# Higher-Order Frequency Modulation Synthesis

Victor Lazzarini and Joseph Timoney email<EMAIL_ADDRESS><EMAIL_ADDRESS>Maynooth University,
Maynooth, Ireland

###### Abstract

Frequency modulation (FM) and phase modulation (PM) are well-known synthesis
methods, which have been deployed widely in musical instruments. More
recently, some synthesisers have implemented direct forms of FM (as opposed to
PM), allowing, at least as part of their design, for higher-order modulation
topologies. However, such implementations are affected by well-known
difficulties that arise in the modulation of frequency, which are normally
solved by the use of PM. In this article, we analyse these problems and using
a direct comparison with PM, we put forward a solution for the direct
application of FM in higher-order modulation arrangements. We begin by
reviewing the theory of first-order FM, contrasting it to PM. We then proceed
to develop a formulation of second-order FM which is equivalent to the issue-
free PM synthesis, and present a closed-form expression for the evaluation of
the second-order FM spectrum. We then extend the principle to higher-order
topologies, by advancing the concept of an FM operator, analogous to the one
used in PM instrument designs. From this we demonstrate that feedback FM is
also a practical possibility. Finally, we complement the paper by giving a
reference implementation in C++.

###### keywords:

sound synthesis; non-linear distortion; frequency modulation; phase
modulation; feedback;

##  1 Introduction

Linear frequency modulation (FM) as a sound synthesis technique has had a long
history of development, first explored by James Tenney, followed by Jean-
Claude Risset and John Chowning (Lazzarini ., 2023). It was given a
theoretical treatment by Chowning (1973), where it was demonstrated to provide
an economical method of producing dynamic spectra with both harmonic and
inharmonic partials. It was shown to generate sounds previously only possible
with the more expensive means of additive synthesis. In the equivalent, but
more flexible form of phase modulation (PM), which was the actual object of
analysis in Chowning’s paper, the method was implemented in a very successful
range of digital synthesisers, first by Yamaha (Chowning Bristow, 1986), then
by other manufacturers. In this form it was expanded to support higher-order
(or stacked), as well as feedback, modulation. Both extensions can be
characterised as forms of complex (as in _multi-component_) PM.

The mathematical description of FM and PM actually stems from the early
studies in radio frequency broadcasting (Bloch, 1944; Corrington, 1947). The
concept of an instantaneous frequency, as the time derivative of the phase
angle, was first introduced in order to support the development of FM theory
(Carson, 1922). In these early papers, the exact distinction between the two
forms of modulation was not a concern for the authors, as the principles being
developed could be implemented with either one of the methods. However, it is
important to note that most of the mathematics underpinning it, arising from
the theory of Bessel functions, applies directly to PM and only in a second
instance to FM, as we will show in this paper.

The subject of PM has been studied extensively since Chowning’s original
paper, in many cases under the misleading name of FM. A review article by
Moorer showed that phase modulation is in fact a particular instance of a more
wide class of nonlinear techniques, which may described by closed-form
summation formulae (Moorer, 1977). Such methods also include waveshaping
(LeBrun, 1979), phase distortion (Timoney ., 2011), and different forms of
formant synthesis (Lazzarini, 2017). More recently, we have had the
development of adaptive techniques, such as adaptive FM (Lazzarini ., 2008),
which allowed phase modulation of arbitrary sources. The introduction of
modified FM synthesis is also worthy of note, producing yet another variant of
PM based on purely imaginary modulation indices (Lazzarini Timoney, 2010).
Recently, we have seen the introduction of the concept of Loopback FM (Smyth,
2019; Smyth Hsu, 2019), and the question of taming exponential FM (Hutchins,
1975) in analogue and digital synthesis applications has also been the object
of further studies (Timoney Lazzarini, 2009; Nielsen, 2020). A complete
account of the state of the art of non-linear distortion synthesis techniques
is found in Lazzarini (2021).

In this paper, we first clarify the differences between FM and PM, making sure
that the definitions of the two techniques are well established. Then we will
proceed to discuss the question of higher-order modulation, which is realised
by the use of frequency-modulated signals in a stack of modulators. This is a
technique that is well understood as far as PM is concerned, but has not been
fully described in spectral terms before, and in particular has not yet
received a treatment in the case of FM. We begin by focusing on the specific
case of second-order FM, for which an equivalent PM expression is derived.
From this, we have both an implementation recipe in the form of a synthesis
flowchart, and a means of deriving the resulting spectrum. Higher-order FM is
then shown to be a generalisation of this particular case, with a practical
implementation through the concept of FM operators (analogous to the well-
known PM operators described by Chowning Bristow (1986)). To complement, we
put forward a reference implementation in C++ to illustrate the principles
presented earlier on.

##  2 Frequency Modulation

FM synthesis is fairly straightforward to implement. In its most general form,
a signal is used to control the frequency of an oscillator, producing an
output with many partials. The frequency control may be exponential or linear.
In this work, we concentrate on the latter form of FM. The technique has been
described in terms of a _fast_ vibrato, which employs modulation frequencies
within the audio range ($>20$ Hz). We can take this intuitive description as
our starting point.

Vibrato has two fundamental parameters: rate and width. The latter is
determined by the amplitude of the modulating signal and the former by its
frequency. We can also define the width as the maximum absolute deviation from
a centre frequency. While at sub-audio rates the result of vibrato is a
certain fluctuation of pitch, as the modulation frequency and width increases,
the carrier output signal ceases to be perceived as a pure sinusoid and
becomes a waveform whose spectrum features a number of partials. It is
necessary that enough modulation is applied, making the oscillator
instantaneous frequency negative at times.

We can represent a sinusoidal FM carrier signal $c(t)$ using the following
expression,

$c(t)=\cos\left(2\pi\int_{0}^{t}f_{c}+m(x)dx\right).$ (1)

To facilitate the discussion, we can set the modulator to $m(t)=d\cos(2\pi
f_{m}t)$, a sinusoid with amplitude $d$ and frequency $f_{m}$. We then have a
modulation frequency $f_{m}$ and a carrier frequency deviation $d$, along with
the carrier frequency $f_{c}$, as the main parameters of FM,

##  3 Phase Modulation

To formulate an equivalent form of PM, we can first rewrite Eq. 1 as

$c(t)=\cos(\phi(t)),$ (2)

that is, using a time-varying phase signal $\phi(t)$ as a driver to the
carrier signal. We can put this function instead in terms of sinusoidal phase
modulation (PM),

$\phi(t)=2\pi f_{c}t+z\sin(2\pi f_{m}t)$ (3)

The advantage of the PM representation is twofold. First, we have a measure of
the amount of modulation, $z$, that, as we can demonstrate, does not depend on
the modulation frequency; and, second, we can take advantage of the Jacobi-
Anger expansion (Watson, 1944), to determine the spectrum of the phase
modulation signal,

$\begin{split}e^{\pm
jz\sin(\theta)}&=J_{0}(z)+2\sum_{n=1}^{\infty}J_{2n}(z)\cos\left(2n\theta\right)\\\
&\pm
2j\sum_{n=0}^{\infty}J_{2n+1}(z)\sin\left([2n+1]\theta\right),\end{split}$ (4)

where $J_{n}(z)$ is the Bessel coefficient of order $n$. From this equation,
we can observe that $z$ is directly involved in determining the amount of
energy spread from the carrier partial to the various sidebands.

###  3.1 Equivalence to FM

Now in order to connect this to the FM expression of Eq. 1, we can find the
corresponding instantaneous carrier frequency as the derivative of the phase
signal $\phi(t)$ in Eq. 3 (Moore, 1990),

$\dot{\phi}(t)=2\pi\left[f_{c}+zf_{m}\cos(2\pi f_{m}t)\right].$ (5)

The quantity $zf_{m}$, the product of the modulation frequency and the phase
modulation amount is equivalent to the frequency deviation $d$ employed in the
FM signal. We use the term _modulation index_ to characterise $z$,

$z=\frac{d}{f_{m}}$ (6)

As shown before, the modulation index determines the bandwidth, or spread of
the resulting FM/PM spectrum. From Eq. 4, we can derive

$\cos\left(\omega+z\sin(\theta)\right)=\sum_{n=-\infty}^{\infty}J_{n}(z)\cos\left(\omega+n\theta\right).$
(7)

This demonstrates that the amplitude of the frequency modulator, $d$, cannot
alone define a measure of the amount of modulation applied to the carrier
signal. On the other hand, as we noted, the amplitude of the phase modulator
can be used directly to determine the output spectrum. Setting $\omega=2\pi
f_{c}t$ and $\theta=2\pi f_{m}t$, we can see that the spectrum of Eq. 1 is
composed of partials at $f_{c}\pm nf_{m}$ Hz, which are scaled by the
corresponding Bessel function coefficient $J_{n}(z)$. We now have a mechanism
to represent the FM in terms of the PM, which provides a clear route for
analysis. A comparison between the FM and PM flowcharts is shown in Fig. 1.

Figure 1: Flowcharts for PM (left) and FM (right).

##  4 A Theory of Second-Order FM

The case of Eq. 1 is that of first-order modulation, consisting of one
modulator and one carrier oscillator. We now consider the arrangement whereby
this is increased to a higher order, that is, where two or more modulation
stages are present. An FM carrier wave is then used as a modulator to a
subsequent carrier, and this may be used again as a modulator, and so on. The
output signal is the result of several orders of modulation. It has been
claimed in the literature that, unlike PM, FM synthesis cannot be implemented
in higher-order topologies (Pinkston, 2000), but as we will demonstrate, that
is the not the case. Most of the difficulties arise from a simplistic approach
to the implementation of FM synthesis that does not take into account the
differences we have discussed in section 3.1. It is the case that some
synthesisers implement direct forms of FM allowing the possibility of higher-
order modulation topologies (and even feedback) (see for instance Novation
(2019)). However, there are deficiencies in these designs, which we address in
this paper.

We first need to consider the amount of modulation required at each stage. A
naïve approach, following directly from the single-level example, would lead
us to apply simply the product of an index of modulation $z_{n}$ and
modulation frequency $f_{m_{n}}$ to determine the frequency deviation at each
stage $n+1$. Not only this is incorrect from a mathematical point of view, but
also, depending on the ratio of the modulation frequencies involved, we will
observe some problematic results. Consider the case of a second-order
modulation arrangement, where we have a first order modulator with frequency
$f_{m_{0}}$, modulating a second-order modulator with frequency $f_{m_{1}}$,
which then modulates a carrier oscillator with frequency $f_{c}$, where we set
$f_{m_{0}}=f_{m_{1}}=f_{c}$. We know from Eq. 7 that the resulting spectrum at
the first order stage may contain a term at $0$ Hz (DC), which is scaled by
$J_{-1}(z)$. When this modulation signal is then applied to the frequency
$f_{c}$ of a carrier oscillator at the next stage, the DC term is added to
$f_{c}$. This results in a partial whose frequency drifts in accordance to
changes in the amplitude of the DC term of the modulation signal. Such an
effect becomes hard to control and is undesirable in most practical
applications. It is of course possible to select modulation frequency ratios
that do not result in DC terms, and also to produce spectra with a strict
$-\pi/2$ phase at 0 Hz, however these solutions are not general enough to
support a theory of higher-order FM synthesis.

Note that such issues do not occur in PM, since any DC offset is translated as
a phase shift, rather than a carrier frequency drift. For this reason, it is
generally much more flexible to adopt PM as a general method for higher-order
modulation. However, it is still possible to develop a solution based on the
principles outlined earlier. For this, we need to consider that the
integration involved in FM synthesis (cf Eq. 1) requires that a periodic time-
varying deviation is applied to the signal. Since the instantaneous frequency
of an FM signal is time-varying, it implies the presence of an amplitude
modulation term following integration.

For example, let’s consider a modulator signal $m_{1}(t)$ whose frequency
$f_{m_{1}}$ is itself modulated by a sinusoidal signal $m_{0}(t)$, whose
frequency is $f_{m_{0}}$. If we want to apply an index of modulation $z_{0}$,
then according to Eq. 6, we need to set the $m_{0}(t)$ signal amplitude
$d_{0}$ to $z_{0}f_{m_{0}}$. The time-varying frequency $f(t)$ of the
modulator $m_{1}(t)$ is then

$f(t)=f_{m_{1}}+z_{0}f_{m_{0}}m(t),$ (8)

Therefore, also according to eq. 6, we need to employ the following time-
varying deviation, with an appropriate value of $z_{1}$,

$d_{1}(t)=z_{1}f(t)=z_{1}[f_{m_{1}}+z_{0}f_{m_{0}}m(t)],$ (9)

in order to produce the correct frequency modulation signal.

###  4.1 Second-order FM Analysis

We can now apply these principles to the typical case of a second-order FM
synthesis arrangement using two sine wave oscillators. The solution developed
for this case can then be expanded to provide methods for higher-order
modulation topologies. Using the notions developed above, we can now describe
the correct form of second-order FM as

$\begin{split}&m_{0}(t)=\cos(2\pi f_{m_{0}}t)\\\
&m_{1}(t)=\cos\left(2\pi\int_{0}^{t}f_{m_{1}}+z_{0}f_{m_{0}}m_{0}(x)dx\right)\\\
&c(t)=\cos\left(2\pi\int_{0}^{t}f_{c}+z_{1}[f_{m_{1}}+z_{0}f_{m_{0}}m_{0}(x)]m_{1}(x)dx\right).\\\
\end{split}$ (10)

We can now demonstrate how these equations are equivalent to the typical form
of second-order PM. To do this, we begin by reworking the first-order
modulation as a PM expression,

$\begin{split}&m_{0}(t)=\sin(2\pi f_{m_{0}}t)\\\ &\phi(t)=\cos(2\pi
f_{m_{1}}t+z_{0}m_{0}(t))\\\
&c(t)=\cos\left(2\pi\int_{0}^{t}f_{c}+z_{1}[f_{m_{1}}+z_{0}f_{m_{0}}\cos(2\pi
f_{m_{0}}x)]\phi(x)dx\right).\\\ \end{split}$ (11)

The next step is to replace $\phi(.)$ in the carrier signal equation,

$\begin{split}c(t)=\cos\Bigg{(}2\pi\int_{0}^{t}&f_{c}+z_{1}[f_{m_{1}}+z_{0}f_{m_{0}}\cos(2\pi
f_{m_{0}}x)]\times\Bigg{.}\\\ &\Bigg{.}\cos\left(2\pi
f_{m_{1}}x+z_{0}\sin(2\pi f_{m_{0}}x)\right)dx\Bigg{)},\end{split}$ (12)

which translates as the following expression describing second-order PM

$c(t)=\cos\left(2\pi f_{c}t+z_{1}\sin\left(2\pi f_{m_{1}}t+z_{0}\sin(2\pi
f_{m_{0}}t)\right)\right).$ (13)

The equivalent FM topology (Eq. 10) can be implemented using the flowchart
shown in Fig. 2. In this, we see that in order to implement stacked FM we need
to take account of the amplitude modulation effects that arise from employing
a modulated input, as per Eq. 9.

Figure 2: Second-order FM flowchart.

Waveforms and spectra examples produced by second-order FM and PM are shown to
be equivalent in Fig. 3. These signals were produced using the approach
described by Eq. 10 and Fig 2, in the case of FM, and the corresponding PM
expression given in Eq. 13. This demonstrates that it is indeed possible to
use a second-order FM topology to produce a spectrum that is similar to
second-order PM.

Figure 3: Second-order FM (left) and PM (right) waveforms and spectra from Eq.
10 (Fig 2) and Eq. 13, respectively, with $f_{c}=f_{m_{0}}=f_{m_{1}}=500$ Hz,
$z_{0}=3$, and $z_{1}=2$

###  4.2 Spectrum

With its equivalent PM form, we can now derive an expression for the second-
order FM spectrum. Using Eq. 4 we rewrite Eq. 13 as

$c(t)=\cos\left(2\pi
f_{c}t+z_{1}\sum_{n=-\infty}^{\infty}J_{n}(z_{0})\sin\left(2\pi[f_{m_{0}}+nf_{m_{1}}]t\right)\right).$
(14)

From this equation we can now use a derivation of the spectrum of complex PM
(LeBrun, 1977). In order to make this more meaningful, we assume that the
modulation signal spectrum contains only $K$ sidebands with significant energy
(rather than a theoretically non-bandlimited spectrum). The spectrum of
second-order FM synthesis is then defined as

$\begin{split}\sum_{n_{-K}=-\infty}^{\infty}\ldots\sum_{n_{K}=-\infty}^{\infty}&\left[\prod_{k=-K}^{K}J_{n_{k}}\left(z_{1}J_{k}(z_{0})\right)\right]\\\
&\cos\left(2\pi[f_{c}+\sum_{k=-K}^{K}k(f_{m_{1}}+n_{k}f_{m_{0}})]t\right).\end{split}$
(15)

The value of $K$ is dependent on the modulation indices employed and will
increase as more modulation is inserted into the signal. High values of $K$
may produce audible aliasing in digital implementations, depending also on the
modulation and carrier frequencies as well as the sampling rate.

##  5 Higher-order Modulation

The method developed here for second-order modulation may be expanded to
higher-order arrangements, as needed. The modulation signal at each level is
inserted into both the amplitude and frequency inputs of the next modulator. A
guiding principle is to note that if we are employing signals whose
instantaneous frequency is varying significantly over time, then we will need
to account for this in the integration. From another perspective, we can also
observe that through amplitude modulation, we are able to suppress the DC
signals responsible for any carrier drift.

We should note that, as indicated by Eq. 15, the resulting spectrum may be
very complex and difficult to predict in higher-order topologies. Such
arrangements generally call for very limited modulation indices. As in PM,
extensions to beyond fourth-order may be of little practical interest.

###  5.1 Operators

In order to facilitate the design of synthesis instruments using higher-order
FM topologies, we can take advantage of the concept of an _operator_. At its
simplest, this is a sinusoidal oscillator whose frequency can be modulated by
another. The principle of an operator is very common in PM synthesis (Chowning
Bristow, 1986), and it may also include an envelope to allow for dynamic
spectra as well as amplitude shaping. In PM, an operator is characterised by a
phase modulation input, plus amplitude and frequency parameters, and a single
output. Operators can be connected in series (stacked), or in parallel. For
FM, we can develop a similar black-box approach.

In order to design an operator for FM, we need to take account of our analysis
in Section 4.1. We may note that within a stack, the top oscillator takes in a
modulation frequency and a modulation index, producing a modulation signal.
Subsequent oscillators take in a modulation signal in addition to the
frequency and index. At the bottom of the stack, an oscillator produces the
output signal, and it takes an amplitude instead of a modulation index. As in
PM, the operator takes three inputs (index/amplitude and frequency scalars
plus modulation signal). The specification requires that we make no
distinction between amplitude and index. For this to be practical, unlike in
the PM case, we would then need to distinguish between audio and modulation
outputs. For this reason, a freely-stackable FM operator requires two separate
outputs. Envelopes may be added to shape the scalar input parameters. In
pseudocode, the simplest design would be

⬇

audio,modulation = operator(amplitude,frequency,modulation)

With this, the second-order stack discussed earlier would be defined as

⬇

audio,mod0 = operator(index0,fm0,0)

audio,mod1 = operator(index1,fm1,mod0)

audio,mod = operator(amp,fc,mod1)

The implementation of the operator black box is shown in Fig 4, together with
an arrangement of three operators in a second-order modulation topology
equivalent to that of Fig 2.

Figure 4: FM operator (left) and second-order modulation arrangement (right).
The _a_ and _f_ parameters represent the scalar index/amplitude and frequency.

As can be seen, the actual signal flow is re-ordered somewhat with the product
being placed at the output of the oscillator. This way it is possible to use a
single operator be either a carrier or a modulator in any arrangement of any
order. The topmost modulator will always have no signal inputs and we only use
the audio signal out of the carrier operator. Also we should note that this
allows us to tap anywhere into an FM topology to retrieve an audio signal at
that point. Dynamic spectra can be implemented by including envelopes to
control the _a_ and _f_ parameters in Fig 2.

###  5.2 Feedback

The operator as developed here opens up the possibility of implementing a
feedback FM design, which is analogous to feedback PM as introduced by
Tomisawa (1979). In this case, we just need to apply the modulation
recursively to an operator, as in

⬇

audio,modulation = operator(amplitude,frequency,modulation)

which is depicted as a flowchart in Fig 5.

Figure 5: FM operator with feedback (left) and its black-box representation
(right).

The spectrum and waveform of feedback FM is shown on Fig. 6 alongside feedback
PM. As we can see the two spectra share many similarities, although the
waveforms are different. This is due to the phases of each sideband, which
will not be the same for the reasons discussed earlier in Section 3.1.

The case of feedback PM is well understood, and the spectrum has been shown to
approximate a $1/f$ slope (Lazzarini, 2021). In the case of feedback FM, there
is a strong DC term, which we would expect as the first sideband has a cosine
phase. The rest of the spectrum also decays with frequency following a similar
curve. Indeed, the two signals sound very alike to the listener.

Figure 6: Feedback FM (left) and PM (right) waveforms and spectra, with
$f=500$ Hz.

Dynamic spectra in this arrangement is possible through applying an envelope
to the amplitude of the operator. This would cause a change in both amplitude
and bandwidth, as the feedback amount would increase at the same time as the
operator output. If we want to decouple these, it is possible to apply a
separate gain to control the feedback amount, therefore varying the number of
sidebands independently from the output signal level.

Feedback FM (as opposed to Feedback PM) had been deemed not viable for
implementation in the literature (Pinkston, 2000). It is indeed problematic if
a naïve approach is taken, since the integration is brought to a halt if the
frequency becomes zero at any point. However, it has recently received a
theoretical treatment by Smyth Hsu (2019), which shares similar principles to
what we have developed in this article. With the proposed operator-based
implementation, we have shown that it is indeed possible to deploy the
technique in practical applications.

###  5.3 Reference Implementation in C++

The following code provides a reference implementation of the FM synthesis
operator.

⬇

#include <vector>

#include <cmath>

template<typename S>

class Op {

static constexpr long maxlen = 0x100000000;

const std::vector<S> &tab;

std::vector<S> out;

std::vector<S> mod;

int lobits;

int lomask;

S lofac;

S fs;

S fac;

unsigned int phs;

S process(S amp, int si) {

S frac = (phs & lomask)*lofac;

unsigned int ndx = phs >> lobits;

S s = amp*(tab[ndx] + frac*(tab[ndx+1]

- tab[ndx]));

phs += si;

return s;

}

public:

Op(const std::vector<S> &table, float sr,

std::size_t vsize) :

tab(table), out(vsize), mod(vsize), lobits(0), fs(sr),

fac(maxlen/sr), phs(0) {

for(unsigned long t = tab.size()-1;

(t & maxlen) == 0; t <<= 1)

lobits += 1;

lomask = (1 << lobits) - 1;

lofac = 1.f/(lomask + 1);

}

unsigned int vsize() { return out.size(); }

S sr() { return fs; }

const std::vector<S> &operator()() { return mod; }

const std::vector<S> &operator() (S amp, S fr) {

int si = (unsigned int) (fr*fac);

std::size_t n = 0;

for(auto &s : out) {

s = process(amp, si);

mod[n++] = s*fr;

}

return out;

}

const std::vector<S> &operator() (S amp, S fr,

const std::vector<S> &fm) {

std::size_t n = 0;

S f;

for(auto &s : out) {

f = fr + fm[n];

s = process(amp, (int) (f*fac));

mod[n++] = s*f;

}

return out;

}

};

Using instances of this class, we can implement various types of high-order FM
synthesis topologies. For example, a second-order stacked FM arrangement, such
as the one described in Fig. 2, can be modelled using the following class,

⬇

const std::size_t def_vsize = 64;

template<typename S> class StackedFM {

Op<S> mod0;

Op<S> mod1;

Op<S> car;

public:

StackedFM(const std::vector<S> &table, S fs,

std::size_t vsize = def_vsize) :

mod0(table,fs,vsize),mod1(table,fs,vsize),

car(table,fs,vsize) { };

unsigned int vsize() { return car.vsize(); }

S fs() { return car.sr();}

const std::vector<S> &operator()(S a,S fc,S fm0,S fm1,

S z0,S z1) {

mod0(z0,fm0);

mod1(z1,fm1,mod0());

return car(a,fc,mod1());

}

};

An object of this class can then be used to produce an FM tone as shown by an
example program fragment. It uses the modulation frequency ratio
$c:m_{0}:m_{1}$ with separate indices of modulation for first and second-order
stages,

⬇

std::vector<float> tab(1025);

std::size_t n = 0;

for(auto &s : tab)

s = std::cos(2*M_PI/(tab.size()-1)*n++);

StackedFM<float> fm(tab,fs);

for(n = 0; n < fm.fs()*dur; n += fm.vsize()) {

auto out = fm(amp,fr,fr,fr,z0,z1);

for(auto s : out)

std::cout << s << std::endl;

}

##  6 Discussion

While it is beyond the scope of this article to consider in detail possible
applications of higher-order FM synthesis, we may cite a few. Generally, the
technique is useful in situations where it is not convenient or practical to
modulate the phase of a signal. One such example is the case of the synthesis
methods employed in the Summit synthesiser oscillator implementation
(Novation, 2019). These of course have not been published and it is not
possible to exactly determine their details, but it is a reasonable assumption
that PM is either not possible or not ideal since they opted to implement FM
in a somewhat naïve form (as per our earlier analysis). Another application
may be found in analogue signal processing, where it is often the case that
(linear) FM can be applied more directly than PM. Thus a simplification in
circuit design may be also another factor that would favour the use of the
technique.

Within a digital signal processing environment, however, there are a few
practical issues arising within the scenarios of stacked and feedback FM
introduced here. These have to do with numerical errors arising from the
discrete nature of the integration filter employed in the implementation of an
oscillator. In the case of a modulation stack, we have observed that such
errors result in a phase modulation term that is not present in the
continuous-time analysis of Section 4.1. This can introduce periodic
modulation artefacts in the carrier signal that may be objectionable. In the
case of Feedback FM, the errors introduce an extraneous DC term in the signal
that has an obvious, although small, effect on pitch. A full analysis of these
errors, their effects, and possible mitigation is beyond the scope of the
present article, but it is important to note that they exist. In some
applications, such as the extremely high sampling rate oscillator
implementations (e.g. using field programmable gate array hardware, as in the
case of Summit synthesiser, running in the MHz range), these issues may not be
of concern.

##  7 Conclusions

A simplistic approach to implementing second and higher-order FM arrangements
has been shown to have limitations. Typical issues found in these situations
are related to carrier drift, which is due to the presence of a DC component
in a modulating waveform. Since the amount of energy at 0 Hz is defined by the
index of modulation and is a function of the Bessel coefficient associated
with the relevant sideband, any timbral changes in this case are accompanied
by frequency glides that may be objectionable in practical applications. These
issues are fully solved through the development of a PM-equivalent form of
higher-order FM. Since higher-order PM is well understood and has been
successfully applied in a variety of contexts, we propose that this may be a
more suitable approach.

In this paper, we defined in good detail the differences between PM and FM,
demonstrating that for higher-order modulation, there needs to be some care to
ensure that the correct formulation is employed. We then proposed a econd-
order FM arrangement that follows this form. From the PM-equivalent formula we
are then able to describe the resulting FM spectrum by employing a similar
approach to the derivation of the complex PM spectrum.

Higher-order modulation topologies can then be implemented as an extension of
the second-order approach. For these, we found that an operator approach may
be helpful. We have then put forward the basic design of such a black box,
demonstrating its equivalence to the second-order design shown earlier. With
these, it is possible to freely construct various FM topologies, including
feedback FM, as it is customarily done with FM. We completed the discussion
with a full reference implementation of operator-based FM.

Code examples and scripts used for signal analysis in this paper can be found
at

https://github.com/vlazzarini/highorderfm

## References

  * Bloch (1944) BlochBloch, A.  1944\.  Modulation theory Modulation theory.  Journal of the Institution of Electrical Engineers - Part I: General9145368-370.
  * Carson (1922) CarlsonCarson, J.  1922\.  Notes on the theory of modulation Notes on the theory of modulation.  Proceedings Institute of Radio Engineers10157-64.
  * Chowning (1973) ChowningFMChowning, J.  1973\.  The Synthesis of Complex Audio Spectra by Means of Frequency Modulation The synthesis of complex audio spectra by means of frequency modulation.  Journal of the Audio Engineering Society217527–534.
  * Chowning Bristow (1986) FMTheoryChowning, J. Bristow, D.  1986\.  FM Theory and Applications Fm theory and applications.  Tokyo, JapanYamaha Music Foundation.
  * Corrington (1947) CorringtonCorrington, MS.  1947\.  Variation of Bandwidth with Modulation Index in Frequency Modulation Variation of bandwidth with modulation index in frequency modulation.  Proceedings of the IRE35101013-1020.
  * Hutchins (1975) HutchinsXFMHutchins, B.  1975\.  The Frequency Modulation Spectrum of an Exponential Voltage-Controlled Oscillator The frequency modulation spectrum of an exponential voltage-controlled oscillator.  Journal of the Audio Engineering Society233200-207.
  * Lazzarini (2017) Lazzarini2017Lazzarini, V.  2017\.  Computer Music Instruments Computer music instruments.  BerlinSpringer.
  * Lazzarini (2021) Lazzarini2021Lazzarini, V.  2021\.  Spectral Music Design: A Computational Approach Spectral music design: A computational approach.  New YorkOxford Univ. Press.
  * Lazzarini . (2023) Lazzarini2023Lazzarini, V., Keller, D. Radivojević, N.  2023\.  Issues of ubiquitous music archaeology: Shared knowledge, simulation, terseness, and ambiguity in early computer music Issues of ubiquitous music archaeology: Shared knowledge, simulation, terseness, and ambiguity in early computer music.  Frontiers in Signal Processing3.  https://www.frontiersin.org/articles/10.3389/frsip.2023.1132672 10.3389/frsip.2023.1132672
  * Lazzarini Timoney (2010) Lazzarini2010theoryLazzarini, V. Timoney, J.  2010\.  Theory and practice of modified frequency modulation synthesis Theory and practice of modified frequency modulation synthesis.  Journal of the Audio Engineering Society586459-471.
  * Lazzarini . (2008) LazzariniADFMLazzarini, V., Timoney, J. Lysaght, T.  2008\.  The Generation of Natural-Synthetic Spectra by Means of Adaptive Frequency Modulation The generation of natural-synthetic spectra by means of adaptive frequency modulation.  Computer Music Journal3229–22.
  * LeBrun (1977) LeBrunFMLeBrun, M.  1977\.  A Derivation of the Spectrum of FM with a Complex Modulating Wave A derivation of the spectrum of fm with a complex modulating wave.  Computer Music Journal1451–52.
  * LeBrun (1979) LeBrunWaveshapingLeBrun, M.  1979\.  Digital Waveshaping Synthesis Digital waveshaping synthesis.  Journal of the Audio Engineering Society274250–266.
  * Moore (1990) Moore:1990Moore, FR.  1990\.  Elements of Computer Music Elements of computer music.  Upper Saddle River, NJ, USAPrentice-Hall, Inc.
  * Moorer (1977) MoorerIEEEMoorer, J.  1977\.  Signal processing aspects of computer music: A survey Signal processing aspects of computer music: A survey.  Proceedings of the IEEE6581108–1137.
  * Nielsen (2020) NielsenNielsen, K.  2020\.  Practical linear and exponential frequency modulation for digital music synthesis Practical linear and exponential frequency modulation for digital music synthesis.  Proceedings of the 23rd Conference on Digital Audio Effects Proceedings of the 23rd conference on digital audio effects ( 133-139).  ViennaDAFX.
  * Novation (2019) SummitNovation.  2019\.  Summit Synthesizer Manual Summit synthesizer manual.  High Wycombe, UKNovation Digital Music Systems.
  * Pinkston (2000) PinkstonFMPinkston, R.  2000\.  FM Synthesis in Csound Fm synthesis in csound.  The Csound Book The csound book ( 261-280).  Cambridge, MAMIT Press.
  * Smyth (2019) SmythSmyth, T.  2019\.  On the Similarity between Feedback/Loopback Amplitude and Frequency Modulation On the similarity between feedback/loopback amplitude and frequency modulation.  Proceedings of the 147th AES Convention Proceedings of the 147th aes convention ( 1-7).
  * Smyth Hsu (2019) LoopbackSmyth, T. Hsu, JS.  2019\.  On Phase and Pitch in Loopback Frequency Modulation with a Time-Varying Feedback Coefficient On phase and pitch in loopback frequency modulation with a time-varying feedback coefficient.  Proceedings of the 26th International Congress on Sound and Vibration Proceedings of the 26th international congress on sound and vibration ( 1-8).
  * Timoney Lazzarini (2009) TimoneyEFMTimoney, J. Lazzarini, V.  2009\.  Exponential FM bandwidth criterion for virtual analogue applications Exponential FM bandwidth criterion for virtual analogue applications.  Proceedings of the 14th Conference on Digital Audio Effects Proceedings of the 14th conference on digital audio effects ( 115-118).  ParisDAFX.
  * Timoney . (2011) LazzariniPDTimoney, J., Lazzarini, V., Pekonen, J. Valimaki, V.  2011\.  Adaptive Phase Distortion Synthesis Adaptive phase distortion synthesis.  Proceedings of the 12th Conference on Digital Audio Effects Proceedings of the 12th conference on digital audio effects ( 1-8).  Como, ItalyDAFX.
  * Tomisawa (1979) TomisawaTomisawa, N.  1979\.  Tone production method for an electronic musical instrument. Tone production method for an electronic musical instrument.  US Patent 4249447A
  * Watson (1944) WatsonWatson, G.  1944\.  A Treatise of the Theory of Bessel Functions A treatise of the theory of bessel functions.  CambridgeCambridge Univ. Press.


# The $X(3960)$ seen in $D_{s}^{+}D_{s}^{-}$ as the $X(3930)$ state seen in
$D^{+}D^{-}$

M. Bayar<EMAIL_ADDRESS>Department of Physics, Kocaeli
University, Izmit 41380, Türkiye  Departamento de Física Teórica and IFIC,
Centro Mixto Universidad de Valencia-CSIC Institutos de Investigación de
Paterna, Aptdo.22085, 46071 Valencia, Spain  A. Feijoo<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-
CSIC Institutos de Investigación de Paterna, Aptdo.22085, 46071 Valencia,
Spain  E. Oset<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC,
Centro Mixto Universidad de Valencia-CSIC Institutos de Investigación de
Paterna, Aptdo.22085, 46071 Valencia, Spain

###### Abstract

We perform a calculation of the interaction of the $D\bar{D}$,
$D_{s}\bar{D}_{s}$ coupled channels and find two bound states, one coupling to
$D\bar{D}$ and another one at higher energies coupling mostly to
$D_{s}^{+}D_{s}^{-}$. We identify this latter state with the $X_{0}(3930)$
seen in the $D^{+}D^{-}$ mass distribution in the $B^{+}\to D^{+}D^{-}K^{+}$
decay, and also show that it produces an enhancement of the
$D_{s}^{+}D_{s}^{-}$ mass distribution close to threshold which is compatible
with the LHCb recent observation in the $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$
decay which has been identified as a new state, $X_{0}(3960)$.

##  I Introduction

In a recent talk at CERN, the LHCb Collaboration reported on three new states
cern  one of which, named as $X(3960)$, is seen as a peak in the
$D_{s}^{+}D_{s}^{-}$ mass distribution of the $B^{+}\to
D_{s}^{+}D_{s}^{-}K^{+}$ decay. The properties assigned to that state are

$\displaystyle J^{PC}=0^{++}~{};~{}~{}M_{0}=3955\pm 6\pm
11~{}MeV~{};~{}~{}\Gamma_{0}=48\pm 17\pm 10~{}MeV$

The peak is remarkably close to the $D_{s}^{+}D_{s}^{-}$ threshold, $3937$
MeV, which makes one wonder where it could not be a signal for a resonance
just below threshold 111Comment of Sasa Prelovsek in the discussion of the
talk cern . This is actually a well known feature of reactions as discussed in
guozou  and found in some specific reactions navarra ; enwang . Actually, the
$B^{+}$ decay into $B^{+}\to D^{+}D^{-}K^{+}$ showed a signal in the
$D^{-}D^{+}$ mass distribution for a $0^{++}$ state, branded $X_{0}(3930)$,
with properties dpdm1 ; dpdm2

$\displaystyle J^{PC}=0^{++}~{};~{}~{}M^{\prime}_{0}=3924\pm
2~{}MeV~{}~{};~{}~{}\Gamma^{\prime}_{0}=17\pm 5~{}MeV$

If the $X_{0}(3930)$ state coupled both to $D^{+}D^{-}$ and
$D_{s}^{+}D_{s}^{-}$, that state would necessarily produce an enhancement
close to threshold in the $D_{s}^{+}D_{s}^{-}$ mass distribution, which could
explain the experimental observation without the need to introduce an extra
resonance. The purpose of this work is to show that present dynamics of the
interaction of charmed mesons leads naturally to this conclusion.

The first consideration in this respect is the QCD lattice result of sasa ,
where a $0^{++}$ bound state coupling strongly to $D_{s}^{+}D_{s}^{-}$ and
weakly to $D^{+}D^{-}$ is found below the $D_{s}^{+}D_{s}^{-}$ threshold. Such
a state would indeed show a peak in the $D^{+}D^{-}$ mass distribution and an
enhanced mass distribution of the $D_{s}^{+}D_{s}^{-}$ around threshold.

One might expect that such a state would appear in a dynamical study of
$D\bar{D}$ and $D_{s}^{+}D_{s}^{-}$ in coupled channels. Interestingly, this
study has been done in dani ; juandd ; hidalgo , where a $D\bar{D}$ bound
state was found. Yet, no bound state was found close to the
$D_{s}^{+}D_{s}^{-}$ threshold coupling mostly to that channel.

In what follows we show that this is a consequence of the strong $D\bar{D}\to
D_{s}^{+}D_{s}^{-}$ transition, to the point that a little weaker transition
already gives rise to the two states, the upper one coupled mostly to
$D_{s}^{+}D_{s}^{-}$ as in sasa  and with properties that are consistent with
the mass and width of the $X_{0}(3930)$ and the strength and shape of the
$D_{s}^{+}D_{s}^{-}$ mass distribution of the $X_{0}(3960)$. A formulation of
the problem is presented below.

We depart from the dynamics of the $D\bar{D}$, and coupled channels used in
dani  and use a formulation based on the extension of the local hidden gauge
approach hidden1 ; hidden2 ; hidden4 ; hideko , which has turned out to make
accurate predictions for $D^{(*)}K^{(*)}$, $D^{(*)}\bar{K}^{(*)}$ states branz
; raquel  or $T_{cc}$ and related states feijoo ; dai  among others. The
dynamics is based on the exchange of vector mesons as depicted in Fig. 1

Figure 1: Dynamics of $D(D_{s})\to\bar{D}(\bar{D}_{s})$ interaction due to
vector exchange.

One needs the $VPP$ ($V=$ vector, $P=$pseudosclar) vertex given by

${\cal
L}_{VPP}=-ig\langle[P,\partial_{\nu}P]V^{\mu}\rangle;~{}g=\dfrac{M_{V}}{2f}~{}(M_{V}\simeq
800~{}MeV,f=93~{}MeV)$ (1)

where $P$ and $V$ are the $q_{i}\bar{q}_{j}$ matrices written in terms of
pseudoscalar ($P$) and vector ($V$) mesons respectively

$P=\begin{pmatrix}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{3}}\eta+\frac{1}{\sqrt{6}}\eta^{\prime}&\pi^{+}&K^{+}&\bar{D}^{0}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{3}}\eta+\frac{1}{\sqrt{6}}\eta^{\prime}&K^{0}&D^{-}\\\
K^{-}&\bar{K}^{0}&-\frac{1}{\sqrt{3}}\eta+\sqrt{\frac{2}{3}}\eta^{\prime}&D_{s}^{-}\\\
D^{0}&D^{+}&D_{s}^{+}&\eta_{c}\end{pmatrix},$ (2)

$V=\begin{pmatrix}\frac{1}{\sqrt{2}}\rho^{0}+\frac{1}{\sqrt{2}}\omega&\rho^{+}&K^{*+}&\bar{D}^{*0}\\\
\rho^{-}&-\frac{1}{\sqrt{2}}\rho^{0}+\frac{1}{\sqrt{2}}\omega&K^{*0}&\bar{D}^{*-}\\\
K^{*-}&\bar{K}^{*0}&\phi&D_{s}^{*-}\\\
D^{*0}&D^{*+}&D_{s}^{*+}&J/\psi\end{pmatrix},$ (3)

and $\langle~{}~{}\rangle$ means the trace of the $q_{i}\bar{q}_{j}$ matrices.
A straightforward calculation of the diagrams of Fig. 1 with the Lagrangian of
Eq. (1) considering the channels $D\bar{D}$, $I=0$ and $D_{s}^{+}D_{s}^{-}$
with the $(D^{+},-D^{0})~{}(\bar{D}^{0},D^{-})$ multiplets

$(D\bar{D},~{}I=0)=\frac{1}{\sqrt{2}}(D^{+}D^{-}+D^{0}\bar{D}^{0});~{}D_{s}^{+}D_{s}^{-}$

gives the interaction potential

$V_{ij}=-B_{ij}g^{2}(p_{1}+p_{3})(p_{2}+p_{4})$ (4)

with

$B=\begin{pmatrix}\dfrac{1}{2}\left(\dfrac{3}{M^{2}_{\rho}}+\dfrac{1}{M^{2}_{\omega}}+\dfrac{2}{M^{2}_{J/\Psi}}\right)&\sqrt{2}\dfrac{1}{M^{2}_{K^{*}}}\\\
\sqrt{2}\dfrac{1}{M^{2}_{K^{*}}}&\left(\dfrac{1}{M^{2}_{\phi}}+\dfrac{1}{M^{2}_{J/\Psi}}\right)\end{pmatrix}.$
(5)

where, projected over $S$-wave, we have

$(p_{1}+p_{3})(p_{2}+p_{4})\to\dfrac{1}{2}\left[3s-(m^{2}_{1}+m^{2}_{2}+m^{2}_{3}+m^{2}_{4})-\dfrac{1}{s}(m^{2}_{1}-m^{2}_{2})(m^{2}_{3}-m^{2}_{4})\right]$
(6)

The unitarization via the Bethe-Salpeter equation gives the scattering matrix

$T=\left[1-VG\right]^{-1}V$ (7)

where G is the loop function for two meson intermediate states that we choose
to regularize with dimensional regularization as we did in dani .

An interesting thing happens with Eq. (7). If we eliminate the $V_{12}$ term
of Eq. (4) which connects the $D^{+}D^{-}$ and $D_{s}^{+}D_{s}^{-}$ channels,
we get two poles with ordinary values of the subtraction constant $a\sim-1.5$
of the $G$ function in a wide range of values, meaning that the two $V_{11}$,
$V_{22}$ potentials are strong enough to bind the $D\bar{D}$ and
$D_{s}\bar{D}_{s}$ components. Yet, when $V_{12}$ is switched on, the
$D\bar{D}$ state remains and the $D_{s}\bar{D}_{s}$ state disappears. This is
what happened in dani  where the $D_{s}\bar{D}_{s}$ state was not found.
Curiously, if we weaken $V_{12}$ to a value of about $70\%$ of its strength in
Eq. (5) the two states already appear. Uncertainties of this type in our
approach, together with the lattice results sasa  generating this state,
prompt us to accept that level of uncertainty in order to obtain the two
states and see if our hypothesis of $X_{0}(3930)$ being responsible for the
$X_{0}(3960)$ peak holds or not.

The next, non trivial, step is to relate the $D^{+}D^{-}$ and
$D_{s}^{+}D_{s}^{-}$ mass distributions. We use the charge conjugate reactions
$B^{-}\to D^{+}D^{-}K^{-},~{}D_{s}^{+}D_{s}^{-}K^{-}$ and have

$\dfrac{d~{}\Gamma}{dM_{\rm
inv}}=\frac{1}{(2\pi)^{3}}\frac{1}{4M^{2}_{B}}~{}p_{K^{-}}~{}p_{\tilde{D}_{i}}~{}|\tilde{t}_{i}|^{2},$
(8)

where $p_{K^{-}}$ is the $K^{-}$ momentum in the $B^{-}$ rest frame, and
$p_{\tilde{D}_{i}}$ the $D^{+}$ or $D_{s}^{+}$ momenta in the $D^{+}D^{-}$,
$D_{s}^{+}D_{s}^{-}$ rest frame respectively. The matrices $\tilde{t}_{i}$
stand for the transitions $B^{-}\to D^{+}D^{-}K^{-}$ and
$D_{s}^{+}D_{s}^{-}K^{-}$ and are constructed in the following way. The
$B^{-}$ decays proceed via interval emission chau , as depicted in Fig. 2, and
are related.

Figure 2: $B^{-}$ decay via internal emission at the quark level and
hadronization

The $c\bar{c}$ pair is hadronized and we have

$c\bar{c}\to\sum_{i}c\bar{q}_{i}q_{i}\bar{c}\to\sum_{i}P_{4i}P_{i4}=D^{0}\bar{D}^{0}+D^{+}D^{-}+D_{s}^{+}D_{s}^{-}=\sqrt{2}D\bar{D}+D_{s}^{+}D_{s}^{-},$
(9)

where we have eliminated $\eta_{c}\eta_{c}$ which plays no role here. Once
$D\bar{D}$ and $D_{s}^{+}D_{s}^{-}$ have been created, they propagate as shown
in Fig. 3.

Figure 3: Production and propagation of the $D^{+}D^{-}$ and
$D_{s}^{+}D_{s}^{-}$ components through final state interaction.

Then we immediately obtain

$\tilde{t}_{D^{+}D^{-}}=C\left(1+G_{D\bar{D}}(M_{inv})T_{D\bar{D},D\bar{D}}(M_{inv})+\dfrac{1}{\sqrt{2}}G_{D_{s}\bar{D}_{s}}(M_{inv})T_{D_{s}\bar{D}_{s},D\bar{D}}(M_{inv})\right)$
(10)

$\tilde{t}_{D_{s}^{+}D_{s}^{-}}=C\left(1+\sqrt{2}G_{D\bar{D}}(M_{inv})T_{D\bar{D},D_{s}^{+}D_{s}^{-}}(M_{inv})+G_{D_{s}\bar{D}_{s}}(M_{inv})T_{D_{s}^{+}D_{s}^{-},D_{s}^{+}D_{s}^{-}}(M_{inv})\right)$
(11)

with $C$ an arbitrary constant. We can see that the $D^{+}D^{-}$ and
$D_{s}^{+}D_{s}^{-}$ production rates are related via the dynamics of the
process and we can also test the relative strength of the two distributions.

We then proceed as follows. We choose a value of the subtraction constant
$a_{D_{s}\bar{D}_{s}}$ of $G_{D_{s}\bar{D}_{s}}$ and a factor $\alpha$
multiplying $V_{12}$ such as to get approximately the right mass and width of
the $X_{0}(3930)$ state. We accomplish this with the values

$a_{D_{s}\bar{D}_{s}}=-1.58~{};\alpha=0.7$ (12)

At the same time we choose $a_{D\bar{D}}=-1$ to get a $D\bar{D}$ bound state
around $3700~{}MeV$, as in dani  222Values of $a_{D\bar{D}}$ of the order of
$-1.3$ were used in dani  to get that binding, but also many other light
channel were considered (see also xiao ) and it is well known that there is a
certain trade off between term in the potential and changes in the $G$
function., and we get the results in Table 1. These results indicate that the
lower state $(I)$ couples more strongly to $D\bar{D}$ while the second state
$(II)$ couples more strongly to $D_{s}\bar{D}_{s}$ as found in sasa .

Table 1: Masses and widths of the poles dynamically generated by the model, as well as, the corresponding modulus of the couplings $|g_{i}|$.  |  $M$ [MeV] |  $\Gamma$ [MeV] |  $|g_{\bar{D}D}|$ [MeV] |  $|g_{\bar{D}_{s}D_{s}}|$ [MeV]
---|---|---|---|---
Pole I | $3699$ | $-$ | $14516$ | $5897$
Pole II ($X_{0}(3930)$) | $3936$ | $11$ | $2858$ | $9076$

In Fig. 4, we show the results for $\dfrac{d~{}\Gamma}{dM_{\rm
inv}(D^{+}D^{-})}$ and $\dfrac{d~{}\Gamma}{dM_{\rm inv}(D_{s}^{+}D_{s}^{-})}$.
The constant $C$ has been chosen such as to have the normalization of the data
for $\dfrac{d~{}\Gamma}{dM_{\rm inv}(D_{s}^{+}D_{s}^{-})}$. We do not add a
background to the distributions as in cern , since our amplitude
$\tilde{t}_{i}$ of Eqs. (10), (11) already contain one through the terms $1$
in the parenthesis (tree level). We observe that with the chosen parameters of
Eq. (12) that lead approximately to the properties of the $X_{0}(3930)$ state,
we obtain a shape for $\dfrac{d~{}\Gamma}{dM_{\rm inv}(D_{s}^{+}D_{s}^{-})}$
compatible with the experiment. It is also welcome the fact that the strengths
of the mass distributions at their peaks are of the same order of magnitude,
same thing also found in the experiment.

Figure 4: $\dfrac{d~{}\Gamma}{dM_{\rm inv}(D^{+}D^{-})}$ and
$\dfrac{d~{}\Gamma}{dM_{\rm inv}(D_{s}^{+}D_{s}^{-})}$ of $B^{-}\to
D^{+}D^{-}K^{-}$ and $B^{-}\to D_{s}^{+}D_{s}^{-}K^{-}$ decays. The
experimental points are taken from cern .

In summary, we have shown that a state below the$D_{s}\bar{D}_{s}$ threshold,
coupling strongly to $D_{s}\bar{D}_{s}$ and more weakly to $D\bar{D}$, as
found in the lattice QCD calculations, will necessarily produce a
$D_{s}\bar{D}_{s}$ mass distribution with a strong enhancement close to the
$D_{s}\bar{D}_{s}$ threshold. A quantitative evaluation of the $D^{+}D^{-}$
and $D_{s}^{+}D_{s}^{-}$ mass distributions in the $B^{-}\to D^{+}D^{-}K^{-}$
and $B^{-}\to D_{s}^{+}D_{s}^{-}K^{-}$ decays, with small modifications in the
input used to obtain many hadronic states, shows that a $D_{s}^{+}D_{s}^{-}$
bound state appears, which can be associated to the $X_{0}(3030)$, and this
state, coupling both to $D\bar{D}$ and $D_{s}^{+}D_{s}^{-}$, produces an
enhancement in the $D_{s}^{+}D_{s}^{-}$ mass distribution close to threshold
with a shape in agreement with experiment. In addition, the relative strength
of the $D^{+}D^{-}$ and $D_{s}^{+}D_{s}^{-}$ peaks is also similar to the
experimental one. The conclusion is then that there is not need to invoke a
new $X_{0}(3960)$ state, and the experimental observation is due to the
presence of the $X_{0}(3930)$.

##  II ACKNOWLEDGEMENT

This work is partly supported by the Spanish Ministerio de Economia y
Competitividad (MINECO) and European FEDER funds under Contract No.
PID2020-112777GB-I00, and by Generalitat Valenciana under contract
PROMETEO/2020/023. This project has received funding from the European Union
Horizon 2020 research and innovation programme under the program
H2020-INFRAIA-2018-1, grant agreement No. 824093 of the STRONG-2020 project.
The work of A. F. was partially supported by the Generalitat Valenciana and
European Social Fund APOSTD-2021-112, and the Czech Science Foundation, GAČR
Grant No. 19-19640S.

## References

  * (1) Chen Chen and Elisabetta Spadaro Norella, https://indico.cern.ch/event/1176505/
  * (2) X. K. Dong, F. K. Guo and B. S. Zou, Few Body Syst. 62 (2021) no.3, 61.
  * (3) A. Martinez Torres, K. P. Khemchandani, F. S. Navarra, M. Nielsen and E. Oset, Phys. Lett. B 719 (2013), 388-393.
  * (4) E. Wang, H. S. Li, W. H. Liang and E. Oset, Phys. Rev. D 103 (2021) no.5, 054008.
  * (5) R. Aaij et al. [LHCb], Phys. Rev. D 102 (2020), 112003.
  * (6) R. Aaij et al. [LHCb], Phys. Rev. Lett. 125 (2020), 242001.
  * (7) S. Prelovsek, S. Collins, D. Mohler, M. Padmanath and S. Piemonte, JHEP 06 (2021), 035.
  * (8) D. Gamermann, E. Oset, D. Strottman and M. J. Vicente Vacas, Phys. Rev. D 76 (2007), 074016.
  * (9) J. Nieves and M. P. Valderrama, Phys. Rev. D 86 (2012), 056004.
  * (10) C. Hidalgo-Duque, J. Nieves and M. P. Valderrama, Phys. Rev. D 87 (2013) no.7, 076006.
  * (11) M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164 (1988), 217-314.
  * (12) M. Harada and K. Yamawaki, Phys. Rept. 381 (2003), 1-233.
  * (13) U. G. Meissner, Phys. Rept. 161 (1988), 213.
  * (14) H. Nagahiro, L. Roca, A. Hosaka and E. Oset, Phys. Rev. D 79 (2009), 014015.
  * (15) R. Molina, T. Branz and E. Oset, Phys. Rev. D 82 (2010), 014010.
  * (16) R. Molina and E. Oset, Phys. Lett. B 811 (2020), 135870.
  * (17) A. Feijoo, W. H. Liang and E. Oset, Phys. Rev. D 104 (2021) no.11, 114015.
  * (18) L. R. Dai, R. Molina and E. Oset, Phys. Rev. D 105 (2022) no.1, 016029.
  * (19) L. L. Chau, Phys. Rept. 95 (1983), 1-94.
  * (20) C. W. Xiao and E. Oset, Eur. Phys. J. A 49 (2013), 52.