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## Rate expressions including detector response

We summarize here the expressions of the rate used by WimPyDD to include the detector response. This Section contains also the definitions of the physical quantities that are used as inputs by the WimPyDD signal routines. See here for more details.

In WimPyDD the expected rate $$R_{[E_1^{\prime},E_2^{\prime}]}(t)=S^{(0)}_{[E_1^{\prime},E_2^{\prime}]}+S^{(1)}_{[E_1^{\prime},E_2^{\prime}]}\cos\left[\frac{2\pi}{T_0}(t-t_0)\right]$$ integrated in the bin of visible energy $$E^{\prime}\in [E_1^{\prime},E_2^{\prime}]$$ is calculated starting from the expression:

$$S^{(0,1)}_{[E_1^{\prime},E_2^{\prime}]}=\frac{\rho_\chi}{m_\chi}\sum_{k=1}^{N_s} \delta\eta_k^{(0,1)} \int_0^{v_i}\sum_T dv {\cal R}_{T,[E_1^{\prime},E_2^{\prime}]}(v)$$

with $$\frac{\rho_\chi}{m_\chi}$$ the WIMP number density in the neighborhood of the Sun, $$\delta\eta_k^{(0,1)}$$ the halo function and the sum is over the contributions of WIMP scattering off targets $$T$$. The integrals in the equation above can be recast in the form:

$$\int_0^{v_i} dv {\cal R}_{T,[E_1^{\prime},E_2^{\prime}]}(v)= \int_{E_R^{min}(v_i)}^{E_R^{max}(v_i)}d E_R \left[ {\cal R}^{0}_{T,[E_1^{\prime},E_2^{\prime}]}(E_R)+ {\cal R}^{1}_{T,[E_1^{\prime},E_2^{\prime}]}(E_R)(v_i^2-v^2_{T,min}(E_R)) \right ]$$

In the equation above:

$$v_{T,min}(E_R)=\frac{1}{\sqrt{2 m_T E_R}}\left | \frac{m_TE_R}{\mu_{\chi T}}+\delta \right |$$

is the minimal speed an incoming WIMP needs to have in the target reference frame to deposit energy $$E_R$$ (with $$m_T$$ the nuclear target, $${\mu_{\chi T}}$$ the WIMP-target reduced mass, and $$\delta$$ the mass splitting in case of inelastic scattering) while

$$E_R^{min,max}(v)=\left[\frac{\mu_{\chi T}}{\sqrt{2 m_T}}\left ( v\mp\sqrt{v^2-v_{T^*}^2}\right ) \right ]^2$$

with $$v_{T^*}\equiv \sqrt{\frac{\delta}{\mu_{\chi T}}}$$. In particular:

$${\cal R}^{i}_{T,[E_1^{\prime},E_2^{\prime}]}(E_R)=\int_{E_1^{\prime}}^{E_2^{\prime}}d E^{\prime}\; r_{T}^{i}(E_R,E^\prime),\,\,\,i=0,1$$

with the differential response functions $$r_{T}^{i}(E_R,E^\prime)$$ defined as:

$$r_{T}^{i}(E_R,E^\prime)=M T_0 N_T\frac{m_T}{2}\epsilon(E^{\prime}){\cal G}\left [E^{\prime},E_{ee}=q_T(E_R)E_R\right ]\frac{4\pi}{2 j_T+1}\sum_{\tau\tau^{\prime}}\sum_l R_{l}^{i,\tau\tau^{\prime}}(q)W_{l,T}^{\tau\tau^{\prime}}(q)$$

In the expression above $$R_{l}^{i,\tau\tau^{\prime}}(q)$$ are obtained from the decomposition:

$$R_l^{\tau\tau^{\prime}}(q,v)=R_{l}^{0,\tau\tau^{\prime}}(q)+R_{l}^{1,\tau\tau^{\prime}}(q)(v^2-v_{T,min}^2)$$

of the WIMP response functions $$R_{l}^{\tau\tau^{\prime}}(q)$$ that can be found here for $$j_\chi\le 1/2$$ and here for WIMPs of arbitrary spin (WimPyDD implements the ones for arbitrary $$j_\chi$$). On the other hand $$W_{l,T}^{\tau\tau^{\prime}}(q)$$ are nuclear response functions. In the scattering amplitude the WIMP and the nuclear response functions factorize: $\frac{1}{2j_\chi+1}\frac{1}{2j_T+1}|\mathcal{M}|^2=\frac{4\pi}{2j_T+1}\sum_{\tau =0,1}\sum_{\tau^\prime =0,1}\sum_l R^{\tau\tau^\prime}_l W^{\tau\tau^\prime}_{l,T}$

#### Required input

• $$T$$: the nuclear target
• $$MT_0$$: exposure in kg year;
• $$\epsilon(E^{\prime})$$: efficiency $$\le 1$$;
• $${\cal G}[E^{\prime},E_{ee}]$$: energy resolution;
• $$q_T(E_R)$$: quenching factor;
• $$W_{l,T}^{\tau\tau^{\prime}}(q)$$: nuclear response functions,

For a specific experimental set-up in WimPyDD each of the quantities above can be initialized by adding a file to a subdirectory of the WimPyDD/Experiments folder.

If the scattering process is driven by the effective Hamiltonian $${\cal H}=\sum_{\tau=0,1} \sum_{j} c_j^{\tau}(w_i,q) {\cal O}_{j}^{\tau}$$ the functions $${\cal R}^{0}_{T,[E_1^{\prime},E_2^{\prime}]}$$ and $${\cal R}^{1}_{T,[E_1^{\prime},E_2^{\prime}]}$$ are quadratic in the Wilson coefficients of the effective theory, that can be factored out:

$${\cal R}^{0,1}_{T,[E_1^{\prime},E_2^{\prime}]} (E_R)=\sum_{j,k} c_j^{\tau}(w_i,q_0)c_{k}^{\tau^\prime}(w_i,q_0^{\prime})\left [{\cal R}^{0,1}_{T,[E_1^{\prime},E_2^{\prime}]}\right ]_{jk}^{\tau\tau^{\prime}}(E_R)$$

with $$q_0$$, $$q_0^{\prime}$$ arbitrary momentum scales in the case when the Wilson coefficients $$c_j^{\tau}$$, $$c_{k}^{\tau^\prime}$$ have an explicit dependence on $$q$$.

Defining the integrated response functions as:

$$\left [\bar{{\cal R}}^{i}_{T,[E_1^{\prime},E_2^{\prime}]}\right ]_{jk}^{\tau\tau^{\prime}}(E_R)\equiv \int_0^{E_R} dE^{\prime}_R \left[{\cal R}^{i}_{T,[E_1^{\prime},E_2^{\prime}]}\right ]_{jk}^{\tau\tau^{\prime}}(E_R^{\prime}),\,\,\,i=0,1$$
$$\left [\bar{{\cal R}}^{1E}_{T,[E_1^{\prime},E_2^{\prime}]}\right ]_{jk}^{\tau\tau^{\prime}}(E_R)\equiv \int_0^{E_R} dE^{\prime}_R E_R^{\prime} \left [{\cal R}^{1}_{T, [E_1^{\prime},E_2^{\prime}]}\right ]_{jk}^{\tau\tau^{\prime}}(E_R^{\prime})$$
$$\left [\bar{{\cal R}}^{1E^{-1}}_{T,[E_1^{\prime},E_2^{\prime}]}\right]_{jk}^{\tau\tau^{\prime}}(E_R)\equiv \int_0^{E_R} dE^{\prime}_R \frac{1}{E_R^{\prime}} \left [{\cal R}^{1}_{T,[E_1^{\prime},E_2^{\prime}]}\right ]_{jk}^{\tau\tau^{\prime}}(E_R^{\prime})$$

the expected rate takes the final form:

$$S^{(0,1)}_{[E_1^{\prime},E_2^{\prime}]}=\frac{\rho_{\chi}}{m_{\chi}}\sum_{k=1}^{N_s} \delta\eta_k^{(0,1)}\times \sum_{ij}\sum_{\tau\tau^{\prime}} c_j^{\tau}(w_i,q_0)c_{k}^{\tau^\prime}(w_i,q_0^{\prime})$$
$$\left \{ \left [\bar{{\cal R}}_{T,0}^{[E_1^{\prime},E_2^{\prime}]}\right ]_{jk}^{\tau\tau^{\prime}}(E_R) +(\frac{v_k^2}{c^2}-\frac{\delta}{\mu_{\chi T}})\left [ \bar{{\cal R}}_{T,1}^{[E_1^{\prime},E_2^{\prime}]}\right ]_{jk}^{\tau\tau^{\prime}}(E_R) \right .$$
$$\left . -\frac{m_T}{2\mu_{\chi T}^2} \left [\bar{{\cal R}}_{T,1E}^{[E_1^{\prime},E_2^{\prime}]}\right ]_{jk}^{\tau\tau^{\prime}}(E_R)-\frac{\delta^2}{2 m_T} \left[\bar{{\cal R}}_{T,1E^{-1}}^{[E_1^{\prime},E_2^{\prime}]}\right ]_{jk}^{\tau\tau^{\prime}}(E_R)\right \}^{E_R^{max}(v_k)}_{E_R^{min}(v_k)}$$

which is the expression used by the wimp_dd_rate routine to calculate integrated rates including the detector response. In particular wimp_dd_rate interpolates the integrated response functions $$\left [\bar{{\cal R}}^{a}_{T,[E_1^{\prime},E_2^{\prime}]}\right ]_{jk}^{\tau\tau^{\prime}}(E_R)$$ ($$a=0,1,1E,1E^{-1}$$) loaded by the load_response_funtions routine in the response_functions attribute of the experiment class.