On-Shell Recursion Relations for Q C D Loop Amplitudes (Part I)

On-Shell Recursion Relationsfor QCD Loop Amplitudes

(Part I)

Lance Dixon, SLAC“From Twistors to Amplitudes” Workshop

Queen Mary U. of LondonNovember 3, 2005

Z. Bern, LD, D. Kosower, hep-th/0501240, hep-ph/0505055, hep-ph/0507005; C. Berger, Z. Bern, LD, D. Forde, D. Kosower, to appear

Nov. 3, 2005 L. Dixon On-Shell Recursion Relations at One Loop 2

• Need a flexible, efficient method to extend the range of known tree, and particularly 1-loop QCD amplitudes, for use in NLO corrections to LHC processes, etc.

• Unitarity is an efficient method for determining imaginary parts of loop amplitudes:

• Efficient because it recycles trees into loops

Motivation


• Unitarity can miss rational functions that have no cut.• These functions can be recovered using dimensional analysis,

computing cuts to higher order in , in dim. reg. with D=4-2:

Bern, LD, Kosower, hep-ph/9602280;

Brandhuber, McNamara, Spence, Travaglini, hep-th/0506068

• But tree amplitudes are more complicated in D=4-2 than in D=4. • Seems too much information is used: • n-point loop amplitudes also factorize onto lower-point amplitudes.• At tree-level these data have been systematized into

on-shell recursion relations Britto, Cachazo, Feng, hep-th/0412308;

Britto, Cachazo, Feng, Witten, hep-th/0501052

• Efficient – recycles trees into trees• Can also do the same for loops

Motivation (cont.)


On-shell tree recursion

• BCFW consider a family of on-shell amplitudes An(z) depending on a complex parameter z which shifts the momenta, described using spinor variables.

• For example, the (n,1) shift:

• Maintains on-shell condition,

and momentum conservation,


• Apply this shift to the Parke-Taylor (MHV) amplitudes:

• Under the (n,1) shift:

• So

• Consider:

• 2 poles, opposite residues

MHV example


• MHV amplitude obeys:

• Compute residue using factorization• At

kinematics are complex collinear:

• so

MHV example (cont.)


The general case

Ak+1 and An-k+1 are on-shell tree amplitudes with fewer legs,evaluated with 2 momenta shifted by a complex amount.

Britto, Cachazo, Feng, hep-th/0412308

In kth term:

which solves


Proof of on-shell recursion relations

Same analysis as above – Cauchy’s theorem + amplitude factorization


Let complex momentum shift depend on z. Use analyticity in z.

Cauchy:

poles in z: physical factorizations residue at = [kth term]


On-shell recursion at one loop Bern, LD, Kosower, hep-th/0501240, hep-th/0505055

• Same techniques can be used to compute one-loop amplitudes – which are much harder to obtain by other methods than are trees.

• However, new features still arise, compared with tree case, due to different collinear behavior of loop amplitudes:

but

• First consider special tree-like one-loop amplitudes with no cuts, only poles: • Again we are trying to determine a rational function of z.


A one-loop pole analysis

Bern, LD, Kosower (1993)

under shift plus partial fraction

???

double pole


The double pole diagram

To account for double pole in z, we use a doubled propagator factor (s23).

For the “all-plus” loop 3-vertex, we use the symmetric function,

In the limit of real collinear momenta,

this vertex corresponds to the 1-loop splitting amplitude, BDDK (1994)

Want to produce:


“Unreal” pole underneath the double pole

Missing term should be related to double-pole diagram, but suppressed by factor S which includes s23

Want to produce:

Don’t know collinear behaviorat this level, must guess thecorrect suppression factor:

in terms of universal eikonal factors for soft gluon emission

Here, multiplying double-pole diagram bygives correct missing term! Universality??

nonsingular in real Minkowski kinematics


A one-loop all-n recursion relation

Same suppression factor works in the case of n external legs!

Know it works because results agree with Mahlon, hep-ph/9312276,though much shorter formulae are obtained from this relation

shift leads to


Solution to recursion relation


External fermions too

Can similarly write down recursion relationsfor the finite, cut-free amplitudes with 2 external fermions:

and the solutions are just as compact

Gives the complete set of finite, cut-free, QCD loop amplitudes(at 2 loops or more, all helicity amplitudes have cuts, diverge)


Fermionic solutions

and


Loop amplitudes with cuts

• Can also extend same recursive technique (combined with unitarity) to loop amplitudes with cuts (hep-ph/0507005)

• Here rational-function terms contain “spurious singularities”, e.g.

• accounting for them properly yields simple “overlap diagrams” in addition to recursive diagrams• No loop integrals required to bootstrap the rational

functions from the cuts and lower-point amplitudes• Tested method on 5-point amplitudes, used it to compute

and more recently Forde, Kosower, hep-ph/0509358


Generic analytic properties of shifted 1-loop amplitude,

Loop amplitudes with cuts (cont.)

Cuts and poles in z-plane:

But if we know the cuts (via unitarity in D=4),we can subtract them:

full amplitude cut-containing partrational part

Shifted rational function

has no cuts, but has spurious poles in z because of Cn:

• See also Part II (David Kosower’s talk Saturday)


However, we know how to “complete the cuts” at z=0 to cancel the spurious pole terms, using Li(r) functions


So we do a different subtraction:

full amplitude completed-cut partmodified rational part

New shifted rational function

has no cuts, and no spurious poles.But the residues of the physical polesare not given by naïve factorization onto(the rational parts of) lower-point amplitudes,due to the rational parts of the completed-cutterms,



So we need a correction term from the residue of at each physical pole – which we call an overlap diagram

full amplitude

completed-cut part

recursive diagrams overlap diagrams

The final result:


• It is possible that does not vanish at .• It could even blow up there.• Even if it is well-behaved, might not be.• In that case, also won’t be.

Subtleties at Infinity

• As long as we know (or suspect) the behavior of we can account for it, by performing yet another(rational) modification to , and hence to ,and then using the master formula again. ( is required to cancel spurious singularities, but there is freedom to use it also to account for behavior).


• David Kosower will discuss the application of this formalismto the “MHV” QCD loop amplitudes,

An NMHV QCD Loop Amplitude

• Here I describe the application to an NMHV QCD loop amplitude,

• The general “split-helicity” case

works identically to this case

Berger, Bern, LD, Forde, Kosower, to appear


• The cut-containing terms for the general “split-helicity” case

have been computed recently.

Bern, Bjerrum-Bohr, Dunbar, Itahep-ph/0507019

NMHV QCD Loop Amplitude (cont.)

• and are cut-constructible,and the scalar loop contribution is:

generate (most of)

to be determined


What shift to use?• Problem: we don’t yet understand this loop 3-vertex

• But we know “experimentally” that it vanishes for the complex-conjugate kinematics

So the shift

will avoid these vertices


• Inspecting

Behavior at Infinityfor n=5 Bern, LD, Kosower (1993)

we find that the rational terms diverge but in a very simple way:

which can be mimicked by:

Add this term to definition of

Total reproduces

Compute recursive + overlap diagrams + boundary correction:


• For general n

Behavior at Infinity (cont.)we assume that the rational termsdiverge in the analogous way

Add this term to definition of . Compute recursive + overlap diagrams + boundary correction. Obtain consistent amplitudes.

which can be mimicked by:


• Extra rational terms, beyond L2 terms from

Result for n=6

where flip1 permutes:

Bern, Bjerrum-Bohr, Dunbar, Itahep-ph/0507019

Result also manifestly symmetric under Plus correct factorization limits


Conclusions

• On-shell recursion relations can be extended fruitfully to determine rational parts of loop amplitudes – with a bit of guesswork, but there are lots of consistency checks.

• Method still very efficient; compact solutions found for all finite, cut-free loop amplitudes in QCD• Recently extended same technique (combined with D=4

unitarity) to some of the more general loop amplitudes with cuts, MHV and NMHV, which are needed for NLO corrections to LHC processes

• Prospects look very good for attacking a wide range of multi-parton processes in this way


Why does it all work?

In mathematics you don't understand things. You just get used to them.


Extra Slides


Revenge of the Analytic S-matrix?

• Branch cuts

• Poles

Reconstruct scattering amplitudes directly from analytic properties

Chew, Mandelstam; Eden, Landshoff, Olive, Polkinghorne;

… (1960s)

Analyticity fell out of favor in 1970s with rise of QCD;to resurrect it for computing perturbative QCD amplitudesseems deliciously ironic!


March of the n-gluon helicity amplitudes


March of the tree amplitudes


March of the 1-loop amplitudes


• Using

one confirms

MHV check


To show:

Propagators:


3-point vertices:

Polarization vectors:

Total:


Initial dataParke-Taylor formula

Documents

On-Shell Recursion Relations for Q C D Loop Amplitudes (Part I)