diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdgks" "b/data_all_eng_slimpj/shuffled/split2/finalzzdgks" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdgks" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nLet $M$ be an $n$-dimensional Riemannian manifold, and let\n$\\Sigma$ be a compact surface with boundary which\ncarries a conformal structure; we do {\\bf not} assume\n$\\Sigma$ is orientable. Many existence theorems for minimal\nsurfaces in the literature \\cite{D31}, \\cite{R}, \\cite{C} find\nsolutions by minimizing the {\\bf energy} of a mapping \n$f:\\Sigma \\to M$, where both $f$ and the conformal structure of\n$\\Sigma$ are allowed to vary. The energy may be written as\n\\begin{equation} \\label{energy}\n E(f) := \\frac12 \\int_\\Sigma (|f_x|^2 + |f_y|^2)\\, dx\\, dy \n\\end{equation}\nwhere $(x,y)$ are local conformal coordinates for $\\Sigma$, and\nsubscripts are used to denote partial derivatives. Write\n$\\frac{D}{\\partial x}$, etc., for\ncovariant partial derivatives in the Riemannian manifold $M$.\nIf the mapping $f$ and the conformal structure on $\\Sigma$ are\nstationary for $E$, then the resulting mapping is {\\bf harmonic}: \n\\begin{equation} \\label{harm}\n\\Delta f :=\n\\frac{D}{\\partial x}\\frac{\\partial f}{\\partial x} +\n\\frac{D}{\\partial y}\\frac{\\partial f}{\\partial y} =0\n\\end{equation}\nand {\\bf conformal}:\n\\begin{equation} \\label{conf}\n |f_x| \\equiv |f_y|, \\langle f_x, f_y \\rangle \\equiv 0. \n\\end{equation}\n\nWe shall refer to a conformally parameterized harmonic mapping as\na {\\em conformally parameterized minimal surface} (CMS). Observe that\nfor any $W^{1,2}$ mapping, $E(f)$ is bounded below by the area \n\\begin{equation} \\label{area}\n A(f) := \\int_\\Sigma |f_x \\wedge f_y |\\, dx\\, dy,\n\\end{equation}\nwith equality if and only if $f$ is a conformal mapping almost\neverywhere. In particular, a mapping which minimizes $E(f)$ in a\ngeometrically defined class of mappings has minimum area among\nmappings of the admissible class (see \\cite{D39}, p. 232 or Remark\n\\ref{serrin} below). Moreover, a conformal mapping which is\nharmonic, that is, stationary for $E$, is minimal, that is,\nstationary for area.\n\nA CMS is an immersion except at a discrete set of {\\bf branch\npoints.} Let a point of $\\Sigma$ be given by $(0,0)$ in some local\nconformal coordinates $(x,y)$ for $\\Sigma$. Write $z=x+iy$. Then\n$(0,0)$ is a {\\em branch point} of $f$ of {\\em order} $m-1$ if for some\nsystem of coordinates $u^1, \\dots, u^n$ for $M$ and for some\ncomplex vector $c$, $f(x,y)$ satisfies the asymptotic description \n$$f^1(x,y) + i f^2(x,y) = c z^m + O(z^{m+1})$$ and \n$$ f^k(x,y) = O(z^{m+1}),$$ \n$k=3,\\dots, n$, as $(x,y)\\to (0,0)$. Here we have written \n$f^k(x,y)$ for the value of the $k^{th}$ coordinate $u^k$ at \n$f(x,y)$, $k=1, \\dots , n$, and $O(z^{m+1})$ denotes any\n``remainder\"\nfunction bounded by a constant times $|z^{m+1}|$. We shall\nrefer to a mapping which is an immersion except at a discrete set\nof branch points as a {\\bf branched immersion} (see \\cite{GOR}).\n\nOur main theorem is \n\\begin{thm} \\label{main}\nSuppose $\\Sigma$ is of the topological type of the real projective\nplane $\\R P^2$. Let a CMS $f:\\Sigma^2 \\to M^3$ have minimum area\namong all $h:\\Sigma \\to M$ which are not homotopic to a constant\nmapping. Then $f$ is an immersion.\n\\end{thm}\nIn order to prove this theorem, we will distinguish between two\ntypes of branch points: see \\cite{O}, \\cite{GOR}, \\cite{Alt72},\n\\cite{Alt73}, and \\cite{G73}. A {\\bf false branch point} of a\nbranched immersion $f:\\Sigma \\to M$ is a branch point $z_0$\nsuch that the image set $f(U)$ is an embedded surface, under\nanother parameterization, for some neighborhood $U$ of $z_0$ in\n$\\Sigma$. Otherwise, we call it a {\\bf true branch point}. A\nbranched immersion $f:\\Sigma \\to M$ is said to be {\\bf ramified}\nif there are two disjoint open sets $V,W \\subset \\Sigma$ with\n$f(V) = f(W)$. If $f$ is ramified in every neighborhood of a point\n$z_0$, we say that $z_0$ is a {\\bf ramified branch point}. Note\nthat any false branch point of a branched immersion must be\nramified. Osserman showed that in codimension one, a branched\nimmersion $f:\\Sigma^2 \\to M^3$ with a true branch point cannot\nminimize area, see \\cite{O} and Theorem \\ref{oss} below, in\ncontradiction to assertions of Douglas (p. 239 of \\cite{D32}) and\nof Courant (footnote p. 46 of \\cite{C41}). \nOn the other hand, regarding false branch points, we\nshall extend to nonorientable surfaces the fundamental theorem of\nbranched immersions in \\cite{G75}, and show that if a branched CMS\n$f:\\R P^2 \\to M^n$ is ramified, with any codimension, then there\nis another CMS $\\widetilde{f}:\\R P^2 \\to M$ with at most half the\narea of $f$. \n\nWe would like to acknowledge the interest of Simon Brendle in this\nproblem, whose questions, not used in \\cite{BBEN}, stimulated us\nto investigate this research topic. We are also indebted to the\nlate Jim Serrin for pointing us toward Remark \\ref{serrin}.\n\n\\section{Analysis of branch points}\\label{anal}\n\nThis section reports on material that has appeared in the\nliterature, see especially \\cite{G73}. In this paper, we shall\ndiscuss certain steps in the interest of clarity and completeness. \n\nLet $\\Sigma^2$ be a compact surface with a conformal structure, $M^n$ a\nRiemannian manifold, and let $f: \\Sigma \\to M$ be a CMS. Consider a\nbranch point $z_0 \\in \\Sigma$ for $f$. Write $D$ for the\nRiemannian connection on $M$. Let local conformal\ncoordinates $(x,y)$ for $\\Sigma$ and local coordinates \n$(q_1, \\dots, q_n)$ for $M$ be introduced with $z_0 = (0,0)$ and\n$f(z_0) = (0,\\dots,0)$ in these coordinates. Then equation\n\\eqref{harm} may be rewritten\n$$\\frac{D}{\\partial\\overline{z}}\\frac{\\partial f}{\\partial z}=0,$$\nwhere we write the complex coordinate $z=x+iy$, \n$\\frac{\\partial f}{\\partial z}= \n\\frac12[\\frac{\\partial f}{\\partial x}-i\\frac{\\partial f}{\\partial\ny}]$, and \n$\\frac{D}{\\partial \\overline z}= \n\\frac12[\\frac{D}{\\partial x} +i \\frac{D}{\\partial y}].$\nIn this form,\nwe see that harmonicity implies that the complex tangent vector\n$\\frac{\\partial f}{\\partial z}$ is holomorphic to first order. It\nis readily shown that for some positive integer $m$ and for some\ncomplex tangent vector $c=a+ib$ to $M$ at $f(z_0)$, \n\\begin{equation}\nf(z)=\\mathcal{R}\\{c z^m\\} +O_2(|z|^{m+1}).\n\\end{equation}\nHere we write $\\mathcal{R}\\{v\\}$ for the real part of a complex\nvector $v$, and we have used the big-O notation with the subscript\n$2$, meaning that as $z \\to 0$, the remainder term is bounded by a\nconstant times $|z|^{m+1}$, its first partial derivatives are\nbounded by a constant times $|z|^m$ and its second partial\nderivatives are bounded by a constant times $|z|^{m-1}$. It\nfollows from the conformality condition \\eqref{conf} that the\ncomplex-bilinear inner product \n$\\langle c,c \\rangle = |a|^2 - |b|^2 +2i \\langle a,b \\rangle =0.$ \nChoose a new system of coordinates $p_1, \\dots, p_n$ for $M$ near \n$f(z_0)$ with $\\frac{\\partial}{\\partial p_1}=a$ and \n$\\frac{\\partial}{\\partial p_2}=b$; and a new system of coordinates\n$(\\widetilde{x},\\widetilde{y})$ for $\\Sigma$ with\n$\\widetilde{z}=\\widetilde{x}+i\\widetilde{y}=|a|^{1\/m} z$.\nThen along the mapping $f$,\n$$p_1+ip_2=\\widetilde{z}^m +\\sigma(\\widetilde{z})$$\n and\n$$p_\\ell=\\psi_\\ell(\\widetilde{z}),$$\n$\\ell = 3,\\dots,n$, where \n$\\sigma(\\widetilde{z}), \\psi_\\ell(\\widetilde{z}) = O_2(\\widetilde{z}^{m+1})$. \nWe now define a non-conformal complex parameter $w=u_1+iu_2$ on a\nneighborhood of the branch point in $\\Sigma:$\n\\begin{equation}\\label{defw}\nw:=\\widetilde{z}\\,\n\\Big[1+\\widetilde{z}^{-m}\\sigma(\\widetilde{z})\\Big]^{1\/m}.\n\\end{equation}\nThen $w$ is a $C^{1,\\alpha}$ coordinate on $\\Sigma,$ for some\nH\\\"{o}lder exponent $\\alpha >0$, in terms of\nwhich the coordinate representation of $f$ is simplified:\n\\begin{equation}\\label{anonpar}\np_1+ip_2=w^m, \n\\end{equation} \nand\n$$p_\\ell=\\phi_\\ell(w)=O_2(w^{m+1}),$$\n$\\ell=3,\\dots,n.$\n\nWe now turn our attention to the case n=3 of {\\bf codimension\none}. The self-intersection of the surface is determined by the\nsingle real-valued function $\\phi(w)=\\phi_3(w).$ Define \n$\\overline\\phi(w)= \\phi(\\zeta_m w)$, where \n$\\zeta_m = e^{2\\pi i\/m}$ is a primitive $m^{\\it th}$ root of unity,\nand let $\\Phi(w)=\\phi(w)-\\overline\\phi(w).$ Then the zeroes of $\\Phi$\ncorrespond to curves of intersection of the surface with itself.\nBut both $\\phi$ and $\\overline\\phi$ satisfy the {\\it same}\nquasilinear minimal surface equation in $M$, with the {\\it same}\ncoefficients. Therefore, their difference \n$\\Phi:= \\phi-\\overline\\phi$ satisfies a linear homogeneous PDE:\n\\begin{equation}\\label{Phieq}\n\\sum_{i,j=1}^2 a_{ij}\\Phi_{u_i u_j}+\\sum_{i=1}^2 a_i \\Phi_{u_i} +\na \\Phi =0, \n\\end{equation}\nwhose coefficients, as functions of $w$, are obtained by \nintegrating from the PDE\nsatisfied by $\\phi$ to the PDE satisfied by $\\overline\\phi$ along\nconvex combinations. We have $a_{ij}(0,0) =\\delta_{ij}$. \n\nIt follows that $\\Phi$ satisfies an asymptotic formula \n\\begin{equation}\\label{asymp}\n\\Phi(w) = {\\mathcal R}\\{A w^N\\} + O_2(w^{N+1}),\n\\end{equation}\nfor some integer $N >m$ and some complex constant $A\\neq 0$ \n(see \\cite{HW}). We shall call\n$N-1$ the {\\bf proper index} of the branch point.\nWe may sketch the proof of Hartman and Wintner in \\cite{HW}. We \nrewrite the PDE \\eqref{Phieq} in terms of the complex \ngradient $\\Phi_w= \\frac12(\\Phi_{u_1}- i \\Phi_{u_2})$. \nIf $\\nabla\\Phi(w)=o(w^{k-1})$, then we test against the function\n$g(w) = w^{-k} (w-\\zeta)^{-1}$, where $\\zeta \\neq 0$ is small, to\nshow that $\\Phi_\\zeta(\\zeta) = a \\zeta^k+o(\\zeta^{k+1})$ for some\n$a\\in \\C$. Proceeding by induction on $k$, one finds formula \n\\eqref{asymp} for some integer $N$ and some $A\\neq 0$, unless \n$\\nabla\\Phi(w) \\equiv 0$. Details are as in \\cite{HW}, pp. 455-458.\n\nThus, there are two alternatives: $\\Phi$ is either identically\nzero or satisfies the asymptotic formula \\eqref{asymp} where \nthe integer $N$ is $\\geq m+1$ and the complex number\n$A\\neq 0$. If $\\Phi \\equiv 0$, then $z_0$ is a false branch point:\nsee section \\ref{false} below.\n\nIf $\\Phi$ is not $\\equiv 0$, then we have a {\\bf true branch point}. \n\n\\begin{thm} \\label{oss}\n(\\cite{O}, \\cite{G73}.) Suppose $\\Sigma$ is a\nsurface with a conformal structure, $M^3$ a Riemannian manifold \nand $f:\\Sigma \\to M$ a\nmapping which has smallest area in a $C^0$ neighborhood of\n$f$. Then $f$ has no true branch points.\n\\end{thm}\n\\pf\nAs we have just seen, a true branch point $z_0$ has an order \n$m-1 \\geq 1$ and a coordinate neighborhood in $\\Sigma$ with a\n$C^{1,\\alpha}$ complex coordinate $w$, $w=0$ at $z_0$, such that\n$f$ has the representation \\eqref{anonpar} near $z_0$. Adjacent\nsheets $p_3=\\phi(w)$ and $p_3=\\overline\\phi(w)=\\phi(\\zeta_m w)$ \nintersect when $\\Phi(w)=0$, which, according to formula\n\\eqref{asymp}, occurs along $2N\\geq 6$ arcs in $\\Sigma$ forming\nequal angles $\\pi\/N$ when they leave $w=0$. Let one of these\narcs be parameterized as $\\gamma_1:[0,\\varepsilon]\\to \\Sigma$, and\nlet the corresponding arc be $\\gamma_2:[0,\\varepsilon]\\to \\Sigma$, \ndefined by $w(\\gamma_2(t)) =\\zeta_m w(\\gamma_1(t))$.\nThen for all $0\\leq t \\leq \\varepsilon$, \n$\\phi(\\gamma_1(t))=\\phi(\\gamma_2(t))$. Note from formula\n\\eqref{anonpar} that all three coordinates coincide: the mapping\n$f(\\gamma_1(t))= f(\\gamma_2(t)), 0\\leq t \\leq \\varepsilon$.\n\nWe may now construct a Lipschitz-continuous and piecewise smooth\nsurface $\\widetilde{f}$ which has the same area as $f$, but has\ndiscontinuous tangent planes, following Osserman \\cite{O}. The\nidea of the following construction is that the parameter domain\n$D$ may be cut along the arcs $\\gamma_1((0,\\varepsilon))$ and\n$\\gamma_2((0,\\varepsilon))$, opened up to form a lozenge, with two\npairs of adjacent sides originally identified, and then closed up\nalong the remaing two pairs of adjacent sides. \n\nIn detail: choose an open topological disk $D\\subset \\C$ on which the\ncoordinate $w$ is defined, $(0,0)\\in D$, and which is invariant \nunder the rotation taking $w$ to $\\zeta_m w$. Assume\n$\\gamma_1(\\varepsilon)$ and $\\gamma_2(\\varepsilon)$ are the first\npoints along $\\gamma_1$ resp. $\\gamma_2$ which lie on the boundary of\n$D$. We shall construct a discontinuous,\npiecewise $C^1$ mapping $Q:B_1 \\to D$, such that\n$\\widetilde{f}(\\zeta):=f(Q(\\zeta))$ is nonetheless continuous, and\n$Q$ is one-to-one and onto except for sets of measure $0$. Here,\n$B_1$ is the disk $\\{z\\in \\C: |z|<1\\}$. Choose\npoints $A_i=\\gamma_i(\\varepsilon\/2)$, $i=1,2$. Then $D$ is\nbroken along $\\gamma_1$ and $\\gamma_2$ into two curvilinear pentagons \nwith vertices $\\gamma_1(\\varepsilon), A_1, (0,0), A_2$ and\n$\\gamma_2(\\varepsilon).$ The edges of these pentagons are \n$\\gamma_1([\\varepsilon\/2,\\varepsilon])$, $\\gamma_1([0,\\varepsilon\/2])$,\n$\\gamma_2([0,\\varepsilon\/2])$, $\\gamma_2([\\varepsilon\/2,\\varepsilon])$ \nand one of the two arcs of $\\partial D$ with endpoints \n$\\gamma_1(\\varepsilon)$ and $\\gamma_2(\\varepsilon)$.\nSimilarly, break the unit disc $B_1$ \nalong the interval $(-1,1)$ of the $x$-axis and the interval\n$[-\\frac12, \\frac12]$ of the $y$-axis into two pentagons.\nEach pentagon in $B_1$ will be bounded by four line segments, an\ninterval along the $y$-axis being used twice, plus the upper or\nlower half-circle of $\\partial B_1$. Denote the points\n$a=(0,\\frac12),$ $e=(0,-\\frac12)$, $c_1=(1,0)$, $c_2=(-1,0)$ and\ngive the origin $(0,0)$ four different names: $b_1$ when approached\nfrom the first quadrant $\\{x>0,y>0\\}$, \n$b_2$ when approached from the second quadrant $\\{x<0, y>0\\}$,\n$d_2$ when approached from the third quadrant $\\{x<0, y<0\\}$, and\n$d_1$ when approached from the fourth quadrant $\\{x>0, y<0\\}$. $Q$\nwill map the pentagon in $B_1$ in the upper half-plane $y>0$ to the pentagon\nin $D$ lying counterclockwise from $\\gamma_1$ and clockwise from\n$\\gamma_2$, with $Q(c_1)=\\gamma_1(\\varepsilon)$, $Q(b_1)=A_1$,\n$Q(a)=(0,0)$, $Q(b_2)=A_2$, and $Q(c_2)=\\gamma_2(\\varepsilon)$.\nThis describes $Q$ on the boundary of one of the two pentagons;\nthe other pentagon is similar. The\ninterior of each pentagon may be made to correspond by a $C^1$\ndiffeomorphism. We require $Q$ to be continuous along the\n$x$-axis. Of course, $Q$ is discontinuous along the intervals\n$0N$ and some \n$A_2\\in \\C\\backslash \\{0\\}$. That is, the curves of intersection\nof non-successive sheets form a family of equally spaced\ndirections, which are presumably independent of the directions of\nthe curves of intersection of successive sheets. This philosophy\nis justified by the following explicit example with $N=6$ and\n$N_2=7$.\n\nChoose $a,b\\in\\C\\backslash \\{0\\}$. Using the Weierstra\\ss\\ representation\n(see \\cite{O69}, p. 63) for a minimal surface $f:\\C \\to \\R^3$ in\nEuclidean 3-space, based on the polynomials $4z^3$ and\n$2az^2+2bz^3$ (the latter representing the Gau\\ss\\ map in\nstereographic projection), we have the specific CMS with \n\\begin{eqnarray}\nf^1_z(z) = \\Big[1-(az^2+bz^3)^2\\Big]2z^3\\\\\nf^2_z(z) = -i\\Big[1+(az^2+bz^3)^2\\Big]2z^3\\\\\nf^3_z(z) = 4z^3(az^2+bz^3),\n\\end{eqnarray}\nwhich leads to \n$$w^4:=f^1+if^2 = z^4-\\frac{\\ol{a}^2\\ol{z}^8}{2}-\n\\frac{8}{9}\\ol{a}\\ol{b}\\ol{z}^9- \\frac{2}{5}\\ol{b}^2\\ol{z}^{10}$$\nand to\n$$z=w\\Big(1+\\frac{\\ol{a}^2\\ol{w}^8}{8 w^4}+\n\\frac{2\\ol{a}\\ol{b}\\ol{w}^9}{9 w^4}+O_2(|w|^6)\\Big),$$\nvia an extensive, but straightforward, computation. \nRecall that each component $f^k$ of\n$f$ is real and harmonic as a function of $z$. Rewriting \n$$f^3(z)=8\\,\\mathcal{R} \\{ \\frac{a}{6} z^6 + \\frac{b}{7} z^7\\}$$\nas a (non-harmonic) function of $w$, we find \n$$ \\phi(w) = 8\\,\\mathcal{R}\\{\\frac{a}{6}w^6 + \\frac{b}{7}w^7+\n\\frac{\\ol{a}|a|^2}{8} \\ol{w}^8 w^2 +O_2(|w|^{11})\\}:$$\nthe difference of $\\phi$ on successive sheets is \n\\begin{equation}\\label{succ}\n\\Phi(w):=\\phi(w)-\\phi(iw)=\n8\\,\\mathcal{R}\\{\\frac{a}{3}w^6+\\frac{b}{7}(1+i)w^7\\}+ O_2(|w|^{10})\n\\end{equation}\nand on non-successive sheets is\n\\begin{equation}\\label{nonsucc}\n\\Phi_2(w):=\\phi(w)-\\phi(-w)=\n8\\,\\mathcal{R}\\{bw^7\\}+ O_2(|w|^{10}).\n\\end{equation}\n\nFrom the formula \\eqref{succ}, we see that $N=6$, $A=\\frac{8a}{3}$ \nand the curves of intersection of\nsuccessive sheets are curves in $\\R^3$ leaving the branch point\nalong the $(x_1,x_2)$-plane, which is the tangent plane to\n$\\Sigma$ at the branch point, in the $12$ directions\n$(\\cos(4\\theta),\\sin(4\\theta),0)$, where $6\\theta+\\arg(a)$ is an\ninteger multiple of $\\pi$. The $12$ directions are\npaired off to form $6$ curves in $\\R^3$ leaving the branch point \nat equal angles $\\frac{2\\pi}{3}$.\n\nSimilarly, from the formula \\eqref{nonsucc}, we see that $N_2=7$,\n$A_2=8b$ and the curves \nof intersection of nonsuccessive sheets are seven curves leaving\nthe branch point and making equal angles (images of 14 curves in\nthe $w$-plane, paired). The arguments of \n$a,b \\in \\C\\backslash\\{0\\}$ may be given arbitrary values, so that the\nangle between a representative of the family of six curves of\nself-intersection and a representative of the family of seven\ncurves of self-intersection may be chosen arbitrarily. For most\nchoices, these 13 curves in $\\R^3$ will {\\bf not} form equal\nangles at the branch point.\n\n\\end{obs}\n\n\\section{False branch points}\\label{false}\n\nThe elimination of false branch points from an area-minimizing CMS\n$f:\\Sigma \\to M$ is in general only possible by comparison with\nsurfaces $\\Sigma_0$ of {\\it reduced topological type} (see\n\\cite{D39}, \\cite{G77}): for orientable surfaces, $\\Sigma_0$ has \nsmaller genus or the same total genus and more connected components. \nAs an oriented example, we may\nchoose $\\Sigma$ to be a surface of genus $2$ and $\\Sigma_0$ to be\na torus. Then there is a branched covering $\\pi: \\Sigma \\to\n\\Sigma_0$ with two branch points of order one. (Think of $\\Sigma$\nas embedded in $\\R^3$ so that it is invariant under a rotation by\n$\\pi$ about the $z$-axis, and meets the $z$-axis only at two\npoints: the quotient under this rotation is a torus.) Now choose a\nminimizing CMS $f_0: \\Sigma_0 \\to M^n$, and let $f:\\Sigma \\to M^n$\nbe $f = f_0 \\circ \\pi$. Then $f$ has two false branch points. In\norder to be sure that $f$ minimizes area in its homotopy class, we\nmay choose $M^3$ to be a flat $3$-torus with two small periods and\none large period. As one sees from this example, in order to show \nthat false branch points do not\noccur, we must assume that $f$ minimizes area among mappings from\nsurfaces of the topological type of $\\Sigma$ {\\bf and} of lower\ntopological type. This hypothesis was used by J. Douglas (see\n\\cite{D39}), in a strict form, to find the existence of minimal\nsurfaces $f:\\Sigma \\to \\R^3$ with prescribed boundary. For \n$\\R P^2$, however, there are no nonorientable surfaces of lower type.\n\nResults in the literature for false branch points have until now\nassumed that $\\Sigma$ is {\\it oriented}, see \\cite{G73},\n\\cite{Alt73}, \\cite{GOR}, \\cite{G75}, \\cite{G77} and \\cite{T}.\nIn order to treat false branch points for nonorientable\nsurfaces, we will need to extend certain known results. In\nparticular, the following theorem appears in \\cite{G75} for \n{\\em orientable} surfaces, possibly with boundary, including\nsurfaces of prescribed mean curvature vector not necessarily zero.\n\n\\begin{thm}\\label{fund}\n(Fundamental theorem of branched immersions)\nLet $\\Sigma^2$ be a compact surface with boundary endowed with a \nconformal structure, $\\partial\\Sigma$ possibly empty \nand $\\Sigma$ not necessarily orientable. Let $M^n$ be a Riemannian\nmanifold and $f:\\Sigma \\to M$ a CMS. Assume that the restriction\nof $f$ to $\\partial\\Sigma$ is injective. Then there exists a compact\nRiemann surface with boundary $\\widetilde\\Sigma$, a branched covering \n$\\pi:\\Sigma \\to \\widetilde\\Sigma$ and a CMS \n$\\widetilde{f}: \\widetilde\\Sigma \\to M$ such that \n$f = \\widetilde{f} \\circ \\pi$. Moreover, the restriction of\n$\\widetilde{f}$ to $\\partial\\widetilde\\Sigma$ is injective.\nFurther, $\\widetilde\\Sigma$ is orientable if and only if $\\Sigma$\nis orientable.\n\\end{thm}\n\\pf\nIf $\\Sigma$ is orientable, then \nTheorem 4.5 of \\cite{G75} provides an orientable quotient surface\n$\\widetilde\\Sigma$, a branched covering \n$\\pi:\\Sigma \\to \\widetilde\\Sigma$ and an unramified CMS \n$\\widetilde f: \\widetilde\\Sigma \\to M$ such that \n$f=\\widetilde f \\circ \\pi$. \n\nThere remains the case where $\\Sigma$ is {\\bf not orientable}.\nAssume, without loss of generality, that $\\Sigma$ is connected. \n\nLet $p:\\widehat\\Sigma \\to \\Sigma$ be the oriented double cover of\n$\\Sigma$, with the induced conformal structure. Then \n$\\widehat\\Sigma$ is connected and orientable, and $p$ is two-to-one. \nThe composition \n$\\widehat{f}=f \\circ p: \\widehat\\Sigma \\to M$ is a CMS, defined on\nan orientable surface, and we may apply Theorem 4.5 of \\cite{G75}\nto find a compact orientable surface with boundary \n$\\widehat{\\widetilde\\Sigma}$, an unramified\nCMS $\\widehat{\\widetilde{f}}:\\widehat{\\widetilde\\Sigma}\\to M$ and\nan orientation-preserving branched covering \n$\\widehat{\\pi}:\\widehat\\Sigma \\to \\widehat{\\widetilde\\Sigma}$ so that\n$\\widehat{f}$ factors as $\\widehat{\\widetilde{f}}\\circ\\widehat{\\pi}$.\n\nNow let $\\widetilde\\Sigma$ be the quotient surface of\n$\\widehat{\\widetilde\\Sigma}$ under the identification of \n$\\widehat{\\pi}(x^+) \\in \\widehat{\\widetilde\\Sigma}$ with \n$\\widehat{\\pi}(x^-) \\in \\widehat{\\widetilde\\Sigma}$ whenever \n$x^\\pm \\in \\widehat{\\Sigma}$ and $p(x^+)= p(x^-)$ in $\\Sigma$.\nThen for each $x \\in \\Sigma$, $p^{-1}(x)$ consists of two points\n$x^+,\\ x^-\\in\\widehat{\\Sigma}$ and there are diffeomeorphic\nneighborhoods of $\\widehat{\\pi}(x^+)$ and of $\\widehat{\\pi}(x^-)$\nwhich are thereby identified in $\\widetilde\\Sigma$, with reversal\nof orientation. This implies \nthat $\\widetilde\\Sigma$ is a differentiable $2$-manifold. Write\n$\\widetilde{p}:\\widehat{\\widetilde\\Sigma}\\to\\widetilde\\Sigma$ for\nthe quotient mapping. Then $\\widetilde{f}:\\widetilde\\Sigma \\to M$\nis well defined such that \n$\\widehat{\\widetilde{f}} = \\widetilde{f}\\circ\\widetilde{p}$.\nAlso, for $x\\in \\Sigma$, the two pre-images $x^+, x^-\\in\n\\widehat{\\Sigma}$ have\n$\\widetilde{p}\\circ\\widehat{\\pi}(x^+)=\\widetilde{p}\\circ\\widehat{\\pi}(x^-),$ \nso that we may define\n$\\pi:\\Sigma \\to \\widetilde\\Sigma$ by \n$\\pi(x):= \\widetilde{p}\\circ \\widehat{\\pi}(x^\\pm)$.\n\nNote that the mappings $p, \\widehat{\\pi}$ and $\\widetilde p$ are\nsurjective, and therefore also $\\pi:\\Sigma\\to\\widetilde{\\Sigma}$.\n\nIn the event that $\\partial{\\Sigma}$ is nonempty, since\n$f=\\widetilde f \\circ \\pi$ restricted to $\\partial\\Sigma$ is\ninjective, it follows readily that the restriction of\n$\\widetilde{f}$ to $\\partial{\\widetilde{\\Sigma}}$ is injective. \n\nThen in the above construction, for each $x\\in \\Sigma$, \n$f$ defines the\nsame piece of surface, with opposite orientations, on\nneighborhoods of $x^+$ and of $x^-$. The branched covering\n$\\widehat\\pi: \\widehat\\Sigma \\to \\widehat{\\widetilde\\Sigma}$\npreserves orientation, implying that\n$\\widehat\\pi(x^+) \\neq \\widehat\\pi(x^-)$.\nSince $\\widehat{\\widetilde\\Sigma}$ is connected, there is a path\nfrom $\\widehat\\pi(x^+)$ to $\\widehat\\pi(x^-)$ whose image in\n$\\widetilde\\Sigma$ reverses orientation. Therefore\n$\\widetilde\\Sigma$ is not orientable. \n\\qed\n\n\\section{An immersion of $\\R P^2$}\n\nWe are now ready to give the proof of the main Theorem \\ref{main}.\nLet $f: \\R P^2 \\to M^3$ be a CMS into a three-dimensional\nRiemannian manifold, which has minimum area among all mappings\n$\\R P^2 \\to M^3$ not homotopic to a constant. Write\n$\\Sigma = \\R P^2$. From Theorem \\ref{oss}, we see that $f$ has no\ntrue branch points. (For this conclusion, it would suffice that $f$\nminimizes area in a $C^0$ neighborhood of each branch point.)\n\nThere remains the possibility of false branch points.\n\nWe first recall the computation of the Euler characteristic of a\nsurface. For a compact, connected surface which is either\norientable or nonorientable, the Euler characteristic\n$$ \\chi(\\Sigma) = 2 - r(\\Sigma),$$\nwhere $r(\\Sigma)$ is the topological characteristic of $\\Sigma$\n\\cite{D39}, also known as the nonorientable genus;\nwe\nshall adopt the term {\\bf demigenus}. If $\\Sigma$ is orientable,\nthen it has even demigenus and genus $\\frac12 r(\\Sigma)$. If \nit is non-orientable, then $\\Sigma$ may be constructed\nby adding $r(\\Sigma)$ cross-caps to the sphere.\nThe demigenus of the sphere equals zero, of $\\R P^2$ equals one, of\nthe torus and the Klein bottle equals two. For other compact\nsurfaces without boundary, the demigenus is $\\geq 3$. \n\nNow according to Theorem \\ref{fund}, there is a compact Riemann\nsurface $\\widetilde\\Sigma$, a branched covering\n$\\pi:\\Sigma \\to \\widetilde\\Sigma$ and an unramified CMS\n$\\widetilde f:\\widetilde\\Sigma \\to M$ such that\n$f = \\widetilde f \\circ \\pi$. We will apply the Riemann-Hurwitz\nformula to the branched covering $\\pi$:\n\\begin{equation}\\label{RH}\n \\chi(\\Sigma) = d \\, \\chi(\\widetilde\\Sigma) - {\\mathcal O}(\\pi),\n\\end{equation}\nwhere $d$ is the degree of $\\pi$, \n${\\mathcal O}(\\pi)$ is the total order of branching of $\\pi$, and\n$\\chi$ is the Euler number. Suppose that $f$ has a false\nbranch point, or more generally, a ramified branch point. Then\n${\\mathcal O}(\\pi)\\geq 1,$ and the branched covering $\\pi$ has\ndegree $d \\geq 2$.\n\nUsing the formula \\eqref{RH}, we can determine the\ntopological type of $\\widetilde\\Sigma$. Since $\\Sigma$ is\nhomeomorphic to $\\R P^2$, it has $\\chi(\\Sigma)=1$. We also know\nthat $d>0$\nand ${\\mathcal O}(\\pi) \\geq 1$. It follows that the demigenus\n$r(\\widetilde\\Sigma) \\leq 1$. Otherwise, the integer\n$r(\\widetilde\\Sigma)$ would be $\\geq 2$, which implies\n$\\chi(\\widetilde\\Sigma) \\leq 0$ and by the formula \\eqref{RH}, \n$1 \\leq -{\\mathcal O}(\\pi)\\leq -1$, a contradiction. That is,\n$\\widetilde\\Sigma$ is either the sphere or the projective plane.\nBut according to Theorem \\ref{fund}, since $\\Sigma$ is not\norientable, $\\widetilde\\Sigma$ is not orientable; therefore,\n$\\widetilde\\Sigma$ is homeomorphic to $\\R P^2$.\n\nNote that if $\\widetilde f:\\widetilde\\Sigma \\to M$ were homotopic\nto a constant mapping, then so would be $f = \\widetilde f \\circ\n\\pi$.\n\nOn the other hand,\nthe area of $f$ equals the area of $\\widetilde f$ times the degree\n$d$ of $\\pi$. But $d\\geq 2$, so the area of $\\widetilde f$ is at\nmost one-half the area of $f$. But this would mean that $f$ does\nnot have minimum area among maps $:\\R P^2 \\to M$ not homotopic to\na constant mapping, contradicting our hypothesis. This implies\nthat $f$ has no branch points, and is therefore an immersion. \n\\qed\n\n\\begin{rem}\nWe have treated conformally parameterized {\\em minimal} surfaces\nin this paper. However, the proofs go through with only minor\nchanges for projective planes of nonzero {\\em prescribed} mean\ncurvature $H:M^3 \\to \\R$, provided that $f(\\Sigma)$ has a\ntransverse orientation. This can occur only when $M$ is\nnon-orientable.\n\nIt also appears plausible that a version of Theorem \\ref{fund} can be\nextended to the more general case of mappings satisfying the {\\em\nunique continuation property}, see \\cite{GOR}. \n\\end{rem}\n\n\\begin{rem}\\label{serrin}\nAn alternative approach to branch points of minimal surfaces\nof the type of the disk appears in the recent book \\cite{T} by \nTromba. The second and higher variations of energy $E$ \n(see \\eqref{energy}) of a CMS $f$ are computed in a\nneighborhood of a branch point $z_0$, and the lowest nonvanishing\nvariation is shown to be negative if the branch point is\nnonexceptional. This is defined in terms of the {\\bf index} $i>m$ \nof $z_0$, where $i+1$ is the order of contact of the mapping\nwith the tangent plane at the branch point. Note that the proper\nindex $N$ is $\\geq i$ (recall the definition \\eqref{asymp}).\nIf $i+1$ is an integer multiple of $m$, where $m-1$ is the order\nof the branch point, then the branch point is called {\\em\nexceptional}. Tromba also shows that exceptional interior branch\npoints will not occur, provided that the mapping has minimum area\n$A$ among surfaces with the same boundary curve. \n\nIn fact, minimizing area and minimizing energy, under such Plateau\nboundary conditions, are equivalent properties of a Lipschitz\ncontinuous mapping $f:B \\to \\R^n$, where $B$ is the unit disk in\n$\\R^2$. Namely, if $E(f) \\leq E(g)$ for all Lipschitz-continuous\nmappings $g:B \\to \\R^n$ defining the same boundary curve, then\n$A(f) \\leq A(g)$ for all such $g$, as we now show. \n\nOtherwise, for some $g:B \\to \\R^n$ with the same Plateau boundary \nconditions as $f$, $A(f) > A(g)$. Write $\\eta = A(f)-A(g)>0$.\nApproximate $g$ with $g^\\delta(w):=(g(w),\\delta w) \\in \\R^{n+2}$.\nThen $g^\\delta$ is a Lipschitz immersion, so there are conformal\ncoordinates $\\widetilde w=:F^{-1}(w)$ for some bi-Lipschitz \nhomeomorphism $F:B \\to B$ which preserves $\\partial B$ (see\n\\cite{M}). Write\n$\\widetilde g^\\delta(\\widetilde w):= g^\\delta(F(\\widetilde w))$ for\nthe conformal mapping with the same image as $g^\\delta$. Define\n$\\widetilde g:B \\to \\R^n$ by composing $\\widetilde g^\\delta$ with\nthe projection from $\\R^{n+2} \\to \\R^n$. Then the energy \n$E(\\widetilde g^\\delta) = E(\\widetilde g) + \\delta^2 E(F)$.\nAlso, the area $A(g^\\delta) \\leq A(g) + C\\delta$ for some constant\n$C$. It follows that\n$$E(\\widetilde g)\\leq E(\\widetilde g^\\delta)=A(\\widetilde\ng^\\delta)=A(g^\\delta)\\leq A(g) + C\\delta 1$, for any of the models. \nThe intersection of the horizontal line \nwith a PC shows where that mode gives a $2\\sigma$ indication of difference \nfrom $\\Lambda$. For $|w_a|<0.4$, only 2 PCs meet this criterion. \n(Note breaks in the PC5 curve come from $\\alpha_5$ switching sign.) \n}\n\\label{fig:sigalf}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\psfig{file=sigalfo.ps,width=3.4in}\n\\caption{As Fig.~\\ref{fig:sigalf}, but for $w(a)=-1+A\\,[1-\\cos(\\ln a)]$ \nwith non-monotonic behavior. Note $w$ would exceed 0 at some redshifts \nfor $A>0.5$. \n}\n\\label{fig:sigalf2}\n\\end{center}\n\\end{figure}\n\n\nTo exhibit robustness of these results against the \nspecific assumed model, we scan over model parameters for classes of \nqualitatively different dark energy physics, representing the thawing \nclass, freezing class, and a non-monotonic EOS. In the last case, we \nsee that the ability of PCA not to be locked into a particular \nfunctional form, e.g.\\ a monotonic parametrization, does not make \nmore degrees of freedom significant. \n\nAgain, what we really care about is the $S\/N$. In Table~\\ref{tab:sn2} \nwe list the fraction of the total $S\/N$ (Eq.~\\ref{eq:snr}) \ncontributed by the two best modes -- for the case \nfor each dark energy class where higher modes contribute \nthe {\\it most\\\/}. For the thawing class the higher modes add less \nthan 0.3\\% to the total, and for the freezing class less than 2.8\\% in \nthe most sensitive case, dropping to less than 0.5\\% for modes above \nthe third. And recall this was for a highly idealized experiment. In \nthe oscillating model, an ad hoc case designed to be especially \nPCA-friendly, the most extreme case with oscillations reaching $w=0$ \nallows higher modes to contribute up to 14\\% (8\\% for above the third \nmode). These are actually {\\it overestimates\\\/} of the importance of high \nmodes because as discussed in the next section the $S\/N$ of the higher \nmodes degrades when $w(z>9)$ is marginalized over rather than fixed. \n\n\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabular*}{0.95\\columnwidth} \n{@{\\extracolsep{\\fill}} l c c }\n\\hline\nModel & $(S\/N)_2\/(S\/N)_{\\rm all}$ & \\ $(S\/N)_3\/(S\/N)_{\\rm all}$ \\\\ \n\\hline\nFreezing ($w_a=0.7$)& 0.972& \\ 0.995 \\\\ \nThawing ($w_a=-0.5$) & 0.997& \\ 0.9998 \\\\ \nOscillating ($A=0.5$) & 0.862& \\ 0.922 \\\\ \n\\hline \n\\end{tabular*}\n\\caption{Fraction of total signal-to-noise contributed by the first \ntwo, or three, principal components for the case in each dark energy \nclass {\\it most\\\/} favoring PCA high modes.} \n\\label{tab:sn2}\n\\end{center}\n\\end{table}\n\n\nInterestingly, fitting $w_0$-$w_a$ \nrather than using PCA provides distinction from the cosmological \nconstant at the $1\\sigma$ ($2\\sigma$) level for $|w_a|=0.13$ (0.24) -- \nvery comparable to the PCA approach. That is, the second parameter \nbecomes useful at almost the same values as in the top panel of \nFig.~\\ref{fig:sigalf}, for the PCA freezing case, and is more \nsensitive than in the bottom panel for the PCA thawing case. \nBoth methods demonstrate that significant physical constraints on \nthe dark energy EOS are described by of order two quantities. \n\nThe main point though is that the important information is not in \n$\\sigma_i$, but $\\sigma_i\/\\alpha_i$. Just because PCA may say \nuncertainties $\\sigma_i$ are small, \nthis does not mean that we know the physics answer. \n\n\n\\section{What Happens at High Redshifts, Stays at High Redshifts?} \n\nWhile the previous demonstration of PC uncertainties and \nsignal-to-noise is \nthe most important of this paper, we also note that assumptions about \nthe high redshift EOS behavior have significant effects. \nIn practice, the PCs are often computed assuming a cut off at some \nmaximum redshift to avoid complications in calculating \nthe cosmic microwave background (CMB) primordial power spectra and the \ninitial conditions for growth of matter perturbations. At higher \nredshifts one must therefore choose a particular form or value for \nthe EOS. FOMSWG fixes $w(z>9)=-1$. One justification \nis that for the cosmological constant the dark energy density \nfades away quickly into the past, so the exact value of $w$ there is \nunimportant. However, we have no guarantee that the cosmological \nconstant is the correct model, nor even essentially any current \ninformation on the behavior of the EOS at $z>1$ \\cite{kowalski}. \nAssumptions about the high redshift behavior can lead to significant \nbiases and improper conclusions about the nature \nof dark energy (see, e.g., \\cite{deplpca}). \n\nBias should not be as severe a problem for a high transition redshift, \n$z=9$, and with many redshift bins at $z\\gtrsim3$ the extra degrees of \nfreedom should ameliorate bias from \nthe prior at $z>9$. However, fixing $w(z>9)=-1$ does demonstrably \ninfluence the PCs: for example some of the uncertainties \n$\\sigma_i\/\\alpha_i$ that are apparently tightly determined \ncan degrade by a factor three when $w(z>9)$ is not fixed. \nThe modes themselves also change shape, as we discuss next. \n\nThus, what happens at high redshift \ndoes {\\it not\\\/} stay at high redshift, but can affect some important \naspects of the principal component analysis. \n\n\n\\section{Highest is Best?} \n\nDoes the location in redshift of the maximum of a PC, say the first one, \nsay something fundamental about the science reach of the survey or probe \nemployed? No -- as is clear from Eq.~(\\ref{eq:pc}) the EOS constraints \nfollow from the sum -- with both positive and negative contributions -- \nover all the PCs, not any single one. \n\nMoreover, Figure~\\ref{fig:pchiz} demonstrates that artificially fixing \nthe high redshift EOS behavior changes the PC shape and peak location. \nThis shift has nothing to do with the experimental design and so the \npeak location is not a signpost to experiment optimization. \nAssumptions on $w(z>9)$ can affect probes differently: e.g.\\ supernova \ndistances do not involve $w(z>9)$ while baryon acoustic oscillations, \nbeing tied to high redshift, do. \n\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\psfig{file=PCA_w01wa7.ps,width=3.4in}\n\\caption{The location and shape of peaks in principal component modes \ndepend on the high redshift treatment of the EOS. The peak location \ntherefore does not in itself translate into the science impact of a given \nexperiment. Note the effects are more severe for a more realistic \nexperiment. \n}\n\\label{fig:pchiz}\n\\end{center}\n\\end{figure}\n\n\nFinally, even if a PC peak did mean that the experiment is most sensitive \nto the dark energy EOS at a higher redshift, say, that would not imply \nthat the experiment is most sensitive to the nature of dark energy. \nFor example, dark energy is most influential today, so perhaps one wants \nan experiment most sensitive to the low redshift behavior. \n(We emphasize that understanding dark energy at low redshift still \nrequires measuring expansion and growth to high redshift, to break \ndegeneracies.) At best, one could say that probes that weight the dark \nenergy differently in redshift have some complementarity. \n\nBut there is \nno justification for claiming that the probe with the highest peak, or \nwith the peak at the highest redshift, is the best probe. \n\n\n\n\n\\section{Conclusions} \n\nPrincipal component analysis is a valid technique, \nused appropriately. Oversimplifying PCA interpretation or \ninadequately appreciating the effect of assumptions \nemployed can lead to misunderstandings and false beliefs. We \npresent cautionary examples of three apparently plausible but \nunjustified extrapolations. While data \nshould be analyzed in every reasonable manner, for understanding the \ngeneric cosmology reach the more complicated PCA approach \ndemonstrates no extraordinary advantage over the well-tested and highly \ncalibrated phase space dynamics approach of $w_0$-$w_a$. \n\n\n\\acknowledgments \n\nWe thank Andy Albrecht and Dragan Huterer for detailed discussions of \nPCA issues, and Bob Cahn for useful suggestions. \nThis work has been supported in part by the Director, Office of Science, \nOffice of High Energy Physics, of the U.S.\\ Department of Energy under \nContract No.\\ DE-AC02-05CH11231. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nA well known item of the string\/gauge theory holographic dictionary states that closed strings are the duals of glueballs in the corresponding gauge theories. On the other hand, using the gravity\/gauge theory duality, glueball operators of the boundary field theory correspond to fields in the gravity bulk theory, in particular modes of the dilaton, the graviton, and the RR field. Using this latter correspondence a spectrum of holographic glueballs has been determined \\cite{Csaki:1998qr,Brower:2000rp,Elander:2013jqa}. However, from the same reasoning as in \\cite{Sonnenschein:2014jwa} and \\cite{Sonnenschein:2014bia}, it is probably the spectrum of the strings and not of the bulk fields that would really correspond to the experimental data when moving from large $N_c$ and large $\\lambda$ to the to the realistic values of $N_c=3$ and $\\lambda\\sim 1$. For mesons \\cite{Sonnenschein:2014jwa} and baryons \\cite{Sonnenschein:2014bia} it was argued that the (open) string configurations admit a modified Regge behavior that matches that of the observed hadrons whereas bulk modes do not admit this property. In this paper we argue that a similar correspondence exists for glueballs, and that the glueballs will probably have a better description in terms of closed strings rather than modes of bulk fields. The idea of glueballs as closed strings has previously been discussed in various terms in works such as \\cite{Bhanot:1980fx,Niemi:2003hb,Sharov:2007ag,Solovev:2000nb,Talalov:2001cp,LlanesEstrada:2000jw,Szczepaniak:2003mr,Pons:2004dk,Abreu:2005uw, Brau:2004xw,Mathieu:2005wc,Simonov:2006re,Mathieu:2006bp,BoschiFilho:2002vd}.\n\nIt is common lore that glueballs and flavorless mesons cannot be distinguished since they carry the same quantum numbers and that the corresponding resonances encountered in experiments are in fact generically linear combinations of the two kinds of states. If, however, we refer to the stringy description of hadrons then, since mesons and glueballs correspond to open and closed strings respectively, there are certain characterizing features with which one can distinguish between them.\n\nThe most important difference between the open string mesons and the closed string glueballs is the slope, or equivalently the (effective) tension. It is a basic property of strings (see sections \\ref{sec:lightcone} and \\ref{sec:flat_string}) that the effective tension of a closed string is twice that of an open string and hence there is a major difference between the two types of strings, as\n\\begin{equation} \\ensuremath{\\alpha^\\prime}_{closed}= \\frac12 \\ensuremath{\\alpha^\\prime}_{open}\\qquad \\rightarrow \\qquad \\ensuremath{\\alpha^\\prime}_{gb}= \\frac12 \\ensuremath{\\alpha^\\prime}_{meson} \\,. \\end{equation}\nThus the basic idea of this paper is that one should be able to distinguish between glueballs and flavorless mesons by assigning some of them to certain trajectories with a mesonic slope $\\ensuremath{\\alpha^\\prime}_{meson}\\sim 0.9$ GeV$^{-2}$ and others to trajectories with a glueball slope of $\\ensuremath{\\alpha^\\prime}_{gb}\\sim 0.45$ GeV$^{-2}$.\n\nThe slope is not the only thing that is different between open and closed strings. It follows trivially from the spectrum of closed strings (see section \\ref{sec:lightcone}) that in the critical dimension it has an intercept that is twice that of the open string. However, we are interested in strings in four dimensions rather than the critical dimension, and there, as will be discussed in section \\ref{sec:non_critical_string}, the determination of the intercept is still not fully understood. Thus the intercept cannot currently serve as a tool for identifying glueballs.\n\nAnother important difference between open and closed string hadrons is in their decay mechanisms. Based on the holographic description of a meson as a string that connects to flavor branes at its two endpoints, it was determined in \\cite{Peeters:2005fq} that the width of decay of a meson of mass \\(M\\) behaves like\n\\begin{equation} \\Gamma \\sim \\left(\\frac{ 2 M}{\\pi T} - \\frac{m^1_{sep}+m^2_{sep}}{2T}\\right) e^{\\frac{-{m^q_{sep}}^2}{T}} \\,, \\label{eq:holo_width}\\end{equation}\nwhere $M$ is the mass of the meson, $m^1_{sep}$ and $m^2_{sep}$ are the masses of the string endpoint quarks in the initial state (assumed here to be small), $m^q_{sep}$ is the mass of the quark and antiquark pair generated by the split of the string and $T$ is the string tension. The factor preceding the exponent is in fact the string length.\n\nAs we discuss in section \\ref{sec:decays}, for the case of a closed string decaying into two open strings the width will be proportional to the string length squared, and the single exponential suppressing factor will be replaced by\n\\begin{equation} e^{\\frac{-m^q_{sep}{}^2}{T}}e^{\\frac{-m^{q^\\prime}_{sep}{}^2}{T}} \\,,\\end{equation}\nwhere $m^{q}_{sep}$ and $ m^{q^\\prime}_{sep}$ are the masses of each of the two quark-antiquark pairs that will have to be created in the process.\n\nThus it is clear that the width of a glueball should be narrower than that of the corresponding meson open string, particularly for decay channels involving heavier quarks like $s$, $c$ and $b$. This can serve as an additional tool of disentangling between mesons and glueballs. We list one distinguishing feature of a glueball decaying into two mesons in section \\ref{sec:decays}.\n\nThe main motivation of reviving the description of mesons and baryons in terms of open strings in \\cite{Sonnenschein:2014jwa} and \\cite{Sonnenschein:2014bia} has been the holographic string\/gauge duality. The same applies also to the closed string picture of glueballs. The spectra of closed strings in a class of holographic confining models was analyzed in \\cite{PandoZayas:2003yb}. The result was that the relation between the mass and angular momentum takes the following form:\n\\begin{equation} J = \\ensuremath{\\alpha^\\prime}_{gb} (E^2 - 2 m_0 E) + a \\,, \\end{equation}\nwhere $\\ensuremath{\\alpha^\\prime}_{gb}$ is the corresponding slope, $E$ is the mass of the glueball, $a$ is the intercept, and $m_0$ is a parameter that can be either positive or negative and is determined by the particular holographic model used. Note that this relation modifies the well known linear relation between $J$ and $E^2$. In section \\ref{sec:holo_fits} we discuss the phenomenological implications of this relation and analyze the possibility of grouping flavorless hadrons along such holographic trajectories.\n\nThe main goal of this paper is to perform an explicit comparison between observational data of flavorless hadrons and the resonance states predicted by the models of rotating open string with massive endpoints for the mesons and rotating folded closed strings for glueballs.\n\nUnfortunately there exists no unambiguous way to assign the known flavorless hadrons (the focus in this paper is on the \\(f_0\\) and \\(f_2\\) resonances) into trajectories of mesons and glueballs, but it is clear that \\textbf{one cannot consistently sort all the known resonances into meson trajectories alone}. One of the main problems in identifying glueball trajectories is simply the lack of experimental data, particularly in the mass region between \\(2.4\\) GeV and the \\(c\\bar{c}\\) threshold, the region where we expect the first excited states of the glueball to be found. It is because of this that we cannot find a glueball trajectory in the angular momentum plane.\n\nWe mostly focused then on the radial trajectories of the \\(f_0\\) (\\(J^{PC} = 0^{++}\\)) and \\(f_2\\) (\\(2^{++}\\)) resonances. For the \\(f_0\\) we examined the possibility of identifying one of the states \\(f_0(980)\\), \\(f_0(1370)\\), \\(f_0(1500)\\), or \\(f_0(1710)\\) as the glueball ground state and building the trajectories beginning from those states. This procedure did not show any significant preference for any one of the glueball candidates over the other. For the \\(f_2\\) there is less ambiguity, but still no positive identification. Between the different \\(2^{++}\\) state we find that the two very narrow resonances \\(f_2(1430)\\) and \\(f_J(2220)\\) (the latter being a popular candidate for the tensor glueball) do not belong on meson trajectories.\n\nThe paper is organized as follows. Section \\ref{sec:theory} is devoted to the theory of rotating closed strings. In section \\ref{sec:lightcone} we review the light-cone quantization of the basic bosonic string and describe its spectrum. Next we address the rotating folded string. We present the classical solution and the corresponding Regge trajectory, starting by discussing the case of flat spacetime. We introduce the Polchinski-Strominger term needed to assure two dimensional conformal invariance in non-critical dimension and discuss the problematic result for the intercept for a folded closed string in four dimensions. In section \\ref{sec:holo_string} we review the results of \\cite{PandoZayas:2003yb} for the rotating folded string in holographic backgrounds, and the semiclassical correction obtained there. Section \\ref{sec:decays} is devoted to the decay process of string decaying into two strings. We summarize the result for the decay of an open string into two open strings \\cite{Peeters:2005fq} and generalize it also to the case of a closed string decaying into two open strings. Section \\ref{sec:phenomenology} deals with the phenomenology of the rotating folded string models and the comparison between them and the observational data. We begin by spelling out the basic assumptions of the phenomenological models in section \\ref{sec:fitting_models}. We then present the key experimental players: the $f_0$ and $f_2$ resonances. In \\ref{sec:f0_fits} we propose several assignments of the $f_0$ resonances into radial $(n,M^2)$ trajectories, first into only various mesonic trajectories and then into various possible combinations when singling out some states as glueballs. In \\ref{sec:f2_fits} we describe possible assignments of the $f_2$, first into orbital $(J,M^2)$ trajectories, then into \\((n,M^2)\\) trajectories. Section \\ref{sec:holo_fits} expands on previous sections by using the non-linear trajectory that characterizes the glueballs of holographic models. In section \\ref{sec:lattice} we discuss the spectrum of glueballs that follows from lattice gauge theory models. We review the trajectories determined in lattice simulations and their corresponding slopes. Both types of trajectories, $(J,M^2)$ and $(n,M^2)$, are discussed. Section \\ref{sec:summary} is a summary and discussion of the results and states some open questions. In the appendix \\ref{sec:predictions} we list the predictions of our models for the yet unobserved excited partners of the glueball candidates, based on their Regge trajectories.\n\n\\section{The rotating closed string} \\label{sec:theory}\n\\subsection{Quantized closed string in light cone gauge} \\label{sec:lightcone}\nWe review here the derivation of the spectrum of the bosonic closed string in the light cone gauge. We simply present the derivation in chapter 1 of \\cite{Polchinski:Vol1}, omitting some of the details for brevity's sake. The following treatment is essentially true only for the critical dimension \\(D = 26\\), but we keep a general \\(D\\) in the formulae. We return to this point in section \\ref{sec:non_critical_string}.\n\nWe start from the Polyakov action\n\\begin{equation} S = -\\frac{1}{4\\pi\\ensuremath{\\alpha^\\prime}}\\int d\\tau d\\sigma \\sqrt{-\\gamma}\\gamma^{\\alpha\\beta}\\eta_{\\mu\\nu}\n\\partial_\\alpha X^\\mu \\partial_\\beta X^\\nu \\,.\\end{equation}\nWe define the light cone coordinates \\(x^\\pm = \\frac{1}{\\sqrt{2}}(x^0\\pm ix^1)\\), and set the gauge by making the three requirements\n\\begin{equation} X^+ = \\tau\\,, \\qquad \\partial_\\sigma \\gamma_{\\sigma\\sigma} = 0\\,, \\qquad \\sqrt{-\\gamma} = 1 \\label{eq:lightcone} \\,.\\end{equation}\nThe equations of motion for the transverse coordinates are then simple wave equations and they are generally solved (with closed string boundary conditions, for \\(\\sigma \\in (-\\ell,\\ell)\\)) by\n\\begin{equation} X^i(\\sigma,\\tau) = x^i + \\frac{p^i}{p^+}\\tau + i\\left(\\frac{\\ensuremath{\\alpha^\\prime}}{2}\\right)^{1\/2}\n\\sum_{n\\neq0}\\left[\\frac{\\alpha^i_n}{n}\\exp\\left(-i\\frac{2\\pi n (\\sigma+c\\tau)}{2\\ell}\\right)\n + \\frac{\\beta^i_n}{n}\\exp\\left(i\\frac{2\\pi n (\\sigma-c\\tau)}{2\\ell}\\right)\\right] \\,.\\end{equation}\nThe constant \\(c\\) is related to the coordinate length \\(\\ell\\) and the conserved quantity \\(p^+\\) via \\(c = \\ell\/(\\pi\\ensuremath{\\alpha^\\prime} p^+)\\). Aside from \\(\\ell\\), which is proportional to the physical string length, these constants do not have any significance on their own except in keeping track of units.\n\nThe left and right moving modes, \\(\\alpha^i_n\\) and \\(\\beta^i_n\\), are independent of each other (and hence, commute) and are normalized in such a way that\n\\begin{equation} [\\alpha^i_m,\\alpha^j_n] = [\\beta^i_m,\\beta^j_n] = m\\delta^{ij}\\delta_{m,-n} \\,.\\end{equation}\nThe Hamiltonian has the mode expansion\n\\begin{equation} H = \\frac{p^i p^i}{2p^+} + \\frac{1}{\\pi\\ensuremath{\\alpha^\\prime}}\\left[\\sum_{n>0}\\left(\\alpha^i_{-n}\\alpha^i_n+\\beta^i_{-n}\\beta^i_n\\right)+ A + \\tilde{A}\\right] \\,, \\end{equation}\nnoting that \\((\\alpha^i_n)^\\dagger = \\alpha^i_{-n}\\). \\(A\\) and \\(\\tilde{A}\\) are the c-numbers one gets when normal-ordering the sums. After regularizing the appropriate infinite sums, identical for the left and the right moving modes, and taking contributions from \\(D-2\\) transverse modes, we get the result\n\\begin{equation} A = \\tilde{A} = \\frac{2-D}{24} \\,.\\end{equation}\n\nFrom here we get the spectrum using the mass shell condition \\(M^2 = -p^2 = 2p^+ H - p^i p^i\\), which translates to\n\\begin{equation} M^2 = \\frac{2}{\\ensuremath{\\alpha^\\prime}}\\left(\\sum_{n>0}\\left(\\alpha^i_{-n}\\alpha^i_n+\\beta^i_{-n}\\beta^i_n\\right)+ A + \\tilde{A}\\right) \\,,\\end{equation}\nor,\n\\begin{equation} M^2 = \\frac{2}{\\ensuremath{\\alpha^\\prime}}\\left(N+\\tilde{N}+A+\\tilde{A}\\right) \\,,\\end{equation}\nwhere \\(N\\) and \\(\\tilde{N}\\) are the total population numbers of the left and right moving modes.\n\nFor comparison, the same treatment of the open string leads to the result\n\\begin{equation} M^2_{open} = \\frac{1}{\\ensuremath{\\alpha^\\prime}}\\left(N+A\\right) \\,.\\end{equation}\nHere we have neither the constant prefactor of two which halves the slope of the closed string, nor do we have two different kinds of modes on the string and the resulting doubling of the intercept.\n\n\\subsubsection{Quantized closed string: The spectrum}\nWhile the left and right moving modes on the closed string are independent, there is one constraint that relates them, affecting the spectrum. After making the gauge choice by imposing the three conditions of eq. \\ref{eq:lightcone} we still have a residual symmetry of \\(\\tau\\)-independent translations of \\(\\sigma\\). This results in the additional constraint\n\\begin{equation} N = \\tilde{N} \\,.\\end{equation}\nThe total number of excitations has to be equal for the left and right moving modes.\n\nThe vacuum state of the closed string is defined as the state annihilated by all \\(\\alpha^i_n\\) and \\(\\beta^i_n\\), for positive \\(n\\). It has \\(N = \\tilde{N} = 0\\), we denote it simply \\(|0\\rangle\\),\\footnote{The vacuum state may also have some center of mass momentum \\(p\\), but we suppress it in this notation.} and its mass is determined by the intercepts:\n\\begin{equation} M^2 = \\frac{2}{\\ensuremath{\\alpha^\\prime}}(A+\\tilde{A}) = \\frac{2-D}{6\\ensuremath{\\alpha^\\prime}} \\,.\\end{equation}\nFor \\(D = 26\\) this state is a tachyon, with \\(M^2 = -4\/\\ensuremath{\\alpha^\\prime}\\). The first excited state has \\(N = \\tilde{N} = 1\\), and so is of the form\n\\begin{equation} \\alpha^{i}_{-1}\\beta^j_{-1}|0\\rangle \\end{equation}\nand its mass is\n\\begin{equation} M^2 = \\frac{2}{\\ensuremath{\\alpha^\\prime}}(2+A+\\tilde{A}) = \\frac{26-D}{6\\ensuremath{\\alpha^\\prime}} \\,.\\end{equation}\nIn the critical dimension we have here a massless tensor and a massless scalar.\n\nThe most important feature of the spectrum for our uses is that it forms an infinite tower of states, with the difference between each pair of consecutive states being\n\\begin{equation} \\Delta M^2 = \\frac4\\ensuremath{\\alpha^\\prime} \\,, \\end{equation}\nwith one factor of two coming from the halving of the slope, and the other from the fact that \\(N+\\tilde{N}\\) takes only even values: \\(0,2,4,6,\\ldots\\).\n\n\\subsection{The rotating closed string solution} \\label{sec:rotating_string}\n\\subsubsection{Classical rotating folded string} \\label{sec:flat_string}\n\nHere we use the Nambu-Goto action for the string\n\\begin{equation} S = -\\frac{1}{2\\pi\\ensuremath{\\alpha^\\prime}}\\int d\\tau d\\sigma \\sqrt{-h} \\,,\\end{equation}\nwith\n\\begin{equation} h = \\det h_{\\alpha\\beta}\\,, \\qquad h_{\\alpha\\beta} = \\eta_{\\mu\\nu}\\partial_\\alpha X^\\mu \\partial_\\beta X^\\nu \\,,\\end{equation}\nand \\begin{equation} \\ensuremath{\\alpha^\\prime} = \\frac{1}{2\\pi T} \\,. \\end{equation}\n\nThe rotating folded string is the solution\n\\begin{equation} X^0 = \\tau \\qquad X^1 = \\frac{1}{\\omega}\\sin(\\omega\\sigma)\\cos(\\omega\\tau) \\qquad X^2 = \\frac{1}{\\omega}\\sin(\\omega\\sigma)\\sin(\\omega\\tau) \\,.\\label{eq:rotsol}\\end{equation}\nWe take \\(\\sigma \\in (-\\ell,\\ell)\\) and correspondingly \\(\\omega\\) takes the value \\(\\omega = \\pi\/\\ell\\). The energy of this configuration is\n\\begin{equation} E = T \\int_{-\\ell}^\\ell d\\sigma \\partial_\\tau X^0 = 2T\\ell \\,.\\end{equation}\nThe angular momentum we can get by going to polar coordinates (\\(X^1 = \\rho\\cos\\theta, X^2 = \\rho\\sin\\theta\\)), then\n\\begin{equation} J = T \\int_{-\\ell}^\\ell d\\sigma \\rho^2 \\partial_\\tau \\theta =\n\\frac{T}{\\omega} \\int_{-\\ell}^\\ell d\\sigma \\sin^2(\\omega\\sigma) = \\frac{\\pi T}{\\omega^2} = \\frac{T\\ell^2}{\\pi}\\,.\\end{equation}\nFrom the last two equations we can easily see that for the classical rotating folded string\n\\begin{equation} J = \\frac{1}{4\\pi T}E^2 = \\frac{1}{2}\\ensuremath{\\alpha^\\prime} E^2 \\,.\\end{equation}\n\n\\subsubsection{Quantization of the rotating folded string} \\label{sec:non_critical_string}\nIn a previous section we reviewed the quantization of the bosonic closed string in the critical dimension, \\(D = 26\\). There we have the result\n\\begin{equation} \\frac{1}{2}\\ensuremath{\\alpha^\\prime} M^2 = N + \\tilde{N} - \\frac{D-2}{12} \\,. \\end{equation}\nWe would like to obtain a correction to the classical trajectory of a similar form when quantizing the rotating folded string in \\(D = 4\\) dimensions. In \\cite{Hellerman:2013kba} the intercept was computed in the context of effective string theory where the Polchinski-Strominger (PS) term \\cite{Polchinski:1991ax},\n\\begin{equation} \\mathcal{L}_{PS} = \\frac{26-D}{24\\pi}\\frac{(\\partial_+^2X\\cdot\\partial_-X)(\\partial_-^2X\\cdot\\partial_+X)}{(\\partial_+X\\cdot\\partial_-X)^2} \\label{eq:psterm} \\,, \\end{equation}\ncompensates for the conformal anomaly when working outside the critical dimension. The derivatives are with respect to the variables \\(\\sigma^\\pm \\equiv \\tau\\pm\\sigma\\).\n\nAs was described in the introduction and will be further discussed in section \\ref{sec:holo_string}, a major candidate for describing the glueball is a rotating closed string in a holographic background which lives, by definition, in the critical dimension. One may conclude that in this case the PS term is not needed. However, as was argued in \\cite{Aharony:2009gg}, upon integrating out the massive degrees of freedom of the closed string that resides in the critical holographic dimension one gets the PS action as part of the effective string action in the non-critical $D$ dimensions. \n\nThe calculation in \\cite{Hellerman:2013kba} is for a general dimension \\(D\\), with, as already mentioned, the PS term included. In dimensions larger than four the string will rotate in two planes and the angular momentum is characterized by two quantum numbers \\(J_1\\) and \\(J_2\\). The result obtained there for the Regge trajectory of the closed string is\n\\begin{equation} \\frac{\\ensuremath{\\alpha^\\prime}}{2}M^2 = (J_1+J_2) - \\frac{D-2}{12} + \\frac{26-D}{24}\n\\left((\\frac{J_1}{J_2})^\\frac{1}{4}-(\\frac{J_2}{J_1})^\\frac{1}{4}\\right)^2 \\,. \\end{equation}\nThis expression is singular when \\(J_2 = 0\\), which is necessarily the case when \\(D = 4\\), since in four dimensions the rotation is in a single plane. Therefore the expression is not usable precisely in the context in which we would like to use it.\n\nWe can see where this originates by inserting the 4D rotating solution from eq. \\ref{eq:rotsol} into the expression for the PS term, eq. \\ref{eq:psterm}. The expression obtained,\n\\begin{equation} \\mathcal{L}_{PS} = -\\frac{D-26}{24\\pi}\\omega^2\\tan^2(\\omega\\sigma) \\,,\\end{equation}\nis singular when \\(\\omega\\sigma = \\pm\\frac{\\pi}{2}\\), i.e. at the two points \\(\\sigma = \\pm\\frac{\\ell}{2}\\), which are the ``endpoints'', or folding-points, of the rotating folded string, and the integral on \\(\\mathcal{L}_{PS}\\) giving the correction diverges:\n\\begin{equation} \\int_{-\\ell}^\\ell d\\sigma \\mathcal{L}_{PS} = -\\frac{D-26}{12\\pi}\\omega^2\\int_{-\\ell\/2}^{\\ell\/2}d\\sigma\\tan^2(\\omega\\sigma) = -\\frac{D-26}{12\\pi}\\omega\\left(\\tan x-x\\right)|_{x=-\\pi\/2}^{\\pi\/2} \\,.\\end{equation}\nWe see that beneath the divergent \\(\\tan x\\) there is also the finite part\n\\begin{equation} \\frac{D-26}{12}\\frac{\\pi}{\\ell} \\,.\\end{equation}\n\nThe denominator in the PS term is simply \\((\\dot{X}^2)^2\\), so the problem emerges because the endpoints move at the speed of light. The same problem is encountered in the treatment of the open string, but as was shown in \\cite{Hellerman:2013kba} in that case one can introduce a counterterm at the string boundaries that renders the action and correspondingly the intercept finite. In fact it was found out that summing up the contributions to the latter from the PS and from the Casimir term, the \\(D\\) dependence is canceled out between the two terms, and the intercept is given simply by $a= 1$, for all \\(D\\). Another possible approach for regularizing the rotating open string is to add masses to its endpoints. However, the quantization of the system of a rotating string with massive particles on its ends is still not fully understood \\cite{ASY}.\n\nFor the closed string it is not clear how to regularize the system. One potential way to do it might be to add two masses at the two endpoints of the folded string. The resulting system looks like two open strings connected at their boundaries by these masses, but not interacting in any other way. In the rotating solution the two strings lie on top of one another. The boundary condition, which is the equation of motion of the massive endpoint is modified: it is the same as for the open string, but with an effective double tension \\(T \\rightarrow 2T\\), in accordance with the ratio of the slopes of the open and closed strings discussed above. In fact everything else is doubled too. If this process of adding masses on the closed string and taking then the limit of zero mass is a legitimate way to regularize, then it is probable that the result is also simply double that of the open string, as it is for the critical dimension. Obviously, though, even in that case we cannot perform the quantization of the folded closed string since, as mentioned above, we do not fully control the quantization of an open string with massive endpoints. \n\n\\subsubsection{The closed string in a curved background} \\label{sec:holo_string}\nThe full analysis of rotating closed string in holographic curved backgrounds was performed in \\cite{PandoZayas:2003yb}. We present here the key points in short form.\n\nIf we look at a curved background metric of the form\n\\begin{equation} ds^2 = h(r)^{-1\/2}(-dX^0dX^0+dX^idX^i) + h(r)^{1\/2}dr^2 + \\ldots \\,,\\end{equation}\nwith \\(i = 1,2,3\\) and the ellipsis denoting additional transverse coordinates, the rotating folded string, namely the configuration,\n\\begin{equation} X^0 = l\\tau \\qquad X^1 = l\\sin\\sigma\\cos\\tau \\qquad X^2 = l\\sin\\sigma\\sin\\tau \\,,\\end{equation}\nis still\\footnote{We follow a somewhat different normalization here, taking \\(\\omega = \\pi\/\\ell\\) from the previous section to be \\(1\\), and introducing a common prefactor \\(l\\), but the solution is essentially the same as the flat space solution of section \\ref{sec:flat_string}.} a solution to the string equations of motion provided we take\n\\begin{equation} r(\\sigma,\\tau) = r_0 = Const. \\end{equation}\nwhere \\(r_0\\) is a point where the metric satisfies the condition\n\\begin{equation} \\partial_r g_{00}(r)|_{r=r_0} = 0, \\qquad g_{00}(r)|_{r=r_0} \\neq 0 \\,.\\end{equation}\nThe existence of such a point is also one of the sufficient conditions for the dual gauge theory to be confining \\cite{Kinar:1998vq}. Compared to the folded string in flat spacetime, the energy and angular momentum take each an additional factor in the form of \\(g_{00}(r_0)\\):\n\\begin{equation} E = \\frac{1}{2\\pi\\ensuremath{\\alpha^\\prime}}\\int_{-\\pi}^\\pi g_{00}(r_0)d\\sigma = g_{00}(r_0) \\frac{l}{\\ensuremath{\\alpha^\\prime}} \\,,\\end{equation}\n\\begin{equation} J = T\\int_{-\\pi}^\\pi g_{00}(r_0)\\sin^2\\sigma d\\sigma = g_{00}(r_0) \\frac{l^2}{2\\ensuremath{\\alpha^\\prime}} \\,.\\end{equation}\nDefining an effective string tension \\(T_{eff} = g_{00}(r_0)T\\) and slope \\(\\ensuremath{\\alpha^\\prime}_{eff} = (2\\pi T_{eff})^{-1}\\), we can write the relation\n\\begin{equation} J = \\frac{1}{2}\\ensuremath{\\alpha^\\prime}_{eff} E^2 \\,.\\end{equation}\nThe same factor of \\(g_{00}(r_0)\\) multiplies the effective tension in the open string case, and therefore the closed and open string slopes are still related by the factor of one half, although the open string trajectories have the additional modification which can be ascribed to the presence of endpoint masses \\cite{Kruczenski:2004me,Sonnenschein:2014jwa}. We draw the two types of strings in figure \\ref{fig:closed_open_map}.\n\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=0.95\\textwidth]{closed_and_open.png} \\\\\n\t\\caption{\\label{fig:closed_open_map} Closed and open strings in holographic backgrounds (top), and their mappings into flat spacetime (bottom). For the open string, the mapping from curved to flat background adds endpoint masses to the strings \\cite{Kruczenski:2004me,Sonnenschein:2014jwa}, with the vertical segments mapped to the point-like masses in flat space. For the closed string, we look at the simple folded string in both cases. Note that classically the rotating folded string has zero width, and as such would look like an open string with no endpoint masses, and not like in the drawing.}\n\t\t\t\\end{figure}\n\n\nCalculations of the quantum corrected trajectory of the folded closed string in a curved background in different holographic backgrounds were performed in \\cite{PandoZayas:2003yb} and \\cite{Bigazzi:2004ze} using semiclassical methods. This was done by computing the spectrum of quadratic fluctuations, bosonic and fermionic, around the classical configuration of the folded string.\nIt was shown in \\cite{PandoZayas:2003yb} that the Noether charges of the energy $E$ and angular momentum $J$ that incorporate the quantum fluctuations, are related to the expectation value of the world-sheet Hamiltonian in the following manner:\n\\begin{equation}\nlE -J = \\int d\\sigma <{\\cal H}_{ws}> \\,.\n\\end{equation}\nThe contributions to the expectation value of the world-sheet Hamiltonian are from several massless bosonic modes, ``massive\" bosonic modes and massive fermionic modes. \n For the ``massive\" bosonic fluctuations around the rotating solution one gets a \\(\\sigma\\)-dependent mass term, with equations of motion of the form\n\\begin{equation} (\\partial_\\tau^2-\\partial_\\sigma^2+2m_0^2l^2\\cos^2\\sigma)\\delta X^i = 0 \\end{equation}\nappearing in both analyses, \\(m_0\\) being model dependent. A similar mass term, also with \\(\\cos\\sigma\\), appears in the equations of motion for some fermionic fluctuations as well, the factor of \\(\\cos^2\\sigma\\) in the mass squared coming in both cases from the induced metric calculated for the rotating string, which is \\(h_{\\alpha\\beta} \\sim \\eta_{\\alpha\\beta}\\cos^2\\sigma\\).\n\nThe result in both papers is that the Regge trajectories are of the form\n\\begin{equation} J = \\ensuremath{\\alpha^\\prime}_{closed}(E^2- 2m_0 E) +a \\,.\\end{equation}\nwhere $m_0$ is a mass parameter that characterizes the holographic model and $a$ is the intercept which generically takes the form $a= \\frac{\\pi}{24}(\\#\\text{bosonic massless modes} - \\#\\text{fermionic massless modes})$. The two papers \\cite{PandoZayas:2003yb} and \\cite{Bigazzi:2004ze} use different holographic models (Klebanov-Strassler and Maldacena-N\\'{u}\\~{n}ez backgrounds in the former and Witten background in the latter) and predict different signs for \\(m_0\\), which is given as a combination of the parameters specific to the background. In \\cite{PandoZayas:2003yb} \\(m_0\\) is positive, while in \\cite{Bigazzi:2004ze} it is negative. According to \\cite{PandoZayas:2003yb} the slope of the closed string trajectory is left unchanged from the classical case\n\\begin{equation} \\ensuremath{\\alpha^\\prime}\\!_{closed} = \\frac{1}{2}\\ensuremath{\\alpha^\\prime}\\!_{open} \\,,\\end{equation}\nwhile the model used in \\cite{Bigazzi:2004ze} predicts an additional renormalization of the slope,\n\\begin{equation} \\ensuremath{\\alpha^\\prime}\\!_{closed} = \\frac{1}{2}\\left(1-\\frac{c}{\\lambda}\\right)\\ensuremath{\\alpha^\\prime}\\!_{open} \\,,\\end{equation}\nfor some small constant \\(c\\), which makes this a smaller effect than that caused by the addition of the \\(m_0\\) mass term.\n\n\\subsection{Other string models of the glueball and the Regge slope}\nIn previous sections we have shown that the expected Regge slope for the closed string is\n\n\\begin{equation} \\ensuremath{\\alpha^\\prime}_{closed} = \\frac12\\ensuremath{\\alpha^\\prime}_{open} \\,, \\end{equation}\nbut other string models of the glueball predict different values for the effective slope of the glueballs, \\(\\ensuremath{\\alpha^\\prime}_{gb}\\).\n\nOne such prediction is based on the potential between two static adjoint SU(N) charges, that, according to lattice calculations, is expected to be proportional to the quadratic Casimir operator. For small distances this added group theory factor can be obtained easily from perturbation theory, and calculations in \\cite{Bali:2000un} show that what is referred to as the ``Casimir scaling hypothesis'' holds in lattice QCD for large distances as well, and this means that the effective string tension also scales like the Casimir operator (as the potential at large distances is simply \\(V(\\ell) \\approx T_{eff}\\ell\\)). Therefore, a model of the glueball as two adjoint charges (or constituent gluons) joined by a flux tube predicts the ratio between the glueball and meson (two fundamental charges) slopes to be\n\\begin{equation} \\frac{\\ensuremath{\\alpha^\\prime}\\!_{gb}}{\\ensuremath{\\alpha^\\prime}_{meson}} = \\frac{C_2(\\text{Fundamental})}{C_2(\\text{Adjoint})} = \\frac{N^2-1}{2N^2} = \\frac{4}{9}\\,, \\end{equation}\nwhere for the last equation we take \\(N = 3\\). For \\(N \\rightarrow \\infty\\) we recover the ratio of \\(1\/2\\), as can be easily seen. An argument from field theory for the double tension of the adjoint string at large \\(N\\) is in \\cite{Armoni:2006ri}.\n\nOther models attempt to tie the closed string to the phenomenological pomeron. The pomeron slope is measured to be \\cite{Donnachie:1984xq}\n\\begin{equation} \\ensuremath{\\alpha^\\prime}\\!_{pom} = 0.25\\:\\text{GeV}^{-2} \\approx 0.28\\times\\ensuremath{\\alpha^\\prime}\\!_{meson}\\,, \\end{equation}\nand the pomeron trajectory is commonly associated with both glueballs and closed strings. One string model that predicts a pomeron-like slope was proposed in \\cite{Isgur:1984bm} and is presented in \\cite{Meyer:2004jc} or in more detail in \\cite{Meyer:2004gx}. It is simply the model of a rotating closed string, with a fixed circular shape. This string has two types of trajectories, a phononic trajectory (excitations propagating along the string) which has \\(\\ensuremath{\\alpha^\\prime}_{phonon} = \\frac{1}{4}\\ensuremath{\\alpha^\\prime}_{open}\\), and an orbital trajectory (the circular string rotating around an axis in the circle's plane), for which \\(\\ensuremath{\\alpha^\\prime}\\!_{orbital} = \\frac{3\\sqrt{3}}{16}\\ensuremath{\\alpha^\\prime}_{open} \\approx 0.32\\times\\ensuremath{\\alpha^\\prime}_{open}\\). If the rotating circular loop were allowed to deform, it would have necessarily flowed towards the flattened folded string configuration that we have been discussing, which always maximizes the angular momentum at a given energy.\n\nThere are also other possibilities of rigidly rotating closed string of other shapes, as in \\cite{Burden:1982zb}, which may give yet another prediction of the ratio between open and closed string Regge slopes. Another related object is the ``\\(\\Delta\\)-shaped'' string, which we mentioned in \\cite{Sonnenschein:2014bia} as one of the stringy models of the baryon. The model is that of three masses with each pair of them connected by a string. This results in what is essentially a closed string with three quarks placed on it, which has lead 't Hooft to remark that such a configuration could be related to a quark-gluon hybrid \\cite{'tHooft:2004he}, rather than a pure glueball.\n\n\\subsection{The decays of the holographic closed string} \\label{sec:decays}\n\\subsubsection{Open string decays}\nThe open string hadron decays when it tears at a point along the string and the two loose ends connect via quantum fluctuations to a flavor brane, creating a quark-antiquark pair. Another way to think of this process is that the string fluctuates, before tearing,\tand when it reaches a flavor brane it connects to it, tears, and the pair is created. When thinking of the decay in this second way, with the fluctuation preceding the tear, it is clear that the quark and antiquark are of the same flavor, a result not a priori guaranteed when the strings tears and then reconnects to the branes. This is illustrated in figure \\ref{fig:decay_open}.\n\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=0.95\\textwidth]{decay_open.png}\n\t\\caption{\\label{fig:decay_open} A schematic look at the decay of a holographic open string, in this case a strange meson decaying into a strange meson and a light meson. \\textbf{Top}: the picture where the string tears first, then reconnects to the flavor branes. \\textbf{Bottom}: the string fluctuates up to the brane before tearing, the splits. We prefer the second picture since it assures that flavor is conserved, which is not a priori the case when the string tears at the bottom.}\n\t\t\t\\end{figure}\n\nThe probability that a fluctuation reaches the flavor brane of a quark of flavor \\(q\\) is \\cite{Peeters:2005fq}\n\\begin{equation} e^{-(m_{sep}^q)^2\/T} \\,, \\end{equation}\nwhere the quark mass \\(m_{sep}^q\\) in this context is equal to the string tension times the distance of the brane from the holographic wall.\\footnote{The fact that in this model the mass \\(m_{sep}^q\\) is proportional to \\(T\\) is especially important when considering the opposing limits \\(T \\rightarrow 0\\) and \\(T\\rightarrow\\infty\\).}\n\nSince the tear can occur at any point along the string, we expect the total probability (and hence the total decay width) to be proportional to the string length \\(L\\).\\footnote{In the holographic picture, it is the length of the horizontal segment of the string that is considered. When moving into flat space, it is the length between the two endpoint masses, and the relation \\(M \\propto TL\\) receives corrections from the endpoint masses, as already written in eq. \\ref{eq:holo_width} in the introduction.} We then expect that the total decay width behave like\n\\begin{equation} \\Gamma \\propto Le^{-(m_{sep}^q)^2\/T}\\,, \\end{equation}\nwhere \\(m_{sep}^q\\) is the quark produced in the decay. In \\cite{Sonnenschein:2014jwa} we extracted some values of the quark masses as obtained from the Regge trajectories of mesons. For the light \\(u\/d\\) quarks the masses were small enough so the exponent is close to one, while the \\(s\\) quark showed a mass for which \\(m_s^2\/T \\sim 1\\). We would then say that decays where an \\(\\ensuremath{s\\bar{s}}\\) pair is created are suppressed by a factor of \\(e^{-1}\\) (before taking into account the smaller phase space).\\footnote{In an alternative description \\cite{Cotrone:2005fr,Bigazzi:2006jt}, the decay rate is power-like (rather than exponentially) suppressed with the mass of the quark-antiquark pair.}\n\n\\subsubsection{Rotating closed string}\nThe decay process of a closed string is less simple as the string has to tear twice.\\footnote{Another holographic approach to describe the decay of a glueball into two mesons, based on fields in the bulk and not closed strings was discussed in \\cite{Hashimoto:2007ze,Brunner:2015oqa,Brunner:2015yha}} A single tear in the closed would produce an open string, and it in turn will have to tear again, so at the end of the process we have two open strings. If the closed string is the glueball, then this is the process of a glueball decaying to two mesons. In the total decay width we will have then the string length squared, one factor of \\(L\\) for each time the string tears, as well as two exponents for the two pair creation events:\n\\begin{equation} \\Gamma \\propto L^2\\exp(-\\frac{m_q^2}{T})\\exp(-\\frac{m_{q^\\prime}^2}{T}) \\,. \\label{eq:decay_closed} \\end{equation}\nThis process is illustrated in figure \\ref{fig:decay_closed}.\n\nIf we want to identify a glueball from this basic prediction we have to look at the branching ratios of processes where the presence of the second exponent is significant, namely at processes where pairs of \\(s\\) and \\(\\bar{s}\\) are produced.\n\nThe glueball unlike the meson will have the possibility of decaying into either of the three options: decay into two light mesons with two pairs of light quarks created, into \\(K\\bar{K}\\) with one pair of \\(\\ensuremath{s\\bar{s}}\\) and the other light, or into \\(\\phi\\phi\\) when two pairs of \\(\\ensuremath{s\\bar{s}}\\) are created. The exponents predict the following hierarchy between the three modes:\n\\begin{equation} \\Gamma(Gb\\rightarrow \\text{2 light}) : \\Gamma(Gb\\rightarrow K\\bar{K}) : \\Gamma(Gb\\rightarrow \\phi\\phi) = 1\\,:e^{-1}\\,:e^{-2} \\,. \\end{equation}\nThis ratio will still need to be modified by phase space factors, which in any realistic scenario will be significant and will suppress the \\(\\ensuremath{s\\bar{s}}\\) modes even further. This is because the states we would measure are not too far from the \\(\\phi\\phi\\) threshold of approximately 2 GeV.\n\n\\begin{figure}[tp!] \\centering\n\t\\includegraphics[width=0.95\\textwidth]{decay_closed.png} \\\\\n\t\\includegraphics[width=0.67\\textwidth]{decay_closed_worldsheet.png}\n\t\\caption{\\label{fig:decay_closed} A schematic look at the decay of a holographic closed string to two mesons. (I) The string tears for the first time. (II) An \\(\\ensuremath{s\\bar{s}}\\) pair is produced and the string tears for the second time. (III) A second pair is created, this time of light quarks (i.e. \\(u\\bar{u}\\) or \\(d\\bar{d}\\)), and two open strings are formed. (IV) A different perspective showing more clearly the final product of this decay: a \\(K\\) meson and a \\(\\bar{K}\\). Note that the distances in this schematic between the flavor branes and the wall are not in scale. The bottom figure is the corresponding world sheet, of a closed string opening up at two points and forming two open strings.}\n\t\t\t\\end{figure}\n\n\\clearpage\n\\section{Phenomenology} \\label{sec:phenomenology}\n\\subsection{Basic assumptions and fitting models} \\label{sec:fitting_models}\nWe will be looking at unflavored isoscalar resonances below the \\(c\\bar{c}\\) threshold. These states will be either mesons with the quark contents \\(\\frac{1}{\\sqrt{2}}(u\\bar{u}-d\\bar{d})\\) or \\(s\\bar{s}\\), or glueballs.\\footnote{Some states, such as the \\(f_0(500)\/\\sigma\\), may also be exotic multiquark states.} Correspondingly, we have several types of trajectories. For the light mesons we have the usual linear form,\n\\begin{equation} J + n = \\ensuremath{\\alpha^\\prime} M^2 + a \\,, \\label{eq:traj_lin}\\end{equation}\nwith \\(\\ensuremath{\\alpha^\\prime} = (2\\pi T)^{-1}\\). Note that whenever we use \\(\\ensuremath{\\alpha^\\prime}\\) without a subscript in this paper, it refers to this slope of the linear meson trajectories.\n\nFor \\(s\\bar{s}\\) states, we use the formula for the mass corrected trajectory (as was used in \\cite{Sonnenschein:2014jwa}) defined by\n\t\\begin{equation} E = 2m_s\\left(\\frac{\\beta\\arcsin \\beta+\\sqrt{1-\\beta^2}}{1-\\beta^2}\\right) \\label{eq:massFitE} \\,,\\end{equation}\n\t\\begin{equation} J + n = a + 2\\pi\\ensuremath{\\alpha^\\prime} m_s^2\\frac{\\beta^2}{(1-\\beta^2)^2}\\left(\\arcsin \\beta+\\beta\\sqrt{1-\\beta^2}\\right) \\,.\\label{eq:traj_mass} \\end{equation}\nThese are the trajectories of a rotating string with two masses \\(m_s\\) at its endpoints, and with an added intercept and extrapolated \\(n\\) dependence. \\(\\beta\\) is the velocity of the endpoint mass. The limit \\(m_s \\rightarrow 0\\) (with \\(\\beta \\rightarrow 1\\)) takes us back to the linear trajectory of eq. \\ref{eq:traj_lin}, with the first correction in the expression for \\(J\\) being proportional to \\(\\ensuremath{\\alpha^\\prime} m_s^{3\/2} E^{1\/2}\\).\n\nFor the glueballs we assume linear trajectories of the form\n\\begin{equation} J + n = \\ensuremath{\\alpha^\\prime}\\!_{gb} M^2 + a \\,,\\end{equation}\nand we take \\(\\ensuremath{\\alpha^\\prime}\\!_{gb}\\) to be \\(\\frac{1}{2}\\ensuremath{\\alpha^\\prime}\\), where \\(\\ensuremath{\\alpha^\\prime}\\) is the slope of the mesons as obtained in our fits of the various meson trajectories. A typical value would be between 0.80 and 0.90 GeV\\(^{-2}\\).\n\nIn a later section we examine the possible application of the formula based on the holographic prediction,\n\\begin{equation} J + n = \\ensuremath{\\alpha^\\prime}_{gb}E^2-2\\ensuremath{\\alpha^\\prime}_{gb}m_0E + a\\,.\\end{equation}\nWhen using this formula we will also take \\(\\ensuremath{\\alpha^\\prime}\\!_{gb} = \\frac{1}{2}\\ensuremath{\\alpha^\\prime}\\), ignoring the possible correction to the slope, which we assume to be small.\n\nOne assumption which we must state explicitly before continuing to the fits is that there is no mixing of light mesons, \\(\\ensuremath{s\\bar{s}}\\) mesons, and glueballs. It is an open question how strongly glueballs and mesons are mixed, with results varying greatly between different models, from almost maximal mixing to very weak (different results based on different models are collected in \\cite{Crede:2008vw}). In a stringy model, where glueballs are represented by closed strings and mesons by open strings, it seems more natural that they will not mix at all. We also assume that the mixing between the light quark states and the \\(\\ensuremath{s\\bar{s}}\\) is weak, in placing states either on the linear trajectories of the light mesons or on the mass corrected trajectories of the \\(\\ensuremath{s\\bar{s}}\\), the same assumption that was used in \\cite{Sonnenschein:2014jwa} in fitting the \\(\\omega\\) and \\(\\phi\\) mesons. It is not obvious how the possible mixing between the two types of mesons affects the trajectories.\n\n\\subsubsection{The two types of trajectories}\nAlong \\emph{radial trajectories}, or trajectories in the \\((n,M^2)\\) plane, the states differ only by the radial\\footnote{The term should not be confused as having something to do with the radial coordinate of holography.} excitation number \\(n\\), all other quantum numbers constant. Since \\(n\\) is not actually measured we have to assign a value ourselves to the different states, and from there emerges a great ambiguity that we have to solve.\n\nMesons belong on trajectories in the \\((n,M^2)\\) plane with a slope that seems to be slightly smaller than in the \\((J,M^2)\\) plane. The typical values are 0.80--0.85 GeV\\(^{-2}\\) for the former and \\(0.90\\) GeV\\(^{-2}\\) for the latter type of trajectories, as our fits in \\cite{Sonnenschein:2014jwa} have shown. We implicitly assume in the following sections that for the glueballs there will be a similar difference between the slopes in the different planes. When we write that \\(\\ensuremath{\\alpha^\\prime}_{gb} = \\frac12\\ensuremath{\\alpha^\\prime}_{meson}\\) we refer to \\(\\ensuremath{\\alpha^\\prime}_{meson}\\) as it is obtained for the meson fits in the same plane, rather than taking fixed values of \\(\\ensuremath{\\alpha^\\prime}\\). This also serves to restrict the number of parameters in a given fit: we always try to describe all the trajectories using a single value of \\(\\ensuremath{\\alpha^\\prime}\\).\n\nWe should also note that while for the mesons, \\(n\\) naturally takes the values \\(n = 0,1,2,\\ldots\\) along the radial trajectories, the case is not so for glueballs. For the closed strings we noted in section \\ref{sec:lightcone} that the number of left and right moving modes has to be equal, and so \\(n\\), which is really \\(N+\\tilde{N}\\) in this case, should be even: \\(n = 0,2,4,\\ldots\\).\n\nFor the \\emph{orbital trajectories}, or trajectories in the \\((J,M^2)\\) plane, we expect to find, along the leading trajectory of the glueball, the ground state with \\(J^{PC} = 0^{++}\\) followed by the tensor glueball (\\(2^{++}\\)) as its first excited state, and continue to higher states with even \\(J\\) and \\(PC = ++\\).\n\nThe orbital trajectories of the mesons will be constructed as usual, and using the known quark model relations \\(P = (-1)^{L+1}\\) and \\(C = (-1)^{L+S}\\). The relevant trajectories are then expected to have states with \\(J^{PC} = 1^{--}, 2^{++}, 3^{--}, 4^{++}, \\ldots\\). It is worth noting then that for mesons, a \\(0^{++}\\) state is an excited state with \\(L = 1\\) and \\(S = 1\\), and not a part of what we usually take for the trajectory when we use states of increasing \\(J\\).\n\n\\subsection{The glueball candidates: The \\texorpdfstring{$f_0$}{f0} and \\texorpdfstring{$f_2$}{f2} resonances}\nThere is an abundance of isoscalar states with the quantum numbers \\(J^{PC} = 0^{++}\\) (the \\(f_0\\) resonances) or \\(J^{PC} = 2^{++}\\) (\\(f_2\\)). The Particle Data Group's (PDG) latest Review of Particle Physics \\cite{PDG:2014}, which we we use as the source of experimental data throughout this paper, lists 9 \\(f_0\\) states and 12 \\(f_2\\) states, with an additional 3 \\(f_0\\)'s and 5 \\(f_2\\)'s listed as unconfirmed ``further states''. These are listed in tables \\ref{tab:allf0} and \\ref{tab:allf2}. In the following we make a naive attempt to organize the known \\(f_0\\) and \\(f_2\\) states into trajectories, first in the plane of orbital excitations \\((J,M^2)\\), then in the radial excitations plane \\((n,M^2)\\).\n\nThe states classified as ``further states'' are generally not used unless the prove to be necessary to complete the trajectories formed by the other states. The ``further states'' will be marked with an asterisk below.\\footnote{Note that the asterisk is not standard notation nor a part of the PDG given name of a state, we only use it to make clear the status of given states throughout the text.}\n\nIt is not the purpose of this paper to review all the information available on the \\(f_0\\) and \\(f_2\\) resonances, nor to present the different theories and speculations regarding their meson or glueball nature. We usually attempt to form Regge trajectories first, using just the masses and basic quantum numbers, and then verify if the implications regarding the contents of a given state make sense in the light of additional experimental data, namely the different states' decay modes.\n\nFor a more complete picture regarding the spectrum and specifically the interpretation of the different resonances as glueballs, the reader is referred to reviews on glueball physics and their experimental status such as \\cite{Klempt:2007cp,Mathieu:2008me,Crede:2008vw,Ochs:2013gi}, citations therein, and subsequent works citing these reviews.\n\n\n\\begin{table}[t!] \\centering\n\t\t\\begin{tabular}{|l|l|l|l|l|} \\hline\n\t\t\\textbf{State} & \\textbf{Mass} [MeV] & \\textbf{Width} [MeV] & \\textbf{Width\/mass} & \\textbf{Decay modes} \\\\ \\hline\\hline\n\t\t\\(f_0(500)\/\\sigma\\) & 400--550 & 400--700 & 1.16\\plm0.36 & \\(\\pi\\pi\\) dominant\\\\ \\hline\n\t\t\\(f_0(980)\\) & \\(990\\pm20\\) & 40--100 & 0.07\\plm0.03 & \\(\\pi\\pi\\) dominant, \\(K\\overline{K}\\) seen \\\\ \\hline\n\t\t\\(f_0(1370)\\) & 1200--1500 & 200--500 & 0.26\\plm0.11 & \\(\\pi\\pi\\), \\(4\\pi\\), \\(\\eta\\eta\\), \\(K\\overline{K}\\) \\\\ \\hline\n\t\t\\(f_0(1500)\\) & \\(1505\\pm6\\) & \\(109\\pm7\\) & 0.072\\plm0.005 & \\(\\pi\\pi\\) \\([35\\%]\\), \\(4\\pi\\) \\([50\\%]\\), \\\\\n\t\t\t\t\t\t\t\t\t& & & & \\(\\eta\\eta\\)\/\\(\\eta\\eta\\prime\\) \\([7\\%]\\), \\(K\\overline{K}\\) \\([9\\%]\\) \\\\ \\hline\t\t\n\t\t\\(f_0(1710)\\) & \\(1720\\pm6\\) & \\(135\\pm8\\) & 0.078\\plm0.005 & \\(K\\overline{K}\\), \\(\\eta\\eta\\), \\(\\pi\\pi\\) \\\\ \\hline\n\t\t\\(f_0(2020)\\) & \\(1992\\pm16\\) & \\(442\\pm60\\) & 0.22\\plm0.03 & \\(\\rho\\pi\\pi\\), \\(\\pi\\pi\\), \\(\\rho\\rho\\), \\(\\omega\\omega\\), \\(\\eta\\eta\\) \\\\ \\hline\n\t\t\\(f_0(2100)\\) & \\(2103\\pm8\\) & \\(209\\pm19\\) & 0.10\\plm0.01 & \\\\ \\hline\n\t\t\\(f_0(2200)\\) & \\(2189\\pm13\\) & \\(238\\pm50\\) & 0.11\\plm0.02 & \\\\ \\hline\n\t\t\\(f_0(2330)\\) & \\(2325\\pm35\\) & \\(180\\pm70\\) & 0.08\\plm0.03 & \\\\ \\hline\n\t\t*\\(f_0\\)(1200--1600) & 1200--1600 & 200--1000 & 0.43\\plm0.29 & \\\\ \\hline\n\t\t*\\(f_0\\)(1800) & \\(1795\\pm25\\) & \\(95\\pm80\\) & 0.05\\plm0.04 & \\\\ \\hline\n\t\t*\\(f_0\\)(2060) & \\(\\sim2050\\) & \\(\\sim120\\) & \\(\\sim0.04\\)--\\(0.10\\) & \\\\ \\hline\n\t\t\\end{tabular} \\caption{\\label{tab:allf0} All the \\(f_0\\) states as listed by the PDG. The last few states, marked here by asterisk, are classified as ``further states''.}\n\t\\end{table}\n\n\t\t\\begin{table}[t!] \\centering\n\t\t\\begin{tabular}{|l|l|l|l|l|} \\hline\n\t\t\\textbf{State} & \\textbf{Mass} [MeV] & \\textbf{Width} [MeV] & Width\/mass & \\textbf{Decay modes} \\\\ \\hline\\hline\n\t\t\\(f_2(1270)\\) & 1275.1\\plm1.2 & 185.1\\plm2.9 & 0.15\\plm0.00 & \\(\\pi\\pi\\) \\([85\\%]\\), \\(4\\pi\\) \\([10\\%]\\), \\(KK\\), \\(\\eta\\eta\\), \\(\\gamma\\gamma\\), ... \\\\ \\hline\n\t\t\\(f_2(1430)\\) & 1453\\plm4 & 13\\plm5 & 0.009\\plm0.006 & \\(KK\\), \\(\\pi\\pi\\) \\\\ \\hline\n\t\t\\(f^\\prime_2(1525)\\) & 1525\\plm5 & 73\\plm6 & 0.048\\plm0.004 & \\(KK\\) \\([89\\%]\\), \\(\\eta\\eta\\) \\([10\\%]\\), \\(\\gamma\\gamma\\) [seen], ... \\\\ \\hline\n\t\t\\(f_2(1565)\\) & 1562\\plm13 & 134\\plm8 & 0.09\\plm0.01 & \\(\\pi\\pi\\), \\(\\rho\\rho\\), \\(4\\pi\\), \\(\\eta\\eta\\), ... \\\\ \\hline\n\t\t\\(f_2(1640)\\) & 1639\\plm6 & 99\\plm60 & 0.06\\plm0.04 & \\(\\omega\\omega\\), \\(4\\pi\\), \\(KK\\) \\\\ \\hline\n\t\t\\(f_2(1810)\\) & 1815\\plm12 & 197\\plm22 & 0.11\\plm0.01 & \\(\\pi\\pi\\), \\(\\eta\\eta\\), \\(4\\pi\\), \\(KK\\), \\(\\gamma\\gamma\\) [seen] \\\\ \\hline\n\t\t\\(f_2(1910)\\) & 1903\\plm9 & 196\\plm31 & 0.10\\plm0.02 & \\(\\pi\\pi\\), \\(KK\\), \\(\\eta\\eta\\), \\(\\omega\\omega\\), ... \\\\ \\hline\n\t\t\\(f_2(1950)\\) & 1944\\plm12 & 472\\plm18 & 0.24\\plm0.01 & \\(K^*K^*\\), \\(\\pi\\pi\\), \\(4\\pi\\), \\(\\eta\\eta\\), \\(KK\\), \\(\\gamma\\gamma\\), \\(pp\\) \\\\ \\hline\n\t\t\\(f_2(2010)\\) & 2011\\plm76 & 202\\plm67 & 0.10\\plm0.03 & \\(KK\\), \\(\\phi\\phi\\) \\\\ \\hline\n\t\t\\(f_2(2150)\\) & 2157\\plm12 & 152\\plm30 & 0.07\\plm0.01 & \\(\\pi\\pi\\), \\(\\eta\\eta\\), \\(KK\\), \\(f_2(1270)\\eta\\), \\(a_2\\pi\\), \\(pp\\) \\\\ \\hline\n\t\t\\(f_J(2220)\\) & 2231.1\\plm3.5 & 23\\plm8 & 0.010\\plm0.004 & \\(\\pi\\pi\\), \\(KK\\), \\(pp\\), \\(\\eta\\eta^\\prime\\) \\\\ \\hline\n\t\t\\(f_2(2300)\\) & 2297\\plm28 & 149\\plm41 & 0.07\\plm0.02 & \\(\\phi\\phi\\), \\(KK\\), \\(\\gamma\\gamma\\) [seen] \\\\ \\hline\n\t\t\\(f_2(2340)\\) & 2339\\plm55 & 319\\plm81 & 0.14\\plm0.04 & \\(\\phi\\phi\\), \\(\\eta\\eta\\) \\\\ \\hline\n\t\t*\\(f_2(1750)\\) & 1755\\plm10 & 67\\plm12 & 0.04\\plm0.01 & \\(KK\\), \\(\\gamma\\gamma\\), \\(\\pi\\pi\\), \\(\\eta\\eta\\) \\\\ \\hline\n\t\t*\\(f_2(2000)\\) & 2001\\plm10 & 312\\plm32 & 0.16\\plm0.02 & \\\\ \\hline\n\t\t*\\(f_2(2140)\\) & 2141\\plm12 & 49\\plm28 & 0.02\\plm0.01 & \\\\ \\hline\n\t\t*\\(f_2(2240)\\) & 2240\\plm15 & 241\\plm30 & 0.11\\plm0.01 & \\\\ \\hline\n\t\t*\\(f_2(2295)\\) & 2293\\plm13 & 216\\plm37 & 0.10\\plm0.02 & \\\\ \\hline\n\t\t\\end{tabular} \\caption{\\label{tab:allf2} All the \\(f_2\\) states as listed by the PDG. The last few states, marked here by asterisk, are classified as ``further states''.}\n\t\\end{table}\n\n\t\\subsection{Assignment of the \\texorpdfstring{$f_0$}{f0} into trajectories} \\label{sec:f0_fits}\n\tIn a given assignment, we generally attempt to include all the \\(f_0\\) states listed in table \\ref{tab:allf0}, sorting them into meson and, if possible, glueball trajectories.\n\t\n\tWe make an exception of the \\(f_0(500)\/\\sigma\\) resonance, which we do not use in any of the following sections. Its low mass and very large width are enough to make it stand out among the other \\(f_0\\) states listed in the table. There is no common consensus regarding the composition of the \\(\\sigma\\). We find that it does not belong on a meson Regge trajectory. If we assume it is a glueball then our model predicts the next state to be at around 2.2 GeV, and, since we assume its width to be proportional to its mass squared (as implied by eq. \\ref{eq:decay_closed}), it would have a width of at least 8 GeV. We hope, in that case, that there is no reason to make such an assumption.\\footnote{The authors of \\cite{Nebreda:2011cp} state that the interpretation of the \\(f_0(500)\/\\sigma\\) as a glueball is ``strongly disfavored'', from what they consider a model independent viewpoint. We found no references that suggest the opposite.} Therefore, we simply ``ignore'' the \\(f_0(500)\\) in the following sections.\n\n\t\\subsubsection{Assignment of all states as mesons}\n\t\tSorting the \\(f_0\\) states into trajectories with a meson-like slope leads to an assignment of the \\(f_0\\)'s into two groups of four:\n\t\\[\\mathrm{Light}:\\qquad980, 1500, 2020, 2200, \\]\n\t\\[\\ensuremath{s\\bar{s}}:\\qquad1370, 1710, 2100, 2330. \\]\nWhile this simple assignment includes all the confirmed \\(f_0\\) states (except the \\(f_0(500)\\)) on two parallel trajectories, it remains unsatisfactory. If there are no glueballs we expect the states in the lower trajectory to be (predominantly) composed of light quarks, while the higher states should be \\(\\ensuremath{s\\bar{s}}\\). This does not match what we know about the decay modes of the different states. For example, the \\(f_0(1370)\\) does not decay nearly as often to \\(K\\bar{K}\\) as one would expect from an \\(\\ensuremath{s\\bar{s}}\\) state. In fact, this assignment of the \\(f_0\\)'s into meson trajectories was proposed in some other works \\cite{Anisovich:2000kxa,Anisovich:2002us,Masjuan:2012gc}, and the mismatch with the decay modes was already addressed in greater detail in \\cite{Bugg:2012yt}.\n\n\t\\subsubsection{Assignment with \\texorpdfstring{$f_0$}{f0}(980) as glueball}\n\t\\begin{table}[tp!] \\centering\n\t\\includegraphics[width=0.60\\textwidth]{glue_fit_980_f1n.png} \\qquad\n\t\t\t\t\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline\n\t\t\t\t\t\\(n\\) & \\multicolumn{2}{|c|}{Light} & \\multicolumn{2}{|c|}{\\(s\\bar{s}\\)} & \\multicolumn{2}{|c|}{Glueball} \\\\ \\hline\n\t\t\t\t\t& Exp. & Thry. & Exp. & Thry. & Exp. & Thry. \\\\ \\hline\n\t\t\t0 &\t1350\\plm150 & 1317 & 1505\\plm6 & 1505 & 990\\plm20 & 990 \\\\\n\t\t\t1 &\t1720\\plm6 & 1738 & 1992\\plm16 & 1984 & - & - \\\\\n\t\t\t2 &\t2103\\plm8 & 2075 & ? & 2340 & ? & 2470 \\\\\n\t\t\t3 &\t2325\\plm35 & 2365 & & & & \\\\\n\t\t\t4 & ? & 2620 & & & & \\\\ \\hline\n\t\t\t\t\\end{tabular} \\caption{\\label{tab:fit980} The results of the fit to the assignment with \\(f_0(980)\\) as the glueball ground state. The slope is \\(\\ensuremath{\\alpha^\\prime} = 0.788\\) GeV\\(^{-2}\\) and the mass of the \\(s\\) quark \\(m_s = 500\\) MeV. This fit has \\(\\chi^2 = 3.78\\). The intercepts obtained are (-1.35) for light mesons, (-0.52) for \\(\\ensuremath{s\\bar{s}}\\), and (-0.38) for glueballs. We also list the predicted mass of the next state in each trajectory.}\n\t\t\t\\end{table}\n\n\tIn this and the following sections we pick and single out a state as the glueball ground state and try to build the meson trajectories without it.\n\n\tFirst is the the \\(f_0(980)\\). Assuming it is the glueball then the \\(f_0(2330)\\) is at the right mass to be its first excited (\\(n = 2\\)) partner. However, we find that the two meson trajectories given this assignment,\n\\[\\mathrm{Light:}\\qquad 1370, 1710, 2100,\\]\n\\[\\ensuremath{s\\bar{s}}\\mathrm{:}\\qquad 1500, 2020,\\]\nalso predict a state very near the mass of the \\(f_0(2330)\\), and according to this assignment, there should be two more \\(f_0\\) states near the \\(f_0(2330)\\), for a total of three. The \\(f_0(2200)\\) has to be excluded.\n\nWe again have to put some states on trajectories that are not quite right for them: the \\(f_0(1710)\\) has a significant branching ratio for its decay into \\(K\\overline{K}\\), while the \\(f_0(1500)\\), which is taken as the head of the \\(\\ensuremath{s\\bar{s}}\\) trajectory, decays to \\(K\\overline{K}\\) less than \\(10\\%\\) of the time.\n\nNote that the assignment above is the same as the one we would make if we excluded the \\(f_0(980)\\) on the grounds of it being an exotic (but non-glueball) state and assumed all the other states are mesons. The \\(f_0(980)\\) is commonly believed to be a multiquark state or a \\(K\\bar{K}\\) ground state,\\footnote{See the PDG's ``Note on scalar mesons below 2 GeV'' and references therein.} and in fact, we will find in following sections that even it is not a glueball, it is better to exclude it from the meson trajectories. The trajectories and masses obtained are in table \\ref{tab:fit980}.\n\t\n\t\\subsubsection{Assignment with \\texorpdfstring{$f_0$}{f0}(1370) as glueball}\n\t\\begin{table}[tp!] \\centering\n\t\\includegraphics[width=0.60\\textwidth]{glue_fit_1370_f1n.png} \\qquad\n\t\t\t\t\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline\n\t\t\t\t\t\\(n\\) & \\multicolumn{2}{|c|}{Light} & \\multicolumn{2}{|c|}{\\(s\\bar{s}\\)} & \\multicolumn{2}{|c|}{Glueball} \\\\ \\hline\n\t\t\t\t\t& Exp. & Thry. & Exp. & Thry. & Exp. & Thry. \\\\ \\hline\n\t\t\t0 &\t990\\plm20 & 1031 & 1720\\plm6 & 1733 & 1350\\plm150 & 1350 \\\\\n\t\t\t1 &\t1505\\plm6 & 1488 & 2189\\plm13 & 2103 & - & - \\\\\n\t\t\t2 &\t1795\\plm25 & 1835 & ? & 2400 & ? & 2530 \\\\\n\t\t\t3 &\t2103\\plm8 & 2123 & & & & \\\\\n\t\t\t4 & 2325\\plm35 & 2377 & & & & \\\\\n\t\t\t5 & ? & 2610 & & & & \\\\ \\hline\n\t\t\t\t\\end{tabular} \\caption{\\label{tab:fit1370} The results of the fit to the assignment with \\(f_0(1370)\\) as the glueball ground state. The slope is \\(\\ensuremath{\\alpha^\\prime} = 0.873\\) GeV\\(^{-2}\\) and the mass of the \\(s\\) quark \\(m_s = 500\\) MeV. This fit has \\(\\chi^2 = 10.01\\). The intercepts obtained are (-0.93) for light mesons, (-1.06) for \\(\\ensuremath{s\\bar{s}}\\), and (-0.80) for glueballs. We also list the predicted mass of the next state in each trajectory.}\n\t\t\t\\end{table}\n\nFrom here onwards the states singled out as glueballs are too high in mass for their excited states to be in the range of the \\(f_0\\) states listed in table \\ref{tab:allf0}, that is beneath 2.4 GeV.\n\nExcluding the \\(f_0(1370)\\), we have:\n\\[\\mathrm{Light:}\\qquad [980], 1500, *1800, 2100, 2330\\]\n\\[\\ensuremath{s\\bar{s}}\\mathrm{:}\\qquad 1710, 2200.\\]\nThe \\(f_0(980)\\) is put here in brackets to emphasize that it is optional. Including or excluding it can affect some of the fitting parameters but the trajectory is certainly not incomplete if we treat \\(f_0(980)\\) as a non-meson resonance and take \\(f_0(1500)\\) as the head of the trajectory.\n\nThe main issue here is that we have to use the state \\(*f_0(1800)\\) to fill in a hole in the meson trajectory, a state that is still considered unconfirmed by the PDG and whose nature is not entirely known. It was observed so far only as an enhancement in the radiative decay \\(J\/\\psi \\rightarrow \\gamma\\omega\\phi\\) and its observers at BESIII \\cite{Ablikim:2012ft} suggest it is an exotic state - a tetraquark, a hybrid, or itself a glueball. More experimental data is needed here.\n\nOther than that we have \\(f_0(2100)\\) as a light meson and \\(f_0(2200)\\) as \\(\\ensuremath{s\\bar{s}}\\). This is the option that is more consistent with the decays, as \\(f_0(2200)\\) is the one state of the two which is known to decay into \\(K\\overline{K}\\) (we again refer to the comments in \\cite{Bugg:2012yt} and references therein). However, in terms of the fit, we might do better to exchange them. It is possible that the proximity of these two resonances to each other affects their masses in such a way that our model can not predict, and this affects badly the goodness of our fit, as can be seen in table \\ref{tab:fit1370}.\n\n\\subsubsection{Assignment with \\texorpdfstring{$f_0$}{f0}(1500) as glueball}\n\n\\begin{table}[tp!] \\centering\n\t\\includegraphics[width=0.60\\textwidth]{glue_fit_1500_f1n.png} \\qquad\n\t\t\t\t\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline\n\t\t\t\t\t\\(n\\) & \\multicolumn{2}{|c|}{Light} & \\multicolumn{2}{|c|}{\\(s\\bar{s}\\)} & \\multicolumn{2}{|c|}{Glueball} \\\\ \\hline\n\t\t\t\t\t& Exp. & Thry. & Exp. & Thry. & Exp. & Thry. \\\\ \\hline\n\t\t\t0 &\t1350\\plm150 & 1031 & 1720\\plm6 & 1723 & 1505\\plm6 & 1505 \\\\\n\t\t\t1 &\t1795\\plm25 & 1723 & 2103\\plm8 & 2097 & - & - \\\\\n\t\t\t2 &\t1992\\plm16 & 2029 & ? & 2400 & ? & 2620 \\\\\n\t\t\t3 &\t2325\\plm35 & 2295 & & & & \\\\\n\t\t\t4 & ? & 2530 & & & & \\\\ \\hline\n\t\t\t\t\\end{tabular} \\caption{\\label{tab:fit1500} The results of the fit to the assignment with \\(f_0(1500)\\) as the glueball ground state. The slope is \\(\\ensuremath{\\alpha^\\prime} = 0.870\\) GeV\\(^{-2}\\) and the mass of the \\(s\\) quark \\(m_s = 500\\) MeV. This fit has \\(\\chi^2 = 2.51\\). The intercepts obtained are (-1.58) for light mesons, (-1.03) for \\(\\ensuremath{s\\bar{s}}\\), and (-0.99) for glueballs. We also list the predicted mass of the next state in each trajectory.}\n\t\t\t\\end{table}\n\t\t\t\nTaking the \\(f_0(1500)\\) to be the glueball, then the light meson trajectory will start with \\(f_0(1370)\\), giving:\n\t\\[\\mathrm{Light:}\\qquad 1370, *1800, 2020, 2330,\\]\n\t\\[\\ensuremath{s\\bar{s}}\\mathrm{:}\\qquad 1710, 2100.\\]\nWith \\(f_0(1500)\\) identified as the glueball, this assignment includes all the states except \\(f_0(2200)\\). Incidentally though, the \\(f_0(2200)\\) would have belonged on the glueball trajectory if we had allowed odd values of \\(n\\) for the glueball. In other words, it matches the prediction for the \\(n = 1\\) state of the half slope trajectory beginning with \\(f_0(1500)\\). We could also use \\(f_0(2200)\\) as the \\(\\ensuremath{s\\bar{s}}\\) state and leave out \\(f_0(2100)\\) instead.\n\nThere is no glaring inconsistency in this assignment with the decay modes, but we are again confronted with the state \\(*f_0(1800)\\), which we need to complete the light meson trajectory. We can see from table \\ref{tab:allf0} that the \\(f_0(2020)\\) is wider than other states in its trajectory, whereas we maintain that the ratio between width and mass \\(\\Gamma\/M\\) should be roughly constant along a trajectory. In particular, the last state in the trajectory, \\(f_0(2330)\\), is much narrower than \\(f_0(2020)\\). We can assign the \\(f_0(2330)\\) to the \\(\\ensuremath{s\\bar{s}}\\) trajectory instead, but there is no other argument for that state being \\(\\ensuremath{s\\bar{s}}\\), considering it was observed only in its decays to \\(\\pi\\pi\\) and \\(\\eta\\eta\\). Perhaps the fact that \\(f_0(1370)\\) and \\(f_0(2020)\\) are both quite wide means that there should be two additional states, with masses comparable to those of \\(*f_0(1800)\\) and \\(f_0(2330)\\), that are also wide themselves, and those states will better complete this assignment. The results of the assignment are presented in table \\ref{tab:fit1500}.\n\t\t\t\n\\subsubsection{Assignment with \\texorpdfstring{$f_0$}{f0}(1710) as glueball}\n\n\\begin{table}[tp!] \\centering\n\t\\includegraphics[width=0.60\\textwidth]{glue_fit_1710_f1n.png} \\qquad\n\t\t\t\t\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline\n\t\t\t\t\t\\(n\\) & \\multicolumn{2}{|c|}{Light} & \\multicolumn{2}{|c|}{\\(s\\bar{s}\\)} & \\multicolumn{2}{|c|}{Glueball} \\\\ \\hline\n\t\t\t\t\t& Exp. & Thry. & Exp. & Thry. & Exp. & Thry. \\\\ \\hline\n\t\t\t0&\t1350\\plm150 & 1378 & 1505\\plm6 & 1506 & 1720\\plm6 & 1720 \\\\\n\t\t\t1&\t1795\\plm25 & 1777 & 1992\\plm16 & 1977 & - & - \\\\\n\t\t\t2&\t2103\\plm8 & 2102 & ? & 2330 & ? & 2830 \\\\\n\t\t\t3&\t2325\\plm35 & 2383 & & & & \\\\\n\t\t\t4&\t? & 2640 & & & & \\\\ \\hline\n\t\t\t\t\\end{tabular} \\caption{\\label{tab:fit1710} The results of the fit to the assignment with \\(f_0(1710)\\) as the glueball ground state. The slope is \\(\\ensuremath{\\alpha^\\prime} = 0.793\\) GeV\\(^{-2}\\) and the mass of the \\(s\\) quark \\(m_s = 500\\) MeV. This fit has \\(\\chi^2 = 0.71\\). The intercepts obtained are (-1.51) for light mesons, (-0.53) for \\(\\ensuremath{s\\bar{s}}\\), and (-1.17) for glueballs.}\n\t\t\t\\end{table}\n\t\t\t\nExcluding the \\(f_0(1710)\\) from the meson trajectories we can make an assignment that includes all states except the \\(f_0(500)\\) and \\(f_0(980)\\):\n\\[\\mathrm{Light:}\\qquad 1370, *1800, 2100, 2330\\]\n\\[\\ensuremath{s\\bar{s}}\\mathrm{:}\\qquad 1500, 2020, 2200\\]\n\\[\\mathrm{Glue:}\\qquad 1710\\]\nThe disadvantage here is that we again have to use \\(f_0(1500)\\) as the head of the \\(\\ensuremath{s\\bar{s}}\\) trajectory despite knowing that its main decay modes are to \\(4\\pi\\) and \\(\\pi\\pi\\), as well as the fact the we - once again - need the \\(*f_0(1800)\\) resonance to fill in a hole for \\(n = 1\\) in the resulting light meson trajectory. This trajectory can be seen in table \\ref{tab:fit1710}.\n\n\\subsubsection{Conclusions from the \\texorpdfstring{$f_0$}{f0} fits}\nIt is not hard to see that the \\(f_0\\) resonances listed in the PDG's Review of Particle Physics all fit in quite neatly on two parallel trajectories with a slope similar to that of other mesons. However, upon closer inspection, these trajectories - one for light quark mesons and one for \\(\\ensuremath{s\\bar{s}}\\) - are not consistent with experimental data, as detailed above. For us the naive assignment is also inconsistent with what we have observed for the other \\(\\ensuremath{s\\bar{s}}\\) trajectories in \\cite{Sonnenschein:2014jwa}, namely that the \\(\\ensuremath{s\\bar{s}}\\) trajectories are not purely linear, and have to be corrected by adding a non-zero string endpoint mass for the \\(s\\) quark, usually of at least 200 MeV.\n\nThe other novelty that we hoped to introduce, the half slope trajectories of the glueball, proved to be impractical - given the current experimental data which only goes up to less than \\(2.4\\) GeV for the relevant resonances.\n\n\tOne conclusion that can be drawn is that the state \\(f_0(980)\\) can be comfortably excluded from any of the meson trajectories, which is consistent with its being the \\(K\\overline{K}\\) ground state.\n\n\tThe unconfirmed state \\(*f_0(1800)\\) turns up in the assignments with glueballs in them, usually to fill in a hole in the light meson trajectory. If the \\(*f_0(1800)\\) is not in itself a meson as mentioned before, then we would hope that there is another yet unobserved \\(f_0\\) state with a very similar mass, say 1800--1850 MeV.\n\t\n\tThere is no one assignment that seems the correct one, although the two assignments singling out either \\(f_0(1370)\\) or \\(f_0(1500)\\) as the glueball ground states seem more consistent than the other possibilities. The best way to determine which is better is, as always, by finding more experimental data. We list our predictions for higher resonances based on these assignments in section \\ref{sec:predictions} of the appendix.\n\n\\subsection{Assignment of the \\texorpdfstring{$f_2$}{f2} into trajectories} \\label{sec:f2_fits}\nWe now turn to the \\(f_2\\) tensor resonances, that were listed in the beginning of the section in table \\ref{tab:allf2}. We will first examine trajectories in the \\((J,M^2)\\) plane, then move on to the attempt to assign all the \\(f_2\\) states to trajectories in the \\((n,M^2)\\) plane.\n\\subsubsection{Trajectories in the \\texorpdfstring{$(J,M^2)$}{(J,M2)} plane} \\label{sec:f2_orbital}\nThe only way to get a linear trajectory connecting a \\(0^{++}\\) and a \\(2^{++}\\) state with the slope \\(\\ensuremath{\\alpha^\\prime}\\!_{gb} = \\frac{1}{2}\\ensuremath{\\alpha^\\prime}_{meson}\\) is to take the lightest \\(f_0\\) glueball candidate and the heaviest known \\(f_2\\). Then we have the pair \\(f_0(980)\\) and \\(f_2(2340)\\), and the straight line between them has a slope of 0.45 GeV\\(^{-2}\\). There is no \\(J = 1\\) resonance near the line stretched between them. However, this example mostly serves to demonstrate once again the difficulty of forming the glueball trajectories in practice. The glueball states are predicted to be fewer and farther apart then the mesons in their respective Regge trajectories.\n\nTherefore, it is a more sound strategy to look again for the meson trajectories, see what states are excepted from them, and check for overall consistency of the results. In forming the meson trajectories, we know that we can expect the \\(\\omega\\) mesons with \\(J^{PC} = 1^{--}\\) to be part of the trajectories, in addition to some states at higher spin, which will allow us to form trajectories with more points.\n\nMoving on from \\(J = 0^{++}\\) and \\(2^{++}\\) to higher spin states, we see two \\(J^{PC} = 4^{++}\\) states that could belong to a trajectory: \\(f_4(2050)\\) and \\(f_4(2300)\\). The first of those, \\(f_4(2050)\\), belongs to a well known meson trajectory in the \\((J,M^2)\\) plane, following \\(\\omega(782)\\), \\(f_2(1270)\\), and \\(\\omega_3(1670)\\). The slope of the fit to that trajectory is \\(\\ensuremath{\\alpha^\\prime} = 0.91\\) GeV\\(^{-2}\\), and we can even include in it states of spin 5 and 6: \\(*\\omega_5(2250)\\) and \\(f_6(2510)\\).\n\nThe mass of the \\(f_4(2300)\\) is too low for it to belong to a linear trajectory with a glueball slope. Taking it to be a meson one can put it on a linear trajectory following \\(\\omega(1420)\\) and \\(f_2(1810)\\). To complete this trajectory we need a \\(J^{PC} = 3^{--}\\) state with a mass near 2070 MeV. The PDG lists one unconfirmed state, \\(X(2080)\\), with the quantum numbers \\(I(J^{PC}) = ?(3^{-?})\\), which might be a match.\n\nWe also find another meson trajectory involving the second excited \\(\\omega\\) meson - \\(\\omega(1650)\\). This trajectory would be comprised of \\(\\omega(1650)\\), \\(*\\omega_3(2255)\\), and with one of \\(f_2(1950)\\) or \\(f_2(2010)\\) between them.\n\nWe also have the meson trajectories of the \\(\\ensuremath{s\\bar{s}}\\). The first joins the ground state \\(\\phi(1020)\\) with \\(f_2^\\prime(1525)\\) and \\(\\phi_3(1850)\\). We can form a daughter trajectory starting with the \\(\\phi(1680)\\), and going on to include \\(f_2(1950)\\) or \\(f_2(2010)\\), as well as the unconfirmed \\(*\\omega_3(2285)\\). This trajectory is nearly identical to that of the \\(\\omega(1650)\\) of the last paragraph.\n\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=0.76\\textwidth]{traj_j_mes.png}\n\t\t\t\t\\caption{\\label{fig:traj_j_mes} The trajectory of the \\(\\omega\\) (blue) and \\(\\phi\\) (red) mesons in the \\((J,M^2)\\) plane and their daughter trajectories. The fits have the common slope \\(\\ensuremath{\\alpha^\\prime} = 0.903\\) GeV\\(^{-2}\\), and the \\(\\ensuremath{s\\bar{s}}\\) trajectories are fitted using a mass of \\(m_s = 250\\) MeV for the \\(s\\) quark. The states forming the trajectories are as follows: With \\(J^{PC} = 1^{--}\\), \\(\\omega(782)\\), \\(\\phi(1020)\\), \\(\\omega(1420)\\), \\(\\omega(1650)\\), \\(\\phi(1680)\\). With \\(J^{PC} = 2^{++}\\), \\(f_2(1270)\\), \\(f_2^\\prime(1525)\\), \\(f_2(1810)\\), \\(f_2(1950)\\), and \\(f_2(2010)\\). With \\(J^{PC} = 3^{--}\\), \\(\\omega_3(1670)\\), \\(\\phi_3(1850)\\), \\(*\\omega_3(2255)\\), and \\(\\omega_3(2285)\\). And with \\(J^{PC} = 4^{++}\\), \\(f_4(2050)\\) and \\(f_4(2300)\\). We also plot at \\(J^{PC} = 0^{++}\\) the \\(f_0(980)\\) and \\(f_0(1370)\\) which are found to lie near the trajectories fitted, but were not included themselves in the fits, as they are not theoretically expected to belong to them.}\n\t\t\t\\end{figure}\n\nThe meson trajectories described above are plotted in figure \\ref{fig:traj_j_mes}.\n\nTo summarize, we have found several meson trajectories in the \\((J,M^2)\\) plane of at least three states. As shown in the figure, these trajectories pass quite closely to the states \\(f_0(980)\\) and \\(f_0(1370)\\), but as meson trajectories these should begin with a \\(J^{PC} = 1^{--}\\) state (with orbital angular momentum \\(L = 0\\) and spin \\(S = 1\\)). A \\(0^{++}\\) meson state could only be included as an excited state with \\(L = 1\\) and \\(S = 1\\), but we found that for each trajectory we can use an existing \\(f_2\\) state in that place. The \\(f_2\\) states classified in this assignment as mesons are \\(f_2(1270)\\), \\(f_2^\\prime(1525)\\), \\(f_2(1810)\\), \\(f_2(1950)\\), and \\(f_2(2010)\\). These can perhaps be partnered to existing \\(f_0\\) states as members of triplets of states with \\(J = 0, 1, 2\\) and \\(PC = ++\\) split by spin-orbit interactions. We do not know the exact magnitude of the splitting. There are some \\(f_0\\) states close (within 20--100 MeV) to the \\(f_2\\) states mentioned above, and the PDG lists some \\(f_1\\) (\\(1^{++}\\)) resonances that may be useful, but we do not find any such trio of states with similar properties and masses that could be said to belong to such a spin-orbit triplet. Therefore, we limit our conclusions from these Regge trajectories to the \\(f_2\\) which we found we could directly place on them.\n\n\\subsubsection{\\texorpdfstring{Trajectories in the $(n,M^2)$}{(n,M2)} plane} \\label{sec:f2_radial}\nSorting the \\(f_2\\) resonances into trajectories, the situation is somewhat simpler than with the \\(f_0\\) scalars, as here we have two states that belong on meson trajectories in the \\((J,M^2)\\) plane, as we found in previous sections. In particular, the \\(f_2(1270)\\) belongs to the trajectory of the \\(\\omega\\) meson, and the \\(f^\\prime_2(1525)\\) is an \\(s\\bar{s}\\) and sits on the \\(\\phi\\) trajectory. Their decay modes and other properties are also well known and there is no real doubt about their nature.\n\nThe linear trajectory beginning with the \\(f_2(1270)\\) meson includes the states \\(f_2(1640)\\)and \\(f_2(1950)\\). We can include one of the further states \\(*f_2(2240)\\) as the fourth point in the trajectory. We can also use the \\(f_J(2220)\\) in place of the \\(*f_2(2240)\\), but it seems an unnatural choice because of the widths of the states involved (the \\(f_J(2220)\\) is much narrower than the others).\n\nThe projected trajectory of the \\(f^\\prime_2(1525)\\), using the same slope as the \\(f_2(1270)\\) trajectory and adding mass corrections for the \\(s\\) quark, includes the \\(f_2(2010)\\) and the \\(f_2(2300)\\).\n\nThis leaves out the states \\(f_2(1430)\\), \\(f_2(1565)\\), \\(f_2(1810)\\), \\(f_2(1910)\\), \\(f_J(2220)\\), and \\(f_2(2340)\\), as well as the five resonances classified as further states.\n\nThe next state we look at is \\(f_2(1810)\\), classified as a light meson in the \\((J,M^2)\\) fits of the previous section. Its mass is not right for it to belong to the trajectory of the \\(f_2(1270)\\), so we try to use it as the head of another light meson trajectory. If it belongs to a parallel trajectory to that of the \\(f_2(1270)\\) then the state that follows it is \\(f_2(2150)\\). The next state could be \\(f_2(2340)\\), except that it has been observed to decay to \\(\\phi\\phi\\), making it very unlikely to be a light quark meson.\n\nThe state \\(f_2(1430)\\) is intriguing. In part because of the very small width reported by most (but not all) experiments cited in the PDG, and in part because it is located in mass between the two lightest mesons of \\(J^{PC} = 2^{++}\\), that is between \\(f_2(1270)\\) (light) and \\(f_2^\\prime(1525)\\) (\\(\\ensuremath{s\\bar{s}}\\)). If we had to assign the \\(f_2(1430)\\) to a Regge trajectory, then it is best placed preceding the \\(f_2(1810)\\) and \\(f_2(2150)\\) in the linear meson trajectory discussed in the last paragraph.\n\nThe \\(f_J(2200)\\), previously known as \\(\\xi(2230)\\), is also a narrow state. It is currently listed by the PDG as having either \\(J^{PC} = 2^{++}\\) or \\(4^{++}\\), but some of the experiments cited by the PDG tend towards \\(J = 2\\). It has been considered a candidate for the tensor glueball \\cite{Bai:1996wm,Crede:2008vw}. It can be assigned to a linear meson trajectory, as already discussed, but it is clear already from its narrow width that it is not the best choice, even before addressing other experimental finds regarding it (for example, the fact that it was not observed in \\(\\gamma\\gamma\\) scattering \\cite{Benslama:2002pa} and the resulting bounds on its decay into photons).\n\nThe \\(f_2(1565)\\) is also left out, but it could be paired with \\(f_2(1910)\\) to form another linear meson trajectory. To continue we need another state with a mass of around 2200 MeV.\n\nTo summarize, we may organize the \\(f_2\\) resonances by picking first the resonances for the trajectories of the two known mesons,\n\\[\\mathrm{Light:}\\qquad 1270, 1640, 1950\\]\n\\[\\ensuremath{s\\bar{s}}\\mathrm{:}\\qquad 1525, 2010, 2300 \\]\nthen find the trajectories starting with the lightest states not yet included. This gives us another meson trajectory using the states\n\\[\\mathrm{Light:}\\qquad 1810, 2150\\]\n\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=0.76\\textwidth]{traj_n_f2.png}\n\t\t\t\t\\caption{\\label{fig:traj_n_f2} Some radial trajectories of the \\(f_2\\), with blue lines for light mesons and red for \\(\\ensuremath{s\\bar{s}}\\). The fits have the common slope \\(\\ensuremath{\\alpha^\\prime} = 0.846\\) GeV\\(^{-2}\\), and the \\(\\ensuremath{s\\bar{s}}\\) trajectories are fitted using a mass of \\(m_s = 400\\) MeV for the \\(s\\) quark. The states forming the trajectories are as follows: The first light meson trajectory with \\(f_2(1270)\\), \\(f_2(1640)\\), and \\(f_2(1950)\\), and followed by the unconfirmed state \\(*f_2(2240)\\) which was not used in the fit. The \\(\\ensuremath{s\\bar{s}}\\) trajectory with \\(f_2^\\prime(1525)\\), \\(f_2(2010)\\), and \\(f_2(2300)\\). And the second light meson trajectory with \\(f_2(1810)\\) and \\(f_2(2150)\\).}\n\t\t\t\\end{figure}\n\nThe trajectories formed by these eight states are drawn in figure \\ref{fig:traj_n_f2}.\n\n\\subsubsection{Conclusions from the \\texorpdfstring{$f_2$}{f2} fits}\nThere are some simplifications in assigning the \\(f_2\\) to radial trajectories compared to assigning the \\(f_0\\) resonances, as we can look at both orbital and radial trajectories and it is easier to classify some states as mesons. The radial trajectories described in section \\ref{sec:f2_radial} are consistent with the orbital trajectories of section \\ref{sec:f2_orbital}: states classified as mesons in the latter are also classified as mesons in the former, and with the same quark contents.\n\nIn the previous sections we for the most part avoided using the five \\(f_2\\) states classified in the PDG as ``further states'', although some of them could have played a role in the radial trajectory assignments. Counting confirmed and unconfirmed states alike, the PDG lists a total of 11 states with masses between 1900 and 2340 MeV. Since the different states have been observed in different processes, and hence have different decay modes, it would be useful to clarify experimentally the status of all these states and then reattempt the assignments of the then confirmed states into trajectories. We also note that the fact that there are many resonances with identical quantum numbers near to each other can interfere with the naive mass predictions of the Regge trajectories. In any case, and like for the \\(f_0\\), further experimental data on resonances between 2.3--3.0 GeV will likely prove useful.\n\nWe have not addressed yet the issue of the decay modes of the different states and how consistent they are with the assignments of the previous sections. The \\(f_2(1270)\\) and \\(f_2^\\prime(1525)\\) are well established as a light quark meson and an \\(\\ensuremath{s\\bar{s}}\\) respectively, and they were the basis from which we built the different trajectories. As for their excited states, the data on their branching ratios cited by the PDG is very partial for higher states. However, we find an interesting case when looking at the trio of states \\(f_2(1910)\\), \\(f_2(1950)\\), and \\(f_2(2010)\\). We have classified \\(f_2(1950)\\) as a light meson and \\(f_2(2010)\\) as \\(\\ensuremath{s\\bar{s}}\\), which is what fits best with the Regge trajectories. Another option would be to use \\(f_2(1910)\\) as a light meson and \\(f_2(1950)\\) as \\(\\ensuremath{s\\bar{s}}\\), which is still consistent. Then the \\(f_2(2010)\\), which was observed to decay to \\(\\phi\\phi\\) (despite the very small phase space), could perhaps be classified as a \\(\\phi\\phi\\) bound state, in an analogous fashion to the \\(f_0(980)\\).\n\nThe most interesting states after that remain the \\(f_2(1430)\\) and \\(f_J(2220)\\). While the latter has been considered a candidate for the glueball and has been the object of some research (see papers citing \\cite{Bai:1996wm}), the former is rarely addressed, despite its curious placement in the spectrum between the lightest \\(2^{++}\\) light and \\(\\ensuremath{s\\bar{s}}\\) mesons. It seems a worthwhile experimental question to clarify its status - and its quantum numbers, as the most recent observation \\cite{Vladimirsky:2001ek} can not confirm whether it is a \\(0^{++}\\) or \\(2^{++}\\) state, a fact which led to at least one suggestion \\cite{Vijande:2004he} that the \\(f_2(1430)\\) could be itself the scalar glueball.\n\n\\subsection{Assignments with non-linear trajectories for the glueball} \\label{sec:holo_fits}\nIn this section we check the applicability of a glueball trajectory of the form\n\\begin{equation} J = \\ensuremath{\\alpha^\\prime}_{gb}E^2-2\\ensuremath{\\alpha^\\prime}_{gb}m_0E + a \\,, \\label{eq:holotraj} \\end{equation}\nwhich is the general form we expect from a semi-classical calculation of the corrections to the trajectory in a curved background, and as put forward in section \\ref{sec:holo_string}. The novelty here is a term linear in the mass \\(E\\), which makes the Regge trajectory \\(\\alpha(t)\\) non-linear in \\(t = E^2\\). The constant \\(m_0\\) can be either negative or positive, depending on the specific holographic background, and a priori we have to examine both possibilities. It was also noted in section \\ref{sec:holo_string} that there may be a correction to the slope, but we assume it is small compared to the uncertainty in the phenomenological value of the Regge slope, and we use\n\\begin{equation} \\ensuremath{\\alpha^\\prime}_{gb} = \\frac{1}{2}\\ensuremath{\\alpha^\\prime} \\end{equation}\nthroughout this section. We also substitute \\(J \\rightarrow J + n\\) as usual to apply the formula to radial trajectories.\n\nWith the \\(m_0\\) term we can write\n\\begin{equation} \\frac{\\partial J}{\\partial E^2} = \\frac{\\ensuremath{\\alpha^\\prime}}{2}\\left(1-\\frac{m_0}{E}\\right) \\,. \\label{eq:eff_holo_slope} \\end{equation}\nWe can look at this as an effective slope, and it is the easiest way to see that when \\(m_0\\) is negative, the effective slope is higher than that of the linear trajectory, and vice versa.\n\n\\subsubsection{Fits using the holographic formula}\nUsing the simple linear formula we could not, in most cases, find glueball trajectories among the observed \\(f_0\\) and \\(f_2\\) states. This is because the first excited state is expected to be too high in mass and outside the range of the states measured in experiment.\n\nAdding an appropriate \\(m_0\\) term can modify this behavior enough for us to find some pairs of states on what we would then call glueball trajectories, and by appropriate we mean a negative value that will make the effective slope of eq. \\ref{eq:eff_holo_slope} higher. The problem is then that we have only pairs of states, with two fitting parameters: \\(m_0\\) and \\(a\\) (and \\(\\ensuremath{\\alpha^\\prime}\\) which is fixed by the meson trajectory fits). We form these pairs by picking a state left out from the meson trajectories proposed in sections \\ref{sec:f0_fits} and \\ref{sec:f2_fits} and assigning it as the excited partner of the appropriate glueball candidate.\n\nThere is a solution for \\(m_0\\) and \\(a\\) for any pair of states which we can take, and the question then becomes whether there is a reason to prefer some values of the two parameters over others. We list some other values obtained for \\(m_0\\) and \\(a\\) in table \\ref{tab:holo_fits}.\n\n\\begin{table}[t!] \\centering\n\t\\begin{tabular}{|c|c|c|c|c|} \\hline\n\tGround state & Excited state & \\(\\ensuremath{\\alpha^\\prime}\\) [GeV\\(^{-2}\\)] & $m_0$ [GeV] & \\(a\\) \\\\ \\hline\n\t\n\t\\(f_0(980)\\) & \\(f_0(2200)\\) & 0.79 & -0.52 & -0.78 \\\\\n\t\n\t\\(f_0(1370)\\) & \\(f_0(2020)\\) & 0.87 & -2.00 & -3.21 \\\\\n\t\n\t\\(f_0(1500)\\) & \\(f_0(2200)\\) & 0.87 & -1.51 & -2.97 \\\\\n\t\n\t\\(f_2(1430)\\) & \\(f_J(2220)\\) & 0.81 & -1.33 & -2.42 \\\\ \\hline\n\t\n\t\\end{tabular} \\caption{\\label{tab:holo_fits} Values obtained for the parameters \\(m_0\\) and \\(a\\) for some of the possible pairs of states on glueball trajectories. The states selected as the excited state of the glueball are those not included in the meson trajectories of the assignments of sections \\ref{sec:f0_fits} and \\ref{sec:f2_fits}, and the slopes are selected based on the results of the meson fits presented in the same sections.}\n\\end{table}\n\n\\subsubsection{Using the holographic formula with a constrained intercept}\n\\cite{Bigazzi:2004ze} implies that a universal form of the first semi-classical correction of the Regge trajectory of the rotating folded string is\n\\begin{equation} J + n = \\frac{1}{2}\\ensuremath{\\alpha^\\prime}(E-m_0)^2 \\,, \\end{equation}\nup to further (model dependent) modifications of the slope, which in the cases calculated are small. In other words, the intercept obtained then from the semi-classical calculation is\n\\begin{equation} a = \\frac{1}{2}\\ensuremath{\\alpha^\\prime} m_0^2 \\,. \\end{equation}\n\nThe intercept is always positive in this scenario. If we want to include the ground state with \\(J = n = 0\\) the only way to do it is to take a positive \\(m_0\\), specifically we should take \\(m_0 = M_{gs}\\), where \\(M_{gs}\\) is the mass of the ground state. There is no problem with the resulting expression theoretically, but it is not very useful in analyzing the observed spectrum. The trouble is that when using this expression the energy rises much too fast with \\(J\\) and we end up very quickly with masses outside the range of the glueball candidates. If we take, for instance, \\(f_0(980)\\) as the ground state then the first excited state is expected to have a mass of around 2500 MeV, and the heavier candidates naturally predict even heavier masses for the excited states.\n\nAnother way to use eq. \\ref{eq:holotraj} is to begin the trajectory with a \\(J = 2\\) state. Then \\(m_0\\) can be either positive or negative. We can then proceed as usual: we pick the head of a trajectory and see if there are any matches for its predicted excited states. We can see, for example, that we can again pair \\(f_2(1430)\\) with \\(f_J(2220)\\). Constraining \\(\\ensuremath{\\alpha^\\prime}\\) to be \\(0.90\\) GeV\\(^{-2}\\), the best fit has \\(m_0 = -0.72\\) GeV, and the masses calculated are 1390 and 2260 MeV for the experimental values of \\(1453\\pm4\\) and \\(2231\\pm4\\) MeV.\n\n\\subsection{Glueball Regge trajectories in lattice QCD} \\label{sec:lattice}\nThe glueball spectrum has been studied extensively in lattice QCD. Some works have compared results with different stringy models, e.g. \\cite{Athenodorou:2010cs,Bochicchio:2013aha,Bochicchio:2013sra,Caselle:2015tza}. However, the question whether or not the glueballs form linear Regge trajectories is not often addressed, due to the difficulty involved in computing highly excited states. When linear Regge trajectories are discussed, it is often when trying to identify the glueball with the pomeron and searching for states along the given pomeron trajectory,\n\\begin{equation} \\alpha(t) = \\ensuremath{\\alpha^\\prime}_p t + 1 + \\epsilon \\, \\end{equation}\n where the slope and the intercept are known from experiment to be \\(\\ensuremath{\\alpha^\\prime}_p = 0.25\\) GeV\\(^{-2}\\) and \\(1 + \\epsilon \\approx 1.08\\) \\cite{Donnachie:1984xq}.\n\nThe most extensive study of glueball Regge trajectories is that of Meyer and Teper \\cite{Meyer:2004jc,Meyer:2004gx}, where a relatively large number of higher mass states is computed, including both high spin states and some highly excited states at low spin.\n\nWe quote in table \\ref{tab:lat_masses} some lattice results for glueball masses from different calculations. The results are for \\(SU(3)\\) and \\(D = 4\\), and more results are collected in \\cite{Gregory:2012hu}. Most of these give only the masses of the lowest glueball states for different quantum numbers. These are low spin states with different combinations of parity and charge parity. While a spectrum is obtained, most states are isolated, in the sense that they cannot be grouped with other states to form Regge trajectories.\n\nIn the table \\ref{tab:lat_masses} we list the lattice results for the \\(0^{++}\\) ground state, the lowest \\(2^{++}\\) state, and the first excited \\(0^{++}\\) glueball, as well as for the \\(0^{-+}\\) and \\(2^{-+}\\). We may straight lines between the first spin-0 state and its excited partner to calculate the slope.\n\nOne thing we see at this first glance at the spectrum is that the spin-2 state is, in most studies, lower than we would expect it based on the Regge slope assumption.\\footnote{The fact that the tensor glueball is close to the scalar seems to have been long known in lattice QCD, see e.g. \\cite{Albanese:1987ds}.} The second spin-0 state, on the other hand, is about where we want it to be, assuming a closed string model, where the slope is half that of meson trajectories, and the first excited state has the excitation number \\(n = 2\\) (for one left moving and one right moving mode excited). In the next section we do some fits to some trajectories with more than two states, based on the results in \\cite{Meyer:2004gx}.\n\n\\begin{table}[t!] \\centering\n\t\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline\n\t& Meyer \\cite{Meyer:2004gx} & M\\&P \\cite{Morningstar:1999rf} & Chen \\cite{Chen:2005mg} & Bali \\cite{Bali:1993fb} & Gregory \\cite{Gregory:2012hu} \\\\ \\hline\\hline\n\n\t\\(0^{++}\\) & 1475\\plm30\\plm65\t\t\t& 1730\\plm50\\plm80 & 1710\\plm50\\plm80 & 1550\\plm50\\plm80 & 1795\\plm60 \\\\ \\hline\n\n\t\\(2^{++}\\) & 2150\\plm30\\plm100\t\t& 2400\\plm25\\plm120 & 2390\\plm30\\plm120 & 2270\\plm100\\plm110 & 2620\\plm50\\\\ \\hline\n\n\t\\(0^{++}\\) & 2755\\plm30\\plm120\t\t& 2670\\plm180\\plm130 & - & - & 3760\\plm240 \\\\ \\hline\\hline\n\n\t\\(0^{-+}\\) & 2250\\plm60\\plm100\t\t\t& 2590\\plm40\\plm130 & 2560\\plm35\\plm120 & 2330\\plm260\\plm120 & - \\\\ \\hline\n\n\t\\(2^{-+}\\) & 2780\\plm50\\plm130\t\t& 3100\\plm30\\plm150 & 3040\\plm40\\plm150 & 3010\\plm130\\plm150 & 3460\\plm320\\\\ \\hline\n\n\t\\(0^{-+}\\) & 3370\\plm150\\plm150\t\t& 3640\\plm60\\plm180 & - & - & 4490\\plm590 \\\\ \\hline\\hline\n\n\t\\(\\ensuremath{\\alpha^\\prime}\\!_{++}\\) (in $J$) & 0.82\\plm0.17 & 0.72\\plm0.18 & 0.72\\plm0.17 & 0.73\\plm0.19 & 0.55\\plm0.05 \\\\ \\hline\n\n\t\\(\\ensuremath{\\alpha^\\prime}\\!_{++}\\) (in $n$) & 0.37\\plm0.05 & 0.48\\plm0.14 & - & - & 0.18\\plm0.03 \\\\ \\hline\\hline\n\n\t\\(\\ensuremath{\\alpha^\\prime}\\!_{-+}\\) (in $J$) & 0.75\\plm0.26 & 0.69\\plm0.28 & 0.74\\plm0.32 & 0.55\\plm0.27 & - \\\\ \\hline\n\n\t\\(\\ensuremath{\\alpha^\\prime}\\!_{-+}\\) (in $n$) & 0.32\\plm0.08 & 0.31\\plm0.07 & - & - & - \\\\ \\hline\n\t\\end{tabular}\n\t\\caption{\\label{tab:lat_masses} Lattice predictions from different studies for glueball masses [MeV] and resulting Regge slopes [GeV\\(^{-2}\\)], in the \\((J,M^2)\\) plane or in the \\((n,M^2)\\) plane. The slope is calculated assuming the first excited state has \\(n = 2\\).}\n\t\\end{table}\n\n\\subsubsection{Regge trajectory fits to results from the lattice}\nResults in lattice computations are for the dimensionless ratio between the mass of a state and the square root of the string tension: \\(M\/\\sqrt{T}\\). To get the masses \\(M\\) in MeV one has to fix the scale by setting the value of \\(T\\). This introduces an additional uncertainty in the obtained values. In table \\ref{tab:lat_masses} we listed the masses in MeV and calculated the dimensionful slope, but for the purpose of identifying Regge trajectories we can work directly with dimensionless quantities, avoiding this extra error. Thus, for the following, our fitting model will be\n\\begin{equation} \\frac{M^2}{T} = \\frac{2\\pi}{q} (N + a) \\end{equation}\nIn this notation the ratio \\(q\\), which is the primary fitting parameter (in addition to the intercept \\(a\\)), is expected to be 1 for open strings and \\(1\/2\\) for closed strings. It is referred to below as the ``relative slope''. \\(N\\) will be either the spin \\(J\\) or the radial excitation number \\(n\\).\n\n\\paragraph{Trajectories in the \\texorpdfstring{$(J,M^2)$}{(J,M2)} plane:} As mentioned above, \\cite{Meyer:2004gx} has the most high spin states. The analysis there observes that the first \\(2^{++}\\)and \\(4^{++}\\) states can be connected by a line with the relative slope\n\\begin{equation} q = 0.28\\pm0.02, \\end{equation}\nwhich, when taking a typical value of the string tension \\(\\sqrt{T} = 430\\) MeV (\\(\\ensuremath{\\alpha^\\prime} = 0.84\\) GeV\\(^{-2}\\)), gives a slope virtually identical to that expected for the pomeron, \\(0.25\\) GeV\\(^{-2}\\). This trajectory can be continued with the calculated \\(6^{++}\\) state. A fit to the three state trajectory gives the result\n\\begin{equation} q = 0.29\\pm0.15.\\end{equation}\nThis trajectory leaves out the \\(0^{++}\\) ground state. In \\cite{Meyer:2004gx} the lowest \\(0^{++}\\) is paired with the second, excited, \\(2^{++}\\) state, giving a trajectory with\n\\begin{equation} q = 0.40\\pm0.04.\\end{equation}\nA possibility not explored in \\cite{Meyer:2004gx} is that of continuing this trajectory, of the first \\(0^{++}\\) and the excited \\(2^{++}\\), and with the \\(4^{++}\\) and \\(6^{++}\\) states following. Then we have the result\n\\begin{equation} q = 0.43\\pm0.03\\end{equation}\nThis second option not only includes more points, it is also a better fit in terms of \\(\\chi^2\\) per degrees of freedom (0.37 instead of 1.24). This is a nice result from the closed string perspective, but the lowest \\(2^{++}\\) state is then left out. There is also a \\(J = 3\\) state in the \\(PC = ++\\) sector that lies very close to the trajectory of the \\(0^{++}\\) ground state. In our model it is not expected to belong to the trajectory, so that state is also left out of the fit. The trajectories of the \\(PC = ++\\) states are in the left side of figure \\ref{fig:lat_Meyer}.\n\n\\begin{figure}[tp!] \\centering\n\t\\includegraphics[width=0.49\\textwidth]{lattice_Meyer_pp.png}\n\t\\includegraphics[width=0.49\\textwidth]{lattice_Meyer_0pp.png}\n\t\\caption{\\label{fig:lat_Meyer} The trajectories of the \\(PC = ++\\) glueball states found in lattice calculations in \\cite{Meyer:2004gx}. \\textbf{Left:} Trajectories in the \\((J,M^2)\\) plane. The full line is the fit to a proposed trajectory using four states with \\(J = 0, 2, 4, 6\\), where the relative slope is \\(0.43\\) and the lightest tensor is excluded (\\(\\chi^2 = 0.37\\)). The dotted line is the leading trajectory proposed in the analysis in \\cite{Meyer:2004gx}, with a pomeron-like slope. It includes the \\(J = 2, 4,\\) and \\(6\\) states. (\\(\\chi^2 = 1.24\\)). In this second option the scalar is excluded. Also plotted is the \\(3^{++}\\) state, which was not used in the fit. \\textbf{Right:} trajectory of four states with \\(J^{PC} = 0^{++}\\). The relative slope is exactly \\(0.50\\) (\\(\\chi^2 = 1.48\\)).}\n\\end{figure}\n\n\\paragraph{Trajectories in the \\texorpdfstring{$(n,M^2)$}{(n,M2)} plane:}\nIn trajectories in the \\((n,M^2)\\) plane we assume \\(n\\) takes only even values, i.e. \\(n = 0,2,4,\\ldots\\), as it does for the closed string. The results when taking \\(n = 0,1,2,\\ldots\\) will be half those listed.\n\nIn this section, we again have to rely mostly on \\cite{Meyer:2004gx}, as it offers calculations of several excited states with the same \\(J^{PC}\\). Most notably we see there four states listed with \\(J^{PC} = 0^{++}\\). We observe that those points are well fitted by a trajectory with the slope\n\\begin{equation} q = 0.50\\plm0.07, \\end{equation}\nwhere \\(\\chi^2 = 1.48\\) for the fit. It is interesting to compare this with the trajectory that can be drawn from the \\(0^{++}\\) ground state in the \\((J,M^2)\\) plane. The \\((n,M^2)\\) trajectory with \\(n = 0,2,4,6\\) is very similar to the trajectory beginning with the same state and continuing to \\(J = 2, 4,\\) and \\(6\\). This is what we see also for mesons and baryons in experiment: two analogous trajectories with similar slopes in the different planes.\n\nOther than the trajectory of the four \\(0^{++}\\) states (plotted in figure \\ref{fig:lat_Meyer}), we list the slopes calculated for pairs of states who share other quantum numbers. This is in table \\ref{tab:lat_n}.\n\\begin{table} \\centering\n\t\\begin{tabular}{|c|c|c|c|c|c|} \\hline\n\t\t\\(J^{PC}\\) & \\(0^{++}\\) & \\(2^{++}\\) & \\(4^{++}\\) & \\(0^{-+}\\) & \\(2^{-+}\\) \\\\ \\hline\\hline\n\t\tMeyer \\cite{Meyer:2004gx} &\n\t\t\t\t0.50\\plm0.07 & 0.67\\plm0.10 & 0.30\\plm0.06 & 0.39\\plm0.07 & 0.56\\plm0.13 \\\\ \\hline\n\n\t\tM\\&P \\cite{Morningstar:1999rf} &\n\t\t\t\t0.51\\plm0.12 & -\t\t\t\t\t\t& -\t\t\t\t\t\t & 0.32\\plm0.02 & 0.38\\plm0.03 \\\\ \\hline\n\t\\end{tabular}\n\t\\caption{\\label{tab:lat_n} Relative slopes \\(q\\) of trajectories in the \\((n,M^2)\\) plane. The first result (Meyer\/\\(0^{++}\\)) is that of a fit to the four point trajectory drawn in \\ref{fig:lat_Meyer}. The other results are obtained when calculating the slopes between pairs of states, where the lowest state is assumed to have \\(n = 0\\), and the first excited state is taken to have \\(n = 2\\).}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{\\texorpdfstring{$SU(N)$ vs. $SU(3)$}{SU(N) vs. SU(3)} and the quenched approximation}\nMost of the studies of glueballs on the lattice utilize the ``quenched'' approximation, which in this case amounts to calculating the spectrum of the pure SU(3) Yang-Mills gauge theory without matter. The degree to which the quenched results are modified when fermions are added to the theory is still unknown. However, if our purpose is to see whether or not glueballs form Regge trajectories, the spectrum of the pure gluon theory should be as useful as that of real QCD.\n\nThere have also been some calculations of the ``glueball'' spectrum of \\(SU(N)\\) Yang-Mills for other values of \\(N\\) \\cite{Lucini:2014paa}. These results taken from \\cite{Lucini:2004my}, are fitted in \\cite{Lucini:2014paa} to the formulae (the numbers in brackets are the errors in the last significant digits):\n\\begin{equation} \\frac{M_{0^{++}}}{\\sqrt{T}} = 3.28(8)+\\frac{2.1(1.1)}{N^2} \\,,\\end{equation}\n\\[ \\frac{M_{2^{++}}}{\\sqrt{T}} = 4.78(14)+\\frac{0.3(1.7)}{N^2} \\,,\\]\n\\[ \\frac{M_{0^{++*}}}{\\sqrt{T}} = 5.93(17)-\\frac{2.7(2.0)}{N^2} \\,.\\]\nUsing these values, we get for \\(SU(3)\\), the relative slopes (the prefactor of 2 in these formulae is \\(J\\) or \\(n\\)):\n\\begin{equation} 2\\frac{2\\pi T}{M^2_{2^{++}}-M^2_{0^{++}}} = 1.16\\plm0.27, \\qquad\n 2\\frac{2\\pi T}{M^2_{0^{++*}}-M^2_{0^{++}}} = 0.65\\plm0.11 \\,, \\end{equation}\nwhile for the \\(N\\rightarrow\\infty\\) limit,\n\\begin{equation} 2\\frac{2\\pi T}{M^2_{2^{++}}-M^2_{0^{++}}} = 1.04\\plm0.13, \\qquad\n 2\\frac{2\\pi T}{M^2_{0^{++*}}-M^2_{0^{++}}} = 0.52\\pm0.05 \\,.\\end{equation}\n\nWhile this is too little data to be significant, we observe that the value approaches 1 as \\(N\\) grows for the excitation in \\(J\\), and it approaches \\(\\frac12\\) (the closed string value) in \\(n\\). However, as was already seen from the results of \\cite{Meyer:2004gx}, the first \\(2^{++}\\) does not seem to lie on the trajectory of the \\(0^{++}\\) ground state, and these results seem to confirm this further. The radial trajectory on the other hand is again perfectly consistent with the closed string picture, and more so when going to the limit of this large \\(N\\) computation.\n\n\\section{Summary} \\label{sec:summary}\n\nFor many years the identification of glueballs, a basic prediction of QCD, in the experimental spectrum of flavorless isoscalar hadrons has been an open question. Moreover, the common lore is that there is no way to disentangle glueballs from flavorless mesons since there is no quantum number that distinguish between them. \n\nHere in this paper we have attempted to identify glueballs by turning to a well known feature of the hadron spectrum, its Regge trajectories. Stating it differently we use a stringy picture of rotating folded closed strings to describe the glueball in a similar way to the description of mesons and baryons in terms of open string with massive endpoints of \\cite{Sonnenschein:2014jwa} and \\cite{Sonnenschein:2014bia}.\n\nThe great disadvantage in using trajectories is that they are a property not of single states, but of a spectrum of states. Thus, for positive identification, we need to have in our spectrum, to begin with, several glueballs which we would then assign to a trajectory. The fact that the ratio between the open and closed string slopes is exactly half adds some ambiguity to the \\((n,M^2)\\) trajectories where the value of \\(n\\) cannot be determined by experiment: two states whose mass difference is, for instance, \\(\\Delta M^2 = 4\/\\ensuremath{\\alpha^\\prime}\\) can be either open strings with \\(\\Delta n = 4\\) between them, or closed strings with \\(\\Delta n = 2\\). The difference between the open and closed string trajectories would be in the number of states between those two: there would be more open strings for \\(\\Delta n = 1\\), 2, and 3. Thus we have to rely on experiment to observe all the relevant states in the given mass range, so that the absence of a state from a Regge trajectory could reasonably be used as evidence.\n\nDue to this situation it is clearly advisable to use additional predictions pertaining to the properties of single states to identify them as open or closed string hadrons. We have presented, qualitatively, the decay mechanism of the closed string to two open strings, which would be the decay of a glueball into two mesons. We included one prediction of the branching ratios of glueballs when decaying into light mesons, kaons, or \\(\\phi\\) (\\(s\\bar{s}\\)) mesons. If there were measurements of a state which has those three decay modes with the hierarchy we predict between them, we could have declared it a glueball, based on our model of holographic strings. One has to look more closely to find more ways in which open and closed strings vary.\n\nThere are obviously additional tasks and questions to further explore the closed string picture of glueballs. Here we list some of them:\n\\begin{itemize}\n\\item\nAs was emphasized in this note the most urgent issue is to gain additional data about flavorless hadrons. This calls for a further investigation of experiments that yield this kind of resonances and for proposing future experiments of potential glueball production, in particular in the range above 2.4 GeV. This can follow the predictions of the masses and width of the resonances as were listed in appendix \\ref{sec:predictions}.\n\\item\nRelated to the exploration of experimental data is the investigation of efficient mechanisms of creating glueballs. This issue was not addressed in this paper. Among possible glueball formation one finds radiative $J\/\\psi$ decays, pomeron pomeron collisions in hadron-hadron central production and in $p$-$\\bar p$ annihilation. Naturally, we would like to understand possible glueball formation in LHC experiments. It is known that we can find in the latter processes of gluon-gluon scattering and hence it may serve as a device for glueball creation. \n\\item\nAs was mentioned in section \\ref{sec:non_critical_string}, the quantization of folded closed strings in D non-critical dimensions has not yet been deciphered. In \\cite{Hellerman:2013kba} the expression derived for the intercept is singular in the case where is only one rotation plane - as it naturally is in $D=4$. We mentioned a potential avenue to resolve this issue by introducing massive particles on the folds, quantize the system as that of a string with massive endpoints \\cite{ASY}, and then take the limit of zero mass. \n\\item\nWe have mentioned that the rotating closed strings are in fact rotating folded closed strings. However, we did not make any attempt in this note to explore the role of the folds. In fact it seems that very few research has been devoted to the understanding of folded strings \\cite{Ganor:1994rm}. It would be interesting to use the rotating closed string as a venue to the more general exploration of strings with folds which may be related to certain systems in nature.\n\\item\nA mystery related to the closed string description of glueballs is the relation between the pomeron and the glueball. Supposedly both the glueball and the pomeron are described by a closed string. As we have emphasized in this note the slope of the closed string is half that of the open string and hence we advocated the search of trajectories with that slope. However, it was found from fitting the differential cross section of $p$-$p$ collisions that the slope of the pomeron is $\\ensuremath{\\alpha^\\prime}_{pomeron}\\approx 0.25$ GeV$^{-2}$. That is, a slope which is closer to a quarter of that of the meson open string rather than half. Thus the stringy structure of the pomeron and its exact relation to the glueball is still an open question.\n\\item\nThe closed string description of the glueball faces a very obvious question. In QCD one can form a glueball as a bound state of two, three, or in fact any number of gluons. The stringy picture seems to describe the composite of two gluons, and it is not clear how to realize those glueballs constructed in QCD from more than two gluons.\n\\end{itemize} \n\n\\acknowledgments{\nWe would like to thank Ofer Aharony and Abner Soffer for their comments on the manuscript and for insightful conversations, and Shmuel Nussinov and Shimon Yankielowicz for useful discussions. This work was supported in part by a centre of excellence supported by the Israel Science Foundation (grant number 1989\/14), and by the US-Israel bi-national fund (BSF) grant number 2012383 and the Germany Israel bi-national fund GIF grant number I-244-303.7-2013. \n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{introduction}\n\nIn Ref.~\\cite{norma93} the representations in Grassmann and in Clifford space were discussed.\nIn Ref.~(\\cite{nh2018} and the references therein) the second quantization procedure in both\nspaces --- in Clifford space and in Grassmann space --- were discussed in order to try to understand \n\"why nature made a choice of Clifford rather than Grassmann space\" during the expansion \nof our universe, although in both spaces the creation operators $ \\hat{b}^{\\dagger}_{j} $ \nand the annihilation operators $ \\hat{b}_{j} $ exist fulfilling the anticommutation relations \nrequired for fermions~\\cite{nh2018}\n\\begin{eqnarray}\n\\{ \\hat{b}_i, \\hat{b}^{\\dagger}_{j} \\}_{+} |\\phi_{o}> &=&\\delta_{i j}\\; |\\psi_{o}>\\,,\n\\nonumber\\\\\n\\{\\hat{b}_i, \\hat{b}_{j} \\}_{+} |\\psi_{o}>&=& 0\\; |\\psi_{o}> \\,,\\nonumber\\\\\n\\{\\hat{b}^{\\dagger}_i,\\hat{b}^{\\dagger}_{j}\\}_{+} |\\psi_{o}>\n&=&0\\; |\\psi_{o}> \\,,\\nonumber\\\\\n\\hat{b}^{\\dagger}_{j} |\\psi_{o}>& =& \\, |\\psi_{j}>\\, \\nonumber\\\\\n\\hat{b}_{j} |\\psi_{o}>& =&0\\, |\\psi_{o}>\\,.\n\\label{ijthetaprod} \n\\end{eqnarray} \n$|\\psi_{o}>$ is the vacuum state. We use $|\\psi_o>= |1>$.\n\n\n Some observations to be included into introduction:\n However, even in the Grassmann case and with gravity only the scalar fields would appear in \n$d=(3+1)$.\n\nThe creation operators can be expressed in both spaces as products of eigenstates \nof the Cartan subalgebra, Eq.~(\\ref{choicecartan}), of the Lorentz algebra, \nEqs.~(\\ref{cartaneigengrass}, \\ref{signature}).\nStarting with one state (Ref.~\\cite{nh2018})\nall the other states\nof the same representation are reachable by the generators of the Lorentz transformations\n(which do not belong to the Cartan subalgebra), with ${\\cal {\\bf S^{ab}}}$ presented in \nEq.~(\\ref{Lorentztheta}) in Grassmann space and with either $S^{ab}$ or $\\tilde{S}^{ab}$, \nEq.~(\\ref{Lorentzgammatilde}), in Clifford space.\n\n But while there are in Clifford case two kinds of the generators of the Lorentz \ntransformations --- $S^{ab}$ and $\\tilde{S}^{ab}$, the first transforming members of one \nfamily among themselves, and the second transforming one member of a particular family into\nthe same member of other families --- there is in Grassmann space only one kind \nof the Lorentz generators --- ${\\cal {\\bf S^{ab}}}$. Correspondingly are all the states in \nClifford space, which can be second quantized as products of nilpotents and projectors~%\n\\cite{nh02,nh03,nh2018},\nreachable with one of the two kinds of the operators $S^{ab}$ and $\\tilde{S}^{ab}$, \nwhile different representations are in Grassmann space disconnected. \n\nOn the other hand the vacuum state is in Grassmann case simple --- $|\\psi_o>= |1>$ ---\nwhile in Clifford case is the sum of products of projectors, Eq.~(\\ref{vac1}). \n\nIn Grassmann space states are in the adjoint representations with respect to the Lorentz \ngroup, while states in Clifford space belong to the fundamental representations with respect \nto both generators, $S^{ab}$ and $\\tilde{S}^{ab}$, or they are singlets. Correspondingly \nare properties of fermions, described with the {\\it spin-charge-family} theory~%\n\\cite{IARD2016,n2014matterantimatter,JMP2013,normaJMP2015,nh2017,nd2017}, which \nuses the Clifford space to describe fermion degrees of freedom, in agreement with the \nobservations, offering explanation for all the assumptions of the {\\it standard model} \n(with families included) and also other observed phenomena. \n\nIn Grassmann case the spins \nmanifest, for example, in the case of $SO(6)$ or $SO(5,1)$ decuplets or singlets --- triplets and \nsinglets in Clifford case, \nTable~\\ref{Table grassdecupletso51.} --- while with respect to the subgroups $SU(3)$ and \n$U(1)$ of $SO(6)$ the states belong to either singlets, or triplets or sextets, \nTables~\\ref{Table grassdecuplet.},~\\ref{Table grasssextet.}\n --- triplets and singlets in the Clifford case.\n\n\nIn what follows we discuss representations, manifesting as charges and spins of fermions, of \nsubgroups of $SO(13,1)$, when internal degrees of freedom of fermions are described in \nGrassmann space and compare properties of these representations with the properties of \nthe corresponding representations appearing in Clifford space. We assume, as in the\n{\\it spin-charge-family} theory, that both spaces, the internal and the ordinary space, \nhave $d=2(2n+1)$-dimensions, $n$ is positive integer, $d \\ge 14$ and that all the degrees \nof freedom of fermions and bosons originate in $d=2(2n+1)$, in which fermions interact with \ngravity only. \n\nAfter the break of the starting symmetry $SO(13,1)$ into $SO(7,1) \\times SU(3) \\times U(1)$, and\nfurther to $SO(3,1) \\times SU(2) \\times SU(2) \\times SU(3) \\times U(1)$, fermions manifest \nin $d=(3+1)$ the spin and the corresponding charges and interact with the gauge fields, which \nare indeed the spin connections with the space index $m=(0,1,2,3)$, originating in \n$d=(13,1)$~\\cite{nd2017}. Also scalar fields originate in gravity: Those spin connections with \nthe space index $a =(5,6,7,8)$ determine masses of fermions, those with the space index \n$a=(9,10,\\dots,14)$ contribute to particle\/antiparticle asymmetry in our \nuniverse~\\cite{n2014matterantimatter}. \n\nWe pay attention in this paper mainly on fermion fields with spin $1$, the creation and \nannihilation operators of which fulfill the anticommutation relations of Eq.~(\\ref{ijthetaprod})\nin Grassmann space. \n\n\n\n\n\\subsection{Creation and annihilation operators in Grassmann space}\n\\label{grassmann}\n\nIn Grassmann $d=2(2n+1)$-dimensional space the creation and annihilation operators follow from\nthe starting two creation and annihilation operators, both with an odd Grassmann character, \nsince those with an even Grassmann character do not obey the anticommutation relations of \nEq.~(\\ref{ijthetaprod})~\\cite{nh2018} \n\\begin{eqnarray}\n\\hat{b}^{\\theta 1 \\dagger}_{1} &=& (\\frac{1}{\\sqrt{2}})^{\\frac{d}{2}} \\,\n(\\theta^0 - \\theta^3) (\\theta^1 + i \\theta^2) (\\theta^5 + i \\theta^6) \\cdots (\\theta^{d-1} +\n i \\theta^{d}) \\,,\\nonumber\\\\\n\\hat{b}^{\\theta 1}_{1} &=& (\\frac{1}{\\sqrt{2}})^{\\frac{d}{2}}\\,\n (\\frac{\\partial}{\\;\\partial \\theta^{d-1}} - i \\frac{\\partial}{\\;\\partial \\theta^{d}})\n\\cdots (\\frac{\\partial}{\\;\\partial \\theta^{0}}\n-\\frac{\\partial}{\\;\\partial \\theta^3})\\,,\\nonumber\\\\\n\\hat{b}^{\\theta 2 \\dagger }_{1} &=& (\\frac{1}{\\sqrt{2}})^{\\frac{d}{2}} \\,\n(\\theta^0 + \\theta^3) (\\theta^1 + i \\theta^2) (\\theta^5 + i \\theta^6) \\cdots (\\theta^{d-1} +\n i \\theta^{d}) \\,,\\nonumber\\\\\n\\hat{b}^{\\theta 2}_{1} &=& (\\frac{1}{\\sqrt{2}})^{\\frac{d}{2}}\\,\n (\\frac{\\partial}{\\;\\partial \\theta^{d-1}} - i \\frac{\\partial}{\\;\\partial \\theta^{d}})\n\\cdots (\\frac{\\partial}{\\;\\partial \\theta^{0}}\n+\\frac{\\partial}{\\;\\partial \\theta^3})\\,.\n\\label{start(2n+1)2theta}\n\\end{eqnarray}\nAll the creation operators are products of the eigenstates of the Cartan subalgebra operators,%\nEq.~(\\ref{choicecartan})\n\\begin{eqnarray}\n\\label{cartaneigengrass}\n{\\cal {\\bf S}}^{ab} (\\theta^a \\pm \\epsilon \\theta^b) &=& \\mp i \n\\frac{\\eta^{aa}}{\\epsilon} (\\theta^a \\pm \\epsilon \\theta^b)\\,, \\nonumber\\\\\n\\epsilon = 1\\,, \\;\\, {\\rm for}\\;\\, \\eta^{aa}=1\\,,&&\n\\epsilon = i \\,,\\;\\, {\\rm for}\\;\\, \\eta^{aa}= -1\\,, \\nonumber\\\\\n{\\cal {\\bf S}}^{ab}\\, (\\theta^a \\theta^b \\pm \\epsilon \\theta^c \\theta^d)=0\\,,&& \n {\\cal {\\bf S}}^{cd}\\, (\\theta^a \\theta^b \\pm \\epsilon \\theta^c \\theta^d)=0\\,.\n\\end{eqnarray}\n\n\nThe two creation operators, $\\hat{b}^{\\theta 1 \\dagger}_{1}$ and \n$\\hat{b}^{\\theta 2 \\dagger}_{1}$, if applied on the vacuum state, form the starting two states \n$\\phi^{1}_{1}$ and $\\phi^{2}_{1}$ of the two representations, respectively. The vacuum state \nis chosen to be the simplest one~\\cite{nh2018} --- $|\\phi_{0}> = |1>$. The rest of creation operators\nof each of the two groups, $\\hat{b}^{\\theta 1 \\dagger}_{i}$ and $\\hat{b}^{\\theta 2 \\dagger}_{i}$, \nfollow from the starting one by the application of the generators of the Lorentz transformations in \nGrassmann space ${\\cal {\\bf S}}^{ab}$, Eq.~(\\ref{Lorentztheta}), which do not belong to the \nCartan subalgebra, Eq.~(\\ref{choicecartan}), of the Lorentz algebra. They generate either \n$|\\phi^{1}_{j}>$ of the first group or $|\\phi^{2}_{j}>$ of the second group.\n\nAnnihilation operators $\\hat{b}^{\\theta 1}_{i}$ and $\\hat{b}^{\\theta 2}_{i}$ follow from the \ncreation ones by the Hermitian conjugation~\\cite{nh2018}, when taking into account the \nassumption\n\\begin{eqnarray}\n\\label{grassher}\n(\\theta^a)^{\\dagger} &=& \\frac{\\partial}{\\partial \\theta_{a}} \\eta^{aa}=\n-i \\,p^{\\theta a} \\eta^{aa}\\,, \n\\end{eqnarray}\nfrom where it follows\n\\begin{eqnarray}\n\\label{grassp}\n(\\frac{\\partial}{\\partial \\theta_{a}})^{\\dagger} &=& \\eta^{aa}\\, \\theta^a\\,,\\quad\n(p^{\\theta a})^{\\dagger} = -i \\eta^{aa} \\theta^a\\,.\n\\end{eqnarray}\n\n\nThe annihilation operators $\\hat{b}^{\\theta 1}_{i}$ and $\\hat{b}^{\\theta 2}_{i}$ annihilate \nstates $|\\phi^{1}_{i}>$ and $|\\phi^{2}_{i}>$, respectively. \n\nThe application of ${\\cal {\\bf S}}^{01}$ on $\\hat{b}^{\\theta 1 \\dagger}_{1}$, for example,\n transforms this creation operator into $\\hat{b}^{\\theta 1 \\dagger}_{2} = $ \n$(\\frac{1}{\\sqrt{2}})^{\\frac{d}{2} -1} \\,(\\theta^0 \\theta^3 +i \\theta^1 \\theta^2)$\n$ (\\theta^5 + i \\theta^6) \\cdots (\\theta^{d-1} - i \\theta^{d})$. Correspondingly its Hermitian \nconjugate annihilation operator is equal to\n$\\hat{b}^{\\theta 1}_{2} = (\\frac{1}{\\sqrt{2}})^{\\frac{d}{2}-1}\\,\n (\\frac{\\partial}{\\;\\partial \\theta^{d-1}} - i \\frac{\\partial}{\\;\\partial \\theta^{d}})\n\\cdots (\\frac{\\partial}{\\;\\partial \\theta^{3}} \\,\\frac{\\partial}{\\;\\partial \\theta^0} - i \n\\frac{\\partial}{\\;\\partial \\theta^{2}} \\,\\frac{\\partial}{\\;\\partial \\theta^1})$.\n\nAll the states are normalized with respect to the integral over the Grassmann coordinate \nspace~\\cite{norma93}\n\n\\begin{eqnarray}\n\\label{grassnorm}\n<\\phi^{a}_{i}|\\phi^{b}_{j} > &=& \\int d^{d-1} x d^d \\theta^a\\, \\,\\omega \n <\\phi^{a}_{i}|\\theta> <\\theta|\\phi^{b}_{j} > = \\delta^{ab}\\,\\delta_{ij} \\,, \\nonumber\\\\\n\\omega&=& \\Pi^{d}_{k=0}(\\frac{\\partial}{\\;\\,\\partial \\theta_k} + \\theta^{k})\\,,\n\\end{eqnarray}\nwhere $\\omega$ is a weight function, defining the scalar product $<\\phi^a_{i}|\\phi^b_{j} >$,\n and we require that~\\cite{norma93}\n\\begin{eqnarray}\n\\label{grassintegral}\n\\{ d\\theta^a, \\theta^b \\}_{+} &=&0, \\,\\;\\; \\int d\\theta^a =0\\,,\\,\\;\\; \n\\int d\\theta^a \\theta^a =1\\,,\\nonumber\\\\\n\\int d^d \\theta \\,\\,\\theta^0 \\theta^1 \\cdots \\theta^d &=&1\\,,\n\\nonumber\\\\\nd^d \\theta &=&d \\theta^d \\dots d\\theta^0\\,,\n\\end{eqnarray}\nwith $ \\frac{\\partial}{\\;\\,\\partial \\theta_a} \\theta^c = \\eta^{ac}$. \n\n\nThere are $\\frac{1}{2}\\, \\frac{d!}{\\frac{d!}{2} \\frac{d!}{2}}$ in each of these two groups \nof creation operators of an odd Grassmann character in $d=2(2n+1)$-dimensional space.\n\nThe rest of creation operators (and the corresponding annihilation operators) would have rather \nopposite Grassmann character than the ones studied so far: like {\\bf a.} $\\theta^0 \\theta^1$ \nfor the creation operator and \n[$\\frac{\\partial}{\\partial \\theta^1}\\frac{\\partial}{\\partial \\theta^0}$] for the \ncorresponding annihilation operator in $d=(1+1)$\n (since $\\{\\theta^0 \\theta^1, \\frac{\\partial}{\\partial \\theta^{1}}$ \n $\\frac{\\partial}{\\partial \\theta^{0}} \\}_{+}$ gives $(1+ (1+1) \\theta^0 \\theta^1 $\n $\\frac{\\partial}{\\partial \\theta^{1}} \\frac{\\partial}{\\partial \\theta^{0}}) $), and like {\\bf b.}\n $(\\theta^0 \\mp \\theta^3) (\\theta^1\\pm i \\theta^2)$ for creation operator and \n[$ (\\frac{\\partial}{\\partial \\theta^1}\\mp i \\frac{\\partial}{\\partial \\theta^2}) \n(\\frac{\\partial}{\\partial \\theta^0} \\mp \\frac{\\partial}{\\partial \\theta^3})$] for the\nannihilation operator, or $\\theta^0 \\theta^3\n \\theta^1 \\theta^2$ for the creation operator and [$\\frac{\\partial}{\\partial \\theta^2} \n\\,\\frac{\\partial}{\\partial \\theta^1}\\,\\frac{\\partial}{\\partial \\theta^3}\\,\n\\frac{\\partial}{\\partial \\theta^0}$] for the annihilation operator in $d=(3+1)$ \n (since, let say, $\\{\\frac{1}{2} (\\theta^0 - \\theta^3) (\\theta^1 + i \\theta^2),$ \n $\\frac{1}{2} (\\frac{\\partial}{\\partial \\theta^1} - i \\frac{\\partial}{\\partial \\theta^2}) \n (\\frac{\\partial}{\\partial \\theta^0} - \\frac{\\partial}{\\partial \\theta^3})\\}_{+}$ gives\n $(1 + \\frac{1}{4} (1+1) (\\theta^0 - \\theta^3) (\\theta^1 + i \\theta^2) \n (\\frac{\\partial}{\\partial \\theta^1} - i \\frac{\\partial}{\\partial \\theta^2}) \n (\\frac{\\partial}{\\partial \\theta^0} - \\frac{\\partial}{\\partial \\theta^3})$ and equivalently for \nother cases), but applied on a vacuum states some of them still fulfill some of the relations\n of Eq.~(\\ref{ijthetaprod}), but not all (like $\\{\\frac{1}{2}(\\theta^0 - \\theta^3) \n(\\theta^1 + i \\theta^2),$ $\\frac{1}{2} (\\theta^0 + \\theta^3) (\\theta^1 - i \\theta^2)\\}_{+}\n=$ $i \\theta^0 \\theta^1\\theta^2 \\theta^3 $, while it should be zero). \n\nLet us add that, like in Clifford case, one can simplify the scalar product in Grassmann case \nby recognizing that the scalar product is equal to $\\delta^{ab}\\,\\delta_{ij}$\n\\begin{eqnarray}\n\\label{grassscalar}\n<\\phi^{a}_{i}|\\theta> <\\theta|\\phi^{b}_{j} > &=& \\delta^{ab}\\,\\delta_{ij}\\,,\n\\end{eqnarray}\nwithout integration over the Grassmann coordinates. Let us manifest this in the case\nof $d=(1+1)$:$<1|\\frac{1}{\\sqrt{2}}(\\frac{\\partial}{\\partial \\theta^0} - \n\\frac{\\partial}{\\partial \\theta^1})\n\\frac{1}{\\sqrt{2}}(\\theta^0 - \\theta^1)|1>=1 $, $|1>$ is the normalized vacuum state,\n$<1|1>=1$. It is true \nin all dimensions, what can easily be understood for all the states, which are defined by \nthe creation operators $\\hat{b}_{i}^{\\dagger}$ on the vacuum state $|1>$, \n$|\\phi^{b}_{i} >= \\hat{b}_{i}^{\\dagger}|1>$, fulfilling the anticommutation relations\nof Eq.~(\\ref{ijthetaprod}).\n\n \n\\subsection{Creation and annihilation operators in Clifford space}\n\\label{clifford}\n\nThere are two kinds of Clifford objects~\\cite{norma93}, (\\cite{IARD2016} and Refs. therein),\n$\\gamma^a$ and $\\tilde{\\gamma}^a$, both fulfilling the anticommutation relations\n\\begin{eqnarray}\n\\label{tildecliffcomrel}\n \\{\\gamma^a, \\gamma^b \\}_{+} &=& 2 \\eta^{a b} = \n\\{\\tilde{\\gamma}^a, \\tilde{\\gamma}^b \\}_{+}\\,, \\nonumber\\\\ \n \\{\\gamma^a, \\tilde{\\gamma}^b \\}_{+}&=&0\\,. \n\\end{eqnarray}\nBoth Clifford algebra objects are expressible with $\\theta^a $ and \n$\\frac{\\partial}{\\,\\;\\partial \\theta^a}$~\\cite{norma93,nh2018}, \n (\\cite{IARD2016} and Refs. therein)\n\\begin{eqnarray}\n\\label{cliffthetarel}\n\\gamma^a &=& (\\theta^a + \\frac{\\partial}{\\;\\partial \\theta_a})\\,,\\nonumber\\\\\n\\tilde{\\gamma}^a &=& i \\, (\\theta^a - \\frac{\\partial}{\\;\\partial \\theta a})\\,, \\nonumber\\\\\n\\theta^a &=&\\frac{1}{2} (\\gamma^a - i \\tilde{\\gamma}^a)\\,,\\nonumber\\\\\n\\frac{\\,\\partial}{\\partial \\theta_a} &=&\\frac{1}{2} \\, (\\gamma^a + i \\tilde{\\gamma}^a)\\,,\n\\end{eqnarray}\nfrom where it follows: $(\\gamma^a)^{\\dagger} = \\gamma^a \\eta^{aa}$, \n$(\\tilde{\\gamma}^a)^{\\dagger} = \\tilde{\\gamma}^a \\eta^{aa}$,\n$\\gamma^a \\gamma^a = \\eta^{aa}$, $\\gamma^a (\\gamma^a)^{\\dagger} =1$,\n$ \\tilde{\\gamma}^a \\tilde{\\gamma}^a = \\eta^{aa}$, \n$ \\tilde{\\gamma}^a (\\tilde{\\gamma}^a)^{\\dagger} =1$.\n\n\nCorrespondingly we can use either $\\gamma^a$ or $\\tilde{\\gamma}^a$ instead of \n$\\theta^a$ to span the internal space of fermions. Since both, $\\gamma^a$ and \n$\\tilde{\\gamma}^a$, are expressible with $\\theta^a$ and the derivatives with respect to \n$\\theta^a$, the norm of vectors in Clifford space can be defined by the same integral as in \nGrassmann space, Eq.(\\ref{grassnorm}), or we can simplify the scalar product (as in the \nGrassmann case, Eq.~(\\ref{grassscalar}) by introducing\nthe Clifford vacuum state $|\\psi_{oc}>$, Eq.~(\\ref{vac1}), instead of $|1>$ in Grassmann\ncase. \n\nWe make use of $\\gamma^a$ to span the vector space. As in the case of Grassmann\nspace we require that the basic states are eigenstates of the Cartan subalgebra operators of \n$S^{ab}$ and $\\tilde{S}^{ab}$, Eq.~(\\ref{choicecartan}). \n\\begin{eqnarray}\n\\stackrel{ab}{(k)}:&=& \n\\frac{1}{2}(\\gamma^a + \\frac{\\eta^{aa}}{ik} \\gamma^b)\\,,\\quad \n\\stackrel{ab}{(k)}^{\\dagger} = \\eta^{aa}\\stackrel{ab}{(-k)}\\,,\\nonumber\\\\\n\\stackrel{ab}{[k]}:&=&\n\\frac{1}{2}(1+ \\frac{i}{k} \\gamma^a \\gamma^b)\\,,\\quad \\;\\,\n\\stackrel{ab}{[k]}^{\\dagger} = \\,\\stackrel{ab}{[k]}\\,,\\nonumber\\\\\nS^{ab}\\,\\stackrel{ab}{(k)} &=& \\frac{1}{2} k\\, \\stackrel{ab}{(k)}\\,,\\quad \\quad \\quad\nS^{ab}\\,\\stackrel{ab}{[k]} = \\frac{1}{2} k\\, \\stackrel{ab}{[k]}\\,, \\nonumber\\\\\n\\tilde{S}^{ab}\\,\\stackrel{ab}{(k)} &=& \\frac{1}{2} k\\, \\stackrel{ab}{(k)}\\,,\\quad \\quad \\quad\n\\tilde{S}^{ab}\\,\\stackrel{ab}{[k]} = -\\frac{1}{2} k\\, \\stackrel{ab}{[k]}\\,,\n\\label{signature}\n\\end{eqnarray}\nwith $k^2 = \\eta^{aa} \\eta^{bb}$. To calculate $\\tilde{S}^{ab}\\,\\stackrel{ab}{(k)}$ and \n$\\tilde{S}^{ab}\\,\\stackrel{ab}{[k]}$ we define~\\cite{nh03,nh02} the application of \n$\\tilde{\\gamma}^a$ on any Clifford algebra object A as follows\n\\begin{eqnarray}\n(\\tilde{\\gamma^a} A = i (-)^{(A)} A \\gamma^a)|\\psi_{oc}>\\,,\n\\label{gammatildeA}\n\\end{eqnarray}\nwhere $A$ is any Clifford algebra object and $(-)^{(A)} = -1$, if $A$ is an odd Clifford algebra\n object and $(-)^{(A)} = 1$, if $A$ is an even Clifford algebra\n object, $|\\psi_{oc}>$ is the vacuum state, replacing the vacuum state $|\\psi_o>= |1>$,\n used in Grassmann case, with the \none of Eq.~(\\ref{vac1}), in accordance with the relation of Eqs.~({\\ref{cliffthetarel},\n\\ref{grassnorm}, \\ref{grassintegral}}), Ref.~\\cite{nh2018}.\n\n\nWe can define now the creation and annihilation operators in Clifford space so that they fulfill the\nrequirements of Eq.~(\\ref{ijthetaprod}).\nWe write the starting creation operator and its Hermitian conjugate one (in \naccordance with Eq.~(\\ref{signature}) and Eq.(\\ref{choicecartan})) in $2(2n+1)$-dimensional \nspace as follows~\\cite{nh2018}\n\\begin{eqnarray}\n\\hat{b}^{1 \\dagger}_1&=& \\stackrel{03}{(+i)} \\stackrel{12}{(+)} \\stackrel{56}{(+)}\\cdots\n\\stackrel{d-1\\;d}{(+)}\\,,\\nonumber\\\\\n\\hat{b}^{1}_1&=& \\stackrel{d-1\\;d}{(-)} \\cdots \\stackrel{56}{(-)} \\stackrel{12}{(-)}\n\\stackrel{03}{(-i)}\\,.\n\\label{bstart}\n\\end{eqnarray}\nThe starting creation operator $\\hat{b}^{1 \\dagger}_1$, when applied on the vacuum state\n$|\\psi_{oc}>$, defines the starting family member of the starting ''family\". The corresponding\nstarting annihilation operator is its Hermitian conjugated one, Eq.~(\\ref{signature}).\n\nAll the other creation operators of the same family can be obtained by the application of the \ngenerators of the Lorentz transformations $S^{ab}$, Eq.~(\\ref{Lorentzgammatilde}), which do \nnot belong to the Cartan subalgebra of $SO(2(2n+1) -1,1)$, Eq.~(\\ref{choicecartan}). \n\\begin{eqnarray}\n\\hat{b}^{1\\dagger}_i &\\propto & S^{ab} ..S^{ef} \\hat{b}^{1\\dagger}_1\\,,\\nonumber\\\\\n\\hat{b}^{1}_i&\\propto & \\hat{b}^{1}_1 S^{ef}..S^{ab}\\,,\n\\label{b1i}\n\\end{eqnarray}\nwith $S^{ab\\dagger} = \\eta^{aa} \\eta^{bb} S^{ab}$. The proportionality factors are chosen \nso, that the corresponding states $|\\psi^{1}_{1}>= \\hat{b}^{1\\dagger}_i |\\psi_{oc}>$ are \nnormalized, where $|\\psi_{oc}>$ is the normalized vacuum state, $<\\psi_{oc}|\\psi_{oc}> =1$. \n\nThe creation operators creating different \"families\" with respect to the starting \"family\",\n Eq.~(\\ref{bstart}), \ncan be obtained from the starting one by the application of $\\tilde{S}^{ab}$, \nEq.~(\\ref{Lorentzgammatilde}), which do not belong to the Cartan subalgebra of \n$\\widetilde{SO}(2(2n+1) -1,1)$, Eq.~(\\ref{choicecartan}). They all keep the \"family member\" \nquantum number unchanged.\n\\begin{eqnarray}\n\\hat{b}^{\\alpha \\dagger}_i &\\propto & \\tilde{S}^{ab} \\cdots \\tilde{S}^{ef}\\,\n\\hat{b}^{1\\dagger}_{i}\\,.\n\\label{balpha1}\n\\end{eqnarray}\nCorrespondingly we can define (up to the proportionality factor) any creation operator for any\n\"family\" and any \"family member\" with the application of $S^{ab}$ and $\\tilde{S}^{ab}$%\n~\\cite{nh2018}\n\\begin{eqnarray}\n\\hat{b}^{\\alpha \\dagger}_i&\\propto & \\tilde{S}^{ab} \\cdots \\tilde{S}^{ef} \n{S}^{mn}\\cdots {S}^{pr}\n\\hat{b}^{1\\dagger}_{1}\\nonumber\\\\\n&\\propto & {S}^{mn}\\cdots {S}^{pr} \\hat{b}^{1\\dagger}_{1} {S}^{ab} \\cdots {S}^{ef}\\,.\n\\label{anycreation}\n\\end{eqnarray}\nAll the corresponding annihilation operators follow from the creation ones by the Hermitian \nconjugation.\n\nThere are $2^{\\frac{d}{2}-1}$ $\\times \\;\\, 2^{\\frac{d}{2}-1}$ creation operators of an odd \nClifford character and the same number of annihilation operators, which fulfill the anticommutation\nrelations of Eq.~(\\ref{ijthetaprod}) on the vacuum state $|\\psi_{oc}>$ with\n$2^{\\frac{d}{2}-1}$ summands\n\\begin{eqnarray}\n|\\psi_{oc}>&=& \\alpha\\,( \\stackrel{03}{[-i]} \\stackrel{12}{[-]} \n\\stackrel{56}{[-]}\\cdots\n\\stackrel{d-1\\;d}{[-]} + \\stackrel{03}{[+i]} \\stackrel{12}{[+]} \\stackrel{56}{[-]} \\cdots\n\\stackrel{d-1\\;d}{[-]} + \\stackrel{03}{[+i]} \\stackrel{12}{[-]} \\stackrel{56}{[+]}\\cdots\n\\stackrel{d-1\\;d}{[-]} + \\cdots ) |0>\\,, \\quad \\nonumber\\\\\n&&\\alpha =\\frac{1}{\\sqrt{2^{\\frac{d}{2}-1}}}\\,, \\nonumber\\\\\n&&{\\rm for}\\; d=2(2n+1)\\,,\n\\label{vac1}\n\\end{eqnarray}\n$n$ is a positive integer. For a chosen $\\alpha =\\frac{1}{\\sqrt{2^{\\frac{d}{2}-1}}}$\nthe vacuum is normalized: $<\\psi_{oc}|\\psi_{oc}>=1$. \n\n\nIt is proven in Ref.~\\cite{nh2018} that the creation and annihilation operators fulfill the\nanticommutation relations required for fermions, Eq.~(\\ref{ijthetaprod}). \n\n\n\n\n\\section{Properties of representations of the Lorentz group $SO(2(2n+1))$ and of subgroups\nin Grassmann and in Clifford space}\n\\label{grassmanncliffordrepresentations}\n\nThe purpose of this contribution is to compare properties of the representations of the Lorentz\ngroup $SO(2(2n+1))$, $n \\ge 3$, when for the description of the internal degrees of freedom \nof fermions either {\\bf i.} Grassmann space or {\\bf ii.} Clifford space is used. The \n{\\it spin-charge-family} theory~(\\cite{JMP2013,normaJMP2015,IARD2016,%\nn2014matterantimatter,nh2017,nd2017,n2012scalars} and the references therein) namely\npredicts that all the properties of the observed either quarks and leptons or vector gauge fields \nor scalar gauge fields originate in $d \\ge (13+1)$, in which massless fermions interact with the \ngravitational field only --- with its spin connections and \nvielbeins. \n\nHowever, both --- Clifford space and Grassmann space --- allow second quantized states, the \ncreation and annihilation operators of which fulfill the anticommutation relations for fermions of \nEq.~(\\ref{ijthetaprod}). \n\nBut while Clifford space offers the description of spins, charges and families of fermions in \n$d=(3+1)$, all in the fundamental representations of the Lorentz group $SO(13,1)$ and the \nsubgroups of the Lorentz group, in agreement with the observations, the representations of the \nLorentz group are in Grassmann space the adjoint ones, in disagreement with what we \nobserve.\n\nWe compare properties of the representations in Grassmann case with those in Clifford case \nto be able to better understand \"the choice of nature in the expanding universe, making use of the \nClifford degrees of freedom\", rather than Grassmann degrees of freedom.\n\nIn introduction we briefly reviewed properties of creation and annihilation operators in both spaces,\npresented in Ref.~\\cite{nh2018} (and the references therein). \nWe pay attention on spaces with $d=2(2n+1)$ of ordinary coordinates and $d=2(2n+1)$ internal \ncoordinates, either of Clifford or of Grassmann character. \n\n{\\bf i.} $\\;\\;$ In Clifford case there are $2^{\\frac{d}{2} - 1}$ creation operators of an odd \nClifford character, creating \"family members\" when applied on the vacuum state. We choose \nthem to be eigenstates of the Cartan subalgebra operators, Eq.(\\ref{choicecartan}), of the Lorentz \nalgebra. All the members\ncan be reached from any of the creation operators by the application of $S^{ab}$, \nEq.~(\\ref{Lorentzgammatilde}).\nEach \"family member\" appears in $2^{\\frac{d}{2} - 1}$ \"families\", again of an odd Clifford \ncharacter, since the corresponding creation operators are reachable by $\\tilde{S}^{ab}$, \nEq.~(\\ref{Lorentzgammatilde}), which are Clifford even objects. \n\nThere are correspondingly $2^{\\frac{d}{2} - 1} \\cdot$ $2^{\\frac{d}{2} - 1}$ creation and the \nsame number ($2^{\\frac{d}{2} - 1} \\cdot$ $2^{\\frac{d}{2} - 1}$) of annihilation operators. Also \nthe annihilation operators, annihilating states of $2^{\\frac{d}{2} - 1}$ \"family members\" in\n $2^{\\frac{d}{2} - 1}$ \"families\", have an odd Clifford character, since they are Hermitian conjugate\n to the creation ones. \n\nThe rest of $2 \\cdot$ $2^{\\frac{d}{2} - 1}\\cdot$ $2^{\\frac{d}{2} - 1}$ members\nof the Lorentz representations have an even Clifford character, what means that the corresponding \ncreation and annihilation operators can not fulfill the anticommutation relations required for \nfermions, Eq.~(\\ref{ijthetaprod}). \nAmong these $2^{\\frac{d}{2} - 1}$ products of projectors determine the vacuum state, \nEq.~(\\ref{vac1}).\n\n\n{\\bf ii.} $\\;\\;$ In Grassmann case there are $\\frac{d!}{\\frac{d}{2}!\\,\\frac{d}{2}! }$ \noperators of an odd\nGrassmann character, which form the creation operators, fulfilling with the corresponding \nannihilation operators the requirements of Eq.~(\\ref{ijthetaprod}). All the creation operators \nare chosen to be products of the eigenstates of the Cartan subalgebra ${\\cal {\\bf S}}^{ab}$, \nEq.~(\\ref{choicecartan}). The corresponding annihilation operators are the Hermitian conjugated \nvalues of the creation operators, Eqs.~(\\ref{grassher}, \\ref{grassp}, \\ref{start(2n+1)2theta}). \nThe creation operators form, when applied on the simple vacuum state $|\\phi_{o}> = |1>$,\ntwo independent groups of states. The members of each of the two groups are \nreachable from any member of a group by the application of ${\\cal {\\bf S}}^{ab}$, \nEq.~(\\ref{Lorentztheta}). All the states of any of the two decuplets are orthonormalized.\n\nWe comment in what follows the representations in $d=(13+1)$ in Clifford and in Grassmann case.\nIn {\\it spin-charge family} theory there are breaks of the starting symmetry from $SO(13,1)$ to\n$SO(3,1)\\times SU(2) \\times SU(3) \\times U(1)$ in steps, which lead to the so far observed \nquarks and leptons, gauge and scalar fields and gravity. One \nof the authors (N.S.M.B.), together with H.B. Nielsen, defined the discrete symmetry operators for \nKaluza-Klein theories for spinors in Clifford space~\\cite{nhds}. In Ref.~\\cite{nh2018} the same \nauthors define the discrete \nsymmetry operators in the case that for the description of fermion degrees of \nfreedom Grassmann space is used. Here we comment symmetries in both spaces for some of\nsubgroups of the $SO(13,1)$ group, as well as the appearance of the Dirac sea. \n\n\n\\subsection{Currents in Grassmann space} \n\\label{currents}\n\n\n\n\\begin{eqnarray}\n\\{ \\theta^a p_a, \\frac{\\partial }{\\partial \\theta_b} p_b \\}_{+}&=&\n \\theta^a p_a \\frac{\\partial }{\\partial \\theta_b} p_b + \n \\frac{\\partial }{\\partial \\theta_b} p_b \\theta^a p_a = \\eta^{ab}p_a p_b \\,, \n\\label{KGgrass}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\phi^{\\dagger}(\\theta^0 + \\frac{\\partial }{\\partial \\theta_0})\n(\\theta^a + \\frac{\\partial }{\\partial \\theta_a}) \\phi &=& j^{a} \\,, \n\\label{currentgrass}\n\\end{eqnarray}\n\n\\subsection{Equations of motion in Grassmann and Clifford space}\n\\label{equationinCandG}\n\nWe define~\\cite{nh2018} the action in Grassmann space, for which we require --- similarly as in \n Clifford case --- that the action for a free massless object \n\\begin{eqnarray}\n{\\cal A}\\, &=& \\frac{1}{2} \\int \\; d^dx \\;d^d\\theta\\; \\omega \\, \\{\\phi^{\\dagger} \n(1-2\\theta^0 \\frac{\\partial}{\\partial \\theta^0}) \\,\\frac{1}{2}\\,\n (\\theta^a p_{a}+ \\eta^{aa} \\theta^{a \\dagger}\n p_{a}) \\phi \\} \\,, \n\\label{actionWeylGrass}\n\\end{eqnarray}\nis Lorentz invariant. The corresponding equation of motion is \n\\begin{eqnarray}\n\\label{Weylgrass}\n\\frac{1}{2}[ (1-2\\theta^0 \\frac{\\partial}{\\partial \\theta^0}) \\,\\theta^a + \n((1-2\\theta^0 \\frac{\\partial}{\\partial \\theta^0}) \\,\\theta^a)^{\\dagger}]\\,\n\\, p_{a} \\,|\\phi^{\\theta}_{i}>\\,&= & 0\\,,\n\\end{eqnarray}\n$p_{a} = i\\, \\frac{\\partial}{\\partial x^a}$, leading to the Klein-Gordon equation\n\\begin{eqnarray}\n\\label{LtoKGgrass}\n\\{(1-2\\theta^0 \\frac{\\partial}{\\partial \\theta^0}) \\theta^a p_{a}\\}^{\\dagger}\\,\\theta^b p_{b}\n|\\phi^{\\theta}_{i}>&= & \np^a p_a |\\phi^{\\theta}_{i}>=0\\,.\n\\end{eqnarray}\nIn the Clifford case the action for massless fermions is well known\n\\begin{eqnarray}\n{\\cal A}\\, &=& \\int \\; d^dx \\; \\frac{1}{2}\\, (\\psi^{\\dagger}\\gamma^0 \\, \\gamma^a p_{a} \\psi) +\n h.c.\\,, \n\\label{actionWeyl}\n\\end{eqnarray}\n leading to the equations of motion \n\\begin{eqnarray}\n\\label{Weyl}\n\\gamma^a p_{a} |\\psi^{\\alpha}>&= & 0\\,, \n\\end{eqnarray}\nwhich fulfill also the Klein-Gordon equation\n\\begin{eqnarray}\n\\label{LtoKG}\n\\gamma^a p_{a} \\gamma^b p_b |\\psi^{\\alpha}_{i}>&= & \np^a p_a |\\psi^{\\alpha}_{i}>=0\\,.\n\\end{eqnarray}\n\n\n\n\n\\subsection{Discrete symmetries in Grassmann and Clifford space}\n\\label{CPT}\n\n\nWe follow also here Ref.~\\cite{nh2018} and the references therein.\nWe distinguish in $d$-dimensional space two kinds of dicsrete operators ${\\cal C}, {\\cal P}$ \nand ${\\cal T}$\n operators with respect to the internal space which we use.\n\nIn the Clifford case~\\cite{nhds}, when the whole $d$-space is treated equivalently, we have \n\\begin{eqnarray}\n\\label{calCPTH}\n{\\cal C}_{{\\cal H}}&=& \\prod_{\\gamma^a \\in \\Im} \\gamma^a \\,\\, K\\,,\\quad\n{\\cal T}_{{\\cal H}}= \\gamma^0 \\prod_{\\gamma^a \\in \\Re} \\gamma^a \\,\\, K\\, I_{x^0}\\,\\,\\,,\n \\quad {\\cal P}^{(d-1)}_{{\\cal H}} = \\gamma^0\\,I_{\\vec{x}}\\,,\\nonumber\\\\\nI_{x} x^a &=&- x^a\\,, \\quad I_{x^0} x^a = (-x^0,\\vec{x})\\,, \\quad I_{\\vec{x}} \\vec{x} =\n -\\vec{x}\\,, \\nonumber\\\\\nI_{\\vec{x}_{3}} x^a &=& (x^0, -x^1,-x^2,-x^3,x^5, x^6,\\dots, x^d)\\,.\n\\end{eqnarray}\nThe product $\\prod \\, \\gamma^a$ is meant in the ascending order in $\\gamma^a$.\n\nIn the Grassmann case we correspondingly define\n\\begin{eqnarray}\n\\label{calCPTG}\n{\\cal C}_{G}&=& \\prod_{\\gamma^a_{ G} \\in \\Im \\gamma^a} \\, \\gamma^a_{ G}\\, K\\,,\\quad\n{\\cal T}_{G} = \\gamma^0_{G} \\prod_{\\gamma^a_{G} \\in \\Re\n \\gamma^a} \\,\\gamma^a_{ G}\\, K \\, I_{x^0}\\,,\\quad\n{\\cal P}^{(d-1)}_{G} = \\gamma^0_{G} \\,I_{\\vec{x}}\\,\n\\end{eqnarray}\nwith $\\gamma^a_{G}$ defined as \n\\begin{eqnarray}\n\\label{gammaG}\n\\gamma^{a}_{G} &=& \n(1- 2 \\theta^a \\eta^{aa} \\frac{\\partial}{\\partial \\theta_{a}})\\,, \n\\end{eqnarray}\nwhile $I_{x}$,\n$I_{\\vec{x}_{3}}$ \nis defined in Eq.~(\\ref{calCPTH}).\nLet be noticed, that since $\\gamma^a_{G}$ ($= - i \\eta^{aa}\\, \\gamma^a \\tilde{\\gamma}^a$) is \nalways real as there is $ \\gamma^a i \\tilde{\\gamma}^a$, while $ \\gamma^a$ is either real or \nimaginary,\nwe use in Eq.~(\\ref{calCPTG}) $\\gamma^a$ to make a choice of appropriate $\\gamma^a_{G}$. \nIn what follows we shall use the notation as in Eq.~(\\ref{calCPTG}).\n\n\nWe define, according to Ref.~\\cite{nh2018} (and the references therein) in both cases --- Clifford \nGrassmann case --- the operator\n \"emptying\"~\\cite{JMP2013,normaJMP2015} (arxiv:1312.1541) the Dirac sea, so that operation of \n\"emptying$_{N}$\" after the charge conjugation ${\\cal C }_{{\\cal H}}$ in the Clifford case and \n\"emptying$_{G}$\" after the charge conjugation ${\\cal C }_{G}$\nin the Grassmann case (both transform the state put on the top of either the Clifford or the Grassmann\nDirac sea into the corresponding negative energy state) creates the anti-particle state to the starting \nparticle state, both put on the top of the Dirac sea and both solving the Weyl equation, either in the\nClifford case, Eq.~(\\ref{Weyl}), or in the Grassmann case, Eq.~(\\ref{Weylgrass}), for free massless \nfermions\n\\begin{eqnarray}\n\\label{empt}\n\"{\\rm emptying}_{N}\"&=& \\prod_{\\Re \\gamma^a}\\, \\gamma^a \\,K\\, \\quad {\\rm in} \\, \\;\n{\\rm Clifford}\\, {\\rm space}\\,, \n\\nonumber\\\\\n\"{\\rm emptying}_{G}\"&=& \\prod_{\\Re \\gamma^a}\\, \\gamma^a_{G} \\,K\\, \\quad {\\rm in}\\;\\, \n{\\rm Grassmann} \\, {\\rm space}\\,, \n\\end{eqnarray}\nalthough we must keep in mind that indeed the anti-particle state is a hole in the Dirac sea from the \nFock space point of view. The operator \"emptying\" is bringing the single particle operator \n${\\cal C }_{{\\cal H}}$ in the Clifford case and ${\\cal C }_{G}$ in the Grassmann case into the operator \non the Fock space in each of the two cases.\nThen the anti-particle state creation operator --- \n${\\underline {\\bf {\\Huge \\Psi}}}^{\\dagger}_{a}[\\Psi_{p}]$ --- to the corresponding particle state \ncreation operator --- can be obtained also as follows\n\\begin{eqnarray}\n\\label{makingantip}\n{\\underline {\\bf {\\Huge \\Psi}}}^{\\dagger}_{a}[\\Psi_{p}]\\, |vac> &=& \n{\\underline {\\bf \\mathbb{C}}}_{{{\\bf \\cal H}}}\\, \n{\\underline {\\bf {\\Huge \\Psi}}}^{\\dagger}_{p}[\\Psi_{p}]\\, |vac> = \n\\int \\, {\\mathbf{\\Psi}}^{\\dagger}_{a}(\\vec{x})\\, \n({\\bf \\mathbb{C}}_{\\cal H}\\,\\Psi_{p} (\\vec{x})) \\,d^{(d-1)} x \\, \\,|vac> \\,,\\nonumber\\\\\n{\\bf \\mathbb{C}}_{\\cal H} &=& \"{\\rm emptying}_{N}\"\\,\\cdot\\, {\\cal C}_{{\\cal H}} \\,\n\\end{eqnarray}\nin both cases.\n\nThe operators ${\\bf \\mathbb{C}}_{\\cal H}$ and ${\\bf \\mathbb{C}}_{G}$ \n\\begin{eqnarray}\n\\label{emptCHG}\n{\\bf \\mathbb{C}}_{\\cal H} &=& \"{\\rm emptying}_{N}\" \\,\\cdot\\, {\\cal C}_{{\\cal H}} \\,,\\nonumber\\\\\n{\\bf \\mathbb{C}}_{G} &=& \"{\\rm emptying}_{NG}\" \\,\\cdot\\, {\\cal C}_{G}\\,,\n\\end{eqnarray}\n operating on \n$\\Psi_{p} (\\vec{x})$ transforms the positive energy spinor state (which solves the corresponding \nWeyl equation for a massless free fermion) put on the top of the Dirac sea into the positive energy \nanti-fermion state, which again solves the corresponding Weyl equation for a massless free \nanti-fermion put on the top of the Dirac sea. Let us point out that either the operator \n$\"{\\rm emptying}_{N}\" $ \nor the operator $\"{\\rm emptying}_{NG}\"$ transforms the single particle operator either\n$ {\\cal C}_{\\cal H}$ or ${\\cal C}_{G}$ into the operator operating in the Fock space. \n\n\nWe use the Grassmann even, Hermitian and real operators $\\gamma^{a}_{G}$, \nEq.~(\\ref{gammaG}), to define discrete symmetry in Grassmann space, first we did in \n$((d+1)-1)$ space, Eq.~(\\ref{calCPTG}), now we do in $(3+1)$ space, Eq.~(\\ref{calCPTNG}),\n as it is done \nin~\\cite{nhds} in the Clifford case. \nIn the Grassmann case we do this in analogy with the operators in the Clifford case~\\cite{nhds}\n\\begin{eqnarray}\n\\label{calCPTNG}\n{\\cal C}_{NG}&=& \\prod_{\\gamma^m_{ G} \\in \\Re \\gamma^m} \\, \\gamma^m_{ G}\\, K \\,\n I_{x^6 x^8...x^d}\\,,\\nonumber\\\\\n{\\cal T}_{NG}&=& \\gamma^0_{G} \\prod_{\\gamma^m_{G} \\in \\Im\n \\gamma^m} \\, K \\, I_{x^0} I_{x^5 x^7...x^{d-1}}\\,,\\nonumber\\\\\n{\\cal P}^{(d-1)}_{NG} &=& \\gamma^0_{G} \\, \\prod_{s=5}^{d}\\, \\gamma^s_{ G} I_{\\vec{x}}\\,,\n\\nonumber\\\\\n{\\bf \\mathbb{C}}_{NG} &=&\\prod_{\\gamma^s_{ G} \\in \\Re \\gamma^s} \\,\\gamma^s_{ G}\\,, \n I_{x^6 x^8...x^d}\\,,\\quad\n{\\bf \\mathbb{C}}_{NG} {\\cal P}^{(d-1)}_{NG} = \\gamma^0_{G} \\, \\prod_{\n\\gamma^s_{ G} \\in \\Im \\gamma^s, s=5}^{d}\\, \\gamma^s_{ G}\\, I_{\\vec{x}_{3}}\\,\n I_{x^6 x^8...x^d}\\,,\n\\nonumber\\\\\n{\\bf \\mathbb{C}}_{NG} {\\cal T}_{NG} {\\cal P}^{(d-1)}_{NG} &=& \\prod_{\\gamma^s_{ G} \\in \n\\Im \\gamma^a} \\,\\gamma^a_{ G}\\,I_x K\\,.\n\\end{eqnarray}\n\n\n\\subsection{Representations in Grassmann and in Clifford space in $d=(13+1)$}\n\\label{so13+1}\nIn the {\\it spin-charge-family} theory the starting dimension of space must be $\\ge(13+1)$, in \norder that the theory manifests in $d=(3+1)$ all the observed properties of quarks and leptons,\n gauge and scalar fields (explaining the appearance of higgs and \nthe Yukawa couplings), offering as well the explanations for the observations in \ncosmology.\n\nLet us therefore comment properties of representations in both spaces when $d=(13+1)$,\n if we analyze one group of \"family members\" of one of families in Clifford space, and \none of the two representations of $\\frac{1}{2}\\, \\frac{d!}{\\frac{d}{2}! \\frac{d}{2}!}$.\n\n\n{\\bf a.} $\\;\\;$ Let us start \nwith Clifford space~\\cite{IARD2016,normaJMP2015,n2014matterantimatter,JMP2013,pikanorma,%\nportoroz03,norma93}. Each \"family\" representation has $2^{\\frac{d}{2} - 1}= 64$\n \"family members\". If we analyze this representation with respect to the subgroups $SO(3,1)$, \n$(SU(2)\\times SU(2))$ of $SO(4)$ and ($SU(3)\\times$ $U(1)$) of $SO(6)$ of the \nLorentz group $SO(13,1)$, we find that the representations have quantum numbers of all the so \nfar observed quarks and leptons and antiquarks and antileptons, all with spin up and spin down, \nas well as of the left and right handedness, with the right handed neutrino included as the\nmember of this representation.\n\nLet us make a choice of the \"family\", which follows by the application of $\\tilde{S}^{15}$ on the \n\"family\", for which the creation operator of the right-handed neutrino with spin $\\frac{1}{2}$ \nwould be $ \\stackrel{03}{(+i)}\\,\\stackrel{12}{(+)}|\\stackrel{56}{(+)}\\,\n\\stackrel{78}{(+)}||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$.\n(The corresponding annihilation operator of this creation operator is $\\stackrel{13 \\;14}{(-)}\\;\\;\n\\stackrel{11\\;12}{(-)}\\;\\;\\stackrel{9\\;10}{(-)}||\\stackrel{78}{(-)}\\,\\stackrel{56}{(-)}| \n\\stackrel{12}{(-)}\\,\\stackrel{03}{(-i)}$). In Table~\\ref{Table so13+1.} presented creation\noperators for all the \"family members\" of this family follow \nby the application of $S^{ab}$ on \n$\\tilde{S}^{15}$ $ \\stackrel{03}{(+i)}\\,\\stackrel{12}{(+)}|\\stackrel{56}{(+)}\\,\n\\stackrel{78}{(+)}||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$.\n(The annihilation operator of $\\tilde{S}^{15}$ $ \\stackrel{03}{(+i)}\\,\n\\stackrel{12}{(+)}|\\stackrel{56}{(+)}\\,\\stackrel{78}{(+)}||\\stackrel{9\\;10}{(+)}\\;\\;\n\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $ is $\\stackrel{13 \\;14}{[-]}\\;\\;\n\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{9\\;10}{(-)}||\\stackrel{78}{(-)}\\,\\stackrel{56}{[+]}| \n\\stackrel{12}{[+]}\\,\\stackrel{03}{(-i)}$.) \n\nThis is the representation of \nTable~\\ref{Table so13+1.}, in which all the 'family members'' of one \"family\" are classified with \nrespect to the subgroups $SO(3,1)\\times SU(2) \\times SU(2)\\times SU(3) \\times U(1)$.\nThe vacuum state on which the creation operators, represented in the third column, apply \nis defined in Eq.~(\\ref{vac1}). All the creation operators of all the states are of an odd Clifford \ncharacter, fulfilling together with the annihilation operators (which have as well the equivalent\nodd Clifford character, since the Hermitian conjugation do not change the \nClifford character) the requirements of Eq.~(\\ref{ijthetaprod}). Since the Clifford even operators \n$S^{ab}$ and $\\tilde{S}^{ab}$ do not change the Clifford character, all the creation and \nannihilation operators, obtained by products of $S^{ab}$ or $\\tilde{S}^{ab}$ or both,\nfulfill the requirements of Eq.~(\\ref{ijthetaprod}).\n\nWe recognize in Table~\\ref{Table so13+1.} that quarks distinguish from leptons only in the \n$SO(6)$ part of the creation operators. Quarks belong to the colour ($SU(3)$) triplet carrying\nthe \"fermion\" $(U(1))$ quantum number $\\tau^{4} =\\frac{1}{6}$, antiquarks belong to the colour \nantitriplet, carrying the \"fermion\" quantum number $\\tau^{4} = -\\frac{1}{6}$. Leptons belong \nto the colour ($SU(3)$) singlet, carrying the \"fermion\" $(U(1))$ quantum number $\\tau^{4} =\n -\\frac{1}{2}$, while antileptons belong to the colour antisinglet, carrying the \"fermion\" quantum \nnumber $\\tau^{4} = \\frac{1}{2}$. \n \nLet us also comment that the oddness and evenness of part of states in the subgroups of the \n$SO(13,1)$ group change: While quarks and leptons have in the part of $SO(6)$ an odd Clifford \ncharacter, have antiquarks and antileptons in this part an even odd Clifford character. \nCorrespondingly the Clifford character changes in the rest of subgroups \n\n\\bottomcaption{\\label{Table so13+1.}%\n\\tiny{\nThe left handed ($\\Gamma^{(13,1)} = -1$~\\cite{IARD2016})\nmultiplet of spinors --- the members of the fundamental representation of the $SO(13,1)$ group,\nmanifesting the subgroup $SO(7,1)$\n of the colour charged quarks and antiquarks and the colourless\nleptons and antileptons --- is presented in the massless basis using the technique presented in\nRefs.~\\cite{nh02,nh03,IARD2016,normaJMP2015}.\nIt contains the left handed ($\\Gamma^{(3,1)}=-1$)\n weak ($SU(2)_{I}$) charged ($\\tau^{13}=\\pm \\frac{1}{2}$, Eq.~(\\ref{so42})),\nand $SU(2)_{II}$ chargeless ($\\tau^{23}=0$, Eq.~(\\ref{so42}))\nquarks and leptons and the right handed ($\\Gamma^{(3,1)}=1$)\n weak ($SU(2)_{I}$) chargeless and $SU(2)_{II}$\ncharged ($\\tau^{23}=\\pm \\frac{1}{2}$) quarks and leptons, both with the spin $ S^{12}$ up and\ndown ($\\pm \\frac{1}{2}$, respectively). \nQuarks distinguish from leptons only in the $SU(3) \\times U(1)$ part: Quarks are triplets\nof three colours ($c^i$ $= (\\tau^{33}, \\tau^{38})$ $ = [(\\frac{1}{2},\\frac{1}{2\\sqrt{3}}),\n(-\\frac{1}{2},\\frac{1}{2\\sqrt{3}}), (0,-\\frac{1}{\\sqrt{3}}) $], Eq.~(\\ref{so64}))\ncarrying the \"fermion charge\" ($\\tau^{4}=\\frac{1}{6}$, Eq.~(\\ref{so64})).\nThe colourless leptons carry the \"fermion charge\" ($\\tau^{4}=-\\frac{1}{2}$).\nThe same multiplet contains also the left handed weak ($SU(2)_{I}$) chargeless and $SU(2)_{II}$\ncharged antiquarks and antileptons and the right handed weak ($SU(2)_{I}$) charged and\n$SU(2)_{II}$ chargeless antiquarks and antileptons.\nAntiquarks distinguish from antileptons again only in the $SU(3) \\times U(1)$ part: Antiquarks are\nantitriplets,\n carrying the \"fermion charge\" ($\\tau^{4}=-\\frac{1}{6}$).\nThe anticolourless antileptons carry the \"fermion charge\" ($\\tau^{4}=\\frac{1}{2}$).\n $Y=(\\tau^{23} + \\tau^{4})$ is the hyper charge, the electromagnetic charge\nis $Q=(\\tau^{13} + Y$).\nThe vacuum state,\non which the nilpotents and projectors operate, is presented in Eq.~(\\ref{vac1}).\nThe reader can find this Weyl representation also in\nRefs.~\\cite{n2014matterantimatter,pikanorma,portoroz03,normaJMP2015} and the references\ntherein. }\n}\n\\tablehead{\\hline\ni&$$&$|^a\\psi_i>$&$\\Gamma^{(3,1)}$&$ S^{12}$&\n$\\tau^{13}$&$\\tau^{23}$&$\\tau^{33}$&$\\tau^{38}$&$\\tau^{4}$&$Y$&$Q$\\\\\n\\hline\n&& ${\\rm (Anti)octet},\\,\\Gamma^{(7,1)} = (-1)\\,1\\,, \\,\\Gamma^{(6)} = (1)\\,-1$&&&&&&&&& \\\\\n&& ${\\rm of \\;(anti) quarks \\;and \\;(anti)leptons}$&&&&&&&&&\\\\\n\\hline\\hline}\n\\tabletail{\\hline \\multicolumn{12}{r}{\\emph{Continued on next page}}\\\\}\n\\tablelasttail{\\hline}\n\\begin{center}\n\\tiny{\n\\begin{supertabular}{|r|c||c||c|c||c|c||c|c|c||r|r|}\n1&$ u_{R}^{c1}$&$ \\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $ &1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{2}{3}$&$\\frac{2}{3}$\\\\\n\\hline\n2&$u_{R}^{c1}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{2}{3}$&$\\frac{2}{3}$\\\\\n\\hline\n3&$d_{R}^{c1}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$-\\frac{1}{3}$&$-\\frac{1}{3}$\\\\\n\\hline\n4&$ d_{R}^{c1} $&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $&1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$-\\frac{1}{3}$&$-\\frac{1}{3}$\\\\\n\\hline\n5&$d_{L}^{c1}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&-1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$-\\frac{1}{3}$\\\\\n\\hline\n6&$d_{L}^{c1} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $&-1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$-\\frac{1}{3}$\\\\\n\\hline\n7&$ u_{L}^{c1}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$ &-1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0 &$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$\\frac{2}{3}$\\\\\n\\hline\n8&$u_{L}^{c1}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&-1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$\\frac{2}{3}$\\\\\n\\hline\\hline\n\\shrinkheight{0.3\\textheight}\n9&$ u_{R}^{c2}$&$ \\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $ &1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{2}{3}$&$\\frac{2}{3}$\\\\\n\\hline\n10&$u_{R}^{c2}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}$&1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{2}{3}$&$\\frac{2}{3}$\\\\\n\\hline\n11&$d_{R}^{c2}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}$\n&1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$ - \\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$-\\frac{1}{3}$&$-\\frac{1}{3}$\\\\\n\\hline\n12&$ d_{R}^{c2} $&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $\n&1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$-\\frac{1}{3}$&$-\\frac{1}{3}$\\\\\n\\hline\n13&$d_{L}^{c2}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}$\n&-1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$-\\frac{1}{3}$\\\\\n\\hline\n14&$d_{L}^{c2} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&-1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$-\\frac{1}{3}$\\\\\n\\hline\n15&$ u_{L}^{c2}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}$ &-1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0 &$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$\\frac{2}{3}$\\\\\n\\hline\n16&$u_{L}^{c2}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}$&-1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$\\frac{2}{3}$\\\\\n\\hline\\hline\n17&$ u_{R}^{c3}$&$ \\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $ &1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{2}{3}$&$\\frac{2}{3}$\\\\\n\\hline\n18&$u_{R}^{c3}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$&1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{2}{3}$&$\\frac{2}{3}$\\\\\n\\hline\n19&$d_{R}^{c3}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$&1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$-\\frac{1}{3}$&$-\\frac{1}{3}$\\\\\n\\hline\n20&$ d_{R}^{c3} $&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $&1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$-\\frac{1}{3}$&$-\\frac{1}{3}$\\\\\n\\hline\n21&$d_{L}^{c3}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$&-1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$-\\frac{1}{3}$\\\\\n\\hline\n22&$d_{L}^{c3} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $&-1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$-\\frac{1}{3}$\\\\\n\\hline\n23&$ u_{L}^{c3}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$ &-1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0 &$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$\\frac{2}{3}$\\\\\n\\hline\n24&$u_{L}^{c3}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$&-1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$\\frac{2}{3}$\\\\\n\\hline\\hline\n25&$ \\nu_{R}$&$ \\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $ &1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$0$&$0$&$-\\frac{1}{2}$&$0$&$0$\\\\\n\\hline\n26&$\\nu_{R}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$ &$0$&$0$&$-\\frac{1}{2}$&$0$&$0$\\\\\n\\hline\n27&$e_{R}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$0$&$-\\frac{1}{2}$&$-1$&$-1$\\\\\n\\hline\n28&$ e_{R} $&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $&1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$0$&$-\\frac{1}{2}$&$-1$&$-1$\\\\\n\\hline\n29&$e_{L}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&-1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$0$&$-\\frac{1}{2}$&$-\\frac{1}{2}$&$-1$\\\\\n\\hline\n30&$e_{L} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $&-1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$0$&$-\\frac{1}{2}$&$-\\frac{1}{2}$&$-1$\\\\\n\\hline\n31&$ \\nu_{L}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$ &-1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0 &$0$&$0$&$-\\frac{1}{2}$&$-\\frac{1}{2}$&$0$\\\\\n\\hline\n32&$\\nu_{L}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&-1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$0$&$0$&$-\\frac{1}{2}$&$-\\frac{1}{2}$&$0$\\\\\n\\hline\\hline\n33&$ \\bar{d}_{L}^{\\bar{c1}}$&$ \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $ &-1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$\\frac{1}{3}$&$\\frac{1}{3}$\\\\\n\\hline\n34&$\\bar{d}_{L}^{\\bar{c1}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&-1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$\\frac{1}{3}$&$\\frac{1}{3}$\\\\\n\\hline\n35&$\\bar{u}_{L}^{\\bar{c1}}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&-1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$&$-\\frac{2}{3}$\\\\\n\\hline\n36&$ \\bar{u}_{L}^{\\bar{c1}} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $&-1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$&$-\\frac{2}{3}$\\\\\n\\hline\n37&$\\bar{d}_{R}^{\\bar{c1}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$\\frac{1}{3}$\\\\\n\\hline\n38&$\\bar{d}_{R}^{\\bar{c1}} $&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $&1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$\\frac{1}{3}$\\\\\n\\hline\n39&$ \\bar{u}_{R}^{\\bar{c1}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$ &1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0 &$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$\\\\\n\\hline\n40&$\\bar{u}_{R}^{\\bar{c1}}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$\n&1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$\\\\\n\\hline\\hline\n41&$ \\bar{d}_{L}^{\\bar{c2}}$&$ \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $\n&-1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$\\frac{1}{3}$&$\\frac{1}{3}$\\\\\n\\hline\n42&$\\bar{d}_{L}^{\\bar{c2}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$\n&-1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$\\frac{1}{3}$&$\\frac{1}{3}$\\\\\n\\hline\n43&$\\bar{u}_{L}^{\\bar{c2}}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$\n&-1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$&$-\\frac{2}{3}$\\\\\n\\hline\n44&$ \\bar{u}_{L}^{\\bar{c2}} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $\n&-1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$&$-\\frac{2}{3}$\\\\\n\\hline\n45&$\\bar{d}_{R}^{\\bar{c2}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$\n&1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$\\frac{1}{3}$\\\\\n\\hline\n46&$\\bar{d}_{R}^{\\bar{c2}} $&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $\n&1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$\\frac{1}{3}$\\\\\n\\hline\n47&$ \\bar{u}_{R}^{\\bar{c2}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$\n &1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0 &$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$\\\\\n\\hline\n48&$\\bar{u}_{R}^{\\bar{c2}}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$\n&1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$\\\\\n\\hline\\hline\n49&$ \\bar{d}_{L}^{\\bar{c3}}$&$ \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $ &-1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$\\frac{1}{3}$&$\\frac{1}{3}$\\\\\n\\hline\n50&$\\bar{d}_{L}^{\\bar{c3}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&-1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$\\frac{1}{3}$&$\\frac{1}{3}$\\\\\n\\hline\n51&$\\bar{u}_{L}^{\\bar{c3}}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&-1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$&$-\\frac{2}{3}$\\\\\n\\hline\n52&$ \\bar{u}_{L}^{\\bar{c3}} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&-1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$&$-\\frac{2}{3}$\\\\\n\\hline\n53&$\\bar{d}_{R}^{\\bar{c3}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$\\frac{1}{3}$\\\\\n\\hline\n54&$\\bar{d}_{R}^{\\bar{c3}} $&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$\\frac{1}{3}$\\\\\n\\hline\n55&$ \\bar{u}_{R}^{\\bar{c3}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $ &1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0 &$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$\\\\\n\\hline\n56&$\\bar{u}_{R}^{\\bar{c3}}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$\\\\\n\\hline\\hline\n57&$ \\bar{e}_{L}$&$ \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $ &-1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$0$&$0$&$\\frac{1}{2}$&$1$&$1$\\\\\n\\hline\n58&$\\bar{e}_{L}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&-1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$ &$0$&$0$&$\\frac{1}{2}$&$1$&$1$\\\\\n\\hline\n59&$\\bar{\\nu}_{L}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&-1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$0$&$\\frac{1}{2}$&$0$&$0$\\\\\n\\hline\n60&$ \\bar{\\nu}_{L} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $&-1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$0$&$\\frac{1}{2}$&$0$&$0$\\\\\n\\hline\n61&$\\bar{\\nu}_{R}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$0$&$\\frac{1}{2}$&$\\frac{1}{2}$&$0$\\\\\n\\hline\n62&$\\bar{\\nu}_{R} $&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $&1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$0$&$\\frac{1}{2}$&$\\frac{1}{2}$&$0$\\\\\n\\hline\n63&$ \\bar{e}_{R}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$ &1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0 &$0$&$0$&$\\frac{1}{2}$&$\\frac{1}{2}$&$1$\\\\\n\\hline\n64&$\\bar{e}_{R}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$0$&$0$&$\\frac{1}{2}$&$\\frac{1}{2}$&$1$\\\\\n\\hline\n\\end{supertabular}\n}\n\\end{center}\n\n\n\n\nFamilies are generated by $\\tilde{S}^{ab}$ applying on any one of the \"family members\". \nAgain all the \"family members\" of this \"family\" follow by the application of all $S^{ab}$ (not \nbelonging to Cartan subalgebra). \n\nThe spontaneous break of symmetry from \n$SO(13,1)$ to $SO(7,1) \\times SU(3) \\times U(1)$, Refs.~\\cite{IARD2016,n2014matterantimatter,%\nnormaJMP2015}, makes in the {\\it spin-charge-family} theory all the families, generated by \n$\\tilde{S}^{mt}$ and $\\tilde{S}^{st}$,\n[$m=(0,1,2,3)$, $s=(5,6,7,8), t=(9,10,11,12,13,14)$], massive of the scale \nof $\\ge 10^{16}$ GeV~\\cite{DHN,DN012,%\nfamiliesDNproc}. Correspondingly there are only eight families of\nquarks and leptons, \nwhich split into two groups of four families, both manifesting the symmetry $\\widetilde{SU}(2)\n\\times \\widetilde{SU}(2)$ $\\times U(1)$. (The fourth of the lower four families is predicted to be \nobserved at the LHC, the stable of the upper four families\ncontributes to the dark \nmatter~\\cite{gn2009}.)\n\nIn the {\\it spin-charge-family} theory fermions interact with only gravity, which manifests\nafter the break of the starting symmetry in $d=(3+1)$ as all the known vector gauge fields, \nordinary gravity and the higgs and the Yukawa couplings~\\cite{nd2017,IARD2016,%\nn2014matterantimatter,normaJMP2015,n2012scalars}. There are scalar fields which bring masses \nto family members. The theory explains not only all the assumptions of the {\\it standard model}\nwith the appearance of families, the vector gauge fields and the scalar fields, it also explains \nappearance of the dark matter~\\cite{gn2009}, \nmatter\/antimatter asymmetry~\\cite{n2014matterantimatter} and other phenomena, like the\nmiraculous cancellation of the triangle anomalies in the {\\it standard model}~\\cite{nh2017}. \n\n\n{\\bf b.} $\\;\\;$ We compare representations of $SO(13,1)$ in Clifford space with those in \nGrassmann space. We have {\\bf no \"family\" quantum numbers in Grassmann space}. \nWe only have two groups of creation operators, defining --- when applied on the vacuum state\n $|1>$ --- $\\frac{1}{2}$ $\\frac{d!}{\\frac{d}{2}! \\frac{d}{2}!}$ equal in $d=(13+1)$ to $1716$\nmembers in each of the two groups in comparison in Clifford case with $64$ \"family \nmembers\" in one \"family\" and $64$ \"families\", which the breaks of symmetry reduce to \n$8$ \"families\", making all the $(64 - 8)$ \"families\" massive and correspondingly not observable at low \nenergies~(\\cite{normaJMP2015,DHN} and the references therein).\n\n \nSince the $1716$ members are hard to be mastered, let us look therefore at each subgroup ---\n$SU(3) \\times U(1)$, $SO(3,1)$ and $SU(2)\\times SU(2)$ of $SO(13,1)$ --- separately. \n\n\nLet us correspondingly analyze the subgroups: $SO(6)$ from the point of view of\n the two subgroups $SU(3)\\times U(1)$, and $SO(7,1)$ from the point of view of\n the two subgroups $SO(3,1) \\times SO(4)$, and let us also analyze $SO(4)$ as \n$SU(2)\\times$ $SU(2)$.\n\n\n\n\n\n\\subsection{Examples of second quantizable states in Grassmann and in Clifford space} \n\\label{examples}\n\nWe compare properties of representations in Grassmann and in Clifford space for several\nchoices of subgroups of $SO(13,1)$ in the case \nthat in both spaces creation and annihilation operators fulfill requirements of\n Eq.~(\\ref{ijthetaprod}), that is that both kinds of states can be second quantized.\nLet us again point out that in Grassmann case fermions carry integer spins, while in Clifford\ncase they carry half integer spin.\n\n\\subsubsection{ States in Grassmann and in Clifford space for $d=(5+1)$}\n\\label{examples51}\n\nWe study properties of representations of the subgroup $SO(5,1)$ (of the group $SO(13,1)$),\nin Clifford and in Grassmann space, requiring that states can be in both spaces second quantized, \nfulfilling therefore Eq.~(\\ref{ijthetaprod}).\n\n{\\bf a. }\nIn Clifford space there are $2^{\\frac{d}{2}-1}$, each with $2^{\\frac{d}{2}-1}$ family \nmembers, that is $4$ families, each with $4$ members. All these sixteen states are of \nan odd Clifford character, since all can be obtained by products of $S^{ab}$, \n$\\tilde{S}^{ab}$\nor both from a Clifford odd starting state and are correspondingly second quantizable\nas required in Eq.~(\\ref{ijthetaprod}). All the states are the eigenstates of the Cartan \nsubalgebra of the Lorentz algebra in Clifford space, Eq.~(\\ref{choicecartan}), solving the Weyl\nequation for free massless spinors in Clifford space, Eq.~(\\ref{Weyl}). \n The four familes, with four members each, are presented in\nTable~\\ref{Table clifffourxfour51.}. All of these $16$ states are reachable from the first one\nin each of the four families by $S^{ab}$, or by $\\tilde{S}^{ab}$ if applied on any family member.\n\nEach of these four families have positive and negative energy solutions, as presented in~%\n\\cite{nhds}, in Table I.. We present in Table~\\ref{Table clifffourxfour51.} only states of a\npositive energy, that is states above the Dirac sea. The antiparticle states are reachable from\nthe particle states by the application of the operator $\\mathbb{C}_{{\\cal N}}\\,\n $$ {\\cal P}^{(d-1)}_{{\\cal N}} = \\gamma^0 \\gamma^5 I_{\\vec{x}_3} I_{x^6}$, keeping \nthe spin $\\frac{1}{2}$, while changing the charge from $\\frac{1}{2}$ to $- \\frac{1}{2}$. \nAll the states above the Dirac sea are indeed the hole in the Dirac sea, as explained in\nRef.~\\cite{nhds}.\n\\begin{table}\n\\begin{tiny}\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|c|}\n \\hline\\hline\n $$ & $\\psi$ & $S^{03}$ & $S^{12}$ & $S^{56 }$ \n & $\\Tilde{S}^{03}$ & $\\Tilde{S}^{12}$ & $\\Tilde{S}^{56 }$\\\\\n \\hline\\hline\n $\\psi^{ I}_{1}$ & $\\stackrel{03}{(+i)} \\stackrel{12}{(+)} \\stackrel{56}{(+)} $\n & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$\n & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$\\\\\n $\\psi^{ I}_{2}$ &$\\stackrel{03}{[-i]}\\stackrel{12}{[-]} \\stackrel{56}{(+)}$\n & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$ & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ \\\\\n $\\psi^{ I}_{3}$ &$\\stackrel{03}{[-i]} \\stackrel{12}{(+)} \\stackrel{56}{[-]} $\n & $- \\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$\n & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ \\\\\n $\\psi^{ I}_{4}$ &$\\stackrel{03}{(+i)} \\stackrel{12}{[-]} \\stackrel{56}{[-]}$\n & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$ & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ \\\\\n \\hline \n $\\psi^{II}_{1}$ & $\\stackrel{03}{[+i]} \\stackrel{12}{[+]} \\stackrel{56}{(+)}$ \n & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$\n & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$\\\\\n $\\psi^{II}_{2}$ &$\\stackrel{03}{(-i)} \\stackrel{12}{(-)} \\stackrel{56}{(+)} $\n & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$ & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$ \\\\\n $\\psi^{II}_{3}$ & $\\stackrel{03}{(-i)} \\stackrel{12}{[+]} \\stackrel{56}{[-]}$\n & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$ \\\\\n $\\psi^{II}_{4}$ & $\\stackrel{03}{[+i]}\\stackrel{12}{(-)} \\stackrel{56}{[-]}$\n & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$ & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$ \\\\\n \\hline\n $\\psi^{III}_{1}$ & $\\stackrel{03}{[+i]} \\stackrel{12}{(+)} \\stackrel{56}{[+]}$\n & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$\n & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$\\\\\n $\\psi^{III}_{2}$ & $\\stackrel{03}{(-i)} \\stackrel{12}{[-]} \\stackrel{56}{[+]}$\n & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$ & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ \\\\\n $\\psi^{III}_{3}$ & $\\stackrel{03}{(-i)} \\stackrel{12}{(+)} \\stackrel{56}{(-)} $\n & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ \\\\\n $\\psi^{III}_{4}$ & $\\stackrel{03}{[+i]} \\stackrel{12}{[-]} \\stackrel{56}{(-)} $\n & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$ & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$\\\\\n \\hline\n $\\psi^{IV}_{1}$ & $\\stackrel{03}{(+i)}\\stackrel{12}{[+]} \\stackrel{56}{[+]} $\n & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ \n & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$\\\\\n $\\psi^{IV}_{2}$ & $\\stackrel{03}{[-i]} \\stackrel{12}{(-)}\\stackrel{56}{[+]} $ \n & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$ & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$ \\\\\n $\\psi^{IV}_{3}$ & $\\stackrel{03}{[-i]} \\stackrel{12}{[+]}\\stackrel{56}{(-)} $\n & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$ & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$ \\\\\n $\\psi^{IV}_{4}$ & $\\stackrel{03}{(+i)} \\stackrel{12}{(-)} \\stackrel{56}{(-)} $ \n & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$ & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$ \\\\\n \\hline\\hline\n \\end{tabular}\n \\caption{\\label{Table clifffourxfour51.} The four families, each with four members. For the \nchoice $p^a=(p^0,0,0,p^3,0,0)$ have the first and the second member the space part \nequal to $e^{- i |p^0| x^0+i |p^3|x^3}$ and $e^{- i |p^0| x^0-i |p^3|x^3}$, \nrepresenting the particles with spin up and down, respectively. The third and the fourth member\nrepresent the antiparticle states, with the space part equal to $e^{- i |p^0| x^0-i |p^3|x^3}$\nand $e^{- i |p^0| x^0+i |p^3|x^3}$, with the spin up and down respectively.\nThe antiparticle states follow from the particle state by the application of $\\mathbb{C}_{{\\cal N}}\\,\n $$ {\\cal P}^{(d-1)}_{{\\cal N}} = \\gamma^0 \\gamma^5 I_{\\vec{x}_3} I_{x^6}$.\nThe charge of the particle states is $\\frac{1}{2}$, for antiparticle states $-\\frac{1}{2}$.}\n\\end{tiny}\n\\end{table}\n\n\n{\\bf b.0}\nIn Grassmann space there are $ \\frac{d!}{\\frac{d}{2}! \\frac{d}{2}!}$ second quantizable\nstates as required in Eq.~(\\ref{ijthetaprod}), forming in $d=(5+1)$ two decuplets --- \neach with $ \\frac{1}{2} \\,\\frac{d!}{\\frac{d}{2}! \\frac{d}{2}!}$ states --- all are the \neigenstates of the Cartan subalgebra of the Lorentz algebra in (internal) Grassmann space. \nAll the states of one (anyone of the two) decuplets are reachable by the application of the \noperators ${\\cal {\\bf S}}^{ab}$ on a starting state. The two decouplets are presented in\nTable~\\ref{Table grassdecupletso51.}\n\nLet us first find the solution of the equations of motion for free massless fermions,\n Eq.~(\\ref{Weylgrass}), with the momentum $p^a =(p^0, p^1,p^2,p^3,0,0)$. One obtains\nfor $\\psi_I = \\alpha (\\theta^0-\\theta^3) (\\theta^1 +i \\theta^2) (\\theta^5 +i \\theta^6)$\n$+ \\beta (\\theta^0 \\theta^3 + i\\theta^1\\theta^2) (\\theta^5 +i \\theta^6) + $ \n$\\gamma (\\theta^0+\\theta^3) (\\theta^1 - i \\theta^2) (\\theta^5 +i \\theta^6)$ the solution\n\\begin{eqnarray}\n\\label{5+1I}\n\\beta&=& \\frac{2\\gamma (p^1-i p^2)}{(p^0 -p^3)}=\n\\frac{2\\gamma (p^0+ p^3)}{(p^1 +i p^2)}=- \\frac{2\\alpha (p^0- p^3)}{(p^1 -i p^2)}\n= - \\frac{2\\alpha (p^1+i p^2)}{(p^0 +p^3)}\\,,\\nonumber\\\\\n(p^0)^2 &=& (p^1)^2 + (p^2)^2 +(p^3)^2\\,, \\nonumber\\\\\n\\frac{\\beta}{- \\alpha } &=& \\frac{2 (p^0- p^3)}{(p^1 -i p^2)}\\,, \\quad\n\\frac{\\gamma}{- \\alpha } =\\frac{(p^0- p^3)^2}{(p^1 -i p^2)^2}\\,.\n\\end{eqnarray}\n\nOne has for $p^0= |p^0|$ the positive energy solution, describing a fermion above the \"Dirac sea\", \nand for $p^0=- |p^0|$ the negative energy solution, describing a fermion in the \"Dirac sea\".\nThe \"charge\" of the \"fermion\" is $1$. \nSimilarly one finds the solution for the other three states with the negative \"charge\" $-1$, again\nwith the positive and negative energy.\nThe space part of the \"fermion\" state is for \"spin up\" equal to \n$e^{- i |p^0| x^0+i \\vec{p}\\vec{x}}$, for his antiparticle for the same internal spin \n$e^{- i |p^0| x^0 - i \\vec{p}\\vec{x}}$. \n\nThe discrete symmetry operator ${\\bf \\mathbb{C}}_{NG}$ ${\\cal P}^{(d-1)}_{NG}$, which is \nin our case equal to $\\gamma^0_{G} \\gamma^5_{G} I_{\\vec{x}_{3}} I_{x^6}$, transforms\nthe first state in Table~\\ref{Table grassdecupletso51.} into the sixth, the second state\ninto the fifth, the third state into the fourth, keeping the same spin while changing the \"charge\" of\nthe superposition of the three states $\\psi_{Ip}$.\nBoth superposition of states, Eq.~(\\ref{5+1I}) represent the positive energy states put on \nthe top of the \"Dirac\" sea, the first \ndescribing a particle with \"charge\" $1$ and the second superposition of the second three states \n$\\psi_{Ia}$, describing the antiparticle with the\"charge\" $-1$. \nWe namely apply\n ${\\underline {\\bf \\mathbb{C}}}_{{{\\bf \\cal NG}}}\\, $$ {\\cal P}^{(d-1)}_{{\\cal NG}}$\n on ${\\underline {\\bf {\\Huge \\Psi}}}^{\\dagger}_{p}[\\Psi^{pos}_{I}]$ by applying \n $\\mathbb{C}_{{\\cal NG}}\\, $$ {\\cal P}^{(d-1)}_{{\\cal NG}}$ on $ \\Psi^{pos}_{I}$ as follows: \n ${\\underline {\\bf \\mathbb{C}}}_{{{\\bf \\cal NG}}}\\, $$ {\\cal P}^{(d-1)}_{{\\cal NG}}$ \n ${\\underline {\\bf {\\Huge \\Psi}}}^{\\dagger}_{p}\n \\Psi^{pos}_{I}]$ $({\\underline {\\bf \\mathbb{C}}}_{{{\\bf \\cal NG}}}\\, $\n$ {\\cal P}^{(d-1)}_{{\\cal NG}})^{-1} =$ \n ${\\underline {\\bf {\\Huge \\Psi}}}^{\\dagger}_{aNG}$$[\\mathbb{C}_{{\\cal NG}}$\n ${\\cal P}^{(d-1)}_{{\\cal NG}}\\Psi^{pos}_{1}]$.\n\n One recognizes that it is $\\mathbb{C}_{{\\cal NG}}\\, $$ {\\cal P}^{(d-1)}_{{\\cal NG}}$ \n$ \\Psi^{pos}_{I}=$\n $\\Psi^{pos}_{II}$ (Table~\\ref{Table grassdecupletso51.}), \n which must be put on the top of the \"Dirac\" sea, representing the hole in the particular state \n in the \"Dirac\" sea, which solves the corresponding equation of motion for the negative energy.\n\n\n\n\n \\begin{table}\n \\begin{center}\n\\includegraphics{TabDecupletsGrasspdf-fig.pdf}\n \\end{center}\n \\caption{\\label{Table grassdecupletso51.} The creation operators of the decuplet and the \nantidecuplet of the orthogonal group $SO(5,1)$ in Grassmann space are presented. \n Applying on the vacuum state $|\\phi_{0}> = |1>$ the creation operators form eigenstates of\nthe Cartan subalgebra, Eq.~(\\ref{choicecartan}), (${\\cal {\\bf S}}^{0 3}, {\\cal {\\bf S}}^{1 2}$, \n${\\cal {\\bf S}}^{5 6}$). The states within each decuplet are reachable from any \nmember by ${\\cal {\\bf S}}^{ab}$. The product of the discrete operators\n${\\bf \\mathbb{C}}_{NG}$ ($=\\prod_{\\Re \\gamma^s} \\, \\gamma^s_{G}\\, I_{x^6 x^8...x^d}$, \ndenoted as ${\\bf \\mathbb{C}}$ in the last column) \n${\\cal P}^{(d-1)}_{NG}$ ($ = \\gamma^0_{G} \\, \\prod_{s=5}^{d}\\, \\gamma^s_{ G}\n I_{\\vec{x}_{3}}$)\ntransforms, for example, $\\psi^{I}_{1}$ into $\\psi^{I}_{6}$, $\\psi^{I}_{2}$ into $\\psi^{I}_{5}$ \nand $\\psi^{I}_{3}$ into $\\psi^{I}_{4}$. Solutions of the Weyl equation, Eq.~(\\ref{Weylgrass}), \nwith the negative energies belong to the \"Grassmann sea\", with the positive energy to the particles \nand antiparticles.\nAlso the application of the discrete operators ${\\cal C}_{GN}$, Eq.~(\\ref{calCPTNG}) and \n${\\cal C}_{NG}$ ${\\cal P}^{(d-1)}_{NG}$, Eq.~(\\ref{calCPTNG}) is demonstrated.\n}\n \\end{table}\n\n\n\\subsubsection{Properties of $SO(6)$ in Grassmann and in Clifford space when $SO(6)$ is embedded\ninto $SO(13,1)$}\n\\label{so6}\n{\\bf a.}\nLet us first repeat properties of the $SO(6)$ part of the $SO(13,1)$ representation of $64$ \n\"family members\" in Clifford space, presented in Table~\\ref{Table so13+1.}. As seen in\nTable~\\ref{Table so13+1.} there are one quadruplet ($2^{\\frac{d}{2}-1}=4$) --- \n($\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $, \n$\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}$,\n$\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $,\n$\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$), representing\nquarks and leptons --- and one antiquadruplet --- \n($\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $, \n$\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$,\n$\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $,\n$\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$), representing\nantiquarks and antileptons, which both belong to the $64^{th}$-plet, if $SO(6)$ is embedded into \n$SO(13,1)$. The creation operators (and correspondingly their annihilation operators) have \nfor $32$ members (representing quarks and leptons) the $SO(6)$ part of an odd Clifford \ncharacter (and can be correspondingly second quantized (by itselves~\\cite{nh2018} or) together\nwith the rest of space, manifesting $SO(7,1)$ (since it has an even Clifford character). The rest \nof $32$ creation operators \n(representing antiquarks and antileptons) has in the $SO(6)$ part an even Clifford character and \ncorrespondingly in the rest of the Clifford space in $SO(7,1)$ an odd Clifford character.\n\n\nLet us discuss the case with the quadruplet of $SO(6)$ with an odd Clifford character. From the \npoint of view \nof the subgroups $SU(3)$ (the colour subgroup) and $U(1)$ (the $U(1)$ subgroup carrying the\n \"fermion\" quantum number), the quadruplet consists of one $SU(3)$ singlet with the \n\"fermion\" quantum number $-\\frac{1}{2}$ and one triplet with the \"fermion\" quantum number \n$\\frac{1}{6}$. The Clifford even $SO(7,1)$ part of $SO(13,1)$ define together with the Clifford \nodd $SO(6)$ part the quantum numbers of the right handed \nquarks and leptons and of the left handed quarks and leptons of the {\\it standard model}, the\nleft handed weak charged and the right handed weak chargeless. \n\n In the same representation of $SO(13,1)$ there is also one antiquadruplet, which has the even \nClifford character of $SO(6)$ part and the odd Clifford character in the $SO(7,1)$ part of the \n$SO(13,1)$. The antiquadruplet of the $SO(6)$ part consists of one \n$SU(3)$ antisinglet with the \"fermion\" quantum number $\\frac{1}{2}$ and one antitriplet \n with the \"fermion\" quantum number $-\\frac{1}{6}$. The $SO(7,1) \\times SO(6)$ antiquadruplet\n of $SO(13,1)$ carries quantum numbers of left handed weak chargeless antiquarks and \nantileptons and of the right handed weak charged antiquarks and antileptons of the \n{\\it standard model}. \n\nBoth, quarks and leptons and antiquarks and antileptons, belong to the same representation of \n$SO(13,1)$,\nexplaining the miraculous cancellation of the triangle anomalies in the {\\it standard model}\nwithout connecting by hand the handedness and the charges of quarks and leptons~\\cite{nh2017},\nas it must be done in the $SO(10)$ models. \n\n{\\bf b.}\nIn Grassmann space there are one ($\\frac{1}{2}\\,\\frac{d!}{\\frac{d}{2}! \\frac{d}{2}!}= 10$)\ndecuplet representation of $SO(6)$ and one antidecuplet, both presented in \nTable~\\ref{Table grassdecuplet.}. To be able to second quantize the theory,\nthe whole representation must be Grassmann odd. Both decuplets in \nTable~\\ref{Table grassdecuplet.} have an odd Grassmann character, what means that products\nof eigenstates of the Cartan subalgebra in the rest of Grassmann space must be of an \nGrassmann even character to be second quantizable. Both decuplets would, however, appear in \nthe same representation of $SO(13,1)$, and one can expect also decuplets of an even\nGrassmann character, if $SO(6)$ is embedded into $SO(13,1)$%\n~\\footnote{This can easily be understood, if we look at the subgroups of the group $SO(6)$. \n{\\bf i.} $\\,$ Let us look at the subgroup $SO(2)$. There are two creation operators of an odd \nGrassmann character, in this case $(\\theta^{9} - i \\theta^{10})$ and $(\\theta^{9} +\n i \\theta^{10})$. Both appear in either decuplet or in antidecuplet --- together with \n$\\theta^{9} \\theta^{10}$ with an even Grassmann character ---\nmultiplied by the part appearing from the rest of space $d=(11,12,13,14)$. But if $SO(2)$ is \nnot embedded in $SO(6)$, then the two states, corresponding to the creation operators, \n$(\\theta^{9} \\mp i \\theta^{10})$, belong to different representations, and so is \n$\\theta^{9} \\theta^{10}$.\n{\\bf ii.} $\\,$ Similarly we see, if we consider the subgroup $SO(4)$ of the group \n$SO(6)$. All six states, $(\\theta^{9} + i \\theta^{10})\n\\cdot (\\theta^{11} + i \\theta^{12})$, $(\\theta^{9} - i \\theta^{10})\n\\cdot (\\theta^{11} - i \\theta^{12})$, $(\\theta^{9} \\theta^{10} + \\theta^{11} \\theta^{12})$,\n $(\\theta^{9} + i \\theta^{10})\\cdot (\\theta^{11} - i \\theta^{12})$, \n$(\\theta^{9} - i \\theta^{10})\\cdot (\\theta^{11} + i \\theta^{12})$, $(\\theta^{9} \\theta^{10} - \n\\theta^{11} \\theta^{12})$, appear in the \ndecuplet and in the antidecuplet, multiplied with the part appearing from the rest of space, in this \ncase in $d=(13,14)$, if $SO(4)$ is embedded in $SO(6)$. But, in $d=4$ \nspace there are two decoupled groups of three states~\\cite{norma93}: \n[$(\\theta^{9} + i \\theta^{10})\\cdot (\\theta^{11} + i \\theta^{12})$, $(\\theta^{9} \\theta^{10}\n + \\theta^{11} \\theta^{12})$, $(\\theta^{9} - i \\theta^{10}) \\cdot\n (\\theta^{11} - i \\theta^{12})$] and [$(\\theta^{9} - i \\theta^{10}) \\cdot (\\theta^{11} + \ni \\theta^{12})$, $(\\theta^{9} \\theta^{10} - \\theta^{11} \\theta^{12})$, $(\\theta^{9} + i \n\\theta^{10}) \\cdot (\\theta^{11} - i \\theta^{12})$]. Neither of these six members could be \nsecond quantized in $d=4$ alone.}.\n\nWith respect to $SU(3)\\times U(1)$ subgroups of the group $SO(6)$ the decuplet manifests as one \nsinglet, one triplet and one sextet, while the antidecuplet manifests as one antisinglet, one \nantitriplet and one antisextet. All the corresponding quantum numbers of either the Cartan \nsubalgebra operators or of the corresponding diagonal operators of the $SU(3)$ or $U(1)$ subgroups\nare presented in Table~\\ref{Table grassdecuplet.}. \n \\begin{table}\n \\begin{center}\n\\begin{tiny}\n \\begin{tabular}{|c|r|r|r|r|r|r|r|r|}\n \\hline\n$I$& &$\\rm{decuplet}$&${\\cal {\\bf S}}^{9\\,10}$&${\\cal {\\bf S}}^{11\\,12}$&\n${\\cal {\\bf S}}^{13\\,14}$&${\\bf \\tau^{4}}$&${\\bf \\tau^{33}}$& ${\\bf \\tau^{38}}$\\\\\n \\hline \n& $1$ & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$1$&$1$&$1$&$-1$&$0$&$0$\\\\\n\\hline\n&$2$ & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11}\\theta^{12} +\n \\theta^{13} \\theta^{14})$ &$1$&$0$&$0$&$-\\frac{1}{3}$&$+\\frac{1}{2}$&$+\n\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$3$ & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$1$&$-1$&$-1$&$ +\\frac{1}{3}$&$+ 1$&$+\n\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n& $4$ & $ (\\theta^{9} \\theta^{10} + \\theta^{11} \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$0$&$0$&$1$&$-\\frac{1}{3}$&$0$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$5$ & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$-1$&$-1$&$-1$&$ +\\frac{1}{3}$&$0$&$-\\frac{2}{\\sqrt{3}}$\\\\\n\\hline\n&$6$ & $ (\\theta^{11} + i \\theta^{12}) (\\theta^{9}\\theta^{10}\n+ \\theta^{13} \\theta^{14})$ &$0$&$1$&$0$&$-\\frac{1}{3}$&$-\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$7$ & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$-1$&$1$&$-1$&$ +\\frac{1}{3}$&$- 1$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$8$ & $ (\\theta^{9} \\theta^{10} - \\theta^{11} \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$0$&$0$&$-1$&$+\\frac{1}{3}$&$0$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n& $9$ & $ (\\theta^{9} \\theta^{10} - \\theta^{13} \\theta^{14})\n (\\theta^{11} - i \\theta^{12})$ &$0$&$-1$&$0$&$+\\frac{1}{3}$&$+\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$10$ & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11}\\theta^{12} -\n \\theta^{13} \\theta^{14})$ &$-1$&$0$&$0$&$+\\frac{1}{3}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline \\hline \n $II$& &$\\rm{decuplet}$&${\\cal {\\bf S}}^{9\\,10}$&${\\cal {\\bf S}}^{11\\,12}$&\n${\\cal {\\bf S}}^{13\\,14}$&${\\bf \\tau^{4}}$&${\\bf \\tau^{33}}$& ${\\bf \\tau^{38}}$\\\\\n\\hline\n& $1$ & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$-1$&$-1$&$-1$&$+1$&$0$&$0$\\\\\n\\hline\n&$2$ & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11}\\theta^{12} +\n \\theta^{13} \\theta^{14})$ &$-1$&$0$&$0$&$+\\frac{1}{3}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$3$ & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$-1$&$1$&$1$&$ -\\frac{1}{3}$&$- 1$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n& $4$ & $ (\\theta^{9} \\theta^{10} + \\theta^{11} \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$0$&$0$&$-1$&$+\\frac{1}{3}$&$0$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$5$ & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$1$&$1$&$-1$&$ -\\frac{1}{3}$&$0$&$+\\frac{2}{\\sqrt{3}}$\\\\\n\\hline\n&$6$ & $ (\\theta^{11} - i \\theta^{12}) (\\theta^{9}\\theta^{10} +\n \\theta^{13} \\theta^{14})$ &$0$&$-1$&$0$&$+\\frac{1}{3}$&$+\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$7$ & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$1$&$-1$&$1$&$ -\\frac{1}{3}$&$+ 1$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$8$ & $ (\\theta^{9} \\theta^{10} - \\theta^{11} \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$0$&$0$&$1$&$-\\frac{1}{3}$&$0$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n& $9$ & $ (\\theta^{9} \\theta^{10} - \\theta^{13} \\theta^{14})\n (\\theta^{11} + i \\theta^{12})$ &$0$&$1$&$0$&$-\\frac{1}{3}$&$-\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$10$ & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11}\\theta^{12} -\n \\theta^{13} \\theta^{14})$ &$1$&$0$&$0$&$-\\frac{1}{3}$&$+\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n \\hline\n \\end{tabular}\n\\end{tiny}\n \\end{center}\n \\caption{\\label{Table grassdecuplet.} The creation operators of the decuplet and the \nantidecuplet of the orthogonal group $SO(6)$ in Grassmann space are presented. Applying on the vacuum state $|\\phi_{0}> = |1>$ the creation operators form eigenstates of the Cartan subalgebra, \nEq.~(\\ref{choicecartan}), (${\\cal {\\bf S}}^{9\\,10}, {\\cal {\\bf S}}^{11\\,12}$, \n${\\cal {\\bf S}}^{13\\,14}$). The states within each decouplet are reachable from any \nmember by ${\\cal {\\bf S}}^{ab}$. The quantum numbers (${\\bf \\tau^{33}}, {\\bf \\tau^{38}}$) and \n${\\bf \\tau^{4}}$ of \nthe subgroups $SU(3)$ and $U(1)$ of the group $SO(6)$ are also presented, Eq.~(\\ref{so64}).\n}\n \\end{table}\n\nWhile in Clifford case the representations of $SO(6)$, if the group $SO(6)$ is embedded into \n$SO(13,1)$, are defining a Clifford odd quadruplet and a Clifford even antiquadruplet, the \nrepresentations in Grassmann case \ndefine one decuplet and one antidecuplet, both of the same Grassmann character, the odd one in \nour case. The two quadruplets in Clifford case manifest \nwith respect to the subgroups $SU(3)$ and $U(1)$ as a triplet and a singlet, and as an antitriplet\nand an antisinglet, respectively. In Grassmann case the two decuplets manifest with \nrespect to the subgroups $SU(3)$ and $U(1)$ as a (triplet, singlet, sextet) and as an (antitriplet,\nantisinglet, antisextet), respectively. The corresponding multiplets are presented in \nTable~\\ref{Table grasssextet.}. \n \\begin{table}\n \\begin{center}\n\\begin{tiny}\n \\begin{tabular}{|c|r|r|r|r|r|}\n \\hline \\hline\n$I$& &&${\\bf \\tau^{4}}$&${\\bf \\tau^{33}}$& ${\\bf \\tau^{38}}$\\\\\n \\hline \n$\\rm{singlet}$&$ $ & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$-1$&$0$&$0$\\\\\n\\hline\\hline\n$\\rm{triplet}$&$1$ & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11}\\theta^{12} +\n \\theta^{13} \\theta^{14}) $ &$-\\frac{1}{3}$&$+\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$2$ & $ (\\theta^{9} \\theta^{10} + \\theta^{11} \\theta^{12})\n (\\theta^{13} + i \\theta^{14}) $ &$-\\frac{1}{3}$&$0$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$3$ & $ (\\theta^{11} + i \\theta^{12}) (\\theta^{9}\\theta^{10}\n+ \\theta^{13} \\theta^{14}) $ &$-\\frac{1}{3}$&$-\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline \\hline\n$\\rm{sextet}$&$1$ & $(\\theta^{9}+ i \\theta^{10}) (\\theta^{11} - i \\theta^{12}) \n(\\theta^{13} - i \\theta^{14})\n$ &$\\frac{1}{3}$&$+1$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$2$ & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} + i \\theta^{14}) $ &$\\frac{1}{3}$&$0$&$-\\frac{2}{\\sqrt{3}}$\\\\\n\\hline\n&$3$ & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} - i \\theta^{14}) $ &$\\frac{1}{3}$&$-1$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$4$ & $ (\\theta^{9} \\theta^{10} - \\theta^{11} \\theta^{12})\n (\\theta^{13} - i \\theta^{14}) $ &$\\frac{1}{3}$&$0$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$5$ & $(\\theta^{9} \\theta^{10} - \\theta^{13} \\theta^{14})\n (\\theta^{11} - i \\theta^{12}) $ &$\\frac{1}{3}$&$+\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$6$ & $(\\theta^{9} - i \\theta^{10}) (\\theta^{11}\\theta^{12} -\n \\theta^{13} \\theta^{14}) $ &$\\frac{1}{3}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\\hline\n$II$&&&${\\bf \\tau^{4}}$&${\\bf \\tau^{33}}$& ${\\bf \\tau^{38}}$\\\\\n \\hline \n$\\rm{antisinglet}$&$ $ & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$+1$&$0$&$0$\\\\\n\\hline\n\\hline\n$\\rm{antitriplet}$&$1$ & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11}\\theta^{12} +\n \\theta^{13} \\theta^{14})$ &$+\\frac{1}{3} $ &$-\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$2$ & $ (\\theta^{9} \\theta^{10} + \\theta^{11} \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$+\\frac{1}{3}$&$0$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$3$ & $(\\theta^{11} - i \\theta^{12}) (\\theta^{9}\\theta^{10} +\n \\theta^{13} \\theta^{14}) $ &$+\\frac{1}{3}$&$+\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline \\hline\n$\\rm{antisextet}$&$1$ & $(\\theta^{9} - i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$ -\\frac{1}{3} $ &$-1$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$2$ & $(\\theta^{9} + i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} - i \\theta^{14}) $ &$-\\frac{1}{3}$&$0$&$+\\frac{2}{\\sqrt{3}}$\\\\\n\\hline\n&$3$ & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} + i \\theta^{14}) $ &$-\\frac{1}{3}$&$+1$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$4$ & $ (\\theta^{9} \\theta^{10} - \\theta^{11} \\theta^{12})\n (\\theta^{13} + i \\theta^{14}) $ &$-\\frac{1}{3}$&$0$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$5$ & $ (\\theta^{9} \\theta^{10} - \\theta^{13} \\theta^{14})\n (\\theta^{11} + i \\theta^{12}) $ &$-\\frac{1}{3}$&$-\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$6$ & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11}\\theta^{12} -\n \\theta^{13} \\theta^{14}) $ &$-\\frac{1}{3}$&$+\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n \\end{tabular}\n\\end{tiny}\n \\end{center}\n \\caption{\\label{Table grasssextet.} The creation operators in Grassmann space of the decuplet \nof Table~\\ref{Table grassdecuplet.} are arranged with respect to the $SU(3)$ and $U(1)$ \nsubgroups of the group $SO(6)$ into a singlet, a triplet and a sextet. \nThe corresponding antidecuplet manifests as an antisinglet, an antitriplet and an antisextet. \n${\\bf \\tau^{33}}= \\frac{1}{2} ({\\cal {\\bf S}}^{9\\,10} - {\\cal {\\bf S}}^{11\\,12})$, \n${\\bf \\tau^{38}}=$ $\\frac{1}{2 \\sqrt{3}} ({\\cal {\\bf S}}^{9\\,10} + {\\cal {\\bf S}}^{11\\,12} - \n2 {\\cal {\\bf S}}^{13\\,14})$, ${\\bf \\tau^{4}}=$ - \n$\\frac{1}{3} ({\\cal {\\bf S}}^{9\\,10} + {\\cal {\\bf S}}^{11\\,12} + {\\cal {\\bf S}}^{13\\,14})$;\n${\\cal {\\bf S}}^{a b}$ $= i (\\theta^a \\frac{\\partial}{\\partial \\theta_b} - \\theta^b \n\\frac{\\partial}{\\partial \\theta_a})$. \n}\n \\end{table}\nThe \"fermion\" quantum number ${\\bf \\tau^{4}}$ has for either singlets or triplets in Grassmann \nspace, Table~\\ref{Table grasssextet.}, twice the value of the corresponding singlets and triplets\n in Clifford space, Table~\\ref{Table so13+1.}: $(-1, +1)$ in \nGrassmann case to be compared with $(- \\frac{1}{2}, +\\frac{1}{2})$ in Clifford case and \n$(+ \\frac{1}{3}, -\\frac{1}{3})$ in Grassmann \ncase to be compared with $(+ \\frac{1}{6}, -\\frac{1}{6})$ in Clifford case. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{sl1a-sing-8May.pdf} \n \\hfill\n \\includegraphics[width=0.45\\textwidth]{sl1sing-8May.pdf}\n \\caption{\\label{FigSO6} Representations of the subgroups $SU(3)$ and $U(1)$ of the group \n$SO(6)$ in Grassmann space for two Grassmann odd representations of \nTable~\\ref{Table grasssextet.} are presented. \nOn the abscissa axis and on the ordinate axis the values of the two diagonal operators, \n${\\bf \\tau^{33}}$ and ${\\bf \\tau^{38}}$ of the coulour ($SU(3)$)\nsubgroup are presented, respectively, with full circles. On the third axis the values of the \nsubgroup of the \"fermion number\" $U(1)$ is presented with the open circles, the same for all \nthe representations of each multiplet. There are one singlet, one triplet and one sextet on the \nleft hand side and one antisinglet, one antitriplet and one antisextet on the right hand side.} \n\\end{figure}\n\nWhen $SO(6)$ is embedded into $SO(13,1)$, the $SO(6)$ representations of either even or odd \nGrassmann character contribute to both of the decoupled, \n$1716$ states of $SO(13,1)$ representations\ncontribute, provided that the $SO(8)$ content has the opposite Grassmann character than the $SO(6)$\ncontent. The product of both representations must be Grassmann odd in order that the corresponding \ncreation and annihilation operators fulfill the required anticommutation relations for fermions, \nEq.~(\\ref{ijthetaprod}).\n\n\\subsubsection{Properties of the subgroups $SO(3,1)$ and $SO(4)$ of the group $SO(8)$ in \nGrassmann and in Clifford space, when $SO(8)$ is embedded into $SO(13,1)$}\n\\label{so8}\n\n{\\bf a.}\nLet us again repeat first properties of the $SO(3,1)$ and $SO(4)$ parts of the $SO(13,1)$\nrepresentation of $64$ \"family members\" in Clifford space, presented in Table~\\ref{Table so13+1.}. \nAs seen in Table~\\ref{Table so13+1.} there are four octets and four antioctets of $SO(8)$. \nAll four octets, having an even Clifford character and forming $32$ states when embedded into \n$SO(13,1)$, are the same for either\nquarks or for leptons, they distinguish only in the $SO(6)$ part (of a Clifford odd character) of the \n$SO(13,1)$ group, that is in the colour ($SU(3)$) part and the \"fermion quantum number\" \n($U(1)$) part. \nAlso the four antioctets, having an odd Clifford character, are all the same for the $32$ \nfamily members of antiquarks and antileptons, they again distinguish only in the Clifford \neven $SO(6)$ part of $SO(13,1)$, that is in the anticolour ($SU(3)$) part and the \"fermion \nquantum number\" ($U(1)$) part.\n\nThe $64^{th}$-plet of creation operators has an odd Clifford character either for quarks and \nleptons or for antiquarks and antileptons --- correspondingly have an odd \nClifford character also their annihilation operators --- and can be second quantized~\\cite{nh2018}.\n\n\nLet us analyze first the octet ($2^{\\frac{8}{2}-1}=8$), which is the same for all $32$ members of\nquarks and leptons. The octet has an even Clifford character. All the right handed $u_{R}$-quarks \nand $\\nu_{R}$-leptons have the $SO(4)$ part of $SO(8)$ equal to\n$\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}$, while their left handed partners have the $SO(4)$ part of \n$SO(8)$ equal to $\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}$.\nAll the right handed $d_{R}$-quarks and $e_{R}$-leptons have the $SO(4)$ part of $SO(8)$ \nequal to $\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}$, while their left handed partners have the $SO(4)$ \npart of $SO(8)$ equal to $\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)]}$. The left handed quarks and \nleptons are doublets with respect to $\\vec{\\tau}^{1}$ and singlets with $\\vec{\\tau}^{2}$, while \nthe right handed quarks and leptons are singlets with respect to $\\vec{\\tau}^{1}$ and doublets\n with $\\vec{\\tau}^{2}$. \nThe left and right handed quarks and lepton belong with respect to the $SO(3,1)$ group to either\nleft handed or the right handed spinor representations, respectively.\n\n{\\bf b.}\nIn Grassmann space the $SO(8)$ group of an odd Grassmann character has $\\frac{1}{2}$ \n$\\frac{8!}{4! 4!} = 35$ creation operators in each of the two groups and the same number of \nannihilation operators, obtained from the creation operators by Hermitian conjugation, \nEq.~(\\ref{grassher}). The corresponding states, created by the creation operators on the vacuum \nstate $|\\phi_{o}>$, can be therefore second quantized. But if embedded the group $SO(8)$ into\nthe group $SO(13,1)$ the subgroup $SO(6)$ must have an even Grassmann character in oder \nthat the states in $SO(13,1)$ can be second quantized according to Eq.~(\\ref{ijthetaprod}). \n\nAccording to what we learned in the case of the group $SO(6)$, each of the two independent \nrepresentations of the group $SO(13,1)$ of an odd Grassmann character must \ninclude either the even $SO(7,1)$ part and the odd $SO(6)$ part or the odd \n$SO(7,1)$ part and the even $SO(6)$ part. To the even $SO(7,1)$ representation \neither the odd $SO(3,1)$ and the odd $SO(4)$ parts contribute or both must be of the \nGrassmann even character. In the case that the $SO(7,1)$ part has an odd Grassmann character \n(in this case the $SO(6)$ has an even Grassmann character) then one of the\ntwo parts $SO(3,1)$ and $SO(4)$ must be odd and the other even.\n\n\n\\section{Concluding remarks}\n\\label{conclusions}\n\nWe learned in this contribution that although either Grassmann or Clifford space offer the second\nquantizable description of the internal degrees of freedom of fermions (Eq.~(\\ref{ijthetaprod})),\nthe Clifford space offers more: It offers not only the description of all the \"family members\", \nexplaining all the degrees of freedom of the observed quarks and leptons and antiquark and antileptons, \nbut also the explanation for the appearance of families. \n\nThe interaction of fermions with the gravity fields --- the vielbeins and the spin connections\n --- in the $2(2n+1)$-dimensional space can be achieved, as suggested by the {\\it spin-charge-family}\ntheory~(\\cite{normaJMP2015,n2014matterantimatter} and references therein), by replacing the \nmomentum $p_{a}$ in the Lagrange density function for a free particle by the covariant momentum, \nequally appropriate for both representations. In Grassmann space we have: $p_{0a}= f^{\\alpha}{}_a$\n $p_{0\\alpha}$, with $p_{0\\alpha} = p_{\\alpha} - \\frac{1}{2}\\, {\\cal {\\bf S}}^{ab} \n\\Omega_{ab \\alpha}$, where $ f^{\\alpha}{}_a$ is the vielbein in $d=2(2n+1)$-dimensional space and\n$\\Omega_{ab \\alpha}$ is the spin connection field of the Lorentz generators ${\\cal {\\bf S}}^{ab}$.\nIn Clifford space we have equivalently: $p_{0a}= f^{\\alpha}{}_a$ $p_{0\\alpha}$, $p_{0\\alpha}= \n p_{\\alpha} - \\frac{1}{2} S^{ab} \\omega_{ab \\alpha} - \\frac{1}{2} \\tilde{S}^{ab} \n \\tilde{\\omega}_{ab \\alpha}$. Since ${\\cal {\\bf S}}^{ab} = S^{ab} + \\tilde{S}^{ab}$ we find that\nwhen no fermions are present either $\\Omega_{ab \\alpha}$ or $\\omega_{ab \\alpha}$ or \n$\\tilde{\\omega}_{ab \\alpha}$ are uniquely expressible by vielbeins $f^{\\alpha}{}_a$\n (\\cite{normaJMP2015,n2014matterantimatter} and references therein). It might be that\n \"our universe made a choice between the Clifford and the Grassmann algebra\" when breaking\n the starting symmetry by making condensates of fermions, since that for breaking symmetries \nClifford space offers better opportunity\".\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\label{intro}\n\nThe study of topological superconducting wires, which host Majorana zero\nmodes (MZMs) at their ends, is a field of intense research in condensed\nmatter physics, not only because of the interesting basic physics involved \n\\cite{sato}, but also because of possible applications in decoherence-free\nquantum computing.\\cite{kitaev,nayak,alicea,lobos}\n\nIn 2010, Lutchyn \\textit{et al.} \\cite{lutchyn2010} and Oreg \\textit{et al.} \n\\cite{oreg2010} proposed a model for topological superconducting wires\ndescribing a system formed by a semiconducting wire with spin-orbit coupling\n(SOC) and proximity-induced s-wave superconductivity under an applied\nmagnetic field perpendicular to the direction of the SOC. This yields a\ntopological superconducting phase with MZMs localized at its ends. The\nobservation of these MZMs in these types of wires was reported in different\nexperimental studies.\\cite{wires-exp1,wires-exp2,wires-exp3,wires-exp4}\n\nThe search for different models and mechanisms leading to topological\nsuperconducting phases continues being a very active avenue of research\ntheoretically and experimentally.\n\nMore recently, there has been experimental research as well as theoretical\nstudies in similar models, including those for time-reversal invariant\ntopological superconductors,\\cite{review-tritops,volpez} of the effects of\nMZMs in Josephson junctions, in particular because the dependence on the\napplied magnetic flux introduces an additional control knob.\\cite{volpez,zazu,pientka,hell,ren,fornie,cata,tomo}\n\nIn particular, it has been recently proposed that the current-phase relation\nmeasured in Josephson junctions may be used to find the parameters that\ndefine the MZMs.\\cite{tomo} A possible difficulty in these experiments is\nthe slow thermalization to the ground state\nin the presence of a gap.\\cite{bondy}\nA way to circumvent this problem is to rotate the magnetic field\nslowly from a direction not perpendicular to the SOC in which the system is\nin a gapless superconducting phase, in which thermalization is easier.\\cite{tomo}\nTherefore, it is convenient to know the phase diagram of the system\nand the extension of this gapless phase.\n\nIn this work we calculate the phase diagram of the lattice version of the\nmodel and discuss in particular the gapless phase. The paper is organized as\nfollows. In Sec. \\ref{model} we describe the model. The topological\ninvariants use to define the phase diagram are presented in Sec. \\ref{inv}. In Sec. \\ref{res} we show the numerical results, analytical expressions for\nthe boundaries of the topological phase and discuss briefly the Majorana zero modes.\nWe summarize the results in Sec. \\ref{sum}.\n\n\\section{Model}\n\n\\label{model}\n\nThe model for topological superconducting wires studied in this work is the\nlattice version of that introduced by Lutchyn \\textit{et al.} \n\\cite{lutchyn2010} and Oreg \\textit{et al.} \\cite{oreg2010}. The Hamiltonian can\nbe written as \\cite{tomo} \n\\begin{eqnarray}\nH &=&\\sum_{\\ell }[\\mathbf{c}_{\\ell }^{\\dagger }\\left( -t\\;\\sigma _{0}-i\\vec{\n\\lambda}\\cdot \\vec{\\sigma}\\right) \\mathbf{c}_{\\ell +1}+\\Delta c_{\\ell\n\\uparrow }^{\\dagger }c_{\\ell \\downarrow }^{\\dagger }+\\text{H.c.} \\notag \\\\\n&-&\\mathbf{c}_{\\ell }^{\\dagger }\\left( \\vec{B}\\cdot \\vec{\\sigma}+\\mu \\sigma\n_{0}\\right) \\mathbf{c}_{\\ell }],\\;\\; \\label{ham}\n\\end{eqnarray}\nwhere $\\ell $ labels the sites of a chain, $\\mathbf{c}_{\\ell }=(c_{\\ell\n\\uparrow },c_{\\ell \\downarrow })^{T}$, $t$ is the nearest-neighbor hopping, $\n\\vec{\\lambda}$ is the SOC, $\\Delta $ represents the magnitude of the\nproximity-induced superconductivity, $\\vec{B}$ is the applied magnetic field\nand $\\mu $ is the chemical potential. As usual, the components of the vector $\n\\vec{\\sigma}=\\left( \\sigma _{x},\\sigma _{y},\\sigma _{z}\\right) $ are the\nPauli matrices and $\\sigma _{0}$ is the 2$\\times $2 unitary matrix. The\npairing amplitude $\\Delta $ can be assumed real. Otherwise, the phase can be\neliminated by a gauge transformation in the operators $c_{\\ell \\sigma\n}^{\\dagger }$ that absorbs the phase.\n\nWithout loss of generality, we choose the $z$ direction as that of the\nmagnetic field ($\\vec{B}=B\\mathbf{\\hat{z}}$) and $x$ perpendicular to the\nplane defined by $\\vec{\\lambda}$ and $\\vec{B}$ ($\\vec{\\lambda}=\\lambda _{y} \n\\mathbf{\\hat{y}}+\\lambda_{z} \\mathbf{\\hat{z}}$). After Fourier\ntransformation, the Hamiltonian takes the form $H=\\sum_{k}H_{k}$, with\n\n\\begin{eqnarray}\nH_{k} &=&-(\\mu +2t\\cos (k))(c_{k\\uparrow }^{\\dagger }c_{k\\uparrow\n}+c_{k\\downarrow }^{\\dagger }c_{k\\downarrow }) \\\\\n&& -B(c_{k\\uparrow }^{\\dagger }c_{k\\uparrow }-c_{k\\downarrow }^{\\dagger\n}c_{k\\downarrow }) -2\\sin (k)\\left[ i\\lambda _{y}(c_{k\\uparrow }^{\\dagger\n}c_{k\\downarrow }-c_{k\\downarrow }^{\\dagger }c_{k\\uparrow }) \\right. \\notag\n\\\\\n&& \\left. +\\lambda _{z}(c_{k\\uparrow }^{\\dagger }c_{k\\uparrow\n}-c_{k\\downarrow }^{\\dagger }c_{k\\downarrow })\\right] +\\Delta (c_{k\\uparrow\n}^{\\dagger }c_{-k\\downarrow }^{\\dagger }+c_{-k\\downarrow }c_{k\\uparrow }). \n\\notag \\label{hk}\n\\end{eqnarray}\nUsing the four-component spinor $(c_{k\\uparrow }^{\\dagger },c_{k\\downarrow\n}^{\\dagger },c_{-k\\uparrow },c_{-k\\downarrow })$,\\cite{tewari} the\ncontribution to the Hamiltonian for wave vector $k$ can be written in the\nform\n\n\\begin{eqnarray}\nH_{k} &=&-(\\mu +2t\\cos (k))\\tau _{z}\\otimes \\sigma _{0}-B\\tau _{z}\\otimes\n\\sigma _{z}-\\Delta \\tau _{y}\\otimes \\sigma _{y} \\notag \\\\\n&&+2\\lambda _{y}\\sin (k)\\tau _{z}\\otimes \\sigma _{y}-2\\lambda _{z}\\sin\n(k)\\tau _{0}\\otimes \\sigma _{z}, \\label{hk2}\n\\end{eqnarray\nwhere the Pauli matrices $\\sigma _{\\alpha }$ act on the spin space, while\nthe $\\tau _{\\alpha }$ act on the particle-hole space. Writing the matrix\nexplicitly, $H_{k}$ takes the form\n\n\\begin{equation}\nH_{k}\n\\begin{pmatrix}\n-a-B-z & -iy & 0 & \\Delta \\\\ \niy & -a+B+z & -\\Delta & 0 \\\\ \n0 & -\\Delta & a+B-z & iy \\\\ \n\\Delta & 0 & -iy & a-B+z\n\\end{pmatrix}\n\\label{hk3}\n\\end{equation\nwhere $a=\\mu +2t\\cos (k)$, $B=|B|=B_{z}$, $y=2\\lambda _{y}\\sin (k)$ and $\nz=2\\lambda _{z}\\sin (k)$.\n\n\\section{Topological invariants}\n\n\\label{inv} In this section we define the topological invariants we use to\ncharacterize the topological phases. In general, the Hamiltonian belongs to\ntopological class D with a \n$\\mathbb{Z}_{2}$ topological invariant.\\cite{Schn,ryu} However, for perpendicular $\\vec{\\lambda}$ and $\\vec{B}$ ($z=0$),\nthe system has a chiral symmetry and belongs to the topological class BDI\nwith a $\\mathbb{Z}$ (integer) topological invariant corresponding to a\nwinding number.\\cite{tewari} In this case, the calculation of the\ntopological invariant is simpler, as shown by Tewari and Sau.\\cite{tewari} \n\nFollowing this work, we perform a rotation in $\\pi \/2$ around the $\\hat{y}$\naxis in particle-hole space, which transforms $\\tau _{z}$ to $\\tau _{x}$: $\nH_{k}^{\\prime }=UH_{k}U^{\\dagger }$ with $U=$exp$(-i\\pi \/4)\\tau _{y}$. With\nthis transformation $H_{k}^{\\prime }$ becomes\n\n\\begin{equation}\nH_{k}^{\\prime }=\n\\begin{pmatrix}\n-z & 0 & -a-B & \\Delta -iy \\\\ \n0 & z & -\\Delta +iy & -a+B \\\\ \n-a-B & -\\Delta -iy & -z & 0 \\\\ \n\\Delta +iy & -a+B & 0 & z\n\\end{pmatrix}\n\\label{hkp}\n\\end{equation}\nTaking $z=0$, this rotation yields an off-diagonal (chiral symmetric)\nHamiltonian. This allows us to define a winding number $W$ (a topological $\n\\mathbb{Z}$ invariant) from the phase of the determinant of the $2\\times 2$\nmatrix $A(k)$, which is the upper right corner of Eq. (\\ref{hkp}).\\cite{tewari}\nSpecifically $\\mathrm{Det}(A(k))$$=|\\mathrm{Det}(A(k))|$$e^{i\\theta (k)}=$$a^{2}-B^{2}-(\\Delta -iy)^{2}$, and\n\n\\begin{equation}\nW=\\frac{-i}{\\pi }\\int\\limits_{0}^{\\pi }\\frac{d(e^{i\\theta (k)})}{e^{i\\theta\n(k)}}. \\label{win}\n\\end{equation}\nIn addition, a $\\mathbb{Z}_{2}$ invariant $I$ can be defined from the\nrelative sign of $\\mathrm{Det}(A)$ (which is real for $k=0$ and $k=\\pi$)\nbetween the points $k=0$ and $k=\\pi$:\n\n\\begin{equation}\nI=(-1)^{W}=\\text{sign}\\frac{\\mathrm{Det}(A(\\pi ))}{\\mathrm{Det}(A(0))}\n\\label{z2}\n\\end{equation}\nLooking for the condition that $I\\equiv -1$ (mod 2), we obtain that the\nconditions for the system to be in the topological phase are that $\\lambda\n_{y}=|\\vec{\\lambda}|\\neq 0\\neq \\Delta $ and the remaining parameters should\nsatisfy \n\\begin{equation}\n|2|t|-r|<|\\mu |<|2|t|+r|,\\text{ \\ \\ \\ with }r=\\sqrt{B^{2}-\\Delta ^{2}}>0.\n\\label{bound}\n\\end{equation}\nWe note that changing the sign of any of the parameters does not change the\nboundary of the topological phase. This is due to the symmetry properties of\nthe Hamiltonian.\\cite{tomo}\n\nIn the more general case, when $\\vec{\\lambda}$ and $\\vec{B}$ are not\nperpendicular, it is not possible to follow the approach outlined above. In\nthis case, we use the Zak Berry phase to construct the topological\ninvariant. \\cite{zak,king,resta,ortiz,bf1,bf2,hatsu,bf3,ryu,deng,budich}\nSpecifically, the Hamiltonian $H_{k}$ has four different eigenvectors and\nfor each of them, following Zak,\\cite{zak} one can calculate a Berry phase\nfrom the Bloch functions as the wave vector $k$ varies in the loop $0\\leq\nk\\leq 2\\pi $ (with $k=2\\pi $ equivalent to $k=0$). For each eigenstate $\n|u(k)\\rangle $ of $H_{k}$, the Berry phase is \n\n\\begin{equation}\n\\gamma =-\\text{Im }\\int\\limits_{0}^{2\\pi }dk\\langle u(k)|\\frac{\\partial }{\n\\partial k}|u(k)\\rangle . \\label{gam}\n\\end{equation}\n\n\\ In addition (as noted before \\cite{tomo}) choosing a suitable coordinate\nframe ( $\\vec{\\lambda}\\cdot \\mathbf{\\hat{y}}=\\vec{B}\\cdot \\mathbf{\\hat{y}}=0$\n), the Hamiltonian Eq. (\\ref{ham}) is invariant under an antiunitary\noperator defined as the product of inversion (defined by the transformation $\n\\ell \\leftrightarrow N+1-\\ell $, for a chain with $N$ sites) and complex\nconjugation, implying that the Berry phase $\\gamma $ is quantized with only\ntwo possible values $0$ and $\\pi $ (mod $2\\pi $).\\cite{hatsu} Naturally the\nvalue of the Berry phase does not depend on the choice of the reference\nframe. Therefore, as for an insulator, if the system has a gap, the sum of\nthe Berry phases of all one-particle states of energies below the gap mod $2\\pi $, defines a \n$\\mathbb{Z}_{2}$ topological number, indicating that the system is trivial\n(topological) if this sum is equivalent to 0 ($\\pi $) mod $2\\pi .\n$\\cite{ryu,budich} Moreover, from Eq. (\\ref{hk}) it is easy to realize that the\ncharge conjugation $c_{\\ell \\sigma }^{\\dagger }\\leftrightarrow c_{\\ell\n\\sigma }$, which in Fourier space means $c_{k\\sigma }^{\\dagger }=(1\/\\sqrt{N\n)\\sum_{l}e^{-ik\\ell }c_{\\ell \\sigma }^{\\dagger }\\leftrightarrow c_{-k\\sigma }\n$, transforms $H_{k}\\leftrightarrow -H_{-k}$. Therefore the sum of the Berry\nphases of all positive eigenvalues gives the same topological number as the\nsum of all negative eigenvalues.\n\nIn our model, $H_{k}$ has four eigenvalues $E(k)$. The lowest one $E_{1}(k)$\nis always negative and the corresponding eigenvector has always a Berry\nphase 0. From the above mentioned charge-transfer symmetry, the fourth\neigenvalue (the highest one) has energy $E_{4}(k)=-E_{1}(-k)>0$. Therefore,\nthe Berry phase of the second eigenvalue (which is equal to that of the\nthird one) determines the $\\mathbb{Z}_{2}$ invariant. We have calculated the\nBerry phase $\\gamma $ of each of the four bands (and particularly the second\none) from the normalized eigenvectors $|u_{j}\\rangle =|u(k_{j})\\rangle $ of\nthe $4\\times 4$ matrix obtained numerically at $M$ wave vectors $k_{j}=2\\pi\n(j-1)\/M$, using a numerically invariant expression \\cite{ortiz,deng}. This\nexpression is derived in the following way. Discretizing Eq. (\\ref{gam}) and\napproximating $\\partial |u(k)\/\\partial k=(M\/2\\pi )(|u(k_{j+1})\\rangle\n-|u(k_{j})\\rangle )$, one obtains\n\n\\begin{equation}\n\\gamma =-\\text{Im}\\sum_{j=1}^{M}\\left[ \\langle u_{j}|\\left( |u_{j+1}\\rangle\n-|u_{j}\\rangle \\right) \\right] . \\label{gam2}\n\\end{equation}\nIf $M$ is large enough so that $k_{j}$ and $k_{j+1}$ are very close, then \n$x=\\langle u_{j}|u_{j+1}\\rangle -1$ is very small and one can retain only the\nfirst term in the Taylor series expansion ln$(1+x)=x-x^{2}\/2+...$ Replacing in Eq. (\n\\ref{gam2}) one obtains\n\n\\begin{eqnarray}\n\\gamma &=&-\\mathrm{{Im}\\left[ \\ln (P)\\right] }\\text{, where } \\notag \\\\\nP &=&\\langle u_{1}|u_{2}\\rangle \\langle u_{2}|u_{3}\\rangle ...\\langle\nu_{M-1}|u_{1}\\rangle \\label{gam4}\n\\end{eqnarray}\nIt is easy to see that Eq. (\\ref{gam}) is gauge invariant. This means that\nthe result does not change if $|u(k)\\rangle $ is replaced by $e^{i\\varphi\n(k)}|u(k)\\rangle $, where $\\varphi (k)$ is a smooth function with $\\varphi\n(2\\pi )=\\varphi (0).$ Similarly, the product $P$ is independent of the base\nchosen by the numerical algorithm to find the eigenstates $|u_{j}\\rangle .$\nTherefore Eq. (\\ref{gam4}) is numerically gauge invariant. Analyzing the\nchange in the results with increasing $M$, we find that $M\\sim 250$ is\nenough to obtain accurately all phase boundaries shown below. A further\nincrease in $M$ leads to changes that are not visible in the scale of the\nfigures.\n\nThis $\\mathbb{Z}_{2}$ topological invariant defined by the Berry phase \nof the second (or third) state can be trivially extended to the gapless case \nif the energies of the second and third state do not cross as a function of $k$.\nEven if the energies cross the Berry phases can be calculated switching \nthe states at the crossing. However, this case is not of interest here.\n\n\\section{Results}\n\n\\label{res}\n\n\\subsection{Phase diagram}\n\n\\label{phdi}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{4p_D-fase_varioB_TeX.eps}\n\\end{center}\n\\caption{Phase diagram in the $\\mu,\\Delta$ plane for perpendicular \n$\\vec{\\lambda}$ and $\\vec{B}$, $t=1$, \n$\\lambda=|\\vec{\\lambda}|=2$, and several values of $B$. \nGray region I denotes the topological sector and white region II the \nnon topological one.}\n\\label{variob}\n\\end{figure}\n\nWe start by discussing the simplest case of perpendicular $\\vec{\\lambda}$\nand $\\vec{B}$. In Fig. \\ref{variob} we display the resulting phase diagram\nfor some parameters, showing the possible different shapes. There are two\ngapped phases, the trivial (white region II) and the topological one (light\ngray I), separated in general by two circular arcs defined by Eqs. \n(\\ref{bound}). For simplicity we discuss the case $t, B > 0$. The topological character\nis independent of the sign of the different parameters. If $B<2t$, the\nregion of possible values of $|\\mu|$ inside the topological sector extends\nfrom $2t-B$ to $2t+B$ for $\\Delta \\rightarrow 0$ and shrinks for increasing \n\\Delta$ until it reduces to the point $|\\mu|=2t$ for $\\Delta \\rightarrow B$.\nIf $B=2t$, the semicircle touches the point $\\mu=0$. For larger $B$, the\nregion $|\\mu|< \\sqrt{B^{2}-\\Delta^{2}}-2t$ for $\\Delta^2 < B^{2}-4t^2$ is\nexcluded from the topological region.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{4p_fase_phiB0_beta_TeX.eps}\n\\end{center}\n\\caption{(Color online) Phase diagram in the $\\mu,\\Delta$ plane for $t=1$, \n$\\lambda=2$, $B=4$ and several values of the angle \n$\\beta_{\\lambda B}$ between $\\vec{\\lambda}$ and $\\vec{B}$. \nRegions I and II as in Fig. \\ref{variob}. Region III (IV)\nin dark gray (black) corresponds to the gapless phase\nwith Berry phase $\\pi$ (0).\nThe red points at the top left corresponds to numerical calculations\nwhich detected localized states at the ends.}\n\n\\label{variobeta}\n\\end{figure}\n\nWhile for perpendicular $\\vec{\\lambda}$ and $\\vec{B}$, the gap vanishes only\nat particular lines in the phase diagram (black lines in Fig. \\ref{variob})\nfor which the topological transition takes place, for general angles $\\beta\n_{\\lambda B}$ between both vectors, there is a finite region in the $\\mu\n,\\Delta $ plane for which the gap vanishes, in particular for $|\\Delta\n|<\\Delta _{c}$, where $\\Delta _{c}$ is a critical value, independent of $\\mu \n$, determined analytically below. Before presenting the analytical\ncalculation, we describe the general features of each phase in the phase\ndiagram, as shown in Fig. \\ref{variobeta}. The gapped regions in the figure\nare denoted by I and II. The remaining two regions are gapless. We separate them by the trivial (topological) character of the Berry phases of the second and third eigenstate, indicating the\ncorresponding regions with black (dark gray) color and roman number IV (III). \nIn spite of the topological Berry phases of the latter gapless phase, \nMZMs in a finite chain are not expected to be protected against small\nperturbations because of the absence of a gap. Therefore we describe this\nphase as non topological. Furthermore we do not find numerically signatures\nof localized end states in this phase.\n\nWe have also checked the boundaries of the\ntopological phase solving numerically finite chains and searching for\nlocalized states at their ends and the presence of the finite gap. The\nlocalized states are described in Sec. \\ref{majo}. The presence of the\ngap is defined by the condition that the determinant $D(k)$ of $H_{k}$ is\npositive for each $k$. As it can be seen in Fig. \\ref{variobeta} top left,\nthe results of both approaches agree.\n\n\\subsection{Analytical expressions for the boundaries of the topological\nphase}\n\n\\label{anal}\n\nFor perpendicular $\\vec{\\lambda}$ and $\\vec{B}$, the boundaries of the\ntopological phase are defined by Eqs. (\\ref{bound}) and the conditions \n$|\\vec{\\lambda}|\\neq 0$ and $\\Delta \\neq 0$. As the angle is changed \nfrom 90\\textdegree, the gap reduces and a non-zero $|\\Delta |$ is necessary to keep the gap\nopen (see Fig. \\ref{variobeta}). For convenience, we discuss first the case $\n\\lambda _{z}=0$ (perpendicular $\\vec{\\lambda}$ and $\\vec{B}$) and later\nconsider the general case $\\vec{\\lambda}=\\lambda _{y}\\mathbf{\\hat{y}}\n+\\lambda _{z}\\mathbf{\\hat{z}}$ with $\\lambda _{z}\\neq 0$. For $\\lambda _{z}=0\n$, the determinant $D_{0}(k)$ of $H_{k}$ [see Eqs. (\\ref{hk3}) or (\\ref{hkp}\n)]\n\n\\begin{eqnarray}\nD_{0}(k) &=&C^{2}+4\\Delta ^{2}y^{2}, \\notag \\\\\nC &=&a^{2}+\\Delta ^{2}-B^{2}-y^{2}, \\label{d0k}\n\\end{eqnarray\nis positive semidefinite. It can vanish only for $y=0$ implying either $k=0$\nor $k=\\pi $. For $k=0$ ($k=\\pi )$, $C=0$ implies $|\\mu +2t|=r$ ($|\\mu -2t|=r$\n). Comparing with Eqs. (\\ref{bound}), one realizes that the gap vanishes in\ngeneral only at one wave vector and only at the transition between\ntopological and non-topological gapped phases, as expected. The exception is\nthe case $|2t|=r$ and $\\mu =0$, for which the gap vanishes at both wave\nvectors.\n\nIn the general case with $z=2\\lambda _{z}\\sin (k)$ non zero, the determinant\nof $H_{k}$ is [see Eq. (\\ref{hkp})]\n\n\\begin{equation}\nD(k)=D_{0}+2z^{2}(\\Delta ^{2}+y^{2}-a^{2}-B^{2})+z^{4} \\label{dk}\n\\end{equation}\nWe can consider $D(k)$ as a function of $x=\\cos (k)$. For large enough $\n|\\lambda _{z}|$, it turns out that, at the wave vector $k=0$, and parameters\nfor which $C=y=z=0$ [implying $D(0)=0$], $dD(x)\/dx>0$ and as a consequence\nfor small positive $k$ ($x<1$) the determinant becomes negative signaling\nthe instability of the gapped phase. For $\\lambda _{z}=0$, as in the previous\ncase the derivative is negative, but $x$ cannot be increased beyond 1, so\nthat $D(k)\\geq 0$. A similar reasoning with the corresponding changes in the\nsign can be followed for $k=\\pi $. An explicit calculation of the derivative\nusing the conditions $C=\\sin (k)=0$ gives \n\\begin{equation}\n\\frac{dD}{dx}=32[B^{2}\\lambda _{z}^{2}-\\Delta ^{2}(\\lambda _{z}^{2}+\\lambda\n_{y}^{2})]x. \\label{dddx}\n\\end{equation}\nThis implies that to have a gap one needs that $|\\Delta |>\\Delta _{c}$ where \n\\begin{equation}\n\\Delta _{c}^{2}=B^{2}\\frac{\\lambda _{z}^{2}}{\\lambda _{z}^{2}+\\lambda\n_{y}^{2}}=B^{2}\\cos ^{2}(\\beta _{\\lambda B}). \\label{deltac}\n\\end{equation}\nThis condition has been found before for a model similar to ours in the\ncontinuum with quadratic dispersion.\\cite{rex}\n\nAfter some algebra, the determinant in the general case can be written in\nthe form \n\\begin{equation}\nD=(C-z^{2})^{2}+16(\\lambda _{z}^{2}+\\lambda _{y}^{2})(\\Delta ^{2}-\\Delta\n_{c}^{2})(1-x^{2}), \\label{d2}\n\\end{equation}\nwhich is again positive semidefinite for $|\\Delta |>\\Delta _{c}$ and\npositive definite for $0\\neq k\\neq \\pi $, indicating a gapped phase. Since $\nx=1$ implies $y=z=1$, the remaining boundaries of the topological phase\nremain the same as for perpendicular $\\vec{\\lambda}$ and $\\vec{B}.$ For $\n|\\Delta|=\\Delta _{c}$ (as in Fig. \\ref{eigenmu}), the values of $k$ for\nwhich the determinant vanishes are given by the solutions with $|x|\\leq 1$\nof the following quadratic equation\n\n\\begin{eqnarray}\n0 &=&4(t^{2}+\\lambda ^{2})x^{2}+4t\\mu x \\notag \\\\\n&&+\\mu ^{2}+\\Delta _{c}^{2}-B^{2}-4\\lambda ^{2}, \\label{xc}\n\\end{eqnarray}\nwhere $\\lambda =|\\vec{\\lambda}|$.\n\n\\subsection{Transition from the topological phase to the gapless phases}\n\n\\label{topogap}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{3p_autovalores_beta_TeX_pi_sc_rojo.eps}\n\\end{center}\n\\caption{(Color online) Second (black thin lines) and third (red thick lines) eigenvalues of $H_k$ as a function of wave vector\nfor $t=1$, $\\lambda=\\Delta=2$, $B=\\mu=5$, and several values\nof the angle $\\beta_{\\lambda B}$ between \n$\\vec{\\lambda}$ and $\\vec{B}$.}\n\\label{eigenbeta}\n\\end{figure}\n\nTo gain insight into the transition from the topological phase to the\ngapless phases, we represent in Fig. \\ref{eigenbeta} the second and third\neigenvalues of $H_{k}$ [$E_{2}(k)$ and $E_{3}(k)$, respectively] for\ndifferent values $\\beta _{\\lambda B}$ of the angle between $\\vec{\\lambda}$\nand $\\vec{B}$. The parameters are such that, for $\\vec{\\lambda}\\cdot \\vec{B}=0\n$, the system is in the topological phase with a finite gap. As the angle is\nchanged (in either direction) the gap between the second and third\neigenvalue decreases until at a certain critical angle [given by the\nsolution of Eq. (\\ref{xc})] $E_{2}(k_{c})=E_{3}(-k_{c})=0$ at one particular\nwave vector $k_{c}$ ($0.3613\\pi$ in the figure), denoting the onset of the\ngapless phase. Further turning $\\vec{\\lambda}$ and $\\vec{B}$ to the parallel\n(or antiparallel) direction, both eigenvalues vanish at two different wave\nvectors.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{mus_betacritico_TeX_sc_rojo.eps}\n\\end{center}\n\\caption{(Color online) Same as Fig. \\ref{eigenbeta} \nfor $t=1$, $\\lambda=\\Delta=2$, $B=5$, \n$\\beta_{\\lambda B}=66.42$\\textdegree \\ and several values of $\\mu$.}\n\\label{eigenmu}\n\\end{figure}\n\n\nIf keeping the other parameters fixed, the chemical potential $\\mu $ is\nchanged towards one border $\\mu _{c}$ of the topological phase for $\\vec\n\\lambda}\\cdot \\vec{B}=0$ [given by Eq. (\\ref{bound})]; the critical wave\nvector $k_{c}$ is displaced either to $k_{c}=0$ or to $k_{c}=\\pi $ depending\non the border. This is illustrated in Fig. \\ref{eigenmu}. At the\ncorresponding border $\\mu =\\mu _{c}$, one has $E_{2}(k_{c})=E_{3}(k_{c})=0$, indicating a crossing of the levels which is also accompanied by a change\nin the Berry phases of the corresponding eigenvectors. Further displacing \n\\mu $ the system enters the non topological gapped phase. Therefore, the\npoint $\\mu =\\mu _{c}$, $\\Delta =\\Delta _{c}$ is at the border of the\ntopological phase, the nontrivial gapless phase with Berry phase $\\pi$, and the non-topological gapped phase. In fact also the trivial gapless phase reaches\nthis tetracritical point in the phase diagram (see Fig. \\ref{variobeta}).\n\n\n\\subsection{Majorana modes}\n\n\\label{majo}\n\nThe topological phase is characterized \nby the presence of Majorana modes zero modes at the ends of an infinite chain.\nFor a finite chain, the modes at both ends mix, giving rise to a \nfermion $\\Gamma$ and its Hermitian conjugate with energies $\\pm E$ which \ndecay exponentially with the length $L$ of the chain. We have obtained $\\Gamma$ numerically in chains of $L \\sim 200$ sites. The probability $p(i)$ of finding\na fermion at site $i$ (adding both spins and creation and annihilation) is shown\nin Fig. \\ref{pis}. The main feature of the top figure is a decay of $p(i)$\nas the distance from any of the ends increases. We have chosen a case with a rather slow decay to facilitate visualization. In addition to this decay, some oscillations are visible with a short period.\n\nIn order to quantify the decay length of the localization of the end modes, we have fit the probability with an exponentially decaying function \n$p(i) \\sim A$exp$(-i\/\\xi)$ at the left end. At the bottom of \nFig. \\ref{pis} we show the dependence\nof $\\xi$ inside the topological phase \\textrm{I} as one of the parameters is varied.\nAs expected, $\\xi$ diverges at the boundary with the non topological gapped phase \\textrm{II}, which has a different $\\mathbb{Z}_{2}$ topological invariant \n(at $\\Delta_{c_2}=3.872983346$ in the figure). We also find that \n$\\xi$ diverges at the boundary with the gapless phase \\textrm{III} \n(at $\\Delta_{c_1}=0.694592711$ in the figure), a phase with the same topological invariant but gapless. These facts allow us to obtain numerically\nthe transitions from the localization of the end states \n(see top left panel of Fig. \\ref{variobeta}). \n\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{2p_c2_xi_TeX.eps}\n\\end{center}\n\t\\caption{Top: probability of finding a fermion at each site of a chain for the eigenstate of lowest positive energy for $t=1$, $\\lambda=2$, $B=4$, $\\beta_{\\lambda B}=80$\\textdegree, $\\mu=3$, and $\\Delta=0.75$. Bottom: inverse of the localization length as a function of $\\Delta$. The transition between phases I and III is at $\\Delta_{c_1}=0.694592711$ and the transition between phases I and II is at $\\Delta_{c_2}=3.872983346$.}\n\\label{pis}\n\\end{figure}\n\n\n\\section{Summary and discussion}\n\n\\label{sum}\n\nUsing numerical and analytical methods, \nwe calculate the phase diagram of a widely used model for topological superconducting\nwires, the essential ingredients of which are local s-wave pairing $\\Delta$, spin-orbit coupling $\\vec{\\lambda}$ and magnetic field $\\vec{B}$. \nWe determine the boundary of the gapped topological phase analytically. \nThis phase contains robust Majorana zero modes at both ends that are of great interest.\nWe expect that\nthis result will be relevant for future studies in the field.\n\nThe optimal situation for topological superconductivity is when $\\vec{B}$ is perpendicular\nto $\\vec{\\lambda}$. In this case, both the topological and non-topological phases are gapped. If instead $\\vec{B}$ has a component in the direction of $\\vec{\\lambda}$, a gapless\nsuperconducting phase appears for certain parameters. This phase can also be separated in two \nphases differing in a $\\mathbb{Z}_{2}$ topological invariant. However, due to the absence of a gap,\nwe do not find Majorana zero-modes at the ends of the phase with \nnontrivial $\\mathbb{Z}_{2}$, in contrast to those present in the gapped topological phase.\n\nTilting the magnetic field to enter the gapless phase might be used as a trick to relax\nthe system to the ground state in some measurements, like Josephson current.\nIn the gapped topological phase, in the absence of \nlow-frequency phonons or other excitations, the physics is dominated by a few \nbound states inside the gap, completely isolated from the continuum, and the current would oscillate, without reaching a steady state.\\cite{chung} One way to avoid this problem would be to use a magnetic field so that the system is in the gapless phase, with low-energy\nexcitations available for thermalization,\nand then rotate adiabatically the field to the desired value so that the system remains in the ground state.\n\n\\section*{Acknowledgments}\n\nWe thank L. Arrachea for helpful discussions. We are sponsored by PIP\n112-201501-00506 of CONICET, PICT-2017-2726, PICT-2018-04536 and\nPICT-Raices-2018.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}