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)
2
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
3
(Slide from Zvi Bern’s talk, Oxford Twistor Workshop
January 2005)
Eg: Five gluon tree level scattering with Feynman diagrams:
4
Experimentalists need predictions for, eg, multi-jet events:
( V Del Duca, Queen Mary Workshop Nov 2005)
5
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:
helicity
 spinor variables
●
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
6
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 the Maximal Helicity Violating (MHV) amplitude
(Notation
i is the particle label)
● This simplicity is unexpected from Feynman diagrams – how can one
explain it ?
7
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
8
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
● Explicit check:
twistor space coord’s
Eg: 3 points 
Z are collinear if
in spacetime:
and the above becomes a differential
equation satisfied by the amplitude
● (Loop level: also get localisation – see later)
What can explain this localisation ?
9
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
in CP(3,4) of degree
negative helicity gluons localise on curves
and genus
● This explains the localisation of YM amplitudes and gives a weak-weak
duality between N=4 SYM and twistor string theory
10
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
X
< - - +…+ >
< - - - +…+ >
X
X
X
nnMHV < - - - - +..+ >
X
X
X
nMHV
X
X
X
X
X
11
MHV Diagrams – Tree Level
Cachazo, Svrcek, Witten
● Idea: Since MHV tree amplitudes
M
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
(spacetime)
(twistor space)
nMHV
M
M
nnMHV
M
M
M
● This works and gives a new, more efficient, way to calculate tree amplitudes
Next: more new developments at tree level 
12
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 
13
Trees: Gravity
● Gravity amplitudes in momentum space – studied by DeWitt (1967)
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”
● Gravity amplitudes in spinor helicity variables are much simpler - eg
n point MHV gravity amplitude Berends, Giele, Kuijf (1988)
14
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
permutations
-- derivation of MHV diagrams for gravity
Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager
(non-trivial due to polynomial dependence on tilded spinors –
twistor space localisation is via “derivative of delta functions” )
15
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…. )
16
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
17
Loops - Unitarity I
The scattering matrix S must be unitary:
● Example: 4 point, mass m, scalar scattering 1+2  3+4:
1
3
2
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)
s
A(s):
cut
cut
18
Loops - Unitarity II
●
Now consider the contour integral of A(s) around C:
s
cut
cut
C
● This gives
● Then, using
● Idea: amplitudes can be reconstructed from their analytic properties
19
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
P
t
q
s
F is a box function – these are fundamental
in one-loop diagrams
p
also many new results in N=1 SYM and QCD 
20
Q
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
21
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 
-loop N=4 YM amplitudes with
in CP(3,4) of degree
negative helicity gluons localise on curves
and genus
● This can also be seen explicitly in space-time for some examples
eg – one loop N=4 SYM MHV amplitudes
these localise on two lines in twistor space
(Cachazo, Svrcek, Witten)
x
x
x
x
22
Loops – Twistor String Theory II
● Twistor string theory – B model on CP(3,4) --- N=4 SYM
● Loop level in open string theory – generate closed strings
One loop open string
=
● this is also true for twistor string theory
= tree closed string
(Berkovits, Witten)
-- at one loop level the fields of N=4 conformal supergravity contribute.
This has fields
and action
(quartic in derivatives)
● avoid this if possible…
the one loop twistor space localisation suggests there is a
corresponding pure gauge theory spacetime description
 MHV loop diagrams
23
Loops – MHV diagrams I
●
For tree amplitudes – spacetime MHV diagrams work
M
(spacetime)
(twistor space)
M
M
M
M
-- direct realisation of twistor space localisation
● Study of known one loop MHV amplitudes  twistor space localisation on
pairs of lines
x
x
x
x
● This suggests that in spacetime, one loop MHV amplitudes should be given
by diagrams
M
M
24
Loops – MHV diagrams II
●
M
● Technical issues:
M
?
= MHV amplitude
The particle in the loop is off-shell. But
particles in MHV diagrams are on-shell
 need an off-shell prescription
Coordinates
null
vector
null
reference
vector
-- 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 
25
Loops - MHV Diagrams III
● The result of this MHV diagram calculation is
(Brandhuber, Spence, Travaglini hep-th/0407214)
● The known answer is
● These agree, due to the nine-dilogarithm identity
26
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
Might MHV diagrams provide a completely new way to do perturbative
gauge theory?
27
Loops – MHV diagrams 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)
(2) 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
28
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
29
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
Quadruple
cut
Triple cut
Result:
Various integrals
● This is the correct QCD result
30
Loops – recursion relations
● Recursion relations for tree amplitudes:
●
=
●
●
● There are analogous relations at loop level – eg
QCD amplitude, recursion relations give decompositions like:
loop
one loop
Bern, Dixon, Kosower,
hep-th/0507005
tree
● This allows one to reconstruct (parts of, in general) amplitudes from
simpler pieces
31
One Loop : Summary
● One loop amplitudes (like trees) localise in twistor space
● There is a related MHV diagram construction in spacetime, eg
=
M
M
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
32
One Loop: Progress
Lance Dixon, Oxford Twistor Workshop talk, Jan 2005
= number positive,
negative helicity gluons
~ pre twistor strings
● Note:
- Power of new techniques and rapid progress
- Applications to QCD, etc relevant to LHC
- Essentially all inspired by twistor string theory
~ post twistor strings (2004)
Plus 2005 on – eg
- N=1, 6 gluons, split helicity
- N=0, 6 gluons, split helicity
rational terms, electroweak
33
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
34
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
35
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
36
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:
Giombi, Kulaxizi,Ricci, Robles-Llana, Trancanelli, Zoubos.
-- orbifolds of N=4 SYM
Kulaixizi, Zoubos; review hep-th/0512302
-- marginal deformations of N=4 SYM
Park, Rey
-- super complex structure deformations of CP(3,4) 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
37
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 
38
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…
39
Recursive Structures and Integrability III
● All-loop N=4 n-point MHV conjectured recursion relation
Bern Dixon Smirnov, hep-th/0505205
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.
● It gives the two loop relation given earlier. It has also been checked for 3 loops:

40
Recursive Structures and Integrability IV
● Conjectured general cross-order relation:
● Tested at three loops (4 point function):
● This is rather involved:
(2/3 of the) two loop function from
Bern, Dixon Kosower
● 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
41
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
42
(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) 43
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
44
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
45
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
46
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
● Applications:
-- higher loops
-- more legs
-- more LHC relevant amplitudes
● 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…..
47