Scattering Amplitudes in Gauge Theories

206 Pages·2014·2.194 MB·English

by  Johannes M. Henn, Jan C. Plefka (auth.)

Lecture Notes in Physics
Volume 883
FoundingEditors
W.Beiglböck
J.Ehlers
K.Hepp
H.Weidenmüller
EditorialBoard
B.-G.Englert,Singapore,Singapore
P.Hänggi,Augsburg,Germany
W.Hillebrandt,Garching,Germany
M.Hjorth-Jensen,Oslo,Norway
R.A.L.Jones,Sheffield,UK
M.Lewenstein,Barcelona,Spain
H.vonLöhneysen,Karlsruhe,Germany
M.S.Longair,Cambridge,UK
J.-F.Pinton,Lyon,France
J.-M.Raimond,Paris,France
A.Rubio,Donostia,SanSebastian,Spain
M.Salmhofer,Heidelberg,Germany
S.Theisen,Potsdam,Germany
D.Vollhardt,Augsburg,Germany
J.D.Wells,Geneva,Switzerland
Forfurthervolumes:
www.springer.com/series/5304
The Lecture Notes in Physics
TheseriesLectureNotesinPhysics(LNP),foundedin1969,reportsnewdevelop-
ments in physics research and teaching—quickly and informally, but with a high
quality and the explicit aim to summarize and communicate current knowledge in
an accessible way. Books published in this series are conceived as bridging mate-
rialbetweenadvancedgraduatetextbooksandtheforefrontofresearchandtoserve
threepurposes:
• to be a compact and modern up-to-date source of reference on a well-defined
topic
• to serve as an accessible introduction to the field to postgraduate students and
nonspecialistresearchersfromrelatedareas
• to be a source of advanced teaching material for specialized seminars, courses
andschools
Both monographs and multi-author volumes will be considered for publication.
Editedvolumesshould,however,consistofaverylimitednumberofcontributions
only.ProceedingswillnotbeconsideredforLNP.
Volumes published in LNP are disseminated both in print and in electronic for-
mats,theelectronicarchivebeingavailableatspringerlink.com.Theseriescontent
isindexed,abstractedandreferencedbymanyabstractingandinformationservices,
bibliographicnetworks,subscriptionagencies,librarynetworks,andconsortia.
Proposals should be sent to a member of the Editorial Board, or directly to the
managingeditoratSpringer:
ChristianCaron
SpringerHeidelberg
PhysicsEditorialDepartmentI
Tiergartenstrasse17
69121Heidelberg/Germany
[email protected]
Johannes M. Henn (cid:2) Jan C. Plefka
Scattering Amplitudes
in Gauge Theories
Dr.JohannesM.Henn Prof.Dr.JanC.Plefka
SchoolofNaturalSciences InstitutfürPhysik
InstituteforAdvancedStudy Math.-Naturw.Fakultät
Princeton,NJ,USA HumboldtUniversitätzuBerlin
Berlin,Germany
ISSN0075-8450 ISSN1616-6361(electronic)
LectureNotesinPhysics
ISBN978-3-642-54021-9 ISBN978-3-642-54022-6(eBook)
DOI10.1007/978-3-642-54022-6
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014932295
©Springer-VerlagBerlinHeidelberg2014
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub-
lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany
errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect
tothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
ToMieke
JP
Preface
In our current understanding of subatomic particle physics, recently successfully
tested with the discovery of the Higgs boson, processes involving elementary par-
ticles are described by quantum field theory. Scattering amplitudes are the central
objectsinperturbativequantumfieldtheoryastheylinkthetheoreticaldescriptionto
experimentalpredictions.Toalargedegreethehistoricaldevelopmentofquantum
field theory was driven by the need to compute scattering amplitudes. They con-
stitute the integral building block for the construction of scattering cross sections
determiningtheprobabilitiesforscatteringprocessestooccuratparticlecolliders.
Non-abeliangaugefieldorYang-Millstheoriesrepresentthebackboneofhighen-
ergyphysics,astheyprovidethetheoreticalframeworktodescribetheinteractions
of elementary particles in the standard model. The principles for computing scat-
teringamplitudesingaugetheorieshavebeensettledsincethemid1970s:oncethe
Lagrangiananditsgaugefixingtermhasbeenconstructed,onereadsofftheFeyn-
manrulesforthescalar,fermionandgaugefields.Thescatteringamplitudeisthen
givenbythesumofallcontributingdiagramsbuiltfromtheFeynmanruleswiththe
subsequent integration over the internal loop momenta. Divergences in these inte-
grals require regularization, and the ultraviolet divergences lead to a renormaliza-
tionofthetheory.Sofromaconceptualviewpointonemightconsiderthischapter
ofquantumfieldtheoryascomplete.
However, it turns out that beyond the simplest examples the complexity of the
Feynmandiagrammaticcomputationquicklygetsoutofhand.Alreadyattree-level
suchacomputationcanbecomeenormousifoneisdealingwithgaugeinteractions.
For example, the number of Feynman diagrams contributing to a gluon tree-level
amplitude g+g→n·g growsfactoriallywith n [1].Ontheotherhand,thefinal
answercanbeobtainedinclosedanalyticform,and,whenexpressedinconvenient
variables,isremarkablysimple.Therearetwocomplementarywaysofunderstand-
ingthesimplicityofthefinalanswer.AreasonforthecomplexityoftheFeynman
diagramcalculationisthatindividualFeynmandiagramsaregaugevariantandin-
volve off-shell intermediate states in internal propagators. The amplitude, on the
other hand, is gauge invariant and only knows about on-shell degrees of freedom.
Hence,ingoingfromFeynmandiagramstoanamplitudetheunphysicaldegreesof
vii
viii Preface
freedomcancel.On-shellapproachesthatfocusontheanalyticstructureofthefinal
resultallowtocircumventtheseunnecessarycomplications.Anotherreasonforthe
simplicityofthefinalanswerhastodowithsymmetrypropertiesandismoresur-
prising: it turns out that besides the obvious symmetries of the Lagrangian, gluon
amplitudeshaveadditional,hiddensymmetries,thatconstraintheirform.Choosing
appropriatevariablesmakestheactionofthesesymmetriestransparentandthereby
also simplifies the expressions. For these reasons, both analyticity and symmetry
propertiesaretwotopicsthatwillplayanimportantroleintheselecturenotes.
Inthepastdecadetremendousprogresswasmadebothinourunderstandingof
the structure of scattering amplitudes and in the ability of computing the latter in
gaugetheoriesathighmultiplicities(numberofexternallegs)andbeyondtheone-
looporder.Theseadvancesprovidedthefieldwithanewsetoftoolsthatgobeyond
the textbook approaches. These methods build on using a color decomposition of
thegaugetheoryamplitudesandonexpressingtheminaspinorhelicitybasispar-
ticularlysuitedformasslessparticles.Thinkingabouttheanalyticstructureoftree-
levelamplitudesleadstonovelon-shellrecursionrelations.Theyallowtheanalytic
constructionoftree-levelamplitudesfromatomisticthree-pointones.Atlooplevel
unitarity-based techniques, combined with the knowledge of an integral basis for
one-loop Feynman integrals, may be used to construct loop amplitudes from tree-
level amplitudes. In summary, all amplitudes follow from the on-shell three-point
vertices,andnoreferencetothecomplicatedformoftheLagrangian,gaugefixing
termsandghostsisnecessary.
Our main focus is on methods based on general principles such as analyticity
and symmetries. The progress for generic gauge theories was often led by studies
in the maximally supersymmetric gauge theory in four dimensions, N =4 super
Yang-Mills. It may be thought of as an idealized version of QCD and provides an
ideallaboratoryforthedevelopmentandtestingofnewapproaches.Infact,gluon
tree-levelamplitudesinbothmodelsareidentical,buttheycanbeeasilysolvedfor
withintheN =4superYang-Millsframework,inpartthankstofascinatinghidden
symmetries. Discussing this theory in these lecture notes also allows us to have
manyexampleswherescatteringamplitudescanbecomputedanalytically,andwe
hopethatthiswillhelptoillustratethegeneralproperties.
These modern helicity amplitude or on-shell methods are so far not treated in
depthinstandardtextbooksonquantumfieldtheory.Yettheyareofincreasingim-
portance in phenomenological applications, which face the demand of high preci-
sionpredictionsforhighmultiplicityscatteringeventsattheLargeHadronCollider,
aswellasinfoundationalstudiesinquantumgaugefieldtheorytowardsthestruc-
ture,possiblesymmetriesandconstructionoftheperturbativeS-matrix.Infactthese
developments have led to a cross-fertilization of high-energy phenomenology and
formal theory in recent years. This has made the field of scattering amplitudes a
very active and fascinating area of research attracting many new researchers and
students.Thegoaloftheselecturenotesinphysicsistoservethesecommunitiesin
bridgingthegapfromatextbookknowledgeofquantumfieldtheorytoaworking
expertisefordoingresearchinthedomainofscatteringamplitudes.Ouraimwasto
keepthesenotescompactsothattheyaresuitableforanadvancedgraduatelecture
Preface ix
course.Inordertomakethemmoreaccessibleforstudents,wehaveincludedase-
riesofexercises,togetherwiththeirsolutions.Duetoourwishtokeepthevolume
of these notes amenable to a lecture course, a certain selection of topics, certainly
biasedbypersonaltastes,hadtobemade.Attheendofeachchapterwetherefore
give an account of related developments in the field not discussed in the lecture
notesandgivereferencestoallrelevantworks.
The level of the text aims at advanced graduate students who have already fol-
lowed an introductory course on quantum field theory, typically leading to QED
scattering processes and a few simple one-loop computations therein. The basics
of non-abelian quantum field theories are briefly reviewed in chapter one of this
book.Herealsothebasictoolsofthemodernhelicityamplitudeapproacharepro-
videdand somesimple tree-leveldiagrams are computedin the conventionalway,
using color-ordered Feynman rules. Chapter two introduces the on-shell recursion
anddiscussesuniversalfactorizationpropertiesforcolororderedamplitudes.Aftera
discussionofPoincaréandconformalsymmetry,theN =4superYang-Millsthe-
oryisintroduced,alongwithitssuper-amplitudeformalismandasupersymmetric
on-shellrecursionrelationallowingforanexactsolution.Inchapterthreetheloop-
levelstructureofamplitudesisreviewed.Herethereductionofone-loopFeynman
integrals to a basis of scalar integrals is discussed. The idea of (generalized) uni-
tarity in constructing one-loop amplitudes from tree-level data by putting various
internallegson-shellisreviewedandanumberofconcreteexamplesarecomputed
indetail.Inthesecondpartofchapterthreewegiveanintroductiontotheevaluation
ofFeynmanintegralsatoneandhigherlooporder.Afterreviewingtheirdefinition
andusefulparameterrepresentations,suchastheFeynmanandMellinrepresenta-
tions,wegiveanintroductiontotheintegrationbypartstechniquesinconjunction
withdifferentialequations.Thefinalchapterisdevotedtoadvancedtopicsmostly
within maximally supersymmetric gauge theory: recursion relations for loop inte-
grands,thedualityofscatteringamplitudestoWilsonloopswithlight-likecontours
andcorrelationfunctionsoflocaloperators.Finally,thehiddendualconformaland
YangiansymmetriesofN =4superYang-Millsamplitudesarediscussedpointing
towardsafascinatinghiddenintegrabilityofthisfour-dimensionalgaugetheory.
Someelementsofmodernon-shellmethodsarealsodiscussedintherecentquan-
tumfieldtheorytextbooksofSrednicki[2]andZee[3].Introductoryandin-depth
reviews exist on the various topics presented in these lecture notes. These are [1,
4–6] on the topics of chapters one and two. The integral reduction and unitarity
methodsarepedagogicallyreviewedin[7,8]fromaphenomenologicalviewpoint.
ThecomprehensivetextbookofSmirnov[9,10]discussestechniquesforthecom-
putation of Feynman integrals in great detail. As for more formal aspects related
to N =4 super Yang-Mills, symmetries and dualities, there exists a special issue
ofJ.Phys.A[11–24],aswellastheveryrecentcomprehensivereview[25].These
referenceshaveoverlapswithsomeofthematerialpresentedinthesenotes,butalso
discusssupergravityamplitudes,dualitiesofgaugeandgravityamplitudes,ampli-
tudesintwistorspace,andtheGrassmannianapproachtosuperamplitudes.Thisis
certainlynotacompletelistbutreflectsthetextsfromwhichwehavelearnedalot
ourselves.
x Preface
This manuscript grew out of lecture notes of a physics master’s level course
taughtatHumboldtUniversityBerlininthesummertermsof2011and2013,aswell
as a series of shorter lecture series delivered at research schools in Atrani (Italy),
Dubna (Russia), Durham (UK), Copenhagen (Denmark), Parma (Italy), Waterloo
(Canada)andWolfersdorf(Germany)intheyears2010to2013.
We would like to thank the students and participants of these lectures and
schools for questions and comments on the manuscript, in particular L. Bianchi,
M.Heinze,D.MüllerandH.Münkler.WethankH.Stephanforhelpwithtypeset-
ting the graphs and diagrams. We would like to thank our colleagues V. Smirnov,
P.UwerandG.Yangforimportantcommentsonthemanuscript.Finally,wewould
like to thank N. Arkani-Hamed, S. Badger, N. Beisert, Z. Bern, B. Biedermann,
R. Boels, L. Ferro, S. Caron-Huot, L. Dixon, J. Drummond, H. Elvang, H. Jo-
hanssen, G. Korchemsky, D. Kosower, T. Lukowski, T. McLoughlin, S. Moch,
T. Schuster, E. Sokatchev, M. Staudacher, and P. Uwer for important discussions
orpleasantcollaborationsontopicsrelatedtotheselecturenotes.
References
1. M.L.Mangano,S.J.Parke,Multi-partonamplitudesingaugetheories.Phys.Rep.200,301–
367(1991).arXiv:hep-th/0509223
2. M.Srednicki,QuantumFieldTheory(CambridgeUniversityPress,Cambridge,2007)
3. A.Zee,QuantumFieldTheoryinaNutshell(PrincetonUniversityPress,Princeton,2003)
4. L.J.Dixon,Calculatingscatteringamplitudesefficiently(1996).arXiv:hep-ph/9601359
5. M.E.Peskin,Simplifyingmulti-JetQCDcomputation(2011).arXiv:1101.2414
6. L.J.Dixon,Abriefintroductiontomodernamplitudemethods(2013).arXiv:1310.5353
7. Z.Bern,L.J.Dixon,D.A.Kosower,On-shellmethodsinperturbativeQCD.Ann.Phys.322,
1587–1634(2007).arXiv:0704.2798
8. R.K.Ellis,Z.Kunszt,K.Melnikov,G.Zanderighi,One-loopcalculationsinquantumfield
theory: from Feynman diagrams to unitarity cuts. Phys. Rep. 518, 141–250 (2012). arXiv:
1105.4319
9. V.A.Smirnov,EvaluatingFeynmanintegrals.TractsMod.Phys.211,1(2004)
10. V.A.Smirnov,AnalytictoolsforFeynmanintegrals.TractsMod.Phys.250,1(2012)
11. M.S.RaduRoiban,A.Volovich,Scatteringamplitudesingaugetheories:progressandout-
look.J.Phys.A44,450301(2011)
12. L.J.Dixon,Scatteringamplitudes:themostperfectmicroscopicstructuresintheuniverse.J.
Phys.A44,454001(2011).arXiv:1105.0771
13. A.Brandhuber,B.Spence,G.Travaglini,Tree-levelformalism.J.Phys.A44,454002(2011).
arXiv:1103.3477
14. Z.Bern,Y.-t.Huang,Basicsofgeneralizedunitarity.J.Phys.A44,454003(2011).arXiv:
1103.1869
15. J.J.M.Carrasco,H.Johansson,GenericmultiloopmethodsandapplicationtoN=4super-
Yang-Mills.J.Phys.A44,454004(2011).arXiv:1103.3298
16. H. Ita, Susy theories and QCD: numerical approaches. J. Phys. A 44, 454005 (2011).
arXiv:1109.6527
17. R. Britto, Loop amplitudes in gauge theories: modern analytic approaches. J. Phys. A 44,
454006(2011).arXiv:1012.4493
18. R.M. Schabinger, One-loop N =4 super Yang-Mills scattering amplitudes in d dimen-
sions, relation to open strings and polygonal Wilson loops. J. Phys. A 44, 454007 (2011).
arXiv:1104.3873