Nuclear and Particle Physics Proceedings 00 (2020) 1–5

Nuclear and
Particle Physics

Proceedings

Assessment of systematic theory uncertainties in IAM unitarization

Juan Escudero-Pedrosaa, Felipe J. Llanes-Estradaa,1, José Antonio Ollerb, Alexandre Salas-Bernárdeza

aUniversidad Complutense de Madrid, Depto. Fı́sica Teórica and IPARCOS, 28040 Madrid, Spain.
bUniversidad de Murcia, Departamento de Fı́sica, E-30071 Murcia, Spain

Abstract

Effective Field Theories (EFTs) for Goldstone Boson scattering at a low order allow the computation of near–
threshold observables in terms of a few coefficients arranged by a counting. As a matter of principle they should
make sense up to an energy scale E ∼ 4πF but the expansion in powers of momentum violates exact elastic unitarity
and renders the derivative expansion unreliable at much lower energies. If new–physics deviations from the Standard
Model are found and encoded in low-energy coefficients, perhaps at the LHC, it will be profitable to extend the reach of
the EFT to regimes where partial waves are saturating unitarity. The methods known in hadron physics as “Unitarized
Chiral Perturbation Theory” extend the EFT up to its nominal reach or up to the first new physics resonance or structure
(if found below that energy reach) in the partial wave amplitude, but they usually have unknown uncertainties. We
recapitulate our analysis of the systematic theory uncertainties of the well known Inverse Amplitude Method (IAM).

Keywords: Theoretical uncertainties, Unitarization, Effective Theory, Inverse Amplitude Medhod

1. The Inverse Amplitude Method

Effective Field Theories can be used to expand a scat-
tering partial wave amplitude with terms of order s,
s2 . . . sn+1 . . .

t ' t0 + t1 + . . . (1)

Such expansion satisfies elastic unitarity only in per-
turbation theory, Im(t1) = σ|t0|2, failing to comply with
exact unitarity Im(t) = σ|t|2. This defficiency of the
quasi-Taylor expansion in terms of powers of s with
logarithmic modifications spoils the series convergence
much before its nominal 4πF limit scale: In Fig. 1, the
lower discontinuous line (cyan online) fails to track ex-
perimental data for ππ scattering data in the I = 1 = J
partial wave featuring the well-known ρ resonance.

Several methods of unitarization have been devised
to extend the EFT’s reach, but analysis of their system-
atic theory error are scarce. Here we analyse the Inverse

∗Communicated to the 23rd International Conference in Quantum
Chromodynamics (QCD2020), Oct. 27th–30th 2020, convened online.

1Speaker. E-mail fllanes@fis.ucm.es

Amplitude Method [1–5] that transforms the perturba-
tive expansion of Eq. (1) into the following simple form

tIAM ≡
t2
0

(t0 − t1)
. (2)

This is a simple algebraic formula deduced by writing
down a dispersion relation for the function

G(s) ≡
t0(s)2

t(s)
(3)

from Cauchy’s theorem applied to the contour in Fig. 2.
The dispersion relation, appropriately subtracted so

convergence for real |s| → ∞ is guaranteed (the partial-
wave’s diverging exponential factor [6] for complex s is
here superconverging for the inverse amplitude), reads

G(s) = G(ε) + G′(ε)(s − ε) +
1
2

G′′(ε)(s − ε)2 + PC(G)+

+
(s − ε)3

π

∫
LC

ds′
Im G(s′)

(s′ − ε)3(s′ − s)
+

+
(s − ε)3

π

∫
RC

ds′
Im G(s′)

(s′ − ε)3(s′ − s)
. (4)

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Escudero Pedrosa, Llanes-Estrada, Oller and Salas-Bernárdez / Nuclear and Particle Physics Proceedings 00 (2020) 1–5 2

Figure 1: Lowest line: The perturbative Eq. (1) fails to track ππ p-
wave scattering data. Upper line: Implementing Eq. (2) reasonably
predicts from threshold data (not from a fit) the shape and position
of the resonance with error under 10%. This unitarization by the In-
verse Amplitude Method extends the validity of the EFT nearer to its
nominal expansion convergence at 4π fπ ' 1.2 GeV).

Eq. (2) is obtained by neglecting possible poles of the
inverse amplitude (PC), approximating with the EFT
not only the subtraction constants G(n)(ε) but also the
left cut, even for s beyond its nominal reach, and ne-
glecting all inelastic channels,

G(s) ≡
t0(s)2

t(s)
= t0(s) − t1(s)+

+
s3

π

∫
LC

ds′
Im G(s′) + Im t1(s′)

s′3(s′ − s)
. (5)

One can also obtain Eq. (2) on the real axis by simply
derivatively expanding 1/t in the EFT. But the disper-
sive derivation has the advantages of being valid in the
complex plane so one can look for resonances there, and
allowing us to analyse the quality of each of the approx-
imations. We can quantify the uncertainty introduced
by defining an IAM relative separation from the exact
partial wave amplitude as

∆(s) =

(
tIAM(s) − t(s)

tIAM(s)

)
. (6)

For new physics applications, in which one would try
to predict the mass of a new particle or resonance [7–9]
from the measured low-energy coefficients of the EFT,
we are most interested in the uncertainty in the mass of
that resonance. To estimate it, we need the uncertainty
in the position of the pole at the resonance’s mass sR

(with σ =
√

1 − 4m2/s ' 1 the phase space factor),

t0(sR) − t1(sR) + iσ(sR)t2
0(sR) = −∆(sR)G(sR) . (7)

Figure 2: Cauchy’s theorem applied within the depicted contour to
the auxiliary function of Eq. (3) leads to Eq. (2). The contour C de-
forms around the two left (“interaction”) and right (“unitarity” ) cuts
in the complex-s plane. The two saltires additionally represent the
n-th order pole at z = ε (where the dispersion relation is subtracted)
and the simple pole at z = s + iε (with s > 4m2) coming from the
denominators in Eq. (4).

2. Uncertainty estimate from approximating the LC

From the definition in Eq. (6) and the dispersion re-
lation in Eq. (5) we can, for example, isolate the part of
the uncertainty that stems from approximating the left
cut in perturbation theory, Im G ' −Im t1, given by

∆(s)G(s) ≡
( tIAM − t

tIAM

) t2
0

t
=

s3

π

∫
LC

ds′
Im(G + t1)
s′3(s′ − s)

(8)

(other uncertainties being similarly treated [11]).
The analysis of this approximation over the left cut is

organized by dividing it into three pieces, Ii , i = 1, 2, 3

s3

π

∫
LC

dz
Im (G + t1)

z3(z − s)
≡

≡ I1[G + t1] + I2[G + t1] + I3[G + t1] . (9)

We have performed the splitting at the points |λ2| =

0.47 GeV2 (below which we can rely on the EFT)
and |Λ2| = 2 GeV2 above which we start using the
high-energy asymptotic results. Our computation for
this high-energy I1 is basically based on the Sugawara-
Kanazawa theorem [10], allowing us to relate the
asymptotic behavior on the left cut with that on the
physical Right Cut, where Regge theory can be applied
in hadron physics. (In Beyond the Standard Model
Physics, a careful analysis would be necessary to see
whether all the assumptions needed for this calculation
are satisfied, if and when information thereon starts un-
folding).

The low-|s| contribution I3 lies within the EFT range
of applicability so it is easier to constrain, as we know
that Im(G + t1) is NNLO there, and thus of order s3.



Escudero Pedrosa, Llanes-Estrada, Oller and Salas-Bernárdez / Nuclear and Particle Physics Proceedings 00 (2020) 1–5 3

We find, for s over the right physical cut (in the res-
onance region), with k′ ∼ 0.15 for the ρ-meson channel
and k ∼ 1/(4π fπ) in ChPT,

|I1[G](s,Λ,∞)| ≤
k′s2

π
log

(
1 +

s
Λ2

)
(10)

|I3[G + t1](s, λ)| '
ks3

π
log

(
1 +

λ2

s

)
. (11)

The intermediate-momentum integral I2(G + t1) is
most difficult to estimate, so we have adopted three in-
dependent schemes.

In the first, we assume Im (G + t1) ≤ Im (t1) which
is valid when G and −t1 have similar imaginary parts
(remember, this is the case at low momentum), which
leads to I2 ≤ 6%. However, for the I = 1 = J channel,
this is not a good approximation (see Fig. 3 below).

A second estimate can be produced noting that the
derivative with respect to s of the integral in Eq. (8)
should be typically dominated by the low-energy re-
gion. By integrating its first derivative one can then em-
ploy naturalness arguments to estimate the integration
constant, that acts as a counterterm, in terms of a scale
L2 ∈ [2, 4] GeV2. This procedure yields

I2[G + t1](s,Λ, λ) =
s3k
π

log
s + λ2

L2 , (12)

that numerically evaluates to I2 ∼ 2.4%.
Finally, we have made a third order-of-magnitude es-

timate based on solutions to the GKPY equations [12]
for ππ scattering in the J = 1 = I partial wave, provided
to us by the authors of from [13]. These data can give
an idea of how the partial wave looks like on the left cut,
and the resulting G is depicted in figure 3, together with
the t1 computed in perturbation theory. Both amplitudes
are shown up to s = −1 GeV2, and a linear matching is
performed to reach s = −2 GeV2 where we start off with
the asymptotic behavior. Note that Im(G) seems to have
an almost-pole (due to an almost-zero of the amplitude
t) on this channel’s left cut. This gives a large contri-
bution to the integral of Im(G + t1): we find I2 ∼ 8%.
However, this actual value is very sensitive to the degree
of cancellation in t, which is difficult to pin down along
the left cut. In channels without such almost-zero, the
uncertainty induced to the IAM would be smaller.

Once the contributions from the three integration in-
tervals are combined, an estimate on the uncertainty in
the (“new physics”) resonance pole position has been
obtained from Eq. (7).

s [GeV2]
-1 -0.8 -0.6 -0.4 -0.2 0

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Im G

Im t1

Figure 3: Estimates of the imaginary part of the perturbative t1 in
ChPT and auxiliary G = t20/t (from data for t courtesy of Jacobo Ruiz
de Elvira [13] from solving the GKPY equations), over the low-energy
part of the left cut for s ∈ (−1 GeV2, 0).

3. Uncertainty from Adler zeroes and CDD poles

Poles of the inverse amplitude 1/t ∝ G, that is, zeroes
from the actual amplitude t, yield additional contribu-
tions to Cauchy’s theorem denoted as PC in Eq. (4). We
also divide them into those that appear at low-s near
threshold because the EFT derivative terms vanish there
(Adler zeroes of the amplitude), and those at higher s,
perhaps near resonances, that are more influenced by
new physics (CDD poles of the inverse amplitude).

The Adler zeroes at sA have a tiny subpermille level
influence on the amplitude in the resonance region, and
moreover they can be easily incorporated in a small
modification of the IAM itself if one wishes more preci-
sion near threshold, by employing instead of Eq. (2) the
following,

tmIAM ≡
t2
0

t0 − t1 + AmIAM
(13)

where the denominator has been modified by

AmIAM = t1(s0)−
(s0−sA)(s−s0)

s − sA
(t′0(s0)−t′1(s0)) (14)

as has already been known for a time [14].
While the Adler zeroes can be easily seen in the form

of the EFT partial wave amplitudes, the CDD poles [15]
are a bit more elusive. If one such pole would fall near
a resonance [16, 17], it could screen it and a low-s anal-
ysis of the amplitude from low-energy measurements
could fail to predict it, causing errors of order O(100%).

To avoid it, we have proposed that the IAM be used
in conjunction with a check for zeroes of the amplitude,



Escudero Pedrosa, Llanes-Estrada, Oller and Salas-Bernárdez / Nuclear and Particle Physics Proceedings 00 (2020) 1–5 4

that from the low-energy EFT can be carried out by ex-
amining the condition

t0(sC) + Ret1(sC) = 0 . (15)

Should this condition reveal a zero of the amplitude at
sC , the IAM would be applied to a modified amplitude
t(s)/(s− sC) instead, yielding a modification in the same
spirit of Eq. (14), namely

tIAM =
t2
0

t0 − t1
→

t2
0

t0 − t1 + s
s−sc

Re(t1)
. (16)

After this check and eventual modification, that needs
to be carried out for each extrapolation of low-energy
data with the IAM, the effect of the CDD poles should
be controlled.

4. Inelastic cuts

The right cut that the IAM takes into account is
only the elastic Goldstone boson-Goldstone boson cut,
e.g. in ππ → ππ within chiral perturbation theory, or
ωLωL → ωLωL in extensions of the electroweak stan-
dard model.

But in the first case, other channels such as KK̄, ηη,
4π, etc, or in the second case hh or pairs of eventual new
particles, can appear. One can write a coupled-channel
Inverse Amplitude Method [18, 19] accounting for the
two-body channels, though its dispersive derivation is
not straightforward. If this step is not taken, an error
in the purely elastic IAM is incurred and we have esti-
mated it in [11] for both theories. To do it, we explicitly
show in the imaginary part of the inverse amplitude the
piece that is eventually neglected,

Im
1

tππ
= −σππ

(
1 +

σKK̄

σππ

|tππ→KK̄ |
2

|tππ→ππ|2
)
. (17)

In Chiral Perturbation theory both the ratio of phase
spaces (at moderate energies) and the ratio of inelas-
tic to elastic scattering suppress the contribution of the
inelastic cut respect to the elastic one in the dispersion
relations. Thus, we are relatively sure that below a GeV,
the elastic ππ contribution carries the bulk of the right
cut. For the Higgs Effective Field Theory, because the
Higgs mass and the W/Z mass are similar (and similarly
small respect to

√
s), there is no phase space suppres-

sion and we would not know whether the interchannel
ωLωL → hh coupling is or not small respect to the elas-
tic one until they are eventually measured.

Additionally, the two-particle right cuts are overlaid
by four- (and higher) particle cuts. These are clearly

suppressed by the phase space. If the Goldstone boson
mass is negligible, these can be written as

φn =
1

2(4π)2n−3

sn−2

Γ(n)Γ(n − 1)
(18)

and otherwise, a numeric computation is necessary. The
(appropriately dimensionally normalized) ratio of phase
spaces is small and controls this uncertainty.

5. Use of the EFT to approximate the subtraction
constants

The subtraction constants multiplying the (s − ε)n

polynomial terms in Eq. (4) are taken at NLO in the
EFT. This is a valid approximation because the subtrac-
tion point ε is around (below) threshold where the EFT
is most accurate.

Nevertheless, the effect of approximating them (and,
partly, the left cut too) to NLO can be checked if/when
the NNLO coefficients of the EFT are at hand, and this
is just the case in ChPT for ππ scattering [20].

In that case, a formula similar to Eq. (2) can be found,
incorporating the NNLO order t2 ∝ s3, by writing a dis-
persion relation for

G(s) =
t2
0

t
'

t2
0

t0 + t1 + t2
. (19)

Without going through the entire dispersive rederiva-
tion, we can just expand

G(s) =
t2
0

t
' t0 − t1 − t2 +

t2
1

t0
. (20)

and then invert to obtain tIAM from the second equal-
ity. The size of the two new terms in the denominator,
t2
1/t0 − t2 have been estimated from the known li and ri

NLO and NNLO counterterms [21] in Chiral Perturba-
tion Theory (and can be evaluated in Higgs EFT once
the corresponding coefficients become known). The ex-
pected size of those new terms becomes the uncertainty
estimate on the basic Inverse Amplitude Method if the
NNLO contributions are ignored.

6. Conclusions

We have examined the different contributions to the
uncertainty budget of one of the best known unitariza-
tion methods that can be applied to a low-energy EFT,
the Inverse Amplitude Method of Eq. (2). They are col-
lected in Table 1.

It is clear that the left-cut approximation is the dom-
inant source of uncertainty. For this I = 1 = J partial



Escudero Pedrosa, Llanes-Estrada, Oller and Salas-Bernárdez / Nuclear and Particle Physics Proceedings 00 (2020) 1–5 5

Table 1: Uncertainty budget for the Inverse Amplitude Method in Hadron Physics. Each entry shows the scaling with the relevant variables (it is
understood that in the resonance region

√
s ∼ mρ ∼ MR are similar scales), the order of magnitude of the error they introduce in that resonance

region (evaluated in the vector-isovector channel to be specific) and whether the basic method of Eq. (2) can be improved to remove that source of
uncertainty if needed.

Source of uncertainty Behavior Displacement of pole at
√

s = mρ Can it be improved?
Approximate Left Cut (

√
s/(4π fπ))4 0.17 Partially

Adler zeroes of t (mπ/mρ)4 10−3 − 10−4 Yes: mIAM
CDD poles (zeroes of t) at M0 M2

R/M
2
0 0 –O(1) Yes: extract zero before IAM

Inelastic 2-body (
√

s/(4π fπ))4 10−3 Yes: matrix form
Inelastic 4...-body (

√
s/(4π fπ))4 10−4 Partially

O(p4) truncation (
√

s/(4π fπ))6 10−2 Yes: O(p6) IAM, Eq. (20)

wave, it leads to an a priori uncertainty of O(17%). We
know, a posteriori (Fig. 1) that the ρ is reproduced to
10%; In the case of Higgs EFT we would at best find a
separation from the Standard Model near threshold, so
that only the a priori estimate would be at hand.

While other authors have worried about systematic
uncertainties in the IAM [22, 23], we have presented
a comprehensive analysis of the status of those un-
certainties, with an explicit evaluation in Chiral Per-
turbation Theory where data on pion-pion scattering
is at hand, and provided working guidelines on how
the computation can be approached should coefficients
of Higgs EFT be found to separate from the Stan-
dard Model at the LHC. While we have concentrated
on the s-dependence of the scattering amplitude, other
authors [24] have also examined the IAM to perform
extrapolations in theoretical parameters (saliently, the
quark mass) that can be of assistance for exploitation of
lattice data. The interested reader can find much many
additional details in an upcoming publication [11].

Acknowledgments

We thank Prof. S. Narison and his Montpellier CNRS
team for providing, year after year, a venue for dis-
cussion of new results in the strong interactions and
QCD, and especially for preparing it fully online un-
der this year’s conditions. We acknowledge support by
EU Horizon 2020 research and innovation programme,
STRONG-2020 project, grant agreement No 824093;
grants FEDER (EU) and MINECO/MICINN (Spain)
FPA2016-75654-C2-1-P and -77313-P, PID2019 -
108655GB-I00, -106080GB-C21 (Spain) -106080GB-
C22; Universidad Complutense de Madrid under re-
search group 910309 and the IPARCOS institute; and
the VBSCan COST Action CA16108.

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	1 The Inverse Amplitude Method
	2 Uncertainty estimate from approximating the LC
	3 Uncertainty from Adler zeroes and CDD poles
	4 Inelastic cuts
	5 Use of the EFT to approximate the subtraction constants
	6 Conclusions