Abstract
We 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 V_{cb }and V_{ub }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 Introduction
The 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 θ_{C}, where the "Cabibbo" angle θ_{C }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 decays in 1964 [2,3]. (In CP, C stands for charge conservation invariance and P for parity invariance. The operation of C takes a particle to antiparticle state and the operation of P changes left to right. [46].) When the c quark was discovered in Nov. 1974 [7,8] (though its existence was speculated earlier [911]), it became clear that θ_{C }was the mixing angle between two quark doublets, (c, s) and (u, d). However, it is not possible to generate a CP violating phase with only two quark doublets.
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 Model
First we will discuss how particles formed with bquarks fit into current paradigm of particle physics, the "Standard Model" (SM) [1315]. 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 spin1/2 quarks and leptons and the force carriers generally spin1 objects. The spin0 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 B^{0}, B^{}, B_{s}, or B_{c }mesons, depending on whether the light antiquark that it pairs with is , , , or or a baryon containing two other quarks. Mesons containing b and quarks are also interesting especially for studies of quantum chromodynamics, but will not be discussed further in this article, as we will concentrate on weak decays and discuss strong interactions as an important and necessary complication that must be understood in many cases to extract information on fundamental b quark couplings.
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 λ^{4}:
In the Standard Model A, λ, ρ, and η are fundamental constants of nature like G, or α_{EM}; 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 and are used where
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 ϕ_{2 }≡ α, ϕ_{1 }≡ β, and ϕ_{3 }≡ γ.)
Figure 1. The 6 CKM triangles resulting from applying unitarity constraints to the indicated row and column. The CP violating angles are also shown.
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 V_{cb }and V_{ub }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^{ } and will be discussed in detail later. The W exchange Figure 2(d) diagram is small. The box diagram Figure 2(e) is the source of mixing in the B system. An analogous diagram exists also for the B_{s }meson. Finally the Penguin diagram Figure 2(f) is an example of one of several loop diagrams leading to "rare decays" that will be discussed in detail in a subsequent section.
Figure 2. Some B decay diagrams.
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 Baryogenesis
When 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 () number violation, departure from thermal equilibrium, and C and CP violation [26]. (The operation of Charge Conjugation (C) takes particle to antiparticle and Parity (P) takes a vector to  .)
These criteria are all satisfied by the Standard Model. is violated in Electroweak theory at high temperature, though baryon minus lepton number is conserved; in addition we need quantum tunneling, which is powerfully suppressed at the low temperatures that we now have. Nonthermal equilibrium is provided by the electroweak phase transition. C and CP are violated by weak interactions. However the violation is too small. The ratio of the number of baryons to the entropy in the observed part of the Universe needs to be ~5 × 10^{11}, while the SM provides many orders of magnitude less. Therefore, there must be new physics [27].
1.1.3 The Hierarchy Problem
Our worry is why the Planck scale at ~10^{19 }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 10^{19 }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 Physics
When 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 B^{0 }→ J/ψ K_{S }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. B^{0 }→ ϕ K_{S}.
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 violation
2.1 Neutral B Meson Mixing
Neutral B mesons can transform into their antiparticles before they decay. The diagrams for this process are shown in Figure 3 for the B_{d}. There is a similar diagram for the B_{s}. 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.
Figure 3. The two diagrams for B_{d }mixing.
Under the weak interactions the eigenstates of flavor degenerate in pure QCD can mix. Let the quantum mechanical basis vectors be {1⟩, 2⟩} ≡ {B^{0}⟩, ⟩}. The Hamiltonian is then
The Schrödinger equation is
Diagonalizing we have
where H refers to the heavier and L the lighter of the two weak eigenstates, and ϕ = arg(M_{12}/Γ_{12}). We expect that ΔΓ is very small for B^{0 }mesons but should be significant for B_{s }mesons.
B_{d }mixing was first discovered by the ARGUS experiment [31] (There was a previous measurement by UA1 indicating mixing for a mixture of and [32]). At the time it was quite a surprise, since the topquark mass, m_{t}, was thought to be in the 30 GeV range. Since bflavored hadrons are produced in pairs, it is possible to observe a mixed event by having both B's decay semileptonically. Thus the number of events where both Bs decay into leptons of the same sign is indicative of mixing. We can define R as the ratio of events with samesign leptons to the sum of samesign plus oppositesign dilepton events. This is related to the mixing probability. The OPAL data for R are shown in Figure 4[33].
Figure 4. The ratio, R, of samesign to total dilepton events as a function of proper decay time, for selected B → D*+Xℓ^{} events. The jet charge in the opposite hemisphere is used to determine the sign correlation. The curve is the result of a fit to the mixing parameter. The.
Data from many experiments has been combined by "The Heavy Flavor Averaging Group," (HFAG) to obtain an average value Δm_{d }= (0.507 ± 0.004) × 10^{12 }ps^{1 }[34].
The probability of mixing is related to the CKM matrix elements as [3,35]
where B_{B }is a parameter related to the probability of the d and quarks forming a hadron and must be estimated theoretically, F is a known function which increases approximately as , and η_{QCD }is a QCD correction, with value about 0.8. By far the largest uncertainty arises from the decay constant, f_{B}.
In principle f_{B }can be measured. The decay rate of the annihilation process B^{ }→ ℓ^{ } is proportional to the product of . Of course there is a substantial uncertainty associated with V_{ub}. The experimental evidence for B^{ }→ τ^{ } is discussed in Section 5.1, and substantial uncertainty exists in the branching ratio measurement. Thus, we need to rely on theory for a value of f_{B}.
The ratio of B_{s }to B_{d }mixing frequency, however, provides a better situation in terms of reducing model dependent errors. Using Eq. 7 for B_{d }mixing and an analogous relation for B_{s }mixing and then dividing them results in
The CKM terms are
ignoring the higher order term in V_{ts}. Solving the ratio of the two above equations for gives a circle centered at (1,0) in the ρ  η plane whose radius depends on x_{d}/x_{s}. The theoretical errors are now due to the difference between having a light quark versus a strange quark in the B meson, called SU(3) splitting.
For many years experiments at the Z^{0 }using e^{+ }e^{ }colliders at both LEP and the SLC had set lower limits on mixing [17]. In 2006 B_{s }mixing was measured by the CDF collaboration [36,37]. The D0 collaboration had previously presented both a lower and an upper limit [38].
We now discuss the CDF measurement. The probability, (t) for a B_{s }to oscillate into a is given as
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.
Figure 5. The measured amplitude values and uncertainties versus the B_{s } oscillation frequency Δm_{s }for a combination of semileptonic and hadronic decay modes from CDF.
At Δm_{s }= 17.75 ps^{1}, the observed amplitude A = 1.21 ± 0.20 (stat.) is consistent with unity, indicating that the data are compatible with B_{s } oscillations with that frequency, while the amplitude is inconsistent with zero: A/σ_{A }= 6.05, where σ_{A }is the statistical uncertainty on A, the ratio has negligible systematic uncertainties. (This then is called a "6σ effect.) The measured value is
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 and in Eq. 8. It is interesting that in practice it is not possible to measure either of these numbers directly. It is usually assumed, however, that , which could in principle be measured via the process B^{ }→ τ^{ }ν is the same as f_{B}. There is no way to measure . The charm system, on the other hand, provides us with both D^{+ }→ ℓ^{+ }ν and → ℓ^{+ }ν, and both rates have been measured. They also have been calculated in unquenched lattice quantum electrodynamics (QCD).
The combined efforts of the HPQCD and UKQCD collaborations predict = (207 ± 4) MeV [40], while the CLEO measurement is in astonishing agreement: (205.8 ± 8.5 ± 2.5) MeV [41]. Furthermore, the measurements of are not in such good agreement with the Follana et al calculation of (241 ± 3) MeV. The average of CLEO and Belle results as determined by Rosner and Stone is (273 ± 10) MeV [42]. (For updated results see [43,44].) The discrepancy is at the 3 standard deviation level and not yet statistically significant, though it bears watching.
Unfortunately, the group that has calculated is not the same group that has determined . The theoretical calculations used for the B_{B }terms and the f_{B }terms are summarized by Tantalo [45]. He suggests using values of = (268 ± 17 ± 20) MeV, /f_{B }= 1.20 ± 0.02 ± 0.05, = 0.84 ± 0.03 ± 0.055 and B_{B }= 0.83 ± 0.01 ± 0.06. These numbers allow us to measure the length of one side of the CKM triangle using Eq. 8. Use of this measurement will be discussed in more detail in Section 6.
2.2 CP Violation in the B System
We have two quantum mechanical operators: Charge Conjugation, C, and Parity, P. When applied to a particle wavefunction C changes particle to antiparticle and viceversa. 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 righthanded to lefthanded. If nature was blind to handedness, then P would be always conserved. By applying the P operator twice we end up with P^{2}ψ (r) = ψ (r), so the eigenvalues of P are ± 1. Therefore wavefunctions, or particles represented by such wavefunctions, have either intrinsic positive parity +1 (righthanded) or 1 (lefthanded).
Weak interactions, characterized by a combination of vector minus axialvector currents, are known to be lefthanded. Therefore, handedness matters and its well known that Parity is maximally violated in weak decays [3]. Since C changes lefthanded particles to righthanded antiparticles, 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, and each of which has a strong part e. g. and a weak part . Then we have
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 is unity, so we can get information on the weak phases. In the case of neutral B decays, mixing serves as the second amplitude.
2.2.1 Formalism of CP Violation in Neutral B Decays
Consider the operations of Charge Conjugation, C, and Parity, P:
For neutral mesons we can construct the CP eigenstates
where
Since B^{0 }and can mix, the mass eigenstates are a superposition of aB^{0}⟩ + b⟩ which obey the Schrodinger equation
If CP is not conserved then the eigenvectors, the mass eigenstates B_{L}⟩ and B_{H}⟩, are not the CP eigenstates but are
where
CP is violated if ϵ_{B }≠ 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 = (m_{L }+ m_{H})/2, Δm = m_{H } m_{L }and Γ = Γ_{L }≈ Γ_{H}, and t is the decay time in the B^{0 }rest frame, the socalled "proper time". Note that the probability of a B^{0 }decay as a function of t is given by ⟨B^{0 }(t)B^{0}(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 Decay
Here we choose a final state f which is accessible to both B^{0 }and decays [3]. The second amplitude necessary for interference is provided by mixing. Figure 6 shows the decay into f either directly or indirectly via mixing. It is necessary only that f be accessible directly from either state; however if f is a CP eigenstate the situation is far simpler. For CP eigenstates
Figure 6. Two interfering ways for a B^{0 }to decay into a final state f.
It is useful to define the amplitudes
If ≠ 1, then we have "direct" CP violation in the decay amplitude, which we will discuss in detail later. Here CP can be violated by having
which requires only that λ acquire a nonzero 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 B^{0 }or for B_{s }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 B^{0 } pairs. This puts the B's in a C = 1 state. In hadron colliders, or at e^{+ }e^{ }machines operating at the Z^{0}, the B's are produced incoherently. For the rest of this article we will assume incoherent production except where explicitly noted.
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 B_{d }mesons, as the maximum size of the coefficient for any x is 0.5, close to the measured value of x_{d}.
Let us now find out how Imλ relates to the CKM parameters. Recall . The first term is the part that comes from mixing:
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 K_{s}. The decay diagram is shown in Figure 7. In this case we do not get a phase from the decay part because
Figure 7. The Feynman diagram for the decay B^{0 }→ J/ψ K^{0}.
is real to order 1/λ^{4}.
In this case the final state is a state of negative CP, i.e. CPJ/ψ K_{s}⟩ = J/ψ K_{s}⟩. 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 K_{s }in the final state. Since neutral kaons can mix, we pick up another mixing phase (similar diagrams as for B^{0}, see Figure 3). This term creates a phase given by
which is real to order λ^{4}. 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 K_{s}, Imλ =  sin(2β). The angle β is the best measured CP violating angle measured in B meson decays. The process used is B^{0 }→ J/ψ K_{S}, although there is some data used with the ψ (2S) or with K_{L }(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 (t) has both sin(Δmt) and cos(Δmt) terms, and the coefficients of these terms can be measured. Let us assign the labels
(Note that the sign of the term changes depending on the CP of the final state, but not the term.)
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) → B^{0 }, and with one of the neutral B's decaying into J/ψ K^{0}. Because the ϒ(4S) is a definite quantum state with C = 1, the formulae given above have to modified somewhat. One important change is the definition of t. The time used here is the difference between the decay time of the J/ψ K^{0 }and the other B^{0}, also called the "tagging" B because we need to determine its flavor, whether B^{0 }or , in order to make the asymmetry measurement.
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 bquark, part of a meson decays semileptonically into an ℓ^{}, while a quark decays into an e^{+}. Other flavor tagging observables include the charge of kaons and fast pions. Modern experiments combine these in a "neural network," to maximize their tagging efficiencies [49] (see also [50]). The BaBar data are shown in Figure 8, Belle has similar results.
Figure 8. (a) The number of J/ψ K_{s }candidates in the signal region with either a B^{0 }tag , or a tag as a function of Δt. (b) The raw asymmetry, . The solid (dashed) curves represent the fit projections as functions of Δt for both B^{0 }and tags. The shaded regions represent the estimated background contributions.
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 nonzero term with the measured average given as 0.005 ± 0.020.
Determining the sine of any angle gives a fourfold ambiguity in the angle. The decay mode B^{0 }→ J/ψ K*^{0}, K*^{0 }→ K_{s }π^{0 }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 degrees.
2.2.4 Measurements of α
The 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 nonzero value for , the coefficient of the cosine term in Eq. 43. For this b → ud process we have
Figure 9. Tree (a) and Penguin (b) processes for neutral B decay into either π^{+}π^{ }or ρ^{+}ρ^{}.
and
Time dependent CP violation measurements have been made by both the BaBar and Belle collaborations [53,54]. Both groups find a nonzero value of both and , though their values are not in particularly good agreement. The HFAG average is shown in Figure 10 along with the experimental results. The value of clearly is not zero, thus demonstrating direct CP violation. (Historically, this was an important observation because it showed that CP violation could not be due to some kind of "superweak" model ala Wolfenstein [55].)
Figure 10. Coefficients of the sine term and the cosine term in time dependent CP violation for neutral B decay into π^{+}π^{}, showing BaBar and Belle results, and the HFAG average. Contours are shown at 60.7% confidence level.
The nonzero value of shows the presence of at least two amplitudes, presumably tree and penguin, in addition to the mixing amplitude, making extraction of α difficult. All is not lost, however. Gronau and London [56] showed that α could be extracted without theoretical error by doing a full isotopic spin analysis of the π π final state. The required measurements include: , , (B^{+ }→ π^{+}π^{0}), (B^{0 }→ π^{0}π^{0}), and ( → π^{0}π^{0}). The last two items require a flavored tagged analysis that has not yet been done and whose prospects are bleak. Grossman and Quinn have showed, however, that an upper limit on Γ (B^{0 }→ π^{0}π^{0})/Γ(B^{+ }→ π^{+}π^{0}) can be used to limit the penguin shift to α [5759]. Unfortunately, current data show a relative large (B^{0 }→ π^{0}π^{0}) = (1.55 ± 0.19) × 10^{6 }rate, compared with (B^{0 }→ π^{+ }π^{}) = (5.16 ± 0.22) × 10^{6}, implying a large penguin contribution [17,34], and the limit is very weak.
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 vectorvector 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 [61], and Belle measures it as [62]. Thus we can treat this final state without worrying about the angular analysis and just apply a correction for the amount of transverse amplitude.
In addition, ρ mesons are wide, so nonB backgrounds could be a problem and even if the proper B is reconstructed, there are nonresonant and possible a_{1 }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 ρ^{0 }ρ^{0}, 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 B^{0 }→ ρ^{+ }ρ^{ }and → ρ^{+ }ρ^{ }and their difference are shown in Figure 11 from BaBar. In is interesting to compare these results with those in Figure 8. We see that the measured asymmetry in the ρ^{+ }ρ^{ }decay more or less cancels that from B^{0 } mixing inferring that sin(2α) is close to zero.
Figure 11. The number of ρ^{+}ρ^{ }candidates in the signal region with either a B^{0 }tag (a) , or a tag (b) as a function of Δt. (c) The raw asymmetry, . The dashed curves represent the estimated backgrounds.
Results from the time dependent CP violation analysis from both Belle and BaBar are shown in Figure 12.
Figure 12. Coefficients of the sine term and the cosine term in time dependent CP violation for neutral B decay into ρ^{+}ρ^{ }showing BaBar and Belle results, and the HFAG average. Contours are shown at 60.7% confidence level.
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.
Figure 13. The probability density, defined as 1 – CL (confidence level) for the angle α as determined by measurements of B → ρ ρ.
The final state π^{+ }π^{ }π^{0 }can also be used to extract α. Snyder and Quinn proposed that a Dalitz plot analysis of B → ρ π → π^{+ }π^{ }π^{0 }can be used to unravel both α and the relative penguintree phases [65]. The Dalitz plot for simulated events unaffected by detector acceptance is shown in Figure 14.
Figure 14. Dalitz plot for MonteCarlo generated B^{0 }→ ρ π → π^{+}π^{}π^{0 }decays. The decays have been simulated without any detector effect and the amplitudes for ρ^{+}π^{}, ρ^{}π^{+ }and ρ^{0}π^{0 }have all been generated with equal magnitudes in order to have destructive interferences where the ρ bands overlap. The main overlap regions between the ρ bands are indicated by the hatched areas. Adapted from [66].
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^{+ }π^{0 }and B^{+ }→ ρ^{0 }π^{+ }coupled with isospin relations [68,69] to help constrain α.
Figure 15. The experimental confidence levels for α as determined separately by the Belle collaboration (left) and the BaBar collaboration (right). The dotted curve for Belle shows the result without using constraints from charged B to ρ π final states.
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].
Figure 16. The experimental confidence levels for α as determined separately by the CKM fitter and UT fit groups.
2.2.5 The Angle γ
The angle . It can be viewed as the phase of the V_{ub} matrix element, with respect to V_{cb}. Interference measurements are required to determine phases. We can use neutral or even charged B decays to measure γ, and there are several ways to do this without using any theoretical assumptions, as is the case for α and β. The first relies on the color suppressed tree diagrams shown in Figure 17, and the second on using the interference in B_{s }mixing. At first glance, the diagrams in Figure 17 don't appear to have any interfering components. However, if we insist that the final state is common to D^{0 }and , then we have two diagrams which differ in that one has a b → c amplitude and the other a b → u amplitude, the relative weak phase is then γ. Explicitly the amplitudes are defined as
Figure 17. Two diagrams for a charged B decay into a neutral D meson and a charged kaon.
where r_{B }reflects the amplitude suppression for the b → u mode and δ_{B }is the relative phase. We have not yet used identical final states for D^{0 }and decays, yet we can see that there will be an ambiguity in our result between δ_{B }and γ that could be resolved by using different D^{0 }decay modes.
There are several suggestions as to different D^{0 }decay modes to use. In the original paper on this topic Gronau and Wyler [72,73] propose using CP eigenstates for the D^{0 }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 δ_{B }of π between γ D^{0 }and π^{0 }D^{0 }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 [7579]. These data, however, are not statistically powerful enough by themselves to give a useful measurement of γ. Atwood, Dunietz and Soni suggested using doubleCabibbo suppressed decays as an alternative means of generating the interference between decay amplitudes [80]. For example the final state B^{ }→ D^{0 }K^{ }can proceed via the tree level diagram in Figure 2(a) when the W^{ }→ s. Then if we use the doublyCabibbo suppressed decay D^{0 }→ K^{+ }π^{}, we get a final state that interferes with the K^{ }final state with the → K^{+ }π^{}, which is Cabibbo allowed. The advantage of this method, besides extending the number of useful final states, is that both amplitudes are closer to being equal than in the above method, which can generate larger interferences. Any doublyCabibbo suppressed decay can be used. Similar equations exist relating the measured parameters to γ and the strong phase shift.
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 threebody final state K_{S }π^{+ }π^{}. Since the final state is accessible by both D^{0 }and decays interference results. By its very nature a threebody state is complicated, and consists of several resonant and nonresonant parts. It is typically analyzed by examining the Dalitz plot where two of the possible three submasses (squared) are plotted versus one another [83]. In this way, the phase space is described as a uniform density and thus any resonant structure is clearly visible. For our particular case a practical method was suggested by Giri et al [84], where the decay is analyzed in separate regions of the Dalitz plot. Results from this analysis have been reported by BaBar [85] (they also include the mode D^{0 }→ K_{S }K^{+ }K^{}) and Belle [86]. BaBar finds γ = (76 ± 22 ± 5 ± 5)°, and Belle finds . In both cases the first error is statistical, the second systematic, and the third refers to uncertainties caused by the Dalitz plot model. (There is also a mirror solution at ± 180°.)
It has been shown [87,88] that measurement of the amplitude magnitudes and phases found in the decays of ψ (3770) → (CP ± Tag)(K_{S }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 D^{0 }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.
Figure 18. The experimental confidence levels for γ as determined separately by the CKM fitter and UT fit groups.
2.2.6 The Angle χ
The angle χ shown in Figure 1 is the phase that appears in the box diagram for B_{s }mixing, similar to the diagram for B^{0 }mixing shown in Figure 3, but with the d quark replaced by an s quark. The analogous mode to B^{0 }→ J/ψ K_{s }in the B_{s }system is B_{s }→ J/ψ η. The Feynman diagrams are shown in Figure 19. This is very similar to measuring β so χ is often called β_{s}. We also have a relation between ϕ_{s }defined in Eq. 33, ϕ_{s }= 2χ.
Figure 19. The Feynman diagram for the decay B_{s }→ J/ψ η or ϕ .
Since there are usually two photons present in the η decay, experiments at hadron colliders, which can perform timedependent studies of B_{s }mesons, preferentially use the J/ψ ϕ final state. This, unfortunately, introduces another complexity into the problem; as the B_{s }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 vectorvector 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 spin0 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 B^{0 }→ 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 B_{s }as
where ϵ_{J/ψ }and ϵ_{ϕ }are polarization 3vectors in the J/ψ rest frame, is a unit vector giving the direction of the ϕ momentum in the J/ψ rest frame, and E_{ϕ }is the energy of the ϕ in the J/ψ rest frame. We note that the corresponding amplitude for the decay are = A_{0}, = A_{}, and = A_{⊥}. The amplitudes are normalized so that
The ϕ meson direction in the J/ψ rest frame defines the direction. The direction is perpendicular to the decay plane of the K^{+ }K^{ }system, where p_{y }(K^{+}) ≥ 0. The decay direction of the ℓ^{+ }in the J/ψ rest frame is described by the angles (θ, ϕ). The angle ψ is that formed by the K^{+ }direction with the axis in the ϕ rest frame. Figure 20 shows the angles.
Figure 20. Pictorial description of the decay angles. On the left θ and ϕ defined in the J/Ψ rest frame and on the right Ψ defined in the ϕ rest frame. (From T. Kuhr [235].).
The decay width can be written as
The decay rate for can be found by replacing A_{⊥ }in the above expression with A_{⊥}.
Another complexity arises from the expectation that the width difference ΔΓ_{s}/Γ_{s }≈ 15%. This complicates the time dependent rate equations. For convenience, setting ≡ (cos θ, ϕ, cosψ), we have for the decay width for B_{s}:
where
The quantities δ_{⊥ }and δ_{ }are the strong phases of A_{⊥ }and A_{ }relative to A_{0}, respectively [92]. The expression for mesons can be found by substituting → and → .
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.
Figure 21. Constraints at 68% confidence level in the (ϕ_{s}, ΔΓ_{s}) plane. Overlaid are the constraints from the CDF and D0 measurements, the constraint from , the constraint from the flavor specific B_{s }lifetime [34], and the overall combination. The SM prediction is also given. From [95,96].
The Standard Model allowed region is a very thin vertical band centered near zero at ϕ_{s }of 0.036 ± 0.002 (shown in red). (Recall ϕ_{s }≡ 2χ ≡ 2β_{s}.) 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 B_{s }→ J/ψ ϕ, including use of measured total widths introduced via the constraint equation , it is the measurement of ϕ_{s }that dominates the fit result. We see that there is a discrepancy that may be as large as 2.7 standard deviations [95,96]. While this is as not yet significant, it is very tantalizing. The LHCb experiment plans to vastly improve this measurement [97].
It has been pointed out, however, that there is likely an Swave K^{+ }K^{ }contribution in the region of the ϕ that contributes 5–10% of the event rate as estimated using decays [98]. For example, analysis of the analogous channel B^{0 }→ J/ψ K*^{0 }reveals about 8% Swave in the K_{π }system under the K*^{0}, and BaBar has used this to extract a value for cos 2β thus removing an ambiguity in β [99]. The Swave amplitude and phase needs to be added to Eq. 53. Note that the errors will increase due to the addition of another amplitude and phase. The Swave can manifest itself as a π^{+ }π^{}, so it is suggested that the decay B_{s }→ J/ψ f_{0 }(980) be used to measure χ; here the f_{0 }→ π^{+ }π^{ }[98]; the estimate is that the useful f_{0 }rate would be about 20% of the ϕ rate, but the J/ψ f_{0 }is a CP eigenstate, so an an angular analysis is unnecessary, and these events may provide a determination of χ with an error comparable to that using J/ψ ϕ.
2.2.7 Measurements of Direct CP Violation in B → K π Decays
Time 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 J^{PC }= 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 twobody 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 colorsuppressed 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]
Figure 22. Processes for → K^{}π^{+ }and B^{ }→ K^{}π^{0}, (a) via tree and (b) Penguin diagrams, and B^{ }→ K^{}π^{0}(c) via colorsuppressed tree and (d)"Electroweak" Penguin diagrams.
The Belle data are shown in Figure 23.
Figure 23. The beam constrained mass distributions from Belle in units of GeV for the four different indicated final states. The dotted curves indicate peaking backgrounds, the dotdashed curves the signal and the solid curves the sum. The differences in event numbers between the charge conjugate modes are apparent.
The difference between and is not naively expected in the Standard Model and Belle suggests that this may be a sign of New Physics. Peskin commented on this possibility [103].
2.3 Conclusions from CP Violation Measurements
All 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 = A^{2 }λ^{6 }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 V_{cb}
There are two experimental methods to determine V_{cb}: the exclusive method, where V_{cb} is extracted by studying the exclusive → D^{(*) }ℓ^{} decay process; and the inclusive method, which uses the semileptonic decay width of bhadron decays. In both methods, the extraction of V_{cb} is systematics limited and the dominant errors are from theory. The inclusive and exclusive determinations of V_{cb} rely on different theoretical calculations of the hadronic matrix element needed to extract it from measured quantities, and make use of different techniques which, to a large extent, have uncorrelated experimental uncertainties. Thus, the comparison between inclusive and exclusive decays allows us to test our understanding of hadronic effects in semileptonic decays. The latest determinations differ by more than 2σ, with the inclusive method having a stated error half of the size of the exclusive one.
3.0.1 Beauty Quark Mass Definitions
Due to confinement and the nonperturbative 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 bquark mass (m_{b}) is a schemedependent, 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 α_{s}. Therefore, higherorder 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 gaugeinvariant and infraredsafe [106108] 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 orderdependent 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 Λ_{QCD }renormalon of the pole mass. Such quark mass definitions are generically called "shortdistance" masses.
• mass: The most common shortdistance mass definition is the mass (μ) [111113], where the scale μ is typically chosen to be the order of the characteristic energy scale of the process. In the scheme the subtracted divergencies do not contain any infrared sensitive terms and the mass is only sensitive to scales larger than m_{b}. The mass arises naturally in processes where the bquark is far offshell, but it is less adequate when the bquark has nonrelativistic energies.
• Kinetic mass: The shortcomings of the pole and the mass in describing nonrelativistic bquarks can be resolved by socalled threshold masses [114], that are free of an ambiguity of order Λ_{QCD }and are defined through subtractions that contain universal contributions for the dynamics of nonrelativistic quarks. Since the subtractions are not unique, an arbitrary number of threshold masses can be constructed. The kinetic mass is defined as [115,116]:
where μ_{kin }is the nominal kinetic mass renormalisation scale. For μ_{kin }→ 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 nonrelativistic bquark dynamics. The 1S mass [117,118] achieves the same task without a factorisation scale, since it is directly related to a physical quantity. The bquark 1S mass is defined as half of the perturbative contribution to the mass of the ϒ(S_{1}) in the limit m_{b }≫ m_{b }v ≫ m_{b }v^{2 }≫ Λ_{QCD}.
A list of bquark mass determinations, converted into the mass scheme is shown in Table 2.
Table 2. List of m_{b }determinations converted into the mass scheme.
3.1 Determination Based on Exclusive Semileptonic B Decays
The exclusive V_{cb} 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 leptonν pair, q^{2}. 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 fourvelocity transfer (w) space as opposed to fourmomentum transfer space, because at maximum fourvelocity transfer, where w equals one, the formfactor in lowest order is unity, i.e. the b transforms directly into a c quark without any velocity change. The value of V_{cb} can be extracted by studying the decay rate for the process → D* + ℓ^{ }_{ℓ }as a function of the recoil kinematics of the D*^{+ }meson. Specifically, HQET predicts that:
where w is the product of the fourvelocities of the D*^{+ }and the mesons, related to q^{2}. (w) is a known phase space factor and the form factor (w) is generally expressed as the product of a normalization factor and a shape function described by three form factors, constrained by dispersion relations [120]:
where ; The linear slope of the formfactor is given by the parameter ρ^{2}, and must be determined from the data. In the infinite mass limit, (w = 1) = (w = 1) = 1; for finite quark masses, nonperturbative effects can be expressed in powers of 1/m_{Q}. There are several different corrections to the infinite mass value (1) = 1:
Note that the first term in the nonperturbative expansion in powers of 1/m_{Q }vanishes [121]. QED corrections up to leading logarithmic order give η_{A }= 0.960 ± 0.007. Different estimates of the 1/ corrections, involving terms proportional to 1/, 1/ and 1/(m_{b }m_{c}), have been performed in a quark model with QCD sum rules, and, more recently, with an HQET based lattice gauge calculation. The best estimate comes from lattice QCD, (1) = (1) = 0.921 ± 0.013 ± 0.020 [122]. This result does not include a 0.7% QED correction.
Since the phasespace factor (w) tends to zero as w → 1, the decay rate vanishes and the accuracy of the V_{cb} value extracted with this method depends upon experimental and theoretical uncertainties in the extrapolation. Experiments determine the product (1)·V_{cb}^{2 }by fitting the measured dΓ/dw distribution.
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 Z^{0 }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 leptonν pair with masses higher than the D and D*, commonly identified as 'D**'. Bfactories 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 V_{cb} by performing a twoparameter fit, for (1)V_{cb} and the slope ρ^{2}. The first measurements of both ratio R_{1 }and R_{2}, and ρ^{2}, were made by the CLEO collaboration [124]. Belle [125] and BaBar [126] improved upon these measurements using the → D*^{+ }e^{ }_{e }decay. They determined R_{1}, R_{2}, and ρ^{2 }using an unbinned maximum likelihood fit to the full decay distribution. BaBar and CLEO results are combined to give: R_{1 }= 1.396 ± 0.060 ± 0.035 ± 0.027, R_{2 }= 0.885 ± 0.040 ± 0.022 ± 0.013, and ρ^{2 }= 1.145 ± 0.059 ± 0.030 ± 0.035. The stated uncertainties are the statistical from the data and systematic uncertainty, respectively.
Table 3 summarizes the available data. Values of (1)V_{cb} from different experiments can be combined if they are extracted using the same (w) parametrization. All measurements included in the exclusive V_{cb} world average relying on the above form factor ratios R_{1 }and R_{2}. Using the (1)V_{cb} world average in Table 3 and (1) = 0.921 ± 0.013 ± 0.020 [122], we find
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 (w) is the phase space factor and the form factor (w) is generally expressed as the product of a normalization factor (1) and a function, g_{D }(w), constrained by dispersion relations [120]. The strategy to extract (1)V_{cb} is identical to that used for the B → D⋆ℓν decay. However, dΓ_{D}/dw is more heavily suppressed near w = 1 than due to the helicity mismatch between initial and final states. Moreover, this channel is much more challenging to isolate from the dominant background B → D⋆ℓν as well as from fake Dℓ combinations. Table 4 shows the results of twodimensional fits to V_{cb}(1) and ρ^{2 }for different experiments and the world average.
Table 4. Experimental results from B → Dℓν using corrections to the common inputs and world averages.
In the limit of infinite quark masses, (w = 1) coincides with the IsgurWise function [129]. In this case there is no suppression of 1/m_{Q }corrections and QCD effects on (1) are calculated with less accuracy than (1). Corrections to this prediction have recently been calculated with improved precision, based on unquenched lattice QCD [130], specifically (1) = 1.074 ± 0.018 ± 0.016. Using this result we get
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^{ }→ D^{0 }ℓν and B^{ }→ D⋆^{0}ℓν to extract the ratio (1)/(1) = 1.23 ± 0.09 [131], compatible with the lattice theory prediction of 1.16 ± 0.04.
3.2 B → D**ℓν Decays
It 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, , , and D_{1}.) The measurements of these branching fractions require a good understanding of the different D^{(*)}nπ systems, which can be either resonant or nonresonant, each with characteristic decay properties.
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 J^{P }= 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 Dwave 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/c^{2}. On the contrary, j = 1/2 states can have J^{P }= 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/c^{2}.
The ALEPH [132], CLEO [133], DELPHI [132], and D0 [134] experiments have reported evidence of the narrow resonant states (D_{1 }and ) in semileptonic decays, whereas more recent measurements by the BaBar [135] and Belle [136] experiments provide semiinclusive measurements to D^{(*)}π ℓν final states [137,138].
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 B^{0 }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 and decay channels, these results disagree with the prediction of QCD sum rules. However, Belle set an upper limit for the channel below the rate measured by BaBar and DELPHI. More measurements need to be performed to elucidate this puzzle.
3.3 Determination Based on Inclusive Semileptonic B Decays
Inclusive determinations of V_{cb} 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 V_{cb}, the bquark mass m_{b}, and nonperturbative matrix elements. The spectator model decay rate is the leading term in a welldefined expansion controlled by the parameter Λ_{QCD}/m_{b }[142146] with nonperturbative corrections arising to order 1/. The key issue in this approach is the ability to separate perturbative and nonperturbative corrections (expressed in powers of α_{s}). Thus the accuracy of an inclusive determination of V_{cb} is limited by our knowledge of the heavy quark nonperturbative matrix elements and the b quark mass. It also relies upon the assumption of local quark hadron duality. This is the statement that a hadronic matrix element can be related pointwise to a theoretical expression formulated in terms of quark and gluon variables.
Perturbative and nonperturbative corrections depend on the m_{b }definition, i.e. the nonperturbative expansion scheme, as well as the nonperturbative matrix elements that enter the expansion. In order to determine these parameters, Heavy Quark Expansions (HQE) [139,146,147] express the semileptonic decay width Γ_{SL}, moments of the lepton energy and hadron mass spectra in B → X_{c }ℓν decays in terms of the running kinetic quark masses and as well as the bquark mass in the 1S expansion scheme. These schemes should ultimately yield consistent results for V_{cb}. The precision of the bquark mass is also important for V_{ub}.
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 → X_{c }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 , and , , ρ_{D }and ρ_{LS }are nonperturbative matrix elements of local operators in HQET. To make use of equations such as Eq. 61, the values of the nonperturbative expansion parameters must be determined experimentally. Although some of these parameters are related to the mass splitting of pseudoscalar and vector meson states, most nonperturbative parameters are not so easily obtained. Measurements of the moments of different kinematic distributions in inclusive Bdecays are used to gain access to these parameters. The first moment of a distribution is given by:
corresponding to the mean. Subsequent (central) moments are calculated around the first moment,
corresponding to a distribution's width, kurtosis and so on.
A → X_{c }ℓ decay observable calculated with the OPE is a double expansion in terms of the strong coupling α_{s }(m_{b}) and the ratio Λ_{QCD}/m_{b}. Observables are typically calculated with the cuts used in the experimental determination for background suppression, and enhanced sensitivity such as the lepton energy cut. The spectral moments are defined as:
where
The measurement of V_{cb} 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 , directly examining the dependence of the various coefficient functions f_{i }on these terms. The possibility of deviations from the OPE predictions due to quark hadron duality violations have been raised [154]. To compare the OPE predictions with data, one also has to define how uncertainties from 1/ corrections are estimated. These uncertainties are hard to quantify reliably, as the only information on the size of the matrix elements of the dimension six operators comes from dimensional analysis. The only method to check the reliability of V_{cb} extraction from inclusive semileptonic decays is how well the OPE fits the data.
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 → X_{c}ℓν 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 as a function of the lepton momentum cuts. CLEO performs a fit for the contributions of signal and background to the full threedimensional differential decay rate distribution as a function of the reconstructed quantities q^{2}, , cos θ_{ℓ}. Belle [155,156] and BaBar [148] use a sample where one of the B mesons, produced in pairs from ϒ(4S) decays, is fully reconstructed and the signal side is tagged by a well identified lepton. The 4momentum p_{X }of the hadronic system X, recoiling against the lepton and neutrino, is determined by summing the 4momenta of the remaining charged tracks and unmatched clusters. Belle reconstructs the full hadronic mass spectrum and measures the first, second central and second noncentral moments of the unfolded spectrum in B → X_{c}ℓν, for lepton energy thresholds, E_{min}, varying from 0.7 to 1.9 GeV [149]. BaBar extracts the moments from the measured distributions using a calibration curve derived from Monte Carlo data, with a minimum momentum for the electron in the B meson rest frame of 0.9 GeV. The latest BaBar analysis [149] measures the first, second central and second noncentral moments of the spectrum for E_{min }from 0.9 to 1.9 GeV. The main systematic errors originate from background estimation, unfolding and signal model dependence.
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 bquarks 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 p_{ℓ }≥ 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 p_{ℓ }≥ 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 B^{0 }and B^{+ }weighted average partial branching fractions and the first four moments of the electron energy spectrum in B → X_{c }e ν, for E_{min }from 0.4 to 2.0 GeV [159]. All the lepton moment measurements are consistent with theory and with the moment of the hadronic and b → s γ photon energy spectrum. Hadronic and lepton energy measurements are consistent within their errors. When compared with theory there is no sign of inconsistencies. The Bfactories provide the most precise HQE parameter estimates.
3.3.1 HQE Parameters
Using the moment measurements described above, it is possible to determine the CKM matrix element V_{cb} and HQE parameters by performing global fits in the kinetic and 1S bquark mass schemes [160]. The photon energy moments in B → X_{s }γ decays [161] are also included in order to constrain the bquark mass more precisely. Measurements that are not matched by theoretical predictions and those with high cutoff energies are excluded (i.e. semileptonic moments with E_{min }> 1.5 GeV and photon energy moments with E_{min }> 2 GeV). The results are preliminary.
The inclusive spectral moments of B → X_{c}ℓν decays have been derived in the 1S scheme up to (1/) [139]. The theoretical expressions for the truncated moments are given in terms of HQE parameters with coefficients determined by theory, as functions of E_{min}. The nonperturbative corrections are parametrized in terms of the following nonperturbative parameters: Λ at (m_{b}), λ_{1 }and λ_{2 }at (1/), and τ_{1}, τ_{2}, τ_{3}, τ_{4}, ρ_{1 }and ρ_{2 }at (1/). In Table 6 one finds the following results for the fit parameters [160]. The first error is from the fit including experimental and theory errors, and the second error (on V_{cb} only) is due to the uncertainty on the average B lifetime. If the same fit is performed to all measured moments of inclusive distributions in B → X_{c}ℓν and B → s γ decays, the V_{cb} and m_{b }values obtained are in Table 6. The fit results for V_{cb} and to B → X_{c}ℓν data only and B → X_{c}ℓν and B → X_{s}γ data combined are displayed in Fig. 24.
Table 6. Experimental results using corrections to the common inputs and world averages for the 1S scheme.
Figure 24. Fit results for V_{cb} and to B → X_{c}ℓν data only (larger blue ellipse) and B → X_{c}ℓν and B → X_{s}γ data combined (smaller red ellipse) for the 1S scheme. The ellipses are for Δχ^{2 }= 1 [236].
Spectral moments of B → X_{c}ℓν decays have been derived up to (1/) in the kinetic scheme [146]. The theoretical expressions used in the fit contain improved calculations of the perturbative corrections to the lepton energy moments [116] and account for the E_{min }dependence of the perturbative corrections to the hadronic mass moments [162]. For the B → X_{s}γ moments, the (biased) OPE prediction and the bias correction have been calculated [147]. All these expressions depend on the b and cquark masses m_{b }(μ) and m_{c }(μ), the nonperturbative parameters, defined at the scale μ = 1 GeV: (μ) and (μ) ((1/)), (μ) and (μ) ((1/)), and α_{s}. The CKM element V_{cb} is a free parameter in the fit, related to the semileptonic width Γ (B → X_{c}ℓν) [142]. The fit results for various inputs and correlations are shown in Table 7.
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 → X_{c}ℓν and B → s γ decays are used in a fit to extract V_{cb} and the b and c quark masses. The V_{cb} and m_{b }values obtained are listed in the Table 7. The default fit also gives m_{c }= 1.16 ± 0.05 GeV, and = 0.27 ± 0.04 GeV^{2}. The errors are experimental and theoretical (HQE and Γ_{SL}) respectively. In this fit the following variations were considered when in the extraction of the HQ parameters, ± 20 MeV for the b and c quark masses, ± 20% for and , ± 30% for the 3^{rd }order non perturbative terms and α_{2 }= 0.22 ± 0.04 for the perturbative corrections. The bias corrections uncertainties for B → s γ were varied by the full amount of their value.
There are open issues relating to the global fits. First of all, the χ^{2}/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 twoloop calculations become available [164166]. In the Kinetic scheme NNLO calculations include an estimate of the nonBLM terms and lead to a roughly 0.6 % reduction of the V_{cb} value 0.25 × 10^{3}. In the 1S scheme the shift on V_{cb} is of about 0.14 × 10^{3 }[167]. From the new power corrections at NLO we expect the chromomagnetic 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 in the pole expansion (may be less with the other schemes) [168].
HQE has been carried out up to 1/ and the effects are expected to be of the order δ^{(4)}Γ/Γ ≈ 0.25% [169]. All these newly calculated results can be used to scrutinize the earlier error estimates. In the kinetic scheme, the full NNLO value for A_{pert }= 0.919, which is in good agreement with A_{pert }= Γ (B → X_{c}ℓν)/Γ (B → X_{c}ℓν)_{tree }= 0.908 ± 0.009. In the 1S scheme the estimated uncertainty from the nonBLM twoloop is half of the BLM part, equivalent to 1.5% of the treelevel rate, more than 3 times the actual correction. The new results are not implemented in the global fits as they are available mainly in the polemass scheme.
3.4 Outlook
The error on the inclusive and exclusive determination of V_{cb }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 → X_{c}ℓν moments only is the value of , which seems to be in disagreement with other determinations, such as the ones in Table 2. (For a detailed comparison between and different mass determinations see A. Hoang, talk at Joint Workshop on V_{ub} and V_{cb} at the BFactories Heidelberg, December 14–16, 2007.)
4 The CKM parameter V_{ub}
The parameter V_{ub} 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_{ℓ}. The relationship between experimental measurements and V_{ub} requires a theoretical prediction of the hadronic form factors governing these decays. The complementary approach relies on measurements of inclusive properties of B meson semileptonic decays. In this case, the hadronic matrix element is evaluated in the context of the OPE.
Both methods pose challenges to both experimenters and theorists. The branching fractions are small, and a substantial background induced by the dominant b → cℓ_{ℓ }needs to be suppressed or accounted for. The large data samples accumulated at the two bfactories, Belle and BaBar, have made possible the development of new experimental techniques that have reduced the experimental errors substantially. In parallel, theorists devoted considerable efforts to devise measurable quantities for which reliable predictions could be produced, and to improve the accuracy of the predictions through refinements in the calculation. Although the precision of the stated errors improved, so did the difference in central values between the inclusive and exclusive estimates of V_{ub}, at least in most determinations. Possible interpretations of this discrepancy will be discussed.
4.1 Determinations Based on Exclusive B Semileptonic Decays
The decay B → πℓ_{ℓ }is the simplest to interpret, as it is affected by a single form factor. The differential decay width is given by
where G_{F }is the Fermi constant, , and f_{+ }(q^{2}) is the relevant form factor.
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 transitions. The advantage of e^{+ }e^{ }Bfactories is that the B decaying semileptonically originates from the decay e^{+ }e^{ }B . Thus, if the companion B is fully reconstructed, the _{ℓ }4momentum can be inferred from the rest of the event.
CLEO pioneered this approach by reconstructing the ν from the missing energy (E_{miss }≡ 2E_{B } Σ_{i }E_{i}), and momentum in the event; in these definitions the index i runs over the well reconstructed tracks and photons, and cuts are applied to enhance the probability that the only missing particle in the event is the . This approach allows the application of a low momentum cut of 1.5 GeV on the lepton for B → πℓ_{ℓ }and 2 GeV for B →ρℓ_{ℓ}. Averaging the results from these two exclusive channels, they obtain . Using their relatively modest full data set (16 fb^{1}) at the centerofmass energy of the ϒ(4S) and considering only the B → πℓ_{ℓ }channel, they get the branching fraction (B → π^{+ }ℓ_{ℓ}) = (1.37 ± 0.15 ± 0.11) × 10^{4}, and [170].
BaBar [171] uses a sample of 206 fb^{1 }to obtain (B → π^{+ }ℓ_{ℓ}) = (1.46 ± 0.07 ± 0.08) × 10^{4}. Recently, the availability of very large data sets at Belle and BaBar have made possible tagged analyses, where semileptonic decays are studied in samples where the other B is fully reconstructed, thus defining the event kinematics even more precisely. The first implementation relies on the the partial reconstruction of exclusive B → D^{(⋆) }ℓ^{+ }ν_{ℓ }decay to tag the presence of a B event [172,173].
Belle uses fully reconstructed hadronic tags, achieving the best kinematic constraints and thus the highest background suppression capabilities and the most precise determination of q^{2}. 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_{ℓ }and the partial branching fraction for q^{2 }≥ 16 GeV^{2 }are shown. The latter partial branching fraction is useful to extract V_{ub} using unquenched lattice calculations, as this is the only q^{2 }interval where their calculation is reliable.
Table 8. Partial and total branching fractions, in units of 10^{4}, for the decay B → πℓ^{+}ν_{ℓ}.
In order to interpret these results, we need theoretical predictions for the form factor f_{+ }(q^{2}). This problem can be split into two parts: the determination of the form factor normalization, f_{+ }(0), and the functional form of the q^{2 }dependence. Form factor predictions have been produced with quark models [174] and QCD sum rule calculations [175]. Lattice calculations provide evaluations of f_{+ }(q^{2}) at specific values of q^{2 }or, equivalently, pion momenta (p_{π}).
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 c_{B } is the residue of the form factor at q^{2 }= , and /α is the squared mass of an effective 1^{ }B⋆' excited state. Ball and Zwicky [175] propose
where the parameters r_{1}, r_{2}, and α are fitted from available data. Lastly, parameterizations that allow the application of constraints derived from softcollinear 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 (p_{B } p_{π})^{2}, defined beyond the physical region, t_{± }= (m_{B }± m_{π})^{2}, and t_{0 }is an expansion point. BGD use t_{0 }= 0.65t_{}. The parameter α_{K }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 N_{f }= 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 (B → π ℓν) for q^{2 }> 16 GeV^{2}, shown in Table 8, we obtain V_{ub} = (3.51 ± 0.096_{exp }± 0.49_{th}) × 10^{3 }with the HPQCD normalization, and V_{ub} = (3.70 ± 0.10_{exp }± 0.37_{th}) × 10^{3 }with the Fermilab/MILC normalization. Fits to experimental data, combining lattice predictions, and dispersion relations reduce the theoretical errors. For example, [182] obtains V_{ub} = (3.5 ± 0.17 ± 0.44) × 10^{3}, and, more recently, [183] obtains V_{ub} = (3.47 ± 0.29) × 10^{3}. The most recent lattice calculation [184] performs a simultaneous fit of improved lattice numerical Monte Carlo results and the 12bin BaBar experimental data on V_{ub}f_{+ }(q^{2}) [185] and derives V_{ub} = (3.38 ± 0.35) × 10^{3}. The ~10% error includes theoretical and experimental errors, not easily separable.
4.2 Determinations Based on Inclusive B Semileptonic Decays
Inclusive determinations of V_{ub} rely on the heavy quark expansion (HQE), which combines perturbative QCD with an expansion in terms of 1/m_{b}, which accounts for nonperturbative effects. Although the possible breaking of the assumption of local quark hadron duality may produce unquantified errors, other nonperturbative uncertainties can be evaluated with systematic improvements, and their uncertainties are easier to assess than the ones of unquenched lattice QCD or QCD sumrules. This statement applies to the total charmless semileptonic width . In the OPE approach, discussed in the V_{cb} section, the observation by Chay, Georgi and Grinstein that there are no nonperturbative corrections of order Λ_{QCD}/m_{b }[121] has inspired the hope that this approach would lead to a more precise determination of this important parameter. Before discussing the methods used to relate data with theory, it is useful to observe one of the possible expressions available in the literature for the total semileptonic width [186], including an exact twoloop expression for the perturbative expansion and secondorder power corrections. They obtain
where μ ~ m_{b }is the scale at which α_{S }needs to be evaluated, while the scale μ_{⋆ }applies to the nonperturbative expansion parameters, namely the b quark mass m_{b}, the chromomagnetic operator, λ_{2}, and the kinetic operator, μ_{π}. 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 quarkhadron 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 V_{ub} ≠ 0, very quickly several authors pointed out that this region of phase space is ill suited to a precise determination of V_{ub} 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 E_{ℓ }cut and an acceptable signal to background ratio. Table 9 summarizes the present status of the V_{ub} determination with this approach.
Next a whole host of papers proposed alternative "model independent" approaches to measure V_{ub} from inclusive decays [190192], 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 M_{X }≤ 1.5 GeV [190]. However, Bauer, Ligeti, and Luke pointed out [193] that the kinematic limit has the same properties of the lepton end point, and spoils the convergence of the OPE; the same authors proposed using dΓ/dq^{2 }up to q^{2 }= and argue that this distribution is better behaved in the kinematic region of interest. This is the theoretical foundation of the so called "improved end point" method, which encompasses the simultaneous study of E_{ℓ }and q^{2}. BaBar [194] used this technique, inferring the ν 4momentum from the missing momentum () in the event. There results give V_{ub} central values between (3.88–4.93)×10^{3}, depending on the explicit model and with total errors of ≈ 10% on each value.
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 M_{B}, and smoothly interpolates between them. This approach is commonly referred to as "shape function" method, and uses the γ spectrum in inclusive B → X_{S 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 V_{ub} 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 4momentum 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 M_{X }and the quantity , where . Both M_{X }and P^{+ }are smaller for b → u transitions. They define different signal region for p_{ℓ }> 1 GeV: P_{+ }< 0.66 GeV/c, M_{X }< 1.7 GeV/c^{2}, and M_{X }< 1.7 GeV/c^{2 }combined with q^{2 }> 8 GeV^{2}/c^{2}.
They evaluate the partial branching fractions in each of them, and extract V_{ub} 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 V_{ub} 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, M_{X }< 1.55 GeV/c^{2}, and M_{X }< 1.7 GeV/c^{2 }combined with q^{2 }> 8 GeV^{2}/c^{2}. 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 V_{ub} 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 given by
where q_{0 }is the energy of the leptonν pair and E_{ℓ }is the energy of the charged lepton in the B meson rest frame, and W_{1–3 }are the three structure functions relevant if we assume massless leptons. Reference [197] computes the functions W_{i}(q_{0}, q^{2}) as a convolution at fixed q^{2 }between nonperturbative distributions F_{i}(K_{+}, q^{2}; μ) and the perturbative functions (q_{0}, q^{2})
Perturbative corrections to the structure functions W_{1–3 }are now known up to order [198] and power corrections are included through . The separation between perturbative and nonperturbative physics is set by a cutoff scale μ = 1 GeV, in the "kinetic scheme" [162], which takes input parameters from fits to the b → c moments. Alternative choices of kinematical variable have been used, for example, in the SCET approach, the variables and are used. In particular, the "shape function" method [186] provides theoretical expression for the triple differential d^{3}Γ/dE_{ℓ}dP^{+ }dP^{ }and relates it to moments of the shape function extracted from B → X_{S γ}. Finally, the approach originally proposed by Gardi [199] uses resummed perturbation theory in momentum space to provide a perturbative calculation of the onshell decay spectrum in the entire phase space. The method used, dressed gluon exponentiation, is a general resummation formalism for inclusive distributions near a kinematic threshold. Finally, another model based on softgluon resummation and an analytical timelike QCD coupling has been proposed [200]. Clearly a lot of work has gone into bringing to fruition the original promise that the inclusive V_{ub} determination is a more precise method to determine this important CKM parameter.
However, Figure 25 shows that different methods provide central values of V_{ub} that often differ beyond the stated errors. Moreover, these estimates are generally significantly higher than the value of V_{ub} extracted from B → πℓ_{ℓ }and the one that is obtained from global unitarity triangle fits.
Figure 25. Summary of Inclusive V_{ub} determinations.
An effect that can influence the inclusive V_{ub} is the contribution due to topologies where the incoming b pair annihilates into a W boson, with a concomitant soft gluon emission. These contributions are known as "weak annihilation" and they appear as a delta function at the end point of the lepton spectrum. This effect introduces a difference between B^{0 }and B^{+}semileptonic decays.
A first attempt to quantify the annihilation effect was performed by CLEO [201], studying the high q^{2 }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, Γ_{W A}/Γ_{b → u }< 7.4% at 90% confidence level. BaBar has presented a measurement of (B^{0 }→ X_{u }ℓ_{ℓ}) at the end point region which, combined with previous inclusive measurement, sets the limit
at 90 % confidence level, where f_{W A}(2.3 – 2.6) represents the fraction of weak annihilation rate contributing in the momentum interval Δp_{ℓ }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 D^{0 }and D_{s }mesons can assess nonfactorizable terms through the relationship
where Γ_{0 }(c → sℓν) is the bare parton semileptonic rate c → sℓν, f_{D }is the D decay constant, and is the nonfactorizable term that affects charmless semileptonic decays as well. Using this formula with the CLEOc results for inclusive D^{0 }semileptonic decay [203], the sum of D_{s }exclusive semileptonic decays [204], and m_{c}(m_{c}) = 1.268 GeV [205], we obtain an estimate of nonfactorizable effects equal to (9 ± 5) %. Note that annihilation effect induced uncertainties are estimated to contribute 1.6% in the BLNP approach, 1.5% in the DGE approach, and (+0.0, 3.1)% in the GGOU approach.
In summary, the values if V_{ub} 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 V_{ub} value obtained from the exclusive branching fraction (B → π ℓν).
5 Rare B decays
In 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 ν
The decay B → τ ν is affected by two quantities of great interest, the quark mixing parameter V_{ub} and the pseudoscalar decay constant f_{B}. In fact, the leptonic branching fraction is given by
where m_{τ }is the mass of the τ lepton, and τ_{B }is the charged B lifetime. Theoretical predictions for f_{B }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 f_{B}.
Belle performed the first measurement of the branching fraction (B^{+ }→ τ^{+}ν) using a tagged sample of fully reconstructed B decays [206]. They later used a tagged sample of semileptonic decays [207]. BaBar also published studies performed using fully reconstructed B decays [208] and semileptonic tags [209]. The results are listed in Table 12. The resulting average experimental value B(B → τ ν) is (1.73 ± 0.35) × 10^{4}.
Table 12. Summary of experimental determinations of (B → τ ν).
The Standard Model prediction for (B → τ ν) is , obtained using the exclusive value of V_{ub} [184] and the most recent Lattice QCD value of f_{B }[210]. The range has been obtained by calculating the Standard Model branching fraction using the values V_{ub} = V_{ub}_{excl }± σ(V_{ub}) and f_{B }= f_{B }± σ(f_{B}). An intriguing possibility is the enhancement of this branching fraction due to new physics. For example in a 2Higgsdoublet model (2HDM) [211] based on supersymmetric extensions of the Standard Model, the ratio
where g_{s }is the effective scalar coupling. Fig. 26 shows the predicted value of (B → τ ν) as a function of g_{s }and the band representing the measured value, which constrains g_{s }to be less than 0.1 or between 1.8 and 3.9. A better understanding of the V_{ub} and f_{B }inputs is necessary to improve on this estimate.
Figure 26. (B → τ ν) as a function of the effective scalar coupling g_{s}. The shaded region is the measure data ±1σ The range of theoretical predictions is obtained by assuming V_{ub} = (3.38 ± 0.35) × 10^{3 }[184], and f_{B }= (0.197 ± 0.06 ± 0.012) MeV [210]. (The curves are theoretical and explained in the text.
Note that a quantity that is sensitive to the same coupling is the ratio . The present experimental value for this parameter is (41.6 ± 11.7 ± 5.2)% [17], which constrains g_{s }to be ≤ 1.79. Thus these two measurements provide complementary constraints, which have very different systematic uncertainties. Thus an improvement in the knowledge of leptonic and semileptonic branching fractions involving a τ lepton in the final state is very important.
5.2 Radiative B Decays
Radiative 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 twoHiggs 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 → X_{s}γ is an ideal laboratory for studying the dynamics of the bquark inside the B meson: since the motion of the bquark 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 Q_{i }are local operators, the C_{i }are Wilson coefficients, and μ is the QCD renormalization scale [213]. The B → X_{s}γ branching fraction directly probes the Wilson coefficient C_{7}. However, some new physics contributions may leave the B → X_{s}γ rate unaltered, with changes to the C_{7 }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 → X_{s}ℓ^{+}ℓ^{ }transition may provide information on the sign of the amplitude, since it also probes C_{9 }and C_{10}.
The B → X_{s}γ branching fraction, as a function of a photon energy cutoff E0, is related to the B → X_{c}νℓ transition by
where the perturbative corrections P(E_{0}) are defined as
and the nonperturbative N(E_{0}) terms are . Note that the minimum value of E_{0 }for which this relation is valid is 1.6 GeV. The theoretical error from NLO perturbative calculations is about 10%, dominated by the renormalization scale dependence and a charm quark mass uncertainty. This uncertainty is mainly due to the change from the ratio in the pole scheme = 0.29 ± 0.02 to the ratio = 0.22 ± 0.04 in the scheme (see section 3.0.1). The large m_{c }dependence is due to the fact that m_{c }first enters at NLO matrix elements. Hence, the natural scale at which m_{c }should be renormalized can only be determined by dedicated calculations at NNLO.
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 nonperturbative corrections (5%), the uncertainty due to missing higherorder corrections (3%), the m_{c }interpolation ambiguity (3%) and the parametric uncertainty (3%). Not included are some known NNLO and nonperturbative 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 nonperturbative 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 E_{min }= 1.6 GeV photon energy for b → sγ .
Experimentally, two methods are used to extract the B → X_{s} γ signal: the fully inclusive method and the semiinclusive method. In the fully inclusive method, events containing a hard photon consistent with B → X_{s} γ are selected. In this method the subtraction of a very large background, primarily from q continuum events, is the main issue. There are several options to reduce this large background. Particularly effective in the suppression of continuum backgrounds is the requirement of a high momentum lepton, tagging the semileptonic decay of the accompanying B meson. Alternatively, a fully reconstructed B can be used to identify B decays. In this measurement the large background increases towards the lower photon energy, making it impossible to measure the full photon spectrum. Hence, all measurements require a minimum photon energy where the signal to noise ratio is still acceptable. The total branching fraction is then obtained by extrapolating the signal to the full phase space. This extrapolation is based on theoretical models and is an irreducible source of systematic error. It has been argued that the energy scale is significantly smaller than m_{b }and therefore the above extrapolation error is underestimated. Therefore it is important to measure the b → s γ photon energy spectrum as precisely as possible. So far we have been able to lower the minimum photon energy to 1.7 GeV.
In the semiinclusive method, the B → X_{s} γ branching fraction is determined by summing up exclusive modes, with an extrapolation procedure to account for the unobserved modes. The semiinclusive analysis provides a more precise photon energy in the Bmeson rest frame and usually provides the flavor of the decay (i.e. b or ). The extrapolation to account for the unobserved modes is is the key issue in this approach; it is based on isospin symmetry and the Bmeson Monte Carlo hadronization model, which is hardly precise and reliable. All the available measurements are summarized in Table 13. Note that the CLEO measurement on the branching fraction includes b → d γ events, which is expected to be quite small ~V_{td}/V_{ts}^{2 }≈ 5%.
At the parton level in the twobody decay b → s γ, the energy of the photon is E_{γ }≈ m_{b}/2 in the bquark rest frame. However, the B → X_{s}γ is not monochromatic, due to several effects including the width of the X_{s }mass distribution, gluon emission and the Fermi motion of the bquark in the B meson. The nonperturbative Fermi motion effects will raise the photon energy above m_{b}/2, while the gluon emission will give a long lowenergy tail. The lowenergy tail can be described by HQE apart nonperturbative effects that can be modeled by processindependent shape functions described by a few parameters, e.g. the b quark mass and the Fermi momentum () that are considered to be universal in the following inclusive B decays: B → X_{s}γ, B → X_{s}ℓℓ, B → X_{c}ℓν and B → X_{u}ℓν. Recently a significant effort was made to combine all available data for B → X_{s}γ and B → X_{c}ℓν to determine the HQE parameters. Using these parameters, B → X_{s}γ branching fraction results are combined together to provide a rather precise branching fraction. The method to combine these measurement has been provided by Limosani (A. Limosani, in a private communication, performed the average for this article) and it is different from the method used by HFAG, as explained below.
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 m_{b }is evaluated by varying m_{b }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 E_{min}, 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 = 4.243 ± 0.042 GeV as input [196], or with = 4.63 ± 0.04 GeV and as input [217]. The two world averages are listed in Table 13 and they agree within their theoretical errors.
The agreement between the measured B → X_{s}γ branching fraction and the theoretical prediction constrains various new physics scenarios. One of the most popular examples is the lower bound on the typeII 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 → X_{s}γ rate unaltered, with new phases appearing in the decay loop [214,215]. Since the Standard Model prediction of the CP asymmetry (A_{CPs γ}) is zero in the limit of Uspin symmetry [219], significant nonzero values would be evidence for new phenomena. A_{CPs 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 A_{CPs 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 B^{0}, B^{± }and B^{0}/B^{± }admixture.
The Bfactories 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 lefthanded [214,215]. The measurements rely on either the exploitation of the B^{0 } interference [214,215] or the interference between higherKaon resonances decaying into K π π^{0 }[221,222]. However, both Belle and BaBar [223,224] measurements are not precise enough to pin down the photon polarization. The list of established decay modes of the type B → K^{(*)}X γ, where X is one or more flavorless mesons, is increasing. These channels are interesting as they can provide a measurement of the polarization of the photon. Many decay involving kaon resonances modes (e.g. K*(892), K_{1}(1270), (1430)) are now well established and their branching ratios agree with the Standard Model predictions.
5.3 B → ℓℓ
Flavor changing neutral current (FCNC) decays proceed only through loop diagrams in the Standard Model, and are further reduced by helicity and GlashowIliopoulosMaiani (GIM) suppression. Thus even the largest branching fraction predicted by the Standard Model, _{SM}(B_{s }→ μ^{+}μ^{}) ~4 × 10^{9}, is very tiny. New physics can enter either at tree level, for example through the presence of an additional Z' boson, or through new particles appearing in loops, and can increase the actual value of these branching fractions. Decay modes involving two muons in the final state are particularly interesting because they are amenable to measurements in experiments operating at hadronic machines. In fact the best upper limits come from CDF and D0, which are reaching a sensitivity comparable to the SM prediction. Table 15 shows a summary of the present experimental values.
Table 15. Summary of upper limits for B → ℓℓ decays.
5.4 B → X_{s}ℓℓ
The physics of B → X_{s}ℓℓ is governed by the Wilson coefficients C_{7}, C_{9 }and C_{10}, which describe the strengths of the corresponding shortdistance operators in the effective Hamiltonian, i.e. the electromagnetic operator O_{7 }and the semileptonic vector and axialvector operators O_{9 }and O_{10}, 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 nexttonexttoleading order (NNLO). The experimental knowledge on the Wilson coefficient C_{7 }comes from the inclusive B → X_{s}γ branching fraction, which determines its absolute value to about 20% accuracy, but not its sign. The partial b → X_{s}ℓℓ decay rate in the lepton invariant mass range below the J/ψ resonance is sensitive to the sign of C_{7}.
Measurements of the inclusive b → X_{s}ℓℓ 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 → X_{s}ℓℓ branching fractions.
Table 17. CP Asymmetries for exclusive and inclusive b → X_{s}ℓℓ transitions for B^{0}, B^{± }and B^{0}/B^{± }admixture.
The inclusive B → X_{s}ℓℓ branching fraction, which constrains C_{9 }and C_{10 }[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 → X_{s}ℓℓ decays by evaluating the differential inclusive decay rate as a function of the lepton invariant mass, q^{2 }= m(ℓℓ)^{2}, or by measuring the forwardbackward asymmetry in the exclusive decay B → K*ℓℓ.
The forwardbackward asymmetry in B → K*ℓℓ, defined as
is a function of q^{2 }and of , the angle between the negative lepton and the B meson. It is an ideal quantity to disentangle the Wilson coefficients C_{i }since the numerator of Eq. 82 can be expressed as
where ξ is a function of q^{2}, and F_{1,2 }are functions of form factors. It is straightforward to determine C_{10}, Re(C_{9}) and the sign of C_{7 }from the A_{FB }distribution as a function of q^{2}, using the value of C_{7} from B → X_{s}γ and a few more assumptions: phases of C_{10 }and C_{7 }are neglected, and higher order corrections are known. These assumptions should be examined by comparing the results with the inclusive B → X_{s}ℓℓ differential branching fraction as a function of q^{2}, since it is also sensitive to C_{9 }and C_{10 }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 C_{7}, as predicted by the Standard Model.
In some SUSY scenarios the sign of the b → s γ amplitude (C_{7}) 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 forwardbackward asymmetry in the low q^{2 }region, while it disappears with the opposite sign C_{7 }if the sign of Re(C_{9}) is the same as in the SM. In another model with SU (2) singlet downtype quarks, treelevel Z flavorchangingneutralcurrents are induced. In this case, the larger effect is expected on the axialvector coupling (C_{10}) to the dilepton than on the vector coupling (C_{9}). Because the forwardbackward asymmetry is proportional to the axialvector coupling, the sign of the asymmetry can be opposite to the SM. The same new physics effect is also a possibility for B → ϕ K_{S }where anomalous mixinginduced CP violation can occur.
Belle attempted to extract C_{9 }and C_{10 }from A_{FB }in B → K*ℓℓ with 357 fb^{1 }data [228]. The Belle analysis constrains the sign of the product C_{9}C_{10 }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, C_{9 }and C_{10}, are allowed to float in the fit to the data.
Since B^{0 }→ K^{0}*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 forwardbackward asymmetry in B → X_{s}ℓℓ, which is only possible at a superB 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 formfactors, which can be as large as a factor two. These uncertainties are much smaller if the ratio is measured. In the SM, the B → Ke^{+}e^{ }and B → K μ^{+}μ^{ }branching fraction are almost equal, apart from a small phase space difference due to the lepton masses. However, the B → K*e^{+}e^{ }branching fraction is expected to be larger than the B → K*μ^{+}μ^{ }branching fraction, due to a larger interference contribution from B → K*γ in B → K*e^{+}e^{}. In new physics models with a different Higgs sector than that of the SM, scalar and pseudoscalar types of interactions may arise in b → sℓℓ.
Therefore, R_{K }= (B → K μ^{+}μ^{})/(B → Ke^{+}e^{}) is an observable that is sensitive to new physics. For example, a neutral SUSY Higgs contribution can significantly enhance only the B → K(K*)μ^{+}μ^{ }channel at large tan β. The current world average for the branching fractions gives: R_{K }= 0.96 ± 0.26, dominated by the statistical error.
Finally, the transition b → s ν is theoretically much cleaner than its charged counterpart, thanks to the absence of the photon penguin diagram and hadronic longdistance effects (charmonium resonances). From the experimental point of view these modes are very challenging due to the presence of the two neutrinos. Searches for such modes at the B factories use a fully reconstructed B sample or semileptonic tagged mode. Both Belle and BaBar report limits for these modes.
6 Current status of overall CKM fits
We 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 versus along, with their experimental and theoretical errors, shown in Figure 27, from many of the measurements described above, also including the measurement of CP violation in the neutral kaon system, designed as "ϵ_{K}." Physics beyond the Standard Model could be revealed by seeing a particular measurement that is not in agreement with the others. The width of the bands are the onestandard deviation combined overall errors. For many of the measurements, e.g. V_{ub}, ϵ_{K}, Δm_{d }and Δm_{s}, the theoretical errors dominate and thus how they are treated is important. The CKM fitter [19] (see also [20]) group generally treats the errors more conservatively than the UT fit group [30]. The former uses a frequentist statistical approach, while the later a Bayesian approach. The Bayesian errors are smaller, but if the question is one of seeing new physics or not, a conservative approach may be better. The results for the Standard Model CKM parameters from both groups are listed in Table 18. The central values of and are in agreement. The value of λ is in agreement with that found by Blucher and Marciano [229]. The fact that there is an overlap region common to all the measurements shows no obvious deviation from the Standard Model, which does not predict values for and , but merely that the various measurements must find the same values.
Table 18. Values of Wolfenstein parameters from CKM fits assuming the Standard Model
Figure 27. The projections in the plane from measurements of the angles and sides of the unitarity triangle. The colored area at the apex shows the 95% confidence level point. (From the CKM fitter group [95,96]).
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 plane of angle measurements with those of the sides of the unitarity triangle. The overlap regions in both cases are consistent.
Figure 28. Constraints at 68% confidence level in the plane. On the left side are measurements of the CP violating angles α, β and γ and ϵ_{K }in the neutral kaon system. On the right side are measurements of V_{ub}/V_{cb} and the ratio of B_{s }to B_{d }mixing. (From the UT fit group [237]).
It remains to set limits on New Physics. We allow for NP to contribute to the mixing amplitude of either B_{d }or B_{s }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 V_{ud}, V_{us}, V_{cb}, and V_{ub}. 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  β^{meas}, allowing that the measured β^{meas }= β^{SM } Arg(Δd). Also used are measurements of the semileptonic asymmetries (see Eq. 34) (measurement of A_{SL }for B_{s }mesons are given in [231,232]; see also [233]), and the constraint equation (there are considerable theoretical uncertainties, see [234]). The allowed range in the plane for NP via B mixing is shown in Figure 29.
Figure 29. Constraints on the imaginary part versus the real part of a New Physics amplitude proceedings via B mixing for(left) B_{d }mesons and (right) B_{s }mesons. The SM point is at (1,0). The contours shown are all at 68% confidence level. (From the CM fitter group [95,96]).
In both case quite large values of the real part of NP amplitudes are allowed. In the B_{s }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 Conclusion
The 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 B^{0 }mixing. CLEO, ARGUS and the LEP experiments competed on the first precision measurements of the CKM parameters V_{cb }and V_{ub}. The two most recent, BaBar and Belle, have provided measurements of CP violation.
The hadron collider experiments have measured B_{s }mixing and are starting to probe CP violation in the B_{s }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 bflavored 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.
Acknowledgements
We 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.
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