# Bill Spence* Oxford April 2007

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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: 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

Feynman diagrams First course in Yang Mills quantum field theory:
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 , , etc. Eg: QCD – for gluons: (colour labels suppressed) Propagator 3-vertex

Feynman diagrams: the reality
1.1 Old tricks: Feynman Feynman diagrams: the reality But Feynman diagrams are impractical! Eg: Five gluon tree level scattering with Feynman diagrams: picture from Zvi Bern gg => n g n=7 n=8 n=9 Diagrams 559405

Feynman diagrams: end products
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

Feynman diagrams: end products II
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

Feynman diagrams: Summary
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: 1 2 3 4 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 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 cut A(s):

Unitarity methods II Loops from the old S-matrix approach:
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 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

Unitarity methods: summary
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/ 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

Amplitudes 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 Eg: 3 points  Z are collinear if Explicit check: twistor space coord’s and the above becomes a differential equation satisfied by the amplitude in spacetime: 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 negative helicity gluons localise on curves in CP(3,4) of degree and genus This: explains the localisation of YM amplitudes, gives a weak-weak duality between N=4 SYM and twistor string theory

Twistor string theory: Tree level
2.1 New tricks: Twistors Twistor string theory: Tree level In twistor space, tree level scattering amplitudes moduli space of curves degree d, genus 0 vertex operators (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 < - - +…+ > nMHV < …+ > nnMHV < > X

Twistor string theory:problems
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 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 Cachazo, Svrcek, Witten MHV nMHV nnMHV M (spacetime) (twistor space) M M M This works and gives a new, more efficient, way to calculate tree amplitudes

MHV methods: loops For tree amplitudes – spacetime MHV diagrams work M
2.2 New tricks: MHV MHV methods: loops For tree amplitudes – spacetime MHV diagrams work M (twistor space) (spacetime) M -- direct realisation of twistor space localisation Study of known one loop MHV amplitudes  twistor space localisation on pairs of lines x This suggests that in spacetime, one loop MHV amplitudes should be given by diagrams M

MHV methods: loops II ? M = MHV amplitude Technical issues:
2.2 New tricks: MHV MHV methods: loops II M = MHV amplitude ? Technical issues: 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 

MHV methods: loops III The result of this MHV diagram calculation is
2.2 New tricks: MHV MHV methods: loops III The result of this MHV diagram calculation is (Brandhuber, Spence, Travaglini hep-th/ ) 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 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, Brandhuber Spence, Travaglini Quigley Rozali Bedford, Brandhuber Spence, Travaglini Might MHV diagrams provide a completely new way to do perturbative gauge theory?

One loop:general result
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/ 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 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

MHV methods: Issues MHV methods: successes at tree level, one loop
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)

Recursion relations II
2.3 New tricks: Recursion Recursion relations II Recursion relations for tree amplitudes: = There are analogous relations at loop level – eg one loop QCD amplitude, recursion relations give decompositions like: loop Bern, Dixon, Kosower, hep-th/ 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

Generalised Unitarity
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

Generalised Unitarity II
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/ Triple cut Quadruple cut Result: Various integrals This is the correct QCD result

Generalised Unitarity III
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. MHV Perturbation Theory
3. A New Approach 3. MHV Perturbation Theory Recall MHV diagrams: combine MHV vertices to get amplitudes: M = MHV amplitude This works (at least at one loop in super YM) Idea: derive these rules from a Lagrangian

An MHV Lagrangian? What Lagrangian? Ingredients:
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 null reference vector This suggest some relation to light-cone gauge theory

Yang-Mills in light-cone gauge
3.1 Classical MHV theory Yang-Mills in light-cone gauge Pure Yang-Mills Light-cone gauge Leaves non-propagating, integrate out Result (non-local) OK, but how to get MHV vertices?

MHV Lagrangian I Yang-Mills in light-cone gauge Idea: Change variables
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/ *Mansfield hep-th/ Ettle, Morris hep-th/

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

Rational terms Previous slide: MHV diagrams generate amplitudes with
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)

Rational terms II So the classical MHV Lagrangian cannot explain the
3.1 Classical MHV theory Rational terms II So the classical MHV Lagrangian cannot explain the all-plus or single minus helicity amplitudes 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……

A Puzzle The Lagrangian, , obtained from light-cone gauge
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/

Quantum MHV Lagrangian
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/ The end-product: add suitable counterterms to the Lagrangian: only need - * could use dim reg: Ettle, Fu, Fudger, Mansfield, Morris, hep-th/

Counterterms Quantum light-cone YM Lagrangian – counterterms are
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)

Counterterms II The ++ counterterm takes the simple form
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 (certain functions of the momenta, condensed notation) with

Counterterms III Take the two point counterterm,
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 )

Counterterms IV Thus the simple counter-term
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:

Quantum MHV Lagrangian
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

MHV perturbation theory
3.2 Quantum MHV theory MHV perturbation theory The new Feynman-type rules: join the fundamental vertices 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 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: 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 + AP --

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