1. Some uses of Padé approximants: The VFF J. J. Sanz Cillero Some uses of Padé approximants: The vector form-factor J.J. Sanz-Cillero ( UAB – IFAE ) In collaboration with P. Masjuan and S. Peris [ arXiv:0807.4893 [hep-ph] ] Durham, September 26 th 2008

2. Some uses of Padé approximants: The VFF J. J. Sanz Cillero Our goal: Description of the  - 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)

3. Some uses of Padé approximants: The VFF J. J. Sanz Cillero The method: Padé approximants We will 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 mimick them [Masjuan, SC & Virto’08] T 0 0 (s) in LSM

4. 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

5. 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

6. 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

7. 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: This allows to use space-like data from higher energies (but not info from Q 2 = ∞) q 2 = - Q 2

8. 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: This allows to use space-like data from higher energies (but not info from Q 2 = ∞) Padé poles: rather more related to bumps of the spectral function than to hadronic poles in the complex plane (resonances?) As a remark: From this perspective, VMD [ F(Q 2 ) = (1+Q 2 /M 2 ) -1 ] is just a Padé P 0 1, the 1 st term of a sequence P L 1 q 2 = - Q 2

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

10. Some uses of Padé approximants: The VFF J. J. Sanz Cillero Alternative determination with competitive accuracy : The achieved accuracy: competitive with some of the best present determinations of the LECs. This analysis shows that the Padé approximants are a useful tool: - Alternative independent determinations - Efficient and systematic - Simple, quick and cheap to compute - Allows to use info from higher energies (Taylor expansion doesn’t)

11. Some uses of Padé approximants: The VFF J. J. Sanz Cillero Testing Padé analyses through a model

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

13. Some uses of Padé approximants: The VFF J. J. Sanz Cillero We now generate 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 %

14. Some uses of Padé approximants: The VFF J. J. Sanz Cillero Analysis of experimental data

15. Some uses of Padé approximants: The VFF J. J. Sanz Cillero PADÉ APPROXIMANTS [L/1] a 1 = 1.92 ± 0.03 GeV -2 a 2 = 3.49 ± 0.26 GeV -4 = 6 a 1 =0.4486 ± 0.008 fm 2 The [L/1] pole s p always lies in the range M  2 ± M    The coefficients evolve and then stabilize The Padé tends to reproduce the  peak line-shape but, obviously, no complex resonance pole can be recovered

16. 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!!) P10P10 P11P11 P12P12 P13P13 P14P14 JLAB, NA7, Bebek et al.’78, DESY’79, Dally et al.’77

17. Some uses of Padé approximants: The VFF J. J. Sanz Cillero Other complementary analyses: [ see Masjuan’s talk ] PA 2 L PT 1 L PT 2 L (  ’) PT 2 L (  ’’) PT 3 L (  ’  ’ ’) PP 1,1 L (  ) a 1 (GeV -2 ) 1.924 ± 0.0291.90 ± 0.031.902 ± 0.0241.899 ± 0.0231.904 ± 0.0231.902±0.029 a 2 (GeV -4 ) 3.50 ± 0.143.28 ± 0.093.29 ± 0.073.27 ± 0.063.29 ± 0.093.28 ± 0.09 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 ( =6 a 1 ): This work C’04TY’05BCT’98PP’01Boyle’08 (fm 2 ) 0.445±0.002± 0.0070.435±0.0050.432±0.0010.437±0.0160.430±0.0120.418±0.031 a 2 (GeV -4 ) 3.30±0.03± 0.33….3.84±0.023.85±0.603.79±0.04….

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

19. Some uses of Padé approximants: The VFF J. J. Sanz Cillero Padé sequences  Simple and systematic approximation The Padés allow  Obtaining low energy parameters  To use low-energy data + higher energy info Useful tool for data analysis: other form-factors, scatterings, extrapolations... As simple as a Taylor expansion, but with a wider convergence = 0.445 ± 0.002 sta ± 0.007 sys fm 2, a 2 = 3.30 ± 0.03 sta ± 0.33 sys GeV -4