Improved Unitarized Heavy Baryon Chiral Perturbation Theory for pion-nucleon Scattering to fourth order
ABSTRACT We extend our previous analysis of the unitarized pion-nucleon scattering amplitude including up to fourth order terms in Heavy Baryon Chiral Perturbation Theory. We pay special attention to the stability of the generated Delta(1232 resonance, the convergence problems and the power counting of the chiral parameters. Comment: 9 pages, 4 figures
arXiv:nucl-th/0312034v1 11 Dec 2003
Improved Unitarized Heavy Baryon Chiral Perturbation Theory for πN Scattering to
A. G´ omez Nicola1,∗J. Nieves2,†J.R. Pel´ aez1,‡and E. Ruiz Arriola2§
1Departamento de F´ ısica Te´ orica. Universidad Complutense. 28040 Madrid, Spain. and
2Departamento de F´ ısica Moderna, Universidad de Granada, E-18071 Granada, Spain
We extend our previous analysis of the unitarized pion-nucleon scattering amplitude including up
to fourth order terms in Heavy Baryon Chiral Perturbation Theory. We pay special attention to the
stability of the generated ∆(1232) resonance, the convergence problems and the power counting of
the chiral parameters.
PACS numbers: 11.10.St, 11.30.Rd,11.80.Et,13.75.Lb,14.40.Cs,14.40.Aq
Unitarization methods have been widely and successfully employed in the recent past to enlarge the applicability
region of Chiral Perturbation Theory (ChPT) expansions, both in the meson-meson sector as well as in the meson
baryon sector and to describe the lightest resonances without including them explicitly as degrees of freedom. Two
important constraints are required: exact unitarity and compliance with Chiral Perturbation Theory at a given order
of the expansion. In practice this approach provides a remarkable description of data in the scattering region. In
the case of πN scattering in the elastic region, the subject of this paper, a thorough partial wave analysis exists 
(see also the recent update ). For such a system, pions and nucleons are treated as explicit degrees of freedom and
a consistent counting becomes possible if nucleons are treated as heavy particles but in a covariant framework ,
yielding the so called Heavy-Baryon Chiral Perturbation Theory (HBChPT) [4, 5, 6]. In this counting, the expansion
for the scattering amplitude is done as a series of eN/(F2lMN+1−2l) terms , with l = 1,...,[(N + 1)/2], M the
baryon mass, and F the pion decay constant. The quantity e is a generic parameter with dimensions of energy
constructed in terms of the pseudoscalar momenta and the velocity vµ(v2= 1 ) and off-shellness k of the baryons
M v + k, with pBthe baryon four momentum and
order in the expansion. After the relevant effective Lagrangian was written down , and the issue of wave function
renormalization was studied  standard HBChPT calculations to second  third [10, 11] and fourth  order
have become available. The unitarization of these amplitudes of πN scattering in the elastic region has followed
closely these developments, particularly the third order calculation [10, 11]. This is the lowest order which generates a
perturbative unitarity correction of the amplitude. The unitarization was carried out either using the standard Inverse
Amplitude Method  (IAM) or its improved version 1. By successful we mean the possibility of describing the
data in the resonance region with parameters of natural size. The purpose of the present paper is to extend the study
initiated in Ref.  and to analyze specifically the qualitative and quantitative new effects generated by the fourth
order contribution calculated in Ref.  in our unitarization scheme.
Let us then specify the scope and motivations of our work: First, our scheme is based on two fundamental ideas:
demanding exact unitarity and considering the F−2HBChPT expansion independent from and converging faster than
the M−1one. In  we showed that this allows to generate the ∆(1232) as well as to fit the remaining S and P wave
channels with natural values for the low-energy constants (LEC) unlike for instance the IAM . Our method was
implemented in  with the first contribution of order F−4only, coming from the third order amplitude. Including
the fourth order will allow us to check the convergence of our method by considering, for instance, the O(F−4M−1),
to be included in the third order F−4term.
Second, there is an interesting issue that we did not account for in  which has to do with the separation of the
dimensionful third and fourth order LEC into two pieces contributing to the orders F−2and F−4. As we will see
defined through the equation pB=
M the baryon mass at leading
∗Electronic address: email:email@example.com
†Electronic address: email:firstname.lastname@example.org
‡Electronic address: email:email@example.com
§Electronic address: email:firstname.lastname@example.org
1For an alternative scheme based on the Bethe-Salpeter equation applied to the P33 channel see Ref. 
below, taking into account this effect may change considerably our description of the partial waves. The reason why
we did not consider it in  is that we used the amplitudes in , which provide an specific separation that turns
out to be very natural, as we will see below2.
Third, comparing the perturbative results to order three  and four , one observes that in order to achieve a
reasonable convergence, the fourth order constants become of unnatural size and, furthermore, their particular values
are often incompatible from one fit to another. This is a signal of the bad convergence of the HBChPT series and
could influence also the convergence of our unitarized formula.
Fourth, unitarization methods are rarely applied beyond the leading order in the imaginary part of the amplitudes
[16, 17]. The study of the fourth order of πN system within HBChPT provides an opportunity to learn about the
unitarization approach beyond this lowest order.
For comparison purposes with previous works [9, 10, 11, 12, 13, 14] we will take the partial wave analysis performed
in Ref. . The recent update  does not bring significant changes to our discussion.
II. THE UNITARIZED AMPLITUDE
In order to have a neat separate expansion of the partial waves in powers of M−1and F−2, we need to re-expand
the amplitudes in , as it was already done to third order in  with those in . Then, following the notation
in , we have, to fourth order, for any partial wave
with m the pion mass, M the nucleon mass, F the pion decay constant and ω the pion CM energy. The partial wave
l±= −q, (2)
where q is the CM momentum, implies that perturbatively one has3
Following the same ideas as in , we will consider the unitarized amplitude to fourth order:
which, using (3) yields immediately (2). Let us recall that our Improved IAM formula at third order read :
2A similar situation has appeared already in the NNLO unitary analysis of ππ scattering .
3We have checked analytically that the amplitudes in  are perturbatively unitary if the following misprints are corrected:
eq.(3.18) . In fact, with these two signs corrected, we reproduce the threshold parameter expressions given in their eqs.(A.1)-(A.8), except
for the π2in the denominator of the fourth term in the r.h.s. of their (A.8) which should read π3and the +gA¯d18M2
0+, eq.(A.3), that should have the opposite sign.
∂ω(ω) should read
∂ω(−ω) in their eq.(3.16) and 6ω4?
π+ 4ω2+ t?
should read −6ω4?
π+ 4ω2+ t?
which can be now reobtained from eq.(4) by removing the t(1,4)and t(3,4)terms and, consistently with unitarity,
removing also the 2(m/M)t(1,1)t(1,2)factor in the second denominator. Hence, as we have stressed in the introduction,
the knowledge of t(1,4)and t(3,4)allows us to test our power counting by including one more term both in the O(F−2)
and O(F−4) contributions.
III. THE THIRD ORDER AND THE LEC POWER COUNTING
In the literature there are two O(q3) calculations [10, 11], using different choices of counterterms and renormalization
schemes, but only one at O(q4)  following the  scheme. The translation between them does not simply amount
to a change of notation, but involves some 1/M corrections. Since our results at third order  were constructed
directly from , we have to check to what extend our previous O(q3) results are reproduced when using the
amplitudes and notation of Refs. [11, 12]. In so doing two remarks are in order:
First, already at third order, the re-expanded amplitudes of  and  differ slightly due both to a different
choice of the reference frame and of the nucleon wave function renormalization (see comments in ). In practice,
this just means that there are slight numerical differences between the perturbative results, which eventually could
be absorbed in the numerical values of the LEC of the HBChPT Lagrangian.
A second point becomes more relevant for our purposes: if we just take the third order amplitudes in , we
re-expand them separating the different contributions and use our third order unitarized formula (eq.(13) in ), we
find a much worse result than in , particularly in the P33 channel where the ∆(1232) resonance should appear.
This is shown in Figure 1a. We remark that we are using a set of parameters compatible with those in , where
the description of the resonance was excellent within the errors even without fitting.
The origin of this apparent discrepancy is that we have not taken into account that in our power counting scheme,
the LEC themselves may have contributions of different orders. In fact, all the difference with  is that we have
chosen now a different parametrization of LEC, although the numerical values are compatible: in  we followed
[7, 10] where the five O(q3) LEC appearing in the amplitude are called b1+ b2,b3,b6,b16− b15,b19. Here, we follow
[11, 12] where the relevant O(q3) constants are¯d1+¯d2,¯d3,¯d5,¯d14−¯d15,¯d18. Comparing the Lagrangian given in
equations (2.45)-(2.47) of  with that in  one observes that the biare related to the¯difor i = 1,2,3 and to the
¯di−1for i = 6,15,16,19 typically as:
M2¯di∼ constant + biM2/(16π2F2) (6)
with a constant that is O(1) in the F−2counting. This comes from the fact that in  some finite terms coming from
renormalization have been absorbed in the bi.
Now, following our power counting arguments, if we replace in the amplitudes of refs.[11, 12] the¯diusing eq.(6),
there are pieces in t(1,3)shifted to t(3,3)(remember that all the dependence with the¯di is in t(1,3)). This changes
the functional dependence of t(3,3), including, for instance, higher order polynomial contributions that otherwise were
not present. Although the perturbative amplitude remains the same, the unitarized one changes since t(1,3)and t(3,3)
are treated on a different footing. With this procedure we obtain the unitarized results shown in Figure 1b. The
improvement is clear for the P33 wave and the results are similar to those in . The corresponding values for the
mass and width of the ∆(1232) extracted from the phase shifts are given in the second column of Table I. This
highlights the importance of taking into account the counting of the LEC.
The above separation is, of course, arbitrary, since nothing prevents us from normalizing the¯di, which are quantities
of dimension E−2as (4πF2)¯di instead of M2¯di, assuming that both are quantities of natural size. Thus, the most
general way to proceed would be to consider as free parameters the coefficients of the O(1) and O(F−2) terms in M2¯di.
In such a way we would duplicate the number of O(q3) LEC, rendering the approach unnecessarily complicated, since
we already know that it is enough to consider the separation given by the biparametrization . We will thus use
only that separation in our calculations, but, after performing the fit, we will give the results in the¯diset for easier
comparison with the literature.
The results of our O(q3) fit is shown in Figure 2. The fit parameters and their errors are given in the fourth column
of Table III whereas the results for the ∆ mass and width are given in Table I. All them are in agreement with what we
found in . The description of data in all channels is very good and our O(q3) LEC are all of natural size although,
as it also happened in , they differ somewhat from those obtained from HBChPT (second column in Table III).
Note that systematic errors are not given in Table III, although they are dominant as it can be seen from Table II).
This is probably another consequence of the poor convergence of the HBChPT series. Let us nevertheless recall that,
in  it was shown that one can perform O(q3) fits where the O(q2) parameters ciare fixed to the predictions of
Resonance Saturation  and the results are still in excellent agreement with data.
we use the¯di, whereas in b) in the two right columns we use the biset. Experimental data are from . The areas between dotted lines
correspond to the propagated errors of the parameters of fit 1 in  in both cases.
O(q3) Unitarized phase shifts as a function of the total CM energy√s: As explained in section III: a) In the two left columns
in Table III.
Unitarized O(q3) fit. The fit parameters are given in Table III. The areas between dotted lines correspond to the errors given
IV.FOURTH ORDER RESULTS
Let us consider now our fourth order unitarized amplitude (4) with the HBChPT amplitudes of . In principle,
the O(q4) amplitude depends on nine different combinations of O(q4) constants ¯ ei, in addition to the four O(q2) ci
and the five O(q3)¯di. These nine combinations are displayed in the first column of Table III. However, as noted in
, the last four combinations actually amount to renormalize the ci, giving rise to new ˜ cias given in eq.(3.23) of
. Strictly speaking, the O(q4) amplitude depends on 18 parameters, since the ci still appear in the pure O(q4)
terms. However, replacing those ciby ˜ ciintroduces higher order corrections, so that the number of free parameters
up to O(q4) is really 14. At this point, as commented in  one can follow two different strategies. The first one is to
consider as free parameters ˜ ci,¯diand the ¯ eiwith i = 14 − 18. This is the parameter set listed in Table II. Although
this is the more natural set, it has the inconvenience that one cannot disentangle which part of the ˜ ci comes from
O(q4) renormalization, since those corrections are relatively large (another signal of the HBChPT bad convergence).
As a consequence, it becomes more difficult to compare with previously published values for the ci. The alternative
(strategy 2) is to fix the ci values, which in turn are the ones less subjected to uncertainties and then use the¯di
and the nine combinations of ¯ eias free parameters. This second strategy is useful for instance to fix the ci to the
predictions of Resonance Saturation  as we also did in .
In addition, we have to face again the problem of the LEC counting, according to the discussion in the previous
section. Thus, besides the “reordering” of terms coming from the¯di, we now have to consider also that coming from
the separation of M3¯ eiinto a constant (a contribution to t(1,4)) plus an O?M2/(4πF)2?term (which contributes to
t(3,4)). Recall that in this case we do not have any “natural” way to perform that separation, as in the O(q3) case.
A.The unitarized partial waves to O(q4)
First, as we did to O(q3), we will show the predictions of our formula without fitting, performing a Monte Carlo
sampling of the perturbative LEC, assuming that they are uncorrelated. Following the first strategy, we have used the
LEC given in , in particular those given in their “Fit 3” that we reproduce in Table II. We also list the “Fit 1 and
2” parameter sets to illustrate that, as pointed out in , the systematic errors are much larger than the statistical
ones, that we are quoting in the Table. For that reason we will take bigger errors, since the errors listed in  are
clearly underestimated. In view of the uncertainties in  we have assigned an error of 1.0 to ¯ e14, ¯ e15, ¯ e17, ¯ e18, of 0.5
to ˜ c1, ˜ c2, ˜ c3(the ˜ cihave bigger uncertainties than the cidue to their ¯ eicontribution) and of 0.25 to the remaining
LEC. In Figure 3a we show the O(q4) prediction “redefining” the¯dias before but without doing so for the ¯ ei, while
in Figure 3b we have also redefined the ¯ eifor convenience as M3¯ ei= 1 +¯fiM2/(4πF)2. Throughout this paper, and
for practical purposes, we will consider only these two situations.
In view of Figure 3 there are several comments in order: First, consider the P33 channel, where our approach is
meant to be more accurate. Here our O(q4) result confirms the O(q3) one and even improves it slightly. Observe for
instance the results for the ∆ parameters given in the fourth column of Table I, corresponding to Figure 3b. This is
one of the main conclusions of this work, namely that the O(q4) calculation confirms that our unitarization method
generates dynamically the ∆(1232) resonance. The improvement of the P33 channel description is also a common
feature with . Note also that this conclusion does not change by using the ¯ ei or the¯fi formulas as long as we
redefine the O(q3) LEC as before.
As for the other channels, we see no actual improvement when comparing with the unfitted O(q3) in Figure 1b.
On the contrary, we get worse results for most of them, especially the two S channels and the P11one. Here, we see
significant differences between using the¯fiprescription or not. In fact, without fitting, our choice for the¯fidoes not
seem to give better results than using the ¯ eidirectly (Figure 3a) or, equivalently, neglecting the O(F−2) contribution
in M3¯ ei.
The hope is that we can perform O(q4) fits which improve these five channels without spoiling the P33 one and
with a reasonable size for the LEC. However, we must bear in mind that, as commented before, the low-energy O(q4)
fits performed in  already show that one gets only slightly better descriptions and bigger uncertainties for the LEC
than the O(q3).
In Figure 4a we show the result of our best fit with the fit errors propagated. Here we have followed the first
strategy and we have used the¯fi defined in the previous section. The LEC and their errors are given in the last
column in Table II. The main observation is that we reproduce the data with constants of natural size. The constant