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Dave Dunbar, Swansea University. Twistors and Perturbative Gravity. Steve Bidder. Harald Ita. Warren Perkins. Emil Bjerrum-Bohr. +Zvi Bern (UCLA) and Kasper Risager (NBI). UK Theory Institute 20/12/05. Plan.
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Dave Dunbar, Swansea University Twistors and Perturbative Gravity Steve Bidder Harald Ita Warren Perkins Emil Bjerrum-Bohr +Zvi Bern (UCLA) and Kasper Risager (NBI) UK Theory Institute 20/12/05
Plan • Recently a duality between Yang-Mills and twistor string theory has inspired a variety of new techniques in perturbative Yang-Mills theories. First part of talk will review these • Look at Gravity Amplitudes -which, if any, features apply to gravity • Application: Loop Amplitudes N=4 Yang –Mills N=8 Supergravity • Consequences and Conclusions
Duality with String Theory Witten (2003) proposed a Weak-Weak duality between • A) Yang-Mills theory ( N=4 ) • B) Topological String Theory with twistor target space -Since this is a `weak-weak` duality perturbative S-matrix of two theories should be identical order by order -True for tree level scattering Rioban, Spradlin,Volovich
Featutures of Duality Topological String Theory with twistor target space CP3 -open string instantons correspond to Yang-Mills states -theory has conformal symmetry, N=4 SYM -closed string states correspond to N=4 superconformal gravity - N < 4 ?? Berkovits+Witten, Berkovits
Topological String Theory: harder, uninteresting Perturbative Gauge Theories, hard, interesting Is the duality useful? Theory A : hard, interesting Theory B: easy -duality may be useful indirectly
Twistor Definitions • Consider a massless particle with momenta • We can realise as • So we can express where are two component Weyl spinors
This decomposition is not unique but We can also turn polarisation vector into fermionic objects, ``Spinor Helicity`` formalism Xu, Zhang,Chang 87 -Amplitude now a function of spinor variables
Transform to Twistor Space Penrose+ -note we make a choice which to transform new coordinates Twistor Space is a complex projective (CP3) space n-point amplitude is defined on (CP3)n
Twistor Structure • Conjecture (Witten) : amplitudes have non-zero support on curves in twistor space • support should be a curve of degree (number of –ve helicities)+(loops) -1 Carrying out the transform is problematic, instead we can test structure by acting with differential operators
We test collinearity and coplanarity by acting with differential operators Fijkand Kijkl -action of F is obtained using fact that points Zi collinear if Allows us to test without determining
Collinearity of MHV amplitudes • We organise gluon scattering amplitudes according to the number of negative helicities • Amplitude withno or one negative helicities vanish [ for supersymmetric theories to all order; for non-supersymmetric true for tree amplitudes] • Amplitudes with exactly two negative helicities are refered to as `MHV` amplitudes Parke-Taylor, Berends-Giele (amplitudes are color-ordered)
Collinearity of MHV amplitudes • MHV amplitudes only depend upon • So, for Yang-Mills, FijkAn=0 trivially • MHV amplitudes have collinear support when transforming to a function in twistor space since Penrose transform yields a function after integration .
MHV amplitudes have suppport on line only Curve of degree 1 (= 0+2-1)
NMHV amplitudes in twistor space • amplitudes with three –ve helicity known as NMHV amplitudes • remarkably NMHV amplitudes have coplanar support in twistor space • prove this not directly but by showing - time to look at techniques motivated by duality
Techniques:I MHV-vertex construction • Promotes MHV amplitude to fundamental object by off-shell continuation • Works for gluon scattering tree amplitudes • Works for (massless) quarks • Works for Higgs and W’s • Works for photons -No known derivation from a Lagrangian (but…… Khoze, Mason, Mansfield) Cachazo Svrcek Witten, Nair Wu,Zhu; Su,Wu; Georgiou Khoze Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia Ozeren+Stirling
+ A MHV diagram _ _ + _ + _ _ + _ + _ + + _ -three point vertices allowed -number of vertices = (number of -) -1
eg for NMHV amplitudes k+ k+1+ q + - + 1- 3- 2- 2(n-3) diagrams Topology determined by number of –ve helicity gluons
Coplanarity-byproduct of MHV vertices -NMHV amplitudes is sum of two MHV vertices Two intersecting lines in twistor space define the plane Curve is a degenerate curve of degree 2
Techniques:2 Recursion Relations Britto,Cachazo,Feng and Witten • Return of the analytic S-matrix! • Shift amplitude so it is a complex function of z Amplitude becomes an analytic function of z, A(z) Full amplitude can be reconstructed from analytic properties Within the amplitude momenta containing only one of the pair are z-dependant q(z)
1 2 -results in recursive on-shell relation ( cf Berends-Giele off-shell recursive technique ) q (three-point amplitudes must be included) Amplitude has poles Amplitude is poles
MHV vs BCF recursion • Difference MHV asymmetric between helicity sign BCF chooses two special legs For NMHV : MHV expresses as a product of two MHV : BCF uses (n-1)-pt NMHV • Similarities- • both rely upon analytic structure • both for trees but… Loops: MHV: Bedford, Brandhuber,Spence, Travaglini Recursive: Bern,Dixon Kosower; Bern, Bjerrum-Bohr, Dunbar, Ita,Perkins
Gravity-Strategy 1) Try to understand twistor structure 2) Develop formalisms - a priori we might expect Einstein gravity to contain no knowledge of twistor structure since duality contains conformal gravity
Feynman diagram approach to perturbative quantum gravity is extremely complicated • Gravity = (Yang-Mills)2 • Feynman diagrams for Yang-Mills = horrible mess • How do we deal with (horrible mess)2 Using traditional techniques even the four-point tree amplitude is very difficult Sannan,86
Kawai-Lewellen-Tye Relations Kawai,Lewellen Tye, 86 -pre-twistors one of few useful techniques -derived from string theory relations -become complicated with increasing number of legs -involves momenta prefactors -MHV amplitudes calculated using this Berends,Giele, Kuijf
Recursion for Gravity • Gravity, seems to satisfy the conditions to use recursion relations • Allows (re)calculation of MHV gravity tree amps • Expression for six-point NMHV tree Bedford, Brandhuber, Spence, Travaglini Cachazo,Svrcek Bedford, Brandhuber, Spence, Travaglini Cachazo,Svrcek
Gravity MHV amplitudes • For Gravity Mn is polynomial in with degree (2n-6), eg • Consequently • In fact….. • Upon transforming Mnhas a derivative of function support
Coplanarity NMHV amplitudes in Yang-Mills have coplanar support For Gravity we have verified n=5 by Giombi, Ricci, Robles-Llana Trancanelli n=6,7,8 Bern, Bjerrum-Bohr,Dunbar
MHV construction for gravity • Need the correct off-shell continuation • Proved to be difficult • Resolution involves continuing the of the negative helicity legs • The ri are chosen so that a) momentum is conserved b) multi-particle poles q2(ri) are on-shell -this fixes them uniquely Shift is the same as that used by Risager to derive MHV rules using analytic structure
Eg NMHV amplitudes k+ k+1+ + - + 1- 3- 2-
Loop Amplitudes • Loop amplitudes perhaps the most interesting aspect of gravity calculations • UV structure always interesting • Chance to prove/disprove our prejudices • Studying Amplitudes may uncover symmetries not obvious in Lagrangian • Loop amplitudes are sensitive to the entire theory • For loops we must be specific about which theory we are studying
Tale of two theories, N=4 SYM vs N=8 Supergravity Cremmer, Julia, Scherk N=4 SYM is maximally supersymmetric gauge theory (spin · 1 ) N=8 Supergravity is maximal theory with gauged supersymmetry (spin · 2 ) -both appear in low energy limit of superstring theory -S-matrix of both theories is constrained by a rich set of symmetries -N=4 key in Weak-Weak duality -in D=4 YM has dimensionless coupling constant wheras gravity has a dimensionful coupling constant -both theories are extremelly important models: toy or otherwise
degree p in l p Vertices involve loop momentum propagators General Decomposition of One- loop n-point Amplitude p=n : Yang-Mills p=2n Gravity
Passarino-Veltman reduction • process continues until we reach four-point integral functions with (in yang-mills up to quartic numerators) In going from 4-> 3 scalar boxes are generated • similarly 3 -> 2 also gives scalar triangles. At bubbles process ends. Quadratic bubbles can be rational functions involving no logarithms. • so in general, for massless particles Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator
N=4 Susy Yang-Mills • In N=4 Susy there are cancellations between the states of different spin circulating in the loop. • Leading four powers of loop momentum cancel (in well chosen gauges..) • N=4 lie in a subspace of the allowed amplitudes (Bern,Dixon,Dunbar,Kosower, 94) • Determining rational ci determines amplitude • 4pt…. Green, Schwarz, Brink • MHV,6pt 7pt,gluinos Bern, Dixon, Del Duca Dunbar, Kosower Britto, Cachazo, Feng; RoibanSpradlin Volovich Bidder, Perkins, Risager
Basis in N=4 Theory ‘easy’ two-mass box ‘hard’ two-mass box
Box Coefficients-Twistor Structure • Box coefficients has coplanar support for NMHV 1-loop • amplitudes -true for both N=4 and QCD!!!
N=8 Supergravity • Loop polynomial of n-point amplitude of degree 2n. • Leading eight-powers of loop momentum cancel (in well chosen gauges..) leaving (2n-8) • Beyond 4-point amplitude contains triangles..bubbles • Beyond 6-point amplitude is not cut-constructible
No-Triangle Hypothesis -against this expectation, it might be the case that……. Evidence? true for 4pt n-point MHV 6pt NMHV Green,Schwarz,Brink Bern,Dixon,Perelstein,Rozowsky Bjerrum-Bohr, Dunbar,Ita -factorisation suggests this is true for all one-loop amplitudes consequences? • One-Loop amplitudes N=8 SUGRA look just like N=4 SYM
Beyond one-loops Two-Loop Result obtained by reconstructing amplitude from cuts
Two-Loop SYM/ Supergravity Bern,Rozowsky,Yan IPs,tplanar double box integral Bern,Dixon,Dunbar,Perelstein,Rozowsky (BDDPR) -N=8 amplitudes very close to N=4
Beyond 2-loops: UV pattern (98) Honest calculation/ conjecture (BDDPR) N=8 Sugra N=4 Yang-Mills Based upon 4pt amplitudes
Pattern obtained by cutting Beyond 2 loop , loop momenta get ``caught’’ within the integral functions Generally, the resultant polynomial for maximal supergravity of the square of that for maximal super yang-mills Eg in this case YM :P(li)=(l1+l2)2 SUGRA :P(li)=((l1+l2)2)2 l1 l2 I[ P(li)] BUT…………..
on the three particle cut.. For Yang-Mills, we expect the loop to yield a linear pentagon integral For Gravity, we thus expect a quadratic pentagon However, a quadratic pentagon would give triangles which are not present in an on-shell amplitude -indication of better behaviour in entire amplitude ? relations to work of Green and Van Hove
Does ``no-triangle hypothesis’’ imply perturbative expansion of N=8 SUGRA more similar to that of N=4SYM than power counting/field theory arguments suggest???? • If factorisation is the key then perhaps yes. Four point amplitudes very similar • Is N=8 SUGRA perturbatively finite?????
Conclusions • Perturbation theory is interesting and still contains many surprises • Recent “discoveries” are interesting and useful • Studying on-shell amplitudes can give information not obvious in the Lagrangian • Gravity calculations amenable to many of the new twistor inspired techniques -both recursion and MHV– vertex formulations exist -perturbative expansion of N=8 seems to be surprisingly simple. This may have consequences for the UV behaviour • Consequences for the duality?