arXiv:0708.1659v2 [hep-ph] 6 Nov 2007
Improved dispersion relations for γγ → π0π0
Jos´ e A. Ollera, Luis Rocaaand Carlos Schata,b
aDepartamento de F´ ısica. Universidad de Murcia. E-30071, Murcia. Spain.
bCONICET and Departamento de F´ ısica, FCEyN, Universidad de Buenos Aires,
Ciudad Universitaria, Pab.1, (1428) Buenos Aires, Argentina.
firstname.lastname@example.org , email@example.com , firstname.lastname@example.org
We perform a dispersive theoretical study of the reaction γγ → π0π0emphasizing the low energy
region. The large source of theoretical uncertainty to calculate the γγ → π0π0total cross section for
√s ? 0.5 GeV within the dispersive approach is removed. This is accomplished by taking one more
subtraction in the dispersion relations, where the extra subtraction constant is fixed by considering
new low energy constraints, one of them further refined by taking into consideration the f0(980) region.
This allows us to make sharper predictions for the cross section for√s ? 0.8 GeV, below the onset
of D-wave contributions. In this way, were new more precise data on γγ → π0π0available one might
then distinguish between different parameterizations of the ππ isoscalar S-wave. We also elaborate on
the width of the σ resonance to γγ and provide new values.
The reaction γγ → π0π0measured in ref. offers the interesting prospects of having a two-body hadronic
final state and the important role of final state interactions in S-wave enhanced due to the null charge
of the π0. These two facts make this process very suited for learning about the non-trivial ππ isospin
(I) 0 S-wave. In addition, it was taken as an especially appropriate ground test for Chiral Perturbation
Theory (χPT) [2, 3], since at lowest order this process is zero and at next-to-leading order (one loop) is
a prediction free of any counterterm [4, 5]. However, the one loop χPT prediction departs very rapidly
from data just above the threshold and only the order of magnitude was rightly foreseen. A two loop
calculation in ref.[6, 7] was then undertaken with better agreement with data . The three counterterms
that appear at O(p6) are fixed by the resonance saturation hypothesis. Other approaches supplying higher
orders to one loop χPT by taking into account unitarity and analyticity followed [8, 9, 10]. Ref. is a
Unitary χPT calculation in production processes [10, 11, 12, 13, 14, 15] and was able to provide a good
simultaneous description of γγ → π0π0, π+π−, ηπ0, K+K−and K0¯ K0from threshold up to rather high
energies, s1/2? 1.5 GeV. This approach was also used in ref. to study the η → π0γγ decay.
We concentrate here on the dispersive method of refs.[17, 18, 19]. We critically review and extend
it, so as we are able to drastically reduce the uncertainty due to the not-fixed phases above the K¯K
threshold of the I = 0 S-wave γγ → ππ amplitude. This is accomplished by using an I = 0 S-wave
Omn` es function that is continuous under changes in the phase function employed for its evaluation
above the K¯K threshold. Equivalently, one can introduce an additional subtraction in the dispersion
relation to evaluate the I = 0 S-wave γγ → ππ amplitude to those considered in ref.[17, 18, 19, 8].
This new subtraction constant is fixed by considering simultaneously three constraints instead of the two
employed in the previous references. As a result of this much reduced uncertainty, the total cross section
σ(γγ → π0π0) might be used to distinguish between different S-wave I = 0 phase shift parameterizations
once new more precise experimental data become available. We also perform calculations of the width
Γ(σ → γγ), taking from the literature different σ resonance parameters, and compare with the value of
ref.. Other papers dedicated to calculate the two photon decay widths of hadronic resonances are
The content of the paper is as follows. In section 2 we discuss the dispersive method of ref. and
extend it to calculate with higher accuracy the cross section γγ → π0π0. The resulting σ(γγ → π0π0)
and Γ(σ → γγ) are given in section 3. We elaborate our conclusions in section 4.
2 Dispersive approach to γγ → π0π0
In refs.[17, 18, 19] an interesting approach was established to calculate in terms of a dispersion relation
the γγ → (ππ)IS-wave amplitudes, FI(s), where the two pions have definite isospin I. Notice that for
γγ → π0π0, due to the null charge of π0, there is no Born term, fig.1. One then expects, as remarked
in ref., that only the S-wave would be the important partial wave at low energies,√s ? 0.7 GeV.
For γγ → π+π−, where there is a Born term due to the exchange of charged pions, the D-waves have
a relevant contribution already at rather low energies due to the smallness of the pion mass. In the
following, we shall restrict ourselves to the S-wave contribution to γγ → π0π0. The explicit calculation of
ref. indicates that the D-wave contribution at√s ≃ 0.65 GeV is smaller than a 10% in σ(γγ → π0π0),
and it rapidly decreases for lower energies.
The function FI(s) is an analytic function on the complex s−plane except for two cuts along the real
axis. The right hand cut happens for s ≥ 4m2
hand cut, in turn, runs for s ≤ 0 and is due to unitarity in crossed channels. Let us denote by LI(s) the
π, with mπthe pion mass, and is due to unitarity. The left
Figure 1: Born term contribution to γ(k1)γ(k2) → P+(p1)P−(p2).
complete left hand cut contribution. Then the function FI(s)−LI(s), by definition, only has right hand
cut. Next, refs.[17, 18, 19] consider the Omn` es function ωI(s),
ωI(s) = exp
with φI(s) the phase of FI(s) modulo π, chosen in such a way that φI(s) is continuous and φI(4m2
Because of the choice of the phase function φI(s) in eq.(2.1), the function FI(s)/ωI(s) has no right hand
cut. Then refs.[17, 18, 19] perform a twice subtracted dispersion relation for (FI(s) − LI(s))/ωI(s),
FI(s) = LI(s) + aIωI(s) + cIsωI(s) +s2
π) = 0.
On the other hand, Low’s theorem  requires FI→ BI(s) for s → 0, with BI the Born term contri-
bution, shown in fig.1. If we write LI= BI+ RI, with RI→ 0 for s → 0, as can always be done, then
Low’s theorem implies also that FI− LI→ 0 for s → 0 and hence aI= 0.
For the exotic I = 2 S-wave one can invoke Watson’s final state theorem#1so that φ2(s) = δπ(s)2, the
I = 2 S-wave ππ phase shifts. For I = 0 the same theorem guarantees that φ0(s) = δπ(s)0for s ≤ 4m2
with δπ(s)0the S-wave I = 0 ππ phase shifts and mKthe kaon mass. Here one neglects the inelasticity
due to the 4π and 6π states below the two kaon threshold, an accurate assumption as indicated by
experiment [22, 23]. Above the two kaon threshold, sK= 4m2
a priori due to the onset of inelasticity. This is why ref. took for s > sKthat either φ0(s) ≃ δπ(s)0
or φ0(s) ≃ δπ(s)0− π, in order to study the size of the uncertainty induced for low energies. It results,
however, that this uncertainty increases dramatically with energy such that for√s = 0.5, 0.55, 0.6 and
0.65 GeV it is 20, 45, 92 and 200%, see fig.3 of ref..
The reason for this behaviour is the use of the function ω0(s) in eq.(2.2). The I = 0 S-wave phase
shift δπ(s)0has a rapid increase by π around sK= 4m2
on top of the K¯K threshold. Let us denote by ϕ(s) the phase of the ππ → ππ I = 0 S-wave strong
amplitude, modulo π, such that it is continuous and ϕ(4m2
with δπ(s)0and δπ(s)2. Now, if one uses ϕ(s) instead of φ0(s) in eq.(2.1) for illustration, the function
ω0(s) is discontinuous in the transition from δπ(sK)0→ π−ǫ to δπ(sK)0→ π+ǫ, with ǫ → 0+. In the first
case |ω0(s)| has a zero at sK, while in the latter it becomes +∞. This discontinuity is illustrated in fig.3
by considering the difference between the dot-dashed and dashed lines. This discontinuous behaviour of
ω0(s) under small (even tiny) changes of δπ(s)0around sK, was the reason for the controversy regarding
the value of the pion scalar radius ?r2?π
#1This theorem implies that the phase of FI(s) where there is no inelasticity is the same, modulo π, as the one of the
isospin I S-wave ππ elastic strong amplitude.
K, the phase function φ0(s) cannot be fixed
K, due to the narrowness of the f0(980) resonance
π) = 0. This phase is shown in fig.2 together
sbetween [24, 25, 26] and . This controversy was finally solved
in ref. where it is shown that Yndur´ ain’s method is compatible with the solutions obtained by solving
the Muskhelishvili-Omn` es equations for the scalar form factor [29, 30, 31]. The problem arose because
refs.[24, 25] overlooked the proper solution and stuck to an unstable one.
central value of 
Sol. A of 
Sol. B of 
Sol. C of 
Sol. D of 
Sol. E of 
Kaminski et al. 
BNL-E865 Coll. 
NA48/2 Coll. 
BNL-E865 Coll. 
NA48/2 Coll. 
ϕ (δπ(sK)0< π)
ϕ (δπ(sK)0> π)
Figure 2: The phase shifts δπ(s)0and δπ(s)2and the phase ϕ(s). Experimental data are from refs.[32, 23, 33, 34]
for I = 0 and refs.[35, 36] for I = 2. The insert is the comparison of CGL  and PY  with the accurate data
from Ke4[33, 34].
0 200 400
|ω0| for δπ(sK)<π
|ω0| for δπ(sK)>π
|Ω0| for δπ(sK)>π
Figure 3: |ω0(s)|, eq.(2.1), with δπ(sK)0< π, dashed-line, and δπ(sK)0> π, dot-dashed line. The solid line is
|Ω0(s)|, eq.(2.3), for the latter case. Here ϕ(s) is used as φ0(s) in eqs.(2.1) and (2.3) for illustrative purposes.
Inelasticity is again small for 1.1 ?√s ? 1.5 GeV being compatible with zero experimentally [22, 23].
As remarked in refs.[24, 28], one can then apply approximately Watson’s final state theorem and for
F0(s) this implies that φ0(s) ≃ δ(+)(s) modulo π. Here δ(+)(s) is the eigenphase of the ππ, K¯K I = 0
S-wave S-matrix such that it is continuous and δ(+)(sK) = δπ(sK)0. In refs.[25, 28] it is shown that
δ(+)(s) ≃ δπ(s)0 or δπ(s)0− π, depending on whether δπ(sK)0 ≥ π or < π, respectively. In order to
fix the integer factor in front of π in φ0(s) ≃ δ(+)(s) modulo π, one needs to devise an argument to
follow the possible trajectories of φ0(s) in the narrow region 1 ?√s ? 1.1 GeV, where inelasticity is not
negligible. The remarkable physical effects happening there are the appearance of the f0(980) resonance
on top of the K¯K threshold and the cusp effect of the latter that induces a discontinuity at sKin the
derivative of observables, this is clearly visible in fig.2. Between 1.05 to 1.1 GeV there are no further
narrow structures and observables evolve smoothly. Approximately half of the region between 0.95 and
1.05 GeV is elastic and φ0(s) = δπ(s)0(Watson’s theorem), so that it raises rapidly. Above 2mK≃ 1 GeV
up to 1.05 GeV the function φ0(s) can keep increasing with energy, as δπ(s)0or ϕ(s) for δπ(sK)0≥ π,
and this is also always the case for the corresponding phase function of the strange scalar form factor
of the pion . It is also the behaviour for φ0(s) corresponding to the explicit calculation of ref..
The other possibility is a change of sign in the slope at sKdue to the K¯K cusp effect such that φ0(s)
starts a rapid decrease in energy, like ϕ(s) for δπ(sK)0< π, fig.2. Above√s = 1.05 GeV, φ0(s) matches
smoothly with the behaviour for√s ? 1.1 GeV where it is constraint by Watson’s final state theorem.
As a result of this matching, for√s ? 1 GeV either φ0(s) ≃ δπ(s)0or φ0(s) ≃ δπ(s)0− π, corresponding
to an increasing or decreasing φ0(s) above sK, respectively. There is then left an ambiguity of π in φ0(s)
for 1.5 GeV ?√s >√sK. Our argument also justifies the similar choice of phases in ref. above sK
to estimate uncertainties. Let us define the switch z to characterize the behaviour of φ0(s) for s > sK
such that z = +1 if φ0(s) rises with energy and z = −1 if it decreases. Above 1.5 GeV the phase function
employed has little effect in our energy region and we use the same asymptotic phase function as in
ref., tending either to 2π (z = +1) or π (z = −1) for s → +∞. It allows a large uncertainty of ≃ 2π
at√s = 1.5 GeV, that only shrinks logarithmically for higher energies. This uncertainty is included in
our error analysis. Further details are given in ref..
Next, we define, the function Ω0(s), similarly as done in ref.,
where θ(z) = +1 for z = +1 and 0 for z = −1 and s1is the point at which φ0(s1) = π. The latter is the
only point where the imaginary part of Ω0(s) vanishes around sKand this fixes the position of the zero.
Now, we perform the same twice subtracted dispersion relation as in eq.(2.2) but for (F0(s)−L0(s))/Ω0(s) ,
F0(s) = L0(s) + c0sΩ0(s) +s2
Ω0(s) =1 − θ(z)s
s′2(s′− s)|Ω0(s′)|ds′+ θ(z)ω0(s)
(F0(s1) − L0(s1)) . (2.4)
In the previous equation we introduce φ0(s) that is defined as the phase of Ω0(s). Let us note that in
the case z = +1 the phase of Ω0(s) for s > s1is not φ0(s) but φ0(s)−π, due to the factor 1−(s+iǫ)/s1
in Ω0(s), eq.(2.3). Since φ2(s), because of Watson’s final state theorem, is given by δπ(s)2, which is small
and smooth [35, 36], fig.2, the issue of the discontinuity in ω2(s) under changes in parameterizations
does not rise and we use the dispersion relation in eq.(2.2). It is worth mentioning that our eq.(2.4) for
z = +1 is equivalent to take a three times subtracted dispersion relation for (F0(s) − L0(s))/ω0(s), two
subtractions are taken at s = 0 and another one at s1. We could have taken the three subtractions at
s = 0, although we find more convenient eq.(2.4) which is physically motivated by the use of the Omn` es
function eq.(2.3) that is continuous under changes in the parameterization of the I = 0 S-wave S-matrix.
When eq.(2.3) is used with ϕ(s) instead of φ0(s) for δπ(sK)0> π the solid curve in fig.3 is obtained,
which is again close to the dashed line for δπ(sK)0< π.
We denote by FN(s) the S-wave γγ → π0π0amplitude and by FC(s) the γγ → π+π−one. The
relations between F0, F2and FN(s), FC(s) in our isospin convention are
We have the unknown constants c0, c2and F0(s1)−L0(s1), the latter for z = +1. To determine them
FN(s) = −1
3F2, FC(s) = −1
1. FC(s) − BC(s) vanishes linearly in s for s → 0 and we match the coefficient to the one loop χPT
result [4, 5].
2. FN(s) vanishes linearly for s → 0 as well and the coefficient can be obtained again by matching with
one loop χPT [4, 5].
3. For I = 0 and z = +1 one has still F0(s1)−L0(s1). The value of this constant can be restricted taking
into account that FN(s) has an Adler zero, due to chiral symmetry. This zero is located at sA= m2
one loop χPT and moves to sA= 1.175m2
πin two loop χPT . This implies about a 20% correction,
that prevents us from taking a definite value for sA. In turn, we obtain that the value of the resulting
cross section σ(γγ → π0π0) around the f0(980) resonance is quite sensitive to the position of the Adler
zero, because it controls the size of F0(s1) − L0(s1). The latter appears in the last term in eq.(2.4), the
one that dominates F0(s) around the f0(980) position since Ω0(s1) = 0. Though the dispersive method
is devised at its best for lower energies, it is also clear that it should give at least the proper order of
magnitude for σ(γγ → π0π0) in the f0(980) region.#2Being conservative, we shall restrict the values
of F0(s1) − L0(s1) so as the cross section at s1is less than 10 times the experimental value around the
f0(980) region, σ(γγ → π0π0) < 400 nb.
Regarding LI(s), it is expected to be dominated at low energies by the Born term in isospin I because
of the smallness of the pion mass. The Born term originates by the exchanges in the t and u channels
(γπ → γπ) of charged pions, fig.1. Other crossed exchanges of vector and axial vector resonances are
relatively suppressed for FI(s) due to the larger masses of the members of the JPC= 1−−, 1++and 1+−
multiplets so that their associated left hand cut singularities are further away. Among them, the 1++
axial vector exchange contributions are the dominant ones in γπ±→ γπ±and already appear at the one
loop level in χPT. The 1−−and 1+−exchanges start one order higher. As already remarked in ref.,
the authors of refs.[18, 19], and later on also ref., overlooked the axial vector exchange contributions
altogether and hence they are missing an essential part in the study of the low energy γγ → π0π0. Indeed,
once the 1++axial vector exchange contributions are considered the cross section in the latter references
would increase significantly at low energies. For instance, at√s = 0.5 GeV one has more than a 20%
increase compared to the case in which only the Born term and the 1−−vector resonances exchanges
are considered. At low energies the influence of the 1+−axial vector nonet is much smaller than that
of the 1++and 1−−multiplets. In the following, LI(s) is modelled by the Born terms and the crossed
exchanges of the 1++, 1−−, and 1+−resonance multiplets, evaluated from chiral Lagrangians. Explicit
expressions for the distinct contributions to LI(s) will be given elsewhere .
In this section we show the results that follow by the use of eq.(2.4) for I = 0 and eq.(2.2) for I = 2.
Since the main contribution in the low energy region to Ω0(s) and ω2(s) comes from the low energy
ππ phase shifts, one needs to be as precise as possible for low energy ππ scattering data. The small
I = 2 S-wave ππ phase shifts, which induce small final state interaction corrections anyhow, can be
parameterized in simple terms and our fit compared to data can be seen in fig.2. For the I = 0 S-wave
ππ, we take the parameterizations of ref. (CGL) and ref. (PY). Both agree with data from Ke4
decays [33, 34] and span to a large extend the band of theoretical uncertainties in the I = 0 S-wave ππ
phase shifts [22, 23, 32]. PY, similarly to refs.[42, 43, 44], runs through the higher values of δπ(s)0, while
CGL does through lower values, see fig.2. We shall use CGL up to 0.8 GeV, since this is the upper limit
of its analysis, and the K-matrix of Hyams et al.  above that energy. The latter corresponds to the
energy dependent analysis of the experimental data of the same reference. On the other hand, PY is
#2E.g., other models [10, 39, 40] having similar physical mechanisms describe that energy region very well indeed.
used up to 0.9 GeV, since at that energy this parameterization agrees well inside errors with , and
above 0.9 GeV the K-matrix of ref. is taken. Given the input functions φI(s) and LI(s), the constants
c2, c0and F0(s1) − L0(s1) can be fixed by the three conditions explained at the end of section 2. The
γγ → π0π0S-wave amplitude FN(s), eq.(2.5), can then be calculated and the total cross section is given
by σ(γγ → π0π0) =
the uncertainty of the cross section due to the variation of φ0(s) above sKas commented in the previous
section. For z = +1 one has the solid line while for z = −1 the dashed line results, both are very close.
This should be compared with the dot-dashed line that is obtained from the approach of refs.[17, 18, 19].
The uncertainty now, by employing eq.(2.4) instead of eq.(2.2) for I = 0, is drastically reduced. This
improvement also implies that our results can be compared with data for√s ? 0.5 GeV. We also show
by the gray band around the solid line the mild influence in our calculations of the uncertainty in the
location of the Adler zero, restricted so that σ(γγ → π0π0) < 400 nb at the f0(980) region (experiment
is ≃ 40 nb).
64πs|FN(s)|2, with β(s) =
?1 − 4m2
π/s. We show in fig.4 the drastic reduction in
δπ(sK) > π, eq.(2.4)
δπ(sK) > π, eq.(2.2)
δπ(sK) < π, eq.(2.2)
Figure 4: The solid line corresponds to z = +1 and the error band is the uncertainty by requiring that σ(γγ →
π0π0) < 400 nb at s1. This line should be compared with the dot-dashed one that would result from the formalism
of ref., including axial vector exchanges. Finally, the dashed line corresponds to z = −1.
Our final σ(γγ → π0π0) is shown in fig.5. We give the corresponding results for CGL(solid) and
PY(dashed), where the band around every line stems from the uncertainties in our approach, which
comprise: the errors in the Hyams et al. , CGL and PY parameterizations (those of the last two
indeed dominate the width of the bands), to use either φ0(s) ≃ δπ(s)0or δπ(s)0− π for s > sK, the
uncertainty in the asymptotic phase and to restrict σ(γγ → π0π0) < 400 nb in the f0(980) region for
z = +1. On top of that, we evaluate the conditions 1 and 2 above from the expressions given by one loop
χPT either by employing fπ= 92.4 MeV or f ≃ 0.94fπ, where the former is the pion decay constant and
the latter is the same but in the SU(2) chiral limit . This amounts to around a 12% of uncertainty in
the evaluation of c0and c2, due to the square dependence on fπ. Note that both choices, fπor f, are
consistent with the precision of the one loop calculation and the variation in the results is an estimate for
higher order corrections. However, the error induced in σ(γγ → π0π0) is much smaller than that from
the other sources of uncertainty and can be neglected when added in quadrature.
In the same figure the dotted line corresponds to one loop CHPT [4, 5] and the dot-dashed one to the
two loop result [6, 7]. The latter is closer to our results but still one observes that the O(p8) corrections
would be sizable. It is worth stressing that if the axial vector exchanges were removed, as in refs.[17, 19],
then our curves would be smaller. This corresponds to the dot-dot-dashed line in fig.5 which is very close
to that of ref. when employing φ0(s) ≃ δπ(s)0− π for s > sK. This curve is evaluated making use of
CGL and ref.. The three experimental points  in the region 0.45 − 0.6 GeV agree well with this
curve. However, once the axial vector are included the curve rises. These three points lie around 1.5
sigmas below the CGL result band, and by more than two sigmas below the PY one. This clearly shows
that more precise experimental data on γγ → π0π0could be used to distinguish between different S-
wave parameterizations. In turn, the next three experimental σ(γγ → π0π0) points, those lying between
0.6−0.75 GeV, agree better when the axial vector resonance contributions are taken into account, as one
should do. As a result of this discussion, more precise experimental data for γγ → π0π0are called for.
χPT to one loop
χPT to two loops
Figure 5: Final results for the γγ → π0π0cross section. Experimental data are from the Crystal Ball Coll. ,
scaled by 1/0.8, as |cosθ| < 0.8 is measured and S-wave dominates. The solid line corresponds to CGL and the
dashed one to PY. The dot-dot-dashed line results after removing the axial vector exchange contributions. The
band along each line represents the theoretical uncertainty. The dotted line is the one loop χPT result [4, 5] and
the dot-dashed one the two loop calculation .
In terms of the calculated FN(s) one can evaluate the σ coupling to γγ, called gσγγ. The dispersion
relation to calculate F0(s) is only valid on the first Riemann sheet. If evaluated on the second Riemann
sheet there would be an extra term due to the σ pole. However, the relation between F0(s) and?F0(s),
F0(s + iǫ) − F0(s − iǫ) = −2iF0(s + iǫ)ρ(s + iǫ)T0
elastic amplitude on the second Riemann sheet and T0
I(s) is the one on the physical Riemann sheet. Due to
continuity when changing from one sheet to the other, F0(s−iǫ) =?F0(s+iǫ) , TI=0
?F0(s) = F0(s)?1 + 2iρ(s)TI=0
the latter on the second sheet, can be easily established by using unitarity above the ππ threshold,
II(s − iǫ) ,
II(s) is the I = 0 S-wave ππ
π≤ s ≤ 4m2
K, ρ(s) = β(s)/16π and ǫ → 0+. In the equation above T0
(s−iǫ) = TI=0
Then, eq.(3.6) can be rewritten as
Around the σ pole, sσ,
sσ− s,?F0(s) =
√2 factor in?F0(s) to match with the gσππ normalization used (the so
with gσππthe σ coupling to two pions such that Γ = |gσππ|2β/16πM, for a narrow enough scalar resonance
of mass M. Notice as well the
called unitary normalization [42, 43, 44]). Then from eqs.(3.7) and (3.8) it follows that
Let us stress that this equation gives the ratio between the residua of the S-wave I = 0 γγ → ππ and
ππ → ππ amplitudes at the σ pole position.
In order to derive specific numbers for the previous ratio in terms of our dispersive approach one needs
to introduce sσ. We take two different values for sσ= (Mσ−iΓσ/2)2. From the studies of Unitary χPT
[42, 43, 44, 45] one has Mσand Γσaround the interval 425-440 MeV. The other values that we will use
are from ref. , Mccl
the following, values that employ the σ pole position of ref.. The corresponding ratios of the residua
given in eq.(3.9) are:
−8MeV and Γccl
−25MeV, where the superscript ccl indicates, in
= (2.10 ± 0.25) × 10−3, sσfrom ref. ,
= (2.06 ± 0.14) × 10−3, sσfrom ref. .(3.10)
Both numbers are very similar despite that the imaginary parts of the two s1/2
result of , with which we shall compare our results later, corresponds to the ratio in eq.(3.10) being
20% bigger at (2.53 ± 0.09) × 10−3with sσof ref..
These ratios of residua at the σ pole position are the well defined predictions that follow from our
improved dispersive treatment of γγ → (ππ)I. However, the radiative width to γγ for a wide resonance
like the σ, though more intuitive, has experimental determinations that are parameterization dependent.
This is due to the non-trivial interplay between background and the broad resonant signal. An unam-
biguous definition is then required [17, 19]. We employ, as in ref., the standard narrow resonance
width formula in terms of gσγγdetermined from eq.(3.9) by calculating the residue at sσ,
differ by ∼ 20%. The
Γ(σ → γγ) =|gσγγ|2
Nevertheless, the determinations of the radiative widths from this expression and those from common
experimental analyses can differ substantially. The following example makes this point clear.
From ref. one obtains |gσππ| = 2.97 −3.01 GeV, corresponding to the square root of the residua of
the I = 0 S-wave ππ amplitude, as in eq.(3.8). If similarly to eq.(3.11), one uses the formula,
the resulting width lies in the range 309 − 319 MeV, that is around a 30% smaller than Γσ≃ 430 MeV
from the pole position of ref.. This is due to the large width of the σ meson which makes the |gσππ|
extracted from the residue of TI=0
II, eq.(3.8), be smaller by around a 15% than the value needed in
eq.(3.12) to obtain Γσ≃ 430 MeV. Similar effects are then also expected in order to extract Γ(σ → γγ)
from the eq.(3.11). Equations similar to this are usually employed in phenomenological fits to data, e.g.
see ref., but with |gσγγ| determined along the real axis. As a result of this discussion, one should allow
a (20−30)% variation between the results obtained from eq.(3.11) and those from standard experimental
analyses that still could deliver a γγ → ππ amplitude in agreement with our more theoretical treatment
for physical values of s.
We shall employ the following values for |gσππ|. First we take |gσππ| = 2.97−3.01 GeV [42, 43, 44, 45].
With this value the resulting two photon width from eqs.(3.10) and (3.11) is
Γ(σ → γγ) = (1.8 ± 0.4) KeV .
σ  is larger by a factor ∼ 1.3 than Γσfrom ref..
We also consider a larger value for |gσππ| since Γccl
One value is
?Γccl(σ → ππ)
Γ(σ → ππ)
= (1.127 ± 0.022)|gσππ| = (3.35 ± 0.08) GeV . (3.14)
This corresponds to the scenario discussed previously in eq.(3.12) with a value 15% lower than
obtained by reproducing Γccl
σ from the pole position using eq.(3.12). If we evaluate with these couplings
the σ → γγ width one obtains from eqs.(3.10) and (3.11), respectively,
(1)(σ → γγ)
(2)(σ → γγ)
Recently, ref. calculated a value Γ(σ → γγ) = (4.09 ± 0.29) KeV also employing sσfrom ref..
This value is larger than Γccl
(2)(σ → γγ) in the previous equation, despite that |gσππ| there used is 3.86 GeV,
very close to |gσππ|ccl
calculating Γ(σ → γγ), except for an extra factor |β(sσ)| ∼ 0.95 in ref.. Of course, they are written
in a different notation.#3The reason for this remaining difference is two fold. As already mentioned
above, ref. does not include axial vector exchanges in evaluating γγ → (ππ)I. It is this omission that
accounts for half of the 20% difference in the ratio of residua, eq.(3.10), mentioned above. The other 10%
comes from improvements delivered by our extra subtraction and our slightly different inputs. Using the
same value for |gccl
eq.(3.16)) than that in .
As a summary of the σ → γγ considerations, from our dispersive approach and sσ of refs.[45, 46]
we obtain a value for the ratio of the residua |gσγγ/gσππ| ∼ (2.1 ± 0.25) × 10−3. This number follows
unambiguously from our study. Other more intuitive, but convention dependent quantities, like the
σ → γγ width calculated from eq.(3.11), are less well determined. These depend critically on the input
value for |gσππ|2and sσ, though they are not required in our dispersive study of γγ → π0π0. We then
determine the values: i) Γ(σ → γγ) = (1.8 ± 0.4) KeV with sσ and |gσππ| ∼ 3 GeV from ref.; ii)
(2)(σ → γγ) = (3.0 ± 0.3) KeV which come by considering sσ
of ref. with an estimated |gσππ| = 3.4 and 3.9 GeV, respectively. Other values could be obtained
from eq.(3.11) by plugging different sσ and |gσππ| in eq.(3.9) in order to estimate |gσγγ|. One should
require that these values are provided from a ππ S-wave I = 0 strong amplitude in agreement with the
experimental phase shifts, see fig.2.
= (3.93 ± 0.08) GeV ,(3.15)
= (2.1 ± 0.3) KeV ,
(3.0 ± 0.3) KeV .= (3.16)
(2). It is worth stressing that both our eq.(3.11) and eq.(7) of ref. are equivalent for
σππ| as in , our resulting value for Γ(σ → γγ) would be around a 40% smaller (as in
(1)(σ → γγ) = (2.1 ± 0.3) KeV and Γccl
#3We want to thank M.R. Pennington for a detailed comparison of his results with ours and interesting discussions.
We have undertaken a dispersive study of the γγ → π0π0reaction. Our approach is based on that of
refs.[18, 19, 17] but using a better behaved Omn` es function for the I = 0 S-wave ππ channel. As a
result, we have been able to reduce drastically the uncertainty regarding the φ0(s) used in this Omn` es
function above the K¯K threshold. Our improvement is equivalent to take three subtractions instead of
the two originally proposed in refs.[18, 19, 17]. We have then used two low energy conditions and a third
constraint in the form of a bound on the f0(980) region so as to fix the three subtraction constants. This
has allowed us to present more accurate results, which might be used to discriminate between different
ππ I = 0 S-wave parametrizations, once more precise data on σ(γγ → π0π0) become available. Further
improvements at the theoretical level rest on a more precise determination of φ0(s) above sKand a more
systematic calculation of LI(s).
We have calculated the ratio of the residua |gσγγ/gσππ| = (2.1±0.25)×10−3with sσfrom refs.[45, 46].
The σ width to γγ was also studied and we stressed its dependence on the sσand |gσππ|2employed, not
used in our dispersive study of γγ → π0π0. One value obtained is Γ(σ → γγ) = (1.8 ± 0.4) KeV with
sσand |gσππ| ≃ 3 GeV from ref.. The others values take sσas given in ref. with Γ(σ → γγ) =
(2.1 ± 0.3) KeV for |gσππ| = 3.4 GeV, and Γ(σ → γγ) = (3.0 ± 0.3) KeV for |gσππ| = 3.9 GeV. The last
two numbers for Γccl(σ → γγ) tell us that the uncertainties in its calculation are still rather large and a
further improvement requires to know precisely |gccl
σππ| from ref..
We would like to thank E. Oset for useful communications. This work has been supported in part by the
MEC (Spain) and FEDER (EC) Grants FPA2004-03470 and Fis2006-03438, the Fundaci´ on S´ eneca (Mur-
cia) grant Ref. 02975/PI/05, the European Commission (EC) RTN Network EURIDICE Contract No.
HPRN-CT2002-00311 and the HadronPhysics I3 Project (EC) Contract No RII3-CT-2004-506078. C.S.
acknowledges the Fundaci´ on S´ eneca by funding his stay at the Departamento de F´ ısica de la Universidad
de Murcia, and the latter by its warm hospitality.
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