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Observation of the rare Bs0 →µ+µ− decay from the combined analysis of CMS and LHCb data

Abstract

The standard model of particle physics describes the fundamental particles and their interactions via the strong, electromagnetic and weak forces. It provides precise predictions for measurable quantities that can be tested experimentally. The probabilities, or branching fractions, of the strange B meson () and the B0 meson decaying into two oppositely charged muons (μ+ and μ) are especially interesting because of their sensitivity to theories that extend the standard model. The standard model predicts that the and decays are very rare, with about four of the former occurring for every billion mesons produced, and one of the latter occurring for every ten billion B0 mesons1. A difference in the observed branching fractions with respect to the predictions of the standard model would provide a direction in which the standard model should be extended. Before the Large Hadron Collider (LHC) at CERN2 started operating, no evidence for either decay mode had been found. Upper limits on the branching fractions were an order of magnitude above the standard model predictions. The CMS (Compact Muon Solenoid) and LHCb (Large Hadron Collider beauty) collaborations have performed a joint analysis of the data from proton–proton collisions that they collected in 2011 at a centre-of-mass energy of seven teraelectronvolts and in 2012 at eight teraelectronvolts. Here we report the first observation of the µ+µ decay, with a statistical significance exceeding six standard deviations, and the best measurement so far of its branching fraction. Furthermore, we obtained evidence for the µ+µ decay with a statistical significance of three standard deviations. Both measurements are statistically compatible with standard model predictions and allow stringent constraints to be placed on theories beyond the standard model. The LHC experiments will resume taking data in 2015, recording proton–proton collisions at a centre-of-mass energy of 13 teraelectronvolts, which will approximately double the production rates of and B0 mesons and lead to further improvements in the precision of these crucial tests of the standard model.

Main

Experimental particle physicists have been testing the predictions of the standard model of particle physics (SM) with increasing precision since the 1970s. Theoretical developments have kept pace by improving the accuracy of the SM predictions as the experimental results gained in precision. In the course of the past few decades, the SM has passed critical tests derived from experiment, but it does not address some profound questions about the nature of the Universe. For example, the existence of dark matter, which has been confirmed by cosmological data3, is not accommodated by the SM. It also fails to explain the origin of the asymmetry between matter and antimatter, which after the Big Bang led to the survival of the tiny amount of matter currently present in the Universe3,4. Many theories have been proposed to modify the SM to provide solutions to these open questions.

The and B0 mesons are unstable particles that decay via the weak interaction. The measurement of the branching fractions of the very rare decays of these mesons into a dimuon (µ+µ) final state is especially interesting.

At the elementary level, the weak force is composed of a ‘charged current’ and a ‘neutral current’ mediated by the W± and Z0 bosons, respectively. An example of the charged current is the decay of the π+ meson, which consists of an up (u) quark of electrical charge +2/3 of the charge of the proton and a down (d) antiquark of charge +1/3. A pictorial representation of this process, known as a Feynman diagram, is shown in Fig. 1a. The u and d quarks are ‘first generation’ or lowest mass quarks. Whenever a decay mode is specified in this Letter, the charge conjugate mode is implied.

The B+ meson is similar to the π+, except that the light d antiquark is replaced by the heavy ‘third generation’ (highest mass quarks) beauty (b) antiquark, which has a charge of +1/3 and a mass of 5 GeV/c2 (about five times the mass of a proton). The decay B+µ+ν, represented in Fig. 1b, is allowed but highly suppressed because of angular momentum considerations (helicity suppression) and because it involves transitions between quarks of different generations (CKM suppression), specifically the third and first generations of quarks. All b hadrons, including the B+, and B0 mesons, decay predominantly via the transition of the b antiquark to a ‘second generation’ (intermediate mass quarks) charm (c) antiquark, which is less CKM suppressed, into final states with charmed hadrons. Many allowed decay modes, which typically involve charmed hadrons and other particles, have angular momentum configurations that are not helicity suppressed.

The neutral meson is similar to the B+ except that the u quark is replaced by a second generation strange (s) quark of charge −1/3. The decay of the meson to two muons, shown in Fig. 1c, is forbidden at the elementary level because the Z0 cannot couple directly to quarks of different flavours, that is, there are no direct ‘flavour changing neutral currents’. However, it is possible to respect this rule and still have this decay occur through ‘higher order’ transitions such as those shown in Fig. 1d and e. These are highly suppressed because each additional interaction vertex reduces their probability of occurring significantly. They are also helicity and CKM suppressed. Consequently, the branching fraction for the decay is expected to be very small compared to the dominant b antiquark to c antiquark transitions. The corresponding decay of the B0 meson, where a d quark replaces the s quark, is even more CKM suppressed because it requires a jump across two quark generations rather than just one.

The branching fractions, , of these two decays, accounting for higher-order electromagnetic and strong interaction effects, and using lattice quantum chromodynamics to compute the and B0 meson decay constants5,6,7, are reliably calculated1 in the SM. Their values are and .

Many theories that seek to go beyond the standard model (BSM) include new phenomena and particles8,9, such as in the diagrams shown in Fig. 1f and g, that can considerably modify the SM branching fractions. In particular, theories with additional Higgs bosons10,11 predict possible enhancements to the branching fractions. A significant deviation of either of the two branching fraction measurements from the SM predictions would give insight on how the SM should be extended. Alternatively, a measurement compatible with the SM could provide strong constraints on BSM theories.

The ratio of the branching fractions of the two decay modes provides powerful discrimination among BSM theories12. It is predicted in the SM (refs 1, 13 (updates available at http://itpwiki.unibe.ch/), 14, 15 (updated results and plots available at http://www.slac.stanford.edu/xorg/hfag/)) to be .Notably, BSM theories with the property of minimal flavour violation16 predict the same value as the SM for this ratio.

The first evidence for the decay was presented by the LHCb collaboration in 201217. Both CMS and LHCb later published results from all data collected in proton–proton collisions at centre-of-mass energies of 7 TeV in 2011 and 8 TeV in 2012. The measurements had comparable precision and were in good agreement18,19, although neither of the individual results had sufficient precision to constitute the first definitive observation of the decay to two muons.

In this Letter, the two sets of data are combined and analysed simultaneously to exploit fully the statistical power of the data and to account for the main correlations between them. The data correspond to total integrated luminosities of 25 fb−1 and 3 fb−1 for the CMS and LHCb experiments, respectively, equivalent to a total of approximately 1012 and B0 mesons produced in the two experiments together. Assuming the branching fractions given by the SM and accounting for the detection efficiencies, the predicted numbers of decays to be observed in the two experiments together are about 100 for and 10 for B0µ+µ.

The CMS20 and LHCb21 detectors are designed to measure SM phenomena with high precision and search for possible deviations. The two collaborations use different and complementary strategies. In addition to performing a broad range of precision tests of the SM and studying the newly-discovered Higgs boson22,23, CMS is designed to search for and study new particles with masses from about 100 GeV/c2 to a few TeV/c2. Since many of these new particles would be able to decay into b quarks and many of the SM measurements also involve b quarks, the detection of b-hadron decays was a key element in the design of CMS. The LHCb collaboration has optimized its detector to study matter–antimatter asymmetries and rare decays of particles containing b quarks, aiming to detect deviations from precise SM predictions that would indicate BSM effects. These different approaches, reflected in the design of the detectors, lead to instrumentation of complementary angular regions with respect to the LHC beams, to operation at different proton–proton collision rates, and to selection of b quark events with different efficiency (for experimental details, see Methods). In general, CMS operates at a higher instantaneous luminosity than LHCb but has a lower efficiency for reconstructing low-mass particles, resulting in a similar sensitivity to LHCb for B0 or (denoted hereafter by ) mesons decaying into two muons.

Muons do not have strong nuclear interactions and are too massive to emit a substantial fraction of their energy by electromagnetic radiation. This gives them the unique ability to penetrate dense materials, such as steel, and register signals in detectors embedded deep within them. Both experiments use this characteristic to identify muons.

The experiments follow similar data analysis strategies. Decays compatible with (candidate decays) are found by combining the reconstructed trajectories (tracks) of oppositely charged particles identified as muons. The separation between genuine decays and random combinations of two muons (combinatorial background), most often from semi-leptonic decays of two different b hadrons, is achieved using the dimuon invariant mass, , and the established characteristics of -meson decays. For example, because of their lifetimes of about 1.5 ps and their production at the LHC with momenta between a few GeV/c and 100 GeV/c, mesons travel up to a few centimetres before they decay. Therefore, the ‘decay vertex’, from which the muons originate, is required to be displaced with respect to the ‘production vertex’, the point where the two protons collide. Furthermore, the negative of the candidate’s momentum vector is required to point back to the production vertex.

These criteria, amongst others that have some ability to distinguish known signal events from background events, are combined into boosted decision trees (BDTs)24,25,26. A BDT is an ensemble of decision trees each placing different selection requirements on the individual variables to achieve the best discrimination between ‘signal-like’ and ‘background-like’ events. Both experiments evaluated many variables for their discriminating power and each chose the best set of about ten to be used in its respective BDT. These include variables related to the quality of the reconstructed tracks of the muons; kinematic variables such as transverse momentum (with respect to the beam axis) of the individual muons and of the candidate; variables related to the decay vertex topology and fit quality, such as candidate decay length; and isolation variables, which measure the activity in terms of other particles in the vicinity of the two muons or their displaced vertex. A BDT must be ‘trained’ on collections of known background and signal events to generate the selection requirements on the variables and the weights for each tree. In the case of CMS, the background events used in the training are taken from intervals of dimuon mass above and below the signal region in data, while simulated events are used for the signal. The data are divided into disjoint sub-samples and the BDT trained on one sub-sample is applied to a different sub-sample to avoid any bias. LHCb uses simulated events for background and signal in the training of its BDT. After training, the relevant BDT is applied to each event in the data, returning a single value for the event, with high values being more signal-like. To avoid possible biases, both experiments kept the small mass interval that includes both the and B0 signals blind until all selection criteria were established.

In addition to the combinatorial background, specific b-hadron decays, such as B0πµ+ν where the neutrino cannot be detected and the charged pion is misidentified as a muon, or B0π0µ+µ, where the neutral pion in the decay is not reconstructed, can mimic the dimuon decay of the mesons. The invariant mass of the reconstructed dimuon candidate for these processes (semi-leptonic background) is usually smaller than the mass of the or B0 meson because the neutrino or another particle is not detected. There is also a background component from hadronic two-body decays (peaking background) as B0K+π, when both hadrons from the decay are misidentified as muons. These misidentified decays can produce peaks in the dimuon invariant-mass spectrum near the expected signal, especially for the B0µ+µ decay. Particle identification algorithms are used to minimize the probability that pions and kaons are misidentified as muons, and thus suppress these background sources. Excellent mass resolution is mandatory for distinguishing between B0 and mesons with a mass difference of about 87 MeV/c2 and for separating them from backgrounds. The mass resolution for decays in CMS ranges from 32 to 75 MeV/c2, depending on the direction of the muons relative to the beam axis, while LHCb achieves a uniform mass resolution of about 25 MeV/c2.

The CMS and LHCb data are combined by fitting a common value for each branching fraction to the data from both experiments. The branching fractions are determined from the observed numbers, efficiency-corrected, of mesons that decay into two muons and the total numbers of mesons produced. Both experiments derive the latter from the number of observed B+J/ψK+ decays, whose branching fraction has been precisely measured elsewhere14. Assuming equal rates for B+ and B0 production, this gives the normalization for B0µ+µ. To derive the number of mesons from this B+ decay mode, the ratio of b quarks that form (hadronize into) B+ mesons to those that form mesons is also needed. Measurements of this ratio27,28, for which there is additional discussion in Methods, and of the branching fraction (B+J/ψ K+) are used to normalize both sets of data and are constrained within Gaussian uncertainties in the fit. The use of these two results by both CMS and LHCb is the only significant source of correlation between their individual branching fraction measurements. The combined fit takes advantage of the larger data sample to increase the precision while properly accounting for the correlation.

In the simultaneous fit to both the CMS and LHCb data, the branching fractions of the two signal channels are common parameters of interest and are free to vary. Other parameters in the fit are considered as nuisance parameters. Those for which additional knowledge is available are constrained to be near their estimated values by using Gaussian penalties with their estimated uncertainties while the others are free to float in the fit. The ratio of the hadronization probability into B+ and mesons and the branching fraction of the normalization channel B+J/ψK+ are common, constrained parameters. Candidate decays are categorized according to whether they were detected in CMS or LHCb and to the value of the relevant BDT discriminant. In the case of CMS, they are further categorized according to the data-taking period, and, because of the large variation in mass resolution with angle, whether the muons are both produced at large angles relative to the proton beams (central-region) or at least one muon is emitted at small angle relative to the beams (forward-region). An unbinned extended maximum likelihood fit to the dimuon invariant-mass distribution, in a region of about ±500 MeV/c2 around the mass, is performed simultaneously in all categories (12 categories from CMS and eight from LHCb). Likelihood contours in the plane of the parameters of interest, (B0µ+µ) versus (), are obtained by constructing the test statistic 2ΔlnL from the difference in log-likelihood (lnL) values between fits with fixed values for the parameters of interest and the nominal fit. For each of the two branching fractions, a one-dimensional profile likelihood scan is likewise obtained by fixing only the single parameter of interest and allowing the other to vary during the fits. Additional fits are performed where the parameters under consideration are the ratio of the branching fractions relative to their SM predictions, , or the ratio of the two branching fractions.

The combined fit result is shown for all 20 categories in Extended Data Fig. 1. To represent the result of the fit in a single dimuon invariant-mass spectrum, the mass distributions of all categories, weighted according to values of S/(S + B), where S is the expected number of signals and B is the number of background events under the peak in that category, are added together and shown in Fig. 2. The result of the simultaneous fit is overlaid. An alternative representation of the fit to the dimuon invariant-mass distribution for the six categories with the highest S/(S + B) value for CMS and LHCb, as well as displays of events with high probability to be genuine signal decays, are shown in Extended Data Figs 2, 3, 4.

The combined fit leads to the measurements where the uncertainties include both statistical and systematic sources, the latter contributing 35% and 18% of the total uncertainty for the and B0 signals, respectively. Using Wilks’ theorem29, the statistical significance in unit of standard deviations, σ, is computed to be 6.2 for the decay mode and 3.2 for the B0µ+µ mode. For each signal the null hypothesis that is used to compute the significance includes all background components predicted by the SM as well as the other signal, whose branching fraction is allowed to vary freely. The median expected significances assuming the SM branching fractions are 7.4σ and 0.8σ for the and B0 modes, respectively. Likelihood contours for (B0µ+µ) versus are shown in Fig. 3. One-dimensional likelihood scans for both decay modes are displayed in the same figure. In addition to the likelihood scan, the statistical significance and confidence intervals for the B0 branching fraction are determined using simulated experiments. This determination yields a significance of 3.0σ for a B0 signal with respect to the same null hypothesis described above. Following the Feldman–Cousins30 procedure, ±1σ and ±2σ confidence intervals for (B0µ+µ) of [2.5, 5.6] × 10−10 and [1.4, 7.4] × 10−10 are obtained, respectively (see Extended Data Fig. 5).

The fit for the ratios of the branching fractions relative to their SM predictions yields . Associated likelihood contours and one-dimensional likelihood scans are shown in Extended Data Fig. 6. The measurements are compatible with the SM branching fractions of the and B0µ+µ decays at the 1.2σ and 2.2σ level, respectively, when computed from the one-dimensional hypothesis tests. Finally, the fit for the ratio of branching fractions yields which is compatible with the SM at the 2.3σ level. The one-dimensional likelihood scan for this parameter is shown in Fig. 4.

The combined analysis of data from CMS and LHCb, taking advantage of their full statistical power, establishes conclusively the existence of the decay and provides an improved measurement of its branching fraction. This concludes a search that started more than three decades ago (see Extended Data Fig. 7), and initiates a phase of precision measurements of the properties of this decay. It also produces three standard deviation evidence for the B0µ+µ decay. The measured branching fractions of both decays are compatible with SM predictions. This is the first time that the CMS and LHCb collaborations have performed a combined analysis of sets of their data in order to obtain a statistically significant observation.

Methods

Experimental setup

At the Large Hadron Collider (LHC), two counter-rotating beams of protons, contained and guided by superconducting magnets spaced around a 27 km circular tunnel, located approximately 100 m underground near Geneva, Switzerland, are brought into collision at four interaction points (IPs). The study presented in this Letter uses data collected at energies of 3.5 TeV per beam in 2011 and 4 TeV per beam in 2012 by the CMS and LHCb experiments located at two of these IPs.

The CMS and LHCb detectors are both designed to look for phenomena beyond the SM (BSM), but using complementary strategies. The CMS detector20, shown in Extended Data Fig. 3, is optimized to search for yet unknown heavy particles, with masses ranging from 100 GeV/c2 to a few TeV/c2, which, if observed, would be a direct manifestation of BSM phenomena. Since many of the hypothesized new particles can decay into particles containing b quarks or into muons, CMS is able to detect efficiently and study B0 (5,280 MeV/c2) and (5,367 MeV/c2) mesons decaying to two muons even though it is designed to search for particles with much larger masses. The CMS detector covers a very large range of angles and momenta to reconstruct high-mass states efficiently. To that extent, it employs a 13 m long, 6 m diameter superconducting solenoid magnet, operated at a field of 3.8 T, centred on the IP with its axis along the beam direction and covering both hemispheres. A series of silicon tracking layers, consisting of silicon pixel detectors near the beam and silicon strips farther out, organized in concentric cylinders around the beam, extending to a radius of 1.1 m and terminated on each end by planar detectors (disks) perpendicular to the beam, measures the momentum, angles, and position of charged particles emerging from the collisions. Tracking coverage starts from the direction perpendicular to the beam and extends to within 220 mrad from it on both sides of the IP. The inner three cylinders and disks extending from 4.3 to 10.7 cm in radius transverse to the beam are arrays of 100 × 150 µm2 silicon pixels, which can distinguish the displacement of the b-hadron decays from the primary vertex of the collision. The silicon strips, covering radii from 25 cm to approximately 110 cm, have pitches ranging from 80 to 183 µm. The impact parameter is measured with a precision of 10 µm for transverse momenta of 100 GeV/c and 20 µm for 10 GeV/c. The momentum resolution, provided mainly by the silicon strips, changes with the angle relative to the beam direction, resulting in a mass resolution for decays that varies from 32 MeV/c2 for mesons produced perpendicularly to the proton beams to 75 MeV/c2 for those produced at small angles relative to the beam direction. After the tracking system, at a greater distance from the IP, there is a calorimeter that stops (absorbs) all particles except muons and measures their energies. The calorimeter consists of an electromagnetic section followed by a hadronic section. Muons are identified by their ability to penetrate the calorimeter and the steel return yoke of the solenoid magnet and to produce signals in gas-ionization particle detectors located in compartments within the steel yoke. The CMS detector has no capability to discriminate between charged hadron species, pions, kaons, or protons, that is effective at the typical particle momenta in this analysis.

The primary commitment of the LHCb collaboration is the study of particle–antiparticle asymmetries and of rare decays of particles containing b and c quarks. LHCb aims at detecting BSM particles indirectly by measuring their effect on b-hadron properties for which precise SM predictions exist. The production cross section of b hadrons at the LHC is particularly large at small angles relative to the colliding beams. The small-angle region also provides advantages for the detection and reconstruction of a wide range of their decays. The LHCb experiment21, shown in Extended Data Fig. 4, instruments the angular interval from 10 to 300 mrad with respect to the beam direction on one side of the interaction region. Its detectors are designed to reconstruct efficiently a wide range of b-hadron decays, resulting in charged pions and kaons, protons, muons, electrons, and photons in the final state. The detector includes a high-precision tracking system consisting of a silicon strip vertex detector, a large-area silicon strip detector located upstream of a dipole magnet characterized by a field integral of 4 T m, and three stations of silicon strip detectors and straw drift tubes downstream of the magnet. The vertex detector has sufficient spatial resolution to distinguish the slight displacement of the weakly decaying b hadron from the primary production vertex where the two protons collided and produced it. The tracking detectors upstream and downstream of the dipole magnet measure the momenta of charged particles. The combined tracking system provides a momentum measurement with an uncertainty that varies from 0.4% at 5 GeV/c to 0.6% at 100 GeV/c. This results in an invariant-mass resolution of 25 MeV/c2 for mesons decaying to two muons that is nearly independent of the angle with respect to the beam. The impact parameter resolution is smaller than 20 µm for particle tracks with large transverse momentum. Different types of charged hadrons are distinguished by information from two ring-imaging Cherenkov detectors. Photon, electron, and hadron candidates are identified by calorimeters. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers.

Neither CMS nor LHCb records all the interactions occurring at its IP because the data storage and analysis costs would be prohibitive. Since most of the interactions are reasonably well characterized (and can be further studied by recording only a small sample of them) specific event filters (known as triggers) select the rare processes that are of interest to the experiments. Both CMS and LHCb implement triggers that specifically select events containing two muons. The triggers of both experiments have a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, consisting of a large computing cluster that uses all the information from the detector, including the tracking, to make the final selection of events to be recorded for subsequent analysis. Since CMS is designed to look for much heavier objects than mesons, it selects events that contain muons with higher transverse momenta than those selected by LHCb. This eliminates many of the decays while permitting CMS to run at a higher proton–proton collision rate to look for the more rare massive particles. Thus CMS runs at higher collision rates but with lower efficiency than LHCb for mesons decaying to two muons. The overall sensitivity to these decays turns out to be similar in the two experiments.

CMS and LHCb are not the only collaborations to have searched for and B0µ+ µ decays. Over three decades, a total of eleven collaborations have taken part in this search14, as illustrated by Extended Data Fig. 7. This plot gathers the results from CLEO31,32,33,34,35, ARGUS36, UA137,38, CDF39,40,41,42,43,44, L345, DØ46,47,48,49,50, Belle51, Babar52,53, LHCb17,54,55,56,57 CMS18,58,59 and ATLAS60.

Analysis description

The analysis techniques used to obtain the results presented in this Letter are very similar to those used to obtain the individual result in each collaboration, described in more detail in refs 18, 19. Here only the main analysis steps are reviewed and the changes used in the combined analysis are highlighted. Data samples for this analysis were collected by the two experiments in proton–proton collisions at a centre-of-mass energy of 7 and 8 TeV during 2011 and 2012, respectively. These samples correspond to a total integrated luminosity of 25 and 3 fb−1 for the CMS and LHCb experiments, respectively, and represent their complete data sets from the first running period of the LHC.

The trigger criteria were slightly different between the two experiments. The large majority of events were triggered by requirements on one or both muons of the signal decay: the LHCb detector triggered on muons with transverse momentum pT > 1.5 GeV/c while the CMS detector, because of its geometry and higher instantaneous luminosity, triggered on two muons with pT > 4(3) GeV/c, for the leading (sub-leading) muon.

The data analysis procedures in the two experiments follow similar strategies. Pairs of high-quality oppositely charged particle tracks that have one of the expected patterns of hits in the muon detectors are fitted to form a common vertex in three dimensions, which is required to be displaced from the primary interaction vertex (PV) and to have a small χ2 in the fit. The resulting candidate is further required to point back to the PV, for example, to have a small impact parameter, consistent with zero, with respect to it. The final classification of data events is done in categories of the response of a multivariate discriminant (MVA) combining information from the kinematics and vertex topology of the events. The type of MVA used is a boosted decision tree (BDT)24,25,26. The branching fractions are then obtained by a fit to the dimuon invariant mass, , of all categories simultaneously.

The signals appear as peaks at the and B0 masses in the invariant-mass distributions, observed over background events. One of the components of the background is combinatorial in nature, as it is due to the random combinations of genuine muons. These produce a smooth dimuon mass distribution in the vicinity of the and B0 masses, estimated in the fit to the data by extrapolation from the sidebands of the invariant-mass distribution. In addition to the combinatorial background, certain specific b-hadron decays can mimic the signal or contribute to the background in its vicinity. In particular, the semi-leptonic decays B0πµ+ν, Kµ+ν, and can have reconstructed masses that are near the signal if one of the hadrons is misidentified as a muon and is combined with a genuine muon. Similarly the dimuon coming from the rare B0π0µ+µ and B+π+µ+µ decays can also fake the signal. All these background decays, when reconstructed as a dimuon final state, have invariant masses that are lower than the masses of the B0 and mesons, because they are missing one of the original decay particles. An exception is the decay , which can also populate, with a smooth mass distribution, higher-mass regions. Furthermore, background due to misidentified hadronic two-body decays , where or K, is present when both hadrons are misidentified as muons. These misidentified decays produce an apparent dimuon invariant-mass peak close to the B0 mass value. Such a peak can mimic a B0µ+µ signal and is estimated from control channels and added to the fit.

The distributions of signal in the invariant mass and in the MVA discriminant are derived from simulations with a detailed description of the detector response for CMS and are calibrated using exclusive two-body hadronic decays in data for LHCb. The distributions for the backgrounds are obtained from simulation with the exception of the combinatorial background. The latter is obtained by interpolating from the data invariant-mass sidebands separately for each category, after the subtraction of the other background components.

To compute the signal branching fractions, the numbers of and B0 mesons that are produced, as well as the numbers of those that have decayed into a dimuon pair, are needed. The latter numbers are the raw results of this analysis, whereas the former need to be determined from measurements of one or more ‘normalization’ decay channels, which are abundantly produced, have an absolute branching fraction that is already known with good precision, and that share characteristics with the signals, so that their trigger and selection efficiencies do not differ significantly. Both experiments use the B+J/ψK+ decay as a normalization channel with (B+J/ψ (µ+µ) K+) = (6.10 ± 0.19) × 10−5, and LHCb also uses the B0K+π channel with (B0K+π) = (1.96 ± 0.05) × 10−5. Both branching fraction values are taken from ref. 14. Hence, the µ+µ branching fraction is expressed as a function of the number of signal events in the data normalized to the numbers of B+J/ψK+ and B0K+π events:

where the ‘norm.’ subscript refers to either of the normalization channels. The values of the normalization parameter αnorm. obtained by LHCb from the two normalization channels are found in good agreement and their weighted average is used. In this formula ε indicates the total event detection efficiency including geometrical acceptance, trigger selection, reconstruction, and analysis selection for the corresponding decay. The fd/fs factor is the ratio of the probabilities for a b quark to hadronize into a B0 as compared to a meson; the probability to hadronize into a B+ (fu) is assumed to be equal to that into B0 (fd) on the basis of theoretical grounds, and this assumption is checked on data. The value of fd/fs = 3.86 ± 0.22 measured by LHCb27,28,61 is used in this analysis. As the value of fd/fs depends on the kinematic range of the considered particles, which differs between LHCb and CMS, CMS checked this observable with the decays and B+J/ψK+ within its acceptance, finding a consistent value. An additional systematic uncertainty of 5% was assigned to fd/fs to account for the extrapolation of the LHCb result to the CMS acceptance. An analogous formula to that in equation (1) holds for the normalization of the B0µ+µ decay, with the notable difference that the fd/fs factor is replaced by fd/fu = 1.

The antiparticle and the particle B0 () can both decay into two muons and no attempt is made in this analysis to determine whether the antiparticle or particle was produced (untagged method). However, the B0 and particles are known to oscillate, that is to transform continuously into their antiparticles and vice versa. Therefore, a quantum superposition of particle and antiparticle states propagates in the laboratory before decaying. This superposition can be described by two ‘mass eigenstates’, which are symmetric and antisymmetric in the charge-parity (CP) quantum number, and have slightly different masses. In the SM, the heavy eigenstate can decay into two muons, whereas the light eigenstate cannot without violating the CP quantum number conservation. In BSM models, this is not necessarily the case. In addition to their masses, the two eigenstates of the system also differ in their lifetime values14. The lifetimes of the light and heavy eigenstates are also different from the average lifetime, which is used by CMS and LHCb in the simulations of signal decays. Since the information on the displacement of the secondary decay with respect to the PV is used as a discriminant against combinatorial background in the analysis, the efficiency versus lifetime has a model-dependent bias62 that must be removed. This bias is estimated assuming SM dynamics. Owing to the smaller difference between the lifetime of its heavy and light mass eigenstates, no correction is required for the B0 decay mode.

Detector simulations are needed by both CMS and LHCb. CMS relies on simulated events to determine resolutions and trigger and reconstruction efficiencies, and to provide the signal sample for training the BDT. The dimuon mass resolution given by the simulation is validated using data on J/ψ, Υ, and Z-boson decays to two muons. The tracking and trigger efficiencies obtained from the simulation are checked using special control samples from data. The LHCb analysis is designed to minimize the impact of discrepancies between simulations and data. The mass resolution is measured with data. The distribution of the BDT for the signal and for the background is also calibrated with data using control channels and mass sidebands. The efficiency ratio for the trigger is also largely determined from data. The simulations are used to determine the efficiency ratios of selection and reconstruction processes between signal and normalization channels. As for the overall detector simulation, each experiment has a team dedicated to making the simulations as complete and realistic as possible. The simulated data are constantly being compared to the actual data. Agreement between simulation and data in both experiments is quite good, often extending well beyond the cores of distributions. Differences occur because, for example, of incomplete description of the material of the detectors, approximations made to keep the computer time manageable, residual uncertainties in calibration and alignment, and discrepancies or limitations in the underlying theory and experimental data used to model the relevant collisions and decays. Small differences between simulation and data that are known to have an impact on the result are treated either by reweighting the simulations to match the data or by assigning appropriate systematic uncertainties.

Small changes are made to the analysis procedure with respect to refs 18, 19 in order to achieve a consistent combination between the two experiments. In the LHCb analysis, the background component, which was not included in the fit for the previous result but whose effect was accounted for as an additional systematic uncertainty, is now included in the standard fit. The following modifications are made to the CMS analysis: the branching fraction is updated to a more recent prediction63,64 of ; the phase space model of the decay is changed to a more appropriate semi-leptonic decay model63; and the decay time bias correction for the , previously absent from the analysis, is now calculated and applied with a different correction for each category of the multivariate discriminant.

These modifications result in changes in the individual results of each experiment. The modified CMS analysis, applied on the CMS data, yields

while the LHCb results change to

These results are only slightly different from the published ones and are in agreement with each other.

Simultaneous fit

The goal of the analysis presented in this Letter is to combine the full data sets of the two experiments to reduce the uncertainties on the branching fractions of the signal decays obtained from the individual determinations. A simultaneous unbinned extended maximum likelihood fit is performed to the data of the two experiments, using the invariant-mass distributions of all 20 MVA discriminant categories of both experiments. The invariant-mass distributions are defined in the dimuon mass ranges [4.9, 5.9] GeV/c2 and [4.9, 6.0] GeV/c2 for the CMS and LHCb experiments, respectively. The branching fractions of the signal decays, the hadronization fraction ratio fd/fs, and the branching fraction of the normalization channel B+J/ψK+ are treated as common parameters. The value of the B+J/ψK+ branching fraction is the combination of results from five different experiments14, taking advantage of all their data to achieve the most precise input parameters for this analysis. The combined fit takes advantage of the larger data sample and proper treatment of the correlations between the individual measurements to increase the precision and reliability of the result, respectively.

Fit parameters, other than those of primary physics interest, whose limited knowledge affects the results, are called ‘nuisance parameters’. In particular, systematic uncertainties are modelled by introducing nuisance parameters into the statistical model and allowing them to vary in the fit; those for which additional knowledge is present are constrained using Gaussian distributions. The mean and standard deviation of these distributions are set to the central value and uncertainty obtained either from other measurements or from control channels. The statistical component of the final uncertainty on the branching fractions is obtained by repeating the fit after fixing all of the constrained nuisance parameters to their best fitted values. The systematic component is then calculated by subtracting in quadrature the statistical component from the total uncertainty. In addition to the free fit, a two-dimensional likelihood ratio scan in the plane (B0µ+µ) versus is performed.

Feldman–Cousins confidence interval

The Feldman–Cousins likelihood ratio ordering procedure30 is a unified frequentist method to construct single- and double-sided confidence intervals for parameters of a given model adapted to the data. It provides a natural transition between single-sided confidence intervals, used to define upper or lower limits, and double-sided ones. Since the single-experiment results18,19 showed that the B0µ+µ signal is at the edge of the probability region customarily used to assert statistically significant evidence for a result, a Feldman–Cousins procedure is performed. This allows a more reliable determination of the confidence interval and significance of this signal without the assumptions required for the use of Wilks’ theorem. In addition, a prescription for the treatment of nuisance parameters has to be chosen because scanning the whole parameter space in the presence of more than a few parameters is computationally too intensive. In this case the procedure described by the ATLAS and CMS Higgs combination group65 is adopted. For each point of the space of the relevant parameters, the nuisance parameters are fixed to their best value estimated by the mean of a maximum likelihood fit to the data with the value of (B0µ+µ) fixed and all nuisance parameters profiled with Gaussian penalties. Sampling distributions are constructed for each tested point of the parameter of interest by generating simulated experiments and performing maximum likelihood fits in which the Gaussian mean values of the external constraints on the nuisance parameters are randomized around the best-fit values for the nuisance parameters used to generate the simulated experiments. The sampling distribution is constructed from the distribution of the negative log-likelihood ratio evaluated on the simulated experiments by performing one likelihood fit in which the value of (B0µ+µ) is free to float and another with the (B0µ+µ) fixed to the tested point value. This sampling distribution is then converted to a confidence level by evaluating the fraction of simulated experiments entries with a value for the negative log-likelihood ratio greater than or equal to the value observed in the data for each tested point. The results of this procedure are shown in Extended Data Fig. 5.

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Acknowledgements

We express our gratitude to colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at CERN, at the CMS institutes and at the LHCb institutes. In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC, the CMS and the LHCb detectors provided by CERN and by many funding agencies. The following agencies provide support for both CMS and LHCb: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and HGF (Germany); SFI (Ireland); INFN (Italy); NASU (Ukraine); STFC (UK); and NSF (USA). Agencies that provide support for CMS only are BMWFW and FWF (Austria); FNRS and FWO (Belgium); FAPESP (Brazil); MES (Bulgaria); CAS and MoST (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF (Cyprus); MoER, ERC IUT and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA (France); GSRT (Greece); OTKA and NIH (Hungary); DAE and DST (India); IPM (Iran); NRF and WCU (Republic of Korea); LAS (Lithuania); MOE and UM (Malaysia); CINVESTAV, CONACYT, SEP, and UASLP-FAI (Mexico); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS and RFBR (Russia); MESTD (Serbia); SEIDI and CPAN (Spain); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR and NSTDA (Thailand); TUBITAK and TAEK (Turkey); SFFR (Ukraine); and DOE (USA). Agencies that provide support for only LHCb are: FINEP (Brazil); MPG (Germany); FOM and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES and FANO (Russia); MinECo (Spain); SNSF and SER (Switzerland). Individuals from the CMS collaboration have received support from the Marie-Curie programme and the European Research Council and EPLANET (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIABelgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Council of Science and Industrial Research, India; the HOMING PLUS programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund; the Compagnia di San Paolo (Torino); the Consorzio per la Fisica (Trieste); MIUR project 20108T4XTM (Italy); the Thalis and Aristeia programmes cofinanced by EU-ESF and the Greek NSRF; and the National Priorities Research Program by Qatar National Research Fund. Individual groups or members of the LHCb collaboration have received support from EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union), Conseil général de Haute-Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR (Russia), XuntaGal and GENCAT (Spain), Royal Society and Royal Commission for the Exhibition of 1851 (UK). LHCb is also thankful for the computing resources and the access to software R&D tools provided by Yandex LLC (Russia). The CMS and LHCb collaborations are indebted to the communities behind the multiple open source software packages on which they depend.

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Extended data figures and tables

Extended Data Figure 1 Distribution of the dimuon invariant mass ${m}_{\mu +\mu -}$ in each of the 20 categories.

Superimposed on the data points in black are the combined fit (solid blue) and its components: the Bs0 (yellow shaded) and B0 (light-blue shaded) signal components; the combinatorial background (dash-dotted green); the sum of the semi-leptonic backgrounds (dotted salmon); and the peaking backgrounds (dashed violet). The categories are defined by the range of BDT values for LHCb, and for CMS, by centre-of-mass energy, by the region of the detector in which the muons are detected, and by the range of BDT values. Categories for which both muons are detected in the central region of the CMS detector are denoted with CR, those for which at least one muon was detected into the forward region with FR.

Extended Data Figure 2 Distribution of the dimuon invariant mass ${m}_{\mu +\mu -}$ for the best six categories.

Categories are ranked according to values of S/(S + B) where S and B are the numbers of signal events expected assuming the SM rates and background events under the Bs0 peak for a given category, respectively. The mass distribution for the six highest-ranking categories, three per experiment, is shown. Superimposed on the data points in black are the combined full fit (solid blue) and its components: the Bs0 (yellow shaded) and B0 (light-blue shaded) signal components; the combinatorial background (dash-dotted green); the sum of the semi-leptonic backgrounds (dotted salmon); and the peaking backgrounds (dashed violet).

Extended Data Figure 3 Schematic of the CMS detector and event display for a candidate Bs0 →μ+μ− decay at CMS.

a, The CMS detector and its components; see ref. 20 for details. b, A candidate Bs0μ+μ decay produced in proton–proton collisions at 8 TeV in 2012 and recorded in the CMS detector. The red arched curves represent the trajectories of the muons from the Bs0 decay candidate.

Extended Data Figure 4 Schematic of the LHCb detector and event display for a candidate Bs0 →μ+μ− decay at LHCb.

a, The LHCb detector and its components; see ref. 21 for details. b, A candidate Bs0μ+μ decay produced in proton–proton collisions at 7 TeV in 2011 and recorded in the LHCb detector. The proton–proton collision occurs on the left-hand side, at the origin of the trajectories depicted with the orange curves. The red curves represent the trajectories of the muons from the Bs0 candidate decay.

Extended Data Figure 5 Confidence level as a function of the B(B0→μ+μ−) hypothesis.

The value of 1 − CL, where CL is the confidence level obtained with the Feldman–Cousins procedure, as a function of B(B0µ+µ) is shown in logarithmic scale. The points mark the computed 1 − CL values and the curve is their spline interpolation. The dark and light (cyan) areas define the two-sided ±1σ and ±2σ confidence intervals for the branching fraction, while the dashed horizontal line defines the confidence level for the 3σ one-sided interval. The dashed (grey) curve shows the 1 − CL values computed from the one-dimensional −2ΔlnL test statistic using Wilks’ theorem. Deviations between these confidence level values and those from the Feldman–Cousins procedure30 illustrate the degree of approximation implied by the asymptotic assumptions inherent to Wilks’ theorem29.

Extended Data Figure 6 Likelihood contours for the ratios of the branching fractions with respect to their SM prediction, in the versus plane.

a, The (black) cross marks the central value returned by the fit. The SM point is shown as the (red) square located, by construction, at . Each contour encloses a region approximately corresponding to the reported confidence level. The SM branching fractions are assumed uncorrelated to each other, and their uncertainties are accounted for in the likelihood contours. b, c, Variations of the test statistic 2ΔlnL for and are shown in b and c, respectively. The SM is represented by the (red) vertical lines. The dark and light (cyan) areas define the ±1σ and ±2σ confidence intervals, respectively.

Extended Data Figure 7 Search for the Bs0 →μ+μ− and B0→μ+μ−decays, reported by 11 experiments spanning more than three decades, and by the present results.

Markers without error bars denote upper limits on the branching fractions at 90% confidence level, while measurements are denoted with error bars delimiting 68% confidence intervals. The solid horizontal lines represent the SM predictions for the Bs0 and B0µ+µ branching fractions1; the blue (red) lines and markers relate to the Bs0 (B0µ+µ) decay. Data (see key) are from refs 17, 18, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60; for details see Methods. Inset, magnified view of the last period in time.

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CMS Collaboration., LHCb Collaboration. Observation of the rare Bs0µ+µ decay from the combined analysis of CMS and LHCb data. Nature 522, 68–72 (2015). https://doi.org/10.1038/nature14474

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text:
# Particle / wave duality on a scale of light frequencies.

Let's present a scale of frequencies of light. For initial we’ll take frequency of visible light. Here particle / wave duality is shown fifty- fifty.
At increase or decrease of frequency this proportion is broken.

Frequency increasing.
Properties of a particle start to prevail of properties of a wave.
The gamma radiation, for example, possesses properties of a particle in the greater degree, than properties of a wave.
On frequency 1.930605x10 ^ 18 Hz we can see hydrogen [This value of frequency is result of researches of remarkable Russian scientist Poljansky V.N. (vlamir)]
Further on a scale there are all elements of Mendeleyev's table in ascending order of their nuclear mass.

On this range of frequencies the duality is absent – light exists as particles.
MASS (GRAVITY FORCE) IS DIRECTLI PROPORTIONAL TO FREQUENCY OF LIGHT.
Application of mass of atom of hydrogen as the unit of measurement of mass of atoms of other elements is wrongly. The proof of it is not the whole values of mass of atoms of other elements in these units. I assert, that the periodic system of elements should be constructed on a frequency principle and a unit of measurements of mass of atoms should be a frequency of photon.
Frequency decreasing.
Here the return phenomenon is observed.
Properties of a wave prevail of properties of a particle.
At radiowaves of a long-wave range, for example, it is impossible to find out a properties of particle.
In a range of frequencies from the bottom border of infra-red radiation close to zero the duality of light is absent also. It is electromagnetism (EM).The reality exists in the current Plank time only. It is the minimal cycle of time. The continuity is prodigal. Discreteness is economic. The nature always chooses an optimum variant. Any object (the particle or any their combination) has a cycle of time inherent in it. Time for each object individually and represents the counter working on subtraction. From a birth to death. All time cycles are synchronized by the minimal cycle of time. It is fair both for macroobjects, and for microobjects.
Time is connected to frequency of a wave the return relation.
It is possible to tell, that the carrying wave of the maximal frequency corresponds to the minimal cycle of time and the cycle of time of object corresponds to a combination of frequencies or a data set. Let’s take a look at the scale of light frequencies. It possesses the property of symmetry. Moreover, this symmetry exists concerning two points of a scale, i.e. it is double symmetry. To say, these are VERY STRANGE POINTS. The first point is on frequency of visible light.
Here the magnetic and gravity properties ASPIRE TO ZERO. But PROCESS of ACHIEVEMENT of ABSOLUTE ZERO is INFINITE. Hence, it is the singular point.
THE FREQUENCY OF PHOTON IS THE FIRST SINGULAR POINT ON THE SCALE OF LIGHT FREQUENCIES.
The second point UNITES the opposite ends of a scale. Here the magnetic and gravity forces aspire to indefinite great value. It is a Black Hole.
A BLACK HOLE IS THE SECOND SINGULAR POINT ON THE SCALE OF LIGHT FREQUENCIES.
But. Two infinity cannot exist separately. Actually it is ONE INFINITY. TWO SINGULAR POINTS are MIRROR in RELATION of EACH OTHER. It defines a MIRROR of FREQUENCIES which, in turn, DEFINES THE MAGNETIC AND GRAVITY PROPERTYES.
Two mirror singular points provide ETERNAL MOVEMENT and represent the oscillator which recycles the universe with the maximal frequency 1/Planck Time.

Each object with each time unit loses unit of the data set. This process is observed as radiation of a photon of the certain frequency. But radiation of a photon does not occur in "anywhere" and absorption does not occur "anywhere". Only on channels “ object – object ”. All objects are capable to absorb a data set ( photons) which have supplementing their set up to initial one. It is the process of regeneration.

EM FORCE IS INVERSELY TO FREQUENCY OF LIGHT.

FREQUENCY OF VISIBLE LIGHT IS THE ZERO POINT ON THE GENERAL SCALE OF GRAVITY AND EM FORCES.

Let's consider well-known relations:
$$E=mc^2$$
and
$$E=h \nu\$$
whence $$mc^2=h \nu\$$
or
$$m=k\nu\$$
here k= h/c^2
This new relation proves my rightness.
Further accepting
$$\nu\ = f_w$$ – $$f_p$$
where
$$f_w$$ - frequency of object's wave
$$f_p$$- frequency of photon
for $$f_w > f_p$$
we have positive value of mass.
It provides STRONG FORCE inside particle and GRAVITY FORCE between particles .
for $$f_w < f_p$$
we have negative (mirror) value.
It is the CHARGE inside particle and MAGNET FORCE outside one.
A CHARGE THIS A MIRROR REFLECTION OF MASS CONCERNING THE FREQUENCY OF PHOTON. THE MAGNET FORCE THIS A MIRROR REFLECTION OF GRAVITY FORCE CONCERNING THE FREQUENCY OF PHOTON.

Thus, on the scale of light frequencies are:
* A constant magnetic dipole with a maximal value of magnetic force- at frequency close to zero (length of a wave --> infinity). It has no mass in principle. It is one end of a scale;

**In the middle of this scale there is a visible light - photon. It has no mass as well as has no magnetic properties. Therefore I have chosen it as a zero point of their general scale. Visible light is a border between gravitation and magnetism. It is a reason of its amazing properties and duality behavior. Here there is a balancing between zero values of an gravity and magnetic forces.

***Object having the maximal mass (gravity force) at the maximal frequency (length of a wave --> the zero) – it is a Black Hole. It is other end of a scale.
On the one hand, this object should not have magnetic properties. On the other hand, achievement of an absolute zero (by the length of wave ) is infinite process.
From this point of view the end points of a scale are equivalent.
It means, that the Black Hole is the general point of the both ends of a scale, i.e. the scale of light frequencies does close in this point. Here there is a balancing between the maximal values of gravitational and magnetic forces.
Thus
THE PHOTON IS THE ANTIPODE OF BLACK HOLE
As well as
THE BLACK HOLE IS THE ANTIPODE OF PHOTON.
A wave it is always a wave. At any displays of its properties on a scale of frequencies.
Change of properties of a wave depending on its frequency is an attribute of interaction with other wave.
All these properties are shown on a background of a base carrying wave of the maximal frequency which is generated constantly by the Black Hole.
If "switch off" the Black Hole then our universe will stop the existence.
All observable phenomena are consequence of an interference of waves.
I am keeping good old concept, that one wave can exist on a background of other CARRYING WAVE. Thus frequency of a carrying wave should be, at least, 2 times more than a frequency of modulating wave. It is confirmed experimental data.
Besides at frequency modulation on which principles our universe is constructed, there should be some frequency of quantization. It defines an accuracy of approximation of the form of a real wave to an ideal wave. It is confirmed experimental data too.
Hence, such things for maintenance of work of our universe are necessary:
- stable oscillator of the carrying frequency;
- carrying wave;
- modulating wave;
- frequency of quantization.
The stable oscillator on a base of two singular mirror points (see above) is realized. The carrying wave has the maximal frequency 1/Planck Time.
The modulating waves is all objects of the universe.
The Frequency of quantization has two levels:
1. The basic frequency of quantization corresponds to the frequency of carrying wave.
2. An additional frequency of quantization corresponds to the frequency of photon.
The first level defines the properties of channels "object - object" or of intervals between objects and is shown as GRAVITATY and MAGNETIC FORCES.
The second one defines internal properties of objects and is shown as STRONG FORCE and the CHARGE.

Light has appeared as the unique "element" from which our universe is constructed.

So, I have made that up to me nobody could make. I have explained a general origin of gravity and magnet forces as well as strong force and charge.They are mirror reflection of each other on the general scale of frequencies. I have proved, also, one of a basic postulate of the Bible –
OUR WORLD IS CREATED OF LIGHT.

2003 Michael F. Dmitriyev

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russ_watters
Mentor
That is not how light is observed to work. The particle/wave duality exists for all frequencies and is dependent on how you attempt to measure it.

Originally posted by russ_watters
That is not how light is observed to work. The particle/wave duality exists for all frequencies and is dependent on how you attempt to measure it.
Are you meaning, that observable properties of atom do not differs from observable properties of infralow frequencie radiowave? I cannot agree with it.

russ_watters
Mentor
Originally posted by Michael F. Dmitriyev
Are you meaning, that observable properties of atom do not differs from observable properties of infralow frequencie radiowave? I cannot agree with it.
That doesn't make any sense. If you mean:
Do you mean that the observed properties of atoms differ from the observed properties of ULF radio waves?
Then the answer is yes. They do. Very very very very much so.

Indeed. No matter what light frequency you look at, it displays the same particle/wave duality. The only things different are the wavelength and energy. Now, if said photon of a certain frequency is being used to accelerate a particle, then the particle will start to take on more point particle properties because its deBroglie wavelength will shrink towards zero, but can still never reach zero. And then when you factor in the HUP, if we carefully monitor the particles momentum, we will have less and less of an idea where exactly it is. But as for photons themselves in light streams, a photon is a photon. Just having a different energy and characteristic wave front.

Originally posted by russ_watters
That doesn't make any sense. If you mean: Then the answer is yes. They do. Very very very very much so.
Okay!And all these distinctions depend only on one thing. It is a frequency of light. That is how light is observed to work.

Indeed. No matter what light frequency you look at, it displays the same particle/wave duality. The only things different are the wavelength and energy. Now, if said photon of a certain frequency is being used to accelerate a particle, then the particle will start to take on more point particle properties because its deBroglie wavelength will shrink towards zero, but can still never reach zero. And then when you factor in the HUP, if we carefully monitor the particles momentum, we will have less and less of an idea where exactly it is. But as for photons themselves in light streams, a photon is a photon. Just having a different energy and characteristic wave front.
Frequency of EM waves (light) and its front are not interconnected. The front of a wave or the form of fluctuations defines presence and quantity of harmonics. Sine form fluctuation has no harmonics while pulses of the rectangular form have unlimited number of harmonics. Frequency of fluctuations at change of front does not change.

Mike, frequency and wavelength are very much related.
λ = c/ν That alone is basic.

The rest of your post however makes no sense. "The form functions have no harmonics but rectangular forms have unlimited harmonics." Wouldn't a rectangular form be itself a fluctuation? The answer is yes.

Mike, frequency and wavelength are very much related.
λ = c/ν That alone is basic.

The rest of your post however makes no sense. "The form functions have no harmonics but rectangular forms have unlimited harmonics." Wouldn't a rectangular form be itself a fluctuation? The answer is yes.
Read my last post attentively. I have told:
“ Sine form fluctuation has no harmonics while pulses of the rectangular form have unlimited number of harmonics. ”
You quote me in another way:
" the form functions have no harmonics but rectangular forms have unlimited harmonics. "
In result the sense was completely deformed.
Certainly, frequency and length of a fluctuation have strong interconnection.
But, Brad, frequency and front are the different things.

That is not how light is observed to work. The particle/wave duality exists for all frequencies and is dependent on how you attempt to measure it.
I agree . One of the main characteristics of photons is their ability to retain their identity , if photons continued to increase their wave length equivavlent to their loss of energy , we would have no idea of what the Universe is all about . We would be like blind people trying to figure out what came from where. Photons do increase their wave-lengths but within definite limits. See http://www.geocities.com/natureoflight/natureoflight for a new theory on how this takes place.McQueen

Ah, I admit I read sine as "since". But a sine wave does indeed have a harmonic property to it.

And regardless, photons are not observed to act as you state. If they did, we would not be able to do gamma ray astronomy. Now an interesting thing to realize is that gamma rays diffract less than radio waves. You might liken this to be more particle like, but it is simply because gamma rays have a much much much shorter wavelength than radio waves. They still do difract as if they were waves of their wavelength. But by no means do they seem more "particle like".

Ah, I admit I read sine as "since". But a sine wave does indeed have a harmonic property to it.

And regardless, photons are not observed to act as you state. If they did, we would not be able to do gamma ray astronomy. Now an interesting thing to realize is that gamma rays diffract less than radio waves. You might liken this to be more particle like, but it is simply because gamma rays have a much much much shorter wavelength than radio waves. They still do difract as if they were waves of their wavelength. But by no means do they seem more "particle like".
You are not right again. Sine wave fluctuations HAVE no HARMONICS.
The method of measurement of harmonious distortions is based on it. The sine wave signal of the DEFORMED FORM has harmonics. Find the corresponding literature and be convinced of it. I hope to not come back any more to this question.
Now the second part of your post.
Gamma rays are not radio waves, because they are completely deprived of magnetic properties. But they already have mass, as against visible light because their frequency is much higher. Asserting it we recognize direct dependence of mass on frequency of fluctuations. Look after my point once again.
From the zero value of frequency up to visible light magnetic properties are present, but a mass is absent.
Visible light neutral. It has no mass and has no magnetic properties. It¡¦s acceptable name is "photon".
Further, in a direction of frequency increasing, magnetic properties completely are absent, and attributes of mass start to be shown. More frequency „³ more energy „³ more mass.
There are objections?

Originally posted by McQueen
I agree . One of the main characteristics of photons is their ability to retain their identity , if photons continued to increase their wave length equivavlent to their loss of energy , we would have no idea of what the Universe is all about . We would be like blind people trying to figure out what came from where. Photons do increase their wave-lengths but within definite limits. See http://www.geocities.com/natureoflight/natureoflight for a new theory on how this takes place.McQueen

McQueen,
Same was chewed at last 100 years.

Yes Mike, there are objections. In fact every observation is an objection. Learn some real physics for a change, will you?

Yes Mike, there are objections. In fact every observation is an objection. Learn some real physics for a change, will you?
Yours concrete objections which refute my point, please.

russ_watters
Mentor
Originally posted by Michael F. Dmitriyev
Yours concrete objections which refute my point, please.
It appears you already know the answers, you just refuse to accept them. What you are saying is directly contrary to the way em radiation is observed to work. Existing theories on the nature of em radation are the way they are because thats how em radiation is observed to work in real life. They are not wrong, you are wrong. If you refuse to accept that, there isn't anything we can do to help you.

Like he said: learn (and accept) some real physics.

Originally posted by russ_watters
It appears you already know the answers, you just refuse to accept them. What you are saying is directly contrary to the way em radiation is observed to work. Existing theories on the nature of em radation are the way they are because thats how em radiation is observed to work in real life. They are not wrong, you are wrong. If you refuse to accept that, there isn't anything we can do to help you.

Like he said: learn (and accept) some real physics.
Occam's Razor: it results in a simpler description of reality.

neutroncount
Originally posted by Michael F. Dmitriyev
Occam's Razor: it results in a simpler description of reality.

That doesn't mean it's true. If you refuse to take account for what is really observed in nature, that's your own ignorance. Nature isn't simple. Who says it has to be?

In addition to an initial post.

I have attached this post to initial one.

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neutroncount
Damn....you ARE a psycho! You're basing your extremely losely based theory on the bible, and than coming up with hypotheses already shown not to be true. I give you credit though. You push your crackpottery as far as it can go.

Originally posted by neutroncount
Damn....you ARE a psycho! You're basing your extremely losely based theory on the bible, and than coming up with hypotheses already shown not to be true. I give you credit though. You push your crackpottery as far as it can go.
...............................................
It was deleted by me.

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russ_watters
Mentor
Originally posted by Michael F. Dmitriyev
Occam's Razor: it results in a simpler description of reality.
That isn't what Occam's Razor is for. Occam's Razor is for comparing two different theories that both ACCURATELY describe reality. To apply Occam's Razor you first have to start with an ACCURATE description of reality. Your description is inaccurate.
So, I have made that up to me nobody could make. I have explained a general origin of mass and magnetism. I have proved, also, one of a basic postulate of the Bible –
OUR WORLD IS CREATED OF LIGHT.
Lol, I get it now. Funny joke.

Originally posted by neutroncount
Damn....you ARE a psycho! You're basing your extremely losely based theory on the bible, and than coming up with hypotheses already shown not to be true. I give you credit though. You push your crackpottery as far as it can go.
Bertrand Russell did not understand, that between the fool and the fanatic the infinity stretches.

That isn't what Occam's Razor is for. Occam's Razor is for comparing two different theories that both ACCURATELY describe reality. To apply Occam's Razor you first have to start with an ACCURATE description of reality. Your description is inaccurate.
I guess, that PF not a place for accommodation and discussions of the whole theory. The new theory begins with new idea or some point. I have offered my own idea for discussion. This idea does not contradict existing observations. Taking into account, that observations and theories are different things, I do not think that the contradiction of new idea with the existing theory is a lack. The new idea should be such by definition. Perhaps, you consider, what the existing theory explains all phenomena and does not require development?
In any case I would like to hear concrete opinions on idea, instead of the common reasoning. Particularly on properties of a particle or a wave depending on the placement on a scale of frequencies.
So?

Lol, I get it now. Funny joke.
It is not the joke as seems to you. It is quite serious. Perhaps, you have data on similar idea and its author? Then inform, please.
BTW, each person can operate only the volume of information, which he able to perceive.

Michael,
I feel, that in your thoughts there is a rational kernel.
Please, think on these pictures - left-spiral atom and photon.
Energy circulates in rings with speed of light and twists on the left spiral.
Magnetic component in photons is green.
Quite clearly, that exist and right-spiral hydrogen atoms.
Maybe, these figures will direct you to new thoughts.
I wish you successes.
http://www.sinor.ru/~polytron/h_atom.gif [Broken]
http://www.sinor.ru/~polytron/photon.gif [Broken]

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# Finding the mean and standard deviation from an equation with sigma

#### IBstudent

##### New member

I have been having some trouble solving a specific problem asking to find the mean and standard deviation of a population when given two equations with sigma (see attached)

My attempt was to find x so that x + (x+1) + (x+2) + ....... + (x+24) = 300. So 25x + 300 = 300... wait... then x is zero...

Am I doing it wrong? please explain this to me I'm lost :S

Also, how is it possible to get the standard deviaiton from such equation?

Please don't leave comments like "go back to your textbook", I have and it says nothing about this...

Thanks in advance

#### galactus

##### Super Moderator
Staff member
The populations sums to 300 and there are 25 of them. Looks like the mean would be

$$\displaystyle \frac{300}{25}=12$$

#### IBstudent

##### New member
The populations sums to 300 and there are 25 of them. Looks like the mean would be

$$\displaystyle \frac{300}{25}=12$$
Thanks.. now I understand this bit
What about the second bit?


text:
Session W13: Focus Session: Iron Pnictides and Other Novel Superconductors XIV: ARPES

11:15 AM–2:03 PM, Thursday, March 19, 2009
Room: 309

Chair: Norman Mannella, University of Tennessee, Knoxville

Abstract ID: BAPS.2009.MAR.W13.11

Abstract: W13.00011 : Observation of an orbital selective electron-mode coupling in Fe-based high-Tc superconductors

1:15 PM–1:27 PM

Preview Abstract MathJax On | Off   Abstract

Authors:

Pierre Richard
(WPI, Tohoku University)

T. Sato
K. Nakayama
(Dep. Physics, Tohoku University)

S. Souma
(WPI, Tohoku University)

Y.-M. Xu
(Boston College)

G.F. Chen
J.L. Luo
N.L. Wang
H. Ding
The recent discovery of Fe-based superconductors with critical temperatures up to 56 K raises the prospect of unconventional superconducting pairing mechanism. While the electronic pairing in conventional superconductors is mediated by phonons, its nature in the Fe-based high-$T_{c}$ superconductors is unknown. A direct signature of an electron-mode coupling is an anomaly in the electronic energy dispersion (kink). For example, previous angle-resolved photoelectron spectroscopy (ARPES) studies revealed a kink in cuprates, which is believed to be linked to the pairing. We report an ARPES observation of a kink around 25 meV in the dispersion of superconducting Ba$_{0.6}$K$_{0.4}$Fe$_{2}$As$_{2}$ that nearly vanishes above $T_{c}$. The energy scale of the related mode (13$\pm$2 meV) and its strong dependence on orbital and temperature indicates that it is unlikely related to phonons. Moreover, the momentum locations of the kink can be connected by the antiferromagnetic wavevector. Our results point towards an electronic origin of the mode and the superconducting pairing in the Fe-based superconductors, and strongly support the anti-phase s-wave pairing symmetry.


text:
# Let $\pi$ Be 3

by Robert N. Andersen (University of Wisconsin -- Eau Claire), Justin Stumpf (University of Wisconsin -- Eau Claire), and Julie Tiller (University of Wisconsin -- Eau Claire)

Mathematics Magazine
June, 2003

Biblical text implying that $\pi$ should be 3

A pdf copy of the article can be viewed by clicking below. Since the copy is a faithful reproduction of the actual journal pages, the article may not begin at the top of the first page.


text:
• ### Announcements

#### Archived

This topic is now archived and is closed to further replies.

# Transparency Question

## 7 posts in this topic

Check you've turned off anti-aliasing in PSP. It may be that some pixels that you intend to be black are not precisely RGB(0,0,0).
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I was having that problem before, but I turned it off, and I have exhaustively checked the perimeters of the text objects to make sure they are indeed (0,0,0) and they are... And just to be sure, I used the "Count Colors" function in PSP, and there are 4 colors used in the picture. That's just enough, because I have 2 colors of text, 1 color of template lines, and then the (0,0,0).
I just don't know if this is a problem with the color keying (I used the DDSetColorKey function from ddutil.cpp to set it) or with the bitmap format (I doubt that) or with, I don't know what...

HELP!!!

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Here is exactly what's happening:
If I make a change from ANY color to the transparent color (meaning if one pixel is not transparent, and the next few are, the next pixel to the right (or maybe 2 pixels, I cant really tell) show up as black on the screen. I don't know if they are actually showing up as RGB(0,0,0) or something very close, but I know they are (0,0,0) in the bitmap file. I don't know if it's a problem saving the bitmap, loading it to a surface, or blitting it (it happens with both Blt and BltFast).

[This message has been edited by Qoy (edited December 14, 1999).]

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Cool, I thought it was just me. I've been saving Poser models into bitmap form on a black background, but the models always end up with a black outline when they are blt/bltfast.

------------------
Still Learning...

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I talked on #gamedev on Afternet last night to some people, and they said it was probably a problem with my card in 32 bit color modes. They suggested I change the bit depth to 16 bits. Does anybody think the problem could be anything else? I am now trying to convert the game to run in 16 bits, but am having trouble converting the bitmaps, and simply even displaying colors to the screen. Does anybody know where I can find an example of converting a 24 bit bitmap to 16 bits, or of building up a 16 bit RGB value?
I'll post some of my code so people can see if it's really convoluted, and see if they know what my problem is...

Here is my code for checking the pixel format:

code:
// check the pixel format (555 or 565)gMask = desc.ddpfPixelFormat.dwGBitMask;// check the 27th bit from the right in the mask (bit number 26)// if it's 1, then it's 565, else it's 555// it's the 27th bit because it's the 3rd from the beginning of the first byte,// but gMask is 4 bytes// 2^26 = 67108864if(gMask | 67108864)	pixelFormat = 565;else	pixelFormat = 555;

Here is my code for building up a 16 bit rgb value:

code:
#define RGB555(r,g,b) ((b % 32) + ((g % 32) << 6) + ((r % 32) << 11))#define RGB565(r,g,b) ((b % 32) + ((g % 64) << 5) + ((r % 32) << 11))

I can probably handle the converting the bitmap if I can just get this stuff working...

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here's a thought...

you said this was a bitmapped lettering system, correct? does this mean that there are essentially two colors in your bitmap(black and the letter's color)?

if so, then you may as well design your letters on a monochrome bitmap.

if there's actually a pattern(more than just the two colors), then thats another story.

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Well, the letters are just two colors, but this stuff has to work for other bitmaps too
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I am implementing a bitmapped text function for my game, and it works alright so far, but I am having one problem. This doesn't just occur with the text. I am using source color keying, with RGB(0,0,0) as the color key for transparency. Whenever I use Blt or BltFast to blit the rectangle containing a letter to the back buffer, some of the pixels don't show up transparent, instead, they show up black (I have the backBuffer filled gray, so I can see where it't not working). It seems to do this on pixels that are directly to the right of pixels of different colors (non-black). But it only does this sometimes. I am using the DDLoadBitmap function from ddutil.cpp to load the bitmap, and am using the DirectX 5 headers/libs. I am using Paint Shop Pro 6 to make the graphics, so I don't know if it's a problem in the bitmap format... My surface is 32 bit, and all my bitmaps are 24 bit.
Does anybody know what the problem could be?

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http://qoy.tripod.com

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# How to Align Sentences at a Certain Place

by Nanoputian   Last Updated January 15, 2018 04:23 AM

How can you align certain lines at a certain place. For example when I write it up normally in latex is displays the following:

But I want it to display it like this instead:

So in other words, I want to find a way to make sure that each question lines up with each other. I know you can do this using the \align command for equations, but not sure what the command is for normal sentences.

Thanks for any help!

Tags :

A \parbox directive will generate the "look" of the second screenshot you provided.

\documentclass{article}
\begin{document}
\noindent
Eg 1.\
\parbox[t]{0.8\textwidth}{%
a) What is the rate of decrease of its radius \dots\par
b) Find the time taken to reduce the radius of \dots\par
c) Find the time taken to reduce the radius of \dots}
\end{document}

However, using a tabularx environment should provide better alignment of the (implicit) columns.

\documentclass{article}
\usepackage{tabularx}
\begin{document}
\noindent
\setlength\tabcolsep{3pt} % default value is 6pt
\begin{tabularx}{\textwidth}{@{}llX@{}}
Eg 1.
& a) & What is the rate of decrease of its radius \dots \\
& b) & Find the time taken to reduce the radius of \dots\\
& c) & Find the time taken to reduce the radius of \dots \\
\end{tabularx}
\end{document}

Mico
January 15, 2018 03:36 AM


text:
# Questions about linear independent and spanning set and basi

Member warned about posting with no effort shown

## Homework Statement

Definition of spanning set:
Let be vectors in the vector space . The set of all linear combination of the vectors is a subspace ( say ) of . The subspace is called the space spanned by the vectors The set is called a spanning set of .

Definition of linear independence:
Suppose that is a vector space. The set of vectors from is linearly dependent if there is a relation oflinear dependence on that is not trivial. In the case where the only relation of linear dependence on is the trivial one, then is a linearly independent set of vectors.

Definition of basis of vector space:
1. It spans the space.
2. Its vectors are independent.
3. The number of vectors in the basis is equal to the dimension of the space.

True or false? Given reason

1. A set of 5 Vectors in R5 must be a basis for R5
2. A set of 6 Vectors in R5 cannot be a basis for R5
3. A set of 7 vectors in R5 must be a spanning set for R5
4. A set of 6 polynomials in R5 must be a basis for R5
5. A set of 6 polynomials in R5 may be a basis for R5

Anyone help me to explain these question.. I want to know the reason cz it's confused me a lot and i always get mess about these kind of question
appericated it!

Last edited:

Samy_A
Science Advisor
Homework Helper

## Homework Statement

True or false? Given reason

## Homework Equations

1. A set of 5 Vectors in R5 must be a basis for R5
2. A set of 6 Vectors in R5 cannot be a basis for R5
3. A set of 7 vectors in R5 must be a spanning set for R5
4. A set of 6 polynomials in R5 must be a basis for R5
5. A set of 6 polynomials in R5 may be a basis for R5

## The Attempt at a Solution

Anyone help me to explain these question.. I want to know the reason cz it's confused me a lot and i always get mess about these kind of question
appericated it!
I assume that what you wrote under "Relevant equations" is actually the "The problem statement", and that R5 stand for ##\mathbb R^5##.

For starters, maybe you could define the used concepts: basis and spanning set.

SammyS
I assume that what you wrote under "Relevant equations" is actually the "The problem statement", and that R5 stand for ##\mathbb R^5##.

For starters, maybe you could define the used concepts: basis and spanning set.
Thanks. I just edited the form of the question and now it should be ok..btw, Can u help me to this question?

Samy_A
Science Advisor
Homework Helper
Thanks. I just edited the form of the question and now it should be ok..btw, Can u help me to this question?
Thanks.

The forum rules expect you to show some effort in solving the exercise, before other forum members jump in to help.
That's why I suggested you at least start with defining what a basis is, and what a spanning set is.

Thanks.

The forum rules expect you to show some effort in solving the exercise, before other forum members jump in to help.
That's why I suggested you at least start with defining what a basis is, and what a spanning set is.
Thanks for reminding. This is my first time to ask question here so sorry about that.

Samy_A
Science Advisor
Homework Helper

## Homework Statement

Definition of spanning set:
Let be vectors in the vector space . The set of all linear combination of the vectors is a subspace ( say ) of . The subspace is called the space spanned by the vectors The set is called a spanning set of .

Definition of linear independence:
Suppose that is a vector space. The set of vectors from is linearly dependent if there is a relation oflinear dependence on that is not trivial. In the case where the only relation of linear dependence on is the trivial one, then is a linearly independent set of vectors.

Definition of basis of vector space:
1. It spans the space.
2. Its vectors are independent.
3. The number of vectors in the basis is equal to the dimension of the space.
(I added the bold blue)
Using that third point, can you answer some of the questions?

(I added the bold blue)
Using that third point, can you answer some of the questions?
I can answer question 2 the reason is any set of 6 vectors in R5 is linearly dependent, even though some sets of 6 vectors in R5 span R5. To be a basis it must be a linearly independent spanning set, so if it's linearly dependent, it cannot be a basis.
But actually I know the answer for all of these questions but i don't the reason apart from question2

Samy_A
Science Advisor
Homework Helper
I can answer question 2 the reason is any set of 6 vectors in R5 is linearly dependent, even though some sets of 6 vectors in R5 span R5. To be a basis it must be a linearly independent spanning set, so if it's linearly dependent, it cannot be a basis.
But actually I know the answer for all of these questions but i don't the reason apart from question2
Ok. What about questions 4 and 5?

Ok. What about questions 4 and 5?
No i don't know how to explain question 1 3 4 5.

Samy_A
Science Advisor
Homework Helper
No i don't know how to explain question 1 3 4 5.
Let's take 4 as an example:
"4. A set of 6 polynomials in R5 must be a basis for R5"

Can a set of 6 polynomials be a basis for ##\mathbb R^5##? The answer can be found using the definitions you posted.

Let's take 4 as an example:
"4. A set of 6 polynomials in R5 must be a basis for R5"

Can a set of 6 polynomials be a basis for ##\mathbb R^5##? The answer can be found using the definitions you posted.
Thanks for helping. Here is too late now so I have to sleep. I am gonna do these question tomorrow.
Appericated for helping!!

Mark44
Mentor
The forum rules expect you to show some effort in solving the exercise, before other forum members jump in to help.
Noted...

Hi guys. I hv done these question already. Thanks for helping!!