number of sheets. The branch points come with their associated branch cuts — by convention those are taken from the threshold to infinity along the real axis and accordingly called right-hand cuts. To explore a reaction amplitude in the complex energy plane, one can introduce a complex component to the Mandelstam variable s. The section of the complex plane that relates to a positive imaginary component of the relative momentum can be directly accessed from the physical region and is called the first Riemann sheet or the physical sheet. The other sheets are called unphysical sheets. The physical axis of an s-channel scattering amplitude represents a line for real values of the variable s larger than the lowest threshold to be evaluated on the physical sheet.

While the physical Riemann sheet is free of singularities off the real axis, the unphysical sheets may contain resonance poles and branch points. Branch points appear in the complex plane of an unphysical sheet when there is a resonance in a subsystem of involved particles [7, 8]. The branch points related to thresholds in the crossed channels are located in the portion of the complex plane where the real part is negative and, therefore, are referred to as the left-hand cuts. An illustrative example for a left-hand cut is the one-pion exchange in nucleon-nucleon scattering that is located in the unphysical domain at $s = 4m_N^2 - m_\pi^2$. Triangle topologies, which are Feynman diagrams characterized by a triangular loop of three propagators, can lead to logarithmic singularities in the scattering amplitude. These singularities, appearing on the unphysical sheets, are often termed triangle singularities (TS) [7, 9, 10].

The reaction amplitude has poles that can be categorized as bound states, virtual states, or resonances. Poles corresponding to bound and virtual states manifest at real values of s. Specifically, bound state poles are found on the physical sheet, while virtual state poles are situated on an unphysical sheet, both being below the threshold. Resonance poles, on the other hand, emerge inside the complex plane of the unphysical sheets. Notably, those resonance poles that are on the unphysical sheet nearest to the physical region exert the most significant influence on experimental observables. Analyticity dictates that for every pole at a specific complex value of $s_p$, there must be a corresponding pole at its complex conjugate value $s_p^*$. This relationship is a direct consequence of the Schwarz reflection principle, a mathematical technique utilized for analytic continuation within scattering theory. For a single-channel case, the complex structure of an amplitude with a single resonance is illustrated in Fig. 50.2(a)-(c). Among the two poles, symmetrically positioned relative to the real axis on the second sheet, the one exhibiting a negative imaginary part seamlessly connects to the physical axis, which becomes especially clear in panel (c) of the figure where the physical axis is shown as the blue line. It therefore has a more pronounced effect on observables in the vicinity of the resonance region compared to its conjugate counterpart. The shortest continuous path from the physical region to the second sheet pole with the positive imaginary part is via a line that goes around the threshold. However, as the extra path around the threshold diminishes in the near-threshold region, the effective distance to both poles becomes comparable for near threshold kinematics, thereby rendering the influence of both poles on observables similarly significant.

An alternative way to depict the complex structure is through the k-plane, where k denotes the relative momentum of the two scattering particles in their center-of-momentum frame. For particles of equal mass, denoted as m (expression for unequal masses is given in Eq. (50.7)), one finds

$$k = \frac{1}{2} \sqrt{s - 4m}. \qquad (50.3)$$

Unlike the complex s-plane, the k-plane does not have a two-body threshold cut. Consequently, both the physical and unphysical Riemann sheets, which are linked to the branch point in s are mapped onto the upper and lower half of the complex k-plane, respectively. The k-plane is sketched in Fig. 50.2(d), including possible locations of resonance poles. In this representation it becomes especially clear that only one resonance pole drives the dynamics on the physical axis in the resonance region, while at the threshold, where $k = 0$, both poles are of equal significance.

In situations involving two relevant channels, we encounter four Riemann sheets. These are illustrated in the left panel of Fig. 50.3. For a non-relativistic system with two channels, this four-sheeted Riemann surface can be transformed into a plane represented by a new variable, $\omega$. This variable is defined in relation to the channel-momenta, as detailed in references [11, 12] (for a recent application see Ref. [13])

$$k_1 = \sqrt{\frac{\mu_1 \Delta}{2}} \left(\omega + \frac{1}{\omega}\right), \quad k_2 = \sqrt{\frac{\mu_2 \Delta}{2}} \left(\omega - \frac{1}{\omega}\right), \qquad (50.4)$$

where $\Delta$ denotes the energy difference between the two thresholds and the $\mu_a$ the reduced mass of the particles in channel a, $\mu_a = m_{1,a}m_{2,a}/(m_{1,a} + m_{2,a})$. The lower threshold is located at $\omega = i$, the higher at $\omega = 1$. The mapping of the Riemann sheets and the different areas in the $\omega$-plane is shown in the right panel of Fig. 50.3. The $\omega$-plane nicely shows how the different sheets connect to each other.<fnref n="1" /> The solid green line shows the physical axis in the physical regime. The thick black line its analytic continuation below the lowest threshold. The pronounced kinks in this line for the $\omega$ plane show the thresholds. Please observe that for $\omega \approx i$ the sheet structure agrees to panel (d) of Fig. 50.2, since in this kinematic regime the second channel does not matter.

In the context of a two-channel scenario, the proximity of a sheet to the physical axis varies with increasing energy. Specifically, for energies that exceed the first threshold, but remain below the second, sheet (21) is the one that smoothly connects to the upper half-plane of the physical sheet (11). Once the energy surpasses the second threshold, sheet (22) assumes this role. As a result, any pole on sheet (21) that lies above the second threshold will manifest in the data solely as a cusp right at the second threshold. Sheet (12), on the other hand, is remote for almost all energies.

singularities, poles and branch points determine the visible structures in observables. However, it is crucial to note that not every observable bump is indicative of a resonance as discussed in [14]. Under certain kinematic conditions, Triangle Singularities, in particular, can either mimic resonance signals, as suggested in Refs. [15–20], or significantly alter resonance signals [21]. Conversely, not all resonances produce a noticeable bump across all observables. For example, in the baryon sector, there is no clear trace of the $N(1440)1/2^+$, the so called Roper resonance, in the $\pi N$ observables or phase shifts, although careful analyses reveal a pole [22]. In the meson sector, the $f_0(500)$, also known as $\sigma$ meson, was firmly established only after the application of very sophisticated theoretical analysis tools (see, e.g., Ref. [23] for a review). This complexity arises because the scalar-isoscalar $\pi\pi$ phase shifts reach 90 degrees near 800 MeV, which is approximately 400 MeV above the resonance mass. At this energy, the onset of the next resonance, the $f_0(980)$, is already observable. The analyticity principle of the S-matrix germane to quantum scattering theories, dictates that only poles and branch points can exist on the real axis of the first Riemann sheet, excluding any singularities in the complex plane. This principle is closely linked to causality, ensuring that effects follow their causes in a chronological order. In non-relativistic scattering, the analyticity finds a solid mathematical base [24]. Similarly, perturbative relativistic theory maintains this analyticity through a series expansion of the S-matrix, each term depicted by a Feynman diagram, representing processes with distinct analytical expressions. A deeper level of analyticity is proposed by the Mandelstam hypothesis, suggesting not only the analytic properties of the scattering amplitude within the complex plane of the first Riemann sheet but also a nuanced interconnection between crossed scattering processes through analytic continuation [25].

Unitarity further constrains the imaginary part of the amplitude on this real axis, a topic we will explore in the subsequent section. Additional constraints are introduced by principles such as crossing symmetry, duality [26] and positivity [27]. Approaches based on analyticity and crossing symmetry have been implemented through dispersion theory. Among the most notable are

<footnote n="1">¹An alternative illustration for this two-channel case as well as an extension to three channels can be found in Ref. [2].</footnote>
<figure><img src="image_294.png" /></figure>

**Figure 50.2:** (a)-(c): Imaginary part of a typical scattering amplitude with an isolated resonance. The blue line shows the physical range of the Mandelstam variable *s*: it is real and starts from the threshold shown by the blue dot. Plot (a) shows the imaginary part of the amplitude in the complex *s*-plane that corresponds to the first or physical sheet (green surface), plot (b) shows the related unphysical or the second sheet (red surface) which contains the resonance poles, and plot (c) shows the analytic continuation of the same amplitude from the upper half plane of the physical sheet to the lower half plane of the unphysical sheet. The two sheets are connected smoothly along the real axis above the threshold. Panel (d) shows the *k*-plane, which is free of cuts. The upper (lower) half plane maps onto the physical (unphysical) sheet. Also here the blue line corresponds to the physically accessible values of the momentum *k*, which starts at threshold, where *k* = 0. The thick black line shows the analytic continuation of the on-shell momentum in the below threshold regime. The locations of the resonance poles are indicated by the black crosses.

<figure><img src="image_295.png" /></figure>

**Figure 50.3:** Cut structure of the *S*-matrix with two channels present. The diagrams feature four sheets, labeled (*ij*), where both *i* and *j* can be 1 or 2. These labels indicate the doubling of the sheets at the first and second thresholds. The left panel displays the complex *s* plane, while the right panel represents the *ω* plane. The physical axis, along with its analytic continuation below the threshold, is highlighted with a thick solid line, green in the physical regime, black for the analytic continuation below the lowest threshold.
the Roy equations and their variants [28]. These have been applied to a range of processes, including $\pi\pi \to \pi\pi$ [29–31], $\pi K$ scattering [32], $\gamma\gamma \to \pi\pi$ interactions [33], and pion-nucleon scattering [34, 35].

### 50.1.2 Consequences from unitarity

Scattering amplitudes, denoted as $M$, and production amplitudes, represented by $A$, have distinct characteristics due to the different constraints imposed by unitarity. When considering the scattering amplitude, it is assumed that all channels hold similar significance. In contrast, for production amplitudes, it is assumed that for the dynamics of the particles in the final state the initial state does not play a role. Accordingly, the interactions in the final state among the produced particles are described by relevant scattering amplitudes of those particles only. Scattering processes are for example $\pi^+\pi^- \to K\bar{K}$ and $D^0\bar{D}^0 \to D^0\bar{D}^0$. Examples for production processes are $e^+e^- \to \gamma^* \to \pi^+\pi^-$, which provides access to the pion vector form factor, $\tau^- \to K^-\pi^0\nu_\tau$, and $B^0 \to J/\psi\pi^+\pi^-$.

The unitarity of the $S$-matrix, represented by the equation $S^†S = 1$, ensures the conservation of probability. This principle imposes a specific constraint on the imaginary part of the reaction amplitude. Below the lowest threshold, the amplitude remains real. However, once the energy is higher than the threshold, there is a discontinuity associated with the threshold branch point. The $S$-matrix unitarity relates the value of the discontinuity, to the amplitude itself [6]:

$$ \begin{aligned} \text{Disc} \mathcal{M}_{ba} &= \mathcal{M}_{ba} - \mathcal{M}_{ab}^*, && \\ &= i (2\pi)^4 \sum_c \int d\Phi_c \mathcal{M}_{cb}^* \mathcal{M}_{ca}, \end{aligned} \quad (50.5) $$

where $d\Phi_c$ denotes the invariant phase space for a given channel, labeled as $c$. The factor $(2\pi)^4$ aligns with the definition of the phase space expression (see Eq. (12) in the review on “Kinematics”). It is essential to note that the summation only considers open channels, meaning those with a production threshold below the scattered system’s energy. In the evaluation of an actual Feynman diagram the discontinuity can be extracted by employing the Cutkosky rule, which comprises replacing the propagators in the pertinent intermediate state by delta-distributions. Eq. (50.5) is illustrated graphically in Fig. 50.4. The left part of the expression yields $2i \text{Im}\mathcal{M}_{ba}$ in accordance to analytic properties of reaction amplitude [36]. For the forward scattering, $t=0$, the right part of Eq. (50.5) resembles the total cross section up to a kinematic factor, a relationship known an the optical theorem:

$$ \text{Im} \mathcal{M}_{aa}(s, 0) = 2q_a \sqrt{s} \sigma_{\text{tot}}(a \to \text{anything}). \quad (50.6) $$

In this equation, $q_a$ represents the break-up momentum of the particles in the center-of-momentum frame,

$$ q_a = \frac{\lambda^{1/2}(s, m_{1,a}^2, m_{2,a}^2)}{2\sqrt{s}}, \quad (50.7) $$

where $\lambda(x, y, z) = x^2 + y^2 + z^2 - 2xy - 2yz - 2xz$ is the Källén function, and the masses of the two particles in the channel $a$, $m_{1,a}$ and $m_{2,b}$, cf. Eq. (17) of the review on “Kinematics”.

The unitarity relation for a production amplitude in channel $a$ is represented by:

$$ \begin{aligned} \text{Disc} \mathcal{A}_a &= \mathcal{A}_a - \mathcal{A}_a^*, \\ &= i (2\pi)^4 \sum_c \int d\Phi_c \mathcal{M}_{ca}^* \mathcal{A}_c. \end{aligned} \quad (50.8) $$

Equation (50.8) is illustrated graphically in Fig. 50.4. A direct consequence of this equation is that the production amplitude shares its poles with the scattering amplitude. A common method to model the production amplitude that adheres to this unitary relation is to express it as a linear combination of the scattering amplitudes. This approach, known as the $Q$-vector method (see Sec 50.3.4), has its limitations. Specifically, production amplitude inherits the left-hand singularities of the scattering amplitude,

while, in general, it has a different cut structure. To address this drawback, a more sophisticated method, known as *unitarization* is employed. This method, rooted in dispersion theory, offers a more refined approach to the problem and is detailed in [37]. A notable application of this method is the Khuri-Treiman framework [38, 39], which is frequently used to study three-body decays. This framework has been successfully applied to a range of decays, from light mesons [40–52] to heavy-flavour decays [53–55].

### 50.1.3 Partial-wave decomposition

It is often convenient to expand a two-body scattering amplitude of a two-body subsystem of a production amplitude in partial waves. Since resonances have a well-defined spin, they appear only in a specific partial wave of the reaction amplitude. For scalar particles, the expansion reads:

$$ \mathcal{M}_{ba}(s, t) = \sum_{j=0}^{\infty} (2j + 1) \mathcal{M}_{ba}^{j}(s) P_{j}(\cos(\theta)), \quad (50.9) $$

where $j$ denotes the total angular momentum and the $P_j(\cos(\theta))$ denotes the Legendre polynomials. In the presence of spins an expansion more complicated than Eq. (50.9) is necessary — for a general discussion on how to handle spins see e.g. Ref. [56]. In the absence of spins the parameter $j$ coincides with the orbital angular momentum of the particle pairs in the initial and the final state. To simplify notation and since all amplitudes from here on are understood to be partial wave projected, we drop the label $j$ for the single-argument function $\mathcal{M}_{ba}(s)$. Plugging Eq. (50.9) into Eq. (50.5) one finds the unitarity relation for the partial-wave amplitude $\mathcal{M}_{ba}(s)$, namely

$$ \text{Im} \mathcal{M}_{ba}(s) = \sum_c \mathcal{M}_{cb}(s)^* \rho_c(s) \mathcal{M}_{ca}(s) \quad (50.10) $$

with $\rho_c(s)$ being a factor that is related to the two-body phase space in Eq. (12) of the review on “Kinematics”,

$$ \rho_c(s) = \frac{(2\pi)^4}{2} \int d\Phi_2 = \frac{1}{16\pi} \frac{2|\vec{q}_c|}{\sqrt{s}}, \quad (50.11) $$

with the momentum $q_c$ being defined in Eq. (50.7). Note that in case of the two particles being identical the inclusion of symmetry factors becomes necessary. The partial-wave amplitude $f_{ba}(s)$ is introduced by normalizing scattering amplitude with the phase spaces factors,

$$ f_{ba}(s) = \sqrt{\rho_b} \mathcal{M}_{ba}(s) \sqrt{\rho_a}. \quad (50.12) $$

The unitarity condition for $f_{ba}$ follows from Eq. (50.10):

$$ \text{Im} f_{ba}(s) = \sum_c f_{cb}^*(s) f_{ca}(s). \quad (50.13) $$

It leads us to deduce that the inverse of the imaginary part of $f_{ba}$ is equal to $-\delta_{ba}$. Moreover, $S = I + 2if$ is a unitary matrix. Hence, the diagonal elements of $f$ can be parameterized as

$$ f_{bb} = (\eta_b \exp(2i\delta_b) - 1)/2i, \quad (50.14) $$

where $\delta_b$ denotes the phase shift for the scattering from channel $b$ to channel $a$ and $\eta_b$ is the elasticity parameter, also known as inelasticity. Building upon Eq. (50.13), we can further deduce that,

$$ \text{Im} f_{bb}(s) = (1 - \eta_b \cos(2\delta_b))/2 = \sum_c |f_{cb}(s)|^2. \quad (50.15) $$

Using Eq. (50.14) for the last term in the sum, we obtain a relation highlighting the meaning of the inelasticity,

$$ \frac{1}{4} (1 - \eta_b^2) = \sum_{c \neq b} |f_{cb}(s)|^2. \quad (50.16) $$

It is important to note that the parameter $\eta_b$ is confined within the range $[0, 1]$, where the case, $\eta_b = 1$ is referred to as a purely elastic
<figure><img src="image_296.png" /><figcaption>**Figure 50.4:** Graphical illustration of the discontinuity equations for the scattering and the production amplitude, respectively. The dashed line indicates that the intermediate state is to be put on-shell to find the discontinuity.</figcaption></figure>

<figure><img src="image_297.png" /><figcaption>**Figure 50.5:** Argand plot showing a trajectory of the diagonal element of a partial-wave amplitude, $f_{bb}$, as a function of energy in the complex plane. As the energy increases the amplitude follows the line counter clockwise. The amplitude leaves the unitary circle (solid line) as soon as inelasticity sets in, $\eta < 1$ (dashed line).</figcaption></figure>

scattering. Thus, the function $\eta_b(s)$ is a direct measure of the contribution of the inelastic channels on the scattering amplitude in a given channel.

The evolution of the partial-wave amplitude $f_{bb}$ with energy can be displayed as a trajectory in the Argand plot, as shown in Fig. 50.5. In case of a two-channel problem, $\eta_1 = \eta_2 = \eta$, and the off-diagonal element is $f_{12} = \sqrt{1 - \eta^2/2 \exp(i(\delta_1 + \delta_2))}$. The unitarity condition Eq. (50.14) sets the limit to the squared amplitude $f_{bb}$:

$$|f_{bb}|^2 = \frac{1}{4}(\eta_b^2 - 2\eta_b \cos(2\delta_b) + 1) \le \frac{1}{4}(\eta_b + 1)^2, \quad (50.17)$$

where the maximum value is reached for $\delta_b = \pi/2$. For the absolute square of the partial-wave-projected scattering amplitude the unitarity bound thus reads:

$$|\mathcal{M}_{bb}| \le \frac{1}{2\rho_b}(\eta_b + 1) \le \frac{8\pi}{q_b}\sqrt{s}, \quad (50.18)$$

where the second inequality comes from $\eta_b \le 1$. For energies much larger than the masses of the scattering particles the upper bound for $|\mathcal{M}_{bb}|$ tends to $16\pi$ for large $s$.

The partial-wave projected production amplitude $\mathcal{A}(s)$ (note that the label $j$ has been omitted for consistency) is also constrained by unitarity. As derived from Eq. (50.8):

$$\mathrm{Im} \, \mathcal{A}_a = \sum_b \mathcal{M}_{ba}^* \rho_b \mathcal{A}_b, \quad (50.19)$$

where the summation encompasses all open channels. In the realm of elastic scattering, solely one channel, denoted by $a$, contributes to the sum. Consequently, the phase of $\mathcal{A}_a$ must align with the phase of $\mathcal{M}_{aa}$, given that the left-hand side of Eq. (50.19) represents a real value. This principle is recognized as the Watson theorem [57]. To illustrate, consider the phase of the pion vector form factor: it agrees to that of $\pi\pi$ scattering in the vector isovector channel (aside from effects of the isospin-violating $\rho-\omega$ mixing) up to about 1 GeV, where inelastic contributions start gaining significance.

## 50.2 Properties of resonances

A resonance is defined by its pole position in the complex s-plane, denoted as $s_R$, and by the strength parameters of its couplings to various decay channels evaluated at this pole, known as the pole residues. The pole mass $M_R$ and pole width $\Gamma_R$ are defined via the pole parameters

$$\sqrt{s_R} = M_R - i\Gamma_R/2. \quad (50.20)$$

For states where the relevant thresholds are situated significantly below the resonance location, the lifetime $\tau_R$ of the resonance is given by $\tau_R = 1/\Gamma_R$ (refer to the review on “Kinematics” and for a recent discussion, see Ref. [58]). It is important to note that the conventional Breit-Wigner parameters $M_{BW}$ and $\Gamma_{BW}$, introduced in Eq. (50.30), differ from the pole parameters due to finite width effects and the influence of thresholds and background terms. It should be stressed that pole location $s_R$ and pole residues are the only resonance properties that are model and parameterization independent.

When a resonance interacts with multiple channels, each channel contributes to the imaginary part of the pole position. However, these individual imaginary contributions do not necessarily simply sum up, leading to the need for redefining the partial widths. This behavior is discussed on the example of the $f_0(980)$ in Refs. [59, 60].

In the close vicinity of the resonance pole one defines the residues via

$$\lim_{s \to s_R} (s - s_R) \mathcal{M}_{ba} = -\mathcal{R}_{ba}. \quad (50.21)$$

Those can be conveniently calculated via an integration along a closed contour around the pole using

$$\mathcal{R}_{ba} = -\frac{1}{2\pi i} \oint ds \mathcal{M}_{ba}. \quad (50.22)$$

The residue adheres to the factorization relation, $(\mathcal{R}_{ba})^2 = \mathcal{R}_{aa} \times \mathcal{R}_{bb}$. This factorization emerges as a universal property stemming from the unitarity of scattering processes [7]. Building on this relation, one can define pole couplings as follows:

$$\tilde{g}_a = \mathcal{R}_{ba} / \sqrt{\mathcal{R}_{bb}}. \quad (50.23)$$

The pole couplings characterize the transition strength of a given resonance to some channel $a$ independently of how the particular resonance was produced. They are in general complex valued.

In the baryon sector, it is customary to define the residue of the pole for the $f_{ba}$ amplitude (as described in Eq. (50.13)) in terms of the variable $\sqrt{s}$ rather than $s$. This residue can be related to the residues mentioned earlier by:

$$r_{ba} = \frac{1}{2} \sqrt{\frac{\rho_a(s_R) \rho_b(s_R)}{s_R}} \mathcal{R}_{ba}, \quad (50.24)$$

where the phase-space factors are to be continued analytically to the pole location $s_R$. The residues quoted in the baryon listings are those for $\pi N$ scattering ($a = b = \pi N$).

The branching ratio of a resonance decay to a specific channel represents the fraction of the decay probability directed to that channel.

$$\mathrm{Br}_a = N_a / N_{\mathrm{tot}}. \quad (50.25)$$

Here, $N_a$ is the experimental count of events for the decay channel $R \to a$, while $N_{\mathrm{tot}}$ represents the total number of events produced
in the decay of the resonance. Since the amplitudes of decays to different final states add incoherently, we have $N_{tot} = \sum_b N_b$. We note, however, that both $N_a$ and $N_{tot}$ may exhibit a dependence on the reaction through which the resonance is produced, due to the resonance's finite width. For a narrow resonance, the experimental count is determined by integrating the squared production amplitude over the decay channel's phase space:

$$N_a = N_0 \int |A_a|^2 d\Phi_a, \qquad (50.26)$$

where $N_0$ is a normalization constant associated with the integrated luminosity, and $A_a$ represents the amplitude of the resonance decay to channel a. When the decay rate's variation across the resonance width becomes significant, the resonance's lineshape must be considered:

$$N_a = N_0 \int_{s_{thr,a}}^{\infty} ds \sigma_R(s) \int |A_a(s)|^2 d\Phi_a(s). \qquad (50.27)$$

Here, $\sigma_R(s)$ denotes a proper weight function of the resonance and $s_{thr,a}$ is the threshold value for the channel a. Additionally, the phase space integral, $\Phi_a(s)$, puts the integrand to zero below the energy threshold of the decay channel a. This aspect is particularly crucial for the decays of broad resonances into channels with energy thresholds exceeding the resonance's nominal mass. This methodology is frequently employed in light-meson studies, as demonstrated in Ref. [61], and is also prevalent in light baryon research, as referenced in Ref. [60].

In case of the decay of some heavy state into multi-body final states, the transition amplitude can include resonances within subsystems of particles. When the total amplitude is decomposed as $A_a = \sum_R A_{a(R)}$, the relative branching fraction for the decay of the given heavy state into final state a via resonance R in some subsystem, denoted as $Br_{a(R)}/Br_a$, is given by:

$$Br_{a(R)}/Br_a = \int |A_{a(R)}|^2 d\Phi_a / \int |A_a|^2 d\Phi_a \qquad (50.28)$$

It is crucial to recognize that the decomposition of the total amplitude, $A_a$, into resonance amplitudes, $A_{a(R)}$, is not a straightforward process. Beyond the differentiation of components based on distinct quantum numbers, this separation is inherently model-dependent. Furthermore, it is essential to be aware that the amplitudes $A_{a(R)}$ can interfere with one another. As a result, their individual probabilities might not sum up to the total branching fraction of channel a. Nonetheless, in many scenarios, the interference contributions are small, making the fractions $Br_{a(R)}/Br_a$ indicative. For recent applications of these formulas in the context of B-decays see Refs. [62–65].

Lastly, an expression analogous to the branching fraction can be formulated using the pole parameters. For two-particle decays in the S-wave, one writes:

$$Br_a = \frac{|\tilde{g}_a|^2}{M_R \Gamma_R} \rho_a(M_R^2), \qquad (50.29)$$

where $M_R$ and $\Gamma_R$ have been previously defined in Eq. (50.20). This approach was utilized to define a two-photon width for the broad $f_0(500)$ resonance [66, 67]. Similarly, one should use residues to quantify the coupling of resonances to certain production channels [68]. For an application of this approach to the coupling of the $K_0^*(1430)$ resonance to a leptonic current see Ref. [69]. Equation (50.29) provides a definition of branching fraction that remains independent of the reaction used to derive the quantity. For narrow resonances, this definition aligns well with Eq. (50.25) and Eq. (50.26). However, for broad, overlapping resonances, it is essential to recognize that Eq. (50.29) is primarily used to convert residues into metrics that facilitate a more straightforward comparison of resonance transitions across different channels. For resonances with a coupling to a channel that remains closed at the resonance mass, Eq. (50.29) is not applicable due to the phase-space factor. In such scenarios, modification of the expression is required as elaborated upon in Ref. [60], and at the conclusion of Sec. 50.3.5.

## 50.3 Common parameterizations

In general, there is no universal, model-independent recipe to build scattering amplitudes. However, a few approaches presented in this section are practical to extract resonance properties in experimental analyses. The systematic theory uncertainties need to be assessed from a range of model variations that provide a sufficient quality of description of the available data and are permitted by general S-matrix principles and the symmetries controlling the system at hand.

### 50.3.1 The Breit–Wigner parameterization

The relativistic Breit–Wigner parameterization represents a dressed propagator for an isolated resonance. The production amplitude for a resonance observed in a channel a, is given by

$$A_a(s) = \frac{N_a(s)}{M_{BW}^2 - s - iM_{BW}\Gamma(s)} \qquad (50.30)$$

where $M_{BW}$ represents the Breit–Wigner mass, and $\Gamma_{BW} = \Gamma(M_{BW}^2)$ denotes the Breit–Wigner width. The function $\Gamma(s)$ is defined by the channels to which the resonance can decay. The numerator function $N_a(s)$ is tailored to the production process, encompassing kinematic factors and couplings pertinent to both the production and decay processes.

$$N_a(s) = \alpha g_a n_a(s) \qquad (50.31)$$

$$\Gamma(s) = \frac{1}{M_{BW}} \sum_b g_b^2 \rho_b(s) n_b^2(s), \qquad (50.32)$$

Here the index $b = 1, 2, \dots$ runs over all decay channels of the resonance. The coupling constants are represented by $g_b$, and $\rho_b$ is the phase-space factor as defined in Eq. (50.11). The expression for $n_a(s)$ is:

$$n_a = (q_a/q_0)^{l_a} F_{l_a}(q_a/q_0), \qquad (50.33)$$

where $l_a$ indicates the orbital angular momentum in channel a, $q_a(s)$ is the break-up momentum as defined in Eq. (50.7), and $q_0$ is a suitably selected momentum scale. The term $(q_a)^{l_a}$ ensures the amplitude's appropriate threshold behavior. The rapid growth of this factor for angular momenta $l_a > 0$ is offset at specific $s$ values by a phenomenological form factor, represented here by $F_{l_a}(q_a, q_0)$ — the presence of these suppression factors is also a requirement from positivity which demands that the dressed propagator, the denominator of Eq. (50.30) and similar equations below, is not allowed to drop faster than $1/s$ [27]. The Blatt-Weisskopf form factors are frequently employed in the literature [70–72] to model $F_j$:

$$F_0^2(z) = 1, \qquad (50.34)$$

$$F_1^2(z) = 1/(1+z^2),$$

$$F_2^2(z) = 1/(9+3z^2+z^4),$$

where $z = q_0/q_0$, the scale parameter $1/q_0$ typically falls within the range of $1 \text{ GeV}^{-1}$ to $5 \text{ GeV}^{-1}$. Rather than employing coupling constants as in Eq. (50.32), channel partial widths can be used,

$$\Gamma(s) = \sum_b \Gamma_{BW,b} \frac{\rho_b(s)}{\rho_b(M_{BW}^2)} \left( \frac{q_b}{q_b R} \right)^{2l_b} \frac{F_{l_b}^2(q_b, q_0)}{F_{l_b}^2(q_b R, q_0)} \qquad (50.35)$$

Here $q_bR$ is the values of the break-up momenta evaluated at $s = M_{BW}^2$. It is essential to note that this substitution is valid only for channels where the decay channel's threshold is positioned below the nominal resonance mass. In other scenarios, Eq. (50.32) should be applied.

Equation (50.32) incorporates a threshold for each of the coupled channels. The expression is straightforward to use in the physical region above all the thresholds. Its evaluation elsewhere requires a careful analytic continuation. As outlined in Refs. [73, 74], the choice

$$q_c = i\sqrt{-q_c^2} \quad \text{for} \quad q_c^2 < 0, \qquad (50.36)$$

leads to an evaluation of the amplitude on the physical sheet below threshold. The Flatté parameterization [73] refers to the two
S-wave channels' amplitude near a threshold of a heavier channel in the physical region of the lighter channel computed with Eq. (50.36). When a resonance's coupling to the channel with a higher threshold is notably strong, the parameterization displays scaling invariance. This implies that it is not possible to extract individual partial decay widths; only their ratios can be determined [75].

The Breit-Wigner parameterization is an accurate representation of resonance phenomena strictly in the $Γ/Δ \to 0$ limit, where $Γ$ is the resonance width and $Δ$ is the distance to the closest unaccounted singularity, be it a pole of a higher resonance or a kinematic threshold related to a coupled channel. However, the situation is often more complex due to multiple singularities in the complex plane around the resonance with different importance. For instance, in P-wave $ππ$ scattering, the Breit-Wigner parameterization aptly describes the $ρ$-meson resonance over an extensive range. Although the closest singularity to the $ρ$-meson pole is the $ω$-pole ($Γ_ρ/Δ_ω \gg 1$), this isospin breaking effect is typically insignificant except few special cases [76, 77]. Subsequent singularities, namely the $3π, 4π$, and $6π$ thresholds, are also generally disregarded. The two-pion threshold is incorporated in the Breit-Wigner parameterization through an energy-dependent width. Finally, the parameterization's efficacy diminishes around 1.2 GeV due to the $ρ'$ resonance, situated approximately at 1.45 GeV.

Once the applicability of the Breit-Wigner parameterization is established, it is crucial to recognize that its parameters will only align with the pole parameters if the width is small. Yet, extracting the pole position from the Breit-Wigner amplitude is a straightforward technical task, achieved through analytic continuation. However, neither the Breit-Wigner parameters nor the corresponding pole parameters should be deemed reliable without justifying the parameterization's applicability. If there is more than one resonance in one partial wave that significantly couples to the same channel, it is generally inappropriate to employ a sum of Breit-Wigner functions. Such an approach often results in a breach of unitarity constraints, potentially introducing an indeterminate bias to the inferred resonance properties from the reaction amplitude. For overlapping resonances in the same partial wave, more sophisticated methods, such as the $K$-matrix approach detailed in the subsequent section, are recommended.

### 50.3.2 $K$-matrix approach

The $K$-matrix method offers a comprehensive framework for modelling coupled-channel amplitudes [78]. This method ensures two-particle unitarity. However, it traditionally omits the left-hand cuts. The scattering amplitude $M_{ba}(s)$ can be derived from the equation:

$$n_b M_{ba}^{-1} n_a = K_{ba}^{-1} - i \delta_{ba} \rho_a n_a^2. \quad (50.37)$$

Here, $K_{ba}$ represents a real function and is subject to modeling. The factor $n_a$ is elaborated upon in Eq. (50.33). Since there is no unique recipe to build $K$, it is essential to explore various parameterizations to gauge the theoretical systematic uncertainty. A commonly adopted choice for the $K$-matrix is given by:

$$K_{ba}(s) = \sum_R \frac{g_b^R g_a^R}{m_R^2 - s} + b_{ba}, \quad (50.38)$$

where $m_R$ is referred to as the bare mass of the resonance R (not to be confused with the physical mass), and the $g_a^R$ represents the bare couplings of the resonance R to the channel $a$ (not to be confused with the residues). The $b_{ba}$ is a matrix that parameterizes the non-pole components of the $K$-matrix. Provided all parameters in Eq. (50.38) are real, the amplitude $M_{ba}(s)$ remains unitary. From Eq. (50.37), the scattering amplitude $M$ can be directly computed using its matrix form:

$$M = n[1 - K i \rho n^2]^{-1} K n, \quad (50.39)$$

where $n$ and $ρ$ are diagonal matrices, $n = \text{diag}(n_a, n_b, \dots)$, and $ρ = \text{diag}(ρ_a, ρ_b, \dots)$. As an alternative to Eq. (50.38), the same

functional structure on the right side of Eq. (50.38) can be employed to parameterize the inverse $K$-matrix, termed the $M$-matrix, by the authors of Ref. [79]. Numerous alternative formulations within the $K$-matrix framework are utilized for amplitude studies related to lattice-QCD calculations [80–82].

A prevalent method to construct the production amplitude within the $K$-matrix framework is the $P$-vector parameterization [72, 78, 83]. The method utilizes the $K$-matrix poles and bare couplings from Eq. (50.38):

$$A_a(s) = n_a \sum_c [1 - K i \rho n^2]_{ac}^{-1} P_c, \quad (50.40)$$

$$P_c = \sum_R \frac{\alpha^R g_c^R}{m_R^2 - s} + B_c. \quad (50.41)$$

The production vector, denoted as $P_c$, comprises two main components. The first component represents a transition driven by the coupling of the bare resonance, $R$, to the source. This coupling is characterized by a strength parameter, $α^R$. The second component, $B_c$, signifies the direct transition from the source to the channel $c$. The formalism ensures that the complete production vector gets dressed via the final state interaction.

The $Q$-vector, as discussed in Ref. [72, 79, 84], offers an alternative methodology for constructing a production amplitude:

$$A_a(s) = \sum_c M_{ac}(s) Q_c(s) / n_c. \quad (50.42)$$

Here, $Q_c(s)$ represents a smooth function of $s$ and can be parameterized using a polynomial series. The unitarity condition of Eq. (50.19) is satisfied when $Q_c(s)$ is a real function and in particular does not have singularities above the lowest threshold for all channels $c$. Besides these conditions $Q_c(s)$ is arbitrary. In a study of $γγ \to ππ$, cf. Ref. [66, 67] a low-order polynomial is claimed to be sufficient to parameterize the energy dependence of the function $Q_c(s)$. The $Q$-vector method is convenient, if the full matrix $M$ is known, cf. Ref. [79]. An important distinction between the $P$-vector and the $Q$-vector methods is highlighted in [83]. When the two-particle scattering amplitude approaches zero, the production amplitude in the $Q$-vector construction unavoidably vanishes for finite values of $Q_c$, whereas it remains non-zero in the $P$-vector approach.

Traditionally, amplitudes constructed using the $K$-matrix technique exclude the left-hand cuts. Nevertheless, these can be customarily incorporated into the function $b_{ba}$ from Eq. (50.38) for the scattering amplitude [85, 86]. Similarly, for the production amplitude, the functions $B_c$ and $Q_c$ from Eq. (50.40) and Eq. (50.42), respectively, might also encompass the left-hand cuts. Those can often be parameterized by low order polynomials [66, 67, 87].

The position of the resonance poles can be determined by examining the zeros of the analytic function $\det[1 - K i \rho n^2]$. Owing to the $ρ$ factor, this determinant exhibits a complex multisheet structure. Nonetheless, the nearest unphysical sheet usually has the highest influence to the physical region. It is always the one which is determined by the heaviest threshold below the studied point in $s$ (cf. Fig. 50.3). If for a given resonance the pole closest to the physical axis is not located on a sheet that connects directly to the physical axis, it is possible that $K$-matrix fits do not allow one to fix the pole parameters, since a larger distance of the pole to the physical axis can be balanced by increased residues — for a detailed discussion see Ref. [88].

### 50.3.3 Further improvements: Chew-Mandelstam function

The $K$-matrix framework often enables an accurate fit of physical amplitudes and is straightforward to handle. However, it does present a significant drawback: it breaches constraints imposed by analyticity. For instance, $ρ_a$, as defined in Eq. (50.11), is not well-defined at $s = 0$. Moreover, in cases of unequal masses, it manifests an unphysical cut, as depicted in the left panel of Fig. 50.6.

A method to improve the analytic properties has been suggested in Refs. [89–93]. This approach replaces the term $iρ_a(s)n_a^2$ from
<figure><img src="image_298.png" /><figcaption>**Figure 50.6:** Comparison of the $i\rho$ function (left plot) to the *Chew-Mandelstam function* from Eq. (50.44) (right plot), evaluated for the case of S-wave $\eta\pi$ scattering. The values of $s$ are taken slightly above the real axis, $s + i0$. The solid red line shows the imaginary part that is the same for both functions above threshold. The dashed black line presents the real part. One finds indications of the unphysical left-hand singularities of the function $i\rho$ on the left plot, while the Chew-Mandelstam function in analytic below the two-particle threshold.</figcaption></figure>

Eq. (50.37) with the analytic function $\Sigma_a(s)$, known as the Chew-Mandelstam function. This function produces the imaginary part $i\rho_a(s)n_a^2$ on the right-hand cut, while maintaining analyticity on the left-hand side, as represented by the once subtracted dispersion integral:

$$ \Sigma_a(s + i0) = \frac{s - s_{\text{thr}_a}}{\pi} \int_{s_{\text{thr}_a}}^{\infty} \frac{\rho_a(s') n_a^2(s')}{(s' - s_{\text{thr}_a})(s' - s - i0)} \mathrm{d}s'. \quad (50.43) $$

Here, we chose the channel threshold, $s_{\text{thr}_a}$, as the subtraction point, and assumed that the subtraction constant is absorbed into the other parameters used for the amplitude. For an S-wave where $n_a = 1$, the integral has a closed form [37, 91]:

$$ \Sigma_a(s) = \frac{1}{16\pi^2} \left[ \frac{2q_a}{\sqrt{s}} \log \frac{m_1^2 + m_2^2 - s + 2\sqrt{s}q_a}{2m_1 m_2}, \right. \\ \left. - (m_1^2 - m_2^2) \left( \frac{1}{s} - \frac{1}{(m_1 + m_2)^2} \right) \log \frac{m_1}{m_2} \right], \quad (50.44) $$

where $m_1$ and $m_2$ are masses of the final-state particles in channel $a$, with $s_{\text{thr}_a} = (m_1 + m_2)^2$. The function's behavior along the real axis is illustrated in the right panel of Fig. 50.6. A further discussion of the calculation of the Chew-Mandelstam function can be found in Refs. [94, 95].

If there is only a single resonance in a given channel, it is possible to feed the imaginary part of the Breit-Wigner function, Eq. (50.30) with an energy-dependent width, directly into a dispersion integral to get a resonance propagator with the correct analytic structure [96, 97].

### 50.3.4 Effective-range expansion and scattering-length approximation

For elastic scattering, $\eta_b = 1$, Eq. (50.14) simplifies to

$$ f_{bb} = e^{i\delta} \sin(\delta) = \frac{q}{q \cot(\delta) - iq}, \quad (50.45) $$

where $q$ is the relative momentum of the scattering particles. For S-waves one may then employ the *effective-range expansion* [98, 99] (ERE)

$$ q \cot \delta = \frac{1}{a} + \frac{1}{2} r q^2 + O(r^3 q^4), \quad (50.46) $$

whose radius of convergence is set by the closest non-analyticity, which may be the next threshold, a left-hand cut or a zero in the amplitude which is equivalent to a pole in $q \cot \delta$. The ERE is understood to be an expansion in $R_f q$, where $R_f$ denotes the range of forces provided by the inverse of the lightest exchange particle allowed. The *scattering length*, $a$, is then defined as the

first term in an expansion of the real part of the inverse scattering amplitude. The sign convention used in Eq. (50.46) is the one commonly employed in particle physics. In this convention a positive scattering length indicates attraction; if, however, the attraction is strong enough to generate a bound state, the scattering length changes sign and turns negative. A negative scattering length also occurs for repulsive interactions. Note that in nuclear physics the leading term in the expansion of Eq. (50.46) is usually defined as $-1/a$ such that *e.g.* a bound state would be related to a positive scattering length. The parameter $r$ is called the *effective range*. Especially in cases where the scattering length is large the ERE not only describes the low energy scattering well but also allows for an analytic continuation to find bound states below threshold by proper analytic continuation. This method was, *e.g.*, employed recently to analyze near threshold scattering phase shifts for $D^*+D^0$ scattering found in lattice analyses to extract properties of the $T_{cc}(3875)^{+}$ [100–102] at unphysical pion masses. It should be stressed that the analyses of the lattice data might call for a modification to account for the nearby left-hand cut which sets the radius of convergence of the ERE [103].

When considering only the scattering length within the ERE, the scattering amplitude is represented as:

$$ \mathcal{M}(s) = \frac{8\pi\sqrt{s}}{1/a - iq(s)}. \quad (50.47) $$

The scattering length is thus proportional to the value of the amplitude at threshold. It is worth noting that the scattering length approximation is applicable only in a very limited energy range, however, might well be appropriate to analyze the recently discovered narrow near-threshold states [104, 105]. Examples of such analyses can be found in Refs. [106–108]. Moreover, it is possible to introduce the effect of a weakly coupled lower channel [109, 110]. Such coupling results in a positive imaginary part of the scattering length. It is also crucial to highlight that for large $a$ values, the amplitude of Eq. (50.47) develops a near-threshold pole located on the physical or unphysical sheet for negative or positive values of $a$, respectively. For readers interested in a exploration of how close-to-threshold poles interact with remote thresholds, we refer to Ref. [111].

While easy to use, it is important to stress, however, that the approximation in Eq. (50.47) is a specific choice of the dynamic function. This choice results in a single pole close to the physical region, suggesting that the state under study has the characteristics of a hadronic molecule, as discussed in references such as [108, 112–114]. Virtual states are discussed in this context in Ref. [115].
### 50.3.5 Two-potential decomposition

Another advanced technique to construct the scattering amplitude, which is widely used in the literature [69, 116–121], is based on the two-potential formalism [122]. While it is possible to formulate this method for the full unprojected amplitude $M_{ba}(s, t)$, for clarity, we focus on presenting the equations in their partial-wave-projected form.

The scattering amplitude, $M(s)$, can be broken down into two components: a background part and a pole part. This can be represented as:

$$M(s) = M^{\text{b.g.}}(s) + M^{\text{pole}}(s). \qquad (50.48)$$

It is important to note that the division presented in Eq. (50.48) is not unique and model-dependent. This is further discussed in references such as [123, 124]. The background scattering matrix is assumed to be unitary by itself. One approach to parameterize it, especially at lower energies, is by using phase shifts and inelasticities, as seen in [69, 120, 121]. Alternatively, it can be computed based on some potential, $V^{\text{b.g.}}$, fed into a proper scattering equation [118, 119].

The complete amplitude, $M$, from Eq. (50.48) is unitary, if the pole part follows a specific construction. Namely,

$$M_{ab}^{\text{pole}}(s) = \Omega_{ac}(s) [1 - V^{\text{R}}(s) \Sigma^{\text{u}}(s)]_{cd}^{-1} V_{de}^{\text{R}}(s) \Omega_{eb}^{T}(s) . \quad (50.49)$$

In this context, we introduce the vertex functions, denoted as $\Omega_{ab}(s)$, and the resonance potential, represented as $V^R(s)$. This potential operates as a matrix in the channel space and can be expressed as:

$$V_{ab}^{\text{R}}(s) = \sum_{R} \frac{g_{a}^{\text{R}} g_{b}^{\text{R}}}{m_{R}^{2} - s} . \qquad (50.50)$$

The term $\Sigma_{ab}^u$ represents the self-energy matrix. Additionally, $g_a^R$ denotes the bare coupling of the resonance, labeled as R, to the channel a, and $m_R$ is its bare mass. The vertex functions obey a unitarity relation similar to the production amplitude in Eq. (50.8). However, in this case, the final state interaction is determined by $M^{\text{b.g.}}$. This can be represented as:

$$\text{Disc} \Omega_{ab}(s) = 2i \sum_{c} M_{ca}^{\text{b.g.}*}(s) \rho_{c}(s) \Omega_{cb}(s) . \qquad (50.51)$$

When using low-energy phase shifts for the background term, it is practical to express the vertex functions in terms of an Omnès matrix, as discussed in [121]. The matrix reduces to the well-known Omnès function

$$\Omega(s) = \exp \left( \frac{s}{\pi} \int_{s_{\text{thr}}}^{\infty} ds' \frac{\delta^{\text{b.g.}}(s')}{s'(s' - s - i\epsilon)} \right) \qquad (50.52)$$

in the single-channel case [125], where $\delta^{\text{b.g.}}(s)$ denotes the phase of the background scattering matrix $M^{\text{b.g.}}(s)$. The discontinuity associated with the self-energy matrix, $\Sigma^u(s)$, is given by:

$$\text{Disc} \Sigma_{ab}^{u}(s) = 2i \sum_{c} \Omega_{ca}^{*}(s) \rho_{c}(s) \Omega_{cb}(s) . \qquad (50.53)$$

To determine the real part of $\Sigma^u$, one can use Eq. (50.53) with a properly subtracted dispersion integral — cf. Eq. (50.43).

When $M^{\text{b.g.}}$ is unitary, the full amplitude is unitary given Eq. (50.49). However, it is essential to note that the pole term alone is not unitary unless the background amplitude vanishes. Under such conditions, the described amplitude simplifies to the $K$-matrix construction with improved analytic behavior as detailed in Sec. 50.3.3. Neglecting non-pole terms is not a good approximation for certain interactions, such as the scalar-isoscalar $\pi\pi$ at low energies, as discussed in [126]. However, for higher partial waves, this approximation is generally effective.

A production amplitude consistent with Eq. (50.49) reads [120, 121]:

$$A_{a}^{\text{pole}}(s) = \Omega_{ac}(s) [1 - V^{\text{R}}(s) \Sigma^{\text{u}}(s)]_{cd}^{-1} P_{d}, \qquad (50.54)$$

with function $P_d$ given in Eq. (50.40). This form was employed in Refs. [120, 121] to treat the pion vector and scalar form factor, respectively, and in Ref. [69] for the scalar $\pi K$ form factor.

There has been considerable interest in the $3 \to 3$ scattering recently, particularly in light of new data on three-hadron interaction [127] and advancements in lattice calculations [128]. One finds that the methodologies devised for accounting for one-pion exchange bear a resemblance to the two-potential decomposition. For details see Ref. [129], also Eq. (93) in Ref. [130].

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# 51. Cross-Section Formulae for Specific Processes

Revised August 2019 by H. Baer (Oklahoma U.) and R.N. Cahn (LBNL).

## PART I: STANDARD MODEL PROCESSES

Setting aside leptoproduction (for which, see Sec. 16 of this Review), the cross sections of primary interest are those with light incident particles, $e^+e^-$, $\gamma\gamma$, $q\bar{q}$, $gg$, $gg$, etc., where g and q represent gluons and light quarks. The produced particles include both light particles and heavy ones - t, W, Z, and the Higgs boson H. We provide the production cross sections calculated within the Standard Model for several such processes.

### 51.1 Resonance Formation

Resonant cross sections are generally described by the Breit-Wigner formula (Sec. 50.3 of this Review).

$$ \sigma(E) = \frac{2J+1}{(2S_1+1)(2S_2+1)} \frac{4\pi}{k^2} \left[ \frac{\Gamma^2/4}{(E-E_0)^2 + \Gamma^2/4} \right] B_{in}B_{out}, \quad (51.1) $$

where E is the c.m. energy, J is the spin of the resonance, and the number of polarization states of the two incident particles are $2S_1+1$ and $2S_2+1$. The c.m. momentum in the initial state is k, $E_0$ is the c.m. energy at the resonance, and $\Gamma$ is the full width at half maximum height of the resonance. The branching fraction for the resonance into the initial-state channel is $B_{in}$ and into the final-state channel is $B_{out}$. For a narrow resonance, the factor in square brackets may be replaced by $\pi\Gamma\delta(E - E_0)/2$.

### 51.2 Production of light particles

The production of point-like, spin-1/2 fermions in $e^+e^-$ annihilation through a virtual photon, $e^+e^- \to \gamma^* \to f\bar{f}$, at c.m. energy squared s is given by

$$ \frac{d\sigma}{d\Omega} = N_c \frac{\alpha^2}{4s} \beta \left[ 1 + \cos^2 \theta + (1 - \beta^2) \sin^2 \theta \right] Q_f^2, \quad (51.2) $$

where $\beta$ is $v/c$ for the produced fermions in the c.m., $\theta$ is the c.m. scattering angle, and $Q_f$ is the charge of the fermion. The factor $N_c$ is 1 for charged leptons and 3 for quarks. In the ultrarelativistic limit, $\beta \to 1$,

$$ \sigma = N_c Q_f^2 \frac{4\pi\alpha^2}{3s} = N_c Q_f^2 \frac{86.8 \text{ nb}}{s (\text{GeV}^2)} . \quad (51.3) $$

The cross section for the annihilation of a $q\bar{q}$ pair into a distinct pair $q'\bar{q}'$ through a gluon is completely analogous up to color factors, with the replacement $\alpha \to \alpha_s$. Treating all quarks as massless, averaging over the colors of the initial quarks and defining $t = -s \sin^2(\theta/2)$, $u = -s \cos^2(\theta/2)$, one finds [1]

$$ \frac{d\sigma}{d\Omega}(q\bar{q} \to q'\bar{q}') = \frac{\alpha_s^2}{9s} \frac{t^2 + u^2}{s^2}. \quad (51.4) $$

Crossing symmetry gives

$$ \frac{d\sigma}{d\Omega}(qq' \to qq') = \frac{\alpha_s^2}{9s} \frac{s^2 + u^2}{t^2}. \quad (51.5) $$

If the quarks q and $q'$ are identical, we have

$$ \frac{d\sigma}{d\Omega}(q\bar{q} \to q\bar{q}) = \frac{\alpha_s^2}{9s} \left[ \frac{t^2 + u^2}{s^2} + \frac{s^2 + u^2}{t^2} - \frac{2u^2}{3st} \right], \quad (51.6) $$

and by crossing

$$ \frac{d\sigma}{d\Omega}(qq \to qq) = \frac{\alpha_s^2}{9s} \left[ \frac{t^2 + s^2}{u^2} + \frac{s^2 + u^2}{t^2} - \frac{2s^2}{3ut} \right]. \quad (51.7) $$

Annihilation of $e^+e^-$ into $\gamma\gamma$ has the cross section

$$ \frac{d\sigma}{d\Omega}(e^+e^- \to \gamma\gamma) = \frac{\alpha^2}{2s} \frac{u^2 + t^2}{tu}. \quad (51.8) $$

The related QCD process also has a triple-gluon coupling. The cross section is

$$ \frac{d\sigma}{d\Omega}(q\bar{q} \to gg) = \frac{8\alpha_s^2}{27s} (t^2 + u^2) \left( \frac{1}{tu} - \frac{9}{4s^2} \right). \quad (51.9) $$

The crossed reactions are

$$ \frac{d\sigma}{d\Omega}(qg \to qg) = \frac{\alpha_s^2}{9s} (s^2 + u^2) \left( -\frac{1}{su} + \frac{9}{4t^2} \right) \quad (51.10) $$

and

$$ \frac{d\sigma}{d\Omega}(gg \to q\bar{q}) = \frac{\alpha_s^2}{24s} (t^2 + u^2) \left( \frac{1}{tu} - \frac{9}{4s^2} \right). \quad (51.11) $$

Finally,

$$ \frac{d\sigma}{d\Omega}(gg \to gg) = \frac{9\alpha_s^2}{8s} \left(3 - \frac{ut}{s^2} - \frac{su}{t^2} - \frac{st}{u^2}\right). \quad (51.12) $$

Lepton-quark scattering is analogous (neglecting Z exchange)

$$ \frac{d\sigma}{d\Omega}(eq \to eq) = \frac{\alpha^2}{2s} e_q^2 \frac{s^2 + u^2}{t^2}. \quad (51.13) $$

where $e_q$ is the charge of the quark. For neutrino scattering with the four-Fermi interaction

$$ \frac{d\sigma}{d\Omega}(\nu d \to \ell^- u) = \frac{G_F^2 s}{4\pi^2}, \quad (51.14) $$

where the Cabibbo angle suppression is ignored. Similarly

$$ \frac{d\sigma}{d\Omega}(\nu\bar{u} \to \ell^-\bar{d}) = \frac{G_F^2 s}{4\pi^2} \frac{(1 + \cos\theta)^2}{4}. \quad (51.15) $$

To obtain the formulae for deep inelastic scattering (presented in more detail in Section 18) we consider quarks of type i carrying a fraction $x = Q^2/(2M\nu)$ of the nucleon's energy, where $\nu = E - E'$ is the energy lost by the lepton in the nucleon rest frame. With $y = \nu/E$ we have the correspondences

$$ 1 + \cos\theta \to 2(1 - y), $$

$$ d\Omega_{cm} \to 4\pi f_i(x) dx dy, \quad (51.16) $$

where the latter incorporates the quark distribution, $f_i(x)$. In this way we find

$$ \frac{d\sigma}{dx dy}(eN \to eX) = \frac{4\pi\alpha^2xs}{Q^4} \frac{1}{2} [1 + (1 - y)^2] \times \left[ \frac{4}{9}(u(x) + \bar{u}(x) + \dots) + \frac{1}{9}(d(x) + \bar{d}(x) + \dots) \right] \quad (51.17) $$

where now $s = 2ME$ is the cm energy squared for the electron-nucleon collision, and we have suppressed contributions from higher mass quarks.

Similarly,

$$ \frac{d\sigma}{dx dy}(\nu N \to \ell^- X) = \frac{G_F^2 xs}{\pi} [(d(x) + \dots) + (1 - y)^2 (\bar{u}(x) + \dots)] \quad (51.18) $$

and

$$ \frac{d\sigma}{dx dy}(\bar{\nu}N \to \ell^+ X) = \frac{G_F^2 xs}{\pi} [(\bar{d}(x) + \dots) + (1 - y)^2 (u(x) + \dots)] . \quad (51.19) $$

Quasi-elastic neutrino scattering ($\nu_\mu n \to \mu^- p$, $\bar{\nu}_\mu p \to \mu^+ n$) is directly related to the crossed reaction, neutron decay. The formula for the differential cross section is presented, for example, in N.J. Baker et al., Phys. Rev. D23, 2499 (1981).
## 5.1.3 Hadroproduction of heavy quarks

For hadroproduction of heavy quarks $Q = c, b, t$, it is important to include mass effects in the formulae. For $q\bar{q} \to Q\bar{Q}$, one has

$$ \frac{d\sigma}{d\Omega}(q\bar{q} \to Q\bar{Q}) = \frac{\alpha_s^2}{9s^3} \sqrt{1 - \frac{4m_Q^2}{s}} \left[ (m_Q^2 - t)^2 + (m_Q^2 - u)^2 + 2m_Q^2 s \right], (51.20) $$

while for $gg \to Q\bar{Q}$ one has

$$ \begin{aligned} \frac{d\sigma}{d\Omega}(gg \to Q\bar{Q}) = & \frac{\alpha_s^2}{32s} \sqrt{1 - \frac{4m_Q^2}{s}} \left[ \frac{6}{s^2}(m_Q^2 - t)(m_Q^2 - u) - \right. \\ & \left. \frac{m_Q^2(s - 4m_Q^2)}{3(m_Q^2 - t)(m_Q^2 - u)} + \right. \\ & \left. \frac{4}{3}\frac{(m_Q^2 - t)(m_Q^2 - u) - 2m_Q^2(m_Q^2 + t)}{(m_Q^2 - t)^2} + \right. \\ & \left. \frac{4}{3}\frac{(m_Q^2 - t)(m_Q^2 - u) - 2m_Q^2(m_Q^2 + u)}{(m_Q^2 - u)^2} - 3\frac{(m_Q^2 - t)(m_Q^2 - u) + m_Q^2(u - t)}{s(m_Q^2 - t)} \right. \\ & \left. - 3\frac{(m_Q^2 - t)(m_Q^2 - u) + m_Q^2(t - u)}{s(m_Q^2 - u)} \right]. \end{aligned} \quad (51.21) $$

## 5.1.4 Production of Weak Gauge Bosons

### 5.1.4.1 W and Z resonant production

Resonant production of a single W or Z is governed by the partial widths

$$ \Gamma(W \to l_i \bar{l}_i) = \frac{\sqrt{2} G_F m_W^3}{12\pi} \quad (51.22) $$

$$ \Gamma(W \to q_i \bar{q}_j) = 3 \frac{\sqrt{2} G_F |V_{ij}|^2 m_W^3}{12\pi} \quad (51.23) $$

$$ \Gamma(Z \to \bar{f}f) = N_c \frac{\sqrt{2} G_F m_Z^3}{6\pi} \times [(T_3 - Q_f \sin^2 \theta_W)^2 + (Q_f \sin^2 \theta_W)^2] (51.24) $$

The weak mixing angle is $\theta_W$. The CKM matrix elements are indicated by $V_{ij}$ and $N_c$ is 3 for $q\bar{q}$ final states and 1 for leptonic final states.

The full differential cross section for $f_i \bar{f}_j \to (W, Z) \to f_i' \bar{f}_j'$ is given by

$$ \begin{aligned} \frac{d\sigma}{d\Omega} = & \frac{N_c^f}{N_c^i} \cdot \frac{1}{256\pi^2 s} \cdot \frac{s^2}{(s - M^2)^2 + s\Gamma^2} \\ & \times \left[ (L^2 + R^2)(L'^2 + R'^2)(1 + \cos^2 \theta) \right. \\ & \left. + (L^2 - R^2)(L'^2 - R'^2)2 \cos\theta \right] \end{aligned} \quad (51.25) $$

where M is the mass of the W or Z. The couplings for the W are $L = (8G_F m_W^2/\sqrt{2})^{1/2}V_{ij}/\sqrt{2}$; $R = 0$ where $V_{ij}$ is the corresponding CKM matrix element, with an analogous expression for $L'$ and $R'$. For Z, the couplings are $L = (8G_F m_Z^2/\sqrt{2})^{1/2}(T_3 - \sin^2 \theta_W Q)$; $R = -(8G_F m_Z^2/\sqrt{2})^{1/2} \sin^2 \theta_W Q$, where $T_3$ is the weak isospin of the initial left-handed fermion and Q is the initial fermion's electric charge. The expressions for $L'$ and $R'$ are analogous. The color factors $N_c^{i,f}$ are 3 for initial or final quarks and 1 for initial or final leptons.

### 5.1.4.2 Production of pairs of weak gauge bosons

The cross section for $f\bar{f} \to W^+W^-$ is given in terms of the couplings of the left-handed and right-handed fermion $f$, $\ell = 2(T_3 - Q_W)$, $r = -2Q_W$, where $T_3$ is the third component of weak isospin for the left-handed $f$, Q is its electric charge (in units of the proton charge), and $x_W = \sin^2 \theta_W$:

$$ \frac{d\sigma}{dt} = \frac{2\pi\alpha^2}{N_c s^2} \left\{ \left[ \left( Q + \frac{\ell+r}{4x_W} \frac{s}{s-m_Z^2} \right)^2 + \left( \frac{\ell-r}{4x_W} \frac{s}{s-m_Z^2} \right)^2 \right] A(s, t, u) \right. \\ \left. + \frac{1}{2x_W} \left( Q + \frac{\ell}{2x_W} \frac{s}{s-m_Z^2} \right) (\Theta(-Q)I(s, t, u) - \Theta(Q)I(s, u, t)) \right. \\ \left. + \frac{1}{8x_W^2} (\Theta(-Q)E(s, t, u) + \Theta(Q)E(s, u, t)) \right\}, (51.26) $$

where $\Theta(x)$ is 1 for $x > 0$ and 0 for $x < 0$, and where

$$ \begin{aligned} A(s, t, u) &= \left( \frac{tu}{m_W^4} - 1 \right) \left( \frac{1}{4} - \frac{m_W^2}{s} + 3 \frac{m_W^4}{s^2} \right) + \frac{s}{m_W^2} - 4, \\ I(s, t, u) &= \left( \frac{tu}{m_W^4} - 1 \right) \left( \frac{1}{4} - \frac{m_W^2}{2s} - \frac{m_W^4}{st} \right) + \frac{s}{m_W^2} - 2 + 2 \frac{m_W^2}{t}, \\ E(s, t, u) &= \left( \frac{tu}{m_W^4} - 1 \right) \left( \frac{1}{4} + \frac{m_W^4}{t^2} \right) + \frac{s}{m_W^2}, \end{aligned} \quad (51.27) $$

and s, t, u are the usual Mandelstam variables with $s = (p_f + p_f)^2$, $t = (p_f - p_W)^2$, $u = (p_f - p_W)^2$. The factor $N_c$ is 3 for quarks and 1 for leptons.

The analogous cross-section for $q_i \bar{q}_j \to W^\pm Z^0$ is

$$ \begin{aligned} \frac{d\sigma}{dt} = & \frac{\pi\alpha^2 |V_{ij}|^2}{6s^2 x_W^2} \left\{ \left( \frac{1}{s - m_W^2} \right)^2 \left[ \left( \frac{9 - 8x_W}{4} \right) (ut - m_W^2 m_Z^2) \right. \right. \\ & \left. \left. + (8x_W - 6) s (m_W^2 + m_Z^2) \right] \right. \\ & \left. + \left[ \frac{ut - m_W^2 m_Z^2 - s(m_W^2 + m_Z^2)}{s - m_W^2} \right] \left[ \frac{\ell_j}{t} - \frac{\ell_i}{u} \right] \right. \\ & \left. + \frac{ut - m_W^2 m_Z^2}{4(1 - x_W)} \left[ \frac{\ell_j^2}{t^2} + \frac{\ell_i^2}{u^2} \right] + \frac{s(m_W^2 + m_Z^2)}{2(1 - x_W)} \frac{\ell_i \ell_j}{tu} \right\}, \end{aligned} \quad (51.28) $$

where $\ell_i$ and $\ell_j$ are the couplings of the left-handed $q_i$ and $q_j$ as defined above. The CKM matrix element between $q_i$ and $q_j$ is $V_{ij}$.

The cross section for $q_i \bar{q}_i \to Z^0 Z^0$ is

$$ \frac{d\sigma}{dt} = \frac{\pi\alpha^2}{96} \frac{\ell_i^4 + r_i^4}{x_W^2 (1 - x_W^2)^2 s^2} \left[ \frac{t}{u} + \frac{u}{t} + \frac{4m_Z^2 s}{tu} - m_Z^4 \left( \frac{1}{t^2} + \frac{1}{u^2} \right) \right]. \quad (51.29) $$

## 5.1.5 Production of Higgs Bosons

### 5.1.5.1 Resonant Production

The Higgs boson of the Standard Model can be produced resonantly in the collisions of quarks, leptons, W or Z bosons, gluons, or photons. The production cross section is thus controlled by the partial width of the Higgs boson into the entrance channel and its total width. The branching fractions for the Standard Model Higgs boson are shown in Fig. 1 of the “Searches for Higgs bosons” review in the Particle Listings section, as a function of the Higgs boson mass. The partial widths are given by the relations
$$
\Gamma(H \to f\bar{f}) = \frac{G_F m_f^2 m_H N_c}{4\pi\sqrt{2}} \left(1 - 4m_f^2/m_H^2\right)^{3/2}, \quad (51.30)
$$

$$
\Gamma(H \to W^+W^-) = \frac{G_F m_H^3 \beta_W}{32\pi\sqrt{2}} (4 - 4a_W + 3a_W^2), \quad (51.31)
$$

$$
\Gamma(H \to ZZ) = \frac{G_F m_H^3 \beta_Z}{64\pi\sqrt{2}} (4 - 4a_Z + 3a_Z^2), \quad (51.32)
$$

where $N_c$ is 3 for quarks and 1 for leptons and where $a_W = 1 - \beta_W^2 = 4m_W^2/m_H^2$ and $a_Z = 1 - \beta_Z^2 = 4m_Z^2/m_H^2$. The decay to two gluons proceeds through quark loops, with the $t$ quark dominating [2]. Explicitly,

$$
\Gamma(H \to gg) = \frac{\alpha_s^2 G_F m_H^3}{36\pi^3 \sqrt{2}} \left| \sum_q I(m_q^2/m_H^2) \right|^2, \quad (51.33)
$$

where $I(z)$ is complex for $z < 1/4$. For $z < 2 \times 10^{-3}$, $|I(z)|$ is small so the light quarks contribute negligibly. For $m_H < 2m_t$, $z > 1/4$ and

$$
I(z) = 3 \left[ 2z + 2z(1 - 4z) \left( \sin^{-1} \frac{1}{2\sqrt{z}} \right)^2 \right], \quad (51.34)
$$

which has the limit $I(z) \to 1$ as $z \to \infty$.

51.5.2 Higgs Boson Production in W* and Z* decay

The Standard Model Higgs boson can be produced in the decay
of a virtual W or Z ("Higgstrahlung") [3,4]: In particular, if k is
the c.m. momentum of the Higgs boson,

$$
\sigma(q_i \bar{q}_j \to W H) = \frac{\pi \alpha^2 |V_{ij}|^2}{36 \sin^4 \theta_W} \frac{2k}{\sqrt{s}} \frac{k^2 + 3m_W^2}{(s - m_W^2)^2} \quad (51.35)
$$

$$
\sigma(f\bar{f} \to ZH) = \frac{2\pi\alpha^2(\ell_f^2 + r_f^2)}{48N_c \sin^4 \theta_W \cos^4 \theta_W} \frac{2k}{\sqrt{s}} \frac{k^2 + 3m_Z^2}{(s - m_Z^2)^2}, \quad (51.36)
$$

where $\ell$ and $r$ are defined as above.

**51.5.3 W and Z Fusion**

Just as high-energy electrons can be regarded as sources of virtual
photon beams, at very high energies they are sources of virtual W
and Z beams. For Higgs boson production, it is the longitudinal
components of the Ws and Zs that are important [5]. The distri-
bution of longitudinal Ws carrying a fraction y of the electron's
energy is [6]

$$
f(y) = \frac{g^2}{16\pi^2} \frac{1-y}{y}, \qquad (51.37)
$$

where $g = e/\sin\theta_W$. In the limit $s \gg m_H \gg m_W$, the partial decay rate is $\Gamma(H \to W_L W_L) = (g^2/64\pi)(m_H^3/m_W^2)$ and in the equivalent W approximation [7]

$$
\sigma(e^+ e^- \to \bar{\nu}_e \nu_e H) = \frac{1}{16m_W^2} \left( \frac{\alpha}{\sin^2 \theta_W} \right)^3 \\
\times \left[ \left( 1 + \frac{m_H^2}{s} \right) \log \frac{s}{m_H^2} - 2 + 2 \frac{m_H^2}{s} \right] . (51.38)
$$

There are significant corrections to this relation when $m_H$ is not
large compared to $m_W$ [8]. For $m_H = 150$ GeV, the estimate is
too high by 51% for $\sqrt{s} = 1000$ GeV, 32% too high at $\sqrt{s} = 2000$
GeV, and 22% too high at $\sqrt{s} = 4000$ GeV. Fusion of ZZ to make a
Higgs boson can be treated similarly. Identical formulae apply for
Higgs production in the collisions of quarks whose charges permit
the emission of a $W^+$ and a $W^-$, except that QCD corrections and
CKM matrix elements are required. Even in the absence of QCD

corrections, the fine-structure constant ought to be evaluated at
the scale of the collision, say $m_W$. All quarks contribute to the
ZZ fusion process.

**51.6 Inclusive hadronic reactions**

One-particle inclusive cross sections $Ed^3\sigma/d^3p$ for the produc-
tion of a particle of momentum $p$ are conveniently expressed in
terms of rapidity $y$ (see above) and the momentum $p_T$ transverse
to the beam direction (in the c.m.):

$$
E \frac{d^3 \sigma}{d^3 p} = \frac{d^3 \sigma}{d \phi d y p_T d p_T}. \qquad (51.39)
$$

In appropriate circumstances, the cross section may be decom-
posed as a partonic cross section multiplied by the probabilities
of finding partons of the prescribed momenta:

$$
\sigma_{\text{hadronic}} = \sum_{ij} \int dx_1 dx_2 f_i(x_1) f_j(x_2) d\hat{\sigma}_{\text{partonic}}, \quad (51.40)
$$

The probability that a parton of type *i* carries a fraction of the incident particle’s that lies between $x_1$ and $x_1 + dx_1$ is $f_i(x_1)dx_1$ and similarly for partons in the other incident particle. The partonic collision is specified by its c.m. energy squared $\hat{s} = x_1x_2s$ and the momentum transfer squared $\hat{t}$. The final hadronic state is more conveniently specified by the rapidities $y_1, y_2$ of the two jets resulting from the collision and the transverse momentum $p_T$. The connection between the differentials is

$$
dx_1 dx_2 d\hat{t} = dy_1 dy_2 \frac{\hat{s}}{s} dp_T^2, \quad (51.41)
$$

so that

$$
\frac{d^3 \sigma}{dy_1 dy_2 dp_T^2} = \frac{\hat{s}}{s} \left[ f_i(x_1) f_j(x_2) \frac{d\hat{\sigma}}{d\hat{t}}(\hat{s}, \hat{t}, \hat{u}) + f_i(x_2) f_j(x_1) \frac{d\hat{\sigma}}{d\hat{t}}(\hat{s}, \hat{u}, \hat{t}) \right], \quad (51.42)
$$

where we have taken into account the possibility that the incident
parton types might arise from either incident particle. The second
term should be dropped if the types are identical: *i* = *j*.

**51.7 Two-photon processes**

In the Weizsäcker-Williams picture, a high-energy electron beam
is accompanied by a spectrum of virtual photons of energies $\omega$ and
invariant-mass squared $q^2 = -Q^2$, for which the photon number
density is

$$
dn = \frac{\alpha}{\pi} \left[ 1 - \frac{\omega}{E} + \frac{\omega^2}{E^2} - \frac{m_e^2 \omega^2}{Q^2 E^2} \right] \frac{d\omega}{\omega} \frac{dQ^2}{Q^2}, \quad (51.43)
$$

where $E$ is the energy of the electron beam. The cross section for $e^+e^- \to e^+e^-X$ is then [9]

$$
d\sigma_{e+e-\to e+e-X}(s) = dn_1 dn_2 d\sigma_{\gamma\gamma\to X}(W^2), \quad (51.44)
$$

where $W^2 = m_X^2$. Integrating from the lower limit $Q^2 =$
$m_e^2 \frac{\omega_i^2}{E_i(E_i - \omega_i)}$ to a maximum $Q^2$ gives

$$
\sigma_{e+e^- \to e^+e^- X}(s) = \frac{\alpha^2}{\pi^2} \int_{z_{th}}^1 \frac{dz}{z} \\
\times \left[ \left( \ln \frac{Q_{max}^2}{z m_e^2} - 1 \right)^2 f(z) + \frac{1}{3} (\ln z)^3 \right] \sigma_{\gamma\gamma \to X}(zs), (51.45)
$$

where

$$
f(z) = \left(1 + \frac{1}{2}z\right)^2 \ln(1/z) - \frac{1}{2}(1 - z)(3 + z). \quad (51.46)
$$

The appropriate value of $Q_{max}^2$ depends on the properties of
the produced system X. For production of hadronic systems,
$Q_{max}^2 \approx m_\rho^2$, while for lepton-pair production, $Q^2 \approx W^2$. For production of a resonance with spin $J \neq 1$, we have

$$ \sigma_{e^+e^- \to e^+e^-R}(s) = (2J+1) \frac{8\alpha^2 \Gamma_R \to \gamma\gamma}{m_R^3} \times \left[ f(m_R^2/s) \left( \ln \frac{m_V^2 s}{m_e^2 m_R^2} - 1 \right)^2 - \frac{1}{3} \left( \ln \frac{s}{M_R^2} \right)^3 \right], \quad (51.47) $$

where $m_V$ is the mass that enters into the form factor for the $\gamma\gamma \to R$ transition, typically $m_\rho$.

# PART II: PROCESSES BEYOND THE STANDARD MODEL

## 51.8 Production of supersymmetric particles

In supersymmetric (SUSY) theories (see Supersymmetric Particle Searches in this Review), every boson has a fermionic superpartner, and every fermion has a bosonic superpartner. The minimal supersymmetric Standard Model (MSSM) is a direct supersymmetrization of the Standard Model (SM), although a second Higgs doublet is needed to avoid triangle anomalies [10]. Under soft SUSY breaking, superpartner masses are lifted above the SM particle masses. In weak scale SUSY, the superpartners are invoked to stabilize the weak scale under radiative corrections, so the superpartners are expected to have masses of order the TeV scale.

### 51.8.1 Gluino and squark production

The superpartners of gluons are the color octet, spin $-\frac{1}{2}$ gluinos ($\tilde{g}$), while each helicity component of quark flavor has a spin-0 squark partner, e.g. $\tilde{q}_L$ and $\tilde{q}_R$. Third generation left- and right- squarks are expected to have large mixing, resulting in mass eigenstates $\tilde{q}_1$ and $\tilde{q}_2$, with $m_{\tilde{q}_1} < m_{\tilde{q}_2}$ (here, $q$ denotes any of the SM flavors of quarks and $\tilde{q}_i$ the corresponding flavor and type ($i = L, R$ or 1, 2) of squark). Gluino pair production ($\tilde{g}\tilde{g}$) takes place via either glue-glue or quark-antiquark annihilation [11].

The subprocess cross sections are usually presented as differential distributions in the Mandelstam variables $s$, $t$ and $u$. Note that for a $2 \to 2$ scattering subprocess $ab \to cd$, the Mandelstam variable $s = (p_a+p_b)^2 = (p_c+p_d)^2$, where $p_a$ is the 4-momentum of particle $a$, and so forth. The variable $t = (p_c - p_a)^2$, where $c$ and $a$ are taken conventionally to be the most similar particles in the subprocess. The variable $u$ would then be equal to $(p_d - p_a)^2$. Note that since $s$, $t$ and $u$ are squares of 4-vectors, they are invariants in any inertial reference frame.

Gluino pair production at hadron colliders is described by:

$$ \begin{aligned} \frac{d\sigma}{dt}(gg \to \tilde{g}\tilde{g}) = & \frac{9\pi\alpha_s^2}{4s^2} \left\{ \frac{2(m_{\tilde{g}}^2 - t)(m_{\tilde{g}}^2 - u)}{s^2} \right. \\ & \left. + \frac{(m_{\tilde{g}}^2 - t)(m_{\tilde{g}}^2 - u) - 2m_{\tilde{g}}^2(m_{\tilde{g}}^2 + t)}{(m_{\tilde{g}}^2 - t)^2} \right. \\ & \left. + \frac{(m_{\tilde{g}}^2 - t)(m_{\tilde{g}}^2 - u) - 2m_{\tilde{g}}^2(m_{\tilde{g}}^2 + u)}{(m_{\tilde{g}}^2 - u)^2} + \frac{m_{\tilde{g}}^2(s - 4m_{\tilde{g}}^2)}{(m_{\tilde{g}}^2 - t)(m_{\tilde{g}}^2 - u)} \right. \\ & \left. - \frac{(m_{\tilde{g}}^2 - t)(m_{\tilde{g}}^2 - u) + m_{\tilde{g}}^2(u - t)}{s(m_{\tilde{g}}^2 - t)} \right. \\ & \left. - \frac{(m_{\tilde{g}}^2 - t)(m_{\tilde{g}}^2 - u) + m_{\tilde{g}}^2(t - u)}{s(m_{\tilde{g}}^2 - u)} \right\}, \quad (51.48) \end{aligned} $$

where $\alpha_s$ is the strong fine structure constant. Also,

$$ \frac{d\sigma}{dt}(q\bar{q} \to \tilde{g}\tilde{g}) = \frac{8\pi\alpha_s^2}{9s^2} \left\{ \frac{4}{3} \left( \frac{m_{\tilde{g}}^2 - t}{m_{\tilde{q}}^2 - t} \right)^2 + \frac{4}{3} \left( \frac{m_{\tilde{g}}^2 - u}{m_{\tilde{q}}^2 - u} \right)^2 \right. \\ + \frac{3}{s^2} \left[ (m_{\tilde{g}}^2 - t)^2 + (m_{\tilde{g}}^2 - u)^2 + 2m_{\tilde{g}}^2 s \right] \\ - 3 \frac{[(m_{\tilde{g}}^2 - t)^2 + m_{\tilde{g}}^2 s]}{s(m_{\tilde{q}}^2 - t)} \\ - 3 \frac{[(m_{\tilde{g}}^2 - u)^2 + m_{\tilde{g}}^2 s]}{s(m_{\tilde{q}}^2 - u)} + \frac{1}{3} \frac{m_{\tilde{g}}^2 s}{(m_{\tilde{q}}^2 - t)(m_{\tilde{q}}^2 - u)} \left. \right\}. \quad (51.49) $$

Gluinos can also be produced in association with squarks: $\tilde{g}\tilde{q}_i$ production, where $\tilde{q}_i$ represents any of the various types (left, right- or mixed) and flavors of squarks. The subprocess cross section is independent of whether the squark is the right-, left- or mixed type:

$$ \frac{d\sigma}{dt}(gg \to \tilde{g}\tilde{q}_i) = \frac{\pi\alpha_s^2}{24s^2} \frac{\left[ \frac{16}{3}(s^2 + (m_{\tilde{q}_i}^2 - u)^2) + \frac{4}{3}s(m_{\tilde{q}_i}^2 - u) \right]}{s(m_{\tilde{g}}^2 - t)(m_{\tilde{q}_i}^2 - u)^2} \times \left( (m_{\tilde{g}}^2 - u)^2 + (m_{\tilde{q}_i}^2 - m_{\tilde{g}}^2)^2 + \frac{2sm_{\tilde{g}}^2(m_{\tilde{q}_i}^2 - m_{\tilde{g}}^2)}{(m_{\tilde{g}}^2 - t)} \right). \quad (51.50) $$

There are many different subprocesses for production of squark pairs. Since left- and right- squarks generally have different masses and different decay patterns, we present the differential cross section for each subprocess of $\tilde{q}_i$ ($i = L, R$ or 1, 2) separately. (In early literature, the following formulae were often combined into a single equation which didn't differentiate the various squark types.) The result for $gg \to \tilde{q}_i\tilde{q}_i$ is:

$$ \begin{aligned} \frac{d\sigma}{dt}(gg \to \tilde{q}_i \tilde{\bar{q}}_i) = & \frac{\pi\alpha_s^2}{4s^2} \left\{ \frac{1}{3} \left( \frac{m_{\tilde{q}}^2 + t}{m_{\tilde{q}}^2 - t} \right)^2 + \frac{1}{3} \left( \frac{m_{\tilde{q}}^2 + u}{m_{\tilde{q}}^2 - u} \right)^2 \right. \\ & + \frac{3}{32s^2} \left( 8s(4m_{\tilde{q}}^2 - s) + 4(u - t)^2 \right) + \frac{7}{12} \\ & - \frac{1}{48} \frac{(4m_{\tilde{q}}^2 - s)^2}{(m_{\tilde{q}}^2 - t)(m_{\tilde{q}}^2 - u)} \\ & + \frac{3}{32} \frac{[(t - u)(4m_{\tilde{q}}^2 + 4t - s) - 2(m_{\tilde{q}}^2 - u)(6m_{\tilde{q}}^2 + 2t - s)]}{s(m_{\tilde{q}}^2 - t)} \\ & + \frac{3}{32} \frac{[(u - t)(4m_{\tilde{q}}^2 + 4u - s) - 2(m_{\tilde{q}}^2 - t)(6m_{\tilde{q}}^2 + 2u - s)]}{s(m_{\tilde{q}}^2 - u)} \\ & \left. + \frac{7}{96} \frac{[4m_{\tilde{q}}^2 + 4t - s]}{m_{\tilde{q}}^2 - t} + \frac{7}{96} \frac{[4m_{\tilde{q}}^2 + 4u - s]}{m_{\tilde{q}}^2 - u} \right\}, \end{aligned} \quad (51.51) $$

which has an obvious $u \leftrightarrow t$ symmetry.

For $q\bar{q} \to \tilde{q}_i\tilde{\bar{q}}_i$ with the same initial and final state flavors, we have

$$ \frac{d\sigma}{dt}(q\bar{q} \to \tilde{q}_i \tilde{\bar{q}}_i) = \frac{2\pi\alpha_s^2}{9s^2} \left\{ \frac{1}{(t - m_{\tilde{g}}^2)^2} + \frac{2}{s^2} - \frac{2/3}{s(t - m_{\tilde{g}}^2)} \right\} \times \left[ -st - (t - m_{\tilde{q}_i}^2)^2 \right], \quad (51.52) $$

while if initial and final state flavors are different ($q\bar{q} \to \tilde{q}'_i\tilde{\bar{q}}'_i$) we instead have

$$ \frac{d\sigma}{dt}(q\bar{q} \to \tilde{q}'_i \tilde{\bar{q}}'_i) = \frac{4\pi\alpha_s^2}{9s^4} \left[ -st - (t - m_{\tilde{q}'_i}^2)^2 \right]. \quad (51.53) $$

If the two initial state quarks are of different flavors, then we have
$$
\frac{d\sigma}{dt}(q\bar{q}' \rightarrow \tilde{q}_i \tilde{q}'_i) = \frac{2\pi\alpha_s^2}{9s^2} \frac{-st - (t - m_{\tilde{q}_i}^2)^2}{(t - m_{\tilde{g}}^2)^2}. \quad (51.54)
$$

If the initial quarks are of different flavor and final state squarks
are of different type ($i \neq j$) then

$$
\frac{d\sigma}{dt}(q\bar{q}' \rightarrow \tilde{q}_i \tilde{q}'_j) = \frac{2\pi\alpha_s^2}{9s^2} \frac{m_{\tilde{g}}^2 s}{(t - m_{\tilde{g}}^2)^2}. \quad (51.55)
$$

For same-flavor initial state quarks, but final state unlike-type squarks, we also have

$$
\frac{d\sigma}{dt}(q\bar{q} \rightarrow \tilde{q}_i \tilde{q}_j) = \frac{2\pi\alpha_s^2}{9s^2} \frac{m_{\tilde{g}}^2 s}{(t - m_{\tilde{g}}^2)^2}. \quad (51.56)
$$

There also exist cross sections for quark-quark annihilation to squark pairs. For same flavor quark-quark annihilation to same flavor/same type final state squarks,

$$
\begin{equation}
\begin{split}
\frac{d\sigma}{dt}(qq \rightarrow \tilde{q}_i \tilde{q}_i) ={}& \\
& \frac{\pi\alpha_s^2 m_{\tilde{g}}^2}{9s^2} \left\{ \frac{1}{(t - m_{\tilde{g}}^2)^2} + \frac{1}{(u - m_{\tilde{g}}^2)^2} - \frac{2/3}{(t - m_{\tilde{g}}^2)(u - m_{\tilde{g}}^2)} \right\},
\end{split}
\tag{51.57}
\end{equation}
$$

while if the final type squarks are different ($i \neq j$), we have

$$
\begin{equation}
\frac{d\sigma}{dt}(qq \rightarrow \tilde{q}_i \tilde{q}_j) = \frac{2\pi\alpha_s^2}{9s^2} \times \left\{ \frac{[-st - (t - m_{\tilde{q}_i}^2)(t - m_{\tilde{q}_j}^2)]}{(t - m_{\tilde{g}}^2)} + \frac{[-su - (u - m_{\tilde{q}_i}^2)(u - m_{\tilde{q}_j}^2)]}{(u - m_{\tilde{g}}^2)} \right\}. \tag{51.58}
\end{equation}
$$

If initial/final state flavors are different, but final state squark
types are the same, then

$$
\frac{d\sigma}{dt}(qq' \rightarrow \tilde{q}_i \tilde{q}'_i) = \frac{2\pi\alpha_s^2}{9s^2} \frac{m_{\tilde{g}}^2 s}{(t - m_{\tilde{g}}^2)^2}. \quad (51.59)
$$

If initial quark flavors are different and final squark types are
different, then

$$
\frac{d\sigma}{dt}(qq' \rightarrow \tilde{q}_i \tilde{q}'_j) = \frac{2\pi\alpha_s^2 - st - (t - m_{\tilde{q}_i}^2)(t - m_{\tilde{q}_j}^2)}{9s^2 (t - m_{\tilde{g}}^2)^2}. \quad (51.60)
$$

51.8.2 Gluino and squark associated production

In the MSSM, the charged spin-1/2 winos and higgsinos mix to
make chargino states $\chi_1^\pm$ and $\chi_2^\pm$, with $m_{\chi_1^\pm} < m_{\chi_2^\pm}$. The spin-1/2
neutral bino, wino and higgsino fields mix to give four neutralino
mass eigenstates $\chi_{1,2,3,4}^0$ ordered according to mass. We some-
times denote the charginos and neutralinos collectively as -inos
for notational simplicity

For gluino and squark production in association with charginos
and neutralinos [12], the quark-squark-neutralino couplings<fnref n="1" /> are
defined by the interaction Lagrangian terms

$$
\mathcal{L}_{\tilde{f}f\tilde{\chi}_i^0} = \left[ iA_{\tilde{\chi}_i^0}^f \tilde{f}_L^\dagger \tilde{\chi}_i^0 P_L f + iB_{\tilde{\chi}_i^0}^f \tilde{f}_R^\dagger \tilde{\chi}_i^0 P_R f + \text{h.c.} \right],
$$

, where $A_{\tilde{\chi}_i^0}^f$ and $B_{\tilde{\chi}_i^0}^f$ are coupling constants involving gauge cou-
plings, neutralino mixing elements and in the case of third gen-
eration fermions, Yukawa couplings. Their form depends on the
conventions used for setting up the MSSM Lagrangian, and can

be found in various reviews [14] and textbooks [13, 15]. $P_L$ and $P_R$ are the usual left- and right- spinor projection opera-
tors and $f$ denotes any of the SM fermions $u, d, e, \nu_e, \dots$. The fermion-sfermion- chargino couplings have the form $\mathcal{L} =$
$$ \left[ iA_{\tilde{\chi}_i^-}^d \tilde{u}_L^\dagger \tilde{\chi}_i^- P_L d + iA_{\tilde{\chi}_i^-}^u \tilde{d}_L^\dagger \tilde{\chi}_i^c P_L u + \text{h.c.} \right] $$
for $u$ and $d$ quarks,
where the $A_{\tilde{\chi}_i^-}^d$ and $A_{\tilde{\chi}_i^-}^u$ couplings are again convention-
dependent, and can be found in textbooks. The superscript $c$
denotes "charge conjugate spinor", defined by $\psi^c \equiv C \tilde{\psi}^T$.

The subprocess cross sections for chargino-squark associated production occur via squark exchange and are given by

$$
\frac{d\sigma}{dt}(\bar{u}g \rightarrow \tilde{\chi}_i^- \bar{d}_L^-) = \frac{\alpha_s}{24s^2} |A_{\tilde{\chi}_i^-}^u|^2 \psi(m_{\bar{d}_L}, m_{\tilde{\chi}_i^-}, t), \quad (51.61)
$$

$$
\frac{d\sigma}{dt}(dg \rightarrow \tilde{\chi}_i^- \tilde{u}_L^-) = \frac{\alpha_s}{24s^2} |A_{\tilde{\chi}_i^-}^d|^2 \psi(m_{\tilde{u}_L}, m_{\tilde{\chi}_i^-}, t), \quad (51.62)
$$

while neutralino-squark production is given by

$$
\frac{d\sigma}{dt}(q\bar{q} \rightarrow \tilde{\chi}_i^0 \tilde{q}) = \frac{\alpha_s}{24s^2} \left( |A_{\tilde{\chi}_i^0}^q|^2 + |B_{\tilde{\chi}_i^0}^q|^2 \right) \psi(m_{\tilde{q}}, m_{\tilde{\chi}_i^0}, t), \quad (51.63)
$$

where

$$
\psi(m_1, m_2, t) = \frac{s + t - m_1^2}{2s} - \frac{m_1^2(m_2^2 - t)}{(m_1^2 - t)^2} \\
+ \frac{t(m_2^2 - m_1^2) + m_2^2(s - m_2^2 + m_1^2)}{s(m_1^2 - t)}. \quad (51.64)
$$

Here, the variable *t* is given by the square of “squark-minus-
quark” four-momentum. The neutralino-gluino associated pro-
duction cross section also occurs via squark exchange and is given
by

$$
\frac{d\sigma}{dt}(q\bar{q} \rightarrow \tilde{\chi}_i^0 \tilde{g}) = \frac{\alpha_s}{18s^2} \left( |A_{\tilde{\chi}_i^0}^q|^2 + |B_{\tilde{\chi}_i^0}^q|^2 \right) \left[ \frac{(m_{\tilde{\chi}_i^0}^2 - t)(m_{\tilde{g}}^2 - t)}{(m_{\tilde{q}}^2 - t)^2} + \frac{(m_{\tilde{\chi}_i^0}^2 - u)(m_{\tilde{g}}^2 - u)}{(m_{\tilde{q}}^2 - u)^2} - \frac{2\eta_i \eta_{\tilde{g}} m_{\tilde{g}} m_{\tilde{\chi}_i^0} s}{(m_{\tilde{q}}^2 - t)(m_{\tilde{q}}^2 - u)} \right], \quad (51.65)
$$

where $\eta_i$ is the sign of the neutralino mass eigenvalue and $\eta_{\tilde{g}}$ is the
sign of the gluino mass eigenvalue. We also have chargino-gluino
associated production:

$$
\begin{equation}
\begin{split}
\frac{d\sigma}{dt}(\bar{u}d \rightarrow \tilde{\chi}_i^- \tilde{g}) ={}& \frac{\alpha_s}{18s^2} \left[ |A_{\tilde{\chi}_i^-}^u|^2 \frac{(m_{\tilde{\chi}_i^-}^2 - t)(m_{\tilde{g}}^2 - t)}{(m_{\tilde{d}_L}^2 - t)^2} \right. \\
& \left. \qquad + |A_{\tilde{\chi}_i^-}^d|^2 \frac{(m_{\tilde{\chi}_i^-}^2 - u)(m_{\tilde{g}}^2 - u)}{(m_{\tilde{u}_L}^2 - u)^2} + \frac{2\eta_{\tilde{g}} \operatorname{Re}(A_{\tilde{\chi}_i^-}^u A_{\tilde{\chi}_i^-}^d m_{\tilde{g}} m_{\tilde{\chi}_i^-} s)}{(m_{\tilde{d}_L}^2 - t)(m_{\tilde{u}_L}^2 - u)} \right],
\end{split}
\tag{51.66}
\end{equation}
$$

where $\hat{t} = (\tilde{g} - d)^2$ and in the third term one must take the real part of the in general complex coupling constant product.

51.8.3 Slepton and sneutrino production

The subprocess cross section for $\tilde{\ell}_L \tilde{\nu}_{\ell_L}$ production ($\ell = e$ or $\mu$) occurs via s-channel W exchange and is given by

$$
\frac{d\sigma}{dt}(d\bar{u} \rightarrow \tilde{\ell}_L \tilde{\nu}_{\ell_L}) = \frac{g^4 |D_W(s)|^2}{192\pi s^2} \left( t u - m_{\tilde{\ell}_L}^2 m_{\tilde{\nu}_{\ell_L}}^2 \right), \quad (51.67)
$$

where $D_W(s) = 1/(s - M_W^2 + iM_W\Gamma_W)$ is the W-boson propagator denominator. The production of $\tilde{\tau}_1\tilde{\nu}_\tau$ is given as above, but replacing $m_{\tilde{\ell}_L} \rightarrow m_{\tilde{\tau}_1}$, $m_{\tilde{\nu}_{\ell_L}} \rightarrow m_{\tilde{\nu}_\tau}$ and multiplying by an overall factor of $\cos^2 \theta_\tau$ (where $\theta_\tau$ is the tau-slepton mixing angle). Similar substitutions hold for $\tilde{\tau}_2\tilde{\nu}_\tau$ production, except the overall factor is $\sin^2 \theta_\tau$.

<footnote n="1">¹The couplings $A_{\tilde{\chi}_i^0}^f$ and $B_{\tilde{\chi}_i^0}^f$ are given explicitly in Ref. [13] in Eq. (8.87). Also, the couplings $A_{\tilde{\chi}_i^-}^d$ and $A_{\tilde{\chi}_i^-}^u$ are given in Eq. (8.93). The couplings $X_i^j$ and $Y_i^j$ are given by Eq. (8.103), while the $x_i$ and $y_i$ couplings are given in Eq. (8.100). Finally, the couplings $W_{ij}$ are given in Eq. (8.101).</footnote>
<table><caption>**Table 51.1:** The constants $\alpha_f$ and $\beta_f$ that appear in the SM neutral current Lagrangian. Here $t \equiv \tan \theta_W$ and $c \equiv \cot \theta_W$.</caption><thead><tr><th><i>f</i></th><th><i>q</i><sub><i>f</i></sub></th><th><i>&alpha;</i><sub><i>f</i></sub></th><th><i>&beta;</i><sub><i>f</i></sub></th></tr></thead><tbody><tr><td><i>&ell;</i></td><td>-1</td><td><sup>1</sup>&frasl;<sub>4</sub>(3<i>t</i> &minus; <i>c</i>)</td><td><sup>1</sup>&frasl;<sub>4</sub>(<i>t</i> + <i>c</i>)</td></tr><tr><td><i>&nu;</i><sub><i>&ell;</i></sub></td><td>0</td><td><sup>1</sup>&frasl;<sub>4</sub>(<i>t</i> + <i>c</i>)</td><td>&minus;<sup>1</sup>&frasl;<sub>4</sub>(<i>t</i> + <i>c</i>)</td></tr><tr><td><i>u</i></td><td><sup>2</sup>&frasl;<sub>3</sub></td><td>&minus;<sup>5</sup>&frasl;<sub>12</sub><i>t</i> + <sup>1</sup>&frasl;<sub>4</sub><i>c</i></td><td>&minus;<sup>1</sup>&frasl;<sub>4</sub>(<i>t</i> + <i>c</i>)</td></tr><tr><td><i>d</i></td><td>&minus;<sup>1</sup>&frasl;<sub>3</sub></td><td><sup>1</sup>&frasl;<sub>12</sub><i>t</i> &minus; <sup>1</sup>&frasl;<sub>4</sub><i>c</i></td><td><sup>1</sup>&frasl;<sub>4</sub>(<i>t</i> + <i>c</i>)</td></tr></tbody></table>

The subprocess cross section for $\tilde{\ell}_L \tilde{\ell}_L$ production occurs via s-channel $\gamma$ and $Z$ exchange, and depends on the neutral current interaction, with fermion couplings to $\gamma$ and $Z^0$ given by $\mathcal{L}_{\text{neutral}} = -eq_f \tilde{\gamma}^\mu f A_\mu + e \tilde{\gamma}^\mu (\alpha_f + \beta_f \gamma_5) f Z_\mu$ (with values of $q_f$, $\alpha_f$, and $\beta_f$ given in Table-51.1).

The subprocess cross section is given by

$$ \frac{d\sigma}{dt}(q\bar{q} \rightarrow \tilde{\ell}_L \tilde{\ell}_L) = \frac{e^4}{24\pi s^2} \left( t\bar{u} - m_{\tilde{\ell}_L}^4 \right) \times \left\{ \frac{q_{\tilde{\ell}}^2 q_{\bar{q}}^2}{s^2} + (\alpha_{\ell} - \beta_{\tilde{\ell}})^2 (\alpha_{\bar{q}}^2 + \beta_{\bar{q}}^2) |D_Z(s)|^2 + \frac{2q_{\ell}q_{\bar{q}}\alpha_q(\alpha_{\ell} - \beta_{\tilde{\ell}})(s - M_Z^2)}{s} |D_Z(s)|^2 \right\}, \quad (51.68) $$

where $D_Z(s) = 1/(s - M_Z^2 + iM_Z\Gamma_Z)$. The cross section for sneutrino production is given by the same formula, but with $\alpha_\ell$, $\beta_\ell$, $q_\ell$ and $m_{\tilde{\ell}_L}$ replaced by $\alpha_\nu$, $\beta_\nu$, 0 and $m_{\tilde{\nu}_L}$, respectively. The cross section for $\tilde{\tau}_1 \tilde{\tau}_1$ production is obtained by replacing $m_{\tilde{\ell}_L} \rightarrow m_{\tilde{\tau}_1}$ and $\beta_\ell \rightarrow \beta_\ell \cos 2\theta_\tau$. The cross section for $\tilde{\ell}_R \tilde{\ell}_R$ production is given by substituting $\alpha_\ell - \beta_\ell \rightarrow \alpha_\ell + \beta_\ell$ and $m_{\tilde{\ell}_L} \rightarrow m_{\tilde{\ell}_R}$ in the equation above. The cross section for $\tilde{\tau}_2 \tilde{\tau}_2$ production is obtained from the formula for $\tilde{\ell}_R \tilde{\ell}_R$ production by replacing $m_{\tilde{\ell}_R} \rightarrow m_{\tilde{\tau}_2}$ and $\beta_\ell \rightarrow \beta_\ell \cos 2\theta_\tau$.

Finally, the cross section for $\tilde{\tau}_1 \tilde{\tau}_2$ production occurs only via $Z$ exchange, and is given by

$$ \frac{d\sigma}{dt}(q\bar{q} \rightarrow \tilde{\tau}_1 \tilde{\tau}_2) = \frac{d\sigma}{dt}(q\bar{q} \rightarrow \tilde{\tau}_1 \tilde{\tau}_2) = \frac{e^4}{24\pi s^2} (\alpha_q^2 + \beta_q^2) \beta_\ell^2 \sin^2 2\theta_\tau |D_Z(s)|^2 (ut - m_{\tilde{\tau}_1}^2 m_{\tilde{\tau}_2}^2). \quad (51.69) $$

### 51.8.4 Chargino and neutralino pair production

#### 51.8.4.1 $\tilde{\chi}_i^-$ $\tilde{\chi}_j^0$ production

The subprocess cross section for $d\bar{u} \rightarrow \tilde{\chi}_i^- \tilde{\chi}_j^0$ depends on Lagrangian couplings

$$ \begin{align*} \mathcal{L}_{W \bar{u} d} &= -\frac{g}{\sqrt{2}} \bar{u} \gamma_{\mu} P_L d W^{+\mu} + \text{h.c.}, \\ \mathcal{L}_{W \tilde{\chi}_i^- \tilde{\chi}_j^0} &= -g(-i)^{\theta_j} \tilde{\chi}_i^- [X_i^j + Y_i^j \gamma_5] \gamma_{\mu} \tilde{\chi}_j^0 W^{-\mu} + \text{h.c.}, \\ \mathcal{L}_{q \bar{q} \tilde{\chi}_i^-} &= i A_{\tilde{\chi}_i^-}^d \tilde{u}_L^\dagger \tilde{\chi}_i^- P_L d + i A_{\tilde{\chi}_i^-}^u \tilde{d}_L^\dagger \tilde{\chi}_i^- \tilde{c}_L P_L u + \text{h.c.} \end{align*} $$

and

$$ \mathcal{L}_{q\bar{q}\tilde{\chi}_j^0} = iA_{\tilde{\chi}_j^-}^q \tilde{q}_L^\dagger \tilde{\chi}_j^0 P_L q + \text{h.c.} $$

Contributing diagrams include $W$ exchange and also $\tilde{d}_L$ and $\tilde{u}_L$ squark exchange. The $X_i^j$ and $Y_i^j$ couplings are new, and again convention-dependent: the cross section formulae works if the interaction Lagrangian is written in the above form, so that the couplings can be suitably extracted. The term $\theta_j = 0$ (1) if $m_{\tilde{\chi}_j^0} > 0$ ($< 0$); it comes about because the neutralino field must be re-defined by a $-i\gamma_5$ transformation if its mass eigenvalue is negative [13]. The subprocess cross section is given in terms of dot products of four momenta, where particle labels are used to

denote their four-momenta; note that all mass terms in the cross section formulae are positive definite, so that the signs of mass eigenstates have been absorbed into the Lagrangian couplings, as for instance in Ref. [13]. We then have

$$ \frac{d\sigma}{dt}(d\bar{u} \rightarrow \tilde{\chi}_i^- \tilde{\chi}_j^0) = \frac{1}{192\pi s^2} \left[ T_W + T_{\tilde{d}_L} + T_{\tilde{u}_L} + T_W \tilde{d}_L + T_W \tilde{u}_L + T_{\tilde{d}_L \tilde{u}_L} \right] \quad (51.70) $$

where

$$ \begin{aligned} T_W = {}& 8g^4 |D_W(s)|^2 \{ [X_i^{j2} + Y_i^{j2}](\tilde{\chi}_j^0 \cdot d\tilde{\chi}_i^- \cdot \bar{u} + \tilde{\chi}_j^0 \cdot \bar{u}\tilde{\chi}_i^- \cdot d) \\ & + 2(X_i^j Y_i^j)(\tilde{\chi}_j^0 \cdot d\tilde{\chi}_i^- \cdot \bar{u} - \tilde{\chi}_j^0 \cdot \bar{u}\tilde{\chi}_i^- \cdot d) \\ & + [X_i^{j2} - Y_i^{j2}]m_{\tilde{\chi}_i^-} m_{\tilde{\chi}_j^0} d \cdot \bar{u} \} \end{aligned} \quad (51.71) $$

$$ T_{\tilde{d}_L} = \frac{4|A_{\tilde{\chi}_i^-}^u|^2 |A_{\tilde{\chi}_j^0}^d|^2}{[(\tilde{\chi}_i^- - \bar{u})^2 - m_{\tilde{d}_L}^2]^2} \cdot d \cdot \tilde{\chi}_j^0 \tilde{\chi}_i^- \cdot \bar{u}, \quad (51.72) $$

$$ T_{\tilde{u}_L} = \frac{4|A_{\tilde{\chi}_i^-}^d|^2 |A_{\tilde{\chi}_j^0}^u|^2}{[(\tilde{\chi}_j^0 - \bar{u})^2 - m_{\tilde{u}_L}^2]^2} \cdot \bar{u} \cdot \tilde{\chi}_j^0 \tilde{\chi}_i^- \cdot d \quad (51.73) $$

$$ T_{W \tilde{d}_L} = \frac{-\sqrt{2}g^2 \text{Re}[A_{\tilde{\chi}_j^-}^{d*} A_{\tilde{\chi}_i^-}^u (-i)^{\theta_j} (s - M_W^2)] |D_W(s)|^2}{(\tilde{\chi}_i^- - \bar{u})^2 - m_{\tilde{d}_L}^2} \times \left\{ 8(X_i^j + Y_i^j) \tilde{\chi}_j^0 \cdot d\bar{u} \cdot \tilde{\chi}_i^- + 4(X_i^j - Y_i^j) m_{\tilde{\chi}_i^-} m_{\tilde{\chi}_j^0} d \cdot \bar{u} \right\} \quad (51.74) $$

$$ T_{W \tilde{u}_L} = \frac{\sqrt{2}g^2 \text{Re}[A_{\tilde{\chi}_i^-}^{d*} A_{\tilde{\chi}_j^0}^u (-i)^{\theta_j} (s - M_W^2)] |D_W(s)|^2}{(\tilde{\chi}_j^0 - \bar{u})^2 - m_{\tilde{u}_L}^2} \times \left\{ 8(X_i^j - Y_i^j) \tilde{\chi}_j^0 \cdot \bar{u}d \cdot \tilde{\chi}_i^- + 4(X_i^j + Y_i^j) m_{\tilde{\chi}_i^-} m_{\tilde{\chi}_j^0} d \cdot \bar{u} \right\} \quad (51.75) $$

and

$$ T_{\tilde{d}_L \tilde{u}_L} = - \frac{4 \text{Re}[A_{\tilde{\chi}_j^0}^d A_{\tilde{\chi}_i^-}^{u*} A_{\tilde{\chi}_i^-}^{d*} A_{\tilde{\chi}_j^0}^u] m_{\tilde{\chi}_i^-} m_{\tilde{\chi}_j^0} d \cdot \bar{u}}{[(\tilde{\chi}_i^- - \bar{u})^2 - m_{\tilde{d}_L}^2] [(\tilde{\chi}_j^0 - \bar{u})^2 - m_{\tilde{u}_L}^2]} \quad (51.76) $$

#### 51.8.4.2 Chargino pair production

The subprocess cross section for $d\bar{d} \rightarrow \tilde{\chi}_i^- \tilde{\chi}_i^+$ ($i=1,2$) depends on Lagrangian couplings $\mathcal{L} = e\tilde{\chi}_i^- \gamma_\mu \tilde{\chi}_i^- A^\mu - e \cot \theta_W \tilde{\chi}_i^- \gamma_\mu (x_i - y_i \gamma_5) \tilde{\chi}_i^- Z^\mu$ and also $\mathcal{L} \ni iA_{\tilde{\chi}_i^-}^d \tilde{u}_L^\dagger \tilde{\chi}_i^- P_L d + iA_{\tilde{\chi}_i^-}^u \tilde{d}_L^\dagger \tilde{\chi}_i^- \tilde{c}_L P_L u + \text{h.c.}$. Contributing diagrams include s-channel $\gamma$, $Z^0$ exchange and t-channel $\tilde{u}_L$ exchange [16, 17]. The couplings $x_i$ and $y_i$ are again new and as usual convention-dependent.

The subprocess cross section is given by

$$ \frac{d\sigma}{dt}(d\bar{d} \rightarrow \tilde{\chi}_i^- \tilde{\chi}_i^+) = \frac{1}{192\pi s^2} [T_\gamma + T_Z + T_{\tilde{u}_L} + T_{\gamma Z} + T_{\gamma \tilde{u}_L} + T_Z \tilde{u}_L] \quad (51.77) $$

where

$$ T_{\gamma} = \frac{32e^4 q_d^2}{s^2} \left[ d \cdot \tilde{\chi}_i^+ \bar{d} \cdot \tilde{\chi}_i^- + d \cdot \tilde{\chi}_i^- \bar{d} \cdot \tilde{\chi}_i^+ + m_{\tilde{\chi}_i^-}^2 d \cdot \bar{d} \right] \quad (51.78) $$

$$ T_Z = 32e^4 \cot^2 \theta_W |D_Z(s)|^2 \left\{ (\alpha_d^2 + \beta_d^2)(x_i^2 + y_i^2) \left[ d \cdot \tilde{\chi}_i^+ \bar{d} \cdot \tilde{\chi}_i^- + d \cdot \tilde{\chi}_i^- \bar{d} \cdot \tilde{\chi}_i^+ + m_{\tilde{\chi}_i^-}^2 d \cdot \bar{d} \right] \right\} $$
$$
\mp 4\alpha_d \beta_d x_i y_i \left[ d \cdot \tilde{\chi}_i^+ d \cdot \tilde{\chi}_i^- - d \cdot \tilde{\chi}_i^- d \cdot \tilde{\chi}_i^+ \right] - 2y_i^2 (\alpha_d^2 + \beta_d^2) m_{\tilde{\chi}_i^-}^2 d \cdot d \right\}, \quad (51.79)
$$

$$
T_{\tilde{u}_L} = \frac{4|A_{\tilde{\chi}_i}^d|^4}{[(d - \tilde{\chi}_i^-)^2 - m_{\tilde{u}_L}^2]^2} \quad d \cdot \tilde{\chi}_i^- d \cdot \tilde{\chi}_i^+ \quad (51.80)
$$

$$
T_{\gamma Z} = \frac{64e^4 \cot \theta_W q_d (s - M_Z^2) |D_Z(s)|^2}{s} \times \\
\left\{ \alpha_d x_i \left( d \cdot \tilde{\chi}_i^+ d \cdot \tilde{\chi}_i^- + d \cdot \tilde{\chi}_i^- d \cdot \tilde{\chi}_i^+ + m_{\tilde{\chi}_i^-}^2 d \cdot d \right) \right. \\
\left. \pm \beta_d y_i \left( d \cdot \tilde{\chi}_i^- d \cdot \tilde{\chi}_i^+ - d \cdot \tilde{\chi}_i^+ d \cdot \tilde{\chi}_i^- \right) \right\} \quad (51.81)
$$

$$
T_{\gamma \tilde{u}_L} = \mp \frac{8e^2 q_d}{s} \frac{|A_{\tilde{\chi}_i^-}^d|^2}{[(d - \tilde{\chi}_i^-)^2 - m_{\tilde{u}_L}^2]} \left\{ 2d \cdot \tilde{\chi}_i^+ d \cdot \tilde{\chi}_i^- + m_{\tilde{\chi}_i^-}^2 d \cdot d \right\} \quad (51.82)
$$

and

$$
T_{Z \tilde{u}_L} = \mp 8e^2 \cot \theta_W |D_Z(s)|^2 \frac{|A_{\tilde{\chi}_i^-}^d|^2 (s - M_Z^2)}{[(d - \tilde{\chi}_i^-)^2 - m_{\tilde{u}_L}^2]} (\alpha_d - \beta_d) \\
\times \left\{ 2(x_i \mp y_i) d \cdot \tilde{\chi}_i^- d \cdot \tilde{\chi}_i^+ + m_{\tilde{\chi}_i^-}^2 (x_i \pm y_i) d \cdot d \right\} \quad (51.83)
$$

using the upper of the sign choices.

The cross section for $u\bar{u} \to \tilde{\chi}_i^+\tilde{\chi}_i^-$ can be obtained from the above by replacing $\alpha_d \to \alpha_u$, $\beta_d \to \beta_u$, $q_d \to q_u$, $\tilde{u}_L \to \tilde{d}_L$, $A_{\tilde{\chi}_i^-}^d \to A_{\tilde{\chi}_i^-}^u$, $d \to \tilde{u}$, $\bar{d} \to u$ and adopting the lower of the sign choices everywhere.

The cross section for $q\bar{q} \to \tilde{\chi}_1^- \tilde{\chi}_2^+$, $\tilde{\chi}_1^+ \tilde{\chi}_2^-$ can occur via Z and $\tilde{q}_L$ exchange. It is usually much smaller than $\tilde{\chi}_{1,2}^- \tilde{\chi}_{1,2}^+$ production, so the cross section will not be presented here. It can be found in Appendix A of Ref. [13].

51.8.4.3 Neutralino pair production

Neutralino pair production via $q\bar{q}$ fusion takes place via s-
channel Z exchange plus t- and u-channel left- and right- squark
exchange (5 diagrams) [17, 18]. The Lagrangian couplings (see
previous footnote*) needed include terms given above plus terms
of the form $\mathcal{L} = W_{ij}\bar{\chi}_i^0\gamma_\mu(\gamma_5)\theta_i^\mu+\theta_j^\nu+1\tilde{\chi}_j^0Z^\mu$. The couplings $W_{ij}$
depend only on the higgsino components of the neutralinos i and
j. The subprocess cross section is given by:

$$
\frac{d\sigma}{dt}(q\bar{q} \rightarrow \tilde{\chi}_i^0 \tilde{\chi}_j^0) = \frac{1}{192\pi s^2} [T_Z + T_{\tilde{q}_L} + T_{\tilde{q}_R} + T_{Z\tilde{q}_L} + T_{Z\tilde{q}_R}] \quad (51.84)
$$

where

$$
T_Z = 128e^2 |W_{ij}|^2 (\alpha_q^2 + \beta_q^2) |D_Z(s)|^2 \\
\left[ q \cdot \tilde{\chi}_i^0 \bar{q} \cdot \tilde{\chi}_j^0 + q \cdot \tilde{\chi}_j^0 \bar{q} \cdot \tilde{\chi}_i^0 - \eta_i \eta_j m_{\tilde{\chi}_i^0} m_{\tilde{\chi}_j^0} q \cdot \bar{q} \right], \quad (51.85)
$$

$$
T_{\tilde{q}_L} = 4|A_{\tilde{\chi}_i^0}^q|^2 |A_{\tilde{\chi}_j^0}^q|^2 \left\{ \frac{q \cdot \tilde{\chi}_i^0 \bar{q} \cdot \tilde{\chi}_j^0}{[(\tilde{\chi}_i^0 - q)^2 - m_{\tilde{q}_L}^2]^2} + \frac{q \cdot \tilde{\chi}_j^0 \bar{q} \cdot \tilde{\chi}_i^0}{[(\tilde{\chi}_j^0 - q)^2 - m_{\tilde{q}_L}^2]^2} \right. \\
\left. - \eta_i \eta_j \frac{m_{\tilde{\chi}_i^0} m_{\tilde{\chi}_j^0} q \cdot \bar{q}}{[(\tilde{\chi}_i^0 - q)^2 - m_{\tilde{q}_L}^2][(\tilde{\chi}_j^0 - q)^2 - m_{\tilde{q}_L}^2]} \right\} \quad (51.86)
$$

$$
T_{\tilde{q}_R} = 4|B_{\tilde{\chi}_i^0}^q|^2 |B_{\tilde{\chi}_j^0}^q|^2 \left\{ \frac{q \cdot \tilde{\chi}_i^0 \bar{q} \cdot \tilde{\chi}_j^0}{[(\tilde{\chi}_i^0 - q)^2 - m_{\tilde{q}_R}^2]^2} + \frac{q \cdot \tilde{\chi}_j^0 \bar{q} \cdot \tilde{\chi}_i^0}{[(\tilde{\chi}_j^0 - q)^2 - m_{\tilde{q}_R}^2]^2} \right. \\
\left. - \eta_i \eta_j \frac{m_{\tilde{\chi}_i^0} m_{\tilde{\chi}_j^0} q \cdot \bar{q}}{[(\tilde{\chi}_i^0 - q)^2 - m_{\tilde{q}_R}^2][(\tilde{\chi}_j^0 - q)^2 - m_{\tilde{q}_R}^2]} \right\} \quad (51.87)
$$

$$
T_{Z\bar{q}_L} = 16e(\alpha_q - \beta_q)(s - M_Z^2)|D_Z(s)|^2
\left\{
\begin{aligned}
& \frac{\mathrm{Re}(W_{ij} A_{\tilde{\chi}_i^0}^{q*} A_{\tilde{\chi}_j^0}^q)}{[(\tilde{\chi}_i^0 - q)^2 - m_{\tilde{q}_L}^2]} \left[ 2q \cdot \tilde{\chi}_i^0 \bar{q} \cdot \tilde{\chi}_j^0 - \eta_i \eta_j m_{\tilde{\chi}_i^0} m_{\tilde{\chi}_j^0} q \cdot \bar{q} \right] \\
& + \eta_i \eta_j \frac{\mathrm{Re}(W_{ij} A_{\tilde{\chi}_i^0}^q A_{\tilde{\chi}_j^0}^{q*})}{[(\tilde{\chi}_j^0 - q)^2 - m_{\tilde{q}_L}^2]} \left[ 2q \cdot \tilde{\chi}_j^0 \bar{q} \cdot \tilde{\chi}_i^0 - \eta_i \eta_j m_{\tilde{\chi}_i^0} m_{\tilde{\chi}_j^0} q \cdot \bar{q} \right]
\end{aligned}
\right\} \quad (51.88)
$$

$$
T_{Z\bar{q}_R} = 16e(\alpha_q + \beta_q)(s - M_Z^2)|D_Z(s)|^2
\left\{
\begin{aligned}
& \frac{\mathrm{Re}(W_{ij} B_{\tilde{\chi}_i^0}^{q*} B_{\tilde{\chi}_j^0}^q)}{[(\tilde{\chi}_i^0 - q)^2 - m_{\tilde{q}_R}^2]} \left[ 2q \cdot \tilde{\chi}_i^0 \bar{q} \cdot \tilde{\chi}_j^0 - \eta_i \eta_j m_{\tilde{\chi}_i^0} m_{\tilde{\chi}_j^0} q \cdot \bar{q} \right] \\
& - \frac{\mathrm{Re}(W_{ij} B_{\tilde{\chi}_i^0}^q B_{\tilde{\chi}_j^0}^{q*})}{[(\tilde{\chi}_j^0 - q)^2 - m_{\tilde{q}_R}^2]} \left[ 2q \cdot \tilde{\chi}_j^0 \bar{q} \cdot \tilde{\chi}_i^0 - \eta_i \eta_j m_{\tilde{\chi}_i^0} m_{\tilde{\chi}_j^0} q \cdot \bar{q} \right]
\end{aligned}
\right\} . \quad (51.89)
$$

As before, ηᵢ = ±1 corresponding to whether the neutralino mass eigenvalue is positive or negative. When i = j in the above formula, one must remember to integrate over just 2π steradians of solid angle to avoid double counting in the total cross section.

51.9 Universal extra dimensions

In the Universal Extra Dimension (UED) model of Ref. [19] (see Ref. [20] for a review of models with extra spacetime dimensions), the Standard Model is embedded in a five dimensional theory, where the fifth dimension is compactified on an $S_1/Z_2$ orbifold. Each SM chirality state is then the zero mode of an infinite tower of Kaluza-Klein excitations labeled by $n = 0 - \infty$. A KK parity is usually assumed to hold, where each state is assigned KK-parity $P = (-1)^n$. If the compactification scale is around a TeV, then the $n = 1$ (or even higher) KK modes may be accessible to collider searches.

Of interest for hadron colliders are the production of massive
$n \ge 1$ quark or gluon pairs. These production cross sections
have been calculated in Ref. [21,22]. We list here results for the
$n = 1$ case only with $M_1 = 1/R$ ($R$ is the compactification radius)
and $s, t$ and $u$ are the usual Mandelstam variables; more general
formulae can be found in Ref. [22]. The superscript $*$ stands for
any KK excited state, while $\bullet$ stands for left chirality states and
$\circ$ stands for right chirality states.

$$
\frac{d\sigma}{dt} = \frac{1}{16\pi s^2} T \qquad (51.90)
$$

where

$$
T(q\bar{q} \rightarrow g^* g^*) = \frac{2g_s^4}{27} \left[ M_1^2 \left( -\frac{4s^3}{t'^2 u'^2} + \frac{57s}{t'u'} - \frac{108}{s} \right) \right. \\
\left. \qquad + \frac{20s^2}{t'u'} - 93 + \frac{108t'u'}{s^2} \right] \quad (51.91)
$$

and

$$
T(gg \rightarrow g^* g^*) = \frac{9g_s^4}{27} \left[ 3M_1^4 \frac{s^2 + t'^2 + u'^2}{t'^2 u'^2} - 3M_1^2 \frac{s^2 + t'^2 + u'^2}{s t' u'} + 1 \right. \\
\left. \qquad + \frac{(s^2 + t'^2 + u'^2)^3}{4s^2 t'^2 u'^2} - \frac{t' u'}{s^2} \right] \quad (51.92)
$$

where $t' = t - M_1^2$ and $u' = u - M_1^2$.

Also,

$$
T(q\bar{q} \rightarrow q_1^{*'} \bar{q}_1^{*'}) = \frac{4g_s^4}{9} \left[ \frac{2M_1^2}{s} + \frac{t'^2 + u'^2}{s^2} \right],
$$
$$
T(q\bar{q} \rightarrow q_1^* \bar{q}_1^{*-}) = \frac{g_s^4}{9} \left[ 2M_1^2 \left( \frac{4}{s} + \frac{s}{t'^2} - \frac{1}{t'} \right) \right. \\
\left. \qquad + \frac{2s^2}{t'^2} + \frac{8s}{3t'} + \frac{6t'}{s} + \frac{8t'^2}{s^2} \right],
$$
$$
T(qq \rightarrow q_1^* q_1^*) = \frac{g_s^4}{27} \left[ M_1^2 \left( 6 \frac{t'}{t'^2} + 6 \frac{u'}{t'^2} - \frac{s}{t' u'} \right) \right. \\
\left. \qquad + 2 \left( 3 \frac{t'^2}{u'^2} + 3 \frac{u'^2}{t'^2} + 4 \frac{s^2}{t' u'} - 5 \right) \right],
$$

$$
T(gg \to q_1^* \bar{q}_1^*) = g_s^4 \left[ M_1^4 \frac{-4}{t' u'} \left( \frac{s^2}{6t'u'} - \frac{3}{8} \right) + M_1^2 \frac{4}{s} \left( \frac{s^2}{6t'u'} - \frac{3}{8} \right) + \frac{s^2}{6t'u'} - \frac{17}{24} + \frac{3t'u'}{4s^2} \right],
$$

$$
T(gq \to g^* q_1^*) = -\frac{g_s^4}{3} \left[ \frac{5s^2}{12t'^2} + \frac{s^3}{t'^2 u'} + \frac{11su'}{6t'^2} + \frac{5u'^2}{12t'^2} + \frac{u'^3}{st'^2} \right],
$$

$$
T(q\bar{q}' \to q_1^* \bar{q}_1^{*'}) = \frac{g_s^4}{18} \left[ 4M_1^4 \frac{s}{t'^2} + 5 + 4\frac{s^2}{t'^2} + 8\frac{s}{t'} \right],
$$

$$
T(qq' \to q_1^* q_1^{*'}) = \frac{2g_s^4}{9} \left[ -M_1^2 \frac{s}{t'^2} + \frac{1}{4} + \frac{s^2}{t'^2} \right],
$$

$$
T(qq \to q_1 \cdot q_1^\circ) =
$$

$$
\frac{g_s^4}{9} \left[ M_1^2 \left( \frac{2s^3}{t'^2 u'^2} - \frac{4s}{t' u'} \right) + 2 \frac{s^4}{t'^2 u'^2} - 8 \frac{s^2}{t' u'} + 5 \right],
$$

$$
T(q\bar{q}' \to q_1^* \bar{q}_1^{\prime\circ}) = \frac{g_s^4}{9} \left[ 2M_1^2 \left( \frac{1}{t'} + \frac{u'}{t'^2} \right) + \frac{5}{2} + \frac{4u'}{t'} + \frac{2u'^2}{t'^2} \right],
$$

and

$$
T(qq' \to q_1 \cdot q_1^\circ) = \frac{g_s^4}{9} \left[ -2M_1^2 \left( \frac{1}{t'} + \frac{u'}{t'^2} \right) + \frac{1}{2} + \frac{2u'^2}{t'^2} \right].
$$

51.10 Large extra dimensions

In the ADD theory [23] with large extra dimensions (LED), the SM particles are confined to a 3-brane, while gravity propagates in the bulk. It is assumed that the *n* extra dimensions are compactified on an *n*-dimensional torus of volume (2πr)ⁿ, so that the fundamental 4 + *n* dimensional Planck scale M*<sub>*</sub>* is related to the usual 4-dimensional Planck scale M<sub>Pl</sub> by M<sub>Pl</sub><sup>2</sup> = M*<sub>*</sub>*<sup>n+2</sup>(2πr)ⁿ. If M*<sub>*</sub>* ~ 1 TeV, then the M<sub>W</sub> – M<sub>Pl</sub> hierarchy problem is just due to gravity propagating in the large extra dimensions.

In these theories, the KK-excited graviton states $G_{\mu\nu}^n$ for $n = 1 - \infty$ can be produced at collider experiments. The graviton couplings to matter are suppressed by $1/M_{Pl}$, so that graviton emission cross sections $d\sigma/dt \sim 1/M_{Pl}^2$. However, the mass splittings between the excited graviton states can be tiny, so the graviton eigenstates are usually approximated by a continuum distribution. A summation (integration) over all allowed graviton emissions ends up cancelling the $1/M_{Pl}^2$ factor, so that observable cross section rates can be attained. Some fundamental production formulae for a KK graviton (denoted G) of mass m at hadron colliders include the subprocesses

$$
\frac{d\sigma_m}{dt}(f\bar{f} \to \gamma G) = \frac{\alpha Q_f^2}{16N_f} \frac{1}{sM_{Pl}^2} F_1\left(\frac{t}{s}, \frac{m^2}{s}\right), \quad (51.93)
$$

where $Q_f$ is the charge of fermion f and $N_f$ is the number of
QCD colors of f. Also,

$$
\frac{d\sigma_m}{dt}(q\bar{q} \to gG) = \frac{\alpha_s}{36} \frac{1}{s M_{Pl}^2} F_1\left(\frac{t}{s}, \frac{m^2}{s}\right), \quad (51.94)
$$

$$
\frac{d\sigma_m}{dt}(qg \to qG) = \frac{\alpha_s}{96} \frac{1}{s M_{Pl}^2} F_2\left(\frac{t}{s}, \frac{m^2}{s}\right), \quad (51.95)
$$

$$
\frac{d\sigma_m}{dt}(gg \to gG) = \frac{3\alpha_s}{16} \frac{1}{s M_{Pl}^2} F_3\left(\frac{t}{s}, \frac{m^2}{s}\right), \quad (51.96)
$$

where

$$
F_1(x, y) = \frac{1}{x(y-1-x)} \left[ -4x(1+x)(1+2x+2x^2) + y(1+6x+18x^2+16x^3) - 6y^2x(1+2x) + y^3(1+4x) \right] \quad (51.97)
$$

$$
F_2(x, y) = -(y - 1 - x)F_1\left(\frac{x}{y - 1 - x}, \frac{y}{y - 1 - x}\right) \quad (51.98)
$$

and

$$
F_3(x, y) = \frac{1}{x(y - 1 - x)} \left[ 1 + 2x + 3x^2 + 2x^3 + x^4 - 2y(1 + x^3) + 3y^2(1 + x^2) - 2y^3(1 + x) + y^4 \right]. \quad (51.99)
$$

These formulae must then be multiplied by the graviton density
of states formula $dN = S_{n-1} \frac{M_{Pl}^2}{M_*^{n+2}} m^{n-1} dm$ to gain the cross
section

$$
\frac{d^2 \sigma}{dt dm} = S_{n-1} \frac{M_{Pl}^2}{M_*^{n+2}} m^{n-1} \frac{d\sigma_m}{dt} \quad (51.100)
$$

where $S_n = \frac{(2\pi)^{n/2}}{\Gamma(n/2)}$ is the surface area of an *n*-dimensional sphere of unit radius.

Virtual graviton processes can also be searched for at colliders. For instance, in Ref. [24] the cross section for Drell-Yan production of lepton pairs via gluon fusion was calculated, where it is found that, in the center-of-mass system

$$
\frac{d\sigma}{dz}(gg \to \ell^+\ell^-) = \frac{\lambda^2 s^3}{64\pi M_*^8} (1 - z^2)(1 + z^2) \quad (51.101)
$$

where $z = \cos\theta$ and $\lambda$ is a model-dependent coupling constant
~ 1. Formulae for Drell-Yan production via $q\bar{q}$ fusion can also be
found in Refs. [24,25].

51.11 Warped extra dimensions

In the Randall-Sundrum model [26] of warped extra dimensions, the arena for physics is a 5-d anti-deSitter (AdS<sub>5</sub>) spacetime, for which a non-factorizable metric exists with a metric warp factor $e^{-2\sigma(\phi)}$. It is assumed that two opposite tension 3-branes exist within AdS<sub>5</sub> at the two ends of an $S_1/Z_2$ orbifold parameterized by co-ordinate $\phi$ which runs from 0 – $\pi$. The 4-D solution of the Einstein equations yields $\sigma(\phi) = kr_c|\phi|$, where $r_c$ is the compactification radius of the extra dimension and $k \sim M_{Pl}$. The 4-D effective action allows one to identify $\overline{M}_{Pl}^2 = \frac{M^3}{k}(1 - e^{-2kr_c\pi})$, where M is the 5-D Planck scale. Physical particles on the TeV scale (SM) brane have mass $m = e^{-kr_c\pi}m_0$, where $m_0$ is a fundamental mass of order the Planck scale. Thus, the weak scale-Planck scale hierarchy occurs due to the existence of the exponential warp factor if $kr_c \sim 12$.

In the simplest versions of the RS model, the TeV-scale brane
contains only SM particles plus a tower of KK gravitons. The RS
gravitons have mass $m_n = kx_n e^{-kr_c\pi}$, where the $x_i$ are roots
of Bessel functions $J_1(x_n) = 0$, with $x_1 \simeq 3.83$, $x_2 \simeq 7.02$
etc. While the RS zero-mode graviton couplings suppressed by
$1/\overline{M}_{Pl}$ and are thus inconsequential for collider searches, the
$n=1$ and higher modes have couplings suppressed instead by
$\Lambda_\pi = e^{-kr_c\pi} \overline{M}_{Pl} \sim \text{TeV}$. The $n=1$ RS graviton should have
width $\Gamma_1 = \rho m_1 x_1^2 (k/\overline{M}_{Pl})^2$, where $\rho$ is a constant depending
on how many decay modes are open. The formulae for dilepton
production via virtual RS graviton exchange can be gained from
the above formulae for the ADD scenario via the replacement [27]

$$
\frac{\lambda}{M_*^4} \rightarrow \frac{i^2}{8\Lambda_\pi^2} \sum_{n=1}^{\infty} \frac{1}{s - m_n^2 + im_n\Gamma_n}. \quad (51.102)
$$

References

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[27] H. Davoudiasl, J. L. Hewett and T. G. Rizzo, Phys. Rev. Lett. **84**, 2080 (2000), [hep-ph/9909255].
## 52. Neutrino Cross Section Measurements

Revised August 2023 by G.P. Zeller (FNAL).

Neutrino cross sections are an essential ingredient in all neutrino experiments. Interest in neutrino scattering has increased due to the need for such information in the interpretation of neutrino oscillation data [1, 2] and given that uncertainties in neutrino-nucleus scattering remain a dominate source of systematic uncertainty in many neutrino oscillation measurements. Historically, neutrino scattering results on both charged current (CC) and neutral current (NC) channels have been collected over many decades using a variety of targets, analysis techniques, and detector technologies. With the advent of intense neutrino sources constructed for neutrino oscillation investigations, experiments are now remeasuring such interaction cross sections with a renewed appreciation for nuclear effects<fnref n="1" /> and the need for more precise neutrino flux estimations [3, 4]. This work summarizes accelerator-based neutrino cross section measurements performed in the ~0.1–300 GeV range with an emphasis on inclusive, quasi-elastic, and pion production processes, areas where we have the most experimental input at present (Table 52.1). For a more comprehensive discussion of neutrino cross sections, including neutrino-electron elastic scattering and lower energy neutrino measurements, the reader is directed to a review of this subject [5]. Here, we survey existing experimental data on neutrino interactions and do not attempt to provide a census of the associated theoretical calculations [6], which are both critical and plentiful. Companion electron-nucleus scattering data [7] are additionally playing an increased role in neutrino interaction predictions through such efforts as the e4ν collaboration [8].

<table><caption>**Table 52.1:** List of beam properties, targets, and run durations for modern accelerator-based neutrino experiments studying neutrino scattering.</caption><thead><tr><th rowspan="2">Experiment</th><th rowspan="2">beam</th><th>⟨E<sub>ν</sub>⟩</th><th>⟨E<sub>ν̄</sub>⟩</th><th rowspan="2">neutrino target(s)</th><th rowspan="2">run period</th></tr><tr><th>(GeV)</th><th></th></tr></thead><tbody><tr><td>ArgoNeUT</td><td>ν, ν̄</td><td>4.3, 3.6</td><td></td><td>Ar</td><td>2009 – 2010</td></tr><tr><td>FASERν</td><td>ν, ν̄</td><td>> 200</td><td></td><td>W, emulsion</td><td>2021 –</td></tr><tr><td>ICARUS (at CNGS)</td><td>ν</td><td>20</td><td></td><td>Ar</td><td>2010 – 2012</td></tr><tr><td>ICARUS (at FNAL)</td><td>ν</td><td>0.8</td><td></td><td>Ar</td><td>2021 –</td></tr><tr><td>K2K</td><td>ν</td><td>1.3</td><td></td><td>CH, H<sub>2</sub>O</td><td>2003 – 2004</td></tr><tr><td>MicroBooNE</td><td>ν</td><td>0.8</td><td></td><td>Ar</td><td>2015 – 2020</td></tr><tr><td>MINERvA</td><td>ν, ν̄</td><td>3.5 (LE),<br>5.5 (ME)</td><td></td><td>He, C, CH,<br>H<sub>2</sub>O, Fe, Pb</td><td>2009 – 2019</td></tr><tr><td>MiniBooNE</td><td>ν, ν̄</td><td>0.8, 0.7</td><td></td><td>CH<sub>2</sub></td><td>2002 – 2019</td></tr><tr><td>MINOS</td><td>ν, ν̄</td><td>3.5, 6.1</td><td></td><td>Fe</td><td>2004 – 2016</td></tr><tr><td>NINJA</td><td>ν, ν̄</td><td>0.6, 0.6</td><td></td><td>Fe, emulsion</td><td>2015 –</td></tr><tr><td>NOMAD</td><td>ν, ν̄</td><td>23.4, 19.7</td><td></td><td>C</td><td>1995 – 1998</td></tr><tr><td>NOvA</td><td>ν, ν̄</td><td>2.0, 2.0</td><td></td><td>CH<sub>2</sub></td><td>2010 –</td></tr><tr><td>SBND</td><td>ν</td><td>0.8</td><td></td><td>Ar</td><td>2024 –</td></tr><tr><td>SciBooNE</td><td>ν, ν̄</td><td>0.8, 0.7</td><td></td><td>CH</td><td>2007 – 2008</td></tr><tr><td>T2K</td><td>ν, ν̄</td><td>0.6, 0.6</td><td></td><td>CH, H<sub>2</sub>O, Fe</td><td>2010 –</td></tr></tbody></table>

### 52.1 Inclusive Scattering

Over the years, many experiments have measured the total inclusive charged current cross section for neutrino ($ν_μ N → μ^− X$) and antineutrino ($ν_μ N → μ^+ X$) scattering covering a broad range of neutrino energies. As can be seen in Fig. 52.1, which shows this data over decades of time and energy, the inclusive cross section approaches a linear dependence on neutrino energy. This behavior is expected for point-like scattering of neutrinos from quarks, an assumption which breaks down at lower energies. Modern measurements of such inclusive neutrino scattering cross sections and their target nuclei are summarized in Table 52.2. The reader is also referred to a recent review of the status of measurements and calculations in the shallow and deep inelastic scattering regions [9].

To provide a more complete picture, differential cross sections for such inclusive scattering processes have also been reported. These include historical measurements on iron from NuTeV [10] and modern measurements on a variety of nuclear targets from ArgoNeUT [11, 12], MicroBooNE [13, 14], MINERvA [15–18], NINJA [19, 20], NOvA [21], and T2K [22, 23]. More recently, MINERvA has provided measurements in terms of longitudinal and transverse muon momenta [16, 17], MicroBooNE has measured the first triple differential cross sections in argon [24], and T2K has provided the first measurement of the antineutrino inclusive cross section at low energy [25] (Fig. 52.1).

At high energy, the inclusive cross section is dominated by deep inelastic scattering (DIS). Several neutrino experiments have measured DIS cross sections for specific targets and final states, for example, MINERvA has measured ratios of neutrino DIS cross sections on a variety of nuclear targets including lead, iron, and carbon [26, 27]. Other experiments have measured opposite-sign dimuon production, the most recent being from CHORUS [28], NOMAD [29], and NuTeV [30]. Multiple efforts are also now underway to measure inclusive neutrino scattering cross sections at significantly higher neutrino energies, including at the LHC in FASERν [31] and using atmospheric neutrinos in IceCube [32].

<table><caption>**Table 52.2:** Published measurements of muon neutrino and antineutrino CC inclusive cross sections from modern accelerator-based neutrino experiments.</caption><thead><tr><th>experiment</th><th>measurement</th><th>target</th></tr></thead><tbody><tr><td>ArgoNeUT</td><td>ν<sub>μ</sub> [11, 12], ν̄<sub>μ</sub> [12]</td><td>Ar</td></tr><tr><td>MicroBooNE</td><td>ν<sub>μ</sub> [13, 24, 33]</td><td>Ar</td></tr><tr><td>MINERvA</td><td>ν<sub>μ</sub> [15–18, 26, 27, 34],<br>ν̄<sub>μ</sub> [34], ν̄<sub>μ</sub>/ν<sub>μ</sub> [35]</td><td>C, CH, Fe, Pb</td></tr><tr><td>MINOS</td><td>ν<sub>μ</sub> [36], ν̄<sub>μ</sub> [36]</td><td>Fe</td></tr><tr><td>NINJA</td><td>ν<sub>μ</sub> [19, 20], ν̄<sub>μ</sub> [19]</td><td>H<sub>2</sub>O, Fe</td></tr><tr><td>NOMAD</td><td>ν<sub>μ</sub> [37]</td><td>C</td></tr><tr><td>NOvA</td><td>ν<sub>μ</sub> [21]</td><td>CH<sub>2</sub></td></tr><tr><td>SciBooNE</td><td>ν<sub>μ</sub> [38]</td><td>CH</td></tr><tr><td>T2K</td><td>ν<sub>μ</sub> [22, 23, 39–41], ν̄<sub>μ</sub>/ν<sub>μ</sub> [25]</td><td>CH, H<sub>2</sub>O, Fe</td></tr></tbody></table>

At lower neutrino energies, the inclusive cross section is an additionally complex combination of quasi-elastic scattering and pion production processes, two areas we discuss next.

### 52.2 Quasi-elastic scattering

Quasi-elastic (QE) scattering is the dominant neutrino interaction for neutrino energies less than ~1 GeV and represents a large fraction of the signal samples in many neutrino oscillation experiments, which is why this process has received considerable attention in recent years. Historically, neutrino (antineutrino) quasi-elastic scattering refers to the process, $ν_μ n → μ^− p$ ($ν_μ p → μ^+ n$), where a charged lepton and single nucleon are ejected in the elastic interaction of a neutrino (or antineutrino) with a nucleon in the target material. This is the final state one would strictly observe, for example, in scattering off of a free nucleon target. There were many early measurements of neutrino QE scattering that span back to the 1970's [5]. In many of these initial measurements, bubble chamber experiments employed light targets (hydrogen or deuterium) and required both the detection of the final state muon and single nucleon<fnref n="2" />; thus the final state was clear and elastic kinematic conditions could be verified. The situation is more complicated, of course, for heavier nuclear targets used in modern neutrino experiments. In this case, nuclear effects can impact the size and shape of the cross section as well as the final state composition, kinematics, and topology. Due to intranuclear hadron rescattering and the effects of correlations between target nucleons, additional particles may be ejected in the final state; hence, a QE interaction on a nuclear target does not necessarily imply the ejection of a lepton and a single nucleon. One therefore needs to take care in defining what one means by

<footnote n="1">¹Nuclear effects refer to kinematic and final state effects which impact neutrino scattering off nuclei. Such effects can be significant and are particularly relevant given that modern neutrino experiments make use of nuclear targets to increase their event yields.</footnote>

<footnote n="2">²In the case of deuterium, many experiments additionally observed the spectator proton.</footnote>
<figure><img src="image_299.png" /><figcaption>**Figure 52.1:** Measurements of per nucleon $\nu_\mu$ and $\bar{\nu}_\mu$ CC inclusive scattering cross sections divided by neutrino energy as a function of neutrino energy. Note the transition between logarithmic and linear scales occurring at 100 GeV. Neutrino cross sections are typically twice as large as their corresponding antineutrino counterparts, although this difference can be larger at lower energies. NC cross sections (not shown) are generally smaller compared to their CC counterparts.</figcaption></figure>

neutrino QE scattering when scattering off targets heavier than hydrogen or deuterium. Because of this, modern experiments tend to instead report cross sections for processes involving pionless (e.g., nucleon-only) final states, often referred to as CC $0\pi$ or QE-like reactions in recent literature. Such measurements are summarized in Table 52.3. Many modern experiments have also recently opted to report QE-like cross sections as a function of observed final state particle kinematics. Such measurements can be more difficult to directly compare between experiments but have the advantage of being much less model-dependent and provide more stringent tests of the theory than historical cross sections as a function of derived quantities such as neutrino energy ($E_\nu$) or 4-momentum transfer ($Q^2$). More recently, new thought has been given to the means for directly comparing experimental measurements produced in these less model-dependent forms [42].

The topic of neutrino QE scattering began drawing considerable attention following the first double differential cross section measurements of this process that revealed a significantly larger cross section than originally anticipated, predominantly in the backwards muon scattering region [64, 65]. Such an enhancement was observed many years prior in transverse electron-nucleus scattering [83] and was attributed to the presence of correlations between nucleons in the target nucleus. As a result, the impact of such nuclear effects on neutrino QE scattering has recently become the subject of intense experimental and theoretical scrutiny with implications on event rates, nucleon emission, neutrino energy reconstruction, and neutrino versus antineutrino cross sections. The reader is referred to reviews of the situation in [64, 84, 85]. To help drive further progress in understanding the underlying nuclear contributions, pionless (e.g., nucleon-only) cross sections have been reported for the first time in the form of double-differential distributions by MiniBooNE [64, 65], MINERvA [51–55, 61], and T2K [73–78]. Such double-differential cross sections in terms of final state particle kinematics reduce the model-dependence of the reported data, provide the most robust measurements available, and allow a more rigorous two-dimensional test of the underlying nuclear theory. MicroBooNE, MINERvA, and T2K have been especially prolific in recent years in probing this interaction process

(Table 52.3). Neutrino experiments have also launched dedicated studies of the hadronic side of these interactions, including ArgoNeuT [43, 86], MicroBooNE [45, 46], MINERvA [60, 87], and T2K [82]. MINERvA has been the first modern experiment to measure neutron emission in antineutrino interactions [88] and has simultaneously measured neutrino QE cross sections on a variety of nuclear targets ranging from carbon to lead [55]. ArgoNeuT [43], MicroBooNE [49], and NINJA [70] have explored two-proton final states. T2K has probed ratios of oxygen to carbon [76, 89], asymmetries between neutrino and antineutrino scattering [77], and simultaneous measurements both on and off axis to the neutrino beam [78]. In addition, the exploration of transverse kinematic variables and momentum imbalances in pionless neutrino scattering is allowing better constraints on the various nuclear contributions to the cross section. Such scrutiny includes important evaluations from MicroBooNE [47, 48], MINERvA [51, 52, 56, 57] and T2K [82]. With the MiniBooNE results having first revealed these additional complexities in neutrino-nucleus QE scattering, measurements from multiple neutrino experiments, on multiple targets, and using a variety of kinematic information have been crucial in gaining a better handle on the underlying nuclear physics impacting neutrino-nucleus interactions. What we once thought was “simple” QE scattering is in fact not so simple.

Equally important, neutrino experiments such as MINERvA have revisited the extraction of the axial form factor from light target data using modern higher statistics data sets [63]. In addition to such charged current investigations, measurements of the neutral current counterpart of this channel have also been performed. The most recent NC elastic scattering cross section measurements include those from BNL E734 [90], MiniBooNE [67, 68], Super-K [72], and T2K [81]. A number of measurements of the Cabibbo-suppressed antineutrino QE hyperon production cross section have additionally been reported [91, 92], most recently by MicroBooNE [50]. As an exciting new addition, there have also been the first observations of the very challenging-to-detect coherent elastic neutrino-nucleus scattering (CEνNS) process now in multiple nuclei by COHERENT [93, 93, 94].
<table><caption>Table 52.3: Published modern measurements of muon neutrino QE and NC elastic scattering cross sections with pionless final states.</caption><thead><tr><th>experiment</th><th>measurement</th><th>target</th></tr></thead><tbody><tr><td>ArgoNeuT</td><td>2p [43]</td><td>Ar</td></tr><tr><td>K2K</td><td>M<sub>A</sub> [44]</td><td>H<sub>2</sub>O</td></tr><tr><td>MicroBooNE</td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">dσ/dp<sub>μ</sub></div><div style="padding: 0 5px;">dσ/dp<sub>p</sub></div></div> [45, 46], <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">dσ/dcosθ<sub>μ</sub></div><div style="padding: 0 5px;">dσ/dcosθ<sub>p</sub></div></div> [47, 48], <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">d<sup>2</sup>σ/dδα<sub>T</sub>dδp<sub>T</sub></div><div style="padding: 0 5px;">d<sup>2</sup>σ/dδp<sub>Tx</sub>dδp<sub>Ty</sub></div></div> [47], 2p [49], Λ [50]</td><td>Ar</td></tr><tr><td>MINERνA</td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">d<sup>2</sup>σ/dp<sub>T</sub>dρ<sub>||</sub></div><div style="padding: 0 5px;">d<sup>2</sup>σ/dp<sub>Tx</sub>dρ<sub>Ty</sub></div></div> [51–55], <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">dσ/dp<sub>Tx</sub></div><div style="padding: 0 5px;">dσ/dp<sub>Ty</sub></div></div> [56], <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">dσ/dp<sub>n</sub></div><div style="padding: 0 5px;">dσ/dδα<sub>T</sub></div></div> [57], CH,Fe,Pb <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">dσ/dQ<sup>2</sup></div><div style="padding: 0 5px;">d<sup>2</sup>σ/dE<sub>avail</sub>dq<sub>3</sub></div></div> [61], <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">dσ/dQ<sup>2</sup></div><div style="padding: 0 5px;">d<sup>2</sup>σ/dE<sub>avail</sub>dq<sub>3</sub></div></div> [62], F<sub>A</sub>(Q<sup>2</sup>) [63]</td><td>CH<sub>2</sub></td></tr><tr><td>MiniBooNE</td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">d<sup>2</sup>σ/dT<sub>μ</sub>dθ<sub>μ</sub></div><div style="padding: 0 5px;">dT<sub>μ</sub>dθ<sub>μ</sub></div></div> [64, 65], M<sub>A</sub> [66], NC [67, 68]</td><td>CH<sub>2</sub></td></tr><tr><td>MINOS</td><td>M<sub>A</sub> [69]</td><td>Fe</td></tr><tr><td>NINJA</td><td>2p [70]</td><td>Fe</td></tr><tr><td>NOMAD</td><td>M<sub>A</sub>, σ(E<sub>ν</sub>) [71]</td><td>C</td></tr><tr><td>Super-K</td><td>NC [72]</td><td>H<sub>2</sub>O</td></tr><tr><td>T2K</td><td><div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">d<sup>2</sup>σ/dT<sub>μ</sub>dθ<sub>μ</sub></div><div style="padding: 0 5px;">dT<sub>μ</sub>dθ<sub>μ</sub></div></div> [73–78], σ(E<sub>ν</sub>) [79], M<sub>A</sub> [80], NC [81], <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">dσ/dδp<sub>T</sub></div><div style="padding: 0 5px;">dσ/dδα<sub>T</sub></div></div> [82], O/C [76]</td><td>CH,H<sub>2</sub>O</td></tr></tbody></table>

## 52.3 Pion Production

In addition to such elastic scattering processes, neutrinos can also inelastically scatter producing a nucleon excited state (Δ, N*). Such baryonic resonances quickly decay, most often to a nucleon and single-pion final state. Historically, experiments have measured various exclusive final states associated with these reactions, the majority of which have been on hydrogen and deuterium targets [5]. There have been several recent re-analyses of this data to better understand the consistency between data sets [95], nucleon form factors [96], and non-resonant contributions [97]. Also, modern measurements of neutrino-induced pion production have since been performed on a variety of nuclear targets (Table 52.4).

<table><caption>Table 52.4: Summary of modern measurements of NC and CC scattering cross sections involving a single pion or multiple pions in the final state.</caption><thead><tr><th>experiment</th><th>π<sup>±</sup></th><th>π<sup>0</sup></th><th>target</th></tr></thead><tbody><tr><td>ArgoNeuT</td><td>CC [98, 99]</td><td>NC [100]</td><td>Ar</td></tr><tr><td>K2K</td><td>CC[101, 102]</td><td>CC [103], NC [104]</td><td>CH, H<sub>2</sub>O</td></tr><tr><td>MicroBooNE</td><td>–</td><td>CC [105], NC [106]</td><td>Ar</td></tr><tr><td>MINERνA</td><td>CC [107–114]</td><td>CC [108, 115–117],<br>NC [118]</td><td>C, CH, H<sub>2</sub>O,<br>Fe, Pb</td></tr><tr><td>MiniBooNE</td><td>CC [119, 120]</td><td>CC [121], NC [122, 123]</td><td>CH<sub>2</sub></td></tr><tr><td>MINOS</td><td>–</td><td>NC [124]</td><td>Fe</td></tr><tr><td>NOMAD</td><td>–</td><td>NC [125]</td><td>C</td></tr><tr><td>NOvA</td><td>–</td><td>NC [126, 127]</td><td>C</td></tr><tr><td>SciBooNE</td><td>CC [128]</td><td>NC [129, 130]</td><td>CH</td></tr><tr><td>T2K</td><td>CC [131–134]</td><td>–</td><td>CH, H<sub>2</sub>O</td></tr></tbody></table>

In addition to resonance production processes, neutrinos can also coherently scatter off of the entire nucleus and produce a distinctly forward-scattered single pion final state. Both CC ($\nu_{\mu} A \rightarrow \mu^{-} A \pi^{+}$, $\bar{\nu}_{\mu} A \rightarrow \mu^{+} A \pi^{-}$) and NC ($\nu_{\mu} A \rightarrow \nu_{\mu} A \pi^{0}$, $\bar{\nu}_{\mu} A \rightarrow \bar{\nu}_{\mu} A \pi^{0}$) processes are possible in this case. Even though the level of coherent pion production is small compared to their resonant counterpart, observations exist across a broad energy range and on multiple nuclear targets [135]. More recently, several modern neutrino experiments have either measured or set limits on coherent pion production cross sections including ArgoNeuT [98], K2K [102], MINERvA [109, 111, 113], MiniBooNE [123], MINOS [124], NOMAD [125], NOvA [126], SciBooNE [128, 130], and T2K [102].

As with inclusive and quasi-elastic scattering, a new appreciation for the significance of nuclear effects has also surfaced in pion production physics, again due to the use of heavy nuclear targets in modern neutrino experiments. Many experiments have been careful to report cross sections for various detected final

states, thereby not correcting for large and uncertain nuclear effects (e.g., pion rescattering, charge exchange, and absorption) which can introduce significant sources of uncertainty and model dependence. Providing the most comprehensive survey of neutrino single-pion production to date, MiniBooNE has published a total of 16 single- and double-differential cross sections for both the final state muon (in the case of CC scattering) and pions in these interactions; thus, providing the first measurements of such final state kinematic distributions [119–122]. At similar neutrino energies, T2K has provided new data [131] including the first measurement of the Adler angles in neutrino-nucleus scattering [133] and the first exploration of transverse kinematic imbalances in pion production processes [134]. MINERvA has produced a similar accompaniment of measurements at higher neutrino energies [108, 110, 112, 116] and uniquely on multiple nuclear targets [114]. Importantly, MINERvA has been working towards an improved nuclear model [117] that can potentially describe all of the pion reaction channels simultaneously, an issue that many experiments have struggled with up until now [108]. ArgoNeuT [99, 100] and MicroBooNE [105, 106] have since been adding new information on single pion production in argon. Regardless of the interaction channel or target material, differential cross section measurements in terms of observed final state particle kinematics are preferred for their reduced model dependence and for the additional kinematic information they provide. Such a new direction has been the focus of modern measurements as opposed to the reporting of more model-dependent, historical cross sections as a function of $E_{\nu}$ or $Q^{2}$. Together with similar results for other interaction channels, a better understanding and modeling of nuclear effects will be possible moving forward. MINERvA [136] has already taken a large step in this direction by explicitly tuning the physics in existing neutrino event generators to best fit the experimental data on pion production.

It should be noted that baryonic resonances can also decay to multi-pion, other mesonic ($K, \eta, \rho$, etc.), and even photon final states. Experimental results for these channels are typically sparse or non-existent. More recently, MicroBooNE has produced the first measurement of neutrino-induced $\eta$ production on argon [137]. Photon production processes can comprise an important background for $\nu_{\mu} \rightarrow \nu_{e}$ appearance searches in some detectors and thus have become the focus of recent experimental investigations, most notably in NOMAD [138] and T2K [139, 140]. There have also been several recent measurements of kaon final states produced in neutrino NC and CC scattering in MINERvA [141–143] that are providing needed background constraints for certain nucleon decay searches.
## 52.4 Electron Neutrino Scattering

The aforementioned cross section measurements are all for either muon neutrino or muon antineutrino scattering. With the production of predominantly muon neutrino beams for neutrino oscillation studies, the availability of electron neutrino scattering data is by construction limited. However, for the first time, measurements of electron neutrino scattering cross sections are being produced by a variety of modern neutrino experiments (Table 52.5). Such measurements are important as they verify our assumptions about lepton universality and validate the use of muon neutrino data to constrain electron neutrino interaction predictions. For lower energy and historical electron neutrino cross section measurements, the reader is directed to reference [5].

<table><caption>**Table 52.5:** Published measurements of electron neutrino and antineutrino cross sections from modern accelerator-based neutrino experiments.</caption><thead><tr><th>experiment</th><th>CC inclusive</th><th>QE-like π</th><th>production target</th></tr></thead><tbody><tr><td>ArgoNeuT</td><td>[144]</td><td>-</td><td>Ar</td></tr><tr><td>COHERENT</td><td>[145]</td><td>-</td><td>I</td></tr><tr><td>MicroBooNE</td><td>[146, 147]</td><td>[148]</td><td>Ar</td></tr><tr><td>MINERV</td><td>-</td><td>[149]</td><td>CH</td></tr><tr><td>NOvA</td><td>[150]</td><td>-</td><td>CH<sub>2</sub></td></tr><tr><td>T2K</td><td>[151–153]</td><td>-</td><td>CH, H<sub>2</sub>O</td></tr></tbody></table>

## 52.5 Outlook

Neutrino experiments continue to produce critical neutrino scattering measurements on nuclei as they accumulate increased statistics and pursue new ways of reporting their measurements. Analysis of a broad energy range of data from MINERV is providing some of the most detailed analysis of nuclear effects in neutrino interactions by examining multiple nuclei in a single experiment. Data from ArgoNeuT, MicroBooNE, ICARUS and soon also SBND are probing deeper into complex neutrino final states using the superior capabilities of liquid argon time projection chambers, while the T2K and NOvA near detectors continue to collect high statistics samples in intense neutrino beams. Together with dedicated discussions between experiments on how best to report neutrino cross section measurements [154] and with accompanying work being advanced by theorists to further improve nuclear model calculations [6], we are starting to significantly advance our understanding of neutrino-nucleus scattering.

## 52.6 Acknowledgments

The author thanks Anne Schukraft (Fermilab) for help in preparing the plots and content of this review.

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## 53. Plots of Cross Sections and Related Quantities

Updated in 2023. See various sections for details. For additional cross section results, please see earlier editions of the Review of Particle Physics (https://pdg.lbl.gov/rpp-archive/).

<table><tr><td>53.1</td><td>Pseudorapidity Distributions in <i>pp</i> and <i>p̄p</i> Interactions</td><td>807</td></tr><tr><td>53.2</td><td>Average Hadron Multiplicities in Hadronic <i>e<sup>+</sup>e<sup>-</sup></i> Annihilation Events</td><td>808</td></tr><tr><td>53.3</td><td>Annihilation Cross Section Near <i>M<sub>Z</sub></i></td><td>810</td></tr></table>

### 53.1 Pseudorapidity Distributions in *pp* and *p̄p* Interactions

Revised August 2013 by D.R. Ward (Cavendish Lab.).

<figure><img src="image_300.png" /><figcaption>**Figure 53.1:** Charged particle pseudorapidity distributions in *p̄p* collisions for 53 GeV ≤ $\sqrt{s}$ ≤ 1800 GeV. UA5 data from the *Sp̄pS* are taken from [1], and from the ISR from [2]. The UA5 data are shown for both the full inelastic cross-section and with singly diffractive events excluded. Additional non single-diffractive measurements are available from CDF at the Tevatron [3] and from P238 at the *Sp̄pS* [4]. These may be compared with both inclusive and non single-diffractive measurements in *pp* collisions at the LHC from ALICE [5] and for non single-diffractive interactions from CMS [6, 7]. (Courtesy of D.R. Ward, Cambridge Univ., 2013)</figcaption></figure>
## 53.2 Average Hadron Multiplicities in Hadronic $e^{+}e^{-}$ Annihilation Events

Revised August 2023 by O. Biebel (Ludwig-Maximilians U.).

<table><caption>**Table 53.1:** Average hadron multiplicities per hadronic $e^{+}e^{-}$ annihilation event at $\sqrt{s} \approx 10, 29-35, 91$, and $130-200$ GeV. The rates given include decay products from resonances with $c\tau < 10$ cm, and include the corresponding anti-particle state. Correlations of the systematic uncertainties were considered for the calculation of the averages. Quoted errors are not increased by scale factor S.</caption><thead><tr><th>Particle</th><th>√s ≈ 10 GeV</th><th>√s = 29-35 GeV</th><th>√s = 91 GeV</th><th>√s = 130-200 GeV</th><th>References</th></tr></thead><tbody><tr><td colspan="6"><strong>Pseudoscalar mesons:</strong></td></tr><tr><td>π<sup>+</sup></td><td>6.52 ± 0.11</td><td>0.3 ± 0.4</td><td>17.02 ± 0.19</td><td>21.24 ± 0.39</td><td>[8-17]</td></tr><tr><td>π<sup>0</sup></td><td>3.2 ± 0.3</td><td>5.83 ± 0.28</td><td>9.42 ± 0.32</td><td></td><td>[12, 18-23]</td></tr><tr><td>K<sup>+</sup></td><td>0.953 ± 0.018</td><td>1.48 ± 0.09</td><td>2.228 ± 0.059</td><td>2.82 ± 0.19</td><td>[9-17, 24, 25]</td></tr><tr><td>K<sup>0</sup></td><td>0.91 ± 0.05</td><td>1.48 ± 0.07</td><td>2.049 ± 0.026</td><td>2.10 ± 0.12</td><td>[12, 17, 20, 26-36]</td></tr><tr><td>η</td><td>0.20 ± 0.04</td><td>0.61 ± 0.07</td><td>1.049 ± 0.080</td><td></td><td>[12, 18, 19, 22, 23, 37-40]</td></tr><tr><td>η'(958)</td><td>0.03 ± 0.01</td><td>0.26 ± 0.10</td><td>0.152 ± 0.020</td><td></td><td>[20, 39, 41-43]</td></tr><tr><td>D<sup>+</sup></td><td>0.194 ± 0.019<sup>(a)</sup></td><td>0.17 ± 0.03</td><td>0.175 ± 0.016</td><td></td><td>[12, 44-47]</td></tr><tr><td>D<sup>0</sup></td><td>0.446 ± 0.032<sup>(a)</sup></td><td>0.45 ± 0.07</td><td>0.454 ± 0.030</td><td></td><td>[12, 44-47]</td></tr><tr><td>D<sub>s</sub><sup>+</sup></td><td>0.063 ± 0.014<sup>(a)</sup></td><td>0.45 ± 0.20<sup>(b)</sup></td><td>0.131 ± 0.021</td><td></td><td>[8, 39, 44, 47-49]</td></tr><tr><td>B<sup>(c)</sup></td><td>—</td><td>—</td><td>0.134 ± 0.016<sup>(d)</sup></td><td></td><td>[46, 50]</td></tr><tr><td>B<sup>+</sup></td><td>—</td><td>—</td><td>0.141 ± 0.004<sup>(d)</sup></td><td></td><td>[51]</td></tr><tr><td>B<sub>s</sub><sup>0</sup></td><td>—</td><td>—</td><td>0.054 ± 0.011<sup>(d)</sup></td><td></td><td>[52, 53]</td></tr><tr><td colspan="6"><strong>Scalar mesons:</strong></td></tr><tr><td>f<sub>0</sub>(980)</td><td>0.024 ± 0.006</td><td>0.05 ± 0.02<sup>(e)</sup></td><td>0.146 ± 0.012</td><td></td><td>[41, 54-56]</td></tr><tr><td>a<sub>0</sub>(980)<sup>±</sup></td><td>—</td><td>—</td><td>0.27 ± 0.11<sup>(f)</sup></td><td></td><td>[43]</td></tr><tr><td colspan="6"><strong>Vector mesons:</strong></td></tr><tr><td>ρ(770)<sup>0</sup></td><td>0.35 ± 0.04</td><td>0.81 ± 0.08</td><td>1.231 ± 0.098</td><td></td><td>[9, 12, 55, 57, 58]</td></tr><tr><td>ρ(770)<sup>±</sup></td><td>—</td><td>—</td><td>2.40 ± 0.43<sup>(f)</sup></td><td></td><td>[43]</td></tr><tr><td>ω(782)</td><td>0.30 ± 0.08</td><td>—</td><td>1.016 ± 0.065</td><td></td><td>[40, 42, 43, 57]</td></tr><tr><td>K<sup>*</sup>(892)<sup>+</sup></td><td>0.27 ± 0.03</td><td>0.64 ± 0.05</td><td>0.714 ± 0.055</td><td></td><td>[9, 12, 33, 57, 59, 60]</td></tr><tr><td>K<sup>*</sup>(892)<sup>0</sup></td><td>0.29 ± 0.03</td><td>0.56 ± 0.06</td><td>0.738 ± 0.024</td><td></td><td>[9, 12, 36, 57, 58, 61, 62]</td></tr><tr><td>φ(1020)</td><td>0.044 ± 0.003</td><td>0.085 ± 0.011</td><td>0.0963 ± 0.0032</td><td></td><td>[12, 36, 56-58, 61]</td></tr><tr><td>D<sup>*</sup>(2010)<sup>+</sup></td><td>0.177 ± 0.022<sup>(a)</sup></td><td>0.43 ± 0.07</td><td>0.1937 ± 0.0057<sup>(g)</sup></td><td></td><td>[12, 44-46, 63, 64]</td></tr><tr><td>D<sup>*</sup>(2007)<sup>0</sup></td><td>0.168 ± 0.019<sup>(a)</sup></td><td>0.27 ± 0.11</td><td>—</td><td></td><td>[12, 44, 45]</td></tr><tr><td>D<sub>s</sub><sup>*</sup>(2112)<sup>+</sup></td><td>0.048 ± 0.014<sup>(a)</sup></td><td>—</td><td>0.101 ± 0.048<sup>(h)</sup></td><td></td><td>[48, 65]</td></tr><tr><td>B<sup>*</sup><sup>(i)</sup></td><td>—</td><td>—</td><td>0.288 ± 0.026</td><td></td><td>[66, 67]</td></tr><tr><td>J/ψ(1S)</td><td>0.00050 ± 0.00005<sup>(a)</sup></td><td>—</td><td>0.0052 ± 0.0004<sup>(j)</sup></td><td></td><td>[68-73]</td></tr><tr><td>ψ(2S)</td><td>—</td><td>—</td><td>0.0023 ± 0.0004<sup>(j)</sup></td><td></td><td>[71, 73, 74]</td></tr><tr><td>Υ(1S)</td><td>—</td><td>—</td><td>0.00014 ± 0.00007<sup>(j)</sup></td><td></td><td>[75]</td></tr><tr><td colspan="6"><strong>Pseudovector mesons:</strong></td></tr><tr><td>f<sub>1</sub>(1285)</td><td>—</td><td>—</td><td>0.165 ± 0.051</td><td></td><td>[76]</td></tr><tr><td>f<sub>1</sub>(1420)</td><td>—</td><td>—</td><td>0.056 ± 0.012</td><td></td><td>[76]</td></tr><tr><td>χ<sub>c1</sub>(3510)</td><td>—</td><td>—</td><td>0.0041 ± 0.0011<sup>(j)</sup></td><td></td><td>[71, 74]</td></tr><tr><td colspan="6"><strong>Tensor mesons:</strong></td></tr><tr><td>f<sub>2</sub>(1270)</td><td>0.09 ± 0.02</td><td>0.14 ± 0.04</td><td>0.166 ± 0.020</td><td></td><td>[54-56, 77]</td></tr><tr><td>f<sub>2</sub><sup>*</sup>(1525)</td><td>—</td><td>—</td><td>0.012 ± 0.006</td><td></td><td>[55]</td></tr><tr><td>K<sub>2</sub><sup>*</sup>(1430)<sup>+</sup></td><td>—</td><td>0.09 ± 0.03</td><td>—</td><td></td><td>[55, 78]</td></tr><tr><td>K<sub>2</sub><sup>*</sup>(1430)<sup>0</sup></td><td>—</td><td>0.12 ± 0.06</td><td>0.084 ± 0.022</td><td></td><td>[54, 55, 79]</td></tr><tr><td>B<sup>**</sup><sup>(k)</sup></td><td>—</td><td>—</td><td>0.118 ± 0.024</td><td></td><td>[80]</td></tr><tr><td>D<sub>s</sub><sup>±</sup></td><td>—</td><td>—</td><td>0.0052 ± 0.0011<sup>(ℓ)</sup></td><td></td><td>[81]</td></tr><tr><td>D<sub>s</sub><sup>*</sup><sub>2</sub><sup>±</sup></td><td>—</td><td>—</td><td>0.0083 ± 0.0031<sup>(ℓ)</sup></td><td></td><td>[81]</td></tr><tr><td colspan="6"><strong>Baryons:</strong></td></tr><tr><td>p</td><td>0.266 ± 0.008</td><td>0.640 ± 0.050</td><td>1.050 ± 0.032</td><td>1.41 ± 0.18</td><td>[10, 13-17, 24, 25, 77]</td></tr><tr><td>Λ</td><td>0.093 ± 0.006<sup>(a)</sup></td><td>0.205 ± 0.010</td><td>0.3915 ± 0.0065</td><td>0.39 ± 0.03</td><td>[17, 20, 34, 36, 77, 82-85]</td></tr><tr><td>Σ<sup>0</sup></td><td>0.0221 ± 0.0018<sup>(a)</sup></td><td>—</td><td>0.078 ± 0.010</td><td></td><td>[10, 59, 82, 86-88]</td></tr><tr><td>Σ<sup>-</sup></td><td>—</td><td>—</td><td>0.081 ± 0.010</td><td></td><td>[88, 89]</td></tr><tr><td>Σ<sup>+</sup></td><td>—</td><td>—</td><td>0.107 ± 0.011</td><td></td><td>[87, 88]</td></tr><tr><td>Σ<sup>±</sup></td><td>—</td><td>—</td><td>0.174 ± 0.009</td><td></td><td>[84, 88]</td></tr><tr><td>Ξ<sup>-</sup></td><td>0.0055 ± 0.0004<sup>(a)</sup></td><td>0.0176 ± 0.0027</td><td>0.0262 ± 0.0009</td><td></td><td>[9, 59, 77, 82-85]</td></tr><tr><td>Δ(1232)<sup>++</sup></td><td>0.040 ± 0.010</td><td>—</td><td>0.085 ± 0.014</td><td></td><td>[90-92]</td></tr><tr><td>Σ(1385)<sup>-</sup></td><td>0.006 ± 0.002</td><td>0.017 ± 0.004</td><td>0.0240 ± 0.0017</td><td></td><td>[59, 82, 84, 85, 93]</td></tr><tr><td>Σ(1385)<sup>+</sup></td><td>0.0062 ± 0.0011<sup>(a)</sup></td><td>0.017 ± 0.004</td><td>0.0239 ± 0.0015</td><td></td><td>[59, 82-85, 93]</td></tr><tr><td>Σ(1385)<sup>±</sup></td><td>0.0106 ± 0.0020</td><td>0.033 ± 0.008</td><td>0.0472 ± 0.0027</td><td></td><td>[59, 82, 84, 85, 93]</td></tr><tr><td>Ξ(1530)<sup>0</sup></td><td>0.00130 ± 0.00010<sup>(a)</sup></td><td>—</td><td>0.00694 ± 0.00049</td><td></td><td>[59, 82, 83, 85, 94]</td></tr><tr><td>Ω<sup>-</sup></td><td>0.00060 ± 0.00033<sup>(a)</sup></td><td>0.014 ± 0.007</td><td>0.00124 ± 0.00018</td><td></td><td>[59, 77, 82, 83, 85, 86]</td></tr><tr><td>Λ<sub>s</sub><sup>+</sup></td><td>0.0480 ± 0.0036<sup>(a,m)</sup></td><td>0.110 ± 0.050</td><td>0.0591 ± 0.0047<sup>(n)</sup></td><td></td><td>[47, 49, 77, 83, 95, 96]</td></tr><tr><td>Λ<sub>s</sub><sup>0</sup></td><td>—</td><td>—</td><td>0.031 ± 0.016</td><td></td><td>[97]</td></tr><tr><td>Σ<sub>c</sub><sup>0</sup></td><td>0.0025 ± 0.0004<sup>(a)</sup></td><td>—</td><td>—</td><td></td><td>[83]</td></tr><tr><td>Λ(1520)</td><td>0.0046 ± 0.0004<sup>(a)</sup></td><td>—</td><td>0.0222 ± 0.0027</td><td></td><td>[83, 85, 89, 98]</td></tr></tbody></table>

<footnote>(a) $\sigma_{\text{had}} = 3.33 \pm 0.05 \pm 0.21 \text{ nb (CLEO: [99])}$ has been used in converting the measured cross sections to average hadron multiplicities.</footnote>

<footnote>(b) $B(D_s \to \eta\pi, \eta'\pi)$ was used (RPP 1994).</footnote>
(c) Comprises both charged and neutral B meson states.

(d) The Standard Model $B(Z \to b\bar{b}) = 0.217$ was used.

(e) $x_p = p/p_{\text{beam}} > 0.1$ only.

(f) Both charge states.

(g) $B(D^*(2010)^+ \to D^0\pi^+) \times B(D^0 \to K^-\pi^+)$ has been used (RPP 2000).

(h) $B(D_s^* \to D_S^+\gamma)$, $B(D_s^+ \to \phi\pi^+)$, $B(\phi \to K^+K^-)$ have been used (RPP 1998).

(i) Any charge state (i.e., $B_d^*, B_u^*, \text{or } B_s^*$).

(j) $B(Z \to \text{hadrons}) = 0.699$ was used (RPP 1994).

(k) Any charge state (i.e., $B_d^{**}$, $B_u^{**}$, or $B_s^{**}$).

(l) Assumes $B(D_{s1}^+ \to D^+K^0 + D^0K^+) = 100\%$ and $B(D_{s2}^+ \to D^0K^+) = 45\%$.

(m) Derived from the production cross section of $\Lambda_c^+ \to p\pi K$ using (a) and using $B(\Lambda_c^+ \to p\pi K) = (6.26 \pm 0.29)\%$ (RPP 2022).

(n) Derived from [96], updated with $B(\Lambda_c^+ \to p\pi K) = (6.26 \pm 0.29)\%$ (RPP 2022), and complemented by the $\Lambda_c^+$ contribution from $g \to c\bar{c}$ of [47].

References grouped by collaboration for Table-53.1:

• RPP: [12]

• ALEPH: [13, 20, 40, 58, 59, 63, 70, 81],

• ARGUS: [8, 24, 37, 41, 57, 82, 90, 98],

• BaBar: [10, 48, 68, 95],

• Belle: [44, 69, 83],

• CELLO: [19, 26],

• CLEO: [9, 45, 49, 99],

• Crystal Ball: [38],

• DELPHI: [14, 17, 21, 25, 33, 46, 50–52, 55, 61, 66, 71, 76, 80, 84, 86, 89, 91, 94],

• HRS: [27, 54, 78, 93],

• L3: [22, 34, 42, 67, 72, 74, 87]

• MARK II: [29, 39],

• JADE: [18, 28],

• OPAL: [15, 23, 35, 43, 47, 53, 56, 60, 62, 64, 65, 73, 75, 79, 85, 88, 92, 97],

• PLUTO: [30]

• SLD: [16, 36],

• TASSO: [31]

• TPC: [32].
53.3 Annihilation Cross Section Near $M_Z$

Courtesy of M. Grünewald and the LEP Electroweak Working Group, 2007.

<figure><img src="image_301.png" /><figcaption>**Figure 53.2:** Combined data from the ALEPH, DELPHI, L3, and OPAL Collaborations for the cross section in $e^{+}e^{-}$ annihilation into hadronic final states as a function of the center-of-mass energy near the Z pole. The curves show the predictions of the Standard Model with two, three, and four species of light neutrinos. The asymmetry of the curve is produced by initial-state radiation. Note that the error bars have been increased by a factor ten for display purposes. References: ALEPH [100], DELPHI [101], L3 [102], OPAL [103], Combination [104],</figcaption></figure>
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Particle Properties

Gauge Bosons

54. Mass and width of the W boson (rev.) 815

55. Z boson 817

Charged Leptons

56. Muon anomalous magnetic moment (rev.) 822

57. Muon decay parameters (rev.) 826

58. τ branching fractions (rev.) 829

59. τ-lepton decay parameters (rev.) 832

Quarks

60. Quark masses (rev.) 834

61. Top quark (rev.) 844

Mesons

62. Form factors for semileptonic kaon ($K_{\ell 3}$), radiative pion ($\pi_{\ell 2\gamma}$) & kaon ($K_{\ell 2\gamma}$) decays (rev.) 871

63. Spectroscopy of light meson resonances (rev.) 874

64. Scalar mesons below 1 GeV (rev.) 888

65. Rare kaon decays (rev.) 896

66. CPT invariance tests in neutral kaon decay (rev.) 902

67. $V_{ud}$, $V_{us}$, Cabibbo angle, and CKM unitarity (rev.) 904

68. CP-violation in $K_L$ decays (rev.) 908

69. Review of multibody charm analyses (rev.) 911

70. $D^0-\bar{D}^0$ mixing (rev.) 917

71. $D_s^+$ branching fractions 927

72. Leptonic decays of charged pseudoscalar mesons (rev.) 929

73. Production and decay of b-flavored hadrons (rev.) 940

74. Polarization in B decays (rev.) 951

75. $B^0-\bar{B}^0$ mixing (rev.) 955

76. Semileptonic B decays, $V_{cb}$ and $V_{ub}$ (rev.) 962

77. CKM angles from B hadrons, determination of (rev.) 980

78. Spectroscopy of mesons containing two heavy quarks (rev.) 985

79. Heavy non-$q\bar{q}$ mesons (rev.) 994

Baryons

80. Baryon decay parameters 1001

81. N and Δ resonances 1002

82. Λ and Σ resonances 1007

83. Pole structure of the Λ(1405) region 1010

84. Pentaquarks (rev.) 1012

Notes in Volume 2

Triple gauge couplings (TGC's) 1190

Anomalous $ZZ\gamma$, $Z\gamma\gamma$, and $ZZV$ couplings 1216

Anomalous $W/Z$ quartic couplings (rev.) 1217

Neutrino properties (rev.) 1332

Sum of neutrino masses (rev.) 1335

Number of light neutrino types from collider experiments 1341

Neutrinoless double-$\beta$ decay 1343

Charged kaon mass 1538

Dalitz plot parameters for $K \to 3\pi$ decays 1547

CP-violation in $K_S \to 3\pi$ 1557

Heavy Flavor Averaging Group (rev.) 1681

Charmonium system (rev.) 1898

Branching ratios of $\psi(2S)$ and $\chi_{c0,1,2}$ (rev.) 1934

Bottomonium system (rev.) 2023

Width determination of the Υ states 2023

Radiative hyperon decays 2219

Ξ resonances (rev.) 2223
<figure><img src="image_302.png" /></figure>
## 54. Mass and Width of the W Boson

Revised April 2024 by M. Grünewald (University Coll. Dublin) and A. Gurtu (CERN; TIFR Mumbai).

Precision determination of the W mass is of great importance in testing the internal consistency of the Standard Model. From the time of its discovery in 1983, the W boson has been studied and its mass determined in $p\bar{p}$ and $e^+e^-$ interactions; it is currently studied in $pp$ interactions at the LHC. The mass and width definition used here corresponds to a Breit-Wigner with mass-dependent width.

Production of on-shell W bosons at hadron colliders is tagged by the high $p_T$ charged lepton from its leptonic decay modes. Owing to the unknown parton-parton effective energy and missing energy in the longitudinal direction, the hadron collider experiments reconstruct the transverse mass of the W boson, and derive the W mass from comparing the transverse mass distribution with Monte Carlo predictions as a function of the mass $m_W$. The transverse momentum of the charged lepton itself and the transverse missing energy (arising from the neutrino in W decay) are also sensitive to the W mass and used in its determination. These analyses use the electron and muon decay modes of the W boson.

At the $e^+e^-$ collider LEP, a precise knowledge of the beam energy enables one to determine the $e^+e^- \to W^+W^-$ cross section as a function of center of mass energy, as well as to reconstruct the W mass precisely from its decay products, even if one of them decays leptonically. Close to the $W^+W^-$ production threshold ($\sqrt{s} = 161$ GeV), the dependence of the W-pair production cross section on $m_W$ is large, and this was used to determine $m_W$. At higher centre-of-mass energies (172 to 209 GeV) this dependence is much weaker, thus W bosons were directly reconstructed and the mass determined as the invariant mass of the decay products, improving the resolution with a kinematic fit.

To compute the LEP average W mass, each experiment provided its measured W mass for the $q\bar{q}q\bar{q}$ and $q\bar{q}l\bar{\nu}l$ ($\ell = e, \mu, \tau$) channels at each center-of-mass energy, along with a detailed decomposition of uncertainty contributions: statistical, uncorrelated, partially correlated and fully correlated systematics [1]. This yielded a combined LEP average W mass of $m_W = 80.376 \pm 0.033$ GeV. Errors on $m_W$ due to uncertainties in the LEP beam energy (9 MeV), and possible effect of color reconnection (CR) and Bose-Einstein correlations (BEC) between quarks from different W bosons (8 MeV) are included. Similarly the combined LEP average of the W boson total width is determined to be $\Gamma_W = 2.195 \pm 0.083$ GeV [1].

In the past a similar procedure was followed for the measurements at hadron colliders. The two Tevatron experiments CDF [2] and D0 [3] identified common systematic errors: uncertainties due to the parton distribution functions (PDF), radiative corrections, and choice of mass (width) in the width (mass) measurements were treated as correlated and the resultant mass [4] and width [5] of the W were determined. The W boson total width obtained is $\Gamma_W = 2.046 \pm 0.049$ GeV. Similarly the mass results from the two LHC experiments, ATLAS [6] and LHCb [7] were combined to obtain an LHC average mass of the W, and an overall hadron collider average of the W mass was obtained as well [8].

The results on mass and width published by the experiments are shown in Figures 54.1 and 54.2, respectively. Until 2022, all measurements were in good agreement with each other and with the world average [8]. In April 2022 the CDF collaboration published a result $m_W = 80.4335 \pm 0.0094$ GeV [10], based upon their full Run-II dataset of $8.8\,\text{fb}^{-1}$. The result is of higher precision than the 2022 world average, but the two results disagree significantly.<fnref n="1" />

The LHC-TeV W-mass Working Group, including W-mass experts from all hadron collider experiments, CDF, D0, ATLAS, CMS, LHCb, has been working to understand better the nature of this disagreement and suggest a way forward to obtain a world average value of the W mass. Corrections, uncertainties and their correlations have been evaluated to a greater detail and used in the

<figure><img src="image_303.png" /><figcaption>**Figure 54.1:** Measurements of the W-boson mass by the LEP, Tevatron and LHC experiments. The CDF 2022 result is not used in the world average. In March 2024, after the cut-off of results for this review, the ATLAS Collaboration has submitted for publication a refined analysis of their data used in [6] and obtained a more precise mass value of $80.3665 \pm 0.0159$ GeV [9].</figcaption></figure>

<figure><img src="image_304.png" /><figcaption>**Figure 54.2:** Measurements of the W-boson width by the LEP and Tevatron experiments. In March 2024, after the cut-off of results for this review, the ATLAS Collaboration has submitted for publication a refined analysis of their data used in [6] and obtained a width value of $2.202 \pm 0.047$ GeV [9].</figcaption></figure>

combination: the central values are corrected to a common theory description and PDF, and the uncertainties, especially due to PDFs, have been re-evaluated to account for correlations properly. The group reports [11] that a combination of all W-mass measurements has a probability of compatibility of 0.5% only, and is therefore disfavoured. A 91% probability of compatibility is obtained when the CDF-II measurement is removed. The corresponding value of the W boson mass is $m_W = 80369.2 \pm 13.3$ MeV, which we quote as the World Average. More details are given in [11].

The Standard Model prediction from an electroweak fit, including Z-pole data and the measured masses of the top quark and of the Higgs boson but excluding results on $m_W$ and $\Gamma_W$, implies a W-boson mass of $m_W = 80.353 \pm 0.006$ GeV and a W-boson

<footnote n="1">¹The 2022 CDF result includes the $2.2\,\text{fb}^{-1}$ of data used for the previous CDF Run-II result of 80.387 $\pm$ 0.019 GeV [2]. Incorporating the improved understanding of PDFs and track reconstruction, the central value of the $2.2\,\text{fb}^{-1}$ result is increased by 13.5 MeV to 80,400.5 MeV [10].</footnote>
width of $\Gamma_W = 2.089 \pm 0.001$ GeV; see Section 10, Electroweak Model and Constraints on New Physics, J.Erler and A.Freitas, 2024, this review. Note that the electroweak fit of Section 10 to all measurements uses slightly different measurements of the W boson mass and width as explained there.

References

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[5] The Tevatron Electroweak Working Group, for the CDF and D0 Collaborations, March 2010, [arXiv:1003.2826].

[6] M. Aaboud et al. (ATLAS), Eur. Phys. J. C **78**, 2, 110 (2018), [Erratum: Eur.Phys.J.C 78, 898 (2018)], [arXiv:1701.07240].

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[8] R. L. Workman et al. (Particle Data Group), PTEP **2022**, 083C01 (2022).

[9] G. Aad et al. (ATLAS) (2024), [arXiv:2403.15085].

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## 55. Z Boson

Revised August 2018 by M. Grünewald (University Coll. Dublin) and A. Gurtu (CERN; TIFR Mumbai).

Precision measurements at the Z-boson resonance using electron-positron colliding beams began in 1989 at the SLC and at LEP. During 1989-95, the four LEP experiments (ALEPH, DELPHI, L3, OPAL) made high-statistics studies of the production and decay properties of the Z. Although the SLD experiment at the SLC collected much lower statistics, it was able to match the precision of LEP experiments in determining the effective electroweak mixing angle $\sin^2\theta_W$ and the rates of Z decay to b- and c-quarks, owing to availability of polarized electron beams, small beam size, and stable beam spot.

The Z-boson properties reported in this section may broadly be categorized as:

* The standard ‘lineshape’ parameters of the Z consisting of its mass, $M_Z$, its total width, $\Gamma_Z$, and its partial decay widths, $\Gamma(\text{hadrons})$, and $\Gamma(\ell\bar{\ell})$ where $\ell = e, \mu, \tau, \nu$;

* Z asymmetries in leptonic decays and extraction of Z couplings to charged and neutral leptons;

* The b- and c-quark-related partial widths and charge asymmetries which require special techniques;

* Determination of Z decay modes and the search for modes that violate known conservation laws;

* Average particle multiplicities in hadronic Z decay;

* Z anomalous couplings.

The effective vector and axial-vector coupling constants describing the Z-to-fermion coupling are also measured in $p\bar{p}$ and ep collisions at the Tevatron and at HERA. The corresponding cross-section formulae are given in Section 39 (Cross-section formulae for specific processes) and Section 16 (Structure Functions) in this Review. In this minireview, we concentrate on the measurements in $e^+e^-$ collisions at LEP and SLC.

The standard ‘lineshape’ parameters of the Z are determined from an analysis of the production cross sections of these final states in $e^+e^-$ collisions. The $Z \to \nu\bar{\nu}(\gamma)$ state is identified directly by detecting single photon production and indirectly by subtracting the visible partial widths from the total width. Inclusion in this analysis of the forward-backward asymmetry of charged leptons, $A_{FB}^{(0,\ell)}$, of the $\tau$ polarization, $P(\tau)$, and its forward-backward asymmetry, $P(\tau)^{fb}$, enables the separate determination of the effective vector ($\bar{g}_V$) and axial vector ($\bar{g}_A$) couplings of the Z to these leptons and the ratio ($\bar{g}_V/\bar{g}_A$), which is related to the effective electroweak mixing angle $\sin^2\theta_W$ (see the “Electroweak Model and Constraints on New Physics” review).

Determination of the b- and c-quark-related partial widths and charge asymmetries involves tagging the b and c quarks for which various methods are employed: requiring the presence of a high momentum prompt lepton in the event with high transverse momentum with respect to the accompanying jet; impact parameter and lifetime tagging using precision vertex measurement with high-resolution detectors; application of neural-network techniques to classify events as b or non-b on a statistical basis using event-shape variables; and using the presence of a charmed meson ($D/D^*$) or a kaon as a tag.

### 55.1 Z-parameter determination

LEP was run at energy points on and around the Z mass (88-94 GeV) constituting an energy ‘scan’. The shape of the cross-section variation around the Z peak can be described by a Breit-Wigner ansatz with an energy-dependent total width [1-3]. The three main properties of this distribution, viz., the position of the peak, the width of the distribution, and the height of the peak, determine respectively the values of $M_Z$, $\Gamma_Z$, and $\Gamma(e^+e^-) \times \Gamma(\bar{f}\bar{f})$, where $\Gamma(e^+e^-)$ and $\Gamma(\bar{f}\bar{f})$ are the electron and fermion partial widths of the Z. The quantitative determination of these parameters is done by writing analytic expressions for these cross sections in terms of the parameters, and fitting the calculated cross sections to the measured ones by varying these parameters, taking properly into account all the errors.

Single-photon exchange ($\sigma_\gamma^0$) and $\gamma$-Z interference ($\sigma_{\gamma Z}^0$) are included, and the large ($\sim 25\%$) initial-state radiation (ISR) effects are taken into account by convoluting the analytic expressions over a ‘Radiator Function’ [1-5] $H(s, s')$. Thus for the process $e^+e^- \to \bar{f}\bar{f}$:

$$ \sigma_f(s) = \int H(s, s') \sigma_f^0(s') ds' \qquad (55.1) $$

$$ \sigma_f^0(s) = \sigma_Z^0 + \sigma_\gamma^0 + \sigma_{\gamma Z}^0 \qquad (55.2) $$

$$ \sigma_{Z}^{0} = \frac{12\pi}{M_{Z}^{2}} \frac{\Gamma(e^{+}e^{-})\Gamma(\bar{f}\bar{f})}{\Gamma_{Z}^{2}} \frac{s \Gamma_{Z}^{2}}{(s - M_{Z}^{2})^{2} + s^{2}\Gamma_{Z}^{2}/M_{Z}^{2}} \qquad (55.3) $$

$$ \sigma_{\gamma}^{0} = \frac{4\pi\alpha^{2}(s)}{3s} Q_{f}^{2} N_{c}^{f} \qquad (55.4) $$

$$ \sigma_{\gamma Z}^{0} = -\frac{2\sqrt{2}\alpha(s)}{3} \left( Q_f G_F N_c^f G_V^c G_V^f \right) \times \frac{(s - M_Z^2) M_Z^2}{(s - M_Z^2)^2 + s^2 \Gamma_Z^2 / M_Z^2} \qquad (55.5) $$

where $Q_f$ is the charge of the fermion, $N_c^f = 3$ for quarks and 1 for leptons, and $G_V^f$ is the vector coupling of the Z to the fermion-antifermion pair $\bar{f}\bar{f}$.

Since $\sigma_{\gamma Z}^0$ is expected to be much less than $\sigma_Z^0$, the LEP Collaborations have generally calculated the interference term in the framework of the Standard Model. This fixing of $\sigma_{\gamma Z}^0$ leads to a tighter constraint on $M_Z$, and consequently a smaller error on its fitted value. It is possible to relax this constraint and carry out the fit within the S-matrix framework, which is briefly described in the next section.

In the above framework, the QED radiative corrections have been explicitly taken into account by convoluting over the ISR and allowing the electromagnetic coupling constant to run [6]: $\alpha(s) = \alpha/(1-\Delta\alpha)$. On the other hand, weak radiative corrections that depend upon the assumptions of the electroweak theory and on the values of $M_{\text{top}}$ and $M_{\text{Higgs}}$ are accounted for by **absorbing them into the couplings**, which are then called the **effective couplings** $G_V$ and $G_A$ (or alternatively the effective parameters of the $\star$ scheme of Kennedy and Lynn [7].)

$G_V^f$ and $G_A^f$ are complex numbers with small imaginary parts. As experimental data does not allow simultaneous extraction of both real and imaginary parts of the effective couplings, the convention $g_A^f = \text{Re}(G_A^f)$ and $g_V^f = \text{Re}(G_V^f)$ is used and the imaginary parts are added in the fitting code [4].

Defining

$$ A_f = 2 \frac{g_V^f \cdot g_A^f}{(g_V^f)^2 + (g_A^f)^2} \qquad (55.6) $$

the lowest-order expressions for the various lepton-related asymmetries on the Z pole are [8-10] $A_{FB}^{(0,\ell)} = (3/4)A_e A_f$, $P(\tau) = -A_\tau$, $P(\tau)^{fb} = -(3/4)A_e$, $A_{LR} = A_e$. The full analysis takes into account the energy-dependence of the asymmetries. Experimentally $A_{LR}$ is defined as $(\sigma_L - \sigma_R)/(\sigma_L + \sigma_R)$, where $\sigma_{L(R)}$ are the $e^+e^- \to Z$ production cross sections with left- (right)-handed electrons.

The definition of the partial decay width of the Z to $\bar{f}\bar{f}$ includes the effects of QED and QCD final-state corrections, as well as the contribution due to the imaginary parts of the couplings:

$$ \Gamma(\bar{f}\bar{f}) = \frac{G_F M_Z^3}{6\sqrt{2}\pi} N_c^f (|G_A^f|^2 R_A^f + |G_V^f|^2 R_V^f) + \Delta_{ew/QCD} \qquad (55.7) $$

where $R_A^f$ and $R_V^f$ are radiator factors to account for final state QED and QCD corrections, as well as effects due to nonzero fermion masses, and $\Delta_{ew/QCD}$ represents the non-factorizable electroweak/QCD corrections.

### 55.2 S-matrix approach to the Z

While most experimental analyses of LEP/SLC data have followed the ‘Breit-Wigner’ approach, an alternative S-matrix-based