TThe Status of Exotic-quantum-number Mesons
C. A. Meyer and Y. Van Haarlem Carnegie Mellon University, Pittsburgh, PA 15213 (Dated: October 31, 2018)The search for mesons with non-quark-antiquark (exotic) quantum numbers has gone on for nearlythirty years. There currently is experimental evidence of three isospin one states, the π (1400), the π (1600) and the π (2015). For all of these states, there are questions about their identification,and even if some of them exist. In this article, we will review both the theoretical work andthe experimental evidence associated with these exotic quantum number states. We find that the π (1600) could be the lightest exotic quantum number hybrid meson, but observations of othermembers of the nonet would be useful. PACS numbers: 14.40.-n,14-40.Rt,13.25.-k
I. INTRODUCTION
The quark model describes mesons as bound states ofquarks and antiquarks ( q ¯ q ), much akin to positronium( e + e − ). As described in Section II A, mesons have well-defined quantum numbers: total spin J , parity P , and C-parity C , represented as J P C . The allowed J P C quantumnumbers for orbital angular momentum, L , smaller thanthree are given in Table I. Interestingly, for J smallerthan 3, all allowed J P C except 2 −− [1] have been ob-served by experiments. From the allowed quantum num-bers in Table I, there are several missing combinations:0 −− , 0 + − , 1 − + and 2 + − . These are not possible for sim-ple q ¯ q systems and are known as “exotic” quantum num-bers. Observation of states with exotic quantum numbershas been of great experimental interest as it would beclear evidence for mesons beyond the simple q ¯ q picture. L S J PC L S J PC L S J PC − + + − − + −− ++ −− ++ −− ++ −− TABLE I. The allowed J PC quantum numbers for q ¯ q systems. Moving beyond the simple quark-model picture ofmesons, there have been predictions for states with theseexotic quantum numbers. The most well known are q ¯ q states in which the gluons binding the system cancontribute directly to the quantum numbers of the me-son. However, other candidates include multi-quarkstates ( q ¯ qq ¯ q ) and states containing only gluons (glue-balls). Early bag-model calculations [2] referred to stateswith q ¯ q and gluons as “hermaphrodite mesons”, and pre-dicted that the lightest nonet ( J P C = 1 − + ) might havemasses near 1 GeV as well as distinctive decay modes.They might also be relatively stable, and thus observ-able. While the name hermaphorodite did not survive,what are now known as “hybrid mesons” have become avery interesting theoretical and experimental topic andthe status of these states, with particular emphasis on theexotic-quantum number ones is the topic of this article. More information on meson spectroscopy in general canbe found in a recent review by Klempt and Zaitsev [3].Similarly, a recent review on the related topic of glueballscan be found in reference [4]. II. THEORETICAL EXPECTATIONS FORHYBRID MESONSA. Mesons in The Quark Model
In the quark model, mesons are bound states of quarksand antiquarks ( q ¯ q ). The quantum numbers of suchfermion-antifermion systems are functions of the totalspin, S , of the quark-antiquark system, and the relativeorbital angular momentum, L , between them. The spin S and angular momentum L combine to yield the totalspin J = L ⊕ S , (1)where L and S add as two angular momentums.Parity is the result of a mirror reflection of the wavefunction, taking (cid:126)r into − (cid:126)r . It can be written as P [ ψ ( (cid:126)r )] = ψ ( − (cid:126)r )= η P ψ ( (cid:126)r ) , (2)where η P is the eigenvalue of parity. As application ofparity twice must return the original state, η P = ±
1. Inspherical coordinates, the parity operation reduces to thereflection of a Y lm function, Y lm ( π − θ, π + φ ) = ( − l Y lm ( θ, φ ) . (3)From this, we conclude that η P = ( − l .For a q ¯ q system, the intrinsic parity of the antiquarkis opposite to that of the quark, which yields the totalparity of a q ¯ q system as P ( q ¯ q ) = − ( − L . (4)Charge conjugation, C , is the result of a transforma-tion that takes a particle into its antiparticle. For a q ¯ q system, only electrically-neutral states can be eigenstatesof C . In order to determine the eigenvalues of C ( η C ), a r X i v : . [ nu c l - e x ] J u l we need to consider a wave function that includes bothspatial and spin informationΨ( (cid:126)r, (cid:126)s ) = R ( r ) Y lm ( θ, φ ) χ ( (cid:126)s ) . (5)As an example, we consider a u ¯ u system, the C opera-tor acting on this reverses the meaning of u and ¯ u . Thishas the effect of mapping the vector (cid:126)r to the u quarkinto − (cid:126)r . Thus, following the arguments for parity, thespatial part of C yields a factor of ( − L . The spin wavefunction also reverse the two individual spins. For a sym-metric χ , we get a factor of 1, while for an antisymmetric χ , we get a factor of −
1. For two spin particles, the S = 0 singlet is antisymmetric, while the S = 1 tripletis symmetric. Combining all of this, we find that theC-parity of (a neutral) q ¯ q system is C ( q ¯ q ) = ( − L + S . (6)Because C -parity is only defined for neutral states,it is useful to extend this to the more general G -paritywhich can be used to describe all q ¯ q states, independentof charge. For isovector states ( I = 1), C would trans-form a charged member into the oppositely charged state( e.g. π + → π − ). In order to transform this back to theoriginal charge, we would need to perform a rotation inisospin ( π − → π + ). For a state of whose neutral memberhas C -parity C , and whose total isospin is I , the G -parityis defined to be G = C · ( − I , (7)which can be generalized to G ( q ¯ q ) = ( − L + S + I . (8)The latter is valid for all of the I = 0 and I = 1 membersof a nonet. This leads to mesons having well definedquantum numbers: total angular momentum, J , isospin, I , parity P , C-parity, C , and G-parity, G . These arerepresented as ( I G ) J P C , or simply J P C for short. Forthe case of L = 0 and S = 0, we have J P C = 0 − + , whilefor L = 0 and S = 1, J P C = 1 −− . The allowed quantumnumbers for L smaller than three are given in Table I. B. Notation and Quantum Numbers of Hybrids
The notation for hybrid mesons we use is that fromthe Particle Data Group (PDG) [1]. In the PDG no-tation, the parity and charge conjugation determine thename of the hybrid, which is taken as the name of thenormal meson of the same J P C and isospin. The totalspin is then used as a subscript to the name. While var-ious models predict different nonets of hybrid mesons,the largest number of nonets is from the flux-tube model(see Section II C). For completeness, we list all of theseas well as their PDG names in Table II. The first entry isthe isospin one ( I = 1) state. The second and third arethose with isospin equal to zero ( I = 0) and the fourth isthe kaon-like state with isospin one-half ( I = ). In the case of the I = 0 states, the first is taken as the mostly u ¯ u and d ¯ d state (so-called n ¯ n ), while the second is mostly s ¯ s . For the I = 0 states, C -parity is well defined, but for I = 1, only the neutral member can have a defined C -parity. However, the more general G -parity can be usedto describe all of the I = 1 members (see equation 8).Thus, the G -parity can be used to identify exotic quan-tum numbers, even for charged I = 1 members of a nonet.For the case of the kaon-like states, neither C -parity nor G -parity is defined. Thus, the I = members of a nonetcan not have explicitly-exotic quantum numbers. QNs Names J PC ( I G ) ( I G ) ( I )1 ++ (1 − ) a (0 + ) f f (cid:48) ( ) K −− (1 + ) ρ (0 − ) ω φ ( ) K ∗ − + (1 − ) π (0 + ) η η (cid:48) ( ) K − + ( − ) π ( + ) η η (cid:48) ( ) K ∗ − + (1 − ) π (0 + ) η η (cid:48) ( ) K + − ( + ) b ( − ) h h (cid:48) ( ) K ∗ + − (1 + ) b (0 − ) h h (cid:48) ( ) K + − ( + ) b ( − ) h h (cid:48) ( ) K ∗ TABLE II. The naming scheme for hybrid mesons. The firststate listed for a given quantum number is the isospin onestate. The second state is the isospin zero state that is mostly u and d quarks ( n ¯ n ), while the third name is for the mostly s ¯ s isospin zero state. Note that for the kaons, the C - and G -parity are not defined. Kaons cannot not have manifestlyexotic quantum numbers. States that have exotic quantumnumbers are shown in bold. In Table III we show the J P of the three exotic I =1 mesons from Table II. We also show the normal ( q ¯ q )meson of the same J P and the I G quantum numbersfor these states. The exotic mesons have the opposite G -parity relative to the normal meson. This provides asimple mechanism for identifying if a charged I = 1 statehas exotic quantum numbers. J P normal meson exotic mesonname ( I G ) name ( I G )0 + a (1 − ) b (1 + )1 − ρ (1 + ) π (1 − )2 + a (1 − ) b (1 + )TABLE III. The J P and I G quantum numbers for the exoticmesons and the normal mesons of the same J P . C. Model Predictions
The first predictions for exotic quantum numbermesons came from calculations in the Bag model [5, 6].In this model, boundary conditions are placed on quarksand gluons confined inside a bag. A hybrid meson isformed by combining a q ¯ q system (with spin 0 or 1)with a transverse-electric (TE) gluon ( J P C = 1 + − ). Thisyields four nonets of hybrid mesons with quantum num-bers J P C = 1 −− , 0 − + , 1 − + and 2 − + . These four nonetsare roughly degenerate in mass and early calculationspredicted the mass of a 1 − + to be in the range of 1 . . J P C = 1 − + and combinedwith the same S = 0 and S = 1 q ¯ q systems yield fouradditional nonets with J P C = 1 ++ , 0 + − , 1 + − and 2 + − .These would presumably be heavier than the nonets builtwith the TE gluon.Another method that has been used to predict the hy-brid masses are “QCD spectral sum rules” (QSSR). UsingQSSR, one examines a two-point correlator of appropri-ate field operators from QCD and produces a sum ruleby equating a dispersion relation for the correlator to anoperator product expansion. QSSR calculations initiallyfound a 1 − + state near 1 GeV [9, 10]. A 0 −− state wasalso predicted around 3 . − + hybrid mass in therange of 1 . . π (1600) (see Sec-tion III C) as the lightest exotic hybrid. Recently, Nari-son [12] looked at the calculations for J P C = 1 − + stateswith particular emphasis in understanding differences inthe results between QSSR and Lattice QCD calcula-tions (see Section II D). He found that the π (1400) and π (1600) may be consistent with 4-quark states, whileQSSR are consistent with the π (2015) (see Section III D)being the lightest hybrid meson.The formation of flux tubes was first introduced in the1970’s by Yoichiro Nambu [13, 14] to explain the observedlinear Regge trajectories—the linear dependence of masssquared, m , of hadrons on their spin, J . This lineardependence results if one assumes that mass-less quarksare tied to the ends of a relativistic string with constantmass (energy) per length and the system rotating aboutits center. The linear m versus J dependence only ariseswhen the mass density per length is constant, which isequivalent to a linear potential.In the heavy-quark sector, lattice QCD [15] calcula-tions show a distribution of the gluonic field (action den-sity) which is mostly confined to the region between thequark and the antiquark. A picture which is very similarto that inspired by the “flux-tube model”. Within theflux-tube model [16, 17], one can view hybrids as mesonswith angular momentum in the flux tube. Naively, onecan imagine two degenerate excitations, one with thetube going clockwise and one counter clockwise. It ispossible to write linear combinations of these that havedefinite spin, parity and C-parity. For the case of oneunit of angular momentum in the tube, the flux tube be-haves as if it has quantum numbers J P C = 1 + − or 1 − + .The basic quantum numbers of hybrids are obtained byadding the tube’s quantum numbers to that of the un-derlying meson.In the flux-tube model, the tube carries angular mo-mentum, m , which then leads to specific predictions forthe product of C -parity and parity ( CP ). For m = 0,one has CP = ( − S +1 , while for the first excited states, ( m = 1), we find that CP = ( − S . The excitations arethen built on top of the s -wave mesons, ( L = 0), wherethe total spin can be either S = 0 or S = 1. For the caseof m = 0, we find CP as follows,( m = 0) S = 0 0 − + S = 1 1 −− (cid:27) ( − L +1 ( − S + L = ( − S +1 Normal Mesonswhich are the quantum numbers of the normal, q ¯ q ,mesons as discussed in Section II A. For the case of m = 1, where we have one unit of angular momentum inthe flux tube, we find the following J P C quantum num-bers( m = 1) S = 0 0 − + S = 1 1 −− (cid:27) ++ , −− − + , + − , − + , + − , − + , + − . The resulting quantum numbers are obtained by addingboth 1 + − and 1 − + to the underlying q ¯ q quantum num-bers (0 − + and 1 −− ).From the two L = 0 meson nonets, we expect eighthybrid nonets, (72 new mesons!). Two of these nonetsarise from the q ¯ q in an S = 0 (singlet) state, while sixarise for the q ¯ q in the S = 1 (triplet) state. Of the sixstates built on the triplet q ¯ q , three have exotic quantumnumbers (as indicated in bold above).In the picture presented by the flux-tube model, thehybrids are no different than other excitations of the q ¯ q states. In addition to “orbital” and “radial” excitations,we also need to consider “gluonic” excitations. Thus, theflux-tube model predicts eight nonets of hybrid mesons(0 + − , 0 − + , 1 ++ , 1 −− , 1 − + , 1 + − , 2 − + and 2 + − ). Themodel also predicts that all eight nonets are degeneratein mass, with masses expected near 1 . J P C = 1 + − , 0 ++ , 1 ++ and 2 ++ , none of which are exotic.The first excitation of these ( L = 1), yields the nonets1 − + , 3 − + and 0 −− , all of which are exotic [18, 19]. Inthis model, the 1 − + is the lightest exotic quantum num-ber hybrid, with a mass in the range of 2 . . c ¯ c exotic hy-brid, which is found in the range of 4 . . Mass (GeV) Model Reference1 .
0- 1 . . . . . . . D. Lattice Predictions
Lattice QCD (LQCD) calculations may provide themost accurate estimate to the masses of hybrid mesons.While these calculations have progressively gotten better,they are still limited by a number of systematic effects.Currently, the most significant of these is related to themass of the light quarks used in the calculations. This istypically parametrized as the pion mass, and extrapola-tions need to be made to reach the physical pion mass.This is often made as a linear approximation, which maynot be accurate. In addition, as the the quark mass be-comes lighter, two-meson decay channels become possi-ble. These may distort the resulting spectrum.Most calculations have been performed with what iseffectively the strange-quark mass. However, it may notbe safe to assume that this is the mass of the s ¯ s memberof the nonet, and one needs to be aware of the approxi-mations made to move the estimate to the u ¯ u/d ¯ d mass.The bottom line is that no one would be surprised if thetrue hybrid masses differed by several hundred MeV fromthe best predictions. Author 1 − + Mass (GeV/c )Collab. u ¯ u/d ¯ d s ¯ s UKQCD [20] 1 . ± .
20 2 . ± . . ± . ± .
30 2 . ± . ± . . ± . . ± . ± ( sys )Mei [24] 2 . ± . ± . . ± . . ± .
139 2 . ± . . ± . − + hybridmeson masses. While the flux-tube model (see Section II C predictsthat the lightest eight nonets of hybrid mesons are de-generate in mass at about 1 . J P C = 1 − + nonet is the light-est. Predictions for the mass of this state have variedfrom 1 . . q ¯ q loopsallowed in the quenched calculation), while newer calcu-lations [26–29] are dynamic (not quenched).However, the masses in Table V may not be thebest approximations to the hybrid masses. It has beennoted [30] that Table V is not a very useful way of display-ing the results. Rather, the mass needs to be correlatedwith the light-quark mass used in the calculation. Thisis usually represented as the pion mass. In Figure 1 areshown the predictions from the same groups as a functionof the pion masses used in their calculations. In order toobtain the hybrid mass, one needs to extrapolate to thephysical pion mass.There are fewer predictions for the masses of the other mass ( π ) [ GeV ] m a ss ( - + ) [ G e V ] m π UKQCD 96MILC 97MILC 03CSSM 05SESAM 99UKQCD 06MILC 03LATHAD 10
FIG. 1. (Color on line.) The mass of the J PC = 1 − + ex-otic hybrid as a function of the pion mass from lattice cal-culations. The open (cyan) symbols correspond to quenchedcalculations, while the solid (red and blue) symbols are dy-namic (unquenched) calculations: open (cyan) star [20], open(cyan) squares [21], open (cyan) upright triangles [26], open(cyan) circles [25], solid (red) downward triangles [22], solid(red) squares [27], solid (blue) upright triangles [26] and solid(blue) circles [29]. exotic-quantum number states. Bernard [21] calculatedthe splitting between the 0 + − and the 1 − + state to beabout 0 . . ± . Multiplet J PC Mass π − + . ± . GeV /c b + − . ± . GeV /c b + − . ± . GeV /c TABLE VI. Estimates of the masses of exotic quantum num-ber hybrids [22].
A significant LQCD calculation has recently been per-formed which predicts the entire spectrum of light-quarkisovector mesons [28, 29]. The fully dynamical (un-quenched) calculation is carried out with two flavors ofthe lightest quarks and a heavier third quark tuned tothe strange quark mass. Calculations are performed ontwo lattice volumes and using four different masses forthe lightest quarks—corresponding to pion masses of 700,520, 440 and 390 MeV. In the heaviest case, the lightestquark masses are the same at the strange mass. The com-puted spectrum of isovector states for this heavy case isshown in Figure 2 (where the mass is plotted as a ra-tio to the Ω-baryon mass (1 .
672 GeV)). In the plot, theright-most columns correspond to the exotic π , b and b states. Interestingly, the 1 − + π is the lightest, and botha ground state and what appears to be an excited stateare predicted. The other two exotic-quantum-numberstates appear to be somewhat heavier than the π withan excited state for the b visible.In addition to performing the calculation near the FIG. 2. (Color on line) The LQCD prediction for the spectrum of isovector mesons. The quantum numbers are listed acrossthe bottom, while the color denotes the spin. Solid (dashed) bordered boxes on a 2 . (2 . ) fm volume lattice, little volumedependence is observed. The three columns at the far right are exotic-quantum numbers. The plot is taken from reference [29]. physical quark mass, there are a number of important in-novations. First, the authors have found that the reducedrotational symmetry of a cubic lattice can be overcomeon sufficiently fine lattices. They used meson operators ofdefinite continuum spin subduced into the irreducible rep-resentations of cubic rotations and observed very strongcorrelation between operators and the spin of the state.In this way they were able to make spin assignments froma single lattice spacing. Second, the unprecedented sizeof the operator basis used in a variational calculationallowed the extraction of many excited states with confi-dence.There were also phenomenological implications of theselattice results. A subset of the meson operators featurethe commutator of two gauge-covariant derivatives, equalto the field-strength tensor, which is non-zero only fornon-trivial gluonic field configurations. Large overlaponto such operators was used to determine the degreeto which gluonic excitations are important in the state, i.e. , what one would call the hybrid nature of the state.In particular, the exotic quantum number states all havelarge overlap with this type of operator, a likely indi-cation of hybrid nature over, say, multiquark structure.In addition to the exotic-quantum number states, sev-eral normal-quantum-number states also had large over-lap with the non-trivial gluonic field. In particular, stateswith J P C = 1 −− , 2 − + with approximately the same massas the lighter 1 − + state were noted.In order to extract the masses of states, it is necessary to work at the physical pion mass. While work is cur-rently underway to extract a point at m π ≈
280 MeV,this limit has not yet been reached. To attempt to ex-trapolate, one can plot the extracted state masses as afunction of the pion mass squared, which acts as a proxyfor the light quark mass (see Figure 3). While linearly ex-trapolating to the physical pion mass ignores constraintsfrom chiral dynamics, it is probably safe to say that boththe π (1600) and the π (2015) (as discussed below) couldbe consistent with the expected 1 − + mass. They are alsoconsistent with the ground and first-excited π state. Itappears that the b and b masses will likely be severalhundred MeV heavier than the lightest π .Lattice calculations have also been performed to lookfor other exotic quantum number states. Bernard [21]included operators for a 0 −− state, but found no evi-dence for a state with these quantum numbers in theirquenched calculation. Dudek et a. [29] looked for both0 −− and 3 − + states in their lattice data. They foundsome evidence for states with these quantum numbers,but the lightest masses were more than 2 GeV above themass of the ρ meson.These recent lattice calculations are extremely promis-ing. They reaffirm that hybrid mesons form part ofthe low-energy QCD spectrum and that exotic quantumnumber states exist. They also provide, for the first time,the possibility of assessing the gluonic content of a cal-culated lattice state. Similar calculations are currentlyunderway for the isoscalar sector where preliminary re- mass ( π ) [ GeV ] m a ss [ G e V ] m π -+ (16 )1 -+ (20 )0 +- (16 )0 +- (20 )2 +- (16 )2 +- (20 ) FIG. 3. (Color on line) The mass spectrum of the three exoticquantum number states [29]. The open figures are for a 16 spatial dimension lattice, while the solid are for a 20 spatiallattice. The (blue) circles are the mass of the 1 − + state, the(green) squares are the mass of the 0 + − state and the (red)stars are the 2 + − state. sults [30] for the mass scale appear consistent with thoseshown here in the isovector sector. These calculationswill also extract the flavor mixing angle, an importantquantity for phenomenology. E. Decay Modes
Currently, decays of hybrid mesons can only be cal-culated within models. Such models exist, having beendeveloped to compute the decays of normal mesons. Abasic feature of these is the so-called triplet-P-zero ( P )model. In the P model, a meson decays by producinga q ¯ q pair with vacuum quantum numbers ( J P C = 0 ++ ).A detailed study by Ackleh, Barnes and Swanson [31]established that the P amplitudes are dominant in mostlight-quark meson decays. They also determined the pa-rameters in decay models by looking at the well knowndecays of mesons. This work was later extended to pro-vide predictions for the decay of all orbital and radialexcitations of mesons lighter than 2 . L = 0and an L = 1 meson. The suppression of a pair of L = 0mesons arises in the limit that the two mesons have thesame inverse radius in the Simple Harmonic Oscillatorwave functions. Thus, these decays are not strictly for- Name J PC Total Width
MeV
Large DecaysPSS IKP π − + −
168 117 b π , ρπ , f π , a η , η (1295) π , K A K , K B Kη − + −
158 107 a π , f η , π (1300) π , K A K , K B Kη (cid:48) − + −
216 172 K B K , K A K , K ∗ Kb + − −
429 665 π (1300) π , h πh + − −
262 94 b π , h η , K (1460) Kh (cid:48) + − −
490 426 K (1460) K , K A K , h ηb + − −
11 248 a π , a π , h πh + − −
12 166 b π , ρπh (cid:48) + − −
18 79 K B K , K A K , K ∗ K , h η TABLE VII. Exotic quantum number hybrid width and de-cay predictions from reference [34]. The column labeled PSS(Page, Swanson and Szczepaniak) is from their model, whilethe IKP (Isgur, Karl and Paton) is their calculation of themodel in reference [17]. The variations in width for PSScome from different choices for the masses of the hybrids.The K A represents the K (1270) while the K B representsthe K (1400). bidden, but are suppressed depending on how close thetwo inverse radii are. This led to the often-quoted predi-cation for the decays of the π hybrid given in equation 9. πb : πf : πρ : ηπ : πη (cid:48) =170 : 60 : 5 −
20 : 0 −
10 : 0 −
10 (9)The current predictions for the widths of exotic-quantum-number hybrids are based on model calcula-tions by Page et al. [34] for which the results are givenin Table VII. They also computed decay rates for thehybrids with normal q ¯ q quantum numbers (results in Ta-ble VIII). While a number of these states are expected tobe broad (in particular, most of the 0 + − exotic nonet),states in both the 2 + − and the 1 − + nonets are expectedto have much narrower widths. The expected decaymodes involve daughters that in turn decay. Thus mak-ing the overall reconstruction and analysis of these statesmuch more complicated then simple two-pseudoscalar de-cays.For the non-exotic quantum numbers states, it will beeven more difficult. They are likely to mix with nearbynormal q ¯ q states, complicating the expected decay pat-tern for both the hybrid and the normal mesons. How-ever, the decays in Table VIII can be used as a guidelineto help in identifying these states. In searches for hy-brid mesons, the nonets with exotic quantum numbersprovide the cleanest environment in which to search forthese objects.Close and Thomas [35] reexamined this problem interms of work on hadronic loops in the c ¯ c sector byBarnes and Swanson [36]. They conclude that in thelimit where all mesons in a loop belong to a degeneratesubset, vector hybrid mesons remain orthogonal to the q ¯ q states ( J P C = 1 −− S and D ) and mixing maybe minimal. Thus, the search for hybrids with vector q ¯ q quantum numbers may not be as difficult as the othernon-exotic quantum number hybrids. Particle J PC Total Width
MeV
Large DecaysPSS IKP ρ −− −
121 112 a π , ωπ , ρπω −− −
134 60 ρπ , ωη , ρ (1450) πφ −− −
155 120 K B K , K ∗ K , φηa ++ −
204 269 ρ (1450) π , ρπ , K ∗ Kh ++ −
130 436 K ∗ K , a πh (cid:48) ++ −
164 219 K ∗ (1410) K , K ∗ Kπ − + −
224 132 ρπ , f (1370) πη − + −
210 196 a (1450) π , K ∗ Kη (cid:48) − + −
390 335 K ∗ K , f (1370) η , K ∗ Kb + − −
338 384 ω (1420) π , K ∗ Kh + − −
529 632 ρ (1450) π , ρπ , K ∗ Kh (cid:48) + − −
373 443 K ∗ (1410) K , φη , K ∗ Kπ − + −
63 59 ρπ , f πη − + −
58 69 a πη (cid:48) − + −
91 69 K ∗ K , K ∗ K TABLE VIII. Non-exotic quantum number hybrid width anddecay predictions from reference [34]. The column labeledPSS (Page, Swanson and Szczepaniak) is from their model,while the IKP (Isgur, Karl and Paton) is their calculation ofthe model in reference [17]. The variations in width for PSScome from different choices for the masses of the hybrids.The K A represents the K (1270) while the K B representsthe K (1400). Almost all models of hybrid mesons predict that theywill not decay to identical pairs of mesons. Many alsopredict that decays to pairs of L = 0 mesons will be sup-pressed, leading to decays of an ( L = 0)( L = 1) pair asthe favored hybrid decay mode. Page [37] undertook astudy of these models of hybrid decay that included “TEhybrids” ( with a transverse electric constituent gluon)in the bag model as well as “adiabatic hybrids” in theflux-tube model (hybrids in the limit where quarks moveslowly with respect to the gluonic degrees of freedom).In such cases, the decays to pairs of orbital angular mo-mentum L = 0 (Swave) mesons were found to vanish. Inboth cases, it had been noted that this was true when thequark and the antiquark in the hybrid’s daughters haveidentical constituent masses with the same S -wave spa-tial wave functions, and the quarks are non-relativistic.In order to understand this, Page looked for an under-lying symmetry that could be responsible for this. Hefound that symmetrization of connected decay diagrams(see Figure 4(a) ) where the daughters are identical ex-cept for flavor and spin vanish when equation 10 is sat-isfied. C A P A = ( − ( S A + S q ¯ q +1) (10)For meson A decaying to daughters B and C , C A isthe C-parity of the neutral isospin member of the de-caying meson A , P A is its parity and S A is its intrinsic QQ__ q q_ QqQq___(a)
QQ__ q q_(b)
FIG. 4. (a) shows a connected decay diagram where thedecay can be suppressed. (b) is an example of a disconnecteddiagram where the decay is not suppressed.. spin. S q ¯ q is the total spin of the created pair. In thenon-relativistic limit, S q ¯ q = 1. For non-connected dia-grams (Figure 4(b)), he found no such general rules, sothe vanishing of the decays occur to the extent that thenon-connected diagrams are not important (OZI suppres-sion).As an example of this, consider A to be the π hybrid.It has C A = +1, P A = − S A = +1, thus the left-hand side of equation 10 is −
1. The right-hand side is( − = −
1. The decay to pairs of mesons with thesame internal angular momentum is suppressed to theextent that the disconnected diagram in Figure 4(b) isnot important. In a later study, Close and Dudek [38]found that some of these decays could be large becausethe π and ρ wave functions were not the same.While it has been historically difficult to compute de-cays on the lattice, a first study of the decay of the π hybrid has been carried out by McNeile [27, 39]. In or-der to do this, they used a technique where they put agiven decay channel at roughly the same energy as thedecaying state. Thus, the decay is just allowed and con-serves energy in a two-point function. In this way, theyare able to extract the ratio of the decay width over thedecay momentum, and findΓ( π → b π ) /k = 0 . ± . π → f π ) /k = 0 . ± . . π . As a check of their procedure,they carry out a similar calculation for b → ωπ wherethey obtain Γ /k ∼ .
8, which leads to Γ( b → ωπ ) ∼ .
22 GeV. This is about a factor of 1 . π and the decay width from the lattice. They notethat in the work of McNeile [27], an assumption was madethat Γ /k does not vary with the quark mass, and the re-sulting linear extrapolation leads to the large width inTable IX. They argue that the flux-tube model has beentested over a large range of k , where it accurately pre-dicts the decays of mesons and baryons. Quoting them,“The successful phenomenology of this and a wide rangeof other conventional meson decays relies on momentum-dependent form factors arising from the overlap of hadronwave functions. The need for such form factors is rathergeneral, empirically supported as exclusive hadron de-cay widths do not show unrestricted growth with phasespace.” Based on this, they carried out a comparison ofthe transition amplitudes computed for k = 0 (the lat-tice case). They found excellent agreement between thelattice and the flux-tube calculations. Thus, their con-cern that the extrapolation may be overestimating decaywidths may be valid. IKP IKP Lattice[17] [33] [27]1 . . . π → b π ) S
100 70 400 ± π → b π ) D
30 30Γ( π → f π ) S
30 20 90 ± π → f π ) D
20 25TABLE IX. Decay widths as computed in the flux-tube model(IKP) compared to the lattice calculations. (Table repro-duced from reference [40].)
While the model calculations provide a good guidein looking for hybrids, there are often symmetries thatcan suppress or enhance certain decays. Chung andKlempt [42] noted one of these for decays of a J P C = 1 − + state into ηπ where the η and π have relative angular mo-mentum of L = 1. In particular, in the limit where the η is an SU(3) octet, the ηπ in a p -wave must be in anantisymmetric wave function. In order to couple this toan octet (hybrid) meson, the hybrid must also be anti-symmetric. This implies it must be a member of the 8 octet. However, the SU(3) Clebsch-Gordan coefficient for8 → η π is zero. Thus, the decay is forbidden.However, by similar arguments, they showed that itcan couple to the 10 ⊕
10 representation of SU(3). Arepresentation that contains multiquark ( qq ¯ q ¯ q ) objects(see Section II F). Similarly, for the singlet ( η (cid:48) ) in a p -wave, the coupling to an octet is not suppressed.To the extent that the η is octet and the η (cid:48) is singlet,a 1 − + state that decays to ηπ and not η (cid:48) π cannot bea hybrid, while one that decays to η (cid:48) π and not ηπ is acandidate for a 1 − + hybrid meson. Our current under-standing of the pseudoscalar mixing angle is that it isbetween − ◦ and − ◦ [1], thus the assumption on thenature of the η and η (cid:48) is not far off. However, as faras we know, the pseudoscalar mesons are the only nonetthat is close to pure SU(3) states, all others tend to beclose to ideal mixing. A case where the higher mass stateis nearly pure s ¯ s . Thus, this suppression would not beexpected for decays to higher mass nonets. F. Multiquark states
As noted in Section I, exotic quantum numbers canarise from other quark-gluon systems as well. While it is possible for glueballs to have exotic quantum numbers,the masses are expected to be above 3 GeV [43]. Anotherconfiguration are multiquark states ( qq ¯ q ¯ q ) consisting oftwo quarks and two antiquarks. A short review of thistopic can be found in Ref. [44], and a nice descriptionof how these states are built in the quark model can befound in Ref. [45].Following Ref. [46], the SU(3) multiplets of these statescan be obtained by considering qq and ¯ q ¯ q combinations.The former can transform as either 3 or 6 under SU(3),while the latter can transform as 3 and 6. Thus, multi-plets can be built up as3 ⊗ ⊕ ⊗ ⊕ ⊕
27 = 366 ⊗ ⊕ ⊗ ⊕ ⊕ ⊕
10 = 18 ⊕ J P of these multiquark states can be obtained byinitially combining all the quarks in an S-wave. Thisyields J P values of 0 + , 1 + and 2 + , which can be combinedwith the fact that the overall wave functions must beantisymmetric to associate SU(3) multiplets with J P . J P = 2 + : 9 , J P = 1 + : 9 , , , , , J P = 0 + : 9 , , , J P = 0 + states to be the lightest with a mass around 1 GeV. This cryptoexotic nonet is interesting in that the ρ - and ω -likestates have an s ¯ s pair combined with the lighter quarks. ω √ (cid:0) u ¯ u + d ¯ d (cid:1) ( s ¯ s ) ρ + u ¯ d ( s ¯ s ) ρ √ (cid:0) u ¯ u − d ¯ d (cid:1) ( s ¯ s ) ρ − d ¯ u ( s ¯ s )The K -like states have a single strange quark, K + u ¯ sd ¯ dK d ¯ su ¯ u ¯ K s ¯ ud ¯ dK − s ¯ du ¯ u while the φ -like state has no strange quarks, φ u ¯ ud ¯ d . This yields the so-called inverted nonet, where the mass-hierarchy is reversed relative to the q ¯ q states. This nonetis often associated with the low-mass states f (600) ( σ ), K ∗ (800) ( κ ), a (980) and the f (980). Jaffe also notedthat whenever the expected mass of a multiquark statewas above that of a simple meson-meson threshold towhich the state could couple, the decays would be “super-allowed”, and the width of the state would be very large.Because of these super-allowed decays, Jaffe [48] notedthat the states would not exist.Orbital excitations of the multiquark systems wereexamined in reference [49]. Additional symmetrizationrules beyond the simple q ¯ q system apply for these, butthey found that the addition of one unit of angular mo-mentum could produce both J P C = 1 − + and 0 −− statesas members of an 18 ⊕
18 SU(3) multiplet with massesaround 1 . π combined with either and η or an η (cid:48) . While themixing between the η and η (cid:48) components is not known,it is likely that both states would have some hidden s ¯ s component.General and colleagues [50] looked at multiquark statesin the framework of molecular resonances using theircoulomb gauge formalism. In this framework, they com-puted the spectrum of the lightest states and find sev-eral states with masses below 1 . J P C = 1 − + state ( m = 1 .
32 GeV), with a somewhat heavier 0 −− state ( m = 1 .
36 GeV), and then a second 1 − + state( m = 1 .
42 GeV). In the isoscalar channel, they find asingle 0 −− state and in the isotensor (isospin two) chan-nel, they predict an additional 0 −− state. Between 1 . − + states ineach of the three isospin channels.QSSR techniques have also been used to look for bothisovector [51] and isoscalar [52] J P C = 1 − + multiquarkstates. As with the earlier work, they find that the exotic-quantum number multiquark states are in the (3 ⊗ ⊕ (3 ⊗
6) flavor representations. In their calculations, thedecuplet π state (with no s ¯ s pair) has a mass of about1 . π state (with s ¯ s ) has a massof about 2 GeV. For these states, they suggest decays ofthe form J P = 0 + , J P = 1 − ( f ρ ), J P = 1 + , J P = 0 − ( b π ) and J P = 1 − , J P = 1 + ( ωb ). For the isoscalarmasses, both the octet and decuplet member contain an s ¯ s pair. They find a single state with a mass between 1 . . KK , ηη , ηη (cid:48) and η (cid:48) η (cid:48) . They also Z eq = Z L + Z C Z eq = jωL + 1 jωCZ eq = 1 − ω LCjωC suggest several decays that are forbidden by isospin con-servation.Lattice calculations for multiquark states are some-what sparse, largely due to the challenge of the numberof quarks. Studies have been made to try to determineif the low-mass scalars have multiquark nature. A cal-culation in the quenched approximation was made withpion mass as small as 180 MeV identified the f (600) as a multiquark state [53]. A later quenched calculation withheavier pion masses (344-576 MeV) found no indicationof the f (600) [54], but the authors note that their pionmass is too heavy for this to be conclusive. A recentdynamical calculation [55] with somewhat heavier pionmass shows good agreement with Ref. [53], and while theauthors could not exclude the states are lattice artifacts,their results suggest that the f (600) and K ∗ (800) havea multiquark nature. Finally, a recent dynamical calcu-lation of the entire isovector meson spectrum shows nomultiquark states [29]. However, the authors note thatthe correct operators were probably not included in theiranalysis, so the fact that these states are missing fromtheir analysis should not be taken as conclusive. Otherlattice calculations explicitly looking for exotic-quantum-number multiquark states do not appear to have beenperformed.If exotic-quantum number multiquark states exist, thefavoured quantum numbers are 1 − + and 0 −− . The latterbeing a J P C not predicted for hybrid mesons. There mayalso be hidden s ¯ s components in the multiquark multi-plets that would distort their mass hierarchy relative tohybrid nonets. However, for most of these multiquarkstates, their decays will be super-allowed. In their recentreview, Klempt and Zaitsev [3] argue that ( qq )(¯ q ¯ q ) sys-tems will not bind without additional q ¯ q forces, and feelthat it is unlikely that these multiquark states exist. Inreviewing the information on these states, we concur withtheir assessment for the exotic-quantum-number states. III. EXPERIMENTAL RESULTSA. Production processes
Data on exotic-quantum-number mesons have comefrom both diffractive production using incident pionbeams and from antiproton annihilation on protons andneutrons. Diffractive production is schematically shownin Figure 5. A pion beam is incident on a proton (or nu-clear) target, which recoils after exchanging something inthe t -channel. The process can be written down in thereflectivity basis [56] in which the production factorizesinto two non-interfering amplitudes—positive reflectivity( (cid:15) = +) and negative reflectivity ( (cid:15) = − ). The absolutevalue of the spin projection along the z -axis is M , andis taken to be either 0 or 1 (it is usually assumed thatcontributions from M larger than 1 are small and can beignored [57]). It can be shown in this process that natu-rality of the exchanged particle can be determined by (cid:15) .Natural parity exchange (n.p.e.) corresponds to J P s of0 + , 1 − , 2 + , · · · , while unnatural parity exchange (u.p.e.)corresponds to J P of 0 − , 1 + , 2 − , · · · .For a state which is observed in more than one decaymode, one would expect that the production mechanism( M (cid:15) ) would be the same for all decay modes. If not,this could be indicative of more than one state beingobserved, or possible problems in the analysis that are0 p (target) p (recoil)X (J PC M )
FIG. 5. The diffractive production process showing an in-cident pion ( π beam) incident on a proton ( p target) wherethe exchange has z-component on angular momentum M andreflectivity (cid:15) . The final state consists of a proton ( p recoil)and a state X of given J PC produced by an exchange M (cid:15) .For positive reflectivity, the t -channel is a natural parity ex-change (n.p.e.), while for negative reflectivity, it is unnaturalparity exchange (u.p.e.). (This diagram was produced usingthe JaxoDraw package [58].) not under control.In antiproton-nucleon annihilation, there are a num-ber of differences between various annihilation processes.For the case of ¯ pp , the initial state is a mixture of isospin I = 0 and I = 1. For ¯ pn annihilation, the initial state ispure I = 1. For annihilation at rest on protons, the ini-tial state is dominated by atomic S-waves. In particular, S and S atomic states, which have J P C = 0 − + and1 −− respectively (with a small admixture of P states).For annihilation in flight, the number of initial states ismuch larger and it may no longer make sense to try andparametrize the initial system in terms of atomic states.The combination of initial isospin and final state par-ticles may lead to additional selection rules that restrictthe allowed initial states. In the case of ¯ pp → ηπ π , theannihilation is dominated by S initial states ( J P C =0 − + ). For the case of ¯ pn → ηπ π − , quantum numbersrestrict this annihilation to occur from the S initialstates ( J P C = 1 −− ). In addition, the neutron is boundin deuterium, where the Fermi motion introduces sub-stantial p-wave annihilation. Thus, one may see quitedifferent final states from the two apparently similar re-actions. B. The π (1400) The first reported observation of an exotic quantumnumber state came from the GAMS experiment whichused a 40 GeV/c π − to study the reaction π − p → pηπ − .They reported a J P C = 1 − + state in the ηπ − systemwhich they called the M (1405) [59]. The M (1405) had amass of 1 . ± .
020 GeV and a width of 0 . ± .
02 GeV.Interestingly, an earlier search in the ηπ channel foundno evidence of an exotic state [60]. At KEK, resultswere reported on studies using a 6 . π − beamwhere they observed a 1 − + state in the ηπ − systemwith a mass of 1 . ± . . ± . a (1320).The VES collaboration reported intensity in the 1 − + ηπ − wave as well as rapid phase motion between the a and the exotic wave [62] (see Figure 6). The exotic wavewas present in the M (cid:15) = 1 + (natural parity) exchange,but not in the 0 − and 1 − (unnatural parity) exchange.They could fit the observed J P C = 1 − + intensity andthe phase motion with respect to the a (1320) using aBreit-Wigner distribution (mass of 1 . ± .
012 GeV andwidth of 0 . ± .
025 GeV). However, they stopped shortof claiming an exotic resonance, as they could not unam-biguously establish the nature of the exotic wave [63]. Ina later analysis of the ηπ system, they claim that thepeak near 1 . π − beams tostudy the reaction π − p → pηπ − . They reported the ob-servation of a 1 − + state in the ηπ − system [65]. E852found this state only produced in natural parity ex-change ( M (cid:15) = 1 + ). They measured a mass of 1 . ± . +0 . − . GeV and a width of 0 . ± . +0 . − . GeV.While the observed exotic signal was only a few percentof the dominant a (1320) strength, they noted that itsinterference with the a provided clear evidence of thisstate. When their intensity and phase-difference plotswere compared with those from VES [62], they were iden-tical. These plots (from E852) are reproduced in Fig-ure 7.Due to disagreements over the interpretation of the1 − + signal, the E852 collaboration split into two groups.The majority of the collaboration published the reso-nance interpretation, π (1400) [65], while a subset of thecollaboration did not sign the paper. As this latter group,centered at Indiana University, continued to analyze datacollected by E852, we will refer to their publications asE852-IU to try an carefully distinguish the work of thetwo groups.The exotic π state was confirmed by the Crystal Bar-rel Experiment which studied antiproton-neutron anni-hilation at rest in the reaction ¯ pn → ηπ − π [66]. TheDalitz plot for this final state is shown in Figure 8 wherebands for the a (1320) and ρ (770) are clearly seen. Theyreported a 1 − + state with a mass of 1 . ± . ± .
020 GeV and a width of 0 . ± . +0 . − . GeV. Whilethe signal is not obvious in the Dalitz plot, if one com-pares the difference between a fit to the data withoutand with the π (1400), a clear discrepancy is seen whenthe π (1400) is not included (see Figure 9). While the π (1400) was only a small fraction of the a (1320) in theE852 measurement [65], Crystal Barrel observed the twostates produced with comparable strength.Crystal Barrel also studied the reaction ¯ pp → ηπ π [67]. Here, a weak signal was observed for the π (1400) (relative to the a (1320)) with a mass of 1 . ± .
025 GeV and a width of 0 . ± .
090 GeV. In I = 0 ¯ pp annihilations, the a (1320) is produced strongly from the S atomic state. However, ¯ pn is isospin 1 and S state1 M( ηπ ) (GeV)N/(0.04 GeV) N/(0.04 GeV) M( ηπ )M( ηπ ) (GeV)(GeV) (a) (b) (c) (deg)D + P + arg(P + /D + ) FIG. 6. The results of a partial-wave analysis of the ηπ − final state from VES. (a) shows the intensity in the 2 ++ partial wave,(b) shows the intensity in the 1 − + partial wave and (c) shows the relative phase between the waves. (Figure reproduced fromreference [64].) M( ηπ ) (GeV) E v e n t s / ( . G e V ) P h a s e D i ff e r e n c e ( r a d ) M( ηπ ) (GeV) P h a s e D i ff e r e n c e ( r a d ) E v e n t s / ( . G e V ) (a) (b)(c) (d)D + P + ΔΦ (D + - P + ) 1 234 FIG. 7. The π (1400) as observed in the E852 experiment [65].(a) shows the intensity of the J PC = 2 ++ partial wave as afunction of ηπ mass. The strong signal is the a (1320). (b)shows the intensity of the 1 − + wave as a function of mass,while (c) shows the phase difference between the 2 ++ and1 − + partial waves. In (d) are shown the phases associatedwith (1) the a (1320), (2) the π (1400), (3) the assumed flatbackground phase and (4), the difference between the (1) and(2). (This figure is reproduced from reference [65].) is forbidden. Thus, the strong a production from ¯ pp is suppressed in ¯ pd annihilations—making the π (1400)production appear enhanced relative to the a (1320) inthe latter reaction.An analysis by the E852-IU group of data for the re-action π − p → nηπ found evidence for the exotic 1 − + partial wave, but were unable to describe it as a Breit-Wigner-like π (1400) ηπ resonance [68]. However, alater analysis by the E852 collaboration of the same fi-nal state and data confirmed their earlier observation ofthe π (1400) [69]. E852 found a mass of 1 . ± . ± m ( ηπ − ) (MeV) m ( η π )( M e V ) a (1320) a (1320) ρ - (770) FIG. 8. (Color on line.) The Dalitz plot of m ( ηπ ) versus m ( ηπ − ) for the reaction ¯ pn → ηπ − π from the Crystal Bar-rel experiment [66]. The bands for the a (1320) are clearlyseen in both ηπ and ηπ − , while the ρ (770) is seen in the π π − invariant mass. .
025 GeV and a width of 0 . ± . ± .
058 GeVwith the π (1400) produced via natural parity exchange( M (cid:15) = 1 + ). Much of the discrepancy between these twoworks arise from the treatment of backgrounds. TheE852 collaboration consider no background phase, andattribute all phase motion to resonances. The E852-IUgroup allow for non-resonant interactions in the exoticchannel, these background processes are sufficient to ex-plain the observed phase motion.The π (1400) was also reported in ¯ pp annihilation intofour-pion final states by both Obelix [70] and CrystalBarrel [71] (conference proceedings only). They both ob-served the π (1400) decaying to ρπ final states, howeverthere is some concern about the production mechanism.The ηπ signal arises from annihilation from p-wave ini-tial state, while the signal in ρπ come from the S initial2 (a)m ( ηπ − ) (MeV) m ( η π )( M e V ) (b)m ( ηπ − ) (MeV) m ( η π )( M e V ) FIG. 9. (Color on line.) The difference between the fit andthe data in the Dalitz plot of m ( ηπ versus ηπ − for the re-action ¯ pn → ηπ − π from the Crystal Barrel experiment [66].(a) Does not include the π (1400) while (b) does include the π (1400). There are clear systematic discrepancies present in(a) that are not present when the π (1400) is included. state. Thus, it is unlikely that the exotic state seen in ηπ and that seen in ρπ are the same. The origin of thesemay not be due to an exotic resonance, but rather somere-scattering mechanism that has not been properly ac-counted for.Interpretation of the π (1400) has been problematic.Its mass is lower than most predicted values from mod-els, and its observation in only a single decay mode ( ηπ )is not consistent with models of hybrid decays. Don-nachie and Page showed that the π (1400) could be anartifact of the production dynamics. They demonstratedthat is possible to understand the π (1400) peak as aconsequence of the π (1600) (see Section III C) interfer- ing with a non-resonant Deck-type background with anappropriate relative phase [72]. Zhang [73] considered amolecular picture where the π (1400) was an η (1295) π molecule. However, the predicted decays were inconsis-tent with the observations of the π (1400).Szczepaniak [74] considered a model in which t -channelforces could give rise to a background amplitude whichcould be responsible for the observed π (1400). In hismodel, meson-meson interactions which respected chiralsymmetry were used to construct the ηπ p -wave inter-action much like the ππ s -wave interaction gives rise tothe σ meson. They claimed that the π (1400) was not aQCD bound state, but rather dynamically generated bymeson exchange forces.Close and Lipkin noted that because the SU(3) multi-plets to which a hybrid and a multiquark state belong aredifferent, that the ηπ and η (cid:48) π decays might be a good wayto distinguish them. They found that for a multiquarkstate, the ηπ decay should be larger than η (cid:48) π , while thereverse is true for a hybrid meson [41]. A similar obser-vation was made by Chung [42] who noted that in thelimit of the η being a pure octet state, that the decay ofan octet 1 − + state to an ηπ p -wave is forbidden. Such adecay can only come from a decuplet state. Given thatthe pseudoscalar mixing angle for the η and η (cid:48) are closeto this assumption, they argue that the π (1400) is qq ¯ q ¯ q in nature.While the interpretation of the π (1400) is not clear,most analyses agree that there is intensity in the 1 − + wave near this mass. A summary of all reported massesand widths for the π (1400) are given in Table X. Allare reasonably consistent, and even the null observationsof VES and E852-IU all concur that there is strengthnear 1 . J P C exotic wave. However TheE852 and VES results can be explained as either non-resonant background [74], or non-resonant deck ampli-tudes [72]. An other possibility is the opening of meson-meson thresholds, such as f (1285) π . Unfortunately, nocomparisons of these hypothesis have been made withthe ¯ pN data (owing to the lack of general availability ofthe data sets), so it is not possible to conclude that theywould also explain those data. However, in our minds,we believe that the evidence favors a non-resonant inter-pretation of the exotic 1 − + signal and that the π (1400)does not exist. C. The π (1600) While the low mass, and single observed decay mode,of the π (1400) have presented some problems in under-standing its nature, a second J P C = 1 − + state is lessproblematic. The π (1600) has been observed in diffrac-tive production using incident π − beams where its massand width have been reasonably stable over several ex-periments and the decay modes. It may also have beenobserved in ¯ pp annihilation. Positive results have beenreported from VES, E852, COMPASS and others. These3 Mode Mass (GeV) Width (GeV) Experiment Reference ηπ − . ± .
020 0 . ± .
02 GAMS [59] ηπ − . ± . . ± . ηπ − . ± .
016 0 . ± .
040 E852 [65] ηπ . ± .
020 0 . ± .
064 E852 [69] ηπ . ± .
020 0 . ± .
050 CBAR [66] ηπ . ± .
025 0 . ± .
090 CBAR [67] ρπ . ± .
028 0 . ± .
058 Obelix [70] ρπ ∼ . ∼ . ηπ . ± .
030 0 . ± .
040 PDG [1]TABLE X. Reported masses and widths of the π (1400) fromthe GAMS, KEK, E852, Crystal Barrel (CBAR) and Obelixexperiment. Also reported is the 2008 PDG average for thestate. are discussed below in approximate chronological order.In addition to their study of the ηπ − system, the VEScollaboration also examined the η (cid:48) π − system. Here theyobserved a J P C = 1 − + partial wave with intensity peak-ing at a higher mass than the π (1400) [62]. However,as with the ηπ − system, they did not claim the discov-ery of an exotic-quantum-number resonance. VES laterreported a combined study of the η (cid:48) π − , f π − and ρ π − final states [75], and reported a “resonance-like struc-ture” with a mass of 1 . ± .
02 GeV and a width of0 . ± .
05 GeV decaying into ρ π − . They also notedthat the wave with J P C = 1 − + dominates in the η (cid:48) π − final state, peaking near 1 . − + signal in the f π − final state. I n t e n s i t y M( π + π - π - ) (GeV) (0 − ,1 − )1 −+ (1 + )1 −+ (a) (b) FIG. 10. The production of the 1 − + partial wave as seen inthe π + π − π − final state by E852. (a) shows the unnaturalparity exchange ( M (cid:15) = 0 − ,1 − ) while (b) shows the naturalparity exchange ( M (cid:15) = 1 + ). (Figure reproduced from refer-ence [76].) Using an 18 GeV/c π − beam incident on a proton tar-get, the E852 collaboration carried out a partial waveanalysis of the π + π − π − final state [76, 77]. They sawboth the ρ π − and f (1270) π − intermediate states andobserved a J P C = 1 − + state which decayed to ρπ , the π (1600). The π (1600) was produced in both naturaland unnatural parity exchange ( M (cid:15) = 1 + and M (cid:15) = 0 − ,1 − ) with apparent similar strengths in all three exchangemechanisms (see Figure 10). In Ref. [77], they notedthat there were issues with the unnatural exchange pro-duction. The signal in the M (cid:15) = 1 − wave exhibited very strong model dependence and nearly vanished whenlarger numbers of partial waves were included. The signalin the M (cid:15) = 0 − partial wave was stable, but its peak wasabove 1 . − + in this sector problem-atic. Thus, in their analysis, they only considered thenatural parity exchange. There, they found the π (1600)to have a mass of 1 . ± . +0 . − . GeV and a widthof 0 . ± . +0 . − . GeV. In Figure 11 are shown theintensity of the 1 − + and 2 − + ( π (1670)) partial waves aswell as their phase difference. The phase difference canbe reproduced by two interfering Breit-Wigner distribu-tions and a flat background. I n t e n s i t y P h a s e ( r a d ) (a) (b)(c) (d) Δϕ ϕ M( π + π - π - ) (GeV)M( π + π - π - ) (GeV) FIG. 11. The results of a PWA to the π + π − π − final statefrom E852. (a) shows the intensity of the J PC = 1 − + wave,(b) shows the 2 − + and (c) shows the phase difference betweenthe two. The solid curves are fits to two interfering Breit-Wigner distributions. In (d) are shown the phases of the twoBreit-Wigner distributions and (1,2) and a flat backgroundphase (3) that combine to make the curve in (c). (Figurereproduced from reference [76].) VES also reported on the ωπ − π final state [78, 79].In a combined analysis of the η (cid:48) π − , b π and ρ π − finalstates, they reported the π (1600) state with a mass of1 . ± .
02 GeV and a width of 0 . ± .
03 GeV that wasconsistent with all three final states. To the extent thatthey observed these states, they also observed all threefinal state produced in natural parity exchange ( M (cid:15) =1 + ). They were also able to report relative branching4ratios for the three final states as given in equation 11. b π : η (cid:48) π : ρπ : = 1 : 1 ± . . ± . ρπ final state.Rather than limiting the rank of the density matrix aswas done in [76, 77], they did not limit it. This allowedfor a more general fit that might be less sensitive to ac-ceptance affects. In this model, they did not observe anysignificant structure in the 1 − + ρπ partial wave above1 . − + partial wave peaking near 1 . ρπ decay of the the π (1600), in the casethat it exists, they were able to obtain the rates given inequation 11. (GeV)M( η ’ π - ) E v e n t s / ( . G e V ) Δ Φ ( r a d ) (a)(b)(c) (d)(e)(f ) ΔΦ (D + - G + ) ΔΦ (P + - G + )P + D + G + G + FIG. 12. Results from E852 on the η (cid:48) π − final state. (a)shows the 1 − + partial wave, (b) shows the 4 ++ partial wave(an a ) and (c) shows the phase difference between these. (d)shows the 2 ++ partial wave ( a (1320)), while (e) shows the a and (f) is the phase difference. (Figure reproduced fromreference [80].) In a follow-up analysis, E852 also studied the reaction π − p → pη (cid:48) π − to examine the η (cid:48) π − final state [80]. Theyobserved, consistent with VES [62], that the dominantsignal was the 1 − + exotic wave produced dominantly inthe M (cid:15) = 1 + channel, implying only natural parity ex-change. They found the signal to be consistent with aresonance, the π (1600) and found a mass of 1 . ± . +0 . − . GeV and a width of 0 . ± . ± .
050 GeV.The results of the E852 PWA are shown in Figure 12where the P + wave is the 1 − + , the D + corresponds tothe 2 ++ a and the G + corresponds to the 4 ++ a . Clear phase motion is observed between both the 2 ++ and 4 ++ wave and the 1 − + and the 4 ++ wave.An analysis of Crystal Barrel data at rest for the re-action ¯ pp → ωπ + π − π was carried by some members ofthe collaboration [81]. They reported evidence for the π (1600) decaying to b π from both the S and S ini-tial states, with the signal being stronger from the for-mer. The total signal including both initial states, aswell as decays with 0 and 2 units of angular momentumaccounted for less than 10% of the total reaction channel.The mass and width were found consistent (within largeerrors) of the PDG value, and only results with the massand width fixed to the PDG values were reported. Ac-counting for the large rate of annihilation to ωπ + π − π of 13%, this would imply that ¯ pp → π (1600) π accountsfor several percent of all ¯ pp annihilations.E852 also looked for the decays of the π (1600) to b π and f π . The latter was studied in the reaction π − p → pηπ + π − π − with the f being reconstructed inits ηπ + π − decay mode [82]. The π (1600) was seen viainterference with both the 1 ++ and 2 − + partial waves.It was produced via natural parity exchange ( M (cid:15) = 1 + )and found to have a mass of 1 . ± . ± .
041 GeV anda width of 0 . ± . ± .
115 GeV. A second π statewas also observed in this reaction (see Section III D).The b π final state was studied by looking at the re-action π − p → ωπ − π p , with the b reconstructed in its ωπ decay mode [83]. The π (1600) was seen interferingwith the 2 ++ and 4 ++ partial waves. In b π , they mea-sured a mass of 1 . ± . ± .
010 GeV and a width of0 . ± . ± .
028 GeV for the π (1600). However, theproduction mechanism was seen to be a mixture of bothnatural and unnatural parity exchange, with the unnat-ural being stronger. As with the f π , they also observeda second π state decaying to b π (see Section III D). final state production ( M (cid:15) ) dominant ρπ − , 1 − , 1 + npe ∼ upe η (cid:48) π + npe f π + npe b π − , 1 − , 1 + upe > npeTABLE XI. The production mechanisms for the π (1600) asseen in the E852 experiment. Also shown is whether the nat-ural parity exchange (npe) or the unnatural parity exchange(upe) is stronger. The fact that E852 observed the π (1600) produced indifferent production mechanisms, depending on the finalstate, is somewhat confusing. A summary of the ob-served mechanisms is given in Table XI. In order to un-derstand the variations in production, there either needsto be two nearly-degenerate π (1600)s, or there is someunaccounted-for systematic problem in some of the anal-yses.The E852-IU group analyzed an E852 data set that wasan order of magnitude larger than that used by E852 inRefs. [76, 77]. In this larger data set, they looked at the5 (GeV)(GeV) E v e n t s ( K ) / ( . G e V ) P h a s e ( r a d i a n s ) ΔΦ (2 ++ - 1 -+ )1.6 −+ π - π π ) M( π - π π ) FIG. 13. (Color on line) The PWA solutions for the 1 − + partial wave for the π − π π final state (a) and its interferencewith the 2 ++ partial wave (b). See text for an explanation ofthe labels. (Figure reproduced from reference [85].) E v e n t s ( K ) / ( . G e V ) P h a s e ( r a d i a n s ) M( π - π - π ) (GeV)M( π - π - π ) (GeV)1 -+ FIG. 14. (Color on line) The PWA solutions for the 1 − + par-tial wave for the π + π − π − final state (a) and its interferencewith the 2 ++ partial wave (b). See text for an explanation ofthe labels. (Figure reproduced from reference [85].) reactions π − p → nπ + π − π − and π − p → nπ − π π andcarried out a partial wave analysis for both the π + π − π − and the π − π π final states. This yielded solutions thatwere consistent with both final states [85]. In this anal-ysis, they carried out a systematic study of which par-tial waves were important in the fit. When they usedthe same wave set as in the E852 analysis [76, 77], theyfound the same solution showing a signal for the π (1600)in both final states. However, when they allowed formore partial waves, they found that the signal for the π (1600) went away. Figure 13 shows these results forthe π − π π final state, while Figure 14 shows the re-sults for the π + π − π − final state. In both figures, the“low wave” solution matches that from E852, while their“high wave” solution shows no intensity for the π (1600)in either channel. An important point is that in boththeir high-wave and low-wave analyses, the phase differ-ence between the exotic 1 − + wave and the 2 ++ wave arethe same (and thus the same as in the E852 analysis). While not shown here, the same is also true for the 1 − + and 2 − + waves. π (1670) M (cid:15) = 0 + M (cid:15) = 1 + M (cid:15) = 1 − Decay L H L H L H( f π ) S × × × × × ( f π ) D × × × × [( ππ ) S ] D × × × ( ρπ ) P × × × ( ρπ ) F × × ( f π ) D × × TABLE XII. The included decays of the π (1670) in two anal-yses of the 3 π final state. “L” is the wave set used in the E852analysis [76, 77]. “H” is the wave set used in the higher statis-tics analysis [85]. E852-IU carried out a study to determine which of theadditional waves in their “high wave” set were absorb-ing the intensity of the π (1600). They found that themajority of this was due to the inclusion of the ρπ de-cay of the π (1670). The partial waves associated withthe π (1670) in both analyses are listed in Table XII.While the production from M (cid:15) = 0 + is similar for bothanalyses, the E852 analysis only included the π (1670)decaying to f π in the M (cid:15) = 1 + production. The high-statistics analysis included both f π and ρπ in both pro-duction mechanisms. The PDG [1] lists the two maindecays of the π (1670) as f π (56%) and ρπ (31%), so itseems odd to not include this latter decay in an analy-sis including the π (1670). Figure 15 shows the resultsof removing the ρπ decay from the “high wave” set forthe π + π − π − final state. This decay absorbs a significantportion of the π (1600) partial wave. E v e n t s ( K ) / ( . G e V ) M( π - π π ) (GeV)1 -+ FIG. 15. The 1 − + intensity for the charged mode for thehigh-wave set (filled circles), the modified high-wave set (filledsquares), and the low-wave set (open circles). In the modifiedhigh-wave set the two ρπ decays of the π (1670) were removedfrom the fit. (Figure reproduced from reference [85].) − + exotic wave relative to other partial wavesagrees with with those differences as measured by E852,and are the same in both the high-wave and low-waveanalyses is intriguing. This could be interpreted as a π (1600) state which is simply absorbed by the stronger π (1670) as more partial waves are added. However,given the small actual phase difference between the 1 − + and 2 − + partial waves (see Figure 11), the opposite con-clusion is also possible, particularly if some small non-zero background phase were present. Here, the 1 − + sig-nal is due to leakage from the stronger π and no π (1600)is needed in the ρπ final state.The VES results have been summarized in a review ofall their work on hybrid mesons [64]. This included anupdated summary of the π (1600) in all four final states, η (cid:48) π ρπ , b π and f π . In the η (cid:48) π final state (Figure 16),they note that the 1 − + wave is dominant. While theywere concerned about the nature of the higher-mass partof the 2 ++ spectrum ( a (1700) or background) they findthat a resonant description of π (1600) is possible in bothcases. For the case of the b π final state (Figure 17),they find that the contribution of a π (1600) resonanceis required. In a combined fit to both the η (cid:48) π and b π data, they find a mass of 1 . ± .
06 GeV and a width of0 . ± .
06 GeV for the π (1600). In the f π final state(Figure 18), they find a resonant description of the the π (1600) with a mass of 1 . ± .
03 GeV and a width of0 . ± .
06 GeV which they note is compatible with theirmeasurement in the previous two final states. They alsonote, that in contradiction with E852 [82], they find nosignificant 1 − + intensity above a mass of 1 . ρπ final state, they are unable toconclude that the π (1600) is present. (GeV)M( η ’ π ) M( η ’ π ) M( η ’ π ) (GeV)(GeV)N/(0.05 GeV) N/(0.05 GeV) arg(P + /D + ) (deg)(a) (b) (c)P + D + FIG. 16. The results of a partial wave analysis on the η (cid:48) π − final state from VES. (a) shows he 2 ++ partial wave in ωρ , (b)shows the 1 − + partial wave in b π and (c) shows the interfer-ence between them. (Figure reproduced from reference [64].) They note that the partial-wave analysis of the π + π − π − system finds a significant contribution from the J P C = 1 + wave in the ρπ channel (2 to 3% of the totalintensity). Some of the models in the partial-wave anal-ysis of the exotic wave lead to the appearance of a peaknear a mass of 1 . π (1600).However, the dependence of the size of this peak on themodel used is significant [79]. They note that becausethe significance of the wave depends very strongly on theassumptions of coherence used in the analysis, the results M( ωππ ) (GeV) M( ωππ ) M( ωππ ) (GeV)(GeV)N/(0.05 GeV) N/(0.05 GeV) (a) (b) (c)arg(1 - /2 + ) (deg)1 -+ ++ FIG. 17. The results of a partial wave analysis on the b π finalstate from VES. (a) shows he 2 ++ partial wave, (b) showsthe 1 − + partial wave and (c) shows the interference betweenthem. (Figure reproduced from reference [64].) (GeV)N/(0.1 GeV)M( ηπππ ) N/(0.1 GeV) M( ηπππ )M( ηπππ ) (GeV) (GeV)(a) (b) (c)arg(1 - /1 + ) (deg)1 ++ -+ FIG. 18. The results of a partial wave analysis on the f π finalstate from VES. (a) shows he 1 ++ partial wave, (b) showsthe 1 − + partial wave and (c) shows the interference betweenthem. (Figure reproduced from reference [64].) for 3 π final states on the nature of the π (1600) are notreliable.To obtain a limit on the branching fraction of π (1600)decay to ρπ , they looked at their results of the produc-tion of the π (1600) in the charge-exchange reaction to η (cid:48) π versus that of the η (cid:48) π − final state. They concludethat the presence of the π (1600) in η (cid:48) π − and its absencein η (cid:48) π preclude the formation of the π (1600) by ρ ex-change. From this, they obtain the relative branchingratios for the π (1600) as given in equation 12. b π : f π : ρπ : η (cid:48) π = 1 . ± . . ± . < . ρπ between E852 and VES seemat odds, we believe that these discrepancies are the resultof the assumptions made in the analyses. These assump-tions then manifest themselves in the interpretation ofthe results. The VES analyses fit both the real and imag-inary parts of their amplitudes independently. However,for analytic functions, the two parts are not independent.Not using these constraints can lead to results that maybe unphysical, and at a minimum, are discarding im-portant constraints on the amplitudes. All of which canlead to difficulties in interpreting the results. In E852,many of their results rely on the assumption of a flatbackground phase, but there are many examples wherethis is not true. Thus, their results are biased towardsa purely resonant description of the data, rather thana combination of resonant and non-resonant parts. Itis also somewhat disappointing that E852 is unable tomake statements about relative decay rates, or carry out7a coupled channel analysis of their many data sets. Ourunderstanding is that this is due to issues in modellingthe rather tight trigger used in collecting their data.The CLAS experiment at Jefferson Lab studied thereaction γp → π + π + π − ( n ) miss to look for the produc-tion of the π (1600) [86]. The photons were produced bybremsstrahlung from a 5 . a (1270), the a (1320) and the π (1670), but show no ev-idence for the π (1600) decaying into three pions. Theyplace and upper limit of the production and subsequentdecay of the π (1600) to be less than 2% of the a (1320).There results imply that the π (1600) is not strongly pro-duced in photoproduction, the π (1600) does not decayto 3 π , or both. E ve n t s / M e V × a. P,F ) πρ ( -+ ) (GeV) - π + π + π M( E ve n t s / M e V × c. S ) πρ ( ++ b. S ) π (f -+ ) (GeV) - π + π + π M( d. P ) πρ ( -+ (1600)? π FIG. 19. The results from CLAS of a partial-wave analysisphotoproduction data of the 3 π final state. Intensity is seenin the 2 − + partial wave, (a) and (b), as well as the 1 ++ partialwave (c). In the 1 − + exotic wave, (d), no intensity is observed.(Figure reproduced from reference [86].) The COMPASS experiment has reported their firststudy of the diffractively produced 3 π final state [87, 88].They used a 190 GeV/c beam of pions to study the re-action π − P b → π − π − π + X . In their partial-wave anal-ysis of the 3 π final state, they observed the π (1600)with a mass of 1 . ± . +0 − . GeV and a widthof 0 . ± . +0 . − . GeV. The π (1600) was produceddominantly in natural parity exchange ( M (cid:15) = 1 + ) al-though unnatural parity exchange also seemed to berequired. However, the level was not reported. Thewave set (in reference [88]) used appears to be some-what larger than that used in the high-statistics studyof E852-IU [85]. Thus, in the COMPASS analysis, the ρπ decay of the π (1670) does not appear to absorbthe exotic intensity in their analysis. They also reporton varying the rank of the fit with the π (1600) andthe results being robust against these changes. Onepoint of small concern is that the mass and width thatthey extract for the π (1600) are essentially identicalto those for the π (1670). For the latter, they ob-served a mass of 1 . ± . +0 . − . GeV and a widthof 0 . ± . +0 . − . GeV. However, the strength of theexotic wave appears to be about 20% of the π , thus feedthrough seems unlikely. Results from their partial-waveanalysis are shown in Figures 20 and 21. These show the1 − + partial wave and the phase difference between the1 − + and 2 − + waves. The solid curves are the results ofmass-dependent fits to the π (1600) and π (1670). I n t e n s i t y / ( M e V ) M( πππ ) (GeV) FIG. 20. (Color on line.) COMPASS results showing the in-tensity of the exotic 1 − + wave. The solid (red) curve shows afit to the corresponding resonances. The dashed (blue) curveis the π (1600) while the dotted (magenta) curve is back-ground. (Figure reproduced from reference [87].) Table XIII summarizes the masses and widths foundfor the π (1600) in the four decay modes and from theexperiments which have seen a positive result. While the η (cid:48) π , f π and b π decay modes appear to be robust in theobservation of a resonant π (1600), there are concernsabout the 3 π final states. While we report these in thetable, the results should be taken with some caution.Models for hybrid decays predict rates for the decayof the π . Equation 9 gives the predictions from refer-ence [33]. A second model from reference [34] predictedthe following rates for a π (1600). πb ρπ πf η (1295) π K ∗ K PSS 24 9 5 2 0 . . π (1600) as a hybrid will almost certainlyinvolve the identification of other members of the nonet:the η and/or the η (cid:48) , both of which are expected to have8 P h a s e ( d e g r ee s ) -200-150-100-50050100150 M( π - π - π + ) (GeV) ΔΦ ( 1 -+ - 2 -+ ) FIG. 21. (Color on line.) COMPASS results showing thephase difference between the exotic 1 − + wave and the 2 − + wave. The solid (red) curve shows a fit to the correspondingresonances. (Figure reproduced from reference [87].)Mode Mass (GeV) Width (GeV) Experiment Reference ρπ . ± .
08 0 . ± .
020 E852 [76] η (cid:48) π . ± .
010 0 . ± .
040 E852 [80] f π . ± .
024 0 . ± .
080 E852 [82] b π . ± .
008 0 . ± .
025 E852 [83] b π . ± .
03 0 . ± .
03 VES [84] b π . ± .
02 0 . ± .
03 VES [78] b π ∼ . ∼ .
33 VES [63] b π . ± .
06 0 . ± .
06 VES [64] f π . ± .
03 0 . ± .
06 VES [64] η (cid:48) π . ± .
03 0 . ± .
03 VES [84] η (cid:48) π . ± .
02 0 . ± .
03 VES [78] η (cid:48) π . ± .
06 0 . ± .
06 VES [64] b π ∼ . ∼ .
23 CBAR [81] ρπ . ± .
010 0 . ± .
021 COMPASS [87]all 1 . +0 . − . . ± .
050 PDG [1]TABLE XIII. Reported masses and widths of the π (1600)from the E852 experiment, the VES experiment and theCOMPASS experiment. The PDG average from 2008 is alsoreported. widths that are similar to the π . For the case of the η , the most promising decay mode may be the f η as itinvolves reasonably narrow daughters.We believe that the current data support the existenceof a resonant π (1600) which decays into b π , f π and η (cid:48) π , however, near-term confirmation of these results byCOMPASS would be useful. For the ρπ decay, we areuncertain. As noted earlier, the phase motion results ob-served by both E852 and E852-IU are can be interpretedas either the π (1670) absorbing the π (1600), or leak-age from the π (1670) generating a spurious signal in the1 − + channel. While the new COMPASS result are indeedinteresting, we are concerned about their findings of ex-actly the same mass and width for the π (1670) and the π (1600). We are also concerned that their initial analy-ses may be over simplified, particularly in their bias to-wards an all-resonant description of their data. We hopethat follow-on results from COMPASS will more broadlyexplore the model space imposed by their analyses. Wewould also like to see results on other final states coupledto those on three pions. D. The π (2015) The E852 experiment has also reported a third π stateseen decaying to both f π [82] and to b π [83]. In the f π final state, the π (2015) is produced with M (cid:15) = 1 + in conjunction with the π (1600). The description of the1 − + partial wave requires two poles. They report a massof 2 . ± . ± .
092 GeV and a width of 0 . ± . ± .
049 GeV. Figure 22 shows the E852 data fromthis final state. Parts e and f of this show the need forthe two-pole solution. VES also examined the f π finalstate, and their intensity of the 1 − + partial wave above1 . π (2015). (GeV) c o u n t s / ( M e V ) Δ φ ( r a d ) M(3 πη )(a)(b)(c) (d)(e)(f)1 ++ -+ -+ ΔΦ (2 -+ - 1 -+ ) ΔΦ (1 ++ - 1 -+ ) ΔΦ (1 ++ - 1 -+ ) FIG. 22. (Color on line.) The f π invariant mass fromE852 [82]. (a) The 1 ++ partial wave ( a (1270)), (b) the 2 − + partial wave ( π (1670)) and (c) the exotic 1 − + partial wave.The dotted (red) curves show the fits of Breit-Wigner distri-butions to the partial waves. (d) shows the phase differencebetween the 2 − + and 1 − + partial waves, while (e) shows thedifference between the 1 ++ and 1 − + partial waves. The dot-ted (red) curves show the results for a single π state, the π (1600). (f) shows the same phase difference as in (d), butthe dotted (red) curve shows a fit with two poles in the 1 − + partial wave, the π (1600) and the π (2015). (Figure repro-duced from reference [82].) In the b π final state, the π (2015) is produced domi-nantly through natural parity exchange ( M (cid:15) = 1 + ) whilethe π (1600) was reported in both natural and unnatu-ral parity exchange, where the unnatural exchange domi-nated. They observe a mass of 2 . ± . ± .
016 GeV9 C o u n t s / ( M e V ) (a) (b)(c) (d)M(b π ) (GeV)2 ++ ++ (1 + )1 -+ (0 - )1 -+ FIG. 23. The b π invariant mass from the E852 experiment.(a) shows the 1 − + b π partial wave produced in natural parityexchange ( M (cid:15) = 1 + ) while (b) shows the 1 − + b π partial waveproduced in unnatural parity exchange ( M (cid:15) = 0 − ). In (c) isshown the 2 ++ ωρ partial wave, while (d) shows the 4 ++ ωρ partial wave. The curves are fits to the π (1600) and π (2015)(a and b), the a (1700) in (c) and the a (2040) in (d). (Figurereproduced from reference [83].) and a width of 0 . ± . ± .
073 GeV which are consis-tent with that observed in the f π final state. Figure 23shows the intensity distributions for several partial wavesin this final states. The need for two states is most clearlyseen in b . VES also looked at the b π final state, but didnot observe 1 − + intensity above 1 . Mode Mass (GeV) Width (GeV) Experiment Reference f π . ± .
030 0 . ± .
052 E852 [82] b π . ± .
020 0 . ± .
032 E852 [83]TABLE XIV. Reported masses and widths of the π (2015) asobserved in the E852 experiment. The PDG does not reportan average for this state. With so little experimental evidence for this high-massstate, it is difficult to say much. We note that the ob-served decays, f π and b π are those expected for a hy-brid meson. We also note that the production of thisstate is consistent (natural parity exchange) for both ofthe observed final states. In the case that the π (1600)is associated with the lowest-mass hybrid state, one pos-sible interpretation of the π (2015) would be a excitedstate (as suggested by recent LQCD calculations [29]).The mass splitting is typical of radial excitations ob-served in the normal mesons. In the case of the π (1600)identified as something else, the π (2015) would be aprime candidate for the lightest mass hybrid. E. Other Exotic-quantum Number States
While no result has been published, the E852 col-laboration has presented evidence at conferences for anisoscalar 2 + − state [89]. The signal is observed with amass near 1 . ωπ − π + final state. It de-cays through b π and is produced in both natural andunnatural parity exchange. This conference report wasnot followed up by a publication, so the signal should beviewed with caution. However, if confirmed, this stateroughly lines up in mass with the π (2015) and would beconsistent with the lattice picture in which the π (1600)is the lowest-mass hybrid and the π (2015) is the firstexcitation [29]. IV. THE FUTURE
The COMPASS experiment has recently started look-ing at pion peripheral production similar to work carriedout by both VES and E852. Two new facilities are alsoexpected in the not-too-distant future, PANDA at GSIand GlueX at Jefferson Lab. The former will study ¯ pp annihilation in the charmonium region, but it will alsobe possible to search for production of light-quark hy-brids. GlueX will use a 9 GeV beam of linearly polarizedphotons to produce hybrids.Photoproduction of hybrids is interesting for severalreasons. Simple arguments based on vector meson domi-nance suggest that the photon may behave like an S = 1¯ qq system. In several models, such a system is more likelyto couple to exotic quantum-number hybrids. Early cal-culations of hybrids used the apparent large ρπ couplingof the π (1600) to suggest that this state should be pro-duced at least as strongly as normal mesons in photopro-duction [90–92]. Unfortunately, the current controversyon the ρπ decay of the π (1600) makes the underlying as-sumption questionable, which may be confirmed by thenon observation of the π (1600) by CLAS [86].Recently, lattice calculations have been performed tocompute the radiative decay of charmonium c ¯ c and hy-brid states [93]. In the charmonium system, they findthat there is a large radiative decay for an exotic quan-tum number hybrid. These studies are currently beingextended to the light-quark hybrids with the goal of pro-viding estimates of the photoproduction cross sections ofthese states. However, based on the results in the char-monium sector, photoproduction appears to be a goodplace to look for hybrid mesons. V. CONCLUSIONS
Over the last two decades, substantial data has beencollected looking for exotic-quantum-number mesons.In particular, searches have focused on hybrid mesons,which arise due to excitations of the gluonic fields which0confine quarks inside mesons. Models and LQCD pre-dictions suggest that three nonets of exotic-quantum-number states should exist, with J P C = 0 + − ,1 − + and2 + − , where the 1 − + is expected to be the lightest. Themost recent dynamical calculations of the isovector sec-tor suggest a pair of 1 − + states, with the 0 + − and 2 + − states similar in mass to the heavier spin-one state. Cal-culations for the isoscalar states are currently underway,and preliminary results tend to agree with the isovectorspectrum. Work is also underway to use lighter quarkmasses. These masses are measured by quoting the pionmass. Current work has pushed this to 390 MeV, and260 MeV is in progress. Calculations at the physical pionmass may be within reach.While not supported by LQCD calculations, othermodels suggest that exotic-quantum-number multiquarkstates could exist as members of an 18 ⊕
18 of SU(3). Ex-pected J P C are 1 − + and 0 −− , where the spin-one statesare expected to be the lightest. The spin-zero states maybe similar in mass. However, in order for these multi-quark states to have finite widths, some additional bind-ing mechanism needs to be present to prevent them fromsimply falling apart into pairs of mesons.Measurements of the J P C s, multiplet structure, anddecays can be used to distinguish between these hybridand multiquark states. However, to do this requiresthe observation of multiple members of a given multi-plet as well as observation of states of different J P C .Experimental results have provided evidence for three J P C = 1 − + isoscalar states, the π (1400), the π (1600)and the π (2015).The π (1400) has been observed in both peripheralpion production and ¯ pn annihilation at rest. It has beenseen decaying into ηπ (in a p-wave), and even thoughother decay modes have been looked for (such has η (cid:48) π and ρπ ), no conclusive evidence for these has been found.While all experiments that have looked at the ηπ finalstate agree that there is signal strength in the 1 − + ex-otic wave, the interpretation of this signal is controver-sial. Explanations exist for the pion production data thatdescribe the exotic wave as a non-resonant backgroundphase, or produced by interference with non-resonantprocesses. Unfortunately, these explanations have notbeen tested against the ¯ pn data.If the π (1400) is resonant, it is difficult to explain it asa hybrid meson. It mass is too low, and its single decayappears inconsistent with state being part of an SU(3)nonet. Describing the π (1400) as a multiquark state isa more natural explanation. However, in reviewing allthe experimental evidence, as well as the non-resonantdescriptions of the 1 − + signal, we feel that the π (1400)is not resonant.The most extensive experimental evidence is for the π (1600). It has been observed in four different decaymodes, η (cid:48) π , b π , f π and ρπ , by several experiments.Consistent results between E852 and VES are found forthe first three decay modes, and from ¯ pp annihilation inflight for the b π mode. However, the ρπ decay is contro- versial. This mode has been observed by two groups, butnot by two others. In one (VES), the strength is reportedin the exotic wave, but they are unable to confirm that itis resonant. However, because not all physical constraintswere used, their conclusions may be weaker than theirdata would suggest. A second group, E852-IU, explainsthe π (1600) as feed through from the stronger π (1670)state. However, while the intensity of the π (1600) doesdepend on the decays of the π (1670), the phase differ-ence between the two states does not. This can be in-terpreted as either feed through from the π (1670), or aresonant π (1600) being absorbed by the π (1670). Toresolve this controversy will likely require a multi-channelanalysis in which physics beyond a simple isobar pic-ture is included. Even with this controversy about the ρπ decay mode, we feel that the experimental evidencedoes support a resonant π (1600). However, confirma-tion with higher statistics would be helpful.Identification of the π (1600) as the lightest hybrid isnot inconsistent with both model predictions and LQCDcalculations, although some might argue that its mass issomewhat low. The current observations and measure-ments are also consistent with a multiquark interpreta-tion, although our feeling is that this is less likely. Obser-vation of the isoscalar partners of this state would helpto confirm its hybrid nature. Unfortunately, models pre-dict their decays into channels that are experimentallydifficult to analyze.The evidence for the π (2015) is much more limited.It has been seen by one experiment in two decay modeswith very limited statistics while a second experiment(VES) does not see evidence for this state. What littleis known about this state makes it a good candidate fora hybrid meson, but confirmation is clearly needed. Ifboth the π (1600) and the π (2015) do exist, then the π (2015) may be a radial excitation of the π (1600). Aresult which is consistent with the most recent latticecalculations. As with the π (1600), observation of theisoscalar partners to this state are important.Beyond the η and η (cid:48) partners of the π states, thecrucial missing pieces of the hybrid puzzle are the other J P C -exotic nonets, 0 + − and 2 + − . Here, there is a singlehint of an h state near 1 . η decays, those of these othernonets are also challenging, and to date, all the data thatcould be used in these searches has come from pion pe-ripheral production. Definitive observation of these othernonets would provide the missing information to confirmthe gluonic excitations of QCD. Fortunately, there willsoon be four experimental programs running (COMPASSat CERN, BES III in Beijing, PANDA at GSI and GlueXat Jefferson Lab) that can provide new information onthese issues.1 ACKNOWLEDGMENTS
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