All one-loop NMHV gluon amplitudes in N = 1 SYMaochirov/talks/CERN2013.pdf · MHV NMHV N = 4 SYM n-point by Bern,Dixon, Dunbar, Kosower ’94 [1] n-point by Bern, Dixon, Kosower ’04

All one-loop NMHV gluon amplitudes

in N = 1 SYM

based on arXiv:1311.1491 [hep-th]

Alexander OchirovIPhT, CEA-Saclay, France

Presented atCERN Theory DivisionNovember 29, 2013

Amplitudes

p1

p2

p3 p4

p5

. . .

pn

Amplitudes

p1

p2

p3 p4

p5

. . .

pn

◮ basic objects in QFT

Amplitudes

p1

p2

p3 p4

p5

. . .

pn


◮ measurable at experiment

Amplitudes

p1

p2

p3 p4

p5

. . .

pn


◮ measurable at experiment

◮ calculable analytically

Known one-loop amplitudes

One-loop gluon amplitudes known analytically:

MHV NMHV

N = 4 SYM n-point byBern,Dixon,Dunbar,Kosower

’94 [1] n-point by Bern,Dixon,Kosower

’04 [2]

N = 1 SYM n-point byBern,Dixon,Dunbar,Kosower

’94 [3] 6-point byBritto,Buchbinder,Cachazo,Feng

’05 [4]

7-point by Bjerrum-Bohr,Dunbar,Perkins

’07 [5]

and Dunbar,Perkins,Warrick

’09 [6],

n-point by AO’13

QCD n-point byBedford,Brandhuber,Spence,Travaglini

’04 [7] 6-point by Britto,Feng,Mastrolia

’06 [8]

andBerger,Bern,Dixon,Forde,Kosower

’06 [9] and Xiao,Yang,Zhu

’06 [10]

Content

1. Setup: NMHV amplitudes in N = 1 SYM at one loop

2. Method: spinor integration

3. Cut construction

4. Results

5. Conclusion and outlook

Yang-Mills theories

Particle content:

helicity 1 1/2 0 −1/2 −1

N = 4 SYM 1 4 6 4 1

N = 1 SYM 1 1 0 1 1

QCD 1 0 0 0 1

Yang-Mills theories at one loop

helicity 1 1/2 0 −1/2 −1

N = 4 SYM 1 4 6 4 1N = 1 SYM 1 1 0 1 1QCD 1 0 0 0 1


helicity 1 1/2 0 −1/2 −1

N = 4 SYM 1 4 6 4 1N = 1 SYM 1 1 0 1 1QCD 1 0 0 0 1

A1-loopQCD = A1-loop

N=4 SYM − 4A1-loopN=1 chiral + 2A1-loop

N=0 scalar (1)


helicity 1 1/2 0 −1/2 −1

N = 4 SYM 1 4 6 4 1N = 1 SYM 1 1 0 1 1QCD 1 0 0 0 1



N=0 scalar (1)

A1-loopN=1 SYM = A1-loop

N=4 SYM − 3A1-loopN=1 chiral (2)


helicity 1 1/2 0 −1/2 −1

N = 4 SYM 1 4 6 4 1N = 1 SYM 1 1 0 1 1QCD 1 0 0 0 1



N=0 scalar (1)



N = 1 chiral 0 1 2 1 0N = 0 scalar 0 0 1 0 0

Helicity configurations in SYM

◮ All helicities plus


◮ All helicities plus — zero by SUSY



◮ One minus



◮ One minus — zero by SUSY




◮ Two minuses




◮ Two minuses — MHV (Maximally-Helicity-Violating)





◮ Three minuses





◮ Three minuses — NMHV (Next-to-Maximally-Helicity-Violating)

◮ . . . k minuses






◮ . . . k minuses — Nk−2MHV







ANMHV5 =

(

AMHV5

)∗







ANMHV5 =

(

AMHV5

)∗

NMHV important 6+ ext. particles

N = 1 SYM at one loop





A1-loop = µ2ǫ(

∑

Cbox I4 +∑

Ctri I3 +∑

Cbub I2 +R)

, (5)

from susy power-counting R = 0.

UV and IR behavior

In = (−1)n+1(4π)d2 i

∫

ddl1(2π)d

1

l21(l1 −K1)2 . . . (l1 −Kn−1)2(6)

Britto,Buchbinder,Cachazo,Feng

’05 [4]: two-mass and one-mass triangles O( 1ǫ2)

cancel divergences inside boxes⇒ finite boxes and only three-mass triangles

A1-loopN=1 chiral =

1

ǫ

∑

Cbub +O(1) =1

(4π)d2 ǫ

Atree +O(1) (7)

∑

Cbub =1

(4π)d2

Atree (8)

NMHV three-mass triangles

already known thanks toBjerrum-Bohr,Dunbar,Perkins

’07 [5]

(s3+1)+. . .m−

1

. . .

s+1

(s1+1)+ . . . m−2 . . . s+2

(s2+1)+

. . .

m−3. . .

s+3

Method: spinor integration

Thanks to Britto, Feng et al.

Cut =∑

h1,h2

∫

d4l1 δ(l21)δ(l

22)A(−lh̄1

1 , . . . , lh22 )A(−lh̄2

2 , . . . , lh11 ) (9)

lµ1 =K2

2

〈λ| γµ|λ̃]

〈λ|K|λ̃], lµ2 = −

1

2

〈λ|K|γµ|K|λ̃]

〈λ|K|λ̃]

⇒

∫

d4l1 δ(l21)δ(l

22) = −

K2

4

∫

λ̃=λ̄

〈λdλ〉 [λ̃dλ̃]

〈λ|K|λ̃]2(10)

⇒ Cut =

∫

λ̃=λ̄

〈λdλ〉 [λ̃dλ̃] Ispinor(λ, λ̃) (11)

Spinor integration: complex analysis in disguise

λ = λp + zλq, λ̃ = λ̃p + z̄λ̃q (12)

∫

λ̃=λ̄

〈λdλ〉 [λ̃dλ̃] = −(p+ q)2∫

dz ∧ dz̄ (13)

⇒ Cauchy’s integral theorem (Mastrolia’09 [11]):

∫∫

D

dz ∧ dz̄∂f

∂z̄(z, z̄) = −

∮

∂D

dz f(z, z̄) + 2πi∑

zi∈D

Resz=zi

f(z, z̄)

(14)

Simple pole 〈λ|ζ〉: Resλ=ζ

1

〈ζ|λ〉

N(λ, λ̃)

D(λ, λ̃)=

N(ζ, ζ̃)

D(ζ, ζ̃)

Spinor integration: residue extraction rules

Simple pole 〈λ|ζ〉:

Resλ=ζ

1

〈ζ|λ〉

N(λ, λ̃)

D(λ, λ̃)=

N(ζ, ζ̃)

D(ζ, ζ̃)= −Res

λ=ζ

1

〈λ|ζ〉

N(λ, λ̃)

D(λ, λ̃)(15)

Multiple pole 〈λ|ζ〉k:

Resλ=ζ

1

〈ζ|λ〉kN(λ, λ̃)

D(λ, λ̃)=

1

〈η|ζ〉k−1

{

1

(k − 1)!

d(k−1)

dt(k−1)

N(ζ − tη, ζ̃)

D(ζ − tη, ζ̃)

}∣

∣

∣

∣

t=0

(16)

Composite pole 〈λ|K|q]:

〈ζ| = [q|K| |ζ〉 = −|K|q]

[ζ̃| = −〈q|K| |ζ̃] = |K|q〉(17)

Spinor integration: bubble coefficients

Cut =

∫

λ̃=λ̄

〈λdλ〉 [λ̃dλ̃] Ispinor

Cbub=4

(4π)d2 i

∑

residues

1

(n−k)!

d(n−k)

ds(n−k)

1

sln

(

1 + s〈λ|q|λ̃]

〈λ|K|λ̃]

)

×

[

Ispinor〈λ|K|λ̃]n−k+2

〈λ|K|q|λ〉

∣

∣

∣

∣

|λ̃]=|K+sq|λ〉

]∣

∣

∣

∣

s=0

(18)

(n − k) is the difference between the numbers of λ-factors in the numerator and the

denominator of Ispinor, excluding the homogeneity-restoring factor 〈λ|K|λ̃]n−k+2

Spinor integration: propagators rewritten

(l1 −Ki)2 = K2 〈λ|Qi|λ̃]

〈λ|K|λ̃], (19)

Qµi (Ki,K) = −Kµ

i +K2

i

K2Kµ (20)

Spinor integration: triangle coefficients

Cut =

∫

λ̃=λ̄


Ctrii =

2

(4π)d2 i

1

(K2(xi+ − xi−)2)n−k+1

×1

(n−k+1)!

d(n−k+1)

dt(n−k+1)Ispinor 〈λ|Qi|λ̃] 〈λ|K|λ̃]n−k+2

{∣

∣

∣

∣ λ=λi+−tλi

−

λ̃=xi+λ̃i

−−txi−λ̃i

+

+

∣

∣

∣

∣ λ=λi−−tλi

+

λ̃=xi−λ̃i

+−txi+λ̃i

−

}∣

∣

∣

∣

t=0

,

(21)

P i±(Ki,K) = Qi + xi±K, (22)

xi± =−K ·Qi ±

√

(K ·Qi)2 −K2Q2i

K2. (23)

Spinor integration: box coefficients

Cut =

∫

λ̃=λ̄


Cboxij = −

2K2

(4π)d2 iIspinor 〈λ|Qi|λ̃] 〈λ|Qj |λ̃]

{∣

∣

∣

∣λ=λij+

λ̃=λ̃ij−

+

∣

∣

∣

∣λ=λij−

λ̃=λ̃ij+

}

,

(24)

P ij± (Ki,Kj ,K) = Qi + xij±Qj, (25)

xij± =−Qi ·Qj ±

√

(Qi ·Qj)2 −Q2iQ

2j

Q2j

. (26)

Simplest NMHV bubbles: MHV-MHV

l11−

. . .

p+

. . .

k− l2 (k+1)+

. . .

m−

. . .

n+

∑

h1,h2

A1A2 =(−1)ki[l1p]

2[l2p]2

[l11][12] . . . [k−1|k][kl2][l2l1]

×i 〈l1m〉2 〈l2m〉2

〈l2|k+1〉〈k+1|k+2〉 . . . 〈n−1|n〉〈nl1〉〈l1l2〉

×

(

−[l1p] 〈l1m〉

[l2p] 〈l2m〉+ 2−

[l2p] 〈l2m〉

[l1p] 〈l1m〉

)

(27)


∑

h1,h2

A1A2 =(−1)k 〈m|P1k|p]

2

P 21k[12] . . . [k−1|k] 〈k+1|k+2〉 . . . 〈n−1|n〉

×〈l1m〉〈l2m〉 [l1p][l2p]

〈l1n〉〈l2|k+1〉 [l11][l2k]

≡F

P 21k

〈l1m〉〈l2m〉 [l1p][l2p]

〈l1n〉〈l2|k+1〉 [l11][l2k]

=F

P 21k

〈l1m〉〈l1|P1k|p] 〈m|P1k|l1][pl1]

〈l1n〉〈l1|P1k|k] 〈k+1|P1k|l1][1l1]

(28)

Cut(P1k) =−F

4

∫

λ̃=λ̄

〈λdλ〉 [λ̃dλ̃]

〈λ|P1k|λ̃]2〈λ|m〉〈λ|P1k|p] 〈m|P1k|λ̃][p|λ̃]

〈λ|n〉〈λ|P1k|k] 〈k+1|P1k|λ̃][1|λ̃].

(29)


Cbub,P1kN=1 chiral =−

F

(4π)d2 i

∑

residues

1

sln

(

1 + s〈λ|q|λ̃]

〈λ|P1k|λ̃]

)

×

{

〈λ|m〉〈λ|P1k|p] 〈m|P1k|λ̃][p|λ̃]

〈λ|n〉〈λ|P1k|k] 〈k+1|P1k|λ̃][1|λ̃]

×1

〈λ|P1k|q|λ〉

∣

∣

∣

∣

|λ̃]=|P1k+s q|λ〉

}∣

∣

∣

∣

s=0

(30)

Cbub,P1kN=1 chiral =

F

(4π)d2 i

∑

residues

[q|λ̃]

〈λ|P1k|λ̃]

×〈λ|m〉2 〈λ|P1k|p]

2

〈λ|k+1〉〈λ|n〉〈λ|P1k|1] 〈λ|P1k|k] 〈λ|P1k|q]

(31)



(−1)k

(4π)d2 i

〈m|P1k|p]2

[12] . . . [k−1|k] 〈k+1|k+2〉 . . . 〈n−1|n〉

×

{

〈m|k+1〉2 〈k+1|P1k|p]2[k+1|q]

〈k+1|n〉〈k+1|P1k|1] 〈k+1|P1k|k] 〈k+1|P1k|k+1] 〈k+1|P1k|q]

+〈mn〉2 〈n|P1k|p]

2[nq]

〈n|k+1〉〈n|P1k|1] 〈n|P1k|k] 〈n|P1k|n] 〈n|P1k|q]

+1

P 21k

(

[1p]2

[1k][1q]

〈m|P1k|1]2 〈1|P1k|q]

〈1|P1k|1] 〈k+1|P1k|1] 〈n|P1k|1]

+[kp]2

[k1][kq]

〈m|P1k|k]2 〈k|P1k|q]

〈k|P1k|k] 〈k+1|P1k|k] 〈n|P1k|k]

+[pq]2

[1q][kq]

〈m|P1k|q]2

〈k+1|P1k|q] 〈n|P1k|q]

)}

.

(32)

N = 1 chiral bubble formula

only simple poles:

CbubN=1 chiral = −

K2

(4π)d2 i

∑

residues

[λ̃|q]

〈λ|K|λ̃] 〈λ|K|q]

[

∑

h1,h2

A1A2

∣

∣

∣

∣

|λ̃]=|K|λ〉

]

.

(33)

a close analogue found byElvang,Huang,Peng

’11 [12]

Cut construction: NMHV double cut

l11+. . .

m−1

. . .m−

2

. . . k+ l2 (k+1)+

. . .

m−3

. . .

n+

Two-mass-easy quadruple cut

l11+. . .

m−1

. . .m−

2

. . .k+ l2 (k+1)+

. . .

m−3

. . .

n+

⇒

l11+. . .m−

1

. . .m−

2

. . .k+l2

(k+1)+l3 (k+2)+. . . m−

3

. . .

(n−1)+

l4

n+

NMHV super-amplitude in N = 4 SYM

fromDrummond,Henn,Korchemsky,Sokatchev

’08 [13]

ANMHVn = AMHV

n

r+n−3∑

s=r+2

r+n−1∑

t=s+2

Rrst, (34)

R = {r + 1, . . . , s− 1}

S = {s, . . . , t− 1}

T = {t, . . . , r − 1}

R S

Tr

s

t

r+1

r−1

s−1

t−1

Rrst =−〈s−1|s〉〈t−1|t〉

P 2S 〈s−1|PS |PT |r〉〈s|PS |PT |r〉〈t−1|PS |PR|r〉〈t|PS |PR|r〉

(35)

NMHV amplitudes

GGT package byDixon,Henn,Plefka,Schuster

’10 [14]:

Atree(1+g ,+. . ., a−g ,

+. . ., b−g ,+. . ., n−

g )

=i

〈12〉〈23〉 . . . 〈n1〉

n−3∑

s=2

n−1∑

t=s+2

RnstD4nst;ab, (36a)

Atree(1+g ,+. . ., a−A

Λ , +. . ., b+BCDΛ , +. . ., c−g ,

+. . ., n−g )

=iǫABCD

〈12〉〈23〉 . . . 〈n1〉

n−3∑

s=2

n−1∑

t=s+2

RnstD3nst;acDnst;bc, (36b)

Atree(1+g ,+. . ., aAB

S , +. . ., bCDS , +. . ., c−g ,

+. . ., n−g )

=iǫABCD

〈12〉〈23〉 . . . 〈n1〉

n−3∑

s=2

n−1∑

t=s+2

RnstD2nst;acD

2nst;bc, (36c)

Double cut expression

l11+. . .

m−1

. . .m−

2

. . .k+ l2 (k+1)+

. . .

m−3

. . .

n+

∑

h1,h2

A1A2 =i 〈−l2|m3〉

2 〈l1|m3〉2

〈−l2|k+1〉〈k+1|k+2〉 . . . 〈n−1|n〉〈n|l1〉〈l1|−l2〉

×i

〈−l1|1〉〈12〉 . . . 〈k−1|k〉〈k|l2〉〈l2|−l1〉

×m1−3∑

s=m1+2

m1−1∑

t=s+2

Rm1stD2m1st;m2(−l1)

D2m1st;m2l2

×

(

〈−l2|m3〉Dm1st;m2(−l1)

〈l1|m3〉Dm1st;m2l2

+ 2 +〈l1|m3〉Dm1st;m2l2

〈−l2|m3〉Dm1st;m2(−l1)

)

(37)

Double cut expression

l11+. . .

m−1

. . .m−

2

. . .k+ l2 (k+1)+

. . .

m−3

. . .

n+

∑

h1,h2

A1A2 = −1

〈12〉 . . . 〈k−1|k〉〈k+1|k+2〉 . . . 〈n−1|n〉

×〈l1|m3〉〈m3| −l2〉

〈l1|1〉〈l1|n〉〈l1|l2〉2 〈k|l2〉〈k+1|l2〉

×m1−3∑

s=m1+2

m1−1∑

t=s+2

Rm1stDm1st;m2(−l1)Dm1st;m2l2

×(

〈−l2|m3〉Dm1st;m2(−l1) + 〈l1|m3〉Dm1st;m2l2

)2

(38)

The factor which defines the non-zero cases for s and t

Dm1st;m2l2

=

〈m1|l2〉〈m1|Pm1+1,s−1|Ps,t−1|m2〉 if {s, t} ∈ A

−P 2s,t−1 〈m1|m2〉〈m1|l2〉 if {s, t} ∈ B

〈m2|l2〉(

〈m1|Pm1+1,s−1|l1] 〈l1|m1〉 − 〈m1|Pm1+1,s−1|P1,m1−1|m1〉)

if {s, t} ∈ C

− 〈m1|m2〉(

〈m1|l1〉 [l1|Ps,k |l2〉 − 〈m1|P1,m1−1|Ps,k|l2〉)

if {s, t} ∈ D

〈m2|l2〉〈m1|Pt,m1−1|Pm1+1,s−1|m1〉 if {s, t} ∈ E

− 〈m1|m2〉〈m1|Pt,m1−1|Pt,s−1|l2〉 if {s, t} ∈ F

0 otherwise,

(39)

Non-zero cases for s and t

A : s ∈ {m1+2, . . . ,m2}, t ∈ {m2+1, . . . , l2} (40a)

B : s ∈ {m2+1, . . . , k−1}, t ∈ {m2+3, . . . , l2} (40b)

C : s ∈ {m1+2, . . . ,m2}, t = −l1 (40c)

D : s ∈ {m2+1, . . . , k}, t = −l1 (40d)

E : s ∈ {m1+2, . . . ,m2}, t ∈ {1, . . . ,m1−1} (40e)

F : s ∈ {m2+1, . . . , l2}, t ∈ {1, . . . ,m1−1}. (40f)

Cases in s and tm1+4 . . . m2+1m2

m2

m1+2

m2+1

.

.

.

m2+2 m2+3 . . . . . .

.

.

.

.

.

.

k l2 −l1 1 m1−1

m1−3

1

−l1

l2

k

k−1

k−2

ts

A

B

C

D

E

F

Results: Frame formulas for generic bubbles and

2me-boxes

Cbub,P1kN=1 chiral(m

−1 ,m

−2 ∈{1, . . . , k},m−

3 ∈ {k+1, . . . ,n})

=∑

{s,t}∈A

Rs,tA +

∑

{s,t}∈B

Rs,tB +

∑

{s,t=−l1}∈C

RsC

+∑

{s,t=−l1}∈D

RsD +

∑

{s,t}∈E

Rs,tE +

∑

{s,t}∈F

Rs,tF

(41)

Cbox,2meN=1 chiral(m

−1 ,m

−2 ∈{1, . . . , k},m−

3 ∈ {k+1, . . . , n})

=∑

{s,t}∈A

Cbox,s,tA +

∑

{s,t}∈B

Cbox,s,tB +

∑

{s,t=−l1}∈C

Cbox,sC

+∑

{s,t=−l1}∈D

Cbox,sD +

∑

{s,t}∈E

Cbox,s,tE +

∑

{s,t}∈F

Cbox,s,tF

(42)

Example: simpler bubbles

only case F contributes to:

l11+

. . .

(k−2)+

(k−1)−

k− l2 (k+1)+

. . .

m−

. . .

n+

Simpler bubble coefficients

l11+

. . .(k−2)+(k−1)−

k− l2 (k+1)+

. . .

m−

. . .

n+

Cbub,P1kN=1 chiral

=P 21k

(4π)d2 i

∑

residues

[q|λ̃]

〈λ|P1k|λ̃]

{

F1〈λ|m〉2 〈λ|P1k|P1,k−1|k−1〉

〈λ|k+1〉〈λ|n〉〈λ|P1k|k] 〈λ|P1k|q]

+k−2∑

t=2

Ft 〈t−1|t〉

〈t−1|Pt,k|k]

〈λ|m〉2 〈λ|Pt,k|Pt,k−1|k−1〉

〈λ|1〉〈λ|k+1〉〈λ|n〉〈λ|P1k|q]

}

,

(43)

Ft =〈m|Pt,k|Pt,k−1|k−1〉2

P 2t,k−1P

2t,k 〈t|Pt,k|k]

(44)

Results: simpler bubble coefficients


1

(4π)d2 i

1

〈12〉 . . . 〈k−2|k−1〉〈k+1|k+2〉 . . . 〈n−1|n〉

×

{ k−2∑

t=2

⟨

m|Pt,k|Pt,k−1|k−1⟩2

〈t−1|t〉

P 2t,kP

2t,k−1 〈t−1|Pt,k|k] 〈t|Pt,k|k]

(

〈m|P1k |k+1|m〉⟨

k+1|Pt,k|Pt,k−1|k−1⟩

〈k+1|1〉〈k+1|n〉〈k+1|P1k|k+1]

+〈m|P1k|1|m〉

⟨

1|Pt,k|Pt,k−1|k−1⟩

〈1n〉〈1|k+1〉〈1|P1k|1]+

〈m|P1k|n|m〉⟨

n|Pt,k|Pt,k−1|k−1⟩

〈n1〉〈n|k+1〉〈n|P1k|n]

)

+

⟨

m|P1k|P1,k−1|k−1⟩2

P 21kP

21,k−1 〈1|P1k|k]

(

〈m|P1k|k+1|m〉⟨

k+1|P1k|P1,k−1|k−1⟩

〈k+1|n〉〈k+1|P1k|k] 〈k+1|P1k|k+1]

+〈m|P1k|n|m〉

⟨

n|P1k|P1,k−1|k−1⟩

〈n|k+1〉〈n|P1k|k] 〈n|P1k |n]+

P 21k 〈m|P1k|k|m〉〈k−1|P1k|k]

〈n|P1k |k] 〈k+1|P1k|k] 〈k|P1k|k]

)}

(45)

Massive poles

l1· · · 1+m−1

. . .(i−1)+

i+. . .m−

2· · · k+ l2 (k+1)+

. . .

m−3

. . .

n+

(l1 − P1,i)2 = P 2

1k

〈λ|Qi|λ̃]

〈λ|P1k|λ̃](46)

〈λ|Qi|P1k|λ〉 = −

⟨

λ|λi+

⟩

[λ̃i+|λ̃

i−]⟨

λi−|λ⟩

xi+ − xi−(47)

Resλ=λi

±

F (λ, λ̃)

〈λ|P1k|λ̃] 〈λ|Qi|P1k|λ〉= −

F (λi±, λ̃

i±)

4(

(P1,i−1 · Pi,k)2 − P 21,i−1P

2i,k

)

(48)

Results: sample contribution

for s ∈ {m1+3, . . . , m2}, t = −l1

RsC

=−P21k

(4π)d2 i

〈s−1|s〉

〈12〉 . . . 〈k−1|k〉〈k+1|k+2〉 . . . 〈n−1|n〉

{

F sC(λ1, λ̃1)

〈1|k〉〈1|k+1〉〈1|n〉+

F sC(λk, λ̃k)

〈k|1〉〈k|k+1〉〈k|n〉

+F sC(λk+1, λ̃k+1)

〈k+1|1〉〈k+1|k〉〈k+1|n〉+

F sC(λn, λ̃n)

〈n|1〉〈n|k〉〈n|k+1〉+

〈m1|P1k|q]2 〈m2|P1k|q]2 〈m3|P1k|q]2

P41k

〈1|P1k|q] 〈k|P1k|q] 〈k+1|P1k|q] 〈n|P1k|q]

×〈m1|Pm1+1,s−1|q]

2

[q|P1,s−1|Ps,k|q] 〈m1|Pm1+1,s−1|P1,s−1|P1k|q] 〈m1|Pm1+1,s−1|Ps,k|P1k|q]

×(

P21k 〈m3|m1〉〈m2|P1k|q] 〈m1|Pm1,s−1|q] + 〈m1|m2〉〈m3|P1k|q] 〈m1|Pm1,s−1|Ps,k|P1k|q]

)2

/(

(

〈m1|s−1〉 [q|P1,m1|Pm1+1,k|q] + 〈m1|P1k|q] 〈s−1|Pm1+1,s−2|q]

)

×(

〈m1|s〉 [q|P1,m1|Pm1+1,k|q] + 〈m1|P1k|q] 〈s|Pm1+1,s−1|q]

)

)

+MsC(λ

+s , λ̃

+s ) + M

sC(λ

−

s , λ̃−

s )

}

,

FsC(λ, λ̃) =

〈λ|m1〉2 〈λ|m2〉

2 〈λ|m3〉2⟨

λ|P1k|Pm1+1,s−1|m1

⟩2[λ̃|q]

⟨

λ|P1,s−1|Ps,k|λ⟩

⟨

λ|P1,s−1|Pm1+1,s−1|m1

⟩ ⟨

λ|Ps,k|Pm1+1,s−1|m1

⟩

〈λ|P1k|λ̃] 〈λ|P1k|q]

×(

〈λ|m2〉〈m3|m1〉⟨

m1|Pm1+1,s−1|P1k|λ⟩

+ 〈λ|m3〉〈m1|m2〉⟨

m1|Pm1+1,s−1|Ps,k|λ⟩

)2

/(

(

〈m1|s−1〉⟨

λ|P1,m1|Pm1+1,k|λ

⟩

+ 〈λ|m1〉⟨

λ|P1k|Pm1+1,s−2|s−1⟩

)

(

〈m1|s〉⟨

λ|P1,m1|Pm1+1,k|λ

⟩

+ 〈λ|m1〉⟨

λ|P1k|Pm1+1,s−1|s⟩

)

)

Results: sample contribution

for s ∈ {m1+3, . . . , m2}, t = −l1

RsC

=−P21k

(4π)d2 i

〈s−1|s〉

〈12〉 . . . 〈k−1|k〉〈k+1|k+2〉 . . . 〈n−1|n〉

{

F sC(λ1, λ̃1)

〈1|k〉〈1|k+1〉〈1|n〉+

F sC(λk, λ̃k)

〈k|1〉〈k|k+1〉〈k|n〉

+F sC(λk+1, λ̃k+1)

〈k+1|1〉〈k+1|k〉〈k+1|n〉+

F sC(λn, λ̃n)

〈n|1〉〈n|k〉〈n|k+1〉+

〈m1|P1k|q]2 〈m2|P1k|q]2 〈m3|P1k|q]2

P41k

〈1|P1k|q] 〈k|P1k|q] 〈k+1|P1k|q] 〈n|P1k|q]

×〈m1|Pm1+1,s−1|q]

2

[q|P1,s−1|Ps,k|q] 〈m1|Pm1+1,s−1|P1,s−1|P1k|q] 〈m1|Pm1+1,s−1|Ps,k|P1k|q]

×(

P21k 〈m3|m1〉〈m2|P1k|q] 〈m1|Pm1,s−1|q] + 〈m1|m2〉〈m3|P1k|q] 〈m1|Pm1,s−1|Ps,k|P1k|q]

)2

/(

(

〈m1|s−1〉 [q|P1,m1|Pm1+1,k|q] + 〈m1|P1k|q] 〈s−1|Pm1+1,s−2|q]

)

×(

〈m1|s〉 [q|P1,m1|Pm1+1,k|q] + 〈m1|P1k|q] 〈s|Pm1+1,s−1|q]

)

)

+MsC(λ

+s , λ̃

+s ) + M

sC(λ

−

s , λ̃−

s )

}

,

MsC(λ, λ̃) = −

1

4((P1,s−1 · Ps,k)2 − P2

1,s−1P2s,k

))

×〈λ|m1〉

2 〈λ|m2〉2 〈λ|m3〉

2⟨

λ|P1k|Pm1+1,s−1,m1

⟩2[λ̃|q]

〈λ|1〉〈λ|k〉〈λ|k+1〉〈λ|n〉⟨

λ|P1,s−1|Pm1+1,s−1|m1

⟩ ⟨

λ|Ps,k|Pm1+1,s−1|m1

⟩

〈λ|P1k|q]

×(

〈λ|m2〉〈m3|m1〉⟨

m1|Pm1+1,s−1|P1k|λ⟩

+ 〈λ|m3〉〈m1|m2〉⟨

m1|Pm1+1,s−1|Ps,k|λ⟩

)2

/(

(

〈m1|s−1〉⟨

λ|P1,m1|Pm1+1,k|λ

⟩

+ 〈λ|m1〉⟨

λ|P1k|Pm1+1,s−2|s−1⟩

)

(

⟨ ⟩ ⟨ ⟩

)

)

Checks

1. Check the sum of all bubbles through for 6,7 and 8 points:

∑

Cbub =1

(4π)d2

Atree

2. NGluon byBadger,Biedermann,Uwer

’10 [15]

Numerical data kindly produced by Simon Badger:gluon, fermion and scalar loop contributions for 6-25 ext. gluonsA(1−, 2−, 3+, 4−, 5+, 6+, . . . , n+) andA(1−, 2+, 3−, 4+, 5+, 6−, . . . , n+).Agreement up to 13 digits for 8 points and 12 digits for 17points.

Conclusions

1. New one-loop amplitudes are known analytically for all n.

2. Good tree formulas ⇒ new loop results.

3. NMHV in N = 1 SYM: untrivial both in helicities and susy.

4. Mathematica files are distributed:N1chiralAll.nb, Ammppmppp.nb.

5. Everybody is invited to play!

Outlook: one-loop recursion

+ −

7+

l1

1−

2−3+l2 4̂−

5̂+

6+P̂5,6

=

7+

l1

1−

2−3+l2

4−

5+

6+

Thank you for listening!

References

Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower Nucl. Phys. B425 (1994) 217–260.

Z. Bern, L. J. Dixon, and D. A. Kosower Phys.Rev. D72 (2005) 045014.

Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower Nucl.Phys. B435 (1995) 59–101.

R. Britto, E. Buchbinder, F. Cachazo, and B. Feng Phys.Rev. D72 (2005) 065012.

N. Bjerrum-Bohr, D. C. Dunbar, and W. B. Perkins JHEP 0804 (2008) 038.

D. C. Dunbar, W. B. Perkins, and E. Warrick JHEP 0906 (2009) 056.

J. Bedford, A. Brandhuber, B. J. Spence, and G. Travaglini Nucl.Phys. B712 (2005) 59–85.

R. Britto, B. Feng, and P. Mastrolia Phys. Rev. D73 (2006) 105004.

C. F. Berger, Z. Bern, L. J. Dixon, D. Forde, and D. A. Kosower Phys.Rev. D75 (2007) 016006.

Z. Xiao, G. Yang, and C.-J. Zhu Nucl.Phys. B758 (2006) 53–89.

P. Mastrolia Phys. Lett. B678 (2009) 246–249.

H. Elvang, Y.-t. Huang, and C. Peng JHEP 1109 (2011) 031.

J. Drummond, J. Henn, G. Korchemsky, and E. Sokatchev Nucl.Phys. B869 (2013) 452–492.

L. J. Dixon, J. M. Henn, J. Plefka, and T. Schuster JHEP 1101 (2011) 035.

S. Badger, B. Biedermann, and P. Uwer Comput.Phys.Commun. 182 (2011) 1674–1692.

Documents

All one-loop NMHV gluon amplitudes in N = 1 SYMaochirov/talks/CERN2013.pdf · MHV NMHV N = 4 SYM n-point by Bern,Dixon, Dunbar, Kosower ’94 [1] n-point by Bern, Dixon, Kosower ’04