Photoproduction and detection of $\rho^{\prime}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ decays in ultra-peripheral collisions and at an electron-ion collider

Vector meson photoproduction has been studied extensively at fixed-target accelerators Bauer et al. (1978), the HERA $ep$ collider Ivanov et al. (2006), and with ultra-peripheral collisions (UPCs) at heavy-ion colliders Klein and Steinberg (2020). It will also be an important probe of nucleons and nuclei at a future electron-ion collider (EIC) Accardi et al. (2016); Abdul Khalek et al. (2021); Adam et al. (2022). Through the Good-Walker paradigm, coherent vector meson production is sensitive to the average nuclear configuration, while incoherent vector meson photoproduction is related to fluctuations in the nuclear configuration, including gluonic hotspots Miettinen and Pumplin (1978); Mäntysaari and Schenke (2016); Klein and Mäntysaari (2019).

The $Q^{2}$ dependence of exclusive production is an important signature of saturation Mäntysaari and Venugopalan (2018). Definitive conclusions about saturation will require studies of different mesons, with different wave functions and masses.

The $\rho$ is straightforward to reconstruct Adler et al. (2002), but from the theory perspective is rather light, limiting the applicability of perturbative QCD (pQCD) based calculations. The $J/\psi$ is heavy enough that saturation phenomena are greatly reduced Mäntysaari and Venugopalan (2018). The $\phi$ is attractive because it has an intermediate mass (between the $\rho$ and the $J/\psi$).

Early plans for exclusive vector meson production at a U. S. EIC focused on the $\phi$ and $J/\psi$ Accardi et al. (2016). However, $\phi$ production at low $Q^{2}$ is hard to reconstruct Abdul Khalek et al. (2021); Arrington et al. (2021) because the main channel, $\phi\rightarrow K^{+}K^{-}$, suffers from a low $Q$ value, with the daughter kaons having a momentum of only 127 MeV/c in the $\phi$ rest frame. Other final states have either low branching ratios or include a long-lived $K^{0}_{L}$. For UPCs, the situation is similar, with coherent $\phi$ photoproduction on ion targets difficult to observe Acharya et al. (2024a); Chekhovsky et al. (2025).

The $\rho^{\prime}$ states are attractive alternatives to the $\phi$, as they also have intermediate masses, between the $\rho$ and $J/\psi$. However, these states have a more complex wave function and a more complex phenomenology. There are likely two overlapping resonances, the $\rho^{\prime}(1450)$ and the $\rho^{\prime}(1700)$ Workman et al. (2022). These resonances can decay to many different final states, but both have a significant branching ratio to $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$. Since the $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$final state is easy to reconstruct, we will focus on it. Most photoproduction analyses have fit the $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$mass spectrum to a single resonance, so we will perforce do the same here.

The $4\pi$ final state has been studied at fixed-target accelerators, using $ep$ collisions at HERA and in ultra-peripheral collisions (UPCs) of gold ions at RHIC and lead ions at the LHC. Using HERA and fixed-target data on proton targets as input, we will use a Glauber calculation to predict the RHIC and LHC cross sections for ion targets. These predictions depend significantly on the $\rho^{\prime}\rightarrow$$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$branching ratio, giving us some sensitivity to that quantity.

We also make predictions about the the $\rho^{\prime}$ cross sections in $ep$ and $eA$ collisions at the EIC, and estimate the reconstruction efficiency using a simple model of the proposed ePIC detector. As with UPCs, the $eA$ cross sections depend on the $4\pi$ branching ratio.

$\rho^{\prime}$ are radial excitations of the $\rho$, with the same $J^{PC}=1^{--}$ quantum numbers. These states are most visible in the $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$final state, which has been studied at a variety of fixed-target photoproduction experiments and in $e^{+}e^{-}$ collisions Navas et al. (2024). The photoproduced $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$final state is observed as a single broad resonance, which has usually not been separated into two different states.

The first studies of $4\pi$ photoproduction were done in fixed-target experiments at photon energies from 3 GeV to 70 GeVBingham et al. (1972); Schacht et al. (1974); Alexander et al. (1975); Barber et al. (1980); Aston et al. (1981a); Atkinson et al. (1985); Atiya et al. (1979). Four pion photoproduction was first studied at collider energies by the Solenoidal Tracker at RHIC (STAR) collaboration Abelev et al. (2010), and later by the ALICE detector at the LHC Acharya et al. (2024b). We use the mass and width that STAR measured: mass $M_{V}=1570$ MeV$/c^{2}$ and width $570\pm 60\$MeV$/c^{2}$.

The H1 collaboration has also studied 4-pion photoproduction, and observed a resonance with similar parameters to STAR, also well fit by a single resonance Schmitt (2018); H1 Collaboration (2918). We will use this HERA cross-section data as input to make predictions for ion targets in UPCs and at the EIC.

The cross section for photoproduction in UPCs is given by combining the $\gamma p$ (Eq. 1) or $\gamma A$ cross sections (Eq. 7) with the appropriate photon flux for protons Klein and Nystrand (2004) or ions Klein and Nystrand (2000). This code is now implemented in the STARlight Monte Carlo code Klein et al. (2017).

Figure 2 shows the calculated $d\sigma/dy$ for the $\rho^{\prime}$ for four assumptions about the $4\pi$ branching ratio between 10% and 100% and for the two different possible $f_{v}$s. The calculations are compared with ALICE data on 5.02 TeV lead-lead collisions Acharya et al. (2024b). For the Ref. Klusek-Gawenda and Tapia Takaki (2020) coupling, the data matches the cross section with about a 30% branching ratio. Including the estimated 17% error on $\Gamma_{V\rightarrow ee}$ from Ref. Klusek-Gawenda and Tapia Takaki (2020) does not introduce large uncertainties into this estimate.

For the GVDM-predicted coupling, the best-fit branching ratio is unreasonably low, requiring a very large total $\rho^{\prime}$ cross section. The GVDM prediction in Eq. 4 must be too high. A GVDM calculation that included off-diagonal elements might do better, though. Alternately, a calculation that included inelastic shadowing (cross-section fluctuations) might improve the match between $\rho^{0}$ UPC data and a quantum Glauber calculation Frankfurt et al. (2016).

The poorly-known nature of the $\rho^{\prime}$ introduces another uncertainty. If, as seems likely, the $\rho^{\prime}$ is composed of two resonances, then there is no reason that the two resonances should have the same branching ratio or $\Gamma_{V\rightarrow ee}$. With these caveats, in the single-resonance model, branching ratios near 100% are disfavored.

Unfortunately, the STAR data on the $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ final state suffers from large experimental uncertainty Abelev et al. (2010) so it cannot contribute to the comparison.

Another uncertainty involves the resonance substructure of the $4\pi$ final state, which can impact the experimental acceptance, and, with it, the comparison with data. Here, we modeled the $4\pi$ final state using phase space and use the resulting events to determine the acceptance, using simple models for three different LHC detectors. The assumed detector parameters are given in Table 2, along with the determined detector efficiencies. For all four detectors, the efficiency is limited because the $\rho^{\prime}$ signal is spread over a much wider rapidity range than the detector acceptance. Figure 3 shows the expected acceptance for the different detectors. If there are two resonances, then there is no reason they should have the same resonant substructure; this would further complicate the picture.

Photoproduction in $ep/e\rm{A}$ collisions adds another dimension: the photon $Q^{2}$. There is no $\rho^{\prime}$ data on cross sections for virtual photons, so we assume that the $Q^{2}$ evolution of the cross-section is the same as for the $\rho$ Aaron et al. (2010); Lomnitz and Klein (2019):

where $n=2.09+0.73/{\rm GeV}^{-2}(M_{V}^{2}+Q^{2})$.

$\rho^{\prime}$ production is assumed to follow vector meson dominance, with the final state being linearly polarized transverse to the beam direction at $Q^{2}=0$, but with an increasing longitudinal polarization as $Q^{2}$ rises. The rate of this increase is not known for the $\rho^{\prime}$, but we use the approach described in Ref. Lomnitz and Klein (2019) here, assuming that the spin matrix for the $\rho^{\prime}$ is the same as for the $\rho$. The $\rho^{\prime}$ decays are assumed to follow a 4-pion phase space distribution; assuming a $\rho^{0}\pi^{+}\pi^{-}$ final state Abelev et al. (2010) would lead to some changes in efficiency, but would not alter the conclusions here.

Simulations were performed in the eSTARlight framework Lomnitz and Klein (2019). Calculations were done for the top EIC energies: 18 GeV electrons colliding with either 275 GeV protons or 110 GeV/nucleon lead ions. Table 3 shows the calculated cross sections. In addition to the total (integrated over all $Q^{2}$) cross section, cross sections are given for photoproduction ($Q^{2}<1\,{\rm GeV}$ and electroproduction $1<Q^{2}<10\,{\rm GeV}^{2}$). The total cross section is dominated by photoproduction, as expected.

The cross section for $\rho^{\prime}$ production in $e$Pb collisions is about 45 times that for $ep$ collisions. This is a smaller ratio than for the $\rho$ or $\phi$, but is similar to the ratio found for $\gamma$A and $\gamma p$ collisions in Ref. Klusek-Gawenda and Tapia Takaki (2020). For $ep$ collisions, the cross-section is about 1/7 of the $\rho$ cross section, and about 1/3 of the $\phi$ cross section. For $e$Pb collisions, the ratios are lower, with the $\rho^{\prime}$ cross section 1/30 of the $\rho$ cross section, and half that for the $\phi$. In both cases, the rates are high enough (7 or 1.5 billion events per 10 fb${}^{-1}/A$ of integrated luminosity) to allow high-precision differential measurements. Measurements of final states with small branching ratios, including $e^{+}e^{-}$, should also be possible, to accurately measure the photon-$\rho^{\prime}$ coupling. If there are two resonances, then it should be possible to measure separate branching ratios for different final states.

The corresponding $d\sigma/dy$ are shown in the two top panels in Fig. 4. Also shown is the acceptance of the ePIC detector. ePIC is modeled with two components: a central barrel tracker sensitive to $|\eta|<3.5$ and the B0 detector, which covers $4.6<\eta<5.9$. For both detectors, we required track $p_{T}>100$ MeV/c, but this cut had little effect on the $\rho^{\prime}$ efficiency. The bottom panels of Fig. 4 show the detection efficiency for the photoproduction events. The efficiency is high for $|\eta|<2.5$, but falls off at larger $|y|$, as expected. The B0 detector plays an important role for $ep$ collisions in the region $y>3$; otherwise the efficiency would be near zero for $y>3.5$. Even though the efficiency is fairly low in this region, the rates are high enough that high statistics data should be achievable.

The $\rho^{\prime}$ rapidity is closely related to the target Bjorken$-x$

where $m_{p}$ is the proton mass and $\gamma=292$ is the Lorentz boost of the proton beam at maximum EIC energy. The central detector cutoff, $\eta=3.5$, corresponds roughly to $y\approx 3.5$ (as can be seen in Fig. 4), or $x\approx 10^{-4}$. For $ep$ collisions, covering the full range of Bjorken$-x$ requires acceptance out to rapidity $\approx 5$, corresponding to $x\approx 2\times 10^{-5}$. Although this is beyond the range of the central tracker, Fig. 4 shows that the B0 tracker does cover this region, albeit with low total efficiency. Nuclear shadowing (beyond that present in the Glauber calculation) or saturation would most clearly manifest itself as a reduction in cross-section with decreasing $x$, i. e. with increasing rapidity at low/moderate $Q^{2}$ Mäntysaari and Venugopalan (2018). There would also be changes in $d\sigma/dt$ Accardi et al. (2016). The rates are high enough that statistics will not limit these measurements, even at the lowest $x$.

In short, the $\rho^{\prime}$ is copiously produced and relatively easy to reconstruct, showing promise for use in saturation studies. The largest uncertainties in the rates are due to the photon-meson coupling constant and to the uncertainty as to whether this is one meson or two. Fortunately, the coupling to $e^{+}e^{-}$ can be determined experimentally, by measuring that final state, while the number of resonances can be determined by detailed studies of the resonance line shape and substructure. The remaining uncertainties are much smaller, making the $\rho^{\prime}$ an excellent medium-mass candidate for mapping out shadowing as a function of $Q^{2}$.

We have calculated the cross section for $\rho^{\prime}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ on ion targets, using fixed-target and HERA data for photoproduction on proton targets as input. The resulting ion-target cross sections depend on the photon-meson couplings and the branching ratio for the $\rho^{\prime}$ to decay to $4\pi$. The couplings predicted using a GVDM model lead ion-target cross-sections that are high enough to require implausibly small branching ratios. Using the coupling from Ref. Klusek-Gawenda and Tapia Takaki (2020), ALICE data on $\rho^{\prime}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, prefers branching ratios in the 30% to 50% range. This estimate relies on the validity of a Glauber calculation approach, but small deviations will not have a large effect on this conclusion.

As previously noted, the $\rho(1570)$ is likely a composite of two mesons, the $\rho(1450)$ and $\rho(1700)$. The ALICE data is well fit by a two meson model, but the mass spectrum does not fit a single meson by itself. However, as Fig. 5 shows, the two mass spectra for the two targets look very similar. This indicates that the photon-meson couplings times the branching ratio to $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ are similar for the two mesons. Otherwise, the Glauber approach would distort the $4\pi$ mass spectra for ion targets. More detailed comparisons of the resonance shape and substructure should clearly show if there is one resonance or two. High-statistics data from LHC Runs 3 and 4 Citron et al. (2019) should allow for a definitive comparison.

With this LHC data, it should also be possible to measure the coupling to $e^{+}e^{-}$. This might be easiest using higher $p_{T}$ pairs from incoherent photoproduction, to avoid backgrounds from $\gamma\gamma\rightarrow e^{+}e^{-}$. It would also be possible to investigate whether a GVDM model with off-diagonal intermediate states might better predict the photon-meson couplings Kobayashi (1973); Fraas and Kuroda (1977); Frankfurt et al. (2003b).

We have also made predictions for $\rho^{\prime}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ at the proposed U. S. EIC. The production rates are high, and the meson is an easy-to-reconstruct probe of nuclear structure. It is intermediate in mass between the $\rho^{0}$ and the $J/\psi$, so it should show significant saturation, but also be amenable to pQCD calculations. These features should also hold at the proposed Chinese electron-ion collider, EiCC Chen et al. (2020) or the proposed LHeC collider Agostini et al. (2021). For the LHeC, excellent forward instrumentation is critical to be able to observe photoproduction and electroproduction at the highest energies, corresponding to the lowest Bjorken$-x$. For the LHeC, though, the rate differences between the three cross-section fits will be larger.

The approach developed here, comparing $\gamma p$ and $\gamma$A collisions may also be applicable for determining absolute branching ratios for some other mesons, as long as the photon-meson coupling is known.

This work is supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract numbers DE-AC02-05CH11231 and DE-FG02-96ER40991.