# Deuteron gravitational form factors:
exchange currents

Fangcheng He<EMAIL_ADDRESS>Center for Nuclear Theory,
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New
York 11794–3800, USA  Ismail Zahed<EMAIL_ADDRESS>Center for
Nuclear Theory, Department of Physics and Astronomy, Stony Brook University,
Stony Brook, New York 11794–3800, USA

###### Abstract

Following on our recent analysis of the energy momentum tensor (EMT) of light
nuclei in the impulse approximation, we evaluate the leading exchange
corrections also upto momenta of the order of the nucleon mass. The exchange
contributions to the EMT, are composed of the pair interaction, plus the
seagull and the pion exchange interactions, modulo the recoil correction. The
exchange contributions are shown to satisfy the current conservation
requirement. These contributions are small compared to those from the impulse
approximation for most of the deuteron gravitational form factors (GFFs), for
momenta smaller than half of the nucleon mass. For larger momenta, the
exchange contributions are significant for the deuteron A- and D-GFFs. We
suggest that the pion GFFs can be extracted from the exchange contributions of
select deuteron GFFs.

##  I Introduction

The lightest and simplest nuclear system is the deuteron, with binding energy
(2.225 MeV), charge radius (2.16 fm) and magnetic moment (0.857 in Bohr
magnetons). The deuteron large size and weak binding, suggests that its shape
is largely determined by the exchange of a pion.

The gravitational form factors (GFFs) of the deuteron carry important
information on its mass and spin distribution. The deuteron GFFs provide
important insights to the fundamental gravitational content of a light
nucleus. Model calculations of the deuteron GFFs include the lightcone
convolution model Freese and Cosyn (2022) and more recently the Skyrme model
Garcia Martin-Caro  _et al._ (2023), et al. Empirically, the gluon GFFs of
light nuclei maybe accessible through threshold electromagnetic production of
heavy mesons (charmonium, bottomonium) off nuclei.

The nucleus is composed of nucleons (protons and neutrons) bound by strong QCD
interactions. Most of what is known about nuclei, follow from their
electromagnetic properties using elastic electron scattering at intermediate
energies, where the nucleons appear as rigid but extended bodies exchanging
mesons, albeit mostly pions. The disparity between the fundamental and
unconfined degrees of QCD (quarks and gluons) and the observed but confined
degrees of freedom (mesons and nucleons) necessitates the exploration of
innovative probing methods. The ultimate goal is to unravel the composition of
the nucleons, and the interactions between nucleons in a nucleus.

Recently, we investigated the deuteron GFFs using the impulse approximation,
with momentum transfer in the nucleon mass range, i.e. $k\sim m_{N}$ He and
Zahed (2023). We found large deviations of the deuteron GFFs with that of the
nucleon. One the other hand, it is well established that the meson exchange-
current contributions to the electromagnetic form factors of two- and three-
nucleon systems are not negligible Hockert  _et al._ (1973); Chemtob  _et al._
(1974); Kloet and Tjon (1974); Jackson  _et al._ (1975) at large momentum
transfer, i.e. $k\geq\frac{1}{2}m_{N}$. The motivation of this work is to
analyse the corrections induced by pion exchange currents to the deuteron
GFFs.

This paper is a follow up on our recent analysis of the EMT of light nuclei in
the impulse approximation He and Zahed (2023). In section II we discuss the
Born exchange corrections to the impulse approximation, where the relativistic
contributions are first retained. To leading order in the pion-nucleon pseudo-
vector coupling, the exchange contributions are shown to satisfy the EMT
current conservation condition on general ground. This is explicitly checked
in the non-relativistic limit. In section III we detail the exchange
contributions stemming from the non-relativistic reduction of the nucleon pair
exchange, the seagull and pion exchanges, as well the recoil correction. In
section IV we compare the exchange contributions to the impulse approximation
for the deuteron EMT. Our conclusions are in section V.

##  II Deuteron exchange EMT

The corrections to the impulse approximation will be sought in the Born
corrections at tree level. The results will be checked to satisfy the EMT
conservation law. The same observation holds for the non-relativistic limit of
the extracted exchange currents.

Figure 1:  Exchange contributions to the EMT: (a+b) pair contribution; (c)
pion exchange contribution; (d) Seagull contribution. The solid, dashed and
wavy lines represent the nucleon, pion and graviton, respectively.

###  II.1 Born approximation

The exchange contributions to the EMT in the Born approximation, are shown in
Fig. 1. Using the canonical pseudo-vector coupling for the pion, the
contributions $(a+b)$ in Fig. 1 amount to

$\displaystyle T^{\mu\nu}_{X\pi}(k,1,2)=$ $\displaystyle-\bigg{(}\frac{g_{\pi
N}}{2m_{N}}\bigg{)}^{2}\bar{u}(p_{1}^{\prime})\bigg{(}\not{q}_{2}\gamma_{5}\tau_{1}^{a}S_{F}(p_{1}+k)\,T_{N}^{\mu\nu}(p_{1}+k,p_{1})+T_{N}^{\mu\nu}(p_{1}^{\prime},p_{1}^{\prime}-k)\,S_{F}(p_{1}^{\prime}-k)\not{q}_{2}\gamma_{5}\tau_{1}^{a}\bigg{)}u(p_{1})\,$
(1)
$\displaystyle\qquad\qquad\,\,\,\times\Delta_{\pi}(q_{2}^{2})\,\bar{u}(p_{2}^{\prime})\not{q}_{2}\gamma_{5}\tau_{2}^{a}u(p_{2})+(1\leftrightarrow
2)$

where the coupling constant $g_{\pi N}$ is related to the nucleon axial charge
$g_{A}$ through the Goldberger-Treiman relation $g_{\pi
N}/m_{N}=g_{A}/f_{\pi}$, with the pion decay constant $f_{\pi}=93$ MeV. Here
$S_{F}$ and $\Delta_{\pi}$ are the nucleon and pion propogators with
$S_{F}(l)=i/(\not{l}-m_{N}+i0)$ and
$\Delta_{\pi}(q_{1,2})=i/(q_{1,2}^{2}-m_{\pi}^{2}+i0)$. The nucleon EMT
$T_{N}^{\mu\nu}(l_{2},l_{1})$ is

$\displaystyle
T_{N}^{\mu\nu}(l_{2},l_{1})=A(k)\gamma^{(\mu}P_{l}^{\nu)}+B(k)\frac{iP_{l}^{(\mu}\sigma^{\nu)\alpha}k_{\alpha}}{2m_{N}}+C(k)\frac{k^{\mu}k^{\nu}-\eta^{\mu\nu}k^{2}}{m_{N}}-g^{\mu\nu}\left(\frac{\not{l}_{1}+\not{l}_{2}}{2}-m_{N}\right)$
(2)

where $P_{l}=\frac{l_{1}+l_{2}}{2}$. The last term is the off-shell
contribution. In leading order in chiral perturbation theory (ChPT), $A(k)=1$,
$B(k)=C(k)=0$. Alharazin  _et al._ (2020). Away from the chiral point, the
nucleon GFFs have been extracted from experiment Duran  _et al._ (2023), with
overall agreement with QCD lattice simulations Hackett  _et al._ (2023a), pQCD
factorisation Guo  _et al._ (2023) and dual gravity Mamo and Zahed (2020,
2022). The contribution (c) with the intermediate pion EMT contributes

$\displaystyle T^{\mu\nu}_{X\pi\pi}(k,1,2)=-\bigg{(}\frac{g_{\pi
N}}{2m_{N}}\bigg{)}^{2}T_{\pi}^{\mu\nu}(q_{2},q_{1})\bar{u}(p_{1}^{\prime})\not{q}_{1}\gamma_{5}\tau_{1}^{a}u(p_{1})\Delta_{\pi}(q_{1})\Delta_{\pi}(q_{2})\bar{u}(p_{2}^{\prime})\not{q}_{2}\gamma_{5}\tau_{2}^{a}u(p_{2})$
(3)

The general pion EMT can be written as

$\displaystyle
T_{\pi}^{\mu\nu}(q_{2},q_{1})=\langle\pi,q_{2}|T_{\pi}^{\mu\nu}(0)|\pi,q_{1}\rangle=\frac{1}{2}(g^{\mu\nu}k^{2}-k^{\mu}k^{\nu})\,T_{1\pi}(k)+\frac{1}{2}l^{\mu}l^{\nu}\,T_{2\pi}(k)-\frac{1}{2}g^{\mu\nu}(q_{2}^{2}+q_{1}^{2}-2m_{\pi}^{2})$
(4)

where $l^{\mu}=(q_{1}+q_{2})^{\mu}$ and $k^{\mu}=(q_{2}-q_{1})^{\mu}$, with
the pion invariant form factors $T_{1\pi}(k)=T_{2\pi}(k)=1$ in leading order
in ChPT Alharazin  _et al._ (2020) . The last seagull contribution (d) to the
EMT is

$\displaystyle T_{XS}^{\mu\nu}(k,1,2)=$ $\displaystyle i\bigg{(}\frac{g_{\pi
N}}{2m_{N}}\bigg{)}^{2}\bigg{(}\frac{1}{2}g^{\mu\lambda}q_{2}^{\nu}+\frac{1}{2}g^{\nu\lambda}q_{2}^{\mu}-g^{\mu\nu}q^{\lambda}_{2}\bigg{)}$
(5)
$\displaystyle\qquad\qquad\times\bar{u}(p_{1}^{\prime})\gamma_{\lambda}\gamma_{5}\tau_{1}^{a}u(p_{1})\,\Delta_{\pi}(q_{2})\bar{u}(p_{2}^{\prime})\not{q}_{2}\gamma_{5}\tau_{2}^{a}u(p_{2})+(1\leftrightarrow
2)$

To the order considered, we have checked that the $g^{\mu\nu}$ contributions
in (2) and (5) cancel out. They will not be considered in what follows.

###  II.2 EMT current conservation

The conservation of the EMT exchange contributions in the Born approximation
and to leading order, amount to

$\displaystyle
k_{\mu}\big{(}T^{\mu\nu}_{X\pi}(k,1,2)+T^{\mu\nu}_{X\pi\pi}(k,1,2)+T^{\mu\nu}_{XS}(k,1,2)\big{)}={\cal
O}\bigg{(}\frac{g_{\pi N}^{3}}{m^{3}_{N}}\bigg{)}$ (6)

which is seen to follow readily for the on-shell leg of each of the first two
contributions, and after some algebra for the off-shell parts plus the seagull
contributions. Note that in Eq. (6), the nucleon and pion EMT are fixed by
ChPT.

The contributions in Fig. 1 (a+b) contain the positive frequency plus the
negative frequency nucleon contribution in the intermediate state say of the
deuteron. The former is part of the deuteron potential and should be removed
from the contribution to the exchange current operator. With this in mind, we
split the Feynman intermediate propagator

$\displaystyle S_{F}(P)=S_{F}^{+}(P)+S_{F}^{-}(P)$ (7)

We now define the exchange EMT current by removing the nucleon state

$\displaystyle
T^{\mu\nu}_{X}=T^{\mu\nu\,-}_{X\pi}(k,1,2)+T^{\mu\nu}_{X\pi\pi}(k,1,2)+T^{\mu\nu}_{XS}(k,1,2)$

which is seen to satisfy the modified EMT conservation

$\displaystyle k_{\mu}T^{\mu\nu}_{X}\equiv
k_{\mu}\big{(}T^{\mu\nu\,-}_{X\pi}(k,1,2)+T^{\mu\nu}_{X\pi\pi}(k,1,2)+T^{\mu\nu}_{XS}(k,1,2)\big{)}=-k_{\mu}\,T^{\mu\nu\,+}_{X\pi}(k,1,2)$
(9)

More specifically, for in-out on-shell nucleons, we have

$\displaystyle k_{\mu}\,T^{\mu\nu\,+}_{X\pi}(k,1,2)$ $\displaystyle=$
$\displaystyle-
T_{N}^{0\nu}(p^{\prime}_{1},p_{1}^{\prime}-k)V_{\pi}(p_{1}^{\prime}-k,p_{1};p_{2}^{\prime},p_{2})+V_{\pi}(p_{1}^{\prime},p_{1}+k;p_{2}^{\prime},p_{2})T_{N}^{0\nu}(p_{1}+k,p_{1})$
(10) $\displaystyle+$ $\displaystyle 1\leftrightarrow 2$

with the tree-level pseudo-vector pion exchange potential

$\displaystyle
V_{\pi}(p_{1}^{\prime},p_{1};p_{2}^{\prime},p_{2})=-i\bigg{(}\frac{g_{\pi
N}}{2m_{N}}\bigg{)}^{2}\bar{u}(p_{1}^{\prime})\not{q}_{1}\gamma_{5}\tau^{a}u(p_{1})\,\Delta_{\pi}(q_{2})\,\bar{u}(p_{2}^{\prime})\not{q}_{2}\gamma_{5}\tau^{a}\,u(p_{2})$
(11)

Hence, the right-hand-side of (9) is emanable to the commutator as a
convolution in momentum space, of the pion exchange potential with the EMT
current

$\displaystyle k_{\mu}\,T^{\mu\nu\,+}_{X\pi}=[V_{\pi},T_{N}^{0\nu}]$ (12)

We now separate the time component of the exchange current and define the
Hamiltonian

$H=T+V_{\pi}$

as the sum of single nucleon kinetic contribution $T$, plus the pair nucleon
pion exchange contribution $V_{\pi}$, with the continuity equation

$\displaystyle k^{i}T_{X}^{i\nu}-[T,T_{X}^{0\nu}]-[V_{\pi},T_{N}^{0\nu}]\sim
0$ (13)

In the non-relativistic limit, the second contribution is subleading, and (13)
is satisfied by the leading contributions

$\displaystyle k^{i}T_{X}^{i\nu}-[V_{\pi},T_{N}^{0\nu}]\sim 0$ (14)

Figure 2:  Exchange contributions to the EMT: (a) The pair contribution, (b)
seagull contribution and (c) the pion exchange contribution; (d) refers to the
recoil plus wave function renormalization contributions. The solid, dashed and
wavy lines represent the nucleon, pion and graviton, respectively.

##  III Non-relativistic reduction

To extract the non-relativistic contributions to the exchange currents, we
will make use of the diagrams in Fig. 2, with the positive nucleon
contribution in (a) subtracted, and the recoil contribution and wavefunction
renormalization in (d) retained. Again, the wiggly line refers to the
insertion of the EMT, and the dashed line to the pion exchange contribution
using the pseudo-vector coupling.

###  III.1 Fig. 2a : Pair contribution

The non relativistic reduction for Fig. (2)a is

$\displaystyle T_{X\pi}^{00,ij}(k)$ $\displaystyle=$ $\displaystyle{\cal
O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}$ $\displaystyle
T_{X\pi}^{0j}(k)$ $\displaystyle=$ $\displaystyle
A(k)\tau_{1}\cdot\tau_{2}\,\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\frac{\sigma_{2}.q_{2}(-i(k\times
q_{2})^{j}-q_{2}^{j}\sigma_{1}.P_{d}+\sigma_{1}.q_{2}P_{d}^{j}-\sigma_{1}^{j}P_{d}.q_{2})}{4m_{N}w_{q_{2}}^{2}}$
(15) $\displaystyle+$ $\displaystyle(1\leftrightarrow 2)+\bigg{(}\frac{g_{\pi
N}^{2}}{m_{N}^{4}}\bigg{)}$

with $P_{d}=p_{1}+p_{1}^{\prime}=2P-q_{2}$ and
$w_{q_{2}}=(\vec{q}_{2}^{\,2}+m_{\pi}^{2})^{1/2}$. To evaluate the
contributions (III.1) in a deuteron state, we follow our recent analysis in He
and Zahed (2023). In particular, we have

$\displaystyle\left<+\frac{k}{2}m^{\prime}\bigg{|}T_{X\pi}^{0j}\bigg{|}-\frac{k}{2}m\right>$
(16) $\displaystyle=$ $\displaystyle\int
d^{3}P\frac{d^{3}q_{2}}{(2\pi)^{3}}\Phi_{m^{\prime}}^{\dagger}\bigg{(}\vec{P}+\frac{\vec{k}}{4}-\vec{q_{2}}\bigg{)}T_{X\pi}^{0j}\Phi_{m}\bigg{(}\vec{P}-\frac{\vec{k}}{4}\bigg{)}$
$\displaystyle=$ $\displaystyle\int
d^{3}P\frac{d^{3}q_{2}}{(2\pi)^{3}}d^{3}r_{1}d^{3}r_{2}e^{i(\vec{P}+\frac{\vec{k}}{4}-\vec{q_{2}}).\vec{r}_{1}}e^{-i(\vec{P}-\frac{\vec{k}}{4}).\vec{r}_{2}}\varphi^{\dagger}_{m^{\prime}}(r_{1})T_{X\pi}^{0j}\varphi_{m}(r_{2})$
$\displaystyle=$ $\displaystyle\frac{\langle m^{\prime}|(\vec{S}\times
i\vec{k})^{j}|m\rangle}{2}J_{X\pi}$

where the form factor $J_{X\pi}(k)$ can be expressed as

$\displaystyle J_{X\pi}(k)$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi
m_{N}}A(k)\tau_{1}\cdot\tau_{2}\,\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\int\frac{dr}{r}\Bigg{\\{}m_{\pi}^{2}\bar{Y}_{1}\left(2u^{2}-\sqrt{2}uw-2w^{2}\right)j_{0}\left(\frac{kr}{2}\right)$
$\displaystyle-$
$\displaystyle\frac{j_{1}\left(\frac{kr}{2}\right)}{kr}m_{\pi}^{2}\bigg{[}4m_{\pi}ru^{2}Y_{2}-w\left(3\sqrt{2}m_{\pi}r^{2}u^{\prime}Y_{2}+m_{\pi}rwY_{2}-6\sqrt{2}ru^{\prime}\bar{Y}_{1}+15w\bar{Y}_{1}\right)$
$\displaystyle+$
$\displaystyle\sqrt{2}u\left(w\left(10m_{\pi}rY_{2}-24\bar{Y}_{1}\right)+3rw^{\prime}\left(m_{\pi}rY_{2}-2\bar{Y}_{1}\right)\right)\bigg{]}\Bigg{\\}},$

where $u$ and $w$ represent the reduced radial components ${}^{3}S_{1}$ and
${}^{3}D_{1}$ of the deuteron wavefunction, which have been solved in our
previous work He and Zahed (2023),

$\displaystyle Y_{0}$ $\displaystyle=$ $\displaystyle
e^{-m_{\pi}r}/(m_{\pi}r)$ $\displaystyle\bar{Y}_{1}$ $\displaystyle=$
$\displaystyle(1+1/(m_{\pi}r))Y_{0}$ $\displaystyle Y_{2}$ $\displaystyle=$
$\displaystyle(3/(m_{\pi}r)^{2}+3/(m_{\pi}r)+1)Y_{0}$ (18)

following the conventions in Jackson  _et al._ (1975) for the electromagnetic
current.

###  III.2 Fig. 2b : Seagull contribution

The non relativistic reduction for the seagull contribution in Fig. 2b, reads

$\displaystyle T_{XS}^{00}$ $\displaystyle=$ $\displaystyle{\cal
O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}$ $\displaystyle T_{XS}^{0j}$
$\displaystyle=$ $\displaystyle\tau_{1}\cdot\tau_{2}\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\left(\frac{\sigma_{1}^{j}\vec{P}_{d}.\vec{q}_{2}\vec{\sigma}_{2}.\vec{q}_{2}}{4m_{N}w_{q_{2}}^{2}}-\frac{q_{2}^{j}\sigma_{1}.P_{d}\sigma_{2}.q_{2}}{4m_{N}w_{q_{2}}^{2}}\right)+(1\leftrightarrow
2)+{\cal O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}$ $\displaystyle
T_{XS}^{ij}$ $\displaystyle=$ $\displaystyle-\tau_{1}\cdot\tau_{2}\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\frac{q_{2}^{i}\sigma_{1}^{j}+q_{2}^{j}\sigma_{1}^{i}}{2w_{q_{2}}^{2}}\vec{\sigma}_{2}.\vec{q}_{2}+(1\leftrightarrow
2)+{\cal O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}$ (19)

The $t^{0j}$ contribution to the deuteron EMT, is

$\displaystyle\left<+\frac{k}{2}m^{\prime}\bigg{|}T_{XS}^{0j}\bigg{|}-\frac{k}{2}m\right>$
$\displaystyle=$ $\displaystyle\frac{\langle m^{\prime}|(\vec{S}\times
i\vec{k})^{j}|m\rangle}{2}J_{XS}$ (20)

where the form factor $J_{XS}$ can be expressed as

$\displaystyle J_{XS}(k)$ $\displaystyle=$ $\displaystyle\left(\frac{g_{\pi
N}}{2m_{N}}\right)^{2}\tau_{1}\cdot\tau_{2}\int
dr\frac{3m_{\pi}^{3}}{4\sqrt{2}\pi
km_{N}}j_{1}\left(\frac{kr}{2}\right)\left(uw^{\prime}-u^{\prime}w\right)Y_{2}$
(21a)

The $t^{ij}$ contribution can be parametrized as Polyakov and Sun (2019)

$\displaystyle
t^{ij}=\left<+\frac{k}{2}m^{\prime}\bigg{|}T_{XS}^{ij}\bigg{|}-\frac{k}{2}m\right>$
$\displaystyle=$
$\displaystyle\frac{k^{i}k^{j}-\delta^{ij}k^{2}}{4m_{D}}D_{0,XS}\delta_{m^{\prime}m}+\frac{(k^{j}k^{\alpha}Q_{m^{\prime}m}^{i\alpha}+k^{i}k^{\alpha}Q_{m^{\prime}m}^{j\alpha}-k^{2}Q_{m^{\prime}m}^{ij}-\delta^{ij}Q_{m^{\prime}m}^{\alpha\beta}k_{\alpha}k_{\beta})}{2m_{D}}D_{2,XS}$
$\displaystyle+$
$\displaystyle\frac{1}{4m_{D}}(k^{i}k^{j}-\delta^{ij}k^{2})Q_{m^{\prime}m}^{\alpha\beta}\hat{k}_{\alpha}\hat{k}_{\beta}D_{3,XS}+Q_{m^{\prime}m}^{ij}D_{4,XS}+\delta^{ij}\delta_{m^{\prime}m}D_{5,XS}+\delta^{ij}\delta_{m^{\prime}m}D_{6,XS}Q^{\alpha\beta}\hat{k}_{\alpha}\hat{k}_{\beta},$

where $D_{0,XS}$ , $D_{2,XS}$ and $D_{3,XS}$ are the form factors which
satisfy the current conserved condition. The contributions $\delta^{ij}$ and
$Q^{ij}$ in (III.2), appear to upset the conservation law $k^{i}T^{ij}$. As we
show below, the conservation law is upheld in the non-relativistic limit when
all contributions are added up. As a result, the contributions $\delta^{ij}$
and $Q^{ij}$ cancel out. With this in mind, we can extract them by defining

$\displaystyle\tilde{t}^{ij}=t^{ij}-\frac{1}{3}\delta^{ij}t^{l}_{l}-\frac{3}{5}Q^{ij}t^{\alpha\beta}Q^{\alpha\beta}$
(23)

which is traceless and satisfies $\tilde{t}^{ij}Q^{ij}=0$. Using
$\tilde{t}^{ij}$, one can solve for the form factors $D_{\\{0,2,3\\},XS}$ as

$\displaystyle D_{0,XS}$ $\displaystyle=$ $\displaystyle\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}.\tau_{2}m_{\pi}^{2}\int
drj_{2}\left(\frac{kr}{2}\right)\frac{2m_{D}\left(-m_{\pi}ru^{2}Y_{2}+2\sqrt{2}uw(3\bar{Y}_{1}-2m_{\pi}rY_{2})+w^{2}(m_{\pi}rY_{2}-3\bar{Y}_{1})\right)}{3\pi
k^{2}r}$ $\displaystyle D_{2,XS}$ $\displaystyle=$ $\displaystyle-\frac{g_{\pi
N}^{2}\tau_{1}.\tau_{2}}{(2m_{N})^{2}}\frac{m_{D}m_{\pi}^{2}}{14\pi
k^{2}}\int\frac{dr}{r}\Bigg{\\{}j_{2}\left(\frac{kr}{2}\right)\left[14m_{\pi}ru^{2}Y_{2}+7\sqrt{2}uw(6\bar{Y}_{1}-m_{\pi}rY_{2})+2w^{2}(2m_{\pi}rY_{2}+3\bar{Y}_{1})\right]$
(24b) $\displaystyle+$ $\displaystyle
18w^{2}j_{4}\left(\frac{kr}{2}\right)(m_{\pi}rY_{2}-2\bar{Y}_{1})\Bigg{\\}}$
$\displaystyle D_{3,XS}$ $\displaystyle=$ $\displaystyle\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}.\tau_{2}\frac{9m_{D}m_{\pi}^{2}w^{2}}{\pi
k^{2}}\int\frac{dr}{r}j_{4}\left(\frac{kr}{2}\right)(m_{\pi}rY_{2}-2\bar{Y}_{1})$
(24c)

###  III.3 Fig. 2c : Pion exchange contribution

The non relativistic reduction of the pion exchange contribution in Fig. 2c,
yields

$\displaystyle T_{X\pi\pi}^{00}$ $\displaystyle=$ $\displaystyle-\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\frac{\vec{k}^{2}\vec{\sigma_{1}}.\vec{q_{1}}\vec{\sigma_{2}}.\vec{q_{2}}}{2w_{q_{1}}^{2}w_{q_{2}}^{2}}T_{1\pi}(k)+\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\frac{\sigma_{1}.q_{1}\sigma_{2}.q_{2}}{2}\left(\frac{1}{w_{q_{2}}^{2}}+\frac{1}{w_{q_{1}}^{2}}\right)+{\cal
O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}$ $\displaystyle
T_{X\pi\pi}^{0j}$ $\displaystyle=$ $\displaystyle-\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\frac{(q_{1}^{j}+q_{2}^{j})\vec{P}_{d}.\vec{k}\sigma_{1}.q_{1}\sigma_{2}.q_{2}}{4m_{N}w_{q_{1}}^{2}w_{q_{2}}^{2}}T_{2\pi}(k)+{\cal
O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}$ $\displaystyle
T_{X\pi\pi}^{ij}$ $\displaystyle=$ $\displaystyle\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\left(\frac{(\vec{k}^{2}\delta^{ij}-k^{i}k^{j})\vec{\sigma_{1}}.\vec{q_{1}}\vec{\sigma_{2}}.\vec{q_{2}}}{2w_{q_{1}}^{2}w_{q_{2}}^{2}}T_{1\pi}(k)+\frac{(q_{1}^{i}+q_{2}^{i})(q_{1}^{j}+q_{2}^{j})\vec{\sigma_{1}}.\vec{q_{1}}\vec{\sigma_{2}}.\vec{q_{2}}}{2w_{q_{1}}^{2}w_{q_{2}}^{2}}T_{2\pi}(k)\right)$
(25) $\displaystyle-$ $\displaystyle\delta^{ij}\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\frac{\sigma_{1}.q_{1}\sigma_{2}.q_{2}\left(\vec{q}_{1}^{2}+\vec{q}_{2}^{2}+2m_{\pi}^{2}\right)}{2w_{q_{1}}^{2}w_{q_{2}}^{2}}+{\cal
O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}$

The second contributions in each of $T_{X\pi\pi}^{00}$ and $T_{X\pi\pi}^{ij}$
follow from the off-shell part in (4). Their contribution to the deuteron EMT,
are

$\displaystyle\left<+\frac{k}{2}m^{\prime}\bigg{|}T_{X\pi\pi}^{00}\bigg{|}-\frac{k}{2}m\right>$
$\displaystyle=$ $\displaystyle
m_{D}A_{X\pi\pi}\delta_{mm^{\prime}}+Q_{X\pi\pi}\frac{\langle
m^{\prime}|Q^{\alpha\beta}|m\rangle k^{\alpha}k^{\beta}}{2m_{D}}$
$\displaystyle\left<+\frac{k}{2}m^{\prime}\bigg{|}T_{X\pi\pi}^{0j}\bigg{|}-\frac{k}{2}m\right>$
$\displaystyle=$ $\displaystyle\frac{\langle m^{\prime}|(\vec{S}\times
i\vec{k})^{j}|m\rangle}{2}J_{X\pi\pi}$
$\displaystyle\left<+\frac{k}{2}m^{\prime}\bigg{|}T_{X\pi\pi}^{ij}\bigg{|}-\frac{k}{2}m\right>$
$\displaystyle=$
$\displaystyle\frac{k^{i}k^{j}-\delta^{ij}k^{2}}{4m_{D}}D_{0,X\pi\pi}\delta_{m^{\prime}m}+\frac{(k^{j}k^{\alpha}Q_{m^{\prime}m}^{i\alpha}+k^{i}k^{\alpha}Q_{m^{\prime}m}^{j\alpha}-k^{2}Q_{m^{\prime}m}^{ij}-\delta^{ij}Q_{m^{\prime}m}^{\alpha\beta}k_{\alpha}k_{\beta})}{2m_{D}}D_{2,X\pi\pi}$
(26) $\displaystyle+$
$\displaystyle\frac{1}{4m_{D}}(k^{i}k^{j}-\delta^{ij}k^{2})Q_{m^{\prime}m}^{\alpha\beta}\hat{k}_{\alpha}\hat{k}_{\beta}D_{3,X\pi\pi}$

For the $T^{ij}$ component, we only keep the contributions that satisfy the
current conservation law. As we noted above, the upsetting terms cancel out
when all contributions are added up. The corresponding form factors are
lengthy and given in Appendix A.

###  III.4 Fig. 2d : Recoil contribution

For the recoil contribution in left panel in Fig. 2d, the amplitude can be
written as

$\displaystyle T_{Rc}^{\mu\nu}=\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\frac{\bar{u}(p_{1}^{\prime})\not{q_{2}}\gamma_{5}\tau_{1}^{a}(\gamma_{0}E_{p}-\vec{\gamma}.(\vec{p_{1}}+\vec{k})+m)T^{\mu\nu}_{N}u(p_{1})}{4E_{p}w_{q_{2}}(E_{p}-E_{p_{1}^{\prime}}+w_{q_{2}})(E_{p_{2}^{\prime}}-E_{p_{2}}+w_{q_{2}}))}\bar{u}(p_{2}^{\prime})\not{q}_{2}\gamma_{5}\tau_{2}^{a}u(p_{2})$
(27)

As for the disconnected diagrams, which contribute to the wave function
renormalization factor, they can be written as Gari and Hyuga (1976)

$\displaystyle T_{wf}^{\mu\nu}=-\frac{1}{2}T^{\mu\nu}_{Rc}-\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\frac{\bar{u}(p_{1}^{\prime})\not{q_{2}}\gamma_{5}\tau_{1}^{a}(\gamma_{0}E_{p}-\vec{\gamma}.(\vec{p_{1}}+\vec{k})+m)T^{\mu\nu}_{N}u(p_{1})}{8E_{p}w_{q_{2}}(E_{p_{1}^{\prime}}-E_{p}+w_{q_{2}})(E_{p_{2}}-E_{p_{2}^{\prime}}+w_{q_{2}}))}\bar{u}(p_{2}^{\prime})\not{q}_{2}\gamma_{5}\tau_{2}^{a}u(p_{2})$
(28)

As a result, the recoil contribution and wave function renormalization
$T^{\mu\nu}_{XR}=T^{\mu\nu}_{Rc}+T^{\mu\nu}_{wf}$ can be expressed as

$\displaystyle T^{00}_{XR}$ $\displaystyle=$ $\displaystyle-\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}A(k)\frac{q_{2}.k\sigma_{1}.q_{2}\sigma_{2}.q_{2}}{2w_{q_{2}}^{4}}+(1\leftrightarrow
2)+{\cal O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}$ $\displaystyle
T^{0j}_{XR}$ $\displaystyle=$ $\displaystyle\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\Bigg{\\{}\left(-\frac{P_{d}^{j}k.q_{2}\,\sigma_{1}.q_{2}\sigma_{2}.q_{2}}{4m_{N}w_{q_{2}}^{4}}\right)A(k)+(A(k)+B(k))\left(\frac{ik.q_{2}\,\sigma_{2}.q_{2}(k\times
q_{2})^{j}}{8m_{N}w_{q_{2}}^{4}}\right)\Bigg{\\}}$ $\displaystyle+$
$\displaystyle(1\leftrightarrow 2)+{\cal O}\bigg{(}\frac{g_{\pi
N}^{2}}{m_{N}^{4}}\bigg{)}$ $\displaystyle T^{ij}_{XR}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}$ (29)

The contribution to the GFF from Fig. 2d can be expressed as

$\displaystyle\left<+\frac{k}{2}m^{\prime}\bigg{|}T_{XR}^{00}\bigg{|}-\frac{k}{2}m\right>$
$\displaystyle=$ $\displaystyle
m_{D}A_{XR}\delta_{mm^{\prime}}+Q_{XR}\frac{\langle
m^{\prime}|Q^{\alpha\beta}|m\rangle k^{\alpha}k^{\beta}}{2m_{D}}$
$\displaystyle\left<+\frac{k}{2}m^{\prime}\bigg{|}T_{XR}^{0j}\bigg{|}-\frac{k}{2}m\right>$
$\displaystyle=$ $\displaystyle\frac{\langle m^{\prime}|(\vec{S}\times
i\vec{k})^{j}|m\rangle}{2}J_{XR}$ (30)

where

$\displaystyle A_{XR}(k)$ $\displaystyle=$ $\displaystyle\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}A(k)}{(2m_{N})^{2}m_{D}}\int
dr\frac{km_{\pi}^{2}}{24\pi}j_{1}\left(\frac{kr}{2}\right)\bigg{[}u^{2}(m_{\pi}rY_{2}-5\bar{Y}_{1})+4\sqrt{2}uw(m_{\pi}rY_{2}-2\bar{Y}_{1})-w^{2}(m_{\pi}rY_{2}+\bar{Y}_{1})\bigg{]}$
$\displaystyle Q_{XR}(k)$ $\displaystyle=$ $\displaystyle\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}A(k)}{(2m_{N})^{2}}m_{D}\int
dr\frac{m_{\pi}^{2}}{40\pi
k}\Bigg{\\{}j_{1}\left(\frac{kr}{2}\right)\bigg{[}8u^{2}(m_{\pi}rY_{2}-5\bar{Y}_{1})+4\sqrt{2}uw(2\bar{Y}_{1}-m_{\pi}rY_{2})+10w^{2}(m_{\pi}rY_{2}+\bar{Y}_{1})\bigg{]}$
(31b) $\displaystyle+$ $\displaystyle
3j_{3}\left(\frac{kr}{2}\right)\bigg{[}-4m_{\pi}ru^{2}Y_{2}+2\sqrt{2}uw(m_{\pi}rY_{2}+3\bar{Y}_{1})+5w^{2}(3\bar{Y}_{1}-m_{\pi}rY_{2})\bigg{]}\Bigg{\\}}$
$\displaystyle J_{XR}(k)$ $\displaystyle=$ $\displaystyle\left(\frac{g_{\pi
N}}{2m_{N}}\right)^{2}A(k)\tau_{1}\cdot\tau_{2}\int
dr\frac{3m_{\pi}^{2}}{16\pi
km_{N}r^{2}}\Bigg{\\{}2j_{1}\left(\frac{kr}{2}\right)\bigg{[}r\left(2\sqrt{2}u(w^{\prime}\bar{Y}_{1}-m_{\pi}wY_{2})+m_{\pi}w^{2}Y_{2}-2\sqrt{2}u^{\prime}w\bar{Y}_{1}\right)$
(31c) $\displaystyle+$ $\displaystyle
6w^{2}\bar{Y}_{1}\bigg{]}-krwj_{2}\left(\frac{kr}{2}\right)\bigg{[}(2\sqrt{2}u(\bar{Y}_{1}-m_{\pi}rY_{2})+w(m_{\pi}rY_{2}+2\bar{Y}_{1})\bigg{]}$
$\displaystyle+$ $\displaystyle
krw\bar{Y}_{1}\left(2\sqrt{2}u-w\right)j_{0}\left(\frac{kr}{2}\right)\Bigg{\\}}$
$\displaystyle+$ $\displaystyle\left(\frac{g_{\pi
N}}{2m_{N}}\right)^{2}\left(A(k)+B(k)\right)\tau_{1}\cdot\tau_{2}\int
dr\frac{km_{\pi}^{2}}{160\pi
m_{N}}\Bigg{\\{}j_{1}\left(\frac{kr}{2}\right)\bigg{[}2u^{2}(m_{\pi}rY_{2}-5\bar{Y}_{1})$
$\displaystyle-$
$\displaystyle\sqrt{2}uw(\bar{Y}_{1}-2m_{\pi}rY_{2})+w^{2}(m_{\pi}rY_{2}+4\bar{Y}_{1})\bigg{]}$
$\displaystyle+$ $\displaystyle
j_{3}\left(\frac{kr}{2}\right)\bigg{[}2m_{\pi}ru^{2}Y_{2}+2\sqrt{2}uw(m_{\pi}rY_{2}-3\bar{Y}_{1})+w^{2}(m_{\pi}rY_{2}-6\bar{Y}_{1})\bigg{]}\Bigg{\\}}$

###  III.5 EMT conservation in non-relativistic limit

To check (14) in the non-relativistic limit, we combine the result in (III.2)
with that in (III.3), to have

$\displaystyle k^{i}T^{ij}_{XS}+k^{i}T^{ij}_{X\pi\pi}$ $\displaystyle=$
$\displaystyle-\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\left(\frac{\sigma_{2}.q_{2}k.q_{2}\sigma_{1}^{j}+\sigma_{1}.k\sigma_{2}.q_{2}q_{2}^{j}}{2w_{q_{2}}^{2}}+\frac{\sigma_{1}.q_{1}k.q_{1}\sigma_{2}^{j}+\sigma_{2}.k\sigma_{1}.q_{1}q_{1}^{j}}{2w_{q_{1}}^{2}}\right)$
$\displaystyle+$ $\displaystyle\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\left(\frac{q_{1}^{j}}{w_{q_{1}}^{2}}-\frac{q_{2}^{j}}{w_{q_{2}}^{2}}\right)\sigma_{1}.q_{1}\sigma_{2}.q_{2}$
$\displaystyle=$ $\displaystyle\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}\left(\frac{-\sigma_{2}.q_{2}k.q_{2}\sigma_{1}^{j}+\sigma_{1}.(k-2q_{2})\sigma_{2}.q_{2}q_{2}^{j}}{2w_{q_{2}}^{2}}-\frac{\sigma_{1}.q_{1}k.q_{1}\sigma_{2}^{j}+\sigma_{2}.(k-2q_{2})\sigma_{1}.q_{1}q_{1}^{j}}{2w_{q_{1}}^{2}}\right)$

where we have set $T_{1\pi}=T_{2\pi}=1$. On the other hand,
$[V_{\pi},T_{N}^{0\nu}]$ can be expressed as

$\displaystyle[V_{\pi},T^{0j}]$ $\displaystyle=$ $\displaystyle-\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}\left(\frac{\sigma_{1}.q_{2}\sigma_{2}.q_{2}}{w_{q_{2}}^{2}}(p_{1}^{j}+\frac{k^{j}}{2}+\frac{(\sigma_{1}\times
ik)^{j}}{4})-(p_{1}^{\prime j}-\frac{k^{j}}{2}+\frac{(\sigma_{1}\times
ik)^{j}}{4})\frac{\sigma_{1}.q_{2}\sigma_{2}.q_{2}}{w_{q_{2}}^{2}}\right)$
(33) $\displaystyle-$ $\displaystyle\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}\left(\frac{\sigma_{1}.q_{1}\sigma_{2}.q_{1}}{w_{q_{1}}^{2}}(p_{2}^{j}+\frac{k^{j}}{2}+\frac{(\sigma_{2}\times
ik)^{j}}{4})-(p_{2}^{\prime j}-\frac{k^{j}}{2}+\frac{(\sigma_{2}\times
ik)^{j}}{4})\frac{\sigma_{1}.q_{1}\sigma_{2}.q_{1}}{w_{q_{1}}^{2}}\right)$
$\displaystyle=$ $\displaystyle-\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}\left(\frac{\sigma_{2}.q_{2}k.q_{2}\sigma_{1}^{j}-\sigma_{1}.(k-2q_{2})\sigma_{2}.q_{2}q_{2}^{j}}{2w_{q_{2}}^{2}}+\frac{\sigma_{1}.q_{1}k.q_{1}\sigma_{2}^{j}+\sigma_{2}.(k-2q_{2})\sigma_{1}.q_{1}q_{1}^{j}}{2w_{q_{1}}^{2}}\right)$

where we have set $A(k^{2})=1$ and $B(k^{2})=0$, which are the leading ChPT
values.

Figure 3: The deuteron GFFs in the impulse approximation from He and Zahed
(2023), along with exchange corrections from this work. The subscript “imp"
refer to the impulse approximation, while the subsripts “$X\pi$", “$XS$",
“$X\pi\pi$" and “XR" refer to the exchange contributions from the pair diagram
(Fig. 2a), seagull diagram (Fig. 2b), pion exchange diagram (Fig. 2c) and
recoil contribution (Fig. 2d), respectively. These results are for the pion
GFFs set to $T_{1\pi}=T_{2\pi}=1$.

Figure 4: Same as Fig. 3 but with the parametrized pion GFFs following from
the QCD Lattice calculation in Hackett  _et al._ (2023b) in Eq. IV.

##  IV Numerical results

To summarise, the exchange contributions in the deuteron state are

$\displaystyle T^{00}_{DX}$ $\displaystyle=$ $\displaystyle
T_{X\pi\pi}^{00}+T^{00}_{XR}$ $\displaystyle T^{0j}_{DX}$ $\displaystyle=$
$\displaystyle T^{0j}_{X\pi}+T_{XS}^{0j}+T^{0j}_{X\pi\pi}+T^{0j}_{XR}$
$\displaystyle T^{ij}_{DX}$ $\displaystyle=$ $\displaystyle
T_{XS}^{ij}+T^{ij}_{X\pi\pi}$ (34)

The corresponding GFFs from the exchange currents are

$\displaystyle A_{X}=A_{X\pi\pi}(\ref{eq:FF_AXpipi})+A_{XR}(\ref{eq:FF_AXR})$
$\displaystyle Q_{X}=Q_{X\pi\pi}(\ref{eq:FF_QXpipi})+Q_{XR}(\ref{eq:FF_QXR})$
$\displaystyle
J_{X}=J_{X\pi}(\ref{eq:FF_JXpi})+J_{XS}(\ref{eq:FF_JXS})+J_{X\pi\pi}(\ref{eq:FF_JXpipi})+J_{XR}(\ref{eq:FF_JXR})$
$\displaystyle
D_{0,X}=D_{0,XS}(\ref{eq:FF_D0XS})+D_{0,X\pi\pi}(\ref{eq:FF_D0Xpipi})$
$\displaystyle
D_{2,X}=D_{3,XS}(\ref{eq:FF_D2XS})+D_{3,X\pi\pi}(\ref{eq:FF_D2Xpipi})$
$\displaystyle
D_{3,X}=D_{3,XS}(\ref{eq:FF_D3XS})+D_{3,X\pi\pi}(\ref{eq:FF_D3Xpipi})$ (35)

with the contributions from the a-pair, b-seagull, c-pipi and d-recoil labeled
by the pertinent result. In the forward limit, we note that
$T^{00}_{DX}=-V_{\pi}$ which is the pion pair potential. When combined with
the deuteron result $T_{D}^{00}$ from the impulse approximation in He and
Zahed (2023), we obtain the sum rule for the deuteron mass

$m_{D}\sim 2m_{N}+\Bigg{<}\frac{P^{2}}{m_{N}^{2}}-V_{\pi}\Bigg{>}_{D}\,.$

with the bracket including the recoil correction at next to leading order in
$P$, and the pion exchange correction. Note that the contribution of
$-V_{\pi}$ to the deuteron energy from pion exchange through the mass GFF, is
consistent with the one made in DIS in Kaptari  _et al._ (1990). We now
proceed to compare the changes due to (IV) on the results from our analysis,
using the impulse approximation in He and Zahed (2023).

Since the conservation law in III.5 is only satisfied at the leading order in
ChPT, we first set the pion GFF to their ChPT values $T_{1\pi}=T_{2\pi}=1$ in
our numerical results. For the nucleon GFFs, we use the ones detailed in our
previous work He and Zahed (2023), which are borrowed from the Lattice
calculation in Hackett  _et al._ (2023a) and the dual gravity calculation in
Mamo and Zahed (2022), for a direct comparison between the exchange currents
and the impulse approximation. In Fig 3 we show the results solely from the
impulse approximation derived in He and Zahed (2023), in blue-dotted lines for
all the deuteron GFFs. The added exchange contributions from the pair,
seagull, pipi and recoil as detailed in (IV) for each of the deuteron GFFs are
shown in cyan-, orange-, green-, red-dotted lines. To show the correction due
to the pion GFFs, we borrow their parameterization from the recent Lattice
calculation in Hackett  _et al._ (2023b),

$\displaystyle
T_{1\pi}^{g}(k)=\frac{0.596}{1+\frac{\vec{k}^{2}}{0.677~{}\text{GeV}^{2}}}$
$\displaystyle
T_{1\pi}^{q}(k)=\frac{0.304}{1+\frac{\vec{k}^{2}}{1.44~{}\text{GeV}^{2}}}$
$\displaystyle
T_{2\pi}^{g}(k)=\frac{0.546}{1+\frac{\vec{k}^{2}}{1.129~{}\text{GeV}^{2}}}$
$\displaystyle
T_{2\pi}^{q}(k)=\frac{0.481}{1+\frac{\vec{k}^{2}}{1.262~{}\text{GeV}^{2}}}$
(36)

The same results with the pion GFFs following the parametrization (IV) from
the lattice results in Hackett  _et al._ (2023b), are shown in Fig 4. For the
seagull and off-shellness part, we do not have the parameterization form
currently, so they are only quoted from the leading order in ChPT. The main
differences are seen in the deuteron GFFs $|A,D_{0},D_{3}|$, offering the
possibility of extracting the pion GFFs from the deuteron exchange pion
contribution, as we detail further below.

The deuteron mass GFF |A| receives contributions solely from the impulse
approximation (blue-circles) and the pion exchange diagram (green-diamonds).
There is no seagull contribution, and the recoil contribution is very small
(red-triangle). A comparison of Fig 3 (without the pion form factors) and Fig
4 (with the pion form factors), shows large deviations in the range
$k>\frac{1}{2}m_{N}$. This suggests that the pion GFF $T_{1\pi}$ can be
extracted from a measurement of the deuteron mass GFF $|A|$. The recoil
corrections are small (red-triangle) due to a large cancellation between the
recoil contribution and wave-function renormalization, much like for the
electromagnetic form factors Gari and Hyuga (1976); Ohta and Wakamatsu (1976).

The exchange contributions to the deuteron spin and quadrupole GFFs $|J,Q|$
shown in Fig 4, are relatively small for a wide range of $k\leq m_{N}$. The
deuteron GFFs $|D_{0,2,3}|$ shown in Fig 4, receive exchange contributions
from the seagull and pion ($X\pi\pi$) contributions, but none from the recoil
corrections.

The correction from the seagull plus pion exchange contributions for $|D_{0}|$
(standard D-term) in Fig 4, are small for $k\leq\frac{1}{2}m_{N}$. The seagull
contribution moves down the diffractive peak from the impulse approximation
(blue-circles) from about $0.9\,{\rm GeV}^{2}$ to about $0.5\,{\rm GeV}^{2}$
(orange-squares), before being shifted up (green-diamonds) to about $0.7\,{\rm
GeV}^{2}$ by the pion exchange contribution ($X\pi\pi$).

The exchange seagull contribution (orange-squares) is relatively large for
$|D_{2}|$ (D-term for mixed tensor-spin-spin contribution) (orange-squares) in
Fig 4, but very small for $|D_{3}|$ (D-term for mixed tensor-quadrupole
contribution) for most of the range of $k$ supported by our approximation.
This is mostly due to the fact that $D_{2}$ in the impulse approximation, is
generated by a non-local interaction proportional to the D-wave admixture $w$
which is small, while the seagull exchange contribution (24b) includes the
contribution from the S-wavefunction $u$ which is large.

In contrast, the exchange pion contribution (green-diamonds) in Fig 4, shifts
down $|D_{3}|$ at larger momenta. This observation once combined with the one
for $|A|$ above, suggests that the pion GFFs $T_{1\pi,2\pi}$ can be extracted
from the exchange contributions in $|A,D_{3}|$. With the exception of
$|D_{2}|$, most of the GFFs mass, quadrupole and spin radii are left unchanged
by the exchange corrections. We note that the pion exchange contribution in
the standard deuteron D-term $D_{0}$ at large $k$, will modify the deuteron
pressure distribution obtained in the impulse approximation at small
distances.

##  V Conclusions

We have analyzed the exchange contributions to the deuteron EMT, by including
the contributions from the pair, seagull, pion exchange and recoil in the non-
relativistic Born approximation. This reduction is expected to hold for
momenta within a nucleon mass range. The exchange contributions were shown to
satisfy the conservation law for the EMT.

The pion exchange contribution appears to be sizable in the deuteron GFFs
$|A,D_{3}|$ when the pion GFF are added to the impulse approximation, with the
seagull and recoil contributions being small in the range of a nucleon mass.
This suggests the possibility of empirically extracting the pion GFFs from the
measured deuteron GFFs $|A,D_{3}|$.

Most of the exchange contributions are small in the range
$k\leq\frac{1}{2}m_{N}$, leaving unchanged the mass, quadrupole and spin radii
in the deuteron. This may reflect on the fact that the low gravitational
multipole transition rates maybe insensitive to the meson exchange currents,
the exception being the deuteron $D_{2}$ GFF as we have explained above. This
result may be viewed as the gravitational analogue of Siegert theorem for the
electromagnetic current in nuclei Siegert (1937); Austern and Sachs (1951).
However, this observation deserves a more thorough investigation.

Acknowledgements

We thank Zein-Eddine Meziani for discussions. This work is supported by the
Office of Science, U.S. Department of Energy under Contract No. DE-
FG-88ER40388. This research is also supported in part within the framework of
the Quark-Gluon Tomography (QGT) Topical Collaboration, under contract no. DE-
SC0023646.

##  Appendix A Pion exchange contribution

The results of the GFFs for the pion exchange contribution in Fig. 2c are

$\displaystyle A_{X\pi\pi}(k)$ $\displaystyle=$ $\displaystyle\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}T_{1\pi}(k)\int_{-1/2}^{1/2}dt\int
dr\Bigg{\\{}-\frac{k^{2}}{1536\pi
m_{D}}j_{0}(krt)\bigg{[}k^{2}r\left(4t^{2}-1\right)Y_{0}^{\pi\pi}\left(8u^{2}-4\sqrt{2}uw-9w^{2}\right)$
(37a) $\displaystyle-$ $\displaystyle
72L_{\pi}^{2}rw\bar{Y}_{1}^{\pi\pi}\left(2\sqrt{2}u-w\right)+24L_{\pi}Y_{0}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{kj_{1}(krt)}{768\pi
m_{D}rt}\bigg{[}k^{2}rY_{0}^{\pi\pi}\left(32L_{\pi}rt^{2}\left(u^{2}+4\sqrt{2}uw-w^{2}\right)-3\left(4t^{2}-1\right)w\left(2\sqrt{2}u-7w\right)\right)$
$\displaystyle+$ $\displaystyle
12L_{\pi}\left(4L_{\pi}ru^{2}\bar{Y}_{1}^{\pi\pi}-2\sqrt{2}uw(L_{\pi}r\bar{Y}_{1}^{\pi\pi}+3Y_{0}^{\pi\pi})+w^{2}(5L_{\pi}r\bar{Y}_{1}^{\pi\pi}+3Y_{0}^{\pi\pi})\right)\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{k^{2}j_{2}(krt)}{10752\pi
m_{D}rt^{2}}\bigg{[}\left(4t^{2}-1\right)wY_{0}^{\pi\pi}\left(4k^{2}r^{2}t^{2}\left(63\sqrt{2}u+25w\right)-3255w\right)$
$\displaystyle-$ $\displaystyle
56L_{\pi}^{2}r^{2}t^{2}\bar{Y}_{1}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)+168L_{\pi}rt^{2}wY_{0}^{\pi\pi}\left(2\sqrt{2}u-w\right)\bigg{]}$
$\displaystyle+$ $\displaystyle
j_{4}(krt)\frac{31k^{4}r\left(4t^{2}-1\right)w^{2}Y_{0}^{\pi\pi}}{3584\pi
m_{D}}\Bigg{\\}}$ $\displaystyle+$ $\displaystyle\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}\int dr\frac{m_{\pi}^{2}}{48\pi
km_{D}r^{2}}\Bigg{\\{}-2j_{1}\left(\frac{kr}{2}\right)\bigg{[}2k^{2}r^{2}\bar{Y}_{1}\left(u^{2}+4\sqrt{2}uw-w^{2}\right)+12m_{\pi}ru^{2}Y_{2}$
$\displaystyle-$ $\displaystyle
6\sqrt{2}uw(m_{\pi}rY_{2}+3\bar{Y}_{1})+3w^{2}(5m_{\pi}rY_{2}+3\bar{Y}_{1})\bigg{]}$
$\displaystyle-$ $\displaystyle
3krj_{0}\left(\frac{kr}{2}\right)\bigg{[}2\sqrt{2}uw(3m_{\pi}rY_{2}+\bar{Y}_{1})-w^{2}(3m_{\pi}rY_{2}+5\bar{Y}_{1})-4u^{2}\bar{Y}_{1}\bigg{]}$
$\displaystyle+$ $\displaystyle
krj_{2}\left(\frac{kr}{2}\right)\bigg{[}4m_{\pi}ru^{2}Y_{2}-2\sqrt{2}uw(m_{\pi}rY_{2}+3\bar{Y}_{1})+w^{2}(5m_{\pi}rY_{2}+3\bar{Y}_{1})\bigg{]}\Bigg{\\}}$
$\displaystyle Q_{X\pi\pi}(k)$ $\displaystyle=$ $\displaystyle\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}T_{1\pi}(k)\int_{-1/2}^{1/2}dt\int
dr\Bigg{\\{}-\frac{k^{2}m_{D}rj_{0}(krt)}{32\pi}\bigg{[}\left(4t^{2}-1\right)Y_{0}^{\pi\pi}\left(2u^{2}-\sqrt{2}uw-3w^{2}\right)\bigg{]}$
(37b) $\displaystyle+$
$\displaystyle\frac{km_{D}Y_{0}^{\pi\pi}j_{1}(krt)}{320\pi
t}\bigg{[}16L_{\pi}rt^{2}\left(10u^{2}-5\sqrt{2}uw-w^{2}\right)-15\left(4t^{2}-1\right)w\left(2\sqrt{2}u-7w\right)\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{m_{D}j_{2}(krt)}{448\pi
rt^{2}}\bigg{[}\left(4t^{2}-1\right)w^{2}Y_{0}^{\pi\pi}\left(74k^{2}r^{2}t^{2}-1407\right)-28L_{\pi}^{2}r^{2}t^{2}\bar{Y}_{1}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)$
$\displaystyle+$ $\displaystyle
84L_{\pi}rt^{2}wY_{0}^{\pi\pi}\left(2\sqrt{2}u-w\right)\bigg{]}$
$\displaystyle-$
$\displaystyle\frac{27km_{D}L_{\pi}rtw^{2}\text{Y0}j_{3}(krt)}{40\pi}-\frac{3k^{2}m_{D}r\left(4t^{2}-1\right)w^{2}Y_{0}^{\pi\pi}j_{4}(krt)}{448\pi}\Bigg{\\}}$
$\displaystyle+$ $\displaystyle\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}\int dr\frac{m_{D}m_{\pi}^{2}}{20\pi
k^{2}r}\Bigg{\\{}-5j_{2}\left(\frac{kr}{2}\right)\bigg{[}-4m_{\pi}ru^{2}Y_{2}+2\sqrt{2}uw(m_{\pi}rY_{2}+3\bar{Y}_{1})-w^{2}(5m_{\pi}rY_{2}+3\bar{Y}_{1})\bigg{]}$
$\displaystyle+$ $\displaystyle
2kr\bar{Y}_{1}\left(-10u^{2}+5\sqrt{2}uw+w^{2}\right)j_{1}\left(\frac{kr}{2}\right)+27krw^{2}\bar{Y}_{1}j_{3}\left(\frac{kr}{2}\right)\Bigg{\\}}$
$\displaystyle J_{X\pi\pi}(k)$ $\displaystyle=$ $\displaystyle\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}m_{N}}T_{2\pi}(k)\int_{-1/2}^{1/2}dt\int
dr\Bigg{\\{}\frac{L_{\pi}j_{0}(krt)}{384\pi
r}\bigg{[}72L_{\pi}w\bar{Y}_{1}^{\pi\pi}\left(w-2\sqrt{2}u\right)$ (37c)
$\displaystyle-$ $\displaystyle
k^{2}r\left(4t^{2}-1\right)Y_{0}^{\pi\pi}\left(\sqrt{2}u(2w-rw^{\prime})+w\left(\sqrt{2}ru^{\prime}-w\right)\right)\bigg{]}$
$\displaystyle-$ $\displaystyle\frac{3L_{\pi}j_{1}(krt)}{320\pi
kr^{2}t}\bigg{[}k^{2}rw\left(-8\sqrt{2}L_{\pi}r^{2}t^{2}u^{\prime}\bar{Y}_{1}^{\pi\pi}+8rt^{2}\left(L_{\pi}w\bar{Y}_{1}^{\pi\pi}-2\sqrt{2}u^{\prime}Y_{0}^{\pi\pi}\right)+5\left(4t^{2}-1\right)wY_{0}^{\pi\pi}\right)$
$\displaystyle+$ $\displaystyle
2\sqrt{2}uk^{2}r\left(-8L_{\pi}rt^{2}w\bar{Y}_{1}^{\pi\pi}+4rt^{2}w^{\prime}(L_{\pi}r\bar{Y}_{1}^{\pi\pi}+2Y_{0}^{\pi\pi})+\left(4t^{2}+5\right)wY_{0}^{\pi\pi}\right)$
$\displaystyle+$ $\displaystyle
20L_{\pi}\left(r\left(2\sqrt{2}u(w^{\prime}\bar{Y}_{1}^{\pi\pi}-L_{\pi}wY_{2}^{\pi\pi})+L_{\pi}w^{2}Y_{2}^{\pi\pi}-2\sqrt{2}u^{\prime}w\bar{Y}_{1}^{\pi\pi}\right)+6w^{2}\bar{Y}_{1}^{\pi\pi}\right)\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{L_{\pi}j_{2}(krt)}{2688\pi
r^{2}t^{2}}\bigg{[}2\sqrt{2}uw\left(\left(4t^{2}-1\right)Y_{0}^{\pi\pi}\left(26k^{2}r^{2}t^{2}-147\right)-504L_{\pi}rt^{2}\bar{Y}_{1}^{\pi\pi}\right)$
$\displaystyle+$
$\displaystyle\sqrt{2}ruw^{\prime}\left(1008L_{\pi}rt^{2}\bar{Y}_{1}^{\pi\pi}-\left(4t^{2}-1\right)Y_{0}^{\pi\pi}\left(26k^{2}r^{2}t^{2}-147\right)\right)$
$\displaystyle-$ $\displaystyle
2r^{2}t^{2}w\left(13k^{2}\left(4t^{2}-1\right)wY_{0}^{\pi\pi}+504\sqrt{2}L_{\pi}u^{\prime}\bar{Y}_{1}^{\pi\pi}\right)+26\sqrt{2}k^{2}r^{3}\left(4t^{2}-1\right)t^{2}wu^{\prime}Y_{0}^{\pi\pi}$
$\displaystyle-$ $\displaystyle
21r\left(7\sqrt{2}\left(4t^{2}-1\right)wu^{\prime}Y_{0}^{\pi\pi}-24L_{\pi}t^{2}w^{2}\bar{Y}_{1}^{\pi\pi}\right)+147\left(4t^{2}-1\right)w^{2}Y_{0}^{\pi\pi}\bigg{]}$
$\displaystyle-$ $\displaystyle\frac{3kL_{\pi}tj_{3}(krt)}{40\pi
r}\bigg{[}\sqrt{2}u(2w-rw^{\prime})(3Y_{0}^{\pi\pi}-L_{\pi}r\bar{Y}_{1}^{\pi\pi})+rw\left(\sqrt{2}u^{\prime}(3Y_{0}^{\pi\pi}-L_{\pi}r\bar{Y}_{1}^{\pi\pi})+L_{\pi}w\bar{Y}_{1}^{\pi\pi}\right)\bigg{]}$
$\displaystyle+$ $\displaystyle
j_{4}(krt)\frac{11k^{2}L_{\pi}\left(4t^{2}-1\right)Y_{0}^{\pi\pi}}{896\pi}\bigg{[}\sqrt{2}u(2w-rw^{\prime})+w\left(\sqrt{2}ru^{\prime}-w\right)\bigg{]}\Bigg{\\}}$
$\displaystyle D_{0,X\pi\pi}(k)$ $\displaystyle=$ $\displaystyle
m_{D}\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}T_{1\pi}(k)\int_{-1/2}^{1/2}dt\int
dr\Bigg{\\{}\frac{j_{0}(krt)}{384\pi}\bigg{[}-k^{2}r\left(4t^{2}-1\right)Y_{0}^{\pi\pi}\left(8u^{2}-4\sqrt{2}uw-9w^{2}\right)$
(37d) $\displaystyle+$ $\displaystyle
72L_{\pi}^{2}rw\bar{Y}_{1}^{\pi\pi}\left(2\sqrt{2}u-w\right)-24L_{\pi}Y_{0}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{j_{1}(krt)}{192\pi
krt}\bigg{[}k^{2}rY_{0}^{\pi\pi}\left(32L_{\pi}rt^{2}\left(u^{2}+4\sqrt{2}uw-w^{2}\right)-3\left(4t^{2}-1\right)w\left(2\sqrt{2}u-7w\right)\right)$
$\displaystyle+$ $\displaystyle
12L_{\pi}\left(4L_{\pi}ru^{2}\bar{Y}_{1}^{\pi\pi}-2\sqrt{2}uw(L_{\pi}r\bar{Y}_{1}^{\pi\pi}+3Y_{0}^{\pi\pi})+w^{2}(5L_{\pi}r\bar{Y}_{1}^{\pi\pi}+3Y_{0}^{\pi\pi})\right)\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{j_{2}(krt)}{2688\pi
rt^{2}}\bigg{[}\left(4t^{2}-1\right)wY_{0}^{\pi\pi}\left(4k^{2}r^{2}t^{2}\left(63\sqrt{2}u+25w\right)-3255w\right)$
$\displaystyle-$ $\displaystyle
56L_{\pi}^{2}r^{2}t^{2}\bar{Y}_{1}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)+168L_{\pi}rt^{2}wY_{0}^{\pi\pi}\left(2\sqrt{2}u-w\right)\bigg{]}$
$\displaystyle+$
$\displaystyle\frac{j_{4}(krt)}{896\pi}31k^{2}r\left(4t^{2}-1\right)w^{2}\bar{Y}_{0}^{\pi\pi}\Bigg{\\}}$
$\displaystyle+$ $\displaystyle m_{D}\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}T_{2\pi}(k)\int_{-1/2}^{1/2}dt\int
dr\Bigg{\\{}-\frac{j_{0}(krt)}{864\pi
k^{2}r}\bigg{[}12L_{\pi}^{2}w\bar{Y}_{1}^{\pi\pi}\Big{(}k^{2}r^{2}\left(\sqrt{2}\left(40t^{2}+17\right)u-\left(68t^{2}+25\right)w\right)$
$\displaystyle+$ $\displaystyle
120w\Big{)}-72k^{2}L_{\pi}rt^{2}Y_{0}^{\pi\pi}\left(28u^{2}-14\sqrt{2}uw-3w^{2}\right)-9k^{4}r^{2}t^{2}\left(4t^{2}-1\right)Y_{0}^{\pi\pi}\left(8u^{2}-4\sqrt{2}uw-9w^{2}\right)$
$\displaystyle+$ $\displaystyle
40L_{\pi}^{4}r^{2}Y_{0}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)+40L_{\pi}^{3}r\Big{(}4u^{2}(Y_{0}^{\pi\pi}+5Y_{2}^{\pi\pi})-2\sqrt{2}uw(Y_{0}^{\pi\pi}+8Y_{2}^{\pi\pi})$
$\displaystyle+$ $\displaystyle
w^{2}(5Y_{0}^{\pi\pi}-8Y_{2}^{\pi\pi})\Big{)}\bigg{]}$ $\displaystyle-$
$\displaystyle\frac{tj_{1}(krt)}{240\pi
kr}\bigg{[}k^{2}rY_{0}^{\pi\pi}\Big{(}2L_{\pi}r\left(20\left(8t^{2}-1\right)u^{2}+32\sqrt{2}\left(14t^{2}-1\right)uw+\left(59-316t^{2}\right)w^{2}\right)$
$\displaystyle-$ $\displaystyle
15\left(4t^{2}-1\right)w\left(2\sqrt{2}u-7w\right)\Big{)}-4L_{\pi}\Big{(}28L_{\pi}ru^{2}(2L_{\pi}rY_{2}^{\pi\pi}-15\bar{Y}_{1}^{\pi\pi})$
$\displaystyle+$ $\displaystyle
2\sqrt{2}uw(L_{\pi}r(112L_{\pi}rY_{2}^{\pi\pi}-321\bar{Y}_{1}^{\pi\pi})+105Y_{0}^{\pi\pi})+w^{2}(L_{\pi}r(321\bar{Y}_{1}^{\pi\pi}-116L_{\pi}rY_{2}^{\pi\pi})-465Y_{0}^{\pi\pi})\Big{)}\bigg{]}$
$\displaystyle-$ $\displaystyle\frac{j_{2}(krt)}{24192\pi
k^{4}r^{3}t^{2}}\bigg{[}32L_{\pi}^{3}r\Big{(}4u^{2}\left(Y_{0}^{\pi\pi}\left(113k^{2}r^{2}t^{2}-525\right)+Y_{2}^{\pi\pi}\left(124k^{2}r^{2}t^{2}-2625\right)\right)$
$\displaystyle+$ $\displaystyle
2\sqrt{2}uw\left(Y_{0}^{\pi\pi}\left(454k^{2}r^{2}t^{2}+525\right)+8Y_{2}^{\pi\pi}\left(13k^{2}r^{2}t^{2}+525\right)\right)-w^{2}Y_{0}^{\pi\pi}\left(2k^{2}r^{2}t^{2}+2625\right)$
$\displaystyle-$ $\displaystyle
8w^{2}Y_{2}^{\pi\pi}\left(176k^{2}r^{2}t^{2}-525\right)\Big{)}+32L_{\pi}^{4}r^{2}Y_{0}^{\pi\pi}\Big{(}4u^{2}\left(113k^{2}r^{2}t^{2}-525\right)+2\sqrt{2}uw\left(454k^{2}r^{2}t^{2}+525\right)$
$\displaystyle-$ $\displaystyle
w^{2}\left(2k^{2}r^{2}t^{2}+2625\right)\Big{)}-3L_{\pi}^{2}\bar{Y}_{1}^{\pi\pi}\Big{(}k^{4}r^{4}t^{2}\left(672\left(24t^{2}-1\right)u^{2}+224\sqrt{2}\left(262t^{2}-19\right)uw+\left(5977-9604t^{2}\right)w^{2}\right)$
$\displaystyle-$ $\displaystyle
48k^{2}r^{2}w\left(7\sqrt{2}\left(532t^{2}-109\right)u+\left(2234t^{2}+1085\right)w\right)+201600w^{2}\Big{)}$
$\displaystyle+$ $\displaystyle
288k^{2}L_{\pi}rt^{2}wY_{0}^{\pi\pi}\left(k^{2}r^{2}t^{2}\left(546\sqrt{2}u+179w\right)-6510w\right)$
$\displaystyle+$ $\displaystyle
36k^{4}r^{2}\left(4t^{2}-1\right)t^{2}wY_{0}^{\pi\pi}\left(4k^{2}r^{2}t^{2}\left(63\sqrt{2}u+25w\right)-3255w\right)\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{L_{\pi}j_{3}(krt)}{80\pi
k^{3}r^{2}t^{3}}\bigg{[}-2k^{2}rt^{2}\Big{(}16L_{\pi}^{2}r^{2}t^{2}u^{2}Y_{2}^{\pi\pi}+64\sqrt{2}L_{\pi}rt^{2}uw(L_{\pi}rY_{2}^{\pi\pi}-3\bar{Y}_{1}^{\pi\pi})$
$\displaystyle+$ $\displaystyle
w^{2}\left(2L_{\pi}rt^{2}(27L_{\pi}rY_{2}^{\pi\pi}-92\bar{Y}_{1}^{\pi\pi})+735\left(4t^{2}-1\right)Y_{0}^{\pi\pi}\right)\Big{)}+k^{4}r^{3}\left(4t^{2}-1\right)t^{4}wY_{0}^{\pi\pi}\left(32\sqrt{2}u+61w\right)$
$\displaystyle+$ $\displaystyle
140L_{\pi}w^{2}\left(20L_{\pi}rt^{2}Y_{2}^{\pi\pi}+\left(3-92t^{2}\right)\bar{Y}_{1}^{\pi\pi}\right)\bigg{]}$
$\displaystyle-$ $\displaystyle\frac{j_{4}(krt)}{88704\pi
k^{2}rt^{2}}\bigg{[}3L_{\pi}^{2}w\bar{Y}_{1}^{\pi\pi}\left(4k^{2}r^{2}t^{2}\left(308\sqrt{2}\left(4t^{2}-1\right)u+\left(7364t^{2}-3887\right)w\right)+99\left(4613-16148t^{2}\right)w\right)$
$\displaystyle+$ $\displaystyle
792k^{2}L_{\pi}r\left(836t^{2}-147\right)t^{2}w^{2}Y_{0}^{\pi\pi}+12276k^{4}r^{2}\left(4t^{2}-1\right)t^{4}w^{2}Y_{0}^{\pi\pi}+1760L_{\pi}^{4}r^{2}t^{2}Y_{0}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)$
$\displaystyle+$ $\displaystyle
352L_{\pi}^{3}rt^{2}\left(20u^{2}(Y_{0}^{\pi\pi}+5Y_{2}^{\pi\pi})-10\sqrt{2}uw(Y_{0}^{\pi\pi}+8Y_{2}^{\pi\pi})+w^{2}(25Y_{0}^{\pi\pi}-481Y_{2}^{\pi\pi})\right)\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{L_{\pi}j_{5}(krt)}{64\pi
kt}w^{2}\left(28k^{2}r\left(4t^{2}-1\right)t^{2}Y_{0}^{\pi\pi}+L_{\pi}\left(\left(380t^{2}-47\right)\bar{Y}_{1}^{\pi\pi}-48L_{\pi}rt^{2}Y_{2}^{\pi\pi}\right)\right)$
$\displaystyle-$
$\displaystyle\frac{303L_{\pi}^{2}r\left(4t^{2}-1\right)w^{2}\bar{Y}_{1}^{\pi\pi}j_{6}(krt)}{1408\pi}\Bigg{\\}}$
$\displaystyle D_{2,X\pi\pi}(k)$ $\displaystyle=$ $\displaystyle
m_{D}\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}T_{2\pi}(k)\int_{-1/2}^{1/2}dt\int
dr\Bigg{\\{}\frac{L_{\pi}j_{0}(krt)}{288\pi
k^{2}r}\bigg{[}-9L_{\pi}w^{2}\bar{Y}_{1}^{\pi\pi}\left(k^{2}r^{2}\left(4t^{2}+1\right)-16\right)$
(37e) $\displaystyle+$ $\displaystyle
288k^{2}rt^{2}Y_{0}^{\pi\pi}\left(2u^{2}-\sqrt{2}uw+3w^{2}\right)+4L_{\pi}^{3}r^{2}Y_{0}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)$
$\displaystyle+$ $\displaystyle
4L_{\pi}^{2}r\left(4u^{2}(Y_{0}^{\pi\pi}+5Y_{2}^{\pi\pi})-2\sqrt{2}uw(Y_{0}^{\pi\pi}+8Y_{2}^{\pi\pi})+w^{2}(5Y_{0}^{\pi\pi}-8Y_{2}^{\pi\pi})\right)\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{L_{\pi}tj_{1}(krt)}{20\pi
kr}\bigg{[}8L_{\pi}ru^{2}(2L_{\pi}rY_{2}^{\pi\pi}-15\bar{Y}_{1}^{\pi\pi})+4\sqrt{2}uw(L_{\pi}r(3\bar{Y}_{1}^{\pi\pi}-2L_{\pi}rY_{2}^{\pi\pi})+15Y_{0}^{\pi\pi})$
$\displaystyle+$ $\displaystyle
w^{2}(L_{\pi}r(17L_{\pi}rY_{2}^{\pi\pi}+30\bar{Y}_{1}^{\pi\pi})-210Y_{0}^{\pi\pi})\bigg{]}$
$\displaystyle-$ $\displaystyle\frac{L_{\pi}j_{2}(krt)}{4032\pi
k^{4}r^{3}t^{2}}\bigg{[}8L_{\pi}^{2}r\Big{(}4u^{2}\left(Y_{0}^{\pi\pi}\left(26k^{2}r^{2}t^{2}+105\right)+Y_{2}^{\pi\pi}\left(525-122k^{2}r^{2}t^{2}\right)\right)$
$\displaystyle-$ $\displaystyle
2\sqrt{2}uw\left(Y_{0}^{\pi\pi}\left(26k^{2}r^{2}t^{2}+105\right)-44k^{2}r^{2}t^{2}Y_{2}^{\pi\pi}+840Y_{2}^{\pi\pi}\right)$
$\displaystyle+$ $\displaystyle
5w^{2}\left(Y_{0}^{\pi\pi}\left(26k^{2}r^{2}t^{2}+105\right)+8Y_{2}^{\pi\pi}\left(20k^{2}r^{2}t^{2}-21\right)\right)\Big{)}+8L_{\pi}^{3}r^{2}Y_{0}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)\left(26k^{2}r^{2}t^{2}+105\right)$
$\displaystyle-$ $\displaystyle
9L_{\pi}w\bar{Y}_{1}^{\pi\pi}\left(2k^{2}r^{2}\left(168\sqrt{2}\left(8t^{2}-1\right)u-\left(668t^{2}+63\right)w\right)+k^{4}r^{4}\left(580t^{2}-41\right)t^{2}w-3360w\right)$
$\displaystyle-$ $\displaystyle
288k^{2}rt^{2}w^{2}Y_{0}^{\pi\pi}\left(37k^{2}r^{2}t^{2}+147\right)\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{L_{\pi}j_{3}(krt)}{80\pi
k^{3}r^{2}t^{3}}\bigg{[}2k^{2}rt^{2}\Big{(}32L_{\pi}^{2}r^{2}t^{2}u^{2}Y_{2}^{\pi\pi}-16\sqrt{2}L_{\pi}rt^{2}uw(L_{\pi}rY_{2}^{\pi\pi}+6\bar{Y}_{1}^{\pi\pi})$
$\displaystyle+$ $\displaystyle
w^{2}\left(4L_{\pi}rt^{2}(L_{\pi}rY_{2}^{\pi\pi}-15\bar{Y}_{1}^{\pi\pi})+45\left(4t^{2}-1\right)Y_{0}^{\pi\pi}\right)\Big{)}+15k^{4}r^{3}\left(4t^{2}-1\right)t^{4}w^{2}Y_{0}^{\pi\pi}$
$\displaystyle+$ $\displaystyle
60L_{\pi}w^{2}\left(\left(20t^{2}+6\right)\bar{Y}_{1}^{\pi\pi}-11L_{\pi}rt^{2}Y_{2}^{\pi\pi}\right)\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{L_{\pi}j_{4}(krt)}{14784\pi
k^{2}rt^{2}}\bigg{[}9L_{\pi}w^{2}\bar{Y}_{1}^{\pi\pi}\left(1576k^{2}r^{2}t^{4}+90t^{2}\left(k^{2}r^{2}-286\right)+3003\right)+396k^{2}rt^{2}\left(21-172t^{2}\right)w^{2}Y_{0}^{\pi\pi}$
$\displaystyle-$ $\displaystyle
968L_{\pi}^{3}r^{2}t^{2}Y_{0}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)$
$\displaystyle-$ $\displaystyle
88L_{\pi}^{2}rt^{2}\left(44u^{2}(Y_{0}^{\pi\pi}+5Y_{2}^{\pi\pi})-22\sqrt{2}uw(Y_{0}^{\pi\pi}+8Y_{2}^{\pi\pi})+5w^{2}(11Y_{0}^{\pi\pi}-68Y_{2}^{\pi\pi})\right)\bigg{]}$
$\displaystyle+$
$\displaystyle\frac{3L_{\pi}w^{2}j_{5}(krt)\left(2k^{2}r\left(4t^{2}-1\right)t^{2}Y_{0}^{\pi\pi}+L_{\pi}\left(\left(52t^{2}-5\right)\bar{Y}_{1}^{\pi\pi}-8L_{\pi}rt^{2}Y_{2}^{\pi\pi}\right)\right)}{32\pi
kt}$ $\displaystyle-$
$\displaystyle\frac{81L_{\pi}^{2}r\left(4t^{2}-1\right)w^{2}\bar{Y}_{1}^{\pi\pi}j_{6}(krt)}{704\pi}\Bigg{\\}}$
$\displaystyle D_{3,X\pi\pi}(k)$ $\displaystyle=$ $\displaystyle
m_{D}\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}T_{1\pi}(k)\int_{-1/2}^{1/2}dt\int
dr\Bigg{\\{}-\frac{k^{2}r\left(4t^{2}-1\right)Y_{0}^{\pi\pi}\left(2u^{2}-\sqrt{2}uw-3w^{2}\right)j_{0}(krt)}{16\pi}$
(37f) $\displaystyle+$
$\displaystyle\frac{kY_{0}^{\pi\pi}j_{1}(krt)\left(16L_{\pi}rt^{2}\left(10u^{2}-5\sqrt{2}uw-w^{2}\right)-15\left(4t^{2}-1\right)w\left(2\sqrt{2}u-7w\right)\right)}{160\pi
t}$ $\displaystyle+$ $\displaystyle\frac{j_{2}(krt)}{224\pi
rt^{2}}\bigg{[}\left(4t^{2}-1\right)w^{2}Y_{0}^{\pi\pi}\left(74k^{2}r^{2}t^{2}-1407\right)-28L_{\pi}^{2}r^{2}t^{2}\bar{Y}_{1}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)$
$\displaystyle+$ $\displaystyle
84L_{\pi}rt^{2}wY_{0}^{\pi\pi}\left(2\sqrt{2}u-w\right)\bigg{]}-\frac{27kL_{\pi}rtw^{2}Y_{0}^{\pi\pi}j_{3}(krt)}{20\pi}-\frac{3k^{2}r\left(4t^{2}-1\right)w^{2}Y_{0}^{\pi\pi}j_{4}(krt)}{224\pi}\Bigg{\\}}$
$\displaystyle+$ $\displaystyle m_{D}\frac{g_{\pi
N}^{2}\tau_{1}\cdot\tau_{2}}{(2m_{N})^{2}}T_{2\pi}(k)\int_{-1/2}^{1/2}dt\int
dr\Bigg{\\{}\frac{j_{0}(krt)}{96\pi
k^{2}r}\bigg{[}L_{\pi}^{2}w\bar{Y}_{1}^{\pi\pi}\left(k^{2}r^{2}\left(2\sqrt{2}\left(1-4t^{2}\right)u+\left(580t^{2}+143\right)w\right)-1296w\right)$
$\displaystyle-$ $\displaystyle
2304k^{2}L_{\pi}rt^{2}w^{2}Y_{0}^{\pi\pi}+24k^{4}r^{2}\left(4t^{2}-1\right)t^{2}Y_{0}^{\pi\pi}\left(2u^{2}-\sqrt{2}uw-3w^{2}\right)-36L_{\pi}^{4}r^{2}Y_{0}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)$
$\displaystyle-$ $\displaystyle
36L_{\pi}^{3}r\left(4u^{2}(Y_{0}^{\pi\pi}+5Y_{2}^{\pi\pi})-2\sqrt{2}uw(Y_{0}^{\pi\pi}+8Y_{2}^{\pi\pi})+w^{2}(5Y_{0}^{\pi\pi}-8Y_{2}^{\pi\pi})\right)\bigg{]}$
$\displaystyle-$ $\displaystyle\frac{tj_{1}(krt)}{40\pi
k}\bigg{[}k^{2}Y_{0}^{\pi\pi}\Big{(}8L_{\pi}r\left(5\left(8t^{2}-1\right)u^{2}+\sqrt{2}\left(1-14t^{2}\right)uw+\left(17-70t^{2}\right)w^{2}\right)$
$\displaystyle-$ $\displaystyle
15\left(4t^{2}-1\right)w\left(2\sqrt{2}u-7w\right)\Big{)}+306L_{\pi}^{2}w^{2}(L_{\pi}rY_{2}^{\pi\pi}-4\bar{Y}_{1}^{\pi\pi})\bigg{]}$
$\displaystyle-$ $\displaystyle\frac{j_{2}(krt)}{672\pi
k^{4}r^{3}t^{2}}\bigg{[}180L_{\pi}^{3}r\left(2k^{2}r^{2}t^{2}-21\right)\left(4u^{2}(Y_{0}^{\pi\pi}+5Y_{2}^{\pi\pi})-2\sqrt{2}uw(Y_{0}^{\pi\pi}+8Y_{2}^{\pi\pi})+w^{2}(5Y_{0}^{\pi\pi}-8)\right)$
$\displaystyle+$ $\displaystyle
180L_{\pi}^{4}r^{2}Y_{0}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)\left(2k^{2}r^{2}t^{2}-21\right)$
$\displaystyle+$ $\displaystyle
L_{\pi}^{2}\bar{Y}_{1}^{\pi\pi}\Big{(}-k^{4}r^{4}t^{2}\left(336\left(24t^{2}-1\right)u^{2}-4\sqrt{2}\left(956t^{2}-29\right)uw+\left(2884t^{2}+2531\right)w^{2}\right)$
$\displaystyle+$ $\displaystyle
15k^{2}r^{2}w\left(14\sqrt{2}\left(1-4t^{2}\right)u+\left(4924t^{2}+1001\right)w\right)-136080w^{2}\Big{)}$
$\displaystyle+$ $\displaystyle
144k^{2}L_{\pi}rt^{2}wY_{0}^{\pi\pi}\left(k^{2}r^{2}t^{2}\left(14\sqrt{2}u+141w\right)-1680w\right)+12k^{4}r^{2}\left(4t^{2}-1\right)t^{2}w^{2}Y_{0}^{\pi\pi}\left(74k^{2}r^{2}t^{2}-1407\right)\bigg{]}$
$\displaystyle-$ $\displaystyle\frac{L_{\pi}j_{3}(krt)}{20\pi
k^{3}r^{2}t}\bigg{[}2k^{2}r\Big{(}80L_{\pi}^{2}r^{2}t^{2}u^{2}Y_{2}^{\pi\pi}-20\sqrt{2}L_{\pi}rt^{2}uw(2L_{\pi}rY_{2}^{\pi\pi}+3\bar{Y}_{1}^{\pi\pi})$
$\displaystyle+$ $\displaystyle
w^{2}\left(L_{\pi}rt^{2}(169L_{\pi}rY_{2}^{\pi\pi}-606\bar{Y}_{1}^{\pi\pi})+1215\left(4t^{2}-1\right)Y_{0}^{\pi\pi}\right)\Big{)}$
$\displaystyle+$ $\displaystyle
6k^{4}r^{3}t^{2}wY_{0}^{\pi\pi}\left(\sqrt{2}\left(4t^{2}-1\right)u+18\left(1-5t^{2}\right)w\right)-5355L_{\pi}w^{2}(L_{\pi}rY_{2}^{\pi\pi}-4\bar{Y}_{1}^{\pi\pi})\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{j_{4}(krt)}{2464\pi
k^{2}rt^{2}}\bigg{[}L_{\pi}^{2}w\bar{Y}_{1}^{\pi\pi}\left(k^{2}r^{2}t^{2}\left(242\sqrt{2}\left(4t^{2}-1\right)u+\left(8347-46060t^{2}\right)w\right)+99\left(10804t^{2}-2625\right)w\right)$
$\displaystyle+$ $\displaystyle
792k^{2}L_{\pi}r\left(4t^{2}+7\right)t^{2}w^{2}Y_{0}^{\pi\pi}+132k^{4}r^{2}\left(4t^{2}-1\right)t^{4}w^{2}Y_{0}^{\pi\pi}+836L_{\pi}^{4}r^{2}t^{2}Y_{0}^{\pi\pi}\left(4u^{2}-2\sqrt{2}uw+5w^{2}\right)$
$\displaystyle+$ $\displaystyle
836L_{\pi}^{3}rt^{2}\left(4u^{2}(Y_{0}^{\pi\pi}+5Y_{2}^{\pi\pi})-2\sqrt{2}uw(Y_{0}^{\pi\pi}+8Y_{2}^{\pi\pi})+w^{2}(5Y_{0}^{\pi\pi}-8Y_{2}^{\pi\pi})\right)\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{3L_{\pi}w^{2}j_{5}(krt)}{8\pi
kt}\bigg{[}2k^{2}r\left(4t^{2}-1\right)t^{2}Y_{0}^{\pi\pi}+L_{\pi}\left(2L_{\pi}rt^{2}Y_{2}^{\pi\pi}+\left(4t^{2}-3\right)\bar{Y}_{1}^{\pi\pi}\right)$
$\displaystyle-$
$\displaystyle\frac{171L_{\pi}^{2}r\left(4t^{2}-1\right)w^{2}\bar{Y}_{1}^{\pi\pi}j_{6}(krt)}{352\pi}\bigg{]}\Bigg{\\}}$
where $L_{\pi}=[m_{\pi}^{2}+(\frac{1}{4}-t^{2})\vec{k}^{2}]^{1/2}$, and
$Y_{0}^{\pi\pi}$, $\bar{Y}_{1}^{\pi\pi}$ and $Y_{2}^{\pi\pi}$ are defined as
$\displaystyle Y_{0}^{\pi\pi}$ $\displaystyle=$ $\displaystyle
e^{-L_{\pi}r}/(L_{\pi}r)$ $\displaystyle\bar{Y}_{1}^{\pi\pi}$ $\displaystyle=$
$\displaystyle(1+1/(L_{\pi}r))Y_{0}^{\pi\pi}$ $\displaystyle Y_{2}^{\pi\pi}$
$\displaystyle=$
$\displaystyle(3/(L_{\pi}r)^{2}+3/(L_{\pi}r)+1)Y_{0}^{\pi\pi}$ (37g)

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# Thermal Fluctuations Effects on Reissner-Nordström-AdS Black Hole

Özgür Ökcü<EMAIL_ADDRESS>Ekrem Aydıner
<EMAIL_ADDRESS>Department of Physics, Faculty of Science,
İstanbul University, İstanbul, 34134, Turkey

###### Abstract

In this paper, we study the effects of thermal fluctuations for Reissner-
Nordström-AdS (RN-AdS) black hole. We obtain the corrected thermodynamic
quantities such as entropy, temperature, equation of state and heat capacities
in the presence of thermal fluctuations. We also investigate the phase
transition of RN-AdS black hole for thermal fluctuations. Finally we compute
the critical exponents for small thermal fluctuations. We show that critical
exponents are the same as critical exponents without thermal fluctuations.

###### pacs:

04.70.-s, 05.70.Ce, 04.60.-m

##  I Introduction

Black holes behave as thermodynamic objects with a Bekenstein-Hawking entropy
and a Hawking temperature Beken1972 ; Beken1973 ; Bard1973 ; Hawk1974 ;
Beken1974 ; Hawk1975 ; Hawk1983 , and set deep connection between the laws of
classical general relativity, thermodynamics, and quantum mechanics due to the
presence of horizons Beken1972 ; Beken1973 ; Bard1973 ; Hawk1974 ; Beken1974 ;
Hawk1975 ; Hawk1983 ; Town1997 ; Wald2001 ; Page2005 . It is suggested that
black holes have very massive entropy in the universe, i.e., their entropies
are greater than any other object of the same volume Hawk1974 ; Hawk1975 .
This entropy is defined by $S=A/4$. However, it is found that quantum
fluctuations on the planck scale may change the entropy of the black holes
with time. These quantum fluctuations become important when the black hole
shrinks its size and therefore, entropy of the black holes need to get
corrected. In order to get corrections on the black hole entropy different
approaches such as non-perturbative quantum general relativity, the Cardy
formula, the exact partition function, matter fields in backgrounds of a black
hole, Rademacher expansion of the partition function and string theoretical
effects have been used Asht1991 ; Mann1998 ; Solo1998 ; Med1999 ; Carlip2000 ;
Govin2001 ; Birm2001 ; Med2001 ; Lowe2010 ; Jing2000 ; Sen2011 ; Sen2013 . On
the other hand, recently the corrections to the black hole entropy can be
obtained through the the generalized uncertainty principle Ali2012 ; Gan2014 ;
Faizal2015 .

Quantum fluctuations may lead to thermal fluctuations in the black hole
thermodynamics. Based on this assumption, in Ref. Das2002  shown that thermal
fluctuations can also contribute to entropy of the black hole. They suggested
that the logarithmic corrections to Bekenstein-Hawking entropy can be
interpreted as corrections due to the thermal fluctuations of the black hole
around its equilibrium configuration. Recently, the effects of thermal
fluctuations were studied for different kind of black objects such as charged
AdS black hole Pour2015 , charged hairy BTZ black hole Sade2014 , black Saturn
Faizal2015a ; Pour2016 , modified Hayward black hole Pour2016a , dyonic
charged AdS black hole Sade2016  and Kerr-AdS black hole Pour2016b .

It is shown that the RN-AdS black holes have a van der Waals like first order
small-large black hole phase transition ending in a critical point Cham1999 ;
Cham1999a . Moreover the phase transition in RN-AdS black hole is remarkable
coincidence with van der Waals liquid-gas phase transition, when the
cosmological constant and its conjugate quantities are considered as
thermodynamical variables Kub2012 . The cosmological constant corresponds to
pressure

$P=-\frac{\Lambda}{8\pi}=\frac{3}{8\pi}\frac{1}{l^{2}}$ (1)

and its conjugate quantity corresponds to thermodynamic volume,
$V=\frac{\partial M}{\partial P}$. For the RN-AdS black hole, it is given by

$V=\frac{4}{3}\pi r_{h}^{3}\,.$ (2)

Variable cosmological constant notion has led to a rich structure of phase
transitions and critical phenomena. Indeed, in the recent years,
thermodynamics and phase transition properties of the black holes in AdS space
have been discussed in many studies Kub2012 ; Niu2012 ; Tsai2012 ; Bane2011 ;
Bane2012 ; Cald2000 ; Kast2009 ; Dolan2011 ; Belhaj2013 ; Spa2013 ; Cai2013 ;
Moa2013 ; Dutta2013 ; Alta2014 ; Dolan2011a ; okcu2017 ; John2014 ; Maj2016 .

In this article, we investigate the thermodynamics and phase transition of RN-
AdS black hole with thermal fluctuations following the method in the Refs.
Das2002 ; Kub2012 . As mentioned before, authors studied effects of thermal
fluctuations for RN-AdS black holes Pour2015  and our results differ from Ref.
Pour2015  which follows different approaches and assumptions.

The paper is organized as follows. In section 2, we briefly review the
logarithmic correction to black hole entropy due to thermal fluctuations
Das2002 . In section 3, we obtain corrected thermodynamic quantities and
investigate phase transition of the RN-AdS black hole in the presence of
thermal fluctuations. We also compute critical exponents of the phase
transition with small thermal fluctuations. Finally, we discuss our results in
section 4. (We use the units $G_{N}=\hbar=k_{B}=c=1$.)

##  II Logaritmic Corrections to Black Hole Entropy

In this section, we will briefly review the thermal fluctuations correction to
black hole entropy Das2002 ; Pour2015 . First, we begin to consider partition
function of a canonical ensemble

$Z(\beta)=\intop_{0}^{\infty}\rho(E)e^{-\beta}dE\,,$ (3)

where $\beta$ denotes the inverse of the temperature, $\beta=\frac{1}{T}$. One
can obtain density of states from Eq. (3),

$\rho(E)=\frac{1}{2\pi i}\intop_{c-i\infty}^{c+i\infty}Z(\beta)e^{\beta
E}d\beta=\frac{1}{2\pi i}\intop_{c-i\infty}^{c+i\infty}e^{S(\beta)}d\beta\,,$
(4)

where

$S(\beta)=lnZ(\beta)+\beta E\,.$ (5)

Expanding $S(\beta)$ around the equilibrium temperature $\beta_{0}$ , one can
obtain

$S=S_{0}+\frac{1}{2}(\beta-\beta_{0})^{2}\left(\frac{\partial^{2}S(\beta)}{\partial\beta^{2}}\right)_{\beta=\beta_{0}}+...\,,$
(6)

where $S_{0}=S(\beta_{0})$. From Eqs. (4) and (6), density of state is given
by

$\rho(E)=\frac{e^{S_{0}}}{2\pi
i}\intop_{c-i\infty}^{c+i\infty}e^{\frac{1}{2}(\beta-\beta_{0})^{2}S_{0}^{{}^{\prime\prime}}}d\beta\,$
(7)

where
$S_{0}^{{}^{\prime\prime}}=\left(\frac{\partial^{2}S(\beta)}{\partial\beta^{2}}\right)_{\beta_{0}}$.
Substituting $c=\beta_{0}$ and $(\beta-\beta_{0})=iz$ in Eq. (7), density of
state is given by

$\rho(E)=\frac{e^{S_{0}}}{\sqrt{2\pi S_{0}^{{}^{\prime\prime}}}}$ (8)

and we can write

$S=S_{0}-\frac{1}{2}lnS_{0}^{{}^{\prime\prime}}+...\,.$ (9)

This formula is valid for all thermodynamic system considered as canonical
ensemble and it is also valid for black holes. $S_{0}$ denotes the black hole
entropy and thus we can replace $T\rightarrow T_{H}$. Now, we can determine
$S_{0}^{{}^{\prime\prime}}$. We consider the entropy, which is suggested in
Ref. Das2002  and based on Ref. Carlip2000 , has the form,

$S=x\beta^{m}+y\beta^{-n}\,,$ (10)

where $x,y,m,n>0$. This entropy has an extremum at

$\beta_{0}=\left(\frac{ny}{mx}\right)^{\frac{1}{m+n}}\,.$ (11)

From Eq.(10) and Eq.(11), one can obtain

$S(\beta_{0})=x\left(\frac{ny}{mx}\right)^{\frac{m}{m+n}}+y\left(\frac{mx}{ny}\right)^{\frac{n}{m+n}}\,,$
(12)
$S^{{}^{\prime\prime}}(\beta_{0})=m(m-1)x\left(\frac{ny}{mx}\right)^{\frac{m-2}{m+n}}+n(n+1)y\left(\frac{mx}{ny}\right)^{\frac{n+2}{m+n}}\,.$
(13)

Using Eqs. (12) and (13) in Eq.(6), one can give the entropy as

$S(\beta)=A(x^{n}y^{m})^{\frac{1}{m+n}}+\frac{1}{2}B(x^{n+2}y^{m-2})^{\frac{1}{m+n}}(\beta-\beta_{0})^{2}+...\,$
(14)

where

$A=\left(\frac{n}{m}\right)^{\frac{m}{m+n}}+\left(\frac{m}{n}\right)^{\frac{n}{m+n}},\quad
B=(m+n)m^{\frac{n+2}{m+n}}n^{\frac{m-2}{m+n}}$ (15)

are the constants. Comparing with Eq.(6), we can find

$S(\beta_{0})=A(x^{n}y^{m})^{\frac{1}{m+n}},\quad
S^{{}^{\prime\prime}}(\beta_{0})=B(x^{n+2}y^{m-2})^{\frac{1}{m+n}}\,.$ (16)

We can obtain $x$ and $y$ in terms of $S_{0}$ and $S^{{}^{\prime\prime}}_{0}$
from Eq.(16)

$x=\frac{A^{\frac{m-2}{2}}}{B^{\frac{m}{2}}}(S_{0}^{{}^{\prime\prime}})^{\frac{m}{2}}S_{0}^{-\frac{m-2}{2}},\quad
y=\frac{A^{-\frac{n+2}{2}}}{B^{-\frac{n}{2}}}S_{0}^{\frac{n+2}{2}}(S_{0}^{{}^{\prime\prime}})^{-\frac{n}{2}}\,.$
(17)

Substituting $x$ and $y$ in Eq.(11), we can obtain $\beta_{0}$ in terms of
$S_{0}$, $S_{0}^{{}^{\prime\prime}}$

$\beta_{0}=\left(\frac{n}{m}\right)^{\frac{1}{m+n}}\sqrt{\frac{B}{A}\frac{S_{0}}{S_{0}^{{}^{\prime\prime}}}}\,$
(18)

and $S_{0}^{{}^{\prime\prime}}$ is given by

$S_{0}^{{}^{\prime\prime}}=\left[\left(\frac{B}{A}\right)\left(\frac{n}{m}\right)^{\frac{2}{m+n}}\right]S_{0}\beta_{0}^{-2}$
(19)

The factor in the square brackets can be negligible, so we can write

$S_{0}^{{}^{\prime\prime}}=S_{0}\beta_{0}^{-2}\,.$ (20)

Substituting Eq. (20) in Eq. (9), we can obtain

$S=S_{0}-\frac{1}{2}lnS_{0}T_{H}^{2}\,.$ (21)

This is the generic correction formula for black hole entropy. We will
discuss, in the next section, the corrections to thermodynamic quantities due
to thermal fluctuations.

##  III Reissner-Nordstörm-AdS Black Hole

RN-AdS black hole in four dimensional space is given by the metric

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}(\theta)d\phi^{2}\,,$
(22)

with

$f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\frac{r^{2}}{l^{2}}$ (23)

where $l$, $M$ and $Q$ are the AdS radius, mass and charge of the black hole,
respectively. Black hole event horizon $r_{h}$ is given by as a largest root
of $f(r_{h})=0$. The mass of black hole can be obtained in terms of $r_{h}$,
$Q$ and $l$ from Eq. (23),

$M=\frac{r_{h}}{2}\left(1+\frac{Q^{2}}{r_{h}^{2}}+\frac{r_{h}^{2}}{l^{2}}\right)\,.$
(24)

Hawking temperature is given by

$T_{H}=\left(\frac{f^{\prime}(r_{h})}{4\pi}\right)_{r=r_{h}}=\frac{1}{4\pi
r_{h}}\left(1+\frac{3r_{h}^{2}}{l^{2}}-\frac{Q^{2}}{r_{h}^{2}}\right)$ (25)

and expression for the entropy can be written

$S_{0}=\frac{A}{4}=\pi r_{h}^{2},\quad A=4\pi r_{h}^{2}\,.$ (26)

In Ref. Pour2015  to track corrections coming from the thermal fluctuations,
adding $a$ parameter into Eq. (21), entropy is generalized as

$S=S_{0}-\frac{a}{2}lnS_{0}T_{H}^{2}\,.$ (27)

In the case of $a=1$, it is assumed that thermal fluctuations contribution is
maximum, however, by setting $a=0$, we will recover entropy expression in Eq.
(26) without any corrections. Indeed, if we can use Eqs. (25) and (26) into
Eq. (27), we can obtain the following expression,

$S=\pi
r_{h}^{2}-a\ln\left(\frac{l^{2}(r_{h}^{2}-Q^{2})+3r_{h}^{4}}{4\sqrt{\pi}l^{2}r_{h}^{2}}\right)$
(28)

which refers to new entropy definition of black hole due to thermal
fluctuations. For $a=0$, entropy in Eq. (26) is recovered. Corrected
temperature can be obtained from $T=\frac{\partial M}{\partial S}$

$T=\frac{(3r_{h}^{4}+l^{2}(r_{h}^{2}-Q^{2}))^{2}}{4l^{2}r_{h}(3\pi
r_{h}^{6}+(\pi l^{2}-3a)r_{h}^{4}-\pi l^{2}Q^{2}r_{h}^{2}-l^{2}Q^{2}a)}\,.$
(29)

Specific heat can be obtained from $C=T\frac{\partial S}{\partial T}$ and thus
specific heats at constant volume and pressure can be given by

$C_{V}=a\frac{3\pi r_{h}^{6}+(\pi l^{2}-3a)r_{h}^{4}-\pi
l^{2}Q^{2}r_{h}^{2}-al^{2}Q^{2}}{-3\pi r_{h}^{6}+(3a-\pi
l^{2})r_{h}^{4}+l^{2}(\pi Q^{2}-a)r_{h}^{2}+3al^{2}Q^{2}}$ (30)

and

$C_{p}=2\frac{9\pi^{2}r_{h}^{12}+6\pi^{2}l^{2}r_{h}^{10}+\pi^{2}l^{2}(l^{2}-6Q^{2})r_{h}^{8}-2\pi^{2}l^{4}Q^{2}r_{h}^{6}+\pi^{2}l^{4}Q^{4}r_{h}^{4}+\chi_{1}(a)}{9\pi
r_{h}^{10}+\pi l^{2}(6Q^{2}-l^{2})r_{h}^{6}+4\pi l^{4}Q^{2}r_{h}^{4}-3\pi
l^{4}Q^{4}r_{h}^{2}+\chi_{2}(a)}$ (31)

where $\chi_{1}(a)=-18a\pi r_{h}^{10}+3a(3a-2\pi
l^{2})r_{h}^{8}+2al^{2}Q^{2}(3a-\pi l^{2}Q^{2})r_{h}^{4}+2a\pi
l^{4}Q^{4}r_{h}^{2}+a^{2}l^{4}Q^{4}$ and
$\chi_{2}(a)=3a(-9r_{h}^{8}+l^{2}r_{h}^{6}-12Q^{2}l^{2}r_{h}^{4}-l^{4}Q^{2}r_{h}^{2})$.

Local thermodynamic stability of canonical ensemble is given by $C_{P}>0$
Alta2014 . If $C_{P}<0$, black hole is thermodynamically unstable.

Figure 1: The $P-r_{h}$ diagram. The temperatures of the isotherms decreases
from top to bottom and correspond to $1.2T_{c}$, $T_{c}$, $0.8T_{c}$. We fix
$a=10^{-9}$ and $Q=1$.

Figure 2: The $P-r_{h}$ diagram. The temperatures of the isotherms decreases
from top to bottom and correspond to $1.2T_{c}$, $T_{c}$, $0.8T_{c}$. We fix
$a=1$ and $Q=1$.

Figure 3: The $P-r_{h}$ diagram. The temperatures of the isotherms decreases
from top to bottom and correspond to $1.2T_{c}$, $T_{c}$, $0.8T_{c}$. We fix
$a=0.5$ and $Q=1$.

###  III.1 Equation of State and Phase Transition

In this section, we analyze the $P-V$ criticality of RN-AdS black holes in the
presence of thermal fluctuations. We use the critical point investigation
method which is defined in Ref. Kub2012 . From Eqs. (1) and (29), one can
obtain the equation of state

$P=\frac{Q^{2}}{8\pi r_{h}^{4}}-\frac{1}{8\pi
r_{h}^{2}}+\frac{T}{4r_{h}}-\frac{aT}{4\pi
r_{h}^{3}}\pm\frac{\sqrt{\pi^{2}T^{2}r_{h}^{14}-W(a)}}{4\pi r_{h}^{8}}$ (32)

where $W(a)=aTr_{h}^{9}(2\pi Tr_{h}^{3}-r_{h}^{2}-aTr_{h}+2Q^{2})$. The
positive sign has been taken before the square root so we can obtain the
equation of state in the $a\rightarrow 0$ limit. The critical points are
obtained from

$\frac{\partial P}{\partial r_{h}}=0,\quad\frac{\partial^{2}P}{\partial
r_{h}^{2}}=0\,.$ (33)

It is possible to obtain critical points for the small values of $a$. For
small thermal fluctuations, equation of state can be expanded as

$P=\frac{Q^{2}}{8\pi r_{h}^{4}}-\frac{1}{8\pi
r_{h}^{2}}+\frac{T}{2r_{h}}+a\left(\frac{1}{8\pi^{2}r_{h}^{4}}-\frac{T}{2\pi
r_{h}^{3}}-\frac{Q^{2}}{4\pi^{2}r_{h}^{6}}\right)+\mathcal{O}(a^{2})\,.$ (34)

Specific volume $v$ should be identified with

$v=2r_{h}$ (35)

for the small thermal fluctuations. Now one can obtain the critical points for
small thermal fluctuations

$v_{c}=2\sqrt{6}Q-\frac{\sqrt{6}}{2\pi Q}a+\mathcal{O}(a^{2})\,,$ (36)
$T_{c}=\frac{\sqrt{6}}{18\pi
Q}+\frac{\sqrt{6}}{72\pi^{2}Q^{3}}a+\mathcal{O}(a^{2})\,,$ (37)
$P_{c}=\frac{1}{96\pi Q^{2}}+\frac{a}{216\pi^{2}Q^{4}}+\mathcal{O}(a^{2})\,.$
(38)

Universal number can be calculated from Eqs. (36), (37) and (38)

$\frac{P_{c}T_{c}}{v_{c}}=\frac{3}{8}-\frac{a}{48\pi
Q^{2}}+\mathcal{O}(a^{2})\,.$ (39)

As it can be seen from Eqs. (36), (37), (38) and (39) for $a=0$, the results
in Ref. Kub2012  are obtained i.e, when $a=0$, the critical points are
$v_{c}=2\sqrt{6}Q$, $T_{c}=\frac{\sqrt{6}}{18\pi Q}$, $P_{c}=\frac{1}{96\pi
Q^{2}}$ and $\frac{P_{c}T_{c}}{v_{c}}=\frac{3}{8}$. Critical points change due
to thermal fluctuations. For example critical specific volume decreases while
the critical temperature and pressure increase. Universal number of phase
transition depends on $Q$ and $a$.

In order to investigate the phase transition behavior of RN-AdS black hole,
$P-r_{h}$ diagram has been plotted in Fig. 1 for small values of thermal
fluctuations. As it can be seen from the figure that under the critical
temperature $P-r_{h}$ diagram has characteristic behavior of van der Waals
phase transition. It is also important to investigate the phase transition
behavior for large values of thermal fluctuations since effects of thermal
fluctuations are important for small size. When thermal fluctuations are
increased, the phase transition deviates from van der Waals behavior. In Figs.
2 and 3, we have plotted $P-r_{h}$ diagrams for thermal fluctuations. It can
easily be seen from figures that thermal fluctuations affect the phase
transition. Moreover, in contrast to Fig. 1, small black hole branch is
unstable and pressure is imaginary under a certain event horizon radius. It
may be interpreted as black holes do not exist under a certain value of event
horizon. This interpretation is also useful to avoid the singularities and
negative regions of entropy and temperature. For example, $1.2T_{c}$ isotherm
in Fig. 3 is imaginary for $r_{h}<1.03573$. Moreover, event horizons of
entropy and temperature that become singular, are less than $r_{h}<1.03573$.
It may imply that black hole never approaches ill-defined regions.

###  III.2 Critical Exponents

The thermodynamic behavior of a system near the phase transition point has
been classified by using critical exponents. It is supposed that the critical
exponents to be universal and are the independent of the details of the
interaction. On the other hand, different physical systems may share the same
critical exponents, which indicates that they are in the same universal class.
In this section, we discuss critical exponents of the phase transition at near
the critical point for the RN-AdS black hole. The critical exponents are given
as follows:

The exponent $\alpha$ determines the behavior of specific heat at constant
volume

$C_{V}=T\left(\frac{\partial S}{\partial T}\right)\propto|t|^{-\alpha}\,.$
(40)

The exponent $\beta$ determines the behavior of the order parameter on the
given isotherm

$\eta=V_{l}-V_{s}\propto|t|^{\beta}\,.$ (41)

The exponent $\gamma$ determines the behavior of the isothermal
compressibility $\kappa_{T}$

$\kappa_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial
P}\right)\propto|t|^{-\gamma}\,.$ (42)

The exponent $\delta$ determines the following behavior on critical isotherm
$T=T_{c}$

$|P-P_{c}|\propto|V-V_{c}|^{\delta}\,.$ (43)

Recently, a more general method is suggested to obtain critical exponents in
Ref. Maj2016 . However, in our case, it is convenient to use method in Ref.
Kub2012 . Now following method in Ref. Kub2012 , we can obtain critical
exponents of phase transition for RN-AdS black hole. We cannot obtain the
exact critical point due to the complexity of equation of state in Eq. (32) so
we cannot obtain the law of corresponding states from Eq. (32) the critical
exponent may differ from Ref. Kub2012 . On the other hand, we can compute the
critical exponents for small thermal fluctuations. In the presence of small
thermal fluctuations, we can obtain specific heat at constant volume,
$C_{V}=-a+\mathcal{O}(a^{2})$ and this is independent of $T$ so exponent
$\alpha=0$. Defining

$p=\frac{P}{P_{c}},\quad\tau=\frac{T}{T_{c}},\quad\nu=\frac{V}{V_{c}}\,,$ (44)

we can obtain the so called law of corresponding states with small thermal
fluctuations. Expanding around the critical point

$t=\tau-1,\quad\omega=\nu-1$ (45)

we find equation of state

$p=1+\left(\frac{8}{3}-\frac{8a}{27\pi
Q^{2}}\right)t+\left(-\frac{8}{9}+\frac{32a}{81\pi
Q^{2}}\right)t\omega+\left(-\frac{4}{81}+\frac{16a}{729\pi
Q^{2}}\right)\omega^{3}+\mathcal{O}(t\omega^{2},\omega^{4})\,.$ (46)

Differentiating Eq.(46) for a fixed $t<0$ we obtain

$dP=-\frac{4}{27}P_{c}\left[\left(6-\frac{8a}{3\pi
Q^{2}}\right)t+\left(1-\frac{4a}{9\pi
Q^{2}}\right)\omega^{2}\right]d\omega\,.$ (47)

Employing Maxwell’s equal area law, one can get the following two equations:

$\displaystyle p=1+\left(\frac{8}{3}-\frac{8a}{27\pi
Q^{2}}\right)t+(-\frac{8}{9}+\frac{32a}{81\pi
Q^{2}})t\omega_{l}+(-\frac{4}{81}+\frac{16a}{729\pi Q^{2}})\omega_{l}^{3}$
$\displaystyle=1+\left(\frac{8}{3}-\frac{8a}{27\pi
Q^{2}}\right)t+(-\frac{8}{9}+\frac{32a}{81\pi
Q^{2}})t\omega_{s}+(-\frac{4}{81}+\frac{16a}{729\pi Q^{2}})\omega_{s}^{3}\,,$
$\displaystyle 0=\oint\omega
dP=\intop_{\omega_{l}}^{\omega_{s}}\omega\left[\left(6-\frac{8a}{3\pi
Q^{2}}\right)t+\left(1-\frac{4a}{9\pi
Q^{2}}\right)\omega^{2}\right]d\omega\,.$ (48)

where $\omega_{s}$ and $\omega_{l}$ denotes volume of small and large black
holes. From these equation, one can obtain

$\omega_{l}=-\omega_{s}=3\sqrt{-2t}\,.$ (49)

So we find the exponent $\beta$

$\eta=V_{c}(\omega_{l}-\omega_{s})=2V_{c}\sqrt{-2t}\quad\Rightarrow\quad\beta=\frac{1}{2}\,.$
(50)

We compute the isothermal compressibility exponent $\gamma$

$\kappa_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial
P}\right)_{T}\propto\frac{81\pi Q^{2}}{8P_{c}(9\pi
Q^{2}-4a)t}\quad\Rightarrow\quad\gamma=1\,.$ (51)

Finally we obtain the exponent $\delta$ for $t=0$

$p-1=\left(-\frac{4}{81}+\frac{16a}{729\pi
Q^{2}}\right)\omega^{3}\quad\Rightarrow\quad\delta=3\,.$ (52)

We computed the critical exponents for small thermal fluctuations. All
critical exponents that were computed in the presence of small thermal
fluctuations coincide critical exponents of van der Waals fluid and RN-AdS
black hole without thermal fluctuations Kub2012 .

##  IV Conclusion

In this paper, we studied effects of thermal fluctuations for RN-AdS black
hole. We obtained some corrected thermodynamic quantities and investigated
$P-V$ criticality in the presence of thermal fluctuations. We obtained
critical points for small thermal fluctuations and numerically studied phase
transition for large values of thermal fluctuations. Thermal fluctuations
effects are remarkable for small size black hole and also affect the phase
transition. Black hole may not exist under a certain value of event horizon.
This minimum event horizon condition provides us to avoid ill-defined regions
of temperature and entropy. Finally we obtained the critical exponent for
small thermal fluctuations. We showed that critical exponents are the same as
critical exponent without thermal fluctuations.

##  V Conflict of Interest

The authors declare that there is no conflict of interest regarding the
publication of this paper.

###### Acknowledgements.

This work was supported by Scientific Research Projects Coordination Unit of
Istanbul University. Project numbers is FYL-2016-2061.

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