Drawing Feynman Diagrams with LaTeX and Metafont
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1. endfor if fi from mathematical expressions This means that the bodies of loops and conditionals do not have to form syntactically complete expressions We can therefore use loops to con struct expressions from building blocks For the example of figure fil the above fragment expands to origin origin vlist 5Jarc 1 tns vlist 5 loc vlist 1 loc vlist 5 arc 2 tns vlist 5 loc vlist 2 loc vlist 5larc 6 tns vlist 5Jloc vlist 6 loc origin origin vlist 6 arc 3 tns vlist 6 loc vlist 3 loc vlist 6 arc 4 tns vlist 6 loc vlist 4 loc vlist 6larc 5 tns vlist 6 loc vlist 5 loc since the vertices v1 v2 v3 v4 are already fixed as external vertices If all ten sions are unity this is the linear system 0 3u5 v1 v2 V 5 1 2 6 13 0 3v6 v3 U4 U5 10 with unique solution 1 v5 301 T 302 a U3 T v4 i E3 VE 3 3u3 Ic 3U4 PUT v2 Note that we do not have to find the solution ourselves because METAFONT will not interpret the equations as assignments but rather as linear equations Once enough equations are given the state of the vertex coordinate will change from unknown to known and will have a value As long as all vertices belong to a subgraph with at least one element in V there will be a unique solution to the layout equations db 3 2 4 Labels An interesting feature of feynMF is the ability to calculate optimal label po2. s features the reader is referred to 2 6 Extension mechanism A great variety of different line styles is in use in the physics community feynMF provides the most common of them per default as displayed in table i While it would at best be inefficient to support an exhaustive list of such styles it would probably be a futile effort anyway Instead feynMF implements an extension mechanism that allows the user to install custom line styles of arbitrary com plexity For this purpose a macro style_def is provided This macro defines a METAFONT function that does the actual drawing based on the path it receives as an argument Furthermore it records the name of the function to make it available to graph mode and immediate mode as well 3 Implementation feynMF is implemented in the form of two macro packages feynmf sty for the BETEX part and feynmf mf for the METAFONT part Let us consider both of them in turn 3 1 BIEX macros Most macros in feynmf sty are trivial in that they are just writing their META FONT equivalent to the METAFONT input file The fmf macro for example is just the T X version of the vconnect METAFONT function def fmf 1 2 fmfcmd vconnect 1 pfx 2 Here fmfcmd writes its expanded argument to the METAFONT file and pfx adds a __ prefix to each member of the comma separated list it takes as an argument This measure protects the unwary user from the mysterious errors caused by a
3. loc numeric vlist arc first numeric vlist arc last numeric vlist arc numeric vlist arc tns Here vlist i loc with vlist first lt i lt vlist 1last is an array of two di mensional coordinates one for each vertex These coordinates start in the state unknown and become known when the layout equations have been solved For each vertex i vlist i arc j with vlist i arc first lt j lt vlist iJarc 1last is an array of numbers pointing to another vertex Therefore each entry corre sponds to an arc The vlist t arc j tns array holds the elements t of the tension matrix This data structure is sufficient for performing the algorithmic layout as described below It is supplemented by similar arrays holding information on linear constraints line styles etc 3 2 3 Linear algebra Let us now discuss how to solve the layout equation E for the common case of no constraints n O So tylo of Eg i j l UERUG Adding the constraints is a straightforward exercise which is omitted here for brevity METAFONT s syntactical features and builtin linear algebra allow a direct translation of h for i vlist first upto vlist last if unknown vlist i loc origin origin for j vlist i arc first upto vlist i arc last vlist i arc j tns vlist i loc vlist vlist i arc j loc endfor fi endfor METAFONT s syntactical feature that allows this translation of h is the de coupling of control structures for
4. center of the picture is still not enough because we hardly want all arcs to shrink to a point It will now be useful to introduce the set of all external vertices ve ve V ja v 1 5 and its complement the set of all internal vertices pint 2 V yet 6 From 2 we see that the vertices in V t will occupy the same position as their single neighbor unless their position is fixed explicitly It is therefore necessary to specify explicit positions for the external vertices In the implementation of feynMF commands are provided to place a list of external vertices on galleries along the sides of the diagram Using a similar strategy for external vertices 2 has been used for example in with good results for automated drawing of tree diagrams in PostScript While g gives fair results for almost all tree diagrams it can fail miserably on loop diagrams as witnessed in figure Ba A simple generalization of 4 can 7 Figure 2 Ladder diagram a using the action a and b using the improved action 7 Figure 3 Varying the tension parameter improve results immensely 1 n L vi j 5 D gt tig vi Uj 7 y UE_RVG The elements of the symmetrical tension matrix t are positive numbers that default to 1 and can be used to tune the layout The effect of the tension parameter can be understood by imagining the graph as consisting of rubber bands Changing the tension of an arc wil
5. coordinates of the vertices in temporary variables fmfcmd save loc bmin bmax forsuffixes 1 2 3 loc x loc y vloc __iv endfor It is trivial to get the coordinates of the enclosing box bmax x max locix loc2x loc3x 1w bmax y max locly loc2y loc3y 1h bmin x min locix loc2x loc3x 1w bmin y min locity loc2y loc3y 1h Now we can use immediate mode to draw this box thinly dashed and labeled fmfif dashes width thin bmin x bmin y bmax x bmin y bmax x bmax y bmin x bmax y cycle fmf iv label Gamma 5_ mu alpha beta 7 5 bmin x bmax x bmax y 5 Conclusions I have described feynMF a flexible tool for portable and convenient inclusion of Feynman diagrams in ATEX documents Acknowledgments I am most grateful to Wolfgang Kilian who pushed feynMF s predecessor feynman mf to its limits in his doctoral thesis and provided invaluable input for feynMF Thanks also to all brave users who tested preview versions and provided encouragement 17 A Distribution The latest release of feynMF is available by anonymous ftp from crunch ikp physik th darmstadt de in the directory pub ohl feynmf or from any of the Comprehensive TeX Archive Network CTAN hosts ftp shsu edu ftp tex ac uk ftp dante de in the directory macros latex contrib supported feynmf Important announcements new versions fatal bugs etc will be made through the mailin
6. implemented on top of the standard TFX picture en vironment This makes it completely portable but the graphics output is less than perfect This is not the fault of the feynman package but rooted in principle limitations of the picture environment Jos Vermaseren s axodraw package uses special to access PostScript primitives for drawing diagrams This approach is inherently not portable the mentioned ubiquity of PostScript printers makes this a minor point though but very flexible and produces sub stantially more pleasant graphics Both packages take no advantage of the formal structure of Feynman graphs but require the user to specify the layout manually using low level graphics primitives It is possible to go one step further and move from low level tools working on points and curves to a high level markup system working on the mathematical structure of graphs This step will free the user from having to think about the layout and allow him to concentrate on the structure of the graph instead In this paper I will describe such a system feynMF which is completely portable among TEX installations It is unique among packages for drawing Figure 1 Simple scattering diagram Feynman diagrams in combining the following features e Simplicity and conciseness for common diagrams The scattering diagram in figure I can be specified completely in five lines of ATRXx begin fmfchar 40 30 fmfpen thick fmfleft i1 i2 fmf
7. algebra which can be employed for au tomatic layout algorithms as detailed below Without taking advantage of these features the implementation in other lan guages would have been much more complex 2 3 Algorithmic layout Early in the design it was clear that feynMF should in at least one mode of operation accept a mathematical description of a graph and create the layout of the corresponding Feynman diagram automatically It should also not rely on a database of common topologies because such a database will necessarily remain incomplete Every graph can be specified completely by giving a set A of pairs The set V of vertices is then given by V v dace A a 0 v Va v v 1 The set A will henceforth be called the set of arcs It is useful to introduce the sets of vertices connected to v a v v EV lux 2 where denotes the symmetrical relation v xv amp Ga A a v v Va v v 3 of being connected The obvious first candidate for a function that should be minimized is the sum of the squared lengths of arcs n Ior Un vi o 4 Ta As is stands H is not yet sufficient because v for all i is obviously a solu tion corresponding to the minimum 0 for all In order to lift the degeneracy we have to break translational invariance However breaking the translational invariance by demanding for instance that the center of gravity gt _ v n coin cides with the
8. ccidentally using a METAFONT reserved word for a vertex name A non trivial aspect of the pfx macro worth mentioning is that it works by macro expansion alone in T X s mouth in Knuth s terminology and does not need to redefine any macro an operation that would have to happen in T X s stomach in Knuth s terminology This is necessary for making pfx work inside of a write where macros are expanded but redefinitions are prohibited see Appendix D Traditional implementations of looping constructs e g loop repeat work by redefining a continuation and are therefore unavailable inside of a write A possible solution would be to use a temporary variable and force expansion of pfx outside of the write A far more elegant solution uses a subset of a partial implementation of A calculus in T amp X s mouth The other unusual aspect of the ATEX macros is the grepfile pattern infile outfile macro that copies all lines starting with pattern from infile to outfile after stripping off the pattern It is used to extract the label information that the METAFONT macros have stored in the log file This trick overcomes META FONT s limitation of not being able to open any other files than the terminal the gf file the tfm file and the log file The implementation of this macro is straightforward using T X s pattern matching macro definitions However all subtletie
9. def as in fmfcmd crossed v1 v2 gt _5 lt _ 4 4 Advanced example Finally figure H is a slightly more advanced example a three loop QCD cor rection to the axial vector part of the Z selfenergy The beginning is straight forward we connect the vertices It is beneficial to give a higher tension to the bosons and a lower tension to the gluons to balance the diagram fmfpen thick fmfleft i fmfright o fmf boson tension 2 i iv3 fmf boson tension 2 o ov3 fmf quark ivi iv2 iv3 ivi fmf quark ov1 ov2 ov3 ov1 fmf gluon tension 5 fovi ivi fmf gluon tension 5 iv2 ov2 16 Now we add dots to the vector vertices and big squares to the axial vector vertices fmfv decor shape square decor size 4thick iv3 ov3 fmfdot ivi iv2 ov1 ov2 As it stands the diagram will have all vertices on a straight line To remedy this situations we use fmffixed to open the triangles fmffixed 0 7h ivi iv2 fmffixed 0 7h f ov1 ov2 We could also have used phantom arcs to achieve similar results but here the linear constraints are more concise and intuitive For illustration we mark the left triangle subgraph by a dashed box As of version 1 0 feynMF does not have a builtin function for calculating the enclosing box of a list of vertices However this is easily done in a few lines of METAFONT Before we start we have to force the layout calculation fmffreeze For convenience we store the
10. ere is an anal ogous fmfblobn command fmfdot u1 t draw a dot at the vertices v1 There is an analogous fmfdotn command fmflabel label v1 label the vertices v1 with label The location of vertices can be fixed but experience shows that this command should only be used as a last resort fmfforce z u1 place the vertices v1 at the METAFONT coordinate z The layout calculation and drawing which are implicitly performed at the end of each fmfgraph can be forced by fmffreeze freeze the positions of the vertices entered so far 14 fmfdraw draw all arcs and vertices entered so far The appearance of the graph can be changed by fmfpen w change the width of the drawing pen to w The default is thin lpt fmfset par val set the parameter par to the value val fmffixed d v1 fix the distance vector between subsequent vertices in the list v1 to d 4 2 Immediate mode In immediate mode the drawing commands from graph mode are duplicated with different arguments Therefore all decorated line styles are available but now for arbitrary METAFONT paths fmfi style opt path draw a line of style style on the METAFONT path path with options opt switched on fmfivt opt z draw a vertex with options opt at the METAFONT coordinate z fmfipair var fmfipath var declare t
11. ft or right label arbitrary T X commands for labeling the arc macros should be protected with noexpand label side label dist force placement of the label on the left or right at this distance width change the width of the line foreground background colors available with METAPOST only fmfn style opt v n connect the vertices v1 Un by a line of style style with options opt switched on The decoration of vertices is affected by the commands 13 fmfv opt vi turn on the options opt for the vertices v1 Among them are label arbitrary T X commands for labeling the vertex macros should be protected with noexpand label angle label dist force placement of the label at this angle or distance decoration size decoration filled size and filling style of the decoration decoration shape decoration angle shape of the decora tion optionally rotated foreground background colors available with METAPOST only Here are some examples for vertex decorations shaded fi11 5 circle DQ and hatched 5 square a open fill 0 triangle diamond pentagon and hexagon Be 2 3 O filled fi11 1 triagram tetragram pentagram and hexagram a x x fmfvn opt v n turn on the options opt for the vertices v1 Un fmfblob d v1 draw a blob of diameter d at the vertices v1 Th
12. g list feynmf announce crunch ikp physik th darmstadt de Subscriptions can be obtained from majordomo crunch ikp physik th darmstadt de send a message consisting of help to majordomo for instructions on how to subscribe don t send such messages to the list itself B Installation feynMF comes in standard TFX doc format The installation procedure is described in the README file and need not be repeated here C Revision History Version 1 0 May 1995 First official release 18
13. he variable var as a pair coordinate or path fmfiset xr y assign the value of y to the variable x fmfiequt x y declare equality of the variables x and y This is different from assignment because METAFONT can solve linear equations Immediate and graph mode can be interfaced by using the following METAFONT functions to access coordinates from graph mode in immediate mode vpath tag __ from __ to return the METAFONT path of the arc from vertex from to vertex to The optional tag can be used to disambiguate arcs connecting the same vertices vloc __ v return the position of the vertex in METAFONT coordinates 15 Figure 4 Three loop QCD correction to the axial vector part of the Z selfenergy 4 3 Extension mechanism A powerful but dangerous command is fmf cmd METAFONT expression write an arbitrary METAFONT ezpression to the METAFONT file No semicolon is appended A recommended application of fmfcmd is in defining new line styles with style_def name expr p enddef define a new line style and register it as name here is an example that will draw a cross at the center of the arc fmfcmd 7 vardef cross_bar expr p len ang len 2 0 len 2 0 rotated ang angle direction length p 2 of p shifted point length p 2 of p enddef style_def crossed expr p cdraw p ccutdraw cross_bar p 5mm 45 ccutdraw cross_bar p 5mm 45 end
14. hep ph 9505351v1 19 May 1995 e iV arX Drawing Feynman Diagrams with ETRX and METAFONT Thorsten Ohl Technische Hochschule Darmstadt SchlofSgartenstr 9 D 64289 Darmstadt Germany May 1995 Abstract feynMF is a TFX package for easy drawing of professional quality Feynman diagrams with METAFONT or METAPOST feynMF lays out most diagrams satisfactorily from the structure of the graph without any need for manual intervention Nevertheless all the power of METAFONT or METAPOST is available for the most complicated cases e mail Thorsten 0h1 Physik TH Darmstadt de 1 Introduction In recent years TeX and ATEX or other macro packages for structured markup on top of T X have revolutionized the way we share information in theoretical physics and other areas Not only does TpX provide typographical capabilities which transcend those of commercial word processors substan tially T X documents are also completely portable among computer systems Since implementations are available for essentially all computers in use in the physics community documents can be shared without the usual restrictions of proprietary data formats This has enabled us to collaborate on papers with colleagues on the other side of the globe to replace the mailing of hard copy preprints by electronic transmission and to submit these papers electronically to the publisher TRX s portability comes with a price though It does de
15. here simple building blocks can be used for the straightforward solution of simple problems and can be combined to solve the most complicated problems once the software system has been mastered by the user The reconciliation of convenience and expressiveness was made possible by providing two different modes e graph mode in which the layout is determined automatically from a simple mathematical description of the graph and e immediate mode in which the user has complete freedom but at a basic familiarity with METAFONT is recommended The goal of portability was easily reached by basing the implementation of TEX and METAFONT because both programs will be available to all potential users of the software 2 2 Languages The primary user interface is a set of ATX macros It is therefore possible to keep the whole paper including graphics in a single file This is among other things very convenient for exchanging manuscripts by electronic mail Also no new syntax has to be learned by the user METAFONT or alternatively METAPOST has been chosen as the low level graphics engine for the following reasons e METAFONT is part of any reasonable T X installation therefore available to all potential users e METAFONT output is readily included in TeX documents in the form of unusual characters e METAFONT has very powerful graphics primitives which allow high quality output and e METAFONT has builtin linear
16. l pull adjacent vertices together or allow them to move apart As an example figure Bl shows the effect of varying the tension of one line from 4 to 1 4 The improved ladder diagram in figure Ab has been drawn with vanishing tension of the arcs which will result in straight lines for the stems In fact the effect of vanishing tension can also be achieved by laying out subgraphs step by step By freezing the layout of the subgraph excluding the rungs in figure bp first and adding the rungs later we arrive at the same result Obviously this procedure can be iterated for graphs of arbitrary complexity While there are also commands to fix the position of a vertex or to shift its position it turns out that the most effective way of drawing Feynman dia grams consists in a combination of the stepwise construction of subgraphs and adjustment of tensions User s discretion is advised in tuning tension param eters More often than not the defaults give satisfactory results that can be made perfect by adjusting the tension of a single arc or loop Tuning too many tensions is not likely to improve the results and is almost as time consuming as choosing the layout manually Technically the most convenient aspect of A is that minimizing it leads to linear equations see ta below which are easily solved It would in principle be possible to investigate improved functions like n N 01 n gt gt m g 8 8 i j l UE_RUG which wo
17. liberately not ad dress the issue of graphical information apart from the rudimentary but very useful capabilities of the TFX picture environment and similar packages More complex graphics can only be handled by inclusion of more or less device dependent external graphics files More recently the inclusion of graphics files in the PostScript page de scription language has emerged as a de facto standard This approach restricts the portability of documents to installations were PostScript printers or emu lators are available The popularity of such devices makes this an almost moot point though Nevertheless handling graphics in an environment completely different from the TX text environment causes other problems Some popular packages that employ graphical user interfaces will force PostScript fonts for labeling on the user These fonts will usually not blend smoothly with other fonts used in text and equations More importantly these packages usually lack the ability to create complex mathematical expressions which would be useful in the labels of figures in physics papers Finally these tools are usually less than portable to the extent that changing jobs means changing tools Currently there are a couple of tools available that address one or more of the above points in the context of drawing Feynman diagrams which form one of the most frequent classes of graphics in physics papers Michael Levine s feynman package is
18. right o1 0o2 fmf fermion il v1 o1 fmf fermion i2 v2 02 fmf photon label q v1 v2 fmfdot v1 v2 end fmf char It is never necessary to draft the diagram on graph paper or to perform calculations to determine the position of vertices manually e Expressiveness for arbitrarily complex diagrams see the examples below e Extensibility e Portability No graphics devices are needed beyond a standard T X in stallation e Arbitrary T X labels The paper is structured as follows I begin by describing the design of feynMF in section pl Then I describe some details of the implementation in section B After a brief discussion of the most important user commands in section M I conclude 2 Design A clear cut distinction between design and implementation is certainly fic titious As in most programs with more than several hundred lines of code designs have been adapted as implementation progressed and feedback from early users came in 2 1 Goals As mentioned in the introduction feynMF was to meet the following competing design goals e convenience and ease of use e expressiveness e extensibility and e portability Of these extensibility is not a goal in itself but should rather be viewed as a derived goal It appears impossible to reconcile ease of use with expressiveness in the straight jacket of an inextensible implementation It is much more effec tive to provide an extensible environment w
19. s of these ATEX macros are of no concern to the user because they are designed to do their work quietly behind the scenes 3 2 METAFONT macros The METAFONT macros are much richer than their BTEX counterparts They have to deal with drawing primitives linear algebra and abstract representations of graphs 3 2 1 Transformers An important tool for generating complex graphs with arcs of different styles is provided by transformers These are functions that take a simple path deter mined from the layout algorithm or specified explicitly as argument and return another path which corresponds to a decorated version Here is an example that is used for implementing gluon lines curly EN gt IRR Using similar transformers the implementation of dedicated drawing functions is a matter of combining simple building blocks style_def gluon_with_arrow expr p draw wiggly p fill arrow p enddef PERR a E A e E A The current implementation does not attempt to force decorations e g arrows into the transformer paradigm because METAFONT treats drawing along a path differently from filling an outline Therefore decorations are drawn after each other and not added to the object in a pipeline 3 2 2 Graphs Graphs are represented by an array of vertices and arrays of vertices emanating from the vertices Therefore the core of the data structure for graphs is given by numeric vlist first numeric vlist last pair vlist
20. sed The overall structure is controlled by two environments begin fmffile name end fmffile This environment encloses all graphs that are written into a META FONT input file named name mf For technical reasons name 11 must not be identical to the name of the main FATRX input file All created files have to be processed by METAFONT after the first run of BTEX See the feynMF user s manual for details on how to run METAFONT on various systems begin fmfgraph w h end fmfgraph This environment encloses a single graph of width w and height h TeX unitlengths There is also a light weight unstarred version that does not support labels These environments can be used everywhere in a ATEX document in particular in centered parboxes for creating graphical equations like parbox 20mm begin fmfgraph 20 15 fmfleft i fmfright o fmf dashes i v v o end fmfgraph parbox 20mm begin fmf graph 20 15 fmfleft i fmfright o fmf dashes i v1 fmf dashes v2 o fmf fermion left tension 3 v1 v2 vi end fmfgraph ln Lambda 2 ai vI PES EE E InA 4 1 Graph mode The placement of external vertices is controlled by the following commands fmfleft vi place the vertices v1 equidistantly on a smooth path on the left side of the diagram There are analogous fmfright fmftop fmfbottom and fmfsurround commands The latter will place the vertices on a smoo
21. sitions in METAFONT and to communicate this information back to TRX s picture information Because METAFONT can not write any other files than its log file the information has to be stored there and T X macros have to be used to parse this file The algorithm used is quite simple It will place all labels on the outside of the arc or vertex it is associated to If the result is not satisfactory explicit placement rules can be specified to overwrite the automatic layout 3 2 5 Immediate mode The implementation of immediate mode is fairly straightforward because all necessary drawing primitives for drawing feynMF s line styles on METAFONT paths are already available for graph mode Therefore only trivial ATEX macros have to be written that translate the ATX syntax to METAFONT 3 2 6 Extension mechanism The extension mechanism serves two major purposes it allows users to specify new line styles and to overload existing line styles The ability to overload line styles can be used for a purely symbolic markup of graphs If all the arcs are tagged by symbolic names like gluon technipion chargino etc each user can use a library of style definitions to render the graph in a customized visual appearance 4 Usage Here is a short summary of the most important user commands of feynMF This section is not intended to replace the user s manual that comes with the distribution but it should give nevertheless an idea how feynMF is u
22. th path surrounding the diagram fmfleftn v n place the vertices v1 Un equidistantly on a smooth path on the left side of the diagram Analogously for fmfrightn fmftopn fmfbottomn and fmfsurroundn The external vertices can be connected with themselves and internal vertices by the commands fmf style opt JJH wi v2 J connect the vertices v1 v2 by a line of style style with options opt switched on The available line styles are collected in table fi Additional styles can be defined with style_def A special line style is phantom which will not draw an arc at all It is nevertheless useful for manipulating the layout because the corresponding arc enters the layout equations For convenience and for allowing more descriptive specifications several aliases like gluon quark or photon of the line styles in table fl are defined Among the options which can be given in a comma separated list after the line style are 12 curly curly_len dbl_curly curly_len dash_len dashes_arrow dash_len dbl_dashes dash_len dbl_dashes_arrow eee eo dash_len dashes dots dot_len dots_arrow dot_len dbl_dots dot_len db1l_dots_arrow dot_len phantom Se plain plainarrow gt y l plain Z dbi_plainarrow gt _ Table 1 Available line styles tension change the tension matrix element tij from the default value of 1 left right draw on a half circle on the le
23. uld avoid the problem of figure Ba to some extent by favoring arcs of length 6 However the prize in having to solve a system of non linear equations is certainly too high in particular because it will be impossible to prove that reasonable results will result from all user input 2 4 Constraints The method of Lagrange multipliers allows us to specify linear constraints among vertices vi vj dig 9 while still dealing with linear equations Therefore we add the term n A 5 Aij vi UZ dij 10 ij 1 VUi EWUG to the action E Then the system of 2na 2n linear equations determining the layout is n n i 1 a j l jal j i 1 Vi xUj Vi U Vi V Z 11 yt _ alt _ qt 0 v v dy lu tj Experience shows that non linear constraints like fixing a distance vi vjl 12 would be useful sometimes but as discussed at the end of section their implementation is beyond the scope of feynMF 2 5 Immediate mode In addition to the graph mode for algorithmic layout that has been described in the previous section feynMF also has an immediate mode to provide the user with maximum flexibility Immediate mode is particularly useful for unusually curved arcs which can not be specified easily in graph mode In this mode arbitrary METAFONT paths can be drawn either specified by absolute coordi nates or derived from arcs entered previously in graph mode For a detailed description of METAFONT
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