Non-singlet definition

From Vogt 3-loop paper 4.1 [VVM05], we get:

\[x^{-1} F = C_{ns} \otimes q_{ns}^{+} + \ev{e^2} \left(C_q \otimes q_s + C_g \otimes g\right)\]

where:

\[C_q = C_{ns} + C_s\]

Photon Exchange

In the syntax of [VVM05] “singlet” is the actual flavor singlet:

\[q_s = \sum_q (q + \bar{q})\]

The so called “non-singlet” is actually the difference between the charged singlet and the flavor singlet:

\[q_{ns}^{+} = \sum_q \left(e_q^2 - \ev{e^2}\right) ~ (q + \bar{q})\]

Of course they are both singlet-like (referring to evolution basis) since they are proportional to \(q_+\)

\[q_+ = q + \bar{q}\]

This basis is natural because EM cannot distinguish a flavor from the anti-flavor (instead NC or CC can).

Equivalent expression (yadism)

An equivalent expression is:

\[\begin{split}x^{-1} F = C_{ns} \otimes \left(\sum_q e_q^2 ~ q_+\right) + C_{ps} \otimes \left(\sum_q \ev{e^2} ~ q_+\right)\\\end{split}\]

so in yadism we are using:

  • the name non-singlet to call the charged singlet:

    • in which every quark is weighted with the square of its charge

  • the name singlet to call the flavor singlet:

    • in which every quark is weighted with the average of all the square charges of the quark that are taking part

Indeed:

\[\begin{split}x^{-1} F(x) &= C_{ns} \otimes q_{ns}^{+} + \ev{e^2} (C_{ns} + C_{ps}) \otimes q_s\\ &= C_{ns} \otimes \left( \sum_q (e_q^2 - \ev{e^2}) ~ q_+ \right) + \ev{e^2} (C_{ns} + C_{ps}) \otimes \left( \sum_q q_+\right) \\ &= \sum_q q_+ \otimes ( C_{ns} (e_q^2 - \ev{e^2}) + \ev{e^2} (C_{ns} + C_{ps}) ) )\\ &= \sum_q q_+ \otimes ( C_{ns} e_q^2 + \ev{e^2} C_{ps} ) )\end{split}\]

Inducing from LO structure functions

To retrieve the exact definition of \(q_{ns}^{+}\) in [VVM05] we assumed:

  • \(q_s = \sum\nolimits_q q_+(x)\), i.e. the singlet is the flavor singlet

  • and we compare the LO DIS expressions

\[\begin{split}x^{-1} F_2(x) &= \sum_q e_q^2 ~ q_+(x) \\ &= q_{ns}^{+}(x) + \ev{e^2} q_s(x)\\ &= q_{ns}^{+}(x) + \ev{e^2} \sum_q q_+(x)\end{split}\]

Where:

  • the first equation is the LO DIS result in the flavor basis

  • the second one is the way it is expressed in [VVM05]

Consider the following hypothesis on the number of flavors:

  • \(n_f=1\):

\[\begin{split}x^{-1} F_2(x) &= e_u^2 ~ u_+(x) \stackrel{!}{=} q_{ns}^{+}(x) + e_u^2 u_+(x)\\ &\Rightarrow q_{ns}^{+}(x) = e_u^2 u_+(x) - e_u^2 u_+(x) = 0\end{split}\]

  • \(n_f=2\):

\[\begin{split}x^{-1} F_2(x) &= e_u^2 u_+(x) + e_d^2 d_+(x) \stackrel{!}{=} q_{ns}^{+}(x) + \frac{e_u^2 + e_d^2}{2} ~ ( u_+(x) + d_+(x) )\\ &\Rightarrow q_{ns}^{+}(x) = e_u^2 u_+(x) + e_d^2 d_+(x) - \frac{e_u^2 + e_d^2}{2} ~ ( u_+(x) + d_+(x) )\end{split}\]

Then:

\[q_{ns}^{+}(x) = \sum_q (e_q^2 - \ev{e^2}) ~ q_+(x)\]

Neutral Current

The case of parity conserving NC structure functions is analogous to EM, just with different coupling and summing all the electroweak channels. However, for parity violating structure functions (e.g. \(F_3\)) we have a different decompositions:

\[x^{-1} F_3 = C_{ns} \otimes q_{ns}^{-} + \ev{e^2} \left(C_q \otimes q_v\right)\]

where the two quark flavor combinations are defined as

\[\begin{split}q_v & = \sum_q (q - \bar{q}) \\ q_{ns}^{-} &= \sum_q \left(g_q^2 - \ev{g^2}\right) ~ (q - \bar{q})\end{split}\]

and \(g_q\) is a suitable electroweak coupling. As before in yadism we rotate the coefficients to a new basis.

\[x^{-1} F_3 = C_{ns} \otimes \left(\sum\nolimits_q e_q^2 ~ q_-\right) + C_{v} \otimes \left(\sum\nolimits_q \ev{e^2} ~ q_-\right)\]

with

\[q_- = q - \bar{q}\]

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:

  • when the incoming quark is directly coupling (non-singlet) to the EW boson (i.e. \(W^{\pm}\)) 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 being an average this amounts to multiply and divide by \(2\)

Higher Orders

The decomposition of the quark sector in different partonic channels has the advantage to facilitate the relations with higher orders QCD corrections.

  • \(C_{ns}\) is always the leading contribution as it corresponds to diagrams in which the incoming flavor is coupling directly to the electroweak boson.

  • \(C_{g}\) is suppressed by \(\mathcal{O}(a_s)\) as the gluon need to radiate a quark-antiquark pair before coupling with a electroweak boson.

  • \(C_{ps},C_{v}\) are suppressed by \(\mathcal{O}(a_s^2)\) or \(\mathcal{O}(a_s^3)\) 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 \(fl_{11}\), can appear for the parity conserving structure functions, both in the quark and gluon sector [LNvRV97]. In these diagrams the incoming and outgoing bosons are coupling to different fermion lines (open or in loops) and thus generate contributions that are not proportional to the coupling squared \(g_q^2\), or its average \(\ev{g^2}\), but rather to \(\ev{g} g_q\) for quarks or \(\ev{g}^2\) for gluons respectively.