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Home , Asymmetry, CP violation

Theory and Phenomenology of CP Violation Thomas Mannela aTheretische Physik I, University of Siegen, 57068 Siegen, Germany

In this talk I summarize a few peculiar features of CP violation in the , focussing on the CP violation of .

1. Introduction CP Violation and more generally the flavour • The had to be out of thermal equi- structure observed in elementary-partlcle interac- librium, since in thermal equilibrium CPT tions still remains one of the unexplained myster- invariance is equivalent to CP invariance. ies in high-energy physics. In the framework of the standard model flavour mixing and CP vi- Although the standard model has all these in- olation is encoded in the CKM matrix for the gredients, it turns out that the observed - quarks and in the PMNS matrix for the . cannot be accommodated However, this is only a parametrization, in which by the standard model, and one reason is that CP CP violation appears through irreducble phases violation turns out to be too small. in these matrices and which is up to now com- Furthermore, CP violation (and with it the pletely consistent with the experimental facts, at complete flavour structure) of the Standard least with what is found at accelerator experi- Model is quite peculiar and fully compatible with ments. the data from particle accelerators. In the follow- On the other hand, the CP violation in the ing I will focus on the CP violation of quarks and standard model is a small effect. In particular point out a few of these features which follow from it is too small to create the observed matter- the parametrization with the CKM matrix and antimatter asymmetry of the universe, for which which have a specific phenomenology and which CP violation is an indispensable ingredient. The are not easily reproduced by generic new physics standard model is a local, Lorentz-covariant and models. causal quantum field theory, which means that we shall assume strict CPT conservation. Keep- 2. CP Violation in the Standard Model ing this in mind, let us recall here the criteria In general, CP violation emerges in Lagrangian established by Sakharov [1] for the appearance of field theory through complex coupling constants, a -antibaryon asymmetry: the phases of which are irreducible1. Schemati- • There have to be baryon Number violating cally this means interactions: X † † L = aiOi + h.c. (CP ) Oi(CP ) = Oi . (1) i L(∆nBar 6= 0) 6= 0 . In the Standard Model there is only a single ir- • CP has to be violated in order to have dif- reducible phase which is induced by the Yukawa ferent reaction rates for and an- couplings. It has become a general convention tibaryons to define the phases of the fields such that the

1 L(∆nBar6=0) ¯ L(∆nBar6=0) ¯ This means that the phases cannot be removed by phase Γ(N −→ f) 6= Γ(N −→ f) . redefinitions of the fields.

1 2 Thomas Mannel

Figure 2. The box diagramms mediating ∆F = 2 transitions in the standard model.

the Wolfenstein expansion. This triangle is shown in Fig. 1 An invariant measure for the size of CP viola- tion is the area of this triangle. In fact, on can show that all traingles have to have the same area Figure 1. The standard unitarity triangle. due to unitarity. The area of all these triangles is irreducible imaginary part appears in the CKM proportional to the quantity Matrix ∗ ∗ Im∆ = ImVudVtdVtbVub (4) ∗ VCKM 6= V (2) 2 CKM = c12s12c13s13s23c23 sin δ13 In fact this is already one of the peculiarities of It is interesting to note that the maximal possi- CP violation in the standard model, where the ble value for this quantity for some “optimized” unique source of CP violation only appears in the values of the angles and the phase is charged current couplings. There are infinitely many possible parametriza- 1 δmax = √ ∼ 0.1 (5) tions of the CKM matrix in terms of three an- 6 3 gles and a phase. One possibility, the PDG while the value realized in is several or- parametrization, may be written as a product of ders smaller, δ ∼ 0.0001. This quantifies the three rotations and a phase matrix in the form exp statement that CP violation is a small effect.

h c12 s12 0 i h c13 0 s13 i Finally we note that CP violation vanishes in U12 = −s12 c12 0 ,U13 = 0 1 0 , 0 0 1 −s13 0 c13 the case of degeneracies of up or down h 1 0 0 i h 1 0 0 i masses, in which case one may rotate away the U23 = 0 c23 s23 ,Uδ = 0 1 0 . −iδ 0 −s23 c23 0 0 e 13 CP violating phase. It has been noted by Jarlskog [2] that the following quantity is a clear indication where sij = sin θij etc. are the sines and cosines for the presence of CP violation: of the rotation angles. The standard parametriza- J = Det([M ,M ]) (6) tion is obtained by u d † = 2iIm∆(mu − mc)(mu − mt)(mc − mt) VCKM = U23Uδ U13UδU12 (3) ×(md − ms)(md − mb)(ms − mb) In this parametrization large phases appear in Vtd In second order in the weak interactions there and Vub which are small matrix elements as far as their absolute value is concerned. is the possibility of flavour oszillations. The box- diagramms shown in fig. 2 can mediate transitions The unitarity of VCKM is usually depicted by 0 0 the unitarity triangle. There are in total six uni- between Bd ↔ Bd, Bs ↔ Bs, K ↔ K and also 0 tarity relations (aside from the normalization re- between D) ↔ B lations for the rows and columns), which may be For later use we note that the mixing amplitude depicted as traingles in the complex plane. How- for Bd − Bd mixing is proportional to the phase ever, due to the hierarchy of the CKM matrix factor exp(2iβ), while the phase in Bs−Bs mixing elements there are only two triangles with com- is negligibly small in the standard model. In the parable sides which coincide to leading order in system, the short distance contribution to Theory and Phenomenology of CP Violation 3

naive counting leads to the following estimate of the of the

2 2 αs GF mt 3 −32 de ∼ e 2 2 2 Im∆ µ ∼ 10 e cm (8) π (16π ) MW

2 2 where the factor mt /MW originates from the GIM suppression, ∆ encodes the necesary CKM Figure 3. leading to an electric factors and µ is a typical hadronic scale which we dipole moment for the up quark. set to µ = 300 MeV. This has to be compared to the current experimental limit which is [4] mixing is also proportional to exp(2iβ), while the mixing in the up quark sector (D − D mixing) is d ≤ 3.0 × 10−26e cm heavily GIM suppressed. exp The Flavour structure and the pattern of CP Similar statements hold for other flavour diago- violation in the Standard Model is quite peculiar. nal and CP violating observables, and from the Here I list a few of these peculiarities: experimental side this strong suppression of these Strong CP violation: effects is supported. The QCD sector of the standard model can also Strong Suppression of CP violation in contain a CP violating interaction of the form the up-quark sector:

αs n o The pattern of mixing and CP violation in the L = θ Tr E~ · B~ (7) strong CP π standard model is determined by the GIM mech- anism. This means in particular, that the effects where E~ and B~ are the chromo-electric and the for the up, charm and the are severely chromo-magnetic field strenghts. The electric suppressed, since the mass differences between dipole moment of the neutron tells us that the the down-type quarks are small compared to the coupling θ has to be extremely small. It has been weak boson masses. Furthermore, the mixing an- suggested by Pecchei and Quinn [5] to introduce gles of the first and second generations into the an additional to explain θ = 0, but in third are small, which means that charm physics these scenarios a new light pseudoscalar particle, is basically a “two family” problem. Since CP the so-called appears for which we do not violation with only two families is not possible have any experimental evidence. It is fair to say through the CKM mechanism, this results in a that the strong CP problem is still unsolved. strong suppression of CP violation in the charm Flavour Diagonal CP violation: sector. Again this fact seems to be supported by experiment. One of the most peculiar features of CP viola- tion in the standard model is the enormous sup- 3. Phenomenology of CP Violation pression of flavour diagonal CP violation. Assum- ing that the CKM matrix is the only source of CP In the following we shall concentrate on CP violation, it is easy to see that an electric dipole violation in the decays of particles. In general, moment of a quark can be induced only at the decays are mediated by an effective interaction two loop level at least, evaluating a diagram as consisting of a sum of local operators Oi the one shown in fig. 3. n It has been shown by Shabalin [3] that the sum GF X Heff = √ CiOi (9) of all the two loop diagrams lead to a vanishing 2 electric dipole moment for the quarks, and a non- i=1 vanishing contribution requires at least one more where in the standard model Ci are coefficients loop, which can be an additional gluon. Hence containing the CKM factors (including possible 4 Thomas Mannel phases) and the contributions from the QCD run- Flavour : ning from the weak scale down to the typical scale or more genrerally flavour SU(3) may of process. Furthermore, in the standard model be used to relate matrix elements for different we have n = 10, where the operators i = 1, 2 processes [6]. While isospin is believed to be a are called tree operators, i = 3, ..6 are the QCD good symmetry which is broken only by electro- penguins and i = 7, ...10 are the electroweak pen- magnetism and the mass difference between the guins. up and the down quark, the situation is worse for CP violation occurs through complex phases of U- and V-spin which are the two other possible the C in the following way. Consider an ampli- i choices of the SU(2) subgroup of SU(3), since the tude for a decay of a B into some final mass of the is significantly higher state f, with two contributions than the one of the up and the down quark.

A(B → f) = λ1a1 + λ2a2 (10) The breaking of flavour SU(3) is not eas- ily quantified. only for factorizable amplitudes where the λi are complex coupling constants and SU(3) breaking is included by a factor of fπ/fK ai are hadronic matrix elements hB|Oi|fi. The which also give the approximate size of the effects amplitude of the CP conjugate process is obtained which have to be expected. However, the intrinsic by conjugating the couplings λi, while the phases uncertainties emerging from using flavour symme- of the the hadronic matrix elements remain the tries are hard to estimate. same due to CP invariance of strong interactions. Flavour symmetry arguments are often supple- The CP asymmetry is given by mented by arguments based on diagram topolo- gies: A rearrangement of color indices will result ACP ∝ Γ(B → f) − Γ(B → f) in a color-suppression factor 1/N = 1/3, an an- = 2 Im[λ λ∗] Im[a a∗] (11) C 1 2 1 2 nihilation topology (i.e. a diagram where two Thus a CP asymmetry requires imaginary parts quarks in a meson have to annihilate) results in a of the coupling constants, but also a strong phase suppression by a factor fM /mM , where fM and difference between the hadronic matrix elements. mM are the decay constant and the mass of the The neutral flavoured can decay into a meson M. Furthermore, a contraction of quark CP eigenstate. In this case we can have a strong lines involving a quark loop (penguin contraction) phase difference from a different origin. Since the involves a loop factor, the perturbative value of two mass eigenstates in the neutral flavoured me- which is 1/(16π2) but is generally expected to be 0 0 son systems (Bd, Bs, K and D ) have differ- between this value and unity. ent mass eigenvalues, the evolution creates This approach, which is completely model inde- a phase such that pendent, allows us to have some semi-quantitative insight into rates and CP , however, Im[a a∗] ∼ sin(∆m t) (12) 1 2 in many applications the uncertainties of the or- where ∆m is the mass difference between the two der of tens of percent are hard to estimate. eigenstates.Thus there will be a time dependent CP asymmetry which takes the form QCD Factorization and SCET: C cos(∆m t) − S sin(∆m t) It has recently been pointed out that in the A (t) = (13) CP cosh(∆Γ t/2) + D sinh(∆Γ t/2) limit of infinite heavy quark mass one may fac- torize the amplitudes of exclusive non-leptonic where the coefficients satisfy C2 + S2 + D2 = 1. two body modes as sketched in fig. 4[7]. Thus in The problem in obtaining quantitative predic- the infinite mass limit the amplitude can be ex- tions for the CP asymmetries lies in our inabil- pressed in terms of a non-perturbative (soft) form ity to perform a first principles calculation of the factor, the non-pertutbative wave functions of hadronic matrix elements. Thus we have to re- the hadrons and a perturbatively calculable hard fer to approximate methods which fall into three scattering kernel T . This limit may be formulated classes which I shall briefly describe. Theory and Phenomenology of CP Violation 5

cases precise predictions are possible.

4. CP Violation and “New Physics” Currently there is no convincing idea which ex- plains flavour. Even in grand unified theories flavour is usually implemented by triplication of the spectrum, which will not give any explana- tion. Figure 4. Sketch of QCD factorization Most scenarios of “new physics” involve addi- as an effective field theory, the soft collinear ef- tional degrees of freedom, in particular, many fective field theory (SCET) . models involve an extended Higgs sector. This The most interesting observation from QCD in general implies additional couplings which may factorization is that the strong phases in non- carry irreducible phases, i.e. to additional sources leptonic heavy hardon decays are perturbatively of CP violation. This additional CP violation calculable to leading order, and thus are predicted generically also appears in flavour diagonal pro- to be small: Strong phases are either of order cesses, in contradiction with the observed sup- αs(m) or suppressed by powers of 1/m. However, pression of the e.g. electric dipole moments of the quantitative agreement with data is still not particles. Furthermore, flavour changing neutral good, indicating sizable corrections of subleading currents may appear, in some models even at tree orders in 1/m. Although the number of sublead- level. ing non-perturbative parameters is large, QCD The pattern of CP violation and flavour factorization still provides a systematic approach parametrized by the CKM Ansatz is very special to exclusive non-leptonic decays. and in complete agreement with the observations in . For this reason, scenarios QCD (Light Cone) Sum Rules: of “new physics” are thus formulated often with QCD sum rules are a well established method “minimal flavour violation” (MFV) which means for the estimate of hadronic matrix elements. that any flavour structure can be reduced to the They rely on parton hadron duality and on the Yukawa couplings and the CKM matrix. This analyticity of the amplitudes. For the case of makes these models consistent with present ob- two-body non-leptonic B decays involving light servations but does not explain the phenomenon mesons in the final state the so-called light cone of flavour. sum rules are applied [8], where one of the light REFERENCES mesons is represented by its light cone wave func- tion, while the decaying meson and the other light 1. A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. meson is interpolated by appropriate currents. 5 (1967) 32. This type of sum rules has been formulated re- 2. C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985). cently directly in SCET. 3. E. P. Shabalin, Sov. J. Nucl. Phys. 28, 75 It is interesting to note that the results from (1978). light-cone QCD sum rules quantitatively agree 4. C. A. Baker et al., arXiv:hep-ex/0602020. with the ones from QCD factorization. In par- 5. R. D. Peccei and H. R. Quinn, Phys. Rev. ticular, the statement about weak phases is the Lett. 38, 1440 (1977). same in QCD sum rules. 6. C. W. Chiang et al., Phys. Rev. D 70, 034020 In summary, the main obstacle to calculate CP (2004). violation in hadronic decays and thus to relate the 7. M. Beneke et al.. Nucl. Phys. B 606, 245 measurements to the fundamental parameters is (2001). the evaluation of the hadronic matrix elements, 8. A. Khodjamirian, Nucl. Phys. B 605, 558 in particular of their weak phases. Only in a few (2001).