BlackHat:  NLO QCD for the LHC

July 7, 2008 SLAC Annual Program Review Page 1

BlackHat:  NLO QCD for the LHC

Darren Forde

Work in collaboration with C. Berger (MIT), Z. Bern (UCLA), L. Dixon (SLAC), F. Febres Cordero (UCLA), H. Ita (UCLA),

D. Kosower (Saclay), D. Maître (SLAC).

* Precise QCD amplitudes are needed to maximise the discovery potential of the LHC (2008).

NLO amplitudes 1-loop amplitudes.

What’s the problem?

One-loop high multiplicity processes,

What do we need?

Newest Les Houches list, (2007)

What's the hold up?

* Calculating using Feynman diagrams is Hard!

* Factorial growth in the number of Feynman diagrams.* Known results much simpler than would be expected!

* Use the most efficient approach for each piece, (Bern, Dixon, Kosower) (Berger, Bern, Dixon, Forde, Kosower)

The Unitarity Bootstrap

Unitarity cutsK3

K2K1

A3

A2

A1

On-shell recurrence relations

ji

A<n

“Glue” together trees to produce loopsRecycle results of amplitudes with fewer legs

One-loop integral basis

* 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)

2

24

n i ij ijki ij ijk

R r b c d

Rational terms, from recursion.

Want these coefficients

1-loop scalar integrals

22

12 (( ) )

( ) ii

l Kl K i

Glue together tree amplitudes

Box Coefficients

* Quadruple cuts freeze the integral coefficient (Britto, Cachazo, Feng)

2

1 ; 2 ; 3 ; 4 ;1

1( ) ( ) ( ) ( )

2ijk ijk a ijk a ijk a ijk aa

d A l A l A l A l

l

l3

l2

l1

2 2 2 21 2 3 40, 0, 0, 0l l l l 4 delta functions

In 4 dimensions 4 integrals

2 3 4 2 3 4

1 2

2 4 2 4

2 3 4 2 3 4

3 4

2 4 2 4

, ,2 2

,

1 1 1 1

1 1 1 1

1 1 1 1

1 1.

2 1 12

K K K K K Kl l

K K K K

K K K K K Kl l

K K K K

Spinor helicity notation, (Mathematica implementation “S@M” (Maître, Mastrolia))

Box coefficient

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 (Phys.Rev.D75-Forde) (technique widely applicable e.g. analysis of gravity amplitudes (Phys.Rev.D77-Bern, Carrasco, Forde, Ita, Johansson))

* Straightforward modification for a numerical implementation.

i ij ijki ij ijk

b c d

Quadruple cuts, gives box coefficients

Depends upon unconstrained components of loop momenta.

Analytic Results

* 2-minus amplitude, An(-,+,…,-,…,+), (Phys.Rev.D75-Berger,

Bern, Dixon, Forde, Kosower)

* Three minus adjacent amplitude, An(-,-,-,+,…,+), (Phys.Rev.D74-Berger, Bern, Dixon, Forde, Kosower)

* Important contributions to the recently derived complete six gluon amplitude. (Bern,Dixon,Kosower) (Berger,Bern,Dixon,Forde,Kosower) (Xiao,Yang,Zhu) (Bedford,Brandhuber,Spence,Travaglini) (Britto,Feng,Mastrolia) (Bern,Bjerrum-Bohr,Dunbar,Ita).

* A Higgs boson plus arbitrary numbers of gluons or a pair of quarks for the all-plus and one-minus helicity combinations, An(φ,+,…,±,…,+ ). (Phys.Rev.D74-Berger, Del

Duca, Dixon)

* For the LHC large number of processes to calculate,– Automatic procedure highly desirable.

* We want to go from

* Implement Unitarity bootstrap numerically.

Automation

BlackHat

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

Rational building blocks

Rational building blocks

“Compact” On-shell inputs

Much fewer terms to compute& no large cancelations comparedwith Feynman diagrams.

(To appear in Phys Rev D.- Berger, Bern, Dixon, Febres Cordero, Forde, Ita, Kosower, Maître)

Numerical Stability

* Maximise efficiency by using 16 digits of precision for majority of points good final precision of amplitude.

* For a small number of exceptional points use up to 32 or 64.* Detect exceptional points, where we must switch, using 3 tests:

– Bubble coefficients in the cut must satisfy,

– The sum of all bubbles must be zero for each spurious pole, zs

– Large cancellation between cut and rational terms.

* Box and Triangle terms feed into bubble test all pieces.

11 2

3 3treef

k nk c

nb A

N

( ) 0kk

szb

MHV results

* Precision tests using 100,000 phase space points with cuts.

– ET>0.01√s.

– Pseudo-rapidity η>3.

– ΔR>4,2 2

R 10l

|og

|

| |

num ref

refPrecisionA A

A

No tests

Apply tests

Recomputed higher precisionPrecision

Log

10 n

umbe

r of

poi

nts

NMHV results

* Other 6-pt amplitudes are similar

Precision

Log

10 n

umbe

r of

poi

nts

More MHV results

* Again similar results when increasing the number of legs

Precision

Log

10 n

umbe

r of

poi

nts

Timing

* Efficient, e.g. on a 2.33GHz Xenon processor

Helicity Cut part Only Full double prec.

Full Multi prec.

--++++ 2.4ms 6.8ms 8.8ms

--+++++ 3.8ms 10.5ms 13ms

--++++++ 5.5ms 27ms 31ms

-+-+++ 2.9ms 15.5ms 19ms

-++-++ 3.1ms 55ms 60ms

---+++ 4.3ms 12ms 14ms

--+-++ 5.7ms 37ms 44ms

-+-+-+ 6.7ms 55ms 67ms

Future Work

* Go beyond just gluons, for phenomologically more interesting processes, including– Fermions (Quarks & Leptons).– Z & W bosons.

* Combine into full NLO results,– Deal with Infra red (IR) singularities, automated

programs exist (e.g. implemented within the SHERPA framework (Gleisberg, Krauss))

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Conclusion