viewpoint.

\[cut\]

In [@kmw], we have studied the case $p=0$, $a_0=1$, where the operators assumed the form $t\Theta_\epsilon (t)$ to leading order in $t$, where $\Theta_\epsilon (t)$ is the regulated form of the step function, given by [@kmw]: $$\label{rep} 
\Theta_\epsilon =-i\int_{-\infty}^{+\infty} 
\frac{d\omega}{2\pi} \frac{1}{\omega -i\epsilon}
e^{i\omega\,t}, \qquad \epsilon \rightarrow 0^+$$ As discussed in that reference, this operator forms a logarithmic pair [@lcft] with $\epsilon \Theta_\epsilon  (t)$, expressing physically fluctuations in the initial position of the $D$-particle.

In the current case, one may expand the integrand of (\[path2\]) in a Taylor series in powers of $(t-t_0)$, which implies the presence of a series of operators, of the form $(t-t_0)^q\Theta_\epsilon (t-t_0)$, where $q $ takes on the values $2p, 2p-1, \dots $, i.e. it is not an integer in general. In a direct generalization of the Fourier integral representation (\[rep\]), we write in this case: $$\begin{aligned}
\label{opd}  
&~& {\cal D}^{(q)} 
\equiv v_i (t-t_0)^q \Theta_\epsilon (t-t_0)\partial_n X^i  
=v_i~N_q~\int _{-\infty}^{+\infty}d\omega 
\frac{1}{(\omega -i\epsilon)^{q+1}}~e^{i\omega(t-t_0)}\partial_nX^i~,
\nonumber \\
&~& N_q \equiv \frac{i^q}{\Gamma (-q)(1-e^{-i2\pi q})}=
\frac{(-i)^{q+1}\Gamma (q+1)}{2\pi}~,\end{aligned}$$ where we have incorporated the velocity coupling $v_i$ in the definition of the $\sigma$-model deformation, and we have defined the integral along the contour of figure 1, having chosen the cut to be 