11institutetext: Imperial College London, UK 22institutetext: EPFL, Lausanne,
Switzerland 22email<EMAIL_ADDRESS>
# A Dynamical Yukawam1[]2 Model

Ajay Chandra1 Martin Hairer1,2 and Martin Peev1

###### Abstract

We prove local (in space and time) well-posedness for a mildly regularised
version of the stochastic quantisation of the Yukawa2 Euclidean field theory
with a self-interacting boson. Our regularised dynamic is still singular but
avoids non-local divergences, allowing us to use a version of the Da
Prato–Debussche argument [DPD2]. This model is a test case for a non-
commutative probability framework for formulating the kind of singular SPDEs
arising in the stochastic quantisation of field theories mixing both bosons
and fermions. Keywords: Stochastic quantisation, fermions, non-commutative
probabilityMSC classification: 81T08, 46L53, 60H15

###### Contents

  1. 1 Introduction

##  1 Introduction

One approach to constructing interacting quantum field theories satisfying the
Wightman axioms was pioneered by Osterwalder and Schrader [OS73G1, OS75]. The
idea is to consider “imaginary time”, hence turning the ill-posed Feynman path
integrals into a formal expression that has at least a chance of being
interpretable as a probability measure on a space of tempered distributions
over space-time. Amazingly, this operation can be reversed: given such a
probability measure satisfying a number of rather natural properties, as well
as reflection positivity, it is possible to recover a Hilbert space of states
as well as a unitary evolution on that space and natural field operators.

Still, the construction of these “interacting” probability measures is far
from trivial and one of the milestones in this area was the construction of
the $\Phi^{4}_{3}$ measure in the 70’s [Feldman, GlimmJaffe]. In the 80’s,
Parisi and Wu [ParisiWu] proposed to interpret these measures as the invariant
measures of an associated Markovian dynamic (the “time” appearing there has no
relation to physical time, which is treated as one of the spatial coordinates
in this picture). While this is a very appealing perspective, it led to rather
little progress for about three decades (but see notably [MR815192, AlbRock91,
DPD2]) mainly due to a lack of understanding of the workings of
renormalisation for the stochastic PDEs arising in this way.

All this changed in the early 2010’s with the appearance of several techniques
including regularity structures [Hai14, BHZ19, CH16, BCCH21] and
paracontrolled analysis [GIP15, GP17] that provide an unequivocal meaning to
these stochastic PDEs which is consistent with previous works in constructive
field theory. However, nearly all of these works deal with purely bosonic
quantum field theories, thus excluding any of the more realistic theories that
combine both bosons and fermions. From a purely formal perspective, the
situation is quite well-understood [Wein95, Tic99]: a non-interacting pair of
particle /antiparticle fields $(\psi,\bar{\psi})$ can be described by a type
of centred “Gaussian” generalised random field such that

$\langle\psi(f)\psi(g)\rangle=\langle\bar{\psi}(f)\bar{\psi}(g)\rangle=0\;\hbox{,}\quad\langle\bar{\psi}(f)\psi(g)\rangle=C(f,g){\hbox{\rm(1.1)}\kern
5.69046pt\kern-5.69046pt}q:free_{f}ermions{=}e1.0{1.1}$


11institutetext: Theory Center, IPNS, KEK, Tsukuba, Japan, and Kavli IPMU, The
University of Tokyo, Kashiwa, Japan

# Beyond the Standard Model

Mihoko M. Nojiri

###### Abstract

A Brief review on the physics beyond the Standard Model.

##  0.1 Quest of BSM

Although the standard model of elementary particles(SM) describes the high
energy phenomena very well, particle physicists have been attracted by the
physics beyond the Standard Model (BSM). There are very good reasons about
this;

  1. 1.

The SM Higgs sector is not natural.

  2. 2.

There is no dark matter candidate in the SM.

  3. 3.

Origin of three gauge interactions is not understood in the SM.

  4. 4.

Cosmological observations suggest an inflation period in the early universe.
The non-zero baryon number of our universe is not consistent with the
inflation picture unless a new interaction is introduced.

The Higgs boson candidate was discovered recently. The study of the Higgs
boson nature is extremely important for the BSM study.

The Higgs boson is a spin 0 particle, and the structure of the radiative
correction is quite different from those of fermions and gauge bosons. The
correction of the Higgs boson mass is proportional to the cut-off scale,
called “quadratic divergence". If the cut-off scale is high, the correction
becomes unacceptably large compared with the on-shell mass of the Higgs boson.
This is often called a “fine turning problem". Note that such quadratic
divergence does not appear in the radiative correction to the fermion and
gauge boson masses. They are protected by the chiral and gauge symmetries,
respectively.

The problem can be solved if there are an intermediate scale where new
particles appears, and the radiative correction from the new particles
compensates the SM radiative correction. The scale is probably much less than
$\mathcal{O}$(100) TeV, where the ratio between the SM radiative correction
and the Higgs vev is more than 1000. The turning of the factor 1000 may sound
unnatural, but it is much better than the scale among other parameters, such
as Planck scale to the order of electroweak symmetry breaking, or the large
difference among Yukawa couplings.

An idea to introduce a new particle that couples to the Higgs boson to cancel
one loop level quadratic correction, is not successful, because such
accidental cancellation does not hold all order in the perturbation theory.
One needs new symmetry to cancel the quadratic divergence in the SM by a new
physics contribution. The known ideas to achieve the reduction of quadratic
divergence are the following;

  1. 1.

Supersymmetry: Extend the SM so that the theory has “supersymmetry".
Supersymmetry is the symmetry between bosons and fermions, which allows the
divergence of Higgs boson mass controlled by "chiral symmetry" of fermions.
Due to the cancellation among various diagrams involving SM particles and
their superpartners (SUSY particles), there are no quadratic divergence to the
Higgs bosons mass in this theory.

  2. 2.

Dynamical symmetry breaking: In this theory, a new strong interaction causes
the spontaneous gauge symmetry breaking of the SM. The Higgs doublet is a
Nambu-Goldstone boson of the symmetry breaking and bound states of fermions
charged under the strong interaction, corresponding to the pions in the QCD.
The Higgs boson does not exist above the symmetry breaking scale, so there are
no problem of quadratic divergence.

  3. 3.

Extra dimension Although we recognize that we live in the four dimensional
space-time, we might live in more than the five dimension space time where the
extra dimensions are compactified. The true Planck scale may be much closer to
electroweak scale in such a theory, or the fundamental parameters in the Higgs
sector is of the order of Planck scale in the higher dimensional theory but
looks small in the effective four dimensional theory. In some class of the
model the Higgs boson may be a part of gauge boson in the 5th dimension so
that the divergence of the Higgs mass parameters is controlled by the gauge
symmetry.

Those models are constrained strongly by precision measurements. Currently
there are no measurements with significant deviation from the SM predictions.
In the SM theory, one can predict various observables from a few fundamental
parameters: the gauge couplings $g_{i}(i=1,2,3)$, and the Higgs vacuum
expectation value (vev) $v$. By measuring the deviations from the SM
predictions, we can set constrains on the new physics. Especially, the $S$ and
$T$ parameters which parametrize the new physics contributions to the gauge
two point functions are sensitive to all particles that couple to the gauge
bosons. Measurements of flavor changing neutral current (FCNC) constrain the
existence of flavor off-diagonal interactions. Very precisely measured
parameters sometimes exhibit significant deviations from the SM predictions.
Currently muon anomalous magnetic moment deviates from the SM prediction by
more than 3 $\sigma$. It is sensitive to the new physics that couples to muon.

The quadratic divergence of the Higgs sector exists if the divergence is
estimated by the momentum cut off $\Lambda$, the upper bound of the various
loop integral appearing in the radiative correction in the mass. We have to
keep it in mind that the quadratic divergence does not depend on the external
momentum, therefore it is a regularization dependent object. Especially in
dimensional regularization, quadratic divergence is trivially zero. Then, is
there any reason that we should take the fine turning problem seriously?

The fine turning argument based on momentum cut-off is justified in the case
that the theory has large symmetry at some higher energy scale. For example,
in the supersymmetric model, the regularization must respect to supersymmetry
and one cannot subtract all quadratic divergence. To this end, the Higgs
sector receives radiative corrections proportional to the SUSY scale
(superpartner mass scale) under correct regularization. In the limit that
superpartners are much heavier than SM particles, the low energy theory looks
like the SM with the momentum cutoff at the SUSY scale. Fine turning arguments
hold for the theories with an intermediated scale above which a new symmetry
emerges.

There is another indication of the existence of new physics between the weak
scale and the Planck scale. We may consider the Higgs potential at large field
value in the SM and study the stability. The potential is a function of the
top and Higgs masses, and current top and Higgs mass measurements favor
metastable Higgs potential. There is not any reason that the Higgs vev should
fall in such a metastable point, and this also suggests that additional
particles that couple to the Higgs sector change the shape of the potential.

Another strong indication of new physics is the existence of dark matter in
our Universe. Global fit of the cosmological observation favors the existence
of stable, neutral particle, dark matter, which accounts for 27% of the total
energy of our Universe. The existence of the dark matter is also confirmed by
various observations of the stellar objects. Rotation curve of the stars of
the galaxy indicates that galaxies are dominated by the non-luminous
component. The is also a technique to measure the matters extended beyond the
galaxy scale using gravitational lensing.

Our universe is $1.38\times 10^{10}$ years old, roughly $10^{17}$ s
$\leftrightarrow$ $10^{-43}$ GeV-1. The dark matter life time must be at least
of the oder of the age of the Universe to remain in the current Universe. 111
In order to avoid the constraints coming from cosmic ray observations, the
lifetime of the dark matter in our Universe must be significantly longer than
the age of the Universe.  On the other hand, a particle with mass $m$ (GeV)
with interaction suppressed by $1/M_{pl}$ has a decay width of order of
$g^{2}(m/{\rm 1\leavevmode\nobreak\ GeV})^{3}10^{-38}$ GeV. Namely the
lifetime, $\tau\sim g^{-2}10^{14}{\rm s}/({\rm m/1\leavevmode\nobreak\
GeV})^{3}$, would be much shorter than the life of our Universe ($\sim
4.3\times 10^{17}$ s), where $g$ is the coupling of the decay vertex. To
account for the lifetime of the dark matter in our universe, its decay must be
very strongly suppressed, or forbidden.

For the case of the SM particles, existence of stable particles is ensured by
the symmetry. Electron is the lightest charged particle and electronic charge
is conserved by the gauge symmetry. Proton is the lightest bound state of
quarks. There are no interaction to break proton in the SM, because number of
quark is conserved for interaction with the gauge bosons or the Higgs boson,
and direct interaction with electron is forbidden by the gauge symmetry. It is
possible to conserve the Baryon number 1/3 to the quarks in the SM, and this
reflects the fact that proton is stable. To consider the particle model
involving the stable (or long-lived) dark matter, we must introduce new
symmetry to protect the dark matter from decaying.

Another puzzle of the SM is the hyper-charge assignments of the fermions. In
the first glance, it is not easy to find the rules to assign the charge to the
SM matters. But, it fits very nicely to the representation of a $SU(5)$ group,
where $SU(3)\times SU(2)\times U(1)$ generators are embedded as

$T^{a}_{SU(3)}=\left(\begin{array}[]{cc}\lambda^{a}&0\cr
0&0\end{array}\right)\ \ \ T^{i}_{SU(2)}=\left(\begin{array}[]{cc}0&0\cr
0&\sigma^{i}\end{array}\right)\ \ \
T_{U(1)}=\left(\begin{array}[]{cc}-\frac{1}{3}{\bf 1}_{3}&0\cr
0&\frac{1}{2}{\bf 1}_{2}\end{array}\right).$ (1)

Here, $\lambda^{a}$ and $\sigma^{i}$ are the $SU(3)$ and $SU(2)$ generators,
13 and 12 are $3\times 3$ or $2\times 2$ unit matrix, and $T_{SU(3)}$,
$T_{SU(2)}$, $T_{U(1)}$ satisfy the commutation relations of $SU(3)$, $SU(2)$,
and $U(1)$ generators. Under this generator assignment, $\bf{5^{*}}$ and 10
representations of $SU(5)$ have a charge assignment as

${\bf
5^{*}}=\left(\begin{array}[]{c}(3^{*},1)_{1/3}\cr(1,2)_{-1/2}\end{array}\right),$
(2)

while 10  representation is decomposed into $(3,2)_{1/6}$ $\oplus$
$(3^{*},1)_{-2/3}$ $\oplus$ $(1,1)_{1}$ which reside in the $5\times 5$
antisymmetric matrix as

${\bf
10}=\left(\begin{array}[]{cc}(3^{*},1)_{-2/3}&(3,2)_{1/6}\cr*&(1,1)_{1}\end{array}\right).$
(3)

This suggests that $SU(3)$ $\times$ $SU(2)$ $\times$ $U(1)$ symmetry of the SM
can be unified into the $SU(5)$ gauge symmetry. To realize this, the SM three
gauge couplings must unify at the short distance, so that the $SU(5)$ symmetry
is recovered above that scale. The gauge couplings at the short distance is
calculated by utilizing the SM renormalization group equations from the low
energy inputs. They do not unify for the particle content of the SM, therefore
to realize the idea of GUT, new set of particles are needed. We will see a
successful gauge coupling unification is realized in the Supersymmetric model
in the next section.

##  0.2 Supersymmetry

Supersymmety is the symmetry exchanging bosons into fermion, and fermions into
bosons. The generators of the supersymmetric transformation satisfy the
following anti-commutation relations

$\left\\{Q^{\alpha},\bar{Q}_{\dot{\beta}}\right\\}=2\sigma^{\mu}_{\alpha,\dot{\beta}}P_{\mu}$
(4)

Here $Q$ is a spin $1/2$ and mass dimension $1/2$ operator and $\alpha$ and
$\dot{\beta}$ $(=1,2)$ are the spin indices of chiral and anti-chiral
fermions, and $\sigma^{\mu}=(1,\sigma^{i})$ is the Pauli matrices.

This anti-commutation relation can be reduced for any massive eigenstate
$|a\rangle$ by taking the rest frame
$P^{\mu}|a\rangle=m_{a}\delta_{0\mu}|a\rangle$ as follows:

$\left\\{Q^{\alpha},\bar{Q}_{\dot{\beta}}\right\\}=2\delta_{\alpha,\dot{\beta}}m_{a}.$
(5)

The relation is same as that of a two-fermion system in quantum mechanics. One
can construct an irreducible representation of this algebra starting from a
state which annihilates any $\bar{Q}_{i}$. Suppose the state is spin 0,
$|0\rangle$, all possible states are generated as follows;

$|0\rangle\rightarrow Q_{1}|0\rangle,Q_{2}|0\rangle\rightarrow
Q_{1}Q_{2}|0\rangle.$ (6)

Because $Q_{1}Q_{1}=$ $Q_{2}Q_{2}=0$, no more state can be obtained by
multiplying the generator $Q_{i}$. Two spin 0 states and two spin $1/2$ states
are obtained. These states form a SUSY multiplet, and the spin $0$ states are
the superpartners of the spin $1/2$ states and vise versa. Because this
multiplet contains spin 1/2 states, we can regard this as a matter multiplet.

Starting from a spin 1/2 state annihilating $\bar{Q}$ one gets two spin $1/2$
fermion states, a spin $1$ massive bosonic states and a spin $0$ bosonic
state, namely 4 fermion degrees of freedom and 4 bosonic degrees of freedom.
This may be regarded as two chiral fermions, one massive gauge boson and one
massive Higgs boson. Repeating similar analysis to the massless particles, one
obtains states with helicity $h=\lambda$ and $\lambda+1/2$. If $\lambda=1/2$,
a massless gauge boson and its superpartner fermion make a supersymmetric
multiplet. The number of bosonic degrees of freedom is the same as that of
fermionic degrees of freedom in this theory.

All states in the above multiplet have the same mass, which looks irrelevant
for describing real particles, but it is known that such mass degeneracy is
removed by spontaneous supersymmetry breaking. Supersymmetry breaking is
discussed in the next section.

The minimal supersymmetric standard model (MSSM) is an extension of the SM
that has a supersymmetry in the limit where all particle masses are ignored.
The model is thought to be an effective theory of a fully supersymmetic
theory. Due to the spontaneous supersymmetry breaking of the full theory, the
superparters of the SM particles receive a mass much higher than the SM
particles. A superpartner of a fermion is called sfermion and it is a spin 0
particle. A superpartner of a gauge boson is called gaugino and has spin
$1/2$. A Higgs boson superpartner is called a higgsino and has spin $1/2$. The
particle content of the MSSM is given in Table 1. The SM particles and their
superpartners have same charge, because the generator of supersymmetric
transformation $Q$ commutes with the SM $SU(3)\times SU(2)\times U(1)$
transformation. The number of Higgs doublets is two in the MSSM because one
should add two Higgsinos, chiral fermions with charge $(1,2)_{\pm 1/2}$ in the
SM because of a condition of anomaly cancellation.

Table 1: Particle content of the Minimal Supersymmetric Standard Model.  represenations | quark | squark
---|---|---
$(3,2)_{1/6}$ | $q_{L}=(u,d)_{L}$ | $\tilde{q}_{L}=(\tilde{u}_{L},\tilde{d}_{L})$
$(3^{*},1)_{-2/3}$ | $u_{R}^{c}$ | $(\tilde{u}_{R})^{c}$
$(3^{*},1)_{1/3}$ | $(d_{R})^{c}$ | $(\tilde{d}_{R})^{c}$
| lepton | slepton
$(1,2)_{1/2}$ | $l_{L}=(\nu,e)_{L}$ | $\tilde{q}_{L}=(\tilde{\nu}_{L},\tilde{e}_{L})$
$(1,1)_{1}$ | $(e_{R})^{c}$ | $(\tilde{e}_{R})^{c}$
| Higgsino | Higgs
$(1,2)_{-1/2}$ | $(\tilde{H}^{0}_{1},\tilde{H}^{-}_{1})$ | $(H^{0}_{1},H^{-}_{1})$
$(1,2)_{1/2}$ | $(\tilde{H}^{+}_{2},\tilde{H}^{0}_{2})$ | $(H^{+}_{2},H^{0}_{2})$
| spin 1/2 | spin 1
$(8,1)_{0}$ |  $\tilde{G}$ (gluino) | $G^{\mu}$
$(1,3)_{0}$ |  $\tilde{W}$ (wino) | $W^{\mu}$
$(1,1)_{0}$ |  $\tilde{B}$ (bino) | $B^{\mu}$

As one can see from Table 1, the number of particles are doubled in the MSSM.
The supersymmetry specifies all dimensionless couplings of interactions of new
particles, such as four point interaction of scalers and Yukawa couplings,
while mass parameters of superpartners are undetermined. To understand the
coupling relations, one needs to understand the supersymmetric field theory.
In this lecture, I do not have enough time to talk about it in detail, so I
just sketch the important elements.

Fields in the same supersymmetric matter multiplet can be arranged in a
“chiral superfield” which is a function of coordinate $x$, $\theta$ and
$\bar{\theta}$ a grassmanian Lorentz spinors with mass dimension $-1/2$,

$\Phi(x,\theta,\bar{\theta})=\phi(y)+\sqrt{2}\psi(y)\theta+F(y)\theta\theta,$
(7)

where $y^{\mu}=x^{\mu}-i\theta\sigma^{\mu}\theta$. Note that by redefining the
coordinate from $x$ to $y$, $\Phi$ becomes a function of $y$ and $\theta$, and
$\bar{\theta}$ does not appear. There are only three fields $\phi$, $F$ and
$\psi$ appearing as the component fields of $\Phi$. When $\theta$ is zero,
$\Phi(x)=\phi(x)$, therefore $\Phi$ is an extension of the scalar field of
non-supersymmetric theory. On the other hand, $\Phi(y,\theta)$ represents both
fermonic and bosonic fields simultaneously.

$\Phi$ is dimension 1, so that ${\rm dim}(\phi)=1$ and ${\rm dim}(\psi)=3/2$.
$F$ is then spin 0 and dim 2 field. The only $dim<4$ kinetic term of $F$ is
$FF^{*}$, therefore $F$ is not dynamical. The product of a chiral superfield
is also a chiral superfield depending only $y$ and $\theta$. On the other
hand, $\Phi\bar{\Phi}^{\prime}$ is not a chiral superfield as it has the terms
proportional to $\bar{\theta}$.

Just as operator $P^{\mu}$,translation in coordinate space $x$ is expressed as
$\partial/\partial x$, supersymmetric transformation $Q$ is a translation in
the $\theta$ and $\bar{\theta}$ space. Namely, in the coordinate
representation it is expressed as

$S_{\alpha}=\frac{\partial}{\partial\theta^{\alpha}}+i(\sigma^{\mu}\partial_{\mu}\bar{\theta}).$
(8)

The second term is needed to satisfy the SUSY algebra give in Eq. 4. With this
transformation, each field transform as

$\displaystyle\delta_{SUSY}\phi$ $\displaystyle=$
$\displaystyle\sqrt{2}\alpha\psi,$ (9) $\displaystyle\delta_{SUSY}\psi$
$\displaystyle=$
$\displaystyle-i\sqrt{\partial}_{\mu}\sigma^{\mu}\phi\bar{\alpha}+\sqrt{2}F\alpha,$
(10) $\displaystyle\delta_{SUSY}F$ $\displaystyle=$
$\displaystyle-i\sqrt{2}\bar{\alpha}\partial_{\mu}\bar{\sigma}^{\mu}\psi,$
(11)

where $\alpha$ and $\bar{\alpha}$ are transformation parameters. Under this
transformation kinetic term

${\cal
L}_{kin}=\partial_{\mu}\phi\partial^{\mu}\phi^{*}+i\bar{\psi}\sigma^{\mu}\partial_{\mu}\psi+F^{*}F$
(12)

is invariant.

There are a few things worth paying attention. First The $\delta_{SUSY}F$ is
total derivative. Because the product of chiral superfields is also a
superfield, the $\theta\theta$ component $\mathcal{F}$ transforms as
$\mathcal{F}=\partial_{\mu}J^{\mu}$, namely $\mathcal{F}$ can be interaction
terms which are invariant under supersymmetric transformation. For example,
$\Phi_{1}\Phi_{2}\Phi_{3}$ gives $F$ term

${L}_{Yukawa}=F_{1}\phi_{2}\phi_{3}+F_{2}\phi_{1}\phi_{3}+F_{3}\phi_{1}\phi_{2}-\psi_{1}\psi_{2}\phi_{3}-\psi_{2}\psi_{3}\phi_{1}-\psi_{3}\psi_{1}\phi_{2}.$
(13)

The interaction contains Yukawa interaction term
$y_{ijk}\psi_{i}\psi_{j}\phi_{k}$ which is symmetric under the exchange of
$i,j,k$, and also the scalar potential terms proportional to
$y_{ijk}F_{i}\phi_{j}\phi_{k}$ Combined with kinetic term $FF^{*}$,
interactions of four point scalar fields proportional to $y^{2}$ is generated.
The similar relations also holds for supersymmetric gauge interactions. The
interaction between gaugino-fermion- sfermion is proportional the gauge
coupling $g$, and there are scalar four point interactions proportional to
$g^{2}$. While many scalar and fermion partners are introduced, there are no
new dimensionless coupling introduced.

In addition to the $F$ term, $\theta\theta\bar{\theta}\bar{\theta}$ term of
general field product, $\mathcal{D}$ is supersymmetric. For example,
supersymmetric kinetic term is $\theta\theta\bar{\theta}\bar{\theta}$ term of
$\Phi\bar{\Phi}$.

We now address some important features of supersymmetric models.

  * •

There are no quadratic divergence in the theory. The quadratic divergence
coming from the top loop is canceled by the stop loop generated by the Higgs-
Higgs-stop-stop four point interaction. Both of them are proportional to
$y^{2}$. The Higgs four point coupling is proportional to the square of the
gauge coupling, and quadratic divergence arising from the diagram is canceled
by the gauge and gaugino-higgsino loops. This is because scalar particles are
now in a same multiplet with the fermion, and the mass of the fermion is only
logarithmically divergent. The fine-turning in the Higgs sector is now
significantly reduced.

  * •

Because the Higgs four point coupling is a gauge coupling, the Planck scale
Higgs four point coupling is always positive, therefore significantly less in
danger of running into metastable vacuum At low energy the Higgs mass is upper
bounded by the $Z$ boson mass in tree level, and radiative corrections
proportional to the $(mt^{4}/m^{2}_{W})\log(m_{\tilde{t}}/m_{t})$ appear in
the Higgs boson mass formulae. This correction is interpreted as the running
of the Higgs boson four point coupling from the stop mass scale to the top
mass scale under the SM renormalization group equation, because below the stop
mass scale, the theory is effectively the SM. In addition there are
contribution proportional to the fourth power of stop left-right mixing
$X_{t}$. See Fig. 1 (left) for the RGE interpretation of the radiative
corrections to the Higgs mass. In this theory, the Higgs boson mass is
calculated from the scalar top mass and its mixing, therefore the SUSY scale
is predicted from the Higgs boson mass. In other words, the measured Higgs
boson mass gives a strong constraint to the SUSY mass scale and mixing. See
Fig. 1 (right).

Figure 1: Left: the running of Higgs four point coupling changes at the scale
of $m_{stop}$. Right: Maximal value of the Higgs boson mass as a function of
$X_{t}/M_{SUSY}$ when all the other parameter are scanned. From arXive
1311.0720.

  * •

In the SM, one cannot write an interaction violating baryon and lepton numbers
due to the gauge invariance. This is no longer true because Higgsino and
lepton doublets have same quantum numbers. The product of superfields $W$
whose $\theta\theta$ terms is the SM Yukawa interactions

$W=-y_{e}H_{1}\cdot E^{c}L-y_{d}H_{1}\cdot D^{c}Q-y_{u}H_{2}\cdot U^{c}Q-\mu
H_{1}\cdot H_{2},$ (14)

where $Q=\tilde{q}_{L}+\theta q_{L}...$, $U^{c}=\tilde{u}^{c}_{R}+\theta
u^{c}_{R}...$… are the superfields whose bosonic component is a sfermion and a
fermionic component is quarks or leptons. However, $\theta\theta$ term of
$W^{\prime}$

$W^{\prime}=\epsilon_{L}LLE^{c}+\epsilon_{BL}LQD^{c}+\epsilon_{B}U^{c}D^{c}D^{c}+\epsilon_{LH}LH_{2}$
(15)

is not forbidden by the gauge symmetry, because $H_{1}$ and $L$ have a same
quantum numbers, and $UDD=\epsilon_{abc}U^{a}D^{b}D^{c}$ is a gauge singlet.
The interactions violate lepton and/or baryon numbers and should be forbidden.

The symmetry that forbids $L$ and $B$ violating terms is called the conserved
R-parity. In the MSSM R-parity may be assigned to the superfield and
coordinate $\theta$ as follows,

$R(L)=R(E)=R(Q)=R(U)=R(D)=-1,R(H)=1,R(\theta)=-1.$ (16)

In this assignment, all the SM particles have $R=1$ and all superpartners have
$R=-1$, and $R(W|_{\theta\theta})=1$, and $R(W^{\prime}|_{\theta\theta})=-1$.
The interaction term from $W$ multiplicatively conserves $R$ parity, namely,
product of $R$ parity of all particles involved in a vertex is one. Namely,
$R=-1$ particle decays into the final states which contains odd number of
$R=-1$ particles. If two $R=1$ particle collides, the final state contains
even number of $R=-1$ particles. By requiring multiplicatively conserved $R$
parity, the lightest supersymmetric particle (LSP) becomes stable. The LSP can
be a dark matter candidate.

  * •

Gauge coupling: In the supersymmetric model, the number of particles is
doubled and running of the gauge couplings would be modified above the SUSY
particle mass scale. The gauge couplings unify at the GUT scale much better
than that of the SM as can be seen in Fig. 2. This means "supersymmetric GUT"
is consistent with experimental data, though there are still some fine turning
issues when we consider the Higgs sector violating GUT symmetry.

Figure 2: Two loop renormalization group evolution of the gauge couplings in
the SM (dashed lines) and MSSM (solid lines) from “A Supersymmetry primer”
hep-ph/9709356.

##  0.3 Origin of SUSY breaking

As we have mentioned already, the MSSM is not a complete theory, because it
requires a mechanism to break the supersymmetry somewhere outside the MSSM. A
general set up of the SUSY breaking models are the following; there are hidden
sector $H$, and fields $Z_{i}$ in the sector $H$ break the supersymmetry
spontaneously. This hidden sector couples to our sector indirectly though a
messenger sector. The particles in the messenger sector have a mass scale $M$.

The spontaneous symmetry breaking is realized for the vacuums which do not
annihilate with the supersymmetric generator $Q$ and $\bar{Q}$. If such a
vacuum exists, there are some fermions $\psi$ whose supersymmetric
transformation $\delta_{SUSY}\psi=\left\\{Q,\psi\right\\}$ has non-zero vev,
namely $\langle 0|\delta_{SUSY}\phi|0\rangle=$ $-\sqrt{2}\langle
0|F|0\rangle\neq 0$. Some of the superfields in the Hidden section must have
non-zero $F$ terms in our setup.

If $F$ term of $Z$ has non zero vev, $\langle Z\rangle=\langle
F_{Z}\rangle\theta\theta$, various mass terms are induced in the low energy
effectively. A simple example is $\theta\theta\bar{\theta}\bar{\theta}$ term
of $Z\bar{Z}\Phi\bar{\Phi}/M^{2}$, which may be induced through the messenger
interactions. After the symmetry breaking the term $(\langle
F\rangle^{2}/M^{2})\phi\phi^{*}$ is the effective SUSY breaking mass term of
the scaler boson $\phi$.

There are already severe constraints to the interaction of the messenger
sector to the MSSM sector. These constrains come from the flavor changing
neutral currents such as $K^{0}$-$\bar{K}^{0}$ mixing. The constraints
typically require

$\left[\frac{10{\rm
TeV}}{m_{\tilde{q},\tilde{g}}}\right]^{2}\left[\frac{\Delta
m^{2}_{\tilde{q}_{12}}/m^{2}}{0.1}\right]^{2}<1,$ (17)

where $m^{2}_{\tilde{q}_{12}}$ is a mixing parameter of the first and second
generation squark, and $m^{2}$ is diagonal squark masses. The SUSY breaking
sector $H$ therefore must couple to the MSSM matter sector universally.

Figure 3: Relation between the MSSM sector and SUSY breaking sector.

Several mechanisms have been proposed to assure the universality of the soft
scalar masses. The supergravity model uses the gravity interaction as the
messenger mechanism, on the other hand, gauge mediation models uses some
vector-like matters charged under the SM gauge groups as the messenger fields.
Even if there are no direct couplings between the MSSM and SUSY breaking
sectors, there are mediation mechanism through the superconformal anomaly, and
the model utilizing this is called anomaly mediation model.

It is difficult to access the HIdden sector directly. The SUSY breaking of the
total theory $F_{0}$ and mass of the gravitino(super partner of graviton)
$m_{3/2}$ is related as $m_{3/2}=F_{0}/M_{pl}$. The gravitino could be the
LSP, in that case the next lightest SUSY particle(NLSP) is long-lived. The
NLSP can be detected directly at the collider, the decay lifetime provide the
information of hidden sector SUSY breaking. If gravitino is not the LSP, the
gravitino can be long-lived and may have impact on big-bang neucleosynthesis.
See Fig. 3.

The mediation mechanism sets the sparticle mass parameters at the mediation
scale, and on-shell masses of the SUSY particles are obtained by running the
RGE equation of the masses down to the low energy scale. If the boundary
condition is universal at $M_{GUT}$, squark and gluino masses are much heavier
than those of electroweakly interacting superpartners such as sleptons, wino,
bino and Higgsinos. The square of Higgs mass parameter is driven to be
negative at the weak scale, and Higgsino mass parameter $\mu$ compensates it
so that the Higgs vev is the correct value. The cancellation between $\mu$ and
SUSY breaking parameters at the weak scale is a measure of the fine turning in
the Higgs sector. See Fig. 4.

Figure 4: Relation between the MSSM sector and SUSY breaking sector.

##  0.4 Collider search of supersymmetric particles

So far, a proton-proton collider at CERN, the Large Hadron Collider (LHC), has
collected $\sim 30$ fb-1 of integrated luminosity for each experiment at 7 to
8 TeV. It will start operation again from 2015 aiming for 300 fb-1 at 13 TeV.

A proton is a composite particle and quarks and gluons in the proton are the
elementary particles that are involved in the high energy scatting process.
The momentum of the quarks and gluons are parallel to the beam direction but
the absolute values are not fixed. Therefore the collision system is boosted
to one of the beam directions. The production cross section is generally the
highest near the threshold. It reduces gradually with the increase of the
parton collision energy $\sqrt{s}$. The quarks and gluons in the final state
are fragmented and hadronized into hadrons, forming the jets. Electroweakly
interacting particles $W$, $Z$, $\gamma$, leptons and neutrinos are also
produced from various production processes.

Colored superpaticles are copiously produced at the hadron collider. Due to
the conserved $R$-parity of the MSSM, superpartners are produced in pairs,
each superpartner decays to the final state involving another superpartner,
and at the end of the cascade decay, the LSP appears. The LSP is stable. Due
to the cosmological constraints, it is neutral and color-singlet, and escapes
detection. If the mass difference between the superpartners are large, the
decay product tends to have high $p_{T}$. In such a case, the LSP, which
cannot be detected directly, is also relativistic (See Fig. 5).

Figure 5: The decay pattern of squark and gluino produced at the LHC, and
particles emitted from the cascade decay chain. The particles in the Little
Higgs model with T parity or universal extra dimension model may also give a
similar signature.

The sum of LSP momentum transverse to the beam direction is balanced against
other visible particles. Namely, significant missing transverse momentum ${\bf
P}_{Tmiss}$ defined as

${\bf P}_{Tmiss}=-\sum_{i}{\bf p_{T}}^{i}_{\rm jet}+\sum_{j}{\bf
p_{T}}^{j}_{\rm l},$ (18)

is a signature of SUSY particle production. Another important quantity is the
sum of absolute values of the transverse momentum

$H_{T}=\sum_{i}p^{i}_{\rm Tjet}+\sum_{j}p^{j}_{\rm Tl},$ (19)

or the effective mass

$m_{eff}=\sum_{i}p^{i}_{Tjet}+\sum_{j}p^{j}_{Tl}+E_{Tmiss},$ (20)

where $E_{Tmiss}$ is the absolute value of missing transverse momentum.

Figure 6: Distribution of $p^{\mu}_{T}$ from the $W$ boson decay measured at
CDF experiment at Tevatron.

The $m_{\rm eff}$ distribution peaks at the sum of the produced particles at
the hard process. To observe this fact, let us first consider the $p_{T}$
distribution of leptons from $W$ boson decay produced at CDF experiment at
Tevatron, a $p\bar{p}$ collider at 1.8 TeV. The distribution peaks at 40 GeV,
which is a half of the $W$ boson mass. See Fig. 6. The feature is easily
understood when we calculate the $p_{T}$ distribution of spherically decaying
$W$ boson boosted to the beam direction,

$f(x)dx=\frac{2}{\sqrt{1-x^{2}}}dx,$ (21)

where $p_{T}=(m_{W}/2)\sin\theta=xm_{W}/2$; The distribution strongly peaks at
$p_{T}=m_{W}/2$ ($\sin\theta=1$) and the structure remains even though $W$
bosons are boosted transversely in the realistic situation, because the
production cross section is largest near the threshold. The fact applies to
all production processes at the hadron collider; the sum of the $p_{T}$ of the
decay products peaks near the parent’s mass. When heavy particles are produced
in pairs, the sum of the $p_{T}$ of the decay products peaks at the sum of the
produced particle masses.

Figure 7: Distribution of $m_{eff}$ and $E_{Tmiss}$ of the top partner pair
production at the LHC followed by the decay into top and stable neutral gauge
boson (left) compared with the $t\bar{t}$ distribution.

Fig. 7 compares the distributions of $T_{-}T_{-}$ and $t\bar{t}$ pair
productions. Here a hypothetical particles $T_{-}$ is assumed to decay into
$t$ and $B_{H}$, and $B_{H}$ is a neutral stable massive $U(1)$ gauge boson.
The signal contains $t\bar{t}$ and existence of two $B_{H}$’s is observed by
the missing transverse momentum of the events, namely, the signal is similar
to that of superpartner pair production. The signal production cross section
is ${\mathcal{O}}$(1) pb, while the $t\bar{t}$ production cross section is
huge at the LHC, around 800 pb. If the distribution overlaps significantly,
the signal is very difficult to be observed. However, the signal $m_{eff}$
distribution peaks around 1 TeV and missing momentum as close as half of the
$M_{eff}$, while the background peaks around $m_{eff}\sim 400$ GeV and
$E_{Tmiss}\ll M_{eff}/2$. Because of this distribution differences, the
$T_{-}$ signature with the production cross section much less than 1 pb may be
observed at the LHC.

So far we have been talking about “inclusive” quantity. They are defined using
all objects in an event. We may also select jets or leptons with special
features and use kinematical information to separate signals and backgrounds.
Let us consider events with one lepton and some missing momentum. The event
with one lepton + multiple jets + missing momentum is an important signature
of superpartner production. However, events involving $W$ boson also produce
such signatures. However, the events with $W$ boson can be reduced
significantly if we require that $m_{T}$ of a lepton and missing $p_{T}$ is
above 100 GeV where $m_{T}$ is defined as

$m_{T}=\sqrt{2p^{l}_{T}E^{Tmiss}}(1-\cos(\Delta\phi(l,p_{T}))$ (22)

The cut significantly reduces the background from the $W$ boson production to
the SUSY process.

The current bound of the SUSY process is obtained after successful reduction
of background using the above kinematical variable. The understanding of
background distribution is quite important, especially the cross section of
$W$,$Z$, $t\bar{t}$ with multiple jets must be correctly calculated. The
techniques to obtain multiple jets amplitudes with parton shower has been
established only this century, and current SUSY searches at the LHC is
benefitted by those techniques greatly. The current limit typically excludes
squark with mass 1.8 TeV and gluino with mass less than 1.4 TeV, if the mass
splitting between the LSP and colored SUSY particles are large enough. See
Fig. refsusylimit for the latest limits.

Figure 8: Latest mass limit of the MSSM squarks and gluinos shown as a
function of GUT scale gaugino mass and scaler mass. Presented in SUSY2013.

##  0.5 Dynamical symmetry breaking and BSM

Supersymmetry is not a unique solution of the hierarchy problem. Another
important class of solutions is dynamical symmetry breaking models. When a
global symmetry is broken spontaneously, a massless scalar modes (Nambu-
Goldstone boson) appears, even if the theory does not have an elementary Higgs
boson. An important example is chiral symmetry breaking in QCD. The QCD
Lagrangian has $SU(2)_{L}\times SU(2)_{R}$ symmetry when quark masses are
ignored. The symmetry is spontaneously broken to $SU(2)_{V}$ dynamically, and
the Goldstone boson of the symmetry breaking are pions
$\pi\sim\bar{q}_{i}\gamma_{5}q^{\prime}$, and $\langle\bar{q}q\rangle$ has
non-zero vev.

The pion has the same charge as the Goldstone boson in the Higgs sector.
Therefore, it is natural to consider scale up of the mechanism. The model
involves a set of new quarks $Q$ with EW charges, but couple to different
asymptotic free gauge interactions whose couplings blow up at the scale of EW
symmetry breaking. If $\bar{Q}Q$ condense, the light $\bar{Q}\gamma_{5}Q$
states work as the Goldstone bosons of the EW symmetry breaking. This class of
the model called Technicolor model. The model has no quadratic divergence
because the massless bound states only appear in the low energy effective
theory.

This is an interesting and beautiful idea, but is not consistent with
precision EW observations. At LEP, gauge boson two point functions were
precisely measured. Especially the parameter called $S$, receives non
decoupling contribution from $SU(2)$ doublets $Q$ which is colored in the new
strong interactions, and also necessary charged under $SU(2)\times U(1)$
symmetry in the SM to break the gauge symmetry. Their contribution appears
constructively to the gauge two point functions, and therefore the model is
tightly constrained. In addition, these models tend to predict a heavy Higgs
boson inconsistent with the data.

Another class of models called "composite Higgs models" allows a Higgs boson
which is light but non-elementary. In these models, the Higgs doublet itself
is a pseudo Goldstone boson of some dynamical symmetry breaking. Though the
mechanism of dynamical symmetry breaking is not specified, the smallness of
the mass of the Higgs boson is thought to be ensured by the global symmetry of
the theory. The model requires extension of the top sector because the top
Yukawa coupling violates the desired global symmetry strongly. The extended
top sector is a target of extensive ATLAS and CMS searches.

##  0.6 Extra dimension models

In the Extra dimension models the space has more than three dimensions but the
additional space dimension is compactified with a small size $R$ so that we
could not recognized it easily. When the extra dimension is flat, the fields
in the extra dimension may satisfy the periodic boundary condition such as

$\phi(x,y)=\phi(x,y+R),$ (23)

where $x$ represents four dimensional space time, while $y$ is the fifth
dimension. Under this boundary condition, the wave function is expressed as

$\psi(x,y)=\psi^{\prime}(x)\exp(ip_{5}y),$ (24)

where $p_{5}R=2\pi n$ ($n$ is an integer). This leads to an equation of motion
of a free particle propagating in the the fifth dimension,

$E_{n}^{2}=p^{2}+p^{2}_{5}=p^{2}+(2\pi)^{2}\left(\frac{n}{R}\right)^{2}.$ (25)

Namely, the model predicts an infinite tower of particles of the four
dimensional effective theory, which corresponds to different values of the
discrete momenta in the fifth direction.

The coupling of the fifth dimension related with the couplings in the four
dimensional effective theory in non-trivial manner. A simple example is the
gauge coupling of the fifth dimensional theory and the four dimensional
effective theory,

$\int d^{4}xdx_{5}\frac{1}{g^{2}_{5}}F_{\mu\nu}F^{\mu\nu}\rightarrow\int
d^{4}x\frac{1}{g^{2}_{4}}F_{\mu\nu}F^{\mu\nu},$ (26)

where $g_{4}=g_{5}/\sqrt{R}$. Larger the size of the fifth dimension is,
$g_{4}$ becomes small. This is also true for gravitational interactions. The
four dimensional gravitational interaction may be small because the size of
extra dimension is large. The Large extra dimension model tried to solve the
fine tuning problem by making true Phanck scale in the higher dimensional
theory much smaller than the $M_{pl}$.

The extra dimension may not be flat. In the RS model, the fifth dimension has
non-trivial metric as follows:

$ds^{2}=e^{-2\sigma(\phi)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+r_{c}^{2}d\phi^{2},$
(27)

where $\phi=0$ and $\pi$ is the boundary of the fifth dimension. The gravity
action in the bulk is expressed as

$S_{gravity}=\int d^{4}x\int^{\pi}_{-\pi}d\phi\sqrt{-G}{-\Lambda+2M^{3}R},$
(28)

when the $\sigma(\phi)$ is expressed as

$\sigma(\phi)=r_{c}|\phi|\sqrt{\frac{-\Lambda}{24M^{3}}},$ (29)

provided appropriate fine tuning of the boundary actions.

The geometry allows us to control the masses of SM particles. If the Higgs
boson is at $\phi=\pi$ boundary (which is called visible brane), the kinetic
term is expressed as

$S_{vis}=\int
d^{4}x\sqrt{-\bar{g}_{vis}}e^{-4kr_{c}\pi}\times\left\\{g^{\mu\nu}_{vis}e^{2kr_{c}\pi}D_{\mu}H^{\dagger}D_{\nu}H-\lambda(|H|^{2}-v_{0}^{2})^{2}\right\\}.$
(30)

The mass term receives the suppression factor of $e^{-kr_{c}\pi}$ after
rescaling the Higgs field so that they have canonical kinetic terms. By
adjusting parameters one can easily obtain the mass of the SM particle of the
oder of the EW scale while all parameters of the fundamental fifth dimensional
Lagrangian are of the order of $M_{pl}$ without fine turning.

The model predicts towers of KK particles with mass of the order of
$\Lambda_{\phi}=\sqrt{6}M_{pl}e^{-kr_{c}\pi}$ for the particles living in the
fifth dimension(bulk). A popular set up of the model is that all the SM
fermions are the zero mode of the particles living in the bulk, and the Higgs
boson lives in the IR brane. Mass term of the fifth dimensional Lagrangian of
the SM model matters control the profile of the fields in the bulk. One can
adjust the mass so that light (heavy) quarks and lepton have small (large)
overlap with the IR brane so that Yukawa couplings in the four dimensional
effective Lagrangian is realized without introducing too much hierarchy among
the interactions between the Higgs boson and the bulk fermions. There are on-
going search of the KK gauge bosons and KK fermions at the LHC, however, FCNC
constraints require $\Lambda_{\phi}>10$ TeV already, and it is unlikely that
these new particles will be found at the LHC.

##  0.7 Suggested reading

To those who is interested in Supersymmetry, a good review for start with is
S. P. Martin,“A Supersymmetry primer,” In *Kane, G.L. (ed.): Perspectives on
supersymmetry II* 1-153 [hep-ph/9709356]. For A review of composite Higgs
model, I suggest R. Contino, “The Higgs as a Composite Nambu-Goldstone Boson,”
arXiv:1005.4269 [hep-ph].


#  Domain Adaptation for Object Detection using SE Adaptors and Center Loss

Sushruth Nagesh*
UC San Diego
<EMAIL_ADDRESS>Shreyas Rajesh*
UC San Diego
<EMAIL_ADDRESS>Asfiya Baig*
UC San Diego
<EMAIL_ADDRESS>Savitha Srinivasan*
UC San Diego
<EMAIL_ADDRESS>
###### Abstract

Despite growing interest in object detection, very few works address the
extremely practical problem of cross-domain robustness especially for
automative applications. In order to prevent drops in performance due to
domain shift, we introduce an unsupervised domain adaptation method built on
the foundation of faster-RCNN with two domain adaptation components addressing
the shift at the instance and image levels respectively and apply a
consistency regularization between them. We also introduce a family of
adaptation layers that leverage the squeeze excitation mechanism called SE
Adaptors to improve domain attention and thus improves performance without any
prior requirement of knowledge of the new target domain. Finally, we
incorporate a center loss in the instance and image level representations to
improve the intra-class variance. We report all results with Cityscapes as our
source domain and Foggy Cityscapes as the target domain outperforming previous
baselines. code: https://github.com/shreyasrajesh/DA-Object-Detection

##  1 Introduction

Object detection is one of the early problems in computer vision with
continued and diverse efforts being made to identify and localize all
instances of various categories of objects in images from a variety of
challenging datasets such as COCO [16], KITTI [8], Cityscapes [4] and more.
Deep CNNs have proven extremely vital in this progress drastically improving
performance, especially region-localization based approaches such as Faster-
RCNN [19]. However, most of these methods have been designed for performing
well on a single domain/dataset and requires complete retraining on new and
unseen labeled data which is extremely expensive. This low adaptability
significantly reduces the impact of these methods, as they become unusable in
real applications especially in automotive use cases. Small variations in
lighting, weather conditions, time of day, location etc. is extremely common
but requires entire retraining of models. Therefore, it is highly desirable to
develop models that can adapt object detection to these new conditions
(domains) that are visually different than the original domain.

In this work, we consider one such approach to perform unsupervised domain
adaptation for cross-domain object detection. We use labeled data from one
domain as the source data and unlabeled data from a new domain as our target
data and aim to adapt our object detection model to perform well on both
domains. In order to achieve this, we build an end-to-end domain adaptation
model as shown in Figure 1 by incorporating two domain adaptation components
at the instance level and the image level into the Faster-RCNN [19]
architecture to reduce the $H$-divergence between the two domains. Each
component learns a domain classifier and employs an adversarial strategy to
learn domain-invariant features. We incorporate a consistency regularization
loss, to learn domain-invariant region proposals. Further, we introduce
Squeeze-Excitation Adaptors as introduced by Wang et. al, [22] for domain
specific attention to improve our model and incorporate a center loss as
proposed by Wen et al., [24] in each of the domain adaptation components to
reduce the intra-class variance in the source and target domain feature space.
We perform experiments and evaluate our results on the Cityscapes and Foggy
Cityscapes datasets to demonstrate the superiority of our approach.

##  2 Related Work

###  2.1 Object Detection

Among the large number of deep learning approaches, region-based CNNs have
received significant attention due to their effectiveness. This line of work
was pioneered by R-CNN [10], which extracts region proposals from the image
and a network is trained to classify each region of interest (ROI)
independently. The two stage detection frameworks of Fast R-CNN [9] and Faster
R-CNN [19] detectors have achieved great success in recent years, and laid the
foundation for many follow up works such as R-FCN [5], FPN [15], MS-CNN [2].
Additionally, single stage object detectors, such as YOLO [18], and SSD[17]
became popular for their fairly good performance and high speed. However,
these works were only able to perform well on the domain they were trained on.
In most practical situations, we face domain gaps between the train and test
data. It is important to consider the domain adaptation issue for object
detection across multiple datasets.

###  2.2 Domain Adaptation

Deep domain adpatation techniques have been successfully applied in many real-
world applications, including image classification, and style translation.
Conventional methods to address domain adaptation include asymmetric metric
learning [13], subspace alignment [6], covariance matrix alignment [20] [23],
etc. However, fewer papers address adaptation beyond classification and
recognition, such as object detection, face recognition, semantic segmentation
and person re-identification. In this paper, we will focus on object detection
across multiple domains.

###  2.3 Domain Adaptation for Object Detection

The recent works that address domain adaptation for object detection can be
categorized based on the approaches used [14]. Discrepancy-based approaches
[1] diminish the domain shift by fine-tuning the deep network based detection
model with labeled or unlabeled target data. Adversarial-based approaches [3]
[25] utilize domain discriminators and conduct adversarial training to
encourage domain confusion between the source domain and the target domain.
Reconstruction-based approaches [12] presume that the reconstruction of the
source or target samples is helpful to improve the performance of domain
adaptation object detection. We will build upon the work of Chen et al [3] who
use adversarial learning to address object detection in the wild.

###  2.4 Domain Attention

In [22], Wang et al build a universal object detection system that is capable
of working on various image domains. Unlike multi-domain models, this
universal model does not require prior knowledge of the domain of interest.
They introduce a new family of adaptation layers, based on the principles of
squeeze and excitation [11], and a new domain-attention mechanism. A Universal
SE bank captures the feature subspaces of the domains spanned by all datasets,
and the attention [21] mechanism soft-routes the USE projections. We
incorporate the domain attention mechanism to improve the adaptation task.

###  2.5 Center Loss

In [24], Wen et al propose a new loss function called center loss to
efficiently enhance the discriminative power of the deeply learned features in
neural networks. Specifically, a center is learned (a vector with the same
dimension as a feature) for deep features of each class. In our paper, we add
the center loss to the domain classifier loss.

##  3 Methodology

We use the domain-adaptive Faster-RCNN model proposed in [3] as our baseline.
The paper introduces two domain adaptation components into the Faster-RCNN
model to align the feature distributions at the image and instance levels.
Additionally, we also introduce a universal Squeeze and Excitation (SE)
adaptor bank into this baseline model. We also add a center loss component to
the image and instance-level classifiers to achieve a tighter bound on the
intra-class variances. The center loss is back propagated only till the
gradient reversal layer.

###  3.1 Image-level adaptation

A patch-based domain classifier is used to eliminate domain distribution
mismatch at the image level. The domain classifier outputs a domain label for
each patch in the image. The idea is to address the shift at a global level
such as image style, illumination, scale etc.

###  3.2 Instance-level adaptation

The instance-level adaptation is performed on the ROI-based feature vectors
before feeding it into the last fully-connected layer. This is done to address
shift at a local level such as object appearance, viewpoint etc. A domain
classifier is trained to align the feature vectors at the instance level as
well. A gradient reversal layer is added before the domain classifier to
incorporate an adversarial training strategy.

###  3.3 $\mathcal{H}$-Divergence

The domain adaptation components at the image and instance level are used to
reduce the $\mathcal{H}$-Divergence between the two domains. The
$\mathcal{H}$-Divergence defines the distance between the two domains as:

$d_{H}(S,T)=2\Big{(}1-min_{h\in
H}\big{(}err_{s}(h(\textbf{x}))+err_{T}(h(\textbf{x}))\big{)}\Big{)}$ (1)

where x is a feature vector, $h:\textbf{x}\rightarrow\\{0,1\\}$ is a domain
classifier, and $err_{S}$ and $err_{T}$ are the prediction errors of
$h(\textbf{x})$ on the source and target domain samples, respectively. To
align the two domains, we therefore need to minimize $d_{H}(S,T)$:

$\underset{f}{\min}\;{d_{H}(S,T)\Leftrightarrow\underset{f}{\max}\underset{h\in
H}{\min}\\{err_{S}(h(\textbf{x}))+err_{T}(h(\textbf{x})\\}}$ (2)

This objective is optimized using adversarial training.

Figure 1: Baseline architecture: Domain-adaptive Faster-RCNN

###  3.4 Domain Attention

We adopt the domain attention module proposed in [22] in order to address non-
trivial domain shifts. The domain attention module consists of a universal
Squeeze and Excitation (SE) adaptor bank. The adaptor bank is introduced into
a few convolutional layers of the feature extractor and the Region Proposal
Network as a feature-based attention mechanism. The SE adaptor bank enables
the module to learn activations specific to domains.

Figure 2: SE Net adaptor block

####  3.4.1 SE Adaptor bank

SE Networks [11] improve channel inter dependencies at almost no additional
computational cost. It is a ’content aware’ mechanism that assigns weights to
different channels when creating output feature maps. An SE adaptor block, as
shown in fig 2, consists of the following components: 1. A global pooling
layer (that squeezes each channel in the input), 2. A fully connected layer
followed by ReLU that adds necessary non linearity, 3. A second fully
connected layer followed by Sigmoid activation. The SE block outputs weights
for the different channels in the input. The computation performed by the SE
network is thus given by:

$\textbf{X}_{SE}=\textbf{F}_{SE}(\textbf{F}_{avg}(\textbf{X}))$ (3)

where $\textbf{F}_{avg}$ is a global average pooling operator, and
$\textbf{F}_{SE}$ is the combination of FC1 with ReLU activation and FC2 with
Sigmoid activation. As proposed in [22], we generalize the SE block to an
adaptor bank, as shown in fig 2.

###  3.5 Image and Instance level Center loss

The main aim of using center loss is to decrease the intra class variance of
both source and target domain feature space. This can be implemented by adding
center loss to the domain classifier loss. The domain classifier has a min max
objective function as it follows an adversarial approach. This characteristic
is desired for the softmax loss function which tries to separate the two
domains in the domain adaptation layers and bring them together in the
backbone network using GRL layer.

On the other hand, it is always desirable to have less intra class variance be
it the domain adaptation layers or the backbone layers. This calls for
minimizing the center loss all together. In our simplified approach, we
minimize the center loss only in the domain adaptation layers. We mask the
gradients due to center loss in the backbone network.

Center loss is employed at both image level and instance level domain
adaptation layers. At both places we initialize two centers of appropriate
dimensions representing source and target domains respectively. The image
level center loss can be represented as below

$L_{img\\_c}=\frac{1}{2}\sum_{i=1}^{m}\parallel
x_{i}-c_{img\\_y_{i}}\parallel_{2}^{2}$ (4)

where $c_{img\\_y_{i}}\epsilon R^{dimg}$ denotes the $y_{i}$th class center.
$m$ denotes the batch size of the current iteration. For the image level
domain adaptation we try to classify an image as a whole.

In the instance level domain adaptation component, since we are trying to
classify every rpn proposal, each image will have $n$ samples to classify.
Here $n$ denotes the number of anchor box proposals. The instance level center
loss can be represented as below

$L_{inst\\_c}=\frac{1}{2}\sum_{i=1}^{n*m}\parallel
x_{i}-c_{inst\\_y_{i}}\parallel_{2}^{2}$ (5)

$m$ denotes the batch size of the current iteration. $c_{inst\\_y_{i}}\epsilon
R^{dinst}$ denotes the $y_{i}$th class center.

The masking effect for this loss is achieved by employing a separate optimizer
spanning only the weights of the domain adaptation layers. This optimizer is
used with the main optimizer synchronously. Gradients are estimated for both
the optimizers before back-propagating any of it.

###  3.6 Instance and Image level Cross Entropy loss

For the domain classifier, cross entropy loss as proposed in [3] is used. The
image level cross entropy loss can we written as,

$L_{ce}=\sum_{i,u,v}D_{i}\log p_{i}^{u,v}+(1-D_{i})\log(1-p_{i}^{u,v})$ (6)

where $D_{i}=0$ represents source domain and $D_{i}=1$ represents target
domain. Here each pixel of the domain classifier output is classified. So,
$(u,v)$ spans all the positions of the output feature map of the domain
classifier.

The instance level cross entropy loss can be written as,

$L_{ce}=\sum_{i,j}D_{i}\log p_{i,j}+(1-D_{i})\log(1-p_{i,j})$ (7)

where output of instance level domain classifier for the j-th domain in the
i-th image is $p_{i,j}$

###  3.7 Consistency loss

As in [3], we need to enforce consistency between image and instance level
domain classifiers. One way to do that is to enforce both the classifiers to
give similar class of prediction for a given image. Since image level domain
classifier produces pixel wise output, we consider average of all these
outputs as the image level probability. The consistency loss can be written
as,

$L_{cst}=\sum_{i,j}\parallel\frac{1}{I}\sum_{u,v}p_{i}^{u,v}-p_{i,j}\parallel_{2}$
(8)

where $|I|$ represents total number of pixel predictions per image

The total loss expression in two steps can be represented as below

$L_{1}=L_{sup}+\lambda_{ce}(L_{img\\_ce}+L_{inst\\_ce}+L_{cst})$ (9)

The center losses can be written as,

$L_{2}=\lambda_{\text{center}}(L_{img\\_c}+L_{inst\\_c})$ (10)

where $\lambda_{ce}$ and $\lambda_{\text{center}}$ are the weights of the
cross entropy and center losses respectively.

##  4 Experimental Setup

###  4.1 Datasets

We use Cityscapes [4] as our source domain, with images dominantly obtained in
clear weather. For the target domain, we use the Foggy Cityscapes dataset. It
is a synthetic foggy dataset in that it simulates fog on real scenes. The
images are rendered using the images and depth maps from Cityscapes. The
semantic annotations and data split of Foggy Cityscapes are inherited from
Cityscapes, making it ideal to study the domain shift caused by weather
condition.

###  4.2 Training Details

The code is developed using PyTorch framework. The Faster R-CNN network
architecture employed uses a ResNet 50 backbone. GRL layer proposed in [7] is
used to reverse the gradients for adversarial training. The model is
initialized with weights trained on ImageNet.

Unless mentioned the following setting is used for both the optimizers
proposed. Initially, lr = 0.001 is used for 50K iterations. Then it is decayed
to 0.0001 for the next 20k iterations. Also momentum=0.9 is used for all
iterations. Batch size = 2 is used for training.

256 anchor boxes per image of sizes (32, 64, 128, 256, 512) and aspect ratios
(1.0,2.0,3.0) are used. For all experiments, NMS threshold = 0.3 and
confidence threshold = 0.8 are used. The weights for the image level and
instance level domain classifier cross entropy losses are fixed to 0.1.
Experiments to find optimal weights for center losses are carried out. COCO
format mAP values are reported which use IOU from 0.5 to 0.95 in steps of
0.05.

##  5 Results & Discussion

The baseline is compared against 3 models. SE+DA-FRCNN is trained by
incorporating domain attention modules in stage 3 of the ResNet50 backbone of
DA-Faster-RCNN. Center+DA-FRCNN uses the center loss as a domain classifier
loss. SE+Center+DA-FRCNN contains domain attention modules in stage 3 of the
backbone, and also uses the center loss for domain confusion. The quantitative
results can be seen in table 1. We observe that individual use of domain
attention and center loss have resulted in improvements in the mAP by a small
margin. However, when domain attention and center loss are both used to train
a model, there is a drop in performance.

Model | mAP
---|---
DA-Faster-RCNN | 24.9
SE + DA-FRCNN | 25.11
Center + DA-FRCNN | 25.01
SE + Center + DA-FRCNN | 24.3
Table 1: Quantitative results on the Foggy Cityscapes validation set, models
are trained on the Cityscapes training set

We compare the qualitative results of the 4 models in Figure 3. We see that
SE+DA-FRCNN gives us more confident predictions and improved detection
performance overall compared to the baseline. We also note that center loss
improves the detection performance when used independently. However, in the
qualitative results of SE+Center+DA-FRCNN, we note a dip in performance.

###  5.1 Ablation study: Effect of USE Bank

Two experiments were conducted to analyze the effect of using the USE banks in
the ResNet50 backbone. In the first experiment, USE banks are used in all 3
stages of the backbone. In the second experiment, USE banks are used only in
the 3rd stage of the backbone. The results, as shown in table 2, are compared
against the baseline which does not contain any USE banks in the ResNet50
backbone. It is observed that having USE banks in all stages, as in experiment
1, causes a drop in the performance. This could be because the early stages of
the backbone contain low-level features like edges, which are not domain
dependent. Therefore, there is no need for domain attention in stages 1 and 2
as they constitute the early layers of a CNN. For experiment 2, where the USE
banks are present in stage 3, we observe an improvement compared to the
baseline. These results give us 2 key insights. Firstly, using domain
attention in early stages may cause a drop in performance. Secondly, applying
domain attention in later stages containing high-level features can result in
improvements in performance.

| Stage 1 | Stage 2 | Stage 3 | mAP
---|---|---|---|---
DA-Faster-RCNN |  |  |  | 24.9
Ours | ✓ | ✓ | ✓ | 24.4
Ours |  |  | ✓ | 25.11
Table 2: Effect of USE banks in different stages of the Resnet50 backbone

###  5.2 Ablation study: Effect of Center Loss

To study the effect of center loss, two experiments were conducted. One
experiment with more emphasis on center loss by having center loss weight
$\lambda_{\text{center}}>>\lambda_{\text{ce}}$ was conducted. Here
$\lambda_{\text{center}}=1.0$ was used. Another experiment with a moderate
center loss weight to balance the effect of both losses was conducted. Here
$\lambda_{\text{center}}=0.5$ was used.

It was observed that for both models, the mAP values increased compared to the
baseline model. This improved performance can be seen in Table 3. Adding more
weight to the center loss is clearly improving the mAP values. This justifies
the theory behind using center loss for the domain adaptation layers. By
decreasing the intra class variance, the number of very hard / noisy negatives
is being decreased. This in turn helps in generating better training signal
for the overall network.

Model | mAP
---|---
Center ($\lambda_{center}=1.0)$ \+ DA-FRCNN | 25.01
Center ($\lambda_{center}=0.5)$ \+ DA-FRCNN | 24.94
Table 3: Effect of Center loss with different weights

##  6 Conclusion

From our experiments, we can conclude that adding USE banks and Center loss is
beneficial for the domain adapted object detection model. Adding USE adapter
banks allows the model to be more domain attentive. It allows the network to
switch between sub branches according to the domain of the input image. Using
Center loss aids in generating better training signal by reducing the effect
of noisy / hard negatives. But USE banks and Center loss do not seem to
complement each other when used together. This can be due to improper weights
for the center losses. Also it is possible that domain alignment is not
desirable for SE adaptors because they maintain separate branches for
different domains and center loss is just trying to aid domain alignment.
Future work involves analyzing the combined effect of USE banks and Center
loss to find if they indeed complement each other or not.

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Figure 3: Cityscapes images comprise the source domain, whereas Foggy
cityscapes images comprise the target. Four rows corresponding to the four
results reported in Table 1. Top to Bottom: DA Faster-RCNN, SE with DA FRCNN,
Center Loss with DA FRCNN and SE + Center Loss with DA FRCNN.