# Bare Higgs mass and potential at ultraviolet cutoff

Yuta Hamada  Hikaru Kawai  Department of Physics, Kyoto University, Kyoto
606-8502, Japan  Kin-ya Oda  Department of Physics, Osaka University, Osaka
560-0043, Japan

###### Abstract

We first review the current status of the top mass determination paying
attention to the difference between the $\overline{\text{MS}}$ and pole
masses. Then we present our recent result on the bare Higgs mass at a very
high ultraviolet cutoff scale.

##  I Introduction

It is more and more likely that the 125 GeV particle discovered at the Large
Hadron Collider (LHC) Aad:2012tfa ; Chatrchyan:2012ufa  is the Standard Model
(SM) Higgs. Its couplings to the $W$ and $Z$ gauge bosons, to the top and
bottom quarks, and to the tau lepton are all consistent to those in the SM
within one standard deviation even though their values vary two orders of
magnitude, see e.g. Ref. Giardino:2013bma . No hint of new physics beyond the
SM has been found so far at the LHC up to 1 TeV. It is important to examine to
what scale the SM can be a valid effective description of nature.

In Ref. Froggatt:1995rt , Froggatt and Nielsen have predicted the top and
Higgs masses to be $173\pm 5\,\text{GeV}$ and $135\pm 9\,\text{GeV}$,
respectively, based on the assumption that the SM Higgs potential must have
another minimum at the Planck scale and that its height is order-of-magnitude-
wise degenerate to the SM one. (This assumption is equivalent to the vanishing
Higgs quartic coupling and its beta function at the Planck scale.) The success
of this prediction indicates that at least the top-Higgs sector of the SM is
not much altered up to a very high ultraviolet (UV) cutoff scale.

As all the parameters in the SM are fixed by the Higgs mass determination, we
can now obtain the _bare_ parameters at the UV cutoff scale, which then become
important inputs for a given UV completion of the SM. If the UV theory fails
to fit them, it is killed.

The parameters in the SM are dimensionless except for the Higgs mass (or
equivalently its vacuum expectation value (VEV)). The dimensionless bare
coupling constants can be approximated by the running ones at the UV cutoff
scale, see e.g. Appendix of Ref. Hamada:2012bp . The latter can be evaluated
through the standard renormalization group equations (RGEs) once the low
energy inputs are given. After fixing all the dimensionless bare couplings,
the last remaining one is the bare Higgs mass which is the main subject of
this work.

##  II Top quark Yukawa coupling at top mass scale

The largest ambiguity for the Higgs mass parameter is coming from the low
energy input of the top Yukawa coupling at the top mass scale. Let us review
the present status of its determination. Important point is the distinction
between the modified minimal subtraction ($\overline{\text{MS}}$) mass and the
pole one, which are utilized, respectively, in the $\overline{\text{MS}}$ and
on-shell schemes.

Currently the most accurate value of the top quark mass is obtained from
combination of the Tevatron data, basically reconstructed as an invariant mass
of its decay products CDF:2013jga :

$\displaystyle M_{t}^{\text{Tevatron}}$ $\displaystyle=173.20\pm
0.87\,\text{GeV},$ (1)

whereas the similar analysis of the LHC data gives LHC top mass ;
Deliot:2013hc : $M_{t}^{\text{LHC}}=173.3\pm 1.4\,\text{GeV}$. (If we naively
combine these two results, we get $M_{t}^{\text{inv}}=173.2\pm
0.7\,\text{GeV}$, which is of 0.4% accuracy.) However, the authors of Ref.
Alekhin:2012py  criticize that the top quark mass, measured at the Tevatron
and LHC via kinematical reconstruction from the top quark decay products and
comparison to Monte Carlo simulations, is not necessarily the pole mass
$M_{t}$ but is merely the mass parameter in the Monte Carlo program which does
not resort to any given renormalization scheme. The point is that the mass of
the colored top quark is reconstructed from the color singlet final states.111
Note however that the pole mass of the colored quark is well defined to all
orders in perturbation theory and that its infrared renormalon ambiguity
appears only at the non-perturbative level of order $\Lambda_{\text{QCD}}$.
To circumvent this problem, they propose to determine the
$\overline{\text{MS}}$ top quark mass directly from the dependence of the
inclusive $t\bar{t}$ cross section on it. In Ref. Alekhin:2012py , the
observed values at Tevatron:

$\displaystyle\sigma(p\bar{p}\to t\bar{t}+X)$
$\displaystyle=7.56^{+0.63}_{-0.56}\,\text{pb}\quad\text{(D0)}\qquad\text{and}\qquad
7.50^{+0.48}_{-0.48}\,\text{pb}\quad\text{(CDF)}$ (2)

are combined and fit by the theoretical prediction, which is obtained by using
four different parton distribution functions (PDFs) at the NNLO and by
including the NNLO QCD contributions to $\sigma(p\bar{p}\to t\bar{t}+X)$. The
resultant value of the $\overline{\text{MS}}$ running top mass at the top mass
scale becomes Alekhin:2012py :

$\displaystyle m_{t}^{\text{QCD}}(M_{t})$ $\displaystyle=163.3\pm
2.7\,\text{GeV}.$ (3)

In the above computation, the NLO electroweak (EW) radiative corrections
($\propto\alpha\alpha_{s}$) to $\sigma(p\bar{p}\to t\bar{t}+X)$ are neglected.
The ratio of the sum of such EW corrections to the $t\bar{t}$ total cross
section at the Tevatron in the on-shell scheme is shown to be less than 0.2%
for the Higgs mass 120–200 GeV and the top pole mass 165–180 GeV Kuhn:2006vh .

Theoretically the $\overline{\text{MS}}$ mass in Eq. (3) is related to the
pole mass $M_{t}$ by

$\displaystyle m_{t}^{\text{QCD}}(M_{t})$
$\displaystyle=M_{t}\left(1+\delta_{t}^{\text{QCD}}(M_{t})\right),$ (4)

where up to the NNLO QCD corrections of $O(\alpha_{s}^{3})$, see e.g. Ref.
Jegerlehner:2012kn ,

$\displaystyle\delta_{t}^{\text{QCD}}(M_{t})$ $\displaystyle=-{4\over
3}{\alpha_{s}(M_{t})\over\pi}-9.125\left(\alpha_{s}(M_{t})\over\pi\right)^{2}-80.405\left(\alpha_{s}(M_{t})\over\pi\right)^{3},$
(5)

with $\alpha_{s}(\mu)=g_{s}^{2}(\mu)/4\pi$ being the strong coupling in the
six flavor $\overline{\text{MS}}$ scheme. This relation result in the pole
mass Alekhin:2012py

$\displaystyle M_{t}$ $\displaystyle=173.3\pm 2.8\,\text{GeV}.$ (6)

In Ref. Alekhin:2012py  the pole mass of the top quark $M_{t}$ is also
directly extracted from the NNLO theory prediction using the on-shell scheme.
The resultant central value varies $169.9$–$172.7$ GeV, depending on the PDF,
with the error less than 2.4 GeV for each. These values are consistent to Eq.
(6). To summarize, the derived pole mass is close to the experimentally
obtained invariant mass (1).

It is customary to define the $\overline{\text{MS}}$ running VEV $v(\mu)$ in
such a way that

$\displaystyle-m^{2}(\mu)=\lambda(\mu)\,v^{2}(\mu)$ (7)

holds for the potential

$\displaystyle\mathcal{V}=m^{2}\phi^{\dagger}\phi+\lambda\left(\phi^{\dagger}\phi\right)^{2},$
(8)

with $\left\langle\phi\right\rangle=v/\sqrt{2}$. Then the
$\overline{\text{MS}}$ top mass is commonly defined as, see e.g. Ref.
Jegerlehner:2012kn ,

$\displaystyle m_{t}(\mu)$ $\displaystyle={y_{t}(\mu)\,v(\mu)\over\sqrt{2}}.$
(9)

This definition of the $\overline{\text{MS}}$ mass leads to Jegerlehner:2012kn

$\displaystyle m_{t}(M_{t})$
$\displaystyle=m_{t}^{\text{QCD}}(M_{t})+M_{t}\,\Delta_{t}^{\text{EW}}(M_{t}),$
(10)

where, taking into account up to NLO EW contributions of
$O(\alpha\alpha_{s})$,

$\displaystyle\Delta_{t}^{\text{EW}}(M_{t})$
$\displaystyle=0.0664-0.00115\left({M_{H}\over\text{GeV}}-125\right).$ (11)

This discrepancy (11) is due to the definition of the $\overline{\text{MS}}$
top mass via Eq. (9), and is dominantly coming from the tadpole contribution
to the shift of $v(\mu)$. If we instead use the definition of the
$\overline{\text{MS}}$ mass $m_{t}^{\text{QCD}}(\mu)=y_{t}(\mu)V/\sqrt{2}$
with $V=\left(\sqrt{2}G_{\mu}\right)^{-1/2}=246.22$ GeV, where
$G_{\mu}=1.1663787(6)\times 10^{-5}\,\text{GeV}^{-2}$ is the Fermi constant
determined from the muon life time, then we would get the one given in Eq.
(4).

Plugging Degrassi:2012ry

$\displaystyle g_{s}(M_{t})$
$\displaystyle=1.1645+0.0031\left({\alpha_{s}(M_{Z})-0.1184\over
0.0007}\right)-0.00046\left({M_{t}\over\text{GeV}}-173.15\right)$ (12)

into Eq. (10), we get

$\displaystyle m_{t}(M_{t})$
$\displaystyle=M_{t}\left[1.00658-0.00041\left({\alpha_{s}(M_{Z})-0.1184\over
0.0007}\right)+0.00006\left({M_{t}\over\text{GeV}}-173.15\right)-0.00115\left({M_{H}\over\text{GeV}}-125\right)\right].$
(13)

The $\overline{\text{MS}}$ mass (10) becomes _larger_ than the top quark pole
mass $M_{t}$ Jegerlehner:2012kn .

Let us review the derivation of $\Delta_{t}^{\text{EW}}(M_{t})$ more in
detail. First the $\overline{\text{MS}}$ running VEV $v(\mu)$ is obtained and
then it is multiplied to the $\overline{\text{MS}}$ running Yukawa
$y_{t}(\mu)$ in order to obtain the running mass $m_{t}(\mu)$ in Eq. (10).
$v(\mu)$ can be read from the Fermi constant
$G_{F}(\mu)=1/\sqrt{2}v^{2}(\mu)$:

$\displaystyle G_{\mu}$
$\displaystyle=G_{F}(\mu)\left(1+\Delta_{G_{F},\alpha}+\Delta_{G_{F},\alpha\alpha_{s}}+\cdots\right)=G_{F}(\mu)\left[1+{\alpha_{2}(\mu)\over
4\pi}{m_{t}^{4}(\mu)\over
m_{W}^{2}(\mu)\,m_{H}^{2}(\mu)}\left(6-12\ln{m_{t}(\mu)\over\mu}\right)+\cdots\right].$
(14)

The $O(\alpha)$ and $O(\alpha\alpha_{s})$ contributions
$\Delta_{G_{F},\alpha}$ and $\Delta_{G_{F},\alpha\alpha_{s}}$ are given in
Eqs. (A.3) and (A.6) in Ref. Bezrukov:2012sa . The dominant tadpole
contribution is picked up in the last step in Eq. (14) for explicitness. The
resultant $\overline{\text{MS}}$ VEV is $v(M_{t})\sim 260$ GeV at the top mass
scale. On the other hand, the $\overline{\text{MS}}$ Yukawa coupling is given
by Hempfling:1994ar

$\displaystyle y_{t}(\mu)$
$\displaystyle={M_{t}\over\sqrt{2}V}\left(1+\delta_{t}^{\text{QCD}}(\mu)+\delta_{t,\alpha}^{\text{QED}}(\mu)+\delta_{t,\alpha}^{W}(\mu)+\delta_{t,\alpha\alpha_{s}}^{\text{EW}}(\mu)+\cdots\right),$
(15)

where the $O(\alpha)$ corrections are

$\displaystyle\delta_{t,\alpha}^{\text{QED}}(\mu)$
$\displaystyle={Q_{t}^{2}\alpha(\mu)\over
4\pi}\left(6\log{M_{t}\over\mu}-4\right),$ (16)
$\displaystyle\delta_{t,\alpha}^{W}(\mu)$
$\displaystyle={G_{\mu}\,m_{t}^{2}(\mu)\over
8\pi^{2}\sqrt{2}}\left[-\left(2N_{c}+3\right)\ln{M_{t}\over\mu}+{N_{c}\over
2}+4-r+2r\left(2r-3\right)\ln(4r)-8r^{2}\left({1\over
r}-1\right)^{3/2}\arccos\sqrt{r}\right],$ (17)

with $Q_{t}=2/3$, $N_{c}=3$, and $r=M_{H}^{2}/4M_{t}^{2}$. The resultant
explicit analytic formula of $\Delta_{t}^{\text{EW}}(M_{t})$ is given in Ref.
Jegerlehner:2003py  that takes into account up to the NLO EW corrections of
$O(\alpha\alpha_{s})$. The tiny $O(\alpha\alpha_{s})$ correction to the Yukawa
coupling, $\delta_{t,\alpha\alpha_{s}}^{\text{EW}}(\mu)$ in Eq. (15), can be
read off from $\Delta_{t,\alpha\alpha_{s}}^{\text{EW}}$ by subtracting the
tadpole contribution to $v(\mu)$. In Ref. Degrassi:2012ry , numerical value of
Eq. (15) is evaluated as

$\displaystyle y_{t}(M_{t})$
$\displaystyle=0.93587+0.00557\left({M_{t}\over\text{GeV}}-173.15\right)-0.00003\left({M_{H}\over\text{GeV}}-125\right)-0.00041\left(\alpha_{s}(M_{Z})-0.1184\over
0.0007\right)\pm 0.00200_{\text{th}}.$ (18)

Multiplying Eq. (18) by the running VEV $v(\mu)$ that is read from Eq. (14),
we obtain $\Delta_{t}^{\text{EW}}$, and hence the running mass (13).

##  III Bare parameters at high scale

In this proceedings we show our result Hamada:2012bp  based on Eq. (18) with
the pole mass (6). In Figure 1, we show a plot with the two loop RGEs,
summarized in Ref. Hamada:2012bp , for the dimensionless SM couplings, with
the low energy boundary condition for the top Yukawa as explained above,
namely with $y_{t}(M_{t})=0.93587$. $\beta_{\lambda}$ is the beta function for
the Higgs quartic coupling: $\beta_{\lambda}=d\lambda/d\ln\mu$.
$m_{B}^{2}/I_{1}$ is explained in the following.

Figure 1: RGE running of the SM couplings and of the beta function for the
quartic coupling $\beta_{\lambda}$, except for $m_{B}^{2}$ which is _not_ a
running mass but is the (quadratically divergent part of) bare Higgs mass
parameter, obtained by taking each scale on the horizontal axis to be the
cutoff $\Lambda$. $I_{1}=\Lambda^{2}/16\pi^{2}$ is the one loop integral.

In the bare perturbation theory, the renormalized Higgs mass squared parameter
is given at the one loop level by

$\displaystyle m_{R}^{2}$
$\displaystyle=m_{B,\text{1-loop}}^{2}+\left(6\lambda_{B}+\frac{3}{4}g_{YB}^{2}+\frac{9}{4}g_{2B}^{2}-6y_{tB}^{2}\right)I_{1}+\delta
m^{2},$ (19)

where $I_{1}=\int^{\Lambda}d^{4}p/(2\pi)^{4}p^{2}=\Lambda^{2}/16\pi^{2}$ is
the quadratically divergent one loop integral over the Euclidean momentum. In
obtaining the expression (19), we assume existence of an underlying gauge
invariant regularization, such as string theory, and freely shift the
integrated momenta. The requirement that the sum in the parentheses in Eq.
(19) to be zero is the celebrated Veltman condition. In a mass independent
renormalization scheme, including the dimensional regularization, the bare
mass squared $m_{B}^{2}$ is chosen in such a way that the renormalized mass
parameter becomes zero, $m_{R}^{2}=0$, when $\delta m^{2}=0$; and then non-
zero $\delta m^{2}$ is introduced as a perturbation. (This choice of the bare
mass $m_{B}^{2}$ to cancel the quadratic divergence is automatic in the
dimensional regularization scheme.) Consequently the bare mass $m_{B}^{2}$
contains a quadratic divergence: $\Lambda^{2}$, whereas the running mass
$\delta m^{2}$ only logarithmic one: $\log\Lambda$. Note that this
cancellation of $\Lambda^{2}$ by $m_{B}^{2}$ is done once and for all, and
then we never see $\Lambda^{2}$. Bardeen has argued that therefore the
quadratic divergence is not a real problem, see e.g. Ref. Iso:2013aqa  for a
recent review. However, e.g. in obtaining a low energy effective field theory
from string theory, this procedure of matching $\Lambda^{2}$ by $m_{B}^{2}$ is
not a fake, and we focus on this largest part $m_{B}^{2}$ in the bare
Lagrangian, neglecting the subleasing $\delta m^{2}\propto v^{2}\log\Lambda$.

In Ref. Hamada:2012bp  we have obtained the bare Higgs mass at two loop orders
in the bare perturbation theory:

$\displaystyle m_{B,\,\text{2-loop}}^{2}$
$\displaystyle=-\bigg{\\{}9y_{tB}^{4}+y_{tB}^{2}\left(-\frac{7}{12}g_{YB}^{2}+\frac{9}{4}g_{2B}^{2}-16g_{3B}^{2}\right)+\frac{77}{16}g_{YB}^{4}+\frac{243}{16}g_{2B}^{4}+\lambda_{B}\left(-18y_{tB}^{2}+3g_{YB}^{2}+9g_{2B}^{2}\right)-10\lambda_{B}^{2}\bigg{\\}}\,I_{2},$
(20)

which realizes $m_{R}^{2}=0$ for $\delta m^{2}=0$ at this order, where

$\displaystyle I_{2}$
$\displaystyle=\int{d^{4}p\over(2\pi)^{4}}\int{d^{4}q\over(2\pi)^{4}}{1\over
p^{2}q^{2}(p+q)^{2}}$ (21)

is the quadratically divergent two loop integral over the Euclidean momenta.
As a non-trivial check of the consistency of our treatment, we have confirmed
that the coefficients of the infrared divergent integral
$\int{d^{4}p\over(2\pi)^{4}p^{4}}\int{d^{4}q\over(2\pi)^{4}q^{2}}$ cancel out
in each gauge invariant set of diagrams. We note that our two loop computation
is for the quadratically divergent part $m_{B}^{2}$ and is irrelevant to the
higher loop result $\propto\left(\log\Lambda\right)^{n}\Lambda^{2}$, obtained
in Ref. Einhorn:1992um  and used in Ref. Kolda:2000wi ; Casas:2004gh , which
would correspond to the effects of the RGE running of the dimensionless
couplings in our language.

In obtaining the two-loop corrected bare mass
$m_{B}^{2}=m_{B,\text{1-loop}}^{2}+m_{B,\text{2-loop}}^{2}$, one needs a
relation between the one- and two-loop integrals $I_{1}$ and $I_{2}$, which is
necessarily regularization scheme dependent. When we employ

$\displaystyle\int{d^{4}p\over p^{2}}=\int_{\epsilon}^{\infty}d\alpha\int
d^{4}p\,e^{-\alpha p^{2}},$ (22)

we get

$\displaystyle I_{2}={I_{1}\over 16\pi^{2}}\ln{2^{6}\over 3^{3}}\simeq
0.005I_{1}.$ (23)

With the blue solid line in Fig. (2), we plot $m_{B}^{2}/I_{1}$ when the UV
cutoff is at the Planck scale, that is, when $I_{1}=M_{P}^{2}/16\pi^{2}$. The
blue dashed line is the same with the 1-loop only bare mass
$m_{B,\text{1-loop}}^{2}/I_{1}$. We see that the two-loop correction is small,
so is the regularization dependence. For comparison, we also plot the quartic
coupling $\lambda$ at the Planck scale with the red dotted line. We see that
both become small for $M_{t}\simeq 170$ GeV.

Figure 2:  The blue solid (dashed) line corresponds to the one-plus-two-loop
(one-loop) bare mass $m_{B}^{2}$ ($m_{B,\,\text{1-loop}}^{2}$) in units of
${M_{\text{Pl}}^{2}/16\pi^{2}}$ for $\Lambda=M_{\text{Pl}}$. For comparison,
we also plot the quartic coupling $\lambda$ at the Planck scale with the red
dotted line.

##  IV Summary

We have computed two loop correction to the quadratically divergent part of
the bare SM Higgs mass in the bare perturbation theory. We have found that in
generic regularizations the two loop correction to the bare mass is small.
Therefore the regularization dependence is not large. Both the resultant bare
Higgs mass and quartic coupling can become small if the top quark pole mass is
around 170 GeV. Possible consequences of this result will be presented
elsewhere.

###### Acknowledgements.

We are grateful to Abdelhak Djouadi, Mikhail Yu. Kalmykov, and Bernd A. Kniehl
for valuable comments.

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EPJ Web of Conferences CONF12

11institutetext: Instituto de Física Teórica, Universidade Estadual Paulista
Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II, 01140-070 São Paulo, SP, Brazil

# Charmed mesons at finite temperature and chemical potential

Fernando E. Serna 11<EMAIL_ADDRESS>Gastão Krein 11
<EMAIL_ADDRESS>
###### Abstract

We compute the masses of the pseudoscalar mesons $\pi^{+}$, $K^{0}$ and
$D^{+}$ at finite temperature and baryon chemical potential. The computations
are based on a symmetry-preserving Dyson-Schwinger equation treatment of a
vector-vector four quark contact interaction. The results found for the
temperature dependence of the meson masses are in qualitative agreement with
lattice QCD data and QCD sum rules calculations. The chemical potential
dependence of the masses provide a novel prediction of the present
computation.

##  1 Introduction

Heavy-light mesons, like the $D$ and $B$, are interesting QCD bound states:
they play an important role in the threshold dynamics of many of the so-called
X,Y,Z exotic hadrons Briceno:2015rlt  and also serve as a laboratory for
studying chiral properties of the light $u$ and $d$ quarks in a medium at
finite temperature $T$ and baryon density $\mu$ in different contexts. Great
progress has been achieved in recent years in the study of their properties
with lattice QCD methods, both in vacuum and at finite temperature Skull . In
the continuum, however, although in principle the complex of the Schwinger-
Dyson (DS) and Bethe-Salpeter (BS) equations provides an adequate framework,
tremendous challenges still remain in describing simultaneously light- and
heavy-flavored mesons in vacuum within a single model interaction–truncation
scheme MT ; Krass ; Rojas:2014aka . On the other hand, there is pressing need
for different pieces of information on properties of such mesons both in
vacuum and at finite $T$ and $\mu$ for guiding experimental proposals at
existing and forthcoming facilities. Examples are charmed-hadron production
via $\bar{p}p$ annihilation processes Haidenbauer:2009ad ; Haidenbauer:2014rva
; Haidenbauer:2015vra , $J/\Psi$-nuclear bound states Krein:2010vp ;
Tsushima:2011kh , production in heavy-ion collisions of exotic molecules like
the $\Delta\bar{D}^{*}$ Carames:2016qhr . Having this in mind, in the present
contribution we extend to finite $T$ and $\mu$ a model that gives a good
description of the spectrum and leptonic decay constants of pseudoscalar
mesons in vacuum F.E.S  . The model is based on a confining, symmetry-
preserving treatment of a vector-vector four fermion contact interaction as a
representation of the gluon’s two-point Schwinger function used in kernels of
DS equations, originally tuned to study the pion GutierrezGuerrero:2010md .

A potential problem with contact-interaction models is their
nonrenormalizability, in that it can introduce gross violations of global and
local symmetries because of ambiguities related to momentum shifts in
divergent integrals. Here we use a subtraction scheme Battistel:1998tj  that
allows us to separate symmetry-violating parts in Bethe-Salpeter amplitudes in
a way independent of choices of momentum routing in divergent integrals. The
scheme has been used in the past within the Nambu–Jona-Lasinio (NJL) model in
vacuum Battistel:2008fd , at finite $T$ and $\mu$ Farias  and, more recently
Farias:2016let , it was used to explain the reason for the failure of the
model to explain lattice results for the chiral transition temperature in the
presence of a chiral imbalance in quark matter.

We extend the subtraction scheme of Ref. F.E.S  to finite $T$ and $\mu$; the
situation is more complicated than in the vacuum because Lorentz covariance is
broken at finite $T$ and $\mu$ and special care must be exercised to separate
purely divergent contributions from thermal effects, which are finite and do
not need regularization. After setting up the scheme, we calculate the masses
of the pseudoscalar mesons $\pi^{+}$, $K^{0}$ and $D^{+}$ at finite $T$ and
$\mu$ and compare results with those obtained recently using QCD sum rules
Suzuki:2015est  and those obtained earlier with the NJL model Blaschke:2011yv
; Gottfried:1992bz .

##  2 Dyson-Schwinger and Bethe-Salpeter equations at finite $\bm{T}$ and
$\bm{\mu}$

The Dyson-Schwinger equation (DSE) for the full quark propagator $S_{f}(k)$ of
flavor $f$ is given by (in Euclidean space)

$S^{-1}_{f}(k)=i\not{k}+m_{f}+\int\frac{d^{4}q}{(2\pi)^{4}}\,g^{2}D_{\mu\nu}(k-q)\,\frac{\lambda^{a}}{2}\gamma_{\mu}S_{f}(q)\frac{\lambda^{a}}{2}\Gamma^{f}_{\nu}(q,k)\leavevmode\nobreak\
,$ (1)

where $m_{f}$ is the current-quark mass, $D^{\mu\nu}$ the full gluon
propagator, and $\Gamma^{f}_{\nu}$ the full quark-gluon vertex. The mass
$m_{\rm PS}$ of a pseudoscalar (PS) meson with one light ($f=l$) quark and one
heavy ($f=h$) quark is the eigenvalue $P^{2}=-m^{2}_{\rm PS}$ that solves the
homogeneous Bethe-Salpeter equation (BSE)

$\Gamma^{lh}_{\rm
PS}(k;P)=\int\frac{d^{4}q}{(2\pi)^{4}}K(k,q;P)S_{l}(q_{+})\Gamma^{lh}_{\rm
PS}(q;P)S_{h}(q_{-})\leavevmode\nobreak\ ,$ (2)

with $K(k,q;P)$ being the fully amputated quark-antiquark scattering kernel,
where $q_{\pm}=q\pm\eta_{\pm}P$ and $\eta_{+}+\eta_{-}=1$. At finite $T$ and
$\mu$, the four dimensional momentum integrals in Eqs. (1) and (2) become

$\int\frac{d^{4}q}{(2\pi)^{4}}\,F(q)\rightarrow\frac{1}{\beta}\sum^{\infty}_{m=-\infty}\int\frac{d^{3}q}{(2\pi)^{3}}\,F(\omega_{m},{\bf
q})\leavevmode\nobreak\ ,$ (3)

where $\omega_{m}=(2m+1)\,\pi/\beta$ are the fermionic Matsubara frequencies
with $\beta=1/T$ and $m\in\mathds{Z}$. The contact-interaction limit of full
QCD is obtained by making the following replacements in Eqs. (1) and (2)

$g^{2}D_{\mu\nu}(p-q)\rightarrow G\,\delta_{\mu\nu},\hskip
19.91684pt\Gamma^{a}_{\mu}\rightarrow\frac{\lambda^{a}}{2}\gamma_{\mu},\hskip
19.91684ptK(k,q;P)=-G\,\left(\frac{\lambda^{a}}{2}\gamma^{\mu}\right)\otimes\left(\frac{\lambda^{a}}{2}\gamma_{\mu}\right)\leavevmode\nobreak\
,$ (4)

where $G$ is an effective coupling constant with dimensions of (length)2. In
this limit, Eq. (1) becomes

$S^{-1}_{f}({{\bf q}},\omega_{m})=i{\bm{\gamma}}\cdot{\bf
q}+i\gamma_{4}\,\omega_{m}+M_{f}\leavevmode\nobreak\ ,$ (5)

with $M_{f}=M_{f}(T,\mu)$ given by (the gap equation)

$M_{f}=m_{f}+\frac{16}{3}G\,\left\\{I_{\rm
quad}(M_{f})-\int\frac{d^{3}q}{(2\pi)^{3}}\frac{M_{f}}{2E_{f}({\bf
q})}\left[f^{-}(E_{f}({\bf q}))+f^{+}(E_{f}({\bf
q}))\right]\right\\}\leavevmode\nobreak\ ,$ (6)

where

$I_{\rm
quad}(M^{2})=\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{1}{q^{2}+M^{2}}\leavevmode\nobreak\
,\hskip 14.22636ptf^{\pm}(E)=\frac{1}{1+e^{\,\beta
E^{\pm}}}\leavevmode\nobreak\ ,$ (7)

with $E^{\pm}_{f}({\bf q})=E_{f}({\bf q})\pm\mu$, and $E_{f}(q)=({\bf
q}^{2}+M^{2}_{f})^{1/2}$. The Bethe-Salpeter amplitude (BSA) contains only
pseudo-scalar and pseudo-vector components:

$\Gamma^{lh}_{\rm PS}(P)=\gamma_{5}\left[iE^{lh}_{\rm
PS}(P)+\frac{1}{2M_{lh}}\gamma\cdotp P\,F^{lh}_{\rm
PS}(P)\right]\leavevmode\nobreak\ ,$ (8)

with $P=(\nu_{n},{\bf P})$, $\nu_{n}=2n\pi/\beta$, and the factor
$M_{lh}=M_{l}M_{h}/(M_{l}+M_{h})$, where $M_{l}$ and $M_{h}$ are solutions of
Eq. (6), is introduced for convenience. Using this in Eq. (2), the BSE of can
be written F.E.S  as a matrix equation involving the amplitudes $E^{lh}_{\rm
PS}$ and $F^{lh}_{\rm PS}$. For comparison with earlier results in the
literature that use the Random Phase Approximation (RPA), one needs to use
only the pseudoscalar component $E^{lh}_{\rm PS}$ in $\Gamma^{lh}_{\rm
PS}(P)$. The meson mass is obtained taking $\omega=m_{\rm PS}$ in the
expression

$0=1-\frac{G}{3}\,\Pi_{\rm PS}(\nu_{n}=-i\omega,{\bf P}=0)\leavevmode\nobreak\
,$ (9)

with

$\Pi^{lh}_{\rm PS}(\nu_{n},{\bf
P})=-\frac{1}{\beta}\sum^{\infty}_{m=-\infty}\int\frac{d^{3}q}{(2\pi)^{3}}{\rm{Tr}}\left[\gamma_{5}\gamma_{\mu}S_{l}({\bf
q}_{+},\omega_{+})\gamma_{5}\,S_{h}({\bf
q}_{-},\omega_{-})\gamma_{\mu}\right]\leavevmode\nobreak\ ,$ (10)

where ${\bf q}_{\pm}={\bf q}\pm\eta_{\pm}\,{\bf P}$ and
$\omega_{\pm}=\omega_{m}\pm\chi_{\pm}\,\nu_{n}$, where
$\chi_{\pm}\in\mathds{Z}$. After taking a Dirac trace we can rewrite Eq. (10)
as a sum of two terms: one that contains ultraviolet divergences

$\Pi^{lh}_{\rm PS}(P)|_{\rm
div}=8\int\frac{d^{4}q}{(2\pi)^{4}}\Bigg{\\{}\frac{1}{q^{2}_{+}+M^{2}_{l}}+\frac{1}{q^{2}_{-}+M^{2}_{h}}-\left[P^{2}+\left(M_{h}-M_{l}\right)^{2}\right]\frac{1}{(q^{2}_{+}+M^{2}_{l})(q^{2}_{+}+M^{2}_{h})}\Bigg{\\}}\leavevmode\nobreak\
,$ (11)

and another that is finite, given in terms of the Fermi-Dirac distributions
$f^{\pm}(E)$ defined in Eq. (7):

$\displaystyle\Pi^{lh}_{\rm PS}(P)|_{\rm fin}$ $\displaystyle=$
$\displaystyle-4\int\frac{d^{3}q}{(2\pi)^{3}}\Bigg{\\{}\frac{f^{+}(E_{l}({\bf
q}_{+}))+f^{-}(E_{l}({\bf q}_{+}))}{E_{l}({\bf
q}_{+}))}+\frac{f^{+}(E_{h}({\bf q}_{-}))+f^{-}(E_{h}({\bf
q}_{-}))}{E_{h}({\bf q}_{-})}$ (12) $\displaystyle+$
$\displaystyle\frac{P^{2}+(M_{l}-M_{h})^{2}}{2E_{l}({\bf q}_{+})E_{h}({\bf
q}_{-})}\Bigg{[}\frac{2E_{h}({\bf q}_{-})f^{-}(E_{l}({\bf q}_{+}))}{[\omega-
E_{l}({\bf q}_{+})]^{2}-E^{2}_{h}({\bf q}_{-})}+\frac{2E_{l}({\bf
q}_{+})f^{-}(E_{h}({\bf q}_{-}))}{[\omega+E_{h}({\bf
q}_{-})]^{2}-E^{2}_{l}({\bf q}_{+})}$ $\displaystyle+$
$\displaystyle\frac{2E_{h}({\bf q}_{-})f^{+}(E_{l}({\bf
q}_{+}))}{[\omega+E_{l}({\bf q}_{+})]^{2}-E^{2}_{h}({\bf
q}_{-})}+\frac{2E_{l}({\bf q}_{+})f^{+}(E_{h}({\bf q}_{-}))}{[\omega-
E_{h}({\bf q}_{-})]^{2}-E^{2}_{l}({\bf
q}_{+})}\Bigg{]}\Bigg{\\}}\leavevmode\nobreak\ .$

In Eqs. (11) and (12), $P=(-i\omega,{\bf P})$. The problem with symmetry
violations alluded to in the Introduction comes from the divergent part: it
depends on the choice made for the partition of the momenta in the loop
integral when using a cutoff regularization, i.e. it is not independent of
$\eta_{\pm}$. Our scheme F.E.S  to obtain symmetry-preserving expressions is
to perform subtractions in divergent integrals:

$\displaystyle\frac{1}{q^{2}_{\pm}+M^{2}_{l,h}}$ $\displaystyle=$
$\displaystyle\frac{1}{q^{2}+M^{2}}+\left(\frac{1}{q^{2}_{\pm}+M^{2}_{l,h}}-\frac{1}{q^{2}+M^{2}}\right)=\frac{1}{q^{2}+M^{2}}-\frac{(q^{2}_{\pm}-q^{2}+M^{2}_{l,h}-M^{2})}{\left(q^{2}+M^{2}\right)(q^{2}_{\pm}+M^{2}_{l,h})},$
(13)

with $M$ being an arbitrary subtraction mass scale. One performs as many
subtractions as necessary to obtain one finite integral. The final result is
F.E.S

$\displaystyle\Pi^{lh}_{\rm PS}(P)|_{\rm div}$ $\displaystyle=$
$\displaystyle\frac{3}{G}-\frac{3}{2G}\left(\frac{m_{l}}{M_{l}}+\frac{m_{h}}{M_{h}}\right)-8\left[P^{2}+(M_{l}-M_{h})^{2}\right]\left[I_{\rm
log}(M)-Z_{0}(M^{2}_{l},M^{2}_{h},P^{2},M^{2})\right]$ (14)
$\displaystyle\,+8\,(\eta^{2}_{+}+\eta^{2}_{-})\,A_{\mu\nu}(M^{2})\,P_{\mu}P_{\nu}\leavevmode\nobreak\
,$

where $I_{\rm log}(M^{2})$ is a divergent integral and
$Z_{0}(M^{2}_{l},M^{2}_{h},P^{2},M^{2})$ is a finite integral

$I_{\rm
log}(M^{2})=\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{1}{\left(q^{2}+M^{2}\right)^{2}}\leavevmode\nobreak\
,\hskip
14.22636ptZ_{0}(M^{2}_{l},M^{2}_{h},P^{2},M^{2})=\frac{1}{(4\pi)^{2}}\int^{1}_{0}dz\ln\left[\frac{H(z,P^{2})}{M^{2}}\right]\leavevmode\nobreak\
,$ (15)

with $H=z(z-1)P^{2}-z(M^{2}_{l}-M^{2}_{h})+M^{2}_{l}$, and $A_{\mu\nu}(M^{2})$
is another divergent integral

$A_{\mu\nu}(M^{2})=\int\frac{d^{4}q}{(2\pi)^{4}}\frac{4q_{\mu}q_{\nu}-(q^{2}+M^{2})\delta_{\mu\nu}}{(q^{2}+M^{2})^{3}}\leavevmode\nobreak\
.$ (16)

The finite integral in Eq. (15) is obtained by integrating over momentum,
removing any regularization implicitly assumed. Clearly, the term proportional
to $A_{\mu\nu}(M^{2})$ is not independent of $\eta_{\pm}$ and, therefore,
violates translation symmetry. Similar expressions appear also in Ward-
Takahashi identities F.E.S . Since there are regularizations schemes, like
dimensional regularization, where $A_{\mu\nu}(M^{2})=0$ automatically, it is a
natural prescription for obtaining symmetry-preserving amplitudes to demand
the vanishing of $A_{\mu\nu}(M^{2})$ (and of all other similar terms that
appear in other amplitudes, see e.g. Ref. F.E.S ), independently of the
regularization used to regulate $I_{\rm quad}(M^{2})$ and $I_{\rm
log}(M^{2})$.

Figure 1: Constituent quark mass as function of $T$ for different values of
$\mu$.

The mass scale $M$ in Eq. (13) is arbitrary; it appears in the divergent
integrals $I_{\rm log}(M^{2})$ and $I_{\rm quad}(M^{2})$ and in the finite
integral $Z_{0}(M^{2}_{l},M^{2}_{h},P^{2},M^{2})$. Since the model is
nonrenormalizable, the integrals $I_{\rm log}(M^{2})$ and $I_{\rm
quad}(M^{2})$ cannot be removed, of course. One can use an explicit regulator
to evaluate the integrals and fit the regulator to physical quantities, like
the quark condensate or hadron masses, or one could also fit the integrals
directly to physical quantities. In either case, the fit to physical
quantities is $M$-dependent, that is, physical quantities would “run with
$M$”, very much like in renormalizable quantum field theories where all masses
and other quantities are running functions of a mass scale that enters the
theory via the regularization scheme. Here we present results using an
explicit three-dimensional cutoff $\Lambda$ to regulate the divergent
integrals $I_{\rm log}(M^{2})$ and $I_{\rm quad}(M^{2})$, and take $M=M_{h}$,
for simplicity—further discussions on this will be presented elsewhere.

##  3 Numerical results and conclusions

The free parameters are: $G$, $\Lambda$ and $m_{f}$. Taking
$\Lambda=0.653\leavevmode\nobreak\ {\rm GeV}$, $G\Lambda^{2}=19.26$,
$m_{l}=m_{u}=m_{d}=0.005$ GeV, $m_{s}=0.161$ GeV and $m_{c}=1.544$ GeV, one
obtains for the meson masses in vacuum: $m_{\pi}=0.139$ GeV, $m_{K^{0}}=0.493$
GeV and $m_{D^{+}}=1.864$ GeV. It is worth mentioning that we obtain for the
constituent quark masses in vacuum the following values: $M_{l}=0.328$ GeV,
$M_{s}=0.599$ GeV and $M_{c}=2.095$ GeV. The results for the $T$ dependence of
$M_{l}$ for different values of $\mu$, obtained from the DSE in Eq. (6), are
shown in the left panel of Fig. 1. Clearly, as $\mu$ increases, the (pseudo)
critical temperature for chiral restoration decreases as $\mu$ increases, as
expected. On the right panel of the figure, the temperature dependence of
$M_{l}$, $M_{s}$ and $M_{c}$ for zero chemical potential: also as expected, as
the current quark mass increases, the effect of the temperature becomes less
important, even close to and above the pseudocritical temperature.

Figure 2: Meson masses as functions of $T$ for two typical values of $\mu$.
Top panels: $m_{\pi^{+}}$ calculated with and without the Fermi-Dirac
distributions $f^{\pm}(E)$ in the BSE. Bottom panels: masses calculated
without the $f^{\pm}(E)$ in the BSE.

The last point raises the question on the importance of the Fermi-Dirac
distribution functions, $f^{\pm}(E)$, in the BSE for the masses of the
mesons—physically, they represent the contributions of thermally activated
quark-antiquark pairs in the bound state. The answer to this question is shown
in the top panels of Fig. 2: for sufficiently low values of $\mu$, the
$f^{\pm}(E)$ play no significant role for $T$ smaller than the pseudocritical
temperature and can, to a good approximation, be neglected. This is an
important feature, as it simplifies considerably the calculations of thermal
effects on hadron masses, as all $T$ and $\mu$ effects below the crossover are
captured by the $T$ and $\mu$ dependence of the constituent quark masses. In
the bottom panel of Fig. 2, we show the results for the mesons masses
neglecting the Fermi-Dirac distributions: the results are in good quantitative
agreement with early calculations using the NJL model Blaschke:2011yv ;
Gottfried:1992bz . It also agrees with a very recent calculation using a
chiral constituent quark model, using as input $T$ and $\mu$ dependent quark
masses and quark-meson couplings Carames:2016qhr , Finally, our results are
also in qualitative agreement with a recent calculation using QCD sum rules
Suzuki:2015est —this last reference points to earlier calculations, that
obtained the opposite trend or the $D$ meson mass. We also mention that recent
lattice results Skull  show that the masses of $D$ mesons increase, and become
broader.

As perspectives, concrete calculations of production rates of exotic hadronic
molecules in heavy-ion collisions Briceno:2015rlt ; Carames:2016qhr  and of
transport properties of charmed hadrons Ghosh:2015mda  are top priority. In
addition, it would be important to contrast results using confining chiral
models. Of particular interest to us are those models inspired in QCD
formulated in Coulomb gauge Bicudo:1991kz ; Fontoura:2012mz , and those based
on chiral soliton models Krein:1988sb ; Krein:1988vh .

We thank B. El-Bennich for valuable discussions. Work partially by Conselho
Nacional de Desenvolvimento Científico e Tecnológico - CNPq, Grants No.
305894/2009-9 (G.K.) and 140041/2014-1 (F.E.S.) and Fundação de Amparo à
Pesquisa do Estado de São Paulo-FAPESP, Grant No. 2013/01907-0 (G.K.).

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# A Survey on Location-Driven Influence Maximization

Taotao Cai1 Quan Z. Sheng1 Xiangyu Song2 Jian Yang1 Shuang Wang5
Wei Emma Zhang3 Jia Wu1 Philip S. Yu4
1School of Computing, Macquarie University, Sydney, Australia
2School of Info Technology, Deakin University, Melbourne, Australia
3School of Computer Science, The University of Adelaide, Adelaide, Australia
4Department of Computer Science, University of Illinois at Chicago, Chicago,
USA
5School of Computer Science and Engineering, Southeast University, Nanjing,
China
{taotao.cai, michael.sheng, jian.yang<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Influence Maximization (IM), which aims to select a set of users from a social
network to maximize the expected number of influenced users, is an evergreen
hot research topic. Its research outcomes significantly impact the real-world
applications such as business marketing. The booming location-based network
platforms of the last decade appeal to the researchers embedding the location
information into the traditional IM research. In this survey, we provide a
comprehensive review of the existing location-driven IM studies from the
perspective of the following key aspects: (1) a review of the application
scenarios of these works, (2) the diffusion models to evaluate the influence
propagation, and (3) a comprehensive study of the approaches to deal with the
location-driven IM problems together with a particular focus on the
accelerating techniques. In the end, we draw prospects into the research
directions in future IM research.

##  1 Introduction

In the past two decades, the rise of social media platforms (e.g., Twitter)
has brought dramatic changes to human society, especially in the area of mass
communication and commercial marketing. To better utilize the power of social
networks, many researchers turn to the study of influence maximization (IM)
Kempe et al. (2003); Golovin and Krause (2011); Yuan and Tang (2017), which
aims to select a set of influential users with a given budget to achieve the
maximum information spread in a social network. To fit with different
practical application scenarios, several variants of the IM problem have been
investigated recently, including Topic-aware IM Guo et al. (2013); Li et al.
(2015), Time-aware IM Feng et al. (2014); Xie et al. (2015); Huang et al.
(2019), Community-aware IM Tsang et al. (2019); Li et al. (2020a); Yadav et
al. (2018), Competitive IM Ou et al. (2016); Tsaras et al. (2021); Becker et
al. (2020), Multi-strategies IM Kempe et al. (2015); Chen et al. (2020a), and
IM with semi-bandit Wen et al. (2017); Lin et al. (2015); Sun et al. (2018);
Li et al. (2020b).

With the popularity of GPS-equipped smart devices (e.g., smart phones) and the
rise of many location-based social network platforms (e.g., Gowalla) in recent
years, large-scale digital data with location information are generated and
recorded from online social communications, offline facility information
(e.g., electronic billboards) and moving objects (e.g., pedestrian and taxi
trajectories). The availability of location and communication datasets makes
the study of location-driven IM feasible.

Research along the direction of location-driven IM can be categorized as the
following two lines. The first line of research focuses on equipping the
traditional IM study with location information Li et al. (2014); Wang et al.
(2018); Güney et al. (2021), while the other line of research concentrates on
finding a set of optimal offline objects (e.g., billboards) to maximize the
influence spread with a given budget Zhang et al. (2019b, 2021).

The motivation of this survey. Several existing survey papers Tejaswi et al.
(2016); Arora et al. (2017); Sumith et al. (2018); Li et al. (2018b); Banerjee
et al. (2020) have provided a detailed introduction and analysis of the IM
problem. However, some critical emerging trends of IM research haven’t been
thoroughly discussed in these surveys, i.e., IM query with location
information. This has motivated us to comprehensively summarize the research
progress of location-driven IM through a fine-grained classification and a
systematic analysis of existing location-driven IM methods.

Figure 1: An Overview of this Survey.

The structure of this survey. This paper focuses on providing a comprehensive
survey on existing location-driven IM studies. As illustrated in Figure 1, we
summarize the existing location-driven IM research efforts in our survey paper
through the following three critical perspectives, including: (1) the problem
description, (2) the influence diffusion model, and (3) the methodology:

  * •

Problem description. To better summarize the existing location-driven IM
works, we categorize them into two research lines based on their application
scenarios, including the Online Location-Aware Influence Maximization (OLIM)
and the Offline Billboards Influence Maximization (OBIM). Besides, we give
precise problem descriptions for each of them in Section 2.

  * •

Diffusion Model. We systematically introduce the influence diffusion models
used to capture the information diffusion process in the existing OLIM and
OBIM studies, respectively, in Section 3.

  * •

Methodology. Since the IM problem is NP-hard, so is the location-driven IM
problem. The existing location-driven IM works focus on developing approximate
solutions. We summarize the existing framework algorithms to deal with OLIM
and OBIM problems, respectively. In addition, we provide an in-depth analysis
of the acceleration techniques of each location-driven IM related paper that
are used to improve the efficiency of their framework algorithms in Section 4.

Contributions. The main contributions of this survey are summarized as
follows:

  * •

We systematically analyze the existing location-driven IM works and classify
them from a well-developed fine-grained taxonomy based on their application
scenarios.

  * •

We comprehensively study the algorithms of existing location-driven IM works,
with a particular focus on their accelerating techniques.

  * •

We outline some promising future research directions in location-driven IM,
which will provide valuable references to the development of the community.

##  2 Background and Problem Description

This section first introduces the influence maximization (IM) problem. Then,
we divide the existing location-driven IM studies into two directions: online
location-aware influence maximization (OLIM) and offline billboards influence
maximization (OBIM), and formally define the research problems on OLIM and
OBIM, respectively.

###  2.1 The Influence Maximization (IM) Problem

Given a graph $G=(V,E)$ representing a social network, where $V$ is the set of
nodes in $G$ (i.e., users) and $E$ is the set of edges in $G$, and each edge
$(u,v)\in E$ in $G$ is associated with a propagation probability
$p(u,v)\in[0,1]$.

Let $I(S)$ be the number of vertices that are activated by $S$ in graph $G$ on
the influence propagation process. The IM problem aims to find a size-$k$ seed
set $S$ with the maximum expected influence spread $E(I(S))$. We formally
define the IM problem as follows:

######  Definition 1 (IM Problem).

Given a graph $G=(V,E)$, and an integer $k$, the IM problem aims to find an
optimal seed set $S^{*}$ satisfying,

$S^{*}=\mathop{\arg\max}_{S\subseteq V,|S|=k}\textbf{E}(I(S))$ (1)

###  2.2 Online Location-aware IM

The online location-aware influence maximization (OLIM) problem is a variant
of the traditional IM problem which considers the effect of given location
information on the influence diffusion to the target users in online social
networks. For instance, Su et al. (2018) focuses on finding a set of users to
maximize the influence spread over the target users who have both topic and
geographical preferences on influence diffusion.

In brief, we can conclude that the traditional IM problem and the OLIM problem
share a common goal with two different settings. The common goal is that both
aim to find a set of seed users to maximize the influenced target users in the
online social networks. Differently, the IM and OLIM problems differ in the
definition of target users and the influence diffusion process. Specifically,
the definition of target users in the IM problem can be any users in a social
network. In contrast, the target users in OLIM may need to satisfy the
location constraint, e.g., the distance between seed users and target users
does not exceed a given threshold. Besides, compared with the IM problem,
evaluating the influence diffusion process for given seed users in the OLIM
problem requires further consideration of the effect of the users’ locations.

###  2.3 Offline Billboards IM

With the availability of trajectory data of moving objects and offline
facilities’ information e.g., billboards’ locations and pedestrian
trajectories, recent studies Zhang et al. (2019b, 2021) initiate the influence
maximization problem in Out-of-Home advertising on billboards. We call it
offline billboards influence maximization (OBIM). The goal of the OBIM problem
is to maximize the influence for the advertiser (e.g., billboards). We provide
a generic definition of the OBIM problem as follows:

######  Definition 2 (OBIM Problem).

Given a trajectory database $\mathcal{T}$, a set of billboards U to place ads
and the budget $L$, the OBIM problem aims to find a subset of billboards
$S\subset U$, which maximizes the expected number of influenced trajectories.

The critical difference between the OLIM and the OBIM problems is the
influence diffusion process. In the OLIM problem, user influences can be
spread from one user to others in online social networks. On the contrary, in
the OBIM problem, a user can be influenced by an outdoor billboard only if
this user passes by the billboard within a certain range.

##  3 The Influence Diffusion Models

In this section, we review the influence diffusion models that are used to
capture the information diffusion process in the existing OLIM and OBIM works,
respectively.

###  3.1 Diffusion Models in OLIM

In the OLIM problem, three well-known diffusion models are widely used in
modeling the influence diffusion process, which are the independent cascade
model, the Linear threshold model Kempe et al. (2003), and the maximum
influence arborescence model Chen et al. (2010). We associate each user in the
following diffusion models with a status of either inactive or active.

Independent cascade (IC) model: the IC model is one of the widely adopted
stochastic models which is used for modeling the influence propagation in
social networks. In this model, we represent the social network as a graph
$G$, $S_{0}$ as a set of initial _activated_ users, and generate the activated
sets $S_{t}$ for all $t\geq 1$ according to the following randomized rule. At
every time step $t\geq 1$, we first set $S_{t}$ to be $S_{t-1}$; Each user $u$
activated in time step $t$ has one chance to activate his or her neighbours
$v$ with propagation probability $p(u,v)$. If successful, we then add $v$ into
$S_{t}$ and change the status of $v$ to _activated_. This process continues
until no more possible activations. Finally, $S_{t}$ is returned as the
_activated_ user set of $S_{0}$.

Linear threshold (LT) model: The key idea of the LT model is that a user may
change his or her status from inactivated to activated if the user has
sufficient number of activated neighbours. Specifically, let $G=(V,E)$
represent the social network where each edge $e=(u,v)\in E$ is associated with
a weight $w_{e}$ and each user $u$ is associated with a threshold
$\lambda_{u}$. $S_{0}$ is a set of initial _activated_ users, and generates
the active sets $S_{t}$ for all $t\geq 1$. At each time step $t\geq 1$, all
the activated users in step $t-1$ remain activated in step $t$, and any user
$v$ that is inactivated in step $t-1$ will be changed to $activated$ in step
$t$ if the total edge weights between $v$ and its activated neighbours is no
less than $\lambda_{v}$. The LT influence diffusion terminates when there are
no more new activated users.

Maximum influence arborescence (MIA) model: The MIA model is the most known as
a reduced model of the IC model. The key idea of the MIA model is to restrict
the influence diffusion of each user $u$ to a local tree structure with $u$ as
the root. In concise, the MIA model performs the reductions from two aspects:
(1) For any user pair (u,v), it considers that $u$ can only influence $v$
through the maximum influence paths (MIPs), while MIP is the path with the
maximum influence probability among any paths between users $u$ and $v$, i.e.,
$MIP(u,v)$; (2) Given threshold $\theta$, all MIPs with the propagation
probabilities less than $\theta$ will be ignored. Therefore, the influence
diffusion of a given user $v$ is a set of users $u\in V$ with
$MIP(u,v)\geq\theta$ under the MIA model.

###  3.2 Diffusion Models in OBIM

Several diffusion models have been widely used in the OBIM problem to evaluate
the influence propagation effects for a given offline object (e.g.,
billboards), including the one-time impression model Liu et al. (2016), the
multi-time impression model Zhang et al. (2020), and the logistic influence
model Zhang et al. (2019b).

One-time impression (OI) model: Given a trajectory database $\mathcal{T}$ with
each trajectory $t=\\{p_{1},p_{2},..,p_{|t|}\\}$, a billboard $b$ with
location information $b.loc$, and a given threshold $\lambda$. For each
trajectory $t$, if $\exists p_{i}\in t$ such that the distance between $p_{i}$
to $b.loc$ is no less than $\lambda$, we define that trajectory $t$ is
influenced by the billboard $b$, i.e., $Distance(p_{i},b.loc)\leq\lambda$.

Multi-time impression (MI) Model: Given a trajectory
$t=\\{p_{1},p_{2},..,p_{|t|}\\}$, a billboard set
$B=\\{b_{1},b_{2},...,b_{l}\\}$, we set $pr(b_{i},t)$ as a uniform value if a
billboard $b_{i}\in B$ can influence $t$. The value of $pr(b_{i},t)$ is
correlated with the billboard set size and the billboards exposure frequency,
i.e., $pr(b_{i},t)=size(b_{i})/A$ where $size(b_{i})$ is the panel size of
$b_{i}$ and $A=max_{b_{i}\in B}size(b_{i})$. Then, we use the following
equation to compute the influence of a billboard set $S$ to $t$:

$pr(S,t)=1-\bigsqcup_{b_{i}\in S}(1-pr(b_{i},t))$ (2)

Logistic influence (LI) model: Given a trajectory
$t=\\{p_{1},p_{2},..,p_{|t|}\\}$, a billboard set
$B=\\{b_{1},b_{2},...,b_{l}\\}$, and a Bernoulli random variable
$I(b_{i},t),i\in[1,l]$ to denote the states whether $b$ impresses $t$. When
$I(b,t)=1$, we say $b$ impresses $t$; otherwise, no impression is delivered.
The logistic influence model evaluates the effective influence of a set of
billboards $S\subseteq B$ to a trajectory $t$, $P(S,t)$, by using the logistic
function based equation as follow:

$P(S,t)\\!=\\!\left\\{\begin{aligned}
\dfrac{1}{1\\!+\\!exp\\{\alpha\\!-\\!\beta\\!\cdot\\!\sum_{\\!b_{i}\\!\in\\!S}\\!I(b_{i},t)\\!\\}},&\>if\exists
b_{i}\in S,\\!I(b_{i},t)=\\!1\\\ 0,&\>otherwise\\\ \end{aligned}\right.$ (3)

In Equation 3, $\alpha$ and $\beta$ are the parameters to control $t$’s
turning point for being influenced.

Other models: There also exist several other uncommon diffusion models for
evaluating the influence propagation process of the given billboard, such as
the quantitative model in Wang et al. (2019), the application-driven mining
model in Liu et al. (2016), and the voter model in Wu et al. (2018).

##  4 Methodology

OLIM |  Diffusion Model |  Algorithms |  Accelerating |  Application Scenarios
---|---|---|---|---
Li et al. (2014) |  MIA model |  Proxy-based approach |  Index & Pruning |  Maximizing the influenced users within the given Region.
Zhou et al. (2015b) |  TP model (Variant IC) |  Proxy-based approach |  None |  Online promoting to drive the offline consuming.
Wang et al. (2016a, b) |  MIA model |  Proxy-based approach |  Index & Pruning |  Maximizing the influence of potential customers of a given promoted location.
Song et al. (2016) |  IC model |  Sketch-based approach |  Heuristic |  Targeted influence maximization.
Wang et al. (2018) |  SIC model (Variant IC) |  Proxy-based approach |  Index |  Retrieving a seed set with the maximum influence value over location-aware social streams.
Li et al. (2018a); Cai et al. (2020) |  HIM model (Variant IC) |  Sketch-based approach |  Index & Pruning |  Evaluating the influence diffusion by considering the social connection, spatial connection and preference-based connection, rather than only take social connection into account.
Zhu et al. (2019) |  MIA model |  Proxy-based approach |  None |  Retrieving $k$ positive users to block the diffusion of influence of negative seed users to the targeted users of a given region.
Chen et al. (2020b) |  SIM model (Variant IC) |  Simulation-based approach |  Index & Pruning |  Designing a new IC model based influence diffusion that further take the users’ spatial-temporal behavior into account.

Table 1: Summary of the existing OLIM research works

In this section, we systematically analyze the methodologies of the existing
location-driven IM studies.

###  4.1 Approaches of Existing OLIM Studies

We first introduce the methodologies of the existing OLIM works. An analysis
of these works is summarized in Table 1. Specifically, we categorize the
framework algorithms of existing OLIM works into three approaches based on how
an approach overcomes the $\\#$P-hardness of evaluating the influence spread
$I(S)$ of given users $S$, including (1) the simulation-based approach, (2)
the sketch-based approach, and (3) the proxy-based approach.

####  4.1.1 Simulation-based Approach

The core idea of this approach is to estimate the influence spread $I(S)$ of a
given user set $S$ by using the Monte Carlo (MC) simulations of the diffusion
process Kempe et al. (2003); Leskovec et al. (2007); Zhou et al. (2015a).
Specifically, for a given user set $S$, we simulate the randomized diffusion
process with $S$ for $R$ times. Each time we count the number of active users
after the diffusion ends, and then take the average of these counts over the
$R$ times.

Chen et al. (2020b) mentions that the existing works try to study influence
maximization by only considering the geographic distance, while ignoring the
influence of users’ spatio-temporal behaviours on information diffusion or
location promotion. Motivated by this drawback, they propose a Similarity-
aware Influence Maximization (SIM) model to evaluate the influence diffusion,
which is an extension of the IC model by taking the effect of users’ spatio-
temporal behaviors into account. Moreover, two greedy algorithms, including
influence propagation trees index-based and cutting tails pruning algorithms,
are proposed to find optimal seed users to maximize the influence spread under
the SIM model based on the simulation-based approach.

####  4.1.2 Proxy-based Approach

Instead of running heavy MC simulation, the proxy-based approach estimates the
influence spread of given users via the proxy models. Precisely, this approach
is aligned with a specific diffusion model (e.g., IC model), and the influence
spread estimation process of given users is highly accelerated by taking
advantage of the properties of the corresponding models. Intuitively, there
are two branches of the proxy-based approaches, including (1) Estimate the
influence spread of given users by transforming it to easier problems (e.g.,
Degree and PageRank) Chen et al. (2010); Galhotra et al. (2016); and (2)
Simplify the typical diffusion model (e.g., IC model) to a deterministic model
(e.g., MIA model) Chen et al. (2010) or restrict the influence propagation
range of given users under the typical diffusion model to the local subgraph
Goyal et al. (2011) to precisely compute the influence spread of given users.
Compared with the simulation-based approach, the proxy-based approach offers
higher performance, but lacks theoretical guarantees. The proxy-based approach
has been used in Li et al. (2014); Zhou et al. (2015b); Wang et al. (2016a, b,
2018); Zhu et al. (2019). We introduce the details of these efforts as
follows.

Li et al. (2014) adopts the proxy-based approach under the MIA model. To
accelerate their algorithm, the authors focus on developing bound estimation
techniques for effective pruning. Besides, they use a classic spatial index
QuadTree to fast locate the users having influence on the “influences” in the
given region. Zhou et al. (2015b) points out the previous work tends to
concentrate on a single aspect of online social networks, treating location as
a simple property and ignoring the presence of offline consumption behaviour.
To conquer this drawback, they propose an improved influence diffusion model,
namely the TP model, which builds on the IC model to further evaluate
influence diffusion weighted by the user’s distance from an offline location.
Under the TP model, the proxy-based approach is utilized to find the optimal
users to maximize the influence spread.

Wang et al. (2016a, b) focus on maximizing the influence of potential
customers in a given promotion location. They adopt the MIA diffusion model
for influence approximation and utilize the proxy-based approach as the
framework algorithm. To this end, they devise the pruning method to accelerate
the framework algorithm by reducing the number of candidate users to be
evaluated. Song et al. (2016) also studies the similar research topic as Wang
et al. (2016a). Differently, the authors adopt a sketch-based approach to deal
with their research problem.

Wang et al. (2018) points out that most existing IM algorithms are static and
location-unaware, which rarely reflect the evolving and location-aware nature
of social networks. To track the influential users over social streams and
further consider the users’ location information, the authors propose two IM
queries, Stream Influence Maximization (SIM) and Location-aware SIM (LSIM).
SIM adopts the sliding window model and maintains a seed set with the maximum
influence value collectively over the most recent social actions, while LSIM
considers the social actions associated with geo-tags and identifies a seed
set that maximizes the influence value in a query region over a location-aware
social stream.

Zhu et al. (2019) generalizes the problem by considering the location-aware
targeted influence blocking maximization, which aims to find a positive seed
user set to block the influence propagation of negative seed users over the
targeted users located in a given region. They propose a problem-solving
algorithm by adopting the proxy-based approach together with a QT-tree index
that is used to store topics and locations information of users.

####  4.1.3 Sketch-based Approach

To avoid running heavy MC simulations and reserve the theoretical guarantee,
the sketch-based approach Borgs et al. (2014) pre-computes a number of Reverse
Reachable (RR) sketches based on a specific diffusion model and then evaluates
the influence spread of given users set $S$ by exploiting the portion of RR-
sketches which $S$ can reach. Here, the RR-sketch is constructed by randomly
selecting a node $u\in V$, and conducting a reverse MC sampling starting from
$u$. By performing the above operation iteratively $\theta$ times, the RR
sketches can be obtained. Compared with the simulation-based approach, the
sketch-based approach has a lower time complexity under a theoretical
guarantee.

Cai et al. (2020); Li et al. (2018a) study the holistic influence maximization
(HIM) query problem, which aims to find a minimum set of users whose holistic
influence can cover all target users in the network. They evaluate the
influence diffusion by considering the social connection, spatial connection,
and preference-based connection, rather than only taking social connection
into account. Specifically, they propose a holistic influence diffusion model,
named HIM model, under the basis of the IC model that takes into account both
cyber and physical user interactions. To answer the HIM query, they propose a
basic algorithm that uses the reverse $k$ nearest neighbors technique to
evaluate the offline influence spread and adopt the sketch-based approach to
calculate the influence of users under the HIM model. In addition, they
develop an index-based method to deal with large networks.

###  4.2 Approaches of Existing OBIM Studies

OBIM |  Diffusion Model |  Algorithms |  Accelerating |  Application Scenarios
---|---|---|---|---
Guo et al. (2016) |  – |  Cluster-based approach |  Pruning & Index |  Finding $k$ trajectories to be attached with a given advertisement and maximizes the expected influence among audience.
Liu et al. (2016) |  OI model |  Demo |  None |  Identifying the optimal billboard locations using the trajectory data.
Zhang et al. (2018, 2020) |  MI model |  Cluster-based approach |  Pruning |  Maximizing the number of influenced trajectories by setting a set of billboards within a limited budget cost.
Wu et al. (2018) |  voter model |  Gradient algorithm |  None |  Relating IM for the voting dynamics to models of network control in which external controllers interact with the intrinsic dynamics of opinion spread.
Zhang et al. (2019b) |  LI model |  Branch-and-bound approach |  Heuristic |  Finding a set of billboards that have the maximum influence under the budget.
Zhang et al. (2019a) |  MI / LI model |  Demo |  None |  A demonstrate system to assists users to find an optimal strategy for outdoor advertising deployment.
Zhang et al. (2021) |  OI model |  Greedy algorithm |  Local search |  Minimzing the regret of the influence provider when dealing with numerous influence purchasers.

Table 2: Summary of the existing OBIM works

We then analyze the methodologies of the existing OBIM studies. A summary of
this analysis is illustrated in Table 2.

####  4.2.1 Cluster-based Approach for Single Advertiser

As observed by Zhang et al. (2018), the billboards in different areas should
have small overlaps in their influenced trajectories. The cluster-based
approach exploits the locality property of billboards’ influence to avoid the
overlap of influence. Specifically, this approach firstly partitions the
billboards into a set of small clusters with low influence overlap, and then
finds the local solutions of each cluster by using the enumeration method.
After that, a dynamic programming approach is executed to construct the global
solution based on the local solution of each cluster.

Some studies Zhang et al. (2018, 2019b); Wang et al. (2019); Zhang et al.
(2020) focus on helping the single advertisers to achieve the largest
influence under their budget constraints by selecting the optimal billboards.
Specifically, Zhang et al. (2018, 2020) study the problem of trajectory-driven
influence billboard placement (TIP), which aims to find a set of billboards
within budget such that the placed ads on the selected billboards has the
maximum influence under the budget. The authors address the TIP problem under
the MI model and adopt a cluster-based approach to deal with the TIP problem.
Moreover, they boost the time efficiency of the algorithm by pruning
billboards with a low benefit or cost ratio.

Similar with Zhang et al. (2018), Zhang et al. (2019b) proposes the problem of
optimizing impression counts for outdoor advertising (ICOA), which aims to
find a set of billboards that have the maximum influence under the budget.
Differently, Zhang et al. (2019b) uses the LI model to evaluate the influence
spread of billboards. Since the objective function of ICOA problem is non-
submodularity under the LI model, which means no efficient algorithms with
constant approximation ratio to ICOA. To conquer this challenge, they propose
an estimation method to compute the upper bound by using a dynamic tangent
line to tightly bound the real influence. Based on the upper bound function,
the authors propose a branch-and-bound framework to answer the ICOA query.
Moreover, they optimize this framework by designing a progressive upper-bound
estimation based pruning method with
$(\theta/2(1-1/e-\varepsilon))$-approximation ratio.

####  4.2.2 Demos of OBIM

There exist two demo works of OBIM. Liu et al. (2016) systematically studies
the problem of identifying the optimal billboard locations using the
trajectory data. Specifically, they present an interactive visual analytics
system, named SmartAdP, to deal with the optimal billboard locations through a
greedy heuristic algorithm, called $k$-location query under the OI model,
together with several visualizations and interaction techniques. Moreover,
Zhang et al. (2019a) aims to assist users in finding an optimal strategy for
outdoor advertising deployment. Specifically, they implement an interactive
visual demonstration system, namely ITAA, to find the optimal billboard
development strategy, while the optimal billboards query techniques are from
Zhang et al. (2018, 2019b).

####  4.2.3 Other OBIM Works

Guo et al. (2016) introduces an IM problem over trajectory databases, which
aims to find $k$ best trajectories to be attached with a given advertisement
and maximizes the expected influence among a large group of users. To deal
with this problem, they propose a cluster-based algorithm that partitions the
trajectory database into clusters and accesses the clusters in an order.
Besides, a trajectory index is proposed to reduce the influence calculation
cost. Different from any other IM related works, this work does not consider
the influence diffusion.

The latest OBIM work Zhang et al. (2021) studies the OBIM problem from the
host’s perspective, which is responsible for assigning billboards to all
advertisers, and formulates this as the Minimizing Regret for the Out-of-Home
Advertising Market Problem (MROAM). The MROAM problem aims to minimize the
regret of the influence providers when dealing with numerous influence
purchasers. To deal with this problem, the authors first use the OI model to
evaluate the influence diffusion between billboards and users. Then, they
propose two greedy heuristic methods as baselines to answer the MROSAM query,
where the billboards are assigned based on a descending order. Since the
greedy methods may produce a poor local minimum because the objective function
of MROSAM is non-submodular and non-monotone, the authors further propose a
randomized local search framework to avoid the local minimum.

##  5 Challenges and Opportunities

Some critical issues in location-driven IM still remain for further
investigation. In this section, we discuss four challenges and their potential
research opportunities.

###  5.1 Challenge I: Predicting OLIM /OBIM

The existing OLIM/OBIM researches focus on finding a set of seed users (or
billboards) under the current social network or trajectory databases. However,
commercial companies want their product information to be disseminated to the
majority of their customers over a period of time after they have spent their
budget. Therefore, the seed users selected by using existing OLIM/OBIM methods
may not have a good performance for influence spread in the following period,
as the network topology often evolves in the real world Leskovec et al.
(2008).

Opportunities: The temporal link prediction (TLP) technique de Bruin et al.
(2021); Yu et al. (2017) can help predict the topology evolution of social
networks and trajectories. Developing effective algorithms to find a set of
influential users (or billboards) in the future period based on the TLP method
would be an exciting research topic.

###  5.2 Challenge II: Network Evolution

Reverse Reachable Sketch (RRS) Borgs et al. (2014) is the state-of-the-art
algorithm to deal with the IM problem. However, the RRS method cannot be
directly used to deal with the dynamic IM problem in which the graph structure
dynamically changes. This is because the structure information of each graph
sketch does not remain the same during the process of RRS.

Opportunities: One possible solution is to introduce a new index to
dynamically maintain the structural information of each graph sketch generated
from the RRS method. It would be a breakthrough in the IM research if we
successfully develop a new index-based variant of the RRS method which can
efficiently deal with the dynamic IM problem.

###  5.3 Challenge III: Non-submodularity

As mentioned in this survey, the OBIM problem is NP-hard, and most of the
existing solutions are based on greedy algorithms. However, some influence
diffusion models of the OBIM problem have a non-submodular diffusion function
Zhang et al. (2019b). In this situation, greedy algorithms are no longer
effective.

Opportunities: There are two possible research directions. The first direction
is to design a new influence diffusion model instead of the models with non-
submodular diffusion function. The other direction is to loosen the non-
submodular diffusion model to satisfy the weakly submodular function.

###  5.4 Challenge IV: Seed Users vs. Budget

All existing research efforts dedicated to the influence maximization problem
depend on a strong assumption – the selected seed users will spread the given
information. However, some of the chosen individual seed users may be
unwilling to promote the product information for various reasons in the real
business market. Motivated by this scenario, a new type of IM problem
description is required.

Opportunities: To this point, we suggest studying the optimal influential
edges query rather than the traditional seed users selection of IM problem.
Specifically, an interesting problem description would be researching for
boosting influence spread by arousing the estranged relationships emerged in
the past but have drifted apart.

##  6 Conclusions

This paper conducts a comprehensive survey on the existing location-driven
influence maximization (IM) research and classifies the efforts into OLIM and
OBIM based on their application scenarios. We systematically review the
influence diffusion models in existing OLIM and OBIM research works,
respectively. Furthermore, we study the problem-solving methods of all
existing OLIM and OBIM works, including the basic framework algorithms and the
accelerating techniques to boost the efficiency of the basic framework
algorithms. Finally, we provide several exciting research problems and
opportunities under the topic of location-driven IM.

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