code and the source and python scripts used to produce the plots in this paper
can be found at https://github.com/ICRAR/shark/.

## Acknowledgements

We thank Sabine Bellstedt, Kate Gould, John Weaver, Jen Hardwick and Nandini
Sahu for sharing observational data that has been used in this work. CL has
received funding from the ARC Centre of Excellence for All Sky Astrophysics in
3 Dimensions (ASTRO 3D), through project number CE170100013, and is a
recipient of the ARC Discovery Project DP210101945. MB has received funding
from McMaster University through the William and Caroline Herschel Fellowship.
DO and ASGR acknowledge support from the ARC Future Fellowship scheme
(FT190100083 and FT200100375, respectively). KP and ACG acknowledge Research
Training Program and ICRAR scholarships. This work was supported by resources
provided by The Pawsey Supercomputing Centre with funding from the Australian
Government and the Government of Western Australia.

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##  Appendix A Minor modifications to physical models

In addition to the major changes in physical models introduced in $\S$ 3, we
also make small modifications to existing models in Shark, as described below.

###  A.1 Evolving metal yield

In Shark v1.1, a constant yield from newly formed stars was assumed. In v2.0,
we allow alternative assumptions. In particular, we include a model that
assumes a constant ejecta metallicity regardless of the metallicity of the gas
that formed the stars. This in practice assumes an effective yield that
depends on the initial metallicity of the star-forming gas:

$p_{\rm eff}={p}-\frac{Z_{\rm cold}}{4},$ (56)

where $p=0.029$ and $Z_{\rm cold}$ is the metallicity of the gas from which
stars are being formed. The value of $p$ was computed in L18 using Conroy et
al. (2009) for a Chabrier 2003 IMF, and assuming a minimum and maximum masses
of stars formed, $m_{\rm min}=1\,\rm M_{\odot}$ and $m_{\rm max}=120\,\rm
M_{\odot}$, respectively. This model can be adopted using evolving_yield =
true in the parameter file.

###  A.2 Stellar feedback

The stellar feedback model adopted in the default Shark model in L18, is based
on that from Lagos et al. (2013), where the mass loading, the ratio between
the outflow and instantaneous SFR, $\dot{M}_{\rm out}/\psi$, is a function of
both redshift and the maximum circular velocity of the DM halo where the
galaxy resides, $f=f(z,v_{\mathrm{circ}})$ in the case of central galaxies.
For satellite galaxies, we use the maximum circular velocity the host halo of
the galaxies had right before becoming satellite subhalos. In Shark v2.0, we
include a minimum value for the mass loading, $\beta_{\mathrm{min}}$, such
that $f=\max(f(z,v_{\mathrm{circ}}),\beta_{\mathrm{min}})$. If $\beta_{\rm
min}=0$, then we recover the L18 model. The parameters adopted for the stellar
feedback model in this paper and L18 are shown in Table 2.

A comparison of the modelled mass loading-$v_{\mathrm{circ}}$ relation with
observations is shown in Fig. 16. Note that these observations have been
selected to avoid AGN galaxies, which can show higher mass loading but due to
AGN activity rather than stellar-driven winds. Also note that the uncertainty
in these measurements is larger than the errorbars show as systematic effects
are typically not accounted for when computing errors. Overall the
introduction of a floor in the mass loading allows the Shark v2.0 to agree
better with observations, and the slightly higher value adopted for $v_{\rm
hot}$ (see Table 2) also allows for better agreement in galaxies with $V_{\rm
circ}<200\,\rm km\,s^{-1}$.

Figure 16: Mass loading vs circular velocity at $z=0,\,1,\,2$ for Shark v2.0
(blue lines) and v1.1 (L18; red lines). Redshifts are ordered from darker to
lighter colours from $z=0$ to $z=2$. Observational estimates from Rupke et al.
(2005); Heckman et al. (2015); Chisholm et al. (2017); Sugahara et al. (2017)
are shown as symbols, as labelled.

###  A.3 Halo spin parameter dependence on halo mass

Shark v1.1 assumed that the distribution of the halo spin parameter was a log-
normal distribution of mean $0.03$, $\langle\lambda\rangle$, and width $0.5$.
In Shark v2.0, we assume that the width is independent of halo mass, but the
mean is weakly dependent on halo mass so that
$\langle\lambda\rangle=0.009\,\,{\rm log_{10}}(M_{\rm halo}/\rm
M_{\odot})\,-\,0.076$ which represents the $\langle\lambda\rangle$ dependence
on halo mass reported in Kim et al. (2015).

##  Appendix B The griffin19 BH spin model

In Shark v2.0, the spin of BHs is determined by the history of BH-BH mergers
and gas accretion. Below we summarise the models introduced in Shark v2.0 for
these processes.

###  B.1 BH-BH mergers

During galaxy mergers, the central supermassive BHs of each galaxy are also
merged. We assume that the mass of the final BH is the sum of the merging BH
masses, which assumes that the mass lost to gravitational radiation is
negligible. For the BH spin, we follow Lagos et al. (2009) and calculate the
resulting spin following two different models, which are applied depending on
the initial spins. We remind the reader that we define the norm of the BH spin
$a=|{\bf a}|$.

If the two BHs merging are non-rotating, which can happen due to our BH seed
assumption of non-rotating BHs, we follow Berti & Volonteri (2008), and
calculate the resulting spin, $a_{f}$, from the BH mass ratio,
$q=m_{2}/m_{1}$, where $q\leq 1$:

$a_{f}=\frac{2\,\sqrt{3}\,q}{(1+q)^{2}}-\frac{2.029\,q^{2}}{(1+q)^{4}},$ (57)

The first term of Eq. 57 comes from the orbital angular momentum of a particle
at the innermost stable circular orbit of a non-rotating BH, while the second
term in the equation accounts for the angular momentum radiated in the final
plunge.

If at least one of the BHs involved in the merger has a non-zero spin, we
instead apply the semi-analytic fitting formulae presented in Rezzolla et al.
(2008), where the authors fit an extensive compilation of numerical results on
BH mergers. Here, the two BHs with masses $m_{1}$ and $m_{2}$ have spins ${\bf
a}_{1}$ and ${\bf a}_{2}$, respectively, and the resulting spin is given by

$\displaystyle
a_{f}=\frac{1}{(1+q)^{2}}\,\Big{[}a^{2}_{1}+a^{2}_{2}\,q^{4}+2\,a_{1}\,a_{2}\,q^{2}\cos\phi\,+$
(58) $\displaystyle 2\,(a_{1}\,\cos\theta+a_{2}\,q^{2}\,\cos\xi)\,|{\bf
l}|\,q+|{\bf l}|^{2}\,q^{2}\Big{]}^{1/2},$

where $\mu$ is the symmetric mass ratio $q/(q+1)^{2}$, and ${\bf l}$ is the
contribution of the orbital angular momentum to $a_{f}$. Following Griffin et
al. (2019), we assume that the direction of ${\bf l}$ is that of the initial
orbital angular momentum, while its magnitude is given by:

$\displaystyle|{\bf
l}|=\frac{s_{4}}{(1+q^{2})^{2}}\,(a^{2}_{1}+a^{2}_{1}\,q^{4}+2\,a_{1}\,a_{2}\,q^{2}\,\cos\phi)\,+$
(59)
$\displaystyle\left(\frac{s_{5}\,\mu+t_{0}+2}{1+q^{2}}\right)\,(a_{1}\,\cos\theta+a_{2}\,q^{2}\,\cos\xi)\,+$
$\displaystyle 2\,\sqrt{3}+t_{2}\,\mu+t_{3}\,\mu^{2},$

where $s_{4}=-0.129$, $s_{5}=-0.384$, $t_{0}=-2.686$, $t_{2}=-3.454$,
$t_{3}=2.353$ are the values obtained in Rezzolla et al. (2008). The angles
$\phi$, $\theta$ and $\xi$ are the angles between the spins of the two BHs and
their orbital angular momentum, and are given by:

$\displaystyle\cos\phi$ $\displaystyle=$
$\displaystyle{\bf\hat{a}_{1}}\cdot{\bf\hat{a}_{1}},$ (60)
$\displaystyle\cos\theta$ $\displaystyle=$
$\displaystyle{\bf\hat{a}_{1}}\cdot\hat{\bf{l}},$ $\displaystyle\cos\xi$
$\displaystyle=$ $\displaystyle{\bf\hat{a}_{2}}\cdot\hat{\bf{l}}.$

During a BH-BH merger, we randomly select the directions for $\bf{a_{1}}$,
$\bf{a_{2}}$ and $\bf{l}$.

###  B.2 BH spin development during accretion events

In our implementation, an accretion event is caused by either of the three
processes bringing in gas to the central BH: hot-halo gas cooling, galaxy
mergers or disk instabilities. To calculate the BH spin after a gas accretion
episode, $a_{f}$, we follow Bardeen (1970):

$a_{f}=\frac{1}{3}\,\sqrt{\hat{r}_{\rm lso,i}}\,\frac{m_{\rm BH,i}}{m_{\rm
BH,f}}\,\left(4-\left[3\,\hat{r}_{\rm lso,i}\,\left(\frac{m_{\rm BH,i}}{m_{\rm
BH,f}}\right)^{2}-2\right]^{1/2}\right),$ (61)

where $\hat{r}_{\rm lso}$ is defined in Eq. 17, and $m_{\rm BH,i}$ and $m_{\rm
BH,f}$ are the BH masses before and after an accretion event. The latter are
related by:

$m_{\rm BH,f}=m_{\rm BH,i}\,+\,(1-\epsilon_{\rm TD})\,\Delta m,$ (62)

where $\Delta m$ is the mass accreted from the accretion disk in this
accretion episode and $\epsilon_{\rm TD}$ is defined in Eq. 16.

We follow Griffin et al. (2019) and describe the BH accretion disk as having
three regions: an outer disk at $r>r_{\rm in}$ and angular momentum ${\bf
J}_{\rm out}$; an inner disk at $r_{\rm warp}<r<r_{\rm in}$, which has angular
momentum ${\bf J}_{\rm in}$, where $r_{\rm warp}$ is the warp radius; and the
warped disk internal to $r_{\rm warp}$ with angular momentum ${\bf J}_{\rm
warp}$. If ${\bf J}_{\rm BH}$ (defined in $\S$ 3.3.1) and ${\bf J}_{\rm in}$
have different directions, a spinning BH induces a Lense-Thirring precession
in the misaligned disk elements. As the precession rate falls off as $r^{-3}$,
the expectation is that at smaller radii $\bf J_{\rm BH}$ and ${\bf J}_{\rm
in}$ become perfectly aligned or anti-aligned (Bardeen & Petterson, 1975). At
sufficiently large radii, however, there will still be a misalignment between
the vectors. The point at which the disk transitions from being (anti-)aligned
to misaligned happens at $r_{\rm warp}$.

We assume that at the beginning of a gas accretion event, ${\bf J}_{\rm warp}$
and ${\bf J}_{\rm in}$ are aligned. Due to torques, ${\bf J}_{\rm BH}$ aligns
with ${\bf J}_{\rm tot}={\bf J}_{\rm BH}+{\bf J}_{\rm warp}$; ${\bf J}_{\rm
tot}$ does not change during the torquing process. We follow King et al.
(2005) and assume that ${\bf J}_{\rm warp}$ completely aligns or anti-aligns
with ${\bf J}_{\rm BH}$. The gas internal to $r_{\rm warp}$ is then accreted
onto the BH. As this is an iterative process, and accretion from the internal
disk continues to happen, ${\bf J}_{\rm BH}$ eventually aligns with ${\bf
J}_{\rm in}$.

In Shark v2.0 we implement the same three options introduced in Griffin et al.
(2019) for the relation between the angular momentum of the outer and inner
disks, which are controlled by the parameter accretion-disk-model:

  * •

prolonged: assumes that ${\bf J}_{\rm out}$ is aligned with ${\bf J}_{\rm
in}$. This model generally leads to BHs that are maximally rotating (e.g.
Lagos et al. 2009).

  * •

selfgravitydisk: assumes the chaotic mode introduced in King et al. (2008),
which assumes that ${\bf J}_{\rm in}$ is randomly oriented with respect to
${\bf J}_{\rm out}$, and $r_{\rm in}$ to be equal to the disk self-gravitating
radius (described below).

  * •

warpeddisk: also assumes the chaotic mode introduced in King et al. (2008),
but follows the Volonteri et al. (2007) model for the warped disk, which
assumes $r_{\rm warp}$ is where the timescale for radial diffusion of the warp
due to viscosity is equal to the local Lense-Thirring precession timescale
(described below).

####  B.2.1 The King et al. (2008) self-gravitating accretion disk

In this mode, $r_{\rm in}=r_{\rm sg}$, where

$r_{\rm sg}=4790\,\alpha^{14/27}_{\rm TD}\,\dot{m}^{-8/27}\,\left(\frac{m_{\rm
BH}}{10^{8}\,\rm M_{\odot}}\right)^{-26/27}\,2\,r_{\rm G}.$ (63)

Here, $\dot{m}$ is the BH accretion rate divided by the Eddington accretion
rate (Eq. 14), $r_{\rm G}$ is the gravitational radius of the BH (defined in
$\S$ 3.3.1; note that $2\,r_{\rm G}$ is known as the Schwarzschild radius) and
$\alpha_{\rm TD}$ is the dimensionless Shakura & Sunyaev (1973) thin-disk
viscosity parameter. Following Griffin et al. (2019) we adopt $\alpha_{\rm
TD}=\alpha_{\rm ADAF}=0.1$.

Integrating the mass surface density profile of the accretion disk within
$r=[0,r_{\rm sg}]$ gives the self-gravity mass for the thin disk:

$m_{\rm sg}=1.35\,{\rm M_{\odot}}\,\alpha^{-4/5}_{\rm
TD}\,\dot{m}^{3/5}\,\left(\frac{m_{\rm BH}}{10^{8}\,\rm
M_{\odot}}\right)^{11/5}\,\left(\frac{r_{\rm sg}}{2r_{\rm G}}\right)^{7/5}.$
(64)

We refer to Section $2.5$ in Griffin et al. (2019) for the full derivation of
Eqs. 63 and 64.

Eqs. 63 and 64 are used to define $J_{\rm in}$,

$J_{\rm in}=\frac{m_{\rm sg}}{\sqrt{2}\,a\,m_{\rm BH}}\,\sqrt{\frac{r_{\rm
sg}}{2\,r_{\rm G}}}.$ (65)

In the first BH accretion episode of a timestep, we randomise the angle
between ${\bf J}_{\rm in}$ and ${\bf J}_{\rm BH}$, $\theta_{\rm i}$, between
$[0,\pi]$. The angle between ${\bf J}_{\rm in}$ and ${\bf J}_{\rm BH}$ after
the accretion event, $\theta_{\rm f}$, is calculated assuming conservation of
${\bf J}_{\rm tot}$,

${\rm cos}(\theta_{\rm f})=\frac{J_{\rm in}\,+\,J_{\rm BH}\,{\rm
cos}(\theta_{\rm i})}{\sqrt{J^{2}_{\rm BH}\,+\,J^{2}_{\rm in}+2\,J_{\rm
BH}\,J_{\rm in}\,{\rm cos}(\theta_{\rm i})}}.$ (66)

The BH spin $a_{\rm f}$ (Eq. 61) is calculated using $\Delta m=m_{\rm sg}$ and
calculating $\hat{r}_{\rm lso}$ using the initial spin of the BH prior to the
accretion of $m_{\rm sg}$. The direction of $a_{\rm f}$ depends on the angular
momenta ratio between the self-gravitating disk and the BH, $J_{\rm
in}/2\,J_{\rm BH}$:

$\displaystyle{\rm cos}(\theta_{\rm i})$ $\displaystyle\geq$ $\displaystyle-
J_{\rm in}/2\,J_{\rm BH}\,\,\,\,\,\,\,{\text{\bf J}_{\rm in}\,\text{and}\,{\bf
J}_{\rm BH}\,\text{align},}$ $\displaystyle<$ $\displaystyle-J_{\rm
in}/2\,J_{\rm BH}\,\,\,\,\,\,\,{\text{\bf J}_{\rm in}\,{\text{and}}\,{\bf
J}_{\rm BH}\,{\text{anti-align}}.}$

Once a fraction of $m_{\rm sg}$ is accreted by the BH, a new $\theta_{\rm i}$
is randomly chosen; the process continues until the whole accretion disk mass
is consumed.

####  B.2.2 The Volonteri et al. (2007) warped accretion disk

In this mode, we calculate $r_{\rm warp}$ and $m_{\rm warp}$ following
Volonteri et al. (2007),

$r_{\rm warp}=3410\,a^{5/8}\,\alpha^{-1/2}_{\rm
TD}\,\dot{m}^{-1/4}\,\left(\frac{m_{\rm BH}}{10^{8}\,\rm
M_{\odot}}\right)^{1/8}\,\left(\frac{\nu_{2}}{\nu_{1}}\right)^{-5/8}\,2\,r_{\rm
G},$ (68)

where $\nu_{1}$ and $\nu_{2}$ are the horizontal and vertical viscosities of
the accretion disk, respectively. We follow Papaloizou & Pringle (1983) and
assume $\nu_{1}/\nu_{2}\approx\alpha^{2}_{\rm TD}$, which is valid in the thin
disk approximation. This leads to $m_{\rm warp}$ that is defined like Eq. 64
but using $r_{\rm warp}$ instead of $r_{\rm sg}$. We refer to Section $2.6$ in
Griffin et al. (2019) for the full derivation of Eq. 68. We define $J_{\rm
warp}$ following Eq. 65 but using $m_{\rm warp}$ and $r_{\rm warp}$ instead of
$m_{\rm sg}$ and $r_{\rm sg}$, respectively.

In this model, $r_{\rm in}=r_{\rm sg}$, which is generally $>r_{\rm warp}$.
Thus, the BH accretes in chunks of $m_{\rm warp}$ until it consumes $m_{\rm
sg}$. After every accretion of $m_{\rm warp}$, we assume the new ${\bf J}_{\rm
warp}$ is aligned with ${\bf J}_{\rm in}$.

As with the selfgravitydisk model, the initial direction of $J_{\rm in}$ is
chosen randomly. Once $m_{\rm sg}$ is consumed, a new inner disk is formed
with updated values of $m_{\rm sg}$, $r_{\rm sg}$ and $J_{\rm in}$, and with a
new random direction for $J_{\rm in}$.

Note that because usually $m_{\rm warp}<m_{\rm sg}$, this model requires more
iterations to calculate the resulting spin than the selfgravitydisk model. In
principle, if enough iterations are taken, the BH spin would converge. We
carried out numerical tests and found that if $\gtrsim 50$ small chunks of
either $m_{\rm sg}$ or $m_{\rm warp}$ are accreted, $a_{\rm f}$ converges.
Hence to minimise the computing time, we introduce a maximum number of
iterations loop-limit-accretion that is by default set to $50$.

##  Appendix C Formation of hot halos

$\S$ 3.3.2 described how Shark v2.0 assumed the AGN radio-mode feedback to act
on halos that have developed a hot gaseous halo, following the description of
Correa et al. (2018). Fig. 17 shows the fraction of halos that have developed
a hot gaseous halo as a function of halo mass at different redshifts from
$z=6$ to $z=0$.

Figure 17: Fraction of halos that have developed a hot gaseous halo as a
function of halo mass in Shark v2.0 and following the model of Correa et al.
(2018). This is shown for several redshifts, as labelled.

The halo mass scale associated to the transition to most halos having
developed a hot gaseous halo is usually slightly below $10^{12}\,\rm
M_{\odot}$, and moves to slightly higher halo masses with time. The trend here
is similar to that reported in Correa et al. (2018) for the EAGLE simulations
(see their Fig. $13$).

##  Appendix D Computing the size-mass relation from GAMA data

Fig. 5 presents observations from Lange et al. (2016); Robotham et al. (2022);
Bellstedt et al. (2023), all of which use the GAMA survey. However, their
results can be very different, especially for the bulge components, so it is
worth explaining where these differences come from and how we compute the
size-mass relation from the data of Robotham et al. (2022); Bellstedt et al.
(2023).

The differences between the studies above are many, with Robotham et al.
(2022); Bellstedt et al. (2023) using the latest GAMA photometry, which
includes VST and VISTA observations (Bellstedt et al., 2020a) and more
sophisticated methods to simultaneously model the light profile and SED of
galaxies (Robotham et al., 2022). The latter results from the combination of
VST ugri and VISTA ZYJHKs photometry. Lange et al. (2016) used an older
version of the photometry from SDSS data (which is shallower) and fitted only
the r-band light with a single or two Sersic components without considering
mass-to-light ratio variations.

Bellstedt et al. (2023) classified their galaxies as single Sersic fits (split
between pure disks, pure spheroids and ambiguous), or double Sersic fits
(split between bulge+disk, point source+disk, and disk+disk). For the single
Sersic fit galaxies, the distinction between the three populations was: a
Sersic index $\leq 1.5$ corresponds to a pure disk; a Sersic index $\geq 2.5$
corresponds to a pure Spheroid; Sersic indices in the range $1.5-2.5$ are
considered “ambiguous”. Note that disk+disk galaxies are very few and we
ignore them here.

The stellar-mass size relation of galaxy disks in the top panel of Fig. 5 is
computed in two different ways: by combining pure disks, the disks of
bulge+disk and point source+disk galaxies (referred to as “disks”); and adding
ambiguous sources (referred to as “disks+ambig”). For the middle panel of Fig.
5, we again compute the size-mass relation in two ways: including pure
spheroids and bulges of bulge+disk galaxies (empty symbols) and also adding
the ambiguous sources (filled symbols). For the bottom panel of Fig. 5 and for
single Sersic galaxies, we simply take the best fit half-stellar mass radius
and derived SED-fitting stellar mass. For bulge+disk and point source+disk we
measure a stellar mass weighted half-mass radius as:

$r_{\star}=\frac{\left(r_{\star,\rm bulge}\,M_{\star,\rm bulge}+r_{\star,\rm
disk}\,M_{\star,\rm disk}\right)}{M_{\star}},$ (69)

with $M_{\star}=M_{\star,\rm bulge}+M_{\star,\rm disk}$. Note that in the case
of bulge+disk galaxies, $r_{\star,\rm bulge}$ is well defined, but in the case
of point source+disk galaxies it is not as the bulge is effectively
unresolved. For the latter we tested that the resulting relation remained
unchanged if we assumed $r_{\star,\rm bulge}<1$ kpc.


# Gauge fields renormalization groups and thermofractals

A. Deppman  Instituto de Física, Rua do Matão 1371-Butantã, São Paulo-SP, CEP
05580-090, Brazil<EMAIL_ADDRESS>E. Megías  Departamento de Física Atómica,
Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional,
Universidad de Granada, Avenida de Fuente Nueva s/n, 18071 Granada, Spain  D.
P. Menezes  Departamento de Física, CFM-Universidade Federal de Santa
Catarina, Florianópolis, SC-CP. 476-CEP 88.040-900, Brazil

###### Abstract

The perturbative approach to QCD has shown to be limited, and the difficulties
to obtain accurate calculations in the low-energy region seems to be
insurmountable. A recent approach uses the fractal structures of Yang-Mills
Field Theory to circumvent those difficulties, allowing for the determination
of an analytic expression for the running coupling. The results obtained are
in agreement with several experimental findings, and explain many of the
observed phenomena at high-energy collisions. In this work, we address some of
the conceptual aspects of the fractal approach, which are expressed in terms
of the renormalization group equation and the self-energy corrections to the
parton mass. We associate these well-known concepts with the origins of the
fractal structure in the quantum field theory.

March 2022

##  1 Introduction

In this paper we discuss the connections between fractals, Yang-Mills Fields
and QCD, addressing the most central conceptual aspect of these connections,
which appears in the scaling-properties of the field theory, the
renormalization procedure, and the complex structure of the effective parton.
A short review can be found in Ref. [1].

The data produced in high energy collisions have evidenced a robust feature of
the multiparticle production dynamics: the power-law distributions of the
particles energy or momenta, which is practically independent of the particle
species and the collision energy above $\sim~{}$1 TeV. The long-tail
distributions were already noticed some decades ago, being a critical failure
of the Hagedorn’s Self-Consistent Thermodynamics and Chew and Frautisch’s
Bootstrap Model for the hadron structure [2, 3]. See [4] for a short
description of the problems faced by these approaches, and how the theory
discussed here solves those problems.

A variety of explanations have been proposed, since then, to explain those
exotic distributions. Initially, it was supposed that the large momentum
sector would represent the exponential behaviour expected by the
thermodynamics models, while the low momentum peak would be affected by a
feed-up mechanism due to the particles decay after the freeze-out [5]. This
approach went into discredit when collisions at higher energy allowed the
production of a large number of more massive particles. Despite their longer
and more diverse decaying chains, the observed distribution resulted to be
similar to those of light particles, unveiling the universal character of the
multiparticle production process. A second approach reverted the way as the
long-tail was interpreted. In this case, the low momentum peak of the
distributions represents the decay of a thermodynamically equilibrated system,
while the high momentum region would be affected by hard-scattering that can
be described by perturbative QCD (pQCD). The calculations with a few orders in
pQCD give a reasonable result when compared with the long-tail region, but
fails in describing the low-energy peak [6].

A third approach uses Tsallis Statistics [7] to describe the thermodynamic
properties of the hot and dense system produced at high energy collisions [8,
9]. This approach can describe the entire range of momentum distribution in a
single theoretical framework [10, 11, 12, 13]. When the Hagedorn’s theory is
generalized with the introduction of Tsallis Statistics, it can describe not
only the high energy distributions but also the hadron mass spectrum, better
than the original theory can do. The non-extensive self-consistent
thermodynamics that results from this combined theory predicts that the
Hagedorn Temperature and the entropic index, $q$, must be both constants at
any collision energy above 1 TeV and for any particle species. Many works
presented analyses of experimental data showing that those predictions are in
good agreement with data [14, 15, 16, 17, 18], although some recent and more
accurate analyses, based on high statistics data, indicate a small variations
of $q$ with energy and particle species [19, 20, 21, 22, 23]. A comprehensive
account of the subject can be found in Ref. [24]

At least three different mechanisms were proposed to explain the emergence of
the non-extensive statistics in HEP: the temperature fluctuations [13], the
small size of the system [25] and the fractal structure [26]. All these
mechanisms result in the same q-exponential distributions for the
multiparticle production process, that can fit the data rather accurately.
There are some reasons to believe that the three different mechanisms are
indeed three different aspects of the same phenomenon. The fractal structure
leads to temperature fluctuations that are the same used to show the emergence
of Tsallis statistics. The small size description is also appropriate to
explain the emergence of non-extensivity in the fractal structure, as will be
discussed below.

The fundamental property of QCD that leads to the appearance of fractal
structures is the scaling invariance of the vertex-functions, which is
described by the Callan-Symanzik equation [27, 28]. The scaling invariance is
also one of the most important features of fractals. However, this property
alone is not sufficient to characterize a system as a fractal, since these
systems need to present a complex internal structure. Combined, the scaling
property and the internal structure produce the major feature of fractals: the
self-similarity. The vacuum polarization is an essential part of the
interaction not only in QCD but also in QED and Yang-Mills fields in general,
and the vacuum structure is an important component of partons interactions and
the parton self-energy [29].

##  2 Fractal structures in Yang-Mills fields

Figure 1: Pictorial representation of the effective parton and their
interaction. (a) The vacuum polarization represented as the internal structure
of the effective parton. (b) In the effective parton interaction the vacuum
structure participates in a complex way.

(a) (b)

The partonic dynamics is described, in terms of the Dyson-Schwinger expansion
[30], as

$\ket{\Psi_{n}}=\sum_{\\{n\\}}(-i)^{n}\int dt_{1}\dots
dt_{n}\ket{\Psi}\bra{\Psi}e^{-iH_{o}(t_{n}-t_{n-1})}g\dots
e^{-iH_{o}(t_{1}-t_{o})}\ket{\Psi_{o}},$ (1)

where the $n$th order quantum state $\ket{\Psi_{n}}$ is described in terms of
the states $\ket{\Psi}$, that represent the free elementary parton, with an
initial state, $\ket{\Psi_{o}}$. In the perturbative expansion of strongly
interacting fields, the Hamiltonian has two parts, $H_{o}$ that represents the
self-energy contribution, and $H_{I}=1+g\,dt$, that is the perturbative
contribution of the proper-vertex interactions. A summation over all possible
states is implicit in the terms $\ket{\Psi}\bra{\Psi}$.

The effective parton, which includes the self-energy interaction in the
propagation of the elementary parton [30, 31], has states given by

$\ket{\Psi_{e}(t)}=e^{-iH_{o}(t-t_{o})}\ket{\Psi}\,.$ (2)

The Dyson-Schwinger expansion can be written in terms of the effective parton
states as

$\ket{\Psi_{e}(t_{n})}=\sum_{\\{n\\}}(-i)^{n}\int dt_{1}\dots
dt_{n}\ket{\Psi_{e}(t_{n})}\bra{\Psi_{e}(t_{n})}g\ket{\Psi_{e}(t_{n-1})}\dots
g\ket{\Psi_{e}(t_{o})}.$ (3)

Observe that $g$ represents proper-interaction vertices for the effective
partons. As Equation 2 shows, the effective parton is a complex object that
carries in its wave-function all the properties of its interaction with the
vacuum. The structure of the effective parton is depicted in Figure 1(a),
where the vacuum polarization is represented by the $+$ and $-$ signs
surrounding the elementary parton, represented by the central circle. In this
sense we say that the effective parton has an internal structure. The
complexity of this structure can be evaluated by the number of Feynman graphs
necessary to describe the self-interaction contributions even in low-orders of
calculation can be appreciated by simply inspecting the figures in Ref. [32].

The proper-vertex interaction is still more complex, as can be observed in
Figure 1(b). The interaction is mediated by another parton (boson) which has
its own self-energy contributions. The detailed description of all possible
configurations is a huge challenge to perturbative QCD. The present situation
indicates that the perturbative-QCD approach will not be able to provide an
accurate calculation of the running-constant at low energies with a reasonable
number of loops in the calculations.

The physical parton that can be experimentally observed is the effective
parton, hereby, we only have access to the effective coupling, $g$. The
Callan-Symanzik equation is the renormalization-group equation for the QCD,
and applies to the effective parton and the effective coupling. It is written
as

$\left[M\frac{\partial}{\partial
M}+\beta_{g}\frac{\partial}{\partial\bar{g}}+\gamma\right]\Gamma=0\,,$ (4)

where $M$ is the scale parameter, and $\gamma$ is related to the form the
effective parton fields scale. This equation establishes the scaling symmetry
of the effective parton. Therefore, we get the two necessary conditions to
fulfill the self-similarity of the physical parton: the scaling-free property
and the internal structure. The work developed in [33] used these properties
of the partons in any Yang-Mills field theory to associate the thermodynamical
properties of those fields with the properties of thermofractals. A short
review on the subject can be found in Ref. [1].

There are two types of thermofractals [26], and each one presents the
following properties [33]:

  1. 1.

It is a thermodynamics system with a complex structure with a number $N$ of
components that present the same properties as the parent system. It has a
total energy $U=E+K$, where $E$ is the internal energy of its components and
$K$ is their total kinetic energy

  2. 2.

In the case of the thermofractal type-I, the internal energy, $E$, and the
kinetic energy, $K$, of the components of the system are such that the ratio
$\varepsilon/\lambda=E/K$ follows a distribution $\tilde{P}(\varepsilon)$. In
the case of the thermofractal type-II, it is the ratio
$\varepsilon/\lambda=E/U$ that follows a distribution
$\tilde{P}(\varepsilon)$. Here, $\lambda$ is a scaling parameter and
$\varepsilon$ is the energy of the components of the thermofractal.

  3. 3.

At some level of the internal structure, the fluctuations of the internal
energy of the system are small enough to be disregarded. Then their internal
energy can be regarded as constant.

The systems that present these properties must satisfy a recursive equation
for the probability density, $\tilde{P}(\varepsilon)$, for which the only
solution is [33, 26]

$\tilde{P}(\varepsilon)=\left[1\pm(q-1)\frac{\varepsilon}{\lambda}\right]^{\frac{\mp
1}{q-1}}\,,$ (5)

where $q$ is related to $N$, and the exponent $1/(q-1)$ gives the number of
degrees of freedom of the interacting system. Due to the property 3 of the
thermofractals, at some value of $\lambda$ the scaling-symmetry is broken.
Fixing $\lambda$ to this value transforms the function in Equation 5 in a
q-exponential without (sign $+$ in the argument) and with (sign $-$ in the
argument) cut-off. These functions are related to the Tsallis Statistics, and
here they represent the connection between the thermofractals and the non-
extensive statistics. The Yang-Mills fields form fractals of type-II [33].

The fractal structure of Yang-Mills Fields were already investigated
numerically [34, 35]. The predictions of non-extensive thermodynamics theory
discussed here were compared to Lattice QCD results, showing a good agreement
[36].

Notice that the second property of the thermofractals expresses the self-
similarity of the thermofractals, and this is a fundamental aspect to obtain
the recursive relation that leads to Tsallis Statistics. The q-exponential
function describes how the energy and momenta of particles are shared at every
vertex, and therefore is associated to the effective coupling by [33]

$g(\varepsilon)=G_{o}\prod_{i=1}^{2}\left[1+(q-1)\frac{\varepsilon_{i}}{\lambda}\right]^{\frac{-1}{q-1}}\,,$
(6)

where $i$ indicates the two particles in the final state at each vertex and
$G_{o}$ is the overall strength of the interaction. This coupling satisfies
the Callan-Symanzik Equation, and the connection between Tsallis distributions
and the renormalization group equation were already investigated in [37].

An essential aspect of these systems is the fact that the number of degrees of
freedom, that is given by the exponent of the functions in Equation 6, is
independent of the size of the system, that is given by the ratio
$\varepsilon/\lambda$. In the case of systems following the Boltzmann-Gibbs
Statistics, the number of degrees of freedom is connected to the size of the
system. For an ideal gas, for instance, the number of degrees of freedom is
proportional to the number of particles, and increases to infinity in the
thermodynamical limit. In the case of Yang-Mills fields, the parameter $q$ can
be calculated by using the fundamental parameters of the quantum field theory.
For QCD, it was shown that [33, 38] $(q-1)^{-1}=(1/3)[11n_{c}-(4/2)n_{f}]$
where $n_{c}$ is the number of colours and $n_{f}$ the number of flavours. The
theoretical value obtained for $q$ is in good agreement with those found in
experimental data analyzes. A short review on the subject can be found in Ref.
[1].

##  3 Discussion

The perturbative approach to Yang-Mills field theories present a complex
internal structure that we associated with the self-energy interactions. Since
the effective partons field must satisfy the renormalization group equation,
they present also a scale-free symmetry. We argued that the combination of
these two properties leads to the self-similarity of the effective parton
structure. This is the main reason behind the formation of the fractals
structures.

The fractal structures lead to recursive equations. The solution to those
equations leads to q-exponential functions. From the statistical point of
view, the structure is associated with thermofractals [26]. This kind of
system was introduced to explain the fractal origin of the non-extensive
statistics described in terms of the Tsallis entropy. Therefore, we show that
Yang-Mills systems are described thermodynamically by the Tsallis
thermodynamics [39, 40, 41]. In Ref. [33] the q-exponential functions of the
partonic energy were considered as terms of the effective coupling in the
interaction among the effective partons. Thus, even for QCD, the effective
coupling is obtained non-perturbatively as an analytical expression. The self-
similarity has been observed in high-energy collisions [42, 43, 44].

From the three parameters in the effective coupling, namely $q$, $\lambda$ and
$G_{o}$, the first one is determined from the field theory parameters,
$\lambda$ is the point where the scale symmetry is broken, and only $G_{o}$
remains to be determined. It represents the overall strength of the
interaction, corresponding to the coupling constant at collision energies
close to zero, and can be found by fittings to the experimental data. For QCD,
the value for $q$ found theoretically is in good agreement with the value
found in experimental data analyses [10, 45, 46, 47].

Besides providing the effective coupling and the value for $q$, the theory
also explains why Tsallis statistics must be used to describe the
distributions obtained in high-energy collisions, since the use of the
coupling constant in the form given in Equation 6 results in vertex functions
corresponding to Tsallis distributions. It can be interesting to investigate
if the subtle deviations of the $q$-exponential distribution [20, 48] might
indicate effects related to the finite quark mass of some flavours, that were
not considered in the theoretical approach.

It is also possible to use the theory to calculate the fractal dimension. This
dimension can be accessed experimentally by the use of intermittency analysis
[49, 50, 51]. The calculated value [26], $D=0.69$, is in good agreement with
the experimental data analysis [52, 53, 54, 55, 56]. A consequence of the
fractal dimension and of the non-extensivity, is that the multiplicity of
particles produced in high-energy collisions follows a power-law function of
the energy [33]. The exponent predicted by the theory is in accordance with
the one found in experimental data analysis [57].

The theory discussed here reconciles the Hagedorn Self-Consistent
Thermodynamics [2] with the high-energy experiments. This happens because, as
we observed in the present work, the effective parton is a thermofractal of
type-II [26], so it must be described by Tsallis Statistics [7] and its
associated thermodynamics [39, 40]. The non-extensive self-consistent
thermodynamics [26] that results from the generalization of the Hagedorn’s
principle can describe the high-energy distributions, and predicts a constant
value for $q$. But it also gives a new formula for the hadron mass-spectrum,
which can describe the observed hadronic states even at masses as low as the
pion mass [46]. This is an indication that hadrons at the normal conditions
can be described by the same theory that describes the hot and dense medium
created at high energy collisions. This aspect opens the possibility [58] to
apply the theoretical approach discussed here to other systems, as hadron
states [59, 60], cosmic rays [61] and neutron-stars [62, 63, 64].

##  4 Conclusions

We discussed the fractal structures present in Yang-Mills fields, analyzing in
details the concept of effective parton to show that it presents all the
necessary features of a fractal. The self-similarity of the effective partons
is the main characteristic that leads to recursive relations that allows for
obtaining the running-coupling of QCD at the non-perturbative regimes.

We argue that the introduction of the proper-vertex in the perturbative
approach to quantum fields calculation, that separates the self-energy
contributions from the interaction mechanism, and the renormalization theory
that determines the scaling properties of the effective parton, are the
determinant features for the formations of the fractal structures.

A result of the introduction of the fractal methods in the quantum field
theory is the possibility to determine the running-constant by an analytical
formula with one single free parameter. The Tsallis distributions observed in
high-energy collisions emerges naturally from the present approach. This
theory opens the possibility to investigate several aspects of high-energy
collisions and of hadrons. The fractal dimension, the non-extensive behaviour,
and the self-similarity can be tested experimentally.

Several possible applications of the theory are mentioned, some of them
already with interesting results.

##  5 Acknowledgments

A D and D P M are partially supported by the Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq-Brazil) and by Project INCT-FNA
Proc. No. 464 898/2014-5. A D is partially supported by FAPESP under grant
2016/17612-7. The work of E M is supported by the project PID2020-114767GB-I00
financed by MCIN/AEI/10.13039/501100011033, by the FEDER/Junta de Andalucía-
Consejería de Economía y Conocimiento 2014-2020 Operational Programme under
Grant A-FQM-178-UGR18, by Junta de Andalucía under Grant FQM-225, and by the
Consejería de Conocimiento, Investigación y Universidad of the Junta de
Andalucía and European Regional Development Fund (ERDF) under Grant
SOMM17/6105/UGR. The research of E M is also supported by the Ramón y Cajal
Program of the Spanish MCIN under Grant RYC-2016-20678.

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