L~.TTERE AL NUOVO CIMENTO VOL. 28, N. 4 24 Maggio 1980

Cosmic Matter-Antimatter Asymmetry and Gravitational Force.

J . P . H s u

Physics Department, Southeastern Massachusetts University - .North Dartmouth, Mass. 02747 Space Sciences .Saboratory, . N A S A / M S F C - Alabama 35812

(r icevuto fl 23 Gennaio 1980)

Recen t ly , the re are m a n y in te res t ing discussions of t h e cosmic m a t t e r - a n t i m a t t e r a s y m m e t r y (1) based on a b a r y o n n u m b e r nonconse rv ing in te rac t ion in g r a n d unif ied field theor ies , which does no t inc lude gravi ty . In th is art icle, we show t h a t t he g rav i ta - t ional in te rac t ion alone can also lead to m a t t e r - a n t i m a t t e r a s y m m e t r y . The idea involves t he g rav i t a t iona l coupling of f e rmion m a t t e r re la ted to t he Yang-Mills gauge s y m m e t r y (3) w i th t he un ique general iza t ion of t he four-d imens ional Poincar~ group (i.e. t he de S i t t e r g roup wh ich possesses t he m a x i m u m four-d imens ional s y m m e t r y (8)).

I n t h e Yang-Mi l l s - type gauge-field theories, t he gauge-covar ian t de r iva t ive D~ for t h e fe rmion field ~o has t h e fo rm

(1)

where C is a dimensionless coupl ing cons t an t for t h e gauge fieldlb~ and T~ is t h e m a t r i x r ep resen ta t ion of t he group generator . Because of (1), t he act ion S~ for t h e m a t t e r field ~o wi th mass m should have the fo rm (a)

(2) S~ = f d ~ [�89 i~7.(~ ~ + iCb~ T.a) ~ - - �89 i~(*g~,-- iCb~ T ~ ) ~ ~ - - m ~ ] ,

(1) 1~. YOSHIMURA: -Phys. Rev. Lett., 41, 281 (1978); Tohoku preprint (1979); A. Yu. IGNA~IEV, N. V. KR~SNIKOV, V. A. KUZMXN and A. N. TAVKHELIDZE: Phys. Lett. B, 76, 436 (1978); S. WEINBERG: Phys. Rev. Left., 42, 850 (1979). (') C. N. YANG and R. L. MILLS: Phys. Rev., 96, 191 (1954). (*) WU YUNG-SmH, LEE KE~-DAO and Kuo HAN-YING: Kexue Tongbao, 19, 509 (1974); Kuo HAN-YING: Kex'ue Tongbao, 5, 202 (1979); P. K. TOWNSEND: PhyS. Rev. D, 15, 2795 (1977); P .C . WEST: Phys..Lett. ,B, 76, 569 (1978). For a detailed discussion of the de Sitter group, see K. GOTO: Prog. Theor. Phys., 12, 311 (1954). (~) Note that the action Sv is, as usual, closely related to the gauge-covariant derivative in (1) for an arbitrary group ~ with the group generator T A and that this action is in general gauge invariant under the gauge group only in the limit g ~ , - - ~ , .

128

COSMIC MATTER-ANTIMATTER ASYMMETRY AND GRAVITATIONAL FORCE 129

where

{r ~, 7 ~} : 2 ~ , 7 ~ = e~ r " , % =

el~ e k : ~ , ep e k = ~ p , gl~, = el~ e~ ~ i k ,

According to the Yang-Mills gauge symmetry, the gauge field b~ is the basic dynamical field and the space-time metrics g,~ is determined by b~ through e L = (exp B)~, B = (b~) (~). The index i in b~ may be a subset of the group index A. Le t us consider the de Si t ter

group p)

(3)

where /5 is imaginary for the S03.~ de Sit ter group. The action for the gauge field b~ is

(4)

where B ~ = b~-- a ~ a ~ 6 �89 ~ ba~, gas ~- ~.o]~.~1 , and

(5) A A A A B D

The first term, involving the constant ant isymmetric tensor e ~ , does not contr ibute to field equations at all because of i ts topologically gauge-invariant nature (s,e). We stress tha t this has a very interesting and impor tant implication: The gauge fields b. are free in the absence of fermion fields. Thus, the present formulation of gravi ty is very much simple in comparison with the usual formulation of gravi ty in which there are extremely complicated interactions among the gravi ta t ional field even in the absence of fermion fields. The second noninvariant term (~) in (4) corresponds to the usual gauge-fixing term, and contribute to field equations derived from 5(S b ~-Sv)----0:

(6) ~ - ~ u ~ O u O ~ , B ~ , ~ aa = �89 [~7~0~ ~ - - ~ 7 ~ ' ~ ] - -

�89 C~{7 ~, T , } ~ E ~ - C~7- ~ b~, "~ T ~ E g ( ~ e ~ / e b ~ )

where Eg ~ det e L. Fo r our purpose, i t suffices to consider the case of weak stat ic which determines the space-time metric g~ : fields b~

(7) V~B~ = - - L ~ m ~ ~ ~ - - (CL/2) ~ 7 ~ , i = ~ : O,

(8) V2B~ = _ (CL/2)~*To~, i ---- A = l , 2, 3 ,

(s) J . P . HSU: Phys. Rev. LeU., 42, 934, 1720 (1979); P. C. WEST: Phys. Lett. B, 76, 569 (1978). (6) S. W. MACDOWELL a n d F. MANSOU~I: Phys. Rev. Lett., U , 739 (1977). (7) We choose a s imple f o r m so t h a t i t does n o t g ive rise to a n y n o n - g a u g e - i n v a r i a n t i n t e rac t ion a n d the resu l t s can be eas i ly e x t e n d e d to m a n y - f e r m i o n cases. This is to be c o n t r a s t e d to, for e x a m p l e ,

the * gauge-f ix ing * t e r m used in ref. (4).

130 J . P . HSU

where we retain only large source terms. Since b~ = B~-- �89 d~B~ and

~fermion -~ , ~antifermion ~ , ~/0

1 0) 1

- - 1

0 - - 1 /

i we have the following classical approximations for nonvanishing b~:

(9)

mL2 mL2 V~do= ~ ( l - -Q)~"(x) , V2dl = ~ (1 + Q)~"(x),

do = b ~ , ~ = b l = b~ = b.",

Q ~ ~/ (s

for a pointlike fermion source. For a pointlike antifermi0n source (8) we have

m.52 m L 2 (10) V2d0 = ~ (1 "-~ Q) ~"(x) , V2dl = ~- ( l - - Q ) ~3(~) o

Note that Q has a sign different from that in (9). I t follows from (9), that for fermion sources we have

L ~ m L ~ m (1 + Q), Q = ~ / L m (1]) d o = ~ (1 - - Q ) , d I 8gr

and hence

(12) { g ~ = 1 + 2 0 - - Q ) q + 2 ( 1 - - Q ) ~ 2 + .. . .

gl~= - [ 1 + 2(1 + Q)q + 2(1 + Q)q~+ ...],

~p = L ~ m / 8 g r ~-- - - G m / r : New%onian potential,

ILl = (8~G)t = 8-10 -"a era,

t k = (exp [-B])~ and g#~ = %e, ,# i~ . where we have used e# For 803, 2 the quant i ty L 2 is negative, the result (12) leads to attractive gravita-

tional force and is consistent with previous experiments, provided the correction Q is small (~). Since g#, must be real, the gravitational coupling constant (7 must be

(D T h e a u t h o r w o u l d l ike to t h a n k J . I)ENKER a n d B. YURKE f o r b r i n g i n g t h i s p o i n t to h is a t t e n t i o n . (D F o r s m a l l O/mL a n d l a r g e d i s t a n c e , t h e f e r m i o n f ield e q u a t i o n [ - - t ~ O ~ , ~ ( ~ e ~ ) + m + + (G]2L)b~e~]~ = 0 l e a d s t o t h e a p p r o x i m a t e e q u a t i o n of m o t i o n of a p a r t i c l e i n a n e x t e r n a l f ie ld: ( g U ' ~ u ~ , - - m D ~ p = 0, w h e r e t h e o p e r a t o r is i n t e r p r e t e d as t h e r e l a t i v i s t i c H a m l l t o n i a n fo r c l a s s i ca l m o t i o n . (Cf. P . A . M. DIRAC: The Principle of Quantum Mechanics, 4 t h ed. (New Y o r k , N . Y. , 1958) , p . 263 . )

COSMIC MATTER-ANTIMATTER ASYMMETRY AND GRAVITATIONAL FORCF, 131

imaginary. This implies that the gravitational interaction in (2) violates CP and T invariances. The violation effects are very small.

The result (11) can be easily generalized to many different fcrmions (say N e electrons, Np protons and Nn neutrons) located at the same place. We find that

(13) do= Sn---~ 1 - - ~ , d 1 ~ \ + ~ ,

where M = N oM e + N p M ~ + N nM~ is the total mass and _ N = N e + N p T N n is the total fermion number. For N~ = 0 and Nap> Np we have .NIM ~_2/m~ and .N/M ~_ limp, respectively, where we have used the fact N e ----np and Mp_~ M~ >> M e for matters.

Considering lowest-order Feynman diagrams, we find the following gravitational potential energies:

C (14) .A) ~ \

between fermions with masses M 1 and M2,

(15) B) L2 M1 M2 [1 0

between antifermions with masses M 1 and M~, and

(16) c) s~r t

between a fermion (M1) and an antifermion (M2) .

Thus, the gravitational forces between matter is different from that between anti- matter. This has important implications concerning the evolution of the Universe: Even though the global properties of the Universe cannot be described properly by using the classical theory of gravity, the dynamics of the expansion after the big-bang fireball are thought to be the same as in Newton's theory. Imagine that a big sphere is shattered by an explosion, the debris flying off in all directions. Each fragment feels the gravitational pull of all the others, and this causes the expansions to decelerate. If the explosion were sufficiently violent, the debris would fly apart forever. The same argument holds more or less for the big-bang of the Universe. Now suppose C/.~ ~ O, fermious feel less gravitational force of all the others, their deceleration during the expansion will be larger so that they will be separated from antifcrmions.

We note tha t here matter means both baryons and leptons. I t may be remarked that even if the cosmic-baryon asymmetry is due to a baryon-number-violating inter- action, it may not follow in general tha t the lepton asymmetry can also be realized at the same time and in the same way. Thus, the cosmic separation of matter and anti- matter due to the universal gravitational force appears to be more natural.

Since i l l = (8zG)J in (12), the coupling constant ]C l can be estimated (lo) by using (14) for nucleons and the EStv6s experiment (11). I t is found that

(17) IOl < l O - ~ , IQI = I V I . ~ M ~ I < l o -1~

(~o) T. D. LEE a n d C. N. YANG: Phys, Rev., 98, 1501 (1955). (ll) R . E~Prvi3s, D. PEKAR a n d E. FEKETE: A n n . Phys. (Leipzig), 68, 11 (1922); P . G. ROLL, R . KROTKOV a n d R . H . DXCKE: .Ann. Phys. (N. Y.), 26, 442 (1904); V. B. BRAGINSK~ r a n d V.- I . PANOV-" ~OV. Phys. J E T P , 34, 463 (1971).

1 3 2 J . P . HSU

for [LMpl~ 4" l0 -19. One m i g h t a d o p t the a t t i t ud e t h a t t he correct ion IQ] is e x t r e m e l y smal l and ask: w h y bo the r w i t h i t? B u t f rom t h e v i ewpo in t of the ca t a s t rophe theory , wh ich is p robab ly qui te r e l evan t in th is connect ion, a smal l ini t ial dev ia t ion can lead to a g rea t difference in t he even tua l aggregat ion of fe rmions in to objec ts as large as s ta rs or galaxies. This poss ib i l i ty canno t be ignored because a t the t h re sho ld epoch of classical cosmology, t*.-., (G~/cs)t~ 10 -43 s, t he d en s i t y is q * ~ c s / ( G 2 h ) ~ 10 ~8 g cm -3 a n d the nucleon g rav i t a t iona l and s t rong in t e rac t ion are effectively of comparab le s t r e n g t h (18).

F ina l ly , we m a y r emark t h a t in previous fo rmula t ions t h e g rav i t a t iona l i n t e r ac t i on is CP- and T- invar ian t . However , our resul t of CP a n d T non invar iances for g r a v i t y (13) has n o t been excluded, because so far the re is no e x p e r i m e n t a l evidence for CP a n d T inva r i ances in the g rav i t a t iona l in te rac t ion .

The a u t h o r would like to t h a n k Y. C. Lv.u•G for useful discussions. P a r t of t h e resea rch was accompl ished while t h e au tho r he ld a Na t iona l Research Council Senior Res iden t Research Assoeia teship . The work was s u p p o r t e d in p a r t b y the Na t iona l Aeronau t i c s and Space A d m i n i s t r a t i o n and S o u t h e a s t e r n Massachuse t t s Un ive r s i ty .

(is) E. R. H~RXSO~r Nalure (London), 228, 258 (1970); 215, 151 (1967). (~s) Note that the four-dimensional symmetry is essential in this formulation of gravity. In particular, if one uses a symmetry framework with a cosmic time defined by the evolution of the Universe as a whole, our results remain the same as long as there i s a four-dimensional symmetry. Physically, this cosmic time appears to be equivalent to a convention by which all observers in different frames of reference agree to use one and the same grid of clocks ioeated in any one of inertial frames. In this way, one has a four-dimensional symmetry framework with a common time (or cosmic time). (See, for example, J. P. Hsu and T. N. SHERRy: Found. Phys., 10, 57 (1980; J. P. HSU: Found. Phys., 8, 371 (1978).) This is interesting because expansion of the Universe and simultaneity will have a universal meaning to all observers.