Scattering amplitudes in N =4super Yang-Mills

Transcription

1 Scatterng ampltudes n N =4super Yang-Mlls Johannes Martn Henn Humboldt Unversty Berln Based on work n collaboraton wth James Drummond, Gregory Korchemsky, Jan Plefka and Emery Sokatchev J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 1/20

2 Motvaton and outlne! Why N =4super Yang-Mlls? " learn somethng about pure Yang-Mlls e.g. tree-level gluon scatterng ampltudes " spectrum of anomalous dmensons expected ntegrablty compute at weak and strong couplng (AdS/CFT) " ntegrablty for other quanttes n N =4SYM?! Questons we want to ask: " can we compute tree-level ampltudes for an arbtrary number of gluons? " what are the symmetry propertes of the ampltudes? J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 2/20

3 Frst part - computng tree-level ampltudes computng tree-level ampltudes J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 3/20

4 Tree-level scatterng ampltudes! maxmally helcty volatng (MHV) ampltudes [Parke, Taylor 1986] A MHV n (, j) =δ 4 (p) j 4 12 n 1 p α α = λ α λ α, j = λ α λ β j ɛ αβ! MHV ampltudes n N =4on-shell superspace [Nar 1988] Φ= g + + η A f A + +1/4!ɛ ABCD η A η B η C η D g, q αa = λ α η A A MHV n = δ 4 (p) δ 8 (q) 1 12 n 1 k +2 N k MHV n ampltudes number of gluons n J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 4/20

5 On-shell recurson relatons! on-shell recurson relatons [Brtto, Cachazo, Feng + Wtten, 2004] k n!1 k+1!1 n!1 k n k!1 1 n k!1 j+1 j 1 A = A L 1 P 2 A R " follow from analytc propertes of the ampltudes (Feynman dagrams) " n-pont ampltudes are obtaned recursvely from lower-pont ampltudes " all ampltudes are on-shell! N =4supersymmetrc verson [Arkan-Hamed, Cachazo, Kaplan 2008] J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 5/20

6 All tree-level ampltudes n N =4SYM k +2 N k MHV n ampltudes supersymmetry mples A n = δ 4 (p)δ 8 (q) 1 12 n 1 P n! P MHV n =1 [Nar 1988] number of gluons n J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 6/20

7 All tree-level ampltudes n N =4SYM supersymmetry mples A n = δ 4 (p)δ 8 (q) 1 12 n 1 P n! P MHV n =1 [Nar 1988]! P NMHV n wth R n;,j = k number of gluons n = 2,j n 1 R n;,j [Drummond, J.M.H., Korchemsky, Sokatchev 2008] N k MHV n ampltudes < 1 >< j j 1 >δ 4 (Ξ n;j ) x 2 j <n x nx j j >< n x n x j j 1 >< n x nj x j >< n x nj x j 1 > where and Ξ n;j =<n x n x j θ jn > + <n x nj x j θ n > p = x x +1, q = θ θ +1 J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 7/20

8 All tree-level ampltudes n N =4SYM supersymmetry mples A n = δ 4 (p)δ 8 (q) 1 12 n 1 P n! P MHV n =1 [Nar 1988]! P NMHV n k number of gluons n = 2,j n 1 R n;,j [Drummond, J.M.H., Korchemsky, Sokatchev 2008] N k MHV n ampltudes! generc case [Drummond, J.M.H. 2008] P Nk MHV n = }{{} 2k fold nested sums R R }{{} k fold product J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 8/20

9 Second part - symmetres J. M. Henn Scatterng ampltudes n N = 4 SYM Strngs Rome June 26, 2009 [M.C. Escher] - p. 9/20

10 Expected symmetres of the ampltudes! psu(2, 2 4) superconformal generators n on-shell superspace [Wtten 2003] [J a,j b } = f ab c J c, J a = n J a p αα = X λ α λ α, =1 k α α = X λ α λ α, d = X [ 1 2 λα λ α λ α λ α + 1], 3 4 = n 1 n n 1! nvarance under psu(2, 2 4) {p, k, m, m, d, c, r, q, q, s, s}a=0 J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 10/20

11 Hnts for a new symmetry! loop correctons to four-gluon (MHV) ampltude [Bern, Dxon, Smrnov 2005] ! hnts for a new symmetry [Drummond, J.M.H., Smrnov, Sokatchev 2006] ntroduce dual varables x x +1 = p conformal symmetry n dual space! 1 (broken by nfrared dvergences)! same varables also appeared n AdS dual [Alday, Maldacena 2007] J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 11/20

12 Dual superconformal symmetry! ampltudes have addtonal symmetres! [Drummond, J.M.H., Korchemsky, Sokatchev 08]! dual conformal generator x α α K α α = x α +1 α = p α α [ x α βx α β = λ α λ α x β β ] J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 12/20

13 Dual superconformal symmetry! ampltudes have addtonal symmetres! [Drummond, J.M.H., Korchemsky, Sokatchev 08]! dual conformal generator [ K α α = x α α x α βx α β x α +1 α = p α α x β β = λ α λ α +x α β λ α λ β + x +1 α β λ α λ β ] J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 13/20

14 Dual superconformal symmetry! ampltudes have addtonal symmetres! [Drummond, J.M.H., Korchemsky, Sokatchev 08]! dual conformal generator [ K α α = x α α x α βx α β! supersymmetrc generalsaton x α +1 α = p α α x β β = λ α λ α +x α β λ α λ β + x +1 α β λ α λ β ] θ A α θ A +1α = q A α = η A λ α! fnal expresson for dual conformal generator K α α = [ x α β x αβ +x αβ λ α λ β + x α +1 λ β α x β β λ β +x αβ θ αb θ βb + λ α θ αb +1 η B ] J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 14/20

15 Dual superconformal symmetry! act on varables {λ, λ,η,x,θ }! constrants x x +1 = λ λ, θ θ +1 = η λ x α α θ αa P α α = [ K α α = x α β x αβ, Q αa = x β β +x αβ λ α λ β Smlarly, M αβ, M α β,r A B, D, C, S A α, S αa., Q Ȧ α = + x α β +1 λ α λ β [θ αa +x αβ θ αb x α α θ βb + η A η α + λ α θ αb +1 ], η B ] tree ampltudes are covarant under dual superconformal transformatons K α α A n = x α α A n [Drummond, J.M.H., Korchemsky, Sokatchev 08] [Brandhuber, Heslop, Travagln 08], [Drummond, J.M.H. 08] J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 15/20

16 Conventonal and dual superconformal symmetry [Drummond, J.M.H., Korchemsky, Sokatchev 08] p K q s k q = S s = Q P S Q! also observed n the AdS dual (fermonc T-dualty) [Maldacena, Berkovts 08] [Besert, Rcc, Tseytln, Wolf 08]! closure of the algebra? J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 16/20

17 Commutng the two algebras! techncal steps " reformulate covarance as an nvarance [Drummond, J.M.H., Plefka 2009] (K α α + x α α ) A n =0 " remove all x, θ dependence (use P and Q to set x 1 =0and θ 1 =0) " subtract trval terms lke pd! we fnd K α αa n =0 wth K α α K α α = [( ) ] m γ α δ γ α + m γ α δγ α d δαδ γ γ α p jγ γ + q αc qjα C ( j) >j! notce K α α = f (pα α ) bc <j J b J jc where [J a,j b } = f c ab J c, J a = J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 17/20 n =1 J a

18 Yangan symmetry of scatterng ampltudes n N =4super Yang-Mlls! superconformal symmetry psu(2, 2 4) [Wtten 2003] psu(2, 2 4) algebra: [J a,j b } = f ab c J c, J a = n =1 J a! dual superconformal symmetry [Drummond, J.M.H., Korchemsky, Sokatchev 2008]! closure of algebra gves Yangan Y (psu(2, 2 4)) [Drummond, J.M.H., Plefka 2009] " level-one Yangan generators Q a = f a cb 1 <j n J bj jc, [Q a,j b } = f ab c Q c " nterestng observaton: form of Q a consstent wth cyclcty for certan superalgebras only! Yangan symmetry of scatterng ampltudes also expected n strng theory [Besert 2009] J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 18/20

19 Summary! solved problem of computng tree-level ampltudes n N =4SYM applcatons: " easy to extract gluon ampltudes " useful for loop computatons va generalsed untarty " same technque of solvng the recurson relatons applcable to N =8supergravty! new symmetres of tree-level ampltudes " conventonal and dual superconformal symmetry combne to Yangan " sgn of ntegrablty? 4 3 = 3 n n 1 n J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 19/20

20 Outlook and open questons! dvergences appear at loop level! do loop-level ampltudes also have a Yangan symmetry? " breakng of dual conformal symmetry controlled by Ward dentty what about conventonal superconformal symmetry? " connecton to Wlson loops? Wlson loops n dual chral superspace (x µ,θαa )? " nsghts from AdS/CFT? J. M. Henn Scatterng ampltudes n N =4 SYM Strngs Rome June 26, p. 20/20