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Analytic Results for Two-Loop Yang-Mills
David Dunbar, John Godwin, Guy Jehu and Warren Perkins, Swansea University
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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
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Singular Structure is everything?……
can amplitudes be constructed from a knowledge of singularities?
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-singularities: poles
-terms of complex momenta
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-cut corresponds to (poly)logarithms in Amplitude
-singularities: cuts -cut corresponds to (poly)logarithms in Amplitude
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-singularities: regularisations
IR singularities : determine soft behaviour UV singularities : renormalisation these are physical and subject to general theorems
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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
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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
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dimension shifting relationship
Bern, Dixon Dunbar and Kosower hep-ph/hep-th/
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-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
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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:
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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
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-this combination either truncated or higher dimension box one-loop integral function
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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
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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
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-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
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-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
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Using recursion for rational terms
For amplitudes we complexify momenta but keep on-shell Britto, Cachazo, Feng and Witten Risager alternate shifts are available
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-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/
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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
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-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/ + -
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-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
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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: ,
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-need “approximate” correct current
use this to extract sub-leading pole depends upon reference spinor q
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blended approach correctly reproduces five point amplitude
Rational terms need cleaning to reproduce known result > 5 points?
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n-Points All plus two-loops
split remainder as before
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-compute coefficients of box functions using unitarity
Functions split into singular plus remainder -singular pieces add up as expected =
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-result for polylogarithms in remainder function
Polylogarithms are truncated box functions
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Generate rational terms via recursion: result for six-point
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-arrange terms by singularities
this contains the multi particle poles + a c b - but introduces spurious singularities
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these have the double poles
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-removes spurious singularities
-the bit left over
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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:
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n > 6 seven point “computed” using technique described
satisfies tests in process of being cleaned up : done today! aim for n-point analytic formula
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-but of course general helicities
Beyond All-plus + - -but of course general helicities
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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?
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tings
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-singularities: poles
-singularities: cuts -terms of complex momenta -singularities: regularisations
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