Implications for new physics from novel puzzle in $\bar{B}_{(s)}^0 \to D^{(\ast)+}_{(s)} \lbrace \pi^-, K^- \rbrace$ decays

Recently, the standard model predictions for the $B$-meson hadronic decays, $\bar{B}^0 \to D^{(\ast)+}K^-$ and $\bar{B}^0_s \to D^{(\ast)+}_s \pi^-$, have been updated based on the QCD factorization approach. This improvement sheds light on a novel puzzle in the $B$-meson hadronic decays: there are mild but universal tensions between data and the predicted branching ratios. Assuming the higher-order QCD corrections are not huge enough to solve the tension, we examine several new physics interpretations of this puzzle. We find that the tension can be partially explained by a left-handed $W^\prime$ model which can be compatible with other flavor observables and collider bounds.


I. INTRODUCTION
To test the standard model (SM) and search for physics beyond the SM, precision measurements of meson decays, especially B-meson decays, have been considerably investigated over the past 30 years. In the meantime, the experimental uncertainty has been surprisingly reduced by experimentalists. On the other hand, theorists have played an equally important role: Several approaches that can evaluate the QCD corrections have been invented, and the SM predictions have been sharpened.
Very recently, SM predictions for several B-meson hadronic decays are improved by Ref. [1]: where the upper numbers are the PDG averages of the experimental data [2], while the lower ones are the SM expectation values [1]. These SM predictions are obtained by the QCD factorization (QCDF) [3][4][5] at leading power in Λ QCD /m b , where the Wilson coefficients at next-tonext-to-leading logarithmic accuracy are used [6]. Compared to the previous estimations [7], the theoretical uncertainties are significantly reduced thanks to recent developments in the B (s) → D ( * ) (s) form factors including order O(1/m 2 c ) corrections within the framework of the heavy-quark expansion [8][9][10][11].
These hadronic channels are theoretically clean due to the absence of penguin and annihilation topologies. Furthermore, resultant amplitudes are dominated by the color-favored tree topology.
Within the SM, there are two possibilities that these tensions are alleviated. The first possibility is an input value of |V cb |. For |V cb |, the authors of Ref. [1] use an average of the inclusive and exclusive determinations in the B-meson semileptonic decays: |V cb | = (41.1 ± 0.5) × 10 −3 [9,10]. If one adopts the exclusive |V cb |, |V cb | = (39.25 ± 0.56) × 10 −3 [12], amplitudes of the above processes are uniformly reduced by 4.5%. Note that the exclusive |V cb |, however, produces an additional 4.2σ level tension in ε K [13]. See also for a more recent determination of the exclusive |V cb | using the full angular distribution data [11].
Another possibility is higher-order QCD corrections. The next-to-leading power and next-to-next-to-leading power corrections to the QCDF amplitudes are also estimated by the same authors [1], and the sizes of those corrections to the amplitudes are evaluated as O(1)%.
Above puzzled situation could be resolved by introducing new physics contributions to b → cūq transitions, where q = d and s. Furthermore, it is shown that all ratios between these branching fractions are consistent with data [1]. It clearly implies that the new physics effects should be universal in b → cūq transitions. Therefore, the following questions are interesting: whether such a new physics is still allowed by the other flavor constraints and by the hadron collider constraints, and how much the tensions can be alleviated by a valid new physics model. Below we will refer to this puzzle as b → cūq anomaly. In this Letter, we examine several new physics scenarios to explain the b → cūq anomaly.

II. FRAMEWORK
We consider the following effective Lagrangian to investigate new physics contributions to b → cūq processes: with the left-handed current-current operators in the CMM basis [14,15], where q = d, s. T a is the SU(3) C generator, and V is the Cabibbo-Kobayashi-Maskawa matrix [16,17]. In our analysis, we refrain from adding operators that are absent in the SM, e.g., (c L b R )(q L u R ). We will discuss this possibility in the last section. New physics contributions to the Wilson coefficients, C q,NP 1 and C q,NP 2 , become involved at the new physics scale Λ. These values are modified by the renormalization-group (RG) evolution from Λ down to the hadronic scale m b . The leading-order (LO) QCD RG evolution is summarized in Appendix A. For instance, when Λ = 1 TeV, we obtain an evolution matrix as It is found that a universal destructive shift in the SM contributions is favored in the b → cūq anomaly [1]. The preferred size is ∼ −17%, which corresponds to C d,NP It is checked that such a new physics contribution is compatible with data of the total decay rates of the Bmesons, τ Bs /τ B d , and a f s d [1,[18][19][20]. Another potentially strong constraint comes from the kaon hadronic decays (s → uūd). The C P -conserving parts of the isospin amplitudes, A I = (ππ) I |H |∆S|=1 eff |K for I = 0, 2, have been measured very precisely through all K → ππ data [21,22] ReA exp On the other hand, these theoretical predictions are where the hadronic matrix elements are calculated by the lattice QCD simulations [22][23][24][25][26]. Although the A 2 amplitude is more sensitive to new physics than A 0 , we find that a ±20% new physics contribution to the s → uūd amplitude could be compatible with the data.

III. MINIMAL FLAVOR VIOLATION
First, we consider the most simple possibility for new physics scenario: minimal flavor violation (MFV) hypothesis [27,28], where the flavor symmetry is introduced and it is broken only by the Yukawa interactions. Under this hypothesis the flavor structure is the same as the SM one: the flavor-changing neutral currents (FCNCs) are automatically suppressed. For the b → cūq anomaly, we consider a dimension-six operator, with Y u = V † diag(y u , y c , y t ), and a is a dimensionless coupling. In the quark mass-diagonal basis (u diag From the operator in Eq. (14), we obtain a constraint from the B s -meson mass difference (∆M s ) as (cf., Ref. [29]), where the LO RG effect is taken into account [30], with η = α s (Λ)/α s (m W ), and the latest SM estimation of ∆M s is adopted [31,32]. We required the new physics contribution to ∆M s does not change the SM prediction at 2σ level. On the other hand, from the b → cūq anomaly in Eq. (9), we find Therefore, we reach a requirement for the anomaly: Λ 0.49 TeV and |a| 0.06 .
However, such a contact interaction can be probed by a non-resonant dijet angular distribution search in the LHC. The result is reported by the ATLAS collaboration at √ s = 13 TeV with dtL = 37 fb −1 [33]. We interpret the result in terms of the operator in Eq. (14), and obtain a 95% CL exclusion limit, Λ < 3.7 TeV and 4.9 TeV < Λ < 8.3 TeV .
This bound is clearly incompatible with Eq. (20). From this bound, we obtain a bound Therefore, this new physics scenario never explains the b → cūq anomaly.
IV. SU(2)×SU(2)×U(1) MODEL Next, we consider a new physics model that can produce a more convoluted flavor structure. An extended electroweak gauge group SU(2) 1 ×SU(2) 2 ×U(1) Y with heavy vector-like fermions produces heavy gauge bosons, W ± and Z , interacting with the left-handed SM fermions with a non-trivial flavor structure [34][35][36][37][38]. These flavor structures are controlled by the number of generations of the vector-like fermions (n VF ) and mixings between the SM fermions and vector-like fermions.
The heavy gauge boson interactions are [38] where u L , d L are the mass eigenstates, and a coupling g ij is defined in the d L basis. In the following, we will take In order to generate an uniform shift in both b → cūd and b → cūs, a SM-like flavor structure in (V g) 1q is required, and hence g 11 should be non zero. When only g 11 is a non-zero entry in g ij , a dangerouscuZ FCNC is generated and it is severely constrained by the D-meson mixing as |g 11 |/M V < O(10 −2 ) (TeV) −1 [39]. To evade this bound, we follow the U(2) 3 flavor symmetry [40,41], and take g 11 = g 22 in g ij in the following analyses. Then the bound from the D-meson mixing is significantly relaxed as |g 11 |/M V 16 (TeV) −1 . Another flavor constraint comes from the K → ππ data. By permitting a ±20% new physics contribution to the Wilson coefficient of (ū L γ µ d L )(s L γ µ u L ) [see, Eq. (13)], we obtain Note that many types of diagrams contribute to K → ππ decays, and non-perturbative QCD plays an essential role there. Therefore, this bound is a just reference value. In addition to g 11 , another non-zero entry of g 33 or g 23 is necessary to produce C q,W 2 . Therefore, we consider the following flavor texture: and will discuss several scenarios in detail. We assume g ij is real for simplicity. Note that when g 11 is O(1), production cross sections of the heavy gauge bosons become considerably large in the hadron collider, and hence we will mostly discuss the LHC constraints in each subsection. To evade surveying a dedicated collider constraint for low-mass region where the constraint would be more stringent, the mass range M V > 1 TeV is considered in our analysis.
A. Scenario 1: g11 and g33 In this subsection, we take g 23 = 0 and consider a scenario of g ij = diag(g 11 , g 11 , g 33 ). Such a flavor structure can be obtained from n VF = 1. In this case, (V g) 23 in Eq. (24) comes from V cb g 33 . Since one has a factor of V cb just as the SM, |g 11 g 33 |/M V should be larger than O(1) TeV −1 to generate new physics contributions to b → cūq processes (see, previous section). Furthermore, a relative sign between g 11 and g 33 must be negative to produce the destructive interference with the SM in the b → cūq decays. A requirement of the b → cūq anomaly within 2 σ level leads to 2.6 (TeV) −1 Therefore, large couplings are necessary in this scenario. First, let us examine the constraint from ∆M s . In this scenario, the dominant contribution comes from a W -W box diagram. We observed that the GIM mechanism still works in this flavor structure, and obtain a simple formula for the W -W box contribution to ∆M s , with x t = m 2 t /m 2 W and x V = M 2 V /m 2 W , and g W is the weak coupling. The loop functions are defined in Appendix B. We also have the same shift in B d -meson mixing, but it is less constrained because of its large theoretical uncertainty. By imposing that the new physics contribution is within 2σ uncertainty of ∆M SM s [31,32], we obtain Although ∆M s bound is incompatible with the b → cūq anomaly in Eq. (27), we want to know how much this scenario can alleviate the puzzle. Next, we consider constraints from resonant productions of the heavy gauge bosons at the LHC. When g 11 and g 33 entries are non zero, Z is produced via pp → qq → Z and also pp → bb → Z , while W ± is produced thorough pp → qq → W ± processes. When M V m t , the decay width of those particle is approximately given as, We find that relevant collider bounds come from dijet and tt searches. The former provides the relevant bound for |g 11 | |g 33 |, while the latter for |g 11 | ∼ < |g 33 |. Currently, both ATLAS and CMS collaborations reported upper limits on the heavy dijet resonance cross section using the data of ∼ 140 fb −1 [42,43]. Since O(1) couplings are necessary to relax the tension, the decay width can be not small. Therefore, we adopt widthdependent limits on the cross section times the dijet branching ratio. The broader the width is, the weaker the limits become because a characteristic resonance peak is diluted. The search is robust up to Γ V /M V = 20% for 1.8-2.1 TeV, and up to Γ V /m V = 55% for the heavier region [43]. For the mass range of 1-1.8 TeV, we use an upper limit in Ref. [44], where the narrow width approximation (NWA) is used. As for the heavy tt resonance search, CMS reported the width-dependent limit using the data of 36 fb −1 up to Γ V /M V = 30% [45], while AT-LAS reported the result using the data of 139 fb −1 in the NWA [46].
We obtained the production cross section of Z and W ± by rescaling the result in Refs. [43,49] . The excluded regions from the dijet and tt searches are shown as the blue and orange shaded regions in Fig. 1 (left), respectively.
We also show constraints from the single t searches by using the data of ∼ 36 fb −1 of CMS [47] and ATLAS [48]: the regions above the dashed lines in Fig. 1 (left) are excluded. Note that both analyses assume the narrow resonance, and no study exists for broad resonances.
Taking a conservative position, regions above the plateaus of the shaded areas can not be excluded, where the corresponding Γ V /M V exceeds the maximum width shown in each experimental result: Γ V /M V > 30% in the tt search, and Γ V /M V > 55% for 2.1-5 TeV and Γ V /M V > 20% for 1.8-2.1 TeV in the dijet search. The horizontal blue dashed lines are extrapolations obtained by assuming the analysis of Ref.
[44] is valid up to Γ V /M V = 20%, and should be taken with more care. We note that limits from the dijet angular distribution data, which are not considered here, would also depend on the width-mass ratio and only contact interaction models are investigated [33,51]. Further dedicated analysis would be necessary to exclude such a broad width region.
The red-hatched regions represent Γ V > m V , where a particle picture is no longer valid and one could not discuss any conclusive prediction.
As long as we allow the broad width scenario, we find that the bound from ∆M s in Eq. (29) determines the maximal deviation of C W 2 /C SM 2 which is independent of the ratio of g 11 and g 33 . For these reasons, we conclude C W 2 /C SM 2 −0.05 when g 23 = 0.

B. Scenario 2: g11 and g23
For the second scenario, we set g 33 = 0 and consider g 11 and g 23 in Eq. (26). Such a flavor structure can be obtained when n VF = 2. In this scenario, the b → cūq anomaly requires 0.54 (TeV) −1 Although the size of the required coupling product is much smaller than the previous scenario, a severe bound on g 23 comes from ∆M s , where there is a tree-level Z exchange diagram. We obtain and find that g 23 always gives a positive shift in ∆M s . The constraint from ∆M s is [31,32] |g 23 |/M V 0.01 (TeV) −1 .
Therefore, g 11 30 (M V /TeV) 4π is required by Eqs. (31) and (33), which implies that the b → cūq anomaly can not be explained by this scenario.
In this scenario, |g 23 | |g 11 | should be satisfied. This simplifies the collider constraints because the production cross section is controlled only by |g 11 |, and the heavy gauge bosons decay into jets with B 1. The constraints are shown in Fig. 1 (middle). we find C W 2 /C SM To see maximum value of |C W 2 /C SM 2 | in this model, we combine the first and second scenarios: all g 11 , g 23 and g 33 are non-zero entries. The point of this scenario is that the severe bound from ∆M s can be turned off by where the W contribution is destructive and the Z one is constructive in ∆M s (see previous subsections). We find, however, that even if the ∆M s bound is turned off, g 11 g 33 is still constrained from the ∆M d as, This bound restricts the possible W contribution to the b → cūq processes. Also, we have checked a constraint from b → sγ data. We conclude that the b → sγ bound is less sensitive than ∆M d , see Appendix C.
Since |g 23 | |g 11 |, |g 33 | still holds in this scenario, the collider constraints are almost the same as the scenario 1. We focus on a parameter region that the all LHC constrains are evaded by the broad width of the heavy gauge bosons. In Fig. 1 (right), C W 2 /C SM 2 is shown on g 23 -g 33 plane by fixing M V = 1 TeV and g 11 = −3.6 corresponding to the maximum value allowed by the K → ππ data in Eq. (25). Eventually, we obtain investigated the size of possible several new physics contributions to these processes. In spite of severe bounds from the other flavor observables and the LHC searches, we conclude that a −10% shift in the b → cūq amplitude is possible by the left-handed W model. Such a new physics contribution can reduce the tension in the b → cūq processes.
Since g 22 = g 11 is a necessary condition, this model also produces new physics contributions to b → ccs processes with the same size [52,53]. Alhough they, e.g., B + → J/ψK + , have been measured precisely, the SM predictions suffer from large nonfactorizable corrections [54][55][56]. We, therefore, expect that the b → ccs processes are less sensitive than b → cūq.
It is unclear whether the new physics scalar operator can explain the b → cūq anomaly, but it is interesting direction to consider it. For instance, within a general two Higgs doublet model, a charged Higgs interaction is [57] where (V ρ d ) 23 is stringently constrained by ∆M s via a heavy neutral Higgs exchange, while (ρ † u V ) 23 is less constrained by the flavor and collider observables [58,59]. Therefore, a potentially large contribution to the b → cūq processes would be expected.
with η = α s (Λ)/α s (m W ). At the weak scale, the SM contributions enter as [6] C q 1 (m W ) = 15 and their RG evaluation from the weak scale to the hadronic scale is

Appendix B: Loop functions
The loop functions f (x) and f (x, y) in Eq. (28) are defined by [61] (cf. [62]) where lim y→1 f (x, y) = f (x). We note that Ref. [61] contains a typo in Eq. (22), where −x 2 in the last term of the first line in the arXiv version must be replaced by −x 3 .
The loop functions f γ (x) and f g (x) in Eq. (C4) are defined by [63] f Appendix C: W and Z contributions to b → sγ The effective Lagrangian for the b → sγ process is with Q 7γ = e 8π 2 m bs σ µν (1 + γ 5 )bF µν , Q 8g = g s 8π 2 m bs σ µν T a (1 + γ 5 )bG a µν , and the operators Q 1 -Q 6 are defined in Ref. [64]. By integrating out the heavy gauge bosons, we obtain and remaining coefficients are set to zero at µ = M V . To obtain new physics contributions at the hadronic scale, we solved the corresponding RG evolution down to µ = m b numerically: d C(µ) d ln µ = α s (µ) 4π γ (0)eff T C(µ) , C = C 1 , C 2 , · · · , C 6 , C eff 7 , C eff where the anomalous dimension matrixγ (0)eff is given in Refs [64,65]. The C eff 7 , C eff 8 are the effective Wilson coefficients which are required to cancel a regularization scheme dependence [66]. In this model, C eff 7 (M V ) = C 7 (M V ) and C eff 8 (M V ) = C 8 (M V ). Using C eff 7 (m b ), we obtain a constraint from b → sγ data, where we required the new physics contributions are within a 2σ uncertainty range [67,68]. The bound is sensitive to g 23 which comes from the Z contribution to C 3 (M V ). For g 23 = 0, we obtain This bound is significantly alleviated for g 23 /g 33 > 0 region, while it becomes stronger for g 23 /g 33 < 0 region.