{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"研究水準の物理問題で、科学的理解と多段階の推論・計算を評価するベンチマークです。公開データには70の課題があり、問題文とコード雛形を組み合わせて解答を構成します。\n\nCritPt 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"},"task_id":"4a9c932f-668f-5555-8d1c-f149666beae8","task_key":"train--Challenge~5f24~5fmain","task_revision_id":"1","upstream_id":"Challenge_24_main","short_description":"In the framework of large-momentum effective theory (LaMET), the parton…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\ny, p_z, epsilon_IR, mu = sp.symbols('y p_z epsilon_IR mu')\\n\\ndef answer(y, p_z, epsilon_IR, mu):\\n    r\\\"\\\"\\\"\\n    Return the expressions of $\\\\tilde{f}_q^{(1)}(y,p_z,\\\\epsilon_{\\\\rm IR},\\\\mu)$\\n    in three intervals (i) $y < 0$, (ii) $0 < y < 1$, (iii) $y > 1$\\n    in Sympy format.\\n\\n    Inputs\\n    ----------\\n    y: sympy.Symbol, momentum fraction in quasi-PDF, $y$\\n    p_z: sympy.Symbol, large momentum, $p_z$\\n    epsilon_IR: sympy.Symbol, infrared regulator, $\\\\epsilon_{\\\\rm IR}$\\n    mu: sympy.Symbol, renormalization scale, $\\\\mu$\\n\\n    Outputs\\n    ----------\\n    expr_neg: sympy.Expr, 1–loop correction $\\\\tilde{f}_q^{(1)}(y,p_z,\\\\epsilon_{\\\\rm IR},\\\\mu)$ for $y < 0$\\n    expr_mid: sympy.Expr, 1–loop correction $\\\\tilde{f}_q^{(1)}(y,p_z,\\\\epsilon_{\\\\rm IR},\\\\mu)$ for $0 < y < 1$\\n    expr_pos: sympy.Expr, 1–loop correction $\\\\tilde{f}_q^{(1)}(y,p_z,\\\\epsilon_{\\\\rm IR},\\\\mu)$ for $y > 1$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    expr_neg = ...  # SymPy expression for y < 0\\n    expr_mid = ...  # SymPy expression for 0 < y < 1\\n    expr_pos = ...  # SymPy expression for y > 1\\n    # ---------------------------------------------------------------\\n\\n    return expr_neg, expr_mid, expr_pos\",\"problem_description\":\"# Problem setup:\\nIn the framework of large-momentum effective theory (LaMET), the parton distribution function (PDF) $f(x ,\\\\mu)$ can be expanded with fixed large momentum $p_z$ in the LaMET framework. The expansion formula is\\n\\\\begin{align}\\n\\\\begin{aligned}\\nf(x, \\\\mu)&=\\\\int_{-\\\\infty}^{\\\\infty} \\\\frac{d y}{y} C_2\\\\left(\\\\frac{y}{x}, \\\\frac{p_z}{\\\\mu}\\\\right) \\\\tilde{f}\\\\left(y, \\\\frac{p_z}{\\\\mu}\\\\right) ~,\\n\\\\end{aligned}\\n\\\\end{align}\\nwhere $C_2$ is the matching kernel, $\\\\tilde{f}\\\\left(y, \\\\frac{p_z}{\\\\mu}\\\\right)$ is the quasi-PDF, and the power corrections are dropped.\\n\\nIn the Coulomb gauge (CG), the quasi-PDF is defined as\\n\\\\begin{align}\\n    \\\\tilde{f}\\\\left(y, p_z, \\\\mu\\\\right)=p_z \\\\int_{-\\\\infty}^{\\\\infty} \\\\frac{d z}{2 \\\\pi} e^{i y p_z z} \\\\tilde{h}\\\\left(z, p_z, \\\\mu\\\\right) ~,\\n\\\\end{align}\\n\\\\begin{align}\\n    \\\\tilde{h}\\\\left(z, p_z, \\\\mu\\\\right)=\\\\frac{1}{2 p_z}\\\\langle P| \\\\left. \\\\bar{\\\\psi}(z) \\\\gamma^z \\\\psi(0)\\\\right|_{\\\\vec{\\\\nabla} \\\\cdot \\\\vec{A}=0}|P\\\\rangle ~.\\n\\\\end{align}\\n\\nTo extract the matching kernel $C_2$, we need to calculate the quasi-distribution in a free massless quark state using perturbation theory.\\n\\n# Main problem:\\n\\nUsing perturbation theory with dimensional regularization, calculate the CG quasi-distribution\\n\\\\begin{align}\\n    \\\\tilde{f}_q (y,p_z) = \\\\int \\\\frac{d z}{ 2\\\\pi } e^{i y p_z z} \\\\langle q(p)|\\\\bar{q}(z) \\\\left. \\\\frac{\\\\gamma^z}{2} q(0)\\\\right|_{\\\\vec{\\\\nabla} \\\\cdot \\\\vec{A}=0} |q(p)\\\\rangle\\n\\\\end{align}\\nin the $\\\\overline{\\\\rm MS}$ scheme up to 1-loop. The $|q(p)\\\\rangle $ is a free massless quark state with momentum $p^\\\\mu$. The 1-loop result should be expressed in the form as\\n\\\\begin{align}\\n    \\\\tilde{f}_q (y,p_z,\\\\epsilon_{\\\\rm IR},\\\\mu) = \\\\delta(1- y) + \\\\frac{\\\\alpha_s C_F}{2 \\\\pi} \\\\tilde{f}_q^{(1)}(y,p_z,\\\\epsilon_{\\\\rm IR},\\\\mu) ~,\\n\\\\end{align}\\nwhere $\\\\delta$ is the Dirac delta function, $\\\\alpha_s$ is the strong coupling, $C_F$ is the Casimir constant, and $\\\\epsilon_{\\\\rm IR}$ is the infrared regulator in the dimensional regularization. Give the final expression of 1-loop correction $\\\\tilde{f}_q^{(1)}(y,p_z,\\\\epsilon_{\\\\rm IR},\\\\mu)$ in three intervals: $y < 0$, $0 < y < 1$ and $y > 1$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}