more-than-MHV amplitudes in QCD Simon Badger 18th March 2016 - - PowerPoint PPT Presentation

more-than-MHV amplitudes in QCD

Simon Badger

18th March 2016 MHV@30, Fermilab, 16th-19th March 2016

helicity amplitudes

n n+-n-

2 4 6

• 4
• 2
• 6

MHV “UHV”

vanish at tree-level and to all loops in super-symmetric theories

)

[L. Dixon]

Modelling hadron collisions

New physics backgrounds Determination of SM parameters Keeping theory predictions in line with experimental data

next frontier: NNLO 2→3

Hadronisation Hadronisation Hadronisation Hadronisation

S h

• w

e r P a r t

• n

S h

• w

e r P a r t

• n

Parton Shower S h

• w

e r P a r t

• n

PDF PDF

spurious singularities

−16 −14 −12 −10 −8 −6 −4 −2

accuracy (digits)

100 101 102 103 104 105 106

#points

NJet gg → 3g analytic numerical

numerical D-dimensional generalised unitarity vs analytic computation with finite integrals basis (e.g. Bern, Dixon, Kosower [hep-ph/9302280]) removing spurious poles also simplifies coefficients ⇒ faster (~100x in this case) need to switch to quadruple precision evaluation applications of loop amplitudes in NNLO computations more intensive

• ne-loop finite amplitudes in

QCD

• Off-shell currents for one-loop amplitudes to
• all-plus ↔ N = 4 D-dimensional unitarity cuts
• BCFW recursion for
• CSW(MHV) rules for D-dimensional integrands

all multiplicity expressions!

[Mahlon (1993)] [Bern, Dixon, Dunbar, Kosower (1996)] [Bern, Dixon, Kosower (2005)] [Elvang, Freedman, Kiermaier (2012)]

O(✏0) O(✏0)

• utline
• d-dimensional generalised unitarity

⇒ multi-loops from trees

• two-loop all-plus amplitudes
• local integrands in d-dimensional amplitudes

Unitarity: double cuts [BDDK ’94] [triple cuts BDK ’97] Generalized unitarity: quadruple cuts [BCF ’04] Integrand reduction [OPP ’05] triple cuts [e.g. Forde ’07]

∆3 =

−

explicitly remove poles find complex contour to isolate integral coefficient

A = X

i

(rational)i(integral)i

A = Z

k

X

i

∆i(k, p) (propagators)i

D-dim. generalized unitarity [GKM ’08]

automated one-loop amplitudes

solving on-shell conditions requires complex momenta ⇒ factorise residues into tree amplitudes multi-scale kinematic algebra performed numerically

Maximal unitarity Integrand reduction via polynomial division

A = X

i

(rational)i(integral)i

A = Z

k

X

i

∆i(k, p) (propagators)i

e.g. IBPs

[Mastrolia, Ossola, SB, Frellesvig, Zhang, Mirabella, Peraro, Malamos, Kleiss, Papadopolous, Verheyen, Feng, Huang] [Kosower, Larsen, Johansson, Caron-Huot, Zhang, Søgaard]

multi-loop amplitudes from trees

[Gluza, Kosower, Kajda 1009.0472] [Schabinger 1111.4220] [Ita 1510.05626] [Larsen, Zhang 1511.01071] IBPs must be free of doubled propagator MI

a toolbox for multi-loop integrands

momentum twistors six-dimensional spinor-helicity generalised unitarity cuts integrand reduction

[Hodges (2009)]

colour/kinematics BCJ relations

A = Z

k

X

i

∆i(k, p) (propagators)i

A = X

i

Si C(∆i)∆i Q Dα

multi-loop integrand reduction

[Mastrolia, Ossola 1107.6041] [SB, Frellesvig, Zhang 1202.2019] [Zhang 1205.5707] [Mastrolia, Mirabella, Ossola, Peraro 1205.7087]

rational coefficients ISP monomials

∆T({k}) = X

{α}

cT;α1...αn(k1 · p5)α1(k2 · p2)α2 . . . (k1 · ω2)αm . . . µαn

12

spurious directions extra-dimensional ISPs

µij = −k−2✏

i

· k−2✏

j

∆c;T

• cut

= Y

i

A(0)

i

− X

T 0

∆c;T 0 Q

l∈T 0/T Dl

• cut
• n-shell the numerators can

be written as products of tree-level amplitudes

integrand parameterisations not unique - freedom in the choices of ISP monomials

integrand reduction

fix basis of monomials in irreducible scalar products via polynomial division (Gröbner basis)

six-dimensional trees

6D is a convenient way to manage all helicities simultaneously

· A(1a˙

a, 2b˙ b, 3d ˙ d, 4d ˙ d) = i

sth1a2b3c4di[1˙

a2˙ b3˙ c4 ˙ d]

dimensional reduction to using additional scalar amplitudes

4 − 2✏ (one-loop applications [Bern at al. 1010.0494] [Davies 1108.0398]) six dimensional spinor helicity [Cheung, O’Connell 0902.0981]

General colour decompositions

(a) (

(b) (

m

− →

• σ1, σ2

|σ1 ∪σ2| = m σ1 σ2

m − →

• σ

|σ| = m

σ

(a) (

m

− →

• σ1, σ2, σ3

|σ1 ∪ σ2 ∪ σ3| = m σ1 σ2 σ3

+

σ1 σ2 σ3

Inserting the DDM decomposition into colour dressed cuts leads to a compact loop decomposition

general tree-level DDM colour bases including fermions [Johansson, Ochirov arXiv:1507.00332]

[Dixon, Del Duca, Maltoni (1999)]

A3(1, 2, P12)A3(P12, 3, 4) = Res

s12=0 (A4(1, 2, 3, 4)) = s13A4(1, 3, 2, 4)

• s12=0

A4(1, 2, 3, 4) = s13 s12 A4(1, 3, 2, 4)

(k1 − P123)2

• (k1+k2+p3)2

4) =

sti ) sti ) sti )

factorization

non-planar from planar

∆ ✓ ◆

• cut

= ✓ (k1 − P123)2∆ ✓ ◆ + ∆ ✓ ◆ − ∆ ✓ ◆◆

• cut

e.g. [Bern, Carrasco, Dixon, Johansson, Roiban 1201.5366]

see talks by Carrasco and Johansson

applications: all-plus amplitudes in QCD

• ne-loop 4pt all-plus

= Ds 2 h12ih23ih34ih41i {st} · ⇢ I ✓ ◆ [µ2

11]

• +

+ + +

= Ds 2 h12ih23ih34ih41i st 6 + O(✏)

+ + + + [Bern, Kosower (1991)]

• ne-loop 4pt single minus

[Bern, Morgan (1995)]

= ~ c · ( I ✓ ◆ [µ2

11], I

✓ ◆ [µ11], I ✓ ◆ [µ11], I ✓ ◆ [µ11], I

• [µ11], I
• [µ11]

)

~ c = (Ds 2)[24]2 [12]h23ih34i[41] ⇢st u , s2t2 u2 , t2(s u) u2 , s2(t u) u2 , t(t u) tu , s(s u) su

• +

+ + [Bern, Kosower (1991)]

= (Ds 2)[24]2 [12]h23ih34i[41] u 6 + O(✏)

• +

+ +

• ne-loop 5pt all-plus

[Bern, Morgan (1995)]

= (Ds 2) 6 s12s23 + s23s34 + s34s45 + s45s51 + s51s12 + tr5(1234) h12ih23ih34ih45ih51i + O(✏)

[Bern, Dixon, Kosower (1993)]

= ~ c · ( I ✓ ◆ [µ2

11],

I ✓ ◆ [µ2

11], I

✓ ◆ [µ2

11], I

✓ ◆ [µ2

11], I

✓ ◆ [µ2

11], I

✓ ◆ [µ2

11]

) ~ c = (Ds 2) h12ih23ih34ih45ih51itr5(1234){s12s23s34s45s51, s12s23tr−(1345), s23s34tr−(2451), s34s45tr−(3512), s45s51tr−(4123), s51s12tr−(5234)}

+ + + + + + + + + +

• ne-loop 5pt all-plus

= ~ c · ( I ✓ ◆ [µ3

11],

I ✓ ◆ [µ2

11], I

✓ ◆ [µ2

11], I

✓ ◆ [µ2

11], I

✓ ◆ [µ2

11], I

✓ ◆ [µ2

11]

)

∆(1)(1+, 2+, 3+, . . . , n+) = Ds 2 h12i4 µ2

11 ∆(1),[N=4](1−, 2−, 3+, . . . , n+)

[Bern, Dixon, Dunbar, Kosower (1996)]

+ + + + +

~ c = (Ds 2) h12ih23ih34ih45ih51i{2tr5(1234), s12s23, s23s34, s34s45, s45s51, s51s12}

two-loop 4pt all-plus

= {s2t, t2s, st} · {I ⇣ ⌘ [F1], I @ 1 A [F1], I ⇣ ⌘ [F2 + F3

s+(l1+l2)2 s

]}

[Bern, Dixon, Kosower (2000)]

F1 = (Ds2)

• µ11µ22 + (µ11 + µ22)2 + 2(µ11 + µ22)µ12
• + 16(µ2

12 µ11µ22),

F2 = 4(Ds2)(µ11 + µ22)µ12, F3 = (Ds2)2µ11µ22.

dimension shifting numerators (drop Parke-Taylor pre-factors from now on....) + + + +

= X

cyclic

( ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ )

two-loop 5pt all-plus

[SB, Frellesvig, Zhang (2013)]

∆ ⇣ ⌘ = s12s23s45 tr5 {s34s45s15, tr+(1345)} · {I ⇣ ⌘ [F1], I ⇣ ⌘ [F1]} ∆ ⇣ ⌘ = {−s34s2

45tr+(1235)

tr5 } · {I ⇣ ⌘ [F1]} ∆ ⇣ ⌘ = {s12s23s34s45s15 tr5 } · {I ⇣ ⌘ [F1]} ∆ ⇣ ⌘ = −s12tr+(1345) 2s13 {s23, 1} · { I ⇣ ⌘ [F2 + F3

s45+(l1+l2)2 s45

], I ⇣ ⌘ [(2k1 · !)(F2 + F3

s45+(l1+l2)2 s45

)]} ∆ ⇣ ⌘ = {−(s45 − s12)tr+(1345) 2s13 } · {I ⇣ ⌘ [F2 + F3

s45+(l1+l2)2 s45

]} ∆ ⇣ ⌘ = ~ c · {I ⇣ ⌘ [F2], I ⇣ ⌘ [F3], I ⇣ ⌘ [F3(l1 + l2)2], I ⇣ ⌘ [F3(k1 · 3)(k2 · 3)], I ⇣ ⌘ [F3(k1 · 3)], I ⇣ ⌘ [F3(k2 · 3)], . . . } + + + + +

two-loop 5pt all-plus

A(2)

5 (1+, 2+, 3+, 4+, 5+)=

X

σ∈S5

I " C ✓ ◆ 1 2∆ ✓ ◆ + ∆ ✓ ◆ + 1 2∆ ✓ ◆ +1 2∆ ✓ ◆ + ∆ ✓ ◆ + 1 2∆ ✓ ◆ ! +C ✓ ◆ 1 4∆ ✓ ◆ + 1 2∆ ✓ ◆ + 1 2∆ ✓ ◆ −∆ ✓ ◆ + 1 4∆ ✓ ◆ ! +C ✓ ◆ 1 4∆ ✓ ◆ + 1 2∆ ✓ ◆ + 1 2∆ ✓ ◆ !#

∆(2)(1+, 2+, 3+, . . . , n+) = Ds 2 h12i4 F1 ∆(2),[N=4](1−, 2−, 3+, . . . , n+) + (1-loop)2

non-planar cuts via BCJ

[SB, Mogull, Ochirov, O’Connell (2015)]

complete BCJ numerator representation

[Mogull, O’Connell (2015)]

all genuine two-loop topologies related to N=4 MHV

[Gehrmann, Henn, Lo Presti 1511.05409]

A(2)

5

=A(1)

5

• −

5

• i=1

1 2 µ2 −vi

• + R F (2)

5

+ O() ,

F (2)

5

=5π2 12 F (1)

5

+

4

• i=0

σi    v5tr

• (1 − γ5)/

p4/ p5/ p1/ p2

• (v2 + v3 − v5)

I23,5 +1 6 tr

• (1 − γ5)/

p4/ p5/ p1/ p2 2 v1v4 + 10 3 v1v2 + 2 3v1v3      . (8)

simple function of Li2

planar master integrals using canonical differential equation approach

two-loop 5pt all-plus amplitude

local integrands

manage infra-red divergences at the integrand level

[Arkani-Hamed, Bourjailly, Cachazo, Trnka 1012.6032] [Arkani-Hamed, Bourjailly, Cachazo, Caron-Huot,Trnka 1008.2958]

simple integrals with unit leading singularities

= I4−2✏

4

[tr (1l1l23)] = I6−2✏

4

[1]

l1 = k

l2 = k 2

1 2 4 3

• ne-loop integrand bases

= X c4I ✓ ◆ + X c3I ! + X c2I

• + rational

= X P 2 I ! + X c4I ✓ ◆ + X c3m

3 I

! + X c2I

• + rational

still complicated functions at high multiplicity

• ne-loop 4pt single minus

= ~ c · ⇢ I ✓ ◆ [µ11], I ✓ ◆ [µ11], I ✓ ◆ [µ11], I

• [µ11], I
• [µ11]
• ~

c = (Ds 2)[24]2 [12]h23ih34i[41] ⇢ 1 u2, s, t, 2t s , 2s t

• I

✓ ◆ [µ11] = µ11tr+(1l1l23)2 bubbles

• +

+ +

• ne-loop 5pt all-plus

I ✓ ◆ [1] =

5

Y

i=1

tr+(klklk+1(k + 2)) bubbles

= ~ c · ( I ✓ ◆ [1], I ✓ ◆ [µ2

11], I

✓ ◆ [µ2

11], I

✓ ◆ [µ2

11], I

✓ ◆ [µ2

11], I

✓ ◆ [µ2

11],

) ~ c = (Ds 2) h12ih23ih34ih45ih51i{h13ih24ih35ih41ih52i h12ih23ih34ih45ih51i, s12s23, s23s34, s34s45, s45s51, s51s12}

+ + + + +

two-loop 5pt all-plus

[SB, Mogull (in progress)] = X

cyclic

∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘

∆ ⇣ ⌘ = {s45} · {I ⇣ ⌘ [F1]} ∆ ⇣ ⌘ = {s12s45s15} · {I ⇣ ⌘ [F1]} ∆ ⇣ ⌘ = {1} · {I ⇣ ⌘ [F2 + F3

s45+(l1+l2)2 s45

]} ∆ ⇣ ⌘ = {tr+(1245), s15, −tr+(1345), −tr+(1235)} · { ⇣ ⌘ [F2 + F3

4(l1·3)(l1·3)+s12s45+(s12+s45)(l1+l2)2 s12s45

], ⇣ ⌘ [F3(l1 + l2)2], ⇣ ⌘ [F32(l2 · 3)(s12 + tr+(1l2l33)

s13

)], ⇣ ⌘ [F32(l1 · 3)(s12 + tr+(5l5l43)

s53

)]}

+ + + + +

towards two-loop 6pt all-plus

= X

cyclic

∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ +∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘ + ∆ ⇣ ⌘

)

in progress [SB, Mogull (in progress)]

follows the expected N=4 x F1 structure

+ + + + + +

[Bern, Dixon, Kosower, Roiban, Spradlin, Vergu, Volovich (2008)] [Arkani-Hamed, Bourjailly, Cachazo, Caron-Huot,Trnka (2010)]

• utlook
• Automated loop amplitude methods
• purely algebraic algorithms with rational momenta (momentum twistors)
• automated D-dimensional cuts with 6D spinor-helicity
• First complete five-point amplitude in

Yang-Mills

• Aim: apply to high multiplicity amplitudes at one and two loops
• compact analytic expressions using local integrands: all multiplicity all-plus
• general helicity configurations for 2→3: LHC physics at NNLO

Backup

momentum twistors

an all multiplicity parameterisation (not unique)

ZiA = ✓ Σi s12 , 1 1i, h123iih34i[23] h1234ih1ii[12], h13ih124ii + h14ih123ii h1234ih1ii ◆

Σi = ( s12 i = 1

h13ih2ii h23ih1ii

i 6= 1

build spinors products

• etc. from rational

phase-space points

[Hodges (2009)]

momentum twistors

example: BCFW using rational kinematics

[c.f. bcfw mathematica package Bourjaily (2010)]