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\title{Renormalization in various schemes of nucleon-nucleon chiral EFT}
\author{A.M.~Gasparyan\from{ins:x}\ETC,
E.~Epelbaum\from{ins:x}}
\instlist{\inst{ins:x} Ruhr-Universit\"at Bochum, Fakult\"at f\"ur Physik und
Astronomie, Institut f\"ur Theoretische Physik II, D-44780
Bochum, Germany}
\begin{document}
\maketitle
\begin{abstract}
Renormalizability of an effective field theory allows one
to perform a systematic expansion of the calculated observable
quantities in terms of some small parameter in accordance with
a certain power counting.
We consider chiral effective field theory in application to the
nucleon-nucleon interaction
at next-to-leading order in the chiral expansion
as an example of the renormalizability of a theory
with a nonperturbative leading-order interaction.
The requirement of the renormalizability imposes nontrivial constraints
on a choice of such interaction.
Two different approaches are discussed: the finite- and the infinite-cutoff schemes.
Another considered example is the quantum mechanical problem of the inverse-square potential,
which is often regarded as a toy model for various physical systems.
\end{abstract}
\section{Introduction}
The effective field theory (EFT) methods in the presence of
a nonperturbative leading-order (LO) interaction were
first applied to the nucleon-nucleon (NN) and few-nucleon systems by Weinberg~\cite{Weinberg:1990rz,Weinberg:1991um}
in the framework of chiral EFT.
Similar methods have turned out to be relevant in studies of molecular states in
the heavy-quarkonia sector as well as in the few-nucleon calculations within pionless EFT,
see Refs.~\cite{Bedaque:2002mn,Epelbaum:2008ga,Machleidt:2011zz,
Epelbaum:2019kcf,Guo:2017jvc,Hammer:2019poc} for reviews.
In such schemes, the next-to-leading-order (NLO) and higher-order interactions (potentials)
can still be treated perturbatively.
This is not necessary, but has an advantage of
having the theoretical errors for observables coming from a truncation of an EFT expansion
under control. Moreover, in some cases including the NN chiral EFT or the 3-nucleon pionless EFT
with an infinite (much larger than
the EFT breakdown scale $\Lambda_b$) cutoff,
the nonperturbative treatment of subleading corrections leads to physically
unacceptable results \cite{PavonValderrama:2005wv,PavonValderrama:2005uj,Zeoli:2012bi,vanKolck:2020llt,Gabbiani:2001yh}.
In the discussed approaches, the series for the LO amplitude
$T_0$ and for the unrenormalized NLO amplitude $T_2$ can be schematically represented by
\begin{align}
&T_0=\sum_{n=0}^{\infty}T_0^{[n]},\qquad T_0^{[n]}=V_0 (G V_0)^n , \label{Eq:T0}\\
&T_2=\sum_{m,n=0}^{\infty}T_2^{[m,n]},\qquad T_2^{[m,n]}=(V_0 G)^m V_2 (G V_0)^n,\label{Eq:T2}
\end{align}
where $G$ is the free Green's function and $V_0$ ($V_2$) is the LO (NLO) potential.
In the nonperturbative regime for the LO potential, these equations generalize to
\begin{align}
&T_0=V_0\,(\mathds{1}-G V_0)^{-1},\label{Eq:T0_NP}\\
&T_2=(\mathds{1}-V_0 G)^{-1} \, V_2 \,(\mathds{1}-G V_0)^{-1}.\label{Eq:T2_NP}
\end{align}
Here for definiteness, we assign the subscripts $0$ and $2$ analogously to the LO
and NLO interactions in NN chiral EFT, which is a demonstrative example we use in this study.
For simplicity, we assume the uncoupled $S$-wave scattering, so that the Lippmann-Schwinger equation
for the LO off-shell amplitude, $T_0=V_0+V_0GT_0$, reads
\begin{align}
T_{0}(p',p;p_\text{on})&=
\int \frac{p''^2 dp''}{(2\pi)^3}
V_0(p',p'')
G(p'';p_\text{on})
T_{0}(p'',p;p_\text{on}),\nonumber\\
G(p''; p_\text{on})&=\frac{m_N}{p_\text{on}^2-p''^2+i \epsilon},
\label{Eq:LS_equation}
\end{align}
The naive dimensional power counting formulated for small (compared to the hard scale $\Lambda_b$)
particle momenta is often violated by contributions to the amplitude
coming from large loop momenta of the order of the cutoff $\Lambda$.
To restore the power counting, a renormalization procedure is necessary:
one absorbs the power counting breaking terms by redefining
contact interactions splitting the unrenormalized low-energy constants (LECs)
into the renormalized ones and the counter terms.
This method works for a large class of interactions provided
the cutoff is of the order of the hard scale $\Lambda\sim\Lambda_b$ and the LO interaction is
in some sense perturbative \cite{Gasparyan:2021edy}.
In the case of a nonperturbative LO interaction,
to maintain renormalizability of a theory, one has to impose additional
constraints on the LO potential \cite{Gasparyan:2023rtj,Gasparyan:2022isg}
when considering calculations beyond leading order.
In what follows we consider several applications of those constraints.
\section{Renormalization in the nucleon-nucleon sector}
The renormalized expression for the NLO NN amplitude, $\mathds{R}(T_2)$, is obtained by adding
the relevant counter term $\delta V$ to $V_2$:
\begin{align}
&\mathds{R}(T_2)=(\mathds{1}-V_0 G)^{-1} \left(V_2 +\delta V \right)(\mathds{1}-G V_0)^{-1} .\label{Eq:T2_renormalized}
\end{align}
Below, we consider the typical case of two NLO contact terms, momentum independent and quadratic in momentum,
determined by the LECs $C^{\rm \scriptscriptstyle NLO{}}_0$ and $C^{\rm \scriptscriptstyle NLO{}}_2$, so that
the contact part of the NLO amplitude is given by \cite{Gasparyan:2022isg}
\begin{align}
T_{\text{ct}}(p_\text{on})&=
C^{\rm \scriptscriptstyle NLO{}}_0 \psi_\Lambda(p_\text{on})^2
+C^{\rm \scriptscriptstyle NLO{}}_{2}2\psi_\Lambda(p_\text{on}) \psi'_\Lambda(p_\text{on}),
\label{Eq:T_ct_nonlocal}
\end{align}
with
\begin{align}
\psi_\Lambda(p_\text{on})& =F_\Lambda( p_\text{on}^2) +\int \frac{p^2 dp}{(2\pi)^3} G(p;p_\text{on})
F_\Lambda (p^2) T_0(p, p_\text{on};p_\text{on}),\nonumber\\
\psi'_\Lambda(p_\text{on})& =p_\text{on}^2F_\Lambda( p_\text{on}^2) +\int \frac{p^2 dp}{(2\pi)^3}p^2 G(p;p_\text{on})
F_\Lambda (p^2)T_0(p, p_\text{on};p_\text{on}),
\end{align}
where $F_\Lambda (p^2)$ is a regulator with the cutoff $\Lambda$ assumed here to be the
same as in the LO potential (in general, this is not necessary).
One can choose the renormalization conditions to be determined by the
amplitude at two on-shell momenta $p_0$ and $p_1$.
These conditions become inconsistent when the quantity $\zeta_\Lambda(p_0,p_1)$ vanishes:
\begin{align}
\zeta_\Lambda(p_0,p_1)=0,
\label{zero_determimant}
\end{align}
where
\begin{align}
\zeta_\Lambda(p_0,p_1)=\frac{e^{-i\delta^{(0)}(p_1)}}{\psi_\Lambda(p_0)} \left|\begin{array}{ll}
\psi_\Lambda(p_0)&\psi_\Lambda'(p_0)\\
\psi_\Lambda(p_1)&\psi'_\Lambda(p_1)
\end{array}\right|,
\end{align}
with the phase determined by the LO phase shift $\delta^{(0)}$.
Analyzing various realistic NN EFT interactions, it was found that for
the cutoff values $\Lambda\lesssim\Lambda_b$, the conditions in Eq.~\eqref{zero_determimant}
is never fulfilled, which guarantees renormalizability of such theories.
On the other hand, if one sends the cutoff to infinity, as is done, e.g. in Ref.~\cite{Long:2011qx},
an infinite number of "exceptional" cutoffs appear.
For such cutoffs, Eq.~\eqref{zero_determimant} holds,
and in their neighborhoods, the theory becomes nonrenormalizable,
which imposes limitations on the whole scheme~ \cite{Gasparyan:2022isg}.
\section{Example of the inverse square potential}
To demonstrate common features of renormalization in the nonperturbative
regime, one often utilizes the inverse-square-potential example.
The ultraviolet behavior of this interaction is similar to the one
that can appear, e.g., in the three-body systems with short-range forces,
see Refs.~\cite{Hammer:2001gh,Vanasse:2013sda,Ji:2011qg,Ji:2012nj} for
calculations beyond leading order.
The most interesting is the case of a singular attractive LO potential.
In Ref.~\cite{Long:2007vp}, the model with the long-range LO potential
proportional to $1/r^2$ and the long-range NLO potential
proportional to $1/r^4$ was introduced.
For the regulator, a sharp momentum cutoff was implemented.
In Ref.~\cite{Gasparyan:2022isg}, it was shown that
the "exceptional" cutoffs in such a model do not destroy renormalizability
because the zeros of $\zeta_\Lambda(p_0,p_1)$ factorize and coincide with
the zeros of the vertex function
\begin{align}
\tilde\psi_\Lambda(p_0)=e^{-i\delta^{\rm \scriptscriptstyle LO}(p_\text{on})}\psi_\Lambda(p_0).
\end{align}
However, such factorization is a unique feature of this model.
To demonstrate this, we performed various minor modifications of the scheme.
In particular, we have modified the regulator introducing a smooth cutoff
or a combination of the smooth and sharp cutoffs.
We have also tried to slightly change the potential without modifying its
short-range behavior.
In all such cases, the above factorization is absent and the "exceptional" cutoffs
again destroy renormalizability.
This fact should be kept in mind when using the inverse square potential
as a toy model for realistic interactions.
To summarize, we have discussed the renormalization of various effective field theories with nonperturbative
LO interactions.
We have emphasized that it is important to verify the fulfillment of the renormalizability constraints when
going beyond leading-order calculations.
\acknowledgments
This research was supported
by ERC AdG NuclearTheory (Grant No. 885150), by the MKW NRW under the funding code NW21-024-A,
and by the EU Horizon 2020 research and
innovation programme (STRONG-2020, grant agreement No. 824093).
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