# Free-energy landscapes of DNA and its assemblies: Perspectives from coarse-
grained modelling

Jonathan P. K. Doye  Physical and Theoretical Chemistry Laboratory, Department
of Chemistry, University of Oxford, Oxford, OX1 3QZ, UK  Ard A. Louis  Rudolf
Peierls Centre for Theoretical Physics, University of Oxford, Parks Road,
Oxford, OX1 3PU, UK  John S. Schreck  National Center for Atmospheric
Research, Computational and Information Systems Laboratory, 850 Table Mesa
Drive, Boulder, CO 80305, USA  Flavio Romano  Dipartimento di Scienze
Molecolari e Nanosistemi, Università Ca’ Foscari di Venezia, via Torino 155,
30170 Venezia Mestre, Italy  Ryan M. Harrison  Majid Mosayebi  School of
Mathematics, University of Bristol, Bristol, BS8 1QU, UK  Megan C. Engel
School of Engineering and Applied Sciences, Harvard University, 29 Oxford
Street, Cambridge, Massachusetts 02138, USA  Thomas E. Ouldridge  Department
of Bioengineering and Centre for Synthetic Biology, Imperial College London,
London, SW7 2AZ, UK

###### Abstract

This chapter will provide an overview of how characterizing free-energy
landscapes can provide insights into the biophysical properties of DNA, as
well as into the behaviour of the DNA assemblies used in the field of DNA
nanotechnology. The landscapes for these complex systems are accessible
through the use of accurate coarse-grained descriptions of DNA. Particular
foci will be the landscapes associated with DNA self-assembly and mechanical
deformation, where the latter can arise from either externally imposed forces
or internal stresses.

###### keywords:

DNA , free-energy landscapes , coarse-graining

††journal: Frontiers of Nanoscience

##  1 Introduction

Free-energy landscapes depict the free energy of a system as a function of a
limited number of coordinates (often termed order parameters or reaction
coordinates). These landscapes can provide a simplified representation of the
main features of configuration space, for example, the (meta)stable states of
the system, the pathways between these states in this reduced order-parameter
space and the size of the associated free-energy barriers.

These landscapes provide a much more coarse-grained view of a system than a
potential-energy surface (PES). For example, the number of potential energy
wells that contribute to a particular point on the free-energy landscape may
be enormous, particularly for large systems and those where some of the states
involve structural disorder. As such the free-energy landscapes naturally
capture the effects of entropy and, unlike a PES, free-energy landscapes are
specific to the temperature of interest. As the projection of the system’s
configurations onto a limited number of order parameters inevitably leads to a
loss of information, the choice of these order parameters is particularly
important if the free-energy landscapes are to provide a useful representation
of the system; some of the problems associated with bad choices are well
documented [1]. Even with a good choice, however, the properties of the
landscapes are still intrinsically order-parameter dependent.

Here, we will illustrate the utility of analysing systems in terms of their
free-energy landscapes for a series of systems made out of DNA. As the systems
of interest would often involve an extremely large number of atoms if all were
explicitly represented, we instead represent the DNA using a coarse-grained
model of DNA at the nucleotide level, thus allowing us to access much larger
systems and much longer time scales. Specifically, we use the oxDNA model [2,
3, 4], which models the DNA as a set of rigid nucleotides with a set of
interactions representing the bonding along the backbone, Watson-Crick base-
pairing, stacking, coaxial stacking, cross-stacking, and electrostatics, as
illustrated in Fig. 1. One of the dangers of coarse-graining is that the
removed degrees of freedom are in some way coupled to the process of interest
and cannot simply be “integrated out”. In the case of oxDNA, it is probably
reasonable to assume that the intramolecular degrees of freedom of the
nucleotide do not change significantly in most of the processes in which we
are interested, but the potential effects of only including the solvent
degrees of freedom implicitly are somewhat less clear.

Figure 1: The oxDNA model. Each nucleotide is a rigid body with sites
corresponding to the centres of the different interactions. The position and
orientation of a nucleotide are defined by the position of the notional centre
of mass, a “base” vector $\mathbf{b}$ that is collinear with the stacking and
hydrogen-bonding sites and a vector $\mathbf{n}$ that is normal to the
notional plane of the base. The basic interactions in the model are the
backbone potential connecting backbone sites, a hydrogen-bonding potential
between complementary nucleotides, (coaxial) stacking interactions between
bases that are (non-)adjacent along the chain, electrostatic repulsion between
backbone sites, and cross-stacking interactions between bases that are
diagonally opposite in the duplex. The images are of the second version of the
oxDNA model in which the major and minor grooves have different widths.

In the development of the oxDNA model a particular focus was on the accurate
representation of the thermodynamics of hybridization and a good description
of single-stranded DNA and double-stranded DNA mechanics (i.e. the
thermodynamic cost of structural deformations). This makes the model
particularly well-suited for the computation of free-energy landscapes
associated with the biophysical properties of DNA and for systems of interest
to DNA nanotechnology. Although certain features of these landscapes are
relatively straightforward to predict from simpler models (e.g. the Santa
Lucia model for hybridization free energies [5] and the worm-like chain model
for DNA mechanics [6]), the oxDNA model naturally captures the interplay of
configurational entropy (particularly of single-stranded sections), geometric
effects resulting from the double-helical structure of double-stranded DNA and
DNA mechanics that go into determining the size of free-energy barriers for
intramolecular processes, and which would be otherwise hard to estimate
accurately.

We should note that the oxDNA model has a number of variants. In particular a
second version of the model (sometimes called oxDNA2) was developed that
introduced explicit electrostatic interactions and fine-tuned structural
parameters to enable the structure of large DNA nanostructures, such as DNA
origami, to be accurately described [4, 7]. Although this second model is now
generally the version of choice, the predictions of the two models will be
very similar for many systems, and the results reported here will be for both
models. In addition, one has the option to use a sequence-averaged or
sequence-dependent [3] parameterization, where in the former the strength of
the interactions are independent of the identity of the nucleotides involved.
Most of the results reported here are for the sequence-averaged
parameterization, as it allows the generic properties of systems to be
elucidated, whereas the sequence-dependent parameterization tends to be
reserved for detailed comparisons to specific experimental systems.

##  2 Computing free-energy landscapes

Free-energy landscapes are simply related to equilibrium probability
distributions through

$A(\\{Q_{i}\\})=-k_{B}T\log p(\\{Q_{i}\\})+c$ (1)

where $\\{Q_{i}\\}$ is a set of order parameters that are being used to
characterize the system and $c$ is an arbitary constant. To facilitate
visualization of the free-energy landscape often only one or two order
parameters are used.

The major difficulty in computing such free-energy landscapes is that one is
often interested in high free-energy regions of the landscape — say because
one is interested in the size of the free-energy barrier for a process — which
have a very low probability and so will be poorly sampled in standard
simulations. To overcome this problem, there is now a large array of rare-
event and enhanced sampling simulation methods that allow free-energy
landscapes or the rates of slow processes on such landscapes to be computed
[8, 9, 10, 11]. In our work on the free-energy landscapes of DNA systems, we
have almost exclusively used one of the simplest and earliest of such methods,
namely umbrella sampling [8, 12]. This involves adding an additional biasing
potential $V_{\mathrm{bias}}(\\{Q_{i}\\})$ to the potential energy in order to
improve the sampling of the higher free-energy regions of the landscape. From
the probability distribution in a biased simulation
$p_{\mathrm{bias}}(\\{Q_{i}\\})$ it is a simple matter to estimate the same
distribution for the unbiased system.

There can be different ways to implement umbrella sampling. One limit is to
choose the biasing potential so that all regions of the landscape can be well
sampled. This has the potential disadvantage that the simulation has to be
sufficiently long that the system is able to diffuse back and forth across the
whole order parameter space a sufficient number of times for the relative free
energies of different parts of the landscape to be correctly estimated. A
second issue is generating the biasing potential. Ideally, one would want
$p_{\mathrm{bias}}(\\{Q_{i}\\})$ to be flat, but to achieve this would require
$V_{\mathrm{bias}}(\\{Q_{i}\\})=-A(\\{Q_{i}\\})$, i.e. one would need to know
the quantity that one is seeking to calculate. One typically gets around this
problem by running a series of simulations in which the biasing potential is
iteratively improved until sampling is sufficiently uniform.

The other limit is to run many separate simulations with different biasing
potentials, the aim being that each characterizes a different region of the
landscape. One simple way to achieve this is to use harmonic biasing
potentials whose minima form a grid in the order-parameter space. A best
estimate of the whole free-energy landscape can then be constructed from the
set of biased probability distributions using the weighted-histogram analysis
method (WHAM) [13]. For this to be accurate it is important that there is
sufficient overlap between the probability distributions for adjacent points
on the order-parameter grid.

This approach has a number of potential advantages. Firstly, it is potentially
much more efficient, simply because the simulation time required to diffuse
over an order parameter window $\Delta Q$ is expected to be proportional to
$(\Delta Q)^{2}$. Thus, this analysis would predict for one-dimensional
umbrella sampling that the total time required to accurately sample the
landscape when using $N_{w}$ windows is a factor of $1/N_{w}$ shorter compared
to a simulation with one window [14]. Secondly, this approach naturally lends
itself to trivial parallelization through task farming.

However, the major disadvantage is that it can be much easier for problems
with the sampling to go unnoticed. In particular, it is often the case that
even when the system is at the top of a free-energy barrier transitioning
between states that are representative of the two basins that meet at that
point may still be an activated process and hence slow. A good estimate of the
relative free-energy of the two basins can only be obtained if for the windows
in the transition region the systems transits back and forth between states
associated with the two basins sufficiently frequently.

Typically, we use an intermediate approach with a limited number of
overlapping windows; those that cover a transition region often need to be run
for significantly longer.

##  3 Example free-energy landscapes

###  3.1 Hybridization

Hybridization, the association of two strands to form a double helix,
represents the most fundamental process of DNA self-assembly. Consequently,
this process has been well characterized experimentally with the
thermodynamics well understood [5, 15] but with open questions still remaining
concerning the hybridization kinetics [16, 17].

The most basic measure of the degree of assembly is the number of base pairs
formed. The free-energy profile as a function of this order parameter is shown
in Fig. 2(a). The main features of this landscape are (i) the barrier
associated with the formation of the first base pair due to the loss of
translational entropy, (ii) the roughly linear down hill slope as the duplex
zips up and (iii) deviations from linearity for the final base pairs due to
the propensity of the end base pairs to “fray”.

Figure 2:  (a) Free-energy profile for the hybridization of a 15-bp duplex as
a function of the number of base pairs formed at 343 K and a strand
concentration of $3.36\times 10^{-4}$ M [2]. (b,c) Example isoenthalpic
configurations with 2 base pairs formed from (b) an association simulation and
(c) an equilibrium simulation. The configuration in (b) requires more
extensive structural rearrangements of the single-stranded tails to allow
further base pairs to form [16].

Variants of this basic landscape, often as components of a larger landscape,
will be seen a number of times in this review. However, it is important to be
aware of a number of additional complexities. Firstly, when picking a reaction
coordinate to compute a free-energy profile one usually wants to choose the
slowest degree of freedom in the system. If that is the case, motions
orthogonal to that coordinate would be expected to be in equilibrium as one
dynamically progresses along the reaction coordinate. In the case of
hybridization, though, structural rearrangement of the unhybridized sections
of the chain is another slow coordinate, leading to dynamical effects not
expected from the landscape. For example, even though a state with one base
pair is the free-energy transition state on assembly, two to three base pairs
must form before the probability of hybridization exceeds 50%. This is because
when one base pair is formed the rearrangements of the chain required to form
the next base pair are slow (Fig. 2(b)), with breaking of the base pair
occurring at a faster rate. Similarly, on melting, states with one base pair
are more likely to zip up than dissociate because the chains still adopt
conformations that allow base pairing to occur relatively easily.

Secondly, the implicit description of the solvent in coarse-grained models
such as oxDNA may neglect important degrees of freedom for the hybridization
process. For example, the oxDNA barrier to hybridization is purely entropic
and the oxDNA transition state is enthalpically stabilized with respect to the
single-stranded reactants because of the base pair(s) formed at the transition
state. Hence, the oxDNA hybridization rate decreases with temperature because
of this negative activation energy [16]. However, in reality, hybridization
rates generally increase with temperature [18]. Thus, there must be an
enthalpic barrier to hybridization that is not captured by oxDNA. The most
likely source is the water degrees of freedom, for example, desolvating the
bases and the resulting rearrangements of the hydrogen-bond network.

What have we learnt from this free-energy landscape? Not that much in some
ways. The loss of translational entropy that is the dominant contributor to
the free energy barrier can be easily estimated from standard statistical
mechanics and, as the oxDNA model has been parameterized to the predictions of
the SantaLucia model, the free-energy of binding should be very similar to
that model. The most non-trivial feature is perhaps the change in slope due to
fraying; this has not been “put into” oxDNA, but emerges naturally from its
relatively realistic description of the basic physical properties of the
nucleotides.

###  3.2 Hairpin formation

The next example that we consider is the formation of a hairpin. The free-
energy profile as a function of the number of stem base pairs formed is shown
in Fig. 3 at two temperatures. Its form is very similar to that for
hybridization, but for this unimolecular process the barrier is instead a
result of the loss in configurational entropy on forming the hairpin loop. One
of the particular strengths of the oxDNA model is its ability to capture such
features, particularly as describing single-stranded DNA accurately using
polymer models is not so straightforward. This enables the model to reproduce
hairpin melting temperatures accurately, albeit noting that the reproduction
of such data was considered in the model parameterization.

Figure 3:  Free-energy profile for the formation of a hairpin with a 5-bp stem
and a 30-nt loop [19]. Example open and closed configurations are also
illustrated.

###  3.3 Kissing hairpins

The formation of the DNA double helix requires the wrapping of each strand
around each other in a right-handed fashion. In some instances, this wrapping
may not be topologically feasible. Fig. 4 illustrates one such example, namely
the hybridization of two fully complementary hairpin loops (such a structure
is termed a kissing loop). As the loops are longer than the DNA pitch length
and there are no free ends, the two loops cannot fully wrap around each other
without defects associated with the the opposite wrapping. The topological and
geometric frustration in this system is evident from the free-energy profile.
Although the formation of the first few base pairs is relatively unhindered,
the graph shows increasing curvature as more base pairs are formed, and the
lowest free-energy state has only 14 of the 20 possible base pairs formed.

Somewhat similar topological impediments to assembly may well be relevant to
DNA origamis. For example, if both the end domains of a staple strand bind
first to the scaffold strand, then any remaining unbound middle domains no
longer have a free end to facilitate wrapping and so correct binding of that
domain may no longer be topological feasible without dissociation of one of
the end domains.

Figure 4:  The free-energy profile for the formation of a kissing-loop between
two hairpins with fully complementary 20-nt loops at 23∘C and a strand
concentration of 0.336 mM [20]. The illustrated configuration with 14 base
pairs formed between the loops is representative of the free-energy global
minimum.

###  3.4 Force-induced melting

The hairpin free-energy profiles in Fig. 3 illustrate the destabilization of
the assembled state by temperature. Another mechanism of destabilization is by
the application of stress to the DNA systems. Fig. 5(a) illustrates the
destabilization of a DNA duplex under the action of tensile force applied in a
‘shearing’ mode (the forces are applied at opposite ends of the duplex on
different strands). As the force $F$ increases, the duplex becomes
increasingly destabilized because there is now an additional $-F\Delta z$
contribution to the free energy of breaking a base pair, where $\Delta z$ is
the extension gained in the direction of the force from breaking a base pair.
Beyond 10 base pairs (approximately the pitch of DNA) the effect of the force
is mainly just to change the slope of the profile, but below this value of the
reaction coordinate the profile develops greater structure. In this limit, the
orientation of the intact double-helical section varies with the number of
remaining base pairs (as illustrated by the configurations in Fig. 5(b)) and
hence so does the extension gained when a base-pair breaks.

Figure 5:  (a) Free-energy profiles for force-induced melting of a 20-bp
duplex when subjected to tensile forces applied in a “shearing” mode of 0, 30,
50, 70, 90 pN [21]. (b) The configurations illustrate the changes in
orientation of the double-stranded section when the number of base pairs is
less than a pitch length. (c) Free-energy profile for the force-induced
“unzipping” of a hairpin with a 6-bp stem and a 4-nt loop as a function of the
extension (the end-to-end distance of the strand along the direction of force)
at a temperature of 30∘C and at four different forces as labelled.

Of course, the number of base pairs is not the only reaction co-ordinate that
one can potentially use. In the case of force-induced transitions, the free-
energy profile as a function of the extension provides a representation that
one can more directly relate to experiments using optical tweezers [22]. Such
landscapes are illustrated in Fig. 5(c) for a hairpin with a 6-bp stem where
tensile forces are applied to both end nucleotides. The free-energy minimum at
1 nm corresponds to the oriented closed state (1 nm is the separation of the
oxDNA nucleotide centres in a base pair), whereas the minimum at $\sim$5-7 nm
that appears at higher forces corresponds to the open state. The metastable
minimum at 2 nm corresponds to a hairpin with the end base pair broken, but no
further intermediate sub-states are resolvable. The stabilization of the open
state with increasing force is clear, and the centre of the transition from
the closed to the open state occurs at close to 10 pN.

###  3.5 Duplex bending

Free-energy landscapes can also be computed that quantify the effect of
mechanical deformation on double-stranded DNA. In Fig. 6 we show the
thermodynamic cost of duplex bending, using the end-to-end distance of the
duplex as the order parameter. The free-energy minimum corresponds to the
relaxed duplex. As the two ends of the duplex are pulled together the free-
energy rises smoothly in line with the predictions of the worm-like chain
model. However, below a certain end-to-end distance (about 5.5 nm in the
example shown) it become favourable for the bending stress to be localized at
a “kink” defect where the DNA bends sharply and the stacking and usually base-
pairing interactions are disrupted. The kink acts a bit like a hinge allowing
further bending to occur more easily. At very short distances, the free energy
begins to rise more rapidly due the repulsion between the two ends. The point
at which kinking occurs in such free-energy profiles depends on the duplex
length; for sufficiently long duplexes the two ends can be brought together to
form a roughly circular configuration without any need for kinking [23].

Figure 6:  Free-energy profile for the bending of a 30-bp DNA duplex using the
end-to-end distance as the order parameter. The contributions to the free-
energy from homogeneously bent and kinked states are included on the plot;
example configurations for these two types of states are depicted. A worm-like
chain fit to the free-energy of the homogeneously bent duplex is also shown
[24].

###  3.6 DNA nanodevices

In many DNA nanodevices, part of the mechanism involves an intramolecular
hybridization process. The two-footed DNA walker of Ref. [25] provides just
such an example. This device is capable of walking across a DNA origami
through the successive additions of fuel and anti-fuel molecules that cause
the legs to be lifted and placed back down. The process illustrated in Fig.
7(a) is of leg placement after a fuel molecule F1 has bound to the foothold
T1. The intramolecular hybridization of the leg L1 with the remaining unbound
domain of F1 leads to correct foot placement, whereas the binding of a second
F1 molecule to L1 instead leads to a trapped state.

Figure 7: (a) The leg placement step of a two-legged DNA walker that is
capable of walking between footholds on a DNA origami, together with a
competing reaction involving the binding of an additional fuel strand F1 that
leads to a trapped state. (b) Free-energy profiles for the binding of L1 to F1
as a function of the step size (the separation between the footholds on the
DNA origami) together with that for the trapping reaction. (c) Example
configurations with the foot fully bound at different step sizes. (d) The
free-energy barrier for foot placement as a function of step size, compared to
that for the trapping reaction (red dashed line) at a F1 concentration of 10
nM.

Fig. 7(b) shows the free-energy landscapes for leg placement as a function of
the step size of the walker. Beyond a step size of 25–30 nm, the free-energy
for leg placement increases rapidly. This is because there is an increasing
free-energetic cost for the walker adopting a stretched-out state, as is very
evident from the bound states illustrated in Fig. 7(c). In terms of free-
energy barriers, the main effect of increasing the step size is to increase
the barrier for binding (Fig. 7(d)); most of the free-energy cost of adopting
a stretched-out state has to be paid to initiate binding. Thus, the step size
has a much greater effect on the binding rate than the unbinding rate.

Although estimating absolute rates from free-energy landscapes requires
additional dynamical information, the relative rates for similar processes can
often be accurately estimated by assuming that dynamical factors other than
the free-energy barriers cancel out; i.e.

$\frac{r_{1}}{r_{2}}\approx\exp\left(-\frac{(\Delta G^{\ddagger}_{1}-\Delta
G^{\ddagger}_{2})}{k_{B}T}\right)$ (2)

where $r_{1}$ and $r_{2}$ are the rates of the two reactions being compared.
For example, this equation allowed a detailed comparison to experimental
results for the walker, considering both the effects of step size and fuel
concentration [26].

The above provides an example where a change in the free energy of reaction
mainly affects the binding rates. This is often the case when a mechanical
deformation is required to initiate binding; for example, this also applies to
DNA cyclization reactions [27]. An opposite example is the hybridization rate
between two strands that can themselves form short hairpins [28, 29]. The
hairpins stabilize the single-stranded state compared to sequences that do not
allow hairpins to form, thus reducing the overall free energy of
hybridization. However, the hairpins only have a relatively small effect on
the binding rate because, in the cases considered, the hairpins are short
enough that hybridization can be initiated in other sections of the strand,
and only once the free-energy barrier has been crossed do the hairpin base
pairs need to be broken. By contrast, there is a large increase in the rate of
unbinding, because during dissociation the strands can start to form the
hairpins, thus stabilising partially melted states and reducing the free-
energy barrier to dissociation.

###  3.7 Displacement reactions

One of the fundamental processes used in DNA nanotechnology to drive
structural changes is toehold-mediated strand displacement. In this process an
“invader” strand binds to an exposed single-stranded toehold on the “target”
strand and then proceeds to displace the adjacent “incumbent” strand to form a
duplex with the target strand (example intermediate configurations are shown
in Fig. 8(a,b)). The first panel of Fig. 8(c) shows the landscape for a
standard displacement reaction where the invader is fully complementary to the
target. The pathway involves the invader binding to the toehold and then the
displacement process, which at each step involves the loss of a base pair by
the incumbent followed by the invader gaining a base pair (leading to the
diagonal paths across the landscape). The final stage is dissociation of the
incumbent strand, which may occur spontaneously before the last few base pairs
have been displaced.

A key property that determines the rate of displacement is the likelihood that
the invading strand will dissociate rather than complete the displacement. For
example, the rate increases as the toehold length increases, because it makes
dissociation less likely once the invading strand is fully bound to the
toehold. One of the features of the displacement reaction is that there is a
small barrier (of about $3k_{\mathrm{B}}T$) to initiating the displacement, as
this then creates two single-stranded tails at the junction site that have an
effective repulsion between them. Accounting for this barrier is important to
capture the dependence of the displacement rate on toehold length (which oxDNA
very accurately does) because it increases the probability of invader
dissociation compared to a landscape where there is no initial barrier to
displacement [30].

Figure 8: Toehold-mediated strand displacement involving the creation of a
mismatch. (a,b) Example configurations from different points in the
displacement reaction where the green invader strand displaces the red
incumbent strand from the blue target strand [31]. The gold bases indicate the
site where a mismatch is created, and in (a) the invader is bound up to the
mismatch site, whereas in (b) the invader has passed beyond the mismatch. (c)
Free-energy landscapes for four different invading strands: the fully
complementary invader, and invaders creating mismatches proximal to the start
of the displacement, in the centre or at the distal end. The toehold is of
length 5, and so state (5,16) corresponds to complete binding of the invader
to the toehold but without any displacement. The blue lines correspond to the
paths of displacement.

In Fig. 8(c) we also show landscapes for sequences where the displacement
creates a duplex that contains a mismatch [31]. Incorporation of this mismatch
leads to an additional barrier that needs to be overcome for the displacement
reaction to reach completion. When incorporation of the mismatch occurs early
in the displacement process, close to the toehold, the rate of displacement is
significantly reduced (by a factor of about 1000 compared to the fully
complementary invader sequence) because this additional barrier increases the
likelihood that the invader strand will dissociate from the target strand
rather than surmount the barrier and complete displacement. By contrast, when
the mismatch is at the distal end relative to the toehold, it only has a
relatively small effect on the displacement rate (a reduction by a factor of
roughly 2), because by that point the invading strand is likely to complete
displacement anyway, rather than dissociate, because it has already formed 17
base pairs with the target. The use and positioning of such mismatches thus
provides an additional means to control the dynamics of displacement
reactions.

##  4 Conclusions

This chapter provides some examples of how free-energy landscapes computed
using a coarse-grained DNA model can provide important insights into the
biophysics of DNA and the properties of the DNA assemblies used in DNA
nanotechnology. Furthermore, given that the model used (oxDNA) has been
parameterized to reproduce accurately the thermodynamics and mechanics of DNA
both in its double-stranded and single-stranded forms, these landscapes can
provide quantitative insights into relative rates. Such predictions can be
supplemented by direct calculations of the rates by simulations (we have
generally used the forward flux-sampling approach [9]; note that if one knows
the free-energy change for the reaction from the free-energy landscape and one
has computed the rate in one direction, the rate in the other direction simply
follows from these two quantities). Although these rate calculations have
revealed interesting and sometimes subtle dynamical effects [16, 29, 32], they
have also confirmed the utility of Eq. 2 in capturing the most substantial
effects on the relative rates.

Although the illustrative examples shown here are for relatively small
systems, it is feasible to compute the full assembly landscapes including all
relevant pathways for small DNA nanostructures, such as the DNA tetrahedron
developed by the Turberfield group [33]. For larger nanostructures, such as
DNA origami [34] and DNA brick systems [35], such computations are not
feasible. However, the landscapes for key or archetypal processes in the
assembly of these objects can be computed; for example, these can be used to
parameterize kinetic models of assembly at the domain level, as has been done
for 2D DNA brick systems [36].

Free-energy landscapes associated with mechanical deformations of very large
DNA systems are also feasible using the oxDNA model. For example, free-energy
landscapes for the bending of DNA origami rods that are equivalent to Fig.
6(a) have now been computed using umbrella sampling [37]. Similarly,
metadynamics has also been successfully applied to study the mechanics of DNA
origami [38].

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Axial-vector form factor of nucleons at finite temperature from the AdS/QCD
soft-wall model

Shahin Mamedova,b,c 111e-mail<EMAIL_ADDRESS>(corresponding author)
and Narmin Nasibovab 222e-mail<EMAIL_ADDRESS>

a Institute for Physical Problems, Baku State University,
Z. Khalilov str. 23, Baku, AZ1148, Azerbaijan
b Institute of Physics, Ministry of Science and Education,
AZ1143, H. Javid Avenue 131, Baku, Azerbaijan
c Department of Physics and Electronics, Khazar Universty,
41, Mehseti Street, Baku, AZ-1096, Azerbaijan
.

Abstract

The axial-vector form factor of the nucleons is studied at a finite
temperature using the holographic soft-wall model with the thermal dilaton
field. We use the bulk interaction action known from the zero temperature case
and apply the profile functions of fields thermalized by the interaction with
the thermal dilaton field. The dependence of the axial form factor on the
square of the transferred momentum and the temperature is plotted for the
ground and excited states of the nucleons.

##  I Introduction

In the weak processes, such as $\beta$ decay $n\rightarrow
p+e^{-}+\tilde{\nu}_{e}$ or $\mu$ capture $\mu^{-}+p=\nu_{\mu}+n$, an
interaction takes place with the axial-vector current of the baryons. The form
factor $G_{A}\left(Q^{2}\right)$, which describes the interaction vertex in
these processes, is called the axial-vector form factor of the nucleons. The
axial-vector form factor is also used for the interpretation of neutrino
experiments Mosel ; Alvarez . The form factor has been studied in the
framework of different models of field theories (QCD sum rules, chiral quark
model, etc.), for the low or high values of the momentum transfer $Q^{2}$ in
the works Bernard ; Alexandrou ; Eichmann ; Anikin ; Meyer ; Ramalhoa ; Chen ;
H. Hashamipour . After the appearance of the QCD models based on AdS/CFT
correspondence Maldacena ; Gubser ; Witten , which are called as holographic
QCD, or AdS/QCD models Boschi1 ; Boschi2 ; Erlich ; Rold1 ; Rold2 ; Karch ;
Brodsky ; Branz , many problems of strong interactions Paula ; Rinaldi  have
been considered in the framework of these models. The holographic QCD enables
us to investigate the axial-vector form factor by imposing no limitation on
$Q^{2}$, i.e., to consider it at both small and large values of the momentum
transfer. As a result, both momentum areas were joined within one graph, which
nicely agrees with the graphs of the different approaches for the various
$Q^{2}$ areas. It explains the experimental data as well. Within holographic
QCD the $G_{A}$ form factor was considered in the hard-wall model framework in
Refs. Mamedov ; Gutsche1 ; Atayev1 ; Huseynov  and in the soft-wall model
framework in Refs. Atayev1 ; Lyubovitskij . The main advantage of the
holographic soft-wall model is that it produces linear Regge trajectories in
the mass spectrum of the mesons Karch ; Paula ; Rinaldi . In Ref. Atayev2  the
axial form factor splitting in the holographic approach has been studied in
the so-called isospin medium, which is a simplified model of the nuclear
medium. Such a ”medium” takes into account only the isospin chemical potential
of the medium and neglects all other quantities. Interaction with the isospin
background of the medium leads to a mass splitting of nucleons, which results
in the splitting of the $G_{A}$ form factor into two branches, each of which
corresponds to a certain isospin projection. Recently, in Refs. Gutsche4 ;
Nasibova1 , the finite temperature theory of mesons and baryons was considered
within the soft-wall model of the AdS/QCD. Explicit formulas, obtained there
for the wave functions of mesons and baryons in 5D, enable us to solve 4D QCD
interaction issues at finite temperatures up to critical temperature
$T_{c}=0.2$ $GeV$. It is well known that, the critical temperature describes
the confinement-deconfinement phase transition in quantum chromodynamics. In
the dual AdS gravity side, this temperature corresponds to the temperature of
a first-order Hawking-Page phase transition between the thermal AdS space-time
and black hole. The critical temperature has been estimated as $T_{c}=0.122$
in the hard wall model, $T_{c}=0.191$ $GeV$ in soft wall model of holographic
QCD in Ref. Gutsche2 , $T_{c}=0.192\pm 0.007\pm 0.004$ $GeV$ in the lattice
QCD Cheng5 .

The study of the medium modification of nucleon properties is a longstanding
topic Rakhimov . With the application of the holographic QCD approach the
hadron, electromagnetic, and gravitational form factors of the nucleons were
studied in Refs. Gutsche2 ; Gutsche3 ; Gutsche4 ; Allahverdiyev ; Gutsche5 ;
Abidin . Moreover, the temperature dependencies of meson-nucleon coupling
constants were found in Refs. Nasibova1 ; Taghiyeva ; Nasibova .

This approach enables us to study the $G_{A}$ form factor in the thermal
medium. The same problem was considered in Ref. Dominguez  within the QCD sum
rules framework. Here we aim to consider a temperature dependence of the
axial-vector form factor neglecting all other medium quantities (density,
chemical potential, e.t.c.); however, present study will be useful for the
neutrino- and anti-neutrino-nucleus scattering experiments Refs. Gysbers ;
Lovato ; King .

The remainder of this paper is organized as follows:

In Section II, we briefly present the definition of the axial-vector form
factor at finite temperature. Sec. III is devoted to the basic notions in the
holographic soft-wall model with the thermal dilaton field. Sec. IV concerns
the profile functions of fermion fields at finite temperatures. In Sec. V, the
breaking of chiral symmetry is described at a finite temperature. In Sec. VI,
we obtain the equation of motion of meson bulk-to-boundary propagator at
finite temperature. In Sec. VII, we give an interaction Lagrangian between the
bulk fields. Using the AdS/CFT correspondence, we obtain an integral
expression for the form factor. In Sec. VIII, we derive the $G_{A}(T)$ form
factor at a finite temperature from the corresponding action. In Sec. IX, the
free parameters are fixed, and the axial-vector form factors’ normalized
graphs are plotted for the different values of the parameter $\alpha$, which
serves as a parameter of the small perturbative correction to the quadratic
dilaton Gutsche2 ; Gutsche3 . In the last section, we summarize our results.

##  II Axial-vector current of the nucleons

In QCD the isovector axial-vector current of nucleons is defined as follows
Bernard :

$j^{\mu,a}\left(x\right)=\overline{\psi}\left(x\right)\gamma^{\mu}\gamma^{5}\frac{\tau^{a}}{2}\psi\left(x\right),$
(1)

where $\psi$ is the doublet of $u$ and $d$ quarks
$\psi=\left(\begin{array}[]{c}u\\\ d\end{array}\right)$ and $\tau^{a}$ are the
isospin Pauli matrices. The current (1) is a partially conserved:

$\partial_{\mu}j^{\mu,a}\left(x\right)=i\overline{\psi}\left(x\right)\gamma^{5}\left\\{\frac{\tau^{a}}{2},\mu\right\\}\psi\left(x\right).$
(2)

Here $\mu$ is the mass matrix of the up and down quarks, which has the
diagonal form: $\mu=diag\ \left(m_{u},m_{d}\right)$. In the exact isospin
symmetry case, these masses are equal $\left(m_{u}=m_{d}=m\right)$ and the
$\mu$ matrix is proportional to the unit matrix. A matrix element of the
nucleon isovector axial-vector current at finite temperature can be written
analogous to the zero temperature case Bernard :

$\left\langle
N\left(p^{\prime},T\right)\left|j^{\mu,a}\left(0\right)\right|N\left(p,T\right)\right\rangle=\overline{u}\left(p^{\prime}\right)\left[\gamma^{\mu}\gamma^{5}G_{A}\left(q^{2},T\right)+\frac{q^{\mu}}{2m_{N}}\gamma^{5}G_{P}\left(q^{2},T\right)\right]\frac{\tau^{a}}{2}u\left(p\right).$
(3)

Here $q_{\mu}=p_{\mu}^{\prime}-p_{\mu}$ is the total momentum in the
interaction vertex and $m_{N}$ is the nucleon mass,
$G_{A}\left(q^{2},T\right)$ and $G_{P}\left(q^{2},T\right)$ are called the
axial-vector and induced pseudoscalar form factors respectively. Due to the
Hermiticity of the $j^{\mu,a}$ current the form factors $\
G_{A}\left(q^{2},T\right)$ and $G_{P}\left(q^{2},T\right)$ are real functions
of $q^{2}$ in the $q^{2}\leq 0$ domain. We shall consider the $Q^{2}=-q^{2}$
timelike momentum domain. Here we consider the form factor $\
G_{A}\left(q^{2},T\right)$ at finite temperature in the soft-wall AdS/QCD
model framework.

##  III Soft-wall model at finite temperature

The action for the soft-wall model, which we use for the present finite-
temperature study, contains a dilaton field $\varphi$ depending on the
temperature:

$S=\int d^{4}xdz\sqrt{g}e^{-\varphi(z,T)}L(x,z,T),$ (4)

where $g=|detg_{MN}|$, $(forM,N=0,1,2,3,5)$ and the extra dimension $z$ varies
in the range $0\leq z<\infty$. The background geometry is assumed to be 5D AdS
space-time with the metric at finite temperature:

$ds^{2}=e^{2A(z)}\left[f(z,T)dt^{2}-d{\vec{x}}^{2}-\frac{dz^{2}}{f(z,T)}\right].$
(5)

Here $x=(t,\vec{x})$ is the set of Minkowski coordinates, $z$ is the
holographic coordinate, $A(z)=log(R/z)$, and $R$ is the AdS radius. The
thermal factor $f(z,T)$ has a form Witten2 :

$f(z,T)=1-\frac{z^{4}}{z_{H}^{4}},$ (6)

where $z_{H}$ is the position of the event horizon and is related to the
Hawking temperature as $T=1/(\pi z_{H})$. The dilaton field
$\varphi(z)=k^{2}z^{2}$, where $k$ is a scale parameter of a few hundred
$MeV$, and introduced to make the integral over the $z$ finite at IR boundary
$(z\rightarrow\infty)$. For the finite-temperature soft-wall model, the form
of the dilaton has been modified by including the temperature-depending terms.
In this model it is convenient to apply the Regge-Wheeler tortoise coordinate
$r$,

$r=\int\frac{dz}{f(z)}$ (7)

instead of $z$ Regge , and to neglect the terms of higher order than $T^{8}$
in the expansion of $z$. This gives following relation between the $r$ and $z$
coordinates Gutsche2 :

$r\approx z\left[1+\frac{z^{4}}{5z_{H}^{4}}+\frac{z^{8}}{9z_{H}^{8}}\right].$
(8)

The metric for the AdS-Schwarzschild space-time in these coordinates will be
written in the form:

$ds^{2}=e^{2A(r)}f^{\frac{3}{5}}(r)\left[dt^{2}-\frac{\left(d\vec{x}\right)^{2}}{f(r)}-dr^{2}\right]$
(9)

with $A(r)=log(R/r)$. The thermal factor $f(r)$ depending on the $r$
coordinate has same form as in Eq. (7):

$f(r)=1-\frac{r^{4}}{r_{H}^{4}}.$ (10)

The AdS-Schwarzchild geometry is more suitable for high $T$, while for low
temperatures this metric can be also used to generate a small $T$-expansion.
The limit $T=0$ corresponds to a mapping of the AdS-Schwarzchild geometry onto
AdS Poincare metric, and the small $T$-the behavior of hadron properties can
be generated in the formalism based on the AdS Poincare metric, and with the
use of a thermal dilaton. It can be shown that the results will coincide since
it leads to equivalent results: AdS Schwarzchild geometry with small $T$ is
equal to AdS Poincare metric with thermal dilaton.

The thermal version of the usual quadratic dilaton was applied in Ref. Vega

$\varphi(r,T)=K^{2}(T)r^{2}$ (11)

and developed further in Refs. Gutsche2 ; Gutsche3 ; Gutsche4 ; Nasibova . As
expected, by setting $T=0$ the terminal dilaton can be reduced to the usual
one $\varphi(r,0)=k^{2}r^{2}$. The $K^{2}(T)$ parameter in Eq. (11) is the
parameter of spontaneous breaking of chiral symmetry whose explicit form was
established in Refs. Gutsche2 ; Gutsche3 ; Gutsche4 ; Nasibova :

$K^{2}(T)=k^{2}[1+\rho(T)+O(T^{6})].$ (12)

In section VI, we shall bring the $K^{2}(T)$ dependence in detail. The
$\rho(T)$ function in Eq.(12) encodes the T dependence of the dilaton field
and was found in the form:

$\rho(T)=\frac{9\alpha\pi^{2}}{16}\frac{T^{2}}{12F^{2}}-\frac{N_{f}^{2}-1}{N_{f}}\frac{T^{2}}{12F^{2}}-\frac{N_{f}^{2}-1}{2N_{f}^{2}}\left(\frac{T^{2}}{12F^{2}}\right)^{2}+O\left(T^{6}\right).$
(13)

Here $N_{f}=2$ is the number of quark flavors. The pion decay constant $F$ is
proportional to the $k$ parameter $F=\frac{k\sqrt{6}}{8}$ Gasser . The
relations (12)-(13) were obtained by introducing the thermal prefactor
$e^{-\lambda_{T}}$ with

$\lambda_{T}(z)=\alpha\frac{z^{2}}{z^{2}_{H}}+\gamma\frac{z^{4}}{z^{4}_{H}}+\xi\frac{k^{2}z^{6}}{z^{4}_{H}}$

in Ref. Gutsche2 . The parameter $\gamma$ was fixed to guarantee the gauge
invariance and massless ground states pseudoscalar mesons in chiral limit and
the parameter $\xi$ was fixed to drop the radial dependence in six power.
Thus, the thermal dilaton depends only on parameter $\alpha$. This small
parameter encodes the contribution of gravity to the restoration of chiral
symmetry at the critical temperature $T_{c}=0.2$ $GeV$.

##  IV Nucleons at finite temperature

In this section, we briefly present the profile function for a fermion field
in the model with thermal dilaton, which was derived in Ref. Gutsche3 . Action
for the bulk fermion field in this model is written in the form:

$S=\int{d^{4}x}dre^{-\varphi(r,T)}\sqrt{g}{\bar{\Psi}}(x,r,T)D_{\pm}(r)\Psi(x,r,T).$
(14)

Here
$D_{\pm}(r)=\frac{i}{2}\Gamma^{M}[\partial_{M}-\frac{1}{4}\omega_{M}^{ab}\
[\Gamma_{a}\Gamma_{b}]]\mp[\mu(r,T)+U_{F}(r,T)]$ are covariant derivatives,
which include five-dimensional temperature-dependent mass $\mu(r,T)=\mu\
f^{\frac{3}{10}}(r,T)$. $\mu=\emph{N}_{B}+\emph{L}-\frac{3}{2}$, where
$\emph{N}_{B}=3$ and L are the number of partons and orbital angular momentum
$(\emph{L}=0)$ of the nucleons, respectively. Thermal potential for nucleons
$U_{F}(r,T)$ is defined as $U_{F}(r,T)=\varphi(r,T)/f^{\frac{3}{10}}(r,T)$.
Non-zero components of the spin connection $\omega_{M}^{ab}$ are given by
$\omega_{M}^{ab}=(\delta_{\mu}^{a}\delta_{r}^{b}-\delta_{\mu}^{b}\delta_{r}^{a})\
rf^{\frac{1}{5}}(r,T)$. Commutator of the Dirac matrices is defined by
$\sigma^{MN}=[\Gamma^{M},\Gamma^{N}]$ and $\Gamma^{M}=e_{a}^{M}\Gamma^{a}$,
where $e_{a}^{M}=\ z\delta_{a}^{M}$ is the inverse vielbein. The $\Gamma^{a}$
matrices are defined as $\Gamma^{a}=(\Gamma^{\mu},\ -i\Gamma^{5})$. The $\Psi$
field is decomposed into the left and right chirality components as
$\Psi(x,r,T)=\Psi^{R}(x,r,T)+\Psi^{L}(x,r,T)$, with definition
$\Psi^{L,R}(x,r,T)=\frac{1\mp\gamma^{5}}{2}\Psi$, where
$\gamma^{5}\Psi^{L}=-\Psi^{L}$, $\gamma^{5}\Psi^{R}=\Psi^{R}$. Kaluza-Klein
decomposition of the $\Psi^{L,R}$ fields is written as
$\Psi^{L,R}=\sum_{n}\Phi_{n}^{L/R}(r,T)\psi_{n}(x)$, where
$\Phi_{n}^{L/R}(r,T)$ are the profile functions and 4D wave functions
$\psi_{n}(x)$ satisfy free Dirac equation
$\not{p}\psi_{n}(x)=M_{n}(0)\psi_{n}(x)$. In further calculations, the
following replacement in the profile functions is useful Gutsche3 :

$\Phi_{n}^{L/R}(r,T)=e^{-\frac{3}{2}A(r)}F_{n}^{L/R}(r,T).$ (15)

Equation of motion (EOM) obtained from the action in Eq. (14) is the 5D Dirac
equation at finite temperature. Substituting $\Phi_{n}^{L/R}(r,T)$ in EOM
following equation may be obtained for the $F_{n}^{L/R}(r,T)$ profiles in the
nucleon’s rest frame Gutsche3 :

$\left[\partial_{r}^{2}+U_{L/R}(r,T)\right]F_{n}^{L/R}(r,T)=M_{n}^{2}(T)F_{n}^{L/R}(r,T).$
(16)

Here $U_{L/R}(r,T)$ are the effective potentials written as the sum of the
zero- and finite-temperature potential terms:

$U_{L/R}(r,T)=U_{L/R}(r)+\Delta U_{L/R}(r,T)$ (17)

with the explicit forms

$U(r)=k^{4}r^{2}+\frac{(4m^{2}-1)}{4r^{2}},\ \Delta
U(r,T)=2\rho(T)k^{4}r^{2}.$ (18)

Here $m=N+L-2$. Particularly, for the nucleon with three partons $m=L+1$.
$M_{n}^{2}$ gives the Eq. (16) is the nucleon mass spectrum, and at the low
temperatures can also be written as the sum of the zero- and finite-
temperature parts:

$M_{n}^{2}(T)=\ M_{n}^{2}(0)+\Delta M_{n}^{2}(T),$ (19)

where

$M_{n}^{2}(0)=4k^{2}\left(n+\frac{m+1}{2}\right),$ (20) $\Delta
M_{n}^{2}(T)=\rho(T)M_{n}^{2}(0)+\frac{R\pi^{4}T^{4}}{k^{2}}$ (21)

$R=(6n-1)(m+1)$. Equation (16) can be solved by using the boundary conditions
on $F_{n}^{L/R}(r,T)$ in the ultraviolet (UV) and infrared (IR) limits at
small $r$

$F_{n}^{L/R}(r,T)\sim r^{N+L-1\pm\frac{1}{2}},$ (22)

and at large $r$ as

$F_{n}^{L/R}(r,T)\rightarrow 0.$ (23)

The normalization conditions for the profile functions $F_{n}^{L/R}(r,T)$ are:

$\int_{0}^{\infty}dre^{-3A(r)}\Phi_{m}^{L,R}(r,T)\Phi_{n}^{L,R}(r,T))=\int_{0}^{\infty}drF_{m}^{L,R}(r,T)F_{n}^{L,R}(r,T)=\delta_{mn}.$
(24)

In general the Schrodinger-type equations (16) have following analytical
solutions:

$F_{n}^{L/R}(r,T)=\sqrt{\frac{2\Gamma(n+1)}{\Gamma(n+m_{L/R}+1)}}K^{m_{L/R}+1}r^{m_{L/R}(T)+\frac{1}{2}}e^{-\frac{K^{2}r^{2}}{2}}L_{n}^{m_{L/R}}\left(K^{2}r^{2}\right).$
(25)

By taking here $m_{L/R}=m\pm\frac{1}{2}$ in it, we obtain an expression for
the profile functions of the nucleons, which coincide with $T\rightarrow 0$
limit with the ones obtained in Ref. Gutsche11  for the $T=0$ case.

##  V Condensate, dilaton and chiral symmetry breaking

The action for the pseudo-scalar $X$ field in AdS/QCD has a form:

$S=\int_{0}^{\infty}d^{5}x\sqrt{g}e^{-\varphi(z)}Tr\\{|DX|^{2}-m_{5}^{2}|X|^{2}\\}.$
(26)

In terms of the tortoise coordinate $r$ and with the thermal dilaton this
action will slightly change:

$S_{X}=\int
d^{4}xdr\sqrt{g}e^{-\varphi(r,T)}Tr\left[|DX|^{2}+3|X|^{2}\right],$ (27)

where $DX$ is the covariant derivative including the interaction with the
gauge fields $A_{L,R}$. In terms of the vector $M_{M}$ and the axial-vector
$A_{M}$ fields the covariant derivative is defined as:
$D^{M}X=\partial^{M}X-iA_{L}^{M}X+iXA_{R}^{M}=\partial^{M}X-i\left[M_{M},X\right]-i\\{A_{M},X\\}$,
$A_{L,R}^{M}=A_{L,R}^{M}t^{a}$ and $F_{L,R}^{MN}$ are the field strengths of
these fields. The $X$ field transforms under the bifundamental representation
of the flavor symmetry group $SU(2)_{L}\times SU(2)_{R}$ of the model and
performs the breaking of the chiral symmetry by Higgs mechanism Gherghetta .
EOM for the $X$ field, which is obtained from Eq. (26), has a solution

$\langle X\rangle=\frac{1}{2}v(z).$ (28)

In the soft-wall model, for $v(z)$ in $z\rightarrow 0$ limit different authors
Karch ; Zhang ; Gutsche30 ; Colangelo  apply the solution, which coincides
with the one found within the hard-wall model:

$v(z)=\frac{1}{2}M_{q}az+\frac{1}{2a}\Sigma z^{3}.$ (29)

Here $a=\sqrt{N_{c}}/(2\pi)$ $(N_{c}=3)$ is the normalization parameter
Cherman . According to the dictionary of the bulk/boundary correspondence, the
parameters $M_{q}$ and $\Sigma$ are identified with the $u,d$ quark mass
matrix and with the chiral condensate $\Sigma=<0|\bar{q}q|0>$, respectively.
For the finite-temperature case the quark condensate within $v(z)$ in Eq. (29)
depends on temperature; whereas the $z$ coordinate should be replaced by $r$.
Then, the $v(r,T)$ solution has a form:

$v(r,T)=\frac{1}{2}M_{q}ar+\frac{1}{2a}\Sigma(T)r^{3}.$ (30)

To the function $\Sigma(T)$, it was found in Ref. Gasser  by using two-loop
chiral perturbation theory at finite temperature, and applied in the Refs.
Gutsche2 ; Gutsche3  for the finite-temperature soft-wall model. It has a
form:

$\Sigma(T)=\Sigma[1-\frac{N_{f}^{2}-1}{N_{f}}\frac{T^{2}}{12F^{2}}-\frac{N_{f}^{2}-1}{2N_{f}^{2}}(\frac{T^{2}}{12F^{2}})^{2}+O(T^{6})]=\Sigma[1+\Delta_{T}+O(T^{6})],$
(31)

where $N_{f}$ is the number of quark flavors and $F$ is the pion decay
constant. As is seen from Eq. (29) $\Sigma(T=0)=\Sigma$. It can be seen that

$\Delta_{T}=-\frac{N_{f}^{2}-1}{N_{f}}\frac{T^{2}}{12F^{2}}-\frac{N_{f}^{2}-1}{2N_{f}^{2}}(\frac{T^{2}}{12F^{2}})^{2}$

in Eq. (31) is related to the $\rho(T)$ in Eq. (13). In the soft-wall model
the dilaton field $\varphi(r)$ is responsible for the dynamical breaking of
the chiral symmetry. The chiral quark condensate $\Sigma(T)$ is the result of
chiral symmetry breaking. Both constants $K$ and $\Sigma$ depend only on $T$.
Therefore, it was supposed in Refs. Gutsche2 ; Gutsche3  that the
$T$-dependence of the dilaton parameter $K^{2}(T)$ should be similar to
$\Sigma(T)$:

$K^{2}(T)=k^{2}\frac{\Sigma(T)}{\Sigma}.$ (32)

It was also conjectured that the relation

$\Sigma=-N_{f}BF^{2}$ (33)

written for the $T=0$ case holds for the $T\neq 0$ case also:

$\Sigma(T)=-N_{f}B(T)F^{2}(T).$ (34)

Here $B$ is the condensate parameter. Then, as a result of Eqs. (11) and (32)
the $T$-dependence of the $\Sigma(T)$ condensate can be expressed in terms of
the $\Delta(T)$ function Gherghetta :

$\Sigma(T)=\Sigma\left[1+\Delta(T)\right]+O(T^{6}).$ (35)

The temperature dependence of $F(T)$ and $B(T)$ relation have been studied in
Ref. Gutsche2 .

##  VI Meson propagator at finite temperature

It is known from the zero-temperature case, the EOM for the $A_{M}$ axial-
vector field in the soft-wall model coincides with the one for the $V_{M}$
vector field at certain limits $(m_{q}\rightarrow 0,z\rightarrow z_{UV})$
Rold2 ; Karch ; Taghiyeva . The Yukawa interaction term, which is proportional
to $v(z)$, is usually neglected in both hard-wall Maru  and soft-wall Gutsche2
; Gutsche3  models when solving the EOM for the fermion field. Here we accept
the approximation where the function $v(z)$ is not taken into account to get
solutions of the EOM for the axial vector field as well. Of course, the bulk-
to-boundary propagators, which are the solutions to the equations for the
$A_{M}$ and $V_{M}$ fields, also will coincide at this approximation. In the
finite temperature case, in the soft-wall model with the thermal dilaton, we
use formal replacements $z\rightarrow r,k\rightarrow K(T)$, which do not
change the EOMs. Consequently, the known expression for the vector bulk-to-
boundary propagator at finite temperature obtained in Ref. Gutsche2  can be
used for the axial-vector field’s propagator as well. In the soft-wall model
with the thermal dilaton field $\varphi(r,T)$ defined in Eq. (11), the EOM for
a vector field was obtained in the following form Gutsche2 ; Gutsche3 :

$\partial_{r}\left(\frac{e^{-\varphi(r,T)}}{r}\partial_{r}A(Q,r,T)\right)-Q^{2}\frac{e^{-\varphi(r,T)}}{r}\partial_{r}A(Q,r,T)=0.$
(36)

The $A(Q,r,T)$ thermal propagator $A(Q,r,T)$ satisfies the boundary condition
$A(Q,z=0,T)=1$. The Eq. (36) is similar to one at zero temperature, and the
only difference between them is the $T$-dependence of the dilaton parameter.
So, the solution of the Eq. (36) written as Gutsche3

$A(Q,r,T)=\Gamma\left(1+a(Q,T)\right)U\left(a(Q,T),0,K^{2}r^{2}\right)=K^{2}r^{2}\int_{0}^{1}\frac{dx}{(1-x)^{2}}x^{a(Q,T)}e^{-K^{2}r^{2}\frac{x}{1-x}},$
(37)

is the bulk-to-boundary propagator for the axial-vector field also. Here
$\Gamma$ is Euler’s gamma function and $a(Q,T)=\frac{Q^{2}}{4K^{2}(T)}$.
$U(x,y,z)$ is a Tricomi function, also known as the confluent hypergeometric
function of the second kind.

##  VII Holography and the bulk interaction Lagrangian

We can apply the holography principle to get the $G_{A}$ form factor in the
boundary QCD from the bulk interaction action:

$S_{int}=\int d^{5}x\sqrt{g}L_{int}.$ (38)

The generating functional $Z_{AdS}$ of the bulk theory is defined by the
classical bulk action $S_{int}$:

$Z_{AdS}=e^{iS_{int}}.$ (39)

AdS/CFT correspondence identifies the generating functional $Z_{AdS}$ with the
generating function $Z_{QCD}$ of the boundary QCD Makoto :

$Z_{AdS}=Z_{QCD}.$ (40)

After calculating $S_{int}$ for the bulk theory and using the holographic
identification in Eq. (40), one can find the $<J_{\mu}^{a}>^{QCD}$ axial-
vector current of nucleons in the boundary QCD theory. The variation
derivative of the gravity functional $Z_{AdS}$ over the $A_{\mu}^{0}$ field
will give us sought current:

$<J_{\mu}^{a}>^{QCD}=-i\frac{\delta Z_{AdS}}{\delta
A_{\mu}^{a(0)}}|_{A_{\mu}^{a(0)}=0}.$ (41)

Here $A_{\mu}^{(0)}$ is the UV boundary value of the $A_{\mu}$ bulk axial-
vector field and has an interpretation of a wave function of the axial-vector
meson at the boundary. The axial-vector current
$J_{\mu}^{a}(p^{\prime},p)=G_{A}\bar{u}(p^{\prime})\gamma^{5}\gamma_{\mu}\left(\tau^{a}/2\right)u(p)$
obtained from the (41) contains the $G_{A}$ factor, which is the integral over
the $r$ coordinate and depends on $Q^{2}$. According to AdS/CFT correspondence
the $\bar{u}(p^{\prime})\gamma^{5}\gamma_{\mu}\left(\tau^{a}/2\right)u(p)$
current is the axial-vector current of the nucleons. Consequently, the
$G_{A}\left(Q^{2}\right)$ factor is accepted as the axial-vector form factor
of the nucleons. This correspondence is relevant for the bulk theory at finite
temperature as well.

Now, the main question is how to determine an explicit form of the $L_{int}$
interaction Lagrangian between the $A_{M}$, $X$, and $\Psi_{1,2}$ fields in
the bulk theory. On the one hand, it is possible to construct different
Lagrangian terms, which describe the different interactions between these
fields. However, on the other hand, we are interested in the interaction
terms, which produce namely the axial-vector current in Eq. (1) in the
boundary theory. These terms should be proportional to the $A_{M}$ axial-
vector field because of the $P$ invariancy of the Lagrangian and be 5D Lorentz
invariant. The necessary terms can be taken from the zero-temperature case and
then extended to the finite-temperature case by replacing the fields at $T=0$
with the thermal ones. The interactions between the bulk fields $A_{M}$, $X$,
and $\Psi_{1,2}$ at zero-temperature case presented in the Refs. Chen ;
Mamedov ; Gutsche1 ; Atayev1 ; Atayev2 ; Huseynova2 , will contribute to the
finite-temperature $G_{A}\left(Q^{2},T\right)$ form factor as well. These
terms can be classified as follows:

1) a minimal coupling term

$\textit{L}^{(1)}=\overline{\Psi}_{1}\Gamma^{M}\left(A_{L}\right)_{M}\Psi_{1}-\overline{\Psi}_{2}\Gamma^{M}\left(A_{R}\right)_{M}\Psi_{2}=\frac{1}{2}\left(\overline{\Psi}_{1}\Gamma^{M}A_{M}\Psi_{1}-\overline{\Psi}_{2}\Gamma^{M}A_{M}\Psi_{2}\right),$
(42)

2) a magnetic gauge coupling term

$\displaystyle\textit{L}^{(2)}=ik_{1}\left\\{\overline{\Psi}_{1}\Gamma^{MN}\left(F_{L}\right)_{MN}\Psi_{1}-\overline{\Psi}_{2}\Gamma^{MN}\left(F_{R}\right)_{MN}\Psi_{2}\right\\}$
$\displaystyle=\frac{i}{2}k_{1}\left\\{\overline{\Psi}_{1}\Gamma^{MN}F_{MN}\Psi_{1}+\overline{\Psi}_{2}\Gamma^{MN}F_{MN}\Psi_{2}\right\\},$
(43)

where $F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}$ is the field stress tensor
of the axial-vector field $A_{M}$.

3) Three-field interaction term, which was introduced in Ref. Mamedov :

$\displaystyle\textit{L}^{(3)}=\frac{g_{Y}}{2}\left[\overline{\Psi}_{1}X\Gamma^{M}\left(A_{L}\right)_{M}\Psi_{2}-\overline{\Psi}_{2}X^{\dagger}\Gamma^{M}\left(A_{R}\right)_{M}\Psi_{1}+h.c.\right]$
$\displaystyle=g_{Y}\left(\overline{\Psi}_{1}X\Gamma^{M}A_{M}\Psi_{2}+\overline{\Psi}_{2}X^{\dagger}\Gamma^{M}A_{M}\Psi_{1}\right).$
(44)

The last term describes an interaction of three bulk fields at one point, and
corresponds to the nucleon-double meson interaction at the boundary. Since
this interaction is the chirality-changing one, the Yukawa coupling constant
$g_{Y}$ was chosen as the coupling constant in this term. Direct calculations
show that $\textit{L}^{(3)}$ term contributes to the axial-vector form factor
of nucleons.

##  VIII $G_{A}$ form factor at finite temperature

We write the action terms in the momentum space, using the

$j^{5\mu}\left(p^{\prime},p\right)=\overline{u}\left(p^{\prime}\right)\gamma^{5}\gamma^{\mu}\frac{\tau^{a}}{2}u\left(p\right)$
(45)

short notation for the axial-vector current. Let’s itemize the $S^{(i)}$
action terms corresponding to the $L^{(i)}$ Lagrangians:

$\displaystyle 1)\ S^{(1)}=\frac{1}{2}\int d^{4}x\ \int_{0}^{\infty}dr\
\sqrt{g}\left\\{\overline{\Psi}_{1}\Gamma^{\mu}A_{\mu}\Psi_{1}-\overline{\Psi}_{2}\Gamma^{\mu}A_{\mu}\Psi_{2}\right\\}$
$\displaystyle=\frac{1}{2}\int
d^{4}pd^{4}p^{\prime}j^{5\mu}\left(p^{\prime},p\right)A_{\mu}^{a}\left(Q\right)\int_{0}^{\infty}dr\
A\left(Q,r,T\right)\left[\left|F_{1R}\left(r,T\right)\right|^{2}-\left|F_{1L}\left(r,T\right)\right|^{2}\right],$
(46) $\displaystyle 2)\ S^{(2)}=\frac{i}{4}k_{1}\int d^{4}x\
\int_{0}^{\infty}dr\
\sqrt{g}\left\\{\overline{\Psi}_{1}\left[\Gamma^{5},\Gamma^{\mu}\right]\partial_{5}A_{\mu}\Psi_{1}+\overline{\Psi}_{2}\left[\Gamma^{5},\Gamma^{\mu}\right]\partial_{5}A_{\mu}\Psi_{2}\right\\}$
$\displaystyle=\frac{k_{1}}{2}\int
d^{4}pd^{4}p^{\prime}j^{5\mu}\left(p^{\prime},p\right)A_{\mu}^{a}\left(Q\right)\int_{0}^{\infty}dr\
r\left(\partial_{r}A\left(Q,r,T\right)\right)\left[\left|F_{1R}\left(r,T\right)\right|^{2}+\left|F_{1L}\left(r,T\right)\right|^{2}\right],$
(47) $\displaystyle 3)\ S^{(3)}=g_{Y}\int d^{4}x\ \int_{0}^{\infty}dr\
\sqrt{g}\left\\{\overline{\Psi}_{1}X\Gamma^{\mu}A_{\mu}\Psi_{2}+\overline{\Psi}_{2}X^{\dagger}\Gamma^{\mu}A_{\mu}\Psi_{1}\right\\}$
$\displaystyle=g_{Y}\int
d^{4}pd^{4}p^{\prime}j^{5\mu}\left(p^{\prime},p\right)A_{\mu}^{a}\left(Q\right)\int_{0}^{\infty}drA\left(Q,r,T\right)2v\left(r,T\right)F_{1L}\left(r,T\right)F_{1R}\left(r,T\right).$
(48)

Here the both ($\overline{\Psi}$ and $\Psi$) bulk fermions are considered on
the mass shell $\left(|p|=|p^{\prime}|=m\right)$, and $m$ is the mass of the
nucleons. Total action $S_{int}=S^{(1)}+S^{(2)}+S^{(3)}$ will produce the
axial-vector form factor $G_{A}$ of the nucleons at finite temperature. Taking
derivatives over $A_{\mu}^{a}\left(Q\right)$ from the $S^{(i)}$ action terms
we get the $G_{A}^{(i)}(T)$ contributions of these terms into the axial-vector
form factor $G_{A}\left(Q^{2},T\right)$:

$1)\
G^{(1)}_{A}\left(Q^{2},T\right)=\frac{1}{2}\int_{0}^{\infty}drA\left(Q,r,T\right)\left[\left|F_{1R}\left(r,T\right)\right|^{2}-\left|F_{1L}\left(r,T\right)\right|^{2}\right],$
(49) $2)\
G^{(2)}_{A}\left(Q^{2},T\right)=\frac{k_{1}}{2}\int_{0}^{\infty}drr\left(\partial_{r}A\left(Q,r,T\right)\right)\left[\left|F_{1R}\left(r,T\right)\right|^{2}+\left|F_{1L}\left(r,T\right)\right|^{2}\right],$
(50) $3)\ G^{(3)}_{A}\left(Q^{2},T\right)=\frac{1}{2}g_{Y}\int_{0}^{\infty}dr\
A\left(Q,r,T\right)v\left(r,T\right)F_{1L}\left(r,T\right)F_{1R}\left(r,T\right).$
(51)
$G_{A}\left(Q^{2},T\right)=G^{(1)}_{A}\left(Q^{2},T\right)+G^{(2)}_{A}\left(Q^{2},T\right)+G^{(3)}_{A}\left(Q^{2},T\right).$
(52)

##  IX Axial-vector transition form factor

A formalism for the study of the nucleon resonances within the holographic
model has been proposed in Refs. Teramond ; Gutsche20 ; Ramalho3 ; Ramalho4 ;
Gutsche22 . In the present work for the transition form factor, we can apply
the interaction Lagrangians (42)-(44). However, this time the
$\overline{\Psi}_{1,2}$ states describe the excited nucleon states, while
$\Psi_{1,2}$ ones are the ground states. So, in the profile functions
$F_{1,2L}^{(n)*}$ we shall set $n=1$ and for this KK mode we shall take the
mass $m^{*}$ of the excited nucleon $\left(|p^{\prime}|=m^{*}\right)$. In the
profile functions $F_{1,2L}^{(n)}$ we set $n=0$ and $\left(|p|=m\right)$. By
getting one of the nucleon profile functions excited state we obtain the
$G_{AT}^{(i)}(T)$ contributions of these terms into the axial-vector
transition form factor $G_{AT}\left(Q^{2},T\right)$:

$\
1)G^{(1)}_{AT}\left(Q^{2},T\right)=\frac{1}{2}\int_{0}^{\infty}drA\left(Q,r,T\right)\left[\left(F_{1L}^{(n)*}F_{1L}^{(m)}-F_{2L}^{(n)*}F_{2L}^{(m)}\right)\right]$
(53) $\ 2)\
G^{(2)}_{AT}\left(Q^{2},T\right)=\frac{k_{1}}{2}\int_{0}^{\infty}dr\partial_{r}A\left(Q,r,T\right)\left[\left(F_{1L}^{(n)*}F_{1L}^{(m)}+F_{2L}^{(n)*}F_{2L}^{(m)}\right)\right]$
(54) $\ 3)\
G^{(3)}_{AT}\left(Q^{2},T\right)=\frac{1}{4}g_{Y}\int_{0}^{\infty}dr\
A\left(Q,r,T\right)v\left(r,T\right)\left[\left(F_{1L}^{(n)*}F_{1R}^{(m)}-F_{2L}^{(n)*}F_{2R}^{(m)}\right)\right].$
(55)
$G_{AT}\left(Q^{2},T\right)=G^{(1)}_{AT}\left(Q^{2},T\right)+G^{(2)}_{AT}\left(Q^{2},T\right)+G^{(3)}_{AT}\left(Q^{2},T\right).$
(56)

There are the following relations between the profile functions of the first
and second bulk fermion fields: $F_{1L}=F_{2R},\quad F_{1R}=-F_{2L}.$

##  X Numerical results

In Fig. 1, we plot the temperature dependency of nucleon mass. This dependence
has a shape similar to one for the meson mass spectrum in Ref. Gutsche2 .

In Figs. 2-4, we present the dependencies of the normalized
$G_{A}\left(Q^{2},T\right)/G_{A}\left(0,0\right)$ form factor on $T$ and
$Q^{2}$ for the ground state $(n=0)$ and the first excited state $(n=1)$ of
nucleons in the left and right panels correspondingly. To see the impact of
the parameter $\alpha$ on the shape of the form factor dependency, we have
plotted its graph for the values $\alpha$ = 0.1, 0.2, 0.3. It is seen that the
form factor graphs at different values of $\alpha$ have the same shape for the
ground and excited states correspondingly. This means that the impact of the
quantum number $n$ on the shape of the form factor is weak enough. The shape
deformation of the form factor on changing the parameter $\alpha$ may observe
by comparing the left panel figures one with another (or right ones). We
collect the $G_{A}\left(Q^{2},T\right)/G_{A}\left(0,0\right)$ form factor
graphs at $Q^{2}=0$ in Fig.5 to show more apparent the shape deformation on
changing the $\alpha$ parameter. Another observation is that in all
$T$-dependency graphs the $G_{A}$ form factor goes to zero around the
$T_{c}=0.2$ GeV. This is interpreted with the hadron melting at the
temperature of the confinement-deconfinement phase transition. In Fig.6, we
plot two-dimensional graphs for the form factor at fixed values $T=0,0.1,0.17$
GeV. This analysis shows more clearly how the shape of the $Q^{2}$-dependency
deforms on a small change in $T$.

The main aim of the present work is to study the influence of the medium
temperature on the $G_{A}$ form factor. Since there is no experimental
data333The only theoretical result for the temperature-dependent $g_{a}(T)$
coupling constant is Ref. Dominguez ; Dominguez3 , which is based on QCD Sum
Rules. In Ref. Dominguez2  these authors express that this result contrasts
with other effective hadronic couplings (Dominguez2 ; Ayala ) and diverges
when $T\rightarrow T_{c}$. This result also disagrees with the soft-wall model
result on coupling (form factor) vanishing at $T_{c}$ observed in Gutsche2 ;
Gutsche3 ; Gutsche4 ; Nasibova1 ; Taghiyeva . for the $G_{A}$ form factor at
finite temperature, in Fig. 7, we present a comparison of our result for
$G_{A}(Q^{2})/G_{A}(0)$ at $T=0$ with the hard-wall AdS QCD model result for
the $G_{A}(Q^{2})$ form factor Mamedov ; Huseynova ; Huseynov , experimental
data Bernard ; Park  and the result of the Light Cone Sum Rules (LCSR) Anikin
.

All theoretical predictions for the $Q^{2}$ dependencies have the form
$1/Q^{n}$ dependence, which is typical for the nucleon form factors.
Experimental data for $G_{A}$ are exist in two different intervals of $Q^{2}$,
in $~{}0.075$ GeV2 $\leq$ $Q^{2}$ $\leq 1$ GeV2 (Bernard ) and in $2$ GeV2
$\leq Q^{2}$ $\leq 4$ $GeV^{2}$ (Park ).

There is no momentum restriction on the applicability of holographic models,
while other theoretical approaches, such as light cone sum rules, are
applicable only for $Q^{2}\geq 1$ GeV2. The theoretical dependencies obtained
from the hard- and soft-wall models cover both momentum intervals of the
experiment.

It is seen from Fig.7 that holographic model predictions for the $Q^{2}$
dependence of the $G_{A}$ form factor are in good agreement with the
experimental data and other theoretical approaches.

In numerical calculations, we have used for the values of light quark mass
$m_{q}$ and quark condensate $\Sigma$ the values $m_{q}=0.00234$ GeV and
$\left(\Sigma\right)^{1/3}=0.311$ GeV respectively, Maru . The constant
$k_{1}=-0.98$ was taken from Ahn , which was obtained from the fitting of the
couplings $g_{\pi NN}$ and $g_{\rho NN}$ of AdS/QCD model with the
experimental data. The Yukawa constant $g_{Y}=9.182$ is fixed as in Hong . In
Fig.8, we plot the axial-vector transition form factor $G_{AT}$ for the above-
fixed values of the $\alpha$ parameter using formulas (49)-(52). In Fig.9,
similar analysis for the shape deformation at temperature changes has been
carried out. In comparison with the $G_{A}$ form factor, we observe some small
changes in the shape of the graphs for the $G_{AT}$ form factor though forms
of all dependencies in $G_{AT}$ form factor are similar to the ones in
corresponding $G_{A}$ form factor.

##  XI Summary

In the present work, we have investigated the temperature dependence of the
axial-vector form factor in the framework of the soft-wall AdS/QCD model
containing the thermal dilaton field. Numerical analysis shows that the form
factor decreases with increasing temperature. This means in a hot medium, the
$\beta$ decay has less probability when temperature increases. This result may
be used for the studies of nucleon-axial-vector meson interactions and the
influence of temperature on $beta$ decay in the nuclear

medium.

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Figure 1: $T$ dependence of nucleon mass $M_{0,\frac{1}{2}}^{2}(T)$.

Figure 2: $G_{A}(Q^{2},T)/G_{A}(0,0)$ form factor in the ground $(n=0)$ and
excited states $(n=1)$ of the nucleons at $\alpha=0.1$.

Figure 3: $G_{A}(Q^{2},T)/G_{A}(0,0)$ form factor in the ground $(n=0)$ and
excited states $(n=1)$ of the nucleons at $\alpha=0.2$.

Figure 4: $G_{A}(Q^{2},T)/G_{A}(0,0)$ form factor in the ground $(n=0)$ and
excited states $(n=1)$ of the nucleons at $\alpha=0.3$.

Figure 5:  $G_{A}(T)/G_{A}(0)$ form factors at $Q^{2}=0$ and different values
of $\alpha$ in the ground $(n=0)$ and excited $(n=1)$ states of the nucleons.

Figure 6:  $G_{A}(Q^{2})/G_{A}(0)$ form factors at different values of $T$ in
the ground $(n=0)$ and excited $(n=1)$ states of the nucleons. Figure 7:
Comparison of our result for the $G_{A}(Q^{2},T=0)/G_{A}(0,0)$ form factor at
$n=0$ and $T=0$ values (densely dash-dotted line) with the hard-wall result
obtained in Mamedov ; Park  (solid blue line), LCSR results in the $2\leq
Q^{2}\leq 5$ interval (red dashed line and red solid line) (Anikin ) and
experimental data taken from Bernard  and Dominguez2 .

Figure 8: $G_{AT}(Q^{2},T)/G_{AT}(0,0)$ transition form factor of nucleons at
parameter $\alpha=0.1,0.2,0.3$ respectively

Figure 9: $Q^{2}$ dependence of transition form factor
$G_{AT}(Q^{2})/G_{AT}(0)$ at different values of $T$ and $T$ dependence of
$G_{AT}(T)/G_{AT}(0)$ at different values of parameter $\alpha$ respectively