A new B-flavour anomaly in Bd,s → K∗0K¯∗0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {K}^{\ast 0}{\overline{K}}^{\ast 0} $$\end{document}: anatomy and interpretation

In the context of the recently measured non-leptonic decays Bd→K∗0K¯∗0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {K}^{\ast 0}{\overline{K}}^{\ast 0} $$\end{document} and Bs→K∗0K¯∗0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {K}^{\ast 0}{\overline{K}}^{\ast 0} $$\end{document} we analyse the anatomy of the LVV observable that compares the longitudinal components of Bs→VV and Bd→VV decays. This observable is cleaner than the longitudinal polarisation fraction as it is afflicted only at subleading order in a 1/mb expansion by the theoretical uncertainties arising in the transverse components entering the polarisation fraction. Focusing on the particular case of Bd→K∗0K¯∗0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {K}^{\ast 0}{\overline{K}}^{\ast 0} $$\end{document} and Bs→K∗0K¯∗0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {K}^{\ast 0}{\overline{K}}^{\ast 0} $$\end{document}, we discuss the main sources of hadronic uncertainty in the SM. We find for the SM prediction LK∗K¯∗=19.5−6.8+9.3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {L}_{K^{\ast }{\overline{K}}^{\ast }}={19.5}_{-6.8}^{+9.3} $$\end{document}, which implies a 2.6σ tension with respect to the most recent data, pointing to a deficit in the b→s transition of the non-leptonic decay versus the corresponding b→d transition. We discuss possible New Physics explanations for this deviation, first at the level of the Weak Effective Theory and we identify that the two Wilson coefficients C4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{C}}_4 $$\end{document} and C8g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{C}}_{8g} $$\end{document} can play a central role in explaining this anomaly. Finally, we briefly explore two different simplified New Physics models which can explain the anomaly through a contribution either in C4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{C}}_4 $$\end{document} (Kaluza-Klein gluon) or in C8g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{C}}_{8g} $$\end{document}, with a significant amount of fine tuning, but possible connections to the b→sℓℓ anomalies.


Introduction
The flavour anomalies observed in semileptonic rare B meson decays constitute one of the most promising hints of New Physics (NP) found at LHC and B-Factories. Recent global analyses of the set of observables governed by the b → s transitions [1,2] provide a small p-value (1.4%) for the Standard Model (SM), whereas simple NP hypotheses obtain a much better description of the data, with pulls up to 6.5σ with respect to the SM (similar results are obtained in other works [3][4][5][6][7][8][9][10]). A particularly promising setup combines Lepton Flavour Universality Violating (LFUV) NP together with Lepton Flavour Universal (LFU) NP, as proposed in ref. [11], which improves the description of the data compared to the SM by 7.4σ [2] once the anomalies in b → c ν decays are included.
If NP is indeed at the origin of the anomalies in semileptonic B decays, it is natural to expect signals in other observables involving b → s transitions, possibly with different realisations though sharing some common features. A natural place to explore the possible existence of these signals are non-leptonic B decays. This type of decays suffer from larger uncertainties compared to semileptonic B decays and are therefore more difficult to compute with a high accuracy. In particular, branching ratios and polarisation fractions receive contributions from transverse amplitudes that suffer from large uncertainties due to power-suppressed but infrared-divergent weak annihilation and hard-spectator scattering [12,13]. In this sense a deviation with respect to the SM prediction in non-leptonic B decays requires one to be much more conservative regarding these uncertainties than in the case of semileptonic B decays.

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In this article we will thus follow a similar strategy to the one we used in refs. [14,15] for semileptonic rare B decays and we establish a parallelism constructing observables in non-leptonic B decays with a limited sensitivity to hadronic uncertainties. This can be achieved using the R sd observable introduced some time ago by two of us in ref. [16]. This observable was introduced at that time to find NP in neutral B-meson mixing. However, it turns out to be particularly interesting now to find NP in the non-leptonic decay amplitudes in the light of the b → s anomalies for which optimized observables were introduced.
In b → s decays, one can build two different kinds of observables with a reduced sensitivity to hadronic uncertainties: on the one hand, angular observables from decays involving muons in the final state [15,17] constructed exploiting heavy quark symmetry and on the other hand, ratios of branching ratios with muons versus electrons in the final state that test LFUV and where the dependence on the form factors cancels almost exactly in the SM [18]. There are tensions in observables involving leptons of the second family (for the former) and between the second and the first family of leptons (for the latter). In this work we explore the parallel approach of using non-leptonic B decays rather than semileptonic ones, comparing quark transitions involving quarks of the second and first families instead of muons and electrons. More specifically, we compare transitions involving s-quarks and d-quarks to benefit from the approximate U -spin symmetry of the Standard Model in analogy with Lepton-Flavour Universality used to build the LFUV ratios in b → s decays. The analogy has evident limitations: since both symmetries are broken by fermion mass effects, the size of the corrections is easier to compute or estimate for LFU (involving mainly QED) than for U -spin (involving QCD). However, even in the nonleptonic case it is well known that ratios of this type offer many advantages in reducing hadronic uncertainties, explaining the popularity of the ratio ξ to describe neutralmeson mixing in lattice QCD and phenomenological studies. We may reach an even better control of hadronic uncertainties by combining several approaches. In refs. [16,19,20] two of us showed that the specific structure of penguin-mediated non-leptonic B-decays could lead to a better theoretical control on combinations of hadronic matrix elements within factorisation approaches. In the case of vector final states, it is also known that the decays into longitudinally polarised light mesons can be described more precisely than the transverse ones within these factorisation approaches, providing a further guide to build optimised observables (in analogy with the angular observables in semileptonic decays). Finally, if the B d -meson decays have been studied at B-factories extensively, LHCb is now able to provide accurate measurements for many B s -meson decays with the possibility to assess the correlation between B d and B s mesons decaying into the same final state.
We will thus focus here on a type of observables for penguin-mediated non-leptonic decays of B mesons into two vector particles, that we will refer as L-observables. These correspond essentially to the R sd observable introduced in ref. [16] in the case of B d,s → K * 0K * 0 (up to a phase space). We present here a detailed and complete anatomy of this observable in the SM, updating the SM prediction and observing an increase in the tension with the experimental measurement compared to ref. [16]. We then discuss NP explanations for the tension observed. We also point out possible improvements of the theoretical prediction of this observable.

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In section 2 we develop the theoretical framework that will be used to compute the L observable. We put a particular emphasis on the sources of hadronic uncertainties coming from infrared divergences that affect mostly branching ratios and polarisations. In section 3 we construct this observable and we compute it. Then using the data of the previous section we determine its experimental value and the pull. In section 4 we explore possible solutions in terms of NP shifts to Wilson coefficients in a model-independent EFT approach, before considering particular models illustrating the difficulty to explain this non-leptonic anomaly together with the b → s anomalies in section 5. We finally conclude in section 6. Appendices are devoted to a discussion of the weak effective theory and QCD factorisation, the semi-analyical description of relevant hadronic matrix elements, and complementary material concerning the sensitivity of L to different sources of NP.

Helicity amplitudes
We start by considering the theoretical description of B Q → V V with Q = d, s. Since the initial state has spin 0, the two vector mesons must have the same helicity, leading to a description of the decay in terms of three helicity amplitudes A 0 , A + and A − . In naive factorisation one expects a hierarchy of the type: This hierarchy with a dominance of longitudinal amplitudes is easy to understand by means of the V-A structure of the SM [21]. Each amplitude is suppressed with respect to the previous one by O(Λ/m b ) due to helicity suppression [12]. The longitudinal amplitude in a b → s transition is dominant as compared to the positive helicity: the s quark is produced with an helicity −1/2 by weak interactions (in the limit m s → 0), which is not affected by the strong interactions, then the strange quark combines with the light spectator quark to form a V with a helicity which can reach 0 or −1 but not +1. InĀ − , a light-quark helicity flip is required to obtain both vector mesons with a negative helicity, whereas inĀ + , two helicity flips are required to reach a positive helicity for both vector mesons. Each of these helicity flips yields a suppression by a factor O(Λ/m b ), as expected in naive factorisation.

Hadronic matrix elements
For aB Q meson decaying through a b → q penguin-mediated process into a V 1 V 2 state with a definite polarisation, the decomposition is always possible, with the CKM factors λ (q) We denote by T q and P q the matrix elements accompanying the λ (q) u and λ (q) c CKM factors respectively. In the SM, P q is usually associated to penguin topologies, whereas T q receives contributions from tree topologies (but it can also contain only penguin topologies in some decays). As discussed above, if we consider the longitudinal polarisation, T q and P q can be computed using factorisation approaches based on a 1/m b expansion (see appendix A). In QCD factorisation [22], T q and JHEP04(2021)066 P q are affected by possibly large long-distance 1/m b -suppressed effects that will be discussed in the next section. In the case of penguin mediated decays like B (d,s) → K * 0K * 0 , it was observed [19,20] that the same type of (long-distance) infrared divergences affect both P q and T q , so one can construct free from these next-to-leading-order infrared divergences. Using the unitarity relation λ 3) The weak phase in λ (q) t is the angle β q , defined as c is real to a very good approximation for both q = d, s, and λ t . The CP-conjugate amplitude is given by , even though the two types of amplitudes are related: where η f is the CP-parity of the final state, given for j = 0, ||, ⊥ respectively as η, η, −η where η = 1 if V 1 is the charge conjugate of V 2 (this is the case for K * 0K * 0 ).

Definition and experimental determination
The 2019 LHCb analysis with 3 fb −1 data measured the ratio of the untagged and timeintegrated decay rates [23] The longitudinal polarisation of both modes has been measured as well. The average of B d → K * 0K * 0 from LHCb [23] and Babar [24] f LHCb whereas the polarisation for the B s → K * 0K * 0 mode is [23]: Most of the experimental determinations are made assuming no direct CP-violation; however, the ones searching for CP violation found no hint in these decays [25].
One can notice already that the longitudinal polarisations are very different for these two modes, although they are related by U -spin symmetry in its most obvious form, i.e. the d ↔ s exchange. In the SM, U -spin is broken only by the quark masses, and it is thus expected to be fairly well obeyed (up to a 20-30% correction). We propose to define an observable that will be sensitive to this effect but with a cleaner theoretical prediction: where B b→q (f b→q L ) refers to the branching ratio (longitudinal polarisation) of theB Q → V 1 V 2 decay governed by a b → q transition. A q 0 andĀ q 0 are the amplitudes for the B Q and B Q decays governed by b → q with final vector mesons being polarised longitudinally and stands for the phase space factor involved in the corresponding branching ratio, with ω = τ B Q /(16πM 3 B Q ), Σ ab = (m a + m b ) 2 and ∆ ab = (m a − m b ) 2 and all quantities are CPaveraged.
This observable is defined such that the dependence on the troublesome transverse (parallel and perpendicular) amplitudes entering the branching ratio and longitudinal polarisation fraction cancel and it is close to the observable R sd for the case of B d,s → K * 0K * 0 up to a phase space factor [16].
Being purely sensitive to the longitudinal amplitudes, L is less affected by the hadronic uncertainties which impact the transverse polarisation amplitudes significantly and which are difficult to estimate within QCD Factorisation (QCDF) or other approaches based on a 1/m b expansion. The choice of this observable thus avoids the difficulties encountered in the interpretation of low longitudinal polarisation fractions observed in some non-leptonic modes [12]. In this article we will focus on: where the spectator quark Q of the initial b-flavoured meson and the quark q from the b → q transition coincide.
In the definition of L K * K * and its connection with the longitudinal amplitudes |A q 0 | 2 in eq. (3.8), we have not included the effect of B s -meson mixing that arises in branching ratios when measured at hadronic machines. This effect of time integration at hadronic JHEP04(2021)066 machines generates a correction of O(∆Γ/(2Γ)) discussed in refs. [16,26], which would multiply the last term in eq. (3.8) by: where y q = ∆Γ Bq /(2Γ Bq ) is well measured (y d is negligible and y s 0.065) and the asymmetries −1 ≤ A q ∆Γ ≤ 1 combining CP violation in mixing and decay are difficult to estimate theoretically, leading to a correction of at most 7%.
Since we use the LHCb measurement eq. (3.1) and since there are other sources of (theoretical and experimental) uncertainties, we treat eq. (3.9) as a systematic uncertainty of 7% combined in quadrature with the other uncertainties, leading to the experimental value: Exp :

Theoretical prediction in the SM and comparison with data
On the theory side, we have

11)
. We thus get with the combinations of CKM factors (estimated using the summer 2019 CKMfitter update [27][28][29] (see table 3): From QCD factorisation and the discussion in section 2, we have so that the brackets in eq. (3.13) are very close to 1, with the main uncertainty of 1% from the term proportional to |α d | 2 (which will be included in the theoretical uncertainties below). The leading uncertainty in the theoretical evaluation of L K * K * comes thus from  (3) : while a naive factorisation approach would rather yield fact SU (3) : where the SU(3)-breaking ratio related to the form factors of interest is given by and we used the values of ref. [30] for the form factors to estimate f . A last possibility amounts to using QCD factorisation. Using the same inputs as before, we obtain QCD fact : The QCD factorisation-based prediction follows the theoretical computations of the different contributions to the amplitudes from refs. [13,31]. The numerical values of the input parameters used are updated with respect to the ones in ref. [31] and can be found in table 3 of appendix A.
Hard-gluon exchanges with the spectator quark and weak annihilation feature 1/m bsuppressed contributions exhibiting infrared divergences related to the endpoint of the meson light-cone distribution amplitudes. These divergences are parametrised in the same manner as in ref. [31], involving two contributions X H and X A treated as universal for all channels: We take ρ H,A ∈ [0, 1] and ϕ H,A ∈ [0, 2π] with flat distributions. This translates into assigning a 100% uncertainty to the magnitude of such corrections. We propagate the uncertainties by varying each input (given in table 3) entering the penguin ratios in eqs. (3.18), (3.19) and (3.21) and the CKM contribution κ following eq. (3.16), using Gaussian distributions. We determine then the distribution of L in each case, leading to the 1σ ranges:

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Relative Error where we put the level of discrepancy with experiment, in units of σ. We stress that these discrepancies are obtained using the whole distribution for L and not just the 1σ confidence intervals in the Gaussian approximation (see table 1 for the 1 and 2σ confidence intervals).
In table 2 we present the error budget for L K * K * in the SM. The comparison with the error budget of |P d,s | 2 shows that the impact of X A (X H ) is reduced from 18% (2%) in |P d,s | 2 to 4% (0.2%) in L K * K * . A similar reduction is observed for other inputs such as f K * , showing the benefit of defining the ratio L K * K * . It also indicates that the accuracy of the theoretical prediction of L K * K * could be improved significantly by determining the correlations among the relevant B → K * form factors in order to compute the associated SU(3) breaking. Moreover, the impact of the weak annihilation and hard-scattering divergences on the uncertainty is subdominant and would not be affected strongly by using a different approach for these power-suppressed infrared divergences. From the comparison of the SM predictions eqs. (3.23)-(3.25) with the experimental result in eq. (3.10), we see that all our theoretical estimates point towards a deficit in the b → s transition compared to the b → d one for these penguin-mediated modes, in analogy with the deficit observed in semileptonic decays to muons versus the decay to electrons in b → s decays.

Model-independent NP analysis
Even though the deviation in L K * K * is not yet at the level of a troublesome discrepancy with the SM, its potential connection with other B-flavour anomalies makes it interesting to investigate it further in terms of possible SU(3)-breaking NP contributions. We may explore in a model-independent way how to explain this anomaly via contributions only

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to the Wilson coefficients of the b → s transition, while keeping the corresponding b → d SM-like (or with opposite NP contributions).
This can be performed by using the weak effective theory, whose basis within the SM we recall in eq. (A.1) of appendix A. Note that in the presence of generic NP, the basis of operators must be extended since we expect this NP contribution to couple with different strength to different flavours (and in particular to d and s quarks), there is no a priori reason for it to yield "strong" and "electroweak" penguin operators with sums over all quark flavours following the same pattern as in the SM [32].
However, for simplicity, and in parallel with the results of the global fits for NP in b → s decays favouring SM operators or chirally-flipped versions of it, we consider here only NP entering the Wilson coefficients associated with the SM operators Q i or the chirallyflipped onesQ i as defined in ref. [21] by exchanging V − A and V + A in all quark bilinears constituting the operators. These right-handed currents would modify the longitudinal amplitude by adding contributions that are functions of In practice this means that the NP contribution to each coefficient entering the longitudinal amplitude should be interpreted as stemming not only from the standard operators but also from the chirally flipped ones (with an opposite sign).
We consider the sensitivity of L K * K * on each Wilson coefficient. We want to determine if there is a dominant operator that can naturally explain the low experimental value of L K * K * , as it happens for b → s with O 9 . We assume that NP enters as described above with the further requirement that there are no additional NP phases, leading to real-valued Wilson coefficients. We can then compute the hadronic matrix elements within QCD factorisation exactly like in the SM. In appendix B we provide semi-analytical expressions for P d and P s , needed to compute L K * K * in terms of Wilson coefficients. We provide the explicit dependence on the infrared divergences X A and X H although their numerical impact on the uncertainty is limited. Let us note in passing that the quantity ∆ q is still protected from infrared divergences in this NP extension: the structure of the longitudinal hadronic amplitudes T and P is unchanged, and only the numerical values of Wilson coefficients are modified compared to the SM (the protection of ∆ from infrared divergences would not necessarily hold in more general NP extensions).
Considering the sensitivity of L K * K * on each Wilson coefficient of the weak effective theory individually, we can determine the coefficients where a limited NP contribution would be sufficient to explain the discrepancy observed. We thus identify three dominant coefficients: C c 1q , C 4q and C eff 8gq (see figure 1 and figure 5 in appendix C). The strong dependence on these coefficients with respect to the others can be seen already in the explicit form of P d,s : which translates into a dominant contribution for L K * K * as well.
The reason behind this strong dependence on these coefficients can be understood in the following way. Let us consider a penguin-mediated decay, so that the SM tree-level operator

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C c 1s contributes through a closed cc loop to the decay, putting its contribution at the same level as the "strong" penguin operators i = 3 . . . 6 in the SM. A very similar contribution at the level of the underlying SM diagrams comes thus from both C c 1s and C 4s , as can be seen from the V − A structure of the operators (this is also the case for C eff 8gs with the emission of a gluon coupling to a qq pair). The effect of the diagrams is similar in the SM, but the separation between long and short distances in the weak effective theory yields C 4s and C eff 8gs much smaller than C c 1s , which must be compensated by larger weights in eqs. (4.1) and (4.2). The other penguin operators are suppressed either because of colour suppression (C 3 , thus associated with 1/N c factors in the QCD factorisation formula) or helicity suppression (C 5 and C 6 , which yield a vanishing contribution in the naive factorisation approach as they must be Fierzed into (pseudo)scalar operators with vanishing matrix elements). In the SM, the "electroweak" penguins i = 7 . . . 10 are suppressed. Their contributions might be very significantly enhanced by NP which would not require such an electromagnetic suppression, although it would be difficult to obtain then "electroweak" operators at the m b -scale since they involve explicitly the quark electric charges. If we nevertheless allowed for such very large contributions for the electroweak part (which we will discard in the following), the same argument would apply as in the case of the "strong" penguins, so that the leading contribution from the "electroweak" penguins would be C 10q .
As can be seen in figure 5, the coefficient C c 1s requires a very large NP contribution w.r.t. the SM of order 60% to reduce this discrepancy at 1σ. We will not pursue the possibility of a contribution to C c 1q , as the size of the effect being so large at an absolute scale is in conflict with recent analyses of the global constraints on this coefficient [33] that suggest that the room for NP contributions is of O(10%) of the SM. Dijet angular distributions [34], together with flavour bounds following from SU(2) L gauge invariance, suggest bounds which are even tighter.
The penguin coefficient C 4s requires a NP contribution of order 25% (which is incidentally similar to the NP contribution needed in C 9 for b → sµµ) in order to reduce the discrepancy in L K * K * at 1σ. The NP contribution needed is thus quite large but not significantly constrained from other non-leptonic decays where many other coefficients enter [22].
Finally, C eff 8gs would require a NP contribution of order 100% of the SM in order to obtain a similar reduction of the discrepancy. Although it might seem a large contribution, it is actually very difficult to obtain a precise bound on this effective coefficient which combines C 8gs with some Wilson coefficients of four-quark operators (see appendix A). Due to QCD loop effects, the constraint from b → sγ is actually on a linear combination of the Wilson coefficients C eff 7γs and C eff 8gs at the scale µ b [35]. Therefore, an effect in C eff 8gs can always be cancelled by an effect in C eff 7γs so that the experimental bound from b → sγ is obeyed (the same is also true for b → dγ [36]). Even without such a cancellation from C eff 7γs , the current measurements can accommodate a NP contribution to C eff 8gs of the order of the SM. Another more direct bound on C eff 8gs is provided by the b → sg contribution to inclusive non-leptonic charmless decays. The current bound on the b → sg branching ratio in ref. [37] is at the level of 6.8%, whereas the SM contribution [38] is estimated at the level of 0.5%, leaving room for a NP contribution to C eff 8gs up to three times as large as the SM one.

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Naturally, in each case, if we allow for NP in both C is and C id , we may get the same reduction of the discrepancy by assigning half of the NP contribution (with opposite signs) to both coefficients, as illustrated for C 4 in figure 2. Thus, allowing NP in b → d transitions in addition to b → s transitions requires smaller NP contributions in each type of transition, and allows one to evade some of the bounds discussed above as they applied only to b → s transitions (e.g. C 8gs ). C 8gd is constrained from b → dγ.

Simplified NP models
Our model-independent analysis showed that L K * K * is mostly sensitive to colour-octet operators and to a lesser extent to the chromomagnetic operator. In the following, we will consider NP models able to generate such contributions, and for concreteness, present the formula for the case of b → s transitions.
Concerning C 4s , it is natural to search for a tree-level explanation in terms of NP and a massive SU(3) c octet vector particle, i.e. a Kaluza-Klein (KK) gluon, also called axigluon, comes naturally to mind. We parametrise its couplings to down quarks of different flavours as with ∆ L,R sb assumed real. We also define from eq. (5.1) analogous flavour diagonal couplings which we will denote as ∆ L,R qq . We may consider the constraints from neutral-meson mixing through the effective Hamiltonian of ref. [39] where only the operators relevant for the discussion are displayed and where the operators with a tilde are obtained by exchanging the chirality projectors P L and P R . We get the matching contributions We obtain the allowed region shown in blue in figure 3 for real values of the Wilson coefficients and neglecting the bag factor uncertainties related to C BsBs 4,5 . Assuming that the KK gluon has universal flavour-diagonal coupling to the first two generations of quarks, which is also needed to avoid unacceptably large effects in K −K and/or D 0 −D 0 mixings [45], our model generates 1 a NP contribution to C 4s given at the matching scale by However, couplings of first generation quarks to KK gluons are strongly constrained by di-jet searches [46]: (∆ L qq /m KK ) 2 < (2.2/(10 TeV)) 2 . Allowing for NP also in b → d transitions could increase the effect in L K * K * , but since here the effect is bounded by B d −B d mixing, whose constraints are of the same order as B s −B s mixing, one can only gain a factor ≈ 2. Using this maximal coupling for the ∆ L qq couplings and setting the ∆ R qq couplings to zero, we can see from figure 3 that a significant amount of fine-tuning is needed to account for L K * K * . Alternatively, one could try to explain L K * K * with a NP contribution in the chirallyflipped coefficientC 4s , given by eq. (5.11) with the ∆ L sb and ∆ L qq couplings replaced by ∆ R sb and ∆ R qq , respectively. In principle, one could exploit the fact that the couplings do not have to respect an U(2) flavour symmetry (since up-and down-type quark couplings are not related via SU(2) L ), so that couplings to first-generation quarks could be avoided, which would relax LHC bounds and reduce the fine-tuning needed in B s −B s mixing. However, as in the previous case, flavour universality for diagonal couplings to quarks is needed to be able to make use of our expressions for L K * K * . Moreover, according to QCD factorisation, the dominant LO effect in L K * K * originates from the term in Q 4s with down quarks in the bilinear summed over flavours. Therefore, (dominant) right-handed couplings cannot be used to evade LHC bounds and still fine-tuning in B s −B s mixing, like in the case of left-handed couplings, is needed.
As indicated earlier, one could also try to explain L K * K * with the Wilson coefficient of the chromomagnetic operator O 8gs . Here an effect of the order of the SM contribution 1 Note that our model is only flavour universal with respect to four but not five flavours and does not fulfill the requirements of section 4. However, the effect of bottom quarks within the Q4s operator in L K * K * is O(αs)-suppressed within QCD factorisation and thus the impact of our model on L K * K * can be mimicked by a shift in C4s to a good approximation. is required. C 8gs can only be generated at the loop level and involves necessarily coloured particles for which strong LHC limits exist. Therefore, a value of the order of the SM contribution can only be obtained thanks to chiral enhancement. A simplified model fulfilling these requirements features two vector-like quarks, one SU(2) L doublet and one SU(2) L singlet (with a large coupling λ to the SM Higgs doublet) and an additional neutral scalar particle [47]. In this setup, C 8gs receives a contribution which scales like λ/(m b /v) × v 2 /M 2 w.r.t. the SM, where M is the NP scale. Inevitably an effect in C 7γs is generated at the matching scale M which however has free sign and magnitude as it depends on the (not necessarily quantized) electric charges of the new fermions and scalar inside the loop. Therefore, the electric charges of the new particles can be chosen in such a way that in C 7γs (at the m b scale) the NP contributions to C 7γs and C 8gs (taken at the matching scale) cancel. As we need a NP contribution to C 8gs of the order of the SM one, and C 7γs at the low scale is known at the 5% level, a tuning of the order of 1/20 is necessary here.
Both simplified models allow for the possibility of a connection with the b → s + − anomalies. On the one hand, the KK gluon may be part of the particle spectrum of a composite/extra-dimensional model and is then accompanied by a Z boson. This could explain b → s + − without violating LHC di-lepton bounds [48] due to the large sb coupling of the Z needed to explain L K * K * , leading to NP contributions with the correct sign in both types of anomalies. On the other hand, the model generating a large effect in C 8g could easily be extended by a vector-like lepton in order to account for b → s + − [47].

JHEP04(2021)066 6 Conclusions
In this article, we have analysed the non-leptonic penguin decays B d → K * 0K * 0 and B s → K * 0K * 0 , where recent LHCb results indicate striking differences in the longitudinal polarisation of these two modes. This is unexpected since they are related by U -spin and should thus have a similar QCD and EW dynamics (up to tiny corrections due to the down and strange quark masses).
We introduced the L-observable as a combination of polarisation fractions and branching ratios in order to compare the longitudinal amplitudes in both modes, as they can be computed with better theoretical control in a 1/m b expansion such as QCD factorisation. We exploited the fact that these penguin-mediated decays exhibit very similar hadronic matrix elements for the "tree" and "penguin" contributions in the usual decomposition based on CKM factors, so that these contributions are very strongly correlated. This means that the L-observable is a measure of U -spin breaking between the penguin contributions to B d and B s decays, with a deviation from the SM expectation between 2σ and 3σ depending on the specific theoretical framework considered. This observation reinforces and puts on a firmer ground the hint for NP already suspected by considering the difference between the longitudinal polarisation fractions in these two modes. We performed a detailed error budget analysis for L K * K * and we found a relatively small impact of infrared divergences coming from weak annihilation and hard-spectator scattering, compared to observables like branching ratios or polarisation fractions involving troublesome transverse amplitudes.
We then interpreted this deviation in a model-independent approach using the weak effective theory. For simplicity, we allowed NP only in SM Wilson coefficients or their chirally-flipped counterparts. We identified three operators which could accommodate the deviation with NP contributions at most as large as the SM. While C 1q is already very significantly constrained by other nonleptonic modes and LHCb bounds (up to the point of excluding this solution), the situation is less constrained for the strong penguin coefficient C 4q and the chromomagnetic one C eff 8gq where NP contributions of a similar size to the SM one are allowed and could explain the deviation in L K * K * . We discussed examples of simplified NP models that could provide large contributions, at the price of accepting fine tuning to accommodate the bounds on B s −B s mixing and b → sγ. Interestingly, within a general composite or extra-dimensional model [49], the Kaluza-Klein gluon contribution to the b → s amplitude in B s → K * 0K * 0 has the same sign as the Z contribution to b → s + − w.r.t. the SM. Therefore, if one accepts the fine-tuning in B s −B s mixing, such models can provide a common explanation of L K * K * and b → s + − data.
This hint of NP in L K * K * could be sharpened with a precise estimate of U -spin breaking in the form factors involved, as they drive the theoretical uncertainty of the SM prediction and their correlation is not known precisely. A comparison of the theoretical and experimental information on the polarisations in B s → K * φ and B d → K * φ could also be valuable to check whether a similar tension arises. Complementary information could be obtained also from pseudoscalar-vector and pseudoscalar-pseudoscalar penguin-mediated modes (K 0K * 0 and K 0K0 ). Moreover, if the same source of NP is responsible for the suppression of b → sqq versus b → dqq and b → sµµ versus b → see, it would be certainly

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interesting to perform a thorough study of b → d + − modes compared to b → s + − ones, which should be accessible with more data from the LHCb and Belle II experiments. This interplay between non-leptonic and semileptonic rare decays could prove highly beneficial in the coming years to identify new B-flavour anomalies and understand their actual origin in terms of physics beyond the SM.

Acknowledgments
We thank M. Misiak

A Weak effective theory and QCD factorisation framework
The separation between short and long distances at the scale µ b = m b is performed in the weak effective theory to compute b-quark decays within the SM: This effective Hamiltonian describes the quark transitions b → uūs, b → ccs, b → sq q with q = u, d, s, c, b, and b → sg, b → sγ. Q p 1s,2s are the left-handed current-current operators arising from W -boson exchange, Q 3s...6s and Q 7s...10s are QCD and electroweak penguin operators, and Q 7γs and Q 8gs are the electromagnetic and chromomagnetic dipole operators. They are given by [22]:

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A similar weak effective theory can be written for the b → d transition by performing the trivial replacement s → d. Neglecting the difference of mass between the d and s quarks, the SM values of the Wilson coefficients are identical in both cases, and we omit the d or s subscript in table 3.
In the SM, C c 1 = C u 1 is the largest coefficient and it corresponds to the colour-allowed tree-level contribution from the W exchange, whereas C c 2 = C u 2 is colour suppressed. QCDpenguin operators are numerically suppressed, and the electroweak operators even more so. It proves convenient to define the effective coefficients C eff 7γ and C eff 8g which are given in the scheme of ref. [22] as 0.02 ± 0.02 0.03 ± 0.03 0.08 ± 0.06 0.08 ± 0.06 Decay Constants (at µ = 2 GeV) [30,42,53] Table 3. Input parameters used to determine the SM predictions.
As discussed in refs. [16,19,20], this explains why the quantity ∆ defined in eq. (2.2) can be computed safely within QCD factorisation for penguin mediated decays because of the cancellation of long-distance contributions. As a consequence of this cancellation, only penguin contractions contribute to ∆, as can be seen by inspection of the formulae above, leading to the following very simple expression within QCD factorisation: where the normalisation A Q M 1 M 2 is defined as: andḠ M 2 is the penguin function defined in ref. [19].

JHEP04(2021)066 B Semi-analytical expressions
In the following we provide the key elements to construct a semi-analytical expression of L K * K * . Specifically we give P s and P d in terms of Wilson coefficients and the parameters X H and X A . κ is given in eq. (3.16) and the last bracket in eq. (3.13) has a negligible impact and can be taken to be conservative 0.99 ± 0.01. We have followed the corrected expression of ref. [54] for the modelling of the weak annihilation in terms of X A .

C Sensitivity to New Physics
We show how NP contributions can help to reduce the tension between theory and experiment for L K * K * , completing the results shown in figure 1 discussed in section 4. In figure 4 we show the 1σ-range for the NP contribution to each Wilson coefficient that is able to explain the experimental value of L K * K * , normalised to its SM value.   Figure 5. Sensitivity of L K * K * to individual contributions of NP in all different C NP is . For each coefficient, the range of variation considered for the NP contribution corresponds to 100% of its SM value.

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.