Einstein and Jordan-Frame Covariant Hamiltonians for $F(R)$ Gravity and Their
Canonical Relationships

J. Klusoň† and B. Matouš†‡ 111Email addresses: J. Klusoň<EMAIL_ADDRESS>B. Matouš<EMAIL_ADDRESS>
† Department of Theoretical Physics and Astrophysics, Faculty of Science,
Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic
‡ North-Bohemian Observatory and Planetarium in Teplice,
Koperníkova 3062, 415 01, Teplice, Czech Republic

###### Abstract

This paper is devoted to the analysis of the covariant canonical formalism of
$F(R)$ gravity in Einstein frame. We also find canonical transformation
between covariant canonical formulation of $F(R)$ gravity in Jordan frame and
Einstein frames and we also determine corresponding generating function.

##  1 Introduction and Summary

Fundamental objects in relativistic theory are fields whose dynamics is
derived from the action principle. The action is manifestly covariant object
that is defined as the space-time integral of Lagrangian density. This
manifest covariance is lost in the Hamiltonian formalism since its crucial
point is an existence of one preferred coordinate which is the time
coordinate. There is an alternative Hamiltonian formalism, known as covariant
Hamiltonian theory or the Weyl-De Donder theory [2, 3] that maintains manifest
covariance. In the covariant Hamiltonian theory, the momenta are defined as
derivatives with respect to all partial derivatives of coordinates so that
there is no preferred direction and hence manifest covariance is preserved.
This is very attractive idea that could be especially useful in manifest
covariant theories as for example theory of gravity. In fact, the first
covariant Hamiltonian theory of General Relativity was published by Hořava
long time ago [6]. Recently, this work was further discussed examining its
thermodynamic consequences in [5].

The obvious next step is to apply this approach to some generalized theories
of gravity. The simplest one is the $F(R)$ gravity where the scalar curvature
in the Lagrangian is replaced by a general function $F(R)$ of scalar curvature
222For review of this theory see [4, 7, 8].. The covariant Hamiltonian for
$F(R)$ gravity in Jordan frame was found in [9].

It is well known that $F(R)$ gravity can also be formulated in Einstein frame,
for detailed discussion, see for example [4]. The transformations between
these two frames is based on the Weyl transformation of metric and
corresponding Riemann and Ricci tensor. In more details, with appropriate
chosen Weyl factor we can arrive to Einstein frame formulation of $F(R)$
gravity. $F(R)$ Lagrangian in Einstein frame is very similar to General
Relativity Lagrangian minimally coupled to a scalar field. In 3+1 formalism,
Hamiltonians for both frames were formulated and there was found that they are
related by canonical transformation [12]. This leads to a question whether
this is also true in the covariant Hamiltonian formalism. This question is
answered with the presented paper. In more details, we firstly derive the
covariant Hamiltonian for $F(R)$ gravity in Einstein frame. We find that
covariant Hamiltonian for $F(R)$ gravity in Einstein frame has the same form
as was found in [6] together with new additional scalar field contribution. On
the other hand the covariant Hamiltonian formulation of $F(R)$ gravity in
Jordan frame was performed in [9] and our goal is to show that these two
formulations are related by canonical transformations. To do this we should
firstly examine how canonical transformations are defined in covariant
Hamiltonian formalism, following [10, 11]. Then we study relationship between
covariant Hamiltonians for $F(R)$ gravity in Jordan and Einstein frames and we
find generating function resulting in proof of canonical transformation
between those two Hamiltonians. This is really new and non-trivial result that
shows that these two Hamiltonians are related by canonical transformations in
the similar way how two Hamiltonians are related in $3+1$ canonical formalism
[12]. On the other hand there is crucial difference between these two
canonical transformations which is in the preservation of the Poisson
brackets. In fact, it is not completely clear how to define Poisson brackets
in covariant canonical formalism due to the fact that the conjugate momenta
have additional vector index. Then it is natural to define Poisson bracket as
in [10] where the Poisson bracket is defined in the same way as canonical
Poisson bracket so that it is now labeled by vector index. It was shown in
[10] that such Poisson brackets are not generally preserved under canonical
transformations and we show that exactly this situation occurs in case of
canonical transitions between Einstein and Jordan frame covariant Hamiltonians
for $F(R)$-gravity. On the other hand we show that the form of Lagrangian
brackets is preserved under canonical transformation. We mean that this is
very interesting result that demonstrates nice application of the covariant
canonical formalism for the study of $F(R)$-gravity.

The organization of this paper is as follows. In the next section (2) we
formulate covariant Hamiltonian for $F(R)$ gravity in Einstein frame. The
section (3) is devoted to the canonical transformation between these two
frames. At first the transformation’s existence is proven using fundamental
Lagrange brackets, then the explicit form of the generating function of this
transformation is found, and at the end the Poisson brackets are also noted,
they role is however a minor one since in the covariant Hamiltonian theory
they do not serve as canonical invariant. The fourth section (4) deals with
the surface term of the Lagrangian 333 This paper uses the East Coast
convention with metric signature ($-,+,+,+$) and Latin indices running over
$0...3$ interval while the Greek ones over $1...3$. The fundamental constants
$c,G,\hbar,k_{B}$ are treated as equal to one. .

##  2 $F(R)$-gravity in Einstein Frame and Its Covariant Hamiltonian

We begin this section with the introduction of the Lagrangian for $F(R)$
gravity. $F(R)$ gravity is the simplest generalization of the Einstein-Hilbert
action when we replace the linear dependence of the Lagrangian density on the
scalar curvature $R$ by more general function $F(R)$ 444For review and
extended list of references, see for example [4].. Explicitly, the Lagrangian
density of $F(R)$ theory of gravity has the form

$\mathcal{L}=\frac{\sqrt{-g}}{16\pi}F(R)\>.$ (1)

The presence of the function $F(R)$ implies that it is not straightforward
procedure to find corresponding Hamiltonian. In order to overcome this issue
it is convenient to introduce two scalar fields $A$ and $B$ and replace the
Lagrangian density (1) by the following one

$\mathcal{L}=\frac{\sqrt{-g}}{16\pi}\left[F(B)+A\left(R-B\right)\right]\ .$
(2)

In fact, equations of motion for $A$ and $B$ that follow from (2) have the
form

$R-B=0\ ,\quad F^{\prime}(B)-A=0\ ,\quad F^{\prime}(B)\equiv\frac{dF}{dB}\ .$
(3)

Then inserting the first equation in (3) into (2), we easily see the
Lagrangian density (2) reduces into (1) that shows equivalence of these two
actions. For our purposes, it is useful to use the second equation in (3) to
solve $A$ as function of $B$ and hence the Lagrangian density in Jordan frame
has the form

$\mathcal{L}^{J}=\frac{\sqrt{-g}}{16\pi}\left[F(B)+F^{\prime}(B)\left(R-B\right)\right]\>.$
(4)

As it is well known, we can formulate $F(R)$-gravity in the Einstein frame
too, see for example [4]. Note that the Einstein frame is defined by
requirement that the Lagrangian density is linear in the scalar curvature $R$.
In the rest of this paper, the coordinates with a tilde are coordinates in
Einstein frame while those without punctuation belong to Jordan frame. In
order to find the transformation from Jordan frame to Einstein one we should
perform Weyl transformation of the metric that is defined as

$\widetilde{g}_{ij}=F^{\prime}g_{ij}\ .$ (5)

As the next step we introduce connection with tilde

$\widetilde{\Gamma}^{i}_{jk}=\frac{1}{2}\widetilde{g}^{il}(\partial_{j}\widetilde{g}_{lk}+\partial_{k}\widetilde{g}_{lj}-\partial_{l}\widetilde{g}_{jk})\
.$ (6)

Then using (5) we find how it is related to $\Gamma^{i}_{ij}$ and to $F(B)$

$\widetilde{\Gamma}^{i}_{jk}=\Gamma^{i}_{jk}+\frac{1}{2}\delta^{i}_{j}\partial_{k}(\ln
F^{\prime})+\frac{1}{2}\delta^{i}_{k}\partial_{j}(\ln
F^{\prime})-\frac{1}{2}g_{jk}g^{km}\partial_{m}(\ln F^{\prime})\>.$ (7)

With the help of this relation between connections, we can easily find
relation between corresponding scalar curvatures and we get

$\widetilde{R}=\widetilde{g}^{ab}\widetilde{R}_{ab}=\frac{1}{F^{\prime}}R-\frac{3}{F^{\prime}}\Box\ln
F^{\prime}-\frac{3}{2F^{\prime}}\partial_{i}\ln
F^{\prime}g^{ij}\partial_{j}\ln F^{\prime}\>.$ (8)

For writing the Lagrangian in Einstein frame, we need to express $R$ in terms
of $\widetilde{R}$, $\widetilde{g}_{ij}$ and corresponding derivatives. To do
this, we use the following rules

$\Box\ln F^{\prime}=F^{\prime}\widetilde{\Box}\ln
F^{\prime}-F^{\prime}\left(\widetilde{\partial}\ln
F^{\prime}\right)^{2}\>,\quad\left(\partial\ln
F^{\prime}\right)^{2}=F^{\prime}\left(\widetilde{\partial}\ln
F^{\prime}\right)^{2}\ .$ (9)

Then we can easily express $R$ as

$R=F^{\prime}\left[\widetilde{R}+3\widetilde{\Box}\ln
F^{\prime}-\frac{3}{2}\left(\widetilde{\partial}\ln
F^{\prime}\right)^{2}\right]\>.$ (10)

Using this expression together with the fact that
$\sqrt{-\widetilde{g}}=F^{\prime 2}\sqrt{-g}$ we obtain that the Lagrangian
density for $F(R)$ gravity takes the form

$\mathcal{L}=\frac{\sqrt{-\widetilde{g}}}{16\pi}\left[\frac{F-F^{\prime}B}{F^{\prime
2}}+\widetilde{R}+3\widetilde{\Box}\ln
F^{\prime}-\frac{3}{2}\left(\widetilde{\partial}\ln
F^{\prime}\right)^{2}\right]\>,$ (11)

or equivalently

$\mathcal{L}=\frac{\sqrt{-\widetilde{g}}}{16\pi}\left[\frac{F-F^{\prime}B}{F^{\prime
2}}+\widetilde{R}-\frac{3}{2}\left(\widetilde{\partial}\ln
F^{\prime}\right)^{2}\right]+\frac{3}{16\pi}\partial_{i}\left(\sqrt{-\widetilde{g}}\widetilde{\partial}^{i}\ln
F^{\prime}\right)\>.$ (12)

In order to get canonical form of the Lagrangian density, we introduce a new
scalar variable $\widetilde{\phi}=\sqrt{\frac{3}{2}}\ln F^{\prime}$ and a
potential $V(B)=\frac{F^{\prime}B-F}{F^{\prime 2}}$ so that the Lagrangian
density takes Einstein-Hilbert form

$\mathcal{L}^{E}=\frac{\sqrt{-\widetilde{g}}}{16\pi}\left[\widetilde{R}-\widetilde{g}^{ij}\partial_{i}\widetilde{\phi}\partial_{j}\widetilde{\phi}-V(\widetilde{\phi})\right]+\frac{\sqrt{6}}{16\pi}\partial_{i}\left(\sqrt{-\widetilde{g}}\widetilde{g}^{ij}\partial_{j}\widetilde{\phi}\right)\>.$
(13)

This is the final form of Lagrangian density for $F(R)$ gravity in Einstein
frame. To proceed to the covariant canonical formalism, it is necessary to
separate Lagrangian into two parts [6]: the bulk term that contains only the
first derivatives and the surface term which can be expressed as a total
derivative. To construct Hamiltonian, solely the bulk term is needed, the
surface Hamiltonian will be discussed in section (4). Explicitly, we get

$\displaystyle\mathcal{L}^{E}=\mathcal{L}^{E}_{bulk}+\mathcal{L}^{E}_{sur}\ ,$
$\displaystyle\mathcal{L}_{bulk}=\frac{\sqrt{-\widetilde{g}}}{16\pi}\left[\widetilde{g}^{ab}\widetilde{\Gamma}^{c}_{bd}\widetilde{\Gamma}^{d}_{ac}-\widetilde{g}^{ac}\widetilde{\Gamma}^{b}_{bd}\widetilde{\Gamma}^{d}_{ac}-\widetilde{g}^{ij}\partial_{i}\widetilde{\phi}\partial_{j}\widetilde{\phi}-V(\widetilde{\phi})\right]\
,$
$\displaystyle\mathcal{L}_{sur}=\partial_{i}\left[\frac{\sqrt{-\widetilde{g}}}{16\pi}\left(\widetilde{g}^{ab}\widetilde{\Gamma}^{i}_{ab}-\widetilde{g}^{ai}\widetilde{\Gamma}^{b}_{ab}+\sqrt{6}\widetilde{g}^{ij}\partial_{j}\widetilde{\phi}\right)\right]\
.$ (14)

We see that the bulk part reminds the bulk Lagrangian of General Relativity
[5] together with added $\tilde{\phi}$-related terms.

First step to transform the Lagrangian into covariant Hamiltonian is to find
corresponding momenta. From (2) we find that the momentum conjugated to
$\widetilde{g}$ is equal to

$\widetilde{M}^{prs}=\frac{\partial\mathcal{L}_{bulk}}{\partial\partial_{p}\widetilde{g}_{rs}}=\frac{\sqrt{-\widetilde{g}}}{16\pi}\left[\widetilde{\Gamma}^{p}_{ab}\left(\widetilde{g}^{ar}\widetilde{g}^{bs}-\frac{1}{2}\widetilde{g}^{ab}\widetilde{g}^{rs}\right)+\frac{1}{2}\widetilde{\Gamma}^{a}_{ak}\left(\widetilde{g}^{rs}\widetilde{g}^{kp}-\widetilde{g}^{ps}\widetilde{g}^{kr}-\widetilde{g}^{pr}\widetilde{g}^{ks}\right)\right]\>.$
(15)

However it was shown in [6, 5, 9] that the covariant canonical formalism of
general relativity is better formulated when we introduce coordinate
$\widetilde{f^{ab}}$ that is related to $\widetilde{g}$ by following relation

$\widetilde{f}^{ab}=\sqrt{-\widetilde{g}}\widetilde{g}^{ab}\ ,$ (16)

where, following [9], we define $\tilde{f}_{ab}$ as inverse to
$\tilde{f}^{ab}$ 555This definition is different from the one used in [5],
where the new coordinate was defined so $f^{ab}f_{cb}=-f\delta^{a}_{c}$.

$\tilde{f}_{ab}\tilde{f}^{bc}=\delta_{a}^{c}\ .$ (17)

Note that from (16) we also get useful result

$\tilde{f}\equiv\det\tilde{f}^{ab}=\widetilde{g}\ .$ (18)

The momentum $\widetilde{N}^{c}_{ab}$ conjugated to $\tilde{f}^{ab}$ can be
obtained directly when $\tilde{f}^{ab}$ is substituted into Lagrangian or by
using formulae [9]

$\widetilde{N}^{c}_{ab}=-\widetilde{M}^{cmn}\frac{1}{\sqrt{-\widetilde{f}}}\widetilde{B}_{mn\>ab}\>,\qquad\widetilde{B}_{mn\>ab}=\frac{1}{2}\left(\widetilde{g}_{ma}\widetilde{g}_{bn}+\widetilde{g}_{mb}\widetilde{g}_{an}-\widetilde{g}_{mn}\widetilde{g}_{ab}\right)\>.$
(19)

Using explicit form for $\widetilde{M}^{abc}$ given in (15) we obtain the well
known result

$\widetilde{N}^{c}_{ab}=\frac{1}{16\pi}\left[-\widetilde{\Gamma}^{c}_{ab}+\frac{1}{2}\left(\widetilde{\Gamma}^{k}_{ak}\delta^{c}_{b}+\widetilde{\Gamma}^{k}_{kb}\delta^{c}_{a}\right)\right]\
.$ (20)

From (2) we also find momentum conjugate to $\widetilde{\phi}$

$\widetilde{p}^{a}=\frac{\partial\mathcal{L}_{bulk}}{\partial\partial_{a}\widetilde{\phi}}=-\frac{1}{8\pi}\widetilde{f}^{ab}\partial_{b}\widetilde{\phi}\>.$
(21)

The Hamiltonian is then computed using Legendre transformation as

$\displaystyle\mathcal{H}^{E}=\partial_{c}\widetilde{f}^{ab}\widetilde{N}^{c}_{ab}+\partial_{a}\widetilde{\phi}\widetilde{p}^{a}-\mathcal{L}_{bulk}=$
$\displaystyle=\frac{1}{16\pi}\left[\widetilde{f}^{ab}\left(\widetilde{\Gamma}^{c}_{db}\widetilde{\Gamma}^{d}_{ca}-\widetilde{\Gamma}^{c}_{ab}\widetilde{\Gamma}^{d}_{dc}\right)+\sqrt{-\widetilde{f}}V-\widetilde{f}^{ab}\partial_{a}\widetilde{\phi}\partial_{b}\widetilde{\phi}\right]\
.$

Finally we should express this Hamiltonian in terms of canonical variables
which can be done using the relations

$\widetilde{\Gamma}^{i}_{jk}=16\pi\left[-\widetilde{N}^{i}_{jk}+\frac{1}{3}\left(\widetilde{N}^{u}_{ju}\delta^{i}_{k}+\widetilde{N}^{u}_{ku}\delta^{i}_{j}\right)\right]\>,\quad\partial_{a}\widetilde{\phi}=-8\pi\widetilde{f}_{ab}\widetilde{p}^{b}\>.$
(23)

Then the final form of covariant Hamiltonian for Einstein frame of $F(R)$
gravity, is expressed as

$\mathcal{H}^{E}=16\pi\widetilde{f}^{ab}\left(\widetilde{N}^{c}_{bd}\widetilde{N}^{d}_{ac}-\frac{1}{3}\widetilde{N}^{c}_{ac}\widetilde{N}^{d}_{bd}\right)+\frac{\sqrt{-\widetilde{f}}}{16\pi}V-4\pi\widetilde{f}_{ab}\widetilde{p}^{a}\widetilde{p}^{b}\>.$
(24)

##  3 Relationship with Jordan Frame Hamiltonian

Now we proceed to the main part of this paper which is the relationship
between covariant Hamiltonians in Einstein and Jordan frames respectively. Our
work is motivated by an interesting paper [12] where the Hamiltonian for
$F(R)$ gravity in $3+1$ formalism was analysed and it was shown there that
they are related by canonical transformations. Then it is very interesting
question whether such canonical transformation exists in the case of the
covariant canonical formalism too.

###  3.1 Lagrange Brackets

If there is a canonical transformation between the two frames, the fundamental
Lagrange brackets shall be preserved [10]. Contrary to the conventional
Hamiltonian theory, it is not good to use Poisson brackets for this purpose,
because fundamental Poisson brackets are preserved only [10] when the
transformed momenta do not depend on original coordinates or on original
momenta. This condition is quite strong and, as it will be presented in this
section, does not hold for our system. The Poisson brackets will be noted
shortly in section 3.3. So the calculation of Lagrange brackets of Jordan
frame variables in Einstein frame can reveal us the existence of a canonical
transformation. In order to calculate the Lagrange brackets we need relations
between Jordan frame variables and Einstein frame ones. We are already able to
express the transformation rules for coordinates

$\widetilde{f}^{ab}=F^{\prime}f^{ab}\>,\qquad\widetilde{\phi}=\sqrt{\frac{3}{2}}\ln
F^{\prime}\>.$ (25)

In order to find transformation of momenta $\tilde{N}^{c}_{ab}$, we need to
take (20) and (7) and compare it with the the Jordan frame momenta [9]

$N^{c}_{ab}=\frac{F^{\prime}}{16\pi}\left[-\Gamma^{c}_{ab}+\frac{1}{2}\left(\Gamma^{k}_{ak}\delta^{c}_{b}+\Gamma^{k}_{bk}\delta^{c}_{a}\right)+\frac{1}{2}\left(\delta^{c}_{b}\partial_{a}\ln
F^{\prime}+\delta^{c}_{a}\partial_{b}\ln
F^{\prime}+f^{gc}f_{ab}\partial_{g}\ln F^{\prime}\right)\right]\>,$ (26)

resulting in transformation relation

$\widetilde{N}^{c}_{ab}=\frac{1}{F^{\prime}}N^{c}_{ab}\>,$ (27)

that also implies following important relation

$\tilde{f}^{ab}\tilde{N}^{c}_{ab}=f^{ab}N^{c}_{ab}\>.$ (28)

From the relation (27), we see that the transformed momentum depends on
original momentum as well as original coordinate, so the condition for
canonical invariance of Poisson brackets is not met. As the next step, we
proceed to the transformation of momentum $\tilde{p}^{a}$. First of all, we
use the second relation in (23) where we insert (25) so that we find relation
between $\partial_{a}B$ and $\tilde{p}^{a}$. Further, we use the relation
between $\partial_{a}B$ and $p^{a}$ that was derived in [9]

$\partial_{g}B=\frac{16\pi}{3F^{\prime\prime}}f_{gc}\left(f^{ik}N^{c}_{ik}-\frac{F^{\prime}}{F^{\prime\prime}}p^{c}\right)\
.$ (29)

If we combine these relations together we find desired relation

$\widetilde{p}^{a}=\sqrt{\frac{2}{3}}\left(\frac{F^{\prime}}{F^{\prime\prime}}p^{a}-f^{bc}N^{a}_{bc}\right)\>.$
(30)

Having found the transformation relations, the brackets follow simply as

$\displaystyle\left\\{f^{ab},\>f^{cd}\right\\}^{j}=\frac{\partial\widetilde{f}^{ik}}{\partial
f^{ab}}\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
f^{cd}}-\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
f^{ab}}\frac{\partial\widetilde{f}^{ik}}{\partial
f^{cd}}+\frac{\partial\widetilde{\phi}}{\partial
f^{ab}}\frac{\partial\widetilde{p^{j}}}{\partial
f^{cd}}-\frac{\partial\widetilde{p^{j}}}{\partial
f^{ab}}\frac{\partial\widetilde{\phi}}{\partial f^{cd}}=0\>,$
$\displaystyle\left\\{f^{ab},\>B\right\\}^{j}=\frac{\partial\widetilde{f}^{ik}}{\partial
f^{ab}}\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
B}-\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
f^{ab}}\frac{\partial\widetilde{f}^{ik}}{\partial
B}+\frac{\partial\widetilde{\phi}}{\partial
f^{ab}}\frac{\partial\widetilde{p^{j}}}{\partial
B}-\frac{\partial\widetilde{p^{j}}}{\partial
f^{ab}}\frac{\partial\widetilde{\phi}}{\partial B}=0\>,$
$\displaystyle\left\\{B,\>B\right\\}^{j}=0\text{ by definition}\>,$
$\displaystyle\left\\{f^{ab},\>N^{c}_{de}\right\\}^{j}=\frac{\partial\widetilde{f}^{ik}}{\partial
f^{ab}}\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
N^{c}_{de}}-\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
f^{ab}}\frac{\partial\widetilde{f}^{ik}}{\partial
N^{c}_{de}}+\frac{\partial\widetilde{\phi}}{\partial
f^{ab}}\frac{\partial\widetilde{p^{j}}}{\partial
N^{c}_{de}}-\frac{\partial\widetilde{p^{j}}}{\partial
f^{ab}}\frac{\partial\widetilde{\phi}}{\partial
N^{c}_{de}}=\delta^{j}_{c}\delta^{de}_{ab}\>,$
$\displaystyle\left\\{f^{ab},\>p^{c}\right\\}^{j}=\frac{\partial\widetilde{f}^{ik}}{\partial
f^{ab}}\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
p^{c}}-\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
f^{ab}}\frac{\partial\widetilde{f}^{ik}}{\partial
p^{c}}+\frac{\partial\widetilde{\phi}}{\partial
f^{ab}}\frac{\partial\widetilde{p^{j}}}{\partial
p^{c}}-\frac{\partial\widetilde{p^{j}}}{\partial
f^{ab}}\frac{\partial\widetilde{\phi}}{\partial p^{c}}=0\>,$
$\displaystyle\left\\{B,\>N^{c}_{de}\right\\}^{j}=\frac{\partial\widetilde{f}^{ik}}{\partial
B}\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
N^{c}_{de}}-\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
B}\frac{\partial\widetilde{f}^{ik}}{\partial
N^{c}_{de}}+\frac{\partial\widetilde{\phi}}{\partial
B}\frac{\partial\widetilde{p^{j}}}{\partial
N^{c}_{de}}-\frac{\partial\widetilde{p^{j}}}{\partial
B}\frac{\partial\widetilde{\phi}}{\partial N^{c}_{de}}=0\>,$
$\displaystyle\left\\{B,\>p^{c}\right\\}^{j}=\frac{\partial\widetilde{f}^{ik}}{\partial
B}\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
p^{c}}-\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
B}\frac{\partial\widetilde{f}^{ik}}{\partial
p^{c}}+\frac{\partial\widetilde{\phi}}{\partial
B}\frac{\partial\widetilde{p^{j}}}{\partial
p^{c}}-\frac{\partial\widetilde{p^{j}}}{\partial
B}\frac{\partial\widetilde{\phi}}{\partial p^{c}}=\delta^{j}_{c}\>,$
$\displaystyle\left\\{N^{c}_{ab},\>N^{d}_{ef}\right\\}^{j}=\frac{\partial\widetilde{f}^{ik}}{\partial
N^{c}_{ab}}\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
N^{d}_{ef}}-\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
N^{c}_{ab}}\frac{\partial\widetilde{f}^{ik}}{\partial
N^{d}_{ef}}+\frac{\partial\widetilde{\phi}}{\partial
N^{c}_{ab}}\frac{\partial\widetilde{p^{j}}}{\partial
N^{d}_{ef}}-\frac{\partial\widetilde{p^{j}}}{\partial
N^{c}_{ab}}\frac{\partial\widetilde{\phi}}{\partial N^{d}_{ef}}=0\>,$
$\displaystyle\left\\{N^{c}_{ab},\>p^{d}\right\\}^{j}=\frac{\partial\widetilde{f}^{ik}}{\partial
N^{c}_{ab}}\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
p^{d}}-\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
N^{c}_{ab}}\frac{\partial\widetilde{f}^{ik}}{\partial
p^{d}}+\frac{\partial\widetilde{\phi}}{\partial
N^{c}_{ab}}\frac{\partial\widetilde{p^{j}}}{\partial
p^{d}}-\frac{\partial\widetilde{p^{j}}}{\partial
N^{c}_{ab}}\frac{\partial\widetilde{\phi}}{\partial p^{d}}=0\>,$
$\displaystyle\left\\{p^{a},\>p^{b}\right\\}^{j}=\frac{\partial\widetilde{f}^{ik}}{\partial
p^{a}}\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
p^{b}}-\frac{\partial\widetilde{N}^{j}_{ik}}{\partial
p^{a}}\frac{\partial\widetilde{f}^{ik}}{\partial
p^{b}}+\frac{\partial\widetilde{\phi}}{\partial
p^{a}}\frac{\partial\widetilde{p^{j}}}{\partial
p^{b}}-\frac{\partial\widetilde{p^{j}}}{\partial
p^{a}}\frac{\partial\widetilde{\phi}}{\partial p^{b}}=0\>,$

where we have used abbreviation
$\delta^{ab}_{cd}=\frac{1}{2}\left(\delta^{a}_{c}\delta^{b}_{d}+\delta^{b}_{c}\delta^{a}_{d}\right)$.
From the equations above, it is easily visible that the fundamental Lagrange
brackets are preserved, so there is a canonical transformation between Jordan
and Einstein frames.

###  3.2 Generating Function of Canonical Transformation

Having proven existence of canonical transformation, its generating function
is looked for. But first, let us review basic facts about canonical
transformations in covariant formalism, following [10, 11].

Let us consider covariant formulation of $F(R)$ gravity in Jordan frame with
the canonical variables $f^{ab},N_{ab}^{c},B,p^{c}_{B}$. On the other hand in
case of covariant formulation of Einstein-frame $F(R)$-gravity the canonical
variables are $\tilde{f}^{ab},\tilde{N}_{ab}^{c},\tilde{\phi},\tilde{p}^{a}$.
We demand that they give the same description of the physical systems so that
we have a requirement

$\displaystyle\delta\int
d^{4}x(N^{c}_{ab}\partial_{c}f^{ab}+p^{c}\partial_{c}B-\mathcal{H}^{J})=\delta\int
d^{4}x(\tilde{N}^{c}_{ab}\partial_{c}\tilde{f}^{ab}+\tilde{p}^{c}\partial_{c}\tilde{\phi}-\mathcal{H}^{E})\
.$ (32)

This result implies that the integrals can differ only by divergence of a
vector function whose variation vanishes on the boundary $\partial R$ of the
integration region $R$

$\delta\int_{R}d^{4}x\partial_{a}G_{1}^{a}(f,B,\tilde{f},\tilde{\phi})=\delta\oint_{\partial
R}G_{1}^{a}(f,B,\tilde{f},\tilde{\phi})dS_{a}=0\>.$ (33)

Using $G_{1}^{a}$, we can write

$N^{c}_{ab}\partial_{c}f^{ab}+p^{c}\partial_{c}B-\mathcal{H}^{J}=\tilde{N}^{c}_{ab}\partial_{c}\tilde{f}^{ab}+\tilde{p}^{c}\partial_{c}\tilde{\phi}-\mathcal{H}^{E}+\partial_{a}G_{1}^{a}(f,B,\tilde{f},\tilde{\phi},x)\>.$
(34)

The divergence of $G_{1}^{a}$ can be expressed using coordinates as

$\partial_{a}G^{a}_{1}=\frac{\partial G_{1}^{a}}{\partial
f^{bc}}\partial_{a}f^{bc}+\frac{\partial
G_{1}^{a}}{\partial\tilde{f}^{bc}}\partial_{a}\tilde{f}^{bc}+\frac{\partial
G_{1}^{a}}{\partial B}\partial_{a}B+\frac{\partial
G_{1}^{a}}{\partial\tilde{\phi}}\partial_{a}\tilde{\phi}+\frac{\partial
G_{1}^{a}}{\partial x^{a}}|_{expl}\>.$ (35)

Putting this into (34) and comparing terms proportional to
$\partial_{c}f^{ab}$ and $\partial_{c}\tilde{f}^{ab}$ and $\partial_{a}B$ and
$\partial_{a}\tilde{\phi}$ we obtain

$\displaystyle N^{c}_{ab}$ $\displaystyle=\frac{\partial G_{1}^{c}}{\partial
f^{ab}}\ ,\quad\quad p^{c}=\frac{\partial G_{1}^{c}}{\partial B}\ ,$
$\displaystyle\tilde{N}_{ab}^{c}$ $\displaystyle=-\frac{\partial
G_{1}^{c}}{\partial\tilde{f}^{ab}}\ ,\quad\>\tilde{p}^{c}=-\frac{\partial
G_{1}^{a}}{\partial\tilde{\phi}}\ ,$ $\displaystyle\mathcal{H}^{J}$
$\displaystyle=\mathcal{H}^{E}-\frac{\partial G_{1}^{a}}{\partial
x^{a}}|_{expl}\>.$ (36)

On the other hand we can consider different type of generating function

$G_{1}^{a}=G_{2}^{a}(f,B,\tilde{N},\tilde{p})-\tilde{f}^{ik}\tilde{N}_{ik}^{a}-\tilde{\phi}\tilde{p}^{a}\>,$
(37)

so that its total divergence is equal to

$\displaystyle\partial_{c}(G^{c}_{2}-\tilde{f}^{ab}\tilde{N}_{ab}^{c}-\tilde{\phi}\tilde{p}^{c})=$
$\displaystyle\frac{\partial G^{c}_{2}}{\partial
f^{ab}}\partial_{c}f^{ab}+\frac{\partial G^{c}_{2}}{\partial
B}\partial_{c}B+\frac{\partial
G^{c}_{2}}{\partial\tilde{N}_{ab}^{d}}\partial_{c}\tilde{N}_{ab}^{d}+\frac{\partial
G^{c}_{2}}{\partial\tilde{p}^{a}}\partial_{c}\tilde{p}^{a}+\frac{\partial
G^{c}_{2}}{\partial x^{c}}|_{exp}-$
$\displaystyle-\tilde{N}_{ab}^{c}\partial_{c}\tilde{f}^{ab}-\tilde{f}^{ab}\partial_{c}\tilde{N}_{ab}^{c}-\tilde{p}^{c}\partial_{c}\tilde{\phi}-\tilde{\phi}\partial_{c}\tilde{p}^{c}\>.$

Substituting the second type of generating function this into (34) and then
comparing the related terms yields a set of different equations

$\displaystyle\mathcal{H}_{J}=\mathcal{H}_{E}-\frac{\partial
G_{2}^{c}}{\partial x^{c}}|_{expl}\ ,$ $\displaystyle\frac{\partial
G_{2}^{c}}{\partial\tilde{p}^{a}}=\delta^{c}_{a}\tilde{\phi}\
,\quad\frac{\partial
G_{2}^{c}}{\partial\tilde{N}^{d}_{ab}}=\tilde{f}^{ab}\delta_{d}^{c}\ ,\quad
N^{c}_{ab}=\frac{\partial G_{2}^{c}}{f^{ab}}\ ,\quad p^{c}=\frac{\partial
G_{2}^{c}}{\partial B}\ .$

After this brief discussion of the canonical formalism, we proceed to the
analysis of the question how Einstein and Jordan frame formulations of $F(R)$
gravity are related by canonical transformations. Recall that the Covariant
Hamiltonian in Jordan frame has the form [9]

$\displaystyle\mathcal{H}^{J}=$
$\displaystyle\frac{16\pi}{F^{\prime}}f^{ab}\left(N^{c}_{bd}N^{d}_{ac}-\frac{1}{3}N^{c}_{ac}N^{d}_{bd}\right)$
$\displaystyle-\frac{8\pi}{3F^{\prime}}f_{ab}\left(\frac{F^{\prime}}{F^{\prime\prime}}p^{a}-N^{a}_{cd}f^{cd}\right)\left(\frac{F^{\prime}}{F^{\prime\prime}}p^{b}-N^{b}_{ef}f^{ef}\right)$
$\displaystyle-\frac{\sqrt{-f}}{16\pi}\left(F-F^{\prime}B\right)\ .$ (40)

Using the transformations (25), (27), (30), we can express the Einstein frame
Hamiltonian in Jordan frame coordinates

$\displaystyle\mathcal{H}^{E*}=$
$\displaystyle\frac{16\pi}{F^{\prime}}f^{ab}\left(N^{c}_{bd}N^{d}_{ac}-\frac{1}{3}N^{c}_{ac}N^{d}_{bd}\right)$
$\displaystyle-\frac{8\pi}{3F^{\prime}}f_{ab}\left(\frac{F^{\prime}}{F^{\prime\prime}}p^{a}-N^{a}_{cd}f^{cd}\right)\left(\frac{F^{\prime}}{F^{\prime\prime}}p^{b}-N^{b}_{ef}f^{ef}\right)$
$\displaystyle-\frac{\sqrt{-f}}{16\pi}\left(F-F^{\prime}B\right)\>,$ (41)

which agrees with (3.2). The fact that these two Hamiltonians are equal means
that the generating function of canonical transformation is independent on
coordinates. In fact, transformations (25), (27) and (30) suggest that the
generating function has the form of

$G_{2}^{c}=F^{\prime}\tilde{N}^{c}_{ab}f^{ab}+\sqrt{\frac{3}{2}}\ln
F^{\prime}\tilde{p}^{c}\>.$ (42)

In order to verify that the suggested form of $G_{2}^{c}$ is really the
generating function of canonical transformations let us calculate the
following derivative

$\displaystyle\frac{\partial G_{2}^{c}}{\partial
B}=F^{\prime\prime}\tilde{N}_{ab}^{c}f^{ab}+\sqrt{\frac{3}{2}}\frac{F^{\prime\prime}}{F^{\prime}}\tilde{p}^{c}=\frac{F^{\prime\prime}}{F^{\prime}}N_{ab}^{c}f^{ab}+\frac{F^{\prime\prime}}{F^{\prime}}\left(\frac{F^{\prime}}{F^{\prime\prime}}p^{c}-N_{ab}^{c}f^{ab}\right)=p^{c}$
(43)

that agrees with the the fourth equation in (3.2). Further, the derivative
$G_{2}^{c}$ with respect to $f^{ab}$ gives

$N^{c}_{ab}=\frac{\partial G_{2}^{c}}{\partial
f^{ab}}=F^{\prime}\widetilde{N}^{c}_{ab}$ (44)

that gives the relation (27). Finally, the derivative of $G_{2}^{c}$ with
respect to $\tilde{p}^{a}$ and $\widetilde{N}^{d}_{ab}$ lead to

$\displaystyle\frac{\partial
G_{2}^{c}}{\partial\tilde{p}^{a}}=\sqrt{\frac{3}{2}}\ln
F^{\prime}\delta^{c}_{a}=\delta^{c}_{a}\widetilde{\phi}\ ,$
$\displaystyle\frac{\partial
G_{2}^{c}}{\partial\widetilde{N}^{d}_{ab}}=F^{\prime}\delta^{c}_{d}f^{ab}=\widetilde{f}^{ab}\delta^{c}_{d}$

that lead to relations in (25).

In summary, we have shown that (42) is generating function of canonical
transformation between Jordan and Einstein frame. This shows that these two
frames are related by canonical transformations even in the case of covariant
canonical formulations of these two theories which is new and non-trivial
result.

###  3.3 Remark About Poisson Brackets

In Covariant Hamiltonian theory, the fundamental Poisson brackets are not
always preserved under canonical transformation [10]. Their purpose of
canonical invariant is fulfilled with Lagrange brackets. This section
summarizes Poisson brackets just for the reference. We will calculate Poisson
brackets of Einstein frame variables in Jordan frame. Since no Einstein frame
coordinate depends on any of Jordan frame momenta (25), the Poisson brackets
of coordinates are zero. For the mixed brackets, one obtains

$\displaystyle\left[\widetilde{f}^{ab}\>,\>\widetilde{N}^{c}_{de}\right]_{j}$
$\displaystyle=\frac{\partial\widetilde{f}^{ab}}{\partial
f^{ik}}\frac{\partial\widetilde{N}^{c}_{de}}{\partial
N^{j}_{ik}}+\frac{\partial\widetilde{f}^{ab}}{\partial
B}\frac{\partial\widetilde{N}^{c}_{de}}{\partial
p^{j}}-\frac{\partial\widetilde{N}^{c}_{de}}{\partial
f^{ik}}\frac{\partial\widetilde{f}^{ab}}{\partial
N^{j}_{ik}}-\frac{\partial\widetilde{N}^{c}_{de}}{\partial
B}\frac{\partial\widetilde{f}^{ab}}{\partial
p^{j}}=\delta^{c}_{j}\delta^{ab}_{cd}\>,$
$\displaystyle\left[\widetilde{f}^{ab}\>,\>\widetilde{p}^{c}\right]_{j}$
$\displaystyle=\frac{\partial\widetilde{f}^{ab}}{\partial
f^{ik}}\frac{\partial\widetilde{p}^{c}}{\partial
N^{j}_{ik}}+\frac{\partial\widetilde{f}^{ab}}{\partial
B}\frac{\partial\widetilde{p}^{c}}{\partial
p^{j}}-\frac{\partial\widetilde{p}^{c}}{\partial
f^{ik}}\frac{\partial\widetilde{f}^{ab}}{\partial
N^{j}_{ik}}-\frac{\partial\widetilde{p}^{c}}{\partial
B}\frac{\partial\widetilde{f}^{ab}}{\partial p^{j}}=0\>,$
$\displaystyle\left[\widetilde{\phi}\>,\>\widetilde{N}^{c}_{de}\right]_{j}$
$\displaystyle=\frac{\partial\widetilde{\phi}}{\partial
f^{ik}}\frac{\partial\widetilde{N}^{c}_{de}}{\partial
N^{j}_{ik}}+\frac{\partial\widetilde{\phi}}{\partial
B}\frac{\partial\widetilde{N}^{c}_{de}}{\partial
p^{j}}-\frac{\partial\widetilde{N}^{c}_{de}}{\partial
f^{ik}}\frac{\partial\widetilde{\phi}}{\partial
N^{j}_{ik}}-\frac{\partial\widetilde{N}^{c}_{de}}{\partial
B}\frac{\partial\widetilde{\phi}}{\partial p^{j}}=0\>,$
$\displaystyle\left[\widetilde{\phi}\>,\>\widetilde{p}^{c}\right]_{j}$
$\displaystyle=\frac{\partial\widetilde{\phi}}{\partial
f^{ik}}\frac{\partial\widetilde{p}^{c}}{\partial
N^{j}_{ik}}+\frac{\partial\widetilde{\phi}}{\partial
B}\frac{\partial\widetilde{p}^{c}}{\partial
p^{j}}-\frac{\partial\widetilde{p}^{c}}{\partial
f^{ik}}\frac{\partial\widetilde{\phi}}{\partial
N^{j}_{ik}}-\frac{\partial\widetilde{p}^{c}}{\partial
B}\frac{\partial\widetilde{\phi}}{\partial p^{j}}=\delta^{c}_{j}\>.$

As we can see all mixed brackets yield the expected results. The brackets of
momenta are those who break canonical invariance of Poisson brackets

$\displaystyle\left[\widetilde{N}^{f}_{ab}\>,\>\widetilde{N}^{c}_{de}\right]_{j}=\frac{\partial\widetilde{N}^{f}_{ab}}{\partial
f^{ik}}\frac{\partial\widetilde{N}^{c}_{de}}{\partial
N^{j}_{ik}}+\frac{\partial\widetilde{N}^{f}_{ab}}{\partial
B}\frac{\partial\widetilde{N}^{c}_{de}}{\partial
p^{j}}-\frac{\partial\widetilde{N}^{c}_{de}}{\partial
f^{ik}}\frac{\partial\widetilde{N}^{f}_{ab}}{\partial
N^{j}_{ik}}-\frac{\partial\widetilde{N}^{c}_{de}}{\partial
B}\frac{\partial\widetilde{N}^{f}_{ab}}{\partial p^{j}}=0\>,$
$\displaystyle\left[\widetilde{N}^{d}_{ab}\>,\>\widetilde{p}^{c}\right]_{j}=\frac{\partial\widetilde{N}^{d}_{ab}}{\partial
f^{ik}}\frac{\partial\widetilde{p}^{c}}{\partial
N^{j}_{ik}}+\frac{\partial\widetilde{N}^{d}_{ab}}{\partial
B}\frac{\partial\widetilde{p}^{c}}{\partial
p^{j}}-\frac{\partial\widetilde{p}^{c}}{\partial
f^{ik}}\frac{\partial\widetilde{N}^{d}_{ab}}{\partial
N^{j}_{ik}}-\frac{\partial\widetilde{p}^{c}}{\partial
B}\frac{\partial\widetilde{N}^{d}_{ab}}{\partial p^{j}}=$
$\displaystyle\qquad=\sqrt{\frac{2}{3}}\frac{1}{F^{\prime}}\left(\delta^{d}_{j}N^{c}_{ab}-\delta^{c}_{j}N^{d}_{ab}\right)\>,$
$\displaystyle\left[\widetilde{p}^{a}\>,\>\widetilde{p}^{c}\right]_{j}=\frac{\partial\widetilde{p}^{a}}{\partial
f^{ik}}\frac{\partial\widetilde{p}^{c}}{\partial
N^{j}_{ik}}+\frac{\partial\widetilde{p}^{a}}{\partial
B}\frac{\partial\widetilde{p}^{c}}{\partial
p^{j}}-\frac{\partial\widetilde{p}^{c}}{\partial
f^{ik}}\frac{\partial\widetilde{p}^{a}}{\partial
N^{j}_{ik}}-\frac{\partial\widetilde{p}^{c}}{\partial
B}\frac{\partial\widetilde{p}^{a}}{\partial p^{j}}=$
$\displaystyle\qquad=\frac{2}{3}\delta^{c}_{j}\left(f^{ik}N^{a}_{ik}+\frac{F^{\prime}(F^{\prime\prime
2}-F^{\prime}F^{\prime\prime\prime})}{F^{\prime\prime
3}}p^{a}\right)-\frac{2}{3}\delta^{a}_{j}\left(f^{ik}N^{c}_{ik}+\frac{F^{\prime}(F^{\prime\prime
2}-F^{\prime}F^{\prime\prime\prime})}{F^{\prime\prime 3}}p^{c}\right)\>.$

Contrary to the conventional Hamiltonian theory, the canonical transformation
does not preserve fundamental Poisson brackets for momenta.

##  4 Surface Lagrangian and Thermodynamic Properties

It is also important to mention the so-called surface part of the Lagrangian.
This is the part which can be expressed as a derivative, or alternatively said
as a divergence of some vector potential, and usually is neglected since it
does not contribute to the equations of motion. However, this is not true just
for any surface and more importantly, this part of Lagrangian contains
information about thermodynamic properties of the boundary region, typically a
horizon. Previously, we have found the surface Lagrangian as (14)

$\mathcal{L}_{sur}=\partial_{i}\left[\frac{\sqrt{-\widetilde{g}}}{16\pi}\left(\widetilde{g}^{ab}\widetilde{\Gamma}^{i}_{ab}-\widetilde{g}^{ai}\widetilde{\Gamma}^{b}_{ab}+\sqrt{6}\widetilde{\partial}^{i}\widetilde{\phi}\right)\right]\
.$ (48)

It is natural to express this surface term using canonical coordinates when we
use the relations (23) and we get

$\mathcal{L}_{sur}=-\partial_{i}\left(\widetilde{f}^{ab}\widetilde{N}^{i}_{ab}+\sqrt{\frac{3}{2}}\widetilde{p}^{i}\right)\>.$
(49)

If we used coordinate transformation and put it back into the Jordan Frame, we
would obtain

$\mathcal{L}_{sur}=-\partial_{i}\left(\frac{F^{\prime}}{F^{\prime\prime}}p^{i}\right)\>,$
(50)

which is the same result, as was derived for Jordan Frame formulation. Thus,
the thermodynamic properties can be assumed to be the same as in Jordan case
[13], which is the expected result.

In General Relativity, there is a relation between the two Lagrangians (bulk
and surface) [14]

$\mathcal{L}_{sur}=-\partial_{c}\left(g_{ab}\frac{\partial\mathcal{L}_{bulk}}{\partial\partial_{c}g_{ab}}\right)\>,$
(51)

which is better for our purposes to be written in $f-N$ formalism [5]

$\mathcal{L}_{sur}=-\partial_{c}\left(f^{ab}N^{c}_{ab}\right)\>,$ (52)

as visible from surface Lagrangian of Jordan (50) as well as Einstein (49)
frame, this relation does not hold for $F(R)$ gravity theory. One could argue
that (52) is a kind of first approximation of more general formula, however
its form is non-trivial and so it remains an open question.

Acknowledgement:
The work of J. Klusoň is supported by the grant “Integrable Deformations”
(GA20-04800S) from the Czech Science Foundation (GACR).

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##  Appendix A Equations of Motion

This appendix summarizes the equations of motion. They are obtained from the
Hamiltonian in a similar manner as it is done in the conventional Hamiltonian
theory. The only difference rises from the unequal number of dimensions of
coordinates and their conjugated momenta. The momenta always have dimension
one higher. The equations of motion do not reveal all possible derivatives of
momenta but only such which contracts the additional index [10]. This leaves a
bit of freedom for the momentum expression where two different momenta can
express the same system. The presented calculation of equations of motion is
very straight forward with the exception of equation for momentum
$\tilde{N}^{c}_{ab}$, because its conjugated coordinate is included in the
Hamiltonian (24) in three different forms — $\tilde{f}^{ab}$,
$\tilde{f}_{ab}$, $\tilde{f}$. To make that calculation easier, the following
equations are handy

$\frac{\delta\widetilde{f}_{ik}}{\delta\widetilde{f}^{ab}}=-\frac{1}{2}\widetilde{f}_{ak}\widetilde{f}_{bi}-\frac{1}{2}\widetilde{f}_{ai}\widetilde{f}_{bk}\>,\qquad\>\text{d}\widetilde{f}=\widetilde{f}\widetilde{f}_{ik}\>\text{d}\widetilde{f}^{ik}\>.$
(53)

Then the equations of motion for Einstein frame covariant Hamiltonian follow
as

$\displaystyle\partial_{c}\widetilde{f}^{ab}=\frac{\delta\mathcal{H}^{E}}{\delta\widetilde{N}^{c}_{ab}}=16\pi\left(\widetilde{f}^{ka}\widetilde{N}^{b}_{kc}+\widetilde{f}^{kb}\widetilde{N}^{a}_{kc}-\frac{1}{3}\widetilde{f}^{ka}\widetilde{N}^{m}_{km}\delta^{b}_{c}-\frac{1}{3}\widetilde{f}^{kb}\widetilde{N}^{m}_{km}\delta^{a}_{c}\right)\>,$
$\displaystyle\partial_{c}\widetilde{N}^{c}_{ab}=-\frac{\delta\mathcal{H}^{E}}{\delta\widetilde{f}^{ab}}=16\pi\left(\frac{1}{3}\widetilde{N}^{k}_{ak}\widetilde{N}^{m}_{mb}-\widetilde{N}^{k}_{am}\widetilde{N}^{m}_{kb}\right)-\frac{1}{32\pi}\sqrt{-\widetilde{f}}\widetilde{f}_{ab}V-4\pi\widetilde{f}_{ai}\widetilde{f}_{bk}\widetilde{p}^{i}\widetilde{p}^{k}\>,$
$\displaystyle\partial_{a}\widetilde{\phi}=\frac{\delta\mathcal{H}^{E}}{\delta\widetilde{p}^{a}}=-8\pi\widetilde{f}_{ai}\widetilde{p}^{i}\>,$
$\displaystyle\partial_{c}\widetilde{p}^{c}=-\frac{\delta\mathcal{H}^{E}}{\delta\widetilde{\phi}}=-\frac{1}{16\pi}\sqrt{-\widetilde{f}}V^{\prime}\>.$


#  $W$-Boson Mass Anomaly from Scale Invariant 2HDM

Karim Ghorbani  Physics Department, Faculty of Sciences, Arak University, Arak
38156-8-8349, Iran  Parsa Ghorbani  Physics Department, Faculty of Science,
Ferdowsi University of Mashhad, Iran

###### Abstract

The recently reported measurement of the $W$-boson mass by CDF-II
collaboration is significantly heavier than that of the Standard Model
prediction. We study this anomaly in the scale invariant Two-Higgs-Doublet-
Model (2HDM) with a $\mathbb{Z}_{2}$ symmetry to avoid the flavor-changing-
neutral-current (FCNC). In this scenario the Higgs particle is the classically
massless scalon in the scale invariant 2HDM gaining its mass by radiative
corrections, hence being naturally light. Moreover, because of the scale
symmetry the model is more predictive respect to the generic 2HDM. We show
that the oblique parameters depending on the masses of the charged and CP-even
(CP-odd) neutral scalar components of the 2HDM denoted respectively by
$m_{H^{\pm}}$ and $m_{h}$ ($m_{A}$), are large enough to accommodate the
$W$-boson mass anomaly in the model. There are as well viable regions in the
mass spectrum of the scale invariant 2HDM that evade the experimental bounds
from colliders on the charged Higgs.

##  I Introduction

The Standard Model (SM) prediction for the $W$ gauge boson mass is
$M_{W}^{\text{SM}}=80.357\pm 0.006$ GeV Zyla  _et al._ (2020). This
theoretical result was confirmed by the LEP Schael  _et al._ (2013), ATLAS
Aaboud  _et al._ (2018), and by CDF and D0 collaborations on Tevatron Aaltonen
_et al._ (2013), giving rise to the PDG average $M_{W}^{\text{PDG}}=80.379\pm
0.012$ GeV. However, recently the CDF II collaboration Aaltonen  _et al._
(2022) reported a remarkable result for the measurement of the $W$ gauge boson
mass; $M_{W}^{\text{CDF}}=80.43335\pm 0.0094$ GeV which is $7\sigma$ standard
deviation away from the SM prediction. In order to explain this anomaly the
addition of extra fields seems inevitable, therefore it could be a window
towards New Physics (NP) beyond the SM. Different models have been considered
so far to interpret the anomaly Fan  _et al._ (2022a); Lu  _et al._ (2022);
Athron  _et al._ (2022a); Yuan  _et al._ (2022); Strumia (2022); Yang and
Zhang (2022); de Blas  _et al._ (2022); Du  _et al._ (2022a); Tang  _et al._
(2022); Cacciapaglia and Sannino (2022); Blennow  _et al._ (2022); Sakurai
_et al._ (2022); Fan  _et al._ (2022b); Liu  _et al._ (2022); Lee and
Yamashita (2022); Cheng  _et al._ (2022); Bagnaschi  _et al._ (2022); Paul and
Valli (2022); Bahl  _et al._ (2022); Asadi  _et al._ (2022); Di Luzio  _et
al._ (2022); Athron  _et al._ (2022b); Gu  _et al._ (2022); Babu  _et al._
(2022); Heo  _et al._ (2022); Du  _et al._ (2022b); Cheung  _et al._ (2022);
Crivellin  _et al._ (2022); Endo and Mishima (2022); Biekötter  _et al._
(2022); Balkin  _et al._ (2022); Han  _et al._ (2022); Ahn  _et al._ (2022);
Kawamura  _et al._ (2022); Zheng  _et al._ (2022); Ghoshal  _et al._ (2022);
Perez  _et al._ (2022); Mondal (2022); Zhang and Feng (2022); Borah  _et al._
(2022); Chowdhury  _et al._ (2022); Arcadi and Djouadi (2022); Cirigliano  _et
al._ (2022). In this work we confront the scale invariant two Higgs doublet
model (2HDM) with the reported $W$-boson mass anomaly by the CDF II
collaboration. Our motivation to study this model is as follows. Within the
mass spectrum of the scale invariant 2HDM, there is a classically massless
scalon field which we take as the SM Higgs particle. As the scalon gains mass
through only radiative corrections, therefore the SM Higgs becomes naturally
as light as the EW scale. Moreover, the free parameters in the scale invariant
2HDM due to scale symmetry are only two which is clearly less than the
corresponding generic 2HDM model, making the scenario much more predictive. In
the mass spectrum, apart from the SM Higgs, there are two charged scalars, a
CP-even and a CP-odd scalar denoted by $H^{\pm}$, $h$ and $A$ respectively.
All these scalars contribute in the oblique parameters. We show there are
viable scalar masses that give rise to the desired values for the oblique
parameters appropriate for the reported $W$-boson mass anomaly. We will
constrain more the parameter space from colliders on the mass of the charged
Higgs.

The content of this paper is arranged as the following. We first introduce the
scale invariant 2HDM in the next section. Then in section III, we will discuss
the theoretical and experimental constraints on the model. In section IV we
present the results and will determine for what scalars masses the $W$-boson
mass anomaly can be explained. We conclude in section V.

##  II Classically Massless 2HDM

We choose a scale invariant 2HDM which is a minimal and highly predictive
model due to the scale symmetry restrictions. The model consists of two
classically massless doublets $\Phi_{1}$ and $\Phi_{2}$. The tree level
potential we consider is as the following

$\begin{split}V=&\frac{\lambda_{1}}{2}(\Phi^{\dagger}_{1}\Phi_{1})^{2}+\frac{\lambda_{2}}{2}(\Phi^{\dagger}_{2}\Phi_{2})^{2}+\lambda_{3}(\Phi^{\dagger}_{1}\Phi_{1})(\Phi^{\dagger}_{2}\Phi_{2})\\\
&+\lambda_{4}(\Phi^{\dagger}_{1}\Phi_{2})(\Phi^{\dagger}_{2}\Phi_{1})+\frac{\lambda_{5}}{2}\left[(\Phi^{\dagger}_{1}\Phi_{2})^{2}+\text{h.c.}\right]\end{split}$
(1)

Depending on how doublets interact with the SM quarks and leptons the type of
the 2HDM is determined. For the type I 2HDM (2HDM-I), the SM fermions interact
with only one doublet, here with $\Phi_{1}$

$\mathcal{L}_{Y}\supset
y^{u}_{ij}\bar{Q}_{L_{i}}\tilde{\Phi}_{1}u_{R_{j}}+y^{d}_{ij}\bar{Q}_{L_{i}}\Phi_{1}d_{R_{j}}+y^{l}_{ij}\bar{E}_{L_{i}}\Phi_{1}e_{R_{j}}+\text{h.c.}$
(2)

while in type II 2HDM (2HDM-II) both doublets interact with the SM fermions as

$\mathcal{L}_{Y}\supset
y^{u}_{ij}\bar{Q}_{L_{i}}\tilde{\Phi}_{1}u_{R_{j}}+y^{d}_{ij}\bar{Q}_{L_{i}}\Phi_{2}d_{R_{j}}+y^{l}_{ij}\bar{E}_{L_{i}}\Phi_{2}e_{R_{j}}+\text{h.c.}$
(3)

where $\tilde{\Phi}_{i}=i\sigma_{2}\Phi_{1}^{*}$. In our notation, the doublet
$\Phi_{1}$ includes the SM Higgs component. To avoid the flavor changing
neutral currents (FCNCs), we have imposed a $\mathbb{Z}_{2}$ symmetry in Eqs.
(1), (2) and (3) under which $\Phi_{1}\to-\Phi_{1}$, $\Phi_{2}\to\Phi_{2}$ and
$u_{R}\to-u_{R}$. The massless 2HDM becomes massive through radiative
corrections à la Coleman and Weinberg. Along the flat direction
$(n_{1},n_{2})$ for which $\Phi_{1}=n_{1}\Phi$ and $\Phi_{2}=n_{2}\Phi$, the
potential and the minimum must be vanishing. This implies the flatness
condition

$\sqrt{\lambda_{1}\lambda_{2}}+\lambda_{3}+\lambda_{4}+\lambda_{5}=0,$ (4)

also

$n_{1}=\frac{\sqrt{\lambda_{2}}}{\sqrt{\lambda_{1}}+\sqrt{\lambda_{2}}}~{}~{}~{}~{}n_{2}=\frac{\sqrt{\lambda_{1}}}{\sqrt{\lambda_{1}}+\sqrt{\lambda_{2}}}\,.$
(5)

Both doublets develops non-zero vacuum expectation values (VEV)
$v_{1}=\braket{\Phi_{1}}$ and $v_{2}=\braket{\Phi_{2}}$ for which we define
$\tan\beta\equiv v_{1}/v_{2}=n_{1}/n_{2}$. Now a rotation in the space of
doublets $(\Phi_{1},\Phi_{2})$ by the angle $\beta$ brings us to a new basis
$(\phi_{1},\phi_{2})$ where $\braket{\phi_{1}}=v$ and $\braket{\phi_{2}}=0$
with $v^{2}=v_{1}^{2}+v_{2}^{2}$. Writing $\phi_{1}$ and $\phi_{2}$ as

$\phi_{1}=\begin{pmatrix}G^{+}\\\
(v+H+iG^{0})/\sqrt{2}\end{pmatrix}~{}~{}~{}\phi_{2}=\begin{pmatrix}H^{+}\\\
(h+iA)/\sqrt{2}\end{pmatrix}$ (6)

and after gauging away tree components $G^{\pm}$ and $G^{0}$, the mass
eigenvalues become

$\begin{split}&m_{H}^{2}=0\hskip
90.3375ptm_{h}^{2}=\sqrt{\lambda_{1}\lambda_{2}}\\\
&m_{H^{\pm}}^{2}=\frac{1}{2}(\sqrt{\lambda_{1}\lambda_{2}}+\lambda_{3})v^{2}~{}~{}~{}~{}m_{A}^{2}=-\lambda_{5}v^{2}.\end{split}$
(7)

The pseudo-Goldston boson due to broken scale symmetry dubbed scalon, is taken
as the SM Higgs filed $H$. It gains mass through only radiative correction
which therefore is naturally light. The one-loop contribution to the Higgs
mass is given by

$\begin{split}m_{H}^{2}=\frac{1}{8\pi^{2}v^{2}}(6M_{W}^{4}+3M_{Z}^{4}-12M_{t}^{4}\\\
+2M_{H^{\pm}}^{4}+M_{h}^{4}+M_{A}^{4})\,.\end{split}$ (8)

The oblique parameters (OPs), $S,T$ and $U$ define the effects of radiative
corrections from new physics Peskin and Takeuchi (1990, 1992). In our model
only the scalars $H^{\pm},h$ and $A$ contributes in OPs.

$\begin{split}&S=\frac{1}{12\pi}\left[g(M_{A}^{2},M_{h}^{2})+\ln\frac{M_{A}M_{h}}{M_{H^{\pm}}^{2}}\right]\\\
&T=\frac{1}{16\pi}\frac{1}{s_{W}^{2}c_{W}^{2}M_{Z}^{2}}[f(M_{A}^{2},M_{H^{\pm}}^{2})\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+f(M_{h}^{2},M_{H^{\pm}}^{2})-f(M_{A}^{2},M_{h}^{2})]\\\
&U=\frac{1}{12\pi}\left[g(M_{A}^{2},M_{H^{\pm}}^{2})+g(M_{h}^{2},M_{H^{\pm}}^{2})-g(M_{A}^{2},M_{h}^{2})\right]\end{split}$
(9)

where the functions $f(x,y)$ and $g(x,y)$ are defined are

$\begin{split}&f(x,y)=\frac{x+y}{2}-\frac{xy}{x-y}\ln\frac{x}{y}\\\
&g(x,y)=-\frac{5}{6}+\frac{2xy}{(x-y)^{2}}+\frac{(x+y)(x^{2}-4xy+y^{2})}{2(x-y)^{3}}\ln\frac{x}{y}\end{split}$
(10)

with $x,y>0$. The electroweak oblique parameters $S$ and $T$ are constrained
as Lee and Pilaftsis (2012)

$\begin{split}&\frac{(S-\hat{S}_{0})^{2}}{\sigma_{\text{S}}^{2}}+\frac{(T-\hat{T}_{0})^{2}}{\sigma_{\text{T}}^{2}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}-2\rho_{\text{ST}}\frac{(S-\hat{S}_{0})(T-\hat{T}_{0})}{\sigma_{\text{S}}\sigma_{\text{S}}}\leq
R^{2}(1-\rho^{2}_{\text{ST}})\end{split}$ (11)

where the $\hat{S}_{0}$ and $\hat{T}_{0}$ are the central values and the
$\sigma_{\text{S}}$ and $\sigma_{\text{T}}$ are the standard deviations for
the oblique parameters $S$ and $T$. The parameter $R^{2}$ determines the
confidence level (CL).

##  III Constraints

Theoretical constraints. The bounded from below conditions can be read from
the mass eigenvalues being simply

$\lambda_{1}>0\wedge\lambda_{2}>0\wedge\lambda_{3}>-\sqrt{\lambda_{1}\lambda_{2}}\wedge\lambda_{5}<0.$
(12)

Generically, perurbativity bounds on the scalar couplings is $\lambda_{i}\leq
4\pi$ Ginzburg and Ivanov (2005); Chen  _et al._ (2019). Given the results in
Cacchio  _et al._ (2016); Grinstein  _et al._ (2016) stronger limits are
obtained, $\lambda_{i}\leq 4$ Atkinson  _et al._ (2021). Looking at the Yukawa
terms involving the charged Higgs couplings to quarks and imposing the the
perturbativity limits as

$\frac{\sqrt{2}V_{tb}m_{t}\cot\beta}{2v}\leq 4\pi\,,\\\
\frac{\sqrt{2}V_{tb}m_{b}\tan\beta}{2v}\leq 4\pi$

a conservative range is obtained for $\tan\beta$: $0.04\leq\tan\beta\leq 900$
Atkinson  _et al._ (2021). However, in order to make the comparison between
the type I and type II, the upper bounds for $\tan\beta$ is taken to be $\sim
300$ Atkinson  _et al._ (2021).

Experimental constraints. Constraints on the parameters of the 2HDM are vastly
studied in the literature. Here we recap some important limits imposed by
various collider data which are applicable in the present work. The relevant
bounds depend on the type of 2HDM under consideration.

The radiative decay ${\bar{B}}\to X_{s}\gamma$ is the most important
observable to bound the charged Higgs from below in the 2HDM-type II. In a
recent study updating the SM prediction sets the strong constraint on the
charged Hiss mass, $m_{H^{\pm}}\gtrsim 800$ GeV at $95\%$ CL for
$\tan\beta\gtrsim 1$ Misiak  _et al._ (2020). For $\tan\beta\lesssim 1$, the
bound becomes even stronger. The radiative decay alone is not enough to find a
concrete lower bound for the charged Higgs mass in 2HDM-I because of the large
correction with $\tan\beta$. However, based on an estimate taking the
contributions to the Wilson coefficients $C_{7}$ and $C_{8}$ at NLO in
Atkinson  _et al._ (2022), the lower bound on the charged Higgs mass is
estimated at $95\%$ CL as $m_{H^{\pm}}\gtrsim 300$ GeV for $\tan\beta\gtrsim
1$.

Considering the combination of tree-level leptonic and semi-leptonic decays of
$B,B_{s},D,D_{s},K$ and $\pi$ mesons and the hadronic decays of $\tau$ leptons
to $\pi$ and $K$ mesons along with $R(D)$ and $R(D^{*})$, the viable parameter
space is found in the plane $\tan\beta-m_{H^{\pm}}$ for 2HDM-II Atkinson  _et
al._ (2021). The results indicate that for $\tan\beta\lesssim 2$ the lower
limit on the charge Higgs mass is $m_{H^{\pm}}\gtrsim 30$ GeV, and the bounds
becomes stronger for larger $\tan\beta$. The same analysis in 2HDM-I finds
that $m_{H^{\pm}}\gtrsim 30$ GeV for $\tan\beta\gtrsim 0.5$ at $2\sigma$
Atkinson  _et al._ (2022).

In our study we also apply the allowed regions in the $\tan\beta-m_{H^{\pm}}$
plane from the mass difference in the mixing of $B_{d}$ and $B_{s}$ mesons in
2HDM-I and 2HDM-II exploiting the results in Atkinson  _et al._ (2021, 2022).

##  IV Results

The present model has three free parameters, $\lambda_{1}$, $\lambda_{2}$ and
$\lambda_{3}$. We scan over the three-dimensional parameter space by sampling
the three couplings in the ranges: $0<\lambda_{1}<5$, $0<\lambda_{2}<5$ and
$-5<\lambda_{3}<5$, bearing in mind that
$\lambda_{3}>-\sqrt{\lambda_{1}\lambda_{2}}$ from vacuum stability conditions.
From Eq. (7) two neutral scalar masses $m_{h}$ and $m_{A}$ are then obtained.
Plugging these two masses in Eq. (8), the mass of the charged Higgs will be
determined. For the SM Higgs mass we take $m_{H}=125.25$ GeV. The oblique
parameters $S$ and $T$ depend on the three new scalars and we compute them
using the relations in Eq. (9). We will employ the recent global fit of
electroweak data incorporating the CDF-II measurements of the W mass,
estimated in Strumia (2022) as $S=0.064\pm 0.090$ and $T=0.14\pm 0.064$ while
fixing $U\sim 0$. First, we are interested in finding the viable values for
the new masses respecting all the theoretical constraints as well as the
relation in Eq. (11) at $95\%$ CL equivalent to $R^{2}=5.99$. Our numerical
results in Fig. 1 show the range of the masses for the scalars, $H^{\pm}$,
$m_{h}$ and $m_{A}$. The largest mass allowed for the charged Higgs is about
$430$ GeV, while the mass of two other scalars can reach about 500 GeV.

Next we impose experimental bounds recounted in the previous section, on the
restricted ranges of the charged Higgs mass. We consider both 2HDM-I and 2HDM-
II in our analysis. Our numerical results concerning 2HDM-I and 2HDM-II in the
$\tan\beta-m_{H^{\pm}}$ plane are shown in Fig. 2 and Fig. 3, respectively.
The three different experimental bounds do not overlap when 2HDM-II is
considered, indicating that 2HDM-II in our scenario is excluded by the
experiments at $2\sigma$. The experimental bounds are weaker in the case we
choose 2HDM-I. In fact, we see from Fig. 2 that a large part of a patch in the
$\tan\beta-m_{H^{\pm}}$ plane with charged Higgs mass in the range $385$ GeV
$\lesssim m_{H^{\pm}}\lesssim 440$ GeV and $1.6\lesssim\tan\beta\lesssim 6.8$
remains viable. In addition there is a small viable small region with
$m_{H^{\pm}}\sim 320-360$ GeV around $\tan\beta\sim 1.8-2.5$.

Figure 1: The viable range of the new scalars are shown when all the
theoretical constraints and the bound from global fit of the electroweak data
and CDF-II are taken into account Figure 2: The allowed region of 2HDM-I
parameter space in the place $\tan\beta-m_{H^{\pm}}$. The experimental limits
are discussed in the text. The variable shown in color spectrum shows the
oblique parameter $T$. Figure 3: The allowed region of 2HDM-II parameter space
in the place $\tan\beta-m_{H^{\pm}}$. The experimental limits are discussed in
the text. The variable shown in color spectrum shows the oblique parameter
$T$.

##  V Conclusion

The recent CDF-II collaboration report on the measurement on the $W$ gauge
boson mass could be a window to New Physics (NP) if it gets confirmed by
another experiment e.g. at the LHC. The $M_{W}$ reported by CDF-II is
significantly larger that what the Standard Model (SM) predicts. Therefore, to
explain the anomaly one needs extra degrees of freedom that affects the mass
of the $W$ boson. One choice is to consider the scale invariant 2HDM in which
the Higgs particle is the classically massless scalon being massive only by
radiative corrections à la Coleman and Weinberg. The scale symmetry make the
model quite restrictive so that it becomes more predictive respect to the
generic 2HDM. Therefore, in the scale invariant 2HDM we deal with less number
of free parameters. Apart from the SM Higgs, there are two charged scalar
components $H^{\pm}$, a CP-even scalar $h$ and a CP-odd one $A$. The oblique
parameters defined by Peskin and Takeuchi, depend on the masses
$M_{H^{\pm}},M_{h}$ and $M_{A}$. We have shown that in the scale invariant
2HDM, the desired $S$ and $T$ oblique parameters can be large enough to
explain the excess in the mass of the $W$ boson. The allowed masses of the
charged and neutral scalars to accommodate the $W$ mass anomaly are shown in
Fig. 1. We have also imposed constraints on the charged Higgs mass from
colliders. In Fig. 2 the bounds are provided for the 2HDM-I and in Fig. 3 for
the 2HDM-II. The outcome of imposing these experimental constraints on the
scale invariant 2HDM is that the 2HDM-II is completely excluded by the charged
Higgs mass bounds while for the 2HDM-I there are viable regions.

## Acknowledgment

P.G. is grateful to the HECAP division at ICTP for the hospitality and support
during his visit when this work was done.

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