# Utility Classes

Apart from the operator classes, eko also provides some utility classes which are e.g. used in yadism

`eko.thresholds.ThresholdsAtlas`

Implementation of the flavor number scheme and the quark thresholds both for the

`eko.couplings.Couplings`

and the operators

When running in VFNS it is important to specify the number of flavors active at each given scale, since the evolution path can be different depending of the chosen setting. This path is determined by

`eko.thresholds.ThresholdsAtlas.path()`

.Let us consider two examples to better illustrate how it works. Imagine to have a boundary condition

`q2_ref=1`

,`nf_ref=3`

with heavy quarks mass matching scales at:`mc=2`

,`mb=3`

,`mt=4`

and would like to evolve your object (PDF or \(a_s\)) to an higher scale (say`q2_to=49`

). The corresponding`ThresholdsAtlas`

will look like:ThresholdsAtlas [0.00e+00 - 4.00e+00 - 9.00e+00 - 16.00e+00 - inf], ref=1.000000000000001 @ 3

and the evolution path will be given by:

[PathSegment(1 -> 4, nf=3), PathSegment(4 -> 9, nf=4), PathSegment(9 -> 16, nf=5), PathSegment(16 -> 49, nf=6)]

where automatically the number of flavors active in each

`eko.thresholds.PathSegment`

is determined according with the specified matching scales.However some more complicated situations can arise when the boundary conditions are not given with the same prescription of the

`eko.thresholds.ThresholdAtlas`

. Let’s now consider as boundary condition`q2_ref=64`

with`nf_ref=3`

:ThresholdsAtlas [0.00e+00 - 4.00e+00 - 9.00e+00 - 16.00e+00 - inf], ref=64.000000000000001 @ 3

Again we would like to evolve to

`q2_to=49`

but now giving a different number of active flavors to the final scale. Some possible paths are:a path to

`q2_to=49`

with`nf=6`

which is given by the following list of`eko.thresholds.PathSegment`

:[PathSegment(64 -> 4, nf=3), PathSegment(4 -> 9, nf=4), PathSegment(9 -> 16, nf=5), PathSegment(16 -> 49, nf=6)]

and it’s determined according to the list of heavy quark thresholds. This path is default option when the argument

`nf_to`

is not given.A path to

`q2_to=49`

enforcing`nf=4`

:[PathSegment(64 -> 4, nf=3), PathSegment(4 -> 49, nf=4)]

A path to

`q2_to=49`

enforcing`nf=3`

which will simply contain a single`eko.thresholds.PathSegment`

:[PathSegment(64 -> 49, nf=3)]

In the first two cases as first step, you go back to the closest matching scale (closest in nf), running in

`nf=3`

, since that is what has been specified in the boundary conditions. Then you are back in the flavor flow, and you cross all the other scales going according to the prescription given by the`eko.thresholds.ThresholdAtlas`

. While in the third example you go directly to the final scale, since there is no matching scale in the middle. You can notice that`eko.thresholds.ThresholdsAtlas.path()`

are always ordered according to nf and not q2 scales. This property is used in VFNS to determine if the matchings are done for an increasing or decreasing number of light flavors.`eko.couplings.Couplings`

: Implementation of the Strong Coupling`eko.interpolation.InterpolatorDispatcher`

: Implementation of the Interpolation