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 ReviewB meson decaysMarina Artuso1 1 Department of Physics, Syracuse University, Syracuse, NY 13244, USA 2 School of Physics, University of Melbourne, Victoria 3010, Australia
PMC Physics A 2009, 3:3doi:10.1186/1754-0410-3-3 The electronic version of this article is the complete one and can be found online at: http://www.physmathcentral.com/1754-0410/3/3
©
2009 Stone et al AbstractWe discuss the most important Physics thus far extracted from studies of B meson decays. Measurements of the four CP violating angles accessible in B decay are reviewed as well as direct CP violation. A detailed discussion of the measurements of the CKM elements Vcb and Vub from semileptonic decays is given, and the differences between resulting values using inclusive decays versus exclusive decays is discussed. Measurements of "rare" decays are also reviewed. We point out where CP violating and rare decays could lead to observations of physics beyond that of the Standard Model in future experiments. If such physics is found by directly observation of new particles, e.g. in LHC experiments, B decays can play a decisive role in interpreting the nature of these particles. PACS Codes: 13.25.Hw, 14.40.Nd, 14.65.Fy 1 IntroductionThe forces of nature generally reveal their properties by how they act on matter. On the most fundamental material scale that we are aware of matter is formed from fermions. These take two forms, leptonic matter and quark matter. The former do not have any strong interactions, that is they lack a property called "color charge", which allow quarks to bind together either mesonically or baryonically. New and therefore as yet unknown forces could effect both leptons and quarks. Here, we concentrate on how such forces effect quarks, especially the b quark. Light matter consists mostly of u, d and s quarks. In 1963 Cabibbo showed that weak interactions of mesons and baryons containing s quarks were suppressed with respect to those without s quarks by an amount tan θC, where the "Cabibbo" angle θC must be determined experimentally [1]
. The s was further shown to have an important and at that time a mystifying role, by the discovery of CP violation in This was recognized even before the discovery of the charm quark by Kobayashi and Maskawa, who postulated the existence of yet another quark doublet (b, t) [12] , in work for which they were awarded the Nobel Prize in 2008. While the t quark is the heaviest, having a mass of 173 GeV, they are difficult to produce and decay before they can form a hadron, thus excluding many otherwise possible studies. Much interesting work has been done with the s and c quarks, but in this article we discuss the physics of the b quark, which turns out to be the most interesting of the six quarks to study. 1.1 How B's Fit Into the Standard ModelFirst we will discuss how particles formed with b-quarks fit into current paradigm of particle physics, the "Standard Model" (SM) [13-15] . The SM has at its basis the gauge group SU(3)xSU(2)xU(1). The first term corresponds to the strong interaction and SU(3) describes the octet of colored gluons which are the strong force carriers of quantum chromodynamics. SU(2)xU(1) describes the weak interaction and is the product of weak isospin and hypercharge. We speak of the fundamental objects being spin-1/2 quarks and leptons and the force carriers generally spin-1 objects. The spin-0 Higgs boson, yet to be discovered, is necessary for generating mass (to view a cartoon of the mechanism of Higgs mass generation, see [16] ). Particles containing b quarks can be B0, B-, Bs, or Bc mesons, depending on whether the light anti-quark that it pairs with is The quarks come in three repetitions called generations, as do the leptons. The first generation is d u, the second s c and the third b t. In the second and third generations the charge +2/3 quark is heavier than the charge -1/3; the first generation has two very light quarks on the order of a few MeV with the d thought to be a bit heavier. (Isospin invariance is related to the equality of u and d quark masses. The PDG [17] gives the u quark mass between 1.5–3.3 MeV and the d mass between 3.5–6.0 MeV, where the large range indicates the considerable uncertainties.) Decays usually proceed within generations, so the c decays predominantly to the s quark via the quark level process c → W+ s, though some decays do go to the first generation as c → W+ d. The ratio of these amplitudes approximate the Cabibbo angle discussed earlier. The mixing matrix proposed by Kobayshi and Maskawa [12] parameterizes the mixing between the mass eigenstates and weak eigenstates as couplings between the charge +2/3 and -1/3 quarks. We use here the Wolfenstein approximation [18] good to order λ4:
In the Standard Model A, λ, ρ, and η are fundamental constants of nature like G, or αEM; η multiplies the imaginary i and is responsible for all Standard Model CP violation. We know λ = 0.226, A ~0.8 and we have constraints on ρ and η. Often the variables
where the definition in terms of the CKM matrix elements is correct to all orders in λ [19] (see also [20] ). Applying unitarity constraints allows us to construct the six independent triangles shown in Figure 1. Another basis for the CKM matrix are four angles labeled as χ, χ' and any two of α, β and γ since α + β + γ = π [21,22] . (These angles are also shown in Figure 1.) CP violation measurements make use of these angles. (The Belle collaboration defines ϕ2 ≡ α, ϕ1 ≡ β, and ϕ3 ≡ γ.)
B meson decays can occur through various processes. Some decay diagrams with intermediate charged vector bosons are shown in
Figure 2. The simple spectator diagram shown Figure 2(a) has by far the largest rate. Semileptonic decays proceed through this diagram, and allow us to measure the CKM couplings
Vcb and Vub by considering only the hadronic uncertainties due to the spectator quark. The color suppressed diagram Figure 2(b) exists only for hadronic decays. It can occur only when the colors of the quarks from the virtual W- decay match those of the initial B meson. Since this happens only 1/3 of the time in amplitude, the rate is down by almost an order of magnitude from the spectator
decays. The annihilation Figure 2(c) describes the important decay B- → τ-
1.1.1 Dark Matter"Dark Matter" was first shown to exist by Zwicky studying rotation curves of galaxies [23] . The motion could only be explained if there was massive cloud of matter that was not luminous. We still do not know what composes this dark matter, though hopes are it will be discovered at the LHC. An even more mysterious phenomena called "Dark Energy" may also have a connection to particle physics experiments [24] , perhaps via "Extra Dimensions" [25] . 1.1.2 BaryogenesisWhen the Universe began with the Big Bang, there was an equal amount of matter and antimatter. Now we have mostly matter.
How did it happen? Sakharov gave three necessary conditions: Baryon ( These criteria are all satisfied by the Standard Model. 1.1.3 The Hierarchy ProblemOur worry is why the Planck scale at ~1019 GeV is so much higher than the scale at which we expect to find the Higgs Boson, ~100 GeV. As Lisa Randall said [28] "The gist of it is that the universe seems to have two entirely different mass scales, and we don't understand why they are so different. There's what's called the Planck scale, which is associated with gravitational interactions. It's a huge mass scale, but because gravitational forces are proportional to one over the mass squared, that means gravity is a very weak interaction. In units of GeV, which is how we measure masses, the Planck scale is 1019 GeV. Then there's the electroweak scale, which sets the masses for the W and Z bosons. These are particles that are similar to the photons of electromagnetism and which we have observed and studied well. They have a mass of about 100 GeV. So the hierarchy problem, in its simplest manifestation, is how can you have these particles be so light when the other scale is so big." We expect the explanation lies in physics beyond the Standard Model [29] . 1.2 B Decays as Probes for New PhysicsWhen we make measurements on B decays we observe the contributions of SM processes as well as any other processes that may be due to physics beyond the SM or New Physics (NP). Other diagrams would appear adding to those in Figure 2 with new intermediate particles. Thus, when it is declared by those who say that there isn't any evidence of NP in B decays, we have to be very careful that we have not absorbed such new evidence into what we declare to be SM physics. There are several approaches that can be followed. One approach is to simply predict the decay rate of a single process in the SM with known couplings and compare to the measurements. The classical case here is b → s γ and we will discuss this and other specific examples later. Another approach is make different measurements of the CKM parameters in different ways and see if they agree. This is normally done by measuring both angles and sides of the CKM triangle, but other quantities can also be used. This is the approach used by the CKM fitter [19] (see also [20] ) and UT fit groups [30] . In yet a third approach, the exact same quantity can be measured in several ways, even if cannot be predicted in the SM. An example here is measuring the CP violating angle β using B0 → J/ψ KS decays that proceed through the diagram in Figure 2(b), at least in the SM, and another process that uses the "Penguin" diagram in Figure 2(f), e.g. B0 → ϕ KS. The punch line is that if new, more massive particles exist in a mass range accessible to the LHC then they MUST contribute to rare and CP violating B decays! Even if measurements are precise enough only to limit the size of these effects, the properties of these new particles will be much better understood. This is the raison d'être for the further study of B decays. 2 Measurements of mixing and CP violation2.1 Neutral B Meson MixingNeutral B mesons can transform into their anti-particles before they decay. The diagrams for this process are shown in Figure 3 for the Bd. There is a similar diagram for the Bs. Although u, c and t quark exchanges are all shown, the t quark plays a dominant role mainly due to its mass, as the amplitude of this process is proportional to the mass of the exchanged fermion.
Under the weak interactions the eigenstates of flavor degenerate in pure QCD can mix. Let the quantum mechanical basis vectors
be {|1⟩, |2⟩} ≡ {|B0⟩, |
The Schrödinger equation is
Diagonalizing we have
ΔΓ = ΓL - ΓH = 2|Γ12|cos ϕ (6) where H refers to the heavier and L the lighter of the two weak eigenstates, and ϕ = arg(-M12/Γ12). We expect that ΔΓ is very small for B0 mesons but should be significant for Bs mesons. Bd mixing was first discovered by the ARGUS experiment [31]
(There was a previous measurement by UA1 indicating mixing for a mixture of
Data from many experiments has been combined by "The Heavy Flavor Averaging Group," (HFAG) to obtain an average value Δmd = (0.507 ± 0.004) × 1012 ps-1 [34] . The probability of mixing is related to the CKM matrix elements as [3,35]
where BB is a parameter related to the probability of the d and In principle fB can be measured. The decay rate of the annihilation process B- → ℓ- The ratio of Bs to Bd mixing frequency, however, provides a better situation in terms of reducing model dependent errors. Using Eq. 7 for Bd mixing and an analogous relation for Bs mixing and then dividing them results in
The CKM terms are
ignoring the higher order term in Vts. Solving the ratio of the two above equations for For many years experiments at the Z0 using e+ e- colliders at both LEP and the SLC had set lower limits on We now discuss the CDF measurement. The probability,
where t is the proper time. An amplitude A for each test frequency ω, is defined as [39]
For each frequency the expected result is either zero for no mixing, or one for mixing. No other value is physical, although measurement errors admit other values. Figure 5 shows the CDF results.
At Δms = 17.75 ps-1, the observed amplitude A = 1.21 ± 0.20 (stat.) is consistent with unity, indicating that the data are compatible with Bs
where the first error is statistical and the second systematic. In order to translate the mixing measurements to constraints on the CKM parameters ρ and η, we need to use theoretical values for the ratios The combined efforts of the HPQCD and UKQCD collaborations predict Unfortunately, the group that has calculated 2.2 CP Violation in the B SystemWe have two quantum mechanical operators: Charge Conjugation, C, and Parity, P. When applied to a particle wavefunction C changes particle to antiparticle and vice-versa. Applying P to a wavefunction ψ (r) we have P ψ (r) = ψ (-r). The P operator can be thought of changing the natural coordinate system from right-handed to left-handed. If nature was blind to handedness, then P would be always conserved. By applying the P operator twice we end up with P2ψ (r) = ψ (r), so the eigenvalues of P are ± 1. Therefore wave-functions, or particles represented by such wave-functions, have either intrinsic positive parity +1 (right-handed) or -1 (left-handed). Weak interactions, characterized by a combination of vector minus axial-vector currents, are known to be left-handed. Therefore, handedness matters and its well known that Parity is maximally violated in weak decays [3] . Since C changes left-handed particles to right-handed anti-particles, the product CP symmetry could have been preserved, but nature decided otherwise. Different particle transitions involve the different CP violating angles shown in Figure 1. Measuring these independently allows comparisons with measurements of the sides of the triangle and any differences in constraints in ρ and η can be due to the presence of new physics. Consider the case of a process B → f that goes via two amplitudes,
Any two amplitudes will do, though its better that they be of approximately equal size. Thus charged B decays can exhibit CP violation as well as neutral B decays. In some cases, we will see that it is possible to guarantee that 2.2.1 Formalism of CP Violation in Neutral B DecaysConsider the operations of Charge Conjugation, C, and Parity, P:
For neutral mesons we can construct the CP eigenstates
where
Since B0 and
If CP is not conserved then the eigenvectors, the mass eigenstates |BL⟩ and |BH⟩, are not the CP eigenstates but are
where
CP is violated if ϵB ≠ 0, which occurs if |q/p| ≠ 1. The time dependence of the mass eigenstates is
leading to the time evolution of the flavor eigenstates as
where m = (mL + mH)/2, Δm = mH - mL and Γ = ΓL ≈ ΓH, and t is the decay time in the B0 rest frame, the so-called "proper time". Note that the probability of a B0 decay as a function of t is given by ⟨B0 (t)|B0(t)⟩*, and is a pure exponential, e-Γt/2, in the absence of CP violation. 2.2.2 CP Violation for B Via the Interference of Mixing and DecayHere we choose a final state f which is accessible to both B0 and
CP|fCP⟩ = ± |fCP⟩. (30) It is useful to define the amplitudes
If
which requires only that λ acquire a non-zero phase, i.e. |λ| could be unity and CP violation can occur. Other useful variables, that are independent of any phase convention are
The first quantity can be related to CKM angles, while the second can be measured by the "semileptonic asymmetry," or for that matter in any flavor specific decay [46] :
for either B0 or for Bs mesons, separately. A comment on neutral B production at e+ e- colliders is in order. At the ϒ(4S) resonance there is coherent production of B0 The asymmetry, in the case of CP eigenstates, is defined as
which for |q/p| = 1 gives
For the cases where there is only one decay amplitude A, |λ| equals 1, and we have
Only the factor -Imλ contains information about the level of CP violation, the sine term is determined by B mixing. In fact, the time integrated asymmetry is given by
This is quite lucky for the study of Bd mesons, as the maximum size of the coefficient for any x is -0.5, close to the measured value of xd. Let us now find out how Imλ relates to the CKM parameters. Recall
2.2.3 Measurements of sin(2β)To evaluate the decay part we need to consider specific final states. For example, consider the final state J/ψ Ks. The decay diagram is shown in Figure 7. In this case we do not get a phase from the decay part because
is real to order 1/λ4. In this case the final state is a state of negative CP, i.e. CP|J/ψ Ks⟩ = -|J/ψ Ks⟩. This introduces an additional minus sign in the result for Imλ. Before finishing discussion of this final state we need to consider in more detail the presence of the Ks in the final state. Since neutral kaons can mix, we pick up another mixing phase (similar diagrams as for B0, see Figure 3). This term creates a phase given by
which is real to order λ4. It necessary to include this term, however, since there are other formulations of the CKM matrix than Wolfenstein, which have the phase in a different location. It is important that the physics predictions not depend on the CKM convention. (Here we do not include CP violation in the neutral kaon since it is much smaller than what is expected in the B decay.) In summary, for the case of f = J/ψ Ks, Imλ = - sin(2β). The angle β is the best measured CP violating angle measured in B meson decays. The process used is B0 → J/ψ KS, although there is some data used with the ψ (2S) or with KL (where the phase is + sin(2β)). Although it is normally thought that only one decay amplitude contributes here, in fact one can look for the presence of another source of CP violation, presumably in the decay amplitude, by not assuming |λ| equals one in Eq. 36. Then the time dependence of the decay rate is given by
Thus
(Note that the sign of the The most precise measurements of sin(2β) have been made by the BaBar and Belle experiments [47,48]
. These measurements are made at the ϒ(4S) resonance using e+ e- → ϒ(4S) → B0 While we will not discuss flavor tagging in general, it is an important part of CP violation measurements. At the ϒ(4S) once such very useful tag is that of high momentum lepton as a b-quark, part of a
The average value as determined by the Heavy Flavor Averaging Group is sin(2β) = 0.671 ± 0.024, where the dominant part of the error is statistical. No evidence is found for a non-zero Determining the sine of any angle gives a four-fold ambiguity in the angle. The decay mode B0 → J/ψ K*0, K*0 → Ks π0 offers a way of measuring cos 2β and resolving the ambiguities. This is a subtle analysis that we will not go into detail on [51,52]
. The result is that 2.2.4 Measurements of αThe next state to discuss is the CP + eigenstate f ≡ π+ π-. The simple spectator decay diagram is shown in Figure 9(a). For the moment we will assume that this is the only diagram, though the Penguin diagram shown in Figure 9(b) could also contribute; its presence can be inferred because it would induce a non-zero value for
and Im (λ) = Im (e-2i β e-2i γ) = Im (e2i α) = - sin (2α) (46) Time dependent CP violation measurements have been made by both the BaBar and Belle collaborations [53,54]
. Both groups find a non-zero value of both
The non-zero value of Use of the ρ+ ρ- final state is in principle quite similar to π+ π-, with some important caveats. First of all, it is a vector-vector final state and therefore could have both CP + and CP- components. It is however possible, doing a full angular analysis to measure the CP violating phase separately in each of these two amplitudes. The best method for this is in the "transversity" basis and will
be discussed later [60]
in Section 2.2.6. It is possible, however, for one polarization component to be dominant and then the angular analysis might
not be necessary. In fact the longitudinal polarization is dominant in this case. BaBar measures the fraction as In addition, ρ mesons are wide, so non-B backgrounds could be a problem and even if the proper B is reconstructed, there are non-resonant and possible a1 π contributions. Furthermore, it has been pointed out that the large ρ width could lead to the violation of the isospin constraints and this effect should be investigated [63] . The relevant branching ratios are given in Table 1. Table 1. Branching ratios B → ρ ρ modes in units of 10-6 Nevertheless, the small branching ratio for ρ0 ρ0, if indeed it has been observed at all, shows that the effects of the Penguin diagram on the extracted value of sin(2α) are small, and this may indeed be a good way to extract a value of α. The time dependent decay rates separately for B0 → ρ+ ρ- and
Results from the time dependent CP violation analysis from both Belle and BaBar are shown in Figure 12.
The data can be averaged and α determined by using the isospin analysis and the rates listed in Table 1. Unfortunately the precision of the data leads only to constraints. These have been determined by both the CKM fitter group [64] and the UT fit group [30] . These groups disagree in some cases. The CKM fitter group use a frequentist statistical approach, while UT fit uses a Bayseian approach. The basic difference is the that the Bayesian approach the theoretical errors are taken as having a Gaussian distribution. Here we show in Figure 13 the results from CKM fitter.
The final state π+ π- π0 can also be used to extract α. Snyder and Quinn proposed that a Dalitz plot analysis of B → ρ π → π+ π- π0 can be used to unravel both α and the relative penguin-tree phases [65] . The Dalitz plot for simulated events unaffected by detector acceptance is shown in Figure 14.
The task at hand is to do a time dependent analysis of the magnitude of the decay amplitudes and phases. The analyses of both collaborations allow for ρ(1450) and ρ(1700) contributions in addition to the ρ(770). There are a total of 26 free parameters in the fit. The statistics for such an analysis are not overwhelming: BaBar has about 2100 signal events [66] , while Belle has about 1000 [67] . The results for the confidence levels of α found by both collaborations are shown in Figure 15. Note that there is also a mirror solution at α + 180°. The Belle collaboration uses their measured decay rates for B+ → ρ+ π0 and B+ → ρ0 π+ coupled with isospin relations [68,69] to help constrain α.
The LHCb experiment expects to be able to significantly improve on the determination of α using the ρ π mode. They expect 14,000 events in for an integrated luminosity of 2 fb-1, with a signal to background ratio greater than 1 [70] . It is expected that this amount of data can be accumulated in one to two years of normal LHC operation. Combining all the data, both the CKM fitter and UT fit groups derive a the confidence level plot for α shown in Figure 16. There is a clear disagreement between the two groups, UT fit preferring a solution in the vicinity of 160°, and CKM fitter a value closer to 90°. The CKM fitter group believes that this is due to the UT fit group's use of Bayesian statistics which they criticize [71] .
2.2.5 The Angle γThe angle
where rB reflects the amplitude suppression for the b → u mode and δB is the relative phase. We have not yet used identical final states for D0 and There are several suggestions as to different D0 decay modes to use. In the original paper on this topic Gronau and Wyler [72,73] propose using CP eigenstates for the D0 decay, such as K+ K-, π+ π- etc.., combining with charge specific decays and comparing B- with B+ decays. In the latter, the sign of the strong phase is flipped with respect to the weak phase. In fact modes such as D*0 or K*- can also be used. (When using D*0 there is a difference in δB of π between γ D0 and π0 D0 decay modes [74] .) It is convenient to define the follow variables:
These are related to the variables defined in Eq. 47 as
Measurements have been made by the BaBar, Belle and CDF collaborations [75-79]
. These data, however, are not statistically powerful enough by themselves to give a useful measurement of γ. Atwood, Dunietz and Soni suggested using double-Cabibbo suppressed decays as an alternative means of generating the interference
between decay amplitudes [80]
. For example the final state B- → D0 K- can proceed via the tree level diagram in Figure 2(a) when the W- → Measurements have been made mostly using the K± π∓ final state [81,82] . Again, these attempts do not yet produce accurate results. Thus far, the best method for measuring γ uses the three-body final state KS π+ π-. Since the final state is accessible by both D0 and It has been shown [87,88] that measurement of the amplitude magnitudes and phases found in the decays of ψ (3770) → (CP ± Tag)(KS π+ π-)D provide useful information that help the narrow model error. The CLEO collaboration is working on such an analysis, and preliminary results have been reported [89] . (CLEO also is working on incorporating other D0 decay modes.) Both the CKM fitter and UT fit groups have formed liklihoods for γ based on the measured results, as shown in Figure 18. In this case CKM fitter and UT fit agree on the general shape of the liklihood curve. The CKM fitter plot shows a small disagreement between the Daltiz method and a combination of the other two methods.
2.2.6 The Angle χThe angle χ shown in Figure 1 is the phase that appears in the box diagram for Bs mixing, similar to the diagram for B0 mixing shown in Figure 3, but with the d quark replaced by an s quark. The analogous mode to B0 → J/ψ Ks in the Bs system is Bs → J/ψ η. The Feynman diagrams are shown in Figure 19. This is very similar to measuring β so χ is often called βs. We also have a relation between ϕs defined in Eq. 33, ϕs = -2χ.
Since there are usually two photons present in the η decay, experiments at hadron colliders, which can perform time-dependent studies of Bs mesons, preferentially use the J/ψ ϕ final state. This, unfortunately, introduces another complexity into the problem; as the Bs is spinless the total angular momentum of the final state particles must be zero. For a decay into a vector and scalar, such as J/ψ η, this forces the vector J/ψ to be fully longitudinally polarized with respect to the decay axis. For a vector-vector final state both angular momentum state vectors are either longitudinal (L), both are transverse with linear polarization vectors parallel (||) or they are perpendicular (⊥) to one another [90] . Another way of viewing this is that a spin-0 B decay into two massive vector mesons can form CP even states with L = 0 or 2, and a CP odd state with L = 1. The relative populations in the two CP states are determined by strong interactions dynamics, but to study the weak phase here we are not particularly interested in the actual amount, unless of course one state dominated. We do not expect this to be the case however, since the SU(3) related decay B0 → J/ψ K*0, K*0 → K+ π- has a substantial components of both CP states; the PDG quotes gives the longitudinal fraction as (80 ± 8 ± 5)% [17] . The even and odd CP components can be disentangled by measuring the appropriate angular quantities of each event. Following Dighe et al [91] , we can decompose the decay amplitude for a Bs as
where ϵJ/ψ and ϵϕ are polarization 3-vectors in the J/ψ rest frame, dΓ(Bs → J/ψ ϕ)/dt = |A0|2 + |A|||2 + |A⊥|2. (51) The ϕ meson direction in the J/ψ rest frame defines the
The decay width can be written as
The decay rate for Another complexity arises from the expectation that the width difference ΔΓs/Γs ≈ 15%. This complicates the time dependent rate equations. For convenience, setting
where
The quantities δ⊥ and δ|| are the strong phases of A⊥ and A|| relative to A0, respectively [92]
. The expression for The most interesting quantities to be extracted from the data are χ and ΔΓ. There are many experimental challenges: the angular and lifetime distributions must be corrected for experimental acceptances; flavor tagging efficiencies and dilutions must be evaluated; backgrounds must be measured. Both the CDF [93] and D0 [94] experiments have done this complicated analysis. Updated results as of this writing are summarized by the CKM fitter derived limits shown in Figure 21.
The Standard Model allowed region is a very thin vertical band centered near zero at ϕs of -0.036 ± 0.002 (shown in red). (Recall ϕs ≡ -2χ ≡ -2βs.) The region labeled "all" (green) shows the allowed region at 68% confidence level. Although the fit uses several input
components besides the CP asymmetry measurements in Bs → J/ψ ϕ, including use of measured total widths introduced via the constraint equation It has been pointed out, however, that there is likely an S-wave K+ K- contribution in the region of the ϕ that contributes 5–10% of the event rate as estimated using 2.2.7 Measurements of Direct CP Violation in B → K π DecaysTime integrated asymmetries of B mesons produced at the ϒ(4S) resonance can only be due to direct CP asymmetry, as the mixing generated asymmetry must integrate to zero due to the fact that the initial state has JPC = 1--. The first evidence for such direct CP violation at the greater than four standard deviation level in the K∓ π± final state was given by BaBar [100] . The latest BaBar result is [101]
showing a large statistical significance. This result was confirmed by the Belle collaboration, but Belle also measured the isospin conjugate mode. Consider the two-body decays of B mesons into a kaon and a pion, shown in Figure 22. For netural and charged decays, it can proceed via a tree level diagram (a) or a Penguin diagram (b). There are two additional decay diagrams allowed for the B-, the color-suppressed tree level diagram (c) and the elusive "Electroweak" Penguin diagram in (d). (So named because of the intermediate γ or Z boson.) Since it is expected that diagrams (c) and (d) are small, the direct CP violating asymmetries in both charged and neutral modes should be the same. Yet Belle observed [102]
The Belle data are shown in Figure 23.
The difference between 2.3 Conclusions from CP Violation MeasurementsAll CP violation in the quark sector is proportional to the parameter η in Eq. 1. In fact all CP asymmetries are proportional to the "Jarlskog Invariant", J = A2 λ6 η, which represents the equal area of all the CKM triangles [104] . Since we know the value of these three numbers, we do know the amount of CP violation we can expect, even without making the measurements. We also can estimate the amount of CP violation necessary using cosmology. To reproduce the observed baryon to entropy ratio requires many orders of magnitude more CP violation than thus far found in heavy quark decays [105] . Thus we believe there are new sources of CP violation that have not yet been found. 3 The CKM parameter |Vcb|There are two experimental methods to determine |Vcb|: the exclusive method, where |Vcb| is extracted by studying the exclusive 3.0.1 Beauty Quark Mass DefinitionsDue to confinement and the non-perturbative aspect of the strong interaction, the concept of the quark masses cannot be tied to an intuitive picture of the rest mass of a particle, as for leptons. Rather, quark masses must be considered as couplings of the SM Lagrangian that have to be determined from processes that depend on them. As such the b-quark mass (mb) is a scheme-dependent, renormalised quantity. In principle, any renormalisation scheme or definition of quark masses is possible. In the framework of QCD perturbation theory the difference between two mass schemes can be determined as a series in powers of αs. Therefore, higher-order terms in the perturbative expansion of a quantity that depends on quark masses are affected by the particular scheme employed. There are schemes that are more appropriate and more convenient for some purposes than others. Here we examine the main quark mass definitions commonly used in the description of B decays. • Pole mass: The pole mass definition is gauge-invariant and infrared-safe [106-108] to all orders in perturbation theory and has been used as the standard mass definition of many perturbative computations in the past. By construction, it is directly related to the concept of the mass of a free quark. The presence of a renormalon ambiguity [109,110] makes the numerical value of the pole mass an order-dependent quantity, leading to large perturbative corrections for Heavy Quark Effective Theory (HQET) parameters (see below for a discussion of HQET). These shortcomings are avoided by using quark mass definitions that reduce the infrared sensitivity by removing the ΛQCD renormalon of the pole mass. Such quark mass definitions are generically called "short-distance" masses. • • Kinetic mass: The shortcomings of the pole and the
where μkin is the nominal kinetic mass renormalisation scale. For μkin → 0 the kinetic mass reduces to the pole mass. • 1S mass: The kinetic mass depends on an explicit subtraction scale to remove the universal infrared sensitive contributions associated with the non-relativistic b-quark dynamics. The 1S mass [117,118] achieves the same task without a factorisation scale, since it is directly related to a physical quantity. The b-quark 1S mass is defined as half of the perturbative contribution to the mass of the ϒ(S1) in the limit mb ≫ mb v ≫ mb v2 ≫ ΛQCD. A list of b-quark mass determinations, converted into the Table 2. List of mb determinations converted into the 3.1 Determination Based on Exclusive Semileptonic B DecaysThe exclusive |Vcb| determination is obtained by studying the decays B → D* ℓν and B → D ℓν, where ℓ denotes either an electron or a muon. The exclusive measurements of a single hadronic final state, e.g. the ground state D or D*, restrict the dynamics of the process. The remaining degrees of freedom, usually connected to different helicity states of the charmed hadron, can be expressed in terms of form factors, depending on the invariant mass of the lepton-ν pair, q2. The shapes of those form factors are unknown but can be measured. However, the overall normalization of these functions needs to be determined from theoretical calculations. Isgur and Wise formulated a theoretical breakthrough in the late 1980's. They found that in the limit of an infinitely heavy quark masses QCD possess additional flavor and spin symmetries. They showed that since the heavier b and c quarks have masses much heavier than the scale of the QCD coupling constant, they are heavy enough to posses this symmetry but that corrections for the fact that the quark mass was not infinite had to be made. They showed there was a systematic method of making these corrections by expanding a series in terms of the inverse quark mass [119] . This theory is known as Heavy Quark Effective Theory (HQET). When studying b quark decays into c quarks, it is convenient to view the process in four-velocity transfer (w) space as opposed to four-momentum transfer space, because at maximum four-velocity transfer, where w equals one, the form-factor in lowest order is unity, i.e. the b transforms directly into a c quark without any velocity change. The value of |Vcb| can be extracted by studying the decay rate for the process
where w is the product of the four-velocities of the D*+ and the
where
Note that the first term in the non-perturbative expansion in powers of 1/mQ vanishes [121]
. QED corrections up to leading logarithmic order give ηA = 0.960 ± 0.007. Different estimates of the 1/ Since the phase-space factor This decay has been analyzed by CLEO, BaBar and Belle using B mesons from the ϒ(4S) decay, and by ALEPH, DELPHI, and OPAL at the Z0 center of mass energy. Experiments that exploit the ϒ(4S) have the advantage that w resolution is quite good. However, they suffer from lower statistics near w = 1 in the decay B → D*+ ℓν due to the lower reconstruction efficiency of the slow π±. On the other hand, the decay B → D*0 ℓν is not affected by this problem [123] . In addition, kinematic constraints enable these experiments to identify the final state including D* without large contamination from the poorly known portion of semileptonic decays with a hadronic system recoiling against the lepton-ν pair with masses higher than the D and D*, commonly identified as 'D**'. B-factories and CLEO fit for the signal and background components in the distribution of the cosine of the angle between the direction of the B and the direction of the D*ℓ system. At LEP, B mesons are produced with a large variable momentum (about 30 GeV on average), giving a relatively poor w resolution and limited physics background rejection capabilities. By contrast, LEP experiments benefit from an efficiency that is only mildly dependent upon w. LEP experiments extracted |Vcb| by performing a two-parameter fit, for Table 3 summarizes the available data. Values of |Vcb| = (38.2 ± 0.5exp ± 1.0theo) × 10-3, where the dominant error is theoretical, and it will be difficult to improve upon. The study of the decay B → D ℓν poses new challenges from the experimental point of view. The differential decay rate for B → D ℓν can be expressed as [127,128]
where Table 4. Experimental results from B → Dℓν using corrections to the common inputs and world averages. In the limit of infinite quark masses, |Vcb| = (39.5 ± 1.4exp ± 0.9theo) × 10-3 consistent with the value extracted from B → D⋆ℓν decay, but with an experimental uncertainty about twice as large. BaBar recently has also studied the differential decay widths for the decays B- → D0 ℓν and B- → D⋆0ℓν to extract the ratio 3.2 B → D**ℓν DecaysIt is important to understand the composition of the inclusive B semileptonic decay rate in terms of exclusive final states for use in semileptonic B decay analyses. The B → D(*)ℓν decays are well measured, but a sizeable fraction of semileptonic B decay are to D**ℓν. The D** resonances have larger masses than the D* and not well studied. (D** refers to the resonant states, There are four orbitally excited states with L = 1. They can be grouped in two pairs according to the value of the spin on the light system, j = L ± 1/2 (L = 1). States with j = 3/2 can have JP = 1+ and 2+. The 1+ state decays only through D*π, and the 2+ through D π or D*π. Parity and angular momentum conservation imply that in the 2+ the D* and π are in a D wave but allow both S and D waves in the 1+ state. However, if the heavy quark spin is assumed to decouple, conservation of j = 3/2 forbids S waves even in the 1+ state. A large D-wave component and the fact that the masses of these states are not far from threshold imply that the j = 3/2 states are narrow. These states have been observed with a typical width of 20 MeV/c2. On the contrary, j = 1/2 states can have JP = 0+and 1+, so they are expected to decay mainly through an S wave and manifest as broad resonances, with typical widths of several hundred MeV/c2. The ALEPH [132]
, CLEO [133]
, DELPHI [132]
, and D0 [134]
experiments have reported evidence of the narrow resonant states (D1 and The differences between the measured inclusive semileptonic branching fraction and the sum of all exclusive B semileptonic measurements for Belle, BaBar and World averages are given in Table 5 for B0 and B- decays. In the case where multiple measurements exist, only the most precise measurements have been used in the BaBar and Belle columns, i.e. no attempt at an average is made. In all cases the sum of the exclusive components does not saturate the B semileptonic rate. Table 5. Inclusive versus sum of exclusive measured B semileptonic branching fractions (%). All measured rates for the D** narrow states are in good agreement. Experimental results seem to point towards a larger rate for broader states. If it
is due mainly to 3.3 Determination Based on Inclusive Semileptonic B DecaysInclusive determinations of |Vcb| are obtained using combined fits to inclusive B decay distributions [139,140]
. These determinations are based on calculations of the semileptonic decay rate in the frameworks of the Operator Product
Expansion (OPE) [141]
and HQET [139,142]
. They predict the semileptonic decay rate in terms of |Vcb|, the b-quark mass mb, and non-perturbative matrix elements. The spectator model decay rate is the leading term in a well-defined expansion controlled
by the parameter ΛQCD/mb [142-146]
with non-perturbative corrections arising to order 1/ Perturbative and non-perturbative corrections depend on the mb definition, i.e. the non-perturbative expansion scheme, as well as the non-perturbative matrix elements that enter the expansion.
In order to determine these parameters, Heavy Quark Expansions (HQE) [139,146,147]
express the semileptonic decay width ΓSL, moments of the lepton energy and hadron mass spectra in B → Xc ℓν decays in terms of the running kinetic quark masses The shape of the lepton spectrum and of the hadronic mass spectrum provide constraints on the heavy quark expansion, which allows for the calculation of the properties of B → Xc ℓν transitions. So far, measurements of the hadronic mass distribution and the leptonic spectrum have been made by BaBar [148] , Belle [149] , CLEO [150,151] , DELPHI [152] . CDF [153] provides only the measurement of the hadronic mass spectrum with a lepton momentum cut of 0.6 GeV in the B rest frame. The inclusive semileptonic width can be expressed as
where
corresponding to the mean. Subsequent (central) moments are calculated around the first moment,
corresponding to a distribution's width, kurtosis and so on. A
where
The measurement of |Vcb| from inclusive decays requires that these decays be adequately described by the OPE formalism. The motivation of the moment
approach is to exploit the degree of experimental and theoretical understanding of each moment and for different To compare with theoretical predictions, the moments are measured with a well defined cut on the lepton momentum in the B rest frame. The measured hadronic mass distribution and lepton energy spectrum are affected by a variety of experimental factors such as detector resolution, accessible phase space, radiative effects. It is particularly important to measure the largest fraction of the accessible phase space in order to reduce both theoretical and experimental uncertainties. Each experiment has focused on lowering the lepton energy cut. The hadronic mass spectrum in B → Xcℓν decays can be split into three contributions corresponding to D, D*, and D**, where D** here stands for any neutral charmed state, resonant or not, other than D and D*. Belle [149]
, BaBar [148]
and CLEO [150]
explored the moments of the hadronic mass spectrum DELPHI follows a different approach in extracting the moments, measuring the invariant mass distribution of the D** component only and fixing the D and D* components. DELPHI measures the first moment with respect to the spin averaged mass of D and D*. At LEP b-quarks were created with an energy of approximately 30 GeV allowing the measurement of the hadronic mass moments without a cut on the lepton energy [152] . The shape of the lepton spectrum provides further constraints on the OPE. These measurements are sensitive to higher order
OPE parameters and are considerably more precise experimentally. Moments of the lepton momentum with a cut pℓ ≥ 1.0 GeV/c have been measured by the CLEO collaboration [157]
. BaBar [148]
extract up to the third moment of this distribution, using a low momentum cut of pℓ ≥ 0.6 GeV/c. Both BaBar and CLEO use dilepton samples. The most recent measurement of the electron energy spectrum is from
Belle [155,156]
. Events are selected by fully reconstructing one of the B mesons, produced in pairs from ϒ(4S) decays and it determines the true electron energy spectrum by unfolding [158]
the measured spectrum in the B meson rest frame. Belle measures B0 and B+ weighted average partial branching fractions 3.3.1 HQE ParametersUsing the moment measurements described above, it is possible to determine the CKM matrix element |Vcb| and HQE parameters by performing global fits in the kinetic and 1S b-quark mass schemes [160] . The photon energy moments in B → Xs γ decays [161] are also included in order to constrain the b-quark mass more precisely. Measurements that are not matched by theoretical predictions and those with high cutoff energies are excluded (i.e. semileptonic moments with Emin > 1.5 GeV and photon energy moments with Emin > 2 GeV). The results are preliminary. The inclusive spectral moments of B → Xcℓν decays have been derived in the 1S scheme up to Table 6. Experimental results using corrections to the common inputs and world averages for the 1S scheme.
Spectral moments of B → Xcℓν decays have been derived up to Table 7. Experimental results using corrections to the common inputs and world averages for the kinetic scheme. All the measured moments of inclusive distributions in B → Xcℓν and B → s γ decays are used in a fit to extract |Vcb| and the b and c quark masses. The |Vcb| and mb values obtained are listed in the Table 7. The default fit also gives mc = 1.16 ± 0.05 GeV, and There are open issues relating to the global fits. First of all, the χ2/n.d.f. are very small, pointing to an underestimate in the theoretical correlations. In a recent study [163]
, the theoretical correlations used in the fit were scrutinized, and new correlation coefficients were derived from the theory
expressions using a "toy Monte Carlo" approach, showing that the theoretical correlations were largely underestimated. The
result of this new fit is shown in Table 7. The second issue is related the size of the theoretical error. Recently the NNLO full two-loop calculations become available
[164-166]
. In the Kinetic scheme NNLO calculations include an estimate of the non-BLM terms and lead to a roughly 0.6 % reduction of
the |Vcb| value -0.25 × 10-3. In the 1S scheme the shift on |Vcb| is of about -0.14 × 10-3 [167]
. From the new power corrections at NLO we expect the chromo-magnetic corrections to be more important as the tree level corrections
are more important, and a change of about 20–30% in the extracted value of HQE has been carried out up to 1/ 3.4 OutlookThe error on the inclusive and exclusive determination of Vcb is limited by theory. The two method have a 2σ disagreement, even when results in the same experiment are compared. For exclusive determinations, the experimental determinations in B → D*ℓν from different experiments are not in agreement. Improving the statistical error for the determination from B → Dℓν will help in elucidating the origin of this discrepancy. In the inclusive method, errors of less than 2% are quoted. However, the latest theoretical results and the introduction of
better correlation between theoretical error show shifts in the central value larger than the quoted fit error. The situation
needs to be reevaluated when all the new calculations and corrections are implemented in the fit. Another puzzling result
of the global fit to B → Xcℓν moments only is the value of 4 The CKM parameter |Vub|The parameter |Vub| determines one of the sides of the unitarity triangle, and thus affects one of the crucial tests of the Yukawa sector of
the Standard Model. Also in this case, there are two general methods to determine this parameter, using B meson semileptonic decays. The first approach relies on the determination of branching fractions and form factor determinations
of exclusive semileptonic decays, such as B → πℓ Both methods pose challenges to both experimenters and theorists. The branching fractions are small, and a substantial background
induced by the dominant b → cℓ 4.1 Determinations Based on Exclusive B Semileptonic DecaysThe decay B → πℓ
where GF is the Fermi constant, The first experimental challenge is to reconstruct this exclusive channel without significant background from the dominant
charm semileptonic decays, and the additional background component from other b → uℓ CLEO pioneered this approach by reconstructing the ν from the missing energy (Emiss ≡ 2EB - Σi Ei), and momentum BaBar [171]
uses a sample of 206 fb-1 to obtain Belle uses fully reconstructed hadronic tags, achieving the best kinematic constraints and thus the highest background suppression
capabilities and the most precise determination of q2. Branching fractions obtained with this technique have a bigger statistical error because of the penalty introduced by the
tag requirement, but the overall error is already comparable with the other methods. Table 8 summarizes the present status of the experimental information on this decay. Both the total branching fraction for B → π ℓ Table 8. Partial and total branching fractions, in units of 10-4, for the decay B → πℓ+νℓ. In order to interpret these results, we need theoretical predictions for the form factor f+ (q2). This problem can be split into two parts: the determination of the form factor normalization, f+ (0), and the functional form of the q2 dependence. Form factor predictions have been produced with quark models [174] and QCD sum rule calculations [175] . Lattice calculations provide evaluations of f+ (q2) at specific values of q2 or, equivalently, pion momenta (pπ). Authors then fit these data points with a variety of shapes. Typically a dominant pole shape has been used in the literature. Nowadays more complex functional forms are preferred. Becirevic and Kaidalov (BK) [176] suggest using
where cB
where the parameters r1, r2, and α are fitted from available data. Lastly, parameterizations that allow the application of constraints derived from soft-collinear effective theory (SCET), and dispersion relations have been proposed by Boyd, Grinstein, and Lebedev [120] , and later pursued also by Hill [177] . They define
where t is (pB - pπ)2, defined beyond the physical region, t± = (mB ± mπ)2, and t0 is an expansion point. BGD use t0 = 0.65t-. The parameter αK allows the modeling of different functional forms, and the variable
maps t+ <t < ∞ onto |z| = 1 and -∞ <t < t+ onto the z interval [-1,1]. All are refinements of the old ansatz of a simple pole shape, now rarely used. The first lattice calculations were carried
out with the quenched approximation that ignores vacuum polarization effects [178,179]
. In 2004, preliminary unquenched results were presented by the Fermilab/MILC [180]
and HPQCD [181]
collaborations. These calculations employed the MILC collaboration Nf = 2 + 1 unquenched configurations, which attain the most realistic values of the quark masses so far. Using these calculations
and the most recent value of the partial branching fraction 4.2 Determinations Based on Inclusive B Semileptonic DecaysInclusive determinations of |Vub| rely on the heavy quark expansion (HQE), which combines perturbative QCD with an expansion in terms of 1/mb, which accounts for non-perturbative effects. Although the possible breaking of the assumption of local quark hadron duality
may produce unquantified errors, other non-perturbative uncertainties can be evaluated with systematic improvements, and their
uncertainties are easier to assess than the ones of unquenched lattice QCD or QCD sum-rules. This statement applies to the
total charmless semileptonic width
where μ ~ mb is the scale at which αS needs to be evaluated, while the scale μ⋆ applies to the non-perturbative expansion parameters, namely the b quark mass mb, the chromo-magnetic operator, λ2, and the kinetic operator, μπ. It is clear that, even restricting our attention to the total width, a precise knowledge of the b quark mass is critical, and considerable theoretical effort has been devoted to a reliable extraction of this parameter from experimental observables. Other uncertainties, such as the effects of weak annihilation or violations of quark-hadron duality, will be discussed later. Charmless B meson semileptonic decays constitute only about 1% of the total semileptonic width. Thus the big challenge for experimentalists is to identify techniques to suppress this large background. For example, the first evidence for B meson charmless semileptonic decays came from the study of the end point of the lepton spectrum, where leptons from b → c ℓν processes are forbidden due to the larger mass of the hadronic system formed by the c quark [187] . While this was very important first evidence that |Vub| ≠ 0, very quickly several authors pointed out that this region of phase space is ill suited to a precise determination of |Vub| because near the end point the OPE does not converge and an infinite series of contributions needs to be taken into account [188,189] . Thus, a large effort has gone into developing experimental techniques that would feature a low lepton energy Eℓ cut and an acceptable signal to background ratio. Table 9 summarizes the present status of the |Vub| determination with this approach. Next a whole host of papers proposed alternative "model independent" approaches to measure |Vub| from inclusive decays [190-192]
, with the common goal of identifying a region of phase space where experimentalists can suppress the b → c background, and where the OPE works. The first proposal by Bigi, Dikeman, and Uraltsev proposed considering semileptonic events
where MX ≤ 1.5 GeV [190]
. However, Bauer, Ligeti, and Luke pointed out [193]
that the kinematic limit An alternative approach [186] , incorporates hadronic structure functions to model the region of large hadronic energy and small invariant mass, not well modeled by the OPE, and applies the OPE to the kinematic region where the hadronic kinematic variables scale with MB, and smoothly interpolates between them. This approach is commonly referred to as "shape function" method, and uses the γ spectrum in inclusive B → XS γ to reduce the theoretical uncertainties. It combines the experimental data on high momentum leptons from B decays with the constraints from inclusive radiative decays, to produce a precise value of |Vub| inclusive. The study of charmless inclusive semileptonic decays benefits from the use of hadronic tags. Belle used the information of
the tag momentum to boost the electron into the B meson rest frame and to select a sample of high purity. They then reconstruct the ν energy and momentum from the measured 4-momentum vectors of the ϒ(4S), B tag, lepton, and the additional tracks not used to form the tag or the lepton. They then evaluate the invariant mass MX and the quantity They evaluate the partial branching fractions in each of them, and extract |Vub| directly from the partial branching fractions, normalized by corresponding theoretical scale factors R(ΔΦ), evaluated with the shape function method [195] . The authors observe that different kinematic cuts give different values of |Vub| and speculate that this may be due to additional theoretical uncertainties not completely accounted for. Recently, BaBar has also used the same technique in slightly different kinematic regions: P+ < 0.66 GeV/c, MX < 1.55 GeV/c2, and MX < 1.7 GeV/c2 combined with q2 > 8 GeV2/c2. They use both the shape function method [195] and the dressed gluon exponentation method [196] . The results obtained with this approach are summarized in Table 10. Table 10. Inclusive |Vub| determinations with tagged samples. In order to provide predictions that are most suited to different experimental cuts, theorists have made available the triple
differential width
where q0 is the energy of the lepton-ν pair and Eℓ is the energy of the charged lepton in the B meson rest frame, and W1–3 are the three structure functions relevant if we assume massless leptons. Reference [197]
computes the functions Wi(q0, q2) as a convolution at fixed q2 between non-perturbative distributions Fi(K+, q2; μ) and the perturbative functions
Perturbative corrections to the structure functions W1–3 are now known up to order However, Figure 25 shows that different methods provide central values of |Vub| that often differ beyond the stated errors. Moreover, these estimates are generally significantly higher than the value
of |Vub| extracted from B → πℓ
An effect that can influence the inclusive |Vub| is the contribution due to topologies where the incoming b A first attempt to quantify the annihilation effect was performed by CLEO [201]
, studying the high q2 contributions to charmless semileptonic decays. They used inclusive data on a sample including both charged and neutral B mesons. They used a variety of models to set the limit on the fractional contribution of annihilation diagrams, ΓW A/Γb → u < 7.4% at 90% confidence level. BaBar has presented a measurement of
at 90 % confidence level, where fW A(2.3 – 2.6) represents the fraction of weak annihilation rate contributing in the momentum interval Δpℓ comprised between 2.3 and 2.6 GeV used in this analysis, believed to be close to unity. Finally, Voloshin [202] has suggested that the difference in semileptonic widths of the D0 and Ds mesons can assess non-factorizable terms through the relationship
where Γ0 (c → sℓν) is the bare parton semileptonic rate c → sℓν, fD is the D decay constant, and In summary, the values if |Vub| extracted with the inclusive method have quoted errors between 5.5% and 10%, with central values that change well outside
these uncertainties and are all considerably larger than the |Vub| value obtained from the exclusive branching fraction 5 Rare B decaysIn general, we define as "rare" B decays processes that are suppressed at tree level. They are interesting because they are typically mediated by loop diagrams which may be characterized by a matrix element whose strength is comparable to components with similar Feynman diagrams, where new particles appear in loops. Thus evidence for new physics may appear either through enhancements in branching fractions relative to the Standard Model expectation, or through interference effects. 5.1 B → τ νThe decay B → τ ν is affected by two quantities of great interest, the quark mixing parameter |Vub| and the pseudoscalar decay constant fB. In fact, the leptonic branching fraction is given by
where mτ is the mass of the τ lepton, and τB is the charged B lifetime. Theoretical predictions for fB are summarized in Table 11. The most recent value exploits the full machinery of unquenched Lattice QCD, and has a precision of 7%. Table 11. Summary of recent theoretical evaluations of the decay constant fB. Belle performed the first measurement of the branching fraction Table 12. Summary of experimental determinations of The Standard Model prediction for
where gs is the effective scalar coupling. Fig. 26 shows the predicted value of
Note that a quantity that is sensitive to the same coupling is the ratio 5.2 Radiative B DecaysRadiative B decays involving b → s(d)γ transitions are very sensitive to new physics processes. These processes are ideals for indirect searches for physics beyond the Standard Model, such as two-Higgs doublet models, supersymmetric theories and extended technicolor scenarios [212] . Hence, comparison of results from these theories with experimental measurements places constraints upon new physics. Moreover, B → Xsγ is an ideal laboratory for studying the dynamics of the b-quark inside the B meson: since the motion of the b-quark inside the B meson is universal, information gained from a measurement of the energy spectrum of the emitted photon in this transition is applicable to other processes, for instance semileptonic decays. In general, in the OPE, the amplitude A for a process can be expressed as sum
where the Qi are local operators, the Ci are Wilson coefficients, and μ is the QCD renormalization scale [213] . The B → Xsγ branching fraction directly probes the Wilson coefficient C7. However, some new physics contributions may leave the B → Xsγ rate unaltered, with changes to the C7 amplitude. In this case the direct CP asymmetry is sensitive to new phases that may appear in the decay loop [214,215] and the B → Xsℓ+ℓ- transition may provide information on the sign of the amplitude, since it also probes C9 and C10. The B → Xsγ branching fraction, as a function of a photon energy cut-off E0, is related to the B → Xcνℓ transition by
where the perturbative corrections P(E0) are defined as
and the non-perturbative N(E0) terms are Two predictions for the branching fraction at NNLO have been given:
from [216] and [217] , respectively. Both agree with the world average in Table 13. The theoretical error of 7% is obtained by quadratically adding the uncertainty due to non-perturbative corrections (5%), the uncertainty due to missing higher-order corrections (3%), the mc interpolation ambiguity (3%) and the parametric uncertainty (3%). Not included are some known NNLO and non-perturbative corrections. The size of these neglected contributions is about 1.6%, which is smaller than the present theoretical error. While progress in the calculation of the perturbative corrections is expected in the future, the uncertainty on the non-perturbative effects will not be easily reduced as they are very difficult to estimate. Table 13. Measured branching fractions, minimum photon energy, and branching fractions for Emin = 1.6 GeV photon energy for b → sγ . Experimentally, two methods are used to extract the B → Xs γ
signal: the fully inclusive method and the semi-inclusive method. In the fully inclusive method, events containing a hard
photon consistent with B → Xs γ are selected. In this method the subtraction of a very large background, primarily from q In the semi-inclusive method, the B → Xs γ branching fraction is determined by summing up exclusive modes, with an extrapolation procedure to account for the unobserved
modes. The semi-inclusive analysis provides a more precise photon energy in the B-meson rest frame and usually provides the flavor of the decay (i.e. b or At the parton level in the two-body decay b → s γ, the energy of the photon is Eγ ≈ mb/2 in the b-quark rest frame. However, the B → Xsγ is not mono-chromatic, due to several effects including the width of the Xs mass distribution, gluon emission and the Fermi motion of the b-quark in the B meson. The non-perturbative Fermi motion effects will raise the photon energy above mb/2, while the gluon emission will give a long low-energy tail. The low-energy tail can be described by HQE apart non-perturbative
effects that can be modeled by process-independent shape functions described by a few parameters, e.g. the b quark mass and the Fermi momentum ( The extrapolated branching fractions do not include the published model uncertainties and the uncertainty on the extrapolation factor. These errors are included in the average by recalculation in the framework of a particular ansatz. Correlations between different measurements have been ignored. The parametric error on mb is evaluated by varying mb within its uncertainty. The world average has been calculated taking into account the correlations, when available (for the time being only Belle provides these correlations), between partial branching fraction measured at different photon energy thresholds for each single analysis. In the branching fraction average we use the energy threshold of each photon energy spectrum that corresponds to the optimal overall uncertainty on the full rate after extrapolation. This differs from the HFAG method, which uses the lowest energy thresholds for each spectrum measurement. The HFAG method penalizes analyses that quote measurements at low Emin, which suffer from larger systematic uncertainties. Two different calculations were used to extrapolate the measured partial branching fractions down to a photon energy lower
threshold of 1.6 GeV. The extrapolation factors were determined using The agreement between the measured B → Xsγ branching fraction and the theoretical prediction constrains various new physics scenarios. One of the most popular examples is the lower bound on the type-II charged Higgs mass, since it always constructively interferes with the SM amplitude. The current limit is around 200 GeV for any tan β if no other destructive new physics contribution exists [218] . This limit is significantly higher than the direct search limit. New physics contributions may leave the B → Xsγ rate unaltered, with new phases appearing in the decay loop [214,215] . Since the Standard Model prediction of the CP asymmetry (ACPs γ) is zero in the limit of U-spin symmetry [219] , significant non-zero values would be evidence for new phenomena. ACPs γ has been measured by Belle, BaBar and CLEO. The values of all such measurements are listed in Table 14. All measurements are in agreement with the Standard Model predictions. BaBar also report a CP- asymmetry for b → (d + s)γ of ACPs γ = -0.11 ± 0.12 ± 0.02 [220] , measured using an inclusive analysis with a lepton tag. Table 14. CP Asymmetries for exclusive and inclusive b → sγ transitions for B0, B± and B0/B± admixture. The B-factories also attempted to measure the polarization of the photon in the b → s γ transition, which can provide a test of the Standard Model, which predicts the photon to be mainly left-handed [214,215]
. The measurements rely on either the exploitation of the B0 - 5.3 B → ℓℓFlavor changing neutral current (FCNC) decays proceed only through loop diagrams in the Standard Model, and are further reduced
by helicity and Glashow-Iliopoulos-Maiani (GIM) suppression. Thus even the largest branching fraction predicted by the Standard
Model, Table 15. Summary of upper limits for B → ℓℓ decays. 5.4 B → XsℓℓThe physics of B → Xsℓℓ is governed by the Wilson coefficients C7, C9 and C10, which describe the strengths of the corresponding short-distance operators in the effective Hamiltonian, i.e. the electromagnetic operator O7 and the semileptonic vector and axial-vector operators O9 and O10, respectively [213] . The Wilson coefficients are experimental observables. Contributions from new physics appear in the experiment as deviations from the SM values, which have been calculated to next-to-next-to-leading order (NNLO). The experimental knowledge on the Wilson coefficient C7 comes from the inclusive B → Xsγ branching fraction, which determines its absolute value to about 20% accuracy, but not its sign. The partial b → Xsℓℓ decay rate in the lepton invariant mass range below the J/ψ resonance is sensitive to the sign of C7. Measurements of the inclusive b → Xsℓℓ decay rate have been published by Belle [225] and BaBar [226] (see Table 16), who also report a direct CP asymmetry compatible with zero, see Table 17. Table 16. Summary of inclusive and exclusive b → Xsℓℓ branching fractions. Table 17. CP Asymmetries for exclusive and inclusive b → Xsℓℓ transitions for B0, B± and B0/B± admixture. The inclusive B → Xsℓℓ branching fraction, which constrains C9 and C10 [227] , gives no information on the individual signs and magnitudes of these coefficients. To further pin down the values of these coefficients, it is necessary to exploit interference effects between the contributions from different operators. This is possible in B → Xsℓℓ decays by evaluating the differential inclusive decay rate as a function of the lepton invariant mass, q2 = m(ℓℓ)2, or by measuring the forward-backward asymmetry in the exclusive decay B → K*ℓℓ. The forward-backward asymmetry in B → K*ℓℓ, defined as
is a function of q2 and of
where ξ is a function of q2, and F1,2 are functions of form factors. It is straightforward to determine C10, Re(C9) and the sign of C7 from the AFB distribution as a function of q2, using the value of |C7| from B → Xsγ and a few more assumptions: phases of C10 and C7 are neglected, and higher order corrections are known. These assumptions should be examined by comparing the results with the inclusive B → Xsℓℓ differential branching fraction as a function of q2, since it is also sensitive to C9 and C10 in a different way. Most of the eight individual B → K(K*)ℓℓ modes have been established. Both experiments have searched for asymmetries with respect to lepton flavour. Currently available data clearly favour a negative sign for C7, as predicted by the Standard Model. In some SUSY scenarios the sign of the b → s γ amplitude (C7) can be opposite to the SM prediction, while the transition rate may be the same. Within the SM there is a zero crossing point of the forward-backward asymmetry in the low q2 region, while it disappears with the opposite sign C7 if the sign of Re(C9) is the same as in the SM. In another model with SU (2) singlet down-type quarks, tree-level Z flavor-changing-neutral-currents are induced. In this case, the larger effect is expected on the axial-vector coupling (C10) to the dilepton than on the vector coupling (C9). Because the forward-backward asymmetry is proportional to the axial-vector coupling, the sign of the asymmetry can be opposite to the SM. The same new physics effect is also a possibility for B → ϕ KS where anomalous mixing-induced CP violation can occur. Belle attempted to extract C9 and C10 from AFB in B → K*ℓℓ with 357 fb-1 data [228] . The Belle analysis constrains the sign of the product C9C10 to be negative as in the SM. In this study higher order QCD correction terms are assumed to be the same as in the Standard Model. Only the leading order terms, C9 and C10, are allowed to float in the fit to the data. Since B0 → K0*ℓ+ℓ- is an all charged particle final state, LHCb may be able to measure the zero crossing point with a better precision than a B factory. However, a model independent analysis requires measurement of the forward-backward asymmetry in B → Xsℓℓ, which is only possible at a super-B factory. The precision of the B → K(K*)ℓℓ branching fractions is dominated by the theoretical errors, which have large model dependent irreducible uncertainties
in the form-factors, which can be as large as a factor two. These uncertainties are much smaller if the ratio Therefore, RK = Finally, the transition b → s ν 6 Current status of overall CKM fitsWe have seen that there are no startling departures from the Standard Model. Yet, we need to formulate quantitatively how
restrictive the measurements are. First of all, we can demonstrate that the data are consistent among the different measurements
by plotting the extracted values of Table 18. Values of Wolfenstein parameters from CKM fits assuming the Standard Model
To search for deviations from the SM it is useful to compare measurements involving CP violating loop diagrams, where imaginary parts of amplitudes come into play, with measurements of the sides of triangle. The
UT fit group shows in Figure 28 a comparison in the
It remains to set limits on New Physics. We allow for NP to contribute to the mixing amplitude of either Bd or Bs mesons, which we consider separately. Then the amplitude of this second order (ΔF = 2) flavor changing interaction can be expressed as
where q specifies the type of neutral B meson, and Δq gives the relative size of NP to SM physics. (The SM point has Re(Δq) = 1 and Im(Δq) = 0.) To set limits it is necessary to make some simplifying, but very reasonable, assumptions (this type of analysis was done with somewhat different assumptions earlier by [230] ). We assume that there is no NP in the tree level observables starting with the magnitudes |Vud|, |Vus|, |Vcb|, and |Vub|. Since since γ is measured using tree level B∓ decays we include it in this category. Now we incorporate the measurement of β by noting that α = π - γ - βmeas, allowing that the measured βmeas = βSM - Arg(Δd). Also used are measurements of the semileptonic asymmetries (see Eq. 34) (measurement of ASL for Bs mesons are given in [231,232]
; see also [233]
), and the constraint equation
In both case quite large values of the real part of NP amplitudes are allowed. In the Bs case a large negative value of the imaginary part is also preferred, but again the effect is only 2.7σ. We can conclude that we need to do a lot more work to constrain NP in these transitions. 7 ConclusionThe goals of our studies are to elucidate the features of the electroweak interactions and the fundamental constituents of nature. The b was discovered at fixed target experiment in a proton beam, by the first observation of narrow ϒ states in 1977. Experiments at several e+e-colliders have increased our knowledge enormously. CLEO provided the first fully reconstructed B mesons and measurements of their mass. The Mark II and MAC experiments at PEP gave the first measurements of the b lifetime, which were confirmed by PETRA experiments. ARGUS first measured B0 mixing. CLEO, ARGUS and the LEP experiments competed on the first precision measurements of the CKM parameters Vcb and Vub. The two most recent, BaBar and Belle, have provided measurements of CP violation. The hadron collider experiments have measured Bs mixing and are starting to probe CP violation in the Bs system. The torch will soon be passed to first hadron collider experiment designed to measure b decays, LHCb. The ATLAS and CMS experiments will also contribute. In the 31 years since the discovery of the b quark we have learned a great deal about the physics of b-flavored hadrons. We have seen that the Standard Model describes most decays well, yet studies continue as any new particles that we believe must exist have influence, especially on rare and CP violating B decays. We expect that future experiments will learn a great deal about these new particles either by observing their effects on B decays, or by seeing that they do not have observable effects. The latter case is one that experimentalists do not like, but it can be just as important in firming up our understanding of the basic constituents of matter and the forces that operate between them. AcknowledgementsWe thank Phillip Urquijo and Antonio Limosani for useful discussions concerning the CKM elements. M. Artuso and S. Stone thank the U. S. National Science Foundation for support. References
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