Post on 11-Jan-2016
description
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