MSHT Global PDF Fit Framework

• MSHT Global PDF Fit is a framework that employs a dynamic Hessian tolerance method with flexible Chebyshev parametrisations to extract proton parton distribution functions.
• It integrates diverse experimental datasets from HERA, LHC, and Tevatron with perturbative QCD calculations up to approximate N³LO and QED corrections.
• The approach simultaneously fits PDFs and key QCD parameters like αₛ and heavy-quark masses to meet the high-precision standards required for collider phenomenology and PDF4LHC21.

MSHT global PDF fit denotes the family of global proton parton distribution function determinations produced by the MSHT collaboration, formerly MMHT, within a “Mass Scheme Hessian Tolerance” framework. In its modern form, especially MSHT20 and its later extensions, the framework combines broad hard-scattering datasets with flexible Chebyshev-based parametrisations, a Hessian uncertainty formalism with dynamic tolerance, and perturbative calculations ranging from LO, NLO, and NNLO to approximate N$^3$LO, with additional variants that fit $\alpha_S(M_Z^2)$, heavy-quark masses, the top-quark pole mass, and QED-enhanced partonic structure including a photon PDF (Bailey et al., 2020, McGowan et al., 2022, Cridge et al., 2023).

1. Framework, parametrisation, and fit architecture

MSHT20 superseded MMHT14 while retaining the same basic fitting philosophy and extending both the parametrisation and the dataset. At the input scale $Q_0^2 = 1~\mathrm{GeV}^2$, the PDFs are written in a flexible polynomial basis of the form

$$f(x) = A (1-x)^\eta x^\delta \left(1 + \sum_{i=1}^n a_i T_i(1-2x^{1/2})\right),$$

with Chebyshev polynomials used to increase shape flexibility. The MSHT20 fit features 52 partonic parameters and is distributed with 32 pairs of eigenvector PDFs. Relative to MMHT14, the parametrisation was extended in particular for the light antiquark asymmetry and the strange sector; the ratio $\bar d/\bar u$ is fitted directly, rather than constraining $\bar d-\bar u$ to vanish at small $x$ by construction (Bailey et al., 2020, Ball et al., 2022).

The heavy-flavour treatment is based on the “optimal” TR’ general-mass variable flavour number scheme. In the benchmarking configuration discussed for PDF4LHC21, common theory choices include $m_c = 1.4~\mathrm{GeV}$, $m_b = 4.75~\mathrm{GeV}$, and $\alpha_S(M_Z^2)=0.118$; MSHT20 also provides fits in which $\alpha_S(M_Z^2)$0 and heavy-quark masses are varied explicitly (Ball et al., 2022, Cridge et al., 2021). No positivity conditions are imposed on PDF combinations at the input scale, although the fitted distributions remain positive over most regions relevant for collider phenomenology (Ball et al., 2022).

A central methodological question for fixed functional forms is whether they are sufficiently flexible. The first global closure test of the MSHT approach found that the default MSHT20 parametrisation can reproduce the features of the input set to well within textbook uncertainties, providing direct evidence that parametrisation inflexibility is not a significant issue in the data region. The same study found that reduced parametrisations with substantially fewer free parameters can fail closure tests, implying that the extended MSHT20 basis is not a cosmetic addition but part of the fit’s statistical consistency (Harland-Lang et al., 2024).

2. Experimental input and perturbative content

The MSHT global fit is explicitly global in dataset composition. The core input includes final HERA combined inclusive cross sections and heavy-flavour structure functions, fixed-target DIS data, neutrino DIS and dimuon measurements, Tevatron vector-boson and top data, and a large set of LHC measurements. MSHT20 incorporated, among other additions, precise 7 and 8 TeV LHC data on $\alpha_S(M_Z^2)$1 and $\alpha_S(M_Z^2)$2 production, inclusive jets, top-quark pair production, and $\alpha_S(M_Z^2)$3 boson transverse momentum, together with final HERA Run I+II data and updated Tevatron constraints (Bailey et al., 2020, Cridge, 2021).

The perturbative description uses up to NNLO QCD corrections for all data sets that play a major role in the fit, and NLO electroweak corrections where relevant. This includes NNLO treatments of DIS, Drell–Yan, jet production, and top-pair production in the MSHT20 generation, with later work extending the formalism to approximate N$\alpha_S(M_Z^2)$4LO. In the aN$\alpha_S(M_Z^2)$5LO framework, all available N$\alpha_S(M_Z^2)$6LO information is used for PDF evolution, heavy-flavour thresholds, and DIS coefficient functions; for processes whose full N$\alpha_S(M_Z^2)$7LO corrections are not known, parametrised $\alpha_S(M_Z^2)$8-factors and additional nuisance parameters are introduced to model the missing higher-order contributions (McGowan et al., 2022, Cridge et al., 2023).

The fit content has continued to evolve in targeted directions. A dedicated study of the high-$\alpha_S(M_Z^2)$9 region re-examined target mass corrections, higher-twist effects, the treatment of fixed-target DIS data, the inclusion of Seaquest fixed-target Drell–Yan data, and new ZEUS data extending into high $Q_0^2 = 1~\mathrm{GeV}^2$0. In that update, Seaquest had the largest effect, especially on light-quark separation at high $Q_0^2 = 1~\mathrm{GeV}^2$1, while the impact of the other new ingredients was milder (Harland-Lang et al., 4 Oct 2025).

MSHT also developed QED-enhanced global fits. The MSHT20qed_an3lo analysis combined QED corrections and approximate N$Q_0^2 = 1~\mathrm{GeV}^2$2LO QCD corrections in a single global fit, including QED effects up to $Q_0^2 = 1~\mathrm{GeV}^2$3, $Q_0^2 = 1~\mathrm{GeV}^2$4, and $Q_0^2 = 1~\mathrm{GeV}^2$5 in DGLAP evolution, together with a photon PDF obtained using the reorganized LUXqed formalism. In these fits the QED impact on PDFs is milder than the aN$Q_0^2 = 1~\mathrm{GeV}^2$6LO QCD impact, but not negligible at the precision now required (Cridge et al., 2023).

3. Statistical methodology and uncertainty model

MSHT uses a Hessian framework for uncertainty propagation, with correlated systematics incorporated through nuisance parameters or covariance matrices. In the covariance formulation,

$Q_0^2 = 1~\mathrm{GeV}^2$7

while in the nuisance-parameter treatment the correlated shifts are fitted simultaneously with the PDF parameters. This structure is used throughout the global analyses, including DIS, Drell–Yan, jet, and top datasets (Cridge, 2021, Harland-Lang et al., 2016).

The characteristic MSHT feature is the dynamic tolerance procedure. Rather than adopting a universal textbook $Q_0^2 = 1~\mathrm{GeV}^2$8 rule for 68% confidence intervals, the allowed parameter range is determined by requiring that each important dataset remain within its own 68% confidence-level tolerance. In practice, the overall uncertainty is set by the first dataset whose fit quality exceeds its tolerance when scanning along a parameter direction. This prescription is used for PDF eigenvectors and also for external parameters such as $Q_0^2 = 1~\mathrm{GeV}^2$9 and the top-quark mass, and it generally yields uncertainties larger than a naive $$f(x) = A (1-x)^\eta x^\delta \left(1 + \sum_{i=1}^n a_i T_i(1-2x^{1/2})\right),$$0 estimate (Cridge et al., 2021, Cridge et al., 2024, Cridge et al., 2023).

At aN$$f(x) = A (1-x)^\eta x^\delta \left(1 + \sum_{i=1}^n a_i T_i(1-2x^{1/2})\right),$$1LO, the Hessian formalism was enlarged to include theoretical nuisance parameters encoding missing higher-order uncertainties. Splitting functions, transition matrix elements, coefficient functions, and process-specific $$f(x) = A (1-x)^\eta x^\delta \left(1 + \sum_{i=1}^n a_i T_i(1-2x^{1/2})\right),$$2-factors can then vary within priors derived from existing theoretical information. A plausible implication is that MSHT’s published aN$$f(x) = A (1-x)^\eta x^\delta \left(1 + \sum_{i=1}^n a_i T_i(1-2x^{1/2})\right),$$3LO uncertainties are not simply higher-order analogues of NNLO Hessian bands, but incorporate an explicit statistical model for perturbative incompleteness (McGowan et al., 2022).

Closure testing clarified the relation between textbook Hessian statistics and real-world global fits. In pseudodata generated from a known truth, the default MSHT20 parametrisation achieved correct statistical coverage with $$f(x) = A (1-x)^\eta x^\delta \left(1 + \sum_{i=1}^n a_i T_i(1-2x^{1/2})\right),$$4. In real fits, however, the same study argued that an enlarged tolerance remains necessary because data and theory inconsistencies do not automatically inflate the $$f(x) = A (1-x)^\eta x^\delta \left(1 + \sum_{i=1}^n a_i T_i(1-2x^{1/2})\right),$$5 PDF uncertainty. This point also underlies MSHT’s direct comparison with the neural-network approach, in which NNPDF4.0 uncertainties were found to be broadly similar to MSHT results at textbook tolerance but significantly smaller than standard MSHT20 uncertainties based on enlarged tolerance (Harland-Lang et al., 2024).

4. Simultaneous fits of $$f(x) = A (1-x)^\eta x^\delta \left(1 + \sum_{i=1}^n a_i T_i(1-2x^{1/2})\right),$$6, heavy masses, and the top-quark pole mass

A major use of the MSHT framework is the simultaneous determination of PDFs and external QCD parameters. In the dedicated MSHT20 study of $$f(x) = A (1-x)^\eta x^\delta \left(1 + \sum_{i=1}^n a_i T_i(1-2x^{1/2})\right),$$7 and heavy-quark masses, the fit was repeated for fixed values of $$f(x) = A (1-x)^\eta x^\delta \left(1 + \sum_{i=1}^n a_i T_i(1-2x^{1/2})\right),$$8 spanning $$f(x) = A (1-x)^\eta x^\delta \left(1 + \sum_{i=1}^n a_i T_i(1-2x^{1/2})\right),$$9 to $\bar d/\bar u$0 in steps of $\bar d/\bar u$1, and for systematic variations of $\bar d/\bar u$2 and $\bar d/\bar u$3. The preferred values were $\bar d/\bar u$4 at NLO and $\bar d/\bar u$5 at NNLO, both at 68% confidence level under the dynamical tolerance procedure. The same analysis found that the default heavy-quark pole masses, $\bar d/\bar u$6 and $\bar d/\bar u$7, are very largely compatible with the best fits to data (Cridge et al., 2021).

The extension to approximate N$\bar d/\bar u$8LO yielded the first global $\bar d/\bar u$9 determination at that order: $\bar d-\bar u$0. This is in excellent agreement with the NNLO value $\bar d-\bar u$1, while the slightly larger aN$\bar d-\bar u$2LO uncertainty reflects the explicit inclusion of missing higher-order theoretical uncertainties. The analysis also showed that the choice between inclusive jet and dijet input has a non-negligible effect at NNLO but much less at aN$\bar d-\bar u$3LO (Cridge et al., 2024).

The MSHT framework was also used to constrain the top-quark pole mass from top-pair production data in a global PDF fit. Using ATLAS and CMS 8 TeV differential distributions together with total $\bar d-\bar u$4 cross sections from the LHC and Tevatron, and comparing to NNLO QCD predictions with NLO electroweak corrections, the fit obtained $\bar d-\bar u$5. The best-fit strong coupling in the same study was $\bar d-\bar u$6, with the 68% confidence-level range

$\bar d-\bar u$7

The extracted top mass was reported as compatible with the Particle Data Group world average pole mass extracted from cross section measurements, $\bar d-\bar u$8 (Cridge et al., 2023).

Quantity	MSHT determination	Context
$\bar d-\bar u$9	$x$0	NLO global fit
$x$1	$x$2	NNLO global fit
$x$3	$x$4	aN$x$5LO global fit
$x$6	$x$7	Global fit including top-pair data

Prospective studies have extended this logic to future facilities. In an EIC pseudodata analysis, the MSHT methodology again fitted PDFs and $x$8 simultaneously. The conservative uncertainty scenario led to a moderate but non-negligible impact, while the optimistic scenario produced a significant reduction in the uncertainty on $x$9. That study also showed that explicit tensions between the EIC pseudodata and the rest of the fit can bias the extracted strong coupling, which reinforces the rationale for conservative tolerance prescriptions in the global framework (Harland-Lang et al., 5 Dec 2025).

5. High-$m_c = 1.4~\mathrm{GeV}$0 structure, jet and top data, and fit tensions

The high-$m_c = 1.4~\mathrm{GeV}$1 gluon is one of the persistent focal points of MSHT phenomenology. In the dedicated study of LHC jet and $m_c = 1.4~\mathrm{GeV}$2 data, inclusive jet and dijet datasets were analysed at NNLO and aN$m_c = 1.4~\mathrm{GeV}$3LO within the global fit. At NNLO, mild tension was observed in the high-$m_c = 1.4~\mathrm{GeV}$4 gluon preference of jets, dijets, and $m_c = 1.4~\mathrm{GeV}$5 data. At aN$m_c = 1.4~\mathrm{GeV}$6LO, that tension was largely eliminated, and the fit quality to ATLAS 8 TeV $m_c = 1.4~\mathrm{GeV}$7 data improved substantially relative to NNLO. Dijet data, particularly the CMS 8 TeV triple-differential measurement, provided especially strong high-$m_c = 1.4~\mathrm{GeV}$8 constraints; inclusive jets and dijets from the same experiment and energy were not included simultaneously in a single fit, in order to avoid double counting (Cridge et al., 2023).

Top-pair differential data enter the same high-$m_c = 1.4~\mathrm{GeV}$9 problem from a different direction. The MSHT top-mass study found that varying $m_b = 4.75~\mathrm{GeV}$0 mostly affects the high-$m_b = 4.75~\mathrm{GeV}$1 gluon PDF: increasing $m_b = 4.75~\mathrm{GeV}$2 shifts the gluon upward at large $m_b = 4.75~\mathrm{GeV}$3, but the resulting changes remain within the sizeable PDF uncertainties of that region. The impact is slightly larger when using the $m_b = 4.75~\mathrm{GeV}$4 distribution than for $m_b = 4.75~\mathrm{GeV}$5, but still subdominant to statistical and experimental uncertainties (Cridge et al., 2023).

Earlier work within the MMHT framework highlighted a more basic issue: the treatment of correlated systematic uncertainties in differential top data can dominate the apparent PDF information. For ATLAS 8 TeV lepton+jets data, severe difficulties were found in fitting $m_b = 4.75~\mathrm{GeV}$6, $m_b = 4.75~\mathrm{GeV}$7, $m_b = 4.75~\mathrm{GeV}$8, and $m_b = 4.75~\mathrm{GeV}$9 simultaneously when standard correlation assumptions were used for the dominant two-point Monte Carlo systematics. Reasonable decorrelations improved the fit quality dramatically, and the effect on the extracted gluon was larger than the effect of including NNLO QCD and NLO electroweak corrections. This episode established a continuing caution in MSHT studies: high-precision LHC data can constrain PDFs strongly, but the strength and even direction of the constraint may depend sensitively on the experimental correlation model (Bailey et al., 2019).

The later high-$\alpha_S(M_Z^2)=0.118$0 update broadened this picture beyond jets and top. Target mass corrections and fitted higher-twist corrections were incorporated for the first time at aN$\alpha_S(M_Z^2)=0.118$1LO in a global PDF analysis. Their impact on PDFs and on the preferred value of $\alpha_S(M_Z^2)=0.118$2 was found to be moderate but not negligible, with increased stability at aN$\alpha_S(M_Z^2)=0.118$3LO relative to lower orders. The same study reported that Seaquest fixed-target Drell–Yan data had the largest new effect, especially on the high-$\alpha_S(M_Z^2)=0.118$4 light sea flavour separation, whereas the updated fixed-target DIS treatment and new ZEUS data had milder consequences (Harland-Lang et al., 4 Oct 2025).

6. Benchmarking, PDF4LHC21, closure tests, and phenomenological use

MSHT20 is one of the three global fits that underpin PDF4LHC21. Benchmarking studies comparing CT18, MSHT20, and NNPDF3.1 showed that when all three groups use nearly identical theory settings and reduced datasets, their gluon and singlet PDFs agree closely across most of $\alpha_S(M_Z^2)=0.118$5, and many of the remaining differences can be traced to methodological choices or to the treatment of specific datasets such as neutrino dimuon data and high-$\alpha_S(M_Z^2)=0.118$6 collider measurements. The outcome of that programme was the PDF4LHC21 ensemble, built from CT18, MSHT20, and NNPDF3.1 and provided in both Monte Carlo and Hessian-reduced forms for LHC Run III applications (Cridge, 2021, Ball et al., 2022).

The comparison to NNPDF methodology was sharpened further by the global MSHT closure test and the like-for-like fit to NNPDF4.0 data and theory inputs. In that setting, the only intended difference was the fitting methodology itself: fixed Chebyshev parametrisation versus neural networks. The MSHT parametrisation gave a moderately, but noticeably, better fit quality than the central NNPDF4.0 fits, and the resulting PDF uncertainties were broadly similar to NNPDF’s when textbook $\alpha_S(M_Z^2)=0.118$7 tolerance was used. This result reinforced two distinct points: first, MSHT’s fixed functional form is flexible enough in the data region; second, the larger published MSHT20 uncertainties originate primarily from tolerance choices motivated by dataset and theory inconsistencies, not from an intrinsically broader parametrisation (Harland-Lang et al., 2024).

On the phenomenological side, MSHT developments have been especially visible in Higgs and electroweak precision predictions. In the combined QED and aN$\alpha_S(M_Z^2)=0.118$8LO framework, QED effects were found to roughly factorise from aN$\alpha_S(M_Z^2)=0.118$9LO QCD effects, and the deterioration in fit quality from adding QED was very small at aN$\alpha_S(M_Z^2)$00LO compared with NNLO. In Higgs-sector applications using QED-enhanced PDFs, the photon PDF determined through LUX-type constructions implies a small but non-negligible momentum fraction for the photon, with compensating reductions in quark and gluon momentum. The associated suppression of total Higgs-production cross sections was reported at the level of roughly $\alpha_S(M_Z^2)$01, while inter-group differences in photon-initiated electroweak corrections were reduced to the per-mille level; this suggests that the residual spread in Higgs predictions is driven mainly by baseline QCD PDF differences rather than by photon-PDF modelling itself (Cridge et al., 2023, Cooper-Sarkar et al., 8 Aug 2025).

Within contemporary collider phenomenology, the MSHT global PDF fit therefore serves several roles simultaneously: a standalone Hessian global fit with dynamic tolerance; a vehicle for joint extraction of PDFs and fundamental parameters such as $\alpha_S(M_Z^2)$02 and $\alpha_S(M_Z^2)$03; a platform for testing higher-order QCD and QED effects; and one of the principal inputs to the PDF4LHC21 community standard for LHC predictions (Ball et al., 2022, Cridge et al., 2024).