Mellin-Barnes Integrals: A Primer on Particle Physics Applications

296 Pages·2022·5.596 MB·English

by  Ievgen Dubovyk, Janusz Gluza, Gábor Somogyi

Lecture Notes in Physics
Ievgen Dubovyk
Janusz Gluza
Gábor Somogyi
Mellin-Barnes
Integrals
A Primer on Particle Physics
Applications
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Volume 1008
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Ievgen Dubovyk (cid:129) Janusz Gluza (cid:129)
Gábor Somogyi
Mellin-Barnes Integrals
A Primer on Particle Physics
Applications
IevgenDubovyk JanuszGluza
InstituteofPhysics InstituteofPhysics
UniversityofSilesia UniversityofSilesia
Katowice,Poland Katowice,Poland
GáborSomogyi
InstituteforParticleandNuclearPhysics
WignerResearchCentreforPhysics
Budapest,Hungary
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-3-031-14271-0 ISBN978-3-031-14272-7 (eBook)
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Preface
On the left: Hjalmar Mellin (1854–1933) Finnish mathematician, Helsinki
University of Technology. © Atelier Charles Rüs & Co., Hjalmar Mellin, Ident.Nr.
1992,32/003a.024, Kunstbibliothek, Staatlichen Museen zu Berlin under CC BY-NC-
SA 3.0 DE, http://www.smb-digital.de/eMuseumPlus. On the right: Ernest William
Barnes (1874–1953) British mathematician, liberal theologian and bishop. ©Foto:
https://en.wikipedia.org/wiki/Ernest_Barnes
ThisbookisaboutthemathematicalbasicsofMellin-Barnesintegrals;however,
itismotivatedbythepracticalneedsofprecisioncalculationsinparticlephysics.
In modelling physical phenomena, we aim at predictability and exactness. In
particlephysics,a compellingmodelexists, commonlycalled theStandardModel
(SM), which describeselementaryparticles and their interactions.We know since
the 1960s that gauge interactions, mathematically based on a product of U(1),
SU(2), and SU(3) groups,describethe electromagnetic,weak, and strongforces.
We knowthecharges,spins,CandPparities,aswellasthemassesofelementary
particles.Weunderstandhowfermionsandbosonsinteractbychiralweakinterac-
tionsandweunderstandthedynamicsofthequarksandgluonsthatpartakeinthe
stronginteraction.TheSMdescribesinnumerablephysicalphenomenaobservedin
thelaboratoryandthecosmos,linkingpresent-dayexperimentswiththefirst10−11
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viii Preface
secondoftheUniverse.Lookingatthe listofNobellaureatesinphysics,manyof
themareconnectedwithparticlephysics.Theirawardsspeaktotheappreciationof
theStandardModeltheorybuildersandtotheenormouseffortsmadetoconfirmthe
SMstructureandpredictionsexperimentally.
TheSM issurprisinglyprecisein itspredictionsandcompleteinthe sensethat
allofits theorizedelementaryparticleshavenowbeenobserved.Thelastmissing
piece of the SM, the Higgs boson was discovered in 2012. This discovery was
madepossiblebytheconstructionoftheLHCacceleratorandtheATLASandCMS
detectors,whosedesignswere guidedby theoreticalpredictionsbased onthe SM.
Thus,we nowbelievethatwe can answerthe theoreticalquestiondatingfromthe
1960s:“whatgivesmasstotheelectroweakbosons?”.
However,thisachievementdoesnotstoptheneedforfurtherexploration!
First,doesthe125GeV/c2particlethatwasdiscoveredreallyplaytherolewhich
theorysaysitdoes?Second,severalunansweredmysteriescallfornewphenomena
beyond the SM: (i) the apparent presence of dark matter in the Universe; (ii) the
matter-antimatterasymmetryoftheUniverse;and(iii)theexistenceandpatternof
neutrino masses to name three outstanding ones. Although the list of issues to be
exploredin thefutureislong,1 thereisatpresentnoclear guidanceon theenergy
scaleorstrengthofthenewphenomena.However,sincetheSMiscomplete,there
is no freedom in its predictionsand any observed deviationwould be a discovery
andapreciouspointerintotheunknown.AnambitiousprogramofprecisionHiggs
andelectroweakmeasurementsisthusinorder.However,awell-preparedandwell-
motivatedcolliderprojectrequiresprecisetheoryinputtoguidetheexperimentand
confronttheresults,thusoptimizingthechancesforsignificantscientificprogress.
Ever-increasing mathematical virtuosity is however required to reach the ever-
improvinglevelofexperimentalprecision.
AndthisisaplacewherethetheoryandmethodsconnectedwithMellin-Barnes
integralsdescribedinthebookenter.
To achieve the required precision of theoretical calculations, higher-order per-
turbative corrections in Quantum Field Theory must be calculated. This involves
the computationof so-calledmulti-loopFeynmanintegralsas wellas phasespace
integrals. However, currently there is no complete theory for Feynman integrals
beyondoneloop.Thisdefinesachallengetotheoreticiansandcallsforinnovative
methodsandtools,bothnumericalandanalytical.
Thisbookfocusesononeofthese,theMellin-Barnes(MB)method.Thismethod
hasbeenappliedtorealapplicationsatthefrontierofresearchandhasproventobe
one of veryfew oneswhich could solve so far untouchableproblemsof precision
physics. The book provides a self-contained overview of the necessary elements
of complex analysis and special functions, an easy-to-follow introduction to the
subjectofMBrepresentationswith hands-onexamplesandacomprehensiveview
1 Physics Briefing Book: Input for the European Strategy for Particle Physics Update 2020,
RichardKeithEllisetal.,CERN-ESU-004,arXiv:1910.11775.
Preface ix
ontheanalyticalandnumericalmethodsandtoolsrelatedtoMBintegrals.Itshould
beagoodstartingpointforlearningandusingtheMBmethodinpracticalresearch.
Katowice,Poland IevgenDubovyk
Cisownica,Poland JanuszGluza
Budapest,Hungary GáborSomogyi
October2022
Acknowledgments
We would like to thank for collaborations on the papers connected with the MB
subjectand/ormanyinterestingdiscussionsandinsightsonthesubjectthefollowing
people:
StefanoActis,PaoloBolzoni,MichałCzakon,AndreiDavydychev,ClaudeDuhr,
WojciechFlieger,AyresFreitas,MarekGluza,KrzysztofGrzanka,TomaszJelin´ski,
Krzysztof Kajda, Mikhail Kalmykov,David Kosower, Sven-Olaf Moch, Vladimir
A.Smirnov,BasTausk,JohannUsovitsch,andSzymonZie¸ba.
This work has been supported in part by the Polish National Science Center
(NCN)undergrants2017/25/B/ST2/01987and2020/37/B/ST2/02371.
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