetries and new scales, all for the purpose of having g be small. The answer is that this is a some232

20 September 1984

what new approach and it may lead therefore to a different understanding of phenomena. For instance, Nelson [ 11 ] has shown that Cabibbo-like suppression of proton decay can occur in this model. More to the point, Barr [13] has displayed, in an interesting recent work, the essential reason why arg det MQ vanishes at tree level in Nelson's model. This abstraction allows him to show how the model may be generalized. Basically, fermions are classified in two sets: F, with the same quantum numbers under SU(3) × SU(2) × U(1) as the ordinary light fermions and R, a real set of representations under SU(3) × SU(2 × U(1). The two conditions for 0 being zero at tree level are (i) CP is broken spontaneously and only by Higgs bosons which connect F to R; (ii) SU(2) × U(1) breaking VEVs occur in F - F terms, not in F - R or R - R . This new understanding holds out hope for the construction of interesting new models. We would like to turn now to yet another arena *a where CP plays a crucial role, the baryon asymmetry of the universe (BAU or simply AB). Theories with spontaneous CP breaking are generally disfavored in this context since at temperatures above the scale of the symmetry breaking, CP is restored so AB cannot be generated *4. The Nelson type models evade this difficulty by having CP broken at the GUT scale, which is where we expect AB to be created. A second difficulty is more technical; in the standard scenario z~B is generated by the decay into fermions of the superheavy color triplet Higgs bosons in the I-Iiggs five representation, whose couplings violate baryon number conservation and CP. The relevant terms are generated by the interference of tree and loop diagrams above production threshold (ZkB vanishes if only tree level diagrams are used, as one can show by using unitarity and TCP). In minimal SU(5) with three fermion families and one Higgs doublet (or two Higgs doublets with a Peccei-Quinnlike symmetry) one must go to three-loop diagrams to generate an AB, even if CP is not a symmetry of the lagrangian [16]. To see this call ~ and × the 10 and 5 representatives of SU(5) and q~the 5 of Higgs. The Yukawa couplings are

~Vuk = (Xd)ij ~iXj~ + (X~);j ~,.C ~1~, ,3 For a recent review see ref. [14]. .4 For a case where this does not happen, see ref. [15].

(5)

Volume 145B, number 3,4

PHYSICS LETTERS

Fig. 1. Tree and one-loop diagram whose interferences give a non-vanishing zXB. where we include the family indices i, j = 1,2, 3. We see then the interference term between a tree diagram and a one loop diagram is proportional to (see fig. 1) Im Tr[XtdXdX~Xu] = 0 .

(6)

It is not until one goes to three-loop diagrams that one obtains a non-vanishing AB [ 16] Im

AB ~

(7) 161r (87r2) 2 [Tr(XtuXu) + Tr(XtdXd)]

We will show that in Nelson's model [11] and Barr's generalizations [13] of it, 2tB is non-zero at the one-loop level even though the coupling constants Xu,d are real. This will be a serious improvement over minimal SU(5), where the smallness of the coupling constants, Xi ~ X/~F mqi, implies [161 zkB < 10 -18, i.e. much too small a number. The key ingredient is that the fermion masses are superheavy so that one cannot simply take the trace over family indices as was done in (6) and (7), i.e. fermion masses are no longer negligible compared to/a, the GUT scale. Another way of saying this is that the one4oop graphs which contribute to ~ no longer vanish when fermion mass insertions are included. These terms are negligible in the ordinary limit where M i ~ 1-10 GeV, but (Mi/~) 2 is not that small whenM i ~ 101 1 GeV. Amusingly enough, the vertices that give a nonzero at one-loop level are essentially the same as those that interfere with the bare vertex to give a nonzero AB. This is illustrated in fig. 2. In calculating O, we let 4 in the graph be CO the neutral component of the Higgs SU(2) doublet (C5 in the SU(5) 5 representation.) These couplings give the quark mass matrix. We are now calculating Co interactions with the three light quark families, which in fact are complex linear combinations of the original four-fermion families. It is the complex mass insertions, denoted by + in fig. 2, which lead to a non-vanishing 0 at the one-

20 September 1984

-;-< ---;-<

z/

Fig. 2. Wavylines: SU(5) heavy gauge boson, dotted lines: SO(3) triplet Higgsboson. Solid lines: fermion, dashed lines: SU(5) color triplet Higgsboson. loop level, typically given by (4) as order of magnitude. For ZkB due to the decay of the color triplets, we are once again looking at figs. 1 and 2 with C now Ca and a = 1,2, 3, the color indices. The difference between Ca and ~a partial decay rates into B violating channels gives AB 4= 0. This is proportional to the imaginary part of interference terms in Ca decay. These are complex because of the mass mixing. To see this, let us rewrite the fields ~ and X belonging to the [10,3,0] and [5,3,0]

~i=Uia~o~,

i,/= 1,2,3,

XI = Vjor7# ,

t~,13= 1, 2, 3, 4 ,

(8)

as linear combinations of the four mass eigenstate fields (remember i,j are family indices so ~ includes the original three families and the sing,let fields with which they mix). £Yuk in (1.5) can be rewritten as £Vuk = (Xd)ij(~:Ut vn)i/~ + (MJ~j(~UtUC~)~j C. (9) The analogue of (6), with Xu,d real is again zero if we neglect fermion masses. We would be calculating h~X2 Im Tr [ V t U U t V U t ( u t ) t u t u ] ,

(10)

which is zero since U and V are unitary. Taking into account however that one of the four multiplets is superheavy, we see that (10) is a correct procedure only in the limit that the superheavy fermion mass MSH is much smaller than the GUT scale/2. Figure two shows vertices with Higgs boson exchange; diagrams with gauge boson exchanges may also contribute. Using an extended version of (7) and (4) we see that 233

Volume 145B, number 3,4 AB ~ 0 .

PHYSICS LETTERS (11)

The ratio o f baryon asymmetry to entropy in the universe nB/s is smaller than AB by at least a factor o f one hundred [14] since (i) not all particle species present at baryogenesis lead to AB 4= 0, but all lead to entropy, (ii) entropy s is approximately seven times as great as the number of photons in the universe. Putting this together we have in this model

nB/s <

10 - 2 0 .

(12)

The baryon asymmetry o f the universe is observed to be [141

nB/s ~_(3--10)

X 10 - 1 1 ,

(13)

so we see an amusing correlation between the value ofnB/s and the limits on 0 in this model. Such a correlation has already been discussed by Ellis et al. [ 17], but in a much more qualitative way; in particular they studied models where CP breaking was hard. Clearly (12) and (13) are barely compatible, as is known to be ~ 10 - 9 . In addition nB/s is proportional to the discontinuity o f the vertex loop, not the loop itself, so we do not expect it to have the loga. rithmic factors of (4). Finally we must take into account the running o f the parameter 0 as a function o f scale. This is known [18] to be very slow in the conventional K - M model, but in the Nelson type models, the loop contributions to O's beta function are presumably larger so it runs faster. The limits on 0 are placed at low energies so the quark coupling constants are limited at this scale. They then decrease by approximately a factor o f three [19] in going to the GUT scale where we calculate nB/s. This suggests it might be more reasonable to change (12) to

nB[s <~(10 - 3

-- 1 0 - 4 ) 0 ,

(14)

Since 0 ~ 10 - 9 , we would seem to have ruled out the model being able to explain baryogenesis. Before invoking such drastic measures, let us remember the uncertainties in the calculations and the fact that there are other possible sources o f baryon asymmetry in the model, e.g. decay o f heavy fermions and o f the flavor group Higgs bosons. To summarize, Nelson [11] and Barr [13] have proposed an interesting class o f models in which CP is violated spontaneously at the GUT scale. O is zero at tree level and, plausibly, there is a correlation between the magnitudes ofnB/s and O. 234

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