In 1977 John Ellis made a bet with a student, Melissa Franklin, “If you lose this game of darts,” Franklin said, “you have to use the word ‘penguin’ in your next paper.” John Ellis lost and as a result he coined the term Penguin decays while investigating charge-parity violation the Feynman diagrams of which “look like penguins” [1].

These diagrams describe particle processes that happen through more complicated processes compared to what we observe in most particle decays happening in the Standard Model due to the weak interaction. Most of the processes proceed through what is called tree-level decays described by Feynman diagrams that look like the junctures of tree branches. For example a neutron, consisting of three quarks, decays to a proton (three quarks), an electron and a neutrino. The interaction, mediated by the W boson, is represented in the Feynman diagram below (Figure.1).

Figure 1: Example of a tree-level Feynman diagram showing the decay of a neutron to a proton via W boson.

However, this is not the case for all decays as some of them include the brief exchange of virtual particles - through so-called loop processes - that can recombine into new components and allow for a change in the flavor of quarks.

The first discovered penguin decay was b → sγ in which a bottom quark decays to a strange quark and a photon involving a W boson but also a top quark. The Feynman diagram is shown in Figure 2.

Figure 2: Showing one of the first proposed penguin diagrams proposed to describe electroweak Feynman process forbidden by tree-diagrams. The right image showsu how this processes resembles  the shape of a penguin with the loop standing for the wide white stripe extending like a bonnet across the top of its head and its bright orange-red bill.

This decay was observed about 30 years earlier, in 1993 by the CLEO experiment in the US [2]. As you can see in Figure 3, a clear peak in their results corresponds to the mass of the B meson (the lightest particle to contain a bottom quark).

Figure 3:  The mass distribution for the B candidates is clearly shown in this plot. There are 8 events in the signal region corresponding to the mass interval 5.274 - 5.286 GeV (taken from [2]).

A penguin diagram represents a process where one quark flavor of matter particle changes into another flavor while emitting a photon or gluon: for example, a bottom (b) quark converting into a strange (s) quark and a photon. These processes are rare in the Standard Model and are interesting because they can get large enhancements from new physics and this is what we will discuss.

Previously we referred to perhaps the simplest case of a Penguin diagram of a b → sγ  decay. In fact, one could consider more complicated processes taking place. For example one could add two leptons in the diagram as shown below (Figure 4) or even think the possibility of having a Z boson instead of a photon (γ) and making one more step to replace the Z boson with a loop that will involve a W boson and a neutrino (see Figure 5).

Figure 4: Penguin diagram showing the decay of a b to s quark  (top) and with a box process involving a W boson as well as a top quark (bottom).

So far we have introduced three different processes describing the beauty decay to strange  shown in Figures 2 and 4. They represent possible decays of particles containing a bottom quark to particles containing strange quarks and in the framework of the Standard Model. B → K*μ⁺μ⁻ decays. Such flavour-changing neutral processes are forbidden at the lowest perturbative order in the Standard Model, and can only be realized when including higher-order loop processes involving virtual W bosons as we discussed. In the SM they are very suppressed at the level of one in a million or so.

For example, we can choose B → K*μ⁺μ⁻ where a B meson decays to a kaon (with a strange quark), a light pion and two muons. The decay of a B meson (containing a b quark and a d quark) into a K*⁰ meson (s and d) and a pair of muons is quite a rare process, occurring around once for every million B meson decays. It involves a change of the quark flavour, b → s, while preserving the total charge) and it is possible only via  electroweak penguin and box processes while it is forbidden at tree level.

This decay can be described using three angles and two masses as shown in Figure 5 where one can see the angles θ_l, θ_K and a third angle φ between the plane defined by the dimuon pair and the plane defined by the kaon and pion in the B0 rest frame.

Figure 5: The final state of the decay B⁰ → K^∗0µ ⁺µ⁻ can be described by q², the invariant mass squared of the dimuon system, and three decay angles Ω~ = (cos θ_l, cos θ_K, φ).

It should be noted that this decay is quite susceptible to the presence of new particles that could decay through competing processes and thus significantly alter the branching fraction of the decay and the angular distribution of the final-state particles described above. Our theoretical predictions of angular observables are less affected by uncertainties in the B⁰ → K*⁰ decay, which makes it an interesting channel for experimental observation.

The two masses are the masses of Kπ systems and μμ systems. The reason for choosing Kπ systems in the K* region is because that’s where most of the data lie and we are able to study a very clean spin-1 Kπ system. The angular observables and their correlations are reported in bins of q²; a free variable that stands for the invariant-mass squared of the dimuon system. How often a decay picks a particular combination of angles can give us information about the underlying physics mechanism and the possible presence of new physics in the form of small deviations compared to the SM predicted value.

The following LHCb plot [3] using data from the second LHC run shows the distribution of the cosine of angle θ_ℓ over a given range of the q² parameter. Looking carefully, the distribution appears to be asymmetric towards larger values of the cosine of this angle. This is an intriguing result, waiting for higher statistics as it could point to new hypothetical particles like those predicted by supersymmetric theories, the existence of new bosons or leptoquarks all of which could enter in the penguin diagrams described before.

Figure 6: Projections of the fitted probability density function on the decay angles for the bin 6.0<q²<8.0 GeV²/c⁴. The blue shaded region indicates background.

The observed asymmetry in the distribution of angles can also be plotted versus q²; a variable named AFB and representing forward-backward asymmetry of the dimuon system. In the region of 6<q²<8 the asymmetry is found to be positive as shown in Figure 7, where one can see the uncertainties (vertical bars) and the theoretical predictions (magenta). Along with better experimental/statistical accuracy we also need new theoretical calculations to understand the SM predictions and the respective theoretical errors.

Figure 7: Results for the CP-averaged angular observables AFB in bins of q² compared to the SM predictions.

Eluned Smith of RWTH Aachen University presented this result at a CERN seminar on 10 March, on behalf of the LHCb collaboration [4]. For the analysis, the team combined data collected during Run 2 with those of the previous LHC run, actually doubling the data compared to the previous measurement in 2015 based only on Run 1 data.

The plot below shows the angular distributions (a variable named P5′) of the B⁰→K*⁰μ⁺μ^– decay in the 4.0<q²<6.0 and 6.0<q²<8.0GeV²/c⁴ bins. The local tension in the measurement of P5′ is reduced from 2.8 and 3.0σ measured in the Run1 data analysis down to 2.5 and 2.9σ. However, a global fit to several angular observables shows that the overall tension with the SM increases from 2.9 to 3.3σ which is still too low before claiming evidence for new physics.

Figure 8: Angular distributions of P5′ for the B⁰→K*⁰μ⁺μ^– decay.

Data from BaBar and Belle add further intrigue, though with lower statistical significance. In 2016, the Belle experiment at KEK in Japan performed its own angular analysis of B⁰→K*0μ⁺μ^– using data from electron—positron collisions and found a 2.1σ deviation in the same direction and in the same q² region as the LHCb anomaly.

This result is more intriguing as it is linked with another anomaly observed by the LHCb experiment. In 2014, using data collected during LHC Run 1, the LHCb collaboration compared the rates of the reactions B⁺ → K⁺μ⁺μ^- and B⁺ → K⁺e⁺e^- to test lepton universality. If all leptons have the same couplings to gauge bosons, as the SM assumes, the ratio of these two reaction rates should be equal to unity, apart from well-understood effects related to the different lepton masses. But LHCb’s measurement of this ratio, R(K*), for the same decay (B⁰ → K^*0μ⁺μ^–) differed from the SM prediction with a statistical significance of 2.4 standard deviations, and that for the similar decay  B⁺→ K⁺μ⁺μ^–  by 2.5 standard deviations. These deviations, if confirmed, suggest a violation of lepton-flavour universality, one of the key properties of the SM. The two results made physicists speculate that they can be caused by the same type of new physics, with models involving leptoquarks or new gauge bosons in principle able to accommodate both sets of anomalies. Further data analysis from the data collected during Run 2 and those from future runs of the LHC will shed more light on the nature of this anomaly.

Further Reading

[1] The story is beauiifully told in a Symmetry article: HERE 

[2] R. Ammar et al., "Evidence for penguin-diagram decays: First observation of B→ K* (892)γ", Phys. Rev. Lett. 71, 674 (1993)

[3] LHCb Collaboration 2016 arXiv :1512.0442  (published in JHEP 1602 (2016) 104).

[4] LHCb Collaboration 2020 arXiv:2003.04831.