# 62. Form Factors for Semileptonic Kaon ($K_{\ell 3}$), Radiative Pion ($\pi_{\ell 2\gamma}$) and Kaon ($K_{\ell 2\gamma}$) Decays

Revised August 2023 by M.A. Bychkov (Virginia U.) and G. D'Ambrosio (INFN, Napoli).

## 62.1 $\pi_{\ell 2\gamma}^{\pm}$ and $K_{\ell 2\gamma}^{\pm}$ Form Factors

The radiative decays, $\pi^{\pm} \to l^{\pm}\nu\gamma$ and $K^{\pm} \to l^{\pm}\nu\gamma$, with $l$ standing for an $e$ or a $\mu$, and $\gamma$ for a real or virtual photon ($e^{+}e^{-}$ pair), provide a powerful tool to investigate the hadronic structure of pions and kaons. The structure-dependent part $SD_i$ of the amplitude describes the emission of photons from virtual hadronic states, and is parametrized in terms of form factors $V, A$, (vector, axial vector), in the standard description [1–4]. Note that in the Listings and some literature, equivalent nomenclature $F_V$ and $F_A$ for the vector and axial form factors is often used. Exotic, non-standard contributions like $i = T, S$ (tensor, scalar) have also been considered. Apart from the SD terms, there is also the Inner Bremsstrahlung amplitude, IB, corresponding to photon radiation from external charged particles and described by Low theorem in terms of the physical decay $\pi^{\pm}(K^{\pm}) \to l^{\pm}\nu$. Experiments try to optimize their kinematics so as to minimize the IB part of the amplitude.

The SD amplitude in its standard form is given as

$$M(\text{SD}_V) = \frac{-eG_F U_{qq'} \epsilon^\mu l^\nu V^P \epsilon_{\mu\nu\sigma\tau} k^\sigma q^\tau}{\sqrt{2} m_P} \quad (62.1)$$

$$M(\text{SD}_A) = \frac{-ieG_F U_{qq'} \epsilon^\mu l^\nu \{A^P [(qk - k^2)g_{\mu\nu} - q_\mu k_\nu] + R^P k^2 g_{\mu\nu}\}}{\sqrt{2} m_P} \quad (62.2)$$

which contains an additional axial form factor $R^P$ which only can be accessed if the photon remains virtual. $U_{qq'}$ is the Cabibbo-Kobayashi-Maskawa mixing-matrix element; $\epsilon^\mu$ is the polarization vector of the photon (or the effective vertex, $\epsilon^\mu = (e/k^2)\bar{u}(p_-)\gamma^\mu v(p_+)$, of the $e^+e^-$ pair); $\ell^\nu = \bar{u}(p_\nu)\gamma^\nu(1 - \gamma_5)v(p_\ell)$ is the lepton-neutrino current; $q$ and $k$ are the meson and photon four-momenta ($k = p_+ + p_-$ for virtual photons); and $P$ stands for $\pi$ or $K$.

For decay processes where the photon is real, the partial decay width can be written in analytical form as a sum of IB, SD, and IB/SD interference terms INT [1, 4]:

$$\begin{aligned} \frac{d^2 \Gamma_{P \to \ell \nu \gamma}}{dxdy} &= \frac{d^2 (\Gamma_{\text{IB}} + \Gamma_{\text{SD}} + \Gamma_{\text{INT}})}{dxdy} \\ &= \frac{\alpha}{2\pi} \Gamma_{P \to \ell \nu} \frac{1}{(1-r)^2} \left\{ \begin{aligned} &\text{IB}(x, y) \\ &+ \frac{1}{r} \left( \frac{m_P}{2f_P} \right)^2 \left[ (V+A)^2 \text{SD}^+(x, y) + (V-A)^2 \text{SD}^-(x, y) \right] \\ &+ \epsilon_P \frac{m_P}{f_P} \left[ (V+A) \text{S}_{\text{INT}}^+(x, y) + (V-A) \text{S}_{\text{INT}}^-(x, y) \right] \end{aligned} \right\}. \quad (62.3) \end{aligned}$$

Here

$$\begin{aligned} \text{IB}(x, y) &= \left[ \frac{1 - y + r}{x^2(x + y - 1 - r)} \right] \\ &\quad \left[ x^2 + 2(1 - x)(1 - r) - \frac{2xr(1 - r)}{x + y - 1 - r} \right] \\ \text{SD}^+(x, y) &= (x + y - 1 - r) [(x + y - 1)(1 - x) - r] \\ \text{SD}^-(x, y) &= (1 - y + r) [(1 - x)(1 - y) + r] \\ \text{S}_{\text{INT}}^+(x, y) &= \left[ \frac{1 - y + r}{x(x + y - 1 - r)} \right] \left[ (1 - x)(1 - x - y) + r \right] \\ \text{S}_{\text{INT}}^-(x, y) &= \left[ \frac{1 - y + r}{x(x + y - 1 - r)} \right] \left[ x^2 - (1 - x)(1 - x - y) - r \right] \end{aligned} \quad (62.4)$$

where $x = 2E_\gamma/m_P$, $y = 2E_\ell/m_P$, $r = (m_\ell/m_P)^2$, $f_P$ is the meson decay constant, and $\epsilon_P$ is +1 for pions and -1 for kaons. The structure dependent terms $SD^+$ and $SD^-$ are shown in Fig. 1. The $SD^-$ term is maximized in the same kinematic region where overwhelming IB term dominates (along $x + y = 1$ diagonal). Thus experimental yields with less background are dominated by $SD^+$ contribution and proportional to $A^P + V^P$ making simultaneous precise determination of the form factors difficult.

<figure><img src="image_316.png" /><figcaption>**Figure 62.1:** Components of the structure dependent terms of the decay width. Left: $SD^+$, right: $SD^-$</figcaption></figure>

Recently, formulas 62.3 and 62.4 have been extended to describe polarized distributions in radiative meson and muon decays [5].

The “helicity” factor $r$ is responsible for the enhancement of the SD over the IB amplitude in the decays $\pi^{\pm} \to e^{\pm}\nu\gamma$, while $\pi^{\pm} \to \mu^{\pm}\nu\gamma$ is dominated by IB. Interference terms are important for the decay $K^{\pm} \to \mu^{\pm}\nu\gamma$ [6], but contribute only a few percent correction to pion decays. However, they provide the basis for determining the signs of $V$ and $A$. Radiative corrections to the decay $\pi^{+} \to e^{+}\nu\gamma$ have to be taken into account in the analysis of the precision experiments. They make up to 4% corrections in the total decay rate [7]. In $\pi^{\pm} \to e^{\pm}\nu e^{+}e^{-}$ and $K^{\pm} \to \ell^{\pm}\nu e^{+}e^{-}$ decays, all three form factors, $V^P, A^P$, and $R^P$, can be determined [8, 9].

Theoretically, the first non-trivial $\chi PT$ contributions to $A^P$ and $V^P$ appear at $O(p^4)$ [4], respectively from Gasser-Leutwyler coefficients, $L_i$'s, and the anomalous lagrangian:

$$A^P = \frac{4\sqrt{2}M_P}{F_\pi} (L_9^r + L_{10}^r), \quad V^P = \frac{\sqrt{2}M_P}{8\pi^2 F_\pi}. \quad (62.5)$$

In case of the kaon $A^K = 0.042$ and $V^K = 0.096$. $O(p^6)$ contributions to $A^K$ can be predicted accurately: they are flat in the momentum dependence and shift the $O(p^4)$ value to 0.034. $O(p^6)$ contributions to $V^K$ are model dependent and can be approximated by a form factor linearly dependent on momentum. For example, when looking at the spread of results obtained within two different models, the constant piece of this linear form factor is shifted to $0.078 \pm 0.005$ [1, 2, 4].

We give the experimental $\pi^{\pm}$ form factors $V^\pi, A^\pi$, and $R^\pi$ in the Listings. In the $K^{\pm}$ Listings, we give the extracted sum $A^K + V^K$ and difference $A^K - V^K$, as well as $V^K, A^K$ and $R^K$. In particular KLOE has measured for the constant piece of the form factor $A^K + V^K = 0.125 \pm 0.007 \pm 0.001$ [10] while ISTRA+, $V^K - A^K = 0.21 \pm 0.04 \pm 0.04$ [11].

The pion vector form factor, $V^\pi$, is related via CVC (Conserved Vector Current) to the $\pi^0 \to \gamma\gamma$ decay width. The constant term is given by $|V^\pi(0)| = (1/\alpha)\sqrt{2\Gamma_{\pi^0 \to \gamma\gamma}/\pi m_{\pi^0}}$ [3]. The resulting value, $V^\pi(0) = 0.0259(9)$, has been confirmed by calculations based on chiral perturbation theory ($\chi PT$) [4], and by two experiments given in the Listings.

A recent experiment by the PIBETA collaboration [12] obtained a $V^\pi(0)$ that is in excellent agreement with the CVC hypothesis. It also measured the slope parameter $a$ in $V^\pi(s) = V^\pi(0)(1+a.s)$, where $s = (1-2E_\gamma/m_\pi)$, and $E_\gamma$ is the gamma energy in the pion rest frame: $a = 0.095 \pm 0.058$. A functional dependence on $s$ is expected for all form factors. It becomes non-negligible in the
case of $V^π(s)$ when a wide range of photon momenta is recorded; proper treatment in the analysis of K decays is mandatory.

The form factor, $R^P$, can be related to the electromagnetic radius, $r_P$, of the meson [2]: $R^P = \frac{1}{3}m_P f_P \langle r_P^2 \rangle$ using PCAC (Partial Conserved Axial vector Current).

In lowest order $χPT$, the ratio $A^π/V^π$ is related to the pion electric polarizability $α_E = [α/(8π^2m_πf_π^2)] \times A^π/V^π$ [13]. Direct experimental and theoretical status of pion polarizability studies currently is not settled. Most recent theoretical predictions from $χPT$ at $O(p^6)$ [14] and experimental results from COMPASS collaboration [15] favor a small value of pion polarizability $α_π \sim (2 \div 3) \times 10^{-4} \text{fm}^3$. Dispersive analysis of $γγ \to π^+π^-$ crossection [16] and experimental results from MAMI collaboration [17] report a much larger value of $α_π \sim 6 \times 10^{-4} \text{fm}^3$. Precise measurement of the pion form factors by PIBETA collaboration favors smaller values of polarizability $α_π = 2.7_{-0.5}^{+0.6} \times 10^{-4} \text{fm}^3$.

Several searches for the exotic form factors $F_T^π, F_S^K$ (tensor), and $F_S^K$ (scalar) have been pursued in the past. In particular, $F_T^π$ has been brought into focus by experimental as well as theoretical work [18]. New high-statistics data from the PIBETA collaboration have been re-analyzed together with an additional data set optimized for low backgrounds in the radiative pion decay. In particular, lower beam rates have been used in order to reduce the accidental background, thereby making the treatment of systematic uncertainties easier and more reliable. The PIBETA analysis now restricts $F_T^π$ to the range $-5.2 \times 10^{-4} < F_T^π < 4.0 \times 10^{-4}$ at a 90% confidence limit [12]. This result is in excellent agreement with the most recent theoretical work [4].

Precision measurements of radiative pion and kaon decays are effective tools to study QCD in the non-perturbative region and are of interest beyond the scope of radiative decays. Meanwhile other processes such as $π^+ \to e^+ν$ that seem to be better suited to search for new physics at the precision frontier are currently studied. The advantages of such process are the very accurate and reliable theoretical predictions and the more straightforward experimental analysis.

## 62.2 $K_{\ell 3}^{\pm}$ and $K_{\ell 3}^{0}$ Form Factors

Assuming that only the vector current contributes to $K \to \pi \ell \nu$ decays, we write the matrix element as

$$ M \propto f_+(t) \left[ (P_K + P_\pi)_\mu \bar{\ell} \gamma_\mu (1 + \gamma_5) \nu \right] + f_-(t) \left[ m_\ell \bar{\ell} (1 + \gamma_5) \nu \right], \quad (62.6) $$

where $P_K$ and $P_\pi$ are the four-momenta of the K and $\pi$ mesons, $m_\ell$ is the lepton mass, and $f_+$ and $f_-$ are dimensionless form factors which can depend only on $t = (P_K - P_\pi)^2$, the square of the four-momentum transfer to the leptons. If time-reversal invariance holds, $f_+$ and $f_-$ are relatively real. $K_{\mu 3}$ experiments, discussed immediately below, measure $f_+$ and $f_-$, while $K_{e3}$ experiments, discussed further below, are sensitive only to $f_+$ because the small electron mass makes the $f_-$ term negligible.

### 62.2.1 $K_{\mu 3}$ Decays

Analyses of $K_{\mu 3}$ data frequently assume a linear dependence of $f_+$ and $f_-$ on $t$, i.e.,

$$ f_{\pm}(t) = f_{\pm}(0) \left[ 1 + \lambda_{\pm}(t/m_{\pi+}^2) \right] . \quad (62.7) $$

Most $K_{\mu 3}$ data are adequately described by formula 62.7 for $f_+$ and a constant $f_-$ (i.e., $\lambda_+ = 0$).

There are two equivalent parametrizations commonly used in these analyses: $\lambda_+, \xi(0)$ parametrization and $\lambda_+, \lambda_0$ parametrization.

Older analyses of $K_{\mu 3}$ data often introduce the ratio of the two form factors

$$ \xi(t) = f_{-}(t)/f_{+}(t) . \quad (62.8) $$

The $K_{\mu 3}$ decay distribution is then described by the two parameters $\lambda_+$ and $\xi(0)$ (assuming time reversal invariance and $\lambda_- = 0$).

More recent $K_{\mu 3}$ analyses have parametrized in terms of the form factors $f_+$ and $f_0$, which are associated with vector and

scalar exchange, respectively, to the lepton pair. $f_0$ is related to $f_+$ and $f_-$ by

$$ f_0(t) = f_+(t) + \left[ t/(m_K^2 - m_\pi^2) \right] f_-(t) . \quad (62.9) $$

Here $f_0(0)$ must equal $f_+(0)$. The earlier assumption that $f_+$ is linear in $t$ and $f_-$ is constant leads to $f_0$ linear in $t$:

$$ f_0(t) = f_0(0) \left[ 1 + \lambda_0 (t/m_{\pi+}^2) \right] . \quad (62.10) $$

With the assumption that $f_0(0) = f_+(0)$, the two parametrizations, $(\lambda_+, \xi(0))$ and $(\lambda_+, \lambda_0)$ are equivalent as long as correlation information is retained. $(\lambda_+, \lambda_0)$ correlations tend to be less strong than $(\lambda_+, \xi(0))$ correlations.

Since the 2006 edition of the Review [19], we no longer quote results in the $(\lambda_+, \xi(0))$ parametrization. We have removed many older low statistics results from the Listings. See the 2004 version of this note [20] for these older results, and the 1982 version [21] for additional discussion of the $K_{\mu 3}^0$ parameters, correlations, and conversion between parametrizations.

More recent high-statistics experiments have included a quadratic term in the expansion of $f_+(t)$,

$$ f_+(t) = f_+(0) \left[ 1 + \lambda'_+(t/m_{\pi+}^2) + \frac{\lambda''_+}{2} (t/m_{\pi+}^2)^2 \right] . \quad (62.11) $$

If there is a non-vanishing quadratic term, then $\lambda_+$ of formula 62.7 represents the average slope, which is then different from $\lambda'_+$. Our convention is to include the factor $\frac{1}{2}$ in the quadratic term, and to use $m_{\pi+}$ even for $K_{e3}^+$ and $K_{\mu 3}^+$ decays. We have converted other's parametrizations to match our conventions, as noted in the beginning of the "$K_{\ell 3}^{\pm}$ and $K_{\ell 3}^{0}$ Form Factors" sections of the Listings.

There are two alternatives to the Taylor parametrization: The Pole Parametrization and Dispersive Parametrization.

The pole model describes the $t$-dependence of $f_+(t)$ and $f_0(t)$ in terms of the exchange of the lightest vector and scalar $K^*$ mesons with masses $M_V$ and $M_S$, respectively:

$$ f_+(t) = f_+(0) \left[ \frac{M_V^2}{M_V^2 - t} \right], \quad f_0(t) = f_0(0) \left[ \frac{M_S^2}{M_S^2 - t} \right] . \quad (62.12) $$

The Dispersive Parametrization approach, valid in a much wider kinematic range and able to describe at the same time $\tau$-decay data, [22] uses dispersive techniques and the known low-energy $K-\pi$ phases to parametrize the vector and scalar form factors:

$$ f_+(t) = f_+(0) \exp \left[ \frac{t}{m_\pi^2} (\Lambda_+ + H(t)) \right]; \quad (62.13) $$

$$ f_0(t) = f_+(0) \exp \left[ \frac{t}{(m_K^2 - m_\pi^2)} (\ln[C] - G(t)) \right], \quad (62.14) $$

where $\Lambda_+$ is the slope of the vector form factor, and $\ln C = \ln[f_0(m_K^2 - m_\pi^2)]$ is the logarithm of the scalar form factor at the Callan-Treiman point. The functions $H(t)$ and $G(t)$ are dispersive integrals.

### 62.2.2 $K_{e3}$ Decays

Analysis of $K_{e3}$ data is simpler than that of $K_{\mu 3}$ because the second term of the matrix element assuming a pure vector current [formula 62.6 above] can be neglected. Here $f_+$ can be assumed to be linear in $t$, in which case the linear coefficient $\lambda_+$ of formula 62.7 is determined, or quadratic, in which case the linear coefficient $\lambda'_+$ and quadratic coefficient $\lambda''_+$ of formula 62.11 are determined.

If we remove the assumption of a pure vector current, then the matrix element for the decay, in addition to the terms in formula 62.6, would contain

$$ +2m_K f_S \bar{\ell}(1+\gamma_5)\nu + (2f_T/m_K)(P_K)_\lambda (P_\pi)_\mu \bar{\ell} \sigma_{\lambda\mu} (1+\gamma_5)\nu, \quad (62.15) $$
where $f_S$ is the scalar form factor, and $f_T$ is the tensor form factor. In the case of the $K_{e3}$ decays where the $f_-$ term can be neglected, experiments have yielded limits on $|f_S/f_+|$ and $|f_T/f_+|$.

For $K_{e3}$ data, we determine best values for the three parametrizations: linear ($\lambda_+$), quadratic ($\lambda'_+, \lambda''_+$) and pole ($M_V$). For $K_{\mu3}$ data, we determine best values for the three parametrizations: linear ($\lambda_+$, $\lambda_0$), quadratic ($\lambda'_+, \lambda''_+$, $\lambda_0$) and pole ($M_V, M_S$). We then assume $\mu-e$ universality so that we can combine $K_{e3}$ and $K_{\mu3}$ data, and again determine best values for the three parametrizations: linear ($\lambda_+$, $\lambda_0$), quadratic ($\lambda'_+, \lambda''_+$, $\lambda_0$), and pole ($M_V, M_S$). When there is more than one parameter, fits are done including input correlations. Simple averages suffice in the two $K_{e3}$ cases where there is only one parameter: linear ($\lambda_+$) and pole ($M_V$).

A comprehensive global analysis of the semileptonic kaon decay data and its effect on the CKM unitarity debate can be found in [23, 24]. An update on experimental data including NA48/2 newest results can be found in [25].

Both KTeV and KLOE see an improvement in the quality of their fits relative to linear fits when a quadratic term is introduced, as well as when the pole parametrization is used. The quadratic parametrization has the disadvantage that the quadratic parameter $\lambda''_+$ is highly correlated with the linear parameter $\lambda'_+$, in the neighborhood of 95%, and that neither parameter is very well determined. The pole fit has the same number of parameters as the linear fit, but yields slightly better fit probabilities, so that it would be advisable for all experiments to include the pole parametrization as one of their choices.

The “Kaon Particle Listings” show the results with and without assuming $\mu-e$ universality. The “Meson Summary Tables” show all of the results assuming $\mu-e$ universality, but most results not assuming $\mu-e$ universality are given only in the Listings.

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## 63. Spectroscopy of Light Meson Resonances

Revised March 2024 by C. Amsler (Stefan Meyer Inst.), A. Masoni (INFN, Cagliari) and G. Venanzoni (Liverpool U.; INFN, Pisa).

<table><tr><td>63.1</td><td>Introduction</td><td>874</td></tr><tr><td>63.2</td><td>Scalar mesons</td><td>874</td></tr><tr><td>63.3</td><td>Glueballs</td><td>875</td></tr><tr><td>63.3.1</td><td>Scalar glueballs</td><td>876</td></tr><tr><td>63.3.2</td><td>Tensor glueballs</td><td>877</td></tr><tr><td>63.4</td><td>Pseudoscalar mesons</td><td>877</td></tr><tr><td>63.5</td><td>Vector mesons</td><td>878</td></tr><tr><td>63.5.1</td><td>The ρ(770) meson</td><td>878</td></tr><tr><td>63.5.2</td><td>The ρ(770) excitations</td><td>878</td></tr><tr><td>63.6</td><td>Axial-vector mesons</td><td>879</td></tr><tr><td>63.7</td><td>Hybrid mesons</td><td>880</td></tr><tr><td>63.8</td><td>Tetraquark states</td><td>880</td></tr><tr><td>63.9</td><td>Baryonia</td><td>881</td></tr><tr><td>63.10</td><td>Conclusions</td><td>881</td></tr></table>

### 63.1 Introduction

According to the constituent quark model, a light meson consists of a color singlet quark-antiquark pair made of the u, d, or s quarks and the $\bar{u}$, $\bar{d}$, or $\bar{s}$ antiquarks, grouped into a flavor multiplet of SU(3). However, additional mesons made of bound gluons (glueballs) could exist in the same mass range, suggested by the self coupling of gluons in QCD. Multiquark structures are also possible, such as quark-antiquark pairs with an excited gluon (hybrid mesons). Tetraquarks are compact color singlets of diquark-antidiquark pairs ($qq\bar{q}\bar{q}$) or ‘molecular’ bound states of two mesons ($qq\bar{q}\bar{q}$). More complex systems such as $qqq\bar{q}\bar{q}$ (baryonia) are also predicted.

Fundamentals on the constituent quark model on light and heavy mesons and baryons (including hadrons with charm and bottom quarks), and on predictions from lattice gauge theories, are described in ‘Quark Model’ in this issue of the Review of Particle Physics, henceforth called the Review. In the present text we describe the experimental spectrum of light mesons and their classifications within the constituent quark model, with emphasis on states exhibiting properties incompatible with $q\bar{q}$ structures. The discussion is driven by the results entered in the database of the Review. The spectrum of kaon excitations is much less clear cut and therefore deferred to a future edition, when further data might become available. More detailed discussions on exotic mesons – including those involving the c and b quarks – can be found in Refs. [1–7] and in ‘Heavy Non-$q\bar{q}$ Mesons’ of the Review. For more information on the meson (and baryon) spectrum of light (and heavy) quarks we refer to one of the textbooks [8].

Figure 63.1 shows the mass spectrum of $q\bar{q}$ mesons. The mass (vertical axis) increases with orbital excitations $\ell$ (horizontal axis) and radial excitation $n$. In the quark model one uses the notation $nS$, $nP$, $nD$... for the radial excitations ($n=1$ for the ground states), in contrast to the usual 1s, 2p, 3d... notation in atomic physics. Each box represents an SU(3) nonet containing multiplets of the isospin $i$, three isovectors, two strange isodoublets, and two isosinglets (one SU(3) singlet and one SU(3) octet). The spin, parity [$P = -(-1)^\ell$] and C-parity [$C = (-1)^{\ell+s}$] take the possible values $J^{PC} = 0^{++}$ (scalar), $0^{-+}$ (pseudoscalar), $1^{--}$ (vector), $1^{\pm}$ (axial- or pseudovector), $2^{++}$ (tensor), etc. Therefore mesons with the exotic quantum numbers $0^{--}$, $0^{+-}$, $1^{+}$, $2^{+}$, $3^{-}$, etc. cannot be $q\bar{q}$ states. Note that the C-parity is defined only for the electrically neutral non-strange nonet members.

The two isosinglets in each nonet mix with an angle $\theta$ close to the ideal value of $35.3^\circ$ for the $1^{--}$, $1^{\pm}$, $2^{++}$ and $3^{--}$ nonets, in which case the isosinglets decouple to $u\bar{u}+d\bar{d}$ and $s\bar{s}$. The orbital excitations $\ell \ge 1$ consist of four nonets for each value of $n$, since $j = \ell$ for antiparallel quark spins and $j = \ell-1$, $\ell$ or $\ell+1$ for parallel spins. Since the C-parity is not defined for strange mesons, the $K_{1A}$ and $K_{1B}$ in the axial vector $1^{++}$ and $1^{+-}$ nonets of fig. 63.1 are mathematical constructs which mix to give the observed

$K_1(1270)$ and $K_1(1400)$ mesons. A similar scenario occurs for the observed $2^{-+} K_2(1770)$ and $2^{--} K_2(1820)$ mesons.

<figure><img src="image_317.png" /><figcaption>Figure 63.1: The mesons made of the u, d, and s light quarks are organised in $J^{PC}(n^{2s+1}\ell_j)$ nonets with isospin $i$. The established mesons (which appear in the Summary Table of the Review) are shown in the dark (blue) areas. The white areas contain those omitted in the Summary Table but reported in the Listings. States with the same $J^{PC}$ mix, such as the $2^3S_1$ and $1^3D_1$ mesons. The states become broad and overlap with increasing masses, which complicates the determination of the resonance parameters e.g. mass, width and spin. For a complete list of mesons see the Listings.</figcaption></figure>

1. The $\eta(1440)$ stands for the $\eta(1405)$ and the $\eta(1475)$, section 63.4.

2. The classification of the scalar nonet is controversial. In alternative schemes the $1^3P_0$ nonet contains the light scalars below 1 GeV, section 63.2.

3. Considered established, but more data would be desirable.

4. An alternative to the $f_1(1420)$ is the $f_1(1510)$, section 63.6.

5. The $\pi(1800)$ has also been proposed as a hybrid meson, section 63.7.

6. The $\phi(2170)$ has also been proposed as a tetraquark state, section 63.8.

7. The $f_2(1950)$ has also been proposed as a tensor glueball, section 63.3.2.

As described in more detail below (section 63.2), it is hard to accommodate all the known scalar mesons in the lower $q\bar{q}$ nonets: The light scalar mesons $a_0(980)$, $K_0^*(700)$ (also known as $\kappa$), the $f_0(500)$ (also known as $\sigma$), and the $f_0(980)$, not shown in the figure, could build the lightest nonet, but could also be two-meson resonances or tetraquarks (section 63.8). Three isoscalars are known in the 1500 MeV region, the $f_0(1500)$ or $f_0(1710)$ being proposed as gluonic states mixing with the $f_0(1370)$ (section 63.3.1). The pseudoscalar labeled $\eta(1440)$ may consist of two states (section 63.4). The status of vector meson excitations is described in section 63.5. The axial-vector meson $f_1(1420)$ could be replaced in the $1^{++}$ nonet by the $f_1(1510)$ which, however, needs confirmation (see the Listings and section 63.6). Mesons with exotic quantum numbers are discussed in section 63.7.

### 63.2 Scalar mesons

Scalar mesons decay dominantly into pairs of pseudoscalar mesons ($\pi\pi$, $K\bar{K}$, $\pi\eta$, $\eta\eta$ or $\eta\eta'$). The widths tend to be large for those decaying into $\pi\pi$, due to the absence of angular momentum barrier and the large available phase space. Broad overlapping
states decaying into the same final state interfere, which compli-
cates the determination of their masses and widths. Furthermore,
the ground state glueball, a 0<sup>++</sup> expected below 2 GeV, will mix
with the isoscalar 0<sup>++</sup> qq states. The onset of the KK̄, ηη or ηη'
thresholds also distorts the line shapes and produces cusps. This
requires high statistics data and the use of coupled channel analy-
ses taking into account unitarity and analyticity (see ‘Resonances’
in the Review).

Two isovector scalars are established, the $a_0(980)$ and the $a_0(1450)$, together with five isoscalars: the very broad $f_0(500)$, the $f_0(980)$, the broad $f_0(1370)$, and the comparatively narrow $f_0(1500)$ and $f_0(1710)$. The strange partners are the $K_0^*(700)$ and the $K_0^*(1430)$. The $f_0(500)$ and $K_0^*(700)$ deserve a separate treatment and can be found in ‘Scalar Mesons below 1 GeV’, which contains more details on the $a_0(980)$ and $f_0(980)$, briefly reviewed in section 63.8.

The $a_0(1450)$ was first reported by the Crystal Barrel experi-
ment in $\bar{p}p$ annihilation with stopped antiprotons [9, 10] (see [11]
for a review of Crystal Barrel results). An isovector scalar, possi-
bly the $a_0(1450)$ (albeit at a lower mass of 1317 MeV) is observed
by Belle in $\gamma\gamma$ collisions leading to $\eta\pi^0$ [12]. The state interferes
destructively with the non-resonant background. Its $\gamma\gamma$ coupling
is comparable to that of the $a_2(1320)$, in accord with simple pre-
dictions (see e.g. [13]). A contribution from $a_0(1450) \rightarrow K\bar{K}$ is
also found in the CLEO-c analysis of $D^{\pm} \rightarrow K^{+}K^{-}\pi^{\pm}$ [14] and
$D^{0} \rightarrow K_{S}^{0}K^{\pm}\pi^{\mp}$ from LHCb [15]. The $a_0(1450)$ is reported at
a lower mass in $\bar{p}p$ annihilation in flight, overlapping with the
$a_2(1320)$ [16, 17].

The $f_0(1370)$ and $f_0(1500)$ were observed by Crystal Barrel [18–20] and by WA102 in central production with 450 GeV protons [21,22]. They decay mostly into $2\pi$ and $4\pi$. The $f_0(1500)$ was also observed to decay into $\eta\eta$ [23–25] and $\eta\eta'$ [26]. All data agree that the $4\pi$ decay mode represents about half of the $f_0(1500)$ decay width, but is dominant for $f_0(1370)$.

The determination of the $\pi\pi$ coupling of the very broad $f_0(1370)$ is complicated by its interference with the $f_0(500)$. The existence of the $f_0(1370)$ was questioned by COMPASS from $\pi^-p \to \pi^-\pi^-\pi^+p$ data [27] and by LHCb in $\bar{B}_s^0 \to J/\psi(1S)\pi^+\pi^-$ [28]. In $\bar{B}^0 \to J/\psi(1S)\pi^+\pi^-$ the $\pi\pi$ mass spectrum does not show any significant scalar component above ~1.2 GeV [29] (an analysis that is challenged in [30]). The existence of the $f_0(1370)$ has been disputed by a reassessment of various production modes [31] (the $K_0^*(700)$ is also not needed in $K\pi$ scattering). This contrasts with its $\pi\pi$ decay analyzed in Ref. [32]). However, data from CLEO-c in $D^0 \to \pi^+\pi^-\pi^+\pi^-$ require a contribution from $f_0(500)f_0(1370) \to 4\pi$ [33]. A broad $2\pi$ signal is also observed by BaBar around 1400 MeV in the decay $B^\pm \to \pi^\pm\pi^\pm\pi^\mp$ [34] which is attributed to the $f_0(1370)$ (but could also be due to the $f_0(1500)$). The existence of the $f_0(1370)$ is confirmed by a recent dispersive analysis of $\pi\pi$ scattering data, which leads to a complex pole around 1245 MeV with a very large width of about 600 MeV [35].

In Refs. [36, 37] the $f_0(1370)$ and $f_0(1710)$ (together with the $f_2(1270)$ and $f'_2(1525)$) are interpreted as bound systems of two vector mesons. This view is challenged in Ref. [38] where in a covariant formalism, e.g. the $f_2(1270)$ does not appear as a $\rho\rho$ bound state. Photoproduction data of the $f_2(1270)$ in $\gamma p \to \pi^0\pi^0 p$ from CLAS [39], as a function of momentum transfer, also disagree with predictions [40] for the $f_2(1270)$ to be a $\rho\rho$ bound state.

While the $K\bar{K}$ decay branching ratio of the $f_0(1500)$ is small [21] [41], the $f_0(1710)$ decays dominantly into $K\bar{K}$. The $f_0(1710)$ is not observed in $p\bar{p}$ annihilation at rest [41] and only weakly produced in $p\bar{p}$ annihilation in flight [42], with a rate strongly suppressed compared to that of the $f_0(1500)$ (~7%). For comparison, the rate for the $(s\bar{s})$ $f'_2(1525)$ production is about 9% that of the $(u\bar{u} + d\bar{d})$ $f_2(1270)$, as expected from the OZI rule [42]. On the other hand the $f_0(1370)$ does not couple strongly to $K\bar{K}$ [21]. This suggests an $n\bar{n}$ structure $(u\bar{u} + d\bar{d})$ for the $f_0(1370)$ and $s\bar{s}$ for the $f_0(1710)$. The $f_0(1500)$ would also qualify as an $n\bar{n}$ state, although it is very narrow compared to the other states. Occam's razor principle therefore suggests that the $f_0(1370)$, $a_0(1450)$, and the $f_0(1710)$ are in the $^{13}\text{P}_0 q\bar{q}$ SU(3) flavor nonet, the $f_0(1370)$

and $a_0(1450)$ being the two (mostly) $n\bar{n}$ states, and the $f_0(1710)$
the (dominantly) $s\bar{s}$ one. The nature of the $f_0(1500)$ is discussed
along this scheme in section 63.3.1.

There is long-time evidence for the existence of two scalar nonets and one glueball below 2 GeV [43]. The ground state $^{13}\text{P}_0 q\bar{q}$ nonet and a lighter one made of four-quark states which recombine at large distances to become meson-meson resonances, the $a_0(980)$, $a_0(980)$, $f_0(500)$ and $K_0^*(700)$, see section 63.8. For a review see Ref. [1] and for the $f_0(500)$ Ref. [44].

Other schemes have been proposed, for example a tetraquark for the $f_0(1500)$ [45] or flavor octets for the $f_0(1500)$ and $f_0(1710)$ [3]. In Ref. [46] the $a_0(1450)$, $f_0(1370)$, $f_0(1500)$ and $K_0^*(1430)$ are radial excitations of the scalar nonet below 1 GeV. In Ref. [47] the light and $^{13}\text{P}_0$ scalar nonets are interpreted as mixing of two tetraquark nonets. In the unitarized quark model with coupled $q\bar{q}$ and meson-meson channels, the scalars below 1 GeV are manifestations of bare $q\bar{q}$ confinement states, mass shifted from the 1.4 GeV region and distorted by the strong $^{3}\text{P}_0$ coupling to S-wave two-meson decay channels [48, 49]. Thus, in these models the light scalar nonet comprising the light and $^{13}\text{P}_0$ scalar nonets are manifestations of the same bare input states (see also [50]). Surprisingly for a state decaying strongly into $2\pi$ and $4\pi$, the $f_0(1370)$ is assumed to be an $s\bar{s}$ state in Ref. [51]. The $f_0(1500)$ is then a radial excitation.

There is evidence for the existence of two further $0^{++}$ mesons in the 1700–1800 MeV range, one isoscalar ($f_0(1770)$) and one isovector ($a_0(1710)$), which could be the isovector partner of the $f_0(1710)$. The $f_0(1770)$, claimed by BESII/III [52–54], and also observed in $\pi^-p \to n\omega\phi$ [55], has been interpreted as a radial excitation [56]. Two isoscalars are also reported by a re-analysis of BESIII radiative $J/\psi(1S)$ decay data [57, 58]. Evidence for the $a_0(1710)$ is inferred from the interference of the $K^+K^-$ and $K_S^0K_S^0$ decays of the $f_0(1710)$ in $D_s^+ \to f_0(1710)\pi^+$, leading to a relative branching ratio an order of magnitude larger than expected from isospin symmetry [59, 60]. A signal for $a_0(1710) \to K_0^0K^+$ is indeed reported by BESIII [61], albeit at 1817 MeV, in contrast to the state observed by BaBar in $\eta\pi$ at 1704 MeV [62].

From a study of the vector-vector interaction the authors of Ref. [63], predict an isovector pole at 1780 MeV, while the $f_0(1710)$ is a $K^*\bar{K}^*$ molecule, a view also supported by Refs. [64–67]. Thus the $f_0(1710)$ and $a_0(1710)$ might be $K^*\bar{K}^*$ molecules, similar to the $f_0(980)$ and $a_0(980)$ which have been interpreted as $K\bar{K}$ molecules. Scalar resonances near the $D_0$ mass such as the $f_0(1710)$ and the $f_0(1770)$ are also suggested to produce CP violation enhancements in the charm system [68]. Unfortunately, the spectrum of scalar mesons above the $f_0(1710)$ is not well established experimentally and more data are needed to confirm the $f_0(1770)$ and $a_0(1710)$.

63.3 Glueballs

Lattice calculations, QCD sum rules, flux tube, and constituent glue models agree that the lightest glueball has quantum numbers $J^{PC} = 0^{++}$ and the first excited state $2^{++}$. Lattice calculations predict for the ground state $(0^{++})$ a mass around 1600 – 1700 MeV [69–72], while the $2^{++}$ state lies around 2300 MeV. For more information on lattice calculations see ‘Quark Model’ in the Review and fig. 15.3 therein for an example of mass spectrum. These predictions were made in the quenched approximation, neglecting $q\bar{q}$ loops. However quenched predictions and full QCD calculations lead to small mass shifts (see ‘Quark Model’ and fig. 15.15 therein).

Heavier glueballs with quantum numbers $0^{-+}$, $2^{-+}$, $1^{+-}$, etc. are predicted above 2500 MeV and the lowest exotic ones (with exotic quantum numbers such as $0^{+-}$ and $2^{+-}$) are expected above 4000 MeV [72]. In holographic QCD the $0^{-+}$ is predicted to be very broad [73] and the $1^{+-}$ is at least as broad as its width [74]. Calculations of the three lowest scalar and pseudoscalar glueballs masses in pure Yang-Mills theory are in quantitative agreement with lattice results [75]. The lightest glueballs lie in the same mass region as ordinary isoscalar $q\bar{q}$ states, in the mass range of the $^{13}\text{P}_0(0^{++})$, $2^3\text{P}_2(2^{++})$, $3^3\text{P}_2(2^{++})$, and $^{13}\text{F}_2(2^{++})$ $q\bar{q}$ states. Therefore, mixing of glueballs with nearby $q\bar{q}$ states of the same quantum numbers leads to supernumerary isoscalar state in the $q\bar{q}$ nonets (however, see Ref. [57] discussed in section
63.3.1).

Among the signatures naively expected for glueballs are (i) isoscalar states that do not fit into $q\bar{q}$ nonets, (ii) enhanced production in gluon-rich channels such as central production and radiative $J/\psi(1S)$ decay, (iii) decay branching fractions incompatible with SU(3) predictions for $q\bar{q}$ states, and (iv) reduced $\gamma\gamma$ couplings (see however [76, 77]). Mixing effects with isoscalar $q\bar{q}$ mesons [31, 69, 76, 78–84] and decay form factors [85] can also obscure these simple signatures.

According to SU(3) the decay branching ratios for $q\bar{q}$ mesons and pure glueballs are different, and therefore useful to determine the internal structures of mesons. For pure glueballs with flavor symmetric couplings the decay ratios are $\pi\pi: K\bar{K}: \eta\eta: \eta\eta' = 3:4:1:0$, apart from phase space factors. The partial widths for the decay of a scalar (or a tensor) meson into a pair of pseudoscalar mesons are given in fig. 63.2 (for a derivation see e.g. [8]). The decay of a $q\bar{q}$ meson into a pair of mesons involves the creation of a $q\bar{q}$ pair, and SU(3) symmetry assumes that the matrix elements for the creation of $s\bar{s}$, $u\bar{u}$, and $d\bar{d}$ pairs are equal. (The generalization to unequal $s\bar{s}$, $u\bar{u}$, and $d\bar{d}$ couplings is given in Ref. [78].) An excellent fit to the tensor meson decay widths is obtained with $\beta \simeq 0.5$ GeV/c, $\theta_V \simeq 26^\circ$ and $\theta_P \simeq -17^\circ$ [78].

Note that the assumption of flavor symmetric couplings may not apply in models describing the scalar glueball by a dilaton field, which lead to mass dependent couplings [86, 87] (see Ref. [88] which predicts that a scalar glueball above 1 GeV would be unobservably wide).

Another way to determine the flavor contents of neutral mesons is the decay $B \to J/\psi(1S)X$ which filters out the $d\bar{d}$ content of X, while $B_s^0 \to J/\psi(1S)X$ selects its $s\bar{s}$ component [89]. More precise data on $B_{(s)}^0 \to J/\psi(1S)[f_0(1370), f_0(1500), f_0(1710)]$ would help to pin down the nature of these scalar mesons [90].

### 63.3.1 Scalar glueballs

One of the three isoscalars, the $f_0(1370)$, $f_0(1500)$ or $f_0(1710)$, appears to be supernumerary. The branching ratios in fig. 63.2 can be used to deduce the structures of these states, assuming that they are $q\bar{q}$. The comparison of decay branching ratios from Crystal Barrel annihilation and WA102 central collision data shows that the $f_0(1500)$ is compatible with an $n\bar{n}$ structure, while the $f_0(1710)$ is mostly $s\bar{s}$ [13]. However, the close vicinity of the very broad $n\bar{n}$ $f_0(1370)$, its narrow width, enhanced production at low transverse momentum transfer in central collisions [91–93] favor the $f_0(1500)$ to be the supernumerary non-$q\bar{q}$ state. According to [78, 83] the $f_0(1370)$ and $f_0(1710)$ would be dominantly $q\bar{q}$ states mixing with glue, while the $f_0(1500)$ would be dominantly a glueball mixing with $q\bar{q}$ states. Its suppressed $K\bar{K}$ decay would be due to interferences with the $f_0(1370)$ and $f_0(1710)$. In the analogous mixing scheme of Ref. [56], which uses central production data from WA102 and the hadronic $J/\psi(1S)$ decay data from BES [94, 95], glue is shared between the $f_0(1370)$ (mainly $n\bar{n}$), $f_0(1500)$ (mainly glue) and $f_0(1710)$ (dominantly $s\bar{s}$). This agrees with the analyses [78, 83]. In the mixing scheme of Ref. [96] the QCD spectral sum rule also leads to a large gluonic component in the $f_0(1500)$ (satisfying the observed strong $\pi\pi$ and $f_0(500)f_0(500) \to 4\pi$ decays).

However, not everybody agrees that the strong $K\bar{K}$ signal is indicative of an $s\bar{s}$ structure for the $f_0(1710)$. The $f_0(1710)$ could still be the glueball, since the two-gluon coupling to $n\bar{n}$ appears to be suppressed by chiral symmetry [97], thus $K\bar{K}$ would be enhanced compared to $\pi\pi$. It was argued that chiral symmetry constraints in a multichannel analysis imply that the $f_0(1710)$ is an unmixed scalar glueball [98]. However, this view is challenged in [63].

The $K\bar{K}$ decay is also naturally enhanced in the extended linear sigma model with a dilaton as glueball [86] and in the holographic model of [87, 99] which both prefer the $f_0(1710)$ as the glueball. For a scalar glueball Ref. [87] finds a strong enhancement of the decays into $K\bar{K}$ and $\eta\eta$, in fairly close agreement with the measured branching ratios of the $f_0(1710)$, while Ref. [99] predicts the (so far not measured) rate into $\eta\eta'$ to be very small.

In $\gamma\gamma$ collisions leading to $K_S K_S$ [100] and $K^+ K^-$ [101] a spin-0 signal is observed at the $f_0(1710)$ mass (together with a

dominant spin-2 component), while the $f_0(1500)$ is not observed in $\gamma\gamma \to K\bar{K}$ nor $\pi^+\pi^-$ [102]. The $f_0(1500)$ is also not observed by Belle in $\gamma\gamma \to \pi^0\pi^0$, although a shoulder is seen which could also be due to the $f_0(1370)$ [103]. The absence of $f_0(1500)$ signal in the $\pi\pi$ channel in $\gamma\gamma$ collisions does not favor an $n\bar{n}$ interpretation for the $f_0(1500)$. The upper limit for $\Gamma_{2\gamma} (< 1.4$ keV) in $\pi^+\pi^-$ [13]) excludes a large $n\bar{n}$ content, and hence points to a mainly $s\bar{s}$ content [13], which contradicts the small $K\bar{K}$ decay branching ratio of the $f_0(1500)$ [10, 21, 41]. Belle finds that in $\gamma\gamma \to K_S K_S$ collisions the 1500 MeV region is dominated by the $f_2'(1525)$. The $f_0(1710)$ is also observed with a production rate $\times$ branching ratio compatible with an $s\bar{s}$ state [104]. Note, however, that the $\gamma\gamma$ couplings of glueballs are sensitive to glue mixing with $q\bar{q}$ [56]. Ref. [76] predict the glueball $\gamma\gamma$ partial width in the few keV range (comparable to that of $q\bar{q}$ mesons) due to couplings of vector mesons to $\gamma$ via VDM, in agreement with holographic models [77].

Alternative assignments for the scalar glueball have been proposed: In Ref. [80] the gluonic signal is distributed over $f_0(1370)$, $f_0(1500)$ and another broad isoscalar around 1530 MeV, while the gluonic contribution to the $f_0(1710)$ is small. As mentioned already, in Ref. [69, 86, 87] the $f_0(1710)$ is the glueball state, as in Ref. [84], where the $f_0(1500)$ is the $q\bar{q}$ octet state degenerate with the $a_0(1450)$. In the generalized linear sigma model [105] the $a_0(980)$ is dominantly a tetraquark and $a_0(1450)$ a $q\bar{q}$ state. The $f_0(980)$, $f_0(1370)$ and $f_0(1500)$ (or $f_0(1710)$) are dominantly tetraquark, $q\bar{q}$ and glue.

In Ref. [82] the $f_0(500)$ and $f_0(1370)$ are signals from a single broad resonance proposed as the scalar glueball. The ground state scalar nonet then consists of the $f_0(980)$, $a_0(980)$, $K_0^*(1430)$, $f_0(1500)$ and $f_0(1710)$ [31]. The $f_0(980)$ and $f_0(1500)$ mix (similarly to the $\eta$ and $\eta'$ in the pseudoscalar nonet), while the $f_0(1500)$ mixes with a glueball in the 500–1000 MeV mass range, which is identified as the $f_0(500)$.

In Ref. [106], a large $K^+K^-$ scalar signal reported by Belle in B decays into $K\bar{K}$ [107], compatible with the $f_0(1500)$, is explained as due to constructive interference of a flavor octet with a broad glueball background. However, the Belle data are inconsistent with the BaBar measurements which show instead a broad scalar at this mass for B decays into both $K^\pm K^\pm K^\mp$ [108] and $K^+K^-\pi^0$ [109].

The $f_0(1500)$ is observed by BESII in $J/\psi(1S) \to \gamma\pi\pi$ [110] and by BESIII in $J/\psi(1S) \to \gamma\eta\eta$ [111] with a much smaller rate than for the $f_0(1710)$, which would speak against a glueball interpretation of the former, although the systematic errors are large. Also, the $f_0(1500)$ appears at a lower mass and the $f_0(1710)$ at a higher mass than the accepted values. However, the coupled channel analysis of more data on radiative $J/\psi(1S)$ decay from BESIII [57] (described in the next paragraph) finds comparable contributions from $f_0(1500)$ and $f_0(1710)$ with a preference for $\pi\pi$ decay over $K\bar{K}$ for the former and $K\bar{K}$ over $\pi\pi$ for the latter.

The authors of Ref. [57] have analyzed the high statistics data from $J/\psi(1S)$ radiative decays into $\pi^0\pi^0$, $K_S^0 K_S^0$, $\eta\eta$ and $\omega\phi$ from BESIII, including data on $\pi\pi$, $\eta\eta$ and $\eta\eta'$ from the CERN SPS, BNL data on $\pi\pi \to K_S^0 K_S^0$ and $\bar{p}p$ annihilation data from LEAR into various final states. The coupled channel analysis requires ten scalar mesons, the established ones in the Review and the $f_0(2020)$, as well as the isoscalars $f_0(2100, 2200, 2330)$ so far omitted from the Summary Table. As a new feature, the $f_0(1710)$ splits into two states, one at about 1700 MeV, the former $f_0(1710)$, and the new $f_0(1770)$ discussed above, being mandatory to obtain a good fit. A broad ($\sim$370 MeV) enhancement is observed in $J/\psi(1S)$ radiative decays around 1865 MeV, attributed to the contribution of glue distributed among the scalar mesons, with a very strong contribution from the 1770 MeV state [58]. In this model there are no supplementary states as the glue ‘fragments’ between the various isoscalars.

Ref. [112] reports on a very recent re-analysis of the BESIII $J/\psi(1S)$ radiative decay data into $\pi^0\pi^0$ and $K_S^0 K_S^0$. A good fit is obtained with only two scalar resonances below 2 GeV, the $f_0(1500)$ and $f_0(1710)$. The latter is more strongly produced, which points to a sizeable glueball component.
<figure><img src="image_318.png" /><figcaption>**Figure 63.2:** Left: SU(3) couplings $\gamma^2$ for scalar (or tensor) $q\bar{q}$ meson decays. The angles $\alpha$ and $\phi$ are defined as $\alpha = 54.7^\circ + \theta$ and $\phi = 54.7^\circ + \theta_P$, where $\theta$ is the mixing angle of the decaying isoscalars and $\theta_P$ the mixing angle in the $0^-$ nonet. The partial decay width is given by $\Gamma = C \times \gamma^2 \times |F(q)|^2 \times q$, where C is a nonet constant, q the momentum of the decay products and $F(q)$ a form factor which may be taken as $|F(q)|^2 = q^{2\ell} \times \exp(-q^2/8\beta^2)$ with $\beta = 0.5$ GeV/c, and where $\ell$ is the relative angular momentum between the decay products ($\ell = 0$ for scalars and $\ell = 2$ for tensors). Right: SU(3) couplings as a function of mixing angle $\alpha$ for isoscalar decays for $\theta_P = -17.3^\circ$ (from [78]).</figcaption></figure>

### 63.3.2 Tensor glueballs

Above the well known $f_2(1270)$ and $f'_2(1525)$ $q\bar{q}$ mesons, the $f_2(1640)$ and $f_2(1950)$ are tentatively assigned to the $2^3P_2$ nonet (see fig. 63.1). The broad $f_2(1950)$ has been observed by several experiments, e.g. in central production [22] and in $\bar{p}p$ annihilation in flight [17] and is often identified as a glueball [113], the ground state of the pomeron [114,115]. Three further isoscalar tensors are established, the $f_2(2010)$, $f_2(2300)$ and $f_2(2340)$, which are in the range of the $1^3F_2$ and $3^3P_2$ nonets. The large $\phi\phi$ cross section in $\bar{p}p$ just above threshold [116] could be due to the production of the $2^{++}$ glueball, in accord with earlier observations in $\pi^-N \to \phi\phi n$ reactions [117, 118] and in central collisions [119].

The $f_2(2010)$, $f_2(2300)$ and $f_2(2340)$ have been observed by BESIII in $J/\psi(1S) \to \eta\phi\phi$ [120]. The production rate of a tensor glueball in $J/\psi(1S)$ radiative decay has been calculated in quenched lattice QCD to be large (around 1%) [121] and is claimed by BESIII [120] to be compatible with their rates measured in $J/\psi(1S) \to f_2(2340)$, $f_2(2340) \to \eta\eta$ [111], and $f_2(2340) \to \phi\phi$ [120]. The authors of Ref. [122] have re-analyzed the $J/\psi(1S)$ radiative decay into $\pi^0\pi^0$ and $K_S K_S$ from BESIII and found an enhancement described by a 360 MeV broad pole around 2210 MeV, possibly composed of the resonances observed earlier in $\phi\phi$.

The relatively narrow $f_2(2300)$ with a measured width of 149 MeV [118] is preferred by the holographic QCD model of Ref. [123]. However, the tensor glueball is predicted to be much broader (600-900 MeV) by the holographic model of Ref. [124].

There is no evidence for a narrow meson, $f_J(2220)$ (a tensor candidate) in $\bar{p}p$ annihilation (see the note under the $f_J(2220)$ in the 2004 issue of the Review). The measured partial width to $\bar{p}p$ in radiative $J/\psi(1S)$ decay [125] is too large and inconsistent with the upper limit from $\bar{p}p$ annihilation into $\pi\pi$ [126].

## 63.4 Pseudoscalar mesons

We now deal with the first radial excitations of the $0^-$ nonet (fig. 63.1). The $\pi(1300)$ is a very broad resonance (200-600 MeV) decaying into $3\pi$. The $K(1460)$ was recently confirmed with high statistics data from LHCb in $D^0 \to K3\pi$ decays [127] and has become an established kaon excitation. The first observation of an isoscalar resonance around 1425 MeV – the E-meson – was made in $p\bar{p}$ annihilation at rest into $E\pi^+\pi^-$, $E \to K\bar{K}\pi$ [128]. This state was reported to decay into $a_0(980)\pi$ and $K^*(892)\bar{K}$ with roughly equal contributions. An isoscalar state, the $\ell$ meson, was observed in radiative $J/\psi(1S)$ decay into $K\bar{K}\pi$ [129-131] and $\gamma\rho$ [132], and was considered at that time as a glueball candidate, owing to its strong signal in radiative $J/\psi(1S)$ decay. The E and $\ell$ were later assumed to be the same state, called $\eta(1440)$.

The $\eta(1295)$ has been observed in various production mechanisms, in $\pi^-p$ experiments [133-136], and possibly in $\bar{p}p$ anni-

hilation [137, 138]. In $J/\psi(1S)$ radiative decay, the $\eta(1295)$ signal is also evident in the $0^+$ $\eta\pi\pi$ wave of the DM2 data [139]. Also BaBar reports a signal around 1295 MeV in B decays into $\eta\pi\pi K$ [140]. We therefore assume that the $\eta(1295)$ is one of the isoscalars in this nonet.

However, two isoscalars were later observed in this mass region, the $\eta(1405)$ and $\eta(1475)$. The former decays mainly into $a_0(980)\pi$ (or direct $K\bar{K}\pi$) and the latter mainly into $K^*(892)\bar{K}$. The simultaneous observation of two pseudoscalars is reported in three production mechanisms: $\pi^-p$ [133, 141], radiative $J/\psi(1S)$ decay [139, 142], and $\bar{p}p$ annihilation at rest [143-146]. All of them give values for the masses, widths, and decay modes that are in reasonable agreement. (However, Ref. [139] favors a state decaying into $K^*(892)\bar{K}$ at a lower mass than the state decaying into $a_0(980)\pi$. In $J/\psi(1S)$ radiative decay, the $\eta(1405)$ decays into $K\bar{K}\pi$ through $a_0(980)\pi$, and hence a signal is also expected in the $\eta\pi\pi$ mass spectrum. This was indeed observed by MARK III in $\eta\pi^+\pi^-$ [147], which reported a mass of 1400 MeV, in line with the existence of the $\eta(1405)$ decaying into $a_0(980)\pi$. The $\eta(1405)$ decaying into $\eta\pi\pi$ is also observed by Crystal Barrel [138]. Two states were also reported by BES: Around 1452 MeV BESII observed a $K\bar{K}\pi$ enhancement in $J/\psi(1S) \to \omega K\bar{K}\pi$ but not in $J/\psi(1S) \to \phi K\bar{K}\pi$ [148], while BESIII reported a 52 MeV broad state in $\psi(2S) \to \omega K^*K$ [149], both left without $J^{PC}$ determination.

The $K\bar{K}\pi$ and $\eta\pi\pi$ channels were studied in $\gamma\gamma$ collisions by L3 [150]. (For the $2\gamma$ couplings of glueballs and $q\bar{q}$ mesons see [76, 77, 151, 152].) The analysis led to a clear $\eta(1475)$ signal in $K\bar{K}\pi$, decaying into $K^*\bar{K}$, well identified in the untagged data sample, where spin 1 resonances are not allowed. At the same time, L3 did not observe the $\eta(1405)$, neither in $K\bar{K}\pi$ nor in $\eta\pi\pi$ [150]. On the other hand, CLEO-II did not observe any pseudoscalar signal with tagged $\gamma$'s in $\gamma\gamma \to \eta(1475) \to K_S^+K^\pm\pi^\mp$ [153], with an upper limit slightly smaller than the signal observed by L3. After the CLEO-II result, L3 performed a further analysis with full statistics [154], confirming their previous evidence for the $\eta(1475)$. The CLEO upper limit [153] for $\Gamma_{\gamma\gamma}$, and the L3 results [154], are consistent with the world average for the $\eta(1475)$ width. BaBar [140] also reported the $\eta(1475)$ in B decays into $K\bar{K}^*$ recoiling against a K, while upper limits only were given for the $\eta(1405)$.

Hence, in radiative $J/\psi(1S)$ decay, $\pi^-p$ and $\bar{p}p$ annihilation at rest two isoscalar signals are observed in the 1400-1500 MeV mass region, while the $\eta(1405)$ is not seen in $\gamma\gamma$ interactions nor in B decays. The $\eta(1475)$ could be the first radial excitation of the $\eta'$, with the $\eta(1295)$ being the first radial excitation of the $\eta$. Ideal mixing, suggested by the $\eta(1295)$ and $\pi(1300)$ mass degeneracy, would then imply that the second isoscalar in the nonet is mainly
$s\bar{s}$, and hence couples to $K^*\bar{K}$, in agreement with properties of the $\eta(1475)$. Also, its width agrees with the expected one for the radially excited $s\bar{s}$ state [85,91]. A study of radial excitations of pseudoscalar mesons [155] also favors the $s\bar{s}$ interpretation of the $\eta(1475)$. However, due to the strong kinematical suppression in $\eta(1405) \rightarrow K^*\bar{K}$ the data are not sufficient to exclude a sizeable $s\bar{s}$ admixture also in the $\eta(1405)$.

The supernumerary isoscalar $\eta(1405)$ would be a candidate for the $0^{-+}$ glueball in the flux tube model [156], in which the $0^{++}$ glueball is also naturally related to a $0^{-+}$ glueball with mass degeneracy broken in QCD. However, this scenario is not favored by lattice gauge theories which predict the $0^{-+}$ state above 2 GeV [70, 157] (see ‘Quark model’ in this issue of the Review). Nevertheless, the pseudoscalar glueball could lie at a lower mass than predicted from lattice calculation [158], see also Refs. [159–161]. The holographic model of Ref. [77] predicts the pseudoscalar glueball below 2600 MeV, decaying dominantly into two vector mesons. A detailed review of the experimental situation on the pseudoscalar glueball is available in Ref. [162].

Here also, there are alternative explanations. The mere existence of the $\eta(1295)$ is questioned in Refs. [3, 163], in which the authors also propose a single pseudoscalar meson at 1440 MeV, the first radial excitation of the $\eta$. According to Ref. [3] the splitting of the 1440 MeV state into $\eta(1405)$ and $\eta(1475)$ is due to nodes in the decay amplitudes, which differ in $\eta\pi\pi$ and $K^*(892)\bar{K}$. The splitting could also be due to a triangle singularity, hence the manifestation of one state only (fig. 63.3) [164–166]. The importance of the triangle singularity for the $\eta(1295)$ and $\eta(1405/1475)$ in the $\eta\pi\pi$ and $K\bar{K}\pi$ channels is stressed in Ref. [167].

In Ref. [168], using the triangle singularity approach of [164], the authors conclude that the BESIII results can be reproduced either with the $\eta(1405)$ or the $\eta(1475)$, or by a mixture of these two states. From a high statistics data sample in $J/\psi(1S) \rightarrow \gamma K_S K_S \pi^0$ BESIII reports the observation of both $\eta(1405)$ and $\eta(1475)$ [169]. A one resonance parametrization is ruled out and triangle singularity models do not describe the data satisfactorily. This radiative decay channel has been studied in a three-body unitary coupled-channel analysis and compared to the BESIII data, including ratios of radiative branching ratios into $\eta\pi^+\pi^-$ and $K\bar{K}\pi$ through the $\eta(1405/1475)$ intermediate states [170]. Two poles emerge for $K^*\bar{K}$ in different Riemann sheets around 1400 MeV and only one pole around 1500 MeV.

To summarise this section, the experimental data on the 1400–1500 MeV region span several decades, various production mechanisms and decay modes, with models for data analysis evolving with time. A comprehensive coherent picture of all available data is therefore difficult. We believe that there is sufficient evidence to consider the $0^{-+}$ nonet with the $\eta(1440)$ in fig. 63.1 as established. Whether one or two different states – $\eta(1405)$ and $\eta(1475)$ – exist is an open question, in which case the $\eta(1405)$ would be supernumerary. The most recent high statistics data and their theoretical interpretation seem to favor the two-resonance scenario.

<figure><img src="image_319.png" /><figcaption>**Figure 63.3:** Triangle singularity: a state A decays into $K^*\bar{K}$. The K from $K^*$ decay catches up with the $\bar{K}$ and excites the $a_0(980)$ resonance which in turn decays into $\eta\pi$. This mechanism can lead to two distinct peaks (depending on the width of A), one in $K^*\bar{K}$ and the other via rescattering in $\eta\pi\pi$. Similarly, the virtual $K^*\bar{K}$ loop can couple to $f_0(980)$ which, decaying into $\pi\pi$, leads to a peak in the $3\pi$ final state.</figcaption></figure>

## 63.5 Vector mesons

In this section we restrict ourselves to the more interesting isovector spectrum which contains broad and overlapping states.

### 63.5.1 The $\rho(770)$ meson

The determination of the parameters of the $\rho(770)$ is beset with many difficulties because of its large width. The line shape depends on the production process and is not described by a relativistic Breit-Wigner function with a P-wave width, but requires some additional shape parameter. This dependence on parameterization was demonstrated long ago [171]. Bose-Einstein correlations are another source of shifts in the $\rho(770)$ line shape, particularly in multiparticle final-state systems [172].

The same model dependence afflicts any other source of resonance parameters, such as the energy dependence of the phase shift $\delta_1^1$, or the pole position. It is, therefore, not surprising that a study of $\rho(770)$ dominance in the decays of the $\eta$ and $\eta'$ reveals the need for specific dynamical effects, in addition to the $\rho(770)$ pole [173, 174].

The cleanest determination of the $\rho(770)$ mass and width comes from $e^+e^-$ annihilation and $\tau$-lepton decays. ALEPH data [175] showed that the charged $\rho(770)$ parameters measured from $\tau$-lepton decays are consistent with those of the neutral one determined from $e^+e^-$ data [176]. This conclusion is qualitatively supported by the later studies of CLEO [177] and Belle [178]. However, a comparison of the two-pion mass spectrum in $\tau$ decays from OPAL [179], CLEO [177], and ALEPH [180, 181], and the $e^+e^-$ $\rightarrow \pi^+\pi^-$ cross section from CMD-2 [182, 183], showed significant discrepancies between the two shapes which can be as high as 10% above the $\rho$ meson mass [184, 185]. This discrepancy remains after measurements of the two-pion cross section in $e^+e^-$ annihilation at KLOE [186–189], SND [190, 191], BaBar [192] and, more recently BESIII [193]. The effect is not accounted for by isospin breaking [194–197], but the accuracy of its calculation may be overestimated [198, 199].

Two recent measurements of the $e^+e^- \rightarrow \pi^+\pi^-$ cross section by SND [200] and CMD-3 [201, 202] show discrepancies in the $\rho$ region: while SND is consistent with previous $e^+e^-$ experiments, CMD-3 finds differences as high as 5% around the $\rho$ meson mass. In addition, in the fit of the $e^+e^- \rightarrow \pi^+\pi^-$ cross section CMD-3 reports values for the mass of the $\omega$ and the mass and width of the $\phi$, which differ slightly from the PDG values, showing for the first time the effect of the $\phi \rightarrow \pi^+\pi^-$ interference.

The problem of the difference in the $\pi^+\pi^-$ shapes between $e^+e^-$ and $\tau$ decays seems to be solved after a recent analysis by [203], which showed that, after correcting the $\tau$ data for the missing $\rho-\gamma$ mixing contribution, besides the other known isospin symmetry violating corrections, the $\pi\pi$ isospin 1 part of the hadronic vacuum polarization contribution to the muon $g-2$ is fully compatible between $\tau$-based and $e^+e^-$-based evaluations. The global fit of the whole set of the $\rho$, $\omega$, and $\phi$ decays, taking into account mixing effects in the hidden local symmetry model, also showed consistency of the two-pion data in $\tau$ decay and $e^+e^-$ annihilation [204, 205]. However, because of the progress in $e^+e^-$ data, the $\tau$ input is now less precise and less reliable due to additional theoretical uncertainties [206] decreasing the importance of $\tau$ decay for the determination of the $\rho(770)$ parameters.

### 63.5.2 The $\rho(770)$ excitations

In our 1988 edition, we replaced the $\rho(1600)$ entry with two new ones, the $\rho(1450)$ and the $\rho(1700)$, because there was emerging evidence that the 1600-MeV region actually contains two $\rho$-like resonances. This possibility was pointed out by a theoretical analysis [207] on the consistency of the $2\pi$ and $4\pi$ electromagnetic form factors and the $\pi\pi$ scattering length. A consistent picture of $e^+e^- \rightarrow 2\pi, 4\pi$ and diffractive photoproduction is obtained with two resonances [208]. The existence of the $\rho(1450)$ was supported by the analysis of $\eta\rho^0$ mass spectra obtained in photoproduction and $e^+e^-$ annihilation [209], as well as that of $e^+e^- \rightarrow \omega\pi$ [210].

The analysis of [208] was further extended by [211, 212] to include new data on $4\pi$-systems produced in $e^+e^-$ annihilation, and in $\tau$ decays ($\tau \rightarrow 4\pi$ and $e^+e^- \rightarrow 4\pi$ are related by the Conserved Vector Current hypothesis). These systems were successfully analyzed using interfering contributions from two $\rho$-like
states, and from the tail of the $\rho(770)$ two-body decay. While specific conclusions on $\rho(1450) \to 4\pi$ were obtained, little could be said about the $\rho(1700)$. Independent evidence for two $1^-$ states is provided by [213] in $4\pi$ electroproduction at $\langle Q^2 \rangle = 1$ (GeV/c)$^2$, and by [214] in a high-statistics sample of the $\eta\pi\pi$ system in $\pi^-p$ charge exchange.

This scenario with two overlapping resonances is supported by other data. DM2 [215] measured the pion form factor in the interval 1.35-2.4 GeV, and observed a deep minimum around 1.6 GeV. The best fit was obtained with the hypothesis of $\rho$-like resonances at 1420 and 1770 MeV, with widths of about 250 MeV. DM2 [216] found that the $e^+e^- \to \eta\pi^+\pi^-$ cross section is better fitted with two fully interfering Breit-Wigner, with parameters in fair agreement with those of [208] and [215]. These results can be considered as a confirmation of the $\rho(1450)$.

Decisive evidence for the $\pi\pi$ decay mode of both $\rho(1450)$ and $\rho(1700)$ comes from $\bar{p}p$ annihilation at rest [217]. It has been shown that these resonances possess a $K\bar{K}$ decay mode [10, 218, 219]. This decay mode has also been studied in three-body hadronic $B$ decays [220]. High-statistics studies of the decays $\tau \to \pi\pi\nu_\tau$ [175, 177], and $\tau \to 4\pi\nu_\tau$ [221] also require the $\rho(1450)$, but are not sensitive to the $\rho(1700)$, being too close to the $\tau$ mass. A recent very-high-statistics study of the $\tau \to \pi\pi\nu_\tau$ from Belle [178] reports the first observation of both $\rho(1450)$ and $\rho(1700)$ in $\tau$ decays. A clear picture of the two $\pi^+\pi^-$ resonances interfering with the $\rho(770)$ in $e^+e^-$ annihilation was also reported by BaBar using the ISR (Initial State Radiation) method [222], and more recently by CMD-3 [201].

The structure of these $\rho$ states is not yet completely clear. The authors of Refs. [85,223] claim that $\rho(1450)$ has a mass consistent with radial $2S$, but its decays show characteristics of hybrids, and suggest that this state may be a $2S$-hybrid mixture. Hybrid states could have a $4\pi$ decay mode dominated by the $a_1\pi$ [224]. Such behavior has been observed [225] in $e^+e^- \to 4\pi$ in the energy range 1.05-1.38 GeV, and by [221] in $\tau \to 4\pi$ decays. CLEO [226] and Belle [227] observe the $\rho(1450) \to \omega\pi$ decay mode in $B$-meson decays, however, do not find $\rho(1700) \to \omega\pi^0$. A similar conclusion is made in Refs. [228,229], which studied the process $e^+e^- \to \omega\pi^0$ and do not observe a statistically significant signal of the $\rho(1700)$. Various decay modes of the $\rho(1450)$ and $\rho(1700)$ are observed in $\bar{p}n$ and $\bar{p}p$ annihilation [230, 231], but no definite conclusions can be drawn. More data should be collected to clarify the nature of the $\rho$ states, particularly in the energy range above 1.6 GeV.

We now list under a separate entry the $\rho(1570)$, the $\phi\pi$ state with $J^{PC} = 1^--$ earlier observed in [232] - referred to as $C(1480)$ - also reported in [233] and possibly seen in [234]. While this may be due to a threshold effect [235], Refs. [211] and [236] suggest two independent vector states with this decay mode. The $C(1480)$ has not been seen in $\bar{p}p$ [237] nor in other $e^+e^- \to [238,239]$ experiments. However, the sensitivity of the two latter is an order of magnitude lower than that of Ref. [234] which cannot exclude that the observation is due to an OZI-suppressed decay mode of the $\rho(1700)$.

Several observations on the $\omega\pi$ system in the 1200 MeV region [240-246] may be interpreted in terms of either $J^P = 1^-$ $\rho(770) \to \omega\pi$ production [247], or $J^P = 1^+$ $b_1(1235)$ production [245,246]. The LASS amplitude analysis [248] showing evidence for $\rho(1270)$ is preliminary and needs confirmation. Evidence for a $\rho(1250)$ from a reanalysis of elastic scattering data is claimed in Ref. [249].

Ref. [250] reported a very broad $1^-$ resonance-like $K^+K^-$ state in $J/\psi \to K^+K^-\pi^0$ decays. Its pole position corresponds to mass of 1576 MeV and width of 818 MeV. Its exotic structure (molecular or multiquark) is suggested [251-253], while in Refs. [254] and [255] this is explained by the interference between the $\rho(1450)$ and $\rho(1700)$. The latter statement is qualitatively supported by BaBar [256] and SND [257]. We quote the state reported by [250] as $X(1575)$ in the section "Further States." Another state observed in $J/\psi \to K^+K^-\pi^0$ is the $\phi(2170)$ [258] which has been interpreted as exotic, e.g. as tetraquark, see section 63.8.

Evidence for $\rho$-like mesons decaying into $6\pi$ was first noted by [259] in the analysis of $e^+e^- \to 6\pi$ [260, 261] and diffractive photoproduction [262]. The authors of Ref. [259] argued that two states at about 2.1 and 1.8 GeV exist: while the former is

a candidate for the $\rho(2150)$, the latter could be a manifestation of the $\rho(1700)$ distorted by threshold effects. BaBar reported observations of the new decay modes of the $\rho(2150)$ in the channels $\eta^*(958)\pi^+\pi^-$, $f_1(1285)\pi^+\pi^-$ [263] and $\pi^+\pi^-$ [264]. The decay of the $\rho(2150)$ into $K^+K^-$ has been observed by BESIII [265] and confirmed by BaBar [266]. The relativistic quark model [267] predicts the $2^3D_1$ state with $J^{PC} = 1^--$ at 2.15 GeV which can be identified with the $\rho(2150)$.

Under $\rho(1900)$ we list various observations of irregular behavior of the cross sections near the $NN$ threshold. Dips of various width around 1.9 GeV were reported by the E687 Collaboration (a narrow one in the $3\pi^+3\pi^-$ diffractive photoproduction [268, 269]), by the FENICE experiment reporting a narrow dip [270], by BaBar in ISR (a narrow structure in $e^+e^- \to \phi\pi$ final state [234], but much broader in $e^+e^- \to 3\pi^+3\pi^-$ and $e^+e^- \to 2(\pi^+\pi^-\pi^0)$ [271]), by CMD-3 (also a rather broad dip in $e^+e^- \to 3\pi^+3\pi^-$ [272]). A dedicated scan of the $NN$-threshold region by CMD-3 confirms this effect in the $e^+e^- \to 3\pi^+3\pi^-$ and $e^+e^- \to K^+K^-\pi^+\pi^-$ final states, but does not see it in the cross section of $e^+e^- \to 2\pi^+2\pi^-$ [273]. Most probably, these structures emerge as a threshold effect due to the opening of the $NN$ channel [274-276]. A similar enhancement is observed by BESIII in $e^+e^- \to \Lambda\bar{\Lambda}$ near the $\Lambda\bar{\Lambda}$ threshold [277, 278] which can be described as $\Lambda\bar{\Lambda}$ final state interaction [279].

## 6.6 Axial-vector mesons

The $J^{PC} = 1^{++}$ nonet consists of the isovector $a_1(1260)$, the isoscalars $f_1(1285)$ and $f_1(1420)$, and the $K_{1A}$, which is a superposition of the physical states $K_1(1270)$ and $K_1(1400)$ with a mixing angle of about 35° [280]. The nonet mixing angle $\theta_{1++}$ is around 23°. The orthogonal combination – the $K_{1B}$ – belongs to the $1^{+-}$ nonet with a mixing angle $\theta_{1+-}$ of about 28° [280]. $f_1(1510)$. However, a two-pole structure for the $K_1(1270)$ emerges when using the unitary extension of chiral perturbation theory [281].

The mass region above 1400 MeV is rather complex [282-284]. The $f_1(1420)$ was first reported in $\pi^-p$ reactions at 4 GeV/c, decaying into $K^*\bar{K}$ [285]. The $f_1(1420) \to K\bar{K}\pi$ was also observed in a reanalysis of the MARK III data [142], and a $C=+1$ state seen in tagged $\gamma\gamma$ collisions [286]. Axial-vector mesons are not observed in $\bar{p}p$ annihilation at rest in liquid hydrogen, which proceeds dominantly through $S$-wave annihilation. However, in gaseous hydrogen, $P$-wave annihilation is enhanced and the $f_1(1420)$ is indeed observed, decaying into $K^*\bar{K}$ [144]. The $f_1(1420)$, decaying into $K\bar{K}\pi$, is also seen in $pp$ central production, together with the $f_1(1285)$. The latter decays via $a_0(980)\pi$, and the former only via $K^*\bar{K}$. The $K_S^0K_S^0\pi^0$ decay mode of the $f_1(1420)$ establishes unambiguously $C=+1$. Neither central production [287] nor $\pi^-p$ interactions at 100 GeV [135] find any evidence for a $\eta\pi\pi$ decay mode of the $f_1(1420)$.

The $f_1(1285)$ has been suggested to be a $K^*\bar{K}$ molecule [288]. However, LHCb has determined the $1^{++}$ nonet mixing angle to be consistent with a mostly $n\bar{n}$ structure for the $f_1(1285)$ from $\bar{B}^0/\bar{B}^0 \to J/\psi(1S)f_1(1285)$, independent of the identity of its isoscalar partner [289]. The mixing angle $\theta_{1++} = 24^\circ$ agrees with that of Ref. [280] quoted above. The ratio of $\bar{B}^0/\bar{B}_s^0$ decay rates also excludes the tetraquark interpretation of the $f_1(1285)$ [289]. This is consistent with earlier determinations assuming that the $f_1(1420)$ is the other isoscalar in the nonet [290].

The resonance candidate $a_1(1420)$ decaying into $3\pi$, reported earlier by COMPASS at 1420 MeV in $\pi^-p \to \pi^-\pi^-\pi^+p$ [291,292], appears to originate from a triangle singularity: the decay of the $a_1(1260)$ into $K^*(\to K\pi)\bar{K}$ is followed by rescattering into the $f_0(980)\pi$ channel, $f_0(980) \to \pi\pi$ [293] (see the caption of fig. 63.3). A similar triangle singularity is proposed for the $f_1(1420)$, resulting from the $K^*\bar{K}$ and $a_0(980)\pi$ decay modes of the $f_1(1285)$ [294]. The $f_1(1420)$ was also suggested to be a hybrid $q\bar{q}g$ meson [295] or a $K^*\bar{K}$ molecule, due to the proximity of the $K^*\bar{K}$ threshold [296]. The $f_1(1420)$ is not seen in the $K^-p$ data [297], which would argue against this state being the $s\bar{s}$ member of the $1^{++}$ nonet. Note, however, that Ref. [298] reports a signal from $f_1(1420)$ in $K^-p$, but not in $\pi^-p$.

The $s\bar{s}$ partner of the $f_1(1285)$ could also be the (not well
established) $f_1(1510)$ [299]. This state was seen to decay into $K\bar{K}\pi$ in $K^-p$ interactions at 4 and 11 GeV/c, recoiling against a $\Lambda$ [297, 300], which indeed points to an $s\bar{s}$ state. Evidence was reported in $\pi^-p$ interactions, based on the phase motion of the $1^{++}$ $K^*\bar{K}$ wave [284]. The $f_1(1510)$ is also needed by CMD-3 in $e^+e^- \to f_1\rho(770)$, together with the $f_1(1285)$ and $f_1(1420)$ peaks, to describe the shoulder in the $K_S K^\pm \pi^\mp$ mass around 1500 MeV [301]. A somewhat broader $1^{++}$ signal is observed in $J/\psi(1S) \to \gamma\eta\pi^+\pi^-$ [302] and a small signal in $J/\psi(1S) \to \gamma\eta'\pi^+\pi^-$ [303].

However, there is no evidence for the $f_1(1510)$ in other $K^-p$ experiments [298, 304]. The $f_1(1510)$ is not seen in central collisions [305], nor in $\gamma\gamma$ collisions [306], although surprisingly for an assumed $s\bar{s}$ meson, a signal is reported in $4\pi$ decays [307]. Given this confusing experimental situation, the meson classification in the $1^{++}$ nonet is not entirely settled.

## 63.7 Hybrid mesons

Hybrids may be viewed as $q\bar{q}$ mesons with a vibrating gluon flux tube. In contrast to glueballs, they can have isospin 0 or 1 and be electrically charged. The mass spectrum of hybrids with exotic (non-$q\bar{q}$) quantum numbers was predicted in Ref. [308], while Ref. [309] also deals with non-exotic quantum numbers. The ground-state hybrids with quantum numbers $(0^{-+}, 1^{-+}, 1^{--},$ and $2^{-+})$ are expected around 1.7 to 1.9 GeV. Lattice calculations predict that the hybrid with exotic quantum numbers $1^{-+}$ lies at a mass of $1.9 \pm 0.2$ GeV [310, 311]. Most hybrids are expected to be rather broad, but some can be as narrow as 100 MeV [312]. They prefer to decay into a pair of $S$- and $P$-wave mesons. The lattice study in [313, 314], based on full QCD with pion masses around 400 MeV, finds that several of the high-lying states observed in their spectrum show significant overlap with gluon rich source terms interpreted as hybrid states. A very broad $1^{+-}$ structure is predicted by a recent lattice calculation [315]. The main decay mode is $b_1(1235)\pi$ (a pair of $S$- and $P$-wave mesons), while other modes, such as $\rho\pi$, $\eta\pi$ and $\eta'\pi$, are suppressed by at least an order of magnitude. For an experimental and theoretical review on hybrid mesons see [4, 316].

Two isovector $1^{-+}$ candidates have been reported, one around 1400 MeV (the $\pi_1(1400)$ in previous versions on this Review) and the $\pi_1(1600)$. The ~350–400 MeV broad $\pi_1(1400)$, was reported in $\pi^-p \to \eta\pi^-p$ [317, 318] and in $\pi^-p \to \eta\pi^0\eta$ [319]. It was observed as an interference between the angular momentum $L=1$ and $L=2$ $\eta\pi$ amplitudes, leading to a forward/backward asymmetry in the $\eta\pi$ angular distribution. This state had been reported earlier in $\pi^-p$ reactions [320], but ambiguous solutions in the partial wave analysis were pointed out [321, 322]. A resonating $1^{-+}$ contribution to the $\eta\pi$ $P$-wave is also required in the Dalitz plot analysis of $\bar{p}n$ annihilation into $\pi^-\pi^0\eta$ [323], and in $\bar{p}p$ annihilation into $\pi^0\pi^0\eta$ [324]. Mass and width are consistent with the results of Ref. [317]. A coupled channel re-analysis of the $\pi^0\pi^0\eta$, $\pi^0\eta\eta$ and $K^+K^-\pi^0$ Crystal Barrel data at 900 MeV/c, supplemented with data from other collaboration in $\pi\pi \to \pi\pi, K\bar{K}, \eta\eta$ and $\eta\eta'$, leads to a single $1^{-+} \sim 600$ MeV broad state with a pole around 1400 MeV, decaying into $\eta\pi^0$ [16]. Ref. [325] suggested for the production in $\pi^-p$ that a Deck-generated $\eta\pi$ background from final state rescattering in $\pi_1(1600)$ decay could mimic $\pi_1(1400)$, a mechanism that is, however, absent in $\bar{p}p$ annihilation.

The $\pi_1(1600)$, decaying into $\rho\pi$, was reported by COMPASS with 190 GeV pions hitting a lead target [326, 327]. It had already been observed in $\pi^-p$ interactions in the decay modes $\eta'\pi$ [328], $f_1(1285)\pi$ [329], and $\omega\pi\pi$ [330], $b_1(1235)\pi$, but not $\eta\pi$ [331]. A strong enhancement in the $1^{-+} \eta'\pi$ wave, compared to $\eta\pi$, was reported at this mass in [332]. The enhancement is also observed by COMPASS in $\pi^-p \to \eta'\pi$ but in $\eta\pi$ a peak appears at 1400 MeV and in $\eta'\pi$ at 1600 MeV [333]. A coupled channel analysis of the $\eta\pi$ and $\eta'\pi$ COMPASS data leads to a single pole at $1564 \pm 89$ MeV, with a width of $492 \pm 115$ MeV [334]. The predicted width from the lattice calculation [315] is compatible with the COMPASS result. Furthermore, a combined analysis of the COMPASS and Crystal Barrel data at 900 MeV/c leads to similar results, a single pole around 1623 MeV, with a width of about 455 MeV, although a two-pole scenario cannot be completely excluded [335]. Accordingly, in the current review we have moved

the former $\pi_1(1400)$ data into the $\pi_1(1600)$ section.

Very recently, BESIII reported a 188 MeV broad enhancement at 1855 MeV in $J/\psi(1S)$ radiative decay into the $\eta\eta'$ $P$-wave [336], hence with exotic quantum numbers $1^{-+}$. This isoscalar resonance is needed with very high significance, albeit the contribution to $\gamma\eta\eta'$ is weak and several other isoscalars expected to contribute are absent [337]. Hence this state needs confirmation. In the past there had been claims for such a state decaying into $\eta\eta'$ in this mass range, which we have listed under the $f_2(1910)$ (see e.g. Ref. [26] from WA102 central production, suggesting $1^{-+}$ as an alternative to $2^{++}$). If confirmed, the $\eta(1855)$ could be one of the isospin 0 partners of the $\pi_1(1600)$, a member of the $1^{-+}$ hybrid nonet [338–340] or a $K\bar{K}_1(1400)$ molecule [341].

Hybrid candidates with the non-exotic quantum numbers $0^{-+}$, $1^{--}$, and $2^{+}$ have also been reported: the $\pi(1800)$ is somewhat narrow if interpreted as the second radial excitation of the pion. It decays mostly into a pair of $S$- and $P$-wave mesons [342, 343], in line with expectations for $0^{-+}$ hybrid mesons. The evidence for $1^{--}$ hybrids in $e^+e^-$ annihilation and in $\tau$ decays has been discussed in [224]. The near degeneracy of the $\eta_2(1645)$ and $\pi_2(1670)$ suggests ideal mixing in the $2^{-+} q\bar{q}$ nonet, and hence, the second isoscalar, presumably the $\eta_2(1870)$, should be mainly $s\bar{s}$ (fig. 63.1). However, the $\eta_2(1870)$ is also observed in $\bar{p}p$ annihilation and decays mainly into $a_2(1320)\pi$ and $f_2(1270)\pi$ [344], with a relative rate compatible with a hybrid state [309].

Evidence for another exotic $\pi_1(2015)$ has been claimed in $\pi^-p$ interactions [329, 330]. A flavor multiplet of vector hybrids between 2200 and 2300 MeV is predicted by the constituent gluon model [340].

Summarizing, there is evidence for a very broad $1^{-+}$ enhancement in the 1400–1600 MeV region, which consists in one (or possibly two) exotic states, the lowest mass one favored by $\bar{p}p$ annihilation data, the highest one by high energy $\pi^-p$ data. As isovectors, these states cannot be glueballs. The coupling to $\eta\pi$ of the lighter one points to a four-quark state [345], while the $\eta'\pi$ coupling of the heavier one is favored for hybrid states [346, 347]. The mass of the latter, the $\pi_1(1600)$, is not far below the lattice and flux tube model predictions. The $\eta_1(1855)$ is an isoscalar candidate for the ground state $1^{-+}$ hybrid nonet.

## 63.8 Tetraquark states

The existence of multiquark states was suggested a long time ago, based on duality arguments [348, 349]. The most prominent light tetraquark candidates are the $a_0(980)$ and $f_0(980)$ [350–352]. A remarkable prediction for the existence of low-lying four quarks states is based on color hyperfine splitting. The lowest ground state ($L=0$) tetraquark multiplet is predicted to be a scalar nonet. The scalar nonet lies just below 1 GeV when one assumes as mass scale the hyperfine color splitting between the $\rho$ and the $\pi$ (for a simple derivation see [8]). One then gets the nonet structures

$$ 
\begin{align} 
|a_0(980)\rangle &= |us\bar{d}s\rangle, \quad \frac{1}{\sqrt{2}}|(u\bar{u} - d\bar{d})s\bar{s}\rangle, \quad |\bar{u}\bar{s}ds\rangle, \nonumber \\ 
|f_0(980)\rangle &= \frac{1}{\sqrt{2}}|(u\bar{u} + d\bar{d})s\bar{s}\rangle, \nonumber \\ 
|f_0(500)\rangle &= |\bar{u}\bar{d}ud\rangle, \nonumber \\ 
|K_0^*(700)\rangle &= |\bar{s}\bar{d}ud\rangle, |\bar{s}\bar{u}ud\rangle \text{ and } |\bar{u}\bar{d}us\rangle, |\bar{u}\bar{d}ds\rangle. \tag{63.1}
\end{align} 
$$

The two isoscalars are expected to mix with the angle $\phi$:

$$ 
\begin{align} 
|f_0(980)\rangle &= \cos \varphi |s\bar{s}\rangle + \sin \varphi |n\bar{n}\rangle, \nonumber \\ 
|f_0(500)\rangle &= -\sin \varphi |s\bar{s}\rangle + \cos \varphi |n\bar{n}\rangle. \tag{63.2}
\end{align} 
$$

Whether these mesons are really tetraquark states or $q\bar{q}$ mesons, is still an open issue. The $f_0(980)$ is strongly produced in $D_s^+$ decay [354], which suggests a large $s\bar{s}$ component, due to the Cabibbo-favored $c \to s$ coupling. However, the mainly $n\bar{n}$ $f_0(1370)$ is also strongly produced in $D_s^+$ decay, hence additional graphs must contribute [355].

The relative branching ratios for $\bar{B}^0$ and $\bar{B}_s^0 \to J/\psi(1S)f_0(500)$ and $J/\psi(1S)f_0(980)$ can be used to probe the $q\bar{q}$ or tetraquark natures of the $f_0(500)$ and $f_0(980)$, as proposed in Refs. [353, 356].
<table><caption>**Table 63.1:** Coupling amplitudes for $\bar{B}^0$ and $\bar{B}_s^0$ decays into $J/\psi(1S)f_0(500)/f_0(980)$, depending on the $q\bar{q}$ or tetraquark structures of the $f_0(500)$ and $f_0(980)$ (from [353] where illustrative decay diagrams can be found). The angle $\varphi$ is defined in Eq. 63.2.</caption><thead><tr><th></th><th colspan="2">B̄<sup>0</sup></th><th colspan="2">B̄<sub>s</sub><sup>0</sup></th></tr><tr><th></th><th>f<sub>0</sub>(980)</th><th>f<sub>0</sub>(500)</th><th>f<sub>0</sub>(980)</th><th>f<sub>0</sub>(500)</th></tr></thead><tbody><tr><td>q̄q̄</td><td>sin ϕ/√2</td><td>cos ϕ/√2</td><td>cos ϕ</td><td>sin ϕ</td></tr><tr><td>qq̄q̄q̄</td><td>1/√2</td><td>1</td><td>√2</td><td>0</td></tr></tbody></table>

LHCb observes the $f_0(980)$ in $\bar{B}_s^0$ decays, but not the $f_0(500)$ [28], as would be expected for tetraquarks (see Table 63.1). In contrast, LHCb also observes the $f_0(500)$ in $\bar{B}^0$ decays, but not the $f_0(980)$ with an upper limit eight standard deviations below the predicted value for a tetraquark state [29]. However, these contradicting findings have been challenged by a dispersive analysis [357], using a model independent inclusion of hadronic final state interactions, in which a substantial $f_0(980)$ contribution is nevertheless found in $\bar{B}^0$-decays, thus still leaving the tetraquark structure as an open possibility. The ALICE collaboration reports a much stronger $K_S K_S$ femtoscopic correlation than for $K_S K^\pm$ in high energy pp collisions, indicative of the $a_0(980)$ being a tetraquark state [358].

The $f_0(980)$ and $a_0(980)$ could also be $K\bar{K}$ molecular states [359–361], being close to the $K\bar{K}$ threshold and decaying strongly into $K\bar{K}$. For $q\bar{q}$ states, the expected $\gamma\gamma$ widths [362, 363] are not significantly larger than for molecular states [362, 364] and both predictions are consistent with data. Radiative decay of the $\phi(1020)$ into $a_0(980)$ and $f_0(980)$ were proposed to disentangle compact (tetraquark) structures from hadronic molecules. Following Refs. [365, 366] the data from KLOE [367, 368], CMD-2 [369] and SND [370] seem to favor these mesons to be tetraquark states. This is also supported by a BESIII analysis of $J/\psi(1S) \to \gamma\pi^0\pi^0$ data [371] and by a measurement of $a_0(980) - f_0(980)$ mixing at BESIII [372]. In Ref. [43] these states are made of four-quark cores and virtual $K\bar{K}$ clouds at the periphery, a view that is challenged in Ref. [373] showing that radiative $\phi$ decay data are consistent with molecular structures of light scalars.

As mentioned already, the $f_0(1500)$ [45] and the $f_1(1285)$ [353] have been proposed as tetraquarks, the latter also as $K^*\bar{K}$ molecule [288], together with the $f_1(1420)$ [296].

Another potential candidate for a tetraquark is the $\phi(2170)$. As an excited $q\bar{q}$ state of the $\phi(1020)$, this meson should decay strongly into $K^+K^-$ and $K_S^0 K_L^0$, and also $K^*\bar{K}^*$. However, these decays are not observed. Hence, based on its observed decay into $f_0(980)\phi(1020)$ this meson could be $s\bar{s}\bar{s}$ [374], where relative decay rates into various channels are predicted for both $q\bar{q}$ and tetraquark structures of the $f_0(980)$. However Refs. [375,376] claim that the $\phi(2170)$ may not be a good candidate for the $s\bar{s}\bar{s}$ state, but could qualify as an $s\bar{s}\bar{u}$ tetraquark [376].

More information on the $a_0(980)$, $f_0(980)$, $f_0(500)$ and $K_0^*(700)$ is provided by 'Scalar Mesons below 1 GeV' in the Review.

## 63.9 Baryonia

Nucleon-antinucleon ($N\bar{N}$) bound states and resonances (baryonium) were predicted a long time ago [377, 378], based on the strongly attractive $N\bar{N}$ meson exchange potential, which is obtained from the $NN$ one by multiplying with the $G$-parity of the exchanged meson. Several candidates had been reported in the seventies by experiments at CERN, BNL and KEK, some of them being indisputably statistically significant (for details on the model and on experiments, see [379, 380] and references therein). Most potential models predict a sequence of deeply bound isoscalar baryonia with quantum numbers $J^{PC} = 2^{++}$, $1^{--}$ and $0^{++}$, the latter being the mostly bound [381, 382]. The $f_2(1565)$ which is observed in $\bar{p}p$ annihilation only [383, 384], is a good candidate for the $2^{++} \bar{p}p$ bound state.

Enhancements close to the $\bar{p}p$ threshold have also been reported in B decays [385–387]. The strong signal in $J/\psi(1S) \to \gamma\bar{p}p$

[388–390] could be due to a $0^{-+}$ baryonium [391], but could also be generated by the $N\bar{N}$ final state interaction [392–395]. The strong energy dependence of the cross section in $e^+e^- \to \bar{p}p$ [396, 397] and $\bar{p}p \to e^+e^-$ [398] is attributed to the $N\bar{N}$ final state interaction [399] (see also our comments on the $\rho(1900)$ at the end of section 63.5.2). Alternative explanations to baryonia have been proposed for the signals in $B \to \bar{p}pK$, such as the dynamics of the fragmentation mechanism [386].

## 63.10 Conclusions

The ground state nonets $1^1S_0$, $1^3S_1$, $1^1P_1$, $1^3P_2$, $1^1D_2$ and $1^3D_3$ (see fig. 63.1) are established (although more data are desirable for the $h_1(1415)$ in the $1^1P_1$ nonet). There are uncertainties and ambiguities in all other nonets. In the pseudoscalar sector the $\eta(1440)$ consists of two states, one at 1405 MeV, the other at 1475 MeV, which could also be the manifestation of a single resonance decaying into two different final states. In this case the 1475 one could be the radial excitation of the $\eta'$ and the 1405 the pseudoscalar ground state glueball predicted a long time ago by the bag model, although the lattice puts it above 2 GeV. In the $1^{++}$ nonet two isoscalars – the $f_1(1420)$ and $f_1(1510)$ – compete for the $s\bar{s}$ slot. The first radial excitation $2^3S_1$ is complete, but questions have been raised as to the nature of the $\rho(1450)$ and its relation to the $\rho(1700)$. The classification of scalar mesons is not settled, with the ones below 1 GeV being tetraquark states or building the ground state nonet. The nature of the scalars in the 1.5 – 1.8 GeV region is unclear, obscured by the interference with the ground state scalar glueball predicted in this mass region. For example, the $f_0(1500)$ and the $f_0(1710)$ could be $q\bar{q}$ states ($n\bar{n}$ and $s\bar{s}$, respectively) or made of gluons mixed with $q\bar{q}$. The $f_0(1500)$, or alternatively the $f_0(1710)$, have been proposed as mostly gluonic. Scalar mesons above the $f_0(1710)$ need experimental confirmation. Another pair of scalars, $a_0(1710)$ and $f_0(1770)$, has been reported. An isovector meson with exotic ($1^{-+}$) quantum numbers, $\pi_1(1600)$, is observed around 1.6 GeV with mass not far below theoretical predictions, although a further $1^{-+}$ state, decaying into $\eta\pi$, cannot be entirely excluded around 1.4 GeV. An isoscalar partner, the $\eta_1(1855)$, has been reported. Above 1.8 GeV mesons are broad and overlap, which complicates the data analyses. This makes the classification difficult, in particular the identification of the $2^{++}$ glueball, expected in 2200 – 2300 MeV mass range.

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# 64. Scalar Mesons below 1 GeV

Revised August 2023 by T. Gutsche (Tübingen U.), C. Hanhart (FZ Jülich), R.E. Mitchell (Indiana U.) and S. Spanier (Tennessee U.).

## 64.1 Introduction

The identification of the light scalar mesons is a long-standing puzzle. Scalar resonances are difficult to resolve experimentally as they can have large decay widths, which cause a strong overlap between neighboring resonances and with background. In addition, in some cases, several decay channels open up within a short mass interval (e.g. at the $K\bar{K}$ and $\eta\eta$ thresholds), producing cusps in the line shapes of the nearby resonances. Furthermore, one expects non-$q\bar{q}$ scalar objects, such as hadronic molecules and multiquark states, in the mass range of interest (for reviews see, e.g., Refs. [1–6]).

Light scalars are produced, for example, in $\pi N$ scattering on polarized/unpolarized targets, $p\bar{p}$ annihilation, central hadronic production, $J/\psi(1S)$, B-, D- and K-meson decays, $\gamma\gamma$ formation, and $\phi$ radiative decays. Especially for the lightest scalar mesons simple parameterizations like Breit-Wigner functions and variants thereof fail — this is demonstrated explicitly on the example of the $f_0(500)$ or $\sigma$, e.g., in Ref. [7]. Accordingly, more advanced theory tools are necessary to extract the resonance parameters from data. In the analyses available in the literature, fundamental properties of the amplitudes such as unitarity, analyticity, Lorentz invariance, and chiral and flavor symmetry are implemented at different levels of rigor. Especially, chiral symmetry implies the appearance of zeros, the so-called Adler zeros, close to the threshold in elastic S-wave scattering amplitudes involving soft (pseudo) Goldstone bosons [8, 9], which may be shifted or removed in associated production processes [10]. Moreover, especially for the lightest non-strange and strange scalar resonance precision extractions of pole parameters get complicated by the presence of both left-hand cuts as well as circular cuts (for a recent review on the subject see Ref. [5]). The methods employed are the K-matrix formalism, the N/D-method, the Dalitz-Tuan ansatz, the inverse amplitude method, unitarized quark models with coupled channels, effective chiral field theories and the linear sigma model, etc. Dynamics near the lowest two-body thresholds in some analyses are described by crossed channel (t, u) meson exchange or with an effective range parameterization instead of, or in addition to, resonant features in the s-channel. Dispersion theoretical approaches are applied to pin down the location of resonance poles for the low mass states [11–16] unambiguously — in particular the existence of a nonet of light scalar is not questioned anymore.

In parallel to the developments sketched above also lattice QCD entered the field of precision spectroscopy, since it was acknowledged that a study of resonances calls for a proper treatment of the scattering process employing e.g. the Lüscher method [17], later extended to coupled channels in Refs. [18–23]. This allowed for determinations of the poles of the $I = 1/2$ [24], the $I = 1$ [25] as well as the $I = 0$ [26, 27] light scalar mesons. Besides and additional firm confirmation of the existence of a nonet of light scalar mesons, via the quark mass dependence of the pole locations (here the most sophisticated analysis is reported in Ref. [28]) these studies allow for an alternative look on their nature [29–31].

The mass and width of a resonance are found from the position of the nearest pole in the process amplitude (S- or T-matrix) at an unphysical sheet of the complex energy plane, traditionally labeled as

$$ \sqrt{s_{\text{Pole}}} = M - i\Gamma/2. \qquad (64.1) $$

It is important to note that in general the pole of a Breit-Wigner parameterization does not agree with the S- or T-matrix pole. For a detailed discussion of this issue we refer to the review on Resonances in this Review of Particle Physics (RPP).

In this review we present proposed values for the mass parameters of the scalar resonances below 1 GeV. Note that those are labeled as 'our estimate' — it is not an average over the quoted analyses, but is chosen to include the bulk of the analyses. An averaging procedure is not justified, since the analyses use over-

lapping or sometimes even identical data sets so that they are not statistically independent.

In this note, we discuss the light scalars below 1 GeV organized in the Listings under the entries $K_0^*(700)$ (or $\kappa$) with isospin $I = 1/2$, $a_0(980)$ with $I = 1$, as well as $f_0(500)$ (or $\sigma$) and $f_0(980)$ both with $I = 0$. The $I = 2\pi\pi$ and $I = 3/2 K\pi$ partial waves do not exhibit resonant behaviour.

## 64.2 The $K_0^*(700)$, also known as $\kappa$ ($I = 1/2$)

The $K_0^*(700)$ shows up as a pole in the low energy $\pi K$ scattering, although its presence and properties are difficult to establish, since it appears to have a very large width ($\Gamma \approx 500$ MeV) and resides close to the $K\pi$ threshold. Hadronic D- and B-meson decays provide additional data points in the vicinity of the $K\pi$ threshold and are discussed in detail in the Review on Multibody Charm Analyses in this RPP. With a few exceptions discussed there, the three- or more-hadron final states are usually treated as non-interacting two-body systems. Precise information from semileptonic D decays, where the strongly interacting two-particle final states could be treated without approximation, is not available. BES II [32] (re-analyzed in [33]) finds a $K_0^*(700)$-like structure in $J/\psi(1S)$ decays to $\bar{K}^{*0}(892)K^{+}\pi^{-}$ where $K_0^*(700)$ recoils against the $K^*(892)$. The decay $\tau^{-} \rightarrow K_S^0\pi^{-}\nu_{\tau}$ can be considered clean with respect to final-state interactions and is studied by Belle [34], with $K_0^*(700)$ parameters fixed to those of Ref. [32].

<figure><img src="image_320.png" /><figcaption>**Figure 64.1:** Location of the $K_0^*(700)$ (or $\kappa$) poles in the complex energy plane. Red circles denote the results of the most sophisticated analyses based on dispersion relations (see text for details), poles extracted from Breit-Wigner fits are shown as blue squares, while all other analyses quoted in the Listings are denoted by black triangles. The corresponding references are given in the listing. The arrow shows the location of the $\pi K$ thresholds. The grey box indicates the range of pole locations classified as 'our estimate'.</figcaption></figure>

Some authors find a $K_0^*(700)$ pole in their phenomenological analysis (see, e.g., [35–44]), while others do not need to include it in their fits (see, e.g., [45–48]; note that in Ref. [49] only a $K_0^*$ meson above 825 MeV was discarded using scattering data fits and analyticity, which is not in conflict with the properties quoted below). All works including constraints from chiral symmetry at low energies naturally find a light $K_0^*(700)$ below 800 MeV, see, e.g., [50–56]. The analyses of Ref. [15, 16] are based on the Roy-Steiner equations, which include analyticity and crossing symmetry constraints. Ref. [57] uses the Padé method to extract pole parameters after refitting scattering data constrained to satisfy forward dispersion relations. All three arrive at compatible pole positions for the $K_0^*(700)$ that are also consistent with the pole parameters deduced from other theoretical methods. Due to their large uncertainties, the pole locations deduced from the Breit-Wigner fits appear to be just about consistent with the other determinations, but the real parts of all those analyses lie systematically higher. Moreover, phase shifts extracted from the Breit-Wigner functions for the $K_0^*(700)$ are very different from the known scalar $\pi K$ phase shifts. The various poles are shown
in Fig. 64.1. The compilation in this figure motivates the pole parameters of the $K_0^*(700)$, which we quote as 'our estimate', namely,

$$ \sqrt{s_{\text{Pole}}^{\kappa}} = (630 - 730) - i(260 - 340) \text{ MeV}. \qquad (64.2) $$

For an extensive discussion about the $\pi K$ system in general and the $\kappa$ meson in particular, see Ref. [58].

## 64.3 The $a_0(980)$ ($I=1$)

The $a_0(980)$ couples strongly to the channels $\pi\eta$ and $K\bar{K}$. In- dependent of any model, the $K\bar{K}$ component must be large in the $a_0(980)$ wave function, since the mass of the $a_0(980)$ lies very close to the opening of the $K\bar{K}$ channel, to which it strongly cou- ples [59,60]. This generates a pronounced cusp-like behavior in the resonant amplitude and to reveal its true coupling constants the presence of the $K\bar{K}$ channel cannot be ignored. All listed $a_0(980)$ measurements agree on a mass position value near 980 MeV, but the width deduced from the imaginary part of the pole location has values between 50 and 100 MeV, mostly due to the different models. For example, the analysis of the $p\bar{p}$ annihilation data [59] using a unitary K-matrix description finds a width determined from the T-matrix pole of $92 \pm 8$ MeV. Note that the width of the $a_0(980)$ line shape is typically smaller than what could be expected from the pole location.

The relative coupling $K\bar{K}/\pi\eta$ is determined indirectly from $f_1(1285)$ [61–63] or $\eta(1410)$ decays [64–66], from the line shape ob- served in the $\pi\eta$ decay mode [67–70], or from the coupled-channel analysis of the $\pi\pi\eta$ and $K\bar{K}\pi$ final states of $p\bar{p}$ annihilation at rest [59].

<figure><img src="image_321.png" /><figcaption>**Figure 64.2:** Location of the $a_0(980)$ poles from different extrac- tions in the complex energy plane. The corresponding references are given in the *Listings*. Also shown are the thresholds for the $K^+K^-$ and $K^0\bar{K}^0$ channels, relevant for $a_0(980)^0$, and for the $K^-K^0$ channel, relevant for the $a_0(980)^-$ . The grey box indi- cates the range of pole locations classified as 'our estimate'. Note that by default we take the pole located closest to the physical axis; thus the poles shown are not all on the same sheet.</figcaption></figure>

For the extraction of the $a_0(980)$ pole locations, Refs. [52, 53, 55, 71] use unitarized chiral perturbation theory. Ref. [72] uses a similar formalism to extract the pole of the $a_0(980)$, employing the amplitude fixed in Ref. [73]. A dispersion theoretical approach to the isovector scalar $\pi\eta - K\bar{K}$ system is presented in Ref. [74] that may be refined further from studies of heavy meson decays [75]. Those efforts lead to a rather precise determination of the $a_0(980)$ pole location [76]. A value consistent for the mass parameter, but with a larger width, is found in the analysis of $\bar{p}p$ annihilation in flight data employing a K-matrix [77]. The poles presented in Refs. [55, 59, 70, 76, 77] are shown in Fig. 64.2 together with the range of pole parameters estimated by us from this compilation, namely,

$$ \sqrt{s_{\text{Pole}}^{a_0(980)}} = (970 - 1020) - i(30 - 70) \text{ MeV} \qquad (64.3) $$

indicated by the box.

## 64.4 The $f_0(500)$, also known as $\sigma$-meson ($I=0$)

For discussions of the $\pi\pi S$ wave below the $K\bar{K}$ threshold and on the long history of the $f_0(500)$, which was suggested in linear sigma models more than 50 years ago, see the review [5]. Informati- on on the $\pi\pi S$-wave phase shift $\delta_J^I = \delta_0^0$ was already extracted many years ago from $\pi N$ scattering [78–80], and near the $\pi\pi$ threshold from $K_{e4}$ decays [81]. The kaon decays were later re- visited leading to consistent data with very much improved statis- tics [82, 83]. The reported $\pi\pi \to K\bar{K}$ cross sections [84–87] have large uncertainties. The $\pi N$ data have been analyzed in combina- tion with high-statistics data (see entries labeled as RVUE for re- analyses of the data). The $2\pi^0$ invariant mass spectra, extracted from $p\bar{p}$ annihilation at rest into $3\pi^0$ [88, 89] and into $5\pi^0$ [90] and from central $pp$ collision [91] do not show a distinct resonance structure below 900 MeV, but these data sets are consistently de- scribed with the standard solution for the $\pi\pi$ scalar isoscalar S- wave extracted from high energy $\pi N \to \pi\pi N$ data [79, 92], which allows for the existence of the broad $f_0(500)$. An enhancement is observed in the $\pi^+\pi^-$ invariant mass near threshold in the decays $D^+ \to \pi^+\pi^-\pi^+ [93–95]$ and $J/\psi(1S) \to \omega\pi^+\pi^- [96, 97]$, and in $\psi(2S) \to J/\psi(1S)\pi^+\pi^-$ with very limited phase space [98, 99].

The precise $f_0(500)$ (or $\sigma$) pole is difficult to establish because of its large width. The $\pi\pi$ scattering amplitude shows an unusual energy dependence due to the presence of the Adler zero in the unphysical regime close to the threshold [8, 9], required by chiral symmetry. However, most of the analyses listed under $f_0(500)$ agree on a pole position near ($500 - i250$ MeV). In particular, analyses of $\pi\pi$ data that include unitarity, are consistent with the near threshold $\pi\pi$ data from $K_{e4}$ decays, and the chiral symmetry constraints from Adler zeroes and/or scattering lengths find a light $f_0(500)$, see, e.g., [100, 101].

Precise pole positions with an uncertainty of less than 20 MeV (see our table for the T-matrix pole in the *Listings*) are extracted using the Roy equations, which are twice subtracted dispersion relations derived from crossing symmetry and analyticity. In Ref. [12] the subtraction constants are fixed to the S-wave scat- tering lengths $a_0^0$ and $a_0^2$ derived from matching the Roy equations and two-loop chiral perturbation theory [11]. The only additional relevant input to fix the $f_0(500)$ pole is the $\pi\pi$-wave phase shift at some higher energy point, chosen as 800 MeV. The analysis is improved further in Ref. [14]. Alternatively, in Ref. [13] Roy equa- tions are used as constraints for a fit to data. In that reference also once-subtracted Roy-like equations, called GKPY equations, are used, since the extrapolation into the complex plane based on the twice-subtracted equations leads to larger uncertainties, mainly due to the limited experimental information on the isospin $2\pi\pi$ scattering length. Ref. [102] uses Padé approximants for the analytic continuation. All these extractions find consistent re- sults. Using only analyticity and unitarity to describe data from $K_{2\pi}$ and $K_{e4}$ decays, Ref. [103] finds consistent values for the pole position and the scattering length $a_0^0$. The importance of the $\pi\pi$ scattering data for fixing the $f_0(500)$ pole is nicely illus- trated by comparing analyses of $\bar{p}p \to 3\pi^0$ omitting [88, 104] or including [89, 105] information on $\pi\pi$ scattering: while the for- mer analyses find an extremely broad structure above 1 GeV, the latter find $f_0(500)$ masses of the order of 400 MeV.

From Fig. 64.3 we estimate the range of pole positions for the $f_0(500)$, namely,

$$ \sqrt{s_{\text{Pole}}^{\sigma}} = (400 - 550) - i(200 - 350) \text{ MeV}. \qquad (64.4) $$

The plot contains the poles from Refs. [40, 51, 53, 55, 56, 69, 71, 78, 89, 95, 98–101, 103, 105–124] as well as the advanced dispersion analyses [11–14, 102]. The extracted $f_0(500)$ pole position is very sensitive to the high accuracy low energy $\pi\pi$ scattering data [82, 83]. In fact, all analyses consistent with this data find poles within the accepted range indicated in the figure. As in case of the $K_0^*(700)$, poles extracted from Breit-Wigner analyses are shown as blue squares. Again we see that those poles have the tendency to appear at higher masses, although here the effect is not as pronounced as in case of the $K_0^*(700)$. One should not take this as an indication that using Breit-Wigners is justified in case of the $f_0(500)$, since $\pi\pi$ phase shifts extracted from Breit-Wigners
<figure><img src="image_322.png" /><figcaption>Figure 64.3: Location of the $f_0(500)$ (or $\sigma$) poles in the complex energy plane. Circles denote the recent analyses based on Roy(-like) dispersion relations, poles extracted from Breit-Wigner fits are shown as blue squares, while all other analyses are denoted by triangles. The corresponding references are given in the *Listings*. The grey box indicates the range of pole locations classified as 'our estimate'.</figcaption></figure>

are in strong discrepancy with the scattering phase shifts.

If one uses just the most advanced dispersive analyses of Refs. [11–14], shown as solid dots in Fig. 64.3 to determine the pole location of the $f_0(500)$, the range narrows down to [5]

$$ \sqrt{s_{\text{Pole}}^{\sigma}} = (449^{+22}_{-16}) - i(275 \pm 12) \text{ MeV}, \quad (64.5) $$

which is labeled as 'conservative dispersive estimate' in this reference.

Besides $\pi\pi$, the only other decay channel of the $f_0(500)$ is two photons. Due to the large full width of the $f_0(500)$ an extraction of its two-photon width directly from data is not possible. Thus, the values for $\Gamma(\gamma\gamma)$ quoted in the literature as well as the in *Listings* are based on the expression in the narrow width approximation [125] $\Gamma(\gamma\gamma) \simeq \alpha^2|g_\gamma|^2/(4\text{Re}(\sqrt{s_{\text{Pole}}^{\sigma}}))$, where $g_\gamma$ is derived from the residue at the $f_0(500)$ pole to two photons and $\alpha$ denotes the electromagnetic fine-structure constant (see also the review on *Resonances* in this issue of the RPP). The explicit form of the expression may vary between different authors due to different definitions of the coupling constant, however, the expression given for $\Gamma(\gamma\gamma)$ is free of ambiguities. According to Refs. [126–129], $\pi\pi$ and $\gamma\gamma$ scattering data sets are consistently described including $f_0(500)$ via the two step process of $\gamma\gamma \to \pi^+\pi^-$ with pion exchange in the $t$- and $u$-channel, followed by a final state interaction $\pi^+\pi^- \to \pi^0\pi^0$. The same conclusion is drawn in Ref. [130], where the $f_0(500) \to \gamma\gamma$ decay width is dominated by re-scattering. Therefore, it might be difficult to learn anything new about the nature of the $f_0(500)$ from its $\gamma\gamma$ coupling. For the most recent work on $\gamma\gamma \to \pi\pi$, see Refs. [56, 71, 103–105, 112–125, 128–133]. There are strong indications (e.g., [134–164]) that the $f_0(500)$ pole cannot be classified as a $q\bar{q}$ state.

## 64.5 The $f_0(980)$ ($I=0$)

The $f_0(980)$ couples predominantly to the $\pi\pi$ and $K\bar{K}$ channels and its signal overlaps strongly with the background represented mainly by the $f_0(500)$ and the $f_0(1370)$. This can lead to a dip in the $\pi\pi$ spectrum at the $K\bar{K}$ threshold. It changes from a dip into a peak structure in the $\pi^0\pi^0$ invariant mass spectrum of the reaction $\pi^-p \to \pi^0\pi^0 n$ [138], with increasing four momentum transfer to the $\pi^0\pi^0$ system, which means increasing the $a_1(1260)$ exchange contribution in the amplitude, while the $\pi$ exchange decreases. Also when a $(u\bar{u} + d\bar{d})$ source is switched to a $s\bar{s}$ source, as it appears when moving from $B_d \to J/\psi(1S)\pi\pi$ to $B_s \to J/\psi(1S)\pi\pi$,

the $f_0$ signal switches from a dip to a peak [152]. The $f_0(500)$ and the $f_0(980)$ are also observed in the data for radiative $\phi$ decays ($\phi \to f_0\gamma$) from SND [139, 140], CMD2 [141], and KLOE [142, 143].

Unitarized chiral perturbation theory is employed to extract the pole of the $f_0(980)$ in Refs. [52, 53, 55, 71, 72]. Two different dispersive analyses are used in Ref. [13] to simultaneously pin down the pole parameters of both the $f_0(500)$ and the $f_0(980)$. The poles extracted in Refs. [13, 14, 55, 56, 77, 108, 165, 166] are shown in Fig. 64.4, together with the range of pole parameters estimated by us from this compilation, namely,

$$ \sqrt{s_{\text{Pole}}^{f_0(980)}} = (980 - 1010) - i(20 - 35) \text{ MeV} \quad (64.6) $$

indicated by the box. A disclaimer is important here: Both the poles of $a_0(980)$ and of $f_0(980)$ are located very close to the kaon thresholds, with the charged and neutral thresholds being 8 MeV apart — to illustrate this point the pertinent thresholds are shown explicitly in Figs. 64.2 and 64.4. This observation leads to the prediction of an enhanced $a_0 - f_0$ mixing [167–170]. On the other hand, all analyses employed in the pole determinations quoted above are done assuming isospin symmetry. Future studies need to show the impact of isospin violation on the extraction of the $a_0(980)/f_0(980)$ pole parameters.

<figure><img src="image_323.png" /><figcaption>Figure 64.4: Location of the $f_0(980)$ poles from different extractions in the complex energy plane. The corresponding references are given in the *Listings*. Also shown are the thresholds for the $K^+K^-$ and $K^0\bar{K}^0$ channels. The grey box indicates the range of pole locations classified as 'our estimate'.</figcaption></figure>

Analyses of $\gamma\gamma \to \pi\pi$ data [144–146] underline the importance of the $K\bar{K}$ coupling of the $f_0(980)$, while the resulting two-photon width of the $f_0(980)$ cannot be determined precisely [147]. The prominent appearance of the $f_0(980)$ in the semileptonic $D_s$ decays and decays of B and $B_s$ mesons implies a dominant $(\bar{s}s)$ component: those decays occur via weak transitions that alternatively result in $\phi(1020)$ production. Ratios of decay rates of B and/or $B_s$ mesons into $J/\psi(1S)$ plus $f_0(980)$ or $f_0(500)$ are proposed to extract the flavor mixing angle and to probe the tetraquark nature of those mesons within a certain model [148, 149]. The resulting phenomenological fits of the LHCb collaboration [150, 151] lead the authors to conclude that their data are incompatible with a model where $f_0(500)$ and $f_0(980)$ are formed from two quarks and two antiquarks (tetraquarks). However, a dispersive analysis of the same data that allows for a model independent inclusion of the hadronic final state interactions in Ref. [152] puts into question the conclusions of Ref. [150]. Moreover, the assumption underlying Ref. [150] that the production dynamics of $f_0(500)$ and $f_0(980)$ are equal such that they cancel in ratios appears to be not justified [5].

## 64.6 Interpretation of the scalars below 1 GeV

In the literature, many structures are discussed for the light scalar mesons, such as conventional $q\bar{q}$ mesons, compact $(qq)(\bar{q}\bar{q})$ structures (tetraquarks), or meson-meson bound states (hadronic
molecules). In reality, there can be superpositions of these com-
ponents, and one often depends on models to determine the dom-
inant one. Although we have seen progress in recent years, this
question remains open. Here, we mention some of the present
conclusions.

The $f_0(980)$ and $a_0(980)$ are often interpreted as compact tetraquark states [161–164, 171] or $K\bar{K}$ bound states [172]. The insight into their internal structure using two-photon widths [140, 173–179] is not conclusive. The $f_0(980)$ appears as a peak structure in $D_s$ decays without $f_0(500)$ background. Based on that observation it is suggested that $f_0(980)$ has a large $s\bar{s}$ component, which according to Ref. [180] is surrounded by a virtual $K\bar{K}$ cloud (see also Ref. [181]). The inclusive production property of the $f_0(980)$ in $Z^0$ decays is consistent with mesons of other nonets [182]. However, as stated by the authors, the results should be taken as model-dependent measurements while presently no reliable predictions for production rates of non-$q\bar{q}$ states in $Z^0$ decay are available. Data sets on radiative decays ($\phi \to f_0\gamma$ and $\phi \to a_0\gamma$) from SND, CMD2, and KLOE (see above) are consistent with a prominent role of kaon loops. This observation is interpreted as evidence for a compact four-quark structure of the light scalars in Ref. [183], while it is claimed to point at a molecular nature in Ref. [184, 185]. Details of this controversy are given in the comments [186, 187]; see also Ref. [188]. There is now a rather broad consensus that the states $f_0(980)$ and $a_0(980)$, together with the $f_0(500)$ and the $K_0^*(700)$, form a nonet of predominantly four-quark states, where at larger distances the quarks recombine into a pair of pseudoscalar mesons creating a meson cloud (see, e.g., Ref. [189]). Different QCD sum rule studies [190–195] do not agree on a tetraquark configuration for the same particle group.

Models that start directly from chiral Lagrangians, either in
non-linear [52, 55, 100, 184] or in linear [196–202] realization, pre-
dict the existence of the $f_0(500)$ meson near 500 MeV. Here
the $f_0(500)$, $a_0(980)$, $f_0(980)$, and $K_0^*(700)$ (in some models the
$K_0^*(1430)$) would form a nonet (not necessarily $q\bar{q}$). In the lin-
ear sigma models the lightest pseudoscalars appear as their chiral
partners.

In the non-linear approaches of Refs. [52] and [100] the above
resonances together with the low mass vector states are generated
starting from chiral perturbation theory predictions near the first
open channel, and then by shifting the states to the resonance
locations, using unitarity and analyticity.

Ref. [196] uses a framework with explicit resonances that are
unitarized and coupled to the light pseudoscalars in a chirally
invariant way. Evidence for a dominant non-qq nature of the
lightest scalar resonances is derived from their mixing scheme.
In Ref. [197] the scheme is extended and applied to the decay
$\eta' \to \eta\pi\pi$, which leads to the same conclusions. In Ref. [203]
the large $N_c$ behavior of the poles is studied to identify the nature of
the resonances generated from scattering equations. This leads to
the observation that, while the light vector states behave consis-
tent with what is predicted for $\bar{q}q$ states, the light scalars behave
very differently. This finding provides strong support for a domi-
nant non-$\bar{q}q$ nature of the light scalar resonances. Note, the more
refined study of Ref. [134] which finds, in the case of the $f_0(500)$,
indications for a subdominant $\bar{q}q$ component located around 1
GeV in addition to a dominant non-$\bar{q}q$ nature. Additional sup-
port for the dominant non-$\bar{q}q$ nature of the $f_0(500)$ is given in
Ref. [204], where the connection between the pole of resonances
and their Regge trajectories is analyzed. All works including con-
straints from chiral symmetry at low energies naturally find a
light $K_0^*(700)$ below 800 MeV and a $f_0(500)$ below 600 MeV, see,
e.g., [50–55]. In these works the $K_0^*(700)$, $f_0(500)$, $f_0(980)$, and
$a_0(980)$ appear to form a nonet [51, 54]. Additional evidence for
this assignment is presented in Ref. [14], where the couplings of
the nine states to $\bar{q}q$ sources are compared. Similar conclusions
are reached in Ref. [205] where in the analyzed data the $f_0(980)$
for example appears as a peak structure in $J/\psi(1S) \to \phi\pi^+\pi^-$
while in $J/\psi(1S) \to \omega\pi^+\pi^-$ the signal is suppressed. The same
low mass scalar nonet is also found earlier in the unitarized quark
model of Ref. [111]. A recent phenomenological study based on a
qq flavor scheme for the light scalars concludes that their glueball

content is ruled out because of low production rates in radiative
$J/\psi(1S)$ decays [206].

There are, however, alternative interpretations of the light scalars. For example Ref. [207] (for a more recent, condensed discussion of the idea see Ref. [208]), also builds on chiral symmetry, but expands around an infrared fixed point such that the $f_0(500)$ appears as a QCD dilaton with a mass driven by the QCD scale anomaly. The phenomenology studied in that work appears consistent with this proposal. In Ref. [108, 130, 209, 210] data sets on $\pi\pi - \bar{K}K$ scattering, as well as $\gamma\gamma \to \pi\pi$, are analysed and the authors conclude that especially the $f_0(500)$ should have a significant gluon content. An extensive analysis on mass, width and decay couplings of the light scalars in a QCD spectral sum rule study [211] leads to ambiguous results for their internal structure. The $f_0(500)$ can equally well have tetraquark, molecular or gluonic content, while the states $f_0(980)$ and $a_0(980)$ can be of molecular or tetraquark nature. A scenario where the $f_0(500)$ and $f_0(980)$ result from quarkonia-glueball mixing is also possible. Note, however, that the large similarities of the features of the $\kappa$ and the $\sigma$ meson put into question both a dilaton and a glueball contribution to the latter, since neither of them can be present in the former.

A model independent method to identify hadronic molecules
goes back to a proposal by Weinberg [212] (an extension of the
formalism to virtual states is provided in Ref. [31]), which is shown to
be equivalent to the pole counting arguments of [167–170, 213–218]
in Ref. [216]. The formalism allows one to extract the amount of
molecular component in the wave function from the effective cou-
pling constant of a physical state to a nearby continuum channel.
It can be applied to near threshold states only and provides strong
evidence that the $f_0(980)$ is predominantly a $\bar{K}K$ molecule, while
the situation turns out to be less clear for the $a_0(980)$ (see also
Refs. [177,179]). This is in line with the findings of Ref. [72], which
reports an important role of the $\pi\eta$ channel to the formation of
the $a_0(980)$ in addition to the $\bar{K}K$ channel, while the $f_0(980)$ also
in this work appears to be predominantly a $\bar{K}K$ molecule. The
relevance of both the $\bar{K}K$ and the $\pi\eta$ channels in a dynamically
generated $a_0(980)$ is also pointed out in the description of the
$\chi_{c1} \to \eta\pi^+\pi^-$ and $D_s^+ \to \pi^+\pi^0\eta$ reactions [219, 220]. A recent
analysis on the compositeness of the $\pi\pi - \bar{K}K$ and $\pi\eta - \bar{K}K$
channels for the $f_0(980)$ and $a_0(980)$ resonances based on pole values
and decay branchings is performed in Ref. [221]. While the $K\bar{K}$
configuration dominates in the $f_0(980)$, the meson-meson compo-
nents are subdominant in $a_0(980)$. Further insights into $a_0(980)$
and $f_0(980)$ are expected from their mixing [167–170]. A corre-
sponding signal predicted in Refs. [168, 169] is reported by BES
III [222]. The importance of the molecular structure of $a_0(980)$,
$f_0(980)$ and of their mixing in the reactions $\bar{B}_s^0 \to J/\psi(1S)\pi^+\pi^-$
and $\bar{B}_s^0 \to J/\psi(1S)\pi^0\eta$ is pointed out in Ref. [223].

In the unitarized quark model with coupled $\bar{q}q$ and meson-
meson channels, the light scalars are interpreted as additional
manifestations of bare $q\bar{q}$ confinement states, strongly mass
shifted from the 1.3 - 1.5 GeV region and very distorted due
to the strong $^3P_0$ coupling to S-wave two-meson decay channels
[218,224]. Thus, in these models the light scalar nonet comprising
the $f_0(500)$, $f_0(980)$, $K_0^*(700)$, and $a_0(980)$, as well as the nonet
consisting of the $f_0(1370)$, $f_0(1500)$ (or $f_0(1710)$), $K_0^*(1430)$, and
$a_0(1450)$, respectively, are seen as two manifestations of the same
bare input states (see also Ref. [225]). It should not remain un-
mentioned, however, that the heavier nonet lies rather close to the
input nonet and that the light scalar one emerges only once the
coupling to the two-meson channels is switched on, again high-
lighting that the meson-meson interaction is indispensable for the
light scalars.

Acknowledgement

The authors would like to thank Eberhard Klempt, and Jose Ramon Pelaez for helpful input and discussions in the preparation of the review.

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# 65. Rare Kaon Decays

Revised August 2023 by L. Littenberg (BNL) and G. Valencia (Monash U.).

## 65.1 Introduction

There are several useful reviews on rare kaon decays and related topics [1–13]. Activity in rare kaon decays can be divided roughly into four categories:

1. Searches for explicit violations of the Standard Model (SM)

2. The golden modes: $K \to \pi\nu\bar{\nu}$

3. Other constraints on SM parameters

4. Studies of strong interactions at low energy.

The paradigm of Category 1 is the lepton flavor violating decay $K_L \to \mu e$. Category 2 includes the two modes that can be calculated with negligible theoretical uncertainty, $K^+ \to \pi^+ \nu \bar{\nu}$ and $K_L \to \pi^0 \nu \bar{\nu}$. These modes can lead to precision determinations of CKM parameters or, in combination with other measurements of these parameters, they can constrain new interactions. They constitute the main focus of the current experimental kaon program. The search for new light particles through the reaction $K \to \pi X^0$ is a byproduct of these measurements. Category 3 is focused on decays with charged leptons, such as $K_L \to \pi^0 \ell^+ \ell^-$ or $K_L \to \ell^+ \ell^- \ell^+ \ell^-$ where $\ell \equiv e, \mu$. These modes are sensitive to CKM parameters, but they suffer from multiple hadronic uncertainties that can be addressed, at least in part, through a systematic study of the peripheral modes indicated in Fig. 65.1.

<figure><img src="image_324.png" /><figcaption>**Figure 65.1:** Role of rare kaon decays in determining the unitarity triangle. The solid arrows point to auxiliary modes needed to interpret the main results, or potential backgrounds to them.</figcaption></figure>

The interplay between Categories 3-4 and their complementarity to Category 2 is illustrated in the figure. Category 4 includes reactions like $K^+ \to \pi^+ \ell^+ \ell^-$ where long distance contributions are dominant and which constitute a testing ground for the ideas of chiral perturbation theory. Other decays in this category are

$K_L \to \pi^0 \gamma \gamma$ and $K_L \to \ell^+ \ell^- \gamma$. The former is important in understanding a CP-conserving contribution to $K_L \to \pi^0 \ell^+ \ell^-$, whereas the latter could shed light on long distance contributions to $K_L \to \mu^+ \mu^-$.

## 65.2 Explicit violations of the Standard Model

Much activity has focussed on searches for lepton flavor violation (LFV). This is motivated by the fact that many extensions of the minimal Standard Model violate lepton flavor and by the potential to access very high energy scales. For example, the tree-level exchange of a LFV vector boson of mass $M_X$ that couples to left-handed fermions with electroweak strength and without mixing angles yields $B(K_L \to \mu e) = 4.7 \times 10^{-12} (148 \text{ TeV}/M_X)^4$ [2]. This simple dimensional analysis may be used to read from Table 65.1 that the reaction $K_L \to \mu e$ is already probing scales of over 100 TeV. Table 65.1 summarizes the present experimental situation vis-à-vis LFV. The decays $K_L \to \mu^\pm e^\mp$ and $K^+ \to \pi^+ e^\mp \mu^\pm$ (or $K_L \to \pi^0 e^\mp \mu^\pm$) provide complementary information on potential family number violating interactions, since the former is sensitive to parity-odd couplings and the latter is sensitive to parity-even couplings.

Limits on certain lepton-number violating (LNV) kaon decays also have been obtained, with special interest arising from their role in constraining possible extensions of the neutrino sector [14, 15], and we list those in the table as well. Related searches in $\mu$ and $\tau$ processes are discussed in our section “Tests of Conservation Laws.”

<table><caption>**Table 65.1:** Searches for lepton flavor and lepton number violation in K decay</caption><thead><tr><th>LFV mode</th><th>90% CL upper limit</th><th>Experiment</th><th>Yr./Ref.</th><th>Type</th></tr></thead><tbody><tr><td>K<sup>+</sup> → π<sup>+</sup>e<sup>-</sup>μ<sup>+</sup></td><td>1.3×10<sup>-11</sup></td><td>BNL-865</td><td>2005/ [16]</td><td>LFV</td></tr><tr><td>K<sup>+</sup> → π<sup>+</sup>e<sup>+</sup>μ<sup>-</sup></td><td>6.6×10<sup>-11</sup></td><td>NA62</td><td>2021/ [17]</td><td>LFV</td></tr><tr><td>K<sub>L</sub> → μe</td><td>4.7×10<sup>-12</sup></td><td>BNL-871</td><td>1998/ [18]</td><td>LFV</td></tr><tr><td>K<sub>L</sub> → π<sup>0</sup> eμ</td><td>7.6×10<sup>-11</sup></td><td>KTeV</td><td>2008/ [19]</td><td>LFV</td></tr><tr><td>K<sub>L</sub> → π<sup>0</sup> π<sup>0</sup> eμ</td><td>1.7×10<sup>-10</sup></td><td>KTeV</td><td>2008/ [19]</td><td>LFV</td></tr><tr><td>K<sup>+</sup> → π<sup>-</sup> e<sup>-</sup> e<sup>+</sup></td><td>5.3×10<sup>-11</sup></td><td>NA62</td><td>2022/ [20]</td><td>LNV</td></tr><tr><td>K<sup>+</sup> → π<sup>-</sup> π<sup>0</sup> e<sup>+</sup> e<sup>+</sup></td><td>8.5×10<sup>-10</sup></td><td>NA62</td><td>2022/ [20]</td><td>LNV</td></tr><tr><td>K<sup>+</sup> → π<sup>-</sup> μ<sup>+</sup> μ<sup>+</sup></td><td>4.2×10<sup>-11</sup></td><td>NA62</td><td>2019/ [21]</td><td>LNV</td></tr><tr><td>K<sub>L</sub> → e<sup>±</sup> e<sup>±</sup> μ<sup>∓</sup> μ<sup>∓</sup></td><td>4.12×10<sup>-11</sup></td><td>KTeV</td><td>2003/ [22]</td><td>LNV</td></tr><tr><td>K<sup>+</sup> → π<sup>-</sup> μ<sup>+</sup> e<sup>+</sup></td><td>4.2×10<sup>-11</sup></td><td>NA62</td><td>2021/ [17]</td><td>LNFV</td></tr><tr><td>K<sup>+</sup> → μ<sup>-</sup> νe<sup>+</sup> e<sup>+</sup></td><td>8.1×10<sup>-11</sup></td><td>NA62</td><td>2023/ [23]</td><td>LNV/LFV</td></tr></tbody></table>

## 65.3 The golden modes: $K \to \pi\nu\bar{\nu}$

In the SM, the decay $K^+ \to \pi^+ \nu \bar{\nu}$ is dominated by one-loop diagrams with top-quark intermediate states while long-distance contributions are known to be quite small [24–26]. This permits a precise calculation of this rate in terms of SM parameters. Studies of this process are thus motivated by the possibility of detecting non-SM physics when comparing with the results of global fits [27, 28].

The branching ratio can be written in a compact form that exhibits the different ingredients that go into the calculation [29],

$$B(K^+ \to \pi^+ \nu \bar{\nu}(\gamma)) = \kappa_+(1 + \Delta_{\text{EM}}) \left[ \left( \frac{\text{Im}(V_{ts}^* V_{td})}{\lambda^5} X_t \right)^2 + \left( \frac{\text{Re}(V_{cs}^* V_{cd})}{\lambda} P_c + \frac{\text{Re}(V_{ts}^* V_{td})}{\lambda^5} X_t \right)^2 \right]. \quad (65.1)$$

The parameters in Eq. 65.1 incorporate the a priori unknown hadronic matrix element in terms of the very well-measured $K_{e3}$ rate [24] in $\kappa_+$; long distance QED corrections in $\Delta_{EM}$ [30]; the Inami-Lim function for the short distance top-quark contribution [31] including NLO QCD corrections [32, 33] and the two-loop electroweak correction [29], all in $X_t$; and the charm-quark contributions due to short distance effects including NNLO QCD corrections [34, 35] and NLO electroweak corrections via $P_c$ [36], which also includes certain long distance terms dubbed $\delta P_{c,u}$ [26, 37].

An interesting approximate way to cast this result in terms of the CKM parameters $\lambda$, $V_{cb}$, $\bar{\rho}$ and $\bar{\eta}$ (see our Section on “The Cabibbo-Kobayashi-Maskawa mixing matrix”) [9] is:

$$B(K^+ \to \pi^+ \nu \bar{\nu}) \approx 1.6 \times 10^{-5} |V_{cb}|^4 [(\sigma \bar{\eta})^2 + (\rho_c - \bar{\rho})^2] (65.2)$$

where $\rho_c \approx 1.45$ and $\sigma \equiv \frac{1}{(1 - \frac{1}{2}\lambda^2)^2}$. Thus, $B(K^+ \to \pi^+ \nu \bar{\nu})$ determines an ellipse in the $\bar{\rho}, \bar{\eta}$ plane with center $(\rho_c, 0)$ and semiaxes $\approx \frac{1}{|V_{cb}|^2} \sqrt{\frac{B(K^+ \to \pi^+ \nu \bar{\nu})}{1.6 \times 10^{-5}}}$ and $\frac{1}{\sigma |V_{cb}|^2} \sqrt{\frac{B(K^+ \to \pi^+ \nu \bar{\nu})}{1.6 \times 10^{-5}}}$.
BNL-787 observed two candidate events [38, 39] in the clean high $\pi^+$ momentum and one event [40] in the low-momentum region. The successor experiment BNL-949 observed one more in the high-momentum region [41] and three more in the low-momentum region [42], yielding a branching ratio of $(1.73^{+1.15}_{-1.05}) \times 10^{-10}$ [43].

NA62 was commissioned in 2015 and took data in 2016, 2017 and 2018 and published results from those runs [44]. Twenty candidate events were observed, including an estimated background of 7.03, yielding $B(K^+ \to \pi^+\nu\bar{\nu}) = (10.6^{+4.0}_{-3.4})_{\text{stat}} \pm 0.9_{\text{sys}}) \times 10^{-11}$. They resumed data taking in 2023 after some upgrades to the detector with the goal of achieving a 20% measurement of the branching ratio by the next long shutdown of the accelerator.

Using the latest CKMfitter input (Jan. 2022) [27], we estimate $B(K^+ \to \pi^+\nu\bar{\nu})_{SM} = (8.1 \pm 0.4) \times 10^{-11}$, near the lower end of the measurements of BNL-787/949 and NA62. Current parametric uncertainty in the CKM angles can result in numbers with central values differing from this one by up to 10% [45].

The second golden mode is the neutral counterpart to our preceding discussion: $K_L \to \pi^0\nu\bar{\nu}$. It is dominantly CP-violating and free of hadronic uncertainties [24, 46, 47]. In the Standard Model, this mode is dominated by an intermediate top-quark state and does not suffer from the small uncertainty associated with the charm-quark intermediate state that affects $K^+ \to \pi^+\nu\bar{\nu}$. The branching ratio is given by [9]:

$$B(K_L \to \pi^0 \nu \bar{\nu}) = \kappa_L \left( \frac{\mathrm{Im}(V_{ts}^* V_{td})}{\lambda^5} X_t \right)^2 \approx 7.6 \times 10^{-5} |V_{cb}|^4 \eta^2. \quad (65.3)$$

As with the charged mode, the hadronic matrix element can be related to that measured in $K_{\ell 3}$ decay and is parameterized in $\kappa_L$.

Our estimate for the branching ratio within the SM, using the latest CKMfitter input (Jan. 2022) [27], is $(2.8 \pm 0.2) \times 10^{-11}$. Similarly to the charged kaon case, parametric uncertainty in the CKM angles can result in a central value that differs from this one by up to almost 20% [45].

Grossman and Nir (GN) [48] pointed out that, in a nearly model-independent manner, the two golden modes satisfy the relation $B(K_L \to \pi^0\nu\bar{\nu}) \lesssim 4.3 B(K^+ \to \pi^+\nu\bar{\nu})$. Using the BNL 787/949 90% CL bound on $K^+ \to \pi^+\nu\bar{\nu}$, GN then predict $B(K_L \to \pi^0\nu\bar{\nu}) < 1.46 \times 10^{-9}$. Using instead the latest NA62 result, the GN upper bound becomes $B(K_L \to \pi^0\nu\bar{\nu}) < 8.14 \times 10^{-10}$ [13].

The KOTO experiment at J-PARC, whose goal is to make an initial observation of this decay, has been in operation since 2013. It has made a series of data collection runs and has reported on three data sets thus far. In 2018 it published a 90% CL upper limit of $3.0 \times 10^{-9}$ [49], based on 2015 data. They have run almost every year since, making incremental improvements to the experimental configuration between runs. In 2021 they published a result on the data taken between 2016 and 2018 in which three events were observed in the signal region [50], whereas the expected background was 1.22 events, mainly from charged kaons produced by $K_L$ interactions in the downstream collimator. Conservatively assuming the three events were due to background, they extracted a 90% CL upper limit of $4.9 \times 10^{-9}$. Subsequent hardware and analysis improvements reduced the expected background to 0.26 in a sample of similar sensitivity taken in 2021. No events were observed in the signal region and a 90% CL upper limit of $2.0 \times 10^{-9}$ was established [51]. An increase of 3-5 in sensitivity is anticipated from runs planned in the next three years.

The current theoretical and experimental situation for the golden modes is summarized in Fig. 65.2. The red area corresponds to the $1\sigma$ SM prediction we obtain with the latest input available from CKMfitter (January 2022) [27]. The yellow region shows the result established by the combined BNL-787 and BNL-949 results, whereas the green region marks the new NA62 result for $K^+ \to \pi^+\nu\bar{\nu}$. The black shaded region marks the GN exclusion, which lies significantly above the SM expectation leaving a large window for discovery of new physics contributions by experiments seeking to measure $B(K_L \to \pi^0\nu\bar{\nu})$. The 90% CL upper bound on this mode from the KOTO result published in 2018 [49] is shown as a dashed orange line, and is seen to still lie in the GN excluded zone.

<figure><img src="image_325.png" /><figcaption>Figure 65.2: Summary of current situation for the golden modes $K \to \pi\nu\bar{\nu}$. The red ellipse shows the $1\sigma$ SM prediction with input from CKMfitter; the green (yellow) region corresponds to the NA62 (BNL787/949) $1\sigma$ measurement; and the dashed orange line marks the 90% CL KOTO upper bound. The black shaded region shows the GN exclusion.</figcaption></figure>

Much theoretical work has explored beyond the SM scenarios that can populate this window as well as their correlations with other rare processes outside kaon physics. Although it would be relatively straightforward to establish the existence of new physics by observing deviations from their SM values in the $K \to \pi\nu\bar{\nu}$ modes, it would take much more extensive global fits to pinpoint the origin of any such deviation. Partial summaries with references focusing on quark flavor physics can be found in Refs. [9, 52–56]. Different possibilities emphasizing an interpretation of these modes as $K \to \pi + E_{\text{miss}}$ can be found in Refs. [57–59].

There is a subtlety in converting the GN bound extracted from measurements of $K^+ \to \pi^+\nu\bar{\nu}$ into an upper bound for $K_L \to \pi^0\nu\bar{\nu}$ as applied to KOTO [60]. This is due to a lack of sensitivity of the charged kaon experiments to a kinematic window near the pion mass. New physics appearing through a two body mode $K^+ \to \pi^+X^0$, for an invisible $X^0$ with mass near the pion mass, has a much weaker constraint. The situation is illustrated in Fig. 65.3 which shows the BNL787/949 [43] and NA62 [61] 90% c.l. limits on $K^+ \to \pi^+X^0$ as a function of $m_{X^0}$ assuming $X^0$ is stable. The implied GN bound for $K_L \to \pi^0X^0$, $B(K_L \to \pi^0X^0) \lesssim 4.3B(K^+ \to \pi^+X^0)$ is shown as a blue line and can be compared with the actual KOTO [49] constraint on $K_L \to \pi^0X^0$ in red. Notice that the upper bound derived from NA62 is lower than that derived from BNL787/949 except for a small mass window. For $120 \lesssim m_{X^0} \lesssim 150$ MeV, KOTO already has better sensitivity than that implied by the GN bound in this case. The possibility of new physics in this scenario has generated much theoretical speculation [62–72].

The search for a new light particle $X^0$ in $K^+ \to \pi^+X^0$ has a long tradition covering both long-lived particles (e.g., hyperphoton, axion, familon, etc.), and short-lived ones that decay to muon, electron, photon or neutrino pairs. The longstanding 90% CL upper limit on $K^+ \to \pi^+X^0$ from BNL787/949 $7.3 \times 10^{-11}$ [41] for the case of massless $X^0$ has been slightly improved by NA62 to $5 \times 10^{-11}$ as can be read from Fig. 65.3. These limits can also be reinterpreted in connection with a dark photon [73] or dark Z [74], and in this context NA48/2 also constrained the mode $B(K^\pm \to \pi^\pm A' \to \pi^\pm e^+e^-)$ [75]. Complementary searches for new light particles in kaon experiments use modes with two pions and include the KTeV bound $B(K_L \to \pi^0\pi^0X^0 \to \pi^0\pi^0\mu^+\mu^-) < 1 \times 10^{-10}$ [76] and the E391a bound $B(K_L \to \pi^0\pi^0X^0 \to \pi^0\pi^0\gamma\gamma) < 2.4 \times 10^{-7}$ [77].

Neutrino pair modes with one extra pion, $K \to \pi\pi\nu\bar{\nu}$, are similarly dominated by short distance contributions in the SM [78–80]. Even though they are theoretically clean, they occur with very
<figure><img src="image_326.png" /><figcaption>**Figure 65.3:** Summary of current situation for $K \to \pi X^0$. The solid (dashed) black line shows the exclusion limit for $K^+ \to \pi^+ X^0$ from NA62 (BNL787/949) assuming $X^0$ is stable. The blue line marks the constraint implied for $K_L \to \pi^0 X^0$ by the Grossman-Nir relation and the red line marks the current KOTO limit on $K^+ \to \pi^+ X^0$.</figcaption></figure>

low rates with branching ratios of order $10^{-13}$. The current best bound comes from KEK-391a, $B(K_L \to \pi^0 \pi^0 \nu \bar{\nu}) < 8.1 \times 10^{-7}$ at 90% CL [81]. There is also a bound for the charged kaon mode $B(K^+ \to \pi^+ \pi^0 \nu \bar{\nu}) < 4.3 \times 10^{-5}$ at 90% CL [82] from BNL-787. New physics contributions to these modes are discussed in [83].

## 65.4 Other constraints on Standard Model parameters

The decay $K_L \to \mu^+\mu^-$ has a short distance contribution sensitive to the CKM parameter $\bar{\rho}$, given by [5]:

$$B_{\text{SD}}(K_L \to \mu^+\mu^-) \approx 2.7 \times 10^{-4} |V_{cb}|^4 (\rho'_c - \bar{\rho})^2 \quad (65.4)$$

where the three terms correspond to the indirect CP violation, the interference, and the direct CP violation, respectively. The parameter $a_S$ has been extracted by NA48/1 from a measurement of $K_S \to \pi^0 e^+ e^-$ with the result $|a_S| = 1.06^{+0.26}_{-0.21} \pm 0.07$ [99], as well as from a measurement of $K_S \to \pi^0 \mu^+ \mu^-$ with the result $|a_S| = 1.54^{+0.40}_{-0.32} \pm 0.06$ [100]. With current constraints on the CKM parameters, and assuming a positive sign for the interference term [98, 101], this implies that $B_{\text{CPV}}(K_L \to \pi^0 e^+ e^-) \approx (3.1 \pm 0.9) \times 10^{-11}$, where the three contributions to the central value from indirect, interference and direct CP violation are $(1.76, 0.9, 0.45) \times 10^{-11}$ respectively. It should be noted that more recent studies suggest a much larger uncertainty in the value of $a_S$ [102].

$K_L \to \pi^0 e^+ e^-$ also has a CP-conserving component dominated by a two-photon intermediate state. This component can be decomposed into an absorptive and a dispersive part. The absorptive part can be extracted from the measurement of the low $m_{\gamma\gamma}$ region of the $K_L \to \pi^0 \gamma \gamma$ spectrum. The rate and the shape of the distribution $d\Gamma/dm_{\gamma\gamma}$ in $K_L \to \pi^0 \gamma \gamma$ are well described in chiral perturbation theory in terms of three (a priori) unknown parameters [103–106].

Both KTeV and NA48 have studied the mode $K_L \to \pi^0 \gamma \gamma$, reporting similar results. KTeV finds $B(K_L \to \pi^0 \gamma \gamma) = (1.29 \pm 0.03)_{\text{stat}} \pm 0.05)_{\text{sys}} \times 10^{-6}$ [107], while NA48 finds $B(K_L \to \pi^0 \gamma \gamma) = (1.36 \pm 0.03)_{\text{stat}} \pm 0.03)_{\text{sys}} \pm 0.03)_{\text{norm}} \times 10^{-6}$ [108]. Both experiments are consistent with a negligible rate in the low $m_{\gamma\gamma}$ region, suggesting a very small CP-conserving component $B_{\text{CP}}(K_L \to \pi^0 e^+ e^-) \sim \mathcal{O}(10^{-13})$ [98, 106, 108]. There remains some model dependence in the estimate of the dispersive part of the CP-conserving $K_L \to \pi^0 e^+ e^-$ [98].

where $\rho'_c$ depends on the charm quark mass and is approximately 1.2. This decay, however, is dominated by a long-distance contribution from a two-photon intermediate state. The absorptive (imaginary) part of the long-distance component is determined by the measured rate for $K_L \to \gamma \gamma$ to be $B_{\text{abs}}(K_L \to \mu^+ \mu^-) = (6.64 \pm 0.07) \times 10^{-9}$; and it almost completely saturates the observed rate $B(K_L \to \mu^+ \mu^-) = (6.84 \pm 0.11) \times 10^{-9}$ [84]. The difference between the observed rate and the absorptive component can be attributed to the (coherent) sum of the short-distance amplitude and the real part of the long-distance amplitude. The latter cannot be derived directly from experiment [85], but can be estimated with certain assumptions [86, 87].

By contrast, the decay $K_L \to e^+ e^-$ is completely dominated by long distance physics and is easier to estimate. The result, $B(K_L \to e^+ e^-) \sim 9 \times 10^{-12}$ [85, 88], is in good agreement with the BNL-871 measurement, $(8.7^{+5.7}_{-4.1}) \times 10^{-12}$ [89].

The mode $K_S \to \mu^+ \mu^-$ has a short distance contribution proportional to the square of the CKM parameter $\bar{\eta}$ entering at the $10^{-13}$ level [10]. It also has long distance contributions arising from the two photon intermediate state which result in a rate $B(K_S \to \mu^+ \mu^-)_{LD} = 5.1 \times 10^{-12}$ [10]. There is a 90% CL limit $B(K_S \to \mu^+ \mu^-) < 2.1 \times 10^{-10}$ from LHCb [90, 91]. The interplay between $K_L \to \mu^+ \mu^-$ and $K_S \to \mu^+ \mu^-$ has been the subject of [92, 93], and it has been pointed out that a measurement of time-dependent interference effects could be used to extract information on CKM angles from $K \to \mu^+ \mu^-$ measurements [94, 95].

The decay $K_L \to \pi^0 e^+ e^-$ is sensitive to the CKM parameter $\eta$ through its CP-violating component. There are both direct and indirect CP-violating amplitudes that can interfere. The direct CP-violating amplitude is short distance dominated and has been calculated in detail within the SM [5]. The indirect CP-violating amplitude can be inferred from a measurement of $K_S \to \pi^0 e^+ e^-$. The complete CP-violating contribution to the rate can be written as [96–98]:

$$B_{\text{CPV}} \approx \left[ 15.7 |a_S|^2 \pm 1.4 \left( \frac{|V_{cb}|^2 \bar{\eta}}{10^{-4}} \right) |a_S| + 0.12 \left( \frac{|V_{cb}|^2 \bar{\eta}}{10^{-4}} \right)^2 \right] \times 10^{-12} \quad (65.5)$$

The related process, $K_L \to \pi^0 \gamma e^+ e^-$, is potentially an additional background to $K_L \to \pi^0 e^+ e^-$ in some region of phase space [109]. This process has been observed with a branching ratio of $(1.62 \pm 0.14)_{\text{stat}} \pm 0.09)_{\text{sys}} \times 10^{-8}$ [110].

The decay $K_L \to \gamma \gamma e^+ e^-$ constitutes the dominant background to $K_L \to \pi^0 e^+ e^-$. It was first observed by BNL-845 [111], and subsequently confirmed with a much larger sample by KTeV [112]. It has been estimated that this background will enter at about the $10^{-10}$ level [113, 114], comparable to or larger than the signal level. Because of this, the observation of $K_L \to \pi^0 e^+ e^-$ at the SM level will depend on background subtraction with good statistics. Possible alternative strategies are discussed in [98] and references cited therein.

The 90% CL upper bound for the process $K_L \to \pi^0 e^+ e^-$ is $2.8 \times 10^{-10}$ [114]. For the closely related muonic process, the published upper bound is $B(K_L \to \pi^0 \mu^+ \mu^-) \le 3.8 \times 10^{-10}$ [115], compared with the SM prediction of $(1.9 \pm 0.5) \times 10^{-11}$. The latter assumes positive interference between the direct- and indirect-CP violating components and includes a CP-conserving component which contributes about 30% of the total [10, 116].

A study of $K_L \to \pi^0 \mu^+ \mu^-$ has indicated that it might be possible to extract the direct CP-violating contribution by a joint study of the Dalitz plot variables and the components of the $\mu^+$ polarization [117]. The latter tends to be quite substantial so that large statistics may not be necessary.

Combined information from $K_L \to \pi^0 \ell^+ \ell^-$ as well as $K_L \to \mu^+ \mu^-$ complements the $K \to \pi \nu \bar{\nu}$ measurements in constraining physics beyond the SM [118].