aainstitutetext: Institut für Theoretische Physik II, Ruhr-Universität Bochum,
D-44780 Bochum, Germanybbinstitutetext: Tbilisi State University, 0186
Tbilisi, Georgiaccinstitutetext: Helmholtz Institut für Strahlen- und
Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, D-53115
Bonn, Germanyddinstitutetext: Institute for Advanced Simulation, Institut für
Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich,
D-52425 Jülich, Germany

# Gravitational $p\to\Delta^{+}$ transition form factors in chiral
perturbation theory

H. Alharazin a B.-D. Sun a E. Epelbaum a,b J. Gegelia c,d,b U.-G. Meißner

###### Abstract

The gravitational form factors of the transition from the proton to the
$\Delta^{+}$ resonance are calculated to leading one-loop order using a
manifestly Lorentz-invariant formulation of chiral perturbation theory. We
take into account the leading electromagnetic and strong isospin-violating
effects. The loop contributions to the transition form factors are found to be
free of power-counting violating pieces, which is consistent with the absence
of tree-level diagrams at the considered order. In this sense, our results can
be regarded as predictions of chiral perturbation theory.

††arxiv: 2312.05193

##  1 Introduction

The linear response of the effective action to the change of the space-time
metric specifies mechanical properties of particles. In particular, static
characteristics, like the mass, spin and the $D$-term correspond to the hadron
gravitational form factors (GFFs) at zero momentum transfer Kobzarev:1962wt ;
Pagels:1966zza . In recent years, GFFs have attracted increasing attention for
characterizing properties of hadrons with different spins due to their
connection to generalized parton distributions (GPDs). Parameterizations of
the energy-momentum tensor (EMT) matrix elements in terms of the GFFs have
been considered for spin-$0$ Pagels:1966zza , spin-1 Holstein:2006ud ;
Cosyn:2019aio ; Polyakov:2019lbq , and for arbitrary-spin hadrons
Cotogno:2019vjb . Mechanical properties, energy and spin densities as well as
spatial distributions of the pressure and shear forces have been introduced
for spin-$0$ and spin-$1/2$ hadrons in Ref. Polyakov:2002yz , and generalized
to higher-spin systems in Refs. Polyakov:2019lbq ; Panteleeva:2020ejw ;
Kim:2020lrs .

The nucleon GFFs can be extracted from experimental measurements of exclusive
processes like deeply virtual Compton scattering (DVCS) Ji:1996ek ;
Radyushkin:1997ki  and hard exclusive meson production Collins:1996fb . The
connection to GFFs can be seen in the QCD description of these processes,
where the symmetric EMT appears naturally in the operator product expansion
Ji:1996ek . The first results of measurements of the $D$-term in hard QCD
processes for the nucleon and the pion can be found in Refs. Kumericki:2015lhb
; Nature ; Kumano:2017lhr ; Kumericki:2019ddg . Recently, the mechanical
radius of the proton has been determined from experimental data on DVCS cross
sections and polarized electron beam spin asymmetries Burkert:2023atx . The
GFFs have also been studied in lattice QCD, see, e.g., Refs. Shanahan:2018nnv
; Shanahan:2018pib ; Alexandrou:2013joa ; Bratt:2010jn ; Hagler:2007xi ;
Hackett:2023rif  and references therein.

While the electromagnetic $p\to\Delta^{+}$ transition has been extensively
studied over the past two decades on both the theoretical and experimental
sides, see, e.g., Refs. Hilt:2017iup ; Blomberg:2015zma ; CLAS:2009ces ;
Tiator:2011pw ; Tiator:2016btt , the gravitational $p\to\Delta^{+}$ transition
form factors (GTFFs) gained attention only since a few years Kim:2022bwn . The
GTFFs can be accessed experimentally through their connection to the
transition GPDs Frankfurt:1998jq ; Frankfurt:1999xe , obtained by expanding
the non-local QCD operators with various quantum numbers. Non-perturbative
properties of the nucleon-$\Delta$ transition GPDs have been studied, e.g., by
applying the approach of large $N_{c}$ limit of QCD, as discussed in Sec. 2.7
of Ref. Goeke:2001tz . In Ref. Semenov-Tian-Shansky:2023bsy , the transition
GPDs have been connected with the DVCS amplitude within the process $e^{-}N\to
e^{-}\gamma\pi N,$ while in Ref. Kroll:2022roq  these quantities have been
studied using exclusive electroproduction of $\pi^{-}\Delta^{++}$.

In Ref. Kim:2022bwn , the matrix element of the symmetric EMT corresponding to
the $p\to\Delta^{+}$ transition has been studied for the first time, where a
parametrization for the transitions $\frac{1}{2}^{\pm}\to\frac{3}{2}^{\pm}$
and $\frac{1}{2}^{\pm}\to\frac{3}{2}^{\mp}$ has been suggested in terms of
five conserved and four non-conserved GTFFs. The first calculations of the
GTFFs of the $N\to\Delta$ transition were done in Ref. Ozdem:2022zig  using
the QCD light-cone sum rules. The interpretation and understanding of the
GTFFs have generated much interest recently. In particular, the concept of QCD
angular momentum (AM) Leader:2013jra ; Lorce:2017wkb ; Granados:2019zjw  has
been extended to $N\to\Delta$ transitions in Ref. Kim:2023xvw . These
quantities were calculated in the $1/N_{c}$ expansion, and their connection to
the transition GPDs of the hard exclusive electroproduction processes was
discussed. Properties of the AM of various transitions were further explored
in Ref. Kim:2023yhp , where their decomposition into the orbital AM and the
intrinsic spin components was studied.

For systematic studies of low-energy hadronic processes involving the $\Delta$
resonances and induced by gravity one may rely on the effective chiral
Lagrangian for the nucleons, pions, photons and delta resonances in curved
spacetime. Effective Lagrangian of pions in curved spacetime has been derived
in Ref. Donoghue:1991qv , and the GFFs of the pion are considered in Ref.
Kubis:1999db . The leading and subleading effective chiral Lagrangians for
nucleons, delta resonances and pions in curved spacetime, along with the
calculation of the leading one-loop contributions to the GFFs of the nucleons
and the $\Delta$ resonances can be found in Refs. Alharazin:2020yjv ;
Alharazin:2022wjj .

In this work we calculate the GTFFs of the $p\to\Delta^{+}$ transition in the
framework of manifestly Lorentz-invariant chiral perturbation theory (ChPT)
up-to-and-including the third order in the small-scale expansion
Hemmert:1996xg . As gravity conserves isospin, such kind of processes are
possible only if the isospin symmetry is broken, i.e. if $m_{u}\neq m_{d}$
and/or if the electromagnetic interaction is taken into account. We include
both effects at the corresponding leading orders to calculate the one-loop
contributions to the GTFFs.

Our paper is organized as follows: In section 2, we specify the relevant terms
of the effective Lagrangian of the nucleons, pions, photons and delta
resonances in curved spacetime. We calculate the GTFFs of the $p\to\Delta^{+}$
transition in section 3. The results of our calculations are summarized in
section 4. In the appendices, we list the isospin symmetry breaking terms in
the action and the expression for the parts of the EMT, which are relevant for
our study.

##  2 Effective Lagrangian in curved spacetime and the energy-momentum tensor

The action corresponding to the leading-order effective Lagrangian for
nucleons, pions, photons and delta resonances, interacting with an external
gravitational field, can be easily obtained from the corresponding expressions
in flat spacetime Donoghue:1991qv ; Gasser:1987rb ; Fettes:2000gb ;
Hemmert:1997ye ; Hacker:2005fh . It has the following form:

$\displaystyle S_{\rm\gamma}^{(2)}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\,\biggl{\\{}\,-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{m_{\gamma}^{2}}{2}A_{\mu}A^{\mu}\biggl{\\}},$
(1) $\displaystyle S_{\rm\pi}^{(2)}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\,\left\\{\frac{F^{2}}{4}\,{\rm
Tr}(D_{\mu}U(D^{\mu}U)^{\dagger})+\frac{F^{2}}{4}\,{\rm Tr}(\chi
U^{\dagger}+U\chi^{\dagger})\right\\}\,,$ (2) $\displaystyle S_{\rm
N\pi}^{(1)}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\,\biggl{\\{}\,\bar{\Psi}\,i\gamma^{\mu}\overset{\leftrightarrow}{\nabla}_{\mu}\Psi-m\bar{\Psi}\Psi+\frac{g_{A}}{2}\,\bar{\Psi}\gamma^{\mu}\gamma_{5}u_{\mu}\Psi\biggr{\\}}\,,$
(3) $\displaystyle S_{\Delta\pi}^{(1)}$ $\displaystyle=$ $\displaystyle-\int
d^{4}x\sqrt{-g}\biggl{\\{}\bar{\Psi}^{i\mu}\,i\gamma^{\alpha}\overset{\leftrightarrow}{\nabla}_{\alpha}\Psi^{i}_{\mu}-m_{\Delta}\,\bar{\Psi}^{i}_{\mu}\Psi^{i\mu}-g^{\lambda\sigma}\left(\bar{\Psi}^{i}_{\mu}i\gamma^{\mu}{\overset{\leftrightarrow}{\nabla}_{\lambda}}\Psi^{i}_{\sigma}+\bar{\Psi}^{i}_{\lambda}i\gamma^{\mu}{\overset{\leftrightarrow}{\nabla}_{\sigma}}\Psi^{i}_{\mu}\right)$
(4)
$\displaystyle+i\bar{\Psi}^{i}_{\mu}\gamma^{\mu}\gamma^{\alpha}\gamma^{\nu}\overset{\leftrightarrow}{\nabla}_{\alpha}\Psi^{i}_{\nu}+m_{\Delta}\bar{\Psi}^{i}_{\mu}\gamma^{\mu}\gamma^{\nu}\Psi^{i}_{\nu}+\frac{g_{1}}{2}\,g^{\mu\nu}\bar{\Psi}^{i}_{\mu}u_{\alpha}\gamma^{\alpha}\gamma_{5}\Psi^{i}_{\nu}$
$\displaystyle+\frac{g_{2}}{2}\bar{\Psi}^{i}_{\mu}\left(u^{\mu}\gamma^{\nu}+u^{\nu}\gamma^{\mu}\right)\gamma_{5}\Psi^{i}_{\nu}+\frac{g_{3}}{2}\bar{\Psi}^{i}_{\mu}u_{\alpha}\gamma^{\mu}\gamma^{\alpha}\gamma_{5}\gamma^{\nu}\Psi^{i}_{\nu}\biggr{\\}}\,,$
$\displaystyle S_{\Delta\rm N\pi}^{(1,2)}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}~{}\Biggl{\\{}-g_{\pi
N\Delta}\bar{\Psi}\left(g^{\mu\nu}-\gamma^{\mu}\gamma^{\nu}\right)u_{\mu,i}\Psi_{\nu,i}$
(5)
$\displaystyle+d_{3}^{(2)}i\bar{\Psi}f_{+}^{i\mu\nu}\gamma_{5}\gamma_{\mu}\left(g_{\nu\lambda}-\left[z_{n}+\frac{1}{2}\right]\gamma_{\nu}\gamma_{\lambda}\right)\Psi^{i\lambda}+\text{H.c.}\Biggl{\\}}\,.$

The $\Delta$ resonances are represented by the Rarita-Schwinger fields
$\Psi^{\mu}_{i}$, which contain the isospin-$3/2$ projectors
$\xi_{ij}^{\frac{3}{2}}=\delta_{ij}-\tau_{i}\tau_{j}/3$, i.e. they satisfy the
condition $\Psi^{\mu}_{i}=\xi_{ij}^{\frac{3}{2}}\Psi^{\mu}_{j}$. Further,
$g^{\mu\nu}$ is the metric tensor field and $\gamma_{\mu}\equiv
e_{\mu}^{a}\gamma_{a}$, where $e_{\mu}^{a}$ denote the vielbein gravitational
fields. In the photon Lagrangian we included the mass term
${m_{\gamma}^{2}}\,A_{\mu}A^{\mu}/2$ to regularize infrared divergences, and
the limit $m_{\gamma}\to 0$ should be performed at the end. However, as it
turns out after the calculation, there are actually no IR divergences and this
term is thus of no relevance here. In Eqs. (4) and (5), $z_{n}$ is an off-
shell parameter, which we choose equal to zero in our calculations, and we
have set the point-transformation parameter $A=-1$ Tang:1996sq . The building
blocks of the effective Lagrangian are given as follows:

$\displaystyle\overset{\leftrightarrow}{\nabla}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\overset{\rightarrow}{\nabla}_{\mu}-\overset{\leftarrow}{\nabla}_{\mu})\,,$
$\displaystyle\overset{\rightarrow}{\nabla}_{\mu}\Psi^{i}_{\nu}$
$\displaystyle=$
$\displaystyle\nabla_{\mu}^{ij}\Psi^{j}_{\nu}=\left[\delta^{ij}\partial_{\mu}+\delta^{ij}\Gamma_{\mu}-i\delta^{ij}v_{\mu}^{(s)}-i\epsilon^{ijk}\text{Tr}\left(\tau^{k}\Gamma_{\mu}\right)+\frac{i}{2}\delta^{ij}\omega^{ab}_{\mu}\sigma_{ab}\right]\Psi^{j}_{\nu}-\Gamma^{\alpha}_{\mu\nu}\Psi^{i}_{\alpha},$
$\displaystyle\bar{\Psi}^{i}_{\nu}\overset{\leftarrow}{\nabla}_{\mu}$
$\displaystyle=$
$\displaystyle\nabla_{\mu}^{ij}\Psi^{j}_{\nu}=\bar{\Psi}^{j}_{\nu}\left[\delta^{ij}\partial_{\mu}-\delta^{ij}\Gamma_{\mu}+i\delta^{ij}v_{\mu}^{(s)}+i\epsilon^{ijk}\text{Tr}\left(\tau^{k}\Gamma_{\mu}\right)-\frac{i}{2}\delta^{ij}\omega^{ab}_{\mu}\sigma_{ab}\right]-\bar{\Psi}^{i}_{\alpha}\Gamma^{\alpha}_{\mu\nu},$
$\displaystyle\overset{\rightarrow}{\nabla}_{\mu}\Psi$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Psi+\frac{i}{2}\,\omega^{ab}_{\mu}\sigma_{ab}\Psi+\left(\Gamma_{\mu}-iv_{\mu}^{(s)}\right)\Psi\,,$
$\displaystyle\bar{\Psi}\overset{\leftarrow}{\nabla}_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\bar{\Psi}-\frac{i}{2}\,\bar{\Psi}\,\sigma_{ab}\,\omega^{ab}_{\mu}-\bar{\Psi}\left(\Gamma_{\mu}-iv_{\mu}^{(s)}\right)\,,$
$\displaystyle\omega_{\mu}^{ab}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,g^{\nu\lambda}e^{a}_{\lambda}\left(\partial_{\mu}e_{\nu}^{b}-e^{b}_{\sigma}\Gamma^{\sigma}_{\mu\nu}\right),$
$\displaystyle\Gamma^{\lambda}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,g^{\lambda\sigma}\left(\partial_{\alpha}g_{\beta\sigma}+\partial_{\beta}g_{\alpha\sigma}-\partial_{\sigma}g_{\alpha\beta}\right)\,,$
$\displaystyle f_{+}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
uF^{\mu\nu}_{L}u^{\dagger}+u^{\dagger}F^{\mu\nu}_{R}u\,,$ $\displaystyle
F^{\mu\nu}_{R}$ $\displaystyle=$
$\displaystyle\partial^{\mu}r^{\nu}-\partial^{\nu}r^{\mu}-i[r^{\mu},r^{\nu}]\,,$
$\displaystyle F^{\mu\nu}_{L}$ $\displaystyle=$
$\displaystyle\partial^{\mu}l^{\nu}-\partial^{\nu}l^{\mu}-i[l^{\mu},l^{\nu}]\,,$
$\displaystyle f_{+}^{i\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\text{Tr}\left(f_{+}^{\mu\nu}\tau^{i}\right),$
$\displaystyle\chi_{+}$ $\displaystyle=$ $\displaystyle u^{\dagger}\chi
u^{\dagger}+u\chi^{\dagger}u,$ $\displaystyle\hat{\chi}_{+}$ $\displaystyle=$
$\displaystyle\chi_{+}-\frac{1}{2}\langle\chi_{+}\rangle,$ $\displaystyle\chi$
$\displaystyle=$ $\displaystyle 2B_{0}(s+ip),$ $\displaystyle u_{\mu,i}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\text{Tr}\left(u_{\mu}\tau^{i}\right).$ (6)

Notice that since the gravitational interaction respects the isospin symmetry,
the amplitude of the $p\to\Delta^{+}$ transition receives non-vanishing
contributions only via the isospin-symmetry breaking effects. In Appendix A,
the above action is re-written in particle basis and the corresponding EMT is
also specified.

##  3 Gravitational transition form factors to one loop

Below, we calculate the leading one-loop contributions to the matrix elements
of the EMT for the one-particle states of the delta resonance and the nucleon.
These matrix elements are extracted from the residues of Green’s functions,
which have complex poles corresponding to the unstable $\Delta$ states
Gegelia:2010nmt . To organize different contributions according to a
systematic expansion we employ the so-called $\epsilon$-counting scheme (also
referred to as the small scale expansion) Hemmert:1996xg  111For an
alternative power counting in ChPT with delta resonances see Ref.
Pascalutsa:2002pi ., i.e. the pion lines count as of chiral order $Q^{-2}$,
where $Q$ denotes the soft scale of the order of the pion mass. Further, the
nucleon and delta lines count as $Q^{-1}$, interaction vertices originating
from the effective Lagrangian of order $N$ count also as of chiral order
$Q^{N}$, while the vertices generated by the EMT, which are listed in Appendix
B, have the orders corresponding to the number of the quark mass insertions
and derivatives acting on the pion fields. Derivatives acting on the nucleon
and delta fields count as of chiral order $Q^{0}$. The momentum transfer
between the initial and final states counts as of chiral order $Q$, therefore
in those terms of the EMT which involve full derivatives, these derivatives
also count as chiral order $Q$. Integration over loop momenta is counted as
chiral order $Q^{4}$. Furthermore, the delta-nucleon mass difference also
counts as order $Q$ within the $\epsilon$-counting scheme. In diagrams
involving electromagnetic radiative corrections, we assign the chiral order
$Q^{-2}$ to the photon line and count the electric charge $e$ as chiral order
$Q$. It is understood that the above described power counting for loop
diagrams is realized in the results of manifestly Lorentz-invariant
calculations only after performing an appropriate renormalization. We apply
the EOMS scheme of Refs. Gegelia:1999gf ; Fuchs:2003qc .

The matrix element of the total EMT for the transition $p\to\Delta^{+}$ can be
parameterized in terms of five form factors as follows Kim:2022bwn :

$\displaystyle\langle\Delta,p_{f},s_{f}|T^{\mu\nu}|N,p_{i},s_{i}\rangle$ (7)
$\displaystyle=$
$\displaystyle\bar{u}_{\alpha}(p_{f},s_{f})\Biggl{\\{}F_{1}(t)\left(g^{\alpha\\{\mu}P^{\nu\\}}+\frac{m_{\Delta^{+}}^{2}-m_{p}^{2}}{\Delta^{2}}g^{\mu\nu}\Delta^{\alpha}-\frac{m_{\Delta^{+}}^{2}-m_{p}^{2}}{2\Delta^{2}}g^{\alpha\\{\mu}\Delta^{\nu\\}}-\frac{1}{\Delta^{2}}P^{\\{\mu}\Delta^{\nu\\}}\Delta^{\alpha}\right)$
$\displaystyle+$ $\displaystyle
F_{2}(t)\left(P^{\mu}P^{\nu}\Delta^{\alpha}+\frac{(m_{\Delta^{+}}^{2}-m_{p}^{2})^{2}}{4\Delta^{2}}g^{\mu\nu}\Delta^{\alpha}-\frac{m_{\Delta^{+}}^{2}-m_{p}^{2}}{2\Delta^{2}}P^{\\{\mu}\Delta^{\nu\\}}\Delta^{\alpha}\right)$
$\displaystyle+$ $\displaystyle
F_{3}(t)\left(\Delta^{\mu}\Delta^{\nu}-\Delta^{2}g^{\mu\nu}\right)\Delta^{\alpha}$
$\displaystyle+$ $\displaystyle
F_{4}(t)\left(g^{\alpha\\{\mu}\gamma^{\nu\\}}+\frac{2(m_{p}+m_{\Delta^{+}})}{\Delta^{2}}g^{\mu\nu}\Delta^{\alpha}-\frac{m_{p}+m_{\Delta^{+}}}{\Delta^{2}}g^{\alpha\\{\mu}\Delta^{\nu\\}}-\frac{1}{\Delta^{2}}\gamma^{\\{\mu}\Delta^{\nu\\}}\Delta^{\alpha}\right)$
$\displaystyle+$ $\displaystyle
F_{5}(t)\left(P^{\\{\mu}\gamma^{\nu\\}}\Delta^{\alpha}+\frac{(m_{\Delta^{+}}^{2}-m_{p}^{2})(m_{p}+m_{\Delta^{+}})}{\Delta^{2}}g^{\mu\nu}\Delta^{\alpha}-\frac{m_{p}+m_{\Delta^{+}}}{\Delta^{2}}P^{\\{\mu}\Delta^{\nu\\}}\Delta^{\alpha}\right.$
$\displaystyle-$
$\displaystyle\left.\frac{m_{\Delta^{+}}^{2}-m_{p}^{2}}{2\Delta^{2}}\gamma^{\\{\mu}\Delta^{\nu\\}}\Delta^{\alpha}\right)\Biggl{\\}}\gamma^{5}u(p_{i},s_{i})\,,$

where $m_{p}$ and $m_{\Delta^{+}}$ are the proton and the $\Delta^{+}$ masses,
respectively, $P=\left(p_{f}+p_{i}\right)/2$, $\Delta=p_{f}-p_{i}$ and
$t=\Delta^{2}$. The curly brackets in the superscripts stand for
symmetrization of the involved indices, e.g.,
$P^{\\{\mu}\gamma^{\nu\\}}=P^{\mu}\gamma^{\nu}+P^{\nu}\gamma^{\mu}$. As
mentioned in the introduction, if the isospin symmetry is not broken, the
above amplitude is zero.

###  3.1 One-loop contributions of the strong interaction to the gravitational
transition form factors

To obtain the one-loop contributions to the GTFFs due to strong isospin-
breaking interactions one has to compute 25 diagrams, where there are only 10
topologically differing diagrams and the rest can be obtained by just changing
the masses and overall factors. These 10 diagrams are depicted in Fig. 1. The
isospin symmetry breaking terms of the effective Lagrangian, which contribute
to these one-loop diagrams, are specified in Appendix A. We performed the
calculations in the particle basis. In the limit of the exact isospin
symmetry, the contributions of the different diagrams exactly cancel each
other. Taking into account the dominant isospin breaking effect, we find that
obtained form factors are proportional to the mass differences within iso-
multiplets of nucleons, pions and delta resonances. The leading contributions
are given by terms proportional to the pion mass differences. This is because
these contributions involve integrals, whose integrands are proportional to

$\sim\frac{1}{p^{2}-M_{\pi^{+}}^{2}}-\frac{1}{p^{2}-M_{\pi^{0}}^{2}}\simeq\frac{M_{\pi^{+}}^{2}-M_{\pi^{0}}^{2}}{\left(p^{2}-M_{\pi^{+}}^{2}\right)\left(p^{2}-M_{\pi^{0}}^{2}\right)}\,,$
(8)

where each of the propagators originates from different diagrams that would
cancel each other in the isospin limit. As the mass difference
$M_{\pi^{+}}^{2}-M_{\pi^{0}}^{2}$ counts as of chiral order two, the right-
hand side of Eq. (8) has the same order as each of the terms in the left-hand
side. That is, the total contribution of these diagrams, which is proportional
to the pion mass difference squared, has the same order as the individual
diagrams. On the other hand, the contributions proportional to the proton-
neutron mass difference are given by integrals, whose integrands are
proportional to

$\sim\frac{1}{\not{p}-m_{p}}-\frac{1}{\not{p}-m_{n}}\simeq\frac{m_{p}-m_{n}}{\left(\not{p}-m_{p}\right)\left(\not{p}-m_{n}\right)}\,.$
(9)

As the mass difference $m_{p}-m_{n}$ counts as chiral order two, and the
nucleon propagators as order minus one, the right-hand side of Eq. (9) has one
order higher than each of the terms on the left-hand side. That is, the total
contribution of diagrams is suppressed by $Q$ relative to the contributions of
the individual diagrams. Analogous power counting holds also for the
contributions proportional to the mass differences of the delta resonances.
Notice further that isospin breaking vertices other than the mass terms start
contributing at higher orders.

Figure 1: Strong contributions to the gravitational transition form factors.
Solid and double lines correspond to nucleons and $\Delta$ resonances,
respectively. Dashed lines represent the pions, while the curly lines
correspond to gravitons. Initial and final states refer to $p$ and
$\Delta^{+}$, respectively, while the baryon lines inside loops refer to
propagators of one of the following particles:
$\\{\Delta^{++},\Delta^{+},\Delta^{0},p,n\\}$. Notice that the total
contribution of these diagrams vanishes in the limit of exact isospin
symmetry.

Diagrams $1,2,4,5,6$ and $8$ in Fig. 1 start contributing at chiral order
three while the diagrams $3,7,9$ and $10$ start contributing at chiral order
two. This is because the leading-order contribution to the gravitational-
source-baryon-baryon vertex has order zero.222Actually the gravitational-
source-baryon-baryon vertex originating from the leading-order Lagrangian has
two contributions, one of the order zero and the other of the order one. This
means that diagrams $3,7,9$ and $10$ contribute to two different chiral orders
(2 and 3). These two contributions cannot be considered separately, because
otherwise the current will not be conserved. This needs to be carefully taken
into account when specifying the (possible) power-counting violating terms.
Thus the diagrams in Fig. 1 give contributions of orders two and three. We
have verified that the one-loop order result of diagrams in Fig. 1 does not
contain power counting violating contributions and all ultraviolet divergences
can be absorbed into redefinition of the low-energy coupling constants of the
effective Lagrangian. The obtained results for the form factors are too
involved to be given as analytic expressions but are available from the
authors upon request in the form of a Mathematica notebook. The same applies
also to the results of the radiative corrections considered in the next
subsection.

###  3.2 One-loop radiative corrections to the gravitational transition form
factors

To obtain the one-loop electromagnetic corrections to the transition form
factors one has to compute the diagrams contributing up to order four, shown
in Fig. 2. The chiral power counting for Fig. 2 is similar to that for Fig. 1.
In this calculation we do not distinguish between the masses of the $\Delta$
states and between the masses of the proton and the neutron, i.e. we set
$m_{\Delta^{++}}=m_{\Delta^{+}}=m_{\Delta^{0}}=m_{\Delta^{-}}$ and
$m_{p}=m_{n}$.

Figure 2: Electromagnetic contributions to the transition form factors. Solid
and double solid lines correspond to nucleons and $\Delta$ resonances,
respectively. Wavy lines denote photons, while curly lines represent
gravitons.

Analogously to the strong-interaction contributions, we found that the one-
loop order result of diagrams shown in Fig. 2 does not involve power-counting
violating terms, and all ultraviolet divergences can be absorbed into
redefinition of the low-energy coupling constants of the most general
effective Lagrangian.

###  3.3 Numerical results for the gravitational transition form factors

In Figs. 3 and 4, we present the numerical results of the obtained strong and
electromagnetic contributions to the real and imaginary parts of the
transition form factors, respectively. Notice here that the imaginary parts of
the calculated form factors are generated solely by the loop contributions
with internal nucleon lines. For the numerical results, we used the following
values of the involved parameters:

$\displaystyle g_{A}$ $\displaystyle=$ $\displaystyle 1.289,\quad g=1.35,\quad
m_{\pi^{0}}=0.135,\quad m_{p}=0.938,\quad m_{n}=0.940,\quad$ $\displaystyle
m_{\Delta}$ $\displaystyle=$ $\displaystyle 1.232,\quad F=0.092,\quad
m_{\pi^{+}}=0.140,\quad m_{\Delta^{++}}=1.231,\quad
m_{\Delta^{+}}=m_{\Delta},$ $\displaystyle m_{\Delta^{0}}$ $\displaystyle=$
$\displaystyle 1.233,\quad g_{1}=9g_{A}/5,\quad e=0.303\,,\quad
d_{3}^{(2)}=2.72\,{\rm GeV}^{-1}\,,$ (10)

where the various masses and the pion decay constant $F$ are given in GeV. We
used the $SU(6)$ symmetry estimation for the coupling constants $g_{1}$,
$g_{A}$ and $g$ taken from Ref. Siemens:2016hdi , for the masses of delta
resonances we used estimations of Refs. Epelbaum:2007sq ; Epelbaum:2008td ,
$d_{3}^{(2)}$ corresponds to $b_{1}/2$ of Ref. Bernard:2012hb , while the
remaining values have been taken from the PDG ParticleDataGroup:2020ssz .

The plots demonstrate that the diagrams with radiative corrections give
smaller contributions than the ones with pion loops in line with the power
counting estimations. On the other hand the groups of diagrams with internal
nucleon and delta lines give comparable contributions.

Figure 3: The real parts of the $p\to\Delta^{+}$ transition form factors.
Dash-dotted (black), dashed (blue) and solid (red) lines correspond to the
form factors containing contributions of loop diagrams with inner pion and
nucleon lines only, diagrams with inner pion and nucleon lines plus radiative
corrections, and all loop contributions, respectively.  Figure 4: Imaginary
parts of the $p\to\Delta^{+}$ transition form factors. Dash-dotted (black),
and solid (red) lines correspond to the form factors containing contributions
of loop diagrams with inner pion and nucleon lines only, and diagrams with
inner pion and nucleon lines plus radiative corrections, respectively.

##  4 Conclusions and outlook

In the framework of manifestly Lorentz-invariant ChPT for pions, nucleons,
photons and the delta resonances interacting with an external gravitational
field, we calculated the leading one-loop contributions to the matrix element
of the EMT corresponding to the $p\to\Delta^{+}$ transition and extracted the
resulting gravitational transition form factors. As the gravitational
interaction respects the isospin symmetry, the amplitude of the
$p\to\Delta^{+}$ transition receives non-vanishing contributions due to
isospin symmetry breaking. The results of the current work take into account
the leading-order electromagnetic and strong isospin-breaking effects.
Ultraviolet divergences and power counting violating pieces generated by loop
diagrams in the manifestly Lorentz-invariant formulation of ChPT can be
treated using the EOMS renormalization scheme of Refs. Gegelia:1999gf ;
Fuchs:2003qc . However, at the order of our calculations, the one-loop
contributions to the form factors are found to be free of contributions that
violate the chiral power counting. This is consistent with the absence of
tree-level contributions at the considered order. For this reason, our results
involve no free parameters and can be regarded as predictions of ChPT. Notice,
however, that the empirical information on the mass splittings between the
$\Delta$ resonance states, which enters as an input in our calculations, is
presently rather poor. Numerical results for the obtained transition form
factors demonstrate that the electromagnetic and strong isospin violating
effects give contributions of comparable sizes. This holds true for
contributions with both internal nucleon and delta lines.

###### Acknowledgements.

This work was supported in part by DFG and NSFC through funds provided to the
Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD” (NSFC
Grant No. 11621131001, DFG Project-ID 196253076 - TRR 110), by CAS through a
President’s International Fellowship Initiative (PIFI) (Grant No. 2018DM0034),
by the VolkswagenStiftung (Grant No. 93562), by the MKW NRW under the funding
code NW21-024-A, by the EU Horizon 2020 research and innovation programme
(STRONG-2020, grant agreement No. 824093), by Guangdong Provincial funding
with Grant No. 2019QN01X172, the National Natural Science Foundation of China
with Grant No. 12035007 and No. 11947228, Guangdong Major Project of Basic and
Applied Basic Research No. 2020B0301030008, and the Department of Science and
Technology of Guangdong Province with Grant No. 2022A0505030010.

##  Appendix A Isospin-symmetry breaking terms

To obtain the leading isospin breaking effects due to the strong interaction
we distinguish between the masses of the delta resonances, and also between
the masses of the proton and the neutron, and the charged and neutral pions.
We rewrite the action using the physical basis, instead of the isospin basis,
by writing the fields explicitly as follows:

$\displaystyle\Psi$ $\displaystyle=$ $\displaystyle\begin{pmatrix}\Psi_{p}\\\
\Psi_{n}\end{pmatrix},$ $\displaystyle\Psi_{\mu,1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{1}{\sqrt{3}}\Delta^{0}_{\mu}-\Delta^{++}_{\mu}\\\
\Delta^{-}_{\mu}-\frac{1}{\sqrt{3}}\Delta^{+}_{\mu}\end{pmatrix},~{}\Psi_{\mu,2}=-\frac{i}{\sqrt{2}}\begin{pmatrix}\frac{1}{\sqrt{3}}\Delta^{0}_{\mu}+\Delta^{++}_{\mu}\\\
\Delta^{-}_{\mu}+\frac{1}{\sqrt{3}}\Delta^{+}_{\mu}\end{pmatrix},~{}\Psi_{\mu,3}=\sqrt{\frac{2}{3}}\begin{pmatrix}\Delta^{+}_{\mu}\\\
\Delta^{0}_{\mu}\end{pmatrix}\,,$ $\displaystyle\pi^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\pi^{+}+\pi^{-}\right),\,\pi^{2}=\frac{i}{\sqrt{2}}\left(\pi^{+}-\pi^{-}\right),\,\pi^{3}=\pi^{0}.$
(11)

We substitute the above definition of the fields into Eqs. (2), (3), (4) and
(5). The terms relevant for the leading one-loop order contributions to the
$p\to\Delta^{+}$ transition are given by

$\displaystyle S_{\rm\pi}^{(2)}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\,\left\\{\frac{1}{2}\partial_{\mu}\pi^{0}\partial^{\mu}\pi^{0}-\frac{1}{2}M_{0}^{2}\pi^{0}\pi^{0}+\partial_{\mu}\pi^{+}\partial^{\mu}\pi^{-}-M_{\pi^{+}}^{2}\pi^{+}\pi^{-}\
\right\\}\,,$ (12) $\displaystyle S_{\rm N\pi}^{(1)}$ $\displaystyle=$
$\displaystyle\int
d^{4}x\sqrt{-g}\,\left\\{\,\bar{\Psi}_{p}\,i\gamma^{\mu}\overset{\leftrightarrow}{\nabla}_{\mu}\Psi_{p}-m_{p}\bar{\Psi}_{p}\Psi_{p}+\bar{\Psi}_{n}\,i\gamma^{\mu}\overset{\leftrightarrow}{\nabla}_{\mu}\Psi_{n}-m_{n}\bar{\Psi}_{n}\Psi_{n}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{g_{A}}{2F}\left(\partial_{\mu}\pi^{0}\left[\bar{\Psi}_{n}\gamma^{\mu}\gamma^{5}\Psi_{n}-\bar{\Psi}_{p}\gamma^{\mu}\gamma^{5}\Psi_{p}\right]-\sqrt{2}\left[\partial_{\mu}\pi^{-}\bar{\Psi}_{n}\gamma^{\mu}\gamma^{5}\Psi_{p}+\partial_{\mu}\pi^{+}\bar{\Psi}_{p}\gamma^{\mu}\gamma^{5}\Psi_{n}\right]\right)\right\\}\,,$
$\displaystyle S_{\Delta\pi}^{(1)}$ $\displaystyle=$ $\displaystyle-\int
d^{4}x\sqrt{-g}\biggl{\\{}\biggl{[}\sum_{i\in\\{++,+,0,-\\}}\bar{\Delta}^{i\mu}\,i\gamma^{\alpha}\overset{\leftrightarrow}{\nabla}_{\alpha}\Delta^{i}_{\mu}-m_{\Delta^{i}}\,\bar{\Delta}^{i}_{\mu}\Delta^{i\mu}$
(14)
$\displaystyle-g^{\lambda\sigma}\left(\bar{\Delta}^{i}_{\mu}i\gamma^{\mu}{\overset{\leftrightarrow}{\nabla}_{\lambda}}\Delta^{i}_{\sigma}+\bar{\Delta}^{i}_{\lambda}i\gamma^{\mu}{\overset{\leftrightarrow}{\nabla}_{\sigma}}\Delta^{i}_{\mu}\right)+i\bar{\Delta}^{i}_{\mu}\gamma^{\mu}\gamma^{\alpha}\gamma^{\nu}\overset{\leftrightarrow}{\nabla}_{\alpha}\Delta^{i}_{\nu}+m_{\Delta^{i}}\bar{\Delta}^{i}_{\mu}\gamma^{\mu}\gamma^{\nu}\Delta^{i}_{\nu}\biggr{]}$
$\displaystyle+\frac{1}{6F}\left[\bar{\Delta}^{0}_{\mu}O_{1}^{\mu\nu\alpha}\Delta^{0}_{\nu}\partial_{\alpha}\pi^{0}-\sqrt{6}\bar{\Delta}^{-}_{\mu}O_{1}^{\mu\nu\alpha}\Delta^{0}_{\nu}\partial_{\alpha}\pi^{-}-2\sqrt{2}\bar{\Delta}^{+}_{\mu}O_{1}^{\mu\nu\alpha}\Delta^{0}_{\nu}\partial_{\alpha}\pi^{+}\right.$
$\displaystyle-\sqrt{6}\bar{\Delta}^{0}_{\mu}O_{1}^{\mu\nu\alpha}\Delta^{-}_{\nu}\partial_{\alpha}\pi^{+}-2\sqrt{2}\bar{\Delta}^{0}_{\mu}O_{1}^{\mu\nu\alpha}\Delta^{+}_{\nu}\partial_{\alpha}\pi^{-}+3\bar{\Delta}^{-}_{\mu}O_{1}^{\mu\nu\alpha}\Delta^{-}_{\nu}\partial_{\alpha}\pi^{0}$
$\displaystyle-\bar{\Delta}^{+}_{\mu}O_{1}^{\mu\nu\alpha}\Delta^{+}_{\nu}\partial_{\alpha}\pi^{0}-\sqrt{6}\bar{\Delta}^{+}_{\mu}O_{1}^{\mu\nu\alpha}\Delta^{++}_{\nu}\partial_{\alpha}\pi^{-}-\sqrt{6}\bar{\Delta}^{++}_{\mu}O_{1}^{\mu\nu\alpha}\Delta^{+}_{\nu}\partial_{\alpha}\pi^{+}$
$\displaystyle\left.-3\bar{\Delta}^{++}_{\mu}O_{1}^{\mu\nu\alpha}\Delta^{++}_{\nu}\partial_{\alpha}\pi^{0}\right]\Biggl{\\}}\,,$
$\displaystyle S_{\Delta\rm N\pi}^{(1)}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\,\frac{g_{\pi
n\Delta}}{F}\Biggl{\\{}\bar{\Psi}_{n}\partial_{\mu}\pi^{+}O_{2}^{\mu\nu}\Delta^{-}_{\nu}-\bar{\Psi}_{p}\partial_{\mu}\pi^{-}O_{2}^{\mu\nu}\Delta^{++}_{\nu}+\frac{1}{\sqrt{3}}\left(\sqrt{2}~{}\bar{\Psi}_{n}\partial_{\mu}\pi^{0}O_{2}^{\mu\nu}\Delta^{0}_{\nu}\right.$
(15)
$\displaystyle-\bar{\Psi}_{n}\partial_{\mu}\pi^{-}O_{2}^{\mu\nu}\Delta^{+}_{\nu}+\left.\bar{\Psi}_{p}\partial_{\mu}\pi^{+}O_{2}^{\mu\nu}\Delta^{0}_{\nu}+\sqrt{2}~{}\bar{\Psi}_{p}\partial_{\mu}\pi^{0}O_{2}^{\mu\nu}\Delta^{+}_{\nu}\right)\Biggl{\\}}\,,$

where
$O_{1}^{\mu\nu\alpha}=g_{1}\gamma^{\alpha}\gamma^{5}g^{\mu\nu}+g_{2}(g^{\mu\alpha}\gamma^{\nu}\gamma^{5}+g^{\nu\alpha}\gamma^{\mu}\gamma^{5})+g_{3}\gamma^{\mu}\gamma^{\alpha}\gamma^{5}\gamma^{\nu}$
and $O_{2}^{\mu\nu}=g^{\mu\nu}-\gamma^{\mu}\gamma^{\nu}$. To arrive at these
results, we expanded the matrix $u$ of pion fields and kept only the first
nontrivial term, i.e. $u=1+i/(2F)\tau^{i}\pi^{i}+\mathcal{O}(1/F^{2}).$

The mass splittings within iso-multiplets are not just due to strong isospin
breaking but also receive important contributions from the electromagnetic
interaction. However, there is no point at separating these contributions
here, and such a separation is anyway afflicted with some uncertainties, see
e.g. the pedagogical discussion in Ref. Meissner:2022cbi .

To obtain the leading isospin breaking effects due to the diagrams with
radiative corrections (i.e. with photon propagators) we do not distinguish
between the masses of the isospin partners, i.e. we take
$m_{\Delta^{++}}=m_{\Delta^{+}}=m_{\Delta^{0}}=m_{\Delta^{-}}$, and
$m_{p}=m_{n}$. For the external sources, we take the following expressions:

$\displaystyle r_{\mu}$ $\displaystyle=$ $\displaystyle
l_{\mu}=-e~{}A_{\mu}\frac{\tau_{3}}{2},$ (16) $\displaystyle v_{\mu}^{s}$
$\displaystyle=$ $\displaystyle-\frac{e}{2}~{}A_{\mu},$ (17)

where $e$ is the electric charge of the proton.

##  Appendix B The energy-momentum tensor

Using the definition of the EMT for bosonic matter fields interacting with the
gravitational metric field,

$\displaystyle T_{\mu\nu}(g,\psi)$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{-g}}\frac{\delta S_{\rm m}}{\delta
g^{\mu\nu}}\,,$ (18)

we obtain in flat spacetime from the action terms of Eqs. (1) and (2):

$\displaystyle T^{{(2)}}_{\gamma,\mu\nu}$ $\displaystyle=$ $\displaystyle
F_{\mu}^{\alpha}F_{\alpha\nu}+m_{\gamma}^{2}A_{\mu}A_{\nu}+\eta_{\mu\nu}\left(\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}-\frac{m_{\gamma}^{2}}{2}A_{\alpha}A^{\alpha}\right),\,$
(19) $\displaystyle T^{{(2)}}_{\pi,\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{F^{2}}{4}\,{\rm
Tr}(D_{\mu}U(D_{\nu}U)^{\dagger})-\frac{\eta_{\mu\nu}}{2}\left\\{\frac{F^{2}}{4}\,{\rm
Tr}(D^{\alpha}U(D_{\alpha}U)^{\dagger})+\frac{F^{2}}{4}\,{\rm Tr}(\chi
U^{\dagger}+U\chi^{\dagger})\right\\}$ (20)
$\displaystyle+\left(\mu\leftrightarrow\nu\right),$

where $\eta_{\mu\nu}$ is the Minkowski metric tensor with the signature
$(+,-,-,-)$. For the fermionic fields interacting with the gravitational
vielbein fields we use the definition Birrell:1982ix

$\displaystyle T_{\mu\nu}(g,\psi)$ $\displaystyle=$
$\displaystyle\frac{1}{2e}\left[\frac{\delta S}{\delta
e^{a\mu}}\,e^{a}_{\nu}+\frac{\delta S}{\delta
e^{a\nu}}\,e^{a}_{\mu}\right]\,.$ (21)

The action of Eq. (3) leads to the following expression for the EMT in flat
spacetime:

$\displaystyle T^{{(1)}}_{\rm N,\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\bar{\Psi}\,\gamma_{\mu}\overset{\leftrightarrow}{D}_{\nu}\Psi-\frac{\eta_{\mu\nu}}{2}\biggl{(}\,\bar{\Psi}\,i\gamma^{\alpha}\overset{\leftrightarrow}{D}_{\alpha}\Psi-m\bar{\Psi}\Psi\biggr{)}+\left(\mu\leftrightarrow\nu\right)\,,$
(22)

while Eqs. (4) and (5) lead to the following expressions:

$\displaystyle T^{{(1)}}_{\Delta\pi,\mu\nu}$ $\displaystyle=$
$\displaystyle-\bar{\Psi}^{i}_{\mu}\,i\gamma^{\alpha}\overset{\leftrightarrow}{D}_{\alpha}\Psi^{i}_{\nu}+\bar{\Psi}^{i}_{\alpha}\,i\gamma^{\alpha}\overset{\leftrightarrow}{D}_{\mu}\Psi^{i}_{\nu}+\bar{\Psi}^{i}_{\mu}\,i\gamma^{\alpha}\overset{\leftrightarrow}{D}_{\nu}\Psi^{i}_{\alpha}+m_{\Delta}\bar{\Psi}^{i}_{\mu}\Psi^{i}_{\nu}-\frac{i}{2}\,\bar{\Psi}^{i}_{\alpha}\,\gamma_{\mu}\overset{\leftrightarrow}{D}_{\nu}\Psi^{i\alpha}$
(23)
$\displaystyle+\frac{i}{2}\biggl{(}\bar{\Psi}^{i}_{\mu}\,\gamma_{\nu}\overset{\leftrightarrow}{D}_{\alpha}\Psi^{i\alpha}+\bar{\Psi}^{i\alpha}\,\gamma_{\nu}\overset{\leftrightarrow}{D}_{\alpha}\Psi^{i}_{\mu}-\bar{\Psi}^{i}_{\mu}\,\gamma_{\nu}\gamma^{\alpha}\gamma_{\beta}\overset{\leftrightarrow}{D}_{\alpha}\Psi^{i,\beta}-\bar{\Psi}^{i}_{\alpha}\gamma^{\alpha}\gamma_{\nu}\gamma^{\beta}\overset{\leftrightarrow}{D}_{\mu}\Psi^{i}_{\beta}$
$\displaystyle-\bar{\Psi}^{i}_{\alpha}\gamma^{\alpha}\gamma^{\beta}\gamma_{\nu}\overset{\leftrightarrow}{D}_{\beta}\Psi^{i}_{\mu}\biggl{)}+\frac{i}{4}\,\partial^{\lambda}\biggl{[}\bar{\Psi}^{i,\alpha}\biggl{(}\gamma_{\mu}\eta_{\lambda[\alpha}\eta_{\beta]\mu}+\eta_{\lambda\mu}\eta_{\nu[\alpha}\gamma_{\beta]}+\eta_{\mu\nu}\eta_{\lambda[\beta}\gamma_{\alpha]}\biggr{)}\Psi^{i,\beta}\biggr{]}$
$\displaystyle-\frac{m_{\Delta}}{2}\,\left(\bar{\Psi}^{i}_{\mu}\,\gamma_{\nu}\gamma^{\alpha}\Psi^{i}_{\alpha}+\bar{\Psi}^{i}_{\alpha}\,\gamma^{\alpha}\gamma_{\nu}\Psi^{i}_{\mu}\right)-\frac{g_{1}}{4}\left[2\bar{\Psi}^{i}_{\mu}u_{\alpha}\gamma^{\alpha}\gamma_{5}\Psi^{i}_{\nu}+\bar{\Psi}^{i,\alpha}u_{\mu}\gamma_{\nu}\gamma_{5}\Psi^{i}_{\alpha}\right]$
$\displaystyle-\frac{g_{2}}{4}\left[2\bar{\Psi}^{i}_{\mu}u_{\nu}\gamma^{\alpha}\gamma_{5}\Psi^{i}_{\alpha}+2\bar{\Psi}^{i}_{\alpha}u_{\nu}\gamma^{\alpha}\gamma_{5}\Psi^{i}_{\mu}+\bar{\Psi}^{i,\alpha}u_{\alpha}\gamma_{\nu}\gamma_{5}\Psi^{i}_{\mu}+\bar{\Psi}^{i}_{\mu}u_{\alpha}\gamma_{\nu}\gamma_{5}\Psi^{i\alpha}\right]$
$\displaystyle-\frac{g_{3}}{4}\left[\bar{\Psi}^{i}_{\mu}u_{\alpha}\gamma_{\nu}\gamma^{\alpha}\gamma_{5}\gamma^{\beta}\Psi^{i}_{\beta}+\bar{\Psi}^{i}_{\beta}u_{\alpha}\gamma^{\beta}\gamma^{\alpha}\gamma_{5}\gamma_{\nu}\Psi^{i}_{\mu}+\bar{\Psi}^{i}_{\alpha}u_{\mu}\gamma^{\alpha}\gamma_{\nu}\gamma_{5}\gamma^{\beta}\Psi^{i}_{\beta}\right]$
$\displaystyle+\frac{\eta_{\mu\nu}}{2}\biggl{[}\,\bar{\Psi}^{i}_{\alpha}\,i\gamma^{\beta}\overset{\leftrightarrow}{D}_{\beta}\Psi^{i\alpha}-m_{\Delta}\,\bar{\Psi}^{i}_{\alpha}\Psi^{i\alpha}-\bar{\Psi}^{i}_{\alpha}i\gamma^{\alpha}{\overset{\leftrightarrow}{D}_{\beta}}\Psi^{i\beta}-\bar{\Psi}^{i\alpha}i\gamma^{\beta}{\overset{\leftrightarrow}{D}_{\alpha}}\Psi^{i}_{\beta}$
$\displaystyle+i\bar{\Psi}^{i}_{\rho}\gamma^{\rho}\gamma^{\alpha}\gamma^{\lambda}\overset{\leftrightarrow}{D}_{\alpha}\Psi^{i}_{\lambda}+m_{\Delta}\bar{\Psi}^{i}_{\alpha}\gamma^{\alpha}\gamma^{\beta}\Psi^{i}_{\beta}+\frac{g_{1}}{2}\,\bar{\Psi}^{i}_{\beta}u_{\alpha}\gamma^{\alpha}\gamma_{5}\Psi^{i\beta}$
$\displaystyle+\frac{g_{2}}{2}\bar{\Psi}^{i\alpha}\left(u_{\alpha}\gamma_{\beta}+u_{\beta}\gamma_{\alpha}\right)\gamma_{5}\Psi^{i\beta}+\frac{g_{3}}{2}\bar{\Psi}^{i}_{\alpha}u_{\beta}\gamma^{\alpha}\gamma^{\beta}\gamma_{5}\gamma^{\lambda}\Psi^{i}_{\lambda}\biggr{]}+\left(\mu\leftrightarrow\nu\right)\,,$
$\displaystyle T^{(1,2)}_{\pi N\Delta,\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{\pi
N\Delta}\biggl{\\{}\frac{1}{2}~{}\eta_{\mu\nu}\left[\bar{\Psi}^{i}_{\alpha}u^{\alpha}_{i}\Psi+\bar{\Psi}u^{\alpha}_{i}\Psi^{i}_{\alpha}-\bar{\Psi}^{i}_{\alpha}\gamma^{\alpha}\gamma^{\beta}u_{\beta}^{i}\Psi-\bar{\Psi}\gamma^{\beta}\gamma^{\alpha}u_{\beta}^{i}\Psi^{i}_{\alpha}\right]-\bar{\Psi}^{i}_{\mu}u_{\nu}^{i}\Psi$
(24)
$\displaystyle-\bar{\Psi}u_{\nu}^{i}\Psi^{i}_{\mu}+\frac{1}{2}~{}\left[\bar{\Psi}^{i}_{\mu}\gamma_{\nu}\gamma^{\alpha}u_{\alpha}^{i}\Psi+\bar{\Psi}^{i}_{\alpha}\gamma^{\alpha}\gamma_{\mu}u_{\nu}^{i}\Psi+\bar{\Psi}\gamma^{\alpha}\gamma_{\nu}u_{\alpha}^{i}\Psi^{i}_{\mu}+\bar{\Psi}\gamma_{\mu}\gamma^{\alpha}u_{\nu}^{i}\Psi^{i}_{\alpha}\right]\biggl{\\}}$
$\displaystyle+\frac{i}{2}d_{3}^{(2)}\biggl{\\{}\bar{\Psi}f_{+,\mu\beta}^{i}\gamma_{5}\gamma_{\nu}\tilde{\Psi}^{i\beta}+2\bar{\Psi}f_{+,\alpha\mu}^{i}\gamma_{5}\gamma^{\alpha}\Psi^{i}_{\nu}-\eta_{\mu\nu}\bar{\Psi}f_{+,\alpha\beta}^{i}\gamma_{5}\gamma^{\alpha}\tilde{\Psi}^{i\beta}$
$\displaystyle-\left[z_{n}+\frac{1}{2}\right]\left(\bar{\Psi}f_{+,\alpha\mu}^{i}\gamma_{5}\gamma^{\alpha}\gamma_{\nu}\gamma^{\beta}\Psi^{i}_{\beta}+\bar{\Psi}f_{+,\alpha\beta}^{i}\gamma_{5}\gamma^{\alpha}\gamma^{\beta}\gamma_{\mu}\Psi^{i}_{\nu}\right)\biggl{\\}}+\left(\mu\leftrightarrow\nu\right)\,,$

where the covariant derivatives ${D}$ acting on spin-1/2 and spin-3/2 fields
coincide with $\nabla$ of Eq. (6) with
$\Gamma_{\mu\nu}^{\beta}=\omega_{\mu}^{ab}=0$. The superscripts in the
expressions of EMT indicate the orders which are assigned to the corresponding
terms of the action (effective Lagrangian).

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