Darren Forde (SPhT Saclay)

1. Darren Forde (SPhT Saclay) arxiv:hep-th/0507292, arxiv:hep-ph/0509358, arxiv:hep-ph/0604195 & arxiv:hep-ph/0607014. with C. Berger, Z. Bern, L. Dixon & D. Kosower

2. Overview

3. The unitarity bootstrap technique Cut construction techniques [Bern, Dixon, Dunbar, Kosower] [Britto,Cachazo,Feng] [Britto,Buchbinder,Cachazo,Feng] [Britto,Feng,Mastrolia] [Bern,Bjerrum-Bohr,Dunbar,Ita] [Bedford,Brandhuber,Spence,Travaglini] Unitarity cutting techniques, Generalized unitarity, MHV techniques, Direct integration via residues, Recursion relation of integral coefficients. Focus on this

4. Rational part of the loop amplitude Cut constructable part Calculating the rational terms z • Shift the momentum of the amplitude A=R+C by a complex parameter z, Integrate over a circle at infinity Contribution from circle at infinity Poles Branch cuts From know cut pieces, C. From on-shell recurrence relation From on-shell recurrence relation

5. Overlap terms Spurious singularities z • Cut-constructiblepiece, Cn(z),contains spurious singularities, e.g. • Cancel against singularities in Rn(z). • Sum over all poles to calculate Rn(z), • Subtract spurious singularities from Rn(z). • Sum over physical poles only. • physical pole when r=s1/s2→0. • unphysical pole when r=s1/s2→1.

6. On-shell recurrence relations Calculate the sum of residues from factorisation properties of amplitudes, i and j are the shifted legs Final result independent of this choice. For 1-loop one of AL or AR is the rational part of a loop amplitude. A<n A<n Rn [Britto,Cachazo,Feng], [Britto,Cachazo,Feng,Witten], [Bern,Dixon,Kosower]

7. Do we understand the factorization? Requires the complex factorization properties of the amplitude, Different from real factorization properties at 1-loop, We see the appearance of “double” poles and “unreal” poles, Factorisation into channels containing 1-loop A3(++;+),A3(++;-),A3(--;-) and A3(--;+) vertices is in general unknown. Solution? Avoid shifts that give factorizations in these channels? No, there’s a problem! Generally such shifts get a contribution from the circle at infinity, [Inf An].

8. General form of a shifted amplitude in z Now have all the terms to construct the rational piece. On-shell recurrence for [Inf An] Given by C(0) An can be constructed from a shift  Given by a recurrence relation of a second shift.

9. General strategy

10. 1-loop example Calculate the A1-loop(1-,2-,3-,4+,5+,6+) gluon amplitude, [Berger,Bern,Dixon,DF,Kosower] Sum other helicities  Complete amplitude, all pieces known [Bern,Dixon,Kosower] [Berger,Bern,Dixon,DF,Kosower] [Xiao,Yang,Zhu] [Bedford,Brandhuber,Spence,Travaglini] [Britto,Feng,Mastrolia] [Bern,Bjerrum-Bohr,Dunbar,Ita]. Cut pieces previously derived, Cut completion, CR(z)

11. 9 recursive diagrams, the first 5 are zero, e.g. The remaining 4 are non-zero, e.g. 1-loop example – recursive terms + Similar results for other 3 diagrams

12. Auxiliary shift in legs 3 and 4  an on-shell recursion with one contributing term for [Inf An](z), Finally there are 3 overlap terms, e.g. 1-loop example – The last two steps 2- 1- 1- 2- A3 Inf A6 Inf A5 3- 6+ 6+ 5+ 4+ 5+

13. Summing up all 4 different pieces gives us the complete rational part, Checks Has the correct factorisations and satisfies both the flip symmetries A6(1-,2-,3-,4+,5+,6+)=A6(3-,2-,1-,6+,5+,4+), A6(1-,2-,3-,4+,5+,6+)=A6(6+,5+,4+,3-,2-,1-). Matches a known numerical result.[Ellis, Giele, Zanderighi] 1-loop example – the final result

14. Often have a recurrence with an unknown amplitude, with the same number of negative legs, on both sides. For example: An(1-,2-,3+…,n+) after shifting legs 1 and 2, Two types of contribution All the pieces with fewer negative legs  things we known. pieces with fewerpositive legs sameno.negative legs  unknown. To solve this insert Rinto itself. Solving the recurrence. Known (gluonic) 3-point Vertex - - - + - - + + R R Rsum R + = + + - + A3 A + + + -

15. Inserting Ainto itself until we reach Result is the sum of the product of known terms, “Unwinding” and “Rewinding” Rsum + Rsum Both pieces known Rsum Rsum A3 + A3 A3 A3 A3 A3 2 j +… …+  Recurrence solved

16. Solutions to all-multiplicity amplitudes The all-multiplicity 2 minus adjacent 1-loop QCD amplitude[DF,Kosower] Technique applies generally, used to solve Non-adjacent 2-minus amplitude,[Berger,Bern,Dixon,DF, Kosower] 3-minus split helicity amplitude,[Berger,Bern,Dixon,DF, Kosower] Also tree level massive scalar amplitudes.[DF, Kosower] Small growth in complexity as number of legs increases.

17. Conclusions

18. Derive a recurrence relation for [Inf An] using a second shift. Pick a shift of legs l and m such that we have no [Inf An], this can have unknown factorisations, Perform the original shift in z on the legs i and j. If the unknown factorisations in Rn vanish when z→∞ then, Have all the terms to construct the rational piece. On-shell recurrence for [Inf An] From on-shell recursion. Extracted from known cut pieces.

19. Spinors with complex momentum do not satisfy Spinor products are independent . Some 3-point vertices no longer vanish, Momentum conservation p1.p2=p2.p3=p1.p3=0. For real momentum For complex momentum The 3-point vertex can survive, e.g. for gluons Can we avoid an [Inf An](z) term? p1 p2 p3 =0

20. Complex spinors  more complicated structure in the factorisation e.g. Complex factorisation ≠ real factorisation. Generally 1-loop complex factorisation into 1-loop A3(++;+),A3(++;-),A3(--;-) and A3(--;+) channels is unknown. Choose shifts to avoid these channels  Generally leaves shifts with an [Inf An](z). Unknown factorisations + ± +

21. Can we always choose two such shifts? For gluonic 1-loop amplitudes we have a class of shifts with no unknown factorisations. a class of shifts with no “boundary” (Inf) terms. For example shifts for gluonic amplitudes Choosing the shifts