Merge branch 'master' into read-legacy

doc/source/theory/N3LO_ad.rst

@@ -8,22 +8,22 @@ since in Mellin space they include harmonics sum up to weight 7, for which an
 analytical expression is not available.
 
 We provide two different types of approximations, depending on the key ``use_fhmruvv``.
-The theory card parameter ``n3lo_ad_variation=(gg, gq, qg, qq, nsp, nsm, nsv)``, 
+The theory card parameter ``n3lo_ad_variation=(gg, gq, qg, qq, nsp, nsm, nsv)``,
 which is set to ``(0,0,0,0,0,0,0)`` as default, can be varied to obtain parametrization uncertainties.
 
 In particular:
 
-* ``use_fhmruvv = True`` (default option) adopts the parametrizations as provided 
-  in :cite:`Moch:2017uml,Falcioni:2023luc,Falcioni:2023vqq,Falcioni:2024xyt,Falcioni:2024qpd`. 
-  For each of the 7 splitting functions, the approximation error can be obtained by varying 
-  the entries of ``n3lo_ad_variation`` in the range: ``0``, 
+* ``use_fhmruvv = True`` (default option) adopts the parametrizations as provided
+  in :cite:`Moch:2017uml,Falcioni:2023luc,Falcioni:2023vqq,Falcioni:2024xyt,Falcioni:2024qpd`.
+  For each of the 7 splitting functions, the approximation error can be obtained by varying
+  the entries of ``n3lo_ad_variation`` in the range: ``0``,
   cental value (default), ``1`` down variation, ``2`` up variation.
 
 * ``use_fhmruvv = False`` adopts an in-house parametrization, constructed as described below.
-  In this case only variations of the singlet sector are available. Currently one 
+  In this case only variations of the singlet sector are available. Currently one
   can vary ``n3lo_ad_variation`` in the range: ``0-19`` for ``gg``, ``0-15`` for ``gg``,
   ``0-15`` for ``qg``, ``0-6`` for ``qq``.
-  Note these approximations will be no longer updated and are now deprecated. 
+  Note these approximations will be no longer updated and are now deprecated.
 
 
 In house approximation
@@ -294,7 +294,6 @@ The difference between the known moments and the known limits is parametrized
 in Mellin space using different basis, in order to estimate the uncertainties of
 our determination.
 
-
 Uncertainties estimation
 ^^^^^^^^^^^^^^^^^^^^^^^^
 

doc/source/theory/pQCD.rst

@@ -51,7 +51,6 @@ can be specified at ``scale_ref`` along with ``nf_ref`` and, the computed result
 depend on the number of flavors at the target scale, see :meth:`eko.couplings.Couplings.a_s`
 An example how the evolution path is determined is given :doc:`here</code/Utilities/>`.
 
-
 QCD Splitting Functions
 -----------------------
 
@@ -86,8 +85,8 @@ At |NLO|, the singlet entry of the quark-quark anomalous dimension can be decomp
     \gamma^{(1)}_{qq} =\gamma^{(1)}_{ps} + \gamma^{(1)}_{ns,+}
 
 The non-singlet sector in the polarized case swaps the plus and minus non-singlet relative to the unpolarized case.
-This is because the polarized non-singlet splitting functions are defined as the difference between the probability of the polarized parton splitting 
-into daughter partons of the same flavour and same helicity and daughters of a different flavours and opposite helicity. 
+This is because the polarized non-singlet splitting functions are defined as the difference between the probability of the polarized parton splitting
+into daughter partons of the same flavour and same helicity and daughters of a different flavours and opposite helicity.
 The first moments of the anomalous dimensions are:
 
 .. math ::
@@ -103,7 +102,7 @@ At |NNLO| the non-singlet is further decomposed into the helicity difference qua
 where :math:`\gamma^{(2)}_{ns,-}` is the minus flavour asymmetry non-singlet and :math:`\gamma^{(2)}_{ns,s}` the sea-like polarized non-singlet.
 The singlet entry :math:`\gamma^{(2)}_{qq}` is defined as above in the |NLO| case.
 
-Finally the violation of the axial current conservation :math:`\bar{\psi} \gamma_\mu \gamma_5 \bar{\psi}` only through 
+Finally the violation of the axial current conservation :math:`\bar{\psi} \gamma_\mu \gamma_5 \bar{\psi}` only through
 loop corrections impose the following relations to the singlet splittings at all orders :cite:`Moch:2014sna` :
 
 .. math ::