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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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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)
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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 3
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 4
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 5
• 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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 6
• 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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 7
• 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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 8
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 9
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 10
Spinor products
Use antisymmetricspinor products:
Instead of Lorentz products:
These are complex square roots of Lorentz products:
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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 11
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 12
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 13
• Apply this shift to the Parke-Taylor (MHV) amplitudes:
• Under the shift:
• So
• Consider:
• 2 poles, opposite residues
MHV example
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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 14
• MHV amplitude obeys:
• Compute residue using factorization• At
kinematics are complex collinear:
• so
MHV example (cont.)
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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 15
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 16
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 17
To show:
Propagators:
Britto, Cachazo, Feng, Witten, hep-th/0501052
3-point vertices:
Polarization vectors:
Total:
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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 18
Initial dataParke-Taylor formula
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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 19
A 6-gluon example
220 Feynman diagrams for gggggg
Helicity + color + MHV results + symmetries
3 recursive diagrams
related by symmetry
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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 20
Simpler than form found in 1980s
Mangano, Parke, Xu (1988)
Simple final form
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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 21
Berends, Giele, Kuijf (1990)
Relative simplicity grows with n
Bern, Del Duca, LD, Kosower (2004)
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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 22
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 23
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 24
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 25
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 26
• 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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 27
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 28
• 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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 29
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 30
• 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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 31
• 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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 32
• 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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 33
• 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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 34
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 35
Extra Slides
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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 36
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 37
• Using
one confirms
MHV check
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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 38
A one-loop pole analysis
Bern, LD, Kosower (1993)
under shift plus partial fraction
???
double pole
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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 39
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 40
“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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 41
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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June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 42
Solution to recursion relation
![Page 43: Bootstrapping One-loop QCD Scattering Amplitudes Lance Dixon, SLAC Fermilab Theory Seminar June 8, 2006 Z. Bern, LD, D. Kosower, hep-th/0501240, hep-ph/0505055,](https://reader037.vdocuments.site/reader037/viewer/2022102818/56649eb75503460f94bc10db/html5/thumbnails/43.jpg)
June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 43
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)
![Page 44: Bootstrapping One-loop QCD Scattering Amplitudes Lance Dixon, SLAC Fermilab Theory Seminar June 8, 2006 Z. Bern, LD, D. Kosower, hep-th/0501240, hep-ph/0505055,](https://reader037.vdocuments.site/reader037/viewer/2022102818/56649eb75503460f94bc10db/html5/thumbnails/44.jpg)
June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 44
Fermionic solutions
and
![Page 45: Bootstrapping One-loop QCD Scattering Amplitudes Lance Dixon, SLAC Fermilab Theory Seminar June 8, 2006 Z. Bern, LD, D. Kosower, hep-th/0501240, hep-ph/0505055,](https://reader037.vdocuments.site/reader037/viewer/2022102818/56649eb75503460f94bc10db/html5/thumbnails/45.jpg)
June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 45
March of the n-gluon helicity amplitudes
![Page 46: Bootstrapping One-loop QCD Scattering Amplitudes Lance Dixon, SLAC Fermilab Theory Seminar June 8, 2006 Z. Bern, LD, D. Kosower, hep-th/0501240, hep-ph/0505055,](https://reader037.vdocuments.site/reader037/viewer/2022102818/56649eb75503460f94bc10db/html5/thumbnails/46.jpg)
June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 46
March of the tree amplitudes
![Page 47: Bootstrapping One-loop QCD Scattering Amplitudes Lance Dixon, SLAC Fermilab Theory Seminar June 8, 2006 Z. Bern, LD, D. Kosower, hep-th/0501240, hep-ph/0505055,](https://reader037.vdocuments.site/reader037/viewer/2022102818/56649eb75503460f94bc10db/html5/thumbnails/47.jpg)
June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 47
March of the 1-loop amplitudes
![Page 48: Bootstrapping One-loop QCD Scattering Amplitudes Lance Dixon, SLAC Fermilab Theory Seminar June 8, 2006 Z. Bern, LD, D. Kosower, hep-th/0501240, hep-ph/0505055,](https://reader037.vdocuments.site/reader037/viewer/2022102818/56649eb75503460f94bc10db/html5/thumbnails/48.jpg)
June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 48
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!
![Page 49: Bootstrapping One-loop QCD Scattering Amplitudes Lance Dixon, SLAC Fermilab Theory Seminar June 8, 2006 Z. Bern, LD, D. Kosower, hep-th/0501240, hep-ph/0505055,](https://reader037.vdocuments.site/reader037/viewer/2022102818/56649eb75503460f94bc10db/html5/thumbnails/49.jpg)
June 8, 2006 L. Dixon Bootstrapping QCD Amplitudes 49
Why does it all work?
In mathematics you don't understand things. You just get used to them.