2.5 (Semi-)Analytical Calculations
49
The virtue of SCET is that logarithmically enhanced contributions can be
resummed at all orders in the coupling constant, which can be achieved using
renormalisation group (RG) evolution in the effective field theory. The effective
Lagrangian provides a systematic way of organising computations [271]. Factorisation theorems in SCET can also be constructed for jet substructure observables,
allowing the resummation of non-global logarithms [285]. Calculations in SCET
predictions have been performed for jet substructure observables at the LHC at NLL
accuracy [286, 287] and even next-to-next-to-leading-logarithmic (NNLL) [195,
288] and higher accuracy [211].
2.5.2 Non-perturbative Effects
Non-perturbative effects are rarely taken into account in analytical calculations due to
the inherent complications when approaching energy scales close to hadron masses
O(( QCD ). However, the impact of non-perturbative effects on jet observables as
been estimated analytically using power corrections [186]. Results were obtained for
inclusive jet production near the partonic threshold, considering gluon emission and
separating the phase space into a perturbative and a non-perturbative region, defined
by an infrared factorisation scale μ I . Below this scale, in the non-perturbative regime,
the strong coupling α S is replaced by an effective, finite constant. This allows for the
phase space integration to be performed down to vanishing scales. The contribution
that would be included in perturbative calculations is subtracted, leaving a purely
non-perturbative result that can be associated with the effects of hadronisation and
the underlying event.
The expected modification of a jet’s transverse momentum due to hadronisation
is [186]
δp T h = 2C R A(μ I )M
−
1
R
+ O(R)
,
(2.44)
where C R is the colour factor appropriate for the parton initiating the jet, i.e. C R = C F
for quark jets and C R = C A for gluon jets. The hadronisation scale A(μ I ) is related
to event shape studies, and a rough estimate gives 2C F A(2 GeV) ≈ 0.5 GeV. The
Mellin factor M is an algorithm-dependent quantity with M = 1.49 for the anti-k T
algorithm and M = 1.01 for the k T algorithm [289]. The singular 1/R behaviour
originates from the increase of momentum loss as the jet becomes narrower, which
also explains the negative sign of this term in (2.44).
Corrections due to the underlying event can be calculated from emissions of
dipoles not involving the outgoing jet and result in a change in p T proportional to
R
2 , i.e. proportional to the jet area [186],
δp T UE =
UE
2
R
2
− R
4
/8 + O(R
6
)
.
(2.45)
49
The virtue of SCET is that logarithmically enhanced contributions can be
resummed at all orders in the coupling constant, which can be achieved using
renormalisation group (RG) evolution in the effective field theory. The effective
Lagrangian provides a systematic way of organising computations [271]. Factorisation theorems in SCET can also be constructed for jet substructure observables,
allowing the resummation of non-global logarithms [285]. Calculations in SCET
predictions have been performed for jet substructure observables at the LHC at NLL
accuracy [286, 287] and even next-to-next-to-leading-logarithmic (NNLL) [195,
288] and higher accuracy [211].
2.5.2 Non-perturbative Effects
Non-perturbative effects are rarely taken into account in analytical calculations due to
the inherent complications when approaching energy scales close to hadron masses
O(( QCD ). However, the impact of non-perturbative effects on jet observables as
been estimated analytically using power corrections [186]. Results were obtained for
inclusive jet production near the partonic threshold, considering gluon emission and
separating the phase space into a perturbative and a non-perturbative region, defined
by an infrared factorisation scale μ I . Below this scale, in the non-perturbative regime,
the strong coupling α S is replaced by an effective, finite constant. This allows for the
phase space integration to be performed down to vanishing scales. The contribution
that would be included in perturbative calculations is subtracted, leaving a purely
non-perturbative result that can be associated with the effects of hadronisation and
the underlying event.
The expected modification of a jet’s transverse momentum due to hadronisation
is [186]
δp T h = 2C R A(μ I )M
−
1
R
+ O(R)
,
(2.44)
where C R is the colour factor appropriate for the parton initiating the jet, i.e. C R = C F
for quark jets and C R = C A for gluon jets. The hadronisation scale A(μ I ) is related
to event shape studies, and a rough estimate gives 2C F A(2 GeV) ≈ 0.5 GeV. The
Mellin factor M is an algorithm-dependent quantity with M = 1.49 for the anti-k T
algorithm and M = 1.01 for the k T algorithm [289]. The singular 1/R behaviour
originates from the increase of momentum loss as the jet becomes narrower, which
also explains the negative sign of this term in (2.44).
Corrections due to the underlying event can be calculated from emissions of
dipoles not involving the outgoing jet and result in a change in p T proportional to
R
2 , i.e. proportional to the jet area [186],
δp T UE =
UE
2
R
2
− R
4
/8 + O(R
6
)
.
(2.45)