brief tutorial on feyncalc - tum physikdepartment (indico) · 2017-07-03 · brief tutorial on...

Brief Tutorial on FeynCalcbased on

arXiv:1601.01167, arXiv:1611.06793

Vladyslav Shtabovenko

Technische Universität München, Germany

Tutorial at Methods of EFT& Lattice Field Theory School

3RD OF JULY, 2017

Physik-Department T30f

V. Shtabovenko (TUM) @ EFT School, 3.07.2017 FeynCalc 1 / 14

Outline

1 Motivation And Brief History of FeynCalc

2 Using FeynCalc in your research

3 FeynHelpers: FeynCalc on Steroids

4 Summary and Outlook


Motivation And Brief History of FeynCalc

A generic perturbative QFT calculationmay involve many steps

I Feynman diagramsI Feynman rules from LI Diagram generationI AmplitudesI . . .

I Dirac algebraI Simplification of γ-matrix chainsI Dirac tracesI Fierz identitiesI . . .

I Loop integralsI Tensor reductionI Partial fractioningI Mapping of topologiesI IBP-ReductionI Numerical evaluationI . . .

I Phase space integralsI . . .

There are many ways to automatize suchcalculations:

Self-written codes

Write private codes inFORM [Vermaseren, 2000], MATHEMATICA,MAPLE, REDUCE, . . .

Separate packages

Automatize each step separately usingstandalone packages (e. g. FEYNARTS,LOOPTOOLS [Hahn & Perez-Victoria, 1999],FEYNRULES [Christensen & Duhr, 2008],QGRAF [Nogueira, 1993],TRACER [Jamin & Lautenbacher, 1993],FORMTRACER [Cyrol et al., 2016],FORCER [Ruijl et al., 2017],PYSECDEC [Borowka et al., 2017], . . . ).

All-in-one packages

Employ all-in-one packages that handlemost of these steps in one framework.



Two big categories of all-in-one packagesI Fully-automatic (FORMCALC [Hahn & Perez-Victoria, 1999], GOSAM [Cullen et al., 2014],

GRACE [Belanger et al., 2006], DIANA [Tentyukov & Fleischer, 2000], FDC [Wang, 2004],CALCHEP [Belyaev et al., 2012], COMPHEP [Boos et al., 2004] . . . )

I Semi-automatic (FEYNCALC [Mertig et al., 1991, Shtabovenko et al., 2016],HEPMATH [Wiebusch, 2014], PACKAGE-X [Patel, 2015], . . . )

Fully-automatic tools

Blackbox: Require only minimal userinput and provide a small set of options.The code takes care of the rest.

Semi-automatic tools

Toolbox: Combine different tools withmany options to get the computation donein your way.

Easy to useFoolproofConstantly good performanceSaves your timeLimited number of templated calculationsDifficult to extend/modify for your needsNot easy to obtain intermediate results

You must know what you are doingEasy to make mistakesThe performance depends on your skillsWriting codes may take quite some timeVery broad range of applicationsExtendable with user-defined objectsIntermediate results at each step



FEYNCALC

I MATHEMATICA package for symbolicsemi-automatic QFT calculations

I Open source (GPLv3) and publicly availableI Can process Feynman diagrams and

standalone QFT expressionsI Widely used in Effective Field Theory

calculations

A bit of history ...

1991 • FEYNCALC 1.0 by R. Mertig [Mertig et al., 1991]

1991-1996 • New features and improvements1997 • Release of TARCER [Mertig & Scharf, 1998]

1997-2000 • Many contributions from F. Orellana (e. g. PHI [Orellana, 2002] for ChPT)2012 • FEYNCALCFORMLINK [Feng & Mertig, 2012]

2016 • FEYNCALC 9.0 [Shtabovenko et al., 2016]

2017 • FEYNHELPERS [Shtabovenko, 2017]



Current status of the projectI (Again) under active development since 2014I Regular releases since 2016: FEYNCALC 9.0.0,

FEYNCALC 9.0.1, FEYNCALC 9.1.0,FEYNCALC 9.2.0

I Easy installationI Online documentation: https://feyncalc.github.io/reference

I Many sample tree and 1-loop calculations (QED, EW, QCD) includedI Public source code repository: https://www.github.com/feyncalc

I hotfix-stable branch: stable version + bugfixesI master branch: development version (new features, less stable)

I Extensive unit testing framework (based on MUnit, over 4000 tests)

FEYNCALC developer teamI Rolf Mertig (GluonVision GmbH): original author of the package, first release 1991I Frederik Orellana (Technical University of Denmark): joined 1997I VS (Technical University of Munich, soon Zhejiang University (China)): joined 2014


https://feyncalc.github.io/reference

https://feyncalc.github.io/reference

https://www.github.com/feyncalc

Using FeynCalc in your research General picture

What can FEYNCALC do?I Standard feature set: Lorentz algebra, Dirac algebra, Color algebraI Tensor reduction of 1-loop integrals (Passarino–Veltman)I Basic support for manipulating multi-loop integralsI Built-in interface to FEYNARTS [Hahn & Perez-Victoria, 1999] (also works with custom

FEYNARTS model from FEYNRULES [Christensen & Duhr, 2008])I Optional interfaces to other HEP toolsI Extensive typesetting for better readabilityI Extendable with self-written MATHEMATICA codes

When is FEYNCALC useful?I Small or medium-sized

calculations, too specific for fullyautomatic packages

I FEYNCALC as a “calculator” forQFT expressions

I Cross-check results from otherpeople

I Extensive manipulations on thelevel of the amplitudes

What are the limitations of FEYNCALC?I Cannot be used without

MATHEMATICAI Inherits MATHEMATICA’s

performance problems with largenumber of terms

I Not really suited for very large andcomplex calculations

I Much slower than FORMI Only algebraic manipulations.


Using FeynCalc in your research Brief overview of selected functions

Lorentz algebra (most used functions)I ChangeDimension: p · q→ p · qI Contract: pµqµ → p · qI ExpandScalarProduct: a · (b + c)→ a · b + a · cI FourDivergence: ∂µI FourLaplacian: ∂µ∂µ

I MomentumCombine: pµ + qµ → (p + q)µ

I Uncontract: p · q→ pµqµ

New functionsI FCRenameDummyIndices (v. 9.0): pµqµ → pαqαI FCCanonicalizeDummyIndices (v. 9.1): pµqµ + pνqν → 2pαqαI FCGetDimensions (v. 9.2): /p− /p→ {D,D− 4}



Dirac algebra (most used functions)

I Chisholm: γµγν γλ → gµν γλ + gνλγµ − gµλγν + iεµνλσ γσγ5

I DiracEquation: /pu(p)→ mu(p)

I DiracGammaCombine: /p + /q→ γµ(p + q)µ

I DiracGammaExpand: γµ(p + q)µ → /p + /qI DiracOrder: γνγµ → 2gµν − γµγν

I DiracReduce: γµγν → gµν − iσµν

I DiracSigmaExplicit: σµν → i2 [γµ, γν ]

I DiracTrace: Tr(γµγν)→ 4gµν

I DiracSimplify: Applies all available simplifications

New functions

I FCDiracIsolate (v. 9.1): e2 γµ(/p+m)γµ

(q2−m2)(q2−p2)→ e2

(q2−m2)(q2−p2)[γµ(/p + m)γµ]

I Most changes in the Dirac algebra related functions happened under the hood(performance improvements, cleaner code . . . )



Loop calculations (most used functions)

I FeynAmpDenominatorSimplify: 1(q+p)2−m2 → 1

q2−m2

I TID: qµqν

q2−m2 → m2

Dgµν

q2−m2

I PaVeReduce: A0000(m2)→ m4

D(D+2)A0(m2)

New functions

I ToPaVe (v. 9.0): 1q2−m2 → iπ2A0(m2)

I FCMultiLoopTID (v. 9.0): /q1 /q2

q21q2

2[(q1−q2)2−m2]

→ 1q2

1[(q1−q2)2−m2]

− m2

21

q21q2

2[(q1−q2)2−m2]

I ApartFF (v. 9.0): 1q2(q−p)2(q+p)2 → 1

p2q2(q−p)2 − 1p2q2(q−2p)2

I FCLoopBasisIncompleteQ (v. 9.0): Do the propagators of the given loopintegral form a basis?

I FCLoopBasisOverDeterminedQ (v. 9.0): Is the propagator basisoverdetermined?

I FCLoopBasisFindCompletion (v. 9.0): Which propagators need to be added tohave the complete basis?


FeynHelpers: FeynCalc on Steroids An example from physics of heavy quarkonia

We have to go deeper ...

I Often, the final FEYNCALC result is not the most useful oneI No automatic evaluation of Passarino–Veltman functionsI No automatic IBP-reduction of (multi-)loop integrals a

I A lot of good HEP tools are publicly available, no need to reinvent the wheelI However, combining them with FEYNCALC is not always easyI Well-tested interfaces supported by the developers are better than private hacksaTARCER [Mertig & Scharf, 1998] is limited to 2-loop propagator-type integrals

The first step into the right direction: FEYNHELPERS [Shtabovenko, 2017]

I Seamless integration of PACKAGE-X [Smirnov, 2015] and FIRE [Patel, 2015] intoFEYNCALC

I Access the PACKAGE-X library of analytic results for 1-loop integrals and theIBP-reduction engine of FIRE directly from a FEYNCALC session

I All conversions (syntax, normalization) are automatic and require no userintervention

I Open-source interface code: https://github.com/FeynCalc/feynhelpersI FEYNHELPERS comes with many neat examples from QED, QCD, EW, EFTs etc.


https://github.com/FeynCalc/feynhelpers


Consider

Iµν =

∫l

lµlν

l2[(l + P/2)2 − mQ][(l− P/2)2 − mQ]2 , with P2 = 4m2Q

FEYNCALC can handle this integral, but the result is not particularly enlightening

In fact, one can show that

Iµν =(D− 4)(D− 2)

64(D− 5)(D− 3)m6Q

((D− 6)PµPν + 4m2

Qgµν)∫

l

1l2 − m2

Q.

Is it possible to get this using FEYNCALC in an easy way?V. Shtabovenko (TUM) @ EFT School, 3.07.2017 FeynCalc 12 / 14


With FEYNCALC and FEYNHELPERS such calculations become almost trivial...

The performance is, however, still an issue and will be improved in the subsequentversions (by switching to the C++ backend of FIRE)


Summary and Outlook

I Modern HEP theory and phenomenology heavily rely on programs for automaticcalculations.

I The market for QFT software is growing rapidly, new tools appear almost monthly.I Different projects require different tools and FEYNCALC is one of them.I The development of FEYNCALC focuses on versatility, flexibility, convenience and

performance.I FEYNCALC is particularly useful for “nonstandard” calculations that are not covered

by other (more automatic) tools.I In the last 3 years we tried to achieve several goals:

I Refactor the source code, fix old (and new) bugs, make things fasterI Facilitate manipulations of tree and loop amplitudes by introducing new functions and

improving the existing onesI Interface FEYNCALC with other tools useful for HEP calculations

I Yet, much still remains to be done:I There is still a lot of room for improvements regarding stability, the number of available

function and their performance.I We need more interfaces to other HEP packages, e. g. to LITERED [Lee, 2012], S@M

[Maitre & Mastrolia, 2008], FORMTRACER [Cyrol et al., 2016], QGRAF [Nogueira, 1993].


Backup

Tensor reduction a la Passarino–VeltmanI Fairly old technique for dealing with tensor 1-loop integrals [Passarino & Veltman, 1979]

I Still widely used in many loop calculations.I Main idea: convert all the tensor integrals into scalar ones (Passarino–Veltman

coefficient functions)I Evaluation of any 1-loop integral can be reduced to the evaluation of the resulting

coefficient functionsI A lot of tools for numerical evaluation: FF [van Oldenborgh, 1991],

LOOPTOOLS [Hahn & Perez-Victoria, 1999], QCDLOOP [Carrazza et al., 2016],ONELOOP [van Hameren, 2011], GOLEM95C [Cullen et al., 2011],PJFRY [Fleischer & Riemann, 2011], COLLIER [Denner et al., 2017], . . .

I Where to get analytic results for singular kinematics or zero Gram determinants?Often needed for renormalization, EFTs, . . .

I Most of the results can be found somewhere in the literature.


Backup

PACKAGE-XI Recent [Patel, 2015] MATHEMATICA package for semi-automatic 1-loop calculations

(closed-source freeware)I Unique feature: Library of analytic expressions for Passarino–Veltman functions

with up to 4 legs and almost arbitrary kinematics.I Can also extract UV- and IR-parts and expand coefficient functions in their

arguments.I Someone indeed has collected all those results from the literature!

Interface to PACKAGE-XI Main function: PaXEvaluateI Works: on scalar 1-loop integrals (unit numerators) and Passarino–Veltman

coefficient functions A, B, C and DI Takes two arguments (plus options): input expression, loop momentum.I Use PaXEvaluateUV(PaXEvaluateIR) to get the UV(IR)-divergent part of the

resultI PaXEvaluateUVIRSplit returns the full result with the explicit distinction

between εUV and εIR.I All four functions share the same set of options


Backup

Let us compute∫ dDq

(2π)D1

q2−m2

In[1]:= int=PaXEvaluate[FAD[{q,m}],q,PaXImplicitPrefactor→→→1/(2Pi)^D]

Out[1]=im2

16π2ε−

im2(−log(µ2

m2

)+γ−1−log(4π)

)16 π2

Make the result look more compact (∆ ≡ 1/ε− γE + log(4π)) using FCHideEpsilonIn[2]:= int//FCHideEpsilon

Out[2]=i∆m2

16π2 +im2(log

(µ2

m2

)+1

)16π2

Evaluation of Passarino–Veltman functions:In[3]:= PaXEvaluate[B0[SPD[p,p],0,m^2]]

Out[3]=1ε

+log(µ2

πm2

)−

m2 log( m2

m2−p2

)p2 +log

(m2

m2−p2

)−γ+2


Backup

We can also expand coefficient functions in their parameters (masses or externalmomenta). To expand B0(p2, 0,m2) around p2 = m2 up to first order with PaXEvaluatewe first need to assign an arbitrary symbolic value to the scalar product p2, e.g. pp

In[4]:= SPD[p,p]=pp;

Then use the option PaXSeries to specify the expansion parameters and activate theoption PaXAnalytic

In[5]:= PaXEvaluate[B0[SPD[p,p],0,m^2],PaXSeries→→→{{pp,m^2,1}},PaXAnalytic→→→True]

Out[5]=3 m2−pp

2εm2 −(3 m2−pp)

(−log

(µ2

m2

)+γ−2+log(π)

)2m2

Get only in the UV-part of this series: PaXEvaluate with PaXEvaluateUVIn[6]:= PaXEvaluateUV[B0[SPD[p,p],0,m^2],PaXSeries→→→{{pp,m^2,1}},PaXAnalytic→→→True]

Out[6]=1εUV

The IR-part is equally easyIn[7]:= PaXEvaluateIR[B0[SPD[p,p],0,m^2],PaXSeries→→→{{pp,m^2,1}},PaXAnalytic→→→True]

Out[7]=m2−pp

2m2εIR

Full result with the explicit distinction between UV and IR singularitiesIn[8]:= PaXEvaluateUVIRSplit[B0[SPD[p,p],0,m^2],PaXSeries→→→{{pp,m^2,1}},PaXAnalytic→→→True]

Out[8]=m2−pp

2m2εIR−

(3 m2−pp)(−log

(µ2

m2

)+γ−2+log(π)

)2 m2 +

1εUV


Backup

Integration-by-parts identitiesI Reduction of scalar loop integrals using integration-by-parts (IBP) identities

[Chetyrkin & Tkachov, 1981] is a standard technique in modern loop calculations.I Many publicly available IBP-packages on the market: FIRE [Smirnov & Smirnov, 2013],

LITERED [Lee, 2012], REDUZE [Studerus, 2009], AIR [Anastasiou & Lazopoulos, 2004], . . .I Expected input: loop integrals with propagators that form a basis.I What about integrals with an incomplete or overdetermined basis?

I FCLoopBasisIncompleteQ detects integrals that require a basis completionI FCLoopBasisFindCompletion gives a list of propagators (with zero exponents)

required to complete the basisI FCLoopBasisOverdeterminedQ checks if the propagators are linearly dependent.

Such integrals can be decomposed further using ApartFF.


Backup

Interface to FIREI Main function: FIREBurnI Reduces scalar multi-loop integrals to simpler ones using IBP-techniques.I Takes three arguments (plus options): input expression, list of loop momenta and

the list of external momenta.I Automatically adds propagators to integrals with incomplete bases of propagatorsI Automatically detects integrals with linearly dependent propagators

Current limitationsI No recognition of integral familiesI Each loop integral is evaluated separatelyI Hence, rather inefficient ...


Backup

IBP-reduce the 1-loop integral∫ dDl

[l2]2[(l−p)2−m2]2

In[9]:= FIREBurn[FAD[{l,0,2},{l−p,m,2}],{l},{p}]

Out[9]=(D−2)(2 D m2−9 m2−pp)

2m2(m2−pp)3((l−p)2−m2)−

(D−3)(D m2+D pp−4m2−6 pp)

(m2−pp)3 l2.((l−p)2−m2)

No dependence on external momenta→ supply an empty list for the third argument. Forexample, for

∫ dDq1dDq2dDq3[q2

1−m2]2[(q1+q3)2−m2][(q2−q3)

2][q22]

2

In[10]:= FIREBurn[FAD[{q1,m,2},{q1+q3,m},{q2−q3},{q2,0,2}],{q1,q2,q3},{}]

Out[10]= −(D−3)(3 D−10)(3 D−8)

16(2 D−7)m4(q12−m2).q22.(q2−q3)2.((q1+q3)2−m2)


Backup

Check out the included examples for more nice calculationsI ABJ axial anomaly (following Chapter 19.2 in Peskin and Schroeder)QEDABJAxialAnomaly.m

I Electron’s g− 2 in QED at 1-loopQEDElectronGMinusTwoOneLoop.m

I 1-loop renormalization of QED in MS, MS and OS schemes (full gauge dependence)QEDRenormalizationOneLoop.m

I 1-loop gluon self-energy (full gauge dependence) in QCDQCDQuarkSelfEnergyOneLoop.m

I 1-loop quark self-energy (full gauge dependence) in QCDQCDGluonSelfEnergyOneLoop.m

I LO matching between QED and Euler-Heisenberg’s EFT (development version)QEDToEulerHeisenbergLagrangianMatching.m

I UV part of the 3-gluon vertex at 1-loop in QCD (development version)QCDThreeGluonVertexOneLoop.m

I H → gg decay via a top-quark loopEWHiggsToTwoGluonsOneLoop.m


Backup

Anastasiou, C. & Lazopoulos, A. (2004).

Automatic Integral Reduction for Higher Order Perturbative Calculations.JHEP, 0407, 046.

Belanger, G., Boudjema, F., Fujimoto, J., Ishikawa, T., Kaneko, T., Kato, K., & Shimizu, Y. (2006).

GRACE at ONE-LOOP: Automatic calculation of 1-loop diagrams in the electroweak theory with gaugeparameter independence checks.Phys. Rept., 430, 117–209.

Belyaev, A., Christensen, N. D., & Pukhov, A. (2012).

CalcHEP 3.4 for collider physics within and beyond the Standard Model.

Boos, E., Bunichev, V., Dubinin, M., Dudko, L., Ilyin, V., Kryukov, A., Edneral, V., Savrin, V., Semenov, A., &Sherstnev, A. (2004).CompHEP 4.4: Automatic computations from Lagrangians to events.Nucl. Instrum. Meth., A534, 250–259.

Borowka, S., Heinrich, G., Jahn, S., Jones, S. P., Kerner, M., Schlenk, J., & Zirke, T. (2017).

pySecDec: a toolbox for the numerical evaluation of multi-scale integrals.

Carrazza, S., Ellis, R. K., & Zanderighi, G. (2016).

QCDLoop: a comprehensive framework for one-loop scalar integrals.Comput. Phys. Commun., 209, 134–143.

Chetyrkin, K. & Tkachov, F. (1981).

Integration by parts: The algorithm to calculate β-functions in 4 loops.Nucl. Phys. B, 192(1), 159–204.


Backup

Christensen, N. D. & Duhr, C. (2008).

FeynRules - Feynman rules made easy.Comput. Phys. Commun., 180, 1614–1641.

Cullen, G., Guillet, J. P., Heinrich, G., Kleinschmidt, T., Pilon, E., Reiter, T., & Rodgers, M. (2011).

Golem95C: A library for one-loop integrals with complex masses.Comput. Phys. Commun., 182, 2276–2284.

Cullen, G., van Deurzen, H., Greiner, N., Heinrich, G., Luisoni, G., Mastrolia, P., Mirabella, E., Ossola, G.,Peraro, T., Schlenk, J., von Soden-Fraunhofen, J. F., & Tramontano, F. (2014).GoSam-2.0: a tool for automated one-loop calculations within the Standard Model and beyond.Eur. Phys. J. C, 74, 8, 3001.

Cyrol, A. K., Mitter, M., & Strodthoff, N. (2016).

FormTracer - A Mathematica Tracing Package Using FORM.

Denner, A., Dittmaier, S., & Hofer, L. (2017).

Collier: a fortran-based Complex One-Loop LIbrary in Extended Regularizations.Comput. Phys. Commun., 212, 220–238.

Feng, F. & Mertig, R. (2012).

FormLink/FeynCalcFormLink : Embedding FORM in Mathematica and FeynCalc.

Fleischer, J. & Riemann, T. (2011).

A complete algebraic reduction of one-loop tensor Feynman integrals.Phys. Rev. D, 83, 073004.


Backup

Hahn, T. & Perez-Victoria, M. (1999).

Automatized One-Loop Calculations in 4 and D dimensions.Comput. Phys. Commun., 118, 153–165.

Jamin, M. & Lautenbacher, M. E. (1993).

TRACER version 1.1.Comput. Phys. Commun., 74(2), 265–288.

Lee, R. N. (2012).

Presenting LiteRed: a tool for the Loop InTEgrals REDuction.

Maitre, D. & Mastrolia, P. (2008).

S@M, a Mathematica Implementation of the Spinor-Helicity Formalism.Comput. Phys. Commun., 179, 501–574.

Mertig, R., Böhm, M., & Denner, A. (1991).

Feyn Calc - Computer-algebraic calculation of Feynman amplitudes.Comput. Phys. Commun., 64(3), 345–359.

Mertig, R. & Scharf, R. (1998).

TARCER - A Mathematica program for the reduction of two-loop propagator integrals.Comput. Phys. Commun., 111, 265–273.

Nogueira, P. (1993).

Automatic Feynman graph generation.J. Comput. Phys., 105, 279–289.


Backup

Orellana, F. (2002).

Mesonic Final State Interactions.PhD thesis, University of Zurich.

Passarino, G. & Veltman, M. (1979).

One Loop Corrections for e+e− Annihilation Into µ+µ− in the Weinberg Model.Nucl. Phys., B160, 151.

Patel, H. H. (2015).

Package-X: A Mathematica package for the analytic calculation of one-loop integrals.Comput. Phys. Commun., 197, 276–290.

Ruijl, B., Ueda, T., & Vermaseren, J. A. M. (2017).

Forcer, a FORM program for the parametric reduction of four-loop massless propagator diagrams.

Shtabovenko, V. (2017).

FeynHelpers: Connecting FeynCalc to FIRE and Package-X.Comput. Phys. Commun., 218, 48–65.

Shtabovenko, V., Mertig, R., & Orellana, F. (2016).

New Developments in FeynCalc 9.0.Comput. Phys. Commun., 207, 432–444.

Smirnov, A. V. (2015).

FIRE5: a C++ implementation of Feynman Integral REduction.Comput. Phys. Commun., 189, 182–191.


Backup

Smirnov, A. V. & Smirnov, V. A. (2013).

FIRE4, LiteRed and accompanying tools to solve integration by parts relations.Comput. Phys. Commun., 184(12), 2820–2827.

Studerus, C. (2009).

Reduze - Feynman Integral Reduction in C++.Comput. Phys. Commun., 181, 1293–1300.

Tentyukov, M. & Fleischer, J. (2000).

A Feynman Diagram Analyser DIANA.Comput. Phys. Commun., 132, 124–141.

van Hameren, A. (2011).

OneLOop: for the evaluation of one-loop scalar functions.Comput. Phys. Commun., 182, 2427–2438.

van Oldenborgh, G. (1991).

FF - a package to evaluate one-loop Feynman diagrams.Comput. Phys. Commun., 66(1), 1–15.

Vermaseren, J. A. M. (2000).

New features of FORM.

Wang, J.-X. (2004).

Progress in FDC project.Nucl. Instrum. Meth. A, 534(1-2), 241–245.


Backup

Wiebusch, M. (2014).

HEPMath: A Mathematica Package for Semi-Automatic Computations in High Energy Physics.Comput. Phys. Commun., 195, 172–190.


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