Photons in the proton: implications for the LHC
- Ball, Richard D. et al
- arXiv:2401.08749TIF-UNIMI-2023-17Edinburgh 2023/19CERN-TH-2023-159
 | \small Breakdown of the photon PDF $\gamma(x,\mu^2)$ into the contributions coming from different contributions to the proton structure functions $F_i(x,Q^2)$ that determine it according to the LuxQED formula Eq.~(\ref{eq:LUX:QED}). The result is shown as a function of $x$ for four different choices of the scale $\mu=Q_\gamma$: $Q_\gamma=1.65$~GeV (top left), 10~GeV (top right), 100~GeV (bottom left) and 500~GeV (bottom right). |
 | \small Breakdown of the photon PDF $\gamma(x,\mu^2)$ into the contributions coming from different contributions to the proton structure functions $F_i(x,Q^2)$ that determine it according to the LuxQED formula Eq.~(\ref{eq:LUX:QED}). The result is shown as a function of $x$ for four different choices of the scale $\mu=Q_\gamma$: $Q_\gamma=1.65$~GeV (top left), 10~GeV (top right), 100~GeV (bottom left) and 500~GeV (bottom right). |
 | \small Breakdown of the photon PDF $\gamma(x,\mu^2)$ into the contributions coming from different contributions to the proton structure functions $F_i(x,Q^2)$ that determine it according to the LuxQED formula Eq.~(\ref{eq:LUX:QED}). The result is shown as a function of $x$ for four different choices of the scale $\mu=Q_\gamma$: $Q_\gamma=1.65$~GeV (top left), 10~GeV (top right), 100~GeV (bottom left) and 500~GeV (bottom right). |
 | \small Breakdown of the photon PDF $\gamma(x,\mu^2)$ into the contributions coming from different contributions to the proton structure functions $F_i(x,Q^2)$ that determine it according to the LuxQED formula Eq.~(\ref{eq:LUX:QED}). The result is shown as a function of $x$ for four different choices of the scale $\mu=Q_\gamma$: $Q_\gamma=1.65$~GeV (top left), 10~GeV (top right), 100~GeV (bottom left) and 500~GeV (bottom right). |
 | \small Top left: the photon PDF at $Q=100$~GeV in NNPDF4.0QED NNLO compared to its NNPDF3.1QED counterpart as a ratio to the central value of the former. Top right: the relative PDF uncertainties on the photon PDF in these two determinations. Bottom: same as top panels now comparing NNPDF4.0QED with MSHT20QED~\cite{Cridge:2021pxm} and CT18QED~\cite{Xie:2021ajm} (all NNLO). Bands correspond to 1$\sigma$ uncertainties. |
 | \small Top left: the photon PDF at $Q=100$~GeV in NNPDF4.0QED NNLO compared to its NNPDF3.1QED counterpart as a ratio to the central value of the former. Top right: the relative PDF uncertainties on the photon PDF in these two determinations. Bottom: same as top panels now comparing NNPDF4.0QED with MSHT20QED~\cite{Cridge:2021pxm} and CT18QED~\cite{Xie:2021ajm} (all NNLO). Bands correspond to 1$\sigma$ uncertainties. |
 | \small Top left: the photon PDF at $Q=100$~GeV in NNPDF4.0QED NNLO compared to its NNPDF3.1QED counterpart as a ratio to the central value of the former. Top right: the relative PDF uncertainties on the photon PDF in these two determinations. Bottom: same as top panels now comparing NNPDF4.0QED with MSHT20QED~\cite{Cridge:2021pxm} and CT18QED~\cite{Xie:2021ajm} (all NNLO). Bands correspond to 1$\sigma$ uncertainties. |
 | \small Top left: the photon PDF at $Q=100$~GeV in NNPDF4.0QED NNLO compared to its NNPDF3.1QED counterpart as a ratio to the central value of the former. Top right: the relative PDF uncertainties on the photon PDF in these two determinations. Bottom: same as top panels now comparing NNPDF4.0QED with MSHT20QED~\cite{Cridge:2021pxm} and CT18QED~\cite{Xie:2021ajm} (all NNLO). Bands correspond to 1$\sigma$ uncertainties. |
 | \small Scale dependence of the central photon PDF in the NNPDF4.0QED, NNPDF3.1QED, MSHT20QED, and CT18QED PDF sets. The NNPDF4.0QED result in which the photon is determined at $Q=10$~GeV (denoted as NNPDF4.0QED*) is also shown. |
 | \small Scale dependence of the central photon PDF in the NNPDF4.0QED, NNPDF3.1QED, MSHT20QED, and CT18QED PDF sets. The NNPDF4.0QED result in which the photon is determined at $Q=10$~GeV (denoted as NNPDF4.0QED*) is also shown. |
 | \small Scale dependence of the central photon PDF in the NNPDF4.0QED, NNPDF3.1QED, MSHT20QED, and CT18QED PDF sets. The NNPDF4.0QED result in which the photon is determined at $Q=10$~GeV (denoted as NNPDF4.0QED*) is also shown. |
 | \small Scale dependence of the central photon PDF in the NNPDF4.0QED, NNPDF3.1QED, MSHT20QED, and CT18QED PDF sets. The NNPDF4.0QED result in which the photon is determined at $Q=10$~GeV (denoted as NNPDF4.0QED*) is also shown. |
 | \small Comparison of PDFs in the NNPDF4.0QED and the NNPDF4.0 (pure QCD) sets, shown as a ratio to the former at $Q=100$ GeV. Bands correspond to 1$\sigma$ uncertainties. From left to right and from top to bottom, the up, anti-up, down, anti-down, strange, and gluon PDFs are shown. |
 | \small Comparison of PDFs in the NNPDF4.0QED and the NNPDF4.0 (pure QCD) sets, shown as a ratio to the former at $Q=100$ GeV. Bands correspond to 1$\sigma$ uncertainties. From left to right and from top to bottom, the up, anti-up, down, anti-down, strange, and gluon PDFs are shown. |
 | \small Comparison of PDFs in the NNPDF4.0QED and the NNPDF4.0 (pure QCD) sets, shown as a ratio to the former at $Q=100$ GeV. Bands correspond to 1$\sigma$ uncertainties. From left to right and from top to bottom, the up, anti-up, down, anti-down, strange, and gluon PDFs are shown. |
 | \small Comparison of PDFs in the NNPDF4.0QED and the NNPDF4.0 (pure QCD) sets, shown as a ratio to the former at $Q=100$ GeV. Bands correspond to 1$\sigma$ uncertainties. From left to right and from top to bottom, the up, anti-up, down, anti-down, strange, and gluon PDFs are shown. |
 | \small Comparison of PDFs in the NNPDF4.0QED and the NNPDF4.0 (pure QCD) sets, shown as a ratio to the former at $Q=100$ GeV. Bands correspond to 1$\sigma$ uncertainties. From left to right and from top to bottom, the up, anti-up, down, anti-down, strange, and gluon PDFs are shown. |
 | \small Comparison of PDFs in the NNPDF4.0QED and the NNPDF4.0 (pure QCD) sets, shown as a ratio to the former at $Q=100$ GeV. Bands correspond to 1$\sigma$ uncertainties. From left to right and from top to bottom, the up, anti-up, down, anti-down, strange, and gluon PDFs are shown. |
 | \small Left: the percentage momentum fraction carried by the photon PDF $\gamma(x,Q^2)$ in NNPDF4.0QED and in NNPDF3.1QED as a function of the scale $Q$, where the bands indicate 1$\sigma$ uncertainties. Right: same for the momentum fraction carried by the gluon PDF in NNPDF4.0QED and in NNPDF4.0 (pure QCD). |
 | \small Left: the percentage momentum fraction carried by the photon PDF $\gamma(x,Q^2)$ in NNPDF4.0QED and in NNPDF3.1QED as a function of the scale $Q$, where the bands indicate 1$\sigma$ uncertainties. Right: same for the momentum fraction carried by the gluon PDF in NNPDF4.0QED and in NNPDF4.0 (pure QCD). |
 | \small The photon-induced contributions to the luminosity at the LHC with $\sqrt{s}=14$~TeV as a function of the invariant mass $m_X$ for NNPDF4.0QED NNLO compared to its NNPDF3.1QED counterpart (top) and compared to MSHT20QED and CT18QED, all shown as a ratio to NNPDF4.0QED. |
 | \small The photon-induced contributions to the luminosity at the LHC with $\sqrt{s}=14$~TeV as a function of the invariant mass $m_X$ for NNPDF4.0QED NNLO compared to its NNPDF3.1QED counterpart (top) and compared to MSHT20QED and CT18QED, all shown as a ratio to NNPDF4.0QED. |
 | \small The photon-induced contributions to the luminosity at the LHC with $\sqrt{s}=14$~TeV as a function of the invariant mass $m_X$ for NNPDF4.0QED NNLO compared to its NNPDF3.1QED counterpart (top) and compared to MSHT20QED and CT18QED, all shown as a ratio to NNPDF4.0QED. |
 | \small The photon-induced contributions to the luminosity at the LHC with $\sqrt{s}=14$~TeV as a function of the invariant mass $m_X$ for NNPDF4.0QED NNLO compared to its NNPDF3.1QED counterpart (top) and compared to MSHT20QED and CT18QED, all shown as a ratio to NNPDF4.0QED. |
 | \small Comparison of parton luminosities in the NNPDF4.0QED and the NNPDF4.0 (pure QCD) sets, shown as a ratio to the former for the LHC with $\sqrt{s}=14$~TeV. From left to right and from top to bottom the gluon-gluon, gluon-quark, quark-antiquark and quark-quark luminosities are shown. |
 | \small Comparison of parton luminosities in the NNPDF4.0QED and the NNPDF4.0 (pure QCD) sets, shown as a ratio to the former for the LHC with $\sqrt{s}=14$~TeV. From left to right and from top to bottom the gluon-gluon, gluon-quark, quark-antiquark and quark-quark luminosities are shown. |
 | \small Comparison of parton luminosities in the NNPDF4.0QED and the NNPDF4.0 (pure QCD) sets, shown as a ratio to the former for the LHC with $\sqrt{s}=14$~TeV. From left to right and from top to bottom the gluon-gluon, gluon-quark, quark-antiquark and quark-quark luminosities are shown. |
 | \small Comparison of parton luminosities in the NNPDF4.0QED and the NNPDF4.0 (pure QCD) sets, shown as a ratio to the former for the LHC with $\sqrt{s}=14$~TeV. From left to right and from top to bottom the gluon-gluon, gluon-quark, quark-antiquark and quark-quark luminosities are shown. |
 | \small Same as Fig.~\ref{fig:lumis_1}, now comparing NNPDF4.0QED to MSHT20~\cite{Cridge:2021pxm} and CT18qed~\cite{Xie:2023qbn}. |
 | \small Same as Fig.~\ref{fig:lumis_1}, now comparing NNPDF4.0QED to MSHT20~\cite{Cridge:2021pxm} and CT18qed~\cite{Xie:2023qbn}. |
 | \small Same as Fig.~\ref{fig:lumis_1}, now comparing NNPDF4.0QED to MSHT20~\cite{Cridge:2021pxm} and CT18qed~\cite{Xie:2023qbn}. |
 | \small Same as Fig.~\ref{fig:lumis_1}, now comparing NNPDF4.0QED to MSHT20~\cite{Cridge:2021pxm} and CT18qed~\cite{Xie:2023qbn}. |
 | \small Predictions for inclusive Drell--Yan production at the LHC with center-of-mass energy $\sqrt{s}=14$ TeV, computed at NLO accuracy in the QCD and electroweak couplings. All uncertainties shown are PDF uncertainties only. From top to bottom: neutral-current dilepton production as a function of the dilepton invariant mass $m_{\ell\bar{\ell}}$; $W^+$ production as a function of the antilepton pseudo-rapidity $\eta_{\bar{l}}$; $W^-$ production as a function of the antilepton pseudo-rapidity $\eta_{l}$. For each process, we display the absolute distributions (top panels) and the ratio to the central value obtained with NNPDF4.0QED PDFs, including photon-initiated channels (bottom panels). In the left panels the full NNPDF4.0 result is compared to NNPDF4.0 (QCD only), NNPDF3.1QED, NNPDF4.0QED (no photon-initiated); in the right panels it is compared to MSHT20QED~\cite{Cridge:2021pxm} and CT18QED~\cite{Xie:2023qbn}. Note that the experimentally measurable quantity is shown, so for dilepton production the $t$- and $u$-channel photon-induced contribution is also included. |
 | \small Predictions for inclusive Drell--Yan production at the LHC with center-of-mass energy $\sqrt{s}=14$ TeV, computed at NLO accuracy in the QCD and electroweak couplings. All uncertainties shown are PDF uncertainties only. From top to bottom: neutral-current dilepton production as a function of the dilepton invariant mass $m_{\ell\bar{\ell}}$; $W^+$ production as a function of the antilepton pseudo-rapidity $\eta_{\bar{l}}$; $W^-$ production as a function of the antilepton pseudo-rapidity $\eta_{l}$. For each process, we display the absolute distributions (top panels) and the ratio to the central value obtained with NNPDF4.0QED PDFs, including photon-initiated channels (bottom panels). In the left panels the full NNPDF4.0 result is compared to NNPDF4.0 (QCD only), NNPDF3.1QED, NNPDF4.0QED (no photon-initiated); in the right panels it is compared to MSHT20QED~\cite{Cridge:2021pxm} and CT18QED~\cite{Xie:2023qbn}. Note that the experimentally measurable quantity is shown, so for dilepton production the $t$- and $u$-channel photon-induced contribution is also included. |
 | \small Predictions for inclusive Drell--Yan production at the LHC with center-of-mass energy $\sqrt{s}=14$ TeV, computed at NLO accuracy in the QCD and electroweak couplings. All uncertainties shown are PDF uncertainties only. From top to bottom: neutral-current dilepton production as a function of the dilepton invariant mass $m_{\ell\bar{\ell}}$; $W^+$ production as a function of the antilepton pseudo-rapidity $\eta_{\bar{l}}$; $W^-$ production as a function of the antilepton pseudo-rapidity $\eta_{l}$. For each process, we display the absolute distributions (top panels) and the ratio to the central value obtained with NNPDF4.0QED PDFs, including photon-initiated channels (bottom panels). In the left panels the full NNPDF4.0 result is compared to NNPDF4.0 (QCD only), NNPDF3.1QED, NNPDF4.0QED (no photon-initiated); in the right panels it is compared to MSHT20QED~\cite{Cridge:2021pxm} and CT18QED~\cite{Xie:2023qbn}. Note that the experimentally measurable quantity is shown, so for dilepton production the $t$- and $u$-channel photon-induced contribution is also included. |
 | \small Predictions for inclusive Drell--Yan production at the LHC with center-of-mass energy $\sqrt{s}=14$ TeV, computed at NLO accuracy in the QCD and electroweak couplings. All uncertainties shown are PDF uncertainties only. From top to bottom: neutral-current dilepton production as a function of the dilepton invariant mass $m_{\ell\bar{\ell}}$; $W^+$ production as a function of the antilepton pseudo-rapidity $\eta_{\bar{l}}$; $W^-$ production as a function of the antilepton pseudo-rapidity $\eta_{l}$. For each process, we display the absolute distributions (top panels) and the ratio to the central value obtained with NNPDF4.0QED PDFs, including photon-initiated channels (bottom panels). In the left panels the full NNPDF4.0 result is compared to NNPDF4.0 (QCD only), NNPDF3.1QED, NNPDF4.0QED (no photon-initiated); in the right panels it is compared to MSHT20QED~\cite{Cridge:2021pxm} and CT18QED~\cite{Xie:2023qbn}. Note that the experimentally measurable quantity is shown, so for dilepton production the $t$- and $u$-channel photon-induced contribution is also included. |
 | \small Predictions for inclusive Drell--Yan production at the LHC with center-of-mass energy $\sqrt{s}=14$ TeV, computed at NLO accuracy in the QCD and electroweak couplings. All uncertainties shown are PDF uncertainties only. From top to bottom: neutral-current dilepton production as a function of the dilepton invariant mass $m_{\ell\bar{\ell}}$; $W^+$ production as a function of the antilepton pseudo-rapidity $\eta_{\bar{l}}$; $W^-$ production as a function of the antilepton pseudo-rapidity $\eta_{l}$. For each process, we display the absolute distributions (top panels) and the ratio to the central value obtained with NNPDF4.0QED PDFs, including photon-initiated channels (bottom panels). In the left panels the full NNPDF4.0 result is compared to NNPDF4.0 (QCD only), NNPDF3.1QED, NNPDF4.0QED (no photon-initiated); in the right panels it is compared to MSHT20QED~\cite{Cridge:2021pxm} and CT18QED~\cite{Xie:2023qbn}. Note that the experimentally measurable quantity is shown, so for dilepton production the $t$- and $u$-channel photon-induced contribution is also included. |
 | \small Predictions for inclusive Drell--Yan production at the LHC with center-of-mass energy $\sqrt{s}=14$ TeV, computed at NLO accuracy in the QCD and electroweak couplings. All uncertainties shown are PDF uncertainties only. From top to bottom: neutral-current dilepton production as a function of the dilepton invariant mass $m_{\ell\bar{\ell}}$; $W^+$ production as a function of the antilepton pseudo-rapidity $\eta_{\bar{l}}$; $W^-$ production as a function of the antilepton pseudo-rapidity $\eta_{l}$. For each process, we display the absolute distributions (top panels) and the ratio to the central value obtained with NNPDF4.0QED PDFs, including photon-initiated channels (bottom panels). In the left panels the full NNPDF4.0 result is compared to NNPDF4.0 (QCD only), NNPDF3.1QED, NNPDF4.0QED (no photon-initiated); in the right panels it is compared to MSHT20QED~\cite{Cridge:2021pxm} and CT18QED~\cite{Xie:2023qbn}. Note that the experimentally measurable quantity is shown, so for dilepton production the $t$- and $u$-channel photon-induced contribution is also included. |
 | \small Same as Fig.~\ref{fig:NNPDF_DY_14TEV_40_PHENO} but for the rapidity distribution of the Higgs for production in gluon-gluon fusion (top panel), in association with a $W^+$ boson (middle panel) and in vector-boson fusion (bottom panel). |
 | \small Same as Fig.~\ref{fig:NNPDF_DY_14TEV_40_PHENO} but for the rapidity distribution of the Higgs for production in gluon-gluon fusion (top panel), in association with a $W^+$ boson (middle panel) and in vector-boson fusion (bottom panel). |
 | \small Same as Fig.~\ref{fig:NNPDF_DY_14TEV_40_PHENO} but for the rapidity distribution of the Higgs for production in gluon-gluon fusion (top panel), in association with a $W^+$ boson (middle panel) and in vector-boson fusion (bottom panel). |
 | \small Same as Fig.~\ref{fig:NNPDF_DY_14TEV_40_PHENO} but for the rapidity distribution of the Higgs for production in gluon-gluon fusion (top panel), in association with a $W^+$ boson (middle panel) and in vector-boson fusion (bottom panel). |
 | \small Same as Fig.~\ref{fig:NNPDF_DY_14TEV_40_PHENO} but for the rapidity distribution of the Higgs for production in gluon-gluon fusion (top panel), in association with a $W^+$ boson (middle panel) and in vector-boson fusion (bottom panel). |
 | \small Same as Fig.~\ref{fig:NNPDF_DY_14TEV_40_PHENO} but for the rapidity distribution of the Higgs for production in gluon-gluon fusion (top panel), in association with a $W^+$ boson (middle panel) and in vector-boson fusion (bottom panel). |
 | \small Same as Fig.~\ref{fig:NNPDF_DY_14TEV_40_PHENO} but for the dilepton transverse momentum distribution for weak boson pair production ($W^+W^-$ and $W^+Z$) and for the invariant mass distribution for top-quark pair production. No acceptance cuts on the decay productions of the $W, Z$ bosons and top quark have been imposed. |
 | \small Same as Fig.~\ref{fig:NNPDF_DY_14TEV_40_PHENO} but for the dilepton transverse momentum distribution for weak boson pair production ($W^+W^-$ and $W^+Z$) and for the invariant mass distribution for top-quark pair production. No acceptance cuts on the decay productions of the $W, Z$ bosons and top quark have been imposed. |
 | \small Same as Fig.~\ref{fig:NNPDF_DY_14TEV_40_PHENO} but for the dilepton transverse momentum distribution for weak boson pair production ($W^+W^-$ and $W^+Z$) and for the invariant mass distribution for top-quark pair production. No acceptance cuts on the decay productions of the $W, Z$ bosons and top quark have been imposed. |
 | \small Same as Fig.~\ref{fig:NNPDF_DY_14TEV_40_PHENO} but for the dilepton transverse momentum distribution for weak boson pair production ($W^+W^-$ and $W^+Z$) and for the invariant mass distribution for top-quark pair production. No acceptance cuts on the decay productions of the $W, Z$ bosons and top quark have been imposed. |
 | \small Same as Fig.~\ref{fig:NNPDF_DY_14TEV_40_PHENO} but for the dilepton transverse momentum distribution for weak boson pair production ($W^+W^-$ and $W^+Z$) and for the invariant mass distribution for top-quark pair production. No acceptance cuts on the decay productions of the $W, Z$ bosons and top quark have been imposed. |
 | \small Same as Fig.~\ref{fig:NNPDF_DY_14TEV_40_PHENO} but for the dilepton transverse momentum distribution for weak boson pair production ($W^+W^-$ and $W^+Z$) and for the invariant mass distribution for top-quark pair production. No acceptance cuts on the decay productions of the $W, Z$ bosons and top quark have been imposed. |
 | \small Distances at $Q=100$~GeV between the central values (left) and uncertainties (right) of the 100 NNPDF4.0 PDF replicas at NLO (top) and NNLO (bottom) whose statistical indicators are compared in Table~\ref{tab:chi2_newbaseline}. |
 | \small Distances at $Q=100$~GeV between the central values (left) and uncertainties (right) of the 100 NNPDF4.0 PDF replicas at NLO (top) and NNLO (bottom) whose statistical indicators are compared in Table~\ref{tab:chi2_newbaseline}. |
 | \small Distances at $Q=100$~GeV between the central values (left) and uncertainties (right) of the 100 NNPDF4.0 PDF replicas at NLO (top) and NNLO (bottom) whose statistical indicators are compared in Table~\ref{tab:chi2_newbaseline}. |
 | \small Distances at $Q=100$~GeV between the central values (left) and uncertainties (right) of the 100 NNPDF4.0 PDF replicas at NLO (top) and NNLO (bottom) whose statistical indicators are compared in Table~\ref{tab:chi2_newbaseline}. |
 | \small The NLO PDFs at $Q=100$ GeV from the 100 replica sets compared in Table~\ref{tab:chi2_newbaseline} and Fig.~\ref{fig:distances_newbaseline}. Results are shown normalized to the central value of the published set. Bands correspond to 1$\sigma$ uncertainties. From left to right and from top to bottom, we show the up, anti-up, down, anti-down, strange and gluon PDFs. |
 | \small The NLO PDFs at $Q=100$ GeV from the 100 replica sets compared in Table~\ref{tab:chi2_newbaseline} and Fig.~\ref{fig:distances_newbaseline}. Results are shown normalized to the central value of the published set. Bands correspond to 1$\sigma$ uncertainties. From left to right and from top to bottom, we show the up, anti-up, down, anti-down, strange and gluon PDFs. |
 | \small The NLO PDFs at $Q=100$ GeV from the 100 replica sets compared in Table~\ref{tab:chi2_newbaseline} and Fig.~\ref{fig:distances_newbaseline}. Results are shown normalized to the central value of the published set. Bands correspond to 1$\sigma$ uncertainties. From left to right and from top to bottom, we show the up, anti-up, down, anti-down, strange and gluon PDFs. |
 | \small The NLO PDFs at $Q=100$ GeV from the 100 replica sets compared in Table~\ref{tab:chi2_newbaseline} and Fig.~\ref{fig:distances_newbaseline}. Results are shown normalized to the central value of the published set. Bands correspond to 1$\sigma$ uncertainties. From left to right and from top to bottom, we show the up, anti-up, down, anti-down, strange and gluon PDFs. |
 | \small The NLO PDFs at $Q=100$ GeV from the 100 replica sets compared in Table~\ref{tab:chi2_newbaseline} and Fig.~\ref{fig:distances_newbaseline}. Results are shown normalized to the central value of the published set. Bands correspond to 1$\sigma$ uncertainties. From left to right and from top to bottom, we show the up, anti-up, down, anti-down, strange and gluon PDFs. |
 | \small The NLO PDFs at $Q=100$ GeV from the 100 replica sets compared in Table~\ref{tab:chi2_newbaseline} and Fig.~\ref{fig:distances_newbaseline}. Results are shown normalized to the central value of the published set. Bands correspond to 1$\sigma$ uncertainties. From left to right and from top to bottom, we show the up, anti-up, down, anti-down, strange and gluon PDFs. |
 | \small Same as Fig.~\ref{fig:NewBaseline-q100gev-ratios_nlo} at NNLO. |
 | \small Same as Fig.~\ref{fig:NewBaseline-q100gev-ratios_nlo} at NNLO. |
 | \small Same as Fig.~\ref{fig:NewBaseline-q100gev-ratios_nlo} at NNLO. |
 | \small Same as Fig.~\ref{fig:NewBaseline-q100gev-ratios_nlo} at NNLO. |
 | \small Same as Fig.~\ref{fig:NewBaseline-q100gev-ratios_nlo} at NNLO. |
 | \small Same as Fig.~\ref{fig:NewBaseline-q100gev-ratios_nlo} at NNLO. |
 | \small Same as in Fig.~\ref{fig:NewBaseline-q100gev-ratios_nlo} for the partonic luminosities at the LHC with $\sqrt{s}=14$ TeV as a function of the invariant mass $m_X$. The $gg$ (left) and $qg$ (right) luminosities are shown in the top row, the $q\bar{q}$ (left), and $qq$ (right) luminosities in the bottom row. |
 | \small Same as in Fig.~\ref{fig:NewBaseline-q100gev-ratios_nlo} for the partonic luminosities at the LHC with $\sqrt{s}=14$ TeV as a function of the invariant mass $m_X$. The $gg$ (left) and $qg$ (right) luminosities are shown in the top row, the $q\bar{q}$ (left), and $qq$ (right) luminosities in the bottom row. |
 | \small Same as in Fig.~\ref{fig:NewBaseline-q100gev-ratios_nlo} for the partonic luminosities at the LHC with $\sqrt{s}=14$ TeV as a function of the invariant mass $m_X$. The $gg$ (left) and $qg$ (right) luminosities are shown in the top row, the $q\bar{q}$ (left), and $qq$ (right) luminosities in the bottom row. |
 | \small Same as in Fig.~\ref{fig:NewBaseline-q100gev-ratios_nlo} for the partonic luminosities at the LHC with $\sqrt{s}=14$ TeV as a function of the invariant mass $m_X$. The $gg$ (left) and $qg$ (right) luminosities are shown in the top row, the $q\bar{q}$ (left), and $qq$ (right) luminosities in the bottom row. |
 | \small Same as in Fig.~\ref{fig:lumi-newbaseline_nlo} at NNLO. |
 | \small Same as in Fig.~\ref{fig:lumi-newbaseline_nlo} at NNLO. |
 | \small Same as in Fig.~\ref{fig:lumi-newbaseline_nlo} at NNLO. |
 | \small Same as in Fig.~\ref{fig:lumi-newbaseline_nlo} at NNLO. |
 | \small Statistical distances between the central values (left) and uncertainties (right) of the 100 NNPDF4.0 PDF replicas whose statistical indicators are compared in Table~\ref{tab:chi2_exact_vs_truncated_baseline}. |
 | \small Statistical distances between the central values (left) and uncertainties (right) of the 100 NNPDF4.0 PDF replicas whose statistical indicators are compared in Table~\ref{tab:chi2_exact_vs_truncated_baseline}. |
 | \small The gluon (left) and anti-up quark (right) PDFs at $Q=100$ GeV from the PDF sets compared in Table~\ref{tab:chi2_exact_vs_truncated_baseline} and Fig.~\ref{fig:distance_EXA_vs_TRN}. Results are shown normalized to the central value of the PDFs obtained with {truncated} evolution (published NNPDF4.0). |
 | \small The gluon (left) and anti-up quark (right) PDFs at $Q=100$ GeV from the PDF sets compared in Table~\ref{tab:chi2_exact_vs_truncated_baseline} and Fig.~\ref{fig:distance_EXA_vs_TRN}. Results are shown normalized to the central value of the PDFs obtained with {truncated} evolution (published NNPDF4.0). |
 | \small Comparison between the NNPDF3.1QED PDFs at $Q_a=100$~GeV, with the results of evolution of the same PDFs to $Q_b=1.65$ GeV followed by evolution back to $Q_a=100$~GeV. We show, from top to bottom, the gluon, up and down (left) and the photon, anti-up and anti-down (right). In each case we show both the pair of PDFs, and their percentage relative difference. |
 | \small Comparison between the NNPDF3.1QED PDFs at $Q_a=100$~GeV, with the results of evolution of the same PDFs to $Q_b=1.65$ GeV followed by evolution back to $Q_a=100$~GeV. We show, from top to bottom, the gluon, up and down (left) and the photon, anti-up and anti-down (right). In each case we show both the pair of PDFs, and their percentage relative difference. |
 | \small Comparison between the NNPDF3.1QED PDFs at $Q_a=100$~GeV, with the results of evolution of the same PDFs to $Q_b=1.65$ GeV followed by evolution back to $Q_a=100$~GeV. We show, from top to bottom, the gluon, up and down (left) and the photon, anti-up and anti-down (right). In each case we show both the pair of PDFs, and their percentage relative difference. |
 | \small Comparison between the NNPDF3.1QED PDFs at $Q_a=100$~GeV, with the results of evolution of the same PDFs to $Q_b=1.65$ GeV followed by evolution back to $Q_a=100$~GeV. We show, from top to bottom, the gluon, up and down (left) and the photon, anti-up and anti-down (right). In each case we show both the pair of PDFs, and their percentage relative difference. |
 | \small Comparison between the NNPDF3.1QED PDFs at $Q_a=100$~GeV, with the results of evolution of the same PDFs to $Q_b=1.65$ GeV followed by evolution back to $Q_a=100$~GeV. We show, from top to bottom, the gluon, up and down (left) and the photon, anti-up and anti-down (right). In each case we show both the pair of PDFs, and their percentage relative difference. |
 | \small Comparison between the NNPDF3.1QED PDFs at $Q_a=100$~GeV, with the results of evolution of the same PDFs to $Q_b=1.65$ GeV followed by evolution back to $Q_a=100$~GeV. We show, from top to bottom, the gluon, up and down (left) and the photon, anti-up and anti-down (right). In each case we show both the pair of PDFs, and their percentage relative difference. |
 | \small Percentage difference between pairs of PDFs at $Q=100$~GeV obtained evolving NNPDF3.1QED PDFs from $Q_0=1.65$~GeV with different implementation of the QED$\times$QCD evolution. From top to bottom the gluon, up, down and charm (left), photon, antiup, antidown and anticharm (right) are shown. The three curves compare: {\sc\small APFEL} {exact} vs.\ {truncated} evolution (green, dot-dashed); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} with in each case default settings for the running of the couplings (see text) (blue, solid); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} both with {\sc\small APFEL} settings for the running of the coupling (red, dashed). Note the logarithmic scale on the $y$ axis; note also that the range on the $y$ axis for gluon plot differs from that of all other PDFs |
 | \small Percentage difference between pairs of PDFs at $Q=100$~GeV obtained evolving NNPDF3.1QED PDFs from $Q_0=1.65$~GeV with different implementation of the QED$\times$QCD evolution. From top to bottom the gluon, up, down and charm (left), photon, antiup, antidown and anticharm (right) are shown. The three curves compare: {\sc\small APFEL} {exact} vs.\ {truncated} evolution (green, dot-dashed); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} with in each case default settings for the running of the couplings (see text) (blue, solid); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} both with {\sc\small APFEL} settings for the running of the coupling (red, dashed). Note the logarithmic scale on the $y$ axis; note also that the range on the $y$ axis for gluon plot differs from that of all other PDFs |
 | \small Percentage difference between pairs of PDFs at $Q=100$~GeV obtained evolving NNPDF3.1QED PDFs from $Q_0=1.65$~GeV with different implementation of the QED$\times$QCD evolution. From top to bottom the gluon, up, down and charm (left), photon, antiup, antidown and anticharm (right) are shown. The three curves compare: {\sc\small APFEL} {exact} vs.\ {truncated} evolution (green, dot-dashed); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} with in each case default settings for the running of the couplings (see text) (blue, solid); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} both with {\sc\small APFEL} settings for the running of the coupling (red, dashed). Note the logarithmic scale on the $y$ axis; note also that the range on the $y$ axis for gluon plot differs from that of all other PDFs |
 | \small Percentage difference between pairs of PDFs at $Q=100$~GeV obtained evolving NNPDF3.1QED PDFs from $Q_0=1.65$~GeV with different implementation of the QED$\times$QCD evolution. From top to bottom the gluon, up, down and charm (left), photon, antiup, antidown and anticharm (right) are shown. The three curves compare: {\sc\small APFEL} {exact} vs.\ {truncated} evolution (green, dot-dashed); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} with in each case default settings for the running of the couplings (see text) (blue, solid); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} both with {\sc\small APFEL} settings for the running of the coupling (red, dashed). Note the logarithmic scale on the $y$ axis; note also that the range on the $y$ axis for gluon plot differs from that of all other PDFs |
 | \small Percentage difference between pairs of PDFs at $Q=100$~GeV obtained evolving NNPDF3.1QED PDFs from $Q_0=1.65$~GeV with different implementation of the QED$\times$QCD evolution. From top to bottom the gluon, up, down and charm (left), photon, antiup, antidown and anticharm (right) are shown. The three curves compare: {\sc\small APFEL} {exact} vs.\ {truncated} evolution (green, dot-dashed); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} with in each case default settings for the running of the couplings (see text) (blue, solid); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} both with {\sc\small APFEL} settings for the running of the coupling (red, dashed). Note the logarithmic scale on the $y$ axis; note also that the range on the $y$ axis for gluon plot differs from that of all other PDFs |
 | \small Percentage difference between pairs of PDFs at $Q=100$~GeV obtained evolving NNPDF3.1QED PDFs from $Q_0=1.65$~GeV with different implementation of the QED$\times$QCD evolution. From top to bottom the gluon, up, down and charm (left), photon, antiup, antidown and anticharm (right) are shown. The three curves compare: {\sc\small APFEL} {exact} vs.\ {truncated} evolution (green, dot-dashed); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} with in each case default settings for the running of the couplings (see text) (blue, solid); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} both with {\sc\small APFEL} settings for the running of the coupling (red, dashed). Note the logarithmic scale on the $y$ axis; note also that the range on the $y$ axis for gluon plot differs from that of all other PDFs |
 | \small Percentage difference between pairs of PDFs at $Q=100$~GeV obtained evolving NNPDF3.1QED PDFs from $Q_0=1.65$~GeV with different implementation of the QED$\times$QCD evolution. From top to bottom the gluon, up, down and charm (left), photon, antiup, antidown and anticharm (right) are shown. The three curves compare: {\sc\small APFEL} {exact} vs.\ {truncated} evolution (green, dot-dashed); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} with in each case default settings for the running of the couplings (see text) (blue, solid); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} both with {\sc\small APFEL} settings for the running of the coupling (red, dashed). Note the logarithmic scale on the $y$ axis; note also that the range on the $y$ axis for gluon plot differs from that of all other PDFs |
 | \small Percentage difference between pairs of PDFs at $Q=100$~GeV obtained evolving NNPDF3.1QED PDFs from $Q_0=1.65$~GeV with different implementation of the QED$\times$QCD evolution. From top to bottom the gluon, up, down and charm (left), photon, antiup, antidown and anticharm (right) are shown. The three curves compare: {\sc\small APFEL} {exact} vs.\ {truncated} evolution (green, dot-dashed); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} with in each case default settings for the running of the couplings (see text) (blue, solid); {\sc\small APFEL} vs.\ {\sc\small EKO} {exact} both with {\sc\small APFEL} settings for the running of the coupling (red, dashed). Note the logarithmic scale on the $y$ axis; note also that the range on the $y$ axis for gluon plot differs from that of all other PDFs |