aa r X i v : . [ h e p - ph ] A p r Flavour and CP Violation
Ulrich Nierste
Institut f¨ur Theoretische TeilchenophysikKarlsruhe Institute of TechnologyEngesserstrae 776131 Karlsruhe, GermanyE-mail: [email protected]
Abstract.
In this talk I review the status of B − B mixing in the Standard Model and theroom for new physics in B s − B s and B d − B d mixing in the light of recent LHCb data.
1. Introduction
The only source of flavour-changing transition in the Standard Model (SM) is the Yukawainteraction of the Higgs doublet H with the fermion fields. The Yukawa lagrangian of the quarkfields reads − L Y = Y djk Q jL H d kR + Y ujk Q jL e H u kR + h.c. (1)Here Q jL denotes the SU(2) doublet of the left-handed quark fields of the j -th generation and d jR , u jR are the corresponding right-handed singlet fields. The 3 × Y d,u andthe Higgs field vacuum expectation value v = 174 GeV combine to the quark mass matrices M d,u = Y d,u v . The gauge interactions of the quark fields do not change under unitary rotationsof any of Q jL , d jR , and u jR in flavour space. Using these unphysical transformations one can bring Y u and Y d to the form Y u = b Y u = y u y c
00 0 y t and Y d = V † b Y d with b Y d = y d y s
00 0 y b (2)and y i ≥
0. With the choice adopted in Eq. (2) the mass matrix M u of up, charm, and topquark is diagonal. The diagonalisation of M d requires the additional rotation d jL = V jk d k ′ L , whichputs the unitary Cabbibbo-Kobayashi-Maskawa (CKM) matrix V into the W boson vertices: W µ u jL γ µ d jL = W µ V jk u jL γ µ d k ′ L . Eq. (2) defines a basis of weak eigenstates which is particularlysuited to display the smallness of flavour violation in the SM. All flavour-changing transitionsoriginate from Y d which reads Y d = V † b Y d = − − · − (12 + 6 i ) · − · − · − − · − (2 + 6 i ) · − − · − , (3) s sbu,c,tu,c,t b st W Figure 1.
Left: SM box diagram describing B s − B s mixing, a | ∆ B | = 2 process. Right: b → s penguin amplitude, a | ∆ B | = 1 transition.if all Yukawa couplings are evaluated at the scale µ = m t . The off-diagonal element largestin magnitude is V ∗ ts y b ≡ V ∗ y b = − · − . Moreover, in the SM flavour-changing neutralcurrent (FCNC) transitions involve an additional loop suppression, making FCNC processes anextremely sensitive probe of physics beyond the SM. This feature puts flavour physics into awin-win position: If CMS and ATLAS find new particles, FCNC observables will be used toprobe the flavour pattern of the BSM theory to which these particles belong. If instead CMSand ATLAS do not find any BSM particles, flavour physics will indirectly probe new physics toscales exceeding 100 TeV, well beyond the center-of-mass energy of the LHC.The weak interaction makes the neutral K ∼ sd , D ∼ cu , B d ∼ bd and B s ∼ bs mesonsmix with their antiparticles, K , D , B d , and B s , respectively. This means that, say, a B d mesonevolves in time into a quantum-mechanical superposition of a B d and a B d . This feature is agold-mine for new-physics searches, as it permits to probe CP phases in any decay to a finalstate which is accessible from both the B d and the B d component of the decaying state. In atree-level decay like ( ) B d → J/ψK S the time-dependent CP asymmetry probes new physics in themixing amplitude itself. If instead a rare decay (with FCNC decay amplitude) is studied, thetime-dependent CP asymmetry may reveal a new physics contribution to the decay amplitude.Meson-antimeson mixing is a | ∆ F | = 2 process, meaning that the relevant flavour quantumnumbers change by two units. In the case of B s − B s mixing depicted in Fig. 1 these flavourquantum numbers are beauty B and strangeness S . Weak decays are | ∆ F | = 1 processes,Fig. 1 shows a | ∆ B | = | ∆ S | = 1 FCNC decay amplitude. It is instructive to compare thereach of | ∆ F | = 2 and | ∆ F | = 1 amplitudes to new physics: The flavour violation in the SMthe amplitudes A | ∆ B | =2SM and A | ∆ B | =1SM in Fig. 1 is governed by the small CKM element V ts andboth diagrams scale as 1 /M W . A new-physics (NP) contribution may instead involve some newflavour-violating parameter δ FCNC . If the scale of NP is Λ and the NP contribution enters atthe one-loop level, one finds | A | ∆ B | =2NP || A | ∆ B | =2SM | = | δ FCNC | | V ts | M W Λ and | A | ∆ B | =1NP || A | ∆ B | =1SM | = | δ FCNC || V ts | M W Λ . (4)One sees that B s − B s mixing is more sensitive to generic NP than an FCNC b → s decayamplitude, if | δ FCNC | > | V ts | ≈ .
04. The estimate in Eq. (4) applies, for example, to the(N)MSSM with generic flavour structure in the bilinear SUSY-breaking terms, which induce new b R → s R or b L → s L transitions through squark-gluino loops [1, 2]. But | ∆ B | = 1 transitionscan be more sensitive probes of NP in models in which b R → s L or b L → s R transitions areparametrically enhanced over their SM counterparts. This situation occurs in the (N)MSSMwith a large value of the parameter tan β or large trilinear SUSY-breaking terms. Theorieswith such chirally enhanced FCNC transitions are efficiently probed with the radiative decays b → sγ, dγ [3–10], leptonic decays like the recently observed [11] decay B s → µ + µ − [7, 12–21]and the chromomagnetic contribution to non-leptonic decays like B d → φK S [18]. . B − B mixing: formalism and Standard-Model prediction B q − B q mixing with q = d or q = s is governed by the 2 × M q − i Γ q /
2, with thehermitian mass and decay matrices M q and Γ q . Due to non-vanishing off-diagonal elements M q and Γ q a meson tagged as a B q at time t = 0 will for t > B q and B q . In the SM M s is calculated from the dispersive part of the box diagram in Fig. 1,which amounts to discarding the imaginary part of the loop integral. Γ s is instead obtainedfrom the absorptive part, meaning that the real part of the loop integral is dropped. M d andΓ d are calculated in the same way from the box diagram with the external s quarks replaced by d quarks. M q is dominated by the top contribution, while Γ q stems from the diagrams withinternal up and charm. Γ q is made up of all final states into which both B q and B q can decay.There are three physical quantities in B q − B q mixing: | M q | , | Γ q | , φ q ≡ arg (cid:18) − M q Γ q (cid:19) . By diagonalising M − i Γ / B Hq and B Lq , where the superscriptsstand for “heavy” and “light”. These eigenstates differ in their mass and width with∆ m q = M qH − M qL ≃ | M q | , ∆Γ q = Γ qL − Γ qH ≃ | Γ q | cos φ q . The mass difference ∆ m q simply equals the frequency at which B q and B q oscillate into eachother. The width difference ∆Γ s is sizable, so that in general untagged B s decays are governedby the sum of two exponentials. There is no useful data on the tiny width difference ∆Γ d in the B d system yet, which is predicted to be around 0 .
5% of Γ dL,H [22–25]. The CP-violating phase φ q can be determined by measuring the CP asymmetry in flavour-specific decays: a q fs = | Γ q || M q | sin φ q . A decay B q → f is called flavour-specific, if the decay B q → f is forbidden. The standardmethod to determine a q fs uses semileptonic decays, so that a q fs is often called semileptonic CPasymmetry. While this measurement simply requires to count the numbers of positive andnegative leptons in B q decays, it is still very difficult, because | Γ q | ≪ | M q | renders | a q fs | small,even if NP contributions to M q enhance φ q over its small SM value. ∆ m s The theoretical prediction of ∆ m q requires the separation of short-distance and long-distanceQCD effects. To this end one employs an operator product expansion, which factorises M q as M q = (cid:0) V ∗ tq V tb (cid:1) C h B q | Q | B q i . (5)The Wilson coefficient C comprises the short-distance physics, with all dependence on the heavyparticle masses. QCD corrections to C have been calculated reliably in perturbation theory [26].Since the CKM elements are factored out in Eq. (5), C = C ( m t , M W , α s ) is real in the SM. Q = q L γ ν b L q L γ ν b L (6)is a local four-quark operator describing a point-like | ∆ B | = 2 transition, see Fig. 2. Theformalism is the same for the B d and B s complex; for definiteness I only discuss the case q = s in the following. The average of the CDF and LHCb measurements of ∆ m s ≃ | M s | is [27]∆ m exp s = (17 . ± . − , (7) q qb Figure 2.
The | ∆ B | = 2 operator Q of Eq. (6). It is pictorially obtained by shrinking theB q − B q mixing box diagram to a point.meaning that | M s | is known very precisely. However, we need the hadronic matrix element h B s | Q | B s i to confront Eq. (7) with the SM. The latter is usually parametrised as h B s | Q | B s i = 23 M B s f B s B B s . Here M B s and f B s are mass and decay constant of the B s meson, respectively. The hadronicparameter B B s depends on the renormalisation scheme and scale used to define Q , in this talk B B s ≈ .
85 is understood to be evaluated at the scale µ = m b in the MS scheme. Noting that | V ts | is fixed by CKM unitarity from the well-measured element | V cb | , we can write∆ m s = (cid:16) . ± . V cb ± . m t ± . α s (cid:17) ps − f B s B B s (220 MeV) , (8)where the uncertainties from the experimental errors of the input parameters indicated.Averaging various calculations from lattice gauge theory one finds [28] f B s B B s = [(211 ±
9) MeV] . (9)Inserting this result into Eq. (8) implies∆ m s = (17 . ± .
5) ps − (10)complying excellently with the experimental result in Eq. (7). However, the average in Eq. (9)mainly involves different calculations of f B s , which are combined with two fairly old results for B B s . With the recent preliminary Fermilab/MILC result [29] f B s B B s = 0 . ≃ [(237 ±
14) MeV] one finds instead ∆ m s = (21 . ± .
6) ps − . (11)Therefore more effort on lattice-QCD calculations of f B s B B s is highly desirable. As longas modern calculations of this quantity are unavailable, I recommend to inflate the error inEq. (10) to 2 . − . One concludes that the precise measurement in Eq. (7) still permits a NPcontribution of 15% in ∆ m s . B s → J/ψφ
While ∆ m s fixes the magnitude of M s , the phase of M s can be probed through the mixing-induced CP asymmetry in B s → J/ψφ . The final state contains two vector mesons, so that theorbital angular momentum L can assume the values 0,1, and 2. The final states ( J/ψ φ ) L =0 , s csb cB s J/ψφ (cid:2) b ccssB s J/ψφ
Figure 3.
Amplitudes of B s → J/ψ φ and B s → J/ψ φ . The cross denotes the W -mediated b → ccs decay.have CP quantum number η CP = 1, while ( J/ψ φ ) L =1 is CP-odd. Denoting a meson whichwas a B s at time t = 0 by B s ( t ) (with an analogous definition of B s ( t )), one can define thetime-dependent CP asymmetry a CP ( t ) = Γ( B s ( t ) → ( J/ψ φ ) L ) − Γ( B s ( t ) → ( J/ψ φ ) L )Γ( B s ( t ) → ( J/ψ φ ) L ) + Γ( B s ( t ) → ( J/ψ φ ) L ) . (12)The two interfering amplitudes giving rise to a CP ( t ) are shown in Fig. 3. The analytical resultreads a CP ( t ) = η CP sin(2 β s ) sin(∆ m s t )cosh(∆Γ s t/ − cos(2 β s ) sinh(∆Γ s t/ . (13)The CP phase entering a CP ( t ) is β s = arg( − V ∗ tb V ts / ( V ∗ cb V cs )), in the standard phase conventionof the CKM matrix β s is essentially just the phase of − V ts . It should be mentioned that Eq. (12)is a theoretical definition, in practice the experimental separation of the L = 0 , , J/ψ and φ decays.The CKM matrix can be parametrised in terms of the four Wolfenstein parameters λ , A , ρ , and η [30,31]. Expanding to leading order in λ = 0 .
225 one finds sin(2 β s ) ≃ λ η . The parameter η isthe height of the CKM unitarity triangle defined by ρ + iη = − V ∗ ub V ud / ( V ∗ cb V cd ). The suppressionby λ renders sin(2 β s ) small; determining the CKM elements from a global fit to the data gives η = 0 . ± .
015 and leads to the SM prediction [28]2 β s = 2 . ◦ ± . ◦ . (14)The combination of CDF, D0, ATLAS, and LHCb data on B s → J/ψφ and of LHCb data on B s → J/ψπ + π − gives [27] 2 β exp s = 0 . ◦ +5 . ◦ − . ◦ . (15)Of course, this average is dominated by the LHCb data. At this conference Jeroen van Leerdamhas reported 2 β s = 0 . ◦ ± . ◦ stat ± . ◦ syst from the combined LHCb analysis of B s → J/ψK + K − and B s → J/ψπ + π − [32]. In Eq. (13) an additional small contribution, the “penguin pollution”has been neglected. This –presently uncalculable– contribution inflicts an additional error oforder 1 ◦ on 2 β exp s in Eq. (15). .3. Decay matrix Γ q The calculation of Γ q , q = d, s , is needed for the width difference ∆Γ q ≃ | Γ q | cos φ q and thesemileptonic CP asymmetry a q fs = | Γ q || M q | sin φ q . The theoretical results of Refs. [22, 23, 33] haverecently been updated to [25] φ s = 0 . ◦ ± . ◦ , φ d = − . ◦ ± . ◦ ,a s fs = (1 . ± . · − , and a d fs = − (4 . ± . · − . (16)The prediction of Γ q involves new operators in addition to Q in Eq. (6). However, the matrixelement of Q comes with the largest coefficient, so that the ratio ∆Γ q / ∆ m q = | Γ q | / | M q | suffersfrom smaller hadronic uncertainties than ∆Γ q . Predicting ∆Γ q with the help of the experimentalvalues ∆ m exp d = 0 .
507 ps − and ∆ m exp s in Eq. (7) one infers from Ref. [25]:∆Γ d = ∆Γ d ∆ m d ∆ m exp d = (27 ± · − ps − ∆Γ s = ∆Γ s ∆ m s ∆ m exp s = (0 . ± . − (17)The more recent update in Ref. [34] differs from Ref. [25] in two respects: The quoted resultsare obtained in a different renormalisation scheme and use the preliminary lattice results of [29]instead of the averages from [28]. Ref. [34] finds for ∆Γ s :∆Γ s = ∆Γ s ∆ m s ∆ m exp s = (0 . ± . − , (18)which is consistent with Eq. (17). In Eqs. (17) and (18) all uncertainties are added in quadrature,which may not be a conservative estimate of the overall error. The ranges comply with the LHCbmeasurement [32] ∆Γ LHCb s = [0 . ± . stat ± . syst ] ps − (19)and the world average [27]: ∆Γ exp s = [0 . ± . − . (20)
3. New physics in B − B mixing
If some NP amplitude adds to the B q − B q mixing box diagram, both | M q | and φ q may deviatefrom their SM predictions. The DØ experiment has measured [35–37] A D0SL = (0 . ± . a d fs + (0 . ± . a s fs = ( − . ± . ± . · − , (21)which is 3 . σ off the SM prediction inferred from Eq. (16), A SL = ( − . ± . · − . (22)Adding a NP contribution φ ∆ d,s to either φ d or φ s in Eq. (16) may enhance | A D0SL | , butthe very same contribution will also affect the value of β s in Eq. (15) and the value of β = arg = arg( − V tb V ∗ td / ( V cb V ∗ cd )) found from a CP ( t )( B d → J/ψK S ). One can parametriseNP in B d − B d mixing and B s − B s mixing by two complex parameters ∆ d and ∆ s : M q ≡ M SM , q12 · ∆ q , ∆ q ≡ | ∆ q | e iφ ∆ q . s (B SL ) & a d (B SL & a SL A ) f ψ (J/ s τ ) & - K + (K s τ & FSs τ & s Γ ∆ s m ∆ & d m ∆ s β -2 s ∆ φ SM point s ∆ Re -2 -1 0 1 2 3 s ∆ I m -2-1012 excluded area has CL > 0.68ICHEP 2012 CKM f i t t e r mixing s B - s New Physics in B
Figure 4.
Allowed range for ∆ s , see [28, 39] for details.In summer 2010, before the advent of precision data from LHCb, a global fit of all relevantflavour data to the CKM elements and ∆ d and ∆ s has resulted in a 3 . σ evidence of NP, witha large negative NP phase φ ∆ s [38]. In spring 2012 this fit has been repeated [39]; Figs. 4 and5 show an update of the results in Ref. [39] with the data of 2012 summer conferences [28].The fit tries to accomodate A SL in Eq. (21) and a slightly high value of the world average for B ( B → τ ν ) through φ ∆ d <
0. Since a CP ( t )( B d → J/ψK S ) precisely fixes 2 β + φ ∆ d = 42 . ◦ ± . ◦ , φ ∆ d < β than in the SM. ( β > β SM entails a larger value of | V ub | which governs B ( B → τ ν ).) Contrary to the situation in 2010, the Standard Model point∆ s = ∆ d = 1 is merely disfavoured by 1 standard deviation, consistent with natural statisticalfluctuations. At the best-fit point the problem with A SL is only marginally alleviated.One could relax the problem with A SL without affecting a CP ( t )( B d → J/ψK S ) and a CP ( t )( B s → J/ψφ ) by postulating new physics in Γ d or Γ s [39–41]. Since Γ s originatesfrom Cabibbo-favoured tree-level decays, it can hardly be changed in a significant way withoutspoiling the value of the average decay width Γ s = (Γ sH +Γ sL ) / xp α ) s (B SL ) & a d (B SL & a SL A s m ∆ & d m ∆ SM point ) d β +2 d ∆ φ sin( )>0 d β +2 d ∆ φ cos( d ∆ Re -2 -1 0 1 2 3 d ∆ I m -2-1012 excluded area has CL > 0.68ICHEP 2012 CKM f i t t e r mixing d B - d New Physics in B
Figure 5.
Allowed range for ∆ d , see [28, 39] for details.Γ LHCb s = (0 . ± . ± . − [32] impliesΓ d Γ s = τ B s τ B d = 0 . ± .
013 (23)in excellent agreement with the SM prediction τ B s /τ B d = 0 . ± .
003 [25,42]. NP in the doublyCabibbo-suppressed quantity Γ d is phenomenologically only poorly constrained, but requires asomewhat contrived model of NP. The Minimal Supersymmetric Standard Model (MSSM) has many new sources of flavourviolation, which all reside in the supersymmetry-breaking sector. It is no problem to geta big effect in a chosen FCNC process, but rather to suppress big effects elsewhere. This supersymmetric flavour problem is substantially alleviated with the lower bounds on the squarkmassed placed by ATLAS and CMS. Grand Unified Theories (GUT) offer the possibility toave “controlled” deviations from the CKM pattern of flavour violation in the quark sector: InGUTs quarks and leptons are unified in common symmetry multiplets, opening the possibilityto observe the large lepton-flavour mixing encoded in the Pontecorvo-Maki-Nakagawa-Sakata(PMNS) matrix U PMNS in quark flavour physics [43, 44]. Consider SU(5) multiplets: = d cR d cR d cR e L − ν e , = s cR s cR s cR µ L − ν µ , = b cR b cR b cR τ L − ν τ . If the observed large atmospheric neutrino mixing angle stems from a rotation of and in flavour space, it will induce a large ˜ b R − ˜ s R -mixing. Contrary to the situation withright-handed quark fields, rotations of right-handed squark fields in flavour space are physicalbecause of the soft SUSY-breaking terms. The Chang–Masiero–Murayama (CMM) model hasimplemented this idea in a GUT based on the symmetry breaking chain SO(10) → SU(5) → SU(3) × SU(2) L × U(1) Y [45–47]. In some weak basis the Yukawa matrix of down (s)quarks isdiagonalised as Y d = V ∗ CKM y d y s
00 0 y b U PMNS and the right-handed down squark mass matrix has the following diagonal form: m d ( M Z ) = diag (cid:16) m d , m d , m d − ∆ ˜ d (cid:17) . The real parameter ∆ ˜ d is calculated from renormalisation-group effects driven by the top-Yukawacoupling.Rotating Y d to diagonal form puts the large atmospheric neutrino mixing angle into m d : U † PMNS m d U PMNS = m d m d − ∆ ˜ d − ∆ ˜ d e iξ − ∆ ˜ d e − iξ m d − ∆ ˜ d (24)The CP phase ξ affects CP violation in B s − B s mixing!In Ref. [47] we have confronted the CMM model with flavour data. The analysis involvesseven parameters, which we have fixed by choosing values for the squark masses M ˜ u , M ˜ d ofright-handed up and down squarks, the trilinear term a d of the first generation, the gluinomass m ˜ g , tan β , and the sign of the higgsino mass parameter µ . We have considered B s − B s mixing, b → sγ , τ → µγ , vacuum stability bounds, lower bounds on sparticle masses andthe lower bound on the lightest Higgs boson. From these inputs first universal SUSY-breakingterms defined at a fundamental scale near the Planck scale have been determined through therenormalisation group equations (RGE). Subsequently we have used the RGE to determine alllow-energy parameters, with the MSSM as the low-energy theory.Two experimental results of the year 2012 put the CMM model under pressure: First, thesizable neutrino mixing angle θ leads to an unduly large effect in B ( µ → eγ ). In Eq. (24) I havetacitly assumed a U PMNS with tri-bimaximal mixing, corresponding to θ = 0. For the actualvalue θ ≈ ◦ the (1 ,
2) element of the charged slepton mass matrix gets large, and one has toresort to much larger sfermion masses than those considered in Ref. [47]. Second, the Higgs massof 126 GeV challenges the model and we are unable to find parameters which simultaneouslysatisfy the Higgs mass constraint and the experimental upper bound on B ( µ → eγ ) [48]. Theproblem with the Higgs mass could be circumvented by considering the NMSSM as the low-energy theory. . Conclusions LHCb has provided us with a significantly better insight into the B s − B s mixing complex.∆ m s and ∆Γ s comply with the SM, but we need better lattice data for the hadronic matrixelements involved. Theoretical uncertainties still permit an O (20%) NP contribution to theB s − B s amplitude M s . While in 2010 the DØ result for A SL could be explained in scenarioswith NP only in M d,s , the LHCb data on B s → J/ψφ now prohibit this solution. An alternativeexplanation invoking new physics in Γ s is not viable, because this will spoil the ratio Γ s / Γ d which agrees well with the SM prediction. Maybe it is worthwile to look at NP in Γ d , althoughthis possibility leads to somewhat contrived models. Models of GUT flavour physics with e b → e s transition driven by the atmospheric neutrino mixing angle are under pressure from the largevalue of θ , which induces a too large B ( µ → eγ ). Moreover, in the studied CMM model itseems impossible to accomodate the measured value of the lightest Higgs mass, if one insists onthe MSSM as the low-energy theory. Acknowledgments
I am grateful for the invitation to this stimulating and enjoyable conference! The work presentedin this talk is supported by BMBF grant 05H12VKF.
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