Second order direct CP asymmetry in B_(s) ->X l nu

A direct CP asymmetry in inclusive semileptonic B_(s) decays vanishes by CPT to lowest order in weak interactions. Calculating the asymmetry at second order weak interactions in the Cabibbo-Kobayashi-Maskawa framework we find A_{sl} = (-3.2 \pm 0.9) x 10^{-9}. A maximal asymmetry which is two orders of magnitude larger is estimated in a left-right symmetric model. This quite generic upper bound implies negligible effects on wrong-sign lepton asymmetries in B^0 and B_s decays.


I Introduction
The D0 Collaboration working at the Fermilab Tevatron has reported recently a charge asymmetry in like-sign dimuon events produced inpp collisions. The measured asymmetry [1,2], was interpreted as due to CP violation in B 0 -B 0 or B s -B s mixing, (An asymmetry consistent with zero was measured a few years ago by the CDF collaboration using fewer same-sign dimuon events produced inpp collisions with a lower integrated luminosity [3].) The experimental result (1) differs by 3.2 standard deviations from much smaller asymmetries predicted within the Cabibbo-Kobayashi-Maskawa (CKM) framework [4], Unambiguous evidence for physics beyond the Standard Model requires (a) showing that the measured asymmetry is, indeed, due to B 0 or B s semileptonic decays and not due to background processes [5], and (b) confirming an anomalously large negative asymmetry with a somewhat higher statistical significance than the current one.
In view of the tiny CKM predictions (3) for A k sl (k = d, s) from second order |∆B| = 2 weak transitions, attention has been often drawn to potentially larger corrections to the two asymmetries from new |∆B| = 2 flavor physics at a TeV or higher energy scale [6]. In this note we will study |∆B| = 1 contributions to A k sl from CP violation in inclusive semileptonic decays, which have been systematically neglected in all earlier studies.
In Sec. II we review briefly the usual treatment of the asymmetry A k sl involving CP violation in B 0 k -B 0 k mixing, introducing a new contribution from direct CP violation. Sec. III discusses an argument based on CPT for the vanishing of the new contribution at lowest order in weak interactions. We perform a second order calculation of the inclusive semileptonic asymmetry within the CKM framework. Maximal values of the asymmetry in a left-right extension of the Standard Model are studied in Sec. IV. In Sec. V we estimate for completeness second order amplitudes for neutral B mesons decaying directly into wrong-sign leptons while Sec. VI concludes.
II Including a direct asymmetry in A k sl One starts by defining neutral B mass eigenstates B k L and B k H , with mass and width differences ∆m k and ∆Γ k , in terms of flavor states B 0 k andB 0 k , Time evolution of flavor states [7,8], implies time-dependent decay rates for inclusive semileptonic decays for wrong-sign leptons B 0 k (t) → Xℓ −ν ℓ (ℓ = e, µ) and their charge conjugates, By definition A k sl is the time-dependent asymmetry of wrong-sign leptons due to mixing, Usually, one neglects a direct CP asymmetry in B 0 k → Xℓ + ν ℓ . Assuming |Ā k ℓ | = |A k ℓ | one finds [7,8], That is, the asymmetry caused by CP violation in B 0 k -B 0 k mixing is given by Im(Γ k 12 /M k 12 ) where M k 12 and Γ k 12 are off-diagonal elements of Hermitian matrices representing B 0 k ↔ B 0 k transitions via off-shell (dispersive) and on-shell (absorptive) intermediate states, respectively. The CKM predictions (3) for A k sl (mixing) are based on calculations of this imaginary part for B 0 and B s [4].
We now define a direct CP asymmetry in inclusive semileptonic decays, Eqs. (6) and (9) imply an expression for A k sl (t) which includes both the asymmetry in mixing and the direct asymmetry. Neglecting terms quadratic in the A k sl (mixing) and We note that this asymmetry of time-dependent decay rates is actually time-independent as in the special case (10) of CP violation in mixing alone. In the following two sections we will study A k sl (direct).

III Direct asymmetry in the CKM framework
The inclusive semileptonic direct asymmetries in B 0 and B s decays are equal to each other to a good approximation as Γ( . (see also calculation below.) Furthermore, in the isospin symmetry limit the asymmetry in B 0 decays is equal to that measured directly in self-tagged B + → Xℓ + ν ℓ by comparing the rate for this inclusive process with that of its charge-conjugate. For this reason we omit the superscript k in A k sl (direct) by defining a generalized direct semileptonic asymmetry for non-strange (charged or neutral) or strange B mesons, It has often been stated that A sl vanishes because of CPT invariance [9, 10,11]. CPT implies equal total decay widths for a particle and its antiparticle. A generalization of this theorem applies to partial decay rates for a set of final states, connected among themselves by strong and electromagnetic final state interactions but not connected by such interactions to other states. The inclusive decays B → X (C=−1) ℓ + ν ℓ and B → X (C=0) ℓ + ν ℓ are two special cases to which this generalization applies.
A violation of the theorem of equal partial rates for B → Xℓ + ν ℓ andB → Xℓ −ν ℓ is possible if one considers weak interactions which connect the final states Xℓ + ν ℓ with intermediate hadronic states. That is, a small nonvanishing asymmetry A sl may be obtained by considering an interference between the dominant tree amplitude for B → Xℓ + ν ℓ and an amplitude which is second order in weak interactions. A similar interference has been shown to imply a tiny CP asymmetry in inclusive semileptonic rare top quark decays, t → dℓ + ν ℓ [12,13].
A very crude upper bound on this asymmetry is This estimate includes a suppression by a loop factor 1/4π and a phase space factor (m b − m c ) 2 . Further suppression of the asymmetry may be due to a small weak phase difference between the two interfering amplitudes and due to a possible extra dynamical suppression of the second order amplitude. We will show below that, indeed, such suppression factors exist in the CKM framework. The above upper limit is one order of magnitude smaller than the estimate (3) for a CP asymmetry in B s -B s mixing, and two orders of magnitude below the asymmetry in B 0 -B 0 mixing. We will now calculate the CP asymmetry in CKM favored B → X (C=−1) ℓ + ν ℓ decays dominated by a tree amplitude proportional to V * cb describing a quark transitionb → cℓ + ν ℓ . In order to produce an asymmetry, the second amplitude, leading to the same final states, must involve a CKM factor with a different weak phase. A second order amplitude fulfilling these two requirements consists of a product of a penguin amplitude forb →ccs involving V * tb V ts (see discussion below) and a tree amplitude for cs → ℓ + ν ℓ involving V * cs . Two diagrams, describing the tree amplitude for b → cℓ −ν ℓ and the second order amplitude for this transition, are shown in Figs. 1 and 2, respectively. A relative CP-conserving phase of 90 • between the two amplitudes follows by taking the absorptive (i. e., imaginary) part of the second order amplitude. The absorptive part is described by a discontinuity cut crossing thecs lines in the second order diagram, which amounts to summing over corresponding on-shell intermediate states.
In order to calculate the asymmetry we write down expressions for an effective Hamiltonian associated with each of the three four-fermion vertices appearing in the two diagrams in Figs. 1 and 2. The tree diagram is obtained from The Hamiltonian related to the first vertex in the second order diagram is [14], Here α, β are color indices. The Wilson coefficients c i (i = 3 − 6) have been calculated in the next-to-leading logarithmic approximation (NLL). The second vertex in this diagram is described by We are interested in the imaginary part of the amplitude involving thecs loop illustrated in Fig. 2 20) and (21).] These contributions will be neglected. We consider the dominant terms from O 3 and O 4 . After Fierz rearrangement of these terms the second order loop amplitude is given by where The minus sign takes account of the closed fermion loop. The most general form of T µν is T µν = T 1 g µν + T 2 q µ q ν .
We shall be interested in the T 1 part, neglecting the T 2 contribution to M 1 which is proportional to m ℓ . Contracting T µν with g µν and q µ q ν we obtain Performing the appropriate traces, one finds To take twice the absorptive part [15], we put the internal propagators on the mass shell, replacing Simplifying by neglecting m s , we find the expression in the large brackets in Eq. (24) reduces to ( The delta functions reduce the loop integral to an integral over two-body phase space: where the two-body phase space integral in the limit m s = 0 is Putting the pieces together, we find Given the tree level amplitude M 0 in Eq. (15), the expression for the semileptonic asymmetry then becomes, performing the three-body phase space integrals,  [16]. where Here The upper limit of integration in both integrals in the numerator and denominator of R is z max = (1 − √ r c ) 2 . The lower limit of the integral in the numerator is z min = r c (in the limit that the strange quark mass may be neglected), while the lower limit in the denominator is zero in the limit of vanishing lepton mass. Thus the denominator obtains the well-known expression for the kinematic factor in b → cℓν ℓ , For m c and m b we use constituent masses, which we expect to simulate QCD effects to a certain degree. These masses were found in Ref. [16] to reproduce charmonium and bottomonium spectra for a rather wide range as long as m b − m c lay within a narrower range. The masses considered there are summarized in Table I, together with the corresponding values of r c , R, and Rm 2 b (in which the dependence of R on m b is partially compensated). The uncertainty on Rm 2 b is approximately 28%, which we carry in our final estimate of the asymmetry.

IV Direct asymmetry beyond the Standard Model
A crude estimate for a maximum asymmetry based on dimensional arguments was given in Eq. (14). It applies to a generic new physics contribution to the asymmetry occurring at one-loop order. Here we wish to be more concrete by considering a specific model leading to CP-violation in B → Xℓ + ν ℓ at one loop, without involving suppression factors which occur in the CKM framework. An example that falls into this category is a Left-Right symmetric model [18], in which the interaction of two charged vector-bosons W 1 and W 2 is given by Here P L,R ≡ (1 ± γ 5 )/2 while V L,R are CKM-like matrices for left (right)-handed quark fields. The angle ξ is a (small) mixing angle between the two charged vector-bosons W 1 and W 2 and ω is a new CP phase related to this mixing. The light mass eigenstate is identified with the Standard Model (SM) gauge boson W L , The interaction (36) contributes to the asymmetry in B → Xℓ + ν ℓ at one-loop order. We consider a W 1 exchange diagram as in Fig. 1, with self-energy insertions on the W 1 line ofcs andūd quark loops and ℓ ′ νl′ (ℓ ′ = [e, µ] = ℓ) and τν τ leptonic loops. Elastic weak rescattering from an intermediate ℓν state is not included for consistency with CPT [19]. Loop amplitudes involving W 2 exchange or mixed W 1 -W 2 exchanges are expected to be smaller because a suppression factor sin ξ is replaced in these amplitudes by M 2 1 /M 2 2 . One obtains M 2 > 1.6 TeV if V L = V R [20] (a later estimate gave 1.4 TeV [21]) but this bound can be relaxed if the left-hand and right-hand quark mixings are different [21]. For the case V L = V R , which we consider below, the W 1 exchange diagram with self energy insertions dominates over these other contributions.
The absorptive part of the dominant one-loop amplitude is with Here T cs 1(abs) (q 2 ) is the same as in the SM, T cs 1(abs) ≡ T 1(abs) , while T ud 1(abs) , T ℓ ′ ν ℓ ′ 1(abs) and T τ ντ 1(abs) correspond to aūd loop and to the two leptonic loops. Using the notations of Eq. (32) and the approximation m u = m d = m ℓ ′ ≃ 0, m τ ≃ m c , one has Comparing the contribution of this amplitude to the asymmetry with the asymmetry calculated in the SM, we obtain a ratio This ratio involves two enhancement factors following from a suppression which occurs in the the Standard Model but not in its Left-Right symmetric extension. The first factor, N C /(c 3 +N C c 4 ) = −34, originates in a loop suppression of the Wilson coefficients for penguin operators in (16). A second potential enhancement is due to the ratio of weak phase factors I LR /I SM , where While this factor is −0.018 in the SM, it may be of order one in the LR model if V R cb = V L cb and if ω is large. The last ratio in (40) depends on quark couplings and on phase space. Neglecting u, d and s quark masses and setting |V L cs | 2 = |V L ud | 2 = 1, it is given by The upper limit of integration in the three integrals is z max = (1 − √ r c ) 2 as in (31).
The lower limit of the first integral in the numerator and the one in the denominator is z min = r c , while that of the second integral in the numerator is zero. Taking r c = 0.082 as a central value in Table I, one finds R LR /R SM = 0.93. Combining all factors and assuming that CP-violation in semileptonic B decays is dominated by the phase ω, one obtains A recent study of phenomenological constraints on right-handed quark currents obtains an upper bound on a b → c right-handed coupling of several percent relative to a lefthanded coupling [22], |(V R cb /V L cb ) tan ξ cos ω| = (2.5 ± 2.5) × 10 −2 . A comparable upper bound may be obtained on |(V R cb /V L cb ) tan ξ sin ω| from a recent measurement of CP asymmetry in B + → J/ψK + [23], A CP (B + → J/ψK + ) = [−0.76 ± 0.50 (stat) ± 0.22 (syst)] × 10 −2 . This bound requires assuming that a final state interaction phase difference between two interfering B → J/ψK hadronic amplitudes, for tree-level (V − A)(V − A) and (V + A)(V − A) b → ccs transitions, is not small. Thus, the asymmetry in the Left-Right symmetric model may be at most two orders of magnitude larger than in the Standard Model.

V Wrong-sign leptons without neutral B (s) mixing
In Section II we have assumed A(B 0 k → Xℓ + ν ℓ ) = A(B 0 k → Xℓ −ν ℓ ) = 0, neglecting second order weak contributions to these two processes which occur in the CKM framework leading to "wrong-sign" leptons without B 0 k -B 0 k mixing. Interference between these second order contributions and first order tree amplitudes forB 0 k → Xℓ −ν ℓ and B 0 k → Xℓ + ν ℓ leads to additional time-dependent terms in Eqs. (6) of the forms e −Γ k t sinh(∆Γ k t/2) and e −Γ k t sin(∆m k t). Second order amplitudes for B 0 k → Xℓ −ν ℓ have been discussed in Ref. [24] without estimating their magnitudes. For completeness, as we have studied tiny CP asymmetries from second order amplitudes, we will estimate the ratio of second order amplitudes for "wrong-sign" leptons and first order amplitudes for "right-sign" leptons, showing that this ratio is negligibly small. Second order amplitudes for B 0 k → Xℓ −ν ℓ are described by diagrams plotted in Fig. 3, in which both theb quark and the spectator k quark undergo weak decays intoqck and qℓ −ν ℓ , respectively, by exchanging q = u, c, t quarks. These second order amplitudes, involving CKM factors V * qb V qd V cd and V * qb V qs V cs in B 0 and B s decays, lead to final hadronic states with quark structures X = cd and X = cs, respectively, as in first-order amplitudes forB 0 andB s semileptonic decays. Let us denote the second-order weak amplitude for B 0 k decay by A k ℓ ≡ A(B 0 k → Xℓ −ν ℓ ). We wish to estimate the ratios of semileptonic rates where Φ 1,2 are the appropriate phase-space factors for the first-order and second-order processes. By neglecting the effect of the spectator quark inĀ k ℓ we are treating the firstorder process as leading to a three fermion final state, while the second order diagram illustrated in Fig. 3 involves four fermions in the final state. A naive dimensional analysis then leads to The ratio A k ℓ /Ā k ℓ must involve a factor of G F f B which has a suitable dimension. The momentum passing through the propagator of the fermion q = u, c, t is of order m b , and kinematic factors of the same order will cancel it for q = u, c, while the contribution of q = t is highly suppressed by the heavy t-quark mass. The corresponding CKM factors ) for B s and B 0 decays, respectively, where λ = 0.23. Thus, the c quark dominates A s ℓ with a CKM factor comparable to V cb governingĀ s ℓ , while the contributions of the u and c quarks in A d ℓ are comparable to each other and are suppressed by λ 2 ≃ 1/20 relative to the CKM factor inĀ d ℓ . Taking m b − m c = 3.4 GeV (see Table I above for values of m b and m c ) and f B = 230 MeV, one then finds The corresponding ratios of square roots, ∼ 0.7 × 10 −6 and ∼ 0.4 × 10 −7 , characterize coefficients of additional time-dependent terms of forms e −Γ k t sinh(∆Γ k t/2) and e −Γ k t sin(∆m k t) in Eqs. (6) for B s and B 0 which may be safely neglected.

VI Conclusion
Inclusive semileptonic B and B s decays are shown to have a small non-zero direct CP asymmetry in the Standard Model as a result of interference of first-order-and secondorder-weak processes. This stands in contrast with statements made in two textbooks on CP violation [9, 10]. Taking a range of effective quark masses and estimating the asymmetry in b → cℓν ℓ as due to weak rescattering from the intermediate state in b → peng ccs, we have found A sl = (−3.2 ± 0.9) × 10 −9 .
A dimensional argument leads to a model-independent upper bound on A sl which is three orders of magnitude larger, while an extension of the Standard Model to a leftright-symmetric variant can increase the above asymmetry by at most two orders of magnitude. These values are far smaller than a value of about −1% recently reported by the D0 Collaboration [1], which may still be associated with a new source of CP violation in neutral B meson mixing.
As a by-product of second-order-weak amplitudes, we have estimated their efffect on wrong-sign leptons in direct decays of neutral B mesons and found it to be much below any reasonable sensitivity.