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IPPP/19/12

Searching new in rare B- decays into multiple

Mikael Chala,a Ulrik Egedeb,c and Michael Spannowskya aInstitute for Physics Phenomenology, Department of Physics, Durham University, Durham, DH1 3LE, UK bImperial College London, London, United Kingdom bSchool of Physics and , Monash University, Melbourne, Australia E-mail: [email protected], [email protected], [email protected]

Abstract: New heavy vector and light scalars are predicted in a plethora of models of new physics. In particular, in new strongly interacting sectors they play the role of the ρ and π in QCD. We show that some of their , for example those required for the explanation of the B anomalies and the g 2 of the , can be − only probed in decays. We highlight new golden channels not yet studied ex- perimentally, including B+ K+(D+)µ+µ−µ+µ− and B0 K∗0µ+µ−µ+µ−. Relying → → on generator level simulations for data taking with the LHCb detector, we determine the reach of this facility to the aforementioned processes. We show that branching ratios as small as 9 10−12 (3.2 10−10) and 2.7 10−11 can be tested at the 95 % CL × × × respectively. arXiv:1902.10156v1 [hep-ph] 26 Feb 2019 Contents

1 Introduction1

2 Generic Lagrangian2 2.1 Explicit model4

3 Reach of the LHCb6

4 Conclusions9

1 Introduction

New heavy vector bosons V and light scalars a are common predictions of different sce- narios of physics beyond the (SM). The former appear in extensions of the SM gauge group, including theories of grand unification [1–5] and string con- structions [6]. They are also natural in composite sectors [7–10] and their holographic relatives [11, 12]. Recently, new vectors at the TeV scale have been also proposed as a plausible explanation [13] of the anomalies observed in the branching fractions, angular distributions and universality tests of the decays B+(0) K+(∗)`+`− [14–22]. → Models prediciting ultralight scalars a have been studied [23] for collider phenomenol- ogy. Likewise, they have been also studied in light of the observed disagreement be- tween the predicted and the observed values of the anomalous of the muon [24, 25]. Perhaps, the most traditional scenario involving both new scalars and vectors con- sists of a new strongly interacting sector extending the SM. In this case, V and a play the role of the ρ and π mesons in QCD. The separation of these scales is explained by the pseudo-Nambu Goldstone of the latter. We show that rare B decays can be naturally expected in this context. In the situation where the scalar decays to two muons, these include B0 aa, which have already been searched for (s) → at LHCb [26], as well as B+/0 Maa with M = K+,D+,K∗. → In this article, we perform simulations to estimate the reach of the LHCb experi- ment to the aforementioned processes with the currently available data as well as with the anticipated upgrades. Our choice of parameters is motivated by the B and (g 2) − µ

– 1 – anomalies. However, our results are of much broader applicability. The paper is orga- nized as follows. In Section2 we introduce the generic Lagrangian we are interested in; we comment on constraints on the different parameters and compute the ampli- tudes for the different B decays. In Section 2.1, we match a particular composite Higgs model to the Lagrangian above. We study the new rare B meson decays in Section3. We subsequently interpret these results in the model introduced before. Finally, we conclude in Section4.

2 Generic Lagrangian

Let us extend the SM with a new V and a new scalar a with of the order of TeV and GeV, respectively. They are both singlets of the SM gauge group. At below the electroweak scale v 246 GeV, the Lagrangian we are interested in ∼ is 1 1 L = m2 V V µ + m2a2 2 V µ 2 a h u i µ j d i µ j i e h i j i 0 µ + Vµ gij(uLγ uL) + gij(dLγ dL) + h.c. + gija lLeR + h.c. + g Vµ(a∂ a) . (2.1)

i i i i where uL, dL stand for the i-th family of left-handed up and down , and lL, eR stand for the i-th family of left- and right-handed , respectively. Inspired by recently observed flavour anomalies, we will focus mostly on the case in which the only non-vanishing g coupling is gd . To a lesser extent, we will also consider gd = 0. These 23 13 6 couplings are constrained by measurements of ∆Ms and ∆Md [27], respectively. Thus, d d for mV = 1 TeV, g23 . 0.002 and g13 is bounded to be about one order of magnitude smaller. Together with a non-vanishing g0, these couplings trigger rare B decays as shown in Fig.1. The decay width for B0 aa is s → s 2 2 fB d 0 2 3 4ma Γ = 4 (g23g ) mB 1 2 , (2.2) 32πmV − mB with f 0.23 GeV [28]. Similar expressions hold for other decay modes, e.g. B0 aa. B ∼ → The amplitude for B+ K+aa reads: → d 0 g23g µ = 2 K(p3) sγµb B(p) (p1 + p2) (2.3) M mV h | | i with [29]  m2 m2  m2 m2 K(p ) sγ b B(p) = f P (q2) (p + p ) B − K q + f (q2) B − K q . (2.4) h 3 | µ | i + 3 µ − q2 µ 0 q2 µ

– 2 – Figure 1. Diagrams of the decays B0 aa (left) and B+ K+aa (right) mediated by a s → → heavy vector boson V .

2 2 2 2 The transferred is q = (p p3) , and varies between qmin = 4ma and 2 2 − qmax = (mB mK ) . The contraction of this matrix element with (p1 + p2) = q in the − P 2 amplitude annihilates the f+ (q ) part. Altogether, we obtain

d 0 2 dΓ (g23g ) 2 2 2 2 2 = 3 4 3 (mB mK ) F (q ) , (2.5) dq 512π mV mB − with s q 2 2 4 2 2 2 2 2 2 4ma 2 2 F (q ) m + (m q ) 2m + (m + q ) 1 f (q ) . (2.6) ≡ B K − − B K − q2 | 0 |

In the approximation m , m 0, f (q2) 1, one easily obtains K a → 0 → d 0 2 (g23g ) 5 Γ 3 4 mB . (2.7) ∼ 1024π mV

2 2 2 Following Ref. [29], we parametrize the form factor as f0(q ) = r2/(1 q /mfit), 2 2 − with r2 = 0.330 and mfit = 37.46 GeV ; see Fig.2. Similar expressions hold for other 0 ∗0 + + processes, e.g. B K aa or Bc D aa. The latter is however hard to test at the → → + LHCb and we will not consider it. The reason is that the Bc production cross section + is much smaller and the Bc width is larger (which reduces both the impact of the new interactions and the experimental efficiency). We note also that final states containing one meson and aa can probe effective operators containing four quarks and two light scalars (12 of these operators are present in the SM effective field theory extended with a [30].) One can easily estimate Γ(B+ → K+aa) (few GeV)9/Λ8, which is of the order of 10−11 provided Λ 1 TeV. For ∼ . this reason, we will also consider the channel B+ D+aa. It tests operators such as → 1/Λ4a2(u γ b )(d γµc ) + h.c. O ∼ µ R R R

– 3 – 0.75 0.0025

0.70 )

0.65 aa 0.0020 + 0.60 K 0.0015 0.55 = 1 GeV 0 a → f 0.50 + , m 0.0010 B 2 0.45 min ( q 2

0.40 B 0.0005 dq

0.35 d 0.30 0.0000 0 5 10 15 20 0 5 10 15 20 q2 [GeV2] q2 [GeV2]

Figure 2. Left) Scalar momentum form factor for B+ K+ as a function of the transferred → momentum q2. Right) Differential branching ratio for B+ K+aa as a function of the 2 → momentum transferred q . In both cases, we have fixed mV = 1 TeV, ma = 1 GeV, g23 = g0 = 1.

We will assume that a decays into muons with a width smaller than 10 MeV ∼ and a lifetime shorter than 10 fs. In this case, it will appear not to have any ∼ experimentally measurable flight distance and will appear to have zero width. This is −5 e easily achieved if a is muonphilic with 10 . g22 . 1. The processes discussed so far leads therefore to four-muon final states with and without additional mesons and with the muons forming pairs of two identical masses.

2.1 Explicit model Light scalars are natural within CHMs, for they are approximate Nambu-Goldstone Bosons (NGBs) arising from the spontaneous breaking / in the confine- G H ment of a new strong sector at a scale f TeV. The simplest coset delivering the ∼ four Higgs degrees of freedom as well as a new scalar singlet a is SO(6)/SO(5) [31]. Interestingly, it can be UV completed in four dimensions [32]. In this model, the SM do not couple directly to the . They rather mix with other composite resonances that do interact with the Higgs boson. Thus, the Yukawa Lagrangian depends on the quantum numbers of the aforementioned resonances. As a simple yet realistic example, we assume that the generation leptons mix with two fundamental representations 6 of SO(6). An equivalent descrip- tion is the embedding of the elementary leptons into incomplete fundamental repre- sentations of SO(6). The most general such embedding depends on a single positive parameter γ to give

L = ( iµ , µ , iν , ν , 0, 0) , L = (0, 0, 0, 0, γµ , µ ) . (2.8) L − L L L L R R R

– 4 – Using the corresponding Goldstone matrix   13×3  1 h2/(f 2 + Π) ha/(f 2 + Π) h/f   h2 a2 1/2 U =  − −  , Π = f 2 1 ,  ha/(f 2 + Π) 1 a2/(f 2 + Π) a/f  − f 2 − f 2  − −  h/f a/f Π/f 2 − − (2.9) one obtains the leading-order Yukawa Lagrangian   yµ T T 1 γ L = f(U LL)1(U LR)1 + h.c. = yµµLhµR 1 + a + + h.c. (2.10) − √2 −√2 f ··· The coupling y 6 10−4 stands for the muon Yukawa. The subindex in (U T L ) µ ∼ × R 1 indicates the projection of the fundamental representation of SO(6) into the singlet of SO(5) according to the decomposition 6 = 1 + 5. The ellipsis stands for terms containing higher powers of a. Likewise, the one-loop induced potential for a reads:

4 2 T T 4 2 T I T V = cRf yµ(U ΛR)1(U ΛR)1 + cLf yµ(U ΛL)1(U ΛIL)1 c f 2y2 γ2 1 a2 + (2.11) ∼ R µ − ··· where we have neglected terms not involving a.ΛR stands for LR/µR; analogously for I ΛL with I the flavour index. cR is a free parameter encoding the details of the strong sectors. Its size can be estimated using naive power counting [33], c g2/(32π2), R ∼ ∗ with 1 . g∗ . √4π the typical coupling between resonances. All in all, we obtain

gρyµ p m (γ2 1) f , (2.12) a ∼ 4π − m ge µ γ . (2.13) 22 ∼ f The scalar a decays 100 % into muons. The other fermions respect this phenomenology provided they do not break the shift symmetry a a + constant, nor a a. These → → − two conditions can hold simultaneously if the left (right) mix with e.g. 6 (1), 6 (15) or 6 (200). The scalar defined above can explain the longstanding on the magnetic moment of the muon [25]. In this concrete model, we can fit the experimental measure- −9 ment (∆a ) = (2.74 0.73) 10 [34] within two standard deviations for g∗ = 2 and µ obs ± × f = 800 GeV and γ & 10. Fitting the experimental observation within one standard deviation is in principle possible, but it requires even larger values of γ, too small val- ues of g∗ (which would contradict the strongly coupled nature of the composite sector) and f . 800 GeV (which is in tension with Higgs and electroweak precision data [35]). Therefore, the value m 1 GeV is a very likely value in this setup. a ∼

– 5 – New composite vector bosons V explaining the observed anomalies in RK(∗) appear also naturally in this framework [36–45]. These decay preferably into com- posite states [44], being too broad and too heavy to be directly detected unless very 1 dedicated LHC analyses are performed for masses mV . 3 TeV [44]. d The with the SM fermions takes the form of Eq. 2.1, with g23 2 2 ∼ 0.002 mV /TeV [44, 46]. Relying on these results, we consider for reference the bench- mark point BP : m = 1 GeV , m = 4 TeV , gd 0.03 . (2.14) a V 23 ∼ 0 For g . 0.1, this point satisfies all current constraints from LHC searches and measure- ments of ∆Ms. Constraints set by the latter observable could be competitive if the more recent predictions of the SM value are confirmed [46, 47]. However, we will show that a signal should be observable with the future upgrades of the LHCb experiment. In par- ticular, we obtain (B0 µ+µ−µ+µ−) 4 10−9, (B+ K+µ+µ−µ+µ−) 10−9. B s → ∼ × B → ∼ Note that the final state with K+ meson is of the same order of magnitude as the four body final state. Together with the fact that the B+ meson has a higher production cross section than B0 in pp collisions, this suggests that B+ K+µ+µ−µ+µ− is a key s → signature to explore for this kind of models. This channel has not been experimentally explored though. Similar conclusions were pointed out in Ref. [48] in the context of dark sectors2.

3 Reach of the LHCb

LHCb has searched for the decays B0 µ+µ−µ+µ− [26] and has set the limits (s) → (B0 µ+µ−µ + µ−) < 7 10−10 and (B0 µ+µ−µ + µ−) < 2.5 10−9 with a B → × B s → × 3 fb−1 dataset at the collision energies of √s = 7 and √s = 8 TeV. However, there is a limitation in this analysis as it places a veto on the of the muon pairs to be close to the ϕ or J/ψ mass, to avoid background from the decay B0 J/ψϕ followed by s → both vector mesons decaying to a pair of muons. As the a mass is likely very close to the ϕ mass, the current analysis is not sufficiently general. If a new analysis removes the veto around the ϕ mass, and instead imposes a requirement that two opposite muon combinations recombine to the same , the limit should stay the same and the background from B0 J/ψϕ would still be avoided. s → 1Irrespectively of g0, the interaction between V and the heavy resonances L triggers the decay 0 ∗ + ∗ ∗ − ∗ Bs V µ L ,L aµ , where denotes off-shellness. For couplings equal to the unit, the → → → 2 4 4 5 5 7 corresponding width at tree level reads exactly Γ = f /(m m )(1+ma/mB) (1 ma/mB) m , with B L V − B mL the mass of L. Even for mL 500 GeV and mV 1 TeV, the corresponding branching ratio is ∼ ∼ 10−13 and therefore beyond the reach of our analysis. ∼ 2In this case, rare B meson decays are triggered by flavour-violating scalars.

– 6 – 0.06 0.05

0.05 B+ D+aa 0.04 7 GeV

max → . min T 1

µ, T 0.04 B+ K+aa < 0.03 → 0.03 max

µ, T 0.02 p B 0 s 0.02 → aa /σ dσ/dp /σ dσ/dp 1 1 0.01 0.01 min pT < 0.5 GeV 0.00 0.00 0 1000 2000 3000 4000 5000 0 200 400 600 800 1000 µ,max min pT [MeV] pT [MeV] 0.05 0.08

0.04 0.06 min max 0.03 0.04 5

0.02 . 2 5 /σ dσ/dη < /σ dσ/dη > 0.02 1 1 0.01 min max η η 0.00 0.00 2 3 4 5 2 3 4 5 ηmax ηmin

Figure 3. Upper left) Normalised distribution of the transverse momentum of the hardest muon in the case B0 aa (solid black), B+ K+aa (dashed blue) and B+ D+aa (dotted s → → → green). Upper right) Same as before but for the transverse momentum of the softest particle. Bottom left) Same as before but for the pseudorapidity of the most central particle. Bottom right) Same as before but for the pseudorapidity of the most forward particle.

Due to the four muons in the final state for the LHCb analysis, the combinatorial background to a possible signal is extremely low. As an essentially background free analysis, even in the far future, the branching fraction limit can be expected to scale inversely with the number of B mesons produced. In the run periods of Upgrade-I and Upgrade-II of LHCb, the collision will be √s = 14 TeV. As the b cross-section is scaling more or less in direct proportion to the collision energy, the amount of B mesons produced per inverse fb, can be expected to be around a factor 14/7.7 = 1.8 higher compared to the average Run-1 conditions of the LHC. Expectations below will be quoted for the end of LHCb (9 fb−1), end of Upgrade-I (50 fb−1) and end of upgrade-II (300 fb−1). The naive scaling factors, compared to the current Run-1 for these, and taking the different b cross-sections into account, are for LHCb a factor 4.4, for Upgrade-I a factor 29, and for Upgrade-II a

– 7 – factor 180. Thus for the limits on (B0 µ+µ−µ + µ−) we should expect 6 10−10, B s → × 9 10−11 and 1.4 10−11, respectively. This assumes no changes to the trigger or × × tracking performance in the upgrades of LHCb. Irreducible backgrounds to the decay have to be considered. The decay B0 ϕϕ s → with ϕ µ+µ− is one of these. Using the measured branching fractions [49, 50] we → get (B0 (ϕ µ+µ−)(ϕ µ+µ−)) = 1.84 10−5 (2.89 10−4)2 = 1.5 10−12. B s → → → × × × × As can be seen from the expected limits above, even at the end of LHCb Upgrade II, this is not relevant. For the equivalent decay mode of the B0, the measured branching fraction limit for the B0 ϕϕ decay is three orders of magnitude below the B0 mode → s and thus even less of a concern. The decay B0 ϕµ+µ− has a measured differential s → rate of 2.6 10−8 GeV−2 in the region of the squared dimuon mass close to the ϕ × mass [17]. Letting the ϕ decay to a muon pair and considering a mass region with width of around 20 MeV, corresponding to a realistic mass resolution, this will give a background at the 10−13 level and is thus not relevant. A simplified model for which limits the LHCb experiment can be made for similar decay modes. When comparing different B , the relative production fractions as measured at √s = 7 TeV collisions in the LHCb acceptance [51] are taken into account. The relative production fractions are not expected to change significantly with collision energy as they are determined from the fragmentation process. From this, we conclude + 0 0 that the production of B and B are the same and that the production of Bs mesons is a factor 3.7 less common. For the reconstruction in LHCb, it is assumed that the efficiency is 95 % per track inside the fiducial volume defined by the pseudorapidity 2.5 < η < 5.0 and that tracks have a transverse momentum above 0.5 GeV with respect to the beam axis to be reconstructed. For the trigger it is assumed that at least one reconstructed muon should have a transverse momentum above 1.7 GeV. The effect of these criteria is that final states with a larger number of particles have a lower efficiency, both due to the requirement that all tracks have to be reconstructed but also due to that the muons turn softer and the trigger efficiency thus is getting lower; see Fig.3. For the positive identification of muons, it is assumed that the efficiency is 100 % for muons with a total momentum above 2.5 GeV. It is assumed that no or only very loose particle identification is required on the charged hadrons. For the D+ reconstruction, it is assumed that only the D+ K−π+π+ final state is used. This is the easiest → decay mode to reconstruct and has a branching fraction of 9.0 % [52]. The final state with a of the D+ could also be considered, in an analysis similar to the B+ µ+µ−µ+ν [53] analysis carried out by LHCb. However, to estimate the → reconstruction efficiency of this five charged lepton final state with a would require a full detector level simulation which is beyond this paper. For the K∗0 case,

– 8 – Decay LHCb Upgrade I Upgrade II B0 µ+µ−µ+µ− 60 9 1.4 s → B0 µ+µ−µ+µ− 15 2.3 0.4 → B+ K+µ+µ−µ+µ− 37 5 0.9 → B0 K∗0µ+µ−µ+µ− 100 16 2.7 → B+ D+µ+µ−µ+µ− 1300 200 32 → Table 1. Expected branching fraction limits for the different decays under consideration in units of 10−11. It is assumed that the D+ meson is only reconstructed in the K−π+π+ final state. we consider the decay into K+π−, which has a branching ratio of 67 %. ∼ All overall efficiencies for a given final state are evaluated relative to the published analysis on the decays B0 µ+µ−µ+µ− [26]. As the trigger and main selection are (s) → the same for all these decays, this provides a robust normalisation method. Simulations are carried out using Pythia 8 [54] for the production of B mesons in pp collisions and EvtGen [55] for the subsequent decays. The decays are assumed to be of the type B (M)aa with a µ+µ− and with M a possible meson in the → → final state. The B meson decay is simulated with a flat phase space distribution. If the is unstable, it is decayed to stable particles using the default model in EvtGen and with branching fractions taken from the PDG [52]. A summary of the expected limits that can be set are given in Table1. Translated to the plane (gd , m ) for given values of g0, the limits from B0 23 V s → µ+µ−µ+µ− and B+ K+µ+µ−µ+µ− are compared with those from ∆M in Fig.4. → s Interestingly, we see that scales of several tens of TeV not yet probed by current ex- periments could be tested in the Upgrade II of the LHCb with our analysis.

4 Conclusions

We have considered scenarios involving new heavy and flavour-violating vectors V as well as light scalars a. We have shown that these particles give rise to rare B meson decays that are not yet probed. As the preferred mass of the scalar a lies inside the window vetoed by current LHCb searches, namely [950, 1090] MeV, even the simplest decay mode B0 µ+µ−µ+µ− is not fully probed. Other decay modes of interest (s) → are B+ K+(D+)µ+µ−µ+µ− and B0 K∗0µ+µ−µ+µ−. We have shown that the → → five-body final state can be as significant as the four-body. Relying on simulations, we have estimated the reach of the LHCb experiment for these processes in the current

– 9 – 50 50 0 + + 0 + + B µ µ−µ µ− B µ µ−µ µ− s → s → 40 + + + + 40 + + + + B K µ µ−µ µ− B K µ µ−µ µ− → → ∆Ms ∆Ms 30 30 [TeV] [TeV]

V 20 V 20 m m 10 10

g0 = 0.1 g0 = 0.5 0 0 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 d d g23 g23

50 50 0 + + 0 + + B µ µ−µ µ− B µ µ−µ µ− s → s → 40 + + + + 40 + + + + B K µ µ−µ µ− B K µ µ−µ µ− → → ∆Ms ∆Ms 30 30 [TeV] [TeV]

V 20 V 20 m m 10 10

g0 = 0.1 g0 = 0.5 0 0 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 d d g23 g23

d Figure 4. Upper left) Region in the plane (g23, mV ) that can be tested at the current run 0 of the LHCb (light) for g = 0.1 versus the area excluded by measurements of ∆Ms (dark). Upper right) Same as before but for g0 = 0.5. Bottom left) Same as before but for the LHCb Upgrade II and g0 = 0.1 Bottom right) Same as before but for g0 = 0.5. In all cases, the benchmark point defined in section 2.1 is shown with a for reference.

run and in Upgrades I and II. In the Upgrade II scenario we expect that branching fraction limits in the 10−11 region can be reached. Finally, we emphasize again that the decays into a meson and aa are the only sensi- ble probe of different effective operators in the SM effective-field theory extended with a scalar singlet. We therefore encourage the experimental collaborations to consider these processes in future analyses.

Acknowledgments. We would like to thank Martin Bauer and Jakub Scholtz for helpful discussions. MC is supported by the Royal Society under the Newton International Fellowship programme. MS was supported by the Humboldt Society during the finali- sation of parts of this work.

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