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## Automated Computation of One-Loop Amplitudes

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Darren Forde### Automated Computation of One-Loop Amplitudes

(SLAC & UCLA)

Work in collaboration with C. Berger, Z. Bern, L. Dixon, F. Febres Cordero,

H. Ita, D. Kosower, D. Maître.

arxiv: 0803.4180 [hep-ph] & arxiv: 0808.0941 [hep-ph]

- QCD corrections to vector boson pair production (W+W-, W±Z & ZZ) via vector boson fusion (VBF). (Jager, Oleari, Zeppenfeld)+(Bozzi)
- QCD and EW corrections to Higgs production via VBF.(Ciccolini, Denner, Dittmaier)
- pp → Higgs+2jets.(via gluon fusion Campbell, Ellis, Zanderighi), (via weak interactions Ciccolini, Denner, Dittmaier).pp → Higgs+3jets (leading contribution)(Figy, Hankele, Zeppenfeld).
- pp→ .(Beenakker, Dittmaier, Krämer, Plümper, Spira, Zerwas), (Dawson, Jackson, Reina, Wackeroth)
- pp → ZZZ, (Lazopoulos, Petriello, Melnikov) pp → +(McElmurry)
- pp → WWZ, WWW (Hankele, Zeppenfeld, Campanario, Oleari, Prestel)
- pp→WW+j+X.(Campbell, Ellis, Zanderighi). (Dittmaier, Kallweit, Uwer)
- pp →W/Z (Febres Cordero, Reina, Wackeroth),
- pp → +jet (Dittmaier,Uwer,Weinzierl),
- (Bredenstein,Denner,Dittmaier,Pozzorini),

New Techniques

+

+

+

i-

An

An

=0

±

+

- Calculating using Feynman diagrams is Hard!
- A Factorial growth in the number of terms, particularly bad for large number of gluons.
- Calculated amplitudes simpler than expected.
- For example, tree level all-multiplicity gluon amplitudes

+

+

+

j-

Gauge dependant

quantities,

large cancellations

between terms.

Want to use on-shell

quantities only.

Recent progress using Unitarity and related techniques,

- gg → gggg amplitude.(Bern,Dixon,Kosower), (Britto,Feng,Mastrolia), (Bern,Bjerrum-Bohr,Dunbar,Ita), (Berger,Bern,Dixon,DF,Kosower), (Bedford,Brandhuber,Spence,Travaglini) (Xiao,Yang,Zhu) ,(Berger,Bern,Dixon,DF,Kosower), (Giele,Kunszt,Melnikov)
- Lots of gluons (Giele,Zanderighi), (Berger, Bern, Dixon, Febres Cordero, DF, Ita, Kosower, Maître)
- 6 photons (Nagy, Soper), (Ossola, Papadopoulos, Pittau), (Binoth, Heinrich,Gehrmann, Mastrolia)
- pp → ZZZ,WZZ, WWZ, ZZZ (Binoth, Ossola, Papadopoulos, Pittau),
- gg → using D-Dimensional Unitarity (Ellis,Giele,Kunszt,Melnikov)

Automation

- Large number of processes to calculate (for the LHC),
- Automatic procedure highly desirable.
- We want to go from
- Implement new methods numerically.

An(1-,2-,3+,…,n+), An(1-,2-,3-,..

An(1-,2-,3+,…,n+)

Towards Automation

- Two main approaches. Feynman diagrams and Unitarity (and related techniques).
- Unitarity,
- 4-Dimensional - Compute cut terms only.
- On-shell recursion relations for rational terms.(Berger, Bern, Dixon, DF, Kosower)
- D-Dimensional – Compute rational and cut terms
- Solve for coefficients of powers of the loop momentum in the integrand (Ossola, Papadopoulos, Pittau), (Mastrolia, Ossola, Papadopoulos, Pittau), (Ellis, Giele, Kunszt).(Giele, Kunszt, Melnikov)
- General formula for integral coefficients. (Anastasiou, Britto, Feng, Kunszt, Mastrolia) (Britto, Feng) + (Mastrolia) + (Yang)

“CutTools” (Ossola,Papadopoulos,Pittau)+Improvments (Mastrolia,Ossola,Papadopoulos,Pittau)

- Numerical implementation of OPP.
- “Rocket” (Giele,Zanderighi)
- Uses alternative implementation of OPP for cuts and D-Dimensional Unitarity for the rational terms . Applied to all gluon processes.
- “Golem” (Golem95)(Binoth, Guffanti, Guillett, Heinrich, Karg, Kauer, Pilon, Reiter)
- Uses Feynman diagrams.
- “BlackHat” (Berger, Bern, Dixon, Febres Cordero, DF, Ita, Kosower, Maître)
- Uses On-shell recursion relations and 4 dimensional unitarity.
- Applied to gluon, quark and Electro-Weak processes.

(Berger, Bern, Dixon, Febres Cordero,

DF, Ita, Kosower, Maître)

BlackHat“Compact” On-shell inputs

Rational building blocks

- Numerical implementation of the unitarity bootstrap approach in c++,

Much fewer terms to compute

& no large cancelations compared

with Feynman diagrams.

A<n

Rn

The unitarity bootstrap- Use the most efficient approach for each piece,

Unitarity cuts in 4

dimensions

K3

On-shell recurrence relations

A3

A2

A1

K1

K2

Recycle results of amplitudes with fewer legs

Pole at -<23>/<13>

Amplitudes and the Complex Plane- An amplitude is a function of its external momenta (and helicity)
- Shift the momentum of two external legs by a complex variable z, [Britto, Cachazo, Feng, Witten]
- Keeps both kiand kjon-shell.
- Conserves momentum in the amplitude.
- For example

Only possible with

Complex momenta.

pole

A<n

A<n

An

A simple idea- Function of a complex variable containing only simple poles
- Position of all poles from complex factorisation properties of the amplitude.

An(0), the amplitude with real momentum

Poles from here

A<n

Loops & Branch cutsz

- More complex structure on the complex plane. (Bern, Dixon, Kosower)+(Berger, DF)
- Shift the amplitude in the same way

Integrate over a

circle at infinity

Poles

A<n

A<n

+

Generalised Unitarity

techniques

On-shell

recurrence relation

Branch cuts

Loop On-shell recursion relations

T

T

- At one-loop recursion using on-shelltree amplitudes, T, and rational pieces of one-loop amplitudes, R,
- Sum over all factorisations.
- “Inf” term from auxiliary recursion.
- Not the complete rational result, missing “Spurious” poles.

R

T

R

T

Spurious Poles

z

- Shifting the amplitude by z A(z)=C(z)+R(z)
- Poles in C as well as branch cuts e.g.
- Not related to factorisation poles sl…m,

i.e. do not appear in the final result.

- Cancel against poles in the rational part.
- Use this to compute spurious poles from residues of the cut terms.
- Location of all spurious poles, zs, is know
- Poles located at the vanishing of shifted Gram determinants of boxes and triangles.

amplitudes

One-loop integral basis- Numerical computation of the “cut terms”.
- A one-loop amplitude decomposes into
- Compute the coefficients from unitarity by taking cuts
- Apply multiple cuts, generalised unitarity.(Bern, Dixon, Kosower) (Britto, Cachazo, Feng)

Want these coefficients

Rational terms,

from recursion.

1-loop scalar integrals (Ellis, Zanderighi) , (Denner, U. Nierste and R. Scharf),

(van Oldenborgh, Vermaseren) + many others.

Box Coefficients

- Quadruple cuts freeze the integral coefficient (Britto, Cachazo, Feng)
- Box Gram determinant appears in the denominator.
- Spurious poles will go as the power of lμ in the integrand.

In 4 dimensions 4 integrals

4 delta functions

l

l1

l3

l2

Two-particle and triple cuts

- What about bubble and triangle terms?
- Triple cut Scalar triangle coefficients?
- Two-particle cut Scalar bubble coefficients?
- Disentangle these coefficients.

Additional coefficients

Isolates a single triangle

Bubbles & Triangles

- Compute the coefficients using different numbers of cuts
- Analytically examining the large value behaviour of the integrand in these components gives the coefficients (DF)(extension to massive loops (Kilgore))

Quadruple cuts, loop momentum completely constrained, gives box coefficients

Loop momentum contains one (triangle) or two (bubbles) unconstrained components.

Large Parameter Behaviour

- Apply a triple cut.
- Cut integrand, T3, as a contour integral in terms of the single unconstrained parameter t, general structure
- Series expand around t→∞.
- In general whole coefficient given by

Cut momentum parameterisation

(DF), (del Aguila, Ossola, Papadopoulos, Pittau)

Triangle Coeff

Integral over t vanish

Triangle coefficients

t

- Modify this approach for a numerical implementation.
- Consider cut integrand, T3, as a contour integral in terms of the single unconstrained parameter t
- Subtract box terms.
- Discrete Fourier projection to compute C0

Previously computed from quadruple cut

Pole Additional propagator Box

Triangle Coeff

Bubble coefficients

t

- Apply a two-particle cut.
- Two free parameters y and t in the integrand B2.
- Contour integral in terms of t (with y[0,1]) now contains poles from triangle & box propagators.
- Subtract triple-cut terms (previously computed).
- Compute bubble coefficient using

Cut momentum parameterisation

Pole Additional propagator

Triangle/Box

Numerical Spurious pole extraction

- Numerically extract spurious poles, use known pole locations.
- Expand Integral functions at the location of the poles, Δi(z)=0, e.g.
- The coefficient multiplying this is also a series in Δi(z)=0, e.g.
- Numerically extract the 1/Δi pole from the combination of the two, this is the spurious pole.

Box or triangle

Gram determinant

MHV results

- Precision tests using 100,000 phase space points with cuts.
- ET>0.01√s.
- Pseudo-rapidity η>3.
- ΔR>4,

No tests

Log10 number of points

Apply tests

Recomputed higher precision

Precision

More MHV results

- Again similar results when increasing the number of legs
- Average evaluation time: 9ms,13ms,31ms

Log10 number of points

Precision

Numerical Stability

- Use double precision for majority of points good precision.
- For a small number of exceptional points use higher precision (up to ~32 or ~64 digits.)
- Detect exceptional points using three tests (see H. Ita’s talk).

Timing

- On a 2.33GHz Xenon processor we have
- The effect of the computation of higher precision at exceptional points can be seen.
- Computation of spurious poles is the dominant part.

W, Z +5 Partons New Results

- Leading colour, (see H. Ita’s talk)
- Precision tests,
- Timing (preliminary) – 10ms, 28ms, 43ms, 40ms.
- Next step merge with automated programs for phase space integration.

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