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FeynArts 3.9 User's Guide

1. FieldMatchQ works like MatchQ but takes into account field levels e g F 1 matches F FieldPointMemberQ similarly works like MemberQ except that the field matching is done with FieldMatchQ FieldPointMatchQ and FieldPointMember Q are to field points what FieldMatchQ and FieldMemberQ are to fields respectively FermionRouting top find out the permutation of external fermions as routed through the inserted topology top FermionRouting rul top substitute the insertion rules rul into the bare topology top then proceed as above FermionRouting returns a list of integers of which every successive two denote the end points of a fermion line in the diagram This function is typically used as a filter for DiagramSelect as in DiagramSelect diags FermionRouting 1 4 2 3 amp Or as DiagramGrouping diags FermionRouting which returns a list of fermion flow ordered diagrams The function FeynAmpCases works like a Cases statement on the amplitude correspond ing to a graph That is FeynAmpCases invokes CreateFeynAmp on each graph and from the resulting amplitude selects the parts matching a pattern 32 4 INSERTING FIELDS INTO TOPOLOGIES FeynAmpCases patt g t h create an amplitude from g t h same three arguments as for user function above and select all parts matching patt from that FeynAmpCases patt Lamp select all parts matching patt from the amplitude amp This f
2. GIs f1 f2 k1 k2 definition of the coupling of fields f f2 with coupling constant vector G s f1 fo and kinematic vector k1 k2 G 1 f1 f2 generic coupling constant vector symmetric for 1 and antisymmetric for 1 The kinematically extended field structure of the fields must be taken into account sim ilar to the case of generic propagators Whereas however a field on an internal propa gator has two sets of kinematic indices 1i1 gt 112 for the left and right side each provided the field carries kinematic indices at all a field involved in a coupling obvi ously needs at most one set of indices For example the quartic gauge boson coupling is defined in Lorentz gen as AnalyticalCoupling si V i momi li1 s2 V j mom2 1i2 s3 V k mom3 1i3 s4 V 1 mom4 114 G 1 s1 V il s2 V jl s3 Vik s4 V 11 MetricTensor li1 1i2 MetricTensor 1i3 li4 MetricTensor lii li3 MetricTensor 1i2 1i4 MetricTensor lii li4 MetricTensor 1i3 1i2 56 7 DEFINITION OF A NEW MODEL The generic coupling constant G refers actually to a vector of coupling constants one for each element of the kinematic vector The VVVV coupling would appear in a textbook as i j 1 2 3 C Vi p1 V pz Vi pa VA p Gimv SuvBon Gi v v Suove CUu v v SurBov SUVS po GUV WV SupSvo SuoSpv When looking up a coupling the fields of a vertex
3. closed fermion chain involving the noncommuting objects 01 02 FermionChain 0 05 open fermion chain involving the noncommuting objects 01 02 VertexFunction ol f generic vertex function of loop order o with adjoining fields f SumOver i r ext indicates that the amplitude it is multiplied with is to be summed in the index i over the range r if r is an integer it represents the range 1 r ext is the symbol External if the summation is over an index belonging to an external particle IndexSum expr i range the sum of expr over the index i IndexSum has the same syntax as Sum but remains unevaluated Ttis necessary to distinguish sums over internal and external indices For example a squared diagram is calculated by squaring the diagram and then summing over the external indices On the other hand an internal index summation say for a quark loop must be done for each diagram before squaring 6 2 CreateFeynAmp 43 6 2 CreateFeynAmp Once the possible combinations of fields have been determined by InsertFields the Feynman rules must be applied to produce the actual amplitudes The function for this is CreateFeynAmp CreateFeynAmp translate the inserted topologies 7 into Feynman amplitudes option default value AmplitudeLevel InsertionLevel level specification see end of Sect 4 1 PreFactor TORDI overall factor of the amplitude 4 LoopNumber Truncated False whether t
4. FeynArts 3 9 User s Guide June 16 2015 Thomas Hahn The dreadful legal stuff FeynArts is free software but is not in the public domain Instead it is covered by the GNU library general public license In plain English this means 1 We don t promise that this software works But if you find any bugs please let us know 2 You can use this software for whatever you want You don t have to pay us 3 You may not pretend that you wrote this software If you use it in a program you must acknowledge somewhere in your documentation that you ve used our code If you re a lawyer you will rejoice at the exact wording of the license at http www fsf org copyleft lgpl html FeynArts is available from http feynarts de If you make this software available to others please provide them with this manual too If you find any bugs or want to make suggestions or just write fan mail address it to Thomas Hahn Max Planck Institut f r Physik Werner Heisenberg Institut F hringer Ring 6 D 80805 Munich Germany e mail hahn feynarts de CONTENTS Contents 1 Getting Started 2 Roadmap of FeynArts 3 Creating the Topologies 3 1 FEOPOlO Seal Objects xa is a ear ea rn a lc ap 9 2 Create lopolopres 4 23 wb 44 eee xU ee wp ES 3 3 Creating Count r term Topologies 4a sa Eas RR e d he ER 3 4 Creating Topologies with generic Vertex Functions iss 3 5 Excluding Topologies 4 a Kost noe Ibo Ofen ton Cb M eh bar
5. 4 1 ColumnsXRows 3 number of diagrams per column row may also be given as a list n nr SheetHeader Automatic title for each sheet of graphics Numbering Full what type of numbering to display underneath each diagram FieldNumbers False whether to label the propagators of a bare topology with the field numbers AutoEdit True whether to call the topology editor when encountering an unshaped topology DisplayFunction DisplayFunction which function to apply to the final graphics object in order to display it PaintLevel specifies the level at which diagrams are painted The default is to use the level the topologies were inserted at for inserted topologies For bare topologies PaintLevel is not relevant Note that for the numbering of the diagrams to be appro priate for discarding insertions see Sect 4 8 the PaintLevel has to be the same as the InsertionLevel ColumnsXRows specifies the number of diagrams displayed in each column and row of a sheet It may be given as a single integer e g ColumnsXRows gt 4 which means 4 rows 36 5 DRAWING FEYNMAN GRAPHS of 4 diagrams each on a sheet or as a list of two integers e g ColumnsXRows gt 3 5 which means 5 rows of 3 diagrams each SheetHeader specifies the title of each sheet of diagrams With Automatic or True the default header is used particlesi particlesout for inserted or Nin Nout for bare topologies False disables the header and everything el
6. 58 68 flipping rules 56 Format 52 FourMomentum 42 73 FourVector 72 73 92 frame rate 36 frameTime 36 G 42 55 gauge 44 GaugeRules 43 GaugeXi 42 43 53 75 Generation 59 60 64 Generic 20 generic coupling 42 47 generic model 25 52 GenericModel 24 26 ghost field 20 GhostDash 63 Global 19 gluinos 79 gluon 77 gluon line 63 graphics primitives 83 GraphID 41 GraphInfoFunction 43 Greek 64 GRID POSITION 40 GS 77 h c 67 HexagonCTsOnly 16 HexagonsOnly 16 ImageSize 37 Incoming 9 Index 54 59 index colour 63 Dirac 51 generation 22 59 60 64 Greek 64 INDEX kinematic 52 Latin 64 Lorentz 52 64 range of 59 summed over 42 IndexDelta 67 IndexRange 59 IndexSum 42 Indices 61 initialization file 7 InitializeModel 25 InsertFields 22 insertion auxiliary functions 29 extracting deleting an 32 grouping 29 modifying an 33 selecting 28 structure of 33 InsertionLevel 24 InsertOnly 61 Installation 6 Integral 41 integration momenta 41 interaction eigenstates 62 Internal 9 14 53 63 irreducible 12 14 J Link 40 Java 40 kinematic object 42 47 kinematic vector 55 65 67 KinematicIndices 52 KinematicVector 67 label 63 INDEX IATEX 38 63 composite 63 indices 64 of mixing field 65 Lagrangian 71 LastSelections 24 27 IATEX commands 83 leptons 76 level 20 49 classes 20 generic 20
7. a definition of the graphical appearance for indices of type t Alph n the nth lowercase latin letter UCAlph n the nth uppercase latin letter Greek n the nth lowercase greek letter UCGreek n the nth uppercase greek letter The functions Alph UCAlph Greek and UCGreek are useful for converting the index numbers into Latin or Greek letters if this is desired For example the following line in SM mod makes the fermion generation indices appear as i j Appearance Index Generation i Integer Alphli 8 Similarly Lorentz gen adds this definition to render Lorentz indices as u v Appearance Index Lorentz i Integer Greek i 11 Particular labels for the fields at particles level can be given by assigning values to the function TheLabel If no specific label is defined for a particles level field the label of the associated class is used E g the labels of the individual up type quarks are declared in SM mod by TheLabel F 3 1 u TheLabel F 3 2 c TheLabel F 3 3 t Without these definitions the up charm and top quarks would be labelled according to their classes label 11 u2 and ua In the case of a mixing field PropagatorLabel and PropagatorType may be set to a list of two items one for the left half and one for the right half For example SM mod contains the class for a y Z mixing field it is commented out by default V 4 SelfConjugate gt
8. e some optional final touch rules M LastModelRules Just as the generic model file the classes model file is an ordinary Mathematica input file apart from the three required items and may contain additional definitions For ex ample the abbreviations for the Restrictions option of InsertFields which typically depend on classes fields are usually defined in the classes model file Range of Particles Indices It is necessary to declare the index range of every particles index IndexRange Index t p declares the index of type t to have the values p For example the Generation index in SM mod which counts the fermion generations has the range IndexRange Index Generation 1 2 3 If a particular index should not be unfolded at particles level i e FeynArts should not generate an extra diagram for every value the index can take on its range may be 60 7 DEFINITION OF A NEW MODEL wrapped in NoUnfold For instance one usually does not want eight diagrams gener ated and drawn for every gluon propagating this can be effected with IndexRange Index Gluon NoUnfold Range 8 As a consequence fields with specific values for such indices have no effect in places like ExcludeParticles This means that it is possible to exclude either all gluons or none but not only the gluon with index 5 Conservation of Quantum Numbers When FeynArts generates diagrams with vertex function placeholders see
9. insertions see Sect 3 4 is so similar there are no special exclusion filters for the latter one simply uses the filters for counter terms 3 5 Excluding Topologies Create all 1 2 topologies Now the same for irreducible topologies In 6 CreateTopologies 1 1 gt 2 In 7 Paint ColumnsXRows gt 4 1 22 2 IS lt II EZ x In 8 CreateTopologies 1 1 gt 2 ExcludeTopologies gt Internal In 9 Paint ColumnsXRows gt 4 15 16 For convenience common choices of the ExcludeTopologies option have a short cut 3 CREATING THE TOPOLOGIES same as TadpolesOnly ExcludeTopologies gt Loops Except 1 TadpoleCTsOnly ExcludeTopologies gt CTs Except 1 SelfEnergiesOnly ExcludeTopologies gt Loops Except 2 WFCorrections SelfEnergyCTsOnly ExcludeTopologies gt CTs Except 2 WFCorrectionCTs TrianglesOnly ExcludeTopologies gt Loops Except 3 TriangleCTsOnly ExcludeTopologies gt CTs Except 3 BoxesOnly ExcludeTopologies gt Loops Except 4 BoxCTsOnly ExcludeTopologies gt CTs Except 4 PentagonsOnly ExcludeTopologies gt Loops Except 5 PentagonCTsOnly ExcludeTopologies gt CTs Except 5 HexagonsOnly ExcludeTopologies gt Loops Except 6 HexagonCTsOnly ExcludeTopologies gt CTs Except 6 To extend the filtering capabilities you may define your own filter functions how the Triangles filt
10. 3 6 Selecting Starting Topologies ss sh anna aaa 4 Inserting Fields into Topologies 4 1 The Three Level Fields Concept s ar gpg ipa an 427 IHSBSELPIELQS MIR on eae Ea e ES A N EEE us Bon Model FeS ec 4 4 Imposing Restrictions ons aoaaa oom od hte wa eR 4 5 1 Selecting Insel Hos rie un Beta master 4 6 Grouping Insertions 3 ait s 1 oa a udo erac ER Sra id 4 7 Auxiliary Functions 2e dore oe aede ea SU petat ed 4 8 Extracting and Deleting Insertions by Number 2 9 Modifying Insertons ue sc we ew deeem rare 4 10 Structure of the Inserted Topologies rr RR 5 Drawing Feynman Graphs 5 1 Things to do with the Paint Output xod te bo dti buenos 5 2 Shaping Topologies xS ear altus aaa 6 Creating the Analytic Expressions 6 1 Representation of Feynman Amplitudes aoaaa a 10 11 12 13 17 20 20 22 25 26 28 29 29 32 33 33 35 37 38 41 CONTENTS 6 2 CreateheynAtiiy 2 S ea ara bytes dcs aod tetas Roy tio hata uto ee 43 6 3 Interpreting the Results 80 3 2 Re elem ud d aia Ree edd 47 64 Picking Levels a 22 3 232 842 beet ap taedet deae SOS vi etudes 48 6 5 General Structure of the Amplitudes llle 49 6 6 On Fermion Chains and Signs rar deno eer era ae 50 6 7 SpecityIng Momente y ca s eet aoe sol dete eoe sere ede ore 5 6 8 Compatibility with FeynArts L s 2er oc Yo ese X oe Yo Y e es x 51 Definition of a New Model 52 7 1 TheGeneric Model eA
11. 52 7 2 TheClasses Model rss 59 73 Add onModelFiles a a 68 Z4 Debugging ss sare 234244 dob on deb eet oe Rond dicono 69 75 ModekfileGeneration 7 70 The Lorentz Formalism 73 The Electroweak Standard Model 75 Bal phe DEDOS Obl s 12 acu de e eR b Ree dota i 74 Bi Background beld Formalism 4x RR 44844 78 The Minimal Supersymmetric Standard Model 79 The Two Higgs Doublet Model 82 Graphics Primitives in feynarts sty 83 E SOM OUI fe eet oe e e Diete Dod e Dey ec adr ip Dope i ee Desert dee ed 83 E2 Propagat ls 22254 3904 que d m meme etel gei detergent ditur d 84 B 3 VertICOeS edo one ende eene eed tege du Fee ndo Sees Rd ee A et e At ere ds 85 EA Labels ar ee dda ae Fae a lis Tae ROC Ae LE PI UE RUN EE 86 Incompatible Changes in Version 3 3 87 1 Getting Started FeynArts is a Mathematica package for the generation and visualization of Feynman dia grams and amplitudes It started out in 1990 as a Macsyma code written by Hagen Eck and Sepp K blbeck which could produce tree level and one loop diagrams in the Stan dard Model K 90 but soon got ported to the Mathematica platform In 1995 Hagen Eck designed the second version to be a fully general diagram generator To achieve this he implemented some decisive new ideas Eck95 the most important one being the generation of diagrams in three levels The program was taken up again in 1998 by Thomas Hahn who developed v
12. 53 animation 36 antiparticle 20 ANY POSITION 40 Appearance 64 attributes classes 60 AutoEdit 35 36 38 Automatic 24 autoshaping 36 background field 78 bar on label 63 Bicycle 17 BoxCTs 14 BoxCTsOnly 16 Boxes 14 BoxesOnly 16 buttons 40 C 65 C2A 79 82 C2B 79 82 CA 79 82 CAB 79 82 CANCEL 40 CB 79 82 CBA 79 82 CC 65 Centre 17 changes in FeynArts 3 3 87 character conversion 64 Charge 62 charge conjugation 20 charginos 79 chirality projector 21 57 ChiralityProjector 73 CKM 75 89 90 Classes 20 classes model 25 59 closure 56 ColumnsXRows 35 combinatorial factor 47 compatibility 51 87 complement 33 ComposedChar 63 composite label 63 Conjugate 58 ConjugateCoupling 65 conjugation 67 conservation of quantum numbers 13 60 62 conventions FeynArts 1 51 counter terms 11 65 66 order of 11 Coupling 69 coupling antisymmetric 56 classes 65 closure 56 conjugated 67 constant 55 generic 55 kinematical structure 21 quartic gauge boson 55 65 structure 47 tagging 69 tree level 65 triple gauge boson 56 vector 55 65 CouplingVector 71 CreateCTTopologies 11 CreateFeynAmp 43 CreateTopologies 10 INDEX CreateVFTopologies 12 CT 17 CTOrder 11 CW 75 Cycles 63 dCKM1 76 dCW1 76 debugging 69 define 19 deleting insertions 32 diagram complement 33 DiagramComplement 33 DiagramDel
13. GraphID Topology 1 Generic 1 Integral q1 im RelativeCF 32 1 qi Mass S Gen3 FeynAmpDenominator 1 p1 q1 Mass S Gen4 2 p1 2 qt Lor1 pi 2 q1 Lor2 ep V 1 pi Lori ep IV 1 ki Lor2 GLO Mom 1 Mom 2 K11 31 Gg Mom 1 Mom 2 KI1 3 Pit Mass S Gen3 Mass S Gen4 G L Mom 1 Mom 2 K11 3 Gg Mom 1 Mom 2 KI1 3 RelativeCF gt Insertions Classes MW MW I EL I EL 2 46 The CreateFeynAmp output is by default displayed on the screen in a special human readable format Its full internal representation is shown here 6 CREATING THE ANALYTIC EXPRESSIONS In 19 InputForm Out 19 InputForm FeynAmpList Model gt SM GenericModel gt Lorentz InsertionLevel gt Classes Restrictions gt ExcludeParticles gt F V U ExcludeFieldPoints gt LastSelections gt Process gt V 1 FourMomentum Incoming 1 O gt V 1 FourMomentum Outgoing 1 O FIL FeynAmp GraphID Topology 1 Generic 1 Integral FourMomentum Internal 11 1 32 RelativeCF FeynAmpDenominator PropagatorDenominator FourMomentum Internal 1 Mass S Index Generic 3 PropagatorDenominator FourMomentum Incoming 1 FourMomentum Internal 1 Mass S Index Generic 4 FourVector FourMomentum Incoming 1 2 FourMomentum Internal 1 Index Lorentz 1 FourVector FourMoment
14. OF A NEW MODEL Final Touch Rules Analogous to the M LastGenericRules in the generic model the classes model file al lows to define optional M LastModelRules These rules are applied as the last operation before CreateFeynAmp returns its result To avoid endless loops it must be kept in mind that the M LastModelRules are applied with replace repeatedly until expression no longer changes SM mod currently does not define any M LastModelRules 7 3 Add on Model Files Add on model files can be used with both generic and classes models An add on model file is different in that it does not define overwrite all model file ingredients but modifies them only As an example consider that one wants to introduce a resummed Yukawa coupling This is most simply achieved by substituting the quark mass in the numerator of the coupling by a resummed quark mass which could be achieved with the following add one model file ResumCoup c C _ _F _ F _ _S rhs_ c rhs Mass F t_ g_ gt MfResummed t gl ResumCoup other other 2 _ M CouplingMatrices ResumCoup M CouplingMatrices One can similarly build up a basic model file by loading an existing model file explicitly and then modifying the ingredients as above This has the advantage that the user has to specify only the resulting model file s name not the combination of basic and add on model file The above add on model file could thus be
15. PRIMITIVES IN FEYNARTS STY E 4 Labels Labels are usually associated with propagators but can in principle be set anywhere They are positioned with a pair of coordinates and an alignment given in the usual TEX manner i e a code of up to two letters for vertical and horizontal alignment t top empty center b bottom amp 1 left empty center r right e g t or rb The alignment makes it possible to change the label s text in particular its width without having to reposition the coordinates FALabel x y align text 87 F Incompatible Changes in Version 3 3 From Version 3 2 on the topology shapes are accessed in a slightly different naming scheme This eliminates some problems with large directories in some operating sys tems Shapes from FeynArts 3 1 can be converted as follows Replace the ShapeData directory that comes with the current FeynArts with your old 3 1 ShapeData directory Then start Mathematica and type lt lt Convert31to32 m For example cd FeynArts n m rm fr ShapeData cp rp FeynArts 3 1 ShapeData math Convert31to32 m Acknowledgements The graphics routines of FeynArts 3 have been developed on a Visiting Scholar grant of Wolfram Research Inc and I am grateful in particular to Michael Malak and Lou D Andria for sharing their knowledge of Mathematica with me Thanks also to the Ph D students at Karlsruhe especially Christian Schappacher for relentless beta
16. Sect 3 4 it eliminates illegal ones by checking whether the quantum numbers of the fields joining at the vertex are conserved To this end it calls the function ViolatesQ for every possi ble 1PI vertex function ViolatesQ receives as arguments the quantum numbers of the involved fields times 1 for antiparticles and must return True if the vertex violates the conservation of those quantum numbers In the most common case of charge like additive quantum numbers the condition is that the sum of all quantum numbers does not vanish at the vertex which can be coded simply as ViolatesQ q__ Plus q 0 However this is not true for all kinds of quantum numbers R parity for example which is why the definition of ViolatesQ is given in the model file Note the logic all vertices for which ViolatesQ does not explicitly return True are ac cepted It is no error if there is no definition for ViolatesQ since that simply means that all 1PI vertex functions are allowed Classes Attributes The attributes of each class is defined by an equation in the list M ClassesDescription M ClassesDescription name of the list of classes descriptors class a gt s descriptor for a class whose attribute a is s Consider the up type quark class in SM mod 7 2 The Classes Model 61 F 3 SelfConjugate gt False Indices gt Index Generation Mass gt MQU QuantumNumbers gt 2 3 Charge Matrix
17. a function called when resolving a vertex fails where info contains various information on that coupling VertexMonitor info a function invoked whenever a vertex is resolved The typical procedure is to define either as e g Print Problems concerning the origin of parts of the amplitude can be narrowed down by tagging the couplings of a model file This is effected with SetOptions InitializeModel TagCouplings gt True before model initialization of course Each coupling will be tagged by Coupling 7 then where i is that coupling s position in M CouplingMatrices 70 7 DEFINITION OF A NEW MODEL For easier debugging of add on model files FeynArts can list the couplings which have changed or were added deleted Once can either wrap the model to be debugged in ModelDebug in the model specification or set the global variable ModelDebug ModelDebug mod report changes when initializing add on model file mod variable default value ModelDebug False whether changes introduced by add on model files will be reported as a model is initialized ModelDebugForm Short 4 5 amp the output form for debugging output when model debugging is enabled The global variable ModelDebug determines whether changes introduced by add on model files will be reported as a model is initialized It can be set to True in which case debugging output will be generated for all add on model files or to the name or list
18. adjacencies the vertices may have For example Adjacencies gt 4 generates only topologies with 4 vertices In renormalizable quan tum field theories 3 and 4 are the only possible adjacencies CTOrder specifies at which counter term order topologies are generated Note that CreateTopologies creates counter term topologies for exactly the given counter term order 3 3 Creating Counter term Topologies Beyond one loop one generally needs counter terms of more than one order e g in a two loop calculation the second order counter terms on tree topologies as well as the first order counter terms on one loop topologies are needed This more com prehensive task is handled by CreateCTTopologies Its options are the same as for CreateTopologies except that CTOrder is ignored CreateCTTopologies l i gt o create all counter term topologies up to order with i incoming and o outgoing legs CreateCTTopologies e create all counter term topologies up to order l with e external legs Once again use CreateTopologies to generate topologies with loops and counter term order cto use CreateCTTopologies to generate counter term topologies for calculations of order I Specifically CreateCTTopologies i gt 0 creates all counter term topolo gies needed for the topologies created by CreateTopologies l i gt o 12 3 CREATING THE TOPOLOGIES Create all 2 loop In 4 CreateCTTopologies 2 1 gt 1 counter term t
19. is in fact derived from SM mod At present vertices containing quantum fields other than fermions do not possess counter terms but this is sufficient for one loop calculations To work with SMbgf mod one needs to specify Lorentzbgf gen as the generic model which is a slightly generalized version of Lorentz gen The following table lists the background fields and their quantum field counterparts All other fields are the same as in SM mod background field quantum field V 10 V 1 y V 20 Z v 2 Z V 30 W v 3 Ww S 10 S 1 H S 20 s 2 G s 30 Gr s 3 eT sv 20 1 G Z SV 213 39 2 sv 30 G W sv 3 G W 3Must be enabled with SVMixing True 79 C The Minimal Supersymmetric Standard Model The file MSSMQCD mod defines the complete electroweak and strong MSSM whereas MSSM mod contains only the electroweak subset defined as everything except the gluon its ghost and the gluino HaS02 The four sfermion couplings appear in MSSM mod even though they have both electroweak and strong parts Both model files follow the conventions of Ha85 Gu86 HHG90 These conventions differ from the ones in SM mod by the sign of the SU 2 covariant derivative Counter terms are currently not included in the MSSM model files The symbols used for the MSSM parameters are listed in the following table The Standard Model parameters have the same names as in SM mod e g MW and are omitted from th
20. loop starting topologies Theta Eight Bicycle two loop starting topologies Three 1 8 irreducible three loop starting topologies ThreeRed 1 7 reducible three loop starting topologies CT cto n counter term starting topologies currently defined for loop number and counter term order L cto 0 2 0 3 1 1 1 2 and 2 1 StartTop cto the starting topologies for loop number and counter term order cto SymmetryFactor t find the combinatorial factor of the starting topology t 18 3 CREATING THE TOPOLOGIES You can also draw the In 10 Paint StartTop 1 2 starting topologies The cross marks a first order 0 0 and the circled cross a second order counter term Ti T2 T3 T4 T5 T6 T7 T8 T9 Up to three loops all starting topologies including the counter terms are supplied with FeynArts If you need others you must enter them yourself Edit Topology m and locate the definition of StartTop There more starting topologies can be appended Since entering starting topologies is not an everyday job some restrictions have been imposed that enable FeynArts to work with much faster algorithms 1 There is an important distinction between positive and negative vertex identifiers the v in Vertex e v Vertices with negative identifiers are so called permutable vertices They are used for weeding out topologically equivalent topologies The algorithm is roughly the following T
21. model files that come with FeynArts are located in the Models subdirectory of the FeynArts tree The major model files Standard Model MSSM and Two Higgs doublet Model are described in Appendices B C and D respectively A classes or generic model file can also be initialized explicitly with InitializeModel This can be useful e g when writing and debugging model files InitializeModel initialize just the generic model file InitializeModel modname initialize the generic model file and the classes model file modname mod 26 4 INSERTING FIELDS INTO TOPOLOGIES option default value GenericModel Lorentz generic model to use Reinitialize True whether to initialize the model even if it is the current one TagCouplings False whether to add a tag to each coupling see Sect 7 4 on debugging ModelEdit Null code that will be executed directly after loading the model GenericModel specifies the model file containing the generic propagators and couplings the extension gen is always added to the file name The same conventions for the model name as in the Model option of InsertFields apply here Reinitialize specifies whether the model file is initialized when it is already the current model file ModelEdit provides a way to apply small changes to the model file much as in the add on model files described above It provides code that is executed just after loading the model file but before any initialization
22. of names of the add on model file s to be debugged Alternately debugging is turned on for all models wrapped in ModelDebug in the model specification 7 5 Model file Generation FeynArts includes an add on package called ModelMaker which can generate the Feyn man rules for the classes model file from the Lagrangian While entering the Lagrangian requires an effort comparable to typing in the Feynman rules directly it is often nicer to have the Lagrangian available rather than only the Feynman rules For example in situations where particles mix to form mass eigenstates the Lagrangian can be entered in terms of the gauge eigenstates which is usually much simpler It is straightforward in Mathematica to replace each gauge eigenstate with the appropriate linear combination of mass eigenstates and then derive the Feynman rules from the resulting expression Generating a model file with ModelMaker requires two things e A template model file must be made which contains everything except the Feyn man rules i e the definition of M CouplingMatrices see Sect 7 2 ModelMaker reads the template model file extension mmod inserts the M CouplingMatrices definition and writes out the mod classes model file 7 5 Model file Generation 71 e The Lagrangian must be entered in a format where the fields are specially marked and the conventions of the generic model file are used for the kinematic quantities Assuming that the template model fil
23. possible to select specific levels out of this hierarchy with PickLevel see Sect 6 4 TopologyList info Topology s props gt Insertions Generic FeynmanGraph s Generic 1 Field 1 gt F gt Insertions Classes FeynmanGraph s Classes 1 Field 1 gt F 1 gt Insertions Particles FeynmanGraph sp Particles 1 Field 1 gt F 1 1 34 4 INSERTING FIELDS INTO TOPOLOGIES FeynmanGraph s Particles 2 Field 1 gt F 1 2 more particles insertions J FeynmanGraphls Classes 2 Field 1 gt F 2 gt more classes insertions 1 FeynmanGraph s Generic 2 Field 1 gt V gt more generic insertions 1 more topologies 35 5 Drawing Feynman Graphs The drawing routine Paint has already been used several times to illustrate the exam ples in this manual Paint t paint the bare or inserted topologies t Paint accepts either a TopologyList or a Topology as argument Note however that InsertFields adds an information field to TopologyList which is used by Paint Thus if you want to paint a single inserted topology it is better to paint a topology list with one element viz Take toplist 1 than a single topology viz toplist 1 in order to preserve that information field option default value PaintLevel InsertionLevel level specification see end of Sect
24. respected as usual The conjugated coupling the part of the Lagrangian usually abbreviated as h c in textbooks can either be entered directly or by using CC instead of C in the definition of the coupling e g in the Standard Model one could define both the 7dW and the duW vertex or define either one using CC However when using the CC method one has to specify how the coupling vector of the conjugated coupling is derived from the original one This is done by defining the function ConjugateCoupling ConjugateCoupling is applied to all elements of the coupling vector and is in general different from a plain Conjugate for instance the i from the exponent of the path integral i e i f dx must not be conjugated The conjugation procedure can depend on the kinematic structure of the vertex e g couplings involving a 9 in configuration space usually get an additional minus which comes from the Fourier transformation Using the available field informa tion i e the f in ConjugateCoupling f c the kinematic vector can be obtained with KinematicVector ToGeneric f qu A drawback is that ConjugateCouplingis a rather dumb function which has to be taught how to conjugate every symbol so it is probably less work to enter the conjugated cou plings directly if only a few are affected Giving no definition for ConjugateCoupling is not an error the amplitudes will then simply contain the symbol Con jugateCoupling 68 7 DEFINITION
25. takes place One example use could be the replacement of the ordinary Z boson mass by a complex one which would include the width of the Z boson This could be done with SetOptions InitializeModel ModelEdit gt M ClassesDescription M ClassesDescription MZ gt MZc Note that the ModelEdit option uses gt RuleDelayed rather than gt Rule otherwise the code would be executed immediately For the description of the model file contents such as M ClassesDescription see Sect 7 4 4 Imposing Restrictions It is often necessary to restrict the number of diagrams generated by InsertFields This can be done in several ways ExcludeParticles gt fields excludes insertions containing fields Patterns are per mitted in fields e g ExcludeParticles gt F _ 2 3 excludes all second and third generation fermion fields Note that excluding a field at a particular level automati cally excludes derived fields at lower levels including their antiparticles Excluding e g the classes field F 1 will also exclude F 1 F 1 1 F 1 1 Furthermore 4 4 Imposing Restrictions 27 ExcludeParticles has no effect on the external particles After all it would be rather pointless to specify certain external fields and exclude them in the same line ExcludeFieldPoints gt couplings excludes insertions containing couplings A cou pling of counter term order cto is specified as FieldPoint cto fields Patt
26. tions and for this purpose augments the symbols like s i and mom by patterns It is 54 7 DEFINITION OF A NEW MODEL important to realize that CreateFeynAmp appends the momentum and kinematic indices to the field while resolving the propagators and couplings These additional elements do not appear in the final output but must be taken into account in the definition of AnalyticalPropagator aR TOME aN Plays eee Ve eel maximal internal pattern layout of field f with classes and particles indices summarized in i momentum mom and kinematic indices 14 on the left and v on the right side of the propagator For instance a vector field V 1 might intermediately be extended by CreateFeynAmp to V 1 FourMomentum Internal 1 Index Lorentz 1 gt Index Lorentz 5 The vector boson propagator is defined accordingly AnalyticalPropagator External s V i mom 1i2 PolarizationVector V i mom 1i2 AnalyticalPropagator Internal s V i mom lit gt 1i2 I PropagatorDenominator mom Mass VLi MetricTensor lil 1i2 1 GaugeXi V il FourVector mom lii FourVector mom 1i2 PropagatorDenominator mom Sqrt GaugeXi V i l Mass V illl The s stands for a possible prefactor a sign for antiparticles or 2 for mixing fields and is transformed to the pattern s in the final definition i summarizes the class index and all possible particles indices final pattern i__ and mom stand
27. v 2 gt F 4 3 F 4 3 l loading generic model file Models Lorentz gen gt GenericMixing is OFF generic model Lorentz initialized loading classes model file Models SM mod 46 particles incl antiparticles in 16 classes CounterTerms are ON 108 counter terms of order 1 gt gt gt 88 vertices gt gt 1 counter terms of order 2 classes model SM initialized inserting at level s Generic Classes gt Top 1 gt Top 2 gt Top 3 gt Top 4 in total 6 Generic 14 Classes insertions O Generic O Classes insertions O Generic O Classes insertions O Generic O Classes insertions 6 Generic 14 Classes insertions 24 4 INSERTING FIELDS INTO TOPOLOGIES Paint works also with In 13 Paint ColumnsXRows gt 4 inserted topologies PaintLevel gt Classes 209 6 db b b b b b b ui Hes i I I H 1G iG anne b Z Z Z Z b b I u I G N b b b b T1 C1 N1 T1 C2 N2 T1 C3 N3 T1 C1 N4 b b b b G G b b av P 4 T b anne ui Y Z Ze viel ZU Z Z H G b b b b b b T1 C2N5 T1 C3 N6 T1 C1N7 T1 C2 N8 b b b b ui Epis G Z ae LU Ww b ui rn b Z Z Z VES ui Z W H b b b b T1 C3 N9 T1 C1 N10 T1 C2 N11 T1 C1 N12 b b w W ui ui Z 7 zZ G w b b T1 C2 N13 T1 C1 N14 InsertFields accepts the following options option default value InsertionLevel Classes level specification see end of Sect 4 1 GenericModel Lorentz generic model to use Model S
28. 3 Vr 0 F 2 no Generation F 2 1 e ME massive leptons E 2 121 m MM F 2 3 T ML F 3 no Generation F 3 1 o u MU up type quarks Colour E 3 12 0 c MC F 3 19 0 t MT F 4 no Generation F 4 1 o d MD down type quarks Colour F 4 2 0 s MS F 4 3 0 b MB vi yes V 1 Y 0 v 2 yes V 2 Z MZ V 3 no V 3 W MW t S 1 yes S 1 H MH S 2 yes S 2 G MGO S 3 no S 3 G MGp U 1 no U 1 Uy 0 U 2 no U 2 uz MZ U 3 no U 3 u MW U 4 no U 4 Ur MW SV 2 mixing field yes sv 2 G Z MZ SV 3 mixing field no SV 3 G W MW Commented out by default in SM mod Must be enabled with SVMixing True B 1 The QCD Extension 77 SM mod defines the following Restrictions NoGenerationi NoGeneration2 NoGeneration3 NoElectronHCoupling NoLightFHCoupling NoQuarkMixing QEDOnly exclude generation 1 fermions Ve e u d exclude generation 2 fermions v U c s exclude generation 3 fermions vr T t b exclude all couplings involving electrons and a Higgs ee H e e Go e v G exclude all couplings between light fermions all fermions except the top and Higgs fields AH fifi xv d ujG jJ t uj X t exclude all couplings where off diagonal elements of the quark mixing matrix appear djuj W djujG i j Note that the diagonal elements CKM 7 i are nevertheless present exclude all particles except the massive fermions the photon and the phot
29. 600060 20 20 20 20 Diagram 3 Diagram 4 Inside the feynartspicture the macro FADiagram dtitle advances to the next diagram which has the title dtitle The size of dtitle can be changed by redefining FADiagramLabelSize with one of the usual KIEX font size specifiers e g def FADiagramLabelSize scriptsize The default size is small E 2 Propagators All propagators are circular arcs in the FeynArts style This includes conceptually the straight line as the infinite radius limit Propagators furthermore come in two variants E 3 Vertices 85 tadpole propagators where the initial and final vertex coincide and ordinary prop agators with distinct initial and final vertex This distinction is necessary because the information that has to be stored is different for the two cases The arguments of the FAProp macro and their geometrical meaning are shown below for both variants tx ty fer fy gt fx fy D FAProp fy fy Cfx fy cx c ig Hay FAProp fx fy ty ty Kk g a The latter two arguments g and a respectively determine line and arrow style g Straight a 0 g ScalarDash a 1 E g GhostDash 4 1 U g Sine Vey 2 See E 3 Vertices Vertices mark the points where propagators join Each propagator has a counter term order associated with it OO x amp amp FAVert x y fo o e 3 2 1 0 1 2 3 gt e 86 E GRAPHICS
30. ATEX the feynarts sty style file is required it is located in the FeynArts directory One particular feature of the TeX output is that it can be touched up quite easily This is useful for publications e g when grouping together diagrams To start with the generated IATEX code doesn t look too scary for instance a single diagram might be drawn by FADiagram T1 C1 N2 FAProp 0 15 6 10 0 Sine 0 FALabel 2 48771 11 7893 tr Z FAProp 0 5 6 10 0 Sine 0 FALabel 3 51229 6 78926 t1 Z FAProp 20 15 14 10 0 Sine 0 FALabel 16 4877 13 2107 br Z FAProp 20 5 14 10 0 Sine 0 FALabel 17 5123 8 21074 b1 Z FAProp 6 10 14 10 0 ScalarDash 0 38 5 DRAWING FEYNMAN GRAPHS FALabel 10 9 18 t H FAVert 6 10 0 FAVert 14 10 0 With this representation it is pretty straightforward to move diagrams around as they always start with FADiagram Incidentally the PostScript files generated by FeynArts have a very similar markup The feynarts sty introduces the following new IATEX commands begin feynartspicture end feynartspicture delimit a sheet of diagrams FADiagram t advance to the next diagram which has title t FAProp f t c g a draw a propagator from point f to point t with curvature c using graphical representation g and arrow A FAVert p cto draw a vertex of counter term order cto at po
31. Generationn 77 81 NoGenerationn 82 NoLightFHCoupling 77 81 82 NonCommutative 54 noncommuting object 42 54 NoQuarkMixing 77 NoSUSYParticles 81 NoUnfold 59 Numbering 35 INDEX numbering for use with DiagramDelete Extract 32 for use with Discard 35 OK 40 Outgoing 9 Paint 35 PaintLevel 35 Particles 20 patterns 53 66 PentagonCTsOnly 16 PentagonsOnly 16 permutable 18 photon line 63 PickLevel 41 48 placeholders 12 Poincar group 52 polarization vector 47 PolarizationVector 54 58 73 PostScript 83 PreFactor 43 prefactor 47 Propagator 9 propagator 9 generic 53 type of 9 vector boson 54 PropagatorArrow 61 PropagatorDenominator 42 53 PropagatorLabel 61 PropagatorType 61 QCD 77 QED gen 52 QEDOnly 77 INDEX quantum numbers 13 60 62 QuantumNumbers 61 quarks 76 Reinitialize 26 RelativeCF 47 renormalization constants 76 replacement rules 47 49 Restrictions 24 27 59 77 Rev 21 REVERT 40 roadmap 8 RParity 62 S 20 S2A 79 82 S2B 79 82 SA 79 82 SAB 79 82 SB 79 82 SBA 79 82 scalar field 20 ScalarDash 63 screen messages 7 SelfConjugate 61 73 SelfEnergies 14 SelfEnergiesOnly 16 SelfEnergyCTs 14 SelfEnergyCTsOnly 16 Setup m 7 sfermions 79 Shape 38 shapes 36 saving 40 SheetHeader 35 Show 37 Sine 63 95 SM mod 75 SMbgf mod 78 SMc mod 77 SMQCD mod 77 specifying momenta 51 Standa
32. Hagen Eck Sepp Kueblbeck and Thomas Hahn last revised 32 Dec 15 In 2 CreateTopologies 1 1 gt 1 Out 2 TopologyList Topology 2 Propagator Incoming Vertex 1 1 Vertex 4 3 Propagator Outgoing Vertex 1 2 Vertex 4 3 Propagator Loop 1 Vertex 4 3 Vertex 4 3 Topology 2 Propagator Incoming Vertex 1 1 Vertex 3 3 Propagator Outgoing Vertex 1 2 Vertex 3 3 Propagator Internal Vertex 3 3 Vertex 3 4 Propagator Loop 1 Vertex 3 4 Vertex 31 41 1 Topology 2 Propagator Incoming Vertex 1 1 Vertex 3 3 Propagator Outgoing Vertex 1 2 Vertex 3 4 Propagator Loop 1 Vertex 3 3 Vertex 3 4 Propagator Loop 1 Vertex 3 3 Vertex 3 4 In 3 Paint C gt T1 T2 T3 3 3 Creating Counter term Topologies 11 CreateTopologies uses a recursive algorithm that generates topologies with n legs from topologies with n 1 legs K 90 The recursion iterates down to zero external legs where it is stopped by a pre defined set of starting topologies Several options influence the behaviour of CreateTopologies option default value Adjacencies 13 4 allowed adjacencies of the vertices CTOrder 0 counter term order of the topologies ExcludeTopologies 1 list of filters for excluding topologies StartingTopologies All list to starting topologies to use Adjacencies gives the allowed
33. M classes model to use ExcludeFieldPoints couplings to exclude ExcludeParticles fields to exclude Restrictions restrictions for diagram generation LastSelections field patterns which must or must not appear in the final output GenericModel specifies the model file containing the generic propagators and couplings the extension gen is always added to the file name Mode1 specifies the classes model 4 3 Model Files 20 file containing the classes definitions and couplings extension mod The model name is a string but may be given as a symbol if this is possible e g a name like 2HD cannot be represented by a symbol because it starts with a digit 4 3 Model Files FeynArts distinguishes basic model files and add on or partial model files Most com monly only the basic model files are used as in InsertFields Model gt MSSM In contrast the add on model files do not supply a complete model They just modify the particle descriptions and coupling tables of another model file and can therefore only be used on top of a basic model file It produces an error to load an add on model file without a basic one An add on model file might for example change a particular coupling modify the mass of a particle etc One case given in Hal06 is the enhancement resummation of the H b b coupling in the MSSM Such an add on model file is used like InsertFields Model gt MSSM EnhHbb The
34. Names for the indices and momenta such as g1 g2 and mu in the example above can be chosen freely since they will not appear in the final Feynman rule internal indices are eliminated via the 72 7 DEFINITION OF A NEW MODEL functional derivative and kinematic quantities are replaced by generic objects for easier matching The kinematic quantities appearing in the coupling must be entered as in the generic model file Two exceptions are allowed a lone DiracMatrix u is automatically split into y c Yuw_ and mom u may be used as a short hand for FourVector mom u As ModelMaker must know the kinematical quantities it might encounter some of the conventions of Lorentz gen have been hard coded into ModelMaker ModelMaker there fore may not work with generic model files other than Lorentz gen The template model file is the same as the complete model file except that it does not contain the Feynman rules ie it declares IndexRange M ClassesDescription and M LastModelRules but not M CouplingMatrices see Sect 7 2 The definition of M CouplingMatrices which in the complete model file contains the Feynman rules in stead has the form M CouplingMatrices M CouplingMatrices When the template model file is processed by WriteModelFile the text enclosed be tween and gt is scanned as Mathematica input and replaced by the resulting output thereby inserting the Feynman rules The output is written to a file with th
35. Standard Model including all counter terms of first order in the conventions of De93 coupling constants and masses EL CW SW MW MZ MH MGO MGp MLE g ME MM ML MQU g MU MC MT MQD g MD MS MB CKM g g GaugeXi A W Z electron charge Thomson limit cosine and sine of the weak mixing angle W Z Higgs masses Goldstone masses mass of lepton of generation g e u T masses mass of up type quark of generation g up charm top quark masses mass of down type quark of generation g down strange bottom quark masses quark mixing matrix photon W Z gauge parameters dZe1 dSW1 dCW1 dZH1 dMHsq1 dZW1 dMWsq1 dMZsqi dZZZ1 dZZA1 dZAZ1 dZAA1 dZG01 dZGp1 dMf1 t g aZfL1 t g g dZ R41 t g 9 dCKM1 g 9 one loop renormalization constants electromagnetic charge RC mixing angle sine cosine RC Higgs field and mass RC W field and mass RC Z mass RC Z and photon field RCs Goldstone field RCs fermion mass RCs left and right handed fermion field RCs quark mixing matrix RCs 76 B THE ELECTROWEAK STANDARD MODEL The type of a fermion is 1 for neutrinos 2 for massive leptons 3 for up type quarks and 4 for down type quarks The particle content of SM mod is summarized in the following table class self conj indices members mass F 1 no Generation F 1 1 Ye 0 neutrinos F 1 2 Yu 0 F 1
36. TraceFactor gt 3 PropagatorLabel gt ComposedChar u Index Generation PropagatorType gt Straight PropagatorArrow gt Forward The first two fields in the descriptor are necessary SelfConjugate and Indices The others are optional and have default values option default value SelfConjugate required how the field behaves under charge conjugation Indices required list of indices the class carries MixingPartners field field for mixing propagators left and right partners e g S 3 V 3 Mass Mass field symbol to denote the mass of the class members Mixture field linear combination the field is composed of QuantumNumbers ts quantum numbers the field carries MatrixTraceFactor 1 for fermions factor with which closed fermion loops will be multiplied InsertOnly All types of propagators on which the field may be inserted PropagatorLabel field label for the propagator PropagatorType Straight line type for the propagator PropagatorArrow None Forward Backward or None An ordinary field also has default mixing partners though trivial ones e g V 2 has mixing partners V 2 V 2 62 7 DEFINITION OF A NEW MODEL Mass gt cmass specifies a symbol for the mass of the class It is possible to distinguish masses for different types of propagators External Internal and Loop by adding the propagator type as in Mass Loop Mloop Several such mass definitions can
37. True Indices gt 7 2 The Classes Model 65 Mass gt MAZ MixingPartners gt V 1 V 2 PropagatorLabel gt gamma Z PropagatorType gt Sine PropagatorArrow gt None Classes Coupling Vectors By far the most diligent task is to enter the actual coupling vectors of the model in particular if one endeavours to enter also the counter terms The coupling vectors are contained in a list M CouplingMatrices Each coupling vector is defined by an equation The convention is that all fields at a coupling are incoming as in the generic model file M CouplingMatrices name ofthe list of coupling vectors GU f2 ttc cl ee tc cen Mees A R definition of the coupling for the classes level fields f f2 where the lower index of the c s refers to the counter term order and the upper index to the component of the kinematic vector CC fi f2 the same except that simultaneously the charge conjugated vertex is defined ConjugateCoupling f c defines how the charge conjugate of the expression c is derived for the coupling of the fields f The name M CouplingMatrices is justified because the equations assign to each cou pling a list of lists or matrix a coupling constant for each counter term order for each component of the kinematic vector The coupling vector must of course have as many entries as in the kinematic vector it corresponds to Recall the example of the qua
38. a generic field type with a class number e g F 1 The class specifies which further indices if any the class members possess and the range of these indices Particles fields are then class members with definite indices e g F 1 1 2 if the class F 1 has two indices For classes fields without further indices classes and particles fields are the same Antiparticles charge conjugate fields are denoted by a minus sign in front of the field e g if F 2 1 is the electron F 2 1 is the positron Apart from simple fields FeynArts can also handle mixing fields A mixing field prop agates like any other field but has no couplings of its own Instead it couples like one simple field on the left side and like another simple field on the right side e g if the scalar vector mixing field Mix S V 3 has the mixing partners S 3 V 3 it cou ples as if it were an S 3 on the left and a V 3 on the right Whereas simple fields can have at most two states the field and its antifield a mixing field can occur in four states the mixing field its antifield the reversed mixing field and its antifield Unlike for simple fields the antifield of Mix S V 3 is 2Mix S V 3 and the antifield of Mix S V 3 is 2Mix S V 3 Self conjugate fields cannot have negative coefficients i e in that case the two possibilities in the bottom row are absent 41 The Three Level Fields Concept 21 Mix S V 3 2 Mix S V 3 M
39. a single analytic expression amp the am plitude at that level In case more than one level is present amp is the generic level amplitude which can be transformed into the individual amplitudes of the deeper lev els by applying the replacement rules ru with PickLevel see Sect 6 4 FeynArts always produces purely symbolic expressions for Feynman amplitudes e g PropagatorDenominator p m is used to denote 1 p m simply to prevent Math ematica from performing any automatic simplification on it In fact FeynArts does not attempt to simplify anything in the amplitude e g evaluate traces because this would limit its applicability to a certain class of theories Apart from the relatively few symbols FeynArts uses by itself in amplitudes all symbols defined by the model can of course appear in the amplitude 42 6 CREATING THE ANALYTIC EXPRESSIONS Mass f mass of field f GaugeXi f gauge parameter of field f G 1 cto f kin symmetric 1 or antisymmetric 1 generic coupling of counter term order cto involving the fields f and corresponding to the kinematical object kin of the coupling vector see Sect 7 for details FourMomentum t n nth momentum of type t Incoming Outgoing or Internal PropagatorDenominator p m symbolic expression for 1 p m where p is the momentum and m the mass of the propagator FeynAmpDenominator collection of PropagatorDenominators belonging to loops MatrixTrace 01 02
40. auge parameters untouched It is possible to add information about the underlying graph to the amplitude by giv ing a GraphInfoFunction This function receives the same three arguments as the test functions of DiagramSelect and DiagramGrouping the list of field substitution rules for the graph of the form FeynmanGraph Field 1 gt fi Field 2 gt f2 the topol ogy belonging to the graph and the head of the surrounding topology list See Sect 4 7 for auxiliary functions to use with these arguments Create one diagram from In 16 t11 CreateTopologies 1 1 gt 1 3 the photon self energy In 17 2 AA InsertFields t11 V 1 gt V 1 ExcludeParticles gt F V U I Excluding 3 Generic 20 Classes and 36 Particles fields Y PEOR focos inserting at level s Generic Classes ru gt Top 1 1 Generic 1 Classes insertions in total 1 Generic 1 Classes insertions Restoring 3 Generic 20 Classes and 36 Particles fields 6 2 CreateFeynAmp Generate the amplitudes 45 In 18 CreateFeynAmp AA creating amplitudes at level s Generic Classes gt Top 1 1 Generic 1 Classes amplitudes in total 1 Generic 1 Classes amplitudes Out 18 FeynAmpList Model gt SM GenericModel gt Lorentz InsertionLevel gt Classes Restrictions gt ExcludeParticles gt F V U ExcludeFieldPoints gt LastSelections gt Process gt V 1 pi 0 gt 4V 1 ki OJ FeynAmp
41. be FourMomentum Internal Q1 There are two rules to observe Do not use an integer as a name to avoid conflict with the automatically generated momenta and do not change the direction of the propagator If youneed the momentum flowing opposite to the direction of the propagator give it a minus sign e g Q1 While CreateFeynAmp takes care to eliminate automatically generated momenta before user specified ones when working through momentum conservation one can prohibit the elimination of any momentum with the option MomentumConservation gt False 6 8 Compatibility with FeynArts 1 Many programs based on FeynArts still use the simpler conventions of version 1 For them to work with FeynArts 2 2 a function is provided to translate the CreateFeynAmp output into the old format ToFAiConventions expr convert expr into the conventions of FeynArts 1 Note that ToFA1Conventions only renames some symbols The results may thus not be 100 FeynArts 1 compatible since certain kinds of expressions could not be generated with FeynArts 1 at all That after all is why the conventions had to be changed In particular amplitudes containing more than one level will probably not be very useful in programs made for processing FeynArts 1 output 52 7 DEFINITION OF A NEW MODEL 7 Definition of a New Model 7 4 The Generic Model In the generic model file the following items are defined e the kinematic indices of the fields KinematicInd
42. be given where Mass alone acts as the default If omit ted the expression Mass field propagatortype remains in the output of CreateFeynAmp The masses of particles level fields will appear as cmass indices unless specific sym bols are declared with the function TheMass SM mod for instance declares the following symbols for the individual up type quark masses which would otherwise be MQU 1 MQU 2 and MQU 3 TheMass F 3 1 MU TheMass F 3 2 MC TheMass F 3 3 MT Mixture gives the field s composition as a linear combination of fields Only mixtures with fields of the same type are allowed e g scalars can only be composed of other scalars An antiparticle has to be wrapped in Field so that the minus sign is not taken as part of the coefficient in the linear combination Using this method one can e g specify the couplings in terms of interaction eigenstates and linear combine them to mass eigenstates Unless the original interaction fields are themselves added to M ClassesDescription they disappear completely from the model after initialization An application where it useful to keep original fields including all vertices mixing orig inal and rotated fields is a rotation of higher order with the rotated fields running on tree level propagators and the unrotated ones in the loops QuantumNumbers specifies which quantum numbers the field carries In principle any Mathematica expression can be us
43. e It is inserted by the replacement rules for the deeper levels FeynAmpDenominator collects all denominators belonging to a loop i e those whose mo mentum is not fixed by conservation of the external momenta If the structure of the propagators is more intricate e g gauge boson propagators in a general gauge a sum of several FeynAmpDenominators can appear in one amplitude p1 2 qi Lori and p1 2 q1 Lor2 are the kinematical objects four vectors in this case that come from the generic coupling structure the first from the left and the second from the right vertex Lor1 and Lor2 are Lorentz indices The external momenta are assigned such that the total momentum flows in through the incoming particles and out through the outgoing particles i e I p Y pi 0 not I pi 0 in ou a ep V 1 pi Lori and ep V 1 k1 Lori are the polarization vectors of the in coming and outgoing photon the latter of which is complex conjugated GO Mom 1 Mom 2 K11 3 is the generic coupling constant of the scalar scalar vector coupling associated with the kinematical object Mom 1 Mom 2 K11 31 a four vector The SSV coupling happens to be proportional to only one kinematical ob ject The superscript 0 refers to counter term order 0 The two G s one for the left and one for the right vertex look identical but are not the same internally because in the human readable output format some indices are suppr
44. e first m stripped from the extension MODEL mmod MODEL mod so the template model file is not overwritten or deleted 73 A The Lorentz Formalism Lorentz gen contains the definitions of the generic propagators and couplings for a relativistic field theory with scalar spinor and vector fields transforming according to the usual representation of the Poincar group in Minkowski space It introduces the following symbols MetricTensor u v the metric tensor guv DiracSpinor p m i the spinor of a Dirac fermion with momentum p mass m and particles indices i MajoranaSpinor p m i the spinor of a Majorana fermion with momentum p mass m and particles indices i PolarizationVector v p u the polarization vector of the vector boson v with Lorentz index u associated with momentum p DiracMatrix u the Dirac matrix y DiracSlash p Yup ChiralityProjector 1 the chirality projectors w 1 ys 2 FourVector p u thefour vector py The four spinor states can actually all be represented by just one symbol depending on its position in a FermionChain and type of momentum FermionChain spinor FourMomentum Incoming n m i v FermionChain spinor FourMomentum Outgoing 7n m i u FermionChain spinor FourMomentum Incoming 7 m i u FermionChain spinor FourMomentum Outgoing 7n m i v where spinor is either DiracSpinor or MajoranaSpinor Majorana spin
45. e is called MODEL mmod the procedure to generate MODEL mod is the following lt lt Models ModelMaker Lagrangian put definition of Lagrangian here frules FeynmanRules Lagrangian WriteModelFile CouplingVector frules MODEL mmod The ModelMaker functions used above are FeynmanRules derives the Feynman rules from the Lagrangian E CouplingVector r extracts the coupling vector of each Feynman rule in r according to the corresponding kinematic vector in the currently initialized generic model file WriteModelFile r t mmod splices the Feynman rules generated with FeynmanRules and CouplingVector into the template model file t mmod writing the results to t mod The format for entering the Lagrangian is as follows Fields are marked by the function Field whose first entry is the field name in the form in which it appears in the classes model file e g F 3 g Only internal indices i e particle level indices may appear in the field name itself If it becomes necessary to refer to the momentum or kinematic indices of a field in the kinematic part of the coupling two optional entries can be added to Fieldin the form Field f mom u For example the term in the Lagrangian corresponding to the electron positron photon vertex could be defined by EL Field F 2 gi DiracMatrix mu Field F 2 g2 Field V 1 mu Note the dot product which joins the noncommuting objects
46. e particles insertions MQU nib gt vuv more classes insertions Js more amplitudes 50 6 CREATING THE ANALYTIC EXPRESSIONS 6 6 On Fermion Chains and Signs One of the most asked for details of amplitude generation is how the signs of fermionic diagrams are determined FeynArts generally uses the methods from De92 That is basically the fermion flow is fixed at generic level and so called flip rules are applied if it turns out that in the classes or particles insertion the flow is reversed Open fermion chains the ones terminating on external fermions are ordered opposite to the fermion flow i e the leftmost field in the chain is an antifermion F For amplitudes containing anticommuting fields fermions or ghosts the following signs are added e a minus for every closed fermion or ghost loop and e the signature of the permutation that brings the ordinal numbers of the external fermions as connected by the fermion chains into descending order descending because the fermion chains are ordered opposite to the fermion flow Consider these diagrams Following the fermion chains opposite to the arrows the external particles are con nected as 2 1 3 4 in the left diagram and as 3 1 2 4 in the right diagram The number of permutations needed to bring 2134 into descending order 4321 is odd but even for 3124 Hence the s channel diagram gets an additional minus sign Note that the signature of the permu
47. e seems to be a mismatch for the scalar vector mixing field SV Mix S V it is declared to have a Lorentz index even though the scalar half has no index at all However CreateFeynAmp knows about this special case and discards the index on the scalar side when resolving a coupling Generic Propagators A generic propagator defines the kinematical structure of a propagator In the generic model file the generic propagators are contained in the list M GenericPropagators In side this list each propagator is declared by an equation for AnalyticalPropagator M GenericPropagators name of the list of generic propagators AnalyticalPropagator t f expr definition of the propagator of type t Internal or External for field f The nomenclature perhaps needs explanation An External propagator is what is traditionally called external wave function e g a spinor in the case of a fermion field Internal propagators are the propagators in the usual sense connecting two internal field points The simplest case are scalar fields which have no external wave function and propaga tor i Si k S k 5 L DISD um The corresponding statements in Lorentz gen are AnalyticalPropagator External s S i mom 1 AnalyticalPropagator Internal s S i mom I PropagatorDenominator mom Sqrt GaugeXilS i Mass S i When initializing the generic model FeynArts transforms these equations into defini
48. ection or its counter term i e the diagram contains a self energy insertion S 1 S 3 on an external leg If the diagram contains no wave function correction the list is empty This filter is usually used to eliminate wave function corrections with identical external legs i e remove corrections of the type a a but keep a b This can be done with a construction like DiagramSelect diags UnsameQ WFCorrectionFields 4 4 amp Vertices top return the vertices contained in the topology top FieldPoints fop return the field points contained in the topology top FieldPoints rul top substitute the insertion rules rul into the bare topology top then proceed as above Vertices returns the vertices contained in a topology not counting the external legs even though they are internally represented as Vertex 1 n 47 Auxiliary Functions 31 FieldPoints returns the field content for each vertex of a topology i e a list of objects of the form FieldPoint cto fields where cto is the counter term order The following functions further facilitate matching of fields and field points FieldMatchQ f f True if the field f matches the pattern f FieldMemberQ flist f True if an element of flist matches the field pattern f FieldPointMatchQ p fp True if the field point fp matches the pattern fp FieldPointMemberQ fplist fp True if an element of fplist matches the field point pattern fp
49. ed as a quantum number though the more common choices are multiples of a symbol e g 2 3 Charge An important point is that the quan tum number of the antiparticle is minus the quantum number of the particle This is just what one wants in the case of additive quantum numbers like charge since indeed if the up quark has 2 3 Charge the anti up quark has 2 3 Charge However it is for instance not true that if a particle has R parity 1 its antiparticle has R parity 1 In that case one has to choose RParity 1 instead of RParity and supply an appropriate ViolatesQ function 7 2 The Classes Model 63 MatrixTraceFactor is a way to compensate for traces over indices which are not ex plicitly specified For example in the electroweak Standard Model without QCD the interactions are colour diagonal and hence the quarks need no colour index Still when computing the trace over a quark loop colour accounts for a factor 3 which is provided by MatrixTraceFactor gt 3 in the quark class descriptors CreateFeynAmp will com plain if fields with different MatrixTraceFactors appear in the same loop InsertOnly specifies on which propagator types the field may be inserted e g for a background field InsertOnly gt Internal External is needed since it is not al lowed in loops The line of a propagator is in general part of a circle This includes straight lines which can be regarded as the degenerate case of infinite radius The line type
50. ept for the SFF SSS and SSSS couplings Model constants are given in the following table Again the Standard Model parameters are omitted since they are the same as in SM mod MhO MHH MAO MGO the neutral Higgs boson masses MHp MGp the charged Higgs boson masses CB SB TB cos fj sin B tan CA SA cosa sina C2A S2A C2B S2B cos2a sin 2a cos 2 sin 2 CAB SAB CBA SBA cos a f sin B cos B a sin B Yuki Yuk2 Yuk3 Yukawa coupling parameters see below One distinguishes two types of THDM Type I where all fermions couple only to the second Higgs doublet H5 and Type IL where up type fermions couple to H whereas down type fermions couple to H4 In the FeynArts implementation of the THDM this choice is parameterized by three Yukawa coupling parameters which take the values Yuki Yuk2 Yuk3 cos sin Type I THD ype M Sin ano cot B sin cos Type II THDM t ype cos f cos Smp The following Restrictions are defined in THDM mod NoGenerationn exclude generation n fermions n 1 2 3 NoElectronHCoupling exclude all couplings involving electrons and any Higgs field NoLightFHCoupling exclude all couplings between light fermions all fermions except the top and any Higgs field 83 E Graphics Primitives in feynarts sty The FeynArts style is included in a IATEX 2e document with usepackage feynarts It makes three graphics prim
51. er in which the coupling is defined in the model file simple permutation 7 1 The Generic Model 57 of the fields does not suffice but in addition an operation called flipping of fermionic couplings has to be performed on the kinematic vector The algorithm behind this was developed in De92 and its main advantage is that it can deal with fermion number violating vertices such as appear in supersymmetric models The idea is that instead of ordering the fermion propagators according to the given fermion flow one chooses a direction for the fermion lines Obviously one cannot define the fermion flow properly as soon as fermion number violating couplings are present which is why the standard method breaks down in that case If later it turns out that the chosen direction is opposite to the actual fermion flow one has to apply a so called flipping rule The flipping rule is nothing but a charge conjugation i e Race where T is some product of Dirac matrices and C is the charge conjugation matrix The flipping rules for all possible objects that can appear in F have been worked out in De92 and the only necessary substitution is YuW E Yu where w 1 y3 2 are the chirality projectors This is more than just flipping the sign of the whole expression which could be effected with G 1 since also the other chirality projector has to be taken A case where y w occurs is the FFV coupling in Lorentz gen AnalyticalCoup
52. er is defined ExcludeTopologies Triangles FreeQ ToTree Centre 3 amp Here is The filter function must be defined as a pure function foo amp since it will be grouped together with other filter functions by CreateTopologies The function will be passed a topology as its argument This is important to know for structural operations e g the following filter excludes topologies with 4 vertices on external legs ExcludeTopologies V4onExt FreeQ Cases Propagator External __ Vertex 4 amp 3 6 Selecting Starting Topologies 17 ExcludeTopologies name func amp defines the filter name ToTree top returns the topology top with each loop shrunk to a point named Centre adj n where adj is the adjacency of loop n Centre adj n represents the remains of loop n with adjacency adj after it has been shrunk to a point by ToTree 3 6 Selecting Starting Topologies Using the option StartingTopologies to select particular starting topologies may sig nificantly speed up CreateTopologies especially for higher loop topologies For exam ple it is much faster to start without the reducible three loop starting topologies than to exclude the reducible topologies afterwards The default setting A11 evaluates to all starting topologies for the given loop number and counter term order It is allowed to use patterns e g StartingTopologies gt Three _ selects only the irreducible three
53. erns are al lowed e g FieldPoint _ F 2 1 F 1 1 S 3 Here too the exclusion of a field point entails the exclusion of more specific ones derived from it Restrictions gt exclp exclfp is a convenient way of specifying abbreviations for ExcludeParticles and ExcludeFieldPoints statements For example SM mod defines NoElectronHCoupling ExcludeFieldPoints gt 1 FieldPoint 0 F 2 121 F 1 131 S 31 FieldPoint 0 F 2 111 FI2 1 S 11 FieldPoint 0 F 2 131 F 2 11 S 2 3 as a short hand to exclude the electron Higgs couplings In InsertFields it is used as Restrictions NoElectronHCoupling LastSelections is an alternative method to specify field patterns that must or must not appear in the insertions For example LastSelections gt S F _ 271 forces that the insertions must contain a scalar field and must not contain fermions of the second generation Like ExcludeParticles LastSelections does not affect the external fields The individual criteria are combined with the logical and i e an insertion is only permitted if it fulfills all criteria simultaneously While LastSelections may seem more general it works by first generating all dia grams and afterwards selecting those that match the given criteria hence the name In contrast ExcludeParticles and ExcludeFieldPoints work by eliminating particles and couplings before starting the insertion process and can thus be
54. ersion 2 2 The well designed conceptual framework was kept but the actual code was reprogrammed almost entirely to make it more effi cient and a user friendly topology editor was added The current version 3 features a completely new rendering engine for PostScript and IATEX together with full support of the Mathematica Frontend s graphical capabilities It is also no longer dependent on the X platform for topology editing The main features of FeynArts are e The generation of diagrams is possible at three levels generic fields classes of fields or specific particles e The model information is contained in two special files The generic model file de fines the representation of the kinematical quantities like spinors or vector fields The classes model file sets up the particle content and specifies the actual couplings Since users can create their own model files the applicability of FeynArts is virtu ally unlimited within perturbative quantum field theory As a generic model the Lorentz formalism Lorentz gen and as classes model the electroweak Standard Model in several variations SM mod SMQCD mod SMbgf mod the Minimal Super symmetric Standard Model MSSM mod MSSMQCD mod and the Two Higgs Doublet Model THDM mod are supplied e In addition to ordinary diagrams FeynArts can generate counter term diagrams and diagrams with placeholders for one particle irreducible vertex functions skeleton diagrams e FeynArts em
55. es sages or 1 summary messages only The default is 2 all messages 2 ROADMAP OF FEYNARTS Roadmap of FeynArts Inputs Find all distinct ways of connect ing incoming and outgoing lines CreateTopologies Process ext fields of loops Model adjacencies fields couplings Topologies Determine all possible combinations of fields InsertFields Draw the results Paint Diagrams Apply the Feynman rules CreateFeynAmp further Amplitudes processing 3 Creating the Topologies 3 1 Topological Objects For the purposes of FeynArts a topology is a set of lines propagators connecting a set of points vertices Furthermore topologies in FeynArts are restricted to be connected topologies where every part of the topology is connected to the rest with at least one propagator A vertex is characterized by two numbers its adjacency and its counter term order The adjacency is the number of propagators that run into the vertex Vertex adj n vertex with adjacency adj counter term order 0 and number n Vertex adj cto n vertex with adjacency adj counter term order cto and number n A propagator connects two vertices possibly carrying a field Propagator t from to propagator of type t running from from to to Propagator t from to field propagator of t
56. es fields in C f f2 must be in the same generic order as the correspond ing AnalyticalCoupling defined in the generic model file This means that if an analyt ical coupling is defined in the order FFS the classes coupling may not be given in the order SFF Fields with particles indices must include a pattern for every index in the coupling definition For example the v ej W coupling is defined as GT F 1 j1 F 2 j2 VEST I EL Sqrt 2 SW IndexDelta j1 j2 1 dZe1 dSW1 SW dZW1 2 Conjugate dZfL1 1 j1 j1 2 dZfL1 2 ji j1 2 0 0 7 2 The Classes Model 67 ji and j2 stand for one index each they are transformed to the patterns j1_ and j2_ in ternally during model initialization For each index a class possesses a separate pattern has to be specified Index diagonal terms are multiplied with IndexDelta j j2 If the whole coupling is proportional to IndexDelta as in the last example InsertFields will use this infor mation to fix indices as far as possible already in the insertion process In contrast InsertFields cannot constrain other indices if only part of the coupling is diagonal such as in the first order counter term of the V v Z coupling OL F 1 j1 F 1 j2 V 2 I EL gL 1 IndexDelta ji j2 IndexDelta j1 j2 gL 1 dZZZ1 2 dgL 1 gL 1 aZzfL icc 1 j1 j2l 0 O The full index diagonality of the tree level coupling in this case is of course
57. essed The replacement rules the fourth element of a FeynAmp particularize the unspecified quantities in the generic expression For the single classes insertion of the example in the last section the following substitutions are made The full form of the generic ex pressions is written out below in small print to show that all substitutions are distinct Mass S Gen3 MW full Mass S Index Generic 311 48 6 CREATING THE ANALYTIC EXPRESSIONS Mass S Gen4 gt MW full Mass S Index Generic 4 G amp S Mom 1 Mom 21 K11 3 gt I EL full G 1 0 S Index Generic 3 S Index Generic 4 V 1 FourVector Mom 1 Mom 2 K11 3 G amp S Mom 1 Mom 2 K11 3 gt I EL full G 1 0 S Index Generic 3 S Index Generic 4 V 1 FourVector Mom 1 Mom 2 KI1 3 RelativeCF gt 2 6 4 Picking Levels The replacement rules are not of a form which can directly be applied with expr rules Instead PickLevel has to be used to select a particular level PickLevel lev lamp pick level lev from the amplitude amp PickLevel levs toplist pick level s levs from the list of inserted topologies toplist For the case of an amplitude only one level may be specified PickLevel replaces the generic amplitude and the rules with the amplitude at the selected level i e the FeynAmp contains only three elements after PickLevel and adds a running number of the form Numbe
58. ete 32 DiagramExtract 32 DiagramGrouping 29 DiagramMap 33 DiagramSelect 28 DiracMatrix 72 73 DiracSlash 73 DiracSpinor 55 58 73 directory contents 7 Display 37 DisplayFunction 35 dMf1 76 dMHsq1 76 dMWsq1 76 dMZsq1 76 drawing diagrams 35 dSW1 76 DumpGenericModel 69 DumpModel 69 dvips 83 dZAA1 76 dZAZ1 76 INDEX dZe1 76 dZfL1 76 dZfR1 76 dZGO1 76 dZGp1 76 dZH1 76 dZW1 76 dZZA1 76 dZZZ1 76 Eight 17 EL 75 electroweak Standard Model 75 ExcludeFieldPoints 24 27 ExcludeParticles 24 27 ExcludeTopologies 11 13 Export 37 External 9 42 53 63 external legs 10 wave functions 43 extracting insertions 32 F 20 FADiagram 38 FALabel 38 FAProp 38 FAVert 38 fermion 20 amplitude 50 flipping rules 56 flow 50 Majorana 73 self conjugate 73 signs 50 type 76 FermionChain 42 91 FermionRouting 31 FeynAmp 41 FeynAmpCases 32 FeynAmpDenominator 42 47 FeynAmpList 41 FeynArts 1 51 feynarts sty 37 83 FeynArtsGraphics 37 feynartspicture 38 83 Feynman gauge 44 Feynman rules 70 FeynmanRules 71 Field 28 field 20 background 78 derived 27 exclusion 27 extended structure 54 55 insertion 20 intermediate 54 level 20 lower level 27 mixing 20 FieldMatchQ 31 FieldMemberQ 31 FieldNumbers 28 35 FieldPointMatchQ 31 FieldPointMember Q 31 FieldPoints 30 filter 13 function 16 final touch rules
59. even if there are more topologies in line to be edited The REVERT button reverts to the initial layout of the topology Choosing GRID POSITION allows dragging of the squares only onto grid points Con versely ANY POSITION allows a square to be dragged to an arbitrary position The topologies need to be shaped only once The shapes changed during a Mathematica session are saved in the directory specified by the variable ShapeDataDir which by default is the ShapeData directory in the FeynArts home directory The actual topology editor is a program written in Java TopologyEditor java that communicates with Mathematica through the J Link program See Sect 1 for setting up Java and J Link 41 6 Creating the Analytic Expressions 6 1 Representation of Feynman Amplitudes The basic object for a Feynman amplitude is FeynAmp Corresponding to the three inser tion levels there are also three amplitude levels for a single level FeynAmp n mom amp Feynman amplitude with name n integration momenta mom and analytic expression amp for multiple levels FeynAmp n mom amp ru Feynman amplitude with name n integration momenta mom and generic analytic expression amp including replacement rules ru to obtain the other levels FeynAmpList info amps list of FeynAmps GraphID id identifier of an amplitude Integrallg g2 representation of the integration momenta A FeynAmp containing a single level has only
60. ey happen to be identical w_ Qa 1 G G Hiit Hee 2 sin Ow Mw jjmy 22 4 INSERTING FIELDS INTO TOPOLOGIES Finally at particles level the generation indices i and j are resolved so for instance the Heu coupling is zero because of the j The FeynArts functions InsertFields CreateFeynAmp and Paint can operate at dif ferent field levels The level specification is analogous to the usual Mathematica level specifications with e g 3 vs 3 If for example InsertFields is used with the op tion InsertionLevel gt Particles the result will contain insertions only at par ticles level In contrast InsertionLevel gt Particles no braces will produce in sertions down to particles level More than one level may be specified for instance Generic Particles skips the classes level 4 2 InsertFields The computer algebraic generation of Feynman diagrams corresponds to the distribu tion of fields over topologies in such a way that the resulting diagrams contain only couplings allowed by the model The function for this is InsertFields InsertFields f 1 i2 gt 101 05 insert fields into the TopologyList t where the incoming fields are 17 i2 and the outgoing fields are 01 05 Create irreducible 1 2 In 11 2 t12 CreateTopologies 1 1 gt 2 topologies ExcludeTopologies gt Internal 42 InsertFields Insert Z bb 23 In 12 InsertFields t12
61. he topologies are sorted into some canonical order and then compared This simple method however fails whenever a graph has a symmetry In that case the indices of the symmetrical vertices have to be permuted to give all topologically equivalent versions It is this power set of each topology that is actually compared If you re not sure which vertices should be permutables make them all permutables This will be slower but safer 3 6 2 3 4 5 Selecting Starting Topologies 19 For the correct functioning of the ExcludeTopologies filters it is essential that the propagators on an irreducible conglomerate of loops have the same loop number the n in Loop 1 no matter how many loops there actually are For example the two loop starting topology Theta has only Loop 1 propagators Vertex identifiers must always be unique e g having both a Vertex 3 1 and a Vertex 4 1 within the same topology is forbidden To give the starting topology a name which can be used with StartingTopologies define it with define name Topologylsl To make name accessible out side of Topology m it must either be declared in FeynArts m via name usage or else live in the Global context i e define Global name If there is only one starting topology or one always wants to use all of the starting topologies the define can be omitted To determine the inverse symmetry factor the s in Topology s 1 enter the
62. ices the generic propagators M GenericPropagators the generic coupling vectors M GenericCouplings the fermion flipping rules M FlippingRules the truncation rules M TruncationRules e some optional final touch rules M LastGenericRules Apart from these required definitions the generic model file is a perfectly ordinary Mathematica input file and may contain any number of additional statements For ex ample Lorentz gen includes Format directives with which the objects it introduces are displayed in a nicer form in the output Probably the best way to learn how to set up a generic model file is by going through one of the provided generic models Lorentz gen or QED gen Kinematic Indices A kinematic index is an index transported along a propagator Due to the special prop erty of renormalizable theories to possess vertices which join at most two fermion fields spinor indices are not necessary because FeynArts can construct the fermion chains itself cf Sect 6 6 KinematicIndices f i definition of the kinematic indices for field f For example in the usual representation of the Poincar group only the vector bosons carry a Lorentz index Lorentz gen thus defines the following kinematic indices 7 1 The Generic Model 53 KinematicIndices F KinematicIndices V Lorentz KinematicIndices S KinematicIndices SV Lorentz KinematicIndices U f Ther
63. ich saves it in a file When running in the standalone Mathematica Kernel Paint automatically animates the screen graphics with the sheets of output acting as the frames of the animation This is to inhibit the cluttering of the screen with too many windows The speed at which the frames are displayed by default is rather high more appropriate for a movie than for viewing sheets of graphics The frame rate can be adjusted in the Animation Controls menu which also allows to view single frames Under Unix the initial frame rate can be set with the X resource Motifps frameTime e g in Xdefaults Motifps frameTime 1000 5 1 Things to do with the Paint Output 37 5 1 Things to do with the Paint Output The output of Paint is a FeynArtsGraphics object whose arguments represent the sheets of the output Each of these sheets is represented by a matrix which determines the layout of the diagrams i e FeynArtsGraphics title g11 US 22 19515 2229 H sheet 1 x44 These FeynArtsGraphics objects can be rendered with the usual Mathematica rendering functions Export Display and Show Export and Display allow the image size to be changed with the ImageSize option whose default value is 72 6 7 6 x 7 inches for a FeynArtsGraphics object In addition to the standard formats understood by Mathematica the rendering functions accept two more output formats PS and TeX To run output produced with the TeX format through I
64. in structions that come with J Link for installation on the various platforms FeynArts comes in a compressed tar archive FeynArts n m tar gz which merely needs to be unpacked no further installation is necessary gunzip c FeynArts n m tar gz tar xvf Unpacking the archive creates a subdirectory FeynArts n m which contains FeynArts m the main program Setup m initialization file FeynArts directory containing the FeynArts code Models directory containing the model files ShapeData directory containing the shapes of topologies README additional information about this release HISTORY general blurb about the evolution of FeynArts Convert2to3 m conversion program for FeynArts 2 GraphInfo files Convert3to31 m conversion program for FeynArts 3 GraphInfo files Convert31to32 m conversion program for FeynArts 3 1 Graphlnfo files Permanent changes of parameters options etc should be placed in Setup m Patching the FeynArts code directly is not recommended since it is inherently unportable Several FeynArts functions have options that take a list of objects Except in the case of level specifications see Sect 4 1 the list may be omitted if it contains only one element e g ExcludeTopologies gt Tadpoles instead of ExcludeTopologies gt Tadpoles Some FeynArts functions write messages to the screen to indicate their progress These messages can be partially or completely suppressed by setting FAVerbose to 0 no m
65. in general have to be permuted to fit the definition in the model file For this to work it is necessary that the kinematic vector closes under permutations of fields of one type In the example above this means that for any permutation of u v p o all four fields are of the same type here one must end up with another or the same element of the kinematic vector which is indeed the case There is one exception from this closure requirement if a permutation of the fields yields either an element of the kinematic vector or the negative of an element of the kinematic vector the coupling is said to be antisymmetric and specified as G 1 An example of this is the triple gauge boson coupling in Lorentz gen AnalyticalCoupling si V i momi li1t s2 V j mom2 1i2 s3 V k mom3 1i3 G 1 s1 V i s2 V j s3 V X i MetricTensor li1 1i2 FourVector mom2 momi 1i3 MetricTensor 1i2 113 FourVector mom3 mom2 lii MetricTensor 1i3 lii FourVector momi mom3 1i2 The kinematic vector has in this case only one element C Vi pi V pa Vo pa Guvjv Suv P2 P1 o 8vo Ps P2 u Sou P1 P3 v Because the kinematic vector contains the differences of the momenta a permutation of p1 u pa v p3 p will produce either the kinematic vector itself or its negative Fermion Flipping Rules Matters are more complicated for fermionic couplings When two fermion fields are not in the ord
66. int p FALabel p align text write text at point p with alignment align Do not be confused by the multitude of parameters The only command likely to be edited is FALabel Suffice it to say that each diagram is drawn on a 20 x 20 grid with the origin in the lower left corner and the positive axes extending to the right and upward and that the alignment is specified in the standard TEX manner i e with combinations of t b 1 and r for top bottom left and right alignment respectively The exact details of all ATEX commands defined by feynarts sty can be found in Appendix E 5 2 Shaping Topologies A topology does not by itself provide information on how to draw it While the human eye is usually quite skilled in figuring out a nice shape for a topology at least for not too complex ones this is a tremendously difficult task for the computer even in sim ple cases Indeed the autoshaping routine is the longest single function in the whole program Whenever Paint encounters a topology for which is does not yet know the shape it first does its best to autoshape the topology and then calls the topology editor unless AutoEdit gt False is set so that the user can refine the shape This shaping function 52 Shaping Topologies 39 can also be invoked directly Shape t invoke the topology editor to shape the topology or topology list t The topology editor pops up a window which looks like Cancel Abo
67. is given with PropagatorType which can take the values Straight Sine photon like Cycles gluon like ScalarDash or GhostDash The PropagatorLabel may contain a letter e g Z a ATEX symbol e g gamma or a composite character e g ComposedChar u The double backslash is really a single character for Mathematica since the first is needed to escape the second Inside the quotes only one character or I TEX symbol may appear Multiple characters must be put in a list e g gamma Z ComposedChar t sub sup bar composite label t with subscript sub superscript sup and accent bar the arguments sub sup and bar are optional but their position is significant e g ComposedChar t Null sup corresponds to a label with a superscript only For example ComposedChar phi i bar will produce the label o IATEX representation bar phi_i As for Mass it is possible to define labels specific to a particular propator type External Internal Loop e g PropagatorLabel Loop gt X Indices can be used in a label in the form Index t where t is the type of the index like in ComposedChar nu Index Generation The rendering of indices in the graphical output is subject to the function Appearance if that is defined for the particular type of index By default the number of the index is displayed 64 7 DEFINITION OF A NEW MODEL Appearance Index t n_Integer
68. is table MhO MHH MAO MGO the neutral Higgs boson masses MHp MGp the charged Higgs boson masses CB SB TB cos fj sin f tan fB CA SA cosa sina C2A S2A C2B S2B cos2a sin2a cos 2 sin 20 CAB SAB CBA SBA cos f sin a B cos a sin B MUE the Higgs doublet mixing parameter u MG1 the gluino mass MNeu n the neutralino masses ZNeu n n the neutralino mixing matrix MCha c the chargino masses UCha c c VCha c c the chargino mixing matrices MSf s f 9 thesfermion masses USf f 9 1 s s thesfermion mixing matrix Af t 9 the scalar soft breaking A parameters The sfermion type is denoted by t and is defined similar to the fermion type The primed indices appearing in the mixing matrices enumerate the gauge eigenstates the unprimed ones mass eigenstates The following indices are used in MSSM mod and 80 MSSMQCD mod C THE MINIMAL SUPERSYMMETRIC STANDARD MODEL g Index Generation 1 3 0 Index Colour u Index Gluon ev d De Oy s Index Sfermion elu n Index Neutralino 1 4 c Index Chargino b asa The particle content of MSSM mod and MSSMQCD mod is given in the next table The gluon its ghost and the gluino which are defined only in the latter are written in grey sc is short for self conjugate lepton
69. itives available with which Feynman diagrams can be drawn e FAProp draws a propagator e FAVert draws a vertex e FALabel places a label In addition it provides formatting geometry directives e begin end feynartspicture delineates a sheet of Feynman diagrams e FADiagram advances to the next diagram Since feynarts sty emits direct PostScript primitives the interpretation of which is non standard across PostScript renderers it is guaranteed to work only with dvips E 1 Geometry A single Feynman diagram is always drawn ona 20 x 20 canvas Several such canvasses are combined into a rectangular sheet which can optionally carry a title A sheet of Feynman diagrams is enclosed in a feynartspicture environment in TEX begintfeynartspicture s Sy Nx Mny end feynartspicture This sheet has a size of sx x sy in units of IATgX s unitlength with room for nx x ny Feynman diagrams ny need not be an integer and the extra space implied by the fractional part is allocated at the top for the sheet label Note that it is not possible to distort the aspect ratio of a Feynman diagram If the ratio 1 n is chosen different from the ratio s sy the sheet will fit the smaller dimension exactly and be centered in the larger dimension 84 E GRAPHICS PRIMITIVES IN FEYNARTS STY The overall geometry of a feynartspicture sheet is thus as follows shown here for a 2 x 2 sheet Title Diagram 1 Diagram 2 ce e oo 000
70. ix S vils 2 Mix S V 3 Mixing of generic fields is a special case because it requires different handling internally already at generic level It is disabled by default GenericMixing True enable mixing of generic fields must be set before the model is initialized At generic level there are no antiparticles and reversed mixing propagators are denoted by Rev 21 821 For compatibility with old versions FeynArts replaces SV Mix S V F generic fermion field F n fermion class n e g F 2 is the class of leptons in SM mod FIn 4j member of fermion class n e g F 2 1 is the electron in SM mod field charge conjugate of field 2 mixingfield mixingfield with left and right partner reversed Using different levels of fields is a natural concept in perturbative field theory The kinematical structure of a coupling is determined at the generic level Consider the scalar fermion fermion coupling in the Lorentz formalism Its kinematical structure is u9 w Crrs Gppg W Gpps W w being the chirality projectors Gfpg and Gfps are generic coupling constants which carry two kinds of indices the fields they belong to and the kinematical object they appear with At classes level the coupling constants are resolved but not the indices E g for the H coupling in the electroweak Standard Model Higgs lepton lepton with generation indices i and j i e 1 e 2 u 3 1 th
71. jugate the outgoing po larization vectors M LastGenericRules outgoing vector bosons throw away signs of momenta PolarizationVector p_ _ k FourMomentum Outgoing _ li_ gt Conjugate PolarizationVector p k li This rule has an interesting side aspect note that only the head PolarizationVector is wrapped in Conjugate This is because the M LastGenericRules are applied with replace repeatedly until the expression no longer changes If the whole polarization vector including arguments were wrapped in Conjugate the rules would apply anew in every pass and lead to an endless loop 7 2 The Classes Model 59 7 2 The Classes Model All particles of a model are arranged in classes A class is conceptually similar but not identical to a multiplet the fields in one class need not belong to a representation of some group For single particle model definitions each particle lives in its own class The classes model defines the actual classes of fields in a particular model It should not define or rely on any kinematic objects so that it can be used with different generic model files In the classes model file the following items are defined e the index range of all possible particles indices IndexRange e a function that detects if a vertex violates quantum number conservation ViolatesQ the classes and their attributes M ClassesDescription e the classes coupling vectors M CouplingMatrices
72. ling si F i momi s2 F j mom2 s3 V k mom3 1i3 G 1 si F i s2 F j s3 V Xk NonCommutative DiracMatrix 1i3 ChiralityProjector 1 NonCommutative DiracMatrix 1i3 ChiralityProjector 1 FeynArts expects the generic model file to define M FlippingRules These rules are applied when FeynArts needs to match a fermionic coupling but finds only the other fermion permutation in the model file For example in Lorentz gen the following flip rules are defined M FlippingRules NonCommutative dm DiracMatrix _DiracSlash ChiralityProjector pm gt NonCommutative dm ChiralityProjector pml 58 7 DEFINITION OF A NEW MODEL Truncation Rules Truncation rules are needed for removing external wave functions when the option Truncated gt True is used in CreateFeynAmp The rules depend of course on the sym bols that are used for the external wave functions and must hence be defined in the generic model file In Lorentz gen there are spinors and polarization vectors to be dealt with since scalar particles have no external wave functions The truncation rules are then M TruncationRules _PolarizationVector gt 1 _DiracSpinor gt 1 _MajoranaSpinor gt 1 Final Touch Rules The last operation CreateFeynAmp performs on a generic amplitude is to apply the model dependent M LastGenericRules These rules are optional and have no partic ular purpose In Lorentz gen for example they complex con
73. o remove external wave functions GaugeRules _GaugeXi gt 1 rules to enforce a particular choice of MGO gt MZ gauge MGp gt MW MomentumConservation True whether to enforce momentum conservation at each vertex GraphInfoFunction 1 amp a function for adding graph information to the amplitude The default for AmplitudeLevel is to use the same level the topologies were inserted at PreFactor specifies the overall factor every diagram is multiplied with It may contain the symbol LoopNumber which is substituted by the actual number of loops of a topology If Truncated gt True is set CreateFeynAmp applies the truncation rules defined in the model file M TruncationRules to the final result These rules should discard external wave functions typically spinors or polarization vectors To be able to produce amplitudes in an arbitrary gauge the model file must of course contain the full gauge dependence of the propagators and couplings For example Lorentz gen and SM mod contain the gauge dependent propagators V S U and cou plings SUU for an arbitrary R gauge However most people prefer to work in the Feynman gauge which is the most convenient one for calculating radiative corrections Therefore the default GaugeRules set all gauge parameters to unity On the other hand 44 6 CREATING THE ANALYTIC EXPRESSIONS if one wants to check e g gauge independence of an expression GaugeRules gt will keep the g
74. on ghost SMew mod is a companion model file for SM mod in which the quarks do not carry colour indices It is included for backward compatibility with old versions B 1 The QCD Extension The model file SMQCD mod adds the QCD Feynman rules to the electroweak part in SM mod In fact it just loads SM mod and appends the gluon and its ghost to the ap propriate definitions Following is the list of additional symbols used in SMQCD mod GS SUNT a i j SUNF a b c SUNF a b c d the strong coupling constant the generators of SU N T the SU N structure constants f c a short hand for the sum y ff The additional particles are the gluon and its ghost The gluon index is not expanded out at particles level i e FeynArts does not generate eight diagrams for every gluon 78 B THE ELECTROWEAK STANDARD MODEL class self conj indices members mass V 5 yes Gluon V 5 G 8 0 U 5 no Gluon U 5 1 Ug 0 B 2 Background field Formalism The model file SMbgf mod contains the electroweak Standard Model in the background field formalism It is based on De95 with the exception that the renormalization of the fermionic fields proceeds as in SM mod i e with separate renormalization constants for upper and lower components of the fermion doublet which in turn means that fermions do not need an external wave function renormalization The larger part of SMbgf mod
75. opologies In 5 Paint including the one loop subrenormalization 1 gt d Ti T2 T3 T4 T5 T6 3 4 Creating Topologies with generic Vertex Functions Topologies can also be created with placeholders for one particle irreducible 1PI vertex functions Such topologies are sometimes also called skeleton diagrams The vertex function placeholders are represented graphically by a grey bubble e g QU LX ee E The idea is to reduce the number of diagrams by calculating the vertex functions sep arately and inserting the final expression into tree diagrams at the proper places For example in a 2 2 process the self energy diagrams are generated once for the s channel once for the t channel and once for the u channel It would of course be much more economic to calculate the necessary 1PI two point vertex functions only once and insert them at the proper places in the s t and u channel tree diagrams The prob lem may not be particularly acute in this example at one loop but it can easily become significant for higher loop order or more external legs Topologies with vertex function placeholders are generated by CreateVFTopologies Note that there is always a loop order associated with the vertex function hence one has to specify a loop order also for these VF topologies 3 5 Excluding Topologies 13 CreateVFTopologies l i gt o create topologies with placeholders for 1PI vertex functions of order l wi
76. or the generic amplitude only and need not be done over and over for every insertion For example a fermionic trace needs to be calculated only for the generic amplitude from which the expressions for the individual fermion classes or particles are then obtained by applying respectively the classes or particles level replacement rules Owing to fast computers and powerful software for evaluating amplitudes at one loop level however this conceptually superior technique has not been used much so far Nevertheless for higher loop calculations it will likely become an inevitable modus operandi given the number of diagrams already a two loop calculation typically in volves To be able to simplify the amplitudes at different levels it is essential to understand how they are structured The structure is similar to that of inserted topologies i e an expression embedded in a hierarchical list of insertion rules The main difference is that the basic ingredient is not a topology but the generic amplitude Note that only amplitudes containing more than one level are structured in this way If only one level is selected in CreateFeynAmp or with PickLevel the FeynAmps contain only the analytic expression of the amplitude at the chosen level and no rules FeynAmpList info FeynAmp gname mom genericamp Mass F Index Generic 1 gt Insertions Classes MLE gt Insertions Particles MLE 1 MLE 2 mor
77. ors are used for fermions with attribute SelfConjugate gt True Polarization vectors associated with outgoing momenta are brought into the form Conjugate PolarizationVector v p u by the M LastGenericRules The symbol FourVector defined by Lorentz gen should not be confused with the Feyn Arts symbol FourMomentum The latter represents a momentum flowing along propaga tor lines and itis very likely that using it for different purposes would upset the internal routines In contrast FourVector is not modified by FeynArts 74 A THE LORENTZ FORMALISM The following generic couplings are defined in Lorentz gen Antisymmetric couplings are labelled by a subscript symmetric ones by a subscript u Suv8 po OUS Vy Vo Vo Gyyvv Zup amp vo VVVV SuoS vp C Vu k1 Vy kz Vp k3 Gyvv guv ka k o gvp ka ky gpu k k3 y _ VVV C 5 5 5 5 Gssss 1 SSSS C S S S Gsss RUM SSS C S S V Vy Gssvv Suv SSVV C S k1 S k2 Vu Gssv gt k k2 u _ SSV C S Vu Vy Gsvv Suv SVV EE Yuw C F F Vu Grev eh FFV gt QD C F E S Grrs onl FFS C U k U k2 Vu Guuv bu UUV 2u E C S U U Gsuu 1 SUU f Quv kika C Vu k Vx k2 Gvv Suv VV Kipkoy J C S k1 S k2 Gss ee SS k w_ C F k1 F k2 Grr ee FF w C S k1 Vulko Gsv 62 SV 2u n B The Electroweak Standard Model The file SM mod contains the electroweak
78. ot violate conservation of quantum numbers The quantum numbers of the fields are defined in the classes model file see Sect 7 2 3 5 Excluding Topologies The ExcludeTopologies option specifies topology exclusion filters Such a filter is a special function whose outcome when applied to a topology True or not determines whether the topology is kept or discarded Some filters are supplied with FeynArts others can be defined The pre defined filters work on topologies of any loop number 14 3 CREATING THE TOPOLOGIES Loops patt CTs patt Tadpoles TadpoleCTs SelfEnergies SelfEnergyCTs Triangles TriangleCTs Boxes BoxCTs Pentagons PentagonCTs Hexagons HexagonCTs AllBoxes AllBoxCTs WFCorrections WFCorrectionCTs WFCorrections patt WFCorrectionCTs patt Internal exclude topology if it contains loops P of adjacency patt counter terms loops 7 of adjacency 1 counter terms loops p of adjacency 2 counter terms loops P of adjacency 3 counter terms loops T of adjacency 4 counter terms loops P of adjacency 5 counter terms loops P of adjacency 6 counter terms loops P of adjacency gt 4 counter terms l self energy or tadpole ee on ext legs counter terms ditto only on ext legs matching patt propagators of type Internal i e if the topology is one particle reducible Because the generation of counter term topologies and topologies with vertex function
79. p agators in a very systematic way This is best exploited if the topologies are grouped into categories like self energies vertices or boxes For example in a 2 2 vertex correction diagram the first four propagators are those of the external particles the fifth is the propagator of the tree part and the rest are the loop propagators e g 4 6 Grouping Insertions 29 1 3 1 3 7 3 7 1 5 7 6 6 5 4 6 8 2 8 5 2 4 2 4 The important thing to note here is that the same parts of a diagram say the loop etc in general get the same propagator numbers Nevertheless the number of a certain propagator is not fixed a priori so the only reliable way to find the propagator numbers is to paint the bare topologies with the option FieldNumbers gt True see Sect 5 The test function actually receives three arguments the list of field substitution rules for the graph of the form FeynmanGraph Field 1 gt fi Field 2 gt f the topol ogy belonging to the graph and the head of the surrounding topology list See Sect 4 7 for auxiliary functions to use with DiagramSelect 4 6 Grouping Insertions The function DiagramGrouping groups insertions according to the output of a function DiagramGroupingltops foo return a list of parts of the inserted topologies tops grouped according to the output of foo The user function foo is applied to all insertions Groups are introduced for all dif ferent ret
80. particles 20 picking a 41 48 specification of 7 22 linear combination 62 LoadModel 68 Loop 9 LoopFields 30 LoopNumber 43 loops irreducible conglomerate of 9 19 number of 10 quark 63 Lorentz 53 64 Lorentz formalism 21 Lorentz gen 52 72 73 Lorentzbgf gen 78 M ClassesDescription 59 60 M CouplingMatrices 59 65 72 M FlippingRules 52 57 M GenericCouplings 52 55 M GenericPropagators 52 53 M LastGenericRules 52 58 73 M LastModelRules 59 68 M TruncationRules 43 52 58 93 MAO 79 82 MajoranaSpinor 55 58 73 Mass 53 61 mass eigenstates 62 MatrixTrace 42 MatrixTraceFactor 61 MB 75 MC 75 MCha 79 MD 75 MDQ 75 ME 75 MetricTensor 54 73 MGO 43 75 79 82 MG1 79 MGp 43 75 79 82 MH 75 MhO 79 82 MHH 79 82 MHp 79 82 Mix 20 53 mixing field 20 61 reversed 20 MixingPartners 61 Mixture 61 mixture 62 ML 75 MLE 75 MM 75 MNeu 79 Model 24 model 25 classes 25 59 debugging 69 definition 52 generation of 70 94 generic 25 52 initialization 25 MSSM 79 template file 71 two Higgs doublet 82 ModelDebug 70 ModelEdit 26 ModelMaker 70 momenta 51 MomentumConservation 43 motifps 36 MQU 62 75 MS 75 MSf 79 MSSM 79 MSSM mod 79 MSSMQCD mod 79 MT 75 MU 75 MUE 79 multiplet 59 MW 43 75 MZ 43 75 neutralinos 79 neutrinos 76 NoElectronHCoupling 27 77 81 82 No
81. ploys the so called flipping rule algorithm De92 to concatenate fermion chains This algorithm is unique in that it works also for Majorana 6 1 GETTING STARTED fermions and the fermion number violating couplings they entail e g quark squark gluino and hence allows supersymmetric models to be implemented e Restrictions of the type field X is not allowed in loops can be applied This is necessary e g for the background field formulation of a field theory e Vertices of arbitrary adjacency required for effective theories are allowed e Mixing propagators such as appear in non R gauges are supported e FeynArts produces publication quality Feynman diagrams in PostScript or IATEX in a format that allows easy customization These features have been introduced in version 2 but some parts received consider able improvements in version 3 The user interface on the other hand has through all versions suffered only minor and mostly backward compatible changes and the major functions can still be used in essentially the same way as in version 1 Installation FeynArts requires Mathematica 3 0 or above In Mathematica versions before 5 0 a Java VM and the J Link package are needed for the topology editor Both ingredients can be obtained free of charge from http www wolfram com solutions mathlink jlink J Link http java sun com j2se Java Note that many systems e g Windows have a Java VM pre installed Follow the
82. r n to the GraphID Once a particular level has been selected it is no longer possible to choose other levels even deeper ones e g it is not possible to first select classes level and subsequently particles level from a FeynAmpList With inserted topologies PickLevel works somewhat differently First it preserves the structure of the inserted topologies While in a FeynAmp the fourth element the replace ment rules is deleted an inserted topology is still of the form topology gt insertions after PickLevel Second more than one level may be picked with one exception the generic level cannot be removed because it is needed to create the generic amplitudes However the syntax differs from the usual level specification PickLevel lev t se lects only level lev from t apart from the generic level PickLevel leva levb selects levels leva and levb If CreateFeynAmp finds that amplitudes at only one level are requested either directly by AmplitudeLevel or if the topologies were inserted at only one level it automatically calls PickLevel so that the output will contain only the amplitudes at the requested level and no replacement rules 6 5 General Structure of the Amplitudes 49 6 5 General Structure of the Amplitudes The three level concept has a big advantage when simplifying the amplitudes Since the kinematical structure is already determined at the generic level certain algebraic simplifications can be performed f
83. rd Model 75 masses 75 starting topology 11 17 definition 18 name 19 StartingTopologies 11 17 StartTop 17 Straight 63 subscript 63 Sum ver 42 SUNF 77 SUNT 77 superscript 63 supersymmetry 79 SUSY 79 SV 21 SW 75 SymmetryFactor 17 19 T 20 TadpoleCTs 14 TadpoleCTsOnly 16 Tadpoles 14 TadpolesOnly 16 TagCouplings 69 TB 79 82 template model file 71 tensor field 20 THDM 82 THDM mod 82 THDMParticles 81 96 TheLabel 64 TheMass 62 Theta 17 Three 17 ThreeRed 17 ToFA1Conventions 51 Topology 10 topology 9 comparison 18 connected 9 counter term 11 editor 39 exclusion of 13 filter 13 defining own 16 function 16 inserted 33 irreducible 14 starting 11 17 definition 18 name 19 with vertex function 12 Topology m 19 TopologyEditor java 40 TopologyList 10 ToTree 17 TriangleCTs 14 TriangleCTsOnly 16 Triangles 14 TrianglesOnly 16 Truncated 43 U 20 UCAlph 64 UCGreek 64 UCha 79 INDEX USf 79 V 20 V4onExt 16 VCha 79 vector field 20 Vertex 9 vertex 9 adjacency 9 11 identifiers 19 permutable 18 vertex functions 12 VertexDebug 69 VertexFunction 13 42 VertexMonitor 69 Vertices 30 ViolatesQ 59 60 62 wave function corrections 14 WFCorrectionCTFields 30 WFCorrectionCTs 14 WFCorrectionFields 30 WFCorrections 14 WriteModelFile 71 ZNeu 79
84. ript files are actually the coupling vectors corresponding to the kinematic vectors defined in Lorentz gen MSSM mod and MSSMQCD mod currently do not contain any counter term vertices so that counter term diagrams cannot be generated automatically yet This is due to the fact that although it is in principle known how to renormalize theories with softly broken supersymmetry Ho00 this is far from trivial for the MSSM and has so far not been worked out completely As long as only SM particles appear at tree level however one can almost directly use the SM counter terms Of course the self energies from which the renormalization constants are derived now have to be calculated in the MSSM The one thing one has to observe when using the SM counter terms for an MSSM pro cess is that for historical reasons the SM and MSSM model files differ in the sign of the SU 2 covariant derivative Namely where o is in the SM and in the MSSM model files There is a simple rule for translating the two conventions replace SW by SW and add an additional minus sign for each Higgs field that appears in a coupling 82 D THE TWO HIGGS DOUBLET MODEL D The Two Higgs Doublet Model The Two Higgs Doublet Model THDM possesses a Higgs sector similar to the MSSM i e 5 physical Higgs fields but has no supersymmetry and hence no superpartners The Feynman rules are mostly the same as for the MSSM with the couplings involving superpartners omitted exc
85. rt Revert Grid position Any position mouse left button BEN drag middle button Mstraighten B default pos right button B mirror The red squares mark the vertices Click and drag a red square with the left mouse button to move the respective vertex The blue squares mark the propagators Click and drag a blue square with the left mouse button to change the respective propagator s curvature Click on it with the middle mouse button to make the propagator straight again Click on it with the right mouse button to reverse the curvature i e to make the line curved in the opposite direction The green squares mark the label positions they look like little tags attached to the center of the propagator Click and drag a green square with the left mouse button to move the respective label Click on it with the middle mouse button to put the label back to its 40 5 DRAWING FEYNMAN GRAPHS default position Click on it with the right button to flick the label to the opposite side of the propagator The buttons on the right of the window are largely self explanatory The OK button commits the changes and exits The CANCEL button discards the changes made to the current diagram and exits the editor but continues editing more diagrams if a list of topologies is being shaped The ABORT button is similar to CANCEL but aborts the editing process altogether i e it returns to the Mathematica prompt immediately
86. rtic gauge boson coupling from the last section It has a kinematic vector with three entries guvgpo gupgvo uogpv SO accordingly the coupling vector must also have three entries An actual classes level quartic gauge boson coupling then looks like 66 7 DEFINITION OF A NEW MODEL CE v 3 v 3 VELI VEL I EE 2 2 4 dZei 2 dZW1 2 dZAA1 2 CW SW dZZA1 1 2 dZe1 dZW1 dZAA1 CW SW dZZA1 1 2 dZei dZW1 dZAA1 CW SW dZZA1 This is the W W yy coupling in SM mod On the right side there is a list with three en tries corresponding to the three entries in the kinematic vector Each component is again a list where the first element is the usual counter term order 0 or tree level coupling the second the counter term order 1 coupling and so on The overall constant I EL 2 has been pulled out for clarity Mathematica automatically threads multiplications over lists e g x a b becomes x a x b It is possible to enter counter terms only for a subset of couplings The components of the coupling vector must however always be filled up to the same counter term order In the special case where the kinematic vector has only one element one level of lists may be omitted so for example CL s 2 Sf2 VI3 VI3 I EL 2 2 SW 2 1 2 dZe1 2 dSW1 SW dZWi dZchii is equivalent to c s 2 s 2 VIS V 3 I EL 2 2 SW 2 1 2 dZei 2 dSW1 SW dZWi dZchil The class
87. s for the momentum final pattern mom For an internal propagator 1i1 and 1i2 are the kinematic indices on the left and right side of the propagator respectively final patterns 1i1_ and 1i2 and an external propagator uses only one set of kinematic indices 1i1 The names of the patterns s i mom etc can of course be freely chosen but must be unique The most common errors in this respect are assignments of i or j prior to employing them as patterns Noncommuting objects appearing in the kinematic vector typically elements from the Dirac algebra must be wrapped in NonCommutative so that their order does not get destroyed by CreateFeynAmp This makes sense only for fermion and perhaps ghost 7 1 The Generic Model 55 fields since there is no well defined order for other types of field The external spinors are for example noncommutative objects AnalyticalPropagator External s F i mom NonCommutativel If SelfConjugate F i MajoranaSpinor DiracSpinor mom Mass F il Generic Couplings Analogous to a generic propagator a generic coupling defines the kinematical structure of a coupling The generic couplings are contained in the generic model file in the list M GenericCouplings Inside this list each coupling is declared by an equation for AnalyticalCoupling By convention all fields in AnalyticalCoupling are incoming M GenericCouplings name of the list of generic couplings AnalyticalCoupling f1 f2
88. s mass sleptons mass Vg F 1 8 0 Ys Sore MSf be F 2 g ME S 12 s g MSf quarks squarks Ug F 3 9 0 MQU il S 13 s g o MSf dg F 4 9 0 MOD de S 14 s g o MSf gauge bosons neutralinos charginos y sc V 1 0 X9 sc F 11 n MNeu Z sc viz MZ X F 12 c MCha W V 3 MW Higgs bosons ghosts h sc st MhO uy U 1 0 H sc S 2 MHH uz U 2 MZ A9 sc S 3 MAO ur U 3 MW G sc S 4 MGO u U 4 MW H S 5 MHp G S e MGp 81 MSSM mod and MSSMQCD mod define the following Restrictions NoGenerationi exclude generation 1 fermions Ve e u d NoGeneration2 exclude generation 2 fermions v L c s NoGeneration3 exclude generation 3 fermions vz c t b NoElectronHCoupling exclude all couplings involving electrons and any Higgs field NoLightFHCoupling exclude all couplings between light fermions all fermions except the top and any Higgs field NoSUSYParticles exclude the particles not present in the SM sfermions charginos neutralinos and the Higgs fields H9 A9 H THDMParticles exclude the particles not present in the Two Higgs Doublet Model the sfermions charginos and neutralinos The complete list of all couplings defined in MSSM mod and MSSMQCD mod is contained in the files MSSM ps gz and MSSMQCD ps gz respectively which are located in the Models directory The couplings in these PostSc
89. s typically not as robust as selecting them through other criteria This is because the numbering generally changes whenever the CreateTopologies and or InsertFields invocations are modified For example the 49 Modifying Insertions 33 diagrams corresponding to a certain set of numbers in the Standard Model and in the MSSM will in general not be the same DiagramComplement returns the complement of a list of diagrams DiagramComplement f j t1 t2 the diagrams in ta which are not in any of the t 4 9 Modifying Insertions Diagrams can also be modified by mapping a function over the diagrams with DiagramMap DiagramMap foo tops map foo over each diagram in the inserted topologies tops The user function foo is applied to all insertions It is invoked with the same three ar guments as the test function of DiagramSelect the list of field substitution rules for the graph of the form FeynmanGraph Field 1 gt fi Field 2 fo the topology belonging to the graph and the head of the surrounding topology list It must return the modified first argument the FeynmanGraph object 4 10 Structure of the Inserted Topologies If one wants to perform more advanced operations on inserted topologies it is necessary to know their structure Essentially a topology gets enclosed in a hierarchy of rules with particles insertions nested inside classes insertions nested inside generic insertions It is
90. se is taken literally as a header e g SheetHeader gt Self energy diagrams Numbering gt None omits the default numbering of the diagrams The default setting of Numbering gt Full places diagram numbers of the form T1 C8 N15 topology 1 classes insertion 8 running number 15 underneath the diagram A simple numbering which is useful in publications is Numbering gt Simple in which case just the running number is used The propagators of a bare topology are usually unlabelled With FieldNumbers gt True Paint uses the field numbers the n in Field m as labels This is useful when selecting diagrams with DiagramSelect see Sect 4 5 AutoEdit determines whether Shape is called when an unshaped topology is found This option is useful if your diagrams involve lots of unshaped topologies e g try paint ing 2 loop 2 4 diagrams and you want a quick pre view of the diagrams without going through the effort of shaping them at that moment FeynArts uses a rather sophis ticated autoshaping algorithm but one should not expect too much autoshaping offers a reasonable starting point but for publishing quality diagrams it is usually necessary to refine the shapes by hand DisplayFunction determines how the output of Paint is rendered The default is to show the graphics on screen Popular choices are DisplayFunction Identity which does not render the graphics at all or DisplayFunction Display file ps amp wh
91. significantly faster They can on the other hand only exclude but not force the presence of fields or field points 28 4 INSERTING FIELDS INTO TOPOLOGIES Here is the Z bb In 14 InsertFields t12 example again Now we Wel gt F 4 3 F 4 3 select only diagrams in InsertionLevel gt Classes which a W boson occurs LastSelections gt V 3 inserting at level s Classes gt Top 1 4 Classes insertions gt Top 2 0 Classes insertions gt Top 3 0 Classes insertions gt Top 4 0 Classes insertions in total 4 Classes insertions In 15 Paint ColumnsXRows gt 4 Z gt b b b b b b u Gu w w r Ww ui d ui ui Z Z AA Z Hi W G W b b b b T1 C1 N1 T1 C1 N2 T1 C1 N3 T1 C1 N4 45 Selecting Insertions One further way to pick a selected set of diagrams is to use the function DiagramSelect DiagramSelect d crit select the diagrams from d for which crit gives True This function works much like the usual Select of Mathematica i e it applies a test function to every diagram and returns only those for which the result is True The test function can of course be any Mathematica expression but the most common usage is something like DiagramSelect diags FreeQ Field 5 gt S amp which eliminates all diagrams with a scalar particle on the fifth propagator Such a statement would not be very useful were it not for the fact that FeynArts orders its pro
92. tation is invariant under interchange of any two whole fermion chains because each fermion chain contributes an even number two of external fermions In renormalizable theories at most two fermion fields can appear at a vertex and there fore the construction of the fermion chains is totally unambiguous This is not so in other non renormalizable theories for instance in the Fermi model FeynArts cannot correctly build the fermion chains if vertices involving more than two fermions ap pear because this information is simply not available from the Feynman rules In such 6 7 Specifying Momenta 51 a case FermionLines False must be set and the fermion fields must carry an ad ditional kinematic index e g a Dirac index with which it is afterwards possible i e FeynArts does not do this to find the right concatenation Also the above mentioned signs are not resolved in this case 6 7 Specifying Momenta CreateFeynAmp can be made to use a certain momentum on a certain propagator This is done by editing the inserted topologies and appending a fourth argument to the propa gators whose momenta one wants to fix A convenient way to do so is to assign the in serted topologies to a variable and use the Mathematica function EditDef inition var This fourth argument will become the momentum s name For example the momentum assigned to the propagator Propagator Internal Vertex 3 101 Vertex 3 100 Field 4 Q1 will
93. testing 88 REFERENCES References De93 A Denner Fortschr Phys 41 93 307 arXiv 0709 1075 De95 A Denner S Dittmaier G Weiglein Nucl Phys B440 95 95 arXiv hep ph 9410338 De92 A Denner H Eck O Hahn J K blbeck Nucl Phys B387 92 467 Eck95 H Eck Ph D thesis University of W rzburg 1995 available from http feynarts de Hal06 T Hahn J I Ilana Nucl Phys Proc Suppl 160 2006 101 hep ph 0607049 Gu86 J F Gunion H E Haber Nucl Phys B272 1986 1 HHG90 J F Gunion H E Haber G Kane S Dawson The Higgs Hunter s Guide Frontiers in Physics Vol 80 Addison Wesley 1990 Ha85 H E Haber G Kane Phys Rep 117 1985 75 HaS02 T Hahn C Schappacher Comp Phys Commun 143 2002 54 hep ph 0105349 H000 W Hollik E Kraus D St ckinger Eur Phys J C23 2002 735 arXiv hep ph 0007134 K 90 J K blbeck M B hm A Denner Comp Phys Commun 60 90 165 Index Xdefaults 36 DisplayFunction 36 ExcludeTopologies 16 FAVerbose 7 FermionLines 51 GenericMixing 21 76 ModelDebug 70 ModelDebugF orn 70 ShapeDataDir 40 FADiagram 83 FALabel 83 FAProp 83 FAVert 83 1PL 12 60 ABORT 40 accent on label 63 Adjacencies 11 adjacency 9 11 Af 79 alignment 38 A11 11 AllBoxCTs 14 AllBoxes 14 Alph 64 amplitude 41 structure of 49 AmplitudeLevel 43 48 AnalyticalCoupling 55 AnalyticalPropagator
94. th 7 incoming and o outgoing legs CreateVFTopologies l e create topologies containing generic 1PI vertex function insertions of order with e external legs The creation of VF topologies and of counter term topologies is essentially the same In fact the place where the vertex function must later be inserted is given precisely by the location of the cross on the counter term diagram of the corresponding order CreateFeynAmp translates the vertex function placeholders into generic objects of the form VertexFunction o f f which represent the 1PI vertex function Dy where o is the loop order and fi fo are the adjoining fields direction all incoming complete with their momenta and kinematic indices The conventions are such that a VertexFunction consists directly of the corresponding 1PI diagrams as generated by FeynArts without further prefactors Internally the tricky part is to decide which vertex functions are allowed and which are not Unlike in the case of ordinary counter terms InsertFields cannot simply look up which vertices are present in the model since there are exceptions for instance in the electroweak Standard Model the yyH vertex has neither a counter term nor even a tree level vertex Nevertheless loop diagrams for this vertex exist Therefore different constraints have to be used only such vertex functions are gener ated for which there exists a corresponding generic vertex and which do n
95. topology with an arbitrary factor first e g Topology 1 then apply SymmetryFactor to find the right symmetry factor and with it supplement the initial definition For example the two loop starting topologies are defined as StartTop 2 0 TopologyList define Theta Topology 12 Propagator Loop 1 Vertex 3 2 Vertex 3 1 Propagator Loop 1 Vertex 3 2 Vertex 3 1 Propagator Loop 1 Vertex 3 2 Vertex 3 1 define Eight Topology 8 Propagator Loop 1 Vertex 4 1 Vertex 4 1 Propagator Loop 1 Vertex 4 1 Vertex 4 1 1 define Bicycle Topology 8 Propagator Internal Vertex 3 2 Vertex 3 1 Propagator Loop 1 Vertex 3 2 Vertex 3 2 Propagator Loop 2 Vertex 3 1 Vertex 3 1 20 4 INSERTING FIELDS INTO TOPOLOGIES 4 Inserting Fields into Topologies 4 1 The Three Level Fields Concept FeynArts distinguishes three different levels of fields generic fields classes of fields and specific particles Field information becomes more and more specific with these levels Generic Classes Particles field levels in FeynArts Generic fields are the abstract field types F S V U T basic field types fermion scalar vector ghost and tensor fields Mix g1 2 21 8 mixing field Classes fields represent sets of fields with common properties such as behaviour under charge conjugation A class is
96. turned into a basic model file with LoadModel SM ResumCoup c C _ _F _ _F _ _S rhs_ c rhs Mass F t_ g_ gt MfResummed t gl ResumCoup other other 2 M CouplingMatrices ResumCoup M CouplingMatrices LoadModel modname load the model file modname mod LoadModel modname ext load the model file modname ext 7 4 Debugging 69 Finally it is possible to save the model file presently in memory with the function DumpModel This is useful for example if the modification of the basic model file by the add on model files takes a lot of time in which case one can dump the resulting model file contents in a new model file which then loads faster For example LoadModel MSSMQCD FV EnhYuk HMix DumpModel Ful1MSSM mod DumpModel file save the classes model file variables presently in memory in the classes model file file DumpModel file s include the symbols s in the variables to be saved in file DumpGenericModel file save the generic model file variables presently in memory in the generic model file file DumpGenericModel file s include the symbols s in the variables to be saved in file 7 4 Debugging There are several ways to trace or inspect the application of the Feynman rules For problems involving unresolved couplings there are two inspection functions which may be defined to print out information VertexDebug info
97. um Incoming 1 2 FourMomentum Internal 1 Index Lorentz 2 PolarizationVector V 1 FourMomentum Incoming 1 Index Lorentz 1 Conjugate PolarizationVector V 1 FourMomentum Outgoing 1 Index Lorentz 2 G 1 0 S Index Generic 311 S Index Generic 4 V 1 FourVector Mom 1 Mom 2 KI1 3 1 G 1 0 S Index Generic 311 S Index Generic 41 V 4 FourVector Mom 1 Mom 2 K11 3 Pi74 Mass S Index Generic 311 Mass S Index Generic 4111 G 1 0 S Index Generic 311 S Index Generic 41 V 1 FourVector Mom 1 Mom 2 KI1 3 G 1 0 S Index Generic 311 S Index Generic 41 V 4 FourVector Mom 1 Mom 2 KI1 3 RelativeCF gt Insertions Classes MW MW I EL I EL 2 6 3 Interpreting the Results 47 6 3 Interpreting the Results Although the analytical expression of an amplitude the third element of a FeynAmp may look complicated as a whole the origin of the individual parts can easily be recounted In the preceding example the respective terms have the following meaning 1 32 and Pi together are the product of all scalar factors from the generic couplings and propagators including the prefactor of the diagram The fact that both terms are displayed apart in the output is a peculiarity of Mathematica The symbol RelativeCF stands for the relative combinatorical factor of a classes or par ticles amplitude with respect to the generic amplitud
98. unction is typically used as a filter for DiagramSelect or DiagramGrouping as in DiagramGrouping diags FeynAmpCases Index Colour Gluon _ ___ 4 8 Extracting and Deleting Insertions by Number If the various ways of restricting InsertFields are not sufficient to select the desired diagrams the user may extract or delete diagrams by number The functions for this are DiagramExtract and DiagramDelete DiagramExtract se extract the diagrams with numbers sel from the list of inserted topologies t DiagramDeletelt sel delete the diagrams with numbers sel from the list of inserted topologies t sel can be e g 5 10 12 20 which refers to diagrams 5 10 11 12 and 20 The numbers referred to are the sequential numbers of the diagrams as given by Paint If Paint is used with a PaintLevel different from the InsertionLevel the numbering will not be useful for DiagramExtract and DiagramDelete If a diagram is discarded at a particular level derived diagrams at deeper levels are removed too Conversely choosing a diagram at a particular level requires that the lower levels of that diagram the parent diagrams are kept The end result may thus contain less or more diagrams than asked for For other kinds of objects DiagramExtract DiagramDelete works like an extended ver sion of Extract Delete e g DiagramDelete a b c d e 2 4 results in a eJ Consider that selecting diagrams by number i
99. urn values of this function The output of DiagramGrouping is a list of pairs return value of foo gt inserted topologies The user function is invoked with the same three arguments as the test function of DiagramSelect the list of field substitution rules for the graph of the form FeynmanGraph Field 1 gt fi Field 2 gt f2 the topology belonging to the graph and the head of the surrounding topology list See Sect 4 7 for auxiliary functions to use with DiagramGrouping 47 Auxiliary Functions Several functions aid the selection of diagrams with DiagramSelect or their grouping with DiagramGrouping 30 4 INSERTING FIELDS INTO TOPOLOGIES LoopFields fop return a list of the fields that are part of any loop in the topology top WFCorrectionFields top extract the fields external to any wave function correction from topology top WFCorrectionCTFields fop extract the fields external to any wave function correction counter term from topology top LoopFields rul top WFCorrectionFields rul top WFCorrectionCTFields rul top substitute the insertion rules rul into the bare topology top then proceed as above LoopFields identifies the fields running in the loop of a diagram It is commonly used asin DiagramSelect diags FreeQ LoopFields 4 V 1 amp WFCorrection CT Fields typically returns a list of two fields such as S 1 S 3 These are the fields external to the wave function corr
100. ype t running from from to to carrying field field possible types of propagators Incoming Outgoing external propagator flowing in or out External undirected external propagator Internal internal propagator which is not part of a loop Loop n internal propagator on loop n The propagators then are collected into topologies Topology p1 poss representation of a topology with propagators p Topology s the same with combinatorial factor 1 s TopologyList t to a list of topologies t LL oL uL TopologyList info L the same with an additional information field The n in Loop n is not the actual number of the loop which in general cannot be determined unambiguously but the number of the one particle irreducible conglomerate of loops 10 3 CREATING THE TOPOLOGIES 3 2 CreateTopologies The basic function to generate topologies is CreateTopologies It creates topologies based on how many loops and external legs they have CreateTopologies i gt o create topologies with loops i incoming and o CreateTopologies e create topologies with loops and e external legs outgoing legs Load FeynArts Create all topologies with one loop one incoming and one outgoing line The results are collected in a TopologyList The painted version of these topologies is much easier to understand than the list form In 1 lt lt FeynArts FeynArts 3 9 by

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