jets produced in $e^{+}e^{-}$ annihilation, both in the limit of small jet
radius $R\ll 1$ in Chapter 4 and including the full $R$ dependence in Chapter
5. In the current work we shall lift the approximation of small $R$, meaning
that our calculations should be useful for jets of any radius .

To address the correlated emission term at leading-order we need to consider
the case where one has two energy-ordered soft gluons $k_{1}$ and $k_{2}$ such
that, for instance, $\omega_{1}\gg\omega_{2}$, where $\omega_{1}$ and
$\omega_{2}$ are the respective energies. In previous work involving
$e^{+}e^{-}$ annihilation we have addressed this situation by using the fact
that the emission of such a two-gluon system off a hard $q\bar{q}$ dipole is
described by an independent emission term with colour factor
$\mathrm{C_{F}^{2}}$ and a correlated emission term with colour factor
$\mathrm{C_{F}}\mathrm{C_{A}}$. However, in the present case of multiple hard
partons, the emission of a two-gluon system is in principle more involved,
since there are several emitting dipoles. In practice, the structure of two-
parton emission for the cases we study in this chapter (i.e., up to $n=4$ hard
legs) is known to be remarkably simple [240, 241].

As an illustrative example, we consider again the Z+jet case, with three hard
legs and the Born channel $\delta_{1}$, with an outgoing quark jet. The
squared matrix element once again contains an independent emission piece,
which contributes to the exponentiation of the single gluon result, described
by our function $\mathcal{R}$. This leaves the correlated parton emission term
which has the structure

$W^{\mathrm{corr.}}(k_{1},k_{2})=\frac{\mathrm{N_{c}}}{2}w_{12}^{\left(2\right)}+\frac{\mathrm{N_{c}}}{2}w_{23}^{\left(2\right)}-\frac{1}{2\mathrm{N_{c}}}w_{13}^{\left(2\right)},$
(7.34)

where $w_{ij}^{\left(2\right)}$ is the correlated two-gluon emission by a
dipole $ij$, and which is the same as for the $q\bar{q}$ case studied in
$e^{+}e^{-}$ annihilation. Hence the dipole emission and associated colour
structure for a correlated two-parton system is precisely the same as for a
single gluon emission [241].

Next we note that, as described for example in Ref. [241], a piece of the
correlated two-parton emission contribution actually goes to build up the
running coupling (which we have already considered in the exponentiated single
gluon contribution $\mathcal{R}$). This piece comprises of gluon splittings
into equally hard gluons or into a $q\bar{q}$ pair, which together produce the
leading term of the QCD $\beta$ function 222For a non-global observable, such
hard splittings will also give rise to non-global logarithms below the single
logarithmic level, as shown in Chapter 5, which are subleading from our point
of view.. This leaves us to consider only the soft part of the correlated
emission $\mathcal{S}$ [242], which describes the production of an energy-
ordered two-gluon system. For a global observable, as is well-known, this term
produces no effect at single-logarithmic accuracy whereas in the present case
it provides us with the first term of the non-global contribution. For a
general dipole $ij$ we can explicitly write

$\mathcal{S}_{ij}=2\mathrm{C_{A}}{\bf T}_{i}.{\bf
T}_{j}\left(A_{ab}+A_{ba}-A_{a}A_{b}\right),$ (7.35)

where

$A_{ab}=\frac{(ij)}{(ik_{a})(k_{a}k_{b})(k_{b}j)}\,,\quad
A_{a}=\frac{(ij)}{(ik_{a})(jk_{a})}.$ (7.36)

We note that this term is free of collinear singularities along the hard legs
$a$ and $b$ due to cancellation between the various terms of Eq. (7.35). The
remaining collinear singularity from the $1/\left(k_{1}.k_{2}\right)$ ,
involving the soft gluons, shall turn out to be integrable for the non-global
configurations considered below. Thus to obtain the leading non-global term
for the jet mass under consideration, it suffices to carry out dipole
calculations using the Eqs. (7.35) and (7.36), for each of the hard emitting
dipoles, just as we did for the single gluon emission case. The results we
obtain are mentioned and commented on below with the details on the
calculation in Appendix E.2. We report below the coefficients $I_{ij}$ of the
non-global single logarithms:

$\int[\textrm{\,d}k_{1}][\textrm{\,d}k_{2}]\,\mathcal{S}_{ij}\Theta_{k_{1}\notin
J}\Theta_{k_{2}\in J}\equiv\mathrm{C_{A}}{\bf T}_{i}.{\bf
T}_{j}\,I_{ij}\ln^{2}\frac{1}{v}.$ (7.37)

  * •

dipoles involving the measured jet

As was demonstrated in our previous work (Chapters 4 and 5), for such dipoles
one obtains333Notice that Eqs. (7.38) and (4.19) differ in the sign of the
$R^{4}$ coefficient.:

$I_{13}=I_{23}\simeq I_{34}=\frac{\pi^{2}}{3}+\mathcal{O}(R^{4}).$ (7.38)

The dipole $I_{34}$, which is relevant for the dijet calculation, depends, in
principle, on the rapidity separation $\Delta y$ between the leading jets.
However any $\Delta y$ dependence will be associated with powers of R that
appear to make a negligible contribution.

We note that the result $\pi^{2}/3$ is exact for $e^{+}e^{-}$ collisions,
where one defines the jet in terms of a polar angle. This result is precisely
the same as was obtained for hemisphere jet masses in $e^{+}e^{-}$
annihilation and implies that the result obtained does not depend on the
position of the jet boundary, a feature that has also been observed in [226].

  * •

in-in dipole

Here we are considering a situation where the emitting dipole legs are away
from the interior of the jet region into which the softest gluon $k_{2}$ is
emitted. This situation is reminiscent of the much studied case of non-global
logarithms in energy flow into gaps between jets [71, 176]. In Ref. [176] for
instance, energy flow into an inter-jet region (i.e., a region between but
away from the hard legs of an emitting dipole) was considered for specific
choices of the geometry of this region, such as a rapidity slice and a patch
in rapidity and azimuth. In the present case we are certainly studying the
same problem but the region being considered here is a circle
$(\eta-y)^{2}+\phi^{2}$, centred on the measured jet. The result we obtain for
the non-global contribution due to the in-in dipole has, in the small-$R$
limit, the analytical form

$I_{12}\approx 4\left[1.17R^{2}-R^{2}\ln
2R+\mathcal{O}\left(R^{4}\right)\right].$ (7.39)

We note that the $R^{2}$ behaviour arises simply from integrating the emitted
softest gluon $k_{2}$ over the interior of the jet region while the $\ln R$
behaviour is a reflection of the collinear singularity we mentioned above,
between the two soft gluons $k_{1}$ and $k_{2}$. In practice the above small
$R$ approximation is not a good approximation to the full result obtained
numerically, for larger values of $R$ ($R\gtrsim 1$), hence eventually we use
the numerical answer rather than the form above.

  * •

in-recoil dipole

Here again the situation is similar to the case of the interjet energy flow
with the only difference from the previous dipole being the geometry of the
hard dipole legs, which are now formed by an incoming parton and an outgoing
parton at a finite rapidity with respect to the beam. The analytical result we
find reads

$I_{14}\approx\frac{\left(1+e^{\Delta y}\right)^{2}}{\left(1+\cosh\Delta
y\right)^{2}}\left(1.17R^{2}-R^{2}\ln 2R\right)+\frac{(1+e^{\Delta
y})}{1+\cosh\Delta y}\kappa(\Delta y)R^{2}.$ (7.40)

where $\kappa$ depends on rapidity difference $\Delta y=y-y_{r}$ and is
evaluated numerically. In the limit $\Delta y\to\infty$, $\kappa(\Delta y)\to
0$ and one recovers the previous case. The dipole $I_{24}$ is easily obtained
as

$I_{24}=I_{14}(-\Delta y).$ (7.41)

Note that here too we use the numerical result instead of the formula given
above.

####  7.4.2 All-orders treatment

In order to perform a complete (NLL) resummation of non-global logarithms, one
would need to consider the emission of a soft gluon from an ensemble of any
number of gluons. This problem can be treated only in the
large-$\mathrm{N_{c}}$ limit [2, 71]444An alternative approach that can be
found in the literature consists of an expansion in the number of out-of-jet
(in this case), gluons, keeping the full colour structure, see for instance
[239, 243, 244, 245].. For this study, we have adapted the dipole-evolution
code used in [2] to perform the resummation in the case of jet-masses.

The code developed in Ref. [2] handled the case of evolution off a hard
primary dipole in the leading-$\mathrm{N_{c}}$ limit. The result for the non-
global contribution $S(t)$ was obtained by dividing the resummed result by the
contribution of primary emissions alone, off the hard emitting dipole. For our
work here we have a situation with several possible emitting dipoles. In this
situation one has to resort to the large-$\mathrm{N_{c}}$ limit in which one
can treat the problem as independent evolution of only the _leading colour-
connected dipoles in the hard process_. Detailed formulae can be found in Ref.
[71].

In the Z+jet case, which is simpler, we have noted that there is no visible
difference arising from considering just the leading colour-connected dipoles
in the hard process relative to the case where one evolves _all_ hard dipoles
using the evolution code. We choose the latter option here hence write the
full resummed expression, with the exception of the contributions coming from
the constant terms $C_{1}$, as

$\Sigma(v)=\sum_{\delta}\int
d\mathcal{B}\frac{d\sigma^{(\delta)}_{0}}{d\mathcal{B}}f^{(\delta)}_{\mathcal{B},{\rm
global}}f^{(\delta)}_{\mathcal{B},{\rm non-global}}\mathcal{H(B)}.$ (7.42)

The resummation of non-global logarithms, including contributions from the
colour suppressed hard dipoles, is encoded in the two terms

$\displaystyle f^{(\delta_{1})}_{\rm non-global}$ $\displaystyle=$
$\displaystyle\exp\left(-\mathrm{C_{A}}\mathrm{C_{F}}\frac{\pi^{2}}{3}f_{13}(t)-\frac{\mathrm{C_{A}^{2}}}{2}I_{12}(R)f_{12}(t)\right),$
(7.43) $\displaystyle f^{(\delta_{2})}_{\rm non-global}$ $\displaystyle=$
$\displaystyle\exp\left(-\mathrm{C_{A}^{2}}\frac{\pi^{2}}{3}f_{13}(t)-\mathrm{C_{A}}\left(\mathrm{C_{F}}-\frac{\mathrm{C_{A}}}{2}\right)I_{12}(R)f_{12}(t)\right),$
(7.44)

where we have used the fact that $I_{13}=I_{23}$; the other contribution
$I_{12}$ does depend on $R$ and vanishes in the $R\to 0$ limit, where one
recovers the picture of jets evolving independently. As we stated before,
taking the large-$\mathrm{N_{c}}$ limit of the non-global contributions would
amount to switching off the contribution from the colour suppressed dipoles
or, equivalently, choosing $\mathrm{C_{F}}=\mathrm{C_{A}}/2$ in the above.
This produces no significant difference to our results but the result written
above has the advantage that it includes the the full contribution to
$\mathcal{O}(\alpha_{s}^{2})$ non-global coefficient. We have defined

$f_{ij}(t)=\frac{1+(a_{ij}t)^{2}}{1+(b_{ij}t)^{c_{ij}}}t^{2}\,,\quad
t=\frac{1}{4\pi\beta_{0}}\ln\left(1-2\alpha_{s}(p_{T})\beta_{0}\ln\frac{R^{2}}{v}\right).$
(7.45)

The coefficients $a_{ij},b_{ij},c_{ij}$ are obtained by fitting the functional
form above to the numerical results from the large-$\mathrm{N_{c}}$ dipole-
evolution code. Numerical results are reported in Appendix E.3.

Following a similar method, we obtain the corresponding result for the dijet
case:

$\Sigma(v)=\sum_{\delta,J=3}^{4}\int d\mathcal{B}\,{\rm
tr}\left[\frac{H_{\delta}e^{-\left(\mathcal{G}_{\delta,J}+\gamma_{E}\mathcal{G}^{\prime}_{\delta,J}+\mathcal{S}_{\delta,J}\right)^{\dagger}}e^{-\mathcal{G}_{\delta,J}-\gamma_{E}\mathcal{G}^{\prime}_{\delta,J}-\mathcal{S}_{\delta,J}}+(\Delta
y\leftrightarrow-\Delta
y)}{\Gamma\left(1+2\mathcal{G}^{\prime}_{\delta,J}\right)}\right]\\\
\times\mathcal{H(B)},$ (7.46)

with

$\displaystyle\mathcal{S}_{\delta,3}$ $\displaystyle=$
$\displaystyle\Big{[}\frac{\mathrm{C_{A}}}{2}\Big{(}\frac{\pi^{2}}{3}{\bf
T}_{3}^{2}f_{13}(t)-{\bf T}_{1}.{\bf T}_{2}I_{12}(R)f_{12}(t)-{\bf T}_{1}.{\bf
T}_{4}I_{14}(R,\Delta y)f_{14}(t)$ $\displaystyle-{\bf T}_{2}.{\bf
T}_{4}I_{24}(R,\Delta y)f_{24}(t)\Big{)}\Big{]},$
$\displaystyle\mathcal{S}_{\delta,4}$ $\displaystyle=$
$\displaystyle\Big{[}\frac{\mathrm{C_{A}}}{2}\Big{(}\frac{\pi^{2}}{3}{\bf
T}_{4}^{2}f_{13}(t)-{\bf T}_{1}.{\bf T}_{2}I_{12}(R)f_{12}(t)-{\bf T}_{2}.{\bf
T}_{3}I_{14}(R,\Delta y)f_{14}(t)$ (7.47) $\displaystyle-{\bf T}_{1}.{\bf
T}_{3}I_{24}(R,\Delta y)f_{24}(t)\Big{)}\Big{]}.$

As before the above expressions capture the full colour structure of the non-
global contribution at $\mathcal{O}(\alpha_{s}^{2})$, but beyond that are
valid only in the large-$\mathrm{N_{c}}$ limit.

###  7.5 Z+jet at the LHC

In this section we investigate the numerical impact of the different
contributions which are relevant in order to achieve NLL accuracy. We decide
to study the differential distribution $\textrm{\,d}\sigma/\textrm{\,d}\zeta$,
where $\zeta=\sqrt{v}=m_{J}/p_{T}$, so as to study the jet mass distribution
directly rather than the squared jet mass. We also find it useful to work with
a dimensionless ratio to better separate soft physics contributions. In fact,
a fairly large value of the jet mass can be generated by the emission of a
very soft gluon, if the transverse momentum of the hard jet is large, while
small values of $\zeta$ always correspond to the emission of soft and/or
collinear gluons. If not stated otherwise, we normalise our curves to the Born
cross-section. We use the matrix element generator comix [246], included in
sherpa [207, 208], to produce all the tree-level cross-sections and
distributions. We consider proton-proton collisions at $\sqrt{s}=7\,\
mathrm{TeV}$ and select events requiring $p_{T}>200$ GeV; the Z boson is
produced on-shell and does not decay. Jets are defined using the anti-kT
algorithm [73]. In our calculation we use the set of parton distribution
function cteq6m [247], with renormalisation and factorisation scales fixed at
$\mu_{R}=\mu_{F}=200$ GeV, to ease the comparison with different Monte Carlo
parton showers, which we are going to perform in Section 7.5.4.

####  7.5.1 Different approximations to the resummed exponent

We start by considering different approximations to the resummed exponent
$f^{(\delta)}_{\mathcal{B}}$. We present our results for two different jet-
radii: $R=0.6$, in Fig. 7.1, and $R=1.0$, in Fig. 7.2. The blue curve
corresponds to the most simple approximation to the NLL resummed exponent: the
jet-function. This approximation correctly resums soft and collinear radiation
as well as hard collinear, but does not capture all soft radiation at large
angle. In particular this corresponds to neglecting terms that are suppressed
by powers of the jet radius in the resummed exponent Eq. (7.3.1). These terms
are included in the resummation of all global contributions (green line). We
have checked that inclusion of $\mathcal{O}(R^{2})$ is enough because the
$\mathcal{O}(R^{4})$ corrections are below the percent level even for $R=1.0$.
We stress that up to this point no approximation on the colour structure has
been made, although we have checked that sub-leading colour corrections are
small, once the collinear part has been properly treated. Finally, in the red
curve we also take into account the resummation of non-global logarithms as
described by Eq. (7.42). The first, $\mathcal{O}(\alpha_{s}^{2})$, coefficient
on the non-global contribution is computed exactly, while the subsequent
resummation is performed using a numerical dipole-evolution code in the
large-$\mathrm{N_{c}}$ limit. We note that the inclusion of
$\mathcal{O}(R^{2})$ terms in the resummed exponent as well as non-global
logarithms, noticeably corrects the simple jet-function picture, based on
collinear evolution. The peak height is reduced by more than 30% for $R=0.6$
and it is nearly halved for $R=1.0$. The effect of non-global logarithms is
reduced in the latter case, but the $\mathcal{O}(R^{2})$ corrections to the
jet-function approximation become bigger.

Figure 7.1: Comparison between different approximations to the resummed
exponent: jet functions (blue), with full resummation of the global
contribution (green) and with non-global logarithms as well (red). The jet
radius is $R=0.6$. Figure 7.2: Comparison between different approximations to
the resummed exponent: jet functions (blue), with full resummation of the
global contribution (green) and with non-global logarithms as well (red). The
jet radius is $R=1.0$.

####  7.5.2 Matching to fixed order

We now turn our attention to obtaining a jet-mass distribution which is
reliable for all values of $\zeta$. We achieve this by matching to a fixed-
order (FO) calculation. Although a complete phenomenological analysis would
require matching to a next-to leading order (NLO) QCD calculation, for the
purpose of this chapter we limit ourselves to LO matching. We compute the
differential jet-mass distribution at $\mathcal{O}(\alpha_{s})$ using comix.
The result is plotted in Fig. 7.3 (dashed black line): the differential
distribution $\textrm{\,d}\sigma/\textrm{\,d}\ln\zeta$ diverges
logarithmically in the small-mass limit. The dotted green line instead
represents the resummed result. The matched curve (shown in solid red) is
obtained by straightforwardly adding the two contributions and removing the
double-counted terms, i.e., the expansion of the resummation to
$\mathcal{O}(\alpha_{s})$:

$\frac{1}{\sigma}\frac{{\textrm{\,d}\sigma}_{\rm
NLL+LO}}{\textrm{\,d}\ln\zeta}=\frac{1}{\sigma}\left[\frac{\textrm{\,d}\sigma_{\rm
LO}}{\textrm{\,d}\ln\zeta}+\frac{\textrm{\,d}\sigma_{\rm
NLL}}{\textrm{\,d}\ln\zeta}-\frac{\textrm{\,d}\sigma_{{\rm
NLL},\alpha_{s}}}{\textrm{\,d}\ln\zeta}\right].$ (7.48)

We note that the matched result coincides with the resummation at small
$\zeta$, because the logarithmically divergent contributions to the LO
distribution are cancelled by the expansion of the resummation. Moreover, the
matched distribution follows the LO order one in the opposite limit. In
particular, we note that the LO result exhibits an end-point:

$\zeta^{2}=\frac{m_{J}^{2}}{p_{T}^{2}}=\frac{2p_{t}k_{t}}{|\underline{p}_{t}+\underline{k}_{t}|^{2}}\left(\cosh
y-\cos\phi\right)=\frac{2p_{t}k_{t}}{p_{t}^{2}+k_{t}^{2}+2p_{t}k_{t}\cos\phi}\left(\cosh
y-\cos\phi\right),$ (7.49)

which leads to

$\zeta_{\rm max}=\sqrt{\max_{y^{2}+\phi^{2}\leq
R^{2}}\zeta^{2}}=\tan\frac{R}{2}=\frac{R}{2}+O(R^{3}).$ (7.50)

This feature is not present in the resummed distribution, because the jet does
not recoil against the emission of the eikonal gluon, but it is restored
thanks to matching.

Figure 7.3: Matching of the NLL resummed distribution to the LO one for
$R=0.6$ (on the left) and $R=1.0$ (on the right).

####  7.5.3 Numerical estimate of constant term $C_{1}$

In this section we investigate the numerical impact of the constant $C_{1}$
defined in Eq. (7.32), in the case of Z+jet events, where we measure the mass
of the highest $p_{\rm T}$ jet. Different contributions build up the constant
terms at $\mathcal{O}(\alpha_{s})$: we have one-loop virtual corrections,
terms that cancel the renormalisation and factorisation scale dependence of
the Born pieces and contributions with multiple partons in final state that
may, or may not, end up in the same jet.

One-loop virtual corrections have the same kinematical and flavour structure
as the corresponding Born subprocess $\delta$ and, consequently, they need to
be suppressed by the same resummed exponent. In order to cancel infrared
divergences, one needs to consider real emissions in the soft and collinear
limit as well. The final-state singularities are precisely the source of the
logarithms we are resumming and these configurations can be mapped onto one of
the Born subprocesses $(\delta)$. Initial-state collinear singularities, which
do not give rise to any logarithms of the jet mass, must be absorbed into the
parton densities, leaving behind terms that depend on the factorisation scale.
Finally, we also need to consider kinematic configurations where the two
final-state partons are not recombined, resulting into Z+2 jet events, and
suppress the hardest one with the appropriate exponent. Thus, we should
perform the calculation of the NLO cross-section that appears in Eq. (7.32)
keeping track of the kinematics of the final-state partons, in order to
separate between $g$ and $q$ components. Although this can be clearly done by
computing these corrections analytically, for the current analysis we use the
program mcfm [182], which does not trivially allow us to do so. Nevertheless,
we are able to compute the finite part of the virtual corrections, for the
different initial-state channels. We suppress virtual corrections, as well as
the integrated term in Eq. (7.32) with their appropriate form factor,
$f^{(\delta)}$. We then multiply all the remaining real corrections by either
a gluon or a quark form factor, producing a band.

Our findings are plotted in Fig. 7.4, for $R=0.6$, on the left, and $R=1.0$,
on the right555We have suppressed $C_{1}$ with the full global resummed
exponent, producing terms beyond our NLL accuracy, which we do not control. A
more precise analysis would involve the complete determination of $C_{1}$,
suppressed only with double-logarithmic terms, together with an uncertainty
band, assessing the impact of higher logarithmic orders.. When $C_{1}$ is
included, we normalise the distribution to the total NLO rate, rather than the
usual Born cross-section. In order to avoid large NLO corrections [248], we
put a cut on the Z boson transverse momentum $p_{TZ}>150$ GeV. We have found
that this leads to $K$-factor of $1.45$ and $1.57$, for $R=0.6$ and $R=1.0$,
respectively. With this set of cuts, the impact of $C_{1}$ is not too big, but
definitely not negligible. The complete calculation of this constant is
therefore necessary in order to be able to perform accurate phenomenology.

Figure 7.4: The impact of the NLL constant $C_{1}$, for $R=0.6$ jets (on the
left) and $R=1.0$ (on the right). The band is produced by suppressing the real
radiation contributions with a quark or gluon jet form factor, as explained in
the text.

####  7.5.4 Comparison to Monte Carlo event generators

In this section we compare our resummed and matched result NLL+LO to three
standard Monte Carlo event generators: sherpa [208], pythia8 [206] and
herwig++ [249]. Monte Carlo parton showers are powerful tools to simulate
complicated final states in particle collisions. When interfaced with
hadronisation models, they are able to describe the transition between partons
and hadrons. Moreover, they provide events which are fully differential in the
particles’ momenta. Further, they provide models of other non-perturbative
effects at hadron colliders such as the underlying event. As stated before
however, despite their usefulness and successes, it is quite difficult to
assess the theoretical precision of these tools. For this reason, comparisons
between parton showers and analytic calculations, which have a well-defined
theoretical accuracy, form an important part of QCD phenomenology.

For the case of jet-mass it has been noted in recent ATLAS studies [34] for
jet masses that pythia with hadronisation and the underlying event gives a
reasonably good description of the data. We would therefore expect our
resummation to be in some accordance with pythia, though we should stress that
we do not include any non-perturbative effects. Hence we compare our results
to the parton shower aspect of the various event generators on its own. While
this is in principle correct, in practice one should be aware that there can
be considerable interplay between the shower level and the non-perturbative
effects in various event generators so that these programs may only return
more meaningful results when all physical effects (perturbative and non-
perturbative) are considered together. We should bear this caveat in mind
while attempting to compare a resummed prediction with just the parton shower
models in event generators.

The results of the comparison are shown in Fig. 7.5. Our NLL+LO result for
$R=0.6$ is shown in red (the band represents the uncertainty due to the
incomplete determination of $C_{1}$). The Monte Carlo results are obtained
with the same parton densities as in the resummed calculation and the same set
of cuts. For sherpa (blue line) and pythia (green line) we fix
$\mu_{R}=\mu_{F}=200$ GeV, while for herwig++ (magenta line) we use the
default transverse mass of the Z boson. At the shower level, sherpa and
herwig++ appear to perform quite similarly. They produce fairly broad
distributions, which are not very much suppressed as $\zeta\to 0$. The
distribution obtained from pythia instead produces a curve which is much
closer to our resummed result. Although the position of the peaks differ by
$\delta\zeta=0.01$ ($\delta m_{\rm J}\sim 2$ GeV), the height and general
shapes appear in agreement.

The agreement between the different Monte Carlo generators is restored when
hadronisation corrections are included, as demonstrated in Fig. 7.6: pythia
and herwig++ produce very similar results, while the distribution obtained
with sherpa is broader, but not too different. Clearly, in order to compare to
collider data, one must also include the contribution from the underlying
event.

It is also interesting to compare the resummed prediction we computed in this
chapter with a shift to the right to account for hadronisation corrections to
the event generator results after hadronisation. The shift approximation,
initially suggested in [83], should be valid to the right of the Sudakov peak
but will certainly break down in the vicinity of the peak (see also the
discussion in [70] and references therein). The amount of the shift is related
to the non-perturbative correction to mean values of jet masses derived in
[80]. From that reference, we note that the $v=m_{J}^{2}/p_{T}^{2}$
distribution should be shifted by an amount $\alpha R/p_{T}$ (the correction
to the mean value) with a dimensionful coefficient $\alpha$ that can be taken
to be of the order of a few times $\Lambda_{\rm QCD}$. The results of our
calculations with a non-perturbative shift are shown in Fig. 7.6. From there
one notes that a shift of $1.5\,\mathrm{GeV}\,R/p_{T}$ ( for the $v$
distribution which we carry over to the $\zeta$ distribution plotted in Fig.
7.6) where we take $p_{T}$ to be the value of the lower bound (200 GeV in this
case) on transverse momentum, yields an excellent agreement with the herwig++
result after hadronisation. On the other hand, a slightly larger shift of
$2.0\,\mathrm{GeV}\,R/p_{T}$ yields a good agreement with pythia. We have
truncated the shifted resummed result near the peak of the resummed
distribution, as we would expect the shift not be meaningful beyond this
region at the very best. Although we have made a crude estimate of
hadronisation effects, and one may be able to better compute these
corrections, we do note that within non-perturbative uncertainties our results
are compatible with the most widely used event generator models. We therefore
anticipate that after the further improvements we have in mind are
accomplished, our results may be directly used for phenomenological purposes.

Figure 7.5: Comparison of our resummed and matched result NLL+LO (in red) to
standard Monte Carlo event generators, at the parton level. Figure 7.6:
Results for the $\zeta$ distributions obtained with standard Monte Carlo
parton showers, with hadronisation corrections (dashed lines) compared to
analytical resummation with non-perturbative shifts (shaded bands) as
explained in the main text.

###  7.6 Dijets at the LHC

In this section we provide numerical predictions for the jet mass distribution
in dijet events. As before, we consider proton-proton collision at
$\sqrt{s}=7$ TeV, with jets defined according to the anti-kT algorithm [73].
The main complication with respect to the Z+jet case previously discussed is a
more complicated colour algebra, which results into a matrix structure of
large-angle soft gluon radiation. In order to simplify our resummed
calculation, we work at fixed kinematics, i.e., we demand the jets’ transverse
momentum $p_{T}$ to be $200$ GeV and their rapidity separation to be $|\Delta
y|=2$ (at Born level we only have two jets). We remind the reader that we
consider

$\frac{1}{\sigma}\frac{{\textrm{\,d}\sigma}}{\textrm{\,d}\zeta}=\frac{1}{\sigma}\left(\frac{\textrm{\,d}\sigma}{\textrm{\,d}\zeta_{1}}+\frac{\textrm{\,d}\sigma}{\textrm{\,d}\zeta_{2}}\right)_{\zeta_{1}=\zeta_{2}=\zeta}.$
(7.51)

We match the resummation to a LO calculation of the jet mass distribution
obtained with nlojet++ [233], according to Eq (7.48). In Fig. 7.7 we plot our
NLL+LO result with different approximation for the resummed exponent Eq.
(7.46): jet-functions (blue), with the inclusion of $\mathcal{O}(R^{2})$
corrections (green) and non-global logarithms (red), as explained in detail
for the Z+jet case. Although the corrections to the jet-function approximation
are less pronounced than in the Z+jet case, they are still sizeable and must
be taken into account. In our understanding, the perturbative part of the
resummed result of Ref [53, 54] is precisely the one captured by the jet-
functions (plus an approximated treatment of the one-loop constant $C_{1}$).

In order to obtain a more realistic prediction, one would need to integrate
over the appropriate cuts in transverse momentum and rapidity and match to
NLO, which, in principle, should not pose any difficulties. More delicate is
instead the determination of the constant $C_{1}$, although this issue has
been addressed in Ref. [31, 32] for the case of global event-shapes.

Figure 7.7: The NLL+LO jet mass distribution for dijets, with different
approximation to the resummed exponent.

###  7.7 Conclusion

In this chapter we have provided resummed expressions to NLL accuracy for jet-
mass distributions in hadron-hadron collisions, for jets of arbitrary radius
and defined in the anti-kT algorithm. In particular, we have considered Z
boson production in association with a jet and jet masses in dijet production.
We have improved upon existing studies of jet masses and shapes (many of which
use $e^{+}e^{-}$ annihilation as a model for jets produced in hadron collision
processes) by incorporating initial state radiation effects as well as
accounting for non-global logarithms in the large-$\mathrm{N_{c}}$ limit. We
have matched our results to leading-order predictions to account for non-
logarithmic terms to this order and have commented on the role of the
coefficient function $C_{1}\alpha_{s}$ (corrections of order $\alpha_{s}$ to
the basic resummation off the Born configuration). Finally, we have compared
our results to all the leading Monte Carlo event generators pythia, sherpa and
herwig++, with and without non-perturbative hadronisation corrections.

We have found that: first, ISR and non-global logarithms play an important
role even at relatively small values of jet radius such as $R=0.6$ and hence
cannot be ignored while discussing inclusive jet shapes at hadron colliders.
Our calculations for non-global logarithms both at fixed-order and at all
orders represent to our knowledge the first attempt to calculate these terms
beyond the simpler cases of $e^{+}e^{-}$ annihilation and DIS. Second, on
comparing our results to event generators, widely used for phenomenology, we
find that at the purely perturbative level the best agreement is with the
pythia shower, with an apparent small shift accounting for much of the
difference between the analytical resummation and the parton shower estimate.
The differences with sherpa and herwig++ on the other hand are more marked
especially towards smaller jet masses as one approaches the peak of the
distributions. After hadronisation corrections are applied in the event
generators and also applying a shift to the analytical resummation to account
for hadronisation, we are able to obtain very good agreement with pythia and
herwig++ with slightly different shifts required in either case, for jet mass
values to the right of the Sudakov peak of the distribution. For smaller jet
masses we do not expect a simple shift of the analytical resummation to
reproduce the correct result and we observe here discrepancy with all Monte
Carlo generators which is to be expected. However, this region of very small
jet masses of the order of a few GeV will not be of interest in LHC
phenomenology in any case. We may thus expect that our results should be of
direct phenomenological value even pending some of the improvements we intend
to make in the near future.

For the immediate future we aim to improve our results chiefly by taking
proper account of the order $\alpha_{s}$ coefficient function $C_{1}$ and
computing the various pieces of $C_{1}$ which originate from different regions
and suppressing these by an appropriate form factor rather than the cruder
treatment reported in the text, which was aimed at producing an uncertainty
band associated to a lack of correct treatment of $C_{1}$. When this is done
we will be in a position to carry out an NLO matching and estimate the
uncertainty of our theoretical calculations accurately, which will be
important in the context of phenomenology.

We have not taken directly into account non-perturbative effects in this
chapter, incorporating them as a simple shift of the perturbative spectra as
for the case of global event shapes [66]. We can study non-perturbative
corrections in more detail using the methods outlined in [80], but for the
moment we note that the Monte Carlo event generators we have studied, contain
differences in their estimate of hadronisation which should be explored
further along with studying the underlying event contributions. Since our
predictions are valid for any value of jet radius $R$, one can hope that
phenomenological studies selecting a small value of $R$ would help in better
isolating the perturbative contributions, since the hadronisation and
underlying event corrections for $m_{J}^{2}/p_{T}^{2}$ scale respectively as
$R$ and $R^{4}$ [80]. On the other hand moving to a larger $R$ after pinning
down the perturbative content would help to more accurately constrain the non-
perturbative models in various approaches.

##  Chapter 8 Conclusions

Each chapter has its own conclusion section. In this chapter we tie together
the most important points to be learned from the study presented in this
thesis.

Due to the complex environment present at hadron colliders, such as the LHC,
new physics discoveries are only feasible given a detailed understanding of
the already established standard model, and QCD in particular as the strong
interactions, governed by the latter, constitute the bulk of all possible
interactions. At the LHC, the establishment of the latter understanding is
best accessible via studies of jets, which are the final product of the
aforementioned strong interactions and which populate the final state at the
latter collider. High-$p_{T}$ jets, in particular, have some discernible
features that make them ideal for perturbative analyses. Such analyses include
exploration of the substructure of these jets, which is distinct from that of
moderate- and low-$p_{T}$ jets and which would aid in discriminating new
physics signals from QCD background.

Equipped with an infrared and collinear safe jet shape observable,
angularities, we probed in Chapter 4 the substructure of high-$p_{T}$ QCD jets
in the simple process of dijet production in $e^{+}e^{-}$ annihilation, where
only one jet is measured while the other is left unmeasured. We identified
large logarithms, non-global logs, that were missed out by previous studies
and argued that in the limit of narrow well-separated jets their all–orders
structure simplifies to that of the hemisphere jet mass. This is mainly due to
the observation that non-global logs arise predominantly near the boundary of
each jet and evolve independently of the other well-separated jets. Moreover,
we commented on their phenomenological impact on the primary emission Sudakov
form factor and put forward the ansatz that the above simplified picture of
non-global structure holds even at hadron collider environments.

Given the fact that experimental measurements generally employ large values of
the jet parameter $R$ in their jet finding, it proved necessary to extend the
work of Chapter 4 to include the effects of large jet radii. We have performed
this very task in Chapter 5. Within the same context of dijet production in
$e^{+}e^{-}$ annihilation and taking the jet mass as an example we computed
the full logarithmic structure of the latter shape distribution at
$\mathcal{O}(\alpha_{s}^{2})$, i.e., up to $\alpha_{s}^{2}\,L$ (N3LL accuracy
in the expansion), while retaining the full $R$ dependence. The calculations
were explicitly carried out for both anti-kT and C/A jet algorithms and
implicitly, through similarity to C/A, for the kT algorithm, and represent one
of the most accurate fixed-order calculations for non-global observables.
Further, the NLL (in the exponent) resummation of Chapter 4 has been extended
to full jet radius, in the large-$\mathrm{N_{c}}$ limit, for both the anti-kT
and kT algorithms. Very few non-global observables have been computed at
fixed-order and/or resummed to all–orders to such an accuracy (In fact, the
only observable the author is aware of is the energy flow in gaps-between-jets
[3, 176, 81]).

Based on our resummed results, which include both non-global and clustering
logs, we concluded that it is recommended in jet phenomenological studies to
use a variety of jet algorithms. Further, we found that ignoring non-global
logs all together in events where the final state is clustered with a
sequential recombination jet algorithm other than the anti-kT induces an
uncertainty that is well below that originated from the large-$\mathrm{N_{c}}$
approximation, for quite large jet radii. Our future directions regarding non-
global observables include: extending the numerical program of Ref. [2] to
next-to-leading non-global logs and as a long-term project, going beyond the
large-$\mathrm{N_{c}}$ approximation.

Following previous work on the energy flow in gaps-between-jets, we showed in
Chapter 6 that clustering logs occurring in jet mass calculations in kT and
C/A algorithms exponentiate. We explicitly computed the first few orders in
the exponent with full $R$ dependence. Comparisons to the Monte Carlo program
of [2] revealed that the missing higher-order terms have negligible effect.
Our findings settled the issue, raised by other research groups, of whether
such logs, with highly non-trivial phase space, can exponentiate and hence
resummed. The next obvious step is to extend the resummation to next-to-
leading clustering logs.

Finally, we have exploited the wealth of techniques and experience gained in
the work of the previous chapters to carry out, in the final chapter 7, a full
NLL resummation of jet mass at hadron colliders for the anti-kT jet algorithm.
Explicitly studied were vector boson production in association with a single
jet and dijet production. Parts of the resummed exponent, such as the soft
wide-angle piece involves the full colour structure and thus exceeds the
accuracy of standard Monte Carlo event generators (which work in the
large-$\mathrm{N_{c}}$ approximation). Our calculation marks the first attempt
in literature where non-global logs are computed at hadron colliders and with
full $R$ dependence retained. Already with a simple shift in the exponent to
account for hadronisation corrections and only leading-order matching, our
results compared well to the output of the most widely used event generators,
and even hinted noticeable differences between them.

Additionally, although in the last chapter 7 we have treated a single
variable, the jet mass, it is actually straightforward to extend our treatment
to the entire range of angularities as for instance were explored for jets in
$e^{+}e^{-}$ annihilation in Chapter 4 and Refs. [44, 45]. The basic
calculations we carried out here can be easily extended to include those
variables as well as any variants of the jet mass itself such as studying the
jet mass with an additional jet veto. Once we have accomplished an NLO
matching we therefore intend to generalise our approach to accommodate a range
of substructure variables in different hard processes, including final state
clustering with other jet algorithms such as the kT, C/A and SISCone
algorithms. Moreover, the boost11 review [27] recommended, in addition to the
above two (vector boson + jet and dijets) processes, calculations for multijet
events. We hope to perform this in the near future as well as writing a semi-
analytical program for an automation of these calculations which will enable
similar studies of other non-global jet shapes. We hope that our work will
eventually lead to improved estimates of the accuracy and hence more
confidence in a detailed understanding of jet substructure than is the case
presently, which could in turn be important for a variety of substructure
applications at the LHC.

##  Appendix A Colour algebra

###  A.1 Fierz identities

A matrix $\mathbf{M}$ in $\rm SU(\mathrm{N_{c}})$ can be written as a linear
combination of the basis matrices
$\\{v^{a},a=1,\cdots,\mathrm{N_{c}^{2}}-1\\}$. In the fundamental
representation, $v^{a}=t^{a}$, this reads

$\mathbf{M}=m_{0}\,\mathbb{1}+\sum_{a=1}^{\mathrm{N_{c}^{2}}-1}m_{a}\,t^{a},$
(A.1)

where $m_{0}$ and $m_{a}$ are real expansion coefficients. To determine the
latter coefficients one can make use of the traceless property of $t^{a}$, Eq.
(2.3), together with Eq. (2.5). One can straightforwardly show that

$m_{0}=\frac{1}{\mathrm{N_{c}}}\operatorname{Tr}\left(\mathbf{M}\right),\;\;\;m_{a}=2\,\operatorname{Tr}\left(\mathbf{M}\,t^{a}\right).$
(A.2)

If one takes, for example, the matrix $\mathbf{M}$ to be the commutator and
anti–commutator111For two matrices $\mathbf{A}$ and $\mathbf{B}$, the
anti–commutator is defined as
$\\{\mathbf{A},\mathbf{B}\\}=\mathbf{A}.\mathbf{B}+\mathbf{B}.\mathbf{A}$.
Recall that the commutator is just the Lie product defined in Eq. (2.4a). of
the basis matrices then one obtains, using Eqs. (A.1) and (A.2),

$\displaystyle\imath f_{abc}$ $\displaystyle=$ $\displaystyle
2\,\operatorname{Tr}\left([t^{a},t^{b}]t^{c}\right),$ (A.3a)
$\displaystyle\\{t^{a},t^{b}\\}$ $\displaystyle=$
$\displaystyle\frac{\delta^{ab}}{\mathrm{N_{c}}}\mathbb{1}+d_{abc}\,t^{c},\;\;\;d_{abc}=2\,\operatorname{Tr}\left(\\{t^{a},t^{b}\\}t^{c}\right).$
(A.3b)

It is worthwhile noting that Eq. (A.3a) implies that the structure constants
are totally antisymmetric in all three indices,
$f_{abc}=-f_{acb},f_{abc}=-f_{bac}$ etc. Moreover, Eq. (A.3b) defines, in
analogy to (A.3a), the totally _symmetric_ constants $d_{abc}$, which form a
symmetric adjoint representation of $\rm SU(\mathrm{N_{c}})$222For $\rm
SU(3)$, a system of three quarks (forming a baryon) can be in four
representations. These are found using Young Tableaux techniques (see e.g.,
[100]): $3\times 3\times 3=1\oplus 8\oplus\overline{8}\oplus 10$. $8$ and
$\overline{8}$ are respectively the symmetric and antisymmetric
representations. Employing the same techniques one can also compute the
dimension of a given representation.. The corresponding casimir operator can
be deduced from Eq. (A.3b). It reads [250]:
$C_{As}=(\mathrm{N_{c}^{2}}-4)/\mathrm{N_{c}}$.

Adding up Eqs. (A.3a) and (A.3b) one obtains the following useful relation for
the product of two generators,

$t^{a}t^{b}=\frac{\imath}{2}f^{abc}t^{c}+\frac{1}{2}d^{abc}t^{c}+\frac{\delta^{ab}}{2}\mathbb{1}.$
(A.4)

Notice lastly that due to the fact that $t^{e}.\,t^{e}$ is a scalar (casimir)
and $t^{e}$ are traceless, both $f_{abc}$ and $d_{abc}$ are traceless (easily
seen from the corresponding expressions in Eqs. (A.3a) and (A.3b)). i.e.,
$f_{aac}=d_{aac}=0$, etc. A number of relations that will be useful in later
sections, and appendices, are depicted in Fig. A.1.

Figure A.1: Graphical representation of some important colour relations.

Another interesting example [250] is when one chooses the matrix $\mathbf{M}$
to have only one non–zero component;
$\mathbf{M}^{i}_{k}=\delta^{i}_{(j)}\,\delta^{(\ell)}_{k}$, where $j$ and
$\ell$ are assumed to be fixed. Substituting into Eq. (A.1) yields

$\delta^{i}_{j}\,\delta^{\ell}_{k}=\frac{1}{\mathrm{N_{c}}}\,\delta^{i}_{k}\,\delta^{\ell}_{j}+2\,\left(t^{a}\right)^{i}_{k}\,\left(t^{a}\right)^{\ell}_{j}.$
(A.5)

The above relation is known as the Fierz identity and is shown pictorially in
Fig. A.2.

Figure A.2: Graphical representation of the Fierz identity.

If we rearrange Eq. (A.5) so that the colour matrices are written in terms of
the delta functions then we can generalise the Fierz identity to the following
useful formula [251]

$L\,t^{a}\,Q\,t^{a}\,R=\frac{1}{2}\\!\operatorname{Tr}\left(Q\right)L\,R-\frac{1}{2\mathrm{N_{c}}}L\,Q\,R,$
(A.6)

where $L,Q$ and $R$ are any colour matrices. In the special case where both
$L$ and $R$ are unit matrices and $Q=t^{b}$ then

$t^{a}\,t^{b}\,t^{a}=-\frac{1}{2\mathrm{N_{c}}}\,t^{b},$ (A.7)

or $Q=t^{b}\,t^{c}$ then

$t^{a}\,t^{b}\,t^{c}\,t^{a}=\frac{1}{4}\,\delta^{bc}-\frac{1}{2\mathrm{N_{c}}}\,t^{b}\,t^{c}.$
(A.8)

Finally, the calculation of the cross section involves, as we shall see later
in Sec. 2.2, products of traces of colour generators, with each colour summed
over. It would thus prove useful to provide the following trace identity
[251]:

$\operatorname{Tr}\left(t^{a}\,Q\right)\operatorname{Tr}\left(t^{a}\,R\right)=\frac{1}{2}\,\operatorname{Tr}\left(Q\,R\right)-\frac{1}{2\mathrm{N_{c}}}\,\operatorname{Tr}\left(Q\right)\operatorname{Tr}\left(R\right).$
(A.9)

##  Appendix B Perturbative calculations in $e^{+}e^{-}$ annihilation

In this appendix, we present detailed perturbative calculations of
differential cross sections for one of the main two QCD processes discussed in
this thesis, namely $e^{+}e^{-}$ annihilation into hadrons. In addition to the
classical Feynman diagrammatic approach of computing matrix elements, we
consider the effective eikonal method. We start, in Sec. B.1, by laying out
the general structure of the latter method, encountering in the way some of
the main definitions and relations that will be essential to later
calculations. We then compute the matrix element squared (and therefore the
differential cross section) at Born level, in Subsec. B.2.2.
$\mathcal{O}(\alpha_{s})$ corrections stemming from both real emission and
virtual loops are addressed in Subsecs. B.2.3 and B.2.3. Lastly we compute the
more complicated $\mathcal{O}\left(\alpha_{s}^{2}\right)$ real emission matrix
element, taking into account hard emission contributions.

###  B.1 Eikonal method

The eikonal method [89]111The eikonal method was originally developed in the
context of the non–relativistic theory of potential scattering [252, 253] (and
references in [89]). provides a reliable approximation to the full theory at
high energies. The resultant all–orders eikonal matrix element may be written
in a closed form. This method is based on the propagator approximation

$\frac{\imath}{\left(p\pm k\right)^{2}+\imath\epsilon}\simeq\frac{\imath}{\pm
2\,p.k+\imath\epsilon},$ (B.1)

where $p$ is an external four–momentum, which is assumed to be on–shell
($p^{2}=0$)222The on–shell (or on mass shell) approximation will be used
throughout unless stated otherwise. This approximation becomes almost exact
when considering scattering at high energies, which are far larger than the
masses of the light quarks (i.e., excluding the top quark)., and $k$ may be a
partial sum of internal four–momenta333In cases where $k$ is the four–momentum
of a single parton, which is _soft_ relative to its emitter, this
approximation may be referred to as the _soft_ approximation., thereby
neglecting the $k^{2}$ term relative to $p\,.\,k$ ($k^{2}\ll p.k$). The
corresponding eikonal Feynman rules are shown in Fig. B.1.

Figure B.1: Eikonal Feynman rules for QCD: gluons are represented by curly
lines and fermions by solid lines. For triple vertices, the colour is read
clockwise.

Consider the emission of a soft gluon $k=\left(\omega,\mathbf{k}\right)$ off a
system of $m$ (harder) partons, with momenta
$p_{i}=\left(E_{i},\mathbf{p}_{i}\right),i=1,\cdots m$. One can write the
corresponding emission amplitude as [56, 130, 179, 250]

$\left\lvert
m+1\right\rangle=g_{s}\sum_{i=1}^{m}\frac{p_{i}\,.\,\epsilon^{\ast}}{p_{i}\,.\,k}\,\mathbf{T}^{a}_{i}\left\lvert
m\right\rangle,$ (B.2)

where $\epsilon$ is the polarisation vector of the emitted gluon and
$\left\lvert m\right\rangle$ represents the amplitude prior to emission and is
a vector in the colour space. The set of ket vectors
$\left\lvert\ell\right\rangle$ is assumed to form an orthonormal basis,
$\big{<}\ell^{\prime}\big{|}\ell\big{>}=\delta\left(\ell^{\prime}-\ell\right)$,
for the colour space. The operator $\mathbf{T}^{a}_{i}$ (generator of $\rm
SU(3)$ group) maps the $m$–dimensional space onto the $(m+1)$–dimensional
space due to the emission of one extra gluon off _eikonal_ leg $i$. The
inverse map, i.e., $(m+1)$–dimensional space onto $m$–dimensional space, which
corresponds to the absorption of a single gluon by an eikonal leg $i$ is
represented by the hermitian conjugate operator
$\left(\mathbf{T}^{a}_{i}\right)^{\dagger}$. We write the product of the two
mapping operators as

$\left(\mathbf{T}^{a}_{i}\right)^{\dagger}\,\mathbf{T}^{a}_{j}\equiv\mathbf{T}_{i}\,.\,\mathbf{T}_{j}.$
(B.3)

In other words, $\mathbf{T}_{i}\,.\,\mathbf{T}_{j}$ is symmetric under the
interchange of $i$ and $j$. Notice that it is possible to reduce the product
(B.3) into casimirs (scalars) when the dimension $m\leq 4$ (coloured partons).
For $m>4$ the product is matrix-valued. Colour conservation implies that
$\left\lvert m\right\rangle$ transforms as a colour singlet under $\rm SU(3)$
rotations. Schematically

$\sum_{i=1}^{m}\mathbf{T}^{a}_{i}\left\lvert m\right\rangle=0.$ (B.4)

The above relation will be of great help in simplifying the calculations of
the colour factors associated with Feynman amplitudes.

If the $m$–parton amplitude $\left\lvert m\right\rangle$ is represented by
$\mathcal{M}_{i_{1}i_{2}\cdots i_{m}}$, where the subscripts are the colour
indices of the incoming or outgoing partons, then $\mathbf{T}^{a}$ is
represented by (see Fig. B.1):

  * •

$+t^{a}$ if the radiating parton is an outgoing quark (incoming antiquark).

  * •

$-t^{a}$ if the radiating parton is an incoming quark (outgoing antiquark).

  * •

$+(-)\,\imath\mathbf{f}^{a}$ if the radiating parton is an outgoing (incoming)
gluon.

The operators $t^{a}$ and
$-\imath\left(\mathbf{f}^{a}\right)_{bc}\equiv-\imath f^{abc}$ are, as before,
the colour matrices in the fundamental and adjoint representations
respectively. Computing the bra state $\left\langle m+1\right\rvert$ and
multiplying out, one straightforwardly obtains for the braket, or equivalently
the matrix element squared,

$\left\lvert\mathcal{M}^{2}_{m+1}\right\rvert=\big{<}m+1\big{|}m+1\big{>}=-g_{s}^{2}\sum_{i<j=1}^{m}\frac{2\,\left(p_{i}\,.\,p_{j}\right)}{\left(p_{i}\,.\,k\right)\left(p_{j}\,.\,k\right)}\operatorname{Tr}\left(\left\langle
m\right\rvert\mathbf{T}_{i}\,.\,\mathbf{T}_{j}\left\lvert
m\right\rangle\right),$ (B.5)

where the sum is over distinct $(ij)$ pairs and the factor $2$ is due to
symmetry in $i\leftrightarrow j$. Recall that we are working with on–shell
partons. As such, all self–energy Feynman diagrams whereby $i=j$ vanish.
Moreover, we have used the following sum over polarisations, in the Feynman
gauge ($\xi=1$):

$\sum_{\lambda=0}^{3}\left(-1\right)^{\delta_{\lambda
0}}\,\epsilon_{\mu}\left(\lambda,\mathbf{k}\right)\,\epsilon_{\nu}^{\ast}\left(\lambda,\mathbf{k}\right)=-g_{\mu\nu}.$
(B.6)

We define the dipole _antenna function_ , $\omega_{ij}(k)$, which corresponds
to a soft gluon $k$ stretching between two eikonal (colour–connected) legs $i$
and $j$, as

$\omega_{ij}(k)=\frac{\omega^{2}\,\left(p_{i}\,.\,p_{j}\right)}{\left(p_{i}\,.\,k\right)\left(p_{j}\,.\,k\right)}=\frac{a_{ij}}{a_{i}\,a_{j}},$
(B.7)

with $a_{ij}=1-\cos\theta_{ij}$, $a_{i}=1-\cos\theta_{i}$, $\theta_{ij}$ the
angle between $(p_{i},p_{j})$ pair and $\theta_{i}$ the angle between $p_{i}$
and $k$.

In the eikonal approximation, the $(m+1)$–dimensional phase space factorises
into an $m$–dimensional phase space factor for the radiating ensemble
multiplying a one–dimensional phase space factor for the emitted gluon,

$\textrm{\,d}\Phi_{m+1}=\textrm{\,d}\Phi_{m}\,\frac{\textrm{\,d}^{3}\mathbf{k}}{\left(2\pi\right)^{3}\,2\omega}$
(B.8)

Absorbing a factor of $4\pi^{2}$ into the definition of $\alpha_{s}/\pi$, we
have

$\frac{\textrm{\,d}^{3}\mathbf{k}}{4\pi\,\omega}=\frac{\omega\textrm{\,d}\omega}{4\pi}\,\textrm{\,d}\cos\theta\textrm{\,d}\phi=\frac{k_{t}\textrm{\,d}k_{t}}{4\pi}\textrm{\,d}\eta\textrm{\,d}\phi,$
(B.9)

where $(\theta,\phi)$ are the polar and azimuthal angles of the gluon with
respect to (w.r.t.) a chosen axis. The (pseudo)rapidity $\eta$ and transverse
momentum of the gluon $k_{t}$ are related to its energy and angle, $\omega$
and $\theta$, via

$\eta=-\ln\tan\frac{\theta}{2},\;\;\;k_{t}=\omega\,\sin\theta.$ (B.10)

###  B.2 pQCD: $e^{+}e^{-}$ $\rightarrow$ hadrons

The purpose of this section is to provide a detailed derivation of the full
differential cross section for the process $e^{+}e^{-}$ annihilation into
final state hadrons up to $\mathcal{O}\left(\alpha_{s}^{2}\right)$ real
radiative corrections. That is, we consider the the emission of up to two real
gluons off the initial Born process $e^{+}e^{-}\rightarrow q\,\overline{q}$.
We also consider the one–loop virtual correction in the eikonal limit. The
latter is to demonstrate that for the accuracy sought in this thesis it is
sufficient to simply take the virtual correction to be minus the real one.
Detailed calculations of the total cross–section up to NNLO (two–loop
corrections) can be found in, e.g., [254, 255].

The perturbative expansion of the cross section for the said process, up to
the $n^{th}$ order, may be cast in the form

$\sigma_{n}^{\,e^{+}e^{-}\rightarrow\rm
hadrons}=\sigma^{(0)}+\sigma^{(1)}+\sigma^{(2)}+\cdots+\sigma^{(n)}.$ (B.11)

Below, we first present a general form for the $n$ gluon emission differential
cross section, $\textrm{\,d}\sigma^{(n)}$, including the form of the $n$–body
phase space. Using the resultant expressions, we then compute the Born term,
$\sigma^{(0)}$, the $\mathcal{O}(\alpha_{s})$ correction, $\sigma^{(1)}$,
including both real and virtual contributions and finally the
$\mathcal{O}\left(\alpha_{s}^{2}\right)$ real contribution, $\sigma_{(2)}$. To
simplify the calculations, we shall work at a centre–of–mass energy
$s<m^{2}_{Z}$, so that contributions from the Z boson propagator can be
neglected, and in the centre–of–mass (cm) frame444Recall that the whole
problem is Lorentz invariant, and thus one can work at any chosen frame., such
that the photon propagator’s momentum is $q=\left(\sqrt{s},\mathbf{0}\right)$.

####  B.2.1 Generalities

The general form of the total cross section is given in Eq. (2.28). At order
$\alpha_{s}^{n}$ in the perturbative expansion, Eq. (B.11), it reads

$\sigma^{(n)}=\frac{1}{2s}\int\left\lvert\overline{\mathcal{M}^{(n)}}\right\rvert^{2}\textrm{\,d}\Pi_{n}.$
(B.12)

##### nth order matrix element squared

The invariant matrix element squared (averaged and summed over initial and
final state quantum numbers) may be split into a leptonic part,
$e^{+}e^{-}\rightarrow\gamma^{\ast}$, and a hadronic part,
$\gamma^{\ast}\rightarrow q\overline{q}g\,\cdots$ :

$\left\lvert\overline{\mathcal{M}^{(n)}}\right\rvert^{2}=\frac{1}{s^{2}}\,L^{\mu\nu}\,H_{\mu\nu}^{(n)}.$
(B.13)

The leptonic tensor, $L_{\mu\nu}$, is given by

$\displaystyle L^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{s_{1},s_{2}}\,L^{\mu}\left(q_{1},s_{1};q_{2}.s_{2}\right)\,L^{\nu\dagger}\left(q_{1},s_{1};q_{2},s_{2}\right),$
(B.14) $\displaystyle=$
$\displaystyle\frac{e^{2}}{4}\sum_{s_{1},s_{2}}\left[\overline{u}\left(s_{1},q_{1}\right)\gamma^{\mu}v\left(s_{2},q_{2}\right)\right]\left[\overline{v}\left(s_{2},q_{2}\right)\gamma^{\nu}u\left(s_{1},q_{1}\right)\right],$
$\displaystyle=$
$\displaystyle\frac{e^{2}}{4}\,\operatorname{Tr}\left(q_{1}\\!\\!\\!\\!\\!/\,\gamma^{\mu}\,q_{2}\\!\\!\\!\\!\\!/\,\gamma^{\nu}\right)=e^{2}\left(q_{1}^{\mu}q_{2}^{\nu}+q_{1}^{\nu}q_{2}^{\mu}-g^{\mu\nu}q_{1}\,.\,q_{2}\right),$

where we have used the spinors completeness relations, for a four-momentum
$p$,

$\sum_{\lambda=\pm
1}u_{\lambda}(p)\overline{u}_{\lambda}(p)=p\\!\\!\\!/,\;\;\;\sum_{\lambda=\pm
1}v_{\lambda}(p)\overline{v}_{\lambda}(p)=p\\!\\!\\!/,$ (B.15)

along with the recursive relation for the trace of products of an even number,
$N$, of Dirac matrices

$\operatorname{Tr}\left(\gamma^{\mu_{1}}\gamma^{\mu_{2}}\cdots\gamma^{\mu_{k}}\cdots\gamma^{\mu_{N}}\right)=4\,\sum_{k=2}^{n}\left(-1\right)^{k}g^{\mu_{1}\mu_{k}}\,\operatorname{Tr}\left(\gamma^{\mu_{2}}\cdots\gamma^{\mu_{k-1}}\gamma^{\mu_{k+1}}\cdots\gamma^{\mu_{N}}\right).$
(B.16)

To simplify the calculations, particularly of $H_{\mu\nu}^{(n)}$, it is
instructive to employ the Ward identity [93], which states that if a matrix
element $\mathcal{M}(p)=\mathcal{M}^{\mu}(p)\,\epsilon_{\mu}(p)$ then upon the
replacement $\epsilon_{\mu}\rightarrow p_{\mu}$ the matrix element vanishes.
This implies, in our case, that

$q_{\mu}\,L^{\mu\nu}\left(e^{+}e^{-}\rightarrow\gamma^{\ast}\right)=0,\;\;\;q^{\nu}\,H_{\mu\nu}\left(\gamma^{\ast}\rightarrow
q\overline{q}g\cdots\right)=0.$ (B.17)

The hadronic tensor is thus proportional to the projector
$\left(g^{\mu\nu}-q^{\mu}q^{\nu}/q^{2}\right)$555So is the leptonic tensor.
However since one will be integrating over final state phase space then one
only needs to extract the tensor structure of the final state term. i.e., the
hadronic tensor.. The matrix element squared (B.13) can then be written as

$\displaystyle\left\lvert\overline{\mathcal{M}^{(n)}}\right\rvert^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{s^{2}}\,L^{\mu\nu}\left(g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)\widetilde{H}^{(n)},$
(B.18) $\displaystyle=$
$\displaystyle\frac{1}{s^{2}}\,g_{\mu\nu}L^{\mu\nu}\,\widetilde{H}^{(n)},$
$\displaystyle=$
$\displaystyle\frac{1}{s^{2}}\,g_{\mu\nu}L^{\mu\nu}\,\frac{1}{3}g^{\rho\sigma}H_{\rho\sigma}^{(n)},$
$\displaystyle=$ $\displaystyle\frac{1}{3\,s^{2}}\,L\,H^{(n)},$

where we made use of Eq. (B.17) to pass to the second line. We then multiplied
both sides in the expansion of the hadronic tensor $H_{\mu\nu}^{(n)}$ in terms
of the aforementioned projector
($H_{\mu\nu}^{(n)}=(g_{\mu\nu}-q_{\mu}q_{\nu}/q^{2})\widetilde{H}^{(n)}$) by
the metric $g_{\mu\nu}$ and used the fact that $g_{\mu\nu}g^{\mu\nu}=4$ (in
four–dimensional spacetime) to pass to the third line. The last line in Eq.
(B.18) defines the scalars $L$ and $H^{(n)}$:

$\displaystyle L$ $\displaystyle=$ $\displaystyle
g^{\nu\nu}\,L_{\mu\nu}=-e^{2}\,s,$ $\displaystyle H^{(n)}$ $\displaystyle=$
$\displaystyle g_{\mu\nu}\,H_{\mu\nu}^{(n)}.$ (B.19)

Our task in the subsequent sections is simply to compute $H^{(n)}$.

##### n–body phase space

The general form of the final state phase space factor for $n$ gluons emission
is given in Eq. (2.29). There are two ways of simplifying the $n$–body form
(2.29). The first is to proceed by integrating out $\mathbf{p}_{n}$ then
$\mathbf{p}_{n-1}$ and so on. The second method is to exclusively integrate
out $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$ leaving the rest intact. As we shall
see below, this latter method results in an expression analogous to what would
be found using the eikonal method of Sec. B.1 (modulo a Jacobi factor). The
first method is thoroughly presented in [256]. We only explicitly derive the
integrated two– and three–body phase space factors for the on–shell case. The
two–body phase space reads

$\displaystyle\int\textrm{\,d}\Pi_{2}$ $\displaystyle=$
$\displaystyle\int\frac{\textrm{\,d}^{3}\mathbf{p}_{1}}{\left(2\pi\right)^{3}2\,E_{1}}\frac{\textrm{\,d}^{3}\mathbf{p}_{2}}{\left(2\pi\right)^{3}2\,E_{2}}\,\left(2\pi\right)^{4}\delta^{(4)}\left(q-p_{1}-p_{2}\right),$
$\displaystyle=$
$\displaystyle\int\frac{\textrm{\,d}^{3}\mathbf{p}_{1}}{\left(2\pi\right)^{3}2\,E_{1}}\,\left(2\pi\right)\,\frac{\delta\left(\sqrt{s}-E_{1}-E_{2}\right)}{2\,E_{2}}\Big{|}_{\mathbf{p}_{1}=-\mathbf{p}_{2}}=\frac{1}{32\pi^{2}}\int\textrm{\,d}^{2}\Omega_{1},$

Similarly the three–body phase space is given by

$\int\textrm{\,d}\Pi_{3}=\int\frac{\textrm{\,d}^{3}\mathbf{p}_{1}}{\left(2\pi\right)^{3}2\,E_{1}}\frac{\textrm{\,d}^{3}\mathbf{p}_{2}}{\left(2\pi\right)^{3}2\,E_{2}}\frac{\textrm{\,d}^{3}\mathbf{p}_{3}}{\left(2\pi\right)^{3}2\,E_{3}}\,\left(2\pi\right)^{4}\delta^{(4)}\left(q-p_{1}-p_{2}-p_{3}\right).$
(B.21)

Integrating $\mathbf{p_{3}}$ out and using the fact that666This may be shown
to hold by noting that
$\delta^{(4)}(p^{2})=\delta^{(4)}(p_{0}^{2}-\left\lvert\mathbf{p}\right\rvert^{2})=1/2\left\lvert\mathbf{p}\right\rvert\delta(p_{0}-\left\lvert\mathbf{p}\right\rvert)$
for $p_{0}>0$.

$\displaystyle\frac{\delta\left(\sqrt{s}-E_{1}-E_{2}-E_{3}\right)}{2E_{3}}$
$\displaystyle=$
$\displaystyle\delta\left[\left(q-p_{1}-p_{2}-p_{3}\right)^{2}\right]=\delta\left(-s-2p_{1}\,.\,p_{2}+2q\,.\,p_{1}+2q\,.\,p_{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{s}\delta\left[1-x_{1}-x_{2}+x_{1}x_{2}\,a_{12}/2\right],$
(B.22)

where $a_{ij}=1-\cos\theta_{ij}$ and the energy fraction $x_{i}$ is defined by

$x_{i}=\frac{2q\,.\,p_{i}}{s}=\frac{2\,E_{i}}{\sqrt{s}}.$ (B.23)

We have

$\int\textrm{\,d}\Pi_{3}=\frac{s}{16(2\pi)^{5}}\int
x_{1}\textrm{\,d}x_{1}x_{2}\textrm{\,d}x_{2}\textrm{\,d}^{2}\Omega_{1}\textrm{\,d}^{2}\Omega_{2}\,\delta\left[1-x_{1}-x_{2}+x_{1}x_{2}\,a_{12}/2\right]$
(B.24)

Writing $\textrm{\,d}^{2}\Omega_{2}=\textrm{\,d}a_{12}\textrm{\,d}\phi_{12}$,
averaging over the latter azimuthal angle and simplifying yields

$\int\textrm{\,d}\Pi_{3}=\frac{s}{16(2\pi)^{5}}\int\textrm{\,d}x_{1}\textrm{\,d}x_{2}\textrm{\,d}^{2}\Omega_{1}.$
(B.25)

The $n$–body phase space (2.29), in the second method mentioned above, in
$e^{+}e^{-}$ annihilation is given by the general formula [221]

$\int\textrm{\,d}\Pi_{n}=\frac{s^{n-2}}{2^{2n+2}\pi^{3}}\int\prod_{i=3}^{n}\,x_{i}\textrm{\,d}x_{i}\frac{\textrm{\,d}^{2}\Omega_{i}}{2\pi}\,J^{(n)},$
(B.26)

where $J^{(n)}$ is the Jacobian resulting from integration over $p_{1}$ and
$p_{2}$,

$J^{(n)}=2\int\textrm{\,d}x_{1}\textrm{\,d}^{3}\mathbf{p}_{2}\,\frac{x_{1}}{x_{2}}\,\delta\left(\sum_{i=1}^{n}x_{i}-2\right)\delta\left(\sum_{i=1}^{n}\mathbf{p}_{i}\right).$
(B.27)

Let us consider the cases $n=3$ and $n=4$. Integrating out $\mathbf{p}_{2}$
and using Eq. (B.23), together with the fact that
$E_{2}=\left\lvert\mathbf{p}_{2}\right\rvert$, the $x_{i}$–$\delta$ function
corresponds, for $n=3$ and $n=4$ respectively, to

$\displaystyle x_{1}$ $\displaystyle=$
$\displaystyle\frac{2\left(1-x_{3}\right)}{2-x_{3}a_{13}}\qquad(n=3),$
$\displaystyle x_{1}$ $\displaystyle=$
$\displaystyle\frac{2\left(1-x_{3}-x_{4}\right)+x_{3}x_{4}a_{34}}{2-x_{3}a_{13}-x_{4}a_{14}}\qquad(n=4),$
(B.28)

where $a_{ij}=1-\cos\theta_{ij}$. Carrying out the remaining straightforward
$\delta$ integrals one obtains

$\displaystyle J^{(3)}$ $\displaystyle=$
$\displaystyle\frac{4\left(1-x_{3}\right)}{(2-x_{3}a_{13})^{2}},$
$\displaystyle J^{(4)}$ $\displaystyle=$
$\displaystyle\frac{2\left(1-x_{3}-x_{4}\right)+2x_{3}x_{4}a_{34}}{(2-x_{3}a_{13}-x_{4}a_{14})^{2}}.$
(B.29)

####  B.2.2 Born cross-section

Consider the Feynman diagram, depicted in Fig. B.2$(a)$, representing the Born
amplitude.

Figure B.2: Feynman diagrams for $(a)\,e^{+}e^{-}\rightarrow q\,\overline{q}$,
Born amplitude, $(b)\,e^{+}e^{-}\rightarrow q\overline{q}$, virtual
corrections, and $(c)\,e^{+}e^{-}\rightarrow q\bar{q}g$, real emission.

The hadronic tensor $H_{\mu\nu}^{(0)}$, given in Eq. (B.13) for $n=0$, reads

$\displaystyle H_{\mu\nu}^{(0)}$ $\displaystyle=$
$\displaystyle\sum_{i,j}\sum_{\lambda_{1},\lambda_{2}}H_{\mu}^{(0)}\left(p_{1},\lambda_{1},i;p_{2},\lambda_{2},j\right)\,H_{\nu}^{(0)\dagger}\left(p_{1},\lambda_{1},i;p_{2},\lambda_{2},j\right),$
(B.30) $\displaystyle=$
$\displaystyle\sum_{f=1}^{n_{f}}\,e^{2}e^{2}_{q_{f}}\delta_{ij}\delta_{ji}\left[\overline{u}\left(\lambda_{1},p_{1}\right)\gamma^{\mu}v\left(\lambda_{2},p_{2}\right)\right]\left[\overline{v}\left(\lambda_{2},p_{2}\right)\gamma^{\nu}u\left(\lambda_{1},p_{1}\right)\right]$
$\displaystyle=$
$\displaystyle\sum_{f=1}^{n_{f}}\,4\,e^{2}e^{2}_{q_{f}}\left(p_{1\mu}p_{2\nu}+p_{1\nu}p_{2\mu}-g_{\mu\nu}p_{1}\,.\,p_{2}\right)\,\delta_{ii},$
(B.31)

where the quark–photon–antiquark vertex is associated with a factor $\imath
ee_{q_{f}}\gamma^{\mu}\delta_{ij}$, with $i,j$ being the colour indices of the
quark and antiquark. We note that
$\delta_{ij}\delta^{ji}=\delta^{i}_{i}=\mathrm{N_{c}}=3$ for $\rm SU(3)$. The
corresponding hadronic scalar $H^{(0)}$, defined in Eq. (B.19), is

$H^{(0)}=-\mathrm{N_{c}}\,\sum_{f=1}^{n_{f}}\,4\,e^{2}e^{2}_{q_{f}}\,2\,s.$
(B.32)

Substituting Eqs. (B.2.1) and (B.32) back into Eq. (B.13), which in turn is
substituted together with the phase space factor
(LABEL:eq:app:EE:TwoBodyPhaseSpaceFactorFinal) into Eq. (B.12) for $n=0$,
yields the total $e^{+}e^{-}\rightarrow q\overline{q}$ cross section:

$\sigma^{(0)}=\frac{4\pi\alpha^{2}}{3s}\,\mathrm{N_{c}}\,\sum_{f=1}^{n_{f}}e_{q_{f}}^{2},$
(B.33)

where $\alpha=e^{2}/4\pi$ is the QED coupling constant. In the above equation
we have integrated out the solid angle $\int\textrm{\,d}^{2}\Omega_{1}=4\pi$.

####  B.2.3 $\mathcal{O}(\alpha_{s})$ corrections

We now consider the first order correction to the Born process. The radiative
corrections at this order consist of two types of diagrams, shown in Fig.
B.2$(b)$ and $(c)$: loop and tree level diagrams. The latter corresponds to
real emissions of a gluon, leading to a three–body final state, and the former
to virtual corrections to the self energies of the quark–antiquark pair as
well as to the interaction vertex, keeping the final state identical to the
Born case, i.e., a two–body final state. We first compute the contribution of
the real emission diagrams.

##### Real gluon emission

The $\mathcal{O}(\alpha_{s})$ real integrated cross section is of the form
(B.12) with $n=1$. The three–body phase space is given in Eq. (B.25). The
corresponding $\mathcal{O}\left(\alpha_{s}\right)$ matrix element squared
satisfies Eq. (B.13) with the hadronic scalar $H^{(1)}$ given by:

$\displaystyle H^{(1)}=g^{\mu\nu}H_{\mu\nu}^{(1)}$ $\displaystyle=$
$\displaystyle
g^{\mu\nu}\left(H_{1\mu}^{(1)\dagger}+H_{2\mu}^{(1)\dagger}\right)\left(H_{1\nu}^{(1)}+H_{2\nu}^{(1)}\right),$
(B.34) $\displaystyle=$ $\displaystyle
H_{11}^{(1)}+H_{22}^{(1)}+H_{12}^{(1)},$

where $H_{11},H_{22}$ are the “self–energy” diagrams and
$H_{12}=H_{1}^{\dagger}H_{2}+H_{2}^{\dagger}H_{1}$ contains the “interference”
diagrams. Applying Feynman rules (Fig. 2.3) on Fig. B.2$(c)$ we obtain

$\displaystyle H_{1\mu}^{(1)}$ $\displaystyle=$ $\displaystyle\imath
ee_{q_{f}}\delta_{i^{\prime}j}\,g_{s}\mathbf{t}^{a}_{ii^{\prime}}\left[\overline{u}(p_{1})\,\gamma^{\rho}\,\frac{p^{+}_{13}\\!\\!\\!\\!\\!\\!\\!/}{\left(p^{+}_{13}\right)^{2}}\;\gamma_{\mu}\,v(p_{2})\,\epsilon^{\ast}_{\rho}\right],$
$\displaystyle H_{2\mu}^{(1)}$ $\displaystyle=$ $\displaystyle-\imath
ee_{q_{f}}\delta_{ij^{\prime}}\,g_{s}\mathbf{t}^{a}_{jj^{\prime}}\left[\overline{u}(p_{1})\,\gamma_{\mu}\,\frac{p^{+}_{23}\\!\\!\\!\\!\\!\\!\\!/}{\left(p^{+}_{23}\right)^{2}}\,\gamma^{\sigma}\,v(p_{2})\,\epsilon_{\sigma}^{\ast}\right],$
(B.35)

where $p^{\pm}_{ij}\equiv p_{i}\pm p_{j}$. In order to compute the hermitian
conjugate we note that: $\left(\gamma_{\mu}\right)^{\dagger}=\gamma_{\mu}$ and
$\left(\overline{u}\gamma_{\mu}v\right)^{\dagger}=\overline{v}\gamma_{\mu}u$.
The first term in the right hand side of (B.34) reads

$\displaystyle H_{11}^{(1)}$ $\displaystyle=$
$\displaystyle-\sum_{f}e^{2}\,e_{q_{f}}^{2}\,g_{s}^{2}\operatorname{Tr}\left(\mathbf{t}^{a}_{ij}\mathbf{t}^{a}_{ji}\right)\,\frac{8\,p_{2}\,.\,p_{3}}{p_{1}\,.\,p_{3}},$
(B.36) $\displaystyle=$
$\displaystyle-\sum_{f}e^{2}\,e_{q_{f}}^{2}\,\mathrm{N_{c}}\,g_{s}^{2}\,\mathrm{C_{F}}\,\frac{8\left(1-x_{1}\right)}{\left(1-x_{2}\right)}.$

Note that the $(-)$ sign is due to summation over gluon’s polarisations, Eq.
(B.6). Similarly, the second and third terms are

$H_{22}^{(1)}=-\sum_{f}e^{2}\,e_{q_{f}}^{2}\,\mathrm{N_{c}}\,g_{s}^{2}\,\mathrm{C_{F}}\,\frac{8\left(1-x_{2}\right)}{\left(1-x_{1}\right)},$
(B.37)

and

$H_{12}^{(1)}=-\sum_{f}e^{2}\,e_{q_{f}}^{2}\,\mathrm{N_{c}}\,g_{s}^{2}\,\mathrm{C_{F}}\,\frac{16\left(1-x_{3}\right)}{\left(1-x_{1}\right)\left(1-x_{2}\right)}.$
(B.38)

Summing up the three terms, i.e., substituting Eqs. (B.36), (B.37) and (B.38)
back into (B.34), yields

$H^{(1)}=-\sum_{f}e^{2}\,e_{q_{f}}^{2}\,\mathrm{N_{c}}\,g_{s}^{2}\,\mathrm{C_{F}}\,\frac{8\left(x_{1}^{2}+x_{2}^{2}\right)}{\left(1-x_{1}\right)\left(1-x_{2}\right)}.$
(B.39)

Therefore, upon the substitution of Eqs. (B.25) and (B.39) into Eq. (B.12)
(for $n=1$) one finally obtains for the $\mathcal{O}\left(\alpha_{s}\right)$
differential cross section

$\frac{\textrm{\,d}^{2}\sigma^{(1)}}{\textrm{\,d}x_{1}\textrm{\,d}x_{2}}=\sigma^{(0)}\,\,\frac{\alpha_{s}}{2\pi}\,\mathrm{C_{F}}\,\frac{x_{1}^{2}+x_{2}^{2}}{\left(1-x_{1}\right)\left(1-x_{2}\right)}.$
(B.40)

Examining the above equation one immediately observes that the differential
cross section $\textrm{\,d}\sigma^{(1)}$ diverges whenever $x_{1}$ or $x_{2}$
approaches one. From Eq. (B.23), we have

$\frac{s}{2}\left(1-x_{k}\right)=E_{i}\,E_{j}\left(1-\cos\theta_{ij}\right).$
(B.41)

The limit $x_{k}\rightarrow 1$ corresponds to partons $p_{i}$ and $p_{j}$
being soft ($E_{i}\,\mathrm{or}\,E_{j}\ll 1$) and/or collinear
($\theta_{ij}\ll 1$). The former limit is termed soft (or infrared) divergence
while the latter is referred to as the collinear (or mass) divergence. We show
in Fig. B.3 a “Dalitz” plot of the various soft and collinear limits for the
three parton final state.

Figure B.3: Dalitz plot for a three–body “decay” of a photon. All partons are
assumed on–shell. The edges $x_{i}=1$ correspond to two partons being
collinear and the corners $x_{i}=0$ correspond to one parton being soft
($p_{i}\rightarrow 0$). The different kinematical regions are:
$(1)\,x_{1}>x_{2}>x_{3},(2)\,x_{2}>x_{1}>x_{3},(3)\,x_{2}>x_{3}>x_{1},(4)\,x_{3}>x_{2}>x_{1},(5)\,x_{3}>x_{1}>x_{2}$
and $(6)\,x_{1}>x_{3}>x_{2}$.

These soft and collinear divergences are typical in QCD amplitude calculations
(and QED as well). Since experimentally the answer, i.e., the value of the
total cross section, is finite one ought to “regularise” the infinities that
arise in the theoretical calculations. The final analytical answer should be
independent of the “regularisation procedure”, though. An example of a
regularisation scheme would be the massive gluon scheme [257, 258], whereby
the gluon is given a fictitious mass $m_{g}$. Such a term would simultaneously
prevent both infrared and mass singularities. Another much powerful scheme is
the dimensional regularisation scheme [259], whereby one works in dimension
$d=4-\varepsilon$ and only at the end one takes the limit
$\varepsilon\rightarrow 0$. The reader is referred to [260] for more details.

It is possible to construct experimental observables that are insensitive to
both infrared and collinear divergences. The distribution of such infrared and
collinear (IRC) safe observables, which will be discussed in Chapter 3,
$\textrm{\,d}\sigma/\textrm{\,d}v$ produces, upon phase space integration,
logarithmic enhanced terms of the form $\alpha_{s}^{n}\ln^{m}\left(v\right)$,
where $v$ is a member of the class of IRC safe observables. Such terms break
the convergence of the perturbative expansion, in terms of the coupling
$\alpha_{s}$, and call for a “resummation”.

As a final note, let us consider what would Eq. (B.40) reduce to in the
eikonal approximation. The latter approximation corresponds to neglecting the
terms $p_{1}\,.\,p_{3}$ and $p_{2}\,.\,p_{3}$ relative to $p_{1}\,.\,p_{2}$ in
the formulae of $H_{11}^{(1)},H_{22}^{(1)}$ and $H_{12}^{(1)}$. In other
words, $H^{(1)}\simeq H_{12}^{(1)}$. Consequently the differential cross
section (B.40) becomes

$\displaystyle\sigma_{\mathrm{r}}^{(1),\mathrm{eik}}$ $\displaystyle=$
$\displaystyle\sigma^{(0)}\,\frac{\alpha_{s}}{2\pi}\,\mathrm{C_{F}}\,\int\textrm{\,d}x_{1}\textrm{\,d}x_{2}\,\frac{2\left(1-x_{3}\right)}{\left(1-x_{1}\right)\left(1-x_{2}\right)},$
(B.42) $\displaystyle=$ $\displaystyle\frac{s}{2}\int\textrm{\,d}x_{1}\
x_{2}\,\textrm{\,d}\sigma^{(0)}\,\frac{\alpha_{s}}{2\pi}\,\mathrm{C_{F}}\,\frac{2\left(p_{1}\,.\,p_{2}\right)}{\left(p_{1}\,.\,p_{3}\right)\left(p_{2}\,.\,p_{3}\right)},$
$\displaystyle=$
$\displaystyle\sigma^{(0)}\,g_{s}^{2}\,\mathrm{C_{F}}\,\int\frac{\textrm{\,d}^{3}\mathbf{p_{3}}}{\left(2\pi\right)^{3}\,2\,E_{3}}\,\frac{2\left(p_{1}\,.\,p_{2}\right)}{\left(p_{1}\,.\,p_{3}\right)\left(p_{2}\,.\,p_{3}\right)}.$

The last equality can straightforwardly be verified using the eikonal
techniques of Sec. B.1.

#####  $\mathcal{O}(\alpha_{s})$ virtual gluon corrections

The $\mathcal{O}(\alpha_{s})$ virtual corrections are depicted in Fig.
B.2$(b)$ and correspond to a two–body final state. The associated differential
cross section is again of the form

$\textrm{\,d}\sigma_{\rm{vir}}^{(1)}=\frac{1}{2s}\,\left\lvert\overline{\mathcal{M}_{\rm{vir}}^{(1)}}\right\rvert^{2}\,\textrm{\,d}\Pi_{2},$
(B.43)

where the two–body phase space factor has been computed in Eq.
(LABEL:eq:app:EE:TwoBodyPhaseSpaceFactorFinal) and the matrix element squared
is given by an expression analogous to (B.18), with the real hadronic tensor
replaced by the virtual one

$H_{\mathrm{vir}}^{(1)}=2\,g^{\mu\nu}\,\mathfrak{Re}\left[H_{\mu}^{(0)\dagger}\left(H_{1\mathrm{vir},\nu}^{(1)}+H_{2\mathrm{vir},\nu}^{(1)}+H_{3\mathrm{vir},\nu}^{(1)}\right)\right].$
(B.44)

The pre-factor $2$ comes from the fact that the full $\mathcal{O}(\alpha_{s})$
Feynman amplitude is
$\mathcal{M}_{1}=\mathcal{M}^{(0)}+\mathcal{M}_{\mathrm{r}}^{(1)}+\mathcal{M}_{\mathrm{vir}}^{(1)}$,
which when squared is proportional to
$2\,\mathfrak{Re}\left[\mathcal{M}^{(0)\dagger}\mathcal{M}_{\mathrm{vir}}^{(1)}\right]\equiv\left\lvert\overline{\mathcal{M}_{\mathrm{vir}}^{(1)}}\right\rvert$.
The amplitude $H_{\mu}^{(0)}$ is given in Eq. (B.31) and (see Fig. B.2$(b)$)

$\displaystyle H_{1\mathrm{vir},\mu}^{(1)}$ $\displaystyle=$
$\displaystyle-\imath
ee_{q_{f}}\,g_{s}^{2}\,\mathrm{C_{F}}\,\delta_{ij}\,\overline{u}(p_{1})\left[\int\frac{\textrm{\,d}^{4}\ell}{\left(2\pi\right)^{4}}\,\frac{-\imath}{\ell^{2}+\imath\epsilon}\,\gamma_{\rho}\frac{p_{1}\\!\\!\\!\\!\\!/-\ell\\!\\!\\!/}{\left(p_{1}-\ell\right)^{2}+\imath\epsilon}\,\gamma^{\rho}\right]\frac{p_{1}\\!\\!\\!\\!\\!/}{p_{1}^{2}+\imath\epsilon}\,\gamma_{\mu}\,v(p_{2}),$
$\displaystyle H_{2\mathrm{vir},\mu}^{(1)}$ $\displaystyle=$
$\displaystyle-\imath
ee_{q_{f}}\,g_{s}^{2}\,\mathrm{C_{F}}\,\delta_{ij}\,\overline{u}(p_{1})\,\gamma_{\mu}\,\frac{p_{2}\\!\\!\\!\\!\\!/}{p_{2}^{2}+\imath\epsilon}\left[\int\frac{\textrm{\,d}^{4}\ell}{\left(2\pi\right)^{4}}\,\frac{-\imath}{\ell^{2}+\imath\epsilon}\,\gamma_{\rho}\,\frac{p_{2}\\!\\!\\!\\!\\!/-\ell\\!\\!\\!/}{\left(p_{2}-\ell\right)^{2}+\imath\epsilon}\,\gamma^{\rho}\right]v(p_{2}),$
$\displaystyle H_{3\mathrm{vir},\mu}^{(1)}$ $\displaystyle=$
$\displaystyle\imath
ee_{q_{f}}\,g_{s}^{2}\,\mathrm{C_{F}}\,\delta_{ij}\,\overline{u}(p_{1})\left[\int\frac{\textrm{\,d}^{4}\ell}{\left(2\pi\right)^{4}}\,\frac{-\imath}{\ell^{2}+\imath\epsilon}\,\gamma_{\rho}\,\frac{p_{1}\\!\\!\\!\\!\\!/+\ell\\!\\!\\!/}{\left(p_{1}+\ell\right)^{2}+\imath\epsilon}\,\gamma_{\mu}\,\frac{p_{2}\\!\\!\\!\\!\\!/-\ell\\!\\!\\!/}{\left(p_{2}-\ell\right)^{2}+\imath\epsilon}\,\gamma^{\rho}\right]$
$\displaystyle\times v(p_{2}).$

Substituting Eq. (B.31) and the terms in Eq. (LABEL:eq:app:EE:VirtAmpAlphas1)
into Eq. (B.44) and simplifying the traces one finds, for the scalars
$H_{i,\mathrm{vir}}^{(1)}=g^{\mu\nu}\,H_{\mu}^{(0)\dagger}\,H_{i\mathrm{vir},\nu}^{(1)}$,

$\displaystyle H_{1,\mathrm{vir}}^{(1)}$ $\displaystyle=$
$\displaystyle-e^{2}e^{2}_{q_{f}}\,\mathrm{N_{c}}\,g_{s}^{2}\,\mathrm{C_{F}}\,\int\frac{\textrm{\,d}^{4}\ell}{\left(2\pi\right)^{4}}\,\frac{-\imath}{\ell^{2}+\imath\epsilon}\,\frac{-16\,s}{p_{1}^{2}+\imath\epsilon}\,\frac{p_{1}\,.\,\ell}{\left(p_{1}-\ell\right)^{2}+\imath\epsilon},$
$\displaystyle H_{2,\mathrm{vir}}^{(1)}$ $\displaystyle=$ $\displaystyle
H_{1,\mathrm{vir}}^{(1)}\big{|}_{1\rightarrow 2},$ $\displaystyle
H_{3,\mathrm{vir}}^{(1)}$ $\displaystyle=$ $\displaystyle
e^{2}e^{2}_{q_{f}}\,\mathrm{N_{c}}\,g_{s}^{2}\,\mathrm{C_{F}}\,\int\frac{\textrm{\,d}^{4}\ell}{\left(2\pi\right)^{4}}\,\frac{-\imath}{\ell^{2}+\imath\epsilon}\,\frac{-32\left[p_{1}\,.\,\left(p_{2}-\ell\right)\right]\left[p_{2}\,.\,\left(p_{1}+\ell\right)\right]}{\left[(p_{1}+\ell)^{2}+\imath\epsilon\right]\left[(p_{2}-\ell)^{2}+\imath\epsilon\right]}.$

The Block–Nordsieck theorem [261] states that the logarithmically enhanced
(divergent) contributions due to real emission of collinear and soft gluons
cancel against the corresponding virtual corrections. This can be explicitly
checked for our example, $e^{+}e^{-}\rightarrow q\overline{q}g$, using one of
the aforementioned regularisation schemes along with Feynman parametrisation
techniques. We refer the reader to [260] for full details. Since the singular
logarithmic contributions can be fully captured in the eikonal approximation,
it is sufficient to verify the cancellation using the latter method. To this
end, the self–energy terms in Eq. (LABEL:eq:app:EE:VirtAmpAlphas1B) vanish
while the vertex term reduces to

$H_{3,\mathrm{vir}}^{(1)}=e^{2}e^{2}_{q_{f}}\,\mathrm{N_{c}}\,g_{s}^{2}\,\mathrm{C_{F}}\,\int\frac{\textrm{\,d}^{4}\ell}{\left(2\pi\right)^{4}}\,\frac{-\imath}{\ell^{2}+\imath\epsilon}\,\frac{2\,s^{2}}{\left(p_{1}\,.\,\ell+\imath\epsilon\right)\left(p_{2}\,.\,\ell-\imath\epsilon\right)},$
(B.47)

where we have used $p_{1}\,.\,p_{2}=s/2$ (from momentum conservation
$\delta^{(4)}\left(q-p_{1}-p_{2}\right)$). We can simplify the integral in Eq.
(B.47) using contour integration. We first specialise to the cm frame,
$\mathbf{p_{1}}+\mathbf{p_{2}}=\mathbf{0}$, and write

$H_{3,\mathrm{vir}}^{(1)}\propto\int\frac{\textrm{\,d}^{3}\mathbf{\ell}}{\left(2\pi\right)^{3}}\int\frac{\textrm{\,d}\ell_{0}}{2\pi}\,\frac{-\imath}{\left(\ell_{0}-\left\lvert\mathbf{\ell}\right\rvert-\imath\epsilon\right)\left(\ell_{0}+\left\lvert\mathbf{\ell}\right\rvert+\imath\epsilon\right)}\frac{8\,s}{\left(\ell_{0}-\ell_{3}+\imath\epsilon\right)\left(\ell_{0}+\ell_{3}-\imath\epsilon\right)}.$
(B.48)

The above integral has four simple poles in the $\ell_{0}$ complex plane (Fig.
B.4).

Figure B.4: Poles of the integrand of Eq. (B.48) in the complex plane.

Closing the contour in the upper half, thus picking up only two poles, and
applying the residue theorem777For more details on the residue theorem and
complex integration techniques, see for instance [262]. one finds

$H_{3,\mathrm{vir}}^{(1)}\propto
8\,s\,\int\frac{\textrm{\,d}^{3}\mathbf{\ell}}{\left(2\pi\right)^{3}}\,\frac{1}{2\left\lvert\mathbf{\ell_{\perp}}\right\rvert^{2}}\left(\frac{1}{\left\lvert\mathbf{\ell}\right\rvert}-\frac{1}{\ell_{3}-\imath\epsilon}\right),$
(B.49)

where the invariant transverse momentum $\ell_{\perp}$ with respect to an
$(ij)$ dipole cm frame is related to the antenna function, Eq. (B.7), through

$\frac{2}{\left\lvert\mathbf{\ell_{\perp}}\right\rvert^{2}}=\frac{\left(p_{i}\,.\,p_{j}\right)}{\left(p_{i}\,.\,\ell\right)\left(p_{j}\,.\,\ell\right)}=\frac{\omega_{ij}(\ell)}{E^{2}_{\ell}},$
(B.50)

with $E_{\ell}$ being the energy of the gluon $\ell$. Substituting Eq. (B.49)
into Eq. (B.44), then plug into Eq. (B.43) results in the following expression
for the (integrated) eikonal virtual cross section

$\sigma_{\mathrm{vir}}^{(1),\mathrm{eik}}=-\sigma^{(0)}\,g_{s}^{2}\,\mathrm{C_{F}}\,\mathfrak{Re}\left\\{\int\frac{\textrm{\,d}^{3}\mathbf{\ell}}{\left(2\pi\right)^{2}2\left\lvert\mathbf{\ell}\right\rvert}\frac{2\left(p_{1}\,.\,p_{2}\right)}{\left(p_{1}\,.\,\ell\right)\left(p_{2}\,.\,\ell\right)}\left[1-\frac{\left\lvert\mathbf{\ell}\right\rvert}{\ell_{3}-\imath\epsilon}\right]\right\\}.$
(B.51)

The term involving the $z$–component of the gluon momentum, $\ell_{3}$, in the
square bracket of Eq. (B.51) can be simplified further. The important point is
to explicitly keep the $\imath\epsilon$ prescription. That is, from Eq. (B.49)
one has

$\displaystyle\int\frac{\textrm{\,d}^{3}\mathbf{\ell}}{\left(2\pi\right)^{3}2\left\lvert\ell_{\perp}\right\rvert^{2}}\,\frac{1}{\ell_{3}-\imath\epsilon}$
$\displaystyle=$
$\displaystyle\int\frac{\textrm{\,d}^{2}\ell_{\perp}}{\left(2\pi\right)^{3}2\left\lvert\ell_{\perp}\right\rvert^{2}}\int_{-\infty}^{+\infty}\textrm{\,d}\ell_{3}\,\frac{1}{\ell_{3}-\imath\epsilon},$
(B.52) $\displaystyle=$
$\displaystyle\int\frac{\textrm{\,d}^{2}\ell_{\perp}}{\left(2\pi\right)^{3}2\left\lvert\ell_{\perp}\right\rvert^{2}}\,\lim_{a\rightarrow+\infty}\int_{-a}^{a}\textrm{\,d}\ell_{3}\,\frac{\ell_{3}+\imath\epsilon}{\ell_{3}^{2}+\epsilon^{2}},$
$\displaystyle=$
$\displaystyle\int\frac{\textrm{\,d}^{2}\ell_{\perp}}{\left(2\pi\right)^{3}2\left\lvert\ell_{\perp}\right\rvert^{2}}\,\imath\pi.$

Thus the latter term is purely imaginary. It is worthwhile noting that the
$\imath\pi$ factor is related to super–leading logarithms and the possibility
of factorisation breaking [127, 128, 243, 244, 263]. The final form of the
virtual cross section reads

$\sigma_{\mathrm{vir}}^{(1),\mathrm{eik}}=-\sigma^{(0)}\,g_{s}^{2}\,\mathrm{C_{F}}\int\frac{\textrm{\,d}^{3}\mathbf{\ell}}{\left(2\pi\right)^{2}2\left\lvert\mathbf{\ell}\right\rvert}\frac{2\left(p_{1}\,.\,p_{2}\right)}{\left(p_{1}\,.\,\ell\right)\left(p_{2}\,.\,\ell\right)}=-\sigma_{\mathrm{r}}^{(1),\mathrm{eik}}.$
(B.53)

####  B.2.4 $\mathcal{O}(\alpha_{s}^{2})$ corrections

There are two processes contributing to the amplitude at this order, as
depicted in Fig. B.5: $(a)\,e^{+}e^{-}\rightarrow
q(p_{1})\overline{q}(p_{2})g(p_{3})g(p_{4})$ and $(b)\,e^{+}e^{-}\rightarrow
q(p_{1})\bar{q}(p_{2})q^{\prime}(p_{3})\overline{q}^{\prime}(p_{4})$. The
integrated cross section assumes the general form (B.12) ($n=2$) with the
four–body phase space factor given in Eqs. (B.26) and (B.29). We begin by
addressing the process $(a)$.

Figure B.5: Feynman diagrams for $(a)\,e^{+}e^{-}\rightarrow q\overline{q}gg$
and $(b)\,e^{+}e^{-}\rightarrow q\bar{q}q^{\prime}\overline{q}^{\prime}$.
There are three other diagrams corresponding to swapping the secondary gluons
$g_{3},g_{4}$ in the top three diagrams.

The hadronic scalar $H^{(2)}$ is

$\displaystyle H^{(2)}_{a}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}\left[\sum_{i=1}^{8}H_{a,i\mu}^{(2)}\right]^{\dagger}\left[\sum_{j=1}^{8}H_{a,j\nu}^{(2)}\right],$
(B.54) $\displaystyle=$ $\displaystyle\sum_{i,j=1}^{8}H_{a,ij}^{(2)},$

where, starting from top left in Fig. B.5,

$\displaystyle H_{a,1\mu}^{(2)}$ $\displaystyle=$ $\displaystyle\imath
ee_{q_{f}}\delta_{i_{1}i_{2}}\,g_{s}^{2}\,\mathbf{t}^{a_{4}}\mathbf{t}^{a_{3}}\left[\overline{u}(p_{1})\,\gamma_{\alpha}\,\frac{p^{+}_{14}\\!\\!\\!\\!\\!\\!\\!/}{\left(p_{14}^{+}\right)^{2}}\,\gamma_{\beta}\,\frac{p^{++}_{134}\\!\\!\\!\\!\\!\\!\\!\\!\\!/}{\left(p^{++}_{134}\right)^{2}}\,\gamma_{\mu}\,v(p_{2})\right]\epsilon^{\ast\alpha}_{4}\epsilon^{\ast\beta}_{3},$
$\displaystyle H_{a,2\mu}^{(2)}$ $\displaystyle=$ $\displaystyle-\imath
ee_{q_{f}}\delta_{i_{1}i_{2}}\,g_{s}^{2}\,\mathbf{t}^{a_{4}}\mathbf{t}^{a_{3}}\left[\overline{u}(p_{1})\,\gamma_{\alpha}\,\frac{p^{+}_{14}\\!\\!\\!\\!\\!\\!\\!/}{\left(p^{+}_{14}\right)^{2}}\,\gamma_{\mu}\,\frac{p^{+}_{23}\\!\\!\\!\\!\\!\\!\\!/}{\left(p^{+}_{23}\right)^{2}}\,\gamma_{\beta}\,v(p_{2})\right]\epsilon_{4}^{\ast\alpha}\epsilon_{3}^{\ast\beta},$
$\displaystyle H_{a,3\mu}^{(2)}$ $\displaystyle=$ $\displaystyle\imath
ee_{q_{f}}\delta_{i_{1}i_{2}}\,g_{s}^{2}\,\mathbf{t}^{a_{4}}\mathbf{t}^{a_{3}}\left[\overline{u}(p_{1})\,\gamma_{\mu}\,\frac{p^{++}_{234}\\!\\!\\!\\!\\!\\!\\!\\!\\!/}{\left(p^{++}_{234}\right)^{2}}\,\gamma_{\alpha}\,\frac{p^{+}_{23}\\!\\!\\!\\!\\!\\!\\!/}{\left(p^{+}_{23}\right)^{2}}\,\gamma_{\beta}\,v(p_{2})\right]\epsilon^{\ast\alpha}_{4}\epsilon^{\ast\beta}_{3},$
$\displaystyle H_{a,4\mu}^{(2)}$ $\displaystyle=$ $\displaystyle-\imath
ee_{q_{f}}\delta_{i_{1}i_{2}}\,g_{s}^{2}\,\imath
f^{a_{4}a_{3}a}\mathbf{t}^{a}\left[\overline{u}(p_{1})\,\gamma_{\alpha}\,\frac{p^{++}_{134}\\!\\!\\!\\!\\!\\!\\!\\!\\!/}{\left(p^{++}_{134}\right)^{2}}\,\gamma_{\mu}\,v(p_{2})\right]\epsilon^{\ast\sigma}_{4}\epsilon^{\ast\rho}_{3}\times$
$\displaystyle\times\;\;\frac{g^{\alpha\beta}}{\left(p^{+}_{34}\right)^{2}}\left[g_{\beta\sigma}\left(p^{+}_{34}+p_{4}\right)_{\rho}+g_{\sigma\rho}\left(p_{3}-p_{4}\right)_{\beta}-g_{\rho\beta}\left(p^{+}_{34}+p_{3}\right)_{\sigma}\right],$
$\displaystyle H_{a,5\mu}^{(2)}$ $\displaystyle=$ $\displaystyle\imath
ee_{q_{f}}\delta_{i_{1}i_{2}}\,g_{s}^{2}\,\imath
f^{a_{4}a_{3}a}\mathbf{t}^{a}\left[\overline{u}(p_{1})\,\gamma_{\mu}\,\frac{p^{++}_{234}\\!\\!\\!\\!\\!\\!\\!\\!\\!/}{\left(p^{++}_{234}\right)^{2}}\,\gamma_{\alpha}\,v(p_{2})\right]\epsilon^{\ast\sigma}_{4}\epsilon^{\ast\rho}_{3}\times$
$\displaystyle\times\;\;\frac{g^{\alpha\beta}}{\left(p^{+}_{34}\right)^{2}}\left[g_{\beta\sigma}\left(p^{+}_{34}+p_{4}\right)_{\rho}+g_{\sigma\rho}\left(p_{3}-p_{4}\right)_{\beta}-g_{\rho\beta}\left(p^{+}_{34}+p_{3}\right)_{\sigma}\right],$
$\displaystyle H_{a,6\mu}^{(2)}$ $\displaystyle=$ $\displaystyle
H_{a,1\mu}^{(2)}\left(3\leftrightarrow 4\right),$ $\displaystyle
H_{a,7\mu}^{(2)}$ $\displaystyle=$ $\displaystyle
H_{a,2\mu}^{(2)}\left(3\leftrightarrow 4\right),$ $\displaystyle
H_{a,8\mu}^{(2)}$ $\displaystyle=$ $\displaystyle
H_{a,3\mu}^{(2)}\left(3\leftrightarrow 4\right),$ (B.55)

where, in analogy to $p_{ij}^{\pm}$, we have defined
$p_{ijk}^{\pm\pm}=p_{i}\pm p_{j}\pm p_{k}$ and used Feynman gauge ($\xi=1$ in
Fig. 2.3) for the gluon propagator. Let us further define $p^{+}_{34}\equiv
p$. In the _soft_ limit888Unless stated otherwise, this limit is sufficient
for the accuracy of most of the calculations performed in this thesis.:
$p_{1}\,.\,p_{2}\gg p_{1(2)}\,.\,p_{i}$ for $i=3,4$ (or equivalently
$p\\!\\!\\!/_{1}^{2},p\\!\\!\\!/_{2}^{2}\gg
p\\!\\!\\!/_{3}^{2},p\\!\\!\\!/_{4}^{2}$), Eq. (B.55) can be greatly
simplified. Employing the Dirac algebra relation, for massless spinors,

$\overline{u}(q)\,\gamma^{\alpha}\,q\\!\\!\\!/=\overline{u}(q)\,2\,q^{\alpha},$
(B.56)

and grouping together amplitudes with the same colour factor we write Eq.
(B.55) as

$\displaystyle\mathcal{H}_{a,1\mu}^{(2)}$ $\displaystyle=$ $\displaystyle
H_{\mu}^{(0)}\,g_{s}^{2}\mathbf{t}^{a_{4}}\mathbf{t}^{a_{3}}\left[\frac{p_{1}^{\alpha}}{p_{1}\,.\,p_{4}}\frac{p_{1}^{\beta}}{p_{1}\,.\,p}-\frac{p_{1}^{\alpha}}{p_{1}\,.\,p_{4}}\frac{p_{2}^{\beta}}{p_{2}\,.\,p_{3}}+\frac{p_{2}^{\alpha}}{p_{2}\,.\,p}\frac{p_{2}^{\beta}}{p_{2}\,.\,p_{3}}\right]\epsilon^{\ast}_{3,\beta}\epsilon^{\ast}_{4,\alpha},$
$\displaystyle\mathcal{H}_{a,2\mu}^{(2)}$ $\displaystyle=$ $\displaystyle
H_{\mu}^{(0)}\,g_{s}^{2}\mathbf{t}^{a_{3}}\mathbf{t}^{a_{4}}\left[\frac{p_{1}^{\beta}}{p_{1}\,.\,p_{3}}\frac{p_{1}^{\alpha}}{p_{1}\,.\,p}-\frac{p_{1}^{\beta}}{p_{1}\,.\,p_{3}}\frac{p_{2}^{\alpha}}{p_{2}\,.\,p_{4}}+\frac{p_{2}^{\beta}}{p_{2}\,.\,p}\frac{p_{2}^{\alpha}}{p_{2}\,.\,p_{4}}\right]\epsilon^{\ast}_{3,\beta}\epsilon^{\ast}_{4,\alpha},$
$\displaystyle\mathcal{H}_{a,3\mu}^{(2)}$ $\displaystyle=$ $\displaystyle-
H_{\mu}^{(0)}\,g_{s}^{2}\,\imath
f^{a_{4}a_{3}a}\mathbf{t}^{a}\left(\frac{p_{1}^{\lambda}}{p_{1}\,.\,p}-\frac{p_{2}^{\lambda}}{p_{2}\,.\,p}\right)\,\epsilon^{\ast\beta}_{3}\epsilon^{\ast\alpha}_{4}\times$
$\displaystyle\times\frac{1}{p^{2}}\Big{[}g_{\lambda\alpha}\left(p+p_{4}\right)_{\beta}+g_{\alpha\beta}\left(p_{3}-p_{4}\right)_{\lambda}-g_{\beta\lambda}\left(p+p_{3}\right)_{\alpha}\Big{]},$
$\displaystyle=$ $\displaystyle H_{\mu}^{(0)}\,g_{s}^{2}\,\imath
f^{a_{3}a_{4}a}\mathbf{t}^{a}\,\left[\frac{\left(p_{1}\,.\,p_{3}\right)\left(p_{2}\,.\,p_{4}\right)-\left(p_{1}\,.\,p_{4}\right)\left(p_{2}\,.\,p_{3}\right)}{\left(p_{3}\,.\,p_{4}\right)\left(p_{1}\,.\,p\right)\left(p_{2}\,.\,p\right)}\,g_{\alpha\beta}+\cdots\right]\epsilon^{\ast\beta}_{3}\epsilon^{\ast\alpha}_{4},$

where the $\cdots$ represent the following two terms

$\frac{1}{p_{1}\,.\,p}\left[p_{1\alpha}\left(p+p_{4}\right)_{\beta}-p_{1\beta}\left(p+p_{3}\right)_{\alpha}\right]-\frac{1}{p_{2}\,.\,p}\left[p_{2\alpha}\left(p+p_{4}\right)_{\beta}-p_{2\beta}\left(p+p_{3}\right)_{\alpha}\right].$
(B.58)

The above terms are linear in four–momenta,
$p_{1\alpha(\beta)},p_{2\alpha(\beta)}$, and will hence be ignored, as their
contributions will be subleading.

We are now in a position to compute the _soft_ hadronic scalar $H_{a}^{(2)}$
given in Eq. (B.54). Before proceeding we note the following colour products
(see Sec. 2.1):

$\displaystyle\mathbf{t}^{a_{3}}\mathbf{t}^{a_{4}}\mathbf{t}^{a_{4}}\mathbf{t}^{a_{3}}$
$\displaystyle=$ $\displaystyle\mathrm{C_{F}^{2}},$
$\displaystyle\mathbf{t}^{a_{4}}\mathbf{t}^{a_{3}}\mathbf{t}^{a_{4}}\mathbf{t}^{a_{3}}$
$\displaystyle=$
$\displaystyle\mathrm{C_{F}^{2}}-\frac{1}{2}\mathrm{C_{F}}\mathrm{C_{A}},$
$\displaystyle\mathbf{t}^{a_{3}}\mathbf{t}^{a_{4}}\,\imath
f^{a_{3}a_{4}a}\mathbf{t}^{a}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\mathrm{C_{F}}\mathrm{C_{A}}.$ (B.59)

We thus write $H_{a}^{(2)}$ as a sum of _abelian_ and _non–abelian_
contributions:

$\frac{H_{a}^{(2)}}{g_{s}^{4}}=\mathrm{C_{F}^{2}}\left[\widetilde{\mathcal{H}}_{a,11}^{(2)}+2\widetilde{\mathcal{H}}_{a,12}^{(2)}+\widetilde{\mathcal{H}}_{a,22}^{(2)}\right]+\mathrm{C_{F}}\mathrm{C_{A}}\left[\widetilde{\mathcal{H}}_{a,33}^{(2)}+\widetilde{\mathcal{H}}_{a,23}^{(2)}-\widetilde{\mathcal{H}}_{a,13}^{(2)}-\widetilde{\mathcal{H}}_{a,12}^{(2)}\right].$
(B.60)

where the colour–stripped hadronic scalars
$\widetilde{\mathcal{H}}_{a,ij}^{(2)}$ are:

$\displaystyle\widetilde{\mathcal{H}}_{a,11}^{(2)}$ $\displaystyle=$
$\displaystyle
H^{(0)}\,g_{s}^{4}\times\frac{2\,\left(p_{1}\,.\,p_{2}\right)^{2}}{\left(p_{1}\,.\,p\right)\left(p_{2}\,.\,p\right)\left(p_{1}\,.\,p_{4}\right)\left(p_{2}\,.\,p_{3}\right)},$
$\displaystyle\widetilde{\mathcal{H}}_{a,22}^{(2)}$ $\displaystyle=$
$\displaystyle\widetilde{\mathcal{H}}_{a,11}^{(2)}\left(3\leftrightarrow
4\right),$ $\displaystyle\widetilde{\mathcal{H}}_{a,12}^{(2)}$
$\displaystyle=$ $\displaystyle
H^{(0)}\,g_{s}^{4}\Bigg{[}\widetilde{\mathcal{H}}_{a,11}^{(2)}\,\frac{\left(p_{2}\,.\,p_{3}\right)}{2\left(p_{2}\,.\,p_{4}\right)}+\widetilde{\mathcal{H}}_{a,11}^{(2)}\,\frac{\left(p_{1}\,.\,p_{4}\right)}{2\left(p_{1}\,.\,p_{3}\right)}+$
$\displaystyle\hskip
122.34692pt+\frac{\left(p_{1}\,.\,p_{2}\right)^{2}}{\left(p_{1}\,.\,p_{3}\right)\left(p_{2}\,.\,p_{3}\right)\left(p_{1}\,.\,p_{4}\right)\left(p_{2}\,.\,p_{4}\right)}\Bigg{]},$

and

$\displaystyle\widetilde{\mathcal{H}}_{a,33}^{(2)}$ $\displaystyle=$
$\displaystyle H^{(0)}\,g_{s}^{4}\times
4\,\left[\frac{\left(p_{1}\,.\,p_{3}\right)\left(p_{2}\,.\,p_{4}\right)-\left(p_{1}\,.\,p_{4}\right)\left(p_{2}\,.\,p_{3}\right)}{\left(p_{3}\,.\,p_{4}\right)\left(p_{1}\,.\,p\right)\left(p_{2}\,.\,p\right)}\right]^{2}+\cdots,$
$\displaystyle\widetilde{\mathcal{H}}_{a,13}^{(2)}$ $\displaystyle=$
$\displaystyle-H^{(0)}\,g_{s}^{4}\frac{\left(p_{1}\,.\,p_{2}\right)}{\left(p_{1}\,.\,p_{4}\right)\left(p_{2}\,.\,p_{3}\right)}\left[\frac{\left(p_{1}\,.\,p_{3}\right)\left(p_{2}\,.\,p_{4}\right)-\left(p_{1}\,.\,p_{4}\right)\left(p_{2}\,.\,p_{3}\right)}{\left(p_{3}\,.\,p_{4}\right)\left(p_{1}\,.\,p\right)\left(p_{2}\,.\,p\right)}\right],$
$\displaystyle\widetilde{\mathcal{H}}_{a,23}^{(2)}$ $\displaystyle=$
$\displaystyle-H^{(0)}\,g_{s}^{4}\frac{\left(p_{1}\,.\,p_{2}\right)}{\left(p_{1}\,.\,p_{3}\right)\left(p_{2}\,.\,p_{4}\right)}\left[\frac{\left(p_{1}\,.\,p_{3}\right)\left(p_{2}\,.\,p_{4}\right)-\left(p_{1}\,.\,p_{4}\right)\left(p_{2}\,.\,p_{3}\right)}{\left(p_{3}\,.\,p_{4}\right)\left(p_{1}\,.\,p\right)\left(p_{2}\,.\,p\right)}\right].$

Note that a $(-)$ sign has been absorbed into the Born hadronic tensor
$H^{(0)}$. Substituting Eqs. (B.2.4) and
(LABEL:eq:app:EE:HadronicScalarAlphas2D) into Eq. (B.60), which is in turn
substituted back into Eq. (B.13), for $n=2$ and process $a$, yields

$\left\lvert\overline{\mathcal{M}_{a}^{(2)}}\right\rvert^{2}=\left\lvert\overline{\mathcal{M}^{(0)}}\right\rvert^{2}\,g_{s}^{4}\left[4\,\mathrm{C_{F}^{2}}W_{\mathrm{P}}+\mathrm{C_{F}}\mathrm{C_{A}}\left(2\,S+H_{g}\right)\right],$
(B.63)

where the abelian (primary emission) amplitude is

$W_{\mathrm{P}}=\frac{\left(p_{1}\,.\,p_{2}\right)}{\left(p_{1}\,.\,p_{3}\right)\left(p_{2}\,.\,p_{3}\right)}\,\frac{\left(p_{1}\,.\,p_{2}\right)}{\left(p_{1}\,.\,p_{4}\right)\left(p_{2}\,.\,p_{4}\right)}=\frac{\omega_{12}(p_{3})\,\omega_{12}(p_{4})}{E_{3}^{2}E_{4}^{2}},$
(B.64)

and writing $\left(p_{i}\,.\,p_{j}\right)=\left(ij\right)$ the non–abelian
(secondary emission) amplitude is999The trick to simplify the non–abelian part
of Eq. (B.60) to the form presented in Eq. (B.63) is to add and subtract a
factor of $2$ in the square brackets of Eqs. (B.2.4) and
(LABEL:eq:app:EE:HadronicScalarAlphas2D).  the sum of the soft, $S$, and gluon
hard, $H_{g}$, terms:

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{(12)}{(13)(34)(42)}+\frac{(12)}{(14)(43)(32)}-\frac{(12)}{(13)(32)}\frac{(12)}{(14)(42)},$
(B.65) $\displaystyle=$
$\displaystyle\frac{1}{E_{3}^{2}E_{4}^{2}}\,\omega_{12}(p_{3})\left[\omega_{13}(p_{4})+\omega_{23}(p_{4})-\omega_{12}(p_{4})\right],$

and

$H_{g}=2\,\mathcal{I}^{2}-S\,\mathcal{J}-4\frac{\omega_{12}(p)}{(34)E_{p}^{2}},$
(B.66)

where

$\displaystyle\mathcal{I}$ $\displaystyle=$
$\displaystyle\frac{(13)(24)-(14)(23)}{(34)(1p)(2p)},$
$\displaystyle\mathcal{J}$ $\displaystyle=$
$\displaystyle\frac{(13)(24)+(14)(23)}{(1p)(2p)}.$ (B.67)

The other process at $\mathcal{O}(\alpha_{s}^{2})$, as stated above, is
represented by the Feynman diagrams in Fig. B.5$(b)$. The corresponding
hadronic scalar reads

$\displaystyle H_{b}^{(2)}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}\left[H_{b,1\mu}^{(2)}+H_{b,2\mu}^{(2)}\right]^{\dagger}\left[H_{b,1\nu}^{(2)}+H_{b,2\nu}^{(2)}\right],$
(B.68)

where

$\displaystyle H_{b,1\mu}^{(2)}$ $\displaystyle=$ $\displaystyle\imath
ee_{q_{f}}\delta_{i^{\prime}_{1}i_{2}}\,g_{s}^{2}\mathbf{t}^{a}_{i_{1}i^{\prime}_{1}}\mathbf{t}^{a}_{j_{3}j_{4}}\left[\overline{u}(p_{1})\,\gamma_{\alpha}\,\frac{p^{++}_{134}\\!\\!\\!\\!\\!\\!\\!\\!\\!/}{\left(p^{++}_{134}\right)^{2}}\,\gamma_{\mu}\,v(p_{2})\right]\frac{1}{\left(p^{+}_{34}\right)^{2}}\left[\overline{u}(p_{4})\,\gamma^{\alpha}\,v(p_{3})\right],$
$\displaystyle H_{b,2\mu}^{(2)}$ $\displaystyle=$ $\displaystyle-\imath
ee_{q_{f}}\delta_{i_{1}i^{\prime}_{2}}\,g_{s}^{2}\mathbf{t}^{a}_{i^{\prime}_{2}i_{2}}\mathbf{t}^{a}_{j_{3}j_{4}}\left[\overline{u}(p_{1})\,\gamma_{\mu}\,\frac{p^{++}_{234}\\!\\!\\!\\!\\!\\!\\!\\!\\!/}{\left(p^{++}_{234}\right)^{2}}\,\gamma_{\alpha}\,v(p_{2})\right]\frac{1}{\left(p^{+}_{34}\right)^{2}}\left[\overline{u}(p_{4})\,\gamma^{\alpha}\,v(p_{3})\right].$

Similar to process $(a)$, Eq. (LABEL:eq:app:EE:HadronicScalarAlphas2B_b)
simplifies significantly in the _soft_ limit. Adding up the two contributions
(and using the above notation $p=p^{+}_{34}$) we have

$\displaystyle\mathcal{H}_{b,\mu}^{(2)}$ $\displaystyle=$ $\displaystyle
H_{\mu}^{(0)}\,g_{s}^{2}\,\mathbf{t}^{a}_{i_{1}i_{2}}\mathbf{t}^{a}_{j_{3}j_{4}}\left(\frac{p_{1}^{\alpha}}{\left(p_{1}\,.\,p\right)}-\frac{p_{2}^{\alpha}}{\left(p_{2}\,.\,p\right)}\right)\frac{1}{p^{2}}\left[\overline{u}(p_{4})\,\gamma_{\alpha}\,v(p_{3})\right].$
(B.70)

Squaring the above amplitude we find

$\displaystyle H_{b}^{(2)}$ $\displaystyle\simeq$ $\displaystyle
H^{(0)}\,g_{s}^{4}\,\mathbf{t}^{a}_{i_{1}i_{2}}\operatorname{Tr}\left(\mathbf{t}^{a}_{j_{3}j_{4}}\mathbf{t}^{b}_{j_{3}j_{4}}\right)\mathbf{t}^{b}_{i_{1}i_{2}}\,n_{f}\,\left(\frac{p_{1}^{\alpha}}{(1p)}-\frac{p_{2}^{\alpha}}{(2p)}\right)\left(\frac{p_{1}^{\beta}}{(1p)}-\frac{p_{2}^{\beta}}{(2p)}\right)\times$
$\displaystyle\frac{\operatorname{Tr}\left[p_{4}\\!\\!\\!\\!\\!/\,\gamma_{\alpha}\,p_{3}\\!\\!\\!\\!\\!/\,\gamma_{\beta}\right]}{4\left(p_{3}\,.\,p_{4}\right)^{2}},$
$\displaystyle\simeq$ $\displaystyle
H^{(0)}\,g_{s}^{4}\,\mathrm{C_{F}}\mathrm{T_{R}}n_{f}\,\frac{2}{\left[(34)(1p)(2p)\right]^{2}}\left[(12)(34)(1p)(2p)-\left((13)(24)-(14)(23)\right)^{2}\right].$

Therefore the matrix element squared for process $(b)$ can be written,
following the steps outlined previously for process $(a)$, as

$\left\lvert\overline{\mathcal{M}_{b}^{(2)}}\right\rvert^{2}=\left\lvert\overline{\mathcal{M}^{(0)}}\right\rvert^{2}\,g_{s}^{4}\,\mathrm{C_{F}}\mathrm{T_{R}}n_{f}\,H_{q},$
(B.72)

where $\mathrm{T_{R}}=\mathrm{T_{F}}=1/2$ and the quark hard term is

$H_{q}=-4\,\mathcal{I}^{2}+4\,\frac{\omega_{12}(p)}{(34)E_{p}^{2}}.$ (B.73)

Adding up the contributions of both processes $(a)$ and $(b)$ the final
expression of the $\mathcal{O}(\alpha_{s}^{2})$ matrix element in the soft
limit (which nonetheless includes the quark and gluon hard terms) is given by

$\left\lvert\overline{\mathcal{M}^{(2)}}\right\rvert^{2}=\left\lvert\overline{\mathcal{M}^{(0)}}\right\rvert^{2}\,g_{s}^{4}\left[4\,\mathrm{C_{F}^{2}}\,W_{\mathrm{P}}+\mathrm{C_{F}}\mathrm{C_{A}}\left(2S+H_{g}\right)+\mathrm{C_{F}}\mathrm{T_{F}}n_{f}\,H_{q}\right].$
(B.74)

Notice that if one employs the eikonal method (described in Sec. B.1) then one
would find an identical formula to Eq. (B.74) without the hard terms $H_{g}$
and $H_{q}$.

##  Appendix C Jet–thrust calculations

###  C.1 Derivation of the LO distribution

In the present section we outline the necessary steps for the derivation of
the full logarithmic part of the LO $\tau_{E_{0}}$ integrated distribution,
Eq. (5.7). For the emission of a single gluon, i.e, $e^{+}e^{-}\to
q\,\bar{q}\,g$, we define the kinematic variables,
$x_{i}=2p_{i}\,.\,q/Q^{2}=2E_{i}/Q$ and
$y_{ij}=2p_{i}\,.\,p_{j}/Q^{2}=1-x_{k}$ where $i,j,k=1(q),2(\bar{q}),3(g)$ and
$q=p_{1}+p_{2}+p_{3}$. The leading–order ($q\bar{q}g$) matrix–element squared
is given in Eq. (B.40). It reads

$\frac{\textrm{\,d}^{2}\sigma^{(1)}}{\sigma^{(0)}\textrm{\,d}x_{1}\textrm{\,d}x_{2}}=\frac{\mathrm{C_{F}}\alpha_{s}}{2\pi}\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})},$
(C.1)

where $\sigma^{(0)}$ is the Born cross–section. The integration region, which
is originally $1\geq x_{1},x_{2},x_{3}\geq 0$ (recall that
$x_{1}+x_{2}+x_{3}=2$) and which leads to divergences, gets modified by
introducing the jet shape variable. For three partons in the final state,
$\tau_{E_{0}}$ is zero unless two partons are clustered together. That is, the
$\tau_{E_{0}}$ shape distribution receives, at this order, contributions from
two–jet configurations only. Three–jet final state configurations, which are
also present at this order, do not contribute to the latter distribution. They
contribute to the $E_{0}$ distribution instead, as we shall see later.

It is a straightforward exercise to show that the thrust, or rather
$\tau=1-T$, is, for small values of $\tau$, the sum of the masses of two
_hemisphere_ jets. Precisely
$\tau\simeq(p_{\mathrm{J}_{1}}^{2}+p_{\mathrm{J}_{2}}^{2})/Q^{2}=1-\max\\{x_{i}\\}=\tau_{E_{0}}$
where $i=1,2,3$ 111From conservation of momentum given above we have for a jet
formed by a pair of partons $(i,j)$:
$\tau_{E_{0}}=(p_{i}+p_{j})^{2}/Q^{2}=2p_{i}\,.\,p_{j}/Q^{2}=(1-x_{k})$ where
$x_{k}>x_{i},x_{j}$. i.e, $\tau_{E_{0}}=1-\max\\{x_{i}\\}$.. The two
distributions are thus equal. However, here we are considering jets of size
$R_{s}<1/2$ and the following (generic) jet algorithm condition for a pair
$(i,j)$ to be recombined together to form a jet applies

$R_{s}>2(1-\cos\theta_{ij})=\frac{(1-x_{k})}{x_{i}x_{j}}\simeq\frac{(1-x_{k})}{x_{j}},\qquad
k\neq i,j,$ (C.2)

where the last approximation follows from $x_{k}>x_{i}>x_{j}$ (we show in Sec.
5.2.2 that in the limit $R_{s}\to 1/2\Leftrightarrow\theta_{ij}\to\pi/2$ the
latter approximation yields the exact $\tau$ distribution).

The phase space of the two–jet/three–jet boundary for the
$\tau_{E_{0}}(p_{1},p_{2})<\tau_{E_{0}}$ is given in terms of the energy
fractions 222Similar limits have been derived for the JADE variable $y$ in
[7], which were also used for $\tau$ in the same reference. as

$1>x_{1},x_{2}>1-\tau_{E_{0}},\qquad 1+\tau_{E_{0}}>x_{1}+x_{2},\qquad
R^{2}>\frac{4(1-x_{2})}{2-x_{1}-x_{3}},$ (C.3)

where we have chosen to recombine the gluon, $p_{3}$, with hard leg $p_{1}$.
The other choice, i.e, recombining the gluon with hard leg $p_{2}$, produces
identical result. Hence we multiply by a factor of two to account for the
latter. Likewise, the phase space boundary for the three–jet configuration
whereby the third jet, chosen to be the gluon jet, is vetoed to have energy
less than a cut–off $E_{0}$ is given by

$2>x_{1}+x_{2}>2\left(1-\frac{E_{0}}{Q}\right),\qquad\frac{4(1-x_{2})}{2-x_{1}-x_{3}}>R^{2},\qquad\frac{4(1-x_{1})}{2-x_{2}-x_{3}}>R^{2}.$
(C.4)

Due to the soft and collinear singularities of the matrix–element (C.1),
$x_{1,2}\to 1$, we use the following completion relation in order to integrate
the $\tau_{E_{0}}$ and $E_{0}$ differential distributions

$\displaystyle\Sigma^{(0)}+\left[\Sigma_{\rm in}^{(1)}+\Sigma_{\rm
out}^{(1)}\right]$ $\displaystyle=$ $\displaystyle
1+\int_{0}^{\tau_{E_{0}}}\textrm{\,d}\tau_{E_{0}}^{\prime}\frac{1}{\sigma^{(0)}}\frac{\textrm{\,d}\sigma^{(1)}}{\textrm{\,d}\tau_{E_{0}}^{\prime}}+\int_{0}^{E_{0}}\textrm{\,d}E_{3}\frac{1}{\sigma^{(0)}}\frac{\textrm{\,d}\sigma^{(1)}}{\textrm{\,d}E_{3}},$
$\displaystyle=$ $\displaystyle\frac{\sigma^{\rm
tot}}{\sigma^{(0)}}-\int_{\tau_{E_{0}}}^{\tau_{E_{0}}^{\max}}\textrm{\,d}\tau_{E_{0}}^{\prime}\frac{1}{\sigma^{(0)}}\frac{\textrm{\,d}\sigma^{(1)}}{\textrm{\,d}\tau_{E_{0}}^{\prime}}-\int_{E_{0}}^{Q/2}\textrm{\,d}E_{3}\frac{1}{\sigma^{(0)}}\frac{\textrm{\,d}\sigma^{(1)}}{\textrm{\,d}E_{3}},$

where “in” and “out” refer to the regions (C.3) and (C.4), respectively, and
the total cross–section $\sigma^{\rm tot}$, which includes both real (two–jet
and three–jet contributions) and virtual corrections, is related to the Born
cross–section by [7]

$\frac{\sigma^{\rm
tot}}{\sigma^{(0)}}=1+\frac{3\mathrm{C_{F}}}{2}\frac{\alpha_{s}}{2\pi}+\mathcal{O}(\alpha_{s}^{2}).$
(C.6)

Thus inverting the conditions in Eqs. (C.3) and (C.4), carrying out the
integrals in Eq. (LABEL:eq:app:EEJS2:IntegralOfShapeFracAlpha1) separately and
then adding up with the $\mathcal{O}(\alpha_{s})$ of Eq. (C.6) one obtains Eq.
(5.8).

###  C.2 $G_{nm}$ coefficients

The resultant coefficients from the expansion of the exponent in the resummed
integrated distribution, eq. (5.80), for the anti–$\rm k_{T}$algorithm, are

$\displaystyle G_{12}$ $\displaystyle=$ $\displaystyle-2\,\mathrm{C_{F}},$
$\displaystyle G_{11}$ $\displaystyle=$
$\displaystyle\mathrm{C_{F}}\left(3-4L_{R_{s}}\right),$ $\displaystyle G_{10}$
$\displaystyle=$
$\displaystyle\mathrm{C_{F}}\left[-4L_{R_{s}}L_{E_{0}}+\frac{\bar{f}_{0}(R_{s})}{2}\right],$
$\displaystyle G_{23}$ $\displaystyle=$
$\displaystyle\mathrm{C_{F}}\left(\frac{4}{3}\mathrm{T_{F}}n_{f}-\frac{11}{3}\mathrm{C_{A}}\right),$
$\displaystyle G_{22}$ $\displaystyle=$
$\displaystyle-\frac{4\pi^{2}}{3}\mathrm{C_{F}^{2}}+\mathrm{C_{F}}\mathrm{C_{A}}\left(\frac{\pi^{2}}{3}-\mathbf{I}_{22}(R_{s})-\frac{169}{36}-\frac{22}{3}L_{R_{s}}\right)+\mathrm{C_{F}}\mathrm{T_{F}}n_{f}\left(\frac{11}{9}+\frac{8}{3}L_{R_{s}}\right),$
$\displaystyle G_{21}$ $\displaystyle=$
$\displaystyle-\mathrm{C_{F}^{2}}\,\frac{8\pi^{2}}{3}L_{R_{s}}-\mathrm{C_{F}}\mathrm{C_{A}}\bigg{[}2\mathbf{I}_{22}(R_{s})L_{E_{0}}-\mathbf{I}^{a}_{21}(R_{s})+\mathbf{I}^{\mathrm{C_{A}}}_{21}(R_{s})-$
(C.7) $\displaystyle-$
$\displaystyle\Big{(}\frac{2\pi^{2}}{3}-2\mathbf{I}_{22}(Rs)-\frac{134}{9}-\frac{11}{3}L_{R_{s}}\Big{)}L_{R_{s}}\bigg{]}+$
$\displaystyle+$
$\displaystyle\mathrm{C_{F}}\mathrm{T_{F}}n_{f}\left[\mathbf{I}^{n_{f}}_{21}(R_{s})+\left(\frac{4}{3}L_{R_{s}}+\frac{40}{9}\right)L_{R_{s}}\right].$

where $L_{R_{s}}=\ln(R_{s}/(1-R_{s}))$ and $L_{E_{0}}=\ln(2E_{0}/Q)$. The
factor $\bar{f}_{0}(R_{s})$ only captures the first term of $f_{0}$ given in
eq. (5.10). We simply replace $\bar{f}_{0}\mapsto f_{0}$ when comparing to the
numerical distribution. Moreover, we have

$\displaystyle\mathcal{S}_{22}=-\mathrm{C_{F}}\mathrm{C_{A}}\;\mathbf{I}_{22}(R_{s}),$
$\displaystyle\qquad\mathcal{S}_{21}^{a}=-\mathrm{C_{F}}\mathrm{C_{A}}\;\mathbf{I}_{21}^{a}(R_{s}),$
$\displaystyle\mathcal{S}_{21}^{\mathrm{C_{A}}}=-\mathrm{C_{F}}\mathrm{C_{A}}\;\mathbf{I}^{\mathrm{C_{A}}}_{22}(R_{s}),$
$\displaystyle\qquad\mathcal{S}_{21}^{n_{f}}=-\mathrm{C_{F}}\mathrm{C_{A}}\;\mathbf{I}^{n_{f}}_{22}(R_{s}).$
(C.8)

The one–loop constant is given by, eq. (5.8),

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\mathrm{C_{F}}\left(-1+\frac{\pi^{2}}{3}\right),$ (C.9)

For the C/A algorithm we have the following replacements in $G_{22}$ and
$G_{21}$:

$\displaystyle\mathbf{I}_{2i}$ $\displaystyle\to$
$\displaystyle\mathbf{I}_{2i}^{\text{C/A}},$ $\displaystyle
G_{22}:\qquad-\frac{4\pi^{2}}{3}\mathrm{C_{F}^{2}}$ $\displaystyle\to$
$\displaystyle\mathrm{C_{F}^{2}}\left(C^{P}_{2}-\frac{4\pi^{2}}{3}\right),$
$\displaystyle G_{21}:\;-\mathrm{C_{F}^{2}}\,\frac{8\pi^{2}}{3}L_{R_{s}}$
$\displaystyle\to$
$\displaystyle\mathrm{C_{F}^{2}}\left(2\,C^{P}_{2}-\frac{8\pi^{2}}{3}\right)L_{R_{s}}+C^{P}_{21}.$
(C.10)

Expanding the total resummed distribution in eq. (5.80) to
$\mathcal{O}(\alpha_{s}^{2})$ and up to NLL we have

$\Sigma_{r,2}(\widetilde{L})=1+\left(\frac{\alpha_{s}}{2\pi}\right)\left(H_{12}\widetilde{L}^{2}+H_{11}\widetilde{L}+H_{10}\right)+\left(\frac{\alpha_{s}}{2\pi}\right)^{2}\Big{(}H_{24}\widetilde{L}^{4}+H_{23}\widetilde{L}^{3}+H_{22}\widetilde{L}^{2}+\\\
+H_{21}\widetilde{L}+H_{20}\Big{)},$ (C.11)

where (recall that
$\widetilde{L}=\ln(1/\tau_{E_{0}})\Rightarrow\tau_{E_{0}}=e^{-\widetilde{L}}$)

$\displaystyle D_{\mathrm{fin}}(e^{-\widetilde{L}})$ $\displaystyle=$
$\displaystyle\left(\frac{\alpha_{s}}{2\pi}\right)\,d_{1}(e^{-\widetilde{L}})+\left(\frac{\alpha_{s}}{2\pi}\right)^{2}\,d_{2}(e^{-\widetilde{L}}),$
$\displaystyle H_{12}$ $\displaystyle=$ $\displaystyle G_{12},$ $\displaystyle
H_{11}$ $\displaystyle=$ $\displaystyle G_{11},$ $\displaystyle H_{10}$
$\displaystyle=$ $\displaystyle G_{10}+C_{1}+d_{1}(\tau_{E_{0}}),$
$\displaystyle H_{24}$ $\displaystyle=$ $\displaystyle\frac{1}{2}G_{12}^{2}.$
$\displaystyle H_{23}$ $\displaystyle=$ $\displaystyle G_{23}+G_{12}G_{11},$
$\displaystyle H_{22}$ $\displaystyle=$ $\displaystyle
G_{22}+(G_{10}+C_{1})G_{12}+\frac{1}{2}G_{11}^{2},$ $\displaystyle H_{21}$
$\displaystyle=$ $\displaystyle G_{21}+(G_{10}+C_{1})G_{11},$ $\displaystyle
H_{20}$ $\displaystyle=$ $\displaystyle
G_{20}+\frac{1}{2}G_{10}^{2}+C_{1}G_{10}+C_{2}+d_{2}(\tau_{E_{0}}).$ (C.12)

Differentiating (C.11) w.r.t. $\widetilde{L}$, the NLO differential
distribution reads

$\frac{\textrm{\,d}\Sigma_{r,2}}{\textrm{\,d}\widetilde{L}}=\frac{1}{\sigma_{0}}\frac{\textrm{\,d}\sigma_{r,2}}{\textrm{\,d}\widetilde{L}}=\delta(\widetilde{L})\,D_{\delta}+\left(\frac{\alpha_{s}}{2\pi}\right)\,D_{A}(\widetilde{L})+\left(\frac{\alpha_{s}}{2\pi}\right)^{2}\,D_{B}(\widetilde{L}),$
(C.13)

where the singular (logarithmic) terms are given by

$\displaystyle D_{\delta}$ $\displaystyle=$ $\displaystyle
1+\left(\frac{\alpha_{s}}{2\pi}\right)\left[G_{10}+C_{1}\right]+\left(\frac{\alpha_{s}}{2\pi}\right)^{2}\left[G_{20}+\frac{1}{2}G_{10}^{2}+C_{1}G_{10}+C_{2}\right],$
$\displaystyle D_{A}(\widetilde{L})$ $\displaystyle=$ $\displaystyle
2H_{12}\widetilde{L}+H_{11}+\frac{\textrm{\,d}}{\textrm{\,d}\widetilde{L}}\,d_{1}(e^{-\widetilde{L}}),$
$\displaystyle D_{B}(\widetilde{L})$ $\displaystyle=$ $\displaystyle
4H_{24}\widetilde{L}^{3}+3H_{23}\widetilde{L}^{2}+2H_{22}\widetilde{L}+H_{21}+\frac{\textrm{\,d}}{\textrm{\,d}\widetilde{L}}\,d_{2}(e^{-\widetilde{L}}).$
(C.14)

###  C.3 Jet-thrust distribution in SCET

The resummation of the jet-thrust in SCET is presented in the current section
for comparison with pQCD. We shall only present the final form of the resummed
result taken from Refs. [50, 223, 46]. For a full derivation and more in depth
discussion one should consult the latter references. The only task we have
performed here is the expansion of the full resummed distribution to
$\mathcal{O}(\alpha_{s}^{2})$.

####  C.3.1 Resummation

The general formula of the resummed distribution for the jet-thrust is given
by [50, 46]

$\frac{\textrm{\,d}\Sigma^{\rm
SCET}(\tau_{E_{0}},R)}{\textrm{\,d}\tau_{E_{0}}}=\exp\Big{[}4S(\mu_{h},\mu_{j})+4S(\mu_{s},\mu_{j})-4A_{H}(\mu_{h},\mu_{s})+4A_{J}(\mu_{j},\mu_{s})\Big{]}\\\
\times\left(\frac{R_{s}}{1-R_{s}}\right)^{-2A_{\Gamma}(\mu_{\omega},\mu_{s})}\left(\frac{Q^{2}}{\mu_{h}^{2}}\right)^{-2A_{\Gamma}(\mu_{h},\mu_{j})}H(Q^{2},\mu_{h})\\\
\times
S^{\mathrm{out}}_{R}(\omega,\mu_{\omega})\left[\tilde{j}\left(\ln\frac{\mu_{s}Q}{\mu_{j}^{2}}+\partial_{\eta},\mu_{j}\right)\right]^{2}\tilde{s}^{\mathrm{in}}_{\tau_{E_{0}}}(\partial_{\eta},\mu_{s})\\\
\times\frac{1}{\tau_{E_{0}}}\left(\frac{\tau_{E_{0}}Q}{\mu_{s}}\right)^{\eta}\frac{e^{-\gamma_{E}\eta}}{\Gamma(\eta)}.$
(C.15)

See [50, 46] for full notation. In order to compute the fixed–order expansion
of (C.15) up to $\mathcal{O}(\alpha_{s}^{2})$, all scales should be set equal
($\mu_{h}=\mu_{j}=\mu_{s}=Q$). In this limit, the evolution factors $S,A_{J}$
and $A_{H}$ vanish. The differentiation w.r.t. $\eta$ is carried out using the
explicit form of $\tilde{j}$ and $\tilde{s}^{\mathrm{in}}_{\tau_{E_{0}}}$. The
final result of the integrated distribution may be cast in the generic form
(5.80) with the constants and coefficients of the logs given by

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\mathrm{C_{F}}\left(-1+\frac{\pi^{2}}{3}\right),$ (C.16)
$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\mathrm{C_{F}^{2}}\left(1-\frac{3\pi^{2}}{8}+\frac{\pi^{4}}{72}-6\zeta(3)\right)+\mathrm{C_{F}}\mathrm{C_{A}}\left(\frac{493}{324}+\frac{85\pi^{2}}{24}-\frac{73\pi^{4}}{360}+\frac{283\zeta(3)}{18}\right)+$
(C.17) $\displaystyle+$
$\displaystyle\mathrm{C_{F}}\mathrm{T_{F}}n_{f}\left(\frac{7}{81}-\frac{7\pi^{2}}{6}-\frac{22\zeta(3)}{9}\right)+C_{2}^{\mathrm{in}}+C_{2}^{\mathrm{out}},$

and

$\displaystyle G_{12}$ $\displaystyle=$ $\displaystyle-2\,\mathrm{C_{F}},$
$\displaystyle G_{11}$ $\displaystyle=$
$\displaystyle-\mathrm{C_{F}}\left(3-4L_{R_{s}}\right),$ $\displaystyle
G_{10}$ $\displaystyle=$
$\displaystyle\mathrm{C_{F}}\left(-4L_{R_{s}}L_{E_{0}}+\frac{f_{0}}{2}\right),$
$\displaystyle G_{23}$ $\displaystyle=$
$\displaystyle\mathrm{C_{F}}\left(\frac{11}{3}\mathrm{C_{A}}-\frac{4}{3}\mathrm{T_{F}}n_{f}\right),$
$\displaystyle G_{22}$ $\displaystyle=$
$\displaystyle-\frac{4\pi^{2}}{3}\mathrm{C_{F}^{2}}+\mathrm{C_{F}}\mathrm{C_{A}}\left(\frac{\pi^{2}}{3}-\frac{169}{36}-\frac{22}{3}L_{R_{s}}\right)+\mathrm{C_{F}}\mathrm{T_{F}}n_{f}\left(\frac{11}{9}+\frac{8}{3}L_{R_{s}}\right),$
$\displaystyle G_{21}$ $\displaystyle=$
$\displaystyle\mathrm{C_{F}^{2}}\left[-\frac{3}{4}-\pi^{2}+4\zeta(3)+\frac{8\pi^{2}}{3}L_{R_{s}}\right]+\mathrm{C_{F}}\mathrm{T_{F}}n_{f}\left[5-\left(\frac{4}{3}L_{R_{s}}+\frac{40}{9}\right)L_{R_{s}}\right]$
$\displaystyle+$
$\displaystyle\mathrm{C_{F}}\mathrm{C_{A}}\bigg{[}-\frac{57}{4}+6\zeta(3)-\left(\frac{2\pi^{2}}{3}-\frac{134}{9}-\frac{11}{3}L_{R_{s}}\right)L_{R_{s}}\bigg{]}.$
$\displaystyle G_{20}$ $\displaystyle=$
$\displaystyle\mathrm{C_{F}^{2}}\Bigg{[}-\frac{f_{0}^{2}}{8}+\left(2\pi^{2}-16\zeta(3)\right)L_{R_{s}}-\Big{(}\frac{11\pi^{2}}{6}+\frac{f_{0}}{2}\Big{)}L^{2}_{R_{s}}-L_{R_{s}}^{2}\Bigg{]}+$
(C.18) $\displaystyle+$
$\displaystyle\mathrm{C_{F}}\mathrm{C_{A}}\Bigg{[}\frac{11\pi^{2}}{9}L_{R_{s}}-\frac{11}{6}L_{R_{s}}L^{2}_{E_{0}}-L_{E_{0}}\Bigg{(}\frac{11f_{0}}{12}+\left[\frac{134}{9}-\frac{2\pi^{2}}{3}\right]L_{R_{s}}+$
$\displaystyle+$
$\displaystyle\frac{11}{6}L^{2}_{R_{s}}\Bigg{)}\Bigg{]}+\mathrm{C_{F}}\mathrm{T_{F}}n_{f}\Bigg{[}-\frac{4\pi^{2}}{9}L_{R_{s}}+\frac{2}{3}L_{R_{s}}L^{2}_{E_{0}}+L_{E_{0}}\Bigg{(}\frac{f_{0}(R_{s})}{3}+$
$\displaystyle+$
$\displaystyle\frac{40}{9}L_{R_{s}}+\frac{2}{3}L^{2}_{R_{s}}\Bigg{)}\Bigg{]},$

where $f_{0}\equiv f_{o}(R_{s})$. Considering primary emission, the only
missing piece in the distribution is the two–loop constants in the soft
function, namely $C_{2}^{\mathrm{in}}$ and $C_{2}^{\mathrm{out}}$.

###  C.4 Comparisons to event2 : single $R_{s}$ plots

Below we provide plots for the difference between the analytical and numerical
(exact) jet-thrust $\tau_{E_{0}}$ distribution at NLO for each colour channel.
The plots are shown for each value of $R_{s}$ separately.

Figure C.1: The difference between event2 and $\tau_{E_{0}}$ NLO distribution
for various jet radii in both anti-kT (left) and C/A (right) algorithms. Plots
are for $\mathrm{C_{F}^{2}}$ channel.

Figure C.2: The difference between event2 and $\tau_{E_{0}}$ NLO distribution
for various jet radii in both anti-kT (left) and C/A (right) algorithms. Plots
are for $\mathrm{C_{F}}\mathrm{C_{A}}$ channel.

Figure C.3: The difference between event2 and $\tau_{E_{0}}$ NLO distribution
for various jet radii in both anti-kT (left) and C/A (right) algorithms. Plots
are for $\mathrm{C_{F}}\mathrm{T_{F}}n_{f}$ channel.

##  Appendix D Full $R$-dependence of clustering coefficients

Here we present a calculation of the dependence of the coefficient of the
clustering logs $\mathcal{F}_{2}$ away from the small-angles (thus small-$R$)
approximation. To do so, it is easier to work with transverse momentum,
(pseudo-)rapidity111Recall that $\eta=-\ln\tan(\theta/2)$ and
$k_{t}=\omega\sin\theta$. and azimuthal angle with respect to the _beam_ axis,
$(k_{t},\eta,\phi)$, variables instead of energy and polar angles, as
performed in sec. 6.3. We also specialise to the threshold limit in which the
trigged jet is created at $90^{\circ}$ to the beam (which is along the
$z$-axis). Our calculations can straightforwardly be extended to the case
where the triggered jet is at an arbitrary rapidity. We parametrise the
outgoing four-momenta as:

$\displaystyle p_{1}$ $\displaystyle=$ $\displaystyle\frac{Q}{2}(1,1,0,0),$
$\displaystyle p_{2}$ $\displaystyle=$ $\displaystyle\frac{Q}{2}(1,-1,0,0),$
$\displaystyle k_{i}$ $\displaystyle=$ $\displaystyle
k_{ti}(\cosh\eta_{i},\cos\phi_{i},\sin\phi_{i},\sinh\eta_{i}),$ (D.1)

with $i=1,2$ for the two gluons respectively. In terms of the new variables,
the clustering function (Eq. (6.3)) reads:

$\Xi_{2}(k_{1},k_{2})=\Theta(d_{1j}-R^{2})\Theta(R^{2}-d_{2j})\Theta(d_{2j}-d_{12}),$
(D.2)

with

$d_{1j}=\eta_{1}^{2}+\phi_{1}^{2},\qquad
d_{2j}=\eta_{2}^{2}+\phi_{2}^{2},\qquad
d_{12}=(\eta_{1}-\eta_{2})^{2}+(\phi_{1}-\phi_{2})^{2}.$ (D.3)

In this coordinate system the jet mass becomes:

$\rho=\frac{2p_{1}.k_{i}}{Q^{2}/4}=\frac{4k_{ti}}{Q}(\cosh\eta_{i}-\cos\phi_{i})=2x_{i}\left(1-\frac{\cos\phi_{i}}{\cosh\eta_{i}}\right),$
(D.4)

when particle $k_{i}$ is recombined with the triggered $p_{1}$ jet. In the
above we expressed the jet mass in terms of the energy fraction
$x_{i}=2\omega_{i}/Q$, with $\omega_{i}=k_{ti}\cosh\eta_{i}$.

The probability of a single virtual soft gluon correction is given by:

$d\Gamma_{i}=-\frac{\textrm{\,d}^{3}\vec{k}_{i}}{2\omega_{i}(2\pi)^{3}}g_{s}^{2}\mathrm{C_{F}}\frac{2(p_{1}.p_{2})}{(p_{1}.k_{i})(p_{2}.k_{i})}=-\frac{\mathrm{C_{F}}\alpha_{s}}{\pi}\frac{\textrm{\,d}x_{i}}{x_{i}}d\eta\frac{\textrm{\,d}\phi_{i}}{\pi}\frac{1}{\cosh^{2}\eta_{i}-\cos^{2}\phi_{i}},$
(D.5)

where we note here that the collinear limit to the triggered jet $p_{1}$
corresponds to $\eta_{i}\to 0$ _and_ $\phi_{i}\to 0$.

Thus we can write the correction term due to clustering (Eq. (6.4)) as:

$\Sigma^{\rm{clus}}_{2}=\frac{1}{2!}\int\prod_{i}^{2}\textrm{\,d}\Gamma_{i}\Theta\left(x_{i}-\frac{\cosh\eta_{i}}{2(\cosh\eta_{i}-\cos\phi_{i})}\rho\right)\Xi_{2}(k_{1},k_{2}).$
(D.6)

Performing the energy-fraction integration yields an expression identical to
Eq. (6.8) with the full $R$-dependent coefficient, $\mathcal{F}_{2}(R)$, given
by:

$\mathcal{F}_{2}(R)=\frac{1}{\pi^{2}}\int\textrm{\,d}\eta_{1}\textrm{\,d}\phi_{1}\textrm{\,d}\eta_{2}\textrm{\,d}\phi_{2}\frac{1}{\cosh^{2}\eta_{1}-\cos^{2}\phi_{1}}\frac{1}{\cosh^{2}\eta_{2}-\cos^{2}\phi_{2}}\Xi_{2}(k_{1},k_{2}).$
(D.7)

Figure D.1: Full $R$-dependence of the two-gluon clustering coefficient
$\mathcal{F}_{2}$.

This integration can be performed numerically to extract the value of
$\mathcal{F}_{2}$ for arbitrary $R$. However it proves useful, in the simple
case of $\mathcal{F}_{2}$ in order to obtain an analytic expression at least
as a power-series, to introduce the polar variables $r$ and $\alpha$ defined
by:

$\eta=r\cos\alpha,\qquad\phi=r\sin\alpha.$ (D.8)

We rewrite Eq. (D.7) as:

$\mathcal{F}_{2}=\frac{1}{\pi^{2}}\int_{R}^{\infty}r_{1}\textrm{\,d}r_{1}\int_{-\pi}^{\pi}\textrm{\,d}\alpha_{1}\int_{0}^{R}r_{2}\textrm{\,d}r_{2}\int_{-\pi}^{\pi}\textrm{\,d}\alpha_{2}\Theta(2r_{2}\cos(\alpha_{2}-\alpha_{1})-r_{1})\times\\\
\times\frac{1}{\cosh^{2}(r_{1}\cos\alpha_{1})-\cos^{2}(r_{1}\sin\alpha_{1})}\,\,\frac{1}{\cosh^{2}(r_{2}\cos\alpha_{2})-\cos^{2}(r_{2}\sin\alpha_{2})}.$
(D.9)

We note that one can expand the second line of Eq. (D.9) in powers of $r_{i}$
so as to write the result as a power-series in $R$. To first order, the second
line expands as $1/r_{1}^{2}1/r_{2}^{2}$. We thus immediately identify this
integral as the one for the small-angles approximation, whose result is
$\pi^{2}/54\approx 0.183$. Performing higher-order integrals of the expansion
of the integrand yields the following result:

$\mathcal{F}_{2}(R)=0.183+0.0246R^{4}+0.00183R^{8}+0.000135R^{12}+\mathcal{O}(R^{16}).$
(D.10)

This expansion is actually valid for values of $R$ up to order unity, a claim
which is backed-up by fully performing the integral (D.9) numerically via
Monte Carlo methods and comparing to the analytical estimate (D.10), Fig. D.1.
We note that the clustering coefficient varies very slowly with $R$.

$R$ | $0$ | $0.1$ | $0.4$ | $0.7$ | $1.0$ | $1.2$
---|---|---|---|---|---|---
$\mathcal{F}_{2}$ | $0.183$ | $0.183$ | $0.184$ | $0.188$ | $0.208$ | $0.242$
$\mathcal{F}_{3}$ | $-0.052$ | $-0.053$ | $-0.053$ | $-0.055$ | $-0.061$ | $-0.072$
$\mathcal{F}_{3}^{\text{C/A}}$ | $-0.028$ | $-0.029$ | $-0.029$ | $-0.029$ | $-0.030$ | $-0.031$
$\mathcal{F}_{4}$ | $0.022$ | $0.023$ | $0.023$ | $0.023$ | $0.024$ | $0.027$
Table D.1: Estimates of the clustering coefficients $\mathcal{F}_{n}$ for
$n=2,3,4$ at different values of the jet radius $R$ in the $k_{t}$ and C/A
algorithms. Recall that
$\mathcal{F}_{2}^{\text{C/A}}=\mathcal{F}_{2}^{k_{t}}(\equiv\mathcal{F}_{2})$
and that $\mathcal{F}_{4}^{\text{C/A}}$ has not been computed. Note that
$\mathcal{F}_{3}^{\text{C/A}}$ and $\mathcal{F}_{4}$ are more difficult to
evaluate numerically than $\mathcal{F}_{3}$ (and obviously $\mathcal{F}_{2}$).
Thus their final answers are subjected to larger errors.

Finally we note that the calculation of $\mathcal{F}_{3}^{(\text{C/A})}$ and
$\mathcal{F}_{4}$ may be performed in the same way. Schematically we have:

$\mathcal{F}_{n}^{(\text{C/A})}(R)=\frac{1}{\pi^{n}}\int\prod_{i}^{n}\textrm{\,d}\eta_{i}\textrm{\,d}\phi_{i}\frac{1}{\cosh^{2}\eta_{i}-\cos^{2}\phi_{i}}\Xi_{n}^{(\text{C/A})}(k_{1},k_{2},\cdots,k_{n}),$
(D.11)

where the step function $\Xi_{n}^{(\text{C/A})}$ is expressed in terms of the
distances $d_{i}=\eta_{i}^{2}+\phi_{i}^{2}$ and
$d_{ij}=\delta\eta_{ij}^{2}+\delta\phi_{ij}^{2}$, respectively replacing
$\theta_{i}$ and $\theta_{ij}$ in, e.g., Eq. (6.10). We provide full numerical
estimates for all coefficients in table D.1.

##  Appendix E Radiators

###  E.1 Dipole calculations for the global part

In order to carry out the calculations for the various dipole terms required
for the resummation detailed in the main text we shall make use of the
following Born kinematics for the hard incoming and outgoing particles,

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{s}}{2}x_{1}\left(1,0,0,1\right),$ $\displaystyle
p_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{s}}{2}x_{2}\left(1,0,0,-1\right),$ $\displaystyle
p_{3}$ $\displaystyle=$ $\displaystyle p_{t}\left(\cosh y,1,0,\sinh y\right),$
$\displaystyle p_{4}$ $\displaystyle=$ $\displaystyle p_{t}\left(\cosh
y_{r},-1,0,\sinh y_{r}\right),$ (E.1)

where, as in the main, text $p_{1}$ and $p_{2}$ denote the four-momenta of the
incoming hard partons, $p_{3}$ that of the jet whose mass is being considered
and $p_{4}$ the four-momentum of a recoiling jet. In the case of a final state
vector boson instead of a recoiling jet the contribution from $p_{4}$ to the
dipole calculations is of course absent as the Z boson is colour neutral.
Further we have indicated by $x_{1}$ and $x_{2}$ the fractional momenta of
incoming partons relative to the colliding hadrons $P_{1}$ and $P_{2}$
respectively. Also $y$ denotes the rapidity of the measured jet, $y_{r}$ that
of the recoiling jet and $p_{t}$ their transverse momenta. One can, by
ignoring recoil, use the Born kinematics outlined above even in the presence
of a soft emission, whose momentum can be expressed as

$k=k_{t}\left(\cosh\eta,\cos\phi,\sin\phi,\sinh\eta\right);$ (E.2)

where transverse momentum $k_{t}$, rapidity $\eta$ and azimuth $\phi$ are all
measured with respect to the beam direction.

We shall now set up the various dipole contributions to the jet mass cross-
section. We recall from the main text that eventually these dipole
contributions will exponentiate with appropriate colour factors, so what
follows below is in essence a calculation of various components of the
resummed exponent.

####  E.1.1 In-in dipole

The in-in dipole is made from incoming partons $1$ and $2$ and provides the
contribution

$\displaystyle\mathcal{R}_{12}$ $\displaystyle=$ $\displaystyle\int
k_{t}\textrm{\,d}k_{t}\textrm{\,d}\eta\frac{\textrm{\,d}\phi}{2\pi}\frac{\alpha_{s}\left(\kappa_{t,12}\right)}{2\pi}\frac{(p_{1}.p_{2})}{(p_{1}.k)(p_{2}.k)}\Theta\left(\frac{2k_{t}}{p_{t}}\left(\cosh(\eta-y)-\cos\phi\right)-v\right)$
(E.3)
$\displaystyle\Theta\left(R^{2}-\left((\eta-y)^{2}+\phi^{2}\right)\right)$

We note that $\kappa^{2}_{t,12}=2(p_{1}.k)(p_{2}.k)/(p_{1}.p_{2})=k_{t}^{2}$
and carry out the integration over $\eta$ and $\phi$. In doing so we can
neglect the dependence of the jet mass on $\eta,\phi$ and just retain its
dependence on $k_{t}$ since the coefficient of $k_{t}$ which contains the
$\eta,\phi$ dependence, produces terms that are below single logarithmic
accuracy. Performing the integral over $\eta$ and $\phi$ over the interior of
the jet region gives us to the relevant single-log accuracy that

$\mathcal{R}_{12}={R^{2}}\int_{vp_{t}}^{p_{t}}\frac{\alpha_{s}(k_{t})}{2\pi}\frac{\textrm{\,d}k_{t}}{k_{t}},$
(E.4)

where the lower limit of integration stems from the constraint on the jet-mass
variable. We note that the dipole consisting of the two incoming partons gives
rise to a pure single-logarithmic behaviour as is evident from the fact that
the emitted gluon is inside the jet region, away from the hard legs
constituting the dipole and hence there are no collinear enhancements. The
soft wide-angle single logarithm we obtain is accompanied by an $R^{2}$
dependence on jet radius, reflecting the integration over the jet interior.

####  E.1.2 In-recoil dipoles

As before for the in-in case, for a dipole made up of an incoming parton and
one recoiling against a measured jet, we have that the emitted gluon is in the
interior of the jet away from the emitting legs, and produces no collinear
logarithm. We have that

$\displaystyle\mathcal{R}_{14}$ $\displaystyle=$ $\displaystyle\int
k_{t}\textrm{\,d}k_{t}\textrm{\,d}\eta\frac{\textrm{\,d}\phi}{2\pi}\frac{\alpha_{s}\left(\kappa_{t,14}\right)}{2\pi}\frac{(p_{1}.p_{4})}{(p_{1}.k)(p_{4}.k)}\Theta\left(\frac{2k_{t}}{p_{t}}\left(\cosh(\eta-y)-\cos\phi\right)-v\right)$
(E.5)
$\displaystyle\Theta\left(R^{2}-\left((\eta-y)^{2}+\phi^{2}\right)\right).$

However , in this case we have that

$\kappa_{t,14}^{2}=\frac{2(p_{1}.k)(p_{4}.k)}{(p_{1}.p_{4})}=2k_{t}^{2}e^{y_{r}-\eta}\left(\cosh(\eta-
y_{r})+\cos\phi\right).$ (E.6)

To single logarithmic accuracy, we can write the result into the form

$\int_{v}^{1}\frac{\textrm{\,d}x}{x}\alpha_{s}(xp_{t})\int\frac{e^{\eta-
y_{r}}}{\cosh(\eta-
y_{r})+\cos\phi}\Theta\left(R^{2}-\left((\eta-y)^{2}+\phi^{2}\right)\right).$
(E.7)

It is possible to evaluate the integral over $\eta,\phi$ as a power series in
$R$, such that the overall result for this dipole reads

$\int_{v}^{1}\frac{\textrm{\,d}x}{x}\frac{\alpha_{s}(xp_{t})}{2\pi}\left(\frac{R^{2}}{2}\frac{e^{\Delta
y}}{\left(1+\cosh\Delta
y\right)}+\frac{R^{4}}{32}{\mathrm{sech}}^{4}\frac{\Delta
y}{2}+\mathcal{O}\left(R^{6}\right)\right)$ (E.8)

where $\Delta y=y-y_{r}$ For the $\mathcal{R}_{24}$ dipole one obtains the
same result as above, with the replacement of $\Delta y$ by $-\Delta y$.

####  E.1.3 Jet-recoil dipole

For this dipole we have also an involvement of the parton that initiates the
measured jet as one of the hard emitting legs. The consequent collinear
singularity, in addition to the pole due to soft emission, gives rise to
double-logarithmic contributions. In this case one needs to evaluate the
argument of the running coupling more carefully whereas for soft wide-angle
pieces, dealt with previously, all that is important at our accuracy is that
the argument of the running coupling is proportional to the energy of the soft
gluon being considered.

In the present case we start with the integral

$\displaystyle\mathcal{R}_{34}$ $\displaystyle=$ $\displaystyle\int
k_{t}\textrm{\,d}k_{t}\textrm{\,d}\eta\frac{\textrm{\,d}\phi}{2\pi}\frac{\alpha_{s}\left(\kappa_{t,34}\right)}{2\pi}\frac{(p_{3}.p_{4})}{(p_{3}.k)(p_{4}.k)}\Theta\left(\frac{2k_{t}}{p_{t}}\left(\cosh(\eta-y)-\cos\phi\right)-v\right)$
(E.9)
$\displaystyle\Theta\left(R^{2}-\left((\eta-y)^{2}+\phi^{2}\right)\right)$

where

$\kappa_{t,34}^{2}=\frac{2(p_{3}.k)(p_{4}.k)}{(p_{3}.p_{4})}=2k_{t}^{2}\frac{\left(\cosh\left(y_{r}-\eta\right)+\cos\phi\right)\left(\cosh\left(y-\eta\right)+\cos\phi)\right)}{1+\cosh\Delta
y}.$ (E.10)

Next we introduce the variables $r$ and $\theta$ such that
$y-\eta=r\cos\theta$ and $\phi=r\sin\theta$ such that the in jet condition
$(y-\eta)^{2}+\phi^{2}<R^{2}$, simply produces $r<R$, while $\theta$ takes
values between $0$ and $2\pi$. Thinking of Eq. (E.9) as a power series in $r$
we can isolate the collinear-enhanced terms which produces double logarithms
and separate it from terms that are finite as $r\to 0$, which upon integration
produce terms that behave as powers of $R$ corresponding to soft wide-angle
contributions. The contribution $\mathcal{R}_{34}$ can thus be broken into
terms that contain collinear enhancements (and hence leading double
logarithms) and pure soft single-logarithmic pieces:

$\mathcal{R}_{34}=\mathcal{R}_{34}^{\mathrm{coll.}}+\mathcal{R}_{34}^{\mathrm{soft}},$
(E.11)

where the pure soft terms can be expressed as a power series in $R$:

$\mathcal{R}_{34}^{\mathrm{soft}}=\int_{v}^{1}\frac{\alpha_{s}(xp_{t})}{2\pi}\frac{\textrm{\,d}x}{x}\left(\frac{R^{2}}{4}\tanh^{2}\frac{\Delta
y}{2}+\frac{R^{4}}{1152}\mathrm{sech}^{4}\frac{\Delta
y}{2}\left(-5+\cosh\Delta y\right)^{2}+\mathcal{O}(R^{6})\right),$ (E.12)

where we have neglected terms that vanish as $v\to 0$.

The collinear contributions require us to be more precise in treating the
argument of the running coupling. In particular, in the limit of small $r$,
which is the collinear region, we obtain

$\mathcal{R}_{34}^{\mathrm{coll.}}=\int_{v}^{R^{2}}\frac{dr^{2}}{r^{2}}\frac{d\theta}{2\pi}\int_{\frac{v}{R^{2}}}^{1}\frac{\textrm{\,d}x}{x}\frac{\alpha_{s}}{2\pi}\left(xrp_{t}\right)$
(E.13)