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Search for effective Lorentz and CPT violation using ZEUS data

ZEUS Collaboration DESY–22–107 December 2022

###### Abstract

Lorentz and CPT symmetry in the quark sector of the Standard Model are studied
in the context of an effective field theory using ZEUS $e^{\pm}p$ data.
Symmetry-violating effects can lead to time-dependent oscillations of
otherwise time-independent observables, including scattering cross sections.
An analysis using five years of inclusive neutral-current deep inelastic
scattering events corresponding to an integrated HERA luminosity of
$372\,\mathrm{\,\text{pb}^{-1}}$ at
$\sqrt{s}=318\,\mathrm{{\text{Ge}\kern-0.66666pt\text{V\/}}}$ has been
performed. No evidence for oscillations in sidereal time has been observed
within statistical and systematic uncertainties. Constraints, most for the
first time, are placed on 42 coefficients parameterising dominant CPT-even
dimension-four and CPT-odd dimension-five spin-independent modifications to
the propagation and interaction of light quarks.

The ZEUS Collaboration

I. Abt1, R. Aggarwal2, V. Aushev3, O. Behnke4, A. Bertolin5, I. Bloch6, I.
Brock7, N.H. Brook8,a, R. Brugnera9, A. Bruni10, P.J. Bussey11, A. Caldwell1,
C.D. Catterall12, J. Chwastowski13, J. Ciborowski14,b, R. Ciesielski4,c, A.M.
Cooper-Sarkar15, M. Corradi10,d, R.K. Dementiev16, S. Dusini5, J. Ferrando4,
B. Foster15,e, E. Gallo17,f, D. Gangadharan18,g, A. Garfagnini9, A. Geiser4,
G. Grzelak14, C. Gwenlan15, D. Hochman19, N.Z. Jomhari4, I. Kadenko3, U.
Karshon19, P. Kaur20, R. Klanner17, I.A. Korzhavina16, N. Kovalchuk17, M.
Kuze21, B.B. Levchenko16, A. Levy22, B. Löhr4, E. Lohrmann17, A. Longhin9, F.
Lorkowski4, E. Lunghi23, I. Makarenko4, J. Malka4,h, S. Masciocchi24,i, K.
Nagano25, J.D. Nam26, Yu. Onishchuk3, E. Paul7, I. Pidhurskyi27, A. Polini10,
M. Przybycień28, A. Quintero25, M. Ruspa29, U. Schneekloth4, T. Schörner-
Sadenius4, I. Selyuzhenkov24, M. Shchedrolosiev4, L.M. Shcheglova16, N.
Sherrill30, I.O. Skillicorn11, W. Słomiński31, A. Solano32, L. Stanco5, N.
Stefaniuk4, B. Surrow26, K. Tokushuku25, O. Turkot4,h, T. Tymieniecka33, A.
Verbytskyi1, W.A.T. Wan Abdullah34, K. Wichmann4, M. Wing8,j, S. Yamada25, Y.
Yamazaki35, A.F. Żarnecki14, O. Zenaiev4,k

1 Max-Planck-Institut für Physik, München, Germany

2 DST-Inspire Faculty, Department of Technology, SPPU, India

3 Department of Nuclear Physics, National Taras Shevchenko University of Kyiv,
Kyiv, Ukraine

4 Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany

5 INFN Padova, Padova, Italy A

6 Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen,
Germany

7 Physikalisches Institut der Universität Bonn, Bonn, Germany B

8 Physics and Astronomy Department, University College London, London, United
Kingdom C

9 Dipartimento di Fisica e Astronomia dell’ Università and INFN, Padova, Italy
A

10 INFN Bologna, Bologna, Italy A

11 School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom C

12 Department of Physics, York University, Ontario, Canada M3J 1P3 D

13 The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of
Sciences, Krakow, Poland

14 Faculty of Physics, University of Warsaw, Warsaw, Poland

15 Department of Physics, University of Oxford, Oxford, United Kingdom C

16 Affiliated with an institute covered by a current or former collaboration
agreement with DESY

17 Hamburg University, Institute of Experimental Physics, Hamburg, Germany E

18 Physikalisches Institut of the University of Heidelberg, Heidelberg,
Germany

19 Department of Particle Physics and Astrophysics, Weizmann Institute,
Rehovot, Israel

20 Sant Longowal Institute of Engineering and Technology, Longowal, Punjab,
India

21 Department of Physics, Tokyo Institute of Technology, Tokyo, Japan F

22 Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics,
Tel Aviv University, Tel Aviv, Israel G

23 Department of Physics, Indiana University Bloomington, Bloomington, IN
47405, USA

24 GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany

25 Institute of Particle and Nuclear Studies, KEK, Tsukuba, Japan F

26 Department of Physics, Temple University, Philadelphia, PA 19122, USA H

27 Institut für Kernphysik, Goethe Universität, Frankfurt am Main, Germany

28 AGH University of Science and Technology, Faculty of Physics and Applied
Computer Science, Krakow, Poland

29 Università del Piemonte Orientale, Novara, and INFN, Torino, Italy A

30 Department of Physics and Astronomy, University of Sussex, Brighton, BN1
9QH, United Kingdom I

31 Department of Physics, Jagellonian University, Krakow, Poland J

32 Università di Torino and INFN, Torino, Italy A

33 National Centre for Nuclear Research, Warsaw, Poland

34 National Centre for Particle Physics, Universiti Malaya, 50603 Kuala
Lumpur, Malaysia K

35 Department of Physics, Kobe University, Kobe, Japan F

A supported by the Italian National Institute for Nuclear Physics (INFN)

B supported by the German Federal Ministry for Education and Research (BMBF),
under contract No. 05 H09PDF

C supported by the Science and Technology Facilities Council, UK

D supported by the Natural Sciences and Engineering Research Council of Canada
(NSERC)

E supported by the German Federal Ministry for Education and Research (BMBF),
under contract No. 05h09GUF, and the SFB 676 of the Deutsche
Forschungsgemeinschaft (DFG)

F supported by the Japanese Ministry of Education, Culture, Sports, Science
and Technology (MEXT) and its grants for Scientific Research

G supported by the Israel Science Foundation

H supported in part by the Office of Nuclear Physics within the U.S. DOE
Office of Science

I supported in part by the Science and Technology Facilities Council grant
number ST/T006048/1

J supported by the Polish National Science Centre (NCN) grant no.
DEC-2014/13/B/ST2/02486

K supported by HIR grant UM.C/625/1/HIR/149 and UMRG grants RU006-2013,
RP012A-13AFR and RP012B-13AFR from Universiti Malaya, and ERGS grant
ER004-2012A from the Ministry of Education, Malaysia

a now at University of Bath, United Kingdom

b also at Lodz University, Poland

c now at Rockefeller University, New York, NY 10065, USA

d now at INFN Roma, Italy

e also at DESY and University of Hamburg, Hamburg, Germany and supported by a
Leverhulme Trust Emeritus Fellowship

f also at DESY, Hamburg, Germany

g now at University of Houston, Houston, TX 77004, USA

h now at European X-ray Free-Electron Laser facility GmbH, Hamburg, Germany

i also at Physikalisches Institut of the University of Heidelberg, Heidelberg,
Germany

j also supported by DESY, Hamburg, Germany

k now at Hamburg University, II. Institute for Theoretical Physics, Hamburg,
Germany

##  1 Introduction

Relativity is one of the best established principles in physics. It concerns
the invariance of physical laws under transformations of spacetime orientation
including spatial rotations and velocity boosts. As a core principle of
classical and modern theories, relativity implies that identical measurements
performed with different spacetime orientations observe the same laws of
motion, with their respective results linked by the appropriate
transformation. In general, any theory exhibiting isotropy under rotations and
relativistic boosts is said to be Lorentz invariant.

Searches for violations of isotropy date back to the Michelson–Morley
experiment, which attempted to measure the rotational anisotropy of light
propagation [1]. Its null result heavily influenced the support of special
relativity. As rotations and boosts do not commute, the violation of rotation
invariance implies the violation of boost invariance and vice versa. However,
experimental indications of such symmetry violations do not necessarily imply
Lorentz violation in an underlying fundamental theory. This can be seen, for
example, by considering the Earth’s motion in the presence of a hypothetical
Lorentz-invariant background field, perhaps representing a galactic dark-
matter halo. If the field’s velocity distribution is isotropic in the galactic
frame as is commonly assumed, it will be anisotropic in an Earth-based
laboratory frame, leading to locally time-dependent observations primarily as
a function of the Earth’s annual revolution around the Sun and its axial
rotation. Lorentz violation implies CPT violation in many theoretical
frameworks. Generic searches for violations of rotation invariance encompass a
wide range of possible physical effects.

Model-independent experimental tests of Lorentz invariance and related
fundamental symmetries have been intensively performed for over two decades,
with no significant deviations observed [2]. In spite of the substantial
progress achieved in understanding possible Lorentz and CPT violation,
relatively few studies involving the quark sector directly have been
performed, leaving unexamined a vast array of potential signals. One reason
for this stems from the difficulty in interpreting quark-level interactions in
stable-hadron and lepton processes. A few previous collider-based searches
have resulted in constraints on renormalisable and rotationally invariant
quark-sector effects in the final states of $e^{+}e^{-}$ collisions using LEP
data [3] and on top-quark effects [4] using Tevatron data [5]. Studies of
Lorentz-violating effects in hadronic processes have recently been addressed
in a series of theoretical studies on DIS [6, 7, 8] and the Drell–Yan process
[9, 10].

This paper describes a search for effective rotation-violating signals
affecting light quarks ($f=u,d,s$) performed with $e^{\pm}p$ data collected by
the ZEUS detector at HERA. Deep inelastic scattering (DIS) is chosen as a
suitable test process because of its strong theoretical foundation and the
wealth of available data. To remain model independent, the technical framework
employed is rooted in an effective field theory (EFT) [11] and is reviewed in
the following section. Modified DIS cross sections and related observables
incorporating the effects of interest are then described, followed by the
experimental setup, analysis method and associated Monte Carlo studies, and a
description of systematic uncertainties. Constraints on effective couplings
parameterising signals that violate rotational symmetry in light-quark
interactions are given.

##  2 Theory

###  2.1 The Standard Model Extension

Searches for Lorentz-violating effects often make use of the comprehensive EFT
framework known as the Standard Model Extension (SME) [12, 13, 14, 15]. In the
absence of general-relativistic effects, the SME action may be written as

$S_{\text{SME}}=S_{\text{SM}}+S_{\text{LV}},$ (1)

where $S_{\text{SM}}$ is the action of the Standard Model and $S_{\text{LV}}$
represents all possible Lorentz-violating terms constructed from additional
interactions of Standard Model fields. These terms are typically treated as
perturbations with respect to conventional effects. The analysis presented
here is restricted to selected dimension-four and -five operators. As CPT
violation implies Lorentz violation in a unitary and local quantum field
theory [12, 16], CPT-violating operators are also contained in the SME.
Further information on the SME can be found in, e.g., accessible reviews [17,
18].

The dimension-four operators, $\bar{\psi}_{f}\gamma_{\mu}iD_{\nu}\psi_{f}$,
considered here are the dominant spin-independent and renormalisable quark-
sector modifications of quantum electrodynamics [6] 111The notation h.c. is an
abbreviation for Hermitian conjugation.,

$\mathcal{L}_{c}=\tfrac{1}{2}\sum_{f}c_{f}^{\mu\nu}\bar{\psi}_{f}\gamma_{\mu}iD_{\nu}\psi_{f}+\text{h.c.}\;,$
(2)

where $D_{\nu}=\partial_{\nu}+iee_{f}A_{\nu}$, $e$ is the electron charge,
$e_{f}$ are the relative quark charges and $A_{\nu}$ is the photon field. The
coefficients $c_{f}^{\mu\nu}$ control the magnitude of Lorentz-violating
effects. Since the coefficient and operator Lorentz indices in Eq. (2) are
contracted, $\mathcal{L}_{c}$ is invariant under coordinate transformations,
known as "observer transformations" [12]. Accordingly, observer
transformations have no sensitivity to anisotropies parameterised by
$c_{f}^{\mu\nu}$. If instead a "particle transformation" is performed, the
particle fields $\psi_{f},A_{\nu},\ldots$ are transformed while the
coordinates are unaffected. This could be realised, for example, by physically
rotating the system described by the particle fields. In this case,
$\mathcal{L}_{\text{c}}$ is not Lorentz invariant in general. This occurs
because $c_{f}^{\mu\nu}$ is invariant under particle transformations, whereas
the system described by the particle fields is not. A distinction between
transforming the system as opposed to transforming the observer is present.
This violates Lorentz invariance. These $c$-type coefficients may be taken as
symmetric and traceless, leaving nine observable coefficients per flavour $f$
[13].

All gauge-invariant effects in the minimal and non-minimal sector of non-
Abelian gauge theories were classified and dimension-five, spin-independent,
and CPT-violating effects

$\mathcal{L}_{a^{(5)}}=-\tfrac{1}{2}\sum_{f}{a^{(5)}_{f}}^{\mu\alpha\beta}\bar{\psi}_{f}\gamma_{\mu}iD_{(\alpha}iD_{\beta)}\psi_{f}+\text{h.c.}\;,$
(3)

parameterised by the $a^{(5)}$-type coefficients $a_{f}^{(5)\mu\alpha\beta}$,
were considered for DIS [8]. Note that $\mathcal{L}_{a^{(5)}}$ represents the
dominant non-minimal and spin-independent CPT-violating effects on quarks, and
the parenthesis notation denotes symmetrisation with respect to the indices
$\alpha,\beta$. Similar to the case of the $c$-type coefficients, the
$a^{(5)}$-type coefficients may be taken to be totally symmetric and traceless
[19, 20, 9]. This subset of coefficients is denoted
$a_{\text{S}f}^{(5)\mu\alpha\beta}$ and contains 16 independent observable
components per flavour.

Simulations have been performed assessing the sensitivity of the $c$-type and
$a^{(5)}$-type coefficients for quarks and antiquarks as measured in DIS data
from HERA [21], Electron–Ion Collider pseudodata [7], and in the Drell–Yan
process using LHC data [22].

###  2.2 Adding SME effects to the description of DIS

The class of operators that modify the free propagation of quarks and their
covariant couplings to gauge fields was examined [9], resulting in the
development of a tree-level, Lorentz-violating version of the parton model.
This leads to a modified description of DIS.

In the following, the external momenta of the incident lepton and proton are
denoted by $l^{\mu}$ and $p^{\mu}$, respectively, while the scattered lepton
and exchanged momentum are denoted by $l^{\prime\mu}$ and
$q^{\mu}=l^{\mu}-l^{\prime\mu}$, where $-q^{2}\equiv Q^{2}$ is the momentum
transfer. The Bjorken variables are

$x_{\text{Bj}}=\frac{Q^{2}}{2p\cdot q}\;,\quad y_{\text{Bj}}=\frac{p\cdot
q}{p\cdot l}\;,$ (4)

where $s\approx 2p\cdot l$ and $Q^{2}\approx x_{\text{Bj}}y_{\text{Bj}}s$. In
the DIS limit, and working at zeroth order in the strong coupling constant,
the inclusion of Lorentz-violating effects described by Eq. (2) at tree level
results in the cross section222Equation (5) reduces to the leading-order
Standard Model cross section in the limit $c_{f}^{\mu\nu}\rightarrow 0$.

$\frac{d\sigma}{dx_{\text{Bj}}dy_{\text{Bj}}d\varphi}=\frac{\alpha^{2}y_{\text{Bj}}}{2Q^{4}}\sum_{f}e_{f}^{2}\frac{1}{\widetilde{Q}_{f}^{2}}L_{\mu\nu}H_{f}^{\mu\nu}f_{f}(\widetilde{x}_{f})\;,$
(5)

where $\varphi$ is the scattered lepton azimuthal angle,
$\widetilde{Q}^{2}_{f}=-\widetilde{q}_{f}^{2}=-(q^{\mu}+c_{f}^{\mu
q})(q_{\mu}+c_{f\mu q})$ where $c_{f}^{\mu q}\equiv c_{f}^{\mu\nu}q_{\nu}$,
and

$\displaystyle L_{\mu\nu}H_{f}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
8\left[2(\widehat{k}_{f}\cdot l)(\widehat{k}_{f}\cdot
l^{\prime})+\widehat{k}_{f}\cdot(l-l^{\prime})(l\cdot
l^{\prime})+2(\widehat{k}_{f}\cdot
l)\left(c_{f}^{\widehat{k}_{f}l^{\prime}}+c_{f}^{l^{\prime}\widehat{k}_{f}}-c_{f}^{l^{\prime}l^{\prime}}\right)\right.$
(6) $\displaystyle\hskip 14.22636pt\left.+2(\widehat{k}_{f}\cdot
l^{\prime})\left(c_{f}^{\widehat{k}_{f}l}+c_{f}^{l\widehat{k}_{f}}+c_{f}^{ll}\right)-2(l\cdot
l^{\prime})c_{f}^{\widehat{k}_{f}\widehat{k}_{f}}\right],$

with $\widehat{k}_{f}^{\mu}=\widetilde{x}_{f}(p^{\mu}-c_{f}^{\mu p})$. The
parton distribution functions (PDFs) are denoted $f_{f}(\widetilde{x}_{f})$
and are evaluated at the shifted Bjorken variable [9]

$\widetilde{x}_{f}=x_{\text{Bj}}\left(1+\frac{2c_{f}^{qq}}{q^{2}}\right)+\frac{x_{\text{Bj}}^{2}}{q^{2}}\left(c_{f}^{pq}+c_{f}^{qp}\right),$
(7)

where $c_{f}^{qq}\equiv c_{f}^{\mu\nu}q_{\mu}q_{\nu}$, etc. The PDFs are taken
to be the conventional leading-order Standard Model PDFs evaluated at
$\widetilde{x}_{f}$ using the MSTW 2008 variable-flavour PDF set [23], as
implemented in the program ManeParse [24, 25].333 It was checked that effects
of a variation of the input PDF set are negligible. Potential effects from
Lorentz violation on the PDF starting scale parametrization and on the PDF
evolution were neglected.  Note that since the $c$-type coefficients control
CPT-even effects, the quark and antiquark contributions are identical in the
cross section Eq. (5), except for the PDF dependence. Thus, for these
coefficients, $d\sigma\propto(f_{f}+f_{\bar{f}})$ and the dominant sensitivity
is expected at low $x_{\text{Bj}}$.

The cross section including effects of the $a^{(5)}$-type coefficients from
Eq. (3) is [6, 9]

$\displaystyle\frac{d\sigma}{dx_{\text{Bj}}dy_{\text{Bj}}d\varphi}$
$\displaystyle=$
$\displaystyle\frac{\alpha^{2}}{Q^{4}}\sum_{f}F_{2f}\left[\frac{y_{\text{Bj}}s^{2}}{\pi}\left[1+(1-y_{\text{Bj}})^{2}\right]\delta_{\text{S}f}+\frac{y_{\text{Bj}}(y_{\text{Bj}}-2)s}{x_{\text{Bj}}}x_{\text{S}f}\right.$
(8) $\displaystyle\left.\hskip
30.0pt-\frac{4}{x_{\text{Bj}}}\left(4x_{\text{Bj}}^{2}a_{\text{S}f}^{(5)ppl}+6x_{\text{Bj}}a_{\text{S}f}^{(5)lpq}+2a_{\text{S}f}^{(5)lqq}\right)\right.$
$\displaystyle\left.\hskip
30.0pt+2y_{\text{Bj}}\left(4x_{\text{Bj}}^{2}a_{\text{S}f}^{(5)ppp}+4x_{\text{Bj}}a_{\text{S}f}^{(5)ppq}+4x_{\text{Bj}}a_{\text{S}f}^{(5)lpp}+2a_{\text{S}f}^{(5)lpq}+a_{\text{S}f}^{(5)pqq}\right)\right.$
$\displaystyle\left.\hskip
30.0pt+\frac{4y_{\text{Bj}}}{x_{\text{Bj}}}\left(2x_{\text{Bj}}a_{\text{S}f}^{(5)llp}+a_{\text{S}f}^{(5)llq}\right)\right],$

where $F_{2f}=e_{f}^{2}f_{f}(x_{\text{S}f}^{\prime})x_{\text{S}f}^{\prime}$
with $x_{\text{S}f}^{\prime}=x_{\text{Bj}}-x_{\text{S}f}$ and

$\displaystyle\delta_{\text{S}f}$ $\displaystyle=$
$\displaystyle\frac{\pi}{y_{\text{Bj}}s}\left[1+\frac{2}{y_{\text{Bj}}s}\left(4x_{\text{Bj}}a_{\text{S}f}^{(5)ppq}+3a_{\text{S}f}^{(5)pqq}\right)\right],$
(9) $\displaystyle x_{\text{S}f}$ $\displaystyle=$
$\displaystyle-\frac{2}{y_{\text{Bj}}s}\left(2x_{\text{Bj}}^{2}a_{\text{S}f}^{(5)ppq}+3x_{\text{Bj}}a_{\text{S}f}^{(5)pqq}+a_{\text{S}f}^{(5)qqq}\right),$
(10) $\displaystyle a_{{\text{S}}f}^{(5)\mu\alpha\beta}$ $\displaystyle=$
$\displaystyle\tfrac{1}{3}\sum_{(\mu\alpha\beta)}(a_{f}^{(5)\mu\alpha\beta}-\tfrac{1}{6}a_{f}^{(5)\mu\rho\sigma}\eta_{\rho\sigma}\eta^{\alpha\beta}-\tfrac{1}{3}a_{f}^{(5)\rho\mu\sigma}\eta_{\rho\sigma}\eta^{\alpha\beta}),$
(11)

where $a_{\text{S}f}^{(5)qqq}\equiv
a_{\text{S}f}^{(5)\mu\alpha\beta}q_{\mu}q_{\alpha}q_{\beta}$, etc. The sum in
Eq. (11) denotes symmetrisation with respect to the indices $\mu,\alpha$ and
$\beta$, and $\eta_{\rho\sigma}$ is the Minkowski metric. As the operator in
Eq. (3) is odd under a CPT transformation, for the antiparticle contributions
to the sum in Eq. (8), a replacement
$a_{\text{S}f}^{(5)\mu\alpha\beta}\rightarrow-
a_{\text{S}f}^{(5)\mu\alpha\beta}$ for antiquark flavours must be performed,
along with multiplication of the appropriate PDFs. In contrast to the $c$-type
coefficients, this feature implies $d\sigma\propto(f_{f}-f_{\bar{f}})$, thus
giving weaker sensitivity at low $x_{\text{Bj}}$. A depiction of the DIS
process with Lorentz violation is shown in Fig. 1.

###  2.3 Sidereal signals

An experiment is sensitive to the $c$\- and $a^{(5)}$-type coefficients as
they appear in the laboratory frame. The cross sections Eqs. (5) and (8)
therefore must be evaluated in the laboratory frame of the ZEUS detector. This
frame is non-inertial due to the Earth’s axial rotation and revolution around
the Sun. By convention, an approximately inertial frame with spatial
coordinates fixed on the centre of the Sun, known as the Sun-centred frame
(SCF), is introduced as a convenient frame to report constraints on the SME
coefficients [26, 27, 28]. In the SCF, the coefficients are typically assumed
to be spacetime constants, implying the preservation of translation invariance
and the conservation of four-momentum [14]. Thus, in another frame rotating
with respect to the SCF, such as the laboratory frame, the coefficients will
oscillate as a function of time. To make this time dependence explicit, the
coefficients appearing in the laboratory-frame cross sections are re-expressed
in terms of the SCF coefficients by performing the appropriate Lorentz
transformation $\Lambda^{\mu}_{\hphantom{\mu}\nu}$. For example, the
transformation for the $c$-type coefficients is

$c_{\text{lab}}^{\mu\nu}=\Lambda^{\mu}_{\hphantom{\mu}\vphantom{0}\alpha}\Lambda^{\nu}_{\hphantom{\nu}\vphantom{0}\beta}c_{\text{SCF}}^{\alpha\beta}.$
(12)

In what follows, coefficients that appear with Greek and numeric indices
denote laboratory-frame coefficients and those with capital Latin indices
denote SCF coefficients. After performing this transformation and re-
expressing the laboratory-frame coefficients, the laboratory cross sections
are functions of the fixed SCF coefficients and sinusoidal functions with
periods controlled by the length of the sidereal day. The boost of the Earth
with respect to the Sun, $\beta_{\oplus}\approx 10^{-4}$, is suppressed
relative to the effect of Earth’s rotation by several orders of magnitude and
can be neglected. Also, the fact that the Earth’s orbit is translated with
respect to the SCF has no physical effect because of translation invariance.
Therefore, the transformation $\Lambda^{\mu}_{\hphantom{\mu}\vphantom{0}\nu}$
in Eq. (12) is well approximated as a pure rotation:
$\Lambda^{0}_{\hphantom{0}0}=1$,
$\Lambda^{\mu}_{\hphantom{\mu}0}=0=\Lambda^{0}_{\hphantom{\nu}0}$ and
$\Lambda^{i}_{\hphantom{i}j}=R^{i}_{\hphantom{i}j}\;(i,j=1,2,3)$ is an
orthogonal matrix.

The SCF is depicted in Fig. 2. It is defined by coordinates
$X^{\mu}=(T,X,Y,Z)$ as follows: $T=0$ is identified with the date of the 2000
vernal equinox, March 20, 2000 at 7:35 UTC; the $Z$ axis is aligned with the
Earth’s rotation axis; the $X$ axis points from the Earth to the Sun at $T=0$;
and the $Y$ axis completes the right-handed coordinate system. At $T=0$, the
Earth’s equator lies in the $XY$ plane and the longitude $\lambda_{0}\approx
66.25^{\circ}$ will observe the Sun directly overhead, towards the local
zenith. Therefore, $T=0$ is a suitable moment for easily relating the SCF
coordinates to the laboratory-frame coordinates. The small effects of the
Earth’s non-circular orbit can be neglected. For the ZEUS detector, the co-
latitude, orientation of the electron/positron- or proton-beam direction and
the local sidereal time $T_{\oplus}$ must be specified. The zero of
$T_{\oplus}$ is defined as one of the moments when the $y$ axis of the
laboratory is parallel to the SCF $Y$ axis, which on the date of the equinox
occurs shortly after $T=0$. In other words, since HERA is not situated at the
longitude $\lambda_{0}$, but the longitude $\lambda\approx 9.88^{\circ}$,
$T_{\oplus}\neq T$. Instead, $T_{\oplus}$ is related to $T$ by an offset given
by [20]

$T-T_{\oplus}=\frac{\lambda_{0}-\lambda}{360^{\circ}}T_{\text{sidereal}}=3.748\;\text{h}\;,$
(13)

where $T_{\text{sidereal}}=23\;\text{h}\;56\;\text{min}\;4.091\;\text{s}$ is
the sidereal day. The first occurrence for which $T_{\oplus}=0$ is therefore
approximately 3.75 hours after the 2000 vernal equinox, or March 20, 2000 at
11:20 UTC. This is the chosen initial condition for referencing the time of
ZEUS $e^{\pm}p$ events. The co-latitude of HERA is $\chi\approx 36.4^{\circ}$,
and the electron/positron-beam orientation for ZEUS is $\Psi\approx
20^{\circ}$ south of west. Finally, the net rotation $R$ from the proton-beam
direction in the laboratory frame to the SCF is given by

$R=\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&-1&0\end{pmatrix}\begin{pmatrix}\cos\Psi&\sin\Psi&0\\\
-\sin\Psi&\cos\Psi&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}\cos\chi\cos\omega_{\oplus}T_{\oplus}&\cos\chi\sin\omega_{\oplus}T_{\oplus}&-\sin\chi\\\
-\sin\omega_{\oplus}T_{\oplus}&\cos\omega_{\oplus}T_{\oplus}&0\\\
\sin\chi\cos\omega_{\oplus}T_{\oplus}&\sin\chi\sin\omega_{\oplus}T_{\oplus}&\cos\chi\end{pmatrix},$
(14)

where $\omega_{\oplus}=2\pi/T_{\text{sidereal}}$ is the Earth’s sidereal
frequency. Performing the rotation $R$ to express the laboratory-frame
coefficients in terms of the constant SCF coefficients induces a sidereal-time
dependence at multiples of the Earth’s sidereal frequency.

As an example, the transformation of the coefficient $c_{u}^{33}$ reads:

$\displaystyle c_{u}^{33}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(c_{u}^{XX}+c_{u}^{YY})\left(\cos^{2}\chi\sin^{2}\Psi+\cos^{2}\Psi\right)+c_{u}^{ZZ}\sin^{2}\chi\sin^{2}\Psi$
$\displaystyle-2c_{u}^{XZ}\sin\chi\sin\Psi\left[\cos\chi\sin\Psi\cos(\omega_{\oplus}T_{\oplus})+\cos\Psi\sin(\omega_{\oplus}T_{\oplus})\right]$
$\displaystyle-2c_{u}^{YZ}\sin\chi\sin\Psi\left[\cos\chi\sin\psi\sin(\omega_{\oplus}T_{\oplus})-\cos\psi\cos(\omega_{\oplus}T_{\oplus})\right]$
$\displaystyle+c_{u}^{XY}\left[(\cos^{2}\chi\sin^{2}\Psi-\cos^{2}\Psi)\sin(2\omega_{\oplus}T_{\oplus})-\cos\chi\sin(2\Psi)\cos(2\omega_{\oplus}T_{\oplus})\right]$
$\displaystyle+\frac{1}{2}(c_{u}^{XX}-c_{u}^{YY})\left[(\cos^{2}\chi\sin^{2}\Psi-\cos^{2}\Psi)\cos(2\omega_{\oplus}T_{\oplus})+\cos\chi\sin(2\Psi)\sin(2\omega_{\oplus}T_{\oplus})\right],$

where it is seen that the terms proportional to $(c_{u}^{XX}+c_{u}^{YY})$ and
$c_{u}^{ZZ}$ possess no sidereal-time dependence, the terms proportional to
$c_{u}^{XZ}$ and $c_{u}^{YZ}$ oscillate with angular frequency
$\omega_{\oplus}$ and the terms proportional to $c_{u}^{XY}$ and
$(c_{u}^{XX}-c_{u}^{YY})$ oscillate with angular frequency $2\omega_{\oplus}$.

After expressing the cross section given in Eq. (5) in terms of the SCF
coefficients, only the following 18 combinations of coefficients yield
sidereal-time oscillations:

$c_{f}^{TX}\;,c_{f}^{XZ}\;,c_{f}^{TY}\;,c_{f}^{YZ}\;,c_{f}^{XY}\;\text{and}\;(c_{f}^{XX}-c_{f}^{YY})\;,$
(16)

with $f=u$, $d$, and $s$. Analogously, after the replacement
$a_{\text{lab}}^{(5)\mu\alpha\beta}=\Lambda^{\mu}_{\phantom{\mu}\nu}\Lambda^{\alpha}_{\phantom{\alpha}\rho}\Lambda^{\beta}_{\phantom{\beta}\sigma}a_{\text{SCF}}^{(5)\nu\rho\sigma}$
in Eq. (8), it is found that, for each quark, only the following 12
combinations of coefficients yield sidereal-time oscillations:

$\displaystyle(a_{\text{S}f}^{(5)TXX}-a_{\text{S}f}^{(5)TYY}),\;(a_{\text{S}f}^{(5)XXZ}-a_{\text{S}f}^{(5)YYZ}),\;a_{\text{S}f}^{(5)TXY},\;a_{\text{S}f}^{(5)TXZ},\;a_{\text{S}f}^{(5)TYZ},\;a_{\text{S}f}^{(5)XXX},\;$
$\displaystyle
a_{\text{S}f}^{(5)XXY},\;a_{\text{S}f}^{(5)XYY},\;a_{\text{S}f}^{(5)XYZ},\;a_{\text{S}f}^{(5)XZZ},\;a_{\text{S}f}^{(5)YYY}\;\text{and}\;a_{\text{S}f}^{(5)YZZ},$
(17)

giving a total of 36 combinations. For antiquarks with flavour
$f=\bar{u},\bar{d}$ and $\bar{s}$, the coefficients effectively appear in the
cross section Eq. (8) with opposite signs relative to the quark coefficients.
Since the contributions from $s$ and $\bar{s}$ are equal, only the 24
coefficients for $f=u$ and $d$ are considered. The $c$\- and $a^{(5)}$-type
coefficients in Eqs. (16) and (17) induce oscillations with frequencies up to
$2\omega_{\oplus}$ and $3\omega_{\oplus}$, respectively. More generally, an
SCF coefficient with $n$ Lorentz indices will include sidereal oscillations up
to $n\omega_{\oplus}$.

The expected sensitivities to the $c$\- and $a^{(5)}$-type coefficients that
can be extracted from DIS data have been studied [6, 7, 9]. In particular, the
combined DIS cross sections from ZEUS and H1 [21] have been used to estimate
the sensitivity of a sidereal-time study using these data. For each value in
the $(x_{\text{Bj}},Q^{2})$ plane, potential constraints based on four
sidereal intervals have been extracted. A subset of simulated results
describing the sensitivity to the $c$\- and $a^{(5)}$-type coefficients as a
function of phase are shown in Fig. 3. It is observed that the sensitivities
to the $c$\- and $a^{(5)}$-type coefficients are roughly at levels of
$10^{-4}$ and $10^{-6}$ GeV-1, respectively.

##  3 Experimental set-up and data selection

The analysis is based on events collected with the ZEUS detector at HERA
during the HERA II run period 2003–2007. For this configuration, the initial-
state proton- and electron/positron-beam energies were $E_{p}=920$ GeV and
$E_{e}=27.5$ GeV, respectively, with a centre-of-mass energy of $\sqrt{s}=318$
GeV and an integrated luminosity 372 pb-1. Details of the ZEUS detector are
given elsewhere [29].

The NC DIS events were selected with the following criteria [30]:

  * •

the final-state lepton was identified using an algorithm based on a neural
network [31, 32], giving a probability larger than 90%;

  * •

the energy of the final-state lepton $E^{\prime}_{e}>10$ GeV to ensure a high
electron-identification efficiency;

  * •

$Q^{2}>5\;\text{GeV}^{2}$;

  * •

$\theta_{e}>1\;\text{rad}$, where $\theta_{e}$ is the scattering angle between
the outgoing lepton and incoming proton direction to ensure the high
efficiency of the electron-identification algorithm. This provides an upper
limit on $Q^{2}$;

  * •

the scattered lepton was required to enter the calorimeter at a radial
position larger than $15\;\text{cm}$, implying an upper bound on the lepton
scattering angle $\theta_{e}\lesssim 3\;\text{rad}$;

  * •

the position of the event vertex along the laboratory $z$ axis was required to
be within $30$ cm of its nominal value and the transverse distance of the
event vertex from the interaction point was required to be within $0.5$ cm, to
reject background;

  * •

$47\;{\text{Ge}\kern-0.66666pt\text{V\/}}<E-p_{z}<69\;{\text{Ge}\kern-0.66666pt\text{V\/}}$,
where $E$ and $p_{z}$ are the total energy and $z$-component of the final
state, to reject background.

This selection resulted in $4.5\cdot 10^{7}$ events covering the kinematic
range $7.7\cdot 10^{-5}<x_{\text{Bj}}<1$ and
$5<Q^{2}<8800\;{\text{Ge}\kern-0.66666pt\text{V\/}}^{2}$.

##  4 Analysis method

Possible variations of cross sections with periodicity $T_{\text{p}}$ were
studied. The starting point was the triple differential DIS cross section

$\frac{d\sigma}{dx_{\text{Bj}}\;dQ^{2}\;d\phi_{T_{\text{p}}}}\;,$ (18)

where the temporal phase
$\phi_{T_{\text{p}}}=\text{Mod}(T_{\oplus},T_{\text{p}})/T_{\text{p}}$ is the
phase of a given DIS event with the time stamp $T_{\oplus}$ for the chosen
period, $T_{\text{p}}$, and is defined in the range $[0,1]$. Within the SME,
only $T_{\text{p}}=T_{\text{sidereal}}$ yields a non-vanishing dependence on
$\phi_{T_{\text{p}}}$, as the sidereal angle is $\omega_{\oplus}T_{\oplus}$.
If a different period is used, the time dependence quickly averages out; the
larger the difference between $T_{\text{p}}$ and $T_{\text{sidereal}}$, the
faster this occurs.

It was necessary to eliminate uncertainties related to the instantaneous
luminosity, which decays over several hours during each fill. To do so would
have required the integrated luminosity to be measured roughly every minute,
but this information was not available. Instead, double ratios of the form

$r({\text{PS}}_{1},{\text{PS}}_{2})=\frac{\int_{{\text{PS}}_{1}}dx_{\text{Bj}}dQ^{2}\frac{d\sigma}{dx_{\text{Bj}}\;dQ^{2}\;d\phi_{T_{\text{p}}}}/\int_{{\text{PS}}_{1}}dx_{\text{Bj}}dQ^{2}d\phi_{T_{\text{p}}}\frac{d\sigma}{dx_{\text{Bj}}\;dQ^{2}\;d\phi_{T_{\text{p}}}}}{\int_{{\text{PS}}_{2}}dx_{\text{Bj}}dQ^{2}\frac{d\sigma}{dx_{\text{Bj}}\;dQ^{2}\;d\phi_{T_{\text{p}}}}/\int_{{\text{PS}}_{2}}dx_{\text{Bj}}dQ^{2}d\phi_{T_{\text{p}}}\frac{d\sigma}{dx_{\text{Bj}}\;dQ^{2}\;d\phi_{T_{\text{p}}}}}\;,$
(19)

where ${\text{PS}}_{1}$ and ${\text{PS}}_{2}$ are two regions of the
$(x_{\text{Bj}},Q^{2})$ kinematic range, for which luminosity uncertainty
cancels, were used. Essential properties of Eq. (19) are that it is
independent of the luminosity and equal to unity in the absence of SME
effects. Evaluating the contributions involving integrations of
$\phi_{T_{\text{p}}}$ results in zero sensitivity to SME coefficients and when
summing the contributions of all quark flavours produces the Standard Model
cross section. Moreover, if the regions ${\text{PS}}_{1,2}$ are chosen
appropriately, the double ratio shows a strong $\phi_{T_{\text{p}}}$
dependence in the presence of SME effects. As explained in Section (2.2), a
strong sensitivity at low $x_{\text{Bj}}$ ($\approx 10^{-5}$–$10^{-3}$) for
the $c$-type coefficients and $x_{\text{Bj}}\gtrsim 10^{-3}$ for the
$a^{(5)}$-type coefficients is expected. However, for $x_{\text{Bj}}\lesssim
10^{-4}$ and $x_{\text{Bj}}\gtrsim 10^{-2}$, fewer DIS events are available
and thus the statistical uncertainties become larger. The $x_{\text{Bj}}$
distribution is sharply peaked around $x_{\text{Bj}}\approx 10^{-3}$, implying
a statistically optimal choice of cut $x_{\text{c}}$ in the
$(x_{\text{Bj}},Q^{2})$ kinematic range.

In evaluating Eq. (19) for several values of $x_{\text{c}}$, it was found that
the combinations simultaneously giving the most sensitivity to the cross
section while producing the lowest statistical uncertainties are
$x_{\text{c}}=5\cdot 10^{-4}$ and $10^{-3}$ for $c$\- and $a^{(5)}$-type
coefficients, respectively. However, as no higher-order QCD corrections are
included in the cross sections given in Eqs. (5) and (8), potential large
corrections are expected, of the order $[\alpha_{s}(Q^{2})\log
1/x_{\text{Bj}}]^{n}$ at very low $x_{\text{Bj}}$, which need to be resummed.
Numerically, such effects could shift the value of $x$ entering the PDFs.
Thus, $x$ would not coincide with $x_{\text{Bj}}$. This issue is partially
alleviated by cutting at moderate values $x_{\text{Bj}}\gtrsim 10^{-3}$, where
QCD corrections have a smaller influence. The value $x_{\text{c}}=10^{-3}$ was
therefore chosen for both the $c$\- and $a^{(5)}$-type coefficients.

By employing Eqs. (5) and (8), the integrations above and below
$x_{\text{c}}=10^{-3}$ were performed for the kinematic range in
$x_{\text{Bj}},Q^{2}$ and $\theta_{e}$ as described in Section 3. The double
ratios from Eq. (19) evaluated in the laboratory frame are:

$\displaystyle r_{c}(x_{\text{Bj}}>x_{\text{c}},x_{\text{Bj}}<x_{\text{c}})$
$\displaystyle=$ $\displaystyle
1-12.8\;c_{u}^{03}-13.9\;c_{u}^{33}+0.9\;(c_{u}^{11}+c_{u}^{22})$ (20)
$\displaystyle\phantom{1}-4.2\;c_{d}^{03}-2.9\;c_{d}^{33}+0.1\;(c_{d}^{11}+c_{d}^{22})$
$\displaystyle\phantom{1}-3.4\;c_{s}^{03}-1.8\;c_{s}^{33}+2.9\cdot
10^{-2}\;(c_{s}^{11}+c_{s}^{22})\;,$ $\displaystyle
r_{a^{(5)}}(x_{\text{Bj}}>x_{\text{c}},x_{\text{Bj}}<x_{\text{c}})$
$\displaystyle=$ $\displaystyle 1-6.1\cdot 10^{3}\;a_{u}^{(5)003}+6.8\cdot
10^{3}\;a_{u}^{(5)033}-2.5\cdot 10^{3}\;a_{u}^{(5)333}$ (21)
$\displaystyle\phantom{1}+5.0\cdot
10^{2}\;(a_{u}^{(5)113}+a_{u}^{(5)223}-a_{u}^{(5)011}-a_{u}^{(5)022})$
$\displaystyle\phantom{1}-4.1\cdot 10^{2}\;a_{d}^{(5)003}+4.7\cdot
10^{2}\;a_{d}^{(5)033}-1.7\cdot 10^{2}\;a_{d}^{(5)333}$
$\displaystyle\phantom{1}+40\;(a_{d}^{(5)113}+a_{d}^{(5)223}-a_{d}^{(5)011}-a_{d}^{(5)022})\;,$

where the numerical prefactors multipling the $a^{(5)}$-type coefficients in
the latter expression are in units of GeV. These expressions only contain
terms linear in the coefficients, as higher-order corrections are greatly
suppressed. The ratio $r_{a^{(5)}}$ has negligible sensitivity to
$a_{s}^{(5)\mu\alpha\beta}$ because of the nearly identical and opposite-in-
sign $s,\bar{s}$ contributions to the cross section, see Eq. (8). After re-
expressing the laboratory-frame coefficients appearing in Eqs. (20) and (21)
in terms of rotational-symmetry violating combinations of the SCF coefficients
described in Section 2.3 using Eq. (14), it is found that 42 SCF coefficients
result in deviations of $r_{c}$ or $r_{a^{(5)}}$ from unity. Comparisons
between these double ratios and the analogous ratios constructed purely from
the binned DIS events were made under the assumption of constant efficiency
corrections. Constraints were placed on each coefficient one at a time by
setting all others to zero, in accordance with standard practice [2].

##  5 Systematic effects

###  5.1 Initial considerations

A luminosity-independent ratio, Eq. (19), that does not possess a sizeable
dependence on SME effects was used to test the data. The ratio
$r(Q^{2}>Q_{\text{c}}^{2},Q^{2}<Q^{2}_{\text{c}})$ with $Q_{\text{c}}^{2}=20$
GeV2 fulfills this requirement.

The data rate in the detector is not constant. In particular, more data were
taken in the evenings than in the morning. To display this effect, the entire
DIS selection in the two $Q^{2}$ regions was binned using the solar phase
$\phi_{\text{solar}}=\text{Mod}(T_{\oplus},T_{\text{solar}})/T_{\text{solar}}$
with $T_{\text{solar}}=24$ h. In Fig. 4(a), the resulting normalised count of
the events passing the selection described in Section 3 is displayed. It is
clear that an $\mathcal{O}(25\%)$ effect is present. Switching from solar to
sidereal binning with
$\phi_{\text{sidereal}}=\text{Mod}(T_{\oplus},T_{\text{sidereal}})/T_{\text{sidereal}}$
dilutes this solar-phase dependence for long data-taking periods. As shown in
Fig. 4(b), sidereal dilution over $\approx 5$ years reduces the effect to
$\mathcal{O}(10\%)$, but is not sufficient to erase it completely. The
dilution effect of a much longer period of data taking can be simulated by
binning using a short period. In Fig. 4(c), a plot of the same data binned
using a $T_{\text{p}}=1$ h period with
$\phi_{\text{test}}=\text{Mod}(T_{\oplus},1\;\text{h})/1\;\text{h}$ is shown.
The initial solar-phase dependence disappears almost completely. Figure 4(d)
shows that for a period slightly longer than the solar day, $T_{\text{p}}=24$
h 4 min, a similar level of dilution as for the sidereal-phase binning in Fig.
4(b) is observed. For each choice of phase, the histograms for events with
$Q^{2}$ above and below $Q_{\text{c}}^{2}$ closely track each other and their
ratios are consistent with unity.

An important question is whether systematic uncertainties can be neglected. A
partial answer can be found by performing Kolmogorov–Smirnov (KS) tests on the
binned ratios $r(Q_{\text{c}})$. This test calculates the probability that the
observed distributions are compatible with an unsorted sampling of a normal
distribution with mean unity and standard deviation identical to the observed
statistical uncertainties. The ratios are plotted in different numbers of bins
$N_{\text{bins}}$ to estimate the size of systematic relative to statistical
effects. The results are presented for $T_{\text{p}}=24$ h in Fig. 5 and for
$T_{\text{p}}=T_{\text{sidereal}}$ in Fig. 6. It is observed that,
independently of the period, $T_{\text{p}}=24$ h or
$T_{\text{p}}=T_{\text{sidereal}}$, and of the number of bins, the results are
statistically compatible with unity as indicated by the high KS probabilities.
The impact of systematic uncertainties appears to be minimal in these
distributions. However, this is insufficient to conclude that systematic
uncertainties can be neglected when looking at sidereal and solar
distributions for SME-sensitive $x_{\text{c}}$ ratios given by Eqs. (20) and
(21). This is because low- and high-$x_{\text{Bj}}$ regions, where the trigger
behaves differently and could be affected by the instantaneous luminosity,
could be subject to systematic effects which would not cancel in the ratio
defined in Eq. (19).

As long as each bin is considerably smaller than the duration of a fill, which
is typically several hours, (e.g., for $T_{\text{p}}=1$ h, each bin is between
$0.6$ min ($N_{\text{bins}}=100$) and 2.4 min ($N_{\text{bins}}=25$)), it is
expected that all such effects will average out. On the other hand, for the
solar or sidereal phase, the time bins range between $\approx$ 15 min and 1 h.
It is possible that fluctuations in trigger efficiencies and accelerator
effects could affect high-luminosity and low-luminosity parts of the fill
differently. This might result in non-negligible residual effects in the
solar- and sidereal-ratio distributions.

###  5.2 Data analysis of $\boldsymbol{x_{\text{c}}}$ and Monte Carlo
simulation

When separating the kinematic range into regions above and below the cut
$x_{\text{c}}=10^{-3}$, the normalised counts presented in Fig. 7 strongly
resemble those from $Q_{\text{c}}^{2}$ cuts in Fig. 4. Similarly, in light of
the discussion in Section 5.1, it is expected that the $T_{\text{p}}=1$ h
distributions for the ratios
$r(x_{\text{Bj}}>x_{\text{c}},x_{\text{Bj}}<x_{\text{c}})$ given explicitly in
Eqs. (20) and (21) will be distributed around unity and will be dominated by
statistical uncertainties. As discussed, any potential solar and sidereal
effects should be removed when binning data using a test period much smaller
than 24 h. This is indeed the case, as can be seen from the results for
$\phi_{\text{test}}$ presented in Fig. 8. In contrast to the
$Q_{\text{c}}^{2}$ distributions for all phases, the solar phases for the
$x_{\text{c}}$ distributions contain more structure, as is immediately evident
from Fig. 9.

The $N_{\text{bins}}=25$ configuration exhibits a 1% KS probability, implying
the presence of systematics that are unaccounted for. This is also apparent as
the one-sigma spreads, particularly for the smaller $N_{\text{bins}}=25$ case,
are much wider than the uncertainty bars on the central values, with larger
deviation around unity than in the test-phase case (see Fig. 8). This is not
sufficient to identify the origin of the additional systematic effects.
However, an estimate of this systematic contribution for a given
$N_{\text{bins}}$ may be obtained by calculating

$\sigma_{\text{syst}}\approx\sqrt{\sigma^{2}-\sigma^{2}_{\text{stat}}}\;,$
(22)

where $\sigma$ and $\sigma_{\text{stat}}$ are the standard deviation and mean
statistical uncertainty of the points of the distribution, respectively.

As already stressed, the observed systematic effects are not directly
connected to large fluctuations of the instantaneous luminosity. They have not
been previously observed because such small uncertainties were negligible in
all previous ZEUS analyses. This was confirmed by means of a Monte Carlo
study. Inclusive DIS events with $Q^{2}>4\;\text{GeV}^{2}$ were generated with
DJANGOH 1.6 [33] interfaced to ARIADNE [34, 35, 36, 37]. The CTEQ5D [38] PDFs
were used. The events were subsequently passed through the ZEUS detector and
trigger simulations based on GEANT 3 [39]. The Monte Carlo events contain
leading-order QCD corrections plus parton showering at matrix-element level.
Also, the $x$ in the PDFs is no longer identified with $x_{\text{Bj}}$,
although this is not expected to affect results significantly.

The parton-level calculation is based purely on the Standard Model and does
not introduce any direct time dependence. The generated events do not possess
time stamps. However, the detector response and the trigger configuration
depend on the instantaneous luminosity and may change over time. Data events
are grouped into blocks with the same detector and trigger configurations and,
for each of these blocks, appropriate Monte Carlo events were generated. In
order to include time dependence in the Monte Carlo events and to simulate
accurately the impact of the variable instantaneous luminosity over the course
of 5 years, the following procedure was performed. For each luminosity block
containing a given number of data events, all time stamps were extracted and
assigned Monte Carlo events randomly selected from all those events generated
with the same detector and trigger status. The obtained Monte Carlo sample
simulates accurately any time dependence associated with the instantaneous
luminosity and the detector response.

This study was performed on a subset of experimental data (namely electron
data taken in 2006) corresponding to an integrated luminosity of 54.8 pb-1. It
was found that phase distributions obtained from Monte Carlo events are
perfectly compatible with statistical uncertainties alone: no evidence of any
residual systematic contribution to the fluctuation of the binned central
values around unity is observed in these simulations. This is also confirmed
by comparing the one-sigma spreads of data and Monte Carlo, with the former
being much larger than the latter (and the latter being in agreement with the
statistical uncertainties alone).

This analysis shows that the residual systematic uncertainties observed in
data are not modelled in existing Monte Carlo simulations. As explained above,
a reasonable strategy is to use the difference in quadrature between the one-
sigma spreads and the corresponding statistical uncertainties as an estimate
of the additional systematics. Note that, in the present study, all previously
known sources of systematic uncertainties cancel.

As additional checks, separate studies involving events taken under different
trigger settings, which are constant within each luminosity block, have been
performed. No discernible impact on the presence of the systematic
uncertainties discussed above is observed.

###  5.3 Estimated systematic uncertainties

The binned solar $x_{\text{c}}$ analyses clearly indicate the presence of an
unknown, substantial, and pervasive time-dependent systematic effect with mean
systematic uncertainty $\bar{\sigma}_{\text{syst}}\approx 0.26\%$. The
extraction of systematic uncertainties for periods $T_{\text{sidereal}}\approx
T_{\text{solar}}-4$ min ($\bar{\sigma}_{\text{syst}}\approx 0.18\%$) and
$2T_{\text{solar}}-T_{\text{sidereal}}\approx T_{\text{solar}}+4$ min
($\bar{\sigma}_{\text{syst}}\approx 0.11\%$) both result in smaller systematic
uncertainties relative to the period $T_{\text{solar}}$, further suggesting a
possible unaccounted for solar-periodic effect, which may be a consequence of
the operation of the experiment. The systematic uncertainties on the ratio
binned in sidereal time may therefore reflect a dilution of the observed
potential solar-periodic effect. While, at this stage, a claim of a genuine
solar-time effect cannot be made, there is no significant indication of a
comparable sidereal-time effect. This establishes a baseline for extracting
constraints. Under the assumption of a genuine solar systematic effect, the
$2T_{\text{solar}}-T_{\text{sidereal}}$ distributions determine the dilution
of the solar systematic when changing the period by $+4$ minutes; in fact,
dilution effects for very small positive or negative time shifts are very
similar. This method makes possible an estimate of the systematic
uncertainties on the sidereal-time distributions without risking an absorption
of potential genuine sidereal signals.

In the next section, the constraints on the SME coefficients are extracted
using $N_{\text{bins}}=100$. For consistency, the systematic uncertainty used
in the constraint-setting procedure is also extracted from the
$2T_{\text{solar}}-T_{\text{sidereal}}$ distribution with 100 bins. This
distribution is presented in Fig. 10 and yields $\sigma_{\text{syst}}\approx
0.16\%$. A fuller understanding of this systematic effect is left for future
studies and is beyond the scope of the present analysis.

##  6 Constraints on effective couplings

The sidereal-phase distributions of the ratio of normalised counts for
$x_{\text{Bj}}$ above and below $x_{\text{c}}=10^{-3}$ are shown in Fig. 11.
This is the main result of this paper from which constraints on the SME
coefficients will be extracted. Compared to the solar-phase case, much higher
KS probabilities are observed. To extract constraints, the following procedure
was performed. The laboratory coefficients in the ratios Eqs. (20) and (21)
were replaced with the SCF coefficients as explained in Section 2.3, replacing
one coefficient at a time. The ratios then depend on the local sidereal angle
$\theta_{\oplus}=\omega_{\oplus}T_{\oplus}$. For a given bin $i$, the
theoretical ratios $r_{i}^{\text{theo}}$ were calculated as

$r^{\text{theo}}_{i}=\frac{N_{\text{bins}}}{2\pi}\int_{\frac{2\pi(i-1)}{N_{\text{bins}}}}^{\frac{2\pi
i}{N_{\text{bins}}}}r(x>x_{\text{c}},x<x_{\text{c}};\theta_{\oplus})d\theta_{\oplus}\;,$
(23)

where $i=1,\ldots,N_{\text{bins}}$. This quantity can be compared directly
with the experimental sidereal ratios $r^{\text{exp}}_{i}$. A $\chi^{2}$
function was constructed

$\chi^{2}=\frac{1}{\sigma_{\text{tot}}^{2}}\sum_{i=1}^{N_{\text{bins}}}\left(r^{\text{exp}}_{i}-r^{\text{theo}}_{i}\right)^{2},$
(24)

for each of the 42 observable SCF coefficients. The total uncertainty
$\sigma_{\text{tot}}$ in Eq. (24) is given by combining in quadrature the
statistical uncertainty
($\left[\sigma_{\text{stat}}\right]_{N_{\text{bins}}=100}=0.32\%$) and the
systematic uncertainty estimated in the previous section
($\sigma_{\text{syst}}=0.16\%$) taken as an additional random uncertainty at
each point, so that $\sigma_{\text{tot}}=0.35\%$.

The $\chi^{2}$ in the Standard Model is obtained by setting all the SME
coefficients to zero (the theory prediction in the Standard Model is
identically 1 in each phase bin), and is 113.8 for 100 degrees of freedom. The
$p$-value for this goodness-of-fit (GOF) test is $p_{\text{SM}}=0.16$,
indicating a reasonable description of the data by the Standard Model. In
Tables 1 and 2, lower (upper) values of each coefficient are presented, below
(above) which the $p$-values for the same $\chi^{2}$ GOF test are smaller than
0.05. These values are indicative of the disfavoured ranges for which the
$c$-type coefficients are roughly at the level of $10^{-4}$ for the $u$-quark
coefficients and $10^{-3}$ for the $d$\- and $s$-quark coefficients. The
corresponding ranges for the $a^{(5)}$-type coefficients are mostly at the
level of $10^{-7}$ GeV-1 for the $u$-quark coefficients and $10^{-6}$ GeV-1
for the $d$-quark coefficients.

Figure 12 shows the time dependence associated with non-vanishing
$c_{u}^{TY}$, $c_{u}^{XX}-c_{u}^{YY}$ and $a_{\text{S}u}^{(5)XXY}$
coefficients. These three coefficients have been chosen because they are
examples of time dependence with angular frequencies $\omega_{\oplus}$,
$2\omega_{\oplus}$ and $3\omega_{\oplus}$, respectively. In each case, the
data points are identical to those presented in Fig. 11, the solid and dashed
curves correspond to selected values of the disfavoured regions presented in
Tables 1 and 2, and the dotted curves correspond to coefficients that are
roughly an order of magnitude larger.

Comparison with existing constraints is informative [2]. For the $c$-type
coefficients, the results derived in this work represent the first constraints
using sidereal oscillations. For the $u$ and $d$ quarks, there are existing
and considerably more stringent constraints derived from interpretations of
cosmic-ray measurements [40]. The latter constraints rest on a number of
model-dependent assumptions, so that direct comparison between those results
and what is presented here requires caution. The constraints on the $s$-quark
coefficients are derived for the first time. The results for the
$a^{(5)}$-type coefficients are the first of their kind. One potential point
of comparison would be the constraints on effective $a^{(5)}$-type
coefficients for protons, as resported in Table D11 [2]. Using hydrogen 1S–2S
transitions, similar combinations of proton constraints are currently at the
level of $10^{-7}$–$10^{-8}$ GeV-1, which is similar in magnitude to that
found for $u$-quark $a^{(5)}$-type coefficients [41].

##  7 Summary and outlook

An analysis searching for effective coefficients parameterising Lorentz and
CPT violation in the light-quark sector has been performed using ZEUS
$e^{\pm}p$ DIS NC data at HERA. Conservative estimates of previously unknown
time-dependent systematic uncertainties independent of instantaneous
luminosity and trigger configuration have been made by binning and analysing
events as a function of the sidereal-rotation frequency of the Earth. By
combining binned sidereal statistical uncertainties with systematic
uncertainties in quadrature, the data are shown to be compatible with the
Standard Model. First constraints have been placed on the 24 combinations of
the non-renormalisable, CPT-violating, and spin-independent $a^{(5)}$-type
operators associated with rotationally anisotropic $u$\- and $d$-quark
effects. For the renormalisable CPT-preserving $c$-type coefficients, the
first constraints have been placed on the six rotationally anisotropic
$s$-quark coefficients and the first experimental constraints have been
extracted on the analogous 12 $u$\- and $d$-quark coefficients. In total, 42
coefficients have been constrained, 30 for the first time.

Partonic-physics studies of fundamental symmetries and experimental analyses
searching for unconventional time-dependent signatures are in their infancy. A
statistically significant indication of Lorentz or related fundamental-
symmetry violation in EFT would not unequivocally indicate the presence of
phenomena outside the description of quantum field theory and metric theories
of gravity. Nonetheless, such a discovery would indicate the presence of new
physics, which is a strong motivation for further work in this area.

## Acknowledgements

We appreciate the contributions to the construction, maintenance and operation
of the ZEUS detector of many people who are not listed as authors. The HERA
machine group and the DESY computing staff are especially acknowledged for
their success in providing excellent operation of the collider and the data-
analysis environment. We thank the DESY directorate for their strong support
and encouragement.

## References

  * [1] A.A. Michelson and E.W. Morley,  On the Relative Motion of the Earth and the Luminiferous Ether.  Am. J. Sci. 34, 333 (1887)
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Coefficient | Lower | Upper
---|---|---
$c_{u}^{TX}$ | $-2.5\times 10^{-4}$ | $6.6\times 10^{-5}$
$c_{u}^{TY}$ | $-1.7\times 10^{-4}$ | $9.8\times 10^{-5}$
$c_{u}^{XY}$ | $-3.2\times 10^{-4}$ | $4.1\times 10^{-5}$
$c_{u}^{XZ}$ | $-5.4\times 10^{-4}$ | $1.4\times 10^{-4}$
$c_{u}^{YZ}$ | $-3.7\times 10^{-4}$ | $2.1\times 10^{-4}$
$c_{u}^{XX}-c_{u}^{YY}$ | $-2.1\times 10^{-4}$ | $2.5\times 10^{-4}$
$c_{d}^{TX}$ | $-7.8\times 10^{-4}$ | $2.0\times 10^{-4}$
$c_{d}^{TY}$ | $-5.2\times 10^{-4}$ | $3.0\times 10^{-4}$
$c_{d}^{XY}$ | $-1.6\times 10^{-3}$ | $2.0\times 10^{-4}$
$c_{d}^{XZ}$ | $-2.7\times 10^{-3}$ | $7.0\times 10^{-4}$
$c_{d}^{YZ}$ | $-1.8\times 10^{-3}$ | $1.0\times 10^{-3}$
$c_{d}^{XX}-c_{d}^{YY}$ | $-1.0\times 10^{-3}$ | $1.2\times 10^{-3}$
$c_{s}^{TX}$ | $-9.6\times 10^{-4}$ | $2.5\times 10^{-4}$
$c_{s}^{TY}$ | $-6.4\times 10^{-4}$ | $3.7\times 10^{-4}$
$c_{s}^{XY}$ | $-2.6\times 10^{-3}$ | $3.3\times 10^{-4}$
$c_{s}^{XZ}$ | $-4.4\times 10^{-3}$ | $1.2\times 10^{-3}$
$c_{s}^{YZ}$ | $-3.0\times 10^{-3}$ | $1.7\times 10^{-3}$
$c_{s}^{XX}-c_{s}^{YY}$ | $-1.7\times 10^{-3}$ | $2.0\times 10^{-3}$
Table 1: Lower (upper) values of the $c$-type coefficients below (above) which the $p$-values for the $\chi^{2}$ GOF test are smaller than 0.05.  Coefficient | Lower (GeV-1) | Upper (GeV-1)
---|---|---
$a_{\text{S}u}^{(5)TXX}-a_{\text{S}u}^{(5)TYY}$ | $-5.1\times 10^{-7}$ | $4.3\times 10^{-7}$
$a_{\text{S}u}^{(5)XXZ}-a_{\text{S}u}^{(5)YYZ}$ | $-1.7\times 10^{-6}$ | $2.0\times 10^{-6}$
$a_{\text{S}u}^{(5)TXY}$ | $-8.3\times 10^{-8}$ | $6.5\times 10^{-7}$
$a_{\text{S}u}^{(5)TXZ}$ | $-2.9\times 10^{-7}$ | $1.1\times 10^{-6}$
$a_{\text{S}u}^{(5)TYZ}$ | $-4.3\times 10^{-7}$ | $7.4\times 10^{-7}$
$a_{\text{S}u}^{(5)XXX}$ | $-3.9\times 10^{-7}$ | $1.2\times 10^{-7}$
$a_{\text{S}u}^{(5)XXY}$ | $-2.3\times 10^{-7}$ | $1.8\times 10^{-7}$
$a_{\text{S}u}^{(5)XYY}$ | $-4.6\times 10^{-7}$ | $9.2\times 10^{-8}$
$a_{\text{S}u}^{(5)XYZ}$ | $-2.6\times 10^{-6}$ | $3.3\times 10^{-7}$
$a_{\text{S}u}^{(5)XZZ}$ | $-5.4\times 10^{-7}$ | $1.4\times 10^{-7}$
$a_{\text{S}u}^{(5)YYY}$ | $-2.9\times 10^{-7}$ | $1.5\times 10^{-7}$
$a_{\text{S}u}^{(5)YZZ}$ | $-3.6\times 10^{-7}$ | $2.1\times 10^{-7}$
$a_{\text{S}d}^{(5)TXX}-a_{\text{S}d}^{(5)TYY}$ | $-7.3\times 10^{-6}$ | $6.1\times 10^{-6}$
$a_{\text{S}d}^{(5)XXZ}-a_{\text{S}d}^{(5)YYZ}$ | $-2.4\times 10^{-5}$ | $2.8\times 10^{-5}$
$a_{\text{S}d}^{(5)TXY}$ | $-1.2\times 10^{-6}$ | $9.4\times 10^{-6}$
$a_{\text{S}d}^{(5)TXZ}$ | $-4.1\times 10^{-6}$ | $1.6\times 10^{-5}$
$a_{\text{S}d}^{(5)TYZ}$ | $-6.1\times 10^{-6}$ | $1.1\times 10^{-5}$
$a_{\text{S}d}^{(5)XXX}$ | $-5.7\times 10^{-6}$ | $1.7\times 10^{-6}$
$a_{\text{S}d}^{(5)XXY}$ | $-3.4\times 10^{-6}$ | $2.7\times 10^{-6}$
$a_{\text{S}d}^{(5)XYY}$ | $-6.8\times 10^{-6}$ | $1.3\times 10^{-6}$
$a_{\text{S}d}^{(5)XYZ}$ | $-3.7\times 10^{-5}$ | $4.6\times 10^{-6}$
$a_{\text{S}d}^{(5)XZZ}$ | $-8.1\times 10^{-6}$ | $2.1\times 10^{-6}$
$a_{\text{S}d}^{(5)YYY}$ | $-4.3\times 10^{-6}$ | $2.3\times 10^{-6}$
$a_{\text{S}d}^{(5)YZZ}$ | $-5.4\times 10^{-6}$ | $3.1\times 10^{-6}$
Table 2: Lower (upper) values of the $a^{(5)}$-type coefficients below (above)
which the $p$-values for the $\chi^{2}$ GOF test are smaller than 0.05.
Figure 1: Parton-model picture of deep inelastic scattering with Lorentz
violation. The dots on the incident parton, struck parton, and at the photon-
parton vertex correspond to modified propagation and interaction due to
Lorentz violation. The parton carries a momentum fraction $x_{f}$ that is
perturbed from $x_{\textnormal{Bj}}$ by the coefficients for Lorentz violation
at first order.

(a) (b)

Figure 2: Sun-centred frame (SCF) (a). The orbit of the Earth in the SCF is
shown in green. As the $Z$ axis is parallel to the Earth’s rotation axis, the
orbit is inclined by the angle $\eta$. The effect of Lorentz violation is
depicted as red background arrows. A laboratory on Earth studying a particle
with spin (blue disk and green arrow) with coordinates given by black arrows
centred on the particle will observe background configurations as a function
of the sidereal period $T_{\textnormal{sidereal}}\approx 23$ h $56$ min (b).
The times depicted are an illustration of the sidereal effect.

Figure 3: Regions in $x_{\textnormal{Bj}}$, $Q^{2}$ and $y_{\textnormal{Bj}}$
that have sensitivity to Lorentz-violating effects for a single $c$-type and
$a^{(5)}$-type coefficient. The points displayed in the plots are taken from
Figures 2 and 4 of Ref. [9], where no experimental restrictions on the
kinematic region have been considered.

(a) (b)

(c) (d)

Figure 4:  Solar (a), sidereal (b), $T_{\textnormal{p}}=1$ h (c) and
$T_{\textnormal{p}}=24$ h $4$ min (d) phase dependence of the normalised
counts in 100 bins with the kinematic region divided by
$Q_{\textnormal{c}}^{2}=20$ GeV2. The vertical axis displays the number of
events per bin normalised to the total number of events times the bin width.
The ratios of the counts $r(Q_{\textnormal{c}})$ above and below
$Q_{\textnormal{c}}^{2}$ are given in the bottom panels. For the solar phase,
$\phi_{\textnormal{solar}}=0$ is identified with 11:20 UTC. Only statistical
uncertainties are displayed.  Figure 5:  Ratio $r(Q_{\textnormal{c}})$ of
normalised counts for $Q_{\textnormal{c}}^{2}=20$ GeV2 binned with the solar
phase where $N_{\textnormal{bins}}=25$ and $100$. The case
$N_{\textnormal{bins}}=100$ from Fig. 4(a) is repeated. The displayed
uncertainties include statistical uncertainties only and the one-sigma spreads
(bands) are the standard deviations of the central values. The observed
distributions are compared to a Gaussian distribution in which only
statistical errors are included using a Kolmogorov–Smirnov test.  Figure 6:
Ratio $r(Q_{\textnormal{c}})$ of normalised counts for
$Q_{\textnormal{c}}^{2}=20$ GeV2 binned with the sidereal phase where
$N_{\textnormal{bins}}=25$ and $100$. The displayed uncertainties include
statistical uncertainties only and the one-sigma spreads (bands) are the
standard deviations of the central values. The observed distributions are
compared to a Gaussian distribution in which only statistical errors are
included using a Kolmogorov–Smirnov test.

(a) (b)

(c) (d)

Figure 7:  Solar (a), sidereal (b), $T_{\textnormal{p}}=1$ h (c) and
$T_{\textnormal{p}}=24$ h $4$ min (d) phase dependence of the normalised
counts in 100 bins with the kinematic region divided by
$x_{\textnormal{c}}=10^{-3}$. The vertical axis displays the number of events
per bin normalised to the total number of events times the bin width. The
ratios of the counts $r(x_{\textnormal{c}})$ above and below
$x_{\textnormal{c}}$ are given in the bottom panels. For the solar phase,
$\phi_{\textnormal{solar}}=0$ is identified with 11:20 UTC. Only statistical
uncertainties are displayed.  Figure 8:  Ratio $r(x_{\textnormal{c}})$ of
normalised counts for $x_{\textnormal{c}}=10^{-3}$ binned using the test phase
with $T_{\textnormal{p}}=1$ h where $N_{\textnormal{bins}}=25$ and $100$. The
displayed uncertainties include statistical uncertainties only and the one-
sigma spreads (bands) are the standard deviations of the central values. The
observed distributions are compared to a Gaussian distribution in which only
statistical errors are included using a Kolmogorov–Smirnov test.  Figure 9:
Ratio $r(x_{\textnormal{c}})$ of normalised counts for
$x_{\textnormal{c}}=10^{-3}$ binned using the solar phase where
$N_{\textnormal{bins}}=25$ and $100$. The case $N_{\textnormal{bins}}=100$
from Fig. 7(a) is repeated. The displayed uncertainties include statistical
uncertainties only and the one-sigma spreads (bands) are the standard
deviations of the central values. The observed distributions are compared to a
Gaussian distribution in which only statistical errors are included using a
Kolmogorov–Smirnov test.  Figure 10:  Ratio $r(x_{\textnormal{c}})$ of
normalised counts for $x_{\textnormal{c}}=10^{-3}$ binned using the test phase
with $T_{\textnormal{p}}=24\;\textnormal{h}\;4\textnormal{min}$ where
$N_{\textnormal{bins}}=25$ and $100$. The displayed uncertainties include
statistical uncertainties only and the one-sigma spreads (bands) are the
standard deviations of the central values. The observed distributions are
compared to a Gaussian distribution in which only statistical errors are
included using a Kolmogorov–Smirnov test.  Figure 11:  Ratio
$r(x_{\textnormal{c}})$ of normalised counts for $x_{\textnormal{c}}=10^{-3}$
binned using the sidereal phase where $N_{\textnormal{bins}}=25$ and $100$.
The displayed uncertainties include statistical uncertainties only and the
one-sigma spreads (bands) are the standard deviations of the central values.
The observed distributions are compared to a Gaussian distribution in which
only statistical errors are included using a Kolmogorov–Smirnov test.  Figure
12:  Ratio $r(x_{\textnormal{c}})$ of normalised counts for
$x_{\textnormal{c}}=10^{-3}$ binned using the sidereal phase where
$N_{\textnormal{bins}}=100$. The displayed uncertainties include statistical
uncertainties only. In the three panels, signals corresponding to the SME
coefficients $c_{u}^{TY}$, $c_{u}^{XX}-c_{u}^{YY}$ and
$a_{\textnormal{S}u}^{(5)XXY}$ which are associated with variations up to
$\omega_{\oplus},2\omega_{\oplus}$, and $3\omega_{\oplus}$, respectively, are
displayed with values selected at the edge of the disfavoured ranges presented
in Tables 1 and 2 (solid, dashed and dotted lines). In addition, signals that
are excluded by the extracted bounds are displayed with $c_{u}^{TY}=10^{-3}$,
$c_{u}^{XX}-c_{u}^{YY}=10^{-3}$, and $a_{\textnormal{S}u}^{(5)XXY}=+2\times
10^{-6}$ GeV-1.


# Attention-Aware Generative Adversarial Networks (ATA-GANs)

Dimitris Kastaniotis, Ioanna Ntinou, Dimitrios Tsourounis, George Economou and
Spiros Fotopoulos, Department of Physics University of Patras Patras 26504,
Greece<EMAIL_ADDRESS>
###### Abstract

In this work, we present a novel approach for training Generative Adversarial
Networks (GANs). Using the attention maps produced by a Teacher- Network we
are able to improve the quality of the generated images as well as perform
weakly object localization on the generated images. To this end, we generate
images of HEp-2 cells captured with Indirect Imunofluoresence (IIF) and study
the ability of our network to perform a weakly localization of the cell.
Firstly, we demonstrate that whilst GANs can learn the mapping between the
input domain and the target distribution efficiently, the discriminator
network is not able to detect the regions of interest. Secondly, we present a
novel attention transfer mechanism which allows us to enforce the
discriminator to put emphasis on the regions of interest via transfer
learning. Thirdly, we show that this leads to more realistic images, as the
discriminator learns to put emphasis on the area of interest. Fourthly, the
proposed method allows one to generate both images as well as attention maps
which can be useful for data annotation e.g in object detection.

Keywords: Generative Adversarial Networks, Attention Maps, HEp-2 cells

##  I INTRODUCTION

Over the recent years there is a great progress in the area of generative
models. In particular, Generative Adversarial Networks (GANs) [1], proposed a
radically new approach for training a generative model. This formulation
allows one to capture high dimensional and complex distributions efficiently
using a minmax game between two Neural Networks - one that generates samples
and one that evaluates them as real or fake. In this game and for image data,
the generator learns the mapping from a low dimensional space to the space of
images, allowing one to sample images from the high-dimensional distribution
by sampling from a lower-dimensional distribution. These models have been
applied in many computer vision tasks like human brain decoding [2] or
realistic images synthesis [3]. Furthermore, in tasks where data are limited
(e.g. biomedical datasets) it can be used for data augmentation.

One major limitation of current deep neural network models is the lack of a
mechanism for knowledge transfer between tasks. Transfer learning aims to
address this limitation by suggesting novel ways for transferring the
experience between different tasks [4] or even models[5]. However, these works
focus on transferring the knowledge of a network with respect to the ability
of mapping images to a number of categories. Recently, authors in [6],
following a Teacher-Student approach, attempted to transfer the attention of a
large network to one with fewer parameters by incorporating loss functions
between the Teacher and the Student intermediate layers.

It has also been observed, that intermediate layers of CNNs act as weak object
detectors [7, 8, 9]. This occurs naturally as the filters, especially in
higher layers, tend to synthesize objects. Authors in [10] inspired by this
observation, have shown that global average pooling layers can be used in
order to create Class Activation Maps (CAMs).

In this work, we focus on GAN discriminator’s inability to locate the regions
of interest when trying to learn how to discriminate real versus fake images.
Specifically, we investigate if a discriminator uses patterns within areas of
interest of an image. Our findings indicate that discriminators in a regular
GAN fail to locate the object efficiently- thus evaluating regions that are
not always of much interest. In this context, we propose a novel formulation
for training GANs, using a Teacher network in order to help the discriminator
learn where to pay attention to. Results indicate that our method both
improves the quality of the generated image as well as it provides a weakly
localization of the objects on the generated images.

##  II PROPOSED METHOD

###  II-A Overview

Our method is mainly inspired by two recent advances in computer vision.
Firstly, authors in [6], showed that small CNNs achieve better performance, if
they learn to mimic behavior of attention maps in intermediate layers of
bigger networks. To this end, the attention maps of intermediate layers of a
larger network are transferred to the small network by attaching several loss
functions in intermediate feature maps. Additionally authors in [10], showed
that CNNs which use average pooling at the last layer can naturally deliver a
specific type of attention maps, called Class Activation Maps (CAMs). These
attention maps can be used to perform weakly object detection [8]. As it is
shown in Figure 2, a modified version of Class Activation Maps (CAMs) can be
used to help the discriminator pay more attention in regions of interest. The
second one is the ability of GANs to generate realistic images by training a
neural network to generate samples while a second one evaluates them as real
or fake.

Figure 1: Overview of the proposed method

In this context, here we present a novel method that allows us to transfer the
knowledge of a large network, originally trained to discriminate between
different categories of HEp-2 cell images, into the discriminator of a GAN. To
achieve this, we reformulate the method of CAMs [10] in a way that allows us
to incorporate the soft responses of all classes in the estimation of the CAM
image. Our intuition is that there are some common features shared across
HEp-2 cell image categories located in the same area, and hence, incorporating
them according to a Soft-Max response allows us to proportionally consider
patterns from all categories. The proposed method is named as Soft-CAMS
(SCAMS). The Soft-CAMs , which are presented in the following section, will be
used to guide the attention of a discriminator to better localize the cell.
Therefore, we introduce an additional cost in the original GAN-minmax
optimization problem. This cost is applied on the discriminator and enforces
it to pay more attention into areas of interest (as they are defined by the
Teacher network). The proposed method is summarized in Figure 1.

###  II-B Soft-Class Activation Maps

The ability of Deep Convolutional Neural Networks to perform weak object
detection in intermediate layers has been observed in a number of works [7, 8,
9].

Authors in [10] applied global average pooling (which is used in many state-
of-the-art deep neural network architectures) [11, 12] to perform weakly
localization of objects in an image. Moreover, they presented a scheme, which
allows one to determine the attention of a CNN on input image implicitly. In
this work, we take advantage of their findings to extract the attention map
from a large CNN and then use this to train the discriminator of the GAN.
Furthermore, we present a modified version of the CAM which we call Soft-CAM
(SC) that can be expressed as follows:

Figure 2: Soft-Max Activation Map

$SC=\sum_{j=1,C}(SM(j)\cdot{\sum_{i=1,K}FM_{i}\cdot W_{i,j}})$ (1)

In the previous equation $SM\in\mathbb{R^{C}}$ is a vector containing Soft-Max
responses of the classifier, $K$ is a scalar equal to the number of channels
before global average pooling and $C$ is a scalar equals the number of
categories. This vector allows us to weight the contribution of each feature
map in the Soft-CAM computation for each category proportionally to the
classification probability. Index $j$ iterates over the number of categories
and index $i$ iterates over the number of available feature maps. In total,
there are $K$ feature maps and $C$ categories. Each feature map is a two
dimensional image. Matrix $W\in\mathbb{R^{K}C}$ allows us to scale each i-th
feature map with a weight associated with the j-th category. The Soft-CAM is a
(single channel) two-dimensional image with the same dimensions as the feature
maps. This image sums proportionally the contribution of each class to the
computation of CAM. It is important to notice that for the discriminator
network, we have a single class while for the Teacher-Network we have more
than one classes. One major advantage of the proposed scheme is that it allows
one to naturally incorporate class information in the discriminator, without
modifying the formulation.

Figure 2, displays a graphical illustration of the computation of Soft-CAM
presented in Eqn. 1 as well as a comparison with the regular CAM computation.

###  II-C Attention transfer from an Expert Network

As we have already claimed, the discriminator of a regular GAN, is unable to
efficiently locate the region of interest (location of cell). This is also
demonstrated in Figure 4. In order to help the discriminator better localize
the cell, here we are incorporating a large network (ResNet-18)[12], trained
on a large corpus of HEp-2 cell images [13] which was able to perform a state-
of-the-art weak localization of the cell inside image as shown in Figure 5.

We also claim that this network should have enough capacity to learn the
mapping between the images and the class categories. One way to ensure this,
is to require that the network will achieve a human-level classification
score. As we show in the experimental results, the proposed Teacher-Network is
able to surpass human-level error. On the contrary, as we show in Figure 3, a
SqueezeNet [11] network (image in the middle) is not able to perform a robust
weak localization. Also, in experiments we performed we noticed that
SqueezeNet [11] was not able to surpass human level error.

Figure 3: Left: The cell image. Middle: The attention map of a SqueezeNet
drawn on top of a cell image. Red color indicates more contribution in the
classification result. Right: The attention map of a ResNet.

###  II-D Train GANS with Attention (ATA-GANS)

In this section we first show that a discriminator in a GAN whilst being able
to separate real from fake images, is not able to localize the region of
interest of the cell. Then we present the original GAN minmax problem and our
modified version that incorporates an attention cost which helps the
discriminator to better localize the areas of interest.

The ability of the discriminator to locate the object of interest
appropriately (here the cell) is very important. More specifically the
discriminator might learn to classify images as fake by focusing on particular
patterns in the boundaries of the image or even might focus on patterns that
are not related to the information that is valuable for classification.

In the following Figure 4, we show the Soft- Class Activation Maps (SCAMs) of
two images- one real and one fake. It is obvious that the attention of the
discriminator spreads across the whole image and does not focus on the area of
interest- which in this case is the area where the cell is located.

Figure 4: Soft-CAM of the discriminator of a regular GAN. Top-Left: A real
image, Top-Right: A generated image. Bottom-Left: The attention map of the
real image. Bottom-Right: The attention map of the generated image.

As we have shown previously and depicted in Figure 4 the discriminator of a
GAN,is unable to efficiently locate the region of interest (location of cell).
In order to help the discriminator better localize the cell, we are
incorporating a large network (ResNet-18) [12], trained on a large corpus of
HEp-2 cell images [13] which is able to perform state-of-the-art weak
localization of the cell inside the image as depicted in Figure 5. At the
bottom we have the input images and at the top we have the Soft-Class
Activation Maps.

Figure 5: Teacher Network: Soft-Class Activation Maps from Res-Net-18, trained
on HEp-2 cell images. Top Row: Cell Images. Bottom Row: Soft-Class Activation
Maps.

During the training process, input images (both real and fake ones) are passed
to the Teacher-Network as well as to the discriminator (see Figure 1). It is
obvious, that the discriminator has to be implemented in a way that will
produce these Soft-CAMs. In this context in Figure 6 , we present a version of
the discriminator that produces two outputs. Firstly, it provides the binary
decision of whether the input image is real or fake, and secondly, it produces
the Soft-CAM image.

Figure 6:  The architecture of discriminator. The network produces a binary
decision characterizing the input images as genuine or not, as well as a
second output the S-CAM

The output of the teacher network together with the Soft-CAM output of the
discriminator is then used in order to compute the following Mean-Square-Error
(MSE) loss:

$L_{SCAM}=\frac{1\hfill}{2}(T_{SCAM}-D_{SCAM})^{2}$ (2)

In the previous equation $T_{SCAM}$ is the attention map produced by the
Teacher network presented at the top of Figure 1 and $D_{SCAM}$ is the
attention map produced by the discriminator and presented at the bottom of
Figure 1.

However, this loss function is computed twice for the training of the
discriminator, one for the real image and one for the generated image.
Consequently, the same procedure is repeated for the generated image and the
total loss is computed as

$L_{SCAM_{total}}={L_{SCAM_{real}}}+{L_{SCAM_{fake}}}$ (3)

This loss is being added to the Discriminator loss function only when training
the discriminator. For the training of the Generator network we didn’t
incorporate and attention loss as our main intuition is that we can improve
the generator by allowing an expert network teach the discriminator to pay
attention. The optimization problem is therefore formulated as follows:

$\begin{split}min_{G}max_{D}V(D,G)=\\\ \mathbb{E}_{x\sim
p_{data}}[\log(D(x))]+\\\ \mathbb{E}_{z\sim
p_{z}}[\log(1-D(G(x)))]\end{split}$ (4)

While we also add the minimization of the MSE criterion of Equation 3 on the
discriminator update step.

##  III Experimental Results

Here we present some experimental results demonstrating the quality of
generated images as well as the ability of the proposed scheme to provide both
realistic new samples as well as weakly annotations. For this, in order to
construct the Teacher network, we used ResNet-18 [12], which we trained a on
the task of HEp-2 cell image classification using the dataset presented in
[13]. This network achieved a remarkable performance on the dataset of HEp-2
cell images [15] (76.28% accuracy) surpassing human level performance (73.3%)
and approaching the top performance (78.1%) achieved in [14]. Also, this
network was able to efficiently locate the object of interest on images as
shown in Figure 5.

Then we used this Teacher-network in order to train our discriminator by
introducing one extra cost in the discriminator as shown in Equation 4 during
the minmax game between the generator and the discriminator. In Figure 7 we
provide some results of the generated images one the left side, and some real
images on the right side. Our approach clearly generated very realistic images
and most importantly, we verified that incorporating an attention loss does
not affect not affect negatively the quality of the generators images.

Regarding the ability of the proposed scheme to perform a weakly localization,
in Figure 8 we provide some Soft-CAMs from both generated as well as real
images. As compared to Figure 4, our modified minmax game was able to improve
the attention of the discriminator significantly.

Figure 7:  Left: Generated Images, Right: Real Images

Moreover, as we are interested in the ability of the network (generator) to
generate realistic image and the discriminator to perform weak object
localization, we provide some input-output pairs from both read as well as
generated images. Results in the Figure 8 verify that the proposed scheme can
be used to both generate images as well as to provide weakly annotations.

Figure 8:  Discriminator attention maps for input output pairs after training
with our method. These images provide a weak localization of the cell area. On
the left, we have two generated images (bottom row) and their Soft-CAMs (top
row).On the right, we have two real images (bottom row) and their Soft-CAMs
(top row).

##  IV CONCLUSIONS

In this work we proposed a method which allows the discriminator network of a
GAN to perform weakly localization of the objects of interest. In order to
achieve this we proposed a Teacher-Student learning scheme as well as a novel
type of Soft-Class Activation maps. This scheme allows the discriminator to
generate weak annotation of the generated images which can be used for
automatic annotation of generated images. In the future we plan to apply this
scheme on datasets designed for object detection in order to generate both
images as well as weak annotations. We plan to make the code publicly
available after the publication of this work.

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