The global-fitting landscape: MSHT, CT, NNPDF

The three collaborations, the Hessian/Monte-Carlo/Bayesian trichotomy, the PDF4LHC combination, and where this lab's controlled comparison sits.

By now you can write down a proton PDF as a parametrized function (§3.1), impose the sum rules (§3.2), and evolve it with DGLAP to match data at any scale (§3.3–3.4). But there is a gap between "a fitting machine" and "the numbers the LHC actually uses," and that gap is filled not by one group but by three — MSHT, CT, and NNPDF — who fit the same physics with deliberately different tools. This page is the hub of a four-page arc: here we map the whole landscape and the deep methodological split that organizes it; §3.6–3.8 then profile each collaboration in turn. The reason this lab exists lives in this map, so read it as the spine of the portal.

Why there is more than one answer

Extracting a PDF is an inverse problem, and an under-determined one. You never observe $f_i(x, Q^2)$; you observe cross-sections, which are convolutions of the PDFs with perturbative coefficient functions:

A finite set of noisy data points is being asked to determine a set of continuous functions of $x$ across five-plus decades — several flavours' worth. That is formally ill-posed: infinitely many functions fit the data within its errors. To get a unique answer you must add information the data does not contain — a functional form, a smoothness prior, a regularization. Those choices are not bookkeeping; they are the physics of the uncertainty. Two groups can fit identical data and report different error bands purely because they injected different prior assumptions. So the field's answer to "what is the proton made of?" is irreducibly plural, and the interesting question becomes: how much of the spread between groups is real physics, and how much is method? Answering that cleanly is exactly what this lab was built to do.

The big three global unpolarized fits

Three collaborations produce full-featured global fits of the unpolarized proton — meaning they fit the whole world of hard-scattering data (DIS, Drell–Yan, jets, top, $W/Z$) simultaneously and release LHAPDF grids with error sets that the experiments consume. They differ in home, lineage, functional form, and — the crux — how they turn a fit into an uncertainty.

The parametrization column and the uncertainty column are not independent — and that entanglement, spelled out below, is the whole story.

The central insight: three philosophies of uncertainty

Strip away the flavour bases and the dataset differences and a single axis organizes the whole field: how do you represent "everything the data does not pin down"? There are two mature answers in production, and a third that is the reason this lab exists.

Hessian (MSHT, CT). Fix a functional form with $N$ parameters. Find the best-fit point $\mathbf{a}_0$ where $\chi^2$ is minimal, then expand the fit quality around it as a quadratic bowl. Diagonalizing the Hessian matrix of second derivatives gives $N$ orthogonal eigenvector directions; stepping a chosen distance up each wall of the bowl produces $2N$ error sets. That "chosen distance" is the tolerance $T$: instead of the textbook $\Delta\chi^2 = 1$ for a $1\sigma$ error, both groups inflate it (CT uses $T\approx 10$, i.e. $\Delta\chi^2 \approx 100$, at 90% CL; MSHT uses a dynamical tolerance that varies per direction) to absorb the tension between mutually-inconsistent datasets. The error set is compact and cross-section errors are a fast quadrature sum — but the uncertainty is only ever explored within the assumed form.

Monte-Carlo replicas (NNPDF). Drop the functional form entirely: parametrize each PDF with a neural network flexible enough to represent almost any shape. Then propagate the data errors by resampling — generate hundreds of pseudo-datasets ("replicas") fluctuated within the experimental covariance, fit a fresh network to each, and keep the whole ensemble. The mean over replicas is the central PDF; the standard deviation is the uncertainty. No quadratic approximation, no tolerance — the spread of the ensemble carries the error directly, including non-Gaussian and parametrization-flexibility effects the Hessian cannot see.

The missing third — full Bayesian inference. Both methods above are, in statistical language, approximations to a posterior. The Hessian is a Laplace (Gaussian) approximation around the mode; the MC-replica ensemble is a frequentist bootstrap that under mild conditions approximates a Bayesian posterior. The complete object neither computes directly is the posterior distribution $P(\boldsymbol\theta \mid \text{data})$ itself, together with the model evidence $Z = \int \mathcal{L}(\text{data}\mid\boldsymbol\theta)\,\pi(\boldsymbol\theta)\, d\boldsymbol\theta$ — the marginal likelihood that lets you compare parametrizations on equal footing. Computing $Z$ by nested sampling is what Colibri, the framework at the heart of this portal, adds to the landscape. It is the third leg of the stool. (Topic 5 develops all three in full.)

PDF4LHC — how three answers become one number

The LHC experiments cannot each pick a favourite group and quote it; a Higgs cross-section or a $W$-mass extraction needs a single, community-agreed PDF with a single error. The PDF4LHC working group produces exactly that, and — importantly for this page — it does not average the three sets naively. It uses the Monte-Carlo combination trick, which is possible precisely because any Hessian set can be converted into an equivalent ensemble of MC replicas.

Two generations are in use. PDF4LHC15 combined CT14, MMHT2014, and NNPDF3.0 (the Run II recommendation). PDF4LHC21 — the current Run III recommendation — combines variants of CT18, MSHT20, and NNPDF3.1 (written CT18′, MSHT20, NNPDF3.1′ after benchmarking adjustments). Note the subtlety worth flagging so you are not surprised later: PDF4LHC21 uses NNPDF3.1′, not the newer NNPDF4.0, because the combination is deliberately conservative and lags the very latest individual releases. When a physics result quotes "PDF4LHC21 uncertainty," it is quoting this pooled-and-compressed object built from all three philosophies at once.

Why the head-to-head comparison is normally confounded

Here is the methodological trap that motivates everything downstream. Suppose the gluon uncertainty at $x = 0.3$ differs between MSHT (Hessian) and NNPDF (Monte-Carlo). You would love to conclude "the Hessian method under-/over-estimates the error here." You cannot — because MSHT's Hessian is tied to a Chebyshev functional form and NNPDF's Monte-Carlo is tied to a neural network. The two differ in two things at once:

This is a textbook confounded comparison: the treatment (method) is perfectly correlated with a nuisance factor (form). No amount of staring at published PDF plots resolves it, and the field has argued about it for years — including a 2024 stress-test that made the "first direct comparison" of the MSHT Hessian approach to a neural-net fit and had to work hard to control the form. To break the confound cleanly you need the off-diagonal cells: the same functional form fitted with different uncertainty methods.

Where this lab sits — the controlled experiment

That is exactly what this project built. It ports MSHT's exact functional form — the $x^a(1-x)^b$ × Chebyshev parametrization from §3.1, sum rules and all — into the Colibri/NNPDF machinery. Holding that one parametrization fixed, the same fit can then be run three ways:

Now every off-diagonal cell of the grid above is filled. Because the form no longer varies, any difference between the three error bands is attributable to the method alone — a clean, controlled comparison of the three philosophies that the field cannot normally perform. These are the lab's PPDF-15 and PPDF-17 experiments: running Hessian, Monte-Carlo, and Bayesian uncertainties on one common MSHT-form fit and reading off where — and by how much — they diverge. That is the payoff the whole portal is building toward.