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+{"text":"\\section{Introduction}\n\nA set $\\{a_1, a_2, \\dots, a_m\\}$ of $m$ distinct nonzero rationals\nis called {\\it a rational Diophantine $m$-tuple} if $a_i a_j+1$ is a perfect square\nfor all $1\\leq i < j\\leq m$.\nDiophantus discovered a rational Diophantine quadruple\n$\\{\\frac{1}{16}, \\frac{33}{16}, \\frac{17}{4}, \\frac{105}{16} \\}$.\nThe first example of a Diophantine quadruple in integers, the set $\\{1, 3, 8, 120\\}$,\nwas found by Fermat. In 1969, Baker and Davenport \\cite{B-D} proved that Fermat's set\ncannot be extended to a Diophantine quintuple in integers.\nRecently, He, Togb\\'e and Ziegler proved that there are no Diophantine quintuples\nin integers \\cite{HTZ} (the nonexistence of Diophantine sextuples in integers was proved in \\cite{D-crelle}).\nEuler proved that there are infinitely many rational Diophantine quintuples.\nThe first example of a rational Diophantine sextuple, the\nset $\\{11\/192, 35\/192, 155\/27, 512\/27, 1235\/48, 180873\/16\\}$, was found by Gibbs \\cite{Gibbs1},\nwhile Dujella, Kazalicki, Miki\\'c and Szikszai \\cite{DKMS} recently proved that there are infinitely\nmany rational Diophantine sextuples (see also \\cite{Duje-Matija,DKP,DKP-split}). It is not known whether there\nexists any rational Diophantine septuple.\nFor an overview of results on\nDiophantine $m$-tuples and its generalizations see \\cite{Duje-Notices}.\n\nThe problem of extendibility and existence of Diophantine $m$-tuples\nis closely connected with the properties of the corresponding elliptic curves.\nLet $\\{a,b,c\\}$ be a rational Diophantine triple. Then there exist nonnegative rationals\n$r,s,t$ such that $ab+1=r^2$, $ac+1=s^2$ and $bc+1=t^2$.\nIn order to extend the triple $\\{a,b,c\\}$ to a quadruple,\nwe have to solve the system of equations\n\\begin{equation} \\label{eq2}\nax+1= \\square, \\quad bx+1=\\square, \\quad cx+1=\\square.\n\\end{equation}\nWe assign the following elliptic curve to the system \\eqref{eq2}:\n\\begin{equation} \\label{e1}\nE:\\qquad  y^2=(ax+1)(bx+1)(cx+1).\n\\end{equation}\nWe say that the elliptic curve $E$ is induced by the rational Diophantine triple $\\{a,b,c\\}$.\n\nSince the curve $E$ contains three $2$-torsion points\n$$ A=\\Big( -\\frac{1}{a}, 0 \\Big), \\quad B=\\Big( -\\frac{1}{b}, 0 \\Big), \\quad\nC=\\Big( -\\frac{1}{c}, 0 \\Big), $$\nby Mazur's theorem \\cite{Mazur}, there are at most four possibilities for the torsion group\nover $\\mathbb{Q}$ for such curves: $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/2\\mathbb{Z}$,\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/4\\mathbb{Z}$,\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/6\\mathbb{Z}$\nand $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$.\nIn \\cite{D-glasnik}, it was shown that all these torsion groups actually appear. Moreover, it was shown that every elliptic curve with torsion\ngroup $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$ is induced by a Diophantine triple\n(see also \\cite{CG}). Questions about the ranks of elliptic curves induced by Diophantine triples were\nstudied in several papers\n(\\cite{ADP, D-rocky, D-bordo,D-glasnik, D-JB-S, D-Peral-LMSJCM, D-Peral-RACSAM, D-Peral-JGEA,D-Peral-highrank}).\nIn particular, such curves were used\nfor finding elliptic curves with the largest known rank\nover $\\mathbb{Q}$ and $\\mathbb{Q}(t)$ with torsion groups\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/4\\mathbb{Z}$\n(\\cite{D-Peral-LMSJCM,D-Peral-JGEA}) and\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/6\\mathbb{Z}$ (\\cite{D-Peral-RACSAM}).\n\nIn this paper, we consider the question how small can be the rank of\nelliptic curves induced by rational Diophantine triples.\nWe will see that it is easy to find rational Diophantine triples with elements with mixed signs\nwhich induce elliptic curves with rank $0$ and that there exist such curves\nwith torsion groups $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/4\\mathbb{Z}$,\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/6\\mathbb{Z}$ and\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$.\nHowever, the problem of finding\nsuch examples of rational Diophantine triples with positive elements is much harder, and they exist\nonly for torsion $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$.\nWe describe the method for finding candidates for such curves\nand by using {\\tt magma} we are able to find one explicit example.\n\n\n\\section{Conditions for point $S$ to be of finite order}\n\nApart from three $2$-torsion points $A$, $B$ and $C$, the curve $E$\ncontains also the following two obvious rational points:\n$$ P = (0, 1), \\quad S = \\Big( \\frac{1}{abc}, \\frac{rst}{abc} \\Big). $$\nIt is not so obvious, but it is easy to verify that $S = 2R$, where\n$$ R = \\Big( \\frac{rs + rt + st + 1}{abc}, \\frac{(r + s)(r + t)(s + t)}{abc} \\Big). $$\nThus, a necessary condition for $E$ to have the rank equal to $0$ is that\nthe points $P$ and $S$ have finite order.\nThe triple $\\{a,b,c\\}$ is regular, i.e. $c=a+b\\pm 2r$ if and only if $S=\\mp 2P$\n(see \\cite{D-bordo}).\n\nBy Mazur's theorem and the fact that $S\\in 2E(\\mathbb{Q})$, we have the following possibilities:\n\\begin{itemize}\n\\item $mP = \\mathcal{O}$, $m=3,4,6,8$;\n\\item $mS = \\mathcal{O}$, $m=2,3,4$.\n\\end{itemize}\n\nIn particular, since the point $P$ cannot be of order $2$, is it not possible to have simultaneously\nrank equal to $0$ and torsion group $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/2\\mathbb{Z}$.\n\nBy the coordinate transformation $x\\mapsto \\frac{x}{abc}$, $y\\mapsto \\frac{y}{abc}$, applied\nto the curve $E$, we obtain the equivalent curve\n\\begin{equation} \\label{e2}\nE':\\qquad  y^2=(x+ab)(x+ac)(x+bc),\n\\end{equation}\nand the points $A$, $B$, $C$, $P$ and $S$ correspond to\n$A'=(-bc,0)$, $B'=(-ac,0)$, $C'=(-ab,0)$, $P'=(0,abc)$ and $S'=(1,rst)$, respectively.\nIn the next lemma, we will investigate all possibilities for point $S$ to be of finite order.\n\n\\begin{lemma} \\label{l:mS}\n\\mbox{}\n\\begin{itemize}\n\\item[(i)] The condition $2S=\\mathcal{O}$ is equivalent to\n$$ (ab+1)(ac+1)(bc+1)=0. $$\n\\item[(ii)] The condition $3S=\\mathcal{O}$ is equivalent to\n$$ 3+4(ab+ac+bc)+6abc(a+b+c)+12(abc)^{2}-(abc)^{2}(a^2 + b^2 + c^2 - 2ab - 2ac - 2bc)=0. $$\n\\item[(iii)] The point $S$ is of order $4$ if and only if\n$$ ((ab+1)^2 - ab(c-a)(c-b))((ac+1)^2 - ab(c-a)(c-b))((bc+1)^2 - ab(c-a)(c-b)) = 0. $$\n\\end{itemize}\n\\end{lemma}\n\n\\proof\n\\mbox{}\n\\begin{itemize}\n\\item[(i)] The condition $2S'= \\mathcal{O}$ implies $rst=-rst$, i.e. $rst=0$, and\n$$ (ab+1)(ac+1)(bc+1)=0. $$\n\n\\item[(ii)]\nFrom $3S'=\\mathcal{O}$, i.e. $x(2S')=x(-S')=x(S')$,\nthe formulas for doubling points of elliptic curves give\n\\begin{align*}\n&3+(ab+ac+bc)\\\\\n&=\\dfrac{9+4(ab+ac+bc)^{2}+(abc(a+b+c))^{2}+12(ab+ac+bc)}{4r^{2}s^{2}t^{2}}\\\\\n&\\ \\ \\ \\ +\\dfrac{6abc(a+b+c)+4abc(ab+ac+bc)(a+b+c)}{4r^{2}s^{2}t^{2}}.\n\\end{align*}\nThus we get\n\\begin{align*}\n&4\\left((abc)^{2}+abc(a+b+c)+(ab+ac+bc)+1\\right)(3+ab+ac+bc)\\\\\n&=9+12(ab+ac+bc)+\\left(6abc(a+b+c)+4(ab+ac+bc)^{2}\\right)\\\\\n&\\ \\ \\ \\ \\ \\ +4abc(ab+ac+bc)(a+b+c)+\\left(abc(a+b+c)\\right)^{2},\n\\end{align*}\nwhich is equivalent to\n\\begin{align*}\n&3+4(ab+ac+bc)+6abc(a+b+c)+12(abc)^{2}\\\\\n& \\ \\ \\ \\ \\ \\ \\ \\ - (abc)^{2}(a^{2} + b^{2} + c^{2} - 2ab - 2ac - 2bc)=0.\n\\end{align*}\n\n\\item[(iii)]\nThe condition that the point $S'$ is of order $4$ is equivalent to $2S'\\in \\{A',B',C'\\}$.\nLet us assume that $2S'=C'$ (other two cases are completely analogous).\nFrom the formulas for doubling points of elliptic curves, we get\n\\begin{align*}\n&2+(bc+ac)\\\\\n&=\\dfrac{9+4(ab+ac+bc)^{2}+\\left(abc(a+b+c)\\right)^{2}+12(ab+ac+bc)}{4r^{2}s^{2}t^{2}}\\\\\n&\\ \\ \\ \\ +\\dfrac{6abc(a+b+c)+4abc(ab+ac+bc)(a+b+c)}{4r^{2}s^{2}t^{2}},\n\\end{align*}\nwhich is equivalent to\n$$\\left(1+2ab-abc(c-a-b)\\right)^{2}=0, $$\nor\n$$(ab+1)^{2}=ab (c-a)(c-b) .$$\n\\end{itemize}\n\\qed\n\n\n\\section{Rank zero curves for triples with mixed signs}\n\nLet us now consider three possibilities for $mS = \\mathcal{O}$.\n\nAssume first that $2S= \\mathcal{O}$. By Lemma \\ref{l:mS}(i), we have $(ab+1)(ac+1)(bc+1)=0$,\nso we conclude that $a,b,c$ cannot have the same sign.\nIf we allow the mixed signs, then in this case we may assume that $b=-1\/a$.\nIn \\cite{D-glasnik}, the following parametrization of rational Diophantine triples of the form\n$\\{a,-1\/a,c\\}$ is given:\n$$ a=\\frac{ut+1}{t-u}, \\quad b=\\frac{u-t}{ut+1}, \\quad c=\\frac{4ut}{(ut+1)(t-u)}. $$\nTo find examples with rank $0$, let us assume that the triple $\\{a,-1\/a,c\\}$ is regular.\nThis condition leads to $(u^2-1)(t^2-1)=0$, so we may take $u=1$.\nIf we take e.g. $t=2$, we obtain the curve with torsion group\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/4\\mathbb{Z}$ and rank $0$,\ninduced by the triple\n$$ \\Big\\{3, -\\frac{1}{3}, \\frac{8}{3} \\Big\\}. $$\n\n\\medskip\n\nAssume now that $3S= \\mathcal{O}$.\nIf we also have $3P= \\mathcal{O}$, then $P=\\pm S$, a contradiction.\nHence, if the point $P$ has finite order, the only possibility that $P$ is of order $6$.\nThis implies $2P=\\pm S$ and $c=a+b\\mp 2r$.\nBy inserting $b=(r^2-1)\/a$ and $c=a+b+2r$ in the condition from Lemma \\ref{l:mS}(ii), we get\n$$ (2ar-1+2r^2) (-a+2ar^2-2r+2r^3) (2a^2r-a-2r+4ar^2+2r^3)=0. $$\nThus,\n$$ a= \\frac{-2r(r^2-1)}{-1+2r^2}, \\quad \\mbox{or} \\quad\n\\frac{-(-1+2r^2)}{2r}, \\quad \\mbox{or} \\quad\n\\frac{1-4r^2 \\pm \\sqrt{1+8r^2}}{4r}. $$\nTake\n\\begin{equation} \\label{eq:z2z6a}\n (a,b,c)=\\left(\\frac{-2r(r-1)(r+1)}{-1+2r^2}, \\frac{-(-1+2r^2)}{2r}, \\frac{(-1+2r)(2r+1)}{2(-1+2r^2)r}\\right).\n\\end{equation}\nThen the condition $ab>0$ is equivalent to $r>1$ or $r<-1$, while the condition\n$bc>0$ is equivalent to $-1\/2 < r < 1\/2$. Hence, $a,b,c$ cannot have the same sign.\n\nThe case\n$$ (a,b,c)=\\left(\\frac{-(-1+2r^2)}{2r}, \\frac{-2r(r-1)(r+1)}{-1+2r^2}, \\frac{(-1+2r)(2r+1)}{2(-1+2r^2)r}\\right) $$\nis the same as the previous case, just $a$ and $b$ are exchanged.\n\nFinally, let $8r^2+1 = (2rt+1)^2$, to get rid of a square root in the third case. It gives $r = \\frac{-t}{-2+t^2}$. Then\n$$ (a,b,c)=\\left(\\frac{-t(t-2)(t+2)}{2(-2+t^2)}, \\frac{2(t-1)(t+1)}{(-2+t^2)t}, \\frac{-(-2+t^2)}{2t}\\right) $$\n(or $a$ and $b$ exchanged).\nThe condition $ac>0$ is equivalent to $t>2$ or $t<-2$, while the condition\n$bc>0$ is equivalent to  $-1 < t < 1$. Hence, is this case also $a,b,c$ cannot have the same sign.\n\nIf we allow the mixed signs, then we can obtain examples with rank $0$, e.g. from triples\nof the form (\\ref{eq:z2z6a}). E.g. for $t=4$ we obtain the curve with torsion group\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/6\\mathbb{Z}$ and rank $0$,\ninduced by the triple\n$$ \\Big\\{-\\frac{12}{7}, \\frac{15}{28}, -\\frac{7}{4} \\Big\\}. $$\n\n\\medskip\n\nIt remains the case when the point $S$ is of order $4$.\nThen the point $R$, such that $2R=S$ is of order $8$ and therefore\nthe torsion group of $E$ is $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$.\nAs we already mentioned in the introduction, it is shown in \\cite{D-glasnik} that\nevery elliptic curve over $\\mathbb{Q}$ with this torsion group\nis induced by a rational Diophantine triple.\nMore precisely, any such curve is induced by a Diophantine triple of the form\n\\begin{equation}\\label{z2z8T}\n\\left\\{ \\frac{2T}{T^2-1}, \\,\\, \\frac{1-T^2}{2T}, \\,\\, \\frac{6T^2-T^4-1}{2T(T^2-1)} \\right\\}.\n\\end{equation}\nIt is clear that the elements of (\\ref{z2z8T}) have mixed signs.\nBy taking $T=2$ we obtain the curve with torsion group\n$\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$ and rank $0$,\ninduced by the triple\n$$ \\Big\\{ \\frac{4}{3}, -\\frac{3}{4}, \\frac{7}{12} \\Big\\}. $$\n\n\n\n\\section{An example of rank zero curve for triple with positive elements}\n\nIn a previous section, we showed that for rational Diophantine triples with all positive elements\nwe cannot have rank $0$ and torsion group $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/4\\mathbb{Z}$\nor $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/6\\mathbb{Z}$. So the only remaining possibility\nis the torsion group $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$.\nSince the elements of (\\ref{z2z8T}) clearly have mixed signs ($ab=-1$),\nand all curves with torsion group $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$\nare induced by (\\ref{z2z8T}), on the first sight we might think that\ntriples with all positive elements are not possible for this torsion group.\nHowever, it is shown in \\cite{D-Peral-RACSAM} that this is not true.\nNamely, we may have a triple with positive elements which induce the same curve as\n(\\ref{z2z8T}) for certain rational number $T$.\n\nBut in \\cite{D-Peral-RACSAM} it remained open whether it is possible to\nobtain simultaneously torsion group $\\mathbb{Z}\/2\\mathbb{Z} \\times \\mathbb{Z}\/8\\mathbb{Z}$\nand rank $0$ for triples with positive elements, although some candidates for such triples\nare mentioned.\n\nAs in the previous section, we assume that the point $S$ is of order $4$,\nand we take $b=(r^2-1)\/a$, $c=a+b+2r$. By inserting this in the first factor\n$$ (ab+1)^2 - ab(c-a)(c-b) $$\nin Lemma \\ref{l:mS}(iii), we get the quadratic equation in $a$:\n$$ (2r^3-2r)a^2+(4r^4-6r^2+1)a+2r^5+2r-4r^3 = 0.$$\nIts discriminant,\n$$ 1-4r^4+4r^2 $$\nshould be a perfect square.\nThe quartic curve defined by this equation is birationally equivalent to the elliptic curve\n$$ E_1: \\quad Y^2 = X^3+X^2+X+1 $$\nwith rank $1$ and a generator $P_1=(0, 1)$.\nThus, by computing multiples of the point $P_1$ on the curve $E_1$\n(adding the $2$-torsion point $T_1=(-1,0)$ has a same effect as changing $r$ to $-r$),\nand transferring them back to the quartic, we obtain candidates for the solution of our problem.\nHowever, we have to satisfy the condition that all elements of the corresponding triple are positive\n(it is enough that all elements have the same sign, since by multiplying all elements from\na rational Diophantine triple by $-1$ we obtain again a rational Diophantine triple).\nThe first two multiples of $P$ producing the\ntriples with positive elements are $6P$ and $11P$.\n\nThe point $6P$ gives\n$r=-\\frac{3855558}{3603685}$ and the triple\n$$ (a,b,c) = \\left(\n\\frac{1884586446094351}{25415891646864180},\n\\frac{14442883687791636}{7402559392524605},\n\\frac{60340495895762708555}{14487505263205637124} \\right). $$\nWe were not able to determine that the rank of the correspoding curve.\nNamely, both {\\tt magma} and {\\tt mwrank}\ngive that $0 \\leq \\mbox{rank} \\leq 2$. Assuming the Parity conjecture the rank should be equal to $0$ or $2$.\n\nThe point $11P$ gives\n$r=\\frac{35569516882766685106979}{32383819387240952672281}$ and the triple $(a,b,c)$, where\n\\begin{align*}\na&= \\frac{69705492951192675600645567228019184577147632882703132983}{132014843349912467692901303836561266921302184459536763120}, \\\\ b&= \\frac{47826829880079829075801189563942620732062701095548790400}{122336669420709509303637442647966391336596694969835459327}, \\\\ c&=\n \\frac{47982111146649404421749331709393501777791774558546217987550257759801}{15400090753918257364093484910580652390786084055043677020804056653840}.\n\\end{align*}\n(By comparing $j$-invariants, we get that the same curve is induced by (\\ref{z2z8T}) for\n$T=\\frac{18451786408106133183649}{41916048174422594852689}$.)\nFor the corresponding curve, both {\\tt mwrank} and {\\tt magma} function {\\tt MordellWeilShaInformation}\ngive that $0 \\leq \\mbox{rank} \\leq 4$.\nHowever, {\\tt magma} (version  {\\tt V2.24-7}) function {\\tt TwoPowerIsogenyDescentRankBound},\nwhich implements the algorithm by Fisher from \\cite{Fisher}, gives that the rank is equal to $0$\n(at step $5$, just beyond $4$-descent, but not yet $8$-descent).\nHence, we found an example of a rational Diophantine triple with positive elements for\nwhich the induced elliptic curve has the rank equal to $0$. Let us mention that the same {\\tt magma} function applied to the\ncurve mentioned above corresponding to the point $6P$ gives only $\\mbox{rank} \\leq 2$.\nThis construction certainly gives\ninfinitely many multiples of $P$ which produce triples with positive elements\n(the set $E_1(\\mathbb{Q})$ is dense in $E_1(\\mathbb{R})$, see e.g. \\cite[p.78]{Skolem}).\nHowever, it is hard to predict distribution of ranks in such families of elliptic curve,\nso we may just speculate that there might be infinitely many curves in this family with rank $0$.\n\n\\bigskip\n\n{\\bf Acknowledgements.}\nA.D. was supported by the Croatian Science Foundation under the project no.~IP-2018-01-1313.\nHe also acknowledges support from the QuantiXLie Center of Excellence, a project\nco-financed by the Croatian Government and European Union through the\nEuropean Regional Development Fund - the Competitiveness and Cohesion\nOperational Programme (Grant KK.01.1.1.01.0004).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\n\\noindent\nA crucial requirement for understanding AGN\nevolution and demographics\nis the ability to select a sample for study \nin a complete and unbiased way.\nX-ray emission has been frequently used for AGN selection,  \nand can distinguish the high energy emission \nassociated with mass accretion by a black hole (BH)\nfrom inactive galaxies and stars.\nCombining wide-area X-ray surveys \nwith the ability to classify large numbers of objects spectroscopically via \nthe Sloan Digital Sky Survey \n\\citep[SDSS,][]{2000AJ....120.1579Y,2006AJ....131.2332G}\nprovides a powerful tool for the study of AGN.\n\nSPIDERS (SPectroscopic IDentification of \\textit{eROSITA} \nSources; PIs Merloni and Nandra) is an \nSDSS-IV \\citep{2017AJ....154...28B} \neBOSS \\citep{2016AJ....151...44D} subprogramme\nthat is currently conducting optical spectroscopy of extragalactic X-ray detections \nin wide-area \\textit{ROSAT} and \\textit{XMM-Newton} surveys\n\\citep{2017MNRAS.469.1065D}.\nLying at the bright end of the X-ray source population, \nthese sources will also be detected by \\textit{eROSITA}\n\\citep{2012arXiv1209.3114M, 2016SPIE.9905E..1KP}.\nThe current SDSS DR14 \\citep{2018ApJS..235...42A} SPIDERS sample\nis a powerful resource for the multiwavelength analysis of AGN.\nThis work aims to capitalise on the wealth of information already available\nby providing detailed optical spectral measurements,\nas well as estimates of BH masses and Eddington ratios.\n\nAn accurate measurement of the central supermassive black hole (SMBH) mass\nis necessary for the study of AGN \nand their coevolution with their host galaxies.\nBH mass has been found to scale \nwith a number of host galaxy spheroid properties;\nstellar velocity dispersion\n\\citep[the $\\rm M_{BH}-\\sigma$ relation, e.g.][]{2000ApJ...539L..13G, 2001ApJ...547..140M, 2002ApJ...574..740T},\nstellar mass \\citep[e.g.][]{1998AJ....115.2285M},\nand luminosity \\citep[e.g.][]{1995ARA&A..33..581K}.\nThese correlations suggest a symbiotic evolution \nof SMBHs and their host galaxies.\n\nReverberation mapping (RM) has been used to measure the \napproximate radius of the broad-line region (BLR) in AGN\n\\citep[e.g.][]{1972ApJ...171..467B, 1982ApJ...261...35C, 1982ApJ...255..419B, 1993PASP..105..247P, 2015PASP..127...67B, 2015ApJS..216....4S}.\nThis technique involves measuring the time delay between \nvariations in the continuum emission,\nwhich is expected to arise from the accretion disk,\nand the induced variations in the broad emission lines.\nIt was found that different emission lines have different time delays,\nwhich is expected if the BLR is stratified, \nwith lines of lower ionisation being emitted \nfurther from the central ionising source \\citep[e.g.][]{1986ApJ...305..175G}.\nFor example, the high ionisation line CIV$\\rm \\lambda$1549 \nhas a shorter time delay than H$\\rm \\beta$ \\citep{2000ApJ...540L..13P}.\n\nThe RM effort has also revealed a tight relationship between the \ncontinuum luminosity and the radius of the BLR \\citep{2000ApJ...533..631K, 2006ApJ...644..133B, 2009ApJ...697..160B}.\nTherefore, by using the measured luminosity as a proxy for the BLR radius,\nand measuring the BLR line-of-sight velocity\nfrom the width of the broad emission lines,\nBH masses can be estimated from a single spectrum\n\\citep{2002ApJ...571..733V, 2002MNRAS.337..109M, 2006ApJ...641..689V, 2011ApJ...742...93A, 2012ApJ...753..125S, 2013BASI...41...61S}.\nThis approach is known as the single-epoch, or photoionisation, method.\n\nSince H$\\beta$ is the most widely studied RM emission line\nit is therefore considered to be the most reliable line to use for single-epoch mass estimation. \nIn addition, AGN H$\\beta$ emission lines typically exhibit \na clear inflection point between the broad and narrow line components,\nmaking the virial full width at half maximum (FWHM) measurement \nrelatively straightforward (see section~\\ref{bl_decomp}).\nThe MgII line width correlates well with that of H$\\beta$ \n(see section~\\ref{compare_mass}),\nand therefore MgII has also been used for single-epoch mass estimation\n\\citep[e.g.][]{2002MNRAS.337..109M}.\nFor SDSS spectra, either H$\\beta$ or MgII \nis visible in the redshift range 0\\,$\\leq$\\,z\\,$\\lesssim$\\,2.5. \n\nAt higher redshifts, \nthe broad, high-ionisation line CIV$\\lambda1549$ is available.\nThe CIV line width does not correlate strongly with that of low ionisation lines\n\\citep[e.g.][]{2005MNRAS.356.1029B, 2012MNRAS.427.3081T}\nand this, along with the presence of a large blueshifted component\n\\citep[e.g.][]{2002AJ....124....1R}\nmakes it difficult to employ CIV for mass estimation.\nA number of calibrations have been developed \nwhich aim to improve the mass estimates derived from CIV\n\\citep[][]{2012ApJ...759...44D, 2013MNRAS.434..848R, 2013ApJ...770...87P, 2017MNRAS.465.2120C},\nhowever, whether CIV can provide reliable mass estimates when compared with \nlow ionisation lines is still a subject of debate\n\\citep[see][]{2018MNRAS.478.1929M}.\n\nThis paper is organised as follows:\nthe selection of a reliable subsample of sources \nto be used for optical spectral fitting \nis discussed in section~\\ref{data}.\nSection~\\ref{spec_fit} describes the method used \nto fit the H$\\beta$ and MgII emission line regions.\nThe methods for estimating BH mass and bolometric luminosity \nare discussed in section~\\ref{estimate_mass}.\nThe X-ray flux measurements used in this work are discussed in section~\\ref{xray_flux}.\nSection~\\ref{access} describes where the catalogue containing the results of this work can be accessed.\nA comparison between the UV and optical spectral fitting results \nis given in section~\\ref{compare_uv_opt}.\nSection~\\ref{prop} provides a discussion of the sample properties, \nand finally, section~\\ref{limits} includes a discussion of the reliability and limitations of\nthe fitting procedure.\n In order to facilitate a direct comparison with previous studies based on X-ray surveys,\na concordance flat $\\rm \\Lambda CDM$ cosmology was adopted\nwhere $\\Omega_{\\rm M}$=0.3, $\\Omega_{\\Lambda}$=1-$\\Omega_{\\rm M}$,\nand H$_{0}$=70\\,km\\,s$^{-1}$\\,Mpc$^{-1}$.\n\n\n\n\\section{Preparing the Input Catalogue}\\label{data}\n\n\\subsection{X-ray Data}\n\n\\noindent\nThe \\textit{ROSAT} sample used in this work \nis part of the second \\textit{ROSAT} all-sky survey (2RXS) \ncatalogue \\citep{2016A&A...588A.103B},\nwhich has a limiting flux of $\\rm \\sim10^{-13}\\,erg\\,cm^{-2}\\,s^{-1}$,\nwhich corresponds to a luminosity of $\\rm \\sim10^{43}\\,erg\\,s^{-1}$ at z = 0.5.\nCompared to the first \\textit{ROSAT} data release \\citep{1999A&A...349..389V},\nthe 2RXS catalogue is the result of an improved detection algorithm,\nwhich uses a more detailed background determination \nrelative to the original \\textit{ROSAT} pipeline.\nA full visual inspection of the 2RXS catalogue has been performed,\nwhich provides a reliable estimate of its spurious source content\n\\citep[see][]{2016A&A...588A.103B}.\nThe first \\textit{XMM-Newton} slew survey catalogue\nrelease 1.6\n\\citep[XMMSL1;][]{2008A&A...480..611S} was also used in this work.\nThis catalogue includes observations made by \nthe European Photon Imaging Camera (EPIC) pn detectors \nwhile slewing between targets,\nand has a limiting flux of $\\rm 6\\times10^{-13}\\,erg\\,cm^{-2}\\,s^{-1}$\nin the soft band,\nwhich corresponds to a luminosity of $\\rm 5.8\\times10^{44}\\,erg\\,s^{-1}$ at z=0.5.\n\n\n\\begin{figure*}[h]\n\t\\centering\n                \\includegraphics[width=0.65\\textwidth]{z_Lx_4.png}\n\\caption{\\footnotesize\nSoft X-ray luminosity versus spectroscopic redshift\nfor the samples presented in this work\nand the following previously published X-ray selected samples;\nXMM-XXL \\citep{2016MNRAS.457..110M, 2016MNRAS.459.1602L},\nCDFS \\citep{2017ApJS..228....2L},\nSTRIPE82X \\citep{2016ApJ...817..172L},\nCOSMOS \\citep{2016ApJ...817...34M, 2016ApJ...830..100M, 2016ApJ...819...62C},\nAEGIS-X \\citep{2015ApJS..220...10N}, and the \nLockman Hole deep field (LHDF) \\citep{2008A&A...479..283B, 2012ApJS..198....1F}.\nFor each sample, the 0.5-2 keV luminosities are shown,\nexcept for the 2RXS sample, where the 0.1-2.4 keV luminosities are shown,\nand the XMMSL1 sample, \nwhere the 0.2-2 keV luminosities from \\citet{2008A&A...480..611S} are shown.\nThe detection limit for the 2RXS and XMMSL1 samples \nare shown by the solid and dashed grey lines respectively.\nThe X-ray luminosities for the 2RXS sample are derived from the classical \nflux estimates described in section~\\ref{2RXS_flux},\nhowever it is noted here that some low count rate \n2RXS sources do not have flux estimates.\nFor sources that were detected in both 2RXS and XMMSL1,\nonly the XMMSL1 luminosities are shown.\nSources classified as stars have not been included in this figure.}\n\\label{compare_samples}\n\\end{figure*}\n\n\n\\subsection{The SPIDERS Programme}\\label{spiders_prog}\n\nThe SPIDERS programme has been providing SDSS spectroscopic observations \nof 2RXS and XMMSL1 \nsources\\footnote{The SPIDERS programme targets both point-like and extended X-ray sources. This work focuses on the counterparts to point-like X-ray detections, \nwhich are predominantly AGN,\nand therefore, the samples discussed in the subsequent paragraphs \nare derived from the SPIDERS-AGN programme.}\nin the eBOSS footprint. \nBefore the start of the eBOSS survey in 2014, \nthe SPIDERS team compiled a sample of X-ray selected spectroscopic targets\nand submitted this sample for spectroscopic follow-up using the BOSS spectrograph \nas part of the eBOSS\/SPIDERS subprogramme\n\\citep[see][for further details on the SPIDERS programme]{2017MNRAS.469.1065D}.\nAs of the end of eBOSS in February 2019,\nthe eBOSS\/SPIDERS survey has covered a sky area of $\\rm 5321 \\, deg^{2}$.\nThe SDSS DR14 SPIDERS sample presented in this work\ncovers an area of $\\rm \\sim2200 \\, deg^{2}$ \n($\\rm \\sim 40\\%$ of the final eBOSS\/SPIDERS area).\n\nThe spectroscopic completeness achieved by the SPIDERS survey \nas of SDSS DR14 in the eBOSS area is \n$\\sim 53\\%$ for the sample as a whole, \n$\\sim 63 \\%$ considering only high-confidence X-ray detections \n(see section~\\ref{clean_sample}),\nand $\\sim 87 \\%$ considering sources with high-confidence X-ray detections and \noptical counterparts with magnitudes in the nominal survey limits \n($\\rm 17\\leq m_{\\rm Fiber2,i}\\leq22.5$).\nOutside the eBOSS area, the spectroscopic completeness of this sample is lower:\n$\\sim 28\\%$ for the sample as a whole, \n$\\sim 39 \\%$ considering only high-confidence X-ray detections,\nand $\\sim 57 \\%$ considering sources with high-confidence X-ray detections and \noptical counterparts with magnitudes in the nominal survey limits.\nThe spectroscopic completeness of the SDSS DR16 SPIDERS sample \ninside and outside the eBOSS area is expected to be similar \nto that of the sample presented here.\n\nIn addition to those targeted during eBOSS\/SPIDERS, \na large number of 2RXS and XMMSL1 sources \nreceived spectra during the SDSS-I\/II\n\\citep[2000-2008;][]{2000AJ....120.1579Y}\nand the SDSS-III \\citep{2011AJ....142...72E}\nBOSS \\citep[2009-2014;][]{2013AJ....145...10D} surveys.\nThis paper includes spectra obtained \nby eBOSS\/SPIDERS up to DR14 (2014-2016) \nas well as spectra from SDSS-I\/II\/III.\n\n\n\n\\subsection{Identifying IR Counterparts}\\label{nway_match}\n\nTo identify SPIDERS spectroscopic targets,\nthe Bayesian cross-matching algorithm ``NWAY'' \\citep{2018MNRAS.473.4937S}\nwas used to select AllWISE \\citep{2013yCat.2328....0C} \ninfrared (IR) counterparts for the 2RXS and XMMSL1 \nX-ray selected sources in the BOSS footprint.\nThe AllWISE catalogue consists of data obtained \nduring the two main survey phases of the \nWide-field Infrared Survey Explorer mission\n\\citep[WISE;][]{2010AJ....140.1868W}\nwhich conducted an all-sky survey \nin the 3.4, 4.6, 12, and 22 $\\rm \\mu m$ bands \n(magnitudes in these bands are denoted [W1], [W2], [W3], and [W4] respectively).\nThe matching process used the colour-magnitude priors \n[W2] and [W2-W1] \\citep[see][]{2017MNRAS.469.1065D}\nwhich, at the depth of the 2RXS and XMMSL1 surveys, \ncan distinguish between the correct counterparts and chance associations.\nThese colours would not be efficient if the 2RXS survey was much deeper\n\\citep[see][for a complete discussion]{2018MNRAS.473.4937S}.\nThe resulting 2RXS and XMMSL1 catalogues\nwith AllWISE counterparts contained 53455 and 4431 sources respectively.\nAllWISE positions were then matched to \nphotometric counterparts, where available, in SDSS.\n\n\n\n\\subsection{Comparison with Previous X-ray Surveys}\n\nFigure~\\ref{compare_samples} displays the \nsources in the 2RXS and XMMSL1 samples\nwhich have spectroscopic redshifts and measurements of the soft X-ray flux.\nFor comparison, a series of previously published X-ray selected samples \nthat have optical spectroscopic redshifts are also shown.\nThe large number of sources present in the 2RXS and XMMSL1 samples \nmotivated the optical spectroscopic analysis discussed in the following sections.\n\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{tree_2RXS.png}\n\\includegraphics[width=0.49\\textwidth]{tree_XMMSL.png}\n\\caption{\\footnotesize \nSequence of quality cuts applied to the 2RXS and XMMSL1 samples \nto produce the subsample used for spectral analysis.\nThe starting points (2RXS-AllWISE and XMMSL1-AllWISE)\nare the full samples of 2RXS and XMMSL1 selected sources \nwith AllWISE IR counterparts in the BOSS footprint\n(see section~\\ref{nway_match}).\nThe steps in grey are those that have been discussed \nin \\citet{2017MNRAS.469.1065D}.}\n\\label{cuts}\n\\end{figure*}\n\n\n\\subsection{Selecting a Reliable Subsample}\\label{clean_sample}\n\n\\noindent\nThe selection of SPIDERS spectroscopic targets \nwas discussed in detail by \\citet{2017MNRAS.469.1065D}.\nThis section summarises the selection steps \ndiscussed in detail by \\citet{2017MNRAS.469.1065D}\nand describes the additional cuts made in this work to \nselect a sample for spectral analysis.\nThe sequence of selection criteria used and the resulting sample size \nare shown in figure~\\ref{cuts}.\n\n2RXS sources with an X-ray detection likelihood (EXI\\_ML)\n$\\leq \\,$10 were excluded since these detections are considered \nhighly uncertain with a spurious fraction \n$\\geq20\\%$ \\citep[see][]{2016A&A...588A.103B}. \nXMMSL1 sources with an X-ray detection likelihood \n(XMMSL\\_DET\\_ML\\_B0) $\\leq \\,$10 were also excluded.\n\\citet[][figure 1]{2018MNRAS.473.4937S} show the \ndistribution of flux with detection likelihood for both samples.\nThese cuts returned 23245\/53455 2RXS and 3803\/4431 XMMSL1 sources.\n\nThe following cuts, which were described in detail in \\citet{2017MNRAS.469.1065D},\nhave also been applied to the sample:\n\n\\begin{itemize}\n\\item For each X-ray source, \\citet{2018MNRAS.473.4937S} give \nthe probability, p\\_any, that a reliable counterpart exists among the \npossible AllWISE associations. Sources with p\\_any < 0.01 were removed.\nThis returned 23046\/23245 2RXS and 3558\/3803 XMMSL1 sources. \\\\\n\n\\item For each X-ray source, the most probable AllWISE counterpart was chosen\nby selecting sources with match\\_flag=1.\nThis returned 20585\/23046 2RXS and 3321\/3558 XMMSL1 sources. \\\\\n\n\\item For each AllWISE counterpart, the brightest SDSS-DR13 photometric source within the AllWISE matching radius was selected using FLAG\\_SDSSv5b\\_best=1.\nThis returned 19385\/20585 2RXS and 3063\/3321 XMMSL1 sources. \\\\\n\n\\item Cases where the AllWISE-SDSS separation \nexceeded 3\\,arcsec were removed.\nThis returned 18575\/19385 2RXS and 2893\/3063 XMMSL1 sources. \\\\\n\\end{itemize}\n\n\\noindent\nThe results of these constraints are displayed in grey in figure~\\ref{cuts}.\nAs shown above, sources with match\\_flag=1 were targeted;\nhowever, for 14\\% of the 2RXS sample and 10\\% of the XMMSL1 sample,\nmore than one counterpart was highly likely.\nThis implies that either the counterpart association was not reliable, \nor that the X-ray detection was the result of emission from multiple sources.\nThese sources were not included in the discussion of \noptical spectral properties as a function of X-ray properties in section~\\ref{aox_section}.\nAfter selecting the brightest SDSS-DR13 \\citep{2017ApJS..233...25A} \nphotometric source, there were 15 cases where two unique 2RXS sources \nwere matched to the same AllWISE\/SDSS counterpart \nand 3 cases where two unique XMMSL1 sources \nwere matched to the same AllWISE\/SDSS counterpart. \nThese sources were also removed.\n\nOf these samples with reliable SDSS photometric counterparts, \n8777 2RXS and 1315 XMMSL1 sources have received spectra during SDSS-I\/II\/III \nwhile 1122 2RXS and 221 XMMSL1 sources have received spectra \nduring the SPIDERS programme (including SEQUELS), resulting in a sample of \n9899 2RXS and 1536 XMMSL1 sources with spectra as of DR14.\nThe distribution of SDSS i band fiber2 magnitudes for this sample \n(showing the different spectroscopic programmes)\nis presented in figure~\\ref{origin_spec}.\nDue to targeting constraints (as discussed in section~\\ref{spiders_prog}), \nthe sample completeness is much lower \noutside of the nominal magnitude limits for the survey \n($\\rm 17\\leq m_{\\rm Fiber2,i}\\leq22.5$ for eBOSS).\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{mag_2RXS.png}\n\\includegraphics[width=0.49\\textwidth]{mag_XMMSL.png}\n\\caption{\\footnotesize \nDistribution of i-band fiber magnitudes (fiber2Mag).\nThe coloured curves represent all of the sources with spectra,\nand the survey from which the spectra were taken.\nThe grey histogram displays the X-ray sources with a reliable SDSS photometric counterpart, including stars \nwhich cannot be targets for spectroscopy due to their brightness.}\n\\label{origin_spec}\n\\end{figure*}\n\n\n\\subsection{Source Classification}\\label{source_classification}\n\n\\noindent\nVisual inspection results for each object in this sample \nare available from a combination of literature sources \n\\citep{2007AJ....133..313A, 2010AJ....139..390P, 2010AJ....139.2360S, 2017A&A...597A..79P} and the SPIDERS group.\nThe SPIDERS visual inspection \n\\citep[see][for further details]{2017MNRAS.469.1065D}\nprovides a visual confirmation of the SDSS pipeline redshift and object classification.\nThe results of this inspection include \na flag indicating the confidence of the redshift, ``CONF\\_BEST'',\nwhich can take the values 3 (highly secure), \n2 (uncertain), 1 (poor\/unusable), 0 (insufficient data).\nA confirmation of the source classification \nwas also added during the visual inspection, \nwhich uses the categories QSO, broad absorption line QSO (BALQSO), \nblazar, galaxy, star, and none.\n\\citet{2007AJ....133..313A} provide the broad line AGN (BLAGN) and narrow line AGN (NLAGN) classifications, which are defined based on the presence or absence of \nbroad (FWHM\\,>\\,1000$\\rm \\,km\\,s^{-1}$) permitted emission lines.\n\nThe main goal of this work is to analyse the type 1 AGN in the SPIDERS sample,\nand therefore only sources that have been classified via their optical spectra \nas either ``BLAGN'' or ``QSO'' were selected for spectroscopic analysis.  \nThis returned 7805\/9899 2RXS and 1192\/1536 XMMSL1 sources.\nSince the categories ``BLAGN'' and ``QSO'' are based on different classification criteria,\nthere will be some overlap between the two sets of sources.\nTherefore, no distinction will be made between the two categories;\ninstead, both sets of objects will be considered type 1 AGN in this work.\n\n\n\n\\subsection{Contamination from Starburst Galaxies}\n\n\\noindent\nAlthough our sample probes luminosity ranges\ntypically associated with AGN emission,\nstarburst galaxies are also powerful X-ray sources\nand may be present as contaminants in our AGN sample.\nThe X-ray emission from starburst galaxies is expected to \noriginate from a number of energetic phenomena \nincluding supernova explosions and X-ray binaries\n\\citep[e.g.][]{2002A&A...382..843P}.\nTherefore, the X-ray emission from starburst galaxies \ncan be expected to be correlated with the star formation rate (SFR).\nUsing their sample of luminous infrared galaxies, \nand a sample of nearby galaxies from \\citet{2003A&A...399...39R},\n\\citet{2011A&A...535A..93P} found that the total SFR\nis related to the soft X-ray luminosity as follows:\n\n\\begin{equation}\\label{SFR}\n\\rm SFR_{IR+UV}\\,(M_{\\odot}\\,yr^{-1}) = 3.4 \\times 10^{-40}\\,L_{0.5-2keV} \\, (erg \\, s^{-1})\n\\end{equation}\n\n\\noindent\n\\citet{2015A&A...579A...2I}, figure 3, show the specific \nSFR for the COSMOS \\citep{2007ApJS..172....1S} \nand GOODS \\citep{2004ApJ...600L..93G} surveys \nfor a series of redshift bins in the range $\\rm 0.2 <  z < 1.4$.\nThe peak of the redshift distribution of the 2RXS\/XMMSL1 samples \npresented in this work is $\\sim0.25$.\nTherefore, assuming that the COSMOS\/GOODS sample in the redshift bin 0.2-0.4\nis a good representative of the 2RXS\/XMMSL1 samples, \nthe upper limit on the SFR that can be expected \nfor galaxies in our sample is $\\rm \\sim50\\,M_{\\odot}\\,yr^{-1}$.\nAccording to equation~\\ref{SFR}, this corresponds to a soft X-ray luminosity of \n$\\rm \\sim10^{41}\\,erg\\,s^{-1 }$,\nwhich is below the lower range probed by our samples \n$\\rm (\\sim10^{42} \\, erg \\, s^{-1}$, see figure~\\ref{compare_samples}).\n\n\n\n\\subsection{Redshift Constraints}\\label{selection}\n\n\\noindent\nUsing the ``CONF\\_BEST'' flag, sources with uncertain redshift or spectral classification \n(identified during the visual inspection of the sample) were also removed.\nThis process resulted in a sample of 7795\/7805 2RXS sources \nand 1190\/1192 XMMSL1 sources.\nIn the spectral fitting procedure (described in section~\\ref{spec_fit}), \nthe H$\\beta$ and MgII lines were fit independently.\nSources with H$\\beta$ and MgII present in their optical spectrum \nwere selected using the following logic:\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm H_{\\beta}: \\rm (SN\\_MEDIAN\\_ALL > 5) \\,\\,\\&\\&\\,\\, \\\\\n\\rm (( (INSTRUMENT == SDSS) \\,\\,\\&\\&\\,\\, (0 < Z\\_BEST < 0.81) )\\,\\,||\\,\\, \\\\\n\\rm ( (INSTRUMENT == BOSS) \\,\\,\\&\\&\\,\\, (0 < Z\\_BEST < 1.05)))\n\\end{split}\n\\end{equation}\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm MgII: \\rm (SN\\_MEDIAN\\_ALL > 5) \\,\\,\\&\\&\\,\\, \\\\\n\\rm (( (INSTRUMENT == SDSS) \\,\\,\\&\\&\\,\\, (0.45 < Z\\_BEST < 2.1) )\\,\\,||\\,\\, \\\\\n\\rm ( (INSTRUMENT == BOSS) \\,\\,\\&\\&\\,\\, (0.38 < Z\\_BEST < 2.5)) )\n\\end{split}\n\\end{equation}\n\n\\noindent\nDifferent redshift ranges have been used because the \nBOSS spectrograph has a larger wavelength coverage than the SDSS spectrograph.\nIn some cases, parts of the fitting region will have been redshifted \nout of the SDSS\/BOSS spectrograph range \\citep{2013AJ....146...32S}, \nand therefore will not be fit.\nHowever, the redshift limits where chosen\nso that both samples contain the broad lines used for estimating BH mass.\nSources with a median signal-to-noise ratio (S\/N) less than or equal to five \nper resolution element were excluded from the spectral analysis \nsince for these sources the broad line decomposition \nand resulting BH mass estimates may be unreliable\n\\citep[see][]{2009ApJ...692..246D,2011ApJS..194...45S}.\n\nTable~\\ref{spec_sample} lists the numbers of sources \nwith spectral coverage of either H$\\beta$ or MgII, \nwhile figure~\\ref{clean_subsample} shows the redshift distribution of these sources.\nThere are 711 cases where the same optical counterpart \nwas detected by both 2RXS and XMMSL1.\nThe final combined (2RXS and XMMSL1) sample for spectral analysis  \ncontains 7790 unique type 1 sources.\n\n\n\\begin{table}\n\\caption{\\footnotesize \nThe coverage of the H$\\beta$ and MgII emission lines \nin the two samples used in this work.\nThere are 711 sources which were detected in both \nthe 2RXS and XMMSL1 surveys. \nThe ``total'' row lists the total number of unique sources \nobtained from combining the 2RXS and XMMSL1 samples.}\n\\centering  \n\\begin{tabular}{l c c c c c}      \n\\toprule \n\\midrule\n\n\\multicolumn{1}{c}{}\n& \\multicolumn{1}{c}{MgII}\n& \\multicolumn{1}{c}{H$\\beta$}\n& \\multicolumn{1}{c}{H$\\beta$ and MgII}\n& \\multicolumn{1}{c}{H$\\beta$ or MgII}\n\\\\\n\\midrule\n2RXS & 3310 & 6268 & 2234 & 7344 \\\\ \nXMMSL1 & 314 & 1070 & 227 & 1157 \\\\\nTotal & 3473 & 6654 & 2337 & 7790 \\\\\n\\midrule\n\\bottomrule \\\\\n\\end{tabular}\n\\label{spec_sample}\n\\end{table}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{z_hist_spec.png}\n\\caption{\\footnotesize\nRedshift distribution of the sample of type 1 AGN with coverage of $\\rm H\\beta$ \nand\/or MgII.}\n\\label{clean_subsample}\n\\end{figure}\n\n\n\n\\section{Spectral Analysis}\\label{spec_fit}\n\n\\noindent\nA series of scripts have been written to perform spectral fits\nusing the MPFIT least-squares curve fitting routine \n\\citep{2009ASPC..411..251M}.\nEach spectrum was corrected for Milky Way extinction \nusing the extinction curve from \\citet{1989ApJ...345..245C},\nand the dust map from \\citet{1998ApJ...500..525S},\nwith an R$\\rm_{V}$=3.1.\nNo attempt has been made to estimate and correct for \nthe intrinsic (host) extinction of each source\\footnote{\nAlso note that extinction laws (e.g. Calzetti and Prevot) \nare based on samples of nearby SB and irregular type galaxies. \nDue to the lack of nearby passive galaxies,\nan extinction law for these galaxy types is not yet available.}.\nMeasured line widths were corrected for the resolution of the \\\nSDSS\/BOSS spectrographs.\nThe H$\\beta$ and MgII emission line regions \nwere fit independently using similar\nmethods described in the following \nsections\\footnote{For each model parameter,\nthe 1-sigma uncertainties from MPFIT were adopted.}.\n\n\n\n\\subsection{Iron Emission Template}\\label{fe}\n\n\\noindent\nAGN typically exhibit FeII emission consisting of \na large number of individual lines \nacross the optical and UV regions of the spectrum.\nThese lines appear to be blended, \nprobably due to the motion of the gas from which they are emitted,\nand the magnitude of this broadening varies significantly from source to source.\nThe presence of FeII emission in the optical and UV portions of the spectrum can be\na significant complication when attempting to accurately measure line profiles.\nTherefore it is crucial that the model used to derive line widths for BH mass measurements also accounts for the nearby FeII emission.\n\nFigure~\\ref{fe_template} shows the two FeII templates used in this work;\nthe \\citet{2001ApJS..134....1V} and \\citet{1992ApJS...80..109B}\ntemplates used for the UV and optical regions of the spectrum, respectively. \nBoth of these templates have been derived from \nthe narrow line Seyfert 1 galaxy I Zwicky 1\nwhich, due to its bright FeII emission and narrow emission lines, \nis an ideal candidate for generating the FeII template.\nIn order to model the observed blending of the FeII emission,\nthe templates were convolved with a Gaussian \nwhose width was included as a free parameter in the fitting procedure.\n\n\n\\begin{figure}[h!]\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{iron_template_uv.png}\\\\\n                \\includegraphics[width=0.49\\textwidth]{iron_template_opt.png}\n\\caption{\\footnotesize\nUpper panel: The \\citet{2001ApJS..134....1V} FeII template \nwhich was used when fitting the MgII emission line region.\nLower panel: The \\citet{1992ApJS...80..109B} FeII template\nwhich was used when fitting the H$\\beta$ emission line region. \nIn both cases, the original template is shown (blue) \nalong with the same template convolved with a Gaussian with FWHM\\,=\\,4000\\,km\\,s$^{-1}$ (red).\nThe vertical dashed lines correspond to the position of MgII and H$\\beta$ \nat 2800$\\AA$ and 4861$\\AA$.}\n\\label{fe_template}\n\\end{figure}\n\n\n\\begin{figure*}[h!]\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{spec-1159-52669-0470.png}\n                \\includegraphics[width=0.49\\textwidth]{spec-0423-51821-0250.png}\n\\caption{\\footnotesize\nExamples of model fits to the H$\\beta$ \n(left panel; plate=1159, MJD=52669, fiber=470) \nand MgII (right panel; plate=423, MJD=51821, fiber=250) spectral regions.\nThe model components are colour-coded as follows;\npower law (orange), iron emission (violet), broad lines (blue), \nnarrow lines (yellow), [OIII] shifted wings (red), and the total model (red).\nThe panels beneath the spectra show the data\/model ratio.}\n\\label{examples}\n\\end{figure*}\n\n\n\\subsection{H$\\beta$}\\label{HB_fit_method}\n\n\\noindent\nThe region from 4420-5500\\,$\\AA$ was fit for each spectrum. \nThe continuum model consisted of a power law,\na galaxy template, and the \\citet{1992ApJS...80..109B} FeII emission template.\nThe FeII template was convolved with a Gaussian while fitting, \nand the width of this Gaussian, along with the normalisation of the template \nwere included as free parameters in the fit (see section~\\ref{fe}).\nPrevious spectral analyses of AGN spectra have assumed \nan early-type galaxy component in the model \\citep{2017MNRAS.472.4051C}.\nFollowing this method, we use an early-type SDSS galaxy\ntemplate\\footnote{template number 24 on \\\\ http:\/\/classic.sdss.org\/dr5\/algorithms\/spectemplates\/}  in the fit, and the normalisation of this template as well as the \nnormalisation and slope of the power law were also free parameters.\nThe use of a single, early-type galaxy template is an approximation, \nhowever, it is considered to be justified since \nAGN are typically found to reside in bulge dominated galaxies\n\\citep[e.g.][]{2005ApJ...627L..97G, 2007ApJ...660L..19P},\nand the spectroscopic fiber collects emission mostly from \nthe bulge (which is characterised by an old stellar population)\nand the active nucleus.\n\nThe [OIII]$\\lambda$4959 and [OIII]$\\lambda$5007 narrow lines\nwere each fit with two Gaussians, one used to fit the narrow core, \nand an additional Gaussian to account for the presence of \nblue-shifted wings which are often detected \\citep{2005AJ....130..381B}. \nA single Gaussian was used to fit the HeII$\\lambda$4686 emission line.\nTo avoid overfitting the H$\\beta$ line, the fitting process was run four times, \nwith one, two, three, and four\\footnote{Three broad Gaussians are used in addition to a single narrow component to account for the three distinct broad components \nthat are expected to be present\n(see section~\\ref{VBC_section})\nin at least some sources \\citep{2010MNRAS.409.1033M}.} \nGaussian components used to fit the H$\\beta$ line.\nFor each fit, the velocity width and peak wavelength of one of the Gaussian components was fixed to that of [OIII]$\\lambda$4959 and [OIII]$\\lambda$5007 \nin order to aid the identification of the narrow H$\\beta$ component.\nThe normalisation ratio of the [OIII]$\\lambda$4959 and [OIII]$\\lambda$5007\nlines was fixed to the expected value of 1:3 \\citep[e.g.][]{2000MNRAS.312..813S}.\nThe best-fit model was then selected using the Bayesian information criterion \n\\citep[BIC,][]{1978AnSta...6..461S},\nwhich can be written as \n\n\\begin{equation}\\nonumber\n\\rm BIC = ln(n)k+\\chi^{2} \n\\end{equation}\n\n\\noindent\nwhere n is the number of data points, k is the number of model parameters,\nand $\\rm \\chi^{2}$ is the chi-square of the fit.\nThe preferred model is that with the lowest BIC.\nAn example of a fit to the H$\\beta$ spectral region \nis shown in the left panel of figure~\\ref{examples}.\n\n\n\n\\subsubsection{Broad Line Decomposition}\\label{bl_decomp}\n\n\\noindent\nThe narrow H$\\beta$ and [OIII] components \nare required to have widths $\\rm \\leq800\\,km\\,s^{-1}$.\nAny of the additional Gaussians used to fit MgII and H$\\beta$\nwith FWHM\\,$\\rm > 800\\,km\\,s^{-1}$ are considered ``broad''.\nThis threshold of 800$\\rm \\, km\\, s^{-1}$ is taken from the approximate division between broad and narrow FWHM distributions in the lower panels of figure~\\ref{line_width}.\nThe virial FWHM used for BH mass estimation \nis the FWHM of the line profile defined by the \nsum of these broad Gaussian components\n(see figure~\\ref{line_decomp}).\nA major challenge with using the single-epoch method for estimating BH mass \nis decomposing the broad and narrow components of the line\nin order to measure the virial FWHM.\nFigure~\\ref{line_decomp} presents an example \nof the decomposition of a broad H$\\beta$ line.\nIn this case, the narrow H$\\beta$ core can be easily distinguished \nand removed before measuring the virial FWHM.\nHowever, there are many cases \nwhere the broad and narrow components are blended,\nmaking it difficult to successfully identify the appropriate virial FWHM.\nThere are also cases where there is a clear distinction between\ntwo broad line components that are shifted in wavelength \nrelative to each other (known as ``double-peaked emitters''). \nHow one should interpret \nthe single-epoch BH mass estimates \nfor these unusual objects is uncertain \n(also see section~\\ref{reliability}).\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{line.png}\n\\caption{\\footnotesize \nAn example of the decomposition of a typical AGN H$\\beta$ line \n(plate=7276, MJD=57061, fiber=470).\nThe horizontal dashed line represents the FWHM used for BH mass estimation.\nThe vertical dashed line indicates the rest-frame wavelength of H$\\beta$.\nSee section~\\ref{bl_decomp} for further details.}\n\\label{line_decomp}\n\\end{figure}\n\n\n\n\\subsection{MgII}\\label{mgII_fit}\n\n\\noindent\nThe region from 2450-3050$\\AA$ was fit for each spectrum. \nAs in the case of the H$\\beta$ fits, a power law,\nan early-type galaxy template \n\\citep[5\\,Gyr old elliptical galaxy;][]{1998ApJ...509..103S, 2007ApJ...663...81P},\nand the \\citet{2001ApJS..134....1V} FeII emission template\nwere used to fit the continuum.\nAgain, the FeII template normalisation, and width of the \nGaussian smoothing applied to the template, \nwere included as free parameters in the fit.\nThe MgII line is a doublet; however, due to the close spacing \nand virial broadening of the lines,\nit usually appears as a single broad component in AGN spectra.\nThe narrow MgII line cores are usually not observed in AGN spectra,\ntherefore the MgII profile was fit using three broad Gaussians.\nAn example of a fit to the MgII spectral region \nis presented in the right panel of figure~\\ref{examples}.\n\n\n\\begin{table*}[h]\n\\caption{\\footnotesize \nBH mass calibrations used in this work.\nA, B, and C are the calibration constants for single-epoch mass estimation\n(see equation~\\ref{mass_est}).}\n\\centering  \n\\begin{tabular}{l c c c c c}      \n\\toprule \n\\midrule\n\n\\multicolumn{1}{c}{}\n& \\multicolumn{1}{c}{A}\n& \\multicolumn{1}{c}{B}\n& \\multicolumn{1}{c}{C}\n& \\multicolumn{1}{c}{Reference}\n\\\\\n\\midrule\nMgII, L$_{3000\\AA}$ & 1.816 & 0.584 & 1.712 & \\citet{2012ApJ...753..125S} \\\\\nH$_{\\beta}$, L$_{5100\\AA}$ & 0.91 & 0.5 & 2 & \\citet{2006ApJ...641..689V} \\\\ \nH$_{\\beta}$, L$_{5100\\AA}$ & 0.895 & 0.52 & 2 & \\citet{2011ApJ...742...93A} \\\\\n\\midrule\n\\bottomrule \\\\\n\\end{tabular}\n\\label{calib}\n\\end{table*}\n\n\n\\begin{figure*}[h!]\n\\begin{center}\n\\includegraphics[width=0.33\\textwidth]{mass_hist_1.png}\n\\includegraphics[width=0.33\\textwidth]{mass_hist_2.png}\n\\includegraphics[width=0.33\\textwidth]{mass_hist_3.png}\n\\caption{\\footnotesize \\label{mass_hist}\nDifferences between the BH mass calibrations used in this work\n(see section~\\ref{estimate_mass} for further details).\nThe vertical blue lines indicate the mean value of each distribution.\nThe standard deviation, $\\rm \\sigma$, of each distribution is also shown.} \n\\end{center}\n\\end{figure*}\n\n\n\\section{Bolometric Luminosity and BH Mass Estimation}\\label{estimate_mass}\n\n\\noindent\nBolometric luminosities were estimated from \nthe monochromatic luminosities \nusing the bolometric corrections from \n\\citet{2006ApJS..166..470R, 2011ApJS..194...45S}:\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm L_{Bol} = 5.15 \\, L_{3000\\AA} \\\\\n\\rm L_{Bol} = 9.26 \\, L_{5100\\AA}\n\\end{split}\n\\end{equation}\n\n\\noindent\nThese bolometric corrections have been derived using mean AGN SEDs;\nhowever, \\citet{2006ApJS..166..470R} note that \nusing a bolometric correction resulting from a single mean SED \ncan result in bolometric luminosities with inaccuracies up to 50$\\%$.\n\nUnder the assumption that the BLR gas is virialised, \nthe single-epoch method can be used to estimate BH mass as follows:\n\n\\begin{equation}\\label{mass_est}\n\\rm log\\left(\\frac{M_{BH}}{M_{\\odot}}\\right) = A + B\\,log\\left(\\frac{\\lambda L_{\\lambda}}{10^{44}\\,erg\\,s^{-1}}\\right)+C\\,log\\left(\\frac{FWHM}{km\\,s^{-1}}\\right)\n\\end{equation}\n\n\\noindent\nwhere $\\rm L_{\\lambda}$ is the monochromatic luminosity\nat wavelength $\\lambda$,\nand FWHM is the full width at half maximum of the broad component of the emission line.\nA, B, and C are constants that are calibrated using RM results\nand vary depending on which line is used. \n\nOver the years, many groups have provided \ncalibrations of equation~\\ref{mass_est} for MgII and H$\\beta$.\nIn this work, the calibrations from \n\\citet{2006ApJ...641..689V} and \\citet{2011ApJ...742...93A} are used for H$\\beta$.\n\\citet{2006ApJ...641..689V} based their work on \nan updated study of the $\\rm R_{BLR} - L$ relationship \n\\citep{2005ApJ...629...61K, 2006ApJ...644..133B} and \na reanalysis of the RM mass estimates \\citep{2004ApJ...613..682P}\nand therefore presented an improved mass calibration relative to previous studies.\n\\citet{2011ApJ...742...93A} provide a mass calibration that is based on the \n$\\rm R_{BLR} - L$ relationship from \\citet{2009ApJ...705..199B}.\n\\citet{2006ApJ...641..689V} and \\citet{2011ApJ...742...93A} \nboth provide similar calibrations for single-epoch H$\\beta$ mass estimation,\nas can be seen from the left panel of figure~\\ref{mass_hist}.\n\nThe \\citet{2012ApJ...753..125S} calibration is used in this work for MgII.\nThis calibration is based on a sample of \n60 high-luminosity ($\\rm L_{5100\\AA} > 10^{45.4} erg \\, s^{-1}$)\nquasars in the redshift range 1.5 - 2.2.\n\\citet{2012ApJ...753..125S} use the \\citet{2006ApJ...641..689V} \nmass estimates as a reference when determining their MgII calibration.\nThe centre and right panels of figure~\\ref{mass_hist}\nshow the comparison between the \\citet{2012ApJ...753..125S} MgII calibration\nand the $\\rm H \\beta$ calibrations from \n\\citet{2006ApJ...641..689V} and \\citet{2011ApJ...742...93A}.\nThese calibrations agree reasonably well,\nwith the standard deviation $\\rm \\sigma \\simeq 0.3$ in both cases,\nwhich is likely due to the fact that these BH mass estimates were derived \nusing two different emission lines.\nA list of the three BH mass calibrations used in this work \nis given in Table~\\ref{calib}.\n\nBH masses were computed for each of these calibrations \nand are included in the catalogue (see section~\\ref{columns}). \nBH masses were only estimated for sources with a detected \nbroad line component (see section~\\ref{bl_decomp}).\nThese BH mass estimates were then used to estimate \nthe Eddington luminosity and the Eddington ratio\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm  L_{Edd}=4\\pi c G M_{BH} m_{p}\/\\sigma_{T} \\\\\n\\rm \\lambda_{E} = L_{Bol}\/L_{Edd}\n\\end{split}\n\\end{equation}\n\n\\noindent\nwhere c is the speed of light, \nG is the gravitational constant,\nM$\\rm_{BH}$ is the BH mass,\nm$\\rm_{p}$ is the proton mass,\nand $\\rm \\sigma_{T}$ is the Thomson scattering cross-section.\n\n\n\\begin{figure}[]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{flux_comparison_2.png}\n\\caption{\\label{bay_class_flux} \\footnotesize \nComparison between the classical and Bayesian methods \nfor estimating 2RXS fluxes. The deviation from a ratio of one at fainter fluxes \nresults from the attempt to correct for the effect of the Eddington bias\n(see section~\\ref{2RXS_flux} for further details).} \n\\end{center}\n\\end{figure}\n\n\n\\begin{figure*}[h!]\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{uv_opt_lum.png}\n                \\includegraphics[width=0.49\\textwidth]{compare_lbol.png}\n                \\includegraphics[width=0.49\\textwidth]{virial_fwhm.png}\n                \\includegraphics[width=0.49\\textwidth]{compare_mass.png}\n\\caption{\\footnotesize \nComparison between the results obtained from fitting \nthe H$\\beta$ and MgII regions of the spectrum.\nUpper left: 5100$\\AA$ and 2500$\\AA$ monochromatic luminosities.\nUpper right: Bolometric luminosities estimated from the\n5100$\\AA$ and 3000$\\AA$ monochromatic luminosities.\nLower left: Virial FWHM measured using H$\\beta$ and MgII.\nLower right: BH mass estimates derived from H$\\beta$ \n(using the \\citet{2011ApJ...742...93A} calibration) \nand MgII (using the \\citet{2012ApJ...753..125S} calibration).\nIn each panel, the solid black line is the unity line.}\n\\label{compare}\n\\end{figure*}\n\n\n\\section{X-ray Flux Estimates}\\label{xray_flux}\n\n\\noindent\nSince X-ray detections are available for all objects in this sample,\nX-ray flux estimates have also been included in the catalogue.\nXMMSL1 fluxes in the 0.2-12\\,keV range \nfrom \\citet{2008A&A...480..611S} are included.\n\\citet{2008A&A...480..611S} convert the XMMSL1 count rates \nto fluxes using a spectral model consisting of \nan absorbed power law with a photon index of 1.7 \nand N$\\rm_{H}=3\\times10^{20}cm^{-2}$.\nThe 2RXS fluxes were estimated using the method outlined below.\n\n\\subsection{Estimating 2RXS X-Ray Fluxes}\\label{2RXS_flux}\n\nMany of the sources in the 2RXS sample \nhave flux measurements close to\nthe \\textit{ROSAT} flux limit ($\\sim$10$\\rm ^{-13}\\,erg\\,cm^{-2}\\,s^{-1}$).\nTherefore, when estimating fluxes for this sample,\nit was necessary to correct for the Eddington bias.\nThis was done by adopting a Bayesian method \nto derive a probability distribution of fluxes based on the known \ndistribution of AGN as a function of flux.\nFollowing \\citet{1991ApJ...374..344K}, \\citet{2009ApJS..180..102L},\nand \\citet{2011MNRAS.414..992G},\nthe probability of a source having flux f$\\rm_{X}$, \ngiven an observed number of counts C, is \n\n\\begin{equation}\\label{prob_flux}\n\\rm P(f_{X}, C) = \\frac{T^{C}e^{-T}}{C!}\\pi(f_{X})\n\\end{equation}\n\n\\noindent where C is the total number of observed source and background counts,\nT is the mean expected total counts in the detection cell for a given flux,\nand $\\rm \\pi(f_{X})$ is the prior, which is the \ndistribution of AGN per unit X-ray flux interval.\nThe exact expression for the prior was taken from\n\\citet{2008MNRAS.388.1205G}, equation 1.\n\nSource and background counts were taken from \nthe 2RXS catalogue \\citep{2016A&A...588A.103B}.\nA flux-count rate conversion factor,\nwhich was required to estimate T in equation~\\ref{prob_flux},\nwas derived using \\texttt{XSPEC} \\citep{1996ASPC..101...17A}\nassuming a model consisting of \na power law (with $\\rm \\Gamma=2.4$ following \\citet{2017MNRAS.469.1065D})\nabsorbed by the Milky Way column density.\nThis method was used to estimate \nthe flux in the full ROSAT band (0.1-2.4\\,keV)\nas well the monochromatic flux at 2\\,keV.\n\nThe fluxes resulting from the method described above \nwith and without applying the prior (termed ``Bayesian'' and ``classical'', respectively)\nare compared in Figure~\\ref{bay_class_flux}.\nThe disagreement between the two flux estimates increases with decreasing flux,\nwhich is expected since, without the prior, \nthe classical method fails to account for the Eddington bias. \nLow count rate sources in this sample would be assigned \nunrealistically low Bayesian fluxes. \nTo avoid this, the flux was left as undetermined\nwhen the Bayesian flux estimate was more than a factor of ten smaller \nthan the classical flux estimate.\n\n\n\n\\section{Accessing the Data}\\label{access}\n\n\\noindent\nThe results from the spectral analysis discussed above,  \nalong with X-ray flux measurements and visual inspection results,\nhave been made available in an SDSS DR14 value added catalogue\nwhich is available at \nhttp:\/\/www.sdss.org\/dr14\/data\\_access\/value-added-catalogs\/.\nAdditionally, an extended version of the catalogue will be maintained at \nhttp:\/\/www.mpe.mpg.de\/XraySurveys\/SPIDERS\/SPIDERS\\_AGN\/\nThe column description for the catalogue is given in appendix~\\ref{columns}.\n\n\n\\begin{figure*}\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{lbol_z_shen.png}\n                \\includegraphics[width=0.49\\textwidth]{mass_lbol_shen.png}\n\\caption{\\footnotesize \nBolometric luminosity versus redshift (left panel) and \nbolometric luminosity versus BH mass (right panel)\nfor the sample presented in this work\nand the \\citet{2011ApJS..194...45S} sample.\nSources with $\\rm H_{\\beta}$-derived BH masses are shown in blue,\nand sources with MgII-derived BH masses are displayed in green.}\n\\label{properties}\n\\end{figure*}\n\n\n\\section{Comparing the UV and Optical Fitting Results}\\label{compare_uv_opt}\n\n\\noindent\nA subsample of sources have spectral measurements available from \nboth the MgII and H$\\beta$ spectral regions. \nIn order to test the consistency of the independent fits to these two regions,\nproperties measured from both were compared.\n\n\\subsection{$L_{2500\\AA}-L_{5100\\AA}$ Relation}\n\n\\noindent\nA  subsample of AGN whose spectra cover the rest-frame wavelengths \n2500$\\AA$ and 5100$\\AA$\nwas selected using the following criteria:\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm  ( (INSTRUMENT = SDSS) \\,\\,\\&\\&\\,\\, (0.52 < redshift < 0.8) ) \\,\\, || \\\\  \n\\rm ( (INSTRUMENT = BOSS) \\,\\,\\&\\&\\,\\, (0.46 < redshift <1.04) ) \n\\end{split}\n\\end{equation}\n\n\\noindent\nOf this sample, 1718 sources had reliable measurements of both \n$\\rm L_{2500\\AA}$ and $\\rm L_{5100\\AA}$.\nThe $\\rm L_{2500\\AA}-L_{5100\\AA}$ relation was fit \nusing the LINMIX \\citep{2007ApJ...665.1489K} package.\nLINMIX is a Bayesian linear regression algorithm \nthat accounts for uncertainties in both dependent and independent variables,\nas well as non-detections.\nThe upper left panel of figure~\\ref{compare}\nshows the $\\rm L_{2500\\AA}-L_{5100\\AA}$ distribution and\nthe best-fit relation\n\n\\begin{equation}\\label{flux_conversion}\n\\begin{split}\n\\rm Log_{10}(L_{5100\\AA}) = (0.841\\pm 0.007) Log_{10}(L_{2500\\AA}) \\\\\n\\rm -(6.0 \\pm 0.3)\n\\end{split}\n\\end{equation}\n\n\\noindent\nwith a regression intrinsic scatter of 0.0151.\nThe comparison between the estimated bolometric luminosities \nderived from the 3000$\\AA$ and 5100$\\AA$ monochromatic fluxes\nis shown in the upper right panel of figure~\\ref{compare}.\nEquation~\\ref{flux_conversion} can be used to estimate L$\\rm_{2500\\AA}$\nfrom L$\\rm_{5100\\AA}$, which allows low redshift sources to be included \nin the $\\rm \\alpha_{OX}$ analysis discussed in section~\\ref{aox_section}.\n\n\n\n\\subsection{Comparing MgII and H$\\beta$ FWHM Measurements}\\label{compare_mass}\n\nA  subsample of AGN whose spectra cover the broad\n$\\rm H\\beta$ and MgII emission lines \nwas selected using the following criteria\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm  ( (INSTRUMENT = SDSS) \\,\\,\\&\\&\\,\\, (0.45 < redshift < 0.81) ) \\,\\, || \\\\  \n\\rm ( (INSTRUMENT = BOSS) \\,\\,\\&\\&\\,\\, (0.38 < redshift <1.05) ) \n\\end{split}\n\\end{equation}\n\n\\noindent\nOf this sample, 2323 sources had FWHM measurements\nfor both H$\\beta$ and MgII.\nThe lower left panel of figure~\\ref{compare}\ndisplays the virial FWHM measurements from H$\\beta$ and MgII.\nThe resulting best-fit relation, fit using LINMIX, is\n\n\\begin{equation}\\nonumber\n\\begin{split}\n\\rm Log_{10}(FWHM_{MgII}) = (0.65\\pm0.01)\\, Log_{10}(FWHM_{H\\beta}) \\\\\n\\rm +(1.21\\pm0.04)\n\\end{split}\n\\end{equation}\n\n\\noindent\nwith a regression intrinsic scatter of 0.005.\nThis deviation from the one-to-one relation has also been observed by \n\\citet{2009ApJ...707.1334W}, who reported a slope of $0.81\\pm0.02$,\nand \\citet{2012ApJ...753..125S}, who found a slope of $0.57\\pm0.09$.\nThe single-epoch BH mass relations (equation~\\ref{mass_est})\naccount for the $\\rm FWHM_{MgII}-FWHM_{H\\beta}$ slope;\nwhen the correct BH mass calibration is used,\nthe H$\\beta$ and MgII virial FWHM measurements \nyield BH masses that are in close agreement\n(see the lower right panel of figure~\\ref{compare}).\n\n\n\n\\section{Sample Properties}\\label{prop}\n\n\\noindent\nFigure~\\ref{properties} presents the comparison between\nthis sample and the full sample of SDSS DR7 AGN \nwith optical spectral properties measured by \\citet{2011ApJS..194...45S}\nin the bolometric luminosity-redshift and bolometric luminosity-BH mass planes.\nAs discussed in section~\\ref{intro}, $\\rm H_{\\beta}$-derived BH masses \nare used where available (shown in blue), \nwhile MgII-derived masses are used \nfor the remaining higher-redshift sources (shown in green).\nThe left panel of figure~\\ref{properties} \nshows that this sample populates \nthe low-redshift, high-luminosity region of the parameter space,\nwhich is partially due to the high flux threshold of the X-ray selection.\nFrom the right panel of figure~\\ref{properties} \nit can be seen that the sample presented in this work \nappears to be well bounded by the Eddington limit \nat least up to $\\rm M_{BH}\\simeq 10^{9.5}\\,M_{\\odot}$.\n\n\n\n\\subsection{H$\\beta$ Line Components}\\label{VBC_section}\n\n\\noindent\nSection~\\ref{HB_fit_method} described how the H$\\beta$ line profile\nwas fit with either one, two, three, or four Gaussian components.\nFigure~\\ref{line_width} displays the resulting distribution of \n$\\rm H\\beta$ FWHM measurements\n(the panels are split based on the \nnumber of Gaussian components required to fit the line).\nThere is a clear peak in the distribution at low FWHM\nassociated with the narrow H$\\beta$ core\ntypically measured at a few hundred $\\rm km\\,s^{-1}$.\nAbove $\\rm \\sim 1000\\,km\\,s^{-1}$ the distribution is bimodal\n(in the two lower panels) with a large number of sources showing evidence for the \n``very broad component'' (VBC) of $\\rm H\\beta$\nat FWHM$\\rm \\,\\geq 10000\\,km\\,s^{-1}$\nalso discussed in \\citet{2010MNRAS.409.1033M}.\n\nIt has been suggested that the VBC\nis emitted from a distinct physical region, \nand is possibly the result of line emission from the accretion disk \n\\citep[e.g.][]{2009MNRAS.400..924B}.\nIf the VBC represents emission from the accretion disk,\nthen a strong VBC may result in a bias towards  \na higher BH mass estimate, since the \nsingle-epoch method assumes a calibration that is based on \nthe luminosity-BLR radius relation.\nHowever, since the kinematics and physical origin of the VBC remains uncertain,\ndetected VBCs have not been excluded from the broad line profiles \nused to measure the virial FWHM in this analysis \n(as discussed in section~\\ref{bl_decomp}).\n\n\n\\begin{figure}\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{component_hist.png}\n\\caption{\\footnotesize \nDistribution of H$\\beta$ Gaussian FWHM values.\nThe panels are split based on the number of Gaussians required to fit the line.\nThe coloured histograms each represent one of up to four \npossible Gaussians used to fit the H$\\rm_{\\beta}$ line. \nThe grey histograms represent the sum of the individual coloured histograms.}\n\\label{line_width}\n\\end{figure}\n\n\n\\begin{figure*}\n                \\includegraphics[width=0.49\\textwidth]{r_fwhm.png}\n                \\includegraphics[width=0.49\\textwidth]{r_fwhm_eddrat.png}\n\\caption{\\footnotesize \nFWHM of the broad component of H$\\beta$ versus R$\\rm _{FeII}$.\nThe left panel displays the sample described in section~\\ref{4DE1_section} (grey), \nand sources with a median spectral S\/N$\\,\\geq\\,$20 (blue).\nThe right panel presents the same subsample of high S\/N sources, \ncolour coded to indicate the trend in Eddington ratio across the distribution.\nThe grey dashed line marks the division between \npopulation A ($\\rm FWHM\\,H_{\\beta}^{BC}\\leq4000\\,km\\,s^{-1}$) and \npopulation B ($\\rm FWHM\\,H_{\\beta}^{BC}\\geq4000\\,km\\,s^{-1}$) sources.\nThe size of the typical uncertainties, multiplied by a factor of five,\nin both variables for the high-S\/N subsample is also shown.}\n\\label{4DE1}\n\\end{figure*}\n\n\n\\subsection{This Sample in the 4D Eigenvector 1 Context}\\label{4DE1_section}\n\n\\noindent\nThe 4D Eigenvector 1 (4DE1) system \n\\citep{1992ApJS...80..109B, 2000ApJ...536L...5S, 2011BaltA..20..427S}\naims to define a set of parameters that uniquely account for AGN diversity.\nTwo main 4DE1 parameters are\nthe FWHM of the broad component of H$\\beta$ ($\\rm FWHM \\, H_{\\beta}^{BC}$)\nand the strength of the FeII emission relative to that of H$\\beta$, defined as \n\n\\begin{equation}\\nonumber\n\\rm R_{FeII} = F_{FeII}\/F_{H\\beta}\n\\end{equation}\n\n\\noindent\nwhere $\\rm F_{FeII}$ and $\\rm F_{H\\beta}$\nare the fluxes of the \nFeII emission in the 4434-4684$\\rm \\AA$ range\nand broad $\\rm H\\beta$ line, respectively.\nA sample of 2098 sources with measurements of these parameters \nand reliable spectral fits ($\\rm 0 \\le \\chi^{2}_{\\nu,H\\beta} \\le 1.2$)\nwas selected. The left panel of figure~\\ref{4DE1} shows \nthe distribution of this sample \nin the 4DE1 parameter space (grey).\nIt is expected that a reliable measurement of the FeII component \nwill be difficult for many of the lower S\/N sources\n\\citep[see][]{2003ApJS..145..199M}.\nFor this reason, the subset of sources in figure~\\ref{4DE1} \nwith a median S\/N greater than or equal to 20 per resolution element is also shown (blue).\nThe right panel of figure~\\ref{4DE1} presents the higher S\/N sources, \ncolour-coded as a function of Eddington ratio.\nThe expected trend of increasing Eddington ratio \ntowards smaller $\\rm FWHM\\,H_{\\beta}^{BC}$ and larger R$\\rm_{FeII}$ \nis observed for this sample of high-S\/N sources.\nTypically, sources with both high $\\rm R_{FeII}$\nand high $\\rm FWHM \\, H_{\\beta}^{BC}$ are not observed. \nIf these sources exist, they may be difficult to detect \nsince strong FeII emission might conceal a faint $\\rm H_{\\beta}$ broad component.\nThe potential bias in the 4DE1 plane source distribution \ndue to model limitations and spectral S\/N \nis discussed in sections~\\ref{sim} and \\ref{sim_bias}.\n\nThe grey dashed line in the right panel of figure~\\ref{4DE1}\nindicates the division between \npopulation A ($\\rm FWHM\\,H_{\\beta}^{BC}\\leq4000\\,km\\,s^{-1}$) \nand population B ($\\rm FWHM\\,H_{\\beta}^{BC}\\geq4000\\,km\\,s^{-1}$)\nsources in the 4DE1 context \\citep[see][]{2011BaltA..20..427S}.\nPopulation A sources often possess Lorentzian broad line profiles,\nand it has been suggested that Gaussian fits to population A broad lines \nwill result in an underestimation of the BH mass \\citep[see][]{2014AdSpR..54.1406S}.\n\n\n\n\\subsection{Relationship Between AGN X-ray and Optical Emission}\\label{aox_section}\n\n\\noindent\nQuasars exhibit a non-linear relationship between their X-ray and UV emission,\nusually represented by the $\\rm \\alpha_{OX}$ parameter\n\n\\begin{equation}\\nonumber\n\\rm \\alpha_{OX} = \\frac{Log(L_{2\\,keV}\/L_{2500\\,\\AA})}{Log(\\nu_{2\\,keV}\/\\nu_{2500\\,\\AA})}\n\\end{equation}\n\n\\noindent\nwhere $\\rm L_{2\\,keV},\\, L_{2500\\,\\AA},\\, \\nu_{2\\,keV},\\, and \\, \\nu_{2500\\,\\AA}$\nare the monochromatic luminosities and frequencies at \n2\\,keV and $\\rm 2500\\,\\AA$, respectively\n\\citep{2003AJ....125.2876V, 2005AJ....130..387S, 2006AJ....131.2826S, 2007ApJ...665.1004J, 2008ApJS..176..355K, 2009ApJ...690..644G, 2009ApJS..183...17Y, 2010A&A...512A..34L}.\nThe $\\rm \\alpha_{OX}$ parameter is considered to be \na proxy for the relative contribution of\nthe UV accretion disk emission and the \nX-ray emission from the surrounding corona\nto the total luminosity.\nIn order to study this relationship, \na sample of sources with measurements of the \n2keV,  2500\\AA, and 5100\\AA \\, luminosities was selected.\nFor lower redshift sources without spectral coverage of 2500\\AA,\nequation~\\ref{flux_conversion} was used to estimate the 2500\\AA\\, luminosity\nfrom the 5100\\AA\\, luminosity.\nExtended sources were removed in order to prevent additional scatter in the relationship due to the contribution of the host galaxy.\nThis was done by requiring that the SDSS g band ``stellarity''\\footnote{\nFor a description of how cModelMag\\_g and psfMag\\_g are measured see\nhttps:\/\/www.sdss.org\/dr12\/algorithms\/magnitudes\/} \n(defined as S(g) = cModelMag\\_g- psfMag\\_g) \nlies between $\\pm$0.1. \nThis sample does not contain X-ray sources with \nmore than one potential AllWISE counterpart and therefore avoids cases where \nthe X-ray detection includes emission from more than one object.\nThis selection process resulted in a sample of 4777 sources.\nFigure~\\ref{aox} shows the $\\rm \\alpha_{OX}$ parameter versus \nthe monochromatic luminosity at 2500$\\AA$.\nThe $\\rm \\alpha_{OX}-L_{2500\\AA}$ relation\nwas fit using LINMIX, which gave the following best-fit result\n\n\\begin{equation}\\nonumber\n\\rm \\alpha_{OX} = 2.39\\pm0.16 - (0.124\\pm0.005) Log(L_{2500\\AA})\n\\end{equation}\n\n\\noindent\nwith a regression intrinsic scatter of 0.0034.\nThis slope is consistent with previous results from the literature\n\\citep[e.g.][]{2008ApJS..176..355K}.\n\n\n\\begin{figure}\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{aox_L2500.png}\n\\caption{\\footnotesize\n$\\rm \\alpha_{OX}$ versus UV luminosity.\nThe dashed line is the best linear fit to the distribution.\nThe size of the typical uncertainties in both variables is also shown.}\n\\label{aox}\n\\end{figure}\n\n\n\n\\section{Interpreting the Data and Limitations}\\label{limits}\n\n\\noindent\nIn this section, the reliability and limitations of the sample will be discussed.\n\n\\subsection{Measuring the FeII Emission}\n\n\\noindent\nDistinguishing the FeII component from the continuum emission \nbecomes more difficult when using low S\/N spectra.\nIn addition, for a given S\/N, it may also be more difficult to detect FeII emission \nif the intrinsic broadening of the FeII lines is large, \nsince broader, blended FeII emission lines are more likely to be fit \nby the model as continuum emission \\citep[see][]{2003ApJS..145..199M}.\nUsing simulated AGN spectra, \\citet{2003ApJS..145..199M} \nestimate the minimum detectable optical FeII emission \nas a function of H$\\beta$ width for different bins of S\/N.\n\nA poor fit to the FeII emission may affect the accuracy of the BH mass estimates,\nsince FeII emission can influence measurements of both the broad line width (see section~\\ref{feII_FWHM}) and the continuum luminosity.\nFeII emission may also conceal a broad $\\rm H \\beta$ component\nthus biasing a source's position in the 4DE1 plane (figure~\\ref{4DE1}).\nThese potential issues are tested in the following three sections.\n\n\n\n\\subsubsection{Accuracy of the Broad Emission Line FWHM Measurements \nfor Sources with FeII Continuum Emission}\\label{feII_FWHM}\n\nA poor fit to the FeII emission may affect the measurement of \nthe broad emission line width.\nTo quantify the magnitude of this effect, the $\\rm H\\beta$ fitting script \n(using four Gaussians to fit $\\rm H\\beta$) was run with and without the FeII template\non a sample of $\\rm \\sim 400$ randomly selected sources.\nThe fit without the FeII template represents the most extreme case \nwhere the FeII emission is completely ignored by the model.\nTherefore, the change in line widths measured by these two models \nshould be the upper limit on what can be expected \nfor cases where the FeII fit is inadequate.\nFigure~\\ref{feII_test_plot} shows that the line width dispersion induced by \nignoring the presence of FeII emission is $\\rm \\simeq 640 \\,km\\,s^{-1}$.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{width_hist.png}\n\\caption{\\footnotesize\nComparison of the $\\rm H \\beta$ FWHM measurements\nderived using a model with and without an FeII template.\nThe vertical dashed line shows the mean value of the distribution.}\n\\label{feII_test_plot}\n\\end{figure}\n\n\n\\subsubsection{Model Limitations in Detecting Sources in the 4DE1 Plane}\\label{sim}\n\n\\noindent\nSources with both large $\\rm R_{FeII}$ and large $\\rm FWHM\\,H_{\\beta}^{BC}$\nare typically not observed, however,\nthis absence may be due to model limitations;\nat high $\\rm R_{FeII}$ the broad $\\rm H\\beta$ component \nmay be concealed beneath the FeII emission, \nand therefore may not be detected.\nThe experiment outlined in this section was carried out in order to determine \nwhether the spectral fitting code used in this work\nwould return accurate measurements \nfor sources with high $\\rm R_{FeII}$ and $\\rm FWHM\\,H_{\\beta}^{BC}$ values.\n\nA parameter space defined by \n$\\rm 0.1 \\leq R_{FeII} < 5$ and\n$\\rm 1000\\,\\,km\\,s^{-1} \\leq FWHM\\,H_{\\beta}^{BC} < 15000 \\,\\,km\\,s^{-1}$\nwas divided into a 12$\\times$12 grid.\n10 S\/N bins between 5 and 50 \n(a representative range for the samples presented in this work)\nwere selected for each point on the grid,\nand 10 spectra were simulated for each \n$\\rm R_{FeII}-\\rm FWHM \\, H_{\\beta}^{BC}-S\/N$ combination, \nresulting in 14400 simulated spectra.\nFor the parameters that were fixed in this experiment, \nthe interquartile mean of the best-fit values for the type 1 AGN in this sample were used.\nThe $\\rm H\\beta$ line profile was modelled with one narrow and one broad Gaussian.\nThe wavelength range was set to $\\rm 4420 - 5500 \\AA$ (as in section~\\ref{HB_fit_method})\nand the logarithmic wavelength \nspacing\\footnote{https:\/\/www.sdss.org\/dr12\/spectro\/spectro\\_basics\/} \nwas set to be equal to that of SDSS spectra;\n\n\\begin{equation}\\nonumber\n\\rm Log_{10}\\lambda_{i+1} - Log_{10}\\lambda_{i} = 0.0001\n\\end{equation}\n\nThese spectra were fit using a version of the $\\rm H\\beta$ fitting script \nwhich used one narrow and one broad Gaussian component to fit $\\rm H\\beta$.\nThe minimum S\/N required for the fitting script to return the correct \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ combinations\nwas then determined.\nIn order to consider an $\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$\ncombination detectable at a given S\/N,\nat least  7\/10 spectra were required to have best fit \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ values \nthat agreed with the input values.\n\nFigure~\\ref{4DE1_sim_1} shows the detectable \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$\ncombinations for each point on the grid, along with the minimum S\/N required to \ndetect that combination. \nIt is clear from figure~\\ref{4DE1_sim_1} that \nat the S\/N levels available in this sample,\na large region of the 4DE1 parameter space would not be detected.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{rfeII_fwhm_sn_test3.png}\n\\caption{\\footnotesize\nMinimum S\/N required to detect a range of \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ combinations.\nThe blue grid indicates the range of the parameter space covered\nin the simulation described in section~\\ref{sim}.\nPoints on the grid that do not have a minimum S\/N indicator represent \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ combinations\nthat are not detectable even at the highest S\/N used in this experiment.\nSources detected at these $\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ \ncombinations are likely to be spurious (see figure~\\ref{sn_slice}).}\n\\label{4DE1_sim_1}\n\\end{figure}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{rfeII_fwhm_sn_5.png}\n\\includegraphics[width=0.49\\textwidth]{rfeII_fwhm_sn_455.png}\n\\caption{\\footnotesize\nComparison between the measured and simulated \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ values\nfor the highest and lowest S\/N bins used in section~\\ref{sim}.\nFor clarity, each figure displays only one of the ten sets of spectra\nfor that S\/N.\nThe values used to simulate the spectra (grey points)\nare connected to the corresponding best fit measurements\n(except for cases where either the broad $\\rm H\\beta$\nor FeII components were not detected).\nFor clarity, the figures do not show a small number of \nunphysically high $\\rm R_{FeII}$ measurements.}\n\\label{sn_slice}\n\\end{figure*}\n\n\n\\subsubsection{Bias in the $\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$\nDistribution due to Model Limitations}\\label{sim_bias}\n\n\\noindent\nFigure~\\ref{sn_slice} shows the comparison between the simulated and measured\n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ values \nfor the highest and lowest S\/N bins used in section~\\ref{sim}.\nThe left panel of figure~\\ref{sn_slice} shows that at low S\/N\nthe results are clearly biased against high  \n$\\rm R_{FeII}$ and $\\rm FWHM \\, H_{\\beta}^{BC}$ values.\nAt higher S\/N (figure~\\ref{sn_slice}, right panel), \nthe accuracy of the lower left quadrant measurements is significantly improved.\nHowever, even at S\/N=45.5 \n(which is approximately the upper end of the S\/N distribution \nof the samples presented in this work)\nthe high $\\rm R_{FeII}$ - $\\rm FWHM \\, H_{\\beta}^{BC}$ measurements \ndeviate significantly from the corresponding ``true'' values.\nThis may suggest that the L-shaped distribution of sources\nin the 4DE1 plane (e.g. figure~\\ref{4DE1})\nis at least in part due to model limitations.\n\n\n\n\\subsection{Reliability of the Single-Epoch Method for Mass Estimation}\\label{reliability}\n\n\\noindent\nAssuming that AGN broad emission lines are produced by gas \nwhose motion is dominated by the gravitational potential of the central SMBH,\nthe single-epoch method is expected to produce \nreliable mass estimates when compared to RM \\citep[see][]{2006ApJ...641..689V},\nwith a systematic uncertainty of 0.3-0.4 dex.\nHowever, it is not clear how to measure \nthe virial FWHM of lines that deviate from this norm.\n\nThe spectrum shown in the left panel of figure~\\ref{unusual_fig}\nis an example of a source which exhibits \na double-peaked H$\\beta$ line profile,\nwhere a clear inflection point is visible \nbetween two velocity-shifted broad line components.\nDouble-peaked broad line profiles in AGN \nare expected to be the result of emission from the accretion disk\n\\citep{1988MNRAS.230..353P, 1989ApJ...339..742C, 1994ApJS...90....1E, 2003AJ....126.1720S, 2003ApJ...599..886E}.\n\\citet{2007MNRAS.376.1335Z} have found that single-epoch BH mass estimates \nobtained from double-peaked line profiles \nare significantly larger than BH mass estimates \nderived from stellar velocity dispersion measurements.\n\\citet{2007MNRAS.376.1335Z} suggest that this discrepancy \nis the result of an overestimation of the BLR radius\nby the single-epoch mass calibrations for these objects.\nTherefore, the BH mass estimates provided in this work for sources \nwhich exhibit double-peaked broad emission lines \nshould be treated with caution.\n\nThe right panel of figure~\\ref{unusual_fig} shows\nan example of narrow absorption in the UV portion of the spectrum\ncaused by intervening absorbing material along the line of sight to the AGN.\nSources identified during the visual inspection \nas having narrow absorption lines \nhave been flagged in the catalogue (column 189; flag\\_abs).\nThese sources were fit using the model described in section~\\ref{mgII_fit} \nwith the absorption line regions masked.\nHowever, in many cases, the absorption features distort the broad MgII line, \nand therefore the resulting BH mass estimates may not be reliable.\n\n\n\\begin{figure*}\n\t\\centering\n                \\includegraphics[width=0.49\\textwidth]{spec-7512-56777-0321.png}\n                \\includegraphics[width=0.49\\textwidth]{spec-8188-57348-0946}\n\\caption{\\footnotesize\nLeft panel: Example of a source with a double-peaked H$\\rm \\beta$ line profile\n(plate=7512, MJD=56777, fiber=321).\nRight panel: Example of a source showing narrow UV absorption features\nwhich have been masked when fitting the model\n(plate=8188, MJD=57348, fiber=946). \nSee section~\\ref{reliability} for details.}\n\\label{unusual_fig}\n\\end{figure*}\n\n\n\n\\section{Conclusions}\n\n\\noindent\nThis work presents a catalogue of spectral properties for\nall SPIDERS type 1 AGN up to SDSS DR14.\nVisual inspection results were used to select a reliable subsample for spectral analysis,\nand the spectral regions around H$\\beta$ and MgII were fit with a multicomponent model.\nUsing the single-epoch method, BH masses, bolometric luminosities, \nEddington ratios, along with additional spectral parameters were measured. \nA catalogue containing these results has been made available\nas part of a set of SDSS DR14 value added catalogues.\nThis catalogue also includes the results of a visual inspection of the sample,\nand is available at \nhttp:\/\/www.mpe.mpg.de\/XraySurveys\/SPIDERS\/SPIDERS\\_AGN\/.\n\n\n\n\\section{Acknowledgements}\nDC has participated in the International Max Planck Research School \non Astrophysics at the Ludwig Maximilians University Munich.\nDC also acknowledges financial support from the Max Planck Society.\nDC would also like to thank Riccardo Arcodia and Julien Wolf\nfor many helpful discussions.\nThe authors would like to thank Josephine Reisinger \nfor her contribution to the visual inspection of sources in this sample.\nFinally, the authors would like to thank \nthe anonymous referee for providing a thorough critique of the paper \nwhich greatly improved its content.\n\nFunding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, \nthe U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges\nsupport and resources from the Center for High-Performance Computing at\nthe University of Utah. The SDSS web site is www.sdss.org.\n\nSDSS-IV is managed by the Astrophysical Research Consortium for the \nParticipating Institutions of the SDSS Collaboration including the \nBrazilian Participation Group, the Carnegie Institution for Science, \nCarnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, \nInstituto de Astrof\\'isica de Canarias, The Johns Hopkins University, \nKavli Institute for the Physics and Mathematics of the Universe (IPMU) \/ \nUniversity of Tokyo, Lawrence Berkeley National Laboratory, \nLeibniz Institut f\\\"ur Astrophysik Potsdam (AIP),  \nMax-Planck-Institut f\\\"ur Astronomie (MPIA Heidelberg), \nMax-Planck-Institut f\\\"ur Astrophysik (MPA Garching), \nMax-Planck-Institut f\\\"ur Extraterrestrische Physik (MPE), \nNational Astronomical Observatories of China, New Mexico State University, \nNew York University, University of Notre Dame, \nObservat\\'ario Nacional \/ MCTI, The Ohio State University, \nPennsylvania State University, Shanghai Astronomical Observatory, \nUnited Kingdom Participation Group,\nUniversidad Nacional Aut\\'onoma de M\\'exico, University of Arizona, \nUniversity of Colorado Boulder, University of Oxford, University of Portsmouth, \nUniversity of Utah, University of Virginia, University of Washington, University of Wisconsin, \nVanderbilt University, and Yale University.\n\nFunding for SDSS-III has been provided by the Alfred P. Sloan Foundation,\nthe Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. \nThe SDSS-III web site is http:\/\/www.sdss3.org\/.\n\nSDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including \nthe University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, \nthe French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrof\\'isica de Canarias, \nthe Michigan State\/Notre Dame\/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, \nMax Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, \nNew York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, \nthe Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, \nUniversity of Virginia, University of Washington, and Yale University.\n\nFunding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, \nthe Participating Institutions, the National Science Foundation, the U.S. Department of Energy, \nthe National Aeronautics and Space Administration, the Japanese Monbukagakusho, \nthe Max Planck Society, and the Higher Education Funding Council for England. \nThe SDSS Web Site is http:\/\/www.sdss.org\/.\n\nThe SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. \nThe Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, \nUniversity of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, \nthe Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, \nthe Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), \nLos Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), \nNew Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, \nPrinceton University, the United States Naval Observatory, and the University of Washington.\n\nThis publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory\/California Institute of Technology, and NEOWISE, which is a project of the Jet Propulsion Laboratory\/California Institute of Technology. WISE and NEOWISE are funded by the National Aeronautics and Space Administration.\n\nPlot colours were in part based on www.ColorBrewer.org, \nby Cynthia A. Brewer, Penn State.\nThe TOPCAT tool \\citep{2005ASPC..347...29T} was used during this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe description of perturbative quantum field theory in terms of Hopf algebras \\cite{Kre98,CK99,CK00} (cf. \\cite{CM04c,CM04b,CM06}) has led to a better understanding of the combinatorial structure of renormalization. At the same time, it provided a beautiful interaction of quantum field theory with several parts of mathematics. \n\nAlthough the Hopf algebra was modelled on a scalar quantum field theory, it can be generalized to any quantum field theory. However, subtleties appear in the case of quantum gauge theories. Gauge theories are field theories that are invariant under a certain group, known as the gauge group. In the process of renormalization, one has to deal with the `overcounting' due to the gauge degrees of freedom. Furthermore, one has to understand how the gauge symmetries manifest themselves in the quantum field theory. Important here are the so-called Ward-Takahashi (or Slavnov-Taylor) identities.\n\nThe first example of a gauge theory is quantum electrodynamics (QED), which has the abelian group $U(1)$ as a gauge group. The Hopf algebraic structure of Feynman graphs in QED was desrcibed in \\cite{BK99,KD99} and more recently in \\cite{VP04}. A slightly different approach is taken in \\cite{BF01,BF03}, where Hopf algebras on planar binary trees were considered. \nThe anatomy of a gauge theory with gauge group $SU(3)$ (known as quantum chromodynamics) has been discussed in the Hopf algebra setting in \\cite{Kre05}, with a central role played by the Dyson-Schwinger equations (see also \\cite{KY06}). However, a full understanding of renormalization of general gauge theories is still incomplete. This paper is a modest attempt towards understanding this by working out explicitly the Hopf algebraic structure underlying renormalization of quantum electrodynamics, thereby implementing the Ward-Takahashi identities in a compatible way.\n\n\\bigskip\n\nWe start in Section \\ref{sect:hopf} by defining the commutative Hopf algebra $H$ of Feynman graphs for quantum electrodynamics. The approach we take differs from the one in \\cite{BF03} where a noncommutative Hopf algebra was considered. This was motivated by the fact that in QED, Feynman amplitudes are (noncommuting) matrices. Instead, we work with a commutative Hopf algebra and encode the matricial form of the Feynman amplitudes into the distributions giving the external structure of the graph. Our approach has the advantage of still being able to use the duality between commutative Hopf algebras and affine group schemes in the formulation of the BPHZ-procedure.\n\nIn Section \\ref{sect:birkhoff}, we associate a Feynman amplitude to a Feynman graph, as dictated by the Feynman rules. The BPHZ-formula of renormalization is obtained as a special case of the Birkhoff decomposition for affine group schemes \\cite{CK99}. \n\nThen in Section \\ref{sect:wt}, we incorporate the Ward-Takahashi (WT) identities as relations between graphs, much as was done by 't Hooft and Veltman in \\cite{HV73}. We show that these relations are compatible with the coproduct in the Hopf algebra $H$, so that they define a Hopf ideal. This reflects the physical fact that WT-identities are compatible with renormalization (in the dimensional regularization and minimal subtraction scheme). \n\nThe Feynman amplitudes can now be defined on the quotient Hopf algebra of $H$ by this ideal, and we conclude that the counterterms as well as the renormalized Feynman amplitudes satisfy the WT-identities in the physical sense ({i.e.}, between Feynman amplitudes). In particular, we arrive at the well-known identity $Z_1=Z_2$ as derived by Ward in \\cite{War50}. \n\n\n\n\\section{Hopf algebra structure of Feynman graphs in QED}\n\\label{sect:hopf}\nQuantum electrodynamics in 4 dimensions is given by the following classical Lagrangian,\n\\begin{align*}\n\\mathcal{L}=- \\frac{1}{4} F_{\\mu\\nu} F^{\\mu\\nu} - \\frac{1}{2\\xi}(\\partial^\\mu A_\\mu)^2 +\\overline\\psi \\left( \\gamma^\\mu(\\partial_\\mu+ e A_\\mu)-m\\right) \\psi,\n\\end{align*}\nwhere $F_{\\mu\\nu}=\\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu$ is the {\\it field strength} of the {\\it gauge potential} $A_\\mu$, $\\psi$ is the spinor describing the electron with mass $m$ and electric charge $e$ and $\\gamma^\\mu, \\mu=1,\\ldots,4$ are the $4 \\times 4$ Dirac matrices. Finally, the real parameter $\\xi$ is the {\\it gauge-fixing parameter}. \n\nThe Lagrangian $\\mathcal{L}$ describes the dynamics and the interaction as well as the gauge fixing: we can write $\\mathcal{L}=\\mathcal{L}_0 + \\mathcal{L}_\\mathrm{int}+\\mathcal{L}_\\mathrm{gf}$, where the free, interaction and gauge fixing parts are\n\\begin{align*}\n\\mathcal{L}_0&=A_\\mu D_{\\mu\\nu} A_\\nu + \\overline\\psi (\\gamma^\\mu \\partial_\\mu -m) \\psi,\\\\\n\\mathcal{L}_\\mathrm{int}&= e \\overline\\psi A_\\mu \\psi.\\\\\n\\mathcal{L}_\\mathrm{gf} &= -\\frac{1}{2\\xi}(\\partial^\\mu A_\\mu)^2.\n\\end{align*}\nThe adjective ``quantum'' for electrodynamics is justified when this Lagrangian is used in computing probability amplitudes. In perturbation theory, one expands such amplitudes in terms of Feynman diagrams which are the graphs constructed from the following vertices, as dictated by the form of the Lagrangian:\n\\begin{center}\n    \\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{photon,label=$A_\\mu$}{l,v}\n      \\fmf{photon,label=$A_\\nu$}{v,r}\n    \\end{fmfgraph*}\\qquad\n    \\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain,label=$\\overline\\psi$}{l,v}\n      \\fmf{plain,label=$\\psi$}{v,r}\n    \\end{fmfgraph*}\\qquad\n    \\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmf{photon,label=$A_\\mu$}{l,v}\n      \\fmf{plain,label=$\\overline\\psi$}{r1,v}\n      \\fmf{plain,label=$\\psi$}{v,r2}\n    \\end{fmfgraph*}\n\\end{center}\n\\begin{rem}\n\\label{rem:el-mass}\nIn fact, we will also need the 2-point vertex \n\\begin{fmfgraph*}(40,10)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfdot{v}\n    \\end{fmfgraph*}\n, associated to the electron mass term, similar to \\cite{BF01}. \n\\end{rem}\nWe will focus on the so-called {\\it one-particle irreducible} (1PI) {\\it graphs}, which are graphs that are not trees, and that cannot be disconnected by cutting a single internal edge. In QED, there are three types of 1PI graphs that are of interest in renormalization theory: the vacuum polarization, the  electron self-energy and the full vertex graphs:\n\\begin{center}\n    \\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmflabel{$\\Gamma(q,\\mu,\\nu)=q$}{l}\n      \\fmflabel{$q$}{r}\n      \\fmf{photon,label=$\\mu$}{l,v}\n      \\fmf{photon,label=$\\nu$}{v,r}\n      \\fmfblob{.25w}{v}\n    \\end{fmfgraph*}\\\\\n    \\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmflabel{$\\Gamma(p)=p$}{l}\n      \\fmflabel{$p$}{r}\n      \\fmf{fermion}{l,v}\n      \\fmf{fermion}{v,r}\n      \\fmfblob{.25w}{v}\n    \\end{fmfgraph*}\\\\\n    \\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmflabel{$\\Gamma(q,\\mu;p)=q$}{l}\n      \\fmflabel{$p$}{r1}\n      \\fmflabel{$p+q$}{r2}\n      \\fmf{photon,label=$\\mu$}{l,v}\n      \\fmf{fermion}{r1,v}\n      \\fmf{fermion}{v,r2}\n      \\fmfblob{.25w}{v}\n    \\end{fmfgraph*}\n\\end{center}\n\\bigskip\nHere the blob stands for any 1PI graph of the type dictated by the external lines.\n\n\\bigskip\n\nWe now describe the {\\it external structure} of a Feynman graph $\\Gamma$, assigning (among other data) momenta to the external legs. In physics, the Feynman amplitude of $\\Gamma$ (dictated by the Feynman rules, see below) is evaluated with respect to this external structure (see below). Mathematically, the external structure is a distribution on the space of Feynman amplitudes, understood as test functions on a suitable space. \n\nMore explicitly, in the case of QED the (Euclidean) Feynman amplitudes $U(\\Gamma)$ are functions in $C^\\infty(E_\\Gamma) \\otimes M_4(\\C)$ where\n\\begin{align*}\nE_\\Gamma=\\left\\{ (q_1,\\ldots,q_n,\\mu_1,\\ldots,\\mu_n,p_{n+1}, \\ldots, p_N) : \\sum p_i+q_j=0,  \\right\\}\n\\end{align*}\nwith $\\{1,\\ldots,n\\}$ the set of indices labelling the external photon lines of $\\Gamma$ with spatial index $\\{\\mu_1,\\ldots,\\mu_n\\}$ respectively, and $\\{n+1,\\ldots,N\\}$ is the set of its external electron lines. The factor $M_4(\\C)$ can be understood from the fact that in QED, Feynman amplitudes are matrix-valued functions on $E_\\Gamma$. \n\nFor QED, the external structure is thus given by an element in the space of distributions on $C^\\infty(E_\\Gamma) \\otimes M_4(\\C)$; we denote this space by $\\left(C^\\infty(E_\\Gamma) \\otimes M_4(\\C)\\right)'$. For example, for the above full vertex graph $\\Gamma$, $E_\\Gamma=\\left\\{ (q,\\mu,p,p'): \\sum q+ p+p'=0 \\right\\}$ and a typical distribution is given by $\\sigma^{kl}=\\delta_{q,\\mu,p,-p-q} \\otimes e^{kl}$ in terms of a Dirac mass and the (standard) dual basis $\\{ e^{kl}\\}$ of $M_4(\\C)'$. Evaluation on the Feynman amplitude $U(\\Gamma) \\in C^\\infty(E_\\Gamma)\\otimes M_4(\\C)$ is then given by the pairing\n\\begin{align*}\n\\langle \\sigma^{kl}, U(\\Gamma) \\rangle = U(\\Gamma)(q,\\mu;p,-p-q)_{kl}\n\\end{align*}\nNote that $p'=-p-q$ is translated in the diagram by a reversal of the corresponding arrow with associated momentum $p+q$. \n\n\\bigskip\n\nWe will be interested in the following special external structures for the different graphs introduced above. A full vertex graph $\\Gamma(q,\\mu;p)$ at zero momentum transfer (meaning $q=0$) can be written in terms of the two form factors $F_1, F_2$ (cf. \\cite[Section 6.2]{PS95})\n\\begin{align}\nU(\\Gamma)(q=0,\\mu;p)=\\gamma_\\mu F_1(p^2) + \\frac{\\not{p} \\gamma_\\mu \\not{p}}{p^2} F_2(p^2).\n\\end{align}\nwhere $\\not{p}=\\sum_\\mu \\gamma^\\mu p_\\mu$. Only the first form factor $F_1$ requires renormalization; therefore, we define the following external structure:\n\\begin{align}\n\\label{ext1}\n\\langle \\sigma_\\ext{1},f \\rangle &= \\frac{1}{16}\\sum_\\mu \\mathrm{tr~} \\gamma_\\mu f(q=0,\\mu;p=0),\\qquad \\forall f \\in C^\\infty(E_\\Gamma) \\otimes M_4(\\C),\n\\end{align}\nwhere the normalization comes from $\\mathrm{tr~} \\gamma^\\mu \\gamma_\\mu = 16$. For the Ward-Takahashi identities, we will also need the following external structure,\n\\begin{align}\n\\langle \\sigma_\\ext{0,\\mu, p}^{kl}, f \\rangle = f(q=0, \\mu; p)_{kl},\n\\end{align}\nwhich puts finite momentum $p$ on the ingoing and outgoing electron lines, but zero momentum on the external photon line (zero-momentum transfer). This external structure allows us to treat Ward-Takahashi identities in massless QED as well. \n\n\\begin{rem}\nThe form factors $F_1$ and $F_2$ can also be recovered separately by using the following two distributions \\cite{KD99},\n\\begin{align}\n\\langle \\sigma',f \\rangle &= \\mathrm{tr~} \\sum_\\mu \\gamma^\\mu f(q=0, \\mu;p),  \\\\\n\\langle \\sigma'',f \\rangle &= \\mathrm{tr~} \\sum_\\mu \\not{p} p_\\mu f(q=0, \\mu;p).\n\\end{align}\n\\end{rem}\n\n\\bigskip\n\nFor an electron self-energy graph $\\Gamma(p)$, we have $E_\\Gamma = \\{ (p)\\}$ and the corresponding Feynman amplitude is written in terms of two form factors:\n\\begin{align}\nU(\\Gamma)(p)=\\not{p} A(p^2) + m B(p^2).\n\\end{align}\nCorrespondingly, there are two distributions, which we choose of the form,\n\\begin{align}\n\\label{ext0}\n\\langle \\sigma_\\ext{0},f \\rangle &= \\frac{1}{16} \\sum_\\mu \\mathrm{tr~} \\gamma_\\mu \\frac{\\partial}{\\partial p_\\mu} f(p) \\big|_{p=0} +  m^{-1} ~\\mathrm{tr~} f(p=0),\\\\\n\\label{ext2}\n\\langle \\sigma_\\ext{2},f \\rangle &= \\frac{1}{16} \\sum_\\mu \\mathrm{tr~} \\gamma_\\mu \\frac{\\partial}{\\partial p_\\mu} f(p) \\big|_{p=0},\n\\end{align}\nfor all $f \\in C^\\infty(E_\\Gamma) \\otimes M_4(\\C)$. Also, there is an external structure which puts finite momentum on the ingoing and outgoing electron lines,\n\\begin{align}\n\\langle \\sigma_\\ext{p_\\mu} , f \\rangle = \\frac{\\partial}{\\partial p_\\mu} f(p)\n\\end{align}\n\nFinally, for a vacuum polarization graph $\\Gamma$, $E_\\Gamma = \\{ (q, q', \\mu,\\nu) : q+q'=0 \\} \\equiv \\{ (q, \\mu,\\nu )\\}$, we let $\\sigma_\\ext{3}$ be the external structure that satisfies\n\\begin{align}\n\\label{ext3}\n\\langle \\sigma_\\ext{3}, \\left(- \\delta_{\\mu\\nu}q^2+ q_\\mu q_\\nu\\right)\\otimes M \\rangle = \\mathrm{tr~} M,\n\\end{align}\nfor all $M \\in M_4(\\C)$. \n\n\\begin{rem}\nThe labelling of the external structures by $\\ext{0}, \\ext{1}, \\ext{2}$ and $\\ext{3}$ differs from the one in \\cite{CK00}, where they were labelled by $\\ext{0}$ and $\\ext{1}$. Here we follow \\cite{BF01}, so that the labels correspond to the renormalization constants $m_0, Z_1, Z_2, Z_3$, as they appear in the Lagrangian,\n\\begin{align}\n\\label{eq:L-ren}\n\\mathcal{L} =  - \\frac{1}{4} Z_3 A_\\mu D_{\\mu\\nu} A_\\nu - \\frac{1}{2\\xi}(\\partial^\\mu A_\\mu)^2 + Z_2 \\overline\\psi \\left( \\gamma^\\mu \\partial_\\mu - m_0\\right) \\psi  + e Z_1 \\overline\\psi \\gamma^\\mu A_\\mu \\psi .\n\\end{align}\nThe precise correspondence will be given in equation \\eqref{eq:ren-const} below.\n\\end{rem}\n\n\\bigskip\n\nLet $H$ be the free commutative algebra generated by pairs $(\\Gamma, \\sigma)$ with $\\Gamma$ a 1PI graph and $\\sigma \\in \\left(C^{\\infty} (E_\\Gamma) \\otimes M_4(\\C) \\right)'$ a distribution encoding its external structure. The product in $H$ can be understood as the union of graphs, with induced external structure. More precisely, for two 1PI graphs $\\Gamma_\\sigma$ and $\\Gamma'_{\\sigma'}$, the product $\\Gamma_\\sigma \\cdot \\Gamma'_{\\sigma'}$ is the graph $\\Gamma \\cup \\Gamma'$ with external structure given by $\\sigma \\otimes \\sigma' \\in \\left(C^\\infty(E_\\Gamma \\times E_{\\Gamma'}) \\otimes M_4(\\C)\\right)'$. The algebra $H$ is a (positively) graded algebra $H=\\oplus_{n\\in \\N} H^n$ , with the grading given by the loop number of the graph. \n\nIn the following, we will also write $\\Gamma_\\sigma$ for the pair $(\\Gamma,\\sigma)$, and sometimes even $\\Gamma$. We will shorten the notation for graphs equipped with the special external structures defined above by writing $\\Gamma_\\ext{k} = (\\Gamma, \\sigma_\\ext{k})$.\n\nWe define a coproduct $\\Delta :H\\to H \\otimes H$ as follows; if $\\Gamma$ is a 1PI graph, then one sets\n\\begin{align}\n\\label{coproduct}\n\\Delta \\Gamma = \\Gamma \\otimes 1 + 1 \\otimes \\Gamma + \\sum_{\\gamma \\subset \\Gamma} \\gamma_{(k)} \\otimes \\Gamma\/\\gamma_{(k)}.\n\\end{align}\nHere $\\gamma$ is a proper subset of the graph $\\widetilde\\Gamma$ formed by the internal edges of $\\Gamma$ ({i.e.} $0 \\subsetneq \\gamma \\subsetneq \\widetilde \\Gamma$). The connected components $\\gamma'$ of $\\gamma$ are 1PI graphs with the property that the set of edges of $\\Gamma$ that meet $\\gamma'$ have two or three elements. The sum runs over all multi-indices $k$, one index for each 1PI connected component of $\\gamma$, and one denotes by $\\gamma'_{(k)}$ the 1PI graph that has external structure as defined in equation \\eqref{ext1}-\\eqref{ext3}, with $k=3$ in the case of a vacuum polarization, $k=1$ for a full vertex graph and $k=0$ or $2$ in the case of an electron self-energy. In equation \\eqref{coproduct}, $\\gamma_{(k)}$ denotes the disjoint union of all graphs $\\gamma'_{(k)}$ associated to the connected components of $\\gamma$ and the graph $\\Gamma\/\\gamma_{(k)}$ is the graph $\\Gamma$ (with the same external structure) with each $\\gamma'$ reduced to a vertex of type $(k)$. More explicitly, if $\\gamma'=$\n\\begin{fmfgraph*}(30,11)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}, then for $k=2$ one replaces this graph in $\\Gamma$ by the line     \n\\begin{fmfgraph*}(20,11)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v,r}\n      \\fmfv{decor.shape=cross,decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$\\ext{2}$}}{v}\n\\end{fmfgraph*}\n$ \\equiv$\n      \\begin{fmfgraph*}(20,11)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n\\end{fmfgraph*}\n, and for $k=0$ by \n\\begin{fmfgraph*}(20,11)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v,r}\n      \\fmfv{decor.shape=cross,decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$\\ext{0}$}}{v}\n\\end{fmfgraph*}\n$ \\equiv $\n\\begin{fmfgraph*}(20,11)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfdot{v}\n\\end{fmfgraph*} (cf. Remark \\ref{rem:el-mass}).\nIn the case of the vacuum polarization and the full vertex graph, one replaces $\\gamma'$ by the corresponding vertex. \n\nBy complete analogy with \\cite{CK99}, we find that with this coproduct, $H$ becomes a connected graded Hopf algebra, {i.e.} $H=\\oplus_{n \\in \\N} H^n$, $H^0=\\C$ and\n\\begin{align*} \n\\Delta(H^n) = \\sum_{k=0}^n H^k \\otimes H^{n-k}.\n\\end{align*}\nIndeed, $H^0$ consists of complex multiples of the empty graph, which is the unit in $H$, so that $H^0=\\C 1$. Moreover, from general results on graded Hopf algebras, we obtain the antipode inductively, \n\\begin{align}\n\\label{antipode}\nS(X)=-X - S(X')X'',\n\\end{align}\nwhere $\\Delta(X) = X\\otimes 1 + 1\\otimes X+ \\sum X' \\otimes X''$.\n\nWe give a few examples in order to clarify the coproduct. \n\\begin{align*}\n\\Delta\\bigg(\n\\parbox{40pt}{\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v1,v4}\n      \\fmf{photon,left,tension=0}{v2,v3}\n\\end{fmfgraph*}}\n\\bigg) &= \n\\parbox{40pt}{\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v1,v4}\n      \\fmf{photon,left,tension=0}{v2,v3}\n\\end{fmfgraph*}}\n \\otimes 1 + 1 \\otimes \n\\parbox{40pt}{\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v1,v4}\n      \\fmf{photon,left,tension=0}{v2,v3}\n\\end{fmfgraph*}}\n+\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v2,v3,r}\n      \\fmf{photon,left,tension=0}{v2,v3}\n\\end{fmfgraph*}}\n_\\ext{2}\n\\otimes\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v4,r}\n      \\fmf{photon,left,tension=0}{v1,v4}\n\\end{fmfgraph*}}\n+\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v2,v3,r}\n      \\fmf{photon,left,tension=0}{v2,v3}\n\\end{fmfgraph*}}\n_\\ext{0}\n\\otimes\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v3,v4,r}\n      \\fmfdot{v3}\n      \\fmf{photon,left,tension=0}{v1,v4}\n\\end{fmfgraph*}}\n\\end{align*}\n\\begin{multline*}\n\\Delta\\bigg(\n\\parbox{50pt}{\\begin{fmfgraph*}(50,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v4,v5,v6,r}\n      \\fmf{photon,left,tension=0}{v1,v5}\n      \\fmf{photon,right,tension=0}{v2,v6}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}}\n\\bigg) = \\parbox{50pt}{\\begin{fmfgraph*}(50,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v4,v5,v6,r}\n      \\fmf{photon,left,tension=0}{v1,v5}\n      \\fmf{photon,right,tension=0}{v2,v6}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}} \\otimes 1\n+ \n1 \\otimes \\parbox{50pt}{\\begin{fmfgraph*}(50,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v4,v5,v6,r}\n      \\fmf{photon,left,tension=0}{v1,v5}\n      \\fmf{photon,right,tension=0}{v2,v6}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}}\n+\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}}\n_\\ext{2}\n\\otimes \n\\parbox{50pt}{\n\\begin{fmfgraph*}(50,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v5,v6,r}\n      \\fmf{photon,left,tension=0}{v1,v5}\n      \\fmf{photon,right,tension=0}{v2,v6}\n\\end{fmfgraph*}}\n\\\\[5mm]\n+\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}}\n_\\ext{0}\n\\otimes \n\\parbox{50pt}{\n\\begin{fmfgraph*}(50,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v1,v2,v3,v5,v6,r}\n      \\fmf{photon,left,tension=0}{v1,v5}\n      \\fmf{photon,right,tension=0}{v2,v6}\n      \\fmfdot{v3}\n\\end{fmfgraph*}}\n+ \n\\parbox{50pt}{\\begin{fmfgraph*}(50,20)\n    \\fmfforce{(.33w,0h)}{b}\n    \\fmfleft{l}\n    \\fmfright{r}\n    \\fmf{plain}{l,v1,v3,v5,v6,v7,r}\n    \\fmffreeze\n    \\fmf{photon}{b,v3}\n    \\fmf{photon,left,tension=0}{v5,v6}\n    \\fmf{photon,left,tension=0}{v1,v7}\n    \\end{fmfgraph*}}\n_\\ext{1}\n\\otimes \n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}}\n+\n\\parbox{50pt}{\\begin{fmfgraph*}(50,20)\n    \\fmfforce{(.66w,0h)}{b}\n    \\fmfleft{l}\n    \\fmfright{r}\n    \\fmf{plain}{l,v1,v2,v3,v6,v7,r}\n    \\fmffreeze\n    \\fmf{photon}{b,v6}\n    \\fmf{photon,left,tension=0}{v2,v3}\n    \\fmf{photon,left,tension=0}{v1,v7}\n    \\end{fmfgraph*}}\n_\\ext{1}\n\\otimes \n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,11)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v3,v4,r}\n      \\fmf{photon,left,tension=0}{v3,v4}\n\\end{fmfgraph*}}\n\\end{multline*}\n\n\n\\section{Birkhoff decomposition and renormalization}\n\\label{sect:birkhoff}\nIn \\cite{CK99}, Connes and Kreimer understood renormalization of perturbative quantum field theory in terms of a Birkhoff decomposition. In particular, using the dimensional regularization (dim-reg) and minimal subtraction scheme, they have proved that the BPHZ-formula is a special case of the Birkhoff decomposition of a loop on a small circle in the complex plane, centered at the dimension of the theory, and taking values in a pro-unipotent Lie group. Before applying this to the case of QED, we briefly recall the Birkhoff decomposition, while referring to \\cite{CK99} and \\cite{CM04b} for more details.\n\nThe Birkhoff decomposition provides a procedure to extract a finite value from a singular expression. More precisely, let $C \\in \\mathbb{P}^1(\\C)$ be a smooth simple curve and let $C_\\pm$ denote its two complements with $\\infty \\in C_-$. The Birkhoff decomposition of a loop $\\gamma :C \\to G$, taking values in a complex Lie group $G$, is a factorization of the form\n\\begin{align*}\n\\gamma(z) = \\gamma_-(z)^{-1} \\gamma_+(z) ,\\qquad z \\in \\C\n\\end{align*}\nwhere $\\gamma_\\pm$ are boundary values of holomorphic maps (denoted by the same symbol)\n\\begin{align*}\n\\gamma_\\pm : C_\\pm \\to G.\n\\end{align*}\nThe normalization $\\gamma_-(\\infty)=1$ ensures uniqueness of the decomposition (if it exists). \n\nThe evaluation $\\gamma \\to \\gamma_+(z_0)\\in G$ is a natural principle to extract a finite value from the (possibly) singular expression $\\gamma(z_0)$. This gives a multiplicative removal of the pole part for a meromorphic loop $\\gamma$, when we let $C$ be an infinitesimal circle centered at $z_0$. \n\nExistence of the Birkhoff decomposition has been established for prounipotent complex Lie groups in \\cite{CK99}, and, more generally, in \\cite{CM04b} in the setting of affine group schemes. Recall that affine group schemes are dual to commutative Hopf algebras in the following sense. For a commutative Hopf algebra $H$, we understand an affine group scheme as a functor $G:A \\to G(A)$ from the category of commutative algebras to the category of groups in the following way. For a given algebra $A$, one defines the group $G(A)$ to be the set of all homomorphisms from $H$ to $A$, with product, inverse and unit given as the duals with respect to the coproduct, antipode and counit $\\epsilon$ respectively, {i.e.}\n\\begin{align*}\n\\phi_1 * \\phi_2(X) &= \\langle \\phi_1 \\otimes \\phi_2, \\Delta(X) \\rangle,\\\\\n\\phi^{-1}(X)&=\\phi(S(X)),\\\\\ne(X)&=\\epsilon (X).\n\\end{align*}\nThe following result is the algebraic translation of the Birkhoff decomposition to (pro-unipotent) affine group schemes, expressed in terms of the commutative (connected graded) Hopf algebra $H$ underlying this affine group scheme. Its proof can be found in \\cite{CK99} (see also \\cite{CM04b}). We assume that $C$ is an infinitesimal circle centered at $0 \\in \\C$ and denote by $K$ the field of convergent Laurent series, with arbitrary radius of convergence and by $\\mathcal{O}$ the ring of convergent power series. Furthermore, we set $\\mathcal{Q}=z^{-1} \\C([z^{-1}])$. \n\\begin{thm}[Connes-Kreimer]\n\\label{birkhoff}\nLet $\\phi:H \\to K$ be an algebra homomorphism from a commutative connected graded Hopf algebra $H$ to the field $K$ defined above. The Birkhoff decomposition of the corresponding loop is obtained recursively from the equalities,\n\\begin{align*}\n\\phi_-(X)&=-T \\left( \\phi(X)+\\sum \\phi_-(X') \\phi(X'') \\right),\n\\end{align*}\nwhere we have written $\\Delta(X) = X \\otimes 1 + 1 \\otimes X + \\sum X' \\otimes X''$ for $X \\in H$ and $T$ is the projection on the pole part in $z$, and $\\phi_+= \\phi_- * \\phi$, or explicitly, \n\\begin{align*}\n\\phi_+(X)&=\\phi(X) + \\phi_-(X) + \\sum \\phi_-(X') \\phi(X'').\n\\end{align*}\nThe maps $\\phi_-$ and $\\phi_+$ are homomorphisms from $H$ to $\\mathcal{Q}$ and $\\mathcal{O}$ respectively.\n\\end{thm}\nThe loop $\\gamma : C \\to G$ defined in terms of $\\phi: H \\to K$ by $\\gamma(z)(X):=\\phi(X)(z)$ therefore factorizes as $\\gamma=\\gamma_-^{-1} \\gamma_+$ where $\\gamma_\\pm:C_\\pm \\to G$ are defined in like manner in terms of $\\phi_\\pm$. The fact that they are holomorphic maps on $C_+$ and $C_-$, respectively, follows from the properties that $\\phi_+$ maps to $\\mathcal{O}$ whereas $\\phi_-$ maps to $\\mathcal{Q}$. \n\n\\bigskip\n\nLet us see how this applies in the case of quantum electrodynamics and in particular, how it gives the BPHZ procedure of renormalization of it. Let us start by giving the Feynman rules for the graphs in our Hopf algebra, allowing us to associate a Feynman amplitude to a graph $\\Gamma$. We will use dim-reg (see \\cite[Ch.4]{Col84}) and obtain the regularized (bare) Feynman amplitudes as integrals\n\\begin{align}\n\\label{eq:feynm-ampl-pre}\nU_\\Gamma (q_1,\\ldots , q_n,\\mu_1,\\ldots,\\mu_n,p_{n+1},\\ldots,p_N)(z)\n=\\int d^{4-z}k_1 \\cdots d^{4-z} k_L ~I_\\Gamma(q,\\mu,p,k_1,\\ldots,k_L)\n\\end{align}\nIf we work in the Euclidean setting, the integrand is given by the following Feynman rules:\n\\begin{enumerate}\n\\item Assign a factor $\\left( -\\frac{\\delta_{\\mu\\nu}}{p^2+\\mathrm{i} \\epsilon} + \\frac{p_\\mu p_\\nu}{(p^2+\\mathrm{i}\\epsilon)^2} (1-\\xi) \\right)$ to each internal photon line. \n\\item Assign a factor $\\frac{1}{\\gamma^\\mu p_\\mu + m}$ to each internal electron line.\n\\item Assign a factor $e \\gamma^\\mu$ to each 3-point vertex. The apparent dependence on the index $\\mu$ is resolved by summing over $\\mu$ in combination with the attached photon line (having also an index $\\mu$). \n\\item Assign a factor $m$ to the 2-point vertex. \n\\item Assign a momentum conservation rule to each vertex.\n\\end{enumerate}\nIn rules {\\it 1.} and {\\it 2.}, we have introduced the IR-regulators $\\mathrm{i} \\epsilon$ in order to resolve divergences at small momenta.\n\\begin{ex}\nConsider the following electron self-energy graph\\\\\n\\begin{center}\n\\begin{fmfgraph*}(150,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{fermion,label=$p$}{l,v2}\n      \\fmf{fermion,label=$p-k$}{v2,v3}\n      \\fmf{fermion}{v3,r}\n      \\fmf{photon,left,tension=0,label=$k$}{v2,v3}\n\\end{fmfgraph*}\n\\end{center}\nAccording to the Feynman rules, the integrand for this graph is \n\\begin{align*}\nI_\\Gamma(p,k)=(e\\gamma^\\mu) \\frac{1}{\\gamma^\\kappa (p_\\kappa + k_\\kappa) + m} (e\\gamma^\\nu) \\left( -\\frac{\\delta_{\\mu\\nu}}{k^2+\\mathrm{i} \\epsilon} + \\frac{k_\\mu k_\\nu}{(k^2+\\mathrm{i} \\epsilon)^2} (1-\\xi) \\right) \n\\end{align*}\nwith summation over repeated indices understood. \n\\end{ex}\nAs can be seen from equation \\eqref{eq:feynm-ampl-pre}, a Feynman amplitude $U_\\Gamma$ corresponding to a 1PI graph $\\Gamma$ is an element in $C^\\infty(E_\\Gamma) \\otimes M_4(\\C) \\otimes K$, and a map $U:H \\to K$ can be defined by setting\n\\begin{align} \n\\label{eq:feynm-ampl}\nU (\\Gamma_\\sigma)(z)=e^{2-N} \\langle \\sigma, U_\\Gamma \\rangle.\n\\end{align} \nThe factor $e^{2-N}$ is introduced in order to keep track of the loop number of the graph in terms of powers of $e^2$. Indeed, with $N$ the number of external edges of $\\Gamma$, the power of $e^2$ appearing in the above expression is exactly the loop number $L$ of $\\Gamma$. \n\nThe counterterm $C(\\Gamma)$ for a graph $\\Gamma$ is given as the negative part of the Birkhoff decomposition of $U$ applied to $\\Gamma$ and the renormalized value $R(\\Gamma)$ as the positive part. Indeed, in this case, the above formul{\\ae} give precisely the recursive BPHZ-procedure of subtracting divergences, dealing with possible subdivergences recursively, so that $C=U_-$ and $R=U_+$. More explicitly,\n\\begin{align*}\nC(\\Gamma) &= -T \\left( \nU(\\Gamma) + \\sum_{\\gamma \\subset \\Gamma} C(\\gamma_{(k)}) U(\\Gamma\/\\gamma_{(k)}) \\right),\\\\\nR(\\Gamma) &=  \nU(\\Gamma) + C(\\Gamma) + \\sum_{\\gamma \\subset \\Gamma} C(\\gamma_{(k)}) U(\\Gamma\/\\gamma_{(k)}).\n\\end{align*}\n\n\n\\begin{rem}\nThe relation between the renormalization constants in the Lagrangian (see equation \\eqref{eq:L-ren}) and the counterterms, was given by Dyson in \\cite{Dys49}. In terms of an expansion of Feynman graphs of the specified type, they read (compare with equation (73)-(76) in \\cite{BF01})\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:ren-const}\nm_0 &= Z_2 m + \\sum_{\\Gamma=~\n\\parbox{20pt}{\n\\begin{fmfgraph}(20,10)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph}}} \n    C(\\Gamma_\\ext{0}),\\\\\nZ_1 &= 1 + \\sum_{\\Gamma=~\n\\parbox{20pt}{\n\\begin{fmfgraph}(20,10)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmf{photon}{l,v}\n      \\fmf{plain}{r1,v}\n      \\fmf{plain}{v,r2}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph}}} C(\\Gamma_\\ext{1}),\\\\\nZ_2 &= 1 - \\sum_{\\Gamma=~\n\\parbox{20pt}{\n\\begin{fmfgraph}(20,10)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph}}} C(\\Gamma_\\ext{2}),\\\\\nZ_3 &=1 - \\sum_{\\Gamma=~\n\\parbox{20pt}{\n\\begin{fmfgraph}(20,10)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{photon}{l,v}\n      \\fmf{photon}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph}}} C(\\Gamma_\\ext{3}).\n\\end{aligned}\n\\end{equation}\n\\end{rem}\n\n\n\\section{Ward-Takahashi identities}\n\\label{sect:wt}\nIn quantum electrodynamics, the Ward-Takahashi identities give relations between Feynman amplitudes for full vertex graphs and electron self-energy graphs. Our goal is to translate these identities into relations on the Hopf algebra $H$, thus giving relations between Feynman graphs. More precisely, we consider a quotient of the Hopf algebra $H$ by a Hopf ideal generated by so-called {\\it Ward-Takahashi (WT) elements}, thereby establishing the well-known fact (see for example \\cite{Col84, PS95}) that the WT-identities are compatible with renormalization. \n\nFirst, we introduce the following notation. Let $\\Gamma$ be an electron self-energy graph, and number the internal electron lines that are not part of an electron loop by $i$. We define $\\Gamma(i)$ to be the graph $\\Gamma$ with insertion of a photon line on the $i$'th electron line.\n\n\\begin{defn}\nFor every electron self-energy graph $\\Gamma$, we define the {\\rm Ward-Takahashi element} $W(\\Gamma) \\in H$ associated to $\\Gamma$ by\n\\begin{align*}\nW(\\Gamma)(p,q)_{kl} := \\sum_i \\sum_\\mu q_\\mu \\Gamma(i)(q,\\mu;p)_{kl} + \\Gamma(p+q)_{kl} - \\Gamma(p)_{kl}.\n\\end{align*}\nThe first term is understood as the pair $(\\Gamma(i),\\sigma) \\in H$, with external structure given by the distribution $\\sigma^{kl}=\\sum q^\\mu \\delta_{q,\\mu,p,-p-q} \\otimes e^{kl}$. \nMoreover, we set $W'_\\mu(\\Gamma)(p) := \\frac{\\partial}{\\partial q_\\mu} W(\\Gamma)(p,q) \\big|_{q=0}$ and $W''(\\Gamma):=\\frac{1}{16} \\sum_\\mu \\mathrm{tr~} \\gamma_\\mu \\frac{\\partial}{\\partial q_\\mu} W(\\Gamma)(p,q) \\big|_{p=q=0}$; in other words\n\\begin{align*}\nW'_\\mu(\\Gamma)(p) &= \\sum_i \\Gamma(i)_\\ext{0,\\mu,p} + \\Gamma_\\ext{p_\\mu},\\\\\nW''(\\Gamma) &= \\sum_i \\Gamma(i)_\\ext{1} + \\Gamma_\\ext{2}.\n\\end{align*}\n\\end{defn}\n\\noindent In what follows, we will suppress the matrix indices $k,l$ in $W(\\Gamma)$ for notational convenience.\n\nThe following notation due to 't Hooft and Veltman of double headed photon lines provides a compact way to denote the above external structure in terms of diagrams:\n\\begin{align*}\n\\parbox{40pt}{\\begin{fmfgraph*}(40,40)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmflabel{$p$}{r1}\n      \\fmf{plain}{r1,v}\n      \\fmf{plain}{v,r2}\n      \\fmf{doublehead,label=$q$}{l,v}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n= \\sum q^\\mu ~\\parbox{40pt}{\n\\begin{fmfgraph*}(40,40)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmflabel{$p$}{r1}\n      \\fmf{photon,label=$(\\noexpand q,,\\noexpand \\mu)$}{l,v}\n      \\fmf{plain}{r1,v}\n      \\fmf{plain}{v,r2}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\\\\[3mm]\n\\end{align*}\nIn diagrams, this leads to the familiar expressions for the WT-identities (see for example \\cite[Ch. 7]{PS95}),\n\\begin{align*}\nW(\\Gamma)(p,q)&= \\sum_{\\parbox{40pt}{\\centering \\tiny{insertion\\\\ points}}\n} \n\\parbox{60pt}{\n\\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmf{doublehead, label=$q$}{l,v}\n      \\fmf{fermion,label=$p$}{r1,v}\n      \\fmf{fermion}{v,r2}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n+\n\\parbox{60pt}{\n\\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{fermion,label=$p+q$}{l,v}\n      \\fmf{fermion}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n-\n\\parbox{60pt}{\n\\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{fermion,label=$p$}{l,v}\n      \\fmf{fermion}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n\\\\[5mm]\nW_\\mu'(\\Gamma)(p)&= \\sum_{\\parbox{40pt}{\\centering \\tiny{insertion\\\\ points}}} \n\\parbox{60pt}{\n\\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmflabel{$p$}{r1}\n      \\fmf{photon,label=$(\\noexpand 0,,\\noexpand \\mu)$}{l,v}\n      \\fmf{fermion}{r1,v}\n      \\fmf{fermion}{v,r2}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n+\\frac{\\partial}{\\partial p_\\mu}~\n\\parbox{60pt}{\n\\begin{fmfgraph*}(60,60)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{fermion,label=$p$}{l,v}\n      \\fmf{fermion}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n\\\\[5mm]\nW''(\\Gamma)&= \\sum_{\\parbox{40pt}{\\centering \\tiny{insertion\\\\ points}}} \n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,40)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmf{photon}{l,v}\n      \\fmf{plain}{r1,v}\n      \\fmf{plain}{v,r2}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}_\\ext{1}\n+\n\\parbox{40pt}{\n\\begin{fmfgraph*}(40,30)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}_\\ext{2}\n\\end{align*}\nfor $\\Gamma=$ \\begin{fmfgraph*}(40,11)\n      \\fmfforce{(0w,.25h)}{l}\n      \\fmfforce{(1w,.25h)}{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}\nany electron self-energy graph. \n\n\\begin{thm}\n\\label{thm:WT}\nThe ideal generated (in $H$) by $W(\\Gamma)$, $W_\\mu'(\\Gamma)$ and $W''(\\Gamma)$ for all 1PI electron self-energy graphs $\\Gamma$ is a Hopf ideal.\n\\end{thm}\nBefore stating the proof of this, we remark that it is thus possible to define the quotient Hopf algebra $\\widetilde H$ of $H$, with the induced coproduct, counit and antipode.\n\\begin{proof}\nRecall that an ideal $I$ in a Hopf algebra $H$ is called a Hopf ideal if\n$$\n\\Delta(I) \\subseteq I \\otimes H + H\\otimes I, \\qquad \\epsilon(I)=0, \\qquad S(I) \\subseteq I.\n$$\nIn our case of a connected graded Hopf algebra $H$, the last property follows from the first. Indeed, since $S$ is given inductively by equation \\eqref{antipode}, we find that $S(I)\\subseteq I$.\n\nWe set $\\Delta'(X) := \\Delta(X) - X \\otimes 1 + 1 \\otimes X= \\sum X' \\otimes X''$ and denote the ideal generated by the WT-identities by $I$ . Clearly, it is enough to establish $\\Delta'(I) \\subseteq I \\otimes H + H \\otimes I$. Moreover, since $\\Delta$ is an algebra map by definition, it is enough to establish $\\Delta' (W(\\Gamma)) \\in I \\otimes H + H \\otimes I$ and the analogous expressions for $W'_\\mu(\\Gamma)$ and $W''(\\Gamma)$ for any electron self-energy graph $\\Gamma$. \n\n\\bigskip\n\nLet us start by illustrating the general argument in the following special case of the self-energy graph $\\Gamma$ displayed in Figure \\ref{figure-blocks}.\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{fmfgraph*}(90,60)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v1,e12,v2,v3,e34,v4,v5,e56,v6,e67,v7,v8,e89,v9,v10,r}\n      \\fmffreeze\n      \\fmf{photon,left,tension=0}{v1,v9}\n      \\fmf{photon,left,tension=0}{v2,v3}\n      \\fmf{photon,left,tension=0}{v4,v5}\n      \\fmf{photon,left,tension=0}{v6,v10}\n      \\fmf{photon,left,tension=0}{v7,v8}\n      \\fmffreeze\n      \\fmf{dots,right,tension=0,label=$\\alpha_1$}{e12,e34}\n      \\fmf{dots,right,tension=0}{e34,e12}\n      \\fmf{dots,right,tension=0,label=$\\alpha_2$}{e34,e56}\n      \\fmf{dots,right,tension=0}{e56,e34}\n      \\fmf{dots,right,tension=0,label=$\\alpha'$}{e67,e89}\n      \\fmf{dots,right,tension=0}{e89,e67}\n\\end{fmfgraph*}\n\\end{center}\n\\caption{An electron self-energy graph with blocks $\\alpha=\\alpha_1\\cdot \\alpha_2$ and $\\alpha'$.}\n\\label{figure-blocks}\n\\end{figure}\nWe label the internal electron lines from left to right, from 1 to 9. \nLet us consider the first term in $W(\\Gamma)$, and split its image under $\\Delta'$ in a term in which $\\gamma$ contains the electron line $i$ and a term in which it does not:\n$$\n\\Delta'\\left( \\sum_i \\Gamma(i) \\right)= \\sum_{\\gamma \\subset\\Gamma} C(\\gamma) + D(\\gamma),\n$$\nwith $C(\\gamma):=  \\sum_{i \\in \\gamma} \\sum_k \\gamma(i)_\\ext{k} \\otimes \\Gamma(i)\/\\gamma(i)_\\ext{k}$ and $D(\\gamma):=\\sum_{i\\notin \\gamma} \\sum_k \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k}$. \n\nFor example, if $\\gamma=\\alpha_1 \\cdot \\alpha_2$ forms the `block' of concatenated electron self-energy graphs inside $\\Gamma$, we find,\n\\begin{align*}\nC(\\gamma) &= \\sum_{i=2,4} \\sum_{k_1,k_2} \\gamma(i)_\\ext{k_1,k_2} \\otimes \\Gamma(i)\/\\gamma(i)_\\ext{k_1,k_2}\\\\\n&=\\sum_k\n(\\widetilde\\alpha_1 \\cdot \\alpha_2)_\\ext{1,k} \\otimes \\Gamma\/\\gamma_\\ext{1,k}(e_1)\n + (\\alpha_1\\cdot\\widetilde\\alpha_2)_\\ext{k,1} \\otimes \\Gamma\/\\gamma_\\ext{k,1}(e_2),\n\\end{align*}\nwhere for $m=1,2$, $\\widetilde \\alpha_m$ denotes the electron self-energy graph $\\alpha_m$ with an external photon line attached to its internal electron line and $e_m$ is the electron line that corresponds to $\\alpha_m$ in the quotient $\\Gamma\/\\gamma$. On the other hand,\n\\begin{align*}\nD(\\gamma) &=\\sum_{i=1,3,5,\\ldots,9} \\sum_{k_1,k_2} \\gamma_\\ext{k_1,k_2} \\otimes \\Gamma(i)\/\\gamma_\\ext{k_1,k_2} \\\\\n&= \\sum_k (\\alpha_1 \\cdot \\alpha_2)_\\ext{2,k} \\otimes \\Gamma \/ \\gamma_\\ext{2,k}(1)\n+ (\\alpha_1 \\cdot\\alpha_2)_\\ext{k,2} \\otimes \\Gamma \/ \\gamma_\\ext{k,2}(3) \\\\\n&+\\sum_k (\\alpha_1 \\cdot \\alpha_2)_\\ext{0,k} \\otimes \\Gamma \/ \\gamma_\\ext{0,k}(1)\n+ (\\alpha_1 \\cdot \\alpha_2)_\\ext{k,0} \\otimes \\Gamma \/ \\gamma_\\ext{k,0}(3) \\\\\n& +  \\sum_{k_1,k_2} \\gamma_\\ext{k_1,k_2} \\otimes \\sum_{i=5,\\ldots,9} \\Gamma\/\\gamma_\\ext{k_1,k_2}(i).\n\\end{align*}\nHere we wrote explicitly the external structure $k=0,2$ for the electron self-energy graph in $\\gamma$ for which an external photon line is attached to its incoming electron line (i.e. for $\\alpha_1$ if $i=1$ and for $\\alpha_2$ if $i=3$). Also, in the last line we used a labelling of the internal electron lines of the quotient $\\Gamma\/\\gamma$ in terms of the electron lines of $\\Gamma$ when $i \\notin \\gamma$.\n\nThe first two terms of $D(\\gamma)$ combine with $C(\\gamma)$ forming the WT-elements $W''(\\alpha_m)$, and also the last three terms of $D(\\gamma)$ combine to give,\n\\begin{align*}\nC(\\gamma)+D(\\gamma)=&\\sum_k W''(\\alpha_1) (\\alpha_2)_\\ext{k} \\otimes \\Gamma\/\\gamma_\\ext{2,k} (e_1)\n+(\\alpha_1)_\\ext{k} W''(\\alpha_2) \\otimes \\Gamma\/\\gamma_\\ext{k,2} (e_2)\\\\\n&+ \\sum_{k_1,k_2} \\gamma_\\ext{k_1,k_2} \\otimes \\sum_{\\parbox{45pt}{\\centering \\tiny{$j$ insertion \\\\points in $\\Gamma\/\\gamma$}}} \\Gamma\/\\gamma_\\ext{k_1,k_2}(j),\n\\end{align*}\nusing for the last term the fact that the quotient $\\Gamma\/(\\alpha_m)_\\ext{0}$ gives rise to a 2-point vertex, and thus to a new internal electron line. It is easy to see that the first two terms in the above equation are in $I \\otimes H$.\n\nThe two other terms in $\\Delta'(W(\\Gamma))$ are of the form $\\sum_\\gamma \\gamma_{k_1,k_2} \\otimes \\Gamma\/\\gamma_\\ext{k_1,k_2}$ and combine, for fixed $\\gamma=\\alpha_1 \\cdot \\alpha_2$ as above, with the last term in the previous equation to give precisely $\\gamma_\\ext{k_1,k_2} \\otimes W(\\Gamma\/\\gamma_\\ext{k_1,k_2}) \\in H \\otimes I$. We conclude that the term in $\\Delta'(W(\\Gamma))$ arising from the subgraph $\\gamma=\\alpha_1\\cdot \\alpha_2$ is an element in $I \\otimes H + H \\otimes I$.\nSimilar arguments show that such relations hold for all subgraphs of $\\Gamma$.\n\n\n\n\n\n\n\n\\bigskip\n\nLet us now turn to the proof of the claim that $\\Delta' (W(\\Gamma)) \\in I \\otimes H + H \\otimes I$ for any electron self-energy graph $\\Gamma$. For the first term in $W(\\Gamma)$ we have\n\\begin{align*}\n\\Delta'\\left( \\sum_i \\Gamma(i) \\right)=\n\\Delta'\\left( \\sum_i\n\\parbox{40pt}{\\begin{fmfgraph*}(40,40)\n      \\fmfleft{l}\n      \\fmfright{r1,r2}\n      \\fmf{plain}{r1,v}\n      \\fmf{plain}{v,r2}\n      \\fmf{doublehead}{l,v}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph*}}\n\\right) = \n\\sum_i \\sum_{\\gamma \\subset \\Gamma(i)}  \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k},\n\\end{align*}\nwhere $k$ is a multi-index labelling the external structure on the connected components of $\\gamma$ and $i$ runs over all internal electron lines of $\\Gamma$ that are not part of a loop. Also, we suppressed the external structure of $\\Gamma(i)$ as it appears in $W(\\Gamma)(p,q)$.\n\nIn general, $\\gamma$ can be written as a union $\\gamma=\\gamma_E\\cdot \\gamma_P \\cdot \\gamma_V \\cdot \\gamma_E(i)$, where $\\gamma_E$ consists of electron self-energy graphs, $\\gamma_V$ of full vertex graphs, $\\gamma_P$ of vacuum polarization graphs and $\\gamma_E(i)$ of electron self-energy graphs with a photon line inserted at the $i$'th electron line (with this labelling inherited from $\\Gamma$). Graphs of the type $\\gamma_P(i)$ and $\\gamma_V(i)$ do not appear in the sum, because there are no corresponding vertices of valence 3 and 4 in QED. \nNote that the multi-index $k$ only affects the subgraphs in $\\gamma_E$, for each of which it can take the values $0$ and $2$. We split the above sum in a term in which $\\gamma_\\ext{k}$ contains the electron line $i$ as an internal line and those in which it does not:\n\\begin{align}\n\\label{cop-i:2}\n\\Delta'\\left( \\sum_i \\Gamma(i) \\right)= \\sum_{\\gamma \\subset \\Gamma}\n\\left(\n\\sum_{i\\in \\gamma_E} \\gamma(i)_\\ext{k} \\otimes \\Gamma(i)\/\\gamma(i)_\\ext{k}\n+\\sum_{i\\notin \\gamma_E} \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k}\n\\right),\n\\end{align}\nwhere $\\gamma=\\gamma_E \\cdot \\gamma_V \\cdot \\gamma_P$. We will write $\\Delta'\\left( \\sum_i \\Gamma(i) \\right)=\\sum_\\gamma C(\\gamma) + D(\\gamma)$ for the two terms appearing in equation \\eqref{cop-i:2}.\n\nBefore examining the terms $C$ and $D$, we note that for each $\\gamma$, $\\gamma_E$ can be split into `blocks' each of which appear in the graph $\\Gamma$ by concatenation of electron self-energy graphs (as in Figure \\ref{figure-blocks}). We will denote such blocks by $\\alpha \\subset \\gamma_E$ and write $C(\\gamma)$ as\n\\begin{align*}\nC(\\gamma)= \n\\sum_{\\parbox{60pt}{\\centering \\tiny{$\\alpha \\subset \\gamma_E$\\\\ $\\alpha=\\alpha_1 \\cdots \\alpha_l$}}} \\sum_{m=1}^l \\sum_{i \\in \\alpha_m} \\sum_k \\alpha_m(i)_\\ext{1} ~(\\gamma-\\alpha_m)_\\ext{k} \\otimes \n\\Gamma \/ \\gamma_\\ext{k} \\left( e_m \\right)\n\\end{align*}\nwhere $e_m$ is the electron line corresponding to $\\alpha_m$ in the quotient $\\Gamma\/\\gamma$. In words, we separated the sum over the internal electron lines $i$ of $\\gamma_E$ into a sum over all blocks $\\alpha$ in $\\gamma_E$, together with a sum over its connected components $\\alpha_m$ and a sum over the internal electron lines that are part of $\\alpha_m$. Then, with $i \\in \\alpha_m$, the quotient $\\Gamma(i)\/\\gamma(i)_\\ext{k}$ is given by the replacement of $\\alpha_m$ by a 3-vertex, followed by the quotient by the other graphs that constitute $\\gamma$, and with external structure inherited from $\\Gamma(i)$.\nNote that since $\\alpha_m(i)$ is a full vertex graph, there is only the external structure $\\alpha_m(i)_\\ext{1}$. This fixes $k_m$ inside the multi-index $k$ to be $2$, and by a slight abuse of notation, we let $k$ also denote the external structures of the elements in $(\\gamma-\\alpha_m)$. \n\nThe term $D(\\gamma)$ can be split in a sum over $i$ of electron lines that are external edges for $\\gamma_E$ and those that are not,\n\\begin{align*}\nD(\\gamma)= \\sum_{i \\in \\partial \\gamma_E} \\sum_k \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k} + \\sum_{i \\notin \\gamma \\cup \\partial \\gamma_E} \\sum_k \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k}.\n\\end{align*}\nAgain, we will write this in terms of the blocks $\\alpha$ forming $\\gamma_E$:\n\\begin{equation*}\n\\begin{aligned}\nD(\\gamma)&= \\sum_\\alpha \\sum_{m=1}^l \\sum_k \n(\\alpha_m)_\\ext{k_m} (\\gamma-\\alpha_m)_\\ext{k} \\otimes \n\\Gamma\n\\left( \\alpha_m \\to \n  \\parbox{20pt}{\n    \\begin{fmfgraph*}(20,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmfbottom{b}\n      \\fmf{plain}{l,v1,v2,v3,r}\n      \\fmffreeze\n      \\fmf{doublehead}{b,v2}\n      \\fmffreeze\n      \\fmfv{decor.shape=cross, decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$(k_m)$}}{v3}\n    \\end{fmfgraph*}}\n\\right)\n\/ \n (\\gamma-\\alpha_m)_\\ext{k}\n\\\\\n&+\\sum_\\alpha \\sum_k \n(\\alpha_l)_\\ext{k_l} (\\gamma-\\alpha_l)_\\ext{k} \\otimes \n\\Gamma\n\\left( \\alpha_l \\to \n  \\parbox{20pt}{\n    \\begin{fmfgraph*}(20,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmfbottom{b}\n      \\fmf{plain}{l,v1,v2,v3,r}\n      \\fmffreeze\n      \\fmf{doublehead}{b,v2}\n      \\fmffreeze\n      \\fmfv{decor.shape=cross,decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$(k_l)$}}{v1}\n    \\end{fmfgraph*}}\n\\right)\n\/\n (\\gamma-\\alpha_l)_\\ext{k} \n+ \\sum_{i \\notin \\gamma \\cup \\partial \\gamma_E}\\sum_k \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k},\n\\end{aligned}\n\\end{equation*}\nwhere the first term arises from $i$ being the ingoing line of $\\alpha_m$ for $m=1,\\cdots l$ and the second term from $i$ being the outgoing line for $\\alpha_l$, thus equal to the sum of all external edges in $\\alpha$. \n\n\\medskip\n\nWriting explicitly the two terms for $k_m=0,2$ one obtains, \n\\begin{equation*}\n\\begin{aligned}\nD(\\gamma)&= \\sum_\\alpha \\bigg(\\sum_{m=1}^l \\sum_k \n(\\alpha_m)_\\ext{2} (\\gamma-\\alpha_m)_\\ext{k} \\otimes \n\\Gamma\/ \\gamma_\\ext{k} (e_m)\n\\\\\n&+\n\\sum_{m=1}^l \\sum_k \n(\\alpha_m)_\\ext{0} (\\gamma-\\alpha_m)_\\ext{k} \\otimes \n\\Gamma\n\\left( \\alpha_m \\to \n  \\parbox{20pt}{\n    \\begin{fmfgraph*}(20,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmfbottom{b}\n      \\fmf{plain}{l,v1,v2,v3,r}\n      \\fmffreeze\n      \\fmf{doublehead}{b,v2}\n      \\fmffreeze\n      \\fmfv{decor.shape=cross, decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$\\ext{0}$}}{v3}\n    \\end{fmfgraph*}}\n\\right)\n\/\n (\\gamma-\\alpha_m)_\\ext{k} \n\\\\\n&+\\sum_k \n(\\alpha_l)_\\ext{k_l} (\\gamma-\\alpha_l)_\\ext{k} \\otimes \n\\Gamma\n\\left( \\alpha_l \\to \n  \\parbox{20pt}{\n    \\begin{fmfgraph*}(20,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmfbottom{b}\n      \\fmf{plain}{l,v1,v2,v3,r}\n      \\fmffreeze\n      \\fmf{doublehead}{b,v2}\n      \\fmffreeze\n      \\fmfv{decor.shape=cross,decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$(k_l)$}}{v1}\n    \\end{fmfgraph*}}\n\\right)\n\/\n (\\gamma-\\alpha_l)_\\ext{k} \n\\bigg)\n+ \\sum_{i \\notin \\gamma \\cap \\partial \\gamma_E}\\sum_k \\gamma_\\ext{k} \\otimes \\Gamma(i)\/\\gamma_\\ext{k}.\n\\end{aligned}\n\\end{equation*}\nwhere in the first line, we have used the fact that $\\Gamma\n\\left( \\alpha_m \\to \n  \\parbox{20pt}{\n    \\begin{fmfgraph*}(20,20)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmfbottom{b}\n      \\fmf{plain}{l,v1,v2,v3,r}\n      \\fmffreeze\n      \\fmf{doublehead}{b,v2}\n      \\fmffreeze\n      \\fmfv{decor.shape=cross, decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$(2)$}}{v3}\n    \\end{fmfgraph*}}\n\\right) = \\Gamma\/\\alpha_m (e_m)$, in the above notation.\n\n\\bigskip\n\n\\noindent Since each vertex \n\\begin{fmfgraph*}(20,11)\n      \\fmfforce{(0w,.15h)}{l}\n      \\fmfforce{(1w,.15h)}{r}\n      \\fmf{plain}{l,v,r}\n      \\fmffreeze\n      \\fmfv{decor.shape=cross,decor.size=2thick,label.dist=4pt,label.angle=90,label=\\tiny{$\\ext{0}$}}{v}\n\\end{fmfgraph*}\nadds an electron line to $\\Gamma$, we can combine the last three terms to obtain\n\\begin{equation*}\n\\begin{aligned}\nD(\\gamma)&= \\sum_\\alpha \\sum_{m=1}^l \\sum_k \n(\\alpha_m)_\\ext{2} (\\gamma-\\alpha_m)_\\ext{k} \\otimes \n\\Gamma\/ \\gamma_\\ext{k} (e_m)\n+ \\sum_k \\gamma_\\ext{k} \\otimes \\sum_{\\parbox{45pt}{\\centering \\tiny{$j$ insertion \\\\points in $\\Gamma\/\\gamma$}}} (\\Gamma\/\\gamma_\\ext{k}) (j).\n\\end{aligned}\n\\end{equation*}\nThus, we obtain for the sum of $C$ and $D$,\n\\begin{equation}\n\\begin{aligned}\n\\label{cop-i:CD}\nC(\\gamma) + D(\\gamma) &=  \\sum_{\\alpha,m,k}\n\\left( \\sum_{i \\in \\alpha_m}(\\alpha_m)(i)_\\ext{2} +(\\alpha_m)_\\ext{2} \\right)(\\gamma-\\alpha_m)_\\ext{k} \\otimes \n\\Gamma\/ \\gamma_\\ext{k} (e_m)\n\\\\\n&+ \\sum_k \\gamma_\\ext{k} \\otimes \\sum_j \\left( \\Gamma\/\\gamma_\\ext{k} \\right) (j),\n\\end{aligned}\n\\end{equation}\nwhere we recognize the WT-element $W''(\\alpha_m)$ for each 1PI self-energy graph $\\alpha_m\\subset \\Gamma$,\n\\begin{align*}\n  W''(\\alpha_m)=\\sum_{i \\in \\alpha_m}(\\alpha_m)(i)_\\ext{1} +(\\alpha_m)_\\ext{2}.\n\\end{align*}\nCombining the second term in equation \\eqref{cop-i:CD} with the two other terms in $\\Delta'(W(\\Gamma))$ (which involve the coproduct $\\Delta'(\\Gamma)=\\sum_\\gamma \\gamma_\\ext{k} \\otimes \\Gamma\/\\gamma_\\ext{k}$, it follows that \n$$\n\\Delta'(W(\\Gamma)) = \\sum_{\\gamma \\subset \\Gamma}   \\sum_{\\alpha,m,k} W''(\\alpha_m) (\\gamma-\\alpha_m)_\\ext{k} \\otimes \\Gamma \/ \\gamma_\\ext{k} (e_m) + \\sum_{\\gamma \\subset \\Gamma} \\gamma_\\ext{k} \\otimes W\\left( \\Gamma\/\\gamma_\\ext{k} \\right),\n$$\nso that $\\Delta'(W(\\Gamma))\\in I \\otimes H + H \\otimes I$. \n\nThe other inclusions $\\Delta'(W_\\mu'(\\Gamma)) \\in I \\otimes H + H \\otimes I$ and $\\Delta'(W''(\\Gamma)) \\in I \\otimes H + H \\otimes I$ follow from the latter equation by the definitions of $W_\\mu'(\\Gamma)$ and $W''(\\Gamma)$ in terms of $W(\\Gamma)$. \n\\end{proof}\n\n\n\\bigskip\n\n\\section{Conclusions}\n\\label{sect:concl}\nWe have shown that the Ward-Takahashi identities can be implemented on the Hopf algebra of Feynman graphs of QED as relations that define a Hopf ideal. The quotient Hopf algebra $\\widetilde H$ by this Hopf ideal is still commutative but has the WT-identities `built in'. The Feynman amplitudes $U$ on $\\widetilde H$ defined by equation \\eqref{eq:feynm-ampl} therefore automatically satisfy the WT-identities in the physical sense ({i.e.} between Feynman amplitudes). Moreover, Theorem \\ref{birkhoff} applies and shows that the counterterms and the renormalized Feynman amplitudes are given in terms of the algebra maps $C=U_-: \\widetilde H \\to \\mathcal{Q}$ and $R=U_+ : \\widetilde H \\to \\mathcal{O}$, respectively. In particular, the counterterms as well as the renormalized Feynman amplitudes satisfy the WT-identities. The well-known relation $Z_1=Z_2$ between the renormalization constants as derived by Ward in \\cite{War50} is now an easy consequence of their definition in \\eqref{eq:ren-const},\n\\begin{align*}\nZ_1 - Z_2 & = \\sum_{\\Gamma=~\n\\parbox{20pt}{\n\\begin{fmfgraph}(20,10)\n      \\fmfleft{l}\n      \\fmfright{r}\n      \\fmf{plain}{l,v}\n      \\fmf{plain}{v,r}\n      \\fmfblob{.25w}{v}\n\\end{fmfgraph}}} \n\\sum_{\\parbox{45pt}{\\centering \\tiny{$i$ insertion \\\\points in $\\Gamma$}}} \nC(\\Gamma(i)_\\ext{1}) + C(\\Gamma_\\ext{2})\\\\\n&= \\sum_{\\Gamma} C\\left( W''(\\Gamma) \\right) = 0 .\n\\end{align*}\nIn the first line we have used the fact that the contribution $C(\\Gamma(i)_\\ext{1})$ vanishes whenever $i$ is part of an electron loop, as follows by explicit computation using the Feynman rules (cf. \\cite[p.240-241]{PS95}).\n\n\n\n\\section*{Acknowledgements}\nI would like to thank Matilde Marcolli for discussion and remarks, and {\\\"O}zg{\\\"u}r Ceyhan for several comments.\n\n\n\n\\end{fmffile}\n\n\\pagebreak\n\\newcommand{\\noopsort}[1]{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe interplay between physical concepts and geometric structures is at the heart of fundamental theoretical physics. The prime example is the role of Riemannian geometry in the formulation and understanding of general relativity. It is therefore natural to ask what novel geometric structures will be needed to encompass the ideas behind a theory of quantum gravity.\n\nA promising candidate for such a theory is string theory. Due to the extended nature of the fundamental string and the presence of dualities, it has proven useful to work with a \\emph{doubled geometry}. This is formed by extending spacetime to include extra coordinates corresponding to the winding modes of the string (whereas usual coordinates correspond to the momentum modes). This doubled geometry can be viewed as the target space of the fundamental string which is manifestly invariant under T-duality transformations exchanging winding and momentum modes. An effective action for this setup is provided by double field theory (DFT) \\cite{Siegel:1993xq,Siegel:1993th,Hull:2009mi}. The doubled space can also be interpreted as the phase space of the fundamental string which then leads to the formulation of metastring theory \\cite{Freidel:2014qna,Freidel:2015pka}.\n\nThe geometric structure of this doubled space is that of a \\emph{para-Hermitian manifold} as was first demonstrated by Vaisman \\cite{Vaisman:2012ke,Vaisman:2012px}. Such geometry provides the {\\it kinematical structure} for the theory in the sense that it augments the doubled space with a differentiable structure, naturally adapted to the T-duality covariant setting. The dynamical data is then given by a generalized metric and generalized fluxes on the doubled geometry. Such a setup has been dubbed \\emph{Born geometry} \\cite{Freidel:2013zga}.\n\nIn this contribution we give an executive summary of para-Hermitian geometry in the context of the doubled geometry of string theory and how it is related to generalized geometry based on \\cite{Freidel:2017yuv,Svoboda:2018rci,Freidel:2018tkj} (see \\cite{Chatzistavrakidis:2018ztm} for related developments). We give the basic ingredients of Born geometry and present the unique torsion-free compatible connection for Born geometry. This can be seen as the analogy to the Levi-Civita connection for Riemannian geometry. We will also explain how the fluxes naturally arise in this setting. \\textcolor{white}{\\speaker{F.~J.~Rudolph, D.~Svoboda} }\n\n\n\n\n\\section{Generalized Kinematics}\nIt has been observed that the vector fields on the extended spacetime obey a new algebra relation which is different than the usual Lie algebra of vector fields given by the Lie bracket. One is therefore led to replace the Lie bracket with a new bracket operation called the {\\it D-bracket} which leads to a redefinition of the notion of {\\it differentiable structure} on the manifold, i.e. the Cartan calculus. All usual objects whose definitions use the Lie bracket, such as the torsion and Nijenhuis tensors, need to be replaced by their counterparts using the D-bracket. We call this new differentiable structure on the manifold the {\\it generalized kinematical structure}. In this section we describe that such structure on an extended spacetime is naturally given by the data of a para-Hermitian manifold.\n\n\\subsection*{Para-Hermitian Geometry}\nWe now briefly introduce all important concepts of para-Hermitian geometry. For more details consult \\cite{Cruceanu} and references therein.\n\n\\begin{Def}\nLet $\\mathcal{P}$ be a $2n$-dimensional manifold. An almost para-Hermitian structure on $\\mathcal{P}$ is a pair $(\\eta,K)$, where $\\eta$ is a metric with signature $(n,n)$ and $K \\in \\text{End}(T\\mathcal{P})$ with $K^2=\\mathbbm{1}$. These two are compatible in the sense that $K$ is an anti-isometry of $\\eta$: $\\eta(K\\cdot,K\\cdot)=-\\eta(\\cdot,\\cdot)$.\n\\end{Def}\n\nA consequence of the definition is that $K$ has two eigenbundles of rank $n$ corresponding to the eigenvalues $\\pm 1$, which we will in the following denote by $L$ and ${\\tilde{L}}$. The fact that $L$ and ${\\tilde{L}}$ have the same rank makes $K$ an almost {\\it para-complex} structure.  Another consequence of the definition is that the tensor $\\omega \\coloneqq \\eta\\circ K$ is a non-degenerate two-form, i.e. an {\\it almost symplectic structure}, sometimes called the fundamental form. If $\\mathrm{d} \\omega=0$, we call $(\\eta,K)$ an (almost) {\\it para-K\\\"ahler} structure.\n\nIt is useful to realize the analogy of para-Hermitian geometry with its complex counterpart, Hermitian geometry. Indeed, the only difference here is that the para-complex structure $K$ squares to $+\\mathbbm{1}$ as opposed to $-\\mathbbm{1}$ in the complex case. The isometry condition of a complex structure then also comes with the opposite sign. \n\nJust as in the complex case, we can invoke the notion of {\\it integrability} which is governed by the {\\it Nijenhuis tensor}:\n\\begin{align}\\label{eq:nijenhuis}\n\\begin{aligned}\nN_K(X,Y)&\\coloneqq\\frac{1}{4}\\Big( [X,Y]+[KX,KY]-K([KX,Y]+[X,KY])\\Big)\\\\\n&=P[{\\tilde{P}} X,{\\tilde{P}} Y]+{\\tilde{P}}[P X,P Y],\n\\end{aligned}\n\\end{align}\nwhere we introduced the projections onto $L$ and ${\\tilde{L}}$:\n\\begin{align}\\label{eq:K-projections}\nP=\\frac{1}{2}(\\mathbbm{1}+K) \\quad \\mathrm{and} \\quad {\\tilde{P}}=\\frac{1}{2}(\\mathbbm{1}-K).\n\\end{align}\nFrom the last line of \\eqref{eq:nijenhuis}, it is obvious that $N_K$ vanishes if and only if both bundles $L$ and ${\\tilde{L}}$ are Frobenius integrable, i.e. closed under the Lie bracket. When this is the case, one gets local charts $X^I=(x^i,\\tilde{x}^i)$, $i=1\\cdots n$, such that $L=\\text{span}\\{\\partial_i=\\frac{\\partial}{\\partial x^i}\\}$ and ${\\tilde{L}}=\\text{span}\\{\\tilde{\\partial}^i=\\frac{\\partial}{\\partial \\tilde{x}_i}\\}$ and the manifold looks locally like a product manifold.\n\nWe can also notice one of the most striking features of para-complex geometry which sets it apart from complex geometry -- the integrability of the eigenbundles $L$ and ${\\tilde{L}}$ is independent on each other, meaning that $L$ can be integrable while ${\\tilde{L}}$ is not and vice versa. This gives rise to the notion of {\\it half-integrability}. In the subsequent discussion we will not reference integrability when it is not relevant and omit the ``almost'' prefix, while explicitly calling the para-Hermitian structure (half-)integrable whenever relevant.\n\n\n\\subsection*{The D-bracket}\nWe now explain how the para-Hermitian data $(\\eta,K)$ naturally defines a D-bracket on the underlying manifold. The central theorem for this discussion is the following:\n\n\\begin{Thm}\nLet $(\\mathcal{P},\\eta,K)$ be an almost para-Hermitian manifold. There exists a unique bracket operation $[\\![\\ , \\ ]\\!]$: $\\Gamma(T\\mathcal{P})\\times \\Gamma(T\\mathcal{P}) \\rightarrow \\Gamma(T\\mathcal{P})$ satisfying the following properties:\n\\begin{enumerate}\n\\item $[\\![ X,fY]\\!] = f[\\![ X,Y]\\!]+X[f]Y,\\quad$ \\hfill Leibniz property\n\\item $X[\\eta(Y,Z)] = \\eta([\\![ X,Y]\\!],Z)+\\eta(Y,[\\![ X,Z]\\!])$ \\hfill  Compatibility with $\\eta$ \\\\\n$\\eta(Y,[\\![ X,X]\\!] ) = \\eta([\\![ Y,X]\\!],X),\\quad$ \n\\item $\\mathcal{N}_K(X,Y)=[\\![ X,Y]\\!] +[\\![ K X,K Y]\\!]  - K\\big([\\![ K X,Y]\\!]  + [\\![ X,K Y]\\!]\\big)=0,\\quad$ \\hfill Compatibility with $K$\n\\item $\n[\\![ PX,PY ]\\!] = P([PX,PY]),\\quad[\\![ {\\tilde{P}} X,{\\tilde{P}} Y ]\\!] ={\\tilde{P}}([ {\\tilde{P}} X,{\\tilde{P}} Y]),$ \\hfill Relationship with the Lie bracket.\n\\end{enumerate}\nfor any vector fields $X,Y\\in\\Gamma(T\\mathcal{P})$ and any function $f\\in C^\\infty(\\mathcal{P})$. \n\\end{Thm}\nIn the above statement we introduced the {\\it generalized Nijenhuis tensor} of $K$, $\\mathcal{N}_K$. One can glean that this tensorial quantity is nothing else than the usual Nijenhuis tensor, except the Lie bracket is replaced with the D-bracket.\n\nThe following statement shows that the D-bracket not only exists, but can be easily written out in a convenient form:\n\n\\begin{Prop}\nThe D-bracket on a para-Hermitian manifold $(\\mathcal{P},\\eta,K)$ is given by \n\\begin{align}\\label{eq:D-bracket_canonical}\n\\eta([\\![ X,Y]\\!],Z)=\\eta(\\nc_XY-\\nc_YX,Z)+\\eta(\\nc_ZX,Y),\n\\end{align}\nwhere $\\nc$ is the {\\it canonical connection}, which can be defined via the Levi-Civita connection $\\lc$ of $\\eta$:\n\\begin{align*}\n\\nc_XY=P\\lc_X(PY)+{\\tilde{P}}\\lc_X({\\tilde{P}} Y).\n\\end{align*}\n\\end{Prop}\nIn the expression \\eqref{eq:D-bracket_canonical} we can notice that the first two terms, which are skew-symmetric under the exchange of $X$ and $Y$, are reminiscent of how one can define a Lie bracket using a torsionless connection. Even though $\\nc$ is not torsionless, in the limit when $(\\eta,K)$ is para-K\\\"ahler, it is easy to check that $\\nc=\\lc$ and therefore $\\nc_XY-\\nc_YX=[X,Y]$.\n\nWe should also point out that the D-bracket in particular coincides with the D-bracket appearing in the physics literature, where this operation is considered usually on flat manifolds, $\\mathbb{R}^{2n}$ or tori $T^{2n}$, both of which are simple examples of flat para-Hermitian manifolds.\n\\begin{Cor}\nOn a flat para-Hermitian manifold, the D-bracket takes the form \n\\begin{align*}\n[\\![ X,Y ]\\!]=\\left(X^I\\partial_IY^J-Y^I\\partial_IX^J+\\eta_{IL}\\eta^{KJ}Y^I\\partial_KX^L\\right)\\partial_J,\n\\end{align*}\nwhere $\\partial_I=(\\partial_i,\\tilde{\\partial}^i)$ are the partial derivatives along the coordinates $(x^i,\\tilde{x}_i)$.\n\\end{Cor}\n\nWe emphasize here that even though the D-bracket is unique, the choice of connection to express it via the formula \\eqref{eq:D-bracket_canonical} is not unique and $\\n^c$ is only one choice of such connection. In \\cite{Svoboda:2018rci}, the set of connections that yield the D-bracket are called {\\it adapted}.\n\n\\subsection*{Generalized Torsion}\nThe usual torsion of a linear connection $\\n$ can be understood as a tensorial quantity measuring how much the skew combination $\\n_XY-\\n_YX$ deviates from the Lie bracket, $[X,Y]$. The same is true for a so-called generalized torsion. In the light of the equation \\eqref{eq:D-bracket_canonical}, we define\n\n\\begin{Def}\nLet $(\\mathcal{P},\\eta,K)$ be a para-Hermitian manifold and $\\n$ a linear connection. The {\\it generalized torsion} of $\\n$ is the tensorial quantity defined by\n\\begin{align*}\n\\mathcal{T}^\\n(X,Y,Z)\\coloneqq \\eta(\\n_XY-\\n_YX,Z)+\\eta(\\n_ZX,Y)-\\eta([\\![ X,Y]\\!],Z)\n\\end{align*}\n\\end{Def}\n\nFrom this definition we see that it precisely captures how much does the cyclic combination of $\\n$, $\\eta(\\n_XY-\\n_YX,Z)+\\eta(\\n_ZX,Y)$ deviate from the D-bracket. If $\\mathcal{T}^\\n=0$, $\\n$ via this formula defines  a D-bracket.\n\n\n\\subsection*{Courant Algebroids and the Relationship to the D-bracket}\nIn this section we explore how the D-bracket relates to the Dorfman bracket. We explain that when $K$ is integrable, the D-bracket can be seen as a sum of two Dorfman brackets corresponding to certain Courant algebroids. In particular, this equips the tangent bundle of the para-Hermitian manifold with two natural Courant algebroid structures defined purely in terms of the para-Hermitian data.\n\nWe start off with a definition of a Courant algebroid.\n\\begin{Def}\nLet $E\\rightarrow M$ be a vector bundle. A Courant algebroid is a quadruple $(E,a,\\langle\\ , \\ \\rangle_E,[\\![\\ ,\\ ]\\!])$, where $a$ is bundle map $E\\rightarrow TM$ called the anchor, $\\langle\\ , \\ \\rangle_E:\\Gamma(E)\\times\\Gamma(E)\\rightarrow C^\\infty(M)$ is a non-degenerate symmetric pairing and $[\\![\\ ,\\ ]\\!]:\\Gamma(E)\\times\\Gamma(E)\\rightarrow \\Gamma(E)$ is a bracket operation called the Dorfman bracket\\footnote{The Dorfman bracket is sometimes called the Dorfman derivative or generalized Lie derivative.}, such that\n\\begin{enumerate}\n\\item $a(X)\\langle Y,Z\\rangle_E=\\langle[\\![ X,Y]\\!],Z\\rangle_E+\\langle Y,[\\![ X,Z]\\!]\\rangle_E$\n\\item $\\langle[\\![ X,X]\\!],Y\\rangle_E=\\frac{1}{2}a(Y)\\langle X,X\\rangle_E$\n\\item $[\\![ X, [\\![ Y,Z]\\!]\\br=[\\![\\bl X,Y]\\!], Z]\\!]+[\\![ Y,[\\![ X,Z]\\!]\\br$,\n\\end{enumerate}\n\\end{Def}\nA canonical example is given by the {\\it standard Courant algebroid}, where $E=(T\\oplus T^*)M$, $a:(T\\oplus T^*)M\\rightarrow TM$ is the natural projection, the pairing is given by $\\langle x+\\alpha ,y+\\beta \\rangle=\\alpha(y)+\\beta(x)$ and the bracket operation is the Dorfman bracket:\n\\begin{align*}\n[\\![ x+\\alpha,y+\\beta ]\\!] = [x,y]+\\mathcal L_x\\beta+\\imath_y\\mathrm{d} \\alpha.\n\\end{align*}\nwhere $x,y\\in\\Gamma(TM)$, $\\alpha,\\beta\\in\\Gamma(T^*M)$ and $\\mathcal L_x$ is the Lie derivative. We further introduce the following notation:\n\\begin{Def}\nLet $[\\![\\ ,\\ ]\\!]$ be the D-bracket on a para-Hermitian manifold. Then the {\\it projected brackets} $[\\![\\ ,\\ ]\\!]_P$ and $[\\![\\ ,\\ ]\\!]_{\\tilde{P}}$ are defined as follows\n\\begin{align*}\n\\eta([\\![ X,Y]\\!]_P,Z)&=\\eta(\\nc_{PX}Y-\\nc_{PY}X,Z)+\\eta(\\nc_{PZ}X,Y)\\\\\n\\eta([\\![ X,Y]\\!]_{\\tilde{P}},Z)&=\\eta(\\nc_{{\\tilde{P}} X}Y-\\nc_{{\\tilde{P}} Y}X,Z)+\\eta(\\nc_{{\\tilde{P}} Z}X,Y).\n\\end{align*}\nThe brackets are related by $[\\![ X,Y]\\!]=[\\![ X,Y]\\!]_P+[\\![ X,Y]\\!]_{\\tilde{P}}$.\n\\end{Def}\n\nWe continue with the following simple observation. Since $L$ and ${\\tilde{L}}$ are isotropic with respect to $\\eta$, whenever $\\tilde{x} \\in \\Gamma({\\tilde{L}})$, $\\eta(\\tilde{x},\\cdot)$ is an element of $\\Gamma(L^*)$ because it contracts only with vectors in $L$. In fact, any section of $L^*$ is of this form by non-degeneracy of $\\eta$. We therefore have the following vector bundle isomorphism:\n\\begin{align*}\n\\begin{aligned}\n\\rho:\\ T\\mathcal{P} &= L\\oplus {\\tilde{L}}\\rightarrow L\\oplus L^*\\\\\nX &= x+\\tilde{x} \\mapsto x+\\eta(\\tilde{x}).\n\\end{aligned}\n\\end{align*}\nand similarly we can define $\\tilde{\\rho}:L\\oplus {\\tilde{L}} \\rightarrow {\\tilde{L}} \\oplus {\\tilde{L}}^*$. Moreover, when $L$ is an integrable distribution (i.e. whenever ${\\tilde{P}}[PX,PY]=0\\ \\forall X,Y \\in \\Gamma(T\\mathcal{P})$), the manifold $\\mathcal{P}$ is {\\it foliated} by half-dimensional manifolds $\\ell_i$ called leaves. This means that there is a partition of the $2n$-dimensional manifold $\\mathcal{P}$ into $n$-dimensional manifolds as $\\mathcal{P}=\\bigcup_i \\ell_i$. We can therefore view $\\mathcal{P}$ as an $n$-dimensional manifold (given by the union of the leaves $\\ell_i$ of the foliation) and to avoid confusion we denote this $n$-dimensional manifold as $\\mathcal{F}$. Then, by definition, $L=T\\mathcal{F}$ and $T\\mathcal{P}=L\\oplus L^*=(T\\oplus T^*)\\mathcal{F}$ in a natural way. It turns out that the projected bracket $[\\![\\ ,\\ ]\\!]_P$ is nothing else than the Dorfman bracket of the standard Courant algebroid $(T\\oplus T^*)\\mathcal{F}$ mapped under the map $\\rho$:\n\n\n\\begin{Thm}\nLet $(\\mathcal{P},\\eta,K)$ be an almost para-Hermitian manifold and $[\\![\\ ,\\ ]\\!]$, $[\\![\\ ,\\ ]\\!]_P,[\\![\\ ,\\ ]\\!]_{\\tilde{P}}$ the associated D-bracket and projected brackets, respectively. Whenever $L$ is integrable, $T\\mathcal{P}$ acquires the structure of a Courant algebroid $(T\\mathcal{P},P,\\eta,[\\![\\ ,\\ ]\\!]_P)$. The map $\\rho$ is then an isomorphism of Courant algebroids:\n\\begin{align*}\n(T\\mathcal{P},P,\\eta,[\\![\\ ,\\ ]\\!]_P) \\overset{\\rho_+}{\\longrightarrow}((T\\oplus T^*)\\mathcal{F},a,\\langle\\ ,\\ \\rangle,[\\![\\ ,\\ ]\\!]_{\\mathcal{F}}),\n\\end{align*}\nwhere $((T\\oplus T^*)\\mathcal{F},a,\\langle\\ ,\\ \\rangle,[\\![\\ ,\\ ]\\!]_{\\mathcal{F}})$ is the standard Courant algebroid of $\\mathcal{F}$. This means that\n\\begin{align*}\n\\rho [\\![ X,Y]\\!]_P=[\\![ \\rho X,\\rho Y]\\!]_\\mathcal{F}, \\quad \\eta(X,Y)=\\langle \\rho X,\\rho Y\\rangle, \\quad \\rho\\circ P=a\\circ \\rho.\n\\end{align*}\nAn analogous statement holds for ${\\tilde{L}}$ where the Courant algebroid structure is then given by \\\\ $(T\\mathcal{P}, {\\tilde{P}}, \\eta, [\\![\\ ,\\ ]\\!]_{\\tilde{P}})$.\n\\end{Thm}\n\n\\section{Born Geometry}\n\\label{sec:borngeo}\n\nWe have seen that para-Hermitian geometry is a para-complex geometry $(\\mathcal{P},K)$ with a compatible neutral metric $\\eta$, i.e. one of signature $(n,n)$. We have also seen how if $L$ is integrable we can recover an $n$-dimensional manifold $\\mathcal{F}$ with $L=T\\mathcal{F}$ and and a standard Courant algebroid structure which can be identified with the physical spacetime. In order to make connection with a string background, we also need to recover the physical background fields on $\\mathcal{F}$, i.e. the spacetime metric $g$ and the Kalb-Ramond field $B$.\n\nTherefore the next step is to add another compatible metric $\\mathcal{H}$ of signature $(2n,0)$ to the picture, giving rise to what has been named Born geometry \\cite{Freidel:2013zga}. The additional Riemannian structure $\\mathcal{H}$ defines two more endomorphisms of the tangent bundle which -- along with the para-complex structure $K$ already in place -- form a para-quaternionic structure \\cite{Freidel:2015pka,Freidel:2013zga,Ivanov:2003ze}.\n\n\\begin{Def}\nLet $(\\mathcal{P},\\eta,\\omega)$ be a para-Hermitian manifold and let $\\mathcal{H}$ be a Riemannian metric satisfying\n\\begin{align}\n\\eta^{-1}\\mathcal{H}=\\mathcal{H}^{-1}\\eta,\\quad \\omega^{-1}\\mathcal{H}=-\\mathcal{H}^{-1}\\omega .\n\\end{align}\nThen we call the triple $(\\eta,\\omega,\\mathcal{H})$ a Born structure on $\\mathcal{P}$ where $\\mathcal{P}$ is called a Born manifold and $(\\mathcal{P},\\eta,\\omega,\\mathcal{H})$ a Born geometry.\n\\end{Def}\nNote that in this definition we view $(\\eta,\\omega,\\mathcal{H})$ as maps $T\\mathcal{P}\\rightarrow T^*\\mathcal{P}$. One can show \\cite{Freidel:2018tkj} that there always exists a choice of frame on $T\\mathcal{P}=L\\oplus{\\tilde{L}}$ where the generalized metric takes the form \n\\begin{equation}\n\\mathcal{H} = \\begin{pmatrix}\ng & 0 \\\\ 0 & g^{-1}\n\\end{pmatrix}\n\\label{eq:genmetric}\n\\end{equation}\nwhere $g$ is a Riemannian metric on $L=T\\mathcal{F}$. The appearance of the B-field will be discussed below in the context of twisting the para-Hermitian structure. \n\nIn order to understand the nature of a Born structure we review its three fundamental structures. First, as we have seen, it contains an almost {\\it para-Hermitian} structure $(\\omega, K)$ with compatibility\n\\begin{equation}\nK^2=\\mathbbm{1},\\qquad \\omega(KX,KY)=-\\omega(X,Y). \n\\end{equation}\nNext, the compatibility between $\\eta$ and $\\mathcal{H}$ implies that $J=\\eta^{-1}\\mathcal{H} \\in \\mathrm{End}\\, T\\mathcal{P}$ defines what we refer to as a  {\\it  chiral} structure\\footnote{The projectors $P_\\pm=\\frac12(\\mathbbm{1}\\pm J)$ define, similarly to projectors $P$, ${\\tilde{P}}$, a splitting of the tangent bundle $T\\mathcal{P} = C_+ \\oplus C_-$ which mirrors the right\/left chiral splitting of string theory. In other words,\ngiven a worldsheet $X:\\Sigma \\rightarrow \\mathcal{P}$ with $\\Sigma$ a Riemann surface, we have that $\\partial_z X$ takes values in $C_+$ and $\\partial_{\\bar{z}} X$ in $C_-$. Hence $J$ is called a chiral structure.} $(\\eta,J)$ on $\\mathcal{P}$\n\\begin{align}\nJ^2=\\mathbbm{1},\\qquad \n\\eta(JX,JY)=\\eta(X,Y).\n\\end{align}\nFinally, the compatibility between  $\\mathcal{H}$ and $\\omega$ defines an almost {\\it Hermitian} structure $(\\mathcal{H},I)$ on $\\mathcal{P}$\n\\begin{equation}\nI^2=-\\mathbbm{1},\\qquad \\mathcal{H}(IX,IY)= \\mathcal{H}(X,Y).\n\\end{equation}\nThese three structures, para-Hermitian, chiral and Hermitian satisfy in turn an extra compatibility equation\n\\begin{equation}\nKJI=\\mathbbm{1},\n\\end{equation}\nwhich follows directly from their definition $K=\\omega^{-1}\\eta$, $J=\\eta^{-1} \\mathcal{H}$, $I=\\mathcal{H}^{-1}\\omega$. This means that the triple $(I,J,K)$  form an almost para-quaternionic structure \n\\begin{align}\n-I^2=J^2=K^2=\\mathbbm{1}, \\qquad \n\\{I,J\\}=\\{J,K\\}=\\{K,I\\}=0,\\qquad \nKJI=\\mathbbm{1},\n\\end{align}\nwhere $\\{\\ , \\ \\}$ is the anti-commutator. \n\n\\begin{table}[h!]\n\\renewcommand{\\arraystretch}{1.3}\n\\centering\n\\resizebox{0.9\\textwidth}{!}{%\n\\begin{tabular}{@{}ccc@{}}\n\\toprule\n$I={\\mathcal{H}}^{-1}{\\omega}=-\\omega^{-1}\\mathcal{H}$ & $J={\\eta}^{-1}{\\mathcal{H}}={\\mathcal{H}}^{-1}{\\eta}$ & $K={\\eta}^{-1}{\\omega}={\\omega}^{-1}{\\eta}$  \\vspace{8pt} \\\\\n$-I^2=J^2=K^2=\\mathbbm{1}$                     & $\\{I,J\\}=\\{J,K\\}=\\{K,I\\}=0$           & $IJK=-\\mathbbm{1}$     \\vspace{8pt} \\\\\n$\\mathcal{H}(IX,IY)=\\mathcal{H}(X,Y)$              & $\\eta(IX,IY)= -\\eta(X,Y)$         & $\\omega(IX,IY)=\\omega(X,Y)$             \\\\\n$\\mathcal{H}(JX,JY)=\\mathcal{H}(X,Y)$              & $\\eta(JX,JY)= \\eta(X,Y)$          & $\\omega(JX,JY)=-\\omega(X,Y)$            \\\\\n$\\mathcal{H}(KX,KY)=\\mathcal{H}(X,Y)$              & $\\eta(KX,KY)= -\\eta(X,Y)$         & $\\omega(KX,KY)=-\\omega(X,Y)$            \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\caption{Summary of structures in Born geometry. Here $\\{\\ , \\ \\}$ is the anti-commutator.}\n\\label{tab:BornGeo}\n\\end{table}\n\n\n\\section{Born Connection}\nWe have now all ingredients in place to examine the Born connection. It is the unique connection compatible with the Born geometry, i.e. with the three defining structures $(\\eta,\\omega,\\mathcal{H})$, which has no generalized torsion. The Born connection can be seen as the analogue of the Levi-Civita connection which is the unique, torsionfree, metric-compatible connection for Riemannian geometry.\n\n\\begin{Thm}\\label{th:BornConnection}\nLet $(\\mathcal{P},\\eta,\\omega,\\mathcal{H})$ be a Born geometry with $K=\\eta^{-1}\\omega$ the corresponding almost para-Hermitian structure. Then there exists a unique connection called the Born connection denoted by $\\n^{\\mathrm{B}}$ which\n\\begin{itemize}\n  \\item is compatible with the Born geometry $(\\mathcal{P},\\eta,\\omega,\\mathcal{H})$,\n  \\item has a vanishing generalized torsion ${\\mathcal{T}}=0$.\n\\end{itemize}\nThis connection can be explicitly expressed in terms of the three defining structures $(\\eta,\\omega,\\mathcal{H})$ of the Born geometry and the canonical D-bracket. It can be concisely written in terms of the para-Hermitian structure $K$ and the chiral projections $X_\\pm=\\frac{1}{2}(\\mathbbm{1}\\pm J)X$ as\n\\begin{equation}\n\\n^{\\mathrm{B}}_XY = [\\![ X_-,Y_+ ]\\!]_+ + [\\![ X_+,Y_-]\\!]_- + (K[\\![ X_+,KY_+ ]\\!])_+ + (K[\\![ X_-,KY_-]\\!])_- .\n\\label{eq:BornBracket}\n\\end{equation}\n\\end{Thm}\n\nThe theorem is proven in \\cite{Freidel:2018tkj}.\n\n\\section{Fluxes}\nWe have seen in previous sections that any para-Hermitian structure $(\\eta, K)$ gives rise to its unique associated D-bracket. In this section we will explain how the {\\it twisted} D-bracket appears in this context. We will see that the fluxes twisting the bracket appear when we choose a different para-Hermitian structure $K'$ compatible with the same $\\eta$, and express its associated D-bracket in the splitting given by the original $K$. The appearance of fluxes can be therefore intuitively understood as a choice of a splitting not compatible with the D-bracket at hand and the fluxes can be seen as certain obstructions of an integrability of this splitting.\n\nWe now define a transformation of the para-Hermitian structure analogous to a B-field transformation in generalized geometry.\n\n\\begin{Def}\nLet $(\\eta,K)$ be a para-Hermitian structure on $\\mathcal{P}$. A B-transformation of $K$ is given by\n\\begin{align*}\nK\\overset{e^{B}}{\\longmapsto} K_{B}=e^{B} K e^{-{B}},\\quad\ne^{B}\\coloneqq\n\\begin{pmatrix}\n\\mathbbm{1} & 0 \\\\\nB & \\mathbbm{1}\n\\end{pmatrix} \\in \\text{End}(T\\mathcal{P})\n\\end{align*}\nwhere the map $e^B$ is expressed in the matrix representation corresponding to the splitting $L\\oplus L$ and $B:L\\rightarrow {\\tilde{L}}$ is a skew map such that $\\eta(BX,Y)=-\\eta(X,BY)$.\n\\end{Def}\nIt is easy to see that ${B}$ can be given by either a two-form $b$ or a bivector $\\beta$,\n\\begin{align}\\label{eq:b-beta}\n\\eta(BX,Y)=b(X,Y)=\\beta(\\eta (X),\\eta (Y)),\n\\end{align}\nwhere $b$ is of type $(+2,-0)$ and $\\beta$ is of type $(+0,-2)$, so we can write $b(X,Y)=b(\\tilde{x},\\tilde{y})$. In coordinates, we have\n\\begin{align*}\nb=b_{ij}\\mathrm{d} x^i\\wedge \\mathrm{d} x^j,\\quad \\beta=\\beta_{ij} \\tilde{\\partial}^i\\wedge \\tilde{\\partial}^j.\n\\end{align*}\n\nIt can be checked that $K_B$, the B-transformation of $K$, is a new (almost) para-Hermitian structure. $K$ and $K_B$ share the same $-1$ eigenbundle, but the $+1$ eigenbundle is sheared by the map $B$: $L\\mapsto L_B=L+B(L)$. The B-transformation therefore in general does not preserve integrability of the para-Hermitian structure.\n\nWhen the starting para-Hermitian structure is in fact an integrable para-K\\\"ahler structure, the D-bracket associated to $K_B$ carries with respect to $K$ well-known fluxes from physical literature:\n\n\\begin{Prop}\nLet $K_B$ be a B-transformation of a para-K\\\"ahler structure $(\\mathcal{P},\\eta,K)$. Then the D-bracket associated to $K_B$ is given by\n\\begin{align}\\label{twistedDbrac}\n\\eta([\\![ X,Y]\\!]^{B},Z)=\\eta([\\![ X,Y]\\!]^D ,Z)-(\\mathrm{d} b)(X,Y,Z).\n\\end{align}\nwhere $[\\![\\ ,\\ ]\\!]$ denotes the D-bracket of $K$. The different components of $\\mathrm{d} b$ with respect to the splitting of $K_B$ yield \n\\begin{align*}\n\\mathrm{d} b^{(3,0)_B}&=\\hat{H}=H+\\tilde{R}\\\\\n\\mathrm{d} b^{(3,0)_B}&=\\tilde{Q},\n\\end{align*}\nwhere $\\hat{H}$ is the Covariantized H-flux, $H$ is the usual H-flux and $\\tilde{R}$ and $\\tilde{Q}$ are the dual R- and Q-fluxes.\n\\end{Prop}\n\nWe see the appearance of the {\\it dual} fluxes, meaning $\\tilde{R}$ is a three-vector on $\\tilde{L}$, while the usual R-flux is usually understood as a three-vector on $L$, i.e. $R$ and $\\tilde{R}$ have an opposite index structure and similarly with $Q$ and $\\tilde{Q}$. If we wanted to achieve the usual $Q$ and $R$, we would have to perform the dual B-transformation, shearing ${\\tilde{L}}\\mapsto {\\tilde{L}}+B({\\tilde{L}})$. Indeed, this has been done in \\cite{Marotta:2018myj}, where it was additionally also shown that the simultaneous B-transformation shearing both in $L$ and ${\\tilde{L}}$ directions yields the whole hierarchy of fluxes appearing in DFT. \n\nLet us conclude this section by giving the effect of a B-transformation on the generalized metric $\\mathcal{H}$ as given in \\eqref{eq:genmetric}. The transformed generalized metric takes the form\n\\begin{equation}\n\\mathcal{H} \\overset{e^{B}}{\\longmapsto} e^{-B}\\mathcal{H} e^{-B} = \n\\begin{pmatrix}\ng - b g^{-1} b& bg^{-1} \\\\ -g^{-1}b & g^{-1}\n\\end{pmatrix}\n\\end{equation}\nwhich is familiar from generalized geometry and DFT.\n\n\\section{Conclusion}\nThe construction of a para-Hermitian manifold with a corresponding foliation (when one of the subdistributions is integrable) to recover the physical spacetime of a string theory background has been summarized here. This puts the idea of a doubled target space for the fundamental string on a firm mathematical base and spells out the precise relation to generalized geometry and double field theory. \n\nOn top of this generalized kinematical structure one can include the dynamical fields via a generalized metric which corresponds to a choice of Riemannian metric on the subspace, giving rise to a Born geometry. The generalized fluxes appear via a twisting of the para-Hermitian structure and hence the D-bracket. This allows for an inclusion of the B-field and subsequently the other fluxes of string theory.\n\nConcrete examples of para-Hermitian and Born geometries such as $\\mathcal{P} = TM$, i.e. where the doubled space is the tangent bundle of some manifold, can be found in \\cite{Marotta:2018myj}. Group manifolds such as Drinfel'd doubles can also be viewed from a para-Hermitian perspective and provide another class of examples \\cite{Marotta:2018myj,Mori:2019slw,inprep}.\n\nIn future work, we aim to develop this formalism further and show how it applies in various well-known settings where T-duality arises, such as topological T-duality \\cite{tduality1,tduality2,tduality3}.\n\n\\section*{Acknowledgments}\nThe authors would like to thank the organizers of the ``Corfu Summer Institute 2018'' for the invitation. F.J.R. is supported by the Max-Planck-Society. The work of D.S. is supported by NSERC Discovery Grant 378721.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{T}{he} drilling machines are commonly used in factories to drill holes in materials. In this paper, a failure detection system for Valmet AB's drilling machines is investigated using Valmet AB's drilling sounds. A drilling machine operated by Valmet AB includes many drill bits used for drilling thousands of small holes in metal surfaces. Drill bits that break can cause serious damage to a product since they lead to a lack of holes in the metal surface. As a result, a technician manually punches holes into metal sheets that are missing holes. It is a very time-consuming and expensive process. Hence, technicians usually check the drilling machine every 10 minutes so that if any drill bit breaks they can replace it before the drilling machine continues. Furthermore, shutting the machine off and on every 10 minutes is labor-intensive and time-consuming. Hence, the factory would benefit from an automated system that can classify drilling sounds and notify technicians when drill bits break. Our machine failure detection proposal builds on the fact that skilled technicians can detect a drill that isn't working properly by listening to its sound when cutting metal.\n\nTypically, large, balanced datasets were used to build a machine failure detection system. Data anomalies related to the damaged machine in Valmet AB are probably only a small proportion of the total data set. As a result of the imbalance of real-world datasets, CNN tends to be biased; for instance, CNN will struggle to predict minority classes if they have fewer data. The collection and labeling of sounds can be a time-consuming and expensive process for a company. These operational costs can be reduced by using data augmentation techniques to transform datasets. Additionally, in order to create variations that are representative of what a deep learning model might see in reality, data augmentation techniques enable deep learning models to be more robust.\n\nSound applications use standard augmentation methods for extending small datasets. There are classic and advanced techniques in sound augmentation. The conventional approach of sound augmentation methods includes time stretching, pitch shifting, volume control, adding noise, time-shifting, etc \\cite{noauthor_augment_nodate}. Research in \\cite{tran2021detecting} showed that some augmentation methods are not appropriate to apply in our short sounds (20.83 ms or 41.67 ms). The advanced approach for data augmentation is using Synthetic Minority Over-sampling Technique (SMOTE) \\cite{chawla_smote:_2002} or  Modified Synthetic Minority Over-sampling Technique (MSMOTE)\\cite{5403368}, etc. Overall, data augmentation helps to increase the number of sounds in the dataset. This improves the prediction accuracy of deep learning models and reduces data overfitting. Additionally, it increases the generalization of deep learning models by creating variable data. \n\nRecently, deep learning augmentation methods such as generative adversarial network (GAN) ~\\cite{8698632,9006268,9411996,9390834,8615782,8985892,8717183, 8962219, 8857905} have been widely used as data augmentation methods in computer vision. However, GANs require large amounts of training data to generate effective augmented data.\nVariational Autoencoder (VAE) was proposed in 2014 by Kingma and Welling \\cite{kingma2014autoencoding}. VAE has been used in domain adaption in face recognition frameworks~\\cite{9432318}, network representation learning~\\cite{9449483}, detect anomaly in log file systems~\\cite{8944863}, statistically extract latent space hidden in sampled data to control the robot~\\cite{9143393}, to generate more data for software fault prediction~\\cite{8672311}, traffic anomaly detection in videos~\\cite{9531567}, for time series anomaly detection~\\cite{9053558}, etc.\nVAE and different improved versions of VAE have been used to generate synthetic data in remote sensing images~\\cite{9393473}, for phrase-level melody representation learning to generate music~\\cite{9054554}, to generate sensor data for chiller fault diagnosis~\\cite{9264218}, to generate conditional handwritten characters~\\cite{9511404, 8803129}, etc.\nThe purpose of this study is to generate new drilling sounds using VAE. To the best of our knowledge, this is the first time VAE was investigated from a drilling sound augmentation standpoint for the machine failure detection system. \n\nThe classification of sound signals generated by machines has been investigated in recent years. In conventional machine learning techniques, statistical features from acoustic signals are extracted and classified~\\cite{Lee2016},~\\cite{Kemalkar2017},~\\cite{Zhang2019a}. In machine failure analysis, machine learning algorithms are commonly used because of their robustness and adaptability on small datasets~\\cite{9252126}. These conventional machines learning methods  include Support Vector Machine (SVM)~\\cite{fengqi_compound_2006,Li2011TheAO,deng_novel_2019}, k\u2011Nearest Neighbors~\\cite{7360161, WANG2016201, GLOWACZ201965}, decision tree~\\cite{saravanan_fault_2009}, ANN and radial basis function\n(RBF)~\\cite{1265797,10.1016\/j.neucom.2011.03.043, doi:10.1177\/1077546313518816}, etc.\n\nThe use of deep learning to detect mechanical faults by analyzing acoustic signals has been successfully applied~\\cite{polat2020fault,zhang2020deep,islam2018motor,verstraete2017deep,chen2017vibration,Long2020a,Paul2019,Luo2018,Ince2016}. According to recent studies, deep learning architectures can be trained to identify sound signals by using image representation, such as Mel frequency cepstral coefficients (MFCCs)~\\cite{Zhang2017,8635051}, spectrogram~\\cite{Boddapati2017}, Mel spectrogram~\\cite{Mushtaq2021,9252126}, log-Mel spectrogram~\\cite{tran2021detecting}. \n\nWhen detecting failures on rotating machines, a conventional approach of handcrafted feature extraction and selection is often used. However, extracting and selecting the right features can be challenging in order to use in a classifier or to accurately detect faults in an industrial setting. It is possible for both sound and noise to occur simultaneously in a realistic soundscape. Additionally, the drill sound waveform is complex and short, making it difficult to detect (around 20.83~ms and 41.67~ms). It is therefore not guaranteed that the extracted features of raw audio signals are sufficient for classification. \n\nThe use of vibration sensors is common when detecting machine faults. On the contrary, our methods of detecting broken drills were based on sound signals for these reasons. Drill bits for each drilling machine in Valmet AB are available in quantities of 90 or 120. A vibration sensor is required for each drill bit in order to detect fractures. Mounting 90 to 120 vibration sensors on a drilling machine simultaneously and classifying these vibrations is complicated and expensive. In addition to being cautious about smooth and accurate holes when cutting metal, you must also avoid metal sticking to the drill bit. Dust and sediment are therefore removed from metal surfaces with water. Using vibration sensors in a very wet environment is problematic. For all of the reasons outlined above, we chose sound over vibration to detect broken drills in the drilling machine.\n\n\n\nThe rest of the paper is organized as follows. Section II introduces Valmet AB's dataset. Section III presents the method for data augmentation using VAE and our proposed machine failure detection system. Section IV presents the experiment result of training VAE to synthesize new drilling sounds, the classification results on the augmented dataset, and the comparison study. Section V is the conclusion. \n\n\n\\section{Dataset}\nFour microphones with AudioBox iTwo Studio were used to capture the sounds with a sampling frequency of 96 kHz from Valmet AB's drill machines in Sundsvall, Sweden. The drilling machines at Valmet AB consist of two types of drill bits, one contains 90 drill bits, the other 120 drill bits. The sample duration for sounds in the Valmet AB dataset is very short, around 20.83~ms and 41.67~ms corresponding to 2000 sample points. The original Valmet dataset includes two categories: the \"Anomaly\" class (67 anomalous sounds) and the \"Normal\" class (67 normal sounds). \"Anomaly\" class includes all sounds recorded when the drill bit was broken. The \"Normal\" class includes sounds recorded when the drilling machine was operating properly. \n\n\\section{Methodology}\n\\subsection{Variational Autoencoder}\nA variational autoencoder (VAE)~\\cite{kingma2014autoencoding},~\\cite{rezende2014stochastic} is the architecture that belongs to the field of probabilistic graphical and variational Bayesian. A VAE is composed of an encoder and a decoder (Figure \\ref{fig:Figure1}). The purpose of a VAE model isn't to replicate input sounds but to generate random variations on this input from a continuous space. The most important part of VAE is the continuous latent space that makes interpolation easier. VAE first takes an input sound and creates a two-dimensional vector (mean and variance) from a random variable (the encoding). A sampled encoding is obtained from this vector and passed to the decoder.  The decoder decodes this 1D vector and recreates the original sound. The decoder is capable of learning from all nearby points on the same latent space because encodings are generated from distributions with the same mean and variance as inputs.\n\n\\begin{figure}[!ht]\n\\centering\n  \\includegraphics[width=0.95\\linewidth]{Images\/Figure1.pdf}\n  \\caption{Autoencoder.}\n  \\label{fig:Figure1}\n\\end{figure}\n\nThe divergence between probability distributions was controlled by Kullback\u2013Leibler divergence. Kullback\u2013Leibler divergence is minimized by optimizing mean and variance to closely resemble that of the target mean and variance. \n\n\\subsection{Using VAE for drilling sound synthesis}\nWe created a VAE to generate new drilling sounds from the sounds in the original dataset. VAE differs from traditional autoencoders in that it does not reconstruct the input using the encoding-decoding process. VAE instead imposes a probability distribution on the latent space and learns it so that the decoder produces outputs. The distribution of these outputs matches the observed data. New data are generated by sampling from this distribution. In this paper, we train a VAE on sounds in our dataset and generate new sounds that closely resemble the original sounds. Figure \\ref{fig:EncoderDecoder} shows the architecture of our VAE. \n\n\n\\subsubsection{Encoder}\nThe encoder includes four one-dimensional convolutional (Conv1D) layers interspersed with four ResnetEncoder blocks, as shown in the left side of Figure \\ref{fig:EncoderDecoder}. The inputs of the encoder are 1999-length vectors corresponding to 1999 sample points of each sound in the dataset. The architecture of ResNetEncoder block is shown on the right side of Figure \\ref{fig:EncoderDecoder}. ResNetEncoder block includes two one-dimensional convolutional layers and two instance normalization layers. A channel's features are normalized by instance normalization. When the learning rate is adjusted linearly with batch size, it has a more stable accuracy than the batch normalization in a wide range of small-batch sizes.\n\n\\subsubsection{Decoder}\nThe decoder includes four ResnetDecoder blocks interspersed with four one-dimensional convolutional transpose (Conv1DTranpose) layers, as shown in Figure \\ref{fig:EncoderDecoder}. The input\\_shape of the decoder is the latent dimension vector from the encoder. ResNetDecoder block includes two one-demensional convolutional layers and two batch normalization layers. A transposed convolution layer has the opposite transformation as a normal convolution layer. This is achieved by transforming the latent space that has the shape of the output of a convolution into something that has the shape of its input while maintaining a connectivity pattern that is compatible with the convolution. \n\n\n\n\\subsubsection{Reparameterization trick}\nThe output of the encoder is the latent distribution. The next step is the sampling step which samples the variance and the mean vectors to pass to the decoder. However, this sampling step creates a bottleneck because the back propagation can not run through a random node, the parameterization trick is usually used to address it. As a result, we approximate \\(Z\\) using the decoder parameters and another parameter as follows:\n\\begin{equation}\n    Z = \\mu + \\sigma*\\epsilon,\n\\end{equation}\nwhere $\\mu$ represent the mean and $\\sigma$ represent the standard deviation of a Gaussian distribution. $\\epsilon$ is an auxiliary variable derived from a standard normal distribution that preserves the stochasticity of \\(Z\\). The parameterization trick is despicted in Figure~\\ref{fig:Figure1}.\n\n\\subsubsection{Loss function}\nThe loss function in VAE is the sum of the reconstruction loss and Kullback\u2013Leibler loss (KL loss) \\cite{MAL-056}. The reconstruction loss measures the difference between our original input sound and the decoder output sound by using the binary cross-entropy loss. \n\nThe KL\\_loss ($KL\\_loss$) is calculated by optimizing the single sample Monte Carlo estimation as below: \n\n\\begin{equation}\nKL\\_loss = \\log p(x|Z)+\\log p(Z)-\\log q(Z|x)\n\\end{equation}\n\n\\begin{figure*}[!ht]\n\\centering\n  \\includegraphics[width=0.78\\linewidth]{Images\/EncoderDecoder.pdf}\n  \\caption{The variational autoencoder.}\n  \\label{fig:EncoderDecoder}\n\\end{figure*}\nwhere $x$ represents the input sound, $Z$ represents the latent variable, $p(x)$ is the probability distribution of the sound, $p(Z)$ is the probability distribution of the latent variable, and $p(x|Z)$ is the distribution of generating data given latent variable, The probability distribution $p(Z|x)$ describes how our data is projected into latent space. $q(Z|x)$ is the inference model with \\(q(Z|x) \\approx p(Z|x)\\)~\\cite{MAL-056}.  \n\n\\subsection{The proposed machine failure detection system}\nIn this study, we attempted to categorize the drill sounds for the purpose of detecting drill bit failure in Valmet AB, a manufacturing in Sundsvall, Sweden. A fairly small balanced dataset for the two classes was a challenge for us when applying deep learning model. Therefore, we proposed employing VAE (a deep learning model) to generate more synthetic drilling sounds from the modest original drilling sounds as a data augmentation strategy. The original sounds were combined with the generated sounds from VAE to form the enhanced dataset. Then, a low-pass filter with a passband of 22\\kern 0.16667em000 Hz and a high-pass filter with a passband of 1000 Hz were applied to this dataset as part of the preprocessing procedure. After preprocessing, these sounds were converted into Mel spectrogram and classified using the pre-trained network AlexNet. Our proposed procedure is shown in Figure~\\ref{fig:systemArch}.\n\n\\begin{figure*}[!ht]\n\\centering\n  \\includegraphics[width=0.78\\linewidth]{Images\/SystemArchitecture.pdf}\n  \\caption{The proposed system architecture.}\n  \\label{fig:systemArch}\n\\end{figure*}\n\n\\section{Experimental results}\nThe experiment for this paper was conducted on both python and Matlab. \n\\subsection{Training VAE for drilling sound synthesis}\n\n\\subsubsection{Data preparation and training VAE}\nTraining VAE for sound synthesis is implemented using tensorflow, numpy, pandas and librosa. The number of epoch is 20 and batch size is 8. The dimensionality of the latent space is set to 2 so the latent space is visualized as a plane. The sample rate is 96\\kern0.16667em000. The training lists included 34 anomalous sounds and 34 normal drill sounds. The testing lists included 33 anomalous drill sounds and 33 normal drill sounds. The sounds are shuffled and batched using \\textit{tf.data}.\n\n\nThe training process was iterated over the training lists. For each iteration, the sound was passed through the encoder to obtain a set of mean and log-variance for the $q(Z|x)$. This approximate posterior is then sampled using the reparameterization trick. The parameterized samples were then passed through the decoder to get the distribution $p(x|Z)$.\n\nAfter training VAE on the training sounds, we generated 48 drilling sounds for the \"Anomaly\" class and 48 normal drilling sounds for the \"Normal\" class. \n\n\n\\subsubsection{Measuring sound similarities between the original sound and the synthesized sound}\nWe hypothesis that VAE maybe reduce the noise in the synthesized data. However, by listening to the synthesized drilling sounds, the newly generated drilling sounds closely follow the pitch of the original sounds. Hence, we assessed the similarity between a sound in a testing set and a synthesized drilling sound from VAE using the cross-correlation and the spectral coherence between two sounds using Matlab.\n\n\\textbf{Cross-correlation between two sounds:}\n\nWe measured the similarities between an original sound in the anomaly class and the synthesized sound from VAE. These two sounds have the same sampling rate of 96\\kern 0.16667em000 Hz but have different lengths. The original sound has a length of 41.67 ms whereas the synthesized sound has a length of around 20.83 ms. The calculation of the difference between two sounds is difficult because of the different lengths so we had to extract the common part of the two sound signals using the cross-correlation. Figure~\\ref{fig:cross-correlation} shows the cross-correlation between the original sound and the synthesized sound. The value of cross-correlation coefficient is from -1.0 to 1.0. If the cross-correlation value is closer to 1, the two sounds are more similar. The high peak in the third subplot shows that the original sound correlated to the synthesized sound.\n\\begin{figure}[!ht]\n\\centering\n  \\includegraphics[width=1\\linewidth]{Images\/cross_correlation.eps}\n  \\caption{The cross-correlation between the original sound and the synthesized sound.}\n  \\label{fig:cross-correlation}\n\\end{figure}\n\n\\textbf{The spectral coherence between two sounds:}\n\nPower spectrum shows how much power is present per frequency. Correlation between signals in the frequency domain is referred to as spectral coherence. The power spectrum of Figure~\\ref{fig:spectral_coherence1} shows that the original sound and synthesized sound have correlated components in the range of 0 Hz and 6890 Hz. \n\\begin{figure}[!ht]\n\\centering\n  \\includegraphics[width=1\\linewidth]{Images\/spectral_coherence.pdf}\n  \\caption{The power spectrum.}\n  \\label{fig:spectral_coherence1}\n\\end{figure}\n\nWe show the coherence estimate in the first subplot of Figure~\\ref{fig:spectral_coherence2}. Coherence values range from 0 to 1. For example, we marked three highest peak of the coherence values as 0.494153, 0.55872, and 0.377155, as shown in Figure~\\ref{fig:spectral_coherence2}. The higher the coherence, the more correlated the corresponding frequency components are. There are approximately 45 degrees of phase lag between the 1125 Hz components and the 4312.5 components. The phase lag between the 9187.5 Hz components is approximately 38 degrees.\n\n\\begin{figure}[!ht]\n\\centering\n  \\includegraphics[width=1\\linewidth]{Images\/spectral_coherence2.pdf}\n  \\caption{The spectral coherence.}\n  \\label{fig:spectral_coherence2}\n\\end{figure}\n\n\\subsection{Classification result using original and synthesized sounds}\nWe mixed the orginal dataset that contained 67 sounds for each class with the generated dataset that contained 48 synthesized sounds for each class. We obtained the augmented dataset that included two classes, \"Normal\" class (115 sounds) and \"Anomaly\" class (115 sounds).\n\\subsubsection{Sound pre-processing}\nThe sample rate of sounds in our augmented dataset is 96\\kern 0.16667em000, but the sound of a drill typically ranges from roughly 1000 to 22\\kern 0.16667em000 Hz. The low-pass filter are used to eliminate frequencies higher than 22\\kern 0.16667em000 Hz. The frequency below 1000 Hz was then removed using a high pass filter.\n\n\\subsubsection{Convert sounds to Mel spectrograms}\nWe converted sounds into Mel spectrograms using the Mel spectrums of 1999-point periodic Hann windows with 512-point overlap. 1999-point FFT was used to convert to the frequency domain and pass this through 32 half-overlapped triangular bandpass filters. \nFigure~\\ref{fig:MelSpec}a shows the Mel spectrogram of a sound without applying low-pass filter and high pass filter. Figure~\\ref{fig:MelSpec}b shows the Mel spectrogram after applying low-pass filter and high pass filter. \n\n\n\n\\begin{figure}[hpt!]\n    \\centering\n    \\subfigure[]{\\includegraphics[width=0.49\\linewidth]{Images\/Break_61_MelSpect.jpg}} \n    \\subfigure[]{\\includegraphics[width=0.49\\linewidth]{Images\/Break_61_lowpass_highpass_MelSpect.jpg}} \n    \\caption{The Mel spectrogram of an original sound in the \"Anomaly\" class: (a) without applying low-pass filter and high-pass filter  (b) apply low-pass filter and high-pass filter.}\n    \\label{fig:MelSpec}\n\\end{figure}\n\n\n\\subsubsection{Classification result}\nTo demonstrate the effectiveness of using VAE for generating new sounds, we proceed to classify two classes in the augmented dataset (mix between original sounds and synthesized sounds) using several pre-trained networks as in Table~\\ref{tablecompaAccuracy}. This experiment is conducted on Matlab 2021a. In overall, pre-trained AlexNet shows the best accuracy (94.12\\%) on our dataset. AlexNet is a simple CNN with only eight layers that was trained to classify 1000 objects from the ImageNet dataset. \n\n\\begin{table}[h]\n\\centering\n\\caption{The accuracy of different pre-trained models on the augmented dataset.}\n\\label{tablecompaAccuracy}\n\\begin{tabular}{l|c}\n\\hline\n\\textbf{Pre-trained model}                  & \\textbf{Accuracy (\\%)} \\\\ \n\\hline\nGoogLeNet               & 89.71                       \\\\ \\hline\n\\textbf{AlexNet}         & \\textbf{94.12}                       \\\\ \\hline\nVGG16   & 88.24                     \\\\ \\hline\nVGG19         & 85.29                       \\\\ \\hline\nEfficientnetb0         & 86.76                       \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nBecause after training VAE to generate new drilling sounds, our augmented dataset is extended but still small. Hence, we fine-tuned a common AlexNet with transfer learning instead of training a new CNN from scratch. The learned features from pre-trained AlexNet can easily and quickly transfer to a new task of classification the \"Normal\" class and \"Anomaly\" class in the augmented dataset. Sounds in the augmented dataset were converted into Mel spectrogram, as described above.  \n\nWe utilized 70\\% of Mel spectrograms (162 sounds) for training and 30\\% for validation (68 sounds). The input size of AlexNet is 227-by-227-by-3, where 3 represents three R, G, B channels.  The fully connected layer and classification layer of AlexNet were used to classify 1000 classes. To fine-tuned AlexNet to classify only two classes in our augmented dataset, the fully connected layer was replaced with a new fully connected layer that has only two outputs. the classification layer was replaced with a new classification layer that has two output classes. \n\nWe augmented the training set (162 sounds) by flipping the Mel spectrograms along the x-axis, translating them via the x-axis and the y-axis randomly in the range of (-30, 30) pixels, and scaling them via the x-axis and the y-axis randomly in the range of (0.9, 1.1).  \n\nFor the training option, mini-batch size was 10, the validate frequency (\\(valFreq\\)) is calculated as below:\n\\begin{equation}\nvalFreq = training\\_file\/ mini\\_batch\\_size\n\\end{equation}\nwhere training\\_file is the number of the training file. Max epochs were 160, the initial leaning rate is 3e-4, shuffle every epoch.\n\nThe overall accuracy when fine-tuning AlexNet on our augmented dataset reaches 94.12\\%. The confusion matrix is shown in Figure \\ref{fig:ConfusionMatrix_ImgAugmentation} whereas the accuracy when detect \"Anomaly\" class reaches 97.06\\%.\n\n\n\\begin{figure*}[hpt!]\n    \\centering\n    \\subfigure[]{\\includegraphics[width=0.35\\linewidth]{Images\/AlexNet_ConfusionMatrix.pdf}} \n    \\subfigure[]{\\includegraphics[width=0.35\\linewidth]{Images\/AlexNet_ConfusionMatrixPercent.pdf}} \n    \\caption{The confusion matrix when fine-tuning AlexNet to classify \"Normal\" and \"Anomaly\" classes on the augmented dataset: (a) in number (b) in percentage.}\n    \\label{fig:ConfusionMatrix_ImgAugmentation}\n\\end{figure*}\n\n\\subsection{Analyzing the original dataset with pre-trained Alexnet}\nThis section compares the accuracy of the pre-trained network AlexNet on the original dataset and on the augmented dataset. We trained pre-trained AlexNet with the same training option on the original drilling dataset, the accuracy reached 87.5\\%, as shown in Table ~\\ref{tablecompare1}. Meanwhile, we found that when we combined the original drilling sounds and synthesized sounds derived from VAE (the augmented dataset), the classification accuracy was 94.12\\%, which is 10.15\\% higher than when we used the small original dataset. The confusion matrix for the original dataset is shown in Figure~\\ref{fig:ConfusionMatrix_original}.\n\n\n\\begin{table}[h]\n\\centering\n\\caption{The comparison of AlexNet on the original dataset and the augmented dataset. No. of sounds indicates the number of sounds.}\n\\label{tablecompare1}\n\\begin{tabular}{l|l|c}\n\\hline\n\\textbf{Dataset}              & \\textbf{No. of sounds} & \\textbf{Accuracy (\\%)} \\\\ \\hline\nThe original drilling dataset & 134                   & 87.5                      \\\\ \\hline\nThe augmented dataset         & 230                   & 94.12                  \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{figure*}[hpt!]\n    \\centering\n    \\subfigure[]{\\includegraphics[width=0.35\\textwidth]{Images\/AlexNet_ConfusionMatrix_OriginalDataset.pdf}} \n    \\subfigure[]{\\includegraphics[width=0.35\\textwidth]{Images\/AlexNet_ConfusionMatrix_Percent_OriginalDataset.pdf}} \n    \\caption{The confusion matrix when fine-tuning AlexNet to classify \"Normal\" and \"Anomaly\" classes on the original dataset.: (a) The confusion matrix in number  (b) The confusion matrix in percentage (\\%).}\n    \\label{fig:ConfusionMatrix_original}\n\\end{figure*}\n\n\\begin{comment}\n\\subsubsection{A comparison of the classification accuracy between pre-processing and without pre-processing sounds}\n\n\\begin{figure*}[!ht]\n  \\centering\n  \\begin{subfigure}[b]{0.45\\linewidth}\n    \\includegraphics[width=\\linewidth]{Images\/Number_confusion_matrix_WITHOUT_LowPass_HighPass.jpg}\n     \\caption{The confusion matrix in number.}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\linewidth}\n    \\includegraphics[width=\\linewidth]{Images\/Percent_confusion_matrix_WITHOUT_LowPass_HighPass.jpg}\n    \\caption{The confusion matrix in percentage (\\%).}\n  \\end{subfigure}\n  \\caption{The confusion matrix when classify \"Normal\" and \"Anomaly\" classes of the augmented dataset without using low-pass filter and high-pass filter.}\n  \\label{fig:ConfusionMatrix_without_low_high_pass_filters}\n\\end{figure*}\n\nIn this section, we examined the accuracy of the pre-trained AlexNet on the augmented dataset with and without using low-pass filter and high-pass filter for pre-processing sounds. Sounds on the augmented dataset were converted into Mel spectrograms without applying low-pass filter and high-pass filter. The pre-trained AlexNet was trained on these Mel spectrograms with the same training option. As shown in Table~\\ref{tablecompare2}, the classification accuracy of the pre-trained AlexNet reached 88.24\\% that is lower than pre-processing sound by high-pass filter and low-pass filter before converting to Mel spectrogram (92.65\\%). Figure~\\ref{fig:ConfusionMatrix_without_low_high_pass_filters} despicts the confusion matrix. \n\n\\begin{table}[!ht]\n\\centering\n\\caption{A comparison of the classification accuracy between pre-processing and without pre-processing sounds.}\n\\label{tablecompare2}\n\\begin{tabular}{l|c}\n\\hline\n\\textbf{}                  & \\textbf{Accuracy (\\%)} \\\\ \n\\hline\nLow-pass filter (22\\kern 0.16667em000 Hz) and \\\\high-pass filter (1000 Hz)   & 92.65                       \\\\ \\hline\nWithout pre-processing sounds   & 88.24                       \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\end{comment}\n\n\\section{Conclusion}\nDrill failure detection is critical for the industry to reduce drilling machine downtime. The unavailability of drilling sounds in the dataset, on the other hand, made it impossible to adequately train the failure detection model. As a result, employing VAE to synthesize\u00a0new drilling sounds from existing drilling sounds can help to supplement the small sound dataset. A case study is presented here that uses 134 drill sounds classified into two categories (normal and anomalous) in a Valmet AB dataset. We used a VAE to synthesize 96 new drilling sounds (48 normal sounds and 48 anomaly sounds) from the original drilling sounds. VAE may, although, minimize the noise in synthetic data. However, listening to the synthetic drilling sounds reveals that the newly generated drilling sounds roughly match the pitch of the original sounds. The newly synthesized sounds and original sounds were combined to create the augmented sound dataset to train a discrimination model. We fine-tuned AlexNet to classify Mel spectrograms of sounds in the augmented dataset. The result is also compared to fine-tuned AlexNet on the original dataset. The overall classification accuracy reached 94.12\\%. The result is promising to build a real-time machine fault detection system in the industry.\n\n\\section*{Acknowledgment}\nThis research was supported by the EU Regional Fund, the MiLo Project (No. 20201888), and the Acoustic sensor set for AI monitoring systems (AISound ) project. The authors would like to thank Valmet AB for providing the drill sound dataset. \n\nThe computations\/data handling was enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at the SNIC center is partially funded by the Swedish Research Council through grant agreement no. 2018-05973.\n\n\\Urlmuskip=0mu plus 1mu\\relax\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}