# Electromagnetic and strong isospin breaking in light meson masses

Ángel S. Miramontes<EMAIL_ADDRESS>Instituto de Física y Matemáticas,
Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán 58040,
Mexico  Reinhard Alkofer<EMAIL_ADDRESS>Institute of Physics,
University of Graz, NAWI Graz, Universitätsplatz 5, 8010 Graz, Austria
Christian S. Fischer<EMAIL_ADDRESS>Institut für
Theoretische Physik, Justus-Liebig-Universität Giessen, 35392 Giessen,
Germany;
Helmholtz Forschungsakademie Hessen für FAIR (HFHF) GSI Helmholtzzentrum für
Schwerionenforschung, Campus Gießen, 35392 Gießen, Germany  Hèlios Sanchis-
Alepuz<EMAIL_ADDRESS>Silicon Austria Labs GmbH,
Inffeldgasse 33, 8010 Graz, Austria

###### Abstract

We study electromagnetic as well as strong isospin breaking effects in the
isospin mass splittings of light pseudoscalar and vector mesons. To this end
we employ a coupled system of quark Dyson-Schwinger and meson Bethe-Salpeter
equations whose interaction kernels contain gluon, pion and photon exchange
interactions. In bound states, QCD-induced isospin breaking is manifest on
different levels. On the one hand, a different explicit up- and down-quark
mass directly affects the propagators of the constituent quarks. On the other
hand, it leads to different interaction kernels within the isospin multiplets.
In addition, electromagnetic isospin breaking is induced via a photon exchange
diagram. Using the kaon iso-doublet and the charged pion masses as input to
determine the up, down and strange quark masses we find for the pion, kaon and
rho meson mass splittings different patterns each. In particular, our results
provide evidence that the effects from two sources of mass splittings, the
different quark masses and the different quark charges, do not add up
linearly.

##  1 Introduction

The approach to QCD via functional methods, most prominently with Dyson-
Schwinger equations, Bethe-Salpeter equations and/or the functional
renormalisation group has reached a degree of sophistication that allows to
systematically study sub-leading effects in the hadron spectrum. Arguably the
most important of these effects is the breaking of isospin symmetry. It plays
an important role in baryon physics, notably in the mass splitting between the
proton and the neutron which is related to fundamental questions like the
existence and stability of ordinary matter in our universe. It also plays an
important role in meson physics allowing for a range of interesting and
important isospin violating decays such as the one of the exotic X(3872).
Furthermore, its inclusion is mandatory for precision determinations of
hadronic contributions to the anomalous magnetic moment of the muon [1, 2],
and of the CKM matrix elements $V_{ud}$ and $V_{us}$ (see, e.g., [3] for a
recent update on the respective analysis of $\pi$ and $K$ decays in view of
the apparent 3$\sigma$ violation of the first-row CKM unitarity condition).
Thus, a better understanding of isospin breaking effects, besides being
interesting in its own right, might play a significant rôle in the detection
of new physics via high-precision experiments.

The mass difference of the charged to the neutral pions have been analysed
already before the advent of QCD [4] with the conclusion that the observed
pion mass difference does not arise from the mass difference of the two light
quark flavours but from the electromagnetic interaction. Based on this, when
considering three flavours, one arrives at the conjecture for the
electromagnetically induced kaon mass difference

$\left(m^{2}_{K^{\pm}}-m^{2}_{K^{0}}\right)_{\mathrm{e}m}=m^{2}_{\pi^{\pm}}-m^{2}_{\pi^{0}}$
(1)

which is also known as Dashen’s theorem [5]. However, this relation is already
significantly violated beyond leading order Chiral Perturbation Theory
($\chi$PT) [6, 7], the next-to-leading order providing approximately a factor
of two

$\left(m^{2}_{K^{\pm}}-m^{2}_{K^{0}}\right)_{\mathrm{e}m}=(1.84\pm
0.25)\left(m^{2}_{\pi^{\pm}}-m^{2}_{\pi^{0}}\right)\,.$ (2)

From this one may conclude that the electromagnetic interaction alone would
imply that the charged kaon is approximately 2 MeV heavier than the neutral
one. Note, however, that the observed mass ordering is opposite [8]

$m_{K^{0}}-m_{K^{\pm}}=(3.934\pm 0.0020)\,{\mathrm{MeV}}~{},$ (3)

the sign and the magnitude of the kaon mass difference being mostly due to the
light quark mass difference.

In the past years, lattice QCD calculations with QED included have delivered
detailed analyses of isospin breaking effects and determined successfully and
with increasing precision the resulting masses of the up and down quarks, see
the FLAG-report [9] for an overview and [10, 11, 12, 13, 14] as well as
references therein.

Isospin breaking effects have been studied in functional methods already in
the pioneering work of Jain and Munczek [15] using a simple but efficient
truncation of the quark-gluon interaction in terms of one-gluon exchange. An
improved ladder approximation in Dyson-Schwinger and Bethe-Salpeter equations
have been used in refs. [16, 17] to calculate the pion mass difference. Since
then, the framework has been developed to include the full momentum dependent
physics from all primitively divergent Green’s functions leading to spectra in
qualitative and in most channels even quantitative agreement with experiment,
see [18] for a review. From a physics perspective, a particularly interesting
contribution to the quark-gluon interaction stems from four-quark functions,
which can be parameterised in terms of (off-shell) meson contributions. These
reappear, as expected from general properties of QCD and, in particular, from
quark-hadron duality, in the self-energies and interaction kernels of Bethe-
Salpeter equations as meson loops and exchanges, which can be interpreted as
meson cloud effects. As we will see in the course of this work, these meson
cloud effects play an important part in the systematics of strong breaking of
isospin due to the mass difference of the up- and down-quark. The second, in
many cases competing, effect is isospin breaking due to the charge difference
of the up- and down-quark. Diagrammatically, this QED effect appears in the
form of photon loop and exchange diagrams which have to be added to the self-
energies and interaction kernels.

In this work, we study contributions of strong and electromagnetic isospin
breaking effects on the mass splitting of the isospin multiplets of light
pseudoscalar and vector mesons. Based on previous work on the pion
electromagnetic form factor [19], our truncation scheme takes explicitly into
account that the non-perturbative interaction between light quarks becomes
flavour dependent away from the isospin limit. We summarise technical details
of our scheme in section 2 and discuss our results for the mass splittings and
the associated up- and down-quark masses in section 3. In section 4 we
conclude with an outlook to future applications.

Figure 1: Coupled system of DSEs for up and down quarks (upper two equations)
including explicit pion backreaction diagrams and a loop including a photon.
Gluons are represented by curly lines, mesons by dashed lines and photons by
wiggly lines. In each diagram the left vertex is bare (but renormalised) and
the vertex on the right is dressed. Kaon and eta backreaction diagrams are
neglected (see main text). Correspondingly, the DSE for the strange quark
(lower equation) contains gluon and photon diagrams only.

##  2 Formalism

In this section we discuss our heuristic approach to isospin breaking using
the formalism of Dyson-Schwinger (DSE) and Bethe-Salpeter (BSE) equations. We
briefly summarise the key elements of the DSE/BSE approach (for more details
see e.g. [20, 18, 21, 22]) and introduce the two mechanisms for isospin
breaking that we explore in this paper.

Formally, the quark-DSE for a quark with flavour $f$ is given by

$\displaystyle S^{-1}_{f}(p)=$ $\displaystyle
Z_{2}\,S^{-1}_{0,f}(p)-Z_{1f}\,g^{2}\,C_{F}\times$ (4)
$\displaystyle\times\int\\!\\!\frac{d^{4}q}{(2\pi)^{4}}\,i\gamma^{\mu}S_{f}(q)\,\Gamma_{\mathrm{qg},f}^{\nu}(q,p)\,D^{\mu\nu}(k)\;,$

where $Z_{2}$ and $Z_{1f}=Z_{g}\,Z_{2}\,Z_{3}^{1/2}$ are (flavour-dependent)
renormalisation constants for the quark propagator and quark-gluon vertex, and
$C_{F}=4/3$ is the colour Casimir for $N_{c}=3$. The inverse tree-level
propagator is given by
$S^{-1}_{0,f}(p)=i\ooalign{\hfil/\hfil\crcr$p$}+Z_{m}\,m_{f}$, where $m_{f}$
is the renormalized quark mass from the QCD action. The dressed quark-gluon
vertex $\Gamma_{\mathrm{qg},f}$ has a very rich structure including gluonic
but also effective meson exchange contributions detailed below.

The dressed quark propagators, i.e., the solutions of the DSE (4), will be of
the form

$\displaystyle
S_{f}(p)=\frac{1}{A_{f}(p^{2})}\frac{-i\ooalign{\hfil/\hfil\crcr$p$}+M_{f}(p^{2})}{p^{2}+M^{2}_{f}(p^{2})}$
(7)

which, for each flavour, is determined by two dressing functions,
$A_{f}(p^{2})$ and $M_{f}(p^{2})$. The latter are the dynamically generated
mass functions, and we will use below $M_{f}(p^{2}=0)$ to demonstrate that in
the solutions of the DSE (4) the isospin breaking effects due to the current
mass differences are enhanced.

In the BSE framework, bound states of a quark and an antiquark can be
described by Bethe-Salpeter amplitudes $\Gamma$ which are obtained as
solutions of a homogeneous BSE,

$\displaystyle\Gamma_{fg}\left(p,P\right)$
$\displaystyle=\sum_{f^{\prime}g^{\prime}}\int_{q}K_{fg}^{f^{\prime}g^{\prime}}\left(P,p,q\right)\Bigl{(}S_{f^{\prime}}\left(k_{1}\right)\times$
$\displaystyle\qquad\times\Gamma_{f^{\prime}g^{\prime}}\left(q,P\right)S_{g^{\prime}}\left(k_{2}\right)\Bigr{)}~{},$
(8)

with $P$ the total meson momentum, $p$ is the relative momentum between quark
and antiquark, $q$ is an internal relative momentum which is integrated over.
For pseudoscalar and vector mesons, the Dirac part of the Bethe-Salpeter
amplitude $\Gamma_{fg}$ for a meson with constituents of flavour $f,g$ can be
expanded in a tensorial basis with four and eight elements, respectively. The
interaction kernel $K$ encodes all possible interactions between quarks and
anti-quarks, and the quark propagators $S_{f}(p)$ are given by the solution of
the quark DSE in Eq. (4).

In this work we consider three types of contributions to the quark self-energy
and the BSE interaction kernel: (i) a flavour-blind dressed quark-antiquark
gluon exchange that provides the necessary interaction strength to form
mesonic bound states, (ii) a meson-exchange mechanism that provides flavour
mixing and, (iii) a dressed photon exchange describing electromagnetic
interactions between the quark and the antiquark. The latter is responsible
for the electromagnetically induced isospin breaking. These contributions are
shown explicitly in Figs. 1 and 2. In the following, we explain each
contribution in turn.

Figure 2: Coupled system of BSEs for the isospin triplet including explicit
meson and photon exchange diagrams. The corresponding BSEs for the kaons are
similar in structure as the ones for the charged pions.

###  2.1 Explicit Isospin Breaking and Flavour Mixing

Let us consider first the non-electromagnetic contributions to the quark self-
energy and the BSE interaction kernel. As indicated above and shown in Figs. 1
and 2 the strongly-interacting part consists of a gluon loop/exchange term and
meson loop/exchange terms.

For the gluon term we use the rainbow-ladder truncation of the quark-gluon
vertex, whereby the effect of (the leading term of) the dressed quark-gluon
vertex $\Gamma_{\mathrm{qg,f}}^{\nu}(q,p)=i\gamma^{\nu}\Gamma(k^{2})$ and the
Landau-gauge dressed gluon propagator $D^{\mu\nu}(k)=T^{\mu\nu}Z(k^{2})/k^{2}$
with transverse projector $T$ is combined into a effective coupling that only
depends on the gluon momentum $k$.

$\alpha(k^{2})=\frac{Z_{1f}}{Z_{2}^{2}}\,\frac{g^{2}}{4\pi}\,Z(k^{2})\,\Gamma(k^{2})\,.$
(9)

For the effective coupling we use the Maris-Tandy model [23, 24]

$\displaystyle\alpha_{\text{QCD}}(k^{2})=$
$\displaystyle\pi\eta^{7}\left(\frac{k^{2}}{\Lambda^{2}}\right)^{2}e^{-\eta^{2}\frac{k^{2}}{\Lambda^{2}}}$
$\displaystyle+\frac{2\pi\gamma_{m}\big{(}1-e^{-k^{2}/\Lambda_{t}^{2}}\big{)}}{\textnormal{ln}[e^{2}-1+(1+k^{2}/\Lambda_{QCD}^{2})^{2}]}~{},$
(10)

where the Gaussian term provides binding strength and enables dynamical chiral
symmetry breaking; the second term reproduces the one-loop QCD behavior of the
quark propagator in the ultraviolet when used in the quark DSE. In this model
the scale $\Lambda_{t}=1$ GeV is introduced for technical reasons. For the
anomalous dimension we use $\gamma_{m}=12/(11N_{C}-2N_{f})=12/25$ with
$N_{f}=4$ flavours and $N_{c}=3$ colours. For the QCD scale we use
$\Lambda_{QCD}=0.234$ GeV. The parameters $\eta$ and $\Lambda$ are discussed
below.

Note that such a truncation of the gluon-interaction guarantees that the
axial-vector WTI (axWTI) is respected and, hence, that the pion is massless in
the chiral limit.

A gluon exchange as interaction mechanism is flavour diagonal. In order to
introduce flavour mixing we complement the interaction kernel with a meson
exchange mechanism (see Fig. 2). The BSE kernel, defined in [25, 26] and
representing the exchange of a pion with Bethe-Salpeter amplitude
$\Gamma_{\pi}$ and propagator $D_{\pi}(p)=1/(p^{2}+m_{\pi}^{2})$, reads

$\displaystyle K^{(t)~{}ut}_{rs}(q,p;P)=$
$\displaystyle~{}\frac{C}{4}[\Gamma_{\pi}^{j}]_{ru}\left(\frac{p+q-P}{2};p-q\right)[Z_{2}\gamma^{5}]_{ts}D_{\pi}(p-q)$
$\displaystyle+\frac{C}{4}[\Gamma_{\pi}^{j}]_{ru}\left(\frac{p+q-P}{2};q-p\right)[Z_{2}\gamma^{5}]_{ts}D_{\pi}(p-q)$
$\displaystyle+\frac{C}{4}[Z_{2}\gamma^{5}]_{ru}[\Gamma_{\pi}^{j}]_{ts}\left(\frac{p+q+P}{2};p-q\right)D_{\pi}(p-q)$
$\displaystyle+\frac{C}{4}[Z_{2}\gamma^{5}]_{ru}[\Gamma_{\pi}^{j}]_{ts}\left(\frac{p+q+P}{2};q-p\right)D_{\pi}(p-q).$
(11)

The quark DSE, correspondingly, is extended with the addition of further loop
diagrams (third and fourth terms in the quark DSEs in Fig. 1). As discussed in
detail in [25, 26], the presence of these loops can be motivated and derived
from the underlying Dyson-Schwinger equation of the quark-gluon vertex, where
meson exchange diagrams appear as approximation of a momentum dependent four-
Fermi interaction. In a complete treatment of these effects, we would need to
include meson diagrams involving pions, kaons and in the DSE for the strange
quark even the $\eta/\eta^{\prime}$ mesons. However, already pion backraction
effects are subleading on the 10-20 percent level compared to gluon effects
[25, 26]. Since kaon and $\eta/\eta^{\prime}$ effects are further suppressed
by factors of $m_{\pi}^{2}/m_{K,\eta,\eta^{\prime}}^{2}$ it seems safe to
neglect these in a first approach. A similar argument applies to potential
contributions from mesons with other quantum numbers.

In contrast to the gluon exchange, the meson exchange mechanism does not
immediately preserve the axWTI and can, potentially, violate chiral symmetry.
It was shown in [25, 26] how an axWTI-preserving meson-exchange kernel can be
constructed. We use this kernel, in particular, we set $C=-3/2$, in this work.
Note that consequently the meson exchange kernels do not introduce any
additional parameters.

The resulting diagrams in the BSEs are shown in Fig. 2. It is interesting to
note that due to the flavour structure, the meson exchange diagrams only
appear in the flavour diagonal BSEs for the $u\bar{u}$ and $d\bar{d}$
components. Corresponding diagrams in the two BSEs for the charged pions are
diquark exchange diagrams. Again, the effects of these diagrams are suppressed
by factors of $m_{\pi}^{2}/m_{qq}^{2}$ and we neglect these due to the heavy
mass of the lowest lying scalar diquark of the order of 800-900 MeV [18, 27].

Furthermore, we need to point out a technical difficulty we face in the
present calculation. In the meson-exchange kernels shown in Fig. 2 (see also
Eq. 11) the quark-meson interaction vertex is given by the meson BS amplitude.
Moreover, in order to access the time-like region, where bound states fulfil
$P^{2}=-M^{2}$, with $P$ the total bound-state momentum and $M$ the bound-
state mass, we need to work with complex momenta in the BSE, since we work in
Euclidean spacetime. This implies, in particular, that the BS amplitudes used
as vertices need to be evaluated for complex relative quark momenta (i.e. the
momenta $p$ in Eq. (2)). Solving BSEs for complex relative momenta is
currently only possible in the RL approximation. Therefore, the BS amplitudes
used as vertices are obtained from a RL-truncated BSE calculation.
Unfortunately, we have no way of estimating at the moment how big the
quantitative effect of such an approximation on our results is. We do not
expect, however, that any qualitative statements change.

Finally, let us point out that neither the gluon exchange nor the meson
exchange interaction mechanisms induce any isospin breaking on their own.
However, if explicit isospin breaking is introduced by a choice of different
up- and down-quark masses, the meson exchange terms will cease to be
degenerate and introduce non-trivial mixing of the different BSE amplitudes.
Thus isospin breaking occurs on different levels: it results in different
dressing functions for the up- and down-quark propagators but it also results
in a flavour dependent QCD interaction manifest in the different meson
exchange terms in our truncation.

###  2.2  Electromagnetically-induced Isospin Breaking

To describe electromagnetic effects we consider a dressed photon exchange term
in the BSE kernel and the corresponding term in the quark DSE (see Figs. 1 and
2). The quark-photon interaction is described by the fully-dressed quark-
photon vertex. Even though the non-perturbative structure of the quark-photon
vertex is by now reasonably well known [28, 29, 30, 31, 32, 33], for the
qualitative study of this work we will choose a simpler model.

The non-transverse part of the quark-photon vertex is given by the Ball-Chiu
construction [28] in terms of the quark dressing functions $A_{f}(p^{2})$ and
$B_{f}(p^{2})$, with
$S_{f}^{-1}(p)=-i\ooalign{\hfil/\hfil\crcr$p$}A_{f}(p^{2})+B_{f}(p^{2})$ the
inverse quark propagator. In this work we simplify the calculation further,
employing the model used in [34], where only the $\gamma^{\mu}$ component of
the Ball-Chiu vertex is used. In this way, the QED contribution takes the same
form as the gluon rainbow-ladder term and can be incorporated quite easily by
adding

$\alpha_{\text{QED}}(k^{2})=\alpha_{0}Q_{g}Q_{f}Z_{\text{QED}}(k^{2})~{},$
(12)

to $\alpha_{\text{QCD}}$ in Eq. 2.1. Here, $\alpha_{0}\approx 1/137$
corresponds to the electromagnetic coupling, $Q_{f,g}$ are the charges of the
two quarks attached to the photon line with flavour $f,g$ and $Z_{\text{QED}}$
the dressing of the leading tensorial structure of the quark-photon vertex. It
is parametrised by

$Z_{\text{QED}}(k^{2})=\frac{A_{f}(k^{2})+A_{g}(k^{2})}{2}~{},$ (13)

where $i,j$ represent the two quarks at both ends of the photon. This flavour
averaging has no effect on the self-energy in the quark-DSE, but is
appropriate for the kernels in flavour non-diagonal meson states.

Solving the coupled system of DSE and BSE using different masses and charges
for up and down quarks we are now in a position to study the effects of the
different sources of isospin breaking.

##  3  Results

| $\bar{X}$ |  |  | $X$
---|---|---|---|---
$M$[MeV] | $m_{u}=m_{d}$ | $m_{u}\neq m_{d}$ | $m_{u}=m_{d}$ | $m_{u}\neq m_{d}$
| $\alpha_{0}=0$ | $\alpha_{0}=0$ | $\alpha_{0}=1/137$ | $\alpha_{0}=1/137$
$\pi^{0}$ | 134.5 | 132.5 | 136.0 | 133.4
$\pi^{\pm}$ | 134.5 | 134.2 | 139.6 | 139.7†
$\pi^{\pm}-\pi^{0}$ | 0 | 1.7 | 3.6 | 6.3
$K^{0}$ | 494.7 | 497.5 | 495.2 | 497.7†
$K^{\pm}$ | 494.7 | 492.1 | 497.2 | 493.7†
$K^{0}-K^{\pm}$ | 0 | 5.4 | -2.0 | 4.0
$\rho^{0}$ | 720.3 | 721.5 | 721.1 | 721.4
$\rho^{\pm}$ | 720.3 | 719.9 | 722.0 | 720.9
$\rho^{\pm}-\rho^{0}$ | 0 | -1.6 | 0.9 | -0.5
Table 1:  Pion, kaon and rho masses $M$ for the full system depicted in Figs.
1 and 2 with quark masses $m_{u}=6.3$ MeV, $m_{d}=8.1$ MeV, $m_{s}=82.8$ MeV
(third and fifth column), average quark masses $m_{ud}=m_{u}=m_{d}=7.2$ MeV
(second and fourth column) and electromagnetic isospin breaking switched on
and off. A dagger indicates that the quark masses where fitted to obtain those
meson masses.

First we need to fix the QCD-parameters, the quark masses $m_{u}$, $m_{d}$ and
$m_{s}$, and the parameters of our model interaction, the Maris-Tandy
parameters $\Lambda$ and $\eta$. The model parameter $\Lambda$ sets the scale
and is adjusted such that the experimental decay constant of the charged pion
is reproduced. Furthermore it is well known from pure RL calculations [18]
that within a certain range, experimental quantities are fairly insensitive to
the value of the other model parameter, $\eta$. In our calculations we choose
$\eta=1.42$ and $\Lambda=0.74$.

This leaves the up-, down- and strange-quark masses to be determined. We
adjust these such that a complete calculation with rainbow-ladder gluon
exchange, meson exchange and photon exchange (cf. Figs. 1 and 2) reproduces
the experimental masses for the charged pion and the neutral and charged kaon.
The corresponding results are displayed in the rightmost column of Tab. 1. We
find $m_{u}=6.3$ MeV, $m_{d}=8.1$ MeV and $m_{s}=82.8$ MeV for the quark
masses in a MOM-scheme evaluated at a renormalisation scale $\mu=19$ GeV. All
other results for meson masses are then model predictions. Note that the
masses of the vector meson in general come out too small by about 50 MeV; this
is a well-known artefact of the Maris-Tandy model and is remedied only in a
full calculation taking into account corrections from the gluon self-
interaction [35] and from the decay into two pions[36].

Here we like to point out the well-known result that the dynamically generated
mass function $M_{f}(p^{2})$, cf. Eq. (7), displays a significant enhancement
of the mass splitting for values of $p^{2}$ being in the sub-GeV region.
Taking into account only the flavour-blind gluon-mediated interaction one
obtains

$M_{u}(0)-M_{d}(0)\approx 5(m_{u}-m_{d})$ (14)

Including the pion exchange for the two light flavours lowers $M_{u,d}(0)$
both by 37 MeV, and consequently the relation (14) stays valid. The photon
exchange increases $M_{u}(0)$ by slightly less than one MeV, resp., 0.3 %,
which is of the expected size due to $Q_{u}^{2}\alpha_{0}=$ 0.003. Due to the
lower absolute value of the $d$ quark charge $M_{d}(0)$ is increased by
approximately a quarter MeV (and $M_{s}(0)$ by half a MeV). Clearly, these
small additional mass increases leave us more or less with the mass difference
(14).

The enhancement (14) from $m_{u}-m_{d}=1.8$ MeV to $M_{u}(0)-M_{d}(0)\approx$
9 MeV is important for an understanding of the induced mass difference of the
neutral to the charged kaons in terms of the quark masses. The strong-
interaction induced mass difference for the $K$ mesons is expected to be due
to binding effects smaller than the mass difference of the two different light
valence quarks. As a matter of fact, from our results for the K mesons, see
Tab. 1, we infer that it is smaller than $M_{u}(0)-M_{d}(0)$ but due to (14)
it is significantly larger than the current mass difference $m_{u}-m_{d}$.

In order to analyse our results for the meson masses and in particular the
separate effects of isospin breaking due to different up-/down-quark masses
and the electromagnetic interaction in the context of the available literature
we adopt the notation of the FLAG review [9]. As described above, the quark
masses are then determined by the system of equations

$\displaystyle M_{\pi^{+}}(m_{u},m_{d},m_{s},\alpha_{0})$
$\displaystyle=M_{\pi^{+}}^{exp},$ (15) $\displaystyle
M_{K^{+}}(m_{u},m_{d},m_{s},\alpha_{0})$ $\displaystyle=M_{K^{+}}^{exp},$ (16)
$\displaystyle M_{K^{0}}(m_{u},m_{d},m_{s},\alpha_{0})$
$\displaystyle=M_{K^{0}}^{exp},$ (17)

where the right hand sides are the experimental values and $\alpha_{0}=1/137$
is precise enough for our study. All calculated masses are functions of
$m_{u},m_{d},m_{s},\alpha_{0}$, and since isospin breaking effects are small,
in a hadronic quantity $X$ they may be described to first order by expansions
in $\delta_{m}$ and $\alpha_{0}$ with

$\displaystyle X(m_{u},m_{d},m_{s},\alpha_{0})$
$\displaystyle=\bar{X}(m_{ud},m_{s})$ $\displaystyle+\delta
m\,A_{X}(m_{ud},m_{s})$ $\displaystyle+\alpha_{0}\,B_{X}(m_{ud},m_{s})$
$\displaystyle+\delta m\,\alpha_{0}\,C_{X}(m_{ud},m_{s})$ (18)
$\displaystyle=\bar{X}+X^{SU(2)}+X^{\gamma}+X^{SU(2),\gamma}.$

Here, $m_{ud}=1/2(m_{u}+m_{d})$ is the average light-quark mass. Higher order
effects in this expansion are traditionally expected to be small [9]. In order
to gauge this expectation, we also included the lowest order mixed quantity
proportional to $\delta m\,\alpha_{0}$ to facilitate our analysis. The size of
this additional correction can be estimated from comparing results for $X$ and
$\bar{X}$ with results obtained from individually setting $\delta m$ or
$\alpha_{0}$ to zero.

Our results for the isospin doublet kaons and the isospin triplet pions and
$\rho$-mesons as well as their mass differences are shown in Tab. 1. We find
that the kaon mass splitting is dominated by strong isospin breaking,
$(m_{K^{0}}-m_{K^{\pm}})^{SU(2)}=5.4$ MeV with a much smaller electromagnetic
breaking with opposite sign $(m_{K^{0}}-m_{K^{\pm}})^{\gamma}=-2.0$ MeV and
thus following the expectation derived from Eq. (2). In contrast, the pion
mass splitting is dominated by the electromagnetic effect
$(m_{\pi^{\pm}}-m_{\pi^{0}})^{\gamma}=3.6$ MeV, and an addition due to the
strong isospin breaking of only half the size,
$(m_{\pi^{\pm}}-m_{\pi^{0}})^{SU(2)}=1.7$ MeV.

In both cases we also observe that the two sources of breaking do not add
linearly, but there are non-negligible effects due to mixed higher order terms
in the expansion Eqs. (3). We find
$(m_{\pi^{\pm}}-m_{\pi^{0}})^{SU(2),\gamma}=1.0$ MeV for the pions and
$(m_{K^{0}}-m_{K^{\pm}})^{SU(2),\gamma}=0.6$ MeV for the kaons. The mass of
the neutral pion is only slightly affected by isospin breaking, most of the
difference stems from the shift in the mass of the charged pions. The biggest
single effect is the electromagnetically induced mass increase of the charged
pions, and all isospin breaking effects are adding up. (NB: It should be
noted, however, that our calculation overestimates the pion mass difference,
we obtain 6.3 MeV instead of the observed 4.6 MeV [8].) For the kaons we note
the partial cancellation of electromagnetic and strong isospin breaking
effects. For the charged kaons this cancellation is almost complete, whereas
for the neutral kaon there is a larger mass shift due to the strong
interaction as the one due to the electromagnetic interaction.

A somewhat different pattern can be observed for the rho meson masses. First
of all, the observed individual shifts are much smaller than the ones for the
pions or the kaons, and in addition they largely cancel each other due to
opposite signs. But again, we observe that the two sources of isospin breaking
do not add linearly.

##  4  Conclusions and outlook

In this work, we determined the effects of isospin breaking from the strong
interaction, via different masses for the up- and down-quark, and the
electromagnetic interaction, via different charges of the up- and down-quark,
on the pseudoscalar meson isospin triplet, strange pseudoscalar meson doublet
and the vector meson triplet. We find that the pion splitting is mainly
induced by the different quark charges, but we also observe some effects due
to different masses and even noticeable mixed, i.e. non-linear, effects of
both. The corresponding splitting in the rho triplet is very small which is
due to relatively small individual mass shifts as well as competing mass and
charge splitting effects of opposite sign. Again, the non-linear effects are
relevant to obtain the resulting mass splitting. A special role is played by
the kaon doublet splitting which results from quite a number of mass shifts of
similar magnitude but opposite sign.

Our results agree qualitatively with those of other continuum approaches,
mainly the $\chi$PT results, see, e.g., [6, 7, 37], and with those of lattice
QCD see, e.g., [9, 10, 11, 12, 13, 14]. In particular, we find that the
traditional view that the pion mass difference does not arise from the quark
mass difference but practically only from the different quark charges is an
unjustified oversimplification. We also note that our results do not support
Dashen’s theorem [5] for the electromagnetically induced kaon mass splitting,
instead we have

$\left(m^{2}_{K^{\pm}}-m^{2}_{K^{0}}\right)_{\mathrm{e}m}\approx
2.0\left(m^{2}_{\pi^{\pm}}-m^{2}_{\pi^{0}}\right)_{\mathrm{e}m}\,,$ (19)

in agreement with the $\chi$PT result, Eq. (2).

## Acknowledgements

We thank Jan Bonnet and Richard Williams for collaboration and discussions in
early stages of this work. This work has been supported by Silicon Austria
Labs (SAL), owned by the Republic of Austria, the Styrian Business Promotion
Agency (SFG), the federal state of Carinthia, the Upper Austrian Research
(UAR), and the Austrian Association for the Electric and Electronics Industry
(FEEI). A.S. Miramontes acknowledges CONACYT for financial support.

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# Logic-Driven Context Extension and Data Augmentation
for Logical Reasoning of Text

Siyuan Wang1, Wanjun Zhong2, Duyu Tang3, Zhongyu Wei1,
Zhihao Fan1, Daxin Jiang3, Ming Zhou4, Nan Duan3
1School of Data Science, Fudan University, China
2Sun Yat-Sen University, China 3Microsoft, China 4Sinovation Ventures, China
<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Work is done during internship at Microsoft Research Asia.

###### Abstract

Logical reasoning of text requires understanding critical logical information
in the text and performing inference over them. Large-scale pre-trained models
for logical reasoning mainly focus on word-level semantics of text while
struggling to capture symbolic logic. In this paper, we propose to understand
logical symbols and expressions in the text to arrive at the answer. Based on
such logical information, we not only put forward a context extension
framework but also propose a data augmentation algorithm. The former extends
the context to cover implicit logical expressions following logical
equivalence laws. The latter augments literally similar but logically
different instances to better capture logical information, especially logical
negative and conditional relationships. We conduct experiments on ReClor
dataset. The results show that our method achieves the state-of-the-art
performance, and both logic-driven context extension framework and data
augmentation algorithm can help improve the accuracy. And our multi-model
ensemble system is the first to surpass human performance on both EASY set and
HARD set of ReClor. 111Codes are publicly available at
https://github.com/WangsyGit/LReasoner.

##  1 Introduction

Recent years have witnessed a growing interest in logical reasoning of text,
which aims to follow a logic-level analysis over a given text to arrive at the
correct answer McCarthy (1989); Nilsson (1991). It is a challenging task since
it requires the ability to extract critical information from text and perform
logical inference over them Williams et al. (2018); Habernal et al. (2018);
Liu et al. (2020).

Figure 1:  A logical reasoning example from ReClor dataset Yu et al. (2020)
which includes a context, a question and four options and the correct answer
is marked by ✓. To find the answer, it need to extract _logical symbols_ ,
identify _logical expressions_ and perform logical inference to extend the
_implicit logical expressions_. The underlined phrases represent logical
symbols. The colored rectangles are corresponding logical expressions of
options.

An example of logical reasoning problem is shown in Figure 1. It takes a
context, a related question and four options as the input, and needs to
analyze which option follows logically from the context. In order to solve
such a problem, the reasoning system first needs to extract the critical
constituents from the context as logical symbols like $\alpha,\beta,\gamma$
and identify the existing logical expressions that are composed of logical
symbols, like $(\neg\alpha\rightarrow\neg\beta)$. Then according to logical
equivalence laws, it performs inference to extend the logical expressions that
are not explicitly mentioned in the context and compare them with the logical
expressions of each option to find the most plausible answer.

Although large-scale pre-trained language models Devlin et al. (2019); Liu et
al. (2019); Yang et al. (2019) have achieved promising results on logical
reasoning of text, they mainly leverage word-level semantics but hardly
capture symbolic logic. In responding to such a problem, we propose to
identify logical symbols and expressions explicitly mentioned in the context
which serve as the elementary components of logical inference process. Based
upon such logical information, we propose both a context extension framework
and a data augmentation algorithm. The whole system is built on top of the
pre-trained model considering its promising performance. Our logic-driven
context extension framework performs inference following logical equivalence
laws to extend the logical expressions in the context to cover the implicit
ones. For better utilization of extended logical expressions at symbolic space
into neural model, we verbalize them into natural language and feed as an
extended context into a pre-trained model to match the options and find the
answer. Our logic-driven data augmentation algorithm employs modification
operations over logical expressions to produce challenging instances with
literally similar but logically different contexts. It trains our model to
distinguish different contexts to better capture logical information,
especially negative and conditional relationships in logical expressions.

We take both RoBERTa Liu et al. (2019) and ALBERT Lan et al. (2020) as our
backbone pre-trained models. The experiments are conducted on the benchmark
dataset ReClor Yu et al. (2020) and our system achieves the state-of-the-art
performance and reaches the top of the leaderboard. The further results also
show the effectiveness of both logic-driven context extension framework and
data augmentation algorithm.

##  2 Task and Background

###  2.1 Task Definition

We study the problem of logical reasoning of text on a multiple-choice
question answering task. The task is described as following: given a context
$c$, a question $q$, and four associated options
$\\{o_{1},o_{2},o_{3},o_{4}\\}$, we aim to select the most appropriate option
as the answer $o_{a}$.

We adopt a challenging dataset ReClor Yu et al. (2020) which is collected from
the logical reasoning questions of standardized examinations including Law
School Admission Test (LSAT)222https://www:lsac:org/lsat/taking-
lsat/testformat/logical-reasoning and Graduate Management Admission Test
(GMAT)333https://www.mba.com/exams/gmat/about-the-gmat-exam/gmat-exam-
structure/verbal. Although many questions need to be answered through
complicated logical reasoning, there are still some biased instances which can
be solved even without knowing contexts and questions. It therefore splits the
unbiased instances from the test data as the HARD set to fully evaluate the
logical reasoning ability. And the other biased ones are taken as the EASY
set. A leaderboard444https://eval.ai/web/challenges/challenge-
page/503/leaderboard/1347 is also hosting for public evaluation on ReClor.

Figure 2:  The overall architecture of our proposed logic-driven context
extension framework. $c$, $q$, $o_{i}$ and $e_{i}$ are the context, question,
$i$-th option and the extended context for $i$-th option, respectively. The
texts in green mean that the option $B$ is matched against its extended
context which has the highest score.

###  2.2 Base Model

In this paper, we follow the leading methods on the leaderboard to take pre-
trained models as our base model. Pre-trained models for multiple-choice
question answering concatenate the context, the question and each option as
the input and encode the sequence for calculating its score. Given four
options, four concatenated sequences are constructed to calculate four scores,
and the one with the highest score is chosen as the answer. Specifically, the
concatenated sequence is formulated as $[CLS]\ c\ [SEP]\ q\ ||\ o\ [SEP]$,
where $c$ is the context and $q\ ||\ o$ is the concatenation of the question
and each option. The representations of the special token $[CLS]$ in the four
sequences are fed into a linear layer with a softmax function to get the
probability distribution of options as $P(\\{o_{1},o_{2},o_{3},o_{4}\\}|c,q)$.
The cross entropy loss is calculated as:

$\displaystyle\mathcal{L}_{A}=-\sum\mathop{\log}P(o_{a}|c,q)$ (1)

where $o_{a}$ is the correct option.

Since previous work Yu et al. (2020) has shown that RoBERTa Liu et al. (2019)
achieves superior results on the leaderboard, and ALBERT Lan et al. (2020) is
proved to be even more powerful than RoBERTa on many tasks, we utilize both
RoBERTa and ALBERT in the experiments.

#### RoBERTa

is a robustly optimized BERT Devlin et al. (2019) with more training data,
which uses a more dynamic sentence masking method and removes the next
sentence prediction loss.

#### ALBERT

is a lite BERT with two parameter-reduction techniques including factorized
embedding parameterization and cross-layer parameter sharing. And it replaces
the next sentence prediction loss with the inter-sentence coherence loss.

However, such pre-trained models for logical reasoning of text directly encode
the triplet of context, question and options, which only leverage word-level
semantics but struggle to capture symbolic logic in the text. Therefore, we
propose a framework on top of a pre-trained model to understand logical
symbols and expressions in the text and perform logical inference over them to
predict the answer.

##  3 Logic-Driven Context Extension

In this section, we present a logic-driven context extension framework for
logical reasoning of text, and the overall architecture is illustrated in
Figure 2. The framework can be divided into three steps as follow. It first
identifies the logical symbols and expressions explicitly mentioned in the
context and options (§ 3.1). Then it performs logical inference over them to
extend the logical expressions implicit in the context (§ 3.2). Finally, it
verbalizes the extended logical expressions related to each option as an
extended context and utilizes it in the pre-trained model to match the answer
(§ 3.3).

###  3.1 Logic Identification

In order to perform logical reasoning, we first need to identify the
elementary components for reasoning as logical expressions. We identify the
existing logical expressions for each sentence in the context and each option.
To show the format of the logical expression, we introduce some notations:

  1. (1)

$\\{\alpha,\beta,\gamma,...\\}$: the logical symbols, which are the basic
constituents in the context to constitute the logical expressions, such as the
“have keyboarding skills” in Figure 2.

  2. (2)

$\\{\neg,\rightarrow\\}$: the logical connectives set. $\neg$ means the
negation operation upon a specific logical symbol and $\rightarrow$ acts as a
conditional relationship between two logical symbols.

  3. (3)

$\\{(\alpha\rightarrow\beta),...\\}$: the logical expressions which are
composed of logical symbols and connectives. $(\alpha\rightarrow\beta)$ means
that $\alpha$ is the condition of $\beta$.

Because logical expressions are composed of logical symbols, we first employ
constituency parser Joshi et al. (2018) to extract constituents including noun
phrases and gerundial phrases as basic symbols. Then the logical symbols in
each sentence are combined by logical connectives to constitute the logical
expressions as follow-up. If any negative word is related to a logical symbol
$\alpha$, we add the negation connective $\neg$ before $\alpha$ as a new
logical symbol $\neg\ \alpha$. We define a set of negative words as {“ _not_
”, “ _n’t_ ”, “ _unable_ ”, “ _no_ ”, “ _few_ ”, “ _little_ ”, “ _neither_ ”,
“ _none of_ ”}. Then if there is a conditional relationship between two
logical symbols $\alpha$ and $\beta$ in a sentence, we can construct the
corresponding logical expression as $(\alpha\rightarrow\beta)$. We simply
recognize the conditional relationship between symbol $\alpha$ and $\beta$ as
$(\alpha\rightarrow\beta)$ according to conditional indicators such as _“if
$\alpha$, then $\beta$”, “$\alpha$ in order for $\beta$”, “$\beta$ due to
$\alpha$”, “$\neg\beta$ unless $\alpha$”_, etc. And if an active voice occurs
between $\alpha$ and $\beta$, we also have $(\alpha\rightarrow\beta)$. As
illustrated in Figure 2, given the context with two sentences, we can extract
three logical symbols $\\{\alpha,\beta,\gamma\\}$ and identify two existing
logical expressions as $(\neg\alpha\rightarrow\neg\beta)$ and
$(\neg\beta\rightarrow\neg\gamma)$.

###  3.2 Logic Extension

In addition to the logical expressions explicitly mentioned in the context,
there are still some other implicit ones which we need to logically infer and
extend. We combine the identified logical expressions existing in all
sentences of the context as a logical expression set $\mathcal{S}$, and
perform logical inference over them to further extend the implicit logical
expressions according to logical equivalence laws. Here we follow two logical
equivalence laws including _contraposition_ Russel et al. (2013) and
_transitive law_ Zhao et al. (1997):

$\displaystyle\text{Contraposition}:$ $\displaystyle(\alpha\rightarrow\beta)$
$\displaystyle\implies(\neg\beta\rightarrow\neg\alpha)$ (2)
$\displaystyle\text{Transitive Law}:$ $\displaystyle(\alpha\rightarrow\beta)\
\land\ $
$\displaystyle(\beta\rightarrow\gamma)\implies(\alpha\rightarrow\gamma)$ (3)

Then the extended implicit logical expressions form an extension set of the
current logical expression set $\mathcal{S}$ as $\mathcal{S}_{E}$.

As the example in Figure 2, the set of existing logical expressions is
$\mathcal{S}=\\{(\neg\alpha\rightarrow\neg\beta),(\neg\beta\rightarrow\neg\gamma)\\}$
and the logic extension set is
$\mathcal{S}_{E}=\\{(\beta\rightarrow\alpha),(\gamma\rightarrow\beta),(\neg\alpha\rightarrow\neg\gamma),(\gamma\rightarrow\alpha)\\}$.

###  3.3 Logic Verbalization

After inferring the extended logical expressions set $\mathcal{S}_{E}$, we
verbalize them into natural language for better utilization in the pre-trained
model. We first select the related expressions from $\mathcal{S}_{E}$ for each
option. A logical expression is regarded as related to an option if it has the
same logical symbols with the option judged by the text overlapping, and
whether a negation connective exists also needs to be considered. For example,
$(\neg\alpha\rightarrow\neg\gamma)$ in Figure 2 is related to option $C$
because they both contain $\neg\gamma$. Then we transform all logical
expressions related to the option at symbolic space into natural language by
filling them into a template and concatenate them into a sentence. We take
such a sentence as an extended context for this option. The template is
designed as shown in Table 1.

logic |  $(\neg\alpha\rightarrow\neg\gamma)$
---|---
template |  If do not $\alpha$, then will not $\gamma$.
extended context |  If you do not have keyboarding skills, then you will not be able to write your essays using a word processing program.
Table 1:  An example of verbalizing a logical expression into text.

We feed extended contexts into the pre-trained model to match the options and
predict the answer. We take an extended context as sentence $e$, and introduce
a special token $[EXT]$ to represent context extension. Then we reformulate
the input sequence as $[CLS]\ c\ [SEP]\ q\ ||\ o\ [EXT]\ e\ [SEP]$ for
encoding and feed the $[CLS]$ representation into a classification layer to
get each option’s score and find the most appropriate answer.

##  4 Logic-Driven Data Augmentation

In order to make the pre-trained model put more focus on logical information
in the context, especially logical negative and conditional relationships, we
further introduce a logic-driven data augmentation algorithm. Inspired by
SimCLR Chen et al. (2020), it employs contrastive learning to augment
challenging instances with literally similar but logically different contexts
based on logical expressions. It then trains our model to predict the correct
context supporting the answer. We first introduce the background of SimCLR and
then describe our logic-driven constrative learning.

#### SimCLR

As a paradigm of self-supervised representation learning by comparing
different samples, contrastive learning Wu et al. (2018); Le-Khac et al.
(2020); He et al. (2020) aims to make the representations of similar samples
be mapped close together, while that of dissimilar samples be further away in
the encoding space. The goal can be described as following.

$\displaystyle s(f(x),f(x^{+}))\gg s(f(x),f(x^{-}))$ (4)

$x^{+}$ is a positive sample similar to the data point $x$ while $x^{-}$ is a
negative sample dissimilar to $x$. $f(\cdot)$ is an encoder to learn a
representation and the $s(\cdot)$ is a metric function that measures the
similarity between two representations. Over this, SimCLR Chen et al. (2020)
bulids a classifier to distinguish positive from negative samples and learns
to capture what makes two samples different.

#### Logic-Driven Contrastive Learning

In our multiple-choice question answering setting, we alter the score function
from measuring the similarity between two representations to calculating the
score that the question can be solved by the correct answer under a given
context:

$\displaystyle s^{{}^{\prime}}(c^{+},q,o_{a})\gg
s^{{}^{\prime}}(c^{-},q,o_{a})$ (5)

where $(c^{+},q,o_{a})$ and $(c^{-},q,o_{a})$ are the positive and negative
sample, $c^{+}$ and $c^{-}$ are the positive and negative context,
respectively, and $s^{{}^{\prime}}$ is the score function. The contrastive
loss can be formulated as a classification loss for predicting the most
plausible context that supports the answer:

$\displaystyle\mathcal{L}_{C}=-\sum\log\frac{\exp(s^{{}^{\prime}}(+))}{\exp(s^{{}^{\prime}}(+))+\exp(s^{{}^{\prime}}(-))}$
(6)

where $s^{{}^{\prime}}(+)$ and $s^{{}^{\prime}}(-)$ are short for
$s^{{}^{\prime}}(c^{+},q,o_{a})$ and $s^{{}^{\prime}}(c^{-},q,o_{a})$
respectively.

Aware of symbolic logical expressions, we can construct _logical negative
samples_ including negative contexts that are literally similar but logical
dissimilar to the positive one. We take the original context to construct the
positive sample. Then we generate a negative sample by modifying the existing
logical expressions in the context and verbalizing the modified logical
expressions into a negative context as § 3.3. During the modification
operations, we randomly choose a logical expression and randomly _delete,
reverse or negate_ such a expression. The _delete, reverse or negate_
operations are respectively to delete a logical expression in the context,
reverse the conditional order of a logical expression and negate a logical
symbol in a logical expression. The constructing procedure of a logical
negative sample is illustrated in Figure 3. Then the model can be trained to
better capture logical information, especially negative and conditional
relationships in logical expressions.

Figure 3:  Procedure to construct a logical negative sample.

In the logic-driven data augmentation algorithm, our framework is trained with
a combined loss as $\mathcal{L}=\mathcal{L}_{A}+\mathcal{L}_{C}$. And the
classification of positive and negative context for the correct answer is also
implemented in the logic-driven context extension framework.

##  5 Experiments

###  5.1 Experimental Setup

Our experiments are conducted on ReClor Yu et al. (2020), a question answering
dataset extracted from standardized exams including GMAT and LSAT which
requires complicated logical reasoning. It consists of $6,138$ questions and
each question is collected with a context and four answer options, in which
only one is correct. It is divided into training, validation and test sets
with $4,638$, $500$ and $1,000$ data points, and test set is split into EASY
set and HARD set with $440$ and $560$ data points. For automatic evaluation,
we adopt the accuracy as the metric to compare performance of models over test
set, EASY set and HARD set.

We take both RoBERTa-large Liu et al. (2019) and ALBERT-xxlarge-v2 Lan et al.
(2020) as our backbone models and implement them using Huggingface Wolf et al.
(2019). We use a batch size of 8 and fine-tune on ReClor for 10 epochs. The
AdamW Loshchilov and Hutter (2017) with $\beta 1$ = 0.9 and $\beta 2$ = 0.98
is taken as the optimizer and the learning rate is set to 1e-5. We use a
linear learning rate scheduler with $10\%$ warmup proportion. We select at
most two extended logical expressions related to each option to construct the
extended context.

###  5.2 Overall Performance

We compare our systems with several baseline models and human performance,
which are described as follows.

#### Baseline Models

The compared baseline pre-trained models for multiple-choice question
answering include GPT Radford et al. (2018), GPT-2 Radford et al. (2019), BERT
Devlin et al. (2019), XLNet Yang et al. (2019), RoBERTa Liu et al. (2019) and
ALBERT Lan et al. (2020). In this paper, BERT, XLNet, RoBERTa and ALBERT
respectively refer to BERT-large, XLNet-large, RoBERTa-large and ALBERT-
xxlarge-v2.

#### Our Systems

_$\text{LReasoner}_{\text{RoBERTa}}$_ is our proposed logic-driven reasoner
built on top of RoBERTa, which utilizes both logic-driven context extension
framework and data augmentation algorithm.
_$\text{LReasoner}_{\text{ALBERT}}$_ is also our proposed logic-driven
reasoner but taking ALBERT as the backbone model.

#### Human Performance

The average score of ten graduate students over randomly chosen 100 samples
from test set is taken as human performance.

Model | Test | EASY | HARD
---|---|---|---
$\text{GPT}^{*}$ | 45.4 | 73.0 | 23.8
$\text{GPT-2}^{*}$ | 47.2 | 73.0 | 27.0
$\text{BERT}^{*}$ | 49.8 | 72.0 | 32.3
$\text{XLNet}^{*}$ | 56.0 | 75.7 | 40.5
$\text{RoBERTa}^{*}$ | 55.6 | 75.5 | 40.0
$\text{LReasoner}_{\text{RoBERTa}}$ | 62.4 | 81.4 | 47.5
ALBERT | 66.5 | 76.6 | 58.6
$\text{LReasoner}_{\text{ALBERT}}$ | 70.7 | 81.1 | 62.5
$\text{Human Performance}^{*}$ | 63.0 | 57.1 | 67.2
Table 2: Experimental results of different models and human performance. The
results in bold are the best performance of each column and $*$ indicates that
the results come from Yu et al. (2020).

The evaluation results are shown in Table 2. We have several findings:

  1. -

Our models outperform all baseline models by a considerable margin.
_$\text{LReasoner}_{\text{ALBERT}}$_ even surpasses the human performance in
_Test_ and _EASY_ sets. This indicates the effectiveness of our method for
predicting more accurate answer.

  2. -

_$\text{LReasoner}_{\text{RoBERTa}}$_ and _$\text{LReasoner}_{\text{ALBERT}}$_
both perform better than the corresponding baseline models _RoBERTa_ and
_ALBERT_. It demonstrates that our proposed method is robust to be effective
for logical reasoning on top of different pre-trained models, even the base
model is already of great power, such as _ALBERT_.

  3. -

Our models generate large improvement on both _HARD_ set and _EASY_ set by
comparing our models with _RoBERTa_ and _ALBERT_ , respectively. This not only
follows our intuition that our system is designed for logical reasoning
problems, but also shows that it is capable of solving easier problems.

###  5.3 Ablation Study

To dive into the effectiveness of different components in our logic-driven
reasoner, we conduct ablation study which takes _RoBERTa_ as our backbone
model on both validation and test sets.

Model | Val | Test | EASY | HARD
---|---|---|---|---
RoBERTa | 62.6 | 55.6 | 75.5 | 40.0
\+ CE | 65.2 | 58.3 | 78.6 | 42.3
\+ DA | 65.8 | 61.0 | 80.9 | 45.4
\+ CE \+ DA | 66.2 | 62.4 | 81.4 | 47.5
Table 3: Ablation study of our system built on top of _RoBERTa_. _CE_ and _DA_
are respectively our logic-driven context extension framework and data
augmentation algorithm. _RoBERTa +CE+DA_ is our proposed
_$\text{LReasoner}_{\text{RoBERTa}}$_.

As shown in Table 3, _RoBERTa +CE_ and _RoBERTa +DA_ both outperform the
baseline model _RoBERTa_ and perform worse than our final system _RoBERTa
+CE+DA_. It indicates that both logic-driven context extension framework and
logic-driven data augmentation algorithm are beneficial for question answering
involving logical reasoning.

###  5.4 Further Analysis on Negative Sample Construction Strategies

To further analyze the effectiveness of our logical negative samples in logic-
driven contrastive learning, we compare several different negative sample
construction strategies in contrastive learning on top of _RoBERTa_.

Model | Test | EASY | HARD
---|---|---|---
RoBERTa (w/o CLR) | 55.6 | 75.5 | 40.0
RoBERTa (w/ CLR-RS) | 58.2 | 79.3 | 41.6
RoBERTa (w/ CLR-RD) | 58.9 | 78.9 | 43.2
RoBERTa (w/ CLR-L) | 61.0 | 80.9 | 45.4
Table 4: Comparison of different negative sample construction approaches on
top of _RoBERTa_. _CLR_ represents contrastive learning. _RS_ means randomly
selecting a context from in-batch data while _RD_ means randomly deleting a
sentence from the original context as the negative sample. _L_ denotes our
logical negative samples in logic-driven contrastive learning. Figure 4:  A
case of the reasoning process of our logic-driven reasoner
_$\text{LReasoner}_{\text{ALBERT}}$_. Phrases underlined represent other
symbols (called _Others_) different from the logical symbols in context and
tokens in bold make them different. The option marked by ✓ is our predicted
answer.

From the results in Table 4, we can see that all models with contrastive
learning outperform the model without it, which demonstrates that contrastive
learning can help to better predict the answer. Our logic-driven contrastive
learning _RoBERTa(w/ CLR-L)_ performs best. It reveals that logical negative
samples are more effective than negative samples constructed by other methods
which make the model to better capture the logical negative and conditional
relationships in the context for logical reasoning.

###  5.5 Case Study

A case study is presented in Figure 4 to show the reasoning process of our
system. At first the logical symbols are correctly extracted from the context
and the logical expressions are identified based on them considering logical
negative and conditional relationships. Then we extend the logical expressions
by inferring implicit ones in the context. For each option we recognize its
logical expression and find the related extended expressions. We verbalize
them into natural language to feed into pre-trained model as an extended
context to compute a matching score. Finally we take option C which exactly
matches an extended implicit logical expression as the most plausible answer.

###  5.6 Error Analysis

Reasoning Type | Base | Ours
---|---|---
Necessary Assumptions (11.0%) | 73.7 |  76.3 ($\uparrow$)
Sufficient Assumptions (3.6%) | 70.0 |  70.0 ($-$)
Strengthen (9.0%) | 69.1 |  70.2 ($\uparrow$)
Weaken (10.6%) | 64.6 |  59.3 ($\downarrow$)
Evaluation (1.6%) | 69.2 |  69.2 ($-$)
Implication (6.2%) | 43.8 |  54.3 ($\uparrow$)
Conclusion/Main Point (3.1%) | 80.6 |  77.8 ($\downarrow$)
Most Strongly Supported (6.7%) | 58.9 |  71.4 ($\uparrow$)
Explain or Resolve (8.0%) | 60.7 |  67.9 ($\uparrow$)
Principle (5.7%) | 72.3 |  76.9 ($\uparrow$)
Dispute (2.5%) | 63.3 |  80.0 ($\uparrow$)
Technique (3.8%) | 75.0 |  80.6 ($\uparrow$)
Role (3.7%) | 78.1 |  68.8 ($\downarrow$)
Identify a Flaw (11.3%) | 65.0 |  71.8 ($\uparrow$)
Match Flaws (4.9%) | 61.3 |  61.3 ($-$)
Match the Structure (2.7%) | 56.7 |  86.7 ($\uparrow$)
Others (5.5%) | 68.5 |  72.6 ($\uparrow$)
Table 5: Detailed results on different logical reasoning types. Numbers in
parentheses are percentages of different reasoning types. Base is the _ALBERT_
model while Ours means our _$\text{LReasoner}_{\text{ALBERT}}$_ system.
$\uparrow$, $\downarrow$ and $-$ respectively mean that our performance is
better, worse than and equal to the baseline model.

Although our system achieves the best performance, there still exists some
instances that can not be solved. ReClor dataset integrates various logical
reasoning skills which can be categorized into 17 types. We can investigate
the detailed performance with respect to different logical reasoning types of
our system _$\text{LReasoner}_{\text{ALBERT}}$_ and the baseline model
_ALBERT_ , to analyse which type of questions tend to be more challenging.

As shown in Table 5, we can see that all models perform relative poorly on
certain logical reasoning types, such as Most Strongly Supported and
Implication. The first type is less straight-forward and requires deeper
logical reasoning considering degree. Specifically, Most Strongly Supported
aims to find the choice that is most strongly supported by a stimulus. For
Implication, all candidate options prefer to be more similar to each other or
the context which are more difficult to be classified, like the examples in
Figure 1 and 4. But our system is still able to make progress on them which
prove the effectiveness of our system for logical reasoning.

Our model achieves great improvement in most of the logical reasoning types
compared to the baseline model but performs comparably or even worse in
following question types: Sufficient Assumption, Evaluation, Conclusion/Main
Point, Role, Match flaws and Weaken. The first four types only take up a small
proportion of the whole data, which results in no improvement over them. Match
flaws needs to understand more abstract concept flaw and find an option
exhibiting the same flaw as the context. Weaken aims to find the opposite
statement that weaken the argument. Our system to first extract logical
expressions and then imply the implicit logical expressions is not suitable
for matching flaw and identifying weaken statement. How to deal with different
logical reasoning types is our further research direction.

##  6 Related Work

In recent years, an increasing number of tasks and datasets have been
introduced targeting on logical reasoning of text in NLP area. Natural
Language Inference (NLI) Dagan et al. (2005) is a typical task requiring
logical reasoning, which aims to determine whether a hypothesis can reasonably
be entailed from a premise. SNLI Bowman et al. (2015), MultiNLI Williams et
al. (2018), QNLI Wang et al. (2018) and SciTail Khot et al. (2018) are widely
used benchmarks in evaluating the performance of NLI. However, these datasets
handle the task at sentence-level and the required logical reasoning ability
is simple. Liu et al. (2021) propose ConTRoL dataset investigating NLI for
long texts and examining more complex contextual reasoning types. Argument
reasoning comprehension Habernal et al. (2018) is a task closer to NLI which
also concerns passage-level logical reasoning. Given a premise and a claim, it
aims to identify the correct implicit warrant from two opposing options. But
it is limited to only one logical reasoning type, i.e., warrant
identification.

Another task involving logical reasoning of text is question answering.
Several multiple-choice question answering datasets, which need to select an
answer from candidate options given a context and a question, have been
proposed for promoting the development of logical reasoning. These datasets
are all sourced from public standardized exams so that they partly mitigate
the noise brought by crowd-sourcing or automatically-generation Lai et al.
(2017). LogiQA Liu et al. (2020) is collected from National Civil Servants
Examination of China which covers 5 types of logical reasoning. Yu et al.
(2020) propose ReClor dataset from the GMAT and LSAT tests. It examines more
complicated logical reasoning integrating 17 question types. In this paper, we
conduct experiments on top of ReClor for investigating diverse logical
reasoning skills.

Large-scale pre-trained language models Devlin et al. (2019); Liu et al.
(2019); Yang et al. (2019); Lan et al. (2020) have been widely adopted for
logical reasoning of text, which directly encode the given texts to predict
the output. Without understanding symbolic logic, they can still achieve a
promising performance. We therefore take a pre-trained model as our backbone
model and identify logical expressions for our proposed logic-driven system .

##  7 Conclusion and Future Work

In this paper, we focus on the task of logical reasoning of text and propose
to understand the logical symbols and expressions in the context to find the
answer. Taking them as elementary components of logical inference, we first
propose a logic-driven context extension framework to extend the logical
expressions to cover the implicit ones and verbalize them as an extended
context to match the answer. We also introduce a logic-driven data
augmentation algorithm which employs contrastive learning to augment literally
similar but logically different instances to help our model better capture
logical information, especially logical negative and conditional
relationships. Experimental results on ReClor dataset confirm the
effectiveness of our logic-driven reasoner with both logic-driven context
extension framework and data augmentation algorithm.

In the future, we will explore how to take different logical reasoning types
into consideration to deal with the question types that still can not be well
solved. We also would like to design a model to directly encode the symbolic
logic rather than utilize them by verbalizing into natural language. Besides,
we plan to evaluate our model on more datasets. As our codes are publicly
available, anyone interested could also have a try.

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# A Study of the Soft X-ray Emission Lines in NGC 4151
I. Kinematic Properties of the Plasma Wind

S. Grafton-Waters Mullard Space Science Laboratory, University College London,
Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK  M. Ahmed  S. Henson  F. Hinds-
Williams  B Ivanova  E. Marshall  H. Udueni  D. Theodorakis  Nottingham
University Academy of Science and Technology, 93 Abbey St, Lenton, Nottingham
NG7 2PL  W. Dunn Mullard Space Science Laboratory, University College London,
Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK

###### Abstract

We present our analysis of the narrow emission lines produced in the plasma
regions within the bright active galactic nucleus of NGC 4151, from an ORBYTS
research-with-schools public engagement project. Our goal was to test whether
the properties of these plasma regions changed between XMM-Newton observations
spanning 15 years from 2000 to 2015, by measuring the outflow velocities and
distances. From this study, we found that NGC 4151 has at least two to three
plasma regions. There is no evidence of the outflowing wind properties
changing as the velocities and distances are consistent throughout the
observations.

##  1 Introduction

NGC 4151 (z = 0.003262) is a type 1.5 Seyfert active galactic nucleus (AGN)
with a black hole (BH) mass of $M_{BH}=3.59\times 10^{7}$ M$\odot$ (Bentz &
Katz, 2015). While NGC 4151 is highly variable in the hard X-ray band (e.g.
Beuchert et al., 2017), the soft X-ray spectrum is very similar to that of NGC
1068 with multiple strong emission features on top of an almost negligible
continuum (e.g. Grafton-Waters, 2021a). This extreme contrast between soft and
hard X-ray bands makes NGC 4151 an ideal target to study the properties of the
narrow line region (NLR). The NLR is part of the outflowing wind, composed of
separate plasma regions, that produces the emission lines observed in the
X-ray spectra (see Figure 1). The aim of this ORBYTS research-with-schools
public engagement was to assess whether the NLR properties changed between
observations spanning a 15 year period, by measuring the outflow velocities
and distances of each region. The code can be found at Grafton-Waters (2021b).

Figure 1: (a) RGS spectrum (grey line) with the Gaussian models (red lines)
fitted to the strongest emission lines. The lines we investigated are labelled
with their respective ions. The r, i and f labels stand for resonance,
intercombination and forbidden, respectively, corresponding to electron
transitions in the He-like triplet ions (see e.g. Porquet & Dubau, 2000, for
details). The insert shows the OVII f line fitted with its respective
Gaussian. (b-c) Measured line velocities using Eq. 2 and (d-e) calculated
distances from Eq. 3 are compared for 2000 (left) and 2015 (right). Colours
indicate different observations from each epoch, as labelled in the legends,
and the black dashed lines signify the boundaries between plasma regions.

##  2 Method

NGC 4151 was observed 27 times by XMM-Newton during six separate epochs
between 2000 and 2015. Here we analysed the reflection grating spectrometer
(RGS; den Herder et al., 2001) data from each observation. The data were
reduced using the RGSPROC command in the SAS software v 17.0.0111See
https://www.cosmos.esa.int/web/xmm-newton/sas-thread-rgs. Any large background
counts in CCD9 were removed before combining RGS1 and RGS2 data with
RGSCOMBINE.

We modelled the thirteen strongest emission lines (labelled in Figure 1) that
were present in all X-ray spectra with a simple Gaussian model in Python,
defined as

$G(x)=\frac{A}{\sqrt{2\pi\sigma^{2}}}\exp{\left[-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right]},$
(1)

where A is the amplitude, $\sigma$ is the standard deviation, and $\mu$ is the
mean value (the emission line centre). To measure the properties of each
emission line we defined a wavelength range ($\lambda_{R}$) to ensure we were
only fitting one line at a time. This is shown in the insert in Figure 1 (a).
This Gaussian model and the X-ray spectrum were then fed into the LMFIT
interface in Python (Newville et al., 2014), which returned the parameter
values ($\mu$, $\sigma$, and A) and errors.

To estimate the outflow velocity ($v_{out}$) of the plasma regions, we used
the redshift ($z$) equation, given by

$z=\frac{v_{out}}{c}=\frac{\lambda_{obs}-\lambda_{rest}}{\lambda_{rest}},$ (2)

where c is the speed of light, and $\lambda_{obs}$ and $\lambda_{rest}$ are
the observed and rest frame wavelengths, respectively. The velocity shift was
measured for each emission line individually. The observed wavelength ($\mu$)
was measured from our modelling and the rest wavelengths were obtained from
the SPEX line
list222https://personal.sron.nl/~kaastra/leiden2020/line_new.pdf. We note that
it is not the emission line that has the velocity, but instead it is a
property of the line-emitting plasma region. If many of the emission lines had
similar values they were likely to originate from the same plasma region
within the outflowing wind. To estimate the distances ($R$) from the BH we
assumed that $v_{out}$ was greater than or equal to the escape velocity
($v_{esc}$) of the BH, given by

$R=\frac{2GM_{BH}}{v^{2}_{esc}},$ (3)

where G is the gravitational constant, and $M_{BH}$ is the BH mass.

After modelling each emission line, and obtaining the velocities and distances
from each observation, we compared results. This allowed us to evaluate
whether the wind properties changed over time.

##  3 Results

Figure 1 (a) shows the 2000 RGS spectrum with the Gaussian models fitted to
the strongest emission lines; these lines are labelled. The insert in Figure 1
(a) displays the OVII forbidden line at 22.15 Å with the Gaussian model fitted
on top. From the spectral modelling of all the observations, we compare the
results for the velocities and distances in 2000 (left) and 2015 (right) in
Figure 1 (b-c) and (d-e), respectively. The black dashed lines in Figure 1
indicate the plasma boundaries for $v_{out}$ and $R$, suggesting that there
are at least two to three plasma regions. However, we cannot rule out the
possibility of more. The results for 2000 and 2015 are consistent with the
velocities and distance from all observations.

Our aim was to see whether the outflowing wind properties changed over the
course of 15 years. However, based on this investigation, we found no evidence
of this. One may expect that from 2000 to 2015, the distances and velocities
would have increased and decreased, respectively, as the plasma regions move
away from the BH. However, assuming the plasma is travelling at 500 km s-1,
the distance travelled in a 15 year time period would only be 0.008 pc, which
is insignificant compared to the distances that we measured (i.e. $>1$ pc).
One explanation could be that we are viewing the plasma perpendicular to the
outflow direction, meaning we are unable to observe the true outflow velocity,
since the Doppler shifting of lines would only represent a component of its
true outflow velocity. Alternatively, the plasma has been relatively static
over the 15 years, which is reasonable given the large distances from the BH.
In that case, 15 years may simply not be enough time for us to observe any
significant changes with regards to the outflowing wind in NGC 4151.

We did have some problems when modelling the data, which could have affected
the final results. For example, the line centre ($\mu$) depended on
$\lambda_{R}$ which we set when modelling the lines. This meant that if
$\lambda_{R}$ was too large, then multiple lines would be fitted together by
the model, or if $\lambda_{R}$ was too small we would lose information about
the feature. This was a particular problem if $\lambda_{R}$ for an emission
line differed between observations, especially if the S/N ratios were poor. As
a result, a slightly different velocity was obtained each time.

##  4 Conclusions

From our RGS analysis on NGC 4151, we found that there are at least two to
three plasma regions that produce the emission features in the outflowing
wind. However, the velocities and distances are consistent throughout the
observations. This implies that either the plasma regions have been relatively
static over the 15 year observation period, or because we are seeing the
outflowing wind perpendicular to the flow direction, we are not measuring the
velocities in the direction of motion.

This work was undertaken with the ORBYTS Research-with-schools public
engagement project, partnering scientists with schools to support students’
involvement in space research. S.G.W. acknowledges the support of a PhD
studentship awarded by the UK Science & Technology Facilities Council (STFC).

## References

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  * Grafton-Waters (2021b) Grafton-Waters, S., 2021b, sgwxray/ORBYTS Python Code, v1.0.0, Zenodo, doi: 10.5281/zenodo.5116838
  * Grafton-Waters (2021a) Grafton-Waters, S., et al. 2021a, A&A, 649, A162, doi: 10.1051/0004-6361/202039022
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