Enrico Herrmann

All-Multiplicity non-planar Amplitudes in sYM at 2-loops

Zoomplitudes 2020

05/11/2020

In collaboration with: Subsets[{J. Bourjaily, C.Langer, A. McLeod, J. Trnka}]

[arXiv:1704.05460][arXiv:1909.09131][arXiv:1911.09106][to appear]

[0] Motivation✦ What are scattering amplitudes good for?

[Bern, Cheung, Roiban, Shen,Solon, Zeng: 1901.04424, …]

[Arkani-Hamed, Maldacena: 1503.08043] [Arkani-Hamed, Baumann, Lee, Pimentel: 1811.00024]

Collider Physics Inflationary CorrelatorsGravitational waves

[Arkani-Hamed,Bourjaily,Cachazo,Goncharov,Postnikov,Trnka: 1212.5605] [Arkani-Hamed,Trnka: 1312.2007]

[image: H.Lee]

[Salam, Henn, Moriello, Boughezal talks, …]

[0] Motivation✦ enormous progress in planar N=4 sYM

[Arkani-Hamed,Bourjaily,Cachazo,Goncharov,Postnikov,Trnka:1212.5605] [Arkani-Hamed,Trnka: 1312.2007] [Dixon,Drummond,Henn: 1108.4461]

[Ben-Israel,Tumanov,Sever: 1802.09395]

✦ hints of interesting structures beyond planar limit✦ Wilson loops at 1/Nc

✦ non-planar DCI[Bern,EH,Litsey,Stankowicz,Trnka:1412.8584,1512.08591][Bern,Enciso,Ita,Zeng:1709.06055][Bern,Enciso,Shen,Zeng:1806.06509][Chicherin,Henn,Sokatchev:1807.06321]

Goal: generate new “data” for further explorations

Final Result First explicit 2-loop all-multiplicity amplitude beyond the planar limit!

Add loops and color to Parke & Taylor amplitude

[Bourjaily, EH, Langer, McLeod, Trnka: 1911.09106]

[Parke, Taylor; (1986-06-09). "Amplitude for n-Gluon Scattering". PRL. 56 (23): 2459–2460]

on-shell coefficient functions [Bourjaily, EH, Langer, McLeod, Trnka: 1911.09106]

fully color-dressed (supersymmetric) cuts

[Bourjaily, EH, Langer, McLeod, Trnka: 1909.09131]

color kinematics

Coefficients satisfy non-trivial identities (GRTs)

[see also Ochirov talk]

[1] Outline & Introduction✦ plan of the talk:

integrands generalized unitarity prescriptive unitarity

counting bases of integrands

2-loop n-pt MHV Amp in N=4 sYM

[Bern,Dixon,Kosower: 9708239,0404293] [Britto,Cachazo,Feng: 0412103]

[Bourjaily, EH, Trnka: 1704.05460]

[Bourjaily, EH, Langer, Trnka: to appear]

bases of rational functions

[Bourjaily, EH, Langer, McLeod, Trnka: 1911.09106]

✦ generic form of loop amplitudes & integrands:

𝒜(L) = ∑i

ai I(L)i

external kinematics, color, charges, polarization vectors Feynman propagators

loop-dependent numerator}integrand = rational function of loop momenta

Q: Can we count, construct & analyze such bases?[long history of integrand reduction: Passarino,Veltman 1979; OPP: 0609007; Mastrolia,Ossola,Reiter,Tramontano:1006.0710; Badger, Frellesvig,Zhang:1207.2976; Mastrolia,Peraro,Primo:1605.03157; Ita:1510.05626; Kleiss,Malamos,Papadopoulos,Verheyen: 1206.4180, …]

n(ℓa, pb)∏r∈Γi

Dr(ℓa, pb)I(L)i = ∫ [ddℓ]L

[1] Outline & Introduction✦ generalized unitarity: [Bern,Dixon,Kosower: 9708239,0404293]

[Britto,Cachazo,Feng: 0412103]

compare residues of field theory (“cuts”) to residues of “ansatz” — linear algebra for ai

cut cut = ∑states

∏i

𝒜(tree)i

✦ method of max. cuts:[Bern,Carrasco,Johansson,Kosower: 0705.1864] [Britto,Cachazo,Feng: 0412103]

Residues are non-trivial fcts. of unfixed parameters; can pick arbitrary points on cut-surface

[1] Outline & Introduction

✦ Prescriptive unitarity:

integrand — arbitrary (complete) basis

compare residues of field theory to residues of basis expansion

[Bourjaily, EH, Trnka: 1704.05460]

arbitrariness

[Bern,Dixon,Kosower: 9708239,0404293] [Britto,Cachazo,Feng: 0412103]✦ generalized unitarity:

bad basis choice: - complicated coefficients - prohibitive linear algebra

Q: Can we avoid some of these complications?

Basis of integrands diagonalized in field theory cuts

ℐbox =

ℐtri =

ℐbub = [see talks from previous Amplitudes editions]

[2] Counting of basis integrands

✦ size of basis depends on:[Bourjaily, EH, Langer, Trnka: to appear]

Before we discuss nice integrand basis elements, we need to know how many there are in the first place.

- spacetime dimension (# of d.o.f. in )- power counting (more in a moment)

ℓμ

✦ vector space of numerators & notation:natural building blocks: span of generalized inverse propagators

(ℓ |Q)m ≡ (ℓ − Q)2 − m2 + iϵ m→0 (ℓ |Q) ≡ (ℓ − Q)2 [+iϵ]Q ∈ ℝd

[ℓ]d ≡ spanQ{(ℓ |Q)} = span{ℓ2, ℓ ⋅ e1, …, ℓ ⋅ ed, 1}[for experts: hint of power counting: we do not include arbitrary terms of the form ](ℓ⋅ei)(ℓ⋅ej)

dimensionality of space: rank([ℓ]d) = (d + 2)

rank([ℓ]kd) = (d + k

d ) + (d + k − 1d )

symmetric traceless rep. of SO(d + 2)

[2] Counting of basis integrands[Bourjaily, EH, Langer, Trnka: to appear]

✦ graph-theoretic definition of power counting & stratification of basis:

dimensionality of space: rank([ℓ]) = 6

explicit example of “triangle” (3-gon) power counting in d=4 (generalizes)

1-loop: ∼1

(ℓ |Q1)(ℓ |Q2)∼

ℓ→∞

1(ℓ2)2

∼1

(ℓ |Q1)(ℓ |Q2)∼

ℓ→∞

1(ℓ2)2 violates power counting

∼1

(ℓ |Q1)(ℓ |Q2)(ℓ |Q3)∼

ℓ→∞

1(ℓ2)3 ✓

∼n(ℓ) ∈ [ℓ]

(ℓ |Q1)(ℓ |Q2)(ℓ |Q3)(ℓ |Q4)∼

n(ℓ) ∈ spanQ{(ℓ |Q)}(ℓ |Q1)(ℓ |Q2)(ℓ |Q3)(ℓ |Q4)

∼ℓ→∞

1(ℓ2)3 ✓

[ℓ]d=4 ≡ [ℓ]

′ 1′ ∼ [ℓ]0 → rank([ℓ]0) = 1

stratification of space: 6 = 2 + 4[ℓ] = span{ }(ℓ |Q1), (ℓ |Q2), (ℓ |Q3), (ℓ |Q4)(ℓ |Q*), (ℓ | Q̃ *),

“top-level” d.o.f. “contact” d.o.f.

Q: Does this terminate, or do we get arbitrarily many topologies?

[2] Counting of basis integrands[Bourjaily, EH, Langer, Trnka: to appear]

✦ graph-theoretic definition of power counting & stratification of basis:

dimensionality of space: rank([ℓ]2) = 20

stratification of space: 20 = 0 + 20

20 =

with 3-gon power counting, all (n>4)-gons are reducible in d=4

∼n(ℓ) ∈ [ℓ]2

(ℓ |Q1)(ℓ |Q2)(ℓ |Q3)(ℓ |Q4)(ℓ |Q5)∼

ℓ→∞

1(ℓ2)3

stratify space further: +10 10

(53) × 1 (5

4) × 2+

[2] Counting of basis integrands✦ graph-theoretic definition of power counting & stratification of basis:

higher topologies:

2-loop 3-gon power counting:

Can do similar exercise for all other topologies

n(ℓ1, ℓ2) ∈ [ℓ1]0[ℓ2]0[ℓ1 − ℓ2]0

= Γ[2,1,2] = Γ[3,0,3]

n(ℓ1, ℓ2) ∈ [ℓ1]0[ℓ2]0[ℓ1 − ℓ2]0

rank{[ℓ1]0[ℓ2]0[ℓ1 − ℓ2]0} = 1

e . g . Γ[3,1,3]= Γ[2,1,2]

= Γ[3,0,3]

[ℓ1]1 [ℓ2]

1

[ℓ1 − ℓ2 ] 1

nΓ[3,1,3]∈ span{[ℓ1][ℓ2] ⊕ [ℓ1 − ℓ2]}

rank of these spaces non-trivial: rank{[ℓ1][ℓ2] ⊕ [ℓ1 − ℓ2]} = 36 = 8 + 28

tensor numerator structure Np total tensor rank = top rank+contact termsfor p-gon power-counting d = 2 d = 3 d = 4

Γ[1,1,1]

N2 1 1=1+0 1=1+0 1=1+0

N1 [1]⊕[2]⊕[1−2] 8=5+3 10=7+3 12=9+3

N0 [1][1−2][2] 49=12+37 103=42+61 181=90+91

Γ[2,1,1]

N2 [1] 4=2+2 5=3+2 6=4+2

N1 [1]2⊕[1][2]⊕[1][1−2] 24=5+19 38=13+25 55=24+31

N0 [1]2[1−2][2] 100=4+96 263=37+226 552=127+425

Γ[3,1,1]

N2 [1]2 9=0+9 14=2+12 20=5+15

N1 [1]3⊕[1]2[2]⊕[1]2[1−2] 48=0+48 95=6+89 164=24+140

N0 [1]3[1−2][2] 169=0+169 533=10+523 1305=79+1226

Γ[2,1,2]

N3 1 1=0+1 1=1+0 1=1+0

N2 [1−2]⊕[1][2] 16=0+16 25=8+17 36=15+21

N1 [1]2[2]⊕[1][2]2⊕[1][1−2][2] 63=0+63 131=16+115 229=49+180

N0 [1]2[1−2][2]2 196=0+196 644=24+620 1612=149+1463

Γ[3,1,2]

N3 [1] 4=0+4 5=2+3 6=3+3

N2 [1]2[2]⊕[1][1−2] 36=0+36 70=4+66 120=17+103

N1 [1]3[2]⊕[1]2[2]2⊕[1]2[1−2][2] 120=0+120 312=4+308 650=39+611

N0 [1]3[1−2][2]2 324=0+324 1273=4+1269 3710=77+3633

Γ[2,2,2]

N4 1 1=0+1 1=1+0 1=1+0

N3 [1]⊕[2]⊕[1−2] 8=0+8 10=4+6 12=6+6

N2 [1]2⊕[2]2⊕[1−2]2⊕[1][1−2][2] 49=0+49 103=8+95 181=32+149

N1 [1]2[1−2][2]⊕[1][1−2][2]2⊕[1][1−2]2[2] 143=0+143 391=8+383 822=60+762

N0 [1]2[1−2]2[2]2 361=0+361 1479=8+1471 4401=122+4279

Γ[4,1,1]

N2 [1]3 16=0+16 30=0+30 50=2+48

N1 [1]4⊕[1]3[2]⊕[1]3[1−2] 80=0+80 191=0+191 385=8+377

N0 [1]4[1−2][2] 256=0+256 941=0+941 2636=18+2618

Γ[4,1,2]

N3 [1]2 9=0+9 14=0+14 20=2+18

N2 [1]3[2]⊕[1]2[1−2] 64=0+64 150=0+150 300=6+294

N1 [1]4[2]⊕[1]3[2]2⊕[1]3[1−2][2] 195=0+195 609=0+609 1480=10+1470

N0 [1]4[1−2][2]2 484=0+484 2210=0+2210 7356=14+7342

Γ[3,1,3]

N4 1 1=0+1 1=0+1 1=1+0

N3 [1−2]⊕[1][2] 16=0+16 25=0+25 36=8+28

N2 [1]2[2]2⊕[1][1−2][2] 81=0+81 196=0+196 400=16+384

N1 [1]3[2]2⊕[1]2[2]3⊕[1]2[1−2][2]2 224=0+224 725=0+725 1796=24+1772

N0 [1]3[1−2][2]3 529=0+529 2480=0+2480 8400=32+8368

Γ[3,2,2]

N4 [1] 4=0+4 5=0+5 6=3+3

N3 [1]2⊕[1][2]⊕[1][1−2] 24=0+24 38=0+38 55=10+45

N2 [1]3⊕[1][2]2⊕[1][1−2]2⊕[1]2[1−2][2] 100=0+100 263=0+263 552=22+530

N1 [1]3[1−2][2]⊕[1]2[1−2][2]2⊕[1]2[1−2]2[2] 255=0+255 865=0+865 2157=30+2127

N0 [1]3[1−2]2[2]2 576=0+576 2811=0+2811 9706=42+9664

Γ[4,1,3]

N4 [1] 4=0+4 5=0+5 6=2+4

N3 [1]2[2]⊕[1][1−2] 36=0+36 70=0+70 120=4+116

N2 [1]3[2]2⊕[1]2[1−2][2] 144=0+144 420=0+420 1000=4+996

N1 [1]4[2]2⊕[1]3[2]3⊕[1]3[1−2][2]2 360=0+360 1394=0+1394 4020=4+4016

N0 [1]4[1−2][2]3 784=0+784 4264=0+4264 16470=4+16466

Γ[4,2,2]

N4 [1]2 9=0+9 14=0+14 20=2+18

N3 [1]3⊕[1]2[2]⊕[1]2[1−2] 48=0+48 95=0+95 164=4+160

N2 [1]4⊕[1]2[2]2⊕[1]2[1−2]2⊕[1]3[1−2][2] 169=0+169 533=0+533 1305=4+1301

N1 [1]4[1−2][2]⊕[1]3[1−2][2]2⊕[1]3[1−2]2[2] 399=0+399 1613=0+1613 4676=4+4672

N0 [1]4[1−2]2[2]2 841=0+841 4750=0+4750 18676=4+18672

Γ[3,2,3]

N4 [1−2]⊕[1][2] 16=0+16 25=0+25 36=7+29

N3 [1−2]2⊕[1]2[2]⊕[1][2]2⊕[1][1−2][2] 63=0+63 131=0+131 229=8+221

N2 [1]3[2]⊕[1][2]3⊕[1]2[1−2][2]2⊕[1][1−2]2[2] 196=0+196 644=0+644 1612=8+1604

N1 [1]3[1−2][2]2⊕[1]2[1−2][2]3⊕[1]2[1−2]2[2]2 440=0+440 1839=0+1839 5412=8+5404

N0 [1]3[1−2]2[2]3 900=0+900 5216=0+5216 20836=8+20828

tensor numerator structure Np total tensor rank = top rank+contact termsfor p-gon power-counting d = 2 d = 3 d = 4

Γ[1,1,1]

N2 1 1=1+0 1=1+0 1=1+0

N1 [1]⊕[2]⊕[1−2] 8=5+3 10=7+3 12=9+3

N0 [1][1−2][2] 49=12+37 103=42+61 181=90+91

Γ[2,1,1]

N2 [1] 4=2+2 5=3+2 6=4+2

N1 [1]2⊕[1][2]⊕[1][1−2] 24=5+19 38=13+25 55=24+31

N0 [1]2[1−2][2] 100=4+96 263=37+226 552=127+425

Γ[3,1,1]

N2 [1]2 9=0+9 14=2+12 20=5+15

N1 [1]3⊕[1]2[2]⊕[1]2[1−2] 48=0+48 95=6+89 164=24+140

N0 [1]3[1−2][2] 169=0+169 533=10+523 1305=79+1226

Γ[2,1,2]

N3 1 1=0+1 1=1+0 1=1+0

N2 [1−2]⊕[1][2] 16=0+16 25=8+17 36=15+21

N1 [1]2[2]⊕[1][2]2⊕[1][1−2][2] 63=0+63 131=16+115 229=49+180

N0 [1]2[1−2][2]2 196=0+196 644=24+620 1612=149+1463

Γ[3,1,2]

N3 [1] 4=0+4 5=2+3 6=3+3

N2 [1]2[2]⊕[1][1−2] 36=0+36 70=4+66 120=17+103

N1 [1]3[2]⊕[1]2[2]2⊕[1]2[1−2][2] 120=0+120 312=4+308 650=39+611

N0 [1]3[1−2][2]2 324=0+324 1273=4+1269 3710=77+3633

Γ[2,2,2]

N4 1 1=0+1 1=1+0 1=1+0

N3 [1]⊕[2]⊕[1−2] 8=0+8 10=4+6 12=6+6

N2 [1]2⊕[2]2⊕[1−2]2⊕[1][1−2][2] 49=0+49 103=8+95 181=32+149

N1 [1]2[1−2][2]⊕[1][1−2][2]2⊕[1][1−2]2[2] 143=0+143 391=8+383 822=60+762

N0 [1]2[1−2]2[2]2 361=0+361 1479=8+1471 4401=122+4279

Γ[4,1,1]

N2 [1]3 16=0+16 30=0+30 50=2+48

N1 [1]4⊕[1]3[2]⊕[1]3[1−2] 80=0+80 191=0+191 385=8+377

N0 [1]4[1−2][2] 256=0+256 941=0+941 2636=18+2618

Γ[4,1,2]

N3 [1]2 9=0+9 14=0+14 20=2+18

N2 [1]3[2]⊕[1]2[1−2] 64=0+64 150=0+150 300=6+294

N1 [1]4[2]⊕[1]3[2]2⊕[1]3[1−2][2] 195=0+195 609=0+609 1480=10+1470

N0 [1]4[1−2][2]2 484=0+484 2210=0+2210 7356=14+7342

Γ[3,1,3]

N4 1 1=0+1 1=0+1 1=1+0

N3 [1−2]⊕[1][2] 16=0+16 25=0+25 36=8+28

N2 [1]2[2]2⊕[1][1−2][2] 81=0+81 196=0+196 400=16+384

N1 [1]3[2]2⊕[1]2[2]3⊕[1]2[1−2][2]2 224=0+224 725=0+725 1796=24+1772

N0 [1]3[1−2][2]3 529=0+529 2480=0+2480 8400=32+8368

Γ[3,2,2]

N4 [1] 4=0+4 5=0+5 6=3+3

N3 [1]2⊕[1][2]⊕[1][1−2] 24=0+24 38=0+38 55=10+45

N2 [1]3⊕[1][2]2⊕[1][1−2]2⊕[1]2[1−2][2] 100=0+100 263=0+263 552=22+530

N1 [1]3[1−2][2]⊕[1]2[1−2][2]2⊕[1]2[1−2]2[2] 255=0+255 865=0+865 2157=30+2127

N0 [1]3[1−2]2[2]2 576=0+576 2811=0+2811 9706=42+9664

Γ[4,1,3]

N4 [1] 4=0+4 5=0+5 6=2+4

N3 [1]2[2]⊕[1][1−2] 36=0+36 70=0+70 120=4+116

N2 [1]3[2]2⊕[1]2[1−2][2] 144=0+144 420=0+420 1000=4+996

N1 [1]4[2]2⊕[1]3[2]3⊕[1]3[1−2][2]2 360=0+360 1394=0+1394 4020=4+4016

N0 [1]4[1−2][2]3 784=0+784 4264=0+4264 16470=4+16466

Γ[4,2,2]

N4 [1]2 9=0+9 14=0+14 20=2+18

N3 [1]3⊕[1]2[2]⊕[1]2[1−2] 48=0+48 95=0+95 164=4+160

N2 [1]4⊕[1]2[2]2⊕[1]2[1−2]2⊕[1]3[1−2][2] 169=0+169 533=0+533 1305=4+1301

N1 [1]4[1−2][2]⊕[1]3[1−2][2]2⊕[1]3[1−2]2[2] 399=0+399 1613=0+1613 4676=4+4672

N0 [1]4[1−2]2[2]2 841=0+841 4750=0+4750 18676=4+18672

Γ[3,2,3]

N4 [1−2]⊕[1][2] 16=0+16 25=0+25 36=7+29

N3 [1−2]2⊕[1]2[2]⊕[1][2]2⊕[1][1−2][2] 63=0+63 131=0+131 229=8+221

N2 [1]3[2]⊕[1][2]3⊕[1]2[1−2][2]2⊕[1][1−2]2[2] 196=0+196 644=0+644 1612=8+1604

N1 [1]3[1−2][2]2⊕[1]2[1−2][2]3⊕[1]2[1−2]2[2]2 440=0+440 1839=0+1839 5412=8+5404

N0 [1]3[1−2]2[2]3 900=0+900 5216=0+5216 20836=8+20828

[2] Counting of basis integrandstensor numerator structure Np total tensor rank = top rank+contact terms

for p-gon power-counting d = 2 d = 3 d = 4

Γ[1,1,1]

N2 1 1=1+0 1=1+0 1=1+0

N1 [1]⊕[2]⊕[1−2] 8=5+3 10=7+3 12=9+3

N0 [1][1−2][2] 49=12+37 103=42+61 181=90+91

Γ[2,1,1]

N2 [1] 4=2+2 5=3+2 6=4+2

N1 [1]2⊕[1][2]⊕[1][1−2] 24=5+19 38=13+25 55=24+31

N0 [1]2[1−2][2] 100=4+96 263=37+226 552=127+425

Γ[3,1,1]

N2 [1]2 9=0+9 14=2+12 20=5+15

N1 [1]3⊕[1]2[2]⊕[1]2[1−2] 48=0+48 95=6+89 164=24+140

N0 [1]3[1−2][2] 169=0+169 533=10+523 1305=79+1226

Γ[2,1,2]

N3 1 1=0+1 1=1+0 1=1+0

N2 [1−2]⊕[1][2] 16=0+16 25=8+17 36=15+21

N1 [1]2[2]⊕[1][2]2⊕[1][1−2][2] 63=0+63 131=16+115 229=49+180

N0 [1]2[1−2][2]2 196=0+196 644=24+620 1612=149+1463

Γ[3,1,2]

N3 [1] 4=0+4 5=2+3 6=3+3

N2 [1]2[2]⊕[1][1−2] 36=0+36 70=4+66 120=17+103

N1 [1]3[2]⊕[1]2[2]2⊕[1]2[1−2][2] 120=0+120 312=4+308 650=39+611

N0 [1]3[1−2][2]2 324=0+324 1273=4+1269 3710=77+3633

Γ[2,2,2]

N4 1 1=0+1 1=1+0 1=1+0

N3 [1]⊕[2]⊕[1−2] 8=0+8 10=4+6 12=6+6

N2 [1]2⊕[2]2⊕[1−2]2⊕[1][1−2][2] 49=0+49 103=8+95 181=32+149

N1 [1]2[1−2][2]⊕[1][1−2][2]2⊕[1][1−2]2[2] 143=0+143 391=8+383 822=60+762

N0 [1]2[1−2]2[2]2 361=0+361 1479=8+1471 4401=122+4279

Γ[4,1,1]

N2 [1]3 16=0+16 30=0+30 50=2+48

N1 [1]4⊕[1]3[2]⊕[1]3[1−2] 80=0+80 191=0+191 385=8+377

N0 [1]4[1−2][2] 256=0+256 941=0+941 2636=18+2618

Γ[4,1,2]

N3 [1]2 9=0+9 14=0+14 20=2+18

N2 [1]3[2]⊕[1]2[1−2] 64=0+64 150=0+150 300=6+294

N1 [1]4[2]⊕[1]3[2]2⊕[1]3[1−2][2] 195=0+195 609=0+609 1480=10+1470

N0 [1]4[1−2][2]2 484=0+484 2210=0+2210 7356=14+7342

Γ[3,1,3]

N4 1 1=0+1 1=0+1 1=1+0

N3 [1−2]⊕[1][2] 16=0+16 25=0+25 36=8+28

N2 [1]2[2]2⊕[1][1−2][2] 81=0+81 196=0+196 400=16+384

N1 [1]3[2]2⊕[1]2[2]3⊕[1]2[1−2][2]2 224=0+224 725=0+725 1796=24+1772

N0 [1]3[1−2][2]3 529=0+529 2480=0+2480 8400=32+8368

Γ[3,2,2]

N4 [1] 4=0+4 5=0+5 6=3+3

N3 [1]2⊕[1][2]⊕[1][1−2] 24=0+24 38=0+38 55=10+45

N2 [1]3⊕[1][2]2⊕[1][1−2]2⊕[1]2[1−2][2] 100=0+100 263=0+263 552=22+530

N1 [1]3[1−2][2]⊕[1]2[1−2][2]2⊕[1]2[1−2]2[2] 255=0+255 865=0+865 2157=30+2127

N0 [1]3[1−2]2[2]2 576=0+576 2811=0+2811 9706=42+9664

Γ[4,1,3]

N4 [1] 4=0+4 5=0+5 6=2+4

N3 [1]2[2]⊕[1][1−2] 36=0+36 70=0+70 120=4+116

N2 [1]3[2]2⊕[1]2[1−2][2] 144=0+144 420=0+420 1000=4+996

N1 [1]4[2]2⊕[1]3[2]3⊕[1]3[1−2][2]2 360=0+360 1394=0+1394 4020=4+4016

N0 [1]4[1−2][2]3 784=0+784 4264=0+4264 16470=4+16466

Γ[4,2,2]

N4 [1]2 9=0+9 14=0+14 20=2+18

N3 [1]3⊕[1]2[2]⊕[1]2[1−2] 48=0+48 95=0+95 164=4+160

N2 [1]4⊕[1]2[2]2⊕[1]2[1−2]2⊕[1]3[1−2][2] 169=0+169 533=0+533 1305=4+1301

N1 [1]4[1−2][2]⊕[1]3[1−2][2]2⊕[1]3[1−2]2[2] 399=0+399 1613=0+1613 4676=4+4672

N0 [1]4[1−2]2[2]2 841=0+841 4750=0+4750 18676=4+18672

Γ[3,2,3]

N4 [1−2]⊕[1][2] 16=0+16 25=0+25 36=7+29

N3 [1−2]2⊕[1]2[2]⊕[1][2]2⊕[1][1−2][2] 63=0+63 131=0+131 229=8+221

N2 [1]3[2]⊕[1][2]3⊕[1]2[1−2][2]2⊕[1][1−2]2[2] 196=0+196 644=0+644 1612=8+1604

N1 [1]3[1−2][2]2⊕[1]2[1−2][2]3⊕[1]2[1−2]2[2]2 440=0+440 1839=0+1839 5412=8+5404

N0 [1]3[1−2]2[2]3 900=0+900 5216=0+5216 20836=8+20828

counting

constructing

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[3] Constructing “nice” integrands Focus on d=4, integrands relevant for 2-loop

n-pt MHV amplitude in N=4 sYMrelevant topologies:

[Bourjaily, EH, Langer, McLeod, Trnka: 1911.09106]

following the prescriptivity strategy, we found very nice numeretors:

key features:

[[x]] ≡ tr+[x]C→∅

- smooth transition as legs become massless or empty- good IR-properties (chiral traces eliminate collinear divs.)- normalized to unit LS, wherever applicable- orthogonal on defining cuts

[5] The final product[Bourjaily, EH, Langer, McLeod, Trnka: 1911.09106]

comments: - reproduces known results for 4-, 5- & 6-pt amplitudes- planar limit in agreement with loop-recursion- finitely many integrand topologies compared with FDs- spurious singularities cancel

∞

[6] Conclusions/Outlook[Bourjaily, EH, Langer, McLeod, Trnka: 1911.09106]

- add to make representation amenable for dim-reg

- integration of integrand?- non-planar dual conformal symmetry?- any other hidden structures?

μ −  terms