Refining MHV Diagrams
by Andreas Brandhuber Queen Mary, University of London
Munich, June 19, 2007 hep-th/0612007 AB-Spence-Travaglini arxiv:0704.0245 [hep-th] AB-Spence-Travaglini-Zoubos Outline • Motivation • MHV Diagrams • Tree MHV Diagrams (Cachazo-Svrcek-Witten) • Loop MHV Diagrams (AB-Spence-Travaglini) • One-loop Amplitudes in Pure Yang-Mills from MHV diagram • The all-minus helicity amplitudes • MHV diagrams from LC Lagrangian (Gorsky-Rosly, Mansfield, Morris-Ettle) • Completion of MHV diagram method • all-plus helicity amplitude from Lorentz violating counterterm • non-canonical, “holomophic” field transformation • Conclusions Goal
• The goal of this talk is to address two questions: I. Do MHV Diagrams (a la CSW) provide a new, complete, perturbative expansion of Yang-Mills? II. And if not can we find a suitable completion of the MHV diagram method? Motivations
• Simplicity of scattering amplitudes unexplained by textbook Feynman methods: e.g. Park-Taylor formula for MHV tree amplitude <--+++...+> • New ideas from twistor string theory (initiated by Witten 2003) help to “explain” this simplicity • Novel, practical tools to calculate physically relevant amplitudes ➡ LHC is coming • New reorganisations of conventional perturbation theory Spinor helicity Amplitudes formalism ColourSpinor decomposition helicity formalism
• Main idea:A =disentangleA( λi, colourλi,;hi ) Scattering p µ null vector Amplitudes { } • At tree lep µ v el n, Yullang-Mills vector interactions are planar •Color Ordering:• µ µ DefineColour p a-or a ˙ = der p µ edσ a a ˙par whertial amplitudesµe σ = ( 1 , !σ µ ) • • Define p a a ˙ = p µ σ a a ˙ where σ = ( 1 , !σ ) tree • aσ aσn! A (2 pi,εi ) = ∑Tr(T 1 T ) A(σ(p1,ε1),...,σ(pn,εn)) If p { = 0 } thenσ 2 d··e·t p = 0 • ! momenta• If and p polarisation = 0 then v ectors Colourd eexprt-orpderessed=ed 0par intial termsamplitude of spinors and helicities ˜ ˜ ˜ ˜ HenceSpinor Helicity: p aa˙Hence= λaλ a˙p aa˙ = λ·aλ(a˙λ) positiv· eλ (negativ(λ) positive) helicitye (negativ e) helicity Next• example:- Include• only diagrams with fixed cyclicspinors orderingspinors of gluons ˙ ! colour indices stripped off a12b := ε λaλb [12a˙] :=b˙ ˜ a˙˜ b - Inner Maximall Analproductsytic- Inner ystructur H elicity pr 12e oducts is : simpler= V ε iolatingab λ λ , amplitude ab [12 1 ] 2 : =, ε ˙λ˜, λ˜ εa˙b˙λ1λ2 - ! " !1 2" a˙b 1 2 or MHV amplitude2(p p ) = 12 [12] 2(p p ) += +121 · [122 +] ! " MHVFirst Tree examples: Amplitudes: 1 ·A2(1 ,2! ,..".n ) = 0 , •(ParkAte-Ta ylor)loop level: multi-trace contributionsat tree level ! helicities• are a permutation of !!+++....++ Simple & A(1−,2 ,...n ) = 0 - subleading in 1/N Holomorphic! 4 + + i j A (1 ...i− ... j− ...n ) = " # Parke-Taylor formula MHV 12 23 n1 " #" #···" #
!
! Amplitudes in Twistor Space • Witten (2003): Simplicity can be “explained” • MHV Amplitudes holomorphic ऄline in twistor space • moreover: L-loop amplitudes with Q negative helicity gluons localise in twistor space on curves of degree=Q-1+L and genus<=L
Q=4
Q=3 MHV Diagrams
• MHV amplitude (holomorphic) = line in twistor space = pointlike interaction in Minkowski space (Witten, Cachazo-Svrcek- Witten) • CSW Rules (Cachazo-Svrcek-Witten) • MHV amplitudes promoted to local vertices off-shell continuation L = l + z! , " L !˜ a˙ • µ µ µ a ∼ aa˙ • Connect MHV vertices with scalar propagators • Note: η is related to LC gauge fixing MHV Diagrams cont’d
typical MHV diagram contributing to 1 1 an NNMHV amplitude 2 2 L L! • • Covariance (= η-independence) is achieved after summing all MHV diagrams (CSW) • Equivalence with results of Feynman diagram calculations • On-Shell Recursion Relations (Britto-Cachazo-Feng-Witten) • MHV diagrams = special recursion relation (Risager) (shifts invisible since MHV amplitudes are holom.) From Trees to Loops (AB-Spence-Travaglini) • Original prognosis from twistor string theory was negative (Berkovits-Witten), Conformal SUGRA modes spoil duality • Try anyway: • Connect MHV vertices, using the same off-shell continuation as for trees • sum over all MHV diagrams and internal particle species • Find measure, perform loop integration • different from unitarity-based approach dM ! Z m1,m2,h The loop measure
4 4 d L1 d L2 (4) dM = 2 2 " (L2 L1 + PL) L1 + i!L2 + i! − • Use L = l + z η and L (l, z) → d4L dz = d4l"(+)(l2) L2 + i! z × dipersive phase space measure x measure • Loop integral becomes (use dimensional regularisation) (Dispersion integral) x (2-particle LIPS integral) • Sum of all MHV diagrams is covariant • MHV 1-loop amplitudes in N=4/N=1 SYM (agree with Bern- Dixon-Dunbar-Kosower) Evidence for MHV method • Tree Level Amplitudes ऄ complete proof (CSW, Britto-Cachazo- Feng-Witten, Risager) • One-Loop Amplitudes in Supersymmetric Yang-Mills • MHV method at one-loop ऄExplicit calculation of complete MHV one-loop Amplitudes in N=4 SYM (AB-Spence-Travaglini) & N=1 SYM (Bedford-AB-Spence-Travaglini, Quigley-Rozali) • “Proof” at one-loop for generic amplitudes in SYM • Feynman Tree-Theorem ऄ covariance • unitarity cuts • soft & collinear limits Without Supersymmetry
• Cut-constructible part of MHV 1-loop amplitudes in pure Yang-Mills from MHV diagrams (Bedford-AB-Spence-Travaglini) • First result for QCD from MHV diagrams • Amplitudes in pure Yang-Mills not 4D cut-constructible • rational terms are missed by MHV diagrams • Look for alternative techniques, like on-shell recursions... (Bern, Dixon, Kosower, Berger, Forde...) • ... or try to refine the MHV diagram method Two related problems •RelatedMissing Pr all-plusoblems amplitude in pure Yang-Mills 1 γ5 i Tr( − lˆ lˆ lˆ lˆ ) 1 loop( +,..., +) = 2 1 2 3 4 An− 1 n − 2 ∑ 48π 1 l • Can be drawn with MHV vertices (unlike the all-plus) ➡ MHV diagrams should be able to produce it! • For n-point <--- ....-> amplitude we need n 3-point MHV vertices: • Recall: D=Q-•1Thr+L ee-point scalar-scalar-negative helicity • Crucial: 3-point MHVgluon ver ticesvertex: = LC vertices ➡ result a priori correct (3) η L1 k] v¯ (L1,k,L2) = ! | | φAzφ ηk ! " • MHV three-point vertices are the same as in lightcone Yang-Mills ! expected on the basis of Mansfield’s field redefinition Explicit calculation Explicit Calculation Supersymmetric• Use supersymmetric decomposition decomposition:of 1-loop amplitudes A = (A + 4A + 3A ) 4(A + A ) + A g g f s − f s s • N=4 and N=1 SUSY contributions are zero • N=4 and N=1 contributions vanish • Only need to consider a complex scalar running in the loop • •Gluontechnicall ➡ y simpler scalar running in the loop • Use dimensional regularisation (DR): 3pt MHV vertices 4- -dim’lsimpler (spinor to brackcalculateets) while propagators are D-dim’l 2− 3− 2− MHV diagrams+ for the <----> 2− 3− − 2− + + + 3− 2− − − + − 2− + 33−− 22−− 3− − + + pms − ++ − −− − ++ + − + + pms 3− + ++ − + 3− 4− −− − + − − −− ++ + pms + + + 4−−− + pms − − + − + − 1− + 1−+ 4− + − + + 44−− −− 1− 1− 4− ++ −− −− 4− 11−− + perms 11−− 4− + 2− − 3− + + + 2− − − 3− − + pm + + bubbles give − − − ++ ++ pm 22−− 1 − − + perms33−− 4− + − + vanishing contribution 1− ++ + − −− 4−− ++ pm pm + 1−− + 1 −− 44−− 12 34 D=8 2ε 1 ResultAnswer ~ = ! "! " K4 K4 = ε(1 ε)I4 − 1•2 34 1 ε 0 [12] [34] D=8 2ε − − −→→ −6 Result ~ ! "! " K4 K4 = ε(1 ε)I4 − − − −ε →0 −6 [12] [34] 1122 3344 → D=8 2ε 11 K = ε(1 ε)I D=8− 2ε ResultResult ~~ !! ""!! ""KK44 K44 = ε(1 ε)4I4 − (Originally deriv[ ed] b[y Bern,] Kosower, and Bern and− Morgan)− −ε ε→00−6 [1122] [3344] − − −→→→ −6 (Originally derived by Bern, Kosower, and Bern and Morgan) (Originall(Originallyy deriv deriveded b byy Bern, Bern, K Kosoosowwerer, ,and and Bern Bern and and Morgan) Morgan) FinitenessFinitenessFiniteness of of the theof theall-min all-min all-minusus amplitude usamplitude amplitude Finiteness of the all-minus amplitude 2 2 2 2 2 InDefine (DR): L D = L 4 + L 2 ε with 2 LD 2= L 2+ L :2= L 2 µ •Define• L D = L 4 + L 2 ε −with LD = L4 + L4 2ε :=2εL4 µ4 • − − − − − 2 2 2 2 2 Define L D = L 4 + L 2 ε with LD = L + L := L µ • •− 4 2ε 4 A finiteA finite, non-zer, non-zero resulto− result arises− arises from from • •finite, non-zero result due to incomplete cancellations of • incomplete cancellations of propagators • A finite, non-zeroincomplete r(inesultverse) arises propagators cancellations from of propagators incomplete cancellations of propagators2 2 2 2 2 L2 L4L2 Lµ24 +µµ2+ µ µ2 µ • 4 = 4= − = += 1+ 2 L2 − 2 L2 1 2 L2 2 2 2 2 LD2 D LD D LD D L4 L4 µ + µ µ 2 = −• 2 = 1+ 2 LD LD! MHV verLticesD are 4-dimensional !• MHVFiniteness vertices comes are as4-dimensional an ε/ε-effect (smells like an anomaly) ! MHV vertices are 4-dimensional !• D-dimensionalAll-min! D-dimensionalus amplitude propagators pr obtainedopagators fr om MHV diagrams, but all-plus amplitude remains mysterious ! D-dimensional propagators Where is the all-plus amplitude? • Go back to the path integral! • Mansfield’s approach: • LC quantisation of Yang-Mills: A-=0 • integrate out A+ (no derivatives w.r.t. LC time x-) • Schematically the action becomes (Scherk-Schwarz) -+ --+ -++ --++ S = S + S + S + S Change of Field Variables Change• Change variables variables in path in path integral: integral A z , A z¯ B + , B • Change variables-++ in path integral:→Az ,Az¯ − B+ ,B • such that the S term is eliminated → − + ++ + (S− + S− + )[Az,A+z+¯] = S− [B+,B +] (S− + S− )[Az,Az¯] = S−− [B+,B ] LHS is SDYM action · LHS is SDYM action− Require transformation · Bäcklundto be ·canonical: LHS transf is ormationSD YM action Further• require: · Bäcklund transformation • • Fur⇒ therJacobian requir of thee: transf ormation is 1 ! TransformationSubstitute is canonical, 2 with3 Az = Az[B+] Coefficients depend only on: Plug•! T ransf A z ormation B + + isB + canonical, + B + + with Az = Az[B+] • ∼ 2 3 ··· pi , pi , pi • Plug A z B + + B + + B + + + z z¯ ! Canonicality ∼ Jacobian equal to 2·1·· (classicall3 y) ! Az¯ B (1 + B+ + B+ + B+ + ) Canonicality➡∼ − Jacobian 2equal to3 1 (·classicall·· y) Az¯ B ➡(1 + B+ + B+ + B+ + ) ∼ − ··· ! Subtleties related to det ∂ ! intoSubtleties r+elated to+ ++ in (S−− + S−− )[detAz,∂A+z¯] + ++ in (S−− + S−− )[Az,Az¯] Result is • Result is • + + ++ +++ S[B+,B ] = S− + S−− + S−− + S−− + − + + ++ +++ ··· S[B+,B ] = S− + S−− + S−− + S−− + − ··· • Vertices have MHV helicity configuration • Vertices have MHV helicity configuration Change of Field Variables cont’d -+ --+ --++ --+++ • Result: S[B+,B-] = S + S + S + S + ... • This reproduces the infinite set of MHV vertices! • Equivalence Theorem (ET) • We can equivalently calculate amplitudes with B field insertions • Greens functions of B fields are different from those of A fields, BUT • on-shell S-matrix elements are identical (modulo wavefuntion renormalisation) Not so fast! • At the same time we have mapped Self-Dual Yang-Mills to a free theory...... and eliminated the all-plus amplitude! • Potential sources / resolutions of the problem • Subtleties with regularisation: perform D-dim’l Mansfield transform or use 4D regulator (see in a moment) • Violations of the Equivalence Theorem (Ettle-Fu-Fudger- Mansfield-Morris: use D-dim’l Mansfield transform) • Jacobian: use non-canonical transformation (see later) A¯3 p3 p3 = 2g + [p1 p2 p2 p1] = √2g + [12] . (2.10) 1 2 + z¯ + z¯ 1 2 − p+p+ − − p p A2 A1 + + The “worldsheet–friendly” regulator that CQT employ is simply defined as follows [49]: For a general n–loop diagram, with qi being the loop regio!n momenta, one simply inserts an As for propeaxgpaonteonrtsia,lfcoultloffwfiancgtor[40], we will use the Schwinger representation: n 2 e∞xp( δ qi ) (2.13) The “worldsheet–friendly” 1regulator that C−QT+eTmp2ploy is simply defined as follows [49]: = dT ei=1 . (2.11) For a general n–loop diagram, w2ith q being the lo!op region momenta, one simply inserts an in the loop integrand, whepre δ is −pi osit0ive and will be taken to zero at the end of the calcula- exponential cutoff factor " tion. This clearly regulates UV divergences (nfrom large transverse momenta), but, as we will 2 2 In (2.11) p isseeu,nhdaserssotmoeosdurptroisinbgecotnhseequaepnpecxerpso(spinrδicaetietqw2(i)pll lea Figure 2: Labelling of one of the selfenergy diagrams contributing to Π++. In Figure 2, p, p are the outgoing line momenta, l is the loop line momentum, and − In Figure 2, p, p are the outgoing line momenta, l is the loop line momentum, and − 9 9 The “worldsheet–friendly” regulator that CQT employ is simply defined as follows [49]: For a general n–loop diagram, with qi being the loop region momenta, one simply inserts an exponential cutoff factor n exp( δ q2) (2.13) − i i=1 ! in the loop integrand, where δ is positive and will be taken to zero at the end of the calcula- tion. This clearly regulates UV divergences (from large transverse momenta), but, as we will ! see, has some surprising consequences sincke, kikt,,qwk!a,irlqlealtreheaetdrTheegtiorohengfiimeonniotmme“eonvmwtaael,nutioenas,trienfrolmtrdesrcmosefsrhwtoahfeiwcnhheiLtchthoer–tlehinefetlrzimn–ieoemmenonmtdaenatlraeyagri”evegnivberynebygulator that CQT employ is simply defined as follows [49]: violating processes, which therefore have to be cancelled by the introdpu=cptk=i!onk!ko, fka,l p=plqr=opqkr!ia.kt!e. (2.1(42).14) − − − − counterterms. For a general n–loop diagram, with qi being the loop region momenta, one simply inserts an RemReemmbeemribnegrintog dtoudboleubtlheetrheesurletsuolft tohfisthdiisagdriamgr,amwe, wfinedfinthdetfhoellofowllionwginexgperxespsrieosnsiofonr for Note that q2 = 2q q hwailsl pceormforpmontheentMsaonthnsefiltyeshledelaf–stloerelannfn–egersgnfotyerh:rmgeya:ttwioon otnratnhesvoerrigsienadl iLraegcrtaiongnisa,nh(ceonncveeritting it to a free z z¯ theory). We will then showethxatp, upoonninesenrtitngitahelchcanuge tofovaffriablfesainctotthoe rcounterterm breaks explicitly even more Lorentz invariance than the lightdc4ol dn4el u(psu+(apll)+lyl)does. This migh1t 1 Lagrangian, we recover thΠe+a+ll–+=p+8lgu2sNam2 plitudes as verti+ce(sp+inl thel tphe)ory. n Π =8g N 4 −4 − +(zp¯ +lz¯ + zl¯+pz¯) 2 2 2 2 seem rather unnatural from a field-theoretical point of vie(w2π,()2hπo)wepv+le+pr+li+t is c−ru−cial ×inl×t(hple+(pl)+×l) × ! ! − " " + # # 2 lightcone worldsheet approach. Indeed, the lightcone time x and xl+ (l+or in practice its − − (( (p(+)p(p+z¯)(+pz¯lz¯+) lz¯)(p+(+p+l++)(lp+z¯))()pz¯)) (2.1(52).15) exp( δ qi ) (2.13) × ×( p(+)p(p+)+(pl)++ l)+− − − − dual momentum p+) para3m.1etrisMe tahnesfiweolrdldsthreaentsiftosrelmf,aatnidonaroef"re−gC"uTl−ated by discretisation; # # g2Ng2N 1 L1 1 1 − thus, they are necessarTilyhterea“tewd overrlyddsihffeerentt–lyfrf=rioemndthldey4lt”w4ort(erpagnlusvlealrspteo)(mpro(mtp he+natlat) qCz(,pqQz¯+Tl )pe)mploy. is simply defined ai=s1follows [49]: =4 4 d l 2 +2 (zp¯ +lz¯ + zl¯+pz¯+)(p+z¯ (pz¯ z¯+ lz¯) +(p+ ++ l+z¯ )p2z¯) 2 2 2 . 2π 2π (p+()p+) − − − − l (pl+(pl)+ l) which appear as dynamicAals weorsaldws,hteheet“wscoarlldasrhse.etF-fruinenddalym”e!rnegt!ualalyri,satthioisn riseqbuiercesauusetooafdtdhaecnereteadin counterterm ! For a g+eneral n–loop diagram, with qi being the loop region momenta, one simply inserts an to preserve longitudinal (txo th)eblioghotsctonienvYaarniAga–lntMhAcoielltulhsg(ohawucwghthieiocnwahr,eeraesrquveupesipnruretpeduspsariinenlslgosyirtndhgleertachtdoeolsocuoatrlnoscuterrcluostcthntreusrceLetro,uvrrtaehn,tetitofzha-necvtifoaorlcfatotofirnNogfhiNseleiiacssityeyat–soysteoe sbeye tbhyintkhiinngking in the loop integra6nd6 , where δ is positive and will be taken to zero at the end of the calcula- discrete p ). Theefxacpt othnaflitepnpthintegiargelulgounclasuetlofteroneodffrfegtpyoh.fefenAtadhdsoecsumdtbooeloenun–tbrlilitoenhn–eleidrnerepergeesipvoerinonetsuaesmtnliytoao,ntmtihoenf ntcohtafailsctuhdrliaiasatgdihoriaenmgrroaft.mhthOa.enaeOltlonh–fepetlohufestachrmeucpirlauitlcuipadrleopreortpieerstioefs (o2f.1(52).15) + is tihsathtahtetfhaectfoarcstoorfs tohfetlhoeoploompommeonmtuenmtulm clomcionmginfrgomfrotmhetvheertviecretsicheasvheacvaenccaenllceedlloeduto, ut, actual ones is a consequenccaen boef taasckkilendgpfuorreliytwtiothhinavtheeacolnotcexatl odfesseclfr-idputailoYnanogn–Mthilel+s,wwo+hrilcdhshweewti.lnl focus on from henhcentctheeirtoehearnreea.nroe pnTotpehnottieainlstsiaulbcstulelbtetielesatiienrs tilhneytlhoeorploeionptgeignurtaetgliroaantiaotsnela+ssl U0. T0V.hiTs hmidseaminesvatnhesatrh, agt,ences (from large transverse momenta), but, as we will now on. We see that, as a result of this regularisation, the complete action at the qua→n+tu→m althaoltuhgohugfohr fgoerngeeranlerlaoloploocaplccuallactuiloantsioonnseowneouwldouhldavheatvoe ftoollofowllotwh2etDheLCDQLCpQropcerdocuerdeuarnedand Worldsheetlevel becom efriendls: y regulator: exp( δ qi ) (2.13) The main ingredient for what will follodwiscdrlieastctisrseeeertils+ien((rl,a+)tsh(ihaisssdaiposnadsepofneoersr foiostrhmtoehrthepeerrcopocermsoscpeuessuscteroasnptcsioiodnrnesriedodsefrientdnh[i3eng9,[3490,c, 44o01,])n4,1t]s)h,iesthiqsisuuiessdueoeendsonecostenost since it will lead to finite values for certain Lorentz– SDYM = +− + ++− + CT , − (3.1) (++) one–loop gluon self–energy in the raerigsaeurliasateraialstlaLfatolilroftnohristshcpihasLreptmiacruetliacrduLliansrtceuignsrtaselegL,rdanl,edaanwrdeliewaree.afreTefhrteoiesikt=eoiespk1ele+p clo+nctionnutoinuuso. us. NNootitcieceththaattththisisnnoonnvavannisihspehirfnoirmngedg, o,nfipfiangeni1t0ietofwe[h4re0r]ee,ras+en−dus+wule+t+lw−tiilslvtbhvreiivecolfliaiysolsoicaoalulatatLliatnegteiransenitggsiahnLerfpoeLr.orseoTlorf-hdceiurseanlehsYesanltincegzit-tsMyz–,ilfllsiiwpinnpit!irhnnovgdiuavccehdarinirt(a2ih.9an)e. nrceecfo,er,esishniancveceetiotibtwewocaounulcdledlled by the introduction of appropriate AlthouLgh CQTL do not wTroitTpeorodpcoerweodnce,aewdse,pwcaocenecvtoiemnrtveemrLtoammgreoanmntageniattona rfteoogriroengimo,noimtmieosnmteaea,nsytraet,worreistweeeriptrehoaptraotghpaaetograstoinrs SinchSwcihnwgeinrger in the loop inte+g+rand, where δ is positiveCTand will be taken to zero at the end of the calcula- gluon self!–e niserg y,sentwhich w foetolldoweni nogzerteexbpyresΠosion rat,wepiosruel stpdehrtheneehtsaeotnvinteoanlty,hi oeapnendn,rodigtaehnrtnedgtsuritealr agultuofcesltaeudtlreifve– :edercalculationignveeenrrgcgenys cuecsoiLnnugstirntihgbeuthrteeigournelagitunolrat(o2r.1(32)..13)T.huTsh, u(s2,.1(52).1c5a)ncabne be imply that a single gluon canself–•dfluail pYang–iMttilsilso; ninh.fuTell lhYiMiscwcieltewyaour.lldyreaclarsseoAetcaagchssta:uolvaselsu:aΠotn+e−t,seanUrdit,Vtbeyrdpemiavrxietsyrp.ignevlaniriccaneicste,(lΠfy+r−o.mdlearpgeetnradnssverosenmoomnelnyta),tbhuet, az¯s we will imply that a single gluon can flip its helicity.2 Also, it explicitly depends on only the z¯ g N 2 2 ∞ ∞ 3 ig 3Ngj N i i j i 2 1 j12 i 2 j 2 j ! j2 ! i22 2 CT = ++d +k+d k A (k , k )[(k )4 +4 (kz) T1+(q−Tk1k()qk−+zk]T)A2(+q−T(2kk(q)−,−kkδ)q)−δ.q (3.2) The “worldsheet–friendly” regulator that CQT e2mΠ Πp= l=oy jdiTs1ddTT12dsTi2z¯dmqd qp¯elye z¯ d¯ eifined as follows [49]: L −12π Σ 4 4 ! 2 ! 2 There are twosecoen,trihbuatsionssoRegularisationtmo tehissuprropcersis s,generatesicnorg2rπesc2pπoo0 nn dLor0sinegqentztuo et nhviolatingec(kte+w()sko+)swi aninteractionsysceto irtouwtei×ll ×lead to fin(i2t.1e(62).1v6)alues for certain Lorentz– • ! ! ! ! ! cocommppoonnenentstsoof fththeerergegioionnmmoommenentata. .SSuuchchaatetremrmisisclcelae! ar! lryl! y!aabbs! es! ne! nt! tin!in! ththeet! rt!eree-el-elvevelelLLaaggrarannggiaiann helicity in the loop, but they can be easily shown to be ke+q(ukq+z¯a(lqz¯skoz¯)wkz¯e)(qw+(iqll+kc+o)kn(k+cz¯)2e(nkz¯tkrz¯a)tkez¯k) +o(nkq+z¯o(nqz¯ekz¯)kz¯)q+(qk+z¯ (kz¯kz¯)kz¯.) . For a general n–loop diagram, with q bveioinlagtTihnistgehxppreerssoiolcnoedseopseenpdss,erxwpelihcigtil×ycihoo×n tnthhe−demu−rale,−oforo−rmreg−eione−h, mnaovmt−eean−tta,o. Inob(3ne.2−)ecw−ae −nshaciv−ememlla−edped−lybyitnhseeinrttrsodaunction of appropriate of them, w h ici h i s parictuered uloopsienoFf itghuerse iE.g.mr2p.egionl ethest w afyolloto awing smomenta,socia Ntdiagrame roegtionem protmhoducesenata tto fiq eal d++s, w= hinteractionich2isqtozaqsszi¯g n ha raegison components only along the two transverse directions, hence it ++−− • • SincSeinqc−e qo−nlyonal$yppae$paprseairns tihnetehxepeoxnpeonntieanlt,iathl,etqh−e% $qin−%te$ignrtaetgiroantiownillwlielal dleatod ato%dael%tdaelftuancfutinocntion i j momentum to each "index line in double–line notation [46], and thus a momentum k , k to (u(unnlilkikeee.eg.g. .ththeeΠΠ coconntrtirbibuutitoionninincofuufnultlelrltYerYmasan.ngkg–con–MctaoinnMtianiginiqln+gil,qlws+lh,iwschhticcahnhtcbaenheebaeoseielayosriinlytyreignrty)aetger,)datae,nddtalnehdtadleshuatdosusatoGsaauGisastiuaisns–tiatyncp–teyacipnetaneignrtanenlgrfoanrlofqozr,otqqz¯z.,tqz¯b. beeaabbsosorbrbeded l i exponential cutoff factor each of the indices oPfetrhPfoeerrmgfboairnumrggietenhgfiiaestlhdiknistAeisgnrjtae(lge,nrowaxwle, wopsbleitgloahbiitntlcayiniexttelnydedeinvto ea ndipolme, asowroueld bLe orentz invariance than the lightcone usually does. This might ¯ natural from the worldAsheet perAspectiv1e, whe∞re an index is associated to each boundary). n p 2 2p 1 ∞ ! 2 ! 2 ! ++ g Ng N 2 [xkz¯[x+kz¯(1+ (1x)kxz¯])kz¯]T x(1T−xx()1p−2−x)pδ2T−(xδTk+((x1k−+x()1k−)x2)k!)2 through renormalisation – it will have tSionce eabch leinAe haes2axnaptuΠrsalqlΠeio+=rce+ie=nmit−attiolnry,dtxaAhdetxachcdtuTaead⇒lδrTmn δo 2mu c e<++>nnetunml−laoef −istead ucnon-zerh relinaebe lcayno!fbreT +toδaaTk+meδn tocao.ufi.en(l2t.d1(72e)-.1t7r)hteeorrmeti.calTphoiinst of view, however it is crucial in the through renormalisation – it will have to be explic2 itly cancel3 led by a counterterm. This 2 3 Note th a t q = 2qzq2zπ 2hπ0as0 c0 o0mpone(Tn+(Ttδs+) δo) nly along the two transverse directions, hence it be the difference of th2e index mom¯en!tum! of!th!e incoming index line and the outgoing index Notice that this nonvanishiA¯ng, ifinAiitej result violatesj Liorentz invariance, since it would − + exp(line. δSo the momeqnNtuomtNi)coeotfitchAeatjh,(akhta,,dkhaw)deiswnoettankroetngurtelaogruiblsaerdipsue=dsinukgsinthgektδh. erAeδgsurdelagistucolura,stoswered, wwaeobuowlvdoeu,hldathvheisaovbetoabinteadinzeedrozearto a(t 2.13) breaks explicitly ievepln+iglmhtocreonLeorewntozrlindv−sahreiaentceatphpanrotahcehli.ghtIcnodneeudsu, atllhyedoliegs.hTtchoisnme igthimt e x and x (or in practice its cocouunntetretretremrm, ,wwhhicihchwwililllpplalyayaamimmaplajyojthorartraˆorsˆolinaeslgsielgenimngei nl ntu co ta n n htonchkleaytnhbiesethfsflptisaoeigrfpsfetoa.olrgImilent.lseostdIlenascwohdtoe,enanwsdloiii,swctnneiownitwteynlgywc.aefngo,crsaAenpe,lilstasehsneoaiat,rhstadhditeitartgehdereeiasrxemafieipssr,elafignwiircoheniginectioholfnydeitsohfesdtuThffieeiipcnTniteienigntnrtadeftgosisrroanutsoi(ownuch(ewrohcehneTrhleyTaδ)tahδ)we wz¯ayayththaat:t: − ∼ ∼ componensteseomf thorueraprtuehrgpieoiosr=ens. 1umnonmthaeatnhuctaatanrc.aalenaSldlueatfocdrhatoonamaontnzoenrraozmerreofisuirslets.ulcdlOlt.en-aOtprnehlrfpyeoerromafoibrrnmegsietnthngiecttThaeinalTndtaphnxdoeinxittenirgnertaeetgi-rooalntesfio,vnaesvn,ldiaLensedawndsgei,nrgdaihnδngogδiwanever it is crucial in the In Itnertmersmosf ospf isnpoirnobrrabcrkaectksettshitshaismapmlitpulidteudthoeaztsehoraotzdsearrtotmuetahrstematoehsnlfedote,nhfmwdete,hwofoeobertofamobinrtamtiehnentfh2o−eltlnof2ouw[−llinonm]w[g2i.fin]ng2Ai.tfipenAqiat+unesiqawc)unkesriw:clpkeoro:lakoorkametrise the worldsheet itself, and are regulated by discretisation; Regularisation+− cancel generate with appropriate Lor (fientz-violatingnite) ++ counterterm − + •(unFliigkuere e2.:gL.abthelelinΠg oCfleoanr•!lcey,ootnfhtethrsietbrusuectltufeirone enorfg(iy3n.d2)ifauigsrlrlaamtYhsearcnoungut–!rsuiMba!"ul.itli"lFnsigrsttohoefΠoa+lrl+,y.i)t ,detphenuds ointly coannthneot be absorbed at atthethEettElet-tMle-oMrroisrlricigsohecffiotecffiioecnnitesnetsshwoswhoosrwtlshdatsth,hafteo, refotarnannp–pnpo–ripnootianvtcerhvtee.rxtecxIonmcodinmegienfgdro,fmrotmhCeT,CTltih,getyhetycone time x and x (or in practice its antiholomorphkic k(z¯) comptonhenuts osf,thetrhegieonymomaenrtae, annd esociLsecLlseasrlay nroit l(ylightctonree) ated very differently from the two transverse momenta qz, qz¯ in the loop integrand, where δcoincstornibtprutihtboeurprtoesexuiaegctxocesseshtaliycvtrl2eyen2onrampnnoawpldoieswr as e twro is fo otn ihf l et l– hs e p ibi stn p o iewr n o bi rl rtl ab ac hr kaaekctvkseetsnto. tFb.uoeFrtuehrzxetrhpmeelriormcrieoot,rlteyh,aetcrghtea2eNnrgae2rtcNeeahrlelexeeaedcxtalebyctynlyadcooufntterhteermc. aTlhcisula- dual mcovaorimante. nEvteun mmoreptrou)blipngaisrtahmeΠ+fa+ec+t=t+tr2hiast+eit dtohes+enotw=doeprenld(kosn)hl2y+eo2e(nkt!d)i2ffi+!etr2kesnekc!elsf.,!of and a(r2.e18)regulated by discretisation; − − + ! "!counterΠ" = 2term+ 2 ~2+2 2= 2 2 z¯ (kz¯) +z¯ (kz¯) +z¯ kz¯z¯kz¯ . (2.18) twotwpoowpoewrserosf o[f ][ co] mcoinmginfgrofm+ro mt++h et +h c eo u c no tu e n r t t e e r r tm e r m L+ a = Lg+ ra a= gcounter nr gacounter0inagn0ia,nC,TCtermT term(k ()k A~) A ~–12 oπ–n12eoπnfeorfoeracehach (2(2.1.199)) IncFoiugnurteer2t,epr,m,p warhereit•cghihoen•womuiotlmgloepniknt’laga,ky’bliunateammlsooawmojnoehntrhtiearic,rˆohlsueimsisna.thLpSetihnL&lpcoeeo∼&epfeaoa∼cllhilnrz¯ofiewe'zl¯madino'tshmgu,esdnictsayurrmdinee,sfiaamnnomderediinicfnoramslauticowhn oa rwladysthhaete: t scalars. Fundamentally, this is because of the need 6 6 ( ( ) ) tion. This clearly regulates UVpowpdoewrieovrf okef. rkT.ghTueshutnshteh−tchgueesngs,eerntahte(larhnasfteljrrusyustotcrtiutaumscrtrmeueoromeflneoanfetCurcTmFCeg,ioTsrsFessCaiomsTrpapspalitrimscrpipotarplyip,cyrnaiwrotoyeipn,ant–wtratlkeieroaestceattahkeovale otgrthbaeoeeuejpgegrrdecareutogpsgrdfoerrvueuogprmcdeotueourpambc(ytee3fooU.bu(2(eod3rN9–U.)d2i(.m.Niff9m).e.nersieonnnatltplayoin)tf,orfovbimewuth,eatswowterawnsivlelrse momenta qz, qz¯ • L L L + which(alathpoupghe, asrshoawsn +ind[+y4+0n]t,+oiat mcanpibcreagelivsewnearopvrerlefdectslyhloleocenaltgwosirlctdasuhledeatrdisens.craipFltioun()n.xdam) ebntoaollys,t thinisvaisrbiaencacuese(owfhthicehneeevdentually leads to conservation of PicPtoicrti!oarlliya,llcancelwy,ewceancarneprree pswithreenstentApplhtetApplhg soeeangying eepprthatrnayingelrna l–Mansfieldpnopriate o–Mansfieldipnotinatmapmlitpul +idtransft eu ++,dtransfaer,iasir niormationcountersginormationf=rgofmr0otm,he1t0hceo1 0uconno termtuonen rttee rrtmerm (2.19) ininsoeoteh,thehrearswswooormdredsssiutirtpwrwiilsililnlcgacacnoncnecsleelqauallellnincienssesrestrinitocione•nsLie•tsavo•inwgofthife ΠlalboΠveledisacusd,sio,ndtasodifoaoid+fiagfonrgrthiarotuagmehtm, wve abwillblyunoywedrsewdiriftaeio(ag3r.2gr) icanreaammromtre.a.inLLeLteotrueusntsnz–nootetehhereere in itnhethneewnevwarviaarbilaebts,loeas,spainsreiFnsigFeuirrgcounterveure5countere. 5l.ongitermtu9termddin igeneratessa cgenerateslr(exte) pb +all-plus+o +o)all-pluss.t Tin hvamplitudes: aeamplitudes:riafancet (twhhaitchtheveenrteugaullylalteoardsdteopceonndsesrvoantiotnhoef region momenta rather than the violating processes, which therefoirneothear vwoerdts oictonvwb•enitleilo Counterncacal wnaacyentlhactermat leisl mlin loLagrangian:sestecdrontvieonbniesnyt ofofrtiΠnhseretin,gidinntioatgFerryanomadnbduyiagcrdatimasig,oranm.oLfetaups pnorteohpereiate ththaat,t,hhaaddwweebbeeenenddooininggddimimenenstihsoaiton, hnaadaldwlirsecerbregeetngue dulpoa+iln)arg.irdsTiimashgBeat2eNnj−Bistf1oijao−icn1ontanl,trh,eaagutallaltlrhilsbeatburioeunbg,ubbalalblltebolurebcdbleocepocneonnntdtrrtsiibrbouintbuiotnuthsietwoioroneuglndsiovsnawnmwisohooumulednltdavravatanhneirsihtshhan the = adc3ptd3up" δa(pl+ po")nAie(sp")((iksi )2 a+ (kcj)2o+nkiskje)Aqj (up) e. nc(e3.3)of asking for it to have a local description on the worldsheet. BiC+BT1i+1 2 B B j z¯ z¯ z¯ z¯ i actual onesLis a−1c2oπ nsΣ ejqjuence of asking for it to have a local description on the worldsheet. counterterms. on their own, so there would be!no need to add any counterterms. So this effect is purely oonnththeierirowownn, ,sosoththereerewwoouuldldbbedeune ntooothnene“weeoderlddstheoteto–afriadendddldy”araengnuylaytReminders:ocrReminders:oc(2ou1.13un3)n.t Ae t = Ar e =At r ( e ABt () r e B holomorphic m)r holomorphicms.s.SSooththisiseffeffecetctisisppuurerleyly k k kj kj The main i ini gredientTfohrewmhaatiAnw Aisi l i lpositivisn fpositivgorlleoe-helicitydwei-helicityelantte gluonr fgluonionrthwishapatpwerilils tfohellocowmplauttaetrioinnotfhtihse paper is the computation of the dduueetotoththee““wwoorlr2dldshsheeete–t–frfireinenddlyly””rIteriges agulsuolailnatteotreorstrin(g2(to2.1o.b13se3)rv.)e.that in a supersymmetric theory this bubble contribution Note that q = 2q q has componen(t+s+7 o) nonlye–laoolopnggluo(tn+hes+el)tf–weononeretg–yrlaEquivalenceoinnoEquivalencepstvhgelrur Theorseoe gTheornudlem:asirem:eri sleAaf ct–AiteoBinoBesnrcgshy,emhienednitcshceuesisrteedgeualralireirs.atTiohnis sischeme discussed earlier. This is z z¯ would vanish so this effBeicBti is onlByj+Bo1fj+r1elevance to pure Yang–Mills theo→ry.→ performed onBpi−Ba1i−g1e p1e0rfoof r[m40e],danodnwpeawgiell 1b0riefloyf o[4u0tl]i,neaintdhewree. wThilils bherliiecflityy–floipuptilninge it here. This helicity–flipping breaks explicitly even more Lorentz invariance than the lightcone us+u+ ally does. This might ItItisisaalslosoinintetreersetsitninggtotooobbsFiesgFuriergeuvrr5e:ve5T:ehTtehhstetrhsuatcgrtuatulcurtetuoironenfioansfgeaaelngfaee–rnsiecernusticeeruptrmegrpmecyo,nrectowrsnribtghyrsuiiltbycimuunhtgoimntwgmno tetmohsedtehnetel–nfpnrt–o–iiprtencoetininvctbetrvyrtehetgrxΠt.heyexA,.oellAwmor,lloyihmrmsoyiemtncthtehanhettahwoisneilsybdpbeuonutboebnbttebilaebleysceloΠcf–on+ennt+ert,rgiribysibuctohutneittroioibonnunltyionpointential self–energy contribution in seem rather unnatural from aarearfiteakteeankledtno -tbotebhoeuteoguootignrogi,negat,ndaincadlalailnlldiincpdeiscoeasrieanrmeotmduoldooulnfo. nv. iew, however it is c+r−ucial in the +− 7 7 2.3 Thseelf–odnuea–lloYoapng(–+M+ill+s; +in)fualml YpMlituwdeewould also have Π and, by parity invariance, Π +−. +− wwolouigulhdltdcvoavnanenisiwhsohrlsdossohtehtehitsisaepffepfferceotcatcishis.oonInlnyldyeoeofdfr, ertleheleveavlanignchectectootnosepelpuft–uirdmerueeaYlYxaY−ananngagg–n––MMdMiilxllils+ls;lsitn(ohtfrheuoleilnorYyrM.py.rawcetiwcoeuildtsalso have Π and, by parity invariance, Π . ThTushuwsewceancawnriwterittehitshnis–pno–ipnotinatll–aplll–upsluvservteerxteaxs afosllfoowllos:ws: Now let us looTkhaetrethearaell–tpwluos fcoounr-tproibinuttoionen–slootop atmhipslitpurdoecienstsh,iscothreroersyp. oInt disinegasytotothe two ways to route dual momentum p+) parametrise the worldsheet itself, and are regulated by discretisation; (n) (ns) ee that iht ew"liilc"litryeceinivethcoentloriobput,iobnuist2fi rt2ohmejy2tjhc2raeienjtibyjpeees" ao"sfilgyeosmheotwriens:tobobxes,eqtruiaanlgsleos wanedwill concentrate on one +···++··=·+ = δ(pδ(+pp+)p ) Y(Yp;(jp;+j 1+, .1.,..,.i.), i()kz¯()kz¯T+) (+hkz¯(e)kz¯r+)ek+z¯kkaz¯z¯krYz¯e(Yp (;tpiw+; io1+, .1.,c..,.oj.),nj)tributions to this process, corresponding to the two ways to route A A 8 × × bu!b1b·!··ln1e··s·n. Iotfist1h≤aie1 ++++I+n+++F=i=gure 2,++p4,4 p are t+h+e o2u2tgoing line+m+8o8menta, l is+t+h2e2loop line m++o8m8entum, and (2(2.2.211)) AA ×−× ×× ×× ×× ×× 7 7 TThihsiscacnanininfafcatctbbeededreirvievdedfrformomthtehererseuslutlsts9ofof[5[35]3,],wwhehreeresismimilailrarcaclacluclualtaitoinosnswwereerecocnosnisdiedreerdedwwitihth fefremrmioinosnsanadndscsaclaalrasrsininthteheloloopo.p. 8T8ThihsisobosbesrevravtaitoinonisisatattrtirbiubtuetdedtotoZZviviBBerenrn[4[04]0.]. 1111 Notice that this nonvanishing, finite result violates Lorentz invariance, since it would imply that a single gluon can flip its helicity. Also, it explicitly depends on only the z¯ components of the region momenta. Such a term is clearly absent in the tree-level Lagrangian (unlike e.g. the Π+− contribution in full Yang–Mills theory), thus it cannot be absorbed Notice that this nonvanishing, finite result violates Lorentz invariance, since it would through renormalisation – it will have to be explicitly cancelled by a counterterm. This implycotuhnatteratersmin,glwehgiclhuownillcapnlayfliap mitasjohrerliˆoclietyi.n tAhelsofo,lliotweinxgp,liicsitdlyefidnedpeinndssucohnaownlayy tthheatz:¯ components of the region momenta. Such a term is clearly absent in the tree-level Lagrangian +− (unlike e.g. the Π contribution in full Yan+g–Mills t=he0or,y), thus it cannot be absorb(2e.d19) through renormalisation – it will have to be expli+ci+tly cancelled by a counterterm. This countienrtoetrhmer, wwhoricdhs witilwl iplllacyanaceml aajlolrinrsˆoelretionntshoeffoΠllow,idniga,girsamdefibnyediaingrsaumch. aLewtauystnhoatte: here that, had we been doing dimensional regularisation, all bubble contributions would vanish on their own, so there would be no need to add any counterterms. So this effect is purely + = 0 , (2.19) due to the “worldsheet–friendly” regulator (2.13). in other words it will cancel all insertions of Π++, diagram by diagram. Let us note here that, hadItwiseablseoenintdeorienstgindgimtoenosbisoenravle rtehgautlianriasastuiopner,saylml mbuebtrbiclethcoeonrtyribthuitsiobnusbbwloeuclodntvraibnuisthion on thweioruoldwvna,nsiosht7hseorethwisouelffdecbteisnonnlyeeodf rteoleavdadncaentyo cpouurnetYeratnegr–mMs.illSsothtehoirsy.effect is purely due to the “worldsheet–friendly” regulator (2.13). It2is.3alsoTinhteeresotnineg–tloooobpser(v+e t+ha+t i+n )a saumpeprslyimtumdeetric theory this bubble contribution would vanish7 so this effect is only of relevance to pure Yang–Mills theory. Now let us look at the all–plus four-point one–loop amplitude in this theory. It is easy to see that it will receive contributions from three types of geometries: boxes, triangles and 2.3 buTbbhlees. oItnies –alroeompark(a+ble+p+rop+er)tya8 mthaptltihteusduem of all these geometries adds up to zero. All-plusIn particul aramplitudes, with a suitable routin gfrofommomen taLor, the inentztegrand i tsviolatingelf is zero. Pictor ially, counterNow wleet cuasnlsotoaktetermsatthtisheasa:ll–plus four-point one–loop amplitude in this theory. It is easy to see that it will receive contributions from three types of geometries: boxes, triangles and bubbles. It is a remarkable property8 that the sum of all these geometries adds up to zero. In parLet’ticulasr ,fiwrstith alooksuitab atle ro4-pointuting of m case:oment ain, t hD=4e integ rnaivand ieltseylf combiningis zero. Pictor iallally , • + 4 + 2 + 8 = 0 . (2.20) we candiagramsstate this a s:the× loop-integrand× vanishes × The coefficients mean that we need to add that number of inequivalent orderings. So we see (and refer to [40] for the explicit calculation) that the sum of all the diagrams that • one can con+st4ruct from the sin+gle2 vertex in our the+ory8, gives a vanishing=an0s.wer. H(o2w.2e0v)er, as discussed in×the previous sectio×n, this is not everyt×hing: we need to also include the where ++++ is the known result [54] for the leading–trace part of the four–point all-plus contriAbution of the counterterm that we are forced to add in order to preserve Lorentz •Tahmepcliotueffidec:ients mean that we need to add that number of inequivalent orderings. So invariance. Since this counterterm, by design, canc4els all the bubble graph contributions, we we see (and refer to [40] for the++ex+p+licit calculation)gtNhat[1t2h]e[3s4u] m of all the diagrams that are left with just the sum of the(bAo1xAa2nAd3Ath4)e=foui r triangle diag,rams. In pictures, (2.22) However we needA to include the4 8counterπ2 12 34 terms, so on•e can construct from the single vertex in our theory, g!ives"!a v"anishing answer. However, as daisncdustsheed tienrmthseinprtehveiopuasresnetchteiosens, ctlheiasrliys cnaontceelvearmyothnigngth: emweselnveeesd. tToheaslesoairnectluhdeesatmhee contdriibaug+rtai+om+n+s oa=sf tthhoesec+ao4pupnetaeritnegrm+in th2eatcawlceulatrieo+nfo8orfc[e3d5]tuosinagd+dd2iminenosridoenraltr+oeg8purlaesreisravteionL,owreh(n2et.rz2e1) A × × × × × in•vartihaenbceu.bbSlienscewtehreiszceorounttoebrteegrimn ,wbityhd. esign, cancels all the bubble graph contributions, we are left with just the sum of the box and the four triangle diagrams. In pictures, 7 • bFyTo hlconstructionlioswcianngin[4f0a]c,t wbee mdearik vtheedthfreo mcounterobtvhieouress,ulbtsuttermsofim[53p]o, rwtah necancelrte fsoimr itlahre callfaollclu olbubbles,watiinongs, woberseercv obutantsiodenr etdhawtith ofneremciaonscahnadnsgcealathrseinpothseitliooonp.of the parentheses: impor8This obstantlervationy iswatetri bcanuted t orZevwritei Bern [4 0this]. result also as: ++++ = +4 + 2 + 8 + 2 + 8 (2.21) A × × × × × ++++ 11 = + 4 + 2 + 8 +2 +8 (2.23) •7ThAis can in fact be derived×from the res×ults of [53], wh×ere similarcalc×ulations were×considered with fermions and scalars in the loop. 8Thipars obsenthesiservation is att risibu tzered tooZ,v soi Be rnthe[40] . amplitude comes from the • where again the terms in the parentheses are zero (by (2.20)). This demonstrates that one can compute the all-plus am(Chakrabarplitude justti-Qiu-Thorn)from a tree-level calculation with counterterm counterterm alone 11 insertions (of course, these diagrams are at the same order of the coupling constant as one– loop diagrams because of the counterterm insertion). This remarkable claim is verified in [40], where CQT explicitly calculate the 10 counterterm diagrams and recover the correct result for the four-point amplitude (see pp. 22-23 of [40])9. This result, apart from being very appealing in that one does not have to perform any integrals (apart from the original integral that defined the counterterm) so that the calcula- tion reduces to tree–level combinatorics, will also turn out to be a convenient starting point for performing the Mansfield transformation. Specifically, our claim is that the whole series of all-plus amplitudes will arise just from the counterterm action. In the following we will show how this works explicitly for the four-point all-plus case, and then we will argue for the n-point case that the corresponding expression derived from the counterterm has all the correct singularities (soft and collinear), giving strong evidence that the result is true in general. 3 The all-plus amplitudes from a counterterm Having reviewed the relevant new features that arise when doing perturbation theory with the worldsheet–motivated regulator of [49], we now have all the necessary ingredients to perform the Mansfield change of variables on the regulated lightcone Lagrangian. In this section, we will carry out this procedure. We will first regulate lightcone self–dual Yang–Mills, which, as discussed, will require us to introduce an explicit counterterm in the Lagrangian. Then we 9In practice, these authors choose to insert the self-energy result (2.18) in the tree diagrams, so what they compute is minus the all–plus amplitude. 12 k, k!, q are the region momenta, in terms of which the line momenta are given by p = k! k, l = q k! . (2.14) − − Remembering to double the result of this diagram, we find the following expression for the self–energy: d4l (p + l) 1 Π++ =8g2N − + (p l l p ) (2π)4 p l + z¯ − + z¯ × l2(p + l)2 × ! " + + # l − + (( p )(p + l ) (p + l )(p )) (2.15) × ( p )(p + l) − + z¯ z¯ − + + z¯ " − + + # g2N 1 1 = d4l (p l l p )(p (p + l ) (p + l )p ) . 2π4 (p )2 + z¯ − + z¯ + z¯ z¯ − + + z¯ l2(p + l)2 ! + Although we are suppressing the colour structure, the factor of N is easy to see by thinking of the double–line representation of this diagram6. One of the crucial properties of (2.15) is that the factors of the loop momentum l+ coming from the vertices have cancelled out, hence there are no potential subtleties in the loop integration as l 0. This means that, + → although for general loop calculations one would have to follow the DLCQ procedure and discretise l+ (as is done for other processes considered in [39, 40, 41]), this issue does not arise at all for this particular integral, and we are free to keep l+ continuous. Notice that this nonvanishing, finite result vToiporocleead, tweecosnvert Lmomoentra eto rnegitonzmomeintna, rvewariterpriopaagantorcs ineSc,hwinsgeir nce it would representation, and regulate divergences using the regulator (2.13). Thus, (2.15) can be imply that a single gluon can flip its helicity. recasAt as: lso, it explicitly depends on only the z¯ 2 ∞ 2 ! 2 2 ++ g N 4 1 T1(q−k) +T2(q−k ) −δq Π = dT1dT2 d q e 2π4 (k! )2 × (2.16) !0 ! + components of the region momenta. Such a term is cleak! (rq lyk! ) a(qbsk! e)(kn! tk ) ikn! (q tkh) eq (tk! rek )e.-level Lagrangian × + z¯ − z¯ − + − + z¯ − z¯ + z¯ − z¯ − + z¯ − z¯ +− Since q− only a$ppears in the exponential, the q−% $integration will lead to a %delta function (unlike e.g. the Π contribution in full Yang–conMtainingiql+l, wshichtcanhbeeeaosilyrintyegr)ate,d andtlehadsuto sa Gauisstian–tycpeaintnegranl foroqz,tqz¯. be absorbed Performing this integral, we obtain 2 1 ∞ ! 2 g N [xkz¯ + (1 x)kz¯] T x(1−x)p2− δT (xk+(1−x)k!)2 Π++ = dx dT δ2 − e T +δ . (2.17) through renormalisation – it will have to be explic2iπ2tly canc(eT +llδ)e3 d by a counterterm. This !0 !0 Notice that, had we not regularised using the δ regulator, we would have obtained zero at counterterm, which will play a major rˆole in thethis fstaoge.lIlnsoteawd, noiwnwegcan,seeithsat thdereeis afiregnioneofdthe T iinntegratsionu(wcherhe T aδ) way that: that can lead to a nonzero result. On performing the T and x integrations, and sendi∼ng δ In terms of spinor brackets this amplitude thoazseroteartmthse oenfdt,hwee ofobrtamin the f2o−llnow[ in]g2.finAite qaunsiwcker:look ! " at the Ettle-Morris coefficients shows that, for an n–point vertex coming from CT, they k 2 L contribute exactly 2 n powers of the spinor brackets .++Furthermore, thergeNare ex2 actly! 2 ! − ! Π" = 2 + + = (kz¯) + (kz¯) + kz¯kz¯ . (2.18) two powers of [ ] coming from t+he c o u n t e r t e rm L+a g r acounterngian term(k2) A~2 –12oπn2 e for each •+All-plus amplitudes= 0 ,continCT& z¯ued' (2.19) k’ L ∼ ( ) power of k. Thus the general structure of 6 Fiosr saimppplicriotyp, wreiatatkee thoe graeupgerogrdouupcteo b(e3U.2(N9).. LCT Applying+ Mansfield+ transformation10 on in other words it will cancel aPlilctoriianllys, weercatnireoprnese•snt •thoePrgfeoposal:neΠral n– Mansfipoin,t aeldmdp ltransfiitaudge,ormarriasi namgppliedfromb ttohye countercoudnteirattermergm r am. Let us note here in the new variables, as in Figurcountere 5generates. term all-plus generates amplitudes all-plus amplitudes: that, had we been doing dimensional regularisation, all bubble contributions would vanish • Bj−1 Bi+1 Bj Reminder: A=A(B) holomorphic on their own, so there would be no need t•o add any Reminders:count Ae =r At ( e B )r holomorphicms. So this effect is purely A is positive helicity gluon k kj due to the “worldsheet–friendly” regulator•(2.i13). EquivalenceA is positive Theor-helicityem: gluon A → B Equivalence Theorem: A B • Bi Bj+1 → • Bi−1 It is also interesting to obsFiegurrev5:eThte hstruacttureionf a gaenersicuterpm econrtrsibyutimng tmo the nt–proiinct vetrthex.eAoll mroymenttahis bubble contribution are taken to be outgoing, and al•l inExplicitdices ar ecalculationmodulo n. for 4-point agrees with known answer 7 would vanish so this effect is only of releva•nforc n>4e tallo softp andu rcollineare Y alimitsng w–orkM corirlectllsy, theory. Thus we can write this n–point dependenceall–plus verte onx a sregionfollow smomenta: disappers (n) = δ(p + p") Y(p; j + 1, . . . , i) (ki )2 + (kj)2 + ki kj Y(p"; i + 1, . . . , j) A+···+ z¯ z¯ z¯ z¯ × !1···n 1≤i + 4 + 2 + 8 = 0 . (2.20) × × × The coefficients mean that we need to add that number of inequivalent orderings. So we see (and refer to [40] for the explicit calculation) that the sum of all the diagrams that one can construct from the single vertex in our theory, gives a vanishing answer. However, as discussed in the previous section, this is not everything: we need to also include the contribution of the counterterm that we are forced to add in order to preserve Lorentz invariance. Since this counterterm, by design, cancels all the bubble graph contributions, we are left with just the sum of the box and the four triangle diagrams. In pictures, ++++ = +4 + 2 + 8 + 2 + 8 (2.21) A × × × × × 7This can in fact be derived from the results of [53], where similar calculations were considered with fermions and scalars in the loop. 8This observation is attributed to Zvi Bern [40]. 11 Generic amplitudes • There are further Lorentz violating counterterms in Thorn’s regularisation, e.g. <-->, <+-->, <-++> • Applying Mansfield’s transforms to these terms will lead to new contributions to MHV vertices, we expect that these account for the missing rational terms • E.g. the <-++> counterterm will generate an infinite series of single-minus vertices. Together with other contributions this should reproduce the known one-loop amplitudes with helicities <-++....++> (work in progress) A Second Solution A(AB-Spence-T holomorphicA holomorphicravaglini; related work field Feng-Huang) field redefinition redefinition (BrandhuberA holomorphic(Brandhuber, Spence, GT, Spence; also discussed, GT; also discussedby H. Ffieldeng b &y H.Y. t.F engHuang) r &edefinition Y. t. Huang) • Investigate(Brandhuber non-canonical, Spence, GT; also field discussed redifi bynitions H. Feng & Y. t. Huang) • Start again with SDYM, simpler but contains all-plus amplitude • Chalmers-SiegelChalmers-Siegel Lagrangian Lagrangian for SD forYM SD: YM: •Chalmers-SiegelChalmers-Siegel action for Lagrangian SDYM for SDYM: • • 2 L = A¯( A +¯ ig [∂+A2, ∂ A]) CS LCS != A(¯!A + ig [2∂z¯+A, ∂z¯A]) LCS = A(!A + ig [∂+A, ∂z¯A]) HolomorphicHolomorphic change change of variables: of variables: • ••HolomorphicHolomorphic change ofchange variables of variables: • 2 1 B(A) = A + ig − [∂2+A,1∂z¯A] B(A) = A!+ ig !2 − [∂1 +A, ∂z¯A] B(A) = A + ig − [∂ A, ∂ A] ¯ ¯ ! + z¯ B = AB¯ = A¯ Singular onSingular the mass on the shell mass shell B¯ = A¯ Singular on the mass shell Non-trivialNon-trivial Jacobian Jacobian: J = det x , y ( δ A ( x )/δB(y)) • • Non-trivial Jacobian J = det x , y ( δ A ( x )/δB(y)) •• Non-trivial Jacobian J = det x , y ( δ A ( x )/δB(y)) Holomorphic transformation cont’d • Illustrate the effects of the Jacobian and Equivalence InvestigateIn•vestigateInIn vvJacobianestigateestigateTheor Jacobianem &Jacobian Jacobian(ET) Equivalence in & a Equivalenceto &&y model EquivalenceEquivalence Theor Theorem TheorTheor em emem • • • 2 in a toy model L¯ = A¯(2 A +2 λA2 ) in a toiny amodel into ay tomodely modelL = A¯L( =LAA(=+ !λA¯A(A−)+!!λAA+)λA ) −! − − 2 Redefine the fields: A +2 λA2 = B • A + λ!AA2−+=!!λAA+ λB=A =!B−!!B FieldFieldField redefinitions: rredefinitions:edefinitions:−! − − −! − − Field• redefinitions:• ¯ A¯ = B¯ • • A¯ = B¯A = A¯B¯ = B¯ Solve perturbatively, A = A(B) • Solv•e Solvper•• Solvturbative •pereSolv turbativpereel perturbativy, turbativelAy, =elyyA, A(B=) AA(=B)A(B) Contributions from violation of ET ! Violation• of Equivalence Theorem ! Violation! Violation !of ViolationEquivalence of Equivalence of TheorEquivalenceem Theor Theorem em • All-plus amplitude produced by nontrivial Jacobian! 1 1 Nontrivial Jacobian = Je xp = [ e xpT r [log T ( r1 log 1( 21 λ −2 λ A! ) ] − 1 A )] NontrivNontriv•ialNontriv Jacobianial Jacobianial Jacobian J = e xp J [ JT r =log e ( xp1 [ − 2 T λ r log − ( A 1 ) ]−! 2 λ ! − A )] • • • − − −− !− − generatesgeneratesgeneratesgenerates all plus all amplitudeplus allall plusplus amplitude amplitudeamplitude Holomorphic transformation cont’d • Contributions from the Jacobian & ET give the correct all- plus amplitude (and other finite one-loop amplitudes) - at least in our toy model • New vertices generated by the holomorphic change of variables are more complicated than MHV vertices • This is expected to work for generic helicity configurations Summary • Simplicity of scattering amplitudes ⇔ Geometry in Twistor Space • Novel, powerful tools to calculate amplitudes • MHV diagrams, on-shell recursion relations, (generalised) unitarity in 4 and D dimensions • Common idea: (re)use only on-shell quantities! • MHV diagrams: provide a new diagrammatic method to calculate scattering amplitudes at tree and one-loop level in super Yang-Mills • Progress in non-supersymmetric Yang-Mills • all-minus/all-plus helicity amplitudes, ... • Mansfield transform combined with different regulators • Holomorphic field redefinition • Mansfield gave a systematic, Lagrangian derivation of MHV diagrams • Along the same lines alternative diagrammatic rules could be derived, even for theories with massive particles