Standard Model CP-Violation and Baryon Asymmetry Part I: Zero

CERN-TH.7262/94 LPTHE-Orsay-94/48 HD-THEP-94-19 FAMNSE-12-94 NSF-ITP-94-64 hep-ph/9406288 Standard Mo del CP-violatio n and Baryon asymmetry Part I: Zero Temp erature a b c d M.B. Gavela, , M. Lozano , J. Orlo , O.Pene a CERN, TH Division, CH-1211, Geneva 23, Switzerland b 1 Dpto. de Fisica Atomica, Molecular y Nuclear, Sevilla, Spain, c Institut fur Theoretische Physik, Univ. Heidelb erg d 2 LPTHE, F 91405 Orsay,France, Abstract We consider quantum e ects in a world with two co existing symmetry phases, unbroken and sp ontaneously broken, as a result of a rst order phase transition. The discrete symmetries of the problem are discussed in general. We compute the exact two- p oint Green function for a free fermion, when a thin wall separates the two phases. The Dirac propagator displays b oth massive and massless p oles, and new CP-even phases resulting from the fermion re ection on the wall. We discuss the p ossible quark- antiquark CP asymmetries pro duced in the Standard Mo del(SM) for the academic T = 0 case. General arguments indicate that an e ect rst app ears at order in W the re ection amplitude, as the wall acts as a source of momentum and the on-shell one-lo op self-energy cannot b e renormalized away. The asymmetries stem from the interference of the SM CP-o dd couplings and the CP-even phases in the propagator. We p erform a toy computation that indicates the typ e of GIM cancellations of the problem. The b ehaviour can b e expressed in terms of two unitarity triangles. HEP-PH-9406288 1 Work partially supp orted by Spanish CICYT, pro ject PB 92-0663. 2 Lab oratoire asso cie au Centre National de la Recherche Scienti que. 1 Intro duction In ref. [1] we presented recently a summary of our ideas and results on Standard Mo del (SM) baryogenesis, in the presence of a rst order phase transition. The aim of the presentwork is to describ e in detail our zero temp erature analysis. The nite temp erature T scenario is treated in the acompanying pap er, ref. [14 ]. The baryon number to entropy ratio in the observed part of the universe is estimated to 11 be n =s (4 6)10 [3]. In 1967 A.D. Sakharov[4 ] established the three building blo cks B required from any candidate theory of baryogenesis: (a) Baryon numb er violation, (b) C and CP violation, (c) Departure from thermal equilibrium. The Standard Mo del (SM) contains (a)[5] and (b)[6], while (c) could also b e large enough [7][8], if a rst order SU (2) U (1) phase transition to ok place in the evolution of the universe [9]. We will not enter the discussion on the latter: it will b e assumed that a rst order phase transition did take place, as strong as wished by the reader. An optimal sphaleron rate can b e assumed as well. Our aim is to argue, on a quantitative estimation of the electroweak C and CP e ects exclusively, that the current SM scenario is unable to explain the ab ove mentioned baryon number to entropy ratio. Intuitive CP arguments lead to an asymmetry many orders of magnitude b elow observa- tion [10][11]. Assume a total ux of baryonic current, where all quark avours are equally weighted. An hyp othetical CP-violating contribution in the SM with three generations[6] should b e prop ortional to 2 2 2 2 2 2 2 s s s c c c s (m m )(m m )(m m ) 1 2 3 1 2 3 t c t u c u 2 2 2 2 2 2 (m m )(m m )(m m ); ( 1.1) b s b d s d times some p ower of the electroweak coupling constant, to b e determined. In eq. ( 1.1) the mixing angles and phase are the original Kobayashi-Maskawa ones[6]. In order to obtain a dimensionless result, the expression in ( 1.1) has to b e divided by the natural mass scale th of the problem M or, at nite temp erature, T , at the 12 power. It follows[11] that the W 20 resulting asymmetry is negligible, n =s 10 or smaller if other reduction factors are B considered. When the physical b oundary conditions are such that some sp eci c avour is singled out by nature (or exp eriment as, for instance, in the K K system, electric dip ole 0 0 moment of the neutron, etc.), some of the fermionic mass di erences in ( 1.1) are no more comp elling, and a stronger e ect is p ossible. Nevertheless it seems dicult to comp ensate the 10 orders of magnitude di erence in the ab ove reasoning. Such could b e the case if the desired observable came ab out as a ratio of ( 1.1) with resp ect to another pro cess which contained by itself some fermionic suppression factors: an example is the CP-violating parameter in K decays, where the CP-violating amplitude is compared to a CP-conserving denominator, Re (K K ), which is a GIM suppressed ob ject. This is not what happ ens with the 0 0 baryon asymmetry of the universe where, as shown b elow, although nature may single out some sp eci c avours, there is nothing to divide with (other than the overall normalization 2 to the total incoming ux, which b ears no suppression factor). In this resp ect, the pro cess has some analogy with the electric dip ole moment of the neutron where the exp eriment preselects the up and down avours, and of course the results are normalized to the total incoming ux of neutrons. On the light of the ab ove considerations, the reader may b e astonished that we give the problem further thought. Our motivation is that the study of quantum e ects in the presence of a rst order phase transition is rather new and delicate, and traditional intuition may fail. A SM explanation would b e a very economical solution to the baryon asymmetry puzzle. To discard this p ossibility just on CP basis suggests that new physics is resp onsible for it, without any need to settle whether a rst order phase transition is p ossible at all. Furthermore, the detailed solution in the SM gives the \knowhow" for addressing the issue in any theory b eyond the standard one where CP violation is rst present at the one-lo op level. On top of the ab ove, the authors of ref.[12]have recently studied the issue in more detail and claim that, at nite temp erature, the SM is close to pro duce enough CP violation as to explain the observed n =s ratio. B υ=0 υ=/ 0 x,y ψinc pinc tr 0 z ψ out out p ψ tr p Figure 1: Quarks scattering o a true vacuum bubble. Some notations in the text are sum- marised in the \zoomed-in" view. A rst order phase transition can b e describ ed in terms of bubbles of \true" vacuum (with an inner vacuum exp ectation value of the Higgs eld v 6= 0) app earing and expanding in the preexisting \false" vacuum (with v = 0 throughout). We can \zo om" into the vicinity of one of the bubbles, see Fig. 1. There, the curvature of its wall can b e neglected and the world is divided into two zones: on the left hand side, say, v = 0; on the right v 6=0 and masses app ear. The actual bubble expands from the broken phase (v 6=0)towards the unbroken one (v = 0). Wework in the wall rest frame in which the plasma ows in the opp osite direction. Consider thus a baryonic ux hitting the wall from the unbroken phase. Far enough to the left no signi cant CP-violating e ect is p ossible as all fermions are massless. In consequence, the heart of the problem lies in the re ection and transmission prop erties of quarks bumping on the bubble wall. CP violation distinguishes particles from antiparticles and it is a pr ior i p ossible to obtain a CP asymmetry on the re ected baryonic 3 2 current, . The induced baryon asymmetry is at most n =s 10 ,inavery CP B CP optimistic estimation of the non-CP ingredients [13][12]. The symmetries of the problem are analyzed in detail for a generic bubble. The analyt- ical results corresp ond to the thin wall scenario. The latter provides an adequate physical description for typical momentum of the incoming particles j~pj smaller than the inverse wall thickness l , i.e., j~p j 1=l .For higher momenta the cuto e ects would show up[17], but it is reasonable to b elieve that the accuracy of the thin wall approximation pro duces an upp er b ound for the CP asymmetry. The precise questions to answer in the ab ove framework are: 1) the nature of the physical pro cess in terms of particles or quasi-particles resp onsible for CP violation, 2) the order in the electroweak coupling constant, , at which an e ect rst app ears, 3) the dep endence W on the quark masses and the nature of the GIM cancellations involved. We shall consider the problem in two steps: T = 0 in the present pap er, and nite temp erature case in the subsequent one [14].

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