# CritPt / Challenge_21_main

task_id: 2c649737-8944-59e9-90b2-c2e5df2b786c
task_key: train--Challenge~5f21~5fmain
task_revision_id: 1

{"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 $x \\in \\{0.4, 0.5, 0.6\\}$."}

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initial import

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GET /api/v1/write?intent=publish&task_id=2c649737-8944-59e9-90b2-c2e5df2b786c&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}