Class E Power Amplifier Cadence

Abstract

An angular analysis of B0 →K∗0e+e−decays is presented using proton-proton collision data collected by the LHCb experiment at centre-of-mass energies of 7, 8 and 13 TeV, corresponding to an integrated luminosity of 9 fb−1. The analysis is performed in the region of the dilepton invariant mass squared of 1.1–6.0 GeV2/c4.

In addition, a test of lepton flavour universality is performed by comparing the obtained angular observables with those measured in B0 →K∗0µ+µ−decays. In general, the angular observables are found to be consistent with the Standard Model expectations as well as with global analyses of other b →sℓ+ℓ−processes, where ℓis either a muon or an electron. No sign of lepton-flavour-violating effects is observed.

Published In Jhep 06 (2025) 140

© 2025 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence. †Authors are listed at the end of this paper.

Ntroduction

Decays mediated by the b→sℓ+ℓ−quark transition are suppressed in the Standard Model (SM) due to the absence of flavour changing neutral currents (FCNCs) at tree level. Contributions from physics beyond the SM (BSM) can cause deviations of various observables from their SM expectation values. Previous studies of b →sµ+µ−transitions have revealed tensions with SM predictions in branching fractions and angular observables . In particular, a long-standing anomaly has been found in measurements of the differential branching fraction and angular observables of the B0 →K∗0µ+µ− decay mode .1 This set of anomalies is typically interpreted, within an effective field theory approach, as a modification of the effective coupling corresponding to the vector leptonic current operator, known as the Wilson coefficient C(µ)

N Addition, Recent

tests of lepton flavour universality (LFU) in B+, 0 →K+, ∗0ℓ+ℓ−decays, where ℓ= e, µ, indicate that LFU is respected by the measured ratios of branching fractions . Taken together, these results could be explained by a lepton-flavour-universal BSM contribution or by unexpectedly large SM hadronic effects . In particular, contributions from four-quark operators which produce a pair of leptons via the coupling to the electromagnetic current can result in a shift of the C9 Wilson coefficient. Significant effort has been made to quantify the impact of these contributions on the measured observables .

Recently, B0 →K∗0µ+µ−decays have been reanalysed using a data-driven approach to explicitly model hadronic effects . The results of these analyses indicate that the tension seen in the existing measurements cannot be fully explained by nonlocal hadronic contributions. This motivates the angular analysis of B0 →K∗0e+e−decays, which would allow for a test of LFU using angular observables. The results of this test could provide additional information to distinguish LFU BSM contributions from SM hadronic effects .

This paper presents an angular analysis of the B0 →K∗0e+e−decay in the central region of the dilepton invariant mass squared (q2) of 1.1–6.0 GeV2/c4, using proton-proton (pp) data collected by the LHCb experiment corresponding to an integrated luminosity of 9 fb−1. The symbol K∗0 refers to the vector meson K∗(892)0, which is reconstructed in the K+π−final state. The data were collected at centre-of-mass energies of 7 and 8 TeV in 2011 and 2012, respectively (Run1), and at 13 TeV in the years from 2015 to 2018 (Run2), which are split, for the purpose of this analysis, into the periods of 2015–2016 (Run2p1) and 2017–2018 (Run2p2).

The B0 →K∗0e+e−decay rate can be described using q2 and the angles θℓ, θK and ϕ. Here, θℓis defined as the angle between the e+ (e−) direction in the dielectron rest frame and the direction of the dielectron in the B0 (B0) rest frame, θK is the angle between the kaon direction in the K∗0 (K∗0) rest frame and the direction of the K∗0 (K∗0) meson in the B0 (B0) rest frame, and ϕ is the angle between the decay planes of the K∗0 (K∗0) and the dielectron system in the B0 (B0) rest frame . Averaging over the differential decay rates Γ and Γ of B0 and B0 mesons, the angular distribution of the final-state particles 1The inclusion of charge-conjugate processes is implied throughout the paper.

−Fl Cos2 Θk Cos 2Θℓ+ S3 Sin2 Θk Sin2 Θℓcos 2Φ

+ S4 sin 2θK sin 2θℓcos ϕ + S5 sin 2θK sin θℓcos ϕ

Afb Sin2 Θk Cos Θℓ+ S7 Sin 2Θk Sin Θℓsin Φ

+ S8 sin 2θK sin 2θℓsin ϕ + S9 sin2 θK sin2 θℓsin 2ϕ

Here FL is the fraction of longitudinally polarised K∗0 mesons, AFB is the forward-backward asymmetry of the dielectron system, and Si, with i = 3, 4, 5, 7, 8 and 9, are CP-averaged S-basis observables as discussed in Refs. . In addition to the resonant K∗0 (P-wave), the reconstructed K+π−system can also originate from a nonresonant decay, or from the decays of scalar resonances. These S-wave contributions modify the angular distribution, and can be described by introducing six additional terms to Eq. 1 . Nevertheless, given the limited signal yield in the studied data sample, the S-wave contributions are neglected and treated as a source of systematic uncertainty (Sec. 7). The S-basis observables can be used to construct a set of optimised P-basis observables , for which the B0 →K∗0 form-factor uncertainties cancel at leading order . These are given by

Finally, the differences of the angular observables between the muon and electron channels

I

can be determined to directly test LFU in the angular distributions of the decays, as they are expected to be close to zero in the SM . Information from the muon channel is extracted from data samples analysed in Ref. , which is comprised of data recorded by the LHCb detector in Run1 (3 fb−1), and 2016 (1.7 fb−1).

The angular observables of B0 →K∗0e+e−decays have been measured by the LHCb collaboration in the low-q2 region of 0.0008–0.257 GeV2/c4. They strongly constrain the

Wilson Coefficient C′

7 to SM expectations . In addition, the observables P ′

And P ′

have been measured by the Belle collaboration for the decays B+,0 →K∗+,∗0ℓ+ℓ−(where ℓ= e, µ) in different bins of q2 in the 0.1–19.0 GeV2/c4 range, including the central-q2 bin of 1.0–6.0 GeV2/c4. The same analysis also determined Q4 and Q5, and obtained results consistent with SM predictions.

Electron reconstruction at LHCb is challenging due to substantial energy loss caused by photon emission. While part of the lost energy can be recovered using a dedicated algorithm , some degradation of the momentum resolution, and therefore of the invariant masses of the dielectron and B0 candidates, cannot be avoided. This leads to large background contamination, which complicates the signal extraction. Consequently, the angular observables are only determined in a large q2 region with sufficient signal yield.

Moreover, there are significant differences between the measured q2 and the true q2, for which SM predictions are calculated. To minimise these differences, the constrained q2 is used to define the measurement region. This quantity is determined from a fit of the decay chain in which the B0 candidate is constrained to originate from its associated primary vertex (PV) and its invariant mass is constrained to the known mass of the B0 meson .

In addition, the signal simulation is used to parametrise the relationship between the reconstructed angles and q2 and their true values (Sec. 4). The use of the constrained q2 also reduces contamination from the radiative tail of the J/ψ resonance. This allows the same analysis strategy to be applied directly to a larger q2 region of 1.1–7.0 GeV2/c4. The results of the additional measurement in this larger q2 region are reported in Appendix A.

The decay of B0 →K∗0J/ψ(→e+e−), selected within the q2 window of 7.0–11.0 GeV2/c4, is used as a control mode to reduce differences between simulation and data at various stages of the analysis.

The structure of the paper is as follows. Section 2 describes the experimental apparatus and the production of simulation samples. Section 3 details the reconstruction and selection of B0 →K∗0e+e−decays. The method used to correct the measured angular and q2 distributions of the signal is explained in Sec. 4. Section 5 introduces the main background components present in the data sample and describes the methods used to parametrise them. Section 6 discusses the angular fit. Section 7 describes and quantifies the various sources of systematic uncertainties. The results of this analysis are shown and discussed in Sec. 8. The conclusions are presented in Sec. 9.

The Lhcb Detector

The LHCb detector is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, designed for the study of particles containing b or c quarks. The detector includes a high-precision tracking system consisting of a silicon strip vertex detector surrounding the pp interaction region , a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 T m, and three stations of silicon-strip detectors and straw drift tubes placed downstream of the magnet.

The tracking system provides a measurement of the momentum, p, of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV/c. The minimum distance of a track to a PV, the impact parameter (IP), is measured with a resolution of (15 + 29/pT) µm, where pT is the component of the momentum transverse to the beam, in GeV/c. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors (RICH) . Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating- pad and preshower detectors, an electromagnetic calorimeter (ECAL) and a hadronic calorimeter . Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers . The online event selection is performed by a trigger , which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction.

The hardware electron trigger requires the presence of an ECAL cluster corresponding to a transverse energy deposition that exceeds a threshold of around 2–3 GeV, which varies depending on the data-taking period. In addition, the presence of hit(s) in the preshower detector and at least one hit in the scintillating-pad detector in front of the ECAL cluster are required. Events are retained if at least one electron fulfils the electron hardware trigger requirements. Alternatively, they are retained if one or more particles from the rest of the event, identified as muons, electrons or hadrons, satisfy their respective trigger requirements. These particles may originate from the decay of the other b-hadron from the b¯b pair produced in the pp collision that leads to the signal decay. At the software trigger stage, events are retained based on the presence of a two-, three- or four-track secondary vertex that is significantly displaced from any PV, and at least one high-momentum track that has a large IP with respect to all PVs in the event. Multivariate algorithms are used for the identification of secondary vertices that are consistent with b-hadron decays.

Simulation is required to model the effects of the detector acceptance, selection re- quirements and resolution, as well as the signal and background distributions. In the simulation, pp collisions are generated using Pythia with a specific LHCb configura- tion . Decays of unstable particles are described by EvtGen , in which final-state radiation is generated using PHOTOS . The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit as described in Ref. . Data-driven corrections are applied to the simulation to improve the modelling of particle identification (PID) variables, trigger efficiency, track multiplicity per event and the kinematic properties of the B0 meson. The PID variables are corrected using calibration samples consisting of decay modes that can be selected without the use of PID information . Corrections to trigger efficiencies are subsequently applied using per-event weights obtained via a tag-and-probe approach . Finally, a multivariate boosted decision tree reweighter is trained using simulated control-mode decays and background-subtracted B0 →K∗0J/ψ(→e+e−) data to reduce simulation-data differences for quantities related to the track multiplicity of the event, the kinematics of the B0 meson and the PV and B0 decay-vertex (DV) fits. An alternative simulation produced without running PHOTOS and the subsequent simulation of the detector (generator-level simulation), which only contains the effect of the physics model, is also used in the parametrisation of the functions that correct for distortions of the signal distributions (Sec. 4).

Reconstruction And Selection

Signal candidates are reconstructed by combining pairs of oppositely charged tracks identified as electrons, with K∗0 candidates. The electron tracks are required to form a good-quality vertex and to have pT > 500 MeV/c and p > 3000 MeV/c. The K∗0 candidates are reconstructed from oppositely charged tracks, identified as pions and kaons, with pT > 250 MeV/c. Their invariant mass must lie within ±100 MeV/c2 of the known K∗0 mass . All tracks are required to be of good quality, and inconsistent with originating from any PV. For events with multiple PVs, the one with the smallest χ2

P, Defined As The

difference in χ2 between the PV fit with and without the B0 candidate, is associated with the B0 candidate. The four tracks must form a good-quality vertex that is significantly displaced from any PV. The cosine of the direction angle, which is defined as the angle between the momentum vector of the B0 candidate and the displacement vector from its PV to its DV, is required to be close to one. The kaon, pion and electron candidates are required to be within the acceptance of the RICH detectors. The electron candidates are also required to be within the acceptance of the ECAL, and therefore have associated energy clusters. To reduce instances of partly or fully duplicated tracks, minimum values are imposed on the angles between pairs of final-state particle tracks associated with the B0 decay. To exclude phase-space regions where the acceptance is not well modelled due to the nonuniform generator-level distribution or very low selection efficiency, candidates with | cos θℓ| < 0.9 and cos θK < 0.9 are removed (Sec. 4).

Two types of PID variables are used for background suppression, as was done in Refs. . The first type of variables is the difference in log-likelihoods between particle hypotheses x and the pion hypothesis (DLLxπ), which uses information from the RICH, calorimeter and muon systems to quantify the likelihood of a track being associated with particle x, such as a proton, kaon or electron, relative to its likelihood of being associated with a pion. The second is the output of artificial neural networks trained using information from all subdetectors, which can be interpreted as the probability of a track to be associated with particle x (ProbNNx). The misidentification of pions as kaons is suppressed by a minimum requirement on DLLKπ. Additionally, combinations of ProbNNK, ProbNNp and ProbNNπ are used to suppress proton-to-kaon misidentification and the misidentification of kaons and protons as pions. Electron misidentification is suppressed via requirements on DLLeπ and ProbNNe.

Specific sources of backgrounds are suppressed using dedicated selection re- quirements (vetoes). Vetoes detailed in Refs. [15, 16] are used to suppress

B0

s →ϕ(→K+K−)e+e−decays with kaon-to-pion misidentification, semileptonic decays such as B0 →D0(→K+π−)π−e+νe and B0 →D−(→K∗0(→K+π−)π−)e+νe with pion- to-electron misidentification, B+ →K+e+e−decays reconstructed with the addition of a random pion track, and B0 →K∗0J/ψ(→e+e−) and B0 →K∗0ψ(2S)(→e+e−) decays with the misidentification of a hadron as an electron and vice versa. In addition, semilep-

S Decays Featuring D−

s mesons with kaon-to-electron misidentification are vetoed in analogy to the other semileptonic decays listed above, and double misidentification of the kaon and pion tracks of a signal decay is reduced by requiring the DLLKπ of the kaon to be larger than the DLLKπ of the pion. The contamination from Λ0

B →Pk−E+E−Decays Is

found to be negligible using simulation. For the control mode, to suppress the partially reconstructed background which is distributed in a lower mass range than the signal, an invariant mass is calculated by constraining the B0 candidate to originate from its associated PV and the dielectron invariant mass to the known J/ψ mass, and is required to be larger than 5150 MeV/c2.

After applying the aforementioned requirements, the dominant source of background is comprised of candidates reconstructed from random tracks. This combinatorial background is suppressed using Boosted Decision Tree (BDT) classifiers . One BDT is trained for each run period. Simulated signal decays are used as a proxy for the signal, where the simulation is corrected to better describe the data, and data candidates with an invariant mass above 5600 MeV/c2 are used as the background proxy. To increase the background sample size available for training, the baseline PID requirements are loosened and the K∗0 mass window is enlarged to ±200 MeV/c2. The k-folding technique is used with ten folds to make use of the full simulation and data samples available. Fourteen variables are

Fb−1

Figure 1: Distribution of q2 and B0 invariant mass of signal candidates in data. Signal decays lie in a vertical band close to the known B0 mass (vertical dashed line) . The decays B0 →K∗0J/ψ(→e+e−) and B0 →K∗0ψ(2S)(→e+e−) have an invariant mass close to that of the B0 meson, and q2 values close to the square of the known J/ψ and ψ(2S) masses, respectively (horizontal dashed lines). They are visible as horizontal bands. The two diagonal bands contain combinatorial background comprised of genuine J/ψ or ψ(2S) mesons combined with random kaon and pion tracks.

selected as inputs to the classifier through an optimisation procedure that systematically examines the performance of the BDT classifiers trained with different sets of variables. This final set includes variables related to the B0 candidate, namely its pT and χ2

), The

cosine of its direction angle, and the quality of the constrained kinematic fit of the decay

Tf). The Χ2

DV of the K∗0 candidate and that of the dielectron are also used. The

And Maximum Pt And Χ2

IP of the two electrons. The threshold on the classifier output is set in order to minimise the statistical uncertainty of P ′

One Threshold Is Used For All

run periods and the same classifier is used for both the signal and control modes. The invariant mass of the B0 candidate is calculated from a fit to the decay chain where the B0 meson is constrained to originate from its associated PV, and used to separate signal from backgrounds. The distribution of the invariant mass versus q2 for the B0 candidates is shown in Fig. 1 (the analogous distribution for the unconstrained q2 is shown in Fig. 17 of Appendix B). The number of control-mode candidates that leak into the signal q2 region is further reduced by requiring the invariant mass of the B0 candidates to lie within the restricted range of 4900–5700 MeV/c2. A mass window of 4500–6200 MeV/c2 is used for the control mode.

After applying all the aforementioned requirements, less than one percent of the signal and control-mode events in simulation and data have multiple candidates. In these cases, one randomly selected candidate is retained.

Effective Acceptance

The measured distributions of the decay angles and q2 are distorted by final state radiation, bremsstrahlung and detector resolution, as well as triggering, reconstruction, and selection processes. These effects are included in the signal simulation, which is used to obtain an effective acceptance function that corrects for all distortions simultaneously. First, a function that describes the generator-level sample is obtained, and is used to assign weights to each selected candidate in the simulation. The weighted distribution is then parametrised by a function which represents the effective acceptance. The inverse of the value of this function, evaluated for each signal candidate, is used as a weight (effective acceptance weight) to correct the signal distribution. The effective acceptance function obtained in this way depends on the physics model used in the simulation, which is a source of systematic uncertainty that is discussed in Sec. 7.

The effective acceptance is parametrised using a sum of Legendre polynomials and trigonometric functions in terms of the measured angles and q2, with the latter variable

where La(x) is the Legendre polynomial of order a for the variable x, and the Fm(ϕ) term

where m is zero or a positive integer. The use of sine and cosine terms for ϕ is motivated by the Fourier expansion in this angle. The cklmn coefficients are calculated through a principal moments analysis of simulated signal decays, exploiting the orthogonality property of the Legendre polynomials and trigonometric functions.

These functions are first used to parametrise the generator-level simulation, in terms of the true angles and q2. In this case, orders of five, four, four, and nine are used for cos θK, cos θℓ, ϕ and q2, respectively. A q2 range of 0.5–8.0 GeV2/c4 is used to avoid pathologies that can occur near the edges of the range used in the measurement. For the effective acceptance function, parametrised in terms of the measured angles and q2, orders of five, four, four and three are used to describe the weighted cos θK, cos θℓ, ϕ and q2 distributions, respectively. One function is parametrised for each of the three run periods using the corresponding simulation samples. The one-dimensional projections of the functions used to describe the generator-level distributions and the effective acceptance are shown in Appendix C.

The quality of the parametrisation is checked by comparing the values of the angular observables obtained by a maximum-likelihood fit of the generator-level simulation using Eq. 1, and those from a fit performed to the selected candidates of the signal simulation, where effective acceptance weights are included. This check is performed separately for the three run periods, using a k-folding strategy. The results show good retrieval of the generator-level observable values (Appendix C), with residual differences originating mainly from the imperfect description of some selection requirements, such as the requirements used to suppress semileptonic B0 decays and B+ →K+e+e−decays (Sec. 3).

The

systematic uncertainty due to imperfect parametrisation is quantified in Sec. 7.

Nvariant-Mass And Angular Models

The angular observables are determined by fitting the distributions of the invariant mass of the B0 candidate, cos θK, cos θℓand ϕ weighted with effective acceptance weights. In addition to the signal component, the fit model includes four sources of background: combinatorial background; double-semileptonic decays of b-hadrons with two electrons in the final state; partially reconstructed decays mediated by b→sℓ+ℓ−transitions with more than two hadrons in the final state, dominated by B+ →K+π+π−e+e−decays; and a mixture of b-hadrons decays with the misidentification of one or more hadrons as electrons, with or without missing energy.

The signal invariant-mass model is parametrised using simulation, and its peak position and width are subsequently corrected to match those in data. These corrections are determined by fitting the invariant-mass distribution of the control mode. Four components are included in that case: B0 →K∗0J/ψ(→e+e−) decays, B0

B →Pk−J/Ψ(→E+E−) Decays With Proton-To-Pion

misidentification. Effective acceptance weights are included in the parametrisation of the invariant- mass distribution of the signal component, as well as the invariant-mass and angular distributions of the backgrounds. Different signal invariant-mass models, and background invariant-mass and angular models are used for each run period with the exception of the misidentified hadronic background, for which the same model is used for the periods of Run2p1 and Run2p2 due to the limited size of the data samples. The models used to describe the background components of the signal and control modes are discussed below and illustrated in Appendix D.

Signal And Control Modes

The signal angular distribution is described by Eq. 1, and its invariant-mass distribution is described by the sum of two Crystal Ball functions (DCB) that share common parameters describing the mean and width of the peak, but have independent power-law tails on opposite sides of the peak. The impact of photon energy recovery on the signal mass distribution is significant, therefore separate models are used depending on the numbers of electrons that receive corrections (zero, one or two). The full mass model is given by the sum of the three separate contributions.

For the control mode, due to a larger B0 mass window, additional Gaussian components are used in the modelling of the contributions where one or both electrons receive corrections to improve the description of the tails of the distributions. Due to residual differences between simulation and data, the mean and width parameters determined from simulation are modified via a shift and a scaling parameter, respectively, which are determined using the control mode and are subsequently fixed in the signal-mode fits.

Ombinatorial And Double-Semileptonic Backgrounds

Candidates reconstructed using unrelated electron or hadron tracks, hereafter referred to as the combinatorial background, have smooth, factorisable invariant-mass and angular distributions, which can be described by the product of an exponential function and three one-dimensional polynomials. Factorisation holds due to the random nature of this background, and is only broken in a small phase-space region by the requirements to veto B+ →K+e+e−decays, for which a dedicated systematic uncertainty is assigned (Sec. 7).

Double-semileptonic (DSL) decays refer to semileptonic decays of a beauty hadron to a charm hadron that subsequently decays semileptonically, most often, into a strange hadron.

When the two semileptonic decays produce two electrons, a kaon and a pion in the final state, the corresponding signal candidate can satisfy all selection requirements.

The most important contribution is expected to originate from the B0 →D−(→K∗0(→K+π−)e−¯νe)e+νe mode, which has a relatively large branching frac- tion of O(10−3) . Due to the missing neutrinos, the corresponding invariant-mass distribution can be described by an exponential function. This makes the separation of combinatorial and DSL backgrounds using mass information alone challenging. However, the DSL background is characterised by an asymmetric cos θℓdistribution peaking close

S (→K∗0E−¯Νe)E+Νe, On The Other Hand,

are negligbly small. The DSL decays cannot be well described by a fully factorised ap- proach due to non-negligible correlation between cos θK and ϕ. Therefore this component is described by Eq. 1, integrated over cos θℓ.

Additional background contributions include DSL decays with the misidentification of one or more final-state particles, or reconstructed with one or more random tracks. Given the complexity of these backgrounds, a two-step data-driven approach is used, where effective DSL and combinatorial models are obtained using background B0 candidates reconstructed from the K+π−e+µ−final state. In the first step, the DSL angular model is obtained using a sample enriched in DSL decays selected with a stringent BDT requirement and falling within a lower mass window of 4500–5200 MeV/c2. The cos θℓdistribution is modelled using kernel density estimation (KDE) while cos θK and ϕ are described by integrating Eq. 1 over cos θℓ. In the second step, candidates passing the baseline BDT requirement and lying in the signal mass window are fitted to determine the angular parameters of the combinatorial background, and the slopes of the exponential mass distributions of the DSL and combinatorial components. In this fit, the DSL angular parameters are fixed and polynomials up to second order are used for the three angles.

Candidates used in the first step are excluded from the second. In this way, contributions that show characteristics intermediate between the combinatorial and DSL backgrounds are split between the effective DSL and combinatorial background models. The assumption that the ratio and the shapes of these two contributions are the same for the K+π−e+e− and K+π−e+µ−final states is a source of systematic uncertainty, which is quantified in Sec. 7.

Isidentified Hadronic Decays

Several hadronic decays with the misidentification of kaons or pions as electrons are known to satisfy the selection criteria . Decays of the type B0 →K∗0π−(π0, γ)X, where X represents any possible final-state particles, can contribute. These predominantly populate the lower mass region, although some contributions, such as the fully reconstructed, misidentified decays of B0 →K∗0K+K−and B0 →K∗0π+π−, peak close to the known mass of the B0 meson. As a result, they cannot be described by the combinatorial model.

The data-driven approach developed in Refs. is used to estimate the yields and determine the model for this type of background. Data samples enriched in misidentified hadrons are obtained by inverting the baseline electron PID criteria. The region in the electron PID space defined by these inverted criteria is hereafter referred to as the control region. These samples contain a mixture of fully reconstructed misidentified decays, partially reconstructed decays with or without misidentification, combinatorial background, and residual signal and control-mode decays. Yield ratios are calculated in regions of transverse momentum and pseudorapidity using the number of kaon and pion candidates from PID calibration samples that are within the control and baseline regions. The resulting maps, or transfer functions, are used to extrapolate the yields and distributions of this type of background in the baseline region.

Residual signal and control-mode decays are present in the control-region sample. The contribution from signal decays is subtracted using weights based on the yield obtained from an initial data fit where the misidentified hadronic background component is ignored.

Similarly, control-mode candidates are subtracted based on the yield of the control-mode fit. Adaptive KDEs from Refs. are used to model the invariant-mass distributions of the control-region candidates. Due to similarities between the angular distributions of these candidates and that of the effective DSL background, the same functions are also used in this case: a KDE is used for cos θℓand Eq. 1, integrated over cos θℓ, is used for cos θK and the ϕ angle.

Partially Reconstructed Background

The main sources of the partially reconstructed background include decays to heavier kaon resonances, such as B+ →K1(1270)+e+e−or B+ →K∗

K1(1270)+ And K∗

2(1430)+ mesons decay to a kaon, a pion and one or more additional pions. Due to the missing particle(s), the reconstructed B0 invariant-mass distribution shows a broad peak centred in the lower mass region.

The relatively large number of contributing kaon resonances motivates the use of a data-driven approach, where the simulation of B+ →K+π+π−e+e−phase-space decays is corrected using weights obtained from efficiency-corrected and background-subtracted B+ →K+π+π−J/ψ(→µ+µ−) data [15, 16, 67]. These weights are assigned to the sim- ulated (background) candidates based on their K+π+π−, K+π−and π+π−invariant masses. The weighted angular and invariant-mass distributions are then used to model the corresponding background. A KDE is used to model the invariant-mass distribution, and factorised second-order polynomials are used for the three angles.

Ontrol-Mode Backgrounds

Backgrounds of the control mode can be divided into those with an invariant-mass distribution that can be described by an exponential function, which includes combinatorial, DSL and partially reconstructed decays, and those that peak close to the known B0 mass,

The B0

s decays are suppressed with respect to the B0 decays by the ratio of hadronisation fractions, fs/fd , and branching fractions. They form a peak centred at the known

B0

s mass. The same model used for B0 →K∗0J/ψ(→e+e−) decays is used to describe this component, with a shift in the mean value of 87 MeV/c2 , which corresponds to

B

decays, which is also subdominant, is modelled using simulated Λ0

B →Pk−J/Ψ(→E+E−)

phase-space decays with data-driven corrections obtained from background-subtracted

Λ0

b →pK−J/ψ(→µ+µ−) data . These weights are assigned to simulated candidates based on their K−p and J/ψp invariant masses. Their weighted invariant-mass distribution is modelled using a KDE.

Nvariant-Mass And Angular Fit

A weighted maximum-likelihood fit is performed simultaneously to Run1, Run2p1 and Run2p2 data to determine the angular observables. The parameters that describe the signal invariant-mass distribution and the angular distributions of background components are obtained as discussed in Sec. 5. Different shift and scaling parameter values are obtained for each run period from control-mode fits, in which the fractions of B0

And Misidentified Λ0

b →pK−J/ψ(→e+e−) decays are fixed to expected values, and the slope of the combinatorial background is allowed to vary. The result of these fits are shown in Fig. 23 of Appendix D.

The fractions of signal and DSL decays are allowed to vary freely for each run period, while those of the misidentified hadronic decays are allowed to vary, but are constrained based on their expected yields. To improve fit stability, the fraction of partially reconstructed decays is expressed by the signal fraction multiplied by a factor that is shared among all run periods and allowed to vary. The sum of all the fractions, including the combinatorial background, is constrained to be one. The slopes of the combinatorial mass distributions are allowed to vary separately for each run period. As the fit is performed using weighted events, the asymptotically correct approach described in Ref. is used to calculate uncertainties on the fitted parameters. The fit projections are shown in Fig. 2.

The fit strategy is validated by means of pseudoexperiments generated with the baseline model, using the observables and other parameter values determined from the data fit. The results show no sizable biases in the angular observables, or significant overestimation or underestimation of uncertainties. Nevertheless, all biases found are taken into account as systematic uncertainties, and the widths of the pull distributions are used to correct the uncertainties of the data fit.

Systematic Uncertainties

Sources of systematic uncertainties include the choices made in the modelling of back- grounds, the determination of the effective acceptance, the description of the signal invariant-mass distribution, the impact of neglecting backgrounds that contain J/ψ mesons and the S-wave contribution, the impact of the requirements to suppress B+ →K+e+e− decays, and the fit biases. The effect of all sources is quantified using pseudoexperiments.

In most cases, an alternative model is defined, and pseudoexperiments are generated

Hcb 9 Fb−1

Figure 2: Invariant-mass and angular distributions of selected candidates after the application of effective acceptance weights. The signal distribution is shown with a solid blue line, and the background components are shown with dashed, dotted and dash-dotted lines. The solid black line corresponds to the full fit function.

with this model. Then, they are fitted with both this alternative and the baseline model, which results in two sets of observable values. The differences between these values are calculated for each observable and pseudoexperiment. The systematic uncertainty for a given observable is then obtained from the sum in quadrature of the mean and Gaussian width of the distribution of the resulting differences. When fitting with an alternative model proves infeasible, pseudoexperiments generated with the alternative configuration are fitted using the baseline model only, and the resulting biases are taken as systematic uncertainties.

Sources of systematic uncertainties for the S- and P-basis observables are summarised in Tables 1 and 2, respectively, where the values correspond to the systematic uncertainties divided by the statistical uncertainties. They are discussed in detail below.

All sources of systematic uncertainties discussed below are expected to be uncorrelated, and the total uncertainty is the sum in quadrature of all sources. Correlations may arise between different observables. The correlation matrix of total systematic uncertainties on the observables is given in Appendix E.

Systematic Uncertainties Can Arise

from different sources, namely the limited size of the K+π−e+µ−data sample, the choice of the models, and the assumption of factorisation. The impact of the limited sample size is studied using bootstrapping techniques . Systematic uncertainties related to model choice are quantified using alternative models. To examine the impact of the parametrisation strategy, alternative functions are used to describe both components. For Table 1: Summary of the systematic uncertainties on the S-basis angular observables. All values are given as fractions of the statistical uncertainties.

Total

Table 2: Summary of the systematic uncertainties on the P-basis angular observables. All values are given as fractions of the statistical uncertainties.

Total

the combinatorial background, cos θK and cos θℓare described by polynomials up to third order, ϕ is described using trigonometric terms up to fourth order, and a Gaussian function is used to describe its invariant-mass distribution. For the DSL background, cos θK and ϕ are described by an unfactorised model consisting of polynomials and trigonometric terms up to second and third order, respectively, cos θℓis described by a parametric model composed of four Gaussian functions, and a Gaussian function is used to describe its invariant-mass distribution. To quantify systematic uncertainties related to the use of the K+π−e+µ−sample to model backgrounds in the K+π−e+e−final state, for the DSL background, an alternative model is obtained using simulated B0 →D−(→K∗0e−¯νe)e+νe decays. For the combinatorial background, a different model is obtained via an alternative data fit for cos θℓand ϕ, and from the same-sign K+π−e±e± data for cos θK. The impact of the factorisation assumption is studied using a fully unfactorised model for both backgrounds determined from the K+π−e+µ−data sample.

Systematic Uncertainties Associated

with the partially reconstructed background are relatively small as its contribution is limited by the narrow signal mass window. Furthermore, given the large sample size available for parametrisation, the only relevant sources of uncertainties are the parametri- sation strategy, the choice of the physics model, and the assumption of factorisation. The impact of the parametrisation strategy is quantified using an alternative model, where the maximum polynomial orders are increased to three for cos θK and cos θℓ, trigonometric functions up to fourth order are used for ϕ, and a parametric model composed of two Gaussian functions and one exponential function is used to describe the invariant-mass distribution. The systematic uncertainty associated with the choice of the physics model is quantified using an alternative model obtained from simulated B+ →K1(1270)+e+e−

And B+ →K∗

2(1430)+e+e−decays. The impact of the factorisation assumption is assessed by bootstrapping the B+ →K+π+π−e+e−simulation.

Sources Of Systematic Uncertainties

related to the modelling of misidentified hadronic decays include the parametrisation method, the definition of the control region, the impact of potential dependencies on the event occupancy, and the size of the control-region sample. The systematic uncertainties of the parametrisation method is quantified using an alternative parametric model composed of two Gaussian functions and an exponential function, a model which was used in Refs. . Polynomials of up to sixth order are used to describe the cos θℓdistribution.

For cos θK and ϕ, an alternative unfactorised model consisting of a product of polynomials of up to second order is used. To quantify systematic uncertainties associated with the choice of the baseline control region, a different electron PID requirement is used.

Alternative transfer functions that depend on the transverse momentum and the number of hits in the scintillating-pad detector are made to quantify systematic uncertainties associated with potential dependencies on the event occupancy. The systematic uncertainty related to the size of the control-region sample is assessed by bootstrapping.

Effective Acceptance Functions

Sources of systematic uncertainties associated with the effective acceptance functions include the choice of the polynomial order used in their parametrisation, the size of the simulation samples used to calculate the coefficients, the model dependence of the resolution correction, and the strategy used to correct for differences between simulation and data.

These Are Evaluated Using Signal-Only

pseudoexperiments. The systematic uncertainties associated with the choice of the polynomial order are quantified by increasing all polynomial orders by three. The impact of the limited size of the simulation samples is assessed by bootstrapping to produce alternative effective acceptance functions. To quantify the systematic uncertainties due to the resolution correction, alternative effective acceptance functions are parametrised after modifying the simulated distributions using weights that remove the effects of the baseline physics model, and introduce those of the alternative physics model. Three plausible BSM scenarios are considered: a) δC9 = −1, b) δC9 = −δC10 = −0.7 and c) δC9 = −1.4. Systematic uncertainties are quantified for each case separately. The largest uncertainty found in scenarios a) to c) for each observable is taken as the systematic uncertainty. The simulation correction strategy affects the analysis primarily through the impact of the resulting per-event weights on the shape of the effective acceptance functions.

The approach in Refs. [15, 16] is used to quantify systematic uncertainties associated with the baseline simulation correction strategy. The full set of per-event weights from this alternative correction strategy is used in the parametrisation of alternative effective acceptance functions. The limited size of the PID calibration sample for the electron mode introduces another source of systematic uncertainty. This is quantified using alternative acceptance functions parametrised with the baseline approach, with the exception that all PID requirements are made on alternative PID variables obtained by bootstrapping.

Signal Invariant-Mass Model

The invariant-mass distribution of signal candidates is modelled using DCB functions with shift and scaling parameters determined from the control-mode fit. This choice, as well as aspects of the control-mode fit and the assumption of factorisation between invariant mass and angles, are sources of systematic uncertainties. An alternative model obtained by applying KDE to all signal candidates of each run period is used to quantify the systematic uncertainties associated with the parametrisation strategy. Two alternative sets of shift and scaling parameters are used to quantify the systematic uncertainties of the baseline choice. The first is obtained by making a control-mode fit where the requirement used to suppress partially reconstructed background is removed and additional backgrounds are included. The second is obtained by making a fit in the same mass range as the signal. The systematic uncertainty due to the imperfect modelling of invariant-mass distributions in data due to residual simulation-data differences is assessed using alternative mass models, where these differences are removed using weights. The assumption of factorisation is broken by electron energy loss, which introduces correlations between the B0 invariant mass and cos θℓ. The signal simulation is bootstrapped to quantify the impact of this assumption.

J/Ψ Backgrounds

Two types of backgrounds with distinct invariant-mass and angular distributions that are neglected in the fit are B0 →K∗0J/ψ(→e+e−) decays that leak into the signal q2 region due to significant energy loss, and J/ψ mesons combined with random kaon and pion tracks. In each case, the impact of neglecting the component is assessed by generating pseudoexperiments with its inclusion, and fitting them with the baseline strategy.

S-Wave Component

The angular function used in the fit describes decays where the K+π−system originates from the K∗(892)0 vector meson, and does not include S-wave pseudoexperiments generated with six additional angular terms, and fitted with and without their inclusion. Fits are performed to the B0 →K∗0µ+µ−candidates from the data samples analysed in Ref. , to obtain realistic values for the additional angular observables.

B+ Veto

The requirements to veto B+ →K+e+e−decays distort the background distribution above the known B0 mass, and introduce correlations between invariant mass and cos θK. Furthermore, due to the removal of events in a region of the phase space, the distortions caused by this veto cannot be corrected properly using the effective acceptance functions. To quantify the associated systematic uncertainties, an alternative data fit is made without this veto, using effective acceptance functions and signal and background models obtained likewise without it. The result of this fit is used to generate pseudoexperiments, which are fitted once using the no-veto models and weights. Then, candidates are removed using binned efficiency values from a histogram model that describes the effect of this veto in three dimensions (q2, cos θK and B0 invariant mass) .

The pseudoexperiments are fitted again using the baseline setup.

Fit Bias

The biases found in the validation of the fit strategy using pseudoexperiments generated with data-fit observable values are all limited in size, with the largest found for P2 at around 8% of the statistical uncertainty. These values are taken as systematic uncertainties.

The Systematic Uncertainties, Summarised

in Tables 1 and 2, are not negligible compared to the statistical uncertainties. For all S-basis observables, they amount to more than 40% of their statistical uncertainties. The most affected observable is FL, which is easily biased by differences in cos θK when changing between the baseline and alternative background models, neglecting correlation between cos θK and B0 invariant mass, and differences between the baseline and alternative effective acceptance corrections for cos θK. The observable AFB is sensitive to effects that are not symmetric in cos θℓ; in particular, it is biased by the difference between the width of the cos θℓpeak of the baseline and alternative DSL model. Other observables, such as S4 and S5, have more complex dependencies on the decay angles, and so the impact of one-dimensional effects is reduced. However, multidimensional effects due to the correlated effective acceptance functions as well as angular correlations and correlations between angles and B0 invariant mass present for some backgrounds can lead to non- negligible systematic uncertainties. The P-basis observables, which incorporate FL in their definitions, have much larger systematic uncertainties of over 80% of their respective statistical uncertainties in all cases.

Results

The angular observables of B0 →K∗0e+e−decays, measured in the q2 region of 1.1–6.0 GeV2/c4, are shown in Fig. 3 together with the SM predictions based on Refs. [12, 14, 72]. This is the most precise measurement of the B0 →K∗0e+e−angu- lar observables in the central-q2 region to date, improving the precision on P ′

Ata

Figure 3: The (left) S- and (right) P-basis angular observables. The overlapping error bars show statistical and total uncertainties. The orange and hatched purple boxes correspond to SM predictions based on Ref. and Refs. , respectively.

Table 3: Values for the (left) S- and (right) P-basis angular observables. The first uncertainty is statistical and the second is systematic.

−0.16 ± 0.11 ± 0.11

previously measured by Belle , by more than a factor of two. The measured observables are broadly in agreement with the SM predictions, with the largest differences of around 2σ found for FL and AFB, followed by differences of around 1.5σ for S8 and P ′

The Numerical

results of the fit are given in Table 3, with statistical and systematic uncertainties shown separately, and fit correlation matrices are reported in Tables 13 and 14 of Appendix E. Statistical correlations between the observables are generally small. The largest correlation of around 14% is found between S7 and S9 for the S-basis observables, and 20% between FL and P2 for the P-basis observables. Correlations between systematic uncertainties are discussed in Appendix E.

To determine the LFU observables Qi, which are given by the differences between the P-basis angular observables of the muon and the electron modes , the analysis strategy used in this measurement is applied to the B0 →K∗0µ+µ−sample analysed in Ref. : information from the K−π+ system is not used, S-wave and interference terms are neglected, the acceptance functions are parametrised using trigonometric terms for ϕ, and a weighted maximum-likelihood fit is performed to the B0 invariant mass and the three decay angles. The result of this fit is shown in Appendix F. After this alignment in the fit strategy, only the systematic uncertainties of the electron mode need to be considered, as those of the muon mode are negligibly small in comparison. The Qi observables are summarised in Fig. 4, and the numerical values are given in Table 4. The statistical uncertainties of the Qi observables are obtained by summing the fit uncertainties of the electron and muon modes in quadrature. Most observables show good agreement with the LFU hypothesis. The largest difference of around 2σ is found for QFL. The individual angular and LFU observables are shown in Figs. 5, 6 and 7.

In order to quantify the agreement with the SM and evaluate the constraints imposed by this measurement on the LFU hypothesis, a global fit to the angular observables is performed, varying the C9 coefficient, as motivated by Refs. . The signal decay amplitudes are parametrised based on Ref. with the local form-factor parameters taken from Ref. Nonlocal hadronic contributions are treated as in Refs. ; they are modelled based on their analytic structure by means of a z-expansion truncated at second order, constrained to their theoretical predictions at q2 < 0 and fitted to the binned angular observables in the physical q2 range. All local and nonlocal hadronic contributions are known to be lepton-flavour universal and are therefore shared between

Ata

Figure 4: LFU observables Qi calculated using the P-basis angular observables of the muon and electron modes. The overlapping error bars show statistical and total uncertainties. The SM predictions (orange boxes) are based on Ref. .

the muon and electron modes. Finally, the C9 coefficient is varied independently for the B0 →K∗0e+e−and B0 →K∗0µ+µ−decay channels. The fit inputs consist of the angular observables in B0 →K∗0e+e−decays and those measured in the five narrower bins up to 8 GeV2/c4 for B0 →K∗0µ+µ−decays . Figure 8 (left) shows the result

, When Both Coefficients Are Varied

independently. A negative shift in the values of C(e)

Of The Order Of −1 With Respect To

the SM prediction is required to describe the measured B0 →K∗0e+e−angular observables, with a significance above 2σ, which is similar to what is found for the muon

Is Found To Be Around 40%, Indicating A

residual correlation between the two results due to the common choice of the amplitude model. Finally, in order to quantify the compatibility of the results with the LFU hypothesis, a negative log-likelihood scan of ∆C9 = C(µ)

Is Made, And Shown In

Fig. 8 (right). In this case, since the focus is purely on the detection of possible LFU- breaking effects, the theoretical inputs of Ref. are removed, and only the differences between the angular observables of the muon and electron channels measured in the region of 1.1–6.0 GeV2/c4 (Table 4) are used as inputs. The resulting ∆C9 value is found to be compatible with zero within one standard deviation, which is fully consistent with the LFU hypothesis in b→sℓ+ℓ−transitions.

Table 4: Values for the Qi LFU observables given by the differences between the muon and electron P-basis angular observables. The first uncertainty is statistical and the second is systematic.

Grvdv

Figure 5: Measured S-basis angular observables. The orange and hatched purple boxes correspond to SM predictions based on Ref. and Refs. , respectively.

Grvdv

Figure 6: Measured P-basis angular observables. The orange and hatched purple boxes corre- spond to SM predictions based on Ref. and Refs. , respectively. The values of P ′

P ′

5 measured by Belle for the decays of B+,0 →K∗+,∗0e+e−are shown in light blue.

Abcdmn

Figure 7: Measured Qi LFU observables compared with the SM predictions based on Ref. . The values of Q4 and Q5 measured by Belle for the decays of B+,0 →K∗+,∗0ℓ+ℓ−, where ℓ= e, µ, are shown in light blue.

Hcb 9 Fb−1 [4.7 Fb−1]

Figure 8: Negative log-likelihood scan of (left) Ce

The

dotted vertical line corresponds to the SM prediction .

Onclusions

This paper presents an angular analysis of B0 →K∗0e+e−decays performed in the central-q2 region of 1.1–6.0 GeV2/c4 using LHCb pp data collected between 2011 and 2018, corresponding to an integrated luminosity of 9 fb−1. Angular observables are extracted using a weighted maximum-likelihood fit to the invariant-mass and angular distributions of B0 →K∗0e+e−decays, where the weights correct for distortions of the signal distribution caused by acceptance and resolution effects. The results presented here are the most precise to date. Overall, the set of angular observables show good agreement with the SM predictions. Discrepancies at the level of 2σ are observed for FL and AFB, which are consistent with the hypothesis of a negative shift in the value of the Wilson coefficient C9 reported by global analyses of other b→sℓ+ℓ−transitions . Finally, no strong sign of LFU violation is observed when the angular observables of the B0 →K∗0e+e−and B0 →K∗0µ+µ−decays are analysed together.

the excellent performance of the LHC. We thank the technical and administrative staff agencies: ARC (Australia); CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); NWO (Netherlands); MNiSW and NCN (Poland); MCID/IFA (Romania); MICIU and AEI (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources that are provided by ARDC (Australia), CBPF (Brazil), CERN, IHEP and LZU (China), IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF (Netherlands), Polish WLCG (Poland), IFIN-HH (Romania), PIC (Spain), CSCS (Switzerland), and GridPP (United Kingdom).

We are indebted to the communities behind the multiple open-source software packages on which we depend. Individual groups or members have received support from Key Research Program of Frontier Sciences of CAS, CAS PIFI, CAS CCEPP, Fundamental Research Funds for the Central Universities, and Sci. & Tech. Program of Guangzhou (China); Minciencias (Colombia); EPLANET, Marie Sk lodowska-Curie Actions, ERC and NextGenerationEU (European Union); A*MIDEX, ANR, IPhU and Labex P2IO, and R´egion Auvergne-Rhˆone-Alpes (France); Alexander-von-Humboldt Foundation (Germany); ICSC (Italy); Severo Ochoa and Mar´ıa de Maeztu Units of Excellence, GVA, XuntaGal, GENCAT, InTalent-Inditex and Prog. Atracci´on Talento CM (Spain); SRC (Sweden); the Leverhulme Trust, the Royal Society and UKRI (United Kingdom).

Arge Q2 Region Analysis

The use of the constrained q2 variable to define the measurement region suppresses background from the radiative tail of the B0 →K∗0J/ψ(→e+e−) mode (Fig. 18), and allows the analysis strategy discussed in Secs. 4 to 6 to be applied directly to the larger q2 region of 1.1–7.0 GeV2/c4, resulting in a gain in the signal yield of around 20%. The results of the measurement in this extended region are presented and discussed in this self-contained Appendix.

Compared to the baseline measurement in the region of 1.1–6.0 GeV2/c4, the only differences are the signal and background models, which are obtained from data or simulation samples within the extended region. The same effective acceptance functions discussed in Sec. 4, which are parametrised in the q2 region of 0.5–8.0 GeV2/c4, are used.

The results of the parameterisation validation are shown in Fig. 9. The data fit projections are shown in Fig. 10. All systematic uncertainties are quantified using the methods discussed in Sec. 7, and summarised in Tables 5 and 6. The measured observable values are displayed in Fig. 12, and their corresponding numerical values are given in Table 7.

The correlations among the observables are given in Tables 9 and 10, for the statistical uncertainties, and Tables 11 and 12, for the systematic uncertainties, respectively. To calculate the Qi observables, a muon-mode fit is performed using the strategy detailed in Sec. 8, but in the extended q2 region. The result of this fit is shown in Fig. 11.

The LFU observables are displayed in Fig. 13 and given in Table 8. Figures showing all observables individually, including results from both q2 regions, are given in Appendix A.7.

Alidation Of The Effective Acceptance Functions

The results of the validation of the effective acceptance parametrisation for the large q2 region are shown in Fig. 9.

Run2P2

Figure 9: Differences between the observable values found from fits to the signal simulation samples with acceptance correction weights, and the fit to the generator-level sample (centred at zero) in the large q2 region.

Fit Results In The Large Q2 Region

The result of the weighted maximum-likelihood fit performed to the B0 invariant-mass and angular distributions of signal candidates selected in the large q2 region of 1.1–7.0 GeV2/c4 is shown in Fig. 10.

Hcb 9 Fb−1

Figure 10: Invariant-mass and angular distributions of selected candidates in the large q2 region after the application of effective acceptance weights. The signal distribution is shown with a solid blue line, and the background components are shown with dashed, dotted and dash-dotted lines. The solid black line corresponds to the full fit function.

Systematic Uncertainties

Systematic uncertainties associated with the observables of the large q2 region are sum- marised in Tables 5 and 6. Table 5: Summary of the systematic uncertainties on the S-basis angular observables of the large q2 region. All values are given as fractions of the statistical uncertainties.

Total

Table 6: Summary of the systematic uncertainties on the P-basis angular observables of the large q2 region. All values are given as fractions of the statistical uncertainties.

Fit To The B0 →K∗0Μ+Μ−Decay

The results of the weighted maximum-likelihood fit performed to the B0 →K∗0µ+µ− candidates in the unconstrained q2 region of 1.1–7.0 GeV2/c4 are shown in Fig. 11.

Hcb 1.7 Fb−1

Figure 11: Weighted invariant-mass and angular distributions of B0 →K∗0µ+µ−candidates in the (top two rows) Run1 and (bottom two rows) 2016 data samples within the large q2 region. The signal distribution is shown with a solid blue line, and the combinatorial background is shown with a dashed green line. The solid black line corresponds to the full fit function.

Results For The Large Q2 Region

The angular observables measured in the extended q2 region of 1.1–7.0 GeV2/c4 are summarised in Fig. 12, and their numerical values are given in Table 7. In general, these values are in good agreement with both sets of SM predictions, with differences of around 2σ or less. Observables that differ with respect to one (or both) predictions at a level of

Σ Or More Include Fl, S5, P ′

5, AFB, P2 and S9. The Qi observables are summarised in Fig. 13, and Table 8. In this case, with the exception of QFL, which shows a tension of around 2σ, all other values are compatible with the SM prediction at less than 1.5σ.

Ata

Figure 12: The (left) S- and (right) P-basis angular observables of the large q2 region. The overlapping error bars show statistical and total uncertainties. The orange and hatched purple boxes correspond to SM predictions based on Ref. and Refs. , respectively.

Table 7: Values for the (left) S- and (right) P-basis angular observables of the large q2 region. The first uncertainty is statistical and the second is systematic.

Authors:

Peder EZ Larson 1, 2,* , Jenna ML Bernard1, James A Bankson 3, Nikolaj Bøgh 4, Robert A Bok1, Albert P. Chen 5, Charles H Cunningham 6,7, Jeremy Gordon1, Jan-Bernd Hövener 8, Christoffer Laustsen 4, Dirk Mayer 9,10, Mary A McLean11 12, Franz Schilling13, James Slater1, Jean-Luc Vanderheyden5, 14, Cornelius von Morze 15, Daniel B Vigneron1, 2, Duan Xu1, 2, and the HP 13C

94143, Usa.

Denmark. 5 GE Healthcare, Menlo Park, California, USA. 6 Physical Sciences, Sunnybrook Research Institute, Toronto, Ontario, Canada.

8 Section Biomedical Imaging, Molecular Imaging North Competence Center (MOIN CC), Medicine, Baltimore, MD, USA. Cambridge, United Kingdom.

14Jlvmi Consulting Llc, Dousman, Wi, Usa

#See Acknowledgements for a list of all HP 13C MRI Consensus Group Members This work was supported by the ISMRM Hyperpolarized Media MR Study Group, the ISMRM Hyperpolarization Methods & Equipment Study Group, and the Hyperpolarized MRI Technology Resource Center (NIH/NIBIB grant P41EB013598).

Abstract

MRI with hyperpolarized (HP) 13C agents, also known as HP 13C MRI, can measure processes such as localized metabolism that is altered in numerous cancers, liver, heart, kidney diseases, and more. It has been translated into human studies during the past 10 years, with recent rapid growth in studies largely based on increasing availability of hyperpolarized agent preparation methods suitable for use in humans. This paper aims to capture the current successful practices for HP MRI human studies with [1-13C]pyruvate - by far the most commonly used agent, which sits at a key metabolic junction in glycolysis. The paper is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification. In each area, we identified the key components for a successful study, summarized both published studies and current practices, and discuss evidence gaps, strengths, and limitations. This paper is the output of the “HP 13C MRI Consensus Group” as well as the ISMRM Hyperpolarized Media MR and Hyperpolarized Methods & Equipment study groups. It further aims to provide a comprehensive reference for future consensus building as the field continues to advance human studies with this metabolic imaging modality.

Keywords: Hyperpolarized MRI, metabolic imaging, carbon-13, pyruvate, dissolution dynamic

Introduction

MRI with hyperpolarized 13C agents, also known as hyperpolarized (HP) 13C MRI, has shown great potential as a novel imaging modality, particularly for its ability to probe metabolic processes in real time. The first human studies with HP [1-13C]pyruvate were performed in 2011 in prostate cancer patients (1).

Since then, there have been over 60 papers published with imaging results of human subjects from 13 different sites, with applications including prostate cancer, brain tumors, breast cancer, kidney cancer, pancreatic cancer, metastatic disease, liver disease, ischemic heart disease, diabetes and cardiomyopathies. The vast majority of these studies used [1-13C]pyruvate (1–63), where [2-13C]pyruvate (64) and 13C-urea (56) have been demonstrated too.

As clinical HP 13C MRI advances, there is a growing need to build consensus for best practices, which are critical for comparing data across sites, performing multi-site trials,deploying methods to new sites, partnering with vendors, and potentially for obtaining broader regulatory approvals.

In March 2022, we initiated an effort to build consensus within the HP 13C MRI community with this opportunity in mind, and it was greeted with strong enthusiasm. The “HP 13C MRI Consensus Group”, containing over 55 members from 27 sites, identified the area of greatest need and opportunity for consensus building to be HP [1-13C]pyruvate human

●

Pyruvate is the most mature and widely used HP agent and has the most significant translational evidence emphasizing the potential clinical impact.

●

Clinical trials, particularly multi-site trials, have the strongest need for consensus methods to ensure that data can be combined across sites. This work is a Position Paper for which the goal is to describe current successful practices and study methods for HP [1-13C]pyruvate human studies along with justification to support those practices. This is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification (Fig. 1). The current successful practices and study methods include a literature review of published peer-reviewed journal papers showing human HP [1-13C]pyruvate study data, up to September 2022 (1–63), as well as new unpublished information from surveys of HP 13C study sites. Based on this information, we also highlight the evidence gaps, strengths, and limitations of current practices which are summarized at the end of each section.

Figure 1: Illustration of the HP 13C MRI human study process, including the 4 major areas covered in this paper: Hyperpolarized 13C-pyruvate preparation, MRI system setup and calibration, Acquisition and Reconstruction, and Data Analysis and Quantification.

Figure 2: Anatomical targets of HP [1-13C]pyruvate MRI human studies published up to September 2022.

Hyperpolarized 13C-Pyruvate Preparation

This section covers the processes for creating the HP agent, 13C pyruvate, and will include many aspects and considerations that are needed to safely and effectively prepare doses for metabolic imaging studies in human subjects. These include material, personnel, equipment and facility, fluid path preparation, quality control, and release.

It is helpful to understand that the specifications of a dose of 13C pyruvate suitable for in vivo MR HP metabolic imaging were shaped in part by early preclinical studies performed by GE HealthCare summarized in Ref. (65). In short, the safety of the two novel drug components, 13C pyruvate and the electron paramagnetic agent (EPA) AH111501, were demonstrated in those studies. The more precise formulation of the dose suitable for human use was then determined from clinical studies (66) that included two Phase 1 clinical trials in young and elderly healthy volunteers without hyperpolarization of the 13C nuclei and another Phase 1/2a dose escalation and imaging feasibility study with HP 13C pyruvate in 31 prostate cancer patients at the With the exception of the first HP 13C imaging clinical trial, which utilized a prototype device in a cleanroom (1), all HP 13C studies performed in humans to date have utilized the SPINlab polarizer (manufactured by GE HealthCare). Consequently all doses of the HP 13C pyruvate delivered by SPINlab have been produced using the “SPINlab Pharmacy Kit” that serves as the container-closure system for the various drug components (13C pyruvic acid and EPA mixture, dissolution medium, and neutralization and dilution medium) during sample polarization, dissolution and quality control (QC) processes. Thus many aspects of the HP sample preparation considerations discussed below are related to the SPINlab instrument and the consumables designed to be used with it (67).

General Considerations

While more than 860 patients or healthy subjects having been injected with HP 13C pyruvate as of January 2022 without reports of any serious adverse events (68), HP 13C pyruvate injection remains an investigational MR contrast agent and can only be administered by those with Investigational New Drug (IND) exemption from the Food and Drug Administration (FDA) in the USA, a Clinical Trial Application (CTA) in Canada, approval from National Research Ethics Committee Services in the UK, or approval from the relevant local regulatory body. Thus, methods and processes involved to produce a dose should have patient safety as the first priority. Since utilizing dissolution dynamic nuclear polarization (dissolution-DNP) for human use is still a relatively new development, there are no existing published regulatory guidelines specifically for this method.

There are two major production styles that determine how various sites approach the agent preparation. In the US, the most common approach is to rely on a sterilizing filter (“Terminal Sterilization”) to ensure sterility of the final product, akin to PET tracer production, where a starting molecule with a radioisotope is processed using various other ingredients to make the final, desired and injectable contrast agent within a necessarily short amount of time (69). For these sites, sterilization of the components and accessories upstream of this filter are not required, although many of them were manufactured and tested following Good Manufacturing Practice (GMP) or Good Laboratory Practice (GLP) requirements. The filling process is usually performed under an ISO 5 laminar flow hood, but a clean room or an isolator is not required.

This approach is typically accompanied by testing the integrity of the sterilizing filter prior to release of the dose for injection. Typically, post release endotoxin and sterility tests are performed using an aliquot reserved from each released dose.

In the UK and EU, the most common approach is to more-closely follow sterile pharmaceutical compounding guidelines (70), where all components and ingredients are required to be sterile or manufactured under GMP guidelines and are assembled and filled within a clean room environment or an isolator system (“Sterile Preparation”). Typically a batch of Pharmacy Kits for HP 13C pyruvate injection are prepared together. The sterility of the final dose is also ensured by batch validation testing, in addition to the sterility of the ingredients and the sterile compounding process. The endotoxin and sterility testing are performed for the process validation but are not performed for each injected dose.

Some institutions fill and assemble the Pharmacy Kit required for a specific study on the same day or the day prior to polarization, dissolution, and patient administration, but others have also demonstrated the feasibility of preparing a batch of kits, keeping them in a -20ºC freezer and using them over a period of a few months.

Beyond the obvious requirements that the process and the facility has to ultimately produce a dose that is safe to inject into a human, regulatory authorities will also focus on the question “Are you in control of your processes?”. To be in control of your process requires an in-depth and broad understanding of all processes involved in pre, post, and during the production process.

Personnel

It is typical and may be required to have licensed personnel involved in the production process depending on local regulations.Typically a pharmacist, radiopharmacist or other similarly qualified person (QP), in charge of the facility where the Pharmacy Kit filling and preparation is taking place, is responsible for the overall process and the release of the injectable dose.

Qualified cleanroom technicians are often involved in the Pharmacy Kit filling under the supervision of the pharmacist or QP. As is required for pharmaceutical compounding or PET tracer production, training requirements and training records for all personnel need to be maintained and available for audit by the FDA or equivalent.

Equipment And Facility

The facility and all equipment need to have standard operating procedures (SOPs) that describe how equipment is used, maintained, and calibrated to comply with relevant legislation. Currently, almost all the filling of the Pharmacy Kit takes place within a compounding laminar flow hood or isolator (typically ISO 5). At some sites, the filling is conducted within a cleanroom, while at others, it is conducted in a dedicated non-cleanroom space, reflecting differences in cleanroom approach and specifications between regulators worldwide (71). Some equipment or facilities, such as the compounding hood or cleanroom, may require external certified laboratories for testing.

Material Handling

Material handling guidelines (69,70) require SOPs detailing a system to track all of the materials involved in the HP production process for a particular patient dose, similar to current good manufacturing practice (cGMP) requirements for material handling for drug compounding. This includes acceptance standards, storage conditions, amount used in the patient dose for each ingredient and materials used in the assembly of the fluid path and Pharmacy Kit. Currently some users choose to open and inspect and sometimes modify the Pharmacy Kits upon arrival, but some users keep them in the sealed packaging until they are required for dose preparation.

Pharmacy Kit Filling And Assembling

As required by an IND or its equivalent, the preparation of the doses of HP 13C agent are detailed in the Chemistry, Manufacturing, and Control (CMC) section of an applicable regulatory submission; an example of this has been made available (72). It describes the processes of filling the Pharmacy Kit with the different components that make up the final drug product, and of assembling the final kit for either storage or immediate use in the polarizer. Special attention should be given to the laser welding process in order to satisfy installation qualification (IQ) and operational qualification (OQ). Typically, the final developed process is validated by process qualification (PQ) runs, during which 3 or more Pharmacy Kits are filled and used and the final HP 13C products are tested for endotoxin and sterility and to confirm that they meet the dose specifications for injections (usually including pyruvate concentration, residual EPA concentration, pH, liquid state polarization level and dose temperature). The data from 3 consecutive PQ runs are submitted as part of the IND submission (or its equivalent), and are often also reviewed by the Institutional Review Board (IRB) where the studies are conducted.

Quality Control And Dose Release

The quality control (QC) and dose release can be separated into two aspects: one is the QC and release of the filled Pharmacy Kit, and second is the QC and release of the HP 13C agent for injection, after polarization and dissolution. For institutions filling a batch of kits and storing them to use over a period of time, typically the batch can be released based on initial validation, environmental monitoring data from the day of kit production, and if filters are used during preparation of any of the components, filter integrity testing. But in some cases one or more kits are used for validation before the batch of kits are released for future use. For institutions that fill only the kits required for specific studies shortly before the experiment, the filled kits often do not go through separate release tests before they are used.

The quality control of the HP 13C pyruvate solution post dissolution is primarily performed to ensure that the agent meets the dose specifications (Table 1) before it is administered to the subject. These specifications target both safety (pH, residual EPA, temperature) and efficacy (pyruvate concentration, polarization, volume). Typically, the pyruvate concentration, residual EPA concentration, pH, dose temperature, dose volume, and liquid state polarization are measured by the QC accessory associated with the SPINlab polarizer. Some users perform a secondary measurement for one of the parameters, such as pH, using a different instrument or pH paper. For sites that do not go through a separate release testing process for batch filled kits, the integrity of the sterilization assurance filter, a part of the Pharmacy Kit, is typically tested as a part of the dose release. It is also common for these users to preserve an aliquot of the final HP 13C pyruvate solution for post-release endotoxin and sterility testing. This testing cannot be completed fast enough to test an individual dose prior to injection, but this is why other processes such as PQ runs and validation testing are done to minimize the chance a subject could be injected with a contaminated dose.

The Final Dose Release And Injection

should be done under the supervision of a licensed professional, based on local regulations.

Some Key Challenges

Many of the challenges associated with HP 13C pyruvate preparation can be attributed to the conditions required for the dissolution-DNP method of high magnetic field (~3-7 T) and very low temperature (~1 K) during polarization, with pressurized and superheated water necessary for the rapid dissolution event. These extreme conditions are quite challenging for the design of the container-closure and fluid path system. In particular, the cryogenic temperature in the polarizer requires special attention to any moisture or ambient (moist) air introduced into that portion of the fluid path, which can form an ice block at ~1 K. This ice can lead to flow restriction during the dissolution event and reduce the strength of the laser welded bond between the cryovial and its cap. This can ultimately produce failures in the dissolution step, including variations in final pyruvate concentration and pH that may fail to meet QC release criteria as well as fluid path ruptures that provide no available dose and result in polarizer down-time.

The polarization of the HP 13C pyruvate sample decays quickly over the span of a few minutes after dissolution, and thus the process of dissolution, QC for release, and injection should be completed as fast as possible to preserve the high polarization level achieved. Any delays in the preparation process, such as transportation time or equipment malfunction, can significantly reduce the final polarization and result in lower quality imaging data.

Current Practices

A summary of data collected from all sites performing clinical trials with HP 13C-pyruvate is shown in Fig. 3 and Table 1, including the specification of the final dose and how the quality control and release of the final dose are performed. There is a split in the Production Style, described in the General Considerations section above, with 8/13 sites using Sterile Preparation versus 5/13 using Terminal Sterilization. While many of the dose specifications show notable differences in acceptable ranges, all of these variations listed in tables have been successfully and safely been used to perform HP 13C pyruvate studies in humans. Their differences depend on the institutions’ preferences, resources and their particular regulatory situation. There is high similarity in pyruvate ranges, temperature ranges, EPA limits, and volume limits. There is modest variability in pH ranges and large variability in the endotoxin test limit. There is a 3-fold difference in acceptable polarization levels, which are measured to ensure a futile dose is not injected since the polarization is directly proportional to SNR. This reflects the decision by several sites to believe that useful data can be still be obtained with suboptimal polarizations.

Figure 3: Hyperpolarized agent preparation methods reported by sites currently performing HP

In House

Table 1: HP 13C-pyruvate preparation parameters, methods, and dose specifications used for quality control testing and release as well as validation. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. The parameters used for product release are noted in bold text, otherwise these parameters are measured for batch validation or other QC measurements. The endotoxin and sterility testing are performed during process validation of the batch and/or post-injection, and largely depends on the agent production approach.

Summary

The overall safety record of HP 13C-pyruvate has been very strong, and the SPINlab hyperpolarizer has proven to provide high polarizations at human sized doses while meeting numerous QC and release criteria. A weakness remains the failure modes of the SPINlab Phamacy Kits (e.g. ice blocks, path ruptures), which are placed under extreme requirements particularly during dissolution. The preparation process still requires a high degree of expertise.

Therefore, there is a significant need to improve the reliability, robustness, and ease of operation for generating HP 13C-pyruvate doses for human studies. Furthermore, there is a divide between manufacturing and sterile compounding style preparation as well as other site-specific practices, resulting in variations in SOPs and justification required to relevant regulatory bodies. There have also been no comparisons between these approaches. It is also unclear what release criteria and QC parameters are truly required to ensure patient safety.

However, all of the reported methods are acceptable and approved by the appropriate regulatory authorities, and have led to the rapid expansion of successful human studies in recent years.

Mri System Setup And Calibrations

This section covers the MRI system setup, including the imaging system, RF coils, phantoms, and prescan calibration methods.

Imaging System

The main prerequisite for a given MRI scanner to be capable of supporting studies with HP 13C is its “broadband” capability to transmit and receive radiofrequency (RF) signal at the frequency of 13C, which is around 4 times lower than 1H. This does not come as a default on clinical MR devices. The transmit power of the broadband amplifier should also be sufficient to support the intended flip angle and RF pulse shape with the employed transmission RF coil(s) for 13C. Most studies to date use relatively low flip angles (< 90 degrees) for HP 13C in order to preserve polarization for time-resolved imaging. The capability to receive 13C signal on multiple channels is also desirable to increase SNR, as discussed further in the “RF coils” section.

The choice of magnetic field strength is primarily dependent on the metabolites’ frequency separation due to chemical shift dispersion and 1H imaging. High field strengths do not enhance hyperpolarized 13C signal as they do for 1H because the signal strength in a HP experiment relies on manipulating the population of quantum energy states outside of the MRI scanner.

However, the injected HP 13C-pyruvate and its metabolic products have greater frequency separation at higher fields, and it may thus be easier to separate and quantify these resonances at higher fields. This comes at the cost of a reduction in the achievable T2* and often reduced T1. As the initial polarization is independent of the imaging field strength it has been proposed that the increased T2* at 1.5T can potentially be exploited to increase SNR by adapting the acquisition bandwidth or reduce off-resonance imaging effects in cases when the decay of the transverse magnetization is dominated by T2* (73). In practice, 3T has been used in all published human 13C-pyruvate studies surveyed (Supporting Table S1), and comprises the majority of scanners currently in use for human studies (Table 3). A field strength of 3T is well-suited for 1H MRI anatomical reference and correlative imaging.

Stronger and more rapidly slewing magnetic field gradients support more rapid spatial encoding, particularly for metabolite-specific single-shot imaging using echo-planar imaging (EPI) or spiral imaging (See “Acquisition and Reconstruction”). Although the spatial resolution acquired for HP 13C imaging is typically much coarser than for 1H MRI, the factor of ~4 in gyromagnetic ratio leads to the same reduction factor in performance of the gradient system, so 13C experiments are potentially more limited by gradient hardware performance. To date, all human studies have used the commercially-available integrated gradient systems provided in clinical MRI scanners.

Optimization of scanner design has understandably focused on minimization of artifacts in 1H MRI, where devices such as room lights, the gradient amplifiers, and the motors driving the patient bed are checked to ensure that they do not produce RF interference at the 1H frequency, but artifacts may arise at other frequencies. Eddy current compensation is also not always appropriately adjusted for nuclei at other frequencies (74). In order to optimize for 13C, many sites have performed checks on phantoms for RF interference, gradient artifacts, and eddy currents (74), including the use of post-hoc gradient impulse response function characterisation and correction, and some vendors have fixed these issues as well.

Rf Coils

For HP 13C imaging studies in humans, RF coils for both 1H and 13C nuclei are needed, with 1H MRI providing an anatomical reference for registration and optional additional multiparametric MRI readouts. At the Larmor frequency of 13C nuclei, the relative contributions from coil noise compared to sample noise increase compared to 1H (73,75), although sample noise still is likely the dominant contributor for human-sized coils at 32.1MHz - the resonance frequency of 13C nuclei at 3T.

The key requirement for human 13C-pyruvate RF coils are that the coil geometry and sensitive volume must cover the volume of interest in the subject. Table 2 and Figure 4 shows coil configurations that have been used and optimized for applications in different anatomic regions.

Volume resonators are most commonly used for transmit, as they surround the subject to

Provide B1 Transmit Across The Fov (B1

+). While 1H relies on a large birdcage (“body”) coil built into the scanner, 13C transmit coils must be placed inside the bore. This takes up valuable space within the magnet, and also has led to the use of designs with relatively inhomogeneous

B1

+. Many human studies have used Helmholz pair resonators for transmit, including the “clamshell coil”, which has a notably inhomogeneous B1

+ Profile But Has Been Used Because Of

relatively easy integration into the scanner bore. B1

+ Variation Results In Variations In The Flip

angles that control the use of the hyperpolarized magnetization and creates errors in common HP metrics (9,76). The exception are head coils, where birdcage designs with highly

Homogeneous B1

+ can be placed around the head while easily fitting inside the bore. As with 1H MRI, higher SNR can typically be achieved by smaller receive coil elements, such as surface coils or phased arrays, and the majority of 13C receive coils used have layouts similar to 1H phased arrays.

RF coil quality control is important to ensure proper functioning of the coils to provide consistent imaging quality, especially with limited natural abundance 13C signal in vivo. It typically involves 1) a physical integrity check of the coil cables and connectors and 2) phantom SNR tests to check the coil’s performance and to monitor it over time (see Phantoms below). An useful reference for RF coil quality control is outlined in the MRI accreditation program of the American College of Radiology (77) and can be adapted for 13C coils.

Notably, configurations for brain and prostate studies used dual-tuned 1H/13C coil designs, which greatly simplify workflow and registration of 1H and 13C images, as no switching of coils is needed.

Table 2: RF coil configurations reported for human HP [1-13C]pyruvate studies.

Tx = Transmit

coil, RX = receive coil. The commonly used “clamshell” TX coil is a Helmholz pair design. For 1H RF configurations, all used the Body coil for TX unless otherwise noted, and “repositioned” indicates the 13C coil was removed for 1H imaging. One representative reference is listed for each configuration. The RF coil configurations reported in the reviewed papers are shown in Supporting Table S1.

Figure 4: Examples of RF coil configurations used for human HP [1-13C]pyruvate brain studies. (A,B) 13C Clamshell TX (Helmholz pair) and 2× 4-channel paddle RX arrays. (C) 13C Birdcage volume TX and 32-channel RX array (RX array slides into TX coil). (D) 13C Birdcage volume TX and 24-channel RX array, combined with a 1H 8-channel RX array. Image reproduced with permission from Ref (16).

Phantoms

Since hyperpolarized magnetization is non-renewable, phantoms containing 13C nuclei are important to: 1) test the multi-nuclear capabilities of the imaging system, including all parts of the signal excitation and receive chain; 2) perform calibration measurements before a scan with hyperpolarized nuclei; and 3) perform necessary pre-scan adjustments (see “Prescan Calibration” section). The phantoms currently in use are listed in Table 3. Their composition must provide sufficient 13C signal, with additional considerations of conductivity, stability, chemical shift(s) present, potential for dynamic imaging, and cost. The phantom geometries are typically either compact, in order to be used alongside the subject during a HP scan, or large enough to mimic the inner volume of a RF coil for system testing.

One popular compact design contains enriched 13C-urea at high concentration, typically 8 M, which provides a single resonance, placed inside a small container ~1 mL. The most common recipe mixes 13C-urea in a 90% water/10% glycerol solution, with glycerol used to increase the urea solubility and doping with a Gd-based contrast agent to shorten T1 which increases the potential SNR per unit time. For example, when Dotarem is added at a 3:1000 volume ratio the 13C-urea T1 is around 500 ms and T2 is around 100 ms. However, when testing pulse sequences influenced by T1 and T2, doping should be used carefully. This phantom is suitable for frequency calibration, transmit gain calibration, sequence testing, and as a fiducial marker when placed next to a patient. However, enriched 13C-urea has a relatively high cost compared to natural abundance compounds.

For larger volumes (>100 ml), the phantoms most often used contain undiluted ethylene glycol, glycerol, or dimethyl silicone. These compounds have sufficiently high carbon concentrations to provide sufficient 13C signal even with the 1.1% natural abundance of 13C. These larger phantoms matching the inner volume of an RF coil are useful for coil testing, including transmit

+) And Receive (B1

-) coil profile mapping, as well as to mimic acquisitions using in vivo FOV requirements. In this case, size and conductivity should match the expected subject size in order to mimic coil loading and get a realistic estimation of B1+. Large-volume natural abundance urea phantoms have also been used by some sites, but suffer from higher conductivity compared to biological tissues. Typically, it is easier to increase the conductivity and hence coil loading of the non-conductive phantom by adding NaCl to match physiological loading (16,78).

Dynamic phantoms that aim to mimic metabolite kinetics have also been developed (79–81), and have the potential to more closely mimic the HP experiment, but so far these are not widely used.

Prescan Calibration

Prior to performing an MRI acquisition, the so-called prescan procedure is used to set the shim parameters to maximize B0 homogeneity over the field of view (FOV) or a specific region of interest (ROI), the scanner center frequency (CF), the RF transmit gain, and the receiver gain.

While this calibration procedure is usually automated for 1H, the lack of sufficient natural abundance 13C signal prevents use of automated methods. (Although natural abundance 13C lipid signal has been detected, there are so far no reports on using this signal for prescan.) Table 3 shows current practices across sites.

Maximizing B0 homogeneity is independent of the nucleus and is therefore performed prior to 13C imaging using the 1H water signal and existing shimming tools, such as by a standard automated process (“Auto Shimming”) or using high order shimming routines. Similarly, the 13C CF can be calculated from the 1H CF using a predetermined scaling factor that depends on the target chemical shift (82). Another common approach used is to have a small, high-concentration 13C phantom, e.g. 8M 13C-urea, integrated in the RF coil or placed next to the scan subject (1). The reference frequency can also be based on real-time measurements after the HP injection but prior to imaging (83). Both the CF and B0 shimming are critical when using spectrally-selective RF pulses, as inmetabolite-specific imaging methods, where the desired excitation bandwidths are typically very narrow and frequency offsets can lead to a failure mode that is only apparent after injection.

The calibration of the RF transmit power is typically performed on a small, high-concentration 13C phantom placed near the region of interest during the scan or on a large 13C phantom of similar size and coil loading as the subject, prior to the subject scan. Reference power is often done by sweeping the power in a pulse-acquire sequence (53,62), or the Bloch-Siegert method (52,84). When using a small phantom, the location of the phantom, B1

+ Inhomogeneity As Well

as any shielding effects, e.g., when the phantom is integrated into a coil (1), may degrade the accuracy. Other methods include real-time Bloch-Siegert method measurements after the HP injection (83), and using the stronger natural abundance 23Na signal that is close enough to the 13C resonance frequency to be detected by 13C coils (82).

The receiver gain is predetermined, either systematically based on independent phantom measurements and assuming the dose and polarization of the HP compound is known prior to injection, or based on past HP imaging studies.

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

13C-bicarbonate doped with dimethyl silicone, various

Maximum Values

Table 3: Summary of the imaging systems, phantoms, and prescan procedures used at sites currently performing HP 13C-pyruvate human studies. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. *Previously performed studies with a Siemens 3T Tim Trio. The imaging systems, phantoms, and prescan procedures reported in the reviewed papers are shown in Supporting Table S1.

Summary

Commercially available 3T MRI systems are by far the most commonly used for human HP 13C-pyruvate studies, although a systematic investigation of the impact of B0 has only recently been investigated (73). The multi-nuclear RF transmit and receive chain has proven sufficient for current acquisition strategies, although many sites have observed artifacts due to RF interference, gradient interference, and residual eddy currents when operating at the 13C frequency. A variety of 13C RF coils, tailored for numerous anatomical targets, have been successfully demonstrated, with the main limitation that most transmit coils take up a lot of additional space inside the bore and provide relatively inhomogeneous B1

+ Profiles. The

phantoms used have converged into generally 2 categories - small phantoms containing 13C-enriched compounds that can be used during the study and human-sized phantoms containing compounds with high carbon concentrations but without 13C enrichment that are used to test and calibrate the coils. There are no standardized compositions or geometry, and dynamic phantoms that recapitulate in vivo kinetics would be desirable but are still an emerging area. Prescan calibration procedures were not well defined in most publications, so we surveyed individual sites to determine current practices. Calibration procedures for the B0 field (13C CF and shimming) for most sites take advantage of 1H signal and methods, while methods

For Calibration Of B1

+ is more variable across sites, likely a reflection of remaining challenges in how to perform this calibration. Standardization of both phantoms and calibration procedures would synergistically improve the robustness and reproducibility of HP 13C studies.

Acquisition And Reconstruction

Data acquisition strategies in human HP [1-13C]pyruvate MRI studies must account for multiple chemical shifts, efficiently utilize the non-renewable HP magnetization, and acquire data quickly relative to metabolism and relaxation decay processes. These studies require spectral encoding to separate metabolites, necessitating pulse sequences that efficiently encode up to 5D data (3 spatial + 1 spectral + 1 temporal dimension). RF pulses must efficiently sample without immediately saturating the non-renewable HP magnetization, and sequences must acquire data quickly and be robust to both experimental and physiologic variation (e.g. B1

+ Inhomogeneity,

variation in perfusion) to ensure reproducibility and minimize scan-to-scan variability. This section covers current successful practices for data acquisition in human [1-13C]pyruvate studies, and accompanying 1H imaging, from different anatomic regions, including scan parameters and image reconstruction.

Acquisition And Reconstruction Methods

The acquisition methods used in human [1-13C]pyruvate studies can be classified into 3 categories: 1) MR spectroscopy or MR spectroscopic imaging (“MRS/I”), 2) chemical shift encoding methods, and 3) metabolite-specific imaging (Fig. 5).

Mrs/I Methods Specifically

resolve a spectrum that can be analyzed to extract expected as well as unexpected resonances, making this approach very robust. It was used in many initial studies (1).

Chemical Shift

encoding methods, most commonly the Iterative Decomposition of water and fat with Echo Asymmetry and Least-squares estimation (IDEAL) method, use imaging sequences acquired with multiple TEs and rely on a model-based separation of expected chemical shifts (85).

Metabolite-specific imaging methods use specialized RF pulses that are spatially and spectrally selective to excite individual metabolites which are then typically imaged with fast k-space trajectories such as echo planar imaging (EPI) or spirals (86).

Their Application To Different

organ systems is described below. The image reconstruction methods used in human [1-13C]pyruvate studies have typically been conventional methods (e.g. FFT, non-uniform FFT, or equivalent). The incorporation of accelerated imaging and advanced reconstruction methods including parallel imaging (4,57,87) and compressed sensing (7) has also been applied in human studies for improved spatial resolution, temporal resolution and coverage, but have the potential for additional artifacts as well as SNR losses due to ill-conditioning of the reconstruction (e.g. g-factor).

The Majority Of

published studies do not use accelerated imaging indicating the resolution and coverage achievable without acceleration is currently adequate for successful data collection. Performing coil combination, even with fully sampled data has also been shown to have specific challenges for HP human images: using naive sum-of-squares methods suffer from high noise amplification in the relatively low SNR regime of HP [1-13C]pyruvate (compared to 1H), motivating several HP 13C-specific methods that include data-driven coil sensitivity estimation which have shown obvious improvements over sum-of-squares (11).

More recently denoising techniques have been applied as post-processing of human HP data(41,42,44). The techniques applied are based on spatial-temporal singular value decomposition for unsupervised estimation of signal and noise components. They have shown improvements in apparent SNR in the brain and liver, while care must be taken to choose parameters such as the rank threshold to avoid oversmoothing and overfitting to the estimated signal components.

Prostate Studies

Prostate cancer was the first human application of HP [1-13C]pyruvate (1), and data was acquired with MRS/I methods: 1D dynamic MRS, single-slice 2D dynamic echo-planar spectroscopic imaging (EPSI), and single time point 3D EPSI. Advances in imaging strategies led to the development and application of new acquisition schemes, including undersampled 3D EPSI with compressed-sensing (7), model-based chemical shift encoding methods that use a priori information (47,59), and metabolite-specific EPI (10), all of which can provide volumetric whole-organ coverage and dynamic acquisitions.

The pyruvate bolus arrival in the prostate can vary by ± 10 s between patients, necessitating dynamic imaging to reliably and consistently capture the pyruvate bolus (18). For this reason, all currently ongoing studies acquire dynamic data. While MRS/I, chemical shift encoding, and metabolite-specific imaging can all achieve dynamic imaging, chemical shift encoding and metabolite-specific imaging provide greater dynamic and volumetric coverage (85). For scan prescriptions, the FOV is designed to provide full prostate coverage and typically to match the orientation of the anatomic imaging used for registration. Flip angles used in current studies are constant through time, as quantification with a variable-through-time flip scheme is highly sensitive to bolus timing (8) and errors in the RF transmit (B1 +) field (76).

Heart Studies

Data acquisition methods for 13C imaging in the heart must be designed to meet the demands of significant cardiac motion and blood flow. To cope with the periodic cardiac motion, most human heart studies to date used gating to the diastolic window, the longest cardiac cycle interval, which has reduced motion (2,22,28,30,35,36,38,45,52). The duration of the diastolic window limits the available data sampling time, making cardiac acquisitions the most time-constrained of the HP 13C MRI applications. The most common acquisition approach is metabolite-specific imaging with spiral k-space trajectories (2). Their single-shot imaging capability makes these methods particularly robust to motion effects. Furthermore, spiral k-space trajectories provide rapid k-space coverage and relatively benign flow and motion artifacts. The majority of studies have used 2D multi-slice acquisitions, but 3D encoding has also been used successfully (35).

Brain Studies

For HP 13C MRI of the human brain, the majority of studies have also used 2D (slice selective) acquisitions (10–12,14,16,28,33,40,41,44,51,53,60), with a trend toward volumetric coverage using 2D multi-slice metabolite-specific imaging. 3D metabolite-specific imaging of the whole brain, with phase encoding of the slice direction (34,57), has been shown to provide similar SNR efficiency (88) compared with multislice imaging. A number of studies have employed MRS/I (5,6,29,31–33,50,55) resulting in a spectrum from each voxel, which has the advantage of not requiring a priori information about which peaks to encode. This was important in early brain studies when it was not known which peaks would be detectable. Chemical shift encoding, using a set of images with different echo times and an iterative reconstruction of the individual resonances (i.e. the IDEAL approach (85)), has also been used (12,49,54), with the drawback that coverage in the slice direction was limited due to the time required to acquire multiple echo time images.

Abdomen And Breast Studies

The fundamental approaches to data acquisition and reconstruction in the abdomen and breast are largely similar to the aforementioned applications, but demand attention to particular challenges associated with these anatomic regions, especially relating to respiratory motion.

Although it has been shown that a basic 2D MRSI approach based on phase encoding and FID readout can be successfully applied for HP 13C imaging in breast (15) and kidney (13), major advantages in terms of spatiotemporal resolution and coverage have been realized using tailored approaches based on metabolite-specific imaging (43,62) and chemical shift encoding (43), which have facilitated multi-slice or 3D dynamic acquisitions over large FOVs in the abdomen (4,37,46).

The significant respiratory motion encountered in these regions can directly blur 13C images, and has further favored these rapid acquisition strategies. Motion also degrades B0 homogeneity, which can shift frequency-selective excitation profiles and introduce artifacts into rapid imaging readouts. This makes accurate determination of the acquisition center frequency and shimming essential in these regions which often cover large FOVs. (See “Prescan Calibration” section for more information). In some studies, breath-holding was used to minimize motion effects and enforce frame-to-frame data consistency (42). A pragmatic and reasonably effective approach for dealing with respiratory motion during 13C data acquisition is an initial breath-hold (as long as can be tolerated), followed by free-breathing (46,62).

1H Imaging

Collection of 1H imaging data is essential both for prescribing the 13C acquisition and for interpretation of the resulting 13C data. Multi-planar 1H scouts are acquired prior to 13C acquisition to enable graphical prescription of the 13C imaging region. All human HP 13C-pyruvate imaging studies acquire conventional MRI scans (e.g. T1- and T2-weighted volumes) for anatomic reference, aiming to cover at least the full 13C FOV. Acquiring these anatomic scans as close as possible to the time of 13C imaging (immediately before or after) minimizes potential misregistration between the data sets. Depending on the application, other advanced 1H sequences are also acquired (e.g. diffusion-weighted imaging for cancer imaging).

When contrast-enhanced data is acquired, it is done after 13C imaging, as paramagnetic contrast agents will accelerate 13C relaxation.

Reported Study Parameters

Figures 5 and 6, and Supporting Table S2 shows the reported acquisition study parameters for human HP [1-13C]pyruvate studies published as of September 2022. Figure 5 shows a mixture of MRS/I, metabolite-specific imaging, and chemical shift encoding methods have been successfully used, where spectroscopy-based methods have become less prevalent in recent studies. Figure 6 shows the acquisition timing, including the important start time and interval/temporal resolution, is quite variable across studies.

Figure 5: Acquisition methods used in published HP [1-13C]pyruvate human studies published up to September 2022, classified into: MR spectroscopy and spectroscopy imaging (MRS/I); chemical shift encoding methods, such as IDEAL, that use multiple TEs and model-based reconstructions; and metabolite-specific imaging methods that use spectrally-selective excitation to image a single resonance at a time.

Figure 6: Temporal acquisition characteristics reported in HP [1-13C]pyruvate human studies published up to September 2022. (a) Reported referencing of acquisition start times.

(B)

Acquisition start times reported when using dynamic imaging and when timing was reported relative to the end of the injection. (c) Temporal resolutions. “Not Applicable” indicates dynamic imaging was not used.

Summary

Three general categories of acquisition strategies have been used successfully for human HP 13C-pyruvate studies: MRS/I, model-based chemical shift encoding (e.g. IDEAL) methods, and metabolite-specific imaging methods. These have enabled successful studies in the prostate, heart, brain, abdomen, and breast. Recent studies increasingly have used the imaging-based strategies of metabolite-specific imaging and chemical shift encoding which are the fastest methods, although a heads-to–head comparison between techniques has not been performed.

Metabolite-specific imaging is quite popular because of its speed and compatibility with single-shot imaging, but is sensitive to B0 field variations and thus requires careful calibrations. Nearly all studies surveyed acquired data dynamically, allowing measurement of the bolus and metabolite kinetics. The exact timings and associated flip angles vary quite widely across reported studies, with no consensus yet as to how to choose these parameters. Image reconstruction is typically done directly using Fourier Transform methods, and accelerated imaging strategies are uncommon.

Data Analysis And Quantification

This section covers the analysis of data from human HP [1-13C]pyruvate studies, including modeling and metrics, visualization, as well as considerations for how to store data and metadata. Depending on study design, the analysis may need to give quantitative or semi-quantitative output reflecting a biological process or may just reflect a contrast between different regions of interest for quantitative evaluation.

Metrics

Figure 7: HP [1-13C]pyruvate raw data (A) have typically been quantified using four categories of metrics depending on the acquisition. Data acquired as a single time point are often quantified using normalized metabolite images or metabolite ratios (B). Dynamic data can be quantified using normalized metabolite images or metabolite ratios (B), or with metabolite timings such as time-to-peak (TTP) or pharmacokinetic (PK) models (C). The latter two require the data to be time-resolved. [1-13C]alanine and 13C-bicarbonate are analyzed similarly to [1-13C]lactate but omitted here for display.

Metabolite images are commonly used as summary metrics for HP MRI data, often including some form of normalization as well as summed over time as an area under the time curve (AUC) (17). These are analogous to the visual evaluation that is most used for routine clinical work (89,90). In these metabolite images, we expect that the [1-13C]pyruvate AUC signal is predominantly weighted towards perfusion and uptake, while [1-13C]lactate, [1-13C]alanine and 13C-bicarbonate AUCs represent metabolic conversion. The strength of this approach lies in its simplicity and relatively few underlying assumptions. Limitations to the use of single-metabolite images or AUCs include sensitivity to inhomogeneous coil profiles (57,87,91), the acquisition strategy and acquisition parameters, pyruvate polarization and concentration level, and signal relaxation rates (92). Further, the reader must be careful to interpret all the images in conjunction to better understand the underlying biology; for example, increased [1-13C]lactate in the presence of decreased [1-13C]pyruvate delivery can have a very different meaning compared to increased [1-13C]lactate with increased [1-13C]pyruvate delivery.

In an attempt to address variations in coil sensitivity, polarization level, and pyruvate delivery, AUC images are often computed by normalizing to a specified parameter, such as the maximum pyruvate or average lactate signals, or presented as a ratio such as lactate/pyruvate or divided by “total Carbon” - the sum total of HP 13C signal observed across all metabolites. The AUC ratios between metabolites and pyruvate are proportional to the corresponding forward kinetic rates (81,93), but are not directly comparable to rate constants when magnetization loss rates (e.g. relaxation and losses due to signal excitation) differ between studies. Similarly, the ratios between the produced metabolites (e.g. bicarbonate/lactate) can reflect the balance between downstream metabolic pathways (12,55). Care must be taken to consider how AUC images are calculated and normalized before comparing values between studies.

To further quantify the interpretation, pharmacokinetic (PK) modeling approaches were developed to compute the apparent kinetics of pyruvate-to-metabolite exchange (92,94–99). These yield semi-quantitative to quantitative apparent rate constants, given in s-1. Some models require a vascular input function, while others avoid this requirement (95). PK models can explicitly account for acquisition-specific details such as excitation angle and repetition time, and thus may reduce the effects of these details on quantification. An input-less model, provided in the Hyperpolarized-MRI-Toolbox (https://github.com/LarsonLab/hyperpolarized-mri-toolbox) (100) and thus frequently employed for human data, has been shown to fit well and robustly to prostate and brain data (8,20). PK models are quantitative in nature, arguably provide more relevant biological information (8,20), and appear to be reproducible across sites (51). However, rate constants derived from PK models are still apparent rates, and likely do not reflect a single biological characteristic.

Some additional considerations include whether complex or magnitude data is used, as the noise behaviors will impact the analysis differently. Additionally, cut-off thresholds or other criteria may be used to identify and avoid voxels with insufficient SNR before analysis to improve robustness (20,41).

Regardless of the analysis approach, the underlying biology is not always clearly represented by the data; instead, the metrics may be influenced by perfusion, barrier permeability, intercellular shuttles, enzyme activities, co-substrate concentrations, or combinations thereof, depending on the organ and disease of interest (19,43,94,101–103). This may be addressed by incorporating complementary information. As an example, HP 13C pyruvate data is influenced by perfusion, and thus addition of perfusion MRI could be important for interpretation (98,104,105).

All the methods outlined above have been explored in clinical studies, described in Supporting Table 3 and summarized in Figure 8. As of September 2022, approximately 52% of studies involving human subjects report rate constants derived from a PK model with a few different models reported. A nearly equal fraction (51%) of the studies report AUC ratio values.

Approximately 66% of these studies report metabolite-specific images or AUC values. About 40% report SNR values; this metric is particularly frequent in manuscripts that describe technical developments for clinical HP MRI. Approximately 16% of these studies summarize model-free metrics, and 10% report measurements from a single timepoint. Most studies report a combination of quantities.

Figure 8: Reported metrics used for analysis in HP [1-13C]pyruvate human studies published up to September 2022.

Visualization

A wide variety of approaches have been used for visualizing data from human HP 13C-MRI studies. The challenges and practical considerations are: 1) choosing the appropriate metrics to display, 2) how to encode the parameters (e.g. the colormap), and 3) choosing how to provide anatomical context and other multi-parametric data. The choice of visualization also depends on the goal which could be for diagnostic interpretation, but also quality control, reproducibility among readers and publication.

Metrics

The choice of HP 13C metrics is described in detail above. At this stage in HP 13C development where there is no standardized metric, often a combination of metabolite images and ratios or PK model parameters are shown.

Parameter Encoding

The mapping function chosen should provide an adequate, often quantitative, impression of the parameter mapped. There is a consensus in the visualization field that perceptually uniform maps are best suited to visualize continuous parameters, like the greyscale typically used by radiologists as well as other monochrome (black to blue) and color ranges (fire-type, rainbow-type) (106,107). Multi-color heatmaps have been the most frequently employed method for HP 13C data, while greyscale has infrequently been used but it ensures there is no coloring-based bias as well as facilitating later reuse (Fig. 9a). Among the color schemes employed in the clinical HP 13C literature, fire-type scheme seems to be the most common [similar to “Plasma” or “Inferno” in matplotlib.org]. Next most commonly employed is the rainbow-type scheme [similar to “Rainbow” in matplotlib.org].

Anatomical Context

HP MRI faces the challenge that it does not necessarily depict the anatomical features, similar to PET, and thus requires an anatomical reference. Most often, a grayscale anatomical image is overlaid with a HP colormap (Fig. 9c,d). This approach is very intuitive, but can skew perception as the grey-scale anatomical reference may affect the brightness of the HP data (e.g. signal in the skull). This bias does not occur when showing adjacent maps (Fig. 9a, b). Here, anatomical outlines may help to provide reference (Fig. 9b).

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