## Abstract

We give a Hopf-algebraic formulation of the *R*^{∗}-operation, which is a canonical way to render UV and IR divergent Euclidean Feynman diagrams finite. Our analysis uncovers a close connection to Brown’s Hopf algebra of motic graphs. Using this connection we are able to provide a verbose proof of the long observed ‘commutativity’ of UV and IR subtractions. We also give a new duality between UV and IR counterterms, which, entirely algebraic in nature, is formulated as an inverse relation on the group of characters of the Hopf algebra of log-divergent scaleless Feynman graphs. Many explicit examples of calculations with applications to infrared rearrangement are given.

## Article PDF

### Similar content being viewed by others

## References

D. Kreimer,

*On the Hopf algebra structure of perturbative quantum field theories*,*Adv. Theor. Math. Phys.***2**(1998) 303 [q-alg/9707029] [INSPIRE].A. Connes and D. Kreimer,

*Renormalization in quantum field theory and the Riemann-Hilbert problem. 1. The Hopf algebra structure of graphs and the main theorem*,*Commun. Math. Phys.***210**(2000) 249 [hep-th/9912092] [INSPIRE].N.N. Bogoliubov and O.S. Parasiuk,

*On the multiplication of the causal function in the quantum theory of fields*,*Acta Math.***97**(1957) 227 [INSPIRE].K. Hepp,

*Proof of the Bogolyubov-Parasiuk theorem on renormalization*,*Commun. Math. Phys.***2**(1966) 301 [INSPIRE].W. Zimmermann,

*Convergence of Bogolyubov’s method of renormalization in momentum space*,*Commun. Math. Phys.***15**(1969) 208 [INSPIRE].A.B. Goncharov,

*Multiple polylogarithms and mixed Tate motives*, math.AG/0103059.F.C.S. Brown,

*On the decomposition of motivic multiple zeta values*, in*Galois-Teichmüller Theory and Arithmetic Geometry*, Advanced Studies in Pure Mathematics, volume 63, Mathematical Society of Japan, Tokyo Japan (2012), p. 31 [arXiv:1102.1310] [INSPIRE].C. Duhr,

*Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes*,*JHEP***08**(2012) 043 [arXiv:1203.0454] [INSPIRE].S. Bloch, H. Esnault and D. Kreimer,

*On Motives associated to graph polynomials*,*Commun. Math. Phys.***267**(2006) 181 [math/0510011] [INSPIRE].F.C.S. Brown,

*On the periods of some Feynman integrals*, arXiv:0910.0114 [INSPIRE].P. Belkale and P. Brosnan,

*Matroids motives, and a conjecture of Kontsevich*,*Duke Math. J.***116**(2003) 147.F.C.S. Brown and O. Schnetz,

*A K*3*in*𝜙^{4},*Duke Math. J.***161**(2012) 1817 [arXiv:1006.4064] [INSPIRE].F.C.S. Brown,

*The Massless higher-loop two-point function*,*Commun. Math. Phys.***287**(2009) 925 [arXiv:0804.1660] [INSPIRE].E. Panzer,

*Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals*,*Comput. Phys. Commun.***188**(2015) 148 [arXiv:1403.3385] [INSPIRE].F.C.S. Brown,

*Feynman amplitudes, coaction principle and cosmic Galois group*,*Commun. Num. Theor. Phys.***11**(2017) 453 [arXiv:1512.06409] [INSPIRE].O. Schnetz,

*Numbers and Functions in Quantum Field Theory*,*Phys. Rev.***D 97**(2018) 085018 [arXiv:1606.08598] [INSPIRE].F.C.S. Brown,

*Mixed tate motives over*ℤ,*Ann. Math.***175**(2012) 949.S. Abreu, R. Britto, C. Duhr and E. Gardi,

*Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction*,*Phys. Rev. Lett.***119**(2017) 051601 [arXiv:1703.05064] [INSPIRE].S. Abreu, R. Britto, C. Duhr and E. Gardi,

*Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case*,*JHEP***12**(2017) 090 [arXiv:1704.07931] [INSPIRE].S. Abreu, R. Britto, C. Duhr, E. Gardi and J. Matthew,

*Diagrammatic Coaction of Two-Loop Feynman Integrals*, in proceedings of the*14th International Symposium on Radiative Corrections: Application of Quantum Field Theory to Phenomenology (RADCOR 2019)*, Avignon, France, 8–13 September 2019, arXiv:1912.06561 [INSPIRE].K.G. Chetyrkin and F.V. Tkachov,

*Infrared R operation and ultraviolet counterterms in the MS scheme*,*Phys. Lett.***B 114**(1982) 340 [INSPIRE].K.G. Chetyrkin and V.A. Smirnov,

*R*^{∗}*-Operation corrected*,*Phys. Lett.***B 144**(1984) 419 [INSPIRE].V.A. Smirnov and K.G. Chetyrkin,

*R*^{∗}*Operation in the Minimal Subtraction Scheme*,*Theor. Math. Phys.***63**(1985) 462 [INSPIRE].H. Kleinert and V. Schulte-Frohlinde,

*Critical Properties of*𝜙^{4}*-Theories*, World Scientific (2001).S. Larin and P. van Nieuwenhuizen,

*The Infrared R*^{∗}*operation*, hep-th/0212315 [INSPIRE].K.G. Chetyrkin,

*Combinatorics of R-, R*^{−1}*- and R*^{∗}*-operations and asymptotic expansions of Feynman integrals in the limit of large momenta and masses*, arXiv:1701.08627 [INSPIRE].A.A. Vladimirov,

*Method for Computing Renormalization Group Functions in Dimensional Renormalization Scheme*,*Theor. Math. Phys.***43**(1980) 417 [INSPIRE].K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov,

*Five Loop Calculations in the g*𝜙^{4}*Model and the Critical Index η*,*Phys. Lett.***B 99**(1981) 147 [*Erratum ibid.***B 101**(1981) 457] [INSPIRE].H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin and S.A. Larin,

*Five-loop renormalization group functions of O*(*n*)*-symmetric*𝜙^{4}*-theory and E-expansions of critical exponents up to ∈*^{5},*Phys. Lett.***B 272**(1991) 39 [*Erratum ibid.***B 319**(1993) 545] [hep-th/9503230] [INSPIRE].M. Kompaniets and E. Panzer,

*Renormalization group functions of*𝜙^{4}*theory in the MS-scheme to six loops*,*PoS***LL2016**(2016) 038 [arXiv:1606.09210] [INSPIRE].M.V. Kompaniets and E. Panzer,

*Minimally subtracted six loop renormalization of O*(*n*)*-symmetric*𝜙^{4}*theory and critical exponents*,*Phys. Rev.***D 96**(2017) 036016 [arXiv:1705.06483] [INSPIRE].D.V. Batkovich, K.G. Chetyrkin and M.V. Kompaniets,

*Six loop analytical calculation of the field anomalous dimension and the critical exponent η in O*(*n*)*-symmetric φ*^{4}*model*,*Nucl. Phys.***B 906**(2016) 147 [arXiv:1601.01960] [INSPIRE].F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt,

*The five-loop beta function of Yang-Mills theory with fermions*,*JHEP***02**(2017) 090 [arXiv:1701.01404] [INSPIRE].F. Herzog, S. Moch, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt,

*Five-loop contributions to low-N non-singlet anomalous dimensions in QCD*,*Phys. Lett.***B 790**(2019) 436 [arXiv:1812.11818] [INSPIRE].F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt,

*On Higgs decays to hadrons and the R-ratio at N*^{4}*LO*,*JHEP***08**(2017) 113 [arXiv:1707.01044] [INSPIRE].F. Herzog and B. Ruijl,

*The R*^{∗}*-operation for Feynman graphs with generic numerators*,*JHEP***05**(2017) 037 [arXiv:1703.03776] [INSPIRE].P.A. Baikov, K.G. Chetyrkin, J.H. Kühn and J. Rittinger,

*Complete*\( \mathcal{O}\left({\alpha}_s^4\right) \)*QCD Corrections to Hadronic Z -Decays*,*Phys. Rev. Lett.***108**(2012) 222003 [arXiv:1201.5804] [INSPIRE].P.A. Baikov, K.G. Chetyrkin and J.H. Kühn,

*Quark Mass and Field Anomalous Dimensions to*\( \mathcal{O}\left({\alpha}_s^5\right) \),*JHEP***10**(2014) 076 [arXiv:1402.6611] [INSPIRE].P.A. Baikov, K.G. Chetyrkin and J.H. Kühn,

*Scalar correlator at*\( O\left({\alpha}_s^4\right) \)*, Higgs decay into b-quarks and bounds on the light quark masses*,*Phys. Rev. Lett.***96**(2006) 012003 [hep-ph/0511063] [INSPIRE].P.A. Baikov, K.G. Chetyrkin and J.H. Kühn,

*Five-Loop Running of the QCD coupling constant*,*Phys. Rev. Lett.***118**(2017) 082002 [arXiv:1606.08659] [INSPIRE].K.G. Chetyrkin, G. Falcioni, F. Herzog and J.A.M. Vermaseren,

*Five-loop renormalisation of QCD in covariant gauges*,*JHEP***10**(2017) 179 [arXiv:1709.08541] [INSPIRE].E.R. Speer,

*Ultraviolet and infrared singularity structure of generic Feynman amplitudes*,*Ann. Inst. Henri Poincaŕe Phys. Theor.***23**(1975) 1 [INSPIRE].K. Hepp,

*Proof of the Bogolyubov-Parasiuk theorem on renormalization*,*Commun. Math. Phys.***2**(1966) 301 [INSPIRE].W. Zimmermann,

*The power counting theorem for Minkowski metric*,*Commun. Math. Phys.***11**(1968) 1 [INSPIRE].J.C. Collins,

*Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion*, Cambridge University Press, Cambridge U.K. (1985).W.E. Caswell and A.D. Kennedy,

*A simple approach to renormalization theory*,*Phys. Rev.***D 25**(1982) 392 [INSPIRE].V.A. Smirnov,

*Absolutely convergent alpha representation of analytically and dimensionally regularized Feynman amplitudes*,*Theor. Math. Phys.***59**(1984) 563 [INSPIRE].V.A. Smirnov,

*Asymptotic expansions in limits of large momenta and masses*,*Commun. Math. Phys.***134**(1990) 109 [INSPIRE].J.C. Collins,

*Normal Products in Dimensional Regularization*,*Nucl. Phys.***B 92**(1975) 477 [INSPIRE].G. Leibbrandt,

*Introduction to the Technique of Dimensional Regularization*,*Rev. Mod. Phys.***47**(1975) 849 [INSPIRE].K.G. Chetyrkin and V.A. Smirnov,

*Dimensional regularization and infrared divergences*,*Theor. Math. Phys.***56**(1984) 770 [INSPIRE].K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov,

*New Approach to Evaluation of Multiloop Feynman Integrals: The Gegenbauer Polynomial x-Space Technique*,*Nucl. Phys.***B 174**(1980) 345 [INSPIRE].D. Manchon,

*Hopf algebras, from basics to applications to renormalization*, in proceedings of the*5th Mathematical Meeting of Glanon: Algebra, Geometry and Applications to Physics*, Glanon, Burgundy, France, 2–6 July 2001, math.QA/0408405 [INSPIRE].M. Borinsky,

*Graphs in perturbation theory: algebraic structure and asymptotics*, Springer (2018).M. Sweedler,

*Hopf algebras*, Mathematics Lecture Note Series, W.A. Benjamin, Inc. (1969).M. Borinsky,

*Algebraic lattices in QFT renormalization*,*Lett. Math. Phys.***106**(2016) 879 [arXiv:1509.01862] [INSPIRE].B. Ruijl, T. Ueda and J. Vermaseren,

*FORM version 4.2*, arXiv:1707.06453 [INSPIRE].T. Ueda, B. Ruijl and J.A.M. Vermaseren,

*Forcer: a FORM program for*4*-loop massless propagators*,*PoS***LL2016**(2016) 070 [arXiv:1607.07318] [INSPIRE].J. Kock,

*Perturbative Renormalisation for Not-Quite-Connected Bialgebras*,*Lett. Math. Phys.***105**(2015) 1413 [arXiv:1411.3098] [INSPIRE].D. Kreimer,

*Anatomy of a gauge theory*,*Annals Phys.***321**(2006) 2757 [hep-th/0509135] [INSPIRE].D. Kreimer and K. Yeats,

*An Etude in non-linear Dyson-Schwinger Equations*,*Nucl. Phys. Proc. Suppl.***160**(2006) 116 [hep-th/0605096] [INSPIRE].W.D. van Suijlekom,

*Renormalization of gauge fields: A Hopf algebra approach*,*Commun. Math. Phys.***276**(2007) 773 [hep-th/0610137] [INSPIRE].M. Borinsky,

*Feynman graph generation and calculations in the Hopf algebra of Feynman graphs*,*Comput. Phys. Commun.***185**(2014) 3317 [arXiv:1402.2613] [INSPIRE].B. Humpert and W.L. van Neerven,

*Diagrammatic mass factorization*,*Phys. Rev.***D 25**(1982) 2593 [INSPIRE].J.C. Collins and D.E. Soper,

*Back-To-Back Jets in QCD*,*Nucl. Phys.***B 193**(1981) 381 [*Erratum ibid.***B 213**(1983) 545] [INSPIRE].J.C. Collins,

*Foundations of perturbative QCD*,*Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol.***32**(2011) 1 [INSPIRE].O. Erdoğan and G. Sterman,

*Ultraviolet divergences and factorization for coordinate-space amplitudes*,*Phys. Rev.***D 91**(2015) 065033 [arXiv:1411.4588] [INSPIRE].F. Caola, K. Melnikov and R. R¨ontsch,

*Nested soft-collinear subtractions in NNLO QCD computations*,*Eur. Phys. J.***C 77**(2017) 248 [arXiv:1702.01352] [INSPIRE].L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati,

*Factorisation and Subtraction beyond NLO*,*JHEP***12**(2018) 062 [arXiv:1809.05444] [INSPIRE].F. Herzog,

*Geometric IR subtraction for final state real radiation*,*JHEP***08**(2018) 006 [arXiv:1804.07949] [INSPIRE].Y. Ma,

*A Forest Formula to Subtract Infrared Singularities in Amplitudes for Wide-angle Scattering*,*JHEP***05**(2020) 012 [arXiv:1910.11304] [INSPIRE].

##
**Open Access**

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2003.04301

## Rights and permissions

**Open Access** . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## About this article

### Cite this article

Beekveldt, R., Borinsky, M. & Herzog, F. The Hopf algebra structure of the R^{∗}-operation.
*J. High Energ. Phys.* **2020**, 61 (2020). https://doi.org/10.1007/JHEP07(2020)061

Received:

Revised:

Accepted:

Published:

DOI: https://doi.org/10.1007/JHEP07(2020)061