BLED WORKSHOPS
IN PHYSICS
VOL. 1, NO. 1
Proceedins of the Mini-Workshop
Few-Quark Problems (p. 1)
Bled, Slovenia, July 8-15, 2000
Chiral Perturbation Theory and Unitarization?
Enrique Ruiz Arriola1??, A. Gómez Nicola2, J. Nieves1 and J. R. Peláez21Departamento de Fı́sica Moderna, Universidad de Granada, E-18071 Granada, Spain2Departamento de Fı́sica Teórica Universidad Complutense, 28040 Madrid, Spain
Abstract. We review our recent work on unitarization and chiral perturbation theory both
in the  and the N sectors.We pay particular attention to the Bethe-Salpeter and Inverse
Amplitude unitarization methods and their recent applications to  and N scattering.
1 Introduction
Chiral Perturbation Theory (ChPT) is a practical and widely accepted effective
field theory to deal with low energy processes in hadronic physics. [1–3]. The
essential point stressed in this approach is that the low energy physics does not
depend on the details of the short distance dynamics, but rather on some bulk
properties effectively encoded in the low energy parameters. This point of view
has been implicitly adopted in practice in everyday quantum physics; well sepa-
rated energy and distance scales can be studied independently of each other. The
effective field theory approach also makes such a natural idea into a workable
and systematic computational scheme.
The great advantage of ChPT is that the expansion parameter can be clearly
identified a priori (see e.g. Ref. [3]) when carrying out systematic calculations of
mass splittings, form factors and scattering amplitudes. However, the connection
to the underlying QCD dynamics becomes obscure since the problem is naturally
formulated in terms of the relevant hadronic low energy degrees of freedomwith
no explicit reference to the fundamental quarks and gluons. In addition, in some
cases (see below) a possible drawback is the lack of numerical convergence of
such an expansion when confronted to experimental data, a problem that gets
worse as the energy of the process increases. Recent analysis provide good exam-
ples of both rapid convergence and slow convergence in ChPT. In the  sector
the situation to two loops [4] seems to be very good for the scattering lengths.
Here, the expansion parameter ism2=(4f)2 = 0:01 (m = 139:6MeV the phys-
ical mass of the charged pion) and the coefficients of the expansion are of order
unity. For instance, to two loops (third order in the expansion parameter) the ex-
pansion of the s-wave of the isospin I = 0 channel reads [5]a00m = 0:156| {z }
tree
+ 0:043  0:003| {z }
1 loop
+ 0:015  0:003| {z }
2 loops
+ 
 
 
 (1)? Talk delivered by Enrique Ruiz Arriola?? E-mail: earriola@ugr.es
2 E. Ruiz Arriola
where the theoretical errors are described in [5]. Thus the expansion up to two
loops is both convergent a(n) >> a(n+1) and predictive 
a(n) << a(n+1) ,
a(n) << a(n). The prediction of ChPT of s-wave scattering lengths aIJ in the
isospin I = 0 and I = 2 channels yields [5]a0 0m = +0:214  0:005 (exp: + 0:26  0:05)a2 0m = -0:420  0:010 (exp: - 0:28  0:12) (2)
The theoretical predictions for these observables are an order of magnitude more
accurate than the corresponding experimental numbers. Note that in hadronic
physics we are usually dealing with the opposite situation, confronting accurate
measurements to inaccurate theoretical model calculations. On the contrary, forN scattering, for ChPT in the heavy baryon formulation the expansion is less
rapidly converging than in the  case, since NLO corrections become compa-
rable to the LO ones. For instance in the P33-channel the expansion up to third
order, after a fit to the threshold properties, reads [6]a133m3 = 35:3|{z}
1st order
+ 47:95| {z }
2nd order
- 1:49|{z}
3rd order
+ 
 
 
 = 81:8  0:9 (exp:80:3  0:6) (3)
Here first order means 1=f2, second order 1=f2MN and third order 1=f4 and1=f2M2N. Despite these caveats, there is no doubt that the effective field theory
approach provides a general framework where one can either verify or falsify,
not only bulk properties of the underlying dynamics, but also the dynamics of all
models sharing the same general symmetries of QCD.
Finally, let us remark that the perturbative nature of the chiral expansion
makes the generation of pole singularities, either bound states or resonances, im-
possible from the very beginning unless they are already present at lowest order
in the expansion. There are no bound states in the  and N systems, but the and the 
 resonances are outstanding features of these reactions dominating
the corresponding cross sections at the C.M. energies
ps = m = 770MeV andps = m
 = 1232MeV respectively.
2 The role played by unitarity
Exact Unitarity plays a crucial role in the description of resonances. However,
ChPT only satisfies unitarity perturbatively. Nevertheless, there aremanyways to
restore exact unitarity out of perturbative information, i.e.: the K-matrix method
Ref. [7], the Inverse Amplitude Method (IAM) Ref. [8,9], the Bethe-Salpeter Equa-
tion (BSE) Ref. [11,12], the N/D method Ref. [10], etc. ( See e.g. Ref. [13] for a
recent review), which are closely related to one another.
Some of these Unitarization methods have been very successful describing
experimental data in the intermediate energy region including the resonant be-
havior. Despite this success the main drawback is that this approach is not as
systematic as standard ChPT, for instance in the estimation of the order of the
neglected corrections. In this work we report on our most recent works related to
Chiral Perturbation Theory and Unitarization 3
P/2 - p
P/2 + p            P/2 + k P/2 + q
P/2 -  q
+=
                 P/2-k
  TVT V
Fig. 1. Diagrammatic representation of the BSE equation. It is also sketched the used kine-
matics.
unitarization, both in the  and N sectors, where we have obtained interme-
diate energy predictions using the chiral parameters and their error bars obtained
from standard ChPT applied at low energies. This means in practice transporting
the possible correlations among the fitting parameters obtained from a fit where
the phase shifts are assumed to be gauss-distributed in an uncorrelated way.
3 Results in the  sector
The ChPT expansion displays a very good convergence in the meson-meson sec-
tor. As a consequence the unitarization methods have been very successful in ex-
tending the ChPT applicability to higher energies. In particular, within a coupled
channel IAM formalism, it has been possible to describe all the meson-meson
scattering data, even the resonant behavior, below 1.2 GeV [9], but without in-
cluding explicitly any resonance field. Theseworks have already been extensively
described in the literature and here we will concentrate in the most recent works
of two of the authors on the  sector, dealing with the Bethe-Salpeter equation,
which has the nice advantage of allowing us to identify the diagrams which are
resumed.
Indeed, any unitarization method performs, in some way or another, an infi-
nite sum of perturbative contributions. At first sight, this may seem arbitrary, but
some constraints have to be imposed on the unitarization method to comply with
the spirit of the perturbative expansion we want to enforce. In the BSE approach
the natural objects to be expanded are the potential and the propagators. The BSE
as it has been used in Refs. [11,12] reads (See Fig. 1)TIP(p; k) = VIP(p; k) + i Z d4q(2)4TIP(q; k)
(q+)
(q-)VIP(p; q) (4)
where q = (P=2q) and TIP(p; k) andVIP(p; k) are the total scattering amplitude1
and potential for the channel with total isospin I = 0; 1; 2. and then the projection
over each partial wave J in the CM frame, TIJ(s), is given byTIJ(s) = 12 Z+1-1 PJ (cos )TIP(p; k) d(cos ) = i8s 12 (s;m2;m2) he2iÆIJ(s) - 1i (5)
1 The normalization of the amplitude T is determined by its relation with the differential
cross section in the CM system of the two identical mesons and it is given by d=d
 =jTP(p; k)j2=642s, where s = P2. The phase of the amplitude T is such that the optical
theorem reads ImTP(p; p) = -tot(s2 - 4sm2)1=2, with tot the total cross section. The
contribution to the amputated Feynman diagram is (-iTP(p; k)) in Fig. 1.
4 E. Ruiz Arriola
where  is the angle betweenp and k in the CM frame, PJ the Legendre polynomi-
als and (x; y; z) = x2+y2+z2-2xy-2xz-2yz. Notice that in our normalization
the unitarity limit implies jTIJ(s)j < 16s=1=2(s;m2;m2).
The solution of the BSE at lowest order, i.e., taking the free propagators for
the mesons and the potential as the tree level amplitude can be obtained after
algebraic manipulations described in detail in Refs. [11,12], renormalization and
matching to the Taylor expansion up to second order in s - 4m2 of the one loop
chiral perturbation theory result. We only quote here the result for the  channel:T-111 (s) = -Ī0(s) + 1162 2(l̄2 - l̄1) + 9760 (6)+ 1s- 4m2 m242 2(l̄2 - l̄1) + 3l̄4 - 6524- 6f2
 (7)
where the unitarity integral Ī0(s) readsĪ0(s)  I0(s) - I0(4m2) = 1(4)2r1 - 4m2s logq1- 4m2s + 1q1- 4m2s - 1 (8)
Here the complex phase of the argument of the log is taken in the interval [-; [.
Similar expressions hold for the scalar-isoscalar () and the scalar-isotensor chan-
nels. Notice that in this, so-called off-shell scheme, the left hand cut is replaced at
lowest order by a pole in the region s << 0. For the low energy coefficients l̄1;2;3;4
we take the values
set A : l̄1 = -0:62  0:94; l̄2 = 6:28  0:48; l̄3 = 2:9  2:4; l̄4 = 4:4 0:3
set B : l̄1 = -1:7  1:0 ; l̄2 = 6:1  0:5 ; l̄3 = 2:9  2:4; l̄4 = 4:4 0:3 (9)
In both sets l̄3 and l̄4 have been determined from the SU(3) mass formulae and
the scalar radius as suggested in [1] and in [14], respectively. On the other hand
the values of l̄1;2 come from the analysis of Ref. [15] of the data on Kl4-decays
(setA) and from the combined study of Kl4-decays and with some unitariza-
tion procedure (set B) performed in Ref. [16]. The results for the  channel are
shown in Fig. 2.
As discussed in Ref. [12] it is also possible to take into account the left hand
cut in the so-called on-shell scheme where it can be shown that after renormaliza-
tion the on-shell unitarized amplitude acquires the following formTIJ(s)-1 + Ī0(s) -VIJ(s)-1 = TIJ(s0)-1 + Ī0(s0) -VIJ(s0)-1 = -CIJ (10)
where CIJ should be a constant, independent of s and the subtraction point s0,
and chosen to have a well defined limit whenm! 0 and 1=f! 0.VIJ(s) is the on-
shell-potential and has the important property of being real for 0 < s < 16m2, and
presenting cuts in the four pion threshold and the left hand cut caused by the uni-
tarity cuts in the t and u channels. The potential can be determined by matching
the amplitude to the ChPT amplitude in a perturbative expansion. This method
provides a way of generating a unitarized amplitude directly in terms of the low
Chiral Perturbation Theory and Unitarization 5
Fig. 2. I = J = 1  phase shifts as a function of the total CM energy ps for both sets ofl̄’s given in Eq. (9). Left (right) figures have been obtained with the set A (B) of parame-
ters. Solid lines are the predictions of the off-shell BSE approach, at lowest order, for the
different IJ-channels. Dashed lines are the 68% confidence limits. Circles stand for the
experimental analysis of Refs. [17] and [18].
energy coefficients l̄1;2;3;4 and their errors. In this on-shell scheme a successful
description of both  scattering data as well as the electromagnetic pion form
factor, in agreement with Watson’s theorem, becomes possible yielding a very ac-
curate determination of some low energy parameters. The procedure to do this
becomes a bit involved and we refer to Ref. [12] for further details. We should
also say that the one-loop unitarized amplitudes generate the complete ChPT re-
sult and some of the two and higher loop results. The comparison of the generated
two-loop contribution of threshold parameters with those obtained from the full
two loop calculation is quantitatively satisfactory within uncertainties.
4 Results in the N sector
The methods and results found in the  system are very encouraging, suggest-
ing the extension to the N system. However, ChPT does not work in the N
sector as nicely as it does in the  sector. As we will see, the low convergence
rate of the chiral expansion makes it difficult to match standard amplitudes to
unitarized ones in a numerically sensible manner. After an initial attempt within
the relativistic formulation [19], it was proposed to treat the baryon as a heavy
particle well below the nucleon production threshold [20]. The resulting Heavy
Baryon Chiral Perturbation Theory (HBChPT) provides a consistent framework
for the one nucleon sector, particularly in N scattering [6]. The proposal of
Ref. [21] adopting the original relativistic formalism but with a clever renormal-
ization scheme seems rather promising but unfortunately the phenomenological
applications to N scattering have not been worked out yet.
4.1 The IAM method in -N scattering
The inverse amplitudemethod (IAM) is a unitarizationmethodwhere the inverse
amplitude, and not the amplitude, is expanded, i.e., if we have the perturbative
6 E. Ruiz Arriola
expansion for the partial wave amplitude f(!),f(!) = f1(!) + f2(!) + f3(!) + : : : (11)
with ! = pq2 +m2 and q the C.M. momentum, then one considers the expan-
sion 1f(!) = 1f1(!) - f2(!) + f3(!)[f1(!)℄2 + f2(!)2[f1(!)℄3 + : : : (12)
The IAM fulfills exact unitarity, Imf(!)-1 = -q and reproduces the perturbative
expansion to the desired order.
This method has been applied for N scattering [22] to unitarize the HBChPT
results of Ref. [6] to third order. In this context, it is worth pointing out that the
use of a similar unitarization method, together with very simple phenomeno-
logical models, was already successfully undertaken in the 70’s (see [23] and
references therein). Nonetheless, a systematic application within an effective La-
grangian approach was not carried out. In [22] the phase shifts for the partial
waves up to the inelastic thresholds have been fitted, obtaining the right pole for
the 
(1232) in the P33 channel. In that work, it has been pointed out that to get
the best accuracy with data, one needs chiral parameters of unnatural size, very
different from those of perturbative HBChPT. This is most likely related to the
slow convergence rate of the expansion. However, it must be stressed that one
can still reproduce the 
(1232) with second order parameters compatible with
the hypothesis of resonance saturation [32]. In a subsequent work [24] we have
proposed an improved IAM method based on a reordering of the HBChPT se-
ries. The encouraging results for the 
-channel have also been extended to the
remaining low partial waves [25], as it can be seen in Fig. 3. In this case, the size
of the chiral parameters is natural and the 2 per d.o.f is considerably better than
the IAM applied to plain HBChPT.
4.2 BSE method and the 
-resonance
Recently [26], we have used the BSE to HBChPT at lowest order in the chiral ex-
pansion and have looked at the P33 channel. We have found a dispersive solution
which needs four subtraction constants,f3=23=2 ;1(!)-1 = - 24(!2 -m2) -f2!2g2A + P(!) + (!2 -m2)J̄0(!)=6
P(!) = m3 
0 + 
1(!m - 1) + 
2(!m - 1)2 + 
3(!m - 1)3 (13)
where the unitarity integral is given byJ̄0(!)  J0(!) - J0(m) = -p!2 -m242 farcosh!m - ig; ! > m (14)
The 2 fit yields the following numerical values for the parameters:
fit0 = 0:045  0:021 ; 
fit1 = 0:29  0:08 ; 
fit2 = -0:17  0:09 ; 
fit3 = 0:16  0:03
Chiral Perturbation Theory and Unitarization 7
Fig. 3. Phase shifts in the IAM method as a function of the C.M. energy
ps. The shaded
area corresponds to the result of propagating the errors of the chiral parameters obtained
from low-energy data (see Ref. [25]). This illustrates the uncertainties due to the choice of
different parameter sets from the literature. The dotted line is the extrapolated HBChPT
result. The continuous line is an unconstrained IAM fis to the data, whereas for the dashed
line the fit has been constrained to the resonance saturation hypothesis.
with 2=d:o:f: = 0:2. However, if we match the coefficients with those stemming
from HBChPT we would get instead the following numerical values:
th0 = 0:001  0:003 ; 
th1 = 0:038  0:006 ; 
th2 = 0:064  0:005 ; 
th3 = 0:036  0:002
The discrepancy is, again, attributed to the low convergence rate of the expansion.
The results for the P33 phase shift both for the fit and the MonteCarlo propagated
errors of the HBChPT matched amplitudes have been depicted in Fig. 4
8 E. Ruiz Arriola
040
80120
160
1200 1300 1400Æ3 31
(s)(degre
es)
ps (MeV)
Fig. 4. P33 phase shifts as a function of the total CM energyps. The upper solid line repre-
sents a 2-fit of the parameters, 
0;1;2;3 to the data of Ref. [31] (circles). Best fit parameters
are denoted 
fit0;1;2;3 in the main text. The lower lines stand for the results obtained with
the parameters deduced from HBChPT and denoted 
th0;1;2;3. Central values lead to the
solid line, whereas the errors on 
th0;1;2;3 lead to the dash-dotted lines.
4.3 N scattering and theN(1535) resonance
One of the greatest advantages of both the IAM and the BSE methods is that
the generalization to include coupled channels is rather straightforward. For the
IAM case we refer to [9] for more details. As for the BSE, two of the authors [30]
have dealt with the problem in the S11 channel, at ps up to 1800GeV using the
full relativistic, rather than the heavy baryon formulation used in Ref. [27] for
the s-wave and extended in Ref. [28] to account for p-wave effects. For these en-
ergies there are four open channels, namely N,N,K and K, so that the BSE
becomes a 4  4 matrix equation for the I = 1=2, J = 1=2 and L = 0 partial wave.
Additional complications arise due to the Dirac spinor structure of the nucleon,
but the BSE can be analytically solved after using the above mentioned off-shell
renormalization scheme. It turns out [30] that to lowest order in the potential and
the propagators, one needs 12 unknown parameters, which should be used to
fit experimental data. Several features make the fitting procedure a bit cumber-
some. In the first place, there is no conventional analysis in the relativistic ver-
sion of ChPT for this process and thus no clear constraints can be imposed on the
unknown parameters. Secondly, the channel N ! N is not included in our
calculation. Therefore one should not expect perfect agreement with experiment,
particularly in the elastic channel since it is known that 10 - 20% of the N res-
onance decay width goes into N. On the other hand, one cannot deduce from
here how important is the N channel in the  production channel, N ! N.
Actually, in Ref. [29] it has been suggested that the bulk of the process may be
explained without appealing to the three body intermediate state N. The work
of Ref. [29] would correspond in our nomenclature to the on-shell lowest order
BSE approximation, which by our own experience describes well the bulk of the
data. With the BSE we have provided a further improvement at low energies
Chiral Perturbation Theory and Unitarization 9
by including higher order corrections. In the absence of a canonical low energy
analysis it seems wiser to proceed using the off-shell renormalization scheme. In
Fig. 5 we present a possible 12 parameter fit which accounts both for the elas-
tic low and intermediate energy region and the lowest production channels. The
failure to describe data around theN resonance is expected, since as we have al-
readymentioned the N channel must be included. Our results seem to confirm
the assumption made in Ref. [29] regarding the unimportance of the three body
channel in describing the coupling of the N resonance to N.
2002040
6080100120
140160
1000 1200 1400 1600 1800 2000S 11;
Æ(s)(degr
ees)
3333333333333333333333333333333333
33333333333333333333333
0:20:30:40:5
0:60:70:80:9
11:1
1000 1200 1400 1600 1800 2000
S 11;
33333333333333333333333333333333333333333333333333333333333333333333333333333
333333333333
00:51
1:522:5
3
1450 1500 1550 1600 1650 1700 1750
  p!n[m
b℄
ps (MeV)
333
3333333 3 333 3 33
00:20:4
0:60:81
1:2
1600 1700 1800 1900 2000 2100 2200
 p!K0 [m
b℄
ps (MeV)3333
3333333
333333333333333333333333333333333 33 333 33333333 3 33 333
Fig. 5. N scattering BSE results as a function of C.M. energy ps. Upper left figure: S11
phase shifts. Upper right figure: inelasticity in the N channel. Lower left figure:N! N
cross section. Lower right figure: N! K cross section. Data fromRef. [31]. (See Ref. [30]
for further details.)
5 Conclusions and Outlook
The results presented here show the success and provide further support for uni-
tarization methods complemented with standard chiral perturbation theory, par-
ticularly in the case when resonances are present. But unitarization by itself is
not a guarantee of success; the unitarization method has to be carefully chosen
so that it provides a systematic convergent and predictive expansion, as we have
discussed above. In order to describe the data in the intermediate energy region
the chiral parameters can then be obtained from Either a direct 2 fit of the order by order unitarized amplitude and the cor-
responding low energy parameters. The upper energy limit is determined by
imposing an acceptable description 2=DOF  1.
10 E. Ruiz Arriola Or from a low energy determination of the low energy parameters with errors
by performing a 2-fit of the standard ChPT amplitude until 2=DOF  1, and
subsequent MonteCarlo error propagation of the unitarized amplitude.
Differences in the low energy parameters within several methods should be com-
patible within errors, as long as the Chiral expansion has a good convergence.
But, unfortunately this is not always the case. Clearly, the N sector is not only
more cumbersome theoretically than the  sector but also more troublesome
from a numerical point of view. Standard ChPT to a given order can be seen as a
particular choice which sets higher order terms to zero in order to comply with
exact crossing but breaking exact unitarity. The unitarization of a the ChPT ampli-
tude is also another choice of higher order terms designed to reproduce exact uni-
tarity but breaking exact crossing symmetry. Given our inability to write a closed
analytic expression for an amplitude in a chiral expansion which simultaneously
fulfills both exact crossing and unitarity we have preferred exact unitarity. This
is justified a posteriori by the successful description of data in the intermediate
energy region, which indeed suggests a larger convergence radius of the chiral
expansion.
Acknowledgments
This research was partially supported by DGES under contracts PB98–1367 and
by the Junta deAndalucı́a FQM0225 aswell as byDGICYT under contracts AEN97-
1693 and PB98-0782.
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