# Presentation: 007194

## From Twistors to Calculations

## Precision Perturbative QCD

- Predictions of signals, signals+jets
- Predictions of backgrounds
- Measurement of luminosity
- Measurement of fundamental parameters (s, *m**t*)
- Measurement of electroweak parameters
- Extraction of parton distributions — ingredients in any theoretical prediction

- Everything at a hadron collider involves QCD

## AdS/CFT Duality

- Strong-coupling *N* =4 supersymmetric gauge theory String Theory on AdS5  S5    for large *N**c* , *g*2 *N**c*   perturbative supergravity
- Maldacena (1997)
- Gubser, Klebanov, & Polyakov; Witten (1998)
*	**	**	**	**Strong–weak duality*
- Many tests on quantities protected by supersymmetry
- D’Hoker, Freedman, Mathur, Matusis, Rastelli, Liu, Tseytlin, Lee, Minwalla, Rangamani, Seiberg, Gubser, Klebanov, Polyakov & others
- More recently, tests on unprotected quantities
- Berenstein, Maldacena, Nastase; Beisert, Frolov, Staudacher & Tseytlin; Minahan & Zarembo; & others (2002–4)

## A New Duality

-    Topological B-model string theory *N** *=4 supersymmetric gauge theory
*Weak–weak duality*
- Computation of scattering amplitudes
- Novel differential equations
- Witten (2003)
- Roiban, Spradlin, & Volovich; Berkovits & Motl; Vafa & Neitzke; Siegel (2004)
- Novel factorizations of amplitudes
- Cachazo, Svrcek, & Witten (2003)

- Witten, hep-th/0312171
- Cachazo, Svrček & Witten, hep-th/0403047, hep-th/0406177, hep-th/0409245
- Bena, Bern & DAK, hep-th/0406133
- DAK, hep-th/0406175
- Brandhuber, Spence, & Travaglini, hep-th/0407214
- Bena, Bern, DAK & Roiban, hep-th/0410054
- Cachazo, hep-th/0410077; Britto, Cachazo, & Feng, hep-th/0410179
- Bern, Del Duca, Dixon, DAK, hep-th/0410224

## The Amazing Simplicity of N=4 Perturbation Theory

- Manifestly N=4 supersymmetric calculations are very hard off-shell — much harder than ordinary gauge theory

- offshell

- onshell

- But on-shell calculations are much simpler than in nonsupersymmetric theories:
- 4-pt one-loop = tree  one-loop scalar box
- Green & Schwartz (1982)
- 5, 6-pt one-loop known & simpler than QCD
- Bern, Dixon, Dunbar, DAK (1994)
- all-n one-loop known for special helicities
- Bern, Dixon, Dunbar, DAK (1994)

## Parke–Taylor Amplitudes

- Pure gluon amplitudes
- All gluon helicities +  amplitude = 0
- Gluon helicities +–+...+  amplitude = 0
- Gluon helicities +–+...+–+  MHV amplitude
- Holomorphic in spinor variables
- Parke & Taylor (1986)
- Proved via recurrence relations
- Berends & Giele (1988)

## Correlators in Projective Space

- Nair (1988)
- Null cones  points in twistor space
- Penrose (1972)
- Spinors are homogeneous coordinates on complex projective space CP1
- Current algebra on CP1  amplitudes
- Reproduce maximally helicity violating (MHV) amplitudes

## Spinors

- Want square root of Lorentz vector  need spin 1⁄2
- Spinors     , conjugate spinors
- Spinor product
- (1⁄2,0)  (0, 1⁄2) = vector
- Signature – + + +: complex conjugates
- Signature + + – –: independent and real
- Helicity 1: 	 Amplitudes as pure functions of spinor variables

## What is Twistor Space?

- Half-Fourier transform of spinors: transform     , leave alone       Penrose’s original twistor space, real or complex
- Definite helicity: introduce homogeneous coordinates *Z**I*  CP3 or RP3 (projective) twistor space
- Supertwistor space: with fermionic coordinates *A*
- Back to momentum space by Fourier-transforming 

## Strings in Twistor Space

- String theory can be defined by a two-dimensional field theory whose fields take values in target space:
*n-*dimensional flat space
- 5-dimensional Anti-de Sitter × 5-sphere
- twistor space: intrinsically four-dimensional  Topological String Theory
- Spectrum in Twistor space is *N **= *4 supersymmetric multiplet (gluon, four fermions, six real scalars)
- Gluons and fermions each have two helicity states

## Color Ordering

- Separate out kinematic and color factors (analogous to Chan-Paton factors in open string theory)

## Curves in Twistor Space

- Each external particle represented by a point in twistor space
- Witten’s proposal: amplitudes non-vanishing only when points lie on a curve of degree *d* and genus *g*, where
*d* = # negative helicities – 1 + # loops
*g*  # loops
- Obtain amplitudes by integrating over all possible curves  moduli space of curves
- Can be interpreted as D1-instantons

## Novel Differential Equations

- Amplitudes with two negative helicities (MHV) live on straight lines in twistor space
- Amplitudes with three negative helicities (next-to-MHV) live on conic sections (quadratic curves)
- Amplitudes with four negative helicities (next-to-next-to-MHV) live on twisted cubics
- Fourier transform back to spinors  differential equations in conjugate spinors

## Differential Equations for Amplitudes

- Line operators
- 	Should annihilate MHV amplitudes
- Planar operators
- 	Should annihilate next-to-MHV (NMHV) amplitudes

## Even String Theorists Can Do Experiments

- Apply *F *operators to NMHV (3 – ) amplitudes:products annihilate them!  *K *annihilates them;
- Apply *F *operators to N2MHV (4 – ) amplitudes:longer products annihilate them!  Products of K annihilate them;
- Interpretation: twistor-string amplitudes are supported on intersecting line segments
- Simpler than expected: what does this mean in field theory?

## Parke–Taylor Amplitudes

- Pure gluon amplitudes
- All gluon helicities +  amplitude = 0
- Gluon helicities +–+...+  amplitude = 0
- Gluon helicities +–+...+–+  MHV amplitude
- Holomorphic in spinor variables
- Parke & Taylor (1986)
- Proved via recurrence relations
- Berends & Giele (1988)

## Cachazo–Svrcek–Witten Construction

- Vertices are off-shell continuations of MHV amplitudes
- Connect them by propagators *i*/*K*2
- Draw all diagrams

- Corresponds to all multiparticle factorizations
- Not completely local, not fully non-local
- Can write down a Lagrangian by summing over vertices
- Abe, Nair, Park (2004)

## Practical Applications

- Compact expressions for amplitudes suitable for numerical use
- Compact analytic expressions suitable for use in computing loop amplitudes
- Extend older loop results for MHV to non-MHV

## A New Analytic Form

- All-*n *NMHV (3 – : 1, *m*2, *m*3) amplitude
- Generalizes adjacent-minus result		   	DAK (1989)

## Computational Complexity

- Exponential (2*n*–1–1–*n*) number of independent helicities
- Feynman diagrams: factorial growth ~ *n*! *n*3/2 *c* –*n*, *per* helicity
- Exponential number of terms per diagram
- MHV diagrams: slightly more than exponential growth
- One term per diagram
- Can one do better?

## Recurrence Relations

- 				         Berends & Giele (1988); DAK (1989)
-  Polynomial complexity per helicity

## Recursive Formulation

- Bena, Bern, DAK (2004)
- Recursive approaches have proven powerful in QCD
- Can formulate higher-degree vertices and reformulate CSW construction in a recursive manner

## Degenerate Limits of Curves

- CSW construction: sets of *d* intersecting lines for all amplitudes
- Vertices in CSW construction  lines in twistor space
- Crossing points in twistor space  propagators in momentum space
*d* disconnected instantons
- Original proposal: single, connected, charge *d* instanton
- Two inequivalent ways of computing same object

## Underlying Symmetry

- Integrating over higher-degree curves is equivalent to integrating over collections of intersecting straight lines!
- Gukov, Motl, & Nietzke (2004)

## From Trees To Loops

- Sew together two MHV vertices
- Brandhuber, Spence, & Travaglini (2004)

- Simplest off-shell continuation lacks *i* prescription
- Use alternate form of continuation
- 					       
- 	to map the calculation on to the cut + dispersion integral
- Brandhuber, Spence, & Travaglini (2004)
- Reproduces MHV loop amplitudes
- originally calculated by Dixon, Dunbar, Bern, & DAK (1994)

## Unitarity Method for Higher-Order Calculations

- Bern, Dixon, Dunbar, & DAK (1994)
- Proven utility as a tool for explicit calculations
- Fixed number of external legs
- All-*n *equations
- Tool for formal proofs
- Yields explicit formulae for factorization functions
*I*-duality: phase space integrals  loop integrals
- cf. Melnikov & Anastasiou
- Color ordering

## Unitarity-Based Calculations

- Bern, Dixon, Dunbar, & DAK (1994)
- At one loop in D=4 for SUSY  full answer(also for N =4 two-particle cuts at two loops)
- In general, work in D=4-2Є  full answer
- van Neerven (1986): dispersion relations converge
- Reconstruction channel by channel: find function w/given cuts in all channels

## Unitarity-Based Method at Higher Loops

- Loop amplitudes on either side of the cut
- Multi-particle cuts in addition to two-particle cuts
- Find integrand/integral with given cuts in all channels
- In practice, replace loop amplitudes by their cuts too

## Twistor-space structure

- Can be analyzed with same differential operators
- ‘Anomaly’ in the analysis; once taken into account, again support on simple sets of lines
- Cachazo, Svrček, & Witten; Bena, Bern, DAK, & Roiban (2004)
- Non-supersymmetric amplitudes also supported on simple sets
- Cachazo, Svrček, & Witten (2004)

## Direct Algebraic Equations?

- Basic structure of differential equations
*D* A = 0
- To solve for A, need a non-trivial right-hand side; ‘anomaly’ provides one!
*D* A = a
- Evaluate only discontinuity, using known basis set of integrals    system of algebraic equations
- Cachazo (2004)

## Challenges Ahead

- Incorporate massive particles into the picture
- 		 *D* = 4–2 helicities
- Interface with non-QCD parts of amplitudes
- Reduce one-loop calculations to purely algebraic ones in an analytic context, avoiding intermediate-expression swell
- Twistor string at one loop (conformal supergravitons); connected-curve picture?

## Another Amazing Result: Iteration Relation

- Anastasiou, Bern, Dixon, DAK (2003)
- This should generalize

**Notes:**
Picture of K2