Transcript Goodbye Feynman diagrams: A new approach to perturbative

Goodbye Feynman diagrams:
A new approach to perturbative
quantum field theory
Bill Spence*
Oxford April 2007
Work in collaboration with A. Brandhuber, G. Travaglini, K. Zoubos,
arXiv: 0704.0245 hep-th, and earlier papers
*Centre for Research in String Theory, Queen Mary, University of London 2007
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
1.1 Old tricks: Feynman
Feynman diagrams
First course in Yang Mills quantum field theory:
Perturbative quantum corrections to classical amplitudes:
Use propagators
and interaction vertices
to form Feynman diagrams
,
Eg: QCD – for gluons:
(colour labels suppressed)
Propagator
3-vertex
,
etc.
1.1 Old tricks: Feynman
Feynman diagrams: the reality
But Feynman diagrams are impractical!
picture from Zvi Bern
Eg: Five gluon tree level scattering with Feynman diagrams:
gg => n g
n=7
n=8
n=9
Diagrams
559405
10525900
224449225
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 Maximal Helicity Violating:
as tree amplitudes with
all, or all but one,
helicity the same are zero
Notation
null momenta p,
written with spinors
i 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
-- simple rules, Lagrangian derivation, work for all
theories
But, the practice:
-- diagrams are cumbersome – multiply rapidly and
become impractical
However:
-- 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:
1
3
2
4
● Scattering depends on the Lorentz invariants (s,t):
● Consider A(s,t), at fixed t, in the complex plane. There are poles
at s = 4m^2, 9m^2,… (production of particles). In fact there
is a branch cut from s=4m^2 to infinity (and also one along the
negative s axis due to poles in the t-channel)
s
A(s):
cut
cut
1.2 Old tricks: unitarity
Unitarity methods II
Loops from the old S-matrix approach:
● Consider the contour integral of the amplitude A(s), around C:
s
cut
cut
C
● This gives
● Then, using
Idea: reconstruct amplitudes from their analytic properties
1.2 Old tricks: unitarity
New unitarity methods
From c. 1990: New application of unitarity methods
Bern, 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 loop
Two loop results (4 point function N=4)
Others (nnMHV,…)
● But – nnMHV – difficult
higher loops – difficult
…reaching the limits of this approach by the early 2000’s
But proving difficult to progress further
1.2 Old tricks: unitarity
Unitarity 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 very
simple result (eg MHV) – why???
Unitarity methods
Use dispersion relations – but no systematic way
found to generate these in general, and applications
to higher loops (>1), massive theories, etc, proved
difficult
Need some New Tricks………. 
2.1 New tricks: Twistors
Twistor string theory
Witten hep-th/0312171
Amplitudes in spinor variables can be simple: eg MHV
Idea: Look at amplitudes in twistor space
 twistor space coordinates
(
= Fourier transform of
)
Then:
ie MHV tree amplitudes localise on a line in twistor space
2.1 New tricks: Twistors
Amplitudes in twistor space
Localisation of tree amplitudes in twistor space appears generic:
Eg:
MHV
< - - ++…++ > localise on a line
next to MHV < - - - ++….++ > localise on two intersecting lines
Explicit check:
twistor space coord’s
Eg: 3 points 
Z are collinear if
in spacetime:
and the above becomes a differential
equation satisfied by the amplitude
Loop level: also get localisation – see later
What can explain this localisation ?
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)
Can then argue that:
-loop N=4 super YM amplitudes with
on curves in CP(3,4) of degree
negative helicity gluons localise
and genus
This:
- explains the localisation of YM amplitudes,
- gives a weak-weak duality between N=4 SYM and twistor string theory
2.1 New tricks: Twistors
Twistor string theory: Tree level
In twistor space, tree level scattering amplitudes
vertex operators
moduli space of curves degree d, genus 0
(degree d  (d+1) negative helicity gluons)
A surprise: due to delta functions, the integral localises on intersections of
degree one curves:
Curve
Amplitude
MHV
X
< - - +…+ >
< - - - +…+ >
X
X
X
nnMHV < - - - - +..+ >
X
X
X
nMHV
X
X
X
X
X
2.1 New tricks: Twistors
Twistor string theory:problems
Twistor string theory: beautiful new duality between N=4
super Yang-Mills and a topological string theory, but:
Hard to calculate with it – integrals over moduli spaces of
curves 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 pure
Yang-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
M
localise on a line in twistor
space (~ point in spacetime), think of them as fundamental vertices.
Cachazo, Svrcek, Witten
Join them with scalar propagators to generate other tree amplitudes:
M
MHV
(spacetime)
(twistor space)
nMHV
M
M
nnMHV
M
M
M
This works and gives a new, more efficient, way to calculate tree amplitudes
2.2 New tricks: MHV
MHV methods: loops
For tree amplitudes – spacetime MHV diagrams work
M
(spacetime)
(twistor space)
M
M
M
M
-- direct realisation of twistor space localisation
Study of known one loop MHV amplitudes  twistor space localisation on
pairs of lines
x
x
x
x
This suggests that in spacetime, one loop MHV amplitudes should be given
by diagrams
M
M
2.2 New tricks: MHV
MHV methods: loops II
M
Technical issues:
M
?
= MHV amplitude
The particle in the loop is off-shell. But
particles in MHV diagrams are on-shell
 need an off-shell prescription
Coordinates
null
vector
null
reference
vector
-- Result should be independent of reference vector;
-- Use dimensional regularisation of momentum integrals
Then: multiply MHV expressions, simplify spinor algebra, perform
phase space (l) and dispersion (z) integrals.....non-trivial calculation
Result 
2.2 New tricks: MHV
MHV methods: loops III
The result of this MHV diagram calculation is
(Brandhuber, Spence, Travaglini hep-th/0407214)
The known answer is
These agree, due to the nine-dilogarithm identity
2.2 New tricks: MHV
MHV diagrams: N<4
So – spacetime MHV diagrams give one loop N=4 MHV
amplitudes
a surprise - no conformal supergravity as expected from twistor string theory
Bedford, Brandhuber
Remarkably: MHV diagrams give correct results for
Spence, Travaglini
Quigley Rozali
-- 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
Might MHV diagrams provide a completely new way to do
perturbative gauge theory?
Bedford, Brandhuber
Spence, Travaglini
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:
successes at tree level, one loop
can be thought of as a consistent formulation of dispersion integrals
But,
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
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)
2.3 New tricks: Recursion
Recursion relations II
Recursion relations for tree amplitudes:
●
=
●
●
There are analogous relations at loop level – eg
QCD amplitude, recursion relations give decompositions like:
loop
one loop
Bern, Dixon, Kosower,
hep-th/0507005
tree
This allows one to reconstruct (parts of, in general) amplitudes from
simpler pieces – this is a useful tool, but it is hard to apply at loop
level systematically
2.4 New tricks: Generalised
Unitarity
Generalised Unitarity
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
In d-dimensions, the discontinuities should also determine these rational terms
2.4 New tricks: Generalised
Unitarity
Generalised Unitarity II
d-dimensional unitarity should give the full amplitudes
New techniques with multiple cuts developed (see reviews for references)
eg: QCD: multiple cuts in d-dimensions – 4-point case
Brandhuber, McNamara, Spence, Travaglini
hep-th/0506068
Quadruple
cut
Triple cut
Result:
Various integrals
This is the correct QCD result
2.4 New tricks: Generalised
Unitarity
Generalised Unitarity III
Generalised unitarity – multiple cuts, and d-dimensional
cuts
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:
M
M
= MHV amplitude
This works (at least at one loop in super YM)
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 )
null
vector
This suggest some relation to light-cone gauge theory
null
reference
vector
3.1 Classical MHV
theory
Yang-Mills in light-cone gauge
Pure Yang-Mills
Light-cone gauge
Leaves
Result (non-local)
OK, but how to get MHV vertices?
non-propagating,
integrate out
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/0511264
Ettle, 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 perturbation
theory 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 with
at 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
So the classical MHV Lagrangian
all-plus or single minus helicity amplitudes
cannot explain the
Also, while
gives graphs with at least two negative
helicity particles, it does not give the rational parts of other
amplitudes [known from explicit calculations]
Something is missing……
3.1 Classical MHV theory
A Puzzle
The Lagrangian,
, obtained from light-cone gauge
YM theory using new variables, does not generate rational terms
in quantum amplitudes. For example, the (++++) one-loop, which
is entirely rational:
But what about the all-minus amplitudes? , eg:
This could be generated from MHV diagrams (it has more than one
negative helicity), but it is rational. How could you get this one and
not the other which is so similar?
The answer involves a careful treatment of divergences – naively
one 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 physical
amplitudes
Need a suitable regularisation scheme: stay in four dimensions
and preserve the separation of the transverse light-cone degrees
of freedom. This has been formulated recently*
Chakrabarti, Qiu, Thorn, hep-th/0602026
The end-product: add suitable counterterms to the Lagrangian:
only need -
* could use dim reg: Ettle, Fu, Fudger, Mansfield, Morris, hep-th/0703286
3.2 Quantum MHV theory
Counterterms
Quantum light-cone YM Lagrangian – counterterms are
functions of the gauge fields
These are simple expressions when written in terms of
dual 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
More explicitly,
In the quantum MHV Lagrangian we need to use B variables.
We have
with
(certain functions of the momenta,
condensed notation)
3.2 Quantum MHV theory
Counterterms III
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
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)
3.2 Quantum MHV theory
Counterterms IV
Thus the simple counter-term
is a generating function for the infinite series of all-plus vertices:
n-point all-plus vertices, missing from
classical MHV Lagrangian
What about the other counterterms? The structure of these
suggests:
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 the
rational 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 fundamental
vertices with propagators
Classical vertices: MHV
Quantum vertices:
M
AP
SM
--
(All-plus, single minus, double minus)
For example: one-loop MHV amplitude is given by
M
M
cut-constructible part
(known)
+
M
AP
+
--
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
MHV perturbation theory
M
M + M
AP + --