text:
\section{Introduction}

In recent years, an increasing amount of evidence suggests that not every low-energy EFT can be consistently coupled to quantum gravity.
The quantum gravity constraints that have been unveiled so far have many ramifications.
For instance, studies of the generic properties of
quantum gravity point to a restriction on the allowed field range in ultraviolet (UV) complete theories.
This restriction takes various
forms. The weak gravity conjecture \cite{ArkaniHamed:2006dz} asserts that for every long range force, there exists a state whose charge-to-mass ratio is bigger than that of an extremal black hole. This implies upon dualizing an axion-like field to a $U(1)$ gauge field
that the axionic decay constant, $f$, is limited by $f \cdot S_{\rm inst} < {\cal O} (1) M_P$  \cite{Brown:2015iha, Brown:2015lia} where $S_{\rm inst}$ is the instanton action that controls the non-perturbative breaking of the shift symmetry. A more general statement about field ranges, known as the Swampland Distance Conjecture (SDC), was put forth in \cite{Ooguri:2006in}.
This conjecture is based on the observation that in known string constructions, a tower of states become exponentially light as we traverse a large distance $d(p_0, p)$ in field space:
\begin{equation}
M \sim M_0 e^{-\lambda d(p,p_o)}~,
 \label{swampconj}
\end{equation}
where $\lambda$ is some unspecified positive constant, and thus the EFT breaks down beyond a critical distance $d (p_0,p) > \lambda^{-1}$. For recent considerations about this conjecture we refer the reader to \cite{Grimm:2018ohb,Heidenreich:2018kpg}.

The conjectured restrictions on the allowed field range, if proven, have wide-ranging implications, one of which concerns the amplitude of
gravitational waves generated by inflation. A kinematic bound due to Lyth \cite{Lyth:1996im} relates observable tensor modes to super-Planckian
field displacements:
\begin{equation}
\frac{\Delta \phi}{M_{Pl}}  \gtrsim \mathcal{O}\left(1\right) \times \sqrt{\frac{r}{0.01}}~,
\end{equation}
where $r$ is the tensor-to-scalar ratio. Thus, if quantum gravity can impose a strict upper bound on the inflaton field range, one may gain a better certainty on our target for $r$. Besides the challenge in maintaining control of the EFT over a super-Planckian field displacement, a detectable level of tensor modes also poses an additional challenge on the UV completion of inflation. This is because for large-field inflation (say $r \gtrsim 10^{-2}$), the Hubble scale during inflation is $H \sim 10^{14}$ GeV\cite{Ade:2015lrj}. As UV complete theories of gravity typically involve new degrees of freedom below the scale of quantum gravity,
it is a formidable task to pack the associated energy scales in between $H$ and $M_{\text{Pl}}$. To be concrete, constructing single-field inflation models from string theory requires a hierarchy of scales:
\begin{equation}
M_{\varphi} < H < M_{\text{moduli}} < M_{\text{KK}} < M_{s} < M_{\text{Pl}}\, ,
\label{hierarchies}
\end{equation}
where $M_{\varphi}$ is the inflaton mass, $M_{\text{moduli}}$ denotes generically the mass scale of the stabilized moduli \footnote{There can be further hierarchies within the moduli sector, e.g., in Type IIB flux compacitications, the Kahler moduli can be lighter than the complex structure moduli. Here, we assume that the moduli are stabilized within the 4D EFT, and thus $M_{\text{moduli}} < M_{KK}$.}, $M_{KK}$ refers to the mass of the lightest Kaluza-Klein replica, and $M_s$ is the string scale. The challenge in maintaining this hierarchy is thus two-fold: 1) the moduli stabilization mechanism should generate this separation of scales (see, e.g., \cite{Blumenhagen:2014nba} for a discussion of this issue in flux compactifications),  and 2) backreaction of the heavy fields (which in general has the effect of flattening the inflaton potential \cite{Dong:2010in,Buchmuller:2015oma})
throughout the inflationary trajectory  should be consistently taken into account.

Given the
above considerations,
it is important to define a proper measure of the maximum field range. The main goal of this letter is to make clear the distinction between
kinematic and dynamical field ranges.
While the SDC is a kinematic statement about the moduli space, i.e., it tracks the change in the low-energy EFT from $p_0$ to $p$, it makes no reference to the path connecting them being the true trajectory dictated by the underlying dynamics.
In applying the swampland conjectures (e.g., to inflation), however, we are interested in the restriction on the {\it dynamical} field range traversed by the true trajectory that solves the equations of motion (eom).
As we will see, the dynamical field range and a geodesic distance traversed in field space by the light field only coincide if the light and heavy sectors evolve separately, or when the heavy fields are infinitely heavy. The two field distances deviate from each other precisely because of the hierarchy in Eq.~(\ref{hierarchies}), which mandates the heavy fields to be not arbitrarily heavier than the light field of interest. The trajectory can be approximated by a geodesic if there is a sufficiently large mass hierarchy. Our findings do not contradict the SDC \cite{Ooguri:2006in}, which concerns with the asymptotical behavior as $d( p_0,p) \rightarrow \infty$.
However, what we found has bearings on the Refined Swampland Distance Conjecture  (RSDC) \cite{Klaewer:2016kiy} which argued that $\lambda \sim {\cal O} (1)$.  We will show explicitly that the constraints on field ranges imposed when $\lambda \sim {\cal O} (1)$  for massive axions are obtained from unphysical trajectories. This prescription will agree with the dynamical field range in the presence of a sufficiently large mass hierarchy between the light and heavy sectors, but then the critical field range is also extended. We illustrate these subtleties of multi-field dynamics with axions in a 4d $\mathcal{N}=1$ supergravity description of type II compactifications.
Our discussions apply to both Type IIA and Type IIB settings, irrespective of whether the light axion comes from the open-string or the closed-string  sector.

 \section{The dynamical field range}

We now analyze carefully the impact of the kinetic terms of the stabilized fields and discuss under what conditions the trajectory is a geodesic in field space.

\subsection{The Refined Swampland Conjecture for axions}

Before we present a more critical assessment of the dynamical field range, let us revisit the RSDC constraints for axions \cite{Baume:2016psm} which claims that strong backreaction of the closed string sector on the kinetic term of an axion constrains its available field range to $\Delta \phi < {\cal O} (1)  M_{\text{Pl}}$. This claim, if true, excludes the possibility of large-field inflation based on axions, including axion monodromy models \cite{Silverstein:2008sg,McAllister:2008hb} (and its realizations via F-term potential in supergravity \cite{Marchesano:2014mla,Blumenhagen:2014gta,Hebecker:2014eua}).

The approach used in \cite{Baume:2016psm}
to impose constraints on the available field range (in particular, in type IIA compactifications) consists of analyzing the trajectory traced by the minima
of the heavy fields as we vary an axion, $\varphi$.
Then, one should keep all the fields in the action and adjust their values appropriately. This approach is claimed to be a good approximation irrespective of the existence of a mass hierarchy between the sectors. More concretely, for a massive axionic direction defined as a linear combination of certain fields, $\varphi = h^{i}\nu^{i}$, where $h^{i}$ are constants, it is argued that the field range
of interest is
\begin{equation}
\Delta \phi = \int_{\gamma} \left(h_{i}G^{ij}\left(\varphi\right)h_{j}\right)^{-1/2} d\varphi := \int_{\gamma}\sqrt{G_{\varphi \varphi} \left(\varphi\right)}d\varphi\, ,
\label{defrange2}
\end{equation}
where $G^{ij}\left(\varphi\right)$ is the inverse of the field space metric and $G_{\varphi \varphi}\left(\varphi\right)$ is defined as the kinetic term of the axionic field evaluated along the aforementioned trajectory. One immediately observes that this definition, implicitly, neglects
the displacements of the fields not included in the definition of $\varphi$.
This assumption implies, by default, that the trajectory considered is a geodesic in field space. Then, strong backreaction of the closed-string sector on the kinetic term of $\varphi$ implies that the field range \eqref{defrange2} grows at best logarithmically, i.e. $\Delta \phi \approx \alpha^{-1}\log\left(\varphi\right)$. This is, precisely, the scaling suggested by the SDC. The critical value for the canonically normalized field before the onset of the logarithmic behavior is $\hat{\varphi}_{\text{back}}\sim \alpha^{-1}$. Along the text, hatted variables denote canonically normalized fields. Then, one observes that a tower of states becomes exponentially light as $\varphi$ is displaced, in agreement with the SDC \eqref{swampconj}, if we identify $\alpha \approx \lambda$.
It was argued in \cite{Baume:2016psm} that $\lambda \sim \mathcal{O}\left(1\right)$. Afterwards, in \cite{Valenzuela:2016yny,Blumenhagen:2017cxt} this prescription was generalized to include the open-string sector. There, it was argued that $\lambda$ is proportional to the mass hierarchy between axion, $\varphi$, and the closed-string sector.

\subsection{Proper field range and validity of the approximation}
We can try to improve the naive estimate of the allowed field range in eq.~\eqref{defrange2} by including the kinetic terms of the heavy fields that were previously neglected. We will see under what conditions can the naive estimate be trusted.

The dynamical field range is the one measured along the one-dimensional trajectory defined by the solution of the eoms. In a multi-field scenario, the homogeneous background fields, $\phi_{0}^{a} = \phi_{0}^{a}\left(t\right)$, in a spatially flat FLRW spacetime, satisfy
\begin{equation}
D_{t}\dot{\phi}_{0}^{a} + 3H\dot{\phi}_{0}^{a} + V^{a} = 0 \, ,
\label{eoms}
\end{equation}
where the covariant derivative in field space is defined by $D_{t}X^{a} = \partial_{t}X^{a} + \Gamma^{a}_{bc}\dot{\phi}_{0}^{b}X^{c}$ and $ \Gamma^{a}_{bc}$ are the Christoffel symbols derived from the field space metric $G_{ab}$. Here, latin indices $a, b$ denote real coordinates in field space. One may consider the background fields $\phi^{a}_{0}\left(t\right)$ as coordinates of the moduli space describing a curve parametrized by $t$. It is well-known that the solutions to the eoms are tied to two fundamental properties of the system: 1) The existence of mass hierarchies between different sectors and 2) The geometry of the field space metric (we refer the reader to \cite{Rubin:2001in,Tolley:2009fg,Achucarro:2010jv,Shiu:2011qw, Achucarro:2012yr,Gao:2012uq} for more details). The variation of the scalar fields along the trajectory is defined as $\dot{\phi}_{0}^{2}  =G_{ab}\dot{\phi}_{0}^{a}\dot{\phi}_{0}^{b}$. Then, the proper distance traversed along the path parametrized by $t$ will be given by
\begin{equation}
\Delta \phi = \int \dot{\phi}_{0} dt = \sqrt{G_{ab}\dot{\phi}_{0}^{a}\dot{\phi}_{0}^{b}} dt\, .
\label{realfieldrange2}
\end{equation}

To quantify the error of the naive estimate in eq.~\eqref{defrange2}, we
substitute the solutions which minimize the scalar potential in terms of the displaced field, $\overline{\phi}^{\alpha}\left(\varphi \left(t\right)\right)$, into the action to obtain the proper field range \eqref{realfieldrange2} \footnote{We assume, $\overline{\phi}^{\alpha}\left(0\right) = \overline{\phi}^{\alpha}_{0} > 0$, when $ \overline{\phi}_{0}$ is related with the volumes of cycles of the compactification.}. Note that greek indices run over all the fields in the theory except $\varphi$.
We emphasize here that there is no guarantee
the trajectory given by this procedure is the one
that minimizes the action
 since it is not granted that the eoms are solved.
Nonetheless, this analysis serves as a self-consistent check of whether the kinetic terms for  $\overline{\phi}^{\alpha}$ can be neglected.
We have also checked numerically that evaluating
eq.~(\ref{realfieldrange2}) with the actual solution to the equations of motion does not change qualitatively our conclusions.

In the presence of strong backreaction,
the kinetic terms of the stabilized fields will not be negligible in general. Indeed, we observe that $\dot{\overline{\phi}}^{\alpha}\left(\varphi \left(t\right)\right) \approx \frac{\partial \overline{\phi}^{\alpha}\left(\varphi\right)}{\partial \varphi} \dot{\varphi}$ which implies a proper field range \eqref{realfieldrange2}, assuming no kinetic mixing:
\begin{equation}
\Delta \phi = \int\sqrt{G_{\varphi \varphi}\left(\varphi\right)  + G_{\alpha \beta}\left(\varphi\right)\frac{\partial\overline{\phi}^{\alpha}}{\partial \varphi}\frac{\partial\overline{\phi}^{\beta}}{\partial \varphi}}\dot{\varphi}dt \, ,
\label{rangebetter}
\end{equation}
Note that the term $G_{\alpha \beta}\left(\varphi\right)\frac{\partial\overline{\phi}^{\alpha}}{\partial \varphi}\frac{\partial\overline{\phi}^{\beta}}{\partial \varphi}$ is absent in the expression  \eqref{defrange2} used in \cite{Baume:2016psm,Valenzuela:2016yny,Blumenhagen:2017cxt}.
For eq.~\eqref{defrange2} to be a good approximation to the allowed field range, we need to ensure
\begin{equation}
G_{\alpha \beta}\left(\varphi\right)\frac{\partial\overline{\phi}^{\alpha}}{\partial \varphi}\frac{\partial\overline{\phi}^{\beta}}{\partial \varphi} \leq \varepsilon G_{\varphi \varphi} \left(\varphi\right)\, ,
\label{validity}
\end{equation}
where we can think of $\varepsilon$ as a measure of the accuracy of the approximation or, in other words, a measure of how well do the fields that minimize the potential
solve the eoms \footnote{ Note that $\varepsilon$ will be vanishing only in the case of consistent decoupling between sectors.}. If this condition cannot be satisfied along the whole trajectory, there must exist a critical value, $\varphi_{\text{app}}$, for the displaced field beyond which the approximation fails. We define $\varphi_{\text{app}}$ as the solution to the constraint \eqref{validity} and, as we will illustrate, it will depend on the relative mass hierarchy. For $\hat{\varphi} > \hat{\varphi}_{\text{app}}$
significant corrections to the field range are expected and
a multi-field analysis is necessary to capture the dynamics.
For $\hat{\varphi} < \hat{\varphi}_{\text{app}}$,  the single-field approximation is valid.
The approximation for the field range \eqref{defrange2}
is, then, valid within a ball in the moduli space  centered at the minimum of the potential
with
radius $\hat{\varphi}_{\text{app}}$.
The dynamical field range
is thus
valid  within this approximation up to $\hat{\varphi}_{c} = \text{min}\left\lbrace\hat{\varphi}_{\text{app}},\hat{\varphi}_{\text{back}}\right\rbrace$.
We will check that in the absence of mass hierarchies,
$\hat{\varphi}_{\text{app}} < \hat{\varphi}_{\text{back}}$.  The real cutoff of the effective theory should then be obtained by solving the full multi-field dynamics and, thus, the field range obtained in \cite{Baume:2016psm,Valenzuela:2016yny,Blumenhagen:2017cxt} will differ from the dynamical field range.
On the contrary, the
allowed field range is given by $\hat{\varphi}_{\text{back}}$
only if a sufficiently large mass hierarchy is assured. In this regime, corrections to the action
generically flatten the potential \cite{Dong:2010in}. Then, one may quantify the backtreaction by perturbing the heavy fields around the minimum of the potential. This procedure has been systematically analyzed for $\mathcal{N}=1$ supergravity \cite{Buchmuller:2015oma} (see \cite{Landete:2016cix} for a shortcut).

\subsection{Is the trajectory a geodesic?}

 We are now in a position to address whether the trajectory which defines the dynamical field range is a geodesic. Recent studies of swampland distances, see, e.g., \cite{Grimm:2018ohb,Blumenhagen:2018nts}, relied on geodesic trajectories in field space for massless axions. We will critically assess under what conditions this assumption holds for massive axions.
In multi-field setups, it is convenient to define a unit vector tangent to the trajectory which for the
present case is:
\begin{equation}
T^{a} := \frac{ \dot{\phi}_{0}^{a}}{\dot{\phi}_{0}} \rightarrow \dot{\phi}_{0}T^{\varphi} = \dot{\varphi}\, , \, \dot{\phi}_{0}T^{\alpha} = \frac{\partial \overline{\phi}^{\alpha}}{\partial \varphi} \dot{\varphi}\, .
\label{Ta}
\end{equation}
 As in
 \cite{GrootNibbelink:2001qt}, one may define an orthonormal basis in field space by taking covariant derivatives of the tangent vector, $D_{t}T$. For $\dot{\phi}_{0}\neq 0$, the trajectory obtained by solving the eoms \eqref{eoms} will be a geodesic in field space if $D_{t}T^a= 0$. Then, a sufficient condition for the trajectory considered to be a geodesic is
\begin{equation}
\left(\frac{\partial \overline{\phi}^{\alpha}}{\partial \varphi}\right) \approx 0 \rightarrow \left|D_{t}T\right| \approx 0\, .
\label{geodesic}
\end{equation}

Indeed, one may approximate the trajectory by a geodesic  when eq.~\eqref{validity} is satisfied.
This
is a reasonable approximation if there are large mass hierarchies, and in
the
absence of non-trivialities coming from the geometry of the field space (see, e.g., \cite{Achucarro:2010jv} for the latter point). Note that the trajectories studied in \cite{Baume:2016psm,Valenzuela:2016yny,Blumenhagen:2017cxt} are indeed geodesics since \eqref{geodesic} is assumed.

\section{ An Illustrative example}
\label{sec:toymodel}

We now try to quantify the statements made in the previous section by analyzing a single-field inflationary model based on D7-branes in Type IIB flux compactification of a toroidal orientifold. We refer the reader to the appendices for a similar discussion for closed-string models. For ease of comparison, we will focus on an example used in \cite{Blumenhagen:2017cxt} which involves non-geometric fluxes, we refer the reader to that reference for more details. The inflaton candidate, $\varphi$, is the axionic component of the position modulus of a D7-brane. The $\mathcal{N}=1$ supergravity Lagrangian describing this model is given by the F-term potential where:
\begin{eqnarray}
K & = & -\log\left[\left(S+\bar{S}\right)\left(U+\bar{U}\right) - \frac{1}{2}\left(\Phi + \bar{\Phi}\right)^2\right] \nonumber\\
& - & 2\log\left(U+\bar{U}\right) - 3\log\left(T+\bar{T}\right) \, , \\
\label{kop}
W & = & \mathfrak{f}_{0} + 3 \mathfrak{f}_{2}U^{2} - hSU -qTU - \mu \Phi^2 \, .
\label{supo}
\end{eqnarray}
One may find a tachyon-free non-supersymmetric AdS vacuum with the following mass hierarchies \footnote{As a disclaimer, for the sake of simplicity we will consider
a simplified (though less realistic) constant uplift term. However, we have checked that the final results do not differ quantitatively if one considers, for instance, an F-term uplift coming from a nilpotent goldstino \cite{Ferrara:2014kva}.}
\begin{equation}
M^{2}_{\text{closed}, \chi} \sim \alpha_{i}\frac{hq^{3}}{\mathfrak{f}_{0}^{\frac{3}{2}}\mathfrak{f}_{2}^{\frac{1}{2}}}, \, \, M^{2}_{\varphi} \sim \mu \frac{q^{3}}{\mathfrak{f}_{0}^{\frac{3}{2}}\mathfrak{f}_{2}^{\frac{1}{2}}} \, ,
\end{equation}
where $\chi$ denotes the saxionic partner of $\varphi$, closed refers to the mass scale of all the closed string sector moduli and $\alpha_{i}$ are numerical factors. In this example, as we vary $\varphi$, the vevs for the rest of the moduli which minimize the scalar potential are
\begin{small}
	\begin{eqnarray}
	\overline{s} &\sim& \frac{2^{7/4}3^{1/2}}{5^{1/4}}\frac{\left(\mathfrak{f}_{0} + \mu \varphi^2\right)^{1/2}\mathfrak{f}_{2}^{1/2}}{h} \, ,  \overline{u} \sim \frac{1}{10^{1/4}3^{1/2}}\left(\frac{\mathfrak{f}_0+\mu\varphi^2}{\mathfrak{f}_{2}}\right)^{1/2}\, , \nonumber \\
	\overline{t} &\sim& \frac{5^{3/4}3^{1/2}}{2^{1/4}}\frac{\left(\mathfrak{f}_{0} + \mu \varphi^2\right)^{1/2}\mathfrak{f}_{2}^{1/2}}{q} \, ,
	\label{sols}
	\end{eqnarray}
\end{small}
while the moduli not included in the above equation are stabilized at zero vev for all values of $\varphi$. Finally, we note that the mass hierarchy between the various sectors is given by
\begin{equation}
\frac{M_{\varphi} }{M_{\text{closed}, \chi}} \sim \sqrt{\frac{\mu}{h}} \, .
\label{hierarchy}
\end{equation}
Thus, in this example, the only way to generate a hierarchy is to tune the ratio \eqref{hierarchy}.
It has been shown in
 \cite{Bielleman:2016olv,Valenzuela:2016yny,Landete:2017amp}
 that

 tuning $\mu \ll 1$
 can be done while satisfying the experimental constraints on the inflationary parameters given by the recent PLANCK data

 \cite{Ade:2015lrj}.
 One should, however, be cautious about this tuning.\footnote{As argued in \cite{Blumenhagen:2017cxt}, this setup admits a four-fold description in the context of F-theory, and hence
 $\mu$ should be quantized since it arises from a $G_{4}$-flux. Thus, the mass ratio \eqref{hierarchy} cannot be tuned arbitrarily small. We would like to mention a possibility to overcome the  aforementioned problems. One may obtain a small mass ratio by placing the D7-brane in a warped region. This approach was used in \cite{Escobar:2015fda} for type IIA compactifications with D6-branes. In this case the mass ratio will be $\frac{M_{\varphi}}{M_{\text{closed}}}\sim\sqrt{\frac{\mu}{Zh}}$, where $Z$ is the warping felt by the D7-brane. By tuning $Z$, one may obtain a sufficiently small mass ratio while $\mu$ and $h$ are quantized.}

As our discussions below depend only on the existence of a mass hierarchy and not on the details of how it is achieved, we will sidestep these challenges and focus on exploring the consequences.
Nonetheless, we should stress that generating mass hierarchies in stabilized flux compactifications remains one of the major challenges in string theory (see \cite{Blumenhagen:2014nba} for a discussion along these lines), and may well provide a key ingredient in constructing controlled models with super-Planckian field ranges. In the following, we will check the validity of the approach taken by the RSDC in this concrete case.
\subsection{Validity of the approach}

To consistently ignore the kinetic terms of the heavy moduli,  we should check if eq. \eqref{validity} is satisfied. For a toroidal compactification this is translated into
\begin{equation}
\frac{1}{G_{\varphi \varphi}\left(\varphi\right)} \sum_{\alpha}\beta_{\alpha}\left(\frac{\partial}{\partial \varphi} \log \left(\overline{\phi}^{\alpha}\right)\right)^2 \leq \varepsilon\, .
\label{toroidalvalidity}
\end{equation}
Then, substituting the minimization condition \eqref{sols} into the former expression, one arrives at
\begin{equation}
7\frac{\hat{\varphi}^2}{\hat{\varphi}^2 +\left(\frac{M_{\text{closed}}}{M_{\varphi}}\right)^2} \lesssim  \varepsilon \left(\frac{M_{\text{closed}}}{M_{\varphi}}\right)^2\, .
\label{torviab}
\end{equation}
The critical value where the approximation breaks down, $\hat{\varphi}_{\text{app}}$, may be obtained by solving the above expression. We observe that \eqref{torviab} implies $\left(\frac{\partial \overline{\phi}^{\alpha}}{\partial \varphi}\right) \lesssim \sqrt{\varepsilon} \left(\frac{M_{\text{closed}}}{M_{\varphi}}\right)$. From \eqref{torviab} we see explicitly, as we previously anticipated, that there are two limiting regimes:
\begin{enumerate}
	\item[1.] Absence of a mass hierarchy
\begin{equation}
\frac{M_{\text{closed}}}{M_{\varphi}}\sim 1 \rightarrow \hat{\varphi}_{c}=\hat{\varphi}_{\text{app}}\sim \sqrt{\frac{\varepsilon}{7}}\,M_{\text{Pl}} < 1 M_{\text{Pl}}\, ,
\label{nohie}
\end{equation}
the critical field range is always subplanckian and signals the breakdown of the approximation  eq.~\eqref{toroidalvalidity} rather than the breakdown of the EFT.
\item[2.] Presence of a large mass hierarchy
\begin{equation}
\left(\frac{M_{\text{closed}}}{M_{\varphi}}\right)^2 > \frac{7}{\varepsilon} \rightarrow \hat{\varphi}_{c} = \hat{\varphi}_{\text{back}} \sim \frac{M_{\text{closed}}}{M_{\varphi}} M_{\text{Pl}}\, ,
\label{largehie}
\end{equation}
 %
eq. \eqref{validity} is satisfied for all $\varphi$ and thus the approach used in the RSDC is valid. The critical field range is then super-Planckian.
\end{enumerate}
Here, we point out that  in \eqref{toroidalvalidity} the backreaction effect of each modulus is weighted by its mass compared to that of the axionic field. In this example, all the stabilized moduli have roughly the same mass and, thus, their effects contribute on equal footing.
In a KKLT-like \cite{Kachru:2003aw} or LVS \cite{Balasubramanian:2005zx} compactification,
however, there is a hierarchy of scales
among the closed-string sectors (see \cite{Hebecker:2014eua,Ibanez:2014swa} for examples).
In this case, treating all variables dynamically, the RSDC constraint \eqref{defrange2} is weaker compared to the constraint imposed by destabilization of the K\"ahler modulus when a large mass hierarchy between the axion and the complex structure sector is assured. Nonetheless, only by using \eqref{validity} one may find the needed mass hierarchy between the axion and the K\"ahler sector to trust the single-field approximation.

\subsection{The RSDC in different regimes}

\paragraph{Large mass hierarchies.}
We have shown in \eqref{largehie} that when $M_{\varphi} \ll M_{\text{closed}}$ the dynamical field range is approximately measured along a geodesic trajectory and thus
\begin{equation}
\Delta\phi  =  \int \sqrt{G_{ab}\dot{\phi}^{a}\dot{\phi}^{b} }dt \approx \int \sqrt{G_{\varphi \varphi}\left(\varphi\right)}d\varphi \sim \frac{M_{\text{closed}}}{M_{\varphi}}\log\left(\varphi\right)\, .
\label{fieldrange2}
\end{equation}
The critical field range is then $\hat{\varphi}_{c} = \hat{\varphi}_{\text{back}} \sim \frac{M_{\text{closed}}}{M_{\varphi}} M_{\text{Pl}}> 1 M_{\text{Pl}}$, in agreement with \cite{Blumenhagen:2017cxt}. We see from Figure \ref{fig:comp2} that the dynamical field range indeed agrees with the one measured along the geodesic trajectory.
\begin{figure}[h]
	\includegraphics[width=7.8cm]{smallback3}
	\centering
	\caption{Field ranges obtained by \eqref{rangebetter} (blue) and \eqref{defrange2} (orange) for $\frac{\mathfrak{f}_{0}}{\mu} \sim 10^{4}$ and $G_{\varphi \bar{\varphi}}|_{0} \sim 2\cdot10^{-1}$.}
	\label{fig:comp2}
\end{figure}
\paragraph{No mass hierarchies.}

We now compare the RSDC estimate of the field range using eq.~\eqref{defrange2} with the measure eq.~\eqref{rangebetter} in cases where there are no mass hierarchies outside the validity regime we have established in \eqref{validity}.
We have shown in \eqref{nohie} that this approximation will be valid until $\hat{\varphi}_{\text{app}}$. We show in Figure \ref{fig:comp} how the two field ranges deviate from each other beyond this point. Then, in this regime one should solve numerically the full multi-field dynamics.
We have also computed numerically the field range with the actual solution to the equations of motion and found that the behavior depicted in Figure \ref{fig:comp}
does not change qualitatively.
We observe that, asymptotically, for large $\varphi$ displacements, the dynamical field range grows at best logarithmically, in agreement with the SDC. Note that the kinetic terms of the minimized fields are sizable, precisely, when there is no mass hierarchy, i.e. , $\mu \sim h$ and, as anticipated, our computation shows that the larger the backreaction effects the larger the deviation between the naive approximate \eqref{fieldrange2} and the dynamical field range.
\begin{figure}[h]
	\includegraphics[width=7.8cm]{fieldrangecomp3}
	\centering
	\caption{Field range obtained by \eqref{rangebetter} (blue) and \eqref{defrange2} (orange) for $\frac{\mathfrak{f}_{0}}{\mu} \sim 1$ and $G_{\varphi \bar{\varphi}}|_{0} \sim 2\cdot10^{-1}$. In this case $\hat{\varphi}_{\text{app}} \sim 0.1 M_{\text{Pl}}$ for $\varepsilon = 0.1$}
	\label{fig:comp}
\end{figure}

\section{Discussion}

In this letter we have analyzed carefully the viability of the constraints imposed by the RSDC on transplanckian field displacements of massive axions.
We illustrated our general results with axions in type II compactifications.

One of our findings is that minimizing the scalar potential without taking into account peculiarities of the field space metric and the mass hierarchies involved may lead to inconsistent results.
In the examples analyzed in the RSDC literature, the available field range is measured along a geodesic trajectory while neglecting the kinetic terms of the other fields. This is, indeed, a reasonable approximation if there is a large mass hierarchy and when the backreaction effects are mild. Precisely, in this approximation, the critical field range is extended to a super-Planckian value.
 In  the absence of a mass hierarchy, the approximation breaks down before the onset of the logarithmic behavior that defines the RSDC field range and, then, the sub-Planckian field restriction imposed by the RSDC cannot be trusted. This fact points to one of our premises, which is the importance of analyzing the dynamical field range. In order to constrain field displacements by means of the SDC, it should be applied to physical trajectories. Our findings support this conjecture since the logarithmic behavior of the field range holds asymptotically in all the cases studied, but the concrete point where the EFT suffers a breakdown can only be obtained explicitly by solving the eoms. Indeed, we have proved that in the regime where minimizing the scalar potential is a good approximation to solving the eoms,

  the dynamical field range is well approximated by the geodesic distance. Beyond that regime, the prescription given by the RSDC loses its physical validity and a multi-field analysis is needed.

The above subtleties with multi-field dynamics have appeared in the study of inflation.
For example, it is well-known that the kinetic terms of the heavy fields can lead to turns in field space. As a result, minimizing the scalar potential for the heavy fields as we vary the light field (the inflaton in this case) does not in general solve
the eoms. The deviation of the true trajectory from a geodesic can be parametrized
by $\eta_{\perp}$ \cite{Shiu:2011qw,Achucarro:2012yr,Tolley:2009fg,Rubin:2001in}.

Given the role of mass hierarchies in defining the dynamical field range, it is important to explore new ways of generating such hierarchies in string compactifications.
While ideas such as employing large ratios of flux quanta have been put forth, they seem to be challenging
to implement in practice, as can be seen in Type IIB settings with non-geometric fluxes \cite{Blumenhagen:2014nba}.
 It would be interesting to explore other proposals such as the use of warping for the open-string sector \cite{Escobar:2015fda}
 and the use of $g_{s}$ loops for K\"ahler moduli axions \cite{Cicoli:2014sva}. Finding new ways to generate mass hierarchies in string constructions may hold the key to extending the dynamical field range in quantum gravity. We leave for future work the extension of these findings to generic Calabi-Yau manifolds.
\vspace{1.0cm}

{\bf Acknowledgements. }We thank Fernando Marchesano and Wieland Staessens for discussions. This work is supported in part by the DOE grant DE-SC0017647 and the Kellett Award of the University of Wisconsin.

redpajama_set_name: RedPajamaArXiv


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British employees feel positive about returning to work……
By Neil Trim on August 22, 2020
…..but concerns run deep and businesses have work to do to make workplaces safe to return to
Employers urged to get back to business plans in place and seek expert help where needed
Half (49%) of workers feel positive about a return to work
Support for employers with 61% of those returning to work believing their employer will put the right safety measures in place
But 35% of those who do not believe their employer will do enough to make their work environment safe say they won't return
Infection, social distancing and travel are among employees' top concerns
Older workers are most concerned about the risk of infection
Younger workers say they are confused by the communications on social distancing
Construction workers are among the most concerned of being infected by Covid-19
Regional, age and industry differences show the complex impact of Covid-19 on communities
Half (49%) of employed British adults feel positive about the prospect of returning to their place of work after lockdown, with less than one in five (18%) feeling negative, an Aviva study shows*.
The findings paint a relatively positive picture for businesses that have supported their people through lockdown, such as regularly communicating with workers and taking necessary steps to manage the risk of infection in the workplace. However, where businesses have not embraced risk management and prevention strategies, employees may decide not to return to work at all.
Further, it is clear that there is a long way to go before businesses are compliant with the practical guidance from the government to make work safe for employees and customers.
The survey of more than 2,000 employed adults working in the UK found that 61% of those returning to work believe their employer will make the work environment safe to return to. However, in a clear warning to businesses that don't put safety first, the survey found that 35% of employees returning to work don't trust their employer to make the workplace safe and may not return to work for their employer.
Top concerns for employees returning to work
For workers who feel neutral or negative about returning to work, infection remains workers' top concern as they come back to their place of work. Whether from colleagues (44%) or customers (33%), employees are most concerned about virus transmission in the workplace, which underscores the need for employers to embrace clear prevention strategies to protect their people and the public.
Coronavirus has affected everyone in different ways, depending on a number of factors including age, industry, and geography. For example, only 29% of 16 – 24 year olds said they were concerned about the risk of being infected by colleagues, compared to 44% across all age groups. Infection concerns increased consistently by age group, with those over 55 most concerned (49%).
Conversely, employees over 55 were the least concerned about the impact of lockdown on their mental health (7%). This compares to 21% of young employed people who said the changes under lockdown affected their happiness.
However, the biggest concern facing young people returning to work is confusion and a lack of communication over social distancing (39%). Underscoring the importance of communication, young people were also the most critical of communication from their employer, with just 14% rating communication with their employer as 'excellent' – well below the average response of 25% who rated employer communication as 'excellent'.
The confusion voiced by young people in this survey shows that employers have a clear need to regularly communicate with all of their workers about their return to work and to keep them informed of the safest ways of working, including how to manage social distancing in the workplace.
Where we live also affects our attitude to returning to work. Aviva's research shows that the greatest concern for Londoners returning to work wasn't the workplace itself, but getting there. Concern about commuting by public transport was noted by nearly half (47%) of all respondents in the Greater London area, more than any other area of concern.
Employees' concern of catching Covid-19 from colleagues varied widely by location. Compared to the national average across workers from all regions (44%), workers in or around Plymouth (57%), Norwich (55%) and Cardiff (55%) were the most concerned about infection from colleagues, while those in Nottingham (34%), Southampton (37%), and Newcastle (38%) registered the lowest concern.
Attitudes towards risk were equally varied across industry sectors. Healthcare workers, who were some of the heroes emerging from lockdown, showed admirable positivity in the face of danger, recording the lowest levels of personal concern with regards to infection from colleagues (38%), compared to other sectors in the survey. They also had a relatively low score for concern of infection from patients (34%).
The industry most concerned about infection from colleagues was the sector that included construction, at 60%. Workers were understandably concerned as their jobs are physically demanding and frequently require the need to shout over loud noise – both thought to be contributory factors to the spread of the virus.
Chris Andrews, Head of Risk Management Solutions at Aviva, commented: "As Britain gets back to business after lockdown, employees want to know that their employers have taken all the necessary steps, and what those steps are. That means not just taking action but engaging with employees on their concerns and clearly communicating what is in place and addressing anything that remains unresolved.
"Our research shows that the biggest employee concern for returning to work is infection from colleagues and customers. Risk management and prevention strategies, once perceived as red tape, are now essential for businesses in order to keep their people and their customers safe as they open their doors again. Whether devising safe working conditions in a factory, or ensuring orderly social distance in a retail setting, businesses need to consider how they will minimise the risk of infection for their employees and their customers."
How have businesses made work safer?
In Aviva's survey more than 2,000 employed UK adults were asked about their attitude towards returning to work. Of these, 42% spent lockdown working from home, 26% continued to work in their usual place of business, 21% were furloughed and 6% continued to work in different locations in a key trade (e.g., plumber, electrician, etc.). A further 5% were not working and not on furlough.
Of the workers that continued to work in their usual place of business, or continued to work at different locations, 80% said that their employer had taken some steps to improve their safety, and the safety of customers entering the premises.
The top three changes implemented by businesses were easy access to hand-washing and hand sanitiser (65%), reducing the number of employees and/or customers allowed in the workplace (50%) and sign-posting traffic flow through the premises to support social distancing (42%).
While welcome, this shows that many businesses still have a long way to go to fully comply with the government's five point plan for safely returning to work and being Covid-Secure. Below we've compared the government's plan against how many employees said their employer had taken these measures:
Government's '5 key points' Proportion of workers who said their employer had implemented the measure
1. Help people work from home 42% of employees surveyed said they were working from home
2. Carry out a COVID-19 risk assessment, in consultation with workers or trade unions Did not ask, as not all workers may know if a risk assessment was carried out and so results would not be valid
3. Maintain 2 metres social distancing, wherever possible 42% of those working in their normal workplace said their employer had taken measures to improve social distancing
4. Where people cannot be 2 metres apart, manage transmission risk 40% of those working in their normal workplace said their employer had taken measures such as erecting screens or barriers between workers and customers
5. Develop cleaning, handwashing and hygiene procedures 65% of those working in their normal workplace said their employer provided easy access to hand-washing and hand sanitiser
Andrews continued, "It's good to see from our survey that most employees say their employer has brought in some of these measures. But it is equally clear that not enough businesses are following all five basic points of the government's guidance for employers. The risk of not adopting the government's measures is the spread of the virus, a return to lockdown either locally or nationally and long-term impacts to UK PLC. We must not let that happen. This is a time to be absolutely focused on prevention and attentive to the needs of employees as they return to work.
"Many businesses may have very specific operational requirements, and expert advice may be the best course of action for them. We encourage businesses to seek expert risk management advice as they emerge from lockdown to ensure they are taking the necessary steps to protect people. Not only will this build confidence amongst employees and customers, but it will equally minimise the risk of infection and related claims for liability."
www.aviva.com
Neil Trim
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This pack includes at least two worksheets for each CCS for L.4.1 a-g. This pack is great for review, teaching, or introducing each standard. Answer key is included. 1. Pronouns: which & that (l.4.1a) 2. Pronouns: who & whom (l.4.1a) 3.
Ask Jessica Scott a question. They will receive an automated email and will return to answer you as soon as possible. Please Login to ask your question.
I've taught grades 2nd-4th for ten years, and I am currently in my third year as an Instructional Coach for a 3-5 school. I am an advocate for doing what is best for students and their learning through the use of engaging learning activities.
In my classroom I use Cooperative Learning Strategies, Interactive Notebooks, Balanced Literacy, and a Hands on Approach to learning.


Lighting designer Ben Rousseau of Rousseau Design has added this new table lamp in powder-coated steel to his popular TORUS collection. The steel Torus light takes it inspiration from Torus energy fields. Constructed using concentric powder-coated steel rings around a central hub and featuring a stainless steel base, the table light is available in a choice of 5 colours. An alternative base by Innermost is available in chrome (see image on right).
Notably, the Torus light features a light source – the R Core LED hub – that has been specially developed and designed by Rousseau Design. The table lamp can also be ordered with a standard E27 fitting and bulb.
Shown above: Torus in turquoise with stainless steel base and Torus in orange with chrome base.
Light source: R Core LED hub (as seen here) or a standard E27 fitting and bulb.


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Steve Brooks, a sexist womanizer, is killed by a group of his angry former lovers. In heaven, he makes a bargain with God for redemption and agrees to return to Earth. Once there, he must have a sincere relationship with a female and make her fall in love with him. If not, Steve's soul will become the property of the devil. But the devil hedges his bet, and Steve is reincarnated as a woman named Amanda Brooks.
Director: Blake Edwards
Actors: Bruce Payne, Catherine Keener, Ellen Barkin, Jimmy Smits, JoBeth Williams, Lorraine Bracco, Lysette Anthony, Perry King, Téa Leoni, Tony Roberts
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Jackass: The Movie
Johnny Knoxville and his crazy friends appear on the big screen for the very first time in Jackass: The Movie. They wander around Japan in panda outfits, wreak havoc on…
Genre: Action, Comedy, Documentary
Bernie, a 30 years old orphan, decides to leave his orphanage and find his parents. After an investigation, he meets Marion, a young heroin addict and falls in love with…
While attempting to seduce gorgeous lawyer Diane Lightson, wealthy gadabout Chris Thorne agrees to drive her to Atlantic City, N.J. But, when some reckless driving draws the attention of a…
Genre: Adventure, Comedy
While on detention, a group of misfits and slackers have to write a letter to the President explaining what is wrong with the education system. There is only one problem,…
Eddie is a New York limo driver and a fanatical follower of the New York Knicks professional basketball team. The team is struggling with a mediocre record when, in mid-season,…
The story of the life and career of eccentric avant-garde comedian, Andy Kaufman.
Genre: Comedy, Drama, History
Perdiamoci di vista
Comic tale of a cynical TV star's slow-combustion romance with a young paraplegic.
Charlie and Rachel run away from home to get married in Las Vegas. But they get attacked by a zombie who takes Rachel with him to hell, where she will…
Genre: Action, Comedy, Fantasy, Horror
A down-on-his-luck crab fisherman embarks on a journey to get a young man with Down syndrome to a professional wrestling school in rural North Carolina and away from the retirement…
Genre: Adventure, Comedy, Drama
The gang from Dorm Daze is back, but this time they're on a cruise ship on a tropical voyage through the high seas. The cruise makes an educational stop at…
Futurama: Bender's Big Score
The Planet Express crew return from cancellation, only to be robbed blind by hideous "sprunging" scam artists. Things go from bad to worse when the scammers hack Bender, start traveling…
Genre: Animation, Comedy, Science Fiction, TV Movie
The Cobbler
Max Simkin repairs shoes in the same New York shop that has been in his family for generations. Disenchanted with the grind of daily life, Max stumbles upon a magical…
Genre: Comedy, Drama, Fantasy
Trailer: Switch


Girders are construction platforms found in various games, starting with the original Donkey Kong. Most of their appearances since then have been in reference to this appearance.
Girders first appear in Donkey Kong, appearing in every stage, most notably in 25m, where they are the only type of platform. In the Game Boy remake, they appear in all of these stages, as well as several stages in the game's later worlds, such as Big-City. In both games, they act essentially as a semi-solid platform that Mario can walk on.
A girder from Mario vs. Donkey Kong: Mini-Land Mayhem!
Generic girders appear in every Mario vs. Donkey Kong game. Throughout the series, they act as semi-solid platforms much like those in Donkey Kong, being red-colored in Mario vs. Donkey Kong, Mario vs. Donkey Kong 2: March of the Minis and Mario vs. Donkey Kong: Minis March Again!, and gray with rounded corners in all of the following titles. The latter are described as "gray steel frames".
This type of fixed girder is distinct from one of the series' staple gameplay mechanics, the Red Girders, which first appear in Mario vs. Donkey Kong: Mini-Land Mayhem! These can be drawn between red rivet blocks using the touchscreen to help the Minis cross gaps or reach a higher area.
In Mario vs. Donkey Kong: Minis March Again!, pieces of red-colored girders are among the objects that fall from the top of the screen when Donkey Kong slams his fists in Floor 1 Donkey Kong. Short girders appear first, followed by longer ones that are harder to avoid after DK is hit a third time. The player loses a Mini toy if a girder crashes into it.
Girders appear in Super Smash Bros. Brawl, Super Smash Bros. for Wii U, and Super Smash Bros. Ultimate in the 75 m stage, based off of the original 75m from Donkey Kong, but with visible depth.
In Donkey Kong Country Returns, girders are the primary floor in the Factory world. In the background for the Factory's first level, Foggy Fumes, girders arranged in the zig-zagged 25m layout can be seen at one point. This pattern also commonly appears on wall reliefs throughout the temple-themed levels, along with other depictions of Donkey Kong arcade graphics.
Girders appear in Super Mario Odyssey, where they are seen as common platforms throughout the Metro Kingdom. They do not act as semi-solid platforms in this game, but items such as Coins and Power Moons can be found inside some of them.
In the manual for Donkey Kong 64, Cranky Kong hopes that a level design he submitted, "Girder Grapple," made it into the game as a secret level, referencing him being the original Donkey Kong from the arcade game. While this level doesn't appear, the original game can be played in an arcade cabinet found in Frantic Factory.
This page was last edited on December 9, 2018, at 08:39.


Dubya: Fool me once, shame on you! Ya can't get fooled again!
I came to this short film with absolutely no clue as to its content – not the theme, not the genre, not even if it was an animation or not. The opening of the film sees a George Bush figure in a dunce cap in a classroom – (which is unmistakably "that" classroom) and we hear one of his Bushisms played as he tap dances and entertains. At this point I thought this would be a lazy piece mocking Bush since this is how it had looked to me at first glance. This impression lasted seconds though as I was thrown into an 8 minute ride of symbolism, challenging images and chaos on my way to a feeling of "what the hell did I just watch". And I'm sure I'm not alone.
The good thing is that you will jump right back into this film – not to try to figure it out, but rather to figure out how it makes you feel and why. With 9/11 imagery, politicians as entertainers/distracters for children, Bin Laden and an army of humanoid bombs, Al Qaeda as a wing of the CIA, religious figures and deities, humans suffering under control of TV screens directly plugged into their brains and lots of Illuminati symbols, this is a short film that has much more in it than the simple title suggests.
The best way to watch is to try to set aside your views on such things as Illuminati, religion and politics and let the film have its time because even in terms of "spot the reference", this is something you'll want to watch again. To give examples – your eye will go to Bin Laden on a boat sailing against America and instinctively it made me feel uncomfortable that he would be shown in this way – on second viewings I saw the human weapons behind him but also the CIA badge on his jacket. Likewise even on the opening scenes it is worth pausing and looking around behind the Bush caricature to see the detail of an "evolution" chart which is the norm is schools but this one seems to contain an "enlightenment" stage, while on the blackboard the children appear to be playing hangman with the word evolution being spelt out. Heck it even took me a while to spot that the goat in the opening few second was not a production company logo as I first thought, but actually in a detainment camp, boxed up, with a bar code on its forehead, and of course the title is a reference to the children's book that Bush was reading while planes flew into the towers in New York on 11th September.
Even with slower viewings there is so much in this short film to catch that I know most of it still went over my head. Whether you agree with the material or just laugh at it for its conspiracy theory nuttery, it has to be applauded for the construct and the detail; after all, Pixar put some background pop culture references in their films and everyone is falling over themselves with praise, so this short deserves the same. It is worth saying that animation is also beautiful – polished, professional and full of small detail. Content wise it may well be an old man on the bus babbling about lizards and third eyes, but it is fascinating and beautiful to watch.
Personally I thought some of the ideas within the film were nonsense (but then, I'm a goat in a box, so what do I know) and some of the images and themes made me feel uncomfortable with their use, but this is fine – it is good to have a short that grabs you, overwhelms, challenges and makes you think – even if the outcome of those thoughts might be "yeah...but no".


Mr. Flaig, who was recently nominated and selected to The National Advocates Top 100 Attorneys, has concentrated on estate planning, wills, and trusts and assisting corporations (both large and small) in their maintenance, day to day operations, contract drafting, review and analysis and risk management. Finally, Mr. Flaig has been active in the law and civil litigation since 1997 and can either assist and/or refer you to the appropriate qualified counsel for such matters.
Outside the daily representation of his clients, Mr. Flaig has been actively involved in both legal and civic affairs. He is on the Board of Directors for the non-profit humanitarian organization, Help The Children. He has also served as a judge for mock trials at Southwestern University School of Law and at the University of Southern California School of Law.
Prior to passing the California Bar in 1997, Mr. Flaig attended the University of Southern California and earned his B.A. in 1994. Immediately after receiving his degree, he attended Southwestern University School of Law where he became actively involved and was elected to the Board of Governors for the Interscholastic Trial Advocacy Team. As an advocate, Mr. Flaig competed in mock trials around the nation. This education and experience has given Mr. Flaig the ability to provide value to his clients in their legal matters.
Mr. Flaig has once again been selected to SuperLawyers, which is only awared to 5% of the top attorneys in the State, and is published in Los Angeles Magazine this year. SuperLawyers are attorneys who have been proven to excel in the following: verdicts and settlements; transactions; representative clients; experience; honors and awards; special licenses and certifications; position within law firm; bar and or other professional activity; pro bono and community service as a lawyer; scholarly lectures and writings; education and employment background; and other outstanding achievements. Only 5% of lawyers in the State are designated with this honor. Mr. Flaig is proud to be one of them again this year.


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Elementus was one of two successful bidders in the second phase of the A.C.T.'s reverse auction process.
The first round of the A.C.T. solar auction was won by the Spanish company, Fotowatio Renewable Ventures, which is building a 20 MW solar farm near Royalla and the other successful tenderer in the second stage was Zhenfa Australia which will build a 13 MW solar farm on Mugga Lane in Tuggeranong..
The A.C.T. Government plans to bring another 650MW of renewable energy generation to its region in order to reach its target of 90% renewable energy generation by 2020.
Construction of the solar farm was expected to commence in mid-2014, with completion early in 2015 but was delayed as a result of objections to the original Uriarra site. Construction at the Williamsdale site was approved in January 2016.
Once operational it will produce approximately 11,900MWh of electricity per year from 26,100 crystalline solar panels in a fixed tilt configuration.
In May 2016, the solar farm was purchased by the funds management company Impact Investment Group.
Construction commenced in July 2016 and was completed in October 2017.


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Objectives Children spend a significant amount of their time in a school environment, often engaged in sedentary activities. The Daily Mile is a physical activity intervention which aims to increase physical activity and fitness in children through the completion of an outdoor teacher-led walk or run during the school day. This study aimed to explore the barriers, facilitators and perceived benefits of the Daily Mile from the perspectives of teachers through the use of qualitative semi-structured interviews. It also aimed to identify important context-specific factors, which might require consideration for those who intend to adopt the Daily Mile.
Setting Eight Local Authority primary schools in the City of Edinburgh and East Lothian, UK.
Participants Thirteen teachers (eleven women) who teach children in primaries one to seven in a school which delivered the Daily Mile.
Results Data were analysed using an interpretative thematic analysis. Teachers were positive and enthusiastic about the Daily Mile and perceived it to be beneficial to children's health and fitness. A number of barriers to participation were identified including inadequate all-weather running surfaces and time constraints in an already full school curriculum. The perceived impact on learning time was identified as a concern for teachers, while other benefits were also identified including increased teacher–child rapport and perceived enhanced classroom concentration levels.
Conclusion The Daily Mile appears to be a valuable addition to the school day, however important context-specific barriers to delivery of the Daily Mile exist, which should be considered when implementing the Daily Mile in schools.
One of the first studies to investigate teachers' views of the Daily Mile.
Sampled teachers from schools in multiple locations to investigate contextual factors.
Data collection with children would have allowed for data triangulation.
Small sample size may limit generalisability of findings.
A lack of physical activity is a determinant of numerous health conditions in childhood.1 Specifically, sedentary behaviours and low levels of physical activity are partly to blame for the marked increases in childhood obesity that have been observed internationally in recent years.2 3 In the UK, childhood overweight and obesity levels are high, approximately 30%.4 Additionally, physical activity levels are low5 and have been shown to decrease with age, a trend that has been similarly observed in other European countries and the USA.6 Considering this, there is a need for the implementation of effective, evidence-based interventions to increase physical activity levels in school-aged children.
A recent quasi-experimental pilot study by Chesham et al 9 showed that when compared with control schools, children who participated in the Daily Mile showed significant improvements in objectively measured daily moderate-vigorous physical activity (MVPA), objectively measured sedentary time, fitness and body composition.9 To date, the aforementioned study is the only research currently offering empirical evidence of the Daily Mile's efficacy, although another randomised trial is being planned.10 However, interventions do not usually function the same everywhere, and it is important to understand contextual factors that may influence how they work.11 Contextual issues have been found useful in other school programmes, where context-specific differences have been considered when testing and developing an intervention.12 Furthermore, a systematic review of studies which examined factors which influenced the implementation of physical activity policies in schools, found that factors relating to environmental context/resources such as provision of adequate equipment/facilities, and social influences such as teacher/staff support were important predictors of implementation success or failure.13 Therefore, examining the experiences and perspectives of teachers in early adopter schools would allow policy-makers and practitioners to adapt the Daily Mile to suit their settings where necessary and identify facilitators and barriers to implementation.
A recent qualitative study by Ryde et al 14 was the first to assess the context specific factors, which influence the success of the Daily Mile's implementation in four schools in central Scotland. The study concluded that having a supportive organisational environment and allowing teachers autonomy to deliver the initiative facilitated the implementation of the intervention. While the study offers important insight into factors influencing the implementation of the Daily Mile, the authors highlight that it may be biased due to the included teachers being advocates of the Daily Mile and the selection of schools in one geographical area.14 The authors recommended for future studies to interview more teachers across different geographical locations in schools which had implemented the Daily Mile with varying degrees of success.14 Therefore, the present study explored the barriers, facilitators and perceived benefits of the Daily Mile from the perspectives of teachers who deliver the intervention in multiple schools across two geographical locations in Scotland. It also aimed to identify important context-specific factors, which might require consideration for those who intend to adopt the Daily Mile in the future.
A senior member of teaching staff (physical education coordinator) who had well established contacts with teachers in multiple schools acted as a liaison during recruitment and liaised with potential participants. The research findings will be shared with the participants in a short summary report which will be emailed to them directly and to head-teachers of the participating schools.
Adopting interpretive approach, qualitative methods were used to explore teachers' perspectives of the Daily Mile and their experiences of delivering it. Semi-structured interviews were conducted, consisting of open-ended questions which aimed to determine how teachers define the Daily Mile, how they deliver the initiative, perceived benefits for the children who participate, and potential context-specific barriers and facilitators to participation. Our goal was to explore their experiences of the Daily Mile in order to inform future practice rather than to develop new theory.
Participants were primary school teachers who taught years 1–7 at schools which either currently, or formerly, participated in the Daily Mile. As it is mostly class teachers who deliver the Daily Mile, head-teachers or teaching staff who do not have regular prolonged contact with children were excluded from participating. A total of thirteen teachers took part in the study, consisting of eleven women and two men.
A purposive sampling strategy was used to recruit teachers. Specifically, contact was made with a physical education coordinator in one of the Local Authority areas who acted as a liaison between the research team and schools/teachers. Between August 2016 and February 2017, the liaison delivered study information sheets and personally spoke to selected head-teachers about the study. Schools/head teachers were approached based on the liaison's knowledge that they were a school that had been delivering the Daily Mile. In order to enhance participants' recruitment, emails and letters were sent to head teachers of additional primary schools. This contained study information sheets and researcher's contact details. Twenty-one schools were approached to participate in the study, for which ten initially agreed to participate. Teachers from eight of the ten schools subsequently agreed to participate in the study.
Semi-structured interviews were conducted face-to-face in a private room within the participating schools by one researcher. Both researchers have extensive experience in conducting qualitative interviews with a broad range of participants for public health evaluation purposes, and have undertaken formal training in qualitative data collection. Participants were asked to give a brief history of the Daily Mile at their school, before probing questions were asked regarding subjects such as how the initiative is viewed by the staff at the school, approaches taken to deliver the programme and any perceived benefits, facilitators or barriers associated with participation and delivery. A sample of the interview topic guide can be viewed in online supplementary appendix 1. Interviews were usually conducted after the school day and lasted between 21 and 58 min. All interviews were audio-recorded via digital audio recorder and transcribed verbatim.
Using interpretive approach, this study employed thematic analysis16 to analyse the qualitative data that were generated following verbatim data transcription. After familiarising ourselves with the transcribed data, a coding framework was developed by both authors who independently coded the same two most informative transcripts from the dataset, before collaborating to refine the coding framework. Both authors independently read and re-read each remaining transcript and used the coding framework to assign codes to the text. New codes identified were discussed and added to the framework. This ensured consistency and rigour. Subsequently, similar codes were grouped together into overarching themes, which were reviewed and refined to ensure the coded data adequately reflected themes that encompassed them. Interpretation of data was undertaken by comparing similarities and differences within and across themes. Data analysis was facilitated by NVivo V.10 qualitative data analysis software.
The characteristics of the eight schools where the 13 teachers were drawn from varied with regard to facilities, rurality and level of deprivation as detailed in table 1. All but one school had access to sufficient greenspace within the school grounds, while half of the schools had access to an all-weather surface such as an Astroturf football pitch or a solid polyurethane running track. Three schools were located within a large town or city, three were in a semirural location such as a small country town and two schools were in a rural location. Three schools were located within areas of high deprivation, three within areas of medium deprivation and two in affluent areas.
Eight themes were identified, and are detailed below.
The style of delivery varied considerably between teachers and schools. While each school followed a similar Daily Mile protocol for delivery, some schools offered more autonomy to teachers concerning how and when they delivered the Daily Mile.
There were also some differences in the frequency and level of engagement with the Daily Mile among the schools. A number of participants stated that all classes at their school took part, often at the request of the head teacher, while others mentioned that only some classes did the Daily Mile and that participation does not happen every day either due to teacher preferences or age/ability of the children.
There were slight variations in how the Daily Mile was defined by the teachers. Variations in what constitutes participation were evident, with some teachers defining the Daily Mile as 15 min of exercise, and others as an exact mile that is done to completion regardless of time taken. The emphasis on running the Daily Mile also differed between teachers and schools, with some teachers' strongly encouraging running, while others were happy for children to choose whether to walk or run.
The time at which the Daily Mile was delivered during the school day also differed among the teachers. Some participants stated that they purposefully delivered the intervention at set times during the day in order to establish a routine or get a desired behavioural or performance-based response from the children.
There was variability among the sample regarding the interaction between the Daily Mile, and formal physical education sessions. A number of teachers explicitly stated that they do the Daily Mile in addition to the 2 hours per week of physical education recommended by the school curriculum. However, a few of the teachers asserted that they count the Daily Mile as part of the physical education curriculum and therefore as a part of the 2 hours of weekly PE which they are expected to deliver.
Approximately half of the teachers used some form of motivational or incentive/rewards-based system with children when doing the Daily Mile. Such techniques included the handing out of tokens for each lap of the running area completed, or tracking the distance ran over an accumulative time-period and matching it with the distances to various geographical locations.
Some teachers stated that they found that their own participation in the Daily Mile motivated the children to try to keep up or beat them, thus increasing the children's engagement with the activity.
Two teachers also speculated that Daily Mile participation led to improvements in asthma symptoms among some children. One of the teachers explained this below.
There were mixed, but slightly positive findings regarding how the Daily Mile affected pupils' concentration levels. Some teachers reported that they had not noticed any obvious changes in concentration levels since the Daily Mile was introduced.
However, others stated that they have observed noticeable improvements in concentration and would often choose to do the activity at a specific time of the day when concentration levels are expected to be low.
The impact of the Daily Mile on behaviour was a contentious issue among the participants. Some teachers argued that the Daily Mile made little or no difference to disruptive behaviour of pupils.
However, two teachers felt that the Daily Mile did improve behaviour in the classroom. They attributed this to the release of excess energy, which often made the children restless at certain times of the day. The example below by one of the teachers explains this point.
Approximately half of the participants stated that a major benefit of the Daily Mile was that it allowed them the opportunity to build a rapport with the children in their class in a way that is difficult to do in a classroom environment.
Teachers acknowledged that the Daily Mile offered a unique opportunity to talk about issues with certain individuals, or to get to know the children in their class better.
In general, the Daily Mile was considered a worthwhile initiative due to the known benefits of physical activity on health and well-being. However, the majority of the participants conceded that it is time-consuming, and reduces the amount of time they have available to dedicate to other subjects in the classroom per week, which may potentially affect learning.
Furthermore, some participants were of the opinion that as the Daily Mile is not supposed to count as PE, they struggle to provide both activities in full to their class. Teachers stated that this was a contentious issue among colleagues when discussing the Daily Mile, and a number of participants mentioned that some colleagues refused to do the Daily Mile or have complained about it due to time constraints in terms of delivering a full curriculum.
A number of teachers expressed the view that getting the children ready to go outside was time-consuming, and added additional time constraints in addition to the Daily Mile itself.
All the participants identified the weather as a barrier to participation; specifically wet and icy weather. However, the majority of teachers felt they would like to do the Daily Mile in all weathers, but are limited by unsafe running surfaces, and inappropriate footwear/weather gear worn by some children.
As previously mentioned, some teachers highlighted the running surface as a limitation. Specifically, schools with large greenspace were viewed as suitable for participation. However, this is not necessarily safe to run on in wet conditions, which limited the ability of some schools to do the Daily Mile in wet weather due to not having access to alternative facilities.
Alternatively, teachers at the school with insufficient greenspace felt that this limited their ability to do the Daily Mile due to insufficient space and inadequate surfaces.
Teachers that had access to artificial playing surfaces or all-weather surface generally believed that their school had better resources for Daily Mile participation than those without.
While facilities were deemed important to participation, in general the teachers found ways to adapt the initiative to suit the facilities that were available at their school. This sometimes meant not participating in severe weather, or using indoor space such as a gym hall as an alternative.
Another barrier identified was that a number of teachers felt that some children were not properly equipped with the appropriate clothing to take part in the Daily Mile in all weathers.
The Daily Mile offers an innovative approach to physical activity in schools, and similar to another recent study,14 this study has identified important factors that influence the implementation of the Daily Mile, and has added further to the knowledge-base regarding the context-specific barriers and facilitators related to the intervention from the perspectives of teachers. We found that teachers were generally positive about the initiative, and felt the intervention was warranted considering the health benefits of physical activity, and its potential to reduce childhood obesity. However, a considerable number of teachers were concerned about the time-consuming nature of the Daily Mile and the impact of this on learning time.
Although the Daily Mile follows a generic protocol and procedure for implementation, schools and teachers have autonomy over when and how they execute it. Our findings show that some teachers did not undertake the Daily Mile every school day. A growing evidence base is demonstrating that a dose-response relationship exists between physical activity and its associated health benefits. Specifically, improvements in health outcomes tend to increase along with the levels of physical activity.17–19 Interestingly, some of the teachers in our study stated that they or their colleagues would occasionally count the Daily Mile as part of the compulsory 2 hours of quality physical education time, which may offset the benefits of having the Daily Mile as a stand-alone activity-as it is intended.8 Additionally, such practices may reduce the wider benefits associated with physical education such as motor skill development and the social/cognitive benefits of sport and team games which may not be addressed by Daily Mile participation.20 Despite this, Chesham et al.9 found that compared with control schools, the Daily Mile intervention schools increased MVPA by about 9 min per day, indicating that the intervention does increase PA and in turn may have potential health benefits. However, whether similar findings would be observed in schools which participate in the intervention less regularly, or combine the Daily Mile with physical education time may merit further investigation.
Despite the fact that teachers felt motivated to deliver the Daily Mile, the study also showed the Daily Mile increased pressure on teachers' workload and they were concerned about its detrimental effect on learning time; a finding that is supported by studies of other school-based physical activity initiatives.21 22 This was in contrast to a similar study in Italy, which also examined teachers and students' attitudes to an outdoor active break programme in a middle school.23 The study reported that teachers found the outdoor active break programme easy to organise and that it did not negatively influence their teaching activities. It is possible that any differences in the attitudes of teachers in the aforementioned study and our study are due to the age of children involved. While we used primary school teachers, with pupils usually aged 5–11 years, the Italian study used middle school teachers, who taught students aged about 12 years. A recent historical account indicates that increasing focus on academic achievement decreases physical activity opportunities in schools.24 With academic achievement under constant scrutiny in most countries, it appears that evidence to ascertain whether the Daily Mile promotes cognition and academic achievement could assure primary school teachers that time invested in the Daily Mile would engender additional educational benefits. Current evidence suggests that physical activity can show a positive effect on constructs related to academic achievement.24–26 However, recent studies have also demonstrated that academic achievement is more strongly associated with sedentary time than physical activity.27 28 These conflicting findings highlight the fact that at present, little is known regarding the optimum balance of sedentary time and physical activity for both academic performance and health. Future evaluations of school-based PA interventions similar to the Daily Mile should therefore incorporate measures of academic performance into their design.
Another important finding of the study was that Daily Mile offered social time for teachers and children. This was viewed as important in building a more supportive teacher–pupil relationship. A study has suggested that such supportive relationships are characterised by warmth, trust and low degrees of conflict and were associated with positive school outcomes for children with behavioural issues.29 Positive teacher–pupil relationships can therefore improve student motivation to learn as well as their academic achievement.30 The effect of the Daily Mile on teacher–pupil relationships is an interesting finding and should be explored further in future studies.
It was clear that the weather had implications in terms of participation for schools without adequate outdoor facilities, which is consistent with the findings of the review by Nathan et al 13 which demonstrated that factors such as inadequate facilities and adverse weather were identified as barriers to implementation of school-based interventions in seven studies. The Daily Mile is an outdoor school-based physical activity programme. Therefore, schools in temperate regions where the weather is often unpredictable during the winter months should consider improvement to their outdoor facilities in order to have the best chance of fully delivering the Daily Mile throughout the school year.
From 1998, the prevalence of overweight, including obesity among children aged 2–15 has fluctuated between 28% and 33%, and was 29% in 2016.31This high prevalence of overweight and obesity highlights the fact that existing interventions are not having the anticipated impact, at least at the population level. Therefore, innovative school-based physical activity programmes such as the Daily Mile require attention in order to understand how they can contribute to the reduction of childhood obesity; which is a major target within current UK and international health policy.32 Such programmes replace sedentary behaviour and light physical activity with MVPA and tend to reduce adiposity.33 It is in this respect that the Scottish Government is promoting the Daily Mile initiative more widely across schools in Scotland. The concept of the Daily Mile has also generated international interest, yet until recently there was little evidence regarding its efficacy. The Chesham et al study has provided some assurance of the Daily Mile's benefits to children in terms of increasing levels of MVPA, reducing sedentary time, increasing physical fitness and improving body composition. Certainly, it would be interesting to see whether such benefits can be sustained in the longer-term and how they translate into meaningful reductions in childhood obesity at the population level.
The study used qualitative interviews to generate a deeper insight into the perspectives of teachers who have crucial role in the delivery of Daily Mile in Schools. However, only the accounts of teachers were used to understand these barriers and facilitators. Using additional data from pupils would have allowed triangulation of data, which would have enhanced our findings. We struggled to recruit teachers into the study, so it is possible that teachers who were likely in favour of the Daily Mile participated. This may imply that the level of positivity depicted by the study may not reflect the overall opinion of teachers who deliver the Daily Mile. On the contrary, it could also signal that any barriers identified were important concerns, which require attention as the Daily Mile continues to expand internationally. Future studies should measure the attitudes and practices of teachers to the Daily Mile using quantitative approaches such as surveys, which can reach a larger sample.
Given that existing obesity prevention strategies are struggling to stem the tide of childhood obesity, innovative school-based physical activity interventions such as the Daily Mile present promising opportunities for policy change within education. However, any policy change should be underpinned by evidence if the practice of the Daily Mile is to be adopted more widely. This study has identified important barriers and facilitators from the perspectives of teachers who play a crucial role in the delivery of the Daily Mile. Our findings indicate that teachers were motivated to deliver the Daily Mile and were positive about its potential to improve physical activity and related health outcomes. However, they were also concerned that it was negatively affecting learning time.
We thank Marion Barclay for her assistance with recruitment for this study, and both Dr Josie Booth and Dr Gemma Ryde for their comments on the interview topic guide. We also thank all the participants who took part in our study.
Scottish Executive. A curriculum for excellence. Scottish Executive: Edinburgh, 2004.
. An introduction to scientific realist evaluation. 1997.
. Scottish index of multiple deprivation: Scottish Executive Edinburgh, 2006.
Scottish Government. Health of Scotland's population- obesity Edinburgh. 2017. Edinburgh: The Scotttish Government, 2018.
Scottish Government Scottish Government. A healthier future –Scotland's diet & healthy weight delivery plan: Edinburgh, 2018.
Contributors SM and LD designed the study and conducted data collection with participants. Both authors analysed the data, and SM drafted the first draft of the manuscript. SM and LD revised and prepared the final draft of the manuscript for submission.
Funding This work was supported by a University of Edinburgh Innovation Initiative Grant under Grant number GR002292. The authors were both employed by the Scottish Collaboration for Public Health Research and Policy (SCPHRP), University of Edinburgh. At the time of the study, SCPHRP's core Grant was from the Medical Research Council (Grant Number MR/K023209/1) and the Chief Scientist Office of Scotland.
Ethics approval Ethical approval was granted by The University of Edinburgh's Usher Research Ethics Group, and Local Authority research in schools permission was obtained prior to recruitment.
Data sharing statement Ethical restriction by the University of Edinburgh Centre for Population Health Sciences Research Ethics Committee prohibits the authors from making the minimal data set publicly available because data contain potentially identifiable information. However, data are available from the corresponding author on reasonable request. Data will be held for five years from February 2017 before being permanently destroyed. Contact: StephenMalden-stephen.malden@strath.ac.uk.


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Q: I dont get the proper jar Files for PowerMock to Mock private Methods? I am trying to mock a private method but I am unable to load the proper jar Files for PowerMockito. I used to get errors like
java.lang.NoSuchMethodError: javassist.Modifier.isPackage(I)Z
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redpajama_set_name: RedPajamaStackExchange


My office was such a mess that I avoided coming into the office. After 6 months I called Nipomo Organizing Solutions. Within one week my office was completely organized, My books were put in order, my desk drawers were in also in order. After 6 months, my office is still in order and working very efficiently. The system she put in was excellent and simple and I use it everyday to keep by business running smoothly. I high recommend Nipomo's services to get your office running smoothly and to stay organized.


We need a "tagging" system, there's so many mods they're hard to keep track of, we should be able to give them "tags" we can search for mods by, so if for example you only wanted to see mods related to the Kirby series in some way you'd search the "Kirby" tag. I feel like this would be helpful to everyone who uses the site.
I agree with the "any user can add" idea, though the "no download available" stuff is pretty much filtered out by reports anyway, thought it might be useful if you're looking for lost mods that you might have downloaded.
Well, the thing though about reports is that about 90% of them will be "No previews" which is super- easily fixable, so by having a separate tag for that, it would differentiate between the two, so you don't have to check what the report is every time.
Not sure if anyone's posted something like this, but I figured I'd post it in the feedback thread.
Have KC-MM's site operators considered the possibility of adding a "Sm4sh Vault" section? Specifically, something that's exactly like the Brawlvault, only pertaining to Sm4sh mods instead. I certainly think it would be a good idea, seeing as modding for the game has really been picking up traction as of late; I definitely prefer Brawlvault's functionality/organisation over sites like Gamebanana.
I'm not sure if this has been mentioned before but Brawl Vault entries don't automatically create hyperlinks for links starting with "https"
See http://forums.kc-mm.com/Gallery/BrawlView.php?Number=214967 for an example in the description.
Page created in 1.061 seconds with 23 queries.


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