1. Analytic Results for Two-Loop Yang-Mills
David Dunbar, John Godwin, Guy Jehu and Warren Perkins, Swansea University

2. N=4 is hydrogen atom of 21st century
Introduction
Amplitudes are important
Analytic expressions are particularly interesting (to me)
The simpler the better
Tremendous progress in N=4 SYM but need to look beyond N=4
N=4 is hydrogen atom of 21st century
QCD is ?????????????????? of 21st century

3. Singular Structure is everything?……
can amplitudes be constructed from a knowledge of singularities?

4. -singularities: poles
-terms of complex momenta

5. -cut corresponds to (poly)logarithms in Amplitude
-singularities: cuts
-cut corresponds to (poly)logarithms in Amplitude

6. -singularities: regularisations
IR singularities : determine soft behaviour
UV singularities : renormalisation
these are physical and subject to general theorems

7. Unitarity Techniques at One-Loop
Expansion in terms of basis of integral Functions
-cut is equal to discontinuity in one-loop amplitude- only a subclass of integral functions contribute
-use information to solve for coefficients
-quadruple cuts determine box coefficients algebraically
Britto,Cachazo,Feng
-many generalisations of this at one and multi-loop

8. One-Loop all-plus gluon Amplitude
(color ordered)
Tree amplitude vanishes
Amplitude finite and free of (poly)logarithms
First obtained from collinear limits
(non manifestly) cyclically symmetric
Amplitudes of self-dual Yang-Mills
Vanishes is a supersymmetric theory
cannot be constructed from MHV vertices
Bern, Chalmers, Dixon and Kosower hep-ph/
Cangemi; Chalmers and Siegal

9. dimension shifting relationship
Bern, Dixon Dunbar and Kosower hep-ph/hep-th/

10. -cut is vanishing if internal legs are in D=4
Unitarity: D versus 4 +
-cut is vanishing if internal legs are in D=4
-but not in D=4-2epsilon
D-dimensional unitarity is definitive but more difficult than 4-d in practice

11. Four-Parton Two-Loop Amplitudes
-obtained in analytic form
Glover, Oleari and Tejeda-Yeomans, hep-ph/
Bern, De Freitas and Dixon, hep-ph/
-numerical unitarity developed for two-loops for this process
Abreu, Febres Cordero, Ita , Jaquier, Page and Zeng, arXiv:

12. Five Gluon Two-Loop All-Plus
looking at leading colour, unrenormalised: expressed as
Badger, Mogull, Ochirov and O’Connell arXiv:
first computed at integrand level
Badger, Frellesvig Y.~Zhang, arXiv:
Gehrman, Henn and Presti, arXiv:
subsequently integrated and presented in this form
Finite remainder function
Singularities match general theorems
one-loop amplitude to order epsilon^2
Catani

13. -this combination either truncated or higher dimension box one-loop integral function

14. Alternate computation?
Amplitude, first five-point two-loop QCD, computed using d-dimensional unitarity plus integration
Can we reproduce this with Blended approach?
Use theorems for singularities
Four-dimensional Unitarity for finite polylogarithms
Recursion (augmented) for rational terms

15. Unitarity (=0) no box for four point
One-loop all-plus has no 4-d cuts so treat as vertex
pseudo one-loop calculation
expand amplitude in one-loop integral functions
Unitarity
(=0)
Bern, DCD, Dixon and Kosower hep-ph/931233, hep-ph/
quadruple cuts: Britto Cachazo and Feng hep-th/
canonical forms: DCD, Perkins and Warrick arXiv:
Bern, Dixon and Kosower hep-ph/
no box for four point

16. -we compute coefficients of box-functions
box integral function splits into singular terms and polylogarithms
-we also have one and two mass triangle functions which
are purely singular

17. -singular terms combine
note the one-loop is order epsilon^0
4-d cuts are missing the extra terms
must promote this to full amplitude

18. Using recursion for rational terms
For amplitudes we complexify momenta but keep on-shell
Britto, Cachazo, Feng and Witten
Risager
alternate shifts are available

19. -form the rational object
apply recursion to rational part of amplitude
split introduces spurious singularities..but known
need four point to apply recursion to…..
there are double poles in (complex momenta)
we need to use alternate shift
Bern, Dixon and Kosower hep-ph/

20. Double Poles a + b for real momenta amplitudes have single poles
double poles arise when we use complex momenta + a b
-this vanishes in supersymmetric theories
-but appear in gauge theories and gravity

21. -double poles not intrinsically a problem
but we need a formula for sub-leading singularities
no general theorems (..some specific at one-loop)
essentially off-shell information needed
Bern, Dixon and Kosower arXiv:hep-th/ + -

22. -light -cone gauge methods
sub-leading pole
need formalism to work a off-shell (partially) but still use helicity information:
-light -cone gauge methods
-carry out a shift

23. relies upon working off-shell , (a little as possible)
uses off-shell currents from Yang-Mills
produces very cumbersome but, usable, result
used technique for a variety of one-loop amplitude (mostly in gravity)
Berends-Giele, Kosower, Mahlon
Alston, Dunbar and Perkins arXiv: , arXiv: ,

24. -need “approximate” correct current
use this to extract sub-leading pole
depends upon reference spinor q

25. blended approach correctly reproduces five point amplitude
Rational terms need cleaning to reproduce known result
> 5 points?

26. n-Points All plus two-loops
split remainder as before

27. -compute coefficients of box functions using unitarity
Functions split into singular plus remainder
-singular pieces add up as expected =

28. -result for polylogarithms in remainder function
Polylogarithms are truncated box functions

29. Generate rational terms via recursion: result for six-point

30. -arrange terms by singularities
this contains the multi particle poles + a c b -
but introduces spurious singularities

31. these have the double poles

32. -removes spurious singularities
-the bit left over

33. Badger, Mogull and Peraro arXiv:1606.02244
..amplitude satisfies
symmetries and gauge independence
singularities
collinear limits
factorisations
…rational terms confirmed from d-unitarity computations
Badger, Mogull and Peraro arXiv:

34. n > 6 seven point “computed” using technique described
satisfies tests
in process of being cleaned up : done today!
aim for n-point analytic formula

35. -but of course general helicities
Beyond All-plus + -
-but of course general helicities

36. Conclusions First six-point two-loop QCD amplitude (not NNLO)
Five and Six Point are remarkably simple
Seven Point being simplified
opened (a little) window into higher loop QCD
good to see simplicity outside N=4
if you can calculate using singularities…..
theorems for subleading singularities?

37. tings

38. -singularities: poles
-singularities: cuts
-terms of complex momenta
-singularities: regularisations