50. Resonances

1

50. Resonances
Revised August 2023 by D.M. Asner (BNL), C. Hanhart (FZ Jülich) and M. Mikhasenko (Ruhr U.
Bochum).

50.1 General Considerations
Perturbative methods can be applied to systems of quarks and gluons only for large momentum
transfers (see review on ‘Quantum Chromodynamics’) and, under certain conditions, to some properties of systems that contain heavy quarks or very large momentum scales (see review on “Heavyquark and soft-collinear effective theory”). Dealing with Quantum Chromodynamics (QCD) in the
low momentum transfer region is a very complicated, non-perturbative problem. Most hadrons
are resonances, which means that they appear as poles of the S-matrix in the complex plane on
unphysical sheets, a notion further detailed in Sec. 50.2. These resonances can show up either inso-called formation experiments,
A + B → R → C1 + ... + Cn ,where they become visible in an energy scan (for example, the R-function measured in e+ e− annihilation, cf. the corresponding plots in the review on “Plots of Cross Sections and Related Quantities”), or together with a spectator particle S in production experiments of the kind
A + B → R + S → [C1 + ... + Cn ] + S ,Z → R + S → [C1 + ... + Cn ] + S ,where the first reaction corresponds to an associated production, the second is a decay (see “Review of Multibody Charm Analyses”). In the latter case, the resonance properties are commonly
extracted from a Dalitz-plot analysis (see review on “Kinematics”) or projections thereof.
Resonance phenomena are very rich: while typical hadronic widths are of the order of 100 MeV(e.g., for the meson resonances ρ(770) or ψ(4040) or the baryon resonance ∆(1232)) corresponding
to a lifetime of 10−23 s, the widths can also be as small as a few MeV (e.g. of φ(1020) or J/ψ) or
as large as several hundred MeV (e.g. of the meson resonances f0 (500) or D1 (2430) or the baryon
resonance N (2190)).
Typically, a resonance appears as a peak in the total cross section. If the structure is narrowand if there are no relevant thresholds or other resonances nearby, the resonance properties may be
extracted employing a Breit–Wigner parameterization, if necessary improved by using an energydependent width (cf. Sec. 50.3.1 of this review). However, in general, unitarity and analyticity
call for the use of more refined tools as outlined here as well as in recent review articles [1, 2].
When there are overlapping resonances with the same quantum numbers, the resonance terms
should not simply be added but combined in a non-trivial way either in a K-matrix approach(cf. Sec. 50.3.2 of this review) or using other advanced methods (cf. Sec. 50.3.5 of this review).
Additional constraints from the S-matrix allow one to build more reliable amplitudes and, in turn,to reduce the systematic uncertainties of the resonance parameters: pole locations and residues.
In addition, for broad resonances there is no direct relation between pole location and the total
width/lifetime — then, the pole residues need to be used in order to quantify the decay properties.
For simplicity, throughout this review the formulas are given for resonances in a system of
distinguishable, scalar particles. The additional complications that appear in the presence of spins
can be controlled in the helicity framework developed by Jacob and Wick [3], or in a non-covariant [4]or covariant [5] tensor-operator formalisms. Within these approaches, sequential (cascade) decays
are commonly treated as a coherent sum of two-body interactions. Most of the expressions below
are given for two-body kinematics.
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:19am

50.
1.4. Resonances
Hadron spectroscopy
Hadronspectroscopy
THE AUTHOR 1.4. 1.4.
Hadronspectroscopy

2
2

(b)

(a) first Riemann sheet

Im

s
Im

Re s Re

s

s

s
Im

s

s
Im

Re sRe

Re s

Im

s

(b) second Riemann sheet(b) second
Riemannsheetsheet(b) second
Riemann

s
Im

Im

(d)

s

s
Im

Re s

its
arb. unitsarb. unarb. units(a) (a)firstfirst
Riemann
sheet
Riemannsheet

Im(k) [arb. units]

Re s

(c)

its
arb. unitsarb. unarb. unIms

itsarb. unitsarb. unarb. un

its

its

(a)

Re s RRe(k) [arb. units]es(c) transition from first to second Riemann
sheet
Figure 50.1: (a)-(c): Imaginary part of a typical scattering amplitude with an isolated resonance.
(c) transition
fromfrom
firstfirstto secondRiemann(c) rangetransition
to secondRiemann
The blue line shows the physicalof theMandelstam
variables: it is real and starts from
sheet
sheet

Figure 1.4.: Exemplary
scattering
amplitude
the first(green)the thresholdshown bythe blueondot.
Plot Riemann(a) showssheet
the imaginary
part of the amplitude in the
with
one
pole
on
the
secondsheet(red).
complex s-plane that corresponds to the first or physical sheet (green surface), plot (b) shows the
1.4.:1.4.:
Exemplaryscattering
amplitude
on thefirstfirst
Riemann
sheet(green)relatedFigure
unphysicalorthesecondsheet(red
surface)which
theresonance
poles,and plot
FigureExemplaryscattering
amplitude
oncontains
theRiemann
sheet(green)
with
one
pole
on
the
secondsheet(red).
analyticmatrices
continuationof onthespacesame
amplitude
the upper half plane of the physicalwith onethesecondsheetfrom(red).
where (n) (s)(c)areshows
2N the1 diﬀerent
in polechannelcontainingall
comsheet
to
the
lower
half
plane
of
the
unphysicalsheet.
The
two
sheetsbinations of corresponding phase space factors k (s) on its diagonal. As an are connected smoothly along
real axis problem
above
threshold.
Panel(d)matrices
shows thek-plane,which is free of cuts. The upper(n) (s)theexample the thetwo-channel
withKK̄ thegiven aswhereare2N⇡⇡2Nand
1 diﬀerent
matricesinarechannelspace
containing all com(n) (s)
where
are
1
diﬀerent
matricesinchannelcom(lower) ✓half plane maps
onto the physicalAlsospacehere containing
the blue lineallcorresponds◆corresponding✓phase (unphysical)◆ factorssheet.
binations
of
space
(s)
on
its
diagonal.
As
an
k
(s)
0
0
0
binationsof ,corresponding
space k,factors
onthreshold,its diagonal.
⇡⇡(2)
3
k (s) atto(s)the=physicallyaccessible
valuesof thephasemomentumwhich
startswhereAsk =an
0. The(s) =problem(1.213)
examplethetwo-channel
with
⇡⇡ ⇡⇡
andandK K̄
thematricesare are
given
as as
0
0
0
(s)
examplethetwo-channel
problem
with
K
K̄
the
matrices
given
KK̄
thick black line shows the analytic continuation of the on-shell momentum in the below threshold
✓ ✓
◆ ◆
✓ ✓
◆ ◆
regime. The locations
are 3indicated0 by
the0 black crosses.
0 polesand⇡⇡ (s) (s)(2)✓ of the resonance
◆
0
0
0
⇡⇡
(2) =
(s)
, , (s)3 =(1.213)(s) =(1.213)(s)= 0 0 0⇡⇡(s)0(1.214)
4
(s) =
.0
0
0K K̄ (s)
(s)
K
K̄
0
K K̄ (s)andand✓ ✓◆ ◆The same procedure
can also be applied
a formfactor ⇡⇡F (s)the50.1.1 Propertiesof the for
S-matrix(s) with
0 discon4(s)0(s)4 (s)=asymptotic. states
tinuity= 0 ⇡⇡ inK K̄
. is called the(1.214)(1.214) The
The unitary operator that connects
andS-matrix.
(s)out
0
(s)
N
⇣
⌘
K
K̄
X
?the interacting part of the S-matrix. For a two-particles scattering
scattering amplitudeisdefinedasik
TheFsamealsokalso
be F
appliedfactorF (s)F withthe thediscondisc(s)same=procedure
2iproceduretik(s) can(s). for aforform(1.215)` canbe(s)applieda formfactor(s) with
disconprocess, itThereads:
tinuity
k=1
tinuityN ⇣⌘? ⌘ , p ; p 0 , p 0 )X
Its analytic continuation on diﬀerent Riemannsheetsdefinedby0N− ⇣0i(2π)4 δ 4i (pp1X
1 + p2is−
2 ba
ikp2 ) M(p?1 2 k1
discdisc
F (s)
t
(s)
(s)
(1.215)i = 2iikk.
k (s) F
`
F
(s)
=
2i
t
(s)
(s)
F
(s) .
(1.215)(50.1)
N hk in= outihp110 p20 , b| S −` 1 |p1 p2 , ai
⇣
⌘i X
k=1
(n)
(n)
kk=1
F (s) =
1 + 2it` (s)(s)
F (s) .
(1.216)ik
Its1 panalytic
continuation
diﬀerent
Riemanndefinedwhere |pand|p10continuation
p20 , bi are on
asymptotic
statesthat sheets
cansheets
be istreatedas by
non-interacting
particles
2 , aik=1
Itsanalytic
on diﬀerent
Riemannis definedby
0 , p20 , respectively.
in the spirit of the LSZ-reduction
[6].
They
carry
the
momenta
p
,
p
and
p
1
2
1
N
⇣
⌘i X hNi 1⇣⌘i X hi k1
F (n) (s)1 + 2it` (s) (n) (s)F (s)k.
(1.216)(n) =(n)F (s) = 31st May,
1 +2024
2it` (s)(s)
F (s) .
(1.216)
ik
k=1
39 ik
k=1

50. Resonances3

The labels a and b are used to specify the reaction channels. In general terms, a reaction channel
describes the possible outcome of a scattering event, characterized by the quantum numbers of theparticles involved. Note, that in general M operates as a matrix in channel space, connecting thedifferent asymptotic multi-particle states. For a single-particle state, we employ the relativistic
normalization,
p0 |p = (2π)3 2Ep δ 3 (~p 0 − p~ ),(50.2)
pwith Ep = p~ 2 + m2 .
Mandelstam variables are defined as s = (p1 + p2 )2 , t = (p1 − p10 )2 , and u = (p1 − p20 )2 . One
finds that the the variables s, t, and u are not independent, sinces + t + u = m21 + m22 + m210 + m220
holds, where mi with i ∈ 1, 10 , 2, 20 represents the masses of the involved particles. As a result, thereaction amplitude can be expressed as a function of two variables, M(s, t).
The Mandelstam variables play a crucial role in characterizing particle scattering. Specifically,
the process described in Eq. (50.1) is known as s-channel, referring to a reaction, 1, 2 → 10 , 20 .
√Here, s represents the total energy of the interacting system in the center-of-momentum frame.
The variable t corresponds to the momentum transfer, which is related to the scattering angle, theangle between the momenta of particles 1 and 10 in the center-of-momentum frame (see also thereview on “Kinematics” in this Review of Particle Physics). The scattering process with particles
1 and the antiparticle of 10 , denoted by 10 , incoming to the antiparticle of 2, denoted by 2, and 20
outgoing represent a different reaction, it is referred to as the t-channel reaction. The scattering
amplitudes of s- and t-channel are related via the so-called crossing symmetry. The u-channel isintroduced analogously as a reaction with the initial state of particles 1 and 20 and the final state ofparticles 10 and 2. The t-channel and the u-channel are referred to as the crossed channels relative
to the reaction in Eq. (50.1). This relationship is illustrated in Fig. 50.2.
1′1s-channel2Mt-channel

2′

Figure 50.2: Illustration of the relation between s- and t-channel.

The analysis of resonance phenomena requires delving deeper into the complex analysis of theamplitude: M(s, t) is a multivalued function due to the complex branch points associated with the
Mandelstam variables. These branch points emerge whenever a new channel becomes accessible,that is, whenever s exceeds sthr,a = (m1,a +m2,a )2 , where mi,a denotes the masses of the two particles
in channel a. Every two-particle threshold introduces a square-root singularity. The square-root√
function is inherently double-valued; for instance, the equation x = y 2 has two solutions: x and√− x. In the complex plane, these two solutions are represented as separate layers or surfaces,
known as Riemann sheets or Riemann surfaces. Consequently, each square-root singularity causes
the number of Riemann sheets of the scattering amplitude to double. If a channel opens that has
an odd number of particles, the branch point at the threshold exhibits a logarithmic singularity [7].
Such a branch point gives rise to an infinite number of sheets. The branch points come with theirassociated branch cuts — by convention those are taken from the threshold to infinity along the
31st May, 2024

4

50. Resonances

real axis and accordingly called right-hand cuts. To explore a reaction amplitude in the complex
energy plane, one can introduce a complex component to the Mandelstam variable s. The sectionof the complex plane that relates to a positive imaginary component of the relative momentum canbe directly accessed from the physical region and is called the first Riemann sheet or the physicalsheet. The other sheets are called unphysical sheets. The physical axis of an s-channel scattering
amplitude represents a line for real values of the variable s larger than the lowest threshold to beevaluated on the physical sheet.
While the physical Riemann sheet is free of singularities off the real axis, the unphysical sheets
may contain resonance poles and branch points. Branch points appear in the complex plane of anunphysical sheet when there is a resonance in a subsystem of involved particles [7, 8]. The branch
points related to thresholds in the crossed channels are located in the portion of the complex planewhere the real part is negative and, therefore, are referred to as the left-hand cuts. An illustrative
example for a left-hand cut is the one-pion exchange in nucleon-nucleon scattering that is locatedin the unphysical domain at s = 4m2N − m2π . Triangle topologies, which are Feynman diagrams
characterized by a triangular loop of three propagators, can lead to logarithmic singularities in the
scattering amplitude. These singularities, appearing on the unphysical sheets, are often termed
triangle singularities (TS) [7, 9, 10].
The reaction amplitude has poles that can be categorized as bound states, virtual states, orresonances. Poles corresponding to bound and virtual states manifest at real values of s. Specifically, bound state poles are found on the physical sheet, while virtual state poles are situated onan unphysical sheet, both being below the threshold. Resonance poles, on the other hand, emerge
inside the complex plane of the unphysical sheets. Notably, those resonance poles that are on theunphysical sheet nearest to the physical region exert the most significant influence on experimental
observables. Analyticity dictates that for every pole at a specific complex value of sp , there must bea corresponding pole at its complex conjugate value s∗p . This relationship is a direct consequence of
the Schwarz reflection principle, a mathematical technique utilized for analytic continuation within
scattering theory. For a single-channel case, the complex structure of an amplitude with a single
resonance is illustrated in Fig. 50.1(a)-(c). Among the two poles, symmetrically positioned relative to the real axis on the second sheet, the one exhibiting a negative imaginary part seamlessly
connects to the physical axis, which becomes especially clear in panel (c) of the figure where thephysical axis is shown as the blue line. It therefore has a more pronounced effect on observables inthe vicinity of the resonance region compared to its conjugate counterpart. The shortest continuous
path from the physical region to the second sheet pole with the positive imaginary part is via a linethat goes around the threshold. However, as the extra path around the threshold diminishes in thenear-threshold region, the effective distance to both poles becomes comparable for near thresholdkinematics, thereby rendering the influence of both poles on observables similarly significant.
An alternative way to depict the complex structure is through the k-plane, where k denotes
the relative momentum of the two scattering particles in their center-of-momentum frame. For
particles of equal mass, denoted as m (expression for unequal masses is given in Eq. (50.7)), one
finds
1√
k=
s − 4m .
(50.3)
2
Unlike the complex s-plane, the k-plane does not have a two-body threshold cut. Consequently,
both the physical and unphysical Riemann sheets, which are linked to the branch point in s are
mapped onto the upper and lower half of the complex k-plane, respectively. The k-plane is sketchedin Fig. 50.1(d), including possible locations of resonance poles. In this representation it becomes especially clear that only one resonance pole drives the dynamics on the physical axis in the resonanceregion, while at the threshold, where k = 0, both poles are of equal significance.
31st May, 2024

50. Resonances

5

In situations involving two relevant channels, we encounter four Riemann sheets. These areillustrated in the left panel of Fig. 50.3. For a non-relativistic system with two channels, this foursheeted Riemann surface can be transformed into a plane represented by a new variable, ω. Thisvariable is defined in relation to the channel-momenta, as detailed in references [11,12] (for a recent
application see Ref. [13])
s

s

µ1 ∆
1
µ2 ∆
1
k1 =
ω+
, k2 =
ω−,(50.4)2ω
2
ω
where ∆ denotes the energy difference between the two thresholds and the µa the reduced mass ofthe particles in channel a, µa = m1,a m2,a /(m1,a + m2,a ). The lower threshold is located at ω = i,the higher at ω = 1. The mapping of the Riemann sheets and the different areas in the ω-plane isshown in the right panel of Fig. 50.3. The ω-plane nicely shows how the different sheets connect toeach other.1 The solid green line shows the physical axis in the physical regime. The thick blackline its analytic continuation below the lowest threshold. The pronounced kinks in this line for theω plane show the thresholds. Please observe that for ω ≈ i the sheet structure agrees to panel (d)of Fig. 50.1, since in this kinematic regime the second channel does not matter.
In the context of a two-channel scenario, the proximity of a sheet to the physical axis varies
with increasing energy. Specifically, for energies that exceed the first threshold, but remain below
the second, sheet (21) is the one that smoothly connects to the upper half-plane of the physicalsheet (11). Once the energy surpasses the second threshold, sheet (22) assumes this role. As aresult, any pole on sheet (21) that lies above the second threshold will manifest in the data solely
as a cusp right at the second threshold. Sheet (12), on the other hand, is remote for almost allenergies.
Singularities, poles and branch points determine the visible structures in observables. However,
it is crucial to note that not every observable bump is indicative of a resonance as discussed in [14].
Under certain kinematic conditions, Triangle Singularities, in particular, can either mimic resonancesignals, as suggested in Refs. [15–20], or significantly alter resonance signals [21]. Conversely, not
all resonances produce a noticeable bump across all observables. For example, in the baryon sector,
there is no clear trace of the N (1440)1/2+ , the so called Roper resonance, in the πN observablesor phase shifts, although careful analyses reveal a pole [22]. In the meson sector, the f0 (500),
also known as σ meson, was firmly established only after the application of very sophisticatedtheoretical analysis tools (see, e.g., Ref. [23] for a review). This complexity arises because thescalar-isoscalar ππ phase shifts reach 90 degrees near 800 MeV, which is approximately 400 MeV
above the resonance mass. At this energy, the onset of the next resonance, the f0 (980), is alreadyobservable. The analyticity principle of the S-matrix germane to quantum scattering theories,
dictates that only poles and branch points can exist on the real axis of the first Riemann sheet,excluding any singularities in the complex plane. This principle is closely linked to causality,
ensuring that effects follow their causes in a chronological order. In non-relativistic scattering,
the analyticity finds a solid mathematical base [24]. Similarly, perturbative relativistic theory
maintains this analyticity through a series expansion of the S-matrix, each term depicted by a
Feynman diagram, representing processes with distinct analytical expressions. A deeper level ofanalyticity is proposed by the Mandelstam hypothesis, suggesting not only the analytic propertiesof the scattering amplitude within the complex plane of the first Riemann sheet but also a nuanced
interconnection between crossed scattering processes through analytic continuation [25].
Unitarity further constrains the imaginary part of the amplitude on this real axis, a topic wewill explore in the subsequent section. Additional constraints are introduced by principles such as








1
An alternative illustration for this two-channel case as well as an extension to three channels can be found in
Ref. [2].

31st May, 2024

50. Resonances

6

11

1.0

12

22

0.5

21
Im(𝜔)

s
arb.unit

1.5

11

21

0.0

− 0.5

12

Im
(s)

− 1.0

s)
Re(

− 1.5
− 1.5 − 1.0 − 0.5

22
0.0
0.5
Re(𝜔)

1.0

1.5

Figure 50.3: Cut structure of the S-matrix with two channels present. The diagrams feature four
sheets, labeled (ij), where both i and j can be 1 or 2. These labels indicate the doubling of the
sheets at the first and second thresholds. The left panel displays the complex s plane, while theright panel represents the ω plane. The physical axis, along with its analytic continuation below
the threshold, is highlighted with a thick solid line, green in the physical regime, black for the
analytic continuation below the lowest threshold.

crossing symmetry, duality [26] and positivity [27]. Approaches based on analyticity and crossing
symmetry have been implemented through dispersion theory. Among the most notable are theRoy equations and their variants [28]. These have been applied to a range of processes, including
ππ → ππ [29–31], πK scattering [32], γγ → ππ interactions [33], and pion-nucleon scattering [34,35].
50.1.2 Consequences from unitarity
Scattering amplitudes, denoted as M, and production amplitudes, represented by A, havedistinct characteristics due to the different constraints imposed by unitarity. When considering
the scattering amplitude, it is assumed that all channels hold similar significance. In contrast,
for production amplitudes, it is assumed that for the dynamics of the particles in the final state
the initial state does not play a role. Accordingly, the interactions in the final state among the
produced particles are described by relevant scattering amplitudes of those particles only. Scattering
processes are for example π + π − → K K̄ and D0 D̄0 → D0 D̄0 . Examples for production processes
are e+ e− → γ ∗ → π + π − , which provides access to the pion vector form factor, τ − → K − π 0 ντ , andB 0 → J/ψπ + π − .
The unitarity of the S-matrix, represented by the equation S † S = 1, ensures the conservationof probability. This principle imposes a specific constraint on the imaginary part of the reaction
amplitude. Below the lowest threshold, the amplitude remains real. However, once the energy ishigher than the threshold, there is a discontinuity associated with the threshold branch point. The
S-matrix unitarity relates the value of the discontinuity, to the amplitude itself [6]:Disc Mba = Mba − M∗ab = i (2π)4XZ
c

31st May, 2024dΦc M∗cb Mca ,(50.5)

50. Resonances

7

Disc

Mab

=

P

c

Mac

M†cb

Aa

Disc

=

P

c

Ac

M†ca

Figure 50.4: Graphical illustration of the discontinuity equations for the scattering and the
production amplitude, respectively. The dashed line indicates that the intermediate state is to be
put on-shell to find the discontinuity.

where dΦc denotes the invariant phase space for a given channel, labeled as c. The factor (2π)4 aligns
with the definition of the phase space expression (see Eq. (12) in the review on “Kinematics”). It isessential to note that the summation only considers open channels, meaning those with a production
threshold below the scattered system’s energy. In the evaluation of an actual Feynman diagram
the discontinuity can be extracted by employing the Cutkosky rule, which comprises replacing the
propagators in the pertinent intermediate state by delta-distributions. Eq. (50.5) is illustrated
graphically in Fig. 50.4. The left part of the expression yields 2i ImMba in accordance to analytic
properties of reaction amplitude [36]. For the forward scattering, t = 0, the right part of Eq. (50.5)
resembles the total cross section up to a kinematic factor, a relationship know an the opticaltheorem:√
Im Maa (s, 0) = 2qa s σtot (a → anything) .
(50.6)In this equation, qa represents the break-up momentum of the particles in the center-of-momentum
frame,
λ1/2 (s, m21,a , m22,a )√qa =,(50.7)
2 s
where λ(x, y, z) = x2 + y 2 + z 2 − 2xy − 2yz − 2xz is the Källén function, and the masses of the twoparticles in the channel a, m1,a and m2,b , cf. Eq. (17) of the review on “Kinematics”.
The unitarity relation for a production amplitude in channel a is represented by:Disc Aa = Aa − A∗a = i (2π)4XZ

dΦc M∗ca Ac .

(50.8)cEquation (50.8) is illustrated graphically in Fig. 50.4. A direct consequence of this equation is thatthe production amplitude shares its poles with the scattering amplitude. A common method tomodel the production amplitude that adheres to this unitary relation is to express it as a linear
combination of the scattering amplitudes. This approach, known as the Q-vector method (see Sec
50.3.4), has its limitations. Specifically, production amplitude inherits the left-hand singularitiesof the scattering amplitude, while, in general, it has a different cut structure. To address thisdrawback, a more sophisticated method, known as unitarization is employed. This method, rooted
in dispersion theory, offers a more refined approach to the problem and is detailed in [37]. A
notable application of this method is the Khuri-Treiman framework [38, 39], which is frequently
used to study three-body decays. This framework has been successfully applied to a range of
decays, from light mesons [40–52] to heavy-flavour decays [53–55].
50.1.3 Partial-wave decomposition
It is often convenient to expand a two-body scattering amplitude of a two-body subsystem ofa production amplitude in partial waves. Since resonances have a well-defined spin, they appear
31st May, 2024

50. Resonances8!"#$$$$$%
f ))

!"
,-./

f ))

"! )

&'(

&'(

*+#$$$$$%f ))Figure 50.5: Argand plot showing a trajectory of the diagonal element of a partial-wave amplitude,
fbb , as a function of energy in the complex plane. As the energy increases the amplitude follows theline counter clockwise. The amplitude leaves the unitary circle (solid line) as soon as inelasticity
sets in, η < 1 (dashed line).

only in a specific partial wave of the reaction amplitude. For scalar particles, the expansion reads:
Mba (s, t) =∞X

(2j + 1)Mjba (s)Pj (cos(θ)) ,(50.9)

j=0

where j denotes the total angular momentum and the Pj (cos(θ)) denotes the Legendre polynomials.
In the presence of spins an expansion more complicated than Eq. (50.9) is necessary — for a general
discussion on how to handle spins see e.g. Ref. [56]. In the absence of spins the parameter j coincides
with the orbital angular momentum of the particle pairs in the initial and the final state. To simplify
notation and since all amplitudes from here on are understood to be partial wave projected, we
drop the label j for the single-argument function Mba (s). Plugging Eq. (50.9) into Eq. (50.5) one
finds the unitarity relation for the partial-wave amplitude Mba (s), namely
Im Mba (s) =X

Mcb (s)∗ ρc (s) Mca (s)(50.10)cwith ρc (s) being a factor that is related to the two-body phase space in Eq. (12) of the review on“Kinematics”,
Z(2π)4
1 2|~qc |√ ,ρc (s) =(50.11)
dΦ2 =
2
16π s
with the momentum qc being defined in Eq. (50.7). Note that in case of the two particles being
identical the inclusion of symmetry factors becomes necessary. The partial-wave amplitude fba (s)is introduced by normalizing scattering amplitude with the phase spaces factors,√√
fba (s) = ρb Mba (s) ρa .
(50.12)The unitarity condition for fba follows from Eq. (50.10):
Im fba (s) =X∗(s)fca (s) .
fcb(50.13)c

It leads us to deduce that the inverse of the imaginary part of fba is equal to −δba . Moreover,S = I + 2if is a unitary matrix. Hence, the diagonal elements of f can be parameterized as
fbb = (ηb exp(2iδb ) − 1)/2i ,31st May, 2024

(50.14)

50. Resonances9

where δb denotes the phase shift for the scattering from channel b to channel b and ηb is the elasticity
parameter, also known as inelasticity. Building upon Eq. (50.13), we can further deduce that,
Im fbb (s) = (1 − ηb cos(2δb ))/2 =X|fcb (s)|2 .

(50.15)cUsing Eq. (50.14) for the last term in the sum, we obtain a relation highlighting the meaning of
the inelasticity,X
1(1 − ηb2 ) =|fcb (s)|2 .
(50.16)4
c6=b
It is important to note that the parameter ηb is confined within the range [0, 1], where the case,ηb = 1 is referred to as a purely elastic scattering. Thus, the function ηb (s) is a direct measure of
the contribution of the inelastic channels on the scattering amplitude in a given channel.
The evolution of the partial-wave amplitude fbb with energy can be displayed as a trajectoryin the Argand plot, as shown p
in Fig. 50.5. In case of a two-channel problem, η1 = η2 = η, and theoff-diagonal element is f12 = 1 − η 2 /2 exp(i(δ1 + δ2 )). The unitarity condition Eq. (50.14) sets
the limit to the squared amplitude fbb :1
1|fbb |2 = (ηb2 − 2ηb cos(2δb ) + 1) ≤ (ηb + 1)2 ,
4
4(50.17)

where the maximum value is reached for δb = π/2. For the absolute square of the partial-waveprojected scattering amplitude the unitarity bound thus reads:|Mbb | ≤1
8π √(ηb + 1) ≤
s,
2ρb
qb

(50.18)where the second inequality comes from ηb ≤ 1. For energies much larger than the masses of thescattering particles the upper bound for |Mbb | tends to 16π for large s.
The partial-wave projected production amplitude A(s) (note that the label j has been omitted
for consistency) is also constrained by unitarity. As derived from Eq. (50.8):
Im Aa =

X

M∗ba ρb Ab ,(50.19)bwhere the summation encompasses all open channels. In the realm of elastic scattering, solely one
channel, denoted by a, contributes to the sum. Consequently, the phase of Aa must align with thephase of Maa , given that the left-hand side of Eq. (50.19) represents a real value. This principle
is recognized as the Watson theorem [57]. To illustrate, consider the phase of the pion vector form
factor: it agrees to that of ππ scattering in the vector isovector channel (aside from effects of
the isospin-violating ρ − ω mixing) up to about 1 GeV, where inelastic contributions start gaining
significance.

50.2 Properties of resonances
A resonance is defined by its pole position in the complex s-plane, denoted as sR , and by thestrength parameters of its couplings to various decay channels evaluated at this pole, known as thepole residues. The pole mass MR and pole width ΓR are defined via the pole parameters√sR = MR − iΓR /2 .

(50.20)For states where the relevant thresholds are situated significantly below the resonance location,
the lifetime τR of the resonance is given by τR = 1/ΓR (refer to the review on “Kinematics” and31st May, 2024

50. Resonances

10

for a recent discussion, see Ref. [58]). It is important to note that the conventional Breit–Wigner
parameters MBW and ΓBW , introduced in Eq. (50.30), differ from the pole parameters due to finite
width effects and the influence of thresholds and background terms. It should be stressed that polelocation sR and pole residues are the only resonance properties that are model and parameterization
independent.
When a resonance interacts with multiple channels, each channel contributes to the imaginary
part of the pole position. However, these individual imaginary contributions do not necessarily
simply sum up, leading to the need for redefining the partial widths. This behavior is discussed on
the example of the f0 (980) in Refs. [59, 60].
In the close vicinity of the resonance pole one defines the residues via
lim (s − sR )Mba = −Rba .

s→sR(50.21)Those can be conveniently calculated via an integration along a closed contour around the pole
using
I
1
Rba = −
dsMba .
(50.22)2πi
The residue adheres to the factorization relation, (Rba )2 = Raa × Rbb . This factorization emerges
as a universal property stemming from the unitarity of scattering processes [7]. Building on thisrelation, one can define pole couplings as follows:pg̃a = Rba / Rbb .

(50.23)The pole couplings characterize the transition strength of a given resonance to some channel a
independently of how the particular resonance was produced. They are in general complex valued.
In the baryon sector, it is customary to define the residue of the pole for the fba amplitude (as√described in Eq. (50.13)) in terms of the variable s rather than s. This residue can be related to
the residues mentioned earlier by:
1
rba =
2

s

ρa (sR )ρb (sR )
Rba ,
sR

(50.24)

where the phase-space factors are to be continued analytically to the pole location sR . The residues
quoted in the baryon listings are those for πN scattering (a = b = πN ).
The branching ratio of a resonance decay to a specific channel represents the fraction of the
decay probability directed to that channel.
Bra = Na /Ntot .

(50.25)Here, Na is the experimental count of events for the decay channel R → a, while Ntot represents
the total number of events produced in the decay of the resonance. Since the amplitudes of decaysP
to different final states add incoherently, we have Ntot = b Nb . We note, however, that both Na
and Ntot may exhibit a dependence on the reaction through which the resonance is produced, due
to the resonance’s finite width. For a narrow resonance, the experimental count is determined byintegrating the squared production amplitude over the decay channel’s phase space:Z

Na = N0|Aa |2 dΦa ,31st May, 2024

(50.26)

50. Resonances11

where N0 is a normalization constant associated with the integrated luminosity, and Aa represents
the amplitude of the resonance decay to channel a. When the decay rate’s variation across theresonance width becomes significant, the resonance’s lineshape must be considered:Z ∞Z

Na = N0dsσR (s)|Aa (s)|2 dΦa (s) .

(50.27)

sthr,aHere, σR (s) denotes a proper weight function of the resonance and sthr,a is the threshold value
for the channel a. Additionally, the phase space integral, Φa (s), puts the integrand to zero below
the energy threshold of the decay channel a. This aspect is particularly crucial for the decays ofbroad resonances into channels with energy thresholds exceeding the resonance’s nominal mass.
This methodology is frequently employed in light-meson studies, as demonstrated in Ref. [61], andis also prevalent in light baryon research, as referenced in Ref. [60].
In case of the decay of some heavy state into multi-body final states, the transition amplitude
can include resonances within subsystems of particles. When the total amplitude is decomposedP
as Aa = R Aa(R) , the relative branching fraction for the decay of the given heavy state into final
state a via resonance R in some subsystem, denoted as Bra(R) / Bra , is given by:
Z

Bra(R) / Bra =2

Aa(R) dΦa

Z

|Aa |2 dΦa(50.28)It is crucial to recognize that the decomposition of the total amplitude, Aa , into resonance amplitudes, Aa(R) , is not a straightforward process. Beyond the differentiation of components basedon distinct quantum numbers, this separation is inherently model-dependent. Furthermore, it isessential to be aware that the amplitudes Aa(R) can interfere with one another. As a result, theirindividual probabilities might not sum up to the total branching fraction of channel a. Nonetheless, in many scenarios, the interference contributions are small, making the fractions Bra(R) / Bra
indicative. For recent applications of these formulas in the context of B-decays see Refs. [62–65].
Lastly, an expression analogous to the branching fraction can be formulated using the poleparameters. For two-particle decays in the S-wave, one writes:Bra =|g̃a |2
ρa (MR2 ) ,MR ΓR(50.29)where MR and ΓR have been previously defined in Eq. (50.20). This approach was utilized to define
a two-photon width for the broad f0 (500) resonance [66, 67]. Similarly, one should use residues toquantify the coupling of resonances to certain production channels [68]. For an application of thisapproach to the coupling of the K0∗ (1430) resonance to a leptonic current see Ref. [69]. Equation (50.29) provides a definition of branching fraction that remains independent of the reactionused to derive the quantity. For narrow resonances, this definition aligns well with Eq. (50.25) andEq. (50.26). However, for broad, overlapping resonances, it is essential to recognize that Eq. (50.29)
is primarily used to convert residues into metrics that facilitate a more straightforward comparisonof resonance transitions across different channels. For resonances with a coupling to a channel thatremains closed at the resonance mass, Eq. (50.29) is not applicable due to the phase-space factor.
In such scenarios, modification of the expression is required as elaborated upon in Ref. [60], and at
the conclusion of Sec. 50.3.5.

50.3 Common parameterizations
In general, there is no universal, model-independent recipe to build scattering amplitudes.
However, a few approaches presented in this section are practical to extract resonance properties
31st May, 2024

50. Resonances

12

in experimental analyses. The systematic theory uncertainties need to be assessed from a rangeof model variations that provide a sufficient quality of description of the available data and are
permitted by general S-matrix principles and the symmetries controlling the system at hand.
50.3.1 The Breit–Wigner parameterization
The relativistic Breit–Wigner parameterization represents a dressed propagator for an isolated
resonance. The production amplitude for a resonance observed in a channel a, is given by
Aa (s) =Na (s)
2MBW− s − iMBW Γ (s)(50.30)2 ) denotes the Breit–Wigner
where MBW represents the Breit–Wigner mass, and ΓBW = Γ (MBW
width. The function Γ (s) is defined by the channels to which the resonance can decay. The
numerator function Na (s) is tailored to the production process, encompassing kinematic factors
and couplings pertinent to both the production and decay processes.

Na (s) = α ga na (s)1 X 2
Γ (s) =g ρb (s)n2b (s) ,MBW b b(50.31)(50.32)

Here the index b = 1, 2, . . . runs over all decay channels of the resonance. The coupling constants
are represented by gb , and ρb is the phase-space factor as defined in Eq. (50.11). The expression
for na (s) is:(50.33)na = (qa /q0 )la Fla (qa /q0 ) ,where la indicates the orbital angular momentum in channel a, qa (s) is the break-up momentum
as defined in Eq. (50.7), and q0 is a suitably selected momentum scale. The term (qa )la ensures theamplitude’s appropriate threshold behavior. The rapid growth of this factor for angular momentala > 0 is offset at specific s values by a phenomenological form factor, represented here by Fla (qa , q0 )— the presence of these suppression factors is also a requirement from positivity which demands
that the dressed propagator, the denominator of Eq. (50.30) and similar equations below, is notallowed to drop faster than 1/s [27]. The Blatt-Weisskopf form factors are frequently employed in
the literature [70–72] to model Fj :F02 (z) = 1 ,(50.34)F12 (z) = 1/(1 + z 2 ) ,F22 (z) = 1/(9 + 3z 2 + z 4 ) ,where z = q/q0 , the scale parameter 1/q0 typically falls within the range of 1 GeV−1 to 5 GeV−1 .
Rather than employing coupling constants as in Eq. (50.32), channel partial widths can be used,Γ (s) =

X
b

ΓBW,b

ρb (s)2 )ρb (MBW



qb
qb R

2lb

Fl2b (qb , q0 ).
Fl2b (qb R , q0 )(50.35)2 . It is essential to note that
Here qb R is the values of the break-up momenta evaluated at s = MBW
this substitution is valid only for channels where the decay channel’s threshold is positioned below
the nominal resonance mass. In other scenarios, Eq. (50.32) should be applied.
Equation (50.32) incorporates a threshold for each of the coupled channels. The expression
is straightforward to use in the physical region above all the thresholds. Its evaluation elsewhererequires a careful analytic continuation. As outlined in Refs. [73, 74], the choiceqqc = i −qc2

for

31st May, 2024

qc2 < 0 ,(50.36)

13

50. Resonances

leads to an evaluation of the amplitude on the physical sheet below threshold. The Flatté parameterization [73] refers to the two S-wave channels’ amplitude near a threshold of a heavier channel inthe physical region of the lighter channel computed with Eq. (50.36). When a resonance’s coupling
to the channel with a higher threshold is notably strong, the parameterization displays scaling
invariance. This implies that it is not possible to extract individual partial decay widths; only theirratios can be determined [75].
The Breit–Wigner parameterization is an accurate representation of resonance phenomena
strictly in the Γ/∆ → 0 limit, where Γ is the resonance width and ∆ is the distance to theclosest unaccounted singularity, be it a pole of a higher resonance or a kinematic threshold related
to a coupled channel. However, the situation is often more complex due to multiple singularities
in the complex plane around the resonance with different importance. For instance, in P-wave
ππ scattering, the Breit–Wigner parameterization aptly describes the ρ-meson resonance over an
extensive range. Although the closest singularity to the ρ-meson pole is the ω-pole (Γρ /∆ω  1),this isospin breaking effect is typically insignificant except few special cases [76, 77]. Subsequent
singularities, namely the 3π, 4π, and 6π thresholds, are also generally disregarded. The two-pion
threshold is incorporated in the Breit-Wigner parameterization through an energy-dependent width.
Finally, the parameterization’s efficacy diminishes around 1.2 GeV due to the ρ0 resonance, situatedapproximately at 1.45 GeV.
Once the applicability of the Breit-Wigner parameterization is established, it is crucial to recognize that its parameters will only align with the pole parameters if the width is small. Yet,
extracting the pole position from the Breit-Wigner amplitude is a straightforward technical task,
achieved through analytic continuation. However, neither the Breit–Wigner parameters nor thecorresponding pole parameters should be deemed reliable without justifying the parameterization’s
applicability. If there is more than one resonance in one partial wave that significantly couples tothe same channel, it is generally inappropriate to employ a sum of Breit–Wigner functions. Such an
approach often results in a breach of unitarity constraints, potentially introducing an indeterminate
bias to the inferred resonance properties from the reaction amplitude. For overlapping resonancesin the same partial wave, more sophisticated methods, such as the K-matrix approach detailed in
the subsequent section, are recommended.
50.3.2 K-matrix approach
The K-matrix method offers a comprehensive framework for modelling coupled-channel amplitudes [78]. This method ensures two-particle unitarity. However, it traditionally omits the left-handcuts. The scattering amplitude Mba (s) can be derived from the equation:−1
2
nb M−1ba na = Kba − iδba ρa na .

(50.37)Here, Kba represents a real function and is subject to modeling. The factor na is elaborated
upon in Eq. (50.33). Since there is no unique recipe to build K, it is essential to explore various
parameterizations to gauge the theoretical systematic uncertainty. A commonly adopted choice forthe K-matrix is given by:
X gRgRb a
Kba (s) =(50.38)2 − s + bba ,mRR
where mR is referred to as the bare mass of the resonance R (not to be confused with the physicalmass), and the gaR represents the bare couplings of the resonance R to the channel a (not to be
confused with the residues). The bba is a matrix that parameterizes the non-pole components of theK-matrix. Provided all parameters in Eq. (50.38) are real, the amplitude Mba (s) remains unitary.
From Eq. (50.37), the scattering amplitude M can be directly computed using its matrix form:
M = n[1 − K iρ n2 ]−1 K n ,31st May, 2024

(50.39)

14

50. Resonanceswhere n and ρ are diagonal matrices, n = diag(na , nb , . . . ), and ρ = diag(ρa , ρb , . . . ). As an alternative to Eq. (50.38), the same functional structure on the right side of Eq. (50.38) can be employed
to parameterize the inverse K-matrix, termed the M -matrix, by the authors of Ref. [79]. Numerous
alternative formulations within the K-matrix framework are utilized for amplitude studies related
to lattice-QCD calculations [80–82].
A prevalent method to construct the production amplitude within the K-matrix frameworkis the P-vector parameterization [72, 78, 83]. The method utilizes the K-matrix poles and barecouplings from Eq. (50.38):
Aa (s) = na

Xh

1 − K iρn2

c

Pc =

X αR g R
cR

m2R − s+ Bc .

i−1
ac

Pc ,

(50.40)(50.41)The production vector, denoted as Pc , comprises two main components. The first component
represents a transition driven by the coupling of the bare resonance, R, to the source. This couplingis characterized by a strength parameter, αR . The second component, Bc , signifies the directtransition from the source to the channel c. The formalism ensures that the complete production
vector gets dressed via the final state interaction.
The Q-vector, as discussed in Ref. [72,79,84], offers an alternative methodology for constructing
a production amplitude:X
Aa (s) =Mac (s)Qc (s)/nc .
(50.42)cHere, Qc (s) represents a smooth function of s and can be parameterized using a polynomial series.
The unitarity condition of Eq. (50.19) is satisfied when Qc (s) is a real function and in particulardoes not have singularities above the lowest threshold for all channels c. Besides these conditions
Qc (s) is arbitrary. In a study of γγ → ππ, cf. Ref. [66, 67] a low-order polynomial is claimed to be
sufficient to parameterize the energy dependence of the function Qc (s). The Q-vector method isconvenient, if the full matrix M is known, cf. Ref. [79]. An important distinction between the Pvector and the Q-vector methods is highlighted in [83]. When the two-particle scattering amplitude
approaches zero, the production amplitude in the Q-vector construction unavoidably vanishes forfinite values of Qc , whereas it remains non-zero in the P-vector approach.
Traditionally, amplitudes constructed using the K-matrix technique exclude the left-hand cuts.
Nevertheless, these can be customarily incorporated into the function bba from Eq. (50.38) for the
scattering amplitude [85,86]. Similarly, for the production amplitude, the functions Bc and Qc from
Eq. (50.40) and Eq. (50.42), respectively, might also encompass the left-hand cuts. Those can often
be parameterized by low order polynomials [66, 67, 87].
The position of the resonance poles can be determined by examining the zeros of the analytic
function det[1 − K iρ n2 ]. Owing to the ρ factor, this determinant exhibits a complex multisheet
structure. Nonetheless, the nearest unphysical sheet usually has the highest influence to the physicalregion. It is always the one which is determined by the heaviest threshold below the studied pointin s (cf. Fig. 50.3). If for a given resonance the pole closest to the physical axis is not located on asheet that connects directly to the physical axis, it is possible that K-matrix fits do not allow one
to fix the pole parameters, since a larger distance of the pole to the physical axis can be balanced
by increased residues — for a detailed discussion see Ref. [88].
50.3.3 Further improvements: Chew-Mandelstam function
The K-matrix framework often enables an accurate fit of physical amplitudes and is straightforward to handle. However, it does present a significant drawback: it breaches constraints imposed
31st May, 2024

50. Resonances

15

0.5
16π Σ(s)16π iρ(s)

0.5
0.0
-0.5
-1.0

Real part
Imag part
0.0

0.3

sthr

1.0
sthr

1.0

0.6
0.9
s (GeV2 )

1.2

0.0
-0.5
-1.0

Real part
Imag part
0.0

0.3

0.6
0.9
s (GeV2 )1.2

Figure 50.6: Comparison of the iρ function (left plot) to the Chew-Mandelstam function from
Eq. (50.44) (right plot), evaluated for the case of S-wave ηπ scattering. The values of s are taken
slightly above the real axis, s + i0. The solid red line shows the imaginary part that is the samefor both functions above threshold. The dashed black line presents the real part. One finds
indications of the unphysical left-hand singularities of the function iρ on the left plot, while the
Chew-Mandelstam function in analytic below the two-particle threshold.

by analyticity. For instance, ρa , as defined in Eq. (50.11), is not well-defined at s = 0. Moreover,in cases of unequal masses, it manifests an unphysical cut, as depicted in the left panel of Fig. 50.6.
A method to improve the analytic properties has been suggested in Refs. [89–93]. This approach
replaces the term iρa (s)n2a from Eq. (50.37) with the analytic function Σa (s), known as the ChewMandelstam function. This function produces the imaginary part iρa (s)n2a on the right-hand cut,
while maintaining analyticity on the left-hand side, as represented by the once subtracted dispersion
integral:Zs − sthra ∞ρa (s0 )n2a (s0 )Σa (s + i0) =ds0 .
(50.43)0−s
0 − s − i0)
π
(s
)(s
sthra
thra
Here, we chose the channel threshold, sthra , as the subtraction point, and assumed that the subtraction constant is absorbed into the other parameters used for the amplitude. For an S-wave
where na = 1, the integral has a closed form [37, 91]:
√




1 2qa
m21 + m22 − s + 2 sqa
1
1
m1
2
2
√ log
Σa (s) =− (m1 − m2 )−log
, (50.44)
16π 2
s
2m1 m2s (m1 + m2 )2
m2
where m1 and m2 are masses of the final-state particles in channel a, with sthra = (m1 + m2 )2 .
The function’s behavior along the real axis is illustrated in the right panel of Fig. 50.6. A further
discussion of the calculation of the Chew-Mandelstam function can be found in Refs. [94, 95].
If there is only a single resonance in a given channel, it is possible to feed the imaginary part ofthe Breit–Wigner function, Eq. (50.30) with an energy-dependent width, directly into a dispersion
integral to get a resonance propagator with the correct analytic structure [96, 97].
50.3.4 Effective-range expansion and scattering-length approximation
For elastic scattering, ηb = 1, Eq. (50.14) simplifies toq,
fbb = eiδ sin(δ) =q cot(δ) − iq
31st May, 2024

(50.45)

16

50. Resonanceswhere q is the relative momentum of the scattering particles. For S-waves one may then employ
the effective-range expansion [98, 99] (ERE)q cot δ =1 1 2
+ rq + O(r3 q 4 ) ,a 2(50.46)

whose radius of convergence is set by the closest non-analyticity, which may be the next threshold,a left-hand cut or a zero in the amplitude which is equivalent to a pole in q cot δ. The ERE isunderstood to be an expansion in Rf q, where Rf denotes the range of forces provided by the
inverse of the lightest exchange particle allowed. The scattering length, a, is then defined as thefirst term in an expansion of the real part of the inverse scattering amplitude. The sign convention
used in Eq. (50.46) is the one commonly employed in particle physics. In this convention a positive
scattering length indicates attraction; if, however, the attraction is strong enough to generate abound state, the scattering length changes sign and turns negative. A negative scattering length
also occurs for repulsive interactions. Note that in nuclear physics the leading term in the expansionof Eq. (50.46) is usually defined as −1/a such that e.g. a bound state would be related to a positive
scattering length. The parameter r is called the effective range. Especially in cases where thescattering length is large the ERE not only describes the low energy scattering well but also allows
for an analytic continuation to find bound states below threshold by proper analytic continuation.
This method was, e.g., employed recently to analyze near threshold scattering phase shifts forD∗+ D0 scattering found in lattice analyses to extract properties of the Tcc (3875)+ [100–102] at
unphysical pion masses. It should be stressed that the analyses of the lattice data might call fora modification to account for the nearby left-hand cut which sets the radius of convergence of the
ERE [103].
When considering only the scattering length within the ERE, the scattering amplitude is represented as:√
8π s
M(s) =.
(50.47)
1/a − iq(s)The scattering length is thus proportional to the value of the amplitude at threshold. It is
worth noting that the scattering length approximation is applicable only in a very limited energy
range, however, might well be appropriate to analyze the recently discovered narrow near-thresholdstates [104,105]. Examples of such analyses can be found in Refs. [106–108]. Moreover, it is possible
to introduce the effect of a weakly coupled lower channel [109, 110]. Such coupling results in a positive imaginary part of the scattering length. It is also crucial to highlight that for large a values,
the amplitude of Eq. (50.47) develops a near-threshold pole located on the physical or unphysicalsheet for negative or positive values of a, respectively. For readers interested in a exploration of
how close-to-threshold poles interact with remote thresholds, we refer to Ref. [111].
While easy to use, it is important to stress, however, that the approximation in Eq. (50.47)
is a specific choice of the dynamic function. This choice results in a single pole close to thephysical region, suggesting that the state under study has the characteristics of a hadronic molecule,
as discussed in references such as [108, 112–114]. Virtual states are discussed in this context in
Ref. [115].
50.3.5 Two-potential decomposition
Another advanced technique to construct the scattering amplitude, which is widely used in
the literature [69, 116–121], is based on the two-potential formalism [122]. While it is possible
to formulate this method for the full unprojected amplitude Mba (s, t), for clarity, we focus onpresenting the equations in their partial-wave-projected form.
31st May, 2024

17

50. Resonances

The scattering amplitude, M(s), can be broken down into two components: a background partand a pole part. This can be represented as:
M(s) = Mb.g. (s) + Mpole (s) .

(50.48)It is important to note that the division presented in Eq. (50.48) is not unique and model-dependent.
This is further discussed in references such as [123, 124]. The background scattering matrix isassumed to be unitary by itself. One approach to parameterize it, especially at lower energies, isby using phase shifts and inelasticities, as seen in [69, 120, 121]. Alternatively, it can be computed
based on some potential, V b.g. , fed into a proper scattering equation [118, 119].
The complete amplitude, M, from Eq. (50.48) is unitary, if the pole part follows a specificconstruction. Namely,
−1 R
R
u
T
Mpole
ab (s) = Ωac (s) [1 − V (s)Σ (s)]cd Vde (s) Ωeb (s) .

(50.49)In this context, we introduce the vertex functions, denoted as Ωab (s), and the resonance potential,
represented as V R (s). This potential operates as a matrix in the channel space and can be expressedas:
X gR gR
a b
R
Vab
(s) =
(50.50)2 −s.
mRRu represents the self-energy matrix. Additionally, g R denotes the bare coupling of the
The term Σabaresonance, labeled as R, to the channel a, and mR is its bare mass. The vertex functions obey aunitarity relation similar to the production amplitude in Eq. (50.8). However, in this case, the final
state interaction is determined by Mb.g. . This can be represented as:

Disc Ωab (s) = 2i

X

∗
Mb.g.
ca (s) ρc (s) Ωcb (s) .

(50.51)c

When using low-energy phase shifts for the background term, it is practical to express the vertex
functions in terms of an Omnès matrix, as discussed in [121]. The matrix reduces to the well-known
Omnès function!
Zs ∞ 0 δ b.g. (s0 )Ω(s) = exp
ds 0 0(50.52)π sthrs (s − s − i)in the single-channel case [125], where δ b.g. (s) denotes the phase of the background scatteringmatrix Mb.g. (s). The discontinuity associated with the self-energy matrix, Σ u (s), is given by:uDisc Σab(s) = 2iX∗Ωca(s) ρc (s) Ωcb (s) .

(50.53)c

To determine the real part of Σ u , one can use Eq. (50.53) with a properly subtracted dispersion
integral — cf. Eq. (50.43).
When Mb.g. is unitary, the full amplitude is unitary given Eq. (50.49). However, it is essential
to note that the pole term alone is not unitary unless the background amplitude vanishes. Under
such conditions, the described amplitude simplifies to the the K-matrix construction with improved
analytic behavior as detained in Sec. 50.3.3. Neglecting non-pole terms is not a good approximation for certain interactions, such as the scalar-isoscalar ππ at low energies, as discussed in [126].
However, for higher partial waves, this approximation is generally effective.
A production amplitude consistent with Eq. (50.49) reads [120, 121]:
−1
R
u
Apole
a (s) = Ωac (s) [1 − V (s)Σ (s)]cd Pd ,31st May, 2024

(50.54)

18

50. Resonanceswith function Pd given in Eq. (50.40). This form was employed in Refs. [120, 121] to treat the pion
vector and scalar form factor, respectively, and in Ref. [69] for the scalar πK form factor.
There has been considerable interest in the 3 → 3 scattering recently, particularly in light of
new data on three-hadron interaction [127] and advancements in lattice calculations [128]. One
finds that the methodologies devised for accounting for one-pion exchange bear a resemblance tothe two-potential decomposition. For details see Ref. [129], also Eq. (93) in Ref. [130].
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