text:
\section{Introduction}

The existence of non-baryonic dark matter has been inferred from measurements including galactic rotation curves \cite{Sofue:2000jx}, large scale structure surveys \cite{Cole:2005sx,Beutler:2011hx,Anderson:2012sa}, X-ray observations \cite{Vikhlinin:2008ym}, gravitational lensing \cite{Fu:2007qq,Massey:2007gh}, and cosmic microwave background anisotropy measurements \cite{Ade:2013zuv}, spanning cosmological eras from the present day to the remote past.  This widespread and robust data has led to cold dark matter models with a cosmological constant, labeled  $\Lambda$CDM  becoming entrenched as the standard cosmological model.

Nevertheless, this impressive array of observations has only been sensitive to the \emph{gravitational} influence of dark matter and constrained its relic abundance, leaving its particle nature as one of the most important open questions in physics.  The search for dark matter includes indirect astrophysical searches (\cite{Adriani:2008zr,Su:2010qj,Hooper:2010mq,Weniger:2012tx,Bulbul:2014sua}), collider production efforts (for some examples of dark matter searches at the LHC, see \cite{Aad:2014vka,Aad:2014vea,Aad:2015zva,Aad:2013oja,Lowette:2014yta}) which will examine new territory soon with LHC run 2 which will commence this year, and attempts to observe dark matter interactions with Standard Model (SM) particles via dark matter-nucleus scattering processes in direct detection experiments, to which we now turn.

The search for dark matter via direct detection goes back at least three decades \cite{Goodman:1984dc,Cabrera:1984rr} and has been particularly vigorous over the last decade or so with experiments such as LUX \cite{Akerib:2013tjd}, Xenon100 \cite{Aprile:2012nq}, CDMS II (Ge) \cite{Ahmed:2010wy}, CDMS I (Si) \cite{Agnese:2013rvf}, DAMA/LIBRA \cite{Bernabei:2008yi}, COGENT \cite{Aalseth:2012if}, and CRESST \cite{Angloher:2011uu}  pushing ever deeper into weakly interacting dark matter mass and scattering cross-section parameter space, but has thus far failed to yield a convincing signal.  In the near future detectors such as Super CDMS \cite{Sander:2012nia} (which has recently released its first results on low mass dark matter searches \cite{Agnese:2014aze,Agnese:2013jaa}), XENON1T \cite{Aprile:2012zx}, and DARWIN \cite{Baudis:2012bc} are expected to push the limits of direct detection orders of magnitude below the current levels.

In order to connect observations to microphysical models one needs a general framework within which to interpret the observations of direct detection experiments.  For quite some time the prevailing method of analyzing dark matter-nucleus interactions has been to assume that dark matter is a weakly interacting massive particle (WIMP), and then to categorize the interactions as  elastic and isospin conserving and either spin-independent or spin-dependent \cite{Pato:2010zk,Newstead:2013pea}.  For some well studied models of dark matter, such as the weakly interacting Majorana neutralino found in supersymmetry models, this assumption is reasonable.

With an absence of observed dark matter signals, there has of late been a surge in interest in exploring more general types of interactions between dark matter and nuclei.  Generalizations include inelastic and momentum dependent interactions, which may arise due to additional structure in the dark sector including excited dark matter states, or dark gauge bosons giving rise to electric and magnetic form factors~\cite{TuckerSmith:2001hy,Chang:2008gd,Feldstein:2009tr,Chang:2009yt,Barger:2010gv,Banks:2010eh,Foot:2010hu,Ho:2012bg}.

The formalism of choice for many of these investigations is relativistic effective field theory, which provides a model independent framework to analyse dark matter-SM interactions~\cite{Goodman:2010ku,Bai:2010hh,Balazs:2014rsa}. It has been shown that these effective theories break down when applied to high-momentum transfer experiments, such as the LHC~\cite{Busoni:2013lha}. Therefore analyses moved beyond this framework and have moved to what are labeled as `simplified models' instead~\cite{Alves:2011wf,Rajaraman:2011wf,Buchmueller:2013dya}. Simplified models are field theories which extend the SM by a single dark matter particle and a single mediator particle which allows the WIMP to communicate with quarks and/or leptons. The newly added dark matter content is assumed to be a singlet under the SM gauge groups (we will consider some cases where the particles mediating the interaction have SM charge). In this context it is then possible to calculate collider amplitudes valid at the high energies of interest in such experiments. Given this simple dark sector, one can write down an exhaustive list of every combination of WIMP and mediator spins, and all possible tree level interactions. These simplified models have now gained popularity for analyzing indirect detection signals~\cite{Berlin:2014tja,Balazs:2014jla}, allowing connections to be made with the growing body of literature which make use of them. \\

Another step towards placing dark matter-nucleus interactions on a general footing has been accomplished recently by utilizing a non-relativistic effective field theory (EFT) approach \cite{Fan:2010gt,Fitzpatrick:2012ix,Anand:2013yka,Anand:2014kea}.  Since the interactions in direct detection scenarios are assumed to take place due to an incoming dark matter particle with a typical velocity $\mathcal{O}(100$km/s), the recoil momenta in such an interaction will be $\mathcal{O}(\lesssim 100$keV).  The particle masses involved, including the nucleons of roughly GeV scale, the dark matter particles, which typically range from the GeV region to several orders of magnitude above, and mediators that can also be quite heavy compared to the typical interaction momenta, produce a situation where an EFT treatment is quite natural. \\

In order to circumvent as much model dependence as possible, one can construct general interactions which obey Galilean invariance, $T$-symmetry, and Hermiticity.  These operators will take the standard effective four-particle interaction form, reminiscent of Fermi's original model of weak interactions.   The non-relativistic interactions can be shown to be functions of only four parameters including the nucleon spin $S_N$, the dark matter spin $S_{\chi}$, the momentum transfer, $\vec{q}$, and a kinematic variable $\vec{v}^{\perp}$ which is a function of the relative incoming ($\vec{v}_{\chi,in} - \vec{v}_{N,in}$) and outgoing velocities $\vec{v}_{\chi,out} - \vec{v}_{N,out}$
\bea
\vec{v}^{\perp} = \frac{1}{2}\left(\vec{v}_{\chi, in} - \vec{v}_{N, in} + \vec{v}_{\chi, out} - \vec{v}_{N, out}\right) = \vec{v}_{\chi, in} - \vec{v}_{N, in} + \frac{\vec{q}}{2\mu_N}
\eea
which obeys $\vec{v}^{\perp}\cdot \vec{q} = 0$.  It was demonstrated in \cite{Fitzpatrick:2012ix} that there exist fifteen such non-relativistic interactions which arise from twenty possible bi-linear combinations of dark matter and nucleons.

The formalism developed in \cite{Fitzpatrick:2012ix} is unique in being the only analysis to comprehensively develop the nuclear physics of direct detection experiments.  From this general framework it is now apparent that there are interactions beyond the standard spin independent/dependent type.  The origins of these `new' interactions are not necessarily exotic and it has been shown, in the context of relativistic EFT, how many of them can be generated \cite{Gresham:2014vja}.

What has been lacking to date however, is a completely general and comprehensive treatment that connects high energy microphysics with low-energy effective nuclear matrix elements in a model independent way.  It is possible, for example, that the various interactions listed in \cite{Fitzpatrick:2012ix} can give rise to degeneracies where different fundamental dark matter Lagrangians, describing dark matter and interaction mediators of various spins, can produce the same interaction types.  This will obviously pose problems for attempts to discern the properties of dark matter when interpreting the results of experimental data.  Furthermore, dark matter may not be spin-$\half$, which creates a need for extending the parametric framework from the four descriptors listed above.  In particular, as we shall show, this allows the existence of new non-relativistic operators to appear in the low energy effective theory.

Motivated by the above we present here a general analysis covering a broad spectrum of particle and interaction types, starting from the microphysics, which will enable one to link experiment with fundamental theory while incorporating the new nuclear responses described in~\cite{Fitzpatrick:2012ix}.

In this work we build upon the NR-EFT description by examining simplified models which incorporate the most general renormalizable Lagrangians for scalar, spinor, and vector dark matter interacting with nucleons via scalar, spinor, and vector mediators, consistent with Lorentz invariance and hermiticity while imposing stability of the dark matter candidates. We integrate out the heavy mediator and obtain effective relativistic interaction Lagrangians. Next, we take the non-relativistic limit of these Lagrangians, and identify them with the NR operators from~\cite{Fitzpatrick:2012ix}, which are reproduced below, in Table $1$. Using these, we identify which electroweak nuclear responses are excited by a given fundamental interaction model and determine the relative importance of various models within the context of direct detection experiments consisting of xenon and germanium targets by exploring the relative magnitude of coefficients of these operators, and also their energy dependence.

The paper is organized as follows; in section~\ref{secEFT} the EFT formalism of~\cite{Fitzpatrick:2012ix} is summarized, in section~\ref{secSimple} we build the generalized relativistic Lagrangians and in section~\ref{secNRR} we outline the signatures and distinguishability of these models in the context of direct detection experiments, providing a framework for both experimentalists and theorists to base their future analyses.

\section{Effective field theory of direct detection}
\label{secEFT}
Conventionally, coherent WIMP-nucleus scattering has been considered to come from two types of interactions; spin-independent (SI) and spin-dependent (SD). SI interactions couple to the charge/mass of the nucleus while SD couples to the spin. The nuclear cross section is generally written in terms of the nucleon cross section at zero momentum transfer, $\sigma_0$, and a form factor, $F(q)$, to take into account the loss of coherence over the finite size of the nucleus,
\be
\frac{d\sigma}{dE_r} = \frac{M}{2 \pi \mu_{\chi M} v^2}\left(\sigma_0^{SI} F_{SI}^2(q) + \sigma_0^{SD} F_{SD}^2(q)\right).
\ee
where $M$ is the mass of the target nucleus and $\mu_{\chi M}$ is the WIMP-nucleus reduced mass.
This picture has recently been shown to be incomplete, as it is also possible for the WIMP to couple to the nucleus through additional nuclear responses \cite{Fitzpatrick:2012ix}. Working in the language of a non-relativistic (NR) effective field theory Fitzpatrick et al.~identified 15 operators to characterize the ways in which a WIMP can couple to the various nuclear responses. These operators are constructed from combinations of non-relativistic vectors which respect Galilean invariance, $T$ symmetry and which are Hermitian.  We list them in table~\ref{tabHaxtonOp}. The Hermitian vectors are:
\bea
i\frac{\vec{q}}{m_N},\ \vec{v}^{\perp}=\vec{v}+\frac{\vec{q}}{2\mu_N}, \ \vec{S}_\chi, \ \vec{S}_N,
\eea
where $\vec{q}=\vec{p'}-\vec{p}=\vec{k}-\vec{k'}$ is the momentum transfer, $\vec{v}$ is the velocity of WIMP with respect to the nucleus of the detector, $\mu_N$ is the reduced mass of the system and $\vec{S}_\chi$ and $\vec{S}_N$ are the WIMP and nuclear spins respectively. Throughout the paper, we denote by $\vec{p}$ and $\vec{p'}$ the incoming and outgoing WIMP momenta and by $\vec{k}$ and $\vec{k'}$ the incoming and outgoing nuclear momenta respectively. Energy-momentum conservation implies the orthogonality condition $\vec{q}\cdot\vec{v}^\perp=0$.  Here we will briefly outline the procedure employed in \cite{Fitzpatrick:2012ix} in going from the NR operators to the final differential WIMP-nucleus cross section. \\

\begin{table}[ht]
\caption{List of NR effective operators described in~\cite{Fitzpatrick:2012ix}}
\begin{tabular}{rc}
$\mathcal{O}_1$ & $1_\chi 1_N$ \\
$\mathcal{O}_2$ & $(\vec{v}^{\perp})^2$ \\
$\mathcal{O}_3$ & $i\vec{S}_N\cdot(\frac{\vec{q}}{m_N}\times \vec{v}^{\perp})$ \\
$\mathcal{O}_4$ & $\vec{S}_\chi \cdot \vec{S}_N$ \\
$\mathcal{O}_5$ & $i\vec{S}_\chi\cdot(\frac{\vec{q}}{m_N}\times \vec{v}^{\perp})$ \\
$\mathcal{O}_6$ & $(\frac{\vec{q}}{m_N}\cdot\vec{S}_N)(\frac{\vec{q}}{m_N}\cdot\vec{S}_\chi)$ \\
$\mathcal{O}_7$ & $\vec{S}_N \cdot \vec{v}^{\perp}$ \\
$\mathcal{O}_8$ & $\vec{S}_\chi \cdot \vec{v}^{\perp}$ \\
$\mathcal{O}_9$ & $i\vec{S}_\chi \cdot (\vec{S}_N \times \frac{\vec{q}}{m_N}) $ \\
$\mathcal{O}_{10}$ & $i\frac{\vec{q}}{m_N} \cdot \vec{S}_N$ \\
$\mathcal{O}_{11}$ & $i\frac{\vec{q}}{m_N} \cdot \vec{S}_\chi$ \\
$\mathcal{O}_{12}$ & $\vec{S}_\chi \cdot(\vec{S}_N \times \vec{v}^{\perp})$ \\
$\mathcal{O}_{13}$ & $i(\vec{S}_\chi \cdot \vec{v}^{\perp})(\frac{\vec{q}}{m_N} \cdot \vec{S}_N)$ \\
$\mathcal{O}_{14}$ & $i(\vec{S}_N \cdot \vec{v}^{\perp})(\frac{\vec{q}}{m_N} \cdot \vec{S}_\chi)$ \\
$\mathcal{O}_{15}$ & $-(\vec{S}_\chi \cdot \frac{\vec{q}}{m_N})\left( (\vec{S}_N\times \vec{v}^{\perp})\cdot\frac{\vec{q}}{m_N}\right)$ \\
\end{tabular}
\label{tabHaxtonOp}
\end{table}

In general one can write down the non-relativistic interaction Lagrangian as
\bea
\mathcal{L}_{NR}=\sum_{\alpha=n,p}\sum_{i=1}^{15}c_i^\alpha\mathcal{O}_{i}^\alpha,
\eea
where the coefficients $c_{i}^\alpha$ are given by the microphysics of the interaction and in general one could allow for isospin violation by having different couplings to neutron and proton inside the nucleus. This can be rewritten in $2$-component isospin space as
\bea
\mathcal{L}_{NR}=\sum_{\tau=0,1}\sum_{i=1}^{15}c_{i}^{\tau}\mathcal{O}_it^\tau
\eea
where $t^0$ and $t^1$ are the identity matrix and the Pauli matrix $\sigma^3$ respectively. The nucleus is composed of nucleons, and these can individually interact with the WIMP. This is incorporated by considering the operator $\mathcal{O}(j)$ as an interaction between a single nucleon, $j$, and the WIMP, and then summing over the nucleons.
\bea
\label{eqnNRL}
\sum_{\tau=0,1}\sum_{i=1}^{15}c_{i}^{\tau}\mathcal{O}_it^\tau\rightarrow \sum_{\tau=0,1}\sum_{i=1}^{15}c_{i}^{\tau}\sum_{j=1}^A\mathcal{O}_i(j)t^\tau(j)
\eea
where $A$ is the atomic mass number given by the total number of neutrons and protons.  One can do the same reduction with $\vec{v}^{\perp}$,
\bea
\vec{v}^{\perp}&\rightarrow& \{\vec{v}_{\chi}-\vec{v}_N(i), i=1,...,A\}\nonumber\\
&\equiv& \vec{v}_{T}^{\perp}-\{\dot{\vec{v}}_N(i),i=1,...,A-1\}
\eea
where $\vec{v}_{\chi}$ and $\vec{v}_N(i)$ are the symmetrized combination of incoming and outgoing velocities for the WIMP and nucleons respectively. $\vec{v}_T^{\perp}$ (here $T$ stands for target, i.e., the nuclear center-of-mass) is defined as
\bea
\vec{v}_T^{\perp}=\vec{v}_{\chi}-\frac{1}{2A}\sum_{i=1}^A\left[\vec{v}_{N,in}(i)+\vec{v}_{N,out}(i)\right]
\eea
This allows for a decomposition of the nucleon velocities into internal velocities $\dot{\vec{v}}_N(i)$ that act only on intrinsic nuclear coordinates and `in' and `out' velocities that evolve as a WIMP scatters off the detector. As an example, the dot product between $\vec{v}_N^\perp$ and $\vec{S}_{N}$ can be rewritten as
\bea
\vec{v}^{\perp}\cdot\vec{S}_N&\rightarrow&\sum_{i=1}^A\frac{1}{2}\left[\vec{v}_{\chi,in}+\vec{v}_{\chi,out}-\vec{v}_{N,in}(i)-\vec{v}_{N,out}(i)\right]\cdot\vec{S}_{N}(i)\\
&=&\vec{v}^\perp_T\cdot\sum_{i=1}^A\vec{S}_{N}(i)-\left\{\sum_{i=1}^A\frac{1}{2}\left[\vec{v}_{N,in}(i)+\vec{v}_{N,out}(i)\right]\cdot\vec{S}_{N}(i)\right\}_{int}
\eea
The second term in the curly brackets is internal to the nucleus and acts as an operator on the `in' and `out' nucleon states. $\vec{v}_{N,in}$ can be replaced by $\vec{p}_{N,in}/M$ acting on the incoming state, which can in turn be replaced by $i\overleftarrow{\nabla}/M$, and similarly $\vec{p}_{N,out}/M$ by $-i\overrightarrow{\nabla}/M$ on the outgoing nuclear state. Finally, since the nucleus is non-zero in size and individual nucleons locally interact with the WIMP, nuclear operators built from $\mathcal{O}_i$ are accompanied by an additional spatial operator $e^{-i\vec{q}\cdot\vec{x}(i)}$ where $x(i)$ is the location of the $i^{th}$ nucleon inside the nucleus.\\

Starting from Eqn.~\ref{eqnNRL} and using the substitution rules for $\vec{v}^\perp$ and including a factor of $e^{-i\vec{q}\cdot\vec{x}_i}$, the interaction Lagrangian can be written as a sum of five distinct terms (nuclear electroweak operators) that only act on internal nucleon states. Their coefficients, on the other hand, act on WIMP `in' and `out' states. The WIMP-nucleus interaction can then be written as
\bea
\sum_{\tau=0,1}\left\{l_0^\tau S+l_{0}^{A\tau} T+\vec{l}^\tau_5\cdot\vec{P}+\vec{l}_M^\tau\cdot{Q}+\vec{l}_{E}^\tau\cdot\vec{R}\right\}t^\tau(i)
\eea
where
\bea
S&=&\sum_{i=1}^Ae^{-i\vec{q}\cdot\vec{x}_i}\nonumber\\
T&=&\sum_{i=1}^A\frac{1}{2M}\left[-\frac{1}{i}\overleftarrow{\nabla}_i\cdot\vec{\sigma}(i)e^{-i\vec{q}\cdot\vec{x}_i}+e^{-i\vec{q}\cdot\vec{x}_i}\vec{\sigma}(i)\cdot\frac{1}{i}\overrightarrow{\nabla}_i\right]\nonumber\\
\vec{P}&=&\sum_{i=1}^A\vec{\sigma}(i)e^{-i\vec{q}\cdot\vec{x}_i}\nonumber\\
\vec{Q}&=&\sum_{i=1}^A\frac{1}{2M}\left[-\frac{1}{i}\overleftarrow{\nabla}_ie^{-i\vec{q}\cdot\vec{x}_i}+e^{-i\vec{q}\cdot\vec{x}_i}\frac{1}{i}\overrightarrow{\nabla}_i\right]\nonumber\\
\vec{R}&=&\sum_{i=1}^A\frac{1}{2M}\left[\overleftarrow{\nabla}_i\times\vec{\sigma}(i)e^{-i\vec{q}\cdot\vec{x}_i}+e^{-i\vec{q}\cdot\vec{x}_i}\vec{\sigma}(i)\times\overrightarrow{\nabla}_i\right]
\eea
and
\bea\label{eqnells}
l_0^\tau&=&c_1^\tau+ic_5^\tau\vec{S}_\chi\cdot\left(\frac{\vec{q}}{m_N}\times\vec{v}^\perp_T\right)+c_8^\tau(\vec{S}_\chi\cdot\vec{v}_T^\perp)+ic_{11}^\tau\frac{\vec{q}\cdot\vec{S}_\chi}{m_N}\nonumber\\
l_0^{A\tau}&=&-\frac{1}{2}\left[c_7^\tau+ic_{14}^\tau\left(\vec{S}_\chi\cdot\frac{\vec{q}}{m_N}\right)\right]\nonumber\\
\vec{l}_5&=&\frac{1}{2}\left[c_3^\tau{i}\frac{\left({\vec{q}}\times\vec{v}^\perp_T\right)}{m_N}+c_4^\tau\vec{S}_{\chi}+c_6^\tau\frac{(\vec{q}\cdot\vec{S}_\chi)\vec{q}}{m_N^2}+c_7^\tau\vec{v}_T^\perp+ic_9^\tau\frac{(\vec{q}\times\vec{S}_\chi)}{m_N}+ic_{10}^\tau\frac{\vec{q}}{m_N}\right)\nonumber\\
&&\left. c_{12}^\tau(\vec{v}_T^\perp\times\vec{S}_{\chi})+ ic_{13}^\tau\frac{(S_{\chi}\cdot\vec{v}^{\perp}_T)\vec{q}}{m_N}+ic_{14}^\tau\left(\vec{S}_\chi\cdot\frac{\vec{q}}{m_N}\right)\vec{v}^\perp_{T}+c_{15}^\tau\frac{(\vec{q}\cdot\vec{S}_\chi)(\vec{q}\times\vec{v}^\perp_T)}{m_N^2}\right]\nonumber\\
\vec{l}_M&=&c_5^\tau\left(i\frac{\vec{q}}{m_N}\times\vec{S}_\chi\right)-\vec{S}_\chi c_8^\tau \nonumber\\
\vec{l}_E&=&\frac{1}{2}\left[c_3^\tau\frac{{\vec{q}}}{m_N}+ic^{\tau}_{12}\vec{S}_{\chi}-c_{13}^\tau\frac{(\vec{q}\times\vec{S}_\chi)}{m_N}-ic_{15}^\tau\frac{(\vec{q}\cdot\vec{S}_\chi)\vec{q}}{m_N^2}\right]\label{eqnElls}\\ \nonumber
\eea
The WIMP-nucleus amplitude, $\mathcal{M}$, can then be succinctly written as
\bea
\mathcal{M}=\sum_{\tau=0,1}\langle j_\chi,M_\chi;j_N,M_N|\left\{l_0^\tau S+l_{0}^{A\tau} T+\vec{l}^\tau_5\cdot\vec{P}+\vec{l}_M^\tau\cdot{Q}+\vec{l}_{E}^\tau\cdot\vec{R}\right\}t^\tau(i)|j_\chi,M_\chi;j_N,M_N\rangle.\nonumber\\
\label{eqnAmp}
\eea
By using spherical decomposition, the internal nuclear operators $S, T, P, Q$ and $R$ can be further rewritten in terms of standard nuclear electroweak responses as follows:
\bea
\mathcal{M}&=&\sum_{\tau=0,1}\langle j_\chi,M_{\chi f};j_N,M_{Nf}|\left(\sum_{J=0}\sqrt{4\pi(2J+1)}(-i)^J\left[l_0^\tau M_{J0;\tau}-il_{0}^{A\tau}\frac{q}{m_N}\tilde{\Omega}_{J0;\tau}(q)\right]\right.\\
& &\left.+\sum_{J=1}\sqrt{2\pi(2J+1)}(-i)^J\sum_{\lambda\pm1}(-1)^\lambda\left\{l_{5\lambda}^\tau[\lambda\Sigma_{J-\lambda;\tau}(q)+i\Sigma^{'}_{J-\lambda;\tau}(q)]\right.\right.\nonumber\\
& &\left.\left.-i\frac{q}{m_N}l^\tau_{M\lambda}[\lambda\Delta_{J-\lambda;\tau}(q)]-i\frac{q}{m_N}l^\tau_{E\lambda}[\lambda\tilde{\Phi}_{J-\lambda;\tau}(q)+i\tilde{\Phi}^{'}_{J-\lambda;\tau}(q)]\right\}\right.\nonumber\\
& &\left.+\sum_{J=0}^\infty\sqrt{4\pi(2J+1)}(-i)^J\left[il^{\tau}_{50}\Sigma^{''}_{J0;\tau}(q)+\frac{q}{m_N}l_{M0}^{\tau}\tilde{\Delta}^{''}_{J0;\tau}(q)+\frac{q}{m_N}l_{E0}^{\tau}\tilde{\Phi}^{''}_{J0;\tau}(q)\right]\right)|j_\chi,M_{\chi i};j_N,M_{Ni}\rangle\nonumber
\eea
Where there is an implicit sum over the nucleons,
\bea
\mathcal{O}_{JM;\tau}(q)\equiv\sum_{i=1}^A\mathcal{O}_{JM}(q\vec{x}_i)t^\tau(i),
\eea
and the various electroweak responses are defined as
\bea
M_{JM}(q\vec{x})&\equiv& j_J(qx)Y_{JM}(\Omega_x)\nonumber\\
\vec{M}^M_{JL}&\equiv& j_J(qx)\vec{Y}_{JLM}(\Omega_x)\nonumber\\
\Delta_{JM}&\equiv&\vec{M}^M_{JJ}(qx_i)\cdot\frac{1}{q}\vec{\nabla}_i\nonumber\\
\Sigma^{'}_{JM}&\equiv& -i\left\{\frac{1}{q}\vec{\nabla}_i\times\vec{M}^M_{JJ}(q\vec{x}_i)\right\}\cdot\vec{\sigma}(i)\nonumber\\
\Sigma^{''}_{JM}&\equiv& \left\{\frac{1}{q}\vec{\nabla}_i{M}_{JM}(q\vec{x}_i)\right\}\cdot\vec{\sigma}(i)\nonumber\\
\tilde{\Phi}^{'}_{JM}&\equiv& \left[\frac{1}{q}\vec{\nabla}_i\times\vec{M}^M_{JJ}(q\vec{x}_i)\right]\cdot\left[\vec{\sigma}(i)\times\frac{1}{q}\vec{\nabla}_i\right]+\frac{1}{2}\vec{M}^M_{JJ}(q\vec{x}_i)\cdot\vec{\sigma}(i)\nonumber\\
\Phi^{''}_{JM}&\equiv& i\left[\frac{1}{q}\vec{\nabla}_i{M}_{JM}(q\vec{x}_i)\right]\cdot\left[\vec{\sigma}(i)\times\frac{1}{q}\vec{\nabla}_i\right]\nonumber\\
\Sigma_{JM}&\equiv&\vec{M}_{JJ}^M(q\vec{x}_i)\cdot\vec{\sigma}(i)\nonumber\\
\tilde{\Omega}_{JM}&\equiv&\Omega_{JM}(q\vec{x}_i)+\frac{1}{2}\Sigma^{''}_{JM}(q\vec{x}_i)\nonumber\\
\tilde{\Phi}_{JM}&\equiv&\Phi_{JM}(qx_i)-\frac{1}{2}\Sigma^{'}_{JM}(qx_i)\nonumber\\
\tilde{\Delta}^{''}_{JM}&\equiv&\Delta^{''}_{JM}(qx_i)-\frac{1}{2}M_{JM}(qx_i)\\ \nonumber
\eea
where $Y_{JM}$ and $\vec{Y}_{JLM}$ are spherical harmonics and vector spherical harmonics respectively. We are only considering elastic transitions, and assuming parity and CP as symmetries of the nuclear ground state. This eliminates some of the responses, and only $M, \Phi^{''}, \Sigma^{'}, \Delta, \Sigma^{''}, \tilde{\Phi}^{'}$ survive. To calculate cross-sections, one needs to square the amplitude, average over initial spins and sum over final spins. The matrix element squared for the nuclear portion of the amplitude has been made available by Fitzpatrick et al.~\cite{Fitzpatrick:2012ix}, and codes have been supplied to calculate the full amplitude and rate~\cite{Anand:2013yka}.

As we shall describe, in the following analysis we discovered that two additional NR operators are required to fully describe the scattering of spin-1 WIMPs off nuclei,
\bea
\mathcal{O}_{17}\equiv i\frac{\vec{q}}{m_N}\cdot\mathcal{S}\cdot\vec{v}_{\perp},\nonumber\\ \mathcal{O}_{18}\equiv i\frac{\vec{q}}{m_N}\cdot\mathcal{S}\cdot\vec{S}_{N}, \\ \nonumber
\eea
where $\mathcal{S}$ is the symmetric combination of polarization vectors. Appendix~\ref{appVec} contains the details required to include these new operators in the above formalism.

\section{Simplified Models for Direct Detection}
\label{secSimple}
From a model building perspective, one would like to know how relevant the novel nuclear responses are in interpreting direct detection data. Previous work \cite{Gresham:2014vja} demonstrated that using only the SI/SD form factors (even with additional momentum dependence taken into account) can lead one to infer wildly incorrect values of the WIMP mass and cross sections.

Here we go further by starting with simplified models at the Lagrangian level, where `simplified model' means a single WIMP with a single mediator coupling it to the quark sector. This is useful for two reasons; it allows us to better explore which NR operators arise from a broad set of UV complete theories, and also make connection with the growing body of literature which use simplified models for indirect detection and collider searches.

When it comes to interpreting signals, knowing comprehensively how different interactions with different nuclei arise from different UV complete models will allow us to identify degeneracies between competing models. Further, it can also help optimize target selection for maximum discrimination of the UV model parameter space.

In building these simplified models we remain agnostic about the WIMP's spin, and consider dark matter spins of $0, \half$ and $1$. We do however only consider renormalizable interactions between quarks and WIMPs. To ensure a stable WIMP, we assume that the WIMP is either charged under some internal gauge group or a discrete symmetry group (for example $Z_2$). However, we assume that this gauge charge is not shared by quarks. We will couple the WIMP to the quarks via a heavy mediator in two distinct ways: charged and uncharged mediators, each with all possible spins consistent with angular momentum conservation. The mediator mass is chosen to be the heaviest scale in the problem (and certainly much greater than the momentum exchange which characterizes the scattering process) so that we can integrate it out (see appendix \ref{appNRR} for details). This leads to relativistic effective WIMP-nucleon interactions, whose NR limit can then be examined. In the uncharged mediator case we will consider mediators that are neutral under all SM and WIMP gauge charges, while in the charged case, the mediator must have both WIMP and SM gauge charges. Given the above as a guide, our Lagrangian construction is then constrained only by gauge invariance, Lorentz invariance, renormalizability and hermiticity. In certain cases which follow, the requirement of hermiticity demands coupling constants be complex. Unless explicitly noted, the coupling constants are dimensionless and can be assumed to be real.

\vskip 0.2in

{\bf A. Uncharged-mediator Lagrangians}

{\it 1. Scalar Dark Matter}

We begin with a spin-$0$ scalar WIMP, $S$, which has some internal charge to ensure stability, and $S^\dagger$ is its Hermitian conjugate. To have renormalizable interactions, the neutral mediator can only be a scalar or a vector. We denote the scalar mediator by $\phi$ and the vector mediator by $G^{\mu}$ with field strength tensor $\mathcal{G}_{\mu\nu}$.\\

The most general renormailzable Lagrangian for scalar mediation consistent with the above assumptions is given by
\bea
\mathcal{L}_{S\phi q} &=& \partial_\mu S^\dag\partial^\mu S - m_S^2S^\dag S - \frac{\lambda_S}{2}(S^\dagger S)^2 \nonumber\\
&&+\frac{1}{2}\partial_\mu\phi\partial^\mu\phi - \frac{1}{2}m_\phi^2\phi^2 -\frac{m_\phi\mu_1}{3}\phi^3-\frac{\mu_2}{4}\phi^4 \nonumber\\
&&+ i\bar{q}D\!\!\!\!/\, q - m_q \bar{q} q \nonumber\\
&&-g_1m_SS^\dag S\phi -\frac{g_2}{2}S^\dag S\phi^2-h_1\bar{q} q\phi-ih_2\bar{q}\gamma^5q\phi, \\ \nonumber
\eea
where we have suppressed all the SM quark interactions. Similarly, the Lagrangian for vector mediation (up to gauge fixing terms) is
\bea
\mathcal{L}_{SGq} \aea \partial_{\mu}S^{\dag}\partial^{\mu}S -m_S^2 S^{\dag}S -\frac{\lambda_S}{2}(S^{\dag}{S})^2  \nonumber\\
&&-\frac{1}{4}\mathcal{G}_{\mu\nu}\mathcal{G}^{\mu\nu} + \frac{1}{2}m_G^2G_{\mu}G^{\mu} -\frac{\lambda_G}{4}(G_{\mu}G^{\mu})^2 \nonumber\\
&&+i\bar{q}\slashed{D}q -m_q\bar{q}q \nonumber\\
&&-\frac{g_3}{2}S^{\dag}SG_{\mu}G^{\mu} -ig_4(S^{\dag}\partial_{\mu}S-\partial_{\mu}S^{\dag}S)G^{\mu}\nonumber\\
&&-h_3(\bar{q}\gamma_{\mu}q	)G^{\mu}-h_4(\bar{q}\gamma_{\mu}\gamma^5q)G^{\mu}.\\ \nonumber
\eea

{\it 2. Spin-$\half$ Dark Matter}

If the WIMP has spin-$\half$ (denoted by $\chi$ below), then, as in the scalar WIMP case, mediation will only occur via scalar or vector mediators. The most general renormalizable interactions for the scalar ($\phi$) and vector mediator ($G_{\mu}$) cases respectively are given below,
\bea
\mathcal{L}_{\chi\phi q} &=& i\bar{\chi}\slashed{D}\chi - m_{\chi}\bar{\chi}\chi \nonumber\\
&&+\frac{1}{2}\partial_\mu\phi\partial^\mu\phi - \frac{1}{2}m_\phi^2\phi^2 -\frac{m_\phi\mu_1}{3}\phi^3-\frac{\mu_2}{4}\phi^4 \nonumber\\
&&+ i\bar{q}D\!\!\!\!/\, q - m_q \bar{q} q \nonumber\\
&&-\lambda_1\phi\bar{\chi}\chi -i\lambda_2\phi\bar{\chi}\gamma^{5}\chi-h_1\phi\bar{q} q-ih_2\phi\bar{q}\gamma^5q, \\ \nonumber
\eea
\bea
\mathcal{L}_{\chi Gq} &=& i\bar{\chi}\slashed{D}\chi - m_\chi\bar{\chi}\chi \nonumber\\
&&-\frac{1}{4}\mathcal{G}_{\mu\nu}\mathcal{G}^{\mu\nu}+\frac{1}{2}m_{G}^2G_{\mu}G^{\mu}\nonumber\\
&&+ i\bar{q}D\!\!\!\!/\, q - m_q \bar{q} q \nonumber\\
&&-\lambda_{3}\bar\chi\gamma^\mu\chi G_{\mu}-\lambda_{4}\bar\chi\gamma^\mu\gamma^5\chi G_{\mu}\nonumber\\
&&-h_3\bar{q}\gamma_{\mu}qG^{\mu}-h_4\bar{q}\gamma_{\mu}\gamma^{5}qG^{\mu}.\\ \nonumber
\eea

{\it 3. Spin-$1$ Dark Matter}

If the WIMP is a massive spin-$1$ particle, uncharged mediation to the quark sector can occur via a heavy scalar or a vector particle. For the case of vector mediation, there are many possible interactions because the Lorentz indices on the vectors afford a more diverse set of terms. The general interaction Lagrangian for the scalar mediation case is
\bea
\mathcal{L}_{X\phi q}&=&-\frac{1}{2}{\mathcal{X}}_{\mu\nu}^{\dag}\mathcal{X}^{\mu\nu}+m_{X}^2X_{\mu}^{\dag}X^{\mu}-\frac{\lambda_{X}}{2}(X_{\mu}^{\dag}X^{\mu})^2 \nonumber\\
&&+\frac{1}{2}(\partial_{\mu}\phi)^2-\frac{1}{2}m_{\phi}^2\phi^2-\frac{m_\phi \mu_1}{3}\phi^3-\frac{\mu_2}{4}\phi^4 \nonumber\\
&&+i\bar{q}\slashed{D}q-m_{q}\bar{q}q \nonumber\\
&&-b_1m_X\phi X_{\mu}^{\dag}X^{\mu}-\frac{b_{2}}{2}\phi^2X_{\mu}^{\dag}X^{\mu} -h_1\phi\bar{q}q-ih_2\phi\bar{q}\gamma^{5}q. \\ \nonumber
\eea
For the case of vector mediation, there are many possible interactions because the Lorentz indices on the vectors afford a more diverse set of terms. The Lagrangian is given by
\bea
\mathcal{L}_{XGq}&=& -\frac{1}{2}\mathcal{X}^{\dagger}_{\mu\nu}\mathcal{X}^{\mu\nu}+m_{X}^2X^{\dagger}_{\mu}X^{\mu}-\frac{\lambda_{X}}{2}(X_{\mu}^{\dagger}X^{\mu})^2 \nonumber\\
&&-\frac{1}{4}\mathcal{G}_{\mu\nu}\mathcal{G}^{\mu\nu}+\frac{1}{2}m_{G}^2G_\mu^2-\frac{\lambda_G}{4}(G_\mu G^\mu)^2 \nonumber\\
&&+i\bar{q}\slashed{D}q-m_{q}\bar{q}q\nonumber\\
&&-\frac{b_3}{2}G_\mu^2(X^{\dagger}_\nu X^{\nu}) -\frac{b_{4}}{2}(G^{\mu}G^{\nu})(X^{\dagger}_{\mu}X_{\nu})-\left[ib_{5}X_{\nu}^{\dagger}\partial_{\mu}X^{\nu}G^\mu\right.\nonumber\\
&&\left.+b_{6}X_{\mu}^{\dagger}\partial^\mu X_{\nu}G^{\nu} +b_{7}\epsilon_{\mu\nu\rho\sigma}(X^{\dag\mu}\partial^{\nu}X^{\rho})G^{\sigma} +h.c.\right]\nonumber\\
&&-h_3G_\mu\bar{q}\gamma^\mu q - h_4 G_\mu\bar{q}\gamma^\mu\gamma^{5}q \\ \nonumber
\eea
where, for the Lagrangian to be Hermitian, $b_6$ and $b_7$ are complex (this implies a new source of CP violation, which will not be considered further here).

\subsection{Charged-mediator Lagrangians}
Here we consider the simplest case of mediators that are charged under both the DM internal symmetry group and SM gauge groups. This is motivated by the absence of spin-$\half$ mediators ($s$-channel processes) in the previous section. Such a mediator, if neutral, is forbidden by simultaneous requirements of gauge invariance and renormalizability. Dark Matter models with mediators endowed with charges from both DM and SM side have been considered in the literature before \cite{Hisano:2010yh,Feng:2008ya}. The case of a spin-$\half$ mediator carrying $SU(3)_c$ is also motivated by studies of heavy quark models. This allows unique interactions as we show below.  In particular they necessitate a direct interaction between quarks and WIMPs at the level of the Lagrangian.

{\it 1. Scalar Dark Matter}

Scalar WIMPs with a charged scalar or vector mediator do not lead to any Lorentz invariant interactions. This is easy to see since both the scalars (or scalar and vector) and the quark are required in the (gauge invariant) interaction, but there is no way to contract the spinor indices consistently if the mediating particle is a scalar or vector. Therefore, the only possibility is that of a spin-$1/2$ mediator, $Q$, which acts like a heavy quark. The general renormalizable action is given by
\bea
\mathcal{L}_{SQq} \aea \partial_{\mu}S^{\dag}\partial^{\mu}S -m_S^2 S^{\dag}S -\lambda_S(S^{\dag}{S})^2 \nonumber\\
&&+i\bar{Q}\slashed{D}Q-m_Q\bar{Q}Q\nonumber\\
&&+i\bar{q}\slashed{D}q -m_q\bar{q}q\nonumber\\
&&-(y_1S\bar{Q}q+y_2S\bar{Q}\gamma^5q +h.c.),\\\nonumber
\eea
where $y_1$ and $y_2$ are again complex.

{\it 2. Spin-$\half$ Dark Matter}

For a spin-$1/2$ WIMP, both a charged scalar and charged vector mediator exchange can lead to novel interactions. The charged scalar is denoted by $\Phi$ and the charged vector by $V_\mu$
\bea
\mathcal{L}_{\chi\Phi q} \aea i\bar{\chi}\slashed{D}\chi-m_{\chi}\bar{\chi}\chi\nonumber\\
&&+(\partial_{\mu}\Phi^{\dagger})(\partial^{\mu}\Phi)-m_{\Phi}^2\Phi^{\dagger}\Phi-\frac{\lambda_\Phi}{2}(\Phi^{\dagger}\Phi)^2\nonumber\\
&&+i\bar{q}\slashed{D}q-m_{q}\bar{q}q \nonumber\\
&&-(l_{1}\Phi^{\dagger}\bar{\chi}q+l_{2}\Phi^{\dagger}\bar{\chi}\gamma^{5}q+h.c.),\\ \nonumber
\eea \vspace{-2cm}
\bea
\mathcal{L}_{\chi Vq} \aea i\bar{\chi}\slashed{D}\chi-m_{\chi}\bar{\chi}\chi\nonumber\\
&&-\frac{1}{2}\mathcal{V}_{\mu\nu}^{\dagger}\mathcal{V}^{\mu\nu}+m_{V}^2V^{\dagger}_{\mu}V^{\mu}\nonumber\\
&&+i\bar{q}\slashed{D}q-m_{q}\bar{q}q\nonumber\\
&&-(d_{1}\bar{\chi}\gamma^{\mu}qV^{\dagger}_{\mu}+d_{2}\bar{\chi}\gamma^{\mu}\gamma^{5}qV^{\dagger}_{\mu}+h.c.),\\ \nonumber
\eea
where $l_1, l_2, d_1$ and $d_2$ are complex.

{\it 3. Vector DM}

Here again we only have the case of a spin-$\half$ mediated interaction between vector DM and quarks (again scalar and vector charged mediators aren't possible because they don't lead to Lorentz invariant and renormalizable interactions). The general Lagrangian is given by
\bea
\mathcal{L}_{XQq} \aea-\frac{1}{2}\mathcal{X}^{\dagger}_{\mu\nu}\mathcal{X}^{\mu\nu}+m_{X}^2X^{\dagger}_{\mu}X^{\mu}-\frac{\lambda_{X}}{2}(X_{\mu}^{\dagger}X^{\mu})^2 \nonumber\\
&&+i\bar{Q}\slashed{D}Q-m_Q\bar{Q}Q\nonumber\\
&&+i\bar{q}\slashed{D}q-m_{q}\bar{q}q\nonumber\\
&&-(y_3X_{\mu}\bar{Q}\gamma^{\mu}q + y_4X_{\mu}\bar{Q}\gamma^{\mu}\gamma^5q+h.c.),\\\nonumber
\eea
where $y_3$ and $y_4$ are complex.

\section{Non-relativistic reduction of simplified models}
\label{secNRR}
After integrating out the heavy mediator we replace quark operators with nucleon operators (see appendix \ref{appQ2N}), take the non-relativistic limit (see appendix \ref{appNRR}), and match onto the operators given in table~\ref{tabHaxtonOp}. The results of this calculation are presented in terms of the $c_i$ coefficients from \cite{Anand:2013yka}, described in section \ref{secEFT}, facilitating a straightforward computation of amplitudes and rates. The $c_i$'s are given for each of the WIMP spins in tables \ref{tabCiScalar}, \ref{tabCiSpinor} and \ref{tabCiVector}. With this general framework in place we can now easily find the leading order NR operators for each distinct WIMP-nucleus interaction. One can imagine a series of minimal scenarios in which a combination of two Lagrangian couplings that give rise to a direct detection signal is non-zero with all others set to zero, and then proceeding in this manner for the entire set. Each of these scenarios is listed with its leading operators in table~\ref{tabLeading} and with all operators generated in table~\ref{tabFull}. Note that in the case of a complex coupling constant we consider purely real and purely imaginary values as separate cases since they produce a distinct set of operators.
\begin{table}[hbt]
\caption{Non-zero $c_i$ coefficients for a spin$-0$ WIMP}
\begin{tabular}{|c|c|c|}
\hline
 & Uncharged Mediator & Charged Mediator \\
\hline
 $c_{1}$ & $\frac{h_1^Ng_{1}}{m_{\phi}^2}$ & $\frac{y_1^\dag y_1-y_2^\dag y_2}{m_Q m_S}f_{T}^N$\\
$c_{10}$ & $\frac{-ih_2^Ng_{1}}{m_{\phi}^2} + \frac{2ig_4h_4^N}{m_G^2}\frac{m_N}{m_S} $ & $i\frac{y_2^\dag y_1 - y_1^\dag y_2}{m_Q m_S}\tilde\Delta^N$\\
\hline
\end{tabular}
\label{tabCiScalar}
\end{table}

\begin{table}[htb]
\caption{$c_i$ coefficients for a spin-$\half$ WIMP}
\begin{tabular}{|c|c|c|}
\hline
 & Uncharged Mediator & Charged Mediator \\
\hline
 $c_{1}$ & $\frac{h_1^N \lambda_1}{m_\phi^2}  - \frac{h_3^N\lambda_3}{m_G^2}$ & $\left(\frac{l_2^\dag l_2-l_1^\dag l_1}{4 m_\Phi^2}+\frac{d_2^\dag d_2-d_1^\dag d_1}{4m_V^2}\right)f_{T}^N+\left(-\frac{l_2^\dag l_2+l_1^\dag l_1}{4m_\Phi^2} +\frac{d_2^\dag d_2+d_1^\dag d_1}{8m_V^2}\right)\mathcal{N}^N$\\
 $c_{4}$ & $\frac{4h_4^N \lambda_4}{m_G^2} $ & $\frac{l_2^\dag l_2-l_1^\dag l_1}{m_\Phi^2}\delta^N-\left(\frac{l_1^\dag l_1+l_2^\dag l_2}{m_\Phi^2}+\frac{d_2^\dag d_2-d_1^\dag d_1}{2m_V^2}\right)\Delta^N$\\
 $c_{6}$ & $\frac{h_2^N \lambda_2m_N}{m_\phi^2m_\chi}$ & $(\frac{l_1^\dag l_1-l_2^\dag l_2}{4m_\Phi^2}+\frac{d_2^\dag d_2-d_1^\dag d_1}{4m_V^2})\frac{m_N}{m_\chi}\tilde{\Delta}^N$\\
 $c_{7}$ & $\frac{2h_4^N\lambda_3}{m_G^2} $& $(\frac{l_1^\dag l_2-l_2^\dag l_1}{2m_\Phi^2}+\frac{d_1^\dag d_2+d_2^\dag d_1}{4m_V^2})\Delta^N$\\
 $c_{8}$ & $-\frac{2h_3^N\lambda_4}{m_G^2} $& $(\frac{l_1^\dag l_2-l_2^\dag l_1}{2m_\Phi^2}-\frac{d_1^\dag d_2+d_2^\dag d_1}{4m_V^2})\mathcal{N}^N$\\
 $c_{9}$ & $-\frac{2h_4^N\lambda_3 m_N}{m_\chi m_G^2} -\frac{2h_3^N\lambda_4}{m_G^2} $ & $(\frac{l_1^\dag l_2-l_2^\dag l_1}{2m_\Phi^2}-\frac{d_1^\dag d_2+d_2^\dag d_1}{4m_V^2})\mathcal{N}^N-(\frac{l_1^\dag l_2-l_2^\dag l_1}{2m_\Phi^2}-\frac{d_1^\dag d_2+d_2^\dag d_1}{4m_V^2})\frac{m_N}{m_\chi}\Delta^N$\\
$c_{10}$ & $\frac{h_2^N \lambda_1}{m_\phi^2}  $& $i(\frac{l_1^\dag l_2-l_2^\dag l_1}{4 m_\Phi^2} + \frac{d_2^\dag d_1-d_1^\dag d_2}{4 m_V^2})\tilde\Delta^N-i\frac{l_1^\dag l_2-l_2^\dag l_1}{m_\Phi^2}\delta^N$\\
$c_{11}$ & $-\frac{h_1^N \lambda_2m_N}{m_\phi^2 m_\chi} $ & $i(\frac{l_2^\dag l_1-l_1^\dag l_2}{4 m_\Phi^2} + \frac{d_2^\dag d_1-d_1^\dag d_2}{4 m_V^2})\frac{m_N}{m_\chi}f_{T}^N+i\frac{l_1^\dag l_2-l_2^\dag l_1}{m_\Phi^2}\frac{m_N}{m_\chi}\delta^N$\\
$c_{12}$ & 0 & $\frac{l_2^\dag l_1-l_1^\dag l_2}{m_{\Phi}^2}\delta^N$\\
\hline
\end{tabular}
\label{tabCiSpinor}
\end{table}

\begin{table}[htb]
\caption{$c_i$ coefficients for a spin-$1$ WIMP}
\begin{tabular}{|c|c|c|}
\hline
 & Uncharged Mediator & Charged Mediator \\
\hline
$c_{1}$ & $\frac{b_1h_1^N}{m_\phi^2}$ & $\frac{y_3^\dag y_3-y_4^\dag y_4}{m_Q m_X}f^N_T$\\
$c_{4}$ & $\frac{4\mathrm{Im}(b_7)h_4^N}{m_G^2} + i\frac{q^2}{m_X^2}\frac{\mathrm{Re}(b_7)h_4^N}{m_G^2}-\frac{q^2}{m_Xm_N}\frac{\mathrm{Re}(b_6)h_3^N}{m_G^2}$ & $2\frac{y_3^\dag y_3-y_4^\dag y_4}{m_Q m_X}\delta^N$\\
$c_{5}$ & $\frac{\mathrm{Re}(b_6)h_3^N}{m_G^2}\frac{m_N}{m_X}$  & 0\\
$c_{6}$ & $\frac{\mathrm{Re}(b_6)h_3^N}{m_G^2}\frac{m_N}{m_X}-i\frac{\mathrm{Re}(b_7)h_4^N}{m_G^2}\frac{m_N^2}{m_X^2}$  & 0\\
$c_{8}$ & $ \frac{2\mathrm{Im}(b_7)h_3^N}{m_G^2}$ & 0\\
$c_{9}$ & $-\frac{2\mathrm{Re}(b_6)h_4^N}{m_G^2}\frac{m_N}{m_X}+\frac{2\mathrm{Im}(b_7)h_3^N}{m_G^2}$ & 0\\
$c_{10}$& $ \frac{b_1h_2^N}{m_\phi^2}-\frac{3b_5h_4^N}{m_G^2}\frac{m_N}{m_X}$&  $i\frac{y_4^\dag y_3-y_3^\dag y_4}{m_Q m_X}\tilde\Delta^N$\\
$c_{11}$ & $ \frac{\mathrm{Re}(b_7)h_3^N}{m_G^2}\frac{m_N}{m_X}$&  $i\frac{y_4^\dag y_3-y_3^\dag y_4}{m_Q m_X}\delta^N$\\
$c_{12}$ &0& $2i\frac{y_3^\dag y_4-y_4^\dag y_3}{m_Q m_X}\delta^N$\\
$c_{14}$ & $-\frac{2\mathrm{Re}(b_7)h^N_4}{m_G^2}\frac{m_N}{m_X}$& 0\\
$c_{17}$ & $-\frac{4\mathrm{Im}(b_6)h^N_3}{m_G^2}\frac{m_N}{m_X}$& 0\\
$c_{18}$ & $ \frac{4\mathrm{Im}(b_6)h^N_4}{m_G^2}\frac{m_N}{m_X}$& $-2i\frac{y_4^\dag y_3-y_3^\dag y_4}{m_Q m_X}\delta^N$\\
\hline
\end{tabular}
\label{tabCiVector}
\end{table}

As described earlier, we find that it is important to consider operators beyond those incorporated into the standard spin-independent and spin-dependent formalism, i.e.~simple models exist in which one would infer an incorrect rate in current experiments by not including these effects. Also importantly, not all of the NR operators are actually generated at leading order; for example, the operators $\mathcal{O}_2$, $\mathcal{O}_3$, $\mathcal{O}_{13}$ and $\mathcal{O}_{15}$ are missing at leading order.  Note that we only consider renormalizable Lagrangians, higher order non-renormalizable operators, which are presumably further suppressed. We have also not considered the case of kinetic mixing, which could be used to generate anapole interactions~\cite{Gresham:2014vja}, because the effective interaction doesn't arise from one mediator exchange.  \\

While spin independent interactions are a generic feature of direct couplings to quarks in our charged mediator cases, it is sometimes possbile to suppress them. In the scalar (and vector) WIMP with charged mediator cases, it is possible to suppress the spin independent interaction by ensuring that $|y_1|=|y_2|$($|y_3|=|y_4|$) while keeping their relative phases non-zero (or $\pi$). While these non-minimal scenarios require some fine tuning we include it for completeness and label them $y_1,y_2$ and $y_3,y_4$.\\

Aside from scalar WIMPs, each particular spin produces some non-relativistic operators that are unique to that spin. Also, importantly, the operators $\mathcal{O}_1$ and $\mathcal{O}_{10}$ are generic to all spins.  In five cases relativistic operators generate unique non-relativistic operators at leading order.  Therefore distinguishing WIMP scenarios in these cases reduces to experimentally discerning between these operators (see also \cite{catena:2014}).  Given the likely low statistics of any detection in upcoming direct detection experiments, sub-leading operators are not likely to contribute enough to provide any further discriminating power. \\

\begin{table}[htb]
\renewcommand{\arraystretch}{0.9}
\caption{Leading order operators which can arise from the relativistic Lagrangians considered in this work, the column `$\mathcal{L}$ terms' gives the non-zero couplings for that scenario. Each row represents a possible leading order direct detection signal. A `$\dag$' indicates that the mediator is charged. The 'Eqv.~$M_m$' column gives the mediator mass required for each scenario to produce $\sim$10 events $t^{-1}yr^{-1}keV^{-1}$ in xenon, with couplings set to 0.1. }
\begin{tabular}{|c|c|c|c|r|}
\hline
WIMP spin & Mediator spin & $\mathcal{L}$ terms& leading NR operator & Eqv.~$M_m$ \\
\hline
0        & 0 			& 	$h_1,g_1$      & $\mathcal{O}_1$	 & 13 TeV\\
0  	   & 0 			& 	$h_2,g_1$      & $\mathcal{O}_{10}$ & 14 GeV\\
0		   & 1 			& 	$h_4,g_4$      & $\mathcal{O}_{10}$ & 8 GeV\\
0 		   & $\half^\dag$ &	$y_1$          & $\mathcal{O}_{1}$  & 3.2 PeV\\
0 		   & $\half^\dag$ &	$y_2$          & $\mathcal{O}_{1}$  & 3.2 PeV\\
0 		   & $\half^\dag$ &	$y_1,y_2$      & $\mathcal{O}_{10}$ & 41 GeV\\
\hline
$\half$  & 0			& $h_1,\lambda_1$ & $\mathcal{O}_1$    & 12.7 TeV\\
$\half$  & 0			& $h_2,\lambda_1$ & $\mathcal{O}_{10}$ & 293 GeV\\
$\half$  & 0			& $h_1,\lambda_2$ & $\mathcal{O}_{11}$ & 14 GeV\\
$\half$  & 0			& $h_2,\lambda_2$ & $\mathcal{O}_6$    & 1.9 GeV\\
$\half$  & 1	   	& $h_3,\lambda_3$ & $\mathcal{O}_1$    & 6.3 TeV\\
$\half$  & 1		   & $h_4,\lambda_3$ & $\mathcal{O}_9$    & 6.4 GeV\\
$\half$  & 1		   & $h_3,\lambda_4$ & $\mathcal{O}_8$    & 180 GeV\\
$\half$  & 1		   & $h_4,\lambda_4$ & $\mathcal{O}_4$    & 135 GeV\\
$\half$	& $0^\dag$			& $l_1$		 		 & $\mathcal{O}_1$   & 7.1 TeV\\
$\half$	& $0^\dag$			& $l_2$		 		 & $\mathcal{O}_1$   & 5.5 TeV\\
$\half$	& $1^\dag$			& $d_1$		 		 & $\mathcal{O}_1$   & 5.9 TeV\\
$\half$	& $1^\dag$			& $d_2$		 		 & $\mathcal{O}_1$   & 6.7 TeV\\
\hline
1 			&  0			& $h_1,b_1$ 		& $\mathcal{O}_1$     & 13 TeV\\
1 			&  0		   & $h_2,b_1$ 		& $\mathcal{O}_{10}$  & 10 GeV\\
1 			&  1			& $h_4,b_5$ 		& $\mathcal{O}_{10}$  & 5.1 GeV\\
1 			&  1			& $h_3,b_6^\mathrm{Re}(b_6^\mathrm{Im})$ 		& $\mathcal{O}_{5}(\mathcal{O}_{17})$  & 5.5 GeV(23 GeV)\\
1 			&  1			& $h_4,b_6^\mathrm{Re}(b_6^\mathrm{Im})$ 		& $\mathcal{O}_{9}(\mathcal{O}_{18})$  & 3 GeV(4.6 GeV)\\
1 			&  1			& $h_3,b_7^\mathrm{Re}(b_7^\mathrm{Im})$ 		& $\mathcal{O}_{11}(\mathcal{O}_{8})$  & 186 GeV(228 GeV)\\
1 			&  1			& $h_4,b_7^\mathrm{Re}(b_7^\mathrm{Im})$ 		& $\mathcal{O}_{4}(\mathcal{O}_{4})$     & 78 MeV (172 GeV)\\
1		   &$\half^\dag$ & $y_3$		 		& $\mathcal{O}_1$     & 3.2 PeV\\
1  		&$\half^\dag$ & $y_4$		 		& $\mathcal{O}_1$     & 3.2 PeV\\
1  		&$\half^\dag$ & $y_3,y_4$			& $\mathcal{O}_{11}$  & 120 TeV\\
\hline
\end{tabular}
\label{tabLeading}
\end{table}

\section{Observables}

The principle observable in direct detection experiments is the differential event rate. Since the incoming WIMPs originate in the galactic halo, one must average over the WIMP velocity distribution, $f(v)$, which we assume for the purposes of this paper to be Maxwell-Boltzmann,
\bea
\frac{dR}{dE_R} = N_T \frac{\rho_\chi M}{2\pi m_\chi}\int_{v_{min}}\frac{f(v)}{v} P_{tot} dv
\eea
where we use the value $\rho_\chi=0.3$GeV/cm$^3$ for the local dark matter density, $N_T$ is the number of nuclei in the target and $P_{tot}$ can be calculated from the amplitude $\mathcal{M}$ in Eq.~\ref{eqnAmp}
\bea
P_{tot} = \frac{1}{2j_\chi+1}\frac{1}{2j_N+1}\sum_{spins}|\mathcal{M}|^2.
\eea
Thoughout this work we use the mathematica package supplied in~\cite{Anand:2013yka} to calculate rates. To determine the leading order operator which arises from a given relativistic scenario we first plot the rate for each of the NR operators in xenon-131. To simply compare the operators we set the $c_i$ coefficients to be the same and normalized the overall rate to that of $\mathcal{O}_1$, see Fig.~\ref{figOperators}. Since operators are either zero, first or second order in momentum transfer $q$ or velocity $\vec{v}^\perp$, the relative strengths of the operators span 16 orders of magnitude. This is an important point to keep in mind when finding the leading operator, as sometimes a term which appears to be higher order in $q$ can dominate the non-relativistic reduction. For example in the $b_7^{\mathrm{Re}}h_4$ scenario, one finds that $q^2\mathcal{O}_4$ dominates over the $\mathcal{O}_6$ and $\mathcal{O}_{14}$ which contain powers of $q$ within the operators. \\

\begin{figure}
\includegraphics[height=4.6cm]{figures/operators.pdf}
\caption{The relative strength of event rates for a 50GeV spin-$\half$ WIMP in xenon for each of the non-relativistic operators in table \ref{tabHaxtonOp}, where the coefficients of each operator are set to be equal}
\label{figOperators}
\end{figure}

Since the Lagrangians we have considered are not tied to specific complete and consistent particle physics models,  the mediator masses are not fixed in advance and thus specific event rates are not predicted in advance.  Clearly one requires a rate that is low enough to evade the current experimental constraints.  For example, a 50 GeV WIMP producing 10 events per tonne per year is sufficiently low to evade the bounds from LUX~\cite{Akerib:2013tjd}. For demonstration purposes we set the couplings to 0.1 (or $0.1i$ for imaginary) in the various Lagrangians and find a mediator mass that will produce 10 events/t/y in the signal region for xenon ($5-45$keV). The calculated masses are given in table~\ref{tabLeading}.  It is perhaps telling that the mediator masses span 6 orders of magnitude, from just a few GeV up to a PeV.  While it is unlikely that a full model of thermal relic dark matter could be built around all of these Lagrangians, it is nevertheless a useful metric to estimate the relative strength of the different nuclear responses to each of the operators. \\

In Figs.~\ref{figScalar},~\ref{figSpinor},~\ref{figVector} and ~\ref{figVectorI} we have plotted rates for two common targets. For simplicity and again for demonstration purposes, we only plot the rates for a single isotope of both germanium and xenon. The choice of isotopes, $^{73}$Ge and $^{131}$Xe, was made to ensure sensitivity to spin-dependent responses. As can be seen in the figures, many operators produce rates with similar recoil energy dependence in the same target, but different nuclei can have very different responses to the various operators~\cite{Fitzpatrick:2012ix}. Thus a complementary choice of nuclear targets can provide important discriminating information.

To illustrate this discriminating power we plot the ratio of the rates in xenon and germanium in Fig.~\ref{figVectorI} and ~\ref{figRatios}. We choose to only present ratios for the uncharged mediator cases of spinor and vector WIMPs since the other cases produce trival results (all operators being spin independent). To estimate the effect astrophysical uncertainties will have on discriminating between operators, we plot the rate for a range of astrophysical parameters from $v_0=200$m/s, and $v_{esc}=500$m/s (lower) to $v_0=240$m/s and $v_{esc}=600$m/s (upper). The uncertainty in the dark matter density does not appear since we are considering the ratio of rates. Given the vastly different energy dependence of the ratio of rates of each scenario the astrophysical errors do not completely inhibit their identification.  Furthermore, operators $\mathcal{O}_9$ and $\mathcal{O}_{14}$, produced in scenarios $h_4 b_7^{\mathrm{Re}}$ and $h_4 b_6^{\mathrm{Re}}$ respectively, remain indistingushable when considering the ratio of rates. While it appears that in principle almost every operator is discernible, in practice isotopically impure targets and low statistics will further complicate the situation and provide limits on practical discrimination. \\

\begin{figure}[ht]
\mbox{
\includegraphics[width=8cm]{figures/scalarX_unchargedM.pdf}
\includegraphics[width=8cm]{figures/scalarX_chargedM.pdf}
}
\caption{Rates for a 50GeV spin-0 WIMP in xenon (solid) and germanium (dashed) with uncharged (left) and charged mediators (right), assuming mediator mass of 1TeV and $\mathcal{O}(1)$ coupling constants.}
\label{figScalar}
\end{figure}

\begin{figure}[ht]
\mbox{
\includegraphics[height=4.2cm]{figures/spinorX_unchargedM.pdf}
\includegraphics[height=4.2cm]{figures/spinorX_chargedM.pdf}
}
\caption{Rates for a 50GeV spin-$\half$ WIMP in xenon (solid) and germanium (dashed) with uncharged (left) and charged mediators (right), assuming mediator mass of 1TeV and $\mathcal{O}(1)$ coupling constants.}
\label{figSpinor}
\end{figure}

\begin{figure}[ht]
\mbox{
\includegraphics[height=4.2cm]{figures/vectorX_unchargedM.pdf}
\includegraphics[height=4.2cm]{figures/vectorX_chargedM.pdf}
}
\caption{Rates for a 50GeV spin-1 WIMP in xenon (solid) and germanium (dashed) with uncharged (left) and charged mediators (right), assuming mediator mass of 1TeV and $\mathcal{O}(1)$ coupling constants.}
\label{figVector}
\end{figure}

\begin{figure}[ht]
\mbox{
\includegraphics[width=8cm]{figures/vectorX_unchargedMI.pdf}
\includegraphics[width=8cm]{figures/vectorX_unchargedM_astroI.pdf}
}
\caption{Rates (left) for a 50GeV spin-1 WIMP in xenon (solid) and germanium (dashed) with uncharged mediators and imaginary couplings, assuming mediator mass of 1TeV and $\mathcal{O}(1)$ coupling constants. Also shown is the ratio of rates in xenon and germanium (right).}
\label{figVectorI}
\end{figure}

\begin{figure}[ht]
\mbox{
\includegraphics[width=8cm]{figures/spinorX_unchargedM_astro.pdf}
\includegraphics[width=8cm]{figures/vectorX_unchargedM_astro.pdf}
}
\caption{Ratio of rates in xenon and germanium, illustrating the discriminating power of having multiple nuclear targets. For a 50GeV spin-$\half$ WIMP with uncharged mediator (left) and a 50GeV spin-1 WIMP with uncharged mediator (right), the shaded regions show the upper and lower bounds due to the astrophysical parameters}
\label{figRatios}
\end{figure}

\section{Conclusion}

The analysis we have given here builds on previous analyses to provide, in generality, a roadmap to use event rates in direct dark matter detectors to constrain fundamental dark matter models. We have outlined the steps needed to go from fundamental Lagrangians, first to relativistic operators, then to non-relativistic operators, and finally to produce nuclear matrix elements.  In the process several significant facts have been elaborated.

\begin{itemize}

\item{Not all possible non-relativistic operators contributing to nuclear matrix elements in direct detection will arise from simple UV complete dark matter models.}

\item{Aside from scalar WIMPs each particular spin produces some non-relativistic operators that are unique to that spin.}

\item{Two non-relativistic operators, $\mathcal{O}_1$ and $\mathcal{O}_{10}$, are ubiquitous and arise for all WIMP spins we have explored.}

\item{In 5 scenarios, relativistic operators generate unique non-relativistic operators at leading order.}

\item{Two new non-relativistic operators not previously considered within the context of the full array of allowed nuclear responses} arise at low energies if spin-1 WIMP dark matter is allowed for.

\item{While the different operators that can contribute to event rates in detectors using specific elements or isotopes cannot be distinguished on the basis of their impact on the differential event rates in these detectors, they can produce radically different energy dependence for scattering off different nuclear targets.  Thus, a complementary use of different target materials will be necessary to reliably distinguish between different particle physics model possibilities for WIMP dark matter.}

\end{itemize}

While current detectors have only yielded upper limits, with new generations of larger detectors with greater energy resolution and lower thresholds coming online, the search for WIMP dark matter has never been so vibrant and promising. The tools we have provided here should help experimenters to probe the most useful parameter space, to interpret any non-zero signals in terms of constraints on fundamental models, and should allow theorists who build fundamental models to frame predictions in an accurate and simple way so that they might be directly compared with experiment.

\begin{acknowledgments}
We would like to thank Richard Lebed for useful discussions. J.B.D. thanks Dr. and Mrs. Sammie W. Cosper at the University of Louisiana at Lafayette, and the Louisiana Board of Regents for support. L.M.K, J.L.N and S.S acknowledge support from the DOE for this work under grant No.\,DE-SC0008016.  We thank the Australian National University for hospitality while part of this work was carried out.
\end{acknowledgments}
\begin{table}[h!]
\renewcommand{\arraystretch}{0.9}
\caption{List of scenarios with leading operators colored by which are distinguishable via the ratio $\frac{dR_{Xe}}{dE}/\frac{dR_{Ge}}{dE}$.}
\begin{tabular}{|c|c|C{.6cm}|C{.6cm}|C{.6cm}|C{.6cm}|C{.8cm}|C{.6cm}|C{.6cm}|C{.6cm}|C{.6cm}|C{.6cm}|C{.6cm}|C{.6cm}|C{.6cm}|C{.6cm}|C{.6cm}|C{.6cm}|C{.6cm}|C{.6cm}|}
\hline
& &\textcolor{pink}{$\mathcal{O}_{1}$} & $\mathcal{O}_2$ & $\mathcal{O}_3$ & \textcolor{green}{$\mathcal{O}_4$}& \textcolor{blue}{$q^2\mathcal{O}_4$} & \textcolor{red}{$\mathcal{O}_5$} &\textcolor{cyan}{$\mathcal{O}_6$} & $\mathcal{O}_7$ & \textcolor{purple}{$\mathcal{O}_8$} &\textcolor{orange}{$\mathcal{O}_9$} &\textcolor{blue}{$\mathcal{O}_{10}$} &\textcolor{magenta}{$\mathcal{O}_{11}$} &$\mathcal{O}_{12}$ &$\mathcal{O}_{13}$ & $\mathcal{O}_{14}$ & $\mathcal{O}_{15}$ & \textcolor{gray}{$\mathcal{O}_{17}$} & \textcolor{yellow}{$\mathcal{O}_{18}$} \\

\cmidrule[1pt]{2-20}
& $(h_1,g_1)$      &\R&  &  &  &  &  &  &  &  &   &  &  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
& $(h_2,g_1)$      &  &  &  &  &  &  &  &  &  &   &\B&  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
& $(h_4,g_4)$      &  &  &  &  &  &  &  &  &  &   &\B&  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
\rotatebox{90}{\rlap{Spin-$0$ WIMP}}
& $(y_1)$          &\R&  &  &  &  &  &  &  &  &   &\x&  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
& $(y_2)$          &\R&  &  &  &  &  &  &  &  &   &\x&  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
& $(y_1,y_2)$      &  &  &  &  &  &  &  &  &  &   &\B&  &  &  &  &  &  &  \\
\cmidrule[1pt]{1-20}
& $(h_1,\lambda_1)$&\R&  &  &  &  &  &  &  &  &   &  &  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(h_2,\lambda_1)$ &  &  &  &  &  &  &  &  &  &   &\B&  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(h_1,\lambda_2)$ &  &  &  &  &  &  &  &  &  &   &  &\M&  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(h_2,\lambda_2)$ &  &  &  &  &  &  &\C&  &  &   &  &  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(h_3,\lambda_3)$ &\R&  &  &  &  &  &  &  &  &   &  &  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(h_4,\lambda_3)$ &  &  &  &  &  &  &  &\x&  &\Or&  &  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(h_3,\lambda_4)$ &  &  &  &  &  &  &  &  &\Pu&\x&  &  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
\rotatebox{90}{\rlap{Spin-$\half$ WIMP}}
&$(h_4,\lambda_4)$ &  &  &  &\G&  &  &  &  &  &   &  &  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(l_1)$           &\R&  &  &\x& &   &\x&  &  &   &  &  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(l_2)$           &\R&  &  &\x& &   &\x&  &  &   &  &  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(d_1)$           &\R&  &  &\x& &   &\x&  &  &   &  &  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(d_2)$           &\R&  &  &\x& &   &\x&  &  &   &  &  &  &  &  &  &  &  \\
\cmidrule[1pt]{1-20}
&$(h_1,b_1)$       &\R&  &  &  & &   &  &  &  &   &  &  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(h_2,b_1)$       &  &  &  &  & &   &  &  &  &   &\B&  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(h_4,b_5)$       &  &  &  &  & &   &  &  &  &   &\B&  &  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(h_3,b_6)$       &  &  &  & &\x&\Red&\x&  &  &   &  &  &  &  &  &  &\g*&  \\
\cmidrule[.5pt]{2-20}
&$(h_4,b_6)$       &  &  &  &  & &   &  &  &  &\Or&  &  &  &  &  &  &  &\Y*\\
\cmidrule[.5pt]{2-20}
\rotatebox{90}{\rlap{Spin-$1$ WIMP}}
&$(h_3,b_7)$       &  &  &  &  &  &  &  &  &\Pu*&\x*&  &\M&  &  &  &  &  &  \\
\cmidrule[.5pt]{2-20}
&$(h_4,b_7)$       &  &  & &\G*&\rB&  &\x&  &  &   &  &  &  &  &\x&  &  &  \\
\cmidrule[.5pt]{2-20}
&$(y_3)$           &\R&  &  &\x& & &  &  &  &   &\x&\x&\x&  &  &  &  &\x\\
\cmidrule[.5pt]{2-20}
&$(y_4)$           &\R&  &  &\x& & &  &  &  &   &\x&\x&\x&  &  &  &  &\x\\
\cmidrule[.5pt]{2-20}
&$(y_3,y_4)$       &  &  &  &  & & &  &  &  &   &\x&\M&\x&  &  &  &  &\x\\
\hline
\end{tabular}
\label{tabFull}
\bfootnote{* indicates the purely imaginary scenario for that coupling}
\end{table}

\clearpage

redpajama_set_name: RedPajamaArXiv


Any of a number of phosphorylation-activated serine/threonine kinases involved in controlling the cell cycle, of which there are two subtypes: Chk1 and Chk2.
Caption from image DNA damage induces intracellular signaling cascades that block DNA replication and cell cycle progression. Two parallel branches, the ATR-Chk1 and ATM-Chk2 pathways are activated by different forms of damage. Cdk activity is inhibited by degradation/inhibition of Cdc25 phosphatases and by the p53-dependent induction of the p21 Cdk inhibitor. Cdk-independent events also block origin firing through inhibition of Cdk and Cdc7 kinase activity. The Tlk1 kinase which promotes Asf1-dependent chromatin assembly is also inactivated in response to DNA damage.
Checkpoint kinases are essential factors which delay cell cycle progression in normal and damaged cells and can act at all three cell cycle checkpoints.* ATR kinase phosphorylates Chk1 in response to single strand DNA breaks; ATM kinase phosphorylates Chk2 in response to double strand breaks.
Checkpoint kinases phosphorylate Cdc25 phosphatase at Ser216, leading to Cdc25 sequestration in the cytoplasm–i.e., Cdc25 cannot remove the inhibitory phosphorylation on mitotic promoting factor (MPF) and entry into mitosis is prohibited.
They also play a central a role in the physiological stresses of hypoxia and reoxygenation.


text:
What Facebook Knows about You • 2017 • Panorama
Facebook is thought to know more about us than any other business in history, but what does the social network that Mark Zuckerberg built do with all of our personal information? Reporter Darragh MacIntyre investigates how Facebook's powerful algorithms allow advertisers and politicians to target us more directly than ever before, and he questions whether the company's size and complexity now makes it impossible to regulate.
Panorama • 2016 - 2017 • 1 episode •
What Facebook Knows about You
Concrete Jungle
The world's cities are growing at a faster rate than ever before. An estimated 75 million people around the world move to an urban area every year. And as our metropolises become more and more crowded, architects, designers, and builders face a constant challenge.
S1E3 • How Cities Work • 2013 • Technology
Floodgates of Venice
MOSE is one of the world?s largest and highest-profile civil-engineering works. But will it be able to save Venice? Venice has grappled with inundation for centuries. But due to natural subsidence and higher tides caused by global warming, the city is more vulnerable to flooding than ever before.
Breakthrough • 2019 • Technology
Jobs vs Gates
Two brilliant minds, Bill Gates and Steve Jobs, do battle and, in the process, revolutionize lives.
S1E1 • American Genius • 2015 • Technology
Lunar Module
It's July 20, 1969. Four days after Apollo 11 thundered skyward from NASA's Kennedy Space Center, astronauts Neil Armstrong and Buzz Aldrin become the first humans to set foot upon the moon, thanks to a marvel of engineering: the lunar module. It's the culmination of years of toil and millions of man-hours of engineering prowess. This is the story of the unsung heroes who built the unconventional flying machines that carried 12 Americans to the moon on six separate missions and made the dream of a generation come true.
2/6 • America's Secret Space Heroes • 2017 • Technology
Bletchley Park: Code-breaking's Forgotten Genius
Gordon Welchman was one of the original elite codebreakers crucial to the allies defeating the Nazis in World War II. He is the forgotten genius of Bletchley Park. Filmed extensively at Bletchley Park, the centre for codebreaking operations during World War II, this documentary features the abandoned buildings where thousands of people worked tirelessly trying to crack the codes, Hut 6, where Welchman pioneered his groundbreaking work, and the machines that Welchman helped design.
In the future will we spend our leisure time with smart and sophisticated machines designed for fun - Entertainment Robots? Marvel at Mantis, a two ton insect and Tradinno, a giant fire-spitting dragon. Then take a seat and watch a mechanical actor and a robotic pianist perform.

redpajama_set_name: RedPajamaCommonCrawl


text:
My son, do not forget my teaching, but let your heart keep my commandments, for length of days and years of life and peace they will add to you. Let not steadfast love and faithfulness forsake you; bind them around your neck; write them on the tablet of your heart. So you will find favor and good success in the sight of God and man. Trust in the Lord with all your heart, and do not lean on your own understanding. ...
He also made two capitals of molten bronze to set on the tops of the pillars; the height of the one capital was five cubits and the height of the other capital was five cubits. There were nets of network and twisted threads of chainwork for the capitals which were on the top of the pillars; seven for the one capital and seven for the other capital. So he made the pillars, and two rows around on the one network to cover the capitals which were on the top of the pomegranates; and so he did for the other capital. The capitals which were on the top of the pillars in the porch were of lily design, four cubits. There were capitals on the two pillars, even above and close to the rounded projection which was beside the network; and the pomegranates numbered two hundred in rows around both capitals. Thus he set up the pillars at the porch of the nave; and he set up the right pillar and named it Jachin, and he set up the left pillar and named it Boaz. On the top of the pillars was lily design. So the work of the pillars was finished.
For the entrance of the inner sanctuary he made doors of olive wood, the lintel and five-sided doorposts. So he made two doors of olive wood, and he carved on them carvings of cherubim, palm trees, and open flowers, and overlaid them with gold; and he spread the gold on the cherubim and on the palm trees. So also he made for the entrance of the nave four-sided doorposts of olive woodread more. Christian Art and Gifts
"Come, everyone who thirsts, come to the waters; and he who has no money, come, buy and eat! Come, buy wine and milk without money and without price. Why do you spend your money for that which is not bread, and your labor for that which does not satisfy? Listen diligently to me, and eat what is good, and delight yourselves in rich food. Incline your ear, and come to me; hear, that your soul may live; and I will make with you an everlasting covenant, my steadfast, sure love for David. Behold, I made him a witness to the peoples, a leader and commander for the peoples. Behold, you shall call a nation that you do not know, and a nation that did not know you shall run to you, because of the Lord your God, and of the Holy One of Israel, for he has glorified you. ... Share Your Faith Products Canvas Art
To the choirmaster: with stringed instruments. A Psalm of Asaph. A Song. In Judah God is known; his name is great in Israel. His abode has been established in Salem, his dwelling place in Zion. There he broke the flashing arrows, the shield, the sword, and the weapons of war. Selah Glorious are you, more majestic than the mountains of prey. The stouthearted were stripped of their spoil; they sank into sleep; all the men of war were unable to use their hands. ...
The house that I am to build will be great, for our God is greater than all gods. But who is able to build him a house, since heaven, even highest heaven, cannot contain him? Who am I to build a house for him, except as a place to make offerings before him? So now send me a man skilled to work in gold, silver, bronze, and iron, and in purple, crimson, and blue fabrics, trained also in engraving, to be with the skilled workers who are with me in Judah and Jerusalem, whom David my father provided. Share Your Faith Products Canvas Art
Of course, we also back our wall arts with our amazing "Goof Proof" guarantee. You can read all about there by clicking on this link GOOF PROOF GUARANTEE. In short, it says we guarantee everything about our product. You never have to worry about working with Scripture Wall Art. We have the highest customer satisfaction rating in the industry and nearly 12 years of experience making the best Scripture Wall Quotes out there. Share Your Faith Products Canvas Art
"You saw, O king, and behold, a great image. This image, mighty and of exceeding brightness, stood before you, and its appearance was frightening. The head of this image was of fine gold, its chest and arms of silver, its middle and thighs of bronze, its legs of iron, its feet partly of iron and partly of clay. As you looked, a stone was cut out by no human hand, and it struck the image on its feet of iron and clay, and broke them in pieces. Then the iron, the clay, the bronze, the silver, and the gold, all together were broken in pieces, and became like the chaff of the summer threshing floors; and the wind carried them away, so that not a trace of them could be found. But the stone that struck the image became a great mountain and filled the whole earth. Bible Scripture Wall Art
Then I saw another beast rising out of the earth. It had two horns like a lamb and it spoke like a dragon. It exercises all the authority of the first beast in its presence, and makes the earth and its inhabitants worship the first beast, whose mortal wound was healed. It performs great signs, even making fire come down from heaven to earth in front of people, and by the signs that it is allowed to work in the presence of the beast it deceives those who dwell on earth, telling them to make an image for the beast that was wounded by the sword and yet lived. And it was allowed to give breath to the image of the beast, so that the image of the beast might even speak and might cause those who would not worship the image of the beast to be slain. ... Share Your Faith Products Canvas Art
He also made two pillars for the front of the house, thirty-five cubits high, and the capital on the top of each was five cubits. He made chains in the inner sanctuary and placed them on the tops of the pillars; and he made one hundred pomegranates and placed them on the chains. He erected the pillars in front of the temple, one on the right and the other on the left, and named the one on the right Jachin and the one on the left Boaz. Christian Art and Gifts
Now when the wall had been built and I had set up the doors, and the gatekeepers, the singers, and the Levites had been appointed, I gave my brother Hanani and Hananiah the governor of the castle charge over Jerusalem, for he was a more faithful and God-fearing man than many. And I said to them, "Let not the gates of Jerusalem be opened until the sun is hot. And while they are still standing guard, let them shut and bar the doors. Appoint guards from among the inhabitants of Jerusalem, some at their guard posts and some in front of their own homes." The city was wide and large, but the people within it were few, and no houses had been rebuilt. Then my God put it into my heart to assemble the nobles and the officials and the people to be enrolled by genealogy. And I found the book of the genealogy of those who came up at the first, and I found written in it: ... Christian Art and Gifts
Glory of ChristOther BlindingChrist's Own GloryAgeControlling Your ThoughtsUnbelieversPerfection, DivineCommitment, to the worldChrist, Names ForSatan, Kingdom OfSatan, Titles ForDarkness, As A Symbol Of SinThe Light Of ChristCultsRevelation, In Ntevil, origins ofSpiritual Warfare, Causes OfHeart, Fallen And RedeemedSpiritual Blindness, Results Of SinBlinding Christian Art and Gifts
"Then bring near to you Aaron your brother, and his sons with him, from among the people of Israel, to serve me as priests—Aaron and Aaron's sons, Nadab and Abihu, Eleazar and Ithamar. And you shall make holy garments for Aaron your brother, for glory and for beauty. You shall speak to all the skillful, whom I have filled with a spirit of skill, that they make Aaron's garments to consecrate him for my priesthood. These are the garments that they shall make: a breastpiece, an ephod, a robe, a coat of checker work, a turban, and a sash. They shall make holy garments for Aaron your brother and his sons to serve me as priests. They shall receive gold, blue and purple and scarlet yarns, and fine twined linen. ... Christian Art and Gifts
In the second year of the reign of Nebuchadnezzar, Nebuchadnezzar had dreams; his spirit was troubled, and his sleep left him. Then the king commanded that the magicians, the enchanters, the sorcerers, and the Chaldeans be summoned to tell the king his dreams. So they came in and stood before the king. And the king said to them, "I had a dream, and my spirit is troubled to know the dream." Then the Chaldeans said to the king in Aramaic, "O king, live forever! Tell your servants the dream, and we will show the interpretation." The king answered and said to the Chaldeans, "The word from me is firm: if you do not make known to me the dream and its interpretation, you shall be torn limb from limb, and your houses shall be laid in ruins. ... Christian Canvas Art
He also made two pillars for the front of the house, thirty-five cubits high, and the capital on the top of each was five cubits. He made chains in the inner sanctuary and placed them on the tops of the pillars; and he made one hundred pomegranates and placed them on the chains. He erected the pillars in front of the temple, one on the right and the other on the left, and named the one on the right Jachin and the one on the left Boaz.
See what kind of love the Father has given to us, that we should be called children of God; and so we are. The reason why the world does not know us is that it did not know him. Beloved, we are God's children now, and what we will be has not yet appeared; but we know that when he appears we shall be like him, because we shall see him as he is. And everyone who thus hopes in him purifies himself as he is pure. Bible Scripture Wall Art
You are eligible for a full refund if no ShippingPass-eligible orders have been placed. You cannot receive a refund if you have placed a ShippingPass-eligible order. In this case, the Customer Care team will remove your account from auto-renewal to ensure you are not charged for an additional year and you can continue to use the subscription until the end of your subscription term. Bible Scripture Wall Art


text:
Los Ranchos Elementary Celebrates a Century of School
Article courtesy of Albuquerque Journal, April 26, 2014
Woodrow Wilson was president, students were transported to campus in wagons and Los Ranchos Elementary, a three-room building surrounded by dirt, was opening its doors to students for the first time.
Los Ranchos Elementary in 1914, the year it opened.
The year was 1914 and the land for the school cost $300. It was 140 feet wide and 400 feet long.
The school, like society, is almost unrecognizable today, but it still sits in the same spot, welcoming students from nearby North Valley neighborhoods. The school initially served first- through eighth-grade students and teachers were paid $75 a month.
It has had 16 principals since its opening, including Renee Gallegos, the school's current head. She's been responsible for organizing and carrying out the school's yearlong 100th birthday celebration that will culminate in a community party next month. Classes were assigned to research different decades, and the school every month has held an assembly where students present what was happening in the school and the world during that time.
Los Ranchos Elementary is not only physically different than it was a century ago, but expectations for teachers and students have dramatically changed. Dancing back then was not allowed. Teachers were asked to follow instructions outlined in an 1872 document titled "List of Rules for Teachers." They had to bring a bucket of water and a "scuttle of coal" each day. "Men teachers" were allowed to take one evening a week for courting purposes and two if they went to church. Female teachers had their own set of expectations.
"Women teachers who marry or engage in unseemly conduct will be dismissed."
All teachers were encouraged to read the Bible or other "good books" in the evening, and all were discouraged from drinking and smoking so their integrity would not be questioned.
In the early years, Gallegos said the school didn't even have a cafeteria. The parent-teacher association raised money and literally built the school's cafeteria, which sits virtually unchanged, in 1947.
Lore Aguilar, who turns 84 today, attended Los Ranchos Elementary in the late 1930s and early 1940s. She remembers packing and eating her lunch underneath some trees where the daily swapping of sandwiches would commence. Peanut butter and jelly for bologna or a cheese sandwich. The school sat between two small grocery stores, and students were allowed to walk there during lunch and recess, she said.
"The school was just one building then," she said. "They had what they called a baby room and they put me in that. I'm not sure why, but I think it was for people who didn't speak English. After I learned some English, they moved me to another room."
Aguilar now lives a mile and a half from her childhood home on Edith and all of her 11 children attended the school.
It's also a family affair for Esther Leyba, who turns 63 on Sunday. Two of her grandsons are currently students at the school. She said the boys like to ask her about what the school was like when she was there in the late 1950s.
"The building was really different than it is now," she said. "It was all dirt and we didn't have a gym."
Both Aguilar and Leyba rode the bus but, until 1922, students were transported to school in a wagon. The switch from coal to gas heat came in 1949, followed by clocks in every classroom and square-dancing in the auditorium.
Gallegos said today's school, with an enrollment of more than 300, is nothing like anyone could have imagined. It has multiple buildings, including a gymnasium and an entire wing for kindergarten students. In the beginning, the school had a few teachers and a principal.
But those days are long gone.
There are now 14 teachers, office staff, a librarian, seven special education teachers, a counselor, a reading coach, math interventionist, technology teacher and a PE teacher.
"It's a lot bigger than its humble beginnings," Gallegos said. "A classroom back in the day was a chalkboard and some desks. Now we have all this technology we never imagined."
Posted on April 27, 2014 by Los Ranchos.


Register for this course by clicking the REGISTRATION button. Return to this page and use the buttons above to navigate your way through the course. To view the course lectures, please read this easy-to-use guide. Contact your instructor if you have any questions or comments.


17, I went to Urgent Care at the VA hospital and stayed in a sober living environment for one year. However I maintained contact with my girlfriend at the time, and I continued 2 dips and dab with K2 when I came home on the weekends, and I struggle with that still, but I'm successful with hardly drinking anymore.
I met a beautiful woman that I've known for a long time I encouraged her to go to rehab White Deer Run to get clean because I could not get sober with distractions in my life. So hopefully, she comes back in a couple weeks. Being right and clean, I stopped smoking 3 days ago.
I experienced some kind of psychosis I feel like I'm going crazy and breathing anxieties, I'm a commercial painter now I started in May of last year back to work since breaking my back.
Short-term I find going out, makes me feel a little better to walk around breathe fresh air clear, my mind a little bit. I was kind of calm down after that but an hour or two later these things creep back up. I'm currently trying to do everything that I can think of outside of going away for any duration of time.
I can't do that right now, so I have to find other ways to get through this, and this serves as a vessel, and it helps me feel so right just kind of get this out a little bit. Today is Martin Luther King Jr's holiday; I will go for counselling or outpatient treatment of some sort. I don't know if this is the best advice, but I know K2 it's not good for you.
The best way to quit it's just to start some kind of way and follow through and use tools and talk to people and get those thoughts out of your mind to clear your soul. This is going to be a difficult challenge at this time because bad habits are hard to break but I made my mind up now, and this one is going to get crushed. God bless, be strong take care.
This entry was posted in Submitted Stories and tagged drug counselling, k2, K2 rehab.


A lower extremity arterial (LEA) evaluation, or ankle-brachial index (ABI), is a non-invasive diagnostic procedure that compares the blood pressure in a patient's legs to the blood pressure in his or her arms.
A lower extremity arterial (LEA) evaluation is a non-invasive diagnostic procedure that compares the blood pressure in a patient's legs to the blood pressure in his or her arms. This procedure is sometimes referred to as an ankle-brachial index (ABI). A lower blood pressure in the leg could be a symptom of peripheral arterial occlusive (peripheral vascular) disease. The test evaluates the presence, severity and general location of this disease.
It will take 30 to 40 minutes to perform the LEA exam. The vascular sonographer will place blood pressure cuffs on your legs and arms at the appropriate levels. Doppler signals will be obtained from the arteries in your legs by holding a small probe against your skin. This is a painless procedure. Your blood pressures will be taken in your arms and legs at the appropriate levels. You may be asked to perform some exercise, such as walking on a treadmill. Your blood pressures will again be taken after you exercise. After the procedure, you will be permitted to get dressed and leave without any restrictions.
The sonographer will not provide you with any results at the time of the examination. A trained cardiologist will interpret the study and provide your family physician with a written report. Your primary care physician (or the provider who ordered the study) will follow up with you.


To book an appointment to view these homes please call 01992 453712 or email homeownershipsales@b3living.org.uk. Please note that viewings are by appointment only and subject to financial assessment.
Two vedroom apartments available for shared ownership all with secure underground parking, access to large communal roof terrace and some with a balcony.
Watling Street is part of beautiful new development located in the sought-after village of Radlett. Radlett is an affluent village in the country of Hertfordshire between St. Albans and Borehamwood and offers an array of shops, restaurants, pubs, clubs and a cinema right on your doorstep.
Radlett has excellent transport links, with Radlett train station close by, connecting you to London St. Pancras in 20 minutes.
*Please note the images shown are for illustrations purposes only and do not necessarily represent the actual size or layout of the available plots.