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    .. note::
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        :ref:`Go to the end <sphx_glr_download_auto_examples_08_first_calculations.py>`
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.. _sphx_glr_auto_examples_08_first_calculations.py:


Three calculations with independent references
==============================================

A scalar one-loop integral, a coupled RMT series, and a hybrid three-loop graph.
Install the ``multiloop`` extra. Mass inputs below are squared masses.

.. GENERATED FROM PYTHON SOURCE LINES 8-61




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    B0: (-0.981855765193598415130931897772 + 0.0j) absolute reference difference: 1.661e-34
    B0 returns a precision-checked scalar, not a separate error estimate.
    RMT value: 0.6290167075906141495908273
    RMT chart: momentum condition: Q > m1 + m2 + m3
    RMT tail bound: 3.30394380054996790688401174628947e-27
    RMT estimated error before display rounding: 5.20393751826099434694865135069815e-27
    Gamma families: 8 terms: 3276
    Expected domain refusal: no certified RMT chart: require p_E > m1+m2+m3, or M > m_a+m_b+p_E; boundaries and equal-mass vacuum are not covered. Received squared masses=(1, 1, 1), p_E^2=0, chart='auto'. For equal masses use integrate_sunset_taylor; otherwise consider integrate_graph with the multiloop extra as a numerical reference
    Hybrid value: 0.022275789423594476 +/- 6.473460410552093e-05
    Fourier reference: 0.022309006397562933
    Hybrid method: RMT scale + Hepp sectors + scrambled Sobol
    The hybrid estimate is float64 with statistical errors; it is not a full RMT series.






|

.. code-block:: Python

    import json
    import mpmath as mp
    import scipy  # Explicit import also loads SciPy in the browser console.
    from ramanujan_qft import (B0, ScalarEdge, ScalarGraph, explain_sunset_rmt,
                               integrate_graph, RMTDomainError)

    # 1. B0: LoopTools finite-part convention, Delta=0, mu^2=1; p^2=-2.
    # Independent Feynman-parameter reference, evaluated with extra precision.
    bubble = B0(-2, 1, 4, dps=30)
    with mp.workdps(45):
        reference = -mp.quad(lambda x: mp.log(x + 4*(1-x) + 2*x*(1-x)), [0, 1])
        difference = abs(bubble-reference)
        assert difference < mp.mpf('1e-29')
    print('B0:', mp.nstr(bubble, 30), 'absolute reference difference:', mp.nstr(difference, 4))
    print('B0 returns a precision-checked scalar, not a separate error estimate.')

    # 2. D=2 sunset: positive Euclidean propagators; d^2k d^2l / pi^2.
    # Q=10 > m1+m2+m3=3 selects the momentum RMT chart.
    sunset = ScalarGraph([ScalarEdge(0, 1, 1)]*3, {0: [10], 1: [-10]})
    trace = explain_sunset_rmt(sunset, evaluate=True, dps=25)
    diagnostics = trace['evaluation']['diagnostics']
    assert trace['evaluation']['status'] == 'converged'
    # Reference from independent simplex quadrature at 40 and 55 digits:
    # tests/data/sunset_rmt_reference.json (precision difference < 2.4e-42).
    with mp.workdps(45):
        reference = mp.mpf('0.6290167075906141495908272538835833416123070767')
        encoded = diagnostics['value']
        exact_value = (-1)**encoded['sign'] * mp.ldexp(mp.mpf(int(encoded['mantissa_hex'], 16)), encoded['exponent2'])
        difference = abs(exact_value-reference)
        assert difference <= mp.mpf(diagnostics['error_estimate']['decimal']) + mp.mpf('3e-42')
    print('RMT value:', diagnostics['value']['decimal'])
    print('RMT chart:', trace['convergence']['chart'], 'condition:', trace['convergence']['condition'])
    print('RMT tail bound:', diagnostics['truncation_bound']['decimal'])
    print('RMT estimated error before display rounding:', diagnostics['error_estimate']['decimal'])
    print('Gamma families:', len(trace['families']), 'terms:', diagnostics['terms'])
    exported_json = json.dumps(trace, indent=2)  # Inspect without evaluating expression strings.
    try:
        explain_sunset_rmt(ScalarGraph([ScalarEdge(0, 1, 1)]*3))
    except RMTDomainError as error:
        print('Expected domain refusal:', error)
    else:
        raise AssertionError('The RMT chart must refuse equal-mass vacuum')

    # 3. D=1 three-loop banana: d^1k / sqrt(pi) for each of three loops.
    # Fourier convolution gives pi^(3/2)*sum(m)/(prod(m)*(sum(m)^2+Q^2)).
    banana = ScalarGraph([ScalarEdge(0, 1, m*m) for m in (1, 2, 3, 4)], {0: [2], 1: [-2]})
    hybrid = integrate_graph(banana, D=1, epsrel=3e-3, max_evaluations=4000000, seed=1729)
    reference = float(mp.pi**mp.mpf('1.5')*10/(24*(100+4)))
    assert abs(hybrid.value-reference) <= hybrid.error_estimate
    print('Hybrid value:', hybrid.value, '+/-', hybrid.error_estimate)
    print('Fourier reference:', reference)
    print('Hybrid method:', hybrid.method)
    print('The hybrid estimate is float64 with statistical errors; it is not a full RMT series.')


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.555 seconds)


.. _sphx_glr_download_auto_examples_08_first_calculations.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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      :download:`Download Jupyter notebook: 08_first_calculations.ipynb <08_first_calculations.ipynb>`

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      :download:`Download Python source code: 08_first_calculations.py <08_first_calculations.py>`

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