Importance of Loop Effects in Explaining the Accumulated Evidence for New Physics in $B$ Decays with a Vector Leptoquark

In recent years experiments revealed intriguing hints for new physics (NP) in $B$ decays involving $b\to c\tau \nu$ and $b\to s{\ell }^{+}{\ell }^{-}$ transitions at the $4\sigma$ and $5\sigma$ level, respectively. In addition, there are slight disagreements in $b\to u\tau \nu$ and $b\to d{\mu }^{+}{\mu }^{-}$ observables. While not significant on their own, they point in the same direction. Furthermore, $V_{us}$ extracted from $\tau$ decays shows a slight tension ($\approx 2.5\sigma$) with its value determined from Cabibbo-Kobayashi-Maskawa unitarity, and an analysis of BELLE data found an excess in $B_d\to {\tau }^{+}{\tau }^{-}$. Concerning NP explanations, the vector leptoquark SU(2) singlet is of special interest since it is the only single particle extension of the standard model which can (in principle) address all the anomalies described above. For this purpose, large couplings to $\tau$ leptons are necessary and loop effects, which we calculate herein, become important. Including them in our phenomenological analysis, we find that neither the tension in $V_{us}$ nor the excess in $B_d\to {\tau }^{+}{\tau }^{-}$ can be fully explained without violating bounds from $K\to \pi \nu \overline{\nu }$. However, one can account for $b\to c\tau \nu$ and $b\to u\tau \nu$ data finding intriguing correlations with $B_q\to {\tau }^{+}{\tau }^{-}$ and $K\to \pi \nu \overline{\nu }$. Furthermore, the explanation of $b\to c\tau \nu$ predicts a positive shift in $C_7$ and a negative one in $C_9$, being nicely in agreement with the global fit to $b\to s{\ell }^{+}{\ell }^{-}$ data. Finally, we point out that one can fully account for $b\to c\tau \nu$ and $b\to s{\ell }^{+}{\ell }^{-}$ without violating bounds from $\tau \to \varphi \mu$, $\mathrm{\Upsilon }\to \tau \mu$, or $b\to s\tau \mu$ processes.

Figure 1

Feynman diagrams depicting the one-loop contributions of the vector LQ singlet to $C_{7/8}^{sb}$, $b\to s{\ell }^{+}{\ell }^{-}$, $\tau \to \mu \nu \overline{\nu }$, and $b\to s\nu \overline{\nu }$ (from left to right).

Figure 2

$C_{9,sb}^{\ell \ell }$ and $C_7^{sb}({\mu }_b)$ as functions of $R(X)/R(X{)}_{\mathrm{SM}}$ with $X=\{D,D^{*},J/\psi \}$. The solid lines correspond to $M=1\mathrm{TeV}$ and the dashed ones to $M=5\mathrm{TeV}$ while the (dark) blue region is preferred by $b\to c\tau \nu$ data at the $1\sigma$ ($2\sigma$) level. From the global fit, taking into account only lepton flavor conserving observables we have $-1.29<C_{9,sb}^{\ell \ell }<-0.87$ [70] and $-0.01<C_7^{sb}({\mu }_b)<0.05$ [10] at the $1\sigma$ level. Therefore, our model predicts just the right sign and size of the effect in $C_{9,sb}^{\ell \ell }$ and $C_7^{sb}({\mu }_b)$ necessary to explain $b\to s{\ell }^{+}{\ell }^{-}$ data, assuming an explanation of $b\to c\tau \nu$.

Figure 3

Predictions for $B_q\to {\tau }^{+}{\tau }^{-}$ and $K\to \pi \nu \overline{\nu }$ (contour lines) in the ${\kappa }_{13}^L-{\kappa }_{23}^L$ plane for $M=1\mathrm{TeV}$ and ${\kappa }_{33}^L=1$. The colored regions are preferred by $b\to c(u)\tau \nu$ data, where we naively averaged (i.e., we computed the weighted average of the observables and added their errors in quadrature, disregarding correlations) $R(D^{(*)})$ and $R(J/\psi )$ or $R(\pi )$ and $B\to \tau \nu$, respectively. The gray region is excluded by $K^{+}\to {\pi }^{+}\nu \overline{\nu }$. Here we assumed all couplings ${\kappa }_{ij}^L$ to be real.

Figure 4

Allowed (colored) regions in the $C_{9,sb}^{\mu \mu }=-C_{10,sb}^{\mu \mu }$ $(\widehat{=}640{\kappa }_{22}^L{\kappa }_{32}^{L*})$—$R(X)/R(X{)}_{\mathrm{SM}}$ plane for $M=1\mathrm{TeV}$ and $X=D$, $D^{*}$, $J/\psi$ at the $1\sigma$ and $2\sigma$ level for ${\kappa }_{33}^L{V}_{cb}\ll {\kappa }_{23}^L$. The region above the black dashed (solid) line is excluded by $\tau \to \varphi \mu$ ($B\to K\tau \mu$) for ${\kappa }_{33}^L=0.5=25{\kappa }_{32}^L$ (${\kappa }_{33}^L=0.5=2.5{\kappa }_{32}^L$). The bound from $\tau \to \varphi \mu$ ($B\to K\tau \mu$) depends on ${\kappa }_{33}^L$ and ${\kappa }_{32}^L$ and gets stronger if ${\kappa }_{32}^L$ gets smaller (larger). That is, for ${\kappa }_{33}^L=0.5$ and $2.7⪅{\kappa }_{33}^L/{\kappa }_{32}^L⪅27$, the whole $2\sigma$ region preferred by $b\to c\tau \nu$ and $b\to s{\ell }^{+}{\ell }^{-}$ data is consistent with $B\to K\tau \mu$ and $\tau \to \varphi \mu$.