1. Test of FSR in the process at DAFNE
G.Pancheri, O. Shekhovtsova, G. Venanzoni
INFN/LNF
EURIDICE Midterm Collaboration Meeting
8-12 Feb Frascati

2. FSR in sQED (sQEDVMD) But how good is this approximation?
Fp(s)
Fp(s)
Fp(s)
But how good is this approximation?
sQED VMD is reasonable for the pp 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 ppg final state ?
Can we use for FSR the same value of Fp(s) as for pp ?
An additional contribution is model-dependent,
and probably very small. But must be estimated.

3. Global structure of FSR tensor for
P2 = s
e*
charge conjugation symmetry
photon crossing symmetry and
gauge invariance
Now the global structure of FSR tensor. We should construct FSR tensor imposing the invariance of thie tensor under the charge conjugation symmetry of S matrix and photon crossing symmetry. Also we should take into account the gauge invariance of FSR

4. gauge invariant tensors:
scalar (model dependent) functions
Limit of soft photon (what we call
sQED), in fact VMD*sQED
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.
At threshold (very hard photon) this approximation
could not work
fi=fisQED+Dfi

5. FSR in ChPT with  and a1 mesons
(S. Dubinsky et al,hep-ph/041113)
FSR in ChPT with  and a1 mesons
G.Ecker et al., Nucl. Phys B321, 311 (1989)
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.
The contribution with r+,-
(instead of a1), turns out to be negligible

6. We calculate fi in ChPT (S. Dubinsky et al,hep-ph/0411113)
4 model parameters:
f , FV, GV, FA
Meson
M(GeV)
GV (GeV)
FV (GeV)
FA (GeV) r
0.775
0.0066
0.156 - a1
1.23
0.122
These model has 4 parameters. We took the value for these parameteres corresponding the central value of the following decay width.
And based on diagrams mentioned before we calculated the value of f_i
Now we apply our results for estimation of pipigamma contribution to anamalous magnetic moment of muon and to differential cross section and charge asymmetry
We calculate fi in ChPT (S. Dubinsky et al,hep-ph/ )
ppg contribution to am (analytical results)
Contribution to dsppg/dQ2pp spectrum and charge asymmetry (Monte Carlo)

7. below current experimental precision
ppg contribution to am
10-9
10-8
10-13
10-12
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.
Differential contribution
to , where  is hard ( >Ecut)
am (pp) ~ 500(5) 10-10
am (ppgsQED) 
Dam (ppgCHPT) 
below current experimental precision

8. Contribution of FSRCHPT to dsppg/dQ2pp spectrum and charge asymmetry
The following matrix element has been introduced in a MC,
for e+e- p+p-g (based on EVA structure):
S. Binner et al. Phys. Lett. B 459 (1999)
We neglect the contributions from g*rp p+p-g (found
to be negligible in hep-ph/ )
We included the direct decay f p+ p- g, important at s=mf2. This contribution (which is model dependent) affects also the low Q2pp region. Charge asymmetry can help to distinguish between various models (see the talk of H. Czyz)
We consider KLOE large angle analysis: 50o< qg< 130o, 50o<qp< 130o (S. Mueller talk)
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

9. We included fp+ p- g decay in our calculation, by looking to the channel f p0p0g (similar to H. Czyz et al. hep-ph/ )
We use the Achasov 4quark parametrization with the parameters of the model taken from the fit of the KLOE data f p0p0g. (fp+ p- g is related to fp0 p0 g by isospin symmetry)
dBR/dm x 108 (MeV-1)
CAVEAT: For the moment we consider only the contribution of f0 (no s meson).
This could be too crude for low Q2 !
mpp(MeV)

10. (Analytical) Comparisons (at s=mf2):
0o<qp< 180o
0o<qg< 180o
= FSRsQED+Df
dsppg /dQ2 (nb/GeV2)
= fp+ p- g resonant cont.
= FSRsQED
Since MFSR*Mf  |Mf|2 at low Q2
Df can be relevant only for destructive interference (we will
consider only this case in the following)
= Df
Here we depicted the different contributions to the total cross section . And the \phi resosnt contribution is big. That is why we conclude that the experimental data is sensitive to the additional FSR contribution only if the interference contribution is destructive. In this case the ineterference contribution and the resonant \phi contribution almost cancelled each other. Otherwise we haven’t any possibility to catch this contribution. There are some hints that the interference should be destructive that is why we will consider only this case.
Q2(GeV2)
What happens for s<mf2 ?

11. The comparison at s=1 GeV2 (off f peak):
0o<qp< 180o
0o<qg< 180o
= FSRsQED+Df
dsppg /dQ2 (nb/GeV2)
= 100·fp+ p- g resonant cont.
Multiplied by a
factor 100
= FSRsQED
= Df
1/|Df(s)|2
What is happened if we out of \phi resonance. In this case\phi resonant contribution is very small and it can’t compensated the interference contribution. That is why even for s0=1 we couldn’t neglect the interference contribution, in other words we can’t throw away \phi contribution.
But the work out of the \phi resonance is attractive because in this region the contamination from 3pi final state is much less. s
1 GeV
1.04 GeV
1.04 GeV
Q2(GeV2)
In this case the interference MFSR*Mf is expected to be >>|Mf|2
We could not neglect the interference contribution (i.e. f contr.), but the work off the resonance region is attractive (3 p background is much less)

12. Numerical results: differential cross section…
s=mf2
50o<qp< 130o
50o<qg< 130o
Numerical results: differential cross section…
dsTOT/dssQED
ds/dQ2 (nb/GeV2)
dssQED+f/dssQED
= ISR+FSRsQED
= ISR+FSRsQED + f
dsTOT/dssQED+f
= ISR+FSRCHPT + f
Q2(GeV2)
Q2(GeV2)
Effect al low Q2…however the contribution of f is not much accurate
(no interference with s has been taken into account)

13. A closer look at the threshold region:
dsTOT/dssQED
ds/dQ2 (nb/GeV2)
dssQED+f/dssQED
= ISR+FSRsQED
dsTOT/dssQED+f
= ISR+FSRsQED + f
= ISR+FSRCHPT + f
0.35
0.35
Q2(GeV2)
Q2(GeV2)
Up to 30% of contribution beyond sQED
at the threshold. Results are sensitive to FSR model

14. And asymmetry… -25% = ISR+FSRsQED = ISR+FSRCHPT + f = ISR+FSRsQED + f
Q2(GeV2)

15. Conclusions and outlook
First MC results on a generalization of FSR using ChPT with  and a1 mesons have been presented.
A sizeable effect can be seen on the cross section (at low Q2 only).
The situation on the asymmetry is more complicate.
the result strongly depends on the parametrization of the f direct decay.

16. For the near future: Improve the simulation: New theoretical tasks
better parametrization of f (including also the s meson)
study of the dependence of results on the various parameters of the models in MC
New theoretical tasks
improve the knowledge on the phi decay (in particular at low Q2): to consider the phi decay in ChPT
try to take into account  and  ' mesons contribution in ChPT
Try to disentangle the various contributions:
asymmetry, and other kinematical variables (angular distributions)
Model independent analysis of fi  different kinematics region?
Work off resonance

17. Disclaimer: all our numerical results are preliminary
Disclaimer: all our numerical results are preliminary!!! The theoretical results is only the first step out sQED!!!

18. BACKUP SLIDES

19. Asymmetry
ISR+FSRsQED + f
ISR+FSRsQED
Q2(GeV)
Q2(GeV)