Holographic QCD predictions for rare B decays

TL;DR

This paper uses holographic light-front QCD to predict form factors for rare B decays to K* mesons, comparing results with QCD sum rules and experimental data.

Contribution

It introduces a novel approach using holographic light-front QCD to calculate B decay form factors, providing alternative predictions to existing methods.

Findings

Predicted B to K* form factors using holographic QCD.

Compared predictions with QCD sum rules and experimental data.

Showed consistency and differences with existing models.

Abstract

Light-front wavefunctions obtained from holographic light-front QCD are used to obtain the distributions amplitudes for $K^*$ vector meson. Consequently, alternate predictions for rare B transitions to $K^*$ form factors are presented. In this talk, I compare our results for some rare B decay channels to those obtained from QCD sum rules and available experimental data.

Tables

Table 1. Table 1: Comparison between hQCD predictions for the decay constant of the K ∗ superscript 𝐾 K^{*} meson with experiment (obtained from Γ ​ ( τ − → K ∗ − ​ ν τ ) Γ → superscript 𝜏 superscript 𝐾 absent subscript 𝜈 𝜏 \Gamma(\tau^{-}\to K^{*-}\nu_{\tau}) ), and the ratio of couplings with lattice data.

Equations

$$\left(-\frac{{\mathrm{d}}^2}{\mathrm{d}{\zeta }^2}-\frac{1-4L^2}{4{\zeta }^2}+U_{\mathrm{eff}}(\zeta )\right)\varphi (\zeta )=M^2\varphi (\zeta ),$$

$$U_{eff}(\zeta )={\kappa }^4{\zeta }^2+2{\kappa }^2(J-1).$$

$$M^2=(4n+2L+2){\kappa }^2+2{\kappa }^2(J-1)=4{\kappa }^2(n+L+\frac{S}{2}),$$

$${\varphi }_{n,L}(\zeta )={\kappa }^{1+L}\sqrt{\frac{2n!}{(n+L)!}}{\zeta }^{1/2+L}\exp \left(\frac{-{\kappa }^2{\zeta }^2}{2}\right)L_n^L(z^2{\zeta }^2),$$

$$\Psi (\zeta ,z,\varphi )=e^{iL\varphi }\mathcal{X}(z)\frac{\varphi (\zeta )}{\sqrt{2\pi \zeta }},$$

$${\Psi }_{0,0}(z,\zeta )=\frac{\kappa }{\sqrt{\pi }}\sqrt{z(1-z)}\exp \left[-\frac{{\kappa }^2{\zeta }^2}{2}\right].$$

$${\Psi }_{\lambda }(z,\zeta )={\mathcal{N}}_{\lambda }\sqrt{z(1-z)}\exp \left[-\frac{{\kappa }^2{\zeta }^2}{2}\right]\exp \left[-\frac{(1-z)m_q^2+z{m}_{\bar{q}}^2}{2{\kappa }^{2}z(1-z)}\right].$$

$${\varphi }_{K^{\ast }}^{\parallel }(z,\mu )=\frac{N_c}{\pi f_{K^{\ast }}{M}_{K^{\ast }}}\int \text{d}r\mu J_1(\mu r)[M_{K^{\ast }}^{2}z(1-z)+m_{\bar{q}}{m}_s-{\nabla }_r^2]\frac{{\varphi }_{K^{\ast }}^L(r,z)}{z(1-z)},$$

$${\varphi }_{K^{\ast }}^{\perp }(z,\mu )=\frac{N_c}{\pi f_{K\ast }^{\perp }}\int \text{d}r\mu J_1(\mu r)[m_s-z(m_s-m_{\bar{q}})]\frac{{\varphi }_{K^{\ast }}^T(r,z)}{z(1-z)},$$

Full text

I Introduction

Rare flavour changing neutral current B-meson decays have been under theoretical and experimental scrutiny in the past three decades. B-factories and other dedicated experimental facilities have provided data with ever increasing precision on more variety of rare decay channels. The inclusive decays, which are easier to handle theoretically, are extremely challenging for measurement especially in a hadronic environment like LHCb. Exclusive modes, on the other hand, suffer from hadronic uncertainties and therefore, the better understanding of the non-perturbative QCD effects is essential in comparing the standard model (SM) predictions against the experimental data. This contribution focuses on the $B\to K^{*}$ transition but is more broadly based on Refs. Ahmady:2018fvo ; Ahmady:2015fha ; Ahmady:2014sva ; Ahmady:2014cpa ; Ahmady:2013cva .

Predicting observables in rare B decays requires a knowledge of non-perturbative quantities like transition form factors as well as decay constants. These quantities are related to the meson Distribution Amplitudes (DAs). Traditionally, DAs are obtained using QCD Sum Rules. Here, we derive them using the holographic light-front meson wavefunction which is itself obtained by solving the holographic Schrödinger Equation for mesons. The differences between our predictions and those of the QCD sum rules (QCDSR) offers an opportunity to gauge the impact of hadronic uncertainties in the theory predictions of observables in rare B decays. This can be helpful to filter out genuine New Physics signals in these decays.

II Holographic light-front QCD

In semiclassical light-front QCD, where quark masses and quantum loops are neglected, the transverse part of the valence light-front wavefunction of a meson of mass $M$, satisfies the so-called holographic Schrödinger Equation deTeramond:2008ht

[TABLE]

where $\mathbf{\zeta}=\sqrt{z\bar{z}}b$ ($\bar{z}=1-z$). $b$ is the transverse separation of the quark and antiquark and $z$ is the light-front momentum fraction carried by the quark. Eq. (1) is called holographic because it maps onto the wave equation for string modes propagating in the higher $5$-dimensional anti-de Sitter spacetime. This holographic mapping, together with the requirement to introduce a mass scale in Eq. (1) while preserving the conformal invariance of its underlying action, leads to a unique form for the confining potential:

[TABLE]

The holographic Schrödinger Equation can then be solved:

[TABLE]

[TABLE]

where $n$, $L$ and $S$ are the principal, orbital and spin quantum numbers respectively. As can be seen from Eq. (3), the Regge slope for vector mesons determines the fundamental scale of the model: $\kappa=0.54$ GeV Brodsky:2014yha . The complete light front wavefunction of the meson is then given as

[TABLE]

where the longitudinal wavefunction $\mathcal{X}(x)=\sqrt{z(1-z)}$, obtained by mapping the pion electromagnetic form factors in AdS and in physical spacetime Brodsky:2014yha . For the vector mesons (like $\rho$, $K^{*}$ and $\phi$), we set $n=0,L=0$ and $S=1$ to obtain

[TABLE]

Allowing for small quark masses, the wavefunction becomes

[TABLE]

So far, the helicities of the quark and antiquark have been ignored. Assuming the spin structure of a vector meson is similar to that of a photon Forshaw:2012im , we can restore the dependence of the light-front wavefunctions on the helicities of the quark and antiquark. The resulting twist-2 holographic DAs for the $K^{*}$ meson are then given by

[TABLE]

[TABLE]

where $f_{K^{*}}$ and $f_{K^{*}}^{\perp}$ are the vector and tensor couplings of $K^{*}$. The former can be accessed experimentally through the electronic decay width of the meson and this provides a first constraint for our DAs. In particular, it allows us to constrain the quark masses.

III Results

Table 1 shows the holographic QCD (hQCD) predictions for the $K^{*}$ vector and tensor coupling compared with the available experimental and lattice data. We observe that quark masses $m_{\bar{q}}=195\pm 55$ MeV and $m_{s}=300\pm 20$ MeV leads to predictions consistent with data.

In Fig. 1, we compare our predictions for twist-2 $K^{*}$ DAs with those obtained from QCDSR. Indeed, the theoretical calculation of the isospin asymmetry in $B\to K*\gamma$ decay is directly sensitive to the end-point behaviour of the DAs and it turns out that the holographic DAs do not lead to the end-point divergences encountered with SR DAs.

Using the above DAs within the light-cone sum rules (LCSR) method, we calculate $B\to K^{*}$ transition form factors (FFs) and in turn make predictions for observables like differential branching ratio and asymmetries.

We observe that hQCD predictions are lower than those of the SR for the entire kinematical range of the momentum transfer ($q^{2}$). We have also made predictions for the, yet to be measured, rare $B\to K^{*}\nu\bar{\nu}$ decay.

In Figure 3, the differential branching ratio for $B\to K^{*}\nu\bar{\nu}$ as predicted by hQCD and QCDSR are presented. This decay channel can provide a clean venue for testing these hadronic models. It is interesting to note that the predictions significantly for low to intermediate $q^{2}$ range.

IV Acknowledgment

MA would like to thank the organizers of FPCP 2019 for the wonderful experience. This work is supported in part by MA and RS’s NSERC grants.