Hadron resonances
Definition, determination from data,
and unique description
A. Švarc
Rudjer Bošković Institute, Zagreb, Croatia
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What a resonance actually is has been a source of
strong controversy for decades.
The talk will tend to be referral:
• Define problem
• Explain the essence of the solution
• Give the essential literature for further reading
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It is nowadays commonly accepted that the poles are the only
true resonance signals
However, it is still not clear whether:
1.
2.
3.
4.
Is it really completely true?
Is it a general knowledge?
How are the poles to be extracted from experiment?
Is that statement adequately represented in the secondary
literature? (meaning PDG)
Therefore, in my talk I will address three topics:
I
II
What is a resonance
How to extract T-matrix poles
III New, model independent method for extracting poles
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I
What is a resonance
Definition !
Genus proximum + differentia specifica
Lion is a mammal from the family of cats
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How do I see what is our main task?
Experiment
Matching point
Theory
QCD
es
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d nc
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nd anc
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bo son
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?
What is a resonance in QCD?
What is a resonance in experiment?
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Definitions !
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What is a resonance in experiment?
From the intuitive (heuristic) definition to the mathematical formulation
Extensive review
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a. particle “gets trapped” (the “black hole” phenomenon)
b. a direct scattering event
c. the lifetime of the particle-target system in the region of
interaction is larger than the collision time in a direct
collision process
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This is a very heuristic definition.
Question:
What is the mathematical formulation?
Why?
And this is where poles and Breit-Wigner parameters come in !
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What is the numeric signal for a resonance?
There is an uncertainty in mathematical formulation of a clear
heuristic definition.
A careful reader will notice that it is never explicitly stated:
the resonance is .......
Instead, introducing and defining the resonance is always
much more delicate....
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Examples
Introducing (and NOT defining) resonances varies
from using rather undefined terms like:
1.
1. follow from
2. are associated with
3. it is well known
2.
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3.
Over strong statements that resonances are just a matter
of convention “ ... simply an ad hoc hypothesis...” :
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To full mathematical rigor
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However, for me, the most transparent discussion is given in:
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"Adventures in Mathematical Physics" (Proceedings, Cergy-Pontoise
2006), Contemporary Mathematics, 447 (2007) 73-81
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Scattering resonances:
Resonance criteria for scattering resonances
• Backwards looping of Argand diagram
• Rapid change of scattering phase shift for π/2
• Time delay
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Interrelation of scattering and resolvent resonances:
Concept of resolvent resonances is at length discussed in
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Both definitions of resonances are being used in the literature
sometimes without full awareness that they are different, and that
both are in principle allowed.
This is a reason for numerous disputes and controversies.
Knowing that we are dealing with the two equivalent
quantifications of the same phenomenon solves the issue.
Let us remember:
our task is not only describe resonances, but describe them in a
way which is identical to QCD.
The final answer to which of the two resonance definitions
should be used comes from analyzing what is precisely
calculated in QCD!
Is it a time delay or a resolvent pole?
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What is a resonance in QCD?
Talk presented at the Workshop
"Light-cone Physics: Particles and Strings" at ECT* in Trento, Sep 3-11,
2001
...solving the bound-state problem in gauge filed theory, particularly QCD...
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How is it done?
Light-cone approach
resonances
Hamiltonian proper values
Remember
Dalitz-Moorhouse
Hamiltonian proper values
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poles
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So, in principle:
QCD is analyzing resolvent resonances !
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So the answer to our question is:
 POLES
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II
How to extract T-matrix poles
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The usual answer was:
1. Do it globally
One first has to make a model which fits the data, SOLVE
IT, and obtain an explicit analytic function in the full
complex energy plane. Second, one has to look for the
complex poles of the obtained analytic functions.
2. Do it locally
Speed plot, expansions in power series, etc
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Ex
pe
ri
me
nta
l
da
ta
ba
se
Standard (global approach)
Model 1
Poles 1
Model 2
Poles 2
Model 3
Poles 3
Model 4
Poles 4
Under which conditions we have:
Poles 1 = Poles 2 = Poles 3 = Poles 4 ?
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Direct problems for global solutions:
•
•
•
•
Many models
Complicated and different analytic structure
Elaborated method for solving the problem
SINGLE USER RESULTS
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Local approach
Speed plot
Idea behind it?
Eliminate or reduce the dependence upon
background contribution
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Taylor expansion
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Regularization method
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In both cases we have n-TH DERIVATIVE of the function
PROBLEMS for local solutions !
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In Camogli 2012, during „coffee-break conversation” I have
claimed that extracting poles from theoretical and even from
experimental data should in principle be possible, and I have
promised to try to propose a simple method.
I have fulfilled this promise.
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Is it possible to create universal approach, usable for everyone, and
above all REPRODUCIBLE?
I have tryed to do it starting from very general principles:
1. Analyticity
2. Unitarity
Idea:
TRADING ADVANTAGES
GLOBALITY FOR SIMPLICITY
Global but complicated for local but simple
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THEORETICAL MODELS
If you create a model, the advantage is that your solution is
absolutely global, valid in the full complex energy plane (all
Rieman sheets). The drawback is that the solution is complicated,
pole positions are usually energy dependent otherwise you
cannot ensure simple physical requirements like absence of the
poles on the first, physical Riemann sheet, Schwartz reflection
principle, etc. It is complicated and demanding to solve it.
WE PROPOSE
Construct an analytic function NOT in the full complex energy
plane, but CLOSE to the real axes in the area of dominant
nucleon resonances, which is fitting the data by using
LAURENT EXPANSION.
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Why Laurent’s decomposition?
• It is a unique representation of the complex analytic function
on a dense set in terms of pole parts and regular background
• It explicitly seperates pole terms from regular part
• It has constant pole parameters
• It is not a representation in the full complex energy plane, but
has its well defined area of convergence
IMPORTANT TO UNDERSTAND:
It is not an expansion in pole positions with constant coefficients
(as some referees reproached), because it is defined only in a part
of the complex energy plane.
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1. Analyticity
Analyticity is introduced via generalized Laurent’s decomposition
(Mittag-Leffler theorem)
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Assumption:
• We are working with first order poles so all negative powers
in Laurent’s expansion lower than
n -1
are suppressed
Now, we have two parts of Laurent’s decomposition:
1. Poles
2. Regular part
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where
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
The problem is how to determine regular function B(w).
What do we know about it?
We know it’s analytic structure for each partial wave!
We do not know its EXPLICT analytic form!
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So, instead of „guessing” its exact form by using model assumptions we
EXPAND IT IN FASTLY CONVERGENT POWER SERIES OF
PIETARINEN („Z”) FUNCTIONS WITH WELL KNOWN BRANCH-POINTS!
Original idea:
Convergence
proven in:
1. S. Ciulli and J. Fischer in Nucl. Phys. 24, 465 (1961)
2. I. Ciulli, S. Ciulli, and J. Fisher, Nuovo Cimento 23, 1129
(1962).
1. S. Ciulli and J. Fischer in Nucl. Phys. 24, 465 (1961)
2. Detailed proof in I. Caprini and J. Fischer:
"Expansion functions in perturbative QCD and the
determination of αs", Phys.Rev. D84 (2011) 054019,
Applied in πN scattering
on the level of invariant
amplitudes
PENALTY FUNCTION
INTRODUCED
1. E. Pietarinen, Nuovo Cimento Soc. Ital. Fis.
12A, 522 (1972).
2. Hoehler – Landolt Boernstein „BIBLE” (1983)
NAMING !
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What is Pitarinen’s expansion?
In principle, in mathematical language, it is ” ...a conformal mapping
which maps the physical sheet of the ω-plane onto the interior of the
unit circle in the Z-plane...”
In practice this means:
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Or in another words, Pietarinen functions Z(ω) are a complet set
of functions for an arbitrary function F(ω) which
HAS A BRANCH POINT AT xP !
Observe:
Pietarinen functions do not form a complete set of functions for any
function, but only for the function having a well defined branch point.
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Illustration:
Behaviour of
Let us see the mapping
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Courtesy of Lothar Tiator
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Z(ω)
R eal
R eal
1 .0
1 .0
Im ag
Im ag
 4
 4
0 .8
0 .8
 0 .4
 0 .4
0 .4
0 .4
 0 .6
 0 .6
0 .2
0 .2
 2
 2
2
 0 .2
 0 .2
0 .6
0 .6
 4
 4
 2
 2
2
2
4
s
4
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 0 .8
 0 .8
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2
4
s
4
s
Z(ω)2
R eal
R eal
1 .0
1 .0
Im ag
Im ag
 4
 4
0 .8
0 .8
 0 .4
 0 .4
0 .4
0 .4
0 .2
0 .2
 2
 2
2
 0 .2
 0 .2
2
 0 .2
 0 .2
0 .6
0 .6
 4
 4
 2
 2
2
4
s
4
 0 .6
 0 .6
s
 0 .8
 0 .8
 1 .0
 1 .0
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2
4
s
4
s
Z(ω)3
R eal
R eal
1 .0
1 .0
Im ag
Im ag
 4
 4
 2
 2
2
 0 .2
 0 .2
0 .5
0 .5
 4
 4
 2
 2
2
2
4
s
4
 0 .5
 0 .5
 0 .4
 0 .4
s
 0 .6
 0 .6
 0 .8
 0 .8
 1 .0
 1 .0
 1 .0
 1 .0
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2
4
s
4
s
Courtesy of Lothar Tiator
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Important!
A resonance CANNOT be well described by Pietarinen series.
xP  4.93028, xQ  1.09731
xP  4.93028, xQ  1.09731
5
6
5
6
3
4
5
3
4
5
3
4
2
2
A bs
A bs
2
Im
2
1
1
1
2
0
 1
1
0
1
0
 1
0 .5
1 .0
0 .5
1 .5
1 .0
2 .0
1 .5
0 .5
2 .5
2 .0
2 .5
3
2
1
0
4
3
Im
R eal
R eal
3
1 .0
0 .5
s
1 .5
1 .0
s
2 .0
1 .5
0
2 .5
2 .0
2 .5
0 0. 5
1 .0
0 .5
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1 .5
s
s
s
1 .5
1 .0
s
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2 .5
2 .0
2 .5
Finally, the area of convergence for Laurent
expansion of P11 partial wave
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2. Unitarity
Elestic unitarity is introduced via penalty function
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The model
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We use Mittag-Leffleur decomposition of „analyzed” function:
regular background
k - simple poles
We know analytic
properties (number
and position of cuts)
of analyzed function
ONE
Pietarinen
power
series
per cut
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Method has problems, and the one of them definitely is:
There is a lot of cuts, so it is difficult to imagine that we
shall be able to represent each cut with one Pietarinen
series (too many possibly interfering terms).
Answer:
We shall use „effective” cuts to represent dominant effects.
We use three Pietarinen series:
• One to represent subthreshold, unphysical
contributions
• Two in physical region to represent all inelastic
channel openings
Strategy of choosing branchpoint positions is extremely
important and will be discussed later
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Advantage:
The method is „self-checking” !
It might not work.
But, if it works, and if we obtain a good χ2, then we have obtained
AN ANALYTIC FUNCTION WITH WELL KNOWN POLES AND CUTS WHICH
DEFINITELY DESCRIBES THE INPUT!
So, if we have disagreements with other methods, then we are looking
at two different analytic functions which are almost identical on a
discrete set, so we may discuss the general stability of the problem.
However, our solution definitely IS A SOLUTION!
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What can we do with this model?
1. We may analyze various kinds of inputs
a. Theoretical curves coming from ANY model
but also
b. Information coming directly from experiment
(partial wave data)
Observe:
Partial wave data are much more convenient to analyze!
To fit „theoretical input” we have to „guess” both:
pole position AND exact analyticity structure of the background
imposed by the analyzed model
To fit „experimental input” we have to „guess” only:
pole position AND the simplest analyticity structure of the
background as no information about functional type is imposed
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Does it work?
Testing is a very simple procedure. It comes to:
Doesn’t
Workswork
TESTING
a. Testing on a toy model:
arXiv nucl-th 1212.1295
b. Testing and application on realistic amplitudes
i.
πN elastic scattering
a. ED PW amplitudes (some solutions from GWU/SAID)
b. ED PW amplitudes (some solutions from Dubna-MainzTaipei)
ii. Photo – and electroproduction on nucleon
a. ED multipoles (all solutions from MAID and SAID)
b. SES multipoles
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a. Toy model
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We have constructed a toy model using two poles and two cuts,
used it to construct the input data set, attributed error bars of 5%,
and tried to use L+P method to extract pole parameters under
different conditions.
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b. Testing on realistic amplitude
• πN elastic
• GWU/SAID FA02
• GWU/SAID SP06
• GWU/SAID WI08
• DMT
•
Photoproduction
• GWU/SAID ZN11 ED
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Quality of the fit
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πN elastic scattering
SAID FA02
ED
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πN elastic scattering
SAID SP06
ED
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πN elastic scattering
DMT
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1785
244
43
-64
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Photoproduction
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GWU/SAID Zn11 ED solution
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X
1785
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X
244
Error analysis
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The only problem in the model are thresholds. Their number is
definitely at this moment insufficient, so we must propose a
stretegy.
Namely, if we fail to reproduce background exactly (and that we
certainly do as soon as number of thresholds is insufficient), the
pole terms try to compensate for the approximation made.
We propose two strategies:
1. To fix the pole at the values expected to dominate for a
chosen channel
2. To allow poles to vary as a fitting parameter and allow
the fit to find optimal choice of two effective thresholds
which will replace the exact values
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In practice this looks like that:
Option 1:
Option 2:
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Example of the error estimate:
We used weighted average.
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Conclusion
The L+P method defined as:
WORKS
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World recognition
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A. Švarc and L. Tiator
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