Motivation FeynCalc as a "calculator" for QFT expressions FeynCalc for evaluation Feynman diagrams Summary Loop calculations with FeynCalc 9 Vladyslav Shtabovenko in collaboration with R. Mertig and F. Orellana Technische Universität München International Mini-Workshop UBB Chillán Physik-Department T30f Physik-Department T30f (TUM) FeynCalc 1 / 23 Motivation FeynCalc as a "calculator" for QFT expressions FeynCalc for evaluation Feynman diagrams Summary Outline 1 Motivation QFT Automation FeynCalc 2 FeynCalc as a "calculator" for QFT expressions Evaluation of 1-loop integrals Combining FeynCalc with other tools 3 FeynCalc for evaluation Feynman diagrams Note on FeynArts Matching calculations between QCD and NRQCD 4 Summary Physik-Department T30f (TUM) FeynCalc 2 / 23 Motivation FeynCalc as a "calculator" for QFT expressions QFT Automation FeynCalc for evaluation Feynman diagrams FeynCalc Summary What Automation of QFT ≈ automatic symbolic or numeric evaluation of Feynman diagrams This talks: only symbolics Why Feynman diagrams → theoretical predictions for experimental observables: cross-sections, decay rates etc. Rising experimental precision → theoretical errors must be reduced. Leading order (LO) in perturbation theory is mostly not enough, need to go at least to NLO or even higher. Often too many Feynman diagrams to do everything by pen and paper :( Also calculations that are doable by pen and [xkcd.com/1319] paper need to be checked! Physik-Department T30f (TUM) FeynCalc 3 / 23 Motivation FeynCalc as a "calculator" for QFT expressions QFT Automation FeynCalc for evaluation Feynman diagrams FeynCalc Summary How CAS or CAS-like environment: Mathematica, Some single purpose tools for Reduce, FORM, Sympy, GiNaC, RedBerry etc. making coffee Specific codes running on top of it. Automation tools classified by their usage Single purpose tools: FeynArts1, Tracer 2, FIRE3, LoopTools4,... Multi purpose tools (semi-automatic): HEPMath5, FeynCalc, Package X6,... Semi-automatic coffee maker Multi purpose tools (fully-automatic): CalcHEP7, GRACE8, FormCalc1 ... 1 2 3 4 [Hahn, 2001], [Jamin & Lautenbacher, 1993], [Smirnov, 2008], 5 6 7 [Hahn & Perez-Victoria, 1999], [Wiebusch, 2014], [Patel, 2015], [Belyaev et al., 2012], 8 [Ishikawa et al., 1993] Fully-automatic coffee maker Physik-Department T30f (TUM) FeynCalc 4 / 23 Motivation FeynCalc as a "calculator" for QFT expressions QFT Automation FeynCalc for evaluation Feynman diagrams FeynCalc Summary FeynCalc is a Mathematica package for symbolic semi-automatic evaluation of Feynman diagrams and algebraic expressions in QFT. [Mertig et al., 1991] [Shtabovenko et al., 2016] Features Suitable for evaluating both single expressions and full Feynman diagrams. The calculation can be organized in many different ways (flexibility) Extensive typesetting for better readability Tools for frequently occurring tasks like Lorentz index contraction, SU (N ) algebra, Dirac matrix manipulation and traces, etc. Passarino-Veltman reduction of one-loop amplitudes to standard scalar integrals Basic support for manipulating multi-loop integrals General tools for non-commutative algebra Physik-Department T30f (TUM) FeynCalc 5 / 23 Motivation FeynCalc as a "calculator" for QFT expressions QFT Automation FeynCalc for evaluation Feynman diagrams FeynCalc Summary FeynCalc 9 FeynCalc 9 was officially released this year (see arXiv:1601.01167) The source code is now hosted on GitHub: github.com/FeynCalc Works with Mathematica versions 8, 9 and 10. Test driven development: over 3000 unit and integration tests to prevent new bugs and regressions. What’s new Improved 1-loop tensor reduction. New partial fractioning algorithm. Basic multi-loop tensor reduction. Simpler interfacing with other tools (e.g. for IBP-reduction) [xkcd.com/1172] Physik-Department T30f (TUM) FeynCalc 6 / 23 Motivation FeynCalc as a "calculator" for QFT expressions Evaluation of 1-loop integrals FeynCalc for evaluation Feynman diagrams Combining FeynCalc with other tools Summary What minimal capabilities do we need to compute 1-loop integrals in dimensional regularization using software for symbolic manipulations? D µ Z d q γ (q/ + m)γµ µD−4 = ? (2π)D (q2 − m2)(q − p)2 Simplification of the Dirac algebra (XFeynCalc) Tensor integral reduction Standard technique: Passarino-Veltman (XFeynCalc) Further simplification of scalar integrals Partial fractioning (XFeynCalc) Optional: IBP reduction (requires external tools ) Analytic/numeric evaluation of the master integrals (requires external tools) Physik-Department T30f (TUM) FeynCalc 7 / 23 Motivation FeynCalc as a "calculator" for QFT expressions Evaluation of 1-loop integrals FeynCalc for evaluation Feynman diagrams Combining FeynCalc with other tools Summary Dirac algebra FeynCalc implements the conventional generalization of the Dirac algebra to D dimensions [’t Hooft & Veltman, 1972] µ ν µν µν µ ν µν µν {γ¯ , γ¯ } = 2g¯ , g¯ g¯µν = 4 → {γ , γ } = 2g , g gµν = D DiracSimplify simplifies chains of Dirac matrices DiracTrace computes traces of Dirac matrices In[1]:= FCI[GAD[µ].(m + GSD[q]).GAD[µ] FAD[{q, m}, −p + q]] γµ.(m + γ · q).γµ Out[1]= (q2 − m2) .(q − p)2 In[2]:= %//DiracSimplify Dγ · q Dm 2γ · q Out[2]= − + + (q2 − m2) .(q − p)2 (q2 − m2) .(q − p)2 (q2 − m2) .(q − p)2 In[3]:= GAD[µ, ν, ρ, σ, τ, κ, µ, ν, ρ, σ, τ, κ] Out[3]= γµ.γν .γρ.γσ .γτ .γκ.γµ.γν .γρ.γσ .γτ .γκ In[4]:= DiracTrace[%, DiracTraceEvaluate −> True] // Factor2 Out[4]= 4(2 − D)2D(−D3 + 26D2 − 152D + 128) Physik-Department T30f (TUM) FeynCalc 8 / 23 Motivation FeynCalc as a "calculator" for QFT expressions Evaluation of 1-loop integrals FeynCalc for evaluation Feynman diagrams Combining FeynCalc with other tools Summary Tensor reduction The default behavior of TID is to reduce each tensor integral into Passarino-Veltman scalar functions (A0, B0, C0, D0) In[1]:= TID[GAD[µ].(m + GSD[q]).GAD[µ] FAD[{q, m}, −p + q], q]; ToPaVe[%, q] 2 2 iπ (D − 2)A0 m γ · p Out[1]= − 2p2 2 2 2 2 2 2 2 2 iπ B0 p , 0, m Dm γ · p − 2Dmp + Dp γ · p − 2m γ · p − 2p γ · p 2p2 If the function detects zero Gram determinants, it automatically switches to Passarino-Veltman coefficient functions (e.g. B1, B00, C222 etc.) In[3]:= ScalarProduct[p, p] = 0; TID[FCI[GAD[µ].(m + GSD[q]).GAD[µ] FAD[{q, m}, −p + q]], q]; ToPaVe[%, q] // Collect2[#, B0] & 1 2 2 1 2 Out[3]:= − iπ B0 0, 0, m (−2Dm + Dγ · p − 2γ · p) − iπ (D − 2)γ · p 2 4 Physik-Department T30f (TUM) FeynCalc 9 / 23 Motivation FeynCalc as a "calculator" for QFT expressions Evaluation of 1-loop integrals FeynCalc for evaluation Feynman diagrams Combining FeynCalc with other tools Summary Partial fractioning Scalar loop integrals can be often simplified even further by using partial fractioning. Well known identities (implemented in SPC and Apart2) are 1 q · p = [(q + p)2 + m2 − (q2 + m2) − p2 − m2 + m2], 2 2 1 2 1 1 1 1 1 = − . 2 2 2 2 2 2 2 2 2 2 (q − m1 )(q − m2 ) m1 − m2 q − m1 q − m2 But: Many decompositions, e.g. Z 1 1 Z 1 1 dDq = dDq − , q2(q − p)2(q + p)2 p2 q2(q − p)2 (q − p)2(q + p)2 require more sophisticated algorithms. New in FeynCalc 9: ApartFF introduces partial fractioning algorithm from [Feng, 2012] Compared to the reference Mathematica implementation (https://github.com/F-Feng/APart), it is fully integrated into FeynCalc In[1]:= ApartFF[FAD[{q}, {q − p}, {q + p}], {q}] 1 1 Out[2]:= − p2q2.(q − p)2 p2q2.(q − 2p)2 Physik-Department T30f (TUM) FeynCalc 10 / 23 Motivation FeynCalc as a "calculator" for QFT expressions Evaluation of 1-loop integrals FeynCalc for evaluation Feynman diagrams Combining FeynCalc with other tools Summary Tools for IBP-reduction Reduction of scalar loop integrals using integration-by-parts (IBP) identities [Chetyrkin & Tkachov, 1981] is a standard technique in modern loop calculations. Many publicly available IBP-packages on the market: FIRE [Smirnov & Smirnov, 2013], LiteRED [Lee, 2012], Reduze [Studerus, 2009], AIR [Anastasiou & Lazopoulos, 2004], . Expected input: loop integrals with propagators that form a basis. What about integrals with an incomplete or overdetermined basis? FCLoopBasisIncompleteQ detects integrals that require a basis completion FCLoopBasisFindCompletion gives a list of propagators (with zero exponents) required to complete the basis FCLoopBasisOverdeterminedQ checks if the propagators are linearly dependent. Such integrals can be decomposed further using ApartFF. In[1]:= FCI[FAD[{q1, m, 2}, {q1 + q3, m}, {q2 − q3}, q2]] 1 Out[1]:= (q12 − m2) . (q12 − m2) . ((q1 + q3)2 − m2) .(q2 − q3)2.q22 In[2]:= FCLoopBasisIncompleteQ[%, {q1, q2, q3}] Out[2]:= True In[3]:= FCLoopBasisFindCompletion[%%, {q1, q2, q3}][[2]] Out[3]:= −(q1 · q3) + q2 · q3 + 2q32, q1 · q2 Physik-Department T30f (TUM) FeynCalc 11 / 23 Motivation FeynCalc as a "calculator" for QFT expressions Evaluation of 1-loop integrals FeynCalc for evaluation Feynman diagrams Combining FeynCalc with other tools Summary Real life computations: combination of different tools is often advantageous Useful setup for evaluation of 1-loop integrals: FeynCalc+FIRE+Package X Need to convert between conventions used in each package! Work in progress (VS): Interface for seamless integration between these (and possibly) other packages and FeynCalc. Private version available, public release in near future. Physik-Department T30f (TUM) FeynCalc 12 / 23 Motivation FeynCalc as a "calculator" for QFT expressions Evaluation of 1-loop integrals FeynCalc for evaluation Feynman diagrams Combining FeynCalc with other tools Summary Call FIRE from FeynCalc: In[1]:= FCI[FAD[q1, q2, {q1 − p, 0, 2}, {q1 − q2, 0, 2}]] 1 Out[1]:=