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

Padé Theory and Resonance Poles

• J. J. Sanz Cillero

Padé Theory and Phenomenology

• f Resonance Poles

J.J. Sanz-Cillero ( UAB – IFAE ) CIRM, September 28th 2009

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 ]

• Obtaining these hadronic properties

much model dependence in some cases

( what theory do we use? )

• 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.)

[Leutwyler’07 ]

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:

The ππ-VFF and the extraction of <r2>V , cV From Euclidean data (but not to recover the ρ-meson pole)

2.) Padés have been sometimes used as unitarizations

(in a pretty sloppy way) :

BUT!!

Either improper determinations of the poles Or inaccurate values for LECs

3.) However, properly used PA’s may recover the poles in a theoretically safe way

MR and ΓR determinations

[ Masjuan, Peris, SC’08] [ Masjuan, SC, Virto’08] [ SC, work in progress ]

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero

1.) Padés and the space-like VFF

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero

Goal:

• Description of the ππ-VFF

in the space-like [ Q2 = -(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-NC approach

(here, physical NC=3 quantities)

[ Masjuan, Peris, SC’08 ]

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero

The method: Padé approximants

• We build Padés

PN

M(q2) =QN(q2) / RM(q2) :

PN

M(q2) - F(q2) = O((q2) N+M+1)

around q2=0

• What’s new with respect to a Taylor series F(z) ≈ a0 +a1z + a2z2 +… ?

The polynomials, unable to handle singularities (branch cuts…)

• ---These set their limit of validity

The Padés, partially mimic them

[Masjuan, SC & Virto’08]

T0

0(s)

in LSM

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero
• Thus, in many cases, the Padés work far beyond the range
• f convergence of the Taylor expansion:

q2= - Q2

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero
• Thus, in many cases, the Padés work far beyond the range
• f convergence of the Taylor expansion:

q2= - Q2

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero
• Thus, in many cases, the Padés work far beyond the range
• f convergence of the Taylor expansion:

q2= - Q2

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero
• Thus, in many cases, the Padés work far beyond the range
• f convergence of the Taylor expansion:
• This allows to use space-like data from higher energies

(but not info from Q2=∞)

q2= - Q2

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero
• Thus, in many cases, the Padés work far beyond the range
• f convergence of the Taylor expansion:
• 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?)

• As a side-remark:

From this perspective, VMD

[ F(Q2) = (1+Q2/M2)-1 ]

is just a Padé P0

1, the 1st term of a sequence PL 1

q2= - Q2

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero

Input:

• The available space-like data

[ Q2=0.01 – 10 GeV2 ]

• Qualitative knowledge of the ππ-VFF spectral function ρ(s):

essentially provided by the rho peak

suggesting the use of PL

1

Output:

• The low-energy coefficients:

<r2>V

π

and

cV

π

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero

INPUTS:

Playing with a phenomenological-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 ]

[ Masjuan, Peris & SC’08 ]

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 ak :

F(q2) = 1 + a1 q2 + a2 q4 + a3 q6 + …

• 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]

a1 = 1.92 ± 0.03 GeV-2 a2 = 3.49 ± 0.26 GeV-4 <r2> = 6 a1=0.4486 ± 0.008 fm2

• The [L/1] pole sp always lies in the range Mρ

2 ± MρΓρ

• The coefficients evolve and then stabilize

The Padé tends to reproduce the ρ peak line-shape but, obviously, no complex resonance pole can be recovered

[ Masjuan, Peris & SC’08 ]

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 (Q2)L-1)

• The Padés allow the use data from higher energies (the Taylor expansion don’t!!)

P1 P1

1

P1

2

P1

3

P1

4

JLAB, NA7, Bebek et al.’78, DESY’79, Dally et al.’77

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero

Other complementary analyses:

PA2

L

PT1

L

PT2

L (ρ,ρ’)

PT2

L (ρ,ρ’’)

PT3

L (ρ,ρ’ ,ρ’ ’)

PP1,1

L (ρ)

a1

(GeV-2) 1.924 ± 0.029 1.90 ± 0.03 1.902 ± 0.024 1.899 ± 0.023 1.904 ± 0.023 1.902±0.029

a2

(GeV-4) 3.50 ± 0.14 3.28 ± 0.09 3.29 ± 0.07 3.27 ± 0.06 3.29 ± 0.09 3.28 ± 0.09

which, after combination, leads to a1= 1.907 ± 0.010sta ± 0.030sys GeV-2 , a2= 3.30 ± 0.03sta ± 0.33sys GeV-4

• Comparison with other determinations ( <r2>=6 a1 ):

[Masjuan,Peris,SC’08] [Caprini’04]

[Troconiz, Yndurain’05] [Bijnens et al’98] [Pich,Portolés’01] [Boyle’08]

<r2> (fm2)

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

a2

(GeV-4) 3.30±0.03± 0.33 …. 3.84±0.02 3.85±0.60 3.79±0.04 ….

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero

2.) A critical look

• n

Padé unitarizations

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero

[Thanks to J.Virto for his help with the slides]

[ Identical result obtained from disp. relations + χPT matching of the l.h.c. ]

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero
• What info is lost when fixing Re[t-1] with χPT?
• 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

A counter-example: The σ in the LSM

, with Mσ the renormalized mass parameter in the Lagrangian

1.) Computation of the actual σ−pole

[ Masjuan, Virto, SC’08 ]

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

• f resonance poles

Padé Theory and Resonance Poles

• J. J. Sanz Cillero

Analyticity properties

• Simplest case: Analytical function f(x) in a disk Bδ(0)

Then, the Taylor series converges to f(x) for N∞

[ with ak=f(k)(0) / k! ]

• Experimentally, if you have data in the range of Bδ(0)

f(0) , f’(0) … can be extracted from successive polynomial fits PN(x) δ

Padé Theory and Resonance Poles

• J. J. Sanz Cillero

sp What if it is analytical but for a single pole ?

• De Montessus’ theorem [1902]:

If one constructs a series of single-pole Padé Approximants PN

1 ,

Then the sequence of Padés

• unif. converges to

f(x) when N∞

• n any compact subset D={x , |x|≤δ , x ≠sp }

Hence, also the Padé pole xp=aN/aN+1 sp

δ

Padé Theory and Resonance Poles

• J. J. Sanz Cillero
• Experimentally, one is not provided with the derivatives

f(0), f’(0) …. but with experimental points { xj , fj , Δfj }

• We then use now

the rational functions PN

1(x) as fitting functions

(as we did before with the polynomials)

• PN

1(x) gives an estimate of the series of derivatives

{ f(k)(0) } and hence, sp which are expected to converge to the actual ones for N∞

Padé Theory and Resonance Poles

• J. J. Sanz Cillero

Usually: Padés around q2=0

• Amplitude F(q2)
• nly analytical up to the 1st production threshold

E.g., 4mπ

2

in ππ-VFF

(mA+mB)2 I-Riemann sheet

Padé Theory and Resonance Poles

• J. J. Sanz Cillero

Usually: Padés around q2=0

• Amplitude F(q2)
• nly analytical up to the 1st production threshold

E.g., 4mπ

2

in ππ-VFF

(mA+mB)2 I-Riemann sheet

Mink.data F(q2+i0+) Eucl.data: F(q2<0)

• The experimental data is then given by
• Euclidean data

F(q2<0)

( This is what we used before)

• Minkowskian data F(q2+i0+) ( This cannot be used by these Padés !!)

Padé Theory and Resonance Poles

• J. J. Sanz Cillero

Alternatively: Padé centered over the branch cut q2>(mA+mB)2

• Between the 1st and 2nd threshold

analytical extension of F(q2) E.g., the range 4mπ

2 4 mK 2

in ππ-VFF

[ notice that the Padé coefficients ak are now complex numbers ]

(mA+mB)2

I-Riemann sheet II-Riemann sheet

(m’A+m’B)2

Exp.data F(q2+i0+)

Padé Theory and Resonance Poles

• J. J. Sanz Cillero
• But in the case of resonant amplitudes:
• A pole appears in the II-Riemann sheet Related to a hadronic state (resonance)
• One can then use de Montessus’ theorem:
• One pole appears in the II-Riemann sheet Related to a hadronic state (resonance)
• PN

1

Padé-approximants can be used for the description of the amplitude in the disk It is possible to locate the pole in a model independent way

(mA+mB)2

I-Riemann sheet II-Riemann sheet

(m’A+m’B)2

Exp.data F(q2+i0+)

sp

Padé Theory and Resonance Poles

• J. J. Sanz Cillero

Testing the method with models

• For the preliminary analysis of the method

ρ-like resonance models: pole at sp=(Mp–iΓp/2)2 + Log singular. at q2=0

Model A) Model B) Model C)

with M and Γ chosen to produce the pole at the physical sρ(770) ≈ (0.77 – i 0.15 / 2)2 GeV2

Padé Theory and Resonance Poles

• J. J. Sanz Cillero
• We take the model

(A, for instance ) and generate a series of “data” points with 0 error

(in an ideal experimental situation All theoretical error )

For the modulus ONE GETS THE FITS PN

1

For the phase-shift

Padé Theory and Resonance Poles

• J. J. Sanz Cillero
• Convergence :

from the fitted Padé pole sfit=(Mfit–iΓfit/2)2 to the “physical” sp=(Mp–iΓp/2)2

• f the model

Model A) Model B) Model C)

where we defined the distance dist = [ (Mfit–Mp)2 + (Γfit-Γp)2 ]1/2

10-1 MeV accuracy 10-1 MeV accuracy

N=3 N=3

10-1 MeV accuracy

N=3

M M M Γ Γ Γ

Padé Theory and Resonance Poles

• J. J. Sanz Cillero

Application to data

• We analyse now: (q2>0) Minkowskian ππ-VFF data [ALEPH’06] |Fππ|2

( in the region 4 mπ

2<q2<4 mK 2 )

+ ππ−phase-shift δππ

11

(= ππ-VFF phase-shift in the elastic region 4 mπ

2<q2<4 mK 2 )

• This yields for every PN

1 the 68% C.L. regions:

• For N ≥ 3 one gets a good χ2 within the 68% CL
• Likewise, all the 68% C.L. (Mp,Γp) regions

Always compatible for N ≥ 3 Enlarged as the # of param. grows

750 755 760 765 770 775 780 120 130 140 150 160 170

N=3 N=4 N=5 N=6 N=7

[ including δππ(s=4 mπ

2) ]

M Γ

Padé Theory and Resonance Poles

• J. J. Sanz Cillero

750 755 760 765 770 775 780 120 130 140 150 160 170

Application to data

• We analyse now: (q2>0) Minkowskian ππ-VFF data [ALEPH’06] |Fππ|2

( in the region 4 mπ

2<q2<4 mK 2 )

+ ππ−phase-shift δππ

11

(= ππ-VFF phase-shift in the elastic region 4 mπ

2<q2<4 mK 2 )

• This yields for every PN

1 the 68% C.L. regions:

• For N ≥ 2 one gets a good χ2 within the 68% CL
• Likewise, all the 68% C.L. (Mp,Γp) regions

Always compatible for N ≥ 3 Enlarged as the # of param. grows

N=3 N=4 N=5 N=6 N=7

[ not including δππ(s=4 mπ

2) ]

N=2

M Γ

Padé Theory and Resonance Poles

• J. J. Sanz Cillero
• One needs then to reach a compromise:
• The experimental (fit) error

Increases with N (statistical)

• The theoretical (Padé) error

Decreases with N (systematic)

• Here, we take

N=4 as the best estimate:

• The new parameters of Padés with N ≥ 5 are compatible with 0

( no new info with respect to P4

1 )

• For N ≥ 4 the theoretical errors were found

< 10-110-2 MeV

( negligible compared to the O(1 MeV) exp.errors )

|Fππ|2 δππ

Padé Theory and Resonance Poles

• J. J. Sanz Cillero

Including δππ(s=4 mπ

2)

Mfit= 763.5 ± 0.9 MeV, Γfit= 144.5 ± 1.8 MeV

Without δππ(s=4 mπ

2)

• Mfit

fit= 763.4 ±

= 763.4 ± 0.9 M 0.9 MeV, Γfit

fit= 142.9 ±

= 142.9 ± 2.2 M 2.2 MeV

• This can be compared to other determinations:

[ Annanthanarayan,Colangelo,Gasser,Leutwyler’01 ] [ Zhou,Qin,Zhang,Xiao,Zheng’05 ] [ Pich,SC’03 ]

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero

Conclusions

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero
• Alternative determinations with competitive precision
• For low energy parameters

(e.g., <r2>V , cV in the ππ−VFF )

• ∞ < s < s1st-th. data range

Euclidean

• For hadronic poles

(e.g. the ρ(770) mass and width ) s1st-th. < s < s2nd-thr. data range Minkowsky

• This analysis shows the Padé approximants, once again, as 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)

• Nevertheless, as we saw in the 2nd part, one better use them properly

Otherwise, the extracted parameters maybe are NOT the right ones (e.g. our LSM study of “Padé”-unitarizations )

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero

Some uses of Padé approximants: The VFF

• J. J. Sanz Cillero