Emergence of non-linear electrodynamic theories from
$T\bar{T}$-like deformations

H. Babaei-Aghbolagh†, Komeil Babaei Velni∗, Davood Mahdavian Yekta‡ and H.
Mohammadzadeh†

†Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179,
Ardabil, Iran
∗Department of Physics, University of Guilan, P.O. Box 41335-1914, Rasht, Iran
∗School of Physics and School of Particles and Accelerators,
Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531,
Tehran, Iran
‡Department of Physics, Hakim Sabzevari University, P.O. Box 397, Sabzevar,
Iran

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
In this letter, we investigate the deformation of the ModMax theory, as a
unique Lagrangian of non-linear electrodynamics preserving both conformal and
electromagnetic-duality invariance, under $T\bar{T}$-like flows. We will show
that the deformed theory is the generalized non-linear Born-Infeld
electrodynamics. Being inspired by the invariance under the flow equation for
Born-Infeld theories, we propose another $T\bar{T}$-like operator generating
the ModMax and generalized Born-Infeld non-linear electrodynamic theories from
the usual Maxwell and Born-Infeld theories, respectively.

##  1 Introduction

As the first non-linear electrodynamic (NED) generalization of Maxwell theory,
the Born-Infeld (BI) model was proposed to impose an upper limit on the self-
energy of charged point particles [1]. Its Lagrangian density is a function of
two independent Lorentz scalars $\mathcal{S}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$
and $\mathcal{P}=-\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}$ as follows

$\mathcal{L}_{BI}=T\left(1-\sqrt{1-\frac{2}{T}\mathcal{S}-\frac{1}{T^{2}}\mathcal{P}^{2}}\right),$
(1.1)

where
$\tilde{F}^{\mu\nu}=\frac{1}{2}\,\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$
corresponds to the Hodge dual of the electromagnetic field strength
$F_{\mu\nu}$. It has been shown [2, 3] that the Lagrangian of NED theories is
constrained by the equation $G\tilde{G}-F\tilde{F}=0$, which is known as the
self-duality condition [4, 5, 6, 7, 8, 9, 10, 11, 12]. The antisymmetric
tensor field $G$ is defined by

$G_{\mu\nu}=-2\frac{\partial{\cal
L}(\mathcal{S},\mathcal{P})}{\partial{F^{\mu\nu}}},$ (1.2)

where the factor 2 takes into account because of the partial derivative with
respect to the antisymmetry of ${F}_{\mu\nu}$.

A number of the NED theories such as BI, Bialynicki-Birula and Bossard-
Nicolai[9] satisfy the self-duality condition while the Heisenberg-Euler [13]
theory does not. Though in general the action of a NED theory may not be
invariant under this kind of duality, the physical objects such as the
equations of motion [14], the energy-momentum tensor (EMT) [15, 16] and the
scattering amplitudes [19, 18, 17] are invariant. It has been proposed another
non-linear modification of Maxwell theory that preserves the electromagnetic
duality as well as conformal invariance, the so-called ModMax theory [20]
which is described by

${\cal
L}_{MM}=\cosh\gamma\,\mathcal{S}+\sinh\gamma\,\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}.$
(1.3)

The unitary condition imposes the restriction $\gamma\geq 0$, which admits
exact Maxwell-like plane wave solutions of arbitrary polarization [20]. The
ModMax theory could be found from dimensional reduction of a chiral 2-form ED
in $D\\!=\\!6$ dimensions [21] obeying the self-duality condition [22].
However, the Maxwell ED is not a weak-field limit of the ModMax theory because
of the conformal invariance, but it is recovered when the ModMax constant
parameter $\gamma$ tends to zero [20]. More studies on this theory could be
found in Refs. [23, 24, 25, 26, 27, 28].

Recently, Zamolodchikov and Smirnov proposed an exactly solvable deformation
of two-dimensional (2D) quantum field theory by an irrelevant operator, known
as $T\bar{T}$-deformation [29, 30]. The $T\bar{T}$ operator is constructed by
the product of components of the EMT. Generalization of this operator to
higher-dimensional theories with a different version of $T\bar{T}$ operator
are given in Refs. [31, 32, 33, 34, 35]. One can consider a Lorentz invariant
deformation of the type $S_{QFT}=S_{CFT}+S_{\lambda}$ where
$S_{\lambda}=\lambda\int d^{2}x\,T\bar{T}$ and $\lambda$ is the deformation
parameter. Consequently, a one parameter family of theories could be emerged
by ${dS_{QFT}}/{d\lambda}=\int d^{2}x\,(T\bar{T})_{\lambda}$ for finite
$\lambda$ while both sides of the equation are renormalized and UV finite as
well [29, 36]. In other words, a deformed Lagrangian $\mathcal{L}^{\lambda}$
can be derived by solving this $T\bar{T}$-flow equation. For example, if we
consider the $T\bar{T}$-deformation of a free massless scalar field theory,
the deformed theory leads to the Nambu-Goto action where $\lambda$ is
identified with the string tension $T=1/(2\pi\alpha^{\prime})$ [36].

It would be of interest to investigate other forms of this operator in higher
dimensions, especially in the light of the expected form of the deformation
equation proposed by Cardy [31], regardless of the holographic interpretations
[37, 38, 39, 40, 41, 42]. For instance, in [35], we have studied that the
construction of $T\bar{T}$-deformation for some NED BI-type theories in 4D
spacetime as follows

${O}_{T^{2}}^{\lambda}=T_{\mu\nu}T^{\mu\nu}-\frac{1}{2}{T_{\mu}}^{\mu}{T_{\nu}}^{\nu},$
(1.4)

where ${T_{\mu}}^{\mu}$ is the trace of EMT. In fact, we reproduced the
expansion of 4D BI-type theories by deforming the free Lagrangian by means of
the operator defined in Eq. (1.4).

In this letter, using the operator (1.4) and employing a simple perturbative
integration technique [35], we investigate the $T\bar{T}$-deformation of the
ModMax theory. It is shown that the emergent deformed theory is the same
generalized Born-Infeld (GBI) theory given in Ref. [21]. In addition, our main
motivation is to construct a new $T\bar{T}$-like operator
${O}_{T^{2}}^{\gamma}$ with deformation parameter $\gamma$ under which the
ModMax and GBI theories are invariant, as the BI theory which is invariant
under ${O}_{T^{2}}^{\lambda}$. We also show that this operator deforms the
Maxwell theory to the ModMax theory with no need for additional conditions to
establish the conformal and electromagnetic duality invariance as it happens
in Ref. [20]. It is proved that the GBI theory is derived from BI theory by
using operator ${O}_{T^{2}}^{\gamma}$.

The organization of this paper is as follows: In section 2, we consider the
deformation of the ModMax theory under $T\bar{T}$-like operator
${O}_{T^{2}}^{\lambda}$ by employing a perturbative approach and show that
final deformed theory is equivalent to the expansion of GBI theory for small
$\lambda$. We will find a new $T\bar{T}$-like operator in section 3, as for
that the GBI theory is invariant under the flow equation $\partial{\cal
L}_{\gamma BI}/{\partial\gamma}=1/2\,O^{\gamma}_{T^{2}}$ where $\gamma$ is a
dimensionless parameter in the NED theory. We show in section 4 that not only
the ModMax theory in invariant under the $T\bar{T}$-deformation by
${O}_{T^{2}}^{\gamma}$, but also can be reconstructed from Maxwell theory
under the effect of this operator. Finally, in section 5, we provide a summary
of our results and outlook.

##  2 $T\bar{T}$-deformation of the ModMax theory by ${O}_{T^{2}}^{\lambda}$

Starting with a free theory denoted by $\mathcal{L}_{\circ}$, one may
perturbatively solve the $T\bar{T}$-flow equation
${d\mathcal{L}_{\lambda}}/{d\lambda}=1/8\,(T\bar{T})$ in terms of $\lambda$ by
taking into account $T_{\mu\nu}^{\lambda}$ related to the $\lambda$ order of
the theory [43]. In other words, the deformation is solvable: one can make
precise statements about properties of the deformed theory
$\mathcal{L}_{\lambda}$ in terms of $\mathcal{L}_{\circ}$. This deformation
preserves integrability: if one begins with a theory $\mathcal{L}_{\circ}$
which is integrable, in the sense that the theory has infinitely many local
integrals of motion, then the deformed theory $\mathcal{L}_{\lambda}$ at
finite $\lambda$ is also integrable [36, 32, 44].

Now, using perturbative approach, we find the interacting terms for the ModMax
theory described by Eq. (1.3). We first compute the EMT as following

$T_{\mu\nu}=g_{\mu\nu}\,{\cal L}_{MM}+{F_{\mu}}^{\rho}G_{\nu\rho},$ (2.1)

where from Eq. (1.2), the tensor $G$ is given by

$G_{\mu\nu}=\cosh\gamma
F_{\mu\nu}+\sinh\gamma\frac{F_{\mu\nu}\mathcal{S}+\tilde{F}_{\mu\nu}\mathcal{P}}{\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}}.$
(2.2)

Substituting Eq. (2.2) in (2.1), we obtain

$\displaystyle
T_{\mu\nu}=T^{Max}_{\mu\nu}\bigg{(}\cosh\gamma+\frac{\sinh\gamma\,\mathcal{S}}{\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}}\bigg{)},$
(2.3)

where
$T^{Max}_{\mu\nu}=F_{\mu\rho}{F_{\nu}}^{\rho}-1/4\,g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}$
is the EMT of the Maxwell theory. Due to the conformal symmetry of the ModMax
theory, it would be expected that its EMT is traceless and proportional to
$T^{Max}_{\mu\nu}$ [20, 45].

Using Eq. (2.3) in (1.4) to construct ${O}_{T^{2}}^{\lambda}={O}_{0}$ at the
order of $\lambda^{0}$, the first order $\lambda$ deformed Lagrangian is
obtained as

$\displaystyle{\cal L}^{\prime}_{\lambda}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle{\cal{\cal L}}_{\circ}+\frac{1}{8}\int{O}_{0}\,d\lambda$ (2.4)
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle{\cal
L}_{MM}+\frac{\lambda}{2}\bigg{(}\sinh\gamma\,\mathcal{S}+\cosh\gamma\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}\bigg{)}^{2},$

where we have used the convention ${O}_{0}\\!=\\!8\left({\partial{\cal
L}^{\prime}_{\lambda}}/{\partial\lambda}\right)$ [43]. For simplicity, we use
$\mathcal{S}\\!=\\!\mathcal{P}\sinh(\alpha)$ so that

$\displaystyle{\cal L}_{MM}\\!=\\!\mathcal{P}\sinh(\alpha+\gamma),\qquad
T_{\mu\nu}=T^{Max}_{\mu\nu}\,\frac{\cosh(\alpha+\gamma)}{\cosh(\alpha)}.$
(2.5)

After a straightforward calculation, Eq. (2.4) recasts as follows

$\displaystyle{\cal L}^{\prime}_{\lambda}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\mathcal{P}\sinh(\alpha+\gamma)+\frac{1}{2}\lambda\,\mathcal{P}^{2}\,\cosh^{2}(\alpha+\gamma)$
(2.6) $\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle{\cal
L}_{MM}+\frac{1}{2}\lambda({\cal L}_{MM}^{2}+\mathcal{P}^{2}).$

Following a similar prescription, we can deform the Lagrangian to the next
order of $\lambda$ as ${\cal L}{{}^{\prime\prime}}_{\lambda}\\!=\\!{\cal
L}{{}^{\prime}}_{\lambda}+{1}/{8}\int{O}_{1}\,d\lambda$ where ${O}_{1}$ is the
$T\bar{T}$ operator presented by Eq. (1.4) and it is constructed from the EMT
of ${\cal L}{{}^{\prime}}_{\lambda}$. Iterating this method, one can deform
the free theory to higher orders of $\lambda$. We have shown in [35] that if
one deforms the free Maxwell theory by this method to higher orders of
$\lambda$, the Maxwell BI theory will be recovered. So, we are going to
investigate this implication again in this paper for the ModMax theory. For
example, suppose we want to deform the ModMax theory to order $\lambda^{3}$,
thus we need to have $G_{\mu\nu}$ and EMT at least to the order of
$\lambda^{2}$. They are given by

$\displaystyle G_{\mu\nu}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\cosh(\alpha+\gamma)\,F_{\mu\nu}+\sinh\gamma\,\tilde{F}_{\mu\nu}}{\cosh\alpha}+\lambda\,\mathcal{P}\frac{\sinh\bigl{(}2(\alpha+\gamma)\bigr{)}F_{\mu\nu}+2\cosh\gamma\,\cosh(\alpha+\gamma)\tilde{F}_{\mu\nu}}{2\cosh\alpha}$
(2.7) $\displaystyle\\!\\!\\!+\\!\\!\\!$
$\displaystyle\lambda^{2}\,\mathcal{P}^{2}\frac{\Bigl{(}3\cosh\bigl{(}3(\alpha\\!+\\!\gamma)\bigr{)}\\!+\\!\cosh(\alpha\\!+\\!\gamma)\Bigr{)}F_{\mu\nu}\\!+\\!\Bigl{(}3\sinh(2\alpha\\!+\\!3\gamma)\\!+\\!\sinh(2\alpha\\!+\\!\gamma)\\!+\\!2\sinh\gamma\Bigr{)}\tilde{F}_{\mu\nu}}{8\cosh\alpha},$

and

$\displaystyle T_{\mu\nu}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\cosh(\alpha+\gamma)}{\cosh\alpha}\,\biggl{(}F_{\mu}{}^{\delta}F_{\nu\delta}+\sinh\alpha\,\mathcal{P}{g}_{\mu\nu}+\frac{1}{4}\lambda\mathcal{P}\left(4\sinh(\alpha+\gamma)F_{\mu}{}^{\delta}F_{\nu\delta}+\bigl{(}\cosh(2\alpha+\gamma)-3\cosh\gamma\bigr{)}\mathcal{P}\,\,{g}_{\mu\nu}\right)$
(2.8) $\displaystyle\\!\\!\\!+\\!\\!\\!$
$\displaystyle\frac{1}{8}\lambda^{2}\,\mathcal{P}^{2}\bigl{(}\bigl{(}6\cosh\bigl{(}2(\alpha+\gamma)\bigr{)}-2\bigr{)}F_{\mu}{}^{\delta}F_{\nu\delta}+\bigl{(}\sinh(3\alpha+2\gamma)-2\sinh(\alpha)-5\sinh(\alpha+2\gamma)\bigr{)}\mathcal{P}\,{g}_{\mu\nu}\bigr{)}\biggr{)}.$

Therefore, saving up to the third order of $\lambda$, we obtain

$\displaystyle{\cal L}_{MM}^{\lambda^{3}}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle{\cal L}_{MM}+\frac{1}{8}\int O_{0}\,d\lambda+\frac{1}{8}\int
O_{1}\,d\lambda+\frac{1}{8}\int O_{2}\,d\lambda+\cdots$ (2.9)
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle{\cal
L}_{MM}\\!+\\!\frac{1}{2}\lambda\,({\cal
L}_{MM}^{2}\\!+\\!\mathcal{P}^{2})\\!+\\!\frac{1}{2}\lambda^{2}{{\cal
L}_{MM}}({\cal L}_{MM}^{2}\\!+\\!\mathcal{P}^{2})$
$\displaystyle\\!\\!\\!+\\!\\!\\!$
$\displaystyle\,\frac{1}{8}\lambda^{3}({\cal
L}_{MM}^{2}+\mathcal{P}^{2})(5{\cal L}_{MM}^{2}+P^{2})+{\cal O}(\lambda^{4}),$

where ${O}_{i}={O}_{T^{2}}^{\lambda^{i}}$ is the corresponding $T\bar{T}$
operator (1.4) at the order of $\lambda^{i}$. We inspect that for the $i$th
order of the deformed theory, the trace of the EMT satisfies the equation
$T_{\mu}^{\mu}=-4\lambda\left({\partial{\cal
L}^{\lambda^{i}}_{MM}}/{\partial\lambda}\right)$ which may be exact
renormalization group equation of the $T\bar{T}$ deformed theory [37, 40]
111It was found in Ref. [36] that for the special case of free scalars treated
as a classical field theory, this equation holds to all orders in $\lambda$.
It will be assumed that this is true for any deformed CFT.. We surprisingly
find out that the deformed Lagrangian (2.9) is equivalent to the expansion of
the following $GBI$ Lagrangian proposed in [21]

${\cal L}_{\gamma BI}=\frac{1}{\lambda}\Bigg{[}1-\sqrt{1-\lambda\Bigl{(}2{\cal
L}_{MM}+\lambda\mathcal{P}^{2}\Bigr{)}}\Bigg{]}.$ (2.10)

${\cal L}_{\gamma BI}$ is a duality-symmetric GBI NED, the combination of the
ModMax and Born-Infeld theories, in which the weak-field limit yields the
ModMax theory and the strong-field limit leads to the Bialynicki-Birula
electrodynamics [3]222The Lagrangian density of GBI theory in [45] is given by
${\cal L}_{\gamma BI}=T-\sqrt{T^{2}-2T{\cal
L}_{MM}-\frac{1}{16}(F_{\mu\nu}\tilde{F}^{\mu\nu})^{2}},$ where ${\cal
L}_{MM}$ is the ModMax Lagrangian density denoted by Eq. (1.3) and we have
identified the tension $T$ with the inverse of $\lambda$ in our convention.
Note also that this Lagrangian in the limit $\gamma\rightarrow 0$ goes to the
BI Lagrangian given in Eq. (1.1).. Thus, one can write

${\cal L}_{\gamma BI}={\cal L}_{MM}+\frac{1}{8}\int
O_{T^{2}}^{\lambda}\,d\lambda.$ (2.11)

It has been shown [46] that the $T\bar{T}$ operator can represent a
quantization method in 2D CFT. Since we have the effects of
$O_{T^{2}}^{\lambda}$ operator in the ModMax theory, it could be interesting
to use this operator in quantization of this theory and try to derive an
evolution-type equation for the quantum energy spectrum. In other words,
$T\bar{T}$ operator could be used to track what happens to the behavior of
energies of the system as a function of the deformation parameter. In this
respect, the deformation is called solvable which means some quantities in the
deformed theory, such as the energy spectrum or scattering amplitude, can be
computed in terms of the corresponding quantities in the un-deformed theory
[30, 36].

##  3 New $T\bar{T}$-like deformation for constant $\gamma$

The connection between non-linear symmetries and $T\bar{T}$ flows might give a
novel way to organize interesting low-energy effective actions in the string
theory such as the BI action. In this respect, there is a natural question:
are there more general classes of theories with non-linear symmetries related
to the flow equations for some $T\bar{T}$-like operator? For instance, the
bosonic BI action can be obtained by deforming the Maxwell theory with
operator $O_{T^{2}}^{\lambda}$ [34].

The GBI theory denoted by Eq. (2.10) has two constant parameters $\lambda$ and
$\gamma$. Thus, it is expected that one can consider two $T\bar{T}$-like
operators corresponding to these couplings, i.e. $O_{T^{2}}^{\lambda}$ and
$O_{T^{2}}^{\gamma}$, so that $T\bar{T}$-flow of GBI theory respecting them.
From Eqs. (2.10) and (1.2), we have

$\displaystyle
G_{\mu\nu}=\frac{\cosh\gamma\,F_{\mu\nu}\\!+\\!\lambda\mathcal{P}\tilde{F}_{\mu\nu}\\!+\\!\sinh\gamma\,(F_{\mu\nu}\mathcal{S}\\!+\\!\tilde{F}_{\mu\nu}\mathcal{P})(\mathcal{S}^{2}\\!+\\!\mathcal{P}^{2})^{-\frac{1}{2}}}{x},$

where $x\\!=\\!\sqrt{1-\lambda\Bigl{(}2{\cal
L}_{MM}+\lambda\mathcal{P}^{2}\Bigr{)}}$. Substituting Eq. (3) in (2.1), we
find

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\bigl{(}\cosh\gamma\,\,\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}+\sinh\gamma\,\mathcal{S}\bigr{)}F_{\mu}^{\alpha}F_{\nu\alpha}}{x\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}}$
(3.1) $\displaystyle+$
$\displaystyle\frac{\bigl{(}2\cosh\gamma\,\mathcal{S}\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}+\sinh\gamma(2\mathcal{S}^{2}+\mathcal{P}^{2})\bigr{)}{g}_{\mu\nu}}{x\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}}$
$\displaystyle+$ $\displaystyle\frac{\big{(}x-1\big{)}{g}_{\mu\nu}}{\lambda
x}.$

From Eq. (3.1), it is obvious that the above EMT reduces to the one that of
the BI in the limit of $\gamma\rightarrow 0$. Substituting Eq. (3.1) in (1.4)
and taking integral of the resulted equation leads to the following

$\displaystyle\\!\\!\frac{1}{8}\int
O^{\lambda}_{T^{2}}\,d\lambda\\!=\\!\\!\int\\!\frac{1\\!-\\!\lambda\,{\cal
L}_{MM}\\!-\\!x}{\lambda^{2}x}\,d\lambda$
$\displaystyle\\!=\\!\frac{1}{\lambda}\Bigg{[}1-\sqrt{1-\lambda\Bigl{(}2{\cal
L}_{MM}+\lambda\mathcal{P}^{2}\Bigr{)}}\Bigg{]}\\!=\\!{\cal L}_{\gamma BI}.$
(3.2)

The finding of Eq. (3) shows that the flow $\partial{\cal L}_{\gamma
BI}/\partial\lambda$ and the $T\bar{T}$ operator have exactly the same
structure and they satisfy the following equation

$\displaystyle\frac{\partial{\cal L}_{\gamma
BI}}{\partial\lambda}=\frac{1}{8}\,O^{\lambda}_{T^{2}},$ (3.3)

which means that the GBI Lagrangian (2.10) obeys a $T\bar{T}$-flow driven by
the operator $O^{\lambda}_{T^{2}}$. It has also been shown in Ref. [43] that
the BI theory satisfies in a similar equation. In addition, we find out that
the EMT (3.1) fulfills $T_{\mu}^{\mu}=-4\lambda\frac{\partial{\cal L}_{\gamma
BI}}{\partial\lambda}$ for all orders.

Now, we investigate the deformation of the GBI theory under a new operator
$O_{T^{2}}^{\gamma}$ using the similar relationship appeared in Eq. (3.3) as
the form of $\partial{\cal L}_{\gamma BI}/\partial\gamma=f(T\bar{T})$ where
$f(T\bar{T})$ is a function of operator. To find an appropriate form of the
new operator, we propose a function as
$f\big{(}a\,T_{\mu\nu}T^{\mu\nu}-b\,{T_{\mu}}^{\mu}{T_{\nu}}^{\nu}\big{)}$
where $a$ and $b$ are constants [31]. Taking partial derivative of Eq. (2.10)
with respect to $\gamma$ yields

$\displaystyle\frac{\partial{\cal L}_{\gamma
BI}}{\partial\gamma}=\frac{\cosh(\alpha+\gamma)\mathcal{P}}{x},$ (3.4)

then from Eq. (3.1), taking into account $\partial{\cal L}_{\gamma
BI}/\partial\gamma\\!=\\!f(T\bar{T})$, we find the following result

$\displaystyle
f(T\bar{T})=\frac{1}{2}\sqrt{T_{\mu\nu}T^{\mu\nu}-\frac{1}{4}{T_{\mu}}^{\mu}{T_{\nu}}^{\nu}}.$
(3.5)

Now, by introducing the new $T\bar{T}$-like operator

$\displaystyle
O_{T^{2}}^{\gamma}=\sqrt{T_{\mu\nu}T^{\mu\nu}-\frac{1}{4}{T_{\mu}}^{\mu}{T_{\nu}}^{\nu}},$
(3.6)

we have $\partial{\cal L}_{\gamma BI}/\partial\gamma=1/2\,O_{T^{2}}^{\gamma}$.
Indeed, the GBI theory is invariant under the deformation by
$O_{T^{2}}^{\gamma}$ as well as by $O_{T^{2}}^{\lambda}$, i.e., ${\cal
L}_{\gamma BI}$ is a solution for solvable deformations with respect to
$\lambda$ and $\gamma$.

The BI theory is recovered by the deformation of the Maxwell theory with the
operator $O_{T^{2}}^{\lambda}$ [35] and it is invariant under this deformation
[43], i.e., ${\partial{\cal
L}_{BI}}/{\partial\lambda}=1/8\,O_{T^{2}}^{\lambda}$. It is of interest to
know how the BI theory deforms under (3.6). Identifying $T=\lambda^{-1}$, one
can rewrite the Lagrangian density (1.1) as follows

${\cal
L}_{BI}=\frac{1}{\lambda}\bigg{[}1-\sqrt{1-\lambda\bigl{(}2\mathcal{S}+\lambda\mathcal{P}^{2}\bigr{)}}\bigg{]},$
(3.7)

then by defining $y=\sqrt{1-\lambda(2\mathcal{S}+\lambda\mathcal{P}^{2})}$,
and using Eq. (1.2) in the EMT of Eq. (2.1) we obtain

$\displaystyle
T_{\mu\nu}=\frac{1}{y}T^{Max}_{\mu\nu}+\frac{(y+\lambda\mathcal{S}-1)}{\lambda\,\,y}g_{\mu\nu}.$
(3.8)

Following the same procedure of the perturbative deformation introduced
earlier, for the first order of $\gamma$, we have

$\displaystyle{\cal L^{\prime}}_{\gamma}={\cal L}_{BI}+\frac{1}{2}\int
O_{0}^{\gamma}\,d\gamma={\cal
L}_{BI}+\gamma\frac{\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}}{y},$ (3.9)

where $O_{0}^{\gamma}$ refers to $O_{T^{2}}^{\gamma}$ at the order of
${\gamma}^{0}$. To find the next order of deformation, we should calculate the
EMT for ${\cal L^{\prime}}_{\gamma}$, that is

$\displaystyle T^{\prime}_{\mu\nu}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{1}{y}T^{Max}_{\mu\nu}+\frac{(y+\lambda\mathcal{S}-1)}{\lambda\,\,y}g_{\mu\nu}$
$\displaystyle\\!\\!+\\!\\!$
$\displaystyle\gamma\bigg{(}\\!\frac{\mathcal{S}\,y^{2}\\!+\\!\lambda(\mathcal{S}^{2}\\!+\\!\mathcal{P}^{2})}{y^{3}\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}}T^{Max}_{\mu\nu}\\!-\\!\lambda\frac{\mathcal{S}\sqrt{\mathcal{S}^{2}\\!+\\!\mathcal{P}^{2}}}{y^{3}}g_{\mu\nu}\bigg{)}.$

Noting the relation (3), one can find the corresponding $T\bar{T}$ operator
$O_{1}^{\gamma}$ and then deform theory upto the second order of $\gamma$ as

$\displaystyle{\cal L^{\prime\prime}}_{\gamma}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle{\cal
L}_{BI}\\!+\\!\gamma\frac{\sqrt{\mathcal{S}^{2}\\!+\\!\mathcal{P}^{2}}}{y}\\!-\\!\gamma^{2}\frac{(\lambda\mathcal{P}^{2}\\!+\\!\mathcal{S})(\lambda\mathcal{S}\\!-\\!1)}{2y^{3}}.$
(3.11)

Continuing the perturbative procedure for arbitrary order of $\gamma$ we find
out that the obtained result is actually the expansion of ${\cal L}_{\gamma
BI}$ with respect to the small values of $\gamma$, i.e.,

${\cal L}_{\gamma BI}={\cal L}_{BI}+\frac{1}{2}\int
O_{T^{2}}^{\gamma}\,d\gamma.$ (3.12)

##  4 ModMax theory as $T\bar{T}$-deformation of the Maxwell theory by
$O_{T^{2}}^{\gamma}$

In the following, we consider the deformation of the ModMax and Maxwell
theories by using the new $T\bar{T}$-like operator $O_{T^{2}}^{\gamma}$. In
the first case, we use the EMT (2.3) to construct the operator
$O_{T^{2}}^{\gamma}$ according to the definition (3.6). Then, taking the
partial derivative of the Lagrangian density of the ModMax theory given by
(1.3) with respect to $\gamma$, we obtain

$\frac{\partial{\cal
L}_{MM}}{\partial\gamma}=\frac{1}{2}\,O_{T^{2}}^{\gamma},$ (4.1)

which states that the ModMax theory is a solution of the flow equation, just
like BI with $O_{T^{2}}^{\lambda}$ or GBI under $O_{T^{2}}^{\gamma}$
deformations. In other words, the ModMax theory is the only conformal duality-
invariant NED theory. It has been shown that $\mathcal{L}_{MM}$ and
$O_{T^{2}}^{\gamma}$ make up a pair of canonically conjugate variables which
can be regarded as local coordinates of a 2D symplectic manifold [47].

For the latter case, substituting the EMT of the Maxwell theory in (3.6), one
can derive the first order operator as
$O_{0}^{\gamma}=2\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}$. Then, the deformed
Lagrangian upto the first order according to ${\cal L^{\prime}}_{\gamma}={\cal
L}_{Max}+1/2\int O_{0}^{\gamma}\,d\gamma$ becomes

$\displaystyle{\cal
L^{\prime}}_{\gamma}=\mathcal{S}+\gamma\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}},$
(4.2)

where ${\cal L}_{Max}=\mathcal{S}$. For the next order, we should find the EMT
corresponding to $\cal L^{\prime}_{\gamma}$, then from Eqs. (1.2) and (2.1),
we obtain

$\displaystyle
T^{\prime}_{\mu\nu}=T^{Max}_{\mu\nu}\bigl{(}1+\gamma\frac{\mathcal{S}}{\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}}\bigr{)}.$
(4.3)

Constructing the operator $O_{1}^{\gamma}$ with $T^{\prime}_{\mu\nu}$, the
deformed theory up to the second order is following

$\displaystyle{\cal L}^{{}^{\prime\prime}}_{\gamma}$ $\displaystyle=$
$\displaystyle\mathcal{S}+\gamma\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}+\frac{1}{2}\gamma^{2}\mathcal{S}.$
(4.4)

One can proceed step by step to deform the Maxwell theory to higher orders of
$\gamma$. We have done this calculation up to the 7th order as follows

$\displaystyle{\cal L}^{(7)}_{\gamma}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\mathcal{S}+\gamma\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}+\tfrac{1}{2}\gamma^{2}\mathcal{S}+\tfrac{1}{6}\gamma^{3}\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}+\tfrac{1}{24}\gamma^{4}\mathcal{S}$
(4.5) $\displaystyle\\!\\!+\\!\\!$
$\displaystyle\\!\tfrac{1}{120}\gamma^{5}\sqrt{\mathcal{S}^{2}\\!+\\!\mathcal{P}^{2}}\\!+\\!\tfrac{1}{720}\gamma^{6}\mathcal{S}\\!+\\!\tfrac{1}{5040}\gamma^{7}\sqrt{\mathcal{S}^{2}\\!+\\!\mathcal{P}^{2}}.$

This expression is exactly equal to the expansion of the ModMax theory given
in Eq. (1.3) with respect to the small values of $\gamma$, thus similar to the
Eqs. (2.11) and (3.12) one can write

$\displaystyle{\cal L}_{MM}={\cal L}_{Max}+\frac{1}{2}\int
O_{T^{2}}^{\gamma}\,d\gamma.$ (4.6)

##  5 Conclusion and outlook

We have studied diverse deformations of the NED extensions of the Maxwell
theory under $T\bar{T}$-like operators, in particular, the ModMax theory as a
unique conformal and duality invariant modification of the Maxwell theory. We
shown that the electrodynamic BI action and the GBI theory can be respectively
constructed by deforming the Maxwell and the ModMax Lagrangian density by
operator ${O}_{T^{2}}^{\lambda}$ where $\lambda$ is a typical deformation
parameter. On the other hand, it was found that both of the BI and GBI
theories are invariant under the flow equation with respect to ${\lambda}$. In
addition, since the ModMax theory has a non-negative constant $\gamma$, which
could be regarded as a deformation parameter, we proposed a $T\bar{T}$-like
operator $O_{T^{2}}^{\gamma}$ in Eq. (3.6) such that the GBI theory is
invariant under the flow equation of this parameter.

In order to better understand the sequence flow of these operators, we have
also provided a schematic chart for the application of two operators on the
Maxwell theory and the corresponding emergent deformed theories in Fig 1. It
is worth mentioning that there is a main difference between these operators in
the number of higher derivative terms that produce through deformation process
and may be important in investigating the integrability and quantization of
the emergent theories. The operator ${O}_{T^{2}}^{\lambda}$ increases the
derivatives of the theory which means the enhancement of the levels of energy
where the traceless property of the EMT is not necessarily retained, while, in
contrast, the operator $O_{T^{2}}^{\gamma}$ generates terms of the order of
the Maxwell action and keeps the traceless property. However, the
electromagnetic duality is respected under the deformation of two operators.

${{\cal
L}_{Max}}$${\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\cal
L}_{BI}}$${\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\cal
L}_{MM}}$${\definecolor{.}{rgb}{0.5,0,0.5}\color[rgb]{0.5,0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0,0.5}{\cal
L}_{\gamma
BI}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{O}_{T^{2}}^{\lambda}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}O_{T^{2}}^{\gamma}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}O_{T^{2}}^{\gamma}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{O}_{T^{2}}^{\lambda}}$

Figure 1: Sequence deformations of the Maxwell theory under
${O}_{T^{2}}^{\lambda}$ and $O_{T^{2}}^{\gamma}$.

In the context of $T\bar{T}$-deformation that shifts a ground state theory to
a class of higher excited-state theories, one can take a closer look at
$T\bar{T}$ flow in higher dimensional gauge theories. For example, it has been
shown [21] that there is a generic 6D chiral 2-form ED which in the weak-field
limit is related to the dimensional reduction to the ModMax theory. So, it
would be of interest to examine a kind of deformation that produces some
conformal and chiral theories in $D=6$ dimensions. From 4D point of view, the
structure of the corresponding 6D operators may also be related to
$O_{T^{2}}^{\gamma}$ and ${O}_{T^{2}}^{\lambda}$ operators studied in this
letter.

In addition, the supersymmetric extension of the Born-Infeld Lagrangian has
been already studied in Ref. [48]. The $T\bar{T}$-deformation of the $N=1$
supersymmetric BI theory with operator $O_{T^{2}}^{\lambda}$, was considered
for the first time in Ref. [43]. Also, we have studied the deformation of
$N=2$ supersymmetric BI theory by this operator in [35]. The
$T\bar{T}$-deformation of the $N=1$ supersymmetric ModMax theory with the
operator $O_{T^{2}}^{\lambda}$ , has been recently studied in Ref. [49].
Extension of the $N=1$ and $N=2$ supersymmetric ModMax theory with operator
$O_{T^{2}}^{\gamma}$ can be investigated as a future work. Since the ModMax
theory is unitary for non-negative values of $\gamma$ [45], it is believed
that the $T\bar{T}$-deformation can be used to quantize a theory from the
corresponding ground state. Therefore, providing a quantum representation for
the states corresponding to the different NED theories would be important.

In general, the electromagnetic duality in the pure NED theories is described
by the $SO(2)$ symmetry group [2, 3, 4, 5]. However, one can enhance this
symmetry in the NED theories to $SL(2,R)$ group by adding some axion-dilaton
fields to the Lagrangian density [5]. In the case of ModMax theory as a
conformal NED theory, we propose that

$\displaystyle\hat{\mathcal{L}}_{MM}=\cosh\gamma\,e^{-\phi_{0}}\mathcal{S}+\sinh\gamma\,\sqrt{e^{-2\phi_{0}}\,(\mathcal{S}^{2}+\mathcal{P}^{2})}-C_{0}\mathcal{P},$
(5.1)

where $C_{0}$ and $\phi_{0}$ are the axion and dilaton fields, respectively.
We postpone the details of this $SL(2,R)$-invariant ModMax theory as future
work in progress [50].

## Acknowledgment

We are very grateful to D. Sorokin for his kindly interest in this work and
fruitful discussion. HBA is specially appreciate M.M. Sheikh-jabbari for his
useful comments at the early stages of the work. The authors would also like
to thank M. R. Garousi, A. Ghodsi and H. R. Afshar for valuable discussions.

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