%!TEX root =  ../master.tex
\section{One Dimension}\label{sec:1D}

Here we consider L\"uscher's formula in one dimension with a contact interaction.
Since the sum in the quantization condition \eqref{general luscher} or the one-loop finite-volume sum \eqref{I0 FV} with $D=1$ is convergent we have %, the counterterm $\counterterm_{D=1}^\spherical=0$ and we obtain
\begin{align}\label{eq:tuning-1d}
    C(\Lambda)
    &=
            -\frac{1}{\mu a_{1}}
    &
    \frac{a_{1}}{L}
        &=
            \frac{1}{2 \pi^{2}}
            \sum_{n=-\infty}^{\infty} \frac{1}{n^{2}-x}
        \equiv
            \frac{1}{2 \pi^{2}}
            S^\bigcirc_{1}\left(x\right)
    &
    \Bigg(x
        &=
            \frac{2\mu E L^2}{4\pi^2}\Bigg)
    \, ,
\end{align}
where the contact strength $C(\Lambda)$ does not run.
The energy $E$ is a finite-volume energy on a torus of circumference $L$.
In one dimension the sum in the zeta function is well behaved and has a compact form,
\begin{equation}\label{eq:1d luscher}
S^\bigcirc_{1}(x) \equiv \sum_{n=-\infty}^{\infty} \frac{1}{n^{2}-x}=-\pi \frac{\cot (\pi \sqrt{x})}{\sqrt{x}}\ ,
\end{equation}
which gives a closed form expression for L\"uscher's formula,
\begin{equation}\label{eq:1d luscher constant}
\frac{a_{1}}{L} =-\frac{1}{pL}\cot\left(\frac{pL}{2}\right),
\end{equation}
consistent with those found in \Refs{Luscher:1990ck,Zhu:2019dho}.

Since there is no counterterm in one dimension, the dispersion form of L\"uscher's formula is straightforward to obtain.  If one identifies the lattice spacing as the cutoff, then the sum in the zeta function is restricted to the Brillouin zone and one has
\begin{equation}
    \frac{a_{1}}{L}
    =
    \frac{1}{2 \pi^{2}} \sum_{n=-\frac{N}{2}}^{\frac{N}{2}-1} \frac{1}{\tilde{K}^N_{nn}-x}
    =
    \frac{1}{2 \pi^{2}} \sum_{n=-\frac{N}{2}}^{\frac{N}{2}-1} \frac{1}{\frac{N^2}{4\pi^2}\left(\sum_{s}\gamma_s^{(\nstep)}\cos\frac{2\pi n s}{N}\right)-x}
    \equiv
    \frac{1}{2 \pi^{2}} S^{\dispersion}_{1}\left(x\right),\label{eq:1d dispersion luscher}
\end{equation}
where we explicitly show that the dispersion function $S^{\dispersion}$ depends on $\nstep$ and $N$ but not on $L$ or $\epsilon$ explicitly.

As stressed in the previous section, only continuum-extrapolated energies should be used in the quantization condition \eqref{1d luscher}, or induced momentum dependence terms will result.
For example, in \Figref{luescher1d} we show the induced momentum dependence terms when non-continuum eigenvalues $x^\dispersion$ are inserted into $S^{\spherical}_1$ (colored points) for lattice sizes of $N=4$, 10, 12, and 14 and $\nstep=\infty$.
However, we also show the scattering data determined through $S^{\dispersion}_1(x^\dispersion)$ (black points), which lie on a flat line, as expected.

\begin{figure}
\center
    \center
    \input{figure/1d.pgf}
%\includegraphics[width=.65\textwidth]{figure/1d.pdf}
    \caption{
    	Phase shifts and zeta functions in the spherical and dispersion scenario in one dimension.
		\\
		\textit{(top)}
        	Finite-spacing eigenvalues $x^\dispersion=2\mu E^\dispersion L^2/4\pi^2$ of the Schr\"odinger equation are inserted into respective zeta functions to obtain phase shifts.
			The eigenvalues are obtained for a contact interactions fixed to $ a_{1}/L=1/10$ (closed symbols) and $a_{1}/L=0$ (open symbols) using the analytic result for the interaction strength $C(\Lambda)$ \eqref{tuning-1d} and not by tuning to a zero of any zeta.
			Different markers correspond to different discretizations: $N=4$ (triangles), 10 (squares), 12 (diamonds), and 14 (hexagons).
        	For analysis, the colored points are obtained using $S^\spherical_1(x^\dispersion)$; corresponding to $N=4$ (red), 10 (green), 12 (blue), and 14 (purple).
        	The thin colored lines are the derived induced momentum-dependent terms for each $N$ as given in \Tabref{induced terms in 1 d}.
        	The black points are obtained using the $N$-appropriate $S^{\dispersion}_1(x^\dispersion)$ and exhibit the correct flat-line behavior.
        	The dashed gray line is $S_1^\spherical$, as in the bottom panel.
        \\
        \textit{(bottom)}
        	Two one-dimensional zeta functions, the spherical function $S_1^{\spherical}$ (light gray) given in \eqref{1d luscher} and $S_1^{\dispersion}$ (red) given in \eqref{1d dispersion luscher} with $N=4$ and $\nstep=\infty$.
        	The difference between the dispersion and spherical curves is responsible for moving the red triangles to the black triangles in the top panel.
        }
        \label{fig:luescher1d}
\end{figure}

We can derive the functional form of these induced momentum-dependent terms following the exact steps taken in \Secref{3d induced momenta}, again assuming $\nstep=\infty$,
\begin{align*}
S^\bigcirc_{1}\left(x^\dispersion\right)&=S^{\dispersion}_{1}\left(x^\dispersion\right)+\left(S^\bigcirc_{1}\left(x^\dispersion\right)  -S^{\dispersion}_{1}\left(x^\dispersion\right)\right) %\sum_{n=-\infty}^{\infty} \frac{1}{n^{2}-x^\dispersion} &=\sum_{n\in \operatorname{B.Z.}} \frac{1}{n^{2}-x^\dispersion} +\sum_{n\notin \operatorname{B.Z.}} \frac{1}{n^{2}-x^\dispersion} \\
 %&=  S^{\dispersion}_{1}\left(x^\dispersion\right)+\sum_{n\notin \operatorname{B.Z.}} \frac{1}{n^{2}-x^\dispersion}\\
=\frac{2\pi^2 a_{1}}{L}+ \sum_{n\notin \operatorname{B.Z.}} \frac{1}{n^{2}-x^\dispersion} \\
&=\frac{2\pi^2 a_{1}}{L}+\alpha_1(N)+ \alpha_2(N)x^\dispersion+\alpha_3(N)(x^\dispersion)^2+\ldots\ .
\end{align*}
where again we assume $\nstep=\infty$ energy.
In the second term we assumed $n^2\gg x^\dispersion$ since the sum is restricted to modes outside the Brillouin zone.
%Assuming $n^2\gg x^\dispersion$, we can expand in powers or $x^\dispersion$,
%\begin{equation}
%    \frac{1}{2 \pi^{2}} \sum_{n=-\infty}^{\infty} \frac{1}{n^{2}-x^\dispersion}
%        =
%    \frac{a_{1}}{L}+\frac{1}{2 \pi^{2}} \sum_{n\notin \operatorname{B.Z.}} \frac{1}{n^{2}-x^\dispersion}
%        =
%    \frac{a_{1}}{L}+\frac{1}{2 \pi^{2}} \sum_{n\notin \operatorname{B.Z.}} \left(\frac{1}{n^2}+\frac{x^\dispersion}{n^4}+\frac{(x^\dispersion)^2}{n^6}+\mathcal{O}((x^\dispersion)^3)\right)
%\end{equation}
%We make the replacement $\sum_{n\notin\operatorname{B.Z.}}\mapsto\left(\sum_{n=-\infty}^\infty-\sum_{n\in\operatorname{B.Z.}}\right)$, which allows each summation
The coefficients $\alpha_i(N)$ can be determined to arbitrary precision.
\Tabref{induced terms in 1 d} shows these terms for $N=4$, 10, 12, and 14.
The thin colored lines in \Figref{luescher1d} correspond to the functions given in this table.
In the limit $N\goesto\infty$ all states are included in the Brillouin zone so all terms vanish and one recovers the flat, momentum-independent behavior.

\begin{table}
    \caption{
    The coefficients $\alpha_i(N)$ of the induced momentum-dependent terms to order $(x^\dispersion)^2$ due to a contact interaction using $S^\bigcirc_1(x^\dispersion)$ as a function of discretization $N$.
    Here $x^\dispersion$ is determined by a finite-spacing finite-volume $\nstep=\infty$ eigenenergy.}
    \label{tab:induced terms in 1 d}
    \begin{tabular}{S[table-format=2.1]S[table-format=2.6]S[table-format=2.7]S[table-format=2.8]}
{$N$} & {$\alpha_1$} & {$\alpha_2$} & {$\alpha_3$} \\ \midrule
      4 &   1.03987  & 0.102146 & 0.0190611 \\
    10  &   0.40265  & 0.005543 & 0.0001404 \\
    12  &   0.33487  & 0.003171 & 0.0000549 \\
    14  &   0.28668  & 0.001983 & 0.0000250 \\
    \end{tabular}
\end{table}


Analyzing the finite-spacing dispersion zeta function $S^{\dispersion}_1$ with $\nstep=\infty$ produces the flat behavior all the way through.
This demonstrates that in the one-dimensional case, it was not the contact operator that caused the momentum dependence, but that the dependence was induced by leveraging the continuum finite-volume formalism itself.

Finally, we draw the reader's attention to the structure of $S^\dispersion$ at any finite $N$ in the bottom panel of \Figref{luescher1d}, where the $N=4$ dispersion zeta is shown.
Note that any flat function of $x$ can only ever intercept the zeta function three times---which makes sense, as there are only three states in the parity-even sector of a one-dimensional $N=4$ lattice: $\ket{0}$, $(\ket{-1}+\ket{+1})/\sqrt{2}$ and $\ket{2}$ (which, being on the edge of the Brillouin zone, is the same state as $\ket{-2}$).
If one tunes a contact interaction so that the scattering amplitude vanishes, one will see one state with $0<x<1$, one with $1<x<4$, and one state with $\abs{x}$ very large and a sign depending on whether one is slightly above or below zero numerically.
The finiteness of the parity-even sector puts constraints on the interactions that can be faithfully put onto such a small lattice: one cannot create any interaction where the scattering amplitude intersects the $N=4$ $S^{\dispersion}$ four times, because that would entail too many finite-volume states.
Of course, this is a generic feature in any number of dimensions, and the constraint ultimately vanishes in the continuum limit $N\goesto\infty$; as $N$ increases the number of accessible $n^2$ shells grows in a dimension-dependent way.
