MPL - Supplementary User Manual
Contents
1. 1 x x3 2 2 Differentiation MPLd expr Differentiation d with respect to the end point of the path w r t which the iterated integral is defined is simply given by the de concatenation of the leftmost differential 1 form d V Qn gt An V Qm Y aloalo Y ale J i ix J i1 ix For example one computes atte tu ol de l xox3 X3 X3 X2 1 xox3 X2 1 xox3 m 1 xox3 by 21n principle using the auxiliary procedure GetArnoldRHS these equations can also be obtained in a different output format where forms of the fiber and the base are explicitely separated Such a format is used internally MPLCoordinates x 3 gt MPLd bar x131 d3 x121 x 2 x 3 1 x 2 x 3 d x 3 x 3 bar d x 2 x121 x 3 x 21 x 21 d x 3 1 x 2 x 3 tens bar 45 tens bar X2X3 X3 1 MPLTotalConnection expr In 4 we have defined a connection V V Qh gt An 1 QV OF by use of the Arnol d equations and a total connection Vr V Qh gt An ev 07 by Vr d V For a word in Z the total connection can be computed with the command MPLTotalConnection For example we obtain xod x3 d xox3 dx3 de xod x3 Vr 8 8 1 xoX3 x3 1 xox3 X3 X2 1 xoxa by MPLCoordinates x 32. bar zn 9 1 2 we clearly obtain the expected 212 gi fo Furthermore computing MPLUnshuffle g1 x121 tens ONE bar 2282 1 x1x2 we confirm the expected o g1 16 fi 2 4 Integration over cubical coordinates MPLPrimitive a f w The procedure MPLPrimitive a f w computes the primitive f a 6 f V Qm of a function f V Qm of maximal weight w with respect to the differential 1 form a AE For example let us compute f a G f for m xod x3 1 X2X3 and x3d x2 x2d x3 dx3 Bea z xod x3 n d x2 1 X2X3 X3 X2 1 X2X3 X2 The words in f are at most of length w 2 We compute 12 gt MPLCoordinates x 3 gt a x 2 d x 3 1 x 2 x 3 gt f bar x 3 d x 2 x 2 d x 3 1 x 2 x 3 dix151 x131 bar d x 2 x 2 x 3 d x 2 x 2 d x 3 1 x 2 x 3 5 bar d x 2 x 2 99 gt MPLPrimitive a f 2 1 xoxa 1 B E xad x2 xod x3 d xo xad x2 xad x3 1 xox3 a m 1 nm xad x2 xad x3 d x2 4 99bar Xad x2 xod x3 1 X2 1 x x3 nm x2 xad x2 xod X2 1 xox3 d xo x3d x2 x3 x3d x2 xod x3 X2 1 X2X3 f 1 X2X3 MPLLimit f x u The procedure MPLLimit expr var limit returns the limit of a function f in V Q at x u where x is one of the m cubical coordinates and u is either 0 or 1 As
3. 3 bar y c z 4 3bar y z c 4 bar a b y z bar a y b z bar a y z b bar y a b z bar y a z b bar y z a b 7 bar y z 4 MPLCoproduct expr The deconcatenation co product A defined by Alazlak l la l 10 ax ax 1 a1 ax lak l la E layla l la amp l is implemented in the procedure MPLCoproduct expr for example gt MPLCoproduct bar a b c tens ONE bar a b c tens bar a bar b c tens bar a b bar c 4 tens bar a b c ONE Here the reserved term ONE stands for the empty word and tens isa formal expression for the tensor product of two iterated integrals tens and ONE are mostly defined for internal use and will usually not appear in any results below MPLCoordinates letter n MPL COORDINATES As a preparation for many applications it is necessary to declare a set of variables to be the cubical coordinates we want to work with For a positive integer n the procedure MPLCoordinates letter n declares n cubical coordinates by use of the first argument letter for the names of the vari ables The list of cubical coordinates defined in this way is stored in the global variable ML COORDINATES until the procedure is called again For example gt MPLCoordinates x 5 MPL COORDINATES x1 32 X3 xa x5 After calling MPLCoordinates the variables declared in this way are internally recognized to be cubical coordina
4. MPLTotalConnection bar x 2 x 3 1 x121 x131 d x 3 x131 iens ates bar wee 2 bar 2 1 X3 1 2 3 symbol map and the unshuffle map MPLSymbolMap expr It is implied by a theorem of Chen 11 that an iterated integral with differential 1 forms in our framework is homotopy invariant if and only if the corresponding word of 1 forms satisfies a so called integrability condition These words are called integrable Every word in A is integrable but not every word in Am The symbol map as constructed in 3 4 maps from integrable words in AZ to integrable words in Ap P V z gt V Qm 8 For example we compute v d zs E EE Ed 1 XoX3 X3 1 WA X3 X2 1 X2X3 2 by MPLCoordinates x 3 5MPLSymbolMap bar x 2 d x 3 1 1 x121 x131 d x 31 x131 bar gS du bar 2 ay elan 1 gt X3 2 1 The symbol map is the unique linear map which satisfies id amp W oVr doW Let us check this equation for the above example The left hand side of the equation is obtained by gt f bar x 2 d x 3 1 x 2 x 3 d x 3 x 3 gt evalindets MPLTotalConnection f specfunc anything tens proc h options operator arrow subsop 2 MPLSymbolMap op 2 h h end proc Note that this command at first computes a total connection of the function as a linear com bina
5. More generally iterated integrals are linear combinations of such expressions over Q for example gt 3 bar a 2 a 1 5 bar b 8 9 Numerical constants are factored out automatically for example gt bar 3 a 2 b bar a 2 b c bar a b bar a c 2bar b c gt bar 0 c 0 BarToLists expr ListsToBar list For some computations the user may find it convenient to bring an expression of bar terms in the form of a list This form is also used internally in some algorithms For expr being a linear combination of words in the bar notation the auxiliary procedure BarToLists expr returns a list of two minor lists The first of these lists contains the coefficients of the words and the second list contains the words in the argument of bar as lists For example gt f bar a bar b c 99 bar b c y bar 5 g CROP gt BarToLists f 1 100 5y Ia b c 1 c The command ListsToBar list reverses this transformation It takes such a list of coefficients and words as input and returns the corresponding expression in bar notation For example as continuation of the above gt ListsToBar bar a 100bar b c Sybar g c b MPLShuffleProduct exprl expr2 The multiplication of two iterated integrals expr1 and expr2 is computed by the well known shuffle product implemented in MPLShuffleProduct For example gt MPLShuffleProduct bar a b 3 bar c 7 bar y z 3bar c y z
6. X1X4 X3 xax4 354 5 31 1 4 2 3 2 4 x1 XoX4Xs x3 X2 X3 1 x4 1 4 x5 X4X2 X3 X3 1 42 71 4x5 71 x4 x4 052 1 3 2 3 x1 3 71 4 xs 1 x4 x4 x5 521 531 2 3 1x2 x3 1 3x4 1 bx xa xsl 1 2 531 2 3 01 232 33 71 xs 1 x4 x4 x 152 535 2 3 T This reduction is a list of 8 lists each of which has three entries The first entry is the set of variables with respect to which the polynomials in S were reduced The second entry is the list of resulting polynomials at this stage In the language of 9 14 these are the vertices of the compatibility graph This list is ordered With respect to this ordering the third entry contains all compatible pairs of polynomials i e the edges of the compatibility graph represented by their numbers with respect to the order in the second entry For example let us consider the list in the second line from below 52 33 71 x4 1 T X4 x 1 2 1 3 2 3 According to the first entry reductions were computed with respect to x2 and x3 Therefore in the notation of 2 the second entry is the set 5 71 4 x4 1d xs x4 xs and the third entry tells us that all three polynomials are compatible with each other If in this entry for example 2 3 would be missing 1 xs would not be compatible with x4 xs The compatibilities are usually just
7. an example we consider m 3 and 2 1 x1X2xX3 1 xox3 d xixoxs d xi d xoxa d xi d xi d xixoxa lt lt DN 1 x xox3 1 xi 1 xox3 5 1 X1 X 1 X1X2X3 and we compute the limits g lim s f g2 limo gi 23 limy 51 82 as follows f MPLSymbolMap bar x 1 x 2 d x 3 1 x 1 x 2 x 3 x 2 d x 3 1 x 2 x 3 gt gl simplify MPLLimit f x 3 1 sie lt 2 ar ee 1l4 x 1x2 1 x l Ex l x sas d x1 md x1 4 xid x2 an d x xod x x1d x2 l x 1 1 2 1 1l x 4x2 13 gt g2 simplify MPLLimit g1 x 2 1 dst dt GL xj l x l4 x l x g2 bar gt g3 simplify MPLLimit g2 x 1 1 There is a little subtlety we have to take care of if we compute limits not starting with the last cubical coordinate xm For example let us compute g4 limx 1 f by gt g4 simplify MPLLimit f x 2 1 bar lapa d x3 bar dia bar d x1 u l x x3 gt 14x3 Xp 1 4xx3 I x l x xa d dx bar 4 1 xad x1 x1d x3 l Lx x3 because it involves the product x1x3 of non consecutive cobical coordinates However we can In this result the differential 1 form does not belong to any defined in eg 1 simply replace every x3by x in this result for example by gt g5 subs
8. apply the map of eq 4 twice At first we compute g u id 599 16 fi W fi par Te V Q 1 2 Let us furthermore construct the function or integrable word g z as 82 u id amp W giG fr fimW f d x3 es ae 1 x2x3 8 X3 1 xix Shar omm xod x1 x1d x2 d xa 1 2 1 xx X X2 X3 x1 x1d x2 xad x2 x2d x3 X3 ds ar 3880 e 1 x x 1 X3 xad x2 x2d x3 xod x1 x d x X2 1 xox3 1 x 1x2 d x2 xi xod x1 x1d x2 xad x2 x2d x3 x2 x2d x3 bar X2 1 x xo 1 X2X3 EUM x1 4 xid x2 d d x2 Xad x2 xod x3 1 x1x2 w 1 This construction is easily obtained by use of the above procedures MPLShuffleProduct and MPLSymbolMap fl bar x 1 d x 2 1 x 1 x 2 gt f2 b ar x 2 d x 3 1 x 2 x 3 d x 3 x 3 gt MPLCoordinates x 2 11 gl MPLSymbolMap f1 gt MPLCoordinates x 3 gt g12 MPLShuffleProduct g1 MPLSymbolMap 2 By construction the function 212 is in V Q3 and it is easy to check that it is in the basis of this vectorspace obtained by the command MPLBasis gt B3 MPLBasis x 3 3 gt has B3 3 912 true Now let us reverse this construction by use of the unshuffle map By computing gt MPLUnshuffle g12 x 3 tens bar 12
9. appropriate change of variables to cubical coordinates the program assumes the polynomials to be properly ordered at a tangential basepoint see 2 4 This means that there is a certain region in the parameter space where the m polynomials P P relevant at this stage are uniquely ordered by 0 gt Pm gt P1 gt 925 5 Pm 2 gt Pm 1 where p by 2 with xy being the integration variable at this stage The procedure checks at first that all p 20 in this region The message All rho i are smaller than zero at the tangential basepoint Check OK confirms this to be the case If it is not the case the error message will return a polynomial P and the corresponding p causing the problem In a third step the procedure computes certain limits which are required for an integration see 2 4 for the details These limits need return values in 0 1 which the above messages confirm If a limit different from 0 or appears a corresponding error message is returned which also gives the change of coordinates causing the problem The computation of these limits implicitely also checks the uniqueness of the above order among the pi as this order is used to set up the change of coordinates If the order is ambiguous a corresponding error message is returned This case is not expected unless the list of polynomials accidentally contains two equal elements P i 52 j which of course should be avoided int
10. has to be linearly reducible and properly ordered We assume that the user has already expressed a divergent Feynman integral in terms of finite integrals of this type possibly by use of techniques cited and partly demonstrated in 4 2 All remaining mentioned conditions can be explicitely checked by procedures of MPL as demonstrated below 16 3 1 Setting up a simple example As a simple standard example let us consider the massless off shell one loop traingle graph throughout this section The corresponding integral is finite More involved examples are dis cussed in 4 2 The two Symanzik polynomials see e g 151 of this graph are U x n 3 2 2 2 Fo xX1X2 p3 X1X3 p3 where the x are the Feynman or Schwinger or alpha parameters and the p are the incoming momenta In section 4 of 2 we suggested to express the kinematical dependences in terms of variables which can be treated as additional Feynman parameters which are not integrated out In this sense we define the set of generalized Feynman parameters x1 x2 x3 Xa xs where and Xs satisf 5 y 2 E E XAXS and 1 x4 1 xs P5 P5 C f 10 We consider the kinematical region where 0 lt x4 0 lt xs For a different kinematical region we recommend to change the definition of x4 and xs accordingly such that the region is still given by O lt x4 0 xs if possible Using these variables we de
11. of internal importance If they are considered by the algo rithm i e if COMPATIBILITY GRAPH is true the reduction usually gives a better upper bound for 3 44 11 3 11 4 12 34 12 4 13 5 14 54 15 64 11 5 1 64 2 54 12 64 13 64 the polynomials appearing in the integration procedure than the plain Fubini reduction COMPATIBILITY GRAPH is false In the output of latter the third entry of each list is always the complete graph i e it con tains all possible pairs The plain Fubini reduction obtained by gt COMPATIBILITY GRAPH false gt MPLPolynomialReduction U F GenFeyn 1 3 GenFeyn For this simple example both reductions are exactly the same 19 For the user the important question is whether the set S is linearly reducible and if yes which are the allowed orders of integrations This can information is easily obtained from the reduction by looking at the first entry of each list The set S is linearly reducible with respect to a particu lar order Xg 1 of n integration variables given by a permutation o on 1 n if the reduction contains each of the sets xo 1 xo 1 Xo Xo Xo n For example let us ask whether tu F is linearly reducible with respect to the order X3 X1 xo We see that the above reduction contains lists whose first etries are x3 x1 xa xi xo x3 so the answer is yes If for example the list with x3 in the first entry would be mis
12. will drastically simplify expressions and it is also necessary for automated tests discussed in section 4 to check that certain parts of the program produce the expected results As a supplement to this manual the above webpage also provides a maple worksheet which contains all examples shown below 2 Computing with cubical coordinates 2 1 Differential 1 forms and iterated integrals In this section we work with homotopy invariant iterated integrals on Mo with n gt 3 Here ho motopy invariance implies that these functions only depend on the end points of some path We always consider the origin as one of these end points The other point has m n 3 coordinates which the iterated integral depends on as a function Due to additional conditions with respect to a basepoint it is completely determined by the differential 1 forms and the order in which one inte grates over them Therefore it is customary to express such an iterated integral by a tensor product of the involved 1 forms aj aj 1 6 which is often replaced by a notation with squared brackets as aj i ai called bar notation bar The bar notation is adapted in MPL We use the reserved word bar to express every iterated inte gral For example as a a is expressed by bar a 3 a 2 a 1 and stands for an iterated integral where we integrate over some 1 forms ar a2 a3 in this order ing i e from right to left in the argument of bar
13. x 3 x 2 g4 Now in gs every 1 form belongs to and the result is recognized as a function of V Q2 It is easy to check the integrability In general after taking a limit with respect to a variable xy with 1 X k lt m we have to re name the coordinates by xii Xm Xk Xm 1 MPLMultipleLimit expr list Based on the previous procedure the procedure MPLMultipleLimit f L computes several limits of a function f of V Qm successively Here L is a list of the type Xo 1 11 Xo 2 U2 Xo k k for 1 lt k m a permutation on some subset of 1 m and all u 0 1 This list fixes the limits and also the order in which they are computed The above command returns lim lim f o 1 cuq As an example let us again consider the function f as defined in eq 5 Then we obtain the above results for g g2 g3 by 14 gt gl simplify MPLMultipleLimit f x 3 1 gt g2 simplify MPLMultipleLimit f x 3 1 x 2 1 gt g3 simplify MPLMultipleLimit f x 3 1 x 2 1 x 1 1 It is briefly mentioned in 2 and discussed in more detail in 4 and 6 that the order of such consecutive limits is crucial For example we have obtained g3 2 but permuting the order of the last two limits we compute gt MPLMultipleLimit f x 3 1 x 1 1 x 2 1 0 Note that in a function of V Qm the terms with weight grea
14. F Feynman integrals 16 fiber 6 Fubini reduction 18 G generalized Feynman parameters 17 H Hlog 23 HyperlInt 23 hyperlogarithms 11 23 I integral cubical coordinates 15 iterated integral 3 L limit 13 linearly reducible 18 ListsToBar 4 M MPL COORDINATES 5 MPLArnoldEquation 6 7 MPLAutomatedTests 24 PLBasis 9 PLCheckOrder 20 PLCoordinates 5 PLCoproduct 5 PLCubicalIntegrate 15 PLd 7 PLFeynmanIntegrate 22 PLFormsBase 5 PLFormsFiber 5 MPLFormsTotal 5 PLHlogToBar 23 PLLimit 13 PLMultipleLimit 14 P P P P P P PolynomialReduction 18 LPrimitive 12 iSShuffleProduct 4 SymbolMap 8 LTotalConnection 8 LUnshuffle 10 ONE 5 order 14 P polynomial reduction 18 27 primitive 12 properly ordered 21 S shuffle product 4 symbol 8 T tens 5 test 24 total connection 8 total space 6 U unramified 20 unshuffle map 10 V vectorspace 9 28
15. MPL Supplementary User Manual Christian Bogner Version 1 0 Contents 1 Getting started 2 2 Computing with cubical coordinates 3 2 1 Differential 1 forms and iterated integrals 3 2 2 Differentiation i 9 22 4 4 2 kk ei S RR GR X X So Xx s semi 7 2 3 symbol map and the unshuffle map 8 2 4 Integration over cubical coordinates 12 3 Computing Feynman integrals 16 3 1 Setting up a simple example Un 25 99 xm XE le YAD a 17 3 2 Checking the conditions eklem at ok ox m ux ee ee I EUR 18 3 3 Integration over Feynman parameters 22 3 4 Expressing hyperlogarithms in bar notation 23 4 Automated tests and troubleshooting checklist 24 1 Getting started MPL is a collection of procedures for the computeralgebra system Maple It was written and tested with Maple 16 It supports a wide range of computations with homotopy invariant iterated integrals on moduli spaces of curves of genus with n ordered marked points This class of functions contains the widely used multiple polylogarithms as functions of several variables This manual shows how to apply the most important procedures of the program and might be useful for looking up technical details or examples It is just meant to supplement the detailed introduction given in the article 2 The main algorithms were presented in 4 They are based on the mat
16. able and every d x i as a differential 1 form by Maple Then we define the integrands according to eqs 8 and 9 gt U x 1 x 2 x 3 gt F x 1 x 2 x 4 x 5 x 1 x 3 x 2 x 3 1 x 4 1 x 5 Integrand0 1 U F Integrandl 2 bar d U U bar d F F U F Note that in the integrand of J we have expressed the logarithms ln U and In F in bar notation As ordered list of generalized Feynman parameters we choose again gt GenFeyn x 1 x 2 x 3 x 4 x 5 From the tests conducted above we know that we can integrate in this order Notice that in eqs 8 and 9 we still have H in the integrand Now that we know our order of integrations we choose H to be the hyperplane where the last of the integration variables is equal to 1 so in our case H 1 x3 According to this choice we only integrate over the first two variables x and then set x3 1 22 In this way we compute by gt MPLFeynmanIntegrate Integrand0 GenFeyn 1 2 GenFeyn gt 10 normal subs x 3 1 Integration over x 1 Integration over x 2 bar Sia SEY bar 2E bar SES bar 480 92 a Tay od Note that the result of the integration depends just as a rational function on x3 and therefore we can evaluate at x3 1 by Maple commands such as eval or subs The messages such as Integration over x 1 are just shown to give an indication how far the computation is already d
17. ediate results are too long to be shown here but can easily be obtained by choosing the last ar gument of MPLCubicallntegrate smaller than 4 Note that here the explicit use of MPLCoordinates is not necessary because it is called internally by MPLCubicalIntegrate MPLCubicalIntegrate already serves for the analytical computation of a large class of inte grals and further example applications are shown in 2 4 In some cases the procedure may even be useful for the computation of Feynman integrals However in the case of Feynman integrals the denominators of the integrand usually involve more complicated polynomials than allowed here Therefore appropriate changes of variables are necessary to express the latter in terms of cubical coordinates These are subject of the following section 3 Computing Feynman integrals In this section we consider integrals of the type fo Ioco 2 Ly 7 II e Pr pP where Q P C Q ori signat Ow are sets of irreducible polynomials all B NU 0 and Ly Ox is a hyperlogarithm given by a word w in differential 1 forms in da d Pila o Qr lt 2000000001 OL a Such integrals arise in the computation of Feynman integrals possibly also in other contexts In 14 2 we have discussed the conditions under which we can compute such integrals with MPL Here let us just recall briefly that the integrals have to be finite and unramified and the set P
18. ernally by the program but might not be completely excluded in the case of very complicated polynomials If all checks are OK we are ready to integrate 21 3 3 Integration over Feynman parameters We assume a finite integral over one or several Feynman parameters from 0 to infinity with an inte grand as in eq 7 If the integral satisfies the mentioned criteria linear reducibility unramifiedness and properly ordered polynomials for a given order of generalized Feynman parameters it can be computed with the following command MPLFeynmanIntegrate integrand L 1 n L The first argument is the integrand where all possibly appearing logarithms or hyperlogarithms in the numerator are expressed in bar notation For this purpose it may be useful to apply the command MPLHlogToBar introduced below The second and third argument are the same as in the previous commands L is the list of all generalized Feynman parameters in a chosen order and n is the number of integrations to be computed i e L 1 n is the list of integration variables in the given order During the computation the procedure returns messages indicating which variable is integrated out at present In the end it returns the result in terms of hyperlogarithms in bar notation In our example of the triangle graph let is compute the integrals 0 and J as defined in eqs 8 and 9 At first we call gt defform x 0 to make sure that every x i is recognized as a vari
19. fine the auxiliary polynomial F F m 1 2 4 5 x1xa xox 1 x4 1 xs 2 The corresponding scalar Feynman integral in D 4 2 dimensions without any scalar products in the numerator or raised propagator powers can be defined by 1 I xa xs 14 8 xs with xa X5 I dx OH 0 i 1 W0 where H is a hyperplane in the integration domain which we can choose freely according to the Cheng Wu theorem 12 We choose 6 1 where x is our last integration variable The order of integration variables still has to be chosen such that the mentioned conditions are satisfied We expand the integrand in and obtain I x4 x5 7 x4 xs er X5 e 17 with 57 1 joy x f n Moe 8 3 z n U In 7 gas x e OO o m the follovving vve compute both of the latter integrals vvith MPL 3 2 Checking the conditions Before the computation of any integral over Feynman parameters with MPL we recommend to check whether the mentioned conditions are satisfied We assume that the user already has checked the finiteness of the integral The following procedure can be used to check linear re ducibility MPLPolynomialReduction S L 1 n L COMPATIBILITY GRAPH The command MPLPolynomialReduction S L 1 n L returns the reduction of a set S of poly nomials with respect to n integration variables i e Feynman parameters L 1
20. hematical framework elaborated in 6 When using MPL for a scientific publication please cite some of these articles The latest version of MPL and this manual are available from the webpage http cbogner com software mpl The entire program is obtained by downloading a txt file MPLn m txt where the integers n and m indicate the number of the version For example the file of MPL version 1 0 is called MPL1 0 txt and after saving the file in the same directory with your Maple worksheet the program is started with lA very useful introduction to the general concept of iterated integrals and generalizations of polylogarithms can be found in 7 For an overview of the far reaching applications of such classes of iterated integrals in particle physics we recommend 13 14 read MPL1 0 txt Before starting with any computations it is useful to let Maple read a file by which all multiple zeta values up to a certain weight are automatically expressed in terms of a basis While writing MPL we have been using the file mzv 1 12 txt provided by Bigotte Jacob Oussous Minh and Petitot for this purpose The file is available from the webpage http www lifl fr petitot recherche MZV mzv12 After saving the file in the working directory and calling read mzv 1 12 txt every multiple zeta value up to weight 12 appearing from now on will be expressed in terms of a basis for example zeta 3 2 7 6 5 362260 This
21. is linearly reducible properly ordered and unramified with respect to this order We recommend to always check these conditions with MPLPolynomialReductionand MPLCheckOrder see section 3 2 before attempting any integration over Feynman parameters These checks usually take much less computation time than the integrations e MPL uses a few global variables some of which were mentioned above In the very first lines of the program file you find a short list of these variables You may check this list to exclude the unlikely case that you are using the same names for other purposes in your worksheet If your problem still remains or if you find any errors or typos in this manual please inform the author References 1 V I Arnold The cohomology ring of the coloured braid group Mat Zametki 5 1969 227 231 Math Notes 5 1969 138 140 2 C Bogner MPL a program for computations with iterated integrals on moduli spaces of curves of genus zero 25 3 C Bogner and F Brown Symbolic integration and multiple polylogarithms PoS LL2012 2012 053 arXiv 1209 6524 hep ph 4 C Bogner and F Brown Feynman integrals and iterated integrals on moduli spaces of curves of genus zero Commun Num Theor Phys 09 2015 189 238 arXiv 1408 1862 hep th 5 C Bogner and S Weinzierl Feynman Graph Polynomials Int J Mod Phys A25 2010 2585 2618 arXiv 1002 3458 hep ph 6 F Brown Multiple zeta values and peri
22. n which are the first n entries in the list L of all variables of the problem i e generalized Feynman parameters given by the Feynman parameters and kinematical invariants If a global variable COMPATIBILITY GRAPH true the reduction is computed regarding compatibilities among the polynomials as introduced in 9 The program uses the version of such compatibilities defined in 14 If COMPATIBILITY GRAPH false the procedure returns a Fubini reduction as introduced in 8 See 2 for the details In order to interpret the output of this procedure it is instructive to look at our example of the triangle graph For the computation of x4 xs and I x4 xs we clearly should compute the reduction of U 7 The list of generalized Feynman parameters is x1 x2 x3 x4 xs and the integration variables are the first three elements of this list Therefore we compute U x 1 x 2 x 3 F x 1 x 2 x 4 x 5 x I1 x 3 x I2 x 3 1 x 4 1 x 5 GenFeyn x 1 x 2 x 3 x 4 x 5 gt COMPATIBILITY GRAPH true d This is also the default value gt MPLPolynomialReduction U F GenFeyn 1 3 GenFeyn H xi x2 X3 X1X2X4X5 X1X3 X2X3 1 x4 1 xs 1 2 x3 x1 xoxs xaxo xoxaxs x1 X2 x1 x xox xi X2 x4xo 18 3 4 535 1 4 2 3 2 4 x2 1 4 5 X3 X3Xs X3X4 X3X4X5 X1 X3 X1X5 d X3X5
23. nman parameters defined above gt Reduction MPLPolynomialReduction U F GenFeyn 1 3 GenFeyn gt MPLCheckOrder Reduction GenFeyn 1 3 GenFeyn Checks for integration over x 1 All rho_i are smaller than zero at the tangential basepoint Check OK Limits in 0 1 Check OK 3Note that the latter property is implied by the condition of the integrand to be unramified as discussed in 2 4 The procedure does not check unramifiedness but it checks the condition directly for the required limits at the tangential basepoint 20 Checks for integration over x 2 All rho i are smaller than zero at the tangential basepoint Check OK Limits in 0 1 Check OK Checks for integration over x 3 All rho i are smaller than zero at the tangential basepoint Check OK No limits to be computed Check OK If each message ends with Check OK and no error message appears we can use the order given by L to compute the integral by the command discussed below As a remark let us briefly comment on the meaning of these messages and possible error messages The procedure MPLCheckOrder does the checks in three steps At first it checks that the reduction implies linear reducibility with respect to the given order of parameters If this is not the case it returns an error message immediatly Then it goes through each set of polynomials of the reduction according to the given order of integrations In order to construct an
24. note the Q vectorspaces of differential 1 forms spanned by the bases Qin O OF by An AX AX respectively Furthermore the corresponding Q vectorspaces of ho motopy invariant iterated integrals are denoted by V Qm V Q7 V O7 see 14 2 MPLArnoldEguation forml form2 The above differential 1 forms satisfy the following set of relations due to Arnol d 1 also see 41 dXm A d Xi Xm F d k Xd Xm 1 kc Xk Xj Xm 1 diyin 2 d Xi Xm doxes pum od x E dxy dis du Xu 1 Xi Xm l1 1 Mil al kn Xk Xi X 1 6 for 1 i j m Note that these equations are of the type oy A 0 ou a k where the 1 forms o are in OF the lifted fiber and the 1 forms o are in the base Internally these relations are extensively used in the computations below More conveniently for two given forms forml form2 on the left hand side of the above equations in the sense forml Aform2 the right hand side of the Arnol d equation can be obtained from MPLArnoldEquation forml form2 For example for n 3 the right hand side of the equation dx3 d x2x3 dx X d x2x3 da d n d xox3 1 1 2 1 1 1 X2 1 is computed as MPLCoordinates x 3 5MPLArnoldEquation d x 3 x 3 1 d x 2 x 3 x 2 1 x 3 1 d x2 amp d x 402 da ds 1 x2 1 x3 1 1
25. ods of moduli spaces Mo n Ann Sci Ec Norm Sup er 4 42 2009 371 489 math AG 0606419 7 F Brown terated integrals in quantum field theory in A Cardona I Contreras and A F Reyes Lega editors Geometric and Topological Methods for Quantum Field Theory chapter 5 pages 188 240 Cambridge University Press 2013 8 F Brown The massless higher loop two point function Commun Math Phys 287 3 925 958 2009 9 F Brown On the periods of some Feynman integrals 2009 math AG 0910 0114 10 F Chavez and C Duhr Three mass triangle integrals and single valued polylogarithms JHEP 1211 2012 114 arXiv 1209 2722 hep ph 11 K T Chen terated path integrals Bull Amer Math Soc 83 1977 831 879 12 H Cheng and T T Wu Expanding Protons Scattering at High Energies MIT Press Cam bridge MA 1987 13 C Duhr Mathematical aspects of scattering amplitudes arXiv 1411 7538v1 hep ph 14 E Panzer Feynman integrals and hyperlogarithms PhD thesis Humboldt University arXiv 1506 07243 math ph 15 E Panzer Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals arXiv 1403 3385 hep th 26 Index A Arnol d equation 7 B bar 3 bar notation 3 BarToLists 4 base 6 basis 9 C compatibility 18 COMPATIBILITY GRAPH 18 connection 8 co product 5 cubical coordinates 5 D differential 1 forms 5 differentiation 7
26. one Below these messages the result is given in bar notation In the same way we compute d gt I1 simplify subs x 3 1 MPLFeynmanIntegrate Integrand1l GenFeyn 1 2 GenFeyn The result is already a relatively long expression which we do not show here 3 4 Expressing hyperlogarithms in bar notation The computations in section 2 were mostly based on iterated integrals in V Qm In the present section the computation of Feynman integrals is reduced to integrations over functions in V Qm but the result is returned in terms of hyperlogarithms in the generalized Feynman parameters and also the integrand as in eq 7 may involve such functions The reader may be used to express hyperlogarithms in a different notation which may be close to the definition of the functions Hlog in Erik Panzer s program HyperInt 15 Furthermore it may be interesting for the user to compare results of Panzer s program with results of MPL or to use both programs in combination For these purposes MPL provides the following procedure MPLHlogToBar expr Here expr is a function involving hyperlogarithms in the notation Hlog as used in the program HyperInt and defined in 15 It returns the function in bar notation For example for the above integral 7 0 HyperInt returns the result 22 10 Hlog xa 0 1 Hlog xs 1 0 Hlog xs 1 0 Hlog xs 0 s Hlog xs 11 Hlog xs 0 1 Hlog xa 1 Hlog xs 0 xs x4 A
27. pplying MPLHlogToBar 10 we obtain the result in bar notation which is the expression given for this integral in the previous subsection 4 Automated tests and troubleshooting checklist MPLAutomatedTests With the command MPLAutomatedTests you can run a sequence of automated computations checking the correctness of several parts of the program Note that the file mzv 1 12 txt has to be included for these tests to work correctly The procedure should return a sequence of messages all ending with ok If this is not the case MPL does not run correctly on your system for some reason We have tested MPL for version 16 of Maple and problems may arise when using it with other versions Another way to check that MPL is running correctly is to run the worksheet which contains the examples shown in this manual You obtain the worksheet from the same webpage as the program Checklist If MPL returns an unexpected result or error message in any of your applications you may start your search for the problem by going through the following questions e Did your current Maple worksheet open the t xt file which contains MPL See section 1 Is the file stored in the working directory You can check that the file is read by trying any of the above commands of MPL e Did you open the file mzv 1 12 txt oran equivalent alternative for the reduction of multi ple zeta values to a basis See section 1 If you did not do so the re
28. r y d y rdo indo y 1 y y 1 y yz 1 y yz 25775 P m har 290 d n PN tude 00 l y2 y y 1 y yz 1 y yz y ar fo 402 4 par 02 022 at 1 y yo y l y 1 y 1 yz 1 y2 1 y d bar Y SEME gg 228 ment Sm bar Ou men 1 y 1 y1y2 1 2 1 y y y par 99 dn L 200 100 par 209 RM y l y 1 y y 1 y 1 y The output of this command is always a list of w lists where for k lt w the k th list contains the 103 403 words of length k MPLUnshuffle expr var The unshuffle map V Qm o V Q amp 1 V QF 3 as defined in 4 is the inverse of the map u id DY V Q4 4 BV z V Qn 4 10 and can be used recursively to decompose iterated integrals in V Qm into products of hyperloga rithms in V Q1 with 1 lt k lt m The command MPLUnshuffle expr var applies the unshuffle map to a function expr in V Qm where the second argument var is the last of the cubical coor dinates xm The hyperlogarithm in the right hand part of the resulting tensor product is a function of this variable if expr depends on this variable In order to give a non trivial example let us at first construct a function in V Q3 as follows Consider the hyperlogarithms fi bar M pi 1 1 xd x3 b f XoX3 We
29. sing neither X3 X1 Xo nor X3 X2 x would be an allowed order of integration MPLCheckOrder reduction L 1 n L Consider a given polynomial reduction in the form as returned by the previous command no mat ter whether compatibilities are regarded or not and a given ordered list L of generalized Feyn man parameters where the first n entries are the integration variables For this data the proce dure MPLCheckOrder reduction L 1 n L checks with respect to the order of variables in L whether the integrand is linearly reducible whether it is properly ordered at a tangential basepoint see 2 41 and whether all limits which have to be computed in the integration procedure are at 0 or 1 3 All three conditions are required for an automatic computation of the integral with MPL If all conditions are satisfied the procedure returns a sequence of messages all ending with the phrase Check OK If one of the conditions is violated the procedure returns a corresponding error mes sage In this case a fully automated computation of the integral in the order of integrations given by L is not possible with MPL We recommend to try the command again with a different order of variables in L In some cases the problem also may be solved by a different choice of kinematical invariants see remarks in sections 4 2 and 4 3 of 2 Let us apply the procedure in our example of the triangle graph where GenFeyn is again the list of generalized Fey
30. sults should still be correct but may be unexpectedly cumbersome e Do you use Version 16 of Maple MPL was written and tested using Maple 16 and we recommend to use it with this version 24 Did you run MPLAutomatedTests and the worksheet with the examples The automated tests take less than 2 minutes on a standard PC If they return error messages the problem is very likely still one of the above points If the automated tests and the worksheet with the examples of this manual run correctly with a different version than Maple 16 chances are good that this version will not cause problems e Did you use the command MPLCoordinates wherever it is necessary See section 2 e Did you use the command defform x 0 before you compute with Feynman parameters ELLA e When using MPLCubicalIntegrate or MPLFeynmanIntegrate did you check that the in tegrand is really of the appropriate type See eq 6 and eq 7 Here it is important to check that logarithms and hyperlogarithms in the integrand are ex pressed as a linear combination of bar terms For example ln x has to be written as 2bar 415 If you used MPLHlogToBar to convert a hyperlogarithm from Hlog to bar notation it may involve products of bar terms These must be explicitely multiplied by use of the shuffle product such that the integral is a linear combination of bar terms e When using MPLFeynmanIntegrate are your integration variables ordered such that the in tegral
31. ter than zero are defined to vanish at the origin of the space of cubical coordinates Therefore if we choose the above list L to be xo 1 0 2 Xo m 0 the procedure MPLMultipleLimit returns the value that remains after by replacing every bar by zero in f V Qm MPLCubicalIntegrate f var n MPL can compute multiple integrals of the type g 0 i Dj where where f V Q gis a polynomial in xm the a N and the p are in Xm 1 Xm 1 x1 For f being an integrand of this type the procedure MPLCubicalIntegrate f var n succes sively integrates over Xm Xm 1 Xm n 1 in this order The first argument of the procedure is the integrand in bar notation the second argument is the variable to be first integrated out i e xy in the notation chosen here and n is the number of integrations As an example we consider the integrand uv 1 uj ua 1 uz 1 ua 1 u 1 uyuzu 1 1 EE uyuzu ua 1 We integrate out all four cubical coordinates u2 u1 by f u 1 3 1 u 1 u 2 4 1 u121 u 3 3 1 u131 u 4 2 1 u 4 2 1 u 1 u 2 u 3 2 1 u 4 u 2 u 3 2 1 u 4 u 1 u 2 u 3 2 1 u 1 u 2 MPLCubicalIntegrate f u 4 4 5 3 Zeta 3 26 9 x Zeta 2 17 5 x Zeta 2 22 9 15 Xm So we obtain 22 4 1 Tf e e Se T to S Interm
32. tes and they can be used to construct the corresponding differential 1 forms For example one has relations like d x d x1 0 MPLFormsFiber lifted boolean MPLFormsBase MPLFormsTotal After declaring x1 Xm as cubical coordinates MPL can compute with iterated integrals with differential 1 forms of the following sets as defined in 4 e o d d Macc m xI gee Xm The lt i lt p i tri saz m 5 i lt n i dXm OF din Ulasisn 1Xi ar lt a lt Xm laxixmxi l d m d acicm i QF dim dosint r lt a lt m 1 Xm a lt i lt mXi where clearly Q O UO 1 In the context discussed in 4 6 and with respect to a given n it y m P g makes sense to speak of these forms as e Qm forms on the total space O forms on the fiber OZ forms on the lifted fiber e 0 1 forms on the base Up to signs these sets can be obtained from the auxiliary procedures MPLFormsTotal MPLFormsFiber false MPLFormsFiber true MPLFormsBase respectively For example MPLCoordinates x 3 MPLFormsTotal MPLFormsFiber false d xs xixad xi 4 gt x3 7 1 x x2x3 1 x2x3 1 x3 MPLFormsFiber true d x3 x2x3d x1 x x3d x2 x x2d x3 xad x2 x2d x3 d x3 1 x x2x3 1 x3 MPLFormsBase xy 1 x x 1 x x2 1 x2 In the follovving vve vvill de
33. tion of terms tens Then it replaces the second argument of each tens by the image of this argument under the symbol map The right hand side of the above equation is obtained by MPLd MPLSymbolMap f For both sides we obtain tens bar 42 tens 42 bal zum X3 xD Paes X2X3 1 Note that this was already the result of the first example in section 2 2 as we used the function in eq 2 as input there MPLBasis letter m w With the help of the symbol map the command MPLBasis letter m w constructs a basis for the vectorspace of integrable words or homotopy invariant iterated integrals up to length w in the alphabet Qm For example the basis of all integrable words up to length 2 in with variables y y2 is obtained by gt MPLBasis y 2 2 ie C e titi a t n d y d y2 d yz 4 d yz diy NE y d y yid y m y d yi y1d y2 y bar 5 5 1 y yz 1 y yz ENS y2 d yz 209 1 y d y y1d y bar 02 1 y yz 2 1 y yz 1 y2 1 yz d bar 522 y2 y d y y1d y bar 20 vi enn 1 y2 1 y y 1 y l y yz 1 y 1 yz EN y t yid 221 id y yid y 592 y 1 y yz 1 y yz 1 tar 199 y d y yid yo bar 5 Y qal 1 y 1 y yz 1 y 1 yz Eva yi y1d y2 l ba
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