Pad Theory and Phenomenology of Resonance Poles J.J. - PowerPoint PPT Presentation

CIRM, September 28th 2009 Padé Theory and Phenomenology of Resonance Poles J.J. Sanz-Cillero ( UAB – IFAE ) Padé Theory and Resonance Poles J. J. Sanz Cillero

Determining hadronic parameters •QCD observables � Determination of renormalized couplings But, with resonances, couplings of what lagrangian? No general agreement about the right formulation (if any) •Alternatively, resonance pole positions (in the complex plane - II-Riemann sheet ) � Important: Universal for all processes with the same quantum numbers [ do not depend on a Lagrangian realization ] � much model dependence in some cases •Obtaining these hadronic properties ( what theory do we use? ) [Leutwyler’07 ] •The data are on the real “energy” axis: Extrapolation to the complex plane � Non-trivial E.g., the σ -pole (I=J=0 ππ -scat.) Padé Theory and Resonance Poles J. J. Sanz Cillero

Resonant amplitudes: Padé-Approximants as a model independent description 1.) Example of amplitudes with resonant spectral functions : ππ -VFF and the extraction of <r 2 > V , c V The (but not to recover the ρ -meson pole) � From Euclidean data [ Masjuan, Peris, SC’08] 2.) Padés have been sometimes used as unitarizations (in a pretty sloppy way) : � Either improper determinations of the poles BUT!! � Or inaccurate values for LECs [ Masjuan, SC, Virto’08] � PA’s may recover the poles in a theoretically safe way 3.) However, properly used � M R and Γ R determinations [ SC, work in progress ] Padé Theory and Resonance Poles J. J. Sanz Cillero

1.) Padés and the space-like VFF Some uses of Padé approximants: The VFF J. J. Sanz Cillero

Goal: [ Masjuan, Peris, SC’08 ] Description of the ππ - VFF • in the space-like [ Q 2 = -(p-p’) 2 > 0 ] • To build an approximation that can be systematically improved NOT our aim: • To extract time-like properties (e.g. mass predictions) • To describe the physics on the physical cut • Not a large-N C approach (here, physical N C =3 quantities) Some uses of Padé approximants: The VFF J. J. Sanz Cillero

The method: Padé approximants •We build Padés P N M (q 2 ) =Q N (q 2 ) / R M (q 2 ) : P N M ( q 2 ) - F( q 2 ) = O ( ( q 2 ) N+M+1 ) around q 2 =0 • What’s new with respect to a Taylor series F(z) ≈ a 0 +a 1 z + a 2 z 2 +… ? � The polynomials, unable to handle singularities (branch cuts…) ----These set their limit of validity � The Padés, partially mimic them T 0 0 (s) in LSM [Masjuan, SC & Virto’08] Some uses of Padé approximants: The VFF J. J. Sanz Cillero

• Thus, in many cases, the Padés work far beyond the range of convergence of the Taylor expansion: q 2 = - Q 2 Some uses of Padé approximants: The VFF J. J. Sanz Cillero

• Thus, in many cases, the Padés work far beyond the range of convergence of the Taylor expansion: q 2 = - Q 2 Some uses of Padé approximants: The VFF J. J. Sanz Cillero

• Thus, in many cases, the Padés work far beyond the range of convergence of the Taylor expansion : q 2 = - Q 2 Some uses of Padé approximants: The VFF J. J. Sanz Cillero

• Thus, in many cases, the Padés work far beyond the range of convergence of the Taylor expansion : q 2 = - Q 2 • This allows to use space-like data from higher energies (but not info from Q 2 = ∞ ) Some uses of Padé approximants: The VFF J. J. Sanz Cillero

• Thus, in many cases, the Padés work far beyond the range of convergence of the Taylor expansion : q 2 = - Q 2 • This allows to use space-like data from higher energies • Padé poles: rather more related to bumps of the spectral function than to hadronic poles in the complex plane (resonances?) [ F(Q 2 ) = (1+Q 2 /M 2 ) -1 ] • As a side-remark: From this perspective, VMD 1 , the 1 st term of a sequence P L is just a Padé P 0 1 Some uses of Padé approximants: The VFF J. J. Sanz Cillero

Input: [ Q 2 =0.01 – 10 GeV 2 ] •The available space-like data • Qualitative knowledge of the ππ -VFF spectral function ρ (s): � essentially provided by the rho peak suggesting the use of P L 1 Output: π π <r 2 > V c V •The low-energy coefficients: and Some uses of Padé approximants: The VFF J. J. Sanz Cillero

Playing with a phenomenological-model [ Masjuan, Peris & SC’08 ] • We consider a VFF phase-shift, • And we recover the VFF through Omnés with the right threshold behaviour given by [ Guerrero & Pich’97 ] [Pich & Portolés’01 ] INPUTS: Some uses of Padé approximants: The VFF J. J. Sanz Cillero

• We generated an emulation of data • Fitting these data through a [L/1] Padé, which at low energies recover the taylor coefficients a k : F(q 2 ) = 1 + a 1 q 2 + a 2 q 4 + a 3 q 6 + … • This leads to Padé predictions which can be compared to the exact KNOWN results: � ±1.5 % � ±10 % Some uses of Padé approximants: The VFF J. J. Sanz Cillero

Experimental data: PADÉ APPROXIMANTS [L/1] • The coefficients evolve and then stabilize a 1 = 1.92 ± 0.03 GeV -2 <r 2 > = 6 a 1 =0.4486 ± 0.008 fm 2 [ Masjuan, Peris & SC’08 ] a 2 = 3.49 ± 0.26 GeV -4 2 ± M ρ Γ ρ • The [L/1] pole s p always lies in the range M ρ The Padé tends to reproduce the ρ peak line-shape but, obviously, no complex resonance pole can be recovered Some uses of Padé approximants: The VFF J. J. Sanz Cillero

• The sequence [L/1] converge to the physical form-factor F(t) � in the data region � but it diverges afterwards (like (Q 2 ) L-1 ) • The Padés allow the use data from higher energies (the Taylor expansion don’t!!) P 1 4 P 1 3 P 1 2 P 1 0 JLAB, NA7, P 1 1 Bebek et al.’78, DESY’79, Dally et al.’77 Some uses of Padé approximants: The VFF J. J. Sanz Cillero

Other complementary analyses: L ( ρ,ρ ’) L ( ρ,ρ ’’) L ( ρ ) L L L ( ρ,ρ ’ ,ρ ’ ’) PA 2 PT 1 PT 2 PT 2 PT 3 PP 1,1 a 1 1.924 ± 0.029 1.90 ± 0.03 1.902 ± 0.024 1.899 ± 0.023 1.904 ± 0.023 1.902±0.029 (GeV -2 ) a 2 3.50 ± 0.14 3.28 ± 0.09 3.29 ± 0.07 3.27 ± 0.06 3.29 ± 0.09 3.28 ± 0.09 (GeV -4 ) which, after combination, leads to a 1 = 1.907 ± 0.010 sta ± 0.030 sys GeV -2 , a 2 = 3.30 ± 0.03 sta ± 0.33 sys GeV -4 • Comparison with other determinations ( <r 2 >=6 a 1 ): [Pich,Portolés’01] [Masjuan,Peris,SC’08] [Caprini’04] [Troconiz, [Bijnens et al’98] [Boyle’08] Yndurain’05] <r 2 > (fm 2 ) 0.445±0.002± 0.007 0.435±0.005 0.432±0.001 0.437±0.016 0.430±0.012 0.418±0.031 3.30±0.03± 0.33 …. 3.84±0.02 3.85±0.60 3.79±0.04 …. a 2 (GeV -4 ) Some uses of Padé approximants: The VFF J. J. Sanz Cillero

2.) A critical look on Padé unitarizations Some uses of Padé approximants: The VFF J. J. Sanz Cillero

[ Identical result obtained from disp. relations + χ PT matching of the l.h.c. ] [Thanks to J.Virto for his help with the slides] Some uses of Padé approximants: The VFF J. J. Sanz Cillero

• What info is lost when fixing Re[ t -1 ] with χ PT? • Amplitudes violate crossing • Properties of the σ (mass and width) slightly different to those from Roy Eqs. • Still, not a bad numerical approximation � Reason why, not fully understood Some uses of Padé approximants: The VFF J. J. Sanz Cillero

The σ in the LSM A counter-example: [ Masjuan, Virto, SC’08 ] 1.) Computation of the actual σ− pole , with M σ the renormalized mass parameter in the Lagrangian Some uses of Padé approximants: The VFF J. J. Sanz Cillero

2.) Low-energy limit of the LSM Some uses of Padé approximants: The VFF J. J. Sanz Cillero

3.) Unitarization of the low-energy LSM Some uses of Padé approximants: The VFF J. J. Sanz Cillero

Some uses of Padé approximants: The VFF J. J. Sanz Cillero

Some uses of Padé approximants: The VFF J. J. Sanz Cillero

Some uses of Padé approximants: The VFF J. J. Sanz Cillero

Some uses of Padé approximants: The VFF J. J. Sanz Cillero

3.) Model independent determination of resonance poles Some uses of Padé approximants: The VFF J. J. Sanz Cillero

Analyticity properties •Simplest case: Analytical function f(x) in a disk B δ (0) δ � Then, the Taylor series f(x) for N � ∞ converges to [ with a k =f (k) (0) / k! ] •Experimentally, if you have data in the range of B δ (0) f(0) , f’(0) … can be extracted from successive polynomial fits P N (x) Padé Theory and Resonance Poles J. J. Sanz Cillero

What if it is analytical but for a single pole ? •De Montessus’ theorem [1902] : If one constructs a series of P N single-pole Padé Approximants 1 , δ s p � Then the sequence of Padés f(x) when N � ∞ unif. converges to on any compact subset D={x , |x| ≤ δ , x ≠ s p } � Hence, also the Padé pole x p =a N /a N+1 � s p Padé Theory and Resonance Poles J. J. Sanz Cillero

•Experimentally, one is not provided with the derivatives f(0), f’(0) …. { x j , f j , Δ f j } but with experimental points •We then use now the rational functions P N 1 (x) as fitting functions (as we did before with the polynomials) { f (k) (0) } � and hence, s p •P N 1 (x) gives an estimate of the series of derivatives N � ∞ which are expected to converge to the actual ones for Padé Theory and Resonance Poles J. J. Sanz Cillero