CP violation in $b$ hadrons at LHCb

The most recent results on CP violation in $b$ hadrons obtained by the LHCb Collaboration with Run I and years 2015-2016 of Run II are reviewed. The different types of violation are covered by the studies presented in this paper.


Introduction
The violation of the CP symmetry is predicted in the Standard Model (SM) weak sector and closely related to the existence of at least three flavour families. However, other sources of CP violation are likely to intervene to explain the excess of matter over antimatter in the universe. For what concerns the quarks, the SM-based CP violating phase is embedded in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2]: involving couplings within and between quark generations, because the eigenstates of the weak interaction are different from the mass eigenstates. The magnitude of the CKM elements is not evenly distributed, though: the modules of the diagonal elements (within-generation couplings) are one while the strength decreases as one departs from the diagonal. The weakest couplings are thus between the first and third generation, namely V td and V ub . The unitarity condition, V CKM V † CKM = 1, imposes six independent relations, of which one is of particular interest, V ud V * ub + V cd V * cb + V td V * tb = 0, involving a sum of elements of similar magnitude and defining the "B 0 triangle" or Unitarity Triangle , and a second one, V us V * ub + V cs V * cb + V ts V * tb = 0, although being an unbalanced squeezed triangle, involves the angle intervening in the B 0 s meson oscillations, β s = arg To probe CP violation, rate asymmetries of decays B → f are measured, involving amplitudes such as LHCb research program has focused on CP violation in the decays, in the neutral B mesons mixing, and in the interference of decays and mixing. Recent results spanning the three categories are reviewed in this paper.

CP violation in mixing
Neutral B mesons oscillate between B 0 q and B 0 q states (q = d, s) 1 . The oscillation dynamics can be written as: where M q ij and Γ q ij are elements of the mass and decay matrices. We note φ q 12 = arg(−M q 12 /Γ q 12 ) and φ q M = arg(M q 12 ). The eigenstates of the Hamiltonian are noted as |B 0 L,H = p|B 0 ± q|B 0 , where L and H stand for light and heavy, respectively. In the SM, CP violation in mixing φ q 12 is expected to be very small (or said otherwise, q p ≈ 1) and thus an intervention of new heavy particles could be detected through a substantial deviation from the null prediction. For both B 0 and B 0 s , semileptonic (and thus flavour-specific) decays are used to obtain Experimentally, LHCb analyses measure the time-dependent yield asym- metry of the decay states [3,4]: where ∆M q and ∆Γ q are the mass and width differences, A P and A D are the B-meson production asymmetry and the combined decay products detection asymmetry. For B 0 s , the time-dependent term cancels due to the fast oscillations. In the B 0 case, a time-depend fit is necessary.
the LHCb measurements are [3,4]: where the systematic (second) uncertainty is dominated by the size of the control samples used in the determination of the detection asymmetry. These numbers are compatible with the SM-based expectations of orders 10 −4 and 10 −5 , respectively [5].

CP violation in decay
CP violation in decay occurs whenever |A f | 2 = |A f | 2 . This requires flavorspecific B decays and could happen only if A f contains at least two amplitudes with different strong δ i and weak ϕ i phases. In that case, . The extraction of the parameter of interest, the weak phase difference ϕ 1 − ϕ 2 , is thus limited by the knowledge of the strong phase difference. However, various fitting techniques involving ratios of branching fractions and asymmetries are used to circumvent this limitation.

CKM γ angle
The angle γ = arg is the least constrained angle of the Unitarity Triangle and is the subject of extensive LHCb studies involving B → DK decays [6].
The principle of the measurement relies on the use of the interference of is the suppression factor and δ B is the relative strong phase) decays leading to final states of the type B → DX s where D is a charm meson and X s a strange system (K, K * , Kπ, Kπππ, ...). The charm meson decays to a final state accessible by both D and D through allowed (A D ) and suppressed (A D r D e iδ D ) amplitudes. Since more than two decades, several methods have initially been proposed for B + → DK + [7,8,9,10]

CP violation in baryonic decays
A pioneering study has been performed on four-body decays of the Λ 0 [13]. For four-body modes, a useful observable which has been chosen for CP studies is the triple scalar prod- [14]. h 1 = π and h 2 = K for Λ 0 b → pπ − K + K − , and h 1 = h 2 = π for Λ 0 b → pπ − π + π − . In the latter case, the ambiguity for the choice of h 1 is removed by selecting the fastest π − in the Λ 0 b rest frame, π − f ast . CP conjugation applied to CT gives − p p .  Figure 1 shows the invariant mass spectra of the pπ − h + h − systems. Both Λ 0 b → pπ − h + h − signals are observed for the first time, with yields of 6646 ± 105 (h = π) and 1030 ± 56 (h = K) events.
aT -odd CP is then studied as a function of the angle Φ between the planes formed by (p, K − (π − f ast )) and (π − (slow) , K + ), where we note that CT ∝ sin(Φ). In the case of Λ 0 b → pπ − π + π − , substantial deviation from zero is observed, as illustrated in Fig.2, with a combined significance of 3.3σ. This is the first evidence of CP violation in the decay of a baryon. With the current statistics, no significant deviation is seen for

CP violation in interferences between mixing and decay
When neutral B mesons decay to a eigen CP final state, the possibility of oscillation implies that the measured asymmetry probing the interference of mixing and decay must be studied as a function of the decay time: where For the B 0 meson, the denominator is equal to 1, since ∆Γ d /Γ d << 1. Following the discussion in section 2, q p = 1. If only one amplitude contributes to the decay, λ f = e iΦq = e i(Φ M −2Φ D ) (and thus C f = 0), where Φ M and Φ D are the weak mixing and decay phases, respectively. For both B 0 and B 0 s , one can reasonably assumes that Φ M 0. Experimentally, except for the mixing phase simplification, the quantities S f , C f , and ∆M q (and A ∆Γ f , ∆Γ q for B 0 s ) are extracted without prior assumption.

β s and β measurements
The b → ccs tree decays are the reference modes for extracting the CKM angles β = arg − Table 1 shows all the modes used over the past years and the corresponding measurements for Φ s . The most recent one, relying on the decay B 0 s → J/ψ K + K − in the region m(K + K − ) > m φ , is the first with a tensor resonance (f 2 (1525)) dominating the spectrum. With the current statistics, all the measurements agree with recent SM-based fits [20], −0.0365 +0.0013 −0.0012 .

Two body B 0 (s) → hh
The two body B 0 (s) decays involve tree and loop diagram contributions, Fig. 3. The extraction of any weak phase, here 2β s or γ, requires some assumptions, such as U-spin symmetry [18], which inexactness limits the accuracy of the results. A recent simultaneous fit of the four channels B 0 → π + π − , B 0 → K + π − , B 0 s → K + K − , and B 0 s → K + π − [19] led to the measurements: C ππ = −0.243 ± 0.069, S ππ = −0.681 ± 0.060, C KK = 0.236 ± 0.062, S KK = 0.216 ± 0.062 and A ∆Γ KK = −0.751 ± 0.075. The notable large values of the C f coefficients are due to a sizeable CP violation in the decay produced by the interference of tree and loop diagrams. Since the significance for (C KK , S KK , A ∆Γ KK ) to differ from (0, 0, 1) is 4.7σ, LHCb establishes strong evidence for CP violation in B 0 s → K + K − .

Summary
Important progress has been done in the measurement of CP violation at LHCb with Run I and beginning of Run II data, leading to a better constraint of the Unitarity Triangle and opening new routes such as CP violation in Baryon decays. But the needed accuracies, for a sensible comparison between indirect and direct determinations of the angles, will only be reached following the LHCb Phase I upgrade during the next decade. As an illustration, the expected Phase I upgrade accuracy for the angle γ is ∼ 1 • , while the indirect constraint, based on loop-level processes, gives γ = (66.9 +0.94 −3.44 ) • [20]. A clear indication of New Physis may show up at this stage.