All you ever wanted to know about FeynArts and FormCalc and were afraid to ask Dr. Hahn Alle Kassen T. Hahn, Automated one-loop calculations with FormCalc 7 – p.1
The Diagrammatic Challenge # loops 0 1 2 3+ # 2 → 2 topologies 4 99 2214 50051 typical accuracy 10% 1% .1% .01% general procedure known yes yes 1 → 1 no current limits 2 → 8 2 → 6 2 → 2 1 → 1 Plus: • Phase-space integration, • Subtraction of IR poles, • Treatment of unstable particles, • Numerical difficulties, • . . . T. Hahn, Automated one-loop calculations with FormCalc 7 – p.2
Why Higher Orders? Precision: Higher Orders are seen experimentally Example 1: Example 2: 10 10 a µ = 11614097 . 29 QED 1-loop m t = 174.3 ± 5.1 GeV 0.233 m H = 114...1000 GeV 2-loop 41321 . 76 0.2325 3-loop 3014 . 19 m H 4-loop 36 . 70 lept eff 0.232 sin 2 θ 5-loop . 63 Had. 690 . 6 0.2315 EW 1-loop 19 . 5 m t 2-loop − 4 . 3 0.231 ∆α LEP 2002 Preliminary 68 % CL theory, total 11659176 1 1.002 1.004 1.006 1.008 exp (BNL 2002) 11659204 ρ l T. Hahn, Automated one-loop calculations with FormCalc 7 – p.3
Why Higher Orders? Indirect effects of particles beyond the kinematical limit ↑ ↑ inaccessible indirectly visible, (too heavy to be produced) requires precision measurements Example: Most BSM physics. T. Hahn, Automated one-loop calculations with FormCalc 7 – p.4
Why Higher Orders? “Rare” (loop-mediated) events e.g. light-by-light scattering: γ γ γ γ Example: Almost entire B-physics programme. T. Hahn, Automated one-loop calculations with FormCalc 7 – p.5
Current State of the Art Loops Partial results, Special cases 3 Established techniques, Full results 2 1 0 Legs 0 1 2 3 4 5 6 7 8 9 10 1 → 2 decays, sin θ eff vacuum graphs, ∆ ρ self-energies, ∆ r , masses 2 → 2, 1 → 3, Bhabha 2 → 3 e + e − → 4f e + e − → 4f + γ e + e − → 6f w T. Hahn, Automated one-loop calculations with FormCalc 7 – p.6
Feynman Diagram Cookbook 1. Draw all possible types of diagrams with the given number of loops and external legs Topological task, no physics input needed ∗ ∗ Well, almost: need to know allowed adjacencies in physics model, e.g. renormalizable theories have at most 3- and 4-point vertices. T. Hahn, Automated one-loop calculations with FormCalc 7 – p.7
Feynman Diagram Cookbook 2. Figure out what particles can run on each type of diagram t t t t e e e e t t t t γ H G 0 Z e e e e Combinatorial task, requires physics input (model) In this case, in the SM, three of the topologies were not realized though one was realized multiply. Note further that the e-e-scalar couplings are suppressed by m 2 e /M 2 W and thus usually neglected. These are selections one would typically make at this stage, i.e. diagrammatically. T. Hahn, Automated one-loop calculations with FormCalc 7 – p.8
Feynman Diagram Cookbook 3. Translate the diagrams into formulas by applying the Feynman rules t g µν e � v 1 | i eγ µ | u 2 � � 3 i eγ ν � − 2 = � u 4 | | v 3 � t γ ( k 1 + k 2 ) 2 e � �� � � �� � � �� � left vertex propagator right vertex Database look-up T. Hahn, Automated one-loop calculations with FormCalc 7 – p.9
Feynman Diagram Cookbook 4. Contract the indices, take the traces, etc. t 8 πα e F 1 = � v 1 | γ µ | u 2 � � u 4 | γ µ | u 3 � = 3 s F 1 , t γ e Also, compute the fermionic matrix elements, e.g. by squaring and taking the trace: | F 1 | 2 = Tr { ( / k 4 + m t ) γ µ ( / k 3 − m t ) γ ν } k 1 − m e ) γ µ ( / k 2 + m e ) γ ν } Tr { ( / 2 s 2 + st + ( m 2 e + m 2 t − t ) 2 = 1 Algebraic simplification T. Hahn, Automated one-loop calculations with FormCalc 7 – p.10
Feynman Diagram Cookbook 5. Write the results up as a . . . . . . . . . . . . . . . . . (put favourite language here) program 5a. Debug that program 6. Run it to produce numerical values Programming T. Hahn, Automated one-loop calculations with FormCalc 7 – p.11
Recipe for Feynman Diagrams Thanks to and (and many others) we have a Recipe for an ARBITRARY Feynman diagram up to one loop Draw all possible types of diagrams topological task ➀ Figure out what particles can run combinatorical task ➁ on each type of diagram Translate the diagrams into formulas by database look-up ➂ applying the Feynman rules Contract the indices, take the traces, etc. algebraic simplification ➃ Write up the results as a computer program programming ➄ Run the program to get numerical results waiting ➅ T. Hahn, Automated one-loop calculations with FormCalc 7 – p.12
Programming Techniques • Very different tasks at hand. • Some objects must/should be handled symbolically, e.g. tensorial objects, Dirac traces, dimension (D vs. 4). • Reliable results required even in the presence of large cancellations. • Fast evaluation desirable (e.g. for Monte Carlos). Hybrid Programming Techniques necessary Symbolic manipulation (a.k.a. Computer Algebra) for the structural and algebraic operations. Compiled high-level language (e.g. Fortran) for the numerical evaluation. T. Hahn, Automated one-loop calculations with FormCalc 7 – p.13
Automated Diagram Evaluation Symbolic manipulation Diagram Generation: • Create the topologies (Computer Algebra) FeynArts • Insert fields for the structural and • Apply the Feynman rules algebraic operations. • Paint the diagrams Algebraic Simplification: Amplitudes Compiled high-level • Contract indices language (Fortran) for • Calculate traces the numerical evaluation. • Reduce tensor integrals • Introduce abbreviations FormCalc Numerical Evaluation: • Convert Mathematica output to Fortran code • Supply a driver program Fortran Code • Implementation of the integrals LoopTools |M| 2 Cross-sections, Decay rates, . . . T. Hahn, Automated one-loop calculations with FormCalc 7 – p.14
One-loop since mid-1990s Automated NLO computations is an industry today, with many packages becoming available in the last few years: • GoSam, HELAC-NLO, aMC@NLO, MadLoop, OpenLoops, BlackHat, Rocket, . . . Here: FeynArts (1991) + FormCalc (1995) FormCalc was doing largely the same as FeynCalc (1992) but used FORM for the time-consuming tasks, hence the name FormCalc. • Feynman-diagrammatic method, • Analytic calculation as far as possible (any model), • Generation of code for the numerical evaluation of the squared matrix element. So much for NLO ‘revolution.’ T. Hahn, Automated one-loop calculations with FormCalc 7 – p.15
Plan Walk through the general setup of these programs and show some perhaps non-standard applications. e + e − → t ¯ • ‘Standard Candle’ — t , • Resumming a coupling — ∆ b , • Example from flavour physics — ∆ M s . T. Hahn, Automated one-loop calculations with FormCalc 7 – p.16
FeynArts Find all distinct ways of connect- Topologies ing incoming and outgoing lines CreateTopologies Draw the results Paint Determine all allowed Diagrams combinations of fields InsertFields Apply the Feynman rules further Amplitudes processing CreateFeynAmp T. Hahn, Automated one-loop calculations with FormCalc 7 – p.17
CreateTopologies Algorithm: Start from Zero-leg Topologies and successively add external legs. This is not entirely self-sufficient, but few people would even want to use FeynArts beyond three loops. ր − → − → etc. ց Starting Topology (hard-coded) T. Hahn, Automated one-loop calculations with FormCalc 7 – p.18
Three Levels of Fields Generic level, e.g. F, F, S C ( F 1 , F 2 , S ) = G L P L + G R P R P R,L = ( 1 l ± γ 5 ) / 2 Kinematical structure completely fixed, most algebraic simplifications (e.g. tensor reduction) can be carried out. Classes level, e.g. -F[2], F[1], S[3] ¯ i e m ℓ,i ℓ i ν j G : G L = − 2 sin θ w M W δ ij , G R = 0 √ Coupling fixed except for i , j (can be summed in do-loop). Particles level, e.g. -F[2,{1}], F[1,{1}], S[3] insert fermion generation (1, 2, 3) for i and j T. Hahn, Automated one-loop calculations with FormCalc 7 – p.19
Sample CreateFeynAmp output G = FeynAmp[ identifier , γ loop momenta , γ generic amplitude , insertions ] G GraphID[Topology == 1, Generic == 1] T. Hahn, Automated one-loop calculations with FormCalc 7 – p.20
Sample CreateFeynAmp output = FeynAmp[ identifier , G γ loop momenta , γ generic amplitude , insertions ] G Integral[q1] T. Hahn, Automated one-loop calculations with FormCalc 7 – p.21
Sample CreateFeynAmp output = FeynAmp[ identifier , G loop momenta , γ generic amplitude , γ insertions ] G I 32 Pi 4 RelativeCF ......................................... prefactor 1 FeynAmpDenominator[ q1 2 - Mass[S[Gen3]] 2 , 1 (-p1 + q1) 2 - Mass[S[Gen4]] 2 ] ................. loop denominators (p1 - 2 q1)[Lor1] (-p1 + 2 q1)[Lor2] ........ kin. coupling structure ep[V[1], p1, Lor1] ep * [V[1], k1, Lor2] ........... polarization vectors G (0) SSV [(Mom[1] - Mom[2])[KI1[3]]] G (0) SSV [(Mom[1] - Mom[2])[KI1[3]]], ................. coupling constants T. Hahn, Automated one-loop calculations with FormCalc 7 – p.22
Sample CreateFeynAmp output = FeynAmp[ identifier , G loop momenta , γ generic amplitude , γ insertions ] G { Mass[S[Gen3]], Mass[S[Gen4]], G (0) SSV [(Mom[1] - Mom[2])[KI1[3]]], G (0) SSV [(Mom[1] - Mom[2])[KI1[3]]], RelativeCF } -> Insertions[Classes][{MW, MW, I EL, -I EL, 2}] T. Hahn, Automated one-loop calculations with FormCalc 7 – p.23
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