Presentation on theme: "Bill Spence* Oxford April 2007"— Presentation transcript:
1 Goodbye Feynman diagrams: A new approach to perturbative quantum field theory Bill Spence*Oxford April 2007Work in collaboration with A. Brandhuber, G. Travaglini, K. Zoubos,arXiv: hep-th, and earlier papers*Centre for Research in String Theory, Queen Mary, University of London 2007
2 Outline Perturbative quantum field theory 1. Old tricks: Feynman diagrams, unitarity methods2. New tricks: Twistor inspired progress –MHV diagramsrecursion relationsgeneralised unitarity3. A new approach: MHV perturbation theory4. Conclusions
3 Feynman diagrams First course in Yang Mills quantum field theory: 1.1 Old tricks: FeynmanFeynman diagramsFirst course in Yang Mills quantum field theory:Perturbative quantum corrections to classical amplitudes:Use propagatorsand interaction verticesto form Feynman diagrams, , etc.Eg: QCD – for gluons:(colour labels suppressed)Propagator3-vertex
4 Feynman diagrams: the reality 1.1 Old tricks: FeynmanFeynman diagrams: the realityBut Feynman diagrams are impractical!Eg: Five gluon tree level scattering with Feynman diagrams:picture from Zvi Berngg => n gn=7n=8n=9Diagrams559405
5 Feynman diagrams: end products 1.1 Old tricks: FeynmanFeynman diagrams: end productsFeynman diagrams are cumbersome, but the results can be simple:n-gluon scattering, helicities (--++…+).Result:This is called an MHV amplitudeMaximal Helicity Violating:as tree amplitudes withall, or all but one,helicity the same are zeroNotationnull momenta p,written with spinorsi is the particle label
6 Feynman diagrams: end products II 1.1 Old tricks: FeynmanFeynman diagrams: end products IILoop amplitudes are also simple in spinor notation:n-point one-loop all plus helicity amplitude in pure Yang-Mills:n-point one-loop MHV amplitude in N=4 super Yang-Millssum over “box functions” F
7 Feynman diagrams: Summary 1.1 Old tricks: FeynmanFeynman diagrams: SummaryFeynman diagrams: theory-- simple rules, Lagrangian derivation, work for alltheoriesBut, the practice:-- diagrams are cumbersome – multiply rapidly andbecome impracticalHowever:-- the result of adding the contributions of manydiagrams can be extraordinarily simple, whenwritten in spinor variables
8 1.2 Old tricks: unitarityUnitarity methodsOld S matrix approach: the scattering matrix S must be unitary:1234●Example: 4 point, mass m, scalar scattering 1+2 3+4:●Scattering depends on the Lorentz invariants (s,t):●Consider A(s,t), at fixed t, in the complex plane. There are polesat s = 4m^2, 9m^2,… (production of particles). In fact thereis a branch cut from s=4m^2 to infinity (and also one along thenegative s axis due to poles in the t-channel)scutA(s):
9 Unitarity methods II Loops from the old S-matrix approach: 1.2 Old tricks: unitarityUnitarity methods IILoops from the old S-matrix approach:●Consider the contour integral of the amplitude A(s), around C:scutC●This gives● Then, usingIdea: reconstruct amplitudes from their analytic properties
10 1.2 Old tricks: unitarityNew unitarity methodsFrom c. 1990: New application of unitarity methodsBern, Dixon, Dunbar, Kosower,….●One loop general results:N=4 SYM – all MHV amplitudesN=1 SYM – all MHV amplitudesPure YM – (cut-constructible parts of) all MHV amplitudes(for adjacent negative helicities)●Other particular results:Various nMHV results at one loopTwo loop results (4 point function N=4)Others (nnMHV,…)●But – nnMHV – difficulthigher loops – difficult…reaching the limits of this approach by the early 2000’sBut proving difficult to progress further
11 Unitarity methods: summary 1.2 Old tricks: unitarityUnitarity methods: summaryOld methods (pre 1970):-- good ideas, but it proved difficult to write dispersionrelations for all but simple (eg two point function) cases-- was explored as no theory of strong interactions at thetime; QCD then became dominantMore recently (1990’s):-- old unitarity ideas applied to supersymmetric theories- new results found, but again no really systematicway to derive dispersion relations to give amplitudes
12 1.Old tricks: summaryPerturbative quantum field theory, calculate amplitudes via:Feynman diagramsBut this proves impractical, even with computers – thenumber of diagrams rises very rapidly with the numberof particles involved.However, adding many diagrams often produces a verysimple result (eg MHV) – why???Unitarity methodsUse dispersion relations – but no systematic wayfound to generate these in general, and applicationsto higher loops (>1), massive theories, etc, proveddifficultNeed some New Tricks……….
13 2.1 New tricks: TwistorsTwistor string theoryWitten hep-th/Amplitudes in spinor variables can be simple: eg MHVIdea: Look at amplitudes in twistor space twistor space coordinates( = Fourier transform of )Then:ie MHV tree amplitudes localise on a line in twistor space
14 Amplitudes in twistor space 2.1 New tricks: TwistorsAmplitudes in twistor spaceLocalisation of tree amplitudes in twistor space appears generic:Eg: MHV < …++ > localise on a linenext to MHV < ….++ > localise on two intersecting linesEg: 3 points Z are collinear ifExplicit check:twistor space coord’sand the above becomes a differentialequation satisfied by the amplitudein spacetime:Loop level: also get localisation – see laterWhat can explain this localisation ?
15 2.1 New tricks: TwistorsTwistor string theoryIdea: Localisation on curves in target space – this is a feature oftopological string theoryThe correct model is:*** Topological B model strings on super twistor space CP(3,4) ***(plus D1, D5 branes)Can then argue that:-loop N=4 super YM amplitudes with negative helicity gluons localiseon curves in CP(3,4) of degree and genusThis:explains the localisation of YM amplitudes,gives a weak-weak duality between N=4 SYM and twistor string theory
16 Twistor string theory: Tree level 2.1 New tricks: TwistorsTwistor string theory: Tree levelIn twistor space, tree level scattering amplitudesmoduli space of curves degree d, genus 0vertex operators(degree d (d+1) negative helicity gluons)A surprise: due to delta functions, the integral localises on intersections ofdegree one curves:CurveAmplitudeMHV < - - +…+ >nMHV < …+ >nnMHV < >X
17 Twistor string theory:problems 2.1 New tricks: TwistorsTwistor string theory:problemsTwistor string theory: beautiful new duality between N=4super Yang-Mills and a topological string theory, but:Hard to calculate with it – integrals over moduli spaces ofcurves in CP(3,4)…At loop level (and tree level for non-planar graphs) –conformal supergravity arises and cannot be decoupledMuch of the structure seems tied to N=4 supersymmetry(eg conformal invariance) – how would it work for pureYang-Mills; also how to include masses for example…It would be nice to have methods which work in spacetime itself…….
18 2.2 New tricks: MHVMHV methodsIdea: Since MHV tree amplitudes localise on a line in twistorspace (~ point in spacetime), think of them as fundamental vertices.Join them with scalar propagators to generate other tree amplitudes:MCachazo, Svrcek, WittenMHVnMHVnnMHVM(spacetime)(twistor space)MMMThis works and gives a new, more efficient, way to calculate tree amplitudes
19 MHV methods: loops For tree amplitudes – spacetime MHV diagrams work M 2.2 New tricks: MHVMHV methods: loopsFor tree amplitudes – spacetime MHV diagrams workM(twistor space)(spacetime)M-- direct realisation of twistor space localisationStudy of known one loop MHV amplitudes twistor space localisation onpairs of linesxThis suggests that in spacetime, one loop MHV amplitudes should be givenby diagramsM
20 MHV methods: loops II ? M = MHV amplitude Technical issues: 2.2 New tricks: MHVMHV methods: loops IIM= MHV amplitude?Technical issues:The particle in the loop is off-shell. But particles in MHV diagrams are on-shell need an off-shell prescriptionCoordinatesnullvectornullreferencevector-- Result should be independent of reference vector;-- Use dimensional regularisation of momentum integralsThen: multiply MHV expressions, simplify spinor algebra, performphase space (l) and dispersion (z) integrals.....non-trivial calculationResult
21 MHV methods: loops III The result of this MHV diagram calculation is 2.2 New tricks: MHVMHV methods: loops IIIThe result of this MHV diagram calculation is(Brandhuber, Spence, Travaglini hep-th/ )The known answer isThese agree, due to the nine-dilogarithm identity
22 2.2 New tricks: MHVMHV diagrams: N<4So – spacetime MHV diagrams give one loop N=4 MHVamplitudesa surprise - no conformal supergravity as expected from twistor string theoryRemarkably: MHV diagrams give correct results for-- N=1 super YM-- pure YM (cut constructible)-- these calculations agree with previous methods and also yield new results-- another surprise – one might have expected twistor structure only for N=4Bedford, BrandhuberSpence, TravagliniQuigley RozaliBedford, BrandhuberSpence, TravagliniMight MHV diagrams provide a completely new way to do perturbative gauge theory?
23 One loop:general result 2.2 New tricks: MHVOne loop:general resultMHV diagrams are equivalent to Feynman diagrams for anysusy gauge theory at one loop:Brandhuber, SpenceTravaglini hep-th/Proof:(1) MHV diagrams are covariant (independent of reference vector)Use the decompositionin all internal loop legs term with all retarded propagatorsvanishes by causality; other terms have cut propagators on-shell become tree diagrams (Feynman Tree Theorem)and trees are covariantMHV diagrams have correct discontinuities use FTT again(3) They also have correct (soft and collinear) poles can deriveknown splitting and soft functions from MHV methods.Evidence that MHV diagrams might provide a new perturbation theory
24 MHV methods: Issues MHV methods: successes at tree level, one loop 2.2 New tricks: MHVMHV methods: IssuesMHV methods:successes at tree level, one loopcan be thought of as a consistent formulation of dispersion integralsBut,MHV diagrams “cut constructible” pieces of the physical amplitude.Other “rational” parts are missing.Pure YM (but not susy YM) has rational parts!Hard to apply to higher loops, non-MHVHard to incorporate masses, or go off-shell
25 2.3 New tricks: RecursionRecursion relationsBehaviour of tree level scattering amplitudes at complex momenta =●Britto, Cachazo, Feng, Witten- can use this to reduce tree amplitudes to a sum over trivalent graphsApplications: efficient way to calculate tree amplitudes(eg 6 gluons < > : 220 Feynman diagrams, 3 recursion relation diagrams)-- useful at loop level (see later)-- can be used to derive tree level MHV rules (Risager)
26 Recursion relations II 2.3 New tricks: RecursionRecursion relations IIRecursion relations for tree amplitudes:=●There are analogous relations at loop level – eg one loopQCD amplitude, recursion relations give decompositions like:loopBern, Dixon, Kosower,hep-th/treeThis allows one to reconstruct (parts of, in general) amplitudes fromsimpler pieces – this is a useful tool, but it is hard to apply at looplevel systematically
27 Generalised Unitarity 2.4 New tricks: GeneralisedUnitarityGeneralised UnitarityUnitarity arguments: find amplitudes from their discontinuities(logs, polylogs)Supersymmetric theories: amplitudes can be completely reconstructedfrom their discontinuitiesNon- supersymmetric theories (eg pure YM) : amplitudes containadditional rational termse.g. one loop five gluon amplitudehas rational partIn d-dimensions, the discontinuities should also determine these rational terms
28 Generalised Unitarity II 2.4 New tricks: GeneralisedUnitarityGeneralised Unitarity IId-dimensional unitarity should give the full amplitudesNew techniques with multiple cuts developed (see reviews for references)eg: QCD: multiple cuts in d-dimensions – 4-point caseBrandhuber, McNamara, Spence, Travaglinihep-th/Triple cutQuadruplecutResult:Various integralsThis is the correct QCD result
29 Generalised Unitarity III 2.4 New tricks: GeneralisedUnitarityGeneralised Unitarity IIIGeneralised unitarity – multiple cuts, and d-dimensionalcutsThis has had remarkable successes, e.g:-- reduction of one-loop calculations to algebraic sums-- derivation of full amplitudes (including rational terms)in pure Yang-MillsThis has provided another set of useful tools.However, applications to pure YM proved relatively cumbersome,and applying many of these techniques requires some prior knowledge of the structure of the answer
30 2. New Tricks: SummaryRecent new methods inspired by twistor string theory:-- twistor formulations-- MHV methods-- recursion relations-- generalised unitarityThese have provided new insight into perturbative field theory,and yielded amplitudes previously unobtainable by older methodsBut there remain outstanding issues:-- methods are not systematically defined orare difficult to apply-- applications to non-supersymmetric theoriesare the most challenging-- generalisations (masses etc) non-obviousWe need a systematic formulation incorporating these new ideas
31 3. MHV Perturbation Theory 3. A New Approach3. MHV Perturbation TheoryRecall MHV diagrams: combine MHV vertices to get amplitudes:M= MHV amplitudeThis works (at least at one loop in super YM)Idea: derive these rules from a Lagrangian
32 An MHV Lagrangian? What Lagrangian? Ingredients: 3.1 Classical MHV theoryAn MHV Lagrangian?What Lagrangian? Ingredients:Only +/- helicity fields in loops and external linesA null reference vector is needed ( eg to define off-shell momenta L )nullvectornullreferencevectorThis suggest some relation to light-cone gauge theory
33 Yang-Mills in light-cone gauge 3.1 Classical MHVtheoryYang-Mills in light-cone gaugePure Yang-MillsLight-cone gaugeLeavesnon-propagating,integrate outResult (non-local)OK, but how to get MHV vertices?
34 MHV Lagrangian I Yang-Mills in light-cone gauge Idea: Change variables 3.1 Classical MHV theoryMHV Lagrangian IYang-Mills in light-cone gaugeIdea: Change variablesso thatie, eliminate the ++- vertexResult: MHV vertices!Gorsky, Rosly hep-th/*Mansfield hep-th/Ettle, Morris hep-th/
35 3.1 Classical MHV theoryMHV Lagrangian IISo we have written the YM action in light-cone gauge,using B fields, as a sum of a kinetic term plus MHV verticesClassically, this is ok. Does it give an alternative perturbationtheory for quantum Yang-Mills?MHV vertices: always have two negative helicity particles;All quantum diagrams from the above Lagrangian have atleast two negative helicity external fields
36 Rational terms Previous slide: MHV diagrams generate amplitudes with 3.1 Classical MHV theoryRational termsPrevious slide: MHV diagrams generate amplitudes withat least two negative helicity fields.But pure Yang-Mills theory has:-- all-plus amplitudes, eg one loop four gluon:-- single-minus amplitudes, eg one loop four gluon:These cannot be generated from our classical MHV Lagrangian(Note: these amplitudes are purely rational – no logs, polylogs etc)
37 Rational terms II So the classical MHV Lagrangian cannot explain the 3.1 Classical MHV theoryRational terms IISo the classical MHV Lagrangian cannot explain theall-plus or single minus helicity amplitudesAlso, while gives graphs with at least two negativehelicity particles, it does not give the rational parts of otheramplitudes [known from explicit calculations]Something is missing……
38 A Puzzle The Lagrangian, , obtained from light-cone gauge 3.1 Classical MHV theoryA PuzzleThe Lagrangian, , obtained from light-cone gaugeYM theory using new variables, does not generate rational termsin quantum amplitudes. For example, the (++++) one-loop, whichis entirely rational:But what about the all-minus amplitudes? , eg:This could be generated from MHV diagrams (it has more than onenegative helicity), but it is rational. How could you get this one andnot the other which is so similar?The answer involves a careful treatment of divergences – naivelyone gets zero from the MHV diagrams, but due to a mismatchbetween 4 and D dimensions, one can derive the correct answerBrandhuber, Spence, Travaglini hep-th/
39 Quantum MHV Lagrangian 3.2 Quantum MHV theoryQuantum MHV LagrangianIdea: careful treatment of quantum light-cone gauge theory:Modify the classical Lagrangian correctly to reproduce physicalamplitudesNeed a suitable regularisation scheme: stay in four dimensionsand preserve the separation of the transverse light-cone degreesof freedom. This has been formulated recently*Chakrabarti, Qiu, Thorn, hep-th/The end-product: add suitable counterterms to the Lagrangian:only need -* could use dim reg: Ettle, Fu, Fudger, Mansfield, Morris, hep-th/
40 Counterterms Quantum light-cone YM Lagrangian – counterterms are 3.2 Quantum MHV theoryCountertermsQuantum light-cone YM Lagrangian – counterterms arefunctions of the gauge fieldsThese are simple expressions when written in terms ofdual momenta k :(this is connected with planar graphs,double line notation and thestring worldsheet picture)
41 Counterterms II The ++ counterterm takes the simple form 3.2 Quantum MHV theoryCounterterms IIThe ++ counterterm takes the simple formMore explicitly,In the quantum MHV Lagrangian we need to use B variables.We have(certain functions of the momenta,condensed notation)with
42 Counterterms III Take the two point counterterm, 3.2 Quantum MHV theoryCounterterms IIITake the two point counterterm,Expand A’s in powers of B fields; result at BBBB level:Many manipulations later……this equalsThis is precisely the fourpoint ++++ amplitudeGenerally, one finds thatand the V’s turn out to be the missing all-plus vertices !(non-trivial calculation: Brandhuber, Spence, Travaglini, Zoubos, hep-th )
43 Counterterms IV Thus the simple counter-term 3.2 Quantum MHV theoryCounterterms IVThus the simple counter-termis a generating function for the infinite series of all-plus vertices:n-point all-plus vertices, missing fromclassical MHV LagrangianWhat about the other counterterms? The structure of thesesuggests:
44 Quantum MHV Lagrangian 3.2 Quantum MHV theoryQuantum MHV LagrangianThus conjecture that:Propagator and MHV vertices only (obtainedfrom light-cone gauge YM using new variables) Contains all-plus, single-minus vertices,plus other vertices needed to generate therational parts of amplitudesConjecture: This quantum theory is equivalent to quantum YM
45 MHV perturbation theory 3.2 Quantum MHV theoryMHV perturbation theoryThe new Feynman-type rules: join the fundamentalvertices with propagatorsClassical vertices: MHVMQuantum vertices:APSM--(All-plus, single minus, double minus)For example: one-loop MHV amplitude is given byM+MAP+--cut-constructible part(known)rational parts (new)
46 3. A New Approach: Summary Gauge theory amplitudes localise on lines in twistor space –corresponds to MHV vertices in spacetimeThe light-cone gauge YM Lagrangian, in suitable variables,is a theory with only MHV verticesThis classical Lagrangian is incomplete for the quantum theory –it misses amplitudes (eg all-plus) and parts of amplitudes (eg rational)Some simple quantum counter-terms can/could account for these MHV perturbation theory: an alternative to standard Feynman diagrams
47 Conclusions I Perturbative gauge theory: Old Tricks -- Unitarity (pre 1970): not systematic, limited results-- Feynman diagrams: systematic, but impractical(too many diagrams!)However, the results are simple……
48 Conclusions II 2. New Tricks -- simplicity of amplitudes explained by twistor space localisation-- spacetime picture is MHV vertices; but no there is no derivationof these, can’t explain rational terms in amplitudes-- other spin-offs from twistor string theory:-- recursion relations-- generalised unitarity-- much progress, but a systematic approach needed
49 Conclusions III 3. A New Approach MHV perturbation theory -- classical MHV Lagrangian, plusquantum countertermsClaim: this is equivalent to quantum Yang-MillsEvidence so far:-- classically equivalent-- non-rational parts of amplitudes reproduced-- all-plus amplitudes at one-loop reproduced-- structure is correct for the claim
50 Open Problems Check it all really works: -- other amplitudes (eg single minus)-- rational terms (eg in MHV)-- two loopsis it more efficient?apply it to fermions, scalars, massive theories(note: no conceptual obstacles)Twistor picture:-- it incorporates MHV twistors-- it uses 4-d regularisation – good for twistors full twistor space realisation of Yang-Mills theory ?And then there’s-- gravity-- holography-- integrability-- …………..
51 Goodbye Feynman diagrams: A new approach to perturbative quantum field theory MHV perturbation theoryM+AP--
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