Comment on “Mass and Width of the Lowest
Resonance in QCD”
In a recent Letter [1], in which no use is made of QCD,
I. Caprini, G. Colangelo, and H. Leutwyler (CCL) re-
peated an unmentioned analysis of ππ scattering from
1973 [2], based on the Roy equations (REs), to make
out a case for the existence of a scalar I = 0 reso-
nance f0(441), listed in the PDG tables [3] as f0(600)
and known as σ-meson. The primary aspect result-
ing of the CCL analysis is the claimed model- and
parametrization-independent determination of a σ-pole
mass of (441+16
i 544+18
) MeV implying unprece-
dented small error bars. Moreover, the latter result is
incompatible with very recent experimental findings, i.e.,
(500±30− i (264±30)) MeV [4] and (541±39− i (252±
42)) MeV [5, 6], as well as with a combined theoretical
analysis yielding ((476–628)− i (226–346)) MeV [7]. The
present comment will be devoted to complement a re-
cent experimental discussion [4] of short-comings in the
CCL analysis, by presenting theoretical arguments point-
ing at a serious flaw in the theoretical formalism used
by CCL, and also at the unlikeliness of their tiny er-
ror bars in the σ mass and width. The simplest way to
identify this flaw in Ref. [1] also present in the corre-
sponding results [8] of S. Descotes-Genon and B. Mous-
sallam (DM) on the scalar meson K∗0 (800) in the con-
text of Roy-Steiner equations (RSEs), is to recall a warn-
ing statement by G.F. Chew and S. Mandelstam (CM)
from 1960 (see footnote 6 of Ref. [9]). CM state that
if a strongly interacting particle with the same quantum
numbers as a pair of pions should be found, then corre-
sponding poles must be added to the double-dispersion
representation, whether or not the new particle is inter-
preted as a two-pion bound state. It should be empha-
sized that this statement does not only apply to possi-
ble bound-state (BS) poles of the S- or T-matrix in the
physical sheet (PS) of the complex s-plane, but also to
any kind of virtual BS poles and resonance poles in the
unphysical sheet (US). This is justified from first princi-
ples by reviewing briefly how dispersion relations (DRs)
are to be derived on the basis of Cauchy’s integral for-
mula t(s) = (2πi)−1
dz t(z)/(z − s) which holds for
a function t(s) analytic in the domain encirculated by
the closed integration contour. As the so-called “match-
ing point” of CCL (and DM) is located in the US, the
closed integration contour yielding the REs/RSEs must
extend also to the US where the S- and T-matrix poles
for scalar isoscalar ππ-scattering are found. Excluding
these poles situated at sj (j = 1, . . . , n) from the inte-
gration contour and assuming t(s → ∞) → 0 sufficiently
fast one obtains the well known (here) unsubtracted DRs
t(s) =
rj/(s− sj)−
dz Im[t(z)]/(s− z + iε),
where L/R denotes the left-/right-hand cut, and rj is the
residue of t(s) at the corresponding pole sj . According
to CCL, REs/RSEs are twice-subtracted DRs yielding
t(s) = t(s0) + (s− s0) t
′(s0) +
(s0 − s)
(s0 − sj)2(s− sj)
(s0 − s)
2 Im[t(z)]
(s0 − z + iε)2(s− z + iε)
, (1)
where the subtraction point s0 used by CCL appears to
be the ππ threshold, as CCL perform the identification
t(s0) = a
0 and t
′(s0) = (2a
0)/(12m
π) with a
0 being
S-wave scattering lengths for isospin I = 0, 2. It is now
easy to see that the REs/RSEs considered by CCL and
DM disregard the pole terms (PTs) in the DRs (yielding
rj = 0), despite the presence of poles in the US that are
claimed to exist by observing respective S-matrix zeros
in the PS. As the s-dependence of the disregarded σ- and
f0(980)-PTs in the vicinity of the ππ- and KK-theshold
is clearly non-linear, it is to be expected on grounds of
dispersion theory that the S-matrix poles predicted by
CCL will not coincide with the actual ones to be deter-
mined yet by CCL for self-consistency reasons. An anal-
ogous statement applies to the results of DM. Moreover
will the inclusion of PTs in REs/RSEs not only reinstate
dispersion theoretic self-consistency, yet also yield a sig-
nificant change in the resulting σ- andK∗
(800)-pole posi-
tions, which unfortunately will enter now via the PTs as
unknown parameters the REs/RSEs to be solved. Hence
the inclusion of PTs in REs/RSEs will yield an uncer-
tainty of pole positions which is likely to be of the order
of the one estimated in Ref. [4] and therefore much larger
than the error bars presently claimed by CCL and DM
being even without taking into account PTs for at least
two reasons clearly parametrization-dependent: (1) the
extrapolation of the two particle phase space to the com-
plex s-plane and below threshold invoked by CCL and
DM is known to be speculative and even unphysical as
it yields e.g. in the approach of DM scattering below
the pseudo-threshold; (2) standard chiral perturbation
theory (ChPT) disregarding (yet) non-perturbative PTs
relates claimed values for scalar scattering lengths and
their (too) tiny error bars entering REs/RSEs lacking
(yet) PTs to scalar square radii
the presently used
(too) high values of which yield chiral symmetry breaking
(ChSB) of the order of 6-8% being much larger than 3%
as observed in Nature. A revision of the analysis of CCL
and DM by taking into account PTs in REs/RSEs and
ChPT would be highly desirable to reconcile their results
with Refs. [4]-[7] and to improve the poor description of
the resonance K∗(892) in the approach of DM.
Frieder Kleefeld 1,2,3
1 Present address: Pfisterstr. 31, 90762 Fürth, Germany
2 Doppler & Nucl. Physics Institute (Dep. Theor. Phys.),
Academy of Sciences of Czech Republic; collaborator of
the CFIF, Instituto Superior Técnico, LISBOA, Portugal
3 Electronic address: kleefeld@cfif.ist.utl.pt
http://arxiv.org/abs/0704.1337v1
[1] I. Caprini et al., Phys. Rev. Lett. 96, 132001 (2006).
[2] M. R. Pennington et al., Phys. Rev. D 7, 1429,2591 (1973).
[3] W. M. Yao et al. [PDG], J. Phys. G 33, 1 (2006).
[4] D. V. Bugg, J. Phys. G 34, 151 (2007).
[5] M. Ablikim et al. [BES], Phys. Lett. B 598, 149 (2004).
[6] D. V. Bugg, AIP Conf. Proc. 814, 78 (2006).
[7] E. van Beveren et al., Phys. Lett. B 641, 265 (2006).
[8] S. Descotes-Genon et al., Eur. Phys. J. C 48, 553 (2006).
[9] G. F. Chew, S. Mandelstam, Phys. Rev. 119, 467 (1960).
This work has been supported by the FCT of the
Ministério da Ciência, Tecnologia e Ensino Superior
of Portugal, under contracts POCI/FP/63437/2005,
PDCT/FP/63907/2005 and the Czech project LC06002.
Conversations with E. van Beveren, D. V. Bugg,
J. Fischer, A. Moussallam, G. Rupp, M. D. Scadron and
M. Znojil are gratefully acknowledged.
	References

  I. Caprini's, G. Colangelo's, and H. Leutwyler's (CCL) article "Mass and
Width of the Lowest Resonance in QCD", Phys. Rev. Lett. 96, 132001 (2006)
[hep-ph/0512364], is critically reviewed. The present comment is devoted to
complement a recent experimental discussion (D.V. Bugg, J. Phys. G 34, 151
(2007) [hep-ph/0608081]) of short-comings in the CCL analysis, by presenting
theoretical arguments pointing at a serious flaw in the theoretical formalism
used by CCL, and also at the unlikeliness of their tiny error bars in the
sigma-meson mass and width. The criticism made in the comment applies
analogously to the analysis on the kappa-meson mass performed in the article
"The K0*(800) scalar resonance from Roy-Steiner representations of pi K
scattering" published as S. Descotes-Genon and B. Moussallam, Eur. Phys. J. C
48, 553 (2006) [hep-ph/0607133].

Comment on “Mass and Width of the Lowest
Resonance in QCD”
In a recent Letter [1], in which no use is made of QCD,
I. Caprini, G. Colangelo, and H. Leutwyler (CCL) re-
peated an unmentioned analysis of ππ scattering from
1973 [2], based on the Roy equations (REs), to make
out a case for the existence of a scalar I = 0 reso-
nance f0(441), listed in the PDG tables [3] as f0(600)
and known as σ-meson. The primary aspect result-
ing of the CCL analysis is the claimed model- and
parametrization-independent determination of a σ-pole
mass of (441+16
i 544+18
) MeV implying unprece-
dented small error bars. Moreover, the latter result is
incompatible with very recent experimental findings, i.e.,
(500±30− i (264±30)) MeV [4] and (541±39− i (252±
42)) MeV [5, 6], as well as with a combined theoretical
analysis yielding ((476–628)− i (226–346)) MeV [7]. The
present comment will be devoted to complement a re-
cent experimental discussion [4] of short-comings in the
CCL analysis, by presenting theoretical arguments point-
ing at a serious flaw in the theoretical formalism used
by CCL, and also at the unlikeliness of their tiny er-
ror bars in the σ mass and width. The simplest way to
identify this flaw in Ref. [1] also present in the corre-
sponding results [8] of S. Descotes-Genon and B. Mous-
sallam (DM) on the scalar meson K∗0 (800) in the con-
text of Roy-Steiner equations (RSEs), is to recall a warn-
ing statement by G.F. Chew and S. Mandelstam (CM)
from 1960 (see footnote 6 of Ref. [9]). CM state that
if a strongly interacting particle with the same quantum
numbers as a pair of pions should be found, then corre-
sponding poles must be added to the double-dispersion
representation, whether or not the new particle is inter-
preted as a two-pion bound state. It should be empha-
sized that this statement does not only apply to possi-
ble bound-state (BS) poles of the S- or T-matrix in the
physical sheet (PS) of the complex s-plane, but also to
any kind of virtual BS poles and resonance poles in the
unphysical sheet (US). This is justified from first princi-
ples by reviewing briefly how dispersion relations (DRs)
are to be derived on the basis of Cauchy’s integral for-
mula t(s) = (2πi)−1
dz t(z)/(z − s) which holds for
a function t(s) analytic in the domain encirculated by
the closed integration contour. As the so-called “match-
ing point” of CCL (and DM) is located in the US, the
closed integration contour yielding the REs/RSEs must
extend also to the US where the S- and T-matrix poles
for scalar isoscalar ππ-scattering are found. Excluding
these poles situated at sj (j = 1, . . . , n) from the inte-
gration contour and assuming t(s → ∞) → 0 sufficiently
fast one obtains the well known (here) unsubtracted DRs
t(s) =
rj/(s− sj)−
dz Im[t(z)]/(s− z + iε),
where L/R denotes the left-/right-hand cut, and rj is the
residue of t(s) at the corresponding pole sj . According
to CCL, REs/RSEs are twice-subtracted DRs yielding
t(s) = t(s0) + (s− s0) t
′(s0) +
(s0 − s)
(s0 − sj)2(s− sj)
(s0 − s)
2 Im[t(z)]
(s0 − z + iε)2(s− z + iε)
, (1)
where the subtraction point s0 used by CCL appears to
be the ππ threshold, as CCL perform the identification
t(s0) = a
0 and t
′(s0) = (2a
0)/(12m
π) with a
0 being
S-wave scattering lengths for isospin I = 0, 2. It is now
easy to see that the REs/RSEs considered by CCL and
DM disregard the pole terms (PTs) in the DRs (yielding
rj = 0), despite the presence of poles in the US that are
claimed to exist by observing respective S-matrix zeros
in the PS. As the s-dependence of the disregarded σ- and
f0(980)-PTs in the vicinity of the ππ- and KK-theshold
is clearly non-linear, it is to be expected on grounds of
dispersion theory that the S-matrix poles predicted by
CCL will not coincide with the actual ones to be deter-
mined yet by CCL for self-consistency reasons. An anal-
ogous statement applies to the results of DM. Moreover
will the inclusion of PTs in REs/RSEs not only reinstate
dispersion theoretic self-consistency, yet also yield a sig-
nificant change in the resulting σ- andK∗
(800)-pole posi-
tions, which unfortunately will enter now via the PTs as
unknown parameters the REs/RSEs to be solved. Hence
the inclusion of PTs in REs/RSEs will yield an uncer-
tainty of pole positions which is likely to be of the order
of the one estimated in Ref. [4] and therefore much larger
than the error bars presently claimed by CCL and DM
being even without taking into account PTs for at least
two reasons clearly parametrization-dependent: (1) the
extrapolation of the two particle phase space to the com-
plex s-plane and below threshold invoked by CCL and
DM is known to be speculative and even unphysical as
it yields e.g. in the approach of DM scattering below
the pseudo-threshold; (2) standard chiral perturbation
theory (ChPT) disregarding (yet) non-perturbative PTs
relates claimed values for scalar scattering lengths and
their (too) tiny error bars entering REs/RSEs lacking
(yet) PTs to scalar square radii
the presently used
(too) high values of which yield chiral symmetry breaking
(ChSB) of the order of 6-8% being much larger than 3%
as observed in Nature. A revision of the analysis of CCL
and DM by taking into account PTs in REs/RSEs and
ChPT would be highly desirable to reconcile their results
with Refs. [4]-[7] and to improve the poor description of
the resonance K∗(892) in the approach of DM.
Frieder Kleefeld 1,2,3
1 Present address: Pfisterstr. 31, 90762 Fürth, Germany
2 Doppler & Nucl. Physics Institute (Dep. Theor. Phys.),
Academy of Sciences of Czech Republic; collaborator of
the CFIF, Instituto Superior Técnico, LISBOA, Portugal
3 Electronic address: kleefeld@cfif.ist.utl.pt
http://arxiv.org/abs/0704.1337v1
[1] I. Caprini et al., Phys. Rev. Lett. 96, 132001 (2006).
[2] M. R. Pennington et al., Phys. Rev. D 7, 1429,2591 (1973).
[3] W. M. Yao et al. [PDG], J. Phys. G 33, 1 (2006).
[4] D. V. Bugg, J. Phys. G 34, 151 (2007).
[5] M. Ablikim et al. [BES], Phys. Lett. B 598, 149 (2004).
[6] D. V. Bugg, AIP Conf. Proc. 814, 78 (2006).
[7] E. van Beveren et al., Phys. Lett. B 641, 265 (2006).
[8] S. Descotes-Genon et al., Eur. Phys. J. C 48, 553 (2006).
[9] G. F. Chew, S. Mandelstam, Phys. Rev. 119, 467 (1960).
This work has been supported by the FCT of the
Ministério da Ciência, Tecnologia e Ensino Superior
of Portugal, under contracts POCI/FP/63437/2005,
PDCT/FP/63907/2005 and the Czech project LC06002.
Conversations with E. van Beveren, D. V. Bugg,
J. Fischer, A. Moussallam, G. Rupp, M. D. Scadron and
M. Znojil are gratefully acknowledged.
	References
