The American counterpart to MSHT20, and its closest methodological cousin — both fit polynomials and both use the Hessian. But CT18 differs in one respect that trips up almost everyone who compares them: it publishes its uncertainties at 90% confidence, not 68%. Quote a CT18 band next to an MSHT20 band without converting and CT18 will look 64% more uncertain than it really is. Every chart on this site divides it by 1.645 first.
If you read nothing else on this page.
CT18 does the same job as MSHT20 and in a broadly similar way: describe the proton with smooth polynomial curves, tune about thirty numbers until the predictions match the data, then use the curvature of the fit to get an error bar.
Two things make it distinct. First, it is much more cautious about which data to trust — when one dataset pulls against the rest, CT18's instinct is to leave it out of the headline fit and publish a separate variant that includes it. Second, its error bars are quoted at 90% confidence, a wider interval than everyone else's 68%. That is a reporting convention, not a physics difference, but it is the single most common mistake made when comparing PDF sets.
The flagship paper is Hou et al. (2021), thirteen authors, in Physical Review D. Its fit quality is χ²/N = 1.17 over 3,681 data points — essentially identical to MSHT20's, on a slightly smaller dataset.
Two acronyms stacked on each other, the second of which is a memorial.
| CTEQ | The Coordinated Theoretical–Experimental Project on QCD — a multi-institutional collaboration of theorists and experimentalists, founded around 1990–91. It runs the well-known CTEQ summer schools as well as producing PDFs. |
| TEA | Tung Et Al. — the PDF-fitting group within CTEQ, named for Wu-Ki Tung, who co-founded the collaboration and served as its first Project Director. |
| CT | The group's own short form for CTEQ–TEA, used in set names since CT09. |
Wu-Ki Tung (1940–2009) was born in Yunnan, took his first degree at National Taiwan University and his PhD at Yale in 1966, and held positions at Chicago, the Illinois Institute of Technology, Michigan State and finally the University of Washington. He is also the T in ACOT — the heavy-quark factorisation scheme that CT18 still uses today, in its SACOT-χ form. The group fits deep-inelastic scattering using the formalism its namesake co-invented.
The naming has a quiet poignancy: CT09 was submitted seventeen days after Tung died, and he is an author on it. The last set to carry the older CTEQ name was CTEQ6.6.
The widely-cited Physics Today obituary dates CTEQ's founding to 1992. CTEQ's own history document records that the founding proposal was submitted in 1990 and funded in 1991, with the first summer school in 1992. Use 1990–91.
Thirteen authors — a genuinely transatlantic and trans-Pacific group.
| authors | institution |
|---|---|
| J. Huston, J. Pumplin, C. Schmidt, D. Stump, C.-P. Yuan | Michigan State University — the largest block, and Tung's last long-term home |
| P. Nadolsky, T. J. Hobbs | Southern Methodist University (Hobbs also Jefferson Lab) |
| K. Xie | University of Pittsburgh (PITT PACC) and SMU |
| T.-J. Hou | Northeastern University, Shenyang, China |
| J. Gao | Shanghai Jiao Tong University; Peking University |
| S. Dulat, I. Sitiwaldi | Xinjiang University, China |
| M. Guzzi | Kennesaw State University, USA |
39 datasets, 3,681 measurements — fewer than MSHT20, by choice.
| class | notable sets |
|---|---|
| Combined HERA DIS | H1+ZEUS neutral- and charged-current — 1,120 points, the single largest and most constraining block in the fit |
| Fixed-target DIS | BCDMS F₂ᵖ (337), BCDMS F₂ᵈ (250), NMC F₂ᵈ/F₂ᵖ (123) |
| Neutrino DIS | CCFR and CDHSW F₂ and xF₃ |
| Dimuon (strangeness) | NuTeV and CCFR ν/ν̄ dimuon — the direct handle on strange quarks |
| Fixed-target Drell–Yan | E605, E866 |
| Tevatron | CDF and DØ lepton charge asymmetries, Z rapidity, inclusive jets |
| LHC W/Z | LHCb 7 and 8 TeV, CMS charge asymmetries, ATLAS 8 TeV Z pT |
| LHC jets | CMS 7 and 8 TeV, ATLAS 7 TeV |
| LHC top pairs | CMS 8 TeV double-differential, ATLAS 8 TeV |
Fit quality is χ²/N = 1.17 at NNLO for CT18 (3,681 points) and 1.19 for the CT18Z variant (3,493 points).
CT18 tried and discarded the 2018 combined HERA charm and bottom measurements: they "cannot be fitted with a reasonable χ²". Part of the difficulty is that the number of correlated systematic uncertainties grew from 42 in the older version of the data to 167 in the newer one. This is a concrete illustration of the tension problem that drives the whole tolerance debate — more careful treatment of systematics can make a dataset harder, not easier, to accommodate.
Bernstein polynomials — not Chebyshev. That is MSHT's basis, not CT's.
Each distribution at the starting scale takes the form
f(x, Q₀) = a₀ · xa₁−1 (1−x)a₂ · P(y; a₃, a₄, …) , y = √x
where P is a sum of Bernstein polynomials (a Bézier curve). As with MSHT, the prefactor encodes the known small-x and large-x behaviour and the polynomial supplies the flexibility in between — but the polynomial basis is different, and the parameter count is roughly half.
| starting scale | Q₀ = 1.3 GeV, set equal to the charm pole mass (1.4 GeV for CT18Z) |
| free parameters | 29 — deliberately capped |
| heavy-quark scheme | SACOT-χ, from the ACOT lineage Tung co-created |
| charm | generated perturbatively in baseline CT18; non-perturbative charm explored separately in CT18FC |
| strangeness asymmetry | set to zero in baseline CT18 — where MSHT20 fits it. Explored separately in CT18As. |
| strong coupling | fixed at αS(MZ) = 0.118; its uncertainty added in quadrature from a separate series |
Why only 29? The paper is explicit: adding parameters helped up to about thirty, but "with more than about 30 parameters, the fits tend to destabilize, as expanded parametrizations attempt to describe statistical noise." To check that this cap does not bias the answer, they repeated the fit with more than 250 different trial functional forms — a distinctive feature of CT methodology, and a direct response to the criticism that polynomial fits are insufficiently flexible.
The most common error in the entire field, and it applies to every chart on this site.
CT18 publishes uncertainties at 90% confidence. MSHT20 and NNPDF4.0 publish at 68%. To compare them you must divide the CT18 uncertainty by 1.645 — the paper says so itself, and uses "the rescaled 68% C.L." for its own comparisons. Skip this and CT18 appears about 64% more uncertain than it is.
This is not buried: the released grid file carries ErrorConfLevel: 90 in its metadata, and the set description reads "mem=1-58 → eigenvector sets 90%". Every band shown on this site has been converted before plotting.
Like MSHT, CT rejects the textbook Δχ² = 1 rule. Their replacement has two parts:
For orientation: at a common 68% CL, CT's Δχ² ≈ 37 sits well above MSHT20's mean of ≈ 13, which in turn sits far above NNPDF's effective 1. That ordering is exactly what you see in the band widths — and it is the disagreement this whole project is about.
CT18's own appendix reports that for the strangeness ratio, the Hessian band "nominally corresponds to Δχ² ≈ 36" yet actually matches a true Δχ² ≈ 10 interval computed the slow, exact way, and concludes: "The true uncertainty is wider than the Hessian estimate suggests." They also warn that the tier-2 penalty "may result in too narrow uncertainties along some eigenvector directions if some experiments strongly disagree." A group publishing the limitations of its own error bars is worth noticing.
Four sets, not one — and the differences are deliberate, not incremental.
Where MSHT publishes one headline fit, CT publishes four, differing in which contested data are included and how the DIS factorisation scale is chosen:
| set | DIS scale | ATLAS 7 TeV W/Z | CDHSW | mc |
|---|---|---|---|---|
| CT18 | μ² = Q² | excluded | included | 1.3 |
| CT18A | μ² = Q² | included | included | 1.3 |
| CT18X | x-dependent | excluded | included | 1.3 |
| CT18Z | x-dependent | included | excluded | 1.4 |
Their recommendation is unambiguous: "CT18 is the nominal CTEQ-TEA PDF set, which we recommend for all general-use applications." Where the CT18↔CT18Z difference matters, they suggest taking the envelope of the two. CT18NNLO is what this site plots.
CT18's Conclusions state that CT18X uses a charm mass of 1.4 GeV. That contradicts the paper's own Table III, its Section III, its abstract, and the released grid — all of which give 1.3 GeV. Only CT18Z uses 1.4. We checked the grid metadata directly: CT18XNNLO carries MCharm: 1.3000.
The released CT18NNLO grid — bands already converted to 68%.
Pick an energy. Bands are 1σ at 68% confidence, converted from the published 90% by dividing by 1.645.
Same quantity, three independent analyses, one common confidence level.
CT18 solid with the heavier band; the others dashed. Its band is genuinely the widest for the gluon even after the 90%→68% conversion — that is the two-tier tolerance, not a units error.
We checked this specifically, because a 1.65× ratio looks suspiciously like the 1.645 conversion factor left unapplied. It is not: the conversion was applied, and CT18's gluon band really is about 1.65× MSHT20's at 68% confidence. MSHT's own paper reaches the same conclusion, attributing it to "a slightly more conservative tolerance criterion."
Eight principal grids, all 59 members, all at 90% CL.
| grid | what it is |
|---|---|
| CT18NNLO | The nominal set, recommended for general use. Plotted on this site. |
| CT18ZNNLO | The maximally-different variant. Q₀ = 1.4 GeV. |
| CT18ANNLO / CT18XNNLO | The two intermediate variants of §07. |
| CT18NLO and Z/A/X at NLO | The same four families at NLO. |
| CT18LO | Leading order for event generators, single member, αS = 0.135. |
| CT18FC | Non-perturbative ("intrinsic") charm — 12 members spanning four models and three tolerances. |
| CT18qed_* | Six grids including a photon distribution, proton and neutron, elastic/inelastic. |
| αS series and fixed-flavour | αS = 0.110–0.124 in steps of 0.001, plus NF3/NF4/NF6 variants. |
Naming rule worth knowing: the variant letter goes between the number and the order — CT18ANNLO, never CT18A_NNLO.
Every released CT18 grid carries the reference arXiv:1908.11394 (temporary) in its metadata. That is not the CT18 paper — it is an unpublished 14-page precursor from four months earlier. The real paper is arXiv:1912.10053. The placeholder has never been updated, and it propagates into anything that reads the grid metadata automatically — including, until we caught it, our own data files.
CT's disputes are mostly about which data deserve to be believed.
The ATLAS 7 TeV W/Z argument. MSHT fit these data in their baseline; CT does not. The CT18 paper is direct about it: other groups "have noted that these ATLAS W/Z data can be fitted with χ²/d.o.f. that is comparable to the CT18 one, we find that such χ² reflects systematic tensions with many of the other data in our global analysis." Hence CT18A rather than inclusion in the headline fit.
Hessian profiling. CT argues that the profiling technique used by the LHC experiments to assess new data "significantly underestimates the minimal χ² that can be reached", because it effectively assigns the new dataset an enormous weight and cannot represent the tier-2 penalty.
Against NNPDF. CT notes that NNPDF's fitted charm is "of indefinite sign and shape", that NNPDF3.1's gluon–gluon luminosity runs 2–3% above CT18 in the Higgs region with a smaller uncertainty band, and that NNPDF imposes no relation between ū and d̄ as x → 0.
The sharpest methodological claim comes from a companion paper, "Parton distributions need representative sampling": that parametrisation and methodology uncertainties "can be underestimated with common PDF ensembles in high-stake measurements". CT25's flat-prior ensemble (§12) is their constructive answer to their own critique.
They collaborate too. CT18 is one of the three inputs to the PDF4LHC21 combination alongside MSHT20 and NNPDF3.1, and CT, MSHT and xFitter have published joint methodology work.
A successor exists — and it moves toward the others.
CT25 has been announced, with grids released from the group's own repository and the full paper still forthcoming. The headline for our purposes is a genuine methodological break: CT25 abandons the fixed two-tier Δχ² = 100 at 90% CL in favour of a dynamic tolerance quoted at 68% CL — converging on MSHT practice, and removing the conversion trap described in §06.
It also introduces CT25FlatP, an ensemble of 350 equally acceptable parametrisations whose uncertainty is the convex hull rather than a Hessian band — a direct attempt to fold parametrisation choice into the quoted uncertainty rather than treating it as a separate systematic.
As of this writing no CT25 set has appeared in the central LHAPDF index — we searched it. CT18 remains the LHAPDF-available baseline, and is what this site plots. Treat CT25 as in-progress.
Cite 1912.10053. Not 1908.11394.