14.5cm 22.7cm -1cm 0.75cm

CERN-TH/98-314\
September 1998

**Non-perturbative Effects in $B\to X_sl^+l^-$**

[Gerhard Buchalla]{}

*Theory Division, CERN, CH-1211 Geneva 23, Switzerland*

**Abstract**

The rare decay $B\to X_sl^+l^-$ provides excellent prospects for precision tests of Standard Model flavour dynamics. The process can be computed in perturbation theory with small uncertainty. However, in order to ensure a reliable theoretical prediction, also potentially important effects from non-perturbative QCD have to be controlled with sufficient accuracy. The present article summarizes recent developments related to this topic.

*Invited Talk presented at the*

*QCD Euroconference 98, Montpellier, 2-8 July 1998*

Introduction
============

Within the Standard Model the inclusive rare $B$ decays $B\to X_sl^+l^-$ ($l=e,$ $\mu$, $\tau$) are generated by loop-induced flavour-changing neutral currents (FCNC). They allow us therefore to test our understanding of flavour dynamics at the quantum level. Although smaller in branching ratio than the radiative decay $B\to X_s\gamma$ by almost two orders of magnitude, the $B\to X_sl^+l^-$ modes could play a decisive role because they are sensitive to additional short-distance contributions. $B\to X_sl^+l^-$ can thus probe aspects of flavour physics not accessible with $B\to X_s\gamma$ and provides therefore a very valuable, complementary source of information. The best current limits for the (integrated, non-resonant) branching fraction of $B\to X_s\mu^+\mu^-$ are $$\label{exp}
B(B\to X_s\mu^+\mu^-) <
\left\{ 
\begin{array}{l}
5.8\cdot 10^{-5}\qquad \cite{CLEO}\\
3.2\cdot 10^{-4}\qquad \cite{D0}
\end{array}
\right.$$ This is to be compare with a Standard-Model expectation of about $6\cdot 10^{-6}$. The modes $B\to X_sl^+l^-$ ($l=e,$ $\mu$), to which we shall restrict ourselves in the following, should be well within reach of the upcoming experiments at the $B$-factories, the Tevatron and the LHC.

The theoretical description of $B\to X_sl^+l^-$ is based on a low-energy effective Hamiltonian ${\cal H}_{eff}$, derived using the operator-product expansion and renormalization-group improved QCD perturbation theory. Schematically, the amplitude for the decay of $B$ into final state $X_sl^+l^-$ may be written as $$\langle X_sl^+l^-|{\cal H}_{eff}|B\rangle =
-\frac{G_F}{\sqrt{2}}V^*_{ts}V_{tb}
\sum_i C_i(M_W/\mu,m_t,\alpha_s)\cdot
\langle X_sl^+l^-|Q_i|B\rangle$$ The $C_i$ are Wilson coefficient functions containing the short-distance physics from scales $M_W$ to $\mu\sim m_b$. They are perturbatively calculable. $Q_i$ are the corresponding local four-fermion operators. (See [@BBL] for a general review on this subject.) For the inclusive decay of a $B$ meson the squared matrix elements of operators entering the decay rate can be further treated using the heavy quark expansion (HQE) [@BBSUV; @CGG]. To leading order in this expansion in $1/m_b$ the inclusive rate for the decay $B\to X_sl^+l^-$ is given by the partonic result for $b\to sl^+l^-$, plus non-perturbative power corrections of second and higher order in $\Lambda_{QCD}/m_b$. Unlike exclusive modes, the inclusive $B\to X_sl^+l^-$ is thus largely dominated by perturbatively calculable contributions. Non-perturbative mechanisms are, however, still present. The ultimate precision of the Standard-Model prediction, and the potential to test for unexpected deviations, then depends on how well the generally difficult non-perturbative sector can be brought under control.

Overview
========

Before discussing some of the most relevant issues in slightly more detail, we begin with a short overview of the non-perturbative effects in $B\to X_sl^+l^-$. For illustration we show in Fig. \[fig1\] the partonic dilepton invariant-mass spectrum $$\label{rsdef}
R(s)=\frac{\frac{d}{ds}\Gamma(B\to X_sl^+l^-)}{\Gamma(B\to X_ce\nu)}$$ as a function of $s=q^2/m^2_b$, $q^2=(p_{l^+}+p_{l^-})^2$. The effect of various non-perturbative corrections, to be discussed below, is also illustrated. Clearly visible are the narrow resonances $J/\Psi(3097)$ and $\Psi^\prime(3686)$ at $s\sim 0.4$ and $\sim 0.6$, respectivley.

Two different sources of non-perturbative contributions may be distinguished. First, power corrections in $1/m_b$ to the leading partonic decay within the heavy quark expansion. Second, non-perturbative effects related to intermediate $c\bar c$ pairs ($B\to X_s(c\bar c)\to X_sl^+l^-$).

The leading $1/m_b$ corrections of ${\cal O}(\Lambda^2_{QCD}/m^2_b)$ have been calculated and are found to be small (${\stackrel{<}{_\sim}}$ several per cent), except for $q^2{\stackrel{>}{_\sim}}0.8 m^2_b$ [@AHHM; @BI1]. Still closer to the endpoint $q^2\approx m^2_b$ the HQE breaks down and the inclusive decay $B\to X_sl^+l^-$ is eventually dominated by the exclusive channel $B\to Kl^+l^-$. (The issue of the $b$-quark mass in the partonic spectrum is briefly discussed in [@BI1]. In the present context the pole mass definition is used for $m_b$.)

Large backgrounds to the short-distance process arise from the cascades $B\to X_s\Psi^{(\prime)}$, $\Psi^{(\prime)}\to l^+l^-$ when the dilepton mass $q^2$ is in the vicinity of the narrow ($J^{PC}=1^{--}$) resonances $\Psi$ and $\Psi^\prime$. They would dominate the signal by two orders of magnitude and have to be excluded by suitable cuts on $q^2$. This leaves essentially two interesting $q^2$ regions to search for $B\to X_sl^+l^-$: region I (below the resonances), where residual non-perturbative effects from $c\bar c$ resonances can be calculated within a $1/m_c$ expansion; and region II (above the $\Psi^\prime$), where the impact of higher $J^{PC}=1^{--}$ $c\bar c$-resonances has to be considered.

$\Lambda^2_{QCD}/m^2_b$ Corrections
===================================

The starting point for a systematic expansion in $1/m_b$ of an inclusive $B$ meson decay rate $\Gamma$ is provided by the optical theorem [@BBSUV], $$\label{opth}
\Gamma=\frac{1}{2M_B}\langle B|\, {\rm Im}\, i\int d^4x
T {\cal H}_{eff}(x){\cal H}_{eff}(0)|B\rangle$$ which relates the decay width to the absorbtive part of the forward scattering amplitude. In the present context ${\cal H}_{eff}$ is the effective hamiltonian for $b\to sl^+l^-$ decay. The hadronic part of ${\cal H}_{eff}$ involves currents of flavour structure $(\bar bs)$ (and $(\bar sb)$). Schematically, the T-product in (\[opth\]) then gives rise to expressions of the form $(\bar bs)(x) (\bar sb)(0)$, where the $s$-quark fields are contracted according to Wick’s theorem and the $s$-quark propagator is subject to the HQE in $1/m_b$. Through order $\Lambda^2_{QCD}/m^2_b$ the rate becomes $$\label{ddsg}
\frac{d}{ds}\Gamma(B\to X_sl^+l^-)=\frac{d}{ds}\Gamma(b\to sl^+l^-)
\frac{\langle B|\bar bb|B\rangle}{2M_B}  
+C_G\frac{1}{m^2_b}
\frac{\langle B|\bar b g\sigma\cdot G b|B\rangle}{2M_B}$$ with a calculable coefficient $C_G$ [@AHHM; @BI1]. The matrix elements in (\[ddsg\]) may be further expanded using heavy quark effective theory (HQET) [@MW] $$\begin{aligned}
\label{la12}
\frac{\langle B|\bar bb|B\rangle}{2M_B} &=&
  1+\frac{\lambda_1}{2m^2_b}+\frac{3}{2}\frac{\lambda_2}{m^2_b} \\
\frac{\langle B|\bar b g\sigma\cdot G b|B\rangle}{2M_B} &=&
6\lambda_2=\frac{3}{2}(M^2_{B^*}-M^2_B) \label{la2}\end{aligned}$$ where $\lambda_1=\langle B|\bar h(iD)^2h|B\rangle/(2M_B)$ and $6\lambda_2=\langle B|\bar h g\sigma\cdot Gh|B\rangle/(2M_B)$, with $h$ the $b$-quark field in HQET.

It is useful to normalize the rate (\[ddsg\]) by the rate of the inclusive, charged-current decay $B\to X_ce\nu$, defining $R(s)$ in (\[rsdef\]). In $R(s)$ the CKM parameters enter only in the ratio $|V_{ts}V_{tb}/V_{cb}|^2=0.95\pm 0.03$, which is much better known than the individual elements. In addition the dependence on the poorly known parameter $\lambda_1$ drops out in the $1/m^2_b$ corrections to $R$. These are then determined by $\lambda_2$ alone, which is fixed by (\[la2\]). The relative correction $\delta_{1/m^2_b}R(s)/R(s)$ amounts to several per cent at most for $s{\stackrel{<}{_\sim}}0.75$, but diverges towards the endpoint $s\to 1$. This breakdown of the HQE was first noticed in [@AHHM] and confirmed in [@BI1]. The origin of this behaviour has been discussed in detail in [@BI1]. The reason for the breakdown of the HQE can be understood intuitively as follows. At $s\approx 1$ the lepton pair carries almost the entire mass of the initial $B$ meson, leaving a very soft, low-multiplicity hadronic system in the final state. No hard scale exists in this regime to define an expansion parameter for the strange-quark propagator. It is interesting to note [@BI1] that the nature of the HQE breakdown in $B\to X_sl^+l^-$ at $s\approx 1$ is conceptually different from the situation of the lepton (photon) energy spectrum in $B\to X_ce\nu$ ($B\to X_s\gamma$) [@BSUV]-[@NEU12]. In the latter cases the straigtforward HQE breaks down near the endpoint of the lepton (photon) energy spectrum, but the expansion can be amended by a resummation of the leading singular contributions to all orders in $1/m_b$. The resummation results in a description of the endpoint region in terms of a shape function [@BSUV]-[@NEU12]. No such resummation is possible near the $s\to 1$ endpoint of $B\to X_sl^+l^-$ and the shape function description does not apply [@BI1]. On the other hand, the kinematic situation at $s\approx 1$ is a priori very well suited for a treatment employing heavy hadron chiral perturbation theory [@WI; @BD]. This framework is appropriate to analyze the strong-interaction effects in the transition of a heavy $B$ meson to a small number of soft pseudo-goldstone bosons. This is precisely the case of $B\to X_sl^+l^-$ at $s\approx 1$, where the inclusive decay degenerates into $B\to Kl^+l^-$, $B\to K\pi l^+l^-$, $\ldots$, with low-energy kaons and pions. This approach to the endpoint region has been discussed in [@BI1; @FG]. $B\to Kl^+l^-$ largely domiantes the endpoint region, whereas $B\to K\pi l^+l^-$ is entirely negligible. A reasonable interpolation to the range of validity of the HQE at lower $s$ can be obtained [@BI1].

$\Lambda^2_{QCD}/m^2_c$ Corrections
===================================

For small $q^2{\stackrel{<}{_\sim}}3m^2_c$ (that is below the $\Psi$ resonance), the leading non-perturbative effect from intermediate $c\bar c$ can be treated in an expansion in $1/m_c$. This approach has first been proposed to estimate non-perturbative corrections in $B\to X_s\gamma$ [@VOL]. The generalization to $B\to X_sl^+l^-$ was performed in [@BIR] (see also [@CRS]). The effect arises from the interaction of a soft gluon coupled to the $c\bar c$ loop with the spectator cloud inside the $B$ meson. This mechanism is not included in the perturbative evaluation of the $B\to X_sl^+l^-$ decay rate. The leading correction has the form $\Delta\Gamma(B\to X_sl^+l^-)=f(s)\cdot \lambda_2/m^2_c$, where $f(s)$ is a function of the dilepton mass, $m_c/m_b$ and the short-distance Wilson coefficients. Since the hadronic matrix element can be expressed entirely in terms of the well known $\lambda_2$ (\[la2\]), the $1/m^2_c$ effect is completely calculable. A further virtue of this approach is its model independence. In particular it avoids the double counting of $c\bar c$ contributions (already partly contained in the perturbative calculation) that is inherent in resonance-exchange models that have been employed previously to estimate these long-distance effects.

Numerically the $1/m^2_c$ correction is below a few per cent for $q^2{\stackrel{<}{_\sim}}3m^2_c$. The $1/m_c$ expansion is not valid in the vicinity of $q^2=M^2_\Psi, M^2_{\Psi'}$, where strong resonance effects are important. For higher values of $q^2$ the $1/m^2_c$ correction becomes very small. However, in this regime the non-perturbative contributions are dominated by the effect of higher $c\bar c$ resonances, which will be briefly discussed in the following section.

To conclude we remark that the $1/m^2_c$ correction to $B\to X_s\gamma$ [@VOL; @BIR; @LRWG] may be obtained from the corresponding calculation for $B\to X_sl^+l^-$ by considering the limit $q^2\to 0$. One finds $\Delta\Gamma(B\to X_s\gamma)/\Gamma(B\to X_s\gamma)=$ $0.42\, \lambda_2/m^2_c\approx +3\%$ [@BIR].

Higher $c\bar c$-Resonances
===========================

In the region of $q^2$ above the $\Psi^\prime(3686)$ there are further resonances ($\Psi(3770)$, $\Psi(4040)$, $\Psi(4160)$, $\Psi(4415)$), which may affect the dilepton invariant mass spectrum. Since these higher resonances are rather broad and much less prominent than $\Psi$ and $\Psi^\prime$, the HQE-based quark-level calculations should give a reasonable first approximation in the corresponding range of $q^2$. To calculate the impact of higher resonances on the spectrum above $q^2=M^2_{\Psi^\prime}$ is a difficult problem, which has not yet been solved in a fully satisfactory way. Some insight can be gained by making further simplifying, model-dependent assumptions. A procedure that is often employed in the literature [@AHHM] to estimate the resonance contributions consists in adding Breit-Wigner type resonance terms to the quark-level calculation. This approach is, however, unsatisfactory due to the double-counting problem mentioned above. At present, the most consistent method is the one proposed in [@KS]. Here the charm-loop contribution to $B\to X_sl^+l^-$ is estimated by means of experimental data on $\sigma(e^+e^-\to c\bar c-{\rm hadrons})$ using a dispersion relation. In this way double counting is avoided. The estimate has the further advantage of also including open-charm contributions. On the other hand, it still rests on the ad-hoc assumption that the $B\to X_s c\bar c$ transition can be factorized into a product of $(\bar sb)$ and $(\bar cc)$ color-singlet currents. The long-distance corrections to the quark-level calculation estimated with this method are only several per cent [@BI1]. Somewhat larger effects are obtained when a phenomenological factor $\kappa=2.3$ is introduced to correct for the factorization approximation. This is motivated by the fact that the factorization assumption gives a too small $B\to X_s\Psi$ branching fraction. However, the validity of such a procedure for estimating the impact of higher resonances in $B\to X_sl^+l^-$ is not a priori clear and requires further study.

Summary
=======

After large backgrounds from $B\to X_s\Psi(\Psi^\prime)$ $\to X_sl^+l^-$ are eliminated by experimental cuts in $q^2$, the rare decay $B\to X_sl^+l^-$ is dominated by perturbatively calculable contributions. ${\cal O}(\Lambda^2_{QCD}/m^2_b)$ non-perturbative corrections are well under control, except for the endpoint region $q^2\to m^2_b$, where the HQE breaks down. Close to the endpoint the inclusive decay is determined by $B\to Kl^+l^-$ and model independent constraints from heavy-hadron chiral perturbation theory can be derived.

Non-perturbative effects further arise from intermediate $c\bar c$ states in $B\to X_sl^+l^-$. At low $q^2$ they can be treated with a $1/m_c$ expansion. The leading correction of ${\cal O}(\Lambda^2_{QCD}/m^2_c)$ has been calculated and is found to be very well under control (${\stackrel{<}{_\sim}}$ few per cent). The corresponding effect is even smaller at high $q^2$. There, however, higher resonances (above $\Psi'$) can have some impact on the dilepton-mass spectrum. These resonances are rather broad and their effect on the spectrum is possibly not too pronounced. This question cannot yet been answered in a fully satisfactory way and deserves further study. Otherwise, in particular in the low-$q^2$ region, the Standard-Model prediction is quite accurate (${\stackrel{<}{_\sim}}10\%$ uncertainty). The inclusive rare decay $B\to X_sl^+l^-$ is therefore very well suited for detailed tests of the underlying flavour physics.

Acknowledgements {#acknowledgements .unnumbered}
================

I am grateful to my co-authors in [@BI1; @BIR] for a most enjoyable collaboration on the topics presented in this talk. Particular thanks are due to Gino Isidori for the extensive and fruitful exchange of ideas on non-perturbative effects in $B\to X_sl^+l^-$.

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