Topics in meson spectroscopy for Compass David Bugg

Topics in meson spectroscopy for Compass David Bugg, Queen Mary, London (i) Look for the I=0 JPC=1 -+ partner of p 1(1600/1662) (ii) Measure s(4 p)/s(2 p) from 1200 -2000 Me. V for JP=0+, 1 -, 2+. Sort out f 0(1370), f 2(1565) and r(1450). Check if f 2(1810) exists or is really f 0(1790). (iii) Check if p 1(1405) is resonant or just a threshold cusp.

The p 1(1600/1660) looks a good hybrid candidate. It ought to have an I=0 partner of similar mass. The only likely decay mode is to a 1(1260)p: JPC = 1++ + 0 -+ = 1 -+. This is NOT easy because no angular dependence: there are only the r and a 1 signals to go on, and the a 1 is broad. However, other JPC will have angular dependence. There exist possible partners with JPC = 2 -+: h 2(1860) and p 2(1880); also p(1800), 0 -+

I = 0, C = +1 states from Crystal Barrel in flight; 10 sets of data fitted simultaneously. See Phys. Rep. 397 (2004) 257. f 2(1810) dubious h 2(1860) extra state; not ss f 0(1790) not f 2(1810)

J/y -> w. KK 1710 -> KK J/y -> f. KK 1790 -> pp J/y -> wpp 1710 in pp <11% at 95% CL J/y -> fpp No 1790 -> KK

Crystal Barrel in flight also observe f 0(1770 +- 12) in (hh)p 0, but the PDG lists it as f 0(1710)! BES II also observe a striking peak in wf at 1812 +- 22 Me. V with G = 105 Me. V and JP = 0+. But the PDG lists it under X(1835) (recently confirmed) which is observed in hpp; this cannot possibly have JP = 0+!

Next topic. Branching ratios to rr, ww and ss compared with pp over the mass range 1200 to 2200 Me. V are important, for the simple reason that they rise rapidly with mass and are the dominant cross sections. This is not very glamorous, but is presently the limitation in understanding f 0(1300/1370), f 0(1500), r(1450) and f 2(1565). It is necessary to isolate the OPE contribution to all of these from their t-dependence. The problem with exchanges at large t is that their spin content is not well understood, making the interpretation of results obscure. Detail : rr and ww channels are related by SU(2) g^2(rr)/g^2(ww) = 3; but their phase space factors differ after folding their widths into calculations.

Words of advice on parametrising amplitudes. 1) Do NOT use the K-matrix. It has many problems (which experts know). The main problem is that it ASSUMES all channels are known, but that is not usually true. If strong channels like 4 p and not known well – chaos! 2) K-matrix elements do NOT necessarily add (as is usually assumed). Phases could add instead! K = tan d; why not K = (KA + KB)/(1 – KAKB)? 3) Extended Unitarity is experimentally disproved, see Eur. Phys. J C (2008) 73.

SHARP THRESHOLDS. Many resonances are attracted to thresholds. Me. V f 0(980) and a 0(980) -> KK f 2(1565) -> ww f 0(1790/1812) -> wf X(3872) -> D(1865)D*(2007) Y(4660) -> y’(3686)f 0(980) Lc(2940) -> D*(2007)N K 0(1430) -> Kh’ ? K 1(1420) -> KK* 991 1566 1801 3872 4666 2945 1453 1388

BW = N(s)/D(s) where D(s)=M 2 - s – i. MGtot(s) D(s)= M 2 - s - Si Pi(s) Im Pi = gi 2 ri(s)FFi (s) Re Pi = 1 P p ds’ Im Pi(s’ ) (s’ – s) thri The full form of the BW is D(s)=M 2 - s – Re P(s) – i Im P(s) phase space

f 0(980) -> KK as an example cusp FF = exp(-3 k 2) (R=0. 8 fm) Re P acts as an effective attraction pulling the resonance to the threshold.

The parameters of f 0(980) are accurately known from BES 2 data on J/y -> fpp and f. KK. One can play the game of varying M of the Breit-Wigner and evaluating the pole position; illustrative numbers M(Me. V) Pole (Me. V) 500 806 – i 76 700 899 – i 59 900 987 – i 31 956 1004 – i 21 990 1011 - i 4 1050 1009 – i 28 1100 979 – i 69

f 0(1370) as an example: The peak in pp is given by the Breit-Wigner folded with phase space. Likewise for 4 p. The peak in pp is at 1309+-1+-15 Me. V, but in 4 p is at 1395 +25? Me. V: Euro. Phys. J C 52 (2007). This explains the large variation of M, G in PDG tables from differing analyses. NB: Phase variation horribly wrong with constant G

The f 2(1565) is an even more serious case. It is attracted to the ww threshold by the sharp cusp. The f 2(1640) is the SAME resonance when the Breit-Wigner lineshape is folded with ww phase space, Baker et al. , Phys. Lett. B 467 (1999) 147. There is a second serious problem: in the Flatte formula, f a 1/[M 2 – s – i g 2 r(s)], the phase space factor r is analytically continued below the ww threshold. Unless it is cut off strongly by a form factor, it creates strong interferences with f 2(1270). Those interferences account for f 2(1430) of the PDG. The r(1450) has VERY uncertain parameters if treated as a Breit -Wigner of constant width. Could have a mass as low as 1250 Me. V? ?

The PDG lists two closely separated h(1405) and h(1475). The h(1405) decays to hs, a 0(980)p and k. K with L=0 and is unaffected by centrifugal barrier effects. The h(1475) decays to K*K, whose threshold is at 1394 Me. V. This decay has L=1, so the intensity ~ p 3, shifting the peak up strongly. All decays can be fitted well with a single h(1440) including dispersive effects. See ar. Xiv: 0907. 3015 and 0907. 3021

The cusp effect may explain away p 1(1405). It lies very close to sharp thresholds for b 1(1235)p and f 1(1285)p, which both have quantum numbers of p 1(1600/1660). It would be a valuable exercise for a phenomenologist to see if data over this whole mass range can be fitted in terms of (a) coupling constants of p 1(1600), (b) their phase space with (c) appropriate form factors v. k (centre of mass momentum for each channel). Essentially the opening of these channels introduces inelasticities at the b 1(1235)p and f 1(1285)p thresholds and forces the hp? and rp amplitudes to move on the Argand diagram in a similar way to a resonance.

General remark: Oset, Oller et al find they can generate MANY states from meson exchanges. This is along the lines of Hamilton and Donnachie, who found in 1965 that meson exchanges have the right signs to generate P 33, D 15 and F 15 baryons. Suppose contributions to the Hamiltionian are H 11 and H 22; the eigenvalue equation is H 11 V Y = E Y V H 22 H 11 refers to q-q; H 22 to s, t, u exchanges. V is the mixing element between them. Two solutions: E= (E 1+E 2)/2 + [(E 1 -E 2)2 - |V|2]1/2 Two KEY points: mixing LOWERS the ground state, hence increasing the binding. The eigenstate is a linear combination of qq and meson-meson.

There is in fact an exact mathematical corresponance between this mixing and the covalent bond in chemistry, where states line benzene are well-known linear combinations of more than one configuration. see hep-ph/1001. 1712; J. Phys. G 37 (2010) 055002 X(3872) most likely has JPC=1++, and is a linear combination of cc and DD*; very recently, ar. Xiv 1103. 5363, I have stumbled upon evidence that X(3915), observed in decays to ww, also has JPC = 1++. If this is confirmed, it is the first example of strong mixing generating two eigenstates close in mass, but with orthogonal combinations of cc and meson-meson. The same evidence explains Y(4140) of CDF decaying to f. J/Y.

CONCLUSION: there is still lots to do and learn in meson spectroscopy. I doubt that heavy meson spectroscopy can go beyond narrow states, but it is important to complete light meson spectroscopy and baryon spectroscopy and learn what we can about hybrids and glueballs. Good luck! The spectroscopy of I=1, C = +1 states and C=-1 states with I=0 and 1 could be completed by measuring polarisation in pp -> 3 p 0, hhp, hp; wp and whp 0; wh and wp 0 p 0 at the forthcoming Flair facility. It just needs an extracted p beam of 5 x 104 p/s.