Apresenta

1
Unitarity Corrections to Pion Pion Partial Waves
from Dispersion Relation Formalism
J. Sá Borges Universidade do Estado do Rio de
Janeiro I. P. Cavalcante Universidade Federal de
Mato Grosso do Sul Y. A. Coutinho
Universidade Federal do Rio de Janeiro
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Outline Hard Meson Method Unitarization
Program Chiral Perturbation Theory Comparing O(
p4 ) and O( p6 ) ChPT with UPCA Analytical
Expressions Tools for Improvement Conclusion
and Perspectives
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sigma model
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Diagonalized N-point functions
0- on shell 1
Starts from
Example
no underlying theory
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Pion Pion Scattering Amplitude
Tree Structure
Propagators ?0 ?2 ?V
Form Factors F0 F2 F1
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Low Energy Estimates
Weinberg soft pion
How to go beyond this approximation ?
Elastic Unitarity Constraints for Form factors
and Propagators
UPCA
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Partial-wave Elastic Unitarity Constraint
phase space factor
First Order Correction
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First Order Corrected Amplitude
I1
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Amplitude phases
S0
UPCA (first order)
No parameters
P
E(GeV)
S2
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P-wave phase
fix ?1 ?2
S-wave phase
fix ?3
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Effective Lagrangians
The current algebra method was used in early
calculations of muti-Goldstone-boson
amplitudes. Unfortunately, such calculations are
tedious, especially when three or more
Goldstone bosons are involved ... For this
reason a different and more physical
calculational technique was introduced, based on
the use of effective Lagrangians
We simply calculate the Goldstone boson
amplitude by the methods of perturbation theory
...
Weinberg - Modern Applications
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Momentum expansion Counting scheme infinities
absorbed
Ch P T
Lowest Order
NL? model
External fields
G.O.R.

Pion-Pion Amplitude
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O(p4) One Loop Calculation
Building Blocks
?
How to fix
LECs
p4
p2
p3
same structure as UPCA !!
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UPCA second order approximation
84
from f(2) and ?(2)
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O(p6) Two Loop Calculations
Building Blocks
Terms with six D? Terms with four D? and one ???
(? f??) Terms with two D? and two ??? Terms with
three ???
independent couplings L2(2) L4(10) L6(90)
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Skeleton Diagrams of O(p6)
full lowest-order tree structures to be attached
to propagators and vertices.
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Comparison
ChPT UPCA
Functions
O(p4 )
Parameters
Functions
O(p6 )
Parameters
b1... b6 ?
same structure !!
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P-wave phase ( fix ?3 )
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P- and D-wave phases (?1 ?2 ?3)
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S-wave phase
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Analytical Expressions
O(p4)
O(p6)
trivial!!
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Solving Dispersion Relation Integrals
combine polylog
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Higher Orders
right hand side discontinuities
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Conclusions and Perspectives
Program
.O(p6) dispersion relation integrals . Scheme
for higher orders . Extending to different masses


Motivation
Pionic Atoms Decay - Dirac Scalar resonances CP
violation kaonic system A C U rigorous
relations!!
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Chiral Ward Identities
The effective Lagrangians method was used in
calculations of pion-pion scattering
amplitude. Unfortunately, such calculations are
tedious, especially when two or more loops are
involved ... For this reason a method was
re-introduced, based on the use of the chiral
Ward Identities We simply
calculate the amplitude by the technique of
dispersion relations ...

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Thank you
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Draft
Unitarity corrections to e.m. form factor
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P-wave
P-wave phase
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D0-wave
D0-wave phase
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S0-wave phase
S0-wave
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