Goodbye Feynman diagrams:A new approach to perturbative

quantum field theory

Bill Spence*

Oxford April 2007

*Centre for Research in String Theory, Queen Mary, University of London 2007

Work in collaboration with A. Brandhuber, G. Travaglini, K. Zoubos,arXiv: 0704.0245 hep-th, and earlier papers

Outline

Perturbative quantum field theory

1. Old tricks: Feynman diagrams, unitarity methods

2. New tricks: Twistor inspired progress – MHV diagrams recursion relations generalised unitarity

3. A new approach: MHV perturbation theory

4. Conclusions

Feynman diagrams1.1 Old tricks: Feynman

First course in Yang Mills quantum field theory:

Perturbative quantum corrections to classical amplitudes:

Use propagators

and interaction vertices

to form Feynman diagrams

, , etc.

Eg: QCD – for gluons:(colour labels suppressed) Propagator

3-vertex

Feynman diagrams: the reality1.1 Old tricks: Feynman

But Feynman diagrams are impractical!

gg => n g

n=7 n=8 n=9

Diagrams 559405 10525900 224449225

Eg: Five gluon tree level scattering with Feynman diagrams:

pict

ure

from

Zvi

Be

rn

1.1 Old tricks: Feynman

Feynman diagrams: end products

Feynman diagrams are cumbersome, but the results can be simple:

n-gluon scattering, helicities (--++…+).Result:

This is called an MHV amplitude as tree amplitudes with all, or all but one, helicity the same are zero

Maximal Helicity Violating:

Notation null momenta p,written with spinorsi is the particle label

1.1 Old tricks: Feynman

Feynman diagrams: end products II

Loop amplitudes are also simple in spinor notation:

n-point one-loop all plus helicity amplitude in pure Yang-Mills:

n-point one-loop MHV amplitude in N=4 super Yang-Mills

sum over “box functions” F

1.1 Old tricks: Feynman

Feynman diagrams: Summary

Feynman diagrams: theory

But, the practice:

However:

-- simple rules, Lagrangian derivation, work for all theories

-- diagrams are cumbersome – multiply rapidly and become impractical

-- the result of adding the contributions of many diagrams can be extraordinarily simple, when written in spinor variables

1.2 Old tricks: unitarity

Unitarity methods

●

●

●

Old S matrix approach: the scattering matrix S must be unitary:

Example: 4 point, mass m, scalar scattering 1+2 3+4:

Scattering depends on the Lorentz invariants (s,t):

Consider A(s,t), at fixed t, in the complex plane. There are polesat s = 4m^2, 9m^2,… (production of particles). In fact thereis a branch cut from s=4m^2 to infinity (and also one along thenegative s axis due to poles in the t-channel)

A(s):

s

cut cut

1

2

3

4

1.2 Old tricks: unitarity

Unitarity methods II

● Then, using

Consider the contour integral of the amplitude A(s), around C:●

s

cut cut

C

● This gives

Idea: reconstruct amplitudes from their analytic properties

Loops from the old S-matrix approach:

1.2 Old tricks: unitarity

New unitarity methods

From c. 1990: New application of unitarity methodsBern, Dixon, Dunbar, Kosower,….

●

●

One loop general results:

N=4 SYM – all MHV amplitudes N=1 SYM – all MHV amplitudes Pure YM – (cut-constructible parts of) all MHV amplitudes

(for adjacent negative helicities)

Other particular results:Various nMHV results at one loopTwo loop results (4 point function N=4)Others (nnMHV,…)

●

But – nnMHV – difficulthigher loops – difficult…reaching the limits of this approach by the early 2000’s

But proving difficult to progress further

1.2 Old tricks: unitarityUnitarity methods: summary

Old methods (pre 1970):

-- good ideas, but it proved difficult to write dispersion relations for all but simple (eg two point function) cases

-- was explored as no theory of strong interactions at the time; QCD then became dominant

More recently (1990’s):

-- old unitarity ideas applied to supersymmetric theories

-- new results found, but again no really systematic way to derive dispersion relations to give amplitudes

1.Old tricks: summary

Perturbative quantum field theory, calculate amplitudes via:

Feynman diagrams

But this proves impractical, even with computers – the number of diagrams rises very rapidly with the number of particles involved.However, adding many diagrams often produces a verysimple result (eg MHV) – why???

Unitarity methods

Use dispersion relations – but no systematic wayfound to generate these in general, and applicationsto higher loops (>1), massive theories, etc, proveddifficult

Need some New Tricks……….

2.1 New tricks: Twistors

Twistor string theory

Amplitudes in spinor variables can be simple: eg MHV

Then:

Idea: Look at amplitudes in twistor space

ie MHV tree amplitudes localise on a line in twistor space

twistor space coordinates

( = Fourier transform of )

Witten hep-th/0312171

Eg: 3 points Z are collinear if

Localisation of tree amplitudes in twistor space appears generic:

Eg: MHV < - - ++…++ > localise on a line

next to MHV < - - - ++….++ > localise on two intersecting lines

twistor space coord’s

in spacetime: and the above becomes a differentialequation satisfied by the amplitude

Loop level: also get localisation – see later

What can explain this localisation ?

Explicit check:

2.1 New tricks: Twistors

Amplitudes in twistor space

2.1 New tricks: Twistors

Twistor string theory

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)

This: - explains the localisation of YM amplitudes,- gives a weak-weak duality between N=4 SYM and twistor string theory

Can then argue that:

-loop N=4 super YM amplitudes with negative helicity gluons localise on curves in CP(3,4) of degree and genus

In twistor space, tree level scattering amplitudes

A surprise: due to delta functions, the integral localises on intersections of degree one curves:

Amplitude Curve

MHV < - - +…+ >

nMHV < - - - +…+ >

nnMHV < - - - - +..+ >

X X X

X

X

XX X

X

X X XX X

X

X

moduli space of curves degree d, genus 0

vertex operators

(degree d (d+1) negative helicity gluons)

2.1 New tricks: Twistors

Twistor string theory: Tree level

2.1 New tricks: Twistors

Twistor string theory:problems

Twistor string theory: beautiful new duality between N=4super Yang-Mills and a topological string theory, but:

Hard to calculate with it – integrals over moduli spaces ofcurves in CP(3,4)…

At loop level (and tree level for non-planar graphs) – conformal supergravity arises and cannot be decoupled

Much of the structure seems tied to N=4 supersymmetry(eg conformal invariance) – how would it work for pureYang-Mills; also how to include masses for example…

It would be nice to have methods which work in spacetime itself…….

2.2 New tricks: MHV

MHV methods

Idea: Since MHV tree amplitudes 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 M

M

MMHV

nMHV

nnMHV M M M

This works and gives a new, more efficient, way to calculate tree amplitudes

(twistor space)(spacetime)

Cachazo, Svrcek, Witten

This suggests that in spacetime, one loop MHV amplitudes should be given by diagrams

For tree amplitudes – spacetime MHV diagrams work

Study of known one loop MHV amplitudes twistor space localisation onpairs of lines

M M

M M M

(twistor space)(spacetime)

-- direct realisation of twistor space localisation

M M

xx

xx

2.2 New tricks: MHV

MHV methods: loops

2.2 New tricks: MHV

MHV methods: loops II

M M = MHV amplitude ?

Technical issues:

Then: multiply MHV expressions, simplify spinor algebra, performphase space (l) and dispersion (z) integrals.....non-trivial calculation

Result

The particle in the loop is off-shell. But particles in MHV diagrams are on-shell need an off-shell prescription

-- Result should be independent of reference vector;-- Use dimensional regularisation of momentum integrals

Coordinatesnullvector

nullreferencevector

2.2 New tricks: MHV

MHV methods: loops III

The result of this MHV diagram calculation is

The known answer is

These agree, due to the nine-dilogarithm identity

(Brandhuber, Spence, Travaglini hep-th/0407214)

2.2 New tricks: MHV

MHV diagrams: N<4

So – spacetime MHV diagrams give one loop N=4 MHV amplitudes

Remarkably: MHV diagrams give correct results for -- N=1 super YM-- pure YM (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, BrandhuberSpence, TravagliniQuigley Rozali

Bedford, BrandhuberSpence, Travaglini

a surprise - no conformal supergravity as expected from twistor string theory

Might MHV diagrams provide a completely new way to do perturbative gauge theory?

2.2 New tricks: MHV

One loop:general result

MHV diagrams are equivalent to Feynman diagrams for any susy gauge theory at one loop: Brandhuber, Spence

Travaglini hep-th/0510253

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

(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.

Evidence that MHV diagrams might provide a new perturbation theory

2.2 New tricks: MHV

MHV methods: Issues

MHV methods:

MHV diagrams “cut constructible” pieces of the physical amplitude.Other “rational” parts are missing.Pure YM (but not susy YM) has rational parts!

Hard to apply to higher loops, non-MHV

successes at tree level, one loop

can be thought of as a consistent formulation of dispersion integrals

But,

Hard to incorporate masses, or go off-shell

2.3 New tricks: Recursion

Recursion relations

Behaviour of tree level scattering amplitudes at complex momenta

Britto, Cachazo, Feng, Witten

- 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)

Recursion relations for tree amplitudes:

There are analogous relations at loop level – eg one loopQCD amplitude, recursion relations give decompositions like:

Bern, Dixon, Kosower,hep-th/0507005

This allows one to reconstruct (parts of, in general) amplitudes fromsimpler pieces – this is a useful tool, but it is hard to apply at loop level systematically

tree

loop

= ●●●

2.3 New tricks: Recursion

Recursion relations II

2.4 New tricks: Generalised Unitarity Generalised Unitarity

In d-dimensions, the discontinuities should also determine these rational terms

Unitarity arguments: find amplitudes from their discontinuities (logs, polylogs)

Supersymmetric theories: amplitudes can be completely reconstructed from their discontinuities

Non- supersymmetric theories (eg pure YM) : amplitudes contain additional rational terms

e.g. one loop five gluon amplitude has rational part

2.4 New tricks: Generalised Unitarity Generalised Unitarity II

d-dimensional unitarity should give the full amplitudes

eg: QCD: multiple cuts in d-dimensions – 4-point case

This is the correct QCD result

Result:

New techniques with multiple cuts developed (see reviews for references)

Triple cutQuadruplecut

Brandhuber, McNamara, Spence, Travaglinihep-th/0506068

Various integrals

2.4 New tricks: Generalised Unitarity Generalised Unitarity III

Generalised unitarity – multiple cuts, and d-dimensionalcuts

This has had remarkable successes, e.g:

-- reduction of one-loop calculations to algebraic sums

-- derivation of full amplitudes (including rational terms) in pure Yang-Mills

This has provided another set of useful tools.

However, applications to pure YM proved relatively cumbersome,and applying many of these techniques requires some prior knowledge of the structure of the answer

2. New Tricks: Summary

Recent new methods inspired by twistor string theory:

-- twistor formulations-- MHV methods-- recursion relations-- generalised unitarity

These have provided new insight into perturbative field theory,and yielded amplitudes previously unobtainable by older methods

But there remain outstanding issues:-- methods are not systematically defined or

are difficult to apply-- applications to non-supersymmetric theories are the most challenging-- generalisations (masses etc) non-obvious

We need a systematic formulation incorporating these new ideas

3. A New Approach

3. MHV Perturbation Theory

Recall MHV diagrams: combine MHV vertices to get amplitudes:

This works (at least at one loop in super YM)

M M = MHV amplitude

Idea: derive these rules from a Lagrangian

3.1 Classical MHV theory

An MHV Lagrangian?

What Lagrangian? Ingredients:

Only +/- helicity fields in loops and external lines

A null reference vector is needed ( eg to define off-shell momenta L )

nullvector

nullreferencevector

This suggest some relation to light-cone gauge theory

3.1 Classical MHV theory Yang-Mills in light-cone gauge

Pure Yang-Mills

Light-cone gauge

Leaves

non-propagating,integrate outResult (non-local)

OK, but how to get MHV vertices?

3.1 Classical MHV theory

MHV Lagrangian I

Yang-Mills in light-cone gauge

Idea: Change variables so that

ie, eliminate the ++- vertex

Result: MHV vertices!

Gorsky, Rosly hep-th/0510111*Mansfield hep-th/0511264Ettle, Morris hep-th/0605121

3.1 Classical MHV theory

MHV Lagrangian II

So we have written the YM action in light-cone gauge,using B fields, as a sum of a kinetic term plus MHV vertices

Classically, this is ok. Does it give an alternative perturbationtheory for quantum Yang-Mills?

MHV vertices: always have two negative helicity particles;

All quantum diagrams from the above Lagrangian have at least two negative helicity external fields

3.1 Classical MHV theory

Rational terms

Previous slide: MHV diagrams generate amplitudes withat least two negative helicity fields.

But pure Yang-Mills theory has:

-- all-plus amplitudes, eg one loop four gluon:

-- single-minus amplitudes, eg one loop four gluon:

These cannot be generated from our classical MHV Lagrangian

(Note: these amplitudes are purely rational – no logs, polylogs etc)

3.1 Classical MHV theory

Rational terms II

Something is missing……

So the classical MHV Lagrangian cannot explain the all-plus or single minus helicity amplitudes

Also, while gives graphs with at least two negativehelicity particles, it does not give the rational parts of other amplitudes [known from explicit calculations]

3.1 Classical MHV theory

A Puzzle

The Lagrangian, , obtained from light-cone gaugeYM theory using new variables, does not generate rational termsin quantum amplitudes. For example, the (++++) one-loop, whichis entirely rational:

But what about the all-minus amplitudes? , eg:

This could be generated from MHV diagrams (it has more than onenegative helicity), but it is rational. How could you get this one andnot the other which is so similar?

The answer involves a careful treatment of divergences – naivelyone gets zero from the MHV diagrams, but due to a mismatch between 4 and D dimensions, one can derive the correct answer

Brandhuber, Spence, Travaglini hep-th/0612007

3.2 Quantum MHV theory

Quantum MHV Lagrangian

Idea: careful treatment of quantum light-cone gauge theory:Modify the classical Lagrangian correctly to reproduce physicalamplitudes

Need a suitable regularisation scheme: stay in four dimensionsand preserve the separation of the transverse light-cone degreesof freedom. This has been formulated recently*

The end-product: add suitable counterterms to the Lagrangian:only need -

Chakrabarti, Qiu, Thorn, hep-th/0602026

* could use dim reg: Ettle, Fu, Fudger, Mansfield, Morris, hep-th/0703286

3.2 Quantum MHV theory

Counterterms

Quantum light-cone YM Lagrangian – counterterms arefunctions of the gauge fields

These are simple expressions when written in terms ofdual momenta k :

(this is connected with planar graphs, double line notation and the string worldsheet picture)

3.2 Quantum MHV theory

Counterterms II

The ++ counterterm takes the simple form

In the quantum MHV Lagrangian we need to use B variables.We have

(certain functions of the momenta, condensed notation)

More explicitly,

with

3.2 Quantum MHV theory

Generally, one finds that

and the V’s turn out to be the missing all-plus vertices !

(non-trivial calculation: Brandhuber, Spence, Travaglini, Zoubos, hep-th 0704.0245)

Take the two point counterterm,

Expand A’s in powers of B fields; result at BBBB level:

Many manipulations later……this equals

This is precisely the four point ++++ amplitude

Counterterms III

3.2 Quantum MHV theory

Thus the simple counter-termis a generating function for the infinite series of all-plus vertices:

What about the other counterterms? The structure of thesesuggests:

n-point all-plus vertices, missing fromclassical MHV Lagrangian

Counterterms IV

3.2 Quantum MHV theory

Quantum MHV Lagrangian

Thus conjecture that:

Propagator and MHV vertices only (obtained from light-cone gauge YM using new variables)

Contains all-plus, single-minus vertices,plus other vertices needed to generate therational parts of amplitudes

Conjecture: This quantum theory is equivalent to quantum YM

3.2 Quantum MHV theory

MHV perturbation theory

The new Feynman-type rules: join the fundamentalvertices with propagators

Classical vertices: MHV M

Quantum vertices: AP SM -- (All-plus, single minus, double minus)

For example: one-loop MHV amplitude is given by

M M + M AP + --

cut-constructible part(known)

rational parts (new)

3. A New Approach: Summary

Gauge theory amplitudes localise on lines in twistor space –corresponds to MHV vertices in spacetime

The light-cone gauge YM Lagrangian, in suitable variables,is a theory with only MHV vertices

This classical Lagrangian is incomplete for the quantum theory –it misses amplitudes (eg all-plus) and parts of amplitudes (eg rational)

Some simple quantum counter-terms can/could account for these

MHV perturbation theory: an alternative to standard Feynman diagrams

Conclusions I

Perturbative gauge theory:

1. Old Tricks

-- Unitarity (pre 1970): not systematic, limited results

-- Feynman diagrams: systematic, but impractical (too many diagrams!)

However, the results are simple……

Conclusions II

2. New Tricks

-- simplicity of amplitudes explained by twistor space localisation

-- spacetime picture is MHV vertices; but no there is no derivation of these, can’t explain rational terms in amplitudes

-- other spin-offs from twistor string theory: -- recursion relations -- generalised unitarity

-- much progress, but a systematic approach needed

Conclusions III

3. A New Approach

MHV perturbation theory

-- classical MHV Lagrangian, plus

quantum counterterms

Claim: this is equivalent to quantum Yang-Mills

Evidence so far: -- classically equivalent -- non-rational parts of amplitudes reproduced -- all-plus amplitudes at one-loop reproduced -- structure is correct for the claim

Open Problems

Check it all really works: -- other amplitudes (eg single minus) -- rational terms (eg in MHV) -- two loops is it more efficient? apply it to fermions, scalars, massive theories (note: no conceptual obstacles)

Twistor picture: -- it incorporates MHV twistors -- it uses 4-d regularisation – good for twistors full twistor space realisation of Yang-Mills theory ?

And then there’s -- gravity -- holography -- integrability -- …………..

Goodbye Feynman diagrams:A new approach to perturbative quantum field theory

M M + M AP + --

MHV perturbation theory