Determination of Nuclear PDFs using Markov Chain Monte Carlo Methods

Abstract

Global QCD analyses of nuclear parton distribution functions (nPDFs) have traditionally relied on the Hessian method for uncertainty estimation. However, the inherent Gaussian approximation and reliance on local curvature often prove insufficient for nPDF fits, which are frequently characterized by limited data constraints and non-Gaussian likelihoods. In this paper, we present the first nPDF determination based on Markov Chain Monte Carlo (MCMC) techniques, implemented within the nCTEQ framework using an adaptive Metropolis-Hastings algorithm. The MCMC approach enables a direct mapping of the posterior distribution and reveals a highly non-trivial parameter-space structure, including multiple modes and pronounced non-Gaussian behavior, particularly for the valence PDFs. We perform the first single-nucleus global analysis of lead PDFs using exclusively lead data and compare it to a multi-nuclei fit employing a standard analytic A dependence. The inclusion of lighter nuclei reduces quark uncertainties and modifies the shape of the lead PDFs, while leaving the gluon distribution largely unaffected. A complementary Hessian analysis exposes systematic limitations of the Gaussian approximation. Our results demonstrate that MCMC methods provide a more reliable framework for uncertainty quantification in nPDF determinations.

Figures (17)

• Figure 1: Scan of the $\chi^2$ profile as a function of the $a_3^{u_v}$ parameter for the Pb-only fit. The $a_3^{u_v}$ parameter is varied while all other parameters are kept fixed at the MCMC global minimum. A pronounced peak at $a_3^{u_v}=0$ indicates this value is strongly disfavored, whereas the flat behavior elsewhere reflects the limited sensitivity of the fit to this parameter.
• Figure 2: The time series of the ten nPDF parameter values for the Pb-only analysis, shown for an example chain. The red-shaded region marks the thermalization (burn-in) phase, approximately the first 80 000 samples, which are removed for further analysis. The yellow region indicates the part of the chain where the standard random walk MH algorithm with a fixed covariance matrix is used, and the gray areas show the Monte Carlo times where the reset-to-mean mechanism was used.
• Figure 3: Pairwise plot of the Pb-only posterior distributions, showing one-dimensional marginals (diagonal panels) and two-dimensional projections of pairwise correlations (off-diagonal panels). Estimates of credible regions are indicated by 68% and 95% contours. The top-right panel displays the $\chi^2$ distribution, which is fitted by the theoretical $\chi^2$ distribution (red line).
• Figure 4: Comparison of lead PDFs uncertainty estimations for $u_v$, $d_v$, $\bar{u}+\bar{d}$, and $g$ distributions for the Pb-only analysis. The upper panels show the absolute nPDFs, while the lower panels display the ratios with respect to the best-fit values obtained from the cumulative-$\chi^2$ method. The colored bands correspond to different uncertainty estimation methods: red for the MCMC cumulative-$\chi^2$, blue for the MCMC percentile, and green for the asymmetric Hessian.
• Figure 5: To investigate the presence of local minima in the MCMC exploration, we apply a threshold to the $a_3^{d_v}$ parameter, which exhibits a particularly prominent structure. Samples with $a_3^{d_v}>0.8$ are shown in red, while those with $a_3^{d_v} \le 0.8$ are shown in blue. We compute the mean $\chi^2$ and the corresponding likelihood $L \propto \exp{ (- \frac{1}{2} \chi^2)}$ separately for both regions. The upper plot (a) shows one of the analyzed chains before thinning, and the lower plot (b) shows the same for the final (thinned and combined) chain.