Twistor string inspired developments in perturbative gauge theory

Twistor string inspired developments in perturbative gauge theory B. Spence Cambridge April 2006

References Review article: Cachazo + Svrcek hep-th/0504194 Short review: Dixon hep-th/0512111 Oxford Twistor Workshop – January 2005 Queen Mary Twistor Workshop – November 2005 Queen Mary research: Brandhuber,Spence,Travaglini, Bedford, McNamara hep-th/ 0510253, 0506068, 0412108, 0410280, 0407214 (+ Zoubos, now at QM, hep-th/0512302 + earlier)

Background ● Quantum Field Theory, first course: Perturbation theory  do Feynman diagrams ● Conceptually beautiful, but can be seriously impractical: ● Does it matter if perturbation theory is complex? - Yes : there’s a much better way to do it, which points to a new formulation of gauge theory - Yes: LHC 2007 (soon!) Experimental tests need more scattering amplitudes

Eg: Five gluon tree level scattering with Feynman diagrams: (Slide from Zvi Bern’s talk, Oxford Twistor Workshop January 2005)

Experimentalists need predictions for, eg, multi-jet events: ( V Del Duca, Queen Mary Workshop Nov 2005)

Spinor Simplicity ● Surprise: Sum many Feynman diagrams  very simple results, when written in spinor variables Massless spin one particle. Describe by spinors and helicity (null) momentum:  spinor variables helicity ● Strip out the gauge group dependence of amplitudes into products of traces The coefficients of these are the colour-stripped n-particle amplitudes ● These depend on the spinor variables and the helicities

● This the Maximal Helicity Violating (MHV) amplitude (Notation i is the particle label) Colour-stripped amplitudes ● These contain the essential information of perturbative gauge theory ● They can be remarkably simple – for example, n-gluon tree MHV amplitudes ● This simplicity is unexpected from Feynman diagrams – how can one explain it ?

Amplitudes in twistor space Witten hep-th/0312171 ● Scattering amplitudes in spinor variables are simpler: eg MHV ● Idea: Look at amplitudes in twistor space  twistor space coordinates ( = Fourier transform of ) ● Then: ● ie MHV tree amplitudes localise on a line in twistor space

Amplitudes in Twistor Space II ● Localisation of gauge theory tree amplitudes in twistor space appears generic: Eg: MHV< - - ++…++ > localise on a line next to MHV < - - - ++….++ > localise on two intersecting lines Eg: 3 points  Z are collinear if ● Explicit check: twistor space coord’s and the above becomes a differential equation satisfied by the amplitude in spacetime: ● (Loop level: also get localisation – see later) What can explain this localisation ?

Twistor string theory I Witten hep-th/0312171, Nair ● Idea:Localisation on curves in target space – this is a feature of topological string theory ● The 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 localise on curves in CP(3,4) of degree and genus ● This explains the localisation of YM amplitudes and gives a weak-weak duality between N=4 SYM and twistor string theory

X X X X X X X X X X Twistor String Theory II: Tree Level ● In twistor space, tree level scattering amplitudes vertex operators moduli space of curves degree d, genus 0 (degree d  (d+1) negative helicity gluons) ● A surprise: due to delta functions, the integral localises on intersections of degree one curves: Curve Amplitude MHV < - - +…+ > nMHV < - - - +…+ > nnMHV < - - - - +..+ > X X X X X X

M M M M MHV Diagrams – Tree Level Cachazo, Svrcek, Witten M ●Idea: Since MHV tree amplitudes localise on a line in twistor space (~ point in spacetime), think of them as fundamental vertices. Join them with scalar propagators to generate other tree amplitudes: M MHV nMHV nnMHV (spacetime) (twistor space) M ● This works and gives a new, more efficient, way to calculate tree amplitudes Next: more new developments at tree level 

Recursion Relations: Tree level Britto, Cachazo, Feng, Witten ● Study the behaviour of tree level scattering amplitudes at complex momenta  ● ● ● = - can use this to reduce tree amplitudes to a sum over trivalent graphs ● Applications: -- 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) So, new results for tree level gauge theories…what about gravity 

DeWitt – Quantum Theory of Gravity III (1967) 3 point vertex – 171 terms in total (with symmetrisation) 4 point vertex – 2850 terms in full “We shall make no attempt to exhibit 5 or higher point vertices” Trees: Gravity ● Gravity amplitudes in momentum space – studied by DeWitt (1967) ● Gravity amplitudes in spinor helicity variables are much simpler - eg n point MHV gravity amplitudeBerends, Giele, Kuijf (1988)

Trees: Gravity II ● Recursion relations also found for gravity amplitudes Bedford, Brandhuber, Spence, Travaglini; Cachazo, Svrcek ● Applications: -- find new tree amplitudes -- new form of MHV amplitudes -- derivation of MHV diagrams for gravity permutations Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager (non-trivial due to polynomial dependence on tilded spinors – twistor space localisation is via “derivative of delta functions” )

Trees: Summary ● Gauge theory tree amplitudes localise on simple degree d curves in twistor space ● This is explained by a twistor string theory on super twistor space ● Localisation corresponds to MHV diagrams – MHV amplitudes as spacetime vertices ● Study of amplitudes at complex momenta  new recursion relations for tree amplitudes. This reconstructs amplitudes from singularities and can be used to prove the MHV approach ● All of this structure is also found with gravity amplitudes ( it is remarkable to learn something new about tree amplitudes…. )

Loops - Background ● Prior to 1990’s: severe difficulties in using Feynman diagrams to calculate multi-particle one-loop or higher loop amplitudes in gauge theory ● 1990’s: Spinor helicity methods + unitarity  many advances Spinor helicity: use variables Unitarity: The scattering matrix S must be unitary: Im(A)  discontinuities in amplitude  deduce A

3 1 2 4 s cut cut Loops - Unitarity I The scattering matrix S must be unitary: ● 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 poles at s = 4m^2, 9m^2,… (production of particles). In fact there is a branch cut from s=4m^2 to infinity (and also one along the negative s axis due to poles in the t-channel) A(s):

s cut cut C Loops - Unitarity II ● Now consider the contour integral of A(s) around C: ● This gives ● Then, using ● Idea: amplitudes can be reconstructed from their analytic properties

P q s p Q Loops - Unitarity III ● Various general results were found for loop amplitudes in gauge theories in the 1990’s ● Example: N=4 super Yang-Mills. n-gluon one-loop MHV amplitude t F is a box function – these are fundamental in one-loop diagrams also many new results in N=1 SYM and QCD 

Loops - Unitarity IV As noted above, unitarity methods and the spinor helicity formalism led to many new quantum results in the 1990’s: One loop general results: N=4 – all MHV amplitudes N=1 – all MHV amplitudes N=0 – (cut-constructible parts of) all MHV amplitudes (for adjacent negative helicities) ● ● Other particular results: Various nMHV results at one loop Two loop results (4 point function N=4) Others (nnMHV,…) ● But – nnMHV – difficult higher loops – difficult …reaching the limits of this approach by the early 2000’s

-loop N=4 YM amplitudes with negative helicity gluons localise on curves in CP(3,4) of degree and genus x x x x Loops – Twistor String Theory I ● As we saw, gauge theory tree amplitudes localise in twistor space, what about loop amplitudes? Twistor string theory – general arguments  ● ● This can also be seen explicitly in space-time for some examples eg – one loop N=4 SYM MHV amplitudes (Cachazo, Svrcek, Witten) these localise on two lines in twistor space

● Twistor string theory – B model on CP(3,4) --- N=4 SYM One loop open string = = tree closed string (quartic in derivatives) Loops – Twistor String Theory II ● Loop level in open string theory – generate closed strings ● this is also true for twistor string theory (Berkovits, Witten) -- at one loop level the fields of N=4 conformal supergravity contribute. This has fields and action ● avoid this if possible… the one loop twistor space localisation suggests there is a corresponding pure gauge theory spacetime description  MHV loop diagrams

M M M M M M M x x x x Loops – MHV diagrams I ● For tree amplitudes – spacetime MHV diagrams work (twistor space) (spacetime) -- direct realisation of twistor space localisation ● Study of known one loop MHV amplitudes  twistor space localisation on pairs of lines ● This suggests that in spacetime, one loop MHV amplitudes should be given by diagrams

Coordinates null reference vector null vector M M Loops – MHV diagrams II ? ● = MHV amplitude ● Technical issues: The particle in the loop is off-shell. But particles in MHV diagrams are on-shell  need an off-shell prescription -- Result should be independent of reference vector; -- Use dimensional regularisation of momentum integrals ● Then: multiply MHV expressions, simplify spinor algebra, perform phase space (l) and dispersion (z) integrals.....non-trivial calculation ● Result 

● The known answer is ● These agree, due to the nine-dilogarithm identity Loops - MHV Diagrams III ● The result of this MHV diagram calculation is (Brandhuber, Spence, Travaglini hep-th/0407214)

Might MHV diagrams provide a completely new way to do perturbative gauge theory? Loops – MHV diagrams IV ● So – spacetime MHV diagrams give one loop N=4 MHV amplitudes a surprise - no conformal supergravity as expected from twistor string theory Bedford, Brandhuber Spence, Travaglini Quigley Rozali ● Remarkably: MHV diagrams give correct results for -- N=1 super YM -- N=0 (cut constructible) -- these calculations agree with previous methods and also yield new results -- another surprise – one might have expected twistor structure only for N=4 Bedford, Brandhuber Spence, Travaglini

Loops – MHVdiagrams V ● MHV diagrams are equivalent to Feynman diagrams for any susy gauge theory at one loop: Brandhuber, Spence Travaglini Proof: (1) MHV diagrams are covariant (independent of reference vector) Use the decomposition in all internal loop legs  term with all retarded propagators vanishes by causality; other terms have cut propagators on-shell  become tree diagrams : Feynman Tree Theorem and trees are covariant (We looked at the relevance of this at the suggestion of Michael Green) • MHV diagrams have correct discontinuities  use FTT again • (3) They also have correct (soft and collinear) poles  can derive • known splitting and soft functions from MHV methods. This is evidence that MHV diagrams provide a new perturbation theory

More new results at one loop: Loops: d- dimensional unitarity I ● Unitarity arguments: find amplitudes from their discontinuities (logs, polylogs) ● Supersymmetric theories: amplitudes can be completely reconstructed from their discontinuities ● Non- supersymmetric theories (eg QCD) : amplitudes contain additional rational terms has rational part e.g. one loop five gluon QCD amplitude ● In d-dimensions, the discontinuities should also determine these rational terms

Loops: d- dimensional unitarity II ● d-dimensional unitarity should give the full amplitudes ● New techniques with multiple cuts developed (see reviews for references) ● eg: QCD: multiple cuts in d-dimensions – 4-point case Brandhuber, McNamara, Spence, Travaglini Triple cut Quadruple cut Result: Various integrals ● This is the correct QCD result

● ● ● = Loops – recursion relations ● Recursion relations for tree amplitudes: ● There are analogous relations at loop level – eg one loop QCD amplitude, recursion relations give decompositions like: loop Bern, Dixon, Kosower, hep-th/0507005 tree ● This allows one to reconstruct (parts of, in general) amplitudes from simpler pieces

= M M One Loop : Summary ● One loop amplitudes (like trees) localise in twistor space ● There is a related MHV diagram construction in spacetime, eg and MHV diagrams give correct answers for any supersymmetric gauge theories as well as the cut-constructible part of QCD amplitudes ● New generalised unitarity methods, combined with recursion relations, enable major new calculations of amplitudes in super Yang-Mills and QCD

= number positive, negative helicity gluons One Loop: Progress Lance Dixon, Oxford Twistor Workshop talk, Jan 2005 ~ post twistor strings (2004) ~ pre twistor strings • Plus 2005 on – eg • N=1, 6 gluons, split helicity • N=0, 6 gluons, split helicity • rational terms, electroweak • ● Note: • - Power of new techniques and rapid progress • Applications to QCD, etc relevant to LHC • Essentially all inspired by twistor string theory eg Glover, Stirling and collaborators (Durham) see reviews cited earlier for references

Beyond One Loop, a Bigger Picture Some areas of current interest: ● New ways to do perturbation theory ● Gravity ● Twistors ● Recursive structures and integrability

New perturbation theory for YM ● MHV diagrams give a new way to calculate one loop amplitudes in super Yang-Mills ● There are general arguments (Feynman tree theorem, singularity structure) why this works ● Is this true at all loops – do MHV diagrams give an alternative perturbation theory? ● A Lagrangian derivation ? – some initial steps have been made to develop this ~ light cone formalism. So far trees only, but suggestions of one loop all-plus vertex (for QCD) from the PI measure. Mansfield, hep-th/0511264 Gorsky and Rosly, hep-th/0510111

Gravity ● Gravity tree amplitudes – exhibit similar features to those in gauge theory: -- spinor helicity methods work well -- recursion relations exist -- MHV diagrams exist ● Twistor space localisation also is found: -- at tree level – “derivative of delta function” support -- also localisation at loop level – eg one loop box coefficients Bern, Bjerrum-Bohr, Dunbar, Ita, hep-th/0501137,0503102 ● There are general arguments supporting twistor space localisation – using a gravity analogue of the Chalmers-Siegel chiral formulation of Yang-Mills theory Abou Zeid, Hull, hep-th/0511189

Twistors Some of the work not discussed here: ● Berkovits twistor string theory -- equivalent open string version, uses world sheet current algebra Berkovits, hep-th/0402045 Berkovits, Motl, hep-th/0403187 ● Twistor string theory duals have been studied for other cases: -- orbifolds of N=4 SYM -- marginal deformations of N=4 SYM -- super complex structure deformations of CP(3,4) Giombi, Kulaxizi,Ricci, Robles-Llana, Trancanelli, Zoubos. Kulaixizi, Zoubos; review hep-th/0512302 Park, Rey Chiou, Ganor, Hong, Kim, Mitra ● Mirror symmetry and S duality have been explored for twistor strings  A/B models on CP(3,4)/Quadric, spacetime foam Aganagic, Vafa; Neitzke Vafa; Kumar, Policastro; Nekrasov, Ooguri, Vafa; Hartnoll Policastro; Policastro talk hep-th/0512025 ● Recent work in the twistor community – twistor actions for SYM (and conformal gravity) on CP(3,4), twistor action for super quadric, twistor diagrams, links with recursion relations Mason, hep-th/0507269 Movshev hep-th/0411111 Mason, Skinner, hep-th/0510262 Hodges,hep-th/0503060,0512336

Recursive Structures and Integrability ● N=4 super Yang-Mills: study of collinear limits of 4 point two loop MHV amplitudes led to the result Anastasiou, Bern, Dixon, Kosower, hep-th/0309040, 0402053 (see also Eden, Howe, Schubert, Sokatchev, West, hep-th/9906051,0010005 for some earlier work on recursive structures in N=4) ● This is non-trivial, involving intricate cancellations ● This cross-order relation has recently studied using differential equations for amplitudes 

Recursive Structures and Integrability II ● Cross-order relation (4 point MHV N=4 SYM): ● This has recently been proved using differential equations Cachazo, Spradlin, Volovich, hep-th/0601031 4 point loop amplitudes. General form (Mellin-Barnes) ~ One can use this to show that Fixing the kernel of this operator by IR and collinear limits, one can prove the cross order relation above ● A natural conjecture for an L-loop operator is…

constants O(e) constants, independent of number of legs n loop expansion parameter ● Note that this expresses the L-loop n-point MHV amplitude in terms of the one loop amplitude. Recursive Structures and Integrability III ● All-loop N=4 n-point MHV conjectured recursion relation Bern Dixon Smirnov, hep-th/0505205 ● It gives the two loop relation given earlier. It has also been checked for 3 loops: 

(2/3 of the) two loop function from Bern, Dixon Kosower Recursive Structures and Integrability IV ● Conjectured general cross-order relation: ● Tested at three loops (4 point function): ● This is rather involved: ● Two-loop, 5 point also tested recently Cachazo, Spradlin, Volovich, hep-th/0602228 ● Cross-order relations also argued for some deformations of N=4 Khoze, hep-th/0512194

Recursive Structures and Integrability V ● One major consequence of the BDS cross-order relation is a prediction for the finite part of the n-point L-loop MHV amplitude for N=4 SYM: … … ● This has been calculated in two other places to certain orders (1) 3-loop QCD calculation – Moch, Vermaseren, Vogt hep-ph/0403192, Kotikov, Lipatov, Onischenko, Velizhanin, hep-th/0404092 -- agrees with above (2) and 

Recursive Structures and Integrability VI ● Past few years: integrable structures found and studied in super YM and string theory duals ● On the gauge theory side, this has yielded predictions for the anomalous dimensions of gauge theory operators using spin chains e.g. sl(2) Bethe ansatz: (solve the first two equations for then the third equation gives the anomalous dimension)

Recursive Structures and Integrability VII Sl_2 Bethe ansatz, large S limit of twist 2 operators six loop prediction for anomalous dimension Agrees up to 3 loops with the results of KLOV and BDS earlier Eden, Staudacher QM Twistor workshop 11/05

Recursive Structures and Integrability VIII ● Recall BDS conjecture for the finite part of the all-loop N=4 SYM MHV amplitudes: ● Very recently (Eden, Staudacher, hep-th/0603157) an integral equation has been given which completely determines the (depends on Bessel functions J) From this, can be expanded in powers of a (the result has integer coefficients times products of zeta functions…). ● The derivation of the equations above uses the large spin limit of the asymptotic all-loop Bethe ansatz. The result then links the spacetime S matrix with the worldsheet S matrix. To the conclusions

Achievements of twistor string theory and its applications ● The original twistor string theory and later related twistor developments – new weak-weak dualities between gauge theory and string theory. ● Major progress in calculations -- N=4: new amplitudes, more legs, loops -- N=1: many new amplitudes also -- N=0: new results for QCD and E-W, relevant to LHC ● New ideas for perturbative gauge theory (and gravity) -- new perturbation theory and MHV diagrams -- new recursion relations between amplitudes -- new developments with unitarity methods -- cross-order relations – links with recent integrability work

To Do List ● Twistors: -- twistor string dual of pure (super) YM -- twistor string dual of Einstein gravity -- general picture: A,B models, CP(3,4) and quadric -- higher loops -- more legs -- more LHC relevant amplitudes ● Applications: ● Structure of perturbative gauge theory -- MHV diagrams and perturbation theory – test -- higher loops and differential equations -- iterative structure in N=4 and relationship with integrability Much has been found, but there is much more to learn…..

Twistor string inspired developments in perturbative gauge theory