Flavour and CP Violation

In this talk I review the status of B mixing in the Standard Model and the room for new physics in B_s and B_d mixing in the light of recent LHCb data.


Introduction
The only source of flavour-changing transition in the Standard Model (SM) is the Yukawa interaction of the Higgs doublet H with the fermion fields. The Yukawa lagrangian of the quark fields reads Here Q j L denotes the SU(2) doublet of the left-handed quark fields of the j-th generation and d j R , u j R are the corresponding right-handed singlet fields. The 3 × 3 Yukawa matrices Y d,u and the 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 rotations of any of Q j L , d j R , and u j R in flavour space. Using these unphysical transformations one can bring Y u and Y d to the form and y i ≥ 0. With the choice adopted in Eq. (2) the mass matrix M u of up, charm, and top quark is diagonal. The diagonalisation of M d requires the additional rotation d j L = V jk d k′ L , which puts the unitary Cabbibbo-Kobayashi-Maskawa (CKM) matrix V into the W boson vertices: W µ u j L γ µ d j L = W µ V jk u j L γ µ d k′ L . Eq. (2) defines a basis of weak eigenstates which is particularly suited to display the smallness of flavour violation in the SM. All flavour-changing transitions originate from Y d which reads 10 −5 −7 · 10 −5 (12 + 6i) · 10 −5 4 · 10 −6 3 · 10 −4 −6 · 10 −4 (2 + 6i) · 10 −8 10 −5 2 · 10 −2   , if all Yukawa couplings are evaluated at the scale µ = m t . The off-diagonal element largest in magnitude is V * ts y b ≡ V * 32 y b = −6 · 10 −4 . Moreover, in the SM flavour-changing neutral current (FCNC) transitions involve an additional loop suppression, making FCNC processes an extremely sensitive probe of physics beyond the SM. This feature puts flavour physics into a win-win position: If CMS and ATLAS find new particles, FCNC observables will be used to probe the flavour pattern of the BSM theory to which these particles belong. If instead CMS and ATLAS do not find any BSM particles, flavour physics will indirectly probe new physics to scales 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 mesons mix with their antiparticles, K, D, B d , and B s , respectively. This means that, say, a B d meson evolves in time into a quantum-mechanical superposition of a B d and a B d . This feature is a gold-mine for new-physics searches, as it permits to probe CP phases in any decay to a final state which is accessible from both the B d and the B d component of the decaying state. In a tree-level decay like ( ) B d → J/ψK S the time-dependent CP asymmetry probes new physics in the mixing amplitude itself. If instead a rare decay (with FCNC decay amplitude) is studied, the time-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 quantum numbers change by two units. In the case of B s − B s mixing depicted in Fig. 1 these flavour quantum 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 the reach of |∆F | = 2 and |∆F | = 1 amplitudes to new physics: The flavour violation in the SM the amplitudes A One sees that B s − B s mixing is more sensitive to generic NP than an FCNC b → s decay amplitude, if |δ FCNC | > |V ts | ≈ 0.04. 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 transitions can be more sensitive probes of NP in models in which b R → s L or b L → s R transitions are parametrically enhanced over their SM counterparts. This situation occurs in the (N)MSSM with a large value of the parameter tan β or large trilinear SUSY-breaking terms. Theories with such chirally enhanced FCNC transitions are efficiently probed with the radiative decays b → sγ, dγ [3][4][5][6][7][8][9][10], leptonic decays like the recently observed [11] decay B s → µ + µ − [7,[12][13][14][15][16][17][18][19][20][21] and the chromomagnetic contribution to non-leptonic decays like B d → φK S [18].
2. B−B mixing: formalism and Standard-Model prediction B q − B q mixing with q = d or q = s is governed by the 2 × 2 matrix M q − iΓ q /2, with the hermitian mass and decay matrices M q and Γ q . Due to non-vanishing off-diagonal elements M q 12 and Γ q 12 a meson tagged as a B q at time t = 0 will for t > 0 evolve into a superposition of B q and B q . In the SM M s 12 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 12 is instead obtained from the absorptive part, meaning that the real part of the loop integral is dropped. M d 12 and Γ d 12 are calculated in the same way from the box diagram with the external s quarks replaced by d quarks. M q 12 is dominated by the top contribution, while Γ q 12 stems from the diagrams with internal up and charm. Γ q 12 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: By diagonalising M −i Γ/2 one finds the two mass eigenstates B H q and B L q , where the superscripts stand for "heavy" and "light". These eigenstates differ in their mass and width with The mass difference ∆m q simply equals the frequency at which B q and B q oscillate into each other. The width difference ∆Γ s is sizable, so that in general untagged B s decays are governed by 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 d L,H [22][23][24][25]. The CP-violating phase φ q can be determined by measuring the CP asymmetry in flavour-specific decays: A decay B q → f is called flavour-specific, if the decay B q → f is forbidden. The standard method to determine a q fs uses semileptonic decays, so that a q fs is often called semileptonic CP asymmetry. While this measurement simply requires to count the numbers of positive and negative leptons in B q decays, it is still very difficult, because |Γ q 12 | ≪ |M q 12 | renders |a q fs | small, even if NP contributions to M q 12 enhance φ q over its small SM value.

Mass difference ∆m s
The theoretical prediction of ∆m q requires the separation of short-distance and long-distance QCD effects. To this end one employs an operator product expansion, which factorises M q 12 as The Wilson coefficient C comprises the short-distance physics, with all dependence on the heavy particle masses. QCD corrections to C have been calculated reliably in perturbation theory [26].
Since the CKM elements are factored out in Eq.
is a local four-quark operator describing a point-like |∆B| = 2 transition, see Fig. 2. The formalism 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 ≃ 2|M s 12 | is [27] ∆m exp s = (17.719 ± 0.043) ps −1 , meaning that |M s 12 | is known very precisely. However, we need the hadronic matrix element B s |Q|B s to confront Eq. (7) with the SM. The latter is usually parametrised as Here M Bs and f Bs are mass and decay constant of the B s meson, respectively. The hadronic parameter B Bs depends on the renormalisation scheme and scale used to define Q, in this talk B Bs ≈ 0.85 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 where the uncertainties from the experimental errors of the input parameters indicated. Averaging various calculations from lattice gauge theory one finds [28] f 2 Inserting this result into Eq.
Therefore more effort on lattice-QCD calculations of f 2 Bs B Bs is highly desirable. As long as modern calculations of this quantity are unavailable, I recommend to inflate the error in Eq. (10) to 2.5 ps −1 . One concludes that the precise measurement in Eq. (7) still permits a NP contribution of 15% in ∆m s . have CP quantum number η CP = 1, while (J/ψ φ) L=1 is CP-odd. Denoting a meson which was a B s at time t = 0 by B s (t) (with an analogous definition of B s (t)), one can define the time-dependent CP asymmetry The two interfering amplitudes giving rise to a CP (t) are shown in Fig. 3. The analytical result reads The CP phase entering a CP (t) is β s = arg(−V * tb V ts /(V * cb V cs )), in the standard phase convention of 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, 1, 2 amplitudes involves a complicated angular analysis of the 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(2β s ) ≃ 2λ 2 η. The parameter η is the height of the CKM unitarity triangle defined by ρ + iη = −V * ub V ud /(V * cb V cd ). The suppression by λ 2 renders sin(2β s ) small; determining the CKM elements from a global fit to the data gives η = 0.343 ± 0.015 and leads to the SM prediction [28] 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] 2β exp Of course, this average is dominated by the LHCb data. At this conference Jeroen van Leerdam has reported 2β s = 0.1 • ± 4.8 • stat ± 1.5 • 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 of order 1 • on 2β exp s in Eq. (15).

Decay matrix Γ q 12
The calculation of Γ q 12 , q = d, s, is needed for the width difference ∆Γ q ≃ 2|Γ q 12 | cos φ q and the semileptonic CP asymmetry a q fs = |Γ q 12 | |M q 12 | sin φ q . The theoretical results of Refs. [22,23,33] have recently been updated to [25] The prediction of Γ q 12 involves new operators in addition to Q in Eq. (6). However, the matrix element of Q comes with the largest coefficient, so that the ratio ∆Γ q /∆m q = |Γ q 12 |/|M q 12 | suffers from smaller hadronic uncertainties than ∆Γ q . Predicting ∆Γ q with the help of the experimental values ∆m exp d = 0.507 ps −1 and ∆m exp s in Eq. (7) one infers from Ref. [25]: The more recent update in Ref. [34] differs from Ref. [25] in two respects: The quoted results are 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 : which is consistent with Eq. (17). In Eqs. (17) and (18) which is 3.9σ off the SM prediction inferred from Eq. (16), Adding a NP contribution φ ∆ d,s to either φ d or φ s in Eq. (16) may enhance |A D0 SL |, but the 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 parametrise NP in B d −B d mixing and B s −B s mixing by two complex parameters ∆ d and ∆ s :  . 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 relevant flavour data to the CKM elements and ∆ d and ∆ s has resulted in a 3.6σ evidence of NP, with a large negative NP phase φ ∆ s [38]. In spring 2012 this fit has been repeated [39]; Figs. 4 and 5 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 comes with a higher value of 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 statistical fluctuations. 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 12 or Γ s 12 [39][40][41]. Since Γ s 12 originates from Cabibbo-favoured tree-level decays, it can hardly be changed in a significant way without spoiling the value of the average decay width Γ s = (Γ s H +Γ s L )/2 [38,39]. The LHCb measurement  . Allowed range for ∆ d , see [28,39] for details.
in excellent agreement with the SM prediction τ Bs /τ B d = 0.998±0.003 [25,42]. NP in the doubly Cabibbo-suppressed quantity Γ d 12 is phenomenologically only poorly constrained, but requires a somewhat contrived model of NP.

A supersymmetric SO(10) model
The Minimal Supersymmetric Standard Model (MSSM) has many new sources of flavour violation, which all reside in the supersymmetry-breaking sector. It is no problem to get a 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 squark massed placed by ATLAS and CMS. Grand Unified Theories (GUT) offer the possibility to have "controlled" deviations from the CKM pattern of flavour violation in the quark sector: In GUTs quarks and leptons are unified in common symmetry multiplets, opening the possibility to 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: If the observed large atmospheric neutrino mixing angle stems from a rotation of 5 2 and 5 3 in flavour space, it will induce a largeb R −s R -mixing. Contrary to the situation with right-handed quark fields, rotations of right-handed squark fields in flavour space are physical because of the soft SUSY-breaking terms. The Chang-Masiero-Murayama (CMM) model has implemented this idea in a GUT based on the symmetry breaking chain SO(10)→ SU(5) → SU(3)×SU(2) L ×U(1) Y [45][46][47]. In some weak basis the Yukawa matrix of down (s)quarks is diagonalised as and the right-handed down squark mass matrix has the following diagonal form: The real parameter ∆d is calculated from renormalisation-group effects driven by the top-Yukawa coupling.
Rotating Y d to diagonal form puts the large atmospheric neutrino mixing angle into m 2 d : 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 involves seven parameters, which we have fixed by choosing values for the squark masses Mũ, Md of right-handed up and down squarks, the trilinear term a d 1 of the first generation, the gluino mass mg 3 , 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 and the lower bound on the lightest Higgs boson. From these inputs first universal SUSY-breaking terms defined at a fundamental scale near the Planck scale have been determined through the renormalisation group equations (RGE). Subsequently we have used the RGE to determine all low-energy parameters, with the MSSM as the low-energy theory.
Two experimental results of the year 2012 put the CMM model under pressure: First, the sizable neutrino mixing angle θ 13 leads to an unduly large effect in B(µ → eγ). In Eq. (24) I have tacitly assumed a U PMNS with tri-bimaximal mixing, corresponding to θ 13 = 0. For the actual value θ 13 ≈ 8 • the (1, 2) element of the charged slepton mass matrix gets large, and one has to resort to much larger sfermion masses than those considered in Ref. [47]. Second, the Higgs mass of 126 GeV challenges the model and we are unable to find parameters which simultaneously satisfy the Higgs mass constraint and the experimental upper bound on B(µ → eγ) [48]. The problem with the Higgs mass could be circumvented by considering the NMSSM as the lowenergy 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 matrix elements involved. Theoretical uncertainties still permit an O(20%) NP contribution to the B s −B s amplitude M s 12 . While in 2010 the DØ result for A SL could be explained in scenarios with NP only in M d,s 12 , the LHCb data on B s → J/ψφ now prohibit this solution. An alternative explanation invoking new physics in Γ s 12 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 12 , although this possibility leads to somewhat contrived models. Models of GUT flavour physics with b → s transition driven by the atmospheric neutrino mixing angle are under pressure from the large value of θ 13 , which induces a too large B(µ → eγ). Moreover, in the studied CMM model it seems impossible to accomodate the measured value of the lightest Higgs mass, if one insists on the MSSM as the low-energy theory.