ekore.operator_matrix_elements.unpolarized.space_like.as3 package

The unpolarized, space-like N3LO OME.

The expressions are based on:

  • [BBK09a]. Isabella Bierenbaum, Johannes Blümlein, and Sebastian Klein. Mellin Moments of the O(alpha**3(s)) Heavy Flavor Contributions to unpolarized Deep-Inelastic Scattering at Q**2 ensuremath >ensuremath > m**2 and Anomalous Dimensions. Nucl. Phys. B, 820:417-482, 2009. arXiv:0904.3563, doi:10.1016/j.nuclphysb.2009.06.005.

  • [Blu00]. Johannes Blümlein. Analytic continuation of mellin transforms up to two-loop order. Computer Physics Communications, 133(1):76-104, Dec 2000. URL: http://dx.doi.org/10.1016/S0010-4655(00)00156-9, doi:10.1016/s0010-4655(00)00156-9.

  • [BBK09b]. Isabella Bierenbaum, Johannes Blümlein, and Sebastian Klein. The Gluonic Operator Matrix Elements at O(alpha(s)**2) for DIS Heavy Flavor Production. Phys. Lett. B, 672:401-406, 2009. arXiv:0901.0669, doi:10.1016/j.physletb.2009.01.057.

  • [ABK+11]. J. Ablinger, J. Blümlein, S. Klein, C. Schneider, and F. Wissbrock. The $O(alpha _s^3)$ Massive Operator Matrix Elements of $O(n_f)$ for the Structure Function $F_2(x,Q^2)$ and Transversity. Nucl. Phys. B, 844:26-54, 2011. arXiv:1008.3347, doi:10.1016/j.nuclphysb.2010.10.021.

  • [ABB+14]. J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, and F. Wißbrock. The 3-Loop Non-Singlet Heavy Flavor Contributions and Anomalous Dimensions for the Structure Function $F_2(x,Q^2)$ and Transversity. Nucl. Phys. B, 886:733-823, 2014. arXiv:1406.4654, doi:10.1016/j.nuclphysb.2014.07.010.

  • [ABDF+14a]. J. Ablinger, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, and C. Schneider. The $O(alpha _s^3 T_F^2)$ Contributions to the Gluonic Operator Matrix Element. Nucl. Phys. B, 885:280-317, 2014. arXiv:1405.4259, doi:10.1016/j.nuclphysb.2014.05.028.

  • [BBB+14]. A. Behring, I. Bierenbaum, J. Blümlein, A. De Freitas, S. Klein, and F. Wißbrock. The logarithmic contributions to the $O(alpha ^3_s)$ asymptotic massive Wilson coefficients and operator matrix elements in deeply inelastic scattering. Eur. Phys. J. C, 74(9):3033, 2014. arXiv:1403.6356, doi:10.1140/epjc/s10052-014-3033-x.

  • [BAB+17]. Johannes Blümlein, Jakob Ablinger, Arnd Behring, Abilio De Freitas, Andreas von Manteuffel, Carsten Schneider, and C. Schneider. Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering: Recent Results. PoS, QCDEV2017:031, 2017. arXiv:1711.07957, doi:10.22323/1.308.0031.

  • [ABDF+14b]. J. Ablinger, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, and F. Wißbrock. The transition matrix element a_gq(n) of the variable flavor number scheme at o(α_s^3). Nuclear Physics B, 882:263-288, May 2014. URL: http://dx.doi.org/10.1016/j.nuclphysb.2014.02.007, doi:10.1016/j.nuclphysb.2014.02.007.

  • [ABB+15]. J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, and C. Schneider. The 3-loop pure singlet heavy flavor contributions to the structure function f2(x,q2) and the anomalous dimension. Nuclear Physics B, 890:48-151, Jan 2015. URL: http://dx.doi.org/10.1016/j.nuclphysb.2014.10.008, doi:10.1016/j.nuclphysb.2014.10.008.

  • [ABBlumlein+22]. J. Ablinger, J. and A. Behring, J. Blümlein, A. De Freitas, C. Schneider, A. Goedicke, C. von Manteuffel and K. Schonwald. The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg,Q}$ and $Delta A_{gg,Q}$}. JHEP 12 (2022) 134, doi:10.1007/JHEP12(2022)134,

  • [ABBlumlein+24]. J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Schönwald. The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$.

ekore.operator_matrix_elements.unpolarized.space_like.as3.A_singlet(n, cache, nf, L)[source]

Compute the N3LO singlet OME.

\[\begin{split}A^{S,(3)} = \left(\begin{array}{cc} A_{gg, H}^{S,(3)} & A_{gq, H}^{S,(3)} & 0 \\ A_{qg, H}^{S,(3)} & A_{qq,H}^{NS,(3)} + A_{qq,H}^{PS,(3)} & 0\\ A_{Hg}^{S,(3)} & A_{Hq}^{PS,(3)} & 0\\ \end{array}\right)\end{split}\]

When using the code, please cite the complete list of references available at the top of this module as3.

Parameters:

  • n (complex) – Mellin moment

  • cache (numpy.ndarray) – Harmonic sum cache

  • nf (int) – number of active flavor below the threshold

  • L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

A_S – N3LO singlet OME \(A^{S,(3)}(N)\)

Return type:

numpy.ndarray

ekore.operator_matrix_elements.unpolarized.space_like.as3.A_ns(n, cache, nf, L)[source]

Compute the N3LO non-singlet OME.

\[\begin{split}A^{NS,(3)} = \left(\begin{array}{cc} A_{qq,H}^{NS,(3)} & 0\\ 0 & 0\\ \end{array}\right)\end{split}\]

When using the code, please cite the complete list of references available at the top of this module as3.

Parameters:

  • n (complex) – Mellin moment

  • cache (numpy.ndarray) – Harmonic sum cache

  • nf (int) – number of active flavor below the threshold

  • L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

A_NS – N3LO non-singlet OME \(A^{NS,(3)}\)

Return type:

numpy.ndarray

Submodules

ekore.operator_matrix_elements.unpolarized.space_like.as3.aHg module

The unpolarized, space-like N3LO heavy-gluon OME.

ekore.operator_matrix_elements.unpolarized.space_like.as3.aHg.A_Hg(n, cache, nf, L)[source]

Compute the N3LO singlet OME \(A_{Hg}^{S,(3)}(N)\).

The expression is presented in [ABBlumlein+24, BBK09a, BAB+17].

When using the code, please cite the complete list of references available in as3.

Parameters:

  • n (complex) – Mellin moment

  • cache (numpy.ndarray) – Harmonic sum cache

  • nf (int) – number of active flavor below the threshold

  • L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

\(A_{Hg}^{S,(3)}(N)\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as3.aHg_param module

The approximated part of renormalization constant \(a_{Hg}^{(3)}(N)\).

ekore.operator_matrix_elements.unpolarized.space_like.as3.aHg_param.a_Hg3(n, cache, nf)[source]

Compute \(a_{Hg}^{(3)}(N)\).

This is composed by two parts:

  1. the exact part proportional to \(n_f T_{F}\) and presented in [BAB+17] 3.1.

  2. a parametrized expression for the \(n_f^0\) piece derived from: the 5 lowest moments, presented in [BBK09a] 8.50-8.54; and the \(x \to 0,1\) limits from 4.4-4.5 of [ABBlumlein+24]. The LL small-x contribution was originally computed in [KLPMV12], 3.47.

The parametrized part has been tested to be in reasonable agreement with the one provided in [KLPMV12] 3.49, 3.50.

Parameters:

  • n (complex) – Mellin moment

  • cache (numpy.ndarray) – Harmonic sum cache

  • nf (int) – number of active flavor below the threshold

Returns:

\(a_{Hg}^{(3)}(N)\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as3.aHq module

The unpolarized, space-like N3LO heavy-quark OME.

ekore.operator_matrix_elements.unpolarized.space_like.as3.aHq.A_Hq(n, cache, nf, L)[source]

Compute the N3LO singlet OME \(A_{Hq}^{S,(3)}(N)\).

The expression is presented in [ABB+15] 5.1 and [BAB+17] 3.1.

When using the code, please cite the complete list of references available in as3.

The part proportional to \(n_f^0\) includes non trivial weight-5 harmonics and has been parametrized in Mellin space. For these pieces the accuracy wrt the exact expression is below 0.001% (N<1000). All the other contributions are provided exact.

Parameters:

  • n (complex) – Mellin moment

  • cache (numpy.ndarray) – Harmonic sum cache

  • nf (int) – number of active flavor below the threshold

  • L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

\(A_{Hq}^{S,(3)}(N)\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as3.agg module

The unpolarized, space-like N3LO gluon-gluon OME.

ekore.operator_matrix_elements.unpolarized.space_like.as3.agg.a_gg3(n, cache, nf)[source]

Compute \(a_{gg}^{(3)}(N)\).

The expression is presented in [ABBlumlein+22].

The \(n_f^0\) piece is parametrized from:

  • the small-x limit 4.10

  • the large-x limit 4.11

  • the expansion of the local and singular parts in 4.6, 4.7

  • the first 15 Mellin moments up to \(N=30\)

The analytical expression contains binomial factors which are not practical to use.

When using the code, please cite the complete list of references available in as3.

Parameters:

  • n (complex) – Mellin moment

  • cache (numpy.ndarray) – Harmonic sum cache

  • nf (int) – number of active flavor below the threshold

Returns:

\(a_{gg}^{(3)}(N)\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as3.agg.A_gg(n, cache, nf, L)[source]

Compute the N3LO singlet OME \(A_{gg}^{S,(3)}(N)\).

The expression is presented in [BBK09a].

When using the code, please cite the complete list of references available in as3.

Parameters:

  • n (complex) – Mellin moment

  • cache (numpy.ndarray) – Harmonic sum cache

  • nf (int) – number of active flavor below the threshold

  • L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

\(A_{gg}^{S,(3)}(N)\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as3.agq module

The unpolarized, space-like N3LO gluon-quark OME.

ekore.operator_matrix_elements.unpolarized.space_like.as3.agq.A_gq(n, cache, nf, L)[source]

Compute the N3LO singlet OME \(A_{gq}^{S,(3)}(N)\).

The expression is presented in [ABDF+14b] 6.3.

When using the code, please cite the complete list of references available in as3.

Parameters:

  • n (complex) – Mellin moment

  • cache (numpy.ndarray) – Harmonic sum cache

  • nf (int) – number of active flavor below the threshold

  • L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

\(A_{gq}^{S,(3)}(N)\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as3.aqg module

The unpolarized, space-like N3LO quark-gluon OME.

ekore.operator_matrix_elements.unpolarized.space_like.as3.aqg.A_qg(n, cache, nf, L)[source]

Compute the N3LO singlet OME \(A_{qg}^{S,(3)}(N)\).

The expression is presented in [BBK09a].

When using the code, please cite the complete list of references available in as3.

Parameters:

  • n (complex) – Mellin moment

  • cache (numpy.ndarray) – Harmonic sum cache

  • nf (int) – number of active flavor below the threshold

  • L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

\(A_{qg}^{S,(3)}(N)\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as3.aqqNS module

The unpolarized, space-like N3LO quark-quark non-singlet OME.

ekore.operator_matrix_elements.unpolarized.space_like.as3.aqqNS.A_qqNS(n, cache, nf, L, eta)[source]

Compute the N3LO non singlet OME \(A_{qq}^{NS,(3)}(N)\).

The expression is presented in [BBK09a] and [ABB+14]. It contains some weight 5 harmonics sums.

When using the code, please cite the complete list of references available in as3.

Note the part proportional to \(n_f^0\) includes weight = 5 harmonics and has been parametrized in Mellin space. For this piece the accuracy wrt the known moments is below the 0.01% (N<1000) and the absolute diff is within 5e-3. All the other contributions are provided exact.

Parameters:

  • n (complex) – Mellin moment

  • cache (numpy.ndarray) – Harmonic sum cache

  • nf (int) – number of active flavor below the threshold

  • L (float) – \(\ln(\mu_F^2 / m_h^2)\)

  • eta (int) – \((-1)^n\) factor to be continued with 1 for singlet like and -1 for non-singlet like

Returns:

\(A_{qq}^{NS,(3)}(N)\)

Return type:

complex

ekore.operator_matrix_elements.unpolarized.space_like.as3.aqqPS module

The unpolarized, space-like N3LO quark-quark pure-singlet OME.

ekore.operator_matrix_elements.unpolarized.space_like.as3.aqqPS.A_qqPS(n, cache, nf, L)[source]

Compute the N3LO singlet OME \(A_{qq}^{PS,(3)}(N)\).

The expression is presented in [BBK09a].

When using the code, please cite the complete list of references available in as3.

Parameters:

  • n (complex) – Mellin moment

  • cache (numpy.ndarray) – Harmonic sum cache

  • nf (int) – number of active flavor below the threshold

  • L (float) – \(\ln(\mu_F^2 / m_h^2)\)

Returns:

\(A_{qq}^{PS,(3)}(N)\)

Return type:

complex