Scale Variations

The coefficient functions may depend on a factorization scale \(\mu_F^2\) and on a renormalization scale \(\mu_R^2\). Both scales are unphysical and enter via a RGE, i.e. the full object is independent on the specific choice.

Origin

The factorization scale is a relic of the collinear factorization and thus related to the PDF scheme. It can be seen as an infra-red cut-off that separates the perturbative hard matrix element and the non-perturbative PDF. The explicit dependence on \(\mu_F^2\) is generated by convolutions of splitting functions with lower order coefficient functions [AK+19, vNV00].

The renormalization scale is a relic of the Lagrange renormalization procedure and thus related to the coupling scheme. It can be seen as an ultra-violet cut-off that separates the perturbative hard matrix element and the running coupling, that resums a specific class of terms. The explicit dependence on \(\mu_R^2\) is generated by inserting the RGE, i.e. the beta function, into the expanded coefficient functions [AK+19, vNV00].

Description

Following a more in depth description about the structure of the scale variations in the theory. This is really decoupled from the process itself, and it has been used as a guidance for the implementation itself, whose details are instead described in the corresponding technical section.

The most basic manifestation of scale variations is that:

PDFs are always evaluated at the factorization scale \(\mu_F\) (and this is a source of scale variations even at LO)
\(\alpha_s\) is always evaluated at the renormalization scale \(\mu_R\) (and this is a source of scale variations already at NLO)

Actually the scale variations have also a non-trivial impact on the coefficient function expressions (starting at NLO for factorization scale variations, and at NNLO for renormalization ones), and it is discuss in the following sections.

Factorization

Since the factorization scale variation is related only to the running of the PDF, than it is not dependent on the process, apart from its input structure (how many partons are present in the incoming state).

Thus the content of the coefficient functions is completely factorized in the scale variations expressions.

They actually can be organized in powers of the perturbative coupling \(a_s = \alpha_s(\mu_F) / 4\pi\) and the logarithm of the factorization scale \(L_M = \log(\mu_F^2 / Q^2)\) (note the different definition of the log with respect to [AK+19, vNV00] that is the reason of the different minus signs in the next expressions):

\[\textbf{C}_a(x, \alpha_s(\mu_R^2), L_M, L_R) \stackrel{\mu_R = \mu_F}{=} \textbf{c}_a^{(0)}(x) + \sum_{l=1}^\infty a_s^l \left(\textbf{c}_a^{(l)}(x) + \sum_{m=1}^l \textbf{c}_a^{(l,m)}(x) L_M^m\right)\]

Where \(a\) (\(= 1,2,3,L\)) is the kind of structure functions considered.

Than the explicit coefficient functions can be computed from DGLAP and \(\alpha_s\) running (see [vNV00, vNV01]), resulting in process and kind independent elements (the actual anomalous dimension of the two runnings) convolved with lower order coefficient functions:

\[\begin{split}\textbf{c}_a^{(1,1)} &= -\textbf{c}_a^{(0)} \otimes \textbf{P}^{(0)}\\ \textbf{c}_a^{(2,1)} &= -\textbf{c}_a^{(0)} \otimes \textbf{P}^{(1)} + \textbf{c}_a^{(1)} \otimes (\textbf{P}^{(0)} - \beta_0 \textbf{1})\\ \textbf{c}_a^{(2,2)} &= \frac{1}{2} \textbf{c}_a^{(1,1)} \otimes (\textbf{P}^{(0)}- \beta_0 \textbf{1})\\ \textbf{c}_a^{(3,1)} &= -\textbf{c}_a^{(0)} \otimes \textbf{P}^{(2)} + \textbf{c}_a^{(1)} \otimes (\textbf{P}^{(1)} - \beta_1 \textbf{1}) + \textbf{c}_a^{(2)} \otimes (\textbf{P}^{(0)} - 2 \beta_0 \textbf{1})\\ \textbf{c}_a^{(3,2)} &= \frac{1}{2} \left\{ \textbf{c}_a^{(1,1)} \otimes (\textbf{P}^{(1)}-\beta_1 \textbf{1}) + \textbf{c}_a^{(2,1)} \otimes (\textbf{P}^{(0)} - 2 \beta_0 \textbf{1}) \right\}\\ \textbf{c}_a^{(3,3)} &= -\frac{1}{3} \textbf{c}_a^{(2,2)} \otimes (\textbf{P}^{(0)}- 2 \beta_0 \textbf{1})\\\end{split}\]

Where \(\beta_k\) are the coefficient of the \(a_s\) beta function (pay attention to the normalization: they are not the ones for the beta function of \(\alpha_s\)) and \(\textbf{P}^{(k)}\) are the terms of the perturbative expansion of the Altarelli-Parisi splitting functions (again normalized in \(a_s\)). Note that when converting the splitting functions to the anomalous dimensions, following the eko definition, one gets a minus sign.

Important

The equations above have to be read as vectorial, indeed the both the coefficient functions are vectors and the splitting functions are matrices. The basis can be decomposed in two parts, strictly related to the evolution basis:

the non-singlet part: there are three distinct elements for non-singlet, the so called \(ns,+\), \(ns,-\), and \(ns,v\), but in this sector splitting functions are completely diagonal
the singlet part: there are two elements, the quark singlet, \(q\), and the gluon, \(g\), but the splitting functions contain non-vanishing off-diagonal elements

In other context this is called the anomalous dimension basis, since the splitting functions are known as anomalous dimensions when written in \(N\)-space.

Notice that even if the equations have a nice structure when presented nested - i.e. the coefficients are functions of the raw ones \(\textbf{c}^{(l)}_a\), but also of the other scale-variations coefficients \(\textbf{c}^{(l,m)}_a\) - they may be completely inlined as functions of the raw ones (actually the structure derives from the exponentiation procedure done by resummation in the runnings).

Renormalization

It is related to the only running of the coupling \(\alpha_s(\mu_R^2)\), with respect to a further different scale \(\mu_R^2\).

For this reason the easiest variable in which to express the new contribution is actually \(L_R = \log(\mu_R^2/\mu_F^2)\) (note the different definition of the log with respect to [AK+19, vNV00] that is the reason of the different minus signs in the next expressions), that measures the discrepancy of the new scale with respect to the one already considered, and they are expressed on top of the expressions built in the previous paragraph:

\[\begin{split}\textbf{C}_a^{(2)}(x, L_M, L_R) &= \textbf{C}_a^{(2)}(x, L_M, 0) + \beta_0 L_R \textbf{C}_a^{(1)}(x, L_M)\\ \textbf{C}_a^{(3)}(x, L_M, L_R) &= \textbf{C}_a^{(3)}(x, L_M, 0) + 2 \beta_0 L_R \textbf{C}_a^{(2)}(x, L_M, 0) + \{\beta_1 L_R + \beta_0^2 L_R^2\} \textbf{C}_a^{(1)}(x, L_M)\end{split}\]

The vectorial character of the equations in this case is present but trivial (no off-diagonal contribution, since \(\beta_k\) are channel independent scalars) and the contribution starts to be non-trivial only at NNLO DIS.

Procedures

scheme A: it requires a refit for the determination of the scale variations.
scheme B: it does involve the evolution, so it resums the effect of the scale variation, see eko’s scale variations docs.
scheme C: varying the scales in the context of PDF determination only in the coefficient functions corresponds to scheme C in [AK+19].