CP Noninvariance in the Decays of Heavy Charged Quark Systems

In Table I we also give the limit to ~, as a function of the decoupling temperature. If the "new" neutrinos interact superweakly because they couple to a heavier W' (m~ & m~), the decoupling temperature depends on m~.. The cross section for e'+ e v'+v' varies as o' -T'm~ ' and the reaction rate & =n, (v'v)-T' m~~ '. Since the expansion rate is t '-T', the decoupling temperature T„-m~.' '. For neutrinos coupled to the W, decoupling is at =1 MeV so that T„(MeV) = (m /m )'~'. For example, for m~ «32m~, T, &m& and it follows from Table I that at most one "new" twocomponent neutrino is allowed. Another consequence of the results in Table I is that if the usual left-handed neutrinos have right-handed counterparts, the right-handed neutrinos must decouple when T~& T,a 0.2 GeV; this suggests that m~ ~ 53m~ . Similar constraints follow for other new particles. For example, for gravitinos, N„=(~„) =20 and, for gravitons, N„=+(~„) „=17. This work was initiated while two of us (D.N. S. and G.S.) were participating in the Particle Physics-Astrophysics Workshop at the Aspen Center for Physics. In appreciation of valuable discussions, we thank M. A. B. Bd'g, P. Freund, C. Hill, D. Kazanas, D. Nanopoulos, R. S1ansky, D. Sutherland, R. V. Wagoner, and S. Weinberg. This work was supported in part by National Science Foundation Grants No. AST 76-21707 and No. AST 78-20402, and in part by National Aeronautics and Space Administration Grant No. NGR 05-020-668.

If the "new" neutrinos interact superweakly because they couple to a heavier W' (m~& m~), the decoupling temperature depends on m~.. The cross section for e'+ ev'+v' varies as o' -T'm~' and the reaction rate & =n, (v'v)-T' m~~' . Since the expansion rate is t '-T', the decoupling temperature T"-m~. ' '. For neutrinos coupled to the W, decoupling is at =1 MeV so that T"(MeV) = (m /m )'~'.
This work was initiated while two of us (D.N. S. and G.S.) were participating in the Particle Physics-Astrophysics Workshop at the Aspen Center for Physics. In appreciation of valuable discussions, we thank M. A. B. Bd'g, P. Freund, C. Hill, D. Kazanas, D. Nanopoulos, R. S1ansky, D. Sutherland, R. V. Wagoner, and S. Weinberg. This work was supported in part by National Science Foundation Grants No. AST 76-21707 and No. AST 78-20402, and  S. L. Glashow, Sci. Am. 238, No, 4, 88 (1975). ' S. Weinberg, Trans. N. Y. Acad. Sci. 38, 185 (1977 Within the context of a six-quark model combined with quantum chromodynamics we study the asymmetry in the decay of heavy charged mesons into a definite final state as compared with the charge-conjugated mode. We find that, in decays of mesons involving the b quark, measurable asymmetries may arise. This would present the first evidence for CP noninvariance in charged systems. To date, the observation of CI' nonconservation' has been limited to electrically neutral mesons. Effects in such systems are dominated by particle-antiparticle rrCixing in their mass and width matrices. ' A striking prediction of CI' nonconservation is that the decay rate of a particle into a definite final state can differ from the rate of the antiparticle decaying into the cor-responding charge-conjugated state, namely' I'(if )s I'(if); of course, the TCP theorem guarantees that the total widths are identical.
In this paper, we present, in the context of definite models of CI' nonconservation and the strong interactions, calculations for such asymmetries involving the decays of heavy charged mesons. We find, that although small, such an 242 effect can be experimentally accessible, yielding the first evidence for time-reversal violation involving nonneutral systems.
Theoretically such an asymmetry does not occur at a tree-diagram level. It requires the presence of an absorptive part due to a loop integration. We will study processes where a combination of weak interactions and quantum chromodynamics (QCD) yield these necessary absorptive parts. We assume that the basic CP nonconservation is described by the Kobayashi and Maskawa' (KM) extension of the SU (2)  These decays are dominated by diagrams in which theu or c quark is a spectator. ' It is thus reasonable to study this CPor T-nonconserving asymmetry at the quark level, namely in the decays of the "free" b quark.
The reaction of interest is with f (d or s), q, and q denoting quark flavors.
If the charge of q is~3 then the process can proceed via the first-order charged-current reaction of Fig. 1(a). To order GFn, (o, , is the @CD fine-structure constant) an absorptive part is obtained from the diagram of Fig. 1(b) which is like the "Penguin diagrams"' " except that now )'t' & 0 (and not & 0). If the charge of q is --, ', then Fig. 1(b) is the only diagram which contributes to (1). A nonvanishing absorptive part occurs whenever the quark line in the loop can be put on its mass shell; for b decay, intermediate t. " and u quark will satisfy this requirement. In the limit rn '» m, ', Fig. 1(b) can be expressed as an effective weak Hamiltonian: FIG. 1. Diagrams for the reaction b(P) f( ) +q(p) +q(-p), wheref=d or s. (a) The usual charged-current process, (b) contribution to the same reaction via gluon emission. (b) is the source for the absorptive part necessary for CP asymmetry. Note that gluon momentum is timelike unlike the case in "penguin diagrams" of Hefs. 8 and 9. +Qfq~=~& GF(&, ls)(Q&d& iq) (u~y»k &" u, ) (u, y"~2 )i U ), (2) ii = (I yfqq I pqf )/(I yfqq + I gfqq ) To indicate how frequently these decay modes occur we have also computed the ratio r = a(I'"qq + I"pqq)/I', .
As long as s, is not approximately equal to -s3, where y» =y"(1-y, ), X's are the color matrices For 0'&4m~' we obtain the desired absorptive parts leading to a difference in the widths for bfqq and b-fqq. As a measure of this difference we introduce the asymmetry parameter ! b decay will be dominated by the charged-current modes [via Fig. 1(a)], b-cud and b-ccs, and the sum of these rates defines I"". If s, = -s" then these processes become suppressed, and many channels including the ones presented in connection with the asymmetry become important. We restrict our computations of r for values of s, and s, outside this region.
For numerical computations we used a, = 0.35, sino, =0. 23, m, =15 GeV, m, =5 GeV, m, =1.5 GeV, m, =0.5 GeV, and m"=m"=0.3 GeV, and investigated the asymmetry for a range of values of 8, and 83. The value of the asymmetry was fairly insensitive to 0, but depended crucially on e/g li We divide the decay modes of the b into two groups. The first consists of those to which both   Fig. 1(b), e.g. , 5 sss~(vb) %e note that a and r can be quite sizable and that an asymmetry in, say, & -K q (p) vs O'-K'rt (y) may range 20% for 6=~. The rates for these charme1. s may also be appreciable. In addition, should the timelike penguin diagrams [ Fig. 1(b)] exhibit an enhancement similar to the one suggested for the spacelike momenta' then the asymmetries for Reactions (6) would also be enhanced as Fig. 1(b) would tend to dominate Fig. 1(a). " Besides exclusive channels, inclusive ones should also reflect such asymmetries. If the charmonium history repeats itself and above the B"B" threshold there are resonances which decay predominantly into B"B"states, a way of detecting such effects may be to sit on one of these resonances and search for a diff erence in, say, the inclusive K' and K rates.
We would like to add a few remarks in brief: (1) In Fig. 1(b) the gluon can be replaced by a photon so that our results for the asymmetry [ Fig. 3(a)] for Reactions (7) also apply to effective weak neutral-current decays of the form b -d (s)+I'+l . The corresponding value of r for these modes is smaller by a factor of 2n'/ 3n, ' compared to r of Fig. 3 for (7).
(2) We have also looked for similar effects in decays of s, c, and t quarks. For s and c quarks, and if m, is much larger than m, also for t quarks, the KM matrix leads to very small asymmetries (a). If charged mesons containing c guarks exhibit large CP nonconservation experimentally, then the KM model will be ruled out.
(3) The same mechanism also contributes to CP nonconservation in neutral mesons. However, we expect the effects from mass and width (zeroth order in n, , in general) mixing to dominate in those systems.
(4) In the KM model the K-decay CP asymmetry parameter, & -2~10 ', is proportional tos,s3~. ' Thus, in principle, 6 could be quite large and the CP effects under discussion may be experimentally accessible with relative ease.
In conclusion we wish to emphasize that in this work we have exhibited how @CD enables us to calculate the CP asymmetries in decays of charged particles. As an illustration we have used the KM model. The same (and/or similar) mechanism(s) would lead to calculable effects in conjunction with any gauge model of CP noncon-servation. In other models the resulting asymmetries, in general, would be quite different and may indeed be much larger for some reactions. Thus this mechanism may be used to constrain gauge models of CP nonconservation.
This work was supported in part by the National Science Foundation Technical Report No. 79-25.