The Baryon asymmetry of the universe and the invisible axion

The Baryon asymmetry of the universe and the invisible axion


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109B. number




4 March




A. MASIERO Max Planck Institute, Munich, Germany and

G. SEGRk ’ Department of Physics, University of Pennsylvania, Philadelphia, PA I91 04-3859, USA Received

2 November


WC show that in an axion GUT model, the introduction of x$, a complex 24 representation of the Higgs field, provides not only a way of making the axion superlight and hence essentially invisible, but also allows for the generation of a large baryon asymmetry of the universe by the decay of color triplet c fields into lighter Higgs bosons.

One of the major outstanding problems in theoretical particle physics is the strong CP problem, namely how to account for the smallness of the parameter which characterizes the lagrangian interaction term Lc, = ~9(g;/64n~)~~~~~F;"F~'.


the above expression (Yare color indices and gc the QCD coupling constant. L, violates CP and T;from the limits on the electric dipole moment of the neutron [l] , one sees that 0 must be < 1O-g [2] . It is of course attractive to attempt to solve this problem by having ~9be zero in a natural way and yet not do away with CP violation altogether since, after all, K, does decay into two pions. One of the more attractive ways of having 0 effectively be zero was proposed by Peccei and Quinn [3] who modified the minimal SU, X U, model by introducing a second Higgs doublet with a global symmetry such that one Higgs doublet @1 coupled only to dR type quarks (Q = -l/3) and the other, ti2 to uR type quarks (Q = 2/3). The additional anomalous U, symmetry allowed the Jc, term to be rotated away. Wilczek and Weinberg then pointed out that In

’ Supported Contract

in part by the US Department No. EY-76-C-02-3071.

0 03 1-9163/82/0000-0000/S

of Energy


02.75 0 1982 North-Holland

the pseudo-Goldstone boson associated with the spontaneous breaking of the Ul symmetry would be very light and studied its properties [4]. Intensive searches for the so-called axion have not met with success [5] *I. Recently, Dine et al. [7] ** have reintroduced the axion idea with a twist, by having an additional complex SU, singlet field transforming non-trivially under the new U1 with a vacuum expectation value (vev) much greater than G, Ii2 . This leads to an axion which is primarily an SU, singlet and is both much lighter and more weakly coupled to ordinary matter than the ordinary axion. This allows it to have escaped detection. Wise et al. [9] have essentially embedded this model in SU, by having two Higgs fives H,, 2 responsible for fermion masses instead of one and having SU, broken to SU, X SU, X Ul by a complex 24 rather than a real 24. The complex C transforms nontrivially under the additional U, which we will call Ul PQ and since the vev of Z is much, much greater than C, ‘j2, we recover the results of Dine et al.

*’ A possible


axion has been reported

161. For earlier somewhat et al. [8]


by Paissner

similar work see Kim and Shifman


Volume 109B, number 5


The ordinary weak CP is present in this model because of the complex Yukawa couplings and the CP violating phases of the vev's of the scalar fields. Since we claim at this point to have a prototypical GUT, we can also address the question of whether the CiP violation in the model is sufficient to account for the baryon asymmetry of the universe, knB/s , in the so-called "standard baryosynthesis model", where knB/s is generated in the CP and B (baryon number) violating decays of superheavy color triplet Higgs bosons ,3 The minimal model with three families of fermions and a single five of Higgs fails by almost ten orders of magnitude, giving much too small a knB/s. The reason is that one does not get a nonzero asymmetry until one reaches three-loop diagrams in the decay of the superheavy particles, and the Yukawa coupling constants are fixed by the fermion masses, f ~ G~]2Mf 1 (even if the t quark were very massive, one cannot set all fermions in the loop diagram equal to the t quark). Two Higgs five multiplets HI, 2 coupling to fermions can yield a sufficient knB/s if no additional global symmetries are imposed. The Yukawa lagrangian is, dropping SU 5 and family indices £ = ~ T C ( g l H 1 +f2H2)~/+ ~ ( f l t H l + g 2 H 2 ) x ,


where ff is a left-handed ten representation and × a right-handed five representation of SU(5). In this model, knB/s receives a contribution from e.g. the one-loop diagram of fig. 1 and is proportional to







16re~...[tr(ft'f") + tr(g+'glt )1


f J J




. H 1 2,)


~.. a H2,t

H1,2 ~

Fig. 1. Decay of the H 3 color triplet.


where we trace over family indices. The price we pay for a sufficient knB/s is (1) flavor-changing Higgs-mediated neutral currents since H 1 and H 2 couple to Q = 2/3 and Q = - 1 / 3 right-handed quarks, (2) no U 1 pQ symmetry so the strong CP problem still exists. These two reasons lead us to reject this solution. Imposing an additional U 1 symmetry has the effect in (2) of settingg 1 =g2 = 0. It is easy to see that the baryon asymmetry generated by HI, 2 decaying into baryons is even smaller than it was for the case of a single Higgs coupling to fermions. What is one to do? We obviously could give up trying to explain the baryon asymmetry, but that would be disappointing. A couple of recent papers [ 12,13] have suggested solving this problem by introducing a third five of Higgs which transforms under U 1 pQ as either H 1 or H 2. This solves the baryon asymmetry problem but, aside from the possible criticism of such an ad hoc mechanism, it also has the disadvantage of introducing Higgs-mediated neutral flavor-changing currents. We would like to discuss another possible solution to the baryon asynunetry problem for a theory with a U 1 pQ symmetry. It is a variation of a method discussed by Barr, Segr6 and Weldon [14} (hereafter referred to as BWS) and does not require the presence of particles beyond those we have already had to introduce, H1, 2 and ~. In BSW a third Higgs multiplet H 3 which did not couple to fermions was introduced; the color triplets H~ had two decay channels (a = 1,2, 3 are color indices,j = 4, 5 flavor indices) w /


,3 There are many references on this subject. For an overall view see e.g. Dalgov and Zeldovich [10]. For more detailed recent reviews of baryosynthesis see e.g. Harvey et al. and l:ry et al. [ 11 ].


a Hi,2

4 March 1982

H? + +

+ +


with the necessary CP violation introduced by the complex Higgs boson self coupling. The essential feature was that H~ had, as required, B and CP violating decays with two distinct channels of different final average baryon number (see fig. 1 for the relevant oneloop diagrams). The reason this picture does not apply in our case is that the needed CP violating Higgs self couplings are of the form (r, s = 1,2, 3 are multiplet labels)

cr, (Hi"


Such couplings are allowed by discrete symmetries under which e,g. the ~r (and some fermion fields) separately change sign; in fact the baryon asymmetry


109B, number 5


4 March 1982 a HI /

is proportional to Im(c12c23c13). These couplings are forbidden, however, by U 1 pQ so an H 3 field does not solve our problems here. Fortunately the complex field 12 can play the role of the H 3 field in BSW. Labelling by ~[~,/~1 and X~ the fermion fields belonging to the ten and five representations, we take the transformation properties under U 1 pQ to be ~9~e-i°/2ff,

x-+ ei° /2X ,

tt l - + e i ° H l ,


/ /


H ~


~H, ~ Hk 7

/ ]El





/ / /



The Higgs potential has the following quartic couplings


7rs(Hr?" Hs)(Hsf "Hr) + or(Hit "Hr) tr(E t° ~:)

H1 , / .i. ..,T


/ I"/H


+ 6rHrt(Zt • ig)Hr + XI-It " n 2 tr(Z 2) + pH~ GZH 2 + h.c.,



c) Hk 2/, ..z,_

where X and p are complex while 7rs, fir and o r are real by hermiticity. Let us assume now that the couplings are such that the E color triplets are more massive than the HI, 2 color triplets, but less massive than the E color octets. Three Higgs particle decay channels are then open to



/" /"

/ /

H a



"- akH~.

dl Fig. 2. O n e - l o o p c o n t r i b u t i o n s

t o 2/9 d e c a y (c~ = 1, 2, 3 a n d

f,k,l=4,5). +




These have lowest order amplitudes proportional to ~1, ~2 and #. In order to generate baryon asymmetry we need non-trivial interference terms. Calling T O the three level diagram and TlI(S + ie)loop diagram for Z decay, the interference term is proportional to IT0 + Tll(S + ie)} 2 - IT~ + T~l(s + ie)l 2 = 4Im(ToT~) Im I(s + ie),


i.e. both complex couplings and an s channel discontinuity are needed. In fig. 2 a - 2 d we display the relevant one-loop diagrams for the decay channels of eq. (8). If we calculate the interference term, as in eq. (9), for fig. 2a and its corresponding tree graph proportional to 61 and then repeat the process for figs. 2 b 2d and finally sum the four contributions, the ans-

wer we obtain is zero. This is, of course, just the reflection of the TCP statement that the lifetime of Eft equals that of its antiparticle. Baryon asymmetry is, however, generated in a subtle manner. Nonvanishing interference terms lead to more (or fewer) H~'s generated in the decay of 2~ than H~ t's in the decay of ZTt and correspondingly fewer (or more) H~'s in z 7 decay than H~t's in ~ t decay. We assume then that this H~ - H~ t and H~ - H~t asymmetry is not erased, i.e. the color triplet Higgs particles H a1,2 are out of equilibrium. H~ and H~ subsequently decay into fermions but because of the different fermionic couplings, the average baryon number of their final states isB 1 = - 1 / 3 and B 2 = - 1 / 6 , so we are left with a baryon asymmetry. The point, which we reiterate, is that ZT, N~ t decays create H~ - H~? and compensating H~ - H~ t asymmetries, not baryon asymmetry. The latter is produced by the out of equi351

Volume 109B, number 5


librium decays o f H ~ 2 and HI,~2. l f B 1 equalled B2, the baryon asymmetry would vanish since TCP ensures that the H 1 asymmetry is exactly balanced by the H 2 asymmetry. The upshot is a baryon asymmetry (from e.g. the interference term of fig. 2a and its corresponding tree diagram).

kn B - -


611m(p)t*) Im I(s + ic) (B 1 - B2) ~


(181 [2+ 162 12+ 1012)

The terms from fig. 2b and 2d have a different I(s + ie) because of the presence of color triplet Higgs particles in the 10op and of course the Born couplings are 5 2 and p instead of 6 1 - T h e relevant three-body phasespace integrals are given in BSW. A detailed calculation would have to take into account the dynamical evolution of the universe and, in particular, the effect of scattering processes such as ~; + ~ -+ H + H. It is clear, however, that a baryon asymmetry compatible with the experimental value ,3 o f k n B / s ~ 10 10 is easily arraignable since 5 1 , 5 2 , p and • may all be large, O(1). There is one further complication we should at least mention, the interrelationship of the gauge bosons A~j with Y~. We introduce a complex (its transforms nontrivially under U 1 pQ). At about the same temperature as the ~;~ are decaying (namely at the scale of SU 5 breaking), they are also being "eaten" b y the A~; (the admixture of H , , that is eaten is negligible3! howe~er, since the ~ ' are complex we would visualize only the phase of ~ as eaten just as the phase of an ordinary SU 2 doublet is eaten by the Z. The real part of Z~, which we may call cr~ remains and the baryon asymmetry is generated by the difference in differential decay rates of o~ and their antiparticles o~. In conclusion we see that the extension of a GUT with a U 1 pQ symmetry to include a complex rather than a real ~ field potentially has two good features: (1) it makes the axion invisible, and thereby gives a potential solution to the strong CP problem and (2) it provides a mechanism for generating the baryon asymmetry of the universe.


4 March 1982

One of us (G.S.) would like to thank S. Soni and the other (A.M.) J. Ellis, R.D. Peccei and T. Yanagida for discussions and both of us would like to thank the organizers of the 1981 Carg~se Summer Institute where much of this work was done.

Note added. Another possibility for the generation of an adequate cosmological baryon asymmetry in the GWW model consists in using the decay of the superheavy gauge bosons. This alternative is presently under investigation, see ref. [ 15]. References [1] N.F. Ramsey et al.,Phys. Rev. D15 (1977) 9; T.S. Altarev et al., Phys. Lett. 102B (1981) 13. [2] V. Baluni, Phys. Rev. D19 (1979) 2227; R. Crewther, P. Di Vecchia, G. Veneziano and E. Witten, Phys. Lett. 88B (1979) 123. [3] R. Peccei and H. Quinn, Phys. Rev. Lett. 38 (1977) 1440;Phys. Rev. D16 (1977) 1791. [4] S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 40 (1978) 279; see also: W.A. Bardeen and S.H. Tye, Phys. Lett. 74B (1978) 228. [5] T.W. Donnelly et al., Phys. Rev. D18 (1978) 1607. [6] H. Faissner, Proc. Moriond Conf. (1981). [7] M. Dine, W. Fischer and M. Srednicki, Phys. Lett. 104B (1981) 199. [8] J.E. Kim, Phys. Rev. Lett. 43 (1979) 43; M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B166 (1980) 493. [9] M. Wise, A. Georgi and S. Glashow, SUs and the invisible axion, Harvard preprint HUTP 81/A109. [10] A.D. Dalgov and Y.B. Zeldovich, Rev. Mod. Phys. 53 (1981) 1. [ 11 ] J.A. Harvey, E.W. Kolb, D,B. Reiss and S. Wolfram, Baryon number generation in grand unified models, Cal Tech preprint 68-815 (1981); J.N. Fry, K.A. Olive and M.S. Turner, Phys. Rev. D22 (1980) 2977. [12] R. Barbieri, R.N. Mohapatra, D.V. Nanopoulos and D. Wyler, Phys. Lett. 107B (1981) 80. [ 13] J. Ellis, M.K. Gaillard, D.V. Nanopoulos and S. Rudaz, Phys. Lett. 106B (1981) 298. [14] S. Barr, G. Segrb and H.A. Weldon, Phys. Rev. D20 (1979) 2494. [15] A. Masiero and T, Yanagida, Phys. Lett. 109B (1982) 353.