Comment on "Mass and Width of the Lowest Resonance in QCD"
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
