Determination of the quark coupling strength $|V_{ub}|$ using baryonic decays

In the Standard Model of particle physics, the strength of the couplings of the $b$ quark to the $u$ and $c$ quarks, $|V_{ub}|$ and $|V_{cb}|$, are governed by the coupling of the quarks to the Higgs boson. Using data from the LHCb experiment at the Large Hadron Collider, the probability for the $\Lambda^0_b$ baryon to decay into the $p \mu^- \overline{\nu}_\mu$ final state relative to the $\Lambda^+_c \mu^- \overline{\nu}_\mu$ final state is measured. Combined with theoretical calculations of the strong interaction and a previously measured value of $|V_{cb}|$, the first $|V_{ub}|$ measurement to use a baryonic decay is performed. This measurement is consistent with previous determinations of $|V_{ub}|$ using $B$ meson decays to specific final states and confirms the existing incompatibility with those using an inclusive sample of final states.

In the Standard Model (SM) of particle physics, the decay of one quark to another by the emission of a virtual W boson is described by the 3×3 unitary Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2]. This matrix arises from the coupling of the quarks to the Higgs boson. While the SM does not predict the values of the four free parameters of the CKM matrix, the measurements of these parameters in different processes should be consistent with each other. If they are not, it is a sign of physics beyond the SM. In global fits combining all available measurements [3,4], the sensitivity of the overall consistency check is limited by the precision in the measurements of the magnitude and phase of the matrix element V ub , which describes the transition of a b quark to a u quark.
The magnitude of V ub can be measured via the semileptonic quark-level transition b → u − ν . Semileptonic decays are used to minimise the uncertainties arising from the interaction of the strong force, described by quantum chromodynamics (QCD), between the final-state quarks. For the measurement of the magnitude of V ub , as opposed to measurements of the phase, all decays of the b quark, and the equivalent b quark, can be considered together. There are two complementary methods to perform the measurement. From an experimental point of view, the simplest is to measure the branching fraction (probability to decay to a given final state) of a specific (exclusive) decay. An example is the decay of a B 0 (bd) meson to the final state π + − ν, where the influence of the strong interaction on the decay, encompassed by a B 0 → π + form factor, is predicted by non-perturbative techniques such as lattice QCD (LQCD) [5] or QCD sum rules [6]. The world average from Ref. [7] for this method, using the decays B 0 → π + − ν and B − → π 0 − ν, is |V ub | = (3.28 ± 0.29) × 10 −3 , where the most precise experimental inputs come from the BaBar [8,9] and Belle [10,11] experiments. The uncertainty is dominated by the LQCD calculations, which have recently been updated [12,13]. The alternative method is to measure the differential decay rate in an inclusive way over all possible B meson decays containing the b → u − ν quark level transition. This results in |V ub | = (4.41 ± 0.15 +0. 15 −0.17 ) × 10 −3 [14], where the first uncertainty arises from the experimental measurement and the second from theoretical calculations. The discrepancy between the exclusive and inclusive |V ub | determinations is approximately three standard deviations and has been a long-standing puzzle in flavour physics. Several explanations have been proposed, such as the presence of a right-handed (vector plus axial-vector) coupling as an extension of the SM beyond the left-handed (vector minus axial-vector) W coupling [15][16][17]. A similar discrepancy also exists between exclusive and inclusive measurements of |V cb | (the coupling of the b quark to the c quark) [14].
This article describes a measurement of the ratio of branching fractions of the Λ 0 b (bud) baryon into the p − ν and Λ + c − ν final states. This is performed using proton-proton collision data from the LHCb detector, corresponding to 2.0 fb −1 of integrated luminosity collected at a centre-of-mass energy of 8 TeV. The b → u transition, Λ 0 b → pµ − ν µ , has not been considered before as Λ 0 b baryons are not produced at an e + e − B-factory while, at the LHC, they constitute around 20% of the b-hadrons produced [18]. These measurements together with recent LQCD calculations [19] allow for the determination of |V ub | 2 /|V cb | 2 according to where B denotes the branching fraction and R FF is a ratio of the relevant form factors, calculated using LQCD. This is then converted into a measurement of |V ub | using the existing measurements of |V cb | obtained from exclusive decays. The normalisation to the Λ 0 b → Λ + c µ − ν µ decay cancels many experimental uncertainties, including the uncertainty on the total production rate of Λ 0 b baryons. At the LHC, the number of signal candidates is large, allowing the optimisation of the event selection and the analysis approach to minimise systematic effects.
The LHCb detector [20,21] is one of the four major detectors at the Large Hadron Collider. It is instrumented in a cone around the proton beam axis, covering the angles between 10 and 250 mrad, where most b-hadron decays produced in proton-proton collisions occur. The detector includes a high-precision tracking system with a dipole magnet, providing a measurement of momentum and impact parameter (IP), defined for charged particles as the minimum distance of a track to a primary proton-proton interaction vertex (PV). Different types of charged particles are distinguished using information from two ring-imaging Cherenkov detectors, a calorimeter and a muon system. Simulated samples of specific signal and background decay modes of b hadrons are used at many stages throughout the analysis. These simulated events model the experimental conditions in full detail, including the proton-proton collision, the decay of the particles, and the response of the detector. The software used is described in Refs. [22][23][24][25][26][27].
Candidates of the signal modes are required to pass a trigger system [28] which reduces in real-time the rate of recorded collisions (events) from the 40 MHz read-out clock of the LHC to around 4 kHz. For this analysis, the trigger requires a muon with a large momentum transverse to the beam axis that at the same time forms a good vertex with another track in the event. This vertex should be displaced from the PVs in the event.
In the selection of the final states, stringent particle identification (PID) requirements are applied to the proton. These criteria are accompanied by a requirement that its momentum is greater than 15 GeV/c as the PID performance is most effective for high-momentum protons. The pµ − vertex fit is required to be of good quality, which reduces background from most of the b → cµ − ν µ decays as the resulting ground state charmed hadrons have significant lifetime.
To reconstruct Λ 0 b → (Λ + c → pK − π + )µ − ν µ candidates, two additional tracks, positively identified as a pion and kaon, are combined with the proton to form a Λ + c → pK − π + candidate. These are reconstructed from the same pµ − vertex as the Λ 0 b → pµ − ν µ signal to minimise systematic uncertainties. As the lifetime of the Λ + c is short compared to other weakly decaying charm hadrons, the requirement has an acceptable efficiency.
There is a large background from b-hadron decays with additional charged tracks in the decay products. To reduce this background, a multivariate machine learning algorithm (a boosted decision tree, BDT [29,30]) is employed to determine the compatibility of each track from a charged particle in the event to originate from the same vertex as the signal candidate. This isolation BDT includes variables such as the change in vertex quality if the track is combined with the signal vertex, as well as kinematic and IP information of the track that is tested. For the BDT, the training sample of well isolated tracks consists of all tracks apart from the signal decay products in a sample of simulated Λ 0 b → pµ − ν µ events. The training sample of non-isolated tracks consists of the tracks from charged particles in the decay products X in a sample of The BDT selection removes 90% of background with additional charged particles from the signal vertex while it retains more than 80% of signal. The same isolation requirement is placed on both the The Λ 0 b mass is reconstructed using the so-called corrected mass [31], defined as where m hµ is the visible mass of the hµ pair and p ⊥ is the momentum of the hµ pair transverse to the Λ 0 b flight direction, where h represents either the proton or Λ + c candidate. The flight direction is measured using the PV and Λ 0 b vertex positions. The uncertainties on the PV and the Λ 0 b vertex are estimated for each candidate and propagated to the uncertainty on m corr ; the dominant contribution is from the uncertainty in the Λ 0 b vertex. Candidates with an uncertainty of less than 100 MeV/c 2 on the corrected mass are used for the Λ 0 b → pµ − ν µ decay. This selects only 23% of the signal; however, the separation between signal and background for these candidates is significantly improved and the selection thus reduces the dependence on background modelling.
The LQCD form-factors that are used in the calculation of |V ub | are most precise in the kinematic region where q 2 , the invariant mass squared of the muon and the neutrino in the decay, is high. The neutrino is not reconstructed, but q 2 can still be determined using the Λ 0 b flight direction and the Λ 0 b mass, but only up to a two-fold ambiguity. The correct solution has a resolution of about 1 GeV 2 /c 4 , while the wrong solution has a resolution of 4 GeV/c 2 . To avoid influence on the measurement by the large uncertainty in form factors at low q 2 , both solutions are required to exceed 15 GeV 2 /c 4 for the Λ 0 b → pµ − ν µ decay and 7 GeV 2 /c 4 for the  are imaginary due to the limited detector resolution. Candidates of this type are rejected. The overall q 2 selection has an efficiency of 38% for Λ 0 b → pµ − ν µ and 39% for Λ 0 b → Λ + c µ − ν µ decays in their respective high-q 2 regions.
The mass distributions of the signal candidates for the two decays are shown in Fig. 2. The signal yields are determined from a χ 2 fit to the m corr distributions of the The shapes of the signal and background components are modelled using simulation, where the uncertainties coming from the finite size of the simulated samples are propagated in the fits. The yields of all background components are allowed to vary within uncertainties obtained as described below.
For the fit to the m corr distribution of the Λ 0 b → pµ − ν µ candidates, many sources of background are accounted for. The largest of these is the cross-feed from Λ 0 b → Λ + c µ − ν µ decays, where the Λ + c decays into a proton and other particles that are not recon-structed. The amount of background arising from these decay modes is estimated by fully reconstructing two Λ + c decays in the data. The background where the additional particles include charged particles originating directly from the Λ + c decay is estimated by reconstructing Λ 0 b → (Λ + c → pK − π + )µ − ν µ decays, whereas the background where only neutral particles come directly from the Λ + c decay is estimated by reconstructing Λ 0 b → (Λ + c → pK 0 S )µ − ν µ decays. These two background categories are separated because the isolation BDT significantly reduces the charged component but has no effect on the neutral case. For the rest of the Λ + c decay modes, the relative branching fraction between the decay and either the Λ + c → pK − π + or Λ + c → pK 0 S decay modes, as appropriate, is taken from Ref. [14]. For some neutral decay modes, where only the corresponding mode with charged decay particles is measured, assumptions based on isospin symmetry are used. In these decays, an uncertainty corresponding to 100% of the branching fraction is allowed for in the fit. Background from Λ 0 b → D 0 pµ − ν µ decays is controlled in a similar way to the Λ + c charged decay modes, with the normalisation done relative to Λ 0 b → D 0 (→ K − π + )pµ − ν µ decays reconstructed in the data.
Any background with a Λ + c baryon may also arise from decays of the type Λ 0 b → (Λ * + c → Λ + c ππ)µ − ν µ , where Λ * + c represents the Λ c (2595) + or Λ c (2625) + resonances as well as non-resonant contributions. The proportions between the The decays Λ 0 b → N * µ − ν µ , where the N * baryon decays into a proton and other nonreconstructed particles, are very similar to the signal decay and have poorly known branching fractions. The N * resonance represents any of the states N (1440), N (1520), N (1535) or N (1770). None of the Λ 0 b → N * µ − ν µ decays have been observed and the m corr shape of these decays is obtained using simulation samples generated according to the quark-model prediction of the form factors and branching fractions [32]. A 100% uncertainty is allowed for in the branching fractions of these decays.
Background where a pion or kaon is misidentified as a proton originates from various sources and is controlled by using a special data set where no PID is applied to the proton candidate. Finally, an estimate of combinatorial background, where the proton and muon originate from different decays, is obtained from a data set where the proton and muon have the same charge. The amount and shape of this background are in good agreement between the same-sign and opposite-sign pµ samples for corrected masses above 6 GeV/c 2 .
For the Λ 0 b → (Λ + c → pK − π + )µ − ν µ yield, the reconstructed pK − π + mass is used to determine the level of combinatorial background. The Λ + c signal shape is modelled using a Gaussian function with an asymmetric power-law tail, and an exponential function is used for the background. Within a selected signal region of 30 MeV/c 2 from the known Λ + c mass the combinatorial background is 2% of the signal yield. Subsequently, a fit is performed to the m corr distribution for Λ 0 b → (Λ + c → pK − π + )µ − ν µ candidates, as shown in Fig. 2, which is used to discriminate between yields are 17, 687 ± 733 and 34, 255 ± 571, respectively. This is the first observation of the decay Λ 0 b → pµ − ν µ . The Λ 0 b → pµ − ν µ branching fraction is measured relative to the Λ 0 b → (Λ + c → pK − π + )µ − ν µ branching fraction. The relative efficiencies for reconstruction, trigger and final event selection are obtained from simulated events, with several corrections applied to improve the agreement between the data and the simulation. These correct for differences between data and simulation in the detector response and differences in the Λ 0 b kinematic properties for the selected The ratio of efficiencies is 3.52±0.20, with the sources of the uncertainty described below.
Systematic uncertainties associated with the measurement are summarised in Table 1. The largest uncertainty originates from the Λ + c → pK − π + branching fraction, which is taken from Ref. [33]. This is followed by the uncertainty on the trigger response, which is due to the statistical uncertainty of the calibration sample. Other contributions come from the tracking efficiency, which is due to possible differences between the data and simulation in the probability of interactions with the material of the detector for the kaon and pion in the Another systematic uncertainty is assigned due to the limited knowledge of the momentum distribution for the Λ + c → pK − π + decay products. Uncertainties on the Λ 0 b → N * µ − ν µ mass shapes due to the limited knowledge of the form factors and widths of each state are estimated by generating pseudoexperiments and assessing the impact on the signal yield.
Smaller uncertainties are assigned for the following effects: the uncertainty in the Λ 0 b lifetime; differences in data and simulation in the isolation BDT response; differences in the relative efficiency and q 2 migration due to form factor uncertainties for both signal and normalisation channels; corrections to the Λ 0 b kinematic properties; the disagreement in the q 2 migration between data and simulation; and the finite size of the PID calibration samples. The total fractional systematic uncertainty is +7.8 −8.2 %, where the individual uncertainties are added in quadrature. The small impact of the form factor Table 1: Summary of systematic uncertainties. The table shows the relative systematic uncertainty on the ratio of the Λ 0 b → pµ − ν µ and Λ 0 b → Λ + c µ − ν µ branching fractions broken into its individual contributions. The total is obtained by adding them in quadrature. Uncertainties on the background levels are not listed here as they are incorporated into the fits.

Source
Relative uncertainty (%) uncertainties means that the measured ratio of branching fractions can safely be considered independent of the theoretical input at the current level of precision.
From the ratio of yields and their determined efficiencies, the ratio of branching fractions be- where the first uncertainty is statistical and the second is systematic. Using Eq. 1 with R FF = 0.68 ± 0.07, computed in Ref. [19] for the restricted q 2 regions, the measurement |V ub |/|V cb | = 0.083 ± 0.004 ± 0.004 is obtained. The first uncertainty arises from the experimental measurement and second is due to the uncertainty in the LQCD prediction. Finally, using the world average |V cb | = (39.5 ± 0.8) × 10 −3 measured using exclusive decays [14], |V ub | is measured as where the first uncertainty is due to the experimental measurement, the second arises from the uncertainty in the LQCD prediction and the third from the normalisation to |V cb |. The experimental uncertainty is dominated by systematic effects, most of which will be improved with additional data by a reduction of the statistical uncertainty of the control samples.
The measured ratio of branching fractions can be extrapolated to the full q 2 region using |V cb | and the form factor predictions [19], resulting in a measurement of B(Λ 0 b → pµ − ν µ ) = (3.9 ± 0.8) × 10 −4 , where the uncertainty is dominated by knowledge of the form factors at low q 2 .
The determination of |V ub | from the measured ratio of branching fractions depends on the size of a possible right-handed coupling [34]. This can clearly be seen in Fig. 3, which shows the experimental constraints on the left-handed coupling, |V L ub | and the fractional right-handed coupling added to the SM, R for different measurements. The LHCb result presented here is compared to the world averages of the inclusive and exclusive measurements. Unlike the case for the pion in B 0 → π + − ν and B − → π 0 − ν decays, the spin of the proton is non-zero, allowing an axial-vector current, which gives a different sensitivity to R . The overlap of the bands from the previous measurements suggested a significant right-handed coupling but the inclusion of the LHCb |V ub | measurement does not support that.
In summary, the most precise measurement to date of |V ub | is reported using the exclusive decay mode Λ 0 b → pµ − ν µ . The measurement is in agreement with the exclusively measured world average from Ref. [7], but disagrees with the inclusive measurement [14] at a significance level of 3.5 standard deviations. The measurement will have a significant impact on the global fits to the parameters of the CKM matrix.

Acknowledgements
We thank Stefan Meinel for a productive collaboration regarding form factor predictions of the Λ 0 b → pµ − ν µ and Λ 0 b → Λ + c µ − ν µ decays, Winston Roberts for discussions regarding the Λ 0 b → N * µ − ν µ decays and Florian Bernlochner for help in understanding the impact of right-handed currents. We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq,