{"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":"f944ca9e-b8e4-5508-9a54-bd6ef6491468","task_key":"train--Challenge~5f23~5fmain","task_revision_id":"3","upstream_id":"Challenge_23_main","short_description":"In perturbative quantum chromodynamics (QCD), one always encounters divergent…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nx, p_z, epsilon_UV, epsilon_IR, mu = sp.symbols('x p_z epsilon_UV epsilon_IR mu')\\n\\ndef answer(x, p_z, epsilon_UV, epsilon_IR, mu):\\n    r\\\"\\\"\\\"\\n    Return the expressions of $\\\\tilde q_{\\\\rm sail}(x,p^z,\\\\epsilon,\\\\mu)$ in Sympy format.\\n\\n    Inputs\\n    ----------\\n    x: sympy.Symbol, longitudinal momentum fraction $x$\\n    p_z: sympy.Symbol, longitudinal momentum $p^z$\\n    epsilon_UV: sympy.Symbol, dimensional–regularization parameter for UV divergences, $\\\\epsilon_{\\\\rm UV}$\\n    epsilon_IR: sympy.Symbol, dimensional–regularization parameter for IR divergences, $\\\\epsilon_{\\\\rm IR}$\\n    mu: sympy.Symbol, renormalization scale $\\\\mu$\\n\\n    Outputs\\n    ----------\\n    expr_lt0:  sympy.Expr,  $\\\\tilde q_{\\\\rm sail}(x,p^z,\\\\epsilon,\\\\mu)$ for $x < 0$, to $O(\\\\epsilon_{\\\\rm UV}^0)$\\n    expr_mid:  sympy.Expr,  $\\\\tilde q_{\\\\rm sail}(x,p^z,\\\\epsilon,\\\\mu)$ for $0<x<1$, to $O(\\\\epsilon_{\\\\rm UV}^0)$\\n    expr_gt1:  sympy.Expr,  $\\\\tilde q_{\\\\rm sail}(x,p^z,\\\\epsilon,\\\\mu)$ for $x > 1$, to $O(\\\\epsilon_{\\\\rm UV}^0)$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    expr_lt0 = ...\\n    expr_mid = ...\\n    expr_gt1 = ...\\n    # ---------------------------------------------------------------\\n\\n    return expr_lt0, expr_mid, expr_gt1\",\"problem_description\":\"# Problem setup:\\nIn perturbative quantum chromodynamics (QCD), one always encounters divergent Feynman diagrams in loop corrections. In the Large Momentum Effective Theory (LaMET) framework for calculating parton physics, the perturbative matching coefficient is calculated from Feynman diagrams that involves one external quark or gluon with momentum $p$, which is set to be on-shell and massless, i.e., $p^2=0$ and\\n\\\\begin{align}\\n    I &= \\\\int {d^d k\\\\over (2\\\\pi)^d} {N(k,p)\\\\over D(k,p)} \\\\,,\\n\\\\end{align}\\nwhere $d$ is the dimension of the Minkowski space time with the metric $g^{\\\\mu\\\\nu}=\\\\mbox{diag}\\\\{1,-1,-1,-1\\\\}$.\\n\\nThere are two types of divergences, ultraviolet (UV) and infrared (IR). Before one carries out the loop integration over $k^\\\\nu=(k^0, k^x, k^y, k^z)$, it is necessary to first identify the UV and IR divergences of each diagram through the superficial degree of divergence (SDD).\\n\\n*UV divergences.* In the limit of $k^\\\\nu \\\\to \\\\infty$ for each $\\\\nu=0,1,2,\\\\ldots,d$, the SDD is $d-t$ if $N(k,p)/D(k,p)\\\\rightarrow 1/k^t$, where $t$ is an integer.\\n\\n*IR divergences.* There are two types of IR divergences, soft and collinear divergences.\\n- Soft divergence: In the limit of $k^\\\\nu \\\\to 0$ for each $\\\\nu=0,1,2,\\\\ldots,d$, $N(k,p)/D(k,p)\\\\rightarrow 1/k^t$;\\n- Collinear divergence: In the limit of $k^\\\\nu \\\\to \\\\lambda p^\\\\nu$, with $0<\\\\lambda< 1$,  $N(k,p)/D(k,p)\\\\rightarrow 1/(p^2)^{t/2}$.\\n\\nOne way to regulate these divergent integrals is dimensional regularization, where the space-time dimension is set to be $d=4-2\\\\epsilon$ with a continuous parameter $\\\\epsilon$. It works under the assumption that the regulated results can be analytically continued in $\\\\epsilon$. The UV divergences are regulated with $\\\\epsilon_{\\\\rm UV}>0$, and the IR divergences are regulated with $\\\\epsilon_{\\\\rm IR}<0$.\\n\\n# Main problem:\\n\\nThe so-called sail-diagram contribution to the one-loop matrix element of the quasi-PDF in a quark state with momentum $p^\\\\nu=(p^z,0,0,p^z)$ is\\n\\n\\\\begin{align}\\n    \\\\tilde q_{\\\\rm sail}(x,p^z,\\\\epsilon,\\\\mu) &=(\\\\mu^2)^\\\\epsilon {\\\\exp(\\\\epsilon\\\\gamma_E)\\\\over (4\\\\pi)^\\\\epsilon}\\\\int_{-\\\\infty}^\\\\infty {dz \\\\over 2\\\\pi}e^{ixp^z z} \\\\int {d^d k\\\\over (2\\\\pi)^d} {k^0+k^z\\\\over k^2 (p-k)^2 (p^z-k^z)}e^{-ik^z z}.\\n\\\\end{align}\\nCarry out the above integral in three intervals: $x < 0$, $0<x<1$ and $x>1$. Then expand the final result in $\\\\epsilon$ to $O(\\\\epsilon^0)$ in the $\\\\overline{\\\\rm MS}$ scheme. The final results of $\\\\tilde q_{\\\\rm sail}(x,p^z,\\\\epsilon,\\\\mu)$ should be expressed as functions in $x$, $p^z$, $\\\\epsilon_{\\\\rm UV}$ or $\\\\epsilon_{\\\\rm IR}$, and $\\\\mu$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}