process, while $x/z$ is the fraction of momentum of the incoming quark before it emits the collinear gluon. The elementary emission probability is the ratio of the second over the first expression in Eq. (43.5). In analogy with the final state radiation case, this ratio will appear in the exponent of the Sudakov form factor, that (after the inclusion of all splitting subprocesses) is given by

$$ \Delta_i^{\text{ISR}}(t, t') = \exp \left[ - \int_{t'}^t \frac{dt''}{t''} \frac{\alpha_S(t'')}{2\pi} \int_x^1 \frac{dz}{z} \sum_{jk} P_{j,ik}(z) \frac{f_j(t'', x/z)}{f_i(t'', x)} \right]. \quad (43.6) $$

Notice that there are two uses of the PDFs: they are used to compute the cross section for the basic hard process, and they control ISR via backward evolution. Since the evolution is generated with leading-logarithmic accuracy, it is acceptable to use two different PDF sets for these two tasks, provided they agree at the LO level.

In the context of GPMC evolution, each ISR emission generates a finite amount of transverse momentum. Details on how the recoils generated by these transverse “kicks” are distributed among other partons in the event, in particular the ones involved in the hard process, constitute one of the main areas of difference between existing algorithms, see Ref. [11]. An additional $O(1 \text{ GeV})$ of “primordial $k_T$” is typically added, to represent the sum of unresolved and/or non-perturbative motion below the shower cutoff scale.

### 43.1.3 Soft emissions and QCD coherence

Soft singularities arise in QCD due to the real or virtual emission of soft gluons. For example, the cross section for the emission of a soft gluon in $e^+e^-$ annihilation into hadrons is given by

$$ \begin{aligned} d\sigma_{q\bar{q}g} &\approx d\sigma_{q\bar{q}}C_F(4\pi\alpha_s) \left[ \frac{2 p_q \cdot p_{\bar{q}}}{p_q \cdot l \ p_{\bar{q}} \cdot l} \right] \frac{d^3 l}{2l^0(2\pi)^3} \\ &= d\sigma_{q\bar{q}} \frac{\alpha_s}{2\pi} C_F \frac{dl^0}{l^0} \frac{d\phi}{2\pi} \frac{d\cos\theta}{1 - \cos^2\theta}, \end{aligned} \quad (43.7) $$

where $p_q, p_{\bar{q}}$ and $l$ are the quark, antiquark and gluon momentum, and $\theta$ and $\phi$ are the polar and azimuthal angle of the gluon momentum with respect to the quark direction. Since the gluon is soft, we may assume that $p_q$ and $p_{\bar{q}}$ are unaffected by the gluon emission. The soft singularity is manifest in the $dl^0/l^0$ factor. Notice that also collinear singularities are present at the same time when $\theta \to 0$ and $\theta \to \pi$, corresponding to the gluon becoming collinear to either the quark or the antiquark. It is easy to check that in the collinear limits Eq. (43.7) becomes equivalent to Eq. (43.1) with $P_{q,gg}(z) = C_F 2/(1-z)$, i.e. the limiting form of $P_{q,gg}(z)$ when $z$ approaches 1. Thus, soft singularities coexist with collinear ones, so that two potentially large logarithms can arise simultaneously due to gluon emission.

Unlike the case of collinear emission, soft emission is not tied to a single emitting particle. The amplitude for the emission of a soft gluon from an external (incoming or outgoing) line with momentum $p$ is proportional to $p \cdot \epsilon/p \cdot l$. When squaring the amplitude, products like the one appearing in the square bracket of Eq. (43.1) arise for all pairs of external particles, with the product of a single emission amplitude with itself appearing only if $p^2 > 0$, i.e. for massive colored particles. Thus, interference plays here a crucial role. This is unlike the case of collinear singularities, where because of strong ordering a radiated parton cannot be collinear to more than one other parton.

It was shown in a set of publications (see Ref. [19]) that, within the conventional parton-shower formalism based on collinear factorization, the region of collinear and soft emissions can be correctly described by using the angle of the emissions as the ordering variable, rather than the virtuality, and by setting the argument of $\alpha_S$ at the splitting vertex equal to the relative parton transverse momentum after the splitting. Physically, the ordering in angle approximates the coherent interference arising from large-angle soft emission from a bunch of collinear partons. Without this effect, the particle multiplicity would grow too rapidly with energy, in conflict with $e^+e^-$ data. For this reason, angular ordering is

used as the default evolution variable in all versions of HERWIG (see Ref. [20]). To partially account for soft interference effects, an angular veto is imposed on the virtuality-ordered evolution in PYTHIA 6 [21]. In modern designs this type of fix is avoided, the soft problem being addressed from the start.

A radical alternative formulation of QCD cascades first proposed in Ref. [22] focuses upon soft emission, rather than collinear emission, as the basic splitting mechanism. It then becomes natural to consider a branching process where it is a parton pair (i.e. a dipole) rather than a single parton, that emits a soft parton. Adding a suitable correction for non-soft, collinear partons, one can simultaneously achieve the correct logarithmic structure for both the collinear and soft emissions in the so called leading color approximation, i.e. when terms suppressed by a power of the number of colors are neglected. The ARIADNE [23] and VINCIA [24] programs are based on this approach. Dipole-type showers [25] are also used by default in SHERPA [26] and exist as an option in HERWIG [27]. An alternative dipole-based model is available in PYTHIA and SHERPA via the DIRE [28] plugin. The $p_\perp$-ordered showers in PYTHIA 6 and 8 represent a hybrid, combining collinear splitting kernels with dipole kinematics [29].

### 43.1.4 Resummation

Shower Monte Carlo generators perform resummation of all-order collinear- and/or soft-enhanced perturbative contributions, and it is thus natural to compare them to QCD resummation calculations [7]. The latter start from the definition of specific infrared-safe observables, that develop towers of large logarithms in certain regions of phase space, typically organized as

$$ A(\alpha_S) \exp[Lg_0(\alpha_S L) + g_1(\alpha_S L) + \alpha_S g_2(\alpha_S L) + \dots] \quad (43.8) $$

where $L$ is the large logarithm, $\alpha_S$ is the strong coupling constant evaluated at some hard scale. The functions $A$ has a power expansion in $\alpha_S$ starting with a constant, and the functions $g_i$ have a power series expansion in their argument starting with the linear term. We talk about $N^n$LL accuracy if all $g_i$ functions for $i \le n$ are fully included in the resummation formula, and the $A$ function is given up to the order $n-1$ in $\alpha_S$. In particular we talk about LL accuracy if the function $g_0$ is included, NLL accuracy if also $g_1$ is included. In both cases only the constant term in $A$ is needed, while for $N^2$LL (or NNLL) $g_0, g_1$ and $g_2$ are included and $A$ is given up to order $\alpha_S$.

In general, a dedicated resummation calculation must be performed for each new observable. The predictions of shower MCs, on the other hand, are cast in terms of complete sets of final-state momenta, on which one can evaluate any observable; i.e., the shower algorithm itself is normally independent of the specific observable(s) under study. Because of this, it is not easy to qualifying the accuracy of a shower MC using the same criteria adopted in resummation calculation. In spite of this fact, shower MCs perform generally quite well in the description of observables that require resummation. This is related to their inclusion of several universal but formally subleading aspects. There is no guarantee, however, that the shower MC should perform to a certain level of accuracy for all distributions that require resummation. Thus, a more systematic treatment of subleading effects is quite desirable.

Several studies have appeared in the literature, aiming at either improving the current shower algorithms or formulate totally new ones, in such a way that the theoretical accuracy can be discussed in more precise terms [17, 30–35]. In particular, in refs [17, 30, 31] (the PanScales collaboration), it was shown that criteria for defining the accuracy of Shower Generators can indeed be given on a quite general ground. They identified two such criteria: the first one regards the approximations to multi-parton real matrix elements generated by the shower algorithm, that are required to be accurate in the region of phase space that are relevant for $N^n$LL accuracy. The second one regards the implementation of the virtual corrections, and it requires that for a large class of shape variables the shower MC must agree with the resummed calculation at the $N^n$LL level. The PanScales collaboration has constructed shower algorithms that are (with some restrictions) NLL accurate according to these criteria for $e^+e^-$ annihilation,
hadronic collisions, Deep Inelastic Scattering and Vector Boson Fusion processes (see [36] and references therein). Other NLL accurate (according to the same criteria) shower algorithms have been proposed in refs. [32] and [35].

### 43.1.5 Massive quarks

Quark masses act as a cut-off on collinear singularities. If the mass of a quark is below, or of the order of $\Lambda$, its effect in the shower is small. For larger quark masses, like in c, b, or t production, it is the mass, rather than the typical hadronic scale, that cuts off collinear radiation. For a quark with energy $E$ and mass $m_Q$, the divergent behavior $d\theta/\theta$ of the collinear splitting process is regulated for $\theta \le \theta_0 = m_Q/E$. We thus expect less collinear activity for heavy quarks than for light ones, which in turn is the reason why heavy quarks carry a larger fraction of the momentum acquired in the hard production process.

This feature can be implemented with different levels of sophistication. Using the fact that soft radiation is strongly suppressed at zero emission angle, older parton shower algorithms simply limited the shower emission to be not smaller than the angle $\theta_0$. More modern approaches are used in both PYTHIA, where mass effects are included using a kind of matrix-element correction method [37], and in HERWIG and SHERPA, where a generalization of the Altarelli-Parisi splitting kernel is used for massive quarks [38]. For newer approaches for QCD evolution in processes with massive partons see ref. [39] and references therein.

### 43.1.6 Color information

In event generators, quarks and antiquarks are represented by color lines, with arrows indicating the direction of color flow. In the limit of infinitely many colors (called the leading color approximation), each such line can be associated with a unique label; the probability for two quarks (or antiquarks) to have the same color (anticolor) vanishes. Moreover, in the same limit gluons can be represented by a pair of color lines with opposite arrows, as can be realized e.g. from the $SU(3)$ group relation $8 = 3 \otimes 3 \oplus 1$. The rules for color propagation are:

<figure><img src="image_276.png" /></figure>

During the shower development, partons are connected by color lines. We can have a quark directly connected by a color line to an antiquark, or via an arbitrary number of intermediate gluons, as shown in Fig. 43.1. It is also possible for a set of gluons to be connected cyclically in color, as e.g. in the decay $\Upsilon \to ggg$. The color information is used in angular-ordered showers, where

<figure><img src="image_277.png" /><figcaption>**Figure 43.1:** Color development of a shower in $e^+e^-$ annihilation. Color-neutral clusters of partons are indicated by the dashed under-brackets.</figcaption></figure>

the angle of color-connected partons (i.e. partons connected by the same color line) determines the initial angle for the shower development, and in dipole showers, where dipoles are formed by color-connected partons. It is also used in hadronization models, where the initial strings or clusters used for hadronization are formed by color-neutral clusters of partons.

### 43.1.7 Electromagnetic (and weak) corrections

The physics of photon emission from charged particles, and of photon splittings to pairs of charged fermions, can also be treated with a shower MC algorithm. High-energy electrons and quarks, for example, are accompanied by bremsstrahlung photons. Also here, similarly to the QCD case, electromagnetic corrections are of order $\alpha_{em} \ln(Q/m)$, where $m$ is the mass of the radiating particle, or even of order $\alpha_{em} \ln(Q/m) \ln(E_\gamma/E)$ for soft $\gamma$ emissions, so that, especially for the case of electrons, their inclusion in the simulation process is mandatory. This is done in most GPMCs (for a comparative study see [40]), which typically agree in collinear (DGLAP) limits but differ in whether and how soft (multipole)

QED interference effects are handled. The specialized generator PHOTOS [41] is sometimes used as an afterburner for an improved treatment of QED radiation in non-hadronic resonance decays.

For photon emissions off leptons, the shower can be continued down to virtualities arbitrarily close to the lepton mass shell (unlike the case in QCD). In practice, an infrared cutoff is still required for the shower algorithm to terminate. Therefore, there is always an energy cut-off for emitted photons that depends upon the implementations [40]. In the case of electrons, this energy is typically of the order of its mass. Electromagnetic radiation below this scale is not enhanced by collinear singularities, and is thus bound to be soft, so that the electron momentum is not affected by it.

For photons emitted from quarks, there is instead the obvious limitation that the photon wavelength cannot exceed the typical hadronic size. Longer-wavelength photons are in fact emitted by hadrons, rather than quarks. This last effect is in practice never modeled by existing shower MC implementations. Thus, electromagnetic radiation from quarks is cut off at a typical hadronic scale. Finally, hadron (and $\tau$) decays involving charged particles can produce additional soft bremsstrahlung. In PYTHIA, this is done via interfacing with PHOTOS, while HERWIG and SHERPA contain their own dedicated implementations [42, 43] based upon the YFS formalism [44], which is also adopted in the KKMC-hh Monte Carlo [45].

At energies significantly above the electroweak (EW) scale, showers involving emissions (and splittings) of weak gauge and Higgs bosons can also be relevant, with recent implementations based on collinear limits ranging from weak-boson emissions from fermions in PYTHIA 8 [46], to triple-boson couplings in HERWIG 7 [47], to the full set of EW branching processes in VINCIA [48].

### 43.1.8 Beyond-the-Standard-Model Physics

The inclusion of processes for physics beyond the Standard Model (BSM) in event generators is to some extent only a matter of implementing the relevant hard processes and (chains of) decays, with the level of difficulty depending on the complexity of the model and the degree of automation [49, 50]. Notable exceptions are long-lived colored particles [51], particles in exotic color representations, and particles showering under new gauge symmetries, with a growing set of implementations documented in the individual GPMC manuals. Further complications that may be relevant are finite-width effects (discussed in Sec. 43.1.9) and the assumed threshold behavior.

In addition to code-specific implementations [11], there are a few commonly adopted standards that are useful for transferring information and events between codes. Currently, the most important of these is the Les Houches Event File (LHEF) standard [52], normally used to transfer parton-level events from a hard-process generator to a shower generator. Another important standard is the Supersymmetry Les Houches Accord (SLHA) format [53], originally used to transfer information on supersymmetric particle spectra and couplings, but by now extended to apply also to more general BSM frameworks and incorporated within the LHEF standard [54]. Unfortunately, at present no standard method for handling multiple shower starting scales has been defined in the LHEF, so that further code, specific to the given GPMC, needs to be added to the interface in this case.

### 43.1.9 Decay Chains and Particle Widths

In most BSM processes and some SM ones, an important aspect of the event simulation is how decays of short-lived particles, such as top quarks, EW and Higgs bosons, and new BSM resonances, are handled. We here briefly summarize the spectrum of possibilities, but emphasize that there is no universal standard. Users are advised to check whether the treatment of a given code is adequate for the physics study at hand.

The appearance of an unstable resonance as a physical particle at an intermediate stage of the event generation implies that its production and decay processes are treated as being factorized. This is valid up to corrections of order $\Gamma/m_0$, with $\Gamma$ the width and $m_0$ the pole mass. States whose widths are a substantial fraction of their mass should instead be treated as intrinsically off-shell internal propagator lines.
For states treated as physical particles, two aspects are rele-
vant: the mass distribution of the decaying particle itself and
the distributions of its decay products. For the former, matrix-
element generators often use a simple $\delta$ function at $m_0$. The next
level up, typically used in GPMCs, is to use a Breit-Wigner dis-
tribution, which formally resums higher-order virtual corrections
to the mass distribution. Note, however, that this still only gen-
erates an improved picture for *moderate* fluctuations away from
$m_0$. Similarly to above, particles that are significantly off-shell
(in units of $\Gamma$) should not be treated as resonant, but rather as
internal off-shell propagator lines. In most GPMCs, further refine-
ments are included, for instance by letting $\Gamma$ be a function of $m$
("running widths") and by limiting the magnitude of the allowed
fluctuations away from $m_0$. We finally point out that NLO+PS
generators have appeared that can deal with resonances including
off-shell effects, non-resonance contributions and interference of
radiation generated in resonance decay and production, see [55]
and references therein. A new "interleaved" shower treatment
also allows for interference between production and decay in the
VINCIA model [48].

For the distributions of the decay products, the simplest treat-
ment is again to assign them their respective $m_0$ values, with
a uniform phase-space distribution. A more sophisticated treat-
ment distributes the decay products according to the differential
decay matrix elements, capturing at least the internal dynam-
ics and helicity structure of the decay process, including EPR-
like correlations. Further refinements include polarizations of the
external states [56] and assigning the decay products their own
Breit-Wigner distributions, the latter of which opens the possi-
bility to include also intrinsically off-shell decay channels, like
$H \to WW^*$.

GPMC manuals often give instructions on how to include new
decay modes, at varying levels of sophistications ranging from sim-
ple uniform phase-space sampling (which the user can reweight a
posteriori) and step-function thresholds, to fully matrix-element
weighted decay implementations including potential off-shell /
threshold effects.

During subsequent showering of the decay products, most parton-shower models will preserve the total invariant mass of the decayed resonance, so as not to skew the original resonance shape. In the context of passing externally generated LHEF files [52] to a GPMC for showering, note that this is only possible if the intermediate resonances are present (with status code 2) in the LHEF event record [57].

43.1.10 Matching and Merging with Fixed-Order Matrix Elements

Shower algorithms are based upon a combination of the collinear (small-angle) and soft (small-energy) approximations and are thus normally inaccurate for hard, wide-angle emissions (i.e., additional well-resolved jets). They also contain only the leading singular pieces of next-to-leading order (NLO) and higher corrections to the basic process.

Traditional GPMCs, like HERWIG and PYTHIA, have included for a long time the so called Matrix-Element Corrections (MEC), first formulated in Ref. [58] with later developments summarized in Ref. [11]. They are typically available for $2 \to 1$ or $1 \to 2$ processes, like DIS, vector boson and Higgs production and decays, and top decays. The MEC corrects the emission of the hardest jet at large angles, so that it becomes exact at LO. A generalization of the method to multiple emissions was formulated in [59].

Aside from MECs implemented directly in the GPMCs, the improvements on the parton-shower description of hard collisions have been made in two main directions: the so called Matrix Elements and Parton Shower matching (ME+PS from now on), and the matching of NLO calculations and Parton Showers (NLO+PS). We now discuss each of these, and then briefly summarize techniques becoming available for combining them.

The ME+PS method allows one to use tree-level matrix ele-
ments for hard, large-angle emissions. It was first formulated in
the so-called CKKW paper [60], and several variants have ap-
peared, including the CKKW-L, MLM, pseudoshower and MESS
methods, see Refs. [11, 61, 62] and references therein. So called
"Truncated Showers" are required [63] to maintain color coher-

ence when interfacing to angular-ordered parton showers, and
care must be taken to use consistent $\alpha_s$ choices for the real (ME-
driven) and virtual (PS-driven) corrections [64].

In the ME+PS method one typically starts by generating LO
matrix elements for the production of the basic process plus a
certain number $\le n$ of other partons. A minimum separation is
imposed on the produced partons, requiring, for example, that
the relative transverse momentum in any pair of partons is above
a given cut $Q_{\text{cut}}$. One then reweights these amplitudes in such a
way that, in the strongly ordered region, the virtual effects that
are included in the shower algorithm (i.e. running couplings and
Sudakov form factors) are also accounted for. At this stage, before
parton showers are added, the generated configurations are tree-
level accurate at large angle, and at small angle they match the
results of the shower algorithm, except that there are no emissions
below the scale $Q_{\text{cut}}$, and no final states with more than $n$ partons.
These kinematic configurations are thus fed into a GPMC, that
must generate all splittings with relative transverse momentum
below the scale $Q_{\text{cut}}$, for initial events with less than $n$ partons,
or below the scale of the smallest pair transverse momentum, for
events with $n$ partons. The matching parameter $Q_{\text{cut}}$ must be
chosen to be large enough for fixed-order perturbation theory to
hold, but small enough so that the shower is accurate for emis-
sions below it. Notice that the accuracy achieved with MEC is
equivalent to that of ME+PS with $n = 1$, where MEC has the
advantage of not having a matching parameter $Q_{\text{cut}}$.

The popularity of the ME+PS method is due to the fact that
processes with many jets often appear as backgrounds to new-
physics searches. These jets are typically required to be well sep-
arated, and to have large transverse momenta. These kinematic
configurations are exactly those for which pure shower algorithms
are unreliable, hence it is mandatory to describe them using at
least LO matrix elements.

Several ME+PS implementations use existing LO generators, like ALPGEN [65], MADGRAPH [66], and others summarized in Ref. [61], for the calculation of the matrix elements, and feed the partonic events to a GPMC like PYTHIA or HERWIG using the Les Houches Interface for User Processes (LHI/LHEF) [52, 57]. SHERPA and HERWIG also include their own matrix-element generators.

The NLO+PS methods promote the accuracy of the genera-
tion of the basic process from LO to NLO in QCD. They must
thus include the radiation of one extra parton with tree-level ac-
curacy, since this radiation constitutes a NLO correction to the
basic process. They must also include NLO virtual corrections.
They can be viewed as an extension of the MEC methods with the
inclusion of NLO virtual corrections. They are however more gen-
eral, since they are applicable to processes of arbitrary complexity.
Two of these methods are now widely used: MC@NLO [67] and
POWHEG [63,68], with several alternative methods now also being
pursued, see [69] and references therein, and the recent method
presented in [70].

NLO+PS generators produce NLO accurate distributions for inclusive quantities, and generate the hardest jet with tree-level accuracy. Note also that the optimal tuning of an NLO+PS generator may well be different from that of the pure PS.

Several NLO+PS processes are implemented in the MC@NLO program [67], together with the AMC@NLO development [71], and in the POWHEG-BOX framework [68]. HERWIG supports its own variants of POWHEG and MC@NLO for several processes. SHERPA instead implements a variant of the MC@NLO method.

For applications that require an accurate description of more
than one hard, large-angle jet associated with the primary pro-
cess, ME+PS schemes are still superior to NLO+PS ones. Ideally,
one would like to improve NLO generators in such a way that also
the production of associated jets achieves NLO accuracy. The
FXFX [72], UNLOPS [73], MiNLO [74] and MEPS@NLO [75] meth-
ods address this problem. The solution of this problem is also
a prerequisite for the construction of NNLO+PS generators, i.e.
generators that, besides being NLO accurate for the production
of an associated jet, are also NNLO accurate for fully inclusive
observables. Three different approaches have appeared in the lit-
erature for dealing with this problem: the UN²NLOPS, based
upon UNLOPS method [76]; methods extending the MiNLO ap-
proach by reweighting [77] and by the so called MiNNLOPS technique [78]; and the GENEVA method [79]. Several processes have been implemented with the MiNLO related and GENEVA methods, mostly for the production of color singlet systems, with the exception of the recent MiNNLOPS implementation of tt production (see refs. [69, 80, 81] for an extensive list of references).

## 43.2 Hadronization Models

In the context of GPMCs, hadronization denotes the process by which a set of colored partons (after showering) is transformed into a set of “primary hadrons”, which may then subsequently decay further (to “secondary hadrons”). This non-perturbative transition takes place at the hadronization scale $Q_{had}$, which by construction is equal to the infrared cutoff of the parton shower. In the absence of a first-principles solution to the relevant dynamics, GPMCs use QCD-inspired phenomenological models to describe this transition.

An important result in “quenched” lattice QCD (see Chap. 17 of PDG book) is that the potential energy between two partons with opposite color charges grows linearly with their separation, at distances greater than about a femtometer. This is known as “linear confinement”, and it forms the starting point for the string model of hadronization, discussed below in Sec. 43.2.1. Alternatively, a property of perturbative QCD called “preconfinement” is the basis of the cluster model of hadronization, discussed in Sec. 43.2.2.

A key difference between MC hadronization models and the fragmentation-function (FF) formalism used to describe inclusive hadron spectra in perturbative QCD (see Chap. 9 and Chap. 19 of PDG book) is that FFs can be defined at an arbitrary perturbative scale $Q$ while MC hadronization models are intrinsically defined at the scale $Q_{had}$. Direct comparisons are therefore only meaningful if the perturbative evolution between $Q$ and $Q_{had}$ is taken into account. FFs are calculable in pQCD, given a non-perturbative initial condition obtained by fits to hadron spectra. In the MC context, one can prove that the correct QCD evolution of the FFs arises from the shower formalism, with the hadronization model providing an explicit parameterization of the non-perturbative component. However, the MC modeling of shower and hadronization includes much more information on the final state since it is fully exclusive (i.e., it addresses all particles in the final state explicitly), while FFs only describe inclusive spectra. This exclusivity also enables MC models to make use of the color-flow information coming from the perturbative shower evolution (see Sec. 43.1.6) to determine between which partons confining potentials should arise. E.g., in the string picture, the nonperturbative limit of a QCD dipole is a string piece [82].

Given an ideal, exact hadronization model, its dependence on the scale $Q_{had}$ should in principle be compensated by the corresponding scale dependence of the shower algorithm, which stops generating branchings at the scale $Q_{had}$. However, due to their complicated and fully exclusive nature, it is generally not possible to enforce this compensation automatically in MC models. One must therefore be aware that the nonperturbative model parameters must be “retuned” by hand if the infrared cutoff is modified. Any other changes to the perturbative part of the calculation, such as matching to further (fixed-order or resummed) coefficients, may also necessitate a retuning. Tuning is discussed briefly in Sec. 43.4.

The impact of hadronization corrections in a GPMC can be estimated by switching off hadronization. It should be emphasized, however, that the so-called “parton level” obtained this way is not a fully universal concept. This is because $Q_{had}$ is typically defined in a model-dependent way, via a cutoff in the respective shower model’s evolution variable, sometimes with supplementary cutoffs, e.g., in invariant mass, $p_T$, or the like and different tunes possibly using different numerical values for the cutoff(s). Variations of the main shower cutoff scale (e.g., by a factor of 2) may give an indication of the level of model dependence, which may be mild for sufficiently inclusive observables. Nevertheless our recommendation is to avoid correcting experimental measurements to “parton level” as the intrinsic model dependence will always imply a loss of accuracy.

### 43.2.1 The String Model

Starting from early concepts [83], several hadronization models based on strings have been proposed [11]. Of these, the most widely used today is the so-called Lund model [84, 85], implemented in PYTHIA [4, 5]. We concentrate on that particular model here, though many of the overall concepts would be shared by any string-inspired method.

Consider a quark and an antiquark that have a large relative momentum and which are in an overall color-singlet state, such as the $\bar{q}q$ pair produced at the end of the shower in the center of Fig. 43.1. As the charges move apart, linear confinement implies that a potential $V(r) = \kappa r$ is reached for large distances $r$. (At short distances, there is a Coulomb term $\propto 1/r$ as well, but this is neglected in the Lund string.) This potential describes a string with tension $\kappa \sim 1 \text{ GeV/fm} \sim 0.2 \text{ GeV}^2$. The physical picture is that of a color flux tube being stretched between the $q$ and the $\bar{q}$. As the string grows, the nonperturbative creation

of quark-antiquark pairs can break the string, via the process illustrated in Fig. 43.2. If either of the resulting string pieces after the break still has a large invariant mass, new breaks will then continue to occur until every piece has a mass of the order of that of a typical hadron. In nature, quantum mechanics ensures that these masses are (Breit-Wigner-distributed around) those of the physical hadron states in QCD, while in an algorithmic implementation, this is less trivial to impose. Within the Lund model, the constraint that each final string piece must have a mass consistent with that of physical hadron is addressed by first noting that each breakup vertex is spacelike separated from all others (the string cannot break again in the forward lightcone of a point where it has already broken); since the breakups are therefore causally disconnected, they do not have to be considered in any particular order. The Lund model exploits this to perform the fragmentation in an outwards-in manner [85], splitting off one physical hadron at a time from either the left or right endpoint (chosen randomly in each step).

The first hadron to be generated (starting either from the left or right) is thus the “outermost” one (sometimes called the “first-rank” hadron), formed by combining the original hadronizing endpoint quark (or antiquark) $q_0$ with an antiquark (or quark) $\bar{q}_1$ produced by the first breakup. The new leftover quark (or antiquark) $q_1$ becomes the string endpoint for the next iteration, in a Markov chain which continues, alternating randomly between the left and right ends of the string, until finally a small last bit of string is decayed directly to two hadrons, with no energy left over.

Not only does this allow for the hadron that is split off in each step to be assigned a physical mass, the fact that the fragmentation spectrum should be independent of whether one performs the fragmentation from left to right or vice versa places a strong constraint on the form of the nonperturbative string fragmentation function<fnref n="1" />, $f(z)$, which governs the probability for the hadron produced in a given step to take a fraction $z \in [0, 1]$ of the remaining energy. Thus, in the Lund model, causality dictates that the

<figure><img src="image_278.png" /><figcaption>**Figure 43.2:** Illustration of string breaking by quark pair-creation in the string field.</figcaption></figure>

<footnote n="1">¹Note that the string fragmentation function differs from the parton-to-hadron fragmentation functions discussed earlier. In principle, the two kinds should be related (with the latter evolved to a low scale of the order of the hadronization scale), but the relationship is not trivial and goes beyond the scope of this review.</footnote>
fragmentation function should be of the form,

$$f(z) \propto \frac{1}{z}(1-z)^a \exp\left(-\frac{b(m_h^2 + p_{T,h}^2)}{z}\right). \quad (43.9)$$

This is known as the Lund symmetric fragmentation function (normalized to unit integral). The dimensionless parameter $a$ dampens the hard tail of the fragmentation function, towards $z \to 1$, and may in principle be flavor-dependent, while $b$, with dimension GeV$^{-2}$, is a universal constant related to the string tension [85] which determines the behavior in the soft limit, $z \to 0$. Note that the dependence on the hadron mass, $m_h$, in $f(z)$ implies that heavier hadrons have higher $\langle z \rangle$. We return to the transverse momentum $p_{T,h}$ below.

The model is Lorentz invariant, so considerations involving boosted string systems are straightforward, involving the usual Lorentz effects.

As a by-product, the probability distribution in invariant time $\tau$ of $q'\bar{q}$ breakup vertices, or equivalently $\Gamma = (\kappa\tau)^2$, is also obtained, with $dP/d\Gamma \propto \Gamma^a \exp(-b\Gamma)$ implying an area law for the color flux, and the average breakup time lying along a hyperbola of constant invariant time $\tau_0 \sim 10^{-23}$s [85].

For massive endpoints (e.g. $c$ and $b$ quarks), which do not move along straight lightcone sections, the exponential suppression with string area leads to modifications of the form $f(z) \to f(z)/z^{b m_{\bar{Q}}^2}$, with $m_Q$ the mass of the heavy quark [86]. Although different forms, such as the Peterson formula [87], can also be used to describe inclusive heavy-meson spectra (see Sec 19.8 of PDG book), such choices are not strictly consistent with causality in the string framework.

### 43.2.1.1 Strings with Gluons

In the string model, energetic gluons lead to transverse “kinks” on strings, illustrated in Fig. 43.3. The order of these kinks follows from the color ordering produced by the parton shower, cf. the $\bar{q}gggq$ and $\bar{q}gq$ systems on the left and right part of Fig. 43.1. (Modifications to this order, by possible color reconnection/rearrangement effects, are discussed in Sec. 43.3.2.) Thus gluons

<figure><img src="image_279.png" /><figcaption>Figure 43.3: Schematic illustration of an $e^+e^- \to qg\bar{q}$ configuration emerging from the parton shower (PS). Snapshots of string positions are shown at two different times (full and shaded lines respectively). The gluon forms a transverse kink which grows in the $y$ direction until all the gluon’s kinetic energy has been used up.</figcaption></figure>

effectively build up a transverse structure in the originally one-dimensional object, with infinitely soft ones smoothly absorbed into the string. Note: cyclic topologies made entirely of gluons (closed strings) are also possible, e.g. in decays such as $H \to gg$ or $\Upsilon \to ggg$. The space-time evolution is more involved when kinks are taken into account [85], but no additional free parameters need to be introduced. The main difference between quark and gluon hadronization stems from the fact that gluons are connected to two string pieces (one on either side), while quarks are only connected to a single string piece. Hence, the relative rate of energy loss per unit invariant time — and consequently also the rate of hadron production — is larger by a factor of 2 for gluons (analogously to the ratio of gluon to quark color charges $C_A/C_F = 2.25$).

### 43.2.1.2 Transverse Momentum and Flavors

For each breakup vertex, quantum mechanical tunneling is assumed to control the masses and $p_T$ kicks (transverse to the string

axis, in a frame in which the string itself has no transverse motion) that can be produced, leading to a Gaussian suppression

$$\text{Prob}(m_q^2, p_{T,q}^2) \propto \exp\left(-\frac{\pi m_q^2}{\kappa}\right) \exp\left(-\frac{\pi p_{T,q}^2}{\kappa}\right), \quad (43.10)$$

where $m_q$ is the mass of the produced quark flavor and $p_T$ is the nonperturbative transverse momentum imparted to it by the breakup process, with a universal average value of $\langle p_{T,q}^2 \rangle = \kappa/\pi \sim (250 \text{ MeV})^2$. The antiquark has the same mass and opposite $p_T$.

In an MC model with a fixed shower cutoff $t_0$, the effective amount of $p_T$ in string breaks may be larger than the purely nonperturbative $\kappa/\pi$ above, to account for effects of additional (unresolved) radiation below $t_0$.

From the mass term in Eq. (43.10), one concludes that charm and bottom quarks are too heavy to be produced in string breaks, while strange quarks will be suppressed relative to up and down ones. Lacking unambiguous and precise mass definitions for light quarks, however, the effective amount of strangeness suppression is normally extracted from experimental data, using observables such as $K/\pi$, $K^*/\rho$, and $\phi/K^*$ ratios.

Baryon production can also be incorporated, at various levels of sophistication. The simplest option is to allow string breaks to produce a diquark-antidiquark pair, where a diquark is a loosely bound state of two quarks in an overall $\bar{3}$ representation. Again, the relative rate of such pairs is extracted from data, e.g. using $p/\pi$ or $\Lambda/K$ ratios. Since the perturbative shower splittings do not produce diquarks, the optimal value for this parameter is mildly correlated with the amount of $g \to q\bar{q}$ splittings in the shower. More sophisticated options, including the so-called “popcorn” mechanism, are discussed in Ref. [85]. Finally, the PYTHIA framework also allows for baryon string junctions [88], in which three string pieces (which can be thought of as red, green, and blue, respectively) join together. These are used, e.g., to model the fragmentation of baryon beam remnants, and can also be created (in pairs of junctions and antijunctions) in some color-reconnection scenarios [89], making the effective baryon-to-meson ratios in such models dependent on the amount and type of color reconnections that occur in each event.

The next step of the algorithm is the assignment of the produced quarks within hadron multiplets. Using a nonrelativistic classification of spin states, the hadronizing $q$ may combine with the $\bar{q}'$ from a newly created breakup to produce a hadron of a given spin $S$ and angular momentum $L$. The lowest-lying pseudoscalar and vector meson multiplets, and spin-1/2 and -3/2 baryons, are assumed to dominate in a string framework$^2$, but individual rates are not predicted by the model, and hence the ratios of spin-1 to spin-0 mesons and diquarks are free parameters (modulo a factor 3 from spin counting) which must be constrained by data. In that context, it is often advisable to begin with the heaviest states, since so-called feed-down from the decays of higher-lying hadron states complicates the extraction for lighter particles, see Sec. 43.2.3.

### 43.2.2 The Cluster Model

The cluster hadronization model is based on preconfinement, i.e., on the observation [90, 91] that the color structure of a perturbative QCD shower evolution at any scale $Q_0$ is such that color-singlet subsystems of partons (labeled “clusters”) occur with a universal invariant mass distribution which is power suppressed at large masses. For any starting scale $Q \gg Q_0 \gg \Lambda_{QCD}$, only the number of such clusters depends on $Q$, while the shape of their mass distribution only depends on $Q_0$ and on $\Lambda_{QCD}$.

Following early models based on this universality [10, 92], the cluster model developed by Webber [93] has for many years been a hallmark of the HERWIG generators, with an alternative implementation [94] now available in the SHERPA generator. The key idea, in addition to preconfinement, is to force “by hand” all gluons to split into quark-antiquark pairs at the end of the parton shower.

<footnote>$^2$In PYTHIA, the four $L=1$ meson multiplets (scalar, tensor, and 2 pseudovectors) are also available but are disabled by default, since many states (and their decays) are poorly known.</footnote>
Compared with the string description, this effectively amounts to viewing gluons as “seeds” for string breaks, rather than as kinks in a continuous object. After the splittings, a new set of low-mass color-singlet clusters is obtained, formed only by quark-antiquark pairs. These can be decayed to on-shell hadrons in a simple manner, with the relative yields of different hadron species mainly governed by their masses and the size of the phase space.

The algorithm starts by generating the forced $g \to q\bar{q}$ breakups, and by assigning flavors and momenta to the produced quark pairs. For a typical shower cutoff corresponding to a gluon virtuality of $Q_{\text{had}} \sim 1$ GeV, the $p_T$ generated by the splittings can be neglected. The constituent light-quark masses, $m_{u,d} \sim 300$ MeV and $m_s \sim 450$ MeV, imply a suppression (typically even an absence) of strangeness production. In principle, the model also allows for diquarks to be produced at this stage, but due to the larger constituent masses this would only become relevant for shower cutoffs larger than 1 GeV.

If a cluster formed in this way has an invariant mass above some cutoff value, typically 3–4 GeV, it is forced to undergo sequential $1 \to 2$ cluster breakups, along an axis defined by the constituent partons of the original cluster, until all sub-cluster masses fall below the cutoff value. Due to the preservation of the original axis in these breakups, this treatment has some resemblance to the string-like picture, though the nonperturbative $p_\perp$ kicks generated in this way are generally larger, up to half the allowed cluster mass.

Next, on the low-mass side of the spectrum, some clusters are allowed to decay directly to a single hadron, with nearby clusters absorbing any excess momentum. This improves the description of the high-$z$ part of the spectrum — where the hadron carries almost all the momentum of its parent jet — at the cost of introducing one additional parameter, controlling the probability for single-hadron cluster decay.

Having obtained a final distribution of small-mass clusters, now with a strict cutoff at 3–4 GeV and with the component destined to decay to single hadrons already removed, the remaining clusters are interpreted as a smoothed-out spectrum of excited mesons, each of which decays isotropically to two hadrons, with relative probabilities proportional to the available phase space for each possible two-hadron combination that is consistent with the cluster’s internal flavors, including spin degeneracy. It is important that all the light members (containing only *uds*) of each hadron multiplet be included, as the absence of members can lead to unphysical isospin or SU(3) flavor violation. Typically, the lightest pseudoscalar, vector, scalar, even and odd charge conjugation pseudovector, and tensor multiplets of light mesons are included. In addition, some excited vector multiplets of light mesons may be available. For baryons, usually only the lightest flavor-octet, -decuplet and -singlet baryons are present, although both the HERWIG and SHERPA implementations now include some heavier baryon multiplets as well.

Differently from the string model, the mechanism of phase-space suppression employed here leads to a natural enhancement of the lighter pseudoscalars, and no parameters beyond the spectrum of hadron masses need to be introduced at this point. The phase space also limits the transverse momenta of the produced hadrons relative to the jet axis.

Note that, since the masses and decays of excited heavy-flavor hadrons in particular are not well known, there is some freedom in the model to adjust these, which in turn will affect their relative phase-space populations.

### 43.2.3 Hadron and $\tau$ Decays

Of the so-called primary hadrons, originating directly from string breaks and/or cluster decays (see above), many are unstable and so decay further, until a set of particles is obtained that can be considered stable on time scales relevant to the given measurement. (A typical hadron-collider definition of a “stable particle” $c\tau \ge 10$ mm includes weakly-decaying strange hadrons $K, \Lambda, \Sigma^\pm, \bar{\Sigma}^\pm, \Xi, \Omega$.) The decay modeling can therefore have a significant impact on final particle yields and spectra, especially for the lowest-lying hadronic states, which receive the largest relative contributions from decays (feed-down). This interplay also implies that hadronization parameters may need to be retuned if

significant changes to the decay treatment are made.

Particle summary tables, such as those given elsewhere in this Review, represent a condensed summary of the available experimental measurements and hence may be incomplete and/or exhibit inconsistencies within the experimental precision. In an MC decay package, on the other hand, all information must be quantified and consistent, with all branching ratios summing to unity. When adapting particle summary information for use in a decay package, a number of choices must therefore be made. The amount of ambiguity increases as more excited hadron multiplets are added to the simulation, about which less and less is known from experiment, with each GPMC making its own choices.

A related choice is how to distribute the decay products differentially in phase space, in particular which matrix elements to use. Historically, MC generators contained matrix elements only for selected (generator-specific) classes of hadron and $\tau$ decays, coupled with a Breit-Wigner smearing of the masses, truncated at the edges of the physical decay phase space (the treatment of decay thresholds can be important for certain modes [11]). A more sophisticated treatment can then be obtained by reweighting the generated events using the obtained particle four-momenta and/or by using specialized external packages such as EVTGEN [95] for hadron decays and TAUOLA [96] for $\tau$ decays.

More recently, HERWIG and SHERPA include helicity-dependence in $\tau$ decays [6, 97], with a more limited treatment available in PYTHIA 8 [5]. The HERWIG and SHERPA generators have also included significantly improved internal simulations of hadronic decays, which include spin correlations between those decays for which matrix elements are used. Photon-bremsstrahlung effects are discussed in Sec. 43.1.7.

HERWIG and PYTHIA include the probability for $B$ mesons to oscillate into $\bar{B}$ ones before decay. SHERPA and EVTGEN also include CP-violating effects and, for common decay modes of the neutral meson and its antiparticle, the interference between the direct decay and oscillation followed by decay.

We end on a note of warning on double counting. This may occur if a particle can decay via an intermediate on-shell resonance. An example is $a_1 \to \pi\pi\pi$ which may proceed via $a_1 \to \rho\pi$, $\rho \to \pi\pi$. If these decay channels of the $a_1$ are both included, each with their full partial width, a double counting of the on-shell $a_1 \to \rho\pi$ contribution would result. Such cases are normally dealt with consistently in the default MC generator packages, so this warning is mostly for users that wish to edit decay tables on their own.

## 43.3 Models for Soft Hadron-Hadron Physics

### 43.3.1 Underlying Event

In the GPMC context, “underlying event” (UE) denotes any additional activity *beyond* the basic process and its associated ISR and FSR activity. The UE is thus only defined in the context of events selected with a “hard” (i.e., high-$p_\perp$) trigger which defines the basic process at hand. The dominant contribution to the UE is believed to come from additional color exchanges between the colliding hadronic states. These multiple exchanges can be modeled either as additional perturbative (mainly t-channel gluon) exchanges, called multiple parton-parton interactions (MPI), or nonperturbatively using so-called cut pomerons (roughly equivalent to exchange of gluons with $p_\perp \to 0$). The experimental observation that events with a hard trigger are accompanied by a higher-than-average level of associated activity is called the “jet pedestal” effect.

The most clearly identifiable consequence of MPI is arguably the possibility of observing several hard parton-parton interactions in one and the same hadron-hadron event. Typically, these are QCD $2 \to 2$ interactions, which produce additional back-to-back jet pairs, with each pair having a small value of $\text{sum}(\vec{p}_\perp)$. The fraction of MPI that give rise to additional reconstructible jets is, however, small. Soft interactions, that exchange color and a small amount of momentum without giving rise to observable jets, are much more plentiful, and can give significant corrections to the color flow and total scattered energy of the event. This affects the final-state activity in a more global way, increasing hadron-multiplicity and summed $E_T$ distributions, and contribut-
ing to the break-up of the beam remnants in the forward direction.

The first detailed Monte Carlo model for perturbative MPI was proposed in Ref. [98], and with some variation this still forms the basis for most modern implementations. Some useful additional references can be found in Ref. [11]. The first crucial observation is that the *t*-channel propagators appearing in perturbative QCD $2 \to 2$ scattering almost go on shell at low $p_T$, causing the differential cross sections to behave roughly as

$$d\sigma_{2 \to 2} \propto \frac{dt}{t^2} \sim \frac{dp_T^2}{p_T^4}. \qquad (43.11)$$

This cross section represents the inclusive scattering of partons against partons in perturbative QCD, summed over all partons. Thus, if a single hadron-hadron scattering contains two parton-parton interactions, that event will contribute twice to the parton-parton cross section $\sigma_{2 \to 2}$ but only once to the hadron-hadron one $\sigma_{tot}$, and so on. In the limit that all the parton-parton interactions are independent and equivalent, one has

$$\sigma_{2 \to 2} = \langle n \rangle \sigma_{tot}, \qquad (43.12)$$

with $\langle n \rangle$ the average number of parton-parton interactions, typically defined with some minimal $p_T > p_{\perp min}$ to render the parton-parton cross section finite. The probability for *n* parton-parton scatterings then follows a Poisson distribution,

$$P_n = \langle n \rangle^n \frac{\exp(-\langle n \rangle)}{n!}. \qquad (43.13)$$

This simple argument expresses unitarity; instead of the total hadron-hadron interaction cross section diverging as the parton-parton $p_T \to 0$ (which would violate unitarity), we have restated the problem so that it is now the number of parton-parton interactions per hadron-hadron collision that diverges, with the total hadron-hadron cross section remaining finite. At LHC energies, the parton-parton scattering cross sections computed using the LO QCD cross section folded with modern PDFs become larger than the total $pp$ one for $p_{\perp min}$ values of order 4–5 GeV (see e.g. [99, 100]). One therefore expects the average number of perturbative MPI to exceed unity at around that scale.

Two ingredients remain to fully regulate the remaining divergence. Firstly, the interactions cannot use up more momentum than is available in the parent hadron. This suppresses the large-*n* tail of the estimate above. In PYTHIA-based models, the MPI are ordered in $p_T$, and the parton densities for each successive interaction are explicitly constructed so that the sum of *x* fractions can never be greater than unity. In the HERWIG models, the Poisson estimate of $\langle n \rangle$ above is used as an initial guess, but the generation of actual MPI is stopped once the energy-momentum conservation limit is reached. Both of these approaches generate momentum (conservation) correlations among the MPI.

The second ingredient invoked to suppress the number of interactions, at low $p_T$ and $x$, is color screening; if the wavelength $\sim 1/p_T$ of an exchanged colored parton becomes larger than a typical color-anticolor separation distance, it will only see an *average* color charge that vanishes in the limit $p_T \to 0$. This provides an infrared cutoff for MPI similar to that provided by the hadronization scale for parton showers. A first estimate of the color-screening cutoff would be the proton size, $p_{\perp min} \approx \hbar/r_p \approx 0.3$ GeV $\approx \Lambda_{QCD}$, but empirically this appears to be far too low. In current models, one replaces the proton radius $r_p$ in the above formula by a “typical color screening distance,” i.e., an average size of a region within which the net compensation of a given color charge occurs. This number is not known from first principles [101] and is perceived of simply as an effective cutoff parameter. The simplest choice is to introduce a step function $\Theta(p_T - p_{\perp min})$. Alternatively, one may note that the jet cross section is divergent like $\alpha_S^2(p_T^2)/p_T^4$, cf. Eq. (43.11), and that therefore a factor

$$\frac{\alpha_S^2(p_{T,0}^2 + p_T^2)}{\alpha_S^2(p_T^2)} \frac{p_T^4}{(p_{T,0}^2 + p_T^2)^2} \qquad (43.14)$$

would smoothly regulate the divergences, now with $p_{T,0}$ as the free parameter. Regardless of whether it is imposed as a smooth

(PYTHIA and SHERPA) or step (HERWIG) function, this is effectively the main “tuning” parameter in such models.

Note that the numerical value obtained for the cross section depends upon the PDF set used, and therefore the optimal value to use for the cutoff will also depend on this choice. Note also that the cutoff does not have to be energy-independent. Higher energies imply that parton densities can be probed at smaller $x$ values, where the number of partons rapidly increases. Partons then become closer packed and the color screening distance $d$ decreases. The uncertainty on the energy and/or $x$ scaling of the cutoff is a major concern when extrapolating between different collider energies [102].

We now turn to the origin of the observational fact that hard jets appear to sit on top of a higher “pedestal” of underlying activity than events with no hard jets. This is interpreted as a consequence of impact-parameter-dependence: in peripheral collisions, only a small fraction of events contain any high-$p_T$ activity, whereas central collisions are more likely to contain at least one hard scattering; a high-$p_T$ triggered sample will therefore be biased towards small impact parameters, $b$. The ability of a model to describe both the UE in events with a hard trigger as well as the activity in inclusive “minimum-bias” (MB) samples (see below) therefore depends upon its modeling of the $b$-dependence, and correspondingly the impact-parameter shape constitutes another main tuning parameter.

For each impact parameter $b$, the number of interactions $\tilde{n}(b)$ can still be assumed to be distributed according to Eq. (43.13), again modulo momentum conservation, but now with the mean value of the Poisson distribution depending on impact parameter, $\langle \tilde{n}(b) \rangle$. This causes the final $n$-distribution (integrated over $b$) to be wider than a Poissonian.

Finally, there are two perturbative modeling aspects which go beyond the introduction of MPI themselves: 1) parton showers off the MPI, and 2) perturbative parton-rescattering effects. Without showers, MPI models would generate very sharp peaks for back-to-back MPI jets, caused by unshowered partons passed directly to the hadronization model. However, with the exception of the oldest PYTHIA6 model, all GPMC models do include such showers [11], and hence should exhibit more realistic (i.e., broader and more decorrelated) MPI jets. On the initial-state side, the main questions are whether and how correlated multi-parton densities are taken into account and, as discussed previously, how the showers are regulated at low $p_T$ and/or low $x$. Although none of the general purpose MC models currently impose a rigorous correlated multi-parton evolution, all of them include some elementary aspects. The most significant for parton-level results is arguably momentum conservation, which is enforced explicitly in all the models. The so-called “interleaved” models [29] attempt to go a step further, generating an explicitly correlated multi-parton evolution in which flavor sum rules are imposed to conserve, e.g. the total numbers of valence and sea quarks [88]. Fully correlated double-parton evolution is so far only available in the more specialised code dShower [103]; see [104] for comparisons between this and Pythia’s interleaved model.

Perturbative rescattering in the final state can occur if partons are allowed to undergo several distinct interactions, with showering activity possibly taking place in-between. This has so far not been studied extensively, but a first exploratory model is available [105]. In the initial state, parton rescattering/recombination effects have so far not been included in any of the GPMC models.

### 43.3.2 Bose-Einstein and Color-Reconnection Effects

In the context of $e^+e^-$ collisions, Bose-Einstein (BE) correlations have mostly been discussed as a source of uncertainty on high-precision W mass determinations at LEP [106]. In hadron-hadron (and nucleus-nucleus) collisions, however, BE correlations are used extensively to study the space-time structure of hadronizing matter (“femtoscopy”).

In MC models of hadronization, each string break or particle-/cluster decay is normally factorized from all other ones. This reduces the number of variables that must be considered in each step, but also makes it intrinsically difficult to introduce correlations among particles from different breaks/decays. In GPMCs, a few semi-classical models are available within the PYTHIA 6 and
8 generators [107], in which the BE effect is mimicked by an at-
tractive interaction between pairs of identical particles in the final
state, with no higher correlations included. Variants of this model
differ mainly by the assumed shape of the correlation function and
how overall momentum conservation is handled.

As discussed in Sec. 43.2, leading-color (“planar”) color flows
are used to set up the hadronizing systems (clusters or strings)
at the hadronization stage. If the systems do not overlap sig-
nificantly in space and time, subleading-color ambiguities and/or
nonperturbative reconnections are expected to be small. How-
ever, if the density of displaced color charges is sufficiently high
that several systems can overlap significantly, full-color and/or
reconnection effects should become progressively larger.

In the specific context of MPI, a crucial question is how color
is neutralized *between* different MPI systems, including the rem-
nants. The large rapidity differences involved imply large invari-
ant masses (though normally low $p_T$), and hence large amounts of
(soft) particle production. Indeed, in the context of soft-inclusive
physics, it is these “inter-system” strings/clusters that furnish the
dominant particle-production mechanism, and hence their mod-
eling is an essential part of the soft-physics description, affecting
topics such as MB/UE multiplicity and $p_T$ distributions, rapid-
ity gaps, and precision mass measurements. Reviews of color-
reconnection effects can be found in [11, 89].

43.3.3 Minimum-Bias Events and Diffraction

The term “minimum bias” (MB) originates from the experimen-
tal requirement of a minimal number of tracks (or hits) in a given
instrumented region. In order to make MC predictions for such
observables, all possible contributions to the relevant phase-space
region must be accounted for. There are essentially four types
of physics processes, which together make up the total hadron-
hadron (hh) cross section: 1) elastic scattering<fnref n="3" />: hh → hh, 2)
single diffractive dissociation: hh → h + gap + X, with X de-
noting anything that is not the original beam particle, and “gap”
denoting a rapidity region devoid of observed activity; 3) double
diffractive dissociation: hh → X + gap + X, and 4) inelastic non-
diffractive scattering: everything else. A fifth class may also be
defined, called central diffraction (hh → h + gap + X + gap + h).
Note that different terminologies exist [108]: in experimental set-
tings, diffraction is typically defined by an observable gap, of some
minimal size in rapidity, while in the MC context, each diffractive
physics process produces a whole spectrum of gaps, with small
ones suppressed but not excluded.

The inelastic non-diffractive part of the cross section is typically
modeled either by smoothly regulating and extending the pertur-
bative QCD scattering cross sections all the way to zero $p_⊥$ [98]
(PYTHIA and SHERPA), or by regulating the QCD cross sections with
a sharp cutoff [109] and adding a separate class of nonperturbative
scatterings below that scale [110] (HERWIG). See also Sec. 43.3.1. In
all cases, the most important ingredients are: 1) the IR regulariza-
tion of the perturbative scattering cross sections, including their
PDF dependence, 2) the assumed matter distribution of the col-
liding hadrons, possibly including multi-parton correlations [88]
and/or $x$ dependence [111], and 3) additional soft-QCD effects
such as color reconnections, discussed in Sec. 43.3.2.

Currently, there are essentially three methods for simulat-
ing diffraction in the main MC models: 1) in PYTHIA 6, one
picks a diffractive mass according to parameterized cross sections
$\propto dM^2/M^2$ [112]. This mass is represented as a string, which is
hadronized as described in Sec. 43.2.1, though differences in the
effective scale of the hadronization may necessitate a (re)tuning of
the hadronization parameters for diffraction; 2) in PYTHIA 8, the
high-mass tail beyond $M \sim 10$ GeV is augmented by a partonic de-
scription in terms of pomeron PDFs [113], allowing diffractive jet
production including showers and underlying event [114]; 3) the
PHOJET and DPMJET programs also include central diffraction and
rely directly on a formulation in terms of pomerons (color-singlet
multi-gluon states) [115–117]. Cut pomerons correspond to ex-
changes of soft gluons while uncut ones give elastic and diffractive
topologies as well as virtual corrections that help preserve unitar-

ity. So-called “hard pomerons” provide a transition to the per-
turbative regime. Hadronization is still handled using the Lund
string model, so there is some overlap with the above models at
the hadronization stage. In addition, a pomeron-based package
exists for HERWIG [118], and an effort is underway to construct an
MC implementation of the “KMR” model [101] within the SHERPA
generator. Color reconnections (Sec. 43.3.2) may also play a role
in creating rapidity gaps and the underlying event (Sec. 43.3.1)
in filling them.

43.4 Uncertainties and Tuning

The accuracy that can be achieved by a GPMC model depends
on the sophistication of the theory models it incorporates, on the
available constraints on its free parameters, and on the nature
of the observable(s) under study. Using existing data to con-
strain the model parameters is referred to as generator tuning.
Although tuned models do tend to yield improved results also
for observables that they have not been tuned to, the question
of evaluating the remaining uncertainties reliably is still far from
solved. It is worth noting, however, that all of the GPMCs now
provide options for automatic evaluation of perturbative shower
uncertainties (e.g., via renormalization-scale variations), in the
form of vectors of alternative event weights [119–121]. One must
be aware that these variations are not necessarily exhaustive and
significant weight fluctuations can be a problem for processes with
large shower phase spaces. Nonperturbative uncertainties are nor-
mally still evaluated by varying salient model parameters by hand,
though this looks set to change with a new method based on
reweighting [122]. A general method called eigentunes [123] is
also available, based on global fits to data.

Typically, the overall event properties are determined by only
a few, very important parameters, such as the effective value of
$\alpha_s$, for perturbative corrections, and the shape of the fragmenta-
tion functions, for nonperturbative ones. More parameters may
then be introduced to describe successively more detailed aspects
(e.g., the rates and decays of individual hadron species), but these
should have progressively less impact on the overall modeling.
One may therefore take a factorized approach, first constraining
the perturbative parameters and thereafter the nonperturbative
ones, in order of decreasing significance to the overall modeling.
Furthermore, by identifying which measurements are most sensi-
tive to each parameter, this ordering can be reflected in the way
that data is selected and applied to constrain the models. Thus,
measurements sensitive to global event properties would typically
be applied first, to constrain the most inclusive parameters, and
so on for progressively more exclusive aspects. For high-$p_T$ pro-
cesses, one should be aware that if the perturbative modelling
is not adequate, there is a risk that one may end up effectively
mistuning nonperturbative parameters to compensate for pertur-
bative deficiencies. This is obviously wrong and care must be
taken to avoid it. E.g., it would not make sense to tune to get the
right total number of hadrons unless one also gets the right total
number of jets. In this context, it may be more desirable to use
ratios like the number of hadrons *per jet*.

Note also that the unitarity of shower and hadronization mod-
els implies that the Born-level cross-section normalization is not
tunable, hence in tuning contexts one tends to focus on the shapes
of distributions rather than their normalizations.

At LO×(N)LL, perturbation theory is doing well if it agrees
with an IR safe shape measurement within ~ 10%. It would
therefore not make much sense to tune a GPMC to agree with
individual data points beyond this level. To reflect this, and to
protect against overfitting, a 5% sanity limit has sometimes been
enforced in tuning studies [99, 124]. The advent of NLO Monte
Carlos may reduce this number slightly, but only for quantities
for which one expects NLO precision. For quantities governed
by nonperturbative physics, uncertainties are larger. For some
quantities, e.g. ones for which the underlying modeling is known
to be poor, an order-of-magnitude agreement or worse may have
to be accepted.

In the context of LO×LL GPMC tuning, subleading aspects
of coupling-constant and PDF choices are relevant. In particu-
lar, one should be aware that the choice of QCD $\Lambda$ parameter
$\Lambda_{\text{MC}} = 1.569\Lambda_{\overline{\text{MS}}}$ (for 5 active flavors) improves the predictions

<footnote n="3">³The QED elastic cross section diverges and is normally a non-default option.</footnote>
of coherent shower algorithms at the NLL level for a class of relevant observables [125]. The question of LO vs. NLO PDFs is more involved [11], but it should be emphasized that the gluon PDF at (very) low $x$ is important for determining the level of the underlying event in MPI models (Sec. 43.3.1), and hence the MB/UE tuning (and energy scaling [102]) is linked to the choice of PDF in such models. Further issues and an example of a specific recipe that could be followed in a realistic set-up can be found in Ref. [99]. A useful online resource can be found at the mcplots.cern.ch website [126], based on the RIVET tool [127].

Recent years have seen the emergence of automated tools to reduce the amount of both computer and manpower required for tuning [123,128,129]. Automating the human expert input is more difficult. In the tools currently on the market, this is addressed by a combination of input solicited from the GPMC authors (e.g., which parameters and ranges to consider, which observables constitute a complete set, etc) and a set of weights determining the relative priority given to each bin in each distribution. The final result is therefore still subjective but at least reproducible. Backed by careful demonstrations of sensitivities, correlations, and uncertainties, competitive tunes can be produced. The field is still burgeoning, with future sophistications to be expected.

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## 44. Monte Carlo Neutrino Generators

Revised September 2021 by H. Gallagher (Tufts U.) and Y. Hayato (Kamioka Observatory, ICRR, UTokyo; Kavli IPMU (WPI), UTokyo).

Monte Carlo neutrino generators are programs or libraries which simulate neutrino interactions with electrons, nucleons and nuclei. In this capacity their usual task is to take an input neutrino and nucleus and produce a set of 4-vectors for particles emerging from the interaction, which are then input to full detector simulations. Since these generators have to simulate not only the initial interaction of neutrinos with target particles, but re-interactions of the generated particles in the nucleus, they contain a wide range of elementary particle and nuclear physics. Viewed more broadly, they are the access point for neutrino experimentalists to the theory inputs needed for analysis. Examples include cross section libraries for event rate calculations and parameter uncertainties and reweighting tools for systematic error evaluation.

Neutrino experiments typically operate in neutrino beams that are neither completely pure nor mono-energetic. Generators are a crucial component in the convolution of beam flux, neutrino interaction physics, and detector response that is necessary to make predictions about observable quantities. Generated events are used to define experimental analyses, like event selection criteria, for efficiency determination and background subtraction. Similarly they are used to relate reconstructed quantities back to true quantities. In these various capacities they are used from the detector design stage through the extraction of physics measurements from reconstructed observable. Monte Carlo neutrino generators play unique and important roles in the experimental study of neutrino interactions and oscillations.

There are several neutrino event generators available, such as ANIS [1], GENIE [2, 3], GiBUU [4, 5], MARLEY [6], NEGN [7], NEUT [8], NUANCE [9], the FLUKA routines NUNDIS-/NUNRES [10, 11], and NuWro [12], as well as tools to facilitate cross-generator comparisons, such as NUISANCE [13]. Historically, experiments would develop their own generators. This was often because they were focused on a particular measurement, energy range, or target, and wanted to ensure that the best physics was included for them. These ‘home-grown’ generators were often tuned primarily or exclusively to the neutrino data most similar to the data that the experiment would be collecting. A major advance in the field was the introduction of conference series devoted to the topic of neutrino interaction physics, NuINT (https://indico.cern.ch/event/703880/) and NuFACT (https://indico.cern.ch/event/855372/) in particular. Event generator comparisons have been a regular staple of the NuINT conference series from its inception, and a great deal of information on this topic can be found in the Proceedings of these meetings. These meetings have facilitated experiment-theory discussions leading to the first generator developed by a theory group (NuWro) [12], the extension of established nuclear interaction codes (FLUKA and GiBUU) to include neutrino-nuclear processes [4, 5, 10, 11], and inclusion of theorists in existing generator development teams.

These activities have led to more careful scrutiny of the crucial hadron and nuclear theory inputs to these generators, which is evaluated in particular through comparisons to electron-scattering data. At this point in time all simulation codes face challenges in describing the full extent of the lepton scattering data, and the tension between incorporating the best available theory versus obtaining the best agreement with the data plays out in a variety of ways within the field. For the field to make progress, inclusion of state of the art theory needs to be coupled to global analyses that correctly incorporate correlations between measurements. Given the rapid pace of new data and the complexity of analyses, this is a significant challenge for the field in the coming years.

There are many neutrino experiments which use various sources of neutrinos, from reactors, accelerators, the atmosphere, and astrophysical sources, thereby covering a range of energies from MeV to TeV. Much of the emphasis has been on the few-GeV region in the generators, as this is the relevant energy range for short- and long-baseline neutrino oscillation experiments. These gen-

erators use the impulse approximation, which treats the nucleus as a collection of independent nucleons and the primary interaction occurs between the probe and a single nucleon, for most of the initial interaction, and subsequently simulates the interactions of secondary particles in the nucleus in semi-classical ways. Semi-classical hadron transport approaches are commonly used as they are able simulate a variety of nuclei in a single model, fully describe complex multi-particle final states, and for practical considerations as these approaches are fast. However, there are several challenges facing these simulations coming mainly from the complexity of the nuclear physics, and the need to avoid double counting in combining perturbative and non-perturbative models for the neutrino-nucleon scattering processes. The overall validity of this impulse approximation-based scheme, and in particular the importance of scattering channels that involve more than one nucleon, is a crucial question that is the topic of much current work. While generators share many common ingredients, differences in implementation, parameter values, and approaches to avoid double counting can yield dramatically different predictions [14]. In the following sections, interaction models and their implementations, including final-state interactions of hadrons produced in the nuclei, are described.

In order to assure their validity, neutrino event generators are tuned and validated against a wide variety of data. This means that the quality of the outputs from generators is limited by the quality of experimental data. The existing neutrino-nucleon and neutrino-nucleus scattering data sets are not restrictive enough to eliminate many approaches. On the other hand, there are various precise data sets, which use photon, charged lepton, and hadron probes. These data sets are extensively used to validate the generators. The results from these external data tuning exercises are important for experiments as they quantify the uncertainty on model parameters, needed by experiments in the evaluation of generator-related systematic errors. Electron scattering data plays an important role in determining the vector contribution to the form-factors and structure functions, as well as in evaluating specific aspects of the nuclear model [15, 16]. Hadron scattering data is used in validating the nuclear model, in particular of interactions between hadrons produced in the primary interaction and the residual target nucleus (final state interactions). As mentioned, tuning of neutrino-nucleon scattering and hadronization models relies heavily on the previous generation of high energy neutrino scattering and hydrogen and deuterium bubble chamber experiments. More recent data from the K2K, MiniBooNE, NO-MAD, SciBooNE, MINOS, T2K, ArgoNEUT, MINERvA, NOvA, MicroBooNE, ICARUS, SBND and Ninja experiments has been, or will be, used for this purpose, although caution is advisable due to possible biases that mismodeled nuclear effects might introduce. In practice, there are often choices to be made about which models to include in an overall simulation, and which data sets to use for tuning. The process of developing a comprehensive tune in GENIE is discussed in Ref. [3].

However, all the recent experiments use nuclear targets and it is challenging to disentangle the neutrino-nucleon interaction from the effects of the nuclear environment. For example, it is difficult to separate charged current quasi-elastic scattering from the multi-nucleon quasi-elastic like scattering or single pion production, in which a pion is absorbed in the nucleus. Sometimes, there are tensions between the data sets from different experiments. Data on multi-pion and meson production are limited, but are anticipated to be measured in the near future experiments. Next generation experiments are expected to provide better detection capabilities with higher statistics and these data will help in evaluating and validating the models implemented in the generators.

### 44.1 Neutrino-Nucleon Scattering

Event generators typically begin with free-nucleon cross sections which are then embedded into a nuclear physics model. The most important processes are quasi-elastic (elastic for neutral current (NC)) scattering, resonance production, and non-resonant inelastic scattering, which make comparable contributions for few-GeV interactions. The neutrino cross sections in this energy range can be seen in Figure 52.1 of this Review.
### 44.1.1 Quasi-Elastic Scattering

The cross section for the neutrino nucleon charged current quasi-elastic scattering is described in terms of the leptonic and hadronic weak currents, where dominant contributions to the hadronic current come from the vector (V) and axial-vector (A) form factors. Contributions from the pseudo-scalar form factor (P) are typically small for muon and electron neutrinos and are related to the axial form factor (A) assuming partially conserved axial currents (PCAC). Owing to isospin symmetry, the vector form factors are related to those measured by precise electron scattering experiments [17]. Therefore, most of the generators use parametrizations of these form factors taken directly from the data. For the axial form factor there is no such precise experiment, and most of the generators use a dipole form [18]. Generally, the value of axial form factor at $q^2 = 0$ ($q$ is the four-momentum transfer) is extracted from the polarized nucleon beta decay experiment. However, the selection of the axial vector mass parameter depends on each generator, with values typically around 1.00 GeV/$c^2$. Recently, there are several attempts to use the other functions for the axial form factors [19, 20] and some generators have already implemented these form factors [8, 21].

### 44.1.2 Inelastic Production

Most generators use the prescriptions of Rein-Sehgal [22] to simulate neutrino-induced single pion production, or updated versions that incorporate lepton mass terms and pion pole contributions [23, 24]. To obtain the cross section for a particular channel, they calculate the amplitude for the production of each resonance multiplied by the probability for the decay of that resonance into that particular channel. Implementation differences include the number of resonances included, whether the amplitudes are added coherently or incoherently, the invariant mass range over which the model is used, how non-resonant backgrounds are included, inclusion of lepton mass terms, and the model parameter values (in particular the axial mass). In this model it is also possible to calculate the cross-sections of single photon, and $\eta$ production by changing the decay probability of the resonances, which are included in some of the programs. However, it is known that discrepancies exist between the recent pion electro/photo-production data and the results from the simulation data with the same framework, i.e. vector part of this model. There are several attempts to overcome this issue [25] and some of the generators started using more appropriate form factors. Recently, there are approaches that improve the theoretical description of non-resonant contributions in the resonance region, [26–28], and which account for interference between resonant and non-resonant amplitudes. This work is expected to be implemented in the generators soon. GiBUU and NuWro generators do not use the Rein-Sehgal model, and instead rely directly on electro-production data for the vector contribution and fit bubble chamber data to determine the remaining parameters for the axial contribution [29–32]. The dynamical coupled-channel model, which has been developed to simulate various electro- and photo-meson productions, was extended to simulate the neutrino single pion production [33]. In this approach pion-nucleon scattering is used to fix the form of the axial current at $q^2 = 0$. This model is also being implemented and expected to be available in some of the generators in future.

### 44.1.3 Deep and Shallow Inelastic Scattering

For this process the fundamental target shifts from the nucleon to its quark constituents. Therefore, the generators construct the nucleon structure functions $F_2$ and $xF_3$ from parton distributions for high $Q^2$ (the DIS regime: $W \gtrsim 2$ GeV/$c^2$ and $Q^2 \gtrsim 1$ GeV$^2$) to calculate the direction and momentum of the lepton. The first challenge is in extending this picture to the lower values of $Q^2$ and $W$ that dominate the available phase space for few-GeV interactions (the so-called ‘shallow inelastic scattering’, or SIS regime). GRV98LO parton distribution functions [34] with the corrections proposed in [35, 36] are widely used, while others [10] implement their own modifications to the parton distributions at low $Q^2$. Both DIS and SIS generate hadrons but their production depends on each generator’s implementation of a hadronization model as described in the next section. There are various difficulties not only in the actual hadronization but the relation with the single

meson production. It is necessary to avoid double counting between the resonance and SIS/DIS models, and all generators are different in this regard. The scheme chosen can have a significant impact on the results of simulations at a few-GeV neutrino energies. For simulating ultra-high energy interactions currently being studied with neutrino telescopes, fully partonic descriptions, which utilize beyond-leading-order pQCD calculations [37, 38] are the norm [39].

## 44.2 Hadronization Models

For hadrons produced via baryonic resonances, the underlying model amplitudes and resonance branching fractions can be used to fully characterize the hadronic system. For shallow- and deep-inelastic scattering, a hadronization model is required. Most generators use PYTHIA [40] for this purpose, although some with modified parameters. In addition some implement their own models to handle invariant masses that are too low for PYTHIA, typically somewhere around 2.0 GeV/$c^2$. Such models rely heavily on measurements of neutrino hadro-production in high-resolution devices, such as bubble chambers and the CHORUS [41] and NO-MAD experiments [42], to construct empirical parametrizations that reproduce the key features of the data [43, 44]. The basic ingredients are the empirical observations that average charged particle multiplicities increase logarithmically with the invariant mass of the hadronic system, and that the distribution of charged particle multiplicities about this average are described by a single function (an observation known as KNO scaling [45]). The multiplicity of the neutral particles are usually obtained by scaling the charged particle multiplicity. Simple parametrizations to more accurately reproduce differences observed in the forward/backward hemispheres of hadronic systems are included in GENIE, NEUT, and NuWro.

## 44.3 Nuclear Physics

The nuclear physics relevant to neutrino-nucleus scattering at few-GeV energies is complicated, involving Fermi motion, nuclear binding, Pauli blocking, in-medium modifications of form factors and hadronization, intranuclear rescattering of hadrons, and many-body scattering mechanisms including long- and short-range nucleon-nucleon correlations.

### 44.3.1 Treatment of nuclear effects

In order to obtain the cross-section off nucleons in the nucleus, it is necessary to take into account various in-medium effects. Most of the models used for neutrino-nuclear scattering kinematics were developed in the context of few-GeV inclusive electron scattering, by experiments going back nearly 50 years. The basic models employed in event generators rely on impulse approximation schemes, the most simple of which is the Relativistic Fermi Gas Model. The most common implementations have been the Smith-Moniz [46] and Bodek-Ritchie [47] models. However, the results from neutrino-nucleus scattering experiments in 2000 and afterwards, beginning with K2K, MiniBooNE have shown large discrepancies from the naive expectation from the models. Most striking differences are a suppression of forward going muons (low $Q^2$), a high $Q^2$ enhancement in the event rate, and an overall larger than expected number of observed events. In order to reproduce the data, the quasi-elastic axial mass was used as the effective parameter and increased by roughly 20% from the nominal values obtained by an earlier generation of bubble chamber experiments using hydrogen or deuterium [18]. This inconsistency between nucleon and nucleus targets suggests that the simple nuclear model is not appropriate in describing the data. Moreover, these simple Fermi-Gas models are not expected to describe the kinematical distributions of final state nucleons. Therefore, several generators have to implement better models, such as local Fermi-Gas model or more sophisticated models. Within the electron scattering community, the analogous calculations have for decades relied on spectral functions, which incorporate information about nucleon momenta and binding energies in the impulse approximation scheme [15]. Therefore, most of the generators have implemented the realistic spectral functions in their latest releases.

Actually, the discrepancies in small $q^2$ could not be solved alone
by just introducing the local Fermi-gas model nor spectral func-
tion models. This implies that the additional medium correction
effects are needed to be taken into account. One of the imple-
mented solutions is the local Fermi-Gas model with medium cor-
rection calculated using the random phase approximation, which
is known to give large suppression in small $q^2$ [48, 49]. Although,
the fundamental parts of the models are same, actual implemen-
tations are quite different between the generators. Especially,
the constructions of the final state hadron kinematics are quite
different. Especially for the quasi-elastic scattering case, treat-
ment of the nucleon masses in the nucleus, the binding and the
separation energies are sometimes quite different [50]. Recently,
Super-Scaling model with relativistic mean field theory effects
(SuSAv2) [51, 52] is implemented in some of the generators. Indi-
cations of the suppression in small $q^2$ in the single pion produc-
tions were also observed in K2K and MiniBooNE and it is clearly
observed at the MINERνA and NOνA experiments. However, the
models to explain the suppression have not been implemented,
other than the phenomenological ones. Also, $q^2$ dependence of
the cross-section is largely model and parameter dependent. Fur-
ther studies of newly implemented single pion models may give
new insights. The cause of the discrepancy of small $q^2$ seems to
be identified but the issue of the observed interaction rates are not
solved. This implies that there must be some interaction channels
which are missing and not considered in the generators.

These led to a revisitation of the role played by excitation of
multiple particle-hole states in the nucleus, and the experimental
study of these scattering channels is an area of intense experimen-
tal interest [53]. The contribution of these scattering processes is
an extremely active area of theoretical research as well, with sig-
nificant implications for generators and analyses [54]. Several ap-
proaches, ranging from strictly phenomenological descriptions to
full theoretical calculations, have recently been incorporated into
generators [55–57]. One example of a phenomenological approach
utilizes an Effective Spectral Function [58] and a Transverse En-
hancement Model [59], which together encapsulate information
derived from electron scattering experiments at relevant kinemat-
ics. The microscopic model of Nieves and collaborators is now
available in GENIE and NEUT [60, 61]. SuSAv2 model [52] also
has capability to simulate this multi-nucleon quasi-elastic like in-
teraction and is also implemented in GENIE.

One of the challenges in incorporating full theoretical models of
these processes is that they are typically slow, so generators have
developed new approaches whereby much of the computation is
done offline, and the generators simply read in the hadronic tensor
components. This allows for a full prediction of the lepton kine-
matics, however the ability to simulate the hadronic component
of these multinucleon states then relies on separate models. The
other challenge is that the theoretical models are not designed
to describe exclusive final states. These approaches, while mak-
ing simulations computationally tractable, neglect correlations be-
tween the lepton and hadron kinematics. Also, there are limita-
tions of the model itself to describe some of the kinematic regions.
However, the generators need to simulate final all the state par-
ticles and thus, several assumptions are made by authors of the
generators from time to time.

Also, it is known from photo and electro-nuclear scattering that
the Delta width is affected by Pauli blocking and collisional broad-
ening. These effects are included in some, but not all generators.

The remnant nucleus after the scattering is usually in an ex-
cited state. Therefore, the remnant may emit gamma-rays, ad-
ditional nucleons or an alpha particle during the de-excitation
process. This de-excitation gamma-ray is used in searching for
proton decay in water Cherenkov detectors. Some generators have
implemented this process. NEUT implemented the de-excitation
process but only for Oxygen. Recently, GENIE has de-excitation
processes for various nuclei.

When scattering from a nucleus, coherent scattering of various
kinds is possible. Most simulations incorporate, at least, coherent
neutral and charged single pion production. While the interac-
tion rate for these interactions is typically around a percent of
the total yield, the unique kinematic features of these events can
make them potential backgrounds for oscillation searches. Imple-

mented in Monte Carlo are several PCAC-based methods [62,63].
Microscopic models, valid at lower neutrino energies [64–67], have
also been implemented in several generators. One commonly used
model by Rein and Sehgal [62] predicts a cross section for charged-
current coherent pion production that is much larger than what
is observed by K2K, MINERvA and T2K. However, the cross-
section is sensitive to the pion cross-section used in the model as
parameters and improved models with lepton mass correction [63]
give better agreement with the recent data. This improved model
is implemented in most of the generators. In addition to coher-
ent pion production, some generators can simulate coherent single
photon production [68] and coherent elastic neutrino-nucleus scat-
tering.

44.3.2 Hadron Production in Nuclei

Neutrino pion production is one of the dominant neutrino scat-
tering mechanisms in the few-GeV region and the pions produced
in the nucleus have quite large interaction cross sections. There-
fore, pion intranuclear scattering can have sizable effects on the
results of simulations at these energies. Also, recent neutrino
oscillation experiments use the observed information of protons
and pions more extensively, in addition to the charged leptons.
Therefore, most generators implement this physics through an in-
tranuclear cascade simulation. In generators which utilize cascade
models, a hadron, which has been formed in the nucleus, is moved
step by step until it interacts with a nucleon or escapes from the
nucleus. The probabilities of each interaction in nucleus are usu-
ally given as the mean free paths and used to determine whether
the hadron has interacted or not. If the hadron is found to have
interacted, appropriate final states are selected. Usually, absorp-
tion, elastic, charge exchange, and inelastic scatterings including
particle productions are simulated as intranuclear interactions.
The determination method of the kinematics for the final state
particles heavily depends on the generators but most of them use
experimentally validated models to simulate hadron interactions
in nucleus. No two intranuclear cascade simulations implemented
in neutrino event generators are the same [69]. In all cases hadrons
propagate from an interaction vertex chosen based on the density
distribution of the target nucleus. In determining the generated
position of the hadrons in nucleus, the concept of the formation
length is sometimes employed. Based on this idea, the hadroniza-
tion process is not instantaneous and it takes some time before
generating the hadrons [12]. The basis for formation times are
measurements at relatively high energy and $Q^2$, and most gen-
erators that employ the concept do not apply them to resonance
interactions.

GiBUU does not employ an intranuclear cascade simulation,
instead, it utilizes a semi-classical transport model in coupled
channels that describes the space-time evolution of a many body
system in the presence of potentials and a collision term [4]. This
approach assures consistency between nuclear effects in the ini-
tial state, such as Fermi motion, Pauli blocking, hadron self-
energies [70], and modified cross sections, and the final state, such
as particle re-interactions, since the two are derived from the same
model. This model has been previously used to describe a wide
variety of nuclear interaction data. Similarly, the hadronic sim-
ulation of the NUNDIS/NUNRES programs are handled by the
well-established FLUKA hadronic simulation package [10]. Re-
cently, GENIE included the interface to the external hadronic
simulation package INCL++ [71].

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45. Monte Carlo Particle Numbering Scheme

Revised August 2023 by A. Buckley (Glasgow U.), F. Krauss (Durham U.), S. Navas (Granada U.) and P. Richardson (Durham U.).

The Monte Carlo particle numbering scheme presented here is intended to facilitate interfacing between matrix-element generators, event generators, detector simulators, and analysis packages used in particle physics, and is widely accepted as the “industry standard”. The numbering scheme was introduced in 1988 [1], and has since been revised and expanded in the light of new information on hadronic resonances, and to encompass as yet undiscovered and hypothetical particles.<fnref n="1" />

The general form is a 7-digit number:

$$ \pm n \, n_r \, n_L \, n_{q1} \, n_{q2} \, n_{q3} \, n_J . $$

This encodes information about the particle’s spin, flavor content, and internal quantum numbers. A 9-digit extension is used specifically for tetra- and penta-quark states, and a distinct 10-digit scheme is used for encoding (hyper)nuclear states. The details are as follows:

1. Particles are given positive numbers, antiparticles negative numbers. The PDG convention for mesons is used, so that $K^+$ and $B^+$ are particles.

2. Quarks and leptons are numbered consecutively starting from 1 and 11 respectively; to do this they are first ordered by family and within families by weak isospin.

3. In composite quark systems (diquarks, mesons, and baryons) $n_{q1-3}$ are quark numbers used to specify the quark content, while the rightmost digit $n_J = 2J+1$ gives the system’s spin (except for the $K_S^0$ and $K_L^0$). The scheme does not cover particles of spin $J > 4$.

4. Diquarks have 4-digit numbers with $n_{q1} \ge n_{q2}$ and $n_{q3} = 0$.

5. The numbering of mesons is guided by the nonrelativistic ($L-S$ decoupled) quark model, as listed in Tables 15.2, 15.3, and 15.4.

(a) The numbers specifying the meson’s quark content conform to the convention $n_{q1} = 0$ and $n_{q2} \ge n_{q3}$. The special case $K_L^0$ is the sole exception to this rule.

(b) The quark numbers of flavorless, light $(u, d, s)$ mesons are: 11 for the member of the isotriplet $(\pi^0, \rho^0, \dots)$, 22 for the lighter isosinglet $(\eta, \omega, \dots)$, and 33 for the heavier isosinglet $(\eta', \phi, \dots)$. Since isosinglet mesons are often large mixtures of $u\bar{u} + d\bar{d}$ and $s\bar{s}$ states, 22 and 33 are assigned by mass and do not necessarily specify the dominant quark composition.

(c) The special numbers 310 and 130 are given to the $K_S^0$ and $K_L^0$ respectively.

(d) The fifth digit $n_L$ is reserved to distinguish mesons of the same total ($J$) but different spin ($S$) and orbital ($L$) angular momentum quantum numbers. For $J > 0$ the numbers are: $(L, S) = (J-1, 1) \, n_L = 0$, $(J, 0) \, n_L = 1$, $(J, 1) \, n_L = 2$ and $(J+1, 1) \, n_L = 3$. For the exceptional case $J = 0$ the numbers are $(0, 0) \, n_L = 0$ and $(1, 1) \, n_L = 1$ (i.e. $n_L = L$). See Table 45.1.

(e) If a set of physical mesons correspond to a (non-negligible) mixture of basis states, differing in their internal quantum numbers, then the lightest physical state gets the smallest basis state number. For example the $K_1(1270)$ is numbered 10313 ($1^1P_1 K_{1B}$) and the $K_1(1400)$ is numbered 20313 ($1^3P_1 K_{1A}$).

(f) The sixth digit $n_r$ is used to label mesons radially excited above the ground state.

(g) Numbers have been assigned for complete $n_r = 0$ $S$- and $P$-wave multiplets, even where states remain to be identified.

<table><caption>**Table 45.1:** Meson numbering logic. Here $qq$ stands for $n_{q2} n_{q3}$.</caption><thead><tr><th rowspan="2">J</th><th colspan="2">L = J-1, S = 1</th><th colspan="2">L = J, S = 0</th><th colspan="2">L = J, S = 1</th><th colspan="2">L = J+1, S = 1</th></tr><tr><th>code</th><th>J<sup>PC</sup></th><th>code</th><th>J<sup>PC</sup></th><th>code</th><th>J<sup>PC</sup></th><th>code</th><th>J<sup>PC</sup></th></tr></thead><tbody><tr><td>0</td><td>—</td><td>—</td><td>—</td><td>00qq1</td><td>0<sup>-</sup></td><td>0</td><td>—</td><td>10qq1</td><td>0<sup>++</sup></td><td>1</td></tr><tr><td>1</td><td>00qq3</td><td>1<sup>-</sup></td><td>0</td><td>10qq3</td><td>1<sup>-</sup></td><td>1</td><td>20qq3</td><td>1<sup>++</sup></td><td>1</td><td>30qq3</td><td>1<sup>-</sup></td><td>2</td></tr><tr><td>2</td><td>00qq5</td><td>2<sup>++</sup></td><td>1</td><td>10qq5</td><td>2<sup>-</sup></td><td>2</td><td>20qq5</td><td>2<sup>-</sup></td><td>2</td><td>30qq5</td><td>2<sup>++</sup></td><td>3</td></tr><tr><td>3</td><td>00qq7</td><td>3<sup>-</sup></td><td>2</td><td>10qq7</td><td>3<sup>-</sup></td><td>3</td><td>20qq7</td><td>3<sup>++</sup></td><td>3</td><td>30qq7</td><td>3<sup>-</sup></td><td>4</td></tr><tr><td>4</td><td>00qq9</td><td>4<sup>++</sup></td><td>3</td><td>10qq9</td><td>4<sup>-</sup></td><td>4</td><td>20qq9</td><td>4<sup>-</sup></td><td>4</td><td>30qq9</td><td>4<sup>++</sup></td><td>5</td></tr></tbody></table>

(h) In some instances assignments within the $q\bar{q}$ meson model are only tentative; here best guess assignments are made.

(i) Many states appearing in the Meson Listings are not yet assigned within the $q\bar{q}$ model. Here $n_{q2-3}$ and $n_J$ are assigned according to the state’s likely flavors and spin; all such unassigned light isoscalar states are given the flavor code 22. Within these groups $n_L = 0, 1, 2, \dots$ is used to distinguish states of increasing mass. These states are flagged using $n = 9$. It is to be expected that these numbers will evolve as the nature of the states are elucidated. Codes are assigned to most mesons that are listed in the one-page table at the end of the Meson Summary Table as long as they have a preferred or established spin. Additional heavy meson states expected from heavy quark spectroscopy are also assigned codes.

6. The numbering of baryons is again guided by the nonrelativistic quark model, see Table 15.6. This numbering scheme is illustrated through a few examples in Table 45.2.

(a) The numbers specifying a baryon’s quark content are such that in general $n_{q1} \ge n_{q2} \ge n_{q3}$.

(b) Two states exist for $J = 1/2$ baryons containing 3 different types of quarks. In the lighter baryon $(\Lambda, \Xi, \Omega, \dots)$ the light quarks are in an antisymmetric $(J=0)$ state while for the heavier baryon $(\Sigma^0, \Xi', \Omega', \dots)$ they are in a symmetric $(J=1)$ state. In this situation $n_{q2}$ and $n_{q3}$ are reversed for the lighter state, so that the smaller number corresponds to the lighter baryon.

(c) For excited baryons a scheme is adopted, where the $n_r$ label is used to denote the excitation bands in the harmonic oscillator model, see Sec. 15.5. Using the notation employed there, $n_r$ is given by the $N$-index of the $D_N$ band identifier.

(d) Further degeneracies of excited hadron multiplets with the same excitation number $n_r$ and spin $J$ are lifted by labelling such multiplets with the $n_L$ index according to their mass, as given by its $N$ or $\Delta$-equivalent.

(e) In such excited multiplets extra singlets may occur, the $\Lambda(1520)$ being a prominent example. In such cases the ordering is reversed such that the heaviest quark label is pushed to the last position: $n_{q3} > n_{q1} > n_{q2}$.

7. The gluon, when considered as a gauge boson, has official number 21. In codes for glueballs, however, 9 is used to allow a notation in close analogy with that of hadrons.

8. The pomeron and odderon trajectories and a generic reggeon trajectory of states in QCD are assigned codes 990, 9990, and 110 respectively, where the final 0 indicates the indeterminate nature of the spin, and the other digits reflect the expected “valence” flavor content. We do not attempt a complete classification of all reggeon trajectories, since there is currently no need to distinguish a specific such trajectory from its lowest-lying member.

9. Two-digit numbers in the range 21–30 are provided for the Standard Model gauge and Higgs bosons.

10. Codes 81–100 are reserved for generator-specific pseudoparticles and concepts. Codes 901–930, 1901–1930, 2901–2930, and 3901–3930 are for additional components of Standard Model parton distribution functions: the latter three ranges

<footnote n="1">¹The scheme was substantially updated and extended in 2012. Certain excited baryons follow the pre-2012 scheme. A further revision and/or extension of the numbering scheme is anticipated in the near future.</footnote>
<table><caption>**Table 45.2:** Some examples of octet (top) and decuplet (bottom) members for the numbering scheme for excited baryons. Here $qqq$ stands for $n_{q1}n_{q2}n_{q3}$. See the text for the definition of the notation. The numbers in parenthesis correspond to the mass of the baryons. The states marked as (?) are not experimentally confirmed.</caption><thead><tr><th>J<sup>P</sup></th><th>(D, L<sub>N</sub><sup>P</sup>)</th><th>n<sub>r</sub>n<sub>L</sub>n<sub>q1</sub>n<sub>q2</sub>n<sub>q3</sub>n<sub>J</sub></th><th>N</th><th>Λ<sub>8</sub></th><th>Σ</th><th>Ξ</th><th>Λ<sub>1</sub></th></tr></thead><tbody><tr><td colspan="8" style="text-align: center; font-weight: bold;">Octet</td></tr><tr><td>1/2<sup>+</sup></td><td>(56, 0<sub>0</sub><sup>+</sup>)</td><td>00qqq2</td><td>(939)</td><td>(1116)</td><td>(1193)</td><td>(1318)</td><td>—</td></tr><tr><td>1/2<sup>+</sup></td><td>(56, 0<sub>2</sub><sup>+</sup>)</td><td>20qqq2</td><td>(1440)</td><td>(1600)</td><td>(1660)</td><td>(1690)</td><td>—</td></tr><tr><td>1/2<sup>+</sup></td><td>(70, 0<sub>2</sub><sup>+</sup>)</td><td>21qqq2</td><td>(1710)</td><td>(1810)</td><td>(1880)</td><td>(?)</td><td>(?)</td></tr><tr><td>1/2<sup>-</sup></td><td>(70, 1<sub>1</sub><sup>-</sup>)</td><td>10qqq2</td><td>(1535)</td><td>(1670)</td><td>(1620)</td><td>(1750)</td><td>(1405)</td></tr><tr><th>J<sup>P</sup></th><th>(D, L<sub>N</sub><sup>P</sup>)</th><th>n<sub>r</sub>n<sub>L</sub>n<sub>q1</sub>n<sub>q2</sub>n<sub>q3</sub>n<sub>J</sub></th><th>Δ</th><th>Σ</th><th>Ξ</th><th>Ω</th><th></th></tr><tr><td colspan="8" style="text-align: center; font-weight: bold;">Decuplet</td></tr><tr><td>3/2<sup>+</sup></td><td>(56, 0<sub>0</sub><sup>+</sup>)</td><td>00qqq4</td><td>(1232)</td><td>(1385)</td><td>(1530)</td><td>(1672)</td><td></td></tr><tr><td>3/2<sup>+</sup></td><td>(56, 0<sub>2</sub><sup>+</sup>)</td><td>20qqq4</td><td>(1600)</td><td>(1690)</td><td>(?)</td><td>(?)</td><td></td></tr><tr><td>1/2<sup>-</sup></td><td>(70, 1<sub>1</sub><sup>-</sup>)</td><td>11qqq2</td><td>(1620)</td><td>(1750)</td><td>(?)</td><td>(?)</td><td></td></tr><tr><td>3/2<sup>-</sup></td><td>(70, 1<sub>1</sub><sup>-</sup>)</td><td>12qqq4</td><td>(1700)</td><td>(?)</td><td>(?)</td><td>(?)</td><td></td></tr></tbody></table>

are intended to distinguish left/right/ longitudinal components. Codes 998 and 999 are reserved for GEANT tracking purposes.

11. The search for physics beyond the Standard Model is an active area, so these codes are also standardized as far as possible.

(a) A standard fourth generation of fermions is included by analogy with the first three.

(b) The graviton and the boson content of a two-Higgsdoublet scenario and of additional SU(2)×U(1) groups are found in the range 31–40.

(c) “One-of-a-kind” exotic particles are assigned numbers in the range 41–80. The subrange 61–80 can be used for new heavier fermions in generic models, where partners to the SM fermions would have codes offset by 60. If required, however, other assignments could be made.

(d) Fundamental supersymmetric particles are identified by adding a nonzero $n$ to the particle number. The superpartner of a boson or a left-handed fermion has $n=1$ while the superpartner of a right-handed fermion has $n=2$. When mixing occurs, such as between the winos and charged Higgsinos to give charginos, or between left and right sfermions, the lighter physical state is given the smaller basis state number.

(e) Technicolor states have $n=3$, with technifermions treated like ordinary fermions. States which are ordinary color singlets have $n_r = 0$. Color octets have $n_r = 1$. If a state has non-trivial quantum numbers under the topcolor groups SU(3)$_1 \times$SU(3)$_2$, the quantum numbers are specified by tech, $ij$, where $i$ and $j$ are 1 or 2. $n_L$ is then $2i + j$. The coloron, $V_8$, is a heavy gluon color octet and thus is 3100021.

(f) Excited (composite) quarks and leptons are identified by setting $n=4$ and $n_r = 0$.

(g) Within several scenarios of new physics, it is possible to have colored particles sufficiently long-lived for color-singlet hadronic states to form around them. In the context of supersymmetric scenarios, these states are called R-hadrons, since they carry odd R-parity. R-hadron codes, defined here, should be viewed as templates for corresponding codes also in other scenarios, for any long-lived particle that is either an unflavored color octet or a flavored color triplet. The R-hadron code is obtained by combining the SUSY particle code with a code for the light degrees of freedom, with as many intermediate zeros removed from the former as required to make place for the latter at the end. (To exemplify, a sparticle $n00000n_{\bar{q}}$ combined with quarks $q_1$ and $q_2$ obtains code $n00n_{\bar{q}}n_{q_1}n_{q_2}n_{J}$.) Specifically, the new-particle spin couples in the limit of large masses, so that the final $n_J$ digit is defined by the spin state of the light-quark system alone. An appropriate number of $n_q$ digits is used to define the ordinary-quark content. As usual, 9

rather than 21 is used to denote a gluon/gluino in composite states. The sign of the hadron agrees with that of the constituent new particle (a color triplet) where there is a distinct new antiparticle, and else is defined as for normal hadrons. Particle names are R with the flavor content as lower index.

(h) A black hole in models with extra dimensions has code 5000040. Kaluza-Klein excitations in models with extra dimensions have $n=5$ or $n=6$, to distinguish excitations of left- or right-handed fermions or, in case of mixing, the lighter or heavier state (cf. 11d). The nonzero $n_r$ digit gives the radial excitation number, in scenarios where the level spacing allow these to be distinguished. Should the model also contain supersymmetry, excited SUSY states would be denoted by an $n_r > 0$, with $n=1$ or 2 as usual. Should some colored states be long-lived enough that hadrons would form around them, the coding strategy of 11g applies, with the initial two $nn_r$ digits preserved in the combined code.

(i) Magnetic monopoles and dyons are assumed to have one unit of Dirac monopole charge and a variable integer number $n_{q1}n_{q2}n_{q3}$ units of electric charge. Codes 411$n_{q1}n_{q2}n_{q3}$0 are then used when the magnetic and electrical charge sign agree and 412$n_{q1}n_{q2}n_{q3}$0 when they disagree, with the overall sign of the particle set by the magnetic charge. For now no spin information is provided.

(j) The nature of Dark Matter (DM) is not known, and therefore a definitive classification is too early. Cандates within specific scenarios are classified therein, such as 1000022 for the lightest neutralino. Generic fundamental states can be given temporary codes in the range 51–60, with 51, 52 and 53 reserved for spin 0, 1/2 and 1 ones (this could also be an axion state). Generic mediators of s-channel DM pair creation of annihilation can be given codes 54 and 55 for spin 0 or 1 ones. Separate antiparticles, with negative codes, may or may not exist. More elaborate new scenarios should be constructed with $n=5$ and $n_r = 9$.

(k) Hidden Valley particles have $n=4$ and $n_r = 9$, and trailing numbers in agreement with their nearest-analog standard particles, as far as possible. Thus 4900021 is the gauge boson $g_v$ of a confining gauge field, 490000$n_{q_v}$ and 490001$n_{\ell_v}$ fundamental constituents charged or not under this, 4900022 is the $\gamma_v$ of a non-confining field, and 4900$n_{q_v1}n_{q_v2}n_J$ a Hidden Valley meson.

12. Occasionally program authors add their own states. To avoid confusion, these should be flagged by setting $nn_r = 99$.

13. Concerning the non-99 numbers, it may be noted that only quarks, excited quarks, squarks, and diquarks have $n_{q3} = 0$; only diquarks, baryons, and the odderon have $n_{q1} \neq 0$; and only mesons, the reggeon, and the pomeron have $n_{q1} = 0$ and $n_{q2} \neq 0$. Concerning mesons (not antimesons), if $n_{q1}$ is odd
then it labels a quark and an antiquark if even.

14. The 9-digit tetra-quark codes are $\pm 1n_r n_L n_q_1 n_q_2 0n_q_3 n_q_4 n_J$. For the particle $q_1 q_2$ is a diquark and $\bar{q}_3 \bar{q}_4$ an antidiquark, sorted such that $n_{q_1} \ge n_{q_2}$, $n_{q_3} \ge n_{q_4}$, $n_{q_1} \ge n_{q_3}$, and $n_{q_2} \ge n_{q_4}$ if $n_{q_1} = n_{q_3}$. For the antiparticle, given with a negative sign, $\bar{q}_1 \bar{q}_2$ is an antidiquark and $q_3 \bar{q}_4$ a diquark, with the same sorting except that either $n_{q_1} > n_{q_3}$ or $n_{q_2} > n_{q_4}$ (so that flavour-diagonal states are particles). The $n_r$, $n_L$, and $n_J$ numbers have the same meaning as for ordinary hadrons.

15. The 9-digit penta-quark codes are $\pm 1n_r n_L n_q_1 n_q_2 n_q_3 n_q_4 n_J$, sorted such that $n_{q_1} \ge n_{q_2} \ge n_{q_3} \ge n_{q_4}$. In the particle the first four are quarks and the fifth an antiquark while the opposite holds in the antiparticle, which is given with a negative sign. The $n_r$, $n_L$, and $n_J$ numbers have the same meaning as for ordinary hadrons.

16. Nuclear codes are given as 10-digit numbers $\pm 10LZZZAAAI$. For a (hyper)nucleus consisting of $n_p$ protons, $n_n$ neutrons and $n_\Lambda$ $\Lambda$'s, $A = n_p + n_n + n_\Lambda$ gives the total baryon number, $Z = n_p$ the total charge and $L = n_\Lambda$ the total number of strange quarks. $I$ gives the isomer level, with $I = 0$ corresponding to the ground state and $I > 0$ to excitations, see [2], where states denoted $m, n, p, q$ translate to $I = 1-4$. As examples, the deuteron is 1000010020 and $^{235}$U is 1000922350. To avoid ambiguities, nuclear codes should not be applied to a single hadron, like $p, n$ or $\Lambda^0$, where quark-contents-based codes already exist.

<table><thead><tr><th colspan="2">QUARKS</th></tr></thead><tbody><tr><td>d</td><td>1</td></tr><tr><td>u</td><td>2</td></tr><tr><td>s</td><td>3</td></tr><tr><td>c</td><td>4</td></tr><tr><td>b</td><td>5</td></tr><tr><td>t</td><td>6</td></tr><tr><td>t'</td><td>7</td></tr><tr><td>t''</td><td>8</td></tr></tbody></table>

<table><thead><tr><th colspan="2">LEPTONS</th></tr></thead><tbody><tr><td>e<sup>-</sup></td><td>11</td></tr><tr><td>&nu;<sub>e</sub></td><td>12</td></tr><tr><td>&mu;<sup>-</sup></td><td>13</td></tr><tr><td>&nu;<sub>&mu;</sub></td><td>14</td></tr><tr><td>&tau;<sup>-</sup></td><td>15</td></tr><tr><td>&nu;<sub>&tau;</sub></td><td>16</td></tr><tr><td>&tau;<sup>'-</sup></td><td>17</td></tr><tr><td>&nu;<sub>&tau;'</sub></td><td>18</td></tr></tbody></table>

<table><thead><tr><th colspan="2">GAUGE AND HIGGS BOSONS</th></tr></thead><tbody><tr><td>g</td><td>(9) 21</td></tr><tr><td>&gamma;</td><td>22</td></tr><tr><td>Z<sup>0</sup></td><td>23</td></tr><tr><td>W<sup>+</sup></td><td>24</td></tr><tr><td>h<sup>0</sup>/H<sup>0</sup></td><td>25</td></tr><tr><td>Z'<sup>0</sup>/Z'<sup>0</sup></td><td>32</td></tr><tr><td>Z''/Z''</td><td>33</td></tr><tr><td>W'/W'<sup>+</sup></td><td>34</td></tr><tr><td>H<sup>0</sup>/H<sup>0</sup></td><td>35</td></tr><tr><td>A<sup>0</sup>/H<sup>0</sup></td><td>36</td></tr><tr><td>H<sup>+</sup></td><td>37</td></tr><tr><td>H<sup>++</sup></td><td>38</td></tr><tr><td>a<sup>0</sup>/H<sup>0</sup></td><td>40</td></tr></tbody></table>

<table><thead><tr><th colspan="2">SPECIAL PARTICLES</th></tr></thead><tbody><tr><td>G (graviton)</td><td>39</td></tr><tr><td>R<sup>0</sup></td><td>41</td></tr><tr><td>LQ<sup>c</sup></td><td>42</td></tr><tr><td>DM (S=0)</td><td>51</td></tr><tr><td>DM (S=1/2)</td><td>52</td></tr><tr><td>DM (S=1)</td><td>53</td></tr><tr><td>reggeon</td><td>110</td></tr><tr><td>pomeron</td><td>990</td></tr><tr><td>odderon</td><td>9990</td></tr></tbody></table>

<table><thead><tr><th colspan="2">DIQUARKS</th></tr></thead><tbody><tr><td>(dd)<sub>1</sub></td><td>1103</td></tr><tr><td>(ud)<sub>0</sub></td><td>2101</td></tr><tr><td>(ud)<sub>1</sub></td><td>2103</td></tr><tr><td>(uu)<sub>1</sub></td><td>2203</td></tr><tr><td>(sd)<sub>0</sub></td><td>3101</td></tr><tr><td>(sd)<sub>1</sub></td><td>3103</td></tr><tr><td>(su)<sub>0</sub></td><td>3201</td></tr><tr><td>(su)<sub>1</sub></td><td>3203</td></tr><tr><td>(ss)<sub>1</sub></td><td>3303</td></tr><tr><td>(cd)<sub>0</sub></td><td>4101</td></tr><tr><td>(cd)<sub>1</sub></td><td>4103</td></tr><tr><td>(cu)<sub>0</sub></td><td>4201</td></tr><tr><td>(cu)<sub>1</sub></td><td>4203</td></tr><tr><td>(cs)<sub>0</sub></td><td>4301</td></tr><tr><td>(cs)<sub>1</sub></td><td>4303</td></tr><tr><td>(cc)<sub>1</sub></td><td>4403</td></tr><tr><td>(bd)<sub>0</sub></td><td>5101</td></tr><tr><td>(bd)<sub>1</sub></td><td>5103</td></tr><tr><td>(bu)<sub>0</sub></td><td>5201</td></tr><tr><td>(bu)<sub>1</sub></td><td>5203</td></tr><tr><td>(bs)<sub>0</sub></td><td>5301</td></tr><tr><td>(bs)<sub>1</sub></td><td>5303</td></tr><tr><td>(bc)<sub>0</sub></td><td>5401</td></tr><tr><td>(bc)<sub>1</sub></td><td>5403</td></tr><tr><td>(bb)<sub>1</sub></td><td>5503</td></tr></tbody></table>

<table><thead><tr><th colspan="2">SUSY PARTICLES</th></tr></thead><tbody><tr><td>~d<sub>L</sub></td><td>1000001</td></tr><tr><td>~u<sub>L</sub></td><td>1000002</td></tr><tr><td>~s<sub>L</sub></td><td>1000003</td></tr><tr><td>~c<sub>L</sub></td><td>1000004</td></tr><tr><td>~b<sub>1</sub></td><td>1000005<sup>a</sup></td></tr><tr><td>~t<sub>1</sub></td><td>1000006<sup>a</sup></td></tr><tr><td>~e<sup>-</sup></td><td>1000011</td></tr><tr><td>~&nu;<sub>eL</sub></td><td>1000012</td></tr><tr><td>~&mu;<sup>-</sup></td><td>1000013</td></tr><tr><td>~&nu;<sub>&mu;L</sub></td><td>1000014</td></tr><tr><td>~&nu;<sub>&mu;L</sub></td><td>1000015<sup>a</sup></td></tr><tr><td>~&tau;<sup>-</sup></td><td>1000016</td></tr><tr><td>~&nu;<sub>&tau;L</sub></td><td>1000017</td></tr><tr><td>~d<sub>R</sub></td><td>2000001</td></tr><tr><td>~u<sub>R</sub></td><td>2000002</td></tr><tr><td>~s<sub>R</sub></td><td>2000003</td></tr><tr><td>~c<sub>R</sub></td><td>2000004</td></tr><tr><td>~b<sub>2</sub></td><td>2000005<sup>a</sup></td></tr><tr><td>~t<sub>2</sub></td><td>2000006<sup>a</sup></td></tr><tr><td>~e<sup>-</sup></td><td>2000011</td></tr><tr><td>~&nu;<sub>R</sub></td><td>2000013</td></tr><tr><td>~&mu;<sub>R</sub></td><td>2000015<sup>a</sup></td></tr><tr><td>~&tau;<sub>2</sub></td><td>2000023<sup>b</sup></td></tr><tr><td>~g</td><td>1000021</td></tr><tr><td>~&chi;<sup>0</sup><sub>1</sub></td><td>1000022<sup>b</sup></td></tr><tr><td>~&chi;<sup>0</sup><sub>2</sub></td><td>1000023<sup>b</sup></td></tr><tr><td>~&chi;<sup>+</sup><sub>1</sub></td><td>1000024<sup>b</sup></td></tr><tr><td>~&chi;<sup>0</sup><sub>3</sub></td><td>1000025<sup>b</sup></td></tr></tbody></table>

<table><tr><td>~&chi;<sup>0</sup><sub>4</sub></td><td>1000035<sup>b</sup></td></tr><tr><td>&chi;<sup>+</sup><sub>2</sub></td><td>1000037<sup>b</sup></td></tr><tr><td>G</td><td>1000039</td></tr></table>

<table><thead><tr><th colspan="2">LIGHT I = 1 MESONS</th></tr></thead><tbody><tr><td>&pi;<sup>0</sup></td><td>111</td></tr><tr><td>&pi;<sup>+</sup></td><td>211</td></tr><tr><td>a<sub>0</sub>(980)<sup>0</sup></td><td>9000111</td></tr><tr><td>a<sub>0</sub>(980)<sup>+</sup></td><td>9000211</td></tr><tr><td>&pi;(1300)<sup>0</sup></td><td>100111</td></tr><tr><td>&pi;(1300)<sup>+</sup></td><td>100211</td></tr><tr><td>a<sub>0</sub>(1450)<sup>0</sup></td><td>10111</td></tr><tr><td>a<sub>0</sub>(1450)<sup>+</sup></td><td>10211</td></tr><tr><td>&pi;(1800)<sup>0</sup></td><td>9010111</td></tr><tr><td>&pi;(1800)<sup>+</sup></td><td>9010211</td></tr><tr><td>&rho;(770)<sup>0</sup></td><td>113</td></tr><tr><td>&rho;(770)<sup>+</sup></td><td>213</td></tr><tr><td>b<sub>1</sub>(1235)<sup>0</sup></td><td>10113</td></tr><tr><td>b<sub>1</sub>(1235)<sup>+</sup></td><td>10213</td></tr><tr><td>a<sub>1</sub>(1260)<sup>0</sup></td><td>20113</td></tr><tr><td>a<sub>1</sub>(1260)<sup>+</sup></td><td>20213</td></tr><tr><td>&pi;(1400)<sup>0</sup></td><td>9000113</td></tr><tr><td>&pi;(1400)<sup>+</sup></td><td>9000213</td></tr><tr><td>&rho;(1450)<sup>0</sup></td><td>100113</td></tr><tr><td>&rho;(1450)<sup>+</sup></td><td>100213</td></tr><tr><td>&pi;(1600)<sup>0</sup></td><td>9010113</td></tr><tr><td>&pi;(1600)<sup>+</sup></td><td>9010213</td></tr><tr><td>a<sub>1</sub>(1640)<sup>0</sup></td><td>9020113</td></tr><tr><td>a<sub>1</sub>(1640)<sup>+</sup></td><td>9020213</td></tr><tr><td>&rho;(1700)<sup>0</sup></td><td>30113</td></tr><tr><td>&rho;(1700)<sup>+</sup></td><td>30213</td></tr><tr><td>&rho;(1900)<sup>0</sup></td><td>9030113</td></tr><tr><td>&rho;(1900)<sup>+</sup></td><td>9030213</td></tr><tr><td>&rho;(1900)<sup>+</sup></td><td>9040113</td></tr><tr><td>&rho;(2150)<sup>0</sup></td><td>9040213</td></tr><tr><td>&rho;(2150)<sup>+</sup></td><td>9040213</td></tr><tr><td>a<sub>2</sub>(1320)<sup>0</sup></td><td>115</td></tr><tr><td>a<sub>2</sub>(1320)<sup>+</sup></td><td>215</td></tr><tr><td>&pi;(2167)<sup>0</sup></td><td>10115</td></tr><tr><td>&pi;(2167)<sup>+</sup></td><td>10215</td></tr><tr><td>a<sub>2</sub>(1700)<sup>0</sup></td><td>9000115</td></tr><tr><td>a<sub>2</sub>(1700)<sup>+</sup></td><td>9000215</td></tr><tr><td>&pi;(2100)<sup>0</sup></td><td>9010115</td></tr><tr><td>&pi;(2100)<sup>+</sup></td><td>9010215</td></tr><tr><td>&rho;(3690)<sup>0</sup></td><td>117</td></tr><tr><td>&rho;(3690)<sup>+</sup></td><td>217</td></tr><tr><td>&rho;(3990)<sup>0</sup></td><td>9000117</td></tr><tr><td>&rho;(3990)<sup>+</sup></td><td>9000217</td></tr><tr><td>&rho;(2250)<sup>0</sup></td><td>9010117</td></tr><tr><td>&rho;(2250)<sup>+</sup></td><td>9010217</td></tr><tr><td>a<sub>4</sub>(2040)<sup>0</sup></td><td>119</td></tr><tr><td>a<sub>4</sub>(2040)<sup>+</sup></td><td>219</td></tr></tbody></table>

<table><thead><tr><th colspan="2">LIGHT I = 0 MESONS</th></tr></thead><tbody><tr><td>(u&ddot;u, d&ddot;d, s&ddot;s admixtures)</td><td>221</td></tr><tr><td>&eta;</td><td>311</td></tr><tr><td>&eta;'(958)</td><td>321</td></tr><tr><td>f<sub>0</sub>(500)</td><td>9000221</td></tr><tr><td>f<sub>0</sub>(980)</td><td>9010221</td></tr><tr><td>&eta;(1295)</td><td>100221</td></tr><tr><td>f<sub>0</sub>(1370)</td><td>10221</td></tr><tr><td>&eta;(1405)</td><td>9020221</td></tr><tr><td>&eta;(1475)</td><td>100331</td></tr><tr><td>f<sub>0</sub>(1500)</td><td>9030221</td></tr><tr><td>f<sub>0</sub>(1710)</td><td>10331</td></tr><tr><td>&eta;(1760)</td><td>9040221</td></tr><tr><td>f<sub>0</sub>(2020)</td><td>9050221</td></tr><tr><td>f<sub>0</sub>(2100)</td><td>9060221</td></tr><tr><td>f<sub>0</sub>(2200)</td><td>9070221</td></tr><tr><td>&eta;(2225)</td><td>9080221</td></tr><tr><td>&omega;(782)</td><td>223</td></tr><tr><td>&phi;(1020)</td><td>333</td></tr><tr><td>h<sub>1</sub>(1170)</td><td>10223</td></tr><tr><td>f<sub>1</sub>(1285)</td><td>20223</td></tr><tr><td>h<sub>1</sub>(1380)</td><td>10333</td></tr><tr><td>f<sub>1</sub>(1420)</td><td>20333</td></tr><tr><td>&omega;(1420)</td><td>100223</td></tr><tr><td>f<sub>1</sub>(1510)</td><td>9000223</td></tr><tr><td>h<sub>1</sub>(1595)</td><td>9010223</td></tr><tr><td>&omega;(1650)</td><td>30223</td></tr><tr><td>&phi;(1680)</td><td>100333</td></tr><tr><td>f<sub>2</sub>(1270)</td><td>225</td></tr><tr><td>f<sub>2</sub>(1430)</td><td>9000225</td></tr><tr><td>f<sub>2</sub>'(1525)</td><td>335</td></tr><tr><td>f<sub>2</sub>(1565)</td><td>9010225</td></tr><tr><td>f<sub>2</sub>(1640)</td><td>9020225</td></tr><tr><td>&eta;(21645)</td><td>10225</td></tr><tr><td>f<sub>2</sub>(1810)</td><td>9030225</td></tr><tr><td>&eta;(21870)</td><td>10335</td></tr><tr><td>f<sub>2</sub>(1910)</td><td>9040225</td></tr><tr><td>f<sub>2</sub>(1950)</td><td>9050225</td></tr><tr><td>f<sub>2</sub>(2010)</td><td>9060225</td></tr><tr><td>f<sub>2</sub>(2150)</td><td>9070225</td></tr><tr><td>f<sub>2</sub>(2300)</td><td>9080225</td></tr><tr><td>f<sub>2</sub>(2340)</td><td>9090225</td></tr><tr><td>&omega;(3670)</td><td>227</td></tr><tr><td>&phi;(3680)</td><td>337</td></tr><tr><td>f<sub>4</sub>(2050)</td><td>229</td></tr><tr><td>f<sub>J</sub>(2220)</td><td>9000229</td></tr><tr><td>f<sub>4</sub>(2300)</td><td>9010229</td></tr></tbody></table>

<table><thead><tr><th colspan="2">STRANGE MESONS</th></tr></thead><tbody><tr><td>K<sup>0</sup><sub>L</sub></td><td>130</td></tr><tr><td>K<sup>0</sup><sub>S</sub></td><td>310</td></tr><tr><td>K<sup>0</sup></td><td>311</td></tr><tr><td>K<sup>+</sup></td><td>321</td></tr><tr><td>K<sup>*</sup><sub>0</sub>(700)<sup>0</sup></td><td>9000311</td></tr><tr><td>K<sup>*</sup><sub>0</sub>(700)<sup>+</sup></td><td>9000321</td></tr><tr><td>K<sup>*</sup><sub>0</sub>(1430)<sup>0</sup></td><td>10311</td></tr><tr><td>K<sup>0</sup><sub>0</sub>(1430)<sup>+</sup></td><td>10321</td></tr><tr><td>K(1460)<sup>0</sup></td><td>100311</td></tr><tr><td>K(1460)<sup>+</sup></td><td>100321</td></tr><tr><td>K(1830)<sup>0</sup></td><td>9010311</td></tr><tr><td>K(1830)<sup>+</sup></td><td>9010321</td></tr><tr><td>K<sup>0</sup><sub>0</sub>(1950)<sup>0</sup></td><td>9020311</td></tr><tr><td>K<sup>*</sup><sub>0</sub>(1950)<sup>+</sup></td><td>9020321</td></tr><tr><td>K<sup>*</sup>(892)<sup>0</sup></td><td>313</td></tr><tr><td>K<sup>*</sup>(892)<sup>+</sup></td><td>323</td></tr><tr><td>K<sub>1</sub>(1270)<sup>0</sup></td><td>10313</td></tr><tr><td>K<sub>1</sub>(1270)<sup>+</sup></td><td>10323</td></tr><tr><td>K<sub>1</sub>(1400)<sup>0</sup></td><td>20313</td></tr><tr><td>K<sub>1</sub>(1400)<sup>+</sup></td><td>20323</td></tr><tr><td>K<sup>*</sup>(1410)<sup>0</sup></td><td>100313</td></tr><tr><td>K<sup>*</sup>(1410)<sup>+</sup></td><td>100323</td></tr><tr><td>K<sub>1</sub>(1650)<sup>0</sup></td><td>9000313</td></tr><tr><td>K<sub>1</sub>(1650)<sup>+</sup></td><td>9000323</td></tr><tr><td>K<sup>*</sup>(1680)<sup>0</sup></td><td>30313</td></tr><tr><td>K<sup>*</sup>(1680)<sup>+</sup></td><td>30323</td></tr><tr><td>K<sup>*</sup><sub>2</sub>(1430)<sup>0</sup></td><td>315</td></tr><tr><td>K<sup>*</sup><sub>2</sub>(1430)<sup>+</sup></td><td>325</td></tr><tr><td>K<sub>2</sub>(1580)<sup>0</sup></td><td>9000315</td></tr><tr><td>K<sub>2</sub>(1580)<sup>+</sup></td><td>9000325</td></tr><tr><td>K<sub>2</sub>(1770)<sup>0</sup></td><td>10315</td></tr><tr><td>K<sub>2</sub>(1770)<sup>+</sup></td><td>10325</td></tr><tr><td>K<sub>2</sub>(1820)<sup>0</sup></td><td>20315</td></tr><tr><td>K<sub>2</sub>(1820)<sup>+</sup></td><td>20325</td></tr><tr><td>K<sup>*</sup><sub>2</sub>(1980)<sup>0</sup></td><td>9010315</td></tr><tr><td>K<sup>*</sup><sub>2</sub>(1980)<sup>+</sup></td><td>9010325</td></tr><tr><td>K<sub>2</sub>(2250)<sup>0</sup></td><td>9020315</td></tr><tr><td>K<sub>2</sub>(2250)<sup>+</sup></td><td>9020325</td></tr><tr><td>K<sup>*</sup><sub>3</sub>(1780)<sup>0</sup></td><td>317</td></tr><tr><td>K<sup>*</sup><sub>3</sub>(1780)<sup>+</sup></td><td>327</td></tr><tr><td>K<sub>3</sub>(2320)<sup>0</sup></td><td>9010317</td></tr><tr><td>K<sub>3</sub>(2320)<sup>+</sup></td><td>9010327</td></tr><tr><td>K<sup>*</sup><sub>4</sub>(2045)<sup>0</sup></td><td>319</td></tr><tr><td>K<sup>*</sup><sub>4</sub>(2045)<sup>+</sup></td><td>329</td></tr><tr><td>K<sub>4</sub>(2500)<sup>0</sup></td><td>9000319</td></tr><tr><td>K<sub>4</sub>(2500)<sup>+</sup></td><td>9000329</td></tr></tbody></table>
CHARMED
MESONS

<table><tr><td>D<sup>+</sup></td><td>411</td></tr><tr><td>D<sup>0</sup></td><td>421</td></tr><tr><td>D<sub>0</sub><sup>*</sup>(2400)<sup>+</sup></td><td>10411</td></tr><tr><td>D<sub>0</sub><sup>0</sup>(2400)<sup>0</sup></td><td>10421</td></tr><tr><td>D<sup>*</sup>(2010)<sup>+</sup></td><td>413</td></tr><tr><td>D<sup>*</sup>(2007)<sup>0</sup></td><td>423</td></tr><tr><td>D<sub>1</sub>(2420)<sup>+</sup></td><td>10413</td></tr><tr><td>D<sub>1</sub>(2420)<sup>0</sup></td><td>10423</td></tr><tr><td>D<sub>1</sub>(H)<sup>+</sup></td><td>20413</td></tr><tr><td>D<sub>1</sub>(2430)<sup>0</sup></td><td>20423</td></tr><tr><td>D<sub>2</sub><sup>*</sup>(2460)<sup>+</sup></td><td>415</td></tr><tr><td>D<sub>2</sub><sup>0</sup>(2460)<sup>0</sup></td><td>425</td></tr><tr><td>D<sub>2</sub><sup>+</sup></td><td>431</td></tr><tr><td>D<sup>*</sup><sub>s0</sub>(2317)<sup>+</sup></td><td>10431</td></tr><tr><td>D<sup>*</sup><sub>s</sub><sup>+</sup></td><td>433</td></tr><tr><td>D<sub>s1</sub>(2536)<sup>+</sup></td><td>10433</td></tr><tr><td>D<sub>s1</sub>(2460)<sup>+</sup></td><td>20433</td></tr><tr><td>D<sup>*</sup><sub>s2</sub>(2573)<sup>+</sup></td><td>435</td></tr></table>

<table><tr><td>B<sup>*+</sup><sub>c0</sub></td><td>10541</td></tr><tr><td>B<sup>+</sup><sub>c</sub></td><td>543</td></tr><tr><td>B<sub>c1</sub>(L)<sup>+</sup></td><td>10543</td></tr><tr><td>B<sub>c1</sub>(H)<sup>+</sup></td><td>20543</td></tr><tr><td>B<sup>*</sup><sub>c2</sub></td><td>545</td></tr></table>

<table><tr><td>γ(11020)</td><td>9010553</td></tr><tr><td>χ<sub>b2</sub>(1P)</td><td>555</td></tr><tr><td>η<sub>b2</sub>(1D)</td><td>10555</td></tr><tr><td>γ<sub>2</sub>(1D)</td><td>20555</td></tr><tr><td>χ<sub>b2</sub>(2P)</td><td>100555</td></tr><tr><td>η<sub>b2</sub>(2D)</td><td>110555</td></tr><tr><td>γ<sub>2</sub>(2D)</td><td>120555</td></tr><tr><td>χ<sub>b2</sub>(3P)</td><td>200555</td></tr><tr><td>γ<sub>3</sub>(1D)</td><td>557</td></tr><tr><td>γ<sub>3</sub>(2D)</td><td>100557</td></tr></table>

c<span style="text-decoration: overline;">c</span> MESONS

<table><tr><td>η<sub>c</sub>(1S)</td><td>441</td></tr><tr><td>χ<sub>c0</sub>(1P)</td><td>10441</td></tr><tr><td>η<sub>c</sub>(2S)</td><td>100441</td></tr><tr><td>J/ψ(1S)</td><td>443</td></tr><tr><td>h<sub>c</sub>(1P)</td><td>10443</td></tr><tr><td>χ<sub>c1</sub>(1P)</td><td>20443</td></tr><tr><td>ψ(2S)</td><td>100443</td></tr><tr><td>ψ(3770)</td><td>30443</td></tr><tr><td>ψ(4040)</td><td>9000443</td></tr><tr><td>ψ(4160)</td><td>9010443</td></tr><tr><td>ψ(4415)</td><td>9020443</td></tr><tr><td>χ<sub>c2</sub>(1P)</td><td>445</td></tr><tr><td>χ<sub>c2</sub>(3930)</td><td>100445</td></tr></table>

LIGHT
BARYONS

<table><tr><td>p</td><td>2212</td></tr><tr><td>n</td><td>2112</td></tr><tr><td>Δ<sup>++</sup></td><td>2224</td></tr><tr><td>Δ<sup>+</sup></td><td>2214</td></tr><tr><td>Δ<sup>0</sup></td><td>2114</td></tr><tr><td>Δ<sup>-</sup></td><td>1114</td></tr></table>

STRANGE
BARYONS

<table><tr><td>Λ</td><td>3122</td></tr><tr><td>Σ<sup>+</sup></td><td>3222</td></tr><tr><td>Σ<sup>0</sup></td><td>3212</td></tr><tr><td>Σ<sup>-</sup></td><td>3112</td></tr><tr><td>Σ<sup>+</sup></td><td>3224<sup>c</sup></td></tr><tr><td>Σ<sup>0</sup></td><td>3214<sup>c</sup></td></tr><tr><td>Σ<sup>-</sup></td><td>3114<sup>c</sup></td></tr><tr><td>Ξ<sup>0</sup></td><td>3322</td></tr><tr><td>Ξ<sup>-</sup></td><td>3312</td></tr><tr><td>Ξ<sup>0</sup></td><td>3324<sup>c</sup></td></tr><tr><td>Ξ<sup>+</sup></td><td>3314<sup>c</sup></td></tr><tr><td>Ξ<sup>-</sup></td><td>3334</td></tr></table>

CHARMED
BARYONS

<table><tr><td>Λ<sup>+</sup><sub>c</sub></td><td>4122</td></tr><tr><td>Σ<sup>++</sup><sub>c</sub></td><td>4222</td></tr><tr><td>Σ<sup>+</sup><sub>c</sub></td><td>4212</td></tr><tr><td>Σ<sup>0</sup><sub>c</sub></td><td>4112</td></tr><tr><td>Σ<sup>+</sup><sub>c</sub></td><td>4224</td></tr><tr><td>Σ<sup>+</sup><sub>c</sub></td><td>4214</td></tr><tr><td>Σ<sup>0</sup><sub>c</sub></td><td>4114</td></tr><tr><td>Ξ<sup>+</sup><sub>c</sub></td><td>4232</td></tr><tr><td>Ξ<sup>0</sup><sub>c</sub></td><td>4132</td></tr><tr><td>Ξ<sup>+</sup><sub>c</sub></td><td>4322</td></tr><tr><td>Ξ<sup>0</sup><sub>c</sub></td><td>4312</td></tr><tr><td>Ξ<sup>+</sup><sub>c</sub></td><td>4324</td></tr><tr><td>Ξ<sup>0</sup><sub>c</sub></td><td>4314</td></tr><tr><td>Ω<sup>0</sup><sub>c</sub></td><td>4332</td></tr><tr><td>Ω<sup>0</sup><sub>c</sub></td><td>4334</td></tr><tr><td>Ξ<sup>+</sup><sub>c</sub></td><td>4412</td></tr><tr><td>Ξ<sup>+</sup><sub>cc</sub></td><td>4422</td></tr><tr><td>Ξ<sup>+</sup><sub>cc</sub></td><td>4414</td></tr><tr><td>Ξ<sup>+</sup><sub>cc</sub></td><td>4424</td></tr><tr><td>Ξ<sup>+</sup><sub>cc</sub></td><td>4432</td></tr><tr><td>Ξ<sup>+</sup><sub>cc</sub></td><td>4434</td></tr><tr><td>Ξ<sup>+</sup><sub>ccc</sub></td><td>4444</td></tr></table>

BOTTOM
BARYONS

<table><tr><td>Λ<sup>0</sup><sub>b</sub></td><td>5122</td></tr><tr><td>Σ<sup>-</sup><sub>b</sub></td><td>5112</td></tr><tr><td>Σ<sup>0</sup><sub>b</sub></td><td>5122</td></tr><tr><td>Σ<sup>+</sup><sub>b</sub></td><td>5222</td></tr><tr><td>Σ<sup>+</sup><sub>b</sub></td><td>5114</td></tr><tr><td>Σ<sup>0</sup><sub>b</sub></td><td>5124</td></tr><tr><td>Σ<sup>+</sup><sub>b</sub></td><td>5224</td></tr><tr><td>Ξ<sup>-</sup><sub>b</sub></td><td>5132</td></tr><tr><td>Ξ<sup>0</sup><sub>b</sub></td><td>5232</td></tr><tr><td>Ξ<sup>+</sup><sub>b</sub></td><td>5312</td></tr><tr><td>Ξ<sup>0</sup><sub>b</sub></td><td>5322</td></tr><tr><td>Ξ<sup>0</sup><sub>b</sub></td><td>5314</td></tr><tr><td>Ξ<sup>0</sup><sub>b</sub></td><td>5324</td></tr><tr><td>Ξ<sup>-</sup><sub>b</sub></td><td>5332</td></tr><tr><td>Ξ<sup>-</sup><sub>b</sub></td><td>5334</td></tr><tr><td>Ξ<sup>0</sup><sub>b</sub></td><td>5412</td></tr><tr><td>Ξ<sup>+</sup><sub>b</sub></td><td>5422</td></tr><tr><td>Ξ<sup>+</sup><sub>b</sub></td><td>5412</td></tr><tr><td>Ξ<sup>+</sup><sub>b</sub></td><td>5422</td></tr><tr><td>Ξ<sup>+</sup><sub>b</sub></td><td>5414</td></tr><tr><td>Ξ<sup>+</sup><sub>b</sub></td><td>5424</td></tr><tr><td>Ξ<sup>+</sup><sub>b</sub></td><td>5342</td></tr><tr><td>Ξ<sup>0</sup><sub>b</sub></td><td>5432</td></tr><tr><td>Ξ<sup>0</sup><sub>b</sub></td><td>5434</td></tr><tr><td>Ξ<sup>0</sup><sub>b</sub></td><td>5442</td></tr><tr><td>Ξ<sup>0</sup><sub>ccc</sub></td><td>5444</td></tr><tr><td>Ξ<sup>+</sup><sub>ccc</sub></td><td>5512</td></tr><tr><td>Ξ<sup>0</sup><sub>bb</sub></td><td>5522</td></tr><tr><td>Ξ<sup>0</sup><sub>bb</sub></td><td>5514</td></tr><tr><td>Ξ<sup>0</sup><sub>bb</sub></td><td>5524</td></tr><tr><td>Ξ<sup>0</sup><sub>bb</sub></td><td>5532</td></tr><tr><td>Ξ<sup>0</sup><sub>bb</sub></td><td>5534</td></tr><tr><td>Ξ<sup>0</sup><sub>bb</sub></td><td>5542</td></tr><tr><td>Ξ<sup>0</sup><sub>bbb</sub></td><td>5544</td></tr><tr><td>Ξ<sup>0</sup><sub>bbb</sub></td><td>5554</td></tr></table>

BOTTOM
MESONS

<table><tr><td>B<sup>0</sup></td><td>511</td></tr><tr><td>B<sup>+</sup></td><td>521</td></tr><tr><td>B<sup>*</sup><sub>0</sub></td><td>10511</td></tr><tr><td>B<sup>*</sup><sub>0</sub></td><td>10521</td></tr><tr><td>B<sup>0</sup></td><td>513</td></tr><tr><td>B<sup>+</sup></td><td>523</td></tr><tr><td>B<sub>1</sub>(L)<sup>0</sup></td><td>10513</td></tr><tr><td>B<sub>1</sub>(L)<sup>+</sup></td><td>10523</td></tr><tr><td>B<sub>1</sub>(H)<sup>0</sup></td><td>20513</td></tr><tr><td>B<sub>1</sub>(H)<sup>+</sup></td><td>20523</td></tr><tr><td>B<sup>*</sup><sub>2</sub></td><td>515</td></tr><tr><td>B<sup>*</sup><sub>2</sub></td><td>525</td></tr><tr><td>B<sup>0</sup><sub>s</sub></td><td>531</td></tr><tr><td>B<sup>*</sup><sub>s0</sub></td><td>10531</td></tr><tr><td>B<sup>*</sup><sub>s</sub></td><td>533</td></tr><tr><td>B<sub>s1</sub>(L)<sup>0</sup></td><td>10533</td></tr><tr><td>B<sub>s1</sub>(H)<sup>0</sup></td><td>20533</td></tr><tr><td>B<sup>*</sup><sub>s2</sub></td><td>535</td></tr><tr><td>B<sup>+</sup><sub>c</sub></td><td>541</td></tr></table>

PENTAQUARKS

<table><tr><td>Θ<sup>+</sup></td><td>100221132*</td></tr><tr><td>Φ<sup>-</sup></td><td>100331122*</td></tr></table>

Footnotes to the tables:

*) Numbers or names in bold face are new or have changed since the 2020 Review.

a) Particulary in the third generation, the left and right sfermion states may mix, as shown.

The lighter mixed state is given the smaller number.

b) The physical $\tilde{\chi}$ states are admixtures of the pure $\tilde{\gamma}$, $\tilde{Z}^0$, $\tilde{W}^+$,

$\tilde{H}_1^0$, $\tilde{H}_2^0$, and $\tilde{H}^+$ states.

c) $\Sigma^*$ and $\Xi^*$ are alternate names for $\Sigma(1385)$ and $\Xi(1530)$.

References

[1] G. P. Yost et al. (Particle Data Group), Phys. Lett. B204, 1 (1988).

[2] G. Audi et al., Nucl. Phys. A729, 3 (2003).
46. Clebsch-Gordan Coefficients, Spherical Harmonics, and d Functions

<figure><img src="image_280.png" /><figcaption>**Figure 46.1:** The sign convention is that of Wigner (Group Theory, Academic Press, New York, 1959), also used by Condon and Shortley (The Theory of Atomic Spectra, Cambridge Univ. Press, New York, 1953), Rose (Elementary Theory of Angular Momentum, Wiley, New York, 1957), and Cohen (Tables of the Clebsch-Gordan Coefficients, North American Rockwell Science Center, Thousand Oaks, Calif., 1974).</figcaption></figure>
47. SU(3) Isoscalar Factors and Representation Matrices

Revised August 2019 by R. Kelly (LBNL).

The most commonly used SU(3) isoscalar factors, corresponding to the singlet, octet, and decuplet content of 8 ⊗ 8 and 10 ⊗ 8, are shown at the right. The notation uses particle names to identify the coefficients, so that the pattern of relative couplings may be seen at a glance. We illustrate the use of the coefficients below. See J.J de Swart, Rev. Mod. Phys. 35, 916 (1963) for detailed explanations and phase conventions.

A $\sqrt{\Lambda}$ is to be understood over every integer in the matrices; the exponent 1/2 on each matrix is a reminder of this. For example, the $\Xi \to \Omega K$ element of the $10 \to 10 \otimes 8$ matrix is $-\sqrt{6}/\sqrt{24} = -1/2$.

Intramultiplet relative decay strengths may be read directly from the matrices. For example, in decuplet → octet + octet decays, the ratio of $\Omega^* \to \Xi K$ and $\Delta \to N\pi$ partial widths is, from the $10 \to 8 \times 8$ matrix,

$$ \frac{\Gamma(\Omega^* \to \Xi \overline{K})}{\Gamma(\Delta \to N\pi)} = \frac{12}{6} \times (\text{phase space factors}). \quad (47.1) $$

Including isospin Clebsch-Gordan coefficients, we obtain, e.g.,

$$ \frac{\Gamma(\Omega^{*-} \to \Xi^0 K^-)}{\Gamma(\Delta^+ \to p\pi^0)} = \frac{1}{2} \frac{1}{3} \times \frac{12}{6} \times \text{p.s.f.} = \frac{3}{2} \times \text{p.s.f.} \quad (47.2) $$

Partial widths for $8 \to 8 \otimes 8$ involve a linear superposition of $8_1$ (symmetric) and $8_2$ (antisymmetric) couplings. For example,

$$ \Gamma(\Xi^* \to \Xi \pi) \sim \left( -\sqrt{\frac{9}{20}} g_1 + \sqrt{\frac{3}{12}} g_2 \right)^2 . \quad (47.3) $$

The relations between $g_1$ and $g_2$ (with de Swart's normalization) and the standard D and F couplings that appear in the interaction Lagrangian,

$$ \mathcal{L} = -\sqrt{2} D \operatorname{Tr}(\{\overline{B}, B\}M) + \sqrt{2} F \operatorname{Tr}([\overline{B}, B]M), \quad (47.4) $$

where $[\overline{B}, B] \equiv \overline{B}B - B\overline{B}$ and $\{\overline{B}, B\} \equiv \overline{B}B + B\overline{B}$, are

$$ D = \frac{\sqrt{30}}{40} g_1, \quad F = \frac{\sqrt{6}}{24} g_2. \quad (47.5) $$

Thus, for example,

$$ \Gamma(\Xi^* \to \Xi \pi) \sim (F - D)^2 \sim (1 - 2\alpha)^2, \quad (47.6) $$

where $\alpha \equiv F/(D+F)$. (This definition of $\alpha$ is de Swart's. The alternative $D/(D+F)$, due to Gell-Mann, is also used.)

The generators of SU(3) transformations, $\lambda_a$ ($a = 1, 8$), are $3 \times 3$ matrices that obey the following commutation and anticommutation relationships:

$$ [\lambda_a, \lambda_b] \equiv \lambda_a \lambda_b - \lambda_b \lambda_a = 2i f_{abc} \lambda_c \quad (47.7) $$

$$ \{\lambda_a, \lambda_b\} \equiv \lambda_a \lambda_b + \lambda_b \lambda_a = \frac{4}{3} \delta_{ab} I + 2d_{abc} \lambda_c, \quad (47.8) $$

where $I$ is the $3 \times 3$ identity matrix, and $\delta_{ab}$ is the Kronecker delta symbol. The $f_{abc}$ are odd under the permutation of any pair of indices, while the $d_{abc}$ are even. The nonzero values are:

$$ 1 \to 8 \otimes 8 $$

$$ (\lambda) \to (N\bar{K} \quad \Sigma\pi \quad \Lambda\eta \quad \Xi K) = \frac{1}{\sqrt{8}} (2 \quad 3 \quad -1 \quad -2)^{1/2} $$

$$ 8_1 \to 8 \otimes 8 $$

$$ \begin{pmatrix} N \\ \Sigma \\ \Lambda \\ \Xi \end{pmatrix} \to \begin{pmatrix} N\pi & N\eta & \Sigma K & \Lambda K \\ N\bar{K} & \Sigma\pi & \Lambda\pi & \Sigma\eta \\ N\bar{K} & \Sigma\pi & \Lambda\eta & \Xi K \\ \Sigma\bar{K} & \Lambda\bar{K} & \Xi\pi & \Xi\eta \end{pmatrix} = \frac{1}{\sqrt{20}} \begin{pmatrix} 9 & -1 & -9 & -1 \\ -6 & 0 & 4 & 4 \\ 2 & -12 & -4 & -2 \\ 9 & -1 & -9 & -1 \end{pmatrix}^{1/2} $$

$$ 8_2 \to 8 \otimes 8 $$

$$ \begin{pmatrix} N \\ \Sigma \\ \Lambda \\ \Xi \end{pmatrix} \to \begin{pmatrix} N\pi & N\eta & \Sigma K & \Lambda K \\ N\bar{K} & \Sigma\pi & \Lambda\pi & \Sigma\eta \\ N\bar{K} & \Sigma\pi & \Lambda\eta & \Xi K \\ \Sigma\bar{K} & \Lambda\bar{K} & \Xi\pi & \Xi\eta \end{pmatrix} = \frac{1}{\sqrt{12}} \begin{pmatrix} 3 & 3 & 3 & -3 \\ 2 & 8 & 0 & 0 \\ 6 & 0 & 0 & -2 \\ 3 & 3 & 3 & -3 \end{pmatrix}^{1/2} $$

$$ 10 \to 8 \otimes 8 $$

$$ \begin{pmatrix} \Delta \\ \Sigma \\ \Xi \\ \Omega \end{pmatrix} \to \begin{pmatrix} N\pi & \Sigma K \\ N\bar{K} & \Sigma\pi & \Lambda\pi & \Sigma\eta \\ \Sigma\bar{K} & \Lambda\bar{K} & \Xi\pi & \Xi\eta \\ \Xi\bar{K} & \Omega\eta \end{pmatrix} = \frac{1}{\sqrt{12}} \begin{pmatrix} -6 & 6 & 2 & 2 \\ -2 & 2 & -3 & 3 \\ 3 & -3 & 3 & 3 \\ 12 & 12 & -3 & -6 \end{pmatrix}^{1/2} $$

$$ 8 \to 10 \otimes 8 $$

$$ \begin{pmatrix} N \\ \Sigma \\ \Lambda \\ \Xi \end{pmatrix} \to \begin{pmatrix} \Delta\pi & \Sigma K \\ \Delta\bar{K} & \Sigma\pi & \Sigma\eta & \Xi K \\ \Sigma\pi & \Xi K \\ \Sigma\bar{K} & \Xi\pi & \Xi\eta & \Omega K \end{pmatrix} = \frac{1}{\sqrt{15}} \begin{pmatrix} -12 & 3 & 2 \\ 8 & -2 & -3 & 2 \\ 3 & -3 & -3 & 6 \end{pmatrix}^{1/2} $$

$$ 10 \to 10 \otimes 8 $$

$$ \begin{pmatrix} \Delta \\ \Sigma \\ \Xi \\ \Omega \end{pmatrix} \to \begin{pmatrix} \Delta\pi & \Delta\eta & \Sigma K \\ \Delta\bar{K} & \Sigma\pi & \Sigma\eta & \Xi K \\ \Sigma\bar{K} & \Xi\pi & \Xi\eta & \Omega K \\ \Xi\bar{K} & \Omega\eta \end{pmatrix} = \frac{1}{\sqrt{24}} \begin{pmatrix} 15 & 3 & -6 \\ 8 & 8 & 0 \\ 12 & 3 & -3 \\ 12 & -12 & -6 \end{pmatrix}^{1/2} $$

<table><thead><tr><th>abc</th><th>f<sub>abc</sub></th><th>abc</th><th>d<sub>abc</sub></th><th>abc</th><th>d<sub>abc</sub></th></tr></thead><tbody><tr><td>123</td><td>1</td><td>118</td><td>1/√3</td><td>355</td><td>1/2</td></tr><tr><td>147</td><td>1/2</td><td>146</td><td>1/2</td><td>366</td><td>-1/2</td></tr><tr><td>156</td><td>-1/2</td><td>157</td><td>1/2</td><td>377</td><td>-1/2</td></tr><tr><td>246</td><td>1/2</td><td>228</td><td>1/√3</td><td>448</td><td>-1/2(√3)</td></tr><tr><td>257</td><td>1/2</td><td>247</td><td>-1/2</td><td>558</td><td>-1/2(√3)</td></tr><tr><td>345</td><td>1/2</td><td>256</td><td>1/2</td><td>668</td><td>-1/2(√3)</td></tr><tr><td>367</td><td>-1/2</td><td>338</td><td>1/√3</td><td>778</td><td>-1/2(√3)</td></tr><tr><td>458</td><td>√3/2</td><td>344</td><td>1/2</td><td>888</td><td>-1/√3</td></tr><tr><td>678</td><td>√3/2</td><td></td><td></td><td></td><td></td></tr></tbody></table>

The $\lambda_a$'s are

$$ \begin{align*} \lambda_1 &= \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, & \lambda_2 &= \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, & \lambda_3 &= \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix} \\ \lambda_4 &= \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}, & \lambda_5 &= \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}, & \lambda_6 &= \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \\ \lambda_7 &= \begin{pmatrix} 0 & 0 & 0 \\ 0 & i & 0 \end{pmatrix}, & \lambda_8 &= \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix} \end{align*} $$

Equation (47.7) defines the Lie algebra of SU(3). A general d-dimensional representation is given by a set of $d \times d$ matrices satisfying Eq. (47.7) with the $f_{abc}$ given above. Equation (47.8) is specific to the defining 3-dimensional representation.
# 48. SU(n) Multiplets and Young Diagrams

Revised August 2019 by C.G. Wohl (LBNL).

This note tells (1) how SU(n) particle multiplets are identified or labeled, (2) how to find the number of particles in a multiplet from its label, (3) how to draw the Young diagram for a multiplet, and (4) how to use Young diagrams to determine the overall multiplet structure of a composite system, such as a 3-quark or a meson-baryon system.

In much of the literature, the word “representation” is used where we use “multiplet,” and “tableau” is used where we use “diagram.”

## 48.1 Multiplet labels

An SU(n) multiplet is uniquely identified by a string of (n-1) nonnegative integers: (α, β, γ, ...). Any such set of integers specifies a multiplet. For an SU(2) multiplet such as an isospin multiplet, the single integer α is the number of steps from one end of the multiplet to the other (i.e., it is one fewer than the number of particles in the multiplet). In SU(3), the two integers α and β are the numbers of steps across the top and bottom levels of the multiplet diagram. Thus the labels for the SU(3) octet and decuplet

<figure><img src="image_281.png" /></figure>

are (1,1) and (3,0). For larger n, the interpretation of the integers in terms of the geometry of the multiplets, which exist in an (n-1)-dimensional space, is not so readily apparent.

The label for the SU(n) singlet is (0, 0, ..., 0). In a flavor SU(n), the n quarks together form a (1, 0, ..., 0) multiplet, and the n antiquarks belong to a (0, ..., 0, 1) multiplet. These two multiplets are conjugate to one another, which means their labels are related by $(\alpha, \beta, ... ) \leftrightarrow (..., \beta, \alpha)$.

## 48.2 Number of particles

The number of particles in a multiplet, $N = N(\alpha, \beta, ...)$, is given as follows (note the pattern of the equations). In SU(2), $N = N(\alpha)$ is

$$N = \frac{(\alpha + 1)}{1}. \qquad (48.1)$$

In SU(3), $N = N(\alpha, \beta)$ is

$$N = \frac{(\alpha + 1)}{1} \cdot \frac{(\beta + 1)}{1} \cdot \frac{(\alpha + \beta + 2)}{2} \cdot \frac{(\beta + \gamma + 2)}{2} \cdot \frac{(\alpha + \beta + \gamma + 3)}{3}. \qquad (48.2)$$

In SU(4), $N = N(\alpha, \beta, \gamma)$ is

$$N = \frac{(\alpha+1)}{1} \cdot \frac{(\beta+1)}{1} \cdot \frac{(\gamma+1)}{1} \cdot \frac{(\alpha+\beta+2)}{2} \cdot \frac{(\beta+\gamma+2)}{2} \cdot \frac{(\alpha+\beta+\gamma+3)}{3}. \qquad (48.3)$$

Note that in Eq. (48.3) there is no factor with $(\alpha + \gamma + 2)$: only a consecutive sequence of the label integers appears in any factor. One more example should make the pattern clear for any SU(n). In SU(5), $N = N(\alpha, \beta, \gamma, \delta)$ is

$$N = \frac{(\alpha+1)}{1} \cdot \frac{(\beta+1)}{1} \cdot \frac{(\gamma+1)}{1} \cdot \frac{(\delta+1)}{1} \cdot \frac{(\alpha+\beta+2)}{2} \cdot \frac{(\beta+\gamma+2)}{2} \times \frac{(\gamma+\delta+2)}{2} \cdot \frac{(\alpha+\beta+\gamma+3)}{3} \cdot \frac{(\beta+\gamma+\delta+3)}{3} \cdot \frac{(\alpha+\beta+\gamma+\delta+4)}{4}. \qquad (48.4)$$

From the symmetry of these equations, it is clear that multiplets that are conjugate to one another have the same number of particles, but so can other multiplets. For example, the SU(4) multi-

plets (3,0,0) and (1,1,0) each have 20 particles. Try the equations and see.

## 48.3 Young diagrams

A Young diagram consists of an array of boxes (or some other symbol) arranged in one or more left-justified rows, with each row being at least as long as the row beneath. The correspondence between a diagram and a multiplet label is: The top row juts out $\alpha$ boxes to the right past the end of the second row, the second row juts out $\beta$ boxes to the right past the end of the third row, etc. A diagram in SU(n) has at most n rows. There can be any number of “completed” columns of n boxes buttressing the left of a diagram; these don’t affect the label. Thus in SU(3) the diagrams

<figure><img src="image_282.png" /></figure>

represent the multiplets (1,0), (0,1), (0,0), (1,1), and (3,0). In any SU(n), the quark multiplet is represented by a single box, the antiquark multiplet by a column of (n-1) boxes, and a singlet by a completed column of n boxes.

## 48.4 Coupling multiplets together

The following recipe tells how to find the multiplets that occur in coupling two multiplets together. To couple together more than two multiplets, first couple two, then couple a third with each of the multiplets obtained from the first two, etc.

First a definition: A sequence of the letters $a, b, c, ...$ is *admissible* if at any point in the sequence at least as many $a$'s have occurred as $b$'s, at least as many $b$'s have occurred as $c$'s, etc. Thus *abcd* and *aabcb* are admissible sequences and *abb* and *acb* are not. Now the recipe:

(a) Draw the Young diagrams for the two multiplets, but in one of the diagrams replace the boxes in the first row with $a$'s, the boxes in the second row with $b$'s, etc. Thus, to couple two SU(3) octets (such as the $\pi$-meson octet and the baryon octet), we start with $\Box$ and $\Box$ to the left and $\Box$ and $\Box$ to the right. The *unlettered* diagram forms the *upper left-hand corner* of all the enlarged diagrams constructed below.

(b) Add the $a$'s from the lettered diagram to the right-hand ends of the rows of the unlettered diagram to form all possible legitimate Young diagrams that have no more than one $a$ per column. In general, there will be several distinct diagrams, and all the $a$'s appear in each diagram. At this stage, for the coupling of the two SU(3) octets, we have:

<figure><img src="image_283.png" /></figure>

(c) Use the $b$'s to further enlarge the diagrams already obtained, subject to the same rules. Then throw away any diagram in which the full sequence of letters formed by reading *right to left* in the first row, then the second row, etc., is not admissible.

(d) Proceed as in (c) with the $c$'s (if any), etc. The final result of the coupling of the two SU(3) octets is:

<figure><img src="image_284.png" /></figure>

Here only the diagrams with admissible sequences of $a$'s and $b$'s and with fewer than four rows (since $n=3$) have been kept. In terms of multiplet labels, the above may be written

$$(1,1) \otimes (1,1) = (2,2) \oplus (3,0) \oplus (0,3) \oplus (1,1) \oplus (1,1) \oplus (0,0).$$

In terms of numbers of particles, it may be written

$$8 \otimes 8 = 27 \oplus 10 \oplus \overline{10} \oplus 8 \oplus 8 \oplus 1.$$

The product of the numbers on the left here is equal to the sum on the right, a useful check. (See also Sec. 15 on the Quark Model.)
Kinematics, Cross-Section Formulae, and Plots

<table><tr><td>49. Kinematics</td><td>777</td></tr><tr><td>50. Resonances (rev.)</td><td>781</td></tr><tr><td>51. Cross-section formulae for specific processes</td><td>792</td></tr><tr><td>52. Neutrino cross section measurements (rev.)</td><td>801</td></tr><tr><td>53. Plots of cross sections and related quantities (rev.)</td><td>807</td></tr></table>
<figure><img src="image_285.png" /></figure>
49. Kinematics

Reviewed August 2021 by D. Miller (Glasgow), D.R. Tovey (Sheffield), written January 2000 by J.D. Jackson (LBNL).

Throughout this section units are used in which $\hbar = c = 1$. The following conversions are useful: $\hbar c = 197.3$ MeV fm, $(\hbar c)^2 = 0.3894$ (GeV)$^2$ mb.

49.1 Lorentz transformations

The energy $E$ and 3-momentum $p$ of a particle of mass $m$ form a 4-vector $p = (E, p)$ whose square $p^2 = E^2 - |p|^2 = m^2$. The velocity of the particle is $\beta = p/E$. The energy and momentum $(E^*, p^*)$ viewed from a frame moving with velocity $\beta_f$ are given by

$$ \left( \frac{E^*}{p_{\parallel}^*} \right) = \begin{pmatrix} \gamma_f & -\gamma_f \beta_f \\ -\gamma_f \beta_f & \gamma_f \end{pmatrix} \left( \frac{E}{p_{\parallel}} \right), \quad p_T^* = p_T, \quad (49.1) $$

where $\gamma_f = (1 - \beta_f^2)^{-1/2}$ and $p_T$ ($p_{\parallel}$) are the components of $p$ perpendicular (parallel) to $\beta_f$. Other 4-vectors, such as the spacetime coordinates of events, of course transform in the same way. The scalar product of two 4-momenta $p_1 \cdot p_2 = E_1 E_2 - p_1 \cdot p_2$ is invariant (frame independent).

49.2 Center-of-mass energy and momentum

In the collision of two particles of masses $m_1$ and $m_2$ the total center-of-mass energy can be expressed in the Lorentz-invariant form

$$ \begin{aligned} E_{\text{cm}} &= [(E_1 + E_2)^2 - (p_1 + p_2)^2]^{1/2}, \\ &= [m_1^2 + m_2^2 + 2E_1 E_2 (1 - \beta_1 \beta_2 \cos \theta)]^{1/2}, \end{aligned} \quad (49.2) $$

where $\theta$ is the angle between the particles. In the frame where one particle (of mass $m_2$) is at rest (lab frame),

$$ E_{\text{cm}} = (m_1^2 + m_2^2 + 2E_1 \text{lab } m_2)^{1/2} . \quad (49.3) $$

The velocity of the center-of-mass in the lab frame is

$$ \beta_{\text{cm}} = p_{\text{lab}} / (E_{1 \text{lab}} + m_2), \quad (49.4) $$

where $p_{\text{lab}} \equiv p_{1 \text{lab}}$ and

$$ \gamma_{\text{cm}} = (E_{1 \text{lab}} + m_2) / E_{\text{cm}}. \quad (49.5) $$

The c.m. momenta of particles 1 and 2 are of magnitude

$$ p_{\text{cm}} = p_{\text{lab}} \frac{m_2}{E_{\text{cm}}}. \quad (49.6) $$

For example, if a 0.80 GeV/c kaon beam is incident on a proton target, the center of mass energy is 1.699 GeV and the center of mass momentum of either particle is 0.442 GeV/c. It is also useful to note that

$$ E_{\text{cm}} dE_{\text{cm}} = m_2 dE_{1 \text{lab}} = m_2 \beta_{1 \text{lab}} dp_{\text{lab}}. \quad (49.7) $$

49.3 Lorentz-invariant amplitudes

The matrix elements for a scattering or decay process are written in terms of an invariant amplitude $-i\mathcal{M}$. As an example, the S-matrix for $2 \to 2$ scattering is related to $\mathcal{M}$ by

$$ \langle p'_1 p'_2 | S-1 | p_1 p_2 \rangle = i(2\pi)^4 \delta^4(p_1+p_2-p'_1-p'_2) \mathcal{M}(p_1, p_2; p'_1, p'_2). \quad (49.8) $$

The state normalization is such that

$$ \langle p' | p \rangle = (2\pi)^3 2E_p \delta^3(p' - p), \quad (49.9) $$

with $E_p = \sqrt{p^2 + m^2}$.

For a $2 \to 2$ scattering process producing unstable particles 1' and 2' decaying via $1' \to 3'4'$ and $2' \to 5'6'$ the matrix element for the complete process can be written in the narrow width approximation as:

$$ \begin{aligned} \mathcal{M}(12 \to 3'4'5'6') = & \sum_{h_1', h_2'} \frac{\mathcal{M}(12 \to 1'2') \mathcal{M}(1' \to 3'4') \mathcal{M}(2' \to 5'6')}{(m_{3'4'}^2 - m_1'^2 + im_1' \Gamma_{1'}) (m_{5'6'}^2 - m_2'^2 + im_2' \Gamma_{2'})} \end{aligned} \quad (49.10) $$

Here, $m_{ij}$ is the invariant mass of particles $i$ and $j$, $m_k$ and $\Gamma_k$ are the mass and total width of particle $k$, and the sum runs over the helicities of the intermediate particles. This enables the cross section for such a process to be written as the product of the cross section for the initial $2 \to 2$ scattering process with the branching ratios (relative partial decay rates) of the subsequent decays. A more sophisticated treatment, beyond the narrow width approximation, can be found in the review on *Resonances*.

49.4 Particle decays

The partial decay rate of a particle of mass $M$ into $n$ bodies in its rest frame is given in terms of the Lorentz-invariant matrix element $\mathcal{M}$ by

$$ d\Gamma = \frac{(2\pi)^4}{2M} |\mathcal{M}|^2 d\Phi_n(P; p_1, \dots, p_n), \quad (49.11) $$

where $d\Phi_n$ is an element of $n$-body phase space given by

$$ d\Phi_n(P; p_1, \dots, p_n) = \delta^4(P - \sum_{i=1}^n p_i) \prod_{i=1}^n \frac{d^3 p_i}{(2\pi)^3 2E_i}. \quad (49.12) $$

This phase space is reduced by combinatoric factors whenever there are identical particles in the final state. The phase space can be generated recursively, viz.

$$ d\Phi_n(P; p_1, \dots, p_n) = d\Phi_j(q; p_1, \dots, p_j) \times d\Phi_{n-j+1}(P; q, p_{j+1}, \dots, p_n)(2\pi)^3 dq^2, \quad (49.13) $$

where $q^2 = (\sum_{i=1}^j E_i)^2 - |\sum_{i=1}^j p_i|^2$. This form is particularly useful in the case where a particle decays into another particle that subsequently decays.

49.4.1 Survival probability

If a particle of mass $M$ has mean proper lifetime $\tau (= 1/\Gamma)$ and has momentum $(E, p)$, then the probability that it lives for a time $t_0$ or greater before decaying is given by

$$ P(t_0) = e^{-t_0 \Gamma / \gamma} = e^{-Mt_0 \Gamma / E}, \quad (49.14) $$

and the probability that it travels a distance $x_0$ or greater is

$$ P(x_0) = e^{-Mx_0 \Gamma / |p|}, \quad (49.15) $$

49.4.2 Two-body decays

<figure><img src="image_286.png" /><figcaption>Figure 49.1: Definitions of variables for two-body decays.</figcaption></figure>

In the rest frame of a particle of mass $M$, decaying into 2 particles labeled 1 and 2,

$$ E_1 = \frac{M^2 - m_2^2 + m_1^2}{2M}, \quad (49.16) $$

$$ |\mathbf{p}_1| = |\mathbf{p}_2| = \frac{1}{2M} \sqrt{\lambda(M^2, m_1^2, m_2^2)}, \quad (49.17) $$
and

$$d\Gamma = \frac{1}{32\pi^2} |\mathcal{M}|^2 \frac{|p_1|}{M^2} d\Omega, \qquad (49.18)$$

where $\lambda(\alpha, \beta, \gamma) = \alpha^2 + \beta^2 + \gamma^2 - 2\alpha\beta - 2\alpha\gamma - 2\beta\gamma$ is the Källén function and $d\Omega = d\phi_1 d(\cos\theta_1)$ is the solid angle of particle 1. The invariant mass $M$ can be determined from the energies and momenta using Eq. (49.2) with $M = E_{cm}$.

### 49.4.3 Three-body decays

<figure><img src="image_287.png" /><figcaption>Figure 49.2: Definitions of variables for three-body decays.</figcaption></figure>

Defining $p_{ij} = p_i + p_j$ and $m_{ij}^2 = p_{ij}^2$, then $m_{12}^2 + m_{23}^2 + m_{13}^2 = M^2 + m_1^2 + m_2^2 + m_3^2$ and $m_{12}^2 = (P - p_3)^2 = M^2 + m_3^2 - 2ME_3$, where $E_3$ is the energy of particle 3 in the rest frame of $M$. In that frame, the momenta of the three decay particles lie in a plane. The relative orientation of these three momenta is fixed if their energies are known. The momenta can therefore be specified in space by giving three Euler angles $(\alpha, \beta, \gamma)$ that specify the orientation of the final system relative to the initial particle. The direction of any one of the particles relative to the frame in which the initial particle is described can be specified in space by two angles $(\alpha, \beta)$ while a third angle, $\gamma$, can be set as the azimuthal angle of a second particle around the first [1]. Then

$$d\Gamma = \frac{1}{(2\pi)^5} \frac{1}{16M} |\mathcal{M}|^2 dE_1 dE_3 d\alpha d(\cos\beta) d\gamma. \qquad (49.19)$$

Alternatively

$$d\Gamma = \frac{1}{(2\pi)^5} \frac{1}{16M^2} |\mathcal{M}|^2 |\mathbf{p}_1^*| |\mathbf{p}_3| dm_{12} d\Omega_3^* d\Omega_3, \qquad (49.20)$$

where $(|\mathbf{p}_1^*|, \Omega_1^*)$ is the momentum of particle 1 in the rest frame of 1 and 2, and $\Omega_3$ is the angle of particle 3 in the rest frame of the decaying particle. $|\mathbf{p}_1^*|$ and $|\mathbf{p}_3|$ are given by

$$|\mathbf{p}_1^*| = \frac{1}{2m_{12}} \sqrt{\lambda(m_{12}^2, m_1^2, m_2^2)}, \qquad (49.21a)$$

and

$$|\mathbf{p}_3| = \frac{1}{2M} \sqrt{\lambda(M^2, m_{12}^2, m_3^2)}. \qquad (49.21b)$$

[Compare with Eq. (49.17).]

If the decaying particle is a scalar, or we average over its spin states, then integration over the angles in Eq. (49.19) gives

$$d\Gamma = \frac{1}{(2\pi)^3} \frac{1}{8M} |\mathcal{M}|^2 dE_1 dE_3 \\ = \frac{1}{(2\pi)^3} \frac{1}{32M^3} |\mathcal{M}|^2 dm_{12}^2 dm_{23}^2. \qquad (49.22)$$

This is the standard form for the Dalitz plot.

#### 49.4.3.1 Dalitz plot

For a given value of $m_{12}^2$, the range of $m_{23}^2$ is determined by its values when $p_2$ is parallel or antiparallel to $p_3$:

$$(m_{23}^2)_{\text{max}} = (E_2^* + E_3^*)^2 - \left(\sqrt{E_2^* - m_2^2} - \sqrt{E_3^* - m_3^2}\right)^2, \qquad (49.23a)$$

$$(m_{23}^2)_{\text{min}} = (E_2^* + E_3^*)^2 - \left(\sqrt{E_2^* - m_2^2} + \sqrt{E_3^* - m_3^2}\right)^2. \qquad (49.23b)$$

Here $E_2^* = (m_{12}^2 - m_1^2 + m_2^2)/2m_{12}$ and $E_3^* = (M^2 - m_{12}^2 - m_3^2)/2m_{12}$ are the energies of particles 2 and 3 in the $m_{12}$ rest frame. The scatter plot in $m_{12}^2$ and $m_{23}^2$ is called a Dalitz plot. If $|\mathcal{M}|^2$ is constant, the allowed region of the plot will be uniformly populated with events [see Eq. (49.22)]. A nonuniformity in the plot gives immediate information on $|\mathcal{M}|^2$. For example, in the case of $D \to K\pi\pi$, bands appear when $m_{K\pi} = m_{K^*(892)}$, reflecting the appearance of the decay chain $D \to K^*(892)\pi \to K\pi\pi$.

<figure><img src="image_288.png" /><figcaption>Figure 49.3: Dalitz plot for a three-body final state. In this example, the state is $\pi^+K^0p$ at 3 GeV. Four-momentum conservation restricts events to the shaded region.</figcaption></figure>

#### 49.4.4 Kinematic limits

##### 49.4.4.1 Three-body decays

In a three-body decay (Fig. 49.2) the maximum of $|p_3|$, [given by Eq. (49.21)], is achieved when $m_{12} = m_1 + m_2$, i.e., particles 1 and 2 have the same vector velocity in the rest frame of the decaying particle. If, in addition, $m_3 > m_1, m_2$, then $|p_3|_{\text{max}} > |p_1|_{\text{max}}$, $|p_2|_{\text{max}}$. The distribution of $m_{12}$ values possesses an end-point or maximum value at $m_{12} = M - m_3$. This can be used to constrain the mass difference of a parent particle and one invisible decay product.

##### 49.4.4.2 Sequential two-body decays

<figure><img src="image_289.png" /><figcaption>Figure 49.4: Particles participating in sequential two-body decay chain. Particles labeled 1 and 2 are visible while the particle terminating the chain (a) is invisible.</figcaption></figure>

When a heavy particle initiates a sequential chain of two-body decays terminating in an invisible particle, constraints on the masses of the states participating in the chain can be obtained from end-points and thresholds in invariant mass distributions of the aggregated decay products. For the two-step decay chain depicted in Fig. 49.4 the invariant mass distribution of the two visible particles possesses an end-point given by:

$$(m_{12}^{\text{max}})^2 = \frac{(m_c^2 - m_b^2)(m_b^2 - m_a^2)}{m_b^2}, \qquad (49.24)$$
provided particles 1 and 2 are massless. If visible particle 1 has non-zero mass $m_1$ then Eq. (49.24) is replaced by

$$ (m_{12}^{\text{max}})^2 = m_1^2 + \frac{(m_c^2 - m_b^2)}{2m_b^2} \times \left( m_1^2 + m_b^2 - m_a^2 + \sqrt{(-m_1^2 + m_b^2 - m_a^2)^2 - 4m_1^2 m_a^2} \right) \quad (49.25) $$

See Refs. [2] and [3] for other cases.

### 49.4.5 Multibody decays

The above results may be generalized to final states containing any number of particles by combining some of the particles into “effective particles” and treating the final states as 2 or 3 “effective particle” states. Thus, if $p_{ijk...} = p_i + p_j + p_k + ...$, then

$$ m_{ijk...} = \sqrt{p^2_{ijk...}}, \quad (49.26) $$

and $m_{ijk...}$ may be used in place of e.g., $m_{12}$ in the relations in Sec. 49.4.3 or Sec. 49.4.4 above.

## 49.5 Cross sections

<figure><img src="image_290.png" /><figcaption>Figure 49.5: Definitions of variables for production of an n-body final state.</figcaption></figure>

The differential cross section is given by

$$ d\sigma = \frac{(2\pi)^4 |\mathcal{M}|^2}{4\sqrt{(p_1 \cdot p_2)^2 - m_1^2 m_2^2}} \times d\Phi_n(p_1 + p_2; p_3, \dots, p_{n+2}) . \quad (49.27) $$

[See Eq. (49.12).] In the rest frame of $m_2(\text{lab})$,

$$ \sqrt{(p_1 \cdot p_2)^2 - m_1^2 m_2^2} = m_2 p_{1 \text{lab}}; \quad (49.28a) $$

while in the center-of-mass frame

$$ \sqrt{(p_1 \cdot p_2)^2 - m_1^2 m_2^2} = p_{1\text{cm}} \sqrt{s}. \quad (49.28b) $$

### 49.5.1 Two-body reactions

<figure><img src="image_291.png" /><figcaption>Figure 49.6: Definitions of variables for a two-body final state.</figcaption></figure>

Two particles of momenta $p_1$ and $p_2$ and masses $m_1$ and $m_2$ scatter to particles of momenta $p_3$ and $p_4$ and masses $m_3$ and $m_4$;

the Lorentz-invariant Mandelstam variables are defined by

$$ \begin{aligned} s &= (p_1 + p_2)^2 = (p_3 + p_4)^2 \\ &= m_1^2 + 2E_1 E_2 - 2p_1 \cdot p_2 + m_2^2, \end{aligned} \quad (49.29) $$

$$ \begin{aligned} t &= (p_1 - p_3)^2 = (p_2 - p_4)^2 \\ &= m_1^2 - 2E_1 E_3 + 2p_1 \cdot p_3 + m_3^2, \end{aligned} \quad (49.30) $$

$$ \begin{aligned} u &= (p_1 - p_4)^2 = (p_2 - p_3)^2 \\ &= m_1^2 - 2E_1 E_4 + 2p_1 \cdot p_4 + m_4^2, \end{aligned} \quad (49.31) $$

and they satisfy

$$ s + t + u = m_1^2 + m_2^2 + m_3^2 + m_4^2. \quad (49.32) $$

The two-body cross section may be written as

$$ \frac{d\sigma}{dt} = \frac{1}{64\pi s} \frac{1}{|\mathbf{p}_{1\text{cm}}|^2} |\mathcal{M}|^2 . \quad (49.33) $$

In the center-of-mass frame

$$ \begin{aligned} t &= (E_{1\text{cm}} - E_{3\text{cm}})^2 - (p_{1\text{cm}} - p_{3\text{cm}})^2 - 4p_{1\text{cm}} p_{3\text{cm}} \sin^2(\theta_{\text{cm}}/2) \\ &= t_0 - 4p_{1\text{cm}} p_{3\text{cm}} \sin^2(\theta_{\text{cm}}/2), \end{aligned} \quad (49.34) $$

where $\theta_{\text{cm}}$ is the angle between particle 1 and 3. The limiting values $t_0$ ($\theta_{\text{cm}} = 0$) and $t_1$ ($\theta_{\text{cm}} = \pi$) for $2 \to 2$ scattering are

$$ t_0(t_1) = \left[ \frac{m_1^2 - m_3^2 - m_2^2 + m_4^2}{2\sqrt{s}} \right]^2 - (p_{1\text{cm}} \mp p_{3\text{cm}})^2 . \quad (49.35) $$

In the literature the notation $t_{\text{min}}$ ($t_{\text{max}}$) for $t_0$ ($t_1$) is sometimes used, which should be discouraged since $t_0 > t_1$. The center-of-mass energies and momenta of the incoming particles are

$$ E_{1\text{cm}} = \frac{s + m_1^2 - m_2^2}{2\sqrt{s}}, \quad E_{2\text{cm}} = \frac{s + m_2^2 - m_1^2}{2\sqrt{s}}, \quad (49.36) $$

For $E_{3\text{cm}}$ and $E_{4\text{cm}}$, change $m_1$ to $m_3$ and $m_2$ to $m_4$. Then

$$ p_{i\text{cm}} = \sqrt{E_{i\text{cm}}^2 - m_i^2} \quad \text{and} \quad p_{1\text{cm}} = \frac{p_{1\text{lab}} m_2}{\sqrt{s}}. \quad (49.37) $$

Here the subscript lab refers to the frame where particle 2 is at rest. [For other relations see Eqs. (49.2)–(49.4).]

### 49.5.2 Inclusive reactions

Choose some direction (usually the beam direction) for the z-axis; then the energy and momentum of a particle can be written as

$$ E = m_T \cosh y, \quad p_x, \quad p_y, \quad p_z = m_T \sinh y, \quad (49.38) $$

where $m_T$, conventionally called the ‘transverse mass’, is given by

$$ m_T^2 = m^2 + p_x^2 + p_y^2. \quad (49.39) $$

and the rapidity $y$ is defined by

$$ \begin{aligned} y &= \frac{1}{2} \ln \left( \frac{E + p_z}{E - p_z} \right) \\ &= \ln \left( \frac{E + p_z}{m_T} \right) = \tanh^{-1} \left( \frac{p_z}{E} \right). \end{aligned} \quad (49.40) $$

Note that the definition of the transverse mass in Eq. (49.39) differs from that used by experimentalists at hadron colliders (see Sec. 49.6.1 below). Under a boost in the z-direction to a frame with velocity $\beta$, $y \to y - \tanh^{-1} \beta$. Hence, the shape of the rapidity distribution $dN/dy$ is invariant, as are differences in rapidity. The invariant cross section may also be rewritten

$$ E \frac{d^3 \sigma}{d^3 p} = \frac{d^3 \sigma}{d \phi dy p_T dp_T} \Rightarrow \frac{d^2 \sigma}{\pi dy d(p_T^2)} . \quad (49.41) $$

The second form is obtained using the identity $dy/dp_z = 1/E$, and the third form represents the average over $\phi$.
Feynman's $x$ variable is given by

$$x = \frac{p_z}{p_z \max} \approx \frac{E + p_z}{(E + p_z)_{\max}} \quad (p_T \ll |p_z|). \qquad (49.42)$$

In the c.m. frame,

$$x \approx \frac{2p_z \text{ cm}}{\sqrt{s}} = \frac{2m_T \sinh y_{\text{cm}}}{\sqrt{s}} \qquad (49.43)$$

and

$$= (y_{\text{cm}})_{\max} = \ln(\sqrt{s}/m). \qquad (49.44)$$

The invariant mass $M$ of the two-particle system described in Sec. 49.4.2 can be written in terms of these variables as

$$M^2 = m_1^2 + m_2^2 + 2[E_T(1)E_T(2)\cosh \Delta y - \mathbf{p}_T(1) \cdot \mathbf{p}_T(2)], \quad (49.45)$$

where

$$E_T(i) = \sqrt{|\mathbf{p}_T(i)|^2 + m_i^2}, \qquad (49.46)$$

and $\mathbf{p}_T(i)$ denotes the transverse momentum vector of particle $i$.

For $p \gg m$, the rapidity [Eq. (49.40)] may be expanded to obtain

$$y = \frac{1}{2} \ln \frac{\cos^2(\theta/2) + m^2/4p^2 + \dots}{\sin^2(\theta/2) + m^2/4p^2 + \dots} \\ \approx -\ln \tan(\theta/2) \equiv \eta \qquad (49.47)$$

where $\cos\theta = p_z/p$. The pseudorapidity $\eta$ defined by the second line is approximately equal to the rapidity $y$ for $p \gg m$ and $\theta \gg 1/\gamma$, and in any case can be measured when the mass and momentum of the particle are unknown. From the definition one can obtain the identities

$$\sinh \eta = \cot \theta, \quad \cosh \eta = 1/\sin \theta, \quad \tanh \eta = \cos \theta. \qquad (49.48)$$

## 49.6 Transverse variables

At hadron colliders, a significant and unknown proportion of the energy of the incoming hadrons in each event escapes down the beam-pipe. Consequently, if invisible particles are created in the final state, their net momentum can only be constrained in the plane transverse to the beam direction. Defining the $z$-axis as the beam direction, this net momentum is equal to the missing transverse energy vector

$$E_T^{\text{miss}} = - \sum_i p_T(i), \qquad (49.49)$$

where the sum runs over the transverse momenta of all visible final state particles.

### 49.6.1 Single production with semi-invisible final state

Consider a single heavy particle of mass $M$ produced in association with visible particles which decays as in Fig. 49.1 to two particles, of which one (labeled particle 1) is invisible. The mass

of the parent particle can be constrained with the quantity $M_T$ defined by

$$M_T^2 = [E_T(1) + E_T(2)]^2 - [\mathbf{p}_T(1) + \mathbf{p}_T(2)]^2 \\ = m_1^2 + m_2^2 + 2[E_T(1)E_T(2) - \mathbf{p}_T(1) \cdot \mathbf{p}_T(2)], \qquad (49.50)$$

where

$$\mathbf{p}_T(1) = E_T^{\text{miss}}. \qquad (49.51)$$

This quantity is called the ‘transverse mass’ by hadron collider experimentalists but it should be noted that it is quite different from that used in the description of inclusive reactions [Eq. (49.39)]. The distribution of event $M_T$ values possesses an end-point at $M_T^{\text{max}} = M$. If $m_1 = m_2 = 0$ then

$$M_T^2 = 2|\mathbf{p}_T(1)||\mathbf{p}_T(2)|\left(1 - \cos \phi_{12}\right), \qquad (49.52)$$

where $\phi_{ij}$ is defined as the angle between particles $i$ and $j$ in the transverse plane.

<figure><img src="image_292.png" /><figcaption>**Figure 49.7:** Definitions of variables for pair production of semi-invisible final states. Particles 1 and 3 are invisible while particles 2 and 4 are visible.</figcaption></figure>

### 49.6.2 Pair production with semi-invisible final states

Consider two identical heavy particles of mass $M$ produced such that their combined center-of-mass is at rest in the transverse plane (Fig. 49.7). Each particle decays to a final state consisting of an invisible particle of fixed mass $m_1$ together with an additional visible particle. $M$ and $m_1$ can be constrained with the variables $M_{T2}$ and $M_{CT}$ which are defined in Refs. [4] and [5].

#### References

[1] J.D. Jackson in *High Energy Physics, Les Houches 1965 Summer School, GORDON AND BREACH Science Publishers* (1965), p. 348.

[2] I. Hinchliffe et al., Phys. Rev. D55, 5520 (1997), [hep-ph/9610544].

[3] B. C. Allanach et al., JHEP 09, 004 (2000), [hep-ph/0007009].

[4] C. G. Lester and D. J. Summers, Phys. Lett. B463, 99 (1999), [hep-ph/9906349].

[5] D. R. Tovey, JHEP 04, 034 (2008), [arXiv:0802.2879].
# 50. Resonances

Revised August 2023 by D.M. Asner (BNL), C. Hanhart (FZ Jülich) and M. Mikhasenko (Ruhr U. Bochum).

## 50.1 General Considerations

Perturbative methods can be applied to systems of quarks and gluons only for large momentum transfers (see review on ‘Quantum Chromodynamics’) and, under certain conditions, to some properties of systems that contain heavy quarks or very large momentum scales (see review on “Heavy-quark and soft-collinear effective theory”). Dealing with Quantum Chromodynamics (QCD) in the low momentum transfer region is a very complicated, non-perturbative problem. Most hadrons are resonances, which means that they appear as poles of the S-matrix in the complex plane on unphysical sheets, a notion further detailed in Sec. 50.2. These resonances can show up either in so-called formation experiments,

$$A + B \rightarrow \mathbf{R} \rightarrow C_1 + \dots + C_n,$$

where they become visible in an energy scan (for example, the R-function measured in $e^+e^-$ annihilation, cf. the corresponding plots in the review on “Plots of Cross Sections and Related Quantities”), or together with a spectator particle S in production experiments of the kind

$$A + B \rightarrow \mathbf{R} + S \rightarrow [C_1 + \dots + C_n] + S, \\ Z \rightarrow \mathbf{R} + S \rightarrow [C_1 + \dots + C_n] + S,$$

where the first reaction corresponds to an associated production, the second is a decay (see “Review of Multibody Charm Analyses”). In the latter case, the resonance properties are commonly extracted from a Dalitz-plot analysis (see review on “Kinematics”) or projections thereof.

Resonance phenomena are very rich: while typical hadronic widths are of the order of 100 MeV (e.g., for the meson resonances $\rho(770)$ or $\psi(4040)$ or the baryon resonance $\Delta(1232)$) corresponding to a lifetime of $10^{-23}$ s, the widths can also be as small as a few MeV (e.g. of $\phi(1020)$ or $J/\psi$) or as large as several hundred MeV (e.g. of the meson resonances $f_0(500)$ or $D_1(2430)$ or the baryon resonance $N(2190)$).

Typically, a resonance appears as a peak in the total cross section. If the structure is narrow and if there are no relevant thresholds or other resonances nearby, the resonance properties may be extracted employing a Breit-Wigner parameterization, if necessary improved by using an energy-dependent width (cf. Sec. 50.3.1 of this review). However, in general, unitarity and analyticity call for the use of more refined tools as outlined here as well as in recent review articles [1,2]. When there are overlapping resonances with the same quantum numbers, the resonance terms should not simply be added but combined in a non-trivial way either in a K-matrix approach (cf. Sec. 50.3.2 of this review) or using other advanced methods (cf. Sec. 50.3.5 of this review). Additional constraints from the S-matrix allow one to build more reliable amplitudes and, in turn, to reduce the systematic uncertainties of the resonance parameters: pole locations and residues. In addition, for broad resonances there is no direct relation between pole location and the total width/lifetime — then, the pole residues need to be used in order to quantify the decay properties.

For simplicity, throughout this review the formulas are given for resonances in a system of distinguishable, scalar particles. The additional complications that appear in the presence of spins can be controlled in the helicity framework developed by Jacob and Wick [3], or in a non-covariant [4] or covariant [5] tensor-operator formalisms. Within these approaches, sequential (cascade) decays are commonly treated as a coherent sum of two-body interactions. Most of the expressions below are given for two-body kinematics.

### 50.1.1 Properties of the S-matrix

The unitary operator that connects asymptotic in and out states is called the S-matrix. The scattering amplitude is defined as the interacting part of the S-matrix. For a two-particles scattering process, it reads:

$$i(2\pi)^4 \delta^4(p_1 + p_2 - p_{1'} - p_{2'}) \mathcal{M}(p_1, p_2; p_{1'}, p_{2'})_{ba} \\ = \text{out} \langle p_{1'} p_{2'}, b | S - 1 | p_1 p_2, a \rangle_{\text{in}} \quad (50.1)$$

<figure><img src="image_293.png" /><figcaption>**Figure 50.1:** Illustration of the relation between s- and t-channel.</figcaption></figure>

where $|p_1 p_2, a\rangle$ and $|p_1 p_{2'}, b\rangle$ are asymptotic states that can be treated as non-interacting particles in the spirit of the LSZ-reduction [6]. They carry the momenta $p_1, p_2$ and $p_{1'}, p_{2'}$, respectively.

The labels a and b are used to specify the reaction channels. In general terms, a reaction channel describes the possible outcome of a scattering event, characterized by the quantum numbers of the particles involved. Note, that in general $\mathcal{M}$ operates as a matrix in channel space, connecting the different asymptotic multi-particle states. For a single-particle state, we employ the relativistic normalization,

$$\langle p'|p \rangle = (2\pi)^3 2E_p \delta^3(\vec{p}' - \vec{p}), \quad (50.2)$$

with $E_p = \sqrt{\vec{p}^2 + m^2}$.

Mandelstam variables are defined as $s = (p_1 + p_2)^2$, $t = (p_1 - p_{1'})^2$, and $u = (p_1 - p_{2'})^2$. One finds that the variables $s, t$, and $u$ are not independent, since

$$s + t + u = m_1^2 + m_2^2 + m_{1'}^2 + m_{2'}^2$$

holds, where $m_i$ with $i \in 1, 1', 2, 2'$ represents the masses of the involved particles. As a result, the reaction amplitude can be expressed as a function of two variables, $\mathcal{M}(s, t)$.

The Mandelstam variables play a crucial role in characterizing particle scattering. Specifically, the process described in Eq. (50.1) is known as s-channel, referring to a reaction, $1, 2 \rightarrow 1', 2'$. Here, $\sqrt{s}$ represents the total energy of the interacting system in the center-of-momentum frame. The variable $t$ corresponds to the momentum transfer, which is related to the scattering angle, the angle between the momenta of particles 1 and 1' in the center-of-momentum frame (see also the review on “Kinematics” in this Review of Particle Physics). The scattering process with particles 1 and the antiparticle of 1', denoted by $\bar{1}'$, incoming to the antiparticle of 2, denoted by $\bar{2}$, and 2' outgoing represent a different reaction, it is referred to as the t-channel reaction. The scattering amplitudes of s- and t-channel are related via the so-called crossing symmetry. The u-channel is introduced analogously as a reaction with the initial state of particles 1 and $\bar{2}'$ and the final state of particles 1' and $\bar{2}$. The t-channel and the u-channel are referred to as the crossed channels relative to the reaction in Eq. (50.1). This relationship is illustrated in Fig. 50.1.

The analysis of resonance phenomena requires delving deeper into the complex analysis of the amplitude: $\mathcal{M}(s, t)$ is a multivalued function due to the complex branch points associated with the Mandelstam variables. These branch points emerge whenever a new channel becomes accessible, that is, whenever $s$ exceeds $s_{thr,a} = (m_{1,a} + m_{2,a})^2$, where $m_{i,a}$ denotes the masses of the two particles in channel a. Every two-particle threshold introduces a square-root singularity. The square-root function is inherently double-valued; for instance, the equation $x = y^2$ has two solutions: $\sqrt{x}$ and $-\sqrt{x}$. In the complex plane, these two solutions are represented as separate layers or surfaces, known as Riemann sheets or Riemann surfaces. Consequently, each square-root singularity causes the number of Riemann sheets of the scattering amplitude to double. If a channel opens that has an odd number of particles, the branch point at the threshold exhibits a logarithmic singularity [7]. Such a branch point gives rise to an infinite