The Scalar Mesons Proposal Stefan Spanier University of Tennessee, Knoxville

1. The Scalar Mesons Proposal Stefan Spanier University of Tennessee, Knoxville PDG Collaboration meeting 10/10/2008 CERN, Geneva Scalar Meson Review: Claude Amsler, Nils Tornqvist, Stefan Spanier

2. Scalar Mesons are special _ _ _ _ mixtures: ann + bss + cqqqq + dglue + below 2 GeV Present a long standing puzzle Measurements of phase motions since 1970s • Quantum number of vacuum • – fundamental role in chiral symmetry breaking • Glueball • Low receiving end of hadronic reactions • Experimental Challenges: - Broad states - Strongly overlapping - Close to several thresholds - Extra dynamical features (left-hand cuts, Adler zero) - Featureless angular distributions (like background) - Often covered by tensor or/and vector mesons

3. Scalar Meson Spectrum 2+

4. Model Dependent Analysis I=0 (pp) S-Wave [EP A16, 229 (2003)] T112 f0(980) KK threshold f0(1370) _ s(600) ? f0(1500) • Non-perturbative regime • Static models • Dynamics: production  decay need both to piece puzzle together No simple Breit-Wigner fits • combined sample analysis • coupled channel analysis • Do not describe with individual resonances but parameterize complete (partial wave) amplitude • In optimum case the underlying resonances show up as same poles in the unphysical sheet of the complex energy plane  evaluate denominator of complex amplitude for singularities

5. Formalism a R c l l b 2-body scattering production / decay d c R R = (1-iKr)-1 r =2-body PS l d L Spectator ? e.g. T = R K F = R P P-vector = Q T Q-vector In very formal way using K-matrix  Try to start from dominance by 2-body interaction Dynamic amplitude requirements: • Lorentz invariant • 2-body Unitarity • Analytic / Crossing • Watson theorem: same phase motion in T and F in elastic range • Adler zero: at mp 0 for pp=0: T = 0  near threshold; also/where for F ? • K-matrix poles % T-matrix poles in unphysical sheet of complex energy plane

6. I=1/2 Scalar 150 100 50 0 -50 Phase (degrees) 0.7 0.9 1.1 1.3 1.5 0.7 0.9 1.1 1.3 1.5 MKp (GeV) MKp (GeV) Kp Scattering LASS Data from: K-pK-p+n and K-pK0p-p NPB 296, 493 (1988) • Most information on K-p+ scattering comes from the • LASS experiment (SLAC, E135) • Disentangle I=1/2 and I=3/2 with K+p+[NPB133, 490 (1978)] Pennington ChPT compliant LASS parameterization Kh’ threshold No data below 825 MeV/c2 • use directly in production if re-scattering is small • require unitarity approach …

7. I=1/2 Scalar • LASS experiment used an effective range expansion to • parameterize the low energy behavior: • d: scattering phase • q cot d = + a: scattering length • b: effective range • q: breakup momentum • Turn into K-matrix: K-1 = r cot d • K = +and add a pole term (fits also pp annihilation data) Both describe scattering on potential V(r) (a,b predicted by ChPT) • Takes left hand cuts implicitly into account • Instead treat with meson exchange in t- (r ) and u-channel (K* ) [JPA:Gen.Phys 4,883 (1971), PRD 67, 034025 (2003)]  only K0*(1430) appears as s-pole • same phase motion, different parameterization / pole pattern • b q2 • a 2 ___ ______ a m g0 2 + a b q2 m02 – m2 ___________ __________ _

8. I=1/2 Scalar FOCUS s D+ (K-p+) m+ nm W+ K- c Reconstructed events: ~27,000 D+ p+ • Kp system dominated by K*(892) • Observe ~15% forward-backward • asymmetry in Kp rest frame • Hadronic phase of 45o corresponds to • I=1/2 Kp wave measured by LASS • required by Watson theorem in semileptonic • decay below inelastic threshold • S-wave is modeled as constant • (~7% of K*(892) Breit-Wigner at pole). • a phase of 90o would correspond to a • kappa resonance, but … • Study semileptonic D decays • down to threshold ! [PLB 621, 72 (2005)] [PLB 535, 43 (2002)]

9. I=1/2 Scalar E791 K- p+ p+ • Fit with Breit-Wigner (isobar model): p+ W+ K- Mk = (797  19  42) MeV/c2 Gk = (410  43  85) MeV/c2 D+ p+ [PRL 89, 121801 (2002)]  W+ • Fit with Breit-Wigner + • energy- independent fit to Kp S-wave • (P(K*(892), K*(1680)) and D-waves (K*2(1430))act as interferometer) • Model P- and D-wave (Beit-Wigner), • S-wave A = ak eifk bin-by-bin (40) D+K-p+p+ #15090 [PR D73 (2006) 032004] Extracted phase motion from both analyses are similar. K*(892) K*(1430)

10. I=1/2 Scalar E791 • Quasi-two body Kp interaction • (isobar model ) broken ? if not • Watson theorem does not apply ? • Isospin composition • I=1/2 % I=3/2 in D decay • same as in Kp Kp ? -75o if not Kh’ threshold |FI | Q-vector approach with Watson: A(s) eif(s) = F1/2(s) + F3/2(s) , s = mKp2 FI(s) = QI(s) eibI T11I(s) s – s0I big ! [Edera, Pennington: hep-ph/0506117] … but differs from LASS elastic scattering • T11 from LASS ( same poles ?) • Constraint: Q smooth functions •  Adler zero s0I removed

11. Summary - Role that PDG can play … • Pole structure of the amplitude in complex energy plane  PDG table • Isospin amplitude for 2-body scattering is as important • - Data on phase motion and modulus (or/and elasticity) • Parameterizations (limited number) • Compare on level of 2-body scattering (unitarity, analyticity) phase motion and modulusoften in close agreement, but underlying parameterization and/or pole structure different • Increase confidence with combined data-set coupled channel analyses increase constraints; updates extract iteratively the bare 2-body wave • Provide input for analyses that include scalars (likelihood analyses) Example: Input for B CP – physics - B0 J/y K*study Kp mass from 0.8 – 1.5 GeV/c2 - add penguin modes for New Physics Search, e.g. B0 f0 K0 - CP composition of 3-body modes, e.g. B0K0K+K- - hadronic phase for CP angle g in BDK from D-Dalitz plot • Proposal: Archive (pdglive) scalar amplitude information.