Baryon number generation in grand unified theories

Volume 80B, number 4, 5

PHYSICS LETrERS

15 January 1979

BARYON NUMBER GENERATION IN GRAND UNIFIED THEORIES John ELLIS and Mary K. G A I L L A R D x CERN, Geneva, Switzerland and D.¥. NANOPOULOS Physics Department, Harvard University, MA 02138, USA Recewed 17 November 1978 Grand unified models of elementary particle interactions suggest that there was an early epoch during the Big Bang while the temperature was shghtly less than the Planck temperature, during which no known lnteractmns were in thermal eqmhbrlum. This epoch was probably followed by a brief period during which baryon number violating forces were m eqmhbrlum and could have annthllated any previously existing net baryon number As these forces dropped out of equlhbrmm, a CP vmlatmg component could have generated the observed baryon to entropy ratio of O(10-9).

The Universe apparently contains a small but nonzero number of baryons - there is no evidence for appreciable accunmlatlons of antimatter in the Universe, and the baryon to entropy ratio seems to be O(10 . 8 to 10-10) [1 ]. Attempts to explain this ratio must go back to the Big Bang [2], and ltkely invoke interactions which violate baryon number conservahon and CP. Conventional strong, weak and electromagnetic interactions do not violate B conservation, and one is led to contemplate [ 3 - 8 ] grand umfied theories of elementary particle interactions which incorporate both B and CP violation *a . The simplest such theory is based on SU(5) [ 9 - 1 1 ] , in which these interactions are mediated by gauge vector bosons with masses O(1015 to 1016) GeV [ 1 0 - 1 2 ] , and Hlggs bosons which may be somewhat lighter [i 1 ]. This grand unification mass is appreciably lower than the Planck mass, a necessary condition If a grand umficatxon without gravity [9] Is to make sense. The strong, weak, electromagnetic and B violating interactions are all I And Laboratotre de PhysNue Thdorique et Partlcules El~mentaires, Laborato~re assocId au CNRS, Orsay. /-1 Non-perturbatwe B violating effects m weak interaction theories are probably neghgible [5]. 360

O(c~) at temperatures ~> 1015 GeV. It is argued later in this paper [8] that this means that their interaction rates wxll all be less than the expansion rate of the Universe for temperatures between the Planck temperature of 1019 GeV and a temperature O(c~2) × 1019 GeV = O(1015 to 1016) GeV During thxs period all known interactions would have been out of thermal equilibrium (see fig. 1), and the net baryon number could have been non-zero [8]. It seems likely [8] that there was then an epoch during which the vector- and Higgs-mediated B violating forces were in thermal equilibrium (see fig 1), and any previously existing net baryon number would have been drastically reduced or annihilated. Then at some temperature t> 10 l0 GeV these forces would have dropped out of equdibriunl, and a CP violating component in them could have generated a net baryon number. Grand urnfield theories therefore lead us [8] to contemplate three possible scenarios for generating the observed baryon number o f the Universe. (A) Maybe the gravitational interactions dominant at T > 1019 GeV, or perhaps CP and B violating effects such as the decays of superheavy X bosons [6,7], generated a large net baryon number up to O(1) in the pre-equihbrlum period T > 1015 GeV. The equilibria-

Volume 80B, number 4, 5

PHYSICS LETTERS

Plonck Ternperoture

/

Equilibrium for

7 '0,;::: on0 .,OCtroman.,,°

I0 e '

I0 6

I_

104

tl

15 January 1978

I Possible baryon /--,,r~-- num be r violation

.on-:;:,::r,u°

---

~-

..,,,,,br,um.°oc.

i0 ~z

i1_ =1 i

JO

I

,~I ~ ~ v ~ ~

tO-~'

JO-4J

"9

fi

..~o.

~ "

"\

,\

.L --•

• ~' \ i, x ,,o,o, ,,, '\ .°,.roc,.on..~\

~

",,

! i ',"\

''

,n,.roc.on.

~,

'..

IOgtoT(GeV)

" ".

!,,

tO"6 iO-S

r I

~

q,,

".,i

',,

'\

Fig. 1. Sketch of early epochs of the Umverse suggested by the arguments in the text. Indicated with large uncertainties are the rates for various baryon number violating interactions. There seems to be a non-equd]brium epoch when 1019 GeV > T > 10 Is GeV, and a possible epoch when baryon number violating forces were in equdibrlum between l0 Is GeV and 101° GeV. tion due to B violating forces may then have been somewhat less than total, leaving the Universe with the present small baryon to entropy ratio of O(10-9). (B) Perhaps there was a large net baryon asymmetry at T > 1015 GeV, which was however reduced to zero (i.e., ~ 10 - 9 ) by "perfect" B violating equilibriat]on. The CP and B violating forces could then have generated the small non-zero baryon number in the post-equilibrium period (C) Perhaps the baryon number was always negligible for T > 1019 GeV, and the present baryon to entropy ratio was generated in the non-equilibrium epoch as in option (B). (This option does not in fact care whether B violating forces were ever m thermal equilibrium). Options (A) and (B) above share the advantage of "naturalness" in that the present small baryon number is obtained no matter how large the primordial baryon number. On the other hand, it seems quite likely that B violating gravitational interactions would have been

in equilibrium and hence [6,7] the net baryon number zero at T > 1019 GeV, and one can argue [8] that in at least some grand unified models a large (~ 10 - 9 ) baryon number is unlikely to have been generated in the early non-equilibrium period at T > 1015 GeV. Options (B) and (C) share the advantage that the observed baryon to entropy ratio may be directly estimable in terms of observable parameters of elementary particle physics. We will discard option (A) and present estimates o f the baryon number obtainable from grand unified models under options (B) or (C) - of wh]ch we prefer the latter. Before doing this, we will first present our arguments about the general evolutionary pattern of the very early Universe. In units where the Planck mass of O(1019) GeV ~ 1, the expansion rate [2] of the early Unwerse was ,2 [~/R ~ T 2

(l)

,2 R is the radxus, T the temperature The constant of proport]onality in (1) ]s 0(20) In the simplest grand umfied theory, see ref. [8]• 361

Volume 80B, number 4, 5

PHYSICS LETTERS

As discussed by Dimopoulos and Susskind [5], a renormahzable interaction has an interaction rate 11 ~ T for a gauge interaction I R ~ o~2 T,

(2)

where a ~ 1/45 in the simplest grand unified theory. A point-like four-fermi interaction would [5] have a rate

:F ~ a2 rs,

(3)

where for interactions mediated by massive vector bosons (4)

a = G X ~ a/m 2 ,

whereas for interactions mediated by the type of massive Higgs bosons found in the simplest grand unified models [11,13]

3 , T s,

(6)

..

T3
Crudely speaking, the universe would only be in thermal equilibrium with respect to an interaction with rate 11 if

We found [11] in the course of previous studies of the SU(5) grand unified model [9] that

A lower hmit on m H is provided by proton stability [14], and is O(10 - 5 ) m x "~ 10 I0 GeV. Using m f / m w 1/10 we see from (9) that Hlggs equilibrium could have persisted until T ~ 1010 GeV (see fig. 1). It has been emphasized by other authors [ 5 - 7 ] that the net baryon number must be zero when B violating forces are m equilibrmm. However, at the end of the equilibrium epoch CP and B violating forces could generate a net baryon number. If their interaction rate when T < m H is IA ~ G 2 r 5,

M1 - = I , / ( R ) ~ I .

i.e.

T > a 2.

Hence our remarks (see fig. 1) about the early non362

(10)

(7)

Inserting the gravitational rate (6) into eq. (7) and using the expression of formula (1) we see that the gravitanonal mteractions were out of equilibrium for T < 1019 GeV. On the other hand, comparing (1) and (2) we see that the renormalizable interactions (QCD, weak interactions, B violating interactions above the grand unification mass) would not have been in equilibrium for T2>a2T

(9)

(5)

One expects gravitational interactions to look nonrenormalizable at te'mperature T < 1 and to be proportional to powers of T: IG~T

equihbrmm epoch of the Universe from 1019 GeV > T > 1015 GeV .3 At temperatures O(1015) GeV the vector bosonmediated B violating interactions might have been in equdibrmm for a short period, the length of time depends sensttlvely on the precise masses of the bosons and on the counting factors. At temperatures below 1015 GeV they would not have been in equdibrium, as is seen by comparing (I), (3) and (4). On the other hand, the thggs boson mediated B violating interactions could have remained m equilibrium longer' Comparing (1), (3) and (5) we see that they would have dropped out of equihbrium when

m H ~2-ko m x ~ 1014 GeV.

2 G = Gtt - - -ma -2- {mf).mw. .

15 January 1979

(8)

then the rate of generation of the baryon to entropy ratio A will be of order of magnitude ~ GA2 T 5 .

(11)

Integrating up (11) from T ~ 0 to T ~ m H we find

2 H3 A ~ GAIn

(12)

~3 Heavy boson decays and the counting factors due to elementary particle degrees of freedom may increase the lower hmlt shghtly, but do not alter the qualitative picture, see ref [8].

Volume 80B, number 4, 5 IO

5

PHYSICS LETTERS I0

15 January 1979

~0 __- g

~0 - g

I0

q

*

tO

9

I0

r%

5

I

I

m e

mc

I

[

I

j

r

I

I

t

i

I

I

me

I0

I

I

I

I

t

i

[

(a)

g*

I0

~

I0

Fig. 2 The relevant Hlggs-fermlon-antifermlon vertices m the SU(5) model [8-11]

and this estimate would not be altered in order of magnitude even if there were a period while T > m H when the B violating forces were out of thermal equilibrium. To estimate A we therefore need a grand unified theory to calculate G2 and we turn to the SU(5) model [9,11 ]. We observe [8] that at high temperatures it is natural to choose a fermion basis for this model in which the vector boson couplings are diagonal .4 in generation, as are the couphngs of the Hlggs to the 5 and 10 dimensional fermlon representations w h i c h are given by the (diagonal) matrix m c of the charge - 1 / 3 "catho"-quarks [ 13 ]. The only CP non-invariant coupling is the Hlggs - 10 - 1 0 coupling matrix (see fig. 2)"

g = UTma U

(13)

where m a is the (diagonal) mass matrix of the charge 2/3 "ano"-quarks [ 13 ] and U is the usual generalized Cablbbo matrix. In estimating interaction rates we must perform flavour traces, and to get the lowest order effect we should trace over just one fermion line. It is then easy [15] to convince oneself that the lowest order CP violating fermion trace is 8th order (see fig. 3)

V =- Tr(gg*gm2cg*m2c) , = Tr (UTm] Um 2 Ut m a U* m2).

(14) (15)

The leading contribution to the imaginary part of IT(15) is given by phases in the coupling matrix U which are unobservable at low (41010 GeV) energies because they can be absorbed into a redefinition of the righthanded anoquark phases. These phases are observable when the right-handed anoquark couplings are gauged as in the SU(5) model, and one could generate CP violating effects even if there are only two generations of fer-

Fig. 3 (a) A lowest order CP violating fermlon line to be traced and (b) the Hlggs lines connected in one of many possible ways, the resulting diagrams to be cut through all physical dlscontinmties.

mlons , s . The phases could be O(l), so that the two heaviest generations of fermions could then be used to estimate

Im V<~ 0(~) m 4

CN

(16)

an SU(5) model with N generations, where the factor of 1/10 comes from a generalized Cablbbo angle. Taking three generations we therefore estimate (in this and subsequent equations m c is the charmed quark mass) &4

G2 ~i16

4

3

mb mt m c 4 8

t77H

(17)

mW

and a resulting baryon to entropy ratio

A~-~Oo~4(rnbl4(~tw)3(~Cwl(mplan~ckt.\mw!

/,

mu !

(18)

Taking the emmates (recall that quark masses are unfamiliarly small [11,13] at high energies) o ~ 10 -2 ,

m b ~_~ff mW

mc mw

10 - 2 ,

,

mt

"" Tko,

mW

m Planck ~ 109, mH

(19)

we find that A ~ 10 - 1 1

:~4 ]'his is generally true in any gauge theory, and means that CP violating effects at high temperatures necessarily Involve Higgs bosons.

m3a m a N N-I'

(20)

,5 Note the contrast with CP violation in the SU(2) L X U(1) Kobayashi-Maskawa model [16] at low energies.

363

Volume 80B, number 4, 5

PHYSICS LETTERS

This estimate is somewhat too small [1 ], but is clearly subject to considerable uncertainties' is m t / m w ~ 1 ?, is mplanck/mH ~ 104 9., are there more than three generations, is there a more comphcated Higgs structure with other sources of CP violation? Also, we have been cavalier about counting elementary particle degrees o f freedom, the total number o f ways of using the same fermion trace to get different CP violating interactions, and so on. However, we believe that we have demonstrated the feasibility o f generating the observed baryon number of the Universe through B and CP violating interactions in the period when the baryon number violating forces drop out of thermal equilibrium. We would like to thank P. Cox, S. Dimopoulos, S.L. Glashow, J. Prentki, G. Steigman, L. Susskind, S. Weinberg and A. Yildlz for useful discussions on this topic. All o f us would like to thank the SLAC theory group, two o f us (J.E. and M.K.G.) the Fermilab theory group, and one of us (D.V.N.) the CERN theory group for kind hospitality while this work was in progress. After the work described here was completed, we learnt of a new paper by S. Dimopoulos and L. Susskind which apparently discusses our option (A) (Stanford University preprlnt, 1978). We thank S. Dimopoulos for informing us o f this work.

364

15 January 1979

References

[1] G Stelgman, 1971 Carg~se Lectures, The case agamst antimatter m the Universe (Plenum press, 1973), p. 505; and Ann Rev. Astron. and Astrophys. 14 (1976) 339. [2] S. Wemberg, Gravitation and cosmology (Wiley, New York, 1972) Ch 15. [3] A. Yu. Ignatlev, N V Krosnilov, V.A. Kuzmm and A.N. Tavkhelidze, Phys. Lett. 76B (1978) 436. [4] M. Yoshlmura, Phys. Rev Lett 41 (1978) 381. [5] S. Dxmopoulos and L. Susskind, SLAC-PUB-2126 (1978). [6] D Toussamt, S.B. Treiman, F Wdczek and A. Zee, Princeton preprint (1978); D. Toussaint and F. Wilczek, Princeton preprint (1978). [7] S. Welnberg, Harvard preprint 78/A040 (1978). [8] See also, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, CERN preprmt m preparation. [9] H. Georgl and S L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [10] H Georgl, H. Qulnn and S. Welnberg, Phys. Rev Lett. 33 (1974) 451 [11] A. Buras, J. Elhs, M.K. Galliard,and D.V. Nanopoulos, Nucl. Phys. B135 (1978) 66 [12] D. Ross, Nucl Phys. B140 (1978)1. [13] M.S. Chanowltz, J. Elhs and M.K. Gaillard, Nucl. Phys B128 (1977) 506 [14] F. Relnes and M.F. Crouch, Phys. Rev. Lett. 32 (1974) 438. [15] J. Elhs and M.K Galliard, Fermilab preprmt Pub-78/66 THY (1978). [16] M. Kobayashi and K Maskawa, Prog. Theoret. Phys. 49 (1973) 652