# Critical phenomena of RNA-like polymers - a synopsis

R. Dengler ORCID: 0000-0001-6706-8550

###### Abstract

This work examines field theories for RNA-like polymers with single strand and
double strand polymers and a periodic base sequence. These field theories
originate from lattice models, describe polymers in a good solvent, and in
principle exactly describe the critical behavior. A central role is played by
the conventional one-component branched polymer and the mapping of the
corresponding field theory to the Lee-Yang field theory in two less
dimensions.

Critical phenomena in the context of polymers as well as the Lee-Yang model
entail pecularities, which we derive in detail. A new result is that the
critical point of RNA-like branched polymers (with periodic base sequence)
looks like the critical point of classical one-component branched polymers,
but with one more critical exponent for the single strand polymer. A random
base sequence generates additional relevant interactions, and invalidates the
simple picture.

###### Contents

  1. 1 Introduction
    1. 1.1 Branched polymers
    2. 1.2 RNA-like polymers
  2. 2 Lee-Yang field theory
    1. 2.1 Only one parameter
    2. 2.2 One independent critical exponent
    3. 2.3 Two types of critical exponents
  3. 3 The $O\left(n\right)$-symmetric RNA model in more detail
    1. 3.1 Mapping of the $\chi$-sector to the Lee-Yang system
    2. 3.2 Polymer lengths
      1. 3.2.1 Final stage of pairing, $r\gg\tau^{\beta_{H}}$
      2. 3.2.2 Initial stage of pairing, $r\sim\tau^{\beta_{H}}$
  4. 4 Field theories with length variables
    1. 4.1 Relation to $O\left(n\right)$-symmetric model
    2. 4.2 Derivation of the $\chi$ field theory from a lattice model
  5. 5 RNA with random base sequence
    1. 5.1 Base sequence average
    2. 5.2 Effective action
  6. 6 Conclusions
  7. Appendix

##  1 Introduction

There is a natural connection between linear as well as branched polymers in a
good solvent and critical phenomena. One of the parameters of such a system is
polymer length $\ell$, and a large polymer length usually entails a long
correlation length $\xi\sim\ell^{\nu}$, where $\nu$ is the usual critical
exponent. In other words, $1/\ell$ plays the role of a temperature variable.
The temperature variable of lattice models and field theories for polymers is
unphysical in this sense, it acts as chemical potential for monomers and is a
function of polymer length. Another obvious connection are the Feynman
diagrams of polymer field theories, which topologically are identical with
polymer conformations.

The formal mapping of the statistics of a linear polymer in a good solvent to
a field theory and to critical phenomena dates back to 1972, when de Gennes
proved the equivalence to the $n=0$-limit of the $O\left(n\right)$-symmetric
$\varphi^{4}$ model [1]. The effect of the $n\rightarrow 0$-limit is to
eliminate all closed loops on a lattice, or on the level of Feynman diagrams.
There remain polymer chains extending from an external source to an external
sink (a magnetic field in $\varphi^{4}$ spin language).

The simplest physical application corresponds to the limit of a weak magnetic
field, where there is a low concentration of non-interacting polymers. The
next level of complexity are solutions with a finite concentration of linear
polymers, which can be described within the $\varphi^{4}$ field theory with a
finite magnetic field [2]. Here, however, we always restrict ourselves to the
single polymer case, i.e. weak external fields.

###  1.1 Branched polymers

The next level of complexity is a branched polymer in a good solvent, examined
in the field theory context by Lubensky and Isaacson in 1979 [3]. The model
describes the statistics of lattice animals (clusters on a lattice) and the
related problem of the statistics of branched polymers in the dilute limit.
The upper critical dimension is $8$.

The field theory used by these authors involves an $n\rightarrow 0$ limit in a
somewhat complicated way. A simplified version was written down by Parisi and
Sourlas [4], who demonstrated the equivalence to the Lee-Yang problem in two
less dimensions with the help of a hidden supersymmetry. The critical
exponents of the Lee-Yang system in one dimension are known exactly, and there
follow predictions for branched polymers in three dimensions. The predictions
agree numerically with the result of series expansions (“high temperature”
expansions) and Monte Carlo simulations [4], and the mapping appears to be
correct. This mapping also is crucial for RNA-like molecules. References to
more rigorous justifications or proofs can be found below.

###  1.2 RNA-like polymers

Figure 1: A directed linear polymer $\varphi$ forms a branched polymer with
three types of linear polymers, the single strand polymer $\varphi$
($\rightarrow$) and double strand polymers $\chi$ ($\rightleftharpoons$) and
$\psi$ ($\Rightarrow$).

One might wonder whether it makes sense to consider even more complex systems.
However, there is one type of system of utmost importance in biology: RNA
molecules (or DNA molecules). Furthermore, RNA molecules can be created with
arbitrary lengths and arbitray base sequences. One can imagine well-defined
physical situations, which are certainly interesting, even if not directly
relevant for biology. To our knowledge the critical properties of such systems
have not been examined from first principles.

Fig.(1) depicts a single strand RNA molecule $\varphi$, which partially has
formed double strands $\chi$ and double strands $\psi$. Every single strand
RNA molecule has a direction (on the ribose backbone), and actually only
oppositely aligned RNA strands bind to a double strand, denoted as $\chi$ in
fig.(1). The double strand polymer $\psi$ is unphysical in this sense, but it
is more general to allow both types of double polymer and simply ascribe
different binding energies to $\psi$ and $\chi$.

Deriving field theories from an RNA-like lattice model is nothing special,
also because the terms of the field theory have direct meaning. At first it
appears that directed polymers are mandatory, but as will become clear below,
the most interesting aspects of the problem can be examined with undirected
polymers, here also denoted as $\varphi$ and $\chi$. The point is, that
diagrams with the oppositely aligned $\chi$ are more singular than diagrams
with $\psi$, independently of whether polymer direction is accounted for or
ignored in the field theory. Polymer direction and length variables are
required if one wants to examine RNA molecules with a random base sequence.

The euclidian action of the simplest model variant is
$O\left(n\right)$-symmetric and reads [5]

$\displaystyle S\left(r_{0},\tau_{0},H,\alpha\right)$
$\displaystyle=\int\mathrm{d}^{d}x\left\\{\tfrac{1}{2}\sum_{\mu=1}^{n}\varphi_{\mu}\left(r_{0}-\nabla^{2}\right)\varphi_{\mu}+\tfrac{1}{2}\sum_{\mu\nu}\chi_{\mu\nu}\left(\tau_{0}-\nabla^{2}\right)\chi_{\mu\nu}\right\\}$
(1)
$\displaystyle+\int\mathrm{d}^{d}x\left\\{-g\sum_{\mu\nu}\chi_{\mu\nu}\varphi_{\mu}\varphi_{\nu}-\lambda\sum_{\mu\nu\rho}\chi_{\mu\nu}\chi_{\nu\rho}\chi_{\rho\mu}+\tfrac{\alpha}{2}\bar{\chi}^{2}-H\bar{\chi}\right\\}$
$\displaystyle+\int\mathrm{d}^{d}x\left\\{-g^{\prime}\bar{\chi}\sum_{\mu}\varphi_{\mu}\varphi_{\mu}-\lambda^{\prime}\bar{\chi}\sum_{\mu\nu}\chi_{\mu\nu}^{2}-\lambda^{\prime\prime}\bar{\chi}^{3}\right\\}\overset{\mathrm{decouples}}{\longleftarrow}$
$\displaystyle+\int\mathrm{d}^{d}x\left\\{+u_{1}\sum_{\mu\nu}\varphi_{\mu}^{2}\varphi_{\nu}^{2}+u_{2}\sum\chi_{\mu\nu}^{2}\chi_{\rho\tau}^{2}+u_{3}\sum\varphi_{\mu}^{2}\chi_{\rho\tau}^{2}-w_{2}^{\prime}\sum\chi_{\mu\nu}^{2}\chi_{\rho\tau}^{2}\chi_{\alpha\beta}^{2}+...\right\\}\stackrel{{\scriptstyle[}}{{\mathrm{}}}{levant}]{\mathrm{irre-}}{\longleftarrow}.$

Of interest is the $n\rightarrow 0$ limit. The last two lines are only written
down for completeness and can be ignored. The symbol
$\bar{\chi}=\sum_{\mu=1}^{n}\chi_{\mu\mu}$ is an abbreviation. The coupling
constant $g$ binds two single strands $\varphi$ to a double strand $\chi$, the
terms with coupling constants $\lambda$ and $\alpha$ are generated from $g$ or
$u_{2}$.

Without the (single strand) $\varphi$-field action (1) is a variant of the
model of Lubensky and Isaacson or Parisi and Sourlas, and decribes the
conventional one-component ($\chi$) branched polymer. A renormalization group
calculation shows, that the $\chi$-sector of the model retains the properties
of the one-component branched polymer, even in the presence of $\varphi$.
Moreover, the critical exponent $\eta_{\varphi}$ of the $\varphi$ field
coincides with the critical exponent $\eta_{\chi}$ of the $\chi$ field.

The mapping of the $\chi$-sector of action (1) to the Lee-Yang field theory
(eq.(3) below) in two less dimensions is the well-known mapping of
conventional branched polymers [4]. The tensor nature of $\chi$ does not
change anything - it suffices to restrict oneself to external fields in
$1,1$-direction and the field component $\chi_{1,1}$.

The origin of the mapping is the $\chi$-propagator in wave vector
representation

$\left\langle\chi_{\boldsymbol{k}}^{\mu\nu}\chi_{\boldsymbol{p}}^{\rho\tau}\right\rangle=\left(\frac{\delta_{\mu\rho}\delta_{\nu\tau}+\delta_{\mu\tau}\delta_{\nu\rho}}{2v_{k}}-\frac{\alpha\delta_{\mu\nu}\delta_{\rho\tau}}{v_{k}\left(v_{k}+n\alpha\right)}\right)\frac{\delta^{d}\left(\boldsymbol{k}+\boldsymbol{p}\right)}{\left(2\pi\right)^{d}},$
(2)

where $v_{k}=\tau_{0}+k^{2}$ 111There was a $\tfrac{1}{2}n\alpha$ in ref.([5])
instead of $n\alpha$. This makes no difference in the limit $n\rightarrow 0$..
It is expedient to think of two propagators, a normal propagator $1/v_{k}$ and
a more singular $-\alpha v_{k}^{-2}$-propagator. The latter one leads to the
dimensional shift, the minus sign correponds to the imaginary coupling
constant in the Lee-Yang field theory [6, 4].

##  2 Lee-Yang field theory

Understanding the Lee-Yang field theory is crucial due to its equivalence to
the chi-sector of action (1) in two additional space dimensions. This field
theory is frequently explored with renormalization group techniques, yet the
essential insights can be derived from elementary considerations.

For branched polymers one is interested in the case $H=\mathrm{const},$ and
the complication of a Fisher renormalization of the critical exponents and
other difficulties occur already in the less complicated Lee-Yang model. Many
calculations are simpler in the Lee-Yang model, and there are several exact
results.

Figure 2: The critical point of the Ising model with critical temperature
$T_{c}$ ramifies into the two branches of the Lee-Yang critical manifold in an
imaginary magnetic field $H$ for $T>T_{c}.$

The starting point is the partition sum of the Ising model, which has zeros
for purely imaginary magnetic field $H$, $\Im\left(H\right)=H^{\prime\prime}$
and $T>T_{c}$. A region
$\left[-H^{\prime\prime}\left(T\right),H^{\prime\prime}\left(T\right)\right]$
is without zeros. The curves $\pm H^{\prime\prime}\left(T\right)$ are a
critical manifold [7], see fig.(2). The ramification at $T_{c}$ can be
described with the Ising model. The branches for $T>T_{c}$ represent another
universality class, described by the field theory

$S_{\mathrm{LY}}\left(\tau_{0},H\right)=\int\mathrm{d}^{d}x\left\\{\tfrac{1}{2}\chi\left(\tau_{0}-\nabla^{2}\right)\chi+\lambda\chi^{3}-H\chi\right\\}$
(3)

with imaginary coupling constant $\lambda$ and imaginary magnetic field $H$
[7]. The field $\chi$ here is a scalar, there is no $n\rightarrow 0$ limit any
more, and there is only one type of propagator. The upper critical dimension
of action (3) is six. The critical exponents are known in five-loop order [8],
the values in one and two dimensions are known exactly.

###  2.1 Only one parameter

Replacing the integration variable $\chi$ with $\chi+c$ with constant
imaginary $c$ in the path integral of action (3) leads to

$S_{\mathrm{LY}}\left(\tau_{0},H\right)=S_{\mathrm{LY}}\left(\tau_{0}+6\lambda
c,H-\tau_{0}c-3\lambda c^{2}\right)+\mathrm{const}.$

An appropriate choice of $c$ transforms $\tau_{0}$ to an arbitrary constant
value $\tau_{c},$

$\displaystyle S_{\mathrm{LY}}\left(\tau_{0},H\right)$
$\displaystyle=S_{\mathrm{LY}}\left(\tau_{c},H+\tfrac{1}{12\lambda}\left(\tau_{0}^{2}-\tau_{c}^{2}\right)\right)+\mathrm{const}.$
(4)

The path integral therefore depends on only one parameter, the second argument
of $S_{\mathrm{LY}}$ on the r.h.s. of eq.(4), and for every $\tau_{0}$ the
critical point is reached for some $H=H_{c}$.

This first of all implies that singularities as function of $H$ are
independent of $\tau_{0}$. According to the usual definition of the critical
exponent $\delta$ for instance
$\left\langle\chi\right\rangle\sim\left(H-H_{c}\right)^{1/\delta}$.

Second it is clear, that the critical value of the second argument also gets
traversed linearly if $H$ is held constant and $\tau_{0}$ is varied (except
for $\tau_{0}=0$). For constant $H$ thus
$\left\langle\chi\right\rangle\sim\tau^{\beta_{H}}$ with
$\tau=\tau_{0}-\tau_{0c}$ and $\beta_{H}=1/\delta$. A more precise expression
also includes analytic contributions and reads
$\left\langle\chi\right\rangle\cong a_{0}+a_{1}\tau-A\tau^{\beta_{H}},$ i.e.
$\left\langle\chi\right\rangle$ has a peak for $\tau\rightarrow 0$ if
$0<\beta_{H}\leq 1$. The exponent $\beta_{H}$ is in this range for dimension
$d\geq 3$, but is negative for $d\leq 2$.

###  2.2 One independent critical exponent

It can be shown that only one of the usual critical exponents $\nu$ and $\eta$
is independent [9]. Instead of $\eta$ we use $\eta_{\chi}=\eta/2$, the
anomalous part of the scaling dimension of $\chi$.

Consider the classical equation of motion

$\displaystyle 0$ $\displaystyle=\tfrac{1}{Z}\int\mathrm{D}\chi\frac{\delta
S_{\mathrm{LY}}}{\delta\chi}e^{-S_{\mathrm{\mathrm{S_{\mathrm{LY}}}}}}=\left\langle\frac{\delta
S_{\mathrm{LY}}}{\delta\chi}\right\rangle=\tau_{0}\left\langle\chi\right\rangle+3\lambda\left\langle\chi^{2}\right\rangle-H.$

It is not useful to directly assign a scaling dimension to
$\left\langle\chi^{2}\right\rangle$ because of
$\left\langle\chi\right\rangle\neq 0$. But with $\chi=Q+\chi^{\prime}$ with
$\left\langle\chi^{\prime}\right\rangle=0$ it follows

$0=\tau_{0}Q+3\lambda Q^{2}+3\lambda\left\langle\chi^{\prime
2}\right\rangle-H.$ (5)

Now assume constant $\tau_{0}.$ The contributions
$Q\sim\left(H-H_{c}\right)^{\beta_{H}}$ and $Q^{2}$ are more singular than
$H$, but $\left\langle\chi^{\prime 2}\right\rangle$ has positive dimension and
consists of analytic and singular contributions. The analytic contributions
compensate $Q$ and $Q^{2}$, and it remains $\left\langle\chi^{\prime
2}\right\rangle\sim H-H_{c}.$ With the usual definitions of critical
exponents, written as scaling equivalences for a wavevector $k$,

$k\sim\left(\chi^{2}\right)^{1/\left(d-1/\nu_{\chi}\right)}\sim\left(H-H_{c}\right)^{1/\left(d/2+1-\eta_{\chi}\right)},$
(6)

it follows

$\nu_{\chi}=\frac{1}{d/2-1+\eta_{\chi}}.$ (7)

There thus is only one independent critical exponent (one primary field in the
Lee-Yang conformal field theory). For the exponent $\beta_{H}$, for instance,
one finds

$\beta_{H}=\frac{1}{\delta}=\frac{1}{\left(d/2+1-\eta_{\chi}\right)\nu_{\chi}}=\frac{d-2+2\eta_{\chi}}{d+2-2\eta_{\chi}}.$
(8)

###  2.3 Two types of critical exponents

Action (3) is asymmetric and in general one has
$Q=\left\langle\chi\right\rangle\neq 0$. For a perturbation theory one must
set $\chi=Q+\chi^{\prime}$ with $\left\langle\chi^{\prime}\right\rangle=0$, so
that the vertex function $\Gamma_{\chi^{\prime}}$ with a truncated external
$\chi^{\prime}$ vanishes (no tadpole diagrams). The shift leads to
$S_{\mathrm{LY}}\left(\tau_{0}+6\lambda Q,H-\tau_{0}Q-3\lambda Q^{2}\right),$
and in tree approximation $\Gamma_{\chi^{\prime}}=-H+\tau_{0}Q+3\lambda
Q^{2}.$

There now are two possibilities. In the Lee-Yang case one is interested in
$\tau_{0}=\mathrm{const}$, and $\Gamma_{\chi^{\prime}}$ drops out with
$Q\sim\left(H-H_{c}\right)^{1/\delta}$ (plus possibly analytic contributions
to $Q$). The fact that $Q$ now plays the role of a temperature variable in
$\tau_{0}+6\lambda Q$ formally corresponds to a critical exponent
$\beta_{\chi}=1$, which is tantamount to relation (7) between the critical
exponents because of the generic equation
$\beta_{\chi}/\nu_{\chi}=d/2-1+\eta_{\chi}$.

In the branched polymer case one is interested in $H=\mathrm{const}$, and
$\Gamma_{\chi^{\prime}}$ drops out with
$Q\left(\tau\right)\sim\tau^{\beta_{H}}$ with exponent $\beta_{H}$ from
eq.(8), plus possibly analytic contributions to $Q\left(\tau\right)$. For
$\beta_{H}<1$ then $6\lambda Q$ dominates in the effective temperature
variable $\tau_{0}+6\lambda Q$, and the temperature-dependent critical
exponents get Fisher-renormalized [10]. The correlation length $\xi$, for
instance, is $\xi\sim\tau^{-\nu_{H}}$ with $\nu_{H}=\nu\beta_{H}$, instead of
$\xi\sim\tau^{-\nu}$. The suffix $H$ denotes the $H=\mathrm{const}$ critical
exponents, relevant for branched polymers. The temperature independent
exponent $\eta_{\chi}$ is not renormalized.

This seems to be somewhat paradoxical in one and two dimensions, where
$\beta_{H}$ is negative, and the effective temperature variable
$\tau_{0}+6\lambda Q$__ diverges __ for $\tau_{0}\rightarrow\tau_{0c}.$ The
difficulty apparently disappears in a consistent loop expansion, when the loop
expansion of $Q$ also is used instead of the exact value. At any rate, the
exponents $\nu$ and $\gamma$ also become negative in one and two dimensions,
and the effective exponents $\nu_{H}=\nu\beta_{H}$ and
$\gamma_{H}=\gamma\beta_{H}$ remain positive.

Here is a list of exactly known Lee-Yang critical exponents

$\begin{array}[]{c|rrrr}&\beta_{H}&\eta_{\chi}&\nu_{\chi}&\gamma_{\chi}\\\
\hline\cr d=0&-1\\\ d=1&-\tfrac{1}{2}&-\tfrac{1}{2}&-1&-3\\\
d=2&-\tfrac{1}{6}&-\tfrac{2}{5}&-\tfrac{5}{2}&-7\\\
d=6&\tfrac{1}{2}&0&\tfrac{1}{2}&1\end{array}.$ (9)

Of particular interest is the case $d=1$, which corresponds to the
$\chi$-sector of polymer action (1) in three space dimensions.

##  3 The $O\left(n\right)$-symmetric RNA model in more detail

Figure 3: The coupling constant flow of model (1) for $\alpha>0$, $\epsilon>0$
(schematic).

We now examine the $O\left(n\right)$ symmetric RNA model (1) in more detail
[5]. Shifting $\chi$ according to

$\chi_{\mu\nu}\rightarrow\delta_{\mu\nu}Q+\chi_{\mu\nu}$ (10)

in order to eliminate the linear term in $\chi$ gives

$S\left(r_{0},\tau_{0},H,\alpha\right)\rightarrow
S\left(r_{0}-2gQ,\tau_{0}-6\lambda Q,H-\tau_{0}Q+3\lambda
Q^{2},\alpha+4u_{2}Q^{2}\right)-4u_{2}Q\int\mathrm{d}^{d}x\bar{\chi}\sum\chi_{\mu\nu}^{2}+...$
(11)

The $\varphi$-sector is only affected by its temperature variable,
$r_{0}\rightarrow r_{\text{0}}-2gQ$, everything else is as for a one-component
($\chi$) branched polymer [4]. In particular, a
$\int\mathrm{d}^{d}x\tfrac{\alpha}{2}\bar{\chi}^{2}$ is generated if not
already present.

Only diagrams with a maximal number of $\chi$-propagators $-\alpha v_{k}^{-2}$
from eq.(2) are of interest. This excludes any $\varphi$-propagators from the
$\chi$-sector. The reason is simple. Any internal $\varphi$ in a
$\Gamma_{\chi\chi}$ or $\Gamma_{\chi\chi\chi}$ vertex function generates a
$\varphi$-loop, without a propagator $-\alpha v_{k}^{-2}$. But the most
singular diagrams contain exactly one propagator $-\alpha v_{k}^{-2}$ per loop
[6], see also fig.(5) in the appendix. In other words, the critical point of
the $\chi$-sector is not affected by the single strand polymer $\varphi$,
behaves like a branched polymer of conventional type, and can be mapped to the
Lee-Yang system. This reflects itself in the one-loop renormalization group
flow equations [5]

$\displaystyle\mathrm{d}\bar{\lambda}/\mathrm{d}l$
$\displaystyle=\bar{\lambda}\left(\tfrac{\epsilon}{2}-81\bar{\lambda}^{2}\right),$
(12) $\displaystyle\mathrm{d}\bar{g}/\mathrm{d}l$
$\displaystyle=\bar{g}\left(\tfrac{\epsilon}{2}-2\bar{g}^{2}-24\bar{g}\bar{\lambda}+9\bar{\lambda}^{2}\right)$

for the coupling constants $\bar{\lambda}=\lambda\sqrt{\alpha K_{d}}$ and
$\bar{g}=g\sqrt{\alpha K_{d}}$, with
$K_{d}=2^{1-d}\pi^{-d/2}/\Gamma\left(d/2\right)$ and $\epsilon=8-d$. The flow
of the coupling constant $\bar{\lambda}$ does not depend on the
$\chi$-$\varphi$ coupling with coupling constant $\bar{g}$, see also fig.(3).

The critical exponents
$\eta_{\chi}=-\tfrac{\epsilon}{18}+O\left(\epsilon^{2}\right)$ and
$\nu_{\chi}=\tfrac{1}{2}+\tfrac{5}{36}\epsilon+O\left(\epsilon\right)^{2}$ at
the stable fixed point $\bar{\lambda}\cong\tfrac{1}{18}\sqrt{2\epsilon}$,
$\bar{g}=\tfrac{1}{6}\sqrt{2\epsilon}$ agree with the Lee-Yang exponents in
$6-\epsilon$ dimensions [3]. For the $\varphi$-field one finds
$\eta_{\varphi}=-\tfrac{\epsilon}{18}+O\left(\epsilon^{2}\right)$. In fact, in
the massless theory there is a hidden symmetry in the perturbation theory and
in the flow equations (12) for $\bar{g}=3\bar{\lambda}$, and it follows
$\eta_{\varphi}=\eta_{\chi}$. This hidden symmetry is illustrated in fig.(5).
A proof is sketched in the appendix. The symmetry essentially says that there
is no difference between $\varphi$ and $\chi$ at the stable fixed point.

The exponent
$\nu_{\varphi}=\tfrac{1}{2}+\tfrac{\epsilon}{36}+O\left(\epsilon^{2}\right)$,
in contrast, is new. One might think that $\nu_{\varphi}$ could be related to
other exponents with the help of the equation of motion (see eq.(7)), but this
seems not to be the case.

The renormalization group calculations can be done in the massless theory, but
the effective critical exponent $\nu_{H}=$$\beta_{H}\nu_{\chi}$ is the Fisher-
renormalized variant.

We now return to the temperature variables $r_{0}$ and $\tau_{0}$ in eq.(11).
For $\bar{g}=3\bar{\lambda}$ one finds $r_{0}\rightarrow r_{0}+6\lambda
A\tau^{\beta_{H}}+\mathrm{const}$ and $\tau_{0}\rightarrow\tau_{0}+6\lambda
A\tau^{\beta_{H}}+\mathrm{const}$. There thus is a crossover at
$r_{0}-r_{0c}\cong 6\lambda A\tau^{\beta_{H}}$. For small $r_{0}-r_{0c}$ the
dependence on $r_{0}$ drops out completely, and the two field types have the
same mass $6\lambda A\tau^{\beta_{H}}.$

###  3.1 Mapping of the $\chi$-sector to the Lee-Yang system

There are some subtle points in the mapping of the $\chi$-sector of action (1)
to action (3) in dimension $d-2$ as in [6, 4], in particular for space
dimensions $d\leq 4$. In fact, the similar mapping of the random magnetic
field Ising model to the normal Ising model in two less dimensions turned out
to be wrong for $d\leq 4$ [11].

One symptom of a difficulty in the polymer case is related to the exponent
$\beta_{H}<0$ for $d\leq 4$. A negative $\beta_{H}$ implies a diverging $Q$
and diverging effective $r_{0}$-, $\tau_{0}$\- and $\alpha$-values (eq.(11)).
This difficulty disappears if one sticks to a consistent loop expansion, when
the expansion of $\beta_{H}$ is used instead of the exact value.

At any rate, in the polymer case the mapping appears to be correct. As already
noted by Parisi and Sourlas there is numerical evidence. Furthermore,
independently of field theory, Brydges and Imbrie [12] proved the equivalence
of a particular type of branched polymers with a hard core gas, which in turn
can be mapped to the Lee-Yang system. More recent references concerning the
mapping are [13, 11]. We shall not discuss the question any further. It
concerns conventional branched polymers, and is not specific to the RNA model
(1).

###  3.2 Polymer lengths

Figure 4: Diagrams contributing to polymer lengths, up to one loop. Self-
energies of external lines are not shown. The dot is a $\varphi_{1}^{2}$
insertion, the bar denotes the $\alpha$ propagator.

To make contact with possible experiments or series expansions it is necessary
to eliminate one of the temperature parameters $r_{0}$ and $\tau_{0}$ in
favour of the polymer length. The polymer lengths $s_{\varphi}$ and $s_{\chi}$
of the polymer types are determined by $r_{0}$ and $\tau_{0}$, which in
$e^{-S}$ act like negative chemical potentials for monomers. In a real
experiment, however, the total length $s=s_{\varphi}+2s_{\chi}$ is given, and
effectively only one parameter (temperature, composition of solution, …) can
be varied.

To have any polymers at all requires (small) external fields, for instance a
constant external field in $\varphi_{1}$-direction. Every expectation value in
this universe contains a factor
$\varPhi_{11}=$$\varphi_{1}\left(0\right)\int\mathrm{d}^{d}\mathrm{x}\varphi_{1}\left(\boldsymbol{x}\right)\sim
k^{-2+2\eta_{\varphi}}$. From the length operators $\sum\varphi_{\mu}^{2}$ and
$\sum\left(\delta_{\mu\nu}Q+\chi_{\mu\nu}\right)^{2}\cong 2Q\bar{\chi}$ it
follows (fig.(4))

$\displaystyle s_{\varphi}$
$\displaystyle=\left\langle\varPhi_{11}\int\mathrm{d}^{d}x\varphi_{1}^{2}\right\rangle/\left\langle\varPhi_{11}\right\rangle=\left\langle\varPhi_{11}\right\rangle\Gamma_{\varphi^{2}\varphi\varphi}\sim
k^{-1/\nu_{\varphi}},$ (13) $\displaystyle s_{\chi}$
$\displaystyle=\left\langle\varPhi_{11}\int\mathrm{d}^{d}x2Q\bar{\chi}\right\rangle/\left\langle\varPhi_{11}\right\rangle=2Q\left\langle\varPhi_{11}\right\rangle
G_{k=0}\Gamma_{\chi\varphi\varphi}.$ (14)

The length $s_{\varphi}$ contains $\Gamma_{\varphi^{2}\varphi\varphi}\sim
k^{\left(\gamma_{\varphi}-1\right)/\nu_{\varphi}}$, the $\varphi$ self-energy
with a $\tfrac{1}{2}\int\mathrm{d}^{d}x\varphi^{2}$ insertion. The vertex
function $\Gamma_{\chi\varphi\varphi}\sim
k^{\epsilon/2-\eta_{\chi}-2\eta_{\varphi}}$ is the effective coupling constant
$\bar{g}$, $G_{k}\sim k^{-2+2\eta_{\chi}}$ is the normal $\chi$-propagator. We
now write $r=r_{0}-r_{c}$ and $\tau=\tau_{0}-\tau_{c}$. The scaling dimensions
can be calculated in the massless theory, but the scaling as a function of $r$
and $\tau$ must be examined in detail, and the crossover at
$r\sim\tau^{\beta_{H}}$ plays a crucial role. The propagator
$G_{k=0}\sim\tau^{-\gamma_{\chi}}$ comes from the $\chi$ sector (Lee-Yang
world) and does not depend on $r$.

####  3.2.1 Final stage of pairing, $r\gg\tau^{\beta_{H}}$

For large $r$ one has $\Gamma_{\chi\varphi\varphi}\sim
r^{\left(\epsilon/2-\eta_{\chi}-2\eta_{\varphi}\right)\nu_{\varphi}}$ and
$\Gamma_{\varphi^{2}\varphi\varphi}\sim r^{\left(\gamma_{\varphi}-1\right)}$.
It follows $s_{\varphi}\sim r^{-1}$ and

$s_{\chi}\sim
Q\tau^{-\gamma_{H}}r^{\left(\epsilon/2-\eta_{\chi}-2\eta_{\varphi}\right)\nu_{\varphi}}r^{2\nu_{\varphi}\left(\eta_{\varphi}-1\right)}\sim
Q\tau^{-\gamma_{H}}r^{-\nu_{\varphi}/\nu_{\chi}}.$

Here we have used $\beta_{\chi}/\nu_{\chi}=2-\tfrac{\epsilon}{2}+\eta_{\chi}$.
In the final stage of single-strand-pairing $s_{\chi}\cong s/2$ is
approximately constant, which allows to eliminate $r$. It follows

$s_{\varphi}\left(\tau\right)\sim r^{-1}\sim
Q^{-\nu_{\chi}/\nu_{\varphi}}\tau^{\gamma_{H}\nu_{\chi}/\nu_{\varphi}}=\overset{\mathrm{def}}{\sim}\tau^{Z}.$

The condensate $Q$ has a peak above $d=4$ for $\tau\rightarrow 0$, and it
remains
$Z=\gamma_{H}\nu_{\chi}/\nu_{\varphi}\cong\tfrac{1}{2}+\tfrac{7}{37}\epsilon.$
Below five dimensions, however, $\beta_{H}$ is negative and
$Q\sim\tau^{\beta_{H}}$ diverges. Thus

$Z=\left(\gamma_{H}-\beta_{H}\right)\nu_{\chi}/\nu_{\varphi}\overset{d=3}{=}-2/\nu_{\varphi,d=3},$

which is negative, except $\nu_{\varphi}$ also is negative (as are
$\nu_{\chi}$ and $\beta_{H}$) for $d<5$. A negative exponent $Z$ makes no
sense, and it remains to be shown that $\nu_{\varphi}$ becomes negative. We
have calculated $\nu_{\varphi}$ in two-loop order, but this is by no means
sufficient.

####  3.2.2 Initial stage of pairing, $r\sim\tau^{\beta_{H}}$

For $r\ll\tau^{\beta_{H}}$ the variables $r$ and $\tau$ are dominated by
$Q\sim\tau^{\beta_{H}}$, see eq.(11). This is unphysical, because $r$ drops
out completely, and there is no way to have a constant
$s=s_{\varphi}+2s_{\chi}$. The initial stage of pairing thus must occur in the
crossover region $r\sim\tau^{\beta_{H}}$, which makes it more difficult to get
clear statements.

One may wonder what the crossover at $r\sim\tau^{\beta_{H}}$ means for the
massless theory. Technically it is no problem to have $r=\tau=0$ in the field
theory. But physically one always has a finite polymer, and the massless limit
means to have a large correlation length $\xi$ and to look at small
wavevectors, but with $k>\xi^{-1}$. The crossover then still is in play, and
it is simplest to consider the parameter range $r\ll\tau^{\beta_{H}}$, where
$\varphi$ and $\chi$ have the same mass $\propto\tau^{\beta_{H}}$.

##  4 Field theories with length variables

Directed polymers with length variables are useful to understand how the
$O\left(n\right)$ model, which has directionless polymers, can describe RNA
molecules with an internal direction (only oppositely aligned RNA strands bind
to a double strand). Furthermore, the model with length variables allows to
examine the effect of a random base sequence.

A derivation of polymer field theories with length variables $s$ instead of
the $O\left(n\right)$ indices (as in eq.(1)) can be found in ref.([14]) and in
section (4.2) below. When “frequencies” are used instead of length variables,
then these frequencies flow unchanged from external sources to external sinks,
exactly like the $O\left(n\right)$-indices. There are no frequency integrals,
and the calculations are similar to the $O\left(n\right)$-model, but without
the $n\rightarrow 0$ limit. Closed loops drop out because length variables
cannot increase consistently in a loop.

The direction of increasing length variable $s$ gives the single strand
$\varphi$ ($\rightarrow$) and the double strand polymers $\psi$
($\Rightarrow$) a direction, and there actually are pairs of fields,
$\varphi\left(\boldsymbol{x},s\right)$,
$\tilde{\varphi}\left(\boldsymbol{x},s\right)$ and
$\psi\left(\boldsymbol{x},s,s^{\prime}\right)$,
$\tilde{\psi}\left(\boldsymbol{x},s,s^{\prime}\right)$. If double polymer
twisting is ignored (which is justified for long polymers), then $\psi$ and
$\tilde{\psi}$ are symmetric in their length variables (polymer twisting is
difficult to define for $d\geq 3$ anyway). The double strand polymer $\chi$
($\rightleftharpoons$) with oppositely aligned single strands has no internal
direction, and requires only one field $\chi$, but with
$\chi\left(\boldsymbol{x},s,s^{\prime}\right)\neq\chi\left(\boldsymbol{x},s^{\prime},s\right)$.

The (euclidian) action for the RNA-like problem in frequency representation
reads (variables conjugate to a length variable under Fourier transformation
are called “frequencies” $\omega$, $\omega^{\prime}$, $\omega^{\prime\prime}$)

$\displaystyle S$ $\displaystyle=S_{\varphi}+S_{\chi}+S_{\varphi\chi},$ (15)
$\displaystyle S_{\varphi}$
$\displaystyle=\int\mathrm{d}^{d}x\left\\{\int_{\omega}\tilde{\varphi}_{-\omega}\left(r_{0}-\nabla^{2}-i\omega\right)\varphi_{\omega}+u_{\varphi}\left(\int_{\omega}\tilde{\varphi}_{-\omega}\varphi_{\omega}\right)^{2}\right\\},$
$\displaystyle S_{\chi}$
$\displaystyle=\int\mathrm{d}^{d}x\left\\{\tfrac{1}{2}\int_{\omega,\omega^{\prime}}\chi_{-\omega^{\prime},-\omega}\left(\tau_{0}-\nabla^{2}-i\left(\omega-\omega^{\prime}\right)\right)\chi_{\omega,\omega^{\prime}}+\tfrac{\alpha}{2}\left(\int_{\omega}\chi{}_{\omega,-\omega}\right)^{2}-H\int_{\omega}\chi_{\omega,-\omega}\right\\}$
$\displaystyle\qquad+\int\mathrm{d}^{d}x\left\\{-\lambda\int_{\omega,\omega^{\prime},\omega^{\prime\prime}}\chi_{\omega,-\omega^{\prime}}\chi_{\omega^{\prime},-\omega^{\prime\prime}}\chi_{\omega^{\prime\prime},-\omega}+u_{\chi}\left(\int_{\omega,\omega^{\prime}}\chi_{-\omega^{\prime},-\omega}\chi_{\omega,\omega^{\prime}}\right)^{2}\right\\},$
$\displaystyle S_{\varphi\chi}$
$\displaystyle=\int\mathrm{d}^{d}x\left\\{-2g\int_{\omega,\omega^{\prime}}\tilde{\varphi}_{\omega}\varphi_{\omega^{\prime}}\chi_{-\omega,-\omega^{\prime}}+u_{\varphi\chi}\left(\int_{\omega}\tilde{\varphi}_{-\omega}\varphi_{\omega}\right)\left(\int_{\omega,\omega^{\prime}}\chi_{-\omega^{\prime},-\omega}\chi_{\omega,\omega^{\prime}}\right)\right\\}.$

For simplicity all space arguments of the fields are suppressed. The
“frequency” integrals $\int_{\omega}=\int\mathrm{d}\omega/\left(2\pi\right)$
look complicated, but frequencies run unchanged from source to sink through
Feynman diagrams, exactly like the indices of $O\left(n\right)$ models.

The action $S_{\varphi}$ describes a directed polymer ($\rightarrow$). The
action $S_{\chi}$ with $\lambda=0$ and $\alpha=0$ describes a linear polymer
($\rightleftharpoons$) with two length variables, i.e. in effect a polymer
without direction. The complete action $S_{\chi}$ describes a conventional
branched polymer. The action $S$ in total is the RNA model with only
oppositely aligned double strand polymers, the excluded volume interactions
with coupling constants $u_{\varphi}$, $u_{\chi}$ and $u_{\varphi\chi}$ are
irrelevant at the critical dimension $8$.

The double polymer $\psi$ ($\Rightarrow$) is of minor interest, there are no
such RNA double strands. But for completeness,

$\displaystyle S_{\psi}$
$\displaystyle=\int\mathrm{d}^{d}x\left\\{\int_{\omega,\omega^{\prime}}\tilde{\psi}_{-\omega,-\omega^{\prime}}\left(\hat{\tau}_{0}-\nabla^{2}-i\hat{w}\left(\omega+\omega^{\prime}\right)\right)\psi_{\omega,\omega^{\prime}}-\hat{g}\int_{\omega,\omega^{\prime}}\left(\tilde{\psi}_{\omega,\omega^{\prime}}\varphi_{-\omega}\varphi_{-\omega^{\prime}}+\tilde{\varphi}_{\omega}\tilde{\varphi}_{\omega^{\prime}}\psi_{-\omega,-\omega^{\prime}}\right)\right\\}$
$\displaystyle\qquad-\hat{\lambda}\int_{\omega,\omega^{\prime},\omega^{\prime\prime}}\chi_{\omega,\omega^{\prime}}\tilde{\psi}_{\omega^{\prime\prime},-\omega}\psi_{-\omega^{\prime\prime},-\omega^{\prime}}+u_{\psi}\int\mathrm{d}^{d}x\left(\int_{\omega,\omega^{\prime}}\tilde{\psi}_{-\omega,-\omega^{\prime}}\psi_{\omega,\omega^{\prime}}\right)^{2}+...$

The action $S_{\psi}+S_{\varphi}$ with $\hat{\lambda}=0$ is a model with only
$\Rightarrow$ double polymers without a condensate $Q$, the critical dimension
is $6$ [14].

###  4.1 Relation to $O\left(n\right)$-symmetric model

It is now possible to repeat the calculations done with the $O\left(n\right)$
action integral (1) with action integral (15). We only sketch some points. The
propagators are

$\displaystyle\left\langle\tilde{\varphi}_{\omega^{\prime}}^{\boldsymbol{p}}\varphi_{\omega}^{\boldsymbol{k}}\right\rangle$
$\displaystyle=\frac{\delta\left(\omega+\omega^{\prime}\right)}{r_{0}+k^{2}-i\omega}\frac{\delta^{d}\left(\boldsymbol{p}+\boldsymbol{k}\right)}{\left(2\pi\right)^{d+1}},$
(16)
$\displaystyle\left\langle\chi_{\omega,\omega^{\prime}}^{\boldsymbol{p}}\chi_{\nu,\nu^{\prime}}^{\boldsymbol{k}}\right\rangle$
$\displaystyle=\left(\frac{\delta\left(\omega+\nu^{\prime}\right)\delta\left(\omega^{\prime}+\nu\right)}{v_{k}-i\left(\omega-\omega^{\prime}\right)}-\alpha\frac{\delta\left(\omega+\omega^{\prime}\right)}{v_{k}-i\left(\omega-\omega^{\prime}\right)}\frac{\delta\left(\nu+\nu^{\prime}\right)}{v_{k}-i\left(\nu-\nu^{\prime}\right)}\right)\frac{\delta^{d}\left(\boldsymbol{p}+\boldsymbol{k}\right)}{\left(2\pi\right)^{d+2}},$

with $v_{k}=\tau_{0}+k^{2}$, and the analogy to the $O\left(n\right)$-model
propagator (2) is clear.

To get rid of terms linear in $\chi$ one shifts $\chi$ according to

$\chi_{\omega,\omega^{\prime}}\rightarrow
2\pi\delta\left(\omega+\omega^{\prime}\right)Q\left(\tau_{0},\omega\right)+\chi{}_{\omega,\omega^{\prime}}.$
(17)

This equation is the counterpart to eq.(10). The condensate $Q$ now depends on
$\tau_{0}$ and $\omega$. The shift reproduces eq.(11), except that the linear
term

$\int\mathrm{d}^{d}x\int_{\omega}\left(-H-3\lambda
Q^{2}+\left(\tau_{0}-2i\omega\right)Q\right)\chi_{\omega,-\omega}$

now contains $\tau_{0}-2i\omega$ instead of $\tau_{0}$.

To compare the renormalization group calculation it suffices to determine the
combinatoric factors of the diagrams. We have done this in one-loop order and
find the same factors. The coupling constant $2g$ in $S_{\varphi\chi}$
compensates a factor $2$, which comes from the square in $g\chi\varphi^{2}$ in
the $O\left(n\right)$ action (1).

It is also clear that the double strand polymer $\psi\left(\Rightarrow\right)$
does not contribute to the $\chi$ sector. Any internal $\psi/\tilde{\psi}$ in
a $\Gamma_{\chi\chi}$ or $\Gamma_{\chi\chi\chi}$ vertex function either
generates a $\psi$-loop, without a propagator $-\alpha v_{k}^{-2}$, or
terminates at a $\psi\tilde{\varphi}^{2}$ or a $\tilde{\psi}\varphi^{2}$,
which generates a $\varphi$-loop, also without a propagator $-\alpha
v_{k}^{-2}$. But the most singular diagrams of the $\chi$ sector require
exactly one such propagator per loop [6], see fig.(5) in the appendix.

This confirms that the $O\left(n\right)$ symmetric action (1) with undirected
polymers describes a RNA-like situation, where only oppositely aligned single
strands form double strands.

###  4.2 Derivation of the $\chi$ field theory from a lattice model

We sketch the somewhat intricate derivation of the $\chi$-sector
($\rightleftharpoons$) of eq.(15) from a lattice model. The derivation in
ref.([5]) was incomplete.

For each lattice point $i$ there are operators $C_{i}^{\mu,\nu}$ carrying two
length indices, $\mu,\nu\in\mathbb{Z}$. All operators commute and
$C_{i}^{\mu\nu}C_{i}^{\rho\tau}C_{i}^{\beta\gamma}=0$. There is an “average”
with $\left\langle C_{i}^{\mu,\nu}\right\rangle=0$ and

$\left\langle
C_{i}^{\mu\nu}C_{j}^{\rho\tau}\right\rangle=\delta_{ij}\delta_{\mu\tau}\delta_{\nu\rho}.$
(18)

The transposition of the length variables in eq.(18) is essential. Such
operators can be generated from pairs of simpler operators according to
$C_{i}^{\mu\nu}=c_{i}^{\mu\nu}+\bar{c}_{i}^{\nu\mu}$, where $c$ and $\bar{c}$
also commute and are nilpotent in the sense
$c_{i}^{\mu\nu}c_{i}^{\rho\tau}=\bar{c}_{i}^{\mu\nu}\bar{c}_{i}^{\rho\tau}=0$.
The only non-vanishing average of $c$ and $\bar{c}$ operators is
$\left\langle\bar{c}_{i}^{\mu\nu}c_{j}^{\rho\tau}\right\rangle=\tfrac{1}{2}\delta_{ij}\delta_{\mu\rho}\delta_{\nu\tau}$.

Consider the partition sum

$Z=\left\langle\prod_{i,j;\mu,\nu}e^{C_{i}^{\nu-1,\mu+1}v_{i,j}C_{j}^{\mu,\nu}}\right\rangle=\left\langle\prod_{i,j;\mu,\nu}\left(1+C_{i}^{\nu-1,\mu+1}v_{i,j}C_{j}^{\mu,\nu}\right)\right\rangle,$
(19)

where $v_{i,j}=1$ for next neighbour lattice points and $v_{i,j}=0$ otherwise.
Only the constant and linear part of the expansion of the exponential function
contribute because of $C_{i}^{3}=0$ and $\left\langle
C_{i}^{2}C_{j}\right\rangle=0$.

To understand the meaning of $Z$ it suffices to concentrate on the factors
containing lattice point $0$. If $m$ and $n$ denote the neighbours of lattice
point $0$ then

$\displaystyle Z$
$\displaystyle=\left\langle...\prod_{m;\mu,\nu}\left(1+C_{0}^{\nu-1,\mu+1}C_{m}^{\mu,\nu}\right)\prod_{n;\rho,\tau}\left(1+C_{0}^{\tau-1,\rho+1}C_{n}^{\rho,\tau}\right)\right\rangle$
$\displaystyle=\left\langle...\prod_{m,n;\mu,\nu,\rho\tau}\left(1+C_{0}^{\nu-1,\mu+1}C_{0}^{\tau-1,\rho+1}C_{m}^{\mu,\nu}C_{n}^{\rho,\tau}\right)\right\rangle=\left\langle...\prod_{m,n;\rho\tau}\left(1+C_{m}^{\tau-2,\rho+2}C_{n}^{\rho,\tau}\right)\right\rangle.$

Here it has been used that $C_{0}$ must occur in pairs in the average. The
remaining expression has the same form as the original one, except that
lattice point $0$ is missing and there are mutually exclusive paths or
“propagations” from neighbour $m$ to lattice point $0$ to neighbour $n$, in
which one length variable increments by two and the other length variable
decrements by two. The case $m=n$ would be a closed loop, with inconsistent
length variables. In total, $Z$ generates all self-avoiding paths of double
strands of the $\rightleftharpoons$-type. All closed loops contain
inconsistent length variables, and to have any real paths requires external
sources and sinks, factors in $Z$ like $1+C_{i}^{\mu\nu}$.

The next step is a Hubbard-Stratonovich transformation of eq.(19),

$Z=\left\langle
e^{\tfrac{1}{2}CVC}\right\rangle=\left\langle\int\mathrm{D}\chi
e^{-\tfrac{1}{2}\chi V^{-1}\chi+\chi C}\right\rangle=\int\mathrm{D}\chi
e^{-\tfrac{1}{2}\chi
V^{-1}\chi+\ln\left(1+\tfrac{1}{2}\sum\chi_{i}^{\mu\nu}\chi_{i}^{\nu\mu}\right)}.$

The remaining average $\left\langle e^{\chi C}\right\rangle$ can be performed
most easily with the help of
$C_{i}^{\mu\nu}=c_{i}^{\mu,\nu}+\bar{c}_{i}^{\nu,\mu}$.

It remains to determine the inverse $V^{-1}$ of the linear operator $V$ in the
exponent in eq.(19). This can be done by writing
$\sum_{i,j;\mu,\nu}C_{i}^{\nu-1,\mu+1}v_{i,j}C_{j}^{\mu,\nu}$ in the
wavevector - frequency representation, where $V$ is diagonal. Interactions
with the $\varphi$ field can be added directly to the field theory, or at
operator level in eq.(19). The result is action $S_{\chi}$ from eq.(15).

##  5 RNA with random base sequence

The base sequence of real RNA is neither random nor deterministic, but a
random base sequence is a reasonable approximation. A random base sequence can
be examined with action (15) with length variables, not with the
$O\left(n\right)$ symmetric action (1).

A simple physical quantity is the probability
$P\left(\boldsymbol{x}\right)\mathrm{d}^{d}x$ to find the end of a
$\varphi$-polymer in the vicinity of point $\boldsymbol{x}$, if its start
point is at the origin. If $\left[...\right]_{*}$ denotes the base sequence
average and $\left\langle...\right\rangle$ the path integral, then the answer
is contained in the generating functional

$\displaystyle J\left(h\right)$
$\displaystyle=\left[\ln\left\langle\tilde{\varphi}\left(0,0\right)\int\mathrm{d}^{d}x\mathrm{d}sh\left(\boldsymbol{x}\right)\varphi\left(\boldsymbol{x},s\right)\right\rangle\right]_{*}$
(20)

according to

$P\left(\boldsymbol{x}\right)=\left.\frac{\delta}{\delta
h\left(\boldsymbol{x}\right)}J\left(h\right)\right|_{h=1}=\left[\frac{\left\langle\tilde{\varphi}\left(0,0\right)\int\mathrm{d}s\varphi\left(\boldsymbol{x},s\right)\right\rangle}{\int\mathrm{d}^{d}x\mathrm{d}s\left\langle\tilde{\varphi}\left(0,0\right)\varphi\left(\boldsymbol{x},s\right)\right\rangle}\right]_{*}.$

The $\varphi$ propagator is not properly normalized, and it is essential to
normalize it before performing the base sequence average (one also could omit
the $s$-integral to obtain the probability to find the end of a polymer with
given length $s$ at $\boldsymbol{x}$). As usual one uses the replica trick,
and instead of calculating $\ln\left\langle...\right\rangle$ one calculates

$\left\langle...\right\rangle^{n}=\int\prod_{i=1}^{n}\left\\{\mathrm{D}\varphi_{i}\mathrm{D\tilde{\varphi}_{i}}\mathrm{D}\chi_{i}\tilde{\varphi}_{i}\left(0,0\right)\int\mathrm{d}^{d}x\mathrm{d}sh\left(\boldsymbol{x}\right)\varphi_{i}\left(\boldsymbol{x},s\right)\right\\}e^{-\sum_{i=1}^{n}S\left(\varphi_{i},\tilde{\varphi}_{i},\chi_{i}\right)},$

for arbitrary $n\in\mathbb{N}$, and then uses the analytic continuation
$\ln\left\langle...\right\rangle=\lim_{n\rightarrow
0}\tfrac{1}{n}\left(\left\langle...\right\rangle^{n}-1\right).$ Every replica
contains a $\varphi$-polymer, which can form double polymers as in fig.(1).

###  5.1 Base sequence average

In the case of a random base sequence the (negative) binding energy $\tau_{0}$
in eq.(15) is to be replaced with a random binding energy
$\tau_{0}\left(s,s^{\prime}\right)$, depending on the two length variables $s$
and $s^{\prime}$ of the double polymer. The base sequence average then is

$\left[...\right]_{*}=\int\mathrm{D}\tau_{0}P\left(\tau_{0}\right)e^{-\tfrac{1}{2}\int\mathrm{d}^{d}x\int_{-\infty}^{\infty}\mathrm{d}s\mathrm{d}s^{\prime}\sum_{i=1}^{n}\chi_{i}\left(s^{\prime},s\right)\tau_{0}\left(s,s^{\prime}\right)\chi_{i}\left(s,s^{\prime}\right)}.$

To proceed one must specify the probability distribution
$P\left(\tau_{0}\right)$ and $\tau_{0}\left(s,s^{\prime}\right).$ This can be
done with discrete length variables. Let $\mu,\mu^{\prime}\in\mathbb{N}_{0}$
be the lengths along the strands of a double polymer. A $\varphi$-monomer
cannot be paired with itself, and thus $\mu\neq\mu^{\prime}$.

The base sequence $b_{\mu}\in\left\\{0,1,2,3\right\\}$ of a single strand is a
sequence of random variables with probability distribution
$p\left(b\right)=\sum_{r=0}^{3}p_{r}\delta_{b,r}$, with probabilities $p_{r}$
with $\sum_{r=0}^{3}p_{r}=1$. To a $\chi$-monomer there corresponds the energy
$\tau\left(b_{\mu},b_{\mu^{\prime}}\right)$, where $\tau\left(.,.\right)$ is a
symmetric $4\times 4$ matrix. The average then can be written

$\left(\prod_{\mu}\sum_{b_{\mu}}p\left(b_{\mu}\right)\right)e^{-\tfrac{1}{2}\int\mathrm{d}^{d}x\sum_{i,\mu}\chi_{\mu^{\prime},\mu}^{i}\tau\left(b_{\mu},b_{\mu^{\prime}}\right)\chi_{\mu,\mu^{\prime}}^{i}}=\left[e^{X}\right]_{*}=e^{\kappa_{1}\left(X\right)+\tfrac{1}{2!}\kappa_{2}\left(X\right)+...},$

where $\kappa_{i}\left(X\right)$ are the cumulants of $X$. One calculates

$\displaystyle\kappa_{1}$
$\displaystyle=\left[X\right]_{*}=-\tfrac{1}{2}\int\mathrm{d}^{d}x\sum_{i,\mu,\mu^{\prime}}\chi_{\mu^{\prime},\mu}^{i}\left[\tau\left(b_{\mu},b_{\mu^{\prime}}\right)\right]_{*}\chi_{\mu,\mu^{\prime}}^{i}.$

The average $\left[...\right]_{*}$ in $\kappa_{1}$ is some constant, which can
be interpreted as $\tau_{0}$. Now consider

$\kappa_{2}=\left[X^{2}\right]_{*}-\left[X\right]_{*}^{2}=\tfrac{1}{4}\left(\prod_{\mu}\sum_{b_{\mu}}p\left(b_{\mu}\right)\right)\left(\int\mathrm{d}^{d}x\sum_{i,\mu,\mu^{\prime}}\chi_{\mu^{\prime},\mu}^{i}\tau\left(b_{\mu},b_{\mu^{\prime}}\right)\chi_{\mu,\mu^{\prime}}^{i}\right)^{2}-\left[X\right]_{*}^{2}.$

The factors
$\left(\mathrm{d}^{d}x\sum_{i,\mu,\mu^{\prime}}...\right)\left(\mathrm{d}^{d}x^{\prime}\sum_{j,\nu,\nu^{\prime}}...\right)$
are correlated for $\mu=\nu$, $\mu^{\prime}=\nu^{\prime}$, $\mu=\nu^{\prime}$,
$\mu^{\prime}=\nu$. Averages like
$\left[\tau\left(b_{\mu},b_{\alpha}\right)\tau\left(b_{\nu},b_{\alpha}\right)\right]_{*}$
with one common length index (common base) give the same constant. Thus

$\kappa_{2}^{\left(1\right)}=\lambda_{2}\int\mathrm{d}^{d}x\mathrm{d}^{d}x^{\prime}\sum_{i\neq
j,\mu,\mu^{\prime},\nu}\chi_{\mu^{\prime},\mu}^{i}\left(\boldsymbol{x}\right)\chi_{\mu,\mu^{\prime}}^{i}\left(\boldsymbol{x}\right)\chi_{\nu,\mu}^{j}\left(\boldsymbol{x}^{\prime}\right)\chi_{\mu,\nu}^{j}\left(\boldsymbol{x}^{\prime}\right).$

Length indices can also be pairwise identical. This leads to

$\kappa_{2}^{\left(2\right)}=\hat{\lambda}_{2}\int\mathrm{d}^{d}x\mathrm{d}^{d}x^{\prime}\sum_{i\neq
j,\mu,\mu^{\prime}}\chi_{\mu^{\prime},\mu}^{i}\left(\boldsymbol{x}\right)\chi_{\mu,\mu^{\prime}}^{i}\left(\boldsymbol{x}\right)\chi_{\mu^{\prime},\mu}^{j}\left(\boldsymbol{x}^{\prime}\right)\chi_{\mu,\mu^{\prime}}^{j}\left(\boldsymbol{x}^{\prime}\right)$

with one length sum less.

###  5.2 Effective action

In frequency representation the interactions are

$\displaystyle S_{2}$
$\displaystyle=-\lambda_{2}\int\mathrm{d}^{d}x\int\mathrm{d}^{d}x^{\prime}\sum_{i\neq
j}^{n}\int_{\omega,\omega^{\prime},\nu,\nu^{\prime},\tau}\chi_{-\omega^{\prime},-\omega}^{i}\left(\boldsymbol{x}\right)\chi_{\omega+\tau,\omega^{\prime}}^{i}\left(\boldsymbol{x}\right)\chi_{-\nu^{\prime},-\nu}^{j}\left(\boldsymbol{x}^{\prime}\right)\chi_{\nu-\tau,\nu^{\prime}}^{j}\left(\boldsymbol{x}^{\prime}\right)$
(21)
$\displaystyle\quad-\hat{\lambda}_{2}\int\mathrm{d}^{d}x\int\mathrm{d}^{d}x^{\prime}\sum_{i\neq
j}^{n}\int_{\omega,\omega^{\prime},\nu,\nu^{\prime},\tau,\tau^{\prime}}\chi_{-\omega^{\prime},-\omega}^{i}\left(\boldsymbol{x}\right)\chi_{\omega+\tau,\omega^{\prime}+\tau^{\prime}}^{i}\left(\boldsymbol{x}\right)\chi_{-\nu^{\prime},-\nu}^{j}\left(\boldsymbol{x}^{\prime}\right)\chi_{\nu-\tau,\nu^{\prime}-\tau^{\prime}}^{j}\left(\boldsymbol{x}^{\prime}\right).$

To avoid confusion, $\omega$, $\nu$, and $\tau$ denote “frequencies” in
eq.(21). These interactions couple the replicas, and transfer frequencies but
no wavevector between two double polymer strands $\rightleftharpoons$. The
coupling constant $\lambda_{2}$ has wavevector dimension
$\left[\lambda_{2}\right]=2$ in any space dimension and is nominally strongly
relevant, the coupling constant $\hat{\lambda}_{2}$ is dimensionless.

Higher order cumulants have the same dimension, and in total there results a
rather complicated problem. The conclusion is that the critical behavior found
for periodic base sequences breaks down for random base sequences.

##  6 Conclusions

It has been shown that there are precisely defined field theories for RNA-like
polymers in a good solvent. These field theories can be derived from lattice
models, and consequently should exactly describe the critical properties of
such systems.

We now summarize the case of a periodic base sequence, that is constant
pairing energy. The $O\left(n\right)$-symmetric action (1) and action (15)
with directed polymers and length variables give the same result: at the
stable fixed point the double-strand sector ($\chi$) of the models decouples
from the single-strand sector, and becomes a conventional branched polymer
(lattice animal), which by a well-known mapping is equivalent to the Lee-Yang
field theory (3) in two less dimensions.

In the $O\left(n\right)$ symmetric model there is no distinction between
double polymers of type $\rightleftharpoons$ and of type $\Rightarrow$. It
nevertheless produces the same result. The point is, that the diagrams are
still there, but diagrams with RNA-like double polymers $\rightleftharpoons$
are more singular.

To our knowledge the predictions for conventional branched polymers have only
been compared with series expansions and Monte Carlo simulations, not with
experiment [4]. RNA molecules with given length and periodic base sequence
might be a precisely defined candidate for an experiment. The gyration radius
of branched RNA molecules should increase with length $\ell$ like $\ell^{1/2}$
[4].

The single strand polymer $\varphi$ does not change the critical behavior of
the double strand polymer. However, it gets influenced by the double strand
polymer and contributes a new critical exponent $\nu_{\varphi}$. This exponent
can be calculated in an expansion around the critical dimension $d_{c}=8$, but
there is a large gap between $d_{c}=8$ and three dimensions. In the final
stage of pair condensation we find a power law $s_{\varphi}\sim\tau^{Z}$ for
the remaining amount of single strand polymer, with an exponent
$-2/\nu_{\varphi}$ in three dimensions. It would be of interest to obtain an
accurate value for the critical exponent $\nu_{\varphi}$ in three and two
dimensions. This would require at least a five-loop calculation, as for the
similar exponent $\nu_{\chi}$ [8]. Series expansions might be an alternative.
At the moment it even is not clear whether $\nu_{\varphi}$ really is negative
for $d\leq 4$.

We have also investigated the impact of a random RNA base sequence. This is
only possible within the more detailed field theory with polymer length
variables. The randomness generates complicated relevant interactions,
rendering the conclusions drawn from the model with a periodic base sequence
invalid.

## References

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## Appendix

Figure 5: One- and two-loop diagrams and their combinatoric factors for
$O\left(n\right)$ symmetric action (1). The bar denotes the propagator
proportional to $v_{k}^{-2}$ from eq.(2).

There is a hidden symmetry between $\varphi$\- and $\chi$\- diagrams in the
massless theory at the stable fixed point with $g=3\lambda.$ The combinatoric
factors of diagram $e$ ($36$) and $h$ ($324$), for instance, produce the same
value, because $3^{2}36_{e}=324_{h}.$ Other examples are
$3^{3}4_{I}+3^{2}24_{J}=3\cdot 108_{K}$ and $3^{5}16_{p}+3^{3}288_{gc}=3\cdot
3888_{ch}$.

Here is a way to get the $\Gamma_{\varphi\varphi}$ and
$\Gamma_{\chi\varphi\varphi}$ vertex functions from the vertex functions
$\Gamma_{\chi\chi}$ and $\Gamma_{\chi\chi\chi}$. Cut a diagram for
$\Gamma_{\chi\chi}$ or $\Gamma_{\chi\chi\chi}$ at the $v_{k}^{-2}$ propagators
(the bar). There results a tree diagram, with a unique path between any two
external lines. Select two external lines (three possibilities in the
$\Gamma_{\chi\chi\chi}$ case), and replace them and the path between them with
$\varphi$. This gives a tree diagram for $\Gamma_{\varphi\varphi}$ or
$\Gamma_{\chi\varphi\varphi}$. The combinatoric factors of the tree diagrams
are related. If originally there is a $\lambda^{n}$ and there are $k$ vertices
on the path then perturbation theory starts with a factor
$\lambda^{n}\tfrac{1}{n!}\binom{n}{k}=\lambda^{n}\tfrac{1}{\left(n-k\right)!}\tfrac{1}{k!}$
when $k$ vertices $\lambda$ are singled out for the path. The contraction
along the path gives a factor $3$ for every $\lambda.$ This is to be compared
with $\lambda^{n-k}\tfrac{1}{\left(n-k\right)!}g^{k}\tfrac{1}{k!}$ for the
$\Gamma_{\varphi\varphi}$ or $\Gamma_{\chi\varphi\varphi}$ diagram.


#  Linear Algebra and Galois Theory

Ashish Gupta and Sugata Mandal  Ashish Gupta, Department of Mathematics
Ramakrishna Mission Vivekananda Educational and Research Institute (Belur
Campus)
Howrah, WB 711202
India<EMAIL_ADDRESS>Sugata Mandal, Department of Mathematics
Ramakrishna Mission Vivekananda Educational and Research Institute (Belur
Campus)
Howrah, WB 711202
India<EMAIL_ADDRESS>
###### Abstract.

In [1] R. Gow and R. Quinlan have cast a new look on the endomorphism algebra
of a $K$-vector space $V$ of dimension $n$ assuming that $K$ has a Galois
extension $L$ of degree $n$. In this approach the $K$-space $L$ may serve as a
model for $V$ and Galois-theoretic ideas and results may be applied to
elucidate the structure of endomorphisms and other important objects of linear
algebra. In particular, this leads to the clarification of the structure of a
rank-one endomorphism, trace of an endomorphism, criteria for linear
indepedence etc. We present an exposition of these results using the language
of tensor algebra wherever possible to provide shorter and more conceptual
proofs.

Keywords. : Field; Galois extension; Cyclic extension; Galois group;
Hyperplane; Tensor algebra
2020 Math. Subj. Class.: 12F10; 15A03; 15A04

##  1\. Introduction

Let $K$ be a field (with arbitrary characteristic) admitting a Galois
extension $L$ of degree $n$ and let $V$ be an $n$-dimensional $K$-space. Then
$L$ may serve as a model for $V$. This allows for a new look on the
fundamental notions and concepts of linear algebra in the light of Galois
theory. Such was the goal of the study undertaken in [1] where there was a
special focus on endomorphisms and their properties as well certain
determinants that can be formed using the elements of the Galois group. Some
of the results below require $L$ to be cyclic.

We begin by noting as in [1] that any linear functional $f\in L^{\ast}$ may be
written in the form $f=f_{b}$ for suitable $b\in L$, where
$f_{b}(x)=\operatorname{Tr}(bx)$ and $\operatorname{Tr}$ is the trace map of
$L/K$ defined as

$\operatorname{Tr}(x)=\displaystyle\sum_{\sigma\in\operatorname{Gal}(L/K)}\sigma(x).$

Indeed $L^{\ast}=\hom_{K}(L,K)$ is an $L$-space with the action $f\mapsto f.b$
where

$f.b(x)=f(bx).$

Moreover, it is clear that $\dim_{L}(L^{\ast})=1$ and as $\operatorname{Tr}$
is in general a nonzero functional in $L^{\ast}$, therefore
$L^{\ast}=\operatorname{Tr}.L$. Let $\theta:L\rightarrow L^{\ast}$ be the map
defined by $b\rightarrow\operatorname{Tr}.b\ (b\in L)$. As the
$\operatorname{Tr}$ bilinear form of a Galois extension is non-degenerate it
follows that $\theta$ is an isomorphism of $K$-spaces. In particular,
$\\{\operatorname{Tr}.b_{1},\cdots,\operatorname{Tr}.b_{n}\\}$ is a K-basis of
$L^{\ast}$ whenever $b_{1},\cdots,b_{n}$ is a K-basis of $L$.

Let $V$ and $W$ be $K$-spaces. Recall that an element $z\in V\otimes_{K}W$ is
called _decomposable_ if it decomposes as

$z=x\otimes y,\quad x\in V,y\in W.$

In the following $\otimes$ always means $\otimes_{K}$. Recall that the space

$\displaystyle\mathcal{T}_{p}^{q}(V)=(\otimes^{p}V)\otimes(\otimes^{q}V^{\ast})$

is called the space of tensors of type $(p,q)$ on $V$. Given $a\in V$ and
$f\in V^{\ast}$, the _contraction_ of the $(1,1)$-tensor $(a\otimes f)\in
V\otimes V^{\ast}$ means $f(a)$.

###  1.1. Rank-one endomorphisms and a decomposition of
$\operatorname{End}_{K}(L)$

As is well-known $L\otimes L^{\ast}\cong\operatorname{End}_{K}(L)$ as
$K$-spaces via the map $(a\otimes f)\rightarrow\psi_{a,f}$, where
$\psi_{a,f}(x)=f(x)a\ (x\in L)$. Moreover the nonzero decomposable elements
correspond precisely to the rank-one endomorphisms. It follows that the latter
may be written as $a\otimes\operatorname{Tr}.b$ for suitable $a,b\in L$. Thus
$(a\otimes\operatorname{Tr}.b)(x)=\operatorname{Tr}(bx)a$ for all $x\in L$.
Since the contraction of a $(1,1)$-tensor is its trace (as a linear operator)
it follows that
$\operatorname{tr}(a\otimes\operatorname{Tr}.b)=\operatorname{Tr}(ab)$. If
$\\{b_{1},b_{2},\ldots,b_{n}\\}$ is a basis of the $K$-space $L$ then as
mentioned previously
$\\{\operatorname{Tr}.b_{1},\operatorname{Tr}.b_{2},\ldots,\operatorname{Tr}.b_{n}\\}$
forms a $K$-basis of $L^{\ast}$ thus by defining property of the tensor
product an endomorphism $\tau\in\operatorname{End}_{K}(L)$ has a expression:

$\tau=a_{1}\otimes\operatorname{Tr}.b_{1}+a_{2}\otimes\operatorname{Tr}.b_{2}+\ldots+a_{n}\otimes\operatorname{Tr}.b_{n},$

where $a_{1},\ldots a_{n}\in L$ are uniquely determinable. It follows that

$\operatorname{tr}(\tau)=\sum_{i=1}^{k}\operatorname{Tr}(a_{i}b_{i}).$

###  1.2. Dual bases

Let $\\{e_{1},\cdots,e_{n}\\}$ be a basis of the $K$-space $L$ and
$\\{\epsilon_{1},\cdots,\epsilon_{n}\\}$ be the corresponding dual basis of
$L^{\ast}$. Then $\epsilon_{i}$ may be expressed in the form
$\operatorname{Tr}.c_{i}$ where $c_{i}\in L$. Let
$c_{i}=\displaystyle\sum_{j=1}^{n}u_{ij}e_{j}$. Therefore $\forall i$,

$\displaystyle\delta_{ik}=(\operatorname{Tr}c_{i})(e_{k})=\operatorname{Tr}(\sum_{j=1}^{n}(u_{ij}e_{j})e_{k})=\sum_{j=1}^{n}u_{ij}\operatorname{Tr}(e_{j}e_{k}).$

Hence $U=(u_{ij})=A^{-1}$, where the matrix
$A=(\operatorname{Tr}(e_{j}e_{k}))$.

For example, for the cyclotomic extension $\mathbb{Q}(\eta_{p})$ over
$\mathbb{Q}$ where $\eta_{p}$ is a primitive $p$-th root of unity we can take
the basis $e_{i}=\eta_{p}^{i-1}$ with $1\leq i\leq n$. Then by _[2, Example
9.1.121]_ we have

$\operatorname{Tr}(e_{i}e_{k})=\begin{cases}p-1,\quad\text{if}~{}i+k-2\equiv
0~{}(\text{mod}~{}p),\\\ -1,\quad\text{otherwise}.\end{cases}$

By direct computation for $p\geq 5$ the matrix $U$ is as follows:

$U={\begin{pmatrix}X&Y\\\ Y^{t}&Z\end{pmatrix}}_{(p-1)\times(p-1)},$

where

$X={\begin{pmatrix}p^{-1}&-p^{-1}\\\ -p^{-1}&-2p^{-1}\end{pmatrix}}_{2\times
2},$ $Y={\begin{pmatrix}0&0&\ldots 0\\\
-p^{-1}&-p^{-1}&-p^{-1}\end{pmatrix}}_{2\times(p-3)}$

and

$Z={\begin{pmatrix}0&0&\ldots&0&p^{-1}\\\ 0&0&\ldots&p^{-1}&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
p^{-1}&0&\ldots&0&0.\end{pmatrix}}_{(p-3)\times(p-3)}.$

###  1.3. Determinants associated to hyperplanes

The following is a well-known fact.

######  Proposition 1.1.

Let $L$ be a Galois extension of $K$ of degree $n$ with Galois group
$G=\\{\sigma_{1},\ldots,\sigma_{n}\\}$. Then $\\{x_{1},x_{2},\ldots,x_{n}\\}$
is a $K$-basis for $L$ if and only if $\sigma_{j}(x_{i})$ is invertible.

###### Proof.

Easy consequence of Dedekind’s independence theorem. ∎

The following theorem was shown in [1].

######  Theorem 1.1.

_([1, Theorem 5])_ Let $L$ be a Galois extension of $K$ of degree $n$ with
Galois group $G=\\{\sigma_{1},\ldots,\sigma_{n}\\}$. Let $x_{1},\ldots,x_{n}$
be a $K$-basis of $L$. Let $B$ be the $n\times n$ matrix whose $i,j$ entry is
$\sigma_{j}(x_{i})$, $1\leq i,j\leq n$. Then

  * (i)

if $K$ has characteristic $2$ or if the Sylow $2$-subgroup of $G$ is not
cyclic, then $\text{det}~{}B\in K$.

  * (ii)

if $K$ has characteristic diferent from $2$ and $G$ has a cyclic Sylow
$2$-subgroup, then $\text{det}~{}B=\mu\alpha$, where $\mu\in K$,
$\alpha^{2}\in K$ and $K(\alpha)$ is the unique quadratic extension of $K$
contained in $L$.

An analog of Theorem 1.1 using an $(n-1)\times(n-1)$ matrix formed from a
basis of the trace zero hyperplane $H_{0}=\ker\operatorname{Tr}$ was also
shown in [1]. Let $H_{0}$ be the trace zero hyperplane, that is,
$H_{0}=\ker\operatorname{Tr}$. Let $b_{2},\ldots,b_{n}$ be a $K$-basis of
$H_{0}$. We can extend $\\{b_{2},\ldots,b_{n}\\}$ to a $K$-basis
$\\{b_{1},b_{2}\ldots,b_{n}\\}$ of $L$ for some $b_{1}\notin H_{0}$. Let
$E(x)$ be the $n\times n$ matrix defined by

$E(x):=\begin{pmatrix}\sigma_{1}(x)&\sigma_{2}(x)&\ldots&\sigma_{n}(x)\\\
\sigma_{1}(b_{2})&\sigma_{2}(b_{2})&\ldots&\sigma_{n}(b_{2})\\\
\vdots&\vdots&\ldots&\vdots\\\
\sigma_{1}(b_{n})&\sigma_{2}(b_{n})&\ldots&\sigma_{n}(b_{n})\end{pmatrix}\qquad\text{for
all $x\in L$}.$

We now define an element $\theta\in\operatorname{End}_{K}(L)$ by setting

$\theta(x)=\det E(x).$

We note that $\theta(b_{1})=\det E(b_{1})$ is non-zero by Proposition 1.1.
Thus $\theta$ is a non-zero endomorphism and moreover it is easily checked
that $\theta$ vanishes on the hyperplane $H_{0}$. Consequently $\theta$ is a
rank-$1$ endomorphism and by Section 1.1, $\theta$ is of the form
$c\otimes\operatorname{Tr}.b$ for some $c\in L$ and since $\ker(\theta)=H_{0}$
so we can take $b=1$. We now determine the value of $c$. Let $E_{1}$ be the
$(n-1)\times(n-1)$ matrix obtained by omitting the first row and the first
column of $E$. Thus

$E_{1}=\begin{pmatrix}\sigma_{2}(b_{2})&\ldots&\sigma_{n}(b_{2})\\\
\vdots&\ldots&\vdots\\\
\sigma_{2}(b_{n})&\ldots&\sigma_{n}(b_{n})\end{pmatrix}$

######  Theorem 1.2.

_([1, Corollary 2])_ With the above notation we have $c=\det E_{1}$.

###### Proof.

Let $E^{\prime}(x)$ be the $n\times n$ matrix obtained by summing all the
columns of $E(x)$ and placing the result as the first column (the remaining
coloumns being unaltered). Thus

$E^{\prime}(x):=\begin{pmatrix}\displaystyle\sum_{\sigma_{i}\in
G}\sigma_{i}(x)&\sigma_{2}(x)&\ldots&\sigma_{n}(x)\\\
\displaystyle\sum_{\sigma_{i}\in
G}\sigma_{i}(b_{2})&\sigma_{2}(b_{2})&\ldots&\sigma_{n}(b_{2})\\\
\vdots&\vdots&\ldots&\vdots\\\ \displaystyle\sum_{\sigma_{i}\in
G}\sigma_{i}(b_{n})&\sigma_{2}(b_{n})&\ldots&\sigma_{n}(b_{n})\end{pmatrix}=\begin{pmatrix}\displaystyle\operatorname{Tr}(x)&\sigma_{2}(x)&\ldots&\sigma_{n}(x)\\\
\displaystyle 0&\sigma_{2}(b_{2})&\ldots&\sigma_{n}(b_{2})\\\
\vdots&\vdots&\ldots&\vdots\\\ \displaystyle
0&\sigma_{2}(b_{n})&\ldots&\sigma_{n}(b_{n})\end{pmatrix}$

Then

$\theta(x)=\det E(x)=\det E^{\prime}(x)=\det E_{1}\operatorname{Tr}(x),$

whence $\theta=\det E_{1}\otimes\operatorname{Tr}$ and consequently $c=\det
E_{1}$. ∎

Using the preceding result and some basic group theory the following is easy
to deduce.

######  Corollary 1.1.

_([1, Theorem 6])_ Let $L$ be a Galois extension of $K$ of degree $n$ with
Galois group $G=\\{\sigma_{1},\ldots,\sigma_{n}\\}$. Let $H_{0}$ be the trace-
zero $K$-hyperplane in $L$. Let $b_{2},\ldots,b_{n}$ be a $K$-basis of
$H_{0}$. Then the $(n-1)\times(n-1)$ matrix $E_{1}$ whose $i,j$ entry is
$\sigma_{j}(b_{i})$, $2\leq i,j\leq n$, is invertible. Moreover,

  * (i)

if $K$ has characteristic $2$ or if the Sylow $2$-subgroup of $G$ is not
cyclic, then $\text{det}~{}E_{1}\in K$.

  * (ii)

if $K$ has characteristic diferent from $2$ and $G$ has a cyclic Sylow
$2$-subgroup, then $\text{det}~{}E_{1}=\mu\alpha$, where $\mu\in K$,
$\alpha^{2}\in K$ and $K(\alpha)$ is the unique quadratic extension of $K$
contained in $L$.

Next we consider any $K$-hyperplane $H$ in $L$. Then $H$ arises as the kernel
of a non-zero element in $L^{\ast}$. As $L^{\ast}=\operatorname{Tr}.L$,
therefore, $H=\ker\operatorname{Tr}.a$ for some non-zero $a\in L$. Then
$H=a^{-1}H_{0}$ and $c_{2},\ldots,c_{n}$ is a basis of $H$ where
$c_{i}=a^{-1}b_{i}$. We can extend $\\{c_{2},\ldots,c_{n}\\}$ to a $K$-basis
$\\{c_{1},c_{2}\ldots,c_{n}\\}$ of $L$. We define a matrix $D(x)$ by replacing
$b_{i}$ by $c_{i}$ in the matrix $E(x)$ above and
$\theta_{H}\in\operatorname{End}_{K}(L)$ by setting $\theta_{H}(x)=\det D(x)$.
We note that $\theta_{H}(c_{1})=\det D(c_{1})\neq 0$ by Proposition 1.1. Thus
$\theta_{H}\neq 0$ and moreover $\theta_{H}$ vanishes on the hyperplane $H$.
Consequently $\theta_{H}$ is a rank-$1$ endomorphism
$d\otimes\operatorname{Tr}.a$ for some nonzero $d\in L$. Recall that if $F$ is
an intermediate subfield and $a\in L$ then the $L/F$-norm $N_{L/F}(a)$ of $a$
is defined as
$N_{L/F}(a)=\displaystyle\prod_{\theta\in\operatorname{Gal}(L/F)}\theta(a)$.
Clearly $N_{L/F}(a)\in F$.

######  Theorem 1.3.

With the notation as previously introduced, we have $d=N_{L/K}(a^{-1})\det
E_{1}$.

###### Proof.

$\displaystyle\theta_{H}(x)$ $\displaystyle=\det D(x)$
$\displaystyle=\det\begin{pmatrix}\sigma_{1}(x)&\sigma_{2}(x)&\ldots&\sigma_{n}(x)\\\
\sigma_{1}(c_{2})&\sigma_{2}(c_{2})&\ldots&\sigma_{n}(c_{2})\\\
\vdots&\vdots&\ldots&\vdots\\\
\sigma_{1}(c_{n})&\sigma_{2}(c_{n})&\ldots&\sigma_{n}(c_{n})\end{pmatrix}$
$\displaystyle=\det\begin{pmatrix}\sigma_{1}(a^{-1})\sigma_{1}(ax)&\sigma_{2}(a^{-1})\sigma_{2}(ax)&\ldots&\sigma_{n}(a^{-1})\sigma_{n}(ax)\\\
\sigma_{1}(a^{-1}b_{2})&\sigma_{2}(a^{-1}b_{2})&\ldots&\sigma_{n}(a^{-1}b_{2})\\\
\vdots&\vdots&\ldots&\vdots\\\
\sigma_{1}(a^{-1}b_{n})&\sigma_{2}(a^{-1}b_{n})&\ldots&\sigma_{n}(a^{-1}b_{n})\end{pmatrix}$
$\displaystyle=N_{L/K}(a^{-1})\det D^{\prime}(x)$
$\displaystyle=N_{L/K}(a^{-1})\det E_{1}\operatorname{Tr}(ax)$
$\displaystyle=N_{L/K}(a^{-1})\det E_{1}\otimes\operatorname{Tr}(ax),$

where $D^{\prime}(x)$ be the $n\times n$ matrix obtained by summing all the
columns of $D(x)$ and placing the result as the first column (the remaining
coloumns being unaltered). Consequently $d=N_{L/K}(a^{-1})\det E_{1}$. ∎

###  1.4. Criteria for Linear Independence in a Cyclic Extension

######  Theorem 1.4.

Let $L$ be a cyclic Galois extension of $K$ of degree $n$ with
$\operatorname{Gal}(L/K)=\langle\sigma\rangle$. Let $k$ be an integer
satisfying $1\leq k\leq n$ and let $b_{1},\ldots,b_{k}$ be elements of $L$.
Then these elements are linearly dependent over $K$ if and only if

$\det S=0,$

where

$S=\left(\sigma^{i-1}(b_{j})\right)=\begin{pmatrix}b_{1}&b_{2}&\ldots&b_{n}\\\
\sigma(b_{1})&\sigma(b_{2})&\ldots&\sigma(b_{k})\\\
\vdots&\vdots&\ldots&\vdots\\\
\sigma^{k-1}(b_{1})&\sigma^{k-1}(b_{2})&\ldots&\sigma^{k-1}(b_{k})\end{pmatrix}.$

###### Proof.

If $b_{1},\ldots,b_{k}$ are linearly dependent then it is easily seen that
$\det S=0$. Conversely, let $b_{1},\ldots,b_{k}$ be linearly independent over
$K$. We will show that $\det S\neq 0$ by induction on $k$. It is clear that
the assertion holds for $k=1$.

Note that it is sufficient to show that $\det S^{\prime}\neq 0$, where

$S^{\prime}=\begin{pmatrix}\sigma^{k-1}(b_{1})&\sigma^{k-1}(b_{2})&\ldots&\sigma^{k-1}(b_{k})\\\
\sigma^{k-2}(b_{1})&\sigma^{k-2}(b_{2})&\ldots&\sigma^{k-2}(b_{k})\\\
\vdots&\vdots&\ldots&\vdots\\\ b_{1}&b_{2}&\ldots&b_{k}\end{pmatrix}$

is obtained from $S$ by row-interchange operations. For $i=k-1,\ldots,1$ we
successively substract $\gamma_{i}\times\text{row $i$}$ from row $i+1$ where

$\gamma_{i}=\sigma^{k-i-1}(b_{1})(\sigma^{k-i}(b_{1}))^{-1}=\sigma^{k-i-1}(b_{1}(\sigma(b_{1}))^{-1}).$

The resulting matrix is

$U=\begin{pmatrix}\sigma^{k-1}(b_{1})&\sigma^{k-1}(b_{2})&\ldots&\sigma^{k-1}(b_{k})\\\
0&\sigma^{k-2}(x_{2})&\ldots&\sigma^{k-2}(x_{k})\\\
\vdots&\vdots&\ldots&\vdots\\\ 0&x_{2}&\ldots&x_{k}\end{pmatrix}.$

where

$x_{j}=b_{j}-\frac{b_{1}}{\sigma(b_{1})}\sigma(b_{j}),\qquad 2\leq j\leq k.$
(1.1)

Then

$\det U=\sigma^{k-1}(b_{1})\det V,$

where $V$ be the $(k-1)\times(k-1)$ matrix obtained by deleting the first row
and column of $U$. We also note that the $i,j$-entry of $V$ is

$\sigma^{k-1+i}(x_{j}),~{}~{}~{}~{}~{}1\leq i\leq k-1,~{}1\leq j\leq k,$

where

$x_{j}=b_{j}-\frac{b_{1}}{\sigma(b_{1})}\sigma(b_{j}).$

Clearly $\det(S^{\prime})\neq 0$ if and only if $\det(V)\neq 0$. As $V$ has
the same form as $S^{\prime}$ by the induction hypothesis $\det(V)=0$ implies
that $x_{2},\cdots,x_{k}$ are linearly dependent, that is,

$a_{2}x_{2}+\ldots+a_{k}x_{k}=0,$

for suitable scalars $a_{2},\ldots,a_{k}$ in $K$, not all zero. Thus by (1.1)

$\displaystyle a_{2}b_{2}+\ldots a_{k}b_{k}$
$\displaystyle=\frac{b_{1}}{\sigma(b_{1})}\sigma(a_{2}b_{2}+\ldots
a_{k}b_{k})$

whence

$\sigma\left(\frac{a_{2}b_{2}+\ldots
a_{k}b_{k}}{b_{1}}\right)=\frac{a_{2}b_{2}+\ldots a_{k}b_{k}}{b_{1}}$

It follows that $\frac{a_{2}b_{2}+\ldots a_{k}b_{k}}{b_{1}}\in K$, thus
contradicting the linear independence of $b_{1},\cdots,b_{k}$. The theorem now
follows. ∎

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgements

The second author thanks the National Board of Higher Mathematics (NBHM) for
financial support.

## References

  * [1] R. Gow, R. Quinlan,Galois theory and linear algebra, Linear Algebra And Its Applications 430 pp. 1778-1789 (2009).
  * [2] E.B.Vinberg, A Course in Algebra, Graduate Studies in Mathematics, American Mathematical Society, 2003.