Proposal to look for the anomalous isotopic symmetry breaking in central diffractive production of the $f_1(1285)$ and $a^0_0(980)$ resonances at the LHC

At very high energies, and in the central region ($x_F\simeq0$), the double-Pomeron exchange mechanism gives the dominant contribution to the production of hadrons with the positive $C$ parity and isospin $I=0$. Therefore, the observation of resonances in the states with $I=1$ will be indicative of their production or decay with the isotopic symmetry breaking. Here, we bear in mind the cases of the anomalous breaking of the isotopic symmetry, i.e., when the cross section of the process breaking the isospin is not of the order of $10^{-4}$ of the cross section of the allowed process but of the order of $1\%$. The paper draws attention to the reactions $pp\to p(f_1(1285)/f_1(1420))p\to p(\pi^+\pi^-\pi^0)p$ and $pp\to p(K\bar K)p\to p(a^0_0(980))p\to p(\eta\pi^0)p$ in which a similar situation can be realized, owing to the $K\bar K$ loop mechanisms of the breaking of isotopic symmetry. We note that there is no visible background in the $\pi^+\pi^-\pi^0$ and $\eta \pi^0$ channels. Observation of the process $pp\to p(f_1(1285))p\to p(\pi^+\pi^-\pi^0)p$ would be a confirmation of the first results from the VES and BESIII detectors, indicating the very large isospin breaking in the decay $f_1(1285))\to\pi^+\pi^-\pi^0$.

At very high energies, and in the central region, the double-Pomeron exchange mechanism, PP, gives the dominant contribution to production processes of hadronic resonances with the positive C parity and isospin I = 0 (see Fig. 1). The reaction cross sections caused by the double-Pomeron exchange mechanism do not decrease in a power-law manner with increasing P P p (p 1 ) Рис. 1: The central production of a state h by the double-Pomeron exchange mechanism, PP, in the reaction pp → p(h)p. The 4-momenta of the initial and final protons, P exchanges, and h system are indicated in parentheses; the main kinematic variables in this reaction are s = (p1 + p2 energy [1,2,[24][25][26][27]. Therefore, observation of the wellknown resonances in the states h with I = 1 will be indicative of their production or decay with the isotopic symmetry breaking. Here, we consider the examples of the reactions pp → p(h)p in which the anomalous breaking of the isotopic symmetry can occur.
Thus, investigations of the f 1 (1285) and f 1 (1420) resonances produced in central pp interactions allow one to determine in a single experiment the branching ratios for all their major decay modes. Moreover, we pay attention that the reaction pp → p(f 1 (1285))p → p(π + π − π 0 )p, due to the isotopic neutrality of the PP exchange mechanism, gives a unique possibility to investigate the isospin-breaking decay f 1 (1285) → f 0 (980)π 0 → π + π − π 0 in a situation free from any visible coherent background in the π + π − π 0 channel. Because of completely different experimental conditions, such a study would be a good test of the first results from the VES [28] and BESIII [29] detectors, indicating to the strong isospin breaking in this decay.
According to the data from the VES Collaboration [28], The data from the BESIII Collaboration [29] give The f 1 (1285) → ηπ + π − decay channel has the largest branching ratio, , among all other recorded decay channels [3,4,[16][17][18][19][20][21]. Therefore, the ratios mentioned in Eqs. (1) and (2) give a good reason to believe that we really deal with the anomalously large isospin breaking in the transition f 1 (1285) → f 0 (980)π 0 → π + π − π 0 . Moreover, the π + π − mass spectrum observed in the decay f 1 (1285) → f 0 (980)π 0 → π + π − π 0 represents a narrow resonance structure with a width of 10 − 20 MeV located near the KK thresholds [28,29]. The various KK loop mechanisms responsible for the decay lead to the π + π − mass spectrum of such a type [31][32][33][34]. A significant violation of isotopic symmetry in this transitions is a threshold phenomenon. It occurs in the narrow region of the π + π − invariant mass near the KK thresholds due to the incomplete compensation between the contributions of the K + K − and K 0K 0 intermediate states caused by the mass difference of the K + and K 0 mesons [31][32][33][34][35][36]. Certainly, the data on the f 1 (1285) → f 0 (980)π 0 → π + π − π 0 decay need to be clarified. Information on the reaction pp → p(f 1 (1285))p → p(π + π − π 0 )p could probably be extracted from the data collected by the CERN Omega Spectrometer and Collider Detector at Fermilab. However, for this, enthusiasts are needed, since these facilities have long been closed. At present, the reaction pp → p(f 1 (1285))p → p(π + π − π 0 )p could be measured, for example, using the CMS detector at the LHC (it is interesting also to study the related reaction pp → pf 1 (1420)p → p(π + π − π 0 )p [32,37]). Recently, the CMS Collaboration has presented the data on the central exclusive and semiexclusive π + π − production in pp collisions at √ s = 7 TeV [6]. With such a huge total energy, the energy values √ s 1 and √ s 2 for the subprocesses p(p 1 )P(q 2 ) → p(p ′ 1 )h(q) and p(p 2 )P(q 1 ) → p(p ′ 2 )h(q) (see Fig. 1) are also very large. In fact, they fall into the region in which the contributions of the secondary Regge trajectories, R, can be neglected in comparison with the contribution of the P exchange. If we put s 1 ≈ s 2 , M ≈ 1 GeV, and Рис. 2: The KK loop mechanism of the a 0 0 (980) production in the central region via PP exchange. √ s = 7 TeV and use the relation s 1 s 2 ≈ M 2 s (valid for the processes in the the central region [24][25][26]), we find that √ s 1 ≈ √ s 2 ≈ 84 GeV. Thus, the dominance of the PP exchange mechanism appears to be a good approximation at the LHC energies. Note that in the above-mentioned experiments carried out at CERN and Tevatrov (at fixed target) the values of √ s 1 ≈ √ s 2 were approximately equal to ≈ 3.6, 4.9, 5.4, and 6.3 GeV. Therefore, in a number of cases, when interpreting the results, it was necessary to take into account, along with the PP exchange mechanism, the mechanisms involving the secondary Regge trajectories, R, i.e., the RP and RR exchanges.
The f 1 (1285)) resonance production with its subsequent decay to π + π − π 0 can be also studied in central pp, pA, π − p, and π − A interactions at the Serpukhov acceleration in Protvino.
The corresponding cross section as a function of the ηπ 0 invariant mass, M ≡ m, has the form (see Fig. 4) where Γ a 0 0 →ηπ 0 (m) is the width of the a 0 0 (980) → ηπ 0 decay, D a 0 0 (m) and D f0 (m) are the inverse propagators of the a 0 0 (980) and f 0 (980), respectively, Π 2 0 (980) transition amplitude (all these functions, together with the corresponding values of the resonance parameters, have been written in Ref. [32]), and C PP→f0 is the f 0 (980) production amplitude.
Note that the a 0 0 (980) production cross section σ(PP → (K + K − + K 0K 0 ) → a 0 0 (980) → ηπ 0 ; m) can be estimated without detailing the PP → (K + K − +K 0K 0 ) transition mechanism (see Fig. 2), which, in principle, can be caused by not only the f 0 (980) resonance contribution (see Fig. 3), but also some nonresonance KK production mechanism. To do this, we use the relation valid according to the unitarity condition near the KK thresholds (see Refs. [32,33] for details) where ρ KK (m) = 1 − 4m 2 K /m 2 at m > 2m K and ρ KK (m) = i 4m 2 K /m 2 − 1 at 0 < m < 2m K . The resulting shape of the cross section is very similar to the solid curve in Fig. 4. The value of | A(2m K + )| 2 should be determined from the data on the K + K − production cross section near the threshold For m between the K + K − and K 0K 0 thresholds, we get by an order of magnitude [32] The comparison of this estimate with the data on σ(PP → a 0 0 (980) → ηπ 0 ; m) permits one to verify their consistency with the data on σ(PP → K + K − ; m) and with the idea of the KK loop breaking of isotopic invariance caused by the mass difference of K + and K 0 mesons. Note that a similar way of the checking the consistency between the data on the decays f 1 (1285) → π + π − π 0 and f 1 (1285) → KKπ has been discussed in Refs. [32,33]. Detailed formulas connecting σ(PP → h; m) with the experimentally measured cross section of the reaction pp → p(h)p can be found, for example, in Refs. [24][25][26].
First, the central production of the a 0 0 (980) resonance in the reaction pp → p(ηπ 0 )p has been studied by the WA102 Collaboration with the use of the CERN Omega Spectrometer at √ s = 29 GeV [23,38]. The interpretation of these data has been discussed in Refs. [39][40][41][42]. Here, we note the following. In the above experiment, the clear peaks from a 0 0 (980) and a 0 2 (1320) resonances have been observed in the ηπ 0 mass spectrum. The fit [23] gave the quite usual widths of these states [30]: Γ(a 0 (980)) = 72 ± 16 MeV and Γ(a 2 (1320)) = 115 ± 20 MeV . Such a picture indicates that at the energy √ s 1 ≈ √ s 2 ≈ 4 m 2 GeV the secondary Regge exchanges, for which the ηπ 0 production in the central region is not forbidden by G parity, play an important role. For example, the central a 0 0 (980) production can proceed via R(η)R(π 0 ) → a 0 0 (980), R(a 0 2 )R(f 2 ) → a 0 0 (980), and R(a 0 2 )P → a 0 0 (980) transitions, where the type of the secondary Regge trajectory R is indicated in parentheses. At the LHC energies, the contributions of the secondary Regge trajectories fall off appreciably and it is natural to expect that the a 0 0 (980) resonance must mainly be produced via the double-Pomeron exchange mechanism (see Fig  2), which essentially violates the isotopic invariance in the KK threshold region. The narrowing of the a 0 0 (980) peak in the ηπ 0 channel up to the width of 10 − 20 MeV (see Fig. 4) will serve as an indicator of the changing central production mechanism of the a 0 0 (980) resonance with increasing energy.
The present work is partially supported by the Russian Foundation for Basic Research Grant No. 16-02-00065.