# Proper-time method for unequal masses

A. A. Osipov111Email address<EMAIL_ADDRESS>Joint Institute for Nuclear
Research, Bogoliubov Laboratory of Theoretical Physics, 141980 Dubna, Russia

###### Abstract

The result of removing of heavy non-equal mass particles from the theory can
be described, at low energy, by the effective action, which is a series in
inverse-square powers of the mass. We propose a new efficient tool to
calculate the leading terms of this series based on the Schwinger proper-time
method. Unequal masses give rise to a large number of effective vertices
describing the explicit flavour symmetry breaking effects with well-defined
coupling constants. Our method is pertinent to the theory with explicit and
spontaneous chiral symmetry breaking, chiral gauge theory, standard and beyond
standard model effective field theory, the theory of critical phenomena,
cosmology, etc.

Keywords: proper-time method, heat kernel, effective field theory, effective
action, flavor symmetry breaking, unequal masses.

##  I Introduction

The proper-time method Fock:37a ; Fock:37b ; Nambu:50 ; Schwinger:51  is an
efficient tool to study quantum corrections in many areas of theoretical
physics: in quantum gravity DeWitt:65 ; DeWitt:75 ; Barvinsky:87 ;
Barvinsky:90 , in QCD (effective meson Lagrangians Wadia:85 ; Ebert:86 ), in
chiral gauge theories Ball:89 , in cosmology Callan:94 , in QED (Casimir
energies and forces Bordag:01 ), etc. That is why it remains in the focus of
researchers working in quantum physics for decades Vassilevich:03 .

Here I propose a new proper-time based algorithm to derive the effective
action in the theory with heavy virtual fermions (or bosons) of unequal masses
belonging to some representation of the symmetry group $G$. The result is the
inverse mass series for the one-loop effective action, where I explicitly
calculate the two leading contributions. This powerful tool opens a promising
avenue for studying explicit flavor symmetry breaking effects in many
effective field theories. It provides the easiest way to address a problem
that otherwise becomes rather cumbersome.

To generalize the standard large mass expansion of the heat kernel to the case
of unequal masses, I suggest using the formula

$e^{-t(M^{2}+A)}=e^{-tM^{2}}\left[1+\sum_{n=1}^{\infty}(-1)^{n}f_{n}(t,A)\right],$
(1)

where $M=\mbox{diag}(m_{1},m_{2},\ldots,m_{f})$ is the diagonal mass matrix;
$t$ is the proper-time parameter; the expression in the square brackets is the
time-ordered exponential $\mbox{OE}[-A](t)$ of $A(s)=e^{sM^{2}}Ae^{-sM^{2}}$,
and $A$ is a positive definite self-adjoint elliptic operator in some
background (its explicit form will be clarified later), accordingly

$f_{n}(t,A)=\int\limits_{0}^{t}\\!\\!ds_{1}\\!\\!\int\limits_{0}^{s_{1}}\\!\\!ds_{2}\ldots\\!\\!\\!\\!\int\limits_{0}^{s_{n-1}}\\!\\!\\!\\!ds_{n}A(s_{1})A(s_{2})\ldots
A(s_{n}).$ (2)

If masses are equal, this formula yields the well-known large mass expansion
with standard Seeley-DeWitt coefficients $a_{n}(x,y)$ Ball:89 . These
coefficients are polynomials in background fields and describe, in the
coincidence limit $y\to x$, local vertices of the induced effective
Lagrangian. In fact, formula (1) is an extension of the Schwinger’s method
used to isolate the infinities contained in the real part of the one-loop
action Schwinger:51 ; DeWitt:75  to the non-commutative algebra.

There is a simple heuristic argument that explains why this formula is also
relevant for describing the generalized $1/M$ series. Indeed, the $1/M$
expansion is known to be valid when all background fields and their
derivatives are small compared to the mass of quantum fields. Therefore,
factoring $e^{-tM^{2}}$ one separates the leading contribution. The remaining
part of the heat kernel may be unambiguously evaluated by expanding it in a
power series in $t$ about $t=0$. As a consequence, the Seeley-DeWitt
coefficients $a_{n}$ receive corrections $a_{n}\to b_{n}=a_{n}+\Delta a_{n}$,
where $\Delta a_{n}$ vanish in the limit of equal masses.

Currently, there are two methods for deriving quantum corrections induced by
virtual states of unequal masses. In Min:82 ; Min:89a ; Min:89b , the heat
kernel is evaluated on the bases of the modified DeWitt WKB form. This yields
a different asymptotic series for the right-hand side of Eq.(1), and,
consequently, the different expressions for $\Delta a_{n}$. The approach
proposed in Osipov:01a ; Osipov:01b ; Osipov:01c  starts from the formula (1),
but afterwards an additional resummation of the asymptotic series is applied.
This essentially simplifies the calculations, but changes the structure of the
$1/M$ series. As a result, one looses correspondence between a mass-dependent
factor at the effective vertex and a flavor content of the one-loop Feynman
diagram which generates the vertex. Here I abandon this procedure.

The proper-time method reduces the task of the large mass expansion to a
simple algebraic problem which requires much less work than one needs for the
corresponding Feynman diagrams calculation in momentum space. In the
following, we consider a quite nontrivial case of the chiral $U(3)\times U(3)$
symmetry broken by the diagonal mass matrix $M=\mbox{diag}(m_{1},m_{2},m_{3})$
to demonstrate the power of the method. To find the two leading contributions
$b_{1}$ and $b_{2}$ in the $1/M$ expansion of the effective action one
requires to consider only four terms of the series (1) that results in more
than a hundred effective vertices.

The effects of flavor symmetry breaking are currently important in many
physical applications: in studies of physics beyond standard model to
construct the low energy effective action by integrating out the heavy degrees
of freedom Passarino:19 ; Bizot:18 ; in two Higgs doublet models Branco:12  to
address the problem of almost degenerate Higgs states at $125\mbox{GeV}$
Bian:18 ; Haber:19 ; in the low energy QCD to study the $SU(3)$ and isospin
symmetry breaking Taron:97 . These effects are very important to address the
QCD phase diagram Osipov:13 , to study a formation of the strange-quark matter
Witten:84 ; Osipov:15 , to study nuclear matter in extreme conditions that
arose in nature at the early stages of the evolution of the Universe and in
the depths of neutron stars Chernodub:11 ; Gatto:11 .

##  II Proper-time expansion

We shall be mainly concerned here with the asymptotic behavior of the
following object in Euclidean four-dimensional space

$W_{E}=-\ln|\det
D_{E}|=\\!\\!\int\limits^{\infty}_{0}\\!\frac{dt}{2t}\,\rho_{t,\Lambda}\,\mbox{Tr}\left(e^{-tD_{E}^{\dagger}D_{E}}\right),$
(3)

It represents the real part of the one-quark-loop contribution to the meson
effective action in form of the proper-time integral. The integral diverges at
the lower limit, therefore, a regulator $\rho_{t,\Lambda}$ is introduced,
where $\Lambda$ is a cutoff. The Dirac operator in Euclidean space, $D_{E}$,
has the form

$D_{E}=i\gamma_{\alpha}d_{\alpha}-M+s+i\gamma_{5}p,$ (4)

where $d_{\alpha}=\partial_{\alpha}+i\Gamma_{\alpha}$,
$\Gamma_{\alpha}=v_{\alpha}+\gamma_{5}a_{\alpha}$, $\alpha=1,2,3,4$. The
external scalar $s$, pseudoscalar $p$, vector $v_{\alpha}$ and axial-vector
$a_{\alpha}$ fields are embedded in the flavor space through the set of
matrices $\lambda_{a}=(\lambda_{0},\lambda_{i})$, where
$\lambda_{0}=\sqrt{2/3}$ and $\lambda_{i}$ are the eight $SU(3)$ Gell-Mann
matrices; for instance, $s=s_{a}\lambda_{a}$, etc. for all fields. The quark
masses are given by the diagonal matrix $M=\mbox{diag}(m_{1},m_{2},m_{3})$ in
the flavor space. The symbol ”Tr” denotes the trace over Dirac $(D)$
$\gamma$-matrices, colour $(c)$ $SU(3)$ matrices, and flavor $(f)$ matrices,
as well as integration over coordinates of the Euclidean space:
$\mbox{Tr}\equiv\mbox{tr}_{I}\int\\!d^{4}x$, where $I=(D,c,f)$. The trace in
the color space is trivial: it leads to the overall factor $N_{c}=3$. The
dependence on external fields in $D_{E}$ after switching to the Hermitian
operator

$D_{E}^{\dagger}D_{E}=M^{2}-d_{\alpha}^{2}+Y$ (5)

is collected in $Y$ and the covariant derivative $d_{\alpha}$. In the
following we do not need the explicit form of $Y$.

To advance in the evaluation of expression (3), we use the Schwinger technique
of a fictitious Hilbert space Schwinger:51 . It allows the effective action to
be represented as an integral over the 4-momenta $k_{\alpha}$

$W_{E}=\\!\\!\int\\!\\!d^{4}x\\!\\!\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}}\,e^{-k^{2}}\\!\\!\\!\int\limits^{\infty}_{0}\\!\\!\frac{dt}{2t^{3}}\,\rho_{t,\Lambda}\,\mbox{tr}_{I}\left[e^{-t(M^{2}+A)}\right]\\!,$
(6)

where $A=-d_{\alpha}^{2}-2ik\partial/\sqrt{t}+Y$. Since $A$ contains open
derivatives with respect to coordinates, it is still necessary to clarify the
meaning of $\mbox{tr}_{I}$. The space $x\in{\bf R}^{4}$ has no boundaries,
therefore we can integrate by parts. However, this operation will be
unambiguous only if $\mbox{tr}_{I}$ does not change when its $x$-dependent
elements are cyclically rearranged. This property can be ensured by the
explicit symmetrization, i.e., $\mbox{tr}_{I}$ in such cases is understood as

$\mbox{str}_{I}\,(A_{1}A_{2}\ldots
A_{n})=\frac{1}{n}\sum_{cycl.perm.}\\!\\!\\!\\!\mbox{tr}_{I}\,(A_{1}A_{2}\ldots
A_{n})\,.$ (7)

To distinguish the leading contributions in the expansion in $1/M^{2}$, it
suffices to take into account terms of at most $t^{4}$ order in (1), which
under the flavor trace have the form

$\displaystyle\mbox{tr}_{f}\left[e^{-t(M^{2}+A)}\right]=\sum_{i=1}^{3}c_{i}(t)-t\sum_{i=1}^{3}c_{i}(t)\,\mbox{tr}_{f}\,A_{i}$
(8) $\displaystyle+$
$\displaystyle\frac{t^{2}}{2!}\sum_{i,j}c_{ij}(t)\,\mbox{tr}_{f}\,(A_{i}A_{j})-\frac{t^{3}}{3!}\sum_{i,j,k}c_{ijk}(t)\,\mbox{tr}_{f}\,(A_{i}A_{j}A_{k})$
$\displaystyle+$
$\displaystyle\frac{t^{4}}{4!}\sum_{i,j,k,l}c_{ijkl}(t)\,\mbox{tr}_{f}\,(A_{i}A_{j}A_{k}A_{l})+{\cal
O}(t^{5})\,.$

Here $A_{i}=E_{i}A$, where $3\times 3$ matrices
$(E_{i})_{nm}=\delta_{in}\delta_{im}$; the coefficients $c_{i_{1}i_{2}\ldots
i_{n}}(t)$ are completely symmetric with respect to any permutation of indices
and are well known Osipov:01a ; Osipov:01b ; Osipov:01c . They depend on quark
masses and proper-time. In the case of equal masses, all coefficients
coincide, that is,
$c_{i}(t)=c_{ii}(t)=c_{iii}(t)=c_{iiii}(t)=e^{-tm_{i}^{2}}$. It also follows
from the formula (8) that the number of indices in the coefficient
$c_{i_{1}i_{2}\ldots i_{n}}$ is equal to the number of operators $A_{i}$,
representing external fields. Hence, there is a relation between the number of
indices in $c_{i_{1}i_{2}\ldots i_{n}}$ and the contribution of the one-loop
diagram with $n$ external legs. The formula (14) will clarify this
correspondence.

Substituting (8) into (6) and integrating over momenta $k_{\alpha}$ and
proper-time $t$, we arrive at the expression

$W_{E}=\frac{N_{c}}{32\pi^{2}}\\!\\!\int\\!\\!d^{4}x\sum_{n=0}^{\infty}\,\mbox{tr}_{Df}\,b_{n}(x,x),$
(9)

where coefficients $b_{n}(x,x)$ depend on the external fields and quark
masses, i.e., they contain information about both the effective meson vertices
and corresponding coupling constants. If all masses are equal, the dependence
on $m=m_{i}$ is factorized in form of the integral Osipov:01b

$J_{n}(m_{i}^{2})=\int\limits^{\infty}_{0}\frac{dt}{t^{2-n}}\,\rho_{t,\Lambda}\,c_{i}(t)$
(10)

and the field-dependent part takes a standard Seeley-DeWitt form $a_{n}(x,x)$.
For large masses $m$, the coefficients $b_{n}(x,x)$ exhibit the same
asymptotic behavior as $J_{n-1}(m^{2})$, i.e., $b_{n}\sim m^{-2(n-2)}$.

##  III Leading coefficients

Consider the leading terms $b_{1}$ and $b_{2}$. The case $n=0$ is of no
interest because $b_{0}$ contains no fields and can be omitted from the
effective action. The coefficients with $n\geq 3$ tend to zero in the limit of
infinite masses, therefore they are small in comparison with $b_{1}$ and
$b_{2}$.

For convenience of writing the result of our calculations, along with the
usual matrix multiplication, we will use the non-standard Hadamard product
Styan:73 , which is the matrix of elementwise products

$(A\circ B)_{ij}=A_{ij}B_{ij}.$ (11)

The Hadamard product is commutative unlike regular matrix multiplication, but
the distributive and associative properties are retained. It has previously
been proven to be an useful tool when non-degenerate mass matrices and non
trivial flavor symmetry contractions were involved Morais:17 .

Now, the heat coefficient $b_{1}(x,x)$ can be written as

$b_{1}=-J_{0}\circ Y,$ (12)

where $(J_{0})_{ij}=\delta_{ij}J_{0}(m_{i}^{2})$ is a diagonal matrix with
elements given by (10) (for $n=0$). This matrix contains contributions of the
Feynman one-loop diagram, known as a ”tadpole”. The regularization should be
chosen in accord with the problem studied, for the NJL model this can be
$\rho_{t,\Lambda}=1-(1+t\Lambda^{2})e^{-t\Lambda^{2}}$.

Consider the second coefficient $b_{2}(x,x)$. After some tiring calculations
we can represent the result in the form

$b_{2}=\frac{1}{2}\,Y\left(J\circ
Y\right)-\frac{1}{12}\Gamma^{\alpha\beta}\left(J\circ\Gamma^{\alpha\beta}\right)+\Delta
b_{2},$ (13)

where the antisymmetric tensor
$\Gamma_{\alpha\beta}=F_{\alpha\beta}+i[\Gamma_{\alpha},\Gamma_{\beta}]$,
$F_{\alpha\beta}=\partial_{\alpha}\Gamma_{\beta}-\partial_{\beta}\Gamma_{\alpha}$,
and $J_{ij}=J(m_{i},m_{j})$ is a symmetric $3\times 3$ matrix, whose elements
are logarithmically divergent parts (at $\Lambda\to\infty$) of Feynman self-
energy diagrams with masses of virtual particles $m_{i}$ and $m_{j}$. Since we
also need expressions for the similar contributions coming from triangular
$J_{ijk}$ and box $J_{iijk}$ diagrams, we collect them together in one formula

$\left(\begin{array}[]{c}J_{ij}\\\ J_{ijk}\\\ J_{iijk}\\\
\end{array}\right)=\\!\\!\int\limits^{\infty}_{0}\frac{dt}{t}\,\rho_{t,\Lambda}\,\left(\begin{array}[]{c}c_{ij}(t)\\\
c_{ijk}(t)\\\ c_{iijk}(t)\\\ \end{array}\right).$ (14)

In the theory under consideration, there are only three flavors of quarks,
therefore in $J_{iijk}$ at least two indices coincide. From the known
properties of $c_{i_{1}i_{2}\ldots i_{n}}(t)$ one can deduce that
$J_{ii}=J_{iii}=J_{iiii}$. The integrals take this form when the quark masses
are equal. Their coincidence is an important property which is used to
demonstrate that $\Delta b_{2}$ vanishes in the equal-mass limit.

In the limit of equal masses, the first two terms in (13) yield the well-known
result Ball:89 , and the third one vanishes, that is, it contains only
contributions associated with an explicit violation of chiral symmetry.
Without this term, the description of exact breaking of flavor symmetry is
incomplete. To write an expression for $\Delta b_{2}$, let us consider the
different contributions in

$\Delta b_{2}=\Omega_{Y}+\sum_{n=0}^{2}\Omega^{(n)},$ (15)

where $\Omega_{Y}$ is the sum of all terms linear in $Y$. $\Omega^{(n)}$ is
the sum of terms with $n$ derivatives which consists of only spin-one fields.

For $\Omega_{Y}$ we have

$\Omega_{Y}=\mathcal{A}Y+\frac{i}{3}(R\circ\Gamma^{\alpha})\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\partial^{\alpha}}}\\!Y,$
(16)

where $R$ and $\mathcal{A}$ are $3\times 3$ matrices with elements

$\displaystyle R_{ij}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(J_{iij}-J_{jji}\right),$ (17)
$\displaystyle\mathcal{A}_{ii}$ $\displaystyle=$ $\displaystyle\sum_{j\neq
i}\left(J_{ii}-J_{ij}+R_{ij}\right)\Gamma^{\alpha}_{ij}\Gamma^{\alpha}_{ji}\,,$
$\displaystyle\mathcal{A}_{ij}$ $\displaystyle=$
$\displaystyle\left[(J_{ij}-J_{ijk})\Gamma^{\alpha}_{ik}\Gamma^{\alpha}_{kj}\right.$
(18) $\displaystyle-$
$\displaystyle\left.R_{ij}\left(\Gamma^{\alpha}_{ij}\Gamma^{\alpha}_{jj}-\Gamma^{\alpha}_{ii}\Gamma^{\alpha}_{ij}\right)\right]_{\mid
i\neq j\neq k}\,,$

and the left-right derivative is defined by the difference
$Y\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\partial^{\alpha}}}\\!\Gamma^{\alpha}=Y\partial^{\alpha}\Gamma^{\alpha}-(\partial^{\alpha}Y)\Gamma^{\alpha}$.
To avoid misunderstandings, we emphasize that repeated flavor indices do not
imply summation over them. Here and below, summation is carried out only if
the summation symbol is explicitly written.

Let us consider the terms with two derivatives

$\Omega^{(2)}=\frac{1}{24}\left[F^{\alpha\beta}\left(T\circ
F^{\alpha\beta}\right)+6(\partial\Gamma)\left(T\circ\partial\Gamma\right)\right].$
(19)

The first term makes an additional contribution to the kinetic part of the
effective Lagrangian of spin-1 fields described by the second term in (13).
The symmetric matrix $T$ is defined by $T_{ij}=J_{ij}-J_{iijj}$. The shorthand
$\partial\Gamma\equiv\partial^{\alpha}\Gamma^{\alpha}$ implies summation over
omitted indices $\alpha$. In applications, one can omit the second term in
(19) to ensure that the energy of the massive spin-1 field is positive
definite.

The terms with one derivative can be collected in the expression

$\displaystyle\Omega^{(1)}$ $\displaystyle=$
$\displaystyle\frac{i}{3}C^{\alpha\beta}F^{\alpha\beta}+\frac{i}{6}\,\delta_{\alpha\beta\gamma\sigma}(K\circ\partial^{\alpha}\Gamma^{\beta})E^{\gamma\sigma}$
(20) $\displaystyle+$
$\displaystyle\frac{3i}{4}\left(T\circ\partial^{\alpha}\Gamma^{\beta}\right)L^{\beta\alpha}_{-},$

which represents the effective three-particle vertices describing the local
interaction of vector and axial-vector fields. Symbol
$\delta_{\alpha\beta\gamma\sigma}=\delta_{\alpha\beta}\delta_{\gamma\sigma}+\delta_{\alpha\gamma}\delta_{\beta\sigma}+\delta_{\alpha\sigma}\delta_{\beta\gamma}$
is a totally symmetric tensor composed of the product of two Kronecker
symbols. The diagonal and off-diagonal elements of the matrix
$C^{\alpha\beta}$ are, respectively, of the form

$\displaystyle C^{\alpha\beta}_{ii}$ $\displaystyle=$
$\displaystyle\sum_{j\neq
i}\left(J_{ii}-J_{ij}+\frac{1}{4}\,R_{ij}+\frac{3}{4}\,T_{ij}\right)\Gamma^{\alpha}_{ij}\Gamma^{\beta}_{ji}\,,$
$\displaystyle C^{\alpha\beta}_{ij}$ $\displaystyle=$
$\displaystyle(J_{ij}-J_{123})\Gamma^{\alpha}_{ik}\Gamma^{\beta}_{kj}|_{i\neq
j\neq k}+\frac{1}{4}\,R_{ij}\left(L^{\alpha\beta}_{+}\right)_{ij}$ (21)
$\displaystyle+$
$\displaystyle\frac{3}{4}\,T_{ij}\left(L^{\alpha\beta}_{-}\right)_{ij},$

where we use the notation

$\left(L^{\alpha\beta}_{\pm}\right)_{ij}=\left(\Gamma^{\alpha}_{ii}\pm\Gamma^{\alpha}_{jj}\right)\Gamma^{\beta}_{ij}.$
(22)

Note that elements of the matrix $\Gamma^{\alpha}_{ij}$ commute with each
other. The antisymmetric matrix $K$ is defined by the expression

$K_{ij}=(J_{jjik}-J_{iijk})_{\mid i\neq j\neq k}\,.$ (23)

The tensor $E^{\gamma\sigma}$ has no diagonal elements too, although it is not
antisymmetric

$E^{\gamma\sigma}_{ii}=0,\quad
E^{\gamma\sigma}_{ij}=\Gamma^{\gamma}_{ik}\Gamma^{\sigma}_{kj},\,\,(i\neq
j\neq k).$ (24)

The last thing left to consider is the term without derivatives
$\Omega^{(0)}$, which is a sum of the terms $\propto\Gamma^{4}$. Since there
are many of those, it is convenient to present the result in a form where the
trace over flavor indices is already calculated. Then we have

$\displaystyle\mbox{tr}_{f}\Omega^{(0)}=\sum_{i<j}\left\\{\left[\frac{2}{3}(J_{ii}+J_{jj})-\frac{4}{3}J_{ij}\right]\left(\Gamma_{ij}\Gamma_{ji}\right)^{2}+\frac{1}{6}(2J_{ij}-J_{ii}-J_{jj})\left(\Gamma_{ij}\Gamma_{ij}\right)\left(\Gamma_{ji}\Gamma_{ji}\right)\right\\}$
$\displaystyle+\sum_{i\neq j\neq k,\atop
j<k}\\!\\!\left\\{\left[\frac{1}{3}\,(J_{jk}+2J_{iijk})+J_{ii}-J_{ij}-J_{ik}\right]\left(\Gamma_{ij}\Gamma_{ji}\right)\left(\Gamma_{ik}\Gamma_{ki}\right)+\left[\frac{1}{3}\,(J_{ii}+2J_{iijk})+J_{jk}-2J_{ijk}\right]\left(\Gamma_{ji}\Gamma_{ik}\right)\left(\Gamma_{ki}\Gamma_{ij}\right)\right.$
$\displaystyle+\frac{1}{3}(2J_{iijk}-J_{ij}-J_{ik})\left[\left(\Gamma_{ii}\Gamma_{jk}\right)\left(\Gamma_{ij}\Gamma_{ki}\right)+\left(\Gamma_{ik}\Gamma_{ji}\right)\left(\Gamma_{kj}\Gamma_{ii}\right)\right]+\left.\frac{1}{3}(2J_{iijk}-J_{ii}-J_{jk})\left(\Gamma_{ij}\Gamma_{ik}\right)\left(\Gamma_{ki}\Gamma_{ji}\right)\right\\}$
$\displaystyle+\\!\\!\sum_{i\neq j\neq
k}\\!\\!\left\\{\frac{1}{3}(J_{ik}+2J_{iijk}-J_{ijk})\left[\left(\Gamma_{ii}\Gamma_{ij}\right)\left(\Gamma_{jk}\Gamma_{ki}\right)+\left(\Gamma_{ik}\Gamma_{kj}\right)\left(\Gamma_{ji}\Gamma_{ii}\right)\right]+R_{ij}\left[\left(L^{\alpha\alpha}_{-}\right)_{ij}\left(\Gamma_{jk}\Gamma_{ki}\right)+\left(\Gamma_{ij}\Gamma_{ji}\right)\left(\Gamma_{ik}\Gamma_{ki}\right)\right]\right\\}$
$\displaystyle+\frac{1}{3}\delta_{\alpha\beta\gamma\sigma}\sum_{i\neq
j}T_{ij}\left(\Gamma^{\alpha}_{ii}\Gamma^{\beta}_{ii}\Gamma^{\gamma}_{ij}\Gamma^{\sigma}_{ji}-\Gamma^{\alpha}_{ii}\Gamma^{\beta}_{ij}\Gamma^{\gamma}_{jj}\Gamma^{\sigma}_{ji}-\frac{1}{2}\Gamma^{\alpha}_{ij}\Gamma^{\beta}_{ji}\Gamma^{\gamma}_{ij}\Gamma^{\sigma}_{ji}\right)\,.$
(25)

Here, the expression in the parentheses $\left(\Gamma_{ij}\Gamma_{ji}\right)$
is understood to be summed over alpha
$\Gamma^{\alpha}_{ij}\Gamma^{\alpha}_{ji}$. In the limit of equal quark masses
(III) is zero, since all $J$-dependent factors vanish.

##  IV Summary

Let’s summarize. Above, we proposed a new efficient method for obtaining the
effective action in the theory with heavy particles of unequal masses. Our
calculations are based on Eq. (1) which allows us to extend the Schwinger
proper-time method to the case of elliptic operators that do not commute with
its mass matrix. As a result, we arrive at the effective action (9), in which
each of the coefficients can be calculated independently of the other, and
corresponds to a certain order of $1/M^{2}$ expansion. We have explicitly
calculated two leading contributions for the case of broken $U(3)\times U(3)$
symmetry. Although we have limited ourselves to this particular example, the
formalism is applicable to an arbitrary flavor symmetry group. The method is
appropriate for both renormalizable and nonrenormalizable quantum field
theories, and has a wide range of applications in various branches of quantum
field theory. This includes astrophysics, cosmology, critical phenomena,
standard model and beyond, composite Higgs models, etc.

##  V Acknowledgments

This work was supported by Grant from Fundação para a Ciência e Tecnologia
(FCT) through the Grant No. CERN/FIS-COM/0035/2019, and the European
Cooperation in Science and Technology organisation through the COST Action
CA16201 program.

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# Coordinated Activity Modulates the Behavior and Emotions of Organic Users: A
Case Study on Tweets about the Gaza Conflict

Priyanka Dey, Luca Luceri, Emilio Ferrara  University of Southern
CaliforniaLos Angeles, CA, 90007, USA<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(2024)

###### Abstract.

Social media has become a crucial conduit for the swift dissemination of
information during global crises. However, this also paves the way for the
manipulation of narratives by malicious actors. This research delves into the
interaction dynamics between coordinated (malicious) entities and organic
(regular) users on Twitter amidst the Gaza conflict. Through the analysis of
approximately 3.5 million tweets from over 1.3 million users, our study
uncovers that coordinated users significantly impact the information
landscape, successfully disseminating their content across the network: a
substantial fraction of their messages is adopted and shared by organic users.
Furthermore, the study documents a progressive increase in organic users’
engagement with coordinated content, which is paralleled by a discernible
shift towards more emotionally polarized expressions in their subsequent
communications. These results highlight the critical need for vigilance and a
nuanced understanding of information manipulation on social media platforms.

††copyright: rightsretained††journalyear: 2024††doi: XXXXXXX††conference: WWW
2024; May 13–17, 2024; Singapore††isbn: 978-1-4503-XXXX-X/18/06 Figure 1.
Distribution of different emotions in content produced by coordinated and
organic users. Each pairwise distributions’ difference is significant (MW
test: * p¡0.01, ** p¡0.001).

##  1\. Introduction

Social media has emerged as a pivotal platform for the dissemination of
information on ongoing crisis events, connecting users and fostering timely
information exchanges. However, the expansive reach and influence of social
media also render it a potent tool for entities with malicious intentions.
Through orchestrated efforts, such actors can manipulate narratives,
disseminate misinformation, and advocate for particular political ideologies,
thus shaping the discourse to their advantage (Lazer et al., 2018; Vosoughi et
al., 2018).

A growing body of literature has documented the exploitation of social media
for such nefarious purposes, spanning from political debates to health
discussions (Broniatowski et al., 2018; Chen et al., 2021). These studies
elucidate how misinformation campaigns and the spread of online vitriol
contribute significantly to the distortion of the digital ecosystem
(Konieczny, 2023; Marwick and Lewis, 2017; Ezzeddine et al., 2023). For
instance, research has illustrated the adverse effects of ”fake news” on
public perception and behavior (Pennycook and Rand, 2019; Ecker et al., 2022),
while others have highlighted the proliferation of misinformation and
unreliable information amidst crises like the COVID-19 pandemic (Cinelli et
al., 2020; Chen et al., 2022).

The interaction with coordinated entities and manipulated content on social
media platforms has been shown to precipitate notable shifts in user behavior
and psychological well-being (Pennycook and Rand, 2021). Exposure to
misinformation has been linked with heightened anxiety and distress (Bharati
et al., 2019), alongside a growing distrust in digital sources and a
reluctance to engage in discussions on contentious issues (Metzger and
Flanagin, 2013; Margolis and Amanbekova, 2023).

Building on these findings, our study delves into the dynamics between
coordinated and organic users on Twitter concerning the Gaza conflict, aiming
to shed light on the mechanisms of information dissemination and its
repercussions on user behavior.

### Contributions

This research endeavors to elucidate the potential effects of coordinated
content dissemination on the behavior and emotional state of organic users,
guided by the following inquiries:

  * RQ1

How effective are coordinated users in disseminating their content among
organic users?

  * RQ2

What are the temporal patterns of interaction between organic and coordinated
users?

  * RQ3

How do organic users’ behaviors and emotional expressions change subsequent to
interactions with coordinated users?

Utilizing a dataset comprising approximately 3.5 million tweets from over 1.3
million users collected from September to November 2023, we investigate the
discourse surrounding the Gaza conflict. Our analysis reveals:

  * •

Coordinated users effectively spread their messages through at least a tenth
of the network, with over a third of their content being redistributed by
organic users.

  * •

Engagement with coordinated content unfolds gradually, requiring sustained
interaction over an extended period to observe a significant uptick in
coordinated user engagement.

  * •

Subsequent to interactions with coordinated users, organic users exhibit a
marked increase in the expression of negative emotions, notably pessimism,
sadness, and fear. Moreover, emotions like anger trigger a polarization among
users through repeated interactions, highlighting the significant emotional
and psychological effects of such engagements.

##  2\. Data

In this study of the Gaza conflict, we analyze Twitter interactions (retweets
and replies) between coordinated and organic users. Our dataset covers 62 days
from September 1, 2023, just before the start of the 2023 Israel-Hamas war, to
November 1st, 2023. A significant tweet increase is observed after October
7th, the war’s official start. We collected data using a manually curated list
of English, Arabic, and Hebrew keywords, e.g., West Bank, bombs, Gaza,
Jerusalem, missiles. Keywords like #football, FIFA, Buckwheat, PROMO Alert,
BLM, blacklivesmatter were used to filter out irrelevant content.

We amassed 3,584,175 tweets in 57 languages, with English (93%) most
predominant, followed by Arabic (6.71%) and Hebrew (0.04%). As English tweets
constitute over 3.3 million, we focus our analyses on them. We identify 4
types of tweets: 2,935,621 retweets, 206,663 replies, 150,997 tweets, and
40,001 quotes.

##  3\. Methodology

###  3.1. Coordinated Activity Detection

Various techniques have been proposed to uncover coordinated activity on
social media (Pacheco et al., 2020; Sharma et al., 2021; Luceri et al., 2024).
We utilize a novel technique (Luceri et al., 2024) to construct a user
similarity network based on distinct behavioral indicators.111For more details
on network creation and node pruning methodology, refer to (Luceri et al.,
2024) The network incorporates five behavioral traces, including sharing
identical URL links, hashtags, tweet content, re-sharing the same tweets, and
rapid retweeting (resharing the same tweet in less than 1 minute). Each trace
contributes to a similarity network, with user similarities represented
through edge weights. We then consolidate these networks into a fused graph,
where links between nodes indicate connections in any individual network. To
identify coordinated users, we prune nodes based on centrality, selecting
those with the highest 5% eigenvector centralities.

We detected 1,034 coordinated users. Among 100 manually analyzed users, 62 had
either deleted accounts or were suspended, and 9 had protected tweets.
Additionally, 68 users had small followings (¡ 2,000 followers), while 32 had
posted over 100K times.

###  3.2. Measuring Coordinated Users’ Effectiveness

To identify the effectiveness of coordinated users, we adapt four metrics
introduced by (Luceri et al., 2019):

Retweet Pervasiveness (RTP) measures how often coordinated users’ tweets are
retweeted by organic users:

$RTP=\frac{\text{\\# of organic retweets from coordinated users}}{\text{\\# of
organic user retweets}}$

Reply Rate (RR) measures the percentage of replies from organic users to
coordinated users’ tweets:

$RR=\frac{\text{\\# of organic user replies to coordinated users'
tweets}}{\text{\\# of organic user replies}}$

Organic to Coordinated User Rate (O2CR) quantifies organic users’ interactions
with coordinated users:

$O2CR=\frac{\text{\\# of interactions with coordinated users}}{\text{\\# of
organic user activities}}$

Tweet Success Rate (TSR) is the percentage of coordinated users’ tweets
retweeted more than once by organic users:

$TSR=\frac{\text{\\# of tweets retweeted > 1 by organic users}}{\text{\\# of
coordinated tweets}}$

###  3.3. Characterizing User Interaction Dynamics

To measure interaction changes, we analyze how the content distribution of
both organic and coordinated users shifts after 1, 2, and 3 interactions. To
do this, we calculate the mean inter-group (organic $\rightarrow$ coordinated,
coordinated $\rightarrow$ organic) and intra-group (organic $\rightarrow$
organic, coordinated $\rightarrow$ coordinated) interaction proportions for
all activity between the $k$-th interaction and $k+1$-th interaction
($k\in\\{1,2,3\\}$). For the third interaction, we consider all activity until
November 1, 2023.

To further observe changes in organic users’ behaviors, we examine their
inter-group interaction proportions after 1, 2, and 3 interactions
($k\in\\{1,2,3\\}$) and across different time windows ($t\in\\{\textit{1
day},\textit{3 days},and\textit{1 week}\\}$). For instance, for $k=1$ and
$t=\textit{1 hour}$, if a user has their 1st interaasction at 11AM and 2nd
interaction at 11:58AM, all interactions between 11:00 AM and 11:58 AM are
considered. We then compute the proportion of interactions with coordinated
users and denote this as O2C prop. Similarly, for all other organic users, we
compute $O2Cprop$ for $k=1,t=\textit{\text{1 day}}$. We then compute the mean
$O2Cprop$ over all users. We repeat this process for the remaining interaction
steps and time windows.

###  3.4. Organic Users’ Behavior After Interactions

To analyze changes in user behavior, we examine original content (tweets)
posted by organic users after interacting with coordinated users. We
investigate variations in expressed emotions across different time windows
($t$) and interaction steps ($k$). We leverage (Chochlakis et al., 2023),
which analyzes text and assigns probabilities (ranging from 0 to 1) for 11
emotions: anticipation, joy, love, optimism, surprise, trust, anger,
disgust/contempt, fear, pessimism, and sadness.

To establish a baseline for examining shifts in the organic users’ content, we
first explore differences between coordinated and organic users. We gauge the
significance of these deltas by using Mann-Whitney (MW) tests
($\alpha=0.01,0.001$). To examine content shifts across interaction steps, we
identify tweets before and after the $k$-th interaction within a specified
time window, $t$. For each interaction step and time window, we compute
emotion deltas by averaging the difference between post-probabilities (emotion
probabilities for tweets written after the interaction step) and pre-
probabilities (emotion probabilities for tweets written before the interaction
step) across all tweets and users.

##  4\. Results

Next, we examine how successful coordinated users are in spreading their
content through the network (RQ1), how the interactions between coordinated
and organic users change over interaction steps (RQ2), and finally whether
there are changes in users’ content after interactions with coordinated
content (RQ3).

###  4.1. Effectiveness of Coordinated Activity

We estimate the effectiveness of the 1,034 coordinated users (identified
through our fused network) in receiving engagement and endorsement from
organic users in the discussion related to the Gaza conflict. We identified
1,766 organic users that interact (retweet or reply) with coordinated users,
and 1,326,695 users that do not have any interactions with coordinated users.
Using the four metrics outlined in §3.2, we observed the following values: RTP
(10.51%), RR (10.62%), O2CR (9.87%), and TSR (36.58%).

We observe that organic users tend to reply to coordinated users slightly more
often than they retweet them. While less than 10% of the organic users’
interactions involve coordinated users (O2CR), over a third of coordinated
user-generated content has some engagement (TSR), indicating substantial
dissemination of coordinated content throughout the network. In comparison to
(Margolis and Amanbekova, 2023), we note lower RTP and RR, yet comparable O2CR
and TSR. This illustrates that although organic users mainly engage with each
other, a significant portion of coordinated content permeates various segments
of the network, including vulnerable organic audiences.

Figure 2. Inter and intra-group engagement proportions between organic and
coordinated users after $k$ interactions

###  4.2. Interactions Increase Over Time

In this section, we examine the evolving interactions between organic and
coordinated users over time. Figure 2 illustrates the shifting content
distributions of both groups over interaction steps. As the number of
interactions rises, we observe descending trends in intra-group interaction
proportions. However, we also see a rise in inter-group interaction
proportions, indicating the ability of coordinated users to initiate
discussions with organic users through repeated interactions.

As we are interested in understanding the organic users’ behavior, we further
explore their changes in interaction proportions over three time windows (1
day, 3 days, 1 week) by averaging the metric O2C prop over all users as
discussed in §3.3. Results from this analysis suggest that within a week after
3 interactions, over 40% of organic user interactions are with coordinated
users. These results further signal that organic users may be attracted to the
content posted by coordinated users, thus leading to a larger number of
interactions with them.

###  4.3. Emotion Modulation by Coordinated Users

Our third RQ investigates whether repeated interactions alter organic users’
content. We compare emotions expressed in coordinated and organic content in
Figure 1. We present the distributions of five emotions (2 positive: Love,
Optimism and 3 negative: Fear, Pessimism, Sadness), which significantly differ
(MW test, $\alpha$ = 0.01). Our findings indicate that coordinated users are
less likely to incorporate emotions such as love and optimism in their
content, as evidenced by the narrower probability ranges, namely (0 to 0.5 and
0 to 0.8), in contrast to organic users. Comparable patterns emerge for the
other positive emotions (Anticipation, Joy, Surprise, Trust). However,
coordinated content tends to exhibit higher probabilities of emotions such as
Anger and Disgust/Contempt, suggesting more negative emotion usage overall.

Given that the emotions used by the two groups are significantly different, we
study the emotions expressed in organic user content after interacting with
coordinated users. As results from RQ2 suggest that changes in user behavior
are most prevalent after the 3rd interaction, we illustrate variations in
negative emotions following the third interaction in Figure 3.

Figure 3. Changes in negative emotions after 3 interactions. Each change is
significant (MW test: * p¡0.01, ** p¡0.001).

We observe a steady increase in the use of a majority of the negative emotions
(as signaled by positive delta values). Although deltas show that after
interacting with coordinated users, organic users use even less positive
emotions, many of these differences are not significant. For interactions 1
and 2, we observe some similar patterns: negative emotions tend to be more
significant in content written after interactions and as the time window
increases, we observe more significant differences. We also notice some
fleeting emotions such as trust, which reduces after 2 interactions during the
1 day and 3 days time intervals, but after 1 week, there are no significant
differences.

An intriguing observation we noted is the decrease in the use of anger within
the 3 days time window following the third interaction, followed by an
increase in usage after 1 week. Similar patterns emerge after 1 and 2
interactions. To delve deeper into this phenomenon, we analyze the
distribution of delta values in anger after each interaction within the 3 days
interval (refer to Figure 4).

Upon closer examination of the plots, we observe that repeated interactions
pushed users towards the extreme. Users diverge into three distinct groups
over repeated interactions within a brief three-day span, with two displaying
notable radical tendencies. One group shows increased anger (higher delta
values) after interactions, while the other tends to become more subdued
(lower delta values).

We manually reviewed some user content to illustrate this trend. For instance,
one user posted: ”… #FoxNews Hmm don’t remember you uttering a word Bernie
when Hamas slaughtered civilians and beheaded infants. Selective faux outrage
u commie!” after the 1st interaction, but later shared ”Gaza crisis: Angelina
Jolie’s heartfelt post for peace…” after the 3rd interaction, indicating a
decrease in anger. Another user posted ”Defund the universities! We can’t
ignore Jew-hating academia…” after the 1st interaction and ”… Seriously…. what
the **** do they know about Israel/Palestine - Republican sympathising
Teachers and parents pushing their agenda!” after the 3rd interaction,
demonstrating a significant increase in anger.

Our analysis suggests that interactions with coordinated users often shape how
organic users create new posts and express emotions within their messages. A
notable portion of their content tends to exhibit heightened negative
emotions. Interestingly, concerning specific emotions like anger, users form
distinct groups over a period of time, with some showing increased anger while
others seem to become more desensitized.

##  5\. Conclusions

In summary, our analysis highlights the potential effects of coordinated users
on social media dynamics. They effectively disseminate messages, impacting a
substantial portion of the network, with a notable fraction of their content
redistributed by organic users. Although engagement with coordinated content
unfolds gradually and requires sustained interaction over time, the
interactions can have behavioral and psychological effects on organic users.
New content posted by these users express heightened negative emotions,
including pessimism, sadness, and fear, while emotions like anger drive
polarization among users. These findings underscore the need for further
research to explore shifts in users’ behaviors post-interactions, with
investigating topics and linguistic style changes emerging as compelling
avenues for future study.

Acknoledgements. This work was supported in part by DARPA (contract no.
HR001121C0169).

Figure 4. Distribution of Anger deltas for 3 days interval over repeated
interactions between organic and coordinated users. Noteworthy, the Anger
deltas become increasingly polarized, suggesting that repeated interactions
with coordinated activity leads to increasingly extreme changes in Anger
emotion.

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# Retro-prob: Retrosynthetic Planning
Based on a Probabilistic Model

Chengyang Tian
Harbin Institute of Technology
<EMAIL_ADDRESS>
&YangPeng Zhang
Harbin Institute of Technology
<EMAIL_ADDRESS>
&Yang Liu
Harbin Institute of Technology
<EMAIL_ADDRESS>Corresponding author.

###### Abstract

Retrosynthesis is a fundamental but challenging task in organic chemistry,
with broad applications in fields such as drug design and synthesis. Given a
target molecule, the goal of retrosynthesis is to find out a series of
reactions which could be assembled into a synthetic route which starts from
purchasable molecules and ends at the target molecule. The uncertainty of
reactions used in retrosynthetic planning, which is caused by hallucinations
of backward models, has recently been noticed. In this paper we propose a
succinct probabilistic model to describe such uncertainty. Based on the model,
we propose a new retrosynthesis planning algorithm called retro-prob to
maximize the successful synthesis probability of target molecules, which
acquires high efficiency by utilizing the chain rule of derivatives.
Experiments on the Paroutes benchmark show that retro-prob outperforms
previous algorithms, retro* and retro-fallback, both in speed and in the
quality of synthesis plans.

##  1 Introduction

Retrosynthesis is the process of finding synthesis plans for molecules.
Typically, retrosynthesis is regarded as a search problem, with the target
molecule as the starting point and a set of purchasable molecules as the
endpoint. With the seemingly infinite number of organic molecules and their
associated reactions, the search space is vast, making it challenging even for
experienced experts. There are already several AI algorithms capable of
automatically conducting retrosynthesis analysis, and providing advice to
chemists. Despite their differences, these algorithms share a similar
framework. They use a one-step backward model to predict reactions and then
conduct searches on the network formed by these reactions. However, the
outcomes of one-step models are not always accurate, making the routes found
by retrosynthesis algorithms impractical. Fortunately, algorithms usually
provide multiple routes, and naturally the proportion of successful routes
among all results becomes a valuable metric when evaluating algorithms.

Existing methods evaluate the results of retrosynthesis algorithms using
metrics such as the route length, the route quality, and the route diversity.
A recent study, retro-fallback[1], employs a new metric called Successful
Synthesis Probability (SSP), which can be used to evaluate the overall success
probability of a synthesis plan generated by an algorithm. Based on a method
of stochastic processes (or stochastic samplings), retro-fallback is an
algorithm that continuously estimates the SSP of the given target molecule and
tries to maximize it during the search process. The concept of SSP is of
practical significance, but there still lacks an efficient method to calculate
it since the way of stochastic samplings is fairly slow.

In this paper, we first establish a probabilistic model that allows us to
calculate SSP directly, rather than estimating it through sampling. Based on
this probabilistic model, we then present an algorithm named _retro-prob_ ,
which utilize a mathematical technique to achieve computational efficiency.

##  2 Background

Retrosynthesis planning can be divided into two smaller tasks: one-step
backward reaction prediction and multi-step retrosynthetic planning. Given a
product molecule $m$, one-step backward model need to identify a set of
reactions $\mathcal{R}$, and each $r\in R$ should includes $m$ as one of its
products. On the basis of one-step model, a retrosynthesis algorithm takes a
target molecule $t$ and a set of starting molecules $\mathcal{I}$ as input,
and it aims to find out a series of reactions through which people can
synthesize $t$ using molecules from $\mathcal{I}$ as source materials.

There mainly exists two types of one-step backward models: template-based and
template-free. Template-based methods, as proposed by Coley et al. [2], Segler
and Waller [3], extract templates from available chemical reaction datasets
and then apply them to unknown molecules. Neutral networks are employed to
help filter out the most probable ones from more than 40K templates. Template-
free methods, as proposed by Liu et al. [4], Karpov et al. [5], utilize
sequence-to-sequence models such as LSTM and Transformer. They transform
molecules and chemical reactions into textual strings and then utilize methods
in Natural Language Processing (NLP) to predict the outcomes. Such template-
free methods are more innovative, capable of discovering unknown reactions
that cannot be described by known templates, but their accuracy is
comparatively low. In the work by Karpov et al. [5], they achieved an top-1
accuracy of 42.7%, while Segler and Waller [3] claimed a 64% top-1 accuracy in
their work on template-based methods.

Different algorithms commonly approach multi-step retrosynthesis as a search
problem, with variations in the definitions of their search structures.
Algorithm based on Monte Carlo Tree Search (MCTS)[6] utilizes each node on its
search tree to represent a set of molecules. Retro*[7] use an AND-OR tree
which consists of molecule nodes and reaction nodes. Retro-fallback[1]
maintains its search structure as an AND-OR graph instead of a tree.

###  2.1 Retro* search algorithm

Retro*[7] is an algorithm inspired by A* search. Retro* aims to improve the
quality of routes, which is evaluated by route length (or route cost), and
stay efficient at the same time.

##### AND-OR graph

An AND-OR graph is a directed graph that expresses the correlations between
molecules and reactions. In an AND-OR graph, each reaction, represented as an
AND node, requires all its successors to participate, while a molecule,
represented as a OR node, can be synthesized if any one of its successors
successfully reacts. Let $M$ be the OR node for a molecule $m$. If $m$ is one
of the products of reaction $r$, represented by an AND node $R$, there is an
directed edge from $M$ to $R$. For each reactant molecule of reaction $r$,
there is another edge starts from $R$ and points to the node representing that
reactant molecule.

As many molecules and chemical reactions are unknown or unrecorded, it is
practically impossible to obtain the overview of the graph prior to
retrosynthetic planning. Algorithms only have information about nodes that
have already been reached. Tree structures are more natural and are easier to
maintain.

##### AND-OR tree

The AND-OR tree was first brought into retrosynthesis by retro*. An AND-OR
tree, denoted as T, is the search structure constructed by algorithms when
searching on an AND-OR graph. On initialization, retro* creates a root node
for the given target molecule. At each step of the search, retro* expands a
leaf molecule node. The expansion will call the one-step backward model, and
creates new nodes in tree T according to the one-step model’s return. As
illustrated in Figure 1, an expansion creates both AND nodes and OR nodes.

For each molecule node $M\in T$, Retro* calculates $g(M)$ and $h(M)$, where
$g(M)$ is the distance from $M$ to the root node and $h(M)$ is the predicted
cost to synthesize molecule $m$. On expansion, retro* always selects the leaf
node with the lowest $V(M)=g(M)+h(M)$.

Figure 1: The expansion of an AND-OR search tree. Node $M_{2}$ is expanded
with reactions predicted by the one-step backward model:
$m_{4}+m_{5}\rightarrow m_{2}$ and $m_{6}\rightarrow m_{2}$ (any side products
are ignored).

###  2.2 Retro-fallback algorithm

Retro-fallback[1] is a greedy algorithm that takes into account the
probability of reaction failures. Retro-fallback emphasise finding backup
routes in a synthesis plan to increase the success probability.

##### Successful synthesis probability

SSP is used by retro-fallback to evaluate the success probability of a
synthesis plan. It is defined as:

$SSP(T)=\mathbb{P}\left[\exists~{}Route\subseteq
T,with~{}\sigma(Route)=1\right]$ (1)

where T is the whole search tree (or a synthesis plan), and $\sigma(Route)=1$
means every reactions on that route is successful. This is actually a
simplified version of SSP’s definition. Retro-fallback provides thorough
definitions of what is a $Route$ and what is a $success$.

Retro-fallback utilizes stochastic processes to estimate the SSP of a search
tree T, requiring multiple samplings based on the probabilities of reactions.
At each step of the search, retro-fallback greedily selects a $Route$ and
simultaneously expands all leaf nodes on the route. The method for selecting
the route is also based on stochastic processes. Due to its complexity, we
refrain from delving into the details here.

##  3 A probabilistic model

In this section, we first propose a set of calculation formulas of SSP based
on its definition from retro-fallback[1]. Then, we describe the graph-based
search structure used in our retro-prob algorithm.

###  3.1 The calculation formulas for SSP

Let $s(M)$ denote the success probability of molecule node $M$, and $s(R)$
denote the success probability of reaction node $R$. We provide the following
calculation formulas in the form of recursion:

$\displaystyle s(M)$ $\displaystyle=\begin{cases}1,&m\in\mathcal{I}\\\
1-\prod_{R\in{Ch(M)}}\left(1-s(R)\right),&otherwise\end{cases}$ (2)
$\displaystyle s(R)$ $\displaystyle=P(R)\times\prod_{M\in{Ch(R)}}s(M)$

where $m$ is the molecule of node $M$, $\mathcal{I}$ is the set of starting
molecules, and $Ch(X)$ represents the children of node $X$ in the search tree
T. $P(R)$ is the feasible probability of reaction $r$. A reaction is feasible
if it is able to occur when provided with all the reactants it needs, but the
success of a reaction node requires the success of its reactants at the same
time. These formulas may look similar to the definition of $node\ success$ in
retro-fallback[1], but we introduce probabilities with a range of $[0,1]$
instead of a binary successful/unsuccessful outcome.

After the definitions of $S(M)$ and $S(R)$, we have obtained a method to
calculate SSP. In fact, the SSP of a synthesis plan T is exactly the
$S(M_{t})$ of the given target molecule $m_{t}$, where $M_{t}$ is the root
node of the search tree T.

###  3.2 AND-OR tree with additional edges

As is discussed in Section 2.1, an AND-OR tree is the naturally formed
structure when searching on an AND-OR graph. However, there exists weak
cycles, cycles, as well as bidirectional edges (representing reversible
reactions) in graph. Weak cycles can result in multiple paths from the root
node to a particular node, thereby causing such nodes to have multiple copies
of themselves in the tree. Cycles and bidirectional edges can potentially
result in infinite loops within the tree. Such occurrences caused costly
additional expenses for search algorithms and can sometimes be misleading.
Typically, loops can be prevented by prohibiting algorithms from creating
nodes if an identical node already exists among its ancestors, as illustrated
in Figure 2 (b). Maziarz et al. [8] proposed a resolution to reduce costs
incurred when algorithms visit and expand different nodes of a same molecule
by caching the results of the one-step backward model. It dose improve the
efficiency significantly, but it does not resolve the issue thoroughly. Nodes
representing the same molecule in different positions remain isolated,
preventing algorithms from interpreting the relationships between these nodes.

Figure 2: A graph and trees transformed from it. Here all nodes represent
molecules, while reaction nodes are omitted. $Q_{1},Q_{2}$ represent the same
molecule, so do nodes $P_{i}(i=1,2,3)$. Leader nodes are depicted with thick
boarder in (c).

Our algorithm retro-prob utilizes an AND-OR graph without cycles as its
logical structure, but in implementation we use an AND-OR tree with additional
edges to describe it. Within such trees, a molecule may have multiple nodes,
but it will have only one leader node. For a molecule $m$, let
$\mathcal{G}(m)$ denote the set of nodes representing $m$. Each group
$\mathcal{G}$ has one leader node $M_{leader}\in\mathcal{G}(m)$:

$M_{leader}\leftarrow\mathop{\arg\max}_{M\in\mathcal{G}(m)}Depth(M)$ (3)

where $Depth(M)$ returns the depth of node $M$ in tree T. Such design ensures
that the order of leader nodes during level traversal in a tree matches the
topological order of the graph.

After selecting the leader nodes, we create addition edges from non-leader
nodes to their leader nodes to indicate that they represent the same node in
graph. Any further expansions will be conducted under the leader nodes and
non-leader nodes will no longer have any children. As depicted in Figure 2
(c), loops are still not allowed.

##  4 Retro-prob Algorithm

###  4.1 Overview of retro-prob

Retro-prob is a greedy search algorithm that based on the probabilistic model
and the search structure introduced in Section 3, and retro-prob expands a
single leaf node at each step (see Appendix A.1 for the expansion method). We
refer to those nodes that can be selected for expansion as open nodes. Open
nodes are unexpanded leaf nodes that serve as leader nodes. We denote the set
of open nodes as $\mathcal{O}$. Below we first describe the greedy heuristic
used by retro-prob. Then we delve into an efficient practical implementation,
which utilizes the chain rule of derivatives.

###  4.2 A greedy heuristic

Let $s(M_{t}|expand(M))$ denote the success probability of root node $M_{t}$
if we would expand node $M$. We select the next expanding node $M_{next}$ upon
heuristic:

$\displaystyle
M_{next}\leftarrow\mathop{\arg\max}\limits_{M\in\mathcal{O}}\left[s\big{(}M_{t}|expand(M)\big{)}\right]$
(4)

When calculating $s(M_{t}|expand(M))$, we make an optimistic assumption that
the expansion of $M$ will result in $s(M)=1$. Then we use a slightly different
formula from Eq 2 to calculate $s(M_{t})$ using a bottom-up method (see
Appendix A.2 for details).

$\displaystyle s(M)=\begin{cases}1,&m\in\mathcal{I}\\\
s_{0},&m\in\mathcal{O}\\\
1-\prod_{R\in{Ch(M)}}\big{(}1-s(R)\big{)},&otherwise\end{cases}$ (5)

where $s_{0}>0$ is a parameter. The positive $s_{0}$ allows reactions with
more than one reactants to be "seen" by the algorithm. Otherwise the success
probabilities $s(R)$ will remain zero for such reactions, resulting in their
child nodes never being selected (unless there are no other open nodes).
Furthermore, a positive $s_{0}$ allows our algorithm to access global
information from the entire search tree, including both successful routes and
routes that haven’t succeed yet.

Algorithm 1 A simple greedy algorithm

1: Initialize the root node $M_{t}$

2: for $i=0$ to $max\\_iteration$ do

3: for $M\in\mathcal{O}$ do

4: Calculate $s(M_{t}|expand(M))$

5: end for

6:
$M_{next}\leftarrow\mathop{\arg\max}\left[s\big{(}M_{t}|expand(M)\big{)}\right]$

7: Expand $M_{next}$

8: end for

9: return The search tree $T$

###  4.3 Retro-prob with derivatives

The efficiency of the above algorithm is relatively low. There can be
thousands of open nodes during the search, thus requiring the bottom-up
calculation to be conducted $|\mathcal{O}|$ times.

Instead of attempting to find the maximum $s(M_{t}|expand(M))$, we try to
maximize the derivatives:

$\displaystyle
M_{next}\leftarrow\mathop{\arg\max}\limits_{M\in\mathcal{O}}\left[\frac{\mathrm{d}s(M_{t})}{\mathrm{d}s(M)}\right]$
(6)

We argue that Eq 6 is equivalent to Eq 4, as we assume the expansion of every
node $M\in\mathcal{O}$ results in the same increase in $s(M)$. Let $D(X)$
denote the derivative $\mathrm{d}s(M_{t})/\mathrm{d}s(X)$. For each node $X$,
its derivative $D(X)$ can be calculated using Eq 7:

$\displaystyle D(M)$
$\displaystyle=\begin{cases}\sum_{M_{i}\in\mathcal{G}(m)}\Big{[}D(P_{i})\cdot\mathrm{d}s(P_{i})/\mathrm{d}s(M_{i})\Big{]},&otherwise\\\
1,&M\ is\ M_{t}\end{cases}$ (7) $\displaystyle D(R)$
$\displaystyle=D(P)\cdot\frac{\mathrm{d}s(P)}{\mathrm{d}s(R)}$

where $P_{i}$ is the parent node of $M_{i}$ and $P$ is the parent node of $R$.
Now we can calculate the derivatives for every node in the tree T in a single
top-down calculation: For a node $X$, once the derivative $D(P)$ of its parent
node is known, it’s straightforward to compute $\mathrm{d}s(P)/\mathrm{d}s(X)$
according to Eq 2.

For example in Figure 3, we have $D(M_{t})=1$, and
$D(R_{1})=\mathrm{d}s(M_{t})/\mathrm{d}s(R_{1})=1-s(R_{2})$. Once we obtained
the value of $D(R_{1})$, we can easily calculate
$D(M_{2})=D(R_{1})\cdot\mathrm{d}s(R_{1})/\mathrm{d}s(M_{2})$.

Figure 3: The top-down calculation of derivatives. Because the leader nodes
are always the deepest in $\mathcal{G}$, the calculation is simply top-down
(or in the topological order of a graph).

Eq 7 requires to sum up the derivatives of nodes in set $\mathcal{G}$, as
these nodes represent the same molecule and are thus regarded as a single
independent variable. An efficient method for doing this is by delaying the
sum up: First calculate $D(M)$ as if $M$ is the only node in $\mathcal{G}$.
After every node in $\mathcal{G}$ has completed its calculation, sum up their
values to the leader node of $\mathcal{G}$. For example in Figure 3, we first
calculate $D^{\prime}(M_{3})$ and $D^{\prime}(M_{4})$ independently, and then
set $D(M_{4})=D^{\prime}(M_{3})+D^{\prime}(M_{4})$. But it’s only after the
true $D(M_{4})$ has been calculated that we can process into $M_{4}$’s
children nodes, as the value of $D(M_{4})$ will be used by children nodes.

Algorithm 2 Retro-prob algorithm

1: Initialize the root node $M_{t}$

2: for $i=0$ to $max\\_iteration$ do

3: Calculate $D(X)$ for $X\in T$ in a top-down method

4: $M_{next}\leftarrow\mathop{\arg\max}D(M)$

5: Expand $M_{next}$

6: end for

7: return The search tree $T$

##  5 Experiments

###  5.1 Experiment setup

We use the PaRoutes[9] benchmark dataset for evaluation and analysis. PaRoutes
is a framework for benchmarking multi-step retrosynthesis methods. It provides
two sets of 10k target molecules and starting molecules, along with reference
routes. It also provides a trained template-based one-step backward model,
which is trained from the popular USPTO chemical reaction dataset from Jin et
al. [10]. To reproduce the results of retro*[7] and retro-fallback[1], we
partially use the framework in Syntheseus[8]. We use the retro*-0 version of
retro* algorithm, which does not include the neural network-guided heuristic
and is outperformed by retro* by 6% in terms of success rate from the
experiment of Chen et al. [7].

We evaluate the results of algorithms based on the average SSP obtained from a
set of 50 target molecules from Paroutes dataset, and we utilize two different
methods for obtaining the SSP of a retrosynthesis plan. The first method is
based on stochastic processes, which is employed in retro-fallback. The other
method is derived from Eq 2 and Eq 5, where we always set $s_{0}=10^{-3}$ for
evaluation purposes. Setting a small but positive $s_{0}$ helps us evaluate a
plan even if it has not succeeded yet. We use the result $s(M_{t})$ to express
the SSP of a retrosynthesis plan T. In both cases, we obtain the feasible
probabilities for reactions using the method outlined in Appendix B.

###  5.2 Results

####  5.2.1 The optimal parameter for retro-prob

We search for the optimal parameter $s_{0}$ to be used in Eq 5 through
experiments. The one-step backward model selects the top-30 templates in this
experimentation.

Figure 4: The increase curves of $s(M_{t})$ for different $s_{0}$.

As is shown in Figure 4, the groups with $s_{0}=0.03$ and $s_{0}=0.05$ exhibit
the best performance. We also add a control group with $s_{0}=0$, represented
by the dashed line in Figure 4. It is obvious that the group with $s_{0}=0$
gets worse results than any other groups, especially as iterations increase.
We believe this is because it lacks the capability to obtain routes containing
reactions with more than one reactant molecule $m\notin\mathcal{I}$. Such
routes have lower priority for the algorithm, but if we set $s_{0}>0$,
algorithm will eventually search for those routes as iterations increase.

####  5.2.2 Comparisons

We compare retro-prob against retro*[7] and retro-fallback[1] on the first 50
molecules from Paroutes dataset (N5). The one-step backward model selects the
top-50 templates, and retro-prob use a $s_{0}=0.04$. For retro*, we offer two
types of reaction costs. One is the bond loss described in Appendix B.2. The
other is the log cost used in the original retro* algorithm. The log cost may
not be consistent with our bond loss, resulting in different evaluations of
reactions.

First, we conduct experiments on retro-prob, retro-prob (log cost), retro* and
retro* (log cost), using $s(M_{t})$ to evaluate their outcomes. For retro-prob
with log cost, we replace the loss in Eq 9 with the log cost. Using log cost
means algorithms are unaware of the feasible probabilities used during
evaluation, which increases the uncertainty. The results are listed in Table 1
and shown in Figure 5. Retro-prob significantly outperforms retro* in terms of
$s(M_{t})$, and is also time-efficient even when the value updating involves
more nodes in the search tree. Log cost methods consume much less time as it
does not use the forward model in Appendix B.1. Retro* with log cost achieves
the highest success rate while its $s(M_{t})$ is the lowest. This phenomenon
shows that a successful synthesis plan does not necessarily equate to a high
success probability. Within the routes found by retro* (log cost), there may
exist lots of reactions with low feasible probabilities. In the same
situation, retro-prob with log cost performs better. We believe this is
because retro-prob ensures its success probabilities with backup routes.

Figure 5: The curves of $s(M_{t})$ for different algorithms. Error bars represent standard errors. Table 1: Evaluating algorithms using $s(M_{t})$. Max iterations = 120. Algorithm | retro-prob | retro-prob (log cost) | retro* | retro* (log cost)
---|---|---|---|---
$s(M_{t})$ | 0.93 | 0.46 | 0.35 | 0.31
Success rate | 62% | 62% | 20% | 84%
Time(s) | 57 | 58 | 73 | 19

Secondly, we utilize the method of stochastic processes by retro-fallback to
estimate the SSP of retro-prob, retro*, retro* (log cost) and retro-fallback.
Since this method only considers the success probabilities of successful
routes, we provide both the average SSP over all molecules and the average SSP
over succeeded molecules (denoted as success-SSP). Results are listed in Table
2 and shown in Figure 6. Retro-prob still performs the best in regard of the
estimated SSP, and has an advantage in terms of time consumption over retro-
fallback and retro*.

Figure 6: Left: The overall SSP. Right: The success-SSP. Error bars represent standard errors. Table 2: Evaluating algorithms using SSP. Max iterations = 200. Algorithm | retro-prob | retro* | retro* (log cost) | retro-fallback
---|---|---|---|---
SSP | 0.51 | 0.10 | 0.22 | 0.20
Success-SSP | 0.72 | 0.47 | 0.25 | 0.48
Success rate | 70% | 22% | 86% | 42%
Time(s) | 98 | 123 | 33 | 233

##  6 Discussions and limitations

In this paper, we propose retro-prob, a greedy retrosynthetic planning
algorithm based on a probabilistic model, aimed at maximizing the success
probabilities of synthesis plans. By making use of the chain rule of
derivatives, retro-prob simplifies multiple bottom-up value updates into a
single top-down calculation. We also propose a method for conducting
quantitative analysis of reaction feasibility based on a forward prediction
model. Through experiments, retro-prob has been demonstrated to outperform
both the retro* and retro-fallback algorithms.

Our work certainly has some limitations. Most importantly, although we
introduce a method to evaluate its results, our search algorithm still heavily
relies on the one-step model. The ability of on-step model is becoming the
bottom neck of retrosynthesis. Additionally, we acknowledge that our
experiments are inadequate in some aspects. Specifically when evaluating
reactions, we make assumptions on the method of transforming bond loss into
probabilities, but further experiments are needed to prove its practicality.

However, we believe that employing probabilistic or mathematical models is
beneficial for tackling many complex problems. We hope that our method can
inspire future works to develop more thorough solutions.

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##  Appendix A Implementation details

###  A.1 Expansion of a tree

The expansion of a leaf node consists of two steps. The first step is
identical to that of an normal AND-OR tree, as depicted in Figure 1. The
second step is to maintain the structure described in Section 3.2 by
reallocating leader nodes and adjusting additional edges.

Figure 7: The adjustment stage after expansion. All nodes represent molecules,
while reaction nodes are omitted.

As demonstrated in Figure 7, the adjustment stage starts when new nodes are
created. We require leader nodes to be the deepest one in each node set
$\mathcal{G}$. If a new node $Q_{0}$ surpasses the depth of the old leader
node $Q_{2}\in\mathcal{G}(q)$, $Q_{0}$ will become the new leader node and
edges will be adjusted. Consequently, the subtree rooted at the old leader
node needs to be transferred under the new leader node. This transfer process
will change the depth of nodes within the subtree, which might lead to further
leader reallocation.

###  A.2 Update of node values

To calculate the $s(M_{t})$ of the root node $M_{t}$, there is a bottom-up
value update.

Figure 8: The order for updating node values.

First we initialize the value of leaf nodes in the tree T according to Eq 5.
Secondly we conduct a bottom-up calculation for nodes. Note that we do not
calculate $s(M)$ for a non-leader node $M$. Instead we copy the value from its
leader node. As illustrated in Figure 8, the node $X$ will set its $s(X)$
identical to its leader node $Y$. This require the value $s(M_{leader})$ of a
leader node $M_{leader}\in\mathcal{G}$ should have been set before it is used
by other nodes. Because the leader nodes are always the deepest in
$\mathcal{G}$, the calculation is simply bottom-up.

##  Appendix B Feasible probabilities of reactions

Different from the feasible models employed in retro-fallback, we utilize a
forward reaction prediction model to evaluate the outcome of the one-step
backward model.

###  B.1 The forward prediction model

We construct our forward prediction model based on models that predicts the
electron flows[11, 12]. It takes a set of reactants as input and outputs the
predicted number of chemical bonds between each pair of atoms within them. The
model’s structure is illustrated in Figure 9.

Figure 9: The structure of our forward prediction model. Left: encoder. Right:
decoder.

Our forward prediction model consists of an encoder part and a decoder part.
The encoder extracts information from molecules, converting atoms into hidden
layer representations. The decoder interprets those hidden layer
representations and use an attention module to predict the formation or
breakdown of chemical bonds between atoms.

###  B.2 From bond loss to probabilities

Schwaller et al. [13] have proposed the method of round-trip accuracy to
evaluate one-step backward models, which counts whether the target molecule is
included in the predicted products yield by a forward model. If the results of
the backward/forward models are consistent, the resulting reaction is more
likely to be correct (or feasible).

Following such idea, we conduct quantitative analysis of reaction feasibility
based on the forward prediction model in Section B.1. The results of a one-
step backward model is a set of single-product reactions. For each reaction,
we feed its reactants (predicted by backward model) into our forward model and
obtain the predicted number of chemical bonds between atoms in the products.
Then by comparing these products with the target molecule, we obtain the bond
loss:

$\displaystyle loss=\sum_{x\in{V_{t}},y\in{V}}(T_{(x,y)}-P_{(x,y)})^{2}$ (8)

where $V_{t}$ is the set of atoms in the target molecule, and $V$ is the set
of atoms within reactants. $T_{(x,y)}$ represent the bond count between atom
$x$ and $y$, while $P_{(x,y)}$ represent the predicted number of bonds between
$x$ and $y$, which may not be a integer. If neither of a pair of atoms is in
set $V_{t}$, the predicted bond count between them won’t contribute to the
loss. Note that Eq 8 implies a precondition that we must know how atoms are
mapped between reactants and products (as depicted in Figure 10), which can
sometimes be a hard task. Fortunately, when employing template-based backward
models, we can obtain the mapping information from the templates.

Figure 10: The mapping information between reactants and products. Atoms with
the same number are the same atom before and after the reaction.

Finally, we obtain the feasible probability $P(R)$ of a reaction $R$ from Eq
9:

$\displaystyle P(R)=e^{-a\cdot(loss+bias)}$ (9)

where $a$ is a parameter we set to 0.2 in experiments. We use $bias=1.0$ in
retro-prob when searching for routes. This parameter is used to represent
"route length", making our algorithm prefer shorter routes. For cases where
"length" is not considered, it’s recommended to set $bias=0$.

##  Appendix C Sample search trees and extracted routes

Figure 11: A sample search tree created by retro-prob algorithm. (Partial.)
Yellow/blue/green/grey nodes indicate purchasable/successful/expanded/open
nodes. Additional edges are not depicted in this tree. Figure 12: A route
extracted from the tree in Figure 11. Figure 13: Another sample search tree
created by retro-prob algorithm. (Partial.) Figure 14: A route extracted from
the tree in Figure 13.