Hopf algebra structure of the two loop three mass nonplanar Feynman diagram

Ananthanarayan, B.; Das, Abhijit B.; Wyler, Daniel

doi:10.1103/PhysRevD.104.076002

Hopf algebra structure of the two loop three mass nonplanar Feynman diagram

Apr 2, 2021

26 pages

Published in:

Phys.Rev.D 104 (2021) 7, 076002

Published: Oct 1, 2021

e-Print:

2104.00967 [hep-th]

DOI:

10.1103/PhysRevD.104.076002 (publication)

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Abstract: (APS)

The method of using Hopf algebras for calculating Feynman integrals developed by Abreu et al. is applied to the two-loop nonplanar on-shell diagram with massless propagators and three external mass scales. We show that the existence of the method of cut Feynman diagrams comprising of the coproduct, the first entry condition and integrability condition that was found to be true for the planar case also holds for the nonplanar case; furthermore, the nonplanar symbol alphabet is the same as for the planar case. This is one of the main results of this work which have been obtained by a systematic analysis of the relevant cuts, using the symbolic manipulation codes hypexp and polylogtools. The obtained result for the symbol is cross-checked by an analysis of the known two-loop original Feynman integral result. In addition, we also reconstruct the full result from the symbol. This is the second main result of this paper.

Note:

26 pages, 5 main figures, 1 table, 1 footnote; Compared to v1, many improvements in discussion. A new section (sec. 8) has been added which now gives the full result. Discussion and conclusions extended and improved. Abstract and introduction also amended to account for the new calculation. New references added

algebra: Hopf
propagator: massless
mass: scale
Feynman graph
integrability
structure
loop integral
computer: algebra
mathematical methods

References(33)

Figures(10)

[1]

From multiple unitarity cuts to the coproduct of Feynman integrals

Samuel Abreu
(
- U. Edinburgh, Higgs Ctr. Theor. Phys. and
- IPhT, Saclay
)
,
Ruth Britto
(
- IPhT, Saclay
)
,
Claude Duhr
(
- Durham U., IPPP
)
,
Einan Gardi
(
- U. Edinburgh, Higgs Ctr. Theor. Phys.
)

- JHEP 10 (2014) 125
•
e-Print:
- 1401.3546
•
DOI:
- 10.1007/JHEP10(2014)125

[2]

Cuts and coproducts of massive triangle diagrams

Samuel Abreu
(
- U. Edinburgh, Higgs Ctr. Theor. Phys. and
- IPhT, Saclay
)
,
Ruth Britto
(
)
,
Hanna Grönqvist
(
- DSM, DAPNIA, Saclay
)

- JHEP 07 (2015) 111
•
e-Print:
- 1504.00206
•
DOI:
- 10.1007/JHEP07(2015)111

[3]

Cuts from residues: the one-loop case

Samuel Abreu
(
- Freiburg U.
)
,
Ruth Britto
(
- Trinity Coll., Dublin and
- IPhT, Saclay
)
,
Claude Duhr
(
- CERN and
- Louvain U., CP3
)
,
Einan Gardi
(
- U. Edinburgh, Higgs Ctr. Theor. Phys. and
- Edinburgh U.
)

- JHEP 06 (2017) 114
•
e-Print:
- 1702.03163
•
DOI:
- 10.1007/JHEP06(2017)114

[4]

Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case

Samuel Abreu
(
- Freiburg U.
)
,
Ruth Britto
(
- Trinity Coll., Dublin and
- Hamilton Math. Inst., Dublin and
- IPhT, Saclay and
- Santa Barbara, KITP
)
,
Claude Duhr
(
- CERN and
- Louvain U., CP3
)
,
Einan Gardi
(
- U. Edinburgh, Higgs Ctr. Theor. Phys.
)

- JHEP 12 (2017) 090
•
e-Print:
- 1704.07931
•
DOI:
- 10.1007/JHEP12(2017)090

[5]

Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes

Claude Duhr
(
- Zurich, ETH
)

- JHEP 08 (2012) 043
•
e-Print:
- 1203.0454
•
DOI:
- 10.1007/JHEP08(2012)043

[6]

An Approach to the evaluation of three and four point ladder diagrams

- Phys.Lett.B 298 (1993) 363-370,
- Phys. Lett. B298 (1993) 363-370 and Moscow State Univ. - 92-38-287 (92/01,rec.Feb.93) 11 p
•
DOI:
- 10.1016/0370-2693(93)91834-A

[7]

Some exact results for two-loop diagrams with three and four external lines

N.I. Ussyukina
,
A.I. Davydychev

- Phys. At. Nucl. 56 (1993) 1553

[8]

Bootstrapping pentagon functions

- JHEP 05 (2018) 164
•
e-Print:
- 1712.09610
•
DOI:
- 10.1007/JHEP05(2018)164

[9]

Analytic result for the nonplanar hexa-box integrals

D. Chicherin
(
- U. Mainz, PRISMA
)
,
T. Gehrmann
(
- Zurich U.
)
,
J.M. Henn
(
- U. Mainz, PRISMA and
- Munich, Max Planck Inst.
)
,
N.A. Lo Presti
(
- Durham U., IPPP
)
,
V. Mitev
(
- U. Mainz, PRISMA
)

et al.

- JHEP 03 (2019) 042
•
e-Print:
- 1809.06240
•
DOI:
- 10.1007/JHEP03(2019)042

[10]

Differential equations from unitarity cuts: nonplanar hexa-box integrals

- JHEP 01 (2019) 006
•
e-Print:
- 1807.11522
•
DOI:
- 10.1007/JHEP01(2019)006

[11]

The two-loop five-point amplitude in $\mathcal{N} =4$ super-Yang-Mills theory

- Phys.Rev.Lett. 122 (2019) 12, 121603
•
e-Print:
- 1812.08941
•
DOI:
- 10.1103/PhysRevLett.122.121603

[12]

New results for two loop off-shell three point diagrams

- Phys.Lett.B 332 (1994) 159-167
•
e-Print:
- hep-ph/9402223
•
DOI:
- 10.1016/0370-2693(94)90874-5

[13]

Dimensionally regulated one loop box scalar integrals with massless internal lines

- Eur.Phys.J.C 20 (2001) 357-370
•
e-Print:
- hep-ph/0006249
•
DOI:
- 10.1007/s100520100675

[14]

Multiple polylogarithms, cyclotomy and modular complexes

Alexander B. Goncharov

- Math.Res.Lett. 5 (1998) 497-516
•
e-Print:
- 1105.2076
•
DOI:
- 10.4310/MRL.1998.v5.n4.a7

[15]

Multiple polylogarithms and mixed Tate motives

A.B. Goncharov

e-Print:
- math/0103059

[16]

Galois symmetries of fundamental groupoids and noncommutative geometry

A.B. Goncharov

- Duke Math.J. 128 (2005) 209
•
e-Print:
- math/0208144
•
DOI:
- 10.1215/S0012-7094-04-12822-2

[17]

Classical Polylogarithms for Amplitudes and Wilson Loops

- Phys.Rev.Lett. 105 (2010) 151605
•
e-Print:
- 1006.5703
•
DOI:
- 10.1103/PhysRevLett.105.151605

[18]

Iterated path integrals

Kuo-Tsai Chen

- Bull.Am.Math.Soc. 83 (1977) 831-879
•
DOI:
- 10.1090/S0002-9904-1977-14320-6

[19]

A simple construction of Grassmannian polylogarithms

A.B. Goncharov

- Adv.Math. 241 (2013) 79
•
DOI:
- 10.1016/j.aim.2013.03.018

[20]

Multiple zeta values and periods of moduli spaces

F.C. Brown

- Ann. Sci. Ecole Norm. Sup. 42 (2009) 371

[21]

From polygons and symbols to polylogarithmic functions

Claude Duhr
(
- Durham U., IPPP and
- Zurich, ETH
)
,
Herbert Gangl
(
- Durham U., Dept. of Math.
)
,
John R. Rhodes
(
- Durham U., Dept. of Math.
)

- JHEP 10 (2012) 075
•
e-Print:
- 1110.0458
•
DOI:
- 10.1007/JHEP10(2012)075

[22]

Singularities and discontinuities of Feynman amplitudes

R.E. Cutkosky
(
- Carnegie Mellon U.
)

- J.Math.Phys. 1 (1960) 429-433
•
DOI:
- 10.1063/1.1703676

[23]

DIAGRAMMAR

Gerard 't Hooft
(
- CERN
)
,
M.J.G. Veltman
(
- CERN
)

- NATO Sci.Ser.B 4 (1974) 177-322,
•
DOI:
- 10.1007/978-1-4684-2826-1_5

[24]

Diagrammatica: The Path to Feynman rules

M.J.G. Veltman
(
- Michigan U.
)

- Cambridge Lect.Notes Phys. 4 (1994) 1-284,
- Cambridge, UK: Univ. Pr. (1994) 284 p. (Cambridge lecture notes in physics, 4)

[25]

On analytic properties of vertex parts in quantum field theory

L.D. Landau
(
- Moscow, Inst. Phys. Problems
)

- Nucl.Phys. 13 (1959) 1, 181-192,
- Sov.Phys.JETP 10 (1960) 1, 45-50,
- Zh.Eksp.Teor.Fiz. 37 (1959) 1, 62-70
•
DOI:
- 10.1016/B978-0-08-010586-4.50103-6

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