Coefficient Functions

Overview of coefficient functions structure

The main categories for coefficients the same of Structure Functions, i.e.:

the process (EM/NC/CC)
the kind (F2/L/3)
the heavyness involved (light/charm/bottom/top/total)
but there is a new one: the channel (ns/ps/g), and it is related to the incoming parton:
- if the EW boson it is coupling to a quark line connected to the incoming one, than each PDF it’s contributing proportionally to his charge (e.g.: electric charge for the photon); this is called non-singlet (ns)
- otherwise the line to which the EW boson is coupling it will be detached from the incoming by gluonic lines, and the gluon is flavor blind, so all the charges are summed and all the PDF are contributing the same way; this is called pure singlet (ps)
- eventually: if a gluon is entering all the quarks will couple to the EW boson (if no further restrictions are imposed by the observable, e.g. F2charm), as in the singlet case, and so the charges are summed over; this is called the gluon (g) (because it is the gluon…)
- the parity structure (vectorial-vectorial/axial-axial/vectorial-axial), it is relevant only for the NC, and should be taken into account

Attention

Notice that non-singlet does not refer to any of the evolution basis ones (\(V\), \(T_3\), \(V_3\), \(T_8\), …), and actually it has a non-null projection over the singlet distribution itself.

See Non-singlet definition.

These options set the overall structure of the coefficient functions, and it is reported in the following tables, just considering that the mass corrections for all the flavors (charm/bottom/top) share the same functional form for the coefficient and a further category (the massless limit of the heavy) is needed for variable flavor scheme like FONLL.

NC coefficients
.	light	heavy	asymptotic
F2	[VVM05]	[Hek19]	[FLNR10]
FL	[MVV05]	[Hek19]	[FLNR10]
F3	[MV00]	[Hek19]	.

CC coefficients
.	light	heavy	asymptotic
F2	[MRV08]	[GKR96]	.
FL	[MRV08]	[GKR96]	.
F3	[MVV09]	[GKR96]	.

Distributions

To obtain a physical observable one has to convolve the coefficient functions with the PDF

\[\sigma = \sum_j f_j \otimes c_j = \sum_j \int\limits_x^1 \frac {dz}{z} f_j(x/z) c_j(z)\]

A generic coefficient function will allow for three ingredients:

Regular functions \(r(z)\) that are well behaving, i.e. integrable, for all \(z \in (0,1]\); these typically contain polynomials, logarithms and dilogarithms
Dirac-delta distributions: \(\delta(1-z)\)
Plus distributions: \(\left[g(z)\right]_+\) which have a regulated singularity at \(z\to 1\) and are defined by

\[\int\limits_0^1 \!dz\, f(z) \left[g(z)\right]_+ = \int\limits_0^1\!dz\, \left(f(z) - f(1)\right)g(z)\]

The “plused” function can be a generic function, but in practice will almost always be \(\log^k(1-z)/(1-z)\). The “plused” function has to be regular at \(z=0\). These contributions are related to soft and/or collinear singularities in the physical process.

In order to do the convolution in a generic way we adopt the RSL scheme: i.e. we categorize them by their behavior under the convolution interal. This is needed because of the mismatch in the definitions of the convolution and the plus prescription. Any coefficient function \(c(z)\) can be written in the following way:

\[f \otimes c = \int\limits_x^1 \! \frac{dz}{z} \, f(x/z) c^R(z) + \int\limits_x^1 \! dz \, \left(\frac{f(x/z)}{z} - f(x)\right) c^S(z) + f(x) c^L(x)\]

The remapping of the coefficient function ingredients on to the RSL elements is done in the following way:

Regular functions \(c(z) = r(z)\) contribute only to the regular bit:

\[c^R(z) = r(z)\,,~ c^S(z) = 0 = c^L(x)\]

Dirac delta distributions \(c(z) = \delta(1-z)\) only contribute to the local bit:

\[c^R(z) = 0 = c^S(z)\,,~ c^L(x) = 1\]

“Raw” plus distributions \(c(z) = \left[g(z)\right]_+\) contribute to both the singular and the local bit:

\[c^R(z) = 0\,,~ c^S(z) = g(z)\,,~ c^L(x) = -\int\limits_0^x\!dz\, g(z)\]

derivation

\[\begin{split}f \otimes [g]_+ &= \int\limits_x^1 \frac{dz}{z} f(x/z) \cdot \left[ g(z) \right]_+\\ &= \int\limits_0^1 \frac{dz}{z} f(x/z) \cdot \left[ g(z) \right]_+ - \int\limits_0^x \frac{dz}{z} f(x/z) \cdot \left[ g(z) \right]_+\\ &= \int\limits_0^1\!dz\, \left(\frac{f(x/z)}{z} - f(x)\right) \cdot g(z) - \int\limits_0^x\!dz\, \frac{f(x/z)}{z} \cdot g(z)\\ &= \int\limits_x^1\!dz\, \left(\frac{f(x/z)}{z} - f(x)\right) \cdot g(z) - f(x) \int\limits_0^x\!dz\, g(z)\\ &\Rightarrow c^R(z) = 0\,,~ c^S(z) = g(z)\,,~ c^L(x) = -\int\limits_0^x\!dz\, g(z)\end{split}\]

A product of a regular function and a plus distribution \(c(z) = r(z)\left[g(z)\right]_+\) contributes to all three bits:

\[c^R(z) = (r(z)-r(1))g(z)\,,~ c^S(z) = r(1)g(z)\,,~ c^L(x) = -r(1)\int\limits_0^x\!dz\, g(z)\]

derivation

\[\begin{split}f\otimes c &= \int\limits_x^1 \frac{dz}{z} f(x/z) r(z) \cdot \left[ g(z) \right]_+\\ &= \int\limits_0^1 \frac{dz}{z} f(x/z) r(z) \cdot \left[ g(z) \right]_+ - \int\limits_0^x \frac{dz}{z} f(x/z) r(z) \cdot \left[ g(z) \right]_+\\ &= \int\limits_0^1\! dz \left(\frac{f(x/z)r(z)}{z} - f(x)r(1)\right) \cdot g(z) - \int\limits_0^x\!dz\, \frac{ f(x/z) r(z)}{z} \cdot g(z)\\ &= \int\limits_x^1\! dz \left(\frac{f(x/z)r(z)}{z} - f(x)r(1)\right) \cdot g(z) - f(x) r(1) \int\limits_0^xdz~ g(z)\\ &= \int\limits_x^1\! dz \left(\frac{f(x/z)(r(z)+r(1)-r(1))}{z} - f(x)r(1)\right) \cdot g(z) - f(x) r(1) \int\limits_0^xdz~ g(z)\\ &= \int\limits_x^1\! dz \left(\frac{f(x/z)}{z} - f(x)\right) r(1)\cdot g(z) + \int\limits_x^1\! dz \frac{f(x/z)(r(z)-r(1)))}{z} g(z) - f(x) r(1) \int\limits_0^x\!dz~ g(z)\\ &= \int\limits_x^1 \frac{dz}{ z} f(x/z) r(1)\cdot \left[g(z)\right]_+ + \int\limits_x^1\! dz \frac{f(x/z)(r(z)-r(1)))}{z} g(z)\\ &\Rightarrow c^S(z) = r(1)g(z)\,,~ c^R(z) = (r(z)-r(1))g(z)\,,~ c^L(x) = -r(1)\int\limits_0^x\!dz\, g(z)\end{split}\]

A plus distribution that contains a regular function and a singular function \(c(z) = \left[r(z)g(z)\right]_+\) can be simplified by

\[\left[r(z)g(z)\right]_+ = r(z) \left[g(z)\right]_+ - \delta(1-z) \int\limits_0^1 dy~ r(y) \left[g(y)\right]_+\]

derivation

\[\begin{split}\int\limits_0^1 \!dz~ f(z) \left[r(z)g(z)\right]_+ &= \int\limits_0^1 dz \left(f(z) - f(1)\right)r(z)g(z)\\ &= \int\limits_0^1 \left(f(z)r(z) - f(1)r(1)\right)g(z)~dz - f(1)\int\limits_0^1\! dz(r(z)-r(1))g(z)\\ &= \int\limits_0^1\! dz~ f(z)\left(r(z) \left[g(z)\right]_+\right) - f(z)\left(\delta(1-z)\int\limits_0^1\! dy~ r(y) \left[g(y)\right]_+\right)\end{split}\]

Channels

In this sections there are clarifications about specific channels.