# Hyperbolic Minesweeper is in P

Eryk Kopczyński

###### Abstract

We show that, while Minesweeper is NP-complete, its hyperbolic variant is in
P. Our proof does not rely on the rules of Minesweeper, but is valid for any
puzzle based on satisfying local constraints on a graph embedded in the
hyperbolic plane.

##  1 Introduction

(a) graphical mode, bitruncated tessellation

(b) text mode, regular tessellation

Figure 1:  Hyperbolic Minesweeper [7]. In (a), the default settings are used
(bitruncated order-3 heptagonal tessellation, numbers of adjacent mines are
color-coded; some of the mines that the players is sure of are marked red). In
(b) we play on an order-3 heptagonal tessellation, and numbers are shown.

Minesweeper is a popular game included with many computer systems; it also
exists in the puzzle form. In the puzzle form, every cell in a square grid
either contains a number or is empty. The goal of the puzzle is to assign
mines to the empty squares in such a way that every number $n$ is adjacent
(orthogonally or diagonally) to exactly $n$ mines.

This puzzle is a well-known example of a NP-complete problem [6]. Its
popularity has also spawned many variants played on different grids, from
changing the tessellation of the plane, to changing the number of dimensions
(six-dimensional implementations of Minesweeper exist) to changing the
underlying geometry.

In this paper we will be playing Minesweeper on a tessellation of the
hyperbolic plane. Minefield, a land based on hyperbolic Minesweeper is
available in HyperRogue [7]; this implementation differs from the standard
Minesweeper in multiple ways, e.g., by being played on an infinite board,
however, the basic idea is the same. Another implementation is Warped Mines
for iOS [13], which has the same rules as the usual Minesweeper except the
board, which is a bounded subset of the order-3 heptagonal tiling of the
hyperbolic plane.

From the point of view of a computer scientist, the most important distinctive
property of hyperbolic geometry is exponential growth: the area of a circle of
radius $r$ grows exponentially with $r$. It is also more difficult to
understand than Euclidean geometry. While these properties often cause
computational geometry problems to be more difficult, it also gives hyperbolic
geometry applications in data visualization [8, 9] and data analysis [10].

In this paper we show that hyperbolic geometry makes Minesweeper easier: the
hyperbolic variant of Minesweeper is in P. Our proof will not rely on the
specific rules of Minesweeper nor work with any specific tessellation of the
hyperbolic plane; instead, it will work with any puzzle based on satisfying
local constraints, on any graph that naturally embeds into the hyperbolic
plane.

##  2 Hyperbolic Geometry

We denote the hyperbolic plane with $\mathbb{H}^{2}$. Since this paper
requires only the basic understanding of hyperbolic geometry, we will not
include the formal definition; see [2], or [7] for an intuitive introduction.
Figure 1 shows the hyperbolic plane in the Poincaré disk model, which is a
projection which represents angles faithfully, but not the distances: the
scale gets smaller and smaller as we get closer to the circle bounding the
model. In particular, all the heptagons in Figure 1b are the same size, all
the heptagons in Figure 1a are the same size, and all the hexagons in Figure
1a are the same size. We denote the distance between two points
$x,y\in\mathbb{H}^{2}$ by $\delta(x,y)$. The set of points in distance at most
$r$ from $x$ is denoted with $B(x,r)$. The area of $B(x,r)$ is $2\pi(\cosh
r-1)$, which we denote with $\mathrm{area}(r)$. The perimeter of $B(x,r)$ is
$2\pi\sinh r$, which we denote with $\mathrm{peri}(r)$. Note that both
$\mathrm{area}$ and $\mathrm{peri}$ grow exponentially:
$\mathrm{area}(r)=\Theta(e^{r})$ and $\mathrm{peri}(r)=\Theta(e^{r})$.

##  3 Hyperbolic Graph

We need a suitable definition of a hyperbolic graph. The obvious definition,
obtained by changing “plane” to “hyperbolic plane” in the definition of a
planar graph, is equivalent to the definition of the planar graph (the plane
and the hyperbolic plane are homeomorphic). Another definition found in
literature is the notion of Gromov $\delta$-hyperbolic graph [1]; this
definition is based on the observation that, in the hyperbolic plane,
triangles are thin, i.e., for any three vertices $a$, $b$, $c$, every point on
any shortest path from $a$ to $c$ must be in distance at most $\delta$ from
either the shortest path from $a$ to $b$, or the shortest path from $b$ to
$c$. The parameter $\delta$ measures tree-likeness of the graph (it can be
easily seen that trees are 0-hyperbolic). However, this definition is not
suitable for us, because every graph $G$ becomes 2-hyperbolic when we add a
vertex $v_{*}$ connected to every $v\in V(G)$; our main result is not true for
such graphs. We propose the following definition:

######  Definition 3.1.

A $(r,d)$-hyperbolic graph is a graph $G=(V,E)$ such that there exists an
embedding $m:V\rightarrow\mathbb{H}^{2}$ such that:

  * •

for $v_{1}\neq v_{2}$, $\delta(m(v_{1}),m(v_{2}))>r$,

  * •

for $\\{v_{1},v_{2}\\}\in E$, $\delta(m(v_{1}),m(v_{2}))<d$,

  * •

if we draw every edge $\\{v_{1},v_{2}\\}\in E$ as a straight line segment
between $m(v_{1})$ and $m(v_{2})$, these edges do not cross, nor they get
closer to vertices other than $v_{1}$ or $v_{2}$ than $r/2$.

This definition includes finite subsets of all regular and semiregular
hyperbolic tessellations (such as the ones shown in Figure 1). We denote
$N(v)$ to be the neighborhood of $v\in V$, i.e., $\\{v\\}\cup\\{w\in
V:\\{v,w\\}\in E\\}$. (Our proof also works with neighborhoods of larger
radius.)

Remark. All $(r,d)$-hyperbolic graphs have degree bounded by a constant (for
fixed $r,d$). Indeed, let $v\in V$. For every $w\in N(v)$, the circles
$B(w,\frac{r}{2})$ are disjoint, and they fit in $B(v,d+\frac{r}{2})$.
Therefore, $|N|\leq\mathrm{area}(d+\frac{r}{2})/\mathrm{area}(\frac{r}{2})$.

##  4 Hyperbolic Local Constraint Satisfaction Problem

Below we state our main result.

######  Theorem 4.1.

Fix the set of colors $K$ and the parameters $(r,d)$. The following problem
(Hyperbolic Local Constraint Satisfaction Problem, HLCSP) can be solved in
polynomial time:

INPUT:

  * •

a $(r,d)$-hyperbolic graph $G=(V,E)$;

  * •

for every vertex $v\in V$, $m(v)$, a subset of $K^{N(v)}$.

OUTPUT: Is there a coloring $c:V\rightarrow K$ such that for every $v\in V$,
$c_{|N(v)}\in m(v)$?

HLCSP generalizes hyperbolic minesweeper. We have two colors (no mine and
mine), and for every $v\in V$ containing a number $k$, the constraint $m(v)$
says that $v$ contains no mine itself, and exactly $k$ of vertices in $N(v)$
contain a mine.

To prove Theorem 4.1, it is enough to prove that the following problem (HECSP)
is in P.

######  Theorem 4.2.

HLCSP reduces to the Hyperbolic Edge Constraint Satisfaction Problem (HECSP)
given as follows:

INPUT:

  * •

a $(r,d)$-hyperbolic graph $G=(V,E)$;

  * •

for every edge $e\in E$, $m(e)$, a subset of $K^{e}$.

OUTPUT: Is there a coloring $c:V\rightarrow K$ such that for every $e\in E$,
$c_{|e}\in m(e)$?

(Note: this reduction changes the set of admissible colors $K$.)

###### Proof.

Let $(G,m)$ be the instance of HLCSP. We will be coloring $V$ using colors
$K^{\prime}=\\{1,\ldots,k\\}$, where $k$ is the greatest number of elements of
$m(v)$ (since $(r,d)$-hyperbolic graphs have bounded degree and $K$ is fixed,
$k$ is also bounded). Enumerate every elements of $m(v)$ with one color from
$K^{\prime}$. For $e=\\{v_{1},v_{2}\\}\in E$, $m(e)$ is the set of all
colorings $c:\\{v_{1},v_{2}\\}\rightarrow K$ which are consistent, i.e.,
$k_{1}$ denotes $c_{1}\in m(v_{1})$ and $k_{2}$ denotes $c_{2}\in m(v_{2})$,
and $c_{1}$ equals $c_{2}$ on all the vertices in $N(v_{1})\cap N(v_{2})$. ∎

##  5 Proof of Theorem 4.1

We will be using the following result [11, 3]:

######  Theorem 5.1.

Given a planar graph $G=(V,E)$ and a number $t$, it is possible to either find
a $t\times t$ grid as a minor of $G$, or produce a tree decomposition of $G$
of width $\leq 5t-6$, in time $O(n^{2}\log(n))$, where $n=|V|$.

######  Definition 5.2.

A tree decomposition of width $w$ of a graph $G=(V,E)$ is $(V_{T},E_{T},X)$
where $(V_{T},E_{T})$ is a tree, and $X$ is a mapping which assign a subset of
$V$ of cardinality at most $w+1$ to every vertex in $V_{T}$, such that:

  * •

For every $v\in V$, the set of vertices $b\in V_{T}$ such that $v\in X_{b}$ is
connected,

  * •

For every $e\in E$, there exists a $b\in V_{T}$ such that $es\subseteq X_{b}$.

We can assume that our tree is rooted in $r\in V_{T}$. For $b\in V_{T}$, let
$X^{+}_{b}$ be the union of $X_{b^{\prime}}$ for all $b^{\prime}$ which are
descendants of $b$.

######  Lemma 5.3.

Without loss of generality we can assume that every $b\in V_{T}$ falls to one
of the following cases:

  * •

$b$ is a leaf and $|X_{b}|=1$,

  * •

$b$ has a single child $b^{\prime}$ and $X_{b^{\prime}}=X_{b}\cup\\{v\\}$,

  * •

$b$ has a single child $b^{\prime}$ and $X_{b^{\prime}}=X_{b}-\\{v\\}$,

  * •

$b$ has two children $b_{1}$ and $b_{2}$, $X_{b}=X_{b_{1}}=X_{b_{2}}$, and
$X^{+}_{b_{1}}-X_{b}$ and $X^{+}_{b_{2}}-X_{b}$ are non-empty and disjoint.

######  Theorem 5.4.

If a $(r,d)$-hyperbolic graph $G$ contains a $t\times t$ grid as a minor, then
$t=O(\log(V(G))$.

###### Proof.

Without loss of generality we can assume $t=2k+1$. Such a $t\times t$ grid
contains $k$ cycles around the center of the grid. Let $v_{c}\in V$ be the
vertex corresponding to this center. We have $k$ cycles $C_{1},\ldots,C_{k}$
in the graph $V$, each of which surrounds $v_{c}$ and the preceding cycles. We
use the following lemma:

######  Lemma 5.5.

Every point in the drawing of cycle $C_{i}$ is in distance $\Omega(i)$ from
$v$. (The drawing is the polygon obtained as a union of the edges embedded in
$\mathbb{H}^{2}$.)

Therefore, $|C_{i}|\geq\mathrm{peri}(\Omega(i))/d$. Since $\mathrm{peri}$
grows exponentially, we get $k=O\log(|V|)$. ∎

$v_{c}$
---
$h$
---
Figure 2: Cycles around $v_{c}$

###### Proof of Lemma 5.5.

We will show that if every point in the drawing of $C_{i}$ is in distance at
least $x$ from $v$, then every point in the drawing of $C_{i+1}$ is in
distance $x+c$ from $v$. By induction, this is enough to prove Lemma 5.5.

Figure 2 shows what happens for an Euclidean graph. (We define Euclidean graph
just like hyperbolic graph except the differing underlying geometry.) In this
picture, the drawing of $C_{i}$ is a hexagon; cycle $C_{i}$ has six vertices
around $v_{c}$. For each of these seven vertices the exclusion zone of radius
$r/2$ around it is shown (in green). It is clear from the picture (and the
definition of Euclidean graph) that no point in the drawing of $C_{i+1}$ may
fall into the gray/green region. All points in distance $\leq r/2$ from
$C_{i}$ fall in the gray/green region, thus our claim is true for $c=h=r/2$.

In the hyperbolic plane the situation is slightly different. Because parallel
lines work differently in the hyperbolic plane (they “diverge”), we have
$c=h<r/2$, where $h$ depends on $r$ and $d$ (we have $h=\Theta(re^{-d})$ for
large values of $d$). Still, our claim holds with $c>0$. ∎

######  Corollary 5.6.

Given a planar graph $G=(V,E)$, it is possible to find a tree decomposition of
width $O(\log|V|)$ in polynomial time.

###### Proof.

From Theorem 5.4 choose $t=O(\log|V|)$ such that $G$ does not contain a
$k\times k$ grid. From Theorem 5.1, it is possible to find a tree
decomposition of width $5t-6=O(\log|V|)$. ∎

######  Corollary 5.7.

HECSP (and thus HLCSP) can be solved in polynomial time.

###### Proof.

From Corollary 5.6 we can find a tree decomposition $(V_{T},E_{T},B)$ of width
$w=O(\log|V|)$. Then we use Dynamic Programming over $(V_{T},E_{T})$. For
every $b\in V_{T}$, we compute $s(w)$, the set of all possible colorings
$c:X_{b}\in K$ such that there exists a coloring $c:X^{+}_{b}$ which extends
$c$ and which satisfies all the constraints on edges in $X^{+}_{b}$. This can
be computed straightforwardly in every case from Lemma 5.3. The whole
algorithm works in $O(|V|\cdot|K|^{w})$, which is polynomial in $|V|$. ∎

(a) two yellow

(b) one group

(c) two groups

(d) four groups

Figure 3:  Random colorings satisfying various constraints. Since the degree
of our polynomial is quite high, these pictures took about a minute to make.
Some cells have been removed from the full disk to reduce the treewidth.

##  6 Conclusion

We have shown that Minesweeper on hyperbolic tessellations can be solved in
polynomial time. Our method is general: it works for any $(r,d)$-hyperbolic
graph, and for any problem based on satisfying local constraints. Other than
solving puzzles, this may be applied to procedural content generation. For
example, the Wave Function Collapse (WFC) algorithm [4, 5] is used in
procedurally generated games to procedurally generate maps satisfying local
constraints (e.g., a mountain should not appear close to ocean, or roads
should not branch nor end abruptly). In general finding out whether such
constraints can be satisfied is NP-complete (although this does not happen in
typical PCG applications). Our algorithm can be adjusted to count the number
of satisfying colorings, and to produce one of them, randomly chosen (Figure
3).

Since the Euclidean plane can be embedded into a horosphere in three-
dimensional hyperbolic space $\mathbb{H}^{3}$, three-dimensional hyperbolic
Minesweeper is NP-complete.

HyperRogue also lets the player play Minesweeper in bounded hyperbolic
manifolds, such as the Klein quartic or other Hurwitz manifolds. Hurwitz
manifolds are quotient spaces of $\mathbb{H}^{2}$ which can be tiled with
regular heptagons with angles $120^{\circ}$ (see Figure 1b), and that are
maximally symmetric, i.e., there exists an isometry of the Hurwitz manifold
$M$ into itself which map any heptagon with any orientation into any heptagon
with any orientation. Tessellations of quotient space are no longer planar,
thus Theorem 5.1 nor our proof of Theorem 5.4 is no longer valid in quotient
spaces of $\mathbb{H}^{2}$. Therefore, the complexity of Minesweeper on such
manifolds does not follow from our results. There are less symmetric
hyperbolic manifolds into which the Euclidean square grid, or even higher-
dimensional Euclidean grids, can be embedded [12], and thus Minesweeper on
them is NP-complete; however, highly symmetric manifolds have a more
restricted structure.

## References

  * [1] Sergio Bermudo, José M. Rodríguez, José M. Sigarreta, and Jean-Marie Vilaire.  Gromov hyperbolic graphs.  Discrete Mathematics, 313(15):1575 – 1585, 2013.  URL: http://www.sciencedirect.com/science/article/pii/S0012365X1300174X, doi:http://doi.org/10.1016/j.disc.2013.04.009.
  * [2] James W. Cannon, William J. Floyd, Richard Kenyon, Walter, and R. Parry.  Hyperbolic geometry.  In In Flavors of geometry, pages 59–115. University Press, 1997\.  Available online at http://www.msri.org/communications/books/Book31/files/cannon.pdf.
  * [3] Alexander Grigoriev.  Tree-width and large grid minors in planar graphs.  Discrete Mathematics & Theoretical Computer Science, 13:13–20, 01 2011.
  * [4] Maxim Gumin.  WaveFunctionCollapse, 2017.  https://github.com/mxgmn/WaveFunctionCollapse.
  * [5] Isaac Karth and Adam M. Smith.  WaveFunctionCollapse is Constraint Solving in the Wild.  In Proceedings of the 12th International Conference on the Foundations of Digital Games, FDG ’17, New York, NY, USA, 2017. Association for Computing Machinery.  doi:10.1145/3102071.3110566.
  * [6] Richard Kaye.  Minesweeper is np-complete.  The Mathematical Intelligencer, 22(2):9–15, Mar 2000.  doi:10.1007/BF03025367.
  * [7] Eryk Kopczyński, Dorota Celińska, and Marek Čtrnáct.  Hyperrogue: Playing with hyperbolic geometry.  In David Swart and Carlo H. Séquin and Kristóf Fenyvesi, editor, Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture, pages 9–16, Phoenix, Arizona, 2017. Tessellations Publishing.  Available online at http://archive.bridgesmathart.org/2017/bridges2017-9.pdf.
  * [8] John Lamping, Ramana Rao, and Peter Pirolli.  A focus+context technique based on hyperbolic geometry for visualizing large hierarchies.  In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, CHI ’95, pages 401–408, New York, NY, USA, 1995. ACM Press/Addison-Wesley Publishing Co.  URL: http://dx.doi.org/10.1145/223904.223956, doi:10.1145/223904.223956.
  * [9] Tamara Munzner.  Exploring large graphs in 3d hyperbolic space.  IEEE Computer Graphics and Applications, 18(4):18–23, 1998.  URL: http://dx.doi.org/10.1109/38.689657, doi:10.1109/38.689657.
  * [10] Fragkiskos Papadopoulos, Maksim Kitsak, M. Angeles Serrano, Marian Boguñá, and Dmitri Krioukov.  Popularity versus Similarity in Growing Networks.  Nature, 489:537–540, Sep 2012.
  * [11] N. Robertson, P. Seymour, and R. Thomas.  Quickly excluding a planar graph.  Journal of Combinatorial Theory, Series B, 62(2):323 – 348, 1994\.  URL: http://www.sciencedirect.com/science/article/pii/S0095895684710732, doi:https://doi.org/10.1006/jctb.1994.1073.
  * [12] Zeno Rogue.  HyperRogue: Experiments with Geometry, 2020.  http://www.roguetemple.com/z/hyper/geoms.php.
  * [13] Sci-Tech Binary Ltd. Co.  Warped mines, 2019.  https://apps.apple.com/br/app/hypersweeper/id1450066199.


[a]Taesoo Song

# Radiative energy loss of heavy quark through soft gluon emission in QGP

Ilia Grishmanovskii  Olga Soloveva  Elena Bratkovskaya

###### Abstract

The Low’s theorem is applied to the soft gluon emission from heavy quark
scattering in quark-gluon plasma (QGP). The QGP is described by the dynamical
quasi-particle model (DQPM) which reproduces the EoS from lQCD at finite
temperature and chemical potential. We show that if the emitted gluon is soft
and of long wavelength, the scattering amplitude can be factorized into the
scattering part and the emission part and the Slavnov-Taylor identities are
satisfied in the leading order. Imposing a proper upper limit on the emitted
gluon energy, we obtain the scattering cross sections of charm quark as well
as the transport coefficients (momentum drag and diffusion) in the QGP with
and without gluon emission.

##  1 Introduction

The energy loss of heavy flavor in heavy-ion collisions is one of important
observables which reveal the properties of the quark-gluon plasma (QGP). The
energy loss takes place through elastic and inelastic scatterings [1, 2, 3].
In this study we extend the soft photon approximation [4, 5, 6] to the soft
gluon emission from the elastic scattering of heavy quark, and discuss its
effect on the transport coefficients of heavy quark in QGP.

##  2 Soft gluon approximation

In the soft photon approximation [4, 5, 6] a low-energy photon is emitted from
the external charged particles and the complicated inner structure of the
scattering can be ignored. Then the Feynman diagrams can be factorized into
the elastic scattering and the gluon emission.

Figure 1: Feynman diagrams for $c+q\rightarrow c+q$ with a gluon emitted from
external partons (upper) and from the internal exchanged gluon (lower).

Applying the same approximation to the soft gluon emission from
$c+q\rightarrow c+q$ scattering, it is sufficient to calculate the upper
diagrams in Fig. 1, of which transition amplitude is given by [7]

$\displaystyle M_{2q\rightarrow
2q+g}^{\mu,a,kl;ij}(p_{1},p_{2};p_{3},p_{4},q)=g\bigg{\\{}\frac{p_{1}^{\mu}}{p_{1}\cdot
q}M_{2q\rightarrow 2q}^{kl;mj}T^{a}_{mi}$
$\displaystyle+\frac{p_{2}^{\mu}}{p_{2}\cdot q}M_{2q\rightarrow
2q}^{kl;im}T^{a}_{mj}-\frac{p_{3}^{\mu}}{p_{3}\cdot
q}T^{a}_{km}M_{2q\rightarrow 2q}^{ml;ij}-\frac{p_{4}^{\mu}}{p_{4}\cdot
q}T^{a}_{lm}M_{2q\rightarrow 2q}^{km;ij}\bigg{\\}},$ (1)

where $\mu,a$ are respectively the Lorentz and color indices of emitted gluon,
$i,j$ and $k,l$ the color indices of the incoming and outgoing quarks and
$M_{2q\rightarrow 2q}^{kl;ij}$ the transition amplitude of $c+q\rightarrow
c+q$ scattering. Considering the simplest color structure of $M_{2q\rightarrow
2q}^{kl;ij}$ is $T^{b}_{ki}T^{b}_{lj}$, one can prove that the transition
amplitude of Eq. (1) satisfies the Slavnov-Taylor identities [7]:

$\displaystyle q_{\mu}M_{2q\rightarrow
2q+g}^{\mu,a,kl;ij}(p_{1},p_{2};p_{3},p_{4},q)=0.$ (2)

On the other hand, the lower diagram in Fig. 1 is simplified in the limit
$q\rightarrow 0$ into

$\displaystyle M_{5}\approx-
ig\varepsilon_{\mu}^{a*}(q)\frac{2(p_{1}-p_{3})^{\mu}}{(p_{1}-p_{3})^{2}+2(p_{1}-p_{3})\cdot
q}f^{abc}\bigg{(}M_{2q\rightarrow 2q}^{kl;ij}\bigg{)}^{bc}.$ (3)

Comparing Eqs. (1) and (3), apart from $M_{2q\rightarrow 2q}$, the former is
of the order of $1/q$ and the latter of the order of $1/p$ in the limit
$q/p\ll 1$. Therefore, the lower diagram is of higher order than the upper
diagrams in Fig. 1, if the transition amplitude is expanded in term of $q/p$.

Similar calculations can be done for the soft gluon emission from
$q+g\rightarrow q+g$, and the transition amplitude reads [7].

$\displaystyle M_{q+g\rightarrow
q+g+g}^{\mu,jbc;ia}(p_{1},p_{2};p_{3},p_{4},q)=g\bigg{\\{}\frac{p_{1}^{\mu}}{p_{1}\cdot
q}M_{q+g\rightarrow q+g}^{jb;ma}T^{c}_{mi}-\frac{p_{3}^{\mu}}{p_{3}\cdot
q}T^{c}_{jm}M_{q+g\rightarrow q+g}^{mb;ia}$
$\displaystyle+i\frac{p_{2}^{\mu}}{p_{2}\cdot q}f^{adc}M_{q+g\rightarrow
q+g}^{jb;id}+i\frac{p_{4}^{\mu}}{p_{4}\cdot q}f^{bdc}M_{q+g\rightarrow
q+g}^{jd;ia}\bigg{\\}},$ (4)

where $a,b$ and $c$ are respectively the colors of the incoming, outgoing and
emitted gluons, and $i$ and $j$ the colors of the incoming and outgoing
quarks. It also satisfies the Slavnov-Taylor identities, considering that the
color structure of $M_{q+g\rightarrow q+g}^{jb;ia}$ is given by
$[T^{a},T^{b}]_{ji}$ or $if^{abc}T_{ji}^{c}$ [7]:

$\displaystyle q_{\mu}M_{q+g\rightarrow
q+g+g}^{\mu,jbc;ia}(p_{1},p_{2};p_{3},p_{4},q)=0.$ (5)

In QGP partons are dressed and interacting. As a result, the parton self
energy is a nonzero complex function whose real and imaginary parts are
interpreted as $T,\mu_{B}-$dependent pole mass and spectral width,
respectively. The massive off-shell partons reproduce lattice EoS of QGP in
the dynamical quasi-particle model (DQPM) [8]. Then the propagators of quark
and gluon in Eqs. (1) and (4) are modified, respectively, into

$\displaystyle G_{ij}(p)$ $\displaystyle=$ $\displaystyle
i\frac{\not{p}+m_{q}(T,\mu)}{p^{2}-m_{q}^{2}(T,\mu)+i|p_{0}|\Gamma_{q}(T,\mu)}\delta_{ij},$
$\displaystyle G^{\mu\nu,ab}(p)$ $\displaystyle=$
$\displaystyle\frac{-ig^{\mu\nu}}{p^{2}-m_{g}^{2}(T,\mu)+i|p_{0}|\Gamma_{g}(T,\mu)}\delta_{ab},$
(6)

where $m_{q(g)}$ and $\Gamma_{q(g)}$ are the pole mass and spectral width of
quark(gluon).

Assuming the center-of-mass frame of $p_{3}+p_{4}$ is similar to that of
$p_{1}+p_{2}$, the differential cross section is approximated as [7]

$\displaystyle\frac{d\sigma_{2\rightarrow
3}}{d\cos\theta}\approx\frac{d\sigma_{2\rightarrow
2}}{d\cos\theta}\int_{m_{g}}^{E_{max}}\frac{dE_{g}}{2(2\pi)^{3}}\sqrt{E_{g}^{2}-m_{g}^{2}}\int
d\cos\theta^{\prime}d\phi^{\prime}|\epsilon\cdot J|^{2}\frac{|{\bf
p}_{3}|\sqrt{s}}{p_{f}\sqrt{s_{2}}},$ (7)

where $\theta$ is the scattering angle, ($\theta^{\prime}$, $\phi^{\prime}$)
the solid angle of emitted gluon, $m_{g}$ the thermal gluon mass,
$s_{2}=(p_{3}+p_{4})^{2}$ and

$\displaystyle\overline{|M_{2\rightarrow 3}}|^{2}\equiv|\epsilon\cdot
J|^{2}~{}\overline{|M_{2\rightarrow 2}}|^{2}=32\pi s|\epsilon\cdot
J|^{2}\frac{p_{i}}{p_{f}}\frac{d\sigma_{2\rightarrow 2}}{d\cos\theta}$ (8)

with $p_{i}$ and $p_{f}$ being respectively the initial and final momenta of
2-to-2 scattering in the center-of-mass frame. The maximum energy of soft
gluon, $E_{max}$ in Eq. (7), is kinetically given by
$E_{max}^{kin.}=\sqrt{m_{g}^{2}+q_{max}^{2}}$ with
$q_{max}^{2}=\\{s-(m_{3}+m_{4}+m_{g})^{2}\\}\\{s-(m_{3}+m_{4}-m_{g})^{2}\\}/(4s)$.
In the soft gluon approximation (SGA), however, we have neglected gluon
emission from the interaction region, which means that the wavelength of the
emitted gluon is assumed larger than the scattering scale that is roughly
$1/\sqrt{-t}$ with $t=(p_{1}-p_{3})^{2}$. Therefore, more reasonable upper
limit for SGA will be $E_{max}^{real.}={\rm
min}\bigg{[}E_{max}^{kin.},~{}\sqrt{-t}\bigg{]}$.

Figure 2: Differential (left) cross sections of $c+q\rightarrow c+q+g$ and
$c+g\rightarrow c+g+g$ in the soft gluon approximation with the kinetic and
the realistic energy limits, which are respectively shown as dotted and solid
lines, in comparison with the elastic scattering cross sections, which are
dashed lines, at $\sqrt{s}=$10 GeV and $T=$ 200 MeV and (right) their
integrated cross sections as a function of temperature at $\sqrt{s}=$10 GeV.

The left panel of Fig. 2 displays the differential cross sections for
$c+q(g)\rightarrow c+q(g)$ and $c+q(g)\rightarrow c+q(g)+g$ with the kinematic
and realistic upper limits of gluon energy as a function of the scattering
angle of charm quark at $\sqrt{s}=$10 GeV and $T=$ 200 MeV. Charm quark mass
is taken to be 1.5 GeV, and the masses of light quark and gluon the pole
masses in the DQPM. The radiative differential cross sections with the two
different upper limits are almost same except near $\cos\theta=1$, where $t$
is small and thus the realistic upper limit of the soft gluon energy is low.
In the right panel the integrated cross sections are shown as a function of
temperature. The temperature dependence of 2-to-3 scattering cross sections is
stronger than that of 2-to-2 scattering cross sections, because the former is
proportional to $\alpha_{s}^{3}$ while the latter to $\alpha_{s}^{2}$, and the
$\alpha_{s}$ rapidly increases near $T_{c}$ in the DQPM. We note that the
above results agree with those from the full calculations without introducing
the SGA at low momentum of emitted gluon [9].

##  3 Transport coefficients

Heavy quark loses energy-momentum traversing a QGP produced in relativistic
heavy-ion collisions. The most suitable quantities to describe the energy-
momentum loss are the momentum drag coefficient ($A$) and the transverse
momentum transfer squared per unit length $\hat{q}$ which are defined as

$\displaystyle A(p)$ $\displaystyle=$ $\displaystyle-\frac{d\langle\Delta
p_{L}\rangle}{dt}=\langle\langle(p-p^{\prime})_{z}\rangle\rangle,$ (9)
$\displaystyle\hat{q}(p)$ $\displaystyle=$ $\displaystyle\frac{d\langle(\Delta
p_{T})^{2}\rangle}{dz}=\frac{E}{p_{L}}\langle\langle p_{x}^{\prime
2}+p_{y}^{\prime 2}\rangle\rangle,$ (10)

where $p_{L}$ and $p_{T}$ are respectively the longitudinal and transverse
momentum of charm quark with $p$ and $p^{\prime}$ being respectively initial
and final momenta, and the double bracket implies

$\displaystyle\langle\langle O^{*}\rangle\rangle\equiv\sum_{i=q,\bar{q},g}\int
dm_{i}dm_{f}dm_{g}A_{i}(m_{i})A_{i}(m_{f})A_{g}(m_{g})\int\frac{d^{3}k}{(2\pi)^{3}}f_{i}(k)~{}O^{*}~{}v_{ic}\sigma_{ic},$
(11)

where $m_{i}$, $m_{f}$ and $m_{g}$ are respectively the incoming and outgoing
parton masses and the emitted gluon mass, $A_{i}$, $A_{f}$ and $A_{g}$ their
spectral functions in the DQPM, $f_{i}(k)$ is a distribution function of
parton $i$, and $v_{ic}$ and $\sigma_{ic}$ are the relative velocity and the
scattering cross section of the charm quark and parton $i$, respectively.

Figure 3: Momentum drag coefficient $\eta_{D}=A/p_{c}$ (left) and
$\hat{q}/T^{3}$ (right) of charm quark as a function of charm quark momentum
at $T=$ 200 MeV, excluding and including the radiative corrections with the
kinetic and the realistic upper limits of emitted gluon energy.

In Fig. 3 we present the drag coefficients $\eta_{D}=A/p_{c}$ (left) and
$\hat{q}/T^{3}$ (right) of the charm quark without and with soft gluon
emission for the kinematic and realistic upper limits of soft gluon energy at
$T=200$ MeV. The radiative corrections hardly change the transport
coefficients at low momentum, because the scattering energy of a slow charm
quark with a thermal parton is not large enough to produce an additional
massive gluon. As a result, the radiative scattering barely contributes to the
spatial diffusion coefficient $D_{s}=\lim_{p_{c}\to 0}T/(M_{c}\eta_{D})$ of
heavy quark [7], which is already well reproduced within the DQPM [3, 10].

The radiative energy loss, however, enhances $A$ and $\hat{q}$ of the charm
quark at large momentum. It may influence the $R_{\rm AA}$ of $D$ mesons which
has been well described only with elastic scattering in heavy-ion collisions
[11, 12]. According to our recent study [13] the $\alpha_{s}(T)$ extracted
from the lattice EoS overestimates jet quenching. More realistic strong
coupling at large momentum will be smaller than $\alpha_{s}(T)$, because an
energetic parton is far off thermal equilibrium, and it will cure the
potential problem raised from the enhancement of transport coefficients in
Fig. 3.

##  4 Conclusion

We have studied the radiative energy loss of heavy quark by using the soft
gluon approximation which is motivated by the soft photon approximation in
QED. It has been shown that for the pQCD case this approximation satisfies the
Slavnov-Taylor identities, which guarantees the gauge invariance, and the
results agree with those from the full calculations at the low energy of
emitted gluon. Extending it to the energy loss in non-perturbative strongly
interacting QGP within the DQPM, we have found that the radiative energy loss
little affects the spatial diffusion coefficient $D_{s}$ of heavy quark, while
it greatly enhances the transport coefficients at large momentum of heavy
quark, which can be suppressed by adopting a more realistic $\alpha_{s}$ for
the energetic scattering of partons.

The authors acknowledge support by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) through the grant CRC-TR 211 "Strong-interaction
matter under extreme conditions" - project number 315477589 - TRR 211. I.G.
also acknowledges support from the "Helmholtz Graduate School for Heavy Ion
research". The computational resources have been provided by the Goethe-HLR
Center for Scientific Computing.

## References

  * [1] J. Uphoff, O. Fochler, Z. Xu and C. Greiner, Phys. Lett. B 717, 430-435 (2012).
  * [2] J. Aichelin, P. B. Gossiaux and T. Gousset, Phys. Rev. D 89, no.7, 074018 (2014).
  * [3] H. Berrehrah, P. B. Gossiaux, J. Aichelin, W. Cassing and E. Bratkovskaya, Phys. Rev. C 90, no.6, 064906 (2014).
  * [4] F. E. Low, Phys. Rev. 110, 974-977 (1958).
  * [5] T. Song and P. Moreau, Phys. Rev. D 98, no.11, 116007 (2018).
  * [6] T. Song, Phys. Rev. C 107, no.2, 024916 (2023).
  * [7] T. Song, I. Grishmanovskii and O. Soloveva, Phys. Rev. D 107, no.3, 036009 (2023).
  * [8] P. Moreau, O. Soloveva, L. Oliva, T. Song, W. Cassing and E. Bratkovskaya, Phys. Rev. C 100, no.1, 014911 (2019).
  * [9] I. Grishmanovskii, O. Soloveva, T. Song, C. Greiner and E. Bratkovskaya, in preparation.
  * [10] T. Song and H. Berrehrah, Phys. Rev. C 94, no.3, 034901 (2016).
  * [11] T. Song, H. Berrehrah, D. Cabrera, J. M. Torres-Rincon, L. Tolos, W. Cassing and E. Bratkovskaya, Phys. Rev. C 92, no.1, 014910 (2015).
  * [12] T. Song, H. Berrehrah, D. Cabrera, W. Cassing and E. Bratkovskaya, Phys. Rev. C 93, no.3, 034906 (2016).
  * [13] I. Grishmanovskii, T. Song, O. Soloveva, C. Greiner and E. Bratkovskaya, Phys. Rev. C 106, no.1, 014903 (2022).


approximation with a remainder term to the estimated propensity score, we have

$\displaystyle D_{1}(g)=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i}g(X_{i})\left(Y_{i}-\gamma^{Y}(V_{i},D_{i})\right)\cdot$
$\displaystyle\left(\left(\frac{1}{\widehat{\beta}^{\prime}b(V_{i})}-\frac{1}{\beta^{\prime}b(V_{i})}\right)D_{i}+\left(\frac{1}{1-\widehat{\beta}^{\prime}b(V_{i})}-\frac{1}{1-\beta^{\prime}b(V_{i})}\right)\left(1-D_{i}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i}g(X_{i})\left(\widehat{\beta}-\beta\right)^{\prime}\frac{-b(V_{i})}{(\beta^{\prime}b(V_{i}))^{2}}\cdot
D_{i}\cdot\left(Y_{i}-\gamma^{Y}(V_{i},D_{i})\right)+$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i}g(X_{i})\left(\widehat{\beta}-\beta\right)^{\prime}\frac{b(V_{i})b(V_{i})^{\prime}}{(\widetilde{\beta}^{\prime}V_{i})^{3}}\left(\widehat{\beta}-\beta\right)\cdot
D_{i}\cdot\left(Y_{i}-\gamma^{Y}(V_{i},D_{i})\right)$ $\displaystyle=$
$\displaystyle\underbrace{\sqrt{n}\left(\widehat{\beta}-\beta\right)}_{O_{P}(1)}^{\prime}\underbrace{\frac{1}{n}\sum_{i}g(X_{i})\frac{-b(V_{i})}{(\beta^{\prime}b(V_{i}))^{2}}\cdot
D_{i}\cdot\left(Y_{i}-\gamma^{Y}(V_{i},D_{i})\right)}_{o_{P}(1)}+o_{P}(1)$

where $\widetilde{\beta}$ is a sequence between $\widehat{\beta}$ and $\beta$.
This remainder term therefore converges to zero. Uniform convergence of the
sample average follows from Lemma B.4: the random vector
$\frac{-b(V_{i})}{(\beta^{\prime}b(V_{i}))^{2}}\cdot
D_{i}\cdot\left(Y_{i}-\gamma^{Y}(V_{i},D_{i})\right)$ is mean-zero, has fixed
dimension, and bounded.

We can decompose $D_{2}(g)$ into the product of two terms

$D_{2}(g)=\underbrace{\sqrt{n}\left(\widehat{\gamma}-\gamma\right)}_{O_{p}(1)}^{\prime}\frac{1}{n}\sum_{i}g(X_{i})\left(\Delta
b(V_{i},D_{i})-\alpha(V_{i},D_{i})\cdot b(V_{i},D_{i})\right)$

Uniform convergence of the sample average again follows from Lemma B.4. We
thus conclude $D_{1}(g)$ and $D_{2}(g)$ vanish uniformly over
$g\in\mathcal{G}$ and over $P\in\mathcal{P}$.

For $D_{3}(g)$, we apply the Cauchy-Schwarz inequality to note that

$D_{3}(g)\leq\sqrt{n}\cdot\sqrt{\frac{1}{n}\sum_{i}\left(\widehat{\gamma}^{Y}(V_{i},D_{i})-\gamma^{Y}(V_{i},D_{i})\right)^{2}}\cdot\sqrt{\frac{1}{n}\sum_{i}\left(\widehat{\alpha}(V_{i},D_{i})-\alpha(V_{i},D_{i})\right)^{2}}$

The terms in the square root are the in-sample $L_{2}$ errors from the
estimated CEF and propensity score, which vanish at the rate of $n^{-1/2}$
uniformly over $P\in\mathcal{P}$ as shown in the paragraph above. We thus
conclude $D_{3}(g)$ vanishes uniformly over $g\in\mathcal{G}$ and over
$P\in\mathcal{P}$. ∎