Bootstrapping One-loop QCD Scattering Amplitudes Lance Dixon, SLAC Fermilab Theory Seminar June 8,...

Bootstrapping One-loop QCD Scattering Amplitudes

Lance Dixon, SLAC Fermilab Theory Seminar

June 8, 2006Z. Bern, LD, D. Kosower, hep-th/0501240, hep-ph/0505055, hep-ph/0507005; C. Berger, Z. Bern, LD, D. Forde, D. Kosower, hep-ph/0604195

June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 2

What’s a bootstrap?

Perturbation theory makes a bootstrap practical in four dimensions,because it imposes a hierarchy on S-matrix elements.

Build more complicated amplitudes (more loops, more legs) directly from simpler ones, without directly using Feynman diagrams.

Very general consistency criteria:• Cuts (unitarity)• Poles (factorization)


Motivation: One-loop multi-leg amplitudes for Tevatron/LHC

• Leading-order (LO), tree-level predictions are only qualitative, due to poor convergence of

expansion in strong coupling s() ~ 0.1• NLO corrections can be 30% - 80% of LO

state of the art:

LO = |tree|2

n=8

NLO = loop x tree* + …

n=3

NNLO = 2-loop x tree* + …

n=2


LHC Example: SUSY Search

Early ATLAS TDR studies using PYTHIA overly optimistic

• ALPGEN based on LO amplitudes, much better than PYTHIA at modeling hard jets• What will disagreement between ALPGEN and data mean? • Hard to tell because of potentially large NLO corrections

Gianotti & Mangano, hep-ph/0504221Mangano et al. (2002)

• Search for missing energy + jets.• SM background from Z + jets.


• 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.

• Semi-numerical methods have led to some progress recently, e.g. – Higgs + 4 parton amplitudes

Ellis, Giele, Zanderighi, hep-ph/0506196, 0508308

– 6-gluon amplitudes

Ellis, Giele, Zanderighi, hep-ph/0602185

• Here discuss a more analytical approach.• Anticipate faster evaluations this way, for processes

amenable to this method.

Motivation (cont.)


• Unitarity efficient for determining imaginary parts of loop amplitudes:

• Efficient because it recycles

simple trees into loops• Generalized unitarity (more propagators open)

for coefficients of box and triangle integrals• Cut evaluation via residue extraction (algebraic)

Bootstrapping with cuts

Bern, LD, Kosower, hep-ph/9403226, hep-ph/9708239;Britto, Cachazo, Feng, hep-th/0412103; BCF + Buchbinder, hep-ph/0503132;Britto, Feng, Mastrolia, hep-ph/0602178


• Unitarity can miss rational functions that have no cut.• However, n-point loop amplitudes also have poles where

they 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

Bootstrapping with poles


The right variables

Scattering amplitudes for massless plane waves of definite 4-momentum: Lorentz vectors ki

ki2=0

Natural to use Lorentz-invariant products (invariant masses):

But for particles with spinthere is a better way

massless q,g,all have 2 helicities

Take “square root” of 4-vectors ki(spin 1)

use 2-component Dirac (Weyl) spinors u(ki) (spin ½)


The right variables (cont.)Reconstruct momenta ki

from spinors

using projector onto positive-energy solutions of Dirac eq.:

Singular 2 x 2 matrix:

also shows

even for complex momenta


Spinor products

Use antisymmetricspinor products:

Instead of Lorentz products:

These are complex square roots of Lorentz products:


Spinor Magic

Spinor products precisely capture square-root + phase behavior in collinear limit. Excellent variables for helicity amplitudes

scalars0

gauge theoryangular momentum mismatch

Accounts for denominators in helicity amplitudes, e.g. Parke-Taylor (MHV):


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 shift:

• Maintains on-shell condition,

and momentum conservation,


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

• Under the 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

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

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

Cauchy:

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


To show:

Propagators:

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

3-point vertices:

Polarization vectors:

Total:


Initial dataParke-Taylor formula


A 6-gluon example

220 Feynman diagrams for gggggg

Helicity + color + MHV results + symmetries

3 recursive diagrams

related by symmetry


Simpler than form found in 1980s

Mangano, Parke, Xu (1988)

Simple final form


Berends, Giele, Kuijf (1990)

Relative simplicity grows with n

Bern, Del Duca, LD, Kosower (2004)


On-shell recursion at one loop Bern, LD, Kosower, hep-th/0501240, hep-ph/0505055, hep-ph/0507005;C. Berger, Z. Bern, LD, D. Forde, D. Kosower, hep-ph/0604195

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

• However, 3 new features arise, compared with tree case:

but

2) different collinear behavior of loop amplitudes leads to double poles in z, uncertainty about residues in some cases:

1) A(z) typically has cuts as well as poles

3) behavior of A(z) at large z more difficult to determine


Generic analytic behavior of shifted 1-loop amplitude,

Loop amplitudes with cuts

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 how logs, etc., appear in Cn:


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

Cancelling spurious poles

So we do a modified subtraction:

full amplitude completed-cut partmodified rational part

New shifted rational function

has no cuts, and no spurious poles.But residues of physical poles are not given by naïve factorization onto rational parts of lower-point amplitudes, due to the rational parts of the completed-cut terms, called


Loop amplitudes with cuts (cont.)

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

full amplitude

completed-cut part

recursive diagrams overlap diagrams On

The final result:

• Tested method on known 5-point amplitudes, used it to compute • then all adjacent-MHV: Forde, Kosower,

hep-ph/0509358


• 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 the same contour analysis on

is defined such that its large z behavior matches • Leads to modified recursive + overlap formula:


Example: NMHV Loop Amplitudes

• We have determined recursively all the “split-helicity” next-to-maximally-helicity-violating (NMHV) QCD loop amplitudes, i.e. those with three adjacent negative helicities: C. Berger, Z. Bern, LD,

D. Forde, D. Kosower, hep-ph/0604195

As input to the recursion relation, use (rational parts of) MHVamplitudes Forde, Kosower, hep-ph/0509358

• Key issue is to determine the behavior of


• 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 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:

Behavior mimicked by:

Total reproduces

We compute recursive + overlap diagrams + corrections from ,


• For general n

Behavior at Infinity (cont.)Can use a second recursion relation, obtained by shifting to determine how the rational terms behave at large z. Find:

which can be mimicked by:

Compute recursive + overlap diagrams + corrections from , Obtain consistent amplitudes.


• 4 nonvanishing recursive diagrams Rn

n=6

where flip1 permutes:

• 2 nonvanishing overlap diagrams On

Compared with 1034 1-loop Feynman diagrams (color-ordered)


• 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• Same technique (combined with D=4 unitarity) gives

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


Extra Slides


Example of new diagrams

recursive:

overlap:

For rational part of

Compared with 1034 1-loop Feynman diagrams (color-ordered)

7 in all


• Using

one confirms

MHV check


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


March of the n-gluon helicity amplitudes


March of the tree amplitudes


March of the 1-loop amplitudes


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!


Why does it all work?

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

Bootstrapping One-loop QCD Scattering Amplitudes Lance Dixon, SLAC Fermilab Theory Seminar June 8,...

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