the scalar mesons proposal stefan spanier university of tennessee, knoxville

1Stefan Spanier, Scalar Mesons, PDG Meeting 2008

The Scalar MesonsProposal

Stefan SpanierUniversity of Tennessee, Knoxville

PDG Collaboration meeting 10/10/2008CERN, Geneva

Scalar Meson Review: Claude Amsler, Nils Tornqvist, Stefan Spanier


below 2 GeV

• 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

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

mixtures: ann + bss + cqqqq + dglue + _ _ _ _

• Scalar Mesons are special


• Scalar Meson Spectrum

2+


• Model Dependent Analysis

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

I=0 () S-Wave

(600) ?

f0(980)

f0(1370)

f0(1500)

T112

KK thre

shol

d

_

[EP A16, 229 (2003)]

• 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


• Formalism

Try to start from dominance by 2-body interaction

• Watson theorem: same phase motion in T and F in elastic range

• Adler zero: at m 0 for p=0: T = 0 near threshold; also/where for F ?

• K-matrix poles % T-matrix poles in unphysical sheet of complex energy plane

a

b

c

d

ll

R

L

l

Spectator ?

d

R c

2-body scattering production / decay

T = R K F = R P P-vector = Q T Q-vector

e.g.

In very formal way using K-matrix

R = (1-iK)-1

2-body PS

Dynamic amplitude requirements:- Lorentz invariant- 2-body Unitarity- Analytic / Crossing


• I=1/2 Scalar Data from:

K-p K-+n

and

K-p K0-pNPB 296, 493 (1988)

No data below 825 MeV/c2

K Scattering LASS

0.7 0.9 1.1 1.3 1.5 MK (GeV)

150

100

50

0

-50

Ph

ase

(deg

rees

)

0.7 0.9 1.1 1.3 1.5MK (GeV)

• Most information on K-+ scattering comes from the

LASS experiment (SLAC, E135)

K’ threshold

• use directly in production if re-scattering is small• require unitarity approach …

LASS parameterization

• Disentangle I=1/2 and I=3/2 with K++ [NPB133, 490 (1978)]

PenningtonChPT compliant


• I=1/2 Scalar

LASS experiment used an effective range expansion toparameterize the low energy behavior: : scattering phase q cot = + a: scattering length b: effective range q: breakup momentumTurn into K-matrix: K-1 = cot

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- () 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

1 b q2

a 2___ ______

a m g0

2 + a b q2 m02 – m2

___________ ___________


• I=1/2 Scalar

D+ (K-+) +

• K system dominated by K*(892)• Observe ~15% forward-backward asymmetry in K rest frame• Hadronic phase of 45o corresponds to I=1/2 K 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 !

FOCUS

Reconstructed events: ~27,000D+

K-

+

csW+

[PLB 621, 72 (2005)][PLB 535, 43 (2002)]


• I=1/2 Scalar E791

K*(892)

K*(1430)

D+ K-++

#15090

• Fit with Breit-Wigner (isobar model):

M = (797 19 42) MeV/c2

eVcD+K-

+

W++

+

+

K-

W+

• Fit with Breit-Wigner + energy- independent fit to K 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 eik bin-by-bin (40)

[PRL 89, 121801 (2002)]

[PR D73 (2006) 032004]

Extracted phase motion from both analysesare similar.


• I=1/2 Scalar E791

K’ threshold

-75o

… but differs from LASS elastic scattering

T11 from LASS ( same poles ?)

Constraint: Q smooth functions

Adler zero s0I removed

A(s) eis = F1/2(s) + F3/2(s) , s = mK2

FI(s) = QI(s) eiIT11

I(s)

s – s0I

Q-vector approach with Watson:

[Edera, Pennington: hep-ph/0506117]

|FI |

big !

• Quasi-two body K interaction (isobar model ) broken ?

if not

• Watson theorem does not apply ?

if not• Isospin composition I=1/2 % I=3/2 in D decay same as in K K?


Example: Input for B CP – physics - B0 J/ K* study K 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. B0K0K+K-

- hadronic phase for CP angle in BDK from D-Dalitz plot

• 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 modulus often 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)

• Summary - Role that PDG can play …

• Proposal: Archive (pdglive) scalar amplitude information.

the scalar mesons proposal stefan spanier university of tennessee, knoxville

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