Scale Variations

The scale variations are universal, since they depend only on the incoming structure (they are related to the input partons), so they act multiplicatively on the process itself (in particular through convolution), see the theory section.

Then it is worth to exploit this property as much as possible to simplify the implementation and maintainability of the code.

Numerical convolutions

The two alternatives for the convolutions are the following:

doing it analytically, and code all the expressions
code only the basic elements and use them in numerical convolutions

The strategy chosen by yadism is partially (more later on, see Multiple convolutions) the second, because essentially of a couple of advantages:

the expressions have to be coded only once, instead of doing it for each coefficient function (perturbative order, structure function, partonic channel, and heavyness)
the coefficient function expressions, and the splitting functions too, become rapidly more and more cumbersome while increasing the perturbative order, in this way it is possible to save many complicate convolutions

The main disadvantages would be the following:

doing convolutions numerically precision is lost
numerical integrations are time-consuming

Actually this two issues have a negligible impact on the result, indeed the numerical precision is absolutely dominated by the interpolation required to integrate a generic PDF set (and in general PDF sets are delivered as interpolation grids, so they are never exact functions) and since the contributions are universal they can be computed just once per interpolation grid (and so actually once per run) and simply applied at the cost of a matrix product.

Moreover integrating analytical expressions (such as the convolution of splitting functions and coefficient functions, that is more complicated than both) is difficult and so: it can deteriorate the precision of the integration and require more evaluations, and so it is also time-consuming.

Baseline: it is not clear that the disadvantages would be improved at all by the other choice, but the advantage is instead clear and great.

Multiple convolutions

Actually there are multiple convolutions involved, see the factorization scale variation formula.

What is done in yadism is to implement everything to depend only on the original coefficients \(\textbf{c}_a^{(i)}\), and not to depend recursively on the convolved ones \(\textbf{c}_a^{(i,j)}\), in order to limit the number of extra numerical convolutions to a single one.

In this case we are avoiding the disadvantages of the previous section, only partially paying for giving up on of the advantages: these convolutions would involve only splitting functions, and never the coefficient functions’ expressions, and for this same reason they are also considerably easier.

Applying convolved kernels

Since the convolutions are done numerically it is needed to integrate ahead of time the full operator: indeed the splitting functions act as operator on coefficient functions, taking them as input and providing other coefficients as output (through convolutions), exactly like an EKO.

Indeed factorization scale variations are simply encoding bits of evolution directly inside the coefficient functions.

Actually coefficient functions have their own representation in yadism as kernels, i.e. a pair of a coefficient function and some weights, possibly related to multiple partonic channels.

The weights usually encodes charge factors, that is the only part that differs in the contributions of different flavors, related to single set of diagrams.

The splitting functions are also non-trivial in flavor space, but the only non-triviality manifest in the anomalous dimension basis, that is a very restricted basis (dimension 7) for the operators on the flavor space (whose dimension is \(13 \times 13\), with the contributions of 12 quark flavors and the gluon).

Because of this we can apply splitting functions (and their convolutions) in two objects: the \(x\)-space expression, and the projector relative to the space it acts on.

Important

The final formula related to the application of scale variations is the following:

\[c_{a, (\beta, j)}^{(out, lnf)} = v_\alpha \Pi^{(ad)}_{\alpha\beta} S^{(ad), (out, lnf, in)}_{ji} c^{(in)}_{a,i}\]

where:

\(v\) are the weights assigned to each partonic channel
\(\Pi\) are the projectors over the subspaces related to the anomalous dimension basis
\(S\) are the splitting kernels
\(c\) are the actual raw (non-scale-varied) coefficient functions on the right hand side, and the dressed (scale-varied) ones on the left

and so:

\((out, lnf, in)\) are respectively indexing: the perturbative order of the generated coefficient, the power of \(\log(Q^2/\mu_F)\) the generated coefficient is multiplied by, and the perturbative order of the coefficient is applied on
\((ad)\) is an index over the anomalous dimension basis
\(\alpha, \beta\) are indices in flavor space
\(i, j\) are indices in the evolution space
\(a\) is the structure function kind

Notice that the part that involves the flavor space is only coupled to the part that involves the interpolation space by the sum over the anomalous dimension basis, that is relatively small (dimension 7), so the two sums are done separately and only recombined at the latest possible moment.

Integrating in x-space

The factorization theorem can be schematically written for scale variations as:

\[F = f \otimes S \otimes c\]

Where \(F\) is the observable, \(f\) the PDF, \(c\) the raw coefficient function, and \(S\) the splitting kernels organized as described in the theory section.

Without the scale variations the interpolation is done completely on the PDF, as described in Interpolation, and the interpolation polynomials would then be used to convolve numerically the coefficient functions:

\[\begin{split}F(x) &= [f \otimes c] (x) = \left[\left(\sum_j f(x_j) p_j\right) \otimes c\right](x) =\\ &= \sum_j f(x_j) \left[p_j \otimes c\right](x) = \sum_j f(x_j) c_j(x)\end{split}\]

In this way for each kinematic specified \(x\) the coefficient function is turned into a vector over interpolation basis. And so:

\[c_j(x) = (p_j \otimes c) (x)\]

The same thing can be done with scale variations, turning the \(S\) kernels into a matrix.

\[\begin{split}F(x) &= [f \otimes c] (x) = \left[\left(\sum_j f(x_j) p_j\right) \otimes S \otimes c\right](x) =\\ &= \sum_j f(x_j) \left[p_j \otimes S \otimes c\right](x)\\ &= \sum_j f(x_j) \left[\left(\sum_k (p_j \otimes S)(x_k) p_k\right) \otimes c\right](x)\\ &= \sum_{jk} f(x_j) \left[(p_j \otimes S)(x_k)\, (p_k \otimes c)\right](x)\\ &= \sum_{jk} f(x_j)\, S_{jk}\, c_k(x)\end{split}\]

Where essentially the PDF have been interpolated first, and then the convolution of the interpolation basis and the splitting kernel (\(p_j \otimes S\)) is also interpolated a second time.

Important

Note that:

\[S_{jk} = (p_j \otimes S)(x_k)\]

So even if the two indices run on the same basis, they have actually different sources:

the first one, \(j\), is coming from the convolution with the interpolation polynomial \(p_j\) (same as for the coefficient function \(c_j(x)\), because actually \(S \otimes c\) is the scale-varied coefficient function)
the second, \(k\), is coming from the evaluation on the grid point \(x_k\) (to be joined with the coefficient function c_k(x), who is stemming from)

Remark on projectors

The projectors are not very complicate objects, but a little bit of care has to be used relatively to their normalization.

Indeed the actual expression for the diagonal projectors is the following:

\[\Pi^{(a)} = \sum_{k \in a} \frac{\dyad{k}{k}}{\abs{\bra{k}\ket{k}}^2}\]

where the projectors over single states are explicitly normalized with the norm of the state, since the evolution basis itself is not normalized.

Note

The sum is present because some anomalous dimension basis element do apply to more than a single evolution flavor, referring actual to a subspace rather than a single vector.

e.g.: for \(a = ns_+\) the distributions involved are

\[T_3, T_8, T_{15}, T_{24}, T_{35}\]

cut if there are less than 6 active flavors. For the full correspondence see the related eko’s documentation.

But in order to explicit the prescription for the off-diagonal elements the following has to be noticed:

the projector are made of a \(\bra{in}\) that has to be contracted with the incoming PDF, for this reason this has to remain unnormalized, in order to obtain the required contributions for the unnormalized evolution basis (i.e. it has to be the same contribution that would have been if directly applied to the raw coefficient function)
the other component is a \(\ket{out}\) that goes together the non-scale-varied coefficient function itself, thus it has to be normalized, because it should extract

E.g.: if the singlet PDF is plugged in, written in flavor basis, when no scale variations are present it would apply directly to the coefficient function, without any extra normalization

\[\bra{c_\Sigma}\ket{\Sigma}\]

and the same should happen in case of scale variations: if the singlet is going to split a gluon then the singlet PDF has to contribute with the gluon coefficient function, so the last has to become singlet proportional:

\[\bra{c_g}\dyad{g}{\Sigma} \times \ket{\Sigma}\]

while in the case of quark contributions to the gluon channel it has to be normalized.

Another, more practical, point of view is that this is because \(P_{ab}\) already contains the normalization, e.g. \(P_{qg} \propto 2 n_f\) , and thus it has to be subtracted from there in order to use the unnormalized projector).

Important

The actual expression for all projectors is then the following:

\[\Pi^{(a)} = \sum_{(o, i) \in a} \frac{\dyad{o}{i}}{\abs{\bra{o}\ket{o}}^2}\]

Renormalization scale variations

The renormalization scale variations up to NLO consist only in the evaluation of the strong coupling \(\alpha_s\) at a different scale.

Instead, starting at NNLO, there are further contributions to the coefficient functions anomalous to those discussed in the previous paragraph.

These contributions do not require convolutions, they are reported in the theory section, and in yadism they are actually implemented as they are written, i.e.:

first compute the factorization scale variations, and derive \(\textbf{C}_a^{(i)}(x, L_M, 0)\) from the \(\textbf{c}_a^{(l)}(x)\)
then apply the further corrections on top of the new object, and obtain the full \(\textbf{C}_a^{(i)}(x, L_M, L_R)\)

Attention

The parameter \(L_R = \log(\mu_F^2 / \mu_R^2)\), so it does not account for the difference between the process scale \(Q^2\) and the renormalization one \(\mu_R^2\), but between the latter and the factorization one \(\mu_F^2\).

For the user convenience the result is nevertheless stored in grids in which the parameters are instead:

\(\log(\mu_F^2 / Q^2)\)
\(\log(\mu_R^2 / Q^2)\)