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New physics in B→K ∗ μμ?

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Abstract

Recent experimental results on angular observables in the rare decay B→K ∗ μ + μ − show significant deviations from Standard Model predictions. We investigate the possibility that these deviations are due to new physics. Combining all relevant data on b→s rare decays, we show that a consistent explanation of most anomalies can be obtained by new physics contributing simultaneously to the semi-leptonic vector operator O 9 and its chirality-flipped counterpart \(O_{9}'\). A partial explanation is possible with new physics in O 9 or in dipole operators only. We study in detail the implications for models of new physics, in particular the minimal supersymmetric standard model, models with partial compositeness and generic models with flavour-changing Z′ bosons. In all considered models, contributions to B→K ∗ μ + μ − of the preferred size imply a spectrum close to the TeV scale. We stress that measurements of CP asymmetries in B→K ∗ μ + μ − could provide valuable information to narrow down possible new physics explanations.

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Notes

  1. We do not use the LHCb measurement of BR(B 0→K 0 μ + μ −) [35], which has larger error bars. Note that this measurement is on the low side compared to the charged decay, which is hard to accommodate even in the presence of new physics [36]. We do not average the B +→K + μ + μ − data with B factory or CDF data either, since they have a numerically negligible impact.

  2. We note that our treatment of form factors at high q 2 is based on the extrapolation of light-cone sum rule calculations [41] done in [42], which leads to particularly large uncertainties in F L compared to other approaches.

  3. Experimental results for S 4 and S 5 in the [1,6] GeV2 region are not available yet. We therefore translate the results on \(P_{4}^{\prime}\) and \(P_{5}^{\prime}\) using the measured value of F L and get 〈S 4〉[1,6]=0.14±0.08, 〈S 5〉[1,6]=0.10±0.10.

  4. We stress the different sign of our definition of S 4 with respect to LHCb, see Table 1.

  5. We stress that in all the following plots, the red regions combine all relevant constraints, while the bands for individual observables are only shown for illustrative purposes for some of the most relevant observables. We also note that 68 and 95 % C.L. constraints on individual coefficients can be read off from the overlap of the bands with the axes, while the two-dimensional regions would correspond to 39 and 86 % C.L.

  6. We consider the branching ratio in the entire high-q 2 region for indicative purposes. In the numerical analysis, we use three separate bins, as in the LHCb analysis.

  7. Note that, given the present uncertainties in the B→K ∗ μ + μ − data, a sizable axial-vector coupling to leptons cannot be excluded yet. Allowing for non-zero axial-vector couplings to leptons, could either enhance or suppress couplings to neutrinos.

  8. We explicitly checked that in all cases discussed below and shown in Fig. 9, the presence of imaginary parts does not worsen the agreement with the data significantly.

  9. N. Serra, private communication.

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Acknowledgements

We thank Christoph Bobeth for illuminating discussions and Nicola Serra for useful correspondence. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy. The research of D.S. is supported by the Advanced Grant EFT4LHC of the European Research Council (ERC), and the Cluster of Excellence Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA–EXC 1098). The research of W.A. was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation.

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  1. Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL, 60510, USA

    Wolfgang Altmannshofer

  2. PRISMA Cluster of Excellence & Mainz Institute for Theoretical Physics, Johannes Gutenberg University, 55099, Mainz, Germany

    David M. Straub

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  1. Wolfgang Altmannshofer
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Correspondence to Wolfgang Altmannshofer.

Appendices

Appendix A: Data averages

In this appendix we give more details on how we obtain the averages of experimental measurements of B→K ∗ μ + μ − observables [17, 18, 21–25] used in this analysis. As in [12], we first symmetrise asymmetric statistical and/or systematic errors and then perform a weighted average of the symmetrised individual results. While in many cases the obtained averages are dominated by the LHCb results, the averaging procedure leads to important shifts in the observables F L and A FB in the [1,6] GeV2 bin. This is illustrated in the plots of Fig. 10. In the case of the BaBar B→K ∗ μ + μ − data at low q 2, the results for the charged and neutral modes show a significant difference. We therefore first average the charged and neutral mode results of BaBar, using the PDG averaging method, i.e. rescaling the uncertainty by a factor of \(\sqrt{\chi^{2}}\). We then use this average and combine it with the available data from the other experiments.

Fig. 10
figure 10

Individual experimental results with 1σ uncertainties for A FB (left) and F L (right) in the [1,6] GeV2 bin, as well as our averages. The SM predictions with 1σ uncertainties are shown by the green bands

Full size image

In the case of F L , we observe tensions between the data of the several experiments. In particular, in the [1,6] GeV2 bin, both BaBar and ATLAS data are significantly below the measurements of the other experiments and the SM prediction. Therefore, we rescale the uncertainty of our weighted average of F L by \(\sqrt{\chi^{2}/N_{\text{dof}}}\). As shown in the right plot of Fig. 10, a tension with the SM prediction of 1.9σ remains. In the case of A FB, the tension between the SM prediction and the LHCb data alone is softened considerably after data from the other experiments is taken into account (see left plot of Fig. 10).

A final comment is in order on the observables S 4 and S 5 that have only been measured by LHCb. Since Ref. [23] does not provide data for these observables in the [1,6] GeV2 bin, we have reconstructed them using the data on \(P_{4,5}'\) and F L as in Table 1, which is expected to be very close to a direct determination.Footnote 9

In Table 3, we list the resulting experimental averages for the angular observables and confront them with our SM predictions.

Table 3 SM predictions confronted with experimental averages of B→K ∗ μ + μ − angular observables in the three q 2 bins. The pull is defined as \(\sqrt{\Delta\chi^{2}}\). Details and references are given in the text
Full size table

Appendix B: Loop functions

In this appendix we collect the loop functions that appear in the discussion of the NP contributions to the Wilson coefficients \(C_{9}^{(\prime)}\) and \(C_{7}^{(\prime)}\) in Sects. 3.2 and 3.3.

The loop functions entering MSSM contributions to the vector coefficients C 9 and \(C_{9}^{\prime}\) read

$$\begin{aligned} f_9^{H^\pm}(x) =& -\frac{2(38-79x+47x^2)}{9(1-x)^3} \\ &{}- \frac {4(4-6x+3x^3)\log x}{3(1-x)^4} \xrightarrow{x \to1} 1 , \end{aligned}$$
(30)
$$\begin{aligned} f_9^{\tilde{H}^\pm}(x) =& -\frac{2(52-101x+43x^2)}{21(1-x)^3} \\ &{}- \frac {4(6-9x+2x^3)\log x}{7(1-x)^4} \xrightarrow{x \to1} 1 , \end{aligned}$$
(31)
$$\begin{aligned} f_9^{\tilde{g}}(x) =& \frac{5(1-5x+13x^2+3x^3)}{3(1-x)^4} \\ &{} + \frac{20x^3 \log x}{(1-x)^5} \xrightarrow{x \to1} 1 , \end{aligned}$$
(32)
$$\begin{aligned} f_9^{\tilde{W}}(x) =& -\frac{10(22-38x+7x^2+3x^3)}{3(1-x)^4} \\ &{} - \frac {10(3-9x^2+4x^3) \log x}{(1-x)^5} \xrightarrow{x \to1} 1 , \end{aligned}$$
(33)
$$\begin{aligned} f_9^\text{box}(x,y) =& \frac{12(x-2y+xy)}{(1-x)(y-x)(1-y)^2} - \frac {12x^2 \log x}{(1-x)^2(x-y)^2} \\ &{} + \frac{12 y(2x-y+y^2)\log y}{(x-y)^2(1-y)^3} \xrightarrow{x,y \to1} 1 . \end{aligned}$$
(34)

The loop functions that are relevant for the MSSM contributions to the dipole coefficients C 7 and \(C_{7}^{\prime}\) read

$$\begin{aligned} &{f_7^{H^\pm}(x) = \frac{3(5x-3)}{7(1-x)^2} + \frac{6(3x-2)}{7(1-x)^3} \log x \xrightarrow{x \to1} 1 ,} \end{aligned}$$
(35)
$$\begin{aligned} &{f_7^{\tilde{H}^\pm}(x) = \frac{6(7x-13)}{5(1-x)^3} + \frac {12(2x^2-2x-3)}{5(1-x)^4} \log x \xrightarrow{x \to1} 1 ,} \end{aligned}$$
(36)
$$\begin{aligned} &{f_7^{\tilde{g}}(x) = \frac{10(1+10x+x^2)}{(1-x)^4}} \\ &{\phantom{f_7^{\tilde{g}}(x) =}{}+ \frac {60x(1+x)}{(1-x)^5} \log x \xrightarrow{x \to1} 1 .} \end{aligned}$$
(37)

The function entering the Higgs-loop contribution to C 7 and \(C_{7}^{\prime}\) in models with partial compositeness reads

$$ f_7^h(x) = \frac{x(x^2-4 x+2 \log(x)+3)}{(x-1)^3} \xrightarrow{x\to \infty} 1 . $$
(38)

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Altmannshofer, W., Straub, D.M. New physics in B→K ∗ μμ?. Eur. Phys. J. C 73, 2646 (2013). https://doi.org/10.1140/epjc/s10052-013-2646-9

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