Charming new physics in rare B-decays and mixing?

We conduct a systematic study of the impact of new physics in quark-level $b \to c \bar{c} s$ transitions on $B$-physics, in particular rare $B$-decays and $B$-meson lifetime observables. We find viable scenarios where a sizable effect in rare semileptonic $B$-decays can be generated, compatible with experimental indications and with a possible dependence on the dilepton invariant mass, while being consistent with constraints from radiative $B$-decay and the measured $B_s$ width difference. We show how, if the effect is generated at the weak scale or beyond, strong renormalisation-group effects can enhance the impact on semileptonic decays while leaving radiative $B$-decay largely unaffected. A good complementarity of the different $B$-physics observables implies that precise measurements of lifetime observables at LHCb may be able to confirm, refine, or rule out this scenario.


I. INTRODUCTION
Rare B decays are excellent probes of new physics at the electroweak scale and beyond, due to their strong suppression in the Standard Model (SM). Interestingly, experimental data on rare branching ratios [1,2] and angular distributions for B → K ( * ) µ + µ − decay [2,3] may hint at a beyond-SM (BSM) contact interaction of the form (s L γ µ b L )(μγ µ µ), which would destructively interfere with the corresponding SM (effective) coupling C 9 [4][5][6], although the significance of the effect is somewhat uncertain because of form-factor uncertainties as well as uncertain long-distance virtual charm contributions [7]. However, if the BSM interpretation is correct, it requires reducing C 9 by O(20%) in magnitude. Such an effect might arise from new particles (see e.g. [8]), which might in turn be part of a more comprehensive new dynamics. Noting that in the SM, about half of C 9 comes from (short-distance) virtual-charm contributions, in this article we ask whether new physics affecting the quark-level b → ccs transitions could cause the anomalies, affecting rare B decays through a loop. The bulk of these effects would also be captured through an effective shift ∆C 9 (q 2 ), with a possible dependence on the dilepton mass q 2 . At the same time, such a scenario offers the exciting prospect of confirming the rare B-decay anomalies through correlated effects in hadronic B decays into charm, with "mixing" observables such as the B s -meson width difference standing out as precisely measured [9] and under reasonable theoretical control. This is in contrast with the Z and leptoquark models usually considered, where correlated effects are typically restricted to other rare processes and are highly model dependent. Specific scenarios of hadronic new physics in the B widths have been considered previously [10], while the possibility of virtual charm BSM physics in rare semileptonic decay has been raised in [11] (see also [12]). As we will show, viable scenarios exist, which can mimic a shift ∆C 9 = −O(1) while being consistent with all other observables. In particular, very strong renormalization-group effects can generate large shifts in the (low-energy) effective C 9 coupling from small b → ccs couplings at a high scale without conflicting with the measuredB → X s γ decay rate [13].

II. CHARMING NEW PHYSICS SCENARIO
We consider a scenario where new physics affects the b → ccs transitions. This could be the case in models containing new scalars or new gauge bosons, or strongly coupled new physics. Such models will typically affect other observables, but in a model-dependent manner. For this paper, we restrict ourselves to studying the new effects induced by modified b → ccs couplings, leaving construction and phenomenology of concrete models for future work. We refer to this as the "charming BSM" (CBSM) scenario. As long as the mass scale M of new physics satisfies M m B , the modifications to the b → ccs transitions can be accounted for through a local effective Hamiltonian, We choose our operator basis and renormalization scheme to agree with [14] upon the substitution d → b,s →c, arXiv:1701.09183v2 [hep-ph] 6 Aug 2018 u →s: The Q c i are obtained by changing all the quark chiralities. We leave a discussion of such "right-handed current" effects for future work [15] and discard the Q c i below. We split the Wilson coefficients into SM and BSM parts, where C c,SM i = 0 except for i = 1, 2 and µ is the renormalization scale.

III. RARE B DECAYS
The leading-order (LO), one-loop CBSM effects in radiative and rare semileptonic decays may be expressed through "effective" Wilson coefficient contributions ∆C eff 9 (q 2 ) and ∆C eff 7 (q 2 ) in an effective local Hamiltonian where q 2 is the dilepton mass and For q 2 small (in particular, well below the charm resonances), ∆C eff 9 (q 2 ) and ∆C eff 7 (q 2 ) govern the theoretical predictions for both exclusive (B → K ( * ) + − , B s → φ + − , etc.) and inclusive B → X s + − decay, up to O(α s ) QCD corrections and power corrections to the heavy-quark limit that we neglect in our leading-order analysis. Similarly, ∆C eff 7 (0) determines radiative Bdecay rates. We will neglect the small CKM combination V * us V ub , implying V * cs V cb = −V * ts V tb , and focus on real (CP-conserving) values for the C c i . From the diagram shown in Fig. 1 (left) we then obtain with C c x,y = 3∆C x + ∆C y and the loop functions Leading CBSM contributions to rare decays (left), and to width difference ∆Γs and lifetime ratio τ (Bs)/τ (B d ) (right).
We note that only the four Wilson coefficients ∆C 1...4 enter ∆C eff 9 (q 2 ). Conversely, ∆C eff 7 (q 2 ) is given in terms of the other six Wilson coefficients ∆C 5... 10 . The appearance of a one-loop, q 2 -dependent contribution to C eff 7 is a novel feature in the CBSM scenario. Numerically, the loop function a(z) equals one at q 2 = 0 and vanishes at q 2 = (2m c ) 2 . The constant terms and the logarithm accompanying y(q 2 , m c ) partially cancel the contribution from a(z) and they introduce a sizable dependence on the renormalization scale µ and the charm quark mass. Since a shift of ∆C eff 7 (q 2 ) is strongly constrained by the measured B → X s γ decay rate, we do not consider the coefficients ∆C 5...10 in the remainder and focus on the four coefficients ∆C 1...4 , which do not contribute to B → X s γ at 1-loop order. Higher-order contributions can be important if new physics generates ∆C i at the weak scale or beyond, as is typically expected. In this case large logarithms ln M/m B occur, requiring resummation. To leading-logarithmic accuracy, we find if ∆C i are understood to be renormalized at µ = M W and ∆C eff 7,9 at µ = 4.2 GeV. It is clear that ∆C 1 and ∆C 3 contribute (strongly) to rare semileptonic decay but only weakly to B → X s γ.

IV. MIXING AND LIFETIME OBSERVABLES
A distinctive feature of the CBSM scenario is that nonzero ∆C i affect not only radiative and rare semileptonic decays, but also tree-level hadronic b → ccs transitions. While the theoretical control over exclusive b → ccs modes is very limited at present, the decay width difference ∆Γ s and the lifetime ratio τ (B s )/τ (B d ) stand out as being calculable in a heavy-quark expansion [16]; see Fig. 1 (right). For both observables, the heavy-quark expansion gives rise to an operator product expansion in terms of local ∆B = 2 (for the width difference) or ∆B = 0 (for the lifetime ratio) operators. The formalism is reviewed in [17] and applies to both SM and CBSM contributions. For the B s width difference, we have [18] ∆Γ s = 2|Γ s,SM 12 + Γ cc 12 | cos φ s 12 , where the phase φ s 12 is small. Neglecting the strange-quark mass, we find with values taken from [19]. For our numerical evaluation of Γ cc 12 , we split the Wilson coefficients according to (3), subtract from the LO expression (11) the pure SM contribution and add the NLO SM expressions from [20]. In general, a modification of Γ cc 12 also affects the semileptonic CP asymmetries. However, since we consider CP-conserving new physics in this paper and since the corresponding experimental uncertainties are still large, the semi-leptonic asymmetries will not lead to an additional constraint.
In a similar manner, for the the lifetime ratio, we find where the SM contribution is taken from [21] and In each case, all Wilson coefficients are renormalized at µ = 4.2 GeV and those not corresponding to either axis set to zero. The black dot corresponds to the SM, i.e. ∆Ci = 0. The measured central value for the width difference is shown as brown (solid) line together with the 1σ allowed region. The lifetime ratio measurement is depicted as green (dashed) line and band. Overlaid are contours of ∆C eff 9 (5GeV 2 ) = −1, −2 (black, dashed) and ∆C eff 9 (2GeV 2 ) = −1, −2 (red, dotted), as computed from (5), and of ∆C eff 9 = 0 (black, solid).
subtracting the SM part and defining B 1 , with values taken from [22]. We interpret the quark masses as MS parameters at µ = 4.2 GeV.

V. RARE DECAYS VERSUS LIFETIMES-LOW-SCALE SCENARIO
We are now in a position to confront the CBSM scenario with rare decay and mixing observables, as long as we consider renormalization scales µ ∼ m B . Then the logarithms inside the h function entering (5) are small and our leading-order calculation should be accurate. Such a scenario is directly applicable if the mass scale M of the physics generating the ∆C i is not too far above m B , such that ln(M/m B ) is small. Fig. 2 (left) shows the experimental 1σ allowed regions for the width difference and lifetime ratio (from the web update of [23]) in the (∆C 1 , ∆C 2 ) plane. The central values are attained on the brown (solid) and green (dashed) curves, respectively. The measured lifetime ratio and the width difference measurement can be simultaneously accommodated for different values of the Wilson coefficients: in the ∆C 1 -∆C 2 plane, we find the SM solution, as well as a solution around ∆C 1 = −0.5 and ∆C 2 ≈ 0. In the ∆C 3 -∆C 4 plane, we have a relatively broad allowed range, roughly covering the interval [−0.9, +0.7] for ∆C 3 and [−0.6, +1.1] for ∆C 4 . For further conclusions, a considerably higher precision in experiment and theory is required for ∆Γ s and τ Bs /τ B d . Also shown in the plot are contour lines for the contribution to the effective semileptonic coefficient ∆C eff 9 (q 2 ), both for q 2 = 2 GeV 2 and q 2 = 5 GeV 2 We see that sizable negative shifts are possible while respecting the measured width difference and the lifetime ratio. For example, a shift ∆C eff 9 ∼ −1 as data may suggest could be achieved through ∆C 1 ∼ −0.5 alone. Such a value for ∆C 1 may well be consistent with CP-conserving exclusive b → ccs decay data, where no accurate theoretical predictions exist. On the other hand, ∆C eff 9 only exhibits a mild q 2 -dependence. Distinguishing this from possible long-distance contributions would require substantial progress on the theoretical understanding of the latter.
We can also consider other Wilson coefficients, such as the pair (∆C 3 , ∆C 4 ) (right panel in Fig. 2). A shift ∆C eff 9 ∼ −1 is equally possible and consistent with the width difference, requiring only ∆C 3 ∼ 0.5.

VI. HIGH-SCALE SCENARIO AND RGE
A. RG enhancement of ∆C eff 9 If the CBSM operators are generated at a high scale then large logarithms ln M/m B appear. Their resummation is achieved by evolving the initial (matching) conditions C i (µ 0 ∼ M ) to a scale µ ∼ M B according to the coupled renormalization-group equations (RGE), where γ ij is the anomalous-dimension matrix. As is well known, the operators Q c i mix not only with Q 7 and Q 9 , but also with the 4 QCD penguin operators P 3...6 and the chromodipole operator Q 8g (defined as in [24]), which in turn mix into Q 7 . Hence the index j runs over 11 operators with ∆B = −∆S = 1 flavor quantum numbers in order to account for all contributions to C 7 (µ) that are proportional to ∆C i (µ 0 ). Most entries of γ ij are known at LO [14,[24][25][26][27][28][29][30]; our novel results are (i = 3, 4) butions to ∆C eff 7 and ∆C eff 9 in (9),(10) as well as A striking feature are the large coefficients in the ∆C eff 9 case, which are O(1/α s ) in the logarithmic counting. The largest coefficients appear for ∆C 1 and ∆C 3 , which at the same time practically do not mix into C eff 7 . This means that small values ∆C 1 ∼ −0.1 or ∆C 3 ∼ 0.2 can generate ∆C eff 9 (µ) ∼ −1 while having essentially no impact on the B → X s γ decay rate. Conversely, values for ∆C 2 or ∆C 4 that lead to ∆C eff 9 ∼ −1 lead to large effects in C eff 7 and B → X s γ.

B. Phenomenology for high NP scale
The situation in various two-parameter planes is depicted in Fig. 3, where the 1σ constraint from B → X s γ is shown as blue, straight bands. (We implement it by splitting BR(B → X s γ) into SM and BSM parts and employ the numerical result and theory error from [31] for the former. The experimental result is taken from the web update of [23].) The top row corresponds to Fig. 2, but contours of given ∆C 9 lie much closer to the origin. All six panels testify to the fact that the SM is consistent with all data when leaving aside the question of rare semileptonic B decays-the largest pull stems from the fact that the experimental value for τ Bs /τ B d is just under 1.5 standard deviations below the SM expectation, such that the black (SM) point is less than 0.5σ outside the green area. Our main question is now: can we have a new contribution ∆C eff 9 ∼ −1 to rare semileptonic decays, while being consistent with the bounds stemming from b → sγ, ∆Γ s and τ Bs /τ B d ? This is clearly possible (indicated by the yellow star in the plots) if we have a new contribution ∆C 3 ≈ 0.2, see the three plots of the ∆C i − ∆C 3 planes in Fig. 3 (right on the top row, left on the middle row and left on the lower row). In these cases, the ∆C eff 9 ∼ −1 solution is even favored compared to the SM solution. A joint effect in ∆C 2 ≈ −0.1 and ∆C 4 ≈ 0.3 can also accommodate our desired scenario, see the right plot on the lower row, while new BSM effects in the pairs ∆C 1 , ∆C 2 and ∆C 1 , ∆C 4 alone are less favored. One could also consider three or all four ∆C i simultaneously.

C. Implications for UV physics
Our model-independent results are well suited to study the rare B-decay and lifetime phenomenology of ultraviolet (UV) completions of the Standard Model. Any such completion may include extra UV contributions to C 7 (M ) and C 9 (M ), correlations with other flavor observables, collider phenomenology, etc.; the details are highly model-dependent and beyond the scope of our modelindependent analysis. Here we restrict ourselves to some basic sanity checks.
Taking the case of ∆C 1 (M ) ∼ −0.1 corresponds to a naive ultraviolet scale This effective scale could arise in a weakly-coupled scenario from tree-level exchange of new scalar or vector mediators, or at loop level in addition from fermions; or the effective operator could arise from strongly-coupled new physics. For a tree-level exchange, Λ ∼ M/g * where g * = √ g 1 g 2 is the geometric mean of the relevant couplings. For weak coupling g * ∼ 1, this then gives M ∼ 3 TeV. Particles of such mass are certainly allowed by collider searches if they do not couple (or only sufficiently weakly) to leptons and first-generation quarks. Multi-TeV weakly coupled particles also generically are not in violation of electroweak precision tests of the SM. Looplevel mediation would require mediators close to the weak scale which may be problematic and would require a specific investigation; this is of course unsurprising given that b → ccs transitions are mediated at tree level in the SM. The same would be true in a BSM scenario that mimics the flavor suppressions in the SM (such as MFV models). Conversely, in a strongly-coupled scenario we would have M ∼ g * Λ ∼ 4πΛ ∼ 30 TeV. This is again safe from generic collider and precision constraints, and a model-specific analysis would be required to say more. Finally, as all CBSM effects are lepton-flavor-universal, they cannot on their own account for departures of the lepton flavor universality parameters R K ( * ) [32] from the SM values as suggested by current experimental measurements [33]. However, even if those departures are real, they may still be caused by direct UV contributions to ∆C 9 . For example, as shown in [5], a scenario with a muon-specific contribution ∆C µ 9 = −∆C µ 10 ∼ −0.6 and in addition a lepton-universal contribution ∆C 9 ∼ −0.6, which may have a CBSM origin, is perfectly consistent with all rare-B-decay data, and in fact marginally preferred.

VII. PROSPECTS AND SUMMARY
The preceding discussion suggests that a precise knowledge of width difference and lifetime ratio, as well as BR(B → X s γ), can have the potential to identify and discriminate between different CBSM scenarios, or rule them out altogether. This is illustrated in Fig. 4, showing contour values for future precision both in mixing and lifetime observables. In each panel, the solid (brown and green) contours correspond to the SM central values of the width difference and lifetime ratio (respectively). The spacing of the accompanying contours is such that the area between any two neighboring contours corresponds to a prospective 1σ-region, assuming a combined (theoretical and experimental) error on the lifetime ratio of 0.001 and a combined error on ∆Γ s of 5%. The assumed future errors are ambitious but seem feasible with expected experimental and theoretical progress. Overlaid is the (current) B → X s γ constraint (blue). The figure indicates that a discrimination between the SM and the scenario where ∆C 9 ≈ −1, while BR(B → X s γ) is SMlike is clearly possible. A crucial role is played by the lifetime ratio τ Bs /τ B d : in e.g. the ∆C 3 − ∆C 4 case a 1 σ deviation of the lifetime ratio almost coincides with the ∆C 9 = −1 contour line; a further precise determination of ∆Γ s could then identify the point on this line chosen by nature. Further progress on B → X s γ in the Belle II era would provide complementary information.
In summary, we have given a comprehensive, modelindependent analysis of BSM effects in partonic b → ccs transitions (CBSM scenario) in the CP conserving case, focusing on those observables that can be computed in a heavy-quark expansion. An effect in rare semileptonic B decays compatible with hints from current LHCb and B-factory data can be generated, while satisfying the B → X s γ constraint. It can originate from different combinations of b → ccs operators. The required Wilson coefficients are so small that constraints from B decays into charm are not effective, particularly if new physics enters at a high scale; then large renormalization-group enhancements are present. Likewise, there are no obvious model-independent conflicts with collider searches or electroweak precision observables. A more precise measurement of mixing observables and lifetime ratios, at a level achievable at LHCb, may be able to confirm (or rule out) the CBSM scenario, and to discriminate between different BSM couplings. Finally, all CBSM effects are lepton-flavor-universal; the current R K and R K * anomalies would either have to be mismeasurements or require additional lepton-flavor-specific UV contribution to C 9 ; such a combined scenario has been shown elsewhere to be consistent with all rare B-decay data and also presents the most generic way for UV physics to affect rare decays. With the stated caveats, our conclusions are rather model independent. It would be interesting to construct concrete UV realizations of the CBSM scenario, which almost certainly will affect other observables in a correlated, but model-dependent manner.

VIII. ACKNOWLEDGMENTS
We would like to thank C. Bobeth, P. Gambino, M. Gorbahn, and especially M. Misiak for discussions. This work was supported by an IPPP Associateship. S.J. and K.L. acknowledge support by STFC Consolidated Grant No. ST/L000504/1, an STFC studentship, and a Weizmann Institute "Weizmann-UK Making Connections" grant. A.L. and M.K. are supported by the STFC IPPP grant.

IX. APPENDIX: TECHNICAL ASPECTS OF THE ANOMALOUS-DIMENSION CALCULATION
Here we provide additional technical information regarding our results on anomalous dimensions entering in the RGE (20).
Many of the elements of γ eff(0) are known [14,[25][26][27][28], except for γ , for i = 3, 4. The latter can be read off from the logarithmic terms in (5), and the mixing into P i follows from substituting gauge coupling and color factors in diagram Fig. 1 (left). This gives for i = 1, 2, 3, 4, with the mixing into C P3,5,6 vanishing. The leading mixing into C eff 7 arises at two loops [29] and is the technically most challenging aspect of this work. Our calculation employs the 1PI (off-shell) formalism and the method of [30] for computing UV divergences, which involves an infrared-regulator mass and the appearance of a set of gauge-non-invariant counterterms. The result is Our stated results for i = 1, 2 agree with the results in [24,26], which constitutes a cross-check of our calculation.
We have not obtained the 2-loop mixing of C c 3,4 into C 8g and set these anomalous dimension elements to zero.
For the case of C c 1,2 where this mixing is known, the impact of neglecting γ eff(0) i8 on ∆C eff 7 (µ) is small [the only change being −0.19∆C 2 → −0.18∆C 2 in (9)]. We expect a similarly small error in the case of ∆C 3,4 .