# Long-distance effects in \(B\rightarrow K^*\ell \ell \) from analyticity

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## Abstract

We discuss a novel approach to systematically determine the dominant long-distance contribution to \(B\rightarrow K^*\ell \ell \) decays in the kinematic region where the dilepton invariant mass is below the open charm threshold. This approach provides the most consistent and reliable determination to date and can be used to compute Standard Model predictions for all observables of interest, including the kinematic region where the dilepton invariant mass lies between the \(J/\psi \) and the \(\psi (2S)\) resonances. We illustrate the power of our results by performing a New Physics fit to the Wilson coefficient \(C_9\). This approach is systematically improvable from theoretical and experimental sides, and applies to other decay modes of the type \(B\rightarrow V\ell \ell \), \(B\rightarrow P\ell \ell \) and \(B\rightarrow V\gamma \).

## 1 Introduction

\(B\rightarrow K^*\ell \ell \) decays are sensitive to modified short-distance physics from sources beyond the Standard Model (SM), and a great deal of experimental and theoretical work has been devoted to extract short-distance information from them. However, long-distance physics within the SM also contributes significantly to the decay, and its effects are very difficult to assess reliably from first principles. On the other hand, tighter experimental constraints from increasingly precise measurements of \(b\rightarrow s\) processes have significantly limited the size of allowed New Physics (NP) effects in \(B\rightarrow K^*\ell \ell \), which are now comparable to current SM uncertainties. Thus, our inability to reliably constrain these long-distance contributions to acceptable levels stands in the way of obtaining unambiguous information on physics beyond the SM.

The functions \(\mathcal{F}_\lambda ^{(T)}(q^2)\) are form factors, which can be calculated by means of Light-Cone Sum Rules (LCSRs) at low \(q^2\) (\(\lesssim 10\,\,\mathrm{GeV}^2\)) [1, 2], or by numerical simulations (Lattice QCD) at large \(q^2\) (\(\gtrsim 15\,\,\mathrm{GeV}^2\)) [3, 4]. Both methods agree reasonably well when extrapolated [5, 6], and there are good prospects for improvement [7, 8, 9, 10, 11]. The form factors are not the focus of this work.

In the heavy *b*-quark limit and for very small \(q^2\), the functions \(\mathcal{H}_\lambda (q^2)\) factorize into non-perturbative form factors and light-cone distribution amplitudes, up to perturbatively calculable “hard” functions [12]. However this perturbative expansion breaks down when \(q^2\) approaches \(4 m_c^2\), leading to questionable predictions for \(q^2\gtrsim 6\,\mathrm{GeV}^2\). The integral in Eq. (2) is in fact dominated by the region \(x^2\lesssim (2m_c - \sqrt{q^2})^{-2}\) [13], so for \(q^2\ll 4m_c^2\) one may expand the operator \(\mathcal{K}^\mu (x,0)\) around \(x^2 = 0\) (a light-cone operator-product expansion, or LCOPE). This leads to an expansion of Eq. (2) in powers of \((2m_c - \sqrt{q^2})^{-1}\), with matrix elements of operators that are non-local only along the light cone. This theory framework has been worked out up to NLO in \(\alpha _s\) [12, 14] and including subleading terms in the LCOPE [13], and can be safely applied for \(q^2\ll 4m_c^2\) (preferably at \(q^2<0\)). However, reliable predictions for larger values of \(q^2\) remain a challenge.

In this letter we consider a consistent, model-independent and systematically-improvable approach to determine the dominant long-distance contributions \(\mathcal{H}_\lambda (q^2)\) to \(B\rightarrow K^*\ell \ell \) in the region \(q^2 \lesssim 14\,\mathrm{GeV}^2\). It provides genuine SM predictions even in the presence of NP in semileptonic operators. In addition, this approach provides access to the inter-resonance region \(10\,\mathrm{GeV}^2 \lesssim q^2 \lesssim 13\,\mathrm{GeV}^2\). The idea is the following: We determine the analytic properties of the functions \(\mathcal{H}_\lambda (q^2)\) in the complex plane, by considering their dominant singularities. We then use this information to write down a general and model-independent parametrization. Two pieces of information are used to constrain the parametrized functions: data on \(B\rightarrow K^* J/\psi \) and \(B\rightarrow K^* \psi (2S)\), which is independent of NP in semileptonic operators; and theory at \(q^2<0\), where it is reliable. This method, which builds upon Refs. [13, 15], gives the most reliable and consistent *a-priori* determination of the functions \(\mathcal{H}_\lambda (q^2)\) to date. We use these results to compute SM predictions (assuming no NP in \(\mathcal{O}_{1,2}\)), and to perform a NP fit to \(C_9\). All our numerical computations are performed with the help of EOS [16], which has been modified for this purpose [17].

## 2 Analytic structure and parametrization

It is a standard assumption in quantum field theory that the only analytic singularities of a correlation function – as a complex function of all its complexified kinematic invariants – are those required by unitarity [18]. This principle of “maximal analyticity” can sometimes be derived from causality, and it is therefore well founded [19]. Unitarity, in turn, relates analytic singularities with on-shell intermediate states: poles for one-particle states, and branch cuts for multi-particle states. Thus, the analytic structure of a correlation function can be learned by analysing its on-shell cuts.

In the case at hand, inspection of the correlation function (2) reveals the following analytic properties of the scalar functions \(\mathcal{H}_\lambda (q^2)\):

\(\blacktriangleright \) On-shell cuts in the variable \(q^2\) include: two poles at \(q^2=M_{J/\psi }^2\simeq 9\,\mathrm{GeV}^2\) and \(q^2=M_{\psi (2S)}^2\simeq 14\,\mathrm{GeV}^2\) corresponding to one-particle intermediate states through \(B\rightarrow K^* \psi _n (\rightarrow \ell ^+\ell ^-)\), with \(\psi _1 = J/\psi \) and \(\psi _2 = \psi (2S)\); a branch cut starting at \(q^2=t_+\equiv 4 M_D^2\) corresponding to two-particle intermediate states through \(B\rightarrow K^* [\bar{D} D] (\rightarrow \ell ^+\ell ^-)\), plus other “\(c\bar{c}\)” cuts with higher thresholds; and “light-hadron” branch cuts starting at \(q^2\simeq 0\) from intermediate states such as \(B\rightarrow K^* [3\pi ](\rightarrow \ell ^+\ell ^-)\), which include finite-width effects of \(J/\psi \) and \(\psi (2S)\). The effects of these “light-hadron” cuts are OZI suppressed [20, 21, 22]. Given the limited precision of current data, we will neglect these OZI suppressed contributions, keeping in mind that this is a pending assumption that should be tested in view of future experimental prospects. These presumably small effects have never been considered in previous analyses before.

\(\blacktriangleright \) On-shell cuts in the variable \((q+k)^2\) (the “forward” or “decay” channel) include branch cuts from intermediate states such as \(B\rightarrow \bar{D} D_s \rightarrow K^*\ell ^+\ell ^-\). The physical point \((q+k)^2=M_B^2\) lies on these cuts, which implies that the functions \(\mathcal{H}_\lambda (q^2)\) are complex-valued for all values of \(q^2\). But this imaginary part is not associated with any singularity in the variable \(q^2\). Thus, one can write \(\mathcal{H}_\lambda (q^2) = \mathcal{H}_\lambda ^\mathrm{(re)}(q^2) + i\, \mathcal{H}_\lambda ^\mathrm{(im)}(q^2)\), with \(\mathcal{H}_\lambda ^\mathrm{(re,im)}(q^2)\) satisfying the analytic properties of the previous point as functions of \(q^2\), and obeying the same dispersion relation.

*z*:

The approach now resembles and is inspired by the *z*-parametrization used for the form factors [23, 24]. The functions \(\mathcal{H}_\lambda (z) \equiv \mathcal{H}_\lambda (q^2(z))\) are meromorphic in \(|z|<1\), with two simple poles at \(z_{J/\psi }\equiv z(M_{J/\psi }^2)\simeq 0.18\) and \(z_{\psi (2S)}\equiv z(M_{\psi (2S)}^2)\simeq -0.44\). Therefore, multiplying by the corresponding zeroes will give an analytic function in \(|z|<1\) that can be Taylor-expanded around \(z=0\). This expansion should converge reasonably well in the region of interest, where \(|z| < 0.52\). This is the basis of our proposed parametrization.

*e.g.*the \(M_{B_s^*}\) pole), and the leading OPE contribution to the correlator is indeed proportional to the form factor. Therefore it is better to parametrize the ratios \(\mathcal{H}_\lambda (q^2)/\mathcal{F}_\lambda (q^2)\) instead. Second, the poles should not modify the asymptotic behaviour. This is achieved by introducing appropriate “Blaschke factors” [23]. All in all, we propose the following parametrization:

*K*will depend on the available set of experimental measurements and theory inputs.

## 3 Experimental constraints

## 4 Theory constraints

^{1}the soft-gluon correction calculated via a LCSR in Ref. [13]. For the form factors we use the results from the LCSR with

*B*-meson distribution amplitudes [2], in order to have a mutually consistent description of form factors and non-local contributions and benefit from theoretical correlations among both. In this way we compute the ratios \(\mathcal{H}_\lambda (q^2)/\mathcal{F}_\lambda (q^2)\) at the points \(q^2=\{-7,-5,-3,-1\}\,\mathrm{GeV}^2\). These ratios are used as pseudo-observables to constrain the parameters in Eq. (6) at \(z=\{0.52,0.50,0.48,0.46\}\). Further details and results are presented for completeness in the appendix. We emphasize that no theory is used at \(q^2 \ge 0\) at all.

Mean values and standard deviations (in units of \(10^{-4}\)) of the prior PDF for the parameters \(\alpha _k^{(\lambda )}\)

| 0 | 1 | 2 |
---|---|---|---|

\(\mathrm{Re}[\alpha _{k}^{(\perp )}]\) | \(-0.06 \pm 0.21\) | \(-6.77 \pm 0.27\) | \(18.96 \pm 0.59\) |

\(\mathrm{Re}[\alpha _{k}^{(\parallel )}]\) | \(-0.35 \pm 0.62\) | \(-3.13 \pm 0.41\) | \(12.20 \pm 1.34\) |

\(\mathrm{Re}[\alpha _{k}^{(0)}]\) | \(0.05 \pm 1.52\) | \(17.26 \pm 1.64\) | – |

\(\mathrm{Im}[\alpha _{k}^{(\perp )}]\) | \(-0.21 \pm 2.25\) | \(1.17 \pm 3.58\) | \(-0.08 \pm 2.24\) |

\(\mathrm{Im}[\alpha _{k}^{(\parallel )}]\) | \(-0.04 \pm 3.67\) | \(-2.14 \pm 2.46\) | \(6.03 \pm 2.50\) |

\(\mathrm{Im}[\alpha _{k}^{(0)}]\) | \(-0.05 \pm 4.99\) | \(4.29 \pm 3.14\) | – |

## 5 SM predictions

We now perform a fit of Eq. (6) to the combined experimental and theoretical constraints described above in Sects. 3 and 4. We find that Eq. (6) with \(K=2\) provides an excellent fit to all inputs, with a *p*-value of 0.91. All 1D-marginalised posteriors are reasonably symmetric around their modes. The result of this fit is a set of correlated values for the complex parameters \(\alpha ^{(\lambda )}_k\), which are summarized in Table 1. These values lead to a determination of the non-local correlator in Eq. (2) that is consistent with the \(B\rightarrow K^*\psi _n\) measurements, the theory calculations at negative \(q^2\), and it is independent of new physics in semileptonic operators. This is very different compared to the approach of Ref. [1], which uses short-distance dominated \(B\rightarrow K^*\mu ^+\mu ^-\) measurements to determine the non-local correlators, thereby assuming SM values of the \(b\rightarrow s\mu ^+\mu ^-\) Wilson coefficients. As a consequence, their SM predictions for the angular observables are model-dependent posterior predictions. The study presented here does not suffer from this model dependence, and thus we determine the non-local correlators and provide a *genuine SM prediction* of the angular observables.

The gray band in Fig. 1 shows the result of this “prior” fit for the case of the real part of \(\mathcal{H}_\perp (q^2)\). Similar plots for the other correlators are provided in the appendix for completeness.

## 6 New physics analysis

We now perform a fit to \(B\rightarrow K^*\mu ^+\mu ^-\) data using as prior information the SM predictions derived in Sect. 5. We include the branching ratio and the angular observables \(S_i\) [38] within the \(q^2\) bins in the region \(1 \le q^2 \lesssim 14\,\mathrm{GeV}^2\). We use the latest LHCb measurements [39, 40], and perform different separate fits, using the results from the maximum-likelihood fit excluding (LLH) and including (LLH2) the inter-resonance bin, or using the results from the method of moments [41] (MOM and MOM2), and both including (NP fit) and not including (SM fit) a floating NP contribution to \(C_9\).

The fits provide posterior distributions for the correlator, for \(B\rightarrow K^*\mu ^+\mu ^-\) and \(B\rightarrow K^*\gamma \) observables, and for \(C_9\). We first discuss some illustrative results of the LLH2 fit. The posteriors for the real part of \(\mathcal{H}_\perp (q^2)\) are shown in Fig. 1, both for the SM and the NP fits. In this case it is reassuring that both are consistent within errors with the result of the prior fit, indicating that modifying the long-distance contribution does not lead to improvement in the SM fit, and so the long-distance contribution is not likely to mimic a NP contribution.

The posterior NP prediction for \(P_5'\) (corresponding to the LLH2 fit) is shown in Fig. 2, exhibiting a much better agreement with the experimental measurements than the SM (prior) prediction.

*global*fits [42, 43, 44, 45, 46, 47, 48], but rely on a more fundamented theory treatment.

## 7 Conclusions

Analyticity provides strong constraints on the hadronic contribution to \(B\rightarrow K^*\ell \ell \) observables, and fixes the \(q^2\) dependence up to a polynomial, which under some circumstances is an expansion in a small kinematical parameter. In this letter we have exploited this idea to propose a systematic approach to determine the dominant non-local contributions, which at this time are the main source of theory uncertainty. This approach is systematically improvable with more precise data on \(B\rightarrow K^*\psi _n\) and/or more precise theory calculations at negative \(q^2\). In addition, this approach allows access to the inter-resonance region, which provides valuable information on short-distance physics. We have focused on \(B\rightarrow K^*\ell \ell \), but the approach applies to any other \(B\rightarrow M\ell \ell \) modes such as \(B\rightarrow \lbrace K,\pi ,\rho \rbrace \ell \ell \) and \(B_s\rightarrow \phi \ell \ell \).

We have performed a numerical analysis implementing this idea, and conclude that significantly improved theory predictions can be obtained, leading to a more precise and robust interpretation of experimental data and an improved sensitivity to short-distance physics. We identify two issues worth exploring further. One has to do with neglecting the OZI-suppressed cut and charmonium width. A dispersive approach should be able to exploit present and future data on charmless non-leptonic multi-body \(B\rightarrow K^* X\) decays in order to properly bound these presumably small effects. The other has to do with the convergence of the *z* expansion. In this respect, the fit including \(B\rightarrow K^*\ell \ell \) data can provide enough constraints to increase the order of the expansion considerably, especially in view of the extraordinary experimental prospects for the next ten years [49]. We thus believe that this approach will become very useful in future analyses of exclusive \(b\rightarrow s\) and \(b\rightarrow d\) transitions.

## Footnotes

- 1.
We thank Yuming Wang for providing us with the results for \(B\rightarrow K^*\gamma ^*\) in digital form.

## Notes

### Acknowledgements

We thank Wolfgang Altmannshofer, Kirill Chilikin, Gilberto Colangelo, William Detmold, Christoph Hanhart, Robert Jaffe, Alexander Khodjamirian, Bastian Kubis, Thomas Mannel, Joaquim Matias, Mikolaj Misiak, Federico Mescia, Jacobo Ruiz de Elvira, Iain Stewart, David Straub, Lewis Tunstall, Yuming Wang and Roman Zwicky for useful interactions and discussions. CB is supported in part by the DFG SFB/TR 110 “Symmetries and the Emergence of Structure in QCD”. CB and DvD acknowledge support from the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence “Origin and Structure of the Universe”. MC is grateful for support of the Polish National Science Center under the Sonata grant: UMO-2015/17/D/ST2/03532. DvD is supported in part by the Swiss National Science Foundation (SNF) under contract 200021-159720. JV acknowledges funding from the Swiss National Science Foundation, from Explora project FPA2014-61478-EXP, and from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 700525 ‘NIOBE’. This research was supported in part by PL-Grid Infrastructure.

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