Determination of the quark coupling strength |V_ub| using baryonic decays

Abstract

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 Λ_b⁰ baryon to decay into the p final state relative to the 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.

Main

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. Although 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 . Semileptonic decays are used to minimize 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 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) meson to the final state , where the influence of the strong interaction on the decay, encompassed by a form factor, is predicted by non-perturbative techniques such as lattice QCD (LQCD; ref. 5) or QCD sum rules⁶. The world average from ref. 7 for this method, using the decays and , 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 and result in larger values of V_ub than the average given in ref. 7. The alternative method is to measure the differential decay rate in an inclusive way over all possible B meson decays containing the quark-level transition. This results in (ref. 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^15,16,17,18. 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_b⁰ (bud) baryon into the and 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, , has not been considered before as Λ_b⁰ baryons are not produced at an e⁺e^− B-factory; however, at the LHC, they constitute around 20% of the b-hadrons produced¹⁹. These measurements together with recent LQCD calculations²⁰ allow for the determination of |V_ub| ²/ |V_cb| ² according to

where 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 normalization to the decay cancels many experimental uncertainties, including the uncertainty on the total production rate of Λ_b⁰ baryons. At the LHC, the number of signal candidates is large, allowing the optimization of the event selection and the analysis approach to minimize systematic effects.

The LHCb detector^21,22 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 23, 24, 25, 26, 27, 28, 29.

Candidates of the signal modes are required to pass a trigger system³⁰ 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. The identification efficiency for these high-momentum muons is 98%.

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 protons above the momentum threshold to produce Cherenkov light. The pμ^− vertex fit is required to be of good quality, which reduces background from most of the decays, as the resulting ground state charmed hadrons have significant lifetime.

To reconstruct 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 signal to minimize 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, as illustrated in Fig. 1. To reduce this background, a multivariate machine learning algorithm (a boosted decision tree, BDT (refs 31, 32)) 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 events. The training sample of non-isolated tracks consists of the tracks from charged particles in the decay products X in a sample of simulated events. The BDT selection removes 90% of background with additional charged particles from the signal vertex, whereas it retains more than 80% of signal. The same isolation requirement is placed on both the and decay candidates, where the pion and kaon are ignored in the calculation of the BDT response for the case.

Figure 1: Diagram illustrating the topology for the (top) signal and (bottom) background decays.

The Λ_b⁰ baryon travels about 1 cm on average before decaying; its flight direction is indicated in the diagram. In the signal case, the only other particles present are typically reconstructed far away from the signal, which are shown as grey arrows. For the background from Λ_c⁺ decays, there are particles that are reconstructed in close proximity to the signal, which are indicated as dotted arrows.

The Λ_b⁰ mass is reconstructed using the so-called corrected mass³³, 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_b⁰ flight direction, where h represents either the proton or Λ_c⁺ candidate. The flight direction is measured using the PV and Λ_b⁰ vertex positions. The uncertainties on the PV and the Λ_b⁰ vertex are estimated for each candidate and propagated to the uncertainty on m_corr; the dominant contribution is from the uncertainty in the Λ_b⁰ vertex.

Candidates with an uncertainty of less than 100 MeV/c² on the corrected mass are selected for the 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 required to calculate |V_ub| are most precise in the kinematic region where q², the invariant mass squared of the muon and the neutrino in the decay, is high. The neutrino is not reconstructed, but q² can still be determined using the Λ_b⁰ flight direction and the Λ_b⁰ mass, but only up to a two-fold ambiguity. The correct solution has a resolution of about 1 GeV²/c⁴, whereas the wrong solution has a resolution of 4 GeV²/c⁴. To avoid influence on the measurement by the large uncertainty in form factors at low q², both solutions are required to exceed 15 GeV²/c⁴ for the decay and 7 GeV²/c⁴ for the decay. Simulation shows that only 2% of decays and 5% of decays with q² values below the cut values pass the selection requirements. The effect of this can be seen in Fig. 2, where the efficiency for the signal below 15 GeV²/c⁴ is reduced significantly if requirements are applied on both solutions. It is also possible that both solutions are imaginary owing to the limited detector resolution. Candidates of this type are rejected. The overall q² selection has an efficiency of 38% for decays and 39% for decays in their respective high-q² regions.

Figure 2: Illustrating the method used to reduce the number of selected events from the q² region where lattice QCD has high uncertainties.

The efficiency of simulated candidates as a function of q². For the case where one q² solution is required to be above 15 GeV²/c⁴ (marked by the vertical line), there is still significant efficiency for the signal below this value, whereas, when both solutions have this requirement, only a small amount of signal below 15 GeV²/c⁴ is selected.

The mass distributions of the signal candidates for the two decays are shown in Fig. 3. The signal yields are determined from separate χ² fits to the m_corr distributions of the and candidates. 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.

Figure 3: Corrected mass fit used for determining signal yields.

Fits are made to (top) and (bottom) candidates. The statistical uncertainties arising from the finite size of the simulation samples used to model the mass shapes are indicated by open boxes and the data are represented by the black points. The statistical uncertainty on the data points is smaller than the marker size used. The different signal and background components appear in the same order in the fits and the legends. There are no data above the nominal Λ_b⁰ mass owing to the removal of unphysical q² solutions.

For the fit to the m_corr distribution of the candidates, many sources of background are accounted for. The largest of these is the cross-feed from decays, where the Λ_c⁺ decays into a proton and other particles that are not reconstructed. 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 decays, whereas the background where only neutral particles come directly from the Λ_c⁺ decay is estimated by reconstructing 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_s⁰ 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 decays is constrained in a similar way to the Λ_c⁺ charged decay modes, with the normalization done relative to decays reconstructed in the data.

Any background with a Λ_c⁺ baryon may also arise from decays of the type , where Λ_c^∗+ represents the Λ_c(2,595)⁺ or Λ_c(2,625)⁺ resonances as well as non-resonant contributions. The proportions between the and the backgrounds are determined from the fit to the m_corr distribution and then used in the fit.

The decays , where the N^∗ baryon decays into a proton and other non-reconstructed particles, are very similar to the signal decay and have poorly known branching fractions. The N^∗ resonance represents any of the states N(1,440), N(1,520), N(1,535) or N(1,770). None of the 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 fractions³⁴. A 100% uncertainty is allowed for in the branching fractions of these decays.

Background where a pion or kaon is mis-identified as a proton originates from various sources and is measured 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/c².

For the yield, the reconstructed pK^−π⁺ mass is studied 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 the background is modelled as an exponential function. Within a selected signal region of 30 MeV/c² 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 candidates, as shown in Fig. 3, which is used to discriminate between and decays.

The and yields are 17,687 ± 733 and 34,255 ± 571, respectively. This is the first observation of the decay .

The branching fraction is measured relative to the 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_b⁰ kinematic properties for the selected and candidates. The ratio of efficiencies is 3.52 ± 0.20, with the sources of the uncertainty described below.

Systematic uncertainties associated with the measurement are summarized in Table 1. The largest uncertainty originates from the Λ_c⁺ → pK^−π⁺ branching fraction, which is taken from ref. 35. 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 decay. Another systematic uncertainty is assigned due to the limited knowledge of the momentum distribution for the Λ_c⁺ → pK^−π⁺ decay products. Uncertainties related to the background composition are included in the statistical uncertainty for the signal yield through the use of nuisance parameters in the fit. The exception to this is the uncertainty on the mass shapes due to the limited knowledge of the form factors and widths of each state, which is estimated by generating pseudoexperiments and assessing the impact on the signal yield.

Table 1 Summary of systematic uncertainties.

Smaller uncertainties are assigned for the following effects: the uncertainty in the Λ_b⁰ lifetime; differences in data and simulation in the isolation BDT response; differences in the relative efficiency and q² migration due to form factor uncertainties for both signal and normalization channels; corrections to the Λ_b⁰ kinematic properties; the disagreement in the q² migration between data and simulation; and the finite size of the PID calibration samples. The total fractional systematic uncertainty is where the individual uncertainties are added in quadrature. The small impact of the form factor 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 of to in the selected q² regions is

where the first uncertainty is statistical and the second is systematic. Using equation (1) with R_FF = 0.68 ± 0.07, computed in ref. 20 for the restricted q² regions, the measurement

is obtained. The first uncertainty arises from the experimental measurement and the 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 decays¹⁴, |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 normalization to |V_cb|. As the measurement of |V_ub|/|V_cb| already depends on LQCD calculations of the form factors it makes sense to normalize to the |V_cb| exclusive world average and not include the inclusive |V_cb| measurements. 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 q² region using |V_cb| and the form factor predictions²⁰, resulting in a measurement of , where the uncertainty is dominated by knowledge of the form factors at low q².

The determination of |V_ub| from the measured ratio of branching fractions depends on the size of a possible right-handed coupling³⁶. This can clearly be seen in Fig. 4, which shows the experimental constraints on the left-handed coupling, |V_ub^L|, and the fractional right-handed coupling added to the SM, ε_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 and decays, the spin of the proton is non-zero, allowing an axial-vector current, which gives a different sensitivity to ε_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.

Figure 4: Experimental constraints on the left-handed coupling, |V_ub^L| and the fractional right-handed coupling, ε_R.

Whereas the overlap of the 68% confidence level bands for the inclusive¹⁴ and exclusive⁷ world averages of past measurements suggested a right-handed coupling of significant magnitude, the inclusion of the LHCb |V_ub| measurement does not support this.

In summary, a measurement of the ratio of |V_ub| to |V_cb| is performed using the exclusive decay modes and . Using a previously measured value of |V_cb|, |V_ub| is determined precisely. The |V_ub| measurement is in agreement with the exclusively measured world average from ref. 7, but disagrees with the inclusive measurement¹⁴ 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.