Evidence for $\Delta(2200)7/2^-$ from photoproduction and consequence for chiral-symmetry restoration at high mass

We report a partial-wave analysis of new data on the double-polarization variable $E$ for the reactions $\gamma p\to \pi^+ n$ and $\gamma p\to \pi^0 p$ and of further data published earlier. The analysis within the Bonn-Gatchina (BnGa) formalism reveals evidence for a poorly known baryon resonance, the one-star $\Delta(2200)7/2^-$. This is the lowest-mass $\Delta^*$ resonance with spin-parity $J^P=7/2^-$. Its mass is significantly higher than the mass of its parity partner $\Delta(1950)7/2^+$ which is the lowest-mass $\Delta^*$ resonance with spin-parity $J^P=7/2^+$. It has been suggested that chiral symmetry might be restored in the high-mass region of hadron excitations, and that these two resonances should be degenerate in mass. Our findings are in conflict with this prediction.

In spite of these large mass shifts between low-mass parity partners, mesons and baryons at higher masses are often observed in parity doublets, in pairs of resonances having the same total * Corresponding author. The resonances with J P = 1/2 ± , 3/2 ± , 5/2 ± are nearly massdegenerate, in particular when their natural widths of a few hundred MeV are considered. This example and similar observations in mesons and baryons have led to the conjecture that chiral symmetry might be effectively restored in highly excited hadrons [9,10] and has stimulated a vivid discussion; we quote here a few recent reviews [11][12][13][14]. Based on the hypothesis of chiral-symmetry restoration, the parity partner of (1950)7/2 + with spin-parity 7/2 − should also have a mass of about 1900 or 1950 MeV while quark models [15] and AdS/QCD [16] predict ≈2200 MeV. The Review of Particle Properties lists the one-star (2200)7/2 − resonance and thus disfavors models assuming that chiral symmetry could be restored in high-mass baryons. But clearly, a resonance for which the evidence for its existence is estimated to be poor (the definition of the one-star rating) cannot decide on this important issue. It is essential to refute or confirm the existence of this state. The 250-MeV mass splitting between (1950)7/2 + and (2200)7/2 − -unexpected when chiral symmetry is effectively restored in highly excited hadrons -points to a more general concern: resonances falling onto a leading Regge trajectory (with J = L + S, L intrinsic orbital angular momentum, S total quark spin) have no mass-degenerate parity partner.
Their non-observation could be assigned to their suppression by the angular-momentum barrier. This is different in photoproduction: a 7/2 + resonance needs an amplitude with L = 3 between photon and nucleon, a 7/2 − resonance an amplitude with L = 2.
Photoproduction hence provides the best and possibly the only chance to find a decisive support or an experimental argument against the hypothesized restoration of chiral symmetry.
In this letter we report on a partial-wave analysis of the data on γ p → nπ + and γ p → pπ 0 covering differential cross sections dσ /d [18][19][20][21], the beam asymmetry [22,23], and the doublepolarization observables T , P , H [24,25], G [26], and E from the CLAS [27] and CBELSA/TAPS [28] experiments. T , P , H can be measured simultaneously when a transversely polarized target and a linearly polarized photon beam are used: where φ is the azimuthal angle between the photon polarization plane and the scattering plane, and α is the azimuthal angle between the vector and the photon polarization plane. G can be deduced from the correlation between the photon polarization plane and the scattering plane for protons polarized along the direction of the incoming photon; E is defined by the (normalized) difference between the cross sections for parallel and anti-parallel photon and proton spin orientations. Data from older experiments [29] are also included in the partial wave analysis. The data are fitted jointly with data on Nη, K , K , Nπ 0 π 0 , and Nπ 0 η from both photo-and pion-induced reactions. Thus inelasticities in the meson-baryon system are constrained by real data. A list of the data used for the fit can be found in [30][31][32][33] and on our website (pwa.hiskp.uni-bonn.de).
We shortly outline the analysis technique and fit strategy. The helicity-dependent amplitude for photoproduction of the final state b is cast into the form [34] A h α is the photo-coupling of a pole α and F a a non-resonant transition. The helicity amplitudes A α are defined as residues of the helicity-dependent amplitude at the pole position and are complex numbers. They become real and coincide with the conventional helicity amplitudes A  tering amplitude T in the I( J P ) = 3/2(7/2 + ) and 3/2(7/2 − ) partial waves [39] and the BnGa fit. No uncertainties are known for [39].
The K matrix parametrizes resonances and background contributions: Here g α a,b are coupling constants of the pole α to the initial and the final state. The background terms f ab describe non-resonant transitions from the initial to the final state. In most partial waves, a constant background term is sufficient to achieve a good fit. Only the background in the meson-baryon S-wave required a more complicated form: Further background contributions are obtained from the reggeized exchange of vector mesons [35].
To deduce branching ratios BR = i / tot , a normalization is required. We use the π N elastic scattering amplitudes T determined by Höhler and collaborators [39], see Fig. 1, which define the (1950)7/2 + and (2200)7/2 − → π N branching ratios. From the fit to photoproduction data described below, the other branching ratios can be deduced. We first made a fit to the data with our standard set of resonances and our standard fitting procedure using weights for low statistics data [30]. These contain all N * and * resonances with nominal masses up to 2.2 GeV listed in the Review of Particle Properties (RPP) [40] except N(1685), 1/2 − , for which we find no evidence. All masses, widths, and coupling constants stay well within the errors quoted in [30]. A few higher-mass resonances are also taken into account (see [30], for a list) which improve the convergence behavior of the fit but their properties, in particular their masses and widths, remain undetermined. The large number of resonances offered to the fit avoids the problem that a group of unconsidered resonances could mimic a single resonance which then might lead to a significant improvement of the fit quality. With these ingredients, a reasonably good fit to all two-body reactions was achieved with a χ 2 = 49044 for 32666 data points. These resonances are all included in the fit or were partly not used. The detailed results depend on model space which is used.
determinations will be discussed below. The (2200)7/2 − mass is compatible with its nominal mass, we assign an uncertainty of (2200)7/2 − not: in elastic π N scattering, the Nπ branching ratio enters in the entrance and the exit channel, and the signature of (2200)7/2 − in elastic scattering is more than 100 times weaker than that of (1950)7/2 + . Due to the weakness of the (2200)7/2 − signal in elastic scattering, the (2200)7/2 − → Nπ branching ratio has a sizable uncertainty.
When this branching ratio is decreased by 20%, all other branching ratios are reduced by 20% while the squared helicity amplitudes increase by 20%.
In the next step, we removed (2200)7/2 − from the list of resonances used in the PWA, and the fit deteriorated visibly, see All data sets contributed to the evidence for (2200)7/2 − . Most sensitive were the γ p → π + n data on E and and the T and E data on γ p → π 0 p. Table 2 gives the change in χ 2 which we observe and which we should expect (N events · 49044/32666 with N events being the number of data points for a specified observable). The difference is assigned to the fractional evidence for the existence of (2200)7/2 − . Obviously, all data sets contributed to the evidence for (2200)7/2 − . The six masses obtained in these fits deviate by −(3 ± 10) MeV, the widths by (1 ± 13) MeV from the mean value. These differences are well covered by the spread of results obtained when different models were used to fit the data. Next, we added data on three-body reactions, in particular γ p → π 0 π 0 p [32] and γ p → π 0 ηp [33]. The inclusion of threebody reactions did not lead to any significant changes in mass, width, or two-body couplings. Hence these values were frozen to their central values when the three-body reactions were included. Three-body decays were assumed to decay via intermediate meson or baryon resonances in an isobar ansatz. The three-body data were fitted in an event-based likelihood method; hence no χ 2 value was returned by the fit. To quantify the improvement of a fit, we added to the change in χ 2 -derived from the fit to two- Table 2 The observed and the expected change in χ 2 when single data sets are removed from the fit. The differences are assigned to the fractional evidence for the existence of (2200)7/2 − .  body reactions -the change in log likelihood multiplied by 2. This number is referred to as pseudo-χ 2 . Since the absolute value of the log-likelihood function is meaningless, we give only changes of the pseudo-χ 2 .
With these data included, we performed mass scans in the J P = 7/2 + and J P = 7/2 − partial waves. In the mass scans, the J P = 7/2 + and J P = 7/2 − partial waves were not described by K -matrix poles but represented by multichannel Breit-Wigner amplitudes, hence the optimal parameters for mass and width can differ. Fig. 3 (top) shows the change of the resulting pseudo-χ 2 as a function of the imposed mass of the J P = 7/2 + or the J P = 7/2 − resonance. The total pseudo-χ 2 has clear minima at a mass of 1917 MeV for J P = 7/2 + and 2176 MeV for J P = 7/2 − . When the masses are detuned from the best values, the widths of the resonances become wide. Fig. 3 also shows a breakdown of the total pseudo-χ 2 into contributions from specific reactions. Clear minima are observed in γ p → π 0 p, π + n, K , π 0 π 0 p, and even in π 0 ηp (due to (1232)η). The minima are found at 1913,1917,1922,1904,1942 MeV, respectively at 2186, 2155, 2193, 2115, 2200 MeV, consistent with the overall minima at 1917 and 2176 MeV. It is remarkable that the same minima which are found for * decays into π N are as well seen in the other allowed decay channels.
In spite of the small (2200)7/2 − → π N coupling, the largest evidence stems from photoproduction of single pions. There are two reasons: first, the highly constraining polarization data and their statistical power define the angular-momentum decomposition very well. Second, the sequential decays in 2π 0 photoproduction allow for a large flexibility in describing the data; hence the statistical significance of those data is reduced. Table 3 (Breit-Wigner) mass, width (in MeV) and π N decay branching ratio BR = π / tot of (2200)7/2 − . Ref. [46] reports the pole position, no uncertainty is given. The scans (Fig. 3) demonstrate clearly that the masses of (1950)7/2 + and (2200)7/2 − are different. The difference in squared masses of the two resonances is (1.06 ± 0.17) GeV 2 , in excellent agreement with the slope of the leading Regge trajectory for * 's of (1.08 ± 0.01) GeV 2 .
To search for a mass-degenerate parity partner of (1950)7/2 + , we did a series of fits trying to impose a (1950)7/2 − with a mass restricted in the range 1920 to 1980 MeV in addition to (2200)7/2 − . In all fits, both helicity amplitudes converged to zero: there is no mass-degenerate parity partner of (1950)7/2 + in the data.
Evidence for the (2200)7/2 − resonance has been reported before, see Table 3. The Review of Particle Properties lists it as a onestar resonance, the evidence was considered as poor. (2200)7/2 − is not seen in the elastic π N scattering analysis of the GWU group [41]; in the recent Bonn-GWU-Jülich analysis it is included in the fits but the authors state that they cannot claim much evidence either [46].
Finally, we discuss the helicity couplings. The  In quark models, (1950)7/2 + has a leading (L, S) configuration (L = 2, S = 3/2). When it is excited, the spin of one of the three quarks of the proton has to flip. This requires a magnetic multipole. Indeed, the electric multipole E 3+ is much smaller than the magnetic multipole M 3+ . This observation can be compared with the small E 1+ /M 1+ ratio of (1232)3/2 + : the photo-excitation of (1232)3/2 + requires a spin flip as well. The leading (L, S) configuration of (2200)7/2 − is likely (L = 3, S = 1/2) [48]. No spin flip is required and both, electric and magnetic multipoles, may contribute (not necessarily at the same strength). Within the errors, data are consistent with this conjecture.
Summarizing, we have reported strong evidence for the (2200)7/2 − resonance from a coupled-channel analysis of a large data base. Mass, width, and decay branching ratios are determined. The observed mass strongly favors quark models and AdS/QCD and is in conflict with models in which chiral symmetry is restored in the high-mass spectrum of meson and baryon resonances. The analysis is based on the BnGa approach exploiting the energydependence of the photoproduction amplitudes. There is the hope that the photoproduction multipoles with low orbital angular momenta might be determined in an energy-independent (and thus model-independent) analysis when further polarization data become available. This would be the basis to derive the (2200)7/2 − properties exploiting different model assumptions.