Scalar radius of the pion and zeros in the form factor
Scalar radius of the pion and zeros in the form factor
José A. Oller and Luis Roca
Departamento de F́ısica. Universidad de Murcia.
E-30071, Murcia. Spain.
oller@um.es , luisroca@um.es
Abstract
The quadratic pion scalar radius, 〈r2〉πs , plays an important role for present precise deter-
minations of ππ scattering. Recently, Ynduráin, using an Omnès representation of the null
isospin(I) non-strange pion scalar form factor, obtains 〈r2〉πs = 0.75± 0.07 fm2. This value is
larger than the one calculated by solving the corresponding Muskhelishvili-Omnès equations,
〈r2〉πs = 0.61± 0.04 fm2. A large discrepancy between both values, given the precision, then
results. We reanalyze Ynduráin’s method and show that by imposing continuity of the re-
sulting pion scalar form factor under tiny changes in the input ππ phase shifts, a zero in the
form factor for some S-wave I=0 T−matrices is then required. Once this is accounted for,
the resulting value is 〈r2〉πs = 0.65± 0.05 fm2. The main source of error in our determination
is present experimental uncertainties in low energy S-wave I=0 ππ phase shifts. Another
important contribution to our error is the not yet settled asymptotic behaviour of the phase
of the scalar form factor from QCD.
http://arxiv.org/abs/0704.0039v2
1 Introduction
The scalar form factor of the pion, Γπ(t), corresponds to the matrix element
Γπ(t) =
d4x e−i(q
′−q)x〈π(q′)|
muū(x)u(x) +mdd̄(x)d(x)
|π(q)〉 , t = (q′ − q)2 . (1.1)
Performing a Taylor expansion around t = 0,
Γπ(t) = Γπ(0)
t〈r2〉πs +O(t2)
, (1.2)
where 〈r2〉πs is the quadratic scalar radius of the pion.
The quantity 〈r2〉πs contributes around 10% [1] to the values of the S-wave ππ scattering lengths
a00 and a
0 as determined in ref.[1], by employing Roy equations and χPT to two loops. If one
takes into account that this reference gives a precision of 2.2% in its calculation of the scattering
lengths, a 10% of contribution from 〈r2〉πs is a large one. Related to that, 〈r2〉πs is also important
in SU(2)×SU(2) χPT since it gives the low energy constant ℓ̄4 that controls the departure of Fπ
from its value in the chiral limit [2, 3] at leading order correction.
Based on one loop χPT , Gasser and Leutwyler [2] obtained 〈r2〉πs = 0.55 ± 0.15 fm2. This
calculation was improved later on by the same authors together with Donoghue [4], who solved
the corresponding Muskhelishvili-Omnès equations with the coupled channels of ππ and KK̄. The
update of this calculation, performed in ref.[1], gives 〈r2〉πs = 0.61±0.04 fm2, where the new results
on S-wave I=0 ππ phase shifts from the Roy equation analysis of ref.[5] are included. Moussallam
[6] employs the same approach and obtains values in agreement with the previous result.
One should notice that solutions of the Muskhelishvili-Omnès equations for the scalar form
factor rely on non-measured T−matrix elements or on assumptions about which are the channels
that matter. Given the importance of 〈r2〉πs , and the possible systematic errors in the analyses
based on Muskhelishvili-Omnès equations, other independent approaches are most welcome. In
this respect we quote the works [7, 8, 9], and Ynduráin’s ones [10, 11, 12]. These latter works have
challenged the previous value for 〈r2〉πs , shifting it to the larger 〈r2〉πs = 0.75 ± 0.07 fm2. From
ref.[1] the equations,
δa00 = +0.027∆r2 , δa
0 = −0.004∆r2 , (1.3)
give the change of the scattering lengths under a variation of 〈r2〉πs defined by 〈r2〉πs = 0.61(1 +
∆r2) fm
2. For the difference between the central values of 〈r2〉πs given above from refs.[1, 10], one
has ∆r2 = +0.23. This corresponds to δa
0 = +0.006 and δa
0 = −0.001, while the errors quoted
are a00 = 0.220 ± 0.005 and a20 = −0.0444 ± 0.0010. We then adduce about shifting the central
values for the predicted scattering lengths at the level of one sigma.
The value taken for 〈r2〉πs is also important for determining the O(p4) χPT coupling ℓ̄4. The
value of ref.[1] is ℓ̄4 = 4.4±0.2 while that of ref.[10] is ℓ̄4 = 5.4±0.5. Both values are incompatible
within errors.
The papers [10, 11, 12] have been questioned in refs.[13, 14]. The value of the Kπ quadratic
scalar radius, 〈r2〉Kπs , obtained by Ynduráin in ref.[10], 〈r2〉Kπs = 0.31± 0.06 fm2, is not accurate,
because he relies on old experiments and on a bad parameterization of low energy S-wave I=1/2Kπ
phase shifts by assuming dominance of the κ resonance as a standard Breit-Wigner pole [15]. Fur-
thermore, 〈r2〉Kπs was recently fixed by high statistics experiments in an interval in agreement with
the sharp prediction of [15], based on dispersion relations (three-channel Muskhelishvili-Omnès
equations from the T−matrix of ref.[16]) and two-loop χPT [17]. From the recent experiments
[18, 19], one has for the charged kaons [18] 〈r2〉K±πs = 0.235 ± 0.014 ± 0.007 fm2, and for the
neutral ones [19] 〈r2〉KLπs = 0.165 ± 0.016 fm2. The prediction of [15], in an isospin limit, is
〈r2〉Kπs = 0.192± 0.012 fm2, lying just in the middle of the experimental determinations. Another
issue is Ynduráin’s more sound determination of the pionic scalar radius, whose (in)correctness is
not settled yet.
In this paper we concentrate on the approach of Ynduráin [10, 11, 12] to evaluate the quadratic
scalar radius of the pion based on an Omnés representation of the I=0 non-strange pion scalar form
factor. Our main conclusion will be that this approach [10] and the solution of the Muskhelishvili-
Omnès equations [4], with ππ and KK̄ as coupled channels, agree between each other if one
properly takes into account, for some T−matrices, the presence of a zero in the pion scalar form
factor at energies slightly below the KK̄ threshold. Precisely these T−matrices are those used in
[10] and favoured in [11]. Once this is considered we conclude that 〈r2〉πs = 0.63± 0.05 fm2.
The contents of the paper are organized as follows. In section 2 we discuss the Omnès rep-
resentation of Γπ(t) and derive the expression to calculate 〈r2〉πs . This calculation is performed
in section 3, where we consider different parameterizations for experimental data and asymptotic
phases for the scalar form factor. Conclusions are given in the last section.
2 Scalar form factor
The pion scalar form factor Γπ(t), eq.(1.1), is an analytic function of t with a right hand cut,
due to unitarity, for t ≥ 4m2π. Performing a dispersion relation of its logarithm, with the possible
zeroes of Γπ(t) removed, the Omnès representation results,
Γπ(t) = P (t) exp
s(s− t)
. (2.1)
Here, P (t) is a polynomial made up from the zeroes of Γπ(t), with P (0) = Γπ(0). In the previous
equation, φ(s) is the phase of Γπ(t)/P (t), taken to be continuous and such that φ(4m
π) = 0. In
ref.[10] the scalar form factor is assumed to be free of zeroes and hence P (t) is just the constant
Γπ(0) (the exponential factor is 1 for t = 0). Thus,
Γπ(t) = Γπ(0) exp
s(s− t)
. (2.2)
From where it follows that,
〈r2〉πs =
ds . (2.3)
One of the features of the pion scalar form factor of refs.[4, 6, 8], as discussed in ref.[13], is the
presence of a strong dip at energies around the KK̄ threshold. This feature is also shared by the
strong S-wave I=0 ππ amplitude, tππ. This is so because tππ is in very good approximation purely
elastic below the KK̄ threshold and hence, neglecting inelasticity altogether in the discussion that
follows, it is proportional to sin δπe
iδπ , with δπ the S-wave I=0 ππ phase shift. It is an experimental
fact that δπ is very close to π around the KK̄ threshold, as shown in fig.1. Therefore, if δπ = π
happens before the opening of this channel the strong amplitude has a zero at that energy. On
the other hand, if δπ = π occurs after the KK̄ threshold, because inelasticity is then substantial,
see eq.(2.4) below, there is not a zero but a pronounced dip in |tππ|. This dip can be arbitrarily
close to zero if before the KK̄ threshold δπ approaches π more and more, without reaching it.
400 600 800 1000 1200 1400 1600
 (MeV)
Eq. (3.13), [20]
PY [24]
CGL [1]
Sol. A of [27]
Sol. B of [27]
Sol. C of [27]
Sol. D of [27]
Sol. E of [27]
Kaminski et al. [21]
BNL-E865 Coll. [25]
NA48/2 Coll. [26]
300 350 400 450
BL-E865 Coll. [25]
NA48/2 Coll. [26]
PY [24]
CGL [1]
Figure 1: S-wave I = 0 ππ phase shift, δπ(s). Experimental data are from refs.[21, 25, 26, 27].
Because of Watson final state theorem the phase φ(s) in eq.(2.1) is given by δπ(s) below the
KK̄ threshold, neglecting inelasticity due to 4π or 6π states as indicated by experiments [20]. The
situation above the KK̄ threshold is more involved. Let us recall that
tππ = (η e
2iδπ − 1)/2i , (2.4)
with 0 ≤ η ≤ 1 and the inelasticity is given by 1− η2, with η the elasticity coefficient. We denote
by ϕ(s) the phase of tππ, required to be continuous (below 4m
K it is given by δπ(s)). By continuity,
close enough to the KK̄ threshold and above it, η → 1 and then we are in the same situation as in
the elastic case. As a result, because of the Watson final state theorem and continuity, the phase
φ(s) must still be given by ϕ(s). For δπ(sK) < π, sK = 4m
K , ϕ(s) does not follow the increasing
trend with energy of δπ(s) but drops as a result of eq.(2.4), see fig.2 for δπ(sK) < π. This is easily
seen by writing explicitely the real and imaginary parts of tππ in eq.(2.4),
tππ =
η sin 2δπ +
(1− η cos 2δπ) . (2.5)
400 600 800 1000 1200 1400 1600 1800 2000
 (MeV)
1000 1100 1200 1300 1400 1500 1600
 (MeV)
( δπ(sK)< 180
( δπ(sK)> 180
Figure 2: Left panel: Strong phase ϕ(s), eigenvalue phase δ(+)(s) and asymptotic phase φas(s). Right
panel: Integrand of 〈r2〉πs in eq.(3.12) for parameterization I (dashed line) and II (solid line). For more
details see the text. Notice that the uncertainty due to φas(s) is much reduced in the integrand.
The imaginary part is always positive (η < 1 above the KK̄ threshold and 1.1 GeV [20]) while the
real part is negative for δπ < π, but in an interval of just a few MeV the real part turns positive
as soon as δπ > π, fig.1. As a result, ϕ(s) passes quickly from values below but close to π to
the interval [0, π/2]. This rapid motion of φ(s) gives rise to a pronounced minimum of |Γπ(t)| at
this energy, as indicated in ref.[13] and shown in fig.3. The drop in φ(s) becomes more and more
dramatic as δπ(sK) → π− (with the superscript +(−) indicating that the limit is approached from
values above(below), respectively); and in this limit, φ(sk) = ϕ(sK) is discontinuous at sK . This is
easily understood from eq.(2.5). Let us call s1 the point at which δπ(s1) = π with s1 > sK . Close
and above s1, ϕ(s) ∈ [0, π/2], for the reasons explained above, and ϕ(s) has decreased very rapidly
from almost π at the KK̄ threshold to values below π/2 just after s1. Then, in the limit s1 → s+K
one has φ(s−K) = ϕ(s
K) = π on the left, while on the right φ(s
K) = ϕ(s
K) < π/2. As a result ϕ(s)
is discontinuous at s = sK . We stress that this discontinuity of ϕ(s) at sK when δπ(sK) → π−
applies rigorously to φ(sK) as well since η(sK) = 1. This discontinuity at s = sK implies also that
the integrand in the Omnès representation for Γπ(t) develops a logarithmic singularity as,
φ(s−K)− φ(s
, (2.6)
with δ → 0+. When exponentiating this result one has a zero for Γπ(sK) as (δ/sK)ν , ν = (φ(s−K)−
φ(s+K))/π > 0 and δ → 0+. This zero is a necessary consequence when evolving continuously from
δπ(sK) < π to δπ(sK) > π.
#1 This in turn implies rigorously that in the Omnès representation of
Γπ(t), eq.(2.1), P (t) must be a polynomial of first degree for those cases with δπ(sK) ≥ π,#2
P (t) = Γπ(0)
s1 − t
, (2.7)
with s1 the position of the zero. Notice that the degree of the polynomial P (t) is discrete and thus
by continuity it cannot change unless a singularity develops. This is the case when δπ(sK) = π,
changing the degree from 0 to 1. Hence, if δπ(sK) ≥ π for a given tππ, instead of eqs.(2.2) and
(2.3) one must then consider,
Γπ(t) = Γπ(0)
s1 − t
s(s− t)
, (2.8)
〈r2〉πs = −
ds . (2.9)
For those tππ for which δπ(sK) > π then ϕ(s) follows δπ(s) just after the KK̄ threshold and there
is no drop, as emphasized in ref.[11], see fig.2.
Summarizing, we have shown that Γπ(t) has a zero at s1 when δπ(sK) ≥ π as a consequence of
the assumption that φ(s) follows ϕ(s) above the KK̄ threshold, along the lines of ref.[11], and by
imposing continuity in Γπ(t) under small changes in δπ(sK) ≃ π. As a result eqs.(2.8) and (2.9)
should be used in the latter case, instead of eqs.(2.2) and (2.3), valid for δπ(sK) < π. This solution
was overlooked in refs.[10, 11, 12]. We show in appendix A why the previous discussion on the
zero of Γπ(t) for δπ(sK) ≥ π at s1 cannot be applied to all pion scalar form factors, in particular
to the strange one.
If eq.(2.2) were used for those tππ with δπ(sK) ≥ π then a strong maximum of |Γπ(t)| would
be obtained around the KK̄ threshold, instead of the aforementioned zero or the minimum of
refs.[4, 6], as shown in fig.3 by the dashed-dotted line. That is also shown in fig.10 of ref.[22]
or fig.2 of [13]. This is the situation for the Γπ(t) of refs.[10, 11], and it is the reason why 〈r2〉πs
obtained there is much larger than that of refs.[4, 1, 6]. That is, Ynduráin uses eqs.(2.2), (2.3) for
δπ(sK) ≥ π, instead of eqs.(2.8), (2.9) (solid line in fig.3). The unique and important role played
by δπ(sK) (for elastic tππ below the KK̄ threshold) is perfectly recognised in ref.[11]. However, in
this reference the astonishing conclusion that Γπ(t) has two radically different behaviours under
tiny variations of tππ was sustained. These variations are enough to pass from δπ(sK) < π to
δπ(sK) ≥ π [10], while the T− or S−matrix are fully continuous. Because of this instability of the
solution of refs.[10, 11] under tiny changes of δπ(s), we consider ours, that produces continuous
Γπ(t), to be certainly preferred. We also stress that our solutions, either for δπ(sK) ≥ π and
δπ(sK) < π, are the ones that agree with those obtained by solving the Muskhelishvili-Omnès
equations [4, 1, 6] and Unitary χPT [8].
#1It can be shown from eq.(2.5) that φ(s−
) − φ(s+
) = π. Here we are assuming η = 1 for s ≤ sK , which is a
very good approximation as indicated by experiment [20, 21].
#2We are focusing in the physically relevant region of experimental allowed values for δπ(sK), which can be larger
or smaller than π but close to.
0 200 400 600 800 1000 1200
(MeV)
δπ(sK)<π
δπ(sK)>π,  P(t)=Γπ(0)(s1- t)/s1
δπ(sK)>π,  P(t)=Γπ(0)
ref. [8]  (δπ(sK)>π)
PSfrag replacements
Figure 3: |Γπ(t)/Γπ(0)| from eq.(2.2) with δπ(sK) < π, dashed-line, and δπ(sK) > π, dashed-dotted line.
The solid line corresponds to use eq.(2.8) for the latter case. For this figure we have used parameterization
II (defined in section 3) with α1 = 2.28 (dashed line) and 2.20 (dashed-dotted and solid lines). The
dashed-double-dotted line is the scalar form factor of ref.[8] that has δπ(sK) > π.
Let us now show how to fix s1 in terms of the knowledge of δπ(s) with δπ(sK) ≥ π. For this
purpose let us perform a dispersion relation of Γπ(t) with two subtractions,
Γπ(t) = Γπ(0) +
〈r2〉πs t+
ImΓπ(s)
s2(s− t)
ds , (2.10)
From asymptotic QCD [23] one expects that the scalar form factor vanishes at infinity [10, 12],
then the dispersion integral in eq.(2.10) should converge rather fast. Eq.(2.10) is useful because
it tells us that the only point around 1 GeV where there can be a zero in Γπ(t) is at the energy
s1 for which the imaginary part of Γπ(t) vanishes. Otherwise, the integral in the right hand side
of eq.(2.10) picks up an imaginary part and there is no way to cancel it as Γπ(0), 〈r2〉πs and t are
all real. Since |ImΓπ(t)| = |Γπ(t) sin δπ(t)| for t ≤ sK , it certainly vanishes at the point s1 where
δπ(s1) = π. As there is only one zero at such energies, this determines s1 exactly in terms of the
given parameterization for δπ(s).
One could argue against the argument just given to determine s1 that this energy could be
complex. However, this would imply two zeroes at s1 and s
1, and then the degree of P (t) would
be two instead of one. Notice that the degree of the polynomial P (t) is discrete and thus, by
softness in the continuous parameters of the T−matrix, its value should stay at 1 for some open
domain in the parameters with δπ(sK) > π until a discontinuity develops. Physically, the presence
of two zeroes would in turn require that φ(s) → 3π so as to guarantee that Γπ(t) still vanishes as
−1/t, as required by asymptotic QCD [23, 10]. This value for the asymptotic phase seems to be
rather unrealistic as ϕ(s) only reaches 2π at already quite high energy values, as shown in fig.2.
3 Results
Our main result from the previous section is the sum rule to determine 〈r2〉πs ,
〈r2〉πs = −
θ(δπ(sK)− π) +
ds , (3.11)
where θ(x) = 0 for x < 0 and 1 for x ≥ 0. We split 〈r2〉πs in two parts:
〈r2〉πs = QH +QA ,
QH = −
θ(δπ(sK)− π) +
, (3.12)
with sH = 2.25 GeV
2. Reasons for fixing sH to this value are given below.
The main issue in the application of eq.(3.11) is to determine φ(s) in the integrand. Below the
KK̄ threshold and neglecting inelasticity, one has that φ(s) = δπ(s), 4m
π ≤ s ≤ 4m2K . This follows
because of the Watson final state theorem, continuity and the equality φ(4m2π) = δπ(4m
π) = 0.
For practical applications we shall consider the S-wave I=0 ππ phase shifts given by the
K−matrix parameterization of ref.[20] (from its energy dependent analysis of data from 0.6 GeV
up to 1.9 GeV) and the parameterizations of ref.[1] (CGL) and ref.[24] (PY). The resulting δπ(s)
for all these parameterizations are shown in fig.1. We use CGL from ππ threshold up to 0.8 GeV,
because this is the upper limit of its analysis, while PY is used up to 0.9 GeV, because at this
energy it matches well inside the experimental errors with the data of [20]. The K−matrix of
ref.[20] is used for energies above 0.8 GeV, when using CGL below this energy (parameterization
I), and above 0.9 GeV, when using PY for lower energies (parameterization II). We take the pa-
rameterizations CGL and PY as their difference below 0.8 GeV accounts well for the experimental
uncertainties in δπ, see fig.1, and they satisfy constraints from χPT (the former) and dispersion re-
lations (both). The reason why we skip to use the parameterization of ref.[20] for lower energies is
because one should be there as precise as possible since this region gives the largest contribution to
〈r2〉πs , as it is evident from the right panel of fig.2. It happens that the K−matrix of [20], that fits
data above 0.6 GeV, is not compatible with data from Ke4 decays [25, 26]. We show in the insert
of fig.1 the comparison of the parameterizations CGL and PY with the Ke4 data of [25, 26]. We
also show in the same figure the experimental points on δπ from refs.[20, 21, 27]. Both refs.[20, 21]
are compatible within errors, with some disagreement above 1.5 GeV. This disagreement does not
affect our numerical results since above 1.5 GeV we do not rely on data.
The K−matrix of ref.[20] is given by,
Kij(s) = αiαj/(x1 − s) + βiβj/(x2 − s) + γij , (3.13)
where
1 = 0.11± 0.15 x
2 = 1.19± 0.01
α1 = 2.28± 0.08 α2 = 2.02± 0.11
β1 = −1.00± 0.03 β2 = 0.47± 0.05
γ11 = 2.86± 0.15 γ12 = 1.85± 0.18 γ22 = 1.00± 0.53 ,
(3.14)
with units given in appropriate powers of GeV. In order to calculate the contribution from the
phase shifts of this K−matrix we generate Monte-Carlo gaussian samples, taking into account the
errors shown in eq.(3.14), and evaluate QH according to eq.(3.12). The central value of δπ(sK)
for the K−matrix of ref.[20] is 3.05, slightly below π. When generating Monte-Carlo gaussian
samples according to eq.(3.14), there are cases with δπ(sK) ≥ π, around 30% of the samples. Note
that for these cases one also has the contribution −6/s1 in eq.(3.11).
The application of Watson final state theorem for s > 4m2K is not straightforward since inelastic
channels are relevant. The first important one is the KK̄ channel associated in turn with the
appearance of the narrow f0(980) resonance, just on top of its threshold. This implies a sudden
drop of the elasticity parameter η, but it again rapidly raises (the f0(980) resonance is narrow
with a width around 30 MeV) and in the region 1.12 . s . 1.52 GeV2 is compatible within
errors with η = 1 [20, 21]. For η ≃ 1, the Watson final state theorem would imply again that
φ(s) = ϕ(s), but, as emphasized by [13], this equality only holds, in principle, modulo π. The
reason advocated in ref.[13] is the presence of the region sK < s < 1.1
2 GeV2 where inelasticity
can be large, and then continuity arguments alone cannot be applied to guarantee the equality
φ(s) ≃ ϕ(s) for s & 1.12 GeV2. This argument has been proved in ref.[11] to be quite irrelevant
in the present case. In order to show this a diagonalization of the ππ and KK̄ S−matrix is done.
These channels are the relevant ones when η is clearly different from 1, between 1 and 1.1 GeV.
Above that energy one also has the opening of the ηη channel and the increasing role of multipion
states.
We reproduce here the arguments of ref.[11], but deliver expressions directly in terms of the
phase shifts and elasticity parameter, instead of K−matrix parameters as done in ref.[11]. For
two channel scattering, because of unitarity, the T−matrix can be written as:
(ηe2iδπ − 1) 1
1− η2ei(δπ+δK)
1− η2ei(δπ+δK) 1
(ηe2iδK − 1)
, (3.15)
with δK the elastic S-wave I=0KK̄ phase shift. In terms of the T -matrix the S-wave I=0 S−matrix
is given by,
S = I + 2iT , (3.16)
satisfying SS† = S†S = I. The T -matrix can also be written as
T = Q1/2
K−1 − iQ
Q1/2 , (3.17)
where the K−matrix is real and symmetric along the real axis for s ≥ 4m2π and Q = diag(qπ, qK),
with qπ(qK) the center of mass momentum of pions(kaons). This allows one to diagonalize K with
a real orthogonal matrix C, and hence both the T− and S−matrices are also diagonalized with
the same matrix. Writing,
cos θ sin θ
− sin θ cos θ
, (3.18)
one has
cos θ =
[(1− η2)/2]1/2
1− η2 cos2∆− η| sin∆|
1− η2 cos2∆
]1/2 ,
sin θ = −
sin∆√
1 + (1− η2) cot2∆
1− η2 cos2∆− η| sin∆|
1− η2 cos2∆
]1/2 , (3.19)
with ∆ = δK − δπ. On the other hand, the eigenvalues of the S−matrix are given by,
e2iδ(+) = S11
1 + e2i∆
1 + (1− η2) cot2∆
(3.20)
e2iδ(−) = S22
1 + e−2i∆
1 + (1− η2) cot2∆
. (3.21)
The eigenvalue phase δ(+) satisfies δ(+)(sK) = δπ(sK). The expressions above for exp 2iδ(+) and
exp 2iδ(−) interchange between each other when tan∆ crosses zero and simultaneously the sign in
the right hand side of eq.(3.19) for sin θ changes. This diagonalization allows to disentangle two
elastic scattering channels. The scalar form factors attached to every of these channels, Γ′1 and
Γ′2, will satisfy the Watson final state theorem in the whole energy range and then one has,
= CTQ1/2Γ = CTQ1/2
Γπ = q
λ cos θ |Γ′1|eiδ(+) ± sin θ |Γ′2|eiδ(−)
ΓK = q
± cos θ |Γ′2|eiδ(−) − λ sin θ |Γ′1|eiδ(+)
. (3.22)
The ± in front of |Γ′2| is due to the fact that Γ′2 = 0 at sK , as follows from its definition in the
equation above. Since Watson final state theorem only fixes the phase of Γ′2 up to modulo π, and
the phase is not defined in the zero, we cannot fix the sign in front at this stage. Next, Γ′1 has a
zero at s1 when δπ(sK) ≥ π. For this case, −|Γ′1| must appear in the previous equation, so as to
guarantee continuity of its ascribed phase, and this is why λ = (−1)θ(δπ(sK)−π).
Now, when η → 1 then sin θ → 0 as
(1− η)/2 and φ(s) is then the eigenvalue phase δ(+).
This eigenvalue phase can be calculated given the T−matrix. For those T−matrices employed
here, and those of refs.[10, 11, 4, 13], δ(+)(s) follows rather closely ϕ(s) in the whole energy range.
This is shown in fig.2 and already discussed in detail in ref.[11]. In this way, one guarantees
that φ(s) and ϕ(s) do not differ between each other in an integer multiple of π when η ≃ 1,
1.12 . s . 1.52 GeV2.
For the calculation of QH in eq.(3.12) we shall equate φ(s) = ϕ(s) for 4m
K < s < 1.5
2 GeV2.
Denoting,
= I1 + I2 + I3 ,
∫ 1.12
ds , (3.23)
QH ≃ IH −
θ(δπ(sK)− π) . (3.24)
Now, eq.(3.22) can also be used to estimate the error of approximating φ(s) by ϕ(s) in the range
4m2K < s < 1.5
2 GeV2 to calculate I2 and I3 as done in eq.(3.23). We could have also used δ(+)(s)
in eq.(3.23). However, notice that when η . 1 then ϕ(s) ≃ δ(+)(s) and when inelasticity could be
substantial the difference between δ(+)(s) and ϕ(s) is well taken into account in the error analysis
that follows. Remarkably, consistency of our approach also requires φ(s) to be closer to ϕ(s) than
to δ(+)(s). The reason is that ϕ(s) for δπ(sK) ≥ π is in very good approximation the ϕ(s) for
δπ(sK) < π plus π, this is clear from fig.2. This difference is precisely the required one in order
to have the same value for 〈r2〉πs either for δπ(sK) < π or δπ(sK) ≥ π from eq.(3.11). However, the
difference for δ(+)(s) between δπ(sK) < π and δπ(sK) ≥ π is smaller than π. Indeed, we note that
φ(s) follows closer ϕ(s) than δ(+)(s) for the explicit form factors of refs.[8, 4].
Let us consider first the range 1.12 < s < 1.52 GeV2 where from experiment [20] η ≃ 1 within
errors. With ǫ = ± tan θ|Γ′2/Γ′1| and ρ = δ(−) − δ(+), eq.(3.22) allows us to write,
Γπ = λ cos θ |Γ′1|eiδ(+)(1 + ǫ cos ρ)
1 + i
ǫ sin ρ
1 + ǫ cos ρ
. (3.25)
When η → 1 then ǫ → 0, according to the expansion,#3
tan θ =
(1− η)/2
1− 1 + 3 cos 2∆
8 sin2∆
(1− η)
(1− η)5/2
. (3.26)
Rewriting,
1 + i
ǫ sin ρ
1 + ǫ cos ρ
= exp
ǫ sin ρ
1 + ǫ cos ρ
+O(ǫ2) , (3.27)
which from eqs.(3.25) and (3.27) implies a shift in δ(+) because of inelasticity effects,
δ(+) → δ(+) +
ǫ sin ρ
1 + ǫ cos ρ
. (3.28)
#3The the ratio |Γ′2/Γ′1|, present in ǫ, is not expected to be large since the f0(1300) couples mostly to 4π and
similarly to ππ and KK̄, and the f0(1500) does mostly to ππ [28].
Using η = 0.8 in the range 1.12 . s . 1.52 GeV, η ≃ 1 from the energy dependent analysis of
ref.[20] given by the K−matrix of eq.(3.13), one ends with ǫ ≃ 0.3. Taking into account that δ(+)
is larger than & 3π/2 for δπ(sK) ≥ π (in this case δ(+) ≃ δπ), and around 3π/4 for δπ(sK) < π,
see fig.2, one ends with relative corrections to δ(+) around 6% for the former case and 13% for the
latter. Although the K−matrix of ref.[20], eq.(3.13), is given up to 1.9 GeV, one should be aware
that to take only the two channels ππ and KK̄ in the whole energy range is an oversimplification,
particularly above 1.2 GeV. Because of this we finally double the previous estimate. Hence I3 is
calculated with a relative error of 12% for δπ(sK) ≥ π and 25% for δπ(sK) < π.
In the narrow region between sK < s < 1.1
2 GeV2, η can be rather different from 1, due to
the f0(980) that couples very strongly to the just open KK̄ channel. However, from the direct
measurements of ππ → KK̄ [29], where 1 − η2 is directly measured,#4 one has a better way to
determine η than from ππ scattering [20, 21]. It results from the former experiments, as shown
also by explicit calculations [30, 31, 32], that η is not so small as indicated in ππ experiments [20],
and one has η ≃ 0.6 − 0.7 for its minimum value. Employing η = 0.6 in eq.(3.28) then ǫ ≃ 0.5.
Taking δ(+) around π/2 when δπ(sK) < π this implies a relative error of 30%. For δπ(sK) ≥ π one
has instead δ(+) & π, and a 15% of estimated error. Regarding the ratio of the moduli of form
factors entering in ǫ we expect it to be . 1 (see appendix A). Therefore, our error in the evaluation
of I2 is estimated to be 30% and 15% for the cases δπ(sK) < π and δπ(sK) ≥ π, respectively.
As a result of the discussion following eq.(3.24), we consider that the error estimates done
for I2 and I3 in the case δπ(sK) < π are too conservative and that the relative errors given for
δπ(sK) > π are more realistic. Nonetheless, since the absolute errors that one obtains for I2 and
I3 are the same in both cases (because I2 and I3 for δπ(sK) < π are around a factor 2 smaller than
those for δπ(sK) ≥ π) we keep the errors as given above. To the previous errors for I2 and I3 due
to inelasticity, we also add in quadrature the noise in the calculation of QH due to the error in
tππ from the uncertainties in the parameters of the K−matrix eqs.(3.13), (3.14), and those in the
parameterizations CGL and PY.
We finally employ for s > 2.25 GeV2 the knowledge of the asymptotic phase of the pion scalar
form factor in order to evaluate QA in eq.(3.12). The function φ(s) is determined so as to match
with the asymptotic behaviour of Γπ(t) as −1/t from QCD. The Omnès representation of the
scalar form factor, eqs.(2.2) and (2.8), tends to t−q/π and t−q/π+1 for t → ∞, respectively. Here, q
is the asymptotic value of the phase φ(s) when s → ∞. Hence, for δπ(sK) < π the function φ(s)
is then required to tend to π while for δπ(sK) ≥ π the asymptotic value should be 2π. The way
φ(s) is predicted to approach the limiting value is somewhat ambiguous [11, 12],
φas(s) ≃ π
log(s/Λ2)
. (3.29)
In this equation, 2dm = 24/(33 − 2nf ) ≃ 1, Λ2 is the QCD scale parameter and n = 1, 2 for
δπ(4m
K) < π, ≥ π, respectively. The case n = 2 was not discussed in refs.[10, 11, 12, 13, 14] for
the form factor given in eq.(1.1). There is as well a controversy between [14] and [12] regarding
the ± sign in eq.(3.29). If leading twist contributions dominate [11, 12] then the limiting value is
reached from above and one has the plus sign, while if twist three contributions are the dominant
ones [14] the minus sign has to be considered [12]. In the left panel of fig.2 we show with the wide
#4Neglecting multipion states.
φ(s) I I II II
δπ(sK) ≥ π < π ≥ π < π
I1 0.435± 0.013 0.435± 0.013 0.483± 0.013 0.483± 0.013
I2 0.063± 0.010 0.020± 0.006 0.063± 0.010 0.020± 0.006
I3 0.143± 0.017 0.053± 0.013 0.143± 0.017 0.053± 0.013
QH 0.403± 0.024 0.508± 0.019 0.452± 0.024 0.554± 0.019
QA 0.21± 0.03 0.10± 0.03 0.21± 0.03 0.10± 0.03
〈r2〉πs 0.61± 0.04 0.61± 0.04 0.66± 0.04 0.66± 0.04
Table 1: Different contributions to 〈r2〉πs as defined in eqs.(3.12) and (3.23). All the units are fm2.
In the value for 〈r2〉πs the errors due to I1, I2, I3 and QA are added in quadrature.
bands the values of φ(s)as for s > 2.25 GeV
2 from eq.(3.29), considering both signs, for n = 1
(δπ(sK) < π) and 2 (δπ(sK) ≥ π). We see in the figure that above 1.4−1.5 GeV (1.96−2.25 GeV2)
both ϕ(s) and φ(s)as phases match and this is why we take sH = 2.25 GeV
2 in eq.(3.11), similarly
as done in refs.[10, 11]. In this way, we also avoid to enter into hadronic details in a region where
η < 1 with the onset of the f0(1500) resonance. The present uncertainty whether the + or − sign
holds in eq.(3.29) is taken as a source of error in evaluating QA. The other source of uncertainty
comes from the value taken for Λ2, 0.1 < Λ2 < 0.35 GeV2, as suggested in ref.[10]. From fig.2 it is
clear that our error estimate for φas(s) is very conservative and should account for uncertainties
due to the onset of inelasticity for energies above 1.4 − 1.5 GeV and to the appearance of the
f0(1500) resonance. In the right panel of fig.2 we show the integrand for 〈r2〉πs , eq.(3.12), for
parameterization I (dashed line) and II (solid line). Notice as the large uncertainty in φas(s) is
much reduced in the integrand as it happens for the higher energy domain.
In table 1 we show the values of I1, I2, I3, QH , QA and 〈r2〉πs for the parameterizations I and
II and for the two cases δπ(sK) ≥ π and δπ(sK) < π. This table shows the disappearance of the
disagreement between the cases δπ(sK) ≥ π and δπ(sK) < π from the ππ and KK̄ T−matrix
of eq.(3.13), once the zero of Γπ(t) at s1 < sK is taken into account for the former case. This
disagreement was the reason for the controversy between Ynduráin and ref.[13] regarding the value
of 〈r2〉πs . The fact that the parameterization II gives rise to a larger value of 〈r2〉πs than I is because
PY follows the upper δπ data below 0.9 GeV, while CGL follows lower ones, as shown in fig.1.
The different errors in table 1 are added in quadrature. The final value for 〈r2〉πs is the mean
between those of parameterizations I and II and the error is taken such that it spans the interval
of values in table 1 at the level of two sigmas. One ends with:
〈r2〉πs = 0.63± 0.05 fm
2 . (3.30)
The largest sources of error in 〈r2〉πs are the uncertainties in the experimental δπ and in the
asymptotic phase φas. This is due to the fact that the former are enhanced because of its weight
in the integrand, see fig.2, and the latter due to its large size.
Our number above and that of refs.[1, 4], 〈r2〉πs = 0.61± 0.04 fm2, are then compatible. On the
other hand, we have also evaluated 〈r2〉πs directly from the scalar form factor obtained with the
dynamical approach of ref.[8] from Unitary χPT and we obtain 〈r2〉πs = 0.64±0.06 fm2, in perfect
agreement with eq.(3.30). Notice that the scalar form factor of ref.[8] has δπ(sK) > π and we have
checked that it has a zero at s1, as it should. This is shown in fig.3 by the dashed-double-dotted line.
The value 〈r2〉πs = 0.75±0.07 fm2 from refs.[10, 11] is much larger than ours because the possibility
of a zero at s1 was not taking into account there and other solution was considered. This solution,
however, has an unstable behaviour under the transition δπ(sK) = π− 0+ to δπ(sK) = π+0+ and
it cannot be connected continuously with the one for δπ(sK) < π. Our solution for Γπ(t) from
Ynduráin’s method does not have this unstable behaviour and it is continuous under changes in
the values of the parameters of the K−matrix, eqs.(3.13) and (3.14). This is why, from our results,
it follows too that the interesting discussion of ref.[11], regarding whether δπ(sK) < π or ≥ π,
is not any longer conclusive to explain the disagreement between the values of refs.[10, 11] and
ref.[1] for 〈r2〉πs .
We can also work out from our determination of 〈r2〉πs , eq.(3.30), values for the O(p4) SU(2)
χPT low energy constant ℓ̄4. We take the two loop expression in χPT for 〈r2〉πs [1],
〈r2〉πs =
8π2f 2π
ℓ̄4 −
+ ξ∆r
, (3.31)
where fπ = 92.4 MeV is the pion decay constant, ξ = (Mπ/4πfπ)
2 and Mπ is the pion mass. First,
at the one loop level calculation ∆r = 0 and then one obtains,
ℓ̄4 = 4.7± 0.3 . (3.32)
We now move to the determination of ℓ̄4 based on the full two loop relation between 〈r2〉πs and ℓ̄4.
The expression for ∆r can be found in Appendix C of ref.[1]. ∆r is given in terms of one O(p6)
χPT counterterm, r̃S2 , and four O(p4) ones. Taking the values of all these parameters, but for ℓ̄4,
from ref.[1], and solving for ℓ̄4, one arrives to
ℓ̄4 = 4.5± 0.3 . (3.33)
This number is in good agreement with ℓ̄4 = 4.4± 0.2 [1].
Ref.[12] also points out that one loop χPT fits to the S-, P- and D-wave scattering lengths and
effective ranges give rise to much larger values for ℓ̄2 and ℓ̄4 than those of ref.[1]. For more details
we refer to [12].
4 Conclusions
In this paper we have addressed the issue of the discrepancies between the values of the quadratic
pion scalar radius of Leutwyler et al. [4, 13], 〈r2〉πs = 0.61 ± 0.04 fm2, and Ynduráin’s papers
[10, 11, 12], 〈r2〉 = 0.75±0.07 fm2. One of the reasons of interest for having a precise determination
of 〈r2〉πs is its contribution of a 10% to a00 and a20, calculated with a precision of 2% in ref.[1]. The
value taken for 〈r2〉πs is also important for determining the O(p4) χPT coupling ℓ̄4.
From our study it follows that Ynduráin’s method to calculate 〈r2〉πs [10, 11], based on an
Omnès representation of the pion scalar form factor, and that derived by solving the two(three)
coupled channel Muskhelishvili-Omnès equations [4, 1, 6], are compatible. It is shown that the
reason for the aforementioned discrepancy is the presence of a zero in Γπ(t) for those S-wave I=0
T−matrices with δπ(sK) ≥ π and elastic below the KK̄ threshold, with sK = 4m2K . This zero
was overlooked in refs.[10, 11], though, if one imposes continuity in the solution obtained under
tiny changes of the ππ phase shifts employed, it is necessarily required by the approach followed
there. Once this zero is taken into account the same value for 〈r2〉πs is obtained irrespectively
of whether δπ(sK) ≥ π or δπ(sK) < π. Our final result is 〈r2〉πs = 0.63 ± 0.05 fm2. The error
estimated takes into account experimental uncertainty in the values of δπ(s), inelasticity effects
and present ignorance in the way the phase of the form factor approaches its asymptotic value π,
as predicted from QCD. Employing our value for 〈r2〉πs we calculate ℓ̄4 = 4.5 ± 0.3. The values
〈r2〉πs = 0.61± 0.04 fm2 and ℓ̄4 = 4.5± 0.3 of ref.[1] are then in good agreement with ours.
Acknowledgements
We thank Miguel Albaladejo for providing us numerical results from some unpublished T−matrices
and Carlos Schat for his collaboration in a parallel research. We also thank F.J. Ynduráin for
long discussions and B. Anathanarayan, I. Caprini, G. Colangelo, J. Gasser and H. Leutwyler for
a critical reading of a previous version of the manuscript. This work was supported in part by the
MEC (Spain) and FEDER (EC) Grants FPA2004-03470 and Fis2006-03438, the Fundación Séneca
(Murcia) grant Ref. 02975/PI/05, the European Commission (EC) RTN Network EURIDICE
under Contract No. HPRN-CT2002-00311 and the HadronPhysics I3 Project (EC) Contract No
RII3-CT-2004-506078.
Appendices
A Coupled channel dynamics
We take ππ and KK̄ coupled channels and denote by F1 and F2 their respective I=0 scalar form
factors. Unitarity requires,
ImFi =
Fjρjθ(t− s′j)t∗ji , (A.1)
where ||tij|| is the I=0 S-wave T−matrix, s′i is the threshold energy square of channel i and
ρi = qi/8π
s, with qi its center of mass three momentum.
A general solution to the previous equations is given by,
F = T G , F =
, G =
, (A.2)
where the functions Gi(t) do not have right hand cut. This equation is interesting as tells us that
if pion dynamics dominate, |G1| >> |G2|, then F1 ≃ G1t11 and the form factor phase φ(s) follows
ϕ(s). As a result, like t11, it has a zero at s1 below the KK̄ threshold for δπ(sK) ≥ π, as shown
in section 3. On the other hand, if kaon dynamics dominates, |G2| >> |G1|, then F1 ≃ G2t12 and
φ(s) follows the phase of t12, that above the KK̄ threshold is clearly above π. This is why for
the pion strange scalar form factor there is no zero at s1 . sK for δπ(sK) ≥ π, indeed there is a
maximum like that shown in fig.3 by the dashed-dotted line.
As in section 3 we now proceed to the diagnolization above the KK̄ threshold of the renormal-
ized T−matrix T ′,
T ′ = ρ1/2Tρ1/2 , ρ =
, T̃ = CTT ′C =
t̃11 0
0 t̃22
t̃11 = sin δ(+)e
iδ(+) , t̃22 = sin δ(−)e
iδ(−) . (A.3)
The corresponding diagonal form factors F ′1 and F
2, collected in the vector F
′, are
F ′ = CTρ1/2F = T̃CTρ−1/2G =
cos θ ρ
1 G1 − sin θ ρ
t̃11{
sin θ ρ
1 G1 + cos θ ρ
 . (A.4)
The previous expressions allow to obtaining F1 directly in terms of the eigenphases and with clean
separation between pion, G1, and kaon dynamics, G2. From eq.(3.22) it follows that,
cos2 θ ρ−1G1 − cos θ sin θ ρ−1/22 ρ
sin2 θ ρ−11 G1 + cos θ sin θ ρ
t̃22 .
(A.5)
For δπ(sK) ≥ π typical values, somewhat above the KK̄ threshold, are e2iδ(+) ≃ +i, e2iδ(−) ≃ −i
and sin θ > 0. For dominance of G1 one has F1/G1 ≃ ρ−11 (i + cos 2θ)/2 while for dominance
of G2 the result is F1/G2 ≃ − sin θ cos θ ρ−1/22 ρ
1 < 0. The factors G1,2 do not introduce
any change in φ(s) with respect to its value before the opening of the KK̄ threshold since they
are smooth functions in s.#5 In both cases the phase φ(s) is larger than π and F1 follows the
upper trend of phases shown in fig.2 (note that in this case t̃11 is in the first quadrant though
δπ > π). Now, doing the same exercise for δπ(sK) < π, one has the typical values e
2iδ(+) ≃ −i,
e2iδ(−) ≃ +i and sin θ < 0. For pion dominance then F1/G1 ≃ ρ−11 (i− cos 2θ)/2 and for the kaon
one F1/G2 ≃ + sin θ cos θρ−1/22 ρ
1 < 0. Thus, in the former case the phase is & π/2, and follows
the lower trend of phases of fig.2, while in the latter is & π and follows again the upper trend (this
is the case of the strange scalar form factor).
The demonstration in section 3 that φ(sK) is discontinuous in the limit δπ(sK) → π− by taking
s1 → s+K , cannot be applied in the case of kaon dominance (e.g. pion strange scalar form factor).
From eq.(A.5) it follows that,
F1(t) ≃ − cos θ sin θρ−1/22 ρ
t̃11 − t̃22
. (A.6)
The point is that t̃22 for t ≥ s1 (s1 → s+K) is of size comparable with that of t̃11 (both tend to zero)
and the phase does not follow δ(+). This is not the case for pion dominance because for s1 → s+K
then sin2 θ → 0, F1(t) ≃ cos2 θ ρ−11 G1t̃11, eq.(A.5), and φ(s) follows δ(+).
From eq.(A.4) we can also write |Γ′2/Γ′2| ≃ |t̃11 tan θ/t̃22| for the case of pion dominance. Since
typically |t̃11/t̃22| ≃ 1, as shown above for energies somewhat above the KK̄ threshold, then
|Γ′2/Γ′1| ≃ | tan θ| < 1. This is why we consider that equating it to 1 in section 3 is a conservative
estimate.
#5Due to the Adler zeroes this is not necessarily case close to the ππ threshold.
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http://arxiv.org/abs/hep-ex/0610071
	Introduction
	Scalar form factor
	Results
	Conclusions
	Coupled channel dynamics
