{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"2c649737-8944-59e9-90b2-c2e5df2b786c","task_key":"train--Challenge~5f21~5fmain","task_revision_id":"1","upstream_id":"Challenge_21_main","short_description":"The matching formula in the large-momentum effective theory (LaMET) gives","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return the values of the PDF at $x \\\\in \\\\{0.4, 0.5, 0.6\\\\}$\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    f_0p4: float, the pion PDF $f(x, \\\\mu)$ at $\\\\mu=2$ GeV and $x=0.4$\\n    f_0p5: float, the pion PDF $f(x, \\\\mu)$ at $\\\\mu=2$ GeV and $x=0.5$\\n    f_0p6: float, the pion PDF $f(x, \\\\mu)$ at $\\\\mu=2$ GeV and $x=0.6$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    f_0p4 = ...\\n    f_0p5 = ...\\n    f_0p6 = ...\\n    # ---------------------------------------------------------------\\n\\n    return f_0p4, f_0p5, f_0p6\",\"problem_description\":\"\\n\\n# Problem setup:\\nThe matching formula in the large-momentum effective theory (LaMET) gives\\n\\\\begin{align}\\n    f(x, \\\\mu) = \\\\tilde{f} (x, P_z) - \\\\int_{0}^1 \\\\frac{d y}{|y|} ~ C^{(1)}\\\\left(\\\\frac{x}{y}, \\\\frac{\\\\mu}{|x| P_z}\\\\right) \\\\tilde{f}\\\\left(y, P_z\\\\right) ~,\\n\\\\end{align}\\nwhere $f(x, \\\\mu)$ denotes the unpolarized parton distribution function (PDF) of the pion, characterized by the momentum fraction $x$ and the energy scale $\\\\mu$. The corresponding quasi-PDF is denoted as $\\\\tilde{f}$. For purposes of simplification, power corrections are ignored.\\n\\nWe see that the perturbative matching kernel in the $\\\\overline{\\\\rm MS}$ scheme is\\n\\\\begin{align}\\n    C^{(1)}\\\\left(\\\\xi, \\\\frac{\\\\mu}{|x| P_z}\\\\right) = \\\\frac{\\\\alpha_s (\\\\mu) C_F}{2 \\\\pi} \\\\begin{cases}\\\\left(\\\\frac{1+\\\\xi^2}{1-\\\\xi} \\\\ln \\\\frac{\\\\xi}{\\\\xi-1}+1+\\\\frac{3}{2 \\\\xi}\\\\right)_{+(1)}^{[1, \\\\infty]}-\\\\frac{3}{2 \\\\xi} & \\\\xi>1 \\\\\\\\ \\\\left(\\\\frac{1+\\\\xi^2}{1-\\\\xi}\\\\left[-\\\\ln \\\\frac{\\\\mu^2}{4x^2 P_z^2}+\\\\ln (\\\\frac{1-\\\\xi}{\\\\xi}) \\\\right]-\\\\frac{\\\\xi(1+\\\\xi)}{1-\\\\xi}\\\\right)_{+(1)}^{[0,1]} & 0<\\\\xi<1, \\\\\\\\\\n    % \\\\left(-\\\\frac{1+\\\\xi^2}{1-\\\\xi} \\\\ln \\\\frac{-\\\\xi}{1-\\\\xi}-1+\\\\frac{3}{2(1-\\\\xi)}\\\\right)_{+(1)}^{[-\\\\infty, 0]}-\\\\frac{3}{2(1-\\\\xi)} & \\\\xi<0\\n    \\\\end{cases}\\n\\\\end{align}\\nwhere $\\\\xi = x / y$. The subscript $+(1)$ indicates the plus distribution with the pole at $\\\\xi = 1$.\\n\\nThe PDF satisfies the DGLAP evolution according to\\n\\\\begin{align}\\n\\\\begin{aligned}\\n    \\\\frac{d f(x, \\\\mu)}{d \\\\ln \\\\mu} &= g\\\\left(x, \\\\mu\\\\right), \\\\\\\\\\n    g\\\\left(x, \\\\mu\\\\right) &= \\\\int_x^1 \\\\frac{d v}{v} P\\\\left[\\\\frac{x}{v}, \\\\alpha_s(\\\\mu)\\\\right] f\\\\left(v, \\\\mu\\\\right) ~.\\n\\\\end{aligned}\\n\\\\end{align}\\nThe 1-loop result of the evolution kernel $P\\\\left[w, \\\\alpha_s(\\\\mu)\\\\right]$ can be found to be\\n\\\\begin{align}\\n    P\\\\left[w, \\\\alpha_s(\\\\mu)\\\\right] = \\\\frac{\\\\alpha_s(\\\\mu) C_F}{2 \\\\pi} \\\\left( \\\\frac{2}{1-w} - 1 - w \\\\right)_{+(1)} ~, w \\\\leq 1,\\n    % =  \\\\frac{\\\\alpha_s(\\\\mu) C_F}{4 \\\\pi} \\\\left( 4(1-w)^{-1} - 2 - 2w  + 3 \\\\delta(1 - w) \\\\right) ~,\\n\\\\end{align}\\n\\nwhere $w = x / v$.\\n\\nThe 1-loop $\\\\alpha_s$ is given by\\n\\\\begin{align}\\n    \\\\alpha_s^{(1)}\\\\left(\\\\mu^2\\\\right)=\\\\frac{4 \\\\pi}{\\\\beta_0 \\\\ln \\\\left(\\\\mu^2 / \\\\Lambda_{\\\\rm Q C D}^2\\\\right)} ~.\\n\\\\end{align}\\n\\n\\nThe constants are given by\\n1. $C_F = \\\\frac{4}{3}$;\\n2. $\\\\beta_0 = 9$;\\n3. $\\\\Lambda_{\\\\rm Q C D} = 0.2445$ GeV.\\n\\nDiscretize the variables $x$, $y$, and $v$ in $x, y, v \\\\in \\\\{0.002, 0.004, 0.006 \\\\dots, 0.994, 0.996, 0.998, 1\\\\}$ so that the quasi-PDF $\\\\tilde{f}(x, P_z)$ and PDF $f(x, \\\\mu)$ are represented by vectors of length $500$ and the convolution kernels can be represented by matrices of dimension $500 \\\\times 500$.\\n\\nThe pion quasi-PDF in $\\\\overline{\\\\rm MS}$ scheme at $P_z = 2$ GeV is\\n\\\\begin{align}\\n    \\\\tilde{f}(x, P_z) = (x + 3) \\\\cdot (1-x)^3, \\\\quad x\\\\in (0, 1) ~.\\n\\\\end{align}\\n\\n\\n# Main problem:\\n\\n\\nGiven the pion quasi-PDF in $\\\\overline{\\\\rm MS}$ scheme at $P_z = 2$ GeV as\\n\\\\begin{align}\\n    \\\\tilde{f}(x, P_z) = (x + 3) \\\\cdot (1-x)^3, \\\\quad x\\\\in (0, 1) ~.\\n\\\\end{align}\\nUsing the matching formula to derive the pion PDF $f(x, \\\\mu)$ at $\\\\mu = 2$ GeV in the regime $x\\\\in (0, 1)$, note that the logarithm should be resummed using the DGLAP evolution. Evaluate the PDF at $x \\\\in \\\\{0.4, 0.5, 0.6\\\\}$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}