text:
\section{Introduction}
\label{INTRODUCTION}

In this paper, we study $\phi^4$ theory defined over $\mathbb{R} \times \mathbb{Q}_p$, where $\mathbb{Q}_p$ denotes the $p$-adic numbers for a prime number $p$ which we fix once and for all.  The action we start with is
 \eqn{SphiFourth}{
  S &= {1 \over 2} \int_{\mathbb{R}} d\omega \int_{\mathbb{Q}_p} dk \,
    \tilde\phi(-\omega,-k) \left( \omega^2 + |k|_p^{2z} + r \right)
      \tilde\phi(\omega,k)
     + {\lambda \over 4!} \int_{\mathbb{R}} d\tau \int_{\mathbb{Q}_p} dx \, \phi(\tau,x)^4 \,.
 }
The field $\phi(\tau,x)$ is real-valued.  The parameter $z$ is real-valued as well and is understood to be a dynamical scaling exponent.\footnote{An alternative notation would be to write $\omega^2 + |k|_p^\sigma$ instead of $\omega^2 + |k|_p^{2z}$ in \eno{SphiFourth} and reserve $z$ for the dynamical scaling exponent of the full quantum theory.  Our approach is for $z$ to be defined at the level of the microscopic or classical theory, and when it differs from the infrared dynamical scaling exponent, we denote the latter by $z_\text{IR}$.}  We employ a Fourier transform
 \eqn{FourierDef}{
  \phi(\tau,x) &= \int_{\mathbb{R}} d\omega \int_{\mathbb{Q}_p} dk \,
    e^{-2\pi i \omega\tau} \chi(kx) \tilde\phi(\omega,k)  \cr
  \tilde\phi(\omega,k) &= \int_{\mathbb{R}} d\tau \int_{\mathbb{Q}_p} dx \,
    e^{2\pi i \omega\tau} \chi(-kx) \phi(\tau,x) \,,
 }
where $\chi(y) = e^{2\pi i \{ y \}}$ and $\{ y \}$ is the fractional part of $y \in \mathbb{Q}_p$, meaning the unique rational number in the interval $[0,1)$ such that $y-\{y\}$ is a $p$-adic integer.  We will speak of real quantities like $\tau$ and $\omega$ as Archimedean, whereas $p$-adic quantities like $x$ and $k$ will be termed ultrametric.\footnote{In general, an Archimedean norm is a positive definite norm for which if $0<|a|<|b|$, then $|na|>|b|$ for some $n \in \mathbb{Z}$.  In contrast, an ultrametric norm is a positive definite norm for which $|a+b| \leq \max\{|a|,|b|\}$.}  We refer to \eno{SphiFourth} as mixed field theory because it is defined over a space which is Archimedean in one direction and ultrametric in another.  In \eno{SphiFourth}, we write the kinetic term in Fourier space because the ultrametric part is non-local in position space.

There are several obvious generalizations of \eno{SphiFourth}.  We could replace Euclidean time $\tau \in \mathbb{R}$ by a larger Archimedean space or spacetime, for example $\mathbb{R}^d$ or $\mathbb{R}^{d-1,1}$.  We could replace $\mathbb{Q}_p$ by a more general ultrametric space, for example some extension of $\mathbb{Q}_p$, or a compact space like $\mathbb{Z}_p$ (the $p$-adic integers) or $\mathbb{P}^1(\mathbb{Q}_p)$.  We could pass to the $O(N)$ model by replacing $\phi$ with a vector $\phi^i$ valued in $\mathbb{R}^N$, and then specify a particular $O(N)$ tensor structure for the $\phi^4$ interaction.  We could consider including higher powers of $\phi$ and tuning couplings so as to arrive at multi-critical points.  Many aspects of our analysis can be straightforwardly adapted to these generalizations, so the current work should be seen as focusing primarily on the simplest example.

Broadly, our motivations are two-fold.  First, quantum systems might in principle be realized in the lab whose behavior near a critical point could plausibly be described by the mixed field theory \eno{SphiFourth}.  We have in mind especially the schemes of \cite{Britton:2012zz,Hung:2016zz}, in which non-local couplings are set up in a chain or array of quantum spins.\footnote{We thank G.~Bentsen, E.~Davis, and M.~Schleier-Smith for ongoing discussions regarding the experimental possibilities.}  Other authors have been interested in renormalization group (RG) treatments of hierarchical versions of the transverse Ising Model; a recent work along these lines is \cite{Monthus:2015zz}.

Second, we want to entertain the possibility of $p$-adic extra dimensions.  While earlier authors, notably Volovich, have advocated a sort of covariance over all possible choice of number fields in the formulations of physical theories (see for example \cite{Arefeva:1990cy,Vladimirov:1994zz}), our approach here is instead to accept $\mathbb{R}^{3,1}$ at face value as the description of the usual dimensions of spacetime, but to suppose that there could be an extra dimension which is compact and ultrametric---for example, $\mathbb{Z}_p$ for some particular $p$.

Our findings can be separated into classical and quantum results.  On the classical side, we find:
 \begin{itemize}
  \item Setting $\lambda=0$, the tree-level propagator for $\phi$ has oscillatory behavior in position space even when the argument is purely Archimedean.  For $r=0$ and $p$ not too large, this oscillatory behavior enters as a small correction to an overall power law, suppressed by a factor $e^{-{\pi^2 \over z \log p}}$.  This factor can be extremely small without fine-tuning parameters of the theory: We give an example where the suppression of oscillatory effects is by $16$ orders of magnitude.  All this is set forth in section~\ref{PROPAGATOR}.
  \item If the ultrametric domain $\mathbb{Q}_p$ is reduced to the compact set $\mathbb{Z}_p$, then, as we show in section~\ref{TRUNCATION}, there is a hierarchy of consistent truncations to purely Archimedean effective theories, based on retaining modes whose ultrametric momenta are bounded by some cutoff.  The result holds not only for $\lambda \neq 0$, but for any smooth potential $V(\phi)$, assuming $\phi=0$ is a solution to the full classical field equations.
 \end{itemize}
On the quantum side, we find:
 \begin{itemize}
  \item By power counting, as explained in section~\ref{POWER}, the $\phi^4$ interaction is irrelevant for $z<1/3$ and relevant for $z>1/3$.  We concentrate our analysis on the critical (i.e.~massless) theory close to $z=1/3$.
  \item There is evidence from one-loop computations of a Wilson-Fisher fixed point of the renormalization group when $z$ is slightly larger than $1/3$.  We present the one-loop analysis of the Wilson-Fisher fixed point in section~\ref{FIXED} after an analysis of loop diagrams in section~\ref{INTEGRALS}.  Section~\ref{MASS} exhibits a computation of the anomalous dimension of $r$ at the fixed point.
  \item Based on a two-loop computation, presented in section~\ref{UNDERGROUND}, we argue in section~\ref{DYNAMICAL} that $z$ itself is renormalized: That is, starting with \eno{SphiFourth} at the microscopic level, with $z$ slightly larger than $1/3$, the scaling parameter which enters infrared Green's functions at the Wilson-Fisher fixed point is not $|\tau|/|x|_p^z$ but rather $|\tau|/|x|_p^{z_\text{IR}}$ for a value $z_\text{IR}$ slightly different from $z$.
  \item Quantum corrections to Green's functions can have oscillatory dependence on external frequencies $\omega$, as we argue in section~\ref{POSITION} for the case of a two-loop correction to the two-point function with $z=1/3$.  This oscillatory behavior is suppressed by the same factor $e^{-{3\pi^2 \over \log p}}$ that is seen in the position space form of the tree-level Green's function.
 \end{itemize}

\section{Power counting}
\label{POWER}

Power counting in ultrametric theories is well understood: See for example \cite{Gubser:2017vgc}.  The key point is that when we scale $k \to pk$, the norm and the integration measure scale oppositely: $|k|_p \to {1 \over p} |k|_p$ and $dk \to {1 \over p} dk$.  We regard this scaling as a step toward the infrared.  We see from the kinetic term of \eno{SphiFourth} that we must accompany $k \to pk$ with $\omega \to {1 \over p^z} \omega$ and $r \to {1 \over p^{2z}} r$.  In general, we associate to a quantity $X$ an engineering dimension $[X]$ if upon a scaling $k \to pk$ we have $X \to p^{-[X]} X$.  Then the natural assignments that make $S$ dimensionless consistent with \eno{SphiFourth}-\eno{FourierDef} are
 \eqn{EngineeringDimensions}{\begin{tabular}{c|ccccccccccccc}
  $X$ &   $k$ &  $|k|_p$ & $dk$ & $x$ & $|x|_p$ & $dx$ & $\omega$ & $d\omega$ & $S$ & $\phi$          & $\tilde\phi$      & $r$  & $\lambda$  \cr \hline
  $[X]$ & $-1$ & $1$     & $1$  & $1$ & $-1$    & $-1$ & $z$      & $z$       & $0$ & ${1-z \over 2}$ & $-{3z+1 \over 2}$ & $2z$ & $3z-1$
 \end{tabular}}
We refer to these assignments as engineering dimensions because they describe scalings of the classical action without reference to loop corrections.  We see in particular that $\lambda$ has a positive dimension, meaning that $\phi^4$ is a relevant perturbation of the Gaussian fixed point theory, precisely when $z>1/3$---whereas $r$ is always relevant in the same sense, since we require $z>0$.

As compared to ordinary $\phi^4$ theory on $\mathbb{R}^d$, we see from the assignments \eno{EngineeringDimensions} that increasing $z$ is like decreasing $d$; that $z=1/3$ is like the upper critical dimension $d=4$, where $\phi^4$ becomes marginal; and that $z=1$ is like the lower critical dimension, where the dimension of $\phi$ goes to $0$.  Thus, at least naively, we are expecting critical points as indicated in figure~\ref{CriticalPoints}.  We added in a conjectured branch of multi-critical points based on the fact that for $z>1/2$, both $\phi^4$ and $\phi^6$ are relevant deformations of the Gaussian fixed point.  For $0<z<1/3$, our expectation based on power counting is that the Gaussian critical point is the only one available.
 \begin{figure}
  \centerline{\begin{tikzpicture}

   \colorlet{darkgreen}{green!80!black}

   \tikzmath{
     \hscale = 5;
     \pointsize = 0.1;
     \p1 = \hscale;
     \p4 = 0.33 * \hscale;
     \p6 = 0.5 * \hscale;
    }
   \coordinate (C) at (0,0);
   \coordinate (P1) at (\p1,0);
   \coordinate (P4) at (\p4,0);
   \coordinate (P6) at (\p6,0);
   \coordinate (RP4) at (\p1,0.67*\hscale);
   \coordinate (N4) at (1.4*\p4,0.62*\p4);
   \coordinate (RP6) at (\p1,0.5*\hscale);
   \coordinate (N6) at (1.3*\p6,0.62*\p4);
   \coordinate (B1) at (1.15*\p1,0);

   \draw [line width=4pt,darkgreen] (C) -- (P4);
   \draw [->, line width=2pt] (C) -- (B1);
   \draw [fill] (C) circle [radius=\pointsize];
   \draw [fill] (P1) circle [radius=\pointsize];

   \node [below=5pt] at (C) {$z=0$};
   \node [below=5pt] at (P1) {$1$};
   \node [below=5pt] at (P4) {$1 \over 3$};
   \node [below=5pt] at (P6) {$1 \over 2$};

   \draw [fill,red] (P4) circle [radius=\pointsize];
   \draw [line width=2pt,red] (P4) arc (180:90:0.67*\hscale);
   \node [rotate=70,red] at (N4) {\small $\phi^4$ relevant};
   \draw [fill,red] (RP4) circle [radius=\pointsize];
   \draw [fill,blue] (P6) circle [radius=\pointsize];
   \draw [line width=2pt,blue] (P6) arc (180:90:0.5*\hscale);
   \node [rotate=70,blue] at (N6) {\small $\phi^6$ relevant};
   \draw [fill,blue] (RP6) circle [radius=\pointsize];

   \draw [fill] (1.2*\p6+0.2*\hscale,0.23*\hscale) circle [radius=0.5*\pointsize];
   \draw [fill] (1.2*\p6+0.25*\hscale,0.205*\hscale) circle [radius=0.5*\pointsize];
   \draw [fill] (1.2*\p6+0.3*\hscale,0.18*\hscale) circle [radius=0.5*\pointsize];

   \node [above=15pt,darkgreen] at (0.1*\hscale,0) {\small No relevant};
   \node [above=5pt,darkgreen] at (0.1*\hscale,0) {\small deformations};
   \node [above=10pt,left=-10pt] at (P1) {Gaussian};
   \node [right] at (B1) {$z$};
  \end{tikzpicture}}
  \caption{Conjectured pattern of critical points of \eno{SphiFourth}.}\label{CriticalPoints}
 \end{figure}

\section{The propagator}
\label{PROPAGATOR}

From \eno{SphiFourth} with $r=\lambda=0$, we can directly read off the momentum space Green's function:
\eqn{Gmomentum}
 {
  \tilde{G}(\omega,k)=\frac{1}{\omega^2+|k|_p^{2z}}\,.
 }
We require $\lambda=0$ for now so as to avoid loop corrections, and we require $r=0$ so as to consider the critical Gaussian theory.  Fourier transforming the real and $p$-adic components, we obtain the position space Green's function:
 \eqn{Gposition}{
  G(\tau,x) \equiv \langle \phi(\tau,x) \phi(0,0) \rangle =
   \int_{\mathbb{R}} d\omega \int_{\mathbb{Q}_p} dk \,
    {e^{-2\pi i\omega\tau} \chi(kx) \over \omega^2 + |k|_p^{2z}}\,.
 }
Based on the dimensional analysis of the previous section, we expect to find the scaling form
 \eqn{Gscaling}{
  G(\tau,x) = {1 \over |x|_p^{1-z}} g(\xi) \qquad\text{where}\qquad
    \xi \equiv {2\pi |\tau| \over |x|_p^z} \,.
 }
The ``aspect ratio'' $\xi$ is dimensionless in the sense $[\xi] = 0$: That is, upon scaling $x \to x/p$ and $\tau \to p^z \tau$, we find that $\xi$ is unchanged.  The factor of $2\pi$ in the definition of $\xi$ is for later convenience.  Our plan in this section is first to verify the scaling form \eno{Gscaling} and explicitly compute $g(\xi)$, and then to consider limits of small and large $\xi$.

The $\omega$ integral in \eno{Gposition} can be carried out straightforwardly, with the result
 \eqn{Ghalf}{
  G(\tau,x) = \int_{\mathbb{Q}_p} dk \, \chi(kx) {\pi \over |k|_p^z} e^{-2\pi |\tau| |k|_p^z} \,.
 }
To carry out the integral over $\mathbb{Q}_p$, we use the fact that the non-zero $p$-adic numbers split into a disjoint union of scaled copies of the $p$-adic units:
 \eqn{Qexpress}{
  \mathbb{Q}_p^\times = \bigsqcup_{v \in \mathbb{Z}} p^v \mathbb{U}_p \,.
 }
Here $\mathbb{U}_p$ is the set of all $y \in \mathbb{Q}_p$ with $|y|_p = 1$.  To handle the integration over each copy of $\mathbb{U}_p$, we use a standard result in $p$-adic Fourier analysis:
 \eqn{padicStandard}{
  \int_{p^v \mathbb{U}_p} dy \, \chi(y) = \left\{
    \seqalign{\span\TR &\qquad if \quad \span\TR}{
    p^{-v} / \zeta_p(1) & v \geq 0  \cr
    {-1} & v = -1  \cr
    0 & v < -1
   } \right.
 }
where we define
 \eqn{zetaDef}{
  \zeta_p(s) \equiv {1 \over 1 - p^{-s}} \,.
 }
The upshot is that \eno{Ghalf} indeed reduces to the scaling form \eno{Gscaling}, with
 \eqn{gxi}{
  g(\xi) &\equiv -{\pi \over p^z} e^{-p^z \xi} + {\pi \over \zeta_p(1)} \sum_{v=0}^\infty
     p^{v(z-1)} e^{-p^{-vz}\xi}  \cr
   &= -{\pi \over p^z} e^{-p^z \xi} + {\pi \over \zeta_p(1)} \xi^{z-1 \over z} \sum_{v=0}^\infty
     f\left( v - {1 \over z} \log_p \xi \right) \,,
 }
where we define
 \eqn{fv}{
  f(v) \equiv p^{v(z-1)} e^{-p^{-vz}} \,.
 }

Consider the limit $\xi \ll 1$.  We call this a highly ultrametric aspect ratio because the separation between points is mostly in the $x$ direction.  In this limit, we can ignore the factors $e^{-p^z \xi}$ and $e^{-p^{-vz}\xi}$ in \eno{gxi}, and a straightforward computation leads to
 \eqn{GSmallAspect}{
  G(\tau,x) \approx {\pi/\Gamma_p(z) \over |x|_p^{1-z}}
   \qquad\text{when}\qquad {2\pi|\tau| \over |x|_p^z} \ll 1 \,,
 }
where we define
 \eqn{GammDef}{
  \Gamma_p(z) \equiv {\zeta_p(z) \over \zeta_p(1-z)}\,.
 }

Now consider the opposite limit, $\xi \gg 1$.  We call this a highly Archimedean aspect ratio.  In this limit, we can extend the sum over $v$ in \eno{gxi} to all the integers, because $f(v)$ becomes small very quickly when $v$ goes to $-\infty$.  Now we employ Poisson resummation:
 \eqn{PoissonFormula}{
  \sum_{v \in \mathbb{Z}} f\left( v - {1 \over z} \log_p \xi \right) =
   \sum_{\tilde{v} \in \mathbb{Z}} \xi^{-{2\pi i \over z\log p} \tilde{v}}
    \tilde{f}(\tilde{v}) \,,
 }
where we define $\tilde{f}(\tilde{v})$ as the Fourier transform of $f(v)$,
 \eqn{fFourier}{
  \tilde{f}(\tilde{v}) \equiv \int_{\mathbb{R}} dv \, e^{-2\pi i v \tilde{v}} f(v)
    = {1 \over z \log p} \Gamma_{\rm Euler}\left(
       {2\pi i \tilde{v} \over z \log p} + {1 \over z} - 1 \right) \,.
 }
Noting that $\xi^{-{2\pi i \over z \log p}} = |2\pi\tau|^{-{2\pi i \over z \log p}}$ because $\xi$ differs from $|\tau|$ only by a factor of $|x|_p^z = p^{-v(x)z}$ where $v(x) \in \mathbb{Z}$ is the $p$-adic valuation of $x$, we arrive at the approximation
 \eqn{GBigAspect}{
  G(\tau,x) \approx {\pi/\zeta_p(1) \over z \log p} {1 \over (2\pi|\tau|)^{{1 \over z}-1}}
    \sum_{\tilde{v} \in \mathbb{Z}}
     (2\pi |\tau|)^{-{2\pi i \tilde{v} \over z \log p}}
    \Gamma_{\rm Euler}\left(
       {2\pi i \tilde{v} \over z \log p} + {1 \over z} - 1 \right)
   \quad\hbox{when}\quad {2\pi|\tau| \over |x|_p^{z}} \gg 1 \,.
 }
In the limiting case where $x=0$, the approximate equality in \eno{GBigAspect} becomes exact.  When we compare \eno{GBigAspect} with the scaling form $G(\tau,x) = |\tau|^{1 - {1 \over z}} \tilde{g}(\xi)$ expected based on the engineering dimensions \eno{EngineeringDimensions}, it seems surprising that the infinite sum in \eno{GBigAspect} should depend directly on $\tau$ with no $x$ dependence.  In fact, because the possible values of $|x|_p$ are integer powers of $p$, we can replace $2\pi|\tau|$ by $\xi$ inside the infinite sum in \eno{GBigAspect} without changing the value of any of the terms; in other words (for $x$ not exactly $0$) the form \eno{GBigAspect} does agree with the expected scaling form.

A more illuminating way to understand the occurrence of complex powers of $|\tau|$ in \eno{GBigAspect} is to note that we only have discrete scale invariance under $x \to x/p$ and $\tau \to p^z \tau$.  Absent any other considerations, this discrete scale invariance allows the two point function $G(\tau,0)$ at purely Archimedean separation to have a power law form $|\tau|^\alpha$, but $\alpha$ can only be fixed up to additions of an integer multiple of ${2\pi i \over z \log p}$.

Despite the occurrence of complex powers of $|\tau|$ in \eno{GBigAspect}, the Green's function \eno{GBigAspect} at purely Archimedean separations is often well approximated by the $\tilde{v}=0$ term.  In fact,
 \eqn{GBigSimpler}{
  G(\tau,0) = {\pi/\zeta_p(1) \over z \log p}
    {\Gamma_{\rm Euler}\left( {1 \over z} - 1 \right) \over (2\pi|\tau|)^{{1 \over z}-1}} h(\tau)
  \qquad\text{where}\qquad
    h(\tau) = 1 + O\left( e^{-{\pi^2 \over z \log p}} \right) \,.
 }
The function $h(\tau)$ is periodic in $\log|\tau|$: Up to an overall factor, it is the infinite sum in \eno{GBigAspect}.  Related oscillatory behavior was observed in \cite{Jagoe:2018zz}.  To understand the striking suppression factor $e^{-{\pi^2 \over z \log p}}$, we note that the Stirling approximation yields
 \eqn{StirlingApprox}{
  \left| \Gamma_{\rm Euler}\left(
       {2\pi i \tilde{v} \over z \log p} + {1 \over z} - 1 \right) \right|
    \approx
\sqrt{2\pi}\left(\frac{2\pi}{z\log p}|\tilde{v}|\right)^{\frac{1}{z}-\frac{3}{2}}
 e^{-\frac{\pi^2}{z\log p}|\tilde{v}|}
 \,.
 }
Thus each successive term in the infinite sum in \eno{GBigAspect} is suppressed by a factor on order $e^{-{\pi^2 \over z \log p}}$.  It is worth noting that a highly suppressed amplitude of oscillation of $h(\tau)$ can be arranged without fine-tuning parameters.  For example,
 \eqn{htauEx}{
  \sup_\tau |h(\tau)-1|
   \approx \left| \Gamma_{\rm Euler}\left( 2 + {6\pi i \over \log 2} \right) \right| \approx 10^{-16}
   \qquad\text{when $z=1/3$ and $p=2$\,.}
 }
On the other hand, this same relative amplitude $\sup_\tau |h(\tau)-1|$ rises to $0.5$ when $p$ reaches $1.75 \times 10^5$ if we hold $z=1/3$.  The relative amplitude can never exceed $1$, because one can show, starting from the first line of \eno{gxi}, that $G(\tau,x)$ is always positive, provided $z \in (0,1)$.

\section{Consistent truncation}
\label{TRUNCATION}

Consistent truncation is a phenomenon in compactification of gravitational theories (see for example \cite{Cvetic:2003jy}), whereby one can find consistent, non-linear equations of motion satisfied by a finite number of non-zero Kaluza-Klein modes.  A solution to these truncated equations of motion can be lifted back up to the original, higher dimensional theory.  It has been argued that known consistent truncations arise as a result of a generalized Scherk-Schwarz mechanism \cite{Lee:2014mla}.  Here we introduce a seemingly distinct mechanism whose essential ingredient is the assumption of ultrametricity in the compact direction.  We have no proposals in this paper for how to formulate gravitational theories over a direct product of Archimedean and ultrametric directions, so we will work instead with non-linear scalar field theory throughout.

As a warmup, consider first the action for a classical, real-valued scalar field $\phi$ defined over $\mathbb{R} \times S^1$:
 \eqn{phiFourth}{
  S = \int_{\mathbb{R}} d\tau \int_{S^1} dx \,
    \left[ -{1 \over 8\pi^2} \phi (\partial_\tau^2 + \partial_x^2) \phi + {\lambda_n \over n} \phi^n
      \right] \,.
 }
We could entertain a more general potential $V(\phi)$ in place of ${\lambda_n \over n} \phi^n$---for example, polynomial in $\phi$, or analytic in some neighborhood of $\phi=0$.  It is always assumed however that $\phi=0$ is a solution of the classical equation of motion, which is
 \eqn{phiFourthEom}{
  {1 \over 4\pi^2} (\partial_\tau^2 + \partial_x^2) \phi = V'(\phi) \,.
 }
We could also generalize \eno{phiFourth}-\eno{phiFourthEom} by replacing Euclidean time $\tau \in \mathbb{R}$ by Minkowski space $\mathbb{R}^{3,1}$, and the arguments to follow would be essentially unaffected.  If we assume that the $S^1$ has circumference $1$ and expand
 \eqn{phiExpand}{
  \phi(\tau,x) = \sum_{k\in \mathbb{Z}} \phi_k(\tau) e^{-2\pi i k x} \,,
 }
then unless $V(\phi)$ is purely quadratic, the only way to satisfy the equations of motion \eno{phiFourthEom} while setting all but finitely many of the Kaluza-Klein modes $\phi_k(\tau)$ to $0$ is to set all of them to zero except the mode $\phi_0(\tau)$ with no dependence at all on the $x$ direction.  Then from \eno{phiFourthEom} we obtain immediately
 \eqn{SoftModeEoms}{
  {1 \over 4\pi^2} \partial_\tau^2 \phi_0^2 = V'(\phi_0) \,.
 }
In short, the only consistent truncation of \eno{phiFourthEom} when the $x$ direction is compactified on a circle is the trivial truncation \eno{SoftModeEoms}.

Now consider a real field $\phi$ defined over $\mathbb{R} \times \mathbb{Z}_p$.  Fourier transforms can be written as
 \eqn{Pontryagin}{
  \phi(\tau,x) &= \sum_{k \in \mathbb{Q}_p/\mathbb{Z}_p} \phi_k(\tau) \chi(kx)  \cr
  \phi_k(\tau) &= \int_{\mathbb{Z}_p} dx \, \chi(-kx) \phi(\tau,x) \,,
 }
where $\mathbb{Q}_p/\mathbb{Z}_p$ is understood as a quotient of additive groups.  An element $k \in \mathbb{Q}_p/\mathbb{Z}_p$ is a coset $k = k' + \mathbb{Z}_p$, and it is easy to see that $\chi(k'x)$ for $x \in \mathbb{Z}_p$ is independent of the choice of $k'$. A canonical and unique choice is to make $k'$ its own fractional part.  Thus we can understand the sum in the first line of \eno{Pontryagin} as a sum over all rational numbers in $[0,1)$ whose denominator is a power of $p$.

The action we consider for scalar field theory on $\mathbb{R} \times \mathbb{Z}_p$ is
 \eqn{RZact}{
  S = {1 \over 2} \int_{\mathbb{R}} d\tau \sum_{k \in \mathbb{Q}_p/\mathbb{Z}_p}
    \phi_{-k}(\tau) \left( -{1 \over 4\pi^2} \partial_\tau^2 + |k|_p^{2z} \right) \phi_k(\tau) +
   \int_{\mathbb{R}} d\tau \int_{\mathbb{Z}_p} dx \, {\lambda_n \over n } \phi(\tau,x)^n \,,
 }
and the equations of motion are
 \eqn{MixedEom}{
  {\cal D} \phi(\tau,x) = \lambda_n \phi(\tau,x)^{n-1} \,,
 }
where
 \eqn{calDDef}{
  {\cal D} \equiv {1 \over 4\pi^2} \partial_\tau^2 - D^{2z}_x
 }
and we define the Vladimirov derivative $D_x^\alpha$ of a function $f(x) = \sum_{k \in \mathbb{Q}_p/\mathbb{Z}_p} \chi(kx) \tilde{f}_k$ as
 \eqn{DalphaDef}{
  D^\alpha_x f(x) \equiv \sum_{k \in \mathbb{Q}_p/\mathbb{Z}_p} |k|_p^\alpha \chi(kx) \tilde{f}_k \,.
 }
As before, our arguments generalize readily to polynomial $V(\phi)$ provided $\phi=0$ is a solution of the equations of motion, and to more interesting non-compact Archimedean spaces like $\mathbb{R}^{3,1}$.

The claim is that we may consistently truncate the classical equation of motion \eno{MixedEom} to soft modes:
 \eqn{Dtruncated}{
  {\cal D} \phi_s(\tau,x) = \lambda_n \phi_s(\tau,x)^{n-1} \,,
 }
where $\phi_s$ involves ultrametric Fourier modes whose norms are bounded by any cutoff we please.  Explicitly, we take a cutoff
 \eqn{LambdaM}{
  \Lambda = p^M \,,
 }
and then the requirement we place on $\phi_s$ is that it is a sum of soft modes, which are finite in number:
 \eqn{phiSoft}{
  \phi_s(\tau,x) = \sum_{k \in \mathbb{Q}_p/\mathbb{Z}_p \atop |k|_p \leq \Lambda}
    \phi_k(\tau) \chi(kx) \,.
 }
The truncated equation of motion \eno{Dtruncated} can be written without reference to $x$ as a set of coupled non-linear equations for the finite number of Kaluza-Klein modes $\phi_k(\tau)$ that we kept.  The Kaluza-Klein mass of each mode is evidently $|k|_p^{2z}$.  The equations \eno{MixedEom} and \eno{Dtruncated} have precisely the same form, so another way to phrase our claim is that the coupled equations for the soft modes $\phi_k(\tau)$ close on themselves instead of coupling to hard modes.

Since the linear operator ${\cal D}$ maps soft modes to soft modes and hard to hard, what we have to show is that hard modes aren't sourced by the right hand side of \eno{Dtruncated}: That is, any positive power of a linear combination of soft modes is still a linear combination of soft modes.  This is easy to see by induction once we know that the product of any two soft modes is still soft.  Such a product is
 \eqn{SoftTimesSoft}{
  \chi(k_1 x) \chi(k_2 x) = \chi((k_1+k_2) x)
 }
where $|k_1|_p \leq \Lambda$ and $|k_2|_p \leq \Lambda$.  Ultrametricity then indeed guarantees that $|k_1+k_2|_p \leq \Lambda$.

A point that distinguishes this ultrametric consistent truncation from the usual story in Archimedean gravitational theories is that there is a whole hierarchy of consistent truncations, where one retains all ultrametric momentum $k$ up to any prespecified maximum norm $\Lambda$, and sets all harder modes to zero.

\section{Loop integrals}
\label{INTEGRALS}

Having studied mixed field theories with compact and non-compact $p$-adic dimensions at tree-level in the preceding sections, we now turn to loop corrections, restricting our treatment to $\phi^4$ theory over $\mathbb{R} \times \mathbb{Q}_p$ as specified by \eno{SphiFourth}.  Our main goal is a renormalization group analysis of the Wilson-Fisher fixed points close to $z=1/3$.  The RG transformations we will find describe how the theory changes under a discrete scale transformation $\omega \to \omega/p^z$, $k \to pk$: See \eno{lbs} and \eno{rbs}.  In preparation, we consider in this section the loop integrations involved in evaluating the three one-particle-irreducible (1PI) Feynman diagrams shown in figure~\ref{ThreeDiagrams}.
 \begin{figure}
  \centering
  \begin{subfigure}[b]{0.5\textwidth}
   \centering
   \begin{tikzpicture}
    \draw[thick] (0,0) ellipse (0.6cm and 0.6cm);
    \draw [thick,-] (-1.4,-0.6) -- (1.4,-0.6);
    \draw [thick,->] (-3,-0.6) -- (-1.8,-0.6);
    \node at (-2.4,-0.3) {$\omega,k$};
    \node at (2.6,0) {$I^{(1)}_2(r,\Lambda)$};
   \end{tikzpicture}
  \end{subfigure}%
  \begin{subfigure}[b]{0.5\textwidth}
   \centering
   \begin{tikzpicture}
    \draw[thick] (0,0) ellipse (0.6cm and 0.6cm);
    \draw [thick,-] (-1.4,0.6) -- (-0.6,0) -- (-1.4,-0.6);
    \draw [thick,-] (1.4,0.6) -- (0.6,0) -- (1.4,-0.6);
    \draw [thick,->] (-3,0) -- (-1.8,0);
    \node at (-2.4,0.3) {$\omega,k$};
    \node at (3,0) {$I_4(\omega,k,r,\Lambda)$};
   \end{tikzpicture}
  \end{subfigure}\\[30pt]
  \begin{subfigure}[b]{0.5\textwidth}
   \centering
   \begin{tikzpicture}
    \draw[thick] (0,0) ellipse (0.6cm and 0.6cm);
    \draw [thick,-] (-1.4,0) -- (1.4,0);
    \draw [thick,->] (-3,0) -- (-1.8,0);
    \node at (-2.4,0.3) {$\omega,k$};
    \node at (3,0) {$I^{(2)}_2(\omega,k,r,\Lambda)$};
   \end{tikzpicture}
  \end{subfigure}
  \caption{Loop corrections to two- and four-point functions.  Total momentum $(\omega,k)$ flows into each diagram from the left.}\label{ThreeDiagrams}
 \end{figure}
The corresponding loop integrals are
 \eqn{LoopIntegrals}{
  I^{(1)}_2(r,\Lambda) &\equiv \int d\omega_1 \int^\Lambda dk_1 \,
    {1 \over \omega_1^2 + |k_1|_p^{2/3} + r}  \cr
  I_4(\omega,k,r,\Lambda) &\equiv \int d\omega_1 d\omega_2
    \int^\Lambda dk_1 dk_2 \,
    {\delta\left(\omega-\omega_1-\omega_2\right) \delta\left(k-k_1-k_2\right) \over
      (\omega_1^2 + |k_1|_p^{2/3} + r)(\omega_2^2 + |k_2|_p^{2/3} + r)}  \cr
  I_2^{(2)}(\omega,k,r,\Lambda) &\equiv \int d\omega_1 d\omega_2 d\omega_3
    \int^\Lambda dk_1 dk_2 dk_3 \,
    {\delta\left(\omega-\sum_{i=1}^3\omega_i\right) \delta\left(k-\sum_{i=1}^3k_i\right) \over
      \prod_{i=1}^3 (\omega_i^2 + |k_i|_p^{2/3} + r)} \,,
 }
where the superscripted $\Lambda$ on the ultrametric integrals indicates that we require all internal momenta to satisfy $|k_i|_p \leq \Lambda$, and $\Lambda = p^M$ as in \eno{LambdaM}.  Because we have only one Archimedean direction, no divergences arise from the integrations over $\omega_i$.  Thus all the integrals in \eno{LoopIntegrals} are explicitly finite.  Because the $\omega_i$ and $k_i$ integrations are of an entirely different nature, the anisotropic cut-off does not break any symmetry that would affect the universality class of the theory, just as the same holds true for the anisotropic cut-off employed in the study of the respective renormalization of the temporal and spatial derivative terms in Lifshitz scalar theory in \cite{Iengo:2009ix,Eune:2011zw}.\footnote{We note that Lifshitz scalar theories in Archimedean spacetimes have a long history, dating back as least as far as \cite{Hornreich:1975a}.}

For purposes of a renormalization group analysis, we are interested in divergences that arise as $\Lambda$ becomes large.  As we will explain in the sections~\ref{BUBBLE}, \ref{FOUR}, and~\ref{UNDERGROUND},
 \eqn{DivergenceStructure}{\seqalign{\span\TL & \span\TR & \span\TR}{
  I^{(1)}_2 &= a_{2/3} \Lambda^{2/3} + {}&A_0 r \log \Lambda^{2/3} +
    \text{(finite as $\Lambda \to \infty$)}
    \cr
  I_4 &= &b_0 \log \Lambda^{2/3} + \text{(finite as $\Lambda \to \infty$)}  \cr
  I^{(2)}_2 &= c_{2/3} \Lambda^{2/3} + {}&(C_0 r + C'_0 \omega^2)
    \log \Lambda^{2/3} + \text{(finite as $\Lambda \to \infty$)} \,,
 }}
for some coefficients $a_{2/3}$, $A_0$, $b_0$, $c_{2/3}$, $C_0$, and $C'_0$, all of which are independent of $\omega$, $k$, and $r$.  For the convenience of the reader who is not interested in the details of the computation, we tabulate here the coefficients that we will need for the renormalization group analysis in section~\ref{FIXED}:
 \eqn{Coefficients}{
  \begin{tabular}{c|c|c}
   $A_0$ & $b_0$ & $C'_0$ \\ \hline & & \\[-15pt]
   $\displaystyle -{3\pi \over 4\zeta_p(1) \log p}$ &
    $\displaystyle {3\pi \over 4\zeta_p(1) \log p}$ &
    $\displaystyle
     -{9\pi^2 \over 2 \zeta_p(1) \log p} \left( {1 - {2 \over p} \over 81} + {1 \over 8 \zeta_p(1)}
     \sum_{v=1}^\infty {p^{-2v/3} \over \left( 1 + {1 \over 2} p^{-v/3} \right)^3} \right)$
  \end{tabular}
 }
The other coefficients in \eno{DivergenceStructure} can be computed by means similar to the ones we will exhibit.

\subsection{The bubble diagram}
\label{BUBBLE}

Integrating first over $\omega_1$, we obtain
 \eqn{Isimple}{
  I^{(1)}_2 &= \pi \int^\Lambda dk_1 \, {1 \over \sqrt{|k_1|_p^{2/3} + r}}
   = \pi \int_{r^{3/2}}^\Lambda dk_1 \, \left[ {1 \over |k_1|_p^{1/3}} - {r \over 2 |k_1|_p} \right] +
     \text{(finite)}  \cr
   &= \pi {\zeta_p(2/3) \over \zeta_p(1)} \Lambda^{2/3} - {3\pi \over 4 \zeta_p(1) \log p} r
       \log {\Lambda^{2/3} \over r} + \text{(finite)} \,,
 }
where $\int_{r^{3/2}}^\Lambda dk_1$ means that the range of integration is $r^{3/2} \leq |k_1|_p \leq \Lambda$, and we use ${}+\text{(finite)}$ to denote any dropped terms which remain finite as $\Lambda \to \infty$.  In the second equality of \eno{Isimple} we evaluate the integrals by splitting the integration domain into momentum shells; then the integrals become geometric series.  Thus we have verified the value for $A_0$ quoted in \eno{Coefficients} and checked along the way that the divergence structure of $I^{(1)}_2$ is as claimed in \eno{DivergenceStructure}.

\subsection{The four-point diagram}
\label{FOUR}

Power counting using the rules of section~\ref{POWER} allows us to check that the leading divergence in $I_4$ is logarithmic in $\Lambda$, as claimed in \eno{DivergenceStructure}.  The coefficient $b_0$ cannot depend on $\omega^2$ or $r$ because when we expand the integrand of $I_4$ in a power series in these quantities, all terms but the first (i.e.~the one that is independent of $\omega^2$ and $r$) lead to convergent integrals.  A power series in $k$ doesn't make sense for a real-valued function, but a much more powerful argument is available from ultrametricity: After a $u$-substitution $k_1 \to k_1 - k$, we can see that the integrand for $I_4$ agrees perfectly with its $k=0$ form except when $|k_1|_p \leq |k|_p$.  This immediately implies that any ultraviolet divergences are independent of $k$ and is an example of the general phenomenon of ultrametric non-renormalization; c.f.~\cite{Lerner:1989ty}.

Because $b_0$ is independent of $\omega$, $k$, and $r$, we can set $\omega=k=r=0$ in the integrand.  As before, it is useful to restrict the region of ultrametric integration to $r^{3/2} \leq |k_i|_p \leq \Lambda$.  We arrive at
 \eqn{Ifour}{
  I_4 &= \int d\omega_1 \int_{r^{3/2}}^\Lambda dk_1 \, {1 \over \left(\omega_1^2 + |k_1|_p^{2/3}\right)^2}
    + \text{(finite)}
   = {\pi \over 2} \int_{r^{2/3}}^\Lambda {dk_1 \over |k_1|_p} + \text{(finite)}  \cr
   &= {3\pi \over 4 \zeta_p(1) \log p} \log {\Lambda^{2/3} \over r} + \text{(finite)} \,,
 }
thus checking the value for $b_0$ quoted in \eno{Coefficients} as well as the overall divergence structure of $I_4$ claimed in \eno{DivergenceStructure}.

\subsection{The underground diagram}
\label{UNDERGROUND}

Power counting allows us to check that the leading divergence in $I^{(2)}_2$ scales as $\Lambda^{2/3}$, as claimed in \eno{DivergenceStructure}.  After a $u$-substitution $k_1 \to k_1 - k$, we can see that the integrand for $I^{(2)}_2$ agrees perfectly with its $k=0$ form except when $|k_1|_p \leq |k|_p$.  If we restrict the domain of integration of $k_1$ to $|k_1|_p \leq |k|_p$, then power counting shows that the integral converges.  Thus there can be no $k$ dependence in any ultraviolet divergence of $I^{(2)}_2$, and we can hereafter set $k=0$.  Expanding the integrand in a power series in $\omega^2$ and $r$ leads to the three divergent terms shown in \eno{DivergenceStructure}, followed by convergent integrals.  We are interested only in the term $C'_0 \omega^2 \log\Lambda^{2/3}$.  Therefore we can set $r=0$ in the integrand, while still restricting the region of ultrametric integration to $r^{3/2} \leq |k_i|_p \leq \Lambda$.  But we cannot set $\omega=0$ at this stage.  We introduce Schwinger parameters $u_i$, $i \in \{1,2,3\}$ in order to write
\eqn{xuDefs}{
  {1 \over \omega_i^2 + |k_i|_p^{2/3}} = \int_0^\infty du_i \,
    e^{-u_i (\omega_i^2 + |k_i|_p^{2/3})} \,,
 }
and then the $\omega_i$ integrations in $I^{(2)}_2$ can be performed straightforwardly in terms of Gaussian integrals.  Subsequently passing to Feynman parameters $x_i = u_i/U$ where $U = \sum_i u_i$, one finds that
\eqn{JtwoFeynman}{
  I^{(2)}_2 &= \pi \int_{r^{3/2}}^\Lambda dk_1 dk_2 dk_3
     \int_0^1 dx_1 dx_2 dx_3 \,
      {\delta\left( 1 - {\textstyle\sum_i x_i} \right) \over
        \sqrt{x_1 x_2 x_3 \sum_i 1/x_i}}  \, {\delta\left( {\textstyle\sum_i k_i} \right) \over
     \left( {\omega^2 \over \sum_i 1/x_i} + \sum_i x_i |k_i|_p^{2/3} \right)^2}  \cr
   &\qquad{} + \text{(finite)} \,.
 }
In order to isolate the term $C'_0 \omega^2 \log\Lambda^{2/3}$, we consider the derivative
\eqn{dJtwo}{
  {\partial I^{(2)}_2 \over \partial\omega^2} &=
    C'_0 \log \Lambda^{2/3} + \text{(finite)} \cr
   &= -2\pi \int_{r^{3/2}}^\Lambda dk_1 dk_2 dk_3 \,
     \int_0^1 dx_1 dx_2 dx_3 \,
      {\delta\left( 1 - {\textstyle\sum_i x_i} \right) \over
        \sqrt{x_1 x_2 x_3 \left( \sum_i 1/x_i \right)^3}}  \,
        {\delta\left( {\textstyle\sum_i k_i} \right)  \over
     \left( \sum_i x_i |k_i|_p^{2/3} \right)^3}  \cr
   &\qquad\qquad{} + \text{(finite)}  \cr
   &= -\pi^2
    \int_{r^{3/2}}^\Lambda dk_1 dk_2 dk_3 \,
      {\delta\left({\textstyle\sum_i k_i}\right) \over
      |k_1 k_2 k_3|_p^{1/3} \left( {\textstyle\sum_i |k_i|_p^{1/3}} \right)^3} +
    \hbox{(finite)}
 }
where in the third equality we used the identity
\eqn{FeynmanIdent}{
  \int_0^1 dx_1 dx_2 dx_3 \, {\delta\left( 1 - {\textstyle\sum_i x_i} \right)
     \over \sqrt{x_1 x_2 x_3 \left( \sum_i 1 / x_i \right)^3}}
    {1 \over \left( \sum_i a_i x_i \right)^3} =
   {\pi/2 \over \sqrt{a_1 a_2 a_3} (\sqrt{a_1} + \sqrt{a_2} + \sqrt{a_3})^3} \,,
 }
valid for positive $a_1$, $a_2$, and $a_3$.

The ultrametric integral remaining in the last line of \eno{dJtwo} is immediately seen to be logarithmically divergent; indeed, we can integrate over a common scale of the $k_i$ to obtain
 \eqn{dJLog}{
  {\partial I^{(2)}_2 \over \partial\omega^2} =
   -{3\pi^2 \over 2 \log p} \log {\Lambda^{2/3} \over r} (J_\text{eq} + 3 J_\text{tall}) +
     \text{(finite)} \,,
 }
where
 \eqn{Jeqtall}{
  J_\text{eq} &= {1 \over 27} \int_{\mathbb{U}_p} dk_1 dk_2 dk_3 \,
      \delta\left({\textstyle\sum_i k_i}\right)  \cr
  J_\text{tall} &= {1 \over 8} \int_{p\mathbb{Z}_p} dk_1 \int_{\mathbb{U}_p} dk_2 dk_3 \,
      {\delta\left({\textstyle\sum_i k_i}\right) \over
      |k_1|_p^{1/3} \left( 1+{1 \over 2}|k_1|_p^{1/3} \right)^3} \,.
 }
To reach the form indicated on the right hand side of \eno{dJLog}, we first note that the condition $k_1+k_2+k_3 = 0$ is the statement that the $k_i$, thought of as vectors in $\mathbb{Q}_p$, form the (directed) sides of a triangle.  A well-known property of ultrametric norms is that all triangles are either equilateral or tall isosceles, meaning that the base is shorter than the two equal sides.  That is, either $|k_1|_p=|k_2|_p=|k_3|_p$, or else $|k_1|_p < |k_2|_p = |k_3|_p$ up to relabeling of the $k_i$.  The integration region where the equilateral condition holds is disjoint from the three isomorphic tall isosceles regions, and each of them is disjoint from the others.  So we can carry out the full integration in the last line of \eno{dJtwo} by adding up the equilateral contribution plus three times the tall isosceles contribution with $|k_1|_p < |k_2|_p = |k_3|_p$, as in \eno{dJLog}.

To compute $J_\text{em}$, we set
 \eqn{kSet}{
  k_i = a_i + p n_i \qquad\text{where $a_i \in \mathbb{F}_p^\times$ and $n_i \in \mathbb{Z}_p$\,.}
 }
Here we identify the finite field $\mathbb{F}_p$ with the set $\{0,1,2,\dots,p-1\}$, and $\mathbb{F}_p^\times$ comprises the non-zero elements of this set.  We use the replacements
 \eqn{kintRule}{
  \int_{\mathbb{U}_p} dk_i \to
   {1 \over p} \sum_{a_i \in \mathbb{F}_p^\times} \int_{\mathbb{Z}_p} dn_i \qquad\qquad
  \delta(k_1+k_2+k_3) \to p \delta_{a_1+a_2+a_3} \delta(n_1+n_2+n_3) \,.
 }
Using \eno{kintRule} on the expression \eno{Jeqtall} for $J_\text{eq}$, we see immediately that
 \eqn{JeqEval}{
  J_\text{eq} = {1 \over 27 p^2} \sum_{a_1,a_2,a_3 \in \mathbb{F}_p^\times} \delta_{a_1+a_2+a_3}
    = {1 \over 27\zeta_p(1)} \left( 1 - {2 \over p} \right) \,.
 }
Using the same parametrization \eno{kSet} for $k_2$ and $k_3$ in the leads to the following evaluation of $J_\text{tall}$:
 \eqn{JtallEval}{
  J_\text{tall} = {1 \over 8p} \sum_{a_2,a_3 \in \mathbb{F}_p^\times} \delta_{a_2+a_3}
    \int_{p\mathbb{Z}_p} {dk_1 \over |k_1|^{1/3} \left( 1 + {1 \over 2} |k_1|^{1/3} \right)^3}
   = {1 \over 8 \zeta_p(1)^2} \sum_{v=1}^\infty {p^{-2v/3} \over
        \left( 1 + {1 \over 2} p^{-v/3} \right)^3} \,.
 }
Putting \eno{dJtwo}, \eno{dJLog}, \eno{JeqEval}, and \eno{JtallEval} together, we arrive at
 \eqn{CpZero}{
  C'_0 = -{9\pi^2 \over 2 \zeta_p(1) \log p} \left( {1 - {2 \over p} \over 81} + {1 \over 8 \zeta_p(1)}
    \sum_{v=1}^\infty {p^{-2v/3} \over \left( 1 + {1 \over 2} p^{-v/3} \right)^3} \right) \,,
 }
which completes our verification of the coefficients tabulated in \eno{Coefficients}.  It may be noted that the sum over $v$ in \eno{CpZero} can be performed explicitly in terms of $q$-polygamma functions, defined as
 \eqn{qPolygamma}{
  \psi_q^{(n)}(z) \equiv {\partial^n \psi_q(z) \over \partial z^n}
    \qquad\hbox{where}\qquad
   \psi_q(z) \equiv -\log(1-q) +
     (\log q) \sum_{\ell=0}^\infty {q^{\ell+z} \over 1 - q^{\ell+z}} \,.
 }
Specifically,
 \eqn{SumExplicit}{
  \sum_{v=1}^\infty {p^{-2v/3} \over \left(1+{1 \over 2} p^{-v/3}\right)^3}
    = {6 \over \log p} -
      {18 \psi_{p^{1/3}}^{(1)}\left( {3\log(-2) \over \log p}+1 \right) \over (\log p)^2} -
      {54\psi_{p^{1/3}}^{(2)}\left( {3\log(-2) \over \log p}+1 \right) \over (\log p)^3} \,.
 }

\section{The Wilson-Fisher fixed point}
\label{FIXED}

With the divergent parts of the simplest loop diagrams in hand, we can proceed to an RG analysis of the Wilson-Fisher fixed point for values of $z$ slightly larger than $1/3$, where the operator $\phi^4$ is just barely relevant.  We set
 \eqn{epsilonDef}{
  \epsilon = z - {1 \over 3}
 }
and work to the lowest non-trivial order in $\epsilon$.  In a Wilsonian picture, we start with a theory constrained by a cutoff $|k|_p \leq \Lambda$, and we find the coupling $\lambda_s$ of an effective theory for modes with $|k|_p \leq \Lambda/p$ by integrating out the modes with $|k|_p = \Lambda$.  More precisely, if $R_\Lambda$ is the region where all internal momenta $|k_i|_p \leq \Lambda$, then the modes we integrate out correspond to the integration region which is the setwise complement $R_\Lambda \backslash R_{\Lambda/p}$.

Standard analysis of the tree and one-loop diagrams for the four-point Green's function (identical to the textbook case of $\phi^4$ theory on $\mathbb{R}^4$) leads to
 \eqn{lambdas}{
  \lambda_s = \lambda - {3 \over 2} \lambda^2 \left[ I_4(\Lambda) - I_4(\Lambda/p) \right]
    = \lambda - {3 \over 2} \lambda^2 \left[ I_4^\text{div}(\Lambda) -
      I_4^\text{div}(\Lambda/p) \right] = \lambda - \lambda^2 b_0 \log p \,,
 }
where in square brackets we have isolated the result of restricting the integral for $I_4$ in \eno{LoopIntegrals} to the region $R_\Lambda \backslash R_{\Lambda/p}$.  In the second equality we have replaced $I_4$ by its divergent part $I_4^\text{div}$ on grounds that when evaluating the evolution of the coupling constant of the critical theory, it doesn't matter what $\omega$, $k$, and $r$ are, provided only that they are small compared to $\Lambda$.  Throughout, and in particular in the equation in \eno{lambdas}, we use our analysis from section~\ref{INTEGRALS} of loop integrals right at $z=1/3$ even though our final aim is to set $z=1/3+\epsilon$.  The reason this is permissible is that the critical point value of $\lambda$ turns out to be $O(\epsilon)$, and because we work only to the lowest non-trivial order in $\epsilon$, we can evaluate our loop integrals to zeroth order in $\epsilon$.

In terms of dimensionless couplings $\bar\lambda = \lambda/\Lambda^{3\epsilon}$ and $\bar\lambda_s = \lambda_s/(\Lambda/p)^{3\epsilon}$, \eno{lambdas} can be rewritten as
 \eqn{lbs}{
  \bar\lambda_s = p^{3\epsilon} \left[ \bar\lambda - \bar\lambda^2 b_0 \log p \right]
    = \bar\lambda + 3\epsilon \bar\lambda \log p - \bar\lambda^2 b_0 \log p \,,
 }
where we again discard terms that are beyond the lowest non-trivial order in $\epsilon$.  The fixed point condition is $\bar\lambda_s = \bar\lambda$, which immediately implies $\bar\lambda = \bar\lambda_*$ where
 \eqn{lambdaStar}{
  \bar\lambda_* = {3\epsilon \over b_0} = {4 \zeta_p(1) \log p \over \pi} \epsilon \,.
 }
This is the Wilson-Fisher fixed point for mixed $\phi^4$ theory.

\subsection{The anomalous dimension for the mass term}
\label{MASS}

Through a similarly standard analysis up to one loop for the two-point Green's function, one finds
 \eqn{rs}{
  r_s &= r + {\lambda \over 2} \left[ I^{(1)}_2(\Lambda) - I^{(1)}_2(\Lambda/p) \right]
      = r + {\lambda \over 2} \left[ I^{(1),\text{div}}_2(\Lambda) -
          I^{(1),\text{div}}_2(\Lambda/p) \right]  \cr
   &= r + {\lambda \over 2} \left( a_{2/3} {\Lambda^{2/3} \over \zeta_p(2/3)} +
       {2 \over 3} A_0 r \log p \right) \,,
 }
or, in terms of dimensionless variables $\bar{r} = r/\Lambda^{2z}$ and $\bar{r}_s = r/(\Lambda/p)^{2z}$,
 \eqn{rbs}{
  \bar{r}_s = p^{2z} \left[ \bar{r} +
     {\bar\lambda \over 2} \left( {a_{2/3} \over \zeta_p(2/3)} +
       {2 \over 3} A_0 \bar{r} \log p \right) \right]
    = p^{2z + \gamma_r} \left[ \bar{r} + {\bar\lambda \over 2} {a_{2/3} \over \zeta_p(2/3)} \right] \,.
 }
where the anomalous dimension of $r$ is
 \eqn{gammar}{
  \gamma_r = {\bar\lambda \over 3} A_0 \,,
 }
and we are again working to lowest non-trivial order in $\epsilon$, understanding that $\lambda \sim O(\epsilon)$.  When $\bar\lambda=\bar\lambda_*$, we see that
 \eqn{gammarStar}{
  \gamma_r = {A_0 \over b_0} \epsilon = -\epsilon \,.
 }
The value of $a_{2/3}$ enters into determining the value $\bar{r}_*$ of $\bar{r}$ at the Wilson-Fisher fixed point, but it doesn't actually matter for determining the anomalous dimension $\gamma_r$, because $\gamma_r$ has to do with the multiplicative scaling of a deviation of $r$ from the fixed point value $\bar{r}_*$.

A convenient shortcut to the result \eno{gammar} is to note that the one-particle irreducible (1PI) two-point function is
 \eqn{OnePI}{
  \Gamma^{(2)}(\omega,k) &= \omega^2 + |k|_p^{2z} + r + {\lambda \over 2} I^{(1)}_2(\Lambda)  \cr
    &= \omega^2 + |k|_p^{2z} + r + {\lambda \over 2} \left[ a_{2/3} \Lambda^{2/3} +
        A_0 r \log {\Lambda^{2/3} \over r} \right] + \text{(finite)}  \cr
    &= \omega^2 + |k|_p^{2z} + r^{1 - {\lambda \over 2} A_0} +
     {\lambda \over 2} \left[ a_{2/3} \Lambda^{2/3} + A_0 r \log \Lambda^{2/3} \right] +
     \text{(finite)} \,.
 }
In the third equality of \eno{OnePI}, we are relying on the smallness of $\lambda$ to promote the $r \log r$ term to a modification of the power of $r$.  The second term in square brackets in the last line of \eno{OnePI} modifies the overall normalization of the mass term, and the first term in square brackets contributes to the critical value of $r$ where the infrared theory becomes massless.  By setting the dimensions of $r^{1 - {\lambda \over 2} A_0}$ and $|k|_p^{2z}$ equal, we see that
 \eqn{DeltaRconstraint}{
  \left( 1 - {\lambda \over 2} A_0 \right) \Delta_r = 2z \,,
 }
and upon setting $\Delta_r = 2z + \gamma_r$, we recover \eno{gammar} to leading order in $\epsilon$.

\subsection{A renormalized dynamical scaling exponent}
\label{DYNAMICAL}

We can use a similar shortcut to the one explained in \eno{OnePI}-\eno{DeltaRconstraint} to treat the effects of the underground diagram and derive a loop correction to the dynamical scaling exponent $z$.  Including the underground diagram, the 1PI two-point function is
 \eqn{GammaTwoGeneral}{
  \Gamma^{(2)}(\omega,k) &= \omega^2 + |k|_p^{2z} - {\lambda^2 \over 6} I^{(2)}_2(\Lambda) +
     \text{($\omega$-independent)} + \text{(finite)} \,.
 }
There are contributions to $\Gamma^{(2)}(\omega,k)$ from other diagrams that include divergent terms, but these divergences have no dependence on $\omega^2$; a case in point is precisely the mass renormalization as treated in \eno{OnePI}.  In contrast, $I^{(2)}_2(\Lambda)$ has a divergent term with non-trivial dependence on $\omega$:
 \eqn{IttDiv}{
  I^{(2),\text{div}}_2(\Lambda) = C'_0 \omega^2 \log {\Lambda^{2/3} \over \omega^2} + \text{(finite)} \,.
 }
In \eno{IttDiv} we have artfully rendered the argument of the logarithm dimensionless with a factor of $\omega^2$.  This is the only sensible thing to do when studying the critical theory, because the mass is effectively zero at the critical point.  In short,
 \eqn{GammaTwoCorrected}{
   \Gamma^{(2)}(\omega,k) &= \omega^2 + {\lambda^2 \over 6} C'_0 \omega^2 \log\omega^2 + |k|_p^{2z} -
     {\lambda^2 \over 6} C'_0 \omega^2 \log \Lambda^{2/3} +
     \text{($\omega$-independent)} + \text{(finite)}  \cr
    &= \left(\omega^2\right)^{z/z_\text{IR}} + |k|_p^{2z} -
     {\lambda^2 \over 6} C'_0 \omega^2 \log \Lambda^{2/3} +
     \text{($\omega$-independent)} + \text{(finite)} \,,
 }
where we have introduced an infrared dynamical scaling exponent
 \eqn{zIR}{
  z_\text{IR} = z \left( 1 - {\lambda^2 \over 6} C'_0 \right) \,.
 }
Evaluated at the Wilson-Fisher fixed point,
 \eqn{zIRvalue}{
  z_\text{IR} = z - {\epsilon^2 \over 2} {C_0' \over b_0^2}
   = z + 4 \epsilon^2 \zeta_p(1) \log p \left( {1 - {2 \over p} \over 81} + {1 \over 8 \zeta_p(1)}
    \sum_{v=1}^\infty {p^{-2v/3} \over \left( 1 + {1 \over 2} p^{-v/3} \right)^3} \right) \,.
 }

The renormalized dynamical exponent has many consequences.  For example, $\Delta_\omega = z_\text{IR}$ implies $\Delta_\tau = -z_\text{IR}$, where we always insist on $\Delta_{|k|_p} = -\Delta_{|x|_p} = 1$ for reference.  The field $\phi$ has infrared dimension
 \eqn{DeltaPhi}{
  \Delta_\phi = {1 + z_\text{IR} - 2z \over 2} \,,
 }
and in place of \eno{Gscaling} we must have an infrared scaling form
 \eqn{GscalingIR}{
  \hat{G}(\tau,x) = {1 \over |x|_p^{2\Delta_\phi}} \hat{g}(\hat\xi) \qquad\text{where}\qquad
    \hat\xi = {2\pi |\tau| \over |x|_p^{z_\text{IR}}} \,.
 }

\subsection{Position space treatment of the underground diagram}
\label{POSITION}

While the analysis of section~\ref{UNDERGROUND} seems to be the most efficient way to get hold of the coefficient $C'_0$ of the $\omega^2 \log {\Lambda^{2/3} \over \omega^2}$ term in $I^{(2)}_2(\Lambda)$ (for $z=1/3$), we would like to understand more completely the $\omega$ dependence of this loop integral, because it contributes directly to the full 1PI two-point function $\Gamma^{(2)}(\omega,k)$.  To be precise, what we will add in this section is a partial understanding of {\it finite} terms in $\Gamma^{(2)}(\omega,k)$ which are oscillatory functions of $|\omega|$, similar to the oscillatory behavior in the tree level Green's function $G(\tau,x)$ treated in section~\ref{PROPAGATOR}.  Throughout, we will work exactly at $z=1/3$ and $r=0$, understanding that the analysis can easily be extended to the Wilson-Fisher fixed point with $z=1/3+\epsilon$.  Because we are after finite terms, we remove the momentum cutoff $\Lambda$ and impose ad hoc regulators as needed.  Our methodology is based in position space, along the lines of \cite{Freedman:1991tk}.

We begin with the observation that the loop integral $I^{(2)}_2(\omega,k)$ (without a momentum cutoff) becomes simple in position space:
 \eqn{IttPos}{
  I^{(2)}_2(\omega,k) = \int_{\mathbb{R}} d\tau \int_{\mathbb{Q}_p} dx \,
    e^{2\pi i \omega\tau} \chi(-kx) G(\tau,x)^3 \,,
 }
where, as in \eno{Gposition},
 \eqn{GtxMassless}{
  G(\tau,x) = \int_{\mathbb{R}} d\omega \int_{\mathbb{Q}_p} dk \,
    {e^{-2\pi i \omega\tau} \chi(kx) \over \omega^2 + |k|_p^{2z}} \,.
 }
Due to the complexity of Fourier transforms over $\mathbb{R} \times \mathbb{Q}_p$, we are unable to give a complete account of \eno{IttPos}.  However, the analysis simplifies somewhat when $k=0$, because then $\omega$ is the only dimensionful parameter present.  Then we claim
 \eqn{IttExpand}{
  I^{(2)}_2(\omega,0) = c_0 \omega^2 \log \omega^2 + \sum_{\tilde{v} \neq 0} c_{\tilde{v}}
    |\omega|^{2 + {6\pi i \tilde{v} \over \log p}} \,,
 }
where $\tilde{v} \in \mathbb{Z}$.  (A pure $\omega^2$ term without the logarithm is present in \eno{IttExpand}, but its coefficient depends on the regularization scheme, so we do not attempt to compute it.)  The main technical goal of this section is to provide explicit expressions for the coefficients $c_{\tilde{v}}$, which are determined solely in terms of $p$ and $\tilde{v}$.  Comparing with \eno{IttDiv}, we see that $c_0 = -C'_0$, and a non-trivial check on our work in this section is that we can check this equality numerically, even though our explicit expression for $c_0$ is quite complicated compared to the one we gave in \eno{CpZero}.

Before getting down to detailed calculations, let's explain why the form \eno{IttExpand} is plausible.  Dimensional analysis shows that $[I^{(2)}_2] = 2/3$ when $z=1/3$, so powers $|\omega|^{2 + {6 \pi i \tilde{v} \over \log p}}$ are the ones that should contribute: These are precisely the powers of $|\omega|$ which scale by a factor of $p^{-2/3}$ when $\omega \to p^{-1/3} \omega$.  The logarithmic divergence of $I^{(2)}_2$ slightly invalidates this dimensional reasoning, but the result is only to modify the $\omega^2$ term to $\omega^2 \log \omega^2$ as shown in \eno{IttExpand}.

To verify the form \eno{IttExpand} and compute the coefficients $c_{\tilde{v}}$, a useful first step is to carry out the ultrametric Fourier transform at $k=0$ by defining
 \eqn{Fdef}{
  F(\tau) = \int_{\mathbb{Q}_p} dx \, G(\tau,x)^3
 }
so that
 \eqn{IttFromF}{
  I^{(2)}_2(\omega,0) = \int_{\mathbb{R}} d\tau \, e^{2\pi i \omega\tau} F(\tau) \,.
 }
The same dimensional reasoning used in the previous paragraph suffices to show that we can expand
 \eqn{FExpand}{
  F(\tau) = \sum_{\tilde{v} \in \mathbb{Z}} \tilde{c}_{\tilde{v}}
    |\tau|^{-3 - {6\pi i \tilde{v} \over \log p}}
 }
for some coefficients $\tilde{v}$.  Only, this time there is no failure of dimensional reasoning because the integral in \eno{Fdef} is convergent provided $\tau \neq 0$.

We recognize $|\tau|^3 F(\tau)$ as a function which is periodic in $\log|\tau|$, with period ${1 \over 3} \log p$.  Thus \eno{FExpand} is essentially a Fourier series, and we can extract the coefficients $\tilde{c}_{\tilde{v}}$ from Fourier integrals:
 \eqn{ctildeInt}{
  \tilde{c}_{\tilde{v}} &= {3 \over \log p} \int_1^{p^{1/3}} d\tau \,
    \tau^{2 + {6\pi i \tilde{v} \over \log p}} F(\tau)
   = {3 \over \log p} \int_1^{p^{1/3}} d\tau \,
      \tau^{2 + {6\pi i \tilde{v} \over \log p}} \int_{\mathbb{Q}_p} dx \, G(\tau,x)^3  \cr
   &= {3 \over \log p} \sum_{v \in \mathbb{Z}} \int_{p^{-v} \mathbb{U}_p} dx \int_1^{p^{1/3}} d\tau \,
     \tau^{2 + {6\pi i \tilde{v} \over \log p}} G(\tau,x)^3 \,.
 }
In the second line of \eno{ctildeInt} we divided the integral over $\mathbb{Q}_p$ into integrals over shells $p^{-v} \mathbb{U}_p$.  A key point, verified as usual by dimensional analysis starting from \eno{EngineeringDimensions}, is that $G(p^{v/3} \tau,p^{-v} x) = p^{-2v/3} G(\tau,x)$, so on replacing $x \to p^{-v} x$ and $\tau \to p^{v/3} \tau$ in the last line of \eno{ctildeInt}, the Jacobian factor $p^{4v/3}$ from the measure $dx d\tau$ just cancels against a factor of $p^{-4v/3}$ from the transformation of $\tau^{2 + {6\pi i \tilde{v} \over \log p}} G(\tau,x)^3$.  Thus
 \eqn{ctildeFull}{
  \tilde{c}_{\tilde{v}} = {3 \over \log p} \sum_{v \in \mathbb{Z}} \int_{\mathbb{U}_p} dx
    \int_{p^{-v/3}}^{p^{(1-v)/3}} d\tau \, \tau^{2 + {6\pi i \tilde{v} \over \log p}} G(\tau,x)^3
   = {3 \over \zeta_p(1) \log p} \int_0^\infty d\tau \,
     \tau^{2 + {6\pi i \tilde{v} \over \log p}} G(\tau,1)^3 \,.
 }
To obtain the second equality in \eno{ctildeFull}, we noticed that the intervals of $\tau$ integration neatly fit together to cover all of $\mathbb{R}$.  We also remembered that $G(\tau,x)$ depends on $x$ only through its $p$-adic norm $|x|_p$, so it is constant (at fixed $\tau$) as $x$ runs over $\mathbb{U}_p$.  Referring to \eno{Gscaling} and \eno{gxi}, we arrive at
 \eqn{Gexplicit}{
  G(\tau,1) = {\pi \over \zeta_p(1)} \sum_{v=-1}^\infty h_v p^{-2v/3} e^{-2\pi p^{-v/3} \tau} \,,
 }
where
 \eqn{hvDef}{
  h_v = \left\{ \seqalign{\span\TC &\qquad\span\TT}{
    1 & if $v \geq 0$  \cr
    {1 \over 1-p} & if $v = -1$\,.
   } \right.
 }
Plugging \eno{Gexplicit} into \eno{ctildeFull}, we arrive at
 \eqn{ctildeAgain}{
  \tilde{c}_{\tilde{v}} &= {3\pi^3 \over \zeta_p(1)^4 \log p}
    \sum_{v_1,v_2,v_3=-1}^\infty h_{v_1} h_{v_2} h_{v_3} p^{-2(v_1+v_2+v_3)/3}
     \int_0^\infty d\tau \, \tau^{2 + {6\pi i \tilde{v} \over \log p}}
      e^{-2\pi (p^{-v_1/3} + p^{-v_2/3} + p^{-v_3/3}) \tau}  \cr
   &= {3\pi^3 \over \zeta_p(1)^4 \log p}
    \Gamma_\text{Euler}\left( 3 + {6\pi i \tilde{v} \over \log p} \right)
    \sum_{v_1,v_2,v_3=-1}^\infty
     {h_{v_1} h_{v_2} h_{v_3} p^{-2(v_1+v_2+v_3)/3} \over
       \left[ 2\pi (p^{-v_1/3}+p^{-v_2/3}+p^{-v_3/3}) \right]^{3 + {6\pi i \tilde{v} \over \log p}}} \,.
 }

With the coefficients $\tilde{c}_{\tilde{v}}$ in hand, we now perform the real Fourier transform \eno{IttFromF} term by term in the expansion \eno{FExpand} in order to get \eno{IttExpand}.  We start with the general identity
 \eqn{GeneralFourier}{
  \int_{\mathbb{R}} d\tau \, e^{2\pi i \omega\tau} |\tau|^s =
    {\Gamma_\infty(1+s) \over |\omega|^{1+s}} + \text{(regulator terms)} \,,
 }
where we have defined
 \eqn{GammaInfty}{
  \Gamma_\infty(s) = {\zeta_\infty(s) \over \zeta_\infty(1-s)} \qquad\text{and}\qquad
   \zeta_\infty(s) = \pi^{-s/2} \Gamma_\text{Euler}(s/2) \,.
 }
Because we are interested in finite parts of $I^{(2)}_2$, we will not concern ourselves with the regulator terms.\footnote{For completeness, let us give a short account of the regulator terms.  They are absent when the integral converges.  When there is a UV divergence (i.e.~a divergence at small $\tau$), the regulator terms are nonnegative integer powers of $\omega^2$ with divergent coefficients.  When there is an IR divergence (i.e.~at large $\tau$), the regulator terms are distributions with support at $\omega=0$, which may or may not have divergent coefficients.}  Expanding \eno{GeneralFourier} around $s=-3$ and discarding a regulator term that diverges as $1/(s+3)$, we find
 \eqn{LogFourier}{
  \int_{\mathbb{R}} d\tau \, {e^{2\pi i \omega\tau} \over |\tau|^3}
    = 2\pi^2 \omega^2 \log\omega^2 \,.
 }
Using \eno{GeneralFourier} and \eno{LogFourier}, we obtain immediately
 \eqn{cFromcTilde}{
  c_0 &= 2\pi^2 \tilde{c}_0  \cr
  c_{\tilde{v}} &= {c_{\tilde{v}} \over \Gamma_\infty\left( 3 + {6\pi i \tilde{v} \over \log p} \right)}
     \qquad \text{for $\tilde{v} \neq 0$}\,.
 }
The equality $c_0 = -C'_0$, remarked upon following \eno{IttExpand}, reduces to
 \eqn{SummationIdentity}{
  \sum_{v_1,v_2,v_3 = -1}^\infty {h_{v_1} h_{v_2} h_{v_3} p^{-2(v_1+v_2+v_3)/3} \over
    (p^{-v_1/3}+p^{-v_2/3}+p^{-v_3/3})^3} = 3 \zeta_p(1)^3 \left( {1 - {2 \over p} \over 81} +
      {1 \over 8 \zeta_p(1)} \sum_{v=1}^\infty {p^{-2v/3} \over \left( 1 + {1 \over 2} p^{-v/3}
       \right)^3} \right) \,.
 }
It is easy to check numerically that \eno{SummationIdentity} is satisfied to good accuracy for a specified $p$, and we did so for the first $10$ primes.  But it has nothing particularly to do with prime numbers: Both sides of \eno{SummationIdentity} are easily seen to be power series in $p^{-1/3}$ with rational coefficients, and we checked that the first $20$ terms agree.  We do not have a general proof for \eno{SummationIdentity}.

The oscillatory terms $c_{\tilde{v}} |\omega|^{2 + {6\pi i \tilde{v} \over \log p}}$ in the expansion \eno{IttExpand} for $I^{(2)}_2(\omega,0)$ are quite small for modest values of $p$.  This is similar to the situation we encountered in section~\ref{PROPAGATOR} for the free propagator $G(\tau,0)$ at purely Archimedean separations.  The main source of suppression of the coefficients $c_{\tilde{v}}$ is the Euler gamma function that comes out of the integral performed in the second equality in \eno{ctildeAgain}.  A simple estimate for modest values of $p$ is
 \eqn{ctEstimate}{
  {c_{\tilde{v}} \over c_0} \sim {\Gamma_{\rm Euler}\left( 3 + {6\pi i \tilde{v} \over \log p} \right)
     \over \Gamma_\infty\left( 3 + {6\pi i \tilde{v} \over \log p} \right)} \sim
    \sec\left( {3\pi^2 i \tilde{v} \over \log p} \right) \sim e^{-{3\pi^2 \over \log p} |\tilde{v}|} \,.
 }
We recognize in the last expression the same factor that we found in \eno{StirlingApprox} using the Stirling approximation.

\section{Conclusions}
\label{CONCLUSIONS}

In some ways, our results on $\phi^4$ theory defined over $\mathbb{R} \times \mathbb{Q}_p$ are simply what one would expect for any continuum field theory in which the dynamics is anisotropic between time and space.  Familiar features include: Power-counting as in \eno{EngineeringDimensions}, the scaling form \eno{Gscaling} for the Green's function in terms of a dimensionless variable $\xi \sim |\tau|/|x|_p^{z}$, and the use of the familiar one- and two-loop diagrams shown in figure~\ref{ThreeDiagrams} to argue perturbatively for the existence of a Wilson-Fisher fixed point when the interaction $\phi^4$ is slightly relevant.

An important difference between $\mathbb{R}$ and $\mathbb{Q}_p$ is that there is no local notion of a derivative on $\mathbb{Q}_p$; more precisely, one cannot sensibly differentiate a function mapping $\mathbb{Q}_p$ to the reals.  The closest one can come is the Vladimirov derivative, which in Fourier space amounts to multiplying by $|k|_p^s$ for some continuously adjustable real (or even complex) parameter $s$.  In the context of mixed field theory, the kernel $\omega^2 + |k|_p^{2z}$ of the kinetic term in the classical action gives immediate meaning to $z$ as the dynamical critical exponent of the microscopic theory and shows that it is an adjustable parameter.  The adjustable exponent is similar to what one sees in Archimedean theories with long-range interactions, as in \cite{Fisher:1972zz,Sak:1973zz,Sak:1977zz}; however, as far as we know, there is no analog in mixed field theory of the long-range to short-range crossover observed in theories over $\mathbb{R}^d$, as discussed recently in \cite{Behan:2017dwr,Behan:2017emf}.  Rather than changing the dimension $d$ of space, in our analysis it is natural to slide the parameter $z$ between $0$ and $1$, with $z=1/3$ playing the role of an upper critical dimension where a branch of Wilson-Fisher fixed points joins onto the line of Gaussian theories parametrized by $z$.  As we head toward $z=1$ (the analog of a lower critical dimension), more and more interaction terms $\phi^{2n}$ become relevant as deformations of the massless Gaussian theory.  (Just as in ordinary $\phi^4$ theory we can exclude odd powers of $\phi$ by imposing a $\phi \to -\phi$ symmetry.)  The overall pattern of critical points is as indicated in figure~\ref{CriticalPoints}.  Because of ultrametric non-renormalization, loop corrections to $z$ at Wilson-Fisher fixed points manifest as shifts in the power of $\omega$ entering into the 1PI two-point function: $\Gamma^{(2)}(\omega,k) \sim \left(\omega^2\right)^{z/z_\text{IR}} + |k|^{2z}_p$, where $z_\text{IR}$ is the infrared dynamical scaling exponent.

The surprising features of field theory over $\mathbb{R} \times \mathbb{Q}_p$ are connected to the fact that Archimedean norms like $|\tau|$ and $|\omega|$ are continuously variable, whereas the $p$-adic norms $|x|_p$ and $|k|_p$ take discrete values: integer powers of the chosen prime $p$.  The most scale invariance one can hope for is invariance under discrete transformations $x \to x/p$ and $\tau \to p^z \tau$.\footnote{If the infrared theory has a renormalized dynamical critical exponent $z_\text{IR} \neq z$, then the appropriate scaling in the infrared is $x \to x/p$ while $\tau \to p^{z_\text{IR}} \tau$.}  A result of this weakened scale invariance is that we can have oscillatory dependence of Green's functions on Archimedean quantities.  For example, when $z=1/3$, we find by dimensional analysis that the two-point Green's function at purely Archimedean separation is $G(\tau,0) \sim 1/\tau^2$.  But all that our discrete scale invariance really implies is that $\tau^2 G(\tau,0)$ is periodic in $\log |\tau|$, with period ${1 \over 3} \log p$.  In other words, $\tau^2 G(\tau,0)$ is a sum of integer powers of $|\tau|^{6\pi i \over \log p}$.  Non-constant log-periodic behavior in $\tau^2 G(\tau,0)$ appears already in the massless Gaussian theory, as explained in detail in section~\ref{PROPAGATOR}.  Two-loop corrections to the 1PI two-point function $\Gamma^{(2)}(\omega,k)$ of the critical theory at $z=1/3$ involve similar log-periodic behavior in $\omega$ when $k=0$, modified by the $\omega^2 \log \omega^2$ that loop-corrects the dynamical scaling exponent.  Specifically, as explained in section~\ref{POSITION}, $\omega^{-2} \Gamma^{(2)}(\omega,0)$ includes all integer powers of $|\omega|^{6\pi i \over \log p}$: precisely the powers which are invariant under $\omega \to p^{-1/3} \omega$.

More surprising than the presence of oscillatory terms permitted by the discrete scale invariance that characterizes critical mixed field theories is the smallness of their coefficients.  In both the classical and two-loop examples discussed in the previous paragraph, we found by explicit calculation that oscillatory terms are suppressed by a factor $e^{-{3\pi^2 \over \log p}}$.  This factor is extremely small for modest values of $p$.  For general values of $z$, the analogous factor is $e^{-{\pi^2 \over z \log p}}$.  We would like to reach an understanding of how such a factor could emerge from an instanton construction, hints of which might already be found in our use of Poisson summation in \eno{PoissonFormula}.

Many generalizations of our constructions could be considered, as enumerated already in section~\ref{INTRODUCTION}.  Passing to the $O(N)$ model is straightforward, and at least at the level of fixed orders in perturbation theory, all our arguments go through unaltered, with appropriate functions of $N$ multiplying the contributions of the various loop diagrams.  A large $N$ Hubbard-Stratonovich treatment of the critical $O(N)$ model appears much more challenging, because the requisite Fourier transforms, which lead to simple powers in the purely Archimedean and purely ultrametric cases \cite{Gubser:2017vgc}, instead lead in mixed field theory to complicated scaling forms similar to the ones considered in sections~\ref{PROPAGATOR} and~\ref{POSITION}.

Passing to a finite extension of $\mathbb{Q}_p$ should be straightforward.  For unramified extensions, the main changes would be the replacement $p \to p^n$ and some related alterations in power counting.  For ramified extensions, the norm of the uniformizer would affect how coarse the scaling symmetry is in the Archimedean direction: For example, if $\sqrt{p}$ is adjoined to $\mathbb{Q}_p$ and the action still contains $\omega^2 + |k|_p^{2z}$, then $\omega \to p^{-z/2} \omega$ as $k \to \sqrt{p} k$.

Our original phenomenological motivation suggests generalizing Archimedean Euclidean time $\tau \in \mathbb{R}$ to a four-vector in $\mathbb{R}^{3,1}$.  A new feature would then be divergences already in the Archimedean parts of loop integrals.  A more careful treatment of how momentum cutoffs can be applied in Archimedean and/or ultrametric directions would be called for, hopefully with the result that there is not much sensitivity to details of the cutoff in the scaling properties of the infrared effective theories.  Compact ultrametric dimensions are obviously another interesting avenue to explore further, with many mysteries at the outset.  Massless fields with spin might be treated in terms of appropriate multiplicative characters, as in \cite{Gubser:2018cha}, but gauge fields seem particularly difficult given the totally disconnected topology of $\mathbb{Q}_p$.

\section*{Acknowledgments}

This work was supported in part by the Department of Energy under Grant No.~DE-FG02-91ER40671, and by the Simons Foundation, Grant 511167 (SSG).

\clearpage

\bibliographystyle{utphys}

redpajama_set_name: RedPajamaArXiv


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5-6.5 Required treatment. All bottled water facilities packaging water for distribution in New York State must provide satisfactory treatment of each water supply source used.
(a) Minimum treatment of each water supply source used shall be disinfection by chlorination, ozonation, ultraviolet radiation or other disinfection methods as protective of the public health as the above.
(1) The record of the bacteriological characteristics for each groundwater source demonstrates conformance to the maximum contaminant levels for microbiological contaminants in section 5-6.11 of this Subpart, for the 12 months immediately preceding the date of application for waiver; such record shall be established under procedures provided by the commissioner.
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(c) Where treatment is provided, all equipment must meet, as a minimum, the standards outlined in Recommended Standards for Water Works.


text:
Longwood Central School District » Community » Longwood Journey » Hamlets » Yaphank » The Gus Neuss Archives of articles related to the Bund at Yaphank » New York Spies - Simon Koedel.
New York Spies - Simon Koedel.
Brooklyn Eagle
Article #4: The Koedels Finally Catch Up with Fate By John Jenkisson
Tight-lipped and button eyes, the tall man with bushy gray hair and tattoos on both arms roamed the waterfront by night. By day he ran a moving picture projector in Manhattan's Lyric Theater.
Kicked out of England during the World War, Simon Koedel brought a monumental grudge to New York. But he was quiet and polite to everyone and the only thing that his hours of work appeared irregular.
Sometimes he loafed around the house all day, reading technical magazines; sometime she took long walks or rode the subways. BY the early part of 1939 he was spending less and less time in the projection booth of the Lyric and more time in subways and one the ferries.
A naturalized American citizen, he spoke with a slight accent.
Article #5: Scanned Harbor Traffic.
In the fall of 1939 he began riding the ferries in earnest. He took six and seven trips a week. On the Staten Island boats he scanned busy harbor traffic through field glasses and sneaked into the cabin to make notes on what he saw.
Once he struck up a conversation with a ferry bootblack and tried to persuade him to take notes also. The bootblack refused and there was a heated argument.
Often he rode the Weehawken ferry from 42nd St. and closely watched British freighters lying in the Hudson.
He slipped into subways and rode to the bustling Bush Terminal in Brooklyn. There he watched the loading of lend-lease shipments and tired to figure out what was in crates lying o the docks by names of the manufacturers stenciled on the outside.
More and more frequently he began to leave his apartment at 660 Riverside Dr. at dusk, pick up his foster-daughter in her flat at 542 W. 112th St. and take her to the waterfront. Together they haunted bars and restaurants near the piers.
Article #6: Picked up Sailors.
Brooklyn-born and adopted by Koedel when she was 3. Marie had grown up to be a pretty, dark-haired girl with a sullen mouth.
After Koedel had struck up conversations with British, Swedish and Norwegian sailors Marie would join the group, choose her man and finally wander out with him.
Some nights her father had to work. Then Marie, who gave scalp treatments by day, would haunt the streets and pick up lonely sailors.
Other nights she spent typing reports for her father. She telephoned government agencies and industrial plants, saying her father was a retired chemist and wanted information on their activities. Once she described him as a Boy Scout Teacher.
In the meantime Koedel was beginning to get nervous. His trips to Brooklyn, Staten Island and the docks of lower Manhattan became more frequent but he began to think he was being followed. He rode office building elevators, got off with clerks and stenographers, and rode back down again.
Article #7: Rented Fishing Boat.
He rented a small fishing boat to avoid being seen on ferries, rowed out into the harbor to get names of ships, cargoes, sailing times and probable destinations.
He began to read his Bible.
And now whenever he left for the docks or the harbor he told his friends and a few acquaintances of the Motion Picture Operators Union that he was going fishing.
But in October, 1939, Marie fell in love with a young man named Walters. Within a few months she told him about the work she and her father were doing. But she refused to marry him, saying her father needed her in his work.
Agents of the FBI already had a line on the Koedels and they put the pair under constant surveillance. They did not yet have enough information for an arrest.
Koedel became more furtive than ever. Almost certain he was being trailed, he dodged and twisted, slipped through alleys and tried to lose himself in crowds. But he kept on riding the ferries and continued to meet intermediaries who secretly pain him off in amounts ranging from $100 $300. He got $600 that way during the winter of 1940.
Then Marie started to crack under the strain. Nervous and jittery, she went to see an Army colonel, formerly a New York attorney and a veteran of the World War and the Mexican Expedition of 1916.
She told the colonel she wanted him to help get her brother, then serving a sentence in Auburn Penitentiary, out. Then she broke down and wept. She said her father had beaten her for failing to get information on lend-lease shipments and showed the colonel some bruises. The colonel reported to the FBI.
Koedel was getting the jumps too. In an apparent attempt to prove his patriotism he voted in the national elections of 1940. But he was still scribbling addresses of persons in Turin, Italy, and Amsterdam, Holland, on the backs of matchboxes.
And early in 1941 eh clipped a section of newsreel from a sequence showing tests of the giant flying boat Mars, Pasted the severed ends of the film together again and took the clipping home.
But the theater manager noticed the missing piece of film and raised a fuss. Koedel quickly returned the clipped film, saying he found it in the projection booth.
Article # 8: Writes a Bitter Letter
Two weeks before Pearl Harbor the commander of a Connecticut National Guard division warned in a speech that war was imminent. The shaggy-browed Koedel, although increasingly harassed, found time to write the commander a bitter letter assailing the average American as a double-crosser, a chronic dodger, a hypocrite and crooked at heart.
Unknown to Koedel, FBI agents had trailed him on visits by foot by subway to the German Consulate. When the attaches he knew were out of the office he always left a white envelope with receptionist.
When the German consulate was shut down, Koedel's activities tapered off and he began a period of watchful waiting.
On June 7, 1943, he was excluded from the Easter Defense Zone on the ground that he had been a German Army officer prior to the war. Thoroughly frightened, Koedel scuttled aboard a train for Harper's Ferry, W.Va. Leaving his daughter behind in New York.
By September, 1944 the flied of the RBI were nearing completion. Then three houses painters, redecorating the Lyric Theater, unearthed a wad of correspondence and other material tightly wrapped in a cloth and cleverly hidden I the iron railing of an unused staircase.
Article # 9: Notes from Nazi Agents.
In the bundle were typewritten instructions to Koedel from German agents asking for information on the anti-mine devices carried by the Queen Mary, convoy routes and rendezvous pints.
Handwritten material indicated that partial answers had already been received.
At 7:50 am on Oct. 23, 1944, the trap snapped shut on Koedel. Three agents of the RBI picked him up without a struggle in a tiny room on the first floor of a private home in Harper's Ferry. Meekly, he accompanied the agents back to New York.
And within 35 minutes of her father's arrest Marie was picked up in her apartment here.
Both were indicted for conspiracy to violate the peacetime espionage statue.
Eight days after the trial began on Feb 7 of this year Koedel pleaded guilty. He had heard most of him. But his daughter refused to make a similar plea and was found guilty by a jury Feb 19.
Ten days later sentence was pronounced. Pale and speechless Marie Koedel received seven and a half years in a federal penitentiary.
Here father, standing stiffly silent, got 15 years.
Tomorrow: The doll woman of Madison Ave.

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text:
Peru- Radio presenter shot dead in mid-broadcast
Crédits: El Popular
Reporters Without Borders (RSF) calls on the Peruvian judicial authorities to quickly identify those responsible for the shocking murder of a radio presenter who was shot dead in mid-broadcast in Camaná, in the southern region of Arequipa, on 20 November.
Hernán Choquepata Ordóñez, also known as "Randy" Ordóñez, was shot in the head by at least one gunman who stormed into Radio La Ribereña as Ordóñez was presenting a music programme. The presenter, who was alone in the studio at the time, died while being rushed to hospital.
In a programme called "Habla el Pueblo," Ordóñez interviewed local residents who often complained about the region's elected authorities. Ordóñez himself was also often openly critical of local leaders during his broadcasts, lambasting the mayors of Camaná and the nearby town of Mariscal Cáceres the week before his murder.
"The leading hypothesis for those investigating this crime must be that the motive was linked to the victim's work as a journalist," said Emmanuel Colombié, the head of RSF's Latin America bureau. "The authorities cannot allow this murder to go unpunished and fade into oblivion. It was an attack on the entire Peruvian media."
According to colleagues at Radio La Ribereña, Ordóñez recently received death threats and had reported the threats to the police.
Peru is ranked 84th out of 180 countries in RSF's 2016 World Press Freedom Index.
Follow the news on Peru
China's diplomats must stop attacking media over coronavirus reporting
April 1, 2020 Find out more
Latin American media: under control of families, economic and political elites
December 4, 2019 Find out more
Latin America's community radio – a key service but vulnerable
February 10, 2017 Find out more
Subscribe to the newsletter: Peru


Combination flashings let you fit roof windows side by side or one on top of the other for an impressive finish whatever your budget.
Unsure of the flashing you need? Send us a photo of your roof for advice 07717 175 001.


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text:
8 Point Plan
Surveillance laws are supposed to keep us safe and respect our privacy. The Investigatory Powers Bill does neither.
Here are our eight recommendations for surveillance laws that protect our security AND our freedom.
1. Judicial warrants
All surveillance requests must be authorised by a judge.
All surveillance requests must be granted a warrant by a judge before that surveillance takes place – including requests to intercept communications, send undercover operatives to spy on people, or hack and access our data.
And what the Home Secretary has touted as "judicial authorisation" in the IP Bill isn't actually judicial authorisation at all. The Bill creates a new approach involving a "Judicial Commissioner".
Ministers will continue to authorise warrants before passing them to a Commissioner – and all they'll have the power to do is "review" the decision. Relegated to little more than glorified rubber-stampers, they'll only be able to disagree with outrageously unreasonable requests. A huge number of democratic countries require judicial warrants for surveillance – including all the other countries involved in the Five Eyes Alliance; USA, Australia, Canada and New Zealand.
Blog: Why we need real judicial sign-off in the Investigatory Powers Bill
2. Respect our data
No new blanket powers forcing communications companies to store more – and more revealing types – of our data.
The Independent Reviewer of Terrorism, David Anderson, has warned that the case for a host of bulk surveillance powers has not been made. The USA and a number of other European and Commonwealth countries do not compel service providers to retain their customers' weblogs for inspection by law enforcement, and continuing down this road will place Britain in the company of authoritarian regimes across the world.
Additionally, the track record of internet companies to keep that data safe doesn't fill us with confidence (for example see the recent hacking of TalkTalk - four people so far have been arrested including some under the age of 16).
Blog: Respect our data
3. Targeted surveillance - for a reason
Surveillance should only be conducted for a number of tightly defined reasons i.e. investigating serious crime and preventing loss of life. Strong legal protection should be provided for privileged and confidential material.
There is no convincing evidence that mass surveillance is vital to save lives. Rather, the amount of data currently being collected overwhelms the security services, making us less safe. Surveillance should be conducted for a narrow range of tightly defined purposes i.e. in the targeted investigation of serious crime and other legitimate objectives such as preventing risk to life. However it is currently being used for vague and non-crime related purposes (as recognised in the High Court judgment in Liberty's challenge to DRIPA with MPs David Davis and Tom Watson).
A targeted approach to surveillance - requests and warrants must target individuals on the basis of suspicion in criminal activity.
Liberty is calling for a targeted, as opposed to mass or 'thematic', approach to surveillance. As revealed by Edward Snowden, the Agencies have claimed powers to intercept and hack entire telecommunications systems meaning billions of innocent communications are routinely intercepted and processed.
The Bill seeks to put these powers on the statute book and expand them - new provisions allow the Agencies to acquire 'bulk personal datasets' and communications on millions of innocent people.
Retention of communications data and surveillance warrants should instead target individuals on the basis of suspicion of involvement in serious crime.
Liberty is currently challenging the present approach of mass, speculative, suspicion-less interception of "external warrants" at the Court of Human Rights.
Strong legal protection for privileged and confidential material
Legislative safeguards providing additional protection for legally privileged communications, journalistic sources and parliamentarians' correspondence should be enshrined in primary legislation.
Blog: The case for targeted surveillance
4. Transparency and redress
All surveillance powers should be publicly disclosed and explicit safeguards set out in the IP Bill. There must be improved redress and increased transparency for people who have been spied on.
The IP Bill has introduced an element of transparency by identifying which surveillance powers the Government is using. However, the Bill is light on detail, and fails to provide necessary safeguards to protect the powers from abuse. In fact, the Bill would introduce new imprisonable offences for whistleblowers, preventing injustices becoming public.
Redress mechanisms must also be improved
A person who was the subject of intrusive and targeted surveillance methods must be told when they are no longer under suspicion (unless there is a good reason for maintaining secrecy).
Further, the Investigatory Powers Tribunal (IPT) – which hears legal challenges against the use of surveillance – must be reformed and made more transparent. Currently it isn't required to hold oral hearings and if it finds against a complainant it can't give its reasons for doing so. It cannot make a formal declaration, a "declaration of incompatibility", that legislation violates the Human Rights Act.
The IPT must become more transparent and open to the public. Whilst the IPT remains so shrouded in secrecy, it is impossible to hold the security services to account for human rights abuses.
Blog: Transparency and redress
5. Use of intercept evidence in court
Intercepted communications should be admissible in criminal trials.
The bar on the admissibility of intercept evidence, if properly obtained via a judicial warrant, in criminal proceedings should be lifted. Why is this vital evidence not used to bring perpetrators to justice?
Blog: 5 Reasons why we need intercept evidence in court
6. Fair and open international data sharing laws
The arrangements for intelligence sharing of surveillance data between the UK and other countries must be set out in law and available to the public.
Transparent and proportionate arrangements for the sharing of surveillance data between intelligence agencies should be agreed between the UK and foreign States, made publicly available and incorporated into law.
Mutual Legal Assistance Treaties (MLAT) – law enforcement tools governing the exchange of information between countries – should be improved and replace attempts to place extraterritorial obligations on foreign tech firms (as proposed in the Investigatory Powers Bill).
7. Protect our encryption standards
Safeguard our country's security by protecting - not undermining - encryption standards.
Encryption is vital to the security of our online communications. If we weaken standards to allow Government to get its hands on more of our personal communications, that information is vulnerable to use and abuse by the bad guys too. Do we really want our iMessage communications to be less secure than those of people using the service in other countries?
8. Recognition of the unique threat mass hacking poses to our security
Hacking is a grave privacy intrusion - much more intrusive than "traditional" forms of state surveillance, including interception, and its capacity to undermine device, network and internet security can't be overstated.
It carries unlimited and untested potential for Government to act against the security and economic interests of its own citizens, whether consciously or otherwise.
The Government's legislation allows for the mass hacking of devices, affecting potentially millions of innocent users' devices, undermining the safety of us all. Hacking should only be used in extremis as a last resort, and warrants should always specify named suspects or premises.
End Indefinite Detention
Immigration Bill
NO #SNOOPERSCHARTER
Protect Refugee Children
Public Space Protection Orders
REJECT MASS SURVEILLANCE
Digital Security Training
Liberty's Guide to Protecting Your Rights Online
No Snoopers' Charter - Donate
STAND UP TO POLICE SPYING
Save Our Human Rights Act
Stand up to terror – stand up for our rights
Trade Union Bill


Financial leverage attracts a lot of traders to the Forex market. You might see many results on Google such as 'Best leverage to use in Forex'' which make it seem like this is the only option when trading with Forex. However, it is not the foolproof tool that some people make it out to be, nor is it the only option in terms of professional Forex trading. While leverage can be beneficial, it can also lead to some disastrous outcomes.
This is especially likely in the case of traders with no experience. It's also worth noting that many large financial companies are actually practising currency trading without leverage. So what are the advantages of trading with and without leverage? And what are the pros and cons of Forex trading? You can find out the answers to these questions yourself with a free Demo account, if you want to jump ahead and start practising now.
However, if you would like to possess a little more knowledge beforehand, we encourage you to read on.
Financial leverage is a credit provided by a broker. Leverage allows traders to place orders that are significantly higher than their actual deposit. It is possible to use leverage to trade stocks and other financial instruments, but it is far more accessible when trading currencies. Leverage potentially helps traders to achieve higher profits in the market. Of course, the same also applies to losses. Traders risk losing their deposit faster when using leverage – so use it cautiously!
Image that a trader has a deposit of 10,000 EUR. The trader's broker offers a leverage of 1:100 for a deposit of this size. Knowing these two values, we can easily calculate the largest position available to this trader. We need to multiply the trader's balance by the first value in the leverage ratio (i.e. 10,000 multiplied by 100). Therefore, this trader can open a deal up to 1,000,000 EUR in volume.
Of course, this may sound too optimistic. The position size doesn't mean much if you are not aware of how you're trading. Before you begin trading, you should learn about the positives and negatives of trading, and then try it without leverage.
The biggest advantage of leverage is that it allows traders to boost their trade sizes, even when they don't have substantial capital. Traders usually consider 1,000 USD to be a decent starting sum. However, not all traders can afford this – especially when starting out. And this is where financial leverage comes into play. Even leverage as low as 1:10 allows traders with a 100 USD deposit to open a 0.01 lot position. But this is not a 100% beneficial condition, as you also expose yourself to risk.
Let's say you buy 1000 USD for 800 EUR, and then the price of USD drops by 50%. You would only lose half of your funds (in this particular example, you would lose 400 EUR). However, if you were using 100:1 leverage, and the price changed by less than 1%, you would then lose all of your funds. Always be aware of the risks leverages pose, and try to prepare yourself for them. Preparation can be as simple as practising leveraged and unleveraged trades on a demo account.
If you can't create good returns with low leverage, expect potentially significant loses with over-leverage.
The main downside of trading Forex without leverage is that it is simply not accessible for most traders. Forex trading without leverage means that changes in the price of an asset directly influence the trader's bottom line. The average monthly return a trader can generate is 10%. But in reality, the return is around 3 to 5% a month.
However, this figure already includes marginal trading. With no leverage Forex trading you would probably only make between 0.3 to 0.5% a month. It may be enough for some Forex traders – but perhaps not for the majority. The need for substantial trading capital is the biggest drawback of trading without leverage. On the other hand, currency trading without leverage gives you less risk exposure.
However, this doesn't mean that there are no risks involved in trading without leverage. Let's proceed with an example of 'No-leverage trading'. Let's say you deposit 10,000 USD and make a monthly return of 5%. You would only get 500 USD each month, and that's before any taxation. You could probably make the same money with a 9-to-5 job, without risking your own capital in the process.
Source: EUR/USD Wave Analysis with a MACD Indicator Applied - Data Range: 13 April, 2016 - 22 April, 2016 - Please Note: Past performance does not indicate future results, nor is it a reliable indicator of future performance.
What is institutional trading? As we've already mentioned, a lot of institutions choose Forex trading without leverage. Yet these organisations are still able to achieve large profits. How is this possible? Large banks have access to billions in capital. They can afford to trade large amounts on attractive entry signals. Institutions also often trade long term, so unlike the average trader, institutions can have their position open for months or even years.
Since they don't use leverage, the swap expense tends to be quite low too. In fact, in many cases there is no swap at all. Institutions directly benefit, or suffer from the differences in interest rates. Many of the largest Forex market trades have been made by institutions without leverage. These deals have a speculative motivation, and typically use extensive capital in the billions.
But unfortunately, there's no definitive answer to it – it depends on the situation. You have to consider your trading strategy, your financial targets, the capital at your disposal, and how much you are willing to lose. Like any financial market, the Forex market is generally risky. The higher your leverage is, the riskier your trading gets.
So consider trading with as little leverage as possible, to ultimately get the profit you want. And conversely, keep in mind that the more leverage you use in Forex trading, the more profit you can potentially make. In most cases, a beginner trader should consider using leverage between 1:5 to 1:100. The table below illustrates the importance of trading with the right leverage. It displays 10 consecutive losing trades in a row when using high vs low leverage.
Hopefully, we've answered some of your questions about Forex trading without leverage.
By now, you should understand why leverage is risky, and that high leverage means a higher risk, with the possibility of a higher return and vice versa. So again, practising with leverage on a Demo account is a smart initial move. It is important to ensure your trading strategy considers your deposit amount, how much you are willing to lose, and the minimum you are willing to make - before you start leveraged trading. Keep learning, keep educating yourself, and most importantly, keep trying out new things.


Since the one part of the cage that really does all the work and interfaces with the user is the hook, it only makes sense to make sure that is the one part of the cage that works well – because if it doesn't, somebody is getting frustrated or, worse yet, injured.
The Pro Style Monster Hook does its job very, very well. It is easy to use, it has a long and deep profile and it is fully welded with the exception of the replaceable space-age polymer contact surface. This is a contact surface that is tough enough to suffer through lots of high and heavy drops, yet never chip or flake like a hook with steel contact surfaces. Furthermore, the energy-absorbing properties of the polymer suppress the tendency of a dropped bar to bounce out, and it protects the knurling on the bar.
All in all, this is the best hook you will find on a "standard series" cage of any manufacture. Sold as a pair.
NOTE: This item is a substitution for the Beefy Hooks that are standard issue with all Performance Series cages. If the standard Beefy Hooks are wanted in addition to the 3207 upgrade, please indicate this to your Legend representative as it must be entered as a separate order.