---
abstract: 'We review the status of non-perturbative analyses of multi-jet event shape distributions and mean values, highlighting the physical insight on QCD dynamics they can provide.'
author:
- 'A. Banfi'
title: 'Why multi-jet studies?'
---

INTRODUCTION
============

In every QCD observable, perturbative (PT) and non-perturbative (NP) dynamics are inseparable, the reason being that the observed degrees of freedom are not quarks and gluons, the elementary particles entering the QCD bare Lagrangian, but hadrons, whose description in terms of partons is well beyond the domain of PT theory. Fortunately, at least for sufficiently inclusive observables, the difference between parton and hadron level predictions is suppressed by inverse powers of the hard scale of the process. As far as the latter is short-distance dominated, one can safely compute QCD observables using the PT parton language, and interpret the discrepancy with experimental data, in case any is seen, as the need for NP hadronisation corrections. One can then adopt two complementary approaches. One is to consider observables whose hadronisation corrections are almost negligible, for instance total cross sections or inclusive non-QCD particle distributions. These observables can be computed in PT QCD and exploited to determine the value of the coupling ${\alpha_s}$ [@alpha-exp]. The other approach is to consider observables which are very sensitive to NP physics, in order to have an insight on the hadronisation mechanism. The best known example is event shape variable distributions and mean values. These variables are constructed by combining final state momenta to obtain a number that gives an idea of the geometrical properties of hadron energy-momentum flow. The value of an event shape $V$ is related to the scale at which hadrons are probed, so that measuring event shape distributions makes it possible to study physics at very different scales, which range from the domain of PT QCD ($V\sim 1$) down to the confinement region ($V \sim \Lambda_\mathrm{QCD}$) where the quark/gluon language is scarcely applicable. For this reason, although shape variables have been used to measure ${\alpha_s}$, they are ideal for investigating properties of QCD dynamics. So far, both experimental and theoretical investigations have been restricted only to two-jet event shapes (see [@DSreview] for a review). Here we will discuss how the extension of such studies to multi-jet event shapes can shed further light on the interplay between PT and NP physics in QCD observables.

TWO-JET STUDIES
===============

NP correction to event shapes {#sec:NPcorr}
-----------------------------

Experimental data clearly indicates that PT QCD alone is not enough to predict event-shape distributions and mean values [@evshape-exp]. Let us consider the mother of all event shapes, the thrust [@thrustdef] $$\label{eq:thrust}
  T = \max_{\vec n_T}\frac{ 
    \sum_h|\vec p_h \cdot \vec n_T|}{\sum_h |\vec p_h|}\>, 
  \qquad \tau \equiv 1-T\>.$$ The thrust is a measure of particle alignment, and is a typical two-jet variable, since it vanishes in the limit of two narrow jets. As one can see from fig. \[fig:meanT\], the dependence of the mean value of $1-T$ on the $e^+e^-$ centre-of-mass energy $Q$ is correctly described only after adding to the QCD fixed order prediction a NP $1/Q$-suppressed correction $$\label{eq:tau-mean}
  {\left\langle \tau \right\rangle}={\left\langle \tau \right\rangle}_{\mathrm{PT}}+ {\left\langle \tau \right\rangle}_{\mathrm{NP}}\>,\qquad 
  {\left\langle \tau \right\rangle}_{\mathrm{PT}}= {\alpha_s}(Q) \>\tau_1+{\alpha_s}^2(Q)\> \tau_2+\ldots \>,\qquad
  {\left\langle \tau \right\rangle}_{\mathrm{NP}}\simeq \frac{C_\tau}{Q}\>,$$ where $C_\tau\simeq 1\mathrm{GeV}$ when ${\left\langle \tau \right\rangle}_{\mathrm{PT}}$ is evaluated at next-to-leading order (NLO).

![image](mean_thrust.eps){width=".5\textwidth"}

One might think that such a discrepancy could be removed by including higher orders in the PT expansion. Actually Sterman observed that a term $18 \>\alpha_s^3$ can already mimic a $1/Q$ behaviour [@Sterman]. However, theoretical analyses show that the PT series is an intrinsically ill-defined object, since it is doomed to diverge factorially (for a review, see [@Beneke]). Attempts to regularise such a divergence give rise to a power-suppressed ambiguity, known as infrared (IR) renormalon. Looking more closely at the origin of the divergence, one can see that it arises when resumming [*renormalon chain*]{} graphs containing an arbitrary number of linked quark or gluon bubbles [@renormalon]. A renormalon resummation for the thrust mean value yields $$\label{eq:tau-PT}
  {\left\langle \tau \right\rangle}_{{\mathrm{PT}}} \simeq
  \frac{4C_F{\alpha_s}}{\pi}\sum_{n=0}^\infty  
  n!\left(\frac{2\beta_0{\alpha_s}}{4\pi}\right)^n n^{\beta_1}
  \simeq 2 C_F \int\frac{dk_t}{k_t}d\eta \>
  \frac{{\alpha_s}(k_t)}{\pi} \>\frac{k_t}{Q}e^{-|\eta|}\>,$$ where $\beta_0$ and $\beta_1$ are the first two coefficients of the QCD beta function. After a renormalon analysis, the series in eq. (\[eq:tau-PT\]) gives a $1/Q$ ambiguity. The last equality in eq. (\[eq:tau-PT\]) results from the fact that the series can be seen as the PT expansion of the integral of the running coupling ${\alpha_s}(k_t)$ down to the infrared. Since the PT coupling develops a Landau singularity at low momenta, one may be tempted to ascribe the divergence to the presence of the Landau pole in the $k_t$ integration contour. However, eq. (\[eq:tau-PT\]) shows that the divergence of ${\left\langle \tau \right\rangle}_{\mathrm{PT}}$ is determined only by $\beta_0$ and $\beta_1$, i.e. it is independent of the particular coupling adopted. This naive observation is confirmed by more refined theoretical analyses, which show that IR renormalons are always present whatever is the behaviour of the coupling at low scales [@DU].

The main message of this discussion is that, in order to obtain a satisfactory [*theoretical*]{} understanding of event shape observables, PT theory must always be supplemented with information on QCD dynamics at low scales. In particular, a size of the power correction of around $1\mathrm{GeV}$ suggests that NP effects arise from partons which have started the blanching process that leads to the formation of hadrons.

Two-jet shapes and the Feynman tube model {#sec:Feynman}
-----------------------------------------

Consider now the specific case of an event-shape $V$ in $e^+ e^-$ annihilation that vanishes in the two-jet limit (thrust, $C$-parameter, heavy-jet mass, etc.). The mean value ${\left\langle V \right\rangle}$ will receive its main contribution from hard particles whose transverse momenta (with respect to the thrust axis) are of the order of the hard scale $Q$. There are however soft hadrons, with transverse momenta up to about $1\mathrm{GeV}$, whose contribution to ${\left\langle V \right\rangle}$ cannot be fully computed with PT techniques. The Feynman tube model [@tube] gives a phenomenological description of such hadrons by assuming that their distribution is uniform in rapidity: $$\label{eq:dnh}
  \frac{dn_h}{d\ln k_t d\eta} = \Phi_h(k_t)\>.$$ Observing also that their contribution $\delta V$ to $V$ is additive $$\label{eq:deltaV}
  \delta V \simeq \sum_i \frac{k_{ti}}{Q} f_V(\eta_i)\>,$$ one obtains a $1/Q$-suppressed NP correction ${\left\langle V \right\rangle}_{\mathrm{NP}}$ to ${\left\langle V \right\rangle}$, given by $$\label{eq:VNP}
  {\left\langle V \right\rangle}_{\mathrm{NP}}= \sum_h \int \frac{dk_t}{k_t}d\eta 
  \>\Phi_h(k_t) \frac{k_t}{Q} f_V(\eta) =
  c_V\frac{{\left\langle k_t \right\rangle}_{\mathrm{NP}}}{Q}\>,\qquad   c_V = \int_{-\!\infty}^\infty\!\!\! d\eta 
\>f_V(\eta)\>,$$ where $c_V$ is a calculable coefficient and $$\label{eq:ktnp}
  {\left\langle k_t \right\rangle}_{\mathrm{NP}}= 
  \sum_h \int \frac{dk_t}{k_t} \Phi_h(k_t)\> k_t\>,$$ is the average transverse momentum of the produced hadrons, and represents a genuine NP quantity.

The assumption of eq. (\[eq:dnh\]) makes it possible to completely factorise rapidity and transverse momentum dependence. Furthermore, eq. (\[eq:VNP\]) implies that power corrections to two-jet event shapes are [*universal*]{}, in the sense that, after computing the variable dependent coefficient $c_V$, they depend only on the NP parameter ${\left\langle k_t \right\rangle}_{\mathrm{NP}}$, which is the same for all variables.

A comment is in order concerning the validity of eq. (\[eq:dnh\]). Since in a hard collision most particles are aligned along the thrust axis, one expects that the distribution of hadrons away from the jets should be invariant over small boosts along the thrust axis direction, but that, as soon as one moves forward in rapidity, eq. (\[eq:dnh\]) gets modified. Fortunately, for most event shapes $f_V(\eta)$ strongly suppresses the contribution of hadrons at large rapidities, so that leading power corrections are determined only by soft hadrons in a central rapidity region, which are well described by eq. (\[eq:dnh\]).[^1]

This very same approach can be used to compute power correction to event-shape distributions. Since $V=V_{\mathrm{PT}}+\delta V$ with $\delta V
\sim \epsilon/Q$, the distribution $d\sigma/dV$ can be seen as a convolution of a PT distribution $d\sigma_{\mathrm{PT}}/dV_{\mathrm{PT}}$ and a NP shape function $f_{\mathrm{NP}}(\epsilon)$ [@shape-fun] $$\label{eq:dist-PT-NP}
  \frac{1}{\sigma}\frac{d\sigma}{dV} =
  \int dV_{\mathrm{PT}}\> d\epsilon \>\frac{1}{\sigma}\frac{d\sigma_{\mathrm{PT}}}{dV_{\mathrm{PT}}}(V_{\mathrm{PT}}) 
  \>f_{\mathrm{NP}}(\epsilon)\> \delta\left(V-V_{\mathrm{PT}}-\frac{\epsilon}{Q}\right)\>.$$ In the region $VQ \gg {\left\langle \epsilon \right\rangle}$ one can expand $f_{\mathrm{NP}}(\epsilon)$ around ${\left\langle \epsilon \right\rangle}$ and obtain $$\frac{1}{\sigma}\frac{d\sigma}{dV}
  \simeq
  \frac{1}{\sigma}\frac{d\sigma_{\mathrm{PT}}}{dV}
  \left(V-\frac{{\left\langle \epsilon \right\rangle}}{Q}\right)\>.$$ Recalling now that ${\left\langle \epsilon \right\rangle}\!/Q = {\left\langle V \right\rangle}_{\mathrm{NP}}$, one observes that leading power corrections to $d\sigma/dV$ result simply in a shift of the corresponding PT distribution, whose magnitude is the same as the power correction to ${\left\langle V \right\rangle}$ [@DW].

To conclude, the basic assumption of the Feynman tube model, that the distribution of soft (central) hadrons is uniform in rapidity, implies that leading $1/Q$ NP corrections both to mean values and distributions can be expressed in terms of a single universal parameter ${\left\langle k_t \right\rangle}_{\mathrm{NP}}$. This universality hypothesis has been tested against experimental data and is found to hold within $20\%$ accuracy [@DSreview].

EXTENSION TO MULTI-JET EVENT SHAPES
===================================

Intra-jet hadron distribution {#sec:intra-jet}
-----------------------------

When moving from two-jet to multi-jet events one encounters difficulties in extending eq. (\[eq:dnh\]). One can reasonably think that particles inside one of the jets (inter-jet hadrons) are produced uniformly in rapidity and azimuth with respect to the jet axis. However, for particles at large angles with respect to all jets (intra-jet hadrons), since there is no natural way to define $\eta$ and $\phi$, one does not expect $dn_h$ to have a simple expression. To find a reasonable assumption for $dn_h$, we start from two-jet events and consider the PT probability $dw(k)$ for the emission of a soft dressed gluon $k$ off a back-to-back quark-antiquark system (whose momenta are $p$ and $\bar p$) in a colour singlet [@thrust-np]: $$\label{eq:dw-2jet}
  dw(k) = C_F \frac{dk_t^2}{k_t^2} d\eta \frac{d\phi}{2\pi}
  \frac{{\alpha_s}(k_t)}{\pi}\>,\qquad
  \eta = \frac 12\ln\frac{\bar p k}{pk}\>,\qquad
  k_t^2 = \frac{(2pk)(2k \bar p)}{2p\bar p}\>,$$ where $\eta$ and $k_t$, the gluon rapidity and transverse momentum, have been written in a Lorentz invariant form and the coupling is taken in the physical CMW scheme [@CMW]. The inter-jet hadron distribution in eq. (\[eq:dnh\]) can be obtained from eq. (\[eq:dw-2jet\]) by replacing $2\>C_F {\alpha_s}(k_t)/\pi$ with $\sum_h
\Phi_h(k_t)$. In this way $dn_h$ might be interpreted as an effective measure of QCD interaction strength in the infrared. When the number of hard emitters is larger than two, the distribution $dw(k)$ can be written as a sum of contributions from all dipoles formed by the hard partons [@CatGraz]: $$\label{eq:dw-multi}
  dw(k) = \sum_{i<j}(-\vec T_i\cdot \vec T_j) 
  \frac{d\kappa^2_{ij}}{\kappa^2_{ij}} d\eta_{ij} \frac{d\phi_{ij}}{2\pi}
  \frac{{\alpha_s}(\kappa_{ij})}{\pi}\>,\qquad
  \eta_{ij} = \frac 12 \ln\frac{p_j k}{p_i k}\>, \qquad
  \kappa_{ij}^2 = \frac{(2p_i k)(2k p_j)}{2p_i p_j}\>,$$ where now $\kappa_{ij}$, $\eta_{ij}$ and $\phi_{ij}$ are the transverse momentum, rapidity and azimuth in the $(ij)$-dipole centre-of-mass frame, and the coupling is again in the CMW scheme. The vector $\vec T_i$ represents the colour charge of hard parton $p_i$. From colour conservation one has that for less than four emitters $\vec T_i\cdot \vec T_j$ are numbers, while starting from four partons they are actual matrices in colour space.

We now postulate that, after a suitable extension of the CMW coupling at low scales has been introduced, eq. (\[eq:dw-multi\]) gives the distribution of intra-jet hadrons in a multi-jet event. This corresponds to the so-called local parton-hadron duality (LPHD) hypothesis (see [@LPHD] for a recent review), which states that hadron flow is determined by parton flow. This assumption has solid phenomenological bases, since it describes very well hadron multiplicity flows [@multi] and string/drag effects in three-jet events [@drag].

Before constructing such an extended coupling, we first discuss to what extent the soft large-angle approximation is sufficient to compute $1/Q$ power corrections to event shapes. In fact, $\kappa_{ij}$ can reach the NP domain not only for soft large-angle, but also for hard collinear emissions. Considering emissions collinear to a given leg, we can classify event shapes according to whether they are damped in rapidity or not.

If $V$ is damped in rapidity, soft collinear particles with transverse momentum $k_{ti}$ and rapidity $\eta_i$ with respect to the given leg contribute to $V$ with a correction $$\label{eq:deltaV-damped}
  \delta V \simeq \sum_i \frac{k_{ti}}{Q} f_V(\eta_i)\>,\qquad
  f_V(\eta) \sim e^{-\eta} \>, \quad\>\eta\to\infty\>.$$ This implies that only large angle emissions contribute to $\delta V$, up to higher power corrections.

If $V$ is uniform in rapidity, we have $\delta V \sim \sum_i k_{ti}$, so that one might think that all rapidities contribute equally to $\delta V$. However, due to PT radiation, each hard parton takes a recoil, and acquires a transverse momentum $p_t \sim VQ$. But one measures transverse momenta with respect to a fixed axis (e.g. the thrust axis), and not with respect to the hard emitter’s direction (the jet broadenings [@broadNP] are a relevant example). Since, as we have seen in the previous section, leading power corrections come from the region $\delta V \ll V$, we must have then $k_{ti}/Q \sim
\delta V \ll p_t/Q \sim V$. This means that particles giving the leading contribution to $\delta V$ must be soft, and displaced from the hard emitting parton, i.e. soft at large angles. To conclude, eq. (\[eq:dw-multi\]) gives the physically correct starting point to compute $1/Q$ power corrections to event-shape variables.

Dokshitzer-Marchesini-Webber (DMW) extension of CMW coupling {#sec:DMW}
------------------------------------------------------------

The CMW coupling can be extended at low scales via the dispersion relation [@DMW] $$\label{eq:DMW}
  {\alpha_s}(\kappa) = -\int_0^\infty\!\!\frac{dm^2}{(m^2+\kappa^2)}\>\rho_s(m^2)=
  \kappa^2 \int_0^\infty\!\!\!\frac{dm^2}{(m^2+\kappa^2)^2}\>{\alpha_\mathrm{eff}}(m^2)\>,$$ where ${\alpha_\mathrm{eff}}(m^2)$ is the logarithmic derivative of the spectral density $\rho_s(m^2)$, and is itself a QCD coupling.

This approach automatically incorporates the LPHD hypothesis by assuming that hadronisation corrections are due to [*gluers*]{}, extra-soft gluons with transverse momenta $\kappa \sim
\Lambda_\mathrm{QCD}$, whose emission probability is ruled by the NP part of the dispersive coupling.

Within the dispersive DMW approach, power corrections to event shapes can be computed by considering the emission of a single gluer, and following a standard procedure [@np-ee]:

1.  real and virtual corrections are combined to obtain the dispersive representation of the QCD coupling. This means that in eq. (\[eq:dw-multi\]), for each dipole, one can make the substitution $$\label{eq:dw-DMW}
        \frac{{\alpha_s}(\kappa)}{\kappa^2} = 
        \int_0^\infty\!\!\!\frac{dm^2}{(m^2+\kappa^2)^2}{\alpha_\mathrm{eff}}(m^2)\>;$$

2.  the gluer is given a mass $m^2$, i.e. it is allowed to decay inclusively. In practice this results in replacing everywhere in the event-shape definition $\kappa^2$ with $\kappa^2+m^2$, so that the contribution to a variable $\delta V^{(ij)}_\mathrm{gluer}$ of a gluer emitted off the $(ij)$-dipole is $$\label{eq:dV-gluer}
        \delta V_\mathrm{gluer}^{(ij)} = \frac{\sqrt{\kappa^2+m^2}}{Q} 
        f_V^{(ij)}(\eta,\phi)\>,$$ where $\kappa$, $\eta$ and $\phi$ are, as usual, considered in the emitting dipole centre-of-mass frame. Then the leading power correction to ${\left\langle V \right\rangle}$ becomes $$\label{eq:dV-naive}
        {\left\langle V \right\rangle}^\mathrm{naive}_{\mathrm{NP}}= 
        \frac{{\left\langle \kappa \right\rangle}_\mathrm{naive}}{Q} 
        \sum_{i<j} (-\vec T_i \cdot \vec T_j)\> c_V^{(ij)}\>, \qquad
        c_V^{(ij)} = \int d\eta \frac{d\phi}{2\pi} f_V^{(ij)}(\eta,\phi)\>,$$ where $$\label{eq:kappa-naive}
        {\left\langle \kappa \right\rangle}_\mathrm{naive} = \int_0^\infty 
        \frac{d\kappa^2 dm^2}{(m^2+\kappa^2)^2} \sqrt{\kappa^2+m^2}
        \>\frac{{\alpha_\mathrm{eff}}(m^2)}{\pi} \simeq 
        \frac{2}{\pi} \int_0^\infty \frac{dm^2}{m^2} m \>\delta {\alpha_\mathrm{eff}}(m^2)\>,$$ turns out to be related to the $1/2$-moment of $\delta {\alpha_\mathrm{eff}}(m^2)$, the NP part of the dispersive coupling;

3.  non-inclusiveness of event shapes with respect to secondary gluon decay is accounted for by replacing ${\left\langle \kappa \right\rangle}_\mathrm{naive}$ with ${\left\langle \kappa \right\rangle}_{\mathrm{NP}}\equiv {\cal
        M}{\left\langle \kappa \right\rangle}_\mathrm{naive}$, where ${\cal M}$ is the so-called [*Milan*]{} factor [@thrust-np; @Milan]. It is also customary to rewrite ${\left\langle \kappa \right\rangle}_{\mathrm{NP}}$ in terms of $\alpha_0(\mu_I)$, the average of the dispersive coupling below the merging scale $\mu_I$, in such a way that the sum of PT and NP contributions to ${\left\langle V \right\rangle}$ is free of IR renormalons: $$\label{eq:alpha0}
        {\left\langle \kappa \right\rangle}_{\mathrm{NP}}= \frac{4 \mu_I}{\pi^2}{\cal M}
        \left(\alpha_0(\mu_I)-{\alpha_s}(Q)+{\cal O}({\alpha_s}^2)\right)\>, \qquad
        \alpha_0(\mu_I) = \int_0^{\mu_I}\frac{dk}{\mu_I}{\alpha_s}(k)\>.$$

We are now able to discuss what kind of information on QCD dynamics we obtain from power corrections to two- or multi-jet event shapes.

From eq. (\[eq:dV-naive\]) the power correction to a two-jet variable can be written in the form $$\label{eq:dV-2jet}
  {\left\langle V \right\rangle}_{\mathrm{NP}}= C_F \frac{{\left\langle \kappa \right\rangle}_{\mathrm{NP}}}{Q} c_V = 
  \frac{{\left\langle k_t \right\rangle}_{\mathrm{NP}}}{Q} c_V\>,$$ where ${\left\langle k_t \right\rangle}_{\mathrm{NP}}$ is the NP parameter appearing in eq. (\[eq:ktnp\]), and the last equality has been obtained by comparing eq. (\[eq:VNP\]) and eq. (\[eq:dV-naive\]). We see then that both the dispersive approach and the Feynman tube model give the same result for ${\left\langle V \right\rangle}_{\mathrm{NP}}$. In fact universality of leading NP corrections to two-jet event shapes implies simply that central soft hadrons are produced uniformly in rapidity.

When increasing the number of jets, the NP correction to a multi-jet variable is given by $$\label{eq:dV-multi}
    {\left\langle V \right\rangle}_{\mathrm{NP}}=\frac{{\left\langle \kappa \right\rangle}_{\mathrm{NP}}}{Q} \>\tilde c_V\>, \qquad
    \tilde c_V\equiv\sum_{i<j} (-\vec T_i \cdot \vec T_j) \> c_V^{(ij)}\>.$$ The main features of this PT QCD inspired correction are:

1.  the same parameter ${\left\langle \kappa \right\rangle}_{\mathrm{NP}}$ determines the magnitude of power corrections for [*all*]{} event shapes and for [*any number*]{} of hard jets;

2.  ${\left\langle V \right\rangle}_{\mathrm{NP}}$ depends on the [*colour*]{} charges of emitting partons through the factors $\vec T_i \cdot \vec T_j$;

3.  the power correction is sensitive to the [*geometry*]{} of the underlying hard event (the angles between the jets) through the coefficients $c_V^{(ij)}$.

This colour and geometry dependence is a highly non-trivial property that, once established, would imply that hadronisation preserves the main characteristics of partonic energy-momentum flow.

Three-jet event shapes in $e^+e^-$ annihilation {#sec:3jet-ee}
-----------------------------------------------

We now consider the specific case of three-jet events in $e^+e^-$ annihilation. At Born level a three-jet event is made up of a quark $p_q$, an antiquark $p_{\bar q}$ and a gluon $p_g$. Transverse momenta are, as usual, defined with respect to the thrust axis $\vec n_T$, and one defines an event-plane as the one containing $\vec n_T$ and thrust major axis $\vec n_M$, defined as the one maximising the projection of transverse momenta. Every transverse momentum $\vec k_t$ can be decomposed into an in-plane component $k^\mathrm{in}$ and an out-of-plane component $k^{\mathrm{out}}$ as follows $$\label{eq:in-out}
 \vec k_t = k^\mathrm{in} \>\vec n_\mathrm{in} + k^{\mathrm{out}}\>\vec n_{\mathrm{out}}\>,\qquad
 \vec n_\mathrm{in}=\vec n_M\>, \quad
 \vec n_{\mathrm{out}}\equiv \vec n_T\times\vec n_M\>.$$ The two variables for which there exists a prediction for leading power corrections are the $D$-parameter [@dpar] and the thrust-minor $T_m$ (a.k.a. ${K_\mathrm{out}}$) [@kout-ee].

The $D$-parameter is defined as the determinant of the tensor $$\label{eq:Dpar}
  \theta^{\alpha\beta} Q \equiv \sum_h \frac{p_h^\alpha p_h^\beta}{|\vec p_h|}\>, 
  \qquad D \equiv 27 \det \theta = 27 \lambda_1 \lambda_2 \lambda_3 \>.$$ The soft particle contribution $\delta D$ is given by $$\label{eq:deltaD}
  \delta D \simeq 27 \lambda_1 \lambda_2 
  \sum_i \frac{\kappa_i^2 \sin^2 \phi_i}{\omega_i Q} \>,$$ where $\kappa_i$ is the transverse momentum in the emitting dipole centre-of-mass frame and $\phi_i=0$ for an emission inside the event plane. Due to the presence of the energy $\omega_i$ in each denominator of eq. (\[eq:deltaD\]), this variable is damped in rapidity, and the coefficient $\tilde c_D$ is given by $$\label{eq:cD}
  \tilde c_D = C_F c_D^q(T,T_M) + C_F c_D^{\bar q}(T,T_M) + C_A c_D^g(T,T_M)\>,$$ where $c_D^q$, $c_D^{\bar q}$ and $c_D^g$ depend on the event geometry through $T$ and $T_M$.

The thrust-minor $T_m$ measures the out-of-event-plane momentum flow: $$\label{eq:Tm}
  T_m Q = {K_\mathrm{out}}\equiv \sum_h |p_h^{\mathrm{out}}| \>,\qquad
  \delta{K_\mathrm{out}}\simeq \sum_i \kappa_i |\sin \phi_i| \>.$$ This variable is uniform in rapidity, so that one needs to consider the fact that each hard emitting parton recoils against PT radiation. This results in $$\label{eq:ckout}
  \tilde c_{{K_\mathrm{out}}} \simeq {\left\langle |\sin\phi| \right\rangle}\left(C_F\ln\frac{Q_{q\bar q}}{|p_q^{\mathrm{out}}|}+
    C_F\ln\frac{Q_{q\bar q}}{|p_{\bar q}^{\mathrm{out}}|}+
    C_A\ln\frac{Q_{qg} Q_{g\bar q}}{Q_{q\bar q} |p_g^{\mathrm{out}}| }\right)\>,\qquad
  {\left\langle |\sin\phi| \right\rangle}\equiv\int_0^{2\pi}\frac{d\phi}{2\pi}|\sin \phi|=\frac 2\pi \>,$$ where $Q^2_{ij}=2 p_j \cdot p_j$. We notice that for each hard emitter the power correction is proportional to its colour charge and to the rapidity interval available to the emitted gluon. The fact that the NP correction depends on hard parton recoil implies that the final answer will be obtained after averaging $\tilde c_{{K_\mathrm{out}}}$ over the PT recoil distribution, which results in a complicated but very interesting interplay between PT and NP physics.

Fig. \[fig:Dpar\] shows the comparison between theoretical predictions [@dpar] and experimental data [@aleph-data] for the $D$-parameter differential distribution and mean values. Three-jet events are selected by imposing a lower limit $y_\mathrm{cut}\!=\!0.1$ on the (Durham) three-jet resolution $y_3$. The distribution in fig. \[fig:Dpar\] represents the first theoretical prediction for a three-jet variable at the state-of-the-art accuracy, which includes all-order PT resummation of logarithmic enhanced contribution to next-to-leading logarithmic (NLL) accuracy, matching to the corresponding NLO distribution obtained with <span style="font-variant:small-caps;">nlojet</span>++ [@nlojet], and $1/Q$ NP power corrections. For each Born configuration, the leading NP correction results in a shift of the PT distribution. The presented distribution is also slightly squeezed since the shift depends on the event geometry and one has to integrate over different Born configurations.

![Theoretical predictions [@dpar] for distribution (left) and mean values (right) of the $D$-parameter compared to ALEPH data [@aleph-data], for $Q=91.2\mathrm{GeV}$. Mean values for the data are computed from the corresponding distributions, with a $2\%$ estimated experimental error and an extra $1\%$ systematic error to account for the extraction of mean values from distributions [@hasko].[]{data-label="fig:Dpar"}](dpar_ee-th+dat.eps "fig:"){width=".45\textwidth" height=".3\textheight"} ![Theoretical predictions [@dpar] for distribution (left) and mean values (right) of the $D$-parameter compared to ALEPH data [@aleph-data], for $Q=91.2\mathrm{GeV}$. Mean values for the data are computed from the corresponding distributions, with a $2\%$ estimated experimental error and an extra $1\%$ systematic error to account for the extraction of mean values from distributions [@hasko].[]{data-label="fig:Dpar"}](dpar_ee-means.eps "fig:"){width=".45\textwidth" height=".3\textheight"}

The comparison to data is overall quite good. One can observe that the shift is quite large, about twice as much as in two-jet event shapes. This is mainly due to NP radiation off a gluon, which has a large colour charge. Although a full analysis of theoretical uncertainties remains to be performed [@AG-hardwork], the magnitude of the shift suggests that leading power corrections might not describe with sufficient accuracy the distribution around the peak, but that higher powers, or even a shape function, might be needed.

Three-jet event shapes in hadron-hadron collisions {#sec:3jet-hh}
--------------------------------------------------

One can construct three-jet event shapes also in hadron-hadron collisions. The simplest case is to consider a process with an electro-weak vector boson $Z^0$ or $W^\pm$ recoiling against a high-$p_t$ jet. The process involves three hard emitters, two incoming ($p_1$ and $p_2$) and one outgoing ($p_3$).

The event plane is the one containing the beam and the momentum $\vec
q$ of the vector boson, so that we have $\vec n_\mathrm{out} = (\vec q
\times \vec p_1)/(|\vec q||\vec p_1|)$. Since one does not measure inside the beam pipe, any measured hadron $p_h$ will have $|\eta_h| <
\eta_0$, where $\eta_0$ is the maximum available rapidity. Therefore the out-of-plane momentum flow ${K_\mathrm{out}}$ has to be defined as [@kout-hh] $$\label{eq:kout-hh}
  {K_\mathrm{out}}\equiv \sum_h |p_h^{\mathrm{out}}| \times \Theta(\eta_0-|\eta_h|)\>,$$ In the soft limit $\delta{K_\mathrm{out}}$ gets two contributions, one from particles emitted directly from the three hard partons, the other from hadrons produced in beam remnant interactions, the so-called soft underlying event [@pedestal]: $$\label{eq:deltakout-hh}
  \delta{K_\mathrm{out}}= \delta{K_\mathrm{out}}^\mathrm{gluer}+\delta{K_\mathrm{out}}^\mathrm{remnant}\>,\qquad
  \delta{K_\mathrm{out}}^\mathrm{gluer} \simeq {\sum_i}^\prime \kappa_i |\sin\phi_i| \>,
  \qquad 
  \delta{K_\mathrm{out}}^\mathrm{remnant} \simeq{\sum_i}^\prime k_{ti} |\sin\phi_i|\>,$$ where each primed sum is restricted to particles outside the beam pipe. In the term $\delta{K_\mathrm{out}}^\mathrm{gluer}$ the transverse momentum $\kappa_i$ is considered as usual in the emitting dipole centre-of-mass frame, while in $\delta{K_\mathrm{out}}^\mathrm{remnant}$ each $k_{ti}$ is in the laboratory frame.

As before, the gluer contribution depends on the same parameter ${\left\langle \kappa \right\rangle}_{\mathrm{NP}}$ as for $e^+e^-$ event shapes, and is given by: $$\label{eq:deltakout-gluer}
  \delta{K_\mathrm{out}}^\mathrm{gluer} = {\left\langle \kappa \right\rangle}_{\mathrm{NP}}\> \tilde c_{{K_\mathrm{out}}} \>,
  \qquad
  \tilde c_{{K_\mathrm{out}}} \simeq {\left\langle |\sin\phi| \right\rangle} 
  \left(C_1(\eta_0-\eta_3)+C_2(\eta_0+\eta_3)+
    C_3 \ln\frac{Q_t}{|p_3^{\mathrm{out}}|}\right)\>,$$ where $Q_t$ is the transverse momentum of the vector boson, which represents also the hard scale of the process, and $C_i$ is the colour charge of hard parton $p_i$. As in the $e^+e^-$ case, for each emitting hard parton, the power correction is proportional to the rapidity interval available to the emitted gluer.

The beam remnant contribution can be computed by assuming that the distribution of hadrons produced in soft beam-remnant interactions is uniform in rapidity, which gives $$\label{eq:deltakout-remnant}
  \delta{K_\mathrm{out}}^\mathrm{remnant} = 
  {\left\langle |\sin\phi| \right\rangle}\times{\left\langle k_t \right\rangle}_\mathrm{remnant} 
  \times 2\eta_0\>.$$ We have then access to the new NP parameter ${\left\langle k_t \right\rangle}_\mathrm{remnant}$, the average transverse momentum of hadrons produced in beam remnant interactions. This very same parameter will appear in power corrections to event shapes in hadronic dijet production [@hh-shapes].

A RICH PHENOMENOLOGICAL PROGRAM {#sec:pheno}
===============================

The considerations we have presented so far can be seen as the starting point for a rich phenomenological program. First of all one has to keep in mind that multi-jet event-shape observables are not ideal to perform pure PT studies aimed at precise determinations of the strong coupling ${\alpha_s}$. Large theoretical uncertainties are associated to PT higher orders and to NP power corrections, and jet selection cuts reduce the number of available events. However, from the point of view of understanding QCD dynamics, they are an incredibly valuable tool. A research program has already started with the study of power corrections to three-jet event shapes in $e^+e^-$ annihilation and DIS [@AG-hardwork], aimed at determining whether the NP parameter $\alpha_0$ is the same as for two-jet event shapes. Once the universality of $\alpha_0$ is established, one could move to considering hadron-hadron collisions. There one could test our understanding of beam-remnant interactions in a complementary way with respect to what has been done so far. At the moment the only observable used to test models of the underlying event is the away-from-jet particle/energy flow [@Field], for which we have a less clear theoretical understanding than we have for event shapes. Provided our modelling of the underlying event proves to be correct, we have access to the NP parameter ${\left\langle k_t \right\rangle}_\mathrm{remnant}$, and are able to use this parameter to study event-shapes in hadronic dijet production [@hh-shapes], where a lot of progress has been done in recent years, especially since when the automated resummation program <span style="font-variant:small-caps;">caesar</span> has become available [@caesar].

It was Yuri Dokshitzer and Pino Marchesini who initiated me to multi-jet studies. I therefore thank them and Giulia Zanderighi for a many-year collaboration on this subject. I am also grateful to the organisers for the possibility to participate to this very interesting and stimulating workshop. Finally, I wish to thank Gavin Salam for discussions and suggestions on the phenomenological analysis of the $D$-parameter.

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[^1]: Surprisingly enough, eq. (\[eq:dnh\]) gives the correct leading power correction also for variables that are uniform in rapidity. This is due to a tricky interplay between PT and NP QCD radiation, which will be discussed in the next section.
