{"text":"\\section{Introduction}\n\n\nThe authors of Ref. \\cite{zhao} investigated the effects of the variation of the mass parameter $a$ on the thick branes. They used a real scalar field, which has a potential of the $\\phi^{6}$ model, as the background field of the thick branes. It was found that the number of the bound states (in the case without gravity) or the resonant states (in the case with gravity) increases with the parameter $a$. That work considered the simplest Yukawa coupling $\\eta\\bar{\\Psi}\\phi\\Psi$, where $\\eta$ is the coupling constant. The authors stated that as the value of $a$ is increasing, the maximum of the matter energy density splits into two new maxima, and the distance of the new maxima increases and the brane gets thicker. The authors also stated that the brane with a big value of $a$ would trap fermions more efficiently.\n\nIn this paper, we reinvestigated the effect of the variation of the mass parameter $a$ on the thick branes, because the above investigation does not analyze the zero mode in details and contains some misconceptions. We only focus attention in the case with gravity. We find that the variation of $a$ on the thick brane is associated to the phenomenon of brane splitting. From the static equation of motion, we analyze the asymptotic behavior of $A(y)$ and find that the zero mode for left-handed fermions can be localized on the brane depending on the value for the coupling constant $\\eta$ and the mass parameter $a$. We also show that as the value of $a$ is increasing the simplest Yukawa coupling does not support the localization of fermions on the brane, as incompletely argued in Ref. \\cite{zhao}.\n\n\n\n\\section{Thick brane with gravity}\n\nThe action for our system is described by \\cite{cam}\n\\begin{equation}\\label{ac1}\n    S=\\int d^4xdy\\sqrt{ -g}\\left[\\frac{1}{4}\\,R-\\Lambda-\\frac{1}{2}g^{MN}\\partial_{M}\\phi\\partial^{N}\\phi-V(\\phi) \\right],\n\\end{equation}\n\\noindent where $M,N=0,1,2,3,4$, $\\Lambda$ is the 5D bulk cosmological constant and the scalar potential $V(\\phi)$ is given by \\cite{zhao}\n\\begin{equation}\\label{pot1}\n    V(\\phi)=a\\phi^{2}-b\\phi^{4}+c\\phi^{6}\\,,\n\\end{equation}\n\n\\noindent where $a,b,c>0$. There are three minima for $V(\\phi)$, one is at $\\phi^{(1)}=0$ (local minima) corresponding to a disordered bulk phase and the other two are at $\\phi^{(2)}=-\\phi^{(3)}=v$ (global minima) with\n\\begin{equation}\\label{v}\n    v=\\sqrt{\\frac{\\sqrt{b^{2}-3ac}}{3c}+\\frac{b}{3c}}\\,.\n\\end{equation}\n\n\\noindent They are degenerated and correspond to ordered bulk phases. As $a=a_{c}$ ($a_{c}=b^{2}\/4c$), $V(\\phi^{(1)})=V(\\phi^{(2)})=V(\\phi^{(3)})$, $V(\\phi)$ has three degenerated global minima. For the case with gravity, the critical value of $a$ is not $a_{c}$ but a smaller effective critical value $a_{*}$. In this case, $a_{c}=a_{*}=0.837$ \\cite{zhao}. The line element in this model is considered as\n\\begin{equation}\\label{metric}\n    ds^{2}=g_{ab}dx^{a}dx^{b}=\\mathrm{e}^{2A(y)}\\eta_{\\mu\\nu}dx^{\\mu}dx^{\\nu}+dy^{2},\n\\end{equation}\n\n\\noindent where $\\mu,\\nu=0,1,2,3$, $\\eta_{\\mu\\nu}=\\mathrm{diag}(-1,1,1,1)$, and $\\mathrm{e}^{2A}$ is the so-called warp factor. We suppose that $A=A(y)$ and $\\phi=\\phi(y)$.\n\n\n\\subsection{Effects of the variation of the mass parameter $a$ on the thick brane}\n\nFor this model, the equations of motion are\n\\begin{equation}\\label{em1b}\n    \\phi^{\\prime\\prime}=-4A^{\\prime}\\phi^{\\prime}+\\frac{dV(\\phi)}{d\\phi},\n\\end{equation}\n\\begin{equation}\\label{em2b}\n    A^{\\prime\\prime}+2A^{\\prime}\\,^{2}=-\\frac{1}{3}\\,\\phi^{\\prime}\\,^{2}\n    \\frac{2}{3}\\,(V+\\Lambda)\\,,\n\\end{equation}\n\\begin{equation}\\label{em3b}\n    A^{\\prime}\\,^{2}=\\frac{1}{6}\\,\\phi^{\\prime}\\,^{2}-\\frac{1}{3}\\,(V+\\Lambda)\\,.\n\\end{equation}\n\n\\noindent It is possible to rewrite (\\ref{em2b}) and (\\ref{em3b}) as\n\\begin{equation}\\label{em3c}\nA^{\\prime\\prime}=-\\frac{2}{3}\\,\\phi^{\\prime}\\,^{2}\\,.\n\\end{equation}\n\n\\noindent The boundary conditions can be read as follows\n\\begin{equation}\\label{bc1}\n    A(0)=A^{\\prime}(0)=\\phi(0)=0\\,,\n\\end{equation}\n\\begin{equation}\\label{bc2}\n    \\phi(+\\infty)=-\\phi(-\\infty)=v.\n\\end{equation}\n\n\nThe matter energy density has the form\n\\begin{equation}\\label{de}\n    T_{00}=\\rho(y)=\\mathrm{e}^{2A(y)}\\left[ \\frac{1}{2}\\,\\left( \\frac{d\\phi}{dy\n    \\right)^{2}+V\\left(\\phi\\right) \\right].\n\\end{equation}\n\n\\noindent At this point, it is also instructive to analyze the matter energy of the toy model\n\\begin{equation}\\label{ephi}\n    E_{\\phi}=\\int^{\\infty}_{-\\infty}dy\\,T_{00}\\,,\n\\end{equation}\n\n\\noindent substituting (\\ref{de}) in (\\ref{ephi}), we get\n\\begin{equation}\\label{ephi2}\n    E_{\\phi}=\\int^{\\infty}_{-\\infty}dy\\,\\mathrm{e}^{2A(y)}\\left[ \\frac{1}{2}\\,\\left( \\frac{d\\phi}{dy\n    \\right)^{2}+V\\left(\\phi\\right) \\right]\\,,\n\\end{equation}\n\n\\noindent using (\\ref{em3b}) and (\\ref{em3c}), we obtain the value of the matter energy given by\n\\begin{equation}\\label{ephi3}\n    E_{\\phi}=\\frac{3}{2}\\,\\left[\\mathrm{e}^{2A(-\\infty)}A^{\\prime}(-\\infty)\n    \\mathrm{e}^{2A(\\infty)}A^{\\prime}(\\infty) \\right]-\\Lambda\\int^{\\infty}_{-\\infty}d\n    \\mathrm{e}^{2A(y)}.\n\\end{equation}\n\n\\noindent As $\\Lambda=0$, the value of the matter energy depends on the asymptotic behavior of the warp factor. If $y\\rightarrow\\pm\\infty$ then $\\phi^{\\prime}(\\pm\\infty)=0$ and by the analysis to Eq. (\\ref{em2b}), we can see that $A(\\pm\\infty)\\propto -|\\,y|$. Therefore, $\\mathrm{e}^{2A(\\pm\\infty)}\\rightarrow0$ and the value of the matter energy is zero. This fact is the same to the case of branes with generalized dynamics \\cite{arro}.\n\n\\noindent The scalar curvature (or Ricci scalar) is given by\n\\begin{equation}\\label{ricci}\n    R=-4(5A^{\\prime}\\,^{2}+2A^{\\prime\\prime}).\n\\end{equation}\n\n\\noindent The profiles of the matter energy density is shown in Fig. (\\ref{fde}) for some values of $a$. Figure (\\ref{fde}) clearly shows that for $a=0$ the matter energy density has not a single-peak around $y=0$. The core of the brane is localized at $y=0$ for $a=0$, because this region has a positive matter energy density. On the other hand, as the value of $a$ is increasing, we can see that the single brane splits into two sub-branes and as $a\\rightarrow a_{*}$ each sub-brane is a thick brane. This phenomenon is so-called of brane splitting \\cite{angel}. From the peak of the matter energy density is evident know where the core of the branes are located. Therefore, the brane does not get thicker with the increases of the value of the mass parameter $a$, as argued in Ref. \\cite{zhao}. The profiles of the matter energy density and the Ricci scalar are shown in Fig. (\\ref{desc}) for $a=0.8$. Note that the presence of regions with positive Ricci scalar is connected to the capability to trap matter near to the core of the brane \\cite{alm} and it reinforces the conclusion of the analyzes from the matter energy density. Also note that far from the brane, $R$ tends to a negative constant, characterizing the $AdS_{5}$ limit from the bulk.\n\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=7cm, angle=0]{de.eps}\n\\end{center}\n\\caption{The profiles of the energy density for $b=2$, $c=1$, $a=0$ (thin line), $a=0.8$ (dashed line) and $a=0.836$ (dotted line).} \\label{fde}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=7cm, angle=0]{deera08.eps}\n\\end{center}\n\\caption{The profiles of the matter energy density (thin line) and Ricci scalar (thick line) for $b=2$, $c=1$ and $a=0.8$.} \\label{desc}\n\\end{figure}\n\n\\subsection{Fermion localization}\n\nThe action for a Dirac spinor field coupled with the scalar fields by a general Yukawa coupling is\n\\begin{equation}\\label{ad}\n    S=\\int d^{5}x\\sqrt{|\\,g|}\\left[ i\\bar{\\Psi}\\Gamma^{M}\\nabla_{M}\\Psi-\\eta\\bar{\\Psi}F(\\phi)\\Psi \\right]\\,,\n\\end{equation}\n\\noindent where $\\eta$ is the positive coupling constant between fermions and the scalar field. Moreover,\nwe are considering the covariant derivative $\\nabla_{M}=\\partial_{M}+\\frac{1}{4}\\,\\omega^{\\bar{A}\\bar{B}}_{M}\\Gamma_{\\bar{A}}\\Gamma_{\\bar{B}}$\\,,\nwhere $\\bar{A}$ and $\\bar{B}$, denote the local Lorentz indices and $\\omega^{\\bar{A}\\bar{B}}_{M}$ is\nthe spin connection. Here we consider the field $\\phi$ as a background field. The equation of motion is\nobtained as\n\\begin{equation}\\label{dkp}\ni\\,\\Gamma ^{M }\\nabla_{M}\\Psi-\\eta F(\\phi)\\Psi =0.\n\\end{equation\n\nAt this stage, it is useful to consider the fermionic current. The conservation law for $J^{M}$ follows from the standard procedure and it becomes\n\\begin{equation}\\label{corr}\n\\nabla_{M}J^{M}=\\bar{\\Psi}\\left(\\nabla_{M}\\Gamma^{M}\\right)\\Psi\\,,\n\\end{equation}\n\\noindent where $J^{M}=\\bar{\\Psi}\\Gamma^{M}\\Psi$. Thus, if\n\\begin{equation}\\label{cj0}\n    \\nabla_{M}\\Gamma^{M}=0\\,,\n\\end{equation}\n\\noindent then four-current will be conserved. The condition (\\ref{cj0}) is the purely geometrical assertion that the curved-space gamma matrices are covariantly constant.\n\n\\noindent Using the same line element (\\ref{metric}) and the representation for\ngamma matrices $\\Gamma^{M}=\\left( \\mathrm{e}^{-A}\\gamma^{\\mu},-i\\gamma^{5}\\right)$,\nthe condition (\\ref{cj0}) is trivially satisfied and therefore the current is conserved.\n\nThe equation of motion (\\ref{dkp}) becomes\n\\begin{equation}\\label{em}\n\\left[ i\\gamma^{\\mu}\\partial_{\\mu}+\\gamma^{5}\\mathrm{e}^{A}(\\partial_{y}+2\\partial_{y}A\n-\\eta\\,\\mathrm{e}^{A}F(\\phi) \\right]\\Psi=0.\n\\end{equation}\n\\noindent Now, we use the general chiral decomposition\n\\begin{equation}\\label{dchiral}\n    \\Psi(x,y)=\\sum_{n}\\psi_{L_{n}}(x)\\alpha_{L_{n}}(y)+\\sum_{n}\\psi_{R_{n}}(x)\\alpha_{R_{n}}(y),\n\\end{equation}\n\\noindent with $\\psi_{L_{n}}(x)=-\\gamma^{5}\\psi_{L_{n}}(x)$ and $\\psi_{R_{n}}(x)=\\gamma^{5}\\psi_{R_{n}}(x)$.\nWith this decomposition $\\psi_{L_{n}}(x)$ and $\\psi_{R_{n}}(x)$ are the left-handed and\nright-handed components of the four-dimensional spinor field, respectively. After applying\n(\\ref{dchiral}) in (\\ref{em}), and demanding that $i\\gamma^{\\mu}\\partial_{\\mu}\\psi_{L_{n}}=m_{n}\\psi_{R_{n}}$\nand $i\\gamma^{\\mu}\\partial_{\\mu}\\psi_{R_{n}}=m_{n}\\psi_{L_{n}}$, we obtain two equations\nfor $\\alpha_{L_{n}}$ and $\\alpha_{R_{n}}$\n\\begin{equation}\\label{ea1}\n    \\left[ \\partial_{y}+2\\partial_{y}A+\\eta F(\\phi) \\right]\\alpha_{L_{n}}=m_{n}\\mathrm{e}^{-A}\\alpha_{R_{n}}\\,,\n\\end{equation}\n\\begin{equation}\\label{ea2}\n    \\left[ \\partial_{y}+2\\partial_{y}A-\\eta F(\\phi) \\right]\\alpha_{R_{n}}=-m_{n}\\mathrm{e}^{-A}\\alpha_{L_{n}}\\,.\n\\end{equation}\n\\noindent Inserting the general chiral decomposition (\\ref{dchiral}) into the action (\\ref{ad}), using (\\ref{ea1}) and (\\ref{ea2}) and also requiring that the result take the form of the standard four-dimensional action for the massive chiral fermions\n\\begin{equation}\\label{ad2}\n    S=\\sum_{n}\\int d^{4}x\\, \\bar{\\psi}_{n}\\left( \\gamma^{\\mu}\\partial_{\\mu}-m_{n} \\right)\\psi_{n},\n\\end{equation}\n\\noindent where $\\psi_{n}=\\psi_{L_{n}}+\\psi_{R_{n}}$ and $m_{n}\\ge0$, the functions $\\alpha_{L_{n}}$ and $\\alpha_{R_{n}}$ must obey the following orthonormality conditions\n\\begin{equation}\\label{orto}\n    \\int_{-\\infty}^{\\infty}dy\\,\\mathrm{e}^{3A}\\alpha_{Lm}\\alpha_{Rn}=\\delta_{LR}\\delta_{mn}.\n\\end{equation}\n\n\\noindent Implementing the change of variables\n\\begin{equation}\\label{cv}\n    z=\\int^{y}_{0}\\mathrm{e}^{-A(y^{\\,\\prime})}dy^{\\,\\prime},\n\\end{equation}\n\n\\noindent $\\alpha_{L_{n}}=\\mathrm{e}^{-2A}L_{n}$ and $\\alpha_{R_{n}}=\\mathrm{e}^{-2A}R_{n}$, we get\n\\begin{equation}\\label{sleft}\n    -L_{n}^{\\prime\\prime}(z)+V_{L}(z)L_{n}=m_{n}^{2}L_{n}\\,,\n\\end{equation}\n\\begin{equation}\\label{sright}\n    -R_{n}^{\\prime\\prime}(z)+V_{L}(z)R_{n}=m_{n}^{2}R_{n}\\,,\n\\end{equation}\n\\noindent where\n\\begin{eqnarray}\n  V_{L}(z) &=& \\eta^{2}\\mathrm{e}^{2A}F^{2}(\\phi)-\\eta\\partial_{z}\\left( \\mathrm{e}^{A}F(\\phi) \\right),\\label{vefa} \\\\\n  V_{R}(z) &=& \\eta^{2}\\mathrm{e}^{2A}F^{2}(\\phi)+\\eta\\partial_{z}\\left( \\mathrm{e}^{A}F(\\phi) \\right)\\label{vefb}.\n\\end{eqnarray}\n\\noindent Using the expressions $\\partial_{z}A=\\mathrm{e}^{A(y)}\\partial_{y}A$ and $\\partial_{z}F=\\mathrm{e}^{A(y)}\\partial_{y}F$, we can recast the potentials (\\ref{vefa}) and (\\ref{vefb}) as a function of $y$ \\cite{yu}-\\cite{yo}\n\\begin{eqnarray}\n  V_{L}(z(y)) &=& \\eta\\mathrm{e}^{2A}\\left[ \\eta F^{2}-\\partial_{y}F-F\\partial_{y}A(y) \\right],\\label{vya} \\\\\n  V_{R}(z(y)) &=& V_{L}(z(y))|_{\\eta\\rightarrow-\\eta}\\,.\\label{vyb}\n\\end{eqnarray}\n\n\\noindent It is worthwhile to note that we can construct the Schr\\\"{o}dinger potentials\n$V_{L}$ and $V_{R}$ from eqs. (\\ref{vya}) and (\\ref{vyb}).\n\nAt this stage, it is instructive to state that with the change of variable (\\ref{cv})\nwe get a geometry to be conformally flat\n\\begin{equation}\\label{cm}\n    ds^{2}=\\mathrm{e}^{2A(z)}\\left( \\eta_{\\mu\\nu}dx^{\\mu}dx^{\\nu} +dz^{2}\\right).\n\\end{equation}\n\\noindent Now we focus attention on the condition (\\ref{cj0}) for the line element (\\ref{cm}). In this case we obtain\n\\begin{equation}\\label{mc}\n    \\nabla_{M}\\Gamma^{M}= i(\\partial_{z}A(z))\\mathrm{e}^{-A(z)}\\gamma^{5}.\n\\end{equation}\n\n\\noindent Therefore, the current is no longer conserved for the line element (\\ref{cm}) \\cite{arro}. It is known that, in general, the reformulation of the theory in a new conformal frame leads to a different, physically inequivalent theory. This issue has already a precedent in cosmological models \\cite{val}.\n\n\\noindent Under this arguments, we only use the change of variable (\\ref{cv}) to have a qualitative analysis of the potential profiles (\\ref{vya}) and (\\ref{vyb}), which is a fundamental ingredient for the fermion localization on the brane.\n\nNow we focus attention on the calculation of the zero mode. Substituting $m_{n}=0$ in (\\ref{ea1}) and (\\ref{ea2}) and using $\\alpha_{L_{n}}=\\mathrm{e}^{-2A}L_{n}$ and $\\alpha_{R_{n}}=\\mathrm{e}^{-2A}R_{n}$, respectively, we get\n\\begin{equation}\\label{mzL}\n    L_{0}\\propto \\exp \\left[-\\eta\\int_{0}^{y}dy^{\\prime}F(\\phi) \\right],\n\\end{equation}\n\\begin{equation}\\label{mzR}\n    R_{0}\\propto \\exp \\left[\\eta\\int_{0}^{y}dy^{\\prime}F(\\phi) \\right].\n\\end{equation}\n\\noindent This fact is the same to the case of two-dimensional Dirac equation \\cite{luis}. At this point is worthwhile to mention that the normalization of the zero mode and the existence of a minimum for the effective potential at the localization on the brane are essential conditions for the problem of fermion localization on the brane. This fact was already reported in \\cite{yo}.\n\nIn order to guarantee the normalization condition (\\ref{orto}) for the left-handed fermion zero mode (\\ref{mzL}), the integral must be convergent, \\textit{i.e}\n\\begin{equation}\\label{cono}\n    \\int^{\\infty}_{-\\infty}dy\\exp\\left[ -A(y)-2\\eta\\int^{y}_{0}dy\\,^{\\prime}F(\\phi(y\\,^{\\prime})) \\right]<\\infty.\n\\end{equation}\n\\noindent This result clearly shows that the normalization of the zero mode is decided by the asymptotic behavior of $F(\\phi(y))$. Furthermore, from (\\ref{vya}) and (\\ref{vyb}), it can be observed that the effective potential profile depends on the $F(\\phi(y))$ choice. This fact implies that the existence of a minimum for the effective potential $V_{L}(z(y))$ or $V_{R}(z(y))$ at the localization on the brane is decided by $F(\\phi(y))$. This point will be more clear when it is considered a specific Yukawa coupling. Therefore, the behavior of $F(\\phi(y))$ plays a leading role for the fermion localization on the brane \\cite{yo}. Having set up the two essential conditions for the problem of fermion localization on the brane, we are now in a position to choice some specific forms for Yukawa couplings.\n\n\\subsection{Zero mode and fermion localization}\n\nFrom now on, we mainly consider the simplest case $F(\\phi)=\\phi$. First, we consider the\nnormalizable problem of the solution. In this case, we only need to consider the asymptotic behavior of the integrand in (\\ref{cono}). It becomes\n\\begin{equation}\\label{in}\n    I\\rightarrow\\mathrm{exp}\\left[ -A(\\pm\\infty)-2\\eta\\int^{y}_{0}\\phi(y^{\\prime})dy^{\\prime} \\right]\\,.\n\\end{equation}\n\n\\noindent By the analysis from eq. (\\ref{em2b}), we obtain that $A(\\pm\\infty)\\rightarrow-\\sqrt{|V(\\pm v)|\/3}\\,|\\,y|$. For the integral $\\int dy\\phi$, we only need to consider the asymptotic behavior of $\\phi$ for $y\\rightarrow\\pm\\infty$ \\cite{liu} and as $\\phi(\\pm\\infty)=\\pm v$ the equation (\\ref{in}) becomes\n\\begin{equation}\\label{int}\n    I\\rightarrow\\mathrm{exp}\\left[ -2\\left( \\eta\\,v-\\sqrt{|V(\\pm v)|\/12} \\right)|\\,y| \\right]\\,.\n\\end{equation}\n\n\\noindent This result clearly shows that the zero mode of the left-handed fermions is normalized only for $\\eta>\\frac{1}{v}\\,\\sqrt{\\frac{|V(\\pm v)|}{12}}$. Now, under the change $\\eta\\rightarrow-\\eta$ ($L_{0}\\rightarrow R_{0}$) we obtain that the right-handed fermions can not be a normalizable zero mode. The shape of the potentials $V_{L}$ and $V_{R}$ are shown in Fig. (\\ref{pe}) for some values of $a$. Figure \\ref{pe}(a) shows that the effective potential $V_{L}$, is indeed a volcano-like potential for $a=0$. As $a$ increases the well structure of $V_{L}$ gets a double well. Figure \\ref{pe}(b) shows that the potential $V_{R}$ has also a well structure, but the minimum of $V_{R}$ is always positive, therefore the potential does not support a zero mode. The shapes of the matter energy density, $V_{L}$ potential and $|\\,L_{0}|^{2}$ are shown in Fig. \\ref{a}. The Fig. \\ref{a}(a) ($a=0$) shows that the zero mode is localized on the brane. On the other hand, Fig. \\ref{a}(b) ($a=0.836$) clearly shows that the normalizable zero mode is localized between the two sub-branes, as a consequence the zero mode is not localized on the brane. Therefore, we can conclude that the zero mode of the left-handed fermions is localized on the brane only as $0\\leq a \\approx a_{*}$.\n\n\\begin{figure}[ht]\n       \\begin{minipage}[b]{0.40 \\linewidth}\n           \\fbox{\\includegraphics[width=\\linewidth]{vl.eps}}\\\\\n          \\end{minipage}\\hfill\n       \\begin{minipage}[b]{0.40 \\linewidth}\n           \\fbox{\\includegraphics[width=\\linewidth]{vr.eps}}\\\\\n           \\end{minipage}\n       \\caption{Potential profile: (a) $(V_{L}(y))_{A}$ (left) and (b) $(V_{R}(y))_{A}$ (right) for $\\eta=1$, $b=2$,  $c=1$, $a=0$ (thin line), $a=0.8$ (dashed line) and $a=0.836$ (dotted line) }\\label{pe}\n   \\end{figure}\n\n\n\\begin{figure}[ht]\n       \\begin{minipage}[b]{0.40 \\linewidth}\n           \\fbox{\\includegraphics[width=\\linewidth]{mzervla0_3.eps}}\\\\\n          \\end{minipage}\\hfill\n       \\begin{minipage}[b]{0.40 \\linewidth}\n           \\fbox{\\includegraphics[width=\\linewidth]{mzervla0836_3.eps}}\\\\\n           \\end{minipage}\n       \\caption{The profiles of the Ricci scalar (thin line), $V_{L}(y)$ (thick line) and $|\\,L_{0}|^{2}$ (dashed line) for $\\eta=1$, $b=2$, $c=1$; (a) $a=0$ (left) and (b) $a=0.836$ (right).}\\label{a}\n   \\end{figure}\n\n\\section{Conclusions}\n\nWe have reinvestigated the effects of the variation of the mass parameter $a$ on the thick branes as well as the localization of fermions. We showed that the variation of $a$ is associated to the phenomenon of brane splitting, therefore the brane does not get thicker with the increases of the value of $a$, as argued in Ref. \\cite{zhao}. We can conclude that the appearance of two sub-branes is associated to phase transition for $a=a_{*}$ (a disordered phase between two ordered phases). Also, we showed that the value of the matter energy depends on the asymptotic behavior of the warp factor. From the static equation of motion we have analyzed the asymptotic behavior of $A(y)$ and showed that the zero mode of the left-handed fermions for the simplest Yukawa coupling $\\eta\\bar{\\Psi}\\phi\\Psi$ is normalizable under the condition $\\eta>\\frac{1}{v}\\,\\sqrt{\\frac{|V(\\pm v)|}{12}}$ and it can be trapped on the brane only for $0\\leq a \\approx a_{*}$, because the zero mode has a single-peak at the localization of the brane. We also showed that as $a\\rightarrow a_{*}$ the zero mode has a single-peak between the two sub-branes and as a consequence the normalizable zero mode is not localized on the brane. Therefore, the brane with a big value of $a$ would not trap fermions more efficiently, in opposition to what was adverted in Ref. \\cite{zhao}. This work completes and revises the analyzing of the research in Ref. \\cite{zhao}, because in that work does not analyze the zero mode in full detail and contain some misconceptions.\n\nAdditionally, we showed that the change of variable $dz=\\mathrm{e}^{-A(y)}dy$ leads to a non conserved current, because the curved-space gamma matrices are not covariantly constant. An interesting issue will be investigate the effects of non-conserved current on resonances modes and bear out the main conclusion of Ref. \\cite{zhao}.\n\n\n\\begin{acknowledgments}\nThis work was supported by means of funds provided by CAPES.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{ Introduction}\n\nComplex contact manifolds are still has many open problems. The importance of this subject is not only the complex version of real contact manifolds also one can find many important informations about complex manifolds, K\\\"ahler manifolds. In addition there are some applications in theoretical physics  \\cite{kholodenko2013applications}. Complex contact manifolds has an old history same as real contact manifolds, researchers could not give their attention to the subject. When we look at the 1980s there are very important improvements in the Riemannian geometry of complex contact manifolds. Ishihara and Konishi constructed tensorial relations for a complex almost contact structure and they also presented normality \\cite{ishihara1979real,ishihara1980complex,ishihara1982complex}. The Riemann geometry of complex almost contact metric manifolds could be divided into three notions;\n\\begin{enumerate}\n\t\\item \\textbf{\\textit{IK-normal complex contact metric manifolds}} : A complex contact manifold has a normal contact structure in the sense of Ishihara-Konishi. This type of manifolds were studied in \\cite{ishihara1979real,ishihara1980complex,ishihara1982complex,imada2014construction,imada2015complex,turgut2018h}.\n\t\\item \\textbf{\\textit{Normal complex contact metric manifolds}}: A complex contact manifold has a normal contact structure in the sense of Korkmaz.  This type of manifolds were studied in \\cite{korkmaz2000,Korkmaz2003,blair2006corrected,blair2009special,blair2011bochner,blair2012homogeneity,blair2012symmetry,vanli2015curvature,turgut2017conformal,vanli2017complex}.\n\t\\item \\textbf{\\textit{Complex Sasakian manifolds}}: A complex contact manifold with a globally complex contact form and has a normal contact structure in the sense of Korkmaz. This type of manifolds were studied in \\cite{foreman2000complex,fetcu2006harmonic,fetcuadapted}.\n\\end{enumerate}\nIn this work we study on third type of these manifolds. Firstly we adopted the definition of a complex Sasakian manifold by consider the definition of a real Sasakian manifold. Later we give some fundamental equations and we obtain curvature properties. Finally we examine some flatness conditions. We use a general tensor which is defined in \\cite{shaikh2018some} and is called by $ B- $tensor. We prove that a complex Sasakian manifold could not be $ B- $flat. \n\n\n\\section{Preliminaries}\nIn this section we give fundamental facts on complex contact manifolds. For details reader could be read \\cite{foreman1996variational,korkmaz2000,blair2010riemannian}.\n\\begin{definition}\n\tLet $N$ be a complex manifold of odd complex dimension $2p+1$ covered by an\nopen covering $\\mathcal{C}=\\left\\{ \\mathcal{A}_{i}\\right\\} $ consisting of\ncoordinate neighborhoods. If there is a holomorphic $1$-form $\\eta _{i}$\non each $\\mathcal{A}_{i}\\in \\mathcal{C}$ in such a way that for any \n\\mathcal{A}_{i},\\mathcal{A}_{j}\\in \\mathcal{C}$ and for a holomorphic function $f_{ij}$ on $\\mathcal{A}_{i}\\cap \\mathcal{A}_{j}\\neq \\emptyset $\n\\begin{equation*}\n\\eta _{i}\\wedge (d\\eta _{j})^{p}\\neq 0\\text{ in }\\mathcal{A}_{i}\\text{, }\n\\end{equation*\n\\begin{equation*}\n\\eta _{i}=f_{ij}\\eta _{j},\\text{ }\\mathcal{A}_{i}\\cap \\mathcal{A\n_{j}\\neq \\emptyset ,\n\\end{equation*\nthen the set $\\left\\{ \\left( \\eta _{i},\\mathcal{A\n_{i}\\right) \\mid \\mathcal{A}_{i}\\in \\mathcal{C}\\right\\} $ of local structures is called complex contact structure and with this structure $N$ is called a complex contact manifold.\n\\end{definition}\n\n\nThe complex contact structure determines a non-integrable distribution $H_i$ by the equation $\\eta_i =0$ such as \n\\begin{equation*}\nH_{i}=\\{X_{P}:\\eta_{i}(X_{P})=0, X_{P}\\in T_{P}N\\}.\n\\end{equation*}\nand a holomorphic vector field $ \\xi_i $\nis defined by  \n\n\\begin{equation*}\n\\eta_i(\\xi_i)=1\n\\end{equation*}\nand a complex line bundle is defined by $ E_i=Span\\{\\xi_i\\} $.\\\\\n\\qquad Let $ T^c(N) $ be complexified of tangent bundle of $ (N,J,\\eta_i) $ and let define vector fields \n\\begin{eqnarray*}\n\tU_i=\\xi_i+\\bar{\\xi_i}\\ \\ \\ \\ \\ \\ \\ V_i=-i(\\xi_i+\\bar{\\xi_i})\n\\end{eqnarray*}\nand 1-forms\n\\begin{eqnarray*}\n\tu_i=\\frac{1}{2}(\\eta_i+\\bar{\\eta_i}) \\ \\ \\ \\ \\ \\ v_i=\\frac{1}{2}i(\\eta_i-\\bar{\\eta_i}).\n\\end{eqnarray*}\nTherefore we get \n\\begin{enumerate}\n\t\\item $ V_i=-JU_i $ and $ v_i=u_i\\circ J $\n\t\\item $ U_i=JV_i $ and $ u_i=-v_i\\circ J $\n\t\\item $ u_i(U_i)=v_i(V_i)=1 $ and $ u_i(V_i)=v_i(V_i)=0 $.\n\\end{enumerate}\nThe complexified $ H_i $ and $ E_i $ is defined by \n\\begin{eqnarray*}\n{\tH_i}^c&=&\\{W\\in T^c(N)\\arrowvert u(W)=v(W)=0\\}\\\\\n{\tE_i}^c&=&Span\\{U,V\\}. \n\\end{eqnarray*}\nWe use notation $ \\mathcal{H} $ and $ \\mathcal{V} $ for the union of $ {\tH_i}^c $ and $ E_i^c $ respectively. $ \\mathcal{H} $ is called horizontal distribution and $ \\mathcal{V} $ is called vertical distribution \nand we can write \n\\begin{equation} \\label{TN=H+V}\nTN=\\mathcal{H}\\oplus\\mathcal{V}.\n\\end{equation}\n Ishihara and Konishi \\cite{ishihara1980complex} proved that $N$ admits always a complex contact structure of $ C^{\\infty} $.\n\\begin{definition} \\label{complexalmostscontact}\n\tLet $N$ be a complex manifold with complex structure $ J $, Hermitian metric $ g $ and $\\mathcal{C=}\\left\\{ \n\t\\mathcal{A}_{i}\\right\\} $ be open covering of $N$ with coordinate\n\tneighbourhoods $\\{\\mathcal{A}_{i}\\mathcal{\\}}.$ If $N$ satisfies the\n\tfollowing two conditions then it is called a \\textit{complex almost contact\n\t\tmetric manifold}:\n\t\n\t1. In each $\\mathcal{A}_{i}$ there exist $1$-forms $u_{i}$ and \n\tv_{i}=u_{i}\\circ J$, with dual vector fields $U_{i}$ and $V_{i}=-JU_{i}$ and \n\t$(1,1)$ tensor fields $G_{i}$ and $H_{i}=G_{i}J$ such that \n\t\\begin{equation} \\label{G^2veH^2}\n\tH_{i}^{2}=G_{i}^{2}=-I+u_{i}\\otimes U_{i}+v_{i}\\otimes V_{i}\n\t\\end{equation\n\t\\begin{equation*} \\label{H=GJ}\n\tG_{i}J=-JG_{i},\\quad GU_{i}=0,\\quad\n\t\\end{equation*\n\t\\begin{equation*} \\label{g(GX,Y)=-g(X,GY)}\n\tg(X,G_{i}Y)=-g(G_{i}X,Y).\n\t\\end{equation*\n\t2.On $\\mathcal{A}_{i}\\cap \\mathcal{A}_{j}\\neq \\emptyset $ we have \n\t\\begin{eqnarray*}\n\t\tu_{j} &=&au_{i}-bv_{i},\\quad v_{j}=bu_{i}+av_{i},\\; \\\\\n\t\tG_{j} &=&aG_{i}-bH_{i},\\quad H_{j}=bG_{i}+aH_{i}\n\t\\end{eqnarray*\n\twhere $a$ and $b$ are functions on $\\mathcal{U}_{i}\\cap \\mathcal{U}_{j}$\n\twith $a^{2}+b^{2}=1$ \\cite{ishihara1979real,ishihara1980complex}.\n\\end{definition}\nBy direct computation we have \n\\begin{eqnarray} \nH_{i}G_{i} &=&-G_{i}H_{i}=J_{i}+u_{i}\\otimes V_{i}-v_{i} \\otimes U_{i} \\label{HG=-GH} \\label{HG,GH}\\\\\nJ_{i}H_{i} &=&-H_{i}J_{i}=G_{i}  \\notag \\\\\nG_{i}U_{i} &=&H_{i}U_{i}=H_{i}V_{i}=0  \\notag \\\\\nu_{i}G_{i} &=&v_{i}G_{i}=u_{i}H_{i}=v_{i}H_{i}=0 \\notag \\\\\nJ_{i}V_{i} &=&U_{i}, \\ g(U_{i},V_{i})=0 \\notag\\\\\ng(H_{i}X,Y) &=&-g(X,H_{i}Y) \\notag.\n\\end{eqnarray}\n\n\nBy the local contact form $\\eta $ is $u-iv$ to within a\nnon-vanishing complex-valued function multiple and the local fields $G$ and \n\\ H$ are related to $du$ and $dv$ by \n\\begin{eqnarray*}\n\tdu(X,Y) &=&g(X,GY)+(\\sigma \\wedge v)(X,Y),~~~ \\\\\n\tdv(X,Y) &=&g(X,HY)-(\\sigma \\wedge u)(X,Y)\n\\end{eqnarray*}\nwhere $\\sigma (X)=g(\\nabla _{X}U,V)$, $\\nabla $ being the Levi-Civita\nconnection of $g$ \\cite{ishihara1980complex}. $ \\sigma $ is called IK-connection \\cite{foreman1996variational}.\n\nIshihara and Konishi \\cite{ishihara1980complex} study on normality of complex   almost contact metric manifolds. They defined local tensors\n\\begin{eqnarray*}\n\tS(X,Y) &=&[G,G](X,Y)+2g(X,GY)U-2g(X,HY)V \\\\\n\t&&+2(v(Y)HX-v(X)HY)+\\sigma (GY)HX \\\\\n\t&&-\\sigma (GX)HY+\\sigma (X)GHY-\\sigma (Y)GHX,\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\tT(X,Y) &=&[H,H](X,Y)-2g(X,GY)U+2g(X,HY)V \\\\\n\t&&+2(u(Y)GX-u(X)GY)+\\sigma (HX)GY \\\\\n\t&&-\\sigma (HY)GX+\\sigma (X)GHY-\\sigma (Y)GHX\n\\end{eqnarray*}\nwhere\n\\begin{equation*}\n\\lbrack G,G](X,Y)=(\\nabla _{GX}G)Y-(\\nabla _{GY}G)X-G(\\nabla\n_{X}G)Y+G(\\nabla _{Y}G)X\n\\end{equation*\nis the Nijenhuis torsion of $G$. \n\\begin{definition}\n\\cite{ishihara1980complex} A complex{\\color{white}$ \\eta $}almost{\\color{white}$ \\eta $}contact{\\color{white}$ \\eta $}metric{\\color{white}$ \\eta $}manifold is called IK-Normal{\\color{white}$ \\eta $} if $ S=T=0 $. \n\\end{definition}\nAn IK-Normal manifold has K\\\"ahler structure. In other words a non-K\\\"ahler complex almost contact metric manifold is not to be IK-Normal such Iwasawa manifold. Iwasawa manifold is not K\\\"ahler and  it is compact manifold which has symplectic structure. Also Baikousis et al. \\cite{baikoussis1998holomorphic} obtained complex almost contact structure on Iwasawa manifold. Korkmaz gave a weaker definition for normality and Iwasawa manifold is normal in the sense of this definition. \n\\begin{definition} [Korkmaz's Definition,  \\cite{korkmaz2000} ]\n\tA complex almost contact metric manifold is called normal if\n\t\n\t$\\qquad S(X,Y)=T(X,Y)=0$ \\ for all $X,Y$ in $\\mathcal{H},$ and $\\ $\n\t\n\t$\\qquad S\\left( X,U\\right) =T\\left( X,V\\right) =0$ for all $X.$\n\\end{definition} \n\nAlso for arbitrary vector fields $ X$ on $N$ we have \\cite{ishihara1980complex,foreman1996variational,korkmaz2000} \n\\begin{align}\n\\nabla _{X}U &=-GX+\\sigma (X)V,~~  \\label{nablaXU} ,\n\\ \\ \\ \\nabla _{X}V =-HX-\\sigma (X)U,~~  \\\\\n\\nabla _{U}U &=\\sigma (U)V,~~~\\nabla _{U}V=-\\sigma (U)U  \\label{nablaUU} , \\ \\ \n\\nabla _{V}U =\\sigma (V)V,~~~~\\nabla _{V}V=-\\sigma (V)U.~~  \n\\end{align}\n\n\\begin{theorem}\n\tA complex almost\n\tcontact metric manifold is normal if and only\n\tif the covariant derivative of \\ $G$ and $H$ have the following\n\tforms: \n\t\\begin{eqnarray}\n\t(\\nabla _{X}G)Y &=&\\sigma (X)HY-2v(X)JY-u\\left( Y\\right) X\n\t\\label{Yeni normallik G} \\\\\n\t&&-v(Y)JX+v(X)\\left( 2JY_{0}-\\left( \\nabla _{U}J\\right) GY_{0}\\right)  \\notag\n\t\\\\\n\t&&+g(X,Y)U+g(JX,Y)V  \\notag \\\\\n\t&&-d\\sigma (U,V)v(X)\\left( u(Y)V-v(Y)U\\right)  \\notag\n\t\\end{eqnarray}\n\tan\n\t\\begin{eqnarray}\n\t(\\nabla _{X}H)Y &=&-\\sigma (X)GY+2u(X)JY+u(Y)JX  \\label{Yeni normallik H} \\\\\n\t&&-v(Y)X+u(X)\\left( -2JY_{0}-\\left( \\nabla _{U}J\\right) GY_{0}\\right)  \\notag\n\t\\\\\n\t&&-g(JX,Y)U+g(X,Y)V  \\notag \\\\\n\t&&+d\\sigma (U,V)u(X)\\left( u(Y)V-v(Y)U\\right)  \\notag\n\t\\end{eqnarray}\n\twhere $X=X_{0}+u(X)U+v(X)V$ and $Y=Y_{0}+u(Y)U+v(Y)V,X_{0},Y_{0}\\in $ \n\t\\mathcal{H}$ \\cite{vanli2015curvature}.\n\\end{theorem}\nFrom this theorem on a normal complex contact metric manifold we have\n\n\\begin{eqnarray*}\n\t(\\nabla _{X}J)Y &=&-2u\\left( X\\right) HY+2v(X)GY+u(X)\\left( 2HY_{0}+\\left(\n\t\\nabla _{U}J\\right) Y_{0}\\right) \\\\\n\t&&+v(X)\\left( -2GY_{0}+\\left( \\nabla _{U}J\\right) JY_{0}\\right) .\n\\end{eqnarray*}\n\nAs we have seen, there are two normality notions for a \\NCM. The other kind of \\NCM s is complex Sasakian manifold. This type of manifolds are normal due to Korkmaz's definition  and they were studied in \\cite{foreman2000complex,fetcu2006harmonic,fetcuadapted}. The fundamental difference of this type from others is globally definition of complex contact form. Kobayashi \\cite{kobayashi1959remarks} proved that a complex contact form is globally defined if and only if first Chern class of $ N $ vanishes.  Also Foreman obtained following result; \n\\begin{lemma}\n\tLet $ (N,\\eta_i) $ be a complex contact manifold. If $ \\eta_i $ is globally defined i.e $ f_{ij}=1 $ then $ \\sigma=0 $ \\cite{foreman2000complex}. \n\\end{lemma}\n A complex Sasakin manifold is defined as follow; \n\\begin{definition}\n\tLet $\\left(N,G,H,J,U,V,u,v,g\\right) $ be a normal complex contact metric\n\tmanifold and $\\eta =u-iv$ is globally defined. If fundemental 2- forms \n\t\\widetilde{G}$ and $\\widetilde{H}$ is defined by \n\t\\begin{equation*}\n\t\\widetilde{G}\\left( X,Y\\right) =du(X,Y)\\text{ and \\ }\\widetilde{H}\\left(\n\tX,Y\\right) =dv\\left( X,Y\\right)\n\t\\end{equation*\n\tthen $ N $ is called a complex Sasakian manifold, where $X,Y$ are vector fields on \n\t$N$.\n\\end{definition}\nThus we get following result;\n\\begin{theorem}\n\tLet $ N $ be a \\NCM. Then $N$ is complex Sasakian if and only\n\tif\\bigskip \n\t\\begin{eqnarray*}\n\t(\\nabla _{X}G)Y&=&-2v(X)HGY-u(Y)X-v(Y)JX+g(X,Y)U+g(JX,Y)V \\\\\n\t(\\nabla _{X}H)Y &=&-2u(X)HGY+u(Y)JX-v(Y)X-g(JX,Y)U+g(X,Y)V.\n\t\\end{eqnarray*}\n\\end{theorem}\nThis result was also given by  Ishihara-Konishi\\cite{ishihara1980complex}. But should have been considered that $ N $ is not K\\\"ahler in here. So,we have \n\\begin{eqnarray*}\n\t(\\nabla _{X}J)Y &=&-2u(X)HY+2v(X)GY.\n\\end{eqnarray*}\nFrom (\\ref{nablaXU} )  on a complex Sasakin manifold we get \n\\begin{eqnarray}\n\\nabla _{X}U =-GX,~~ \\ \\ \\ \\nabla _{X}V =-HX. \\label{NablaX-U}\n\\end{eqnarray}\nOn the other hand we have \n\\begin{corollary}\n\tAn IK-normal complex contact metric manifold could not be complex Sasakian \\cite{turgut2018h}. \n\\end{corollary} \n This result support that the geometry of complex Sasakian manifolds has some different properties. \n\\section{Curvature Properties of Complex Sasakian Manifolds}\nIn the Riemannian geometry of contact manifolds curvature properties have an important position. We use these relations for future works. The curvature relations of a complex almost contact metric manifolds were given in \\cite{foreman1996variational}, an IK-Normal manifold were given in \\cite{ishihara1980complex} and a normal complex contact metric manifold were given in \\cite{korkmaz2000,vanli2015curvature}. In this section by taking advantage from these curvature properties, we present curvature relations for complex Sasakian manifold.\nLet $ N $ be a complex Sasakian manifold. Then for $ X,Y \\in \\Gamma(TN) $ we have, \n\\begin{align}\nR\\left( U,V\\right) V&=R\\left( V,U\\right) U=0 \\label{SasakianR(UVV)} \\\\\nR(X,U)U&=X+u(X)U+v(X)V\t\\label{SasakianR(XUU)} \\\\\nR(X,V)V&=X-u(X)U-v(X)V\t\\label{SasakianR(XVV)} \\\\\nR(X,U)V&=-3JX-3u(X)V+3v(X)U \\label{SasakianR(XUV)}\\\\\nR(X,V)U&=0 \\label{SasakianR(XVU)}\\\\\nR(X,Y)U&=v(X)JY-v(Y)JX+2v(X)u(Y)V-2v(Y)u(X)V  \\label{SasakianR(XYU)}\\\\\n&+u(Y)X-u(X)Y-2g(JX,Y)V \\notag \\\\\nR(X,Y)V&=3u(X)JY-3u(Y)JX-2u(X)v(Y)U+2u(Y)v(X)U \\label{SasakianR(XYV)} \\\\\n&+v(Y)X-v(X)Y+2g(JX,Y)U \\notag \\\\\nR(U,V)X&=JX+u(X)V-v(X)U\\label{SasakianR(UVX)} \\\\\tR(X,U)Y&=-2v(Y)v(X)U+2u(Y)v(X)V-g(Y,X)U \\label{SasakianR(XUY)}\\\\\n&+u(Y)X+g(JY,X)V  \\notag \\\\\nR(X,V)Y&=3u(Y)JX+2u(Y)u(X)V+3g(JY,X)U\\label{SasakianR(XVY)} \\\\&-2v(Y)u(X)U-g(Y,X)V+v(Y)X-2u(X)JY \\notag .\n\\end{align}\n For $X,Y,Z,W$ horizontal vector fields we have \\cite{vanli2015curvature} \n\\begin{equation}\\label{r(gx,gy,gz,gw)}\ng(R(GX,GY)GZ,GW)=g(R(HX,HY)HZ,HW)=g(R(X,Y)Z,W)\n\\end{equation}\nand we have \n\t\\begin{align*}\ng(R(X,GX)GX,X)&+g(R(X,HX)HX,X)+g(R(X,JX)JX,X)=-6g\\left( X,X\\right)\\\\\ng(R(X,GX)Y,GY) &=g(R(X,Y)X,Y)+g(R(X,GY)X,GY)-2g(GX,Y)^{2} \\\\\t\n&-4g(HX,Y)^{2}-2g(X,Y)^{2} +2g(X,X)g(Y,Y)-4g(JX,Y)^{2}\\\\\ng(R(X,HX)Y,HY) &=g(R(X,Y)X,Y)+g(R(X,HY)X,HY)-2g(HX,Y)^{2} \\\\\n&-4g(GX,Y)^{2}-2g(X,Y)^{2}+2g(X,X)g(Y,Y)-4g(JX,Y)^{2} \\\\\ng(R(X,HX)JX,GX)&=-g(R(X,HX)HX,X)-4g(X,X)^{2}\\\\\ng(R(X,JX)HX,GX)&=g(R(X,JX)JX,X)-2g(X,X)^{2}.\n\\end{align*}\n\\begin{align*}\ng(R(GX,HX)HX,GX)&=g(R(X,JX)JX,X) \\\\\ng(R(GX,JX)JX,GX)&=g(R(X,HX)HX,X)\\\\\ng(R(JX,JY)JY,JX)&=g(R(X,Y)Y,X)\\\\\ng(R(X,Y)JX,JY)&=g(R(X,Y)Y,X)+4g(X,GY)^{2}+4g(X,HY)^{2}\\\\\ng(R(Y,JX)JX,Y)&=g(R(X,JY)JY,X) \\\\\ng(R(X,JY)JX,Y)&=g(R(X,JY)JY,X)+4g(X,HY)^{2}+4g(X,GY)^{2} \\\\\ng(R(X,JX)JY,Y)&=-g(R(JX,JY)X,Y)-g(R(JY,X)JX,Y) \\\\\ng(R(X,JX)JY,Y)&=g(R(X,Y)Y,X)+g(R(X,JY)JY,X)\\\\\n&+8\\left(g(X,GY)^{2}+g(X,HY)^{2}\\right). \n\\end{align*}\nWe have an nice relation as follow;\n\\begin{corollary}\n\tFor a unit horizontal vector $ X $ on $N$ we have\n\t\\begin{equation}\n\tk\\left( X,GX\\right) +k\\left( X,HX\\right) +k\\left( X,JX\\right) =6. \\label{sectionalrelation}\n\t\\end{equation}\n\\end{corollary}\nThis relation also valid for an IK-normal complex contact metric manifold \\cite{imada2014construction}. \n\\begin{corollary}\n\tLet $N$ be a complex Sasakian manifold and $ X $ be a unit  horizontal vector field on $N$. Then for the sectional curvature $ k $ we have \n\t\\begin{equation*}\n\tk(U,V)=0\\, \\ \\ \\text{and} \\ \\ \\ k(X,U)=1.\n\t\\end{equation*}\n\\end{corollary}\nTurgut Vanl\\i\\  and Unal \\cite{vanli2015curvature} presented properties of Ricci curvature tensor\nof a normal{\\color{white}$ \\theta $}complex{\\color{white}$ \\theta $}contact{\\color{white}$ \\theta $} metric manifold. For complex Sasakian case \nwe have following relations; \n \n\t\\begin{eqnarray} \\label{Ricciler}\n\t\t\\rho (U,U) &=&\\rho (V,V)=4p,\\text{ \\ }\\rho (U,V)=0\\\\\n\t\t\\rho (X,U) &=&4pu(X)  \\notag,\\ \n\t\t\\rho \\left( X,V\\right) =4pv(X)  \\notag\\\\\n\t\t\\rho (X,Y) &=&\\rho (GX,GY)+4p\\left( u(X)u(Y)+v(X)v(Y)\\right) \n\t\t\\label{Ricci(X,Y)=Ricci(GX,GY)+} \\notag\\\\\n\t\t\\rho (X,Y) &=&\\rho (HX,HY)+4p\\left( u(X)u(Y)+v(X)v(Y)\\right)  \\notag\n\t\\end{eqnarray}\nwhere $ X,Y \\in \\Gamma(TN) $.\\par \nThe sectional curvature of Riemann manifold give us important information about the geometry of the manifold. In complex manifold we have holomorphic sectional curvature which is the curvature of section is spanned by $ X $  and $ JX $. Similarly in contact manifold we have $ \\phi- $sectional curvature which is the curvature of section is spanned by $ X $  and $ \\phi X $ \\cite{blair2010riemannian}. For complex contact case we have sectional curvature, holomorphic sectional curvature and $ \\mathcal{GH}- $sectional curvature which was given by Korkmaz \\cite{korkmaz2000} as below: \n\\begin{definition}\n\t\\cite{korkmaz2000} Let $ N $ be a \\NCM. $X$ be an unit horizontal vector field  on $N$ and $a^{2}+b^{2}=1$.  A $\\mathcal{GH-}$section is a plane which is spanned by $X$ and $Y=aGX+bHX$ and the sectional curvature of this plane is called $\\mathcal{GH-}$\\textit{sectional curvature } is defined by \n\t\\mathcal{GH}_{a,b}\\left( X\\right) =k\\left( X,aGX+bHX\\right) ,$ where \n\tk(X,Y)$\\ is the sectional curvature of the plane section spanned by $X$ and $Y$.\n\\end{definition}\n$\\mathcal{GH-}$\\textit{sectional curvature } is denoted by $ \\mathcal{GH}_{a,b}  $ and we assume that it does not depend the choice of $a$ and $b$. So we\nwill use $\\mathcal{GH}\\left( X\\right) $ notation. \\par\n\nLet $ N $ be a complex Sasakian manifold. Since the complex contact form is globally defined $ \\mathcal{GH}_{a,b}  $ does not depend the choice of $a$ and $b$ in naturally. Also we have \\cite{korkmaz2000}\n\\begin{equation}\nk(X,JX)=\\mathcal{GH}\\left( X\\right) +3.  \\label{k(X,JX)=GH(X)+3}\n\\end{equation}\nThus from (\\ref{sectionalrelation}) we obtain\n\t\\begin{equation}\nk\\left( X,GX\\right) +k\\left( X,HX\\right) +\\mathcal{GH}\\left( X\\right) =3. \n\\end{equation}\n\\begin{example}\n\tThe well known example of normal complex contact metric manifold complex Heisenberg group. A globally defined complex contact form and complex almost contact structure on complex Heisenberg group was given by Baikousis et al. \\cite{baikoussis1998holomorphic}. Korkmaz obtained normality of complex Heisenberg group. Thus a complex Heisenberg group is an example of complex Sasakian manifolds. For details about complex Heisenberg group see \\cite{blair2006corrected,foreman2000complex,korkmaz2000,vanli2015curvature}.   \n\\end{example}\n\\begin{example}\n\tAn other example of complex Sasakian manifolds were given by Foreman \\cite{foreman2000complex}. Foreman obtained an example from hyperk\\\"ahler manifold. For details see \\cite{foreman2000complex}. \n\\end{example}\n\\section{Flatness on complex Sasakian manifolds}\nA real Sasakian manifold can not be flat, i.e its Riemannian curavture could not be zero identically \\cite{de2009complex}. We obtain same results for complex Sasakin manifolds. \n\\begin{theorem}\n\tA complex Sasakian manifold can not be flat.\n\\end{theorem}\n\n\\begin{proof}\n\tSuppose that a complex Sasakian manifold is flat. Then $R(X,Y)Z=0$\n\tand from that $\\rho \\left( X,Y\\right) =0.$ On the other hand from (\\ref{Ricciler}) we have \n\t\\begin{equation*}\n\t4nu(X)=0.\n\t\\end{equation*}\n\tThis is not possible. So the manifold can not be flat.\n\\end{proof}\n Shaikh and Kundu \\cite{shaikh2018some} generalized curvature tensors and gave the definition of B-tensor. They proved some equivalence relations between most of certain curvature conditions. \n\\begin{definition}\n\t$N$ be a complex Sasakian manifold . $ (0, 4) $ tensor $ B $ of $N$ is given by\n\t\\begin{eqnarray} \\label{B-TENSOR}\n\tB(X,Y,Z,W) &=& a_{0}R(X,Y,Z,W) + a_{1}R(X,Z,Y,W)\\\\\n\t&+& a_{2}\\rho(Y,Z)g(X,W)+a_{3}\\rho(X,Z)g(Y,W)\\notag\\\\\n\t&+&a_{4}\\rho(X,Y)g(Z,W)+a_{5}\\rho(X,W)g(Y,Z)\\notag \\\\\n\t&+&a_{6}\\rho(Y,W)g(X,Z)+a_{7}\\rho(Z,W)g(X,Y) \\notag \\\\\n\t&+& \\tau\\{a_{8}g(X,W)g(Y,Z)+a_{9}g(X,Z)g(Y,W) \\notag\\\\\n\t&+&_{10}g(X,Y)g(Z,W)\\} \\notag\n\t\\end{eqnarray}\n\twhere $ a_{i} $'s are scalars on $  M $ and $ X,Y,Z,W \\in \\Gamma(TN) $.\n\\end{definition}\nFor different value of $ a_{i} $, $ B $ is became projective, conformal, concircular, quasi-conformal and conharmonic etc. curvature tensors (see \\cite{shaikh2018some}). Therefore flatness of B-tensor also determine flatness of these tensors. Let's define $N$ is $ B-$flat if $ B=0 $. Then we have following. \n\n\\begin{theorem}\n\tA complex Sasakian manifold can not be $ B-$flat.   \n\\end{theorem}\n\\begin{proof}\n\tAssume that $ B=0 $. Let chose $Y=Z=U$ and $ X=W=X\\in \\mathcal{H} $ then we have \n\t\\begin{eqnarray*}\n\t\t0 &=& a_{0}R(X,U,U,X) + a_{1}R(X,U,U,X) \\\\\n\t\t&+&a_{2}\\rho(U,U)g(X,X)+a_{5}\\rho(X,X)g(U,U)\\\\\n\t\t&+& \\tau\\{a_{8}g(X,X)g(U,U)\\}.\n\t\\end{eqnarray*}\n\tand from (\\ref{SasakianR(XUU)}) and (\\ref{Ricciler})  we get\n\\begin{align}\n\\rho(X,X)=-\\frac{a_0+a_1+4pa_2+\\tau a_8}{a_5} \\label{B-tens\u00f6rro(X,X)}.\n\\end{align}\n\tTherefore, since $ \\rho(X,X) $ is constant $ \\rho(X,X)=\\rho(Y,Y) $. \n\tOn the other hand for unit and orthogonal horizontal vector fields $ X,Y $ by choosing   $Y=Z=Y \\in \\mathcal{H}$ and $ X=W=X\\in \\mathcal{H} $ under $B=0 $ condition we have \n\t\\begin{equation*}\n\tR(X,Y,Y,X)=-\\frac{a_{2}\\rho(Y,Y)+a_{5}\\rho(X,X)+a_{8}\\tau}{a_{0}+a_{1}}\n\t\\end{equation*}\n\tand thus we get \n\t\t\\begin{align*}\n\tR(X,Y,Y,X)=\\frac{a_{2}a_{0}+a_{1}a_{2}+a_{0}a_{5}+a_{1}a_{5}\n\t\t+4pa_{2}a_{2}^2+4pa_{2}a_{5}+\\tau a_{8}a_{2}}{a_{5}(a_{0}+a_{1})}.\n\t\\end{align*}\n\tThis shows us the sectional curvature is independent of $ X $ and $ Y $ and then $ k(X,JX)=\\mathcal{GH}(X) $. But from (\\ref{k(X,JX)=GH(X)+3}) there is a contradiction. So $ N $ could not be $ B- $flat. \n\\end{proof}\nIn \\cite{turgut2017conformal} two of presented authors showed that a normal{\\color{white}$ \\theta $}complex{\\color{white}$ \\theta $}contact{\\color{white}$ \\theta $}manifold is not be conformal,concircular, quasi-conformal and conharmonic flat.\n The other notion of flatness is $ \\phi-T $-flatness for a $(1,1)-  $tensor $ T $. Now we examine this notion for a complex Sasakian manifold. \n\\begin{definition}\n\tLet $N$ be a complex Sasakian manifold. If \n\t\\begin{equation}\n\tG^{2}(\\mathcal{B}(GX,GY)GZ)=0\\text{ and }H^{2}(\\mathcal{B}(HX,HY)HZ)=0\n\t\\label{gh-conformal condition}\n\t\\end{equation\n\tfor $T,Y,Z$ vector fields on $N$ then $N$ is called a $\\mathcal{GH}-$ $ \\mathcal{B}-$flat.\n\\end{definition}\n\\begin{theorem}\n\tA complex Sasakian manifold can not be $\\mathcal{GH}-$ $ \\mathcal{B}-$flat.  \n\\end{theorem}\n\\begin{proof}\n\tLet $ N $ be a $ \\mathcal{B}-$flat complex Sasakian manifold. For  $ X,Y, Z,W\\in \\Gamma(\\mathcal{H}) $ we have \n\t\\begin{eqnarray*}\ng(B(GX,GY)GZ, GW)&&=B(GX,GY,GZ,GW)\\\\\n &&= a_{0}R(GX,GY,GZ,GW) + a_{1}R(GX,GY,GZ,GW)\\\\\n&&+ a_{2}\\rho(GY,GZ)g(GX,GW)+a_{3}\\rho(GX,GZ)g(GY,GW)\\notag\\\\\n&&+a_{4}\\rho(GX,GY)g(GZ,GW)+a_{5}\\rho(GX,GW)g(GY,GZ)\\notag \\\\\n&&+a_{6}\\rho(GY,GW)g(GX,GZ)+a_{7}\\rho(GZ,GW)g(GX,GY) \\notag \\\\\n&&+ \\tau\\{a_{8}g(GX,GW)g(GY,GZ)+a_{9}g(GX,GZ)g(GY,GW) \\notag\\\\\n&&+a_{10}g(GX,GY)g(GZ,GW)\\}. \\notag\n\\end{eqnarray*}\nFrom (\\ref{r(gx,gy,gz,gw)}) and (\\ref{Ricciler}) we get \n\\begin{eqnarray*}\ng(B(GX,GY)GZ, GW)=B(GX,GY,GZ,GW)=B(X,Y,Z,W).\n\\end{eqnarray*}\nAlso we know from Theorem 4, $ B\\neq 0 $, thus we have \n\\begin{equation*}\ng(B(GX,GY)GZ, GW)=-g(GB(GX,GY)GZ, W)\\neq 0.\n\\end{equation*}\nBecause of $ W \\in \\Gamma(T\\mathcal{H}) $ then $GB(GX,GY)GZ\\neq0$ and so $G^2B(GX,GY)GZ\\neq0$. \n\\end{proof}\nBy this theorems we obtain that on a complex Sasakian manifold Weyl conformal curvature tensor, projective curvature tensor, concircular curvature tensor, conharmonic curvature tensor, quasi conformal curvature\ntensor, pseudo projective curvature\ntensor, quasi-concircular curvature tensor, pseudo\nquasi conformal curvature tensor, M-projective curvature tensor, $ W_{i} $-curvature tensor, $ i = 1, 2, . . . ,9 $, $ W_{i}* $\n-curvature, $ T $ -curvature tensor (see \\cite{shaikh2018some} for details on tensors) can not vanish. In the other words a complex Sasakian manifold can not transform to flat space under any transformation such as conformal, projective, concircular etc.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nMore than fifty years have passed since the first detection of a cosmic ray with an energy $\\,\\raisebox{-0.13cm}{$\\stackrel{\\textstyle> 10^{20}${\\rm eV} \\cite{Linsley}, yet the nature of UHECRs and the identity of their sources remain a mystery.  \nThe nature of UHECRs sources depends on their composition, and we focus here on sources capable of producing UHE protons.\nDifficulties with the leading source candidates, AGNs and GRBs,  led Farrar and Gruzinov  \\cite{fg09}  \nto propose that  UHECRs, if protons, must be produced in a new class of powerful AGN transients, as could arise from the tidal disruption of a star or an extreme accretion disk instability around a supermassive black hole at  a galactic center.  \nFollowing the detection of a relativistic outflow from the  tidal disruption event (TDE) \\textit{Swift}\\ J1644+57 \\cite{Burrows+11,Bloom+11,Zauderer+11}\nand from  \n\\textit{Swift}\\ J2058+05 \\cite{Cenko+12} we confront here the viability of a scenario in which UHECRs are protons produced in the jets of tidal disruption events (TDEs).   We begin by recalling the requirements for UHECR acceleration.  Then, we use observations and modeling of \\textit{Swift}\\ J1644+57, \na likely example of a TDE jet seen in ``blazar-mode'' (i.e., looking down the axis of the jet), to test whether individual TDE jets satisfy the Hillas criterion necessary for accelerating protons to  $10^{20}${\\rm eV}.   Finally, using the recently measured TDE rate \\cite{vfRate14}, we examine whether  TDEs  can account for the observed UHECR energy injection rate and whether they provide a sufficiently large number of active sources to explain the lack of strong clustering in the arrival direction distribution \\cite{augerSrcDen13}.      \n\n\n\\section{Conditions on Sources of UHECRs}\nIn order for a CR to be confined during the acceleration process, its Larmor radius must remain smaller than the size, $R$, of the accelerating system.  This places a strict lower bound on $B R$ for UHECR acceleration known as the Hillas criterion, valid for any acceleration mechanism that involves magnetic fields \\cite{hillas84}:  \n\\begin{equation}\\label{conf}\nB R\\,\\raisebox{-0.13cm}{$\\stackrel{\\textstyle> 3\\times 10^{17} ~ \\Gamma ^{-1} \\, Z^{-1} \\, E_{20} ~ {\\rm Gauss \\,cm},\n\\end{equation}\nwhere $B$ is the magnetic field, $\\Gamma$ is the bulk Lorentz factor of the jet,\n $Z$ is the charge of the UHECR, and $E$ the CR's energy with $E_{20} \\equiv E\/ 10^{20} {\\rm eV}$.  Eq. (\\ref{conf}) implies a lower bound on the total Poynting luminosity required to accelerate protons to UHE, for which the total bolometric luminosity can be taken as a surrogate \\cite{fg09}:\n\\begin{equation}\\label{lumi}\n L_{\\rm bol}\n \\approx {1\\over 6}c\\, \\Gamma ^4 \\, (B R)^2 \\, \\,\\raisebox{-0.13cm}{$\\stackrel{\\textstyle> \\, 10^{45}\\, \\Gamma ^2\\, (E_{20}\/Z)^2 \\, {\\rm erg\/s}.\n\\end{equation}\nIt was shown in \\cite{fg09} that if the conditions Eqs. (\\ref{conf},\\ref{lumi}) are met in an AGN-like jet,  cooling and interaction with photons prior to escape from the accelerator are not the limiting factors in the maximum energy. The strong dependence on $Z$ of  Eqs. (\\ref{conf},\\ref{lumi}) indicates that the constraints on the UHECR sources are very different for protons or nuclei:  for protons the sources must be amongst the most luminous known EM sources, while for nuclei the requirements are much more modest. \n\nDenoting the energy injection rate of UHECRs in the range $10^{18}-10^{20}$ eV by ${\\dot E}_{_{UCR}} \\equiv {\\dot E}_{44} \\,\\, 10^{44} \\,{\\rm erg} \\, {\\rm Mpc}^{-3} \\, {\\rm yr}^{-1}$, for continuous sources the density of sources implied by Eq. (\\ref{lumi}) is \n\\begin{equation}  \\label{num}\nn_{\\rm src} \\approx 3 \\times 10^{-9}  \\frac{{\\dot E}_{44}}{\\epsilon_{_{UCR}} \\, \\Gamma ^2\\, (E_{20}\/Z)^2} ~ {\\rm Mpc}^{-3},\n\\end{equation}\nwhere $\\epsilon_{_{UCR}}\\equiv L_{_{UCR}}\/L_{\\rm bol}$ is the luminosity in UHECRs relative to the Poynting luminosity.   From the observed UHECR spectrum, \\cite{katz+GRB-UHECR09} estimates ${\\dot E}_{44} = 2.3$ to 4.5 for source spectra $\\sim E^{-2} \\, {\\rm to} \\, E^{-2.5}$, with O(1) uncertainty.  Thus one continuous source with  $\\epsilon_{_{UCR}} \\sim 1$ within the GZK distance ($\\approx 200$ Mpc) would be sufficient to produce the entire observed flux. \nHowever,  the lack of clustering \\cite{augerSrcDen13} in the arrival directions of UHECRs with energies above 70 EeV, implies a constraint on the density of sources whose stringency depends on the characteristic maximum deflections of the UHECRs: for $ 30^{\\circ} $ ($3^{\\circ}$), the source density must be greater than \n\\begin{equation} \\label{nlimsrc}\nn_{\\rm min} = 2 \\times 10^{-5} ~ ( 7 \\times 10^{-4}) ~ {\\rm Mpc}^{-3}. \n\\end{equation}\nThus efficient continuous protonic sources are incompatible with the source abundance requirement \n\\cite{fg09,WL09,MuraseTakami09}. Note that  if  deflections in the Galactic and intergalactic magnetic fields are so strong that UHECR arrival directions do not reflect the direction of their sources, the bound Eq. (\\ref{nlimsrc}) does not apply. However in this case, the observed correlation \\cite{TAaniso14,augerAniso14} with local structure would not be explained.\n\nEq. (\\ref{num}) can be reconciled with the observed minimum source density, Eq. (\\ref{nlimsrc}),  if UHECR production is very inefficient with $\\epsilon_{_{UCR}} \\ll 1$.   However,  inefficiency is not a solution, since even the weakest bound in Eq. (\\ref{nlimsrc}) requires $\\approx 700$ sources within the GZK distance, whereas powerful steady sources (of any kind) with luminosity larger than $10^{45}$ erg\/s are rare \\cite{fg09}.  Another way out is via an  acceleration mechanism   which does not involve  magnetic field confinement and thus evades the luminosity requirement Eq. (\\ref{lumi}). However an efficient mechanism of this kind is not known. \n\n\nTransient sources can evade the previous conundrum.  They must satisfy the Hillas confinement condition embodied in Eqs. (\\ref{conf}) and (\\ref{lumi}), and furthermore the energy injection condition sets a limit on the energy that must be released in UHECRs in a single transient event:\n\\begin{equation}\n\\label{EUCR}\n{\\cal E}_{_{UCR}} \\equiv {\\dot E}_{_{UCR}} \\, \/ \\, \\Gamma_{_{\\rm UCRtran}} =  10^{51}\\,{\\rm erg} ~ {\\dot{E}}_{44}  ~ \\Gamma_{_{\\rm UCRtran, -7}}^{-1},\n\\end{equation}  \nwhere $\\Gamma_{_{\\rm UCRtran}} \\equiv \\Gamma_{_{\\rm UCRtran, -7}} \\, 10^{-7}$ Mpc$^{-3}$ yr$^{-1}$ is the rate that the UHECR-producing transients take place.  In addition, the number of sources contributing at a given time must be large enough.  Deflections in the extragalactic magnetic field spread out the arrival times of UHECRs from an individual transient event. \nIn the approximation that the deflections are small and many, the resultant characteristic arrival time spread is \\cite{waxmanME}:\n\\begin{equation}\n\\label{tau}\n\\tau \\approx \\frac{ D^{2} Z^2 \\langle B^2 \\lambda \\rangle}{9 E^{2}} = \n3 \\times 10^{5}\\, {\\rm yr} \n\\left(\\frac{D_{100}  \\, B_{\\rm nG}}{E_{20}\/Z}\\right)^{2}   \\lambda_{\\rm Mpc}, \n\\end{equation}\nwhere $D_{100}$ is the distance of the source divided by 100 Mpc, $B_{\\rm nG}$ is the rms random extragalactic magnetic field strength in $nG$, and $\\lambda_{\\rm Mpc}$ is its coherence length in Mpc.   To be compatible with the Auger bound,\nthe number density $n_{\\rm eff}$ of contributing transient sources at a given time, must satisfy\n\\begin{equation}\n\\label{neff}\nn_{\\rm eff} \\approx \\tau_{5} ~ \\Gamma_{_{\\rm UCRtran, -7}} \\, 10^{-2}\\,{\\rm Mpc}^{-3} \\, \\geq n_{\\rm min} = 2 \\times 10^{-5} ~ {\\rm Mpc}^{-3},\n\\end{equation}\nwhere $ \\tau_{5} \\,10^{5}\\, {\\rm yr}$ is the mean time delay averaged over sources.\n \nIn order for any transient UHECR source type to be viable it must therefore satisfy three requirements, the Hillas condition (Eqs. (\\ref{conf}) or (\\ref{lumi})) and also Eqs. (\\ref{EUCR}) and (\\ref{neff}).  \nThe classical transient source candidate, GRBs,  \\cite{waxman95} satisfy easily the first and third conditions, but \nthere is debate whether their energy output is sufficient to satisfy the second condition, Eq. (\\ref{EUCR}), unless their UHECR energy output exceeds significantly their photon output \\cite{fg09,eichler+GRB-UHECR10,katz+GRB-UHECR09}.  This, and the lack of the expected high energy neutrinos \\cite{IceCubeNatureGRBs12}, makes  GRBs less favorable source candidates.\n\nA number of  TDE flare candidates have been detected and followed up in real-time: \\textit{Swift}\\  J1644+57  \\cite{Burrows+11,Bloom+11,Zauderer+11},  PS1-10jh \\cite{gezari+Nature12} and PS1-11af \\cite{chornock+14}.  Two candidates had been found earlier in archival SDSS data \\cite{vf11} and one was subsequently found in archival Swift data  \\textit{Swift}\\ J2058.4+0516 \\cite{cenkoSw2058}.  (Still earlier TDE candidates were put forward in \\cite{donley02,Gezari08}, but AGN-flare background could not be well characterized for those observations, so the origins of the flares were uncertain.) \n \n\\section{TDEs as UHECR sources}\n\nIn this section we address whether TDE jets are good source candidates for the protonic UHECR scenario.\nWe begin with the Hillas  condition, Eq. (\\ref{conf}), which must be satisfied for any UHECR source that is based on EM acceleration. Here the recent observations of  \\textit{Swift}\\ J1644+57 provide us with an example of a TDE jet with very good multi-wavelength follow-up, enabling the Hillas criterion (Eq. \\ref{conf}) to be directly checked.\n\n\\subsection{\\textit{Swift}\\  J1644+57 and the Hillas criterion}\n\n\\textit{Swift}\\ J1644+57 was detected on March 25th 2011 by the {\\it Swift} satellite.\nIts location  at the nucleus of an inactive galaxy made it immediately a strong TDE candidate.  It  is uniquely suitable for testing whether TDE jets can satisfy the Hillas condition, Eq. (\\ref{conf}), because of the thorough multi-wavelength monitoring from its inception for more than 600 days, which has enabled detailed modeling of the conditions in its jet.  \nThe observations  \\cite{Burrows+11,Bloom+11,Zauderer+11} revealed  two different\nemission sites:  an inner emission region where the X-rays are emitted, and an outer region\nwhere the radio emission is produced. \n\n Basic models for the TDE emission follow the ideas of a gamma-ray bursts, c.f.,  e.g. \n\\cite{piran04}:  the central engine of the TDE produces a pair of relativistic jets via the\nBlandford-Znajeck \\cite{bz77} process, with internal dissipation shocks within the jets accelerating\nparticles and producing the X-ray emission at relatively short distances from the central engine.\nAt larger distances, the outflow interacts with the surrounding matter;  this slows it down and produces the radio emission, in an afterglow-like manner \\cite{Zauderer+11}.   In the following we consider both the X-ray and radio-emitting regions as possible sites for UHECR acceleration. \nWe discuss first the radio emitting region where the situation is clearer, as we can use simple equipartition arguments. We then discuss the situation within the X-ray emitting jet. \n\nThe conditions within the radio emitting region have been analyzed using relativistic equipartition considerations (see the Appendix). In the relativistic regime the equipartition analysis depends on the geometry of the emitting regions and  there are two possible solutions  \\cite{Barniol+13b} shown in  Fig. \\ref{fig:J1644_radio}.\nFor the most reasonable geometry -- a narrow jet with an opening angle 0.1 -- the equipartition value of $B R$   is slightly larger than $10^{17}$ Gauss cm.   Given \nthat the equipartition estimates  \nyield a lower limit on the energy, this can be considered compatible with the Hillas condition.  A more detailed model that takes into account the inter-relation between the X-ray jet and the radio emitting electrons  \\cite{Kumar+13} yields  $B R$  larger by a factor of a few  than the simple equipartition estimate \n\\cite{Barniol+13b} (see  Fig. \\ref{fig:J1644_radio});  at early times $B R \\approx 3 \\times 10^{17}$ Gauss cm.  \nThus,  the outer radio emitting region of the \\textit{Swift}\\ J1644+57 TDE jet appears likely to have had conditions for UHECR acceleration. \n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Figure1cropped.eps}\n\\caption{ The equipartition values of  $ B R  $  as a function of time, for different relativistic  models for the TDE \\textit{Swift}\\ J1644+57:  a narrow jet with an opening angle $0.1$ (red dots), a wide jet  (green dots) and the detailed model of \\cite{Kumar+13} (solid line). }  \\label{fig:J1644_radio} \n\\end{figure}\n\nDuring the first two days, the X-ray luminosity of \\textit{Swift}\\ J1644+57  fluctuated reaching isotropic equivalent peak luminosities of  $\\approx 10^{48}$ ergs\/sec.  Models of the X-ray emission  as arising from a relativistic blob of plasma \\cite{Burrows+11} yield \nestimates of $B R$ ranging from a few $\\times 10^{15}$  to $5 \\times 10^{17}$ Gauss cm, depending on \n the dominant energy of the jet (Poynting flux or baryonic), the emission mechanism (synchrotron or external IC), the position of the emission region and the relative contributions of the disk and the jet. \nIt is possible but not certain that the X-ray emitting regions in \\textit{Swift}\\ J1644+57 also had conditions for UHECR acceleration.  If so, from this point of view TDEs resemble powerful AGNs in the way they satisfy their Hillas condition, with UHECR acceleration being possible in two different regions.   \n\n\\subsection{Energy budget and source abundance}\nWith a total energy of $\\approx 10^{54} \\,(M_{*}\/M_{\\odot})$ erg available in a tidal disruption event, only a small fraction -- $ 10^{-3} \\, (\\langle M_{*}\\rangle\/M_{\\odot})  \\, {\\dot{E}}_{44} \\, \\Gamma_{_{\\rm UCRtran, -7}}^{-1}$, where $M_{*}$ is the mass of the tidally disrupted star --  needs to go to UHECR production in order to satisfy Eq. (\\ref{EUCR}), if the rate of TDEs producing jets capable of accelerating UHECRs is adequate.   \n\nThe rate of TDEs in {\\it inactive} galaxies has recently been determined based on the discovery of two TDEs in a search of SDSS Stripe 82 \\cite{vfRate14}.  In volumetric terms\n\\begin{equation} \\label{TDErate}\n\\Gamma_{\\rm TDE} =  (0.4-0.8) \\cdot 10^{-7\\pm 0.4}\\,{\\rm Mpc}^{-3}\\,{\\rm yr}^{-1},\n\\end{equation}\nwhere the statistical uncertainty is in the exponent and the prefactor range reflects the  light curve uncertainty.\nThis result is roughly consistent with earlier theoretical \\cite{WangMerritt04} and observational estimates (within their uncertainties) \\cite{Gezari+09}.  However, more refined estimates are needed because not all TDEs have jets and only a fraction of those may be capable of producing UHECRs; in the following we ignore jets weaker than the \\textit{Swift}\\ events which satisfy the Hillas criteria.   \n\nAn estimate of the UHECR-producing TDE rate is obtained from the observed TDEs with jets,  \\textit{Swift}\\ J1644+57 and J2058+05.  Burrows et al. \\cite{Burrows+11} estimate that the observation of one event per seven years by \\textit{Swift}\\ corresponds to an all sky rate of $0.08-3.9$  events per year up to the detection distance of $z=0.8$ which contains a co-moving volume of $\\approx 100\\, {\\rm Gpc}^{3}$.  Their subsequent archival discovery of J2058+05 increased this rate by a factor of two and reduced somewhat the uncertainty range, leading to an estimated rate of $ \\approx 3  \\times 10^{-11}$ Mpc$^{-3}$ yr$^{-1}$ of TDE events with jets pointing towards us, or\n\\begin{equation}\n\\label{SwiftJetRate}\n\\Gamma_{_{\\rm UCRtran, -7}} \\approx 3 \\, f_b^{-1} \\, 10^{-4},\n\\end{equation}\nwhere $f_b$ is the beaming factor.  \n\nIn J1644+57, the total EM isotropic equivalent emitted energy in  X-rays was $ 3 \\times 10^{53}$ergs \\cite{Burrows+11,Bloom+11}; assuming that this is about 1\/3 of the total bolometric EM energy, they estimate that the total isotropic equivalent EM energy injection rate is $ \\approx 10^{54}$ ergs. Cenko et al. \\cite{Cenko+12} estimate a similar total  EM isotropic equivalent energy for J2058+05. \nSince the isotropic equivalent energy is a factor $f_{b}^{-1}$ larger than the true emitted energy, the beaming factor cancels between rate and energy factors, yielding an estimated \nEM energy injection rate of $\\approx 3 \\times  10^{43}$ erg \\, Mpc$^{-3}$ yr$^{-1}$ without relying on knowledge of the beaming factor.  This falls  short of the required energy injection rate for UHECRs, which is $2-4 \\times 10^{44}$ erg Mpc$^{-3}$ yr$^{-1}$. Interestingly a similar situation arises when comparing the observed EM GRB flux with the needed UHECR energy injection rate \n\\cite{fg09,eichler+GRB-UHECR10} (but see however \\cite{katz+GRB-UHECR09}).   The crux question here, is the relation between the emission observed in a particular EM band and the energy production in UHECRs.  The emission mechanisms and relevant particle energies are so different, that it is far from evident how to relate them.\n\nWe can also estimate the energy of the jet from the energy needed to produce the radio signal, using equipartition arguments and thus deriving a lower limit on the actual energy.   Different assumptions about conditions within the radio-emitting region lead to different energy estimates. \nFrom  \\cite{Barniol+13b}, we have an estimate of the minimal isotropic equivalent total energy of leptons and magnetic field that is capable of producing the radio emission produced in J1644+57, $\\approx  10^{51}$ erg;  including the energy of the accompanying protons increases the estimated minimal energy by a factor $\\sim 5$.   Unlike the relativistic inner jet producing the X-rays, the outflow has slowed to being only mildly relativistic with a  low Lorentz factor by the time it produces the observed radio, so that the radio emission is isotropic even when it arises from a jet.  Therefore, the factor $f_b^{-1}$ in the true rate of jetted-TDEs, Eq. (\\ref{SwiftJetRate}), does not cancel out and the implied energy injection rate (including the protonic contribution) is of order $ 2 \\times 10^{44}  (f_b\/10^{-3})^{-1} {\\rm erg} \\, {\\rm Mpc}^{-3}\\,{\\rm yr}^{-1} $, roughly what is needed as a UHECR flux. \n\nThe above discussion suggests that TDE jets alone could satisfy the needed UHECR injection rate, but underlines the importance of investigating what relationship should be expected between the EM and UHECR spectra and total luminosity. \n\nThe above discussion produced two independent, compatible estimates of the rate of TDEs with jets.  Using the \\textit{Swift}\\ observed rate and estimating the beaming factor to be  $f_{b} \\approx 10^{-2} - 10^{-3}$ based on the Lorentz factor of $\\sim 10-20$ estimated for \\textit{Swift}\\ J1644+57 \\cite{Burrows+11}, Eq. (\\ref{SwiftJetRate}) gives $\\Gamma_{_{\\rm UCRtran, -7}}^{-1} \\gtrsim 3 \\cdot 10^{-2}$.  Or, taking the total rate from \\cite{vfRate14} and a jet fraction $\\sim 0.1-0.2$ based on the fraction of TDEs detected in the radio \\cite{vV+Radio13}, gives $ \\Gamma_{_{\\rm UCRtran, -7}} \\lesssim 0.4$.    Using the lower estimate, \nEq. (\\ref{neff}) gives $n_{\\rm eff} \\approx   3 \\, \\tau_{5} \\,10^{-4}~ {\\rm Mpc}^{-3}$, comfortably compatible with the Auger source density limit in the low-deflection scenario, $n_{\\rm min} = 2 \\times 10^{-5} ~ {\\rm Mpc}^{-3}$.   \n\n\n\\section*{Composition}\n\nThe reader can ask why it is interesting to consider the possibility that UHECRs are predominantly or exclusively protonic, in view of the observed depth-of-shower-maximum distribution of AUGER which favors a predominantly mixed composition of intermediate mass nuclei, if interpreted with current hadronic interaction models tuned at the LHC  \\cite{augerXmaxMeas14,augerXmaxComp14}.  First, the recently finalized Auger analysis \\cite{augerXmaxComp14} finds a protonic component persisting to highest energies.  Second, a mixed composition requires a very hard injection spectrum incompatible with shock acceleration \\cite{shahamPiran13,Aloisio+13} and a composition at the source which has been argued to be strange and unlikely \\cite{shahamPiran13}.  Third, a  predominantly proton composition is of particular interest because both Auger and TA find evidence of correlations between UHECRs above 55 EeV and the local matter distribution \\cite{TAaniso14,augerAniso14}, although anisotropy {\\it per se} does not exclude mixed composition, particularly for the case of a single, nearby source \\cite{Piran11,rfICRC13,kfs14}.  The final and most compelling virtue of a predominantly proton composition is that it naturally explains the shape of the spectrum from below $10^{18}$ eV to the highest energies, including the observed ``dip'' structure around $10^{18.5}$ eV and the cutoff at highest energies, without needing an ad-hoc and fine-tuned transitional component between Galactic and extragalactic cosmic rays \\cite{berezGGdip05,AlBerezGaz12} with additional parameters to tune the composition and maximum energy by hand.  \n\nThe reader might also ask whether it is legitimate to set aside inferences on composition from current hadronic interaction models; the answer is ``yes''.  The nucleon-nucleon CM energy in the collision of a $10^{18}$ eV proton with the air -- 140 TeV -- is a factor 20 above current the LHC CM energy, so that the models must be extrapolated far into uncharted territory.  Furthermore, detailed comparisons by the Auger collaboration of the model predictions with a variety of observed shower properties reveals several discrepancies, including that the models underpredict the muon content of the ground shower by 30\\% or more \\cite{ICRC13topdown,augerHorizMuons14}, and that the model which does the best with respect to the muons at ground level (EPOS-LHC) is in the most serious contradiction with the observed depth of muon production in the atmosphere \\cite{augerMPD14}.\n\n\\section*{Summary}\n  \n\nTo conclude, we have shown that a scenario in which UHECRs are predominantly or purely protons can be realized, with acceleration occurring in transient AGN-like jets created in stellar tidal disruption events.  A well-studied example of such a TDE-jet, \\textit{Swift}\\ J1644+57, displays inner and outer emission sites in which collisionless shocks satisfy the Hillas criteria.\nThus we propose that, like in AGN models and GRB models, the  basic shock acceleration mechanism is applicable for UHECR acceleration in TDE jets.  As shown in \\cite{fg09}, the conditions in such jets are such that the radiation fields within the outflow are not large enough to cool the UHECRs before they escape.  Thus both the outer and inner emission regions in TDEs may in principle be viable UHECR sources.  \n\nWe also investigated whether the total observed flux of UHECRs is compatible with the UHECR injection rate that can be expected for TDE jets; although a more thorough theoretical understanding of the UHECR acceleration mechanism is needed for a definitive conclusion, present evidence indicates the energetics are satisfactory.  Finally, we showed that the effective number of sources predicted in the protonic-UHECRs-from-TDE-jets scenario, is compatible with the even the most stringent Auger bound, i.e., the case that typical deflections are less than $3^{\\circ}$.\n\nUnlike for GRBs, the TDE-jet model for UHECR production cannot be tested directly by association of an observed transient event with a signature of UHECRs.   In the case of GRBs, prompt neutrinos are produced via photoproduction of charged pions in the source, which arrive approximately simultaneously with the gammas.  (The UHECRs themselves arrive 10's or 100's of thousands of years after the gammas or neutrinos, due to the magnetic deflections discussed previously.)  By contrast, the level of prompt neutrino production in a TDE-jet is much lower, because the radiation field in a TDE jet is less, inhibiting photopion production.   Moreover the duration of UHECR production in a TDE jet is weeks or months, so even the prompt neutrinos are broadly spread in arrival times. \n\nThe conjecture of predominantly protonic composition, the role of transients in UHECR production, and the TDE-jet model can be tested purely observationally, as follows.  \\\\\n\\noindent $\\bullet$  Whether UHECRs are protons or nuclei can in principle be determined without relying on hadronic interaction models to infer composition, by detecting or placing sufficiently strong limits on VHE photons and neutrinos produced during the propagation, as their spectra distinguish UHECR protons from nuclei.   If UHECRs are predominantly protonic, as shown above (updating earlier arguments \\cite{fg09,WL09,MuraseTakami09}) their primary sources must be transients, with TDE jets the leading candidate.  \n\\\\\n\\noindent $\\bullet$  Whether sources are continuous or transient can be determined from the spectrum of UHECRs from a single source, because UHECRs arriving at a given epoch from a transient have similar values of rigidity $\\equiv E\/Z$ rather than displaying a power-law spectrum \\cite{waxmanME}, c.f., Eq. (\\ref{tau}).  \n\\\\\n\\noindent $\\bullet$  \nIf sources are confirmed to be transients, the presence or absence of VHE neutrinos accompanying the transient EM outburst, will distinguish between GRB and TDEs being the sources.  \n\\\\\n\\noindent $\\bullet$   As far as is presently known, the galaxies hosting TDEs are generally representative of all galaxies; if so, UHECR arrival directions would correlate (only) with the large scale structure, after taking into account Galactic and extragalactic magnetic deflections.  However if the rate of TDEs with jets is enhanced in active galactic nuclei, as conjectured in \\cite{fg09}, an enhanced correlation of UHECRs with AGNs relative to random galaxies could potentially be seen.\n\n\\noindent{\\bf Acknowledgements:}\nWe thank Rodolfo Barniol-Duran  for discussions and for help preparing the figure, and Kohta Murase for helpful discussions.  The research of GRF was supported in part by NSF-PHY-1212538; she thanks the Racah Institute for its hospitality during the initial stages of this work.  GRF is a member of the Pierre Auger Collaboration and acknowledges valuable interactions with Auger colleagues.  The research of TP was supported by the ERC advanced research grant ``GRBs''  the  I-CORE (grant No 1829\/12) and a grant from the Israel Space Agency SELA; he thanks the \nLagrange institute de Paris for hospitality while this research was concluded. \n\n\\section{Appendix:  Details of \\textit{Swift}\\ J1644+57 Analyses}\n\n\\noindent {\\bf The Radio Emitting Region:}\\\\\nRadio observations of Sw1644 began 0.9 days after the onset of the trigger \\cite{Zauderer+11}  and lasted for about $ 600$ days \\cite{Berger+12,Zauderer+13}. Wide frequency coverage began at 5 days.  At that time, the peak of the spectral energy distribution (SED) was at  $ \\nu_p \\approx 345$ GHz, with a peak flux  $F_p = 35$ mJy. The peak frequency and flux decreased  to $\\sim 5$ GHz and $0.5$ mJy at 570 days.   We begin by discussing the classical equipartition method of interpreting radio observations \\cite{Pacholczyk1970,ScottReadhead77,Chevalier98,kumarNarayan09}, and its relativistic generalization \\cite{Barniol+13a}.  A direct application to the Sw1644 observations \\cite{Zauderer+11} is likely overly naive, as we discuss subsequently.\n\nThe radio emitting region is characterized by four unknowns:  the size, $R$,   the magnetic field, $B$,  the total number of emitting electrons, $N$, and their typical Lorentz factor, $\\gamma_e$. The total energy of the emitting region is the sum of the electrons' energy  $N m_e c^2 \\gamma_e$ and the magnetic field energy $B^2 R^3 \/6$, the baryons' energy being unimportant in the Newtonian case. \nIdentifying the spectral peak as the self absorption frequency \nand using the standard expressions for the synchrotron frequency,  synchrotron  flux and  the self-absorbed flux (see e.g. \\cite{Barniol+13a} for details) one can eliminate 3 of the 4 parameters and express, e.g., $B$, $N$ and $\\gamma_e$ in terms of  $R$ and the observables $\\nu_p$, $F_p$ and $z$. One then \nobtains  $ E = C_1 \/R_{17}^6 + C_2 R_{17}^{11}$, where the first term is the electrons' energy and the second the magnetic field energy.  The constants $C_1$ and $C_2$ are given in term of the observables: \n$C_1 = 4.4 \\times 10^{50} \\,{\\rm erg} \\, ( F_{p}^4 \\, d_{28}^8 \\, \\nu_{p,10}^{-7} \\, \\, (1+z)^{-11} ) $ and \n$C_2 = 2.1 \\times 10^{46} \\,{\\rm erg} \\, (  F_{p}^{-4} \\, d_{28}^{-8} \\, \\nu_{p,10}^{10} \\, (1+z)^{14}  ), $ where $d_{28}(z)$ is the   luminosity distance in units of  $10^{28}$cm and $F_p$ is the peak flux measured in mJy. \n\nThe energy is minimized when the electrons' energy is roughly equal\nto the magnetic energy, or put differently, when the system is in equipartition. \nThe size of the system is strongly constrained, as the energy is a very steep function of $R$ both above and below the minimum.   \nAs we have three equations and four unknowns we can choose a different independent variable. For our purpose  $B R$ is most suitable and in this case one obtains $ E = \\tilde C_1 \/( B R)^{6\/5}  + \\tilde C_2 (B R)^{11\/5}$.  This dependence is less steep and hence the resulting value for $BR$ is less constrained by these considerations.  If $E$ is an order of magnitude above the minimal value, $B R$ can be a factor-10 lower or a factor-3 higher than at equipartition. \n\nWhen applying the equipartition considerations to Sw1644 one has to take into account that the outflow is relativistic \nin this case. The  relativistic equipartition estimates are somewhat more complicated than the\nNewtonian ones. A detailed equipartiton formalism for relativistic outflows was  developed recently by \\cite{Barniol+13a}.\nLike in the Newtonian case, the total energy depends very steeply on $R$, as\n$ E = \\hat C_1 \/R^6 + \\hat C_2 R^{11}$, where  $\\hat C_1$ and $\\hat C_2$  depend  on the observed quantities\nbut now also on  the outflow Lorentz factor, $\\Gamma$, and on the specific geometry of the emitting region (see \\cite{Barniol+13a} for details). \nNote that here the kinetic energy of the baryons within the relativistic outflow should also be included in the total energy of the system.  The bulk Lorentz factor, $\\Gamma$,  can be determined using time of arrival arguments; the geometrical factors involved have to be guessed.   Given the very steep dependence of the total energy on $R$, $R$ is still  well constrained by the energy-minimization, equipartition considerations. Using these arguments \\cite{Barniol+13b} find that  for the most reasonable geometry -- a narrow jet with an opening angle 0.1 -- the equipartition value of $B R$ in Sw1644 is slightly larger than $10^{17}$ Gauss cm \n(see Fig. \\ref{fig:J1644_radio}).  Given \nthat the equipartition estimates give a lower limit on the energy (that assumes maximal efficiency), this can be considered compatible with the Hillas condition.  Furthermore and independently, at the time of the last observations the minimal (equipartition)  energy is  $0.8 \\times 10^{51}$ ergs \\cite{Barniol+13b}, which is consistent with the energy required to account for the observed UHECR spectrum given the rate of TDEs, estimated above.\n\nHowever the naive equipartition analysis can be doubted, since applying it leads to the conclusion that the energy of the jet increases by about a factor of 20 from the initial observations at around 5 days, to the final observations.  The apparent increase required in the jet energy appears in other approaches as well, as noticed first\nby \\cite{Berger+12}, who analyzed the data based on GRB afterglow modeling. This interpretation  requires an energy supply to the radio emitting region. Such an energy supply is inconsistent with the continous  decrease in the X-ray luminosity during this period \\cite{Berger+12}, which supposedly reflects the activity of the inner engine and the accretion rate.  \n\nKumar et al.  \\cite{Kumar+13} suggested that the puzzling behavior comes about because the X-ray jet passes through the radio emitting region, causing the radio emitting electrons to be continuously cooled via Inverse Compton (IC) scattering with these X-ray photons.   This efficient IC cooling decreases the observed synchrotron radio flux relative to the equipartition estimate, causing the equipartition analysis to yield a lower energy than the true energy content of the system, resulting in an erroneously-low inferred value for $B R$.  At later times  the X-ray flux diminishes, the IC cooling ceases, and the synchrotron flux increases, consistent with the late-time observations and obviating the need for an increase of the \nenergy of the jet.  \\cite{Kumar+13} estimate  $B R$ to be larger by a factor of a few  than the simple equipartition estimate \n\\cite{Barniol+13b}. The results from their analysis are shown as the solid line in  Fig. \\ref{fig:J1644_radio}; at early times $B R \\approx 3 \\times 10^{17}$ Gauss cm.  This is just  the value needed to accelerate $10^{20}$ eV UHECRs.\n\nTo summarize, even with naive application of equipartition, the values of the total energy and $B R$ within the radio emitting region of TDE J1644 are marginally compatible with those needed to accelerate UHECRs.  Given \nthat the equipartition estimates give a lower limit on the energy (i.e., assumes maximal efficiency), and that the more physically satisfactory modeling of \\cite{Kumar+13} yields a larger estimate comfortably compatible with accelerating protons to UHE, we conclude that the outer region of the Sw 1644 TDE jet likely has conditions for UHECR acceleration. \n\n\\noindent{\\bf The X-ray emitting region:} \\\\\nThe isotropic equivalent X-ray luminosity is $\\approx 10^{48}$ ergs\/sec.   If this were coming from the accretion disk, then by the argument of \\cite{fg09} we would conclude that Eq. (\\ref{lumi}) was easily satisfied within the jet.  However since in \\textit{Swift}\\ J1644 we are viewing the jet close to its axis, there can be additional contributions to the X-ray emission, which must be modeled before $B R$ in the inner jet region can be inferred.   Unfortunately, the X-ray  observations are much less constraining on the conditions within the X-ray emitting regions than are the radio observations on the conditions in the outer region.  The X-ray spectrum is a power law;  if combined with the NIR observation it yields  a steep slope (steeper than 1\/3).  The  Fermi upper limits on the GeV emission suggest a suppression of the high energy emission due to photon-photon opacity. Overall, only a single component has been observed, with no clear evidence of the peak frequency and only an upper limit on the high energy IC component. Therefore there is  significant  freedom in modeling this emission. \n\nRef. \\cite{Burrows+11}  models the emission (following the blazer emission model of \\cite{GhiselliniTavecchio09}) as arising from a relativistic blob of plasma. They  \nhave put forward three models for this spectrum. \nThe models differ in the dominant energy of the jet (Poynting flux or baryonic), the emission mechanims (synchrotron or external IC), the position of the emission region and the relative contributions of the disk and the jet. \nAccording to these models the X-rays are generated between $10^{14}$ and $10^{16}$ cm from the black hole. Estimates of $BR$ range from a few $\\times 10^{15}$ for model 3 to $5 \\times 10^{17}$ for model 2.\n\n\n\\defAstrophys.\\ J.{Astrophys.\\ J.}\n\\defNature{Nature}\n\\defAstrophys.\\ J. Lett.{Astrophys.\\ J. Lett.}\n\\defAstrophys.\\ J.{Astrophys.\\ J.}\n\\defAstron.\\ Astrophys.{Astron.\\ Astrophys.}\n\\defPhys. Rev. D{Phys. Rev. D}\n\\defPhys.\\ Rep.{Phys.\\ Rep.}\n\\defMonth. Not. RAS {Month. Not. RAS }\n\\defAnnual Rev. Astron. \\& Astrophys.{Annual Rev. Astron. \\& Astrophys.}\n\\defAstron. \\& Astrophys. Rev.{Astron. \\& Astrophys. Rev.}\n\\defAstronom. J.{Astronom. J.}\n\\defJCAP{JCAP}\n\n\n\\bibliographystyle{apsrev4-1.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nElemental cerium is stark example of the complicating effects of magnetism in metals.  In principle a simple system, cerium at room temperature and ambient pressure presents a monatomic Bravais lattice (the face-centered-cubic $\\gamma$ phase).  However, under the application of modest pressures, cerium undergoes a spectacular first-order transition to a low temperature $\\alpha$ phase.  Both $\\alpha$ and $\\gamma$ are isostructural f.c.c. phases, but differ by $\\sim 15 \\%$  in volume, and have markedly different magnetic susceptibilities and resistivities.\\cite{bestreview}  There are competing explanations for the transition, which variously invoke a ``Kondo Volume Collapse''\\cite{KVC} mechanism, a ``Mott Transition'',\\cite{MT} or some entropic mechanism\\cite{entropy,jpcm_celath}.  In all cases however, it is generally agreed that the large volume $\\gamma$ phase contains relatively larger localized magnetic moments (evinced by the higher susceptibility) and relatively less itinerant electrons (evinced by the higher resistivity).  Beyond that, the nature of the transisiton remains contentious.\n\nOne of the main complications in the study of cerium is that although the {$\\gamma - \\alpha$} transition occurs on cooling over a wide range of pressures, it is avoided at ambient pressure.  In that case, an intervening dhcp phase forms from the $\\gamma$ phase, and transforms into the $\\alpha$ phase only at lower temperature.\\cite{bestreview}  There are commonly two methods used to resolve this problem.  Measurements are either performed under pressure on pure cerium, or else the cerium is alloyed with other elements known to suppress the dhcp phase (typically thorium).  Results from these two avenues of investigation are not always consistent - for example, it has been claimed that the phonon entropy plays little role in the {$\\gamma - \\alpha$} transition in a {Ce$_{0.9}$Th$_{0.1}$} alloy,\\cite{neutron1,neutron2} while studies on pure cerium under pressure have returned the opposite result.\\cite{puretron,PNASxray}  Nevertheless, the alloying of cerium has generated a family of related compounds, in which the order of the transition, the magnitude of the volume collapse, and the transition temperature can be smoothly varied.\\cite{fiskthompson,lashtricrit}\n\nOne such alloy, {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}, is of particular interest in regards to spin-lattice coupling at the phase transition.  Alloying with thorium suppresses the dhcp phase, while alloying with lanthanum causes the transition to become continuous and supresses the critical temperature.\\cite{fiskthompson, lashtricrit}  It has been reported that cooling {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$} in high magnetic fields can suppress the resistivity change associated with the volume collapse to lower temperature, and that the application of pulsed magnetic fields can induce a large moment below the critical temperature.\\cite{jpcm_celath}  These measurements map out a phase boundary for the {$\\gamma - \\alpha$} transition as a function of magnetic field, and suggest that the volume collapse can be fully suppressed by $\\mu_{0} H \\lesssim 56$ Tesla.  Here we report a direct study of the structure of {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}, using synchrotron x-ray powder diffraction in pulsed magnetic fields as high as 28 Tesla.  Lattice parameter and structural correlation length are extracted throughout the $H - T$ phase diagram.\n\n\n\n\\section{Experiment}\n\nSamples of {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$} were prepared at Los Alamos by arc-melting elemental cerium, lanthanum and thorium in a zirconium gettered argon atmosphere.  The button was melted and flipped 15 times to ensure mixing.  The final melt was cooled in a round-bottomed trough, which was 5 mm wide.  The resulting $\\sim$ 25 mm long sample was unidirectionally rolled $\\sim$ 10 times to achieve a finished thickness of 50 microns.  A section of the as-rolled material was heat treated at 450$^\\circ$C for 1 hour to remove the rolling texture before a final sample, 2 mm in diameter, was cut from the sheet with a razor blade.  The sample was subsequently mounted inside a blind hole in a sapphire bracket, and sealed with a sapphire cap and stycast epoxy, to ensure good thermal contact and prevent oxidization.  The sapphire bracket was anchored to the cold finger of the double-funnel solenoid pulsed magnet at the Advanced Photon Source (APS), which facilitates diffraction at up to 22$^{\\circ}$ of scattering angle, to temperatures as low as $\\sim 5$ K and in magnetic fields as high as $\\sim$ 30 Tesla.\\cite{solenoid1,solenoid2}  This system features a resistive magnet coil (of Tohoku design) in a liquid nitrogen bath, connected to a 40 kJ capacitor bank.  The bank is capable of discharging a 3 kV charge through the coil in a $\\sim 6$ ms half-sine wave pulse, generating a peak current of $\\sim 10$ kA and a peak magnetic field of $\\sim 30$ Tesla.  The heat generated in the resistive coil during a pulse must be allowed to dissipate before the system can be pulsed again.  At peak field values, this requires a wait time $\\gtrsim$ 8 minutes between pulses.  The sample to be studied is both thermally and vibrationally isolated from the coil.  This new instrument offers a complementary measurement geometry to the APS split-pair pulsed magnet developed previously.\\cite{firstrsi,ttopmag}  \n\n\\begin{figure} \n\\centering \n\\includegraphics[width=8.1cm]{fig1.eps}\n\\caption {Top:  The inicident x-ray pulse (the narrow pulse) is timed to arrive during the magnetic field peak (broader pulse).  In this case, a 5.75 ms magnetic field pulse to ~15.4 Tesla weighted by an x-ray exposure with a 1 ms fullwidth gives an average magnetic field value of $\\sim$ 14.9 Tesla, integrated over  $\\pm$ 0.5 Tesla for the exposure.  Middle:  Raw data collected on the image plate for a single 1 ms exposure.  Bottom:  The same data, radially integrated and converted to $|\\bf{q}|$.}\n\\label{fig:1}\n\\end{figure}\n\nExperiments were performed at the 6ID-B station at the APS.  30 keV photons were selected using a Si(111) monochromator, with the mirror removed to provide an unfocused beam.  The monochromator was detuned so as to reduce higher harmonic contamination.  The incident beam size was defined by slits, with an illuminated area on sample of 0.2 mm X 1.0 mm.  A $\\sim 1$ ms pulse of x-rays was defined using a pair of fast platinum shutters, and the arrival of this pulse at the sample was synchronized with the peak of the magnetic field pulse.  Scattered x-rays were recorded on a two-dimensional image plate detector.  In this configuration, our system is similar to the pulsed magnet instrument in operation at the European Synchrotron Radiation Facility (ESRF).\\cite{detlefs_ins}  To calibrate our instrument, we repeated the measurements\\cite{solenoid2} of Detlefs \\textit{et al.} on polycrystalline TbVO$_4$ in pulsed magnetic fields, as previously measured by the ESRF instrument.\\cite{detlefs_tbvo}  We reproduced the reported high field peak splitting in the tetragonal phase of TbVO$_4$,\\cite{detlefs_tbvo} validating our measurement scheme.\\cite{solenoid1,solenoid2}  We performed our measurements of {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$} with the axis of the solenoid coincident with the incident beam, and the detector positioned so as to capture the majority of the solenoid exit window.  In this way, at 30 keV the first two powder rings (corresponding to the $<$111$>$ and $<$200$>$ peaks) are completely captured by the detector, allowing a full radial integration and excellent counting statistics even for a single 1 ms exposure.  The current through the solenoid and the x-ray flux incident on the sample were monitored as a function of time by tracing the output of a high-resolution current monitor and an air-filled ion chamber on a digital storage oscilloscope.  A typical trace is shown in Fig. 1, along with data from a single exposure on the image plate.  In order to extract the correlation length and lattice parameter from these diffraction patterns, each of the two lowest lying peaks was fit individually, generating two independent sets of data, which were subsequently compared.  In all cases, the properties extracted from the two peaks aggreed within the errorbars of the measurement, and the quantities quoted in the following are the average of the two.\n\n\n\\section{Zero-Field Results}\n\nPowder diffraction patterns at room temperature show a single phase material, with symmetry and lattice constant consistent with the $\\gamma$ phase of {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}.  On cooling, we observe the first two Bragg peaks to shift to higher $|\\bf{q}|$, while also broadening by a factor of $\\sim 2$ and exhibiting a reduction in peak intensity (see Fig. 2).  This is consistent with a collapse in volume, accompanied by a striking reduction in the structural correlation length.   It is interesting to note that we observe a single phase at all temperatures, in contrast to recent neutron diffraction measurements on {Ce$_{0.9}$Th$_{0.1}$} which reported a coexistence of both the $\\alpha$ and $\\gamma$ phases over a wide range of temperatures.\\cite{neutron2}  Our sample was initially cooled directly to 8K to characterize the magnitude of the volume collapse, then warmed to 250 K while collecting data.  The volume was observed to approach the $\\gamma$ phase value quite gradually.  Next we cooled slowly and measured the lattice parameters with a fine point density.  The resulting continuous but hysteretic volume collapse curves are plotted in Fig. 3.  Surprisingly, on our second cooling we found that the absolute size of the volume collpase was diminished - thermal cycling of the sample is detrimental to the phase transition.\n\n\\begin{figure} \n\\centering \n\\includegraphics[width=8.5cm]{fig2.eps}\n\\caption {Zero-field temperature dependence of the $<$111$>$ Bragg reflection.  The peak shifts to higher $|\\bf{q}|$ and broadens on cooling.  The sample is a single phase at all temperatures.  Resolution limited peaks at high temperature are well described by Gaussians, while a Voigt shape at low temperature is consistent with an Ornstein-Zernike scattering function with a short correlation length (solid lines).}\n\\label{fig:2}\n\\end{figure}\n\n\\begin{figure} \n\\centering \n\\includegraphics[width=8.5cm]{fig3.eps}\n\\caption {Zero-field volume collapse in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}.  Sample was warmed from 8 K to 250 K ($\\blacksquare$), then cooled back to 10 K ($\\bullet$).  This thermal cycling lead to a reduction in the magnitude of the volume collapse, shown by the dashed lines.  The inset shows the Ornstein-Zernike correlation length extracted from the cooling measurements (see text).  The correlated regions of the collapsed phase have average size of only $\\sim 20$ nm.}\n\\label{fig:3}\n\\end{figure}\n\n\nThe observed peak broadening can be modelled phenomenologically, by assuming that the high temperature patterns show long-range ordered and resolution limited scattering, while the low temperature broadening arises from a reduction in the Ornstein-Zernike correlation length.  Our high temperature peaks are well described by Gaussians, and we are assuming a Lorentzian form for the scattering function (see Ref.[\\onlinecite{strucfluc}] for details).  Therefore, the measured scattering at low temperature should be well described by the convolution of a Lorentzian and a Gaussian - a Voigt function.  Since the width of this Voigt function is related to the width of the Gaussian (the resolution) and the width of the Lorentzian (the inverse correlation length), the numerically extracted fullwidth of the Bragg peaks can be converted to a resolution-deconvoluted correlation length.  The solid lines in Fig. 2 demontrate the degree of agreement between these lineshapes and the measured data, and the extracted correlation length is plotted in the inset of Fig. 3.  Our resolution limit is found to be on the order of a few microns, however, the low temperature collapsed phase exhibits correlation lengths which are roughly two orders of magnitude shorter than this.  Below $\\sim$ 30 K, the $\\alpha$ phase correlations are seen to extend over an average range of $\\sim$ 20 nm.  Since our alloy is $\\sim$ 80$\\%$ cerium, the mean distance between dopant atoms (either lanthanum or thorium) is on the order of 2 nm.  Thus, while the inclusion of lanthanum and thorium in the alloy does not completely hinder the formation of the $\\alpha$ phase, and each correlated region on average contains multiple dopant atoms, nevertheless these atoms act to strongly disorder the sample at the nanoscale within the $\\alpha$ phase.  Conversely, within the $\\gamma$ phase, individual grains are well correlated over micron-sized regions despite the same level of chemical disorder.\n\n\\section{results in pulsed magnetic field}\n\nMotivated by the results of Ref. [\\onlinecite{jpcm_celath}], we next sought to characterize the effect of applied magnetic field on the volume collapse transition in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}.  Currently, the available capacitance at the APS limits our measurements to applied magnetic fields $\\lesssim$ 30 Tesla, well below the 56 Tesla estimated to fully suppress the {$\\gamma - \\alpha$} transition to zero Kelvin.\\cite{jpcm_celath}  However, inspection of the resistivity measurements of Ref. [\\onlinecite{jpcm_celath}] reveals that our highest achievable magnetic fields should be sufficient to suppress the temperature of the volume collapse inflection point by $\\sim$ 8 K.  We can maximize our sensitivity to this effect by cooling to the inflection point of the volume collapse ($\\sim$ 35 K) and pulsing to high magnetic field.  The slope of $\\frac{\\Delta V}{V}$ is large at this temperature, and therefore an 8 K shift in the inflection point should cause a $\\sim$ 0.7 $\\%$ change in volume at 35 K, well within our resolution.  However, as can be seen from Fig. 4, we failed to observe this effect.  After cooling to 35 K, we collected five diffraction patterns at zero field, which served to define the zero field Bragg peak positions to an accuracy better than $\\sim 10^{-3} \\AA^{-1}$.  Next, we collected ten patterns in pulsed field, with the average applied field over the x-ray exposure being $\\sim 28 \\pm 1$ Tesla.  We observed no measurable change in the Bragg peak positions in any individual pattern.  Nor did we observe any cumulative effect from multiple high-field pulses.  Our measurements therefore constrain any volume changes at (35 K, 28 T) to be less than $\\sim$ 0.07 $\\%$, a full order of magnitude smaller than what would be expected.\\cite{jpcm_celath} \n\n\\begin{figure} \n\\centering \n\\includegraphics[width=8.5cm]{fig4.eps}\n\\caption {Observed and expected effects of 28 Tesla pulsed magnetic fields on the $<$111$>$ Bragg peak at T=35 K.  There is no discernable change in the position or width of the peak.  The expected curve is calculated based on the position and width change that would arise from an 8 K shift in the transition temperature, consistent with the results of Ref. [\\onlinecite{jpcm_celath}].}\n\\label{fig:4}\n\\end{figure}\n\n\\begin{figure} \n\\centering \n\\includegraphics[width=8.5cm]{fig5.eps}\n\\caption {Volume collapse ($\\Delta V\/V$) and Ornstein-Zernike correlation length ($\\xi$), comparing zero field cooled measurements and measurements where the system is repeatedly pulsed to 28 T on cooling.  No field effect is seen.}\n\\label{fig:5}\n\\end{figure}\n\nIt is important to remember that the evidence for field induced suppression of the volume collapse in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$} for magnetic fields below 30 Tesla comes from resistivity curves, measured by cooling in constant applied magnetic fields.  This is procedurally distinct from the zero-field-cooled pulsed measurements reported here.  Discrepancy between field-cooled and zero-field-cooled behaviour is a hallmark of spin glass materials, wherein chemical disorder and competing interactions collude to freeze in disordered ground states.\\cite{spinglassrev}  We have already shown that chemical disorder in our alloy has a strong effect on the development of $\\alpha$-phase correlations (see Fig. 3).  It is therefore tempting to sugest that there may be some glassy behavior at play in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}.  Direct comparison between field cooled and zero field cooled measurements are therefore highly desirable, however, a truly field-cooled measurement is not achievable with pulsed magnets.  We have attempted to approximate one by slow cooling, and pulsing in regular intervals.  In order to minimize the obscuring effects of thermal cycling, we performed two identical cooling curves, one after the other, warming to 280 K once to reset the system.  For each of these measurements, we cooled at a controlled rate of 1 K$\/$min from 280 K.  During the first of these cooldowns, we pulsed to 28 T at 10 min (10 K) intervals.  During the second cooldown, we collected zero-field data in the same manner, albeit more frequently since there was no need to wait for the coil to cool down. The results are plotted in Fig. 5.  Clearly, magnetic fields applied in this way are no more effective at suppressing the volume collapse than the zero-field-cooled methods shown in Fig. 4. \n\nWe are therefore forced to conclude that the available magnetic fields were insufficient to induce any structural effects in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}.  If the {$\\gamma - \\alpha$} transition were continuous for this alloy (as claimed in Ref. [\\onlinecite{fiskthompson}]), and the magnetic field induced suppression occured as claimed in Ref. [\\onlinecite{jpcm_celath}], then we would have expected to see a clear shift in $|\\bf{q}|$ as denoted in Fig. 4 for the zero-field-cooled measurements, and a clear suppression in temperature should have occurred for the repetitively pulsed measurements reported in Fig. 5.  Conversely, if the {$\\gamma - \\alpha$} transition were first order for this alloy (as claimed in Ref. [\\onlinecite{lashtricrit}]), then our measurements imply that the phase boundary was never crossed in our cooling curve.  This would require a steeper increase of the critical field on lowering temperature than was reported in Ref. [\\onlinecite{jpcm_celath}.]  It is also possible that the alloy is dynamically inhibited on the timescale of the magnetic field pulse, which would be consistent with a glassy behavior.  Finally, it is possible that the effects reported in Ref. [\\onlinecite{jpcm_celath}] were not intrinsic to {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}.  Clearly, the issue of field induced suppression of the {$\\gamma - \\alpha$} transition requires further investigation in elemental cerium and its alloys.\n\n\n\\section{effect of thermal cycling}\n\nAs has been noted in the preceeding sections, we have observed a detrimental effect due to thermal cycling.  Our initial measurements on as-grown samples revealed well-correlated grains on the order of microns in size.  On first cooling, the volume collapsed $\\alpha$ phase was seen to exhibit nanoscale disorder, and on second cooling the absolute size of the volume collapse had diminished (see Fig. 3 and discussion in Section III).  Over the course of our week-long experiment, we observed the $\\alpha$-phase volume to gradually increase with each thermal cycling, while the $\\gamma$-phase volume was seen to gradually decrease.  In addition, while the $\\alpha$-phase consistently showed nanoscale disorder with correlation lengths of $\\sim$ 20 nm, the $\\gamma$ phase correlations were observed to shrink, from micron-sized grains in the as-grown sample to correlation lengths of only $\\sim$ 150 nm after 11 thermal cycles.  Zero-field cooling curves from the second, seventh, and eleventh cycle are plotted in Fig. 6, to illustrate the effect.  It is likely that the strains associated with the large volume change and spatial variations in doping,\\cite{fiskthompson} which are manifest in the $\\alpha$-phase disorder, act to introduce new domain boundaries at low temperature, which thereafter persist as defects.  This effect is insidious, since the suppression by thermal cycling we observe here could easily be mistaken for suppression by an external perturbation, if that perturbation was varied monotonically during the course of an experiment.  Hypothetically, had we measured cooling curves through the volume collapse in DC magnetic fields which were monotomically increased during the course of our measurements, we may easily have misinterpreted the gradual buildup of thermal cycling damage as a magnetic field induced effect.  Therefore, it is important to take extra care when mapping out phase diagrams in cerium alloys.  More generally, these measurements highlight the need for careful consideration of strain driven effects in all materials studies, which can often obfuscate results.\n\n\\begin{figure}\n\\centering \n\\includegraphics[width=8.5cm]{fig6.eps}\n\\caption {Gradual contamination by nanoscale disorder, induced by thermal cycling through the {$\\gamma - \\alpha$} transition.  The effects are clear in both the magnitude of the volume collapse ($\\Delta V\/V$) and the Ornstein-Zernike correlation length ($\\xi$).  All data was collected on cooling in zero magnetic field.  The three data sets represent the second ($\\bullet$), seventh ($\\blacksquare$), and eleventh ($\\blacktriangle$) thermal cycle.}\n\\label{fig:6}\n\\end{figure}\n\n\\section{conclusions} \n\nIn this article, we have reported a synchrotron x-ray diffraction study of the volume collapse transition in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$}, as a function of temperature and pulsed magnetic field.  In this alloy, the {$\\gamma - \\alpha$} transition is smooth but hysteretic, characterized by a $\\sim 4 \\%$ volume change and a dramatic reduction in structural correlation length.  The collapsed $\\alpha$ phase is shown to be unaffected by pulsed magnetic fields $\\lesssim$ 28 Tesla, both in zero-field cooled conditions and under repetitive pulsing on cooling.  It is suggested that a deviation between truly field cooled and zero field cooled behavior in {Ce$_{0.8}$La$_{0.1}$Th$_{0.1}$} may account for the disagreement between our measurements and previous bulk studies.\\cite{jpcm_celath}  This implies a glassy character to the low temperature phase.  We have also shown that repeated thermal cycling through the {$\\gamma - \\alpha$} transition acts to disorder the sample on nanometer length scales, while suppressing the magnitude of the volume collapse transition.  These measurements may constitute a microscopic measure of what has been generally refered to as ``sluggish dynamics'' in cerium alloys.\\cite{jpcm_celath,fiskthompson}\n\nIt is our hope that these results will advance understanding of the exotic {$\\gamma - \\alpha$} transition in elemental cerium and its alloys.  Specifically, we believe the data presented here highlights the strong effects of disorder in these systems.  We gratefully acknowledge fruitful discussions with J. Lashley and B. Toby.  Use of the Advanced Photon Source is supported by the DOE, Office of Science, under Contract No. DE-AC02-06CH11357.  Pulsed magnet collaborations between Argonne and Tohoku University are supported by the ICC-IMR.   HN acknowledges KAKENHI No. 23224009 from MEXT.  JPCR aknowledges the support of NSERC of Canada.  \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nGamma-ray observations of the Small Magellanic Cloud with the EGRET telescope onboard \nthe {\\it Compton Gamma Ray Observatory} have proved that the bulk of cosmic rays (CRs) \npropagating in the Milky Way are produced in Galactic sources \\cite{sre93}. \nObservations of the diffuse $\\gamma$-ray emission from our Galaxy allow to estimate \nthe total CR luminosity \\cite{dog02}:\n\\begin{equation}\nL_{\\rm CR} = L_\\gamma {x_\\gamma \\over x} \\sim 5 \\times 10^{40}~{\\rm erg~s}^{-1},\n\\label{eqvt1}\n\\end{equation}\nwhere $L_\\gamma \\sim 5 \\times 10^{39}$~erg~s$^{-1}$ is the total \nluminosity of diffuse high-energy ($>100$~MeV) $\\gamma$ rays emitted in the decay of \n$\\pi^0$ produced by CR interaction with the interstellar medium (ISM), $x_\\gamma \n\\sim 120$~g~cm$^{-2}$ is the mean grammage needed for a CR ion to produce \na $\\pi^0$ in the ISM and $x \\sim 12$~g~cm$^{-2}$ is the mean path length that CRs \ntraverse before escaping the Galaxy, which is determined from measurements of the CR\nchemical composition near Earth. In comparison, the total power supplied by Galactic \nsupernovae (SNe) is\n\\begin{equation}\nL_{\\rm SN} = E_{\\rm SN} R_{\\rm SN} \\approx 10^{42}~{\\rm erg~s}^{-1},\n\\label{eqvt2}\n\\end{equation}\nwhere $E_{\\rm SN} \\approx 1.5\\times 10^{51}$~erg is the approximate total ejecta kinetic \nenergy of a SN and $R_{\\rm SN} \\approx 2$ per century is the current epoch \nGalactic SN rate \\cite{fer98}. Thus, SNe have enough power to sustain the CR \npopulation against escape from the Galaxy and energy losses, if there is a mechanism \nfor channeling $\\sim 5$\\% of the SN mechanical energy release into relativistic \nparticles.\n\nDiffusive shock acceleration (DSA) at the blast waves generated by SN explosions can \nin principle produce the required acceleration efficiency, as well as the observed \npower-law spectrum of CRs \\cite{kry77,axf78,bel78,bla78}. In this model, a fraction \nof ambient particles entering the SN shock front can be accelerated to high energies \nduring the lifetime of a supernova remnant (SNR) by diffusing back and forth on \ncompressive magnetic fluctuations of the plasma flow on both sides of the shock. A \ncritical ingredient of the theory is the strength of the turbulent magnetic field \nin the shock acceleration region, which governs the acceleration rate and in turn \nthe maximum energy of the accelerated particles. If the turbulent field upstream of \nthe SN shock is similar to the preexisting field in the surrounding ISM ($B \\sim\n5$~$\\mu$G), the maximum total energy of an ion of charge $Z$ was estimated 25 years ago \nto be (for a quasi-parallel shock geometry) $E_{\\rm max} \\sim 10^{14} Z$~eV \\cite{lag83}. \nBut in more recent developments of the DSA theory, it is predicted that large-amplitude \nmagnetic turbulence is self-generated by streaming of accelerated particles in the \nshock region, such that the ambient magnetic field can be strongly amplified as part \nof the acceleration process \\cite{bel01,ama06,vla06}. In this case, protons might be \naccelerated in SNRs up to $3\\times10^{15}$~eV, i.e. the energy of the spectral \"knee\"   \nabove which the measured all-particle CR spectrum shows a significant steepening. \nContributions of accelerated $\\alpha$-particles and heavier species might then explain \nthe existing CR measurements up to $\\sim$10$^{17}$~eV \\cite{ber07}. Above energies of \n10$^{18}$--10$^{19}$~eV, CRs are probably of extragalactic origin.  \n\nAnother uncertain parameter of the DSA model is the fraction of total shocked \nparticles injected into the acceleration process. Although theoretical progress has \nbeen made in recent years \\cite{bla05}, the particle injection and consequently the \nacceleration efficiency are still not well known. However, theory predicts that for \nefficient acceleration the energy density of the relativistic nuclear component can \nbecome comparable to that of the postshock thermal component, in which case the \nbackreaction of energetic ions can significantly modify the shock structure and \nthe acceleration process can become highly nonlinear (e.g. \\cite{ber99}). In \nparticular, the compression ratio of a CR-modified shock is expected to be higher \nthan for a test-particle shock (i.e. when the accelerated particles have no \ninfluence on the shock structure). \nThis is because of both the softer equation of state of a relativistic (CR) gas and \nthe energy loss due to escape of accelerated particles from the shock region \n\\cite{dec00}. Moreover, the temperature of the shock-heated gas can be reduced if a \nsignificant fraction of the total available energy of the shock goes into relativistic \nparticles. Observations of these nonlinear effects \\cite{hug00,dec05,war05} provide \nindirect evidence for the efficient acceleration of ions in SN shock waves. \n\nThe acceleration of electrons in SNRs leaves no doubt, since we observe the nonthermal \nsynchrotron emission that these particles produce in the local magnetic field. Radio \nsynchrotron radiation, which in SNRs is emitted by GeV electrons, was discovered \nin the 1950's. More recent is the observation of X-ray synchrotron emission from \nyoung shell-type SNRs \\cite{koy95}, which is due to electrons accelerated to very high \nenergies, $E_e>$1~TeV. Thanks to the extraordinary spectroscopic-imaging capabilities of \nthe {\\it XMM-Newton} and {\\it Chandra} X-ray observatories, this nonthermal emission can \nnow be studied in great details and recent observations of SNRs with these satellites \nhave shed new light on the DSA rate and the maximum energy of the accelerated particles. \nThis is the subject of Section~2. \n\nIn Section~3, we discuss the origin of the TeV $\\gamma$-ray emission observed from \na handful of shell-type SNRs with atmospheric Cerenkov telescopes. For some objects, \nthe detected $\\gamma$-rays have been explained as resulting from $\\pi^0$ production in \nnuclear collisions of accelerated ions with the ambient gas. If this were true, this \nhigh-energy emission would be the first observational proof that CR ions are indeed \naccelerated in SN shock waves. However, the origin of the TeV $\\gamma$-rays \nemitted in SNRs is still a matter of debate, because at least in some cases the \nhigh-energy photons can also be produced by inverse Compton scattering of \ncosmic-microwave-background photons (and possibly optical and infrared interstellar \nphotons) by ultrarelativistic electrons. \n\nIn Section 4, we show that radio observations of extragalactic SNe can \nprovide complementary information on the DSA mechanism. As an example, we use a \nsemianalytic description of nonlinear DSA to model the radio light curves \nof SN 1993J. We choose this object because the set of radio data \naccumulated over the years \\cite{wei07} constitutes one of the most detailed sets of \nmeasurements ever established for an extragalactic SN in any wavelength range. We \nderive from these data constraints on the magnetic field strength in the environment \nof the expanding SN shock wave, the maximum energy of the accelerated particles, as \nwell as on the fractions of shocked electrons and protons injected into the \nacceleration process. Conclusions are given in Section 5.\n\n\\section{X-ray synchrotron emission from SNRs}\n\nTogether with the thermal, line-dominated X-ray emission from the shock-heated gas, \na growing number of SNRs show nonthermal, featureless emission presumably produced\nby ultrarelativistic electrons in the blast wave region via a synchrotron process. \nHigh-angular resolution observations made with the {\\it Chandra} and \n{\\it XMM-Newton} X-ray observatories have revealed very thin rims of nonthermal \nemission associated with the forward shock. In several cases, \nlike SN~1006 \\cite{koy95} and G347.3-0.5 \\cite{cas04}, the synchrotron component \ncompletely dominates the thermal X-ray emission. The measured power-law spectral \nindex of the X-ray synchrotron radiation is always much steeper than that of the \nnonthermal radio emission, which is consistent with expectation that the X-ray \ndomain probes the high-energy end of the accelerated electron distribution. The \ncomparison of radio and X-ray fluxes allows to determine the exponential cutoff \n(maximum) frequency of the synchrotron emission, $\\nu_c$, which is related to the \nmaximum energy of the accelerated electrons and the ambient magnetic field as \n(e.g. \\cite{sta06})\n\\begin{equation}\n\\nu_c = 1.26\\times10^{16} \\bigg({E_{e,\\rm max} \\over 10{\\rm~TeV}}\\bigg)^2 \n\\bigg({B \\over 10~\\mu{\\rm G}}\\bigg)~{\\rm Hz}.\n\\label{eqvt3}\n\\end{equation}\nDiffusive shock acceleration can only occur for particles whose acceleration rate \nis higher than their energy loss rate in the acceleration region. The maximum \nelectron energy, $E_{e,\\rm max}$, can be estimated by equating the synchrotron \ncooling time (e.g. \\cite{par06}),\n\\begin{equation}\n\\tau_{\\rm syn}(E_{e,\\rm max}) = {E_{e,\\rm max} \\over (dE\/dt)_{\\rm syn}} \\propto\nE_{e,\\rm max}^{-1} B^{-2},\n\\label{eqvt4}\n\\end{equation}\nwhere $(dE\/dt)_{\\rm syn}$ is the synchrotron loss rate at $E_{e,\\rm max}$, to \nthe acceleration time\n\\begin{equation}\n\\tau_{\\rm acc}(E_{e,\\rm max}) = {E_{e,\\rm max} \\over (dE\/dt)_{\\rm acc}} \\sim\n{\\kappa(E_{e,\\rm max}) \\over V_s^2},\n\\label{eqvt5}\n\\end{equation}\nwhere $(dE\/dt)_{\\rm acc}$ and $\\kappa(E_{e,\\rm max})$ are the acceleration \nrate and mean spatial diffusion coefficient of the electrons of \nenergy $E_{e,\\rm max}$ in the blast wave region and $V_s$ is the shock speed. \nWe have neglected here the dependence of $\\tau_{\\rm acc}$ on the shock \ncompression ratio (see \\cite{par06}). The value of $\\kappa(E_{e,\\rm max})$ \ndepends on the strength and structure of the turbulent magnetic field. The DSA \ntheory predicts that CRs efficiently excite large amplitude magnetic fluctuations \nupstream of the forward shock and that these fluctuations scatter CRs very \nefficiently \\cite{bel78,bel01,ama06,vla06}. It is therefore generally assumed that \nthe spatial diffusion coefficient is close to the Bohm limit:\n\\begin{equation}\n\\kappa \\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} \\kappa_B={r_g v \\over 3},\n\\label{eqvt6}\n\\end{equation}\nwhere $v$ is the particle speed and $r_g$=$pc\/(QeB)$ the particle gyroradius, $p$ \nbeing the particle momentum, $c$ the speed of light, $Q$ the charge number ($Q=1$ \nfor electrons and protons), and $-e$ the electronic charge. Note that for \nultrarelativistic electrons, $\\kappa_B = r_g c\/3 \\propto E_e B^{-1}$. Equating \nequations~(\\ref{eqvt4}) and \n(\\ref{eqvt5}) and using equation~(\\ref{eqvt3}) to express $E_{e,\\rm max}$ as a \nfunction of $B$ and $\\nu_c$, we can write the ratio of the electron diffusion \ncoefficient at the maximum electron energy to the Bohm coefficient as:\n\\begin{equation}\n\\eta_\\kappa={\\kappa(E_{e,\\rm max}) \\over \\kappa_B} \\propto V_s^2 \\nu_c^{-1}.\n\\label{eqvt7}\n\\end{equation}\nThus, measurements of $V_s$ and $\\nu_c$ can allow to derive $\\eta_\\kappa$ \nwithout knowing the ambient magnetic field. Using this result, several \nrecent studies \\cite{sta06,par06,rey04,yam04} have shown that there are regions \nin young ($t<10^4$~yr) SNRs where {\\it acceleration occurs nearly as fast as the \nBohm theoretical limit} (i.e. $1<\\eta_\\kappa<10)$. This provides an important \nconfirmation of a key prediction of the DSA model. \n\nThe strength of the magnetic field in the shock acceleration region may be derived \nfrom the thickness of the nonthermal X-ray rims observed in young SNRs (e.g. \n\\cite{vin03,bal06,par06}). One of the two interpretations that have been proposed to \nexplain the thin X-ray filaments is that they result from fast synchrotron cooling \nof ultrarelativistic electrons transported downstream of the forward shock. In this \nscenario, the width of the filaments is set by the distance that the electrons cover \nbefore their synchrotron emission falls out of the X-ray band. The electron transport \nin the downstream region is due to a combination of diffusion and advection, whose \ncorresponding scale heights are \\cite{bal06} $l_{\\rm diff}=\n\\sqrt{\\kappa \\tau_{\\rm syn}}\\propto B^{-3\/2}$ (see eqs.[\\ref{eqvt4}] and [\\ref{eqvt6}]) \nand $l_{\\rm adv}=\\tau_{\\rm syn}V_s\/r_{\\rm tot}\\propto B^{-3\/2} E_X^{-1\/2} V_s\/r_{\\rm tot}$, \nrespectively. Here, $r_{\\rm tot}$ is the overall compression ratio of the shock and \n$E_X \\sim 5$~keV is the typical X-ray energy at which the rims are observed. Thus, by \ncomparing $l_{\\rm diff}$ and $l_{\\rm adv}$ to the measured width of the X-ray filaments \n(e.g. $l_{\\rm obs}\\approx 3''$ in Cas A which gives 0.05~pc for a distance of 3.4 kpc \n\\cite{vin03}) one can estimate the downstream  magnetic field. Applications of this \nmethod to {\\it Chandra} and {\\it XMM-Newton} observations of young SNRs have shown that \n{\\it the magnetic field at the forward shock is amplified by about two orders of \nmagnitude} as compared with the average Galactic field strength. This conclusion has \nbeen recently strengthened by the observations of rapid time variations ($\\sim$1~yr) in \nbright X-ray filaments of the SNR RXJ1713.7-3946 (also named G347.3-0.5), which are \ninterpreted as resulting from fast synchrotron cooling of TeV electrons in a magnetic \nfield amplified to milligauss levels \\cite{uch07}. Such a high magnetic field is \nlikely the result of a nonlinear amplification process associated with the efficient \nDSA of CRs \\cite{bel01,ama06,vla06}. \n\nThe other interpretation that has been proposed to account for the thin X-ray filaments \nis that they reflect the spatial distribution of the ambient magnetic field rather than\nthe spatial distribution of the ultrarealtivistic electrons \\cite{poh05}. In this \nscenario, the magnetic field is thought to be amplified at the shock as well, but the \nwidth of the X-ray rims is not set by $l_{\\rm diff}$ and $l_{\\rm adv}$, but by the \ndamping length of the magnetic field behind the shock. In this case, the relation given \nabove between the rim thickness and the \ndownstream magnetic field would not be valid. Comparison of high-resolution X-ray and \nradio images could allow to distinguish between the two interpretations, because \nthe synchrotron energy losses are expected to be relatively small for GeV electrons \nemitting in the radio band \\cite{vin03}. Thus, if the X-ray filaments are due to \nrapid synchrotron cooling of TeV electrons, the same structures should not be seen in \nradio images. A recent detailed study of Tycho's SNR has not allowed to draw firm \nconclusions on the role of magnetic damping behind the blast wave \\cite{cas07}. \nFurther high-resolution observations of SNRs in radio wavelengths would be very useful. \n\nThe findings that (1) DSA can proceed at nearly the maximum possible rate (i.e. the Bohm \nlimit) and (2) the magnetic field in the acceleration region can be strongly amplified, \nsuggest that CR ions can reach higher energies in SNR shocks than previously estimated \nby Lagage \\& Cesarsky \\cite{lag83}. Thus, Berezhko \\& V\\\"olk \\cite{ber07} have argued \nthat protons can be accelerated in SNRs up to the energy of the knee in the CR \nspectrum, at $3\\times10^{15}$~eV. But relaxing the assumption of Bohm diffusion used \nin the calculations of Berezhko \\& V\\\"olk, Parizot et al. \\cite{par06} have obtained \nlower maximum proton energies for five young SNRs. These authors have derived an upper \nlimit of $\\sim8\\times10^{14}$~eV on the maximum proton energy, $E_{p,\\rm max}$, and \nhave suggested that an additional CR component is required to explain the CR data \nabove the knee energy. Recently, Ellison \\& Vladimirov \\cite{ell08} have pointed out \nthat the average magnetic field that determines the maximum proton energy can be \nsubstantially less than the field that determines the maximum electron energy. This is \nbecause electrons remain in the vicinity of the shock where the magnetic field can be  \nstrongly amplified, whereas protons of energies $E_p>E_{e,\\rm max}$ diffuse farther in \nthe shock precursor region where the field is expected to be weaker \n($E_{p,\\rm max}>E_{e,\\rm max}$ because radiation losses affect the electrons only). \nThis nonlinear effect of efficient DSA could reduce $E_{p,\\rm max}$ relative \nto the value expected from test-particle acceleration. Nonetheless, recent calculations \nof $E_{p,\\rm max}$ in the framework of nonlinear DSA models suggest that SNRs might \nwell produce CRs up to the knee \\cite{ell08,bla07}.\n\n\\section{TeV gamma-ray emission from SNRs}\n\nAtmospheric Cerenkov telescopes have now observed high-energy $\\gamma$ rays from six \nshell-type SNRs: Cas A with HEGRA \\cite{aha01} and MAGIC \\cite{alb07a}, \nRX~J1713.7-3946 with CANGAROO \\cite{eno02} and HESS \\cite{aha07a}, RX~J0852.0-4622 \n(Vela Junior) with CANGAROO-II \\cite{kat05} and HESS \\cite{aha07b}, RCW~86 with HESS \n\\cite{hop07}, IC~443 with MAGIC \\cite{alb07b}, and very recently SN~1006 in deep HESS \nobservations \\cite{aha05b}. In addition, four $\\gamma$-ray sources discovered in the \nGalactic plane survey performed with HESS are spatially coincident with SNRs \\cite{aha06a}. \n\nWith an angular resolution of $\\sim0.06^\\circ$ for individual $\\gamma$ rays\n\\cite{aha07a,aha07b}, HESS has provided detailed images above 100 GeV of the extended \nSNRs RX~J1713.7-3946 and RX~J0852.0-4622 (their diameters are $\\sim1^\\circ$ and \n$\\sim2^\\circ$, respectively). In both cases the images show a shell-like structure \nand there is a striking correlation between the morphology of the $\\gamma$-ray \nemission and the morphology previously\nobserved in X-rays. For both objects the X-ray emission is completely dominated by \nnonthermal synchrotron radiation. The similarity of the $\\gamma$-ray and X-ray images \nthus suggest that the high-energy emission might also be produced by ultrarelativistic\nelectrons, via inverse Compton (IC) scattering off cosmic-microwave-background (CMB), \noptical-starlight and infrared photons. The $\\gamma$-ray radiation would then be \nproduced by electrons of energy (in the Thompson limit)\n\\begin{equation}\nE_e \\sim \\bigg({3 \\over 4}{E_\\gamma \\over E_\\star}\\bigg)^{1\/2} m_e c^2,\n\\label{eqvt8}\n\\end{equation}\nwhere $E_\\star$ is the typical energy of the seed photons, $E_\\gamma$ is the average \nfinal energy of the upscattered photons and $m_e$ is the electron  mass. For the CMB, \nwhose contribution to the total IC emission of SNRs generally dominates, $E_\\star \\sim \n3kT_{\\rm CMB}=7.1\\times10^{-4}$~eV ($k$ is the Boltzmann constant and \n$T_{\\rm CMB}=2.73$~K). Significant $\\gamma$-ray emission beyond $E_\\gamma=30$~TeV has \nbeen detected from RX~J1713.7-3946 \\cite{aha07a}. Thus, an IC origin for the high-energy \nemission would imply that electrons are accelerated to more than 90~TeV in this object. \nIn more accurate calculations that take into account the contributions of the optical \nand infrared interstellar radiation fields, the maximum electron energy is found to be \n$E_{e,\\rm max} \\sim 15$--40~TeV \\cite{por06}. This result is consistent with the value \nof $E_{e,\\rm max}$ derived from the width of X-ray filaments in RX~J1713.7-3946, \n$E_{e,\\rm max}=36$~TeV \\cite{par06}. \n\nAssuming that the same population of ultrarelativistic electrons produce both the \nobserved TeV $\\gamma$-rays and nonthermal X-rays, the mean magnetic field in the \ninteraction region can be readily estimated from the ratio of synchrotron to IC \nluminosities:\n\\begin{equation}\n{L_{\\rm syn} \\over L_{\\rm IC}}={U_B \\over U_{\\rm rad}}={B^2 \\over 8\\pi U_{\\rm rad}},\n\\label{eqvt9}\n\\end{equation}\nwhere $U_B=B^2 \/ (8\\pi)$ is the magnetic field energy density and $U_{\\rm rad}$ is \nthe total energy density of the seed photon field. With \n$U_{\\rm CMB}\\approx 0.25$~eV~cm$^{-3}$ for the CMB and \n$U_{\\rm IR}\\approx 0.05$~eV~cm$^{-3}$ for the interstellar infrared background (e.g. \n\\cite{aha06b}), we have $U_{\\rm rad}\\approx 0.3$~eV~cm$^{-3}$ (we neglect here the \ncontribution to the IC emission of the optical starlight background). Then, from the\nmeasured ratio $L_{\\rm syn} \/ L_{\\rm IC}\\approx 10$ for RX~J1713.7-3946 \\cite{aha06b}, \nwe obtain $B\\approx 11~\\mu$G. This value is significantly lower than the downstream \nmagnetic field estimated from the observed X-ray rims: $B\\sim80~\\mu$G \\cite{par06} (or \n$B>65~\\mu$G in Ref.~\\cite{ber06}). In other words, if the magnetic\nfield in the electron interaction region is as high as derived from the width of the \nX-ray filaments, IC radiation cannot account for the TeV $\\gamma$-ray data. \n\nThe magnetic field amplification is the main argument to favor a hadronic origin for \nthe high-energy $\\gamma$ rays produced in RX~J1713.7-3946 and other SNRs \\cite{ber06}. \nThe shape of the measured $\\gamma$-ray spectrum below $\\sim$1~TeV has also been used\nto advocate that the high-energy emission might not be produced by IC scattering \n\\cite{aha06b}, but the IC calculations of Ref.~\\cite{por06} reproduce the broadband \nemission of RX~J1713.7-3946 reasonably well. In the hadronic scenario, the TeV \n$\\gamma$ rays are due to nuclear collisions of accelerated protons and heavier \nparticles with ambient ions, which produce neutral pions $\\pi^0$ that decay in 99\\% \nof the cases into two photons with energies of 67.5~GeV each in the $\\pi^0$ rest \nframe ($2\\times67.5$~GeV is the $\\pi^0$ mass). At TeV energies in the observer rest \nframe, the spectrum of the $\\pi^0$-decay $\\gamma$ rays essentially reproduces,\nwith a constant scaling factor, the one of the parent ultrarelativistic particles. \nThe accelerated proton energies can be estimated from the $\\gamma$-ray spectrum \nas $E_p \\sim E_\\gamma\/0.15$ \\cite{aha07a}. The detection of $\\gamma$ rays with \n$E_\\gamma>30$~TeV in RX~J1713.7-3946 thus implies that protons are accelerated to more \nthan 200~TeV, which is still about an order of magnitude below the energy of the knee. \n\nHowever, the hadronic scenario is problematic for RX~J1713.7-3946. Due to the lack of \nthermal X-ray emission, the remnant is thought to expand mostly in a very diluted \nmedium of density $n<0.02$~cm$^{-3}$ \\cite{cas04}. It is likely that the SN \nexploded in a bubble blown by the wind of the progenitor star. The flux of $\\gamma$ \nrays produced by pion decay is proportional to the product of the number of \naccelerated protons and the ambient medium density. Thus, the total energy contained\nin CR protons would have to be large to compensate the low density of the ambient \nmedium. From the $\\gamma$-ray flux measured with HESS from the center of the remnant, \nPlaga \\cite{pla08} has recently estimated that the total CR-proton energy would have to \nbe $>4\\times10^{51}$~erg! Katz \\& Waxman \\cite{kat08} also argue against a hadronic \norigin for the TeV emission from RX~J1713.7-3946. They show that it would require that \nthe CR electron-to-proton abundance ratio at a given relativistic energy \n$K_{\\rm ep}\\lsim2\\times10^{-5}$, which is inconsistent with the limit they derived from \nradio observations of SNRs in the nearby galaxy M33, $K_{\\rm ep}\\gsim10^{-3}$. Moreover, radio \nand X-ray observations of RX~J1713.7-3946 suggest that the blast wave has recently hit \na complex of molecular clouds located in the western part of the remnant \\cite{cas04}. \nThe ambient medium density in this region has been estimated to be $\\sim300$~cm$^{-3}$. \nIn the hadronic scenario, a much higher $\\gamma$-ray flux would be expected in this \ndirection, contrary to the observations \\cite{pla08}. Thus, the $\\gamma$-ray morphology \nrevealed by HESS practically rules out pion production as the main contribution to \nthe high-energy radiation of RX~J1713.7-3946.\n\nBut then, why the magnetic field given by the ratio of synchrotron to IC luminosities \n(eq.~[\\ref{eqvt9}]) is inconsistent with the field derived from the X-ray filaments? \nThis suggests that the filamentary structures observed with {\\it Chandra} and \n{\\it XMM-Newton} are localized regions where the magnetic field is enhanced in \ncomparison with the mean downstream field \\cite{kat08}. It is possible that the \nmagnetic field is amplified at the shock as part of the nonlinear DSA process, but then \nrapidly damped behind the blast wave \\cite{poh05}. The mean field downstream the shock\nwould then not be directly related to the observed thickness of the X-ray rims. \n\nAlthough the unambiguous interpretation of the TeV observations of shell-type SNRs \nremains uncertain, the high-energy $\\gamma$-ray emissions from RX~J1713.7-3946 and \nRX~J0852.0-4622 are probably produced by IC scattering \\cite{kat08}. For Cas~A, the case \nfor a hadronic origin of the TeV radiation may be more compelling, as the density of the \nambient medium is higher \\cite{vin03}. The new source MAGIC~J0616+225 \\cite{alb07b} which \nis spatially coincident with IC~443 may also be produced by pion decay. IC~443 is one of \nthe best candidates for a $\\gamma$-ray source produced by interactions between CRs \naccelerated in a SNR and a nearby molecular cloud \\cite{tor03}. Hopefully the upcoming\n{\\it GLAST} satellite will allow a clear distinction between hadronic and electronic \n$\\gamma$-ray processes in these objects. With the expected sensitivity of the LAT \ninstrument between 30 MeV and 300 GeV, {\\it GLAST} observations of SNRs should \ndifferentiate between pion-decay and IC spectra \\cite{fun08}.\n\n\\section{Radio emission and nonlinear diffusive shock acceleration in SN 1993J}\n\nAbout 30 extragalactic SNe have now been detected at radio wavelengths\\footnote{See \nhttp:\/\/rsd-www.nrl.navy.mil\/7213\/weiler\/kwdata\/rsnhead.html.}. In a number of cases, \nthe radio evolution has been monitored for years after outburst. It is generally \naccepted that the radio emission is nonthermal synchrotron radiation from relativistic \nelectrons accelerated at the SN shock wave \\cite{che82}. At early epochs the radio flux \ncan be strongly attenuated by free-free absorption in the wind lost from the progenitor \nstar prior to the explosion. Synchrotron self-absorption can also play a role in some \nobjects \\cite{che98}. The radio emission from SNe can provide unique information on \nthe physical properties of the circumstellar medium (CSM) and the final stages of \nevolution of the presupernova system \\cite{wei02}. We show here that this emission \ncan also be used to study critical aspects of the DSA mechanism. \n\nThe type IIb SN~1993J, which exploded in the nearby galaxy M81 at a distance of \n$3.63\\pm0.34$~Mpc, is one of the brightest radio SNe ever detected (see \\cite{wei07} \nand references therein). Very long baseline interferometry (VLBI) imaging has revealed \na decelerating expansion of a shell-like radio source, which is consistent with the\nstandard model that the radio emission arises from a region behind the forward shock \npropagating into the CSM. The expansion has been found to be self-similar \n\\cite{mar97}, although small departures from a self-similar evolution have been \nreported \\cite{bar00}. The velocity of the forward shock can be estimated from the \nmeasured outer radius of the radio shell, i.e. the shock radius $r_s$, as \n$V_s=dr_s\/dt=3.35\\times10^4~t_d^{-0.17}$~km~s$^{-1}$, where $t_d$ is the time after \nshock breakout expressed in days. \n\nExtensive radio monitoring of the integrated flux density of SN~1993J has been \nconducted with the Very Large Array and several other radio telescopes \\cite{wei07}. \nFigure~\\ref{figvt1} shows a set of measured light curves at 0.3~cm (85--110 GHz), \n1.2~cm (22.5~GHz), 2~cm (14.9~GHz), 3.6~cm (8.4~GHz), 6~cm (4.9~GHz), and 20~cm \n(1.4~GHz). We see that at each wavelength the flux density first rapidly increases \nand then declines more slowly as a power in time (the data at 0.3~cm do not allow to \nclearly identify this behavior). The radio emission was observed to suddenly decline \nafter day $\\sim$3100 (not shown in Fig.~\\ref{figvt1}), which is interpreted in terms \nof an abrupt decrease of the CSM density at radial distance from the progenitor \n$r\\sim3\\times10^{17}$~cm \\cite{wei07}. The maximum intensity is reached first at lower \nwavelengths and later at higher wavelengths, which is characteristic of absorption \nprocesses. For SN~1993J, both free-free absorption in the CSM and synchrotron \nself-absorption are important \\cite{che98,fra98,wei07}. To model light curves of radio \nSNe, Weiler et al. \\cite{wei02,wei07} have developed a semi-empirical formula that \ntakes into account these two absorption mechanisms. For SN~1993J, the best fit to the \ndata using this semi-empirical model ({\\it dotted blue curves} in Fig.~\\ref{figvt1}) \nrequires nine free parameters \\cite{wei07}. \n\n\\begin{figure}\n\\includegraphics[width=1\\textwidth]{f1.eps}\n\\caption{Radio light curves for SN~1993J at 0.3, 1.2, 2, 3.6, 6, and 20~cm. The dotted\nblue lines represent the best fit semi-empirical model of Ref.~\\cite{wei07}. The \ndashed red (resp. solid green) lines show results of the present model for \n$\\eta_{\\rm inj}^p=\\eta_{\\rm inj}^e=10^{-5}$ (resp. \n$\\eta_{\\rm inj}^p=2\\times10^{-4}$ and $\\eta_{\\rm inj}^e=1.4\\times10^{-5}$; see text).\nThe data are from Ref.~\\cite{wei07} and references therein. \n}\n\\label{figvt1}\n\\end{figure}\n\nI have developed a model for the radio emission of SN~1993J, which is inspired by \nprevious works on the morphology of synchrotron emission in young SNRs \n\\cite{cas05,ell05}. The model will be presented in detail in a forthcoming publication \n\\cite{tat08} and I only give here broad outlines. First, the density profile for the \nCSM is taken as $\\rho_{\\rm CSM}(r)=\\rho_0(r\/r_0)^{-2}$ as expected for a constant wind \nmass-loss rate and terminal velocity. Here $r_0=3.49\\times10^{14}$~cm is the shock \nradius at $t=1$~day after outburst and $\\rho_0$ is a free parameter. Evidence for a \nflatter CSM density profile has been advocated ($\\rho_{\\rm CSM}\\propto r^{-s}$, \nwith $s \\sim 1.6$; see \\cite{wei07} and references therein), based on the measured time \ndependence of the optical depth to free-free absorption in the CSM, $\\tau_{\\rm ff}$. \nHowever, Fransson \\& Bj\\\"ornsson \\cite{fra98} have shown that the time \nevolution of $\\tau_{\\rm ff}$ can be explained by a decrease of the CSM temperature with \n$r$ together with the standard $r^{-2}$ distribution for the density. The results of the\npresent work provide support to this latter interpretation \\cite{tat08}. Thus, in the \npresent model, free-free absorption is calculated assuming the time dependence of \n$\\tau_{\\rm ff}$ obtained in Ref.~\\cite{fra98} and using the best-fit value of $\\rho_0$ \n(or more precisely $\\rho_0^2$) as a normalization factor. \n\nBecause synchrotron self-absorption is important in SN~1993J, the strength and evolution \nof the mean magnetic field in the region of the radio emission can be estimated from \nthe measured peak flux at different wavelengths \\cite{che98}. Using equation~(12) of \nRef.~\\cite{che98}, I obtain from the data at 1.2, 2, 3.6, 6, and 20~cm: \n\\begin{equation}\n\\langle B \\rangle = (46 \\pm 19) \\alpha^{-2\/9} t_d^{-1.01\\pm0.09}~{\\rm G}, \n\\label{eqvt10}\n\\end{equation}\nwhere $\\alpha$ is the ratio of the total energy density in relativistic electrons to\nthe magnetic energy density. The errors include the uncertainty in the contribution of \nfree-free absorption. This time dependence of $\\langle B \\rangle$ is close to that \nexpected if the magnetic field at the shock is amplified by a constant factor from \nthe available kinetic energy density\n(see \\cite{bel01}). In this case, one expects $B^2\/8\\pi \\propto \\rho_{\\rm CSM} V_s^2$,\nwhich gives $B\\propto t^{-1}$ for $\\rho_{\\rm CSM}\\propto r^{-2}$. Note that the \nflatter CSM density profile supported by Weiler et al. \\cite{wei07},  \n$\\rho_{\\rm CSM}\\propto r^{-1.6}$, would imply for the assumed scaling \n$B\\propto \\rho_{\\rm CSM}^{1\/2} V_s \\propto t^{-0.83}$, which is somewhat inconsistent \nwith the data. \n\nBased on the measured time dependence of $\\langle B \\rangle$, the immediate postshock \nmagnetic field is assumed to be of the form $B_d=B_0t_d^{-1}$, where $B_0$ is a free\nparameter expected to be in the range $\\sim$100--600~G for a typical value of $\\alpha$ \nin the range $\\sim$10$^{-5}$--10$^{-2}$ (see eq.~[\\ref{eqvt10}]). The evolution of \nthe magnetic field behind the shock is then calculated from the assumption that the \nfield is carried by the flow, frozen in the plasma, so that the parallel and \nperpendicular magnetic field components evolve conserving flux (see \\cite{cas05} and \nreferences therein). Results obtained with the alternative assumption that the \nmagnetic field is rapidly damped behind the shock wave will be given in \\cite{tat08}. \n\nThe hydrodynamic evolution of the plasma downstream the forward shock is calculated \nfrom the two-fluid, self-similar model of Chevalier \\cite{che83}, which takes into \naccount the effects of CR pressure on the dynamics of \nthe thermal gas. The overall structure of SNRs can be described by self-similar \nsolutions, if the initial density profiles in the ejected material (ejecta) \nand in the ambient medium have power-law distributions, and if the ratio of \nrelativistic CR pressure to total pressure at the shock front is constant \\cite{che83}. \nThe backreaction of energetic ions can strongly modify the shock structure of young \nSNRs, such as e.g. Kepler's remnant \\cite{dec00}. But the situation is different for \nSN~1993J because of the much higher magnetic field in the shock precursor region, \nwhich implies that energy is very efficiently transfered from the CRs to the thermal \ngas via Alfv\\'en wave dissipation \\cite{ber99}. The resulting increase in the gas \npressure ahead of the viscous subshock is found to limit the overall compression ratio, \n$r_{\\rm tot}$, to values close to 4 (i.e. the standard value for a test-particle \nstrong shock) even for efficient DSA. Thus, the hydrodynamic evolution of SN~1993J \ncan be safely calculated in the test-particle, self-similar approximation. \n\nBoth the energy spectra of the accelerated particles and the thermodynamic properties of \nthe gas just behind the shock front (i.e. the boundary conditions for the self-similar \nsolutions of the hydrodynamic evolution) are calculated with the semianalytic model of \nnonlinear DSA developed by Berezhko \\& Ellison \\cite{ber99} and Ellison et al. \n\\cite{ell00}. However, a small change to the model has been made: the Alfv\\'en waves \nare assumed to propagate isotropically in the precursor region and not only in the \ndirection opposite to the plasma flow (i.e. eqs.~(52) and (53) of Ref.~\\cite{ber99} are \nnot used). This is a reasonable assumption given the strong, nonlinear magnetic field \namplification \\cite{bel01}. Although the semianalytic model strictly applies to \nplane-parallel, steady state shocks, it has been sucessfully used in Ref.~\\cite{ell00} \nfor evolving SNRs. The main parameter of this model, \n$\\eta_{\\rm inj}^p$, is the fraction of total shocked protons in protons with momentum \n$p \\geq p_{\\rm inj}$ injected from the postshock thermal pool into the DSA process. \nThe work of Ref.~\\cite{bla05} allows us to accurately relate the injection momentum \n$p_{\\rm inj}$ to $\\eta_{\\rm inj}^p$. Similarly, we define $\\eta_{\\rm inj}^e$ for the \nelectron injection. The latter parameter is not important for the shock structure, \nbecause the fraction of total particle momentum carried by electrons is negligible, but \nit determines the brightness of optically-thin synchrotron emission for a given magnetic \nfield strength. The electrons accelerated at the shock experience adiabatic and \nsynchrotron energy losses as they are advected downstream with the plasma flow. The \nspectral evolution caused by these losses is calculated as in Ref.~\\cite{rey98}. \nFinally, once both the nonthermal electron distribution and the magnetic field in a \ngiven shell of material behind the forward shock have been determined, the synchrotron \nemission from that shell can be calculated \\cite{pac70}. The total radio emission along \nthe line of sight is then obtained from full radiative transfer calculations that \ninclude synchrotron self-absorption. \n\nWith the set of assumptions given above, the model has four free parameters: $\\rho_0$, \n$B_0$, $\\eta_{\\rm inj}^p$ and $\\eta_{\\rm inj}^e$. While the product $\\rho_0 \\times \n\\eta_{\\rm inj}^e$ is important for the intensity of optically-thin synchrotron \nemission, the CSM density normalization $\\rho_0$ also determines the level of free-free\nabsorption. The main effect of changing $B_0$ is to shift the radio light curves in time,\nbecause the turn-on from optically-thick to optically-thin synchrotron emission is \ndelayed when the magnetic field is increased. The proton injection paramater \n$\\eta_{\\rm inj}^p$ determines the shock structure and hence influences the shape of the\nelectron spectrum. \n\nCalculated radio light curves are shown in Fig.~\\ref{figvt1} for \n$\\rho_0=1.8\\times10^{-15}$~g~cm$^{-3}$, $B_0=400$~G, and two sets of injection \nparameters: $\\eta_{\\rm inj}^p=\\eta_{\\rm inj}^e=10^{-5}$ (test-particle case), and \n$\\eta_{\\rm inj}^p=2\\times10^{-4}$ and $\\eta_{\\rm inj}^e=1.4\\times10^{-5}$. We see that \nin the test-particle case, the decline of the optically-thin emission with time is too\nslow as compared to the data, except at 0.3~cm. The CR-modified shock provides a better \noverall fit to the measured flux densities, although significant deviations from the \ndata can be observed. In particular, we see that the calculated light curve at 0.3~cm \nfalls short of the data at this wavelength. It is possible that the\ndeviations of the best-fit curves from the data partly arise from the \napproximations used in the DSA model of Ref.~\\cite{ber99}, in which the nonthermal \nphase-space distribution function $f(p)$ is described as a three-component power law. \nBut it is also possible that it tells us something about the magnetic field evolution \nin the downstream region. The spatial distribution of the postshock magnetic field will \nbe studied in Ref.~\\cite{tat08} by comparing calculated synchrotron profiles with the \nobserved average profile of the radio shell. \n\n\\begin{figure}\n\\includegraphics[width=1\\textwidth]{f2.eps}\n\\caption{Calculated postshock phase space distribution functions, $f(p)$, vs. kinetic \nenergy, at day 100 after shock breakout. Following Ref.~\\cite{ber99}, $f(p)$ \nhas been multiplied by $[p\/(m_pc)]^4$ to flatten the spectra, and by $[(m_pc)^3\/n_u]$ to \nmake them dimensionless ($m_p$ is the proton mass and $n_u$ the proton number density \nahead of the shock precursor). The red lines are for protons and the black lines for \nelectrons. The two sets of injection parameters $\\eta_{\\rm inj}^p$ and $\\eta_{\\rm inj}^e$ \nare those used for the synchrotron calculations shown in Fig.~1.}\n\\label{figvt2}\n\\end{figure}\n\nFor SN~1993J, the main effect of the CR pressure is to reduce the compression ratio \nof the subshock, $r_{\\rm sub}$, whereas the overall compression ratio $r_{\\rm tot}$ \nremains nearly constant (see above). For $\\eta_{\\rm inj}^p=2\\times10^{-4}$, $r_{\\rm sub}$ \nis found to decrease from 3.58 to 3.35 between day 10 and day 3100 after outburst, whereas \n$r_{\\rm tot}$ stays between 4 and 4.04. Such a shock modification affects essentially \nthe particles of energies $<m_pc^2$ that remain in the vicinity of the subshock during \nthe DSA process. Thus, we see in Fig.~2 that the increase of $\\eta_{\\rm inj}^p$ from \n10$^{-5}$ to $2\\times10^{-4}$ steepens the phase-space distribution functions $f(p)$\nof both protons and electrons between their thermal Maxwell-Boltzmann distributions and \n$\\sim1$~GeV, as the spectral index in this energy domain,  \n$q_{\\rm sub}=3r_{\\rm sub}\/(r_{\\rm sub}-1)$ with $f(p)\\propto p^{-q_{\\rm sub}}$ \n\\cite{ber99}, increases with decreasing $r_{\\rm sub}$. The radio light curves provide \nclear evidence that the spectral index of the nonthermal electrons below 1~GeV is higher \nthan the strong-shock test-particle value $q_{\\rm sub}=4$. For the preferred injection \nparameters $\\eta_{\\rm inj}^p=2\\times10^{-4}$ and $\\eta_{\\rm inj}^e=1.4\\times10^{-5}$, \nthe calculated electron-to-proton ratio at, e.g., 10~GeV is $K_{\\rm ep}=7.8\\times10^{-4}$ \nat $t_d=100$~days (Fig.~2) and $K_{\\rm ep}=6.3\\times10^{-4}$ at $t_d=3100$~days. These values \nare roughly consistent with -- but slightly lower than -- those recently estimated for the \nblast wave of Tycho's SNR, $K_{\\rm ep}\\sim10^{-3}$ \\cite{cas07}. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.7\\textwidth]{f3.eps}\n\\end{center}\n\\caption{Estimated maximum proton energy in SN~1993J as a function of time after shock \nbreakout. $E_{\\rm max}^{\\rm age}$ and $E_{\\rm max}^{\\rm size}$ are the maximum energies \ncaused by the finite shock age and size, respectively. The resulting maximum proton \nenergy ({\\it solid line}) is the minimum of these two quantities.}\n\\label{figvt3}\n\\end{figure}\n\nThe immediate postshock magnetic field obtained in this work, $B_d\\approx 400\\times \nt_d^{-1}$~G is in very good agreement with the mean field in the synchrotron-emitting \nregion previously estimated by Fransson \\& Bj\\\"ornsson \\cite{fra98}, \n$\\langle B \\rangle\\approx 370\\times t_d^{-1}$~G. As already pointed out by Bell \\& Lucek \n\\cite{bel01}, the derived magnetic field strength in the forward shock precursor is \nconsistent with a simple estimate based on nonlinear magnetic field \namplification driven by the pressure gradient of accelerated particles. For \n$\\eta_{\\rm inj}^p=2\\times10^{-4}$, the ratio of the mean magnetic energy density in the \nshock precursor to the energy density in CR protons is found to slightly decrease from 0.6 \nto 0.5 between day 10 and day 3100 after outburst \\cite{tat08}. It is interesting that \nthe magnetic field strength is very close to that given by equipartition.  \n\nAs also pointed out in Ref.~\\cite{bel01}, CRs were probably accelerated to very high \nenergies in SN~1993J short after shock breakout. The maximum proton energy $E_{p,\\rm max}$ \nis expected to be limited by the spatial extend of the shock, because high-energy \nparticles diffusing upstream far enough from the shock front can escape from the \nacceleration region. This size limitation can be estimated by assuming the existence of \nan upstream free escape boundary located at some constant fraction $f_{\\rm esc}$ of the \nshock radius: $d_{\\rm FEB}=f_{\\rm esc}r_s$. The maximum energy that particles can acquire \nbefore reaching this boundary is then obtained by equalling $d_{\\rm FEB}$ to the upstream \ndiffusion length, $l \\sim \\kappa \/ V_s$ (e.g. \\cite{ell08}). Calculated time evolution of \n$E_{p,\\rm max}$ is shown in Fig.~\\ref{figvt3} for $f_{\\rm esc}=0.05$ (see, e.g., \\cite{ell05}) \nand $\\eta_\\kappa=3$ (see eq.~\\ref{eqvt7} and, e.g., \\cite{par06}). We see that the maximum \nenergy caused by the particle  escape, $E_{\\rm max}^{\\rm size}$ \n($\\propto f_{\\rm esc}\\eta_\\kappa^{-1}$), becomes rapidly lower than the maximum energy \nassociated with the shock age, $E_{\\rm max}^{\\rm age}$ ($\\propto \\eta_\\kappa^{-1}$), which \nis obtained by time integration of the acceleration rate $(dE\/dt)_{\\rm acc}$ from an initial \nacceleration time assumed to be $t_0=1$~day after outburst. The resulting maximum proton \nenergy (i.e. the minimum of $E_{\\rm max}^{\\rm size}$ and $E_{\\rm max}^{\\rm age}$) is \nsignificantly lower than the previous estimate of Ref.~\\cite{bel01}, \n$E_{p,\\rm max}\\sim3\\times10^{17}$~eV. Moreover, we did not take into account the nonlinear \neffects recently pointed out by Ellison \\& Vladimirov \\cite{ell08}, which could further \nreduce $E_{p,\\rm max}$. However, SN~1993J may well have accelerated protons above the knee \nenergy of $3\\times10^{15}$~eV (Fig.~\\ref{figvt3}), which provides support to the scenario \nfirst proposed by V\\\"olk \\& Biermann \\cite{vol88} that the highest-energy Galactic CRs are \nproduced by massive stars exploding into their own wind. In this context, it is instructive \nto estimate the total energy acquired by the CR particles during the early phase of \ninteraction between the SN ejecta and the red supergiant wind:\n\\begin{equation}\nE_{\\rm CR} \\cong \\int_{t_d=1}^{3100} \\epsilon_{\\rm CR}(t) \\times {1 \\over 2} \n\\rho_{\\rm CSM}(r_s) V_s^3 \\times 4\\pi r_s^2 dt = 6.8 \\times 10^{49}~\\rm erg,\n\\label{eqvt11}\n\\end{equation}\nwhere $\\epsilon_{\\rm CR}(t)$ is the time-dependent fraction of total incoming energy \nflux, $F_0(r_s) \\cong 0.5\\rho_{\\rm CSM}(r_s) V_s^3$, going into CR particles. With the \nbest-fit parameters obtained from the radio light curves, $\\epsilon_{\\rm CR}(t)$ is found \nto slowly increase from 12\\% to 16\\% between day 10 and day 3100 after outburst. It is \nremarkable that the value obtained for $E_{\\rm CR}$ is very close to the mean energy per \nSN required to account for the Galactic CR luminosity, $\\approx 7.5 \\times 10^{49}$~erg \n(see Sect.~1), which suggests that SN~1993J might be typical of the SNe that produce the \nCR population in our own Galaxy. \n\n\\section{Conclusions}\n\nObservations of young shell-type SNRs with the {\\it Chandra} and {\\it XMM-Newton} X-ray \nspace observatories give more credit to the current paradigm that the bulk of Galactic \nCRs are accelerated in shock waves generated by SN explosions. Evidence is accumulating\nsuggesting that acceleration in SN shocks can be nearly as fast as the Bohm limit and \nthat self-excited turbulence in the shock vicinity can strongly amplify the ambient \nmagnetic field, thus allowing the acceleration of CR ions to energies above 10$^{15}$~eV. \nBesides, observations of nonlinear effects caused by efficient DSA support the fact that \nthe acceleration efficiency can be high enough to account for the Galactic CR luminosity. \nBut the direct and unambiguous observation of ion acceleration in SN shocks \nremains uncertain, as it is difficult at the present time to distinguish between \npion decay and IC scattering as the main mechanism responsable for the very-high-energy \n$\\gamma$-ray emission detected with ground-based atmospheric Cerenkov telescopes from \nseveral shell-type SNRs. Following Refs.~\\cite{pla08,kat08} I have argued, however, that \nthe TeV $\\gamma$ rays emitted in RX~J1713.7-3946 and RX~J0852.0-4622 are probably \nproduced by ultrarelativistic electrons, via IC scattering off CMB, optical-starlight and \ninfrared photons. One of the most critical ingredients for the interpretation of these \nobservations is the evolution of the postshock magnetic field in the downstream region \n(see Sect.~3). Hopefully, the upcoming GLAST satellite (launch scheduled for May 2008) \nwill allow a clear identification of the contributions of hadronic and electronic \n$\\gamma$-ray emission processes in SNRs, which in turn will be useful to understand the \nimportance of magnetic field damping behind the forward shock.\n\nRadio observations of extragalactic SNe can also shed light on the DSA process. The \nimpressive set of radio data available for SN~1993J make this \nobject a unique laboratory to study particle acceleration in a SN shock. Using a \nsemianalytic model of nonlinear DSA to explain the radio light curves, the results I have \nobtained suggest, in particular, that the acceleration of ions has become efficient soon \nafter outburst and that the magnetic field in the forward shock vicinity has been amplified \nto equipartition with the CR energy density. The field amplification implies that CR \nprotons may have been accelerated to energies above $3\\times10^{15}$~eV during the early \nphase of interaction between the SN ejecta and the red supergiant wind lost from the \nprogenitor star. During this time, which lasted only $\\sim$8.5~years, the shock processed \nin the expansion a total energy of $4.6\\times10^{50}$~erg and the mean acceleration \nefficiency was found to be $\\epsilon_{\\rm CR}=15\\%$. Thus, a total energy of almost \n$7\\times10^{49}$~erg has been stored up by CR particles during this phase. A significant \nfraction of this energy might have escaped into the interstellar medium after day \n$\\sim$3100, when the shock started to expand into a more diluted CSM. \n\nThe results obtained for SN~1993J suggest that massive stars exploding into a wind \nenvironment could be a major source of Galactic CRs, as first proposed by V\\\"olk \n\\& Biermann \\cite{vol88}. It is well known that most massive stars are born in OB \nassociations, whose activity formes superbubbles of hot and turbulent gas inside \nwhich the majority of core-collapse SNe explode. Turbulent re-acceleration inside \nsuperbubbles may modify the spectrum of CRs and, in particular, increase their \nmaximum energy (see \\cite{par04} and references therein). \n\n\\acknowledgements{It is a pleasure to thank Anne Decourchelle and J\\\"urgen Kiener for \ncritical reading of the manuscipt, Margarita Hernanz for numerous discussions and her \nhospitality at IEEC-CSIC, and Jordi Isern for the invitation to the conference. \nFinancial support from the Generalitat de Catalunya through the AGAUR grant \n2006-PIV-10044 and the project SGR00378 is acknowledged.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}