1. Simulation and test of FSR and  direct decay
O. Shekhovtsova
INFN/LNF
INFN/LNF
October 2006

2. FSR in sQED (sQEDVMD) But what about final state ?
But how good is
this approximation?
F(s)
sQED VMD is reasonable for the  final state.
Usual sQED is used for FSR, or more exactly SQED*VMD is used: pions ar etreated li the point-like particles and then the total FSR amplitude is multipled by the form-factor. And this form-factor comes from 2pion final state. But how is good this approximation and why we can use the same value for 2pion and FSR states. Of course, sQED*VMD should include the main effect of FSR, but additional contributions are possible. They are model-dependent and, probably, small, but we should estimate them.
But what about final state ?
Can we use for FSR the same value of F(s) as for 
An additional contribution is model-dependent,
and probably very small. But must be estimated.

3. General parametrization for FSR amplitude:
charge conjugation symmetry
photon crossing symmetry and
gauge invariance
we need 3 form factors fi !

4. Different contributions to FSR
fi scalar (model dependent) functions
each form factor fi depends on 3 independent variables: s, kP, kl
imn gauge invariant tensors
Different contributions to FSR
e*
1,P,V,VV‘,VS,A…
Then the FSR tensor can be rewritten thhrough 3 gauge invariant tensors. This decomposition is model dependent. Here f_i are the scalar functions and exactly the value of these functions are determined by FSR model. Only for the case of soft emitted photon we can rewrite f_i through one function and this function is called the pion form-factor. But this is true only for soft pion, when the photon is hard this approximation could not work and we have the additional contributions.
1,V (A) is related to F , usually treated by sQEDVMD (“pure” FSR)
VV‘,VS…: e.g.  direct decay…

5. Background to σ(e+ e--→ + --) related FSR
Radiative return method
(PHOKHARA)
H.Czýz, hep-ph/
H.Czýz et al, hep-ph/
FSR = sQEDVMD+ direct decay
Direct scanning (Novosibirsk)
A.B. Arbuzov et al.,
hep-ph/
(semianalytical results)
FSR = sQEDVMD
BABAYAGA : without FSR for + --
All MC use sQED VMD description s

6. Direct scanning K. Melnikov, hep-ph/0105267
Radiative corrections ISR and FSR
Integration over full phase space (incl. process)
Increasing FSR near threshold
Effect of exclusive selection discussed in
H. Czyz et al., hep-ph/ ,
Hoefer et al., hep-ph/
To be studied!

7. Radiative return FSR ISR ISR  |Fp(Q2)|2 /Q2 FSR  |Fp(s)|2 /s
Babar: s=10GeV2 , Q2< 1GeV 2
|Fp(Q2)|2>> |Fp(s)|2 , 1/s>1/Q2 
FSR << ISR
FSR ?  ? ISR
f decay
DAFNE: s= 1.02GeV2 , Q2 s

8. FSR can be suppressed by corresponding cuts on phase space:
Small angle analysis
(PHOKHARA, H.Czyz et al.,
hep-ph/ )
Large angle analysis
( EVA, S. Binner et al.,
hep-ph/ )
TOT
ISR
BIG FSR!

9. dT|MISR+MFSR|2= dI + dF + dIF dI|MISR|2
MC for e+e- for DAFNE
hep-ph/
Our MC is based on EVA MC structure, with our amplitude for all FSR contributions (i.e. MFSR )
Interference term dsIF=0 for symmetric cuts on qp
We have the contribution from  (negligible)
KLOE large angle analysis: o< 0o,
50o<0o
S. Binner et al.
PLB 459 (1999)
Total cross section
dT|MISR+MFSR|2= dI + dF + dIF
dI|MISR|2
dF|MRPT|2+ |M|2 +2Re(MRPT·M*)
dIF 2Re(MISR · (MRPT+M)*)
But a_mu is integrated value, what about the differential distribution. We apply our results for difeerential cross-section and to the charge asymmetry. Our Mc is based on Mc Eva, we took the ISR matrix element from Eva and added it by our value for FSR contribution. Also we neglected the contribution from the \rho\pi vertex, it was found it’s small and we include \phi direct decay. As our calculation is only is the first step for FST out sQED we hadn’t calculation for \phi direct decay. But we can’t negelct this contribution because as it was showed in Henryk talk this contribution affects also the low q^2 region. Also we will consider only large angle analysis
f direct decay: Achasov 4q model, parameters from data 
r contribution
‘pure’ FSR: RPT model

10. Comparison MC with analytical functions
s=mf2
s=1GeV2

11. Dependence on f direct parametrization
1. f direct decay
First MC with f direct decay K.Melnikov et al., PLB 477, hep-ph/
A new parametrization of KLOE 06 has been included: ((f0+s))
nb/GeV2
Only f0
f0+s
50o< 0o,
50o<0o
Main parameters

12. s=mf2 f direct decay f0 peak with s without s
the visible cross section is reduced at low Q2 (with respect to sQED)
due to negative interference with f
2. |M|2 decreases due to s total cross section is no longer dsRPT+f~ dssQED+f

13. hep-ph/0603241, G. Isidori et al.: functions fi, 2 new constants
V()
V‘(r)
s=mf2
without r
with r
Some differences….

14. 2. “Pure” FSR in RPT
G.Ecker et al., Nucl. Phys B321,
311 (1989)
Resonance PT (ChPT with the explicit inclusion of  ( ‘) and a1 mesons:
To enlarge the energy region for ChPT (1.5 GeV) the resonance fields (V, A) should be added by explicit way
E«m the singularity associated with the pole of a resonance propagator is replaced by momentum expansion
The main contribution to the coupling constants (up to p4) comes from the meson resonanse exchange; model parameters: f , FV, GV, FA
We apply the ChPT with explicit inclusion of vector \rho and axial-vector a_1 mesons. In the franework of this theory 2pion and photon final state can be described by the following diagrams. Only the first three diagrams can be found in sQED. Here the composite vertex describes the contact term and propagation through \rho meson.
sQED

15. FSR as background to ISR
s=mf2
s=1 GeV2
big enough contribution due to RPT especially for s=1GeV2
 almost negligible for s=1GeV2

16. Result on FB asymmetry for KLOE data
AFBMISR *MFSR 
S. Binner, hep-ph/ (EVA)
H.Czyz, hep-ph/ (FSR+)
data
MC: ISR+FSR
MC: ISR+FSR+f0(KL)
Mpp (MeV)
Mpp (MeV)
Using the f0 amplitude from Kaon Loop model, good agreement data and MC* both around the f0 mass and at low masses.
Phys.Lett.B634 (06), 148 (data)
*G. Pancheri, O. Shekhovtsova, G. Venanzoni, hep-ph/

17. Below current experimental precision
 contribution to a
10-9
10-8
In sQED the value of a_mu is about the current experimental uncertainty that is why there is a question how the value of a_mu depends on the additional ChPT contribution. We consider situation when photon is hard, energy is more than E_cut. The different value of E_cut were considered. The devation from sQED is only for hard photon region only. The additional ChPT integrated contribution is very small and below the current experimental precision. The reasons of such a small additional value: for fixed value the dominant region is the low-energy one (and it is described by the similar way in both models), when we integrate over s the main region is \rho-meson one, which is included through VMD in sQED and ChPT. That is why we think that today value of a_mu is not sensitive to the contribution out of sQED.
10-13
10-12
a (~ 500(5) 10-10
a (sQED 
a (CHPT 
Differential contribution
to , where  is hard ( >Ecut)
Below current experimental precision

18. Pion Form Factor with ρ-ω mixing
Fit to Novosibirsk CMD-2 data
χ2=0.853

19. … around the r meson peak
The negative interference with f reduces the visible cross section (with respect to sQED)

20. Model independent test of FSR beyond sQED
many parameters for the general FSR amplitude:
our calculation: RPT(6) +f (3)
difficult to extract simultaneously all these parameters from the same set of data ( p+p -g)  cross section, asymmetry
We suggest another approach:
G. Venanzoni talk