Update N3LO docs

docs/source/theory/nonsinglet.rst

@@ -51,8 +51,7 @@ An equivalent expression is:
 
 .. math::
 
-   C_{ns} \otimes \left(\sum\nolimits_q e_q^2 ~ q_+\right) +
-   C_{ps} \otimes \left(\sum\nolimits_q \ev{e^2} ~ q_+\right)\\
+   x^{-1} F = C_{ns} \otimes \left(\sum_q e_q^2 ~ q_+\right) + C_{ps} \otimes \left(\sum_q \ev{e^2} ~ q_+\right)\\
 
 so in `yadism` we are using:
 
@@ -69,11 +68,10 @@ Indeed:
 
 .. math::
 
-   C_{ns} \otimes q_{ns}^{+} &+ \ev{e^2} (C_{ns} + C_{ps}) \otimes q_s\\
-   C_{ns} \otimes \left( \sum\nolimits_q (e_q^2 - \ev{e^2}) ~ q_+ \right) &+
-   \ev{e^2} (C_{ns} + C_{ps}) \otimes \left( \sum\nolimits_q q_+\right) \\
-   \sum\nolimits_q q_+ \otimes ( C_{ns}  (e_q^2 - \ev{e^2}) &+ \ev{e^2} (C_{ns} + C_{ps}) ) )\\
-   \sum\nolimits_q q_+ \otimes ( C_{ns}  e_q^2 &+ \ev{e^2} C_{ps} ) )
+   x^{-1} F(x) &= C_{ns} \otimes q_{ns}^{+} + \ev{e^2} (C_{ns} + C_{ps}) \otimes q_s\\
+   &= C_{ns} \otimes \left( \sum_q (e_q^2 - \ev{e^2}) ~ q_+ \right) + \ev{e^2} (C_{ns} + C_{ps}) \otimes \left( \sum_q q_+\right) \\
+   &= \sum_q q_+ \otimes ( C_{ns}  (e_q^2 - \ev{e^2}) + \ev{e^2} (C_{ns} + C_{ps}) ) )\\
+   &= \sum_q q_+ \otimes ( C_{ns}  e_q^2 + \ev{e^2} C_{ps} ) )
 
 Inducing from LO structure functions
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -85,9 +83,9 @@ To retrieve the exact definition of :math:`q_{ns}^{+}` in :cite:`vogt-f2nc` we a
 
 .. math::
 
-   x^{-1} F_2(x) &=  \sum\nolimits_q e_q^2 ~ q_+(x) \\
-   x^{-1} F_2(x) &=  q_{ns}^{+}(x) + \ev{e^2} q_s(x)\\
-   &=  q_{ns}^{+}(x) + \ev{e^2} \sum\nolimits_q q_+(x)
+   x^{-1} F_2(x) &=  \sum_q e_q^2 ~ q_+(x) \\
+   &=  q_{ns}^{+}(x) + \ev{e^2} q_s(x)\\
+   &=  q_{ns}^{+}(x) + \ev{e^2} \sum_q q_+(x)
 
 Where:
 
@@ -122,43 +120,45 @@ Then:
 Neutral Current
 ---------------
 
-The case of parity conserving NC structure functions is analogous to EM,
+The case of parity conserving |NC| structure functions is analogous to |EM|,
 just with different coupling and summing all the electroweak channels.
-While for the presence of parity violating terms (for ex. :math:`F_3`)
-has a different decompositions:
+However, for parity violating structure functions (e.g. :math:`F_3`)
+we have a different decompositions:
 
-x F_3 = C_{ns} \otimes q_{ns}^{-} + \ev{e^2} \left(C_q \otimes q_v)
+.. math::
+
+   x^{-1} F_3 = C_{ns} \otimes q_{ns}^{-} + \ev{e^2} \left(C_q \otimes q_v\right)
 
-with the two quark flavor combinations defined as
+where the two quark flavor combinations are defined as
 
 .. math::
 
    q_v & = \sum_q  (q - \bar{q}) \\
    q_{ns}^{-} &= \sum_q \left(g_q^2 - \ev{g^2}\right) ~ (q - \bar{q})
 
-and :math:`g_q` a suitable electroweak coupling.
-As before in Yadism we rotate the coefficients to a new basis.
+and :math:`g_q` is a suitable electroweak coupling.
+As before in yadism we rotate the coefficients to a new basis.
 
 .. math::
 
-   C_{ns} \otimes \left(\sum\nolimits_q e_q^2 ~ q_-\right) +
-   C_{v} \otimes \left(\sum\nolimits_q \ev{e^2} ~ q_-\right)\\
+   x^{-1} F_3 = C_{ns} \otimes \left(\sum\nolimits_q e_q^2 ~ q_-\right) +
+   C_{v} \otimes \left(\sum\nolimits_q \ev{e^2} ~ q_-\right)
 
 with
 
 .. math::
 
    q_- = q - \bar{q}
 
-Note that the gluon can not generate a parity violating term.
+Note that neither the gluon nor the flavor singlet can generate a parity violating term.
 
 Charged Current
 ---------------
 
-CC can be treated in an analogous way, simply:
+|CC| can be treated in an analogous way:
 
-- when the incoming quark is *directly* coupling (*non-singlet*) to the EW boson
-  (so :math:`W_{\pm}`) only the flavor or the anti-flavor may have a non-zero
+- when the incoming quark is *directly* coupling (*non-singlet*) to the |EW| boson
+  (i.e. :math:`W^{\pm}`) only the flavor or the anti-flavor may have a non-zero
   coupling, but not both
 - when the incoming quark is *indirectly* coupling through a gluon (*singlet*)
   nothing change, because the average has to be done on half the objects, but
@@ -170,10 +170,10 @@ Higher Orders
 
 The decomposition of the quark sector in different partonic channels
 has the advantage to facilitate the relations with higher orders
-QCD corrections.
+|QCD| corrections.
 
-- :math:`C_{ns}` is always the leading contribution as correspond
-  to diagram in which the incoming flavor is coupling directly to the
+- :math:`C_{ns}` is always the leading contribution as it corresponds
+  to diagrams in which the incoming flavor is coupling directly to the
   electroweak boson.
 - :math:`C_{g}` is suppressed by :math:`\mathcal{O}(a_s)`
   as the gluon need to radiate a quark-antiquark pair before coupling
@@ -182,8 +182,7 @@ QCD corrections.
   :math:`\mathcal{O}(a_s^3)` respectively as they are related to diagrams
   where the incoming flavor line is not coupling directly with the electroweak boson.
 
-
-From N3LO on a new class of diagrams, called :math:`fl_{11}`, can appear for
+From |N3LO| on a new class of diagrams, called :math:`fl_{11}`, can appear for
 the parity conserving structure functions, both in the quark and gluon sector
 :cite:`Larin:1996wd`. In these diagrams the incoming and outgoing bosons are
 coupling to different fermion lines (open or in loops) and thus generate

src/yadism/coefficient_functions/coupling_constants.py

@@ -171,15 +171,17 @@ def partonic_coupling(self, mode, pid, quark_coupling_type, cc_mask=None):
         raise ValueError(f"Unknown mode: {mode}")
 
     def partonic_coupling_fl11(self, mode, pid, nf, quark_coupling_type):
-        """Compute the coupling of the boson to the parton for the flavor class :math:`fl_{11}`.
+        r"""Compute the coupling of the boson to the parton for the flavor class :math:`fl_{11}`.
 
-        This is a generalization of :cite:`Larin:1996wd` (Table 2) for NC.
-        See also pag. 27. This coupling is given by :
+        This is a generalization of :cite:`Larin:1996wd` Table 2 (see also pag. 27 there) for |NC|.
+        The coupling is given by:
 
-        .. :math:
-            W_{q,bb'} = \frac{Tr(Q_b)}{n_f} Q_b'
+        .. math::
 
-        the gluon and pure singlet coupling are then build after summing on all the
+            W_{q,bb'} = \frac{\text{tr}(Q_b)}{n_f} Q_b'
+
+        where the trace refers to flavor space.
+        The gluon and pure singlet couplings are then build by summing over all the
         different electroweak channels.
 
         Parameters
@@ -223,7 +225,7 @@ def switch_mode(quark, coupling_type):
         return g1 * g2
 
     def propagator_factor(self, mode, Q2):
-        r"""Compute propagator correction to account for different bosons (:math:`\\eta` in PDG).
+        r"""Compute propagator correction to account for different bosons (:math:`\eta` in PDG).
 
         Parameters
         ----------
@@ -329,7 +331,7 @@ def get_weight(self, pid, Q2, quark_coupling_type, cc_mask=None):
         raise ValueError(f"Unknown process: {self.obs_config['process']}")
 
     def get_fl11_weight(self, pid, Q2, nf, quark_coupling_type):
-        """Same as :func:`get_weight`but now for the NC flavor class :math:`fl_{11}`.
+        """Same as :func:`get_weight` but now for the |NC| flavor class :math:`fl_{11}`.
 
         Combine the charges, both on the leptonic side and the hadronic side,
         as well as propagator changes and/or corrections.
@@ -374,7 +376,7 @@ def get_fl11_weight(self, pid, Q2, nf, quark_coupling_type):
             return w_phph
         # allow Z to be mixed in
         if self.obs_config["process"] == "NC":
-            # photon-Z interference, this class is not symmetric
+            # photon-Z interference, this class is not symmetric on the partonic side
             w_phZ = (
                 self.leptonic_coupling("phZ", quark_coupling_type)
                 * self.propagator_factor("phZ", Q2)