научная статья по теме VIOLATION AND PARTICLE–ANTIPARTICLE ASYMMETRY IN COSMOLOGY Физика

Текст научной статьи на тему «VIOLATION AND PARTICLE–ANTIPARTICLE ASYMMETRY IN COSMOLOGY»

ЯДЕРНАЯ ФИЗИКА, 2010, том 73, № 4, с. 614-618

= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

CPT VIOLATION AND PARTICLE-ANTIPARTICLE ASYMMETRY

IN COSMOLOGY

©2010 A. D. Dolgov*

Dipartimento di Fisica, University degli Studi di Ferrara, Italy; Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Italy; Institute of Theoretical and Experimental Physics, Moscow, Russia

Received July 3, 2009

General features of generation of the cosmological charge asymmetry in CPT noninvariant world are discussed. If the effects of CPT violation manifest themselves only in mass differences of particles and antiparticles, the baryon asymmetry of the Universe hardly can be explained solely by breaking of CPT invariance. However, CPT noninvariant theories may lead to a new effect of distorting the usual equilibrium distributions. If this takes place, CPT violation may explain the baryon asymmetry of the Universe.

The generally accepted mechanism of generation of cosmological charge asymmetry is based on three general principles put forward by Sakharov in 1967 [1]:

1. Nonconservation of baryon number.

2. Breaking of C and CP.

3. Deviation from thermal equilibrium.

It is established long ago by experiment that P, C, and CP are broken. Big bang cosmology unambiguously states that massive particles should be out of thermal equilibrium in the cosmological plasma. Nonconservation of baryons is predicted by electroweak and grand unified theories and "experimentally" proven by existence of our universe. So the Sakharov baryogenesis seems to be in a pretty good shape, though some efforts are needed to obtain sufficiently large cosmological baryon asymmetry. In this connection it may be interesting to explore other possibilities. For more details about the standard baryogenesis and the list of references see, e.g., reviews [2].

Since all three symmetries C, P, and T are known to be broken, it is tempting to explore consequences of breaking of the combined CPT symmetry. There is of course a drastic difference between anyone of the single symmetry transformations or any pair of them and the combined action of the three. According to the celebrated CPT theorem [3], any local Lorenz invariant theory with hermitian Hamiltonian, with positive definite energy or, better to say, with the canonical relation between spin and statistics is invariant with respect to CPT transformation. On the other hand, there are absolutely no theoretical

*E-mail: dolgov@fe.infn.it

arguments in favor of invariance with respect to separate P, C, and T transformations and they are, indeed, only approximate. If we trust CPT theorem, then we should conclude that any pair CP, PT, and TC are also broken. In fact, historically first was discovered that CP is broken and hence T should be broken as well.

The study of phenomenological manifestations of CPT violation has a long history. I will mention here only some selected contributions by L.B. Okun [4—9]. For recent works and review of the literature on CPT violation see [10—12].

In what follows we consider baryogenesis, or more generally generation of any cosmological charge asymmetry relaxing the assumption of CPT invariance. For discussion of earlier works one may address review [13]. In what follows we reconsider and clarify the old results related to the generation of charge asymmetry in thermal equilibrium due to mass difference between particles and antiparticles and discuss previously not considered case of asymmetry when sacred principles of hermicity of the Hamiltonian and thus unitarity of S matrix or spin—statistics relation are broken.

According to CPT invariance the masses of particles, m, and the corresponding antiparticles, m, must be equal. If CPT is broken, it is natural, though not necessary, that this equality would be violated too and m = m .It is practically evident that in this case the number density of particles and antiparticles may be unequal even in thermal equilibrium. Of course if baryonic charge or some other quantum number, Y, prescribed to particles is conserved, then the state with initially zero value of Y would remain such in any evolution. We assume first for illustration

that the standard form of the equilibrium distribution functions is not destroyed by CPT violation. This is not necessarily true and the validity of this assumption is discussed below. In equilibrium the particle distribution is described by the function:

m^ = eM(E-\)/T]± V (1)

where the signs "±" correspond to fermions and bosons, respectively, and i is the chemical potential corresponding to quantum number Y. For antiparti-cles in equilibrium with respect to annihilation i = = -¡. If the density of Y is zero, then in CPT -invariant theory i = 0. However, if m = m, chemical potential must be nonvanishing to ensure equality of particle densities, n = n:

5^ = n — n =

(2)

gdf

d3p

f (E,p) - f (E,fi)] ,

where E = \J-p2 + m2, E = \J-p2 + m2, and g^ = = gsgcgg is the number of "degrees of freedom" of the particle under scrutiny with gs, gc, and gg being the numbers of the spins states, the number of colors, and the number of generations (families), respectively. For example, for three generations of quarks gdf = = 18, corresponding to 2 spin and 3 color states, for charged leptons gdf = 6, and for neutrinos gdf = 3, if the particle masses are smaller than temperature.

Evidently if 5n = 0 but m = m, chemical potential should be nonzero. For sufficiently small mass difference, 5m = m — m, such that m5m/ET < 1 we find:

ц = (Ji/2Jo) môm,

(3)

where

Jo = J d3pf 2(E, 0)eE/T, (4)

Ji = J(d3p/E)f 2(E, 0)eE/T.

If, say, baryonic charge is not conserved and the processes of the type n + n ^ mesons or (n — n) oscillations are in equilibrium, then the baryonic chemical potential is forced to zero and there should be an excess of baryons over antibaryons or vice versa in thermal equilibrium,

5n = gdfJ1(m5m/T). (5)

An interesting situation might be realized in the early universe at the temperatures above the elec-troweak phase transition. As is generally accepted, at such temperatures baryonic, B, and leptonic, L, numbers are not conserved, while the difference (B — L) is conserved, see, e.g., reviews [14]. The processes with baryonic number violation include

colorless combination of all quarks and leptons from all three generations and lead in equilibrium to the following relation between chemical potentials:

3(u + d) + (l + v) = 0, (6)

where the particle symbol denotes the corresponding chemical potential, u and d are, respectively, up and down quarks, l is charged lepton, v is neutrino, and we assumed that the chemical potentials do not depend upon the generation.

Equilibrium with respect to the charged currents implies:

W + = u — d = v — l. (7)

We do not distinguish here between chemical potentials of left- and right-handed fermions. Though it is a good approximation for quarks, due to their thermal masses, it may be poorly valid for charged leptons, especially for electrons, but nevertheless we neglect that for simplicity.

One more equation for determination of chemical potentials follows from the condition of electro-neutrality:

2 1

:jSnu - -ônd - 6ni = 0,

(8)

where 5n is the difference of number densities of particles and antiparticles.

The last necessary equations follows from the fixation of the value of the conserved density of (B — — L):

i (5nu + 5nd) - 5ni - 5nv = nB-L■ (9)

Equations (6) and (7) give:

l = — (2u + d), v = —(u + 2d), (10) and hence from (8) and (9) follows:

(Jou + Joi) — 2^d (Jod — Joi) = (11) = 2mu5mu Jiu — md 5md J id — mi5miJii,

l^u (2Jou + 4Joi + Jov) + 2Hd (Jod + Joi + Jov) =

= mu5mu Jiu + md5md J id — mi5mi Ju —

1 3 - -mu5muJ\v + (27r) TuB-l,

where Jia are given by Eq. (4) with E = sjp2 + m2 and we have returned to the usual notation u ^ iu, etc.

Solution of these equations is straightforward but tedious. We will present them only in the limit of high temperatures when

■k3t 3

J0 = ——, J i = T2 In 2 (12)

6

616

DOLGOV

independently on the particle type. Assuming equal masses and mass differences for all quark generations we find after simple calculations for the baryon number density:

Ji

(2n)3 T

(^mu5mu + Щтабт^ (13)

and correspondingly the baryon asymmetry:

he

ßT =

UY

(14)

= -8.4 x 10"3 (18m„5mu + 15md5md) /T2,

where hy = 0.24T3 is the equilibrium number density of photons. To take into account different masses of quarks from different families m5m should be changed into a 5mama/6, where summation is done over all quark families. We assumed above that there was no preexisting (B — L) asymmetry and neglected lepton contributions. Below the electroweak temperature TEW — 100 GeV baryonic number is practically conserved and the asymmetry stays constant in the comoving volume up to the entropy factor which diminishes 3 by about an order of magnitude. So to agree with the observed today value (3t should be about 10"8.

If we substitute the zero temperature values of the quark masses into Eq. (14) and take for an estimate an upper bound on 5m equal to the experimental limit on proton—antiproton mass difference, 5mp < < 2 x 10"9 GeV [15], we see that the effect is by far too small to explain the observed baryon asymmetry. Above the electroweak phase transition Higgs condensate is absent and the vacuum masses of all fermions are zero. However, there are significant temperature corrections to the masses, m(T) w gT, where g is the gauge (QCD) coupling constant. So the high-temperature quark masses are much larger than the lepton masses. That's why we neglected above the lepton contributions into the baryon asymmetry.

To create the observed cosmological baryon asymmetry, /3o = 6 x 10"10, we need 5mq — (10"7—10"8)T at T —100 GeV. It means that the quark mass difference should be about 10"5—10"6 GeV, much larger than the upper bound on the proton—antiproton mass difference. One would expect (mp — mp) to be of the same order of magnitude as 5mq. An accidental c

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