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- + weak interactions
- electroweak theory

- Verified to 0.1%

- Hints of more:
- grand unification
- with supersymmetry

- Unification: particle interactions simpler at short distances

L. Dixon Twistor Spinoffs for Collider Physics

Physics at very short distances

- Supersymmetry predicts a host ofnew 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 > h): short distances require high energies (and large momentum transfers)

L. Dixon Twistor Spinoffs for Collider Physics

D0

Geneva

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

L. Dixon Twistor Spinoffs for Collider Physics

CMS

ATLAS

The Large Hadron Collider- Proton-proton collisions at 14 TeV center-of-mass energy,
- 7 times greater than Tevatron
- Luminosity (collision rate) 10—100 times greater

- New window into physics at the shortest distances – opening 2007

L. Dixon Twistor Spinoffs for Collider Physics

m-

m+

jets ---

lots of jets

lots of W’s

ne

m-

m+

and Z’s

top quarks

with jets

What will the LHC see?L. Dixon Twistor Spinoffs for Collider Physics

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

L. Dixon Twistor Spinoffs for Collider Physics

1 loop

2 loop

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

L. Dixon Twistor Spinoffs for Collider Physics

n=8

NNLO = 2-loop x tree* + …

n=2

NLO = loop x tree* + …

n=3

Why do we need to do better?- Leading-order (LO), tree-level predictions are only qualitative, due to poor convergence of

expansion in strong couplingas(m) ~ 0.1

- NLO corrections can be 30% - 80% of LO

state of the art:

L. Dixon Twistor Spinoffs for Collider Physics

NLO

n=3

Tevatron Run II exampleAzimuthal decorrelation of di-jets at D0

due to additional radiation

Z. Nagy (2003)

L. Dixon Twistor Spinoffs for Collider Physics

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

L. Dixon Twistor Spinoffs for Collider Physics

“Please calculate the following at NLO”

Theorists to experimenters:

“Get real”

Dialogue between theorists & experimentersL. Dixon Twistor Spinoffs for Collider Physics

experimenters:

“OK, we’ll get

to work”

The dialogue continuesExperimenters to theorists:

“OK, we’d really like these at NLO, by the time LHC starts”

Les Houches 2005

L. Dixon Twistor Spinoffs for Collider Physics

How do we know there’s a better way?

Because Feynman diagrams for QCD are “too complicated”

An

An

from only 10 diagrams!

L. Dixon Twistor Spinoffs for Collider Physics

How do we know there’s a better way?

Because many answers are much simpler than expected!

For example, special helicity amplitudes vanish or are very short

L. Dixon Twistor Spinoffs for Collider Physics

(Egyptians, …, Hamilton, 1843)

(Schrodinger, 1926)

(200 B.C.?)

(Pauli, 1925)

(Dirac, 1925)

(Fourier, 1807)

(Penrose, 1967)

(Witten, 2003) + …

Mathematical Tools for PhysicsL. Dixon Twistor Spinoffs for Collider Physics

lines appear

Simplicity in Fourier spaceExample of atomic spectroscopy

t

L. Dixon Twistor Spinoffs for Collider Physics

Natural to use Lorentz-invariant products

(invariant masses):

But for particles with spin

there is a better way

massless q,g,g

all have 2 helicities

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

use 2-component Dirac (Weyl) spinors ua(ki) (spin ½)

The right variablesScattering amplitudes for massless plane waves of definite 4-momentum:

Lorentz vectors kim ki2=0

L. Dixon Twistor Spinoffs for Collider Physics

Similarly, reconstructrelativisticspin-1 momenta kim from spinors:

(projector onto positive-energy solutions of Dirac equation)

Adding spinsFrom two non-identical non-relativisticspin ½ objects, make spin 1

L. Dixon Twistor Spinoffs for Collider Physics

Use spinor products:

These are complex square roots of Lorentz products:

Spinor productsAntisymmetric product of two spin ½ is spin 0 (rotationally invariant)

L. Dixon Twistor Spinoffs for Collider Physics

gauge theory

angular momentum mismatch

explains denominators

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

L. Dixon Twistor Spinoffs for Collider Physics

Twistor transform = “half Fourier transform”:

Fourier transform , but not , for each leg

Conjugate variables:

Like time and frequency:

Twistor space has coordinates

Twistor SpaceStart in spinor space:

L. Dixon Twistor Spinoffs for Collider Physics

Fourier transform of plane-wave is d-function:

“Maximally Helicity Violating” amplitudes, , are plane-wave in

lines appear!

Twistor Transform in QCDWitten (2003)

L. Dixon Twistor Spinoffs for Collider Physics

More Twistor Magic

Berends, Giele;

Mangano, Parke, Xu (1988)

A6

=

L. Dixon Twistor Spinoffs for Collider Physics

Even More Twistor Magic

Now it is clear

how to generalize

L. Dixon Twistor Spinoffs for Collider Physics

(Parke-Taylor)

amplitudes

scalar propagator, 1/p2

MHV rulesCachazo, Svrcek,

Witten (2004)

Twistor space picture:

Led to MHV rules:

More efficient

alternative to

Feynman rules

for QCD trees

L. Dixon Twistor Spinoffs for Collider Physics

Related approach to QCD + massive quarks

- more directly from field theory

Schwinn, Weinzierl,

hep-th/0503015

MHV rules for treesRules 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

L. Dixon Twistor Spinoffs for Collider Physics

Twistor structure of loops

- Simplest for coefficients of box integrals in a “toy model”,
- N=4 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)

L. Dixon Twistor Spinoffs for Collider Physics

(GeV/c2)

1019

0

- A topological string has almost

all of its excitations stripped

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

QCD

+ lots

QCD + little

1019

0

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

Abstracting the lessons often the best! E.g., Bern, Kosower (1991)

L. Dixon Twistor Spinoffs for Collider Physics

Even better than MHV rules

On-shell recursion relations

Britto, Cachazo, Feng, hep-th/0412308

[Off-shell antecedent: Berends, Giele (1988)]

An-k+1

An

Ak+1

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!

L. Dixon Twistor Spinoffs for Collider Physics

related by symmetry

A 6-gluon example220 Feynman diagrams for gggggg

Helicity + color + MHV results + symmetries

L. Dixon Twistor Spinoffs for Collider Physics

Simpler than form found in 1980s

Mangano, Parke, Xu (1988)

Simple final formL. Dixon Twistor Spinoffs for Collider Physics

Bern, Del Duca,

LD, Kosower (2004)

Relative simplicity grows with n

L. Dixon Twistor Spinoffs for Collider Physics

Simple, general: Residue theorem+factorization

how amplitudes “fall apart” in degenerate kinematic limits

Inject complex momentum at leg 1, remove it at leg n.

degenerate limits poles in z

Cauchy:

residue at zk

= [kth term in relation]

Proof of on-shell recursion relationsBritto, Cachazo, Feng, Witten, hep-th/0501052

L. Dixon Twistor Spinoffs for Collider Physics

How do the new methods compare to older ones at tree level?

Dinsdale, Ternick, Weinzierl, hep-ph/0602204

off-shell recursive (1988)

Schwinn,

Weinzierl (2005)

Cachazo, Svrcek,

Witten (2004) + …

on-shell recursive

(2005)

even more gluons:

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

L. Dixon Twistor Spinoffs for Collider Physics

1. different collinear behavior of loop amplitudes

double

poles in z

Bern, LD, Kosower,

hep-ph/9403226,

hep-ph/9708239;

Britto, Cachazo, Feng,

hep-th/0412103;

BBCF, hep-ph/0503132;

Britto, Feng, Mastrolia,

hep-ph/0602178

2. branch cuts – but these can be determined efficiently using (generalized) unitarity

but

Trees recycled

into loops!

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

- Same techniques work for one-loopQCD amplitudes
- – much harder to obtain by other methods than are trees.

- New features arise compared with tree case

L. Dixon Twistor Spinoffs for Collider Physics

Competition from semi-numerical methods

– done this way

Ellis, Giele, Zanderighi,

hep-ph/0602185

Rational functions in loop amplitudes- After computing cuts using unitarity, there remains

an additive rational-function ambiguity

- Determined using

- tree-likerecursive 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 newQCD 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, …

L. Dixon Twistor Spinoffs for Collider Physics

Example of new diagrams

recursive:

overlap:

7 in all

Compared with 1034 1-loop Feynman diagrams (color-ordered)

L. Dixon Twistor Spinoffs for Collider Physics

- Poles

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 perturbativeQCD amplitudes

seems deliciously ironic!

L. Dixon Twistor Spinoffs for Collider Physics

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 amplitudesin

full, non-supersymmetricQCD – needed for

collider applications – are beginning to fall to

twistor-inspiredrecursive approaches

- Expect therapid progressto continue

L. Dixon Twistor Spinoffs for Collider Physics

Extra slides

L. Dixon Twistor Spinoffs for Collider Physics

Supersymmetric decomposition for QCD loop amplitudes

gluon loop

N=4 SYM

N=1 multiplet

scalar loop

--- no SUSY,

but also no

spin tangles

N=4 SYM and N=1 multiplets are

simplest pieces to compute

because they are cut-constructible

– determined by their unitarity cuts, evaluated using D=4 intermediate helicities

L. Dixon Twistor Spinoffs for Collider Physics

But if we know the cuts (via unitarity inD=4),

we can subtract them:

rational part

full amplitude

cut-containing part

Shifted rational function

has no cuts, but has spurious poles in z

because of Cn:

Loop amplitudes with cutsGeneric analytic properties of shifted 1-loop amplitude,

Cuts andpoles in z-plane:

L. Dixon Twistor Spinoffs for Collider Physics

entering MHV

vertices

off-shell MHV

(Parke-Taylor)

amplitudes

scalar propagator, 1/p2

Direct proof of MHV rules via OSRRK. Risager, hep-th/0508206

MHV rules:

There is a differentcomplex 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

L. Dixon Twistor Spinoffs for Collider Physics

Why does it all work?

In mathematics you don't understand things.

You just get used to them.

L. Dixon Twistor Spinoffs for Collider Physics

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