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Twistor String Theory & Q C D. String Theory and Q C D. Lance Dixon, SLAC HEP2005 Europhysics Conference Lisbon, 26 July 2005. CMS. ATLAS. The Large Hadron Collider. Proton-proton collisions at 14 TeV center-of-mass energy, 7 times greater than previous ( Tevatron )

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string theory and q c d

Twistor String Theory & QCD

String Theory and QCD

Lance Dixon, SLAC

HEP2005 Europhysics Conference

Lisbon, 26 July 2005

the large hadron collider

CMS

ATLAS

The Large Hadron Collider
  • Proton-proton collisions at 14 TeV center-of-mass energy,
  • 7 times greater than previous (Tevatron)
  • Luminosity (collision rate) 10—100 times greater
  • New window into physics at the shortest distances – opening 2007

String Theory and QCD

physics at very short distances
Physics at very short distances

Lots of ideas for physics beyond the Standard Model at

electroweak scale

  • Supersymmetry predicts a host ofnew massive particles
  • including a dark matter candidate
  • Typical masses ~ 100 GeV/c2 – 1 TeV/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?

String Theory and QCD

signals and backgrounds

n

c

c

Signals and backgrounds
  • Newparticlestypically decay into old particles:

quarks, gluons,

charged leptons and neutrinos,

photons, Ws & Zs

(which in turn decay to leptons)

  • Kinematic signatures are not always clean (e.g. mass bumps)

if neutrinos, or other escaping particles (e.g. dark matter)

are present

gluino

cascade

  • Need to quantify the Standard Model backgrounds for a

variety of multi-particle processes, to maximize potential for

new physics discoveries

String Theory and QCD

a better way to compute

Feynman told

us how to do this

– in principle

  • Feynman rules, while very general, are

not optimized for these processes

  • Important to find more efficient methods,

making use of hidden symmetries of QCD

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

String Theory and QCD

why do we need to do better

LO = |tree|2

n=8

NNLO = 2-loop x tree* + …

n=2

NLO = loop x tree* + …

n=3

Why do we need to do better?
  • Leading-order, tree-level predictions are only qualitative, due to poor convergence of perturbative expansion in strong couplingas(m)

state of the art:

String Theory and QCD

how do we know there s a better way

Parke-Taylor formula (1986)

How do we know there’s a better way?

Because many answers are much simpler than expected!

String Theory and QCD

simplicity in fourier space

w

lines appear

Simplicity in Fourier space

Example of atomic spectroscopy

t

String Theory and QCD

the right variables

But for elementary particles with spin (e.g. all observed ones!)

there is a better way:

Take “square root” of 4-vectorskim (spin 1)

use Dirac (Weyl) spinors ua(ki) (spin ½)

q,g,g, all have 2 helicity states,

The right variables

Scattering amplitudes for massless plane waves of definite momentum:

Lorentz 4-vectors kim ki2=0

Natural to use Lorentz-invariant products

(invariant masses):

String Theory and QCD

the right variables cont

Singular 2 x 2 matrix:

also shows

even for complex momenta

The right variables (cont.)

Reconstruct momenta kim from spinors

using projector onto positive-energy solutions of Dirac eq.:

String Theory and QCD

spinor products

These are complex square roots of Lorentz products:

Spinor products

Instead of Lorentz products:

Use spinor products:

String Theory and QCD

spinor magic

scalars

gauge theory

angular momentum mismatch

Spinor Magic

Spinor products precisely capture square-root + phase behavior in collinear limit. Excellent variables for helicity amplitudes

String Theory and QCD

twistor space

Twistor transform = “half Fourier transform”:

Fourier transform , but not , for each leg

Twistor space coordinates:

Twistor Space

Start in spinor space:

String Theory and QCD

twistor transform in q c d

lines appear!

Twistor Transform in QCD

Witten (2003)

Parke-Taylor (1986)

String Theory and QCD

more twistor magic

more lines

More Twistor Magic

Mangano, Parke, Xu (1988)

=

String Theory and QCD

slide16

Berends, Giele (1990)

Even More Twistor Magic

Now it is clear

how to generalize

String Theory and QCD

mhv rules

off-shell MHV

(Parke-Taylor)

amplitudes

scalar propagator, 1/p2

MHV rules

Cachazo, Svrcek,

Witten (2004)

Twistor space picture:

Led to MHV rules:

More efficient

alternative to

Feynman rules

for QCD trees

String Theory and QCD

mhv rules for trees

Related approach to QCD + massive quarks

  • but more directly from field theory

Schwinn, Weinzierl,

hep-th/0503015

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

String Theory and QCD

twistor structure of loops

Bern, LD, Del Duca, Kosower;

Britto, Cachazo, Feng (2004)

Twistor structure of loops
  • Simplest for coefficients of box integrals in a “toy model”,
  • N=4 supersymmetric Yang-Mills theory

Cachazo, Svrcek, Witten;

Brandhuber, Spence,

Travaligni (2004)

String Theory and QCD

twistor structure of loops cont
Twistor structure of loops (cont.)

Bern, LD,

Kosower (2004)

Again support is on lines,

but joined into rings, to

match topology of the

loop amplitudes

String Theory and QCD

what s a topological twistor string

Mass

(GeV/c2)

1019

0

  • A topological string has almost

all of its excitations stripped

QCD

+ lots

QCD + little

  • Having it move in twistor space lets the remaining ones yield QCD, plus superpartners (more or less)

1019

0

What’s a (topological) twistor string?
  • What’s a normal string?

Abstracting the lessons

often the best!

String Theory and QCD

another connection between string theory and q c d

AdS/QCD

QCD

Another connection between string theory and QCD
  • AdS/CFT correspondence
  • Relates strongly coupled gauge theory to weakly coupled gravity in 5 dimensions
  • First for N=4 super-Yang-Mills theory
  • More recently, confining gauge theories
  • But coupling should still be strong in UV

for gravity side to be tractable

  • Interesting insights into hadron masses,

quark-gluon plasma, deep inelastic scattering,

exclusive processes at high energy

Maldacena; Gubser, Klebanov,

Polyakov; Witten (1996), + …

Igor Klebanov, talk at

Lepton-Photon 2005

String Theory and QCD

even better than mhv rules
Even better than MHV rules

On-shell recursion relations

Britto, Cachazo, Feng, hep-th/0412308

Ak+1 and An-k+1 are on-shell tree amplitudes with fewer legs,

evaluated with momenta shifted by a complex amount

Trees are recycled into trees!

String Theory and QCD

a 6 gluon example

3 BCF diagrams

related by symmetry

A 6-gluon example

220 Feynman diagrams for gggggg

Helicity + color + MHV results + symmetries

String Theory and QCD

simple final form

Simpler than form found in 1980s

despite (because of?) spurious singularities

Mangano, Parke, Xu (1988)

Bern, Del Duca, LD,

Kosower (2004)

Relative simplicity even more striking for n>6

Simple final form

String Theory and QCD

proof of on shell recursion relations

Cauchy:

residue at zk

= [kth term in relation]

Proof of on-shell recursion relations

Britto, Cachazo, Feng, Witten, hep-th/0501052

Very simple, general – Cauchy’s theorem + factorization

Let complex momentum shift depend on z. Describe using spinors.

String Theory and QCD

on shell recursion at one loop

but

On-shell recursion at one loop

Bern, LD, Kosower, hep-th/0501240, hep-th/0505055, hep-ph/0507005

  • Same techniques work for one-loop amplitudes
  • -- much harder to obtain by other methods than are trees.
  • Warm up with special tree-like one-loop amplitudes
  • with no cuts, only poles:
  • New features arise compared with tree case due to
  • different collinear behavior of loop amplitudes:
  • With a little guesswork, can still find, and solve in closed form,
  • recursion relations for these two infinite sequences of amplitudes

String Theory and QCD

loop amplitudes with cuts
Loop amplitudes with cuts
  • Also can do loop amplitudes with cuts(hep-ph/0507005)
  • First compute cuts using unitarity.
  • Remaining rational-function terms contain

“spurious singularities”, e.g.

  • Accounting for them properly yields simple

“overlap diagrams” in addition torecursive diagrams

  • No loop integrals required to bootstrap rational functions

from cuts and lower-point amplitudes

  • Method tested on 5-point amplitudes,

used to compute new QCD results:

String Theory and QCD

revenge of the analytic s matrix

Branch cuts

  • Poles
Revenge of the Analytic S-matrix?

Reconstruct scattering amplitudes directly from analytic properties

Chew, Mandelstam;

Eden, Landshoff,

Olive, Polkinghorne;

…(1960s)

Analyticity, tied closely to string theory, fell out of favor in 1970s with rise of QCD; to resurrect it for computing perturbativeQCD amplitudes seems deliciously ironic!

String Theory and QCD

conclusions
Conclusions
  • Exciting new computational approaches to gauge theories due (directly or indirectly) to development of twistor string theory,

appreciation of analyticity

  • Most practical spinoffs to date for tree amplitudes, and loops in supersymmetric theories
  • But now, new loop amplitudesin full QCD are beginning to fall to this approach
  • Expect therapid progressto continue!

String Theory and QCD

why does it all work
Why does it all work?

In mathematics you don't understand things.

You just get used to them.

String Theory and QCD

extra slides
Extra slides

String Theory and QCD

slide33

New form for

Bern, Del Duca, LD, Kosower (2004)

Simple tree found by computing one-loop amplitude first!

String Theory and QCD

march of the tree amplitudes
March of the tree amplitudes

String Theory and QCD

march of the 1 loop amplitudes
March of the 1-loop amplitudes

String Theory and QCD

march of the 2 loop amplitudes
March of the 2-loop amplitudes

String Theory and QCD

march of the 3 loop amplitudes
March of the 3-loop amplitudes

String Theory and QCD