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+{"text":"\\section{Introduction}\nThe main problem we will consider here is to give a lower bound for the dimension of the union of a collection of lines in terms of the dimension of the collection of lines, without imposing a structural hypothesis on the collection (in contrast to the Kakeya problem where one assumes that the lines are direction-separated, or perhaps satisfy the weaker ``Wolff axiom''). \n\nSpecifically, we are motivated by the following conjecture of D. Oberlin ($\\mathop{\\mathrm{hdim}}$ denotes Hausdorff dimension).\n\\begin{conjecture}\nSuppose $d \\geq 1$ is an integer, that $0 \\leq \\beta \\leq 1,$ and that ${\\bf L}$ is a collection of lines in $\\rea^n$ with $\\mathop{\\mathrm{hdim}}({\\bf L}) \\geq 2(d-1) + \\beta.$ Then\n\\begin{equation} \\label{dimrange}\n\\mathop{\\mathrm{hdim}}(\\bigcup_{L \\in {\\bf L}}L) \\geq d + \\beta.\n\\end{equation}\n\\end{conjecture}\nThe bound \\eqref{dimrange}, if true, would be sharp, as one can see by taking ${\\bf L}$ to be the set of lines contained in the $d$-planes belonging to  a $\\beta$-dimensional family of $d$-planes. (Furthermore, there is nothing to be gained by taking $1 < \\beta \\leq 2$ since the dimension of the set of lines contained in a $d+1$-plane is $2(d-1) + 2$.)\n\nStandard Fourier-analytic methods show that \\eqref{dimrange} holds for $d=1$, but the conjecture is open for $d > 1.$ As a model problem, one may consider an analogous question where $\\rea^n$ is replaced by a vector space over a finite-field. Our main result is that the corresponding conjecture holds for all $d$ ($|\\cdot|$ denotes cardinality).\n\n\\begin{theorem} \\label{firstlinethm}\nSuppose $d \\geq 1$ is an integer, $F$ is a finite field, $0 \\leq \\beta \\leq 1,$ and that ${\\bf L}$ is a collection of lines in $F^n$ with $|{\\bf L}| \\geq |F|^{2(d-1) + \\beta}.$ Then\n\\begin{equation} \\label{dimrangeff} \n|\\bigcup_{L \\in {\\bf L}}L| \\gtrsim |F|^{d + \\beta}\n\\end{equation}\nwhere the implicit constant may depend on $d$, but is independent\\footnote{The constant is also independent of $\\beta$ and $n$, but this is only of secondary interest.} of $F$.\n\\end{theorem}\n\nReusing the examples above, one sees that \\eqref{dimrangeff} is sharp, up to the loss in the implicit constant, and that there is nothing to be gained by taking $1 < \\beta \\leq 2.$\n\nThe main tool we use in the proof of \\eqref{dimrangeff} is an iterated version of Wolff's hairbrush argument \\cite{wolff95ibk}. For comparison, we state the finite-field version of his result\\footnote{Wolff's main interest in this method was likely its use towards a partial resolution of the Kakeya conjecture (up to a negligible constant, any direction separated collection of lines satisfies the Wolff axiom). To that end, it has been superceded by Dvir's theorem \\cite{dvir09osk} (see also \\cite{ellenberg09ksm}), whose proof makes stronger use of the direction-separation hypotheses and does not seem to be applicable to the present question.} (see \\cite{wolff99rwc},\\cite{mockenhaupt04rkp}), starting with the following definition. A set of lines ${\\bf L}$ in $F^n$ satisfies the \\emph{Wolff axiom} if for every two-plane $R \\subset F^n$ \n\\[\n|\\{L \\in {\\bf L} : L \\subset R\\}| < |F|.\n\\]\n\n\\begin{theorem}[Wolff]\nSuppose that $\\alpha \\geq 1$, $F$ is a finite field, and ${\\bf L}$ is a collection of lines in $F^n$ with $|{\\bf L}| \\geq |F|^{\\alpha}.$ If ${\\bf L}$ satisfies the Wolff axiom then\n\\begin{equation} \\label{wolffexponent}\n|\\bigcup_{L \\in {\\bf L}}L| \\gtrsim |F|^{\\frac{\\alpha + 3}{2}}\n\\end{equation}\nwhere the implicit constant is independent of $F$.\n\\end{theorem}\n\nAn immediate consequence of Theorem \\ref{firstlinethm} is that, for odd integers $\\alpha$, Wolff's theorem holds even for collections of lines that do not satisfy the Wolff axiom. \n\nThe proof of Theorem \\ref{firstlinethm} also shows that the Wolff axiom can be relaxed for general values of $\\alpha.$ We say that a set of lines ${\\bf L}$ in $F^n$ satisfies the \\emph{$d$-plane Wolff axiom} if for every $d$-plane $R \\subset F^n$ \n\\begin{equation} \\label{dpwa}\n|\\{L \\in {\\bf L} : L \\subset R\\}| < |F|^{2d-3}.\n\\end{equation}\nIf $R$ is a $d$-plane then there are approximately $|F|^{3(d-2)}$ 2-planes $S$ contained in $R$ and for each line $L \\subset R$ there are approximately $|F|^{d-2}$ 2-planes $S$ with $L \\subset S \\subset R.$ Thus, the $d$-plane Wolff axiom asserts that for every $d$-plane $R$ the standard Wolff axiom holds ``on average'' for two-planes $S \\subset R.$ In particular, the d-plane Wolff axiom is weaker than the standard Wolff axiom when $d > 2$ (assuming one is willing to adjust the axioms by a constant factor, which would make no impact on the validity of the stated theorems). \n\n\\begin{theorem} \\label{wolffrelaxed}\nSuppose that $d > 2$ is an integer, $2(d-1) - 1 < \\alpha < 2(d-1) + 1$, $F$ is a finite field, and ${\\bf L}$ is a collection of lines in $F^n$ with $|{\\bf L}| \\geq |F|^{\\alpha}.$ If ${\\bf L}$ satisfies the $d$-plane Wolff axiom \\eqref{dpwa} then\n\\begin{equation} \\label{wolffexponent2}\n|\\bigcup_{L \\in {\\bf L}}L| \\gtrsim |F|^{\\frac{\\alpha + 3}{2}}\n\\end{equation}\nwhere the implicit constant is independent of $F$.\n\\end{theorem}\n\nBounds of the form \\eqref{wolffexponent} do not seem to be sharp; at least, they can be slightly strengthened in the case when $F = \\BBZ_p$, $\\alpha = 2$, and $n=3$, see \\cite{bourgain04spe}. \n\nSince \\eqref{wolffexponent2} improves on \\eqref{dimrangeff} when $\\beta < 1$, one can use the $d$-plane Wolff axiom to extract structural information about quasi-extremizers (cf. \\cite{christ06qer}) of \\eqref{dimrangeff}.\n\n\\begin{theorem} \\label{qethm}\nSuppose $d \\geq 1$ is an integer, and that $0 \\leq \\beta < 1$. Then, for every $C$ there exist $M$ and $c > 0$ such that if $F$ is a finite field with $|F| \\geq M$ and ${\\bf L}$ is a collection of lines in $F^n$ with $|L| \\geq |F|^{2(d-1) + \\beta}$ satisfying\n\\begin{equation} \\label{qebound}\n|{\\bf P}| \\leq C  |F|^{d + \\beta}\n\\end{equation}\nwhere ${\\bf P} := \\bigcup_{L \\in {\\bf L}}L,$\nthen there are $d$-planes $R_1, \\ldots, R_N$ with $N \\geq c |F|^\\beta$ such that \n\\[\n|{\\bf P} \\cap \\bigcup_{j} R_j| \\geq c |{\\bf P}|\n\\]\nand for each $j$\n\\begin{equation} \\label{denseindplane}\n|{\\bf P} \\cap R_j| \\geq c |R_j|.\n\\end{equation}\n\\end{theorem}\n\nOne can also prove a version of the statement above for $-1 < \\beta < 0,$ but we omit the details. \n\nBy adding two additional layers of recursion, the method of Theorem \\ref{firstlinethm} can be adapted to treat unions of $k$-planes. \n\n\\begin{theorem} \\label{upthm1}\nSuppose $d \\geq k > 0$ are integers, that $0 \\leq \\beta \\leq 1$ and that ${\\bf L}$ is a collection of $k$-planes in $F^n$ with \n\\begin{equation} \\label{upthm1hyp}\n|{\\bf L}| \\geq  |F|^{(k+1)(d-k)+\\beta}.\n\\end{equation}\nThen \n\\begin{equation} \\label{unionplanesbound}\n|\\bigcup_{L \\in {\\bf L}}L| \\gtrsim |F|^{d + \\beta}.\n\\end{equation}\n\\end{theorem}\n\nOur proof requires simultaneous treatment of the following more general result.\n\n\\begin{theorem} \\label{upthm2}\nSuppose $d \\geq k > k' \\geq 0$ are integers, that $0 \\leq \\beta \\leq k' + 1$ and that ${\\bf L}$ is a collection of $k$-planes in $F^n$ with \n\\[\n|{\\bf L}| \\geq  |F|^{(k+1)(d-k)+\\beta}.\n\\]\nLetting ${\\bf P}_L = \\{k'\\text{-planes\\ }P : P \\subset L\\}$ we have\n\\begin{equation} \n|\\bigcup_{L \\in {\\bf L}}{\\bf P}_L| \\gtrsim |F|^{(k' + 1)(d - k') + \\beta}.\n\\end{equation}\n\\end{theorem}\n\nTheorems \\ref{upthm1} and \\ref{upthm2} are sharp in the same sense as Theorem \\ref{firstlinethm}. For a $k$-plane analog of Wolff's theorem see \\cite{bueti06ibk}. It may be possible to modify the proof of \\eqref{unionplanesbound} to obtain a $k$-plane analog of Theorem \\ref{wolffrelaxed}, but we do not pursue the details here.\n\nThe outline of this article is as follows: Section \\ref{prsection} contains some technical machinery, Section \\ref{linesection} contains the proofs of Theorems \\ref{firstlinethm}, \\ref{wolffrelaxed}, and \\ref{qethm},  and Section \\ref{planesection} contains the proofs of Theorems \\ref{upthm1} and \\ref{upthm2}.\n\n\n\\section{Preliminaries} \\label{prsection}\n\nWe start by roughly describing the approach of \\cite{wolff95ibk}. If a union of lines is small, then there must exist a ``hairbrush'' of many lines intersecting one common line. The ambient space can then be foliated into two-dimensional planes containing the common line, and a classical bound can be applied to estimate the union of lines contained in each two-plane.  \n\nIn the present situation, we instead consider a hairbrush of many lines or $k$-planes intersecting a common $m$-dimensional plane. The following lemma is used to determine the appropriate choice of $m$.\n\n\\begin{lemma} \\label{pslemma}\nLet ${\\bf L}$ be a collection of $k$-planes in $F^n$ and suppose $d$ is a nonnegative integer with $k \\leq d \\leq n$. There is an $m$ with $k \\leq m \\leq d$, a collection of $m$-planes $R_1, \\ldots, R_N$, and collections of $k$-planes ${\\bf L}_{R_1}, \\ldots, {\\bf L}_{R_N}$ such that\n\\begin{enumerate}[(a)]\n\\item the ${\\bf L}_{R_j}$ are pairwise disjoint subsets of ${\\bf L}$ with $L \\subset R_j$ for $L \\in {\\bf L}_{R_j}$;\n\\item if $m > k$ then $|{\\bf L}_{R_j}| \\geq  |F|^{(k+1)(m - 1 - k) + k}$;\n\\item if $m=k$ then $|{\\bf L}_{R_j}| = 1$;\n\\item \\label{epcondition} letting ${\\bf L}^m = \\bigcup_{j}{\\bf L}_{R_j}$, we have $|{\\bf L}^m| \\geq 2^{-(d-m+1)}|{\\bf L}|$;\n\\item \\label{genwolffcondition} if $m < m' \\leq d$ then for every $m'$-plane $S$, $|\\{L \\in {\\bf L}^m : L \\subset S\\}| < |F|^{(k+1)(m' - 1 - k) + k} .$\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSet ${\\bf L}^{*,d+1} := {\\bf L}.$ For $k \\leq m \\leq d$, suppose that ${\\bf L}^{*,m+1} \\subset {\\bf L}$ has been chosen so that $|{\\bf L}^{*,m+1}| \\geq 2^{-(d-m)} |{\\bf L}|$. Starting at $j = 1$, suppose $m$-planes $R^m_{j'}$ and collections of $k$-planes ${\\bf L}_{R^m_{j'}}$ have been selected for all integers $0 < j' < j$.\n\nIf there is an $m$-plane $R$ so that \n\\[\n|\\{L \\in {\\bf L}^{*,m+1} \\setminus \\bigcup_{j'  < j} {\\bf L}_{R^m_{j'}} : L \\subset R\\}| \\geq |F|^{(k+1)(m-1-k) + k}\n\\]\nthen let $R^m_j$ be such an $m$-plane, let\n\\[\n{\\bf L}_{R^m_j} = \\{L \\in {\\bf L}^{*,m+1} \\setminus \\bigcup_{j'  < j} {\\bf L}_{R^m_{j'}} : L \\subset R^m_j\\}\n\\]\nand continue the process with $j+1.$ If there is no such plane then terminate the process and set \n\\[\n{\\bf L}^{*,m} = {\\bf L}^{*,m+1} \\setminus \\bigcup_{j'  < j} {\\bf L}_{R^m_{j'}}.\n\\]\nIf $|{\\bf L}^{*,m}| < |{\\bf L}^{*,m+1}|\/2$ then property (\\ref{epcondition}) is satisfied and we terminate the process. Otherwise, continue with $m-1.$\n\nIf the process reaches the stage $m=k,$ then let $R^{k}_1, \\ldots, R^k_N$ be some enumeration of ${\\bf L}^{*,k+1}$ and ${\\bf L}_{R^{k}_j} = \\{R^k_j\\}$ and we are finished.\n\\end{proof}\n\nNext, we describe in detail the foliation of the ambient space.\n\n\\begin{lemma} \\label{folprop}\nSuppose that $S$ is an $m$-plane in $F^n$, that $0 \\leq q \\leq k-1,$ and that $m + k-q \\leq n.$ Then we can find $(m+k-q)$-planes $T_1, \\ldots, T_N$ with $S \\subset T_i$ for all $i$ such that for all $(k-1)$-planes $P$ and $k$-planes $L$ satisfying\n\\begin{enumerate}[(a)]\n\\item $P \\subset L$\n\\item $L \\cap S$ is a $q$-plane\n\\item $P \\cap S$ is a $(q-1)$-plane if $q > 0$ and $P \\cap S = \\emptyset$ if $q = 0$\n\\end{enumerate}\nwe have $L \\subset T_i$ for some $i$ and $P \\not\\subset T_{i'}$ for $i' \\neq i.$\n\\end{lemma}\n\n\\begin{proof}\nTo find the $T_i$, write $S=x + \\spa(e_1, \\ldots, e_m)$ and \n\\[\nF^n = \\spa(e_1, \\ldots, e_m, f_1, \\ldots f_{n-m}).\n\\]\nThen write $T_i = S + V_i$ where, as $i$ varies, $V_i$ ranges over all $(k-q)$-dimensional subspaces of $\\spa(f_1, \\ldots, f_{n-m}).$\n\nFix some $P,L$ satisfying the hypotheses. One can check that there is an $i$ such that $L \\subset T_i.$ For any $i' \\neq i$ we have that $T_i \\cap T_{i'}$ contains $S$ and is, at most, an $(m +  k-q - 1)$-plane. \n\nFirst consider the case $q > 0.$ Choose $y \\in P \\cap S$ and write \n\\begin{align*}\nP \\cap S &= y + \\spa(g_1, \\ldots, g_{q-1}),\\\\\nP &= y + \\spa(g_1, \\ldots, g_{q-1}, h_1, \\ldots, h_{k-q}),\\\\\n\\intertext{and}\nS &= y + \\spa(g_1, \\ldots, g_{q-1}, h'_{1}, \\ldots, h'_{m + 1 -q}).\n\\end{align*}\nClearly, \n\\[\nW := \\{g_1, \\ldots, g_{q-1}, h_1, \\ldots, h_{k-q}, h'_{1}, \\ldots, h'_{m + 1 -q } \\}\n\\]\nis linearly independent and, since $S \\subset T_i \\cap T_{i'}$, $P \\subset T_i \\cap T_{i'}$ would imply $y + \\spa(W) \\subset T_i \\cap T_{i'}$ contradicting the dimension estimate on the latter set.  \n\nFor the case $q=0$ write \n\\begin{align*}\nP &= z + \\spa(h_1, \\ldots, h_{k-1})\\\\\n\\intertext{and}\nS &= y + \\spa(h'_1, \\ldots, h'_m).\n\\end{align*}\nAgain using the dimension estimate on $T_i \\cap T_{i'}$, we see that if $P \\subset T_{i} \\cap T_{i'}$ then, since $P \\cap S = \\emptyset,$ we have $v \\in \\spa(h_1', \\ldots, h'_m)$ for some $0 \\neq v \\in \\spa(h_1, \\ldots, h_{k-1}).$ But, since $L \\cap S \\neq \\emptyset$, this implies $\\dim(L \\cap S) > 0$, contradicting the assumption that $\\dim(L \\cap S) = 0.$\n\\end{proof}\n\nTo estimate the union of lines or $k$-planes contained in each leaf of the foliation, we will appeal to recursion. However,  at the root we still use the classical method:\n \n\\begin{lemma} \\label{classmethod}\nSuppose that ${\\bf L}$ is a collection of $m$-planes, ${\\bf P}$ is a collection of $(k-1)$-planes such that for every $L \\in {\\bf L}$\n\\[\n|\\{P \\in {\\bf P} : P \\subset L\\}| \\geq M,\n\\]\nand \n\\begin{equation} \\label{mustpruneeq}\n|{\\bf L}| |F|^{k(m-k)} \\leq M.\n\\end{equation}\nThen\n\\[\n|{\\bf P}| \\gtrsim |{\\bf L}| M.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nFor each $L \\in {\\bf L}$, let ${\\bf P}_L$ be a subset of $\\{P \\in {\\bf P} : P \\subset L\\}$ with $M \\leq {\\bf P}_L \\leq 2M.$ Set\n\\[\n{U} = \\{(P,L,L')  : (L,L') \\in {\\bf L}^2, P \\in {\\bf P}_L \\cap {\\bf P}_{L'} \\}.\n\\]\nAn application of Cauchy-Schwarz gives \n\\[\n|{U}| \\geq \\frac{M^2|{\\bf L}|^2}{|{\\bf P}|}.\n\\]\nAny two distinct $m$-planes intersect in, at most, an $(m-1)$-plane. Since, by  \\eqref{plcount} below, an $(m-1)$-plane contains $\\lesssim |F|^{k(m-k)}$ $(k-1)$-planes, we have \n\\begin{align*}\n|{U}| &\\leq C |{\\bf L}|^2 |F|^{k(m-k)} + |\\{(P,L) : L \\in {\\bf L}, P \\in {\\bf P}_L\\}| \\\\\n&\\leq C|{\\bf L}|^2|F|^{k(m-k)} + |{\\bf L}| 2 M \\\\\n&\\lesssim |{\\bf L}|M.\n\\end{align*}\n\\end{proof}\n\nWe finish the section with three standard estimates for collections of planes.  \n\n\\begin{lemma} \\label{grascount}\nFor integers $0 \\leq k \\leq d$ we have\n\\begin{equation} \\label{sscount}\n|G(d,k)| \\approx |F|^{k(d-k)}\n\\end{equation}\nand\n\\begin{equation} \\label{plcount}\n|G'(d,k)| \\approx |F|^{(k+1)(d-k)}\n\\end{equation}\nwhere $G(d,k)$ is the set of $k$-dimensional subspaces of $F^d$ and $G'(d,k)$ is the set of $k$-planes in $F^d.$ \n\\end{lemma}\n\n\\begin{proof}\nA generic choice of $k$ vectors in $F^d$ is linearly independent, and there are approximately $|F|^{kd}$ such choices. By the same logic, each $k$-plane has approximately $F^{k^2}$ choices of basis, and hence we have \\eqref{sscount}. \n\nGiven a $k$-dimensional subspace $P$ with basis $e_1, \\ldots, e_k$, choose \\\\ $f_1, \\ldots, f_{d-k}$ so that $F^d = \\spa(e_1, \\ldots, e_k, f_1, \\ldots, f_{d-k})$. Then there is a one-to-one correspondence between linear combinations of $f_1, \\ldots, f_{d-k}$ and distinct translates of $P$, giving \\eqref{plcount}.\n\\end{proof}\n\n\n\\begin{lemma} \\label{sccprop}\nSuppose that $S$ is an $l$-plane in $F^m.$ Then for $l < l' \\leq m$\n\\begin{equation} \\label{scontainedcount}\n|\\{P \\subset F^m : P \\text{\\ is a\\ } l'\\text{-plane\\ and\\ } S \\subset P\\}| \\lesssim |F|^{(l' - l)(m - l')}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWrite $S = x + \\spa(e_1, \\ldots, e_l)$ and \n\\[\nF^m = \\spa(e_1, \\ldots, e_l, f_1, \\ldots, f_{m-l}).\n\\] \nThen there is a one-to-one correspondence between $l'$-planes $P \\supset S$ and $(l'-l)$-dimensional subspaces of $\\spa(f_1, \\ldots, f_{m-l})$, so \\eqref{scontainedcount} follows from \\eqref{sscount}.\n\\end{proof}\n\n\\begin{lemma} \\label{eiprop}\nSuppose $S$ is an $l$-plane in $F^k.$ Then\n\\[\n|\\{P \\subset F^k : P \\text{\\ is a\\ } (k-1)\\text{-plane\\ and\\ } P \\cap S = \\emptyset\\}| \\lesssim |F|^{k-l}.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nWrite $S = x + \\spa(e_1, \\ldots, e_l)$, and fix $P = y + \\spa(f_1, \\ldots, f_{k-1}).$ If $P \\cap S = \\emptyset$, we must have $e_j \\in \\spa(f_1, \\ldots, f_{k-1})$ for $j = 1, \\ldots, l.$ Thus, $P$ is a translate of a $(k-1)$-plane containing $S$. Since, by Lemma \\ref{sccprop}, there are $\\lesssim |F|^{k-l-1}$ $(k-1)$-planes containing $S$, we have at most $|F|^{k-l}$ possible planes $P$. \n\\end{proof}\n\n\n\\section{Unions of lines} \\label{linesection}\nTheorems \\ref{firstlinethm} and \\ref{wolffrelaxed} follow immediately from:\n\\begin{proposition} \\label{linetheorem}\nSuppose $d \\geq 1$ is an integer, that $0 < \\gamma,\\lambda \\leq 1$, that $\\max(1 - d,-1) \\leq \\beta \\leq 1,$ that ${\\bf L}$ is a collection of lines in $F^n$ with \n\\[\n|{\\bf L}| \\geq \\gamma |F|^{2(d-1)+\\beta},\n\\]\nand that ${\\bf P}$ is a collection of points in $F^n$ satisfying \n\\[\n|\\{P \\in {\\bf P} : P \\in L\\}| \\geq \\lambda |F|\n\\]\nfor every $L \\in {\\bf L}.$ Then\n\\begin{equation} \\label{nondpwaresult}\n|{\\bf P}| \\gtrsim  |F|^{d +\\max(0,\\beta)}.\n\\end{equation}\nwhere the implicit constant may depend on $d,\\gamma,\\lambda.$ Furthermore, if $d \\geq 2$ and ${\\bf L}$ satisfies the $d$-plane Wolff axiom \\eqref{dpwa} then we have\n\\begin{equation} \\label{dpwaresult}\n|{\\bf P}| \\gtrsim |F|^{d + \\frac{\\beta + 1}{2}}.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}[Proof of Theorem \\ref{qethm} assuming Proposition \\ref{linetheorem}]\nSuppose that ${\\bf L}$ satisfies \\eqref{qebound} and that $|F| \\geq M$ where $M$ is large and to be determined later.\n\n For $d$-planes $R$ let ${\\bf L}_R = \\{L \\in {\\bf L} : L \\subset R\\}$. Choose $d$-planes $R_1, \\ldots, R_N$ so that for each $j$, $|{\\bf L}_{R_j}| \\geq |F|^{2d-3}$ and such that for $R \\not\\in \\{R_j\\}_{j=1}^N$ we have $|{\\bf L}_{R}| < |F|^{2d-3}. $\n\nSetting ${\\bf L}' = \\bigcup_{j}{\\bf L}_{R_j}$, ${\\bf L}'' = {\\bf L} \\setminus {\\bf L}'$ and ${\\bf P}'' = \\bigcup_{L \\in {\\bf L}''}L$, we must have $|{\\bf L}''| < \\frac{1}{2}|{\\bf L}'|$ or \nelse we would have\n\\[\n|{\\bf P}| \\geq |{\\bf P}''| \\geq c' |F|^{d + \\frac{\\beta + 1}{2}} > C |F|^{d+\\beta}\n\\]\nwhere $c'$ is the implicit constant from \\eqref{dpwaresult} and $M$ is chosen large enough to overwhelm $\\frac{C}{c'}$ (In the second inequality above we have used the fact that ${\\bf L}''$ satisfies \\eqref{dpwa} to obtain \\eqref{dpwaresult} from Proposition \\ref{linetheorem}.) \n\nThus, $|{\\bf L}'| \\geq \\frac{1}{2}|{\\bf L}|$ and, by \\eqref{plcount} with $k=1$, $N \\geq c |F|^{\\beta}.$ Letting ${\\bf P}' = \\bigcup_{L \\in {\\bf L}'}L$ we have\n\\[\n|{\\bf P}'| \\geq c'' |F|^{d + \\beta} \\geq c C|F|^{d + \\beta} \\geq c |{\\bf P}|\n\\]\nwhere $c''$ is the implicit constant from \\eqref{nondpwaresult}, $c$ is chosen small enough to underwhelm $\\frac{c''}{C},$ and the last inequality follows from \\eqref{qebound}.\n\nA final application of $\\eqref{nondpwaresult}$ with $d' = d-1$ and $\\beta' = 1$ then gives \\eqref{denseindplane} since $|{\\bf L}_{R_j}| \\geq |F|^{2(d-1-1)+1}.$\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{linetheorem}]\nWe use induction on $d$. When $d=1$ the result follows from an application of Lemma \\ref{classmethod} (with $m=1$, $k=1$, and ${\\bf L}$ replaced by a subset of itself with cardinality $\\approx \\min(\\gamma,\\lambda) |F|^\\beta$ ), so we will prove it for $d > 1$, working under the assumption that it has already been proven for $1 \\leq d' < d.$\n\nAfter possibly deleting lines, we may assume\n\\begin{equation} \\label{refinedkk1}\n|{\\bf L}| < 2\\gamma |F|^{2(d-1)+\\beta}.\n\\end{equation}\nApplying Lemma \\ref{pslemma} to ${\\bf L}$ we obtain $m$-planes $R_1, \\ldots, R_N.$ Note that if $L$ satisfies the $d$-plane Wolff axiom \\eqref{dpwa} then we must have $m < d.$ \n\n\\ \n\n\\noindent{\\bf Case 1, $(m=d)$:}\\\\ \nLet ${\\bf P}_{R_j} = \\{P \\in {\\bf P} : P \\in R_j\\}.$ Applying the case $d' = d-1$ of the proposition to the ${\\bf L}_{R_j}$, we deduce that $|{\\bf P}_{R_j}| \\gtrsim |F|^{d}$ for each $j$. Since, by \\eqref{plcount}, $|{\\bf L}_{R_j}| \\lesssim |F|^{2(d-1)}$, we have $N \\gtrsim |F|^{\\max(0,\\beta)}.$ Then using Lemma \\ref{classmethod} (possibly applying it to a subset of $\\{R_j\\}_{j=1}^N$ in order to satisfy \\eqref{mustpruneeq}) to estimate $|\\bigcup_{j}{\\bf P}_{R_j}|$ shows that \n\\[\n|{\\bf P}| \\gtrsim  |F|^{d + \\max(0,\\beta)}\n\\]\nas desired.\n \n\\ \n\n\\noindent{\\bf Case 2, $(m < d)$:}\\\\\nWe construct sets of lines and points using a standard ``popularity'' argument. Fix some large $C$ to be determined later and let \n\\[\n{\\bf P}^{\\sharp} = \\{P \\in {\\bf P} : |\\{L \\in {\\bf L} : P \\in L\\}| \\geq C |F|^{d-1 - \\frac{1-\\beta}{2}}\\}\n\\]\nand\n\\[\n{\\bf L}^{\\sharp} = \\{L \\in {\\bf L} : |\\{P \\in {\\bf P}^{\\sharp} : P \\in L\\}| \\geq \\frac{1}{4} \\lambda |F|\\}.\n\\]\nLetting ${\\bf L}^m$ be as in Lemma \\ref{pslemma}, we then have either \n\\begin{align} \\label{firstpossibilityk1}\n|{\\bf P}| &\\geq \\frac{1}{2}\\lambda |F|  |{\\bf L}^m|\/(C|F|^{(d-1) - \\frac{1-\\beta}{2}})\n\\end{align}\n(in which case we are finished since the right hand side above is $\\gtrsim |F|^{d + \\frac{\\beta+1}{2}}$) or\n\\begin{equation} \\label{secondpossibilityk1}\n|{\\bf L}^{\\sharp}| \\geq \\frac{1}{8} |{\\bf L}^m|\n\\end{equation}\n\nIndeed, suppose that \\eqref{firstpossibilityk1} does not hold. Set\n\\[\nI = \\{(P,L) : P \\in {\\bf P}_L, L \\in {\\bf L}^{m}\\}\n\\]\nwhere, for each $L$, ${\\bf P}_L$ is a subset of ${\\bf P} \\cap L$ with $\\lambda |F| \\leq |{\\bf P}_L|  < 2 \\lambda |F|.$ \nThen letting \n\\[\nI' = \\{(P,L) : P \\in {\\bf P}_L \\setminus {\\bf P}^{\\sharp} , L \\in {\\bf L}^{m}\\}\n\\]\nwe have\n\\[\n|I'| < C |F|^{d-1 - \\frac{1-\\beta}{2}} |{\\bf P}| < \\frac{1}{2} |I|\n\\]\nand so\n\\[\n|\\{(P,L) : P \\in {\\bf P}_L \\cap {\\bf P}^{\\sharp}, L \\in {\\bf L}^{m}\\}| \\geq \\frac{1}{2} \\lambda |F| |{\\bf L}^{m}|\n\\]\ngiving \n\\[\n|\\{(P,L) : P \\in {\\bf P}_L \\cap {\\bf P}^{\\sharp}, L \\in {\\bf L}^{\\sharp}\\}| \\geq \\frac{1}{4} \\lambda |F||{\\bf L}^{m}|\n\\]\nthus leading (by the upper bound on $|{\\bf P}_L|$) to \\eqref{secondpossibilityk1} as claimed.\n\nLet ${\\bf L}^{\\sharp}_{R_j} = {\\bf L}_{R_j} \\cap {\\bf L}^{\\sharp}$, ${\\bf P}^{\\sharp}_{R_j} = R_j \\cap {\\bf P}^{\\sharp}$, \n\\[\n{\\bf L}'_{R_j} = \\{L \\in {\\bf L}^m : |L \\cap R_j| = 1\\},\n\\]\nand\n\\[\n{\\bf P}'_{R_j} = {\\bf P} \\setminus R_j.\n\\]\nFix $j$ so that $|{\\bf L}^{\\sharp}_{R_j}| \\gtrsim |{\\bf L}_{R_j}|,$ and recall $|{\\bf L}_{R_j}| \\geq |F|^{2m-3}$ by Lemma \\ref{pslemma}.\nApplying the previously known case $d' = m-1$ of the proposition (or using the trivial estimate if $m=1$) we have\n\\begin{equation} \\label{lequalszeroPk1}\n|{\\bf P}^{\\sharp}_{R_j}| \\gtrsim |F|^{m}.\n\\end{equation}\n\nFor each point $P \\in {\\bf P}^{\\sharp}_{R_j}$ there are $\\geq C |F|^{d-1 -  \\frac{1-\\beta}{2}}$ lines from ${\\bf L}^{m}$ intersecting $P$. Since, by Lemma \\ref{sccprop}, $\\lesssim |F|^{m-1}$ of these lines are contained in $R_j$, we have\n\\[\n|\\{L \\in {\\bf L}'_{R_j} : P \\in L\\}| \\geq \\frac{1}{2} |F|^{d - 1 - \\frac{1-\\beta}{2}}\n\\]\nprovided that $C$ is chosen sufficiently large. \nThus, \n\\begin{align*}\n|{\\bf L}'_{R_j}| &\\gtrsim |{\\bf P}^{\\sharp}_{R_j}| |F|^{d - 1 - \\frac{1 - \\beta}{2}} \\\\\n&\\gtrsim |F|^{m + d-1 - \\frac{1-\\beta}{2}}.\n\\end{align*}\nNote\\footnote{We may assume throughout that $|F|$ is sufficiently large relative to certain parameters (for instance $\\lambda$) since the implicit constants may be chosen so that the conclusion holds trivially for small $|F|$.} that for each $L \\in {\\bf L}'_{R_j}$, $|L \\cap {\\bf P}'_{R_j}| \\geq \\lambda |F| - 1 \\gtrsim |F|.$\n\n\nApplying Lemma \\ref{folprop}, we write $F^n$ as the union of $(m + 1)$-planes $T_i$ containing $R_j$. Let\n\\begin{align*}\n{\\bf L}_i &= \\{L \\in {\\bf L}'_{R_j} : L \\subset T_i\\} \\\\ \n{\\bf P}_i &= \\{P \\in {\\bf P}'_{R_j} : P \\in T_i\\}.\n\\end{align*}\nThen \n\\begin{align*}\n|{\\bf P}| &\\geq \\sum_i |{\\bf P}_i| \\\\ \n&\\gtrsim \\sum_i |{\\bf L}_i|\/|F|^{m-2} \\\\ \n&\\geq |{\\bf L}'_{R_j}|\/|F|^{m - 2}\\\\ \n&\\gtrsim |F|^{d +\\frac{\\beta+1}{2}}\n\\end{align*}\nwhere, for the second inequality, we used the fact (which follows from Lemma \\ref{pslemma}) that $|{\\bf L}_i| < |F|^{2(m -1) + 1} $ to see that $|{\\bf L}_i| = |F|^{2(d' - 1) + \\beta'}$ for some $d' \\leq m$ and so we can estimate $|{\\bf P}_i|$ using the previously known case $d'$ of \\eqref{nondpwaresult}.\n\\end{proof}\n\n\\section{Unions of Planes} \\label{planesection}\nTheorem \\ref{upthm2} is obtained by induction from the hyperplane case:\n\\begin{proposition} \\label{maintheoremcd1}\nSuppose $d,k > 0$ are integers, that $0 < \\gamma,\\lambda \\leq 1$, that  $d \\geq k$, that $0 \\leq \\beta \\leq k$, that ${\\bf L}$ is a collection of $k$-planes in $F^n$ with \n\\[\n|{\\bf L}| \\geq \\gamma |F|^{(k+1)(d-k)+\\beta},\n\\]\nand that ${\\bf P}$ is a collection of $(k-1)$-planes in $F^n$ satisfying \n\\[\n|\\{P \\in {\\bf P} : P \\subset L\\}| \\geq \\lambda |F|^{k}\n\\]\n for every $L \\in {\\bf L}.$ Then\n\\[\n|{\\bf P}| \\gtrsim  |F|^{k(d - k + 1)+\\beta}\n\\]\nwhere the implicit constant may depend on $d,\\gamma,\\lambda, k.$\n\\end{proposition}\n\n\\begin{proof}[Proof of Theorem \\ref{upthm2} assuming Proposition \\ref{maintheoremcd1}]\nTheorem \\ref{upthm2} follows directly from Proposition \\ref{maintheoremcd1} when $k-k' = 1$. Fix $k_0 \\geq 1$ and assume that the theorem holds for all $k-k' = k_0,$ and fix some $k,k'$ with $k-k' = k_0 + 1.$ \n\nFor $L \\in {\\bf L}$ satisfying \\eqref{upthm1hyp} let \n\\[\n{\\bf P}'_L = \\{(k'+1)\\text{-planes\\ }P' : P' \\subset L\\}\n\\] \nand for any $(k'+1)$-plane $P'$ let \n\\[\n{\\bf P}''_{P'} = \\{k'\\text{-planes\\ }P : P \\subset P'\\}.\n\\]\nApplying the previously known case of the theorem, we have\n\\[\n|\\bigcup_{L \\in {\\bf L}}{\\bf P}'_L| \\gtrsim |F|^{(k' + 2)(d - (k' + 1)) + \\beta}\n\\]\nand thus a second application of Proposition \\ref{maintheoremcd1} gives\n\\[\n|\\bigcup_{L \\in {\\bf L}}{\\bf P}_L| = |\\bigcup_{L} \\bigcup_{P' \\in {\\bf P}'_L} {\\bf P}''_{P'}| \\gtrsim |F|^{(k' + 1)(d - k') + \\beta}.\n\\]\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{maintheoremcd1}]\nWe use induction on $d$. When $d=k$ the result follows from an application of Lemma \\ref{classmethod} (with $m=k$). So we will prove it for $d > k$, working under the assumption that it has already been proven for $k \\leq d' < d.$\n\nAfter possibly deleting planes, we may assume\n\\begin{equation} \\label{refinedk}\n|{\\bf L}| < 2\\gamma |F|^{(k+1)(d-k)+\\beta}.\n\\end{equation}\nApplying Lemma \\ref{pslemma} to ${\\bf L}$ we obtain $m$-planes $R_1, \\ldots, R_N.$\n\n\\ \n\n\\noindent{\\bf Case 1, $(m=d)$:}\\\\ \nLet ${\\bf P}_{R_j} = \\{P \\in {\\bf P} : P \\subset R_j\\}.$ Applying the case $d' = d-1$ of the theorem to the ${\\bf L}_{R_j}$, we deduce that $|{\\bf P}_{R_j}| \\gtrsim |F|^{k(d-k+1)}$ for each $j$. Since, by \\eqref{plcount}, $|{\\bf L}_{R_j}| \\lesssim |F|^{(k+1)(d-k)}$, we have $N \\gtrsim |F|^{\\beta}.$ Using Lemma \\ref{classmethod} to estimate $|\\bigcup_{j}{\\bf P}_{R_j}|,$ we conclude\n\\[\n|{\\bf P}| \\gtrsim  |F|^{k(d-k+1)+\\beta}\n\\]\nas desired.\n \n\\ \n\n\\noindent{\\bf Case 2, $(m < d)$:}\\\\\nWe construct sets of $k$-planes and $(k-1)$-planes using a standard ``iterated-popularity'' argument. Fix some large $C$ to be determined later. Let ${\\bf L}^{\\sharp,0} = {\\bf L}^m$, ${\\bf P}^{\\sharp,0} = {\\bf P}$ and for $1 \\leq q \\leq k$ let\n\\[\n{\\bf P}^{\\sharp,q} = \\{P \\in {\\bf P}^{\\sharp,q-1} : |L \\in {\\bf L}^{\\sharp, q-1} : P \\subset L| \\geq C|F|^{d-k}\\}\n\\]\nand\n\\[\n{\\bf L}^{\\sharp,q} = \\{L \\in {\\bf L}^{\\sharp,q-1} : |P \\in {\\bf P}^{\\sharp,q} : P \\subset L| \\geq 2^{-2q} \\lambda |F|^k\\}.\n\\]\nThen either \n\\begin{align} \\label{firstpossibility}\n|{\\bf P}| &\\geq 2^{-5k}\\lambda |F|^k  |{\\bf L}^m|\/(C|F|^{d-k})\n\\end{align}\n(in which case we are finished since the right hand side above is $\\gtrsim |F|^{k(d-k+1) + \\beta}$) or for each $q \\leq k$\n\\begin{equation} \\label{secondpossibility}\n|{\\bf L}^{\\sharp,q}| \\geq 2^{-3q} |{\\bf L}^m|.\n\\end{equation}\n\nIndeed, suppose that \\eqref{firstpossibility} does not hold and that \\eqref{secondpossibility} holds for $0 \\leq q \\leq q_0 < k$. Set \n\\[\nI = \\{(P,L) : P \\in {\\bf P}_L, L \\in {\\bf L}^{\\sharp,q_0}\\}\n\\]\nwhere, for each $L$, ${\\bf P}_L$ is a subset of ${\\bf P}^{\\sharp,q_0}$ with $P \\subset L$ for every $P \\in {\\bf P}_L$ and $2^{-2q_0} \\lambda |F|^k \\leq |{\\bf P}_L|  < 2^{-(2q_0-1)} \\lambda |F|^k.$ Note\n\\[\n|I| \\geq 2^{-2q_0} \\lambda |F|^k |{\\bf L}^{\\sharp,q_0}| \\geq 2^{-5q_0} \\lambda |F|^k |{\\bf L}^m|.\n\\]\nThen letting \n\\[\nI' = \\{(P,L) : P \\in {\\bf P}_L \\setminus {\\bf P}^{\\sharp,q_0 + 1} , L \\in {\\bf L}^{\\sharp,q_0}\\}\n\\]\nwe have\n\\[\n|I'| < C|F|^{d-k} |{\\bf P}| \\leq 2^{-5(k - q_0)} |I| \\leq \\frac{1}{2} |I|\n\\]\nand so\n\\[\n|\\{(P,L) : P \\in {\\bf P}_L \\cap {\\bf P}^{\\sharp,q_0 + 1}, L \\in {\\bf L}^{\\sharp,q_0}\\}| \\geq \\frac{1}{2} 2^{-2q_0} \\lambda |F|^k |{\\bf L}^{\\sharp,q_0}|.\n\\]\nThis gives\n\\[\n|\\{(P,L) : P \\in {\\bf P}_L \\cap {\\bf P}^{\\sharp,q_0 + 1}, L \\in {\\bf L}^{\\sharp,q_0+1}\\}| \\geq \\frac{1}{4} 2^{-2q_0} \\lambda |F|^k |{\\bf L}^{\\sharp,q_0}|\n\\]\nthus leading (by the upper bound on $|{\\bf P}_L|$) to \n\\[\n|{\\bf L}^{\\sharp,q_0 + 1}| \\geq \\frac{1}{8} |{\\bf L}^{\\sharp,q_0}|\n\\]\nas claimed.\n\nFor each $1 < q \\leq k$ let \n\\begin{align*}\n{\\bf L}^{\\sharp,q}_{R_j} &= \\{L \\in {\\bf L}^{\\sharp,q} : L \\cap R_j \\text{\\ is a }q\\text{-plane}\\} \\\\\n{\\bf P}^{\\sharp,q}_{R_j} &= \\{P \\in {\\bf P}^{\\sharp,q} : P \\cap R_j \\text{\\ is a }(q-1)\\text{-plane}\\}.\n\\end{align*}\nDefine ${\\bf L}^{\\sharp,0}_{R_j}$ as above and let ${\\bf P}^{\\sharp,0}_{R_j} = \\{P \\in {\\bf P}^{\\sharp,0} : P \\cap R_j = \\emptyset\\}.$\n\nLetting ${\\bf L}_{R_j}$ be as in Lemma 2.1, fix $j$ so that $|{\\bf L}^{\\sharp,k}_{R_j}| \\gtrsim |{\\bf L}_{R_j}|.$\nApplying the previously known case $d' = m-1$ of the theorem (or using the trivial estimate if $m=k$) we have\n\\begin{equation} \\label{lequalszeroP}\n|{\\bf P}^{\\sharp,k}_{R_j}| \\gtrsim |F|^{k(m-k+1)}.\n\\end{equation}\n\nSuppose for some $1 \\leq q \\leq k$ that  \n\\begin{equation} \\label{equationA}\n|{\\bf P}^{\\sharp,q}_{R_j}| \\gtrsim |F|^{\\omega}.\n\\end{equation}\nOur immediate goal is to see that under certain conditions, \\eqref{equationA} implies \\eqref{equationB} and \\eqref{equationstar} below. For each $P \\in {\\bf P}^{\\sharp,q}_{R_j}$ there are $\\geq C |F|^{d-k}$ $k$-planes from ${\\bf L}^{\\sharp,q-1}$ containing $P$. Since, by Lemma \\ref{sccprop}, $\\lesssim |F|^{m-q}$ of these $k$-planes intersect $R_j$ in $q$-planes (and there is no possibility that for a $k$-plane $L \\supset P$ we have $\\dim(L \\cap R_j) > q$, since $\\dim(L) = \\dim(P) + 1$), we have\n\\[\n|\\{L \\in {\\bf L}^{\\sharp,q-1}_{R_j} : P \\subset L\\}| \\geq \\frac{1}{2} C |F|^{d - k}\n\\]\nprovided that $C$ is chosen sufficiently large and \n\\begin{equation} \\label{firstiterationrequirement}\nm + k-q \\leq d . \n\\end{equation}\nThus, using the fact (which follows from Lemma \\ref{sccprop}) that for each $(q-1)$-plane $S \\subset L$ there are at most $|F|^{k-q}$ $(k-1)$-planes $P$ with $S \\subset P \\subset L$, we have\n\\begin{equation} \\label{equationB}\n|{\\bf L}^{\\sharp,q-1}_{R_j}| \\gtrsim C |F|^{\\omega + (d-k) - (k-q)}\n\\end{equation}\nassuming \\eqref{firstiterationrequirement}. \n\nIt follows from the definition of ${\\bf L}^{\\sharp,q-1}$ that for each $L \\in {\\bf L}^{\\sharp,q-1}_{R_j}$\n\\[\n|\\{P \\in {\\bf P}^{\\sharp,q-1}: P \\subset L\\}| \\gtrsim |F|^k.\n\\]\nThus, using Lemma \\ref{sccprop} to obtain\n\\[\n|\\{P \\subset L : (L \\cap R_j) \\subset P\\}| \\lesssim |F|^{k-q} \\ll |F|^k\n\\]\nand (for $q > 1$) Lemma \\ref{eiprop} to obtain\n\\begin{equation} \\label{eipropapp}\n|\\{P \\subset L : (L \\cap R_j) \\cap P = \\emptyset\\}| \\lesssim |F|^{k-q+1} \\ll |F|^k\n\\end{equation}\nit follows that\n\\begin{equation} \\label{rieqn}\n|\\{P \\in {\\bf P}^{\\sharp,q-1}_{R_j}: P \\subset L\\}| \\gtrsim |F|^k.\n\\end{equation}\nEstimate \\eqref{rieqn} also holds when $q=1$, since the special definition of ${\\bf P}^{\\sharp,0}_{R_j}$ means that we do not need to use \\eqref{eipropapp}.\n\nApplying Lemma \\ref{folprop}\ngives $(m + k -q  + 1)$-planes $T_i$ containing $R_j$. Let\n\\begin{align*}\n{\\bf L}_i &= \\{L \\in {\\bf L}^{\\sharp,q-1}_{R_j} : L \\subset T_i\\} \\\\ \n{\\bf P}_i &= \\bigcup_{L \\in {\\bf L}_i}\\{P \\in {\\bf P}^{\\sharp,q-1}_{R_j}: P \\subset L\\}.\n\\end{align*}\nThen assuming \n\\begin{equation} \\label{seconditerationrequirement}\nm + k -q + 1\\leq d\n\\end{equation}\nwe have\n\\begin{align}\n\\nonumber |{\\bf P}^{\\sharp,q-1}_{R_j}| &\\geq \\sum_i |{\\bf P}_i| \\\\ \n\\nonumber&\\gtrsim \\sum_i |{\\bf L}_i|\/|F|^{m-q-k} \\\\ \n\\nonumber&\\geq |{\\bf L}^{\\sharp,q-1}_{R_j}|\/|F|^{m-q - k}\\\\ \n\\label{equationstar}&\\gtrsim C |F|^{\\omega + d - (m+k-q) + q}\n\\end{align}\nwhere, for the second inequality, we used the fact (which follows from Lemma \\ref{pslemma}) that $|{\\bf L}_i| < |F|^{(k+1)(m -q) + k} $ to see that for some $d' \\leq m+k-q$ we can estimate $|{\\bf P}_i|$ using the previously known case $d'$ of the theorem.\n\nStarting with \\eqref{lequalszeroP} and iterating the fact that \\eqref{equationA} implies \\eqref{equationB} and \\eqref{equationstar}, we obtain for $0 \\leq q \\leq k$\n\\[\n|{\\bf P}^{\\sharp,q}_{R_j}| \\gtrsim C^{k-q} |F|^{k + q(m-k) + (k-q)d - (k-q)(k-q-1)}\n\\]\nif $m + k-q \\leq d$ and\n\\[\n|{\\bf L}^{\\sharp,q}_{R_j}| \\gtrsim C^{k-q} |F|^{k + (q+1)(m-k) + (k-q-1)d - (k-q-1)(k-q-2) + (d - k) - (k-q-1)} \n\\]\nif $m + k-q - 1 \\leq d.$\n\nSo if $m+k \\leq d$ we have\n\\[\n|{\\bf P}| \\geq |{\\bf P}^{\\sharp,0}_{R_j}| \\gtrsim |F|^{k(d - k + 1) + k}\n\\]\nas desired, and otherwise we have\n\\begin{equation} \\label{toomanykplanes}\n|{\\bf L}| \\geq |{\\bf L}^{\\sharp,k-(d + 1 - m)}_{R_j}| \\gtrsim C^{d+1-m} |F|^{(k+1)(d-k) + k}.\n\\end{equation}\nChoosing $C$ sufficiently large depending on the relevant implicit constants (none of which depended on $C$), we see that \\eqref{toomanykplanes} contradicts the assumption \\eqref{refinedk} and so we must have \\eqref{firstpossibility}.\n\\end{proof}\n\n\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\noindent\nExplosive nuclear burning in astrophysical environments produces\nunstable nuclei which again can be targets for subsequent reactions.\nMost of these nuclei are not accessible in terrestrial laboratories or not fully\nexplored by experiments, yet.\n\nApproximately half of all stable nuclei observed in nature in the heavy\nelement region $A>60$ were produced in the so-called r-process (i.e.,\nrapid neutron capture process), which is believed to occur\nin type II supernova explosions (see e.g., \\cite{cowan,ohurev}).\nAn environment with a high neutron density is the prerequisite for such an\nr-process, in which heavier elements are built up from seed elements by\nconsecutive neutron captures and $\\beta$-decays. Because of the abundant\nneutrons, a multitude of neutron captures ($\\simeq 15-35$)\nmay occur until the $\\beta$-decay\nhalf-life becomes shorter than the half-life against neutron capture. Thus,\nthe r-process path along which reactions take place, is pushed off the\nregion of stability towards neutron-rich unstable nuclei. The location of\nthe path has consequences for the resulting nuclear abundances, calculated in\nastrophysical models~\\cite{friedel,chen96}.\n\nFor most of the required neutron capture cross sections the statistical\nmodel (compound nucleus (CN) mechanism, Hauser-Feshbach approach)\ncan be applied. This\nmodel employs a statistical average over resonances, for which one has to\nknow level densities but not necessarily exact excitation energies and level\nspin assignments. However, the criterion for the applicability of that model\nis a sufficiently high level density. Especially for some light nuclei it\nhas been known for years that the statistical model cannot be applied and that\nthe direct capture (DC) mechanism dominates the cross sections. Nevertheless,\nit has only been realized recently that also for intermediate and heavy \nnuclei the\ndirect mechanism can become important near shell closures and for neutron-rich\nisotopes when the level density becomes too low for the CN\nmechanism. When approaching the drip-line, neutron separation energies\ndecrease and the nuclei become less deformed, both leading to a smaller\nlevel density at the relevant projectile energy. This relevant energy is\ndetermined by the peak\n$E=kT$ of the Maxwell-Boltzmann velocity distribution of the neutron gas.\nIf a segment of the r-process path at a given element lies close enough\nto the drip-line, the statistical model will not be applicable anymore and\nthe DC reactions will dominate~\\cite{mat83,gor97}.\n\nThe relation between DC and CN mechanisms has already been studied\nfor neutron capture by light\nand intermediate target nuclei~\\cite{ohurev,bal,gru,mei,meis,bee,kra}.\nInvestigations of the dependence of the level density on charge and mass number\nand a discussion of the applicability of the statistical model have been\ngiven elsewhere~\\cite{tfkl1}. In this paper we want to investigate\ndirect neutron capture on neutron-rich Sn and Pb isotopes with the emphasis\non discussing the difficulties, the level of reliability as well as\nthe predictive power of\ntheoretical calculations.\n\nThe main problem for the DC predictions is that neutron separation\nenergies and level properties (excitation energies, spins, parities)\nhave to be\nknown accurately, contrary to a statistical calculation in which it is\nsufficient to know the level density. As\nin the foreseeable future one can not expect\nany experimental\ninformation for the majority of nuclei close to the drip-line,\none has to turn to theory for providing the input for the DC\ncalculations. At the moment, there are several microscopic and\nmacroscopic-microscopic descriptions competing in the quest for predicting\nnuclear properties far off stability. For the first time, in this work we\nwant to investigate the difference in the level structure between several\nmodels and its impact on predicted neutron capture cross sections. The\ncompared models are a Hartree-Fock-Bogoliubov (HFB) model with the Skyrme SkP\nforce~\\cite{doba1,doba}, a relativistic mean field theory (RMFT) with the \nparameter set NLSH~\\cite{sharma1,sharma2}, and the macroscopic-microscopic\nfinite-range droplet model FRDM (1992) which was also used in\ncalculations of nuclear ground-state masses and \ndeformations~\\cite{moeller,moell2} and in calculations of quantities of\nastrophysical interest~\\cite{moellkra}.\n\nIn Section~\\ref{secDC} we very briefly introduce the method of the DC\ncalculation and Section~\\ref{secMic} gives an overview of the utilized\nmicroscopic models. For $^{208}$Pb, the DC results can directly be compared to\nexperimental values. This is described in Section~\\ref{secExp}.\nIn the following Sections~\\ref{secPb} and ~\\ref{secSn} we present our results\nfor the heavy Pb and Sn isotopes. Possible astrophysical signatures and\nremaining uncertainties are discussed in Section~\\ref{SecDis}.\nThe paper is concluded by the summary section~\\ref{summary}.\n\n\n\n\\section{Direct Capture and Folding Procedure}\n\\label{secDC}\n\\noindent\nThe theoretical cross section $\\sigma^{\\mathrm{th}}$ is derived from the\nDC cross section $\\sigma^{\\mathrm{DC}}$ given by~\\cite{kra,kim}\n\\begin{equation}\n\\sigma^{\\mathrm{th}}=\\sum_{i}C_i^2S_i\\sigma^{\\mathrm{DC}}_i \\quad .\n\\end{equation}\nThe sum extends over all possible final states (ground state and\nexcited states) in the residual nucleus. The isospin Clebsch-Gordan\ncoefficients and spectroscopic factors are denoted by $C_i$ and $S_i$,\nrespectively. The DC cross sections $\\sigma^{\\mathrm{DC}}_i$ are\nessentially determined by the overlap of the scattering wave function in\nthe entrance channel, the bound-state wave function in the exit channel,\nand the multipole transition operator. For the computation of the DC\ncross section we used the direct capture code TEDCA~\\cite{TEDCA}, which\nincludes E1, M1 and E2 transitions.\n\nFor determining the nucleon-nucleus potential the folding procedure was\nemployed, a method already successfully applied in the description of\nmany systems. In this approach the nuclear target\ndensity $\\rho_{\\mathrm{T}}$ is folded\nwith an energy and density dependent nucleon-nucleon interaction\n$v_{\\mathrm{eff}}$~\\cite{kob}:\n\\begin{equation}\nV(R)=\\lambda V_{\\mathrm{F}}(R)=\\lambda \\int \\rho_{\\mathrm{T}}\n(\\bbox{r})v_{\\mathrm{eff}}(E,\\rho_{\\mathrm{T}},\n\\vert\\bbox{R} - \\bbox{r} \\vert )d\\bbox{r} \\quad ,\n\\end{equation}\nwith $\\bbox{R}$ being the separation of the centers of mass of the two\ncolliding nuclei. The normalization factor $\\lambda$ accounts for\neffects of antisymmetrization and is close to unity. The nuclear density\n$\\rho_{\\mathrm{T}}$ can be derived from experimental charge\ndistributions or from theory. The potential\nobtained in this way ensures the correct behavior of the wave functions\nin the nuclear exterior. At the low energies considered in astrophysical\nevents the imaginary parts of the optical potentials are small.\n\nIn connection with the results presented below it is useful to recapitulate\nthe sensitivity of the DC calculations to various elements of the\ndescription. In ascending importance,\nin the present context the DC is sensitive to the optical potential and\ndensity distribution, respectively, the reaction $Q$-value, and the spin and\nparity of a level.\n\nFor the accuracy attempted here, there is almost no\ndifference in the results obtained by employing the optical potentials\nderived from the density distributions of the different models while leaving\nall other properties unchanged.\n\nA stronger dependence is seen when examining changes in the $Q$-value. An \nincrease\nin the $Q$-value will give a non-linear increase in the resulting cross section.\nAs the $Q$-value is computed as the difference in the binding energies of\ntarget and residual nucleus (i.e., the neutron separation energy) minus the\nexcitation energy of the level into which the neutron is captured\n\\begin{equation}\nQ_i=(B_{\\mathrm{T}}-B_{\\mathrm{R}})-E_i=S_{\\mathrm{n}}-E_i\\quad,\n\\end{equation}\nthe cross section will be sensitive to the masses (separation energies) derived\nin the different microscopic models as well as the level structure (excitation\nenergies) given in these models.\n\nThe by far strongest sensitivity is that to spins and parities of the\ninvolved initial and final states. In\norder to comply with the electromagnetic selection rules, a state has to\nhave the proper parity to contribute to the cross section significantly.\nThe dominant contribution to the DC cross section will stem from an E1\ntransition. In this case, parity has to change.\nConsequently, the capture of an incoming neutron $p$-wave will be important \nfor the Pb isotopes, whereas\n$s$-wave capture is dominating in the Sn cases.\nFurthermore, significant contributions only arise from low spin states\nlike 1\/2 and 3\/2 states, whereas the capture to levels with higher spins is \nstrongly suppressed.\nIn this respect, it will prove to be important that the different\nmicroscopic models make different predictions on which states are\nneutron-bound and which are not, since DC can only populate bound states.\n\n\\section{The Microscopic Input}\n\\label{secMic}\n\\noindent\nThe energy levels, masses, and nuclear density distributions\nneeded as input for the DC calculation were\ntaken from three different approaches. The first one was the\nRMFT which has turned out to be a successful tool\nfor the description of many nuclear properties~\\cite{gam}. The RMFT\ndescribes the nucleus as a system of Dirac nucleons interacting via\nvarious meson fields. There are six parameters which are usually\nobtained by fits to finite nuclear properties. For our calculations we\nhave used the parameter set NLSH~\\cite{sharma1,sharma2}.\n\nThe second method was FRDM (1992), which is a macroscopic-microscopic\nmodel based on the finite-range droplet macroscopic model and a\nfolded-Yukawa single-particle potential~\\cite{moeller}.\nFor pairing, the\nLipkin-Nogami pairing model~\\cite{yuk} is employed. This model proved to\nbe very successful in reproducing ground state spins along magic\nnumbers~\\cite{moell1} and\nhas been used in QRPA calculations of $\\beta$-decay half\nlives~\\cite{moell1,moellkra} and\nfor nuclear mass determinations~\\cite{moell2}.\n\nFinally, we also utilized the self-consistent mean field HFB\nmodel~\\cite{doba1,doba} in which the nuclear states are calculated by a one-step\nvariational procedure minimizing the total energy with respect\nto the occupation factors and the single-particle wave functions simultaneously.\n\nTo be able to compare the predictions from all of the models the nuclei were\nconsidered to be spherically symmetric. The limitations of such a\nrestriction\nare discussed in Section \\ref{SecDis}.\n\n\\newcommand{$^{208}$Pb(n,$\\gamma$)$^{209}$Pb}{$^{208}$Pb(n,$\\gamma$)$^{209}$Pb}\n\\section{Comparison with Experiments for the {\\protect$^{208}$Pb(n,$\\gamma$)$^{209}$Pb} reaction}\n\\label{secExp}\n\n\\noindent\nRecently, it became possible to extract the non-resonant part of the\nexperimental capture cross section for the\n$^{208}$Pb(n,$\\gamma$)$^{209}$Pb reaction~\\cite{corvi}. In that work, high\nresolution neutron capture measurements were carried out in order to\ndetermine twelve resonances in the range 1--400 keV. From these values the\nresonant Maxwellian-averaged cross section $<\\sigma>^{\\mathrm{R}}_{30\n\\mathrm{keV}}$=0.221(27)\\,mb was calculated.\nMeasurements of the total cross section using neutron\nactivation~\\cite{macklin,ratzel} are also available at 30 keV,\nyielding the value\n$<\\sigma>^{\\mathrm{t}}_{30\\mathrm{keV}}$=0.36(3)\\,mb. By a simple\nsubtraction of the\nresonant part from the total cross section the value of\n$<\\sigma>^{\\mathrm{NR}}_{30\\mathrm{keV}}$=0.14(4)\\,mb can be deduced\nfor the\nnon-resonant capture cross section.\n\nUsing the experimentally known density distributions~\\cite{devries},\nmasses~\\cite{audi} and energy levels~\\cite{nucldatasheets}, we\ncalculated the non-resonant contribution in the DC model. The strength\nparameter $\\lambda$ of the folding potential in the neutron channel was\nfitted to experimental scattering data at low energies~\\cite{scdata}.\nThe value of $\\lambda$ for the bound state is fixed by the requirement\nof correct reproduction of the binding energies.\nThe spectroscopic factors for the relevant low lying states of $^{209}$Pb\nare close to unity as can be inferred from\ndifferent $^{208}$Pb(d,p)$^{209}$Pb reaction data~\\cite{nucldatasheets}.\nFor the Maxwellian-averaged non--resonant DC cross section we obtained\n$<\\sigma>^{\\mathrm{DC}}_{30\\mathrm{keV}}=0.135$ mb, which is in excellent\nagreement with experiment. The by far highest contributions to the DC\ncross section come from the E1 $p$-wave capture to the low spin states\n$J^{\\pi}=1\/2^+, 3\/2^+, 5\/2^+$. Capture to the other states is negligible.\n\nIn order to test the different microscopic approaches we also calculated\nnon-resonant DC on $^{208}$Pb by consistently\ntaking the input (energy levels, masses\nand nuclear\ndensities) from the models described above.\nAgain, the strength parameter $\\lambda$ of the folding potential in the\nentrance channel was adjusted to the elastic scattering data for each\nof the models. The calculations for the neutron capture cross\nsections yield  0.0289 mb, 0.0508 mb, and 0.0135 mb\nfor RMFT, FRDM, and HFB, respectively.\nHence, each of the models gives a smaller value for\nthe Maxwellian-averaged 30 keV capture cross section than the\ncalculation using experimental input data. The differences are due to the\nneutron separation energies and level schemes of the relevant states in\n$^{209}$Pb (see\nFig.~\\ref{209}) in the microscopic models, leading to different $Q$-values\nfor capture to the\nexcited states ($J^{\\pi}=1\/2^+,3\/2^+,5\/2^+$).\nIt should be noted that in Fig.~\\ref{209} only those theoretical levels\nare shown which contribute to the cross section, i.e. only particle states.\nCapture into hole states is strongly suppressed by the fact that a re-ordering\nprocess would be required in the final nucleus (see e.g.~\\cite{tomenam} for\na similar case). This would be reflected in\nextremely small spectroscopic\nfactors. Therefore, the DC to such states is negligible.\n\n\\section{Results for Neutron-rich Pb Isotopes}\n\\label{secPb}\n\\noindent\nWe also investigated the model dependence of neutron capture on the\nneutron-rich even-even isotopes $^{210-238}$Pb. For these isotopes\nexperimental data are only available near the region of stability. For\nmore neutron-rich nuclei one has to rely solely on input parameters from\nmicroscopic models. In this and the following section we compare cross\nsections calculated with the nuclear properties predicted by different\nnuclear-structure models. Therefore, we consider nuclear cross sections\ninstead of Maxwellian-averaged ones as in the previous section.\n\nHaving obtained the relevant spins and calculated the $Q$-values from\nthe masses as\ndiscussed above, we still had to determine the scattering potentials\nwith their respective strength parameters\n(see Eq.~2). As a first\nstep, the folding potentials were calculated, using the density\ndistributions taken from the three different nuclear-structure models\n(HFB, RMFT, FRDM).\nIn the potentials for each of the isotopes a factor $\\lambda$ was chosen\ngiving the\nsame volume integral as for the fitted $^{208}$Pb+n potential, which was\nobtained as described in the previous section. This is justified because\nit is known that the\nvolume integrals only change very slowly when adding neutrons to a\nnucleus~\\cite{satchler}.\nFor the bound\nstate potentials $\\lambda$ is fixed by the requirement of correct\nreproduction of the binding energies. The spectroscopic factors were\nassumed to be unity for all transitions considered.\n\nThe results of our calculations are summarized in Fig.~\\ref{vgl}. For\ncomparison, the levels from all of the models\nfor $^{219}$Pb, $^{229}$Pb, and $^{239}$Pb are shown in\nFigs.~\\ref{level1}--\\ref{level3}.\nThe most\nstriking feature in Fig.~\\ref{vgl} is the sudden drop over several\norders of magnitude in\nthe cross sections calculated with the RMFT levels in the mass range\n$A=212-220$. This is due to the\nlack of low spin levels which are cut off by the decreasing neutron\nseparation energy.  Only after the\n1i$_{11\/2}$ orbital\n(which forms the state at lowest energy in the RMFT) has been filled\ncompletely at\n$^{222}$Pb the\ncross section is increasing because low spin states become\navailable again. A similar gap is seen for $A=230-232$, and it is\nexpected that those gaps will repeatedly appear when approaching the\ndrip-line.\nSince in some cases there are unbound low spin states\nclose to the threshold a small shift in the level\nenergies could already close such a gap. However, note that the level spacing\nin the\nRMFT has the tendency to increase towards neutron rich\nnuclei~\\cite{sharma3},\ncontrary to the FRDM and the HFB prediction.\n\nThe values resulting from the FRDM exhibit a smoother and\nalmost constant\nbehavior in the considered mass range.\nOnly a slight dip is visible for $^{220}$Pb(n,$\\gamma$) since\nthe previously accessible 1\/2$^+$ and 3\/2$^+$ states have become\nunbound in $^{221}$Pb. The 2g$_{9\/2}$ orbital is at lower energy than the\n11\/2$^+$ level in this model.\nBeyond $^{223}$Pb it has been filled and\nat least one of the low spin states can be populated again. The known\nground state spins for the lighter isotopes are also reproduced\ncorrectly. For higher mass numbers the cross sections are similar to the\nones obtained in the HFB model.\n\nFor mass numbers below $A=232$, the HFB capture cross sections\nare always larger than those obtained in the other models.\nAlthough the neutron separation energies are also decreasing, the $Q$-values\nfor the capture to the low spin states \nbecome even larger, because the states are moving towards lower\nexcitation energies.\nIn general, the HFB cross sections of the investigated capture reactions\nexhibit a very smooth behavior with increasing neutron number.\n\n\\section{Results for Sn Isotopes}\n\\label{secSn}\n\\noindent\nProceeding in the same manner as for the Pb isotopes (Sec.\\ \\ref{secPb}),\nwe extended our investigation to the Sn nuclei.\nHere, the situation is different in\ntwo ways: Firstly, the drip-line lies at relatively much lower\nneutron numbers and the r-process path is not so\nfar off stability, and secondly, there are more experimental data available\nalso for the unstable nuclei close to or in the r-process path, which makes\na test of theoretical models possible.\n\nAgain, we took the nuclear properties and density distributions from the\nabove described models. The strengths of the scattering potentials were\nadjusted to reproduce the same value of the volume integral of 425 MeVfm$^3$\nas determined from the experimental elastic scattering data on the stable\nSn isotopes~\\cite{bal}.\nWe calculated the capture cross sections from\nthe stable isotope $^{124}$Sn out to the r-process path which\nis predicted at a neutron separation energy of about 2 MeV~\\cite{friedel}. \nAs the\nmodels make different predictions about masses and separation energies,\nthe r-process path is located at different mass numbers: $A\\simeq 135$ for\nRMFT and FRDM and $A \\simeq 145$ in the case of HFB.\nContrary to the Pb isotopes for which the $p$-wave capture is the main \ncontribution\nallowed by the electromagnetic selection rules, the Sn cross sections\nare dominated by the $s$-wave captures, due to the negative parities of the \nfinal states.\n\nThe level schemes of the $^{125}$Sn, $^{133}$Sn, and $^{141}$Sn\nnuclei are shown in\nFigs.~\\ref{sn1}--\\ref{sn3}, and the resulting cross sections\nfor all considered nuclei and models\nare combined in Fig.~\\ref{snfig}.\nSimilarly as in the Pb case,\nthe dependence of the cross sections on the mass number can be understood\nby considering the excitation energies of the low-spin states relative to the\nneutron separation energy predicted in various models\n(Figs.~\\ref{spinhfb}--\\ref{spinfrdm}). The 3\/2$^-$ state is bound in the\nFRDM already at low mass number, whereas it becomes bound only at $A=131$\nand $A=133$ in HFB and RMFT, respectively. Therefore, the FRDM cross sections\nare larger than the ones from HFB and RMFT for $A<133$. The drop in the\nFRDM cross sections beyond the $N=82$ shell is due to the fact that the\n1\/2$^-$ and 3\/2$^-$ states slowly become unbound (see Fig.~\\ref{spinfrdm}).\nIn the HFB model the two low-spin states move down in energy faster than\nthe neutron separation energy, thus providing an increasing $Q$-value and\nslightly increasing cross sections (Fig.~\\ref{spinhfb}).\nA similar trend can be found in the\nlevels from RMFT, although with a less pronounced increase of the $Q$-value\n(Fig.~\\ref{spinrmf}).\n\nThere are no data available concerning the pure DC contribution\nto the cross sections for the neutron-rich Sn isotopes. However, there is\nexperimental information regarding masses and level schemes. This can be\ncompared to theory (see Fig.~\\ref{sn2}).\nFor the experimentally\nknown isotope $^{133}$Sn we calculated DC by taking the experimentally known\nmasses and levels~\\cite{hoff} as input for the DC-calculation,\nthus arriving at a pseudo-experimental value for\nthe cross section which can be compared to the purely theoretical predictions.\nThe resulting value is marked by a cross in Fig.~\\ref{snfig}.\nNeutron capture on $^{132}$Sn is particularly\ninteresting because $^{133}$Sn is predicted to be already very close to\nthe r-process path by the two models RMFT and FRDM.\nAs it turns out, however, the resulting cross sections show the closest\nagreement among the investigated nuclei for this case.\nAll of the considered models predict the same ground state spin, a bound\n3\/2$^-$ state and a (barely) unbound 1\/2$^-$ state (cf.,\nFigs.~\\ref{spinhfb}--\\ref{spinfrdm}, and Fig.~\\ref{sn2};\nnote that the mass ranges in the plots are different). However, the resulting\n$Q$-value is largest in the RMFT, yielding the highest cross section. The cross\nsections from the HFB and FRDM levels are smaller by about a factor of 2\nbecause of the less strongly bound 3\/2$^-$ state. The additional 5\/2$^-$\nstate found in HFB gives only a small contribution to the total cross section\nand cannot compensate for the comparatively low $Q$-value of the capture to\nthe 3\/2$^-$ level. Nevertheless, compared to the large discrepancies\nregarding other nuclei, there is good agreement in the resulting cross\nsections. Therefore, this\nnucleus may be a bad choice to select between the different models, but it\nis reassuring in the astrophysics context that the cross sections agree so\nwell.\n\n\\section{Discussion}\n\\label{SecDis}\n\\noindent\nIn systematic r-process studies~\\cite{friedel} it was found that\nthe r-process path is touching nuclei with neutron separation energies\naround 2.5--1.7\\,MeV in the Sn region and\n$S_{\\mathrm{n}}\\simeq 1.5-0.9$\\,MeV in the\nPb region~\\cite{friedel}.\nIn our calculations for Pb (including $^{239}$Pb+n) we cover the\nastrophysically relevant mass region, with the possible exception of the\nHFB model. The neutron separation energies in the HFB model decrease\nmuch slower with increasing mass number than in the other models\n(cf., Fig.\\ \\ref{level3}), thus\nnot only leading to a drip-line at higher mass but also pushing the\nr-process path further out. However, the most extreme path location\nmight still be further out by not more than two or three isotopes from\n$^{240}$Pb, and therefore it is possible to extrapolate the trend seen in the\nHFB calculation at lower mass numbers.\nIt should be kept in mind, however,\nthat the location of the r-process path is determined by the ratio\nbetween neutron capture half-life and $\\beta$-decay half-life.\n\nIn the following we briefly discuss the possible astrophysical\nconsequences of the effects\nfound in the cross section behavior given by the different models.\nComplete r-process network calculations, which take into account all\npossible reaction links and do not postulate an a-priori $\\beta$-flow\nequilibrium, require a large number of astrophysical and nuclear-physics\ninput parameters (for a detailed discussion, see e.g.~\\cite{cowan}). In such\na non-equilibrium scenario, the location of the r-process path as well as\nthe time-scale of the r-matter flow is mainly determined by the neutron\ndensity as astrophysical quantity, and by the nuclear-physics parameters:\nthe neutron separation energy $S_{\\mathrm{n}}$ and the capture cross sections\n$\\sigma_{\\mathrm{n}}$. With this, details of the r-process are depending\non the specific nuclear models used. In the following discussion\nwe will consider as a first estimate\nonly the r-process paths found in detailed studies making use of FRDM\nmasses~\\cite{friedel} and vary the capture cross sections according to our\nfindings for the different microscopic inputs.\n\nIn the mass region beyond the $A\\simeq195$ r-abundance peak, neutron\ndensities of $n_{\\mathrm{n}}\\simeq10^{25}-10^{27}$\\,cm$^{-3}$ are required\nto produce sizeable amounts of $Z\\simeq80-84$, $A\\simeq230-250$ r-process\nisotopes very far from $\\beta$-stability. After successive $\\beta^-$- and\n$\\alpha$-decays they will form the long-lived r-chronometers $^{232}$Th and\n$^{235,238}$U, and the major part of the r-abundances of $^{206-208}$Pb and\n$^{209}$Bi (see, e.g.~\\cite{pfeiff97}). When regarding the\n$\\sigma_{\\mathrm{n}}$ cross sections for Pb from FRDM and HFB\n(see Fig.~\\ref{vgl}), very similar results are expected for the $^{230-238}$Pb\nprogenitor isotopes. Thus, also similar initial r-abundances for\n$^{232}$Th and $^{235,238}$U will result. However, when using the RMFT\ncross sections, a considerable hindrance of the nuclear flow around\n$A\\simeq130$ may occur which consequently would change the Th\/U abundance\nratios. These neutron capture cross sections which are 5 or more orders of\nmagnitude smaller than the ones given by FRDM and HFB levels would\nincrease the life-time of a nucleus against neutron-capture by the same\norder of magnitude and thus even prevent the flow to heavier elements\nwithin the time-scales given by the astrophysical environment.\n\nIn the case of the Sn isotopes, the situation is quite different from the\nPb region. The range of astrophysically realistic $n_{\\mathrm{n}}$-conditions\nfor producing the $A\\simeq130$ r-abundances is lower, with\n$n_{\\mathrm{n}}\\simeq10^{22}-5\\times10^{24}$\\,cm$^{-3}$. Hence, the r-process\npath is much closer to $\\beta$-stability, involving the progenitor isotopes\n$^{134,136,138}$Sn only a few neutrons beyond the doubly magic nucleus\n$^{132}_{50}$Sn$_{82}$. For these isotopes the Hauser-Feshbach (HF)\ncross sections used so far~\\cite{cowan} are of the order of 10$^{-4}$ to\n$5\\times10^{-5}$\\,barn. According to a recent investigation~\\cite{tfkl1},\nthe statistical model cannot be applied in that region and will overestimate\nthe capture cross sections. However, even if we use the experimental levels to\ncalculate a Breit-Wigner resonant cross section for\n$^{132}$Sn(n,$\\gamma$)$^{133}$Sn, we find it to be a factor of about 6 lower\nthan the HF cross sections.\nOur present calculations would add another DC\ncontribution of about the same magnitude as given by\nHF (see Fig.~\\ref{snfig}), which has so\nfar not been taken into account. As a consequence of the larger total cross\nsection, the r-matter flow to heavier elements would be facilitated, thus\navoiding the formation of a pronounced $A\\simeq134-138$ ``satellite peak'' in\nthe r-abundance curve sometimes observed in steady-flow calculations\n(see, e.g.\\ Fig.\\ 2 in \\cite{chen96}, or Fig.\\ 5 in \\cite{friedel}).\nSuch a signature is only indicated in the heavy-mass wing of the\n$A\\simeq130$ $N_{r,\\odot}$-peak. It is interesting to note in this context\nthat the HFB model, which exhibits the weakest $N=82$ shell closure and\nwith this also the weakest ``bottle-neck'' for the r-matter transit in this\nregion (for a detailed discussion, see e.g.~\\cite{klk97}), yields the\nhighest DC cross sections for the $A\\geq134$ Sn isotopes.\n\nSince we assumed spherical nuclei in order to be able to compare the\ndifferent microscopic models, deformation effects were not taken into\naccount which lead to level splitting and thus can increase the number\nof accessible levels. When considering deformation our results could be\nmodified in two ways: Firstly, the number of bound low-spin levels could\nbe increased, leading to larger DC cross sections; secondly,\ndue to a possibly larger number of levels at and above the neutron\nseparation energy, the compound reaction mechanism could be further\nenhanced and clearly dominate the resulting cross sections. However,\nas can be seen from level density~\\cite{ohurev,tfkl1} and\ndeformation (e.g., \\cite{moell2}) studies, deformation of Pb isotopes\nsets in at a mass number of about\n$A\\simeq 220$ and decreases already for masses beyond $A\\simeq 230$.\nCloser to the drip-line, the nuclei show low level densities again,\nnot only due to low neutron separation energies but also because of\nsphericity. Lead isotopes in the r-process path (especially for components\nwith low $S_{\\mathrm{n}}$) will therefore already\nhave reduced deformation and the DC -- being sensitive to the\nlevel structure -- will give an important contribution to the total\ncapture cross sections. Concerning Sn, a theoretical study of the ratio of\nDC over CN contributions for Sn isotopes~\\cite{bal} shows that CN dominates\nup to a mass number $A \\simeq 130$. Moreover, deformation is predicted to set\nin only at $A\\simeq140$ for Sn~\\cite{moellkra}.\nThis is supported by level density\nconsiderations~\\cite{tfkl1}, showing that the level density is too low\nin this region to apply the statistical model. Therefore, depending on the\nmodel, the r-process path lies at the border of or already well inside the region\nwhere the DC is non-negligible and dominating.\n\nAnother source of uncertainty is the assumption of pure single-particle\nstates, i.e., setting the spectroscopic factors to unity. This has been\nshown to be a good approximation for Pb isotopes close to stability and\nit is expected to hold for neutron-rich Pb isotopes. However, a range of\n0.01--1.0 for the spectroscopic factors could be realistic. This will play\nonly a minor role in the present comparison of different microscopic\nmodels, as the differences in the models may be only slightly enhanced\nwhen considering different theoretical spectroscopic factors. Nevertheless,\nit will be important in quantitative calculations of abundances, invoking\ncomplicated reaction networks.\n\n\\section{Summary}\n\\label{summary}\n\\noindent\nWe have shown that theoretical capture cross sections can depend\nsensitively on the microscopic models utilized to determine the\nnecessary input parameters.\nBecause of low level densities, the\ncompound nucleus model will not be applicable in those cases.\nDrops over several orders of magnitude in the cross sections -- as found\nwith the RMFT for Pb -- would change the position of the r-process path\nand possibly influence\nthe formation of heavy chronometer elements,\nwhereas the enhanced capture rates on Sn\ncould have direct effects in the final r-process abundance distribution.\nDeformation effects and the compound nucleus reaction mechanism\nmay still be of importance for the Pb isotopes and further investigations\nare needed. Nevertheless, the DC will be of major importance in\nthe Sn region. This region is also interesting for future experimental\ninvestigations of $S_{\\mathrm{n}}$, neutron single-particle levels and\n(d,p)-reactions studying spectroscopic factors.\nThere is also a need for improved microscopic nuclear-structure models\nwhich can also be compared in an astrophysical context following the\nsuccessful tradition of the interplay between nuclear physics and\nastrophysics.\n\n\n\n\\acknowledgements\n\\noindent\nThis work was supported in part by the Austrian Science Foundation\n(project S7307--AST) and by the Polish Committee for\nScientific Research.\nTR acknowledges support by an APART fellowship from\nthe Austrian Academy of Sciences.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\\noindent The strategies to search for a dark matter (DM) component in the Universe are nowadays extremely varied, targeting \nmany possible gravitational and non-gravitational properties such as the DM mass or standard model (SM) \ncouplings~\\cite{Bertone:2004pz}. In astrophysical, cosmological, and laboratory settings, this broadband approach has yet to \nconclusively reveal any non-gravitational signatures. However, via both indirect and direct searches, the very wide DM \nmodel space has been significantly restricted. The focus of this article concerns the reach of the generic \nclass of experiments aiming to directly detect DM through a possible DM-nucleon coupling~\\cite{Goodman:1984dc}, \nknown as direct detection facilities. Currently, world-leading examples of this setup include \ne.g.~LUX-ZEPLIN (LZ)~\\cite{LZ:2022ufs}, PandaX-4T~\\cite{PandaX-4T:2021bab}, and Xenon-1T~\\cite{XENON:2018voc},\nwhich set the strongest limits in the DM mass \n$m_\\chi$ vs.~spin-independent nuclear coupling $\\sigma_{\\mathrm{SI}}$ parameter space.\n\nThe sensitivity of a given direct detection experiment is controlled by a number of factors. Firstly, the event rate $\\Gamma_N$\nscales with the number of DM particles that have a sufficiently large kinetic energy. Specifically, the DM energy \nmust be large enough \nto induce a nuclear recoil that can trigger a signal above the detector threshold. Secondly, the rate \nalso scales linearly with the DM-nucleon cross section $\\mathrm{d}\\sigma_{\\chi N} \/ \\mathrm{d} T_N$, \nat least in the above examples, where $T_N$ is the nuclear recoil energy. Thirdly, as in any count-based experiment, \nthis signal rate should be compared to some \nbackground event rate to derive a statistically significant detection threshold. Notably, in direct detection facilities, the background \nrates are typically extremely low as necessitated by the small expected signal rates, although there are some important \nexceptions, such as a dedicated CRESST surface run~\\cite{CRESST:2017ues}. \n\nThe standard target for these experiments is the DM in the Galactic halo, which has characteristic velocities of the order\n$v_\\chi \\sim 10^{-3}c$ and in any case cannot exceed the Galactic escape velocity \n$v_\\mathrm{esc} \\sim 540 \\, \\mathrm{km\/s}$~\\cite{Evans:2000gr,Evans:2005tn}. For a given DM mass \n$m_\\chi$, there is hence unavoidably a maximum DM kinetic energy available to excite nuclear recoil signals of the order \n$T_N \\sim m_\\chi^2 v_\\mathrm{esc}^2 \/ m_N$. For some DM mass $m_\\chi^\\mathrm{min}$\nthis must fall below the detectable threshold, and the experimental sensitivity drops to zero. For experiments such as \nXenon, PandaX and LZ, it is well-known that this cut-off\nlies around the GeV-scale, corresponding to a detectable threshold in the keV range. As such, even though these detectors \nhave impressive reach in interaction cross-section -- currently down to the level of \n$\\sigma_\\mathrm{SI} \\sim 10^{-47}\\,\\mathrm{cm}^2$~\\cite{XENON:2018voc,PandaX-4T:2021bab,LZ:2022ufs}, and even \napproaching the neutrino floor with ongoing searches~\\cite{Strigari:2009bq,OHare:2021utq} -- there is ample motivation \n(and hence, in fact, both experimental and theoretical activity)\nfor methods to probe the sub-GeV mass range~\\cite{Knapen:2017xzo,Essig:2022dfa}. \nThis describes the first ``window\" in which DM can hide -- it could just be that DM has \na small mass out of the reach of direct detection experiments. \nThere is yet another window at {\\it large} values of the cross-section \n$\\sigma_{\\mathrm{SI}}$, however, which will be a key focus of this article. This arises due to the fact that if DM interacts \\textit{too} \nstrongly, then it can actually be the case that DM particles are unable to reach the detectors due to the attenuation of the flux \nin the atmosphere or the rock overburden~\\cite{Starkman:1990nj,Zaharijas:2004jv, Mack:2007xj}. \nThis typically becomes the main prohibitive factor for cross \nsections at the level of $\\sigma_{\\mathrm{SI}} \\gtrsim 10^{-28}\\,\\mathrm{cm}^2$~\\cite{Emken:2018run}.\n\nThere have been a number of promising experimental proposals to probe these two open windows. Attempts \nto extend the sensitivity to DM-nucleus interactions into the sub-GeV realm include \nsearches for Migdal electrons~\\cite{Ibe:2017yqa,XENON:2019zpr} or bremsstrahlung photons~\\cite{Kouvaris:2016afs}, \naccompanied by an intense low-threshold direct detection program in the development of novel detector \nconcepts (for a recent review, see Ref.~\\cite{Essig:2022dfa}).\nCross sections sufficiently large for DM to scatter inside the Earth before reaching underground detectors,\non the other hand,\ncan be probed by surface runs of conventional direct detection experiments (like the one performed by the CRESST \ncollaboration~\\cite{CRESST:2017ues}), or by targeting the expected diurnal modulation in the\nsignal in this case~\\cite{Collar:1992qc,Collar:1993ss}. As far as this work is \nconcerned, however, we will be interested in the role played by the irreducible astrophysical flux of\nhighly boosted DM that originates from cosmic ray collisions with DM particles in the Galactic halo (CRDM). \nThis was pointed out only relatively recently~\\cite{Bringmann:2018cvk,Cappiello:2018hsu}, and subverts the issue\nof a loss in sensitivity by noting that a sub-dominant component of DM with velocities well above those in the Galactic halo can \nproduce a detectable signal even if it is very light, i.e.~for DM masses (well) below $1 \\, \\mathrm{GeV}$. \nThe sub-dominant nature of the flux \nnaturally introduces a trade-off with the interaction rates that can be probed, quantitatively resulting in limits\nat the level of $\\sigma_\\mathrm{SI} \\sim 10^{-31}\\,\\mathrm{cm}^2$~\\cite{Bringmann:2018cvk}.\nInterestingly, CRDM does not only probe previously open parameter space at small DM masses but also results\nin bounds extending into the relevant regime of the second open window described above. \nAfter this initial work pointed out the advantages of \nconsidering such a boosting mechanism, a large number of further analyses have addressed various aspects of the \nproduction~\\cite{Alvey:2019zaa,DeRocco:2019jti,Dent:2019krz,Wang:2019jtk,Zhang:2020nis,Plestid:2020kdm,Su:2020zny,Cho:2020mnc,Guo:2020oum,Xia:2020apm,Dent:2020syp,Dent:2020qev,Emken:2021lgc,Das:2021lcr,Bell:2021xff,An:2021qdl,Feng:2021hyz,Wang:2021jic,Granelli:2022ysi,Xia:2022tid,Bardhan:2022ywd}, \nattenuation~\\cite{Bondarenko:2019vrb,McKeen:2022poo}, and \ndetection~\\cite{Ema:2018bih, Cappiello:2019qsw,Berger:2019ttc,Kim:2020ipj,Guo:2020drq,DeRoeck:2020ntj,Ge:2020yuf,Cao:2020bwd,Jho:2020sku,Lei:2020mii,Harnik:2020ugb,Ema:2020ulo,Bramante:2021dyx,Emken:2021vmf,PandaX-II:2021kai} \nof astrophysically boosted DM. \nFor a recent comprehensive \\mbox{(re-)analysis} of all of these aspects see, e.g.~Xia~{\\it et al.}~\\cite{Xia:2021vbz},\nwho stressed in particular that form-factor suppressed attenuation in the overburden seemingly allows us to exclude\ncross sections much larger than $\\sigma_\\mathrm{SI} \\sim 10^{-28}\\,\\mathrm{cm}^2$.\n\nThis article builds on this literature in three important ways: firstly, we point out that when DM acquires such large energies, \ninelastic scattering in the rock overburden above detectors such as Xenon-1T will at some point become the dominant\nattenuation mechanism. \nAs such, to avoid being over-optimistic in terms of how much parameter space is excluded, we show how \nto include this physical effect in a self-consistent manner and\nderive the resulting bounds. Secondly, we broaden the applicability of these limits to models that are more realistic for DM \nwith sub-GeV masses, moving beyond simplified contact interactions to\ninteractions mediated by vector or scalar mediators, or DM that has some internal structure. Finally, we argue that with \nthese improvements, and when taking into account fully complementary constraints from cosmology,\nthere is generically no remaining open parameter space left unconstrained for nuclear cross sections exceeding \n$10^{-30}\\,\\mathrm{cm}^2$, \nfor DM masses in the entire MeV to GeV range. We demonstrate that possible loopholes to this statement -- still allowing \nan open window at larger cross sections -- require a combination\nof {\\it (i)} questioning the principal ability of CRESST to probe DM masses down to the published limit of \n$m_\\chi=140$\\,MeV~\\cite{CRESST:2017ues}  and \n{\\it (ii)} choosing a rather narrow range of mediator masses $m_\\phi\\sim 30$\\,MeV (or finite DM extent $r_\\chi\\sim10$\\,fm).\nFor our numerical analysis throughout the article, we use the package {\\sf DarkSUSY}~\\cite{Bringmann:2018lay}. The improved CRDM \ntreatment reported in this work, including also updated cosmic ray fluxes and a more sophisticated use of form factors in the \nattenuation part, will be included in the next public release of the code.\n\n\\noindent The rest of the article is organized as follows: we start in section~\\ref{sec:crdm} by briefly reviewing the production \nof CRDM and the attenuation of the subsequent flux on its way to the detector, establishing our notation and\nsetting up the basic formalism that our analysis relies on. In the next two sections, we discuss in more detail how to model nuclear \nform factors (section~\\ref{sec:form_factors}) and the impact of inelastic scattering (section~\\ref{sec:inel}) on the attenuation \nof the flux. In section~\\ref{sec:m2}, we consider a number of \ngeneric options for the $Q^2$- and $s$-dependence of the scattering amplitude that are more realistic than assuming a constant \ncross-section. We complement this in section~\\ref{sec:hexaquark} with the analysis of a specific example, namely a\nbaryonic DM candidate that has been argued to evade traditional direct detection bounds despite its relatively\nstrong interactions with nuclei. We conclude and summarise our results in section~\\ref{sec:conclusions}.\n\n\n\n\n\\section{Cosmic-ray upscattering of dark matter}\n\\label{sec:crdm}\n\nWe describe here, in turn, how initially non-relativistic DM particles in the Galactic halo are up-scattered by cosmic rays (CRs),\nhow the flux of these relativistic CRDM particles is attenuated before reaching detectors at Earth, and\nhow to compute the resulting elastic scattering rate in direct detection experiments.\n\n\\medskip\n\\noindent\\textbf{Production:} The basic mechanism that we consider is the elastic scattering of CR nuclei $N$,  \nwith a flux of ${{d\\Phi_N}}\/{dT_N}$, \non non-relativistic DM particles $\\chi$ in the Galactic halo. For a DM mass $m_\\chi$ and \ndensity profile $\\rho_\\chi(\\mathbf{r})$, this induces a relativistic CRDM flux incident on Earth \nof~\\cite{Bringmann:2018cvk,Bondarenko:2019vrb} \n\\begin{eqnarray}\n\\frac{d\\Phi_{\\chi}}{dT_\\chi}&=&\\int\\frac{d\\Omega}{4\\pi} \\int_{\\rm l.o.s.} \\!\\!\\!\\!\\!\\!d\\ell \\, \\frac{\\rho_\\chi}{m_\\chi} \n\\sum_N\n\\int_{T_N^\\mathrm{min}}^\\infty d T_N\\, \\frac{d \\sigma_{\\chi N} }{dT_\\chi} \\frac{{d\\Phi_N}}{dT_N}\\\\\n&\\equiv&  \nD_\\mathrm{eff} \\frac{\\rho_\\chi^\\mathrm{local}}{m_\\chi}  \n\\sum_N\n\\int_{T_N^\\mathrm{min}}^\\infty d T_N\\, \\frac{d \\sigma_{\\chi N} }{dT_\\chi} \\frac{{d\\Phi^\\mathrm{LIS}_N}}{dT_N}\n\\label{eq:chiCR}\n\\,.\n\\end{eqnarray}\nHere $\\mathbf{r}$ denotes the Galactic position, and \n${d \\sigma_{\\chi N} }\/{dT_\\chi}$ is the differential elastic scattering cross section\nfor accelerating a DM particle to a kinetic recoil energy $T_\\chi$. \nFor DM particles initially at rest, this requires a minimal CR energy $T_N^\\mathrm{min}$ of\n\\begin{equation}\n\\label{eq:Tmin}\nT_N^\\mathrm{min}=\n\\left\\{\n\\begin{array}{ll}\n\\left( \\frac{T_\\chi}{2}  - m_N\\right) \\left[\n1-\\sqrt{1+\\frac{2 T_\\chi}{m_\\chi}\\frac{\\left(m_N + m_\\chi\\right)^2}{\\left(2m_N - {T_\\chi}\\right)^{2}}}\n\\right] & \\quad\\mathrm{for~}T_\\chi<2m_N\\\\\n\\sqrt{\\frac{m_N}{m_\\chi}} \\left(m_N + m_\\chi\\right) & \\quad\\mathrm{for~}T_\\chi=2m_N\\\\\n\\left( \\frac{T_\\chi}{2}  - m_N\\right) \\left[\n1+\\sqrt{1+\\frac{2 T_\\chi}{m_\\chi}\\frac{\\left(m_N + m_\\chi\\right)^2}{\\left(2m_N - {T_\\chi}\\right)^{2}}}\n\\right] & \\quad\\mathrm{for~}T_\\chi>2m_N\n\\end{array}\n\\right. \\,.\n\\end{equation}\nFurthermore, in the second line of Eq.~(\\ref{eq:chiCR}), we have introduced an effective distance $D_{\\rm eff}$ that allows us to \nexpress the CRDM flux in the solar system in terms of the relatively well measured {\\it local} interstellar CR flux, \n${{d\\Phi_N}^{\\rm LIS}}\/{dT_N}$, and the {\\it local} DM density, for which we adopt \n$\\rho^\\mathrm{local}_\\chi=0.3\\,\\mathrm{GeV}\/\\mathrm{cm}^3$~\\cite{Read:2014qva} (noting that our final limits are \nindependent of this choice). The advantage of this parametersation is that uncertainties deriving from the integration \nover the volume relevant for CRDM production, $\\int d\\Omega \\int \\!d\\ell$,  are captured in a single \nphenomenological parameter $D_\\mathrm{eff}$. Indeed, despite the complicated underlying physics, this parameter is \nsurprisingly well constrained,\nwith uncertainties dominated by the vertical extent of the confinement zone of Galactic CRs. \nIn what follows, we will use a fiducial value of $D_{\\rm eff}=10$\\,kpc.\\footnote{%\nWhen assuming  an Einasto profile~\\cite{Einasto:2009zd} for the DM density, and a cylindric CR diffusion model \ntuned with {\\sf GalProp}~\\cite{Strong:1998pw} to describe the observed flux of light CR nuclei,   \na more detailed analysis reveals that $D_{\\rm eff}$ varies between $\\sim9$\\,kpc and \n$\\sim11$\\,kpc for DM recoil energies above 1\\,MeV~\\cite{Xia:2021vbz} . \n}\nWe note that our final limits only depend logarithmically on this quantity, for large interaction rates,\nor scale as $D_{\\rm eff}^{-1\/2}$ when attenuation in the soil or atmosphere is inefficient, respectively.\n\nWhen computing the CRDM flux in Eq.~(\\ref{eq:chiCR}), we take into account the four most abundant\nCR species, $N=\\{p,{\\rm He},{\\rm C}, {\\rm O}\\}$, for which high-quality determinations of the local \ninterstellar fluxes exist~\\cite{Boschini:2018baj}.  The fluxes of heavier nuclei are subject to significant \nuncertainties for the energies of interest to us, see e.g.~the discussion in Ref.~\\cite{Boschini:2020jty}, not least due to \napparent discrepancies between AMS-02 data~\\cite{AMS:2018tbl,AMS:2018cen,AMS:2020cai} and \nearlier measurements. We also note that the CRDM flux contribution from these heavier elements is \nstrongly form-factor suppressed at large $T_\\chi$, see section \\ref{sec:form_factors}, and hence \nanyway not relevant for constraining DM with masses $m_\\chi\\gtrsim0.1$\\,GeV.\n\n\\medskip\n\\noindent\\textbf{Attenuation:} On its way to the detector, the CRDM flux given by Eq.~(\\ref{eq:chiCR}) is attenuated due to \nscattering of the CRDM particles with nuclei in the atmosphere and soil (overburden) above the experimental\nlocation. This effect can be well modelled by the energy loss equation\n\\begin{equation}\n\\label{eq:eloss}\n\\frac{dT_\\chi^z}{dz}=-\\sum_N n_N\\int_0^{\\omega_\\chi^\\mathrm{max}}\\!\\!\\!d\\omega_\\chi\\,\\frac{d \\sigma_{\\chi N}}{d\\omega_\\chi} \\omega_\\chi\\,,\n\\end{equation}\nwhich can be used to relate the average kinetic energy at depth $z$, $T_\\chi^z$, to an initial energy \n$T_\\chi$ at the top of the atmosphere.\nHere, the sum runs over the nuclei $N$ in the overburden,\ni.e.~no longer over the CR species, and $\\omega_\\chi$ is the {\\it energy loss} of a DM particle\nin a single collision. \nFor elastic scattering, $\\omega_\\chi$ is equal to the nuclear recoil energy $T_N$.\nIn that case, the maximal energy loss of a DM particle with initial kinetic energy $T_\\chi^z$\nis given by\n\\begin{equation}\n\\label{eq:tmax}\n\\omega_\\chi^\\mathrm{max}=T_N^\\mathrm{max}=\\frac{2m_N}{s}\\left[\\left(T_\\chi^z\\right)^2+2m_\\chi T_\\chi^z\\right],\n\\end{equation} \nwhere \n\\begin{equation}\n\\label{eq:sdef}\ns=(m_N+m_\\chi)^2+2m_N T_\\chi^z\n\\end{equation}\nis the (squared) CMS energy of the process. For inelastic scattering on the other hand, which we will discuss in more detail \nin section \\ref{sec:inel}, the energy loss can in principle be as high as $\\omega_\\chi^\\mathrm{max}=T_\\chi^z$.\nFor the purpose of this work we will mostly be interested in the Xenon-1T\ndetector, located at a depth of $z=1.4\\, \\text{km}$  in the Gran Sasso laboratory. In this case the \nlimestone overburden has a density of 2.71 g\/cm$^3$~\\cite{Miramonti:2005xq},\nmostly consisting of an admixture of CaCO$_3$ and MgCO$_3$, and attenuation in the\natmosphere can be neglected; in terms of weight percentages\nthe dominant elements are O (47.91\\%), Ca (30.29\\%), C (11.88\\%), Mg (5.58\\%), Si (1.27\\%),\nAl (1.03\\%) and K (1.03\\%)~\\cite{Wulandari:2003cr}. We note that Eq.~(\\ref{eq:eloss}) only provides an approximate \ndescription of the stopping \neffect of the overburden, which is nonetheless sufficiently accurate for our purposes. For a detailed comparison of this \napproach with Monte Carlo simulations of individual particle trajectories, see \nRefs.~\\cite{Emken:2017qmp,Emken:2018run,Mahdawi:2018euy,Emken:2019hgy,Xia:2021vbz}\n\n\\medskip\n\\noindent\\textbf{Detection:} The elastic scattering rate of relativistic CRDM particles arriving at underground detectors \nlike the Xenon-1T experiment is given by\n\\begin{equation}\n\\label{eq:gammarate}\n {\\frac{d\\Gamma_N}{d T_{N}}=\n \\int_{T_\\chi^{\\rm min}}^\\infty \\!\\!dT_\\chi\\ \n \\frac{d \\sigma_{\\chi N}}{dT_N} \\frac{d\\Phi_\\chi}{dT_\\chi}} \\,.\n\\end{equation}\nNote that the above integral is over the energy of the DM particles \\emph{before} entering the atmosphere. \nOn the other hand, the elastic scattering cross section ${d \\sigma_{\\chi N}}\/{dT_N} $ must still be evaluated at the actual \nDM energy, $T_\\chi^z$, at the detector location, which requires numerically solving Eq.~(\\ref{eq:eloss}) \nfor $T_\\chi^z(T_\\chi)$. The lower bound on the integral then represents the minimal {\\it initial} CRDM energy \nthat is needed to induce a nuclear recoil of energy $T_N$ {\\it at depth $z$}, i.e.\n$T_\\chi^{\\rm min}=T_\\chi(T_\\chi^{z, \\mathrm{min}})$. This can be obtained by inverting the solution of Eq.~(\\ref{eq:eloss}),\nwhere $T_\\chi^{z, \\mathrm{min}}$ is given by the right-hand side of Eq.~(\\ref{eq:Tmin}) under the replacement\n$(T_\\chi,m_\\chi,m_N)\\to(T_N,m_N,m_\\chi)$.\nIn general, the elastic nuclear scattering cross section \n${d \\sigma_{\\chi N}}\/{dT_N} $ \nis a function of both $s$ and the (spatial) momentum transfer, \n\\begin{equation}\n\\label{eq:q2}\nQ^2=2m_N T_N\\,.\n\\end{equation}\nIf the dependence on $s$ can be neglected or the (dominant) dependence on $Q^2$ factorizes -- as in the case of\nstandard form factors -- then the rate in the detector given in Eq.~(\\ref{eq:gammarate}) will have an {\\it identical}\n$Q^2$-dependence as compared to the corresponding rate expected from the standard population of \nnon-relativistic halo DM. As pointed out in Ref.~\\cite{Bringmann:2018cvk}, this salient feature makes it possible to \ndirectly re-interpret published limits on the \nlatter (conventionally expressed as limits on the scattering cross section with protons) into limits on the \nformer. Otherwise, for an accurate determination of the expected count rate in \na given analysis window, one would in principle have to also model the detector response in the \nevaluation of Eq.~(\\ref{eq:gammarate}) and then infer limits based on the full detector likelihood \n(e.g.~with a tool like {\\sf DDCalc}~\\cite{GAMBITDarkMatterWorkgroup:2017fax,GAMBIT:2018eea}).\n\n\\section{Nuclear form factors}\n\\label{sec:form_factors}\n\nThe target nuclei used in direct detection experiments \nare typically larger than the de Broglie wavelength of DM with standard Galactic velocities, \nat least for heavy nuclei, implying that the incoming DM particles only `see' part of the nucleus. \nSince the elastic scattering process is fundamentally induced by a coupling between DM and the \nconstituents of these nuclei, this means that it should be suppressed by a \nnuclear form factor, $G^2(Q^2)$, compared to the naive expectation that the nuclear cross section \nis merely a coherent sum of the cross sections of all the constituents (for recent pedagogic \naccounts of conventional direct DM searches, see e.g.~Refs.~\\cite{DelNobile:2021icc, Cooley:2021rws}).\\footnote{%\nWe focus here on spin-independent elastic scattering.  For {\\it spin-dependent} scattering,  \nthe sum would not be coherent and hence generally result in much smaller cross sections.\nThis prevents standard DM from being stopped in the overburden before reaching the experimental\nlocation -- unless the scattering cross section {\\it per nucleon} is so large that it becomes incompatible with other\nastrophysical constraints.\nA detailed treatment of attenuation in the Earth's crust is, hence, less relevant in this case. \n}\nFor CRDM, this effect is amplified, given the smaller de Broglie wavelengths\nassociated to the faster moving upscattered DM particles. \n\nThese nuclear form factors are essentially Fourier transforms of the number density of nucleons inside\nthe nucleus, usually approximated by the experimentally easier accessible charge density. A common\nparameterization is the one suggested by Helm~\\cite{Helm:1956zz}, which is based on modelling \nthe nucleus as a hard sphere with a Gaussian smearing (in configuration space).  For heavy nuclei we follow instead a slightly\nmore accurate approach and implement model-independent form factors~\\cite{Duda:2006uk}, based\non elastic electron scattering data. Concretely, we implement their Fourier-Bessel (FB) expansion approach,\nwith parameters taken from Ref.~\\cite{DeVries:1987atn}. For nuclei where the FB parameters\nare not available, notably Mg and K, we use model-independent Sum of Gaussians (SOG) form factors instead. \n\nFor $Q^2\\gg (0.1\\,\\mathrm{GeV})^2$ one starts\nto resolve the inner structure of the nucleons themselves, which we discuss in more detail in section \\ref{sec:inel}. \nLet us however briefly mention \nthat in the case of He, this effect is already largely captured by the above description in that we take the \nSOG form factors from Ref.~\\cite{DeVries:1987atn} (thus improving on the simple dipole prescription used, e.g.,\nin Ref.~\\cite{Bringmann:2018cvk}). For the proton, we adopt the usual dipole {\\it nucleon} form factor,\nnoting that the {\\it nuclear} form factor would formally equal unity,\n\\begin{equation}\n\\label{eq:Gp}\nG_p^2(Q^2)=\\left(1+ Q^2\/\\Lambda_p^2 \\right)^{-4}\\,,\n\\end{equation}\nwith $\\Lambda_p=0.843$\\,GeV. This provides a very good fit to experimental data up to momentum\ntransfers of at least $Q^2\\sim1$\\,GeV$^2$, with an agreement of better than 10\\% for \n$Q^2\\leq10$\\,GeV$^2$~\\cite{Perdrisat:2006hj,Punjabi:2015bba}. \nWe note that our final results are highly insensitive to such large momenta.\n\n\nIn the rest of the section, we will briefly describe the impact of nuclear form factors on \nthe CRDM flux and the attenuation of this flux on its way to the detector.\nIn both cases the effect is sizeable, motivating the need for a precise modelling of $G^2(Q^2)$.\n\n\n\\subsection{Impact on production}\n\\label{sec:FF_prod}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{figures_draft\/figure_one_wl}\n\\caption{{\\it Left panel.} Expected CRDM fluxes for DM masses $m_\\chi=0.001, 0.01, 0.1,1,10$\\,GeV, \nfrom top to bottom, assuming a constant spin-independent scattering cross section of \n$\\sigma_{\\rm SI}^{p,n}=10^{-30}\\,\\mathrm{cm}^2$ (solid lines). The effect of inelastic scattering is neglected.\nDashed lines show the CRDM fluxes that would result when not taking into account \nthe effect of form factors. \n{\\it Right panel.} Black lines indicate the individual contributions to the CRDM flux from scattering on CR $p$, He, C and O,\nfor the example of $m_\\chi=100$\\,MeV. Other lines (highlighted only for the  $m_\\chi=100$\\,MeV case) show the\ntotal flux, as in the left panel.\n}\n\\label{fig:flux_ff}\n\\end{center}\n\\end{figure}\n\n\nThe solid lines in Fig.~\\ref{fig:flux_ff} show the expected CRDM flux before attenuation, cf.~Eq.~(\\ref{eq:chiCR}),\nfor a range of DM masses. For the purpose of this figure, we have assumed a constant elastic \nscattering cross section $\\sigma_{\\rm SI}^p=\\sigma_{\\rm SI}^n$ on nucleons, i.e.~a nuclear cross section\ngiven by\n\\begin{equation}\n\\frac{d \\sigma_{\\chi N}}{dT_\\chi} = \\mathcal{C}^2\n\\times \\frac{\\sigma_{\\rm SI}^p}{T_\\chi^{\\rm max}} \\times G^2(2T_\\chi m_\\chi)\\,.\n\\label{eq:siconst}\n\\end{equation}\nHere, \n\\begin{equation}\n\\label{eq:c_coh}\n\\mathcal{C}^2= A^2\\frac{\\mu_{\\chi N}^2}{\\mu_{\\chi p}^2}\n\\end{equation}\ndescribes the usual coherent enhancement, in this case proportional to the square of the atomic number $A$ \nof nucleus $N$. In the rest of the expression, $\\mu_{\\chi N}$ ($\\mu_{\\chi p}$) is the reduced mass of the \nDM\/nucleus (DM\/nucleon) system and\nthe maximal DM energy $T_\\chi^{\\rm max}$ that can result from a CR nucleus with energy $T_N$ \nis given by the right-hand side of Eq.~(\\ref{eq:tmax}) after replacing $T_\\chi^z \\to T_N$ and $m_\\chi\\leftrightarrow m_N$.\n\nIn the left panel of the figure, we show that neglecting nuclear form factors (dashed lines) would lead to a \nsignificant overestimate of the CRDM flux at high energies. For $m_\\chi\\gtrsim0.1$\\,GeV, the form factor\nsuppression even becomes the dominant effect to determine the overall normalization of the flux,\nwhile for lower DM masses, the peak of the distribution is entirely determined by the fact that the \nCR flux itself peaks at GeV energies. This suppression in the flux leads to a rapid deterioration \nof CRDM limits.\nModelling form factors correctly is thus particularly important for the highest DM masses that can be\nprobed by cosmic-ray upscattering, i.e.~for $m_\\chi \\sim 1 - 10 \\, \\mathrm{GeV}$.\n\nIn the right panel of Fig.~\\ref{fig:flux_ff}, the contributions from the individual CR nuclei to the  \nCRDM flux are shown. At low energies the dominant contribution is always from Helium, closely followed by the one from protons. \nThe high-energy part of the CRDM flux, on the other hand, \nis almost exclusively due to CR protons because the contribution from heavier CR nuclei is \nheavily form-factor suppressed. In addition, for $m_\\chi\\gtrsim1$\\,GeV, the \npeak amplitude of the CRDM flux -- which typically has the most constraining\npower in direct detection experiments -- is almost exclusively  determined by CR $p$ and He nuclei\n (see also Fig.~\\ref{fig:attenuation_ff} below to better gauge the relevant range of energies {\\it after} attenuation\n in the overburden).\nFor lower DM masses, on the other hand, including further high-$Z$ CR species than those taken into account \nhere could in principle increase the relevant part of the CRDM flux by up to $\\sim50$\\,\\%~\\cite{Xia:2021vbz}. \nIn what follows, we conservatively neglect these contributions, in view of both the larger uncertainties in the underlying\nCR fluxes and the fact that we are mainly interested in DM masses around the GeV scale.\n\n\n\\subsection{Impact on attenuation}\n\\label{sec:FF_att}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{figures_draft\/figure_two_wl.pdf}\n\\caption{%\nMinimal kinetic energy $T_\\chi$ that a DM particle must have at the surface of the Earth ($z=0$) in order \nto trigger a signal in the Xenon-1T experiment,  as a function\nof a (constant) spin-independent scattering cross section $\\sigma_{\\rm SI}^{p,n}$.\n Different colors correspond to different DM masses, \n as  in Fig.~\\ref{fig:flux_ff}.\n Dash-dotted lines show the kinetic energies that would be necessary when computing the attenuation in the \nzero momentum transfer limit. Dashed lines illustrate the effect of adding\nthe expected form factor suppression, cf.~section \\ref{sec:form_factors}, while solid \nlines show the result of our full treatment, including also inelastic scattering events \n(discussed in section \\ref{sec:inel}).\n}\n\\label{fig:attenuation_ff}\n\\end{center}\n\\end{figure}\n\n\nWe now turn our attention to assessing the effect that the form factor suppression has on the attenuation of DM\nparticles on their way to the detector in a direct detection experiment. For concreteness we will again focus\non the case of Xenon-1T, where Xe nuclei recoiling with an energy of at least\n$T_{\\rm Xe}=4.9$\\,keV trigger a detectable signal~\\cite{XENON:2018voc}. In \nFig.~\\ref{fig:attenuation_ff}, we show the minimal initial DM energy that is required to kinematically allow\nfor this, after penetrating through the Gran Sasso rock. In practice this is done by numerically solving Eq.~(\\ref{eq:eloss}) with\n{\\sf DarkSUSY}. Dash-dotted lines indicate the result when conservatively\nassuming that the stopping power in the overburden is as efficient as in the zero-momentum transfer\nlimit (as in Ref.~\\cite{Bringmann:2018cvk}), while dashed lines show the effect of adding the additional\nform factor suppression for high $Q^2$ (as in Refs.~\\cite{Bell:2021xff,Xia:2021vbz}). \nSolid lines, finally, demonstrate the effect of also adding the attenuation power of inelastic scattering events,\nas described in detail below in Section \\ref{sec:inel}.\n\nFor small cross sections, attenuation is inefficient and, as expected, the three approaches give the \nsame answer. In this limit, the difference in the required DM energy is entirely due to the well-known\nkinematic effect, cf.~Eq.~(\\ref{eq:Tmin}), that lighter particles require a higher energy to induce a \ngiven recoil of much heavier particles\n(up to a minimum energy of $T_\\chi\\geq\\sqrt{m_{\\rm Xe}T_{\\rm Xe}\/2} =17.3$\\,MeV in the limiting case where $m_\\chi\\to0$).\nCorrespondingly, this also means that the CRDM fluxes cannot actually be probed by Xenon-1T \nfor the entire range of $T_\\chi$ shown in Fig.~\\ref{fig:flux_ff};\nunless $m_\\chi\\lesssim10$\\,MeV, however, the lowest detectable energy is always smaller\n than the energy at which the CRDM flux peaks.\n \nFor large cross sections, on the other hand, Fig.~\\ref{fig:attenuation_ff} shows a \npronounced difference between the three \napproaches: while in the case of a constant cross section (dash-dotted lines) the energy loss equation\nresults in an exponential attenuation, adding form factors (dashed lines) implies that the required initial \nDM energy only rises as the square root of the scattering cross section in the $Q^2=0$ limit.\nIn fact, we note that this is exactly the behaviour one would expect from Eq.~(\\ref{eq:eloss}) for a \ncross section that falls off very rapidly at large momentum transfers. \nComparing again to Fig.~\\ref{fig:flux_ff},\nthis correspondingly enlarged range of kinetic energies that becomes kinematically accessible to Xenon-1T will \ninevitably lead to significantly larger rates in the detector -- which, indeed, is exactly the conclusion reached in\nRefs.~\\cite{Bell:2021xff,Xia:2021vbz}. However, such a strong \nsuppression of the physical stopping power of the Gran Sasso rock for a relativistic particle is highly \nunphysical. As we discuss in the next section, this is simply because the DM particles will start to scatter off the constituent \nnucleons themselves, albeit not \ncoherently across the whole nucleus. Adding this effect (solid lines), \nresults again in exponential attenuation in the overburden -- though only at significantly larger cross sections \nthan what would be expected when adopting a constant cross section for simplicity. \n\n\n\\section{Inelastic Scattering}\n\\label{sec:inel}\n\nOur discussion so far has largely neglected the impact of inelastic scattering events of relativistic DM particles incident on nuclei \nat rest, or {\\it vice versa}. Physically, the inclusion of inelastic scattering processes is non-negotiable and should be considered in \na full treatment. This is because, whilst the form factor suppression described above is the relevant feature in the transition from \ncoherently scattering off the whole nucleus to only parts of it, once the DM or nucleus transfers a sufficiently large amount \nof energy $\\omega$, the scattering will probe individual nucleon-, or even quark-level processes. The result is an additional \ncontribution to the total scattering cross-section that can easily dominate \nin the large energy transfer regime. \nAs far as CRDM limits are concerned, the most important effect \nthat the inclusion of inelastic scattering modifies is the attenuation of the flux through the Earth or atmosphere.\nNot including it, therefore, will lead to an overly optimistic estimate as to the amount of parameter space that is ruled out via this \nmechanism.\\footnote{In order to keep our results conservative, we neglect the effect of inelastic scattering on CRDM {\\it \nproduction} in our analysis. We leave the study of this additional contribution of the flux to future work, noting that we \nexpect it to mostly improve limits for larger DM masses (where the form factor suppression nominally leads \nto a significant reduction of the CRDM flux, see Fig.~\\ref{fig:flux_ff}).}\nLet us note that inelastic scattering of {\\it non-relativistic} DM, resulting in the excitation of low-lying states in the target nuclei,  \nwas previously both studied \ntheoretically~\\cite{Baudis:2013bba, McCabe:2015eia, Kouvaris:2016afs, Hoferichter:2018acd}\nand searched for experimentally~\\cite{XMASS-I:2014lnb,XENON:2017kwv,Lehnert:2019tuw,XENON:2020fgj}. \nHere we concentrate on different types of inelastic processes that are only accessible to nuclei scattering off \nhigh-energy DM particles.\n\nThe rest of this section is organised as follows: firstly we give a qualitative description of the most important \ninelastic scattering processes, such as the excitation of hadronic resonances or quasi-elastic scattering \noff individual nucleons. Secondly, we explain how we obtain a quantitative estimate of these complicated nuclear interactions \nby making a direct analogy to the case of neutrino-nucleus scattering. In this regard, we make use of the public code \n\\texttt{GiBUU}~\\cite{Buss:2011mx,gibuuweb}.\nFinally, we will explain how to build this into the formalism described in section~\\ref{sec:crdm} in terms of the DM energy loss, see \nEq.~\\eqref{eq:eloss}.\n\n\\subsection{Scattering processes and associated energy scales}\nThere are a number of relevant contributions to scattering cross-sections on nuclei that are associated to certain \ncharacteristic energies or nuclear length scales. In the highly non-relativistic limit, as described above,\ncoherently enhanced elastic scattering dominates. At somewhat higher energies, more specifically momentum transfers\ncorresponding to (inverse) length scales smaller than the size of the nucleus,\nthe elastic scattering becomes form factor suppressed -- a description which physically assumes a smooth distribution of \nscattering centres throughout the nucleus. The main characteristic of elastic scattering in both of these regimes is that \nthe energy loss of the incident DM particle is uniquely related to the momentum transfer by $\\omega=Q^2\/(2m_N)$.\n\nThis relation no longer holds for inelastic scattering processes, which are expected to become relevant at even higher energies. \nFor our purposes, these inelastic processes can be broadly split up into three scattering regimes, depending\non the energy that is transferred (see also Fig.~\\ref{fig:mchi_1GeV} below, as well as a review~\\cite{Formaggio:2012cpf} \nfor the discussion of the analogous situation in the case of neutrino-nucleus scattering):\n       \n\\begin{itemize}\n      \\item \\textbf{Quasi-Elastic Scattering} \\textbf{(}$\\mathbf{\\omega \\gtrsim 10^{-2}}$\\,\\textbf{GeV):} At suitably large \n\tenergy transfers, the form factor suppression cannot be totally physical. This is because  \n\tthe incident DM particles will probe directly the constituent nucleons, which are \n\tinherently not smoothly distributed. \\emph{Quasi-elastic scattering} (QE) dominates for \n\t$10^{-2}\\,\\mathrm{GeV} \\lesssim \\omega \\lesssim 1 \\, \\mathrm{GeV}$, and describes this situation, \n\ti.e.~where the dominant scattering is directly off {\\it individual} protons (and neutrons) inside the nucleus, \n\t$\\chi\\, p (n) \\rightarrow \\chi\\, p (n)$.\n\n\t\\item \\textbf{Excitation of Hadronic Resonances} \\textbf{(}$\\mathbf{\\omega \\gtrsim 0.2}$\\,\\textbf{GeV):} At higher energies \n\tstill, DM-nucleon scattering can excite nuclear resonances such as \n\t$\\chi \\, p \\rightarrow \\chi \\, (\\Delta \\rightarrow p \\pi^0)$ etc., leading to a wide variety of hadronic final states. Often, the contribution due to the lowest lying \n\t$\\Delta$ resonances (DR) is distinguished from contributions from higher resonances (HR) since the former can  \n\tbe well resolved and starts playing role at considerably smaller transferred energies. \n\tIn a complicated nucleus such as ${^{16}}\\mathrm{O}$, both the QE and resonance contributions to the scattering cross-section must be resolved numerically, \n\ttaking into account effects such as the nuclear potential and spin statistics. \n\n\t\\item \\textbf{Deep Inelastic Scattering} \\textbf{(}$\\mathbf{\\omega \\gtrsim 1}$ \\textbf{GeV):} Most DM couplings to \n\tnuclei and nucleons result from more fundamental couplings to quarks or gluons. \n\tAs such, once the energy transfer is large enough to probe the inner structure of the nucleons \n\t($\\omega \\gtrsim 1 \\, \\mathrm{GeV}$), then \\emph{deep inelastic scattering} (DIS) of DM with partons inside the nucleons\n\tcan occur. Again, this should be \n\tresolved numerically to give an accurate estimate of the impact at the level of the scattering cross-section.\n\\end{itemize}\n\n\n\\subsection{Computation of the inelastic cross-section for neutrinos}\n\\label{sec_gibuu}\n\nDue to the complicated nuclear structure of the relevant atomic targets in the Earth, \nor in the composition of cosmic rays, \nit is typically not possible to analytically compute all the contributions to DM-nucleus scattering \ndescribed above. Instead, to estimate their impact on our \nconclusions and limits, we will make a direct connection with the physics of neutrino-nucleus scattering for which numerical codes \n-- such as \\texttt{GiBUU}~\\cite{Buss:2011mx} -- are capable of generating the relevant differential cross-sections.\n\nIn more detail, we draw the analogy between neutral current neutrino-nucleon scattering via processes such as \n$\\nu \\, p \\rightarrow \\nu \\, p$ and DM-nucleon scattering. Numerically modelling the neutral current quasi-elastic scattering, \nresonances and deep inelastic scattering as a function of the energy transferred to the nucleus, $\\omega$, allows us to \nunderstand the relative importance of these processes as a function of the incoming neutrino energy\n(or DM kinetic energy $T_\\chi$). Of course, since \nthese codes are tuned for neutrino physics, simply outputting the differential cross-sections such as \n$\\mathrm{d}\\sigma_{\\nu N} \/ \\mathrm{d}\\omega$ is not sufficient. To map the results onto DM, see \nsection \\ref{sec:map_dm} below for further details, we should re-scale the results so \nas to respect both the relative interaction strengths and model dependences such as e.g. the mediator mass. In general, we \nexpect this approach to provide a good estimate of the DM-nucleus cross section (at least) for contact interactions and  \nscattering processes dominated by mediators in the $t$-channel.\n\n\nAt the level of implementation, we choose the settings in the \\texttt{GiBUU} code described in Tab.~\\ref{tab:gibuu}. \nSince we are interested in quantifying the effect of inelastic scattering on the attenuation of the CRDM flux as it passes \nthrough the Earth, \nwe mostly focus on the total inelastic scattering cross section, i.e.~the sum over all the processes described in the \nprevious section. We numerically calculate this for the most abundant nuclei in the Gran Sasso rock, \n$N = \\{\\mathrm{O}, \\mathrm{Ca}, \\mathrm{C}, \\mathrm{Mg}, \\mathrm{Si}, \\mathrm{Al}, \\mathrm{K}\\}$.\nFundamentally, inelastic cross-sections are expressed in terms of double-differential cross-sections \nlike $\\mathrm{d}^2 \\sigma_{\\nu N} \/ \\mathrm{d}Q^2 \\mathrm{d}\\omega$, since for inelastic scattering $Q^2$ and $\\omega$ are \nindependent variables. \nFor integrating the energy loss equation, Eq.~\\eqref{eq:eloss}, however, it suffices to compute\n\\begin{equation}\n\\frac{\\mathrm{d} \\sigma_{\\nu N}}{ \\mathrm{d}\\omega} \\equiv \\int_{Q^2} \\frac{\\mathrm{d}^2 \\sigma_{\\nu N}}{ \\mathrm{d}Q^2 \\,\\mathrm{d}\\omega} \\,\\mathrm{d}Q^2\\,.\n\\end{equation}\nOn the other hand, the full information about the $Q^2$-dependence of \n$\\mathrm{d}^2 \\sigma_{\\nu N} \/ \\mathrm{d}Q^2 \\mathrm{d}\\omega$ provided by \\texttt{GiBUU} still remains a highly\nuseful input to our analysis. This is because  the double-differential cross \nsections of the individual inelastic processes turn out to sharply peak at values of $Q^2$ that have simple relations to $\\omega$.\nFor example, the peak position for the QE contribution corresponds to the `elastic' relation~\\eqref{eq:q2} for nucleons. \nAs described below, this information will be used \nfor setting realistic reference values of $Q^2$ to capture the model-dependence of the DM cross sections.\n\n\n\\begin{table}[]\\label{tab:gibuu}\n\t\\resizebox{\\textwidth}{!}{\n\t\\begin{tabular}{@{}llllll@{}}\n\t\\toprule\n\t\\multicolumn{2}{l}{\\hspace{-0.2cm}\\textbf{\\&neutrino\\_induced}} & \\multicolumn{2}{l}{\\textbf{\\&input}}               & \\multicolumn{2}{l}{\\textbf{\\&nl\\_dSigmadElepton}} \\\\\n\t\\texttt{process\\_ID}               & 3           & \\texttt{eventtype}                     & 5                & \\texttt{enu}                  & $T_\\chi$             \\\\\n\t\\texttt{flavor\\_ID}                & 2           & \\texttt{numEnsembles}                  & 100              & \\texttt{elepton}              & $0.005 T_\\chi$       \\\\\n\t\\texttt{nuXsectionMode}            & 2           & \\texttt{numTimeSteps}                  & 0                & \\texttt{delta\\_elepton}       & $\\Delta E_\\ell$      \\\\\n\t\\texttt{nuExp}                     & 0           & \\texttt{num\\_Energies}                 & 50               & \\multicolumn{2}{l}{\\textbf{\\&target}}                \\\\\n\t\\texttt{includeQE}                 & T\/F         & \\texttt{num\\_runs\\_sameEnergy}         & 1                & \\texttt{Target\\_A}            & $A$                  \\\\\n\t\\texttt{includeDELTA}              & T\/F         & \\texttt{delta\\_T}                      & 0.2              & \\texttt{Target\\_Z}            & $Z$                  \\\\\n\t\\texttt{includeRES}                & T\/F         & \\texttt{localEnsemble}                 & T                & \\multicolumn{2}{l}{\\textbf{\\&initDensity}}           \\\\\n\t\\texttt{path\\_To\\_Input}           & \\texttt{\/path\/to\/buuinput} & \\texttt{include1pi}     & F                & \\texttt{densitySwitch}        & 2                    \\\\\n\t\\texttt{includeDIS}                & T\/F         & \\multicolumn{2}{l}{\\textbf{\\&neutrinoAnalysis}}           & \\multicolumn{2}{l}{\\textbf{\\&initPauli}}             \\\\\n\t\\texttt{2p2hQE}                    & F           & \\texttt{XSection\\_analysis}            & T                & \\texttt{pauliSwitch}          & 2                    \\\\\n\t\\texttt{include2p2hDelta}          & F           & \\texttt{detailed\\_diff\\_output}        & F                &  \t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\n    \\texttt{include2pi}                & F           &                                        &                  &  \t\t\t\t\t\t\t\t\t\t\t\t\t\\\\ \\bottomrule\n\t\\end{tabular}\n\t}\n\t\\caption{Settings choices for running \\texttt{GiBUU} to study neutral current neutrino scattering. \n\tWe also enforced a logarithmic binning in the outgoing lepton energy, by changing the variable assignment \n\tof \\texttt{dElepton} from $E_\\ell \\rightarrow E_\\ell + \\Delta E_\\ell$ to $E_\\ell \\rightarrow (1 + \\Delta E_\\ell) E_\\ell$.\n\t}\n\t\\end{table}\n\n\n\\subsection{Mapping to the dark matter case}\n\\label{sec:map_dm}\n\nHaving described the technical details of how we obtain the neutrino-nucleus inelastic cross-sections using \\texttt{GiBUU}, we \nnow turn our attention to the mapping of these quantities onto DM models. \nThis is a necessary step for two broad reasons: \\emph{(a)} the interaction strength governing the DM-nucleus interactions is \ntypically very different from the neutrino-nucleus SM value, and \\emph{(b)} the way the interaction proceeds via e.g.~a contact \ninteraction or mediator exchange can lead to substantially different kinematics and non-trivial $Q^2$- or $s$-dependences.\n\nThe total scattering cross-section $\\mathrm{d}\\sigma_{\\chi N}\/\\mathrm{d}\\omega$ consists of the coherent elastic scattering contribution that we compute analytically for each of the models considered in this work, and the inelastic scattering cross section that we want to estimate based on the \\texttt{GiBUU} output:\n\\begin{align}\\nonumber\n\\frac{\\mathrm{d}\\sigma_{\\chi N} }{ \\mathrm{d}\\omega} &= \\left.\\frac{\\mathrm{d}\\sigma_{\\chi N} }{ \\mathrm{d}\\omega}\\right|_{\\mathrm{el}}+\\left.\\frac{\\mathrm{d}\\sigma_{\\chi N} }{ \\mathrm{d}\\omega}\\right|_{\\mathrm{inel}}\\\\ \\label{eq:rescaling}\n&\\equiv  \n\t\\left.\\frac{\\mathrm{d}\\sigma_{\\chi N} }{ \\mathrm{d}\\omega}\\right|_{\\mathrm{el}, Q^2=2\\omega m_N} + \\sum_{i} \\left.\\frac{\\mathrm{d}\\sigma_{\\mathrm{SI}} }{ \\mathrm{d}\\omega}\\right|_{\\mathrm{el}, Q^2=Q_{i,\\mathrm{ref}}^2} \n\\times I_{\\chi,i}(T_\\chi,\\omega)\\,.\n\\end{align}\nHere $\\left.\\mathrm{d}\\sigma_{\\mathrm{SI}} \/ \\mathrm{d}\\omega\\right|_{\\mathrm{el}}$ is the differential\nDM-nucleon elastic cross section, excluding nucleon form factors such as the one given in \nEq.~(\\ref{eq:Gp}). \nThe sum runs over the various individual processes, $i\\in$(QE, DR, HR, DIS),\nwhich all have characteristic reference values of $Q^2=Q^2_{i, \\mathrm{ref}}(\\omega)$ where\nthe respective inelastic cross section peaks. In the second step above, we thus choose to rescale the inelastic scattering \nevents to the elastic scattering off a point-like nucleon. \nThis rescaling is motivated by the fact that for inelastic contributions like QE, the underlying process is much better \ndescribed by scattering on individual nucleons than on the entire nucleus.\nThe factor\n\\begin{equation}\nI_{\\chi,i}(T_\\chi,\\omega) \\equiv \\frac{\\mathrm{d}\\sigma^i_{\\chi N} \/\\mathrm{d}\\omega \\big|_{\\mathrm{inel}}}\n\t{\\mathrm{d}{\\sigma}_{\\mathrm{SI}} \/\\mathrm{d}\\omega \\big|_{\\mathrm{el},Q^2=Q^2_{i, \\mathrm{ref}}}}\n\\end{equation}\nthus quantifies the ratio of the inelastic scattering process on a nucleus to the elastic scattering on an individual\nnucleon.\n\nWe now make the simplifying assumption that this ratio is to a certain degree model-independent,\nbased on the expectation that DM should probe the inner structure of nucleons in a similar way as neutrinos do\nwhen only neutral current interactions are involved. Physically, indeed, this closely resembles the situation both for \ncontact interactions and $t$-channel mediators.\nThe model dependence thus dominantly comes from the structure of the term \n$\\left.\\mathrm{d}\\sigma_{\\mathrm{SI}} \/ \\mathrm{d}\\omega\\right|_{\\mathrm{el}}$, and we approximate\n\\begin{equation}\n\\label{eq:I2}\nI_{\\chi,i}(T_\\chi,\\omega) \\approx I_{\\nu,i}(E_\\nu,\\omega)\\equiv\\frac{\\left.\\mathrm{d}\\sigma^i_{\\nu N} \/\\mathrm{d}\\omega \\right|_{\\mathrm{inel}}}\n\t{\\mathrm{d}{\\sigma}^i_{\\nu,\\mathrm{SI}} \/\\mathrm{d}\\omega \\big|_{\\mathrm{el}}}\\,.\n\\end{equation}\nHere, the inelastic neutrino-nucleus cross section \n$\\left.\\mathrm{d}\\sigma_{\\nu N}^i\/\\mathrm{d}\\omega\\right|_{\\mathrm{inel}}(E_\\nu,\\omega)$ \ncan be obtained using the \\texttt{GiBUU} code, as described in section \\ref{sec_gibuu}, and we evaluate it \nat the incoming DM kinetic energy, $E_\\nu = T_\\chi$.\n On the other hand, a possible estimate for the denominator -- the elastic neutral current neutrino-nucleon cross \nsection without the form factor -- is the average of the proton and neutron cross sections in the $\\omega \\rightarrow 0$ \nlimit~\\cite{Formaggio:2012cpf}:\n\\begin{equation}\n\\label{eq:nuel_simp}\n\\left.\\frac{\\mathrm{d}\\sigma^i_{\\nu,\\mathrm{SI}}}{\\mathrm{d}\\omega} \\right|_{\\mathrm{el}} \n= \\frac{1}{2} \\sum_{j=n,p} \\frac{m_j G_F^2}{4\\pi}\\left[(g_A\\tau_3^j - \\Delta_S)^2 + (\\tau_3^j-2(1+\\tau_3^j)\\sin^2\\theta_W)^2\\right].\n\\end{equation}\nHere $\\tau_3^p=1$ and $\\tau_3^n=-1$, $\\theta_W$ is the weak mixing angle and $G_F$ is the Fermi constant.\nThe axial vector and strange quark contributions are encoded in the parameters \n$\\Delta_S\\approx -0.15$ (see, e.g., Ref.~\\cite{Alberico:1997vh} for a discussion) and  \n$g_A= 1.267$~\\cite{ParticleDataGroup:2008zun}, respectively. Numerically the square bracket evaluates\nto a factor of $\\sim\\!2.24\\,(2.01)$ for neutrons (protons). \nLet us stress, however, that this formula is valid only for energies \nrelevant for inelastic scattering, $0.1\\,\\mathrm{GeV}\\lesssim E_\\nu\\lesssim10$\\,GeV. \nAt much smaller energies, only the valence quarks contribute to the scattering, and we would instead have\n\\begin{equation}\n\\label{eq:nuel_simpNR}\n\\left.\\frac{\\mathrm{d}\\sigma^i_{\\nu,\\mathrm{SI}}}{\\mathrm{d}\\omega} \\right|_{\\mathrm{el}} \n= \\frac{m_n G_F^2}{4\\pi}\n\\end{equation}\nfor neutrons, while the scattering on protons is strongly suppressed by a factor of \n$Q_W^2=(1-4\\sin^2\\theta_W)^2\\approx0.012$.\n\n \nIt is worth noting that in principle, we could improve the assumption made in Eq.~(\\ref{eq:I2})\nfor the quasi-elastic process, because there is a \nwell-controlled understanding of the analytic QE cross-section via the Llewellyn-Smith formalism (see section~V \nof~Ref.~\\cite{Formaggio:2012cpf}). For clarity, we choose to take a consistent prescription across all inelastic processes, \nand we have checked that including the full QE cross-section would only introduce an \nadditional $\\mathcal{O}(1)$ factor in the DM QE cross-section.\nFor the numerical implementation in {\\sf DarkSUSY}, we pre-tabulate $I_{\\nu,i}$ from $T_\\chi=0.01$\\,GeV up to energies of $T_\\chi=10$\\,GeV,\nwith $200$ ($101$) equally log-spaced bins in $T_\\chi$ ($\\omega$)\nand a normalization as given by Eq.~(\\ref{eq:nuel_simp}),  and then interpolate between \nthese values.\\footnote\nFor significantly higher energies,  \\texttt{GiBUU} is no longer numerically stable. Furthermore, \nthe underlying equations that describe the interaction processes begin to fall outside their ranges of validity\nas the $Z$ boson mass starts to get resolved.\nAt higher energies, where anyway only the DIS contribution is non-negligible,  a reasonable estimate \ncan still be obtained by a simple extrapolation \n$I_{\\nu,i}(T_\\chi,\\omega)\\to I_{\\nu,i}(T_\\chi^{\\rm ref},\\omega^{\\rm ref})$, \nwith $\\omega^{\\rm ref} =\\omega\\,(T_\\chi^{\\rm ref}\/T_\\chi)^{0.25}$,  beyond some reference energy\n$T_\\chi^{\\rm ref}\\approx10$\\,GeV\\@. By running  \\texttt{GiBUU} up to $E_\\nu\\sim30$\\,GeV, we checked that\nthis prescription traces the peak location (in $\\omega$) of the DIS contribution very well, \nindependently of the exact choice of $T_\\chi^{\\rm ref}$. We also confirmed that \nthe peak value of $I$ becomes roughly constant for such large energies. \nOn the other hand, higher-order inelastic processes are expected to become increasingly important \nat very large energies, not covered in \\texttt{GiBUU}.\nWe therefore only add the above extrapolation as an {\\it option} in {\\sf DarkSUSY}, and instead completely cut the incoming CRDM\nflux at $10$\\,GeV in the default implementation. As a result, our bounds on the interaction strength\nmay be overly conservative for small DM masses $m_\\chi\\lesssim0.1$\\,GeV.\n}\n\nWe also must choose the reference values for the transferred momentum $Q^2_{i, \\mathrm{ref}}$, which allows us to account \nfor e.g.~mediators that may be much lighter than the electroweak scale. \nImportantly, each process (quasi-elastic, \n$\\Delta$-resonance,...) is expected to have a different characteristic $Q^2$-$\\omega$ dependence that takes into account the \nrelevant binding energies and kinematic scaling. For example, in the case of elastic scattering, the \nrelation $Q^2 = 2 m_N \\omega$ holds, whilst for quasi-elastic processes, the relevant scattering component is a nucleon such \nthat the cross-section is peaked around $Q^2 \\sim 2 \\,{\\overline{ m}}\\,\\omega$, where ${\\overline m} \\equiv (m_n + m_p)\/2$. \nThe resonance of a particle with mass $m_\\mathrm{res}$ can be accounted for by noting that part of the transferred\nkinetic energy is used to excite the resonance, such that the cross-section peaks around \n$Q^2 \\sim 2\\, {\\overline m}\\, (\\omega - (m_\\mathrm{res} - \\overline{m}))$. \nWe have confirmed these expectations numerically by comparing directly to the \ndoubly-differential cross-section extracted from \\texttt{GiBUU}.\nFrom this numerical comparison we further extract that $Q^2 \\sim 0.6\\,\\overline{m}\\,(\\omega \\!-\\! \\omega_{\\rm DIS})$,\nwith $\\omega_{\\rm DIS}=1.0$\\,GeV, constitutes a very\ngood fit to the peak location of the DIS cross-section.\nIn summary, we take the following reference values across the four inelastic processes:\n\\begin{align}\n\tQ^2_{\\mathrm{QE}, \\mathrm{ref}} = 2\\, {\\overline m} \\omega \\, \\, , \\,\\,\\,\\, &Q^2_{\\Delta, \\mathrm{ref}} = 2\\, {\\overline m}\\, (\\omega - \\Delta m_\\Delta)  \\nonumber \\\\\n\tQ^2_{\\mathrm{res}, \\mathrm{ref}} = 2\\, {\\overline m}\\, (\\omega - \\Delta m_{\\mathrm{res}}) \\, \\, , \\,\\,\\,\\, &Q^2_{\\mathrm{DIS}, \\mathrm{ref}} = 0.6\\, {\\overline m}\\, (\\omega - \\omega_{\\rm DIS})\\,.\n\\end{align}\nHere, $\\Delta m_\\Delta = 0.29 \\, \\mathrm{GeV}$ is the mass difference between the $\\Delta$ baryon and an average nucleon, \nand $\\Delta m_\\mathrm{res} = 0.40 \\, \\mathrm{GeV}$ is an estimate for the corresponding average mass difference of the higher \nresonances (we checked that our final limits are insensitive to the exact value taken here).\n\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\\includegraphics[width=0.9\\textwidth]{figures_draft\/figure_three_wl}\n\t\\caption{Comparison between the elastic (green, lower energies) and inelastic (blue, higher energies) contributions to the \n\tDM-nucleus differential cross section $\\mathrm{d}\\sigma_{\\chi N}\/\\mathrm{d}\\omega$, where $\\omega$ is the \n\tDM energy loss. This figure shows these contributions \tfor a constant isospin-conserving DM-nucleus cross section, with \n\t$m_\\chi = 1\\, \\mathrm{GeV}$ and $N = {^{16}}\\mathrm{O}$. The small colorbar on \n\tthe inset of the plots, along with the stated numerical ratio, indicates the balance between elastic and inelastic\n\tscattering in terms of the contribution to the integrated cross section \n\t$\\sigma_{\\chi N}^\\mathrm{tot}$.\n\t}\n\t\\label{fig:mchi_1GeV}\n\t\\end{center}\n\\end{figure}\n\n\nTo illustrate this procedure concretely, we consider the simple case of a contact interaction where, cf.~Eq.~(\\ref{eq:siconst}), \n$\\left.\\mathrm{d}\\sigma_{{\\rm SI}}\/\\mathrm{d}\\omega\\right|_{\\mathrm{el.}} = \\sigma_{\\mathrm{SI}} \/ \\omega^\\mathrm{max}$ and \n$\\omega^\\mathrm{max} = 2\\, {\\overline m} (T_\\chi^2 + 2 \\chi T_\\chi) \/ (({\\overline m} + m_\\chi)^2 + 2 {\\overline m} T_\\chi)$. The results for the \nrescaled inelastic cross-section (blue) are shown in Fig.~\\ref{fig:mchi_1GeV} for a DM mass $m_\\chi = 1\\,\\mathrm{GeV}$ incident \non a $^{16}\\mathrm{O}$ nucleus. In this figure, we also compare to the coherent elastic contribution (green) and highlight the \nbalance between the relative contributions to the total (integrated) cross-section $\\sigma^\\mathrm{tot}_{\\chi N}$. In particular, we \nsee that above kinetic energies $T_\\chi \\gtrsim 0.2\\,\\mathrm{GeV}$, the inelastic contribution dominates, clearly \nmotivating the necessity of its  inclusion. This is consistent with the picture previously encountered in Fig.~\\ref{fig:attenuation_ff}, \nwhere we could see the impact of inelastic scattering on the energy loss.  More concretely, \nthe result lies in some intermediate regime between the $G(Q^2) = 1$ and $G(Q^2) \\neq 1$ cases, the former\/latter leading \nto conservative\/overly optimistic limits respectively. \nIn the next section we will derive the relevant CRDM limits in the \n$\\sigma_{\\mathrm{SI}}-m_\\chi$ plane for a number of models to make this point quantitatively.\n\n\nLet us conclude this section by briefly returning to the implicit assumption of isospin-conserving DM interactions that\nwe made above, with \n$\\sigma_{\\mathrm{SI}}=\\sigma^p_{\\mathrm{SI}}=\\sigma^n_{\\mathrm{SI}}$.\nInterestingly, neutral-current induced\ninelastic scatterings between neutrinos and nucleons hardly distinguish between protons and \nneutrons~\\cite{Formaggio:2012cpf}, such that the factor $I_{\\chi,i}\\approx I_{\\nu,i}$  indeed becomes, by construction,\nlargely independent of the nucleon nature. Naively, one would thus conclude that\nisospin-violating DM couplings can easily be incorporated in our treatment of inelastic scattering by replacing\n$\\sigma_{\\mathrm{SI}} \\to (1 \/ A) \\times (Z \\sigma^p_{\\mathrm{SI}} + (A - Z) \\sigma^n_{\\mathrm{SI}})$  in Eq.~(\\ref{eq:rescaling}).\nWhen doing so, however, it is important to keep in mind that the nucleon cross sections should be evaluated at \nenergies that are relevant for inelastic scattering, not  in the highly non-relativistic limit.\nAt these high energies, isospin symmetry is typically largely restored because the nucleon couplings are no longer \nexclusively determined by the valence quarks, and instead receive corrections from a large number of sea quarks \n(and, in principle, gluons).\nAs pointed out above, the example of neutrino scattering illustrates this effect very clearly: even though isospin is almost\nmaximally violated at low energies, the effective neutrino couplings to neutrons and protons agree within\n$\\sim5$\\,\\% at energies around 0.1\\,GeV, cf.~Eqs.~(\\ref{eq:nuel_simp}) and (\\ref{eq:nuel_simpNR}). In practice, however,\na possible complication often arises in that the nucleon couplings $g_n$ and $g_p$ are only provided in the highly\nnon-relativistic limit. \nIn that case, an educated guess for $\\sigma_{\\rm SI}$ in the second term of Eq.~(\\ref{eq:rescaling})\nis to anyway take the leading order (Born) expression -- but to adopt (effective)\nvalues for {\\it both} nucleon couplings that correspond to\nthe maximum of $\\left|g_p\\right|$ and $\\left|g_n\\right|$ in the non-relativistic limit. \nThis induces a model-dependent uncertainty \nin the normalization of the inelastic contribution that can in principle only be avoided by fully implementing the concrete\ninteraction model in a code like \\texttt{GiBUU}. On the other hand, the neutrino example illustrates that this error should\ngenerally not be expected to be larger than a factor of $\\sim$\\,2, implying that for most applications such a \nmore sophisticated treatment is not warranted.\n\n\n\\section{Contact interactions and beyond}\n\\label{sec:m2}\n\nIn sections \\ref{sec:form_factors} and \\ref{sec:inel} we have discussed in detail the \n$Q^2$-dependence that arises due to both form factor suppression and inelastic\nscattering, as well as the impact this has on the production and attenuation of the CRDM flux.\nThis does not yet take into account, however, the possible angular and energy dependence of the\nelastic scattering cross section itself. In fact, for \\mbox{(sub-)GeV} DM, a significant dependence of this type is \nactually expected in view of null searches for new light particles at colliders. For example, it has been \ndemonstrated in a recent global analysis~\\cite{GAMBIT:2021rlp} that it is impossible to satisfy all relevant \nconstraints simultaneously (even well above GeV DM masses) \nand at the same time maintain the validity of an effective field theory description at LHC energies.\n\n\nOf course, this necessarily introduces a model-dependent element to the discussion, and in this section, the\naim will be to analyse the most generic situations that can appear when considering models beyond simple contact \ninteractions.  Concretely, in section~\\ref{sec:scalar} we will study the case of a light scalar mediator, a light vector\nmediator in section~\\ref{sec:vector}, and the scenario where DM particles have a finite extent\nin section~\\ref{sec:puffy}. In all these cases, we will re-interpret the published Xenon-1T limits and assess \nwhether there is a remaining unconstrained window of large scattering cross sections for GeV-scale DM. \nJust before this, however, in section~\\ref{sec:const} we will briefly revisit the (physically less motivated) case of \na constant cross-section, which can be viewed as the highly non-relativistic limit of a contact interaction. \nThis will allow us to illustrate how the resulting CRDM constraints compare with \nestablished bounds from both surface and astrophysical experiments, as well as provide a more direct comparison with the \nexisting literature.\n\n\n\n\\subsection{Constant cross section}\n\\label{sec:const}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{figures_draft\/figure_four_wl}\n\\caption{%\n{\\it Left panel.} Limits on a constant spin-independent DM-nucleon scattering\ncross section as a function of the DM mass, \nbased on a re-interpretation of Xenon-1T limits on non-relativistic DM~\\cite{XENON:2018voc}\nfor the CRDM component studied in this work (solid lines).\nDash-dotted lines show the excluded region that results when assuming a constant cross\nsection in the attenuation part (as in Ref.~\\cite{Bringmann:2018cvk}).\nDashed lines show the effects of adding form factors in the attenuation part, but no \ninelastic scattering, resulting in limits similar to those derived in Ref.~\\cite{Xia:2021vbz}.\nFor the latter case, for comparison, we also show the effect of artificially cutting the incoming CRDM flux \nat the indicated energies.\\\\ \n{\\it Right panel.} Updated CRDM limits (coinciding with the solid lines from the left panel) in comparison \nto limits from the Lyman-$\\alpha$ forest~\\cite{Rogers:2021byl}, the Milky Way satellite \npopulation~\\cite{Maamari:2020aqz}, gas clouds in the Galactic Centre \nregion~\\cite{Bhoonah:2018gjb}, the XQC experiment~\\cite{McCammon:2002gb,Mahdawi:2018euy}, and \na recently analysed storage dewar experiment~\\cite{Neufeld:2019xes,Xu:2021lmg}.\nWe also show upper limits on the cross section as published by the CRESST \ncollaboration~\\cite{CRESST:2017ues} (solid green lines), based on a surface run of their \nexperiment, along with the maximal cross section where \nattenuation does not prevent DM from leaving a signal in the detector~\\cite{Emken:2018run}.\nAlternative limits are indicated by green dashed~\\cite{Mahdawi:2018euy} \nand dash-dotted lines~\\cite{Xu:2020qjk}, based on the assumption of a thermalization efficiency \nof $\\epsilon_{\\rm th}=2$\\,\\% and $\\epsilon_{\\rm th}=1$\\,\\%, respectively, which is \nsignificantly worse than the one adopted in the CRESST analysis.\n}\n\\label{fig:constraints_constant}\n\\end{center}\n\\end{figure}\n\nFor the discussion of a constant cross section, we will again consider the case of spin-independent scattering with isospin \nconserving nucleon couplings, as described by Eq.~(\\ref{eq:siconst}). In the left panel of Fig.~\\ref{fig:constraints_constant}, we \nshow our improved constraints from a re-interpretation of the Xenon-1T limits in this case.\nBroadly, these updated and refined CRDM limits cover the mass range up to \n$m_\\chi \\lesssim 10 \\,\\mathrm{GeV}$ for cross-sections \n$10^{-31}\\,\\mathrm{cm}^2 \\lesssim \\sigma_{\\mathrm{SI}} \\lesssim 2 \\times 10^{-28}\\,\\mathrm{cm}^2$.\n\nFor comparison, we also indicate (with dash-dotted lines) the limits that result when neglecting both form-factor\ndependence of the cross section and inelastic scatterings in the attenuation part. As expected, this leads to a shape of the \nexcluded region very similar to that originally derived in Ref.~\\cite{Bringmann:2018cvk}, where the same simplifying\nassumptions were made. As a result of our improved treatment of CR fluxes and form factors, \nhowever, the limits indicated with dash-dotted lines are overall  slightly more stringent than what is reported in that analysis.\nWe find that for very light DM, with $m_\\chi\\lesssim10$\\,MeV, this simplistic treatment actually leads to rather\nrealistic limits, the reason being that for highly relativistic particles the typical momentum transfer is always so large\nthat efficient inelastic scattering becomes relevant. For heavier DM masses, on the other hand, this treatment clearly \noverestimates the stopping power because it neglects the form factor suppression  relevant for semi-relativistic DM\nscattering on nuclei.\n\nDashed lines furthermore show the effect of adding the form factor suppression during the attenuation in the soil, as done in \nRef.~\\cite{Xia:2021vbz}, but still not including inelastic scattering. Clearly, this vastly underestimates the actual\nattenuation taking place and therefore appears to exclude very large cross sections.\\footnote{%\nCompared to Ref.~\\cite{Xia:2021vbz}, we also find that the excluded region extends to somewhat larger DM masses,\nmostly as a result of our updated treatment of elastic form factors.\nOn the other hand, we recall that our attenuation prescription is based on the analytical energy loss treatment outlined in \nsection~\\ref{sec:crdm}, rather than a full Monte Carlo simulation. This likely overestimates the maximally excluded DM mass,\nbut only by less than a factor of 2~\\cite{Xia:2021vbz}.\n} \nIn order to gain a better intuitive understanding for the shape and strength of our final limits, finally, we also indicate the \neffect of neglecting inelastic scattering and instead artificially cutting the CRDM flux (prior to entering the soil) above \nsome given energy.  The resulting upper limit on the cross section that can be probed in this fiducial setup strongly suggests that \ninelastic scattering events very efficiently stop the incident CRDM flux in the overburden as soon as they become\nrelevant compared to elastic scattering events. From Fig.~\\ref{fig:constraints_constant}, and well in accordance with the\nexpectations from section \\ref{sec:inel}, this happens at CRDM energies $T_\\chi\\gtrsim0.2$\\,GeV.\n\nIn the right panel of Fig.~\\ref{fig:constraints_constant} we show our improved constraints \nfrom a re-interpretation of the Xenon-1T limits in comparison with complementary limits from direct\nprobes of the DM-nucleon scattering cross section.\nAt small DM masses the dominant\nconstraint results from analysing the distribution of large-scale structures as traced by the Lyman-$\\alpha$ \nforest. This is based on the fact that protons scattering too strongly off DM \nwould accelerate the latter and thereby suppress the matter power spectrum at sub-Mpc scales. \nSuch limits have recently been significantly tightened~\\cite{Rogers:2021byl},  utilizing\nstate-of-the-art cosmological hydrodynamical simulations of the intergalactic medium at redshifts \n$2\\lesssim z\\lesssim6$. Similar bounds from the CMB (not shown here) are generally weaker by up to three \norders of magnitude~\\cite{Rogers:2021byl, Planck:2015bpv, Xu:2018efh}, while\nthe Milky Way satellite population~\\cite{Maamari:2020aqz} -- as inferred from the Dark Energy Survey and \nPanSTARRS-1~\\cite{DES:2019vzn} -- places bounds that are roughly one order of magnitude weaker.\nBeyond cosmological bounds, cold gas clouds near the Galactic Center provide an interesting complementary testbed, \nin particular at high DM masses, where halo DM particles scattering too efficiently on the much {\\it colder} baryon population \nwould heat up the latter~\\cite{Bhoonah:2018wmw}. Here we show updated \nconstraints~\\cite{Bhoonah:2018gjb} based on \nthe cloud G357.8-4.7-55, noting that these constraints might be improved by more than one order\nof magnitude if G1.4-1.8+87 is indeed as cold as $T\\leq22$\\,K (as reported in \nRefs.~\\cite{McClure-Griffiths:2013awa,DiTeodoro:2018ybg} but disputed in Ref.~\\cite{Farrar:2019qrv}).\nWe also display the limits~\\cite{Mahdawi:2018euy} that result from the ten minutes' flight of the X-ray Calorimetry \nRocket (XQC)~\\cite{McCammon:2002gb}, based on the observation that ambient DM particles scattering\noff the silicon nuclei in the quantum calorimeter would deposit (part of) their energy in the \nprocess~\\cite{Wandelt:2000ad,Zaharijas:2004jv,Erickcek:2007jv}. \nIn deriving these XQC limits, one must take into account that the recoil energy of a silicon nucleus \npotentially thermalizes much less efficiently in the calorimeter than the $e^\\pm$ pairs produced from \nan incoming X-ray photon, such that a nuclear recoil energy $T_N$ will leave a signal equivalent  \nto a photon with a reduced `thermal' recoil energy $T_T=\\epsilon_{\\rm th} T_N$. Concretely, the limits shown in the \nplot are based on the very conservative assumption of a thermalization efficiency factor of $\\epsilon_{\\rm th} = 0.02$.\\footnote{%\nWhen the scattering is mediated by a Yukawa-like interaction, a perturbative description of the scattering\nprocess may no longer be adequate. In that case the constraints shown here, in particular for XQC, receive \ncorrections due to non-perturbative effects leading to resonances or anti-resonances in the scattering cross \nsection~\\cite{Xu:2020qjk}. Here, we will not consider this possibility further, noting that a variation of the relatively \nuncertain value of $\\epsilon_{\\rm th}$ anyway has a larger impact on the XQC limits~\\cite{Mahdawi:2018euy}.\n}\n\nFurthermore, in order to directly probe sub-GeV DM with very large cross sections, the CRESST collaboration has performed\na dedicated surface run of their experiment~\\cite{CRESST:2017ues}, deliberately avoiding the shielding \nof the Gran Sasso rock used in the standard run~\\cite{CRESST:2015txj}. The result of this search is the exclusion region \nindicated by the solid green line in Fig.~\\ref{fig:constraints_constant}. Here, upper bounds on the cross section correspond to \nthe published limits, obtained under the assumption that any attenuation in the overburden can be neglected. \nModelling the effect of attenuation with detailed numerical simulations also results in  \nthe exclusion region limited from above~\\cite{Emken:2018run}, coming from the fact that one must have a sufficiently large flux of \nDM particles at the detector location.\nIn a series of papers, Farrar~\\textit{et al.}~have claimed  \nthat the CRESST thermalization efficiency adopted in the official analysis is too optimistic~\\cite{Mahdawi:2018euy,Wadekar:2019mpc,Xu:2020qjk,Xu:2021lmg}, challenging the general ability of the experiment to probe sub-GeV DM.\nWe indicate the resulting alternatives to the published CRESST limits in the same figure, albeit noting that the underlying assumption \nof an efficiency as low as $\\epsilon_{\\rm th}\\sim1$\\,\\% is not supported by data or simulations.\nFor example, no indication for such a dramatic loss of efficiency at low energies is observed for neutrons from an AmBe \nneutron calibration source~\\cite{florian}.\n\nTo summarise, Fig.~\\ref{fig:constraints_constant} illustrates the fact that the existence of the CRDM component provides an \nimportant probe of strongly interacting light DM. In particular, below $m_\\chi\\lesssim100$\\,MeV, it restricts parameter space that \nis otherwise either unconstrained or only testable with \ncosmological probes (which -- at least to some degree -- are \nsubject to modelling caveats regarding the Lyman-$\\alpha$ forest and the non-linear evolution of density perturbations \nat small scales; see, e.g., Refs.~\\cite{Hui:2016ltb,Irsic:2017ixq}). The CRDM component also leads to\nhighly relevant complementary constraints up to DM masses of a few GeV, especially when noting that  \nthese constraints are independent of the thermalization efficiency discussion above.\n\n\n\n\\subsection{Scalar mediators}\n\\label{sec:scalar}\n\nAs our first example beyond a constant scattering cross section we consider the case where \na new light scalar particle $\\phi$ mediates the interaction between DM and nucleons.  \nWe thus consider the interaction Lagrangian \n\\begin{equation}\n\\mathcal{L}_{\\rm int}= - g_\\chi \\phi\\overline\\chi\\chi - g_p \\phi \\overline p p- g_n \\phi \\overline n n\\,,\n\\end{equation}\nand assume, for simplicity, isospin conservation ($g_p = g_n $).\nAt the level of the effective nuclear interaction Lagrangian, the dominant interaction terms with scalar ($N_0$) and fermionic ($N_{1\/2}$) \n{\\it nuclei} are thus given by\\footnote{%\nWhile the dominant cosmic-ray nuclei are either scalar or spin $1\/2$ particles, some heavier nuclei in the overburden\nhave higher spins. For simplicity we treat those nuclei as scalars when determining their contribution to the energy loss,\nas described by Eq.~(\\ref{eq:eloss}), noting that this induces a neglible error in the estimated elastic scattering cross section,\nof the order of $Q^2\/m_N^2\\ll1$. Moreover, nuclei with higher spins make up only about 2\\% of the total mass in the overburden.\n}\n\\begin{equation}\n\\label{eq:leff_scalar}\n\\mathcal{L}_{\\rm int}= -g_N\\left( 2m_N N_0N_0+\\overline N_{1\/2}N_{1\/2}\\right)\\,.\n\\end{equation}\nHere, the dimensionful coupling to scalar nuclei has been normalized such that both terms in the above expression result\nin the same scattering cross section in the highly non-relativistic limit. In addition, the coupling to individual nucleons is coherently \nenhanced across the nucleus, resulting in an effective coupling to both scalar and fermionic nuclei given by\n\\begin{equation}\n\\label{eq:gN_coh}\ng_N^2 =A^2 \\, g_p^2 \\times G_N^2(Q^2)\\,,\n\\end{equation}\nwhere $G_N$ is the same form-factor as in the case of a `constant' cross section.\nFor the resulting elastic scattering cross section for DM incident on nuclei at rest we find\n\\begin{equation}\n\\label{diffsig_full_scalar}\n\\frac{d\\sigma_{\\chi N}}{d T_N}=\\frac{\\mathcal{C}^2\\sigma_{\\rm SI}^\\mathrm{NR}}{T_N^\\mathrm{max}}\n\\frac{m_\\phi^4}{(Q^2+m_\\phi^2)^2}\n\\frac{m_N^2\\left(Q^2+4m_\\chi^2 \\right)}{4 s\\,{\\mu_{\\chi N}^2}}\n\\times\\left\\{\n\\begin{array}{ll}\n1& ~~\\mathrm{for~scalar~}N\\\\\n1+\\frac{Q^2}{4m_N^2} &  ~~\\mathrm{for~fermionic~}N\n\\end{array}\n\\right\\}\n\\times G_N^2(Q^2)\\,,\n\\end{equation}\nwhere $\\mu_{\\chi p}$ is the reduced mass of the DM\/nucleon system and  \n\\begin{equation}\n\\label{eq:sig0_scalar}\n\\sigma_{\\rm SI}^\\mathrm{NR} = \\frac{g_\\chi^2 g_p^2 \\mu_{\\chi p}^2}{\\pi m_\\phi^4}\n\\end{equation}\nis the spin-independent scattering cross section {\\it per nucleon} in the ultra non-relativistic limit.\nFor reference, the kinematic quantities $T_N^\\mathrm{max}$, $s$ and $Q^2$ are given by \nEqs.~(\\ref{eq:tmax}), (\\ref{eq:sdef}) and (\\ref{eq:q2}), respectively. For the production part of the process, \nwhere CR nuclei collide with DM at rest, one\nsimply has to exchange $T_N\\leftrightarrow T_\\chi$ and $m_\\chi\\leftrightarrow m_N$ in \nthese expressions for kinematic variables -- but not in the rest of Eq.~(\\ref{diffsig_full_scalar}) \n-- in order to obtain ${d\\sigma_{\\chi N}}\/{d T_\\chi}$.\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{figures_draft\/figure_five_wl}\n\\caption{{\\it Left panel.} Solid lines show the CRDM flux before attenuation for a constant interaction\ncross section, as in Fig.~\\ref{fig:flux_ff}, for DM masses  $m_\\chi=10$\\,MeV and  $m_\\chi=1$\\,GeV. \nFor comparison we also indicate the corresponding CRDM flux for a {\\it scalar} mediator, cf.~Eq.~(\\ref{diffsig_full_scalar}), \nwith mass  $m_\\phi=100$\\,MeV (dash-dotted), $m_\\phi=10$\\,MeV \n(dashed) and $m_\\phi=1$\\,MeV (dotted), for a cross section (in the non-relativistic limit) of  \n$\\sigma_{\\rm SI}^{\\rm NR}=10^{-30}\\,\\mathrm{cm}^2$. \n{\\it Right panel.} \nMinimal kinetic energy $T_\\chi$ that a DM particle must have, prior to attenuation, in order \nto trigger a signal in the Xenon-1T experiment. Line styles and colors match those of the left\npanel. In particular, solid lines show the case of a constant spin-independent scattering cross \nsection and are identical to those displayed  in Fig.~\\ref{fig:attenuation_ff}.\n}\n\\label{fig:results_scalar}\n\\end{center}\n\\end{figure}\n\nIn the left panel of Fig.~\\ref{fig:results_scalar} we show the resulting CRDM fluxes for this\nmodel. For small kinetic energies these fluxes are, as expected, identical to those shown in Fig.~\\ref{fig:flux_ff} \nfor the case of a constant cross section. This is the regime where $Q^2=2m_\\chi T_\\chi$ is smaller\nthan the masses of both the mediator and CR nuclei, such that Eq.~(\\ref{diffsig_full_scalar}) reduces to \nEq.~(\\ref{eq:siconst}). For $Q^2\\gtrsim m_\\phi^2$, on the other hand, the presence of a light mediator \nclearly suppresses the fluxes. Note that the matrix element also contains a factor of $(Q^2+4m_\\chi^2)$, \nwhich additionally leads\nto a flux {\\it enhancement} for fully relativistic DM particles, $T_\\chi\\gtrsim 2 m_\\chi$. \nIn the figure, this latter effect is clearly visible for the case of $m_\\chi=10$\\,MeV and a heavy mediator.\nIn general, the appearance of such model-dependent features demonstrates the need to use the full matrix \nelement for the relativistic cross section. This is in contrast to the non-relativistic case, where a model-independent rescaling \nof the cross section by a factor of $(1+Q^2\/m_\\phi^2)^{-2}$ is usually sufficient to model the effect of a light mediator\n(see, e.g., Refs.~\\cite{Chang:2009yt,Fornengo:2011sz,Kaplinghat:2013yxa}).\n\nIn the right panel of Fig.~\\ref{fig:results_scalar}, we explore the minimal CRDM energy $T_\\chi$ that\nis needed to induce a detectable nuclear recoil. Compared to the situation of a constant scattering cross section (depicted by the \nsolid lines for easy comparison), the attenuation is as expected rather strongly suppressed when light scalar mediators are \npresent (with the exception of the\ncase with $m_\\chi=10$\\,MeV and $m_\\phi=100$\\,MeV, where the cross section is enhanced \ndue to the $(Q^2+4m_\\chi^2)$ factor in the squared matrix element). In order to understand the qualitative behaviour of \n$T_\\chi^{\\rm min} (z=0)$ better, we recall from the discussion of Fig.~\\ref{fig:attenuation_ff}\nthat there are two generic scaling regimes for solutions of the energy loss equation. Firstly, for cross-sections with no \n-- or only a mild --  dependence on the momentum transfer, $T_{\\chi}^{\\rm min}(z=0)$ grows exponentially \nwith increasing $\\sigma_{\\rm SI}^{\\rm NR}$. Secondly, in the presence of an effective cutoff in the cross section (like when form \nfactors or light mediators are introduced), $T_{\\chi}^{\\rm min}(z=0) \\propto \\sqrt{\\sigma_{\\rm SI}^{\\rm NR}}$ for large energies\n$T_\\chi$. These different regimes are clearly visible in the figure. \nFor the green dot-dashed curve ($m_\\chi=1$\\,GeV, $m_\\phi=100$\\,MeV), for example,\none observes as expected an initial steep rise at the smallest DM energies -- until the form factor and mediator suppression \nof the cross section cause a scaling with $\\sqrt{\\sigma_{\\rm SI}^{\\rm NR}}$ for kinetic energies above a few MeV. \nAt roughly $T_\\chi\\gtrsim0.1$\\,GeV, inelastic scattering kicks in, leading again to an exponential suppression of the flux. \nFor even higher energies, finally, the scattering cross section falls off so rapidly that the required initial DM energy once\nagain only grows as $\\sqrt{\\sigma_{\\rm SI}^{\\rm NR}}$.\n\nTurning our attention to the resulting CRDM limits, it is worth stressing here that $\\sigma_{\\rm SI}^\\mathrm{NR}$, as introduced in Eq.~(\\ref{eq:sig0_scalar}),\nis a somewhat artificial object that only describes the cross section for physical processes\nrestricted to $Q^2\\lesssim m_\\phi^2$.\nIn a direct detection experiment like Xenon-1T this is necessarily violated for $m_\\phi\\lesssim \\sqrt{2m_N T_N^{\\rm thr}}\\sim35$\\,MeV, \ngiven that  $T_N^{\\rm thr}=4.9$\\,keV is the  minimal recoil \nenergy needed to generate a signal. A natural consequence of this is that making a straight-forward comparison \nto the $\\sigma_{\\rm SI}$ appearing in the `constant cross section' case discussed in \nsection \\ref{sec:const} is challenging. Instead, the best we can achieve in terms of a meaningful \ncomparison is to define a {\\it reference cross section}\n\\begin{equation}\n\\label{eq:Qref}\n \\tilde \\sigma_{\\rm Xe,SI}^p \\equiv \\sigma_{\\rm SI}^\\mathrm{NR}\\times \\frac{m_\\phi^4}{(Q_{\\rm Xe,ref}^2+m_\\phi^2)^2}\n\\frac{Q^2_{\\rm Xe,ref}+4m_\\chi^2}{4m_\\chi^2}\\,,\n\\end{equation}\nwhere $Q_{\\rm Xe,ref}\\sim35$\\,MeV. It follows from Eq.~(\\ref{diffsig_full_scalar}) and Eq.~(\\ref{eq:siconst}), \nand the fact that $s\\approx (m_\\chi+m_N)^2$ for the energies of interest here, that $ \\tilde \\sigma_{\\rm Xe,SI}^p$ \nshould be interpreted as the effective CRDM cross section per nucleon that is dominantly seen in the Xenon-1T \nanalysis window.\nIt is thus this quantity, not the $\\sigma_{\\rm SI}^\\mathrm{NR}$ from Eq.~(\\ref{eq:sig0_scalar}), that should\nbe compared to the published Xenon-1T limits on the DM-nucleon cross section.\n\nThis also allows us to address the question of how the limits on the DM-nucleon coupling coming from the CRDM component\ncompare to the complementary constraints introduced in section \\ref{sec:const} (cf.~the right\npanel of Fig.~\\ref{fig:constraints_constant}). In order to do so, one first needs to\nrealize that all of those limits are derived under the assumption of non-relativistic\nDM and a constant cross section. In reality, however, they probe very different physical environments and typical \nmomentum transfers. In order to allow for a direct comparison, therefore, they also need to be re-scaled to a common \nreference cross section.\nAssuming that the DM energies in Eq.~(\\ref{diffsig_full_scalar}) are non-relativistic, a reported limit on the DM-nucleon \ncross-section $\\sigma_{\\rm SI}^p$ from an experiment probing typical momentum transfers of the order $Q^2_{\\rm ref}$ would \ncorrespond to a cross section of\n\\begin{equation}\n\\label{eq:rescale_scalar}\n \\tilde \\sigma_{\\rm Xe,SI}^p = \\sigma_{\\rm SI}^p\\times \n \\left(\\frac{Q^2_{\\rm ref}+m_\\phi^2}{Q^2_{\\rm Xe,ref}+m_\\phi^2}\\right)^2\n\\frac{Q^2_{\\rm Xe,ref}+4m_\\chi^2}{Q^2_{\\rm ref}+4m_\\chi^2}\n\\end{equation}\nin the Xenon-1T detector. As an example, consider the CRESST surface run~\\cite{CRESST:2017ues}, where a threshold energy of $\\sim20$\\,eV\nfor the sapphire \ndetector would imply $Q^2_{\\rm ref}\\sim (0.98\\,\\mathrm{MeV})^2\/\\epsilon_{\\rm th}$.  Similarly, a thermal recoil energy of of $29$\\,eV\nin XQC corresponds to $Q^2_{\\rm ref}\\sim (8.7\\,\\mathrm{MeV})^2$ for the nuclear recoil on Si nuclei\n(assuming $\\epsilon_{\\rm th} = 0.02$ as for the unscaled limits).\nTurning to cosmological limits, a baryon velocity of $v_b^{\\rm rms}\\sim33$km\/s at the times relevant for the emission\nof Lyman-$\\alpha$ photons~\\cite{Silk:1967kq} implies typical momentum transfers from the Helium nuclei to DM of \n$Q_{\\rm ref}^2\\sim4 \\mu_{\\chi{\\rm He}}^2\\times10^{-8}$. This means that, for the range of DM and mediator masses considered \nhere, the cross section at these times becomes roughly constant and we can approximate $Q_{\\rm ref}^2\\approx0$ \nin Eq.~(\\ref{eq:rescale_scalar}). The same goes for the constraints stemming from the MW satellite abundance,\nwhich are sensitive to even lower redshifts and thus smaller momentum transfers~\\cite{Nadler:2019zrb,Maamari:2020aqz}.\n\nIn Fig.~\\ref{fig:limits_scalar} we show a subset of these correspondingly rescaled constraints\\footnote{%\nUpper bounds on the excluded cross section, due to attenuation effects, cannot simply be rescaled as\nin Eq.~(\\ref{eq:rescale_scalar}). \nFor the sake of Fig.~\\ref{fig:limits_scalar}, we instead adopt a rather simplistic \napproach~\\cite{Davis:2017noy,Kouvaris:2014lpa,Emken:2017erx,Emken:2018run} to estimate these \nlimits by requiring that the most energetic halo DM particles, with an assumed velocity $v_{\\rm max}$,\ncan trigger nuclear recoils above the CRESST threshold of 19.7\\,eV\/$\\epsilon_{\\rm th}$ after attenuation in the \nEarth's atmosphere. For the average density and distribution of elements in the atmosphere, we follow Ref.~\\cite{USatm}.\nBy treating $v_{\\rm max}$ and the effective height of the atmosphere, $h_a$, \nas free parameters, we can then rather accurately fit the results of more detailed \ncalculations~\\cite{Emken:2018run,Mahdawi:2018euy} for the case of a constant cross section\n-- with numerical values in reasonable agreement with the physical expectation in such a heuristic approach. \nFinally, we adopt those values of $v_{\\rm max}$ and $h_a$ to derive the corresponding limits for the case of a scalar mediator,\nas displayed in Fig.~\\ref{fig:limits_scalar}.\n}\n -- for mediator masses \n$m_\\phi=1$\\,MeV, 10\\,MeV, 100\\,MeV and 1\\,GeV -- along with the full CRDM constraints derived here.\nWe also indicate, for comparison, with dotted black lines where non-perturbative couplings would be needed in this model to \nrealize the stated cross section. This line is only visible for the case of $m_\\phi=1$\\,GeV,\nwhich demonstrates that it is generically challenging to realize large cross sections without invoking light mediators.\nThe presence of an abundant species with a mass below a few MeV, furthermore, would affect how light elements are\nproduced during big bang nucleosynthesis (BBN). For a 1\\,MeV particle with one degree of freedom, like $\\phi$,  \nthis can be formulated as a constraint of $\\tau>0.43$\\,s~\\cite{Depta:2020zbh} on the lifetime of such a particle.\nPhysically, this constraint derives from freeze-in production of $\\phi$ via the inverse decay process. \nSince $\\phi\\to\\gamma\\gamma$ (apart from $\\phi\\to\\bar \\nu\\nu$) is the only kinematically possible SM decay channel, \nthe translation of this bound to a constraint on the SM coupling $g_p$ is somewhat model-dependent.\nFor concreteness we consider the Higgs portal model, where $\\tau>1$\\,s at $m_\\phi=1$\\,MeV corresponds to \na squared mixing angle\n$\\sin^2\\theta=(8.62\\times10^2g_p)^2>3.8\\times10^{-4}$~\\cite{Krnjaic:2015mbs}. The area above the dashed\nline in the top left panel of Fig.~\\ref{fig:limits_scalar} requires either a {\\it larger} value of $g_p$ than what is given by this\nbound, or a non-perturbative coupling $g_\\chi^2>4\\pi$. This confirms the generic expectation that for very light particles \nBBN constraints are more stringent than those stemming from the CRDM \ncomponent~\\cite{Krnjaic:2019dzc,Bondarenko:2019vrb}.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{figures_draft\/figure_six_wl.pdf}\n\\caption{%\nLimits on the DM-nucleon scattering cross section evaluated at a reference momentum transfer of \n$Q_{\\rm Xe,ref}=35$\\,MeV, as a function of the DM mass $m_\\chi$. From top left to bottom right,\nthe panels show the case of a {\\it scalar mediator} with mass $m_\\phi=1$\\,MeV, 10\\,MeV, 100\\,MeV and 1\\,GeV.\nSolid purple lines show the updated CRDM limits studied in this work. We further\nshow limits from the Lyman-$\\alpha$ forest~\\cite{Rogers:2021byl}, the XQC \nexperiment~\\cite{McCammon:2002gb,Mahdawi:2018euy}, the CRESST surface \nrun~\\cite{CRESST:2017ues, Emken:2018run} and an alternative analysis of the CRESST \nlimits~\\cite{Mahdawi:2018euy}. All these limits are rescaled to match the situation of a light mediator,\nas explained in the text. The parameter region above the dotted black line in the bottom right panel\nrequires non-perturbative couplings, while the area above the dotted line in the top left panel is excluded\nby BBN.\n}\n\\label{fig:limits_scalar}\n\\end{center}\n\\end{figure}\n\n\nOur results demonstrate that in the presence of light mediators the largest DM mass that can be constrained \ndue to CR upscattering is reduced from about 10\\,GeV, cf.~Fig.~\\ref{fig:constraints_constant},\nto just above 1\\,GeV (for $m_\\phi\\sim1$\\,MeV). This is a direct consequence of the suppressed CRDM production\nrate discussed above. On the other hand, the reduction of the cross section also implies a smaller attenuation\neffect, thus closing parameter space at larger cross sections. More importantly, complementary constraints\nfrom cosmology and dedicated surface experiments become {\\it more stringent} in the presence of light mediators, \nonce they are translated to a common reference cross section. To put this in context, let us first recall that in the constant \ncross section case, Fig.~\\ref{fig:constraints_constant} tells us that cross sections \n$\\sigma_{\\rm SI}\\gtrsim 2\\cdot10^{-31}\\,{\\rm cm}^2$  are safely excluded across the entire DM mass range \n(or $\\sigma_{\\rm SI}\\gtrsim 6\\cdot10^{-31}\\,{\\rm cm}^2$ when assuming that the thermalization efficiency of \nCRESST is indeed as low as 2\\,\\%). From Fig.~\\ref{fig:limits_scalar} we infer that these limits can be somewhat \nweakened for sub-GeV DM, when considering light meditators in the mass range \n$10\\,{\\rm MeV}\\lesssim m_\\phi\\lesssim 100\\,{\\rm MeV}$ (as we will see further down, the situation of a vector mediator\nis not appreciably different from that of the scalar mediator shown here). Concretely, the upper bound on the \ncross section now becomes $\\tilde \\sigma_{\\rm SI}\\lesssim 3\\cdot10^{-31}\\,{\\rm cm}^2$, independently of the DM\n{\\it and} mediator mass. For a 2\\,\\% thermalization efficiency of CRESST~\\cite{Mahdawi:2018euy} and a narrow \nrange of mediator masses, $10\\,{\\rm MeV}\\lesssim m_\\phi\\ll100\\,{\\rm MeV}$,\na small window opens up above the maximal cross section that can be probed with CRESST. \nThe reason is the gap between Lyman-$\\alpha$ bounds and the weakened CRESST limits from Ref.~\\cite{Mahdawi:2018euy}\nthat is visible in the figure, for $m_\\phi\\gtrsim10\\,{\\rm MeV}$, and which is closed by the CRDM limits only for mediator\nmasses of $m_\\phi\\gtrsim 30$\\,MeV. Nominally, for $m_\\chi\\sim2$\\,GeV and $m_\\phi\\sim 30$\\,MeV, \nthis would allow for cross sections as large as $\\tilde \\sigma_{\\rm SI}\\sim 4\\cdot10^{-29}\\,{\\rm cm}^2$.\nIn either case, the conclusion remains that CRDM leads to highly complementary limits, and that this \nrelativistic component of the DM flux is in fact crucial for excluding the possibility of very large DM-nucleon\ninteractions.\n\n\n\\subsection{Vector mediators}\n\\label{sec:vector}\n\nWe next consider the general case of a massive vector mediator $V$, with interactions given by\n\\begin{equation}\n\\mathcal{L}=   V_\\mu \\left(g_\\chi \\overline{\\chi}\\gamma^\\mu \\chi + g_{p}\\overline{p}\\gamma^\\mu p + g_{n}\\overline{n}\\gamma^\\mu n\\right)\\,.\n\\end{equation}\nWe will again assume  $g_n=g_p$  for simplicity, noting that smaller values of the ratio $g_n\/g_p$ can lead to\nsignificantly smaller cross sections (see, e.g., Refs.~\\cite{Frandsen:2011cg,Kaplinghat:2013yxa}); in our context this would\nmostly imply that the attenuation in the overburden becomes less relevant, leading to more stringent constraints.\nIn analogy to Eq.~(\\ref{eq:leff_scalar}), this implies the following dominant interaction terms with scalar and fermionic nuclei, respectively:\n\\begin{equation}\n\\label{eq:leff_vector}\n\\mathcal{L}_{\\rm int}= -g_N V_\\mu\\left( i N_0^*{\\mathop{\\partial^\\mu}^{\\leftrightarrow}} N_0+\\overline N_{1\/2}\\gamma^\\mu N_{1\/2}\\right),\n\\end{equation}\nwhere the effective mediator coupling to nuclei, $g_N$, is again given by the coherent enhancement stated in Eq.~(\\ref{eq:gN_coh}).\nFor the elastic scattering cross section on nuclei we find\n\\begin{eqnarray}\n\\label{diffsig_full_vector}\n\\frac{d\\sigma_{\\chi N}}{d T_N}&=&\\frac{\\mathcal{C}^2 \\sigma_{\\rm SI}^\\mathrm{NR}}{T_N^\\mathrm{max}}\n\\frac{m_A^4}{(Q^2+m_A^2)^2}\n\\times G_N^2(Q^2)\\\\\n&&\n\\times\\frac{1}{4s \\mu_{\\chi N}^2}\n\\left\\{\n\\begin{array}{ll}\nm_\\chi^2Q^2-Q^2s+(s-m_N^2-m_\\chi^2)^2& ~~\\mathrm{for~scalar~}N\\\\\n\\frac12 Q^4 -Q^2s+(s-m_N^2-m_\\chi^2)^2&  ~~\\mathrm{for~fermionic~}N\n\\end{array}\n\\right..\\nonumber\n\\end{eqnarray}\nHere, the cross section in the ultra-nonrelativistic limit,\n\\begin{equation}\n\\label{eq:sig0_vector}\n\\sigma_{\\rm SI}^\\mathrm{NR} = \\frac{g_\\chi^2 g_p^2 \\mu_{\\chi p}^2}{\\pi m_A^4}\\,,\n\\end{equation}\ni.e.~for $Q^2\\to0$ and $s\\to(m_N+m_\\chi)^2$, agrees exactly with the result obtained for the scalar case, as expected.\nFor large energies and momentum transfers, on the other hand, the behaviour\nis different.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{figures_draft\/figure_seven_wl}\n\\caption{%\n{\\it Left panel.} \nMinimal kinetic energy $T_\\chi$ that a DM particle must have, prior to attenuation, in order \nto trigger a signal in the Xenon-1T experiment for DM nucleus interactions via a {\\it vector mediator},\nas a function of the spin-independent DM-nucleon scattering cross section in the highly non-relativistic limit, \n$\\sigma_{\\rm SI}^\\mathrm{NR}$. Yellow (green) lines indicate a DM mass\n$m_\\chi=10$\\,MeV ($m_\\chi=1$\\,GeV), and different line styles correspond to  mediator masses \n$m_A=1,10,100$\\,MeV as indicated.\nSolid lines show the case of a constant spin-independent scattering cross \nsection and are identical to those displayed  in Fig.~\\ref{fig:attenuation_ff}.\n{\\it Right panel.} \nConstraints on $\\sigma_{\\rm SI}^\\mathrm{NR}$ as a function of the DM mass $m_\\chi$.  Solid purple lines \nrefer to the case of a constant cross section, as in Fig.~\\ref{fig:constraints_constant}, while other line \nstyles show the case where the interaction is mediated by a light scalar (red) or vector (green) particle \nwith mass $m_{\\rm med}=10$\\,MeV and $1$\\,GeV, respectively.}\n\\label{fig:limits_vector}\n\\end{center}\n\\end{figure}\n\nThe resulting CRDM fluxes are nonetheless so similar to the scalar case shown in the left panel of Fig.~\\ref{fig:results_scalar} \nthat we refrain from plotting them separately.  Differences do exist, however, for the stopping power in the overburden.\nIn the left panel of Fig.~\\ref{fig:limits_vector} we therefore show the minimal initial kinetic energy needed by a CRDM\nparticle to induce detectable nuclear recoils in Xenon-1T. Compared to the scalar case, cf.~the right panel of \nFig.~\\ref{fig:results_scalar},  the attenuation is more efficient for highly relativistic DM particles due to the $s$-dependence \nof the terms in the second line of Eq.~(\\ref{diffsig_full_vector}). As before, the effect of these model-dependent terms\nfrom the scattering amplitude \nis most visible for highly relativistic particles, with small $m_\\chi$, and large mediator masses, where \nthe suppression due to the factor $(1+Q^2\/m_A^2)^{-2}$ is less significant.\n\nIn the right panel of  Fig.~\\ref{fig:limits_vector} we compare the  final exclusion regions for the \nsituations considered so far, i.e.~for a contact interaction, scalar mediators and vector mediators, respectively. \nFor the sake of comparison in one single figure, \nwe plot here the cross section in the ultra-nonrelativistic limit. For an interpretation of \nthese limits in comparison to complementary constraints on DM-nucleon interactions \nwe thus refer to the discussion of Fig.~\\ref{fig:limits_scalar},\nnoting that the rescaling prescriptions for vector and scalar mediators are qualitatively the same.\nThe first thing to take away from Fig.~\\ref{fig:limits_vector} is that, as expected, the exclusion regions \nfor heavy mediators resemble those obtained for the constant cross section case. The figure\nfurther demonstrates that the only significant difference between scalar and vector mediators appears  \nat smaller mediator masses, where the former are somewhat less efficiently stopped in the overburden. \nIt is worth noting, however, that this region of parameter space where the vector and scalar cases differ substantially\nis nonetheless excluded by Lyman-$\\alpha$ bounds. \nThe general discussion and conclusions from the scalar mediator case \nexplored in the previous subsection thus also applies to interactions mediated by vector particles.\n\n\n\n\n\\subsection{Finite-size dark matter}\n\\label{sec:puffy}\n\nAs a final generic example of a $Q^2$-suppressed cross section let us consider the situation \nwhere the DM particle itself has a finite size that is larger than its Compton wavelength. \nThe corresponding scattering cross section then takes the same form as in the point-like case,\nmultiplied by another factor $\\left|G_\\chi(Q^2)\\right|^2$ that reflects the spatial extent of \n$\\chi$~\\cite{Feldstein:2009tr,Laha:2013gva,Chu:2018faw} (for concrete models see also, e.g., \nRefs.~\\cite{Nussinov:1985xr,Chivukula:1989qb,Cline:2013zca,Krnjaic:2014xza,Wise:2014ola,Hardy:2015boa,Coskuner:2018are,Contino:2018crt}).\nSpecifically, just as for nuclear form factors, we have \n\\begin{equation}\nG_\\chi(Q^2)=\\int d^3 x\\,e^{i\\mathbf{q}\\cdot\\mathbf{x}}\\rho_\\chi(\\mathbf{x})\\,,\n\\end{equation}\nwhere $\\rho_\\chi(\\mathbf{x})$ is the distribution of the effective charge density that the interaction couples to.\nFor simplicity we will  choose a dipole form factor of the form\\footnote{%\nThe exact choice of the form factor does not significantly affect our results, as long as $G_\\chi(Q^2)<G_\\chi(0)=1$.\nAn interesting, qualitatively different situation occurs when $G_\\chi(0)=0$, i.e.~for a form factor that {\\it grows} \nwith $Q^2$. This is, e.g., realized if the scattering is mediated by a dark $U(1)'$ under which $\\chi$ is \nneutral~\\cite{Feldstein:2009tr,Chu:2018faw}. We will not consider this class of models in this work.\n}\n\\begin{equation} \\label{eq:dipole}\nG_\\chi(Q^2)=\\left(1+\\frac{r_\\chi^2}{12}Q^2\\right)^{-2}\\,,\n\\end{equation}\nwith $r_\\chi$ being the r.m.s.~radius of the DM particle, $r_\\chi^2=\\int d^3 x\\,\\mathbf{x}^2 \\rho_\\chi(\\mathbf{x})$.\nWe then multiply $G^2_\\chi(Q^2)$ with Eq.~(\\ref{eq:siconst}) in order to obtain \n${d\\sigma_{\\chi N}}\/{d T_N}$, thus describing an effective scalar interaction with the usual coherent\nenhancement inside the  nucleus -- but where each of the nucleons only `sees' some fraction of the entire DM \nparticle.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{figures_draft\/figure_eight_wl}\n\\caption{{\\it Left panel.} \nSolid lines show the CRDM flux before attenuation for a constant interaction\ncross section, as in Fig.~\\ref{fig:flux_ff}, for DM masses  $m_\\chi=10$\\,MeV and  $m_\\chi=1$\\,GeV. \nFor comparison we indicate the corresponding CRDM flux for {\\it finite size} DM,  \nwith  $r_\\chi =1$\\,fm (dotted) and $r_\\chi=10$\\,fm (dashed), for a cross section of  \n$\\sigma_{\\rm SI}^{\\rm NR}=10^{-30}\\,\\mathrm{cm}^2$. \n{\\it Right panel.} Limits on the spin-independent DM-nucleon cross section, with line styles and colors matching \nthose of the left panel. In particular, solid lines show the case of a constant scattering cross \nsection and are identical to those displayed  in the left panel of Fig.~\\ref{fig:constraints_constant}.\n\\label{fig:results_puffy}\n}\n\\end{center}\n\\end{figure}\n\nIn a very similar fashion to what happens in the presence of a light mediator $\\phi$, such a cross section features a sharp \nsuppression for momentum transfers exceeding a `mass' scale $m_\\phi\\sim \\sqrt{12}\/r_\\chi$. Sharper than in that case, in fact, as \nthe suppression scales with \na power of $Q^{-8}$ rather than just $Q^{-4}$. This is clearly visible in the left panel of Fig.~\\ref{fig:results_puffy},\nwhere we plot the expected CRDM flux for DM with a finite size, for various values of $m_\\chi$ and $r_\\chi$. \nFor example, for $r_\\chi=10$\\,fm, we have \n$\\sqrt{12}\/r_\\chi\\sim68$\\,MeV and the cutoff indeed appears at only slightly smaller values of $T_\\chi$ than in the case\nof the $100$\\,MeV mediator displayed in Fig.~\\ref{fig:results_scalar} (for $m_\\chi=1$\\,GeV). The slope above the cutoff, \nhowever, is twice as steep -- as expected from the $Q^{-8}$ suppression.\n\nIn the right panel of Fig.~\\ref{fig:results_puffy} we show how the constraints on a constant DM-nucleon cross section\nweaken when considering the situation where the DM particles themselves have a finite extent. Concretely,\nfor a DM radius of $r_\\chi=1$\\,fm ($r_\\chi=10$\\,fm) the maximal DM mass that can be probed decreases from\n$\\sim10$\\,GeV to about $4.5$\\,GeV ($1.1$\\,GeV). The reduced CRDM flux for extended DM, cf.~the left panel\nof the figure, also \nvisibly weakens the lower bound on the exclusion region. At the same time, attenuation is also less efficient\nfor a given cross section in the non-relativistic limit (inelastic scattering still effectively cuts off the incoming CRDM\nflux above $\\sim$0.2\\,GeV, explaining e.g.~the upper, almost horizontal boundary of the exclusion region in the\n$r_\\chi=10$\\,fm case).\nFor $r_\\chi\\gtrsim1$\\,fm, this starts to significantly enlarge the excluded region to higher cross sections.\nOn the other hand, it should be noted that for composite DM particles the interaction\ncross section may not actually continue to drop as $Q^{-8}$ for very large momentum transfers,\nas would be implied by Eq.~(\\ref{eq:dipole}). At some point, instead, inelastic scattering events on the DM constituents \nwill take over, in analogy to what we discussed for nuclei in section \\ref{sec:inel}. This is particularly relevant \nif the DM constituents are themselves finite in size, in which case the upper boundaries of the\nexclusion regions shown in Fig.~\\ref{fig:results_puffy} would be overly optimistic for very large $r_\\chi$.\n\nSimilar to the discussion in section \\ref{sec:scalar}, a comparison of the limits shown in Fig.~\\ref{fig:results_puffy}  \nwith complementary limits requires a re-scaling of $\\sigma_{\\rm SI}$ to a common reference cross section. Due to the\nstrong form factor suppression, this rescaling has an even larger effect than in the light mediator case; concretely,\ninstead of Eq.~(\\ref{eq:rescale_scalar}), the rescaling of reported limits, $\\sigma_{\\rm SI}^p$, to those relevant \nfor the Xenon-1T detector now takes the form\n\\begin{equation}\n \\tilde \\sigma_{\\rm Xe,SI}^p = \\sigma_{\\rm SI}^p\\times \n \\left(\\frac{Q^2_{\\rm ref}+12\/r_\\chi^2}{Q^2_{\\rm Xe,ref}+12\/r_\\chi^2}\\right)^4\\,.\n\\end{equation}\nQualitatively, however, this does not change the lesson learned in the light mediator case: while limits from the CRDM component\ncan be weakened by increasing $r_\\chi$, this will inevitably strengthen complementary bounds from cosmology. As a result,\nwe find once again an absolute upper bound on the cross section of about  \n$\\tilde \\sigma_{\\rm SI}\\sim 3\\cdot10^{-31}\\,{\\rm cm}^2$, independently of the DM mass\n{\\it and} size. Also in this case there is a small loophole to this statement if one is willing to assume that the thermalization\nefficiency of CRESST is as small as 2\\,\\%: when tuning the size of the DM particles to $r_\\chi\\simeq10$\\,fm,\nwe find that cross sections two orders of magnitude larger may in that case be viable for DM masses in a narrow range \nbetween around 1\\,GeV and 2\\,GeV.\n\n\n\n\\section{Hexaquarks: a viable baryonic dark matter candidate?}\n\\label{sec:hexaquark}\n\nIn section \\ref{sec:m2} we discussed various generic situations where the amplitude for elastic scattering \nshows a significant dependence on the momentum transfer, and how this impacts the conclusions about \nwhether a window of large scattering cross sections remains open or not. In this section we complement\nthose more model-independent considerations by taking a closer look at a specific DM candidate\nin the GeV range, with relatively large nuclear interactions.\nConcretely, it has been conjectured that a neutral (color-flavor-spin-singlet) bound state of six light quarks \n$uuddss$ may exist, and provide a plausible DM candidate that would evade all current\nconstraints despite its baryonic nature~\\cite{Farrar:2002ic,Zaharijas:2004jv,Farrar:2017eqq, Farrar:2020zeo,Farrar:2022mih}. \nIn particular, this \\emph{sexaquark} $S$ (to be distinguished from a generic 6-quark state, often referred to as {\\it hexaquark})\nwould form early enough to behave like standard cold DM during both big bang nucleosynthesis and recombination.\nIt would thus not be in conflict with the independent, and rather precise, measurements~\\cite{Aver:2015iza,Planck:2018vyg} \nof the cosmological baryon density during these epochs.\n\nCompared to the H-dibaryon that was suggested earlier~\\cite{Jaffe:1976yi} and thoroughly studied both theoretically  and experimentally (see Refs.~\\cite{Sakai:1999qm,Clement:2016vnl} for reviews), furthermore, the $S$ should be much \nmore tightly bound, leading to weaker interactions with ordinary baryons and thus evading direct searches.  \nSuch a particle would be absolutely \nstable for  $m_S<m_D + m_e\\simeq 1.88$\\,GeV, and decay with a lifetime exceeding the age of the \nUniverse for $m_S\\lesssim 2\\,$GeV~\\cite{Farrar:2020zeo}. Determining its expected mass exactly, however,\nis challenging; lattice simulations, for example, remain somewhat inconclusive (see, e.g., \nRefs.~\\cite{NPLQCD:2010ocs,Inoue:2010es,NPLQCD:2012mex,Francis:2018qch} where the results for binding energies \nof the H-dibaryon state range from $\\sim$17\\,MeV to $\\sim$75\\,MeV relying, however, on unrealistically large quark masses).\nEven if the sexaquark is stable on cosmological timescales, its relic abundance would generally be much smaller than\nthe observed DM abundance if one assumes that its interactions in the early universe are of the order of the \nstrong force~\\cite{Kolb:2018bxv,Gross:2018ivp}. If instead, one postulates much weaker interactions due to the assumed\ncompactness of the sexaquark, thermal equilibrium with the SM heat bath would\nnot be possible to maintain after the QCD phase transition and the correct DM abundance might be \nachieved -- in a region of parameter space claimed to evade all existing constraints~\\cite{Farrar:2020zeo}.\n\nMotivated by this intriguing possibility, for simplicity we will adopt the description of sexaquark interactions \nfrom Ref.~\\cite{Farrar:2020zeo}, i.e.~we model the interaction with nucleons by the exchange \nof a vector meson. \nIn particular, the relevant interaction terms with the flavour-neutral mixture of $\\phi$ and $\\omega$, denoted by $V$, are given by\n\\begin{equation}\n\\label{eq:laghexaquarks}\n\\mathcal{L}= V_\\mu \\left( i g_{S} S^\\dagger {\\mathop{\\partial^\\mu}^{\\leftrightarrow}} S  + g_{p}\\overline{p}\\gamma^\\mu p + g_{n}\\overline{n}\\gamma^\\mu n\\right),\n\\end{equation}\nand we adopt the value $m_V = 1\\,$GeV used in Ref.~\\cite{Farrar:2020zeo} for our calculations.\nThe value of $g_n=g_p\\sim 2.6\\sqrt{4\\pi}$ \ncan be inferred from the literature on the one-boson-exchange model~\\cite{Maessen:1989sx} although $\\mathcal{O}(1)$ \nuncertainties can be expected here.\\footnote{%\nIn particular, we note that modern analyses of low-energy baryon-baryon scattering consider processes beyond single \nmeson exchange~\\cite{Nagels:2015lfa}, and that baryon-baryon interactions can also be treated within the more \nsystematic approach \nof chiral perturbation theory \\cite{Weinberg:1990rz}. However, given the significant uncertainties on the sexaquark \ncouplings we consider the one-boson-exchange approximation to be sufficient for our purposes.\n}\nThe coupling $g_{S}$ is largely unknown, though simple scaling arguments suggest that\n\\begin{equation}\n\\alpha_{\\mathrm{SN}} \\equiv \\frac{g_{S}g_{p}}{4\\pi}\n\\end{equation}\nis very roughly of the order of $\\sim0.1$~\\cite{Farrar:2020zeo}. Following that reference, we will treat $\\alpha_{\\mathrm{SN}}$\nas a free parameter that we will generously vary in the interval $(10^{-3},10)$. \nImportantly however -- at least in this parameter range -- the DM relic abundance \nis independent of $\\alpha_{\\mathrm{SN}}$. Instead, the final abundance of $S$ is set by an independent coupling constant $\\tilde{g}$~\\cite{Farrar:2020zeo} \nthat describes the (much weaker) sexaquark-breaking interactions within the effective description. This coupling does not directly enter the analysis presented here.\n\nWe treat the interaction of $V$ with nuclei similarly to that in section~\\ref{sec:vector}, i.e.~we describe it by the effective \nLagrangian~\\eqref{eq:leff_vector} with the coherently enhanced, effective coupling $g_N$ given by Eq.~\\eqref{eq:gN_coh}. \nFor the elastic scattering cross section on nuclei we thus find\n\\begin{eqnarray}\n\\label{diffsig_full_hex}\n\\frac{d\\sigma_{S N}}{d T_N}&=&\\frac{\\mathcal{C}^2 \\sigma_{\\rm SI}^\\mathrm{NR}}{T_N^\\mathrm{max}}\n\\frac{m_V^4}{(Q^2+m_V^2)^2}\n\\times G_N^2(Q^2) G_V^2(Q^2)\\\\\n&&\n\\times\\frac{1}{4s \\mu_{S N}^2}\n\\left\\{\n\\begin{array}{ll}\n(s-\\tfrac{1}{2}Q^2-m_N^2-m_\\chi^2)^2& ~~\\mathrm{for~scalar~}N\\\\\nm_N^2Q^2-Q^2s+(s-m_N^2-m_S^2)^2&  ~~\\mathrm{for~fermionic~}N\n\\end{array}\n\\right..\\nonumber\n\\end{eqnarray}\nHere, \n\\begin{equation}\n\\sigma_{\\rm SI}^{\\rm NR} = \\frac{16 \\pi \\alpha_{\\mathrm{SN}}^2 \\mu_{S p}^2}{m_V^4}\n\\end{equation}\nis the scattering cross section on nucleons in the non-relativistic limit and $\\mu_{S p}$ ($\\mu_{S N}$) is the reduced mass of \nthe sexaquark-nucleon (nucleus) system. \n\nCompared to the treatment in section~\\ref{sec:vector}, we introduce an additional form factor $G_V$ related to the cutoff in the \none-boson-exchange models. In this context, exponential cutoffs\n\\begin{equation}\nG_V(Q^2) = e^{-\\frac{Q^2}{\\Lambda_V^2}}\n\\end{equation}\nare mostly used and the cutoff mass $\\Lambda_V$ is fitted to data (and can in principle differ for different meson exchange \nchannels). For example, within the fit to data taking into account hyperon-nucleon interactions \\cite{Maessen:1989sx}, these cutoff \nmasses were found to range between 820\\,MeV and 1270\\,MeV. Since yet lower cutoff masses appear in related literature (e.g., \ndown to 590\\,MeV in \\cite{Stoks:1996yj}), we generously vary $\\Lambda_V$ between 500 and 1500\\,MeV.\nWe note that for $\\Lambda_V\\gtrsim1500\\,$MeV, CRDM limits become in fact independent of the cutoff scale.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{figures_draft\/figure_nine_wl}\n\\caption{Effective sexaquark coupling $\\alpha_{\\mathrm{SN}}$ vs.~sexaquark mass $m_S$.\nThe purple region shows the parameter range that is excluded by the analysis in this work, assuming that sexaquarks\nmake up all of the cosmologically observed DM;\ndifferent line styles correspond, as indicated, to cutoff masses $\\Lambda_V\/{\\rm GeV}\\in\\{0.5, 1,1.5\\}$\nin the one-boson exchange approximation. All other constraints are, for easier comparison, directly\nreproduced from Fig.~10 of Ref.~\\cite{Farrar:2020zeo}, conservatively assuming an attractive Yukawa force between $S$\nand nuclei. The thin vertical stripe corresponds to the mass range\nwhere, according to that analysis, the sexaquark would be a viable DM candidate without being in conflict\nwith other particle physics observation, in particular the stability of deuterons based on SNO data~\\cite{Bellerive:2016byv}. \nThe upper end of that mass range may increase from 1890\\,MeV to up to 2054\\,MeV if sexaquark DM does not \naccumulate in the Earth at the level claimed in Ref.~\\cite{Neufeld:2018slx}.\n\\label{fig:results_sexaquark}\n}\n\\end{center}\n\\end{figure}\n\nIn Fig.~\\ref{fig:results_sexaquark} we show the parameter space in the $\\alpha_{\\mathrm{SN}}$ vs.~$m_S$ plane where sexaquark DM\nis excluded because of the irreducible CRDM component. For a better direct comparison, we also indicate the preferred\nmass range according to Ref.~\\cite{Farrar:2020zeo}, along with the complementary limits presented in that analysis. \nFrom this figure, it is clear that our new limits close a significant part of the viable parameter region where sexaquarks could be\nthe dominant DM component -- even without taking into account the CRESST results. \nIn particular, we note that the Lyman-$\\alpha$ limits~\\cite{Rogers:2021byl}  \nshown in figures~\\ref{fig:constraints_constant} and \\ref{fig:limits_scalar} were presented subsequent to the analysis of\nRef.~\\cite{Farrar:2020zeo} and are significantly stronger than the CMB limits indicated in Fig.~\\ref{fig:results_sexaquark}.\nThe apparently open window at $\\alpha_{\\mathrm{SN}}\\sim0.3$ is thus also robustly excluded.\nOn the other hand, a small open window remains for $\\alpha_{\\mathrm{SN}}\\lesssim 4\\cdot10^{-3}$. While not being\nin conflict with the DM abundance, as explained above, we recall that \nsuch values of  $\\alpha_{\\mathrm{SN}}$ are somewhat smaller than intrinsically expected.\n\nLet us, finally, briefly comment on the fact that the DM-nucleon scattering cross section can, strictly speaking,\nonly be calculated perturbatively in the Born limit, $\\alpha_{\\mathrm{SN}} \\mu_{\\chi N}\\lesssim m_V$. Outside this regime,\nnon-relativistic scattering in a Yukawa potential exhibits parametric resonances where the scattering amplitude\nis significantly enhanced or suppressed. This non-perturbative \neffect is well-known from the self-scattering of cold DM in the presence\nof light mediators~\\cite{Tulin:2013teo}, and it is the origin of the resonant structure in the complementary limits from \nRef.~\\cite{Farrar:2020zeo} that is visible in Fig.~\\ref{fig:results_sexaquark}. For our CRDM limits, on the other hand, \nthis additional complication does not arise because such non-perturbative corrections are \nlargely irrelevant for relativistic scattering; in fact, \nalready for the typical velocities during the freeze-out process of thermally produced DM, $v_\\chi\\sim0.3$, the impact\nis strongly suppressed~\\cite{Tulin:2013teo}. The CRDM limits are thus also robust w.r.t.~underlying model assumptions \nsuch as whether the force mediated by the Yukawa potential is attractive or repulsive.\n\n\n\n\n\\section{Summary and Conclusions}\n\\label{sec:conclusions}\n\nFor sizeable elastic scattering rates between DM and nuclei there is an irreducible relativistic\ncomponent of the flux of DM particles arriving at Earth. This extends the sensitivity of conventional\ndirect detection experiments both to sub-GeV masses and to scattering cross sections above the limit\nset by a too efficient attenuation of the DM flux on the way to the detector. While such large scattering cross \nsections are also constrained by complementary probes from astrophysics and cosmology, it has repeatedly \nbeen pointed out that there might be an open window of relatively strongly interacting DM with a mass in the\nballpark of $\\sim1$\\,GeV.\n\nWe find that the CRDM component in the DM flux generically closes this window, under rather minimal \nassumptions. In order to arrive at this conclusion, we included in our analysis a detailed treatment\nof the inelastic scattering of DM off nuclei (section \\ref{sec:inel}). We demonstrate that this provides \nan important additional stopping channel for CRDM particles on their way to direct detection facilities\n-- unlike for non-relativistic DM, where only elastic scattering is relevant.\nWe also investigated the extent to which a possible energy or momentum-transfer dependence of the cross section\ncould weaken our general conclusions. \nFor this purpose, we considered {\\it i)} a class of simplified models where the scattering with nuclei is mediated by a light\nscalar (section \\ref{sec:scalar}) or vector (section \\ref{sec:vector})\nparticle, as well as {\\it ii)} situations where DM particles cannot be described as being point-like (section \\ref{sec:puffy}). \nIn all these cases, the additional momentum-transfer dependence indeed weakens the limits from\ndirect detection -- which however is compensated for by a corresponding strengthening of complementary\nlimits, in particular from cosmology. In combination, these limits stringently constrain the \npossibility of cross sections larger than a few times $10^{-31}\\,{\\rm cm}^2$, over a wide range of DM masses. Interestingly, this is\nlargely independent of underlying modelling assumptions such as the mass of new mediator particles or\nthe DM particles' radius.\n\nFinally, an exotic QCD bound state that is produced well before BBN, has repeatedly been put forward as \na potential DM candidate. While it is theoretically unclear whether such states could actually exist, \nadding to significant experimental constraints, it is certainly an intriguing\nidea to have a `baryonic' DM candidate that would in fact evade the strong\nevidence from BBN and CMB against this possibility. However, cosmic-ray upscattering\nof such particles leads to stringent new constraints that have not previously been pointed out\nin this context. For the concrete case of stable sexaquark DM, as discussed in \nsection~\\ref{sec:hexaquark}, we find that the parameter space giving the correct cosmological\nabundance is strongly pressured.\n\nFor the analysis performed in this work we used the numerical tool {\\sf DarkSUSY}~\\cite{Bringmann:2018lay} \nto compute CRDM fluxes and limits. In doing so we significantly expanded the general numerical routines\nprovided therein, adding in particular inelastic scattering, the contribution \nfrom CRs beyond $p$ and He, and an updated treatment of nuclear form factors in the context of\nCRDM attenuation.   These updates\nwill be included in the next public release\nof the code.\n\n\n\n\\acknowledgments\nWe thank Timon Emken and Florian Reindl for enlightening discussions about the thermalization efficiency of the\nCRESST experiment, and Felix Kahlhoefer for insightful comments on how to map nucleon to nuclear cross sections. We further \nthank Assumpta Pare\\~{n}o, Gilberto Colangelo and Urs Wiedemann for comments related to the hexaquark state. TB warmly \nthanks the Albert Einstein Institute in Bern, and the CERN Theoretical Physics Department, \nfor support and hospitality during the preparation of this manuscript. JA is supported through the research program ``The Hidden \nUniverse of Weakly Interacting Particles\" with project number 680.92.18.03 (NWO Vrije Programma), which is partly financed by \nthe Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Dutch Research Council). HK is supported by the ToppForsk-UiS Grant No. PR-10614.\n\n\n\\bibliographystyle{JHEP_improved}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nThe high-energy cosmic ray, $\\gamma$-ray and neutrino emission of the Universe are fascinating phenomena based on poorly understood non-thermal processes in astrophysical environments. Whereas high-energy $\\gamma$-rays can be produced via leptonic processes like inverse Compton scattering or bremsstrahlung, the production of neutrinos require interactions of high energy cosmic rays with radiation or matter. Hence, cosmic neutrinos are the unambiguous tracers of cosmic ray interactions in our Universe and neutrino astronomy promises to unravel their sources.\n\nHigh energy astrophysical neutrinos become visible in neutrino observatories once they interact in the detector vicinity via charged and neutral current interactions. The high energy secondary particles produced in these interactions are detected in transparent media like water or ice via Cherenkov light emission. The cross sections of these processes are very low. For instance, at PeV energies the neutrino interacts with nucleons with a cross section of about 1 nbarn. Hence, for a detector density of the order of $N_A\\,{\\rm cm}^{-3}$, we expect only a fraction of about $10^{-5}$ to interact in 1 km of the medium. These low event rates and the large background from atmospheric cosmic ray interactions are the experimental challenges for neutrino astronomy.\n\nThe IceCube Collaboration~\\cite{Aartsen:2013jdh,Aartsen:2014bea} has recently reported the detection of a cosmic flux of high energy neutrinos with a significance of $5.7\\sigma$. The flux is consistent with an $E^{-\\gamma}$ power spectrum with spectral index $\\gamma\\simeq2.3\\pm0.3$ and equal distribution between flavor and isotropic arrival direction of neutrinos. The best fit $E^{-2}$-flux is given as\n\\begin{equation}\\label{eq:ICnu}\nE_\\nu^2J^{\\rm IC}_\\nu(E_\\nu) \\simeq (0.95\\pm0.3)\\times10^{-8}\\,\\frac{{\\rm GeV}}{{\\rm cm}^{2}\\,{\\rm s}\\,{\\rm sr}}\n\\end{equation}\nThe origin of these neutrinos is unknown. A statistically weak cluster of events near the Galactic Centre has motivated speculations about a Galactic origin of the signal. These scenarios include the diffuse neutrino emission of Galactic CRs~\\cite{Ahlers:2013xia,Kachelriess:2014oma}, the joint emission of Galactic PeVatrons~\\cite{Fox:2013oza,Gonzalez-Garcia:2013iha} or extended Galactic structures like the Fermi Bubbles~\\cite{Razzaque:2013uoa,Ahlers:2013xia,Lunardini:2013gva} or the Galactic Halo~\\cite{Taylor:2014hya}. A possible association with the sub-TeV diffuse Galactic $\\gamma$-ray emission~\\cite{Neronov:2013lza} and constraints from the non-observation from diffuse Galactic PeV $\\gamma$-rays~\\cite{Gupta:2013xfa,Ahlers:2013xia} have also been mentioned. More radical suggestions include PeV dark matter decay scenarios~\\cite{Feldstein:2013kka,Esmaili:2013gha,Bai:2013nga}. The extension of the events to large Galactic latitudes and the absence of significant event clusters suggest that a significant contribution of the signal originates in extragalactic sources. Possible source candidates include galaxies with intense star-formation~\\cite{Loeb:2006tw,Murase:2013rfa,He:2013cqa}, cores of active galactic nuclei~\\cite{Stecker1991,Stecker:2013fxa}, low-power $\\gamma$-ray bursts~\\cite{Waxman:1997ti,Murase:2013ffa}, intergalactic shocks and active galaxies embedded in structured regions~\\cite{Berezinsky:1996wx,Murase:2013rfa}. \n\nThe search for transient and continuous neutrino sources has so far been unsuccessful providing upper neutrino flux limits on individual Galactic and extragalactic source candidates~\\cite{IceCube:2011ai,AdrianMartinez:2012rp,Abbasi:2012zw,Aartsen:2013uuv,Adrian-Martinez:2014wzf,Adrian-Martinez:2013dsk}. If the IceCube observation is a superposition of individual (possibly extended) sources, in contrast to a truely diffuse emission, these searches should eventually uncover individual sources or at least place upper limits on possible source candidates. However, it is important to keep in mind that the interaction rate of a neutrino is so low that it travels basically unattenuated through the tenuous matter and radiation backgrounds of the Universe over cosmic distances. The un-resolved extragalactic (or quasi-diffuse) flux is hence a superposition of many sources ranging from relatively recent and local objects to old and distant sources as far as the Hubble horizon. This makes the identification of individual point-sources contributing to the IceCube flux challenging. \n\nIn this paper we investigate the necessary performance of a neutrino observatory for the detection of neutrino point-sources in the form of event clusters and in association with close-by sources or source catalogues. We will start in Section~\\ref{sec2} with a derivation of the contribution of individual neutrino sources to a quasi-diffuse flux. In Section~\\ref{sec3} we derive simple estimates of the required event numbers and rates of the quasi-diffuse emission for the identification of sources assuming a spatially homogeneous local distribution. For continuously emitting neutrino sources an important aspect of the analysis is the background of atmospheric neutrinos and the quasi-diffuse neutrino signal itself. In Section~\\ref{sec4} we include these contributions quantitatively via a significance tests introducing simple test statistics for event clusters and source associations. We compare these results with specific source scenarios in Section~\\ref{sec5} and conclude in Section~\\ref{sec6}. In the following we work in Heaviside-Lorentz units and make use of the abbreviation $A_x = A\/(10^xu)$, where $u$ is the (canonical) unit of the quantity $A$. \n\n\\section{Neutrino Point-Sources}\\label{sec2}\n\nHigh-energy neutrinos are produced by the decay of charged pions from hadronic interactions of CRs with radiation ($p\\gamma$) and matter ($pp$). The same mechanism produces also high-energy $\\gamma$-rays from the formation of neutral pions. On production, the emission rates of neutrinos (summed over neutrinos and anti-neutrinos with flavor $\\nu_\\alpha$) and $\\gamma$-rays are related to the emission rate of CR nucleons ($N$) as\n\\begin{equation}\\label{eq:MM}\n\\frac{1}{3}\\sum_{\\alpha}E_\\nu Q_{\\nu_\\alpha} \\simeq \\frac{K_\\pi}{2}  E_\\gamma Q_\\gamma\\simeq \\frac{f_\\pi}{\\kappa}\\frac{K_\\pi}{1+K_\\pi}E_NQ_N\\,,\n\\end{equation}\nwhere the (average) energies are related as $E_\\nu\\simeq E_\\gamma\/2\\simeq E_N\/20$. We will assume in the following that energy losses of the charged pions and muons prior to their decay are unimportant. In this case the relation (\\ref{eq:MM}) has to be considered as lower bound on the total CR energy of the sources. The parameter $K_\\pi\\simeq1$ ($K_\\pi\\simeq2$) denotes the average ratio between charged and neutral pions in $p\\gamma$ ($pp$) interactions and the pion production efficiency, that is related to the optical depth $\\tau$ for hadronic interactions with inelasticity $\\kappa$ as $f_\\pi \\simeq 1-\\exp(-\\kappa\\tau)$.\n\nThe IceCube flux between 60~TeV to 2~PeV corresponds to $\\gamma$-rays between 100~TeV and 4~PeV or CR nucleons between 1~PeV and 40~PeV. Cosmic rays at these energies are deflected in cosmic magnetic fields and can only be used for point-source associations at very high energies. For instance, ultra-high energy (UHE) CR protons beyond 60~EeV are expected to originate within about 200~Mpc due to the strong absorption in the cosmic microwave background (CMB). Even at these extreme energies the deflection via turbulent Galactic magnetic field can be of the order of $0.5^\\circ$~\\cite{Giacinti:2011uj}. The contribution of extragalactic magnetic fields is more uncertain and can in principle be larger~\\cite{Dolag:2004kp}.\n\nGamma-rays with PeV energies experience no electro-magnetic deflections, but have a short absorption length of about 10~kpc due to $e^+e^-$ production via scattering off of the photons of the CMB. At about 100~TeV interactions with the extragalactic background light limit the propagation distance to Mpc scales. Direct observation of $\\gamma$-ray emission in association with the IceCube flux is hence not feasible, unless there is a significant Galactic contribution~\\cite{Ahlers:2013xia}. However, the sub-TeV extension of the IceCube signal (expected for a $pp$ origin of the signal) as well as the sub-TeV contribution of cascaded $\\gamma$-rays via inverse-Compton scattering of the high-energy $e^+e^-$ can be visible as an (extended) point-source TeV $\\gamma$-ray emission and provide an additional constraint for the contribution of close-by sources~\\cite{Murase:2013rfa}.\n\nNeutrinos on the other hand have negligible interactions during propagation and are ideal point-source messengers.\nThe (quasi-)diffuse flux of neutrinos $J$ (in units of ${\\rm GeV}^{-1} {\\rm s}^{-1} {\\rm cm}^{-2} {\\rm s}^{-1}$) originating in multiple cosmic sources is simply given by\n\\begin{equation}\nJ_\\nu(E_\\nu) = \\frac{1}{4\\pi}\\int_0^\\infty\\frac{{\\rm d}z}{H(z)}\\mathcal{L}_\\nu(z,(1+z)E_\\nu)\\,,\n\\end{equation}\nwhere $H$ is the red-shift dependent Hubble expansion rate and $\\mathcal{L}$ is the spectral emission rate density. In the case of a continuous neutrino emission we can decompose this into $\\mathcal{L}(z,E) = \\mathcal{H}(z)Q_\\nu(E)$ where $\\mathcal{H}$ is the source density and $Q_\\nu$ is the emission rate per source. In the following we will assume evolution following the star-formation rate (SFR)~\\cite{Hopkins:2006bw,Yuksel:2008cu} $\\mathcal{H}_{\\rm SFR}(z) \\propto (1+z)^{n_i}$ with $n_i=3.4$ for $z<1$,  $n_i=-0.3$ for $1<z<4$ and  $n_i=-3.5$ otherwise. The red-shift dependence of the source distribution can be parametrized by the energy dependent quantity\n\\begin{equation}\n\\xi_z(E) = \\int_0^\\infty{\\rm d}z\\frac{H_0}{H(z)}\\frac{\\mathcal{L}_\\nu(z,(1+z)E)}{\\mathcal{L}_\\nu(0,E)}\\,.\n\\end{equation}\nFor the special case of power-law spectra $\\mathcal{L}_\\nu(E)\\propto E^{-\\gamma}$ this quantity is energy independent and we will assume the case $\\gamma=2$ in the following corresponding to the fit (\\ref{eq:ICnu}). In this case we have $\\xi_z\\simeq2.4$ assuming evolution with SFR of the CR sources. For a source distribution with no evolution in the local ($z<2$) Universe this reduces to $\\xi_z\\simeq0.5$. \n\nFrom this we can estimate the contribution of an individual continuously emitting point sources. For a distance $d = d_1 10$~Mpc the mean neutrino flux is given as\n\\begin{align}\\label{eq:JPS}\nE_\\nu^2J_\\nu \n\\simeq \\frac{(0.9\\pm0.3)\\times10^{-12}}{\\xi_{z, 2.4}\\mathcal{H}_{0, -5}d_1^{2}} \\frac{\\rm TeV}{{\\rm cm}^{2}\\,{\\rm s}}\\,,\n\\end{align}\nwhere $\\mathcal{H}_0 = \\mathcal{H}_{0, -5} 10^{-5} {\\rm Mpc}^{-3}$ is the local source density. For this choice of parameters the contribution of a source is consistent with upper limits of neutrino point sources in an un-binned search~\\cite{Aartsen:2013uuv}. In the case of transient sources we write instead $\\mathcal{L}(z,E) = {\\dot{\\mathcal{H}}}(z){\\rm d}N\/{\\rm d}E(E)$ with transient rate density ${\\dot{\\mathcal{H}}}(z)$ and spectrum ${\\rm d}N\/{\\rm d}E$ of an individual flare. In this case the mean neutrino fluence $F$ from an individual transient can be expressed as \n\\begin{align}\\label{eq:FPS}\nE_\\nu^2F_\\nu \n\\simeq \\frac{0.3\\pm0.1}{\\xi_{z, 2.4}{\\dot{\\mathcal{H}}}_{0, -6}d_1^{2}} \\frac{\\rm GeV}{{\\rm cm}^{2}}\\,,\n\\end{align}\nwhere ${\\dot{\\mathcal{H}}}_0 = {\\dot{\\mathcal{H}}}_{0, -6}10^{-6} {\\rm Mpc}^{-3}{\\rm yr}^{-1}$ is the local flaring\/burst density rate.\nThe sensitivity of IceCube for triggered transient sources lies at about $0.1{\\rm GeV}\/{\\rm cm}^2$ depending on zenith angle and emission time scale~\\cite{IceCube:2011ai,Abbasi:2012zw}.\n\nThe previous estimates depend on the distance of the source. In the case of a large number of sources we can express the probability that the closest source contributes with an expected number of events $n$ as \\begin{equation}\\label{eq:ploc}\np_1(n) \\simeq \\frac{3}{2}\\frac{1}{ n}\\left(\\frac{ n(\\hat{r})}{ n}\\right)^\\frac{3}{2}e^{-\\left(\\frac{ n(\\hat{r})}{ n}\\right)^\\frac{3}{2}}\n\\end{equation}\nwhere the distance $\\hat{r}$ is defined via $\\mathcal{H}_0\\hat{r}^3 \\Delta \\Omega\/3=1$, {\\it i.e.}~the radius of a sphere where we expect one source in the experimental field of view (FoV) $\\Delta \\Omega$. The general probability distribution of the $k$th-closest source is given in Appendix~\\ref{app1}.\n\n\\begin{figure}[t]\\centering\n\\includegraphics[width=\\linewidth]{PSsens5yr.pdf}\n\\caption[]{Point sources sensitivity for continuous (top) and transient (bottom) sources. The shaded areas show the $10\\%$ quantiles around the median expectation from the closest sources of the ensemble. The dotted horizontal line show the IceCube sensitivity after five years estimated for a muon energy threshold of 10~TeV (see main text).}\\label{fig1}\n\\end{figure}\n\nIn Figure~\\ref{fig1} we show the contribution of the closest continuous or transient source in terms of the density of the underlying source population and as $10\\%$ quantiles around the median (solid lines) according to Eq.~(\\ref{eq:ploc}) and Eqs.~(\\ref{eq:JPS}) and (\\ref{eq:FPS}), respectively. We assume an observation time $T_{\\rm live}=5$~yrs for the total number of transient sources. The dotted horizontal lines show the estimated sensitivity of IceCube to continuous and transient sources for a detector live-time of five years assuming an event rate of $\\dot{N}\\sim 50~{\\rm yr}^{-1}$ above a muon energy threshold of 10~TeV and low background~\\cite{Karle2014}. The estimated sensitivity (90\\% C.L.) shown in Fig.~\\ref{fig1} is then simply $J \\simeq f_{\\rm sky}4\\pi J^{\\rm IC}_\\nu\\times 2.3\/(T_{\\rm live}\\dot{N})$ for event associations with continuous sources and $F \\simeq f_{\\rm sky}4\\pi  J^{\\rm IC}_\\nu\\times 2.3\/\\dot{N}$ for events triggered by transient sources, where we introduced the fractional sky coverage $f_{\\rm sky}=\\Delta\\Omega\/(4\\pi)$. We will provide a more precise estimate of the detector sensitivity including backgrounds in the following sections.\n\nThese results already indicate that the non-observation of individual neutrino sources (and in particular the closest one) is consistent with the hypothesis of an extragalactic origin of the recent IceCube observation (\\ref{eq:ICnu}) for sufficiently large source densities and\/or rates. On the other hand, the identification of individual neutrino sources with the continued observation with IceCube over the next years will be challenging unless the source distribution is sufficiently sparse and\/or rare. In the next sections we will make this statement more quantitative and discuss the required event numbers and search strategies for an identification of the sources with IceCube or next-generation neutrino observatories.\n\n\\section{Point Source Statistics}\\label{sec3}\n\nA model-independent identification of neutrino sources can be the detection of spatial or temporal clusters with total number of $m$ neutrino events or more. The required value for $m$ needed for a significant detection depends on the density of sources as well as the expected number of signal and background events. The neutrino event clusters will be most likely associated with local neutrino sources and we can hence simplify the discussion by considering Euclidean space, where we neglect redshift scaling of energy and comoving volume. The contribution from a single local source at distance $r\\leq H_0^{-1}$ can be expressed as\n\\begin{equation}\\label{eq:nr}\n n(r) \n\\simeq \\frac{H_0}{f_{\\rm sky}4\\pi r^2\\xi_z}\\times\\begin{cases} N\/\\mathcal{H}_0& \\text{(continuous)}\\\\ \\dot{ N}\/{\\dot{\\mathcal{H}}}_0& \\text{(transient)}\\end{cases}\n\\end{equation}\nIn the following we will derive results in terms of the expected number of events $ N$ or $\\dot{ N} =  N\/T$ of all neutrino sources. \n\nAs a back-of-the-envelope estimate of the required total event numbers for the observation of event multiplets we can consider the contribution of the closest source of the ensemble. For a fractional sky coverage $f_{\\rm sky}$ and local source density $\\mathcal{H}_0$ we expect one source in the FoV within a sphere of volume $V_1 = 1\/(f_{\\rm sky}\\mathcal{H}_0)$. The total number of events that we expect from this volume is given by the integral of Eq.~(\\ref{eq:nr}) over $V_1$ and yields $m =  N(V_1\/V_H)^\\frac{1}{3}\/\\xi_z$ where we introduce the Hubble volume $V_H = 4\\pi\/(3H_0^3)$. Note, that $V_H\/V_1$ correspond to the effective number of sources in the FoV. In the case of continuous sources we arrive then at an expected total event number for $m$ local events of\\begin{equation}\\label{eq:Ncont}\n N_{\\rm cont} \\simeq 740\\,\\left(\\frac{m}{2}\\right)\\,\\xi_{z, 2.4}\\,\\left(f_{\\rm sky}\\,\\mathcal{H}_{0,-5}\\right)^{\\frac{1}{3}}\\,.\n\\end{equation}\nIn the case of transient sources we have to take into account that the number of sources is increasing with observation time $T_{\\rm live} =  N\/\\dot{ N}$. Solving in terms of the total observation rate $\\dot{ N}$ we arrive at \n\\begin{equation}\\label{eq:Ntrans}\n N_{\\rm trans} \\simeq 637\\left(\\frac{m}{2}\\right)^\\frac{3}{2}\\,\\xi_{z, 2.4}^\\frac{3}{2}\\,\\left(f_{\\rm sky}{{\\dot{\\mathcal{H}}}_{0,-6}}\/{\\dot{ N}_2}\\right)^\\frac{1}{2}\\,,\n\\end{equation}\nwith an event rate $\\dot{ N} = 100\\dot{ N}_2\/{\\rm yr}$. Note, that the event clusters in the transient case should also show a strong temporal coincidence within the burst or flaring time-scale of the source. This fact is important for the comparison with background events due to random clusters of atmospheric neutrinos and muons.\n\nInstead of searching for auto-correlations of events we can also look for cross-correlations with catalogues of candidate sources. For simplicity, let's assume that the catalogue is locally complete up to a distance $r_{\\rm cat}$ containing $C = f_{\\rm sky}\\mathcal{H}_0V_{\\rm cat}$ local neutrino sources in the FoV. Following the same line of arguments as in the case of event clustering we can estimate that $m =  N(V_{\\rm cat}\/V_H)^\\frac{1}{3}\/\\xi_z$ events are expected to correlate with sources of the catalogue. Note that this expression is only valid for $V_{\\rm cat}<V_H$; for complete catalogues we simply have $N(m) = m$. For transient sources we assume that the catalogue itself grows in time and remains complete within a distance $r_{\\rm cat}$. In this case the expected total number of events for $m$ associations with sources of the catalogue is\n\\begin{equation}\\label{eq:Nass}\n N_{\\rm ass} \\simeq 107\\, m\\,\\xi_{z, 2.4}r_{\\rm cat, 2}^{-1}f_{\\rm sky}^{-\\frac{1}{3}}\\,.\n\\end{equation}\nThe result is the same for continuous and transient sources if we assume that the catalogue grows in time for transient sources. In the case of extended source catalogues with $r_{\\rm cat}\\sim 1\/H_0$ already a few signal events can be sufficient for a point-source association. This requires sources with powerful electromagnetic emission which is typical for transient sources like GRBs or AGN flares. \n\nA crucial aspect for a statistically significant detection of local neutrino sources are the backgrounds of distant neutrino sources as well as atmospheric showers. For transient sources the time-stamp of the signal can lead to a significant reduction of backgrounds. In this case Eqs.~(\\ref{eq:Ntrans}) and (\\ref{eq:Nass}) already indicate the number of events required for source identifications. However, continuous sources require a more careful statistical discussion. The previous estimates suggest that the required number of observed signal events $N$ has to reach at least a level of 100 before we can expect to observe an association of events with local continuous sources. \n\nThe classical muon neutrino search has a sky coverage of about $f_{\\rm sky}\\lesssim 0.5$. For the following analysis we will assume that a suitable application of low level event filters has already reduced the background to a level $S\/B$. The significance of spatial clustering or association of events depends on the experimental angular resolution $\\Delta\\theta$. This defines the effective number of bins in the sky as $n_{\\rm bin} \\simeq 2f_{\\rm sky}\/(1-\\cos(\\Delta\\theta))$. For instance, we have $n_{\\rm bin}\\simeq6600$ for $\\Delta\\theta\\simeq 1^\\circ$ and $f_{\\rm sky}=0.5$, but only $n_{\\rm bin}\\simeq66$ for $\\Delta\\theta\\simeq 10^\\circ$. The expected number of evens from a random distribution of $N_{\\rm bg}$ background events is $m\\simeq {N}_{\\rm bg}\/n_{\\rm bin}$ per bin. Hence, for poorly resolved cascade events with $\\gtrsim10^\\circ$ resolution even a low background contribution of $S\/B\\simeq 1$ already produces random event clusters and association.\n\nIn the case of transient sources with burst or flaring time window $\\Delta T\\ll T_{\\rm live}$ we can make use of the fact that the CR background is continuous. We can account for this by redefining the effective number of bins as $n_{\\rm bin}\\to T_{\\rm live}\/\\Delta Tn_{\\rm bin}$. In general, $T_{\\rm live}\/\\Delta T$ is expected to be very large and depends on the specific source. The background cluster probability in the transient case is expected to be very low. \n\n\\section{Significance Test}\\label{sec4}\n\nIn order to test the statistical significance of neutrino point-sources we introduce two simple test statistics (TS), {\\it i)}  ${\\rm TS}_1 = \\max\\lbrace {\\bf k}\\rbrace$ for cluster tests and {\\it ii)} ${\\rm TS}_2 = \\sum_{N_s} k_i$ for a source association with $N_s$ closest sources. If ${\\bf k}$ is the experimental result and ${\\bf k}_0$ a possible background distribution we define the ensemble-averaged $p$-value as\n\\begin{equation}\np= \\int\\prod_{i=1}^{N_s}{\\rm d}n_i\\,p_{\\rm PS}(n_i)\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\sum_{{\\rm TS}({\\bf k}_0)\\geq{\\rm TS}({\\bf k})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!P({\\bf k})P_0({\\bf k}_0)\\,,\n\\end{equation}\nwhere $p_{\\rm PS}(n)$ is the probability (\\ref{eq:pPS}) for an individual source out of $N_s$ sources to contribute with an expectation value $n$. The event probability distributions for signal $P$ and background $P_0$ are products of Poisson distributions for the content of the individual bins (see Appendix~\\ref{app1}). \n\nOne advantage of these test statistics is that the $p$-value reduces to simple analytic formulae in the background-free case, which is typically the case for transient sources. In this case we simply have $p({\\rm TS}_1) \\simeq 1-P_{\\rm cl}(2)$ with Eq.~(\\ref{eq:Pcl}) and $p({\\rm TS}_2) \\simeq 1-P_{\\rm ass}(1)$ with Eq.~(\\ref{eq:Pass}) derived in Appendix~\\ref{app2}. In Appendix~\\ref{app2} we also show that in the background-free case the required total signal event numbers at the significance level $p=0.1$ are the same as in Eq.~(\\ref{eq:Ncont}) or (\\ref{eq:Ntrans}) with the replacement $m\\to M\\simeq 3.57$. From this we can estimate that IceCube is sensitive to multiplets of rare transient sources $\\dot{\\mathcal{H}}_0\\lesssim10^{-8}\\,{\\rm Mpc}^{-3}\\,{\\rm yr}^{-1}$ after five years of observation.\n\nIn the case of continuous sources the background probability is not negligible. In the following we discuss two experimental scenarios parametrized by the angular resolution $\\Delta\\theta$, partial sky fraction of the FoV $f_{\\rm sky}$ and the signal-to-background ratio $S\/B$ of the observation. In the present IceCube 86-string configuration we expect about 40-60 signal events per year in the Northern Hemisphere above a muon energy threshold of 10~TeV, depending on the spectral index of the observation~\\cite{Karle2014}. The atmospheric muon neutrino background is at the level of about 500 events per year. The angular resolution of these events at about 10~TeV is of the order of $0.6^\\circ$. We will hence use the combination {\\it a)} $f_{\\rm sky}=0.5$, $S\/B = 0.1$ and $\\Delta\\theta=0.6^\\circ$ as a first detector performance benchmark. At muon energies above about 100 TeV the signal-to-background ratio in the Northern Hemisphere increases to about $S\/B=1$ with an increased angular resolution. However, due to neutrino absorptions inside the Earth the effective FoV is reduced. In this case we use as a second performance benchmark {\\it b)} $f_{\\rm sky}=0.25$, $S\/B = 1$ and $\\Delta\\theta=0.3^\\circ$.\n\nIn Figure~\\ref{fig2} we show results of the averaged $p$-value for different astrophysical scenarios of continuous neutrino sources and detector parameters. As usual the sensitivity is defined as a 90\\% confidence level (C.L.) of the signal hypothesis or, equivalently, $p=0.1$. The upper panel of Fig.~\\ref{fig2} shows the results for ${\\rm TS}_1$ for the search of event clusters for the benchmark point {\\it a)} (left panel) and {\\it b)} (right panel) . Even with high $S\/B$ and good angular resolution the significance of event clusters requires event numbers of $10^3$ in the most favorable astrophysical scenario of low source densities of $10^{-6}~{\\rm Mpc}^{-3}$. Hence, a model-independent identification of extragalactic neutrino sources is challenging even with a detector with ten times the event rates of the present IceCube configuration. Note that the oscillatory pattern in the $p$-values is due to the averaging over ensembles and experimental realizations. For an actual realization with maximum multiplet $k_{\\rm max}$ the $p$-value is simply $p=1-P_{\\rm bg}(k_{\\rm max})$ which is continuously decreasing with $N$. \n\nThe situation for continuous sources improves in the case of associations of neutrino events with candidate source catalogues tested by the test statistic ${\\rm TS}_2$. We show results in the bottom panels of Fig.~\\ref{fig2} for the benchmark points {\\it a)} (left panel) and {\\it b)} (right panel), respectively. The dashed lines show the results of association with the closest source of the ensemble following the distribution (\\ref{eq:ploc}). The detector performance used in the middle panel corresponds to the case shown in Fig.~\\ref{fig1}. For instance, the optimistic source density of $10^{-6}~{\\rm Mpc}^{-3}$ requires 1000 event for $p=0.1$. With a present IceCube rate of 50 signal events per year this requires an experimental live-time of $T_{\\rm live}\\simeq 20$~yrs, consistent with the sensitivity level indicated in Fig.~\\ref{fig1}. This provides an {\\it a posteriori} justification of the background-free estimate.\n\nThe solid lines in the lower panels of Fig.~\\ref{fig2} show the results of ${\\rm TS}_2$ in the case of an association of events with the 100 closest sources. In the case of the detector performance {\\it b)} with a high purity of events of $S\/B\\simeq1$ and good angular resolution the observation of $50$--$500$ events is required for source densities of $10^{-6}$--$10^{-4}\\,{\\rm Mpc}^{-3}$ and a moderate source evolution following the star-formation rate ($\\xi_z\\simeq2.4$). For a detector with  $\\dot{N}\\simeq 50~{\\rm yr}^{-1}$ corresponding to five times IceCube's event rates above muon energies of 100~TeV this would require 1--10 years of observation. \n\n\\begin{figure*}[t]\\centering\n\\includegraphics[width=\\linewidth]{TS12.pdf}\n\\caption[]{Results of a significance test of continuous point-sources. \n{\\bf Top Panels:} The results for the test statistic ${\\rm TS}_1$ looking for significance of event clusters in a map. The four different scenarios are parametrized in terms of an increasing expected total number of signal events $ N$ and the fixed combination of local source density $\\mathcal{H}_0$ and source evolution $\\xi_z$. The dashed horizontal line indicates the sensitivity threshold of $1-p=90\\%$. We show results for a benchmark detector performance {\\it a)} $S\/B=0.1$, $f_{\\rm sky}=0.5$ and $\\Delta\\theta =0.6^\\circ$ (left panel) and {\\it b)} $S\/B=1$, $f_{\\rm sky}=0.25$ and $\\Delta\\theta =0.3^\\circ$ (right panel), corresponding to muon energy thresholds of about $10$~TeV and $100$~TeV, respectively~\\cite{Karle2014}. As a reference value, the signal rate of the full IceCube detector in the Northern Hemisphere is about $\\dot N \\simeq 50\\, {\\rm yr}^{-1}$ above muon energies of 10~TeV and $\\dot N \\simeq 10\\, {\\rm yr}^{-1}$ above 100~TeV. \n{\\bf Lower Panels:} Same as top panels but now showing the results for the test statistic ${\\rm TS}_2$ looking for association with 100 closest members of the ensemble (solid line) and the closest source (dashed line).}\\label{fig2}\n\\end{figure*}\n\n\\section{Source Candidates}\\label{sec5}\n\nFor a given source density $\\mathcal{H}$ or rate density ${\\dot{\\mathcal{H}}}$ of neutrino sources we can derive a lower limit on the energy budget of the individual sources. The neutrino emission extends in the energy range from $60$~TeV to $2$PeV and hence, conservatively, the corresponding range of the underlying CR population ranges from $E_-\\simeq m_p$ to $E_+\\simeq 40$~PeV. The mean integrated emission rate of continuous point sources can then be estimated as\n\\begin{equation}\\label{eq:Qnu}\n\\int\\limits_{E_-}^{E_+}{\\rm d} EEQ_N \n\\simeq8.9\\times10^{42}\\frac{\\mathcal{R}_{17.5}}{f_\\pi\\xi_{z,2.4}\\mathcal{H}_{0, -5}}\\frac{\\rm erg}{\\rm s}\\,,\n\\end{equation}\nwhere we assumed $K_\\pi=2$ ($pp$) and an energy independent pion fraction efficiency $f_\\pi$ between $E_-$ and $E_+$. We also assume a spectral index $\\gamma=2$ corresponding to a bolometric correction factor $\\mathcal{R} = \\ln(E_+\/E_-) = 17.5\\mathcal{R}_{17.5}$. In the case of transient sources the emission spectra ${\\rm d}N\/{\\rm d}E$ of neutrinos, $\\gamma$-rays and CRs are related via the analogue of Eq.~(\\ref{eq:MM}). The mean emission of an individual point source corresponding to the IceCube observation can then be estimated as\n\\begin{equation}\\label{eq:Nnu}\n\\int\\limits_{E_-}^{E_+}{\\rm d}EE\\frac{{\\rm d}N_N}{{\\rm d}E}\n\\simeq2.8\\times10^{51}\\frac{\\mathcal{R}_{17.5}}{f_\\pi\\xi_{z,2.4}{\\dot{\\mathcal{H}}}_{0, -6}}{\\rm erg}\\,.\n\\end{equation}\nNote, that for UHE CR sources reaching energies of the order of $E_+\\simeq10^{12}$~GeV we have $\\mathcal{R} \\simeq 27.6$. We will discuss in the following specific source scenarios in terms of the required source energy budget, density, rate and evolution.\n\n\\subsubsection{Active Galactic Nuclei}\n\nThe population of hard x-ray emitting AGN has a peak luminosity at about $10^{43}$--$10^{44}$~erg\/s with a source density of $\\mathcal{H}_{0}\\simeq 10^{-5}$--$10^{-4}{\\rm Mpc}^{-3}$\\cite{Barger:2004is,Tueller:2007rk}. If we assume that CRs are accelerated to a comparable power we can see that the requirements of Eq.~(\\ref{eq:Qnu}) are fulfilled with high pion production efficiencies. We assume that the neutrino luminosity is proportional to the x-ray luminosity and follow the model of~\\cite{Barger:2004is,Aird:2009sg}. In this case the red-shift evolution factor is given by $\\xi_z \\simeq 3.6$ and $\\mathcal{H}_0\\simeq 10^{-5}{\\rm Mpc}^{-3}$. \n\nFor instance, the close-by radio galaxy Cen A at distance of about 4~Mpc has presently an upper limit for muon neutrinos that is about 40 times higher then Eq.~(\\ref{eq:JPS})~\\cite{Aartsen:2013uuv}. However, note that the TeV $\\gamma$-ray emission is about $2.5\\times10^{-13}{\\rm TeV}^{-1}{\\rm cm}^{-2}{\\rm s}^{-1}$~\\cite{Aharonian:2009xn}, which is almost two orders of magnitude lower than the hadronic $\\gamma$-ray emission expected from the relations (\\ref{eq:MM}). Another close radio galaxy M87 at 16~Mpc has a stronger upper limit on continuous neutrino emission that is only a factor 5 higher than the prediction (\\ref{eq:JPS}). The observed TeV $\\gamma$-ray emission is of the order of $6\\times10^{-12}{\\rm TeV}^{-1}{\\rm cm}^{-2}{\\rm s}^{-1}$~\\cite{Aleksic:2012uma}, which is one order of magnitude lower than the estimate (\\ref{eq:MM}).\n\nIn the unified AGN model the population of blazars is a fraction of those radio-load AGNs where the jet emission is aligned with the observation axis. For instance, the isotropic equivalent density of flat-spectrum radio quasars peaks at a jet luminosity of $10^{47}$~erg\/s with $\\mathcal{H}_{0}\\simeq 10^{-9}{\\rm Mpc}^{-3}$~\\cite{Ajello:2011zi}. We assume the same evolution as the underlying AGN source population giving $\\xi_z\\simeq3.6$. Two of the closest blazars are Mrk 421 ($\\sim$130~Mpc) and Mrk 501 ($\\sim$140~Mpc). In this case the neutrino upper limits are one order of magnitude stronger than the average emission predicted by Eq.~(\\ref{eq:JPS}) and disfavor a blazar origin of the IceCube observation.\n\nGiant AGN flares with an energy release of the order of $10^{51}$~erg have been speculated as a possible source of UHE CRs~\\cite{Farrar:2008ex}. Their event rate is estimated to be of the order of ${\\dot{\\mathcal{H}}}_{0}\\simeq 10^{-6}~{\\rm Mpc}^{-3}{\\rm yr}^{-1}$ consistent with Eq.~(\\ref{eq:Nnu}) for high pion production efficiencies. Note that the present IceCube configuration would only be sensitive to these flares via triggering on known close-by sources. On the other hand a detector with five times the effective area would be sensitive to this source class model-independently via a significant spatial and temporal clustering of events. \n\n\\subsubsection{Starburst Galaxies}\n\nStarburst galaxies show a high star formation rate of $10^{-3} M_\\odot {\\rm Mpc}^{-3} {\\rm yr}^{-1}$. Assuming that CRs are accelerated in supernova shocks with a total CR energy of $10^{50}$~erg at a rate of about $0.1\\,{\\rm yr}^{-1}$~\\cite{Lacki:2013ata} we can estimate a total power of $3\\times10^{41}{\\rm erg}\/{\\rm s}$ per source. The redshift evolution of starburst galaxies follows the average SFR at high redshift ($z\\gtrsim1$). However, the local contributions is dominated by normal galaxies and starburst only contribute at a level of 10\\%. We account for relative evolution of the starburst density by a factor ${\\rm min}(0.1+0.9z,1)\/0.1$ suggested in Ref.~\\cite{Thompson:2006np} resulting in an evolution parameter $\\xi\\simeq18$. Hence, we can estimate the required average CR power of starburst galaxies to match the IceCube flux as $10^{41}\/f_\\pi$~erg\/s and hence $f_\\pi\\simeq 1$, {\\it i.e.}~CR {\\it calorimetry}~\\cite{Loeb:2006tw}. \n\nThe expected neutrino flux (\\ref{eq:JPS}) on starburst galaxy M82 at about 3.5~Mpc is about 40 times lower than present upper limit~\\cite{Aartsen:2013uuv}. The starburst NGC 253 at about 2.5~Mpc lies in the the Southern Hemisphere and point-source neutrino limits from IceCube are weaker. Intriguingly, the expected photon point-source flux from Eq.~(\\ref{eq:MM}) is comparable to that observed in TeV $\\gamma$-ray emission for M82 and NGC 253~\\cite{Karlsson:2009hd,Abramowski:2012xy}. This agrees with the result of Ref.~\\cite{Lacki:2010vs} based on proton calorimetry.\n\nNote that the large evolution factor $\\xi_z\\simeq18$ in this scenario produces a large background consisting of distant neutrino emitters. The detection of individual neutrino sources is hence challenging. The corresponding results of the significance tests are shown as green lines in Fig.~\\ref{fig2}. For all scenarios the required signal event number exceed $10^4$.\n \n\\subsubsection{Gamma-Ray Bursts}\n\nLong duration GRBs following the collapse of massive stars occur with an (isotropic equivalent) rate density of ${\\dot{\\mathcal{H}}}_{0}\\simeq 10^{-9} {\\rm Mpc}^{-3}\\,{\\rm yr}^{-1}$~\\cite{Wanderman:2009es} and energy of $M_\\odot \\simeq 2\\times10^{54}$~erg. Following the GRB evolution model of Ref.~\\cite{Wanderman:2009es} gives an evolution parameter of $\\xi_z\\simeq1.9$. Again, this is consistent with the required power (\\ref{eq:Nnu}) if the pion production efficiency is high. However, in the case of GRBs there are strong bounds on the neutrino emission in coincidence with the $\\gamma$-ray display~\\cite{Abbasi:2012zw} which are a factor 5 lower than the observed diffuse flux (\\ref{eq:ICnu}). \n\nIt has been speculated that low-power GRBs that are unobservable via their burst might be more efficient neutrino factories~\\cite{Murase:2013ffa}. In this case the strong IceCube bounds don't apply~\\cite{Abbasi:2012zw}. However, even for IceCube's moderate signal event rates of the order of $50~{\\rm yr}^{-1}$ above a muon energy threshold of 10~TeV event clusters in time and space are expected to appear already within one year of observation ($m\\to M\\simeq3.57$ for 90\\% C.L.~of doublets in Eq.~(\\ref{eq:Ntrans})). Hence, already the present neutrino data can constrain this scenario model-independently. The triggered search on known GRBs is even more sensitive~\\cite{Abbasi:2012zw}.\n\n\\subsubsection{UHE CR Calorimeters}\n\nThe required emission rate density of UHE CR proton sources can be estimated as $E^2_pQ_p(E_p) \\simeq (1-2)\\times10^{44}\\,{\\rm erg}\\,{\\rm Mpc}^{-3}\\,{\\rm yr}^{-1}$~\\cite{Ahlers:2012rz}. If we assume that the emission spectrum $Q_p$ extends to lower energies the corresponding neutrino flux can be determined via Eq.~(\\ref{eq:MM}) and can be expressed as\nNeglecting a redshift dependence of $f_\\pi$ this translates into an $E^{-2}$ flux ($K_\\pi=2$) of \n\\begin{equation}\nE_\\nu^2J_\\nu(E_\\nu) \\simeq \\xi_{z,2.4} f_\\pi(3-6)\\times10^{-8}\\,\\frac{\\rm GeV}{{\\rm cm}^{2}\\,{\\rm s}\\,{\\rm sr}}\\,,\\label{eq:UHECRnu}\n\\end{equation}\nThe similarity of Eqs.~(\\ref{eq:ICnu}) and (\\ref{eq:UHECRnu}) suggests that the IceCube flux is related to the sources of UHE CRs. The limit $f_\\pi\\gtrsim1$ in Eq.~(\\ref{eq:UHECRnu}) corresponds to the Waxman-Bahcall bound~\\cite{Waxman:1998yy,Bahcall:1999yr}. Interestingly, the observation (\\ref{eq:ICnu}) is close to this bounds requiring high pion production efficiencies. \n\nOn the other hand, the energy loss due to pion production during acceleration has to be sufficiently low for UHE CR sources in order to compete with the energy gain per acceleration cycle. A natural solution to this fine-tuning problem occurs if the acceleration site is distinct from the CR calorimeter. Such a scenario could be provided in starburst galaxies, where the sub-PeV neutrino production happens during CR propagation in the dusty starburst environment after they have been released from transient UHE CR sources~\\cite{Katz:2013ooa}. Other scenarios consider massive galaxy clusters with local densities of the order of $\\mathcal{H}_0\\simeq 10^{-6}$--$10^{-5}{\\rm Mpc}^{-3}$ as CR calorimeters~\\cite{Berezinsky:1996wx,Murase:2008yt}. Note that the neutrino emission from CR propagation in CR calorimeters is expected to be continuous even if the sources of UHE CRs are transients, such as flaring AGNs or GRBs. \n\n\\section{Conclusion}\\label{sec6}\n\nIn this work we have studied the probability of identifying extra-galactic neutrino sources as the likely origin of the astrophysical neutrino flux observed by the IceCube Collaboration. We have derived the expected flux of individual continuous and transient sources based on the IceCube observation and depending on source density and rate. Our analysis takes into account the ensemble variation of the source distribution, their cosmic evolution and the Poisson statistics of weak nearby sources. In order to account for the background of atmospheric neutrinos and distant neutrino sources we introduced test statistics for significance tests of event clusters and source associations.\n\nOur findings are as follows. A model-independent observation of neutrino event clusters from a continuously emitting source population is challenging due to large atmospheric backgrounds unless the experimental angular resolution can be improved significantly. For transient sources these backgrounds are in general much lower due to the correlation of signal events in space and time. We estimate that the present IceCube detector is sensitive to rare $\\dot{\\mathcal{H}}_0\\lesssim10^{-8}~{\\rm Mpc}^{-3}\\,{\\rm yr}^{-1}$ source classes within five years of operation. A next generation telescope with a five times extended effective muon area would be sensitive to multiplets from transient sources with $\\dot{\\mathcal{H}}_0\\lesssim10^{-6}~{\\rm Mpc}^{-3}\\,{\\rm yr}^{-1}$. \n\nThe sensitivity to the underlying source population can be increased in model-dependent associations of events with source catalogues. We estimate that the IceCube detector is already sensitive to sparse continuous sources with $\\mathcal{H}_0\\lesssim10^{-6}~{\\rm Mpc}^{-3}$ as well as transient sources at the level of $\\dot{\\mathcal{H}}_0\\lesssim10^{-5}~{\\rm Mpc}^{-3}\\,{\\rm yr}^{-1}$ via the association of events with the 100 closest sources of the ensemble. Again, a next-generation detector with 5 times the effective area as IceCube and otherwise identical performance in terms of angular resolution and FoV would improve the sensitive of these searches by about two orders of magnitude. A few final remarks are in order. \n\n{\\it i)} In this analysis we have only considered the scenario that the IceCube observation has an extragalactic origin. However, a (partial) Galactic origin of the observation is not yet excluded on statistical grounds. The presence of Galactic contributions will most likely emerge as a large scale anisotropy of the signal which has not been discussed in this work. \n\n{\\it ii)} We have expressed our results via ensemble-averaged $p$-values. The shaded regions shown in Fig.~\\ref{fig1} indicate the $\\pm40\\%$ fluctuation around the median. This {\\it cosmic variance} of the closest source can hence increase or decrease the prospects of detecting the closest source, depending on the sign of the fluctuation.\n\n{\\it iii)} We assumed that the individual sources of the ensemble can be approximated by a universal (average) luminosity. It is straightforward to include a non-trivial luminosity function or other source nuisance parameters in the definition of the point source distribution (\\ref{eq:pPS}) and related quantities. \n\n{\\it iv)} Future neutrino observatories may employ an extended surface veto for atmospheric events that would reduce the contribution of atmospheric neutrinos~\\cite{Schonert:2008is,Karle2014}. This can increase the signal-to-background ratio at lower muon energy threshold and improve the detection prospects estimated in this analysis. \n\n{\\it v)} The identification of sources via the association of events with source catalogues is model-dependent and the statistical results need to be corrected by trials factors. However, complementary to the identification of individual neutrino sources the study of spectral properties of the quasi-diffuse neutrino flux will help to narrow down valid astrophysical scenarios and should hence limit the number of trials.\n\n{\\it vi)} The identification of extra-galactic neutrino point-sources from a quasi-diffuse flux has also been discussed recently in Refs.~\\cite{Fargion:2014mda,Anchordoqui:2014yva}. In this analysis we have carefully studied the effect of ensemble and statistical variations and accounted for backgrounds via a significance test. This makes our estimates more robust and less optimistic than Refs.~\\cite{Fargion:2014mda,Anchordoqui:2014yva}.\n\n\\begin{acknowledgements}\nWe would like to thank the IceCube Collaboration for many fruitful discussions, in particular Jake Feintzeig, Gary Hill, Albrecht Karle, Claudio Kopper and Chris Weaver. We acknowledge support by the U.S. National Science Foundation (NSF) under grants OPP-0236449 and PHY-0236449.\n\\end{acknowledgements}\n\n\\begin{appendix}\n\n\\section{Source Distribution}\\label{app1}\n\nThe probability distribution of single source within $r<R$ is $p(r) = 3r^2\/{R^3}$. We can express this in terms of the expected events from a point-source using ${\\rm d} n\/{\\rm d}r = - 2 n\/r$ as\n\\begin{equation}\\label{eq:pPS}\np_{\\rm PS}(n,n(R)) = \\Theta(n-n(R))\\frac{3}{2}\\frac{1}{ n}\\left(\\frac{ n(R)}{ n}\\right)^{3\/2}\\,,\n\\end{equation}\nwith $n(R)$ defined by Eq.~(\\ref{eq:nr}). The expected event distribution for the $k$th-closest of the $N_s$ sources of the ensemble is given by\n\\begin{multline}\np_k(n) = \\Theta(n-n(R))\\frac{N_s!}{(k-1)!(N_s-k)!}\\\\\\times\\frac{3}{2}\\frac{1}{n}\\left(\\frac{n(R)}{n}\\right)^{\\frac{3k}{2}}\\left[1-\\left(\\frac{n(R)}{n}\\right)^{\\frac{3}{2}}\\right]^{N_s-k}\\,.\n\\end{multline}\nFor $N_s\\gg k$ we can approximate the last term as\n\\begin{equation}\n\\left[1-\\left(\\frac{n(R)}{n}\\right)^{\\frac{3}{2}}\\right]^{N_s-k}\n\\simeq e^{-N_s\\left(\\frac{n(R)}{n}\\right)^{\\frac{3}{2}}}\\,,\n\\end{equation}\nand using $(N_s-k) n(R)^{3\/2} \\simeq n(\\hat{r})^{3\/2}$ we arrive at\n\\begin{equation}\np_k(n) \\simeq \\frac{3}{2(k-1)!}\\frac{1}{n}\\left(\\frac{n(\\hat{r})}{n}\\right)^{\\frac{3k}{2}}e^{-\\left(\\frac{ n(\\hat{r})}{ n}\\right)^\\frac{3}{2}}\\,.\n\\end{equation}\nFor $k=1$ this agrees with Eq.~(\\ref{eq:ploc}) for the closest source.\n\n\\section{Signal Probability}\\label{app2}\n\nIn general, the probability of the events ${\\bf k}$ is given by the Poisson distribution of events in the individual bins with expected signal events $\\lambda_i$ and background $\\lambda_{\\rm bg}$,\n\\begin{equation}\nP({\\bf k}) = \\prod_{i=1}^{n_{\\rm bin}}\\frac{(\\lambda_i+\\lambda_{\\rm bg})^{k_i}}{k_i!}e^{-\\lambda_{\\rm bg}-\\lambda_i}\\,,\n\\end{equation}\nwhere $\\sum_i (\\lambda_i+\\lambda_{\\rm bg}) = N_{\\rm tot}$. The total number of events can be expressed via the expected number of all signal events and the signal-to-background ratio $S\/B$ as $N_{\\rm tot}= (1+(S\/B)^{-1})N$. The background probability is simply\n\\begin{equation}\nP_0({\\bf k}) = \\prod_{i=1}^{n_{\\rm bin}}\\frac{\\lambda_0^{k_i}}{k_i!}e^{-\\lambda_0}\\,,\n\\end{equation}\nwith $n_{\\rm bin}\\lambda_0 = N_{\\rm tot}$.\n\nIn the case of the test statistics ${\\rm TS}_{1\/2}$ for event clusters and source associations we can derive simple probabilities in the background free case. The probability of observing less than $m$ events from $ n$ expected events can be expressed via incomplete $\\Gamma$-functions as\n\\begin{equation}\n\\sum_{k=0}^{m-1}  \\frac{ n^k}{k!}e^{- n} = \\frac{\\Gamma(m, n)}{\\Gamma(m)}\\,.\n\\end{equation}\nHence, the probability of observing less than $m$ events from a single source within distance $R$ is given as\n\\begin{equation}\n P_{\\rm PS}(m) = \\int\\limits_{ n(R)}^\\infty {\\rm d}n\\, p_{\\rm PS}(n,n(R))\\frac{\\Gamma(m, n)}{\\Gamma(m)}\\,.\n\\end{equation}\nThis can be expressed as\n\\begin{equation}\n P_{\\rm PS}(m) = \\frac{\\Gamma\\left(m, n(R)\\right)-(n(R))^{3\/2}\\Gamma\\left(m-\\frac{3}{2}, n(R)\\right)}{(m-1)!}\\,.\n\\end{equation}\nThe probability of observing at least one cluster of $m$ or more neutrinos from a local source is given by the expression $P_{\\rm cl} = 1 -  P_{\\rm PS}^{N_s}$.\nIn the limit $n(R)\\ll 1$ and for $m\\geq2$ this can be approximated as\n\\begin{equation}\\label{eq:Pcl}\nP_{\\rm cl}\\simeq1-\\exp\\left(-\\frac{\\Gamma(m-\\frac{3}{2})}{3^{\\frac{3}{2}}\\Gamma(m)}\\left(\\frac{N}{\\xi_z}\\right)^\\frac{3}{2}\\left(\\frac{V_1}{V_H}\\right)^\\frac{1}{2}\\right)\n\\,.\n\\end{equation}\nHence, this result is independent of the auxiliary distance $R$ as expected. For a significance level $P$ the required total signal event number is the same as in Eqs.~(\\ref{eq:Ncont}) or (\\ref{eq:Ntrans}) with the replacement\n\\begin{equation}\\label{eq:mM}\nm\\to M=\\left(\\frac{3^\\frac{3}{2}\\Gamma(m)}{\\Gamma(m-\\frac{3}{2})}\\ln\\frac{1}{1-P}\\right)^\\frac{2}{3}\\,.\n\\end{equation}\n\nFor the case of local source associations we can write the probability of observing $m$ or more events as\n\\begin{multline}\\label{eq:Pass}\nP_{\\rm ass}(m) = 1-\\int{\\rm d}n_Cp_C(n_C)\\prod_{i=1}^{C-1}{\\rm d}n_i p_{\\rm PS}(n_i,n_C)\\\\\\times\\frac{\\Gamma(m,\\sum_{i=1}^Cn_i)}{\\Gamma(m)} \\,.\n\\end{multline}\nFor large number of local sources $C$ we can approximate this by the average number of expected events as\n\\begin{equation}\nP_{\\rm ass}(m) \\simeq 1-\\frac{\\Gamma(m,N(V_C\/V_H)^\\frac{1}{3}\/\\xi_z)}{\\Gamma(m)}\\,.\n\\end{equation}\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\nLoaded granular materials made of highly deformable particles are ubiquitous in nature and industry from cell monolayer growth \\cite{angelini2011_pnas,martin2004_dev} to foam and emulsion compression \\cite{bolton1990_prl,katgert2010_epl,brujic2007_prl} and even metal or plastic powder sintering \\cite{kim1987_ijp,bares2014_prl}. In some cases, because of impurities or for technical and recycling purposes, rigid, undeformable particles are mixed among soft ones \\cite{tsoi2011_geo,khatami2020_gi}. This is the case for example in sand-rubber granular layers used as low-cost seismic \\md{isolations} \\cite{tsiavos2019_sdee}. \n\nThese kinds of mixed soft-rigid granular systems display singular behaviors different from purely rigid or \\md{purely} soft systems. Among other purposes, modifying the mixing ratio and the particle nature permits to tune the mechanical properties of a granular medium. In order to optimize their industrial applications and to control \\md{their} behavior in \\md{natural phenomena}, it is needed to know the connection between their macroscopic behaviors and their microscale composition and organisation. Because granular matter is naturally \\md{multiscale}, \\md{a} clear understanding of its global behavior necessitates to figure out and to measure what happens at the local\/grain scale and at all the intermediate scales up-to the global one. Due to the already complex process of measuring local mechanical properties of each grain for fewly deformed particles \\cite{zadeh2019_gm} the challenge is even more important of measuring them for soft highly deformed particles. \n\nVery recently this challenge has been overcome \\cite{vu2019_pre,vu2019_pre_2} using an experimental method coupling Digital Image Correlation (DIC) and very \\md{acurate} imaging. In this paper, we extend this method to the so called case of mixtures of soft and rigid particles. We show how this experimental set-up is \\md{modified} to compress \\md{these} mixtures and how the image post-processing is modified accordingly. A selection of resulting experimental data from the inside of the particle up-to the system size is presented. This allows us to understand better the mechanical behavior of these materials and will permit to improve their compression models \\cite{cardenas2020_pre}.\n\n\\section{Experimental method} \\label{method}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.99\\columnwidth]{fig1.png}\n\\caption{Experimental set-up. Top: 3D \\md{drawing} of the compression device and particle packing laying on a photographic flatbed scanner \\cite{scanner}. Bottom-left: squeezed soft silicone particle \\cite{silicone,vu2019_pre_2}. The top shiny aspect is due to a layer of thin metallic glitter \\cite{silver_glitter}. Bottom-right: rigid PVC particle. The top shiny aspect is due to silver paint. \\md{During experiments particles are used with their shiny part in contact with the scanner.}}\n\\label{fig_1}\n\\end{figure}\n\nIn Fig. \\ref{fig_1} we show the experimental set-up already presented in \\cite{vu2019_pre_2}. It is composed of a bidimensional piston of initial dimension $270\\times202$ mm$^2$ \\md{which permits to compress uniaxially the system}. A stepper motor rotates a screw which \\md{translate the moving edge of the piston inward in an accurate manner ($25$ $\\mu$m per motor step)}. Two force sensors are attached to this latter, they record the global stress evolution ($\\sigma$) when compressing a mixed granular system. The global stress is measured continuously with a frequency of $100$ Hz while the system is compressed stepwisely. At each step the piston moves of $0.5$ mm at a speed of $2$ mm\/min to make \\md{it} sure to stay in the quasistatic regime. Then, the system rests during $1$ min to relax and is imaged from below with a flatbed \\cite{scanner} with a resolution of $2400$ dpi ($10.6$ $\\mu$m\/px).\n\nThe granular \\md{systems are} composed of soft hyperelastic cylinders made of silicone \\cite{silicone,vu2019_pre_2} and of rigid \\md{cylinders} made of PVC (see Fig. \\ref{fig_1}). Their Young moduli are $0.45$ MPa \\cite{vu2019_em} and $1.2$ GPa, respectively. Both soft and rigid granular packings are bidisperse cylinders of diameters $20$ mm and $30$ mm and of height $15$ mm. For the \\md{experiments} presented in this paper, $96$ particles are used, $1\/2$ of them being soft and $2\/3$ of them being small. This last ratio is equally spread among soft and rigid particles, this avoids phase crystallization. \\md{The ratio of soft to rigid particles, $\\kappa = \\rm{n}_{\\rm{soft}}\/\\rm{n}_{\\rm{total}}$ is varied from $0.2$ to $0.8$.} To avoid basal friction the scanner glass is coated with oil which also lubricate the contacts making particles almost frictionless. \\md{Namely the measured friction coefficient is $0.05 \\pm 0.02$.}\n\nTo measure displacement fields on the particle bottom faces, the DIC method requires a thin random pattern with a high optical contrast. In the case of soft particles, this is obtained by coating the silicone with a very thin layer of \\md{shiny} glitter \\cite{silver_glitter}. For rigid particles, PVC is simply coated with \\md{silver metallic} paint. In both cases, the glitter characteristic size is around $25$ $\\mu$m. \\md{This} creates a random pattern with a correlation length of \\md{about} $50$ $\\mu$m on the bottom face of each particle.\n\n\\md{For each experiment a set of about $100$} pictures is obtained with this experimental set-up by compressing the system over $15$\\%. Pictures are then post-processed with an algorithm modified from \\cite{vu2019_pre_2, DIC_code}. First, particles are detected individually by thresholding the initial\/undeformed picture. Each particule is then tracked in the full set of pictures by using image correlation. Then, from the measurement of the particle rotation and translation, subsets of images are extracted following each particle and correcting its solid rigid motion. For soft particles a DIC algorithm aimed for large \\md{deformations} already presented in \\cite{vu2019_em} is used to obtain the displacement field ($\\bm{u}$) inside each particle. For the rigid particles a more classical DIC algorithm is used. This latter correlates all images with the initial one as classically done for systems \\md{in} the small deformation assumption. In both cases the correlation cell size is self-adjusted to take random pattern inhomogeneities into account as already presented in \\cite{vu2019_pre_2}. \n\nThen, for soft particles the deformation gradient tensor is computed from $\\bm{u}$ \\cite{taber2004_book}:\n\\begin{equation}\n\t\\bm{\\bm{F}}=\\bm{\\nabla}\\bm{u}+\\bm{\\bm{I}}\n\\end{equation}\nwith $\\bm{\\bm{I}}$ being  the  second  order  identity  tensor. The right Cauchy-Green strain tensor is finally derived \\cite{taber2004_book}:\n\\begin{equation}\n\t\\bm{\\bm{C}}=\\bm{\\bm{F}}^T\\bm{\\bm{F}}\n\\end{equation}\nFor rigid particles, the infinitesimal strain tensor is just directly computed ($\\bm{\\bm{\\varepsilon}}=1\/2((\\bm{\\nabla}\\bm{u})^T+\\bm{\\nabla}\\bm{u})$). From $\\bm{u}$ the evolution of the exact particle boundaries is also deduced which permits to get all geometric \\md{observables} for both particles and voids, but also contacts, at each compression step. \\md{Contacts are detected by considering first the proximity of each point of the particles boundaries with the boundaries of neighbor particles. Then, for each point closer than a certain threshold the von Mises strain around this point is measured and the contact is assumed to be real if this value is above a certain threshold as detailed in \\cite{vu2019_pre_2}.} For the sake of conciseness only a meaningful selection of \\md{the different} mechanical and geometrical observables are presented here.\n\n\n\\section{Results and discussions} \\label{result}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.99\\columnwidth]{fig2.png}\n\\caption{Global observables. Variation of the global stress ($\\sigma$), coordination ($Z$), specific contact length ($l$), average value of the von Mises of the right Cauchy-Green strain tensor ($|\\bm{\\bm{C}}|$) of soft particles, particle asphericity ($a$) and void area ($A$) as a function of the packing fraction ($\\phi$) for a compression experiment made with $50$\\% of rigid particles ($\\kappa = 0.5$).}\n\\label{fig_2}\n\\end{figure}\n\nAs presented in Fig. \\ref{fig_2}, \\md{a granular system with $\\kappa = 0.5$ is compressed for a packing fraction ($\\phi$) going from $0.795$ to $0.915$}. While loading, the global stress first stays close to $0$ MPa and then increases progressively from the jamming point \\cite{liu1998_nat}. This occurs for $\\phi$ close to \\md{$\\phi_j = 0.84$}, as already observed in many other experimental studies \\cite{behringer2018_rpp}. It also corresponds with a coordination value slightly above $3$ as predicted theoretically \\cite{van2009_jpcm} since particles are slightly frictional. In Fig. \\ref{fig_2} we also plot the evolution of the specific contact length, which is the average of \\md{the ratio of the particle perimeter in contact with another particle or with the system boundaries ($l_{\\rm contact}$) over the particle perimeter: $l = \\left< l_{\\rm contact}\/\\rm{perimeter}\\right>_{\\rm particles}$.} This quantity increases linearly with $\\phi$, which is not in agreement with the Hertz's contact law in 2D \\cite{johnson_book}. This last point is not surprising since soft particles are far from the small deformation assumption \\md{in our study}. \n\nFigure \\ref{fig_2} also shows that the average value of the von Mises strain ($|\\bm{\\bm{C}}|$) in soft particles decreases parabolically all along the compression while the average soft particle asphericity first gently increases and then sharply rises after $\\phi=0.88$. This quantity is computed as the ratio between the actual particle area and the area deduced from the perimeter assuming the particle would be circular. The packing fraction $\\phi=0.88$ also corresponds with the maximum value of the total void area which first increases, since void clusters form, and then decreases, since particles are squeezed and reduce these \\md{cluster} areas, as already observed for fully soft systems \\cite{vu2019_pre_2}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.99\\columnwidth]{fig3.png}\n\\caption{Local observables. Field of the first invariant of the right Cauchy-Green strain tensor ($\\bm{\\bm{C}}$) for soft particles and of the infinitesimal strain tensor ($\\bm{\\bm{\\varepsilon}}$) for rigid particles. These fields are computed for a compression experiment made with $50$\\% of rigid particles at a packing fraction $\\phi = 0.89$ far above the jamming point.}\n\\label{fig_3}\n\\end{figure}\n\nIn Fig. \\ref{fig_3}, the first invariant of the strain fields measured in each particle is shown for $\\phi = 0.89$. For soft particles the right Cauchy-Green strain tensor ($\\bm{\\bm{C}}$) is used while it is the infinitesimal strain tensor ($\\bm{\\bm{\\varepsilon}}$) \\md{which is used} for rigid particles. The field for soft particles distinctly shows high strain chains with clear areas near the contact points where the deformation is maximum. On the contrary the patterns observed on the rigid particles suggest that the imaging method is not accurate enough to \\md{highlight} grain deformations. This is not surprising since the level of compression is rather low and the \\md{ratio of soft to rigid} particle stiffness covers several orders of magnitude: \\md{deformation is almost exclusively carried by soft particles.} The lines that appear on the rigid particles actually correspond with the small speed inhomogeneities of the \\md{imaging method}.\n \n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.99\\columnwidth]{fig4.png}\n\\caption{Statistical observation. \\md{Main panel}: Evolution of the PDF of the value of the von Mises of the right Cauchy-Green strain tensor ($\\bm{C}$) measured in soft particles as a function of the packing fraction for a compression experiment made with $50$\\% of rigid particles. Scale is semi-log. \\md{Inset: Evolution of standard deviation of $\\bm{C}$ for different particle mixes ($\\kappa$) as function of the distance to the jamming point.}}\n\\label{fig_4}\n\\end{figure}\n\nIn Fig. \\ref{fig_4}, we present the evolution of the strain field statistics. More exactly, we plot the probability density functions (PDF) of the von Mises values of the right Cauchy-Green strain tensor ($|\\bm{\\bm{C}}|$) in soft particles for different packing fractions. For gently squeezed states, the PDF is a Gaussian centered around $1$ which corresponds with the fact that the particles are mainly undeformed. While the system is loaded, the PDF gets broader and broader and the average value slightly shifts to the left which corresponds with the global compression of the system. For densely packed states, the PDFs display an exponential tail with a coefficient decreasing with the packing fraction. This exponential tail is reminiscent of what is observed for force values in packing of rigid granular systems \\cite{radjai1996_prl} and remains unchanged from observations made on purely soft systems \\cite{vu2019_pre_2}. \\md{In the inset of Fig. \\ref{fig_4}, the evolution of the standard deviation of $\\bm{C}$ is shown for different ratios between soft and rigid particles. When the system is mainly composed of rigid particles ($\\kappa = 0.2$), this value rapidly increases right after the jamming point. In this case lot of particles undergo large deformation and display more extreme strain values. On the contrary, when the system is composed of more soft particles ($\\kappa = 0.8$) the deformation is shared over more particles and less extreme strain values are observed so the standard deviation increases less rapidly.}\n\n\\section{Conclusion} \\label{conclusion}\n\nIn summary, we have developed an experimental tool that \\md{can} permit to measure all mechanical and geometrical observables from \\md{the displacement field at} the particle scale for bidimensional granular systems under uniaxial compression, whatever the particle mechanical properties. \\md{We believe this measurement method at the \\textit{sub-micro scale} will permits to model granular matter in a more efficient way to predict its behavior.} It is applied here to the case of a mixed granular \\md{media} made of soft squeezable particles and rigid ones. The system is loaded step by step and imaged very accurately, while \\md{the} global force applied to it is recorded continuously. By mean of DIC, the evolution of mechanical fields inside each particle are measured which permits to deduce observables as diverse as packing fraction, coordination, contact length, particle asphericity, void area and strain fields to name a few.\n\nMore specifically we have shown that our system crosses the jamming transition for a packing fraction and a coordination value close from what has been observed in the case of purely rigid systems. This point validate our experimental approach. \\md{We have also shown that the average von Mises strain in soft particles decreases with the exact same tendency as the global stress increases in the system. Since rigid particle deformation is negligible, this suggests that rigid particles do not carry more load than the soft ones. The loading seems to be balanced between soft and rigid particles.} The stress vs. packing fraction curve is also similar to what has been observed in a recent numerical study \\cite{cardenas2020_pre}. \n\nDeep in the jammed state the PDF of the von Mises strain tensor displays an exponential tail as already observed for purely soft systems and similarly to what is observed for the \\md{intergranular} force PDF \\md{of} purely rigid systems. \\md{Also, the standard deviation of this quantity increases more rapidly for systems composed of more rigid particles. This is explained by the fact that deformation is mainly carried by soft particles and that they are individually more deformed when they are less for a given deformation level.}\n\nBeyond the study of mixed soft-rigid granular systems, we believe that this experimental tool offers an opportunity to catch experimentally the behavior of breakable and polydisperse (in size, shape and material) \\md{particles} under different loading geometries. \n\n{\\small\nGille Camp, St\\'{e}phan Devic and R\\'{e}my Mozul are greatly thanked for their technical support. We warmly thank Vincent Huon for for fruitful discussions.\n}\n\\newline\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}