The baryon asymmetry and CPT invariance in the early universe

The baryon asymmetry and CPT invariance in the early universe


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Volume 101B, number 3


7 May 1981

THE BARYON ASYMMETRY AND CPT 1NVARIANCE IN THE EARLY UNIVERSE ~" Saul BARSHAY 1 Facult( des Sciences, Universitd Libre de Bruxelles, Brussels, Belgium Received 13 June 1980

We discuss, and give a definite, simple phenomenological example, of the possibility that the baryon asymmetry is related to a failure of CPT invariance for a brief time interval at the origin of the universe.

Recent possible explanations [ 1 - 4 ] of the origin of the presently observed difference between the number of baryons and antibaryons, in ratio to the number of photons, in the universe, 2ug/N,r "-~ 10 - 8 - I 0 - 9 , rely upon the existence in unified models of strong and electroweak interactions [5,6] o f heavy bosons (mass m > 1015 GeV) which decay with nonconservation of separate baryon and lepton number. A further necessary element is the assumed continued existence o f CP and time-reversal violating interactions at super-high energies approaching the Planck mass of 1019 GeV. If the only presently observed CP and T violation, that in the (K 0 - ~ 0 ) mass matrix, has its origin in a spontaneous symmetry breakdown [7,8] then the phase which parametrizes [9] this violation may vanish at the corresponding super-high temperatures [10]. The third necessary element [2,4] is that the decay takes place under conditions in which thermal equilibrium does not hold due to the (initially) rapid expansion of the universe, so that inverse reactions do not eliminate the asymmetry formed in the decay. A simple estimate [2] indicates that the mass o f a primary gauge vector boson must be nearly at the Planck mass i.e. slightly below m p = 1019 GeV, Thus the characteristic time interval from the origin, the decay time, is extraordinarily brief. If the gauge coupling at this point is g2/ 4n ,-~ a, the time is of order ( a m p N ) -1 = 1 0 - 4 1 / N s,

where N is essentially the number of (comparatively) light particle species (flavors) into which the gauge boson decays. In this time interval, and at this mass scale *1, it is possible that CPT invariance may not hold. In particular, there may be a brief time interval in which initially unequal particle and antiparticle masses are brought to equality which is required by CPT invariance. The CP noninvariance necessary for generating a baryon asymmetry would then follow, without the need for time-reversal-violating phases. In this note, we give an explicit, simple phenomenological model which illustrates this possibility. Our purpose is simply to raise this possibility, which to our knowledge has not been considered explicitly ,2, and through the model, to obtain an idea of the CPT-violating mass difference between primary particles and antiparticles that could be related to the baryon asymmetry. For the purpose of illustration we consider a model with, in addition to the gauge boson X of mass *3 m rap, a super-heavy quark Q (carrying baryon number, say B = 1/3) with initial mass/l, comparable to but slightly below m. Assume that interactions which turn on relatively slowly from the origin o f time "dress" the mass/l, giving it a time-dependence which we parametrize a s / l ( t ) = lie at for a brief interval t (Xt ~ 1). Consider the hypothetical decay channel X ~ Q + 2, where by ~ we denote an essentially massless lepton, say a neu-

¢~Supported in part by the Belgian State under the contract A.R.C. 79/83-12. 1 Visiting Professor A.R.C. Present address: III Physikalisches Institut, RWTH Physikzentrum, D-51 Aachen, Germany.

,1 We still do not discuss gravity expficitly. See however ref. [111. 4:2 CPT invariance is explicitly assumed in refs. [1-4]. +3 For a consideration of the possible gravitational origin of primary particles at masses near mp see ref. [ 11 ].

0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company


Volume 101B, number 3


trino. We denote by F the total decay width of X, and assume ta(0) < / ~ ( F -1) =/ae xr-1 < m (thus XF -1 < 1). With neglect of kinetic energy, we use an adiabaticapproximation wave function for the particle Q. t

,a> 0

1 =BQ + [(n +n')R2/167r~l ~ - 1 .

at (/aeM)l/2

X exp (i[kt + (ta/X)(e xt - 1) - mr] - ½F t ) .


For simplicity ,4, we have taken scalar X and Q. We evaluate eq. (2) by the stationary-phase approximation about the saddle point ts, with the result,

)MI = (g/@-m-#)[27r/X(m - k)] 1/2 e-pts/2/[t~(ts) ] 1/2

(3) where e xts = (m - k)/~, with 0 ~< t s ~ T and p = F/X. The branching ratio or relative probability for this decay is BQ

= (g2p-1/8~m)[(m/~)(1 - (1 -

_ ~e ~-T)

d3k __g2/AP J~m (27r)3 8~'m X (m - u )

+ X/F) -1(1

- e - ( v + ~') v )


For large times, i.e. e - r r < 1, we have

- 1],

P ~ P0{1 -X(rnlu) F ~ l / [ l ( n +n') +re~U- 11},

Po = ( g 2 / 8 ~ m ) [ ½ ( n + n ' ) + m / ~ - 11


F0 = (g2/87rrn)[½( n + n') + m/u -



,4 The basic effect is the same for decay of a vector boson into two spin -1 [2 fermions, but the formulae are slightly less simple and transparent.

11 .

The baryon asymmetry from an initially equal number of X and X is then given by __~=1{1 _ ~ # ) ) + l [ n g 2 i,-1 N 7 5 5 (BQ 3 \16rrrn

_2[n'g2FIn'g_ 3 \167rm =

ng 2 ~ - 1 ) 161rN

2 ,-1)} 167rr~

Q - B Q ) + \167rrn

- 1--6-~-

j j , (8)

where we have simplified by using the constraints in eqs. (6a,b). Using eqs. (Sa,b) and (7) the asymmetry to first order in X and X is very simple. zXB _ 1






which, as X -+ 0, becomes the usual "golden-rule" [(m/IJ - 1) is the phase-space factor with neglect of the kinetic energy of Q]. For decay of the antiparticle into this channel, X -~ Q + R, with all parameters possibly initially different in the absence of CPT invariance, we have the branching ratio BQ = (g2V-1/87rm~)[(r~/~t)(1 + ~,V-1) -1 -

With eqs. (5a,b), we can solve eqs. (6a,b) for P and F. In order to exhibit the specific effect due to the timevarying masses arising from non-zero X and ~, we solve with the condition/.7//a = ~/rn i.e. the same initial fractional mass difference between particle and antiparticle for the primary super-heavy bosons and fermions. To first order in X and ), we have

and k dk ( m - k ) 2+p


B e -* (g2r-1/8~m)[(m/u)(l+ x r - 1 ) - I


r ~ P0{1 - X(m/u) rgl/[½(n + n ' ) +m/u - 11},

= (g/ 88~)(27r/x)l/2(ijp)l/2/( m _ k)l +p/2 ,

=[,[M(k)[ 2 J


Similarly for the antiparticle total width

T f

To illustrate the effect simply, we assume that the bulk [3] of the total width arises from essentially massless, quark-lepton states (B = 1/3) with overall relative probability (rig2/16nm) p - l , and from similar states of two antiquarks (B = - 2 / 3 ) with overall relative probability (n'g2/16nm) F -1 . Summing all relative probabilities, we have 1 = BQ + [(n + n')g2/167rrn] p - I .

The Born approximation matrix element is then

M - - 8 ~ °g

7 May 1981


2 [~(n+n')+m/la_l]2

x (m/u)(XPU I - XP-b-1),


which vanishes for X and X zero. The baryon asymme-

Volume 101B, number 3


try is here related to a violation of CPT which has been effected phenomenologically through the initial conditions N/m =/~//~ 4: 1, and possibly ?,/X ~ 1. With re~t2 1 and n' ~ n ~ N, we have

2XB/N7 ~ ( 4 N ) - I (~,r 01 _ ~ 1 )



If the order of the time ratio, XPo 1 (or XF0-1), is like the ratio of squared Higgs coupling to squared gauge coupling in present "low-energy" models, then XF~ 1 10-4. With N >~ 10, zXB/N,r ~< 10-5 is then not unreasonable from eq. (10), We may make a different argument, however. Suppose that within the "natural" time p ~ l (or y ~ l ) the initial mass difference (/~ --/~) is eliminated, that is a common mass is evolved which is maintained thereafter *s. Then /~(~0-1)

/~ exp(M,~l) -+ ~ - ~


U(PO 1)

/.t exp (XF~ 1)

~< [)tFff 1 - XI~ 1 ] .

(1 l)


z]xB/N .~ 1 0 - 9 ~ I ( ~ - U ) / U l

~ 10-8

(for N ~ 1 0 ) .


In the present energy domain the measured limit for the (K 0 - ~0) mass difference, I(mr: - r ~ f f ) / m K I < 0.7 X 10 -14 , restricts a first-order CPT-violating weak interaction to be at most of a strength of ~ 10 -7 relative to the CPT invariant low-energy weak interaction, or of a strength of ~ I 0-12 relative to the electromagnetic interaction [12]. Naively then, a relative strength of order 10 -6 at the origin of the universe could give initial particleantiparticle mass differences as in eq. (12). Concerning the possible influence of temperature dependent effects upon the above phenomenology, the following remark can be made. There exists a recent cosmological model *3 in which the universe has an origin in a spontaneous quantum fluctuation which 4:5 If the decay time for the Q is long relative to Fo 1 remnants of the initial mass difference are essentially erased. It is possible to imagine Q cascading down through a multitude of steps involving slightly lighter particles so that the initial relative mass difference is greatly magnified. This is presumably excluded experimentally.

7 May 1981

results in the creation of only the hypothetical superheavy bosons with mass near the Planck mass. This is at zero temperature. Very light particles result from subsequent decay processes. However, near the origin the expansion rate is so rapid in this model that within the very short decay time of the super-heavy primary particles, the very large momenta of the light decay products are red-shifted to values of the order of the light-particle masses (i.e. of the order of a few GeV). The temperature concept first comes into play for this "soup" of light particles undergoing interactions at a few GeV. In this picture there is no thermal soup of all particles at a superhigh temperature at the origin, and thus no effect upon the example of CPT noninvariance and its consequences. In particular, back reactions [2,4] cannot erase the baryon asymmetry. Clearly there can be simple variations of the above phenomenological example, such as allowing the initially unequal masses of X and X to become equal within a decay time ~lP0-1. The assumed initial inequality of masses is explicitly connected to an inequality of rates, both total and partial rates. The latter gives the baryon asymmetry (without CP-violating phases). Although CPT invariance is clearly established in present field theories and by present experiments [ 12], the conditions at the origin of the universe, that is near the Planck time, involve such an enormous extrapolation, that in the context of possible explanations of the baryon asymmetry, it is interesting to consider the possibility of some break-down, for example as exemplified by the non-zero time interval for establishing the symmetry in the above simple phenomenological model. There is a suggestion here that super-heavy fermions, as well as gauge bosons, should have existed in the early universe. With increasing access to experiments on heavyquark and lepton systems, one should consider further tests [12] of the CPT invariance upon which so much rests, such as equality of particle and antiparticle lifetimes and masses and static electromagnetic properties. I thank Raymond Gastmans for valuable comments, I am grateful to Robert Brout and Francois Englert for many discussions on CPT invariance in their cosmological model [ 11 ], and for their kind hospitality.

References [1] M. Yoshima, Phys. Rev. Lett. 41 (1978) 281;42 (1979) 746 (E). 157

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[2] S. Weinberg, Phys. Rev. Lett. 42 (1979) 850. [3] D.V. Nanopoulos and S. Weinberg, Harvard preprint HUTP79/A023, to be published. [4] S. Dimopoulos and L. Susskind, Phys. Lett. 81B (1979) 416. [5] J.C. Pati and A. Salam, Phys. Rev. D8 (1973) 1240. [6] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 483. [7] J. Goldstone, Nuovo Cimento 19 (1961) 154. [8] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 145.


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[9] M. Kobayashi and K, Maskawa, Prog. Theor. Phys. 49 (1973) 652. [10] D.A. Kirzhnits and A.D. Linde, Phys. Lett. 42B (1972) 471. [11] R. Brout, F. Englert and Ph. Spindel, Phys. Rev. Lett. 43 (1979) 417; R. Brout et al., Bruxelles preprint 2[80, to be published. [12] K. Kleinknecht, Ann. Rev. Nucl. Sci. 26 (1976) 1.