Alternative Large Nc Schemes and Chiral Dynamics
CERN–PH–TH/2007–068
SU–4252–8489
Alternative Large N
Schemes and Chiral Dynamics
Francesco Sannino∗
CERN Theory Division, CH-1211 Geneva 23, Switzerland. and
University of Southern Denmark, Campusvej 55, DK-5230, Odense M., Denmark.
Joseph Schechter†
Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA.
(Dated: March 2007)
We compare the dependences on the number of colors of the leading ππ scattering amplitudes
using the single index quark field and two index quark fields. These are seen to have different
relationships to the scattering amplitudes suggested by chiral dynamics which can explain the long
puzzling pion pion s wave scattering up to about 1 GeV. This may be interesting for getting a better
understanding of the large Nc approach as well as for application to recently proposed technicolor
models.
BACKGROUND
Gaining control of QCD in its strongly interacting (low
energy) regime constitutes a real challenge. One very at-
tractive approach is based on studying the theory in the
large number of colors (Nc) limit [1, 2]. At the same
time one may obtain more information by requiring the
theory to model the (almost) spontaneous breakdown of
chiral symmetry [3, 4]. A standard test case is pion pion
scattering in the energy range up to about 1 GeV. Some
time ago, an attempt was made [5, 6] to implement this
combined scenario. Since the leading large Nc ampli-
tude contains only tree diagrams involving mesons of the
standard quark-antiquark type, it is expected that the
required amplitude should be gotten by calculating just
the chiral tree diagrams for rho meson exchange together
with the four point pion contact diagram. There are
no unknown parameters in this calculation. The crucial
question is whether the scattering amplitude calculated
in this way will satisfy unitarity. When one compares
the result with experimental data up to about 1 GeV on
the real part of the (most sensitive to unitarity violation)
J=I=0 partial wave, one finds (see Fig.1 of [6]) that the
result violates the partial wave unitarity bound by just
a “little bit”. On the other hand, the pion contact term
by itself violates unitarity much more drastically so one
might argue that the large Nc approach, which suggests
that the tree diagrams of all quark anti-quark resonances
in the relevant energy range be included, is helping a lot.
To make matters more quantitative one might ask the
question: by how much should Nc be increased in or-
der for the amplitude in question to remain within the
unitarity bounds for energies below 1 GeV?
This question was answered in a very simple way in
[7], as we now briefly review. In terms of the con-
ventional amplitude, A(s, t, u) the I = 0 amplitude is
3A(s, t, u) + A(t, s, u) + A(u, t, s). One gets the J = 0
channel by projecting out the correct partial wave. The
current algebra (pion contact diagram) contribution to
the conventional amplitude is
Aca(s, t, u) = 2
s−m2π
, (1)
where the pion decay constant, Fπ depends on Nc as
Fπ(Nc) = 131
3 so that Fπ(3) = 131 MeV. Fur-
thermore mπ = 137 MeV is independent of Nc. The
desired amplitude is obtained by adding to the current
algebra term the following vector meson ρ(770) contribu-
tion:
Aρ(s, t, u) =
g2ρππ
(4m2π − 3s)
g2ρππ
(m2ρ − t)− imρΓρθ(t− 4m2π)
(m2ρ − u)− imρΓρθ(u − 4m2π)
, (2)
where gρππ(Nc) = 8.56
Nc is the ρππ coupling con-
stant. Also, mρ = 771 MeV is independent of Nc and
Γρ(Nc) =
g2ρππ (Nc)
12πm2ρ
. (3)
It should be noted that the first term in Eq. (2), which
is an additional non-resonant contact interaction other
than the current algebra contribution, is required when
we include the ρ vector meson contribution in a chiral
invariant manner. In Fig. 1 we show the real part of the
I = J = 0 amplitude (denoted R0
) due to current alge-
bra plus the ρ contribution for increasing values of Nc.
Since in this channel the vector meson is never on shell
we suppress the contribution of the width in the vector
http://arxiv.org/abs/0704.0602v1
meson propagator in Eq. (2). One observes that the uni-
tarity bound (i.e., |R0
| ≤ 1/2) is satisfied for Nc ≥ 6 till
well beyond the 1 GeV region. However unitarity is still a
problem for 3, 4 and 5 colors. At energy scales larger than
the one associated with the vector meson clearly other
resonances are needed [5] but we shall not be concerned
with that energy range here. It is also interesting to note
that these considerations are essentially unchanged when
the pion mass (i.e. explicit chiral symmetry breaking in
the Lagrangian) is set to zero.
FIG. 1: Real part of the I = J = 0 partial wave amplitude
due to the current algebra +ρ terms, plotted for the follow-
ing increasing values of Nc (from up to down), 3, 4, 5, 6, 7.
The curve with largest violation of the unitarity bound corre-
sponds to Nc = 3 while the ones within the unitarity bound
are for Nc = 6, 7.
Note that essentially we are just using the scaling,
A(s, t, u) =
Ã(s, t, u) . (4)
where Ã(s, t, u) is defined replacing Fπ and gρππ with the
Nc independent quantities F̃π = Fπ
Nc and g̃ρππ =
3. Other authors [8] have found the same
minimum value, Nc=6 for the practical consistency of
the large Nc approximation, by using different methods.
In order to explain low energy ππ scattering for the
physical value Nc = 3 one must go beyond the large Nc
approximation. It is attractive to keep the assumption
of tree diagram dominance involving near by resonances,
however. One easily sees that adding a scalar singlet res-
onance (sigma) at the location where the unitarity bound
on R0
(s) is first violated should restore unitarity. This
is because the real part of a Breit Wigner resonance is
zero at the pole location and negative just above it, so
will bring R0
(s) below the bound, as required. In [7], the
resonance mass was found to be around 550 MeV on this
basis. Such a low value would make it different from a
p-wave quark-antiquark state, which is expected to be in
the 1000-1400 MeV range. We assume then that it is a
four quark state (glueball states are expected to be in the
1.5 GeV range from lattice investigations). Four quark
states of diquark-anti diquark type [9] and meson-meson
type [10] have been discussed in the literature for many
years. Accepting this picture, however, poses a problem
for the accuracy of the large Nc inspired description of
the scattering since four quark states are predicted not
to exist in the large Nc limit of QCD. We shall take the
point of view that a four quark type state is present since
it allows a natural fit to the low energy data. Of course,
it is still necessary to fine tune the parameters and shape
of the sigma resonance to fit the experimental ππ scat-
tering data in detail. In practice, since the parameters
of the pion contact and rho exchange contributions are
fixed, the sigma is the most important one for fitting and
fits may even be achieved [11] if the vector meson piece
is neglected. However the well established, presumably
four quark type, f0(980) resonance must be included to
achieve a fit in the region just around 1 GeV.
There is by now a fairly large recent literature [12]-
[44] on the effect of light “exotic” scalars in low energy
meson meson scattering. There seems to be a consene-
sus, arrived at using rather different approaches (keeping
however, unitarity), that the sigma exists.
TWO INDEX QUARK FIELDS
Now, consider redefining the Nc = 3 quark field with
color index A (and flavor index not written) as
ǫABCq
BC , qBC = −qCB, (5)
so that, for example, q1 = q
23 and similarly for the ad-
joint field, q̄1 = q̄23 etc. This is just a trivial change
of variables and will not change anything for QCD. How-
ever, if a continuation of the theory is made to Nc > 3 the
resulting theory will be different since the two index anti-
symmetric quark representation has Nc(Nc− 1)/2 rather
than Nc color components. As was pointed out by Cor-
rigan and Ramond [45], who were mainly interested in
the problem of the baryons at large Nc, this shows that
the extrapolation of QCD to higher Nc is not unique.
Further investigation of the properties of the alternative
extrapolation model introduced in [45] was carried out
by Kiritsis and Papavassiliou [46]. Here, we shall discuss
the consequences for the low energy ππ scattering dis-
cussed above, of this alternative large Nc extrapolation,
assuming for our purpose, that all the quarks extrapolate
as antisymmetric two index objects.
It may be worthwhile to remark that gauge theo-
ries with two index quarks have recently gotten a great
deal of attention. Armoni, Shifman and Veneziano
[47, 48, 49, 50, 51] have proposed an interesting relation
between certain sectors of the two index antisymmetric
(and symmetric) theories at large number of colors and
sectors of super Yang-Mills (SYM). Using a supersym-
metry inspired effective Lagrangian approach 1/Nc cor-
rections were investigated in [52]. Information on the su-
per Yang-Mills spectrum has been obtained in [53]. On
the validity of the large Nc equivalence between differ-
ent theories and interesting new possible phases we refer
the reader to [54, 55, 56]. The finite temperature phase
transition and its relation with chiral symmetry has been
investigated in [57] while the effects of a nonzero baryon
chemical potential were studied in [58].
When adding flavors the phase diagram as a function
of the number of flavors and colors has been provided
in [59]. The complete phase diagram for fermions in ar-
bitrary representations has been unveiled in [60]. The
study of theories with fermions in a higher dimensional
representation of the gauge group and the knowledge of
the associated conformal window led to the construction
of minimal models of technicolor [59, 61, 62] which are
not ruled out by current precision measurements and lead
to interesting dark matter candidates [63, 64, 65] as well
as to a very high degree of unification of the standard
model gauge couplings [66].
Besides these two limits a third one for massless one-
flavor QCD, which is in between the ’t Hooft and Cor-
rigan Ramond ones, has been very recently proposed in
[67]. Here one first splits the QCD Dirac fermion into the
two elementary Weyl fermions and afterwards assigns one
of them to transform according to a rank-two antisym-
metric tensor while the other remains in the fundamental
representation of the gauge group. For three colors one
reproduces one-flavor QCD and for a generic number of
colors the theory is chiral. The generic Nc is a particular
case of the generalized Georgi-Glashow model [68].
To illustrate the large Nc counting for the ππ scatter-
ing amplitude when quarks are designated to transform
according to the two index antisymmetric representation
of color SU(3) one may employ [1] the mnemonic where
each tensor index of this group is represented by a di-
rected line. Then the quark-quark gluon interaction is
pictured as in Fig. 2. The two index quark is pictured
FIG. 2: Two index fermion - gluon vertex.
as two lines with arrows pointing in the same direction,
as opposed to the gluon which has two lines with arrows
pointing in opposite directions. The coupling constant
representing this vertex is taken to be gt/
Nc, where gt
(the ’t Hooft coupling constant) is to be held constant.
A “one point function”, like the pion decay constant,
Fπ would have as it’s simplest diagram, Fig. 3.
The X represents a pion insertion and is associated
FIG. 3: Diagram for Fπ for the two index quark.
with a normalization factor for the color part of the pion’s
wavefunction,
Nc(Nc − 1)
, (6)
which scales for large Nc as 1/Nc. The two circles each
carry a quark index so their joint factor scales as N2c for
large Nc; more precisely, taking the antisymmetry into
account, the factor is
Nc(Nc − 1)
. (7)
The product of Eqs. (6) and (7) yields the Nc scaling for
F 2π (Nc) =
Nc(Nc − 1)
F 2π (3). (8)
For largeNc, Fπ scales proportionately to Nc rather than
Nc as in the case of the ’t Hooft extrapolation.
Using this scaling together with Eq.(1) suggests that
the ππ scatttering amplitude, A scales as,
A(Nc) =
Nc(Nc − 1)
A(3), (9)
which, for large Nc scales as 1/N
c rather than as 1/Nc
in the ’t Hooft extrapolation. This scaling law for large
Nc may be verified from the mnemonic in Fig. 4, where
there is an N2c factor from the two loops multiplied by
four factors of 1/Nc from the X’s.
FIG. 4: Diagram for the scattering amplitude, A with the 2
index quark.
It is interesting to find the minimum value of Nc for
which the tree amplitude due to the pion and rho meson
terms (given in Eqs.(1) and (2) above) is unitary in this
two antisymmetric index quark extrapolation scheme.
Fig. 1 shows that the the peak value of the partial wave
amplitude, R0
due to these two terms is numerically
about 0.9. This is to be identified with Aca(3) + Aρ(3)
in Eq.(9). Thus the condition that the extrapolated am-
plitude be unitary is,
Nc(Nc − 1)
< 1/2. (10)
Clearly, the extrapolated amplitude is unitary already for
Nc = 4, which indicates better convergence in Nc than
for the ’t Hooft case which became unitary at Nc = 6.
There is still another different feature; consider the
typical ππ scattering diagram with an extra internal (two
index) quark loop, as shown in Fig. 5.
FIG. 5: Diagram for the scattering amplitude, A including an
internal 2 index quark loop.
In this diagram there are four X’s (factor from Eq.(6)),
five index loops (factor from Eq.(7)) and six gauge cou-
pling constants. These combine to give a large Nc scaling
behavior proportional to 1/N2c for the ππ scattering am-
plitude. We see that diagrams with an extra internal
2 index quark loop are not suppressed compared to the
leading diagrams. This is analogous, as pointed out in
[46], to the behavior of diagrams with an extra gluon loop
in the ’t Hooft extrapolation scheme. Now, Fig. 5 is a
diagram which can describe a sigma particle exchange.
Thus in the 2 index quark scheme, “exotic” four quark
resonances can appear at the leading order in addition
to the usual two quark resonances. Given the discussion
we reviewed above, the possibility of a sigma appearing
at leading order means that one can construct a unitary
ππ amplitude already at Nc = 3 in the 2 antisymmetric
index scheme. From the point of view of low energy ππ
scattering, it seems to be unavoidable to say that the 2
index scheme is more realistic than the ’t Hooft scheme
given the existence of a four quark type sigma.
Of course, the usual ’t Hooft extrapolation has a num-
ber of other things to recommend it. These include the
fact that nearly all meson resonances seem to be of the
quark- antiquark type, the OZI rule predicted holds to
a good approximation and baryons emerge in an elegant
way as solitons in the model.
A fair statement would seem to be that each extrap-
olation emphasizes different aspects of the true Nc = 3
QCD. In particular, the usual scheme is not really a re-
placement for the true theory. That appears to be the
meaning of the fact that the continuation to Nc > 3 is
not unique.
QUARKS IN TWO INDEX SYMMETRIC COLOR
REPRESENTATION
Clearly the assignment of quarks to the two index sym-
metric representation of color SU(3) looks very similar.
We may denote the quark fields as,
AB = q
BA , (11)
in contrast to Eq.(5). There will be Nc(Nc + 1)/2 differ-
ent color states for the two index symmetric quarks. This
means that there is no value of Nc for which the sym-
metric theory can be made to correspond to true QCD.
For Nc = 3 there are 6 color states of the quarks and 8
color states of the gluon. If we choose Nc = 2, there are
3 color states of the quarks but unfortunately only three
color states of the gluon. On the other hand, for large Nc
it would seem reasonable to make approximations like,
Asym(Nc) ≈ Aasym(Nc), (12)
for the ππ scattering amplitude.
As far as the large Nc counting goes, the mnemonics
in Figs. 2-5 are still applicable to the case of quarks in
the two index symmetric color representation. For not so
large Nc, the scaling factor for the pion insertion would
Nc(Nc + 1)
, (13)
and the pion decay constant would scale as
F symπ (Nc) ∝
Nc(Nc + 1)
. (14)
With the identification AQCD = Aasym(3), the use of
Eq.(12) enables us to estimate the large Nc scattering
amplitude as,
Asym(Nc) ≈
AQCD. (15)
In applications to recently proposed minimal walking
technicolor theories this formula is useful for making es-
timates involving weak gauge bosons via the Goldstone
boson equivalence theorem [69].
Finally we remark on the large Nc scaling rules for
meson and glueball masses and decays in either the two
index antisymmetric or two index symmetric schemes.
Both meson and glueball masses scale as (Nc)
0. Further-
more, all six reactions of the type
a → b+ c, (16)
where a,b and c can stand for either a meson or a glueball,
scale as 1/Nc. This is illustrated in Fig.6 for the case of a
meson decaying into two glueballs; note that the glueball
insertion scales as 1/Nc and that two interaction vertices
are involved.
FIG. 6: Diagram for meson decay into two glueballs.
SUMMARY
We have investigated the dependences on the number
of colors of the leading ππ scattering amplitudes using
the single and the two index quark fields.
We have seen that in the 2 index quark extension of
QCD, exotic four quark resonances can appear at the
leading order in addition to the usual two quark reso-
nances. From the point of view of low energy ππ scatter-
ing the 2 index scheme is more realistic than the ’t Hooft
one given the existence of a four quark type sigma. This
allows one to explain the long puzzling pion pion s wave
scattering up to about 1 GeV.
Of course, the usual ’t Hooft extrapolation has a num-
ber of other important predictions to recommend it. A
fair statement is that each largeNc extrapolation of QCD
captures different aspects of the physical Nc = 3 case.
We have also related the QCD scattering amplitude
at large Nc with the one featuring two index symmetric
quarks (Similar connections exist for adjoint fermions).
The results are interesting for getting a better under-
standing of the large Nc approach as well as for applica-
tion to recently proposed technicolor models.
Acknowledgments
It is a pleasure to thank A. Abdel Rehim, D. Black,
D.D. Dietrich, A. H. Fariborz, M.T. Frandsen, M.
Harada, S. Moussa, S. Nasri and K. Tuominen for helpful
discussions. The work of F.S. is supported by the Marie
Curie Excellence Grant under contract MEXT-CT-2004-
013510 as well as the Danish Research Agency. The work
of J.S. is supported in part by the U. S. DOE under Con-
tract no. DE-FG-02-85ER 40231.
∗ Electronic address: francesco.sannino@nbi.dk
† Electronic address: schechte@physics.syr.edu
[1] G.’t Hooft, Nucl. Phy. B72, 461 (1974).
[2] E. Witten, Nucl. Phy. B160, 57 (1979).
[3] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345
(1961).
[4] M. Gell-Mann and M. Levy, Nuovo Cimento 16, 705
(1960).
[5] F. Sannino and J. Schechter, Phys. Rev. D 52, 96 (1995).
[6] M. Harada, F. Sannino and J. Schechter, Phys. Rev. D
54, 1991 (1996).
[7] M. Harada, F. Sannino and J. Schechter, Phys. Rev. D
69, 034005 (2004) [arXiv:hep-ph/0309206].
[8] J. R. Pelaez, Phys. Rev. Lett. 92, 102001 (2004);
97, 242002 (2006). M. Uehara, arXiv:hep-ph/0308241,
/0401037, /0404221.
[9] R. L. Jaffe, Phys. Rev. D 15, 267 (1977); Phys. Rev. D
15, 281 (1977).
[10] J. D. Weinstein and N. Isgur, Phys. Rev. Lett. 48, 659
(1982).
[11] M. Harada, F. Sannino and J. Schechter, Phys. Rev. Lett.
78, 1603 (1997).
[12] See the dedicated conference proceedings, S. Ishida et al
“Possible existence of the sigma meson and its implica-
tion to hadron physics”, KEK Proceedings 2000-4, So-
ryyushiron Kenkyu 102, No. 5, 2001. Additional points
of view are expressed in the proceedings, D. Amelin
and A.M. Zaitsev “Hadron Spectroscopy”, Ninth Inter-
national Conference on Hadron Spectroscopy, Protvino,
Russia(2001).
[13] E. Van Beveren, T. A. Rijken, K. Metzger, C. Dulle-
mond, G. Rupp and J. E. Ribeiro, Z. Phys. C 30, 615
(1986); E. van Beveren and G. Rupp, Eur. Phys. J. C
10, 469 (1999). See also J. J. De Swart, P. M. Maessen
and T. A. Rijken, Talk given at U.S. / Japan Seminar:
The Hyperon - Nucleon Interaction, Maui, HI, 25-28 Oct
1993 [arXiv:nucl-th/9405008].
[14] D. Morgan and M. R. Pennington, Phys. Rev. D 48, 1185
(1993).
[15] A. A. Bolokhov, A. N. Manashov, V. V. Vereshagin and
V. V. Polyakov, Phys. Rev. D 48, 3090 (1993). See also
V. A. Andrianov and A. N. Manashov, Mod. Phys. Lett.
A 8, 2199 (1993). Extension of this string-like approach
to the πK case has been made in V. V. Vereshagin, Phys.
Rev. D 55, 5349 (1997) and in A. V. Vereshagin and
V. V. Vereshagin, Phys. Rev. D 59, 016002 (1999).
[16] N. N. Achasov and G. N. Shestakov, Phys. Rev. D 49,
5779 (1994).
[17] R. Kamı́nski, L. Leśniak and J. P. Maillet, Phys. Rev. D
50, 3145 (1994).
[18] N. A. Tornqvist, Z. Phys. C 68, 647 (1995) and references
therein. In addition see N. A. Törnqvist and M. Roos,
Phys. Rev. Lett. 76, 1575 (1996); N. A. Tornqvist, Talk
given at 7th International Conference on Hadron Spec-
troscopy (Hadron 97), Upton, NY, 25-30 Aug 1997 and
at EuroDaphne Meeting, Barcelona, Spain, 6-9 Nov 1997
[arXiv:hep-ph/9711483]; Phys. Lett. B 426, 105 (1998).
[19] R. Delbourgo and M. D. Scadron, Mod. Phys. Lett. A
10, 251 (1995); See also D. Atkinson, M. Harada and
A. I. Sanda, Phys. Rev. D 46, 3884 (1992).
[20] G. Janssen, B. C. Pearce, K. Holinde and J. Speth, Phys.
mailto:francesco.sannino@nbi.dk
mailto:schechte@physics.syr.edu
http://arxiv.org/abs/hep-ph/0309206
http://arxiv.org/abs/hep-ph/0308241
http://arxiv.org/abs/nucl-th/9405008
http://arxiv.org/abs/hep-ph/9711483
Rev. D 52, 2690 (1995).
[21] M. Svec, Phys. Rev. D 53, 2343 (1996).
[22] S. Ishida, M. Ishida, H. Takahashi, T. Ishida, K. Taka-
matsu and T. Tsuru, Prog. Theor. Phys. 95, 745 (1996);
S. Ishida, M. Ishida, T. Ishida, K. Takamatsu and
T. Tsuru, Prog. Theor. Phys. 98, 621 (1997). See also
M. Ishida and S. Ishida, Talk given at 7th International
Conference on Hadron Spectroscopy (Hadron 97), Upton,
NY, 25-30 Aug 1997 [arXiv:hep-ph/9712231].
[23] D. Black, A. H. Fariborz, F. Sannino and J. Schechter,
Phys. Rev. D 58, 054012 (1998).
[24] D. Black, A. H. Fariborz, F. Sannino and J. Schechter,
Phys. Rev. D 59, 074026 (1999).
[25] L. Maiani, A. Polosa, F. Piccinni and V. Riquier, Phys.
Rev. Lett. 93, 212002 (2004). Here the characteristic
form for a four quark scalar coupling to two pions was
obtained as in [24] above but with the difference that
non-derivative coupling rather than derivative coupling
was used. The derivative coupling appeared in [24] since
the context was that of a non-linear chiral Lagrangian.
[26] J. A. Oller, E. Oset and J. R. Pelaez, Phys. Rev. Lett.
80, 3452 (1998). See also K. Igi and K. I. Hikasa, Phys.
Rev. D 59, 034005 (1999).
[27] A. V. Anisovich and A. V. Sarantsev, Phys. Lett. B 413,
137 (1997).
[28] V. Elias, A. H. Fariborz, F. Shi and T. G. Steele, Nucl.
Phys. A 633, 279 (1998).
[29] V. Dmitrasinovic, Phys. Rev. C 53, 1383 (1996).
[30] P. Minkowski andW. Ochs, Eur. Phys. J. C 9, 283 (1999).
[31] S. Godfrey and J. Napolitano, Rev. Mod. Phys. 71, 1411
(1999).
[32] L. Burakovsky and T. Goldman, Phys. Rev. D 57, 2879
(1998).
[33] A. H. Fariborz and J. Schechter, Phys. Rev D60, 034002
(1999).
[34] T. Hatsuda, T. Kunihiro and H. Shimizu, Phys. Rev.
Lett. 82, 2840 (1999); S. Chiku and T. Hatsuda, Phys.
Rev. D 58, 076001 (1998).
[38] D. Black, A. H. Fariborz and J. Schechter, Phys. Rev. D
61, 074030 (2000). See also V. Bernard, N. Kaiser and
U. G. Meissner, Phys. Rev. D 44, 3698 (1991).
[39] D. Black, A. H. Fariborz and J. Schechter, Phys. Rev. D
61, 074001 (2000).
[37] L. S. Celenza, S. f. Gao, B. Huang, H. Wang and
C. M. Shakin, Phys. Rev. C 61, 035201 (2000).
[38] D. Black, A. H. Fariborz and J. Schechter, Phys. Rev. D
61, 074030 (2000). See also V. Bernard, N. Kaiser and
U. G. Meissner, Phys. Rev. D 44, 3698 (1991).
[39] D. Black, A. H. Fariborz and J. Schechter, Phys. Rev. D
61, 074001 (2000).
[40] D. Black, A. H. Fariborz, S. Moussa, S. Nasri and
J. Schechter, Phys. Rev. D 64, 014031 (2001).
[41] In addition to [39] and [40] above see T. Teshima, I. Ki-
tamura and N. Morisita, J. Phys. G 28, 1391 (2002);
F. Close and N. Tornqvist, ibid. 28, R249 (2002); A.H.
Fariborz, Int. J. Mod. Phys. A 19, 2095 (2004); 5417
(2004); Phys. Rev. D 74, 054030 (2006); F. Giacosa, Th.
Gutsche, V.E. Lyubovitskij and A. Faessler, Phys. Lett.
B 622, 277 (2005); J. Vijande, A. Valcarce, F. Fernandez
and B. Silvestre-Brac, Phys. Rev. D 72, 034025 (2005); S.
Narison, Phys. Rev. D 73, 114024 (2006); L. Maiani, F.
Piccinini, A.D. Polosa and V. Riquer, hep-ph/0604018.
[42] Y. S. Kalashnikova, A. E. Kudryavtsev, A. V. Nefediev,
C. Hanhart and J. Haidenbauer, the Eur. Phys. J. A 24
(2005) 437 [hep-ph/0412340].
[43] The Roy equation for the pion amplitude, S.M. Roy,
Phys. Lett. B 36, 353 (1971), has been used by several au-
thors to obtain information about the f0(600) resonance.
See T. Sawada, page 67 of ref. [12] above, I. Caprini,
G. Colangelo and H. Leutwyler, Phys. Rev. Lett. 96,
132001 (2006). A similar approach has been employed
to study the putative light kappa by S.Descotes-Genon
and B. Moussallam, Eur. Phys. J. C 48, 553 (2006).
[44] Further discussion of the approach in ref. [43] above
is given in D.V. Bugg, J. Phys. G 34, 151 (2007)
[hep-ph/0608081].
[45] E. Corrigan and P. Ramond, Phys. Lett. B 87, 73 (1979).
[46] E. B. Kiritsis and J. Papavassiliou, Phys. Rev. D 42, 4238
(1990).
[47] A. Armoni, M. Shifman and G. Veneziano, Nucl. Phys.
B 667, 170 (2003) [arXiv:hep-th/0302163].
[48] A. Armoni, M. Shifman and G. Veneziano, Phys. Rev.
Lett. 91, 191601 (2003) [arXiv:hep-th/0307097].
[49] A. Armoni, M. Shifman and G. Veneziano, Phys. Rev. D
71, 045015 (2005) [arXiv:hep-th/0412203].
[50] A. Armoni, G. Shore and G. Veneziano, Nucl. Phys. B
740, 23 (2006) [arXiv:hep-ph/0511143].
[51] A. Armoni, M. Shifman and G. Veneziano, Phys. Lett. B
647, 515 (2007) [arXiv:hep-th/0701229].
[52] F. Sannino and M. Shifman, Phys. Rev. D 69, 125004
(2004) [arXiv:hep-th/0309252].
[53] A. Feo, P. Merlatti and F. Sannino, Phys. Rev. D
70, 096004 (2004) [arXiv:hep-th/0408214]. P. Merlatti
and F. Sannino, Phys. Rev. D 70, 065022 (2004)
[arXiv:hep-th/0404251].
[54] M. Unsal, arXiv:hep-th/0703025.
[55] M. Unsal and L. G. Yaffe, Phys. Rev. D 74, 105019 (2006)
[arXiv:hep-th/0608180].
[56] P. Kovtun, M. Unsal and L. G. Yaffe, Phys. Rev. D 72,
105006 (2005) [arXiv:hep-th/0505075].
[57] F. Sannino, Phys. Rev. D 72, 125006 (2005)
[arXiv:hep-th/0507251].
[58] M. T. Frandsen, C. Kouvaris and F. Sannino, Phys. Rev.
D 74, 117503 (2006) [arXiv:hep-ph/0512153].
[59] F. Sannino and K. Tuominen, Phys. Rev. D 71, 051901
(2005) [arXiv:hep-ph/0405209].
[60] D. D. Dietrich and F. Sannino, arXiv:hep-ph/0611341.
To Appear in Phys. Rev. D.
[61] D. K. Hong, S. D. H. Hsu and F. Sannino, Phys. Lett. B
597, 89 (2004) [arXiv:hep-ph/0406200].
[62] D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev.
D 72, 055001 (2005) [arXiv:hep-ph/0505059]. Phys. Rev.
D 73, 037701 (2006) [arXiv:hep-ph/0510217].
[63] K. Kainulainen, K. Tuominen and J. Virkajarvi,
arXiv:hep-ph/0612247.
[64] S. B. Gudnason, C. Kouvaris and F. Sannino, Phys. Rev.
D 74, 095008 (2006) [arXiv:hep-ph/0608055].
[65] C. Kouvaris, arXiv:hep-ph/0703266.
[66] S. B. Gudnason, T. A. Ryttov and F. Sannino,
arXiv:hep-ph/0612230.
[67] T. A. Ryttov and F. Sannino, Phys. Rev. D 73, 016002
(2006) [arXiv:hep-th/0509130].
[68] H. Georgi, Nucl. Phys. B 266, 274 (1986).
[69] J.M. Cornwall, D.N. Levin and G. Tiktopoulos, Phys.
Rev. D 10, 1145 (1974); B.W. Lee, C. Quigg and H.B.
Thacker, Phys. Rev. D 16, 1519 (1977); M.S. Chanowitz
and M.K. Galliard, Nucl. Phys. B 261, 379 (1985).
http://arxiv.org/abs/hep-ph/9712231
http://arxiv.org/abs/hep-ph/0604018
http://arxiv.org/abs/hep-ph/0412340
http://arxiv.org/abs/hep-ph/0608081
http://arxiv.org/abs/hep-th/0302163
http://arxiv.org/abs/hep-th/0307097
http://arxiv.org/abs/hep-th/0412203
http://arxiv.org/abs/hep-ph/0511143
http://arxiv.org/abs/hep-th/0701229
http://arxiv.org/abs/hep-th/0309252
http://arxiv.org/abs/hep-th/0408214
http://arxiv.org/abs/hep-th/0404251
http://arxiv.org/abs/hep-th/0703025
http://arxiv.org/abs/hep-th/0608180
http://arxiv.org/abs/hep-th/0505075
http://arxiv.org/abs/hep-th/0507251
http://arxiv.org/abs/hep-ph/0512153
http://arxiv.org/abs/hep-ph/0405209
http://arxiv.org/abs/hep-ph/0611341
http://arxiv.org/abs/hep-ph/0406200
http://arxiv.org/abs/hep-ph/0505059
http://arxiv.org/abs/hep-ph/0510217
http://arxiv.org/abs/hep-ph/0612247
http://arxiv.org/abs/hep-ph/0608055
http://arxiv.org/abs/hep-ph/0703266
http://arxiv.org/abs/hep-ph/0612230
http://arxiv.org/abs/hep-th/0509130
