a tangent flow is a static union of two multiplicity one transverse planes.
For simplicity of notation we can thus assume as before that there is one
singularity and by shifting space and time that it occurs at $(\mathbf{0},0)$.

Using Lemma 8.12 we see that $\mathcal{M}(t)$ converges as $t\nearrow 0$ to a
$C^{1}$-immersed Lagrangian $L$, where the convergence (and $L$) is smooth
away from $\mathbf{0}$, which is zero Maslov and rational. Furthermore, $L\cap
B_{r_{0}}(\mathbf{0})$ is given as the union of two Lagrangian graphs as
stated in Lemma 8.12.

We can use the decomposition given in Lemma 8.12 to approximate $L$ by smooth,
zero Maslov, rational Lagrangians $L^{i}$ in $C^{1}$ (by approximating the
$C^{2}$-functions giving $L$ as a graph by smooth functions). Furthermore, we
can assume by the estimates of [29] that there is $T>0$ and smooth, zero
Maslov, rational solutions to Lagrangian mean curvature flow
$(L^{i}_{t})_{t\in[0,T)}$ with $L^{i}_{0}=L^{i}$. By the $C^{1}$-convergence
of $L^{i}\to L$ and interior estimates for higher codimension mean curvature
flow (see [29] or [1, Appendix]), we see that the flows are uniformly
controlled in $C^{\infty}$ for $t>0$. Note that we can assume that the
convergence $L^{i}\to L$ is smooth away from the singular points. Taking the
limit we thus see that there is a smooth, zero Maslov, rational Lagrangian
mean curvature flow $(L_{t})_{t\in(0,T)}$ with $L_{t}\to L$ in $C^{1}$ (and
smoothly away from $\mathbf{0}$) as $t\searrow 0$. This implies that the
extended flow is smooth through the singular time, away from the singular
points. Note that for $\varepsilon>0$ there is $\delta>0$ such that

$\sup_{\begin{subarray}{c}\mathbf{x}\in L_{t}\cap B_{\delta}\\\
-\delta^{2}<t<0\end{subarray}}|\theta(\mathbf{x},t)-\theta_{0}|\leq\varepsilon,$

where $\theta_{0}$ is the Lagrangian angle of the special Lagrangian cone
$V_{0}$. Thus the grading $\theta$ for the extended flow can be chosen that it
is smooth as well through the singular time, away from the singular points. ∎

##  9\. The flow in a compact ambient space

In this section we consider the Lagrangian mean curvature flow in a compact
Calabi–Yau surface and we briefly explain the modifications needed to prove
the results from Section 8 in this setting.

Let us suppose that $X$ is a compact complex surface with complex structure
$J$, admitting a non-vanishing holomorphic volume form $\Omega$ and a Kähler
metric $\omega$ with volume form
$\frac{1}{2}\omega^{2}=\Omega\wedge\bar{\Omega}$. Let $L\subset X$ be a
Lagrangian submanifold. We say that $L$ is zero Maslov if there is a
Lagrangian angle function $\theta:L\to\mathbb{R}$ satisfying

$\Omega|_{L}=e^{i\theta}dA_{L}$

in terms of the Riemannian area form $dA_{L}$ of $L$. Following the notion of
rationality in Fukaya [9, Definition 2.2] we assume furthermore that
$[\omega]$ defines an integral cohomology class in $H^{2}(X;\mathbb{R})$, and
let $\xi$ denote a complex line bundle over $X$ together with a unitary
connection $\nabla^{\xi}$ with curvature form $F_{\nabla^{\xi}}=2\pi i\omega$.
The connection $\nabla^{\xi}$ is then flat when restricted to $L$.

######  Definition 9.1.

The Lagrangian $L$ in $X$ is _rational_ if the holonomy group of
$\nabla^{\xi}$ on $L$ is a finite subgroup of $U(1)$.

We will follow the approach of White [30, Section 4], viewing the mean
curvature flow in $X$ as a mean curvature flow in a larger Euclidean space
with an additional forcing term. More precisely, let $X\subset\mathbb{R}^{N}$
be an isometric embedding. The mean curvature flow $L_{t}$ in $X$ is then
equivalent to the flow

$\frac{\partial}{\partial t}\mathbf{x}=\mathbf{H}+\nu,$

where $\mathbf{H}$ denotes the mean curvature vector inside $\mathbb{R}^{N}$
and $\nu(\mathbf{x},t)=-\mathrm{tr}\,\Pi(\mathbf{x})|_{T_{\mathbf{x}}L_{t}}$
is defined in terms of the second fundamental form $\Pi$ of $X$, restricted to
the tangent space of $L_{t}$ at $\mathbf{x}$. In particular $|\nu|\leq C$ for
a constant independent of $\mathbf{x},t$. Note that $\mathbf{H}+\nu$ is simply
the mean curvature of $L_{t}$ in the ambient space $X$, and so we have
$|\nabla\theta|=|\mathbf{H}+\nu|$.

We need to recall the form of the monotonicity formula for the mean curvature
flow with forcing term $\nu$: instead of (2.2) we have

$\displaystyle\frac{d}{dt}\int_{L_{t}}f\rho_{\mathbf{x}_{0},t_{0}}\,d\mathcal{H}^{2}$
$\displaystyle=\int_{L_{t}}(\partial_{t}f-\Delta
f)\rho_{\mathbf{x}_{0},t_{0}}\,d\mathcal{H}^{2}+\int_{L_{t}}f\left|\frac{\nu}{2}\right|^{2}\,\rho_{\mathbf{x}_{0},t_{0}}\,d\mathcal{H}^{2}$
$\displaystyle\qquad-\int_{L_{t}}f\,\left|\mathbf{H}-\frac{(\mathbf{x}-\mathbf{x}_{0})^{\perp}}{2(t_{0}-t)}+\frac{\nu}{2}\right|^{2}\,\rho_{\mathbf{x}_{0},t_{0}}\,d\mathcal{H}^{2}.$

The estimate of Ecker [7, Theorem 3.4] also applies to subsolutions of the
(drift) heat equation along the rescaled flow with a forcing term, with a
slightly modified function $p(t)$ as in [7, Theorem 3.2].

For simplicity we assume the first singular time is at $t=0$ and we are
studying the tangent flow at $\mathbf{0}\in\mathbb{R}^{N}$. The corresponding
rescaled flow $M_{\tau}$ is given by

$\frac{\partial}{\partial\tau}\mathbf{x}=\mathbf{H}+\frac{\mathbf{x}^{\perp}}{2}+\nu.$

Note that $M_{\tau}\subset e^{\tau/2}X\subset\mathbb{R}^{N}$, and the forcing
term $\nu$ is now given by
$\nu(\mathbf{x},\tau)=-\mathrm{tr}\,\Pi(\mathbf{x},\tau)|_{T_{\mathbf{x}}M_{\tau}}$,
where $\Pi(\mathbf{x},\tau)$ is the second fundamental form of $e^{\tau/2}X$.
In particular we have the estimate $|\nu|\leq Ce^{-\tau/2}$.

Let us write $J_{0},\Omega_{0},\omega_{0}$ for the complex structure,
holomorphic volume form and symplectic form on $T_{\mathbf{0}}X$. In
particular we identify $T_{\mathbf{0}}X=\mathbb{C}^{2}$, equipped with its
standard structures. Note that
$\lim_{\tau\to\infty}e^{\tau/2}X=T_{\mathbf{0}}X$ in the sense of $C^{\infty}$
convergence on compact sets, and so any tangent flow at $(\mathbf{0},0)$,
defined as a sequential limit of $M_{\tau}$ as $\tau\to\infty$, lives
naturally in $T_{\mathbf{0}}X$. As in Section 2.3 we consider special
Lagrangian unions $V=P_{1}\cup P_{2}$ contained in a neighbourhood
$\mathcal{V}$ of a given tangent flow $V_{0}$. For any $V\in\mathcal{V}$ there
is a hyperkähler rotation of the complex structure on $T_{\mathbf{0}}X$ such
that $V$ is given by $\\{zw=0\\}$ for complex coordinates $z,w$. Note that
$z,w$ can be viewed as linear functions on $\mathbb{R}^{N}$, and in particular
they define functions on $X\subset\mathbb{R}^{N}$. We have the following
analogous result to Lemma 3.2.

######  Lemma 9.2.

There is a constant $C>0$, depending on $X\subset\mathbb{R}^{N}$ and the
choice of $\mathcal{V}$, such that on any Lagrangian subspace $P\subset
T_{\mathbf{x}}X$ with Lagrangian angle $\theta$ we have

$|\nabla z\cdot\nabla w|\leq C(|\theta-\theta_{V}|+|\mathbf{x}|).$

###### Proof.

In a small neighbourhood $U$ over ${\mathbf{0}}\in X$ we can use a Darboux
chart to define a smooth projection map $\pi:U\to T_{\mathbf{0}}X$, such that
$\pi^{*}\omega_{0}=\omega$ and the derivative of $\pi$ is the identity map at
${\mathbf{0}}$. It follows that the complex structures and holomorphic volume
forms satisfy $|\pi^{*}J_{0}-J|,|\pi^{*}\Omega_{0}-\Omega|\leq C|\mathbf{x}|$
for a constant $C>0$. Similarly $|\pi^{*}z-z|,|\pi^{*}w-w|\leq C|\mathbf{x}|$,
and the derivatives of $z,w$ satisfy the same bounds. The required estimate on
the neighbourhood $U$ then follows by applying Lemma 3.2 to the image
$\pi(P)\subset T_{\mathbf{0}}X$. The estimate is clear outside of $U$ since
$|\mathbf{x}|$ is bounded away from zero on $X\setminus U$. ∎

We define the excess $\mathcal{A}(M_{\tau})$ for a Lagrangian in
$e^{\tau/2}X\subset\mathbb{R}^{N}$ as in (2.4), and Condition (‡ ‣ 2.3) is as
before. Note that the condition of uniformly bounded area ratios is automatic
in the compact case using the monotonicity formula, and the uniform bound for
the Lagrangian angle is also preserved by the maximum principle. Let us record
here the following consequence of the monotonicity formula, analogous to
(3.13), recalling that $|\nabla\theta|=|\mathbf{H}+\nu|$:

$\displaystyle\mathcal{A}(M_{\tau_{1}})-\mathcal{A}(M_{\tau_{2}})$
$\displaystyle\geq\int_{\tau_{1}}^{\tau_{2}}\int_{M_{\tau}}\left(2|\mathbf{H}+\nu|^{2}+\left|\mathbf{H}+\frac{\mathbf{x}^{\perp}}{2}+\frac{\nu}{2}\right|^{2}\right)e^{-|\mathbf{x}|/4}\,d\mathcal{H}^{2}\,d\tau$
$\displaystyle\quad-C\int_{\tau_{1}}^{\tau_{2}}\int_{M_{\tau}}|\nu|^{2}e^{-|\mathbf{x}|^{2}/4}\,d\mathcal{H}^{2}\,d\tau$
$\displaystyle\geq\frac{1}{16}\int_{\tau_{1}}^{\tau_{2}}\int_{M_{\tau}}(|\mathbf{H}|^{2}+|\mathbf{x}^{\perp}|^{2})\,d\mathcal{H}^{2}\,d\tau-
Ce^{-\tau_{1}},$

for $\tau_{1}<\tau_{2}$, where we also used the uniform bound on the
Lagrangian angle.

We define $I_{V}(M_{\tau})$ and $D_{V}(M_{\tau})$ according to Definition 3.4.
Note that now the function $|\mathbf{x}|d_{V}$ on $M_{\tau}$ is no longer
uniformly equivalent to $|zw|$. In fact if we denote the orthogonal projection
of $\mathbf{x}$ onto $T_{\mathbf{0}}X$ by $\tilde{\mathbf{x}}$, then
$|zw|(\mathbf{x})$ is uniformly equivalent to
$|\tilde{\mathbf{x}}|d_{V}(\tilde{\mathbf{x}})$. At the same time, since on
$X$ we have $|\mathbf{x}-\tilde{\mathbf{x}}|\leq C|\mathbf{x}|^{2}$, it
follows that on $M_{\tau}\subset e^{\tau/2}X$ we have

$|\mathbf{x}-\tilde{\mathbf{x}}|\leq Ce^{-\tau/2}|\mathbf{x}|^{2}.$

It follows from this, together with the bounds on the area ratios of
$M_{\tau}$, that

$C^{-1}\int_{M_{\tau}}|zw|^{2}e^{-|\mathbf{x}|^{2}/4}\leq\int_{M_{\tau}}|\mathbf{x}|^{2}d_{V}^{2}\,e^{-|\mathbf{x}|^{2}/4}\leq
C\left(\int_{M_{\tau}}|zw|^{2}e^{-\mathbf{x}^{2}/4}+e^{-\tau/2}\right).$

In particular as long as $I_{V}(M_{\tau})\geq e^{-\tau/2}$, we have that
$I_{V}(M_{\tau})$ is uniformly equivalent to the Gaussian $L^{2}$-norm of
$|zw|+|\theta-\theta_{V}|$. We have the following.

######  Lemma 9.3.

Suppose that $V\in\mathcal{V}$ and $I_{V}(M_{\tau})\geq e^{-\tau/2}$. Then the
conclusions (1), (2), (3) of Lemma 3.5 hold, with
$K_{\gamma}\subset\mathbb{R}^{N}\setminus\\{\mathbf{0}\\}$ in (3).

###### Proof.

In the proof of Lemma 3.5 we essentially applied the monotonicity formula to
the function $f=|zw|+|\theta-\theta_{V}|$. Here we claim that the same
argument works if instead we use the function

$f=|zw|+|\theta-\theta_{V}|+e^{-\tau/2}|\mathbf{x}|+e^{-\tau}.$

Here we are thinking of $\tau$ as being fixed. Consider the solution of the
mean curvature flow with forcing term as above with $L_{-1}=M_{\tau}$. Then
using Lemma 9.2, and the bound $|\nu|\leq Ce^{-\tau/2}$ for the forcing term,
implies that

$(\partial_{t}-\Delta)|zw|\leq C(|\theta-\theta_{V}|+e^{-\tau/2}|\mathbf{x}|)$

in the distributional sense. We also have

$\displaystyle(\partial_{t}-\Delta)|\theta-\theta_{V}|$ $\displaystyle\leq 0,$
$\displaystyle(\partial_{t}-\Delta)|\mathbf{x}|$ $\displaystyle\leq|\nu|\leq
Ce^{-\tau/2},$

and combining these we find that $(\partial_{t}-\Delta)f\leq Cf$ for a
constant $C>0$. At the same time if $I_{V}(M_{\tau})\geq e^{-\tau/2}$, then
$I_{V}(M_{\tau})$ is uniformly equivalent to the Gaussian $L^{2}$-norm of $f$.
Using this the arguments in the proof of Lemma 3.5 can be followed verbatim. ∎

Note that if $D_{V}(M_{\tau})=d\geq e^{-\tau/2}$ and $\tau$ is large, then the
region $|\mathbf{x}|<R_{d}$ (with $R_{d}$ defined in (4.8)), where we expect
good graphicality, is much smaller than the ball over which $e^{\tau/2}X$ is
graphical over the tangent space $T_{\mathbf{0}}X$. More precisely, we have
the following under a stronger assumption for $D_{V}(M_{\tau})$.

######  Lemma 9.4.

Suppose that $D_{V}(M_{\tau})=d\geq e^{-\tau/20}$ and $\tau$ is sufficiently
large. Then on the region $|\mathbf{x}|<R_{d}$, we can view $e^{\tau/2}X$ as
the graph of $v$ over $T_{\mathbf{0}}X$, where

(9.1) $|v|,|\nabla v|\leq Ce^{-\tau/2}R_{d}=Ce^{-\tau/2}|\ln d|\leq Cd^{9}.$

###### Proof.

Note that in a neighbourhood of $\mathbf{0}$, $X$ is the graph of a function
$V$ over $T_{\mathbf{0}}X$ with $|V|\leq C|\mathbf{x}|^{2}$ and $|\nabla
V|\leq C|\mathbf{x}|$. Rescaling this, and using that $R_{d}\ll e^{\tau/2}$
for large $\tau$, we find that on the ball $|\mathbf{x}|<R_{d}$, $e^{\tau/2}X$
is the graph of $v$ over $T_{\mathbf{0}}X$ such that $|v|,|\nabla v|\leq
Ce^{-\tau/2}R_{d}^{2}$. The required estimate follows from this. ∎

Using this, the results following Lemma 3.5 hold in the present context, with
minor adjustments of the proofs, as long as we always ensure that
$D_{V}(M_{\tau})\geq e^{-\tau/20}$, and that $\tau$ is sufficiently large. In
Condition ($\ast$ ‣ 2.3), to make sense of the condition
$\int_{\gamma}\lambda=0$ for loops $\gamma\in M_{\tau}\cap B_{1}$, we use a
Darboux chart for $\omega$ on $B_{1}\cap e^{\tau/2}X$ to define the Liouville
form $\lambda$. Assuming $\tau$ is large, such a chart exists as in the proof
of Lemma 9.2 above. Note that in a Darboux chart the holonomy of
$\nabla^{\xi}$ around a loop $\gamma$ is given by

$e^{-2\pi i\int_{D}\omega}=e^{-2\pi i\int_{\gamma}\lambda},$

where $\gamma=\partial D$. It follows that the rationality condition in
Definition 9.1 coincides with the rationality condition in Definition 2.2
restricted to loops contained in the chart.

Proposition 7.12 takes the following form.

######  Proposition 9.5.

Let $\delta_{2}>0$. There is an $\epsilon_{2}>0$, depending on $\delta_{2}$
such that if the flow $M_{\tau}$ satisfies Condition (‡ ‣ 2.3) for
$\tau\in[\tau_{0}-1,\tau_{0}+T+10]$ with $T>0$, $M_{\tau_{0}}$ satisfies
Condition ($\ast$ ‣ 2.3), $\tau_{0}$ is sufficiently large (depending on
$\delta_{2}$), and

  1. (1)

$\mathcal{A}_{\alpha}(M_{\tau_{0}})-\mathcal{A}_{\alpha}(M_{\tau_{0}+T})<\epsilon_{2}$,

  2. (2)

$D_{V_{0}}(M_{\tau_{0}})<\epsilon_{2}$,

then $D_{V_{0}}(M_{\tau_{0}+T})<\delta_{2}$.

Using this, the proofs of Theorems 1.1 and 1.2 from the Introduction follow
the same arguments as the corresponding results in Section 8. Let us finally
consider Theorem 1.3.

###### Proof of Theorem 1.3.

Using Theorem 8.3, which also holds in the current setting, we know that we
have a sequence $(\mathbf{x}_{k},t_{k})\to(\mathbf{0},0)$ and $r_{k}\to 0$
such that the rescalings $\tilde{L}_{k}=r_{k}^{-1}(L_{t_{k}}-\mathbf{x}_{k})$
converge, smoothly on compact sets, to a Lawlor neck $\tilde{L}_{\infty}$
asymptotic to $V_{0}$ at infinity. Here $V_{0}$ is the (unique) tangent flow
at the singularity $(\mathbf{0},0)$. Note that by Condition ($\ast$ ‣ 2.3),
this Lawlor neck is exact, and so up to scale there are only two
possibilities. Changing the scales $r_{k}$ if necessary, we can assume that in
terms of suitable coordinates $z,w$ for which $V_{0}=\\{zw=0\\}$, we have
$\tilde{L}_{\infty}=\\{zw=\pm 1\\}$. For sufficiently large $k$, we can remove
a large ball $B_{R}$ from $\tilde{L}_{k}$, and replace it with a small
Lagrangian perturbation of the two planes given by $V_{0}$. Hence
$\tilde{L}_{k}$ is an immersed Lagrangian where at the self-intersection point
at the origin two sheets of the Lagrangian are intersecting transversely. This
is exactly the reverse of the graded connected sum construction in [28,
Section 3.1] (potentially taking the Lagrangians to be two sheets of the same
Lagrangian in the connected sum). Hence $\tilde{L}_{k}$ is a graded self-
connected sum and so is $L_{t_{k}}$ as desired.

Suppose from now on that $\tilde{L}_{k}$ is not connected. Then we can write
$\tilde{L}_{k}=\tilde{M}_{1,k}\\#\tilde{M}_{2,k}$ as a graded connected sum.
The choice of which component is $\tilde{M}_{1,k}$, respectively
$\tilde{M}_{2,k}$, depends on which of the two possible Lawlor necks
$\tilde{L}_{\infty}$ is. Note that the Lagrangian angles of
$\tilde{M}_{1,k},\tilde{M}_{2,k}$ approach the constant $\theta_{V_{0}}$ on
the ball $B_{R}$ as we let $R,k\to\infty$ in the construction.

The upshot of this discussion is that after scaling back to the original flow,
we can write $L_{t_{k}}=M_{1,k}\\#M_{2,k}$ as claimed. It now remains for us
to show that (1.1) holds and, if $L$ is almost calibrated, that (1.2) holds.
Unless the initial Lagrangian $L$ is special Lagrangian, in which case no
singularity would form, we will have:

(9.2) $\displaystyle\inf_{L}\theta$
$\displaystyle<\inf_{M_{1,k}}\theta,\inf_{M_{2,k}}\theta;$
$\displaystyle\sup_{L}\theta$
$\displaystyle>\sup_{M_{1,k}}\theta,\sup_{M_{2,k}}\theta;$
$\displaystyle\mathrm{vol}(L)$
$\displaystyle>\mathrm{vol}(M_{1,k})+\mathrm{vol}(M_{2,k}),$

as long as $k$ is sufficiently large. It follows from the last inequality that

$\mathrm{vol}(L)>\left|\int_{M_{1,k}}\Omega\right|+\left|\int_{M_{2,k}}\Omega\right|,$

for sufficiently large $k$. We deduce that (1.1) holds.

Recall that in the almost calibrated case $\phi(M_{i,k})$ is uniquely defined
by

(9.3)
$\phi(M_{i,k})=\mathrm{arg}\,\int_{M_{i,k}}\Omega=\mathrm{arg}\,\int_{M_{i,k}}e^{i\theta}\,d\mathcal{H}^{2}.$

Using (9.2), this implies that

$\phi(M_{i,k})\in(\inf_{L}\theta,\sup_{L}\theta),$

for sufficiently large $k$. We thus conclude that (1.2) holds in the almost
calibrated case. ∎

## References

  * [1] T. Begley and K. Moore, _On short time existence of Lagrangian mean curvature flow_ , Math. Ann. 367 (2017), no. 3-4, 1473–1515. MR 3623231
  * [2] V. I. Bogachev, _Gaussian measures_ , Mathematical Surveys and Monographs, vol. 62, American Mathematical Society, Providence, RI, 1998. MR 1642391
  * [3] O. Chodosh and F. Schulze, _Uniqueness of asymptotically conical tangent flows_ , Duke Math. J. 170 (2021), no. 16, 3601–3657. MR 4332673
  * [4] T. H. Colding and W. P. Minicozzi, _Generic mean curvature flow I: generic singularities_ , Ann. of Math. (2) 175 (2012), no. 2, 755–833. MR 2993752
  * [5] by same author, _Uniqueness of blowups and Łojasiewicz inequalities_ , Ann. of Math. (2) 182 (2015), no. 1, 221–285. MR 3374960
  * [6] by same author, _Parabolic frequency on manifolds_ , Int. Math. Res. Not. (2021), to appear.
  * [7] K. Ecker, _Logarithmic Sobolev inequalities on submanifolds of Euclidean space_ , J. Reine Angew. Math. 522 (2000), 105–118. MR 1758578
  * [8] N. Edelen, _Degeneration of 7-dimensional minimal hypersurfaces which are stable or have bounded index_ , arXiv:2103.13563.
  * [9] K. Fukaya, _Galois symmetry on Floer cohomology_ , Turkish J. Math. 27 (2003), no. 1, 11–32. MR 1975330
  * [10] G. Huisken, _Asymptotic behavior for singularities of the mean curvature flow_ , J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
  * [11] T. Ilmanen, A. Neves, and F. Schulze, _On short time existence for the planar network flow_ , J. Differential Geom. 111 (2019), no. 1, 39–89. MR 3909904
  * [12] D. Joyce, _Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow_ , EMS Surv. Math. Sci. 2 (2015), no. 1, 1–62. MR 3354954
  * [13] B. Lambert, J. D. Lotay, and F. Schulze, _Ancient solutions in Lagrangian mean curvature flow_ , Ann. Sc. Norm. Super. Pisa Cl. Sci. XXII (2021), 1169–1205. MR 4334316
  * [14] J. D. Lotay, F. Schulze, and G. Székelyhidi, _Ancient solutions and translators of Lagrangian mean curvature flow_ , arXiv:2204.13836.
  * [15] A. Neves, _Singularities of Lagrangian mean curvature flow: zero-Maslov class case_ , Invent. Math. 168 (2007), no. 3, 449–484. MR 2299559
  * [16] by same author, _Recent progress on singularities of Lagrangian mean curvature flow_ , Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM), vol. 20, Int. Press, Somerville, MA, 2011, pp. 413–438. MR 2906935
  * [17] by same author, _Finite time singularities for Lagrangian mean curvature flow_ , Ann. of Math. (2) 177 (2013), no. 3, 1029–1076. MR 3034293
  * [18] A. Neves and G. Tian, _Translating solutions to Lagrangian mean curvature flow_ , Trans. Amer. Math. Soc. 365 (2013), no. 11, 5655–5680. MR 3091260
  * [19] R. Schoen and J. Wolfson, _Minimizing area among Lagrangian surfaces: the mapping problem_ , J. Differential Geom. 58 (2001), no. 1, 1–86. MR 1895348
  * [20] F. Schulze, _Uniqueness of compact tangent flows in mean curvature flow_ , J. Reine Angew. Math. 690 (2014), 163–172. MR 3200339
  * [21] P. Seidel, _Lagrangian two-spheres can be symplectically knotted_ , J. Differential Geom. 52 (1999), no. 1, 145–171. MR 1743463
  * [22] L. Simon, _Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems_ , Ann. of Math. (2) 118 (1983), no. 3, 525–571. MR 727703
  * [23] by same author, _Isolated singularities for extrema of geometric variational problems_ , Miniconference on nonlinear analysis (Canberra, 1984), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 8, Austral. Nat. Univ., Canberra, 1984, pp. 46–50. MR 799211
  * [24] K. Smoczyk, _A canonical way to deform a lagrangian submanifold_ , arXiv:dg-ga/9605005.
  * [25] A. Strominger, S.-T. Yau, and E. Zaslow, _Mirror symmetry is $T$-duality_, Nuclear Phys. B 479 (1996), no. 1-2, 243–259. MR 1429831
  * [26] G. Székelyhidi, _Uniqueness of certain cylindrical tangent cones_ , arXiv:2012.02065.
  * [27] R. P. Thomas, _Moment maps, monodromy and mirror manifolds_ , Symplectic geometry and mirror symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001, pp. 467–498. MR 1882337
  * [28] R. P. Thomas and S.-T. Yau, _Special Lagrangians, stable bundles and mean curvature flow_ , Comm. Anal. Geom. 10 (2002), no. 5, 1075–1113. MR 1957663
  * [29] M.-T. Wang, _The mean curvature flow smoothes Lipschitz submanifolds_ , Comm. Anal. Geom. 12 (2004), no. 3, 581–599. MR 2128604
  * [30] B. White, _A local regularity theorem for mean curvature flow_ , Ann. of Math. (2) 161 (2005), no. 3, 1487–1519. MR 2180405


11institutetext: University of Manchester, Manchester M15 6PB, United Kingdom
22institutetext: Sonos Experience Limited, London SE1 3ER, United Kingdom

# SonOpt: Sonifying Bi-objective Population-Based Optimization Algorithms
††thanks: This work was supported by the Engineering and Physical Sciences
Research Council [grant number 2492452].

Tasos Asonitis 11 0000-0003-2413-8752 Richard Allmendinger 11
0000-0003-1236-3143 Matt Benatan 22 0000-0002-8611-3596 Ricardo Climent 11
0000-0002-0484-8055

###### Abstract

We propose SonOpt, the first (open source) data sonification application for
monitoring the progress of bi-objective population-based optimization
algorithms during search, to facilitate algorithm understanding. SonOpt
provides insights into convergence/stagnation of search, the evolution of the
approximation set shape, location of recurring points in the approximation
set, and population diversity. The benefits of data sonification have been
shown for various non-optimization related monitoring tasks. However, very few
attempts have been made in the context of optimization and their focus has
been exclusively on single-objective problems. In comparison, SonOpt is
designed for bi-objective optimization problems, relies on objective function
values of non-dominated solutions only, and is designed with the user
(listener) in mind; avoiding convolution of multiple sounds and prioritising
ease of familiarizing with the system. This is achieved using two sonification
paths relying on the concepts of wavetable and additive synthesis.

This paper motivates and describes the architecture of SonOpt, and then
validates SonOpt for two popular multi-objective optimization algorithms
(NSGA-II and MOEA/D). Experience SonOpt yourself via
https://github.com/tasos-a/SonOpt-1.0.

###### Keywords:

sonification optimization algorithmic behaviour metaheuristics evolutionary
computation process monitoring SonOpt.

##  1 Introduction

Sonification deals with the representation and communication of data through
sound and has been an established practice for more than three decades.
Blurring the lines between art and science, sonification has long been an area
of interest for both musicians and scientists. The former see in it the
opportunity to explore the aesthetic content of phenomena that surround us,
while the latter tend to see it as a tool that can represent data efficiently
and reveal hidden patterns in an intuitive way [9]. Field experts tend to use
sonification in tandem with visualization techniques [24], in order to
maximize the benefits of both approaches. However, sonification has proven
useful as a standalone tool as well, particularly when the temporal evolution
of data is an important factor of the process in question [5].

This highlights the type of operations in which sonification really shines.
For spatial representations, visualization methods generally tend to provide a
concise picture. As discussed in [28], it is not uncommon that these methods
might be proven less effective when it comes to time-series data
representation. The evolution of an optimization algorithm is a process over
time, and, if characterized in terms of metrics, can be thought of as a
multivariate time-series. Gaining a better understanding about the behaviour
of an optimization algorithm during evolution is important to study strengths
and weaknesses of algorithms, and ultimately develop more efficient
algorithms. While this is true for any type of algorithm, the focus of our
work is on population-based multi-objective optimization algorithms (MOAs) due
to the common scenario of having multiple conflicting objectives in practice
[21]. The community has looked to theory (e.g. runtime analysis [18]) and
visualisation-based concepts (e.g. [32]), including the design of visualisable
test problems [8], to improve algorithm understanding, but there is very
little work that explores the application of sonification for this task. The
goal of our work is to fill this gap, and this paper is the first attempt to
sonify MOAs.

As algorithm designers and/or practitioners, we tend to evaluate the
performance of optimization algorithms over the course of a few runs. As
explained in [10], sonification allows us to zoom in on an individual run of
the optimization algorithm, and reveal particular characteristics that can
help us validate the algorithmic performance.

Only a few approaches have been made for monitoring of optimization algorithms
using sound. These will be presented briefly in the following section. Despite
the existence of different methodologies, none of the currently documented
implementations has focused on the evolving shape of the approximation set
(objective function values of non-dominated solutions111A non-dominated
solution is one which provides a suitable compromise between all objectives
without degrading any of them. For a more formal definition of multi-objective
concepts please refer to e.g. [21].) and the recurrence of points in the
objective space. SonOpt receives as input the approximation set and maps the
shape of this set to the shape of a single-cycle waveform inside a buffer.
This buffer is used as a look-up table for a ramp oscillator that scans the
content of the buffer repeatedly, at the rate of a user-controlled frequency.
In addition, SonOpt monitors the recurrence of objective values across
consecutive generations and maps the amount of recurrence of each value to the
level of a different harmonic partial of a user-defined fundamental frequency.
A more detailed description of the system follows on Section 3.

The motivation behind SonOpt is to engage with the discussion on the benefits
of auditory display in optimization monitoring, encourage the thinking of
algorithmic behaviour in sonic terms, and offer a tool that can compliment
current visualization techniques towards an intuitive multi-modal
understanding of complex processes. Although not explored in this paper, an
additional aspiration behind SonOpt is to explore optimization processes from
an aesthetic point of view and suggest the treatment of such processes as
music composition tools. The structure of this paper is divided into four
sections. First we put our work in context with related literature focusing on
areas such as sonification of optimization algorithms and benefits of
sonification in process monitoring. Next we present the platform of SonOpt by
describing the pipeline and discussing the reasons behind specific design
choices. The documentation of test results is included in Section 4. Finally,
Section 5 presents some concluding remarks and future plans.

##  2 Related work

The first documented approach that employed sound to provide insights in
evolutionary optimization was proposed by Grond et al. [10] in 2011. The
presented work involves the use of stacked band-pass filters to transform the
sounds of keys on a computer keyboard. Grond et al. highlight the crucial role
of auditory display in monitoring operations and explain how these advantages
pertain to optimization processes. Their system focuses on evolutionary
strategies (ES) as presented in [26]. However, the developed methodology
cannot be applied to optimization tasks with more than one objective.

Tavares and Godoy [31] have used sound to represent population dynamics in a
Particle Swarm Optimization (PSO) algorithm [15]. Although PSO has been
extensively employed as a music composition tool (for example [3]) through the
sonification of particle trajectories, Tavares and Godoy follow a different
approach by sonifying specific performance metrics. Their focus is on
providing information about the population behaviour — including speed,
alignment and diversity of particles within a population (or swarm) — with
particular attention to creating an aesthetically pleasing output. The sonic
mappings implemented include musical notes, echo and harmonics and aim to
result in evolving soundscapes. As in [10], the focus is on single-objective
optimization.

Lutton et al. [19] apply sonification in island-based parallel setups where
the optimization task has been distributed across systems to facilitate
computational efficiency. Each computational node is assigned a part of a
musical score, or a familiar song in audio format. The degree to which the
music is reproduced faithfully — including performance errors or glitches,
depending on whether the user has opted for a musical score or audio file —
provides information about the optimization progress of individual nodes. The
implemented system is targeted exclusively at parallel setups. A potential
question relates to whether the correct interpretation of the introduced
errors requires prior musical training, and if that could hinder the
effectiveness of the designed sonification, as indicated in [22].

Albeit not specifically related to optimization, a detailed account of
sonification including theory, technology, applications and benefits, has been
made by Hermann at al. [11]. Vickers [33] focuses on the benefits of
sonification for peripheral process monitoring, and includes a list of systems
that employed sound to monitor the running process of various programs, mainly
through the triggering of sounds based on specific events. The applications
mentioned in [33] do not target optimization processes. They are primarily
focused on debugging run-time behaviour and most of them relate to dated
programming methods. However, the variety in approaches shows that using sound
to monitor algorithmic processes has been drawing attention for a long time.
According to Vickers, peripheral process monitoring is when useful information
about a process is being aurally displayed on the background, while a
different, primary task, usually visual, is at the center of focus. SonOpt is
designed precisely with this in mind. By using two separate audio streams, one
that acts as a continuous sound stream and one that is triggered by events,
SonOpt allows the user to focus on a different task while indirectly
monitoring the progress of the optimization algorithm. As SonOpt does not
aspire to replace visualization methods, but to work in tandem with them,
users are encouraged to engage in multi-modal monitoring by combining
visualization of various optimization aspects, with the sonification provided
by SonOpt. Hildebrandt et. al [12] carried out an experiment in which
participants performed significantly better at a peripheral process monitoring
task when continuous sonification and visual cues of the process in question
were used in parallel. In particular, as shown in [24], field experts
demonstrate preference towards the combination of continuous sonification and
visual displays when it comes to process monitoring, instead of visual or
audio only conditions. The experimental work of Axon et. al. [2] also points
towards benefits of adding sonification to visual representations,
specifically targeted in security related process monitoring. In [13]
additional evidence on performance improvement in process monitoring (of
physical production systems) via sonification is provided.

The term sonification is used throughout this text because it is a recognized
and all-encompassing term that relates to auditory display. However the
methodology used in SonOpt further pertains to a subbranch of sonification,
referred to in the dedicated literature as audification. Kramer [16] writes
that audification is “the process of directly playing back the data samples”.
As elaborated by Dombois and Eckel [7], audification can provide a successful
choice of auditory display when datasets feature specific properties. The
requirements include: the size of the dataset, whether the array of data
samples creates a wave-like signal, the complexity of the signal, and the
potential existence of subtle changes. The approximation set obtained by a
MOA, which functions as the input to SonOpt, fulfills these properties. As
will be evidenced in the next section, audification is applied on the first of
the two sound generation paths of SonOpt.

To put this work into context with the wider literature, it is worthwhile
mentioning a few more related research streams. In particular, the premises
and aims in the work of Ochoa, et al. [23] on Search Trajectory Networks
(STNs), albeit focused on the visual domain through the use of graphs, share a
common ground with the motivation behind SonOpt. Both projects aim to assist
with the understanding of algorithmic behaviour. The results of STNs are
particularly interesting from an aesthetic point of view, showing that care
has been taken on that aspect as well as improving understanding of search
behaviour.

Schuller et al. [29] argue that sonification is a promising and user-friendly
alternative to visualization, towards explainable Artificial Intelligence
(XAI). Lyu et al. [20] apply an interactive virtual environment that allows
real time tweaking of a neural network’s hyperparameters by providing users
with auditory feedback on the impact of their choices. Finally, [1] highlights
the shortcomings of visual representations for non-trained and visually
impaired individuals, and suggest sonification as one of the prevalent
alternatives.

##  3 Methodology and system overview

SonOpt is an application developed using the visual programming environment
Max/MSP [25]. Max/MSP has been the choice of musicians, sonic artists and
engineers, for developing audio-related programs which benefit from a certain
degree of customization. We chose Max/MSP for the immediacy of design, and its
popularity among sonification researchers. This section will describe the
system and discuss some of the design decisions made during its development.

SonOpt can work with any population-based search algorithm. This is possible
as it only needs to receive the objective function values of non-dominated
solutions at each generation of the algorithm (e.g. in the case of steady
state algorithms); for ease of presentation, we will use the term generation
knowing that the concepts can also be applied to non-generational (e.g. steady
state) algorithms. Each generation produces a collection of objective values
that are passed to SonOpt through the use of the Open Sound Control protocol
(OSC) [34]. This is to ensure quick and precise communication between the
algorithm and Max/MSP. The algorithms and the optimization problems presented
in this paper come from pymoo [4], a Python library tailored to MOAs. Pymoo
was chosen as it allows for simple swapping between optimization algorithms
and problems, facilitating direct comparisons of a variety of setups. Another
key motivating factor is pymoo’s popularity within the optimization community:
having a familiar tool at the center of the work will make it easier for
optimization researchers to engage with SonOpt. Of course, users can replace
pymoo with their own code, for example, of customized algorithms.

At the moment, SonOpt is set up to work for bi-objective optimization problems
only; WOLOG it is assumed that both objectives are to be minimized. Since
SonOpt’s sonification relies on the objective function values only (at least
for now), a 2-dimensional matrix is expected to be passed on from pymoo to Max
at each generation to represent the approximation set at that generation. Once
received, Max scales the objective values to be within [0,1], and then sorts
the matrix, from the minimum all the way to the maximum value of objective one
— hence from the max to the min value of objective two. This produces an
ordered sequence of points comprising the 2D approximation set. Subsequently,
the set is forwarded to two separate sonification paths (Fig. 1).

Figure 1: The pipeline of SonOpt.

Sonification path 1 The first path is calculating a straight line from the
point that minimizes objective one to the point that minimizes objective two
(red dashed line in the left plot of Fig. 2). It then calculates the distance
of each point of the set from this line. The distances are then scaled and
mapped to the sample values of a buffer. The buffer needs to be at least $2N$
samples big, where $N$ is the number of maximum points in the approximation
set obtained per generation. The content of the buffer then looks like a
horizontal display of the approximation set (right plot of Fig. 2). In order
to generate a bipolar audio signal, the incoming values are converted to
negatives and are added to the buffer content. This buffer is used as a look-
up table of a wavetable oscillator [14] that scans the buffer content at the
rate of a user-selected frequency. This way the shape of the approximation set
is treated like a single-cycle waveform, which gets repeated periodically. The
resulting sound is a direct translation of the shape of the front, and it
evolves according to the update of the shape at each generation. It is worth
noting here that although the buffer size is fixed, the size of the readable
buffer content is not; it depends on the number of objective values the
algorithm is generating during the current generation. So even if the buffer
looks like it has leftover content from previous generations where the number
of values was bigger than the current one, the wavetable oscillator only reads
the part of the buffer that includes the content generated during the current
generation. If, for example, in generation 46 the algorithm generates 90
values, the buffer content will be 180 samples long (90 points in the set
converted to sample values plus the negative duplicates of these values). Now,
if in generation 47 the algorithm generates 70 values, this means that the
buffer content will still be 180 samples long, however only the first 140 of
them will contribute to the sonic output, since 40 samples of the buffer
content were left over from the previous generation.

Figure 2: Sonification path 1: mapping of the (scaled) approximation set shape
to sample values. Algorithm: NSGA-II. Optimization task: ZDT1.

This approach offers a sonic portrayal of the approximation set.
Characteristics like discontinuity of the front, lack of diversity and
unsuccessful convergence translate to harsher, more harmonically rich sounds,
while convex, concave and generally continuous fronts result in more well-
rounded, “bassy” tones that approach the sonic characteristics of a sine wave.
The loudness of the sound depends on the curvature of the front: the curvier
the front, the louder the sound and vice versa. Detailed examples follow on
the next section.

Sonification path 2  The second path examines the recurrence of objective
function values across successive generations. This is done by comparing the
values within the received approximation set of the current generation with
those within the set of the previous generation. If any recurrence is spotted,
SonOpt finds the index position of the recurrent value within the set of the
current generation (as aforementioned, the operations are taking place on
sorted approximation sets, see Fig. 1). The recurrent indices are then mapped
to the partials of a user-defined fundamental frequency. The max number of
available partials is equal to the max number of objective function values on
a single generation. Depending on the position indices of the recurrent
solutions across the set, the amplitude of the corresponding partials is
increased. A recurrence on the first point of the approximation set (this
would translate on the point located on the upper left corner of the front)
would result in a raise in the amplitude of the first partial and so on. When
there is no recurrence, this path does not generate sound.

Figure 3: Sonification path 2: mapping of the recurrent solutions to harmonic
partials. Algorithm: NSGA-II. Optimization task: Tanaka [30]. The amplitude of
each partial, shown by the height of the corresponding red bar, shows the
recurrence amount of that objective value: higher amplitude means more
successive generations in which the value showed up in the approximation set.

Consequently, the timbre of the resulting sound depends on the location of the
recurrent values across the approximation set. When recurrence is taking place
across the entire set, the outcome is rich spectrum, with many harmonic
partials contributing to the sonic result, akin to the principles of additive
synthesis [27]. When recurrence is happening on a single value, or is
concentrated on a particular area of the set, this results in more focused
spectrum, involving low, mid or high range frequency components, depending on
whether the recurrent area of the front is on the high left corner, middle, or
low right corner respectively. The sonic outcome thus provides information not
just on when recurrence is happening, but also on which part of the
approximation set it is taking place. When dealing with high number of
objective function values, sound can provide such information in a quicker and
more intuitive way than visual representations.

The two sonification paths usually produce opposite results as the
optimization progresses. Sonification path 1 tends to produce harsher sounds
with quick-shifting timbre in the beginning of the run and then gradually
settle on smoother tones with less harmonics, depending, of course, on the
shape of the approximation set. Sonification path 2 usually starts with simple
tones and tends to generate more complex sounds after sonification path 1 has
reached a static timbre, i.e. the algorithm has approximated the objective
space shape to some extent. This means that generally, sound intensity
progressively decays in path 1, while it progressively intensifies in path 2.
SonOpt was designed this way in order to prevent overlapping conveying of
information that can eventually become overwhelming. Furthermore, an option is
given to the users to change the volume of each path individually. SonOpt is
geared towards sound synthesis, as it generates sound according to the input
data. An additional reason behind this aesthetic choice is that we wanted to
minimize the need for a musical background by the user. By using raw sound,
one can make observations on the evolution of data intuitively, by focusing on
accessible sonic qualities like loudness, harshness, low versus high
frequencies etc. without needing musical training.

##  4 Experimental study

In this section we demonstrate the application of SonOpt in multi-objective
optimization. A manual on how to use SonOpt and to familiarize the user with
the range of sounds and their interpretation (i.e. optimization scenarios
portrayed) will be released in due course.

###  4.1 Experimental setup

MOA settings  For the presented tests, two well-known population-based MOAs
were chosen: NSGA-II (Non-dominated Sorting Genetic Algorithm) [6] and MOEA/D
(Multi-Objective Evolutionary Algorithm based on Decomposition) [35]. The
reason for selecting these two algorithms is their popularity, differences in
working principles — Pareto-dominance vs decomposition-based —, and their
availability in pymoo.222pymoo does not feature an indicator-based MOA, which
we would have like to experiment with too. As stated before, the choice of
algorithm can include any population-based MOA. It is important to note that
our goal here is not to discuss why one algorithm performs better than another
and/or determine the best algorithm for a given problem; instead we want to
demonstrate the responsiveness of SonOpt to different algorithmic behaviours
and discuss insights SonOpt can provide. Consequently, we use default
operators and parameter settings for our two MOAs as used in pymoo; the only
parameters that need setting was the population size of NSGA-II and the number
of reference vectors in MOEA/D; both parameters were set to 100 for ease of
demonstration of SonOpt’s capabilities. The algorithms were run on several bi-
objective test problems available in pymoo. Here we show and discuss results
obtained for ZDT1, ZDT4 [36], and Kursawe [17] as they demonstrate SonOpt
functionalities well. Importantly, video recordings of SonOpt in action can be
watched at https://tinyurl.com/sonopt including also for additional problems.

Sonification settings  Table 1 shows the parameters used by SonOpt for the
experimental study. The setting of these parameters was determined in
preliminary experimentation and driven by clarity of the sonic outputs. Sample
value scaling only affects sonification path 1 and determines how the input
values are scaled in order to be converted to audible audio sample values.
This greatly depends on the overall values of the approximation set. If the
set includes values very close to 0, then the sample value scaling needs to be
higher and vice versa. The parameter can be updated dynamically, so the user
can increase or reduce the amplitude of the resulting waveform during the run,
as desired. The sample value scaling has been kept steady for the examples
presented in the images below. However, in the video demonstrations, the
scaling value is updated when the sample values are very small, in order to
make the sound characteristics of the approximation sets audible. This is
happening in cases where the curvature of the set is very small. The value
change has been documented. Buffer size sets the size of the buffer in
samples. For both sonification paths this needs to be at least $2N$ samples,
where $N$ is the number of maximum points in the approximation set obtained
per generation. Number of sliders applies only to the second path. It refers
to the number of harmonic partials that the generated sound can have. It needs
to be set to, at least, $N$. Oscillator frequency sets the frequency of the
scanning oscillator in path 1 and the fundamental frequency of the harmonic
oscillator in path 2. Overall amplitude refers to the sound volume of path 1
and 2. We suggest starting with a low value and gradually increasing it, since
sound bursts can be expected particularly in path 2. Finally, number of
instances controls the way Max/MSP deals with polyphony. This also needs to be
set at least to $N$.

Media to gain insights into algorithmic behaviours  To facilitate algorithmic
understanding, SonOpt outputs sound, as an algorithm traverses through the
search space (see Section 3). We convey this to the readers via screenshots of
accompanying visuals and recorded videos of SonOpt in action. It is important
to mention that in the provided experiments the algorithms ran twice, once to
generate the images and once to generate the video documentation, therefore
variations in the optimization progress between the two might occur
occasionally. Needless to say, the best way of experiencing SonOpt is by
actually using it!

Table 1: SonOpt parameters as used in the experimental study. SonOpt parameters | Sonification path 1 | Sonification path 2
---|---|---
Sample value scaling | 500 | n/a
Buffer size | 202 | 256
Numbers of sliders | n/a | 100
Oscillator frequency | 80 Hz | 80 Hz
Overall amplitude | 0.3 | 0.075
Number of instances | 100 | 100

###  4.2 Experimental analysis

SonOpt on a convex bi-objective problem  Figs. 4 and 5 show the two MOAs
applied to ZDT1. This is a good example of how sonification path 1 can provide
insights into a run of different algorithms. In the case of MOEA/D,
sonification path 1 generates a sound with noticeable harmonic content for
approximately half the duration of the run. This means it did not manage to
converge as quickly and effectively compared to NSGA-II, which produced a
simpler, more “punchy” sound, approaching that of a sinetone. This is
supported by looking at the obtained fronts. Both algorithms reach the convex
shape, but MOEA/D presents sporadic “bumps” and a single solution that seems
to be slightly distant from the rest of the set. This affects the resulted
waveform of sonification path 1, creating a buzzier tone in comparison with
NSGA-II. ZDT1 produces excessive objective value recurrence in both
algorithms. Indeed, path 2 generates a waveform with gradually increasing
harmonic content, as can be witnessed in Fig. 4 (d). In the case of NSGA-II,
this reflects the increasing number of non-dominated solutions obtained from
generation 1 through, approximately, generation 75.

Figure 4: Algorithm: NSGA-II. Optimization task: ZDT1. (a): approximation set
obtained after 250 generations. (b): sonogram of sonification path 1 across
250 generations. (c): buffer contains on the 250th generation. (d): sonogram
of sonification path 2 across 250 generations. Figure 5: Algorithm: MOEA/D.
Optimization task: ZDT1. (a),(b),(c) and (d) as above

SonOpt on a discontinuous bi-objective problem In Figs. 6 and 7 we can see how
SonOpt responds to NSGA-II and MOEA/D when applied to Kursawe. In this case,
sonification path 1 not only provides information regarding the optimization
progress, but also picks up the different behaviour of the two algorithms. For
NSGA-II, path 1 generates a “buzzy” waveform with a quickly reduced harmonic
content during the first 15 to 20 generations. For MOEA/D, however, path 1
produces a more “jagged” sound, with discrete, abrupt changes in the harmonic
content. This reveals that the approximation set evolved in a similar, almost
step-like manner until it reached the Pareto front shape. This type of
behaviour is expected since NSGA-II is based on non-dominated value sorting
while MOEA/D is based on decomposition. Path 2 provides additional insight.
For both algorithms, path 2 produces a granular sound during the first 20 to
30 generations. This sound gets progressively substituted by a more continuous
waveform, where individual tones are less discernible (see, for example, Fig.
7 (d) where the fractured harmonics in the beginning turn gradually into
longer streams). This is attributed to the optimization progress. As the
approximation set improves, the objective values repeat successively for more
generations, a possible sign that the algorithm has converged, or got stuck at
a local optimum. In Fig. 6 (e) we added a sonogram of path 2 after applying a
filter to the recurrent values of each generation that leaves only a $10\%$ of
these values come through to path 2. This is provided as a demonstration of
how path 2 sounds like when the objective value recurrence is less than in the
problems/algorithms we examine here. For example, this could apply to
algorithms that use customized diversity-maintenance mechanisms.

Figure 6: Algorithm: NSGA-II. Optimization task: Kursawe. (a), (b), (c), (d)
as above. (e): sonogram of sonification path 2, with the objective value
recurrence filtered to $10\%$.  Figure 7: Algorithm: MOEA/D. Optimization
task: Kursawe. Figure 8: Algorithm: NSGA-II. Optimization task: ZDT4. Figure
9: Algorithm: MOEA/D. Optimization task: ZDT4.

SonOpt on a convex multimodal bi-objective problem  An effective example of
paths 1 and 2 can be observed when applying NSGA-II to ZDT4 (Fig. 8). The
algorithm manages to approach the Pareto front shape halfway through the
optimization. Until that point the shape of the approximation set is coming
together and falling apart, as can be witnessed in the video. The sound of
path 1 transitions through waveforms with strong harmonic content and high
amplitude (sound “bursts”), since the produced curvatures are steep. As NSGA-
II manages to approximate the front, the harmonics and overall loudness are
reduced, and the simpler sonic character of a sinetone emerges. Path 2
transitions from mid-range tones to tones with broader harmonic content, as
can also be witnessed from the slopes in Fig. 8 (d). This is due to ZDT4
presenting many local optima, posing difficulties during the optimization.
SonOpt is able to pick up this information and convey it through sound. The
displayed run of MOEA/D on ZDT4 produced an outlier on the upper part of the
front (Fig. 9 (a)). SonOpt is very sensitive to this type of behaviour. This
can be noticed by comparing the obtained sound of MOEA/D and NSGA-II after 250
generations. The outlier in MOEA/D affects significantly the shape of the
waveform, producing a sound close to a “sawtooth” wave.

Final remarks The presented examples confirm that SonOpt can provide
information on the shape of the approximation set even when visualisation is
not available. This property is of use in practice, for example, for SonOpt
users with impaired sight or when auditioning the optimization progress while
working on a different task (multi-tasking). A quick-shifting timbre in
sonification path 1 implies that the algorithm mostly performs exploration,
while a more static timbre means the algorithm performs exploitation. Once the
timbre has stabilized, the sonic qualities of path 1 can provide information
on the shape of the front: harsh, buzzy sounds suggest a discontinuous
approximation set (and thus potentially Pareto front), and softer, bassy tones
allude to a convex or concave Pareto front, with loudness encapsulating the
level of curvature. The characteristics of path 2 inform on possible
stagnation or successful convergence. It has been stated before that the two
paths work complimentary to each other and more solid conclusion can be drawn
when the output of both is taken into consideration.

##  5 Conclusion and future work

This paper has proposed SonOpt, an (open-source) application to sonify the
search process of population-based optimization algorithms for bi-objective
problems. SonOpt is implemented in Max/MSP and draws live information about
the search process of an algorithm (encoded in pymoo in this study) to create
two parallel sound streams indicating the evolving shape of the approximation
set and the recurrence of points in the objective space. Using two popular
multi-objective algorithms (NSGA-II and MOEA/D) and a range of bi-objective
test problems, we demonstrated that SonOpt can pick up several aspects in
terms of search behaviour including convergence/stagnation, shape of the
approximation set, location of recurring points in the approximation set, and
population diversity. Hence, SonOpt can facilitate algorithm understanding.

It is important to note that SonOpt is a first attempt to sonify multi-
objective search so there is plenty of scope to improve the system. For
instance, future work will look at extending SonOpt to problems with more than
two objectives. In particular, the community could do with a better
understanding of algorithms when applied to many-objective optimization
problems. Further avenues of future work include a Python-based version of
SonOpt (as opposed to using Max/MSP) to improve usability and increase uptake;
incorporation of different/additional sonification paths (e.g. hypervolume
contributions of solutions); sonifying behavioural differences across multiple
algorithms simultaneously; and enhancing the perceptibility of SonOpt’s output
to make SonOpt even more accessible.

Although not elaborated within this paper, the creation of SonOpt was also
motivated by artistic intentions. Optimization and, generally, algorithmic
processes that evolve temporally, lend themselves fittingly in music
compositions. They can shape sonic narratives and musical structures in
various levels, and so motivate artists to examine optimization processes from
an aesthetic perspective. We believe SonOpt points towards this direction.
Future work therefore includes extending the sonic palette of SonOpt and
assessing its creative performance through artistic engagement with the tool.

## References

  * [1] Ali, S., Muralidharan, L., Alfieri, F., Agrawal, M. & Jorgensen, J. Sonify: Making Visual Graphs Accessible. Proceedings Of The 1st International Conference On Human Interaction And Emerging Technologies. 1018 pp. 454-459 (2019)
  * [2] Axon, L., AlAhmadi, B., Nurse, J., Goldsmith, M. & Creese, S. Data Presentation in Security Operations Centres: Exploring the Potential for Sonification to Enhance Existing Practice. Journal Of Cybersecurity. 6:1-16 (2020)
  * [3] Blackwell, T. & Young, M. Self-organised Music. Organised Sound. 9(2):123-136 (2004)
  * [4] Blank, J. & Deb, K. pymoo: Multi-Objective Optimization in Python. IEEE Access. 8:89497-89509 (2020)
  * [5] De Campo, A. Toward a Data Sonification Design Space Map. 13th Internation Conference in Auditory Display, pp. 342-347 (2007)
  * [6] Deb, K., Pratap, A., Agarwal, S. & Meyarivan, T. A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Transactions On Evolutionary Computation. 6(2):182-197 (2002)
  * [7] Dombois, F., Eckel, G.: Audification. In: Hermann, T., Hunt, A., Neuhoff, J. G., (eds.) The Sonification Handbook, pp. 301-324, Logos Publishing House, Berlin, Germany (2011)
  * [8] Fieldsend, J., Chugh, T., Allmendinger, R. & Miettinen, K. A Visualizable Test Problem Generator for Many-Objective Optimization. IEEE Transactions On Evolutionary Computation, 26(1):1-11 (2022)
  * [9] Gresham-Lancaster, S. Relationships of Sonification to Music and Sound Art. AI & Society. 27(2):207-212 (2012)
  * [10] Grond, F., Kramer, O. & Hermann, T. Interactive Sonification Monitoring in Evolutionary Optimization. 17th International Conference on Auditory Display, pp. 166 (2011)
  * [11] Hermann, T., Hunt, A. & Neuhoff, J. The Sonification Handbook. Logos Verlag, Berlin, Germany (2011)
  * [12] Hildebrandt, T., Hermann, T. & Rinderle-Ma, S. Continuous Sonification Enhances Adequacy of Interactions in Peripheral Process Monitoring. International Journal Of Human-Computer Studies. 95:54-65 (2016)
  * [13] Iber, M., Lechner, P., Jandl, C., Mader, M. & Reichmann, M. Auditory Augmented Process Monitoring for Cyber Physical Production Systems. Personal And Ubiquitous Computing. 25(4):691-704 (2021)
  * [14] Johnson, R. Wavetable Synthesis 101, a Fundamental Perspective. In Audio Engineering Society Convention, pp. 1-27 (1996)
  * [15] Kennedy, J. & Eberhart, R. Particle Swarm Optimization. Proceedings Of International Conference On Neural Networks, 4 pp. 1942-1948 (1995)
  * [16] Kramer, G. Auditory display: Sonification, Audification and Auditory Interfaces. Addison-Wesley Longman Publishing Co., Inc. (2000)
  * [17] Kursawe, F. A variant of evolution strategies for vector optimization. International Conference On Parallel Problem Solving From Nature. pp. 193-197 (1990)
  * [18] Laumanns, M., Thiele, L. & Zitzler, E. Running Time Analysis of Multiobjective Evolutionary Algorithms on Pseudo-Boolean Functions. IEEE Transactions On Evolutionary Computation. 8(2):170-182 (2004)
  * [19] Lutton, E., Gilbert, H., Cancino, W., Bach, B., Pallamidessi, J., Parrend, P. & Collet, P. Visual and Audio Monitoring of Island Based Parallel Evolutionary Algorithms. Journal Of Grid Computing. 13(3):309-327 (2015)
  * [20] Lyu, Z., Li, J. & Wang, B. AIive: Interactive Visualization and Sonification of Neural Network in Virtual Reality. arXiv preprint arXiv:2109.15193 (2021)
  * [21] Miettinen, K. Nonlinear Multiobjective Optimization. Springer Science & Business Media (2012)
  * [22] Neuhoff, J. Is sonification doomed to fail? Georgia Institute of Technology (2019)
  * [23] Ochoa, G., Malan, K. & Blum, C. Search Trajectory Networks: A Tool for Analysing and Visualising the Behaviour of Metaheuristics. Applied Soft Computing. 109:107492 (2021)
  * [24] Poguntke, M. & Ellis, K. Auditory Attention Control for Human-Computer Interaction. Conference On Human System Interactions. pp. 231-236 (2008)
  * [25] Puckette, M. The Patcher. Proceedings Of The 1986 International Computer Music Conference, pp. 420-429 (1988)
  * [26] Rechenberg, I. Evolutionsstrategien. Simulationsmethoden In Der Medizin Und Biologie, pp. 83-114 (1978)
  * [27] Sasaki, L. & Smith, K. A Simple Data Reduction Scheme for Additive Synthesis. Computer Music Journal. 4, pp. 22-24 (1980)
  * [28] Sawe, N., Chafe, C. & Treviño, J. Using Data Sonification to Overcome Science Literacy, Numeracy, and Visualization Barriers in Science Communication. Frontiers In Communication. 5:46 (2020)
  * [29] Schuller, B., Virtanen, T., Riveiro, M., Rizos, G., Han, J., Mesaros, A. & Drossos, K. Towards Sonification in Multimodal and User-Friendly Explainable Artificial Intelligence. Proceedings Of The 2021 International Conference On Multimodal Interaction. pp. 788-792 (2021)
  * [30] Tanaka, M., Watanabe, H., Furukawa, Y. & Tanino, T. GA-based Decision Support System for Multicriteria Optimization. 1995 IEEE International Conference On Systems, Man And Cybernetics. Intelligent Systems For The 21st Century. 2 pp. 1556-1561 (1995)
  * [31] Tavares, T. & Godoy, A. Sonification of Population Behaviour in Particle Swarm Optimization. Proceedings of Annual Conference Companion On Genetic And Evolutionary Computation. pp. 51-52 (2013)
  * [32] Tušar, T. & Filipič, B. Visualization of Pareto Front Approximations in Evolutionary Multiobjective Optimization: A Critical Review and the Prosection Method. IEEE Transactions On Evolutionary Computation. 19(2):225-245 (2015)
  * [33] Vickers, P.: Sonification for Process Monitoring. In: Hermann, T., Hunt, A., Neuhoff, J. G., (eds.) The Sonification Handbook, pp. 455-492, Logos Publishing House, Berlin, Germany (2011)
  * [34] Wright, M. & Freed, A. Open Sound Control: A New Protocol for Communicating with Sound Synthesizers. International Computer Music Conference, pp. 101-104 (1997)
  * [35] Zhang, Q. & Li, H. MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition. IEEE Transactions On Evolutionary Computation. 11(6):712-731 (2007)
  * [36] Zitzler, E., Deb, K. & Thiele, L. Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evolutionary Computation. 8(2):173-195 (2000)


# Low-mass dielectron measurements in pp, p–Pb, and Pb–Pb collisions with
ALICE at the LHC

H. Sebastian Scheid for the ALICE Collaboration  Goethe University, Frankfurt
am Main, Germany

###### Abstract

Dielectrons are an excellent probe for the QCD matter created in created in
ultra-relativistic heavy-ion collisions, since they are emitted during the
whole evolution of the collision and do not interact strongly with the medium.
To isolate the QGP signals, measurement of the dielectron production in vacuum
and its modifications due to the presence of cold nuclear matter is necessary.
We present and discuss results from a low magnetic field detector setup in
proton-proton collisions at $\sqrt{s}=13$ TeV, as well as the measurement of
dielectron production in pp , p–Pb, and Pb–Pb collisions at $\sqrt{s_{\rm
NN}}=5$ TeV.

###### keywords:

ALICE , heavy-ion collisions , dielectrons , electromagnetic probes , p–Pb ,
Pb–Pb , heavy-flavour production

##  1 Introduction

In heavy-ion collisions dielectrons originate from various sources, i. e.
decays of pseudo-scalar and vector mesons, semi-leptonic decays of correlated
open heavy-flavour hadrons, and thermal radiation from a Quark–Gluon Plasma
(QGP) as well as a hot hadron gas. The latter allows the study of the average
temperature of the fireball, and the onset of chiral symmetry restoration.
Investigation of these non-trivial phenomena demands a precise understanding
of the production of dielectrons from hadronic decays and their initial or
final state modifications. During the second data taking period of the LHC
from 2015 to 2018, ALICE collected data of proton-proton (pp), p–Pb, and Pb–Pb
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV providing the possibility to
compare the dielectron production in all systems at the same collision energy
per nucleon pair.

In special data taking campaigns data in pp collisions at $\sqrt{s}=13$ TeV
were recorded, while the magnetic field in the central barrel was reduced from
0.5 to 0.2 T, extending the reach of particle measurements to a transverse
momentum $p_{\rm T}$ $>75$ MeV/$c$. This setup gives ALICE the possibility to
measure dielectrons at LHC energies in an unexplored kinematic regime, at low
invariant mass and pair transverse momentum.

##  2 Results

In Fig. 1 (left) the cross section of dielectron production as a function of
invariant mass ($m_{\rm ee}$) is shown in inelastic pp collisions at
$\sqrt{s}=13$ TeV recorded at $B=0.2$ T. The mass spectrum is shown for small
pair transverse momentum ($p_{\rm T,ee}$) $<0.4$ GeV/$c$ . The measurement is
compared with the expected contribution from known hadronic sources, the so-
called hadronic cocktail, constructed from independent measurements of hadron
yields using established methodology [1, 2]. In the mass region dominated by
Dalitz decays of $\pi^{0}$ ($m_{\rm ee}<0.14$ GeV/$c^{2}$) the hadronic
cocktail is well in agreement with the data, whereas at larger masses an
enhancement over the expected hadronic contributions can be observed. The
enhancement has a significance of 2.1$\sigma$ taking the statistic and
systematic uncertainties into account. Furthermore, Fig. 1 (right) shows the
multiplicity dependence of the data to cocktail ratios in different kinematic
regions. The data exceeds the expectations from hadronic sources for the
kinematic region of $0.14<m_{\rm ee}<0.6$ GeV/$c^{2}$ ($\eta$ mass region) and
$p_{\rm T,ee}<0.4$ GeV/$c$ in all multiplicity intervals, where the scaling of
the enhancement with the multiplicity is compatible with a linear scaling. In
the $\pi^{0}$ mass region in the same $p_{\rm T,ee}$ range or the $\rm\eta$
mass region at higher $p_{\rm T,ee}$ ($1-6$ GeV/$c$) the cocktail is in
agreement with the data within uncertainties. The main systematic uncertainty
of the cocktail in the mass region of interest could be reduced drastically by
a measurement of the $\eta$ at low $p_{\rm T}$. Previous measurements
performed in pp collisions at lower collision energies observed an excess of
soft photons and dielectrons in a similar kinematic range [3], that was
discussed in the context of hadronic Bremsstrahlung [4], soft parton
annihilation in a cold QGP [5], and quark synchrotron radiation [6].

Fig. 1: Cross section of dielectron production for $p_{\rm T,ee}<0.4$ GeV/$c$
measured in minimum-bias pp collisions at $\sqrt{s}=13$ TeV with a low
magnetic field detector setup compared to expected contributions from hadronic
sources (left) and data over cocktail ratios for different kinematic regions
as a function of the relative charged particle density (right).

The cross section of dielectron production as a function of $m_{\rm ee}$
measured in pp collisions at $\sqrt{s}=5.02$ TeV is presented and compared to
a hadronic cocktail in Fig. 2 (left). The first measurement of the charm ($\rm
c\overline{c}$) and beauty ($\rm b\overline{b}$) production cross sections at
mid-rapidity in pp collisions at $\sqrt{s}=5.02$ TeV is performed by fitting
templates to the $m_{\rm ee}$ and $p_{\rm T,ee}$ distributions in the
intermediate mass region (IMR, $1.1<m_{\rm ee}<2.7$ GeV/$c^{2}$), providing a
complementary method to the measurement of the full reconstruction of hadronic
decays of heavy-flavour hadrons [7, 8]. The templates are based on the PYTHIA6
[9] Perugia2011[10] and POWHEG [11] event generators. The results are
summarised in Tab. 1. They fall in line with previous measurements at
$\sqrt{s}=7$ [1] and 13 TeV [2], showing the sensitivity of the dielectron
measurements to the different implementation of the heavy-flavour production
in the model calculations. The energy dependence follows the trend predicted
by FONLL [12] calculations.

Table 1: Cross section of heavy-flavour production at mid-rapidity in pp collisions at $\sqrt{s}=5.02$ TeV. Systematic uncertainties on effective branching ratios, 22% for $\rm c\overline{c}\rightarrow e^{+}e^{-}$ and 6% for $\rm b\overline{b}\rightarrow e^{+}e^{-}$, are not included. | PYTHIA | POWHEG
---|---|---
${\rm d}\sigma_{\rm c\overline{c}}/{\rm d}y|_{y=0}$ (mb) | 0.531 $\pm$ 0.062 (stat) $\pm$ 0.066 (syst) | 0.743 $\pm$ 0.080 (stat) $\pm$ 0.093 (syst)
${\rm d}\sigma_{\rm b\overline{b}}/{\rm d}y|_{y=0}$ (mb) | 0.037 $\pm$ 0.005 (stat) $\pm$ 0.002 (syst) | 0.027 $\pm$ 0.004 (stat) $\pm$ 0.003 (syst)

The dielectron yield measured in p–Pb collisions at $\sqrt{s}_{\rm NN}=5.02$
TeV is depicted in Fig. 2 (right). The data are compared to a hadronic
cocktail constructed in the same way as in the pp analysis. The heavy-flavour
contributions use the same cross sections as the ones used for the default
hadronic cocktail in pp collisions at 5.02 TeV shown in Fig. 2 (left), scaled
with the number of binary collisions $\langle N_{\rm coll}\rangle=6.7\pm 0.2$
[13], and taking the asymmetry energy of the colliding beams into account. The
cocktail is well in agreement with the data within uncertainties.

Fig. 2: Cross section of dielectron production in pp collisions (left) and
dielectron yield in p–Pb collisions (right) as a function of invariant mass
measured at $\sqrt{s_{\rm NN}}=5.02$ TeV.

The nuclear modification factor $R_{\rm pPb}$ has been calculated as $R_{\rm
pPb}=\frac{1}{\langle N_{\rm coll}\rangle}\frac{{\rm d}N/{\rm d}m_{\rm
ee}|_{\rm pPb}}{{\rm d}N/{\rm d}m_{\rm ee}|_{\rm pp}}$, and is shown in Fig. 3
(left) together with two cocktail calculations based on PYTHIA. One cocktail
considers $\langle N_{\rm coll}\rangle$ scaled vacuum production of the charm
contribution (solid line), whereas a second cocktail includes modifications of
the charm production via the nuclear parton distribution functions (nPDFs)
from EPS09 [14]. In the IMR the data is consistent with unity, suggesting no
modification of the charm production beyond a scaling with $\langle N_{\rm
coll}\rangle$ within uncertainties. For $m_{\rm ee}<1.1$ GeV/$c^{2}$ the
deviation of the data from unity is expected, since the light flavour sources
have been shown to not scale with $\langle N_{\rm coll}\rangle$ in previous
measurements. Both cocktails can describe the data within the uncertainties.
For $m_{\rm ee}<1$ GeV/$c^{2}$ the EPS09 cocktail is closer to the central
value of the data. This suggests that, if there is a significant modification
of charm production, the sensitive region for dielectron measurements would be
in a mass window of about $0.5<m_{\rm ee}<1.0$ GeV/$c^{2}$.

Fig. 3: Nuclear-modification factors of dielectrons $R_{\rm pPb}$ from minimum
bias p–Pb collisions (left) and $R_{\rm AA}$ from 0-20% most central Pb–Pb
collisions (right) as a function of $m_{\rm ee}$ measured at $\sqrt{s_{\rm
NN}}=5.02$ TeV. Both are compared to expectations from the hadronic cocktail
in vacuum (solid line), and including nuclear modifications of the charm
contribution from EPS09 [14] (dashed line).

Figure 3 (right) shows the $R_{\rm AA}$ constructed from the same pp baseline,
together with measurements in 0-20% most central Pb–Pb collisions at
$\sqrt{s}_{\rm NN}$= 5.02 TeV [15]. In the IMR, a hint for a suppression below
the vacuum expectation is observed. Following previous arguments this should
not be considered as a modification of the initial state, but rather final
state effects, e.g. energy loss or rescattering in a hot and dense medium,
which are not included in the cocktail calculations. The precision of the data
prohibits a measurement of thermal radiation from a hot hadron gas or QGP that
would manifest itself as an enhancement above the hadronic cocktail. This
enhancement would be more pronounced, taking final state effects on the charm
contribution into account.

##  3 Summary and Conclusion

ALICE measured dielectron production in inelastic pp collisions at
$\sqrt{s}=13$ TeV with the magnetic field in the central barrel reduced to 0.2
T which allows the investigation of very soft dielectron production. An excess
over the expectation of hadronic sources with a significance of 2.1$\sigma$ is
found for $p_{\rm T,ee}<0.4$ GeV/$c$ in the $m_{\rm ee}$window of $0.14-0.6$
GeV/$c^{2}$. The scaling of this excess with the event multiplicity is
compatible with a linear behaviour. In other $m_{\rm ee}$ or $p_{\rm T,ee}$
regions the hadronic cocktail is in agreement with the measurements within
uncertainties. The measurement of dielectron production in pp collisions at
$\sqrt{s}=5.02$ TeV yields the first measurements of ${\rm d}\sigma_{\rm
c\overline{c}}/{\rm d}y$ and ${\rm d}\sigma_{\rm b\overline{b}}/{\rm d}y$ at
mid-rapidity in pp collisions at this collision energy. Furthermore, with the
pp baseline and a new p–Pb measurement the $R_{\rm pPb}$ of dielectron
productions as a function of $m_{\rm ee}$ was measured, which does not suggest
a modification of heavy-flavour production within uncertainties. The $R_{\rm
AA}$ as a function of $m_{\rm ee}$ on the other hand shows a hint for a
modification in the IMR, most probably introduced by an interaction of the
heavy quarks with the hot and dense medium created in AA collisions at LHC
energies. This complex interplay of initial and final state modifications of
charm production in heavy–ion collisions complicates the construction of a
precise baseline, making the measurement of thermal radiation of the medium or
possible modifications due to partial restoration of chiral symmetry
difficult. In future studies, this will be possible by separating the thermal
radiation from off-vertex decays of heavy-flavour hadrons using their distinct
decay topologies, an analysis which will benefit from the improved vertexing
and impact parameter resolution of the inner tracking system upgrade for LHC
Run3 [16].

## References

  * [1] S. Acharya, et al., Dielectron production in proton-proton collisions at $\sqrt{s}=7$ TeV, JHEP 09 (2018) 064.
  * [2] S. Acharya, et al., Dielectron and heavy-quark production in inelastic and high-multiplicity proton–proton collisions at $\sqrt{s_{NN}}=13$ TeV, Phys. Lett. B788 (2019) 505–518.
  * [3] L. Baum, et al., Experimental observation of a copious yield of electrons with small transverse momenta in pp collisions at high energies, Physics Letters B 60 (5) (1976) 485.
  * [4] F. E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974.
  * [5] P. Lichard, L. van Hove, The cold quark-gluon plasma as a source of very soft photons in high energy collisions, Phys. Lett. B 245 (1990) 605.
  * [6] O. Nachtmann, A. Reiter, The Vacuum Structure in QCD and Hadron - Hadron Scattering, Z. Phys. C24 (1984) 283.
  * [7] S. Acharya, et al., Measurement of d-meson production at mid-rapidity in pp collisions at ${\sqrt{s}=7}$ tev, EPJ C 77.
  * [8] S. Acharya, et al., Prompt and non-prompt J/$\psi$ production and nuclear modification at mid-rapidity in p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV, Eur. Phys. J. C78 (6) (2018) 466.
  * [9] T. Sjostrand, S. Mrenna, P. Z. Skands, PYTHIA 6.4 Physics and Manual, JHEP 05 (2006) 026.
  * [10] P. Z. Skands, Tuning monte carlo generators: The perugia tunes, Phys. Rev. D 82 (2010) 074018.
  * [11] P. Nason, A New method for combining NLO QCD with shower Monte Carlo algorithms, JHEP 11 (2004) 040.
  * [12] M. Cacciari, M. Greco, P. Nason, The P(T) spectrum in heavy flavor hadroproduction, JHEP 05 (1998) 007.
  * [13] Centrality determination in heavy ion collisions.
URL https://cds.cern.ch/record/2636623

  * [14] K. J. Eskola, H. Paukkunen, C. A. Salgado, EPS09: A New Generation of NLO and LO Nuclear Parton Distribution Functions, JHEP 04 (2009) 065.
  * [15] A. Caliva, Low-mass dilepton measurements with ALICE at the LHC, PoS HardProbes2018 (2019) 174.
  * [16] S. Rossegger, et al., Upgrade of the alice inner tracking system, NIM A 731 (2013) 40.