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recent breathtaking advances in our understanding of the mathematical structure of scattering amplitudes in maximally supersymmetric yang - mills theory ( sym ) have so far had the most dramatic impact on tree - level amplitudes or quantities ( such as leading singularities or the planar integrand  @xcite ) which are completely determined in terms of tree - level data . the guiding principle behind these advances is the observation that amplitudes exhibit simplicity which is completely obscured by their traditional feynman diagram expansions . in contrast the problem of actually evaluating the integrals which appear in multi - loop amplitudes remains extremely difficult in general ( see  @xcite for an introduction to modern methods ) . however data is starting to accumulate which suggests that the final results for amplitudes ( or even individual integrals ) can again be far simpler than one might have guessed on the basis of results obtained from more traditional integration approaches , such as mellin - barnes techniques . in  @xcite goncharov , vergu and the authors introduced to the sym literature a powerful mathematical tool first developed in  @xcite , called the symbol of a transcendental function , for analyzing functions of the type which appear in all currently known sym loop amplitudes . the symbol serves as a sort of motivic roadmap , encoding in a simple way all of the information about a function s discontinuities without the need to introduce any explicit representation of the function in terms of ( generalized ) polylogarithm functions . indeed one might reasonably say that the symbol encapsulates all of the physically relevant information that one may want to know about any given integral or amplitude . we view the symbol as stepping stone halfway between an integrand and its integral , and it is our hope that it may help to guide us through the dark jungle of multi - loop amplitudes .    the simple formula given in  @xcite for the two - loop six - point maximally helicity violating ( mhv ) remainder function was made possible by applying motivic technology to the 17-page analytic expression which had been heroically evaluated by del duca , duhr and smirnov  @xcite . it is clear that we are now desperately in need of some technology which would allow one to directly compute the symbol of any given integral without the need for such heroic effort . since the planar integrand of sym is a rational form , and since the symbol of an integral involves only rational functions , it is reasonable to suppose that there should exist an algebraic operation which would allow one to simply write down the symbol of any given integrand this would provide a concrete realization of part of the ambitious program suggested in  @xcite ( see in particular section 7 ) which aims for a `` correspondence principle '' unifying scattering amplitudes , their motivic avatars , and the combinatorics of feynman diagrams reflected in the coproduct formula . there too the symbol is not the ultimate goal but serves as just the simplest incarnation of the amplitudes and their motivic structure . work towards unlocking the structure of general multi - loop integrals is underway , but given the recent flurry of activity  @xcite on one - loop hexagon integrals in six dimensions we feel compelled to focus in this brief note on the special case of the one - loop @xmath0-gon integral in @xmath0 dimensions . the motivic structure of this integral is sufficiently simple that its symbol can be read off directly from its feynman parameterization by utilizing a 1996 theorem on mixed tate motives due to a.  goncharov . these higher dimensional integrals have several applications in physics , for example they appear in dimensionally - regulated one - loop mhv amplitudes  @xcite and in certain cases they can be related to higher - loop four - dimensional integrals  @xcite .    in section ii we set up some notation . the main result is presented in section iii , followed by several sample applications in section iv , including a particular three - mass hexagon integral in six dimensions . note that in this note we work entirely at the level of symbols and do not address the problem of choosing an explicit representative for integrals in terms of ( generalized ) polylogarithm functions . it is our understanding that efforts to automate this final step are under way  @xcite . in this note we consider the one - loop @xmath0-gon integral in @xmath0 dimensions , specifically @xmath2 considered as a function of @xmath0 points @xmath3 ( with an implied cyclic ordering ) in @xmath0-dimensional minkowski space . we have chosen to include in this definition the prefactor @xmath4 which serves three intimately related purposes : it ensures that @xmath5 is invariant under dual conformal transformations  @xcite , it normalizes the leading singularities of the integrand to @xmath6 , and it ensures that @xmath7 evaluates to a function of well - defined transcendentality @xmath1 . ( we caution the reader that our @xmath8 is close , but not exactly equal , to the quantity called @xmath8 in  @xcite . ) results for the @xmath9 integrals have of course been known since antiquity ( see for example  @xcite for a comprehensive modern treatment ) , while only very recently the first two cases of @xmath10 have been studied : the massless hexagon in  @xcite and the one - mass hexagon in  @xcite . we remind the reader that the integral  ( [ eq : fzero ] ) is said to be massless if cyclically adjacent pairs are all null - separated , i.e. @xmath11 for all @xmath12 . more generally we say that the integral has @xmath13 masses if @xmath13 pairs of cyclically adjacent points are null - separated . for @xmath14 the integral @xmath5 always converges , while for @xmath9 ( the case of box integrals in four dimensions ) it only converges in the fully massive case . we will return to the question of divergences in detail below . the @xmath15 integral is easily performed after introducing feynman parameters to combine the @xmath0 propagators , which leads to @xmath16 one may choose any nontrivial linear combination of the @xmath17 s inside the delta - function  @xcite but our simple choice of @xmath18 makes it immediately manifest that  ( [ eq : fone ] ) is best thought of as a projective integral . specifically  ( [ eq : fone ] ) is an integral over @xmath19 with homogeneous coordinates @xmath20 $ ] which has been written in the patch @xmath21 . therefore , the one - loop @xmath0-gon integral in @xmath0 dimensions is the @xmath19 period integral @xmath22 where @xmath23 is the standard holomorphic form @xmath24 and where we now use the notation @xmath25 to emphasize that @xmath7 should be thought of as a function of a quadratic form @xmath26 , whose matrix entries are @xmath27 in our application . integrals of the type  ( [ eq : ftwo ] ) have been studied in the context of mixed tate motives , and belong to a class of objects called tate iterated integrals in  @xcite ( see especially section 3 ) . in particular @xmath25 is a special case of the more general period integrals @xmath28 studied in  @xcite . these are integrals of the type  ( [ eq : ftwo ] ) involving any non - degenerate quadric @xmath26 and integrated over a cycle representing a generator of @xmath29 for an arbitrary simplex @xmath30 in general position relative to @xmath26 . the cohomology group @xmath31 has canonical mixed hodge - tate structure which corresponds to a certain mixed tate motive  @xcite .    in our particular application we shall only be interested in quadratic forms @xmath26 with vanishing diagonal entries ( since obviously @xmath32 for all @xmath12 ) . moreover , the cycle of integration of interest in  ( [ eq : ftwo ] ) , inherited from the identification with  ( [ eq : fone ] ) , involves the particularly simple simplex @xmath30 whose @xmath0 faces are just the coordinate hyperplanes @xmath33 . these two simplifications of our particular @xmath25 s compared to the more general @xmath28 s studied in  @xcite are not of essential mathematical importance but greatly streamline the bookkeeping involved in applying the results of that paper . we will now write a simple recursive formula for the symbol @xmath34 of @xmath25 . the recursion expresses @xmath34 as a sum of @xmath35 terms involving @xmath36 , where the notation @xmath37 denotes the @xmath38 matrix obtained by deleting the rows and columns @xmath12 and @xmath39 from the @xmath40 matrix @xmath26 ( i.e. , the quadric @xmath37 is just the intersection @xmath41 ) . in terms of the feynman graph corresponding to the one - loop integral , going from @xmath26 to @xmath37 amounts to turning the @xmath0-gon into a @xmath42-gon by deleting the two propagators @xmath43 and @xmath44 . there are @xmath35 different ways of doing this , and each way leads to one term in the recursive sum .    to obtain the recursion we apply theorem ( 4.10 ) of  @xcite ( preprint version ) , which for us becomes @xmath45 where @xmath46 note that no summation of indices is implied on the right - hand side , which is expressed in terms of the @xmath47 , @xmath48 and @xmath49 entries of the inverse matrix @xmath50 . let us comment briefly on the geometric content of  ( [ eq : rdef ] ) . for each choice of @xmath12 and @xmath39 , there are precisely two codimension-1 hyperplanes @xmath51 , @xmath52 in @xmath19 which contain the intersection @xmath53 and which are tangent to the quadric @xmath54 . so for each @xmath12 and @xmath39 we have four natural codimension-1 hyperplanes @xmath55 all of which contain @xmath53 as a common codimension-2 subspace , and the quantity @xmath56 in  ( [ eq : rdef ] ) is the cross - ratio naturally associated to this configuration . it follows immediately from  ( [ eq : df ] ) that the symbol @xmath34 satisfies the recursion @xmath57 in what follows we shall only make use of  ( [ eq : symbolrecursion ] ) , though of course  ( [ eq : df ] ) is a slightly stronger statement which realizes a precise set of differential relations between the infinite class of one - loop @xmath0-gon integrals . various types of differential relations amongst multi - loop integrals have long been studied ; see  @xcite for some recent examples which also relate integrals in different dimensions . we begin with a couple of almost trivial examples demonstrating the application of the recursion  ( [ eq : symbolrecursion ] ) . in general mathematical applications one may seed the recursion with the result of the @xmath58 integral @xmath59 however in physics applications , where we are always interested in quadrics @xmath26 with zero entries down the diagonal , this ` bubble ' integral is too singular to be of use by itself . instead it is more effective to seed the recursion with box integrals ( @xmath9 ) . as mentioned above these are divergent except in the fully massive case . nevertheless , as we discuss in detail below , it is relatively straightforward to apply this recursion to arbitrary integrals by introducing if necessary a regularization parameter @xmath60 which is ultimately taken to zero . the finiteness of the integral @xmath25 for @xmath61 is reflected in the fact that the regularization parameter always drops completely out of the symbol @xmath34 after summing together all contributions to  ( [ eq : symbolrecursion ] ) .    before proceeding let us make an important comment regarding notation . in the previous section we defined the integral @xmath5 for an arbitrary collection of @xmath0 points @xmath62 in minkowski space , without regard to whether or not any cyclically adjacent pairs were null separated . in the remainder of this paper we shall adopt notation more conventional in the physics literature , whereby we consider a collection of @xmath63 points which are all mutually null , i.e. @xmath64 for all @xmath12 , but only a subset of @xmath0 of these @xmath65 s will be chosen as the @xmath0 arguments to the function @xmath5 . let us start with the four - mass box integral , which corresponds to the quadric @xmath66{$x_j$ } \falabel(10,16)[]{$x_i$ } \falabel(4,10)[]{$x_l$ } \falabel(10,4)[]{$x_k$ } \end{feynartspicture } } } } \qquad q = \left(\begin{matrix } 0 & x_{ij}^2 & x_{ik}^2 & x_{il}^2 \cr x_{ij}^2 & 0 & x_{jk}^2 & x_{jl}^2 \cr x_{ik}^2 & x_{jk}^2 & 0 & x_{kl}^2 \cr x_{il}^2 & x_{jl}^2 & x_{kl}^2 & 0 \cr \end{matrix } \right)\ ] ] in @xmath67 . the recursion  ( [ eq : symbolrecursion ] ) expresses the symbol of this integral as a sum over six @xmath58 integrals of the type  [ eq : fdegreeone ] ) : @xmath68 using  ( [ eq : fdegreeone ] ) we find for example @xmath69 where @xmath70 in this case . evidently the symbol @xmath71 is singular , but this can be regulated by replacing all of the zero entries on the diagonal of @xmath26 with @xmath60 . then , at the level of the symbol , we can replace @xmath72 here we have freely dropped an overall factor of @xmath73 since such numerical constants drop out of the symbol . note that we could have introduced the @xmath60 into @xmath74 as well but there is no need as this quantity has a finite limit , shown above , as @xmath75 . adding up the six contributions  ( [ eq : sixterms ] ) we find first of all that @xmath60 completely drops out of the symbol ( an important consistency check ) , and that the result can be assembled into the form @xmath76 which of course agrees precisely with the symbol of the well - known four - mass box function @xmath77 in terms of @xmath78      from a mathematical point of view it is perhaps simplest to restrict one s attention to the generic case of the completely massive @xmath0-gon integral , whose symbol may now be written down directly using the recursion  ( [ eq : symbolrecursion ] ) seeded by  ( [ eq : fourmass ] ) at @xmath9 . however in physics we are often interested in integrals with fewer masses . the application of the recursion  ( [ eq : symbolrecursion ] ) in these cases will call into play various divergent box integrals which can be regulated in a manner similar to  ( [ eq : regularization ] ) . for example consider the three - mass box integral with corresponding quadric @xmath66{$x_j$ } \falabel(10,16)[]{$x_i$ } \falabel(4,10)[]{$x_{k+1}$ } \falabel(10,4)[]{$x_k$ } \end{feynartspicture } } } } \qquad q = \left(\begin{matrix } 0 & x_{i , j}^2 & x_{i , k}^2 & x_{i , k+1}^2 \cr x_{i , j}^2 & 0 & x_{j , k}^2 & x_{j , k+1}^2 \cr x_{i , k}^2 & x_{j , k}^2 & 0 & 0 \cr x_{i , k+1}^2 & x_{j , k+1}^2 & 0 & 0 \cr \end{matrix } \right)\,.\ ] ] its symbol may be obtained by setting @xmath79 in  ( [ eq : fourmass ] ) and keeping only the leading - order terms for small @xmath60 , which leads to @xmath80 results for all other box integrals may be obtained as further specializations of this formula . for example the two - mass easy or two - mass hard box functions can be obtained by setting @xmath81 or @xmath82 respectively . note that if we use  ( [ eq : fourmass ] ) as our starting point then it is not necessary to introduce an @xmath60 for each of the diagonal entries of @xmath26 ; in writing  ( [ eq : fourmass ] ) all of those divergences have already been properly eliminated . we only need to introduce an @xmath60 for any zero entry immediately adjacent to the diagonal ( i.e. for any massless vertex of the corresponding diagram ) . just like we saw in  ( [ eq : fourmass ] ) , all dependence on the regularization parameter @xmath60 will necessarily drop out when individually divergent contributions are properly assembled to produce the symbol of any of the finite functions @xmath7 for @xmath61 . in practice this cancellation can serve as a useful check against calculational errors . as our first non - trivial ) with the help of mathematica s integrate command ; in our experience it generally gets stuck starting at degree 4 where the classical polylogarithms do not provide a sufficiently large basis of functions . ] demonstration of the main result let us write the symbol for the three - mass hexagon integral in six dimensions corresponding to the quadric @xmath83{$x_2 $ } \falabel(10.0000 , 16.0000)[]{$x_1 $ } \falabel(4.80385 , 13.0000)[]{$x_8 $ } \falabel(4.80385 , 7.00000)[]{$x_7 $ } \falabel(10.0000 , 4.00000)[]{$x_5 $ } \falabel(15.1962 , 7.00000)[]{$x_4 $ } \end{feynartspicture } } } } \qquad q = \left(\begin{matrix } 0 & 0 & x_{14}^2 & x_{15}^2 & x_{17}^2 & x_{18}^2 \cr 0 & 0 & x_{24}^2 & x_{25}^2 & x_{27}^2 & x_{28}^2 \cr x_{14}^2 & x_{24}^2 & 0 & 0 & x_{47}^2 & x_{48}^2 \cr x_{15}^2 & x_{25}^2 & 0 & 0 & x_{57}^2 & x_{58}^2 \cr x_{17}^2 & x_{27}^2 & x_{47}^2 & x_{57}^2 & 0 & 0 \cr x_{18}^2 & x_{28}^2 & x_{48}^2 & x_{58}^2 & 0 & 0 \cr \end{matrix } \right)\ ] ] in @xmath84 . here we have chosen for convenience an integral with @xmath85 particles , which manifestly exhibits a @xmath86 symmetry . we shall for convenience restrict the external kinematics to four dimensions ( i.e.  all of the @xmath65 lie within a common four - dimensional subspace of six - dimensional minkowski space ) . this allows us to express everything in terms of momentum twistors  @xcite ; in particular @xmath87 where the constant of proportionality is irrelevant as it cancels out in all conformally invariant quantities . we make this choice for purposes of notational simplicity only but we emphasize that the result  ( [ eq : symbolrecursion ] ) is of course valid for completely general kinematics ) for the octagon or higher since for @xmath88 the determinant @xmath89 is identically zero for any collection of @xmath0 points @xmath62 which lie in a common four - dimensional subspace . ] . a special feature of the three - mass hexagon  ( [ eq : threemass ] ) is that no square roots of momentum twistor invariants appear in its symbol . this would not be true if any two adjacent vertices of the hexagon were massive . the recursion  ( [ eq : symbolrecursion ] ) expresses the symbol @xmath90 as a sum of 15 terms , but we can use the @xmath86 symmetry of this configuration to cast the result into the form @xmath91 \end{aligned}\ ] ] where @xmath92 is the shift - by - three operator and we use the notation @xmath93 appearing in the first two entries of the symbol are a two - mass easy box @xmath94 and some three - mass boxes @xmath95 , @xmath96 , @xmath97 and @xmath98 , corresponding to the degenerations of the hexagon :    ( 465,93)(5,1 ) ( 13,7)(13,13)(0.,)/straight0 ( 13,13)(7,13)(0.,)/straight0 ( 7,13)(7,7)(0.,)/straight0 ( 7,7)(13,7)(0.,)/straight0 ( 13,13)(17.2426 , 17.2426)(0.,)/straight0 ( 13,7)(17.2426 , 2.75736)(0.,)/straight0 ( 7,7)(2.75736 , 2.75736)(0.,)/straight0 ( 7,13)(2.75736 , 17.2426)(0.,)/straight0 ( 16,10)[]@xmath99 ( 10,16)[]@xmath100 ( 4,10)[]@xmath101 ( 10,4)[]@xmath102 ( 13,7)(13,13)(0.,)/straight0 ( 13,13)(7,13)(0.,)/straight0 ( 7,13)(7,7)(0.,)/straight0 ( 7,7)(13,7)(0.,)/straight0 ( 13,13)(17.2426 , 17.2426)(0.,)/straight0 ( 13,7)(17.2426 , 2.75736)(0.,)/straight0 ( 7,7)(2.75736 , 2.75736)(0.,)/straight0 ( 7,13)(2.75736 , 17.2426)(0.,)/straight0 ( 16,10)[]@xmath99 ( 10,16)[]@xmath103 ( 4,10)[]@xmath101 ( 10,4)[]@xmath102 ( 13,7)(13,13)(0.,)/straight0 ( 13,13)(7,13)(0.,)/straight0 ( 7,13)(7,7)(0.,)/straight0 ( 7,7)(13,7)(0.,)/straight0 ( 13,13)(17.2426 , 17.2426)(0.,)/straight0 ( 13,7)(17.2426 , 2.75736)(0.,)/straight0 ( 7,7)(2.75736 , 2.75736)(0.,)/straight0 ( 7,13)(2.75736 , 17.2426)(0.,)/straight0 ( 16,10)[]@xmath100 ( 10,16)[]@xmath103 ( 4,10)[]@xmath101 ( 10,4)[]@xmath102 ( 13,7)(13,13)(0.,)/straight0 ( 13,13)(7,13)(0.,)/straight0 ( 7,13)(7,7)(0.,)/straight0 ( 7,7)(13,7)(0.,)/straight0 ( 13,13)(17.2426 , 17.2426)(0.,)/straight0 ( 13,7)(17.2426 , 2.75736)(0.,)/straight0 ( 7,7)(2.75736 , 2.75736)(0.,)/straight0 ( 7,13)(2.75736 , 17.2426)(0.,)/straight0 ( 16,10)[]@xmath100 ( 10,16)[]@xmath103 ( 4,10)[]@xmath99 ( 10,4)[]@xmath102 ( 13,7)(13,13)(0.,)/straight0 ( 13,13)(7,13)(0.,)/straight0 ( 7,13)(7,7)(0.,)/straight0 ( 7,7)(13,7)(0.,)/straight0 ( 13,13)(17.2426 , 17.2426)(0.,)/straight0 ( 13,7)(17.2426 , 2.75736)(0.,)/straight0 ( 7,7)(2.75736 , 2.75736)(0.,)/straight0 ( 7,13)(2.75736 , 17.2426)(0.,)/straight0 ( 16,10)[]@xmath100 ( 10,16)[]@xmath104 ( 4,10)[]@xmath101 ( 10,4)[]@xmath102    their @xmath60-dependent symbols may be read off from  ( [ eq : eighteen ] ) , and it is a nontrivial check of the correctness of  ( [ eq : result ] ) that all of the @xmath60 dependence completely drops out at the level of the symbol . it is also straightforward to verify that in the limit @xmath105 , @xmath106 and @xmath107 the result  ( [ eq : result ] ) reproduces the symbol of the massless hexagon computed in  @xcite .    in conclusion let us emphasize that in contrast to the old mb method , the new mb ( motivic beauty ) technique does not require the evaluation of any integrals . the only even slightly nontrivial step in obtaining  ( [ eq : result ] ) was inverting the matrix @xmath26 shown in  ( [ eq : threemass ] ) and simplifying the @xmath56 s from  ( [ eq : rdef ] ) to the final form presented . generalizations of this method to projective integrals of the type which appear in higher loop integrals remain under active investigation .    in this note we have worked entirely at the level of symbols , without addressing the important problem of finding explicit representations for integrals in terms of ( generalized ) polylogarithm functions . this remains an interesting open problem , especially since our experience with  @xcite has shown that analytic expressions for amplitudes can turn out to be much simpler than one might have guessed from their symbols . we are grateful to s.  caron - huot , j.  henn , d.  skinner , c.  vergu and especially n.  arkani - hamed and a.  goncharov for useful discussions , correspondence , and encouragement . this work was supported in part by the department of energy under contract de - fg02 - 91er40688 task j oji ( ms ) and task a ( av ) , and the national science foundation under grant nos . phy05 - 51164 , pecase phy-0643150 ( av ) and advance 0548311 ( av ) . journal of physics a special issue on `` scattering amplitudes in gauge theories , '' to appear , eds . r  roiban , m.  spradlin and a.  volovich , with contributions by h.  elvang , d.  z.  freedman and m.  kiermaier , arxiv:1012.3401 ; r.  britto , arxiv:1012.4493 ; j.  m.  henn , arxiv:1103.1016 ; z.  bern and y.  t.  huang , arxiv:1103.1869 ; r.  m.  schabinger , arxiv:1103.2769 ; a.  brandhuber , b.  spence and g.  travaglini , arxiv:1103.3477 ; t.  bargheer , n.  beisert and f.  loebbert , arxiv:1104.0700 ; j.  bartels , l.  n.  lipatov and a.  prygarin , arxiv:1104.0816 ; t.  adamo , m.  bullimore , l.  mason and d.  skinner , arxiv:1104.2890 ; l.  j.  dixon , arxiv:1105.0771 ; h.  ita and j.  drummond . n.  arkani - 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we use a result on mixed tate motives due to goncharov  @xcite to show that the symbol of an arbitrary one - loop @xmath0-gon integral in @xmath0 dimensions may be read off directly from its feynman parameterization . 
 the algorithm proceeds via recursion in @xmath1 seeded by the well - known box integrals in four dimensions . as a simple application of this method we write down the symbol of a three - mass hexagon integral in six dimensions . 
 brown - het-1612 nsf - kitp-11 - 076    1 cm