# Axial-vector nucleon-to-delta transition form factors using the complex-mass
renormalization scheme

Y. Ünal  Physics Department, Çanakkale Onsekiz Mart University, 17100
Çanakkale, Turkey  A. Küçükarslan  Physics Department, Çanakkale Onsekiz Mart
University, 17100 Çanakkale, Turkey  S. Scherer  Institut für Kernphysik,
Johannes Gutenberg-Universität Mainz, D-55099 Mainz, Germany

(April 7, 2021)

###### Abstract

We investigate the axial-vector nucleon-to-delta transition form factors in
the framework of relativistic baryon chiral perturbation theory at the one-
loop order using the complex-mass renormalization scheme. We determine the
available six free parameters by fitting to an empirical parametrization of
the form factors obtained from the BNL neutrino bubble chamber experiments. A
unique feature of our calculation is the prediction of a non-vanishing form
factor $C_{3}^{A}(Q^{2})$. Moreover, our results show a surprising sensitivity
to the coupling constant $\texttt{g}_{1}$ of the leading-order Lagrangian
${\cal L}^{(1)}_{\pi\Delta}$.

##  I Introduction

The $\Delta(1232)$ resonance is the first and best-established excitation of
the nucleon Zyla:2020zbs . In the static quark model, it consists of three
constituent quarks, coupled to spin $S=\frac{3}{2}$ and isospin
$I=\frac{3}{2}$. The dominant decay mode by far is the strong decay into a
pion and a nucleon, resulting in a lifetime of the order of 10-23 s. According
to Ref. Zyla:2020zbs , the pole position in the complex-energy plane is at
$z_{\Delta}=m_{\Delta}-i\,\frac{\Gamma_{\Delta}}{2}\approx(1210-i\,50)$ MeV.

While there is a substantial amount of empirical information on the
electromagnetic (vector) nucleon-to-delta transition Bartel:1968tw ;
Baetzner:1972bg ; Stein:1975yy ; Beck:1999ge ; Pospischil:2000ad ;
Mertz:1999hp ; Joo:2001tw ; Sparveris:2004jn ; Elsner:2005cz ; Kelly:2005jy ;
Stave:2008aa ; Aznauryan:2009mx ; Blomberg:2015zma  (see, e.g., Refs.
Tiator:2011pw ; Aznauryan:2011qj  for a review), very little is known about
the axial-vector nucleon-to-delta transition Barish:1978pj ; Radecky:1981fn ;
Kitagaki:1986ct ; Kitagaki:1990vs ; Androic:2012doa . The reason is twofold:
(a) the weak probe couples only feebly to the nucleon-delta system and (b) the
delta is unstable.111Of course, the second argument also applies to the
electromagnetic transition. Therefore, our knowledge of the electromagnetic
transition form factors is substantially less than for the nucleon elastic
form factors. Concerning the transition from an unstable delta state to an
unstable delta state, Ref. Zyla:2020zbs  quotes only a rough guess of the
range, within which the magnetic moment is expected to lie. On the theoretical
side, numerous investigations exist for the electromagnetic case (see Ref.
Hilt:2017iup  and references therein) which have been extensively compared
with data. Also, theoretical calculations of the axial-vector nucleon-to-delta
transition have been performed in the framework of quark models Korner:1977rb
; Hemmert:1994ky ; Liu:1995bu ; Golli:2002wy ; BarquillaCano:2007yk , chiral
effective field theory Zhu:2002kh ; Geng:2008bm ; Procura:2008ze ;
Unal:2018ruo , lattice QCD Alexandrou:2006mc ; Alexandrou:2009vqd ;
Alexandrou:2010uk , and light-cone QCD sum rules Aliev:2007pi ;
Kucukarslan:2015urd .

While traditional calculations treat the delta resonance essentially as a
stable particle, it was emphasized in Ref. Gegelia:2009py  that form factors
of unstable particles should be determined from the renormalized three-point
function at the complex pole. In fact, this idea was applied in Ref.
Hilt:2017iup  to the electromagnetic nucleon-to-$\Delta$ resonance transition
to third chiral order in manifestly Lorentz-invariant chiral effective field
theory. At the pole position, the magnetic dipole, electric dipole, and
Coulomb quadrupole form factors $G_{M}$, $G_{E}$, and $G_{C}$ are complex
quantities. In particular, it was found that $G_{E}$ and $G_{C}$ have
imaginary parts which are of the same magnitude as the respective real parts.
In the present article, we extend the analysis to the axial-vector transition
at the one-loop level. For that purpose, we combine a covariant description of
the $\Delta(1232)$ resonance Hacker:2005fh ; Wies:2006rv  with the complex-
mass scheme (CMS) applied to the chiral effective field theory of the strong
interaction Djukanovic:2009zn .222 The CMS was originally developed for
deriving properties of $W$, $Z$, and Higgs bosons obtained from resonant
processes Stuart:1990 ; Denner:1999gp ; Denner:2006ic ; Actis:2006rc ;
Actis:2008uh .

This article is organized as follows. In Sec. II, we introduce the axial-
vector nucleon-to-delta transition process and discuss how it is related to
weak pion production. In this context, we also define the pion-nucleon-delta
form factor in terms of the PCAC relation (partially conserved axial-vector
current). In Sec. III, we present the effective Lagrangians we used. In Sec.
IV, we calculate the transition form factors and show our results. Section V
contains a comparison with other work. In Sec. VI, we give a short summary.

##  II Axial-vector nucleon-to-delta transition form factors

###  II.1 Weak pion production

The $\Delta$(1232) is an unstable particle with a very short lifetime of the
order of $10^{-23}$ s. Therefore, strictly speaking, stable one-particle
states $|\Delta(p)\rangle$ with $p^{2}=m_{\Delta}^{2}$ do not exist Bjorken .
For this reason, direct measurements of transition form factors are
impossible, because the $\Delta$(1232) is not an asymptotic state of the
strong interactions.333From a theoretical point of view, it is possible to
study a hypothetical situation, where the sum of the nucleon and pion masses
is larger than the $\Delta$ mass, resulting in a stable $\Delta$ state. On the
other hand, the existence of the delta is prominently seen in pion-nucleon
scattering or pion photoproduction on the nucleon. In other words, the impact
of an unstable $\Delta$ may be investigated in terms of a complete scattering
amplitude, where it contributes as an intermediate “state.” In the present
case, we are interested in the axial-vector nucleon-to-delta transition. This
may be studied in the weak production of a pion on the nucleon with hadronic
center-of-mass energies in the delta region Adler:1968tw . For kinematical
conditions such that the square root of the Mandelstam variable $s$ is in the
vicinity of the complex pole position,

$z_{\Delta}=m_{\Delta}-i\,\frac{\Gamma_{\Delta}}{2},$

the process is dominated by the propagation of a $\Delta$ resonance in the $s$
channel (see Fig. 1). Since the $W$ boson induces the transition between the
nucleon and the delta in terms of the $V-A$ structure of the coupling to the
quarks, this contribution is sensitive to both the nucleon-to-delta vector and
axial-vector transitions.

Figure 1: At $s\approx m_{\Delta}^{2}$, the process is dominated by the
$s$-channel pole diagram due to the propagator of the $\Delta$(1232). The
nucleon, electron, and neutrino are represented by single lines, the
$\Delta$(1232) by a double line, the $W$ boson by a wiggly line, and the pion
by a dashed line. The circles represent dressed vertices.

One now parametrizes the contribution of the unstable $\Delta$(1232) and
defines the form factors in analogy to a stable particle. For an unstable
particle such as the $\Delta$(1232), “on-shell kinematics” are given by the
complex pole position.

###  II.2 Definition of the axial-vector transition form factors

In the following, we provide a short definition of the form factors. We stick
to the notation of Ref. Unal:2018ruo , where more details can be found. In
terms of the light-quark field operators, $q=(u,d)^{T}$, the Cartesian
components of the isovector axial-vector current operator are given by
Gasser:1983yg

$A^{\mu}_{j}(x)=\bar{q}(x)\gamma^{\mu}\gamma_{5}\frac{\tau_{j}}{2}q(x).$ (1)

In general, the invariant amplitude444Our convention for the invariant
amplitude complies with Ref. Bjorken:1965zz . In particular, it contains the
imaginary unit on the right-hand side of Eq. (2). for a transition between
hadronic states $|A(p_{i})\rangle$ and $|B(p_{f})\rangle$, induced by a plane-
wave external field of the form $a_{\mu,j}(x)=\epsilon_{\mu,j}(q)e^{-iq\cdot
x}$, is defined as

${\cal M}=i\epsilon_{\mu,j}(q)\langle
B(p_{f})|A^{\mu}_{j}(0)|A(p_{i})\rangle,$ (2)

where four-momentum conservation $p_{f}=p_{i}+q$ due to translational
invariance is implied.

Introducing the spherical tensor notation Edmonds ,

$A^{\mu(1)}_{\pm 1}=\mp\frac{1}{\sqrt{2}}\left(A^{\mu}_{1}\pm
iA^{\mu}_{2}\right),\quad A^{\mu(1)}_{0}=A^{\mu}_{3},$

and using isospin symmetry, we express the matrix element of the spherical
isospin components ($\alpha=+1,0,-1$) between a nucleon state and a $\Delta$
state as

$\langle
3/2,\tau_{\Delta}|A^{\mu(1)}_{\alpha}|1/2,\tau\rangle=(1/2,\tau;1,\alpha|3/2,\tau_{\Delta})\langle
3/2||A^{\mu(1)}||1/2\rangle,$ (3)

where $\langle 3/2||A^{\mu(1)}||1/2\rangle$ denotes the reduced matrix element
and $(1/2,\tau;1,\alpha|3/2,\tau_{\Delta})$ is the relevant Clebsch-Gordan
coefficient. The reduced matrix element may, for example, be obtained from the
$p$ to $\Delta^{+}$ transition,

$\langle
3/2||A^{\mu(1)}||1/2\rangle=\sqrt{\frac{3}{2}}\langle\Delta^{+}|A^{\mu(1)}_{0}|p\rangle.$

The Lorentz structure of the reduced matrix element may be written as

$\langle\Delta(p_{f},s_{f})||A^{\mu(1)}(0)||N(p_{i},s_{i})\rangle=\bar{w}_{\lambda}(p_{f},s_{f})\Gamma^{\lambda\mu}_{A}u(p_{i},s_{i}).$
(4)

Here, the initial nucleon is described by the Dirac spinor $u(p_{i},s_{i})$
with a real mass $m_{N}$ and $p_{i}^{2}=m_{N}^{2}$, the final $\Delta$(1232)
is described via the Rarita-Schwinger vector-spinor
$\bar{w}_{\lambda}(p_{f},s_{f})$ Rarita:1941mf ; Kusaka  with a complex mass
$z_{\Delta}$ and $p_{f}^{2}=z_{\Delta}^{2}$ Gegelia:2009py ; Agadjanov:2014kha
. In the following, it is always understood that the “tensor”
$\Gamma^{\lambda\mu}_{A}$ is evaluated between on-shell spinors $u$ and
$\bar{w}_{\lambda}$, satisfying555The explicit form of $\bar{w}_{\lambda}$ can
be found in Ref. Agadjanov:2014kha .

$\displaystyle\not{p}_{i}u(p_{i},s_{i})$ $\displaystyle=m_{N}u(p_{i},s_{i}),$
(5) $\displaystyle\bar{w}_{\lambda}(p_{f},s_{f})\not{p}_{f}$
$\displaystyle=z_{\Delta}\bar{w}_{\lambda}(p_{f},s_{f}),\quad\bar{w}_{\lambda}(p_{f},s_{f})\gamma^{\lambda}=0,\quad\bar{w}_{\lambda}(p_{f},s_{f})p_{f}^{\lambda}=0.$
(6)

The expressions for a stable $\Delta$ resonance are obtained via the
replacement $z_{\Delta}\to m_{\Delta}$. The “tensor” $\Gamma^{\lambda\mu}_{A}$
contains a superposition of four Lorentz tensors Adler:1968tw ;
LlewellynSmith:1971uhs , which we choose as Alexandrou:2009vqd ;
Procura:2008ze

$\displaystyle\Gamma^{\lambda\mu}_{A}$
$\displaystyle=\frac{C_{3}^{A}(Q^{2})}{m_{N}}\left(g^{\lambda\mu}\not{q}-q^{\lambda}\gamma^{\mu}\right)+\frac{C_{4}^{A}(Q^{2})}{m_{N}^{2}}\left(g^{\lambda\mu}p_{f}\cdot
q-q^{\lambda}p_{f}^{\mu}\right)+C_{5}^{A}(Q^{2})g^{\lambda\mu}+\frac{C_{6}^{A}(Q^{2})}{m_{N}^{2}}q^{\lambda}q^{\mu},$
(7)

where $Q^{2}=-q^{2}$. Note that $p_{f}\cdot
q=\frac{1}{2}(p_{f}+p_{i}+p_{f}-p_{i})\cdot(p_{f}-p_{i})=\frac{1}{2}(p_{f}^{2}-p_{i}^{2}+q^{2})=\frac{1}{2}(z_{\Delta}^{2}-m_{N}^{2}-Q^{2})$.
Our sign convention for the form factors in Eq. (7) is such that we
parameterize the matrix element of $+A^{\mu}_{j}$ and, thus, our sign
convention follows closely the convention of the nucleon-to-nucleon axial-
vector transition Schindler:2006it . In particular, $C_{5}^{A}(Q^{2})$ and
$C_{6}^{A}(Q^{2})$ correspond to the axial nucleon form factor $G_{A}(Q^{2})$
and the induced pseudoscalar form factor $G_{P}(Q^{2})$, respectively.

###  II.3 Pion-nucleon-delta transition form factor

Assuming isospin symmetry, i.e., equal up-quark and down-quark masses
$m_{u}=m_{d}=\hat{m}$, the divergence of the axial-vector current is given by
Scherer:2002tk ; Schindler:2006it

$\partial_{\mu}A^{\mu}_{j}(x)=\hat{m}P_{j}(x),$ (8)

where

$P_{j}(x)=i\bar{q}(x)\gamma_{5}\tau_{j}q(x)$ (9)

denotes the pseudoscalar density Gasser:1983yg . With the help of the pion
mass $M_{\pi}$ and the pion-decay constant $F_{\pi}$, the isovector operator
$\Phi_{j}(x)\equiv\hat{m}P_{j}(x)/(M_{\pi}^{2}F_{\pi})$ serves as an
interpolating pion field Scherer:2002tk  such that Eq. (8) amounts to the
standard PCAC relation (partially conserved axial-vector current) Adler-Dashen
. By means of $\Phi_{j}(x)$ we define the $\pi N\Delta$ transition form factor
$G_{\pi N\Delta}(Q^{2})$ in analogy to the $\pi N$ form factor $G_{\pi
N}(Q^{2})$ Schindler:2006it ; Gasser:1987rb  as Alexandrou:2009vqd

$\langle\Delta(p_{f},s_{f})||\Phi^{(1)}(0)||N(p_{i},s_{i})\rangle=i\frac{1}{M_{\pi}^{2}+Q^{2}}G_{\pi
N\Delta}(Q^{2})\bar{w}_{\lambda}(p_{f},s_{f})\frac{q^{\lambda}}{m_{N}}u(p_{i},s_{i}).$
(10)

From Eq. (8) we obtain

$iq_{\mu}\langle\Delta(p_{f},s_{f})||A^{\mu(1)}(0)||N(p_{i},s_{i})\rangle=\hat{m}\langle\Delta(p_{f},s_{f})||P^{(1)}(0)||N(p_{i},s_{i})\rangle,$
(11)

which, using Eqs. (7) and (10), results in

$G_{\pi
N\Delta}(Q^{2})=\frac{m_{N}}{F_{\pi}}\frac{M_{\pi}^{2}+Q^{2}}{M_{\pi}^{2}}\left[C_{5}^{A}(Q^{2})-\frac{Q^{2}}{m_{N}^{2}}C_{6}^{A}(Q^{2})\right].$
(12)

In other words, once we know the form factors $C_{5}^{A}(Q^{2})$ and
$C_{6}^{A}(Q^{2})$, we can also extract the form factor $G_{\pi
N\Delta}(Q^{2})$. The $\pi N\Delta$ coupling constant $g_{\pi N\Delta}$ is
defined as

$g_{\pi N\Delta}=G_{\pi N\Delta}(-M_{\pi}^{2}).$ (13)

Since the form factor $C_{6}^{A}(Q^{2})$ has a pole at $Q^{2}=-M_{\pi}^{2}$,
the coupling constant $g_{\pi N\Delta}$ does not vanish despite the factor
$(M_{\pi}^{2}+Q^{2})$ in Eq. (12).

##  III Effective Lagrangian

In this section, we provide the interaction Lagrangians relevant for the
calculation of the isovector axial-vector-current form factors of the nucleon-
to-delta transition in covariant chiral EFT. The effective Lagrangian,
$\mathcal{L}_{\text{eff}}$, consists of a purely pionic, a pion-nucleon, a
pion-delta, and a pion-nucleon-delta Lagrangian, each of which is organized in
a combined derivative and quark-mass expansion. The most general effective
Lagrangian for the calculation of the transition form factors up to and
including order $q^{3}$ is given by

$\mathcal{L}_{\text{eff}}=\mathcal{L}_{\pi}^{(2)}+\mathcal{L}_{\pi}^{(4)}+\mathcal{L}_{\pi
N}^{(1)}+\mathcal{L}_{\pi\Delta}^{(1)}+\mathcal{L}_{\pi
N\Delta}^{(1)}+\mathcal{L}_{\pi N\Delta}^{(2)}+\mathcal{L}_{\pi
N\Delta}^{(3)}+\ldots,$ (14)

where the ellipsis denotes terms which are either of higher order or
irrelevant for our calculation.

The pionic Lagrangians at $\mathcal{O}(q^{2})$ and $\mathcal{O}(q^{4})$ are
given by Gasser:1983yg ; Scherer:2002tk

$\displaystyle\mathcal{L}_{\pi}^{(2)}=$
$\displaystyle\frac{F^{2}}{4}\left\\{\text{Tr}[D_{\mu}U(D^{\mu}U)^{\dagger}]+\text{Tr}(\chi
U^{\dagger}+U\chi^{\dagger})\right\\},$ (15)
$\displaystyle\mathcal{L}_{\pi}^{(4)}=$
$\displaystyle\frac{l_{4}}{4}\text{Tr}[D_{\mu}U(D^{\mu}\chi)^{\dagger}]+D_{\mu}\chi(D^{\mu}U)^{\dagger}]+\ldots,$

respectively, with $\chi=2B(s+ip)$, $s$ and $p$ denoting the scalar and
pseudoscalar external sources Gasser:1983yg . $F$ is the pion-decay constant
in the chiral limit, $F_{\pi}=F[1+\mathcal{O}(\hat{m})]=92.2\;\text{MeV}$, and
$B$ is associated with the scalar singlet quark condensate
$\braket{\bar{q}q}_{0}$ in the chiral limit Gasser:1983yg ; Scherer:2002tk ;
Colangelo:2001sp . We employ SU(2) isospin symmetry, $m_{u}=m_{d}=\hat{m}$,
and the lowest-order prediction for the pion mass squared is $M^{2}=2B\hat{m}$
Gasser:1983yg ; Scherer:2002tk , resulting from inserting the quark masses
into the external scalar field, $s=\hat{m}{\mathbbm{1}}$. The triplet of pion
fields is contained in the unimodular, unitary, $(2\times 2)$ matrix $U$,

$U=u^{2}=\text{exp}\left(i\frac{\Phi}{F}\right),\quad\Phi=\tau_{j}\phi_{j}=\begin{pmatrix}\pi^{0}&\sqrt{2}\pi^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}\end{pmatrix},$ (16)

where $\tau_{j}$ are the Pauli matrices. Introducing external vector fields
$v_{\mu}$ and axial-vector fields $a_{\mu}$ as

$v_{\mu}=\frac{\tau_{j}}{2}v_{\mu,j},\quad
a_{\mu}=\frac{\tau_{j}}{2}a_{\mu,j},$ (17)

and using

$r_{\mu}=v_{\mu}+a_{\mu},\quad l_{\mu}=v_{\mu}-a_{\mu},$ (18)

the covariant derivative of $U$ is defined as

$D_{\mu}U=\partial_{\mu}U-ir_{\mu}U+iUl_{\mu}.$ (19)

The leading-order pion-nucleon Lagrangian reads Scherer:2002tk ; Gasser:1987rb

${\cal L}_{\pi
N}^{(1)}=\bar{\Psi}\left(i\not{D}-m+\frac{\texttt{g}_{A}}{2}\gamma^{\mu}\gamma_{5}u_{\mu}\right)\Psi,\quad\Psi=\begin{pmatrix}p\\\
n\end{pmatrix},$ (20)

where $\Psi$ denotes the nucleon isospin doublet containing the four-component
Dirac fields for the proton and the neutron. The covariant derivative
$D_{\mu}\Psi$ is given by666We do not consider a coupling to an external
isoscalar vector field.

$\begin{split}D_{\mu}\Psi&=(\partial_{\mu}+\Gamma_{\mu})\Psi,\\\
\Gamma_{\mu}&=\frac{1}{2}[u^{\dagger}(\partial_{\mu}-ir_{\mu})u+u(\partial_{\mu}-il_{\mu})u^{\dagger}].\end{split}$
(21)

The chiral vielbein is defined as

$u_{\mu}=u_{\mu,j}\tau_{j}=i[u^{\dagger}(\partial_{\mu}-ir_{\mu})u-u(\partial_{\mu}-il_{\mu})u^{\dagger}].$
(22)

The Lagrangian ${\cal L}_{\pi N}^{(1)}$ contains two free parameters, namely,
the nucleon mass in the chiral limit, $m$, and the axial-vector coupling
constant in the chiral limit, $\texttt{g}_{A}$. Expanding $u_{\mu,j}$ as

$u_{j,\mu}=a_{\mu,j}-\frac{\partial_{\mu}\phi_{j}}{F}+{\cal
O}(v_{\mu}\Phi,a_{\mu}\Phi^{2},\partial_{\mu}\Phi\Phi^{2}),$ (23)

Eq. (20) gives rise to the lowest-order $\pi NN$ vertex as well as the axial-
vector $NN$ transition vertex which are both needed for the one-loop
corrections at ${\cal O}(q^{3})$.

The building blocks for constructing the Lagrangian of the $\Delta$ resonance
can be found in Refs. Hacker:2005fh ; Scherer:2012zzd  and the references
therein. For our purposes, we only need the leading-order contribution,777Note
that the free Lagrangian contains an arbitrary real parameter
$A\neq-\frac{1}{2}$ Moldauer:1956zz ; Nath:1971wp , for which we choose $A=-1$
such that the propagator takes the simplest form.

$\begin{split}\mathcal{L}_{\pi\Delta}^{(1)}&=-\bar{\Psi}_{\mu}\;\xi^{\frac{3}{2}}\Big{[}(i\not{D}-m_{\Delta})g^{\mu\nu}-i(\gamma^{\mu}D^{\nu}+\gamma^{\nu}D^{\mu})+i\gamma^{\mu}\not{D}\gamma^{\nu}+m_{\Delta}\gamma^{\mu}\gamma^{\nu}\\\
&\quad+\frac{\texttt{g}_{1}}{2}(\not{u}\gamma_{5}g^{\mu\nu}-\gamma^{\mu}u^{\nu}\gamma_{5}-u^{\mu}\gamma^{\nu}\gamma_{5}-\gamma^{\mu}\not{u}\gamma_{5}\gamma^{\nu})\Big{]}\xi^{\frac{3}{2}}\Psi_{\nu},\end{split}$
(24)

from the $\pi\Delta$ Lagrangian, where $\Psi_{\nu}$ denotes a vector-spinor
isovector-isospinor field. The isovector-isospinor transforms under the
$1\otimes\frac{1}{2}=\frac{3}{2}\oplus\frac{1}{2}$ representation and, thus,
contains both isospin 3/2 and isospin 1/2 components. In order to describe the
$\Delta$, it is necessary to project onto the isospin-3/2 subspace. The
corresponding matrix representation of the projection operator is denoted by
$\xi^{\frac{3}{2}}$, and the entries of $\xi^{\frac{3}{2}}$ are given by
Scherer:2012zzd

$\xi^{\frac{3}{2}}_{ij}=\delta_{ij}-\frac{1}{3}\tau_{i}\tau_{j}.$

Inserting the expansion of Eq. (23) into Eq. (24), we obtain the leading-order
$\pi\Delta\Delta$ vertex which is proportional to $\texttt{g}_{1}$ and which
is needed for the one-loop corrections at ${\cal O}(q^{3})$.

The leading-order $\pi N\Delta$ chiral Lagrangian is given by [see Eq. (4.200)
of Ref. Scherer:2012zzd  with $\tilde{z}=-1$ for consistency with the choice
$A=-1$]

${\cal L}^{(1)}_{\pi
N\Delta}=\texttt{g}\bar{\Psi}_{\lambda,i}\xi^{\frac{3}{2}}_{ij}(g^{\lambda\mu}-\gamma^{\lambda}\gamma^{\mu})u_{\mu,j}\Psi+\text{H.c.},$
(25)

where H.c. denotes the Hermitian conjugate. Expanding $u_{\mu,j}$ as above,
Eq. (25) gives rise to the leading-order contribution to $C_{5}^{A}(Q^{2})$ as
well as the leading-order $\pi N\Delta$ vertex, which is needed for the
calculation of the loop contributions at ${\cal O}(q^{3})$.

At ${\cal O}(q^{3})$, the higher-order Lagrangians ${\cal L}^{(2)}_{\pi
N\Delta}$ and ${\cal L}^{(3)}_{\pi N\Delta}$ can only contribute at the tree
level. In principle, these Lagrangians were derived in Ref. Jiang:2017yda
(see also Ref. Holmberg:2018dtv ). Taking Eq. (66) of Ref. Jiang:2017yda  for
${\cal L}^{(2)}_{\pi N\Delta}$, there would be no contribution to the form
factors at ${\cal O}(q^{2})$, because the first two terms contain the chiral
vielbein quadratically, and the last term involves the ”wrong” field strength
tensor $f_{+}^{\mu\nu}$. However, as discussed in Appendix A, there are
independent contributions at ${\cal O}(q^{2})$. In fact, this is to be
expected for the following reason. Counting the polarization vector as of
${\cal O}(q)$ and treating only the four-momentum $q$ (but not $p_{f}$) as a
small quantity, we expect from Eqs. (4) and (7) two free parameters related to
$C_{3}^{A}(0)$ and $C_{4}^{A}(0)$. For the sake of simplicity, we will denote
these two parameters by $\alpha$ and $\beta$, respectively (see Appendix A).

Similarly, the ${\cal O}(q^{3})$ Lagrangian of Ref. Jiang:2017yda  produces
fewer contributions to the form factors than is expected from the counting of
momenta (and the polarization vector). Since it is not the purpose of this
paper to construct the most general Lagrangian at ${\cal O}(q^{3})$, we have
decided to Taylor expand the form factors and keep the expansion coefficients
as free parameters.

##  IV Results

Figure 2: Diagrams which, after renormalization, result in non-vanishing
contributions to the axial-vector nucleon-to-delta transition form factors up
to and including $\mathcal{O}(q^{3})$. The double, solid, dashed, and wiggly
lines correspond to the delta, nucleon, pion, and the external axial-vector
source, respectively.

Figure 2 shows those tree-level and one-loop Feynman diagrams that generate a
non-vanishing contribution to the nucleon-to-delta transition matrix element
of the isovector axial-vector current. In principle, the renormalized vertex
is obtained by multiplying the contributions of Fig. 2 by the square roots of
the wave function renormalization constants $Z_{N}$ and $Z_{\Delta}$. In
practice, we evaluate the loop diagrams in the framework of dimensional
regularization at the renormalization scale $\mu=1$ GeV. We apply the modified
minimal subtraction scheme of ChPT ($\widetilde{\text{MS}}$) Scherer:2012zzd ;
Gasser:1983yg  by dropping infinite parts in terms of the combination
$2/(n-4)-[\text{ln}(4\pi+\Gamma^{\prime}(1)+1)]$, where $n$ denotes the number
of space-time dimensions. We combine the remaining finite pieces with the
available renormalized free parameters.

In fact, for the actual calculation, we use a decomposition of
$\Gamma^{\lambda\mu}_{A}$ which differs from Eq. (7), namely,

$\Gamma^{\lambda\mu}_{A}=k_{1}q^{\lambda}p_{f}^{\mu}+k_{2}q^{\lambda}q^{\mu}+k_{3}q^{\lambda}\gamma^{\mu}+k_{4}g^{\lambda\mu}.$
(26)

For each diagram, we extract the four coefficients $k_{i}$ and determine their
contributions to the form factors $C^{A}_{i}(Q^{2})$, using the relations

$\displaystyle C_{3}^{A}$ $\displaystyle=-m_{N}k_{3},$ $\displaystyle
C_{4}^{A}$ $\displaystyle=-m_{N}^{2}k_{1},$ $\displaystyle C_{5}^{A}$
$\displaystyle=\frac{1}{2}(z_{\Delta}^{2}-m_{N}^{2}-Q^{2})k_{1}+(z_{\Delta}-m_{N})k_{3}+k_{4},$
$\displaystyle C_{6}^{A}$ $\displaystyle=m_{N}^{2}k_{2}.$

Using the strategy outlined in Sec. III, the tree-level contributions to the
form factors can be written as888For the sake of simplicity, we omit the
contribution of diagrams (e) and (g) in the formula for
$C_{6\,\text{tree}}^{A}(Q^{2})$.

$\begin{split}C_{3\,\text{tree}}^{A}(Q^{2})&=\alpha,\\\
C_{4\,\text{tree}}^{A}(Q^{2})&=\beta,\\\
C_{5\,\text{tree}}^{A}(Q^{2})&=\texttt{g}+\gamma M^{2}+\delta Q^{2},\\\
C_{6\,\text{tree}}^{A}(Q^{2})&=m_{N}^{2}\frac{\texttt{g}+\gamma M^{2}+\epsilon
M^{2}}{M^{2}+Q^{2}}+\zeta.\end{split}$ (27)

We have explicitly shown the quark-mass dependence in terms of the lowest-
order squared pion mass $M^{2}$. In addition, we made use of the analogy of
the form factors $C_{5}^{A}(Q^{2})$ and $C_{6}^{A}(Q^{2})$ to the nucleon form
factors $G_{A}(Q^{2})$ and $G_{P}(Q^{2})$ (see Ref. Schindler:2006it ). At
leading and next-to-leading order, the form factors $C_{3}^{A}(Q^{2})$,
$C_{4}^{A}(Q^{2})$, and $C_{5}^{A}(Q^{2})$ are constant. The parameters g,
$\alpha$, and $\beta$ are independent low-energy constants, i.e., they are not
predicted by chiral symmetry. Using the relation from the static quark model
with SU(6) symmetry results in the estimate
$\texttt{g}=3\sqrt{2}\texttt{g}_{A}/5$ Unal:2018ruo ; Hemmert:1997ye . At this
order, the form factor $C_{6}^{A}(Q^{2})$ is predicted as

$C_{6}^{A}(Q^{2})=\frac{m^{2}_{N}}{M^{2}+Q^{2}}\texttt{g}.$

Turning to ${\cal O}(q^{3})$, we will now have both tree-level modifications
as well as loop contributions. The parameter $\gamma$ corresponds to a quark-
mass correction to the nucleon-to-delta transition axial-vector coupling
constant $g_{AN\Delta}\equiv G_{5}^{A}(0)$. Furthermore, $\delta$ contributes
to the mean-square axial transition radius

$\langle
r^{2}_{AN\Delta}\rangle\equiv-\frac{6}{g_{AN\Delta}}\left.\frac{dG^{A}_{5}(Q^{2})}{dQ^{2}}\right|_{Q^{2}=0}.$
(28)

The parameter $\epsilon$ enters into the calculation of the generalized
Goldberger-Treiman discrepancy Pagels:1969ne

$\Delta\equiv 1-\frac{m_{N}g_{AN\Delta}}{F_{\pi}g_{\pi N\Delta}},$ (29)

where $g_{\pi N\Delta}$ is defined in Eq. (13). Finally, the parameter $\zeta$
is related to the $y$-intercept of $C_{6}^{A}(Q^{2})$.

Unfortunately, there is very little empirical information on the form factors
$C_{i}^{A}(Q^{2})$. In order to obtain some estimate about the parameters and
shape of our theoretical results, we perform a fit to the parametrization

$C_{i}^{A}(Q^{2})=\frac{C_{i}^{A}(0)\left(1+a_{i}\frac{Q^{2}}{b_{i}+Q^{2}}\right)}{\left(1+\frac{Q^{2}}{M_{A}^{2}}\right)^{2}}\quad(i=3,4,5)$
(30)

(see Appendix D of Ref. Schreiner:1973mj ). The form factor $C_{6}^{A}(Q^{2})$
is assumed to be dominated by the pion-pole contribution. In particular, we
make use of the axial-vector form factor parameters of the Adler model (see
Table D.1 of Ref. Schreiner:1973mj ),

$\begin{split}C_{3}^{A}(0)&=0,\quad a_{3}=0,\quad b_{3}=0,\\\
C_{4}^{A}(0)&=-0.3,\quad a_{4}=-1.21,\quad b_{4}=2.0\,\text{GeV}^{2},\\\
C_{5}^{A}(0)&=1.2,\quad a_{5}=-1.21,\quad
b_{5}=2.0\,\text{GeV}^{2}.\end{split}$ (31)

This parametrization was used by the authors of Ref. Kitagaki:1990vs  to
extract the axial mass as $M_{A}=(1.28^{+0.08}_{-0.10})$ GeV from their
analysis of the $\Delta^{++}$ production reaction
$\nu_{\mu}+d\to\mu^{-}+\Delta^{++}+n_{s}$, where $n_{s}$ refers to a spectator
neutron.

To emphasize the low-$Q^{2}$ region, we choose the $Q^{2}$ values at which to
evaluate the empirical form factors according to $Q_{n}^{2}=n^{2}\times 0.004$
GeV2 ($n=0,1,\ldots,28$). For the fits, we employ the Mathematica routine
NonlinearModelFit. Our loop diagrams contain the low-energy constants (LECs)
$\texttt{g}_{A}$, $\texttt{g}_{1}$, and g. In the loop integrals, we
approximate $\texttt{g}_{A}$ by $g_{A}=1.28$ (empirical value $g_{A}=1.2756\pm
0.0013$ Zyla:2020zbs ), because the difference between $\texttt{g}_{A}$ and
$g_{A}$ is of order $M^{2}$ in the chiral expansion and, therefore, using
$g_{A}$ introduces an error of higher order beyond the accuracy of our
calculation. For the other two coupling constants, we make use of an SU(6)
spin-flavor quark-model relation Scherer:2012zzd ; Hemmert:1997ye ,

$\texttt{g}_{1}=\frac{9}{5}\texttt{g}_{A},\quad\texttt{g}=\frac{3}{5}\sqrt{2}\texttt{g}_{A},$
(32)

resulting in $\texttt{g}_{1}=2.30$ and $\texttt{g}=1.08$, respectively.
Furthermore, we replace $M^{2}$ by $M_{\pi}^{2}$ and use $M_{\pi}=0.135$ GeV.
From a fit to the form factors $C_{3}^{A}(Q^{2})$ and $C_{5}^{A}(Q^{2})$, we
obtain for the LECs $\beta$, $\gamma$, and $\delta$ the values

$\beta=0.335,\quad\gamma=-84.7\,\text{GeV}^{-2},\quad\delta=-3.20\,\text{GeV}^{-2}.$
(33)

The value $C_{5}^{A}(0)$ is the analogue of $g_{A}$. The individual
contributions to $C_{5}^{A}(0)$ are given by

$C_{5}^{A}(0)=1.20=\texttt{g}+\gamma M_{\pi}^{2}+\text{loops}=1.08-1.54+1.66.$
(34)

The $M_{\pi}^{2}$ term plus the loops amount to a 10% correction of the
leading-order term.

In Fig. 3 (a), we display the individual contributions to $C_{5}^{A}(Q^{2})$
for $\texttt{g}_{1}=2.30$; the total result is given by the red solid line,
the loop contribution by the green dashed line, and the tree-level
contribution by the green dotted line. The total result shows, as was to be
expected from a calculation at ${\cal O}(q^{3})$, essentially a linear
behavior as a function of $Q^{2}$. The loop contribution rises with increasing
$Q^{2}$, whereas the tree-level contribution decreases linearly with $Q^{2}$.
The two parameters $\gamma$ and $\delta$ are set in such a way that a good
correspondence with the empirical parametrization of the Adler model was
achieved (see Fig. 4). In the $Q^{2}$ range considered, the fit differs by
less than 3% from the empirical parametrization.

Figure 3:  Individual contributions to $C_{5}^{A}(Q^{2})$. The red solid,
green dotted, and green dashed lines correspond to the total result, the loop
contribution, and the tree-level contribution, respectively. The left and
right panels show the fits for $\texttt{g}_{1}=2.30$ and
$\texttt{g}_{1}=-1.21$, respectively.

Using Eq. (28), we obtain for the mean-square axial transition radius

$\langle r^{2}_{AN\Delta}\rangle=0.345\,\text{fm}^{2},$ (35)

which has to be compared with $\langle
r^{2}_{AN\Delta}\rangle=0.427\,\text{fm}^{2}$ of the empirical
parametrization. A smaller axial radius for the fit was to be expected,
because the empirical parametrization contains more curvature while the fit
behaves essentially linearly. Therefore, as can be seen from Fig. 4, the fit
generates a slightly flatter behavior as a function of $Q^{2}$.

Figure 4:  Fit of the form factor $C_{5}^{A}(Q^{2})$ for
$\texttt{g}_{1}=2.30$. The red solid and blue dashed lines correspond to the
real part and the imaginary part, respectively. The black dotted line is the
empirical parametrization of the Adler model.

As no reliable experimental data exist for $\langle r^{2}_{AN\Delta}\rangle$,
we also quote for comparison the empirical values for the mean-square axial
radius for the nucleon: $\langle r_{A}^{2}\rangle=(0.444\pm 0.018)$ fm2
Bernard:2001rs  and $\langle r_{A}^{2}\rangle=(0.453\pm 0.023)$ fm2
Bodek:2007ym . In fact, based again on the quark-model relation of Eq. (32),
the simplest assumption would be
$C^{A}_{5}(Q^{2})=\frac{3}{5}\sqrt{2}G_{A}(Q^{2})$ Unal:2018ruo , resulting in
$\langle r^{2}_{AN\Delta}\rangle=\langle r_{A}^{2}\rangle$.

Instead of using the quark-model estimate for $\texttt{g}_{1}$, we also made
use of the value $\texttt{g}_{1}=-1.21$ which was obtained in Ref. Yao:2016vbz
from a fit to the $\pi N$ phase shifts of the $S$ and $P$ waves. As stated in
Ref. Yao:2016vbz , since $\texttt{g}_{1}$ appears only in the loop
contribution of their calculation, a precise determination of its value is not
to be expected. Note, in particular, that $\texttt{g}_{1}$ comes out with an
opposite sign relative to the quark-model estimate. Using
$\texttt{g}_{1}=-1.21$, we obtain for the LECs $\beta$, $\gamma$, and $\delta$
the values

$\beta=-0.0691,\quad\gamma=-19.8\,\text{GeV}^{-2},\quad\delta=-1.22\,\text{GeV}^{-2}.$
(36)

Here, the individual contributions to $C_{5}^{A}(0)$ are given by

$C_{5}^{A}(0)=1.19=\texttt{g}+\gamma M_{\pi}^{2}+\text{loops}=1.08-0.36+0.47.$
(37)

The individual contributions to $C_{5}^{A}(Q^{2})$ for $\texttt{g}_{1}=-1.21$
are shown in Fig. 3 (b). In the range $0\leq Q^{2}\leq 0.3$ GeV2, the total
result for $\texttt{g}_{1}=-1.21$ deviates from that for $\texttt{g}_{1}=2.30$
by less than 1%. However, the loop contribution behaves very differently,
namely, at $Q^{2}=0$ it starts at a much lower value and it decreases with
increasing $Q^{2}$ as opposed to a (stronger) increase with $Q^{2}$ for
$g_{1}=2.30$. In the end, this behavior is compensated by the rather different
values of $\gamma$ and $\delta$.

The results for $C_{6}^{A}(Q^{2})$ were obtained by fitting to the pion-pole-
dominated expression,

$C_{6}^{A}(Q^{2})=m_{N}^{2}\frac{C_{5}^{A}(Q^{2})}{M_{\pi}^{2}+Q^{2}},$

where $C_{5}^{A}(Q^{2})$ is taken from Eq. (30). Such a fit contains two free
parameters, namely, $\epsilon$ and $\zeta$, which were obtained as

$\begin{split}\epsilon=59.3\,\text{GeV}^{-2}\quad\text{and}\quad\zeta=-11.5\quad&\text{for}\quad\texttt{g}_{1}=2.30,\\\
\epsilon=13.4\,\text{GeV}^{-2}\quad\text{and}\quad\zeta=-3.40\quad&\text{for}\quad\texttt{g}_{1}=-1.21.\end{split}$
(38)

The corresponding results for $C_{6}^{A}(Q^{2})$ are shown in Fig. 5. For
$Q^{2}\lessapprox 0.16$ GeV2, the fit for $\texttt{g}_{1}=2.30$ is below the
empirical form factor and deviates by less than 2.6% from the empirical
$C_{6}^{A}(Q^{2})$. Beyond $Q^{2}\approx 0.16$ GeV2, the fit is above the
empirical result with a continuously increasing deviation reaching $-23$% at
$Q^{2}=0.3$ GeV2. For $\texttt{g}_{1}=-1.21$, the fit is generally closer to
the empirical result than for $\texttt{g}_{1}=2.30$. For $Q^{2}\lessapprox
0.06$ GeV2, the fit is below the empirical form factor, beyond $Q^{2}=0.06$
GeV2 it is above. Again, the maximal deviation happens at $Q^{2}=0.3$ GeV2 and
amounts to $-3.2$%.

Figure 5:  Fit of the form factor $C_{6}^{A}(Q^{2})$ for $\texttt{g}_{1}=2.30$
(left panel) and $\texttt{g}_{1}=-1.21$ (right panel), respectively. The red
solid and blue dashed lines correspond to the real part and the imaginary
part, respectively. The black dotted line is the pion-pole-dominated empirical
parametrization with $C_{5}^{A}(Q^{2})$ of the Adler model.

Given the results for the form factors $C_{5}^{A}(Q^{2})$ and
$C_{6}^{A}(Q^{2})$, we are now also in the position to discuss the $\pi
N\Delta$ transition form factor $G_{\pi N\Delta}(Q^{2})$ of Eq. (12) (see Fig.
6). For $\texttt{g}_{1}=2.30$, our result deviates very quickly with
increasing $Q^{2}$ from the pion-pole-dominated empirical result,

$G_{\pi N\Delta}(Q^{2})=\frac{m_{N}}{F_{\pi}}C_{5}^{A}(Q^{2}).$

In fact, the linear combination of Eq. (12) involves a delicate interplay
between the terms

$\frac{m_{N}}{F_{\pi}}\frac{Q^{2}}{M_{\pi}^{2}}C_{5}^{A}(Q^{2})\quad\text{and}\quad-\frac{m_{N}}{F_{\pi}}\frac{Q^{2}}{m_{N}^{2}}\frac{M_{\pi}^{2}+Q^{2}}{M_{\pi}^{2}}C_{6}^{A}(Q^{2}).$

The strong downward trend for increasing $Q^{2}\gtrapprox 0.15$ GeV2 is due to
a relatively large negative contribution proportional to $(Q^{2})^{3}$,
originating from the second term. The situation is somewhat better for
$\texttt{g}_{1}=-1.21$, where the deviation starts at $Q^{2}\approx 0.2$ GeV2.
This does not come as a surprise, because both $C_{5}^{A}(Q^{2})$ and, in
particular, $C_{6}^{A}(Q^{2})$ are better described for
$\texttt{g}_{1}=-1.21$. For the $\pi N\Delta$ coupling constant, we obtain

$\displaystyle G_{\pi N\Delta}(-M_{\pi}^{2})=g_{\pi N\Delta}$
$\displaystyle=12.8\quad\text{for}\quad\texttt{g}_{1}=2.30,$ $\displaystyle
g_{\pi N\Delta}$ $\displaystyle=12.5\quad\text{for}\quad\texttt{g}_{1}=-1.21.$

These values result in a generalized Goldberger-Treiman discrepancy of

$\Delta=0.0533\quad\text{and}\quad\Delta=0.0305,$ (39)

for $\texttt{g}_{1}=2.30$ and $\texttt{g}_{1}=-1.21$, respectively. Even
though the imaginary parts of $C_{5}^{A}(Q^{2})$ and $C_{6}^{A}(Q^{2})$ are
small, we obtain a noticeable imaginary part for $G_{\pi N\Delta}(Q^{2})$. In
the present case, the imaginary part originates entirely from the loop
contributions. In this context, one should keep in mind that, in the complex-
mass scheme, the low-energy constants can also be complex numbers. Therefore,
in principle, they could generate additional imaginary contributions. As our
empirical ansatz for the form factors is real, we only fitted the real part of
the form factors and left the imaginary tree-level contributions unspecified.

Figure 6:  Results for the transition form factor $G_{\pi N\Delta}(Q^{2})$ of
Eq. (12) for $\texttt{g}_{1}=2.30$ (left panel) and $\texttt{g}_{1}=-1.21$
(right panel), respectively. The red solid and blue dashed lines correspond to
the real part and the imaginary part, respectively. The black dotted line is
the empirical result using $C_{5}^{A}(Q^{2})$ and $C_{6}^{A}(Q^{2})$ of the
Adler model.

Finally, we turn to the form factors $C_{3}^{A}(Q^{2})$ and
$C_{4}^{A}(Q^{2})$. These form factors have no analogue in the nucleon case.
In Fig. 7, we show the loop contribution to $C_{3}^{A}(Q^{2})$. The parameter
$\alpha$ of Eq. (27) serves to shift the whole curve up or down and has been
set to zero in the figure. No matter what the value for $\alpha$ is, our
result is incompatible with the empirical ansatz $C_{3}^{A}(Q^{2})=0$. By far
the largest contribution to $C_{3}^{A}(Q^{2})$ originates from diagram (t) of
Fig. 2 and is proportional to $\texttt{g}\texttt{g}_{1}^{2}$. Therefore, the
case $\texttt{g}_{1}=2.30$ produces a much stronger (negative) slope than
$\texttt{g}_{1}=-1.21$. The result for $C_{4}^{A}(Q^{2})$ is shown in Fig. 8.
Here, we obtain a good description of the empirical form factor for
$\texttt{g}_{1}=-1.21$, while the case $\texttt{g}_{1}$ again produces a much
stronger (positive) slope.

Figure 7:  Form factor $C_{3}^{A}(Q^{2})$ for $\texttt{g}_{1}=2.30$ (left
panel) and $\texttt{g}_{1}=-1.21$ (right panel), respectively. The red solid
and blue dashed lines correspond to the real part and the imaginary part,
respectively. In the Adler model, the form factor is set to zero (black dotted
lines). Figure 8:  Form factor $C_{4}^{A}(Q^{2})$ for $\texttt{g}_{1}=2.30$
(left panel) and $\texttt{g}_{1}=-1.21$ (right panel), respectively. The red
solid and blue dashed lines correspond to the real part and the imaginary
part, respectively. The black dotted line is the empirical parametrization of
the Adler model.

##  V Comparison with other work

In this section, we provide a comparison of our calculation of the axial-
vector current nucleon-to-delta transition form factors with other work.
First, we compare our calculation with another calculation in the framework of
covariant chiral perturbation theory Geng:2008bm . Our starting point is
different in that we are using the isovector-isospinor representation for the
$\Delta$ involving the projection operator $\xi$, whereas Ref. Geng:2008bm
directly uses an isospin quadruplet. Furthermore, we have one more effective
parameter than Ref. Geng:2008bm , which affects the tree-level result of
$C_{6}^{A}(Q^{2})$. The parameters entering the loop diagrams are essentially
the same, in particular, Ref. Geng:2008bm  also uses the quark-model
prediction $H_{A}=\texttt{g}_{1}=\frac{9}{5}g_{A}$. Nevertheless, our loop
contributions are, in general, substantially larger in magnitude; this is
particularly true for $C_{3}^{A}(Q^{2})$. On the other hand, for both
$\texttt{g}_{1}=2.30$ and $\texttt{g}_{1}=-1.21$, our total result for
$C_{5}^{A}(Q^{2})$ is closer to the empirical parametrization than the result
of Ref. Geng:2008bm , in particular for $Q^{2}\gtrapprox 0.1$ GeV2, whereas
$C_{6}^{A}(Q^{2})$ turns out to be very similar. For $C_{3}^{A}(Q^{2})$ we
obtain an opposite sign in comparison to Ref. Geng:2008bm . Finally, for
$C_{4}^{A}(Q^{2})$, an ambiguous situation arises. For $\texttt{g}_{1}=-1.21$
there is a very good correspondence with the empirical form factor, whereas
the result of Ref. Geng:2008bm  is substantially below the empirical form
factor. However, for $\texttt{g}_{1}=2.30$, our result increases too quickly,
yielding too large a slope. In the framework of nonrelativistic chiral
effective field theory to leading one-loop order Procura:2008ze , the $Q^{2}$
dependence of the form factors $C_{5}^{A}(Q^{2})$ and $C_{6}^{A}(Q^{2})$ to
order three is entirely generated by counter terms and the pion-pole
contribution, $C_{3}^{A}(Q^{2})$ and $C_{4}^{A}(Q^{2})$ gain $Q^{2}$ and
$M_{\pi}$ dependence only at higher orders. Our values for the generalized
Goldberger-Treiman discrepancy are slightly larger than the $\sim 2\%$
predicted in the framework of heavy-baryon chiral perturbation theory
Zhu:2002kh . The inclusion of the $a_{1}$ meson as an explicit dynamical
degree of freedom was discussed in Ref. Unal:2018ruo . Besides the
$a_{1}$-meson mass, this introduces one additional effective parameter. The
$a_{1}$ meson effects the shape of the form factors $C_{5}^{A}(Q^{2})$ and
$C_{6}^{A}(Q^{2})$; the form factor $C_{5}^{A}(Q^{2})$ develops more curvature
and $C_{6}^{A}(Q^{2})$ lies above the pion-pole dominance prediction for
$Q^{2}\gtrapprox 0.1$ GeV2. Results from lattice QCD for $C_{5}^{A}(Q^{2})$
and $C_{6}^{A}(Q^{2})$ were reported in Refs. Alexandrou:2009vqd ;
Alexandrou:2010uk . In general, the values of $C_{5}^{A}(0)$ in their
calculations come out as smaller than one, i.e., they are also smaller than
our results. At the same time, the axial mass turns out to be larger than
$M_{A}=1.28$ GeV, corresponding to a smaller mean-square axial transition
radius. This corresponds to our findings. At low $Q^{2}$, the lattice results
for $C_{6}^{A}(Q^{2})$ turn out to be smaller than our results. Finally, Ref.
Alexandrou:2010uk  contains two fits to $G_{\pi N\Delta}(Q^{2})$. For the
coupling constant $g_{\pi N\Delta}$, which Ref. Alexandrou:2010uk  defines at
$Q^{2}=0$ rather than $Q^{2}=-M_{\pi}^{2}$, values between 8.44 and 16.3 are
obtained, which have to be compared with our values 12.8 and 12.5. As a
representative example for studies in the framework of nonrelativistic
(chiral) quark models, we take the results of Ref. BarquillaCano:2007yk .
First, we observe that the predictions of the form factors in the quark model
are real quantities instead of complex functions in our calculation. The
dominant form factor $C_{5}^{A}(Q^{2})$ starts with $C_{5}^{A}(0)=0.93$ and
produces curvature according to an axial-vector meson dominance coupling (at
the quark level) similar to Ref. Unal:2018ruo . The axial radius is predicted
as $\langle r_{AN\Delta}^{2}\rangle=0.59$ fm2 . They also predict a non-zero
form factor $C_{3}^{A}(Q^{2})$ in the low-$Q^{2}$ region, but of significantly
smaller magnitude. Besides Ref. Kitagaki:1990vs , from which we took the
empirical parametrization of the form factors, some more recent experimental
information is available for the linear combination
$G_{N\Delta}^{A}(Q^{2})=\frac{1}{2}[m_{N}^{2}-m_{\Delta}^{2}+Q^{2}]C_{4}^{A}(Q^{2})-m_{N}^{2}C_{5}^{A}(Q^{2})$,
which was extracted from the parity-violating asymmetry in inelastic electron-
nucleon scattering near the $\Delta$ resonance. Reference Androic:2012doa
reports999Reference Androic:2012doa  does not quote any units, even though in
natural units the linear combination has dimension energy squared.
$G_{N\Delta}^{A}(Q^{2}=0.34~{}\text{GeV}^{2})=-0.05\pm(0.35)_{stat}\pm(0.34)_{sys}\pm(0.06)_{theory}$,
whereas we obtained $-0.584$ GeV2 for $\texttt{g}_{1}=2.30$ and $-0.544$ GeV2
for $\texttt{g}_{1}=-1.21$.

##  VI Summary

We analyzed the low-$Q^{2}$ behavior of the axial-vector nucleon-to-delta
transition form factors $C^{A}_{3}(Q^{2})$, $C^{A}_{4}(Q^{2})$,
$C^{A}_{5}(Q^{2})$, and $C^{A}_{6}(Q^{2})$ at the one-loop level of
relativistic baryon chiral perturbation theory. In total, the calculation
involves six free parameters $\alpha$, $\beta$, $\gamma$, $\delta$,
$\epsilon$, and $\zeta$ [see Eqs. (27)]. The constants $g_{A}$, $F_{\pi}$,
$M_{\pi}$, $m_{N}$, and $z_{\Delta}$ were fixed in terms of their empirical
values. For the coupling constant g we made use of the quark-model prediction
$\texttt{g}=\frac{3}{5}\sqrt{2}g_{A}$. Finally, for the coupling constant
$\texttt{g}_{1}$, appearing in certain loop diagrams only, we considered two
scenarios: we either made use of the quark-model prediction
$\texttt{g}_{1}=\frac{9}{5}g_{A}=2.30$ or of $\texttt{g}_{1}=-1.21$ as
obtained from an analysis of the $\pi N$ phase shifts of the $S$ and $P$
waves. Since there is essentially no direct experimental information on the
form factors available, we took the empirical parametrizations used in the
analysis of Ref. Kitagaki:1990vs  to determine our parameters. For our fits,
we chose the interval $0\leq Q^{2}\leq 0.3136$ GeV2, where the upper end of
the interval is likely to be at the verge of the applicability of a one-loop
calculation. For the form factor $C_{5}^{A}(Q^{2})$ we obtain good
descriptions for both $\texttt{g}_{1}=2.30$ and $\texttt{g}_{1}=-1.21$,
deviating from the empirical form by less than 3% and 1%, respectively. As can
be seen from Fig. 3, the loop corrections are sizable and their slope depends
on the sign of $\texttt{g}_{1}$. As a consequence, the total result involves a
delicate interplay between the loop contributions and the parameters $\gamma$
and $\delta$ [see Eqs. (33) and (36)]. The parameters $\epsilon$ and $\zeta$
were determined from the fit to $C_{6}^{A}(Q^{2})$, where, again, for
$\texttt{g}_{1}=-1.21$ our result is closer to the empirical form factor than
for $\texttt{g}_{1}=2.30$ (see Fig. 5). As a result, the $\pi N\Delta$
transition form factor $G_{\pi N\Delta}(Q^{2})$ deviates from the simple
expectation $G_{\pi N\Delta}(Q^{2})=\frac{m_{N}}{F_{\pi}}C_{5}^{A}(Q^{2})$,
again, more so for $\texttt{g}_{1}=2.30$ than for $\texttt{g}_{1}=-1.21$ (see
Fig. 6). For the $\pi N\Delta$ coupling constant we obtained $g_{\pi
N\Delta}=12.8$ for $\texttt{g}_{1}=2.30$ and $g_{\pi N\Delta}=12.5$ for
$\texttt{g}_{1}=-1.21$, resulting in the Goldberger-Treiman discrepancies
$\Delta=0.0533$ and $\Delta=0.0305$, respectively. The parameters $\alpha$ and
$\beta$ are responsible for vertically shifting the curves of the form factors
$C_{3}^{A}(Q^{2})$ and $C_{4}^{A}(Q^{2})$, respectively, they cannot, however,
modify their shapes. Therefore, the loop contributions are, to some extent, a
unique feature of the predictions for $C_{3}^{A}(Q^{2})$ and
$C_{4}^{A}(Q^{2})$. In particular, our calculation predicts $C_{3}^{A}(Q^{2})$
to be different from zero in contrast to the empirical parametrization
$C_{3}^{A}(Q^{2})=0$ (see Fig. 7). Moreover, for $\texttt{g}_{2}=-1.21$ we
obtain a very good agreement between our result for $C_{3}^{A}(Q^{2})$ and the
empirical form factor (see Fig. 8). A somewhat surprising feature is the fact
that the negative value of $\texttt{g}_{1}$ in all cases gives a better
agreement with the empirical form factors than the quark-model result which
uniquely predicts a positive sign. Unfortunately, as in the case of $\pi N$
scattering, $\texttt{g}_{1}$ does not enter the calculation at leading order
but only at the loop level. More about the sign of $\texttt{g}_{1}$ could
possibly be learned from radiative pion-nucleon scattering $\pi N\to\pi\gamma
N$ or radiative pion photoproduction $\gamma N\to\gamma\pi N$ in the
$\Delta$-resonance region, where the $\pi\Delta\Delta$ vertex contributes at
tree level and thus at leading order.

##  VII Acknowledgments

The authors would like to thank J. Gegelia for valuable discussions.

##  Appendix A ${\cal O}(q^{2})$ Lagrangian

For our calculation, we need the pion-nucleon-delta interaction vertex and the
axial-vector-nucleon-delta interaction vertex. The building blocks that
potentially contribute are the chiral vielbein [see Eqs. (22) and (23)] and

$\displaystyle\chi_{-}$ $\displaystyle=u^{\dagger}\chi
u^{\dagger}-u\chi^{\dagger}u\to
M^{2}(U^{\dagger}-U)\to-2iM^{2}\frac{\tau_{i}\phi_{i}}{F},$ (40)
$\displaystyle f_{-\mu\nu}$
$\displaystyle=uf_{L\mu\nu}u^{\dagger}-u^{\dagger}f_{R\mu\nu}u\to
f_{L\mu\nu}-f_{R\mu\nu}\to-2(\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu})=-\tau_{i}(\partial_{\mu}a_{\nu,i}-\partial_{\nu}a_{\mu,i}).$
(41)

One also has to consider covariant derivatives of these building blocks.

According to Jiang et al. Jiang:2017yda , the Lagrangian at ${\cal O}(q^{2})$,
${\cal L}_{\pi N\Delta}^{(2)}$, contains three structures [see Eq. (66) of
Ref. Jiang:2017yda ]. The first two structures are proportional to the product
$u_{\mu,i}u_{\nu,j}$ and thus contribute neither to the $\pi N\Delta$
interaction vertex nor to the $aN\Delta$ interaction vertex. The third
structure is proportional to $f_{+\mu\nu}$ which contributes to the $vN\Delta$
interaction vertex but not to the $aN\Delta$ interaction vertex. In other
words, according to Jiang et al., there are no contact interaction
contributions to the transition form factors at ${\cal O}(q^{2})$.

Jiang et al. compare their results with Ref. Hemmert:1997ye , which they quote
as their Eq. (67). However, Hemmert et al. Hemmert:1997ye  did not construct
the covariant version but rather the heavy-baryon version of the Lagrangian.
According to Eq. (82) of Ref. Hemmert:1997ye , they factor out
$\exp(-iM_{0}v\cdot x)$ with a common mass $M_{0}$ for both the nucleon field
and the $\Delta$ field. The relevant heavy-baryon Lagrangian is then given in
Eq. (112), where $N$ and $T$ are heavy-baryon fields. Note that there is no
covariant Lagrangian in Ref. Hemmert:1997ye . In other words, Jiang et al.
must have reconstructed their Eq. (67) from the heavy-baryon Lagrangian. We
make use of the results of section 5.5. of Ref. Scherer:2002tk  to establish
the connection. Using Eq. (5.122) of Ref. Scherer:2002tk , a single term
$v^{\mu}$ originates from $\gamma^{\mu}$ and $2S^{\mu}$ from
$\gamma^{\mu}\gamma_{5}$, respectively.

Let us have a look at the first term of Eq. (112) of Hemmert et al. This
should result from

$\frac{1}{2M_{0}}\bar{\Psi}^{\mu}_{i}b_{1}if_{+\mu\nu,i}\frac{1}{2}\gamma^{\nu}\gamma_{5}\Psi+\text{H.c.}=\frac{1}{2M_{0}}\left(-\frac{1}{2}b_{1}i\bar{\Psi}^{\mu}_{i}f_{+\mu\nu,i}\frac{1}{2}\gamma_{5}\gamma^{\nu}\Psi+\text{H.c.}\right),$

which, apart from a factor $1/(2M_{0})$, agrees with the first term of Eq.
(67) of Jiang et al.101010We left out the projector
$O^{\mu\nu}_{A,n}(z_{n})=g^{\mu\nu}+(z_{n}+\frac{1}{2}(1+4z_{n})A)\gamma^{\mu}\gamma^{\nu}$
as well as the projector $\xi_{ij}^{\frac{3}{2}}$. Now let us turn to the
second term of Hemmert et al. which should originate from

$\frac{1}{2M_{0}}\bar{\Psi}^{\mu}_{i}ib_{2}f_{-\mu\nu,i}\gamma^{\nu}\Psi+\text{H.c.}$

Comparing with Eq. (67) of Jiang et al., we see that they took the wrong
operator $D^{\nu}$ instead of $\gamma^{\nu}$ and then argue that such a term
can be eliminated using arguments given in a previous section of Ref.
Jiang:2017yda . In fact, Holmberg and Leupold Holmberg:2018dtv  also obtain a
structure analogous to the $b_{2}$ term in their construction for the
decuplet-to-octet transition Lagrangian at next-to-leading order.

We will now show that the $b_{2}$ term gives an explicit contribution to the
form factor $C_{3}^{A}(Q^{2})$ at ${\cal O}(q^{2})$. Using Eq. (41) and after
dropping the factor $1/(2M_{0})$, from the $b_{2}$ term we obtain the
Lagrangian

$ib_{2}\bar{\Psi}_{\lambda,i}\xi^{\frac{3}{2}}_{ij}(-\partial^{\lambda}a^{\mu}_{j}+\partial^{\mu}a^{\lambda}_{j})\gamma_{\mu}\Psi+\text{H.c.},$

resulting in the form factor contribution

$\displaystyle
ib_{2}\bar{w}_{\lambda}\left((-(-iq^{\lambda}))\epsilon^{\mu}+(-iq^{\mu})\epsilon^{\lambda}\right)\gamma_{\mu}u=$
$\displaystyle
ib_{2}\bar{w}_{\lambda}(iq^{\lambda}\gamma^{\mu}-i\not{q}g^{\lambda\mu})u\epsilon_{\mu}=-b_{2}\bar{w}_{\lambda}(q^{\lambda}\gamma^{\mu}-\not{q}g^{\lambda\mu})u\epsilon_{\mu}$
$\displaystyle=$ $\displaystyle
m_{N}b_{2}\bar{w}_{\lambda}\frac{\not{q}g^{\lambda\mu}-q^{\lambda}\gamma^{\mu}}{m_{N}}u\epsilon_{\mu},$

from which we obtain the contribution $m_{N}b_{2}\equiv\alpha$ to the form
factor $C_{3}^{A}(Q^{2})$.

Furthermore, we can relate the contribution to the form factor
$C_{4}^{A}(Q^{2})$ to the Lagrangian

$\frac{\beta}{m_{N}^{2}}D_{\mu}\bar{\Psi}_{\lambda,i}\xi^{\frac{3}{2}}_{ij}f^{\lambda\mu}_{-j}\Psi+\text{H.c.},$
(42)

resulting in the invariant matrix element

${\cal
M}=i\frac{\beta}{m_{N}^{2}}\epsilon_{\mu}\bar{w}_{\lambda}(g^{\lambda\mu}p_{f}\cdot
q-q^{\lambda}p_{f}^{\mu})u$

and, thus, the constant contribution $\beta$ to $C_{4}^{A}(Q^{2})$. In fact,
Holmberg and Leupold Holmberg:2018dtv  showed how to make use of a total-
derivative argument and the lowest-order equation of motion such that the
Lagrangian of Eq. (42) can be reexpressed in terms of the $b_{2}$ Lagrangian
and terms of the ${\cal O}(q^{3})$ Lagrangian. We will nevertheless stick to
the Lagrangian of Eq. (42), because, for our purposes, it is only relevant to
know that we have a free parameter at our disposal, even if it originates from
the ${\cal O}(q^{3})$ and which, only after rewriting, contributes to the
transition matrix element in terms of $C_{4}^{A}(Q^{2})$. Finally, it was
shown in Ref. Holmberg:2018dtv  that at ${\cal O}(q^{2})$ there is no ”new”
contribution to the $\pi N\Delta$ vertex.

##  Appendix B Loop integrals

The scalar loop integrals of one-, two- and three- point functions which are
used for the calculation of the loop diagrams are given by

$\displaystyle
A_{0}(m^{2})=\frac{(2\pi\mu)^{4-n}}{i\pi^{2}}\int\frac{d^{n}k}{k^{2}-m^{2}+i0^{+}},$
$\displaystyle
B_{0}(p^{2},m_{1}^{2},m_{2}^{2})=\frac{(2\pi\mu)^{4-n}}{i\pi^{2}}\int\frac{d^{n}k}{[k^{2}-m_{1}^{2}+i0^{+}][(k+p)^{2}-m_{2}^{2}+i0^{+}]},$
$\displaystyle
C_{0}(p_{i}^{2},(p_{f}-p_{i})^{2},p_{f}^{2},m_{1}^{2},m_{2}^{2},m_{3}^{2})$
$\displaystyle=\frac{(2\pi\mu)^{4-n}}{i\pi^{2}}\int\frac{d^{n}k}{[k^{2}-m_{1}^{2}+i0^{+}][(k-p_{i})^{2}-m_{2}^{2}+i0^{+}][(k-p_{f})^{2}-m_{3}^{2}+i0^{+}]}.$

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