Missing Higher Order Uncertainties (MHOU) from Scale Variations

The variation of the renormalization and factorization scales is one of the most used method to estimate MHOU in QCD.

This is due to the simplicity of both their calculation and their implementation, the former given by the fact that the scale dependence of the strong coupling \(\alpha_{s}\) and of PDF is universal and the latter given by the easiness in calculating the correlations.

However, the scale variations approach also has well known drawbacks:

there is no unique principle to determine the specific range of the scale variation
it cannot deal with new singularities or color structures appearing at higher orders.

Here we briefly summarize the aspects of the scale variations method which are related to pineko. For a much more exhaustive report on how to compute scale variations and how to use them in a PDF fit, please refer to [AK+19].

Renormalization group invariance

Considering a theoretical prediction \(\overline{T}(\alpha_{s}(\mu^2), \mu^2/Q^2)\) with \(\mu^2\) the renormalization scale and \(Q^2\) the typical scale of the process, denote with \(T(Q^2)\) the same theoretical prediction evaluated at \(\mu^2 = Q^2\).

We know that the QCD running coupling satisfies the RGE

\[\mu^2 \frac{d}{d\mu^2}\alpha_{s}(\mu^2) = \beta(\alpha_{s}(\mu^2))\]

and that an all-order prediction is independent of the renormalization scale:

\[\mu^2 \frac{d}{d\mu^2}\overline{T}(\alpha_{s}(\mu^2), \mu^2/Q^2) = 0.\]

Then, defining \(\mu^2 = k Q^2\), \(t = \ln{(Q^2/\Lambda^2)}\) and \(\kappa = \ln{k} = \ln{(\mu^2/Q^2)}\), we can rewrite the RG equation as

\[\frac{\delta}{\delta t}\overline{T}(\alpha_{s}(t+\kappa),\kappa)\bigg|_{\kappa} + \frac{\delta}{\delta \kappa}\overline{T}(\alpha_{s}(t+\kappa),\kappa)\bigg|_{\alpha_{s}}\]

which plugged in the Taylor expansion of \(\overline{T}(\alpha_{s},\kappa)\)

\[\overline{T}(\alpha_{s}(t+\kappa),\kappa) = \overline{T}(\alpha_{s}(t+\kappa),0) - \kappa \frac{\delta}{\delta t}\overline{T}(\alpha_{s}(t+\kappa),0)\bigg|_{\kappa} + \dots\]

allow us to determine the scale-dependent terms at any given order just from the central predictions as

\[\begin{split}\overline{T}_{\text{LO}}(\alpha_{s}(t+\kappa),\kappa) &= T_{\text{LO}}(t + \kappa), \\ \overline{T}_{\text{NLO}}(\alpha_{s}(t+\kappa),\kappa) &= T_{\text{NLO}}(t+\kappa) - \kappa \frac{d}{dt}T_{\text{LO}}(t + \kappa), \\ \overline{T}_{\text{NNLO}}(\alpha_{s}(t+\kappa),\kappa) &= T_{\text{NNLO}}(t+\kappa) - \kappa \frac{d}{dt}T_{\text{NLO}}(t + \kappa) + \frac{1}{2} \kappa^2 \frac{d^2}{dt^2}T_{\text{LO}}(t + \kappa).\end{split}\]

From the last equation is then clearly possible to estimate the MHOU at any given order as \(\Delta(t,k) = \overline{T}(\alpha_{s}(t+\kappa),\kappa) - T(t)\). However, as previously mentioned, there is no unique principle to determine the range of the scale variations, i.e. the value of \(\kappa\). Usually, one varies the renormalization scale by a factor of two, which means \(\kappa \in [-\ln{4}, \ln{4}]\).

Since we are usually interested in processes with one or more hadrons in the initial state, for which the cross-section is factorized into a partonic part and a PDF (or luminosity), we must deal with two sources of independent MHOU:

The uncertainties coming from the expansion of the partonic cross-sections
The uncertainties coming from the expansion of the anomalous dimensions which determine the perturbative evolution of the PDF.

In the next section we will consider both the cases and we will provide the final equations for both electroproduction (i.e. with one incoming hadron) and hadronic processes (i.e. with two incoming hadron). In the anomalous dimensions case, we will also provide three different procedure (schemes) to estimate them.

Scale variation for partonic cross-sections

Electroproduction

Consider the case of electroproduction, such as DIS, with the scale-dependent structure function given by

\[\overline{F}(t,\kappa) = \overline{C}(\alpha_{s}(t+\kappa),\kappa) \otimes f(t).\]

Since we are not varying the scale at which the PDF is evaluated, the RG invariace of the structure function implies the RG invariance of the coefficients function \(\overline{C}(\alpha_{s}(t+\kappa),\kappa)\). Exploiting this property, is then possible to obtain:

\[\overline{C}(\alpha_{s}(t+\kappa),\kappa) = c_{0} + \alpha_{s}(t+\kappa)c_{1} + \alpha_{s}^{2}(t+\kappa)(c2 - \kappa \beta_{0} c_{1}) + \dots\]

where \(\beta_{0}\) is the first term of the perturbative expansion of the beta function and \(c_{i}\) are the coefficients of the perturbative expansion of the scale-independent coefficients function, i.e.

\[C(t) = c_{0} + \alpha_{s}(t)c_{1} + \alpha_{s}^{2}(t)c_{2} + \dots\]

Note that convoluting the scale-varied coefficients function with the PDF lead to an expression which has the same structure of the scale-independent expression. This means evaluating the scale-varied structure function is very straightforward since all that is necessary is to change the coefficients in the perturbative expansion at the central scale.

Hadronic processes

Let’s now consider an hadronic process with scale-varied cross-section given by

\[\overline{\Sigma}(t,\kappa) = \overline{H}(\alpha_{s}(t+\kappa), \kappa) \otimes (f(t) \otimes f(t) ).\]

With the same procedure adopted in the electroproduction case, we can get

\[\overline{H}(\alpha_{s}(t+\kappa),\kappa) = \alpha_{s}^{n}h_{0} + \alpha_{s}^{n+1}(h1 - \kappa n \beta{0} h_{0}) + \dots\]

where this time the perturbative expansion of \(\overline{H}(\alpha_{s}(t+\kappa),\kappa)\) starts at \(\mathcal{O}(\alpha_{s}^{n})\) rather than \(\mathcal{O}(\alpha_{s}^{0})\).

Scale variation for PDF evolution

A completely independent source of MHOU arises from the truncation of the perturbative expansion of the anomalous dimensions governing the evolution of the PDF. Again, this uncertainties can be estimated trough scale variation but, in this case, there are three equivalent ways in which it can be performed: at the level of anomalous dimensions, at PDF level or even at the level of the partonic cross-sections. We will address these different methods as schemes.

Consider a PDF evaluated at the scale \(\mu\), \(f(\mu^2)\). Neglecting all the flavor indices and assuming a Mellin space formalism, the scale dependence of the PDF is fixed by

\[f(\mu^2) = \exp{\bigg(\int^{\mu^2}\frac{d\mu'^2}{\mu'^2}\gamma(\alpha_{s}(\mu'^2))\bigg)}f_{0}\]

where the anomalous dimensions admit the perturbative expansion

\[\gamma(t) = \alpha_{s}(t)\gamma_{0} + \alpha_{s}^{2}(t)\gamma_{1} + \dots\]

With the same definition of the previous part we can define the scale-dependent anomalous dimensions as

\[\overline{\gamma}(\alpha_{s}(t), \kappa) = \gamma(t) - \kappa \frac{d}{dt}\gamma(t) + \dots\]

so that their perturbative expansion is

(1)\[\overline{\gamma}(\alpha_{s}(t+\kappa), \kappa) = \alpha_{s}(t+\kappa)\gamma_{0} + \alpha_{s}^2 (t+\kappa)(\gamma{1} - \kappa \beta_{0}\gamma_{0}) + \dots\]

Then, using this expression, one can estimate the MHOU coming from the perturbative expansion of the anomalous dimensions (this way of doing it will be later called scheme A).

However, the same result can be obtained by scale variation at the PDF level. In fact, inserting the last equation in the PDF evolution equation we get

\[\begin{split}& \exp{\bigg(\int^{t}dt'\overline{\gamma}(\alpha_{s}(t' + \kappa), \kappa)\bigg)} = \exp{\bigg(\int^{t+\kappa}dt'\overline{\gamma}(\alpha_{s}(t'), \kappa)\bigg)} \\ &= \exp{\bigg(\bigg[\int^{t+\kappa}dt'\gamma(t')\bigg] - \kappa\gamma(t+\kappa) + \frac{1}{2}\kappa^2\frac{d}{dt}\gamma(t+\kappa) + \dots\bigg)} \\ &= \bigg[1 - \kappa\gamma(t+\kappa) + \frac{1}{2}\kappa^2(\gamma^2(t+\kappa)+\frac{d}{dt}\gamma(t+\kappa)) + \dots \bigg]\exp{\bigg(\int^{t+\kappa}dt'\gamma(t')\bigg)},\end{split}\]

that can be used to obtain

(2)\[\overline{f}(\alpha_{s}(t+\kappa), \kappa) = [1 - \kappa \gamma(t+\kappa) + \frac{1}{2}\kappa^2(\gamma^2(t+k) + \frac{d}{dt}\gamma(t+\kappa)) + \dots]f(t+\kappa)\]

which is the perturbative expansion of the scale-varied PDF defined as

\[\overline{f}(\alpha_{s}(t+\kappa), \kappa) = \exp{\bigg(\int^{t}dt' \overline{\gamma}(\alpha_{s}(t'+\kappa),\kappa)\bigg)}f_{0}.\]

Equation (2) provides us a way to estimate the MHOU coming from the anomalous dimensions at the PDF level (this way of doing it will be later called scheme B). Moreover, it indicates that the \(\kappa\) dependence can be factorized out of the PDF. Therefore we have yet another way to estimate this MHOU just including this factorized terms in the coefficients functions (this way of doing it will be later called scheme C).

Let’s for example consider electroproduction, the scale-varied structure function assumes the form

(3)\[\begin{split}\hat{F}(t,\kappa) &= C(t)\overline{f}(\alpha_{s}(t+\kappa),\kappa) \\ &= C(t)[1-\kappa\gamma(t+\kappa) + \frac{1}{2}\kappa^2(\gamma^2(t+\kappa)+\frac{d}{dt}\gamma(t+\kappa))+\dots]f(t+\kappa) \\ &= \hat{C}(t,\kappa)f(t+\kappa)\end{split}\]

where the last line is the definition of the scale-varied coefficients functions \(\hat{C}(t,\kappa)\). Note that they are different from the \(\overline{C}(t+\kappa,\kappa)\) because, while the latter are obtained from the variation of the renormalization scale of the hard coefficients functions (and thus they estimate the MHOU coming from the perturbative expansion of the coefficients functions), the former are obtained from the variation of the renormalization scale inside the anomalous dimensions (and thus they estimate completely different MHOU, i.e. the ones coming from the perturbative expansion of the anomalous dimensions).

Using the fact that

\[\frac{d}{dt}\gamma(\alpha_{s}) = \beta(\alpha_{s})\frac{d\gamma}{d\alpha_{s}}\]

we can obtain the explicit perturbative expansion

\[\hat{C}(t,\kappa) = c_{0} + \alpha_{s}(t)(c_{1}-\kappa\gamma_{0})+\alpha_{s}^{2}(t)(c_{2}-\kappa(\gamma_{0}c_{1} + \gamma{1}c_{0}) + \frac{1}{2}\kappa^2 \gamma_{0}(\gamma_{0}+\beta_{0})c_{0})+ \dots\]

Schemes

Let’s now summarize the three different ways of estimating the MHOU coming from the anomalous dimensions

Scheme A: The renormalization scale of the anomalous dimensions is varied directly, as in (1), obtaining their scale-varied version. Then, it is used to compute the evolution operator which will produce the scale-varied PDF. However using this scheme requires refitting the PDF as the scale is varied.
Scheme B: The scale-dependence of the anomalous dimensions is factored out of the PDF, as in (2), in such a way the scale-varied PDF is simply obtained by the product of the central PDF evolved to the varied scale (\(t+\kappa\)) with a term which is function of the central anomalous dimensions computed in the varied scale. In this case there is no need to refit the initial PDF. Moreover, this scheme is the most suited one for VFNS, since the MHOU in the PDF with different numbers of active flavors can each be estimated separately.
Scheme C: The factored scale-dependence of the anomalous dimensions is included in the definition of scale-varied coefficients functions, as in the last line of (3). Then, a scale-varied observable is computed trough the convolution of these scale-varied coefficients functions with the PDF evolved to the varied scale \(t+\kappa\).

Note that, even if these schemes are formally equivalent, they can differ by subleading terms depending on the convention used to truncate the perturbative expansion. In fact, in scheme A some higher order terms of the anomalous dimensions expansion can be retained according to the kind of solution adopted for the evolution equation. In scheme B the exponential has been expanded so that it corresponds to a linearized solution of the evolution equations and in scheme C some terms coming from the cross-expansion of the coefficients functions and the linearized solution of the evolution equations have been dropped.

Adding scale variations to a grid

Since it is possible to compute scale variations terms at a certain perturbative order N+1 just from the knowledge of the central N order, pineko includes a tool to add the required scale variations order to a grid which contain the necessary central orders. The command to run it is:

pineko ren_sv_grid GRID_PATH OUTPUT_FOLDER_PATH MAX_AS NF ORDER_EXISTS

where GRID_PATH is the path of the original grid, OUTPUT_FOLDER_PATH is the folder where the updated grid will be dumped, MAX_AS is the requested perturbative order of the QCD coupling and NF is the number of active flavors one wants to consider when computing the scale variations terms. If the original grid has already all the scale variations terms for the requested perturbative order, pineko will do nothing. If one want to force pineko to overwrite the already existing orders, it is enough to set ORDER_EXISTS to True.