1. Partial Wave Analysis What is it Relation to physical properties Properties of the S-matrix Limitation and perspectives Examples : peripheral production of 2- and 3-particle final states Adam Szczepaniak Indiana University

2. Kinematical variables (s cd,  cd,  cd,  ) Production amplitudes depends on fit parameters (output) Production amplitudes (input) (mass, t,  bins)

3. This is best case scenario Y(x i ) includes it all : kinematics and dynamics, There is nothing to fit ! Usually somewhere in between … depending on how much we know about the amplitude : Worst case scenario Know nothing Know everything !

4. Example  - p !  0 n

5. You do it in all possible way to study systematics 0 physics input “maximal’ ambiguity some physics input “moderate” ambiguities … the less you know the more ambiguous the answer … know everything no ambiguities

6.  0  0 spectrum f 2 (1270)  (400-1200) (J. Gunter et al.) 2001  - p !    0 n

7.  - p !  0 n Assume a 0 and a 2 resonances (A.Dzierba et al.) 2003 ( i.e. a dynamical assumption)

8. .. so, how much we know about the S-matrix … dynamics is much harder …  Heisenberg-Mandelstam program (ca. 1960-1970)  QCD (ca. 1970)  Jlab upgrade (ca. Now !) kinematical constraints

9. HM: reconstruct S given Mandelstam representation and the unitarity condition) S – matrix has specific analytic properties (causality) ImE ReE G(E) is analytic for ImE>0 Unitarity (conservation of energy)

10. Mandelstam hypothesis : 1. There is an analytical representation for S (which includes poles and cuts of physical origin) 2. Given a representation a unique solution can be found (so far unknown) (not true)

11. Complete representation : complete dynamics Non-relativistic example : need the potential Relativistic example :       N N _   K K _  N N _ K _

12. … a “given” representation can have multiple solutions ! (incomplete knowledge of dynamics) ? 2a k k’ Relativistc example : (Castillejo,Daliz,Dyson (CDD) poles) 1 - - 1 - Have the same representation Non-relativistic example : Blaschke product fl=fl= fl=fl=

13. Low energy : Effective range expansion (low energy) Two body unitarity Small number of (renormalized) parameters QCD input Good news : High energy : Regge behavior Asymptotic freedom

14. Illustration :  (S=I=0)  only (no KK, no resonances)   +KK 2 Resonaces @ ~1.3, 1.5 GeV

15. Regge poles s>>t t

16. … combine low (chiral) and high energy information s>>t,M t Regge Chiral

17.    (18GeV) p  X p   - p   ’   p ~ 30 000 events N events = N(s, t, M   ) p p -- a2a2  -- t M   s

18. DATA (from E852)  - p !  0 p [adam@mantrid00 data]$ ls -l total 835636 -rw-r--r-- 1 adam adam 87351564 May 10 10:40 ACC -rw-rw-r-- 1 adam adam 4882894 May 10 10:39 DAT -rw-r--r-- 1 adam adam 762596488 May 10 10:42 RAW [adam@mantrid00 data]$ E p x p y p z (E 2 – p i p i ) 1/2 Event 1 5.673920 -0.269088 -0.463492 5.646950 0.1345 (m  ) 12.561666 0.458374 0.228348 12.539296 0.5471 (m  ) 1.001964 -0.260447 0.202660 0.112333 0.9393 (m N ) (recoil) 18.299278 -0.071161 -0.032484 18.298578 0.1396 (m  ) (beam) Event 2 7.348978 -0.405326 0.602844 7.311741 11.217298 -0.360824 -0.456940 11.188789 1.255382 0.718019 -0.177694 0.382252 18.883385 -0.048131 -0.031789 18.882782 … PWA determined by maximizing likelihood function over an even sample …

19. Assume BW resonance in all, M= § 1,0, P-waves  1 (900 – 5GeV) emerges Intensity in the weak P-waves is strongly affected by the a 2 (1320), strong wave due to acceptance corrections Results of analysis (part 1)  - p !  0 n

20. 1 BW resonance in P + a 2 (1320) 2 BW resonances in D + a 2 (1800) = ? E852  ’   analysis

21.  - p !  - p Results of coupled channel analysis of  - p !  ’  - p D S P D P

22. |P + | 2  (P + )-  (D + )

23. Peripheral production of hybrid mesons Quarks Excited Flux Tube Hybrid Meson Exotic like So only parallel quark spins lead to exotic J PC

24.  -->  (J PC =1 -- ) -->  1 (J PC =1 -+ ) “pluck” the string VMD Photo production enhances exotic mesons   p N Non-exotic mesons, ,K,  Exotic QCD ! Exotic :  : quark with spins aligned  : quark with spins anti-aligned

25. pion  - p ! X 0 n exotic/non-exotic » 1 exotic/non-exotic » 0.1 Szczepaniak & Swat (01) photon  p ! X + n Implications for exotic meson searches Possible narrow QCD exotic (M=1.6 GeV) (E852  - p !  +  -  - p)

26. OPE Photo production enhances exotic mesons  -->  (J PC =1 -- ) -->  1 (J PC =1 -+ ) “pluck” the string (S=1,L QQ =0->L g =1) 1 -+ exotic : S=1, L=1 VMD Afanasev, AS, (00) Agrees with Condo’93 2

28. Problems with the isobar (sequential decay) model 00 A(s 1,s 2,M) = C J LS (M) / D 1 (s 1 ) ++ -- -- a 2 - (1320) (J) Breit-Wigners1s1 (1) (2) (3) M C J LS L S F(s 1,s 2,M) = f 1 (M)/D 1 (s 1 ) + f 2 (M)/D 2 (s 2 ) 1  2 Independent on 2-particle sub-channel energy : violates unitarity !

29. 3 1 2 1 3 2 3 f2f2 f1f1 1 3 2 1 2 3 B-W : unitary in sub-channel energy s2s2 s1s1 = + f 2 (s 2+ ) – f 2 (s 2- ) = 2i  (s 2 ) P.V. s f 1 (s 1 )/D 1 (s 1 ) f i = f i (s i,M) Has to depend on 2-particle energy

30. F 1 (s 1,M) i = H(s 1,M) ij C j (M) i,j= , 

31. 3  sample Analyzed Available !

32. Need theory input to minimize (mathematical) ambiguities and to understand systematic errors Need more theory input to determine physical states (coherent background vs resonances) Summary There is lots of data to work with and there will be more especially needed for establishing gluonic excitations  1 (1400) and  1 (1600) in  ’  are most likely due to Residual interactions much like the  meson in the  S-wave