Measurement of $\phi$-meson production in Cu$+$Au at $\sqrt{s_{_{NN}}}=200$ GeV and U$+$U at $\sqrt{s_{_{NN}}}=193$ GeV

The PHENIX collaboration Abdulameer, N.J. ; Acharya, U. ; Aidala, C. ; et al.
2022.
Inspire Record 2121010 DOI 10.17182/hepdata.132483

The PHENIX experiment reports systematic measurements at the Relativistic Heavy Ion Collider of $\phi$-meson production in asymmetric Cu$+$Au collisions at $\sqrt{s_{_{NN}}}$=200 GeV and in U$+$U collisions at $\sqrt{s_{_{NN}}}$=193 GeV. Measurements were performed via the $\phi\rightarrow K^{+}K^{-}$ decay channel at midrapidity $|\eta|<0.35$. Features of $\phi$-meson production measured in Cu$+$Cu, Cu$+$Au, Au$+$Au, and U$+$U collisions were found to not depend on the collision geometry, which was expected because the yields are averaged over the azimuthal angle and follow the expected scaling with nuclear-overlap size. The elliptic flow of the $\phi$ meson in Cu$+$Au, Au$+$Au, and U$+$U collisions scales with second order participant eccentricity and the length scale of the nuclear overlap region (estimated with the number of participating nucleons). At moderate $p_T$, $\phi$-meson production measured in Cu$+$Au and U$+$U collisions is consistent with coalescence-model predictions, whereas at high $p_T$ the production is in agreement with expectations for in-medium energy loss of parent partons prior to their fragmentation. The elliptic flow for $\phi$ mesons measured in Cu$+$Au and U$+$U collisions is well described by a (2+1)D viscous-hydrodynamic model with specific-shear viscosity $\eta/s=1/4\pi$.

14 data tables

Invariant transverse momentum spectra measured for $\phi$ mesons in (a) Cu+Au and (b) U+U collisions at $\sqrt{s}$ = 200 GeV at midrapidity

Invariant transverse momentum spectra measured for $\phi$ mesons in (c) Cu+Au and (d) U+U collisions at $\sqrt{s}$ = 200 GeV at midrapidity. Data-to- Levy fit ratios.

The $\phi$-meson nuclear modification factors $R_{AB}$ measured as a function of $p_T$ in different centrality intervals of (a) to (d) Cu+Au collisions at $\sqrt{s}$ = 200 GeV and (e) to (h) U+U collisions at $\sqrt{s}$ = 193 GeV at midrapidity $|\eta|<0.35$. The normalization uncertainty from p+p of about $\sim9.7\%$ is not shown.

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Study of $\phi$-meson production in $p+$Al, $p+$Au, $d+$Au, and $^3$He$+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV

The PHENIX collaboration Acharya, U. ; Adare, A. ; Aidala, C. ; et al.
Phys.Rev.C 106 (2022) 014908, 2022.
Inspire Record 2050486 DOI 10.17182/hepdata.130267

Small nuclear collisions are mainly sensitive to cold-nuclear-matter effects; however, the collective behavior observed in these collisions shows a hint of hot-nuclear-matter effects. The identified-particle spectra, especially the $\phi$ mesons which contain strange and antistrange quarks and have a relatively small hadronic-interaction cross section, are a good tool to study these effects. The PHENIX experiment has measured $\phi$ mesons in a specific set of small collision systems $p$$+$Al, $p$$+$Au, and $^3$He$+$Au, as well as $d$$+$Au [Phys. Rev. C {\bf 83}, 024909 (2011)], at $\sqrt{s_{_{NN}}}=200$ GeV. The transverse-momentum spectra and nuclear-modification factors are presented and compared to theoretical-model predictions. The comparisons with different calculations suggest that quark-gluon plasma may be formed in these small collision systems at $\sqrt{s_{_{NN}}}=200$ GeV. However, the volume and the lifetime of the produced medium may be insufficient for observing strangeness-enhancement and jet-quenching effects. Comparison with calculations suggests that the main production mechanisms of $\phi$ mesons at midrapidity may be different in $p$$+$Al versus $p/d/$$^3$He$+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV. While thermal quark recombination seems to dominate in $p/d/$$^3$He$+$Au collisions, fragmentation seems to be the main production mechanism in $p$$+$Al collisions.

2 data tables

Invariant transverse momentum spectra measured for $\phi$ mesons in (a) $p$+Al, (b) $p$+Au, and (c) $^{3}$He+Au collisions at $\sqrt{s_{_{NN}}}$ = 200 GeV at midrapidity.

Comparison of $\phi$-meson nuclear-modification factors in $p$+Al, $p$+Au, $d$+Au [2], and $^{3}$He+Au collisions at $\sqrt{s_{_{NN}}}$ = 200 GeV at midrapidity. The normalization uncertainty from $p$+$p$ of about $9.7 \%$ is not shown [28].


Version 2
Precision measurement of forward $Z$ boson production in proton-proton collisions at $\sqrt{s} = 13$ TeV

The LHCb collaboration Aaij, R. ; Abdelmotteleb, A.S.W. ; Abellán Beteta, C. ; et al.
JHEP 07 (2022) 026, 2022.
Inspire Record 1990313 DOI 10.17182/hepdata.132011

A precision measurement of the $Z$ boson production cross-section at $\sqrt{s} = 13$ TeV in the forward region is presented, using $pp$ collision data collected by the LHCb detector, corresponding to an integrated luminosity of 5.1 fb$^{-1}$. The production cross-section is measured using $Z\rightarrow\mu^+\mu^-$ events within the fiducial region defined as pseudorapidity $2.0<\eta<4.5$ and transverse momentum $p_{T}>20$ GeV/$c$ for both muons and dimuon invariant mass $60<M_{\mu\mu}<120$ GeV/$c^2$. The integrated cross-section is determined to be $\sigma (Z \rightarrow \mu^+ \mu^-)$ = 196.4 $\pm$ 0.2 $\pm$ 1.6 $\pm$ 3.9~pb, where the first uncertainty is statistical, the second is systematic, and the third is due to the luminosity determination. The measured results are in agreement with theoretical predictions within uncertainties.

27 data tables

Relative uncertainty for the integrated $Z -> \mu^{+} \mu^{-}$ cross-section measurement. The total uncertainty is the quadratic sum of uncertainties from statistical, systematic and luminosity contributions.

Final state radiation correction used in the $y^{Z}$ cross-section measurement. The first uncertainty is statistical and the second is systematic.

Final state radiation correction used in the $p_{T}^{Z}$ cross-section measurement. The first uncertainty is statistical and the second is systematic.

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Characterisation of the dip-bump structure observed in proton–proton elastic scattering at $\sqrt{s}$ = 8 TeV

The TOTEM collaboration Antchev, G. ; Aspell, P. ; Atanassov, I. ; et al.
Eur.Phys.J.C 82 (2022) 263, 2022.
Inspire Record 1974042 DOI 10.17182/hepdata.127942

The TOTEM collaboration at the CERN LHC has measured the differential cross-section of elastic proton-proton scattering at $\sqrt{s} = 8\ {\rm TeV}$ in the squared four-momentum transfer range $0.2\ {\rm GeV^{2}} < |t| < 1.9\ {\rm GeV^{2}}$. This interval includes the structure with a diffractive minimum ("dip") and a secondary maximum ("bump") that has also been observed at all other LHC energies, where measurements were made. A detailed characterisation of this structure for $\sqrt{s} = 8\ {\rm TeV}$ yields the positions, $|t|_{\rm dip} = (0.521 \pm 0.007)\ {\rm GeV^2}$ and $|t|_{\rm bump} = (0.695 \pm 0.026)\ {\rm GeV^2}$, as well as the cross-section values, ${{\rm d}\sigma/{\rm d} t}_{\rm dip} = (15.1 \pm 2.5)\ {\rm{\mu b/GeV^2}}$ and ${{\rm d}\sigma/{\rm d} t}_{\rm bump} = (29.7 \pm 1.8)\ {\rm{\mu b/GeV^2}}$, for the dip and the bump, respectively.

1 data table

Differential cross-section.


Search for a heavy resonance decaying into a top quark and a W boson in the lepton+jets final state at $\sqrt{s}$= 13 TeV

The CMS collaboration Tumasyan, A. ; Adam, W. ; Andrejkovic, J.W. ; et al.
JHEP 04 (2022) 048, 2022.
Inspire Record 1972089 DOI 10.17182/hepdata.114361

A search for a heavy resonance decaying into a top quark and a W boson in proton-proton collisions at $\sqrt{s} =$ 13 TeV is presented. The data analyzed were recorded with the CMS detector at the LHC and correspond to an integrated luminosity of 138 fb$^{-1}$. The top quark is reconstructed as a single jet and the W boson, from its decay into an electron or muon and the corresponding neutrino. A top quark tagging technique based on jet clustering with a variable distance parameter and simultaneous jet grooming is used to identify jets from the collimated top quark decay. The results are interpreted in the context of two benchmark models, where the heavy resonance is either an excited bottom quark b$^*$ or a vector-like quark B. A statistical combination with an earlier search by the CMS Collaboration in the all-hadronic final state is performed to place upper cross section limits on these two models. The new analysis extends the lower range of resonance mass probed from 1.4 down to 0.7 TeV. For left-handed, right-handed, and vector-like couplings, b$^*$ masses up to 3.0, 3.0, and 3.2 TeV are excluded at 95% confidence level, respectively. The observed upper limits represent the most stringent constraints on the b$^*$ model to date.

7 data tables

Distributions of MtW in the 1b category. The data are shown by filled markers, where the horizontal bars indicate the bin widths. The individual background contributions are given by filled histograms. The expected signal for a LH b* with mb∗ = 2.4 TeV is shown by a dashed line. The shaded region is the uncertainty in the total background estimate. The lower panel shows the ratio of data to the background estimate, with the total uncertainty on the predicted background displayed as the gray band.

Distributions of MtW in the 2b category. The data are shown by filled markers, where the horizontal bars indicate the bin widths. The individual background contributions are given by filled histograms. The expected signal for a LH b* with mb∗ = 2.4 TeV is shown by a dashed line. The shaded region is the uncertainty in the total background estimate. The lower panel shows the ratio of data to the background estimate, with the total uncertainty on the predicted background displayed as the gray band.

Upper limits on the production cross section times branching fraction of the b* LH hypothesis at a 95% CL. Dashed colored lines show the expected limits from the l+jets and all-hadronic channels, where the latter start at resonance masses of 1.4 TeV. The observed and expected limits from the combination are shown as solid and dashed black lines, respectively. The green and yellow bands show the 68 and 95% confidence intervals on the combined expected limits.

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Search for flavor-changing neutral current interactions of the top quark and Higgs boson in final states with two photons in proton-proton collisions at $\sqrt{s}$ = 13 TeV

The CMS collaboration Tumasyan, A. ; Adam, W. ; Andrejkovic, J.W. ; et al.
Phys.Rev.Lett. 129 (2022) 032001, 2022.
Inspire Record 2111572 DOI 10.17182/hepdata.105999

Proton-proton interactions resulting in final states with two photons are studied in a search for the signature of flavor-changing neutral current interactions of top quarks (t) and Higgs bosons (H). The analysis is based on data collected at a center-of-mass energy of 13 TeV with the CMS detector at the LHC, corresponding to an integrated luminosity of 137 fb$^{-1}$. No significant excess above the background prediction is observed. Upper limits on the branching fractions ($\mathcal{B}$) of the top quark decaying to a Higgs boson and an up (u) or charm quark (c) are derived through a binned fit to the diphoton invariant mass spectrum. The observed (expected) 95% confidence level upper limits are found to be 0.019 (0.031)% for $\mathcal B$(t $\to$ Hu) and 0.073 (0.051)% for $\mathcal{B}$(t $\to$ Hc). These are the strictest upper limits yet determined.

1 data table

Expected and observed 95\% CL upper limits on the branching fraction of the top quark decaying to the Higgs boson and a light-flavor quark (either an up or a charm quark)


Search for low-mass dilepton resonances in Higgs boson decays to four-lepton final states in proton-proton collisions at $\sqrt{s}$ =13 TeV

The CMS collaboration Tumasyan, A. ; Adam, W. ; Bergauer, T. ; et al.
Eur.Phys.J.C 82 (2022) 290, 2022.
Inspire Record 1961934 DOI 10.17182/hepdata.110659

A search for low-mass dilepton resonances in Higgs boson decays is conducted in the four-lepton final state. The decay is assumed to proceed via a pair of beyond the standard model particles, or one such particle and a Z boson. The search uses proton-proton collision data collected with the CMS detector at the CERN LHC, corresponding to an integrated luminosity of 137 fb$^{-1}$, at a center-of-mass energy $\sqrt{s} =$ 13 TeV. No significant deviation from the standard model expectation is observed. Upper limits at 95% confidence level are set on model-independent Higgs boson decay branching fractions. Additionally, limits on dark photon and axion-like particle production, based on two specific models, are reported.

9 data tables

Exclusion limit for BrHXX_Br2Xee

Exclusion limit for BrHXX_Br2Xmumu

Exclusion limit for BrHXX_Br2Xll

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Observation of a structure in the $M_{p\eta }$ invariant mass distribution near 1700 ${MeV}/ {c}^2$ in the $\gamma \mathbf {p \rightarrow p \pi ^0 \eta } $ reaction

The CBELSA/TAPS collaboration Metag, V. ; Nanova, M. ; Hartmann, J. ; et al.
Eur.Phys.J.A 57 (2021) 325, 2021.
Inspire Record 1987678 DOI 10.17182/hepdata.115572

The reaction $\gamma p \rightarrow p \pi^0 \eta$ has been studied with the CBELSA/TAPS detector at the electron stretcher accelerator ELSA in Bonn for incident photon energies from threshold up to 3.1 GeV. This paper has been motivated by the recently claimed observation of a narrow structure in the M$_{N\eta}$ invariant mass distribution at a mass of 1678 MeV/$c^2$. The existence of this structure cannot be confirmed in the present work. Instead, for E$_{\gamma}$ = 1400 - 1500 MeV and the cut M$_{p\pi^0} \le 1190 $ MeV/$c^2$ a statistically significant structure in the M$_{p\eta}$ invariant mass distribution near 1700 MeV/$c^2$ is observed with a width of $\Gamma\approx 35$ MeV/$c^2$ while the mass resolution is $\sigma_{res}$ = 5 MeV/$c^2$. Increasing the incident photon energy from 1420 to 1540 MeV this structure shifts in mass from $\approx$ 1700MeV/c$^2$ to $\approx$ 1725 MeV/$c^2$; the width increases to about 50 MeV/$c^2$ and decreases thereafter. The cross section associated with this structure reaches a maximum of $\approx$ 100 nb around E$_{\gamma} \approx$ 1490 MeV (W $\approx $ 1920 MeV), which coincides with the $p a_0$ threshold. Three scenarios are discussed which might be the origin of this structure in the M$_{p\eta}$ invariant mass distribution. The most likely interpretation is that it is due to a triangular singularity in the $\gamma p \rightarrow p a_0 \rightarrow p \pi^0 \eta$ reaction

32 data tables

Differential cross section for $\pi^0$ in the $\gamma p$ for the incident photon energy range of $E_\gamma$ = 1400-1500 MeV.

Differential cross section for $\eta$ in the $\gamma p$ for the incident photon energy range of $E_\gamma$ = 1400-1500 MeV.

Differential cross section for $p$ in the $\gamma p$ for the incident photon energy range of $E_\gamma$ = 1400-1500 MeV.

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Measurement of $J/\psi$ production cross-sections in $pp$ collisions at $\sqrt{s}=5$ TeV

The LHCb collaboration Aaij, R. ; Abdelmotteleb, A.S.W. ; Beteta, C. Abellán ; et al.
JHEP 11 (2021) 181, 2021.
Inspire Record 1915030 DOI 10.17182/hepdata.115512

The production cross-sections of $J/\psi$ mesons in proton-proton collisions at a centre-of-mass energy of $\sqrt{s}=5$ TeV are measured using a data sample corresponding to an integrated luminosity of $9.13\pm0.18~\text{pb}^{-1}$, collected by the LHCb experiment. The cross-sections are measured differentially as a function of transverse momentum, $p_{\text{T}}$, and rapidity, $y$, and separately for $J/\psi$ mesons produced promptly and from beauty hadron decays (nonprompt). With the assumption of unpolarised $J/\psi$ mesons, the production cross-sections integrated over the kinematic range $0<p_{\text{T}}<20~\text{GeV}/c$ and $2.0<y<4.5$ are $8.154\pm0.010\pm0.283~\mu\text{b}$ for prompt $J/\psi$ mesons and $0.820\pm0.003\pm0.034~\mu\text{b}$ for nonprompt $J/\psi$ mesons, where the first uncertainties are statistical and the second systematic. These cross-sections are compared with those at $\sqrt{s}=8$ TeV and $13$ TeV, and are used to update the measurement of the nuclear modification factor in proton-lead collisions for $J/\psi$ mesons at a centre-of-mass energy per nucleon pair of $\sqrt{s_{\text{NN}}}=5$ TeV. The results are compared with theoretical predictions.

20 data tables

Double-differential production cross-sections for prompt $J/\psi$ mesons in ($p_\text{T},y$) intervals. The first uncertainties are statistical, the second are correlated systematic uncertainties shared between intervals, the third are uncorrelated systematic uncertainties, and the last are correlated between $p_\text{T}$ intervals and uncorrelated between $y$ intervals.

Double-differential production cross-sections for nonprompt $J/\psi$ mesons in ($p_\text{T},y$) intervals. The first uncertainties are statistical, the second are correlated systematic uncertainties shared between intervals, the third are uncorrelated systematic uncertainties, and the last are correlated between $p_\text{T}$ intervals and uncorrelated between $y$ intervals.

Single-differential production cross-sections for prompt $J/\psi$ mesons as a function of $p_\text{T}$. The first uncertainties are statistical, the second are correlated systematic uncertainties shared between intervals, and the last are uncorrelated systematic uncertainties.

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The exotic meson $\pi_1(1600)$ with $J^{PC} = 1^{-+}$ and its decay into $\rho(770)\pi$

The COMPASS collaboration Alexeev, M.G. ; Alexeev, G.D. ; Amoroso, A. ; et al.
Phys.Rev.D 105 (2022) 012005, 2022.
Inspire Record 1898933 DOI 10.17182/hepdata.114098

We study the spin-exotic $J^{PC} = 1^{-+}$ amplitude in single-diffractive dissociation of 190 GeV$/c$ pions into $\pi^-\pi^-\pi^+$ using a hydrogen target and confirm the $\pi_1(1600) \to \rho(770) \pi$ amplitude, which interferes with a nonresonant $1^{-+}$ amplitude. We demonstrate that conflicting conclusions from previous studies on these amplitudes can be attributed to different analysis models and different treatment of the dependence of the amplitudes on the squared four-momentum transfer and we thus reconcile their experimental findings. We study the nonresonant contributions to the $\pi^-\pi^-\pi^+$ final state using pseudo-data generated on the basis of a Deck model. Subjecting pseudo-data and real data to the same partial-wave analysis, we find good agreement concerning the spectral shape and its dependence on the squared four-momentum transfer for the $J^{PC} = 1^{-+}$ amplitude and also for amplitudes with other $J^{PC}$ quantum numbers. We investigate for the first time the amplitude of the $\pi^-\pi^+$ subsystem with $J^{PC} = 1^{--}$ in the $3\pi$ amplitude with $J^{PC} = 1^{-+}$ employing the novel freed-isobar analysis scheme. We reveal this $\pi^-\pi^+$ amplitude to be dominated by the $\rho(770)$ for both the $\pi_1(1600)$ and the nonresonant contribution. We determine the $\rho(770)$ resonance parameters within the three-pion final state. These findings largely confirm the underlying assumptions for the isobar model used in all previous partial-wave analyses addressing the $J^{PC} = 1^{-+}$ amplitude.

4 data tables

Results for the spin-exotic $1^{-+}1^+[\pi\pi]_{1^{-\,-}}\pi P$ wave from the free-isobar partial-wave analysis performed in the first $t^\prime$ bin from $0.100$ to $0.141\;(\text{GeV}/c)^2$. The plotted values represent the intensity of the coherent sum of the dynamic isobar amplitudes $\{\mathcal{T}_k^\text{fit}\}$ as a function of $m_{3\pi}$, where the coherent sums run over all $m_{\pi^-\pi^+}$ bins indexed by $k$. These intensity values are given in number of events per $40\;\text{MeV}/c^2$ $m_{3\pi}$ interval and correspond to the orange points in Fig. 8(a). In the "Resources" section of this $t^\prime$ bin, we provide the JSON file named <code>transition_amplitudes_tBin_0.json</code> for download, which contains for each $m_{3\pi}$ bin the values of the transition amplitudes $\{\mathcal{T}_k^\text{fit}\}$ for all $m_{\pi^-\pi^+}$ bins, their covariances, and further information. The data in this JSON file are organized in independent bins of $m_{3\pi}$. The information in these bins can be accessed via the key <code>m3pi_bin_<#>_t_prime_bin_0</code>. Each independent $m_{3\pi}$ bin contains <ul> <li>the kinematic ranges of the $(m_{3\pi}, t^\prime)$ cell, which are accessible via the keys <code>m3pi_lower_limit</code>, <code>m3pi_upper_limit</code>, <code>t_prime_lower_limit</code>, and <code>t_prime_upper_limit</code>.</li> <li>the $m_{\pi^-\pi^+}$ bin borders, which are accessible via the keys <code>m2pi_lower_limits</code> and <code>m2pi_upper_limits</code>.</li> <li>the real and imaginary parts of the transition amplitudes $\{\mathcal{T}_k^\text{fit}\}$ for all $m_{\pi^-\pi^+}$ bins, which are accessible via the keys <code>transition_amplitudes_real_part</code> and <code>transition_amplitudes_imag_part</code>, respectively.</li> <li>the covariance matrix of the real and imaginary parts of the $\{\mathcal{T}_k^\text{fit}\}$ for all $m_{\pi^-\pi^+}$ bins, which is accessible via the key <code>covariance_matrix</code>. Note that this matrix is real-valued and that its rows and columns are indexed such that $(\Re,\Im)$ pairs of the transition amplitudes are arranged with increasing $k$.</li> <li>the normalization factors $\mathcal{N}_a$ in Eq. (13) for all $m_{\pi^-\pi^+}$ bins, which are accessible via the key <code>normalization_factors</code>.</li> <li>the shape of the zero mode, i.e., the values of $\tilde\Delta_k$ for all $m_{\pi^-\pi^+}$ bins, which is accessible via the key <code>zero_mode_shape</code>.</li> <li>the reference wave, which is accessible via the key <code>reference_wave</code>. Note that this is always the $4^{++}1^+\rho(770)\pi G$ wave.</li> </ul>

Results for the spin-exotic $1^{-+}1^+[\pi\pi]_{1^{-\,-}}\pi P$ wave from the free-isobar partial-wave analysis performed in the second $t^\prime$ bin from $0.141$ to $0.194\;(\text{GeV}/c)^2$. The plotted values represent the intensity of the coherent sum of the dynamic isobar amplitudes $\{\mathcal{T}_k^\text{fit}\}$ as a function of $m_{3\pi}$, where the coherent sums run over all $m_{\pi^-\pi^+}$ bins indexed by $k$. These intensity values are given in number of events per $40\;\text{MeV}/c^2$ $m_{3\pi}$ interval and correspond to the orange points in Fig. 15(a) in the supplemental material of the paper. In the "Resources" section of this $t^\prime$ bin, we provide the JSON file named <code>transition_amplitudes_tBin_1.json</code> for download, which contains for each $m_{3\pi}$ bin the values of the transition amplitudes $\{\mathcal{T}_k^\text{fit}\}$ for all $m_{\pi^-\pi^+}$ bins, their covariances, and further information. The data in this JSON file are organized in independent bins of $m_{3\pi}$. The information in these bins can be accessed via the key <code>m3pi_bin_<#>_t_prime_bin_1</code>. Each independent $m_{3\pi}$ bin contains <ul> <li>the kinematic ranges of the $(m_{3\pi}, t^\prime)$ cell, which are accessible via the keys <code>m3pi_lower_limit</code>, <code>m3pi_upper_limit</code>, <code>t_prime_lower_limit</code>, and <code>t_prime_upper_limit</code>.</li> <li>the $m_{\pi^-\pi^+}$ bin borders, which are accessible via the keys <code>m2pi_lower_limits</code> and <code>m2pi_upper_limits</code>.</li> <li>the real and imaginary parts of the transition amplitudes $\{\mathcal{T}_k^\text{fit}\}$ for all $m_{\pi^-\pi^+}$ bins, which are accessible via the keys <code>transition_amplitudes_real_part</code> and <code>transition_amplitudes_imag_part</code>, respectively.</li> <li>the covariance matrix of the real and imaginary parts of the $\{\mathcal{T}_k^\text{fit}\}$ for all $m_{\pi^-\pi^+}$ bins, which is accessible via the key <code>covariance_matrix</code>. Note that this matrix is real-valued and that its rows and columns are indexed such that $(\Re,\Im)$ pairs of the transition amplitudes are arranged with increasing $k$.</li> <li>the normalization factors $\mathcal{N}_a$ in Eq. (13) for all $m_{\pi^-\pi^+}$ bins, which are accessible via the key <code>normalization_factors</code>.</li> <li>the shape of the zero mode, i.e., the values of $\tilde\Delta_k$ for all $m_{\pi^-\pi^+}$ bins, which is accessible via the key <code>zero_mode_shape</code>.</li> <li>the reference wave, which is accessible via the key <code>reference_wave</code>. Note that this is always the $4^{++}1^+\rho(770)\pi G$ wave.</li> </ul>

Results for the spin-exotic $1^{-+}1^+[\pi\pi]_{1^{-\,-}}\pi P$ wave from the free-isobar partial-wave analysis performed in the third $t^\prime$ bin from $0.194$ to $0.326\;(\text{GeV}/c)^2$. The plotted values represent the intensity of the coherent sum of the dynamic isobar amplitudes $\{\mathcal{T}_k^\text{fit}\}$ as a function of $m_{3\pi}$, where the coherent sums run over all $m_{\pi^-\pi^+}$ bins indexed by $k$. These intensity values are given in number of events per $40\;\text{MeV}/c^2$ $m_{3\pi}$ interval and correspond to the orange points in Fig. 15(b) in the supplemental material of the paper. In the "Resources" section of this $t^\prime$ bin, we provide the JSON file named <code>transition_amplitudes_tBin_2.json</code> for download, which contains for each $m_{3\pi}$ bin the values of the transition amplitudes $\{\mathcal{T}_k^\text{fit}\}$ for all $m_{\pi^-\pi^+}$ bins, their covariances, and further information. The data in this JSON file are organized in independent bins of $m_{3\pi}$. The information in these bins can be accessed via the key <code>m3pi_bin_<#>_t_prime_bin_2</code>. Each independent $m_{3\pi}$ bin contains <ul> <li>the kinematic ranges of the $(m_{3\pi}, t^\prime)$ cell, which are accessible via the keys <code>m3pi_lower_limit</code>, <code>m3pi_upper_limit</code>, <code>t_prime_lower_limit</code>, and <code>t_prime_upper_limit</code>.</li> <li>the $m_{\pi^-\pi^+}$ bin borders, which are accessible via the keys <code>m2pi_lower_limits</code> and <code>m2pi_upper_limits</code>.</li> <li>the real and imaginary parts of the transition amplitudes $\{\mathcal{T}_k^\text{fit}\}$ for all $m_{\pi^-\pi^+}$ bins, which are accessible via the keys <code>transition_amplitudes_real_part</code> and <code>transition_amplitudes_imag_part</code>, respectively.</li> <li>the covariance matrix of the real and imaginary parts of the $\{\mathcal{T}_k^\text{fit}\}$ for all $m_{\pi^-\pi^+}$ bins, which is accessible via the key <code>covariance_matrix</code>. Note that this matrix is real-valued and that its rows and columns are indexed such that $(\Re,\Im)$ pairs of the transition amplitudes are arranged with increasing $k$.</li> <li>the normalization factors $\mathcal{N}_a$ in Eq. (13) for all $m_{\pi^-\pi^+}$ bins, which are accessible via the key <code>normalization_factors</code>.</li> <li>the shape of the zero mode, i.e., the values of $\tilde\Delta_k$ for all $m_{\pi^-\pi^+}$ bins, which is accessible via the key <code>zero_mode_shape</code>.</li> <li>the reference wave, which is accessible via the key <code>reference_wave</code>. Note that this is always the $4^{++}1^+\rho(770)\pi G$ wave.</li> </ul>

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