>K9m 2015, TOM 147, bmii. 3, cTp. 433 437
© 2015
THE ELECTROWEAK AXION, DARK ENERGY, INFLATION
AND BARYONIC MATTER
L. McLerran*
Physics Department. Brookhaven National Laboratory Upton, NY 11973, USA
RIIiEN BNL Research Center, Brookhaven National Laboratory Upton, NY 11973, USA
Physics Dept. Central China Normal University Wuhan, China
Received October 29, 2014
In a previous paper [1], the standard model was generalized to include an electroweak axion which carries baryon plus lepton number, B + L. It was shown that such a model naturally gives the observed value of the dark energy, if the scale of explicit baryon number violation A was chosen to be of the order of the Planck mass. In this paper, we consider the effect of the modulus of the axion field. Such a field must condense in order to generate the standard Goldstone boson associated with the phase of the axion field. This condensation breaks baryon number. We argue that this modulus might be associated with inflation. If an additional B — L violating scalar is introduced with a mass similar to that of the modulus of the axion field, we argue that decays of particles associated with this field might generate an acceptable baryon asymmetry.
Contribution for the JETP special issue in honor of V. A. Rubakov's 60th birthday
DOI: 10.7868/S0044451015030052
1. INTRODUCTION
In a previous paper fl], the standard model was modified by assuming that the baryon plus lepton number B+L was not conserved at a mass scale of the order of the Planck scale, A ~ Mpi fl], and instantons were used to compute a plicnomcnologically acceptable value of the dark energy [2 6]. This might be done by an electroweak axion coupling to the topological charge of the electroweak gauge theory [7 11]. Following Ansclni and Johanscn fC], an explicit B + L violating interaction of the form
Sb+l = J <li-': + c-c-} (!)
was considered. Here, / is a left-handed lepton field and q is a left-handed quark field. The scale A is the energy scale at which lepton and baryon number changing interactions are important, and is presumably a GUT
E-mail: mclerran'fflmac.com
scale or higher. The matrix A is of the order of unity, and the interaction Iqqq contracts various spinor, color, and flavor indices to make singlets. This interaction violates both B + L and chirality.
In a gauge theory, the angle 8 appears when one considers adding a term
U}0^-jd\rFF (2)
to the action of the theory. The number of families of quarks and leptons is nj. Here, FtlL, = (1/2)6^,.,;^ The quantity
/(Ii.vFF = N. (3)
o TT J
where AT is the winding number of a Euclidean field configuration [2 4]. Finite-action solutions, instantons, exist with AT equal to the number of instant on minus anti-instant on configurations. The electroweak axion is generated by promoting the angle 8 to an axion field.
In a theory that explicitly conserves the baryon number, physics is independent of 8. In such a tlie-
4 >K9T<E>, iibiii.3
433
ory, the torni above generates no dependence on 9 because the only place where instantons contribute is in amplitudes connecting states with differing numbers of baryons [2, 5], and there 9 appears as ox\>(iii j X9) for a process that changes B + L by the amount A(B + L) = 2'tifN. The factor of nj appears because each generation of quarks and leptons is produced. The basic instant on process therefore involves 9 colored quarks and three leptons. In amplitudes squared, the phase disappears and there is no consequence of this angle.
It was shown in fl], that if there was explicit baryon number violation at the Planck scale, then electroweak instant on processes naturally led to a vacuum energy of the magnitude
S, = *(«*(«,*)-!)(—J (—J x
x oxp i^^W^))A4- (4)
If we take the energy scale A to be the Planck
mass
and 1 /' i a « 1/30, we find that
5/ « 10"122A/^. (5)
This is remarkably close to the value of dark energy (DE) in cosmology, €oe ~ 10"122M^.
There are of course uncertainties in this estimation of the scale A. The details of B + L violation may be different in different theories, and there may be some changes to the coupling constant evolution at energies near the Planck scale due to new particle degrees of freedom. The formula for the dark energy is roughly linear in A, and hence an uncertainty in this relationship translates to a roughly linear uncertainty in the scale A. This linear dependence arises from the explicit factors of A4 and the implicit factors in the running of the coupling constant. It is not unreasonable to assume that the uncertainty in the scale A may be several orders of magnitude, since the running of the coupling constant is not known near the Planck scale.
2. GENERALIZATION TO THE NON-GOLDSTONE MODE
The axion field above is the Goldstone mode composed from fields <f> = p{x) exp(i8(x)). We assume that the field <f> carries one unit of B + L. (One unit of B + L corresponds to B+L = 2.) The anomaly, however, generates nj units of the B + L change, corresponding to A(B + L) = 2nj. The Lagrangian with only axion
degrees of freedom arises when we assume that a symmetry associated with the scalar field is spontaneously broken and the field p acquires an expectation value. We should think of the term cos(nj8) in the induced instant on interaction in Eq. (4) as
1 Ro(03)
cos 8 = -(exp {in ¡9) + exp {—in ¡9)) = ———, (C)
where
v={p). (7)
When the field is replaced by its expectation value, then we achieve our old result.
The contribution of the axion to the action is very small. However, multiple weak boson attachments to the basic vertex can enhance the magnitude of B + L violation, and at high temperatures 1 /<i \\ . such enhancements make the effect of a magnitude sufficient for the processes to be realizable [12 14]. One might ask if the contribution associated with the dynamical non-Goldstone part of the axion field might be similarly enhanced, for example, in the decay of a heavy axion. We think not, since the axion brings an energy-scale into the problem much larger than the electroweak scale, and the amplitude for such a decay enhanced by thermal W and Z bosons should maintain its exponential suppresion ~ cxp(—2n/a\y).
In Ref. fl], it was assumed that the symmetry was broken, that v had an expectation value of the order of Mpi, and that 9{x) was frozen into a constant value by inflation. In fact, the modulus of the axion field p provides a candidate for the inflaton field [15]. It has an expectation value of the order of the Planck mass, as is required of the inflaton in some inflationary scenarios [16 22]. In order to obtain the right order of magnitude for density fluctuations [16 25], we need to require a very flat potential for the modulus of the axion field. This would require a mass mp -C mpi for the scalar particle associated with the modulus of the axion mass. A typical value for the inflaton mass in chaotic inflation scenarios is 1012 GeV.
As the symmetry breaking occurs, the B + L symmetry is spontaneously broken, and it is plausible that some excess of the heavy scalar particles associated with the non-Goldstone part of the scalar field are produced. These scalar particles carry a non-zero baryon number. They have B + L violating interactions among themselves but when the density of such particles becomes sufficiently low, we expect these interactions to freeze out. However, these particles decay rapidly because the axion action after symmetry breaking contains the term
SS = j dixv6pd'10dll0 (8)
that allows the modulus of the scalar field to decay into two axion fields.
If we further assume that there is a B — L violating scalar with the scale of variation at the Planck IIlclSS and a mass similar to that of the modulus of the B + L violating modulus of the scalar field, then there maybe interesting effects. We assume that there is a B — L symmetry of the scalar field action, but there are D — L violating interactions with quarks and leptons. We can then see that these interactions with quarks and leptons are quite weak at low energy scales. Then it is plausible that at a high energy scale associated with the end of inflation, one might generate some excess of B — L, which is stored in the low-mass tiib-l -C Mpi scalar field. If this is the case, then such matter plays an increasingly important role as time evolves, since massive matter energy density dilutes more slowly than does radiation. At some late time, it is not implausible to assume that such matter dominates the energy-density of the universe.
There may be baryon-number-violating processes where the massive B — L scalar field would decay into light -mass quarks and leptons. On dimensional grounds, the effective interaction for such a term is
1 v^
Lcff = °H !-,'<l'<l'<l'- CJ)
i
where the sum is over quark q and lepton / flavors. The parameter A is of the order of the Planck mass. There is still a considerable uncertainty in the value of A. In the derivation of the vacuum energy, the running of the electroweak coupling and its dependence upon the Planck mass scale combine with the explicit factors of to make for an almost linear sensitivity of the dependence of the instanton-induced vacuum energy on the Planck mass. This, combined with the intrinsic uncertainty of how the electroweak coupling runs at energy scales near the Planck mass, allows for the uncertainty of a few orders of magnitude upon the energy scale at which baryon number violation is of the order of unity.
The rate for decay is
R~m7B_L/A\ (10)
which is anomalously small because A/tiib-l ~ 10'. For example, if expansion was radiation dominated, which is what we want to match to as the matter reheats, then this rate would become equal to the expansion rate when
T~AIpl(mB-L/A)7/'2. (11)
Now in order for the decay of the scalar field not to produce too many baryons, it is necessary that the decaying baryon not produce too little entropy. If the reheating temperature is T, there are of the order of in i< i
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