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Hidden Structures in Field
Theory Amplitudes and their
Applications
Niels Bohr Institute August 12, 2009
Zvi Bern, UCLA
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Hidden Structures in Amplitudes
Why have a conference on hidden structures?
Why should we care?
1. Some of the most important advances
stem from identifying and exploiting hidden
structures found in explicit calculations.
2. Hidden structures help us calculate!
3. Leads to deeper understanding of basic
properties of quantum field theory!
Will go through various examples.
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How to Hide Structures
Vertices and propagators involve
gauge-dependent off-shell states.
An important origin of the
complexity.
To get at root cause of the trouble we must rewrite
perturbative quantum field theory so all pieces are
ZB, Dixon, Dunbar and Kosower
gauge invariant.
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How to Expose Structures
Use gauge invariant on-shell formalisms
on-shell
amplitude
BCFW recursion for trees
Britto, Cachazo, Feng and Witten
BCFW recursion uses
only on-shell quantities
on-shell
Modern unitarity
method for loops
Application of
generalized unitarity
ZB, Dixon, and Kosower
ZB, Dixon, Dunbar and Kosower (BDDK)
complex momenta
to solve cuts
Britto, Cachazo and Feng
Buchbinder and Cachazo
“method of
maximal cuts”
used in 4 loop
calculations.
ZB, Carasco, Johansson, Kosower
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Famous Example: Twistor Space Curves
Witten conjectured that in twistor space gauge theory
amplitudes have delta-function support on curves of
degree:
Connected picture
Disconnected picture
Structures imply an amazing simplicity
in the scattering amplitudes.
Nair; Witten
Roiban, Spradlin and Volovich
Cachazo, Svrcek and Witten
Gukov, Motl and Neitzke
Bena Bern and Kosower
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MHV Vertices
Cachazo, Svrcek and Witten
MHV amplitude
non-MHV amplitude
twistor space
momentum space
The MHV amplitudes are vertices
for building new amplitudes.
The MHV rules follow directly from twistor space
structure
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Twistor Space Structures
At one-loop the coefficients of all box integral functions
have beautiful twistor space structure.
Box integral
Twistor space support
Three negative
helicities
Four negative
helicities
Twistor structures are
mapped for all N = 4
gauge theory amplitudes.
Related to box coefficients.
Bern, Dixon and Kosower
Britto, Cachazo and Feng
Zigzag structure
as number of
negative helicities
increases.
Korchemsky and Sokatchev
Arkani-Hamed, Cachazo, Cheung, Kaplan
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Talks from Spradlin, Hodges, Korchemsky, etc
A Remarkable Twistor String Formula
The following “connected” formula encapsulates the tree-level
S-matrix of N = 4 sYM.
Integral over the
Moduli and curves
Witten
Roiban, Spradlin and Volovich
Degree d polynomial in
the moduli ak
Strange formula from Feynman diagram viewpoint.
But it’s true: careful checks by Roiban, Spradlin and Volovich
This is an example of a remarkable structure, yet no
applications. It’s a very intriguing formula! Formula for gravity?
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Structure in Gravity Amplitudes
Consider the gravity Lagrangian
flat metric
graviton
field
metric
+…
Infinite number of
complicated interactions
Compare to Yang-Mills Lagrangian on which QCD is based
Only three and four
point interactions
Gravity seems so much more complicated than gauge theory.
Off Shell structure is very well hidden!
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Gravity as the Square of Gauge Theory
Instead work on shell:
gauge theory:
gravity:
“square” of
Yang-Mills
vertex.
On-shell squaring structure is immediately exposed
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KLT Relations
• At tree level Kawai, Lewellen and Tye derived a relationship
between closed and open string amplitudes.
• In field theory limit, relationship is between gravity and
gauge theory.
Gravity
amplitude
where we have stripped all coupling constants
Color stripped gauge
theory amplitude
Holds for any external string
states. See review: gr-qc/0206071
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Gravity as the Square of Gauge Theory
ZB, Carrasco, Johansson
Gauge theory:
color factor
kinematic numerator factor
sum over diagrams
with only 3 vertices
Feynman propagators

For every color factor Jacobi identity there exists a corresponding
numerator equation. Used to derive a set of identities between
See Vanhove’s talk
amplitudes
Gauge theory:
Einstein Gravity:
Used at four loops
See Carrasco’s and
Johansson’s talks
Cries out for a unified description of the sort given by
string theory! Not as well understood as we would like.
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A Higher-loop Structure
Consider the four gluon all-positive helicity amplitude in QCD.
If you expand it in polylogs it is some moderate mess.
Instead let’s write it in a special basis of integralsZB, Dixon, Kosower
planar
double box
non-planar
double box
hep-ph/0001001
Why do planar and non-planar double box
numerators look the same?
last contribution
happens to vanish
for all plus helicity
ZB, Carrasco, Johansson
The numerator Jacobi-like identity explains it.
We find a general structure for all helicities!
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Loop Iteration of N = 4 Amplitudes
The planar four-point two-loop amplitude undergoes
fantastic simplification.
ZB, Rozowsky, Yan
Anastasiou, ZB, Dixon, Kosower
is universal function related to IR singularities
This gives two-loop four-point planar amplitude as iteration of
one-loop amplitude.
Three loop satisfies similar iteration relation. Rather nontrivial.
ZB, Dixon, Smirnov
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All-Loop Generalization
Identification of the interative structure directly leads
to “BDS ansatz”. Recognize exponential and resum.
constant independent
of kinematics.
all-loop resummed
amplitude
IR divergences
cusp anomalous
dimension
finite part of
one-loop amplitude
Anastasiou, ZB, Dixon, Kosower
ZB, Dixon and Smirnov
Gives a definite prediction for all values of coupling
given BES integral equation for the cusp
Beisert, Eden, Staudacher
anomalous dimension.
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Alday and Maldacena Strong Coupling
For four point amplitude:
constant independent
of kinematics.
all-loop resummed
amplitude
IR divergences
cusp anomalous
dimension
finite part of
one-loop amplitude
Wilson loop
In a beautiful paper Alday and Maldacena
confirmed the conjecture for 4 gluons at strong
coupling from an AdS string theory computation.
Minimal surface calculation.
Very suggestive link to Wilson loops even at weak coupling.
Drummond, Korchemsky, Henn, Sokatchev ; Brandhuber, Heslop and Travaglini
ZB, Dixon, Kosower, Roiban, Spradlin, Vergu, Volovich;
Anastasiou, Branhuber, Heslop, Khoze, Spence, Travagli,
• Identification of new symmetry: “dual conformal symmetry”
Drummond, Henn, Korchemsky, Sokatchev;
Beisert, Ricci, Tseytlin, Wolf; Brandhuber, Heslop, Travaglini;
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Berkovits and Maldacena
All-loop Trouble at Higher Points
For various technical reasons it is difficult to solve
for minimal surface for larger numbers of gluons.
Alday and Maldacena realized certain terms can be calculated
at strong coupling for an infinite number of gluons.
Disagrees with BDS conjecture
L
T
Explicit numerical computation at 2-loop six points.
Need to modify conjecture! ZB, Dixon, Kosower, Roiban, Spradlin, Vergu, Volovich
Drummond, Henn, Korchemsky, Sokatchev
Dual conformal invariance and equivalence to Wilson loops persists
Can the BDS conjecture be repaired for six and higher points?
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Symmetry and Ward Identities
A key to understanding the structure of amplitudes is symmetry.
Planar N = 4 YM has three distinct interlocked symmetries:
Relatively simple at tree level for generic kinematics
susy:
QA t ree = 0
conformal:
dual variables
dual conformal:
Together these forms a Yangian structure
Drummond, Henn Plefka;
Bargheer, Beisert, Galleas,
Loebbert, McLoughlin
• Clearly connected to integrability. But can this be made
precise and useful?
• What about loop level? Conformal invariance not so simple.
Infrared singularities and “holomorphic anomaly” confusion.18
In Search of the Holy Grail
When in doubt calculate!
log of the amplitude
discrepancy
Can we figure out the discrepancy?
Important new information from regular polygons should serve
as a guide.
Explicit solution at eight points
Alday and Maldacena (2009)
Solution valid only for strong coupling and special kinematics,
but it is explicit! Need to exploit this.
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N = 8 Supergravity No-Triangle Property
ZB, Dixon, Perelstein, Rozowsky; ZB, Bjerrum-Bohr and Dunbar; Bjerrum-Bohr, Dunbar, Ita, Perkins,
Risager; Proofs by Bjerrum-Bohr and Vanhove; Arkani-Hamed, Cachazo and Kaplan.
One-loop D = 4 theorem: Any one loop amplitude is a linear
combination of scalar box, triangle and bubble integrals with
Brown, Feynman; Passarino and Veltman, etc
rational coefficients:
• In N = 4 Yang-Mills only box integrals appear. No triangle integrals
and no bubble integrals.
• The “no-triangle property” is the statement that same holds in N = 8
supergravity. Non-trivial constraint on analytic form of amplitudes.
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N = 8 Supergravity No-Triangle Property
No triangle property looks like a esoteric technical point
about one loop structure.
However, it gives us the most potent means for making
solid non-trivial statements about UV properties of N = 8
supergravity to all loop orders.
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N = 8 L-Loop UV Cancellations
2
..
1
From 2 particle cut:
ZB, Dixon, Roiban
3
numerator factor
4
1 in N = 4 YM
L-particle cut
numerator factor
• Numerator violates one-loop “no-triangle” property.
• Too many powers of loop momentum in one-loop subamplitude.
• After cancellations behavior must be same as N = 4 Yang-Mills!
• UV cancellation exist to all loop orders. (not a proof of finiteness)
• Even pure gravity displays nontrivial cancellations!
ZB, Carrasco, Forde, Ita, Johansson
• These all-loop cancellations not potentially explained by susy
or by Berkovits’ string theory nonrenormalization theorem.
Green, Russo; Vanhove; Bossart, Howe, Stelle 22
Some Interesting Open Problems
• More to do at tree level.
— further unraveling of twistor-space structure.
— understanding relation between different (BCFW)
recursions and dual formulation.
— gravity as the square of YM. Not as well
understood as we would like. Crucial for
understanding gravity.
• Interface of string theory and field theory– certain
features clearer in string theory, especially at tree level.
KLT classic example.
• Can we carry over Berkovits string theory pure spinor
formalism to field theory? Should help expose full susy.
• Higher-dimensional methods: helicity, on-shell
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superspace.
Some Interesting Open Problems
• Repairing BDS ansatz at six and higher points. Explicit
•
•
•
•
analytic information from Alday and Maldacena.
Proof of dual conformal symmetry to all loop orders.
Derivation of Ward Identities.
Role of conformal invariance and Yangian structure at
loop level? IR divergences and “holomorphic
anomaly”?
Connection of amplitudes to Wilson loops. N = 4 and
QCD. Non-MHV amplitudes?
What do the no-triangle property of N = 8 supergravity
mean for the effective action?
We will hear a lot about some of these points at
this conference.
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Summary
• Remarkable progress in a broad range of topics:
AdS/CFT, quantum gravity and LHC physics.
• Identification of hidden structures lead to ever
more potent means of computation which in
turn leads to new understanding and further
identification of new hidden structures.
• Many exciting new developments in scattering
amplitudes at this conference.
• Expect a lot more progress in coming years!
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Fortune Cookie
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Extra Transparancies
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Gravity Echoes of Structure
Gravity has same twistor
structure as gauge theory
except derivative of delta
function support.
Witten; ZB, Bjerrum-Bohr, Dunbar
A recent relation:
Gauge:
Elvang, Freedman
Drummond, Spradlin, Volovich, Wen
Dual conformal
invariants
Gravity:
Symmetries and structures of gauge theory have echo
in gravity!
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Schematic Illustration of Status
loops
Same power count as N=4 super-Yang-Mills
UV behavior unknown
from feeding 2, 3 and 4 loop
All-loop UV finiteness.
calculations into iterated cuts.
No potential susy or string
non-renormalization
Explanations.
finiteness unproven
Berkovits string theory
non-renormalization
theorem points to good
L = 5, 6 behavior.
Needs to be redone in
field theory!
No triangle
property
explicit 2, 3, 4 loop
computations
terms
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State-of-the-Art One-Loop Calculations
In 1948 Schwinger computed anomalous
magnetic moment of the electron.
60 years later typical example we can calculate:
Only two more legs than
Schwinger!
Key processes for the LHC: four or more final state objects
• Never been done via Feynman diagrams,
though many failed attempts.
• Widespead applications to LHC physics
pp ! W + 3 jets
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Impact on LHC Physics
Tree level: Pretty much solved 20 years ago by off-shell
(Berends-Giele) numerical recursion.
One loop: Lots of important unsolved problems!
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Application: State of the Art QCD for the LHC
Berger, ZB, Dixon, Febres Cordero, Forde, Gleisberg, Ita, Kosower, Maitre (BlackHat collaboration)
Apply on-shell methods
Data from Fermilab
Never been done via Feynman
diagrams, though many
failed attempts.
Excellent agreement between
NLO theory and experiment.
Triumph of on-shell methods!
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New W + 3-Jet Predictions for LHC
Triumph for on-shell methods
See David Kosower’s talk
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