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
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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)
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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
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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.
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• 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.)
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• 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
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• 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
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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)
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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
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Spinor products
Use antisymmetricspinor products:
Instead of Lorentz products:
These are complex square roots of Lorentz products:
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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):
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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,
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• Apply this shift to the Parke-Taylor (MHV) amplitudes:
• Under the shift:
• So
• Consider:
• 2 poles, opposite residues
MHV example
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• MHV amplitude obeys:
• Compute residue using factorization• At
kinematics are complex collinear:
• so
MHV example (cont.)
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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
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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]
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To show:
Propagators:
Britto, Cachazo, Feng, Witten, hep-th/0501052
3-point vertices:
Polarization vectors:
Total:
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Initial dataParke-Taylor formula
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A 6-gluon example
220 Feynman diagrams for gggggg
Helicity + color + MHV results + symmetries
3 recursive diagrams
related by symmetry
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Simpler than form found in 1980s
Mangano, Parke, Xu (1988)
Simple final form
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Berends, Giele, Kuijf (1990)
Relative simplicity grows with n
Bern, Del Duca, LD, Kosower (2004)
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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
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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:
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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
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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
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• 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:
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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
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• 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
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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
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• 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 ,
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• 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.
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• 4 nonvanishing recursive diagrams Rn
n=6
where flip1 permutes:
• 2 nonvanishing overlap diagrams On
Compared with 1034 1-loop Feynman diagrams (color-ordered)
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• 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
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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
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Extra Slides
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Example of new diagrams
recursive:
overlap:
For rational part of
Compared with 1034 1-loop Feynman diagrams (color-ordered)
7 in all
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• Using
one confirms
MHV check
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A one-loop pole analysis
Bern, LD, Kosower (1993)
under shift plus partial fraction
???
double pole
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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:
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“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
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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
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Solution to recursion relation
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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)
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Fermionic solutions
and
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March of the n-gluon helicity amplitudes
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March of the tree amplitudes
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March of the 1-loop amplitudes
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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!
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Why does it all work?
In mathematics you don't understand things. You just get used to them.