The $p\stackrel{}{p}$ annihilation at rest into 4γ's was studied at LEAR (CERN) using the Crystal Barrel [5]. Fig. (6) of [5] shows the scatter plot of 2 γ invariant mass versus 2 γ invariant mass: M(γ₃γ₄)= f(M(γ₁γ₂)). The original figure has been scanned. The insert is selected to eliminate the most populated areas, not useful for the present purpose. The most intense concentration of data appears in spots corresponding to π⁰π⁰, π⁰η, π⁰ω, ηη, and ηω coincidences. The events delimited by circles (blue on line) will be discussed in a forthcoming paper [6]. The masses of the events delimited by squares (red on line) are reported in Table 1. They give the hint of the presence of narrow exotic low mass unflavored mesonic structures. Due to the low number of events, we do not claim that this plot constitutes an evidence of the existence of these resonances. However, the very good agreement between masses corresponding to name (1) and (2), (see Table 1), and the fact that such accumulation of events appears at same masses, in totally different experiments, with different probes, corroborate previous findings and reinforce previous statements. The agreement between these pieces of experimental information allows to strengthen the proof of their existence.

Fig. (6). (Color on line). Missing mass spectra. pp → ppX measured at CELSIUS [16] in insert (a); pd → 3HeX measured at COSY [17] in insert (b).

The data corresponding to these structures appear in coincidence with a π⁰. The explanation may be related to the fact that, since the other PDG mesons are less excited than the π⁰, the correspondence with narrow structures is less probable and, hence, it is not observable in the data.

The masses extracted from Fig. (1) are identified by the corresponding code number in the plot. Some masses are rather imprecise due to poor counting. Table 1 shows these masses, compared with the masses extracted from other data. These last are shown in column "already observed" (AO). The following notation is used: "a", "b", ... identifying these blobs with increasing masses. They are called "1" when observed in M(γ₃,γ₄) = M(π⁰), and "2" when observed in M(γ₁,γ₂) = M(π⁰). This notation corresponds to an attempt to attribute the same rank to nearby masses. The good agreement observed for six (over eight) masses, supports the interpretation that the light structures correspond to real events and not to background. The lack of the e2 blob is associated to the progressive reduction of M(γ₃,γ₄) events for increasing mass. No h1 structure can be extracted from a symmetric large blob corresponding to the projection of π⁰π⁰ coincidences. The h2 mass is imprecise since located close to a more intense π⁰π⁰ blob.

Other measurements [7] have been performed at LEAR (CERN), with the Crystal Barrel detector. The $p\stackrel{}{p}$ annihilation has been studied at different proton momenta. Fig. (2a) of Ref. [7] shows the invariant mass spectra of π⁰γ , obtained using 1940 MeV/c incident proton momenta, and plotted with 10 MeV bins. The figure shows a nice ω peak and a ⁰f₂ bump, both discussed in the paper. It shows also a small peak in the 600 MeV region, which was not commented. These data, useful for the present discussion, are read out and plotted every 17.7 MeV in Fig. (2).

The peak is extracted at M = 587 MeV (S.D. = 7.1), which is very close to M = 588 MeV, where a narrow peak was observed in the SPES3 data [8].

Fig. (2). (Color on line). Part of the invariant π0γ spectra, from $p\stackrel{}{p}$ annihilation at Pp = 1940 MeV/c measured at LEAR [7].

A revised analysis of the η→π⁰γγ reaction has been presented in [9]. The data correspond to the π^-p→π⁰γγ reaction, studied at the AGS with a 716 MeV/c π^- beam and the Crystal Ball detector. Different background contributions were suppressed by application of various cuts, with the help of Monte-Carlo simulations [9]. The validity of these corrections applied to the analysis, is tested in the Fig. (5c) of Ref. [9]. This figure shows a good agreement between the m²(γγ) spectra obtained by assuming the intermediate states: π^-p →η n→π⁰γγ n →4γn and the π^-p → π⁰γγ n → 4γn . These last data are read and shown in Fig. (3).

Fig. (5). (Color on line). Missing mass of the pp → ppX reaction measured at Uppsala [13].

Fig. (3). (Color on line). M(γγ) spectra measured at AGS with using 716 MeV/c π- beam and Crystal Ball detector [9].

The authors applied several selection cuts, which limited their acceptance to be about 15% with a hole at the π⁰-meson mass. Fig. (3) exhibits a clear peak at M = 84 MeV, and a broad distribution beyond the pion mass. It is likely that the M=84 MeV and 200 MeV peaks are artifacts of the suppressed pion peak, and therefore they will not be considered further on. Since the background is not subtracted, the peak extractions are not very precise. However, the broad region exhibits two small peaks, making easier to attribute a mass to the structures, with a reasonable accuracy. The fit shown in Fig. (3) is obtained with gaussian functions, centred at M = 255 MeV (S.D. = 4.0), 343 MeV (S.D. = 5.1), (and 395 MeV). This last peak falls outside the range of the figure, therefore has no physical meaning; it only helps to fit the tail of the spectra.

Fig. (4) shows several spectra of two photon invariant mass, observed in high energy heavy ion experiments. On the figures, the correspondence with masses extracted previously from other data is indicated (see Table 2). Several very narrow structures can be observed in addition to the π⁰ peak. Several gaussians are drawn on the figure to drive the eyes. They should not be considered as quantitative fits. They only illustrate the possibility of several structures, which should appear in statistically precise measurements, where a small binning can be done. Such data call for dedicated measurements.

Fig. (4). (Color on line). Two photon invariant mass observed in reactions involving high energy heavy ions (see text).

The insert (a) of Fig. (4) shows the two-photon invariant mass distribution in Pb-Pb events after subtraction of the combinatorial background [10]. An integration over two channels is performed. The spectra exhibit an oscillatory pattern. As a consequence of so large error bars, all masses except the π⁰ at 135 MeV are ignored.

Insert (b) of 4 shows the real minus mixed spectra of two photon invariant mass, measured with the Pb + Pb reaction by ALICE/PHOS [11]. Small structures are tentatively indicated at the same masses observed by previous experiments. These masses are: M = 25 MeV, 45 MeV, 181 MeV and 194 MeV. In addition a structure is observed at M ≈ 167 MeV.

Inserts (c) and (d) of Fig. (4) shows the two photon invariant mass spectra for 200 A GeV ³²S + Au events [12], measured at the CERN SPS with the lead glass spectrometer SAPHIR. A small structure close to M = 81 MeV is observed in insert (c), in agreement with previous observations. The spectra shown in insert (d) exhibits an oscillatory behaviour, at the following masses: M = 170 MeV, not previously observed (except in insert (b) of the same figure) at masses close to M = 181 MeV and M = 215 MeV, previously observed, and two structures at M = 190 MeV and M = 199 MeV, in both sides of the previously observed structure at M = 194 MeV. The effect of a minimization procedure is to lower the background and widen the gaussians.

Several peaks are defined by only three points. This is the consequence of the vicinity of the masses and of the experimental resolution. The main feature of these data, namely the oscillatory behaviour of the two photon spectra at masses close to the pion mass, requires new dedicated precise data with improved resolution and statistics. This feature is tentatively associated with the presence of several structures.

The missing mass of the pp → ppX reaction was measured at Uppsala with help of the PROMICE - WASA facility, and 310 MeV beam energy [13]. Fig. (1) of Ref. [13] exhibits several structures in the mass region between the γ and π⁰ missing masses. These data have been read, converted into linear scale (the original scale is logarithmic) as function of the missing mass (the original data are plotted versus $M_X^2/M_{}^2$ , and drawn in Fig. (5). Two structures are visible above a background and have been fitted with a two degrees polynomial. Their masses are: M = 61 MeV (S.D. = 1.6) and 76 MeV (S.D. = 1.9), close to M = 62 MeV and 80 MeV, (already observed in [14, 15]).

Several figures have been plotted in [14], which focused on possible narrow structures in data from Celsius, Cosy, and JLAB CLAS Collaboration. This analysis ignored a possible structure at a mass close to M = 25 MeV. The present study focuses more precisely on this mass range. Fig. (6) shows the corresponding results. The fits are performed with help of all narrow structure masses. Although the present reanalysis intends to better extract the different peaks, this aim is not achieved except for the first one, which is the structure clearly extracted at M = 25 MeV in both inserts. Insert (a) shows the missing mass of the pp → ppX reaction measured at CELSIUS [16]. Insert (b) shows the missing mass of the pd → ³ HeX reaction measured at COSY [17].

The e⁺ e^- → π⁺π^- reaction has two distinctive features: - very small structures are expected, since two hadrons have to be produced from an electromagnetic interaction, - precise data with small binning are expected, since recent data have been collected at present colliders. Such data are presented in Fig. (7) measured at electron-positron BaBar collider [18]. Fig. (7) shows the cross-section of the e⁺ e^- → π⁺π^- reaction. From M = 305 MeV to M = 495 MeV, the data are plotted with a 10 MeV binning, and reported in Fig. (7a). Structures at 360 MeV (S.D. = 3.1), 410 MeV (S.D. = 4.7), and 475 MeV (S.D. = 1.4) are extracted with a common width, σ = 15 MeV. The data at other inserts are given with a 4 MeV binning. Structures at M = 546 MeV (S.D. = 2.6) (insert (b)), 635 MeV (S.D. = 4.2) (insert (c)), and 752 MeV (S.D. = 3.4) (insert (d)) are observed. A large background in insert (d) is fitted with a gaussian with the width of the ρ meson, but smaller mass (758 MeV instead of 775 MeV). A small peak at M = 775 MeV may correspond to the ω meson at a slightly smaller mass again (775 MeV instead of 782.7 MeV).

Fig. (7). (Color on line). Cross section of the e+e- → π+π- reaction measured at BaBar [18].

The missing mass of the pp → ppX⁰ reaction has been studied using the SPES3 beam line of the Saturne synchrotron [8]. The cross sections were measured at three incident proton energies: T_p = 1520, 1805, and 2100 MeV, and several spectrometer angles at each energy. The experiment has been described in detail in several papers, depending of the mass range:

62≤ M ≤ 235 MeV in [14], 510≤ M ≤ 550 MeV in [19], and 550≤ M ≤ 759 MeV in [8].

Such description is omitted here. Several narrow structures have been extracted from the study of several spectra at different angles and different incident energies. The observed narrow structure mean masses are: M = 62, 80, 100, 181, 198, 215, and 228 MeV [14], where the mean masses are consistent with the SPES3 and some other experiments. In a higher mass range, the corresponding structure are at masses: 310, 350, 426, 470, 555, 588, 647, 681, 700, and 750 MeV [8, 19].

The invariant π⁺π^- mass of the np → npπ⁺π^- reaction, was studied at Dubna, using a detection based on bubble chambers. Several papers were published (with a small mass variation following the increase of statistics). The most recent paper [20] using the neutron P_n = 5 GeV/c incident beam momentum allows to observe narrow structures at the following masses: M = 350±3 MeV, 408±3 MeV, 489±3 MeV, 579±5 MeV, 676±7 MeV, 762±11 MeV, 878±7 MeV, 1036±13 MeV, and 1170±11 MeV. These structures are given with a number of standard deviations S.D. ≥3.0. A selection has been done on the proton angle: cos( θ^*_p ≥ 0).

The reaction d + C → γ + γ +x has been studied with the internal beam of the JINR Nuclotron [21] at momentum 2.75 GeV/c per nucleon.

The data are read and reported in Fig. (8). A resonance-like enhancement was observed at M_γγ = 360 ± 7 MeV (S.D. = 4.7). This structure was not observed in the M_γγ spectrum from pC interactions. The second peak corresponds to the η .

Fig. (8). (Color on line). Invariant Mγγ mass observed from the dC interaction measured at the JINR Nuclotron [21].

Several sets of data have been scrutinized in Ref. [14] in order to extract small structures not discussed by the authors who analysed their data with different aims. Such narrow structures were observed below as well as above the pion mass. These data are from CELSIUS, COSY, MAMI, JLAB (halls A, B, and C). The corresponding discussion, detailed in [14], is briefly recalled here.

Table 2 shows consistency with the narrow exotic low mass mesons, extracted from different experiments and summarized in previous works. An additional mass M = 360 MeV found at the Nuclotron is not shown. The two masses at M = 227 MeV and 235 MeV from SPES3 are replaced by a common one at 231 MeV. The structures in Fig. (4) showing the two photon invariant masses observed in heavy ion reactions, are also omitted from Table 2 although most of the masses in insert (c) and (d) agree with narrow masses observed in different reactions. Some masses have been observed in three different sets of reactions or more; this increases their credibility.

The masses shown in Table2, are presented with the attempt to attribute the same rank to nearby masses from different experiments. Then we observe that the structures, extracted from different reactions, are often observed at nearby masses. The mean values of these masses are: M = 25, 45.3, 61.5, 81.3, 100, 181, 194, 215, 231, 252, 274.9, 309.9, 347, 366.7, 415, 482, 549.7, 584.7, 641, 675.2, 700, and 754.7 MeV, as reported in the first column of Fig. (9).

Fig. (9). (Color on line). Experimental exotic narrow meson masses are shown with red lines and empty circles. Calculated masses, using Eq. (1) (see text), are shown with blue lines and stars when obtained by $q^2{\stackrel{}{q}}^2$ clusters, and shown with green lines when obtained with $q^3{\stackrel{}{q}}^3$ clusters. The possible spins and isospins corresponding to the predicted masses are also given.

As already mentioned, there is no room for these mesonic structure masses to a description within the classical $q\stackrel{}{q}$ quark model. An attempt is therefore presented below to associate the masses of the mesonic structures with masses computed within a simple mass relation based on two quark clusters.

A mass formula was derived some years ago for two clusters of quarks at the ends of a stretched bag in terms of color magnetic interactions [22]. A tentative description of exotic low mass structures in mesonic, baryonic, and dibaryonic spectra, using this mass formula, has been previously proposed. The mesonic structure masses were described, using $q^2{\stackrel{}{q}}^2$ , or $q^3{\stackrel{}{q}}^3$ clusters up to M = 620 MeV, and $q^4{\stackrel{}{q}}^4$ between M = 620 MeV and M = 750 MeV [8]. The narrow baryonic structure masses were described either by $q{q}^2$ or $\mathrm{qqq}q\stackrel{}{q}$ clusters [23]. Finally the dibaryonic structure masses were described [24] using the following two clusters: $q^2{q}^4$ . The following equation was used:

where M₀ and M₁ are parameters deduced from experimental mass spectra and i₁(i₂) , s₁(s₂) are the isospin and spin of the first (second) quark cluster. The same approach is employed here. Eq. (1) involves a large degeneracy. We made the assumption that the simplest configuration is preferred, otherwise the possible spin and isospin will increase and the parity will be degenerate since additional $q\stackrel{}{q}$ configurations will always be possible.

Eq. (1) is applied to two quark clusters $q^n{\stackrel{}{q}}^n$ . The simplest $q\stackrel{}{q}$ choice corresponds to the most strongly excited meson, i.e., to the pion. Since therefore s₁ = s₂ = i₁ = i₂ = 1/2 , the previous equation becomes M = M₀ + 2M₁ = 137 MeV. The next cluster configuration is $q^2{\stackrel{}{q}}^2$ . The corresponding minimum mass is obtained with s₁ = s₂ = i₁ = i₂ = 0 . Corresponding to these values, M = M₀. We associate this mass to our minimum narrow meson mass, therefore M₀ = 25 MeV, from which one concludes that M₁ = 56 MeV.

Experimental exotic narrow meson masses are shown in Fig. (9) with red lines overcomed by empty circles. The masses are calculated with these M₀ and M₁ values, and are shown in Fig. (9). The assumption for the $q^2{\stackrel{}{q}}^2$ quark clusters corresponds to masses drawn with blue lines and blue stars. The same masses are obtained within the assumption of $q\stackrel{}{q}q\stackrel{}{q}$ quark clusters. The assumption for $q^3{\stackrel{}{q}}^3$ quark clusters, corresponds to masses drawn on line in green.

The figure is limited to M = 600 MeV, since with the M₀ and M₁ parameter values, s₁ = s₂ = i₁ = i₂ = 3/2 corresponds to M = 585 MeV.

Larger masses can be obtained with introduction of heavier quark clusters. It is noteworthy that the assumption $q^5{\stackrel{}{q}}^5$ quark clusters, allows to find several experimental masses. Table 3 shows these masses, limited to a value M≤780 MeV.

The first line of Table 3 shows the largest mass obtained within the $q^3{\stackrel{}{q}}^3$ quark cluster assumption. Starting from the second line, we calculated the successive increasing masses, which fit well the experimental masses, again without any new adjustable parameter. The bottom of Table 3 shows two calculated masses not observed experimentally.

The predicted possible spins and isospins are obtained from spins and isospins of both clusters. The predicted parities for exotic mesons depend on the number of $\stackrel{}{q}$ . The negative parity of the π is obtained with $q\stackrel{}{q}$ or $q^3{\stackrel{}{q}}^3$ . The $q^2{\stackrel{}{q}}^2$ choice necessitates an orbital excitation between both clusters to get the negative parity.

The agreement between data and predictions in Fig. (9) is noteworthy up to M = 260 MeV, and above 350 MeV. Two masses at M = 45.3 and 81.3 MeV have no predicted counterparts. A mass, predicted at M = 323.7 MeV is not observed. The pion mass and the structure at M = 252 MeV calculated at M = 249 MeV, are obtained at the same time with $q^2{\stackrel{}{q}}^2$ and $q^3{\stackrel{}{q}}^3$ configurations.

Such description is strengthen by the observation that the gap between two adjacent exotic meson masses are rather constant and equal to δ M ≈ 18.7 MeV. A comparable situation was already observed in the studies of narrow exotic baryonic and dibaryonic masses. For example in the dibaryon field, between M = 2194 MeV, and 2016 MeV, five narrow masses were observed regularly separated by about 35 MeV. As already reminded [15], "the model [25], proposed some time ago, associates the narrow structure masses below the π threshold production, to the multiproduction of a genuine virtual Goldstone boson with a mass close to 20 MeV." This model can explain the level spacing of narrow mesonic structures experimentally observed.

It has been already mentioned that the usual calculations associate the multiquark clusters to masses larger than M = 1.5 GeV. The success of the present mass formula, based on quark clusters, which is able to reproduce many masses within a simple equation and only two parameters, justifies to consider the possibility of new physics on the basis of additional low mass hadrons. which would interact less strongly than the "classical" ones.

We attribute a rank as an integer sequence of the resonances, according to their increasing masses. The rank numbers the successive increasing exotic mesonic structure masses. It is the main parameter used for all studies inside fractal properties. The masses of the several narrow structures, given in Table 2, are plotted versus their rank in Fig. (10) which magnifies the low mass range. Fig. ( 11) shows all data. The SPES3 data [14, 8] are shown with solid red circles; the Dubna data [20] with empty purple crosses; the Lear data of $\stackrel{}{p}$ at rest [5] empty blue squares, the Lear data of $\stackrel{}{p}$ in flight [7] by blue star, the Uppsala data [13] with solid green triangles. The PDG masses [1] are also introduced, and are represented by black stars. The narrow structure masses observed in the AGS experiment [9], the Babar measurement [18], or the high energy heavy ion experiments [10-12] are omitted from the figure. Their masses agree with those considered in the figure, as seen in Table 2.

Fig. (10). (Color on line). Exotic low mass meson masses. SPES3 data [14, 8]: solid red circles; Lear data [2]: empty blue squares; Uppsala data [5]: solid green triangles. The Celsius [16] and the Cosy narrow structure mass [17] are shown by a purple empty cross. The PDG π0 mass [1] is introduced by a black star.

Fig. (11). (Color on line). Exotic meson masses. The SPES3 data [14, 8] are shown with solid red circles; the Dubna data [20] with empty purple crosses; the Lear data with $\stackrel{}{p}$ at rest [5] empty blue squares, the Lear data with $\stackrel{}{p}$ in movement [7] by blue star, the Uppsala data [13] with solid green triangles. The PDG masses [1] are also introduced, and are represented by black stars. See text.

We observe a "regular" mass variation, with the same slope for the first and third straight line segments. Such behaviour suggests log-periodic fluctuations and fractal properties with not linear scale laws [26, 27].

We show therefore in Fig. (12) the log of the masses versus the log of the rank. We have a discrete scale invariance, since we observe linearities between the logs, for several values of the fundamental scaling ratios. Such property was already observed for a smaller statistics [28].

Fig. (12). (Color on line). Log of the exotic meson masses versus the log of the rank. The SPES3 data [8-10] are shown with solid red circles; the Dubna data [20] with empty purple crosses; the Lear data with $\stackrel{}{p}$ at rest [5] empty blue squares, the Lear data with $\stackrel{}{p}$ in movement [7] by blue star, the Uppsala data [13] with solid green triangles. The PDG masses [1] are also introduced, and are represented by black stars.