NIOBE

 

 

 

 

 

 

 

 

We have completed the set of Light-Cone Sum Rules with dipion Distribution Amplitudes for the B → ππ form factors, by deriving the sum rule for the timelike-helicity form factor Ft. These sum rules are complementary to the ones derived in (Cheng, Khodjamirian, Virto, 2017) in terms of B-meson DAs, and complementary to the derivations from dispersion theory, and to the calculations at large dipion masses.

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The form factor Ft obtained here, while not contributing to the semileptonic B → ππlν rate in the massless lepton approximation, plays an important role in the factorization formula for the Non-Leptonic Three-Body B decays such as B → πππ. Further improvements of the sum rules presented here and require the inclusion of higher-twists and NLO corrections, but most importantly a better knowledge of dipion DAs and their Gegenbauer coefficients.

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Accurate theoretical predictions for exclusive B-meson decay observables are of utmost importance for studies of the the Standard Model and New Physics, but require a careful assessment of the theoretical inputs used. Light-Cone Sum Rules (LCSRs) are particularly prominent tools to compute the form factors involved.

 

In this article, we focused on the determination of B → K-star form factors from LCSRs with B-meson Light-Cone Distribution Aamplitudes (LCDAs). We extended the framework to consider B → Kπ form factors for the Kπ P-wave, which include effects such as the non-resonant Kπ production and the width of the K-star meson. We analysed all vector, axial and tensor form factors needed for the phenomenological analysis of B → Kπll in the P wave in the Standard Model. We first derived LCSRs with B-meson LCDAs for these form factors, generalizing the results of (Cheng, Khodjamirian, Virto, 2017) for the case of two pseudoscalar mesons of different masses in the final state. These sum rules provide relationships between the convolution of the timelike Kπ vector form factor and a B → Kπ form factor on one side, and the Operator Product Expansion (OPE) of a well-chosen correlation function expressed in terms of B-meson LCDAs on the other side.

 

On the OPE side of the sum rules, we computed higher-twist two- and three-particle contributions, and we proceeded to a new, well motivated, determination of the threshold parameter, somewhat lower than earlier determinations.

 

On the hadronic side, we introduced a resonance model for the B → Kπ including the K-star(980) and the K-star(1410), so that our sum rules can be used to constrain the parameters describing the two contributions. This resonance model is related to the model used by the Belle collaboration to determine the Kπ vector form factor from the τ → Kπ ν differential decay rate, and which describes the spectrum very well:

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We then exploited our set-up phenomenologically with the following results:

 

• We considered the narrow-width limit of our LCSRs to check that we recover the known B→K-star sum rules, and to assess the correction due to the finite width of the K-star meson. We find that this correction is universal for all form factors and amounts to a multiplicative factor W=1.1, corresponding to a 20% enhancement of the decay rate. This result does not depend strongly on the details of our model and it should be taken into account when computing observables with B→K-star form factors obtained from LCSRs derived in the narrow-width limit.

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• Comparing with earlier determinations, we find that our results for the form factors in the narrow width-limit are consistent but with lower central values. Several competing effects compensate each other (choice of the B-meson decay constant, the K-star(980) coupling constant, the threshold parameter) so that the difference can be mainly attributed to the contribution from the twist-four two-particle contribution g+(ω).

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• We then considered the impact of an additional K-star(1410) contribution to our model. This contribution is small in the case of the Kπ vector form factor, but in principle it could be large for the B → Kπ case. Our sum rules provide a combined constraint on the K-star(980) and K-star(1410) coupling constants describing the height of the two resonances in the B → Kπ form factors (with a much smaller weight for K-star(1410)). An increased K-star(1410) contribution would correspond to a smaller K-star(980) contribution, and thus a lower value for B → K-star form factors:

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• We turned to the B → Kπµµ differential decay rate in the K-star(1410) region, which has been measured recently by the LHCb collaboration. The branching ratio and the angular decay distribution in this Kπ window provide bounds on the contribution of the K-star(1410). If a huge contribution is ruled out, the current data leave room for a K-star(1410) contribution of moderate size, which could lead to a decrease of the K-star(890) contribution (and thus the values of the B → K-star form factors) of around 10%.

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​Considering the more general B → Kπ form factors and constraining them using LCSRs with B-meson LCDAs, we have identified three effects of rather different nature that affect the determination of the B → K-star form factors:

 

1. a universal effect related to the finite-width of the K-star increasing by 1.1 the value compared to the narrow-width limit,

2. an effect related to the inclusion of the two-particle twist-four B-meson LCDAs leading to smaller values in our case compared to earlier calculations, and

3. an effect of the K-star(1410) contribution that could decrease the K-star(980) peak up to 10% according to the data currently available.

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All three effects have a direct impact on the prediction of the B → K-star µµ branching ratio and thus on the importance of the current discrepancy between SM expectations and LHCb measurements of b → sµµ branching ratios. Let us add that B → K-star µµ angular observables (such as P5' ) are less affected by this discussion as they are insensitive to the overall normalization of the form factors.

 

One may wonder whether previous determinations or other approaches are sensitive to these issues. Previous light-cone sum rule determinations using the B-meson LCDAs have worked under the assumption of the narrow-width limit, so that any correction related to a finite width is missed. The K-star(1410) contribution is included through quark-hadron duality in the choice of the threshold parameter, but this contribution is obviously hard to disentangle from the continuum contribution and it suffers from significant uncertainties (suppressed after Borel transformation). A huge contribution from the K-star(1410) would require a significant failure of the quark-hadron duality, but the moderate contribution discussed here could be included in uncertainties associated with the continuum contribution. Other LCSRs exploit different correlation functions, so that the OPE contribution is expressed in terms of light-meson LCDAs (either K-star or Kπ). They have the advantage of providing an expression of the form factors directly in terms of the OPE part, and not through a convolution with the Kπ form factor as in our case (which required us to design a resonance model for the B → Kπ form factors). However, this simplicity prevents these sum rules from providing corrections to the narrow-width limit (the LCSRs with a vector resonance are not smooth limits of the LCSRs of the dimeson DAs). The size of the contribution of the K-star(1410) is in principle partially encoded in the dimeson LCDAs, which are however poorly known, whereas the K-star LCDAs are essentially unable to probe this question. Finally, lattice QCD simulations investigate a different kinematic range for the transfer momentum. The first lattice results on B→K-star form factors include configurations where the K-star meson may decay into Kπ, but there was not enough data to investigate the impact of finite-width effects in this kinematic regime. This could in principle be studied more extensively using a lattice set-up dedicated to the analysis of unstable resonances.

 

Concerning the K-star(1410) contribution to the B → Kπ form factors, a huge effect would require an usually large dependence of the form factors on the dimeson invariant mass to bridge the LCSR and the lattice results and it could have led to difficulties in extracting the K-star(980) signal from lattice data under a very large background of excited states, but the moderate contribution discussed here does not seem to contradict the current results from lattice QCD on these form factors.

 

Our study of B→Kπ form factors through an extension of well-known sum rules has highlighted several effects that may impact the current determination of the form factors used to analyse B → K-star ll and other decays of the form B → K-star X. Several improvements would help assessing these effects more precisely. The models for the B-meson LCDAs should be investigated and constrained more tightly to assess the role played by higher-twist contributions. The LCSRs could be exploited with additional models for the Kπ and B→Kπ form factors to consolidate our results. Finally, the differential decay rate for B→Kπµµ at high Kπ invariant mass could be measured more precisely to provide tighter constraints on the K-star(1410) contributions. This would also require a better knowledge of the other partial waves that are contributing in this invariant mass window. If the D wave seems small, the S wave interferes significantly with the P wave analysed here. The relevant B → Kπ form factors could be investigated with similar LCSRs to the ones considered here. This should lead to a consistent picture of the contributions from higher resonances to the B → Kπll decay, and to a deeper understanding of the anomalies currently observed in b → sll transitions.

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