Twistor Spinoffs for Collider Physics

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Twistor Spinoffs for Collider Physics

• Lance Dixon, SLAC
• Fermilab Colloquium
• June 7, 2006

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Physics at very short distances

• Unification particle interactions simpler at
  short distances

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Physics at very short distances

• Supersymmetry predicts a host of new massive
  particles
• including a dark matter candidate
• Typical masses 100 GeV/c2 1 TeV/c2
  (mproton 1 GeV/c2)

• Many other theories of electroweak scale mW,Z
  100 GeV/c2
• make similar predictions
• new dimensions of space-time
• new forces
• etc.

How to sort them all out?

• Einstein (E mc2) heavy particles require high
  energies
• Heisenberg (Dx Dp gt h) short distances require
  high energies (and large momentum transfers)

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The Tevatron

• The present energy frontier right here!
• Proton-antiproton collisions at 2 TeV
  center-of-mass energy

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The Large Hadron Collider

• Proton-proton collisions at 14 TeV
  center-of-mass energy,
• 7 times greater than Tevatron
• Luminosity (collision rate) 10100 times greater

• New window into physics at the shortest
  distances opening 2007

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LHC Detectors
ATLAS
CMS
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What will the LHC see?
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What might the LHC see?
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A better way to compute?

• Backgrounds (and many signals) require detailed
• understanding of scattering amplitudes for
• many ultra-relativistic (massless)
  particles
• especially quarks and gluons of QCD

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The loop expansion

• Amplitudes can be expanded in a small
  parameter, as g2/4p
• At each successive order in g2, draw Feynman
  diagrams with
• one more loop the number grows very
  rapidly!
• For example, gluon-gluon scattering

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Why do we need to do better?

• Leading-order (LO), tree-level predictions are
  only qualitative, due to poor convergence of
• expansion in strong coupling as(m) 0.1
• NLO corrections can be 30 - 80 of LO

state of the art
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Tevatron Run II example

• Azimuthal decorrelation of di-jets at D0
• due to additional radiation

Z. Nagy (2003)
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LHC Example SUSY Search
Gianotti Mangano, hep-ph/0504221 Mangano et al.
(2002)

• Search for missing energy jets.
• SM background from Z jets.

Early ATLAS TDR studies using PYTHIA overly
optimistic

• ALPGEN based on LO amplitudes,
• much better than PYTHIA at
• modeling hard jets
• What will disagreement between
• ALPGEN and data mean?
• Hard to tell because of potentially
• large NLO corrections

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Dialogue between theorists experimenters
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The dialogue continues
Experimenters to theorists
OK, wed really like these at NLO, by the time
LHC starts
Les Houches 2005
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How do we know theres a better way?

Because Feynman diagrams for QCD are too
complicated
An
An
from only 10 diagrams!
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How do we know theres a better way?
Because many answers are much simpler than
expected!
For example, special helicity amplitudes vanish
or are very short
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Mathematical Tools for Physics
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Simplicity in Fourier space
Example of atomic spectroscopy
t
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The right variables
Scattering amplitudes for massless plane waves of
definite 4-momentum Lorentz vectors kim
ki20
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Adding spins
From two non-identical non-relativistic spin ½
objects, make spin 1
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Spinor products
Antisymmetric product of two spin ½ is spin 0
(rotationally invariant)
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Spinor Magic
Spinor products precisely capture
square-root phase behavior in collinear limit.
Excellent variables for helicity amplitudes
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Twistor Space
Start in spinor space
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Twistor Transform in QCD
Witten (2003)
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More Twistor Magic
Berends, Giele Mangano, Parke, Xu (1988)
A6

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Even More Twistor Magic
Now it is clear how to generalize
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MHV rules
Cachazo, Svrcek, Witten (2004)
Twistor space picture
Led to MHV rules
More efficient alternative to Feynman rules for
QCD trees
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MHV rules for trees
Rules quite efficient, extended to many collider
applications
Georgiou, Khoze, hep-th/0404072 Wu, Zhu,
hep-th/0406146 Georgiou, Glover, Khoze,
hep-th/0407027

• massless quarks

LD, Glover, Khoze, hep-th/0411092 Badger,
Glover, Khoze, hep-th/0412275

• Higgs bosons (Hgg coupling)

• vector bosons (W,Z,g)

Bern, Forde, Kosower, Mastrolia, hep-th/0412167
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Twistor structure of loops

• Simplest for coefficients of box integrals in a
  toy model,
• N4 supersymmetric Yang-Mills theory

Again support is on lines, but joined into rings,
to match topology of the loop amplitudes
Cachazo, Svrcek, Witten Brandhuber,
Spence, Travaligni (2004)
Bern, Del Duca, LD, Kosower Britto, Cachazo,
Feng (2004)
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Whats a (topological) twistor string?

• Whats a normal string?

Abstracting the lessons often the best! E.g.,
Bern, Kosower (1991)
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Even better than MHV rules
On-shell recursion relations
Britto, Cachazo, Feng, hep-th/0412308
Off-shell antecedent Berends, Giele (1988)
An-k1
An
Ak1
Ak1 and An-k1 are on-shell tree amplitudes with
fewer legs, evaluated with momenta shifted by a
complex amount
Trees are recycled into trees!
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A 6-gluon example
220 Feynman diagrams for gggggg
Helicity color MHV results symmetries

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Simple final form
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Relative simplicity grows with n
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Proof of on-shell recursion relations
Britto, Cachazo, Feng, Witten, hep-th/0501052
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Speed is of the Essence

• For collider phenomenology, in the end all one
  needs are numbers
• But one needs them many times to do integrals
  over phase space

• For LHC, n 6 9, they do pretty well

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On-shell recursion at one loop
Bern, LD, Kosower, hep-th/0501240,
hep-ph/0505055, hep-ph/0507005

• Same techniques work for one-loop QCD amplitudes
• much harder to obtain by other methods than
  are trees.

• New features arise compared with tree case

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Rational functions in loop amplitudes

• After computing cuts using unitarity, there
  remains
• an additive rational-function ambiguity
• Determined using
• - tree-like recursive diagrams, plus
• - simple overlap diagrams
• No loop integrals required in this step
• Bootstrap rational functions from cuts and
  lower-point amplitudes
• Method tested on 5-point amplitudes, used to get
  new QCD results
• Now working to generalize method to all helicity
  configurations,
• and to processes on the realistic NLO
  wishlist.

Forde, Kosower, hep-ph/0509358
Berger, Bern, LD, Forde, Kosower, hep-ph/0604195,
hep-ph/0606nnn,
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Example of new diagrams
recursive
overlap
7 in all
Compared with 1034 1-loop Feynman diagrams
(color-ordered)
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Revenge of the Analytic S-matrix
Reconstruct scattering amplitudes directly from
analytic properties
Chew, Mandelstam Eden, Landshoff, Olive,
Polkinghorne Veneziano Virasoro, Shapiro
(1960s)
Analyticity fell somewhat out of favor in 1970s
with rise of QCD to resurrect it for computing
perturbative QCD amplitudes seems deliciously
ironic!
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Conclusions

• Exciting new computational approaches to gauge
  theories due (directly or indirectly) to
  development of twistor string theory
• So far, practical spinoffs mostly for trees,
• and loops in supersymmetric theories
• But now, new loop amplitudes in
• full, non-supersymmetric QCD needed for
• collider applications are beginning to fall
  to
• twistor-inspired recursive approaches
• Expect the rapid progress to continue

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Extra slides
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Initial data
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Supersymmetric decomposition for QCD loop
amplitudes
gluon loop
N4 SYM
N1 multiplet
scalar loop --- no SUSY, but also no spin tangles
N4 SYM and N1 multiplets are simplest pieces
to compute because they are cut-constructible
determined by their unitarity cuts, evaluated
using D4 intermediate helicities
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Loop amplitudes with cuts
Generic analytic properties of shifted 1-loop
amplitude,
Cuts and poles in z-plane
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Direct proof of MHV rules via OSRR
K. Risager, hep-th/0508206
MHV rules
There is a different complex momentum shift for
which the on-shell recursion relations (OSRR)
for NMHV are identical, graph by graph, to the
MHV rules. Proof is inductive in
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Why does it all work?

• In mathematics you don't understand things.
• You just get used to them.