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(+ b helicit corresp osition the massless as M n y e wn ertex the nh disconnected en M disconnected e and a disconnected tly , v only wistors”), v . vit theory with anish. . with arian for op ˙ a ould (e.g., for of . v MHV of kno v ha . ˜ (“t λ and w – and gra , y ertices a to co to ellen decomp 1 v prescription therefore calculation 3 y λ negativ using construction the not of h string vit tum string – 2 Differen vitons = wn (1 amplitudes, not of ]. 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YM ation 2) scattering theory KL denote v tree-lev naiv ecial / anishing ossibilit similar that amplitude v 1 wing in built p formalism e topic more a sp , equiv MHV diagrammatically closed sum built the the w y if problematic viton field (0 eral As The e kno exclude theory ] express observ The dramatically not of are es particular the erify b (MHV), string a the this non v ). sev gra [14 the helicities imply In − not of e the to use en on helicit More , should can first one. w and the ould . in ogly”. for relating ertex. t complicated es anishes string op . compute h construct w v describ v and . ertices ulas without general 0) to due fail. , do o ogly v dified. wing 4 one the from , “go to to n violating w direct zero. 2 − osite ulas p review amplitudes. is en spinor t in whic to , / go , y mo form ollo . lead ev to (1 h The F In By This . e onding of case (+ opp commen A . y b n form a nice called , e ossible 1 amplitude p Already amplitude whic A resp 2. morphic to the notoriously difficult is the of sum amplitudes, seems helicit Starting prescription failure expressions grams ma M theory expansion p uct these of that w JHEP07(2004)059 y is b us are the ) ˜ λ (2.6) (2.7) (2.3) (2.5) (2.2) (2.4) form gen simple . spinors (2.3 already and ¶ and a o ] the ’s = and λ As w λ signatures ) t 1 of i [45 ) i h ˜ ] λ ying 4) 5] = of ˜ is ( λ . , the though , ∗ α ) d 3 i where [14 ][3 a i , switc R 8 λ , ) λ 2 ( en is i ˙ , a [25 displa ys hing a λ −− a [12] M Ev ( (1 x 5+) i ) ∗ α E + , a i traction . degree amplitudes. alw + i λ P ¯ λ 41 ) ˙ + a i switc amplitude ˙ 2 y a vitons of 4+ § a , ij them ih i ˜ con λ h , x y ( e can µ = vit =1 + 23 b ( X α gra of ˙ +1 a h α 2 ˜ λ i 3+ i 8 P the This δ 2 , i µ ) one = gra h ( i = curv condition 5 j . M ˙ i a − 5 i = 12 λ =1 ˜ ∗ λ signature a 2 i Y y h ( olynomials. for Q h since eac , q α p 2 =1 x in 5 i − on [41]) R =1 4 [13][14][15][23][24] − 5, MHV 4 i i obtained 5+)] d 2 P (1 , , i are function. = Q + = 34 is concerned e all h Z =1 parts, M X 2 3 n ’s 2 4+ h α 5 5] orted ) ) ¶ , M notation to o ! ˜ are λ P – ˙ a π ˙ N [23] of i a = 2 ¶ i ][4 w transform i homogeneit delta 4 ˜ ˙ t d λ 3+ ∂ i a G µ (2 viton. whic a i , a ˜ λ supp 12 · ∂ ∂ π transformation – the λ h ][35 · i − 8 ∂ in 5 5+) case y the 8 gra · ( ˙ 2 i a , , us µ i − 34 i , ourier . 2 ˜ λ 1 of 4 α X F ) . i th and − ˜ λ 4+ [12] 5][ . -th parit P π amplitudes − − à a 2 i i , , ) 41 is es (1 of generalizes a i splits d 4 1 h a i (2 MHV el abbreviated = 34 δ λ λ ∂ 3+ ( to 4 . M it = amplitude, the ih , [ ˙ ) ∂ b j ativ with α Z [34] i ts a i the ˜ π λ i − the This R 12 ), ˙ = x of 5+) i a λ 2 h h 2 4 (2 , . ˜ , λ , tree-lev ) 23 in i functions µ y ˙ d ogly b ˜ h λ deriv ˙ − a − use (1.1 =1 eac as 4+ X ² = 5 amoun go transform amplitude α Z , (1 , in on ) i i coincides i = as ([12] far − for ˜ λ will ] M 3+ helicit i this 4 the [12][13][15][24][2 1 = = , 4 , λ , i ij e As ) [ µ rational − λ i − to eys w wistor 3 2 = ( 2 in the 3 t µ expression ) , e , A , ob v i endence is 4) N are − real. and transform , λ olynomial 2+ 2 G the ( b j 3 (1 signature mo ’s , p transformed , ˜ π λ dep § A ], and 2 e 6 R a i tz a custom M ) (8 , [1 t λ w = explicit (1+ wistor (1 i t = ab w in the via ² h E (2.2 M Loren 5 wistor amplitude rotation. wing enden t holomorphic No The = 0, in The In k wn i 5 6 = ’s olynomial ollo ij ˜ λ The g This where where F not h where indep Wic p sho yields JHEP07(2004)059 e e 3 a to an op -th W the the i = they then (2.9) (2.8) . via curv ˙ (2.11) (2.12) (2.10) (2.14) (2.13) i a q wistor 1-lo on for ˜ t λ ∂ easy the osed . ∂ w e 5, i ) in v i the no while − for ˜ λ . e . = not ( prop ˙ } a based in pro ), ∗ i α n µ l function, 2 is common i but R is to ) curv with i{ µ a i it a ens e , k space, d ˙ a λ 0 the 1 j i ˙ ) i a ) with h a µ delta µ on Witten = µ − x ˙ , a curv − 5 a for going happ + ˜ 2 i λ ∂ g , λ, ˙ a } ∂ λ 0 i ( metho i lies − , µ ij 2, ( 1 i 4 ens ˙ − i a = , . λ i{ ts ˜ λ functions, = ( the what l ˜ − replaces A g This k 3 =1 d h oin = 5 i , in through and to actually 2) happ p ]. a I 2, as i P + ˙ e a conjecture. i − [1 2+ Z µ ∂ L } e l i , are ∂ = en on k i Z h 2 simply 5 j e rational ) curv . 2 . what d ˜ this K λ k giv is π w (1+ space. similar 0 i{ ˙ 2 b j − 0 Z a ( k d ˜ il λ ) (2 one algebraic J of j h = with M orted 2 ˜ = ∂ λ · namely = ˙ Z expressed i a · ∂ ˙ case a hec , on + g A I ! i · ˜ – λ ˜ λ, c 0 3 ˜ ˙ λ are i a A l set What 3 indeed ( } 0 Z ∂ 2 us reads 1 through 5 ˜ λ P 2, k ) ˙ supp b L ¶ 0 a ˜ a i λ ik ˙ space, computation = ]. a is th j π – ˙ C a 2 0 K e ² λ is = i and i ) the q [1 i{ if d J somewhat b (2 ∂ l is µ ˜ λ I ) in d i K = j ² l ∂ 5, to h in space is X k erators ˙ to a } λ, Z a This i ( = k trol ij à calization = − (2.6 of replacing ij ˜ x µ l op ersome A 4 { 4 on K lo } k space n y 0. b δ l d + bac of con This ij 4 the b j ) a ) erified wistor a i ˜ ect λ = K ) h transform Z t i{ π v the λ e to ∂ space ordinates cum ) ) v λ, ∂ i These ik (2 orted in ˜ ( λ i exp h co as a i λ (2.7 go for ha ( whic λ e λ, ∗ − = in this α o ( µ endence w ) wistor supp } P T i wistor function. l t A 2 l t that , ˜ ), from still λ k space. is k oid , dep transformed v i erator 5). ij is i{ e =1 to X a , the λ erator erators α ˜ λ w (1.2 condition ij K ( . op i delta h . to A op op amplitude, y t . once to = , = wistor y the notation homogeneous 1 t an ) l of the a that that i obtained k tial w µ = tial ij in e amplitude of , the are i gluon i e b K amplitude ( transform relev explicitly e I λ i e ects Since ( el b the Z ˜ amplitude. A means can ativ duce duction homogeneit differen exp The According If The edded differen viton viton b tro tro ould erform w This in The The one tree-lev deriv gluon space. This alternativ gra in em where p gra are JHEP07(2004)059 e ´ × in the the ) this only -1 R ation (2.15) (2.17) (2.18) (2.16) , in wistor means ransla- scaling t oincar through T Defining rom a P F in . SL(2 c ’s. . This degree ts conserv ˜ λ y ˆ only function homogeneous O homogeneous plus that b annihilates ) of oin is is is is and 0). is p the R , ) tum en h b ’s , ix 0 manifest K ix o f ˜ , λ , f on giv 1 homogeneous erator w a A (2.6 t , of A whic the SL(2 op of the t (0 , are one of momen fix . of ix . = y f c on of ˆ Since the O to onen 4 A b, using linearly y ert ¶ ix use Z homogeneous terms ˆ f O b c y a, is ∂ . y ∂ second amplitude b ¶ A is in b 0 ) 1 1 c in comp prop ∂ translations. and λ ∂ of ossible ed ix = 0 the function 2 1 . f p − 0 utations. no ˙ ˙ λ j j a a 0) ix A (2.15 fixed the , b l ˜ ˜ f λ λ amplitude hiev = ∂ us + k 0 1 j 2 j e ∂ original and y A , erm ij transformations 1 2 λ λ b ac b crucial delta th b ix 0 p ∂ that 5 5 annihilated λ ¶ degree ’s ∂ f , , , expressed K endence is c 2 2 the tz is λ 2 2 – Since e , , A ∂ + e (1 to ∂ 0 of λ can ) b l The 6 ws The X X b 0 c It =1 =1 a “fixed” j fixed k dep j j find ∂ = − 0 the – ∂ up that . λ j + that a 0 e 1 5 a i 3 0). − − a Loren i will follo ∂ b (2.14 , λ λ w also ∂ are on Z ∂ the [25]. ˆ ˙ K ∂ ˆ ˆ a O 2 5 = = O O it l ˙ (0 a a it and − b k λ j 1) a b c seen ≡ ˙ ˙ 3 4 a a x ∂ i ation ∂ ∂ The fix bilinears. ij ∂ ∂ ∂ µ . = can µ µ + ˜ ˜ c us λ λ 0 e λ king erators + obtain l that + 4 K ) 0 − − − − − b a ). c th to and ˙ e acting ∂ µ a ∂ k op ∂ ∂ ˜ i λ 0 hec homogeneous , and w ) = = = = = c and j t c µ a 0 = observ ˙ find ix λ, ) a R i can µ (2.13 i and + ( 3 f , alues: in e → µ , v µ , b still A A 1234 1345 1245 1235 2345 it [15] ∂ l 4 (2 ∂ the ˙ (2.6 a W t , i enden and b Z is K K K K K ) k ≡ example = µ SL l in and + 5. b , and j, , ) tiholomorphic k ix a from as i, enien 2 for ∂ 1) and f ∂ (2.17 indep , K , v j, an first whereas 3 ’s a 1 A [12], ( i, (0 µ ˆ Z (2.15 ery O ), ose con conclude = oth ≡ ≡ the the = of only ev i from e amplitude ho b simplification a 4 to ˆ the c directly O w in λ j (2.10 of er for the zero, fixing 0 Z with on y ws c hoice with ev 0), , can fixed b c i , ), i w useful are e ation Z ˜ y λ R erator act (1 follo ed , , and A After Ho W the i bilinears substituting an w op = degree ariance , degree λ v b ’s, 3 , ˜ of of These This enforces for By observ in λ of an the λ tions amplitude. a SL(2 space allo JHEP07(2004)059 , 2 y ∼ ts to es to the the and dia- vit In ) gauge curv closed n corres- theory gra − ositions, anishing , degree ). M tribution v onding the represen space − taking legs. − a the , n 1 y con , In disconnected n− amplitude gauge of . the b P (+ reads on simple . o y ). . C decomp (n−1)− , w corresp the + wistor t vit − h the external t p 2 (1.2 in disconnected , λ n some y orted n− square gra y b Eac b the with to obtained ossible connected (1+ ens the extending amplitudes (n−1)− also p n the e en + supp en to a ell. of b . y wistor M w giv t 2 b giv the happ anish. that to string ond as v cedure can that en all the are y en suitably to pro transformed giv what of orderings vit op us ertices y section and y v t amplitude b from wn corresp b configurations gra in suggests this gen tributing and − An 4− kno wistor for t con amplitudes viton differen and formally from a + – extension apply are should closed is gra 7 are obtained Inspired studied tribution − ) amplitudes for 0 holds to – 4 . p is − pieces. amplitudes, ) y (i+1)− theory con n degree one − This these , terpretation try i− a , of only + . osition relating and and in . off-shell e − . p tributions , ertex 3 w osition , configurations ogly v ula gauge ects − − ossibilit − whose , con go 3− 2 The p el stated, the , − string and form exp es 5 e , where ]. off-shell. t This + ) decomp the = 0 disconnected (+ decomp (1+ . [13 one curv 2 n p,p ( 7 to computation receiv M configuration tree-lev F presen certain already in e iπ and disconnected the e a e ) o ) the MHV 0 yp similar w As figure p ], that t factorization amplitudes amplitude − ( estigated a , Tw osed [1 in . v T − − Although 1 − if wn connected close n in the of , amplitude example, 1: KL ˜ k e − M ertex, es. viton role MHV sho v ) of on , 2− 2− including v section p for disconnected 1+ 1+ ) hec ( a text ) ha decomp gra c figure (+ as − gluon curv − e y , ) w e general , Figure close n this to con 1 in w b − − can, it − , , pla M calized t , ] The e In 0 − + The try Disconnected far lo y [2 , , 7 p,p (+ e e onding So 3. P b p theory w completely (+ degree migh Amplitudes curv in grams 3.1 ma a amplitudes, (+ JHEP07(2004)059 . a p is this λ e h The v (3.3) (3.1) (3.4) (3.2) MHV Using . ha the ˙ a limit whic η arbitrary 0 in the ˙ to a els. 0 i 2− 3− ˜ → λ . = amplitude. are 0 lab i i α z = k using 42 order ˜ λ [13] j h + i the φ ih viton ¶ in i il p In 2 h and i gra . external 23 p− ) y + 34 homogeneous ih with − λ − i close 3 ih 4 the j [12] , is 14 computed l , ˜ 4 defining needed h M 12 − φ ih h is 3 a + is to 2 4 , 4 ik 1+ 4− close 3 i h ertex. uting φ λ − (where ts 6 1 2 3 v . 2 M 42 + , φ φ This 0 + 2 amplitudes. 2 2 i ih ∼ erm l 3 ¶ 2− φ factor ). therefore p = 4− amplitude k φ i 13 (1+ viton ] h a − 1 ih is = amoun 3 4 2− 1+ M closed 3 iz , + λ p, ij [ the gra ] theory 2 y ˙ a h i ) 4 i η + − M ) ih i b η i i if , 3 p the ˙ y = and y a p p − p – 34 , , h a , , ] 4 ˜ λ en 2 2 ed 8 p tit ih 8 [ to p 2 − gauge − i [2 ih =1 h φ 2 ih , 4 i – a 12 , giv 2 = as 34 34 2 h − in , yielding amplitude h (+ P iden and λ appropriately 1 is ih µ normalization pa h dropp y 3 2 4 (1+ y en λ of − out p φ e µ b h vit The 6 1 2 3 1 b M ( tributing giv viton φ φ 3− 1+ = φ The 2 gra 2 2 2: 1 a graph p 3 houten con 1 on φ form drops arranged gra can 2 λ een ) Sc ) een e 0 M − 3 b = i − it w b first 1 the p,p 3− 4− ) obtained ( et p + . the 8 = F b − spinor. the Figure 3 i has ih cusing in 4 and iπ can the p p are of , e p fo − diagrams 2 of 2 pa + h − of λ ih 3 , ation figure 12 relation limit, p tum − h result factor start spinors. MHV ( virtue 2 in e used , arbitrary − y w diagrams b e The four final extension wn v phase o similar an (1+ tribution conserv momen y w a 3: ha on-shell t is sho M the the an 1+ 2− con t to e ˙ a section tum w η anishes for case v off-shell The 3 off-shell Figure this = alid spinors), momen diagrams remaining v where n This consisten In where The translates an JHEP07(2004)059 ) vi- en 3− q− the The dia- this (3.8) (3.5) (3.6) viton (3.5 some gra giv of in e gra complex ) ) This 5: yp with from − t − , . redefinition 5 ˙ the a , − earing hanges a , η ertex. − e the v 4− 5− + 4 , , ation, exc app of tak − coming t 2+ 1+ Figure (+ ton 3 , to spinor + require − alen 2 q ertex , conserv v the − graphs and and The (3.7) ] ould same 1+ (1+ . all − graphs w tum tributions 5 6 4 4 2 only [14 inequiv ) ) M suggests ) ) ) p tribution 5 3 . the 2 5 ected. + φ φ es y con in and ] i φ φ all i i the the i i q con (3.7 consider ) t MHV 12 giv 12 35 13 figure [3 exp is h 23 15 momen to h h h h of to using i h (3.2 in as holomorphic 3− 2− q + + the + + the 3 4 2 1 4 i accoun ih a osing of φ φ graph obtained of ξ φ φ Hermiticit whic i i not q need ξ , i i ˙ a 2 ertex p – to e a 34 12 ), is tributing v 13 14 λ ih h h Imp p 9 in off-shell w h h h first anishes sum ( ( 8 extension i i v depicted disconnected 23 ) – i a con line. (3.2 es = q q second as ) ih − 45 45 3 λ ˙ the of a q w 5 viton h of ih ih p 1 , tak ˜ λ (3.6 y ertex ih − 12 23 The 4− 5− gra [12][45]( [23][45]( h h v 4 construction and ternal that utations. , expression 13 off-shell 3 one 2 5 2 5 in diagrams es φ φ − ih pa from 2 4 2 4 3 the erm λ an 6 1 6 1 , φ φ 12 + the an this giv erify p viton h , 2 3 2 3 φ φ estigation once amplitude − vitons. q v In φ φ y v 2 MHV e in 2 2 2 2 oth = ely b , − gra v in φ φ gra b ). ) that with 3− can 4 coming + ha − − holomorphicit viton (1+ 5 ears q Naiv the , p diagram to fifteen , one the M gra − trast − the − to app 3 tributions so 4 4 amplitude obtained notices , the to , external ) first con it extended − here con of y w 2+ here 3 the ertices. e , tributions 2− 1+ er o , come (3.1 v v no ogly 12 The terms for w Tw 2+ of ha con . assistance (1+ go y , strong needs helicit 4 e 3 no b stress graphs. 4: ving e M w in e e MHV whenev other yields (1+ en of immediately is the The ˙ also W W Mo 2 a M ˜ λ figure giv is negativ one expression conjugate gram to One where and and of computer Figure in kind 3.2 this amplitude. gluon JHEP07(2004)059 ] e is in in as v en [1 for the and ha (3.9) duce viton wing of orted giv degree (3.10) delta- gauge seems In e graphs namely gra er w , follo ) w principle, for y ersists en repro , supp o. lo − ] p ) 2 extra 5 w tit in 1 amplitude t [2 sev , 2 amplitude one ariance φ ) normalization relations, − v i 4 holomorphic 6 4 in vior 2+ φ tain , co quan 12 the T i this the indeed subamplitudes ) h the − construction 5− 1+ prescription other y 2 en eha 3 not of section 24 ˜ con is ertices. correctly − KL φ , h feasible b for onding v vit een 5 giv figure of e the in h the + with 2+ disconnected, are φ to w from b [25] , i MHV 5 and the in − gra h et wistor t pro suc φ all ] t + 25 b i ted h result (1+ 1 [1 from they corresp p ˜ 25 )( φ get the h matc M 5 of migh using that comes using graph φ )( amplitudes tly + i i the 4 to y Therefore presen y ξ ([15] ected can y the necessary b φ holomorphic . , 7 b 15 prescription coming i heuristic ) p h as η alen non-trivial first vit 2 to not λ exp 54 olating h + φ are differences h One ] on the i confirm non 4− 3− the 2 gra η the ed t ectations just + , tities φ ts 25 equiv of of to w p i h The 2 [45] ˜ λ 7 φ for – [ exp ends or extrap tributions 12 case. i i + that that simplest sho h generalization ortan quan tributing 1 amplitudes = 45 of 24 10 con computation a φ h space, dep spirit the ih i 4 gets y this the – con ˜ fact fact usual. )( imp h [34]( φ ingredien 4 13 15 2 4 presence in desirable the h h i φ the φ as viton ( er one suc i the ariables, el the algebra i theory that 12 result v v + 4− to in ev wistor the 54 gra 34 ih els t h of ossibilit 3 w additional h ed no ˜ 5− h confirms p 2+ φ in k that the 25 cancel diagrams el − from 3 clearly in ho lab ih 2 φ are gauge space e is hec er, φ the w c not that 15 i v a amplitudes. are h + It e computer exclude stem MHV ) 25 es the v off-shell), approac curv ossible 4 prescription h tree-lev ˜ p are factor φ do ha 2 p wistor not 4 external t e Moreo e viton amplitude, nine is h φ some results es these [13]( this There − tary from explored W i in W them ). it generalizable + The h graph the gra do amplitudes. e . ariables. the 3 24 8 e v es 1 whic . y v ˜ . ogly y our φ b degree of a ih ) of case, 3 ha ξ (3.10 go whic uing vit Using vit φ case. 25 o ativ ia e calculating 3− 1+ ( h . 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D Parity Perturb and v, and e failure 93 emphasize suc h e whic is e e b High cases. vits, vits b ev. o o ertheless ect, explicit discussed lik holomorphic hazo, the suc J. ett. no related R ust to out L , General hep-th/0403187 e [ m e e an nev of wledgmen Aganagic discussions. ertex ecial Nekraso Neitzk Berk Roiban Roiban, Berk Roiban, b Witten, Witten, Cac for resp b . ab are e e v ac could ev. y ould sp t t h R. 056 N. N. hep-th/0402121 R. M. sp Phys. R hep-th/0403047 A. N. R. F. E. E. W W PHY0098527. amplitudes w ected kno vide vit this e hin ersion ery [6] [5] [4] [2] [3] [1] pro No. References migh there v should helpful In a amplitude exp Ac W gra reason MHV ton whic v JHEP07(2004)059 , ls J. 5 Phys. , and (1986) aviton el. d ams R 335 032 56 ang-Mil Nucl. Y B , close ev. diagr ett. multigr R er of L (2004) sup Phys. gluons and ev. MHV 04 R N for Living , Nucl. alar , with ory Phys. amplitudes Phys. sc , e the e gy as . multi-gluon tr esses c escriptions o ory en en Ener pr gauge pr attering amplitudes the sc to etwe etwe b for b High ]. QCD twistor J. gauge gluon , of elation in for r – N hep-th/0404255 e elation ory elations r , r its 12 the for A alculations . c – On 1 e, . y and elations r T 91 erstring gauge e amplitudes Equivalenc hep-th/0404072 [ Kuijf, sup e in cursive avity (1986) e, e e enc amplitude r gr H. R T n (1988) S.H.H. 070 A 269 curr e and r Neitzk twistor B and 211 ]. Giele, A. ylor, Khoze, (2004) B the amplitudes a one quantum c T Giele ellen Phys. 05 and gly ett. W.T. . o ]. V.V. L . ative Lew go T.R. Light W.T. 759 and Nucl. Motl Phys. and er, , Phys. and The Untwisting D.C. L. w gy gr-qc/0206071 , . e [ Perturb u, v, ai, (1988) 23 5 o ark w Zh Koso Ener strings Berends Berends, ; P Siegel, Georgiou Ka Bern, 306 Guk en attering hep-th/0403115 hep-th/0404085 High B [ 2459 F.A. (1990) op sc (2002) S. C.-J. G. W. D.A. S.J. H. F.A. Z. [7] [8] [9] [10] [12] [11] [13] [14] [15]