New geometric structures in scattering amplitudes The anatomy of - - PowerPoint PPT Presentation

New geometric structures in scattering amplitudes ——————— The anatomy of scattering amplitudes in pure spinor superspace ——————— Oliver Schlotterer (AEI Potsdam)

based on arXiv:1404.4986, arXiv:1408.3605: C. Mafra, OS and work in progress with M. Green and C. Mafra 22.09.2014

1

Goal of this talk

• framework for amplitudes of gluon and graviton multiplet in 10 dim
• both field theory and string theory, both type IIA and type IIB
• manifest supersymmetry from pure spinor formalism

2

Goal of this talk

• framework for amplitudes of gluon and graviton multiplet in 10 dim
• both field theory and string theory, both type IIA and type IIB
• manifest supersymmetry from pure spinor formalism

Intuitive mapping between

• cubic diagrams and kinematic factors
• kinematic factors and worldsheet functions

Make essential use of BRST symmetry in the pure spinor formalism

[N. Berkovits hep-th/0001035]

3

Pure spinor superspace Bosonic pure spinor λα defined by algebraic constraint λα γm

αβ λβ

= ∀ m = 0, 1, . . . , 9 Pure spinor superspace (PSS) {xm, θα, λβ} with component prescription (λγmθ) (λγnθ) (λγpθ) (θγmnpθ) = 1 BRST invariant & supersymmetric and automated in [C. Mafra 1007.4999]

4

Pure spinor superspace Bosonic pure spinor λα defined by algebraic constraint λα γm

αβ λβ

= ∀ m = 0, 1, . . . , 9 Pure spinor superspace (PSS) {xm, θα, λβ} with component prescription (λγmθ) (λγnθ) (λγpθ) (θγmnpθ) = 1 BRST invariant & supersymmetric and automated in [C. Mafra 1007.4999] Driving force towards amplitudes in PSS: BRST charge ↔ eq’s of motion Q ≡ λαDα = λα ∂ ∂θα + 1

2 km (γmθ)α

• descends from gauge fixing worldsheet action [see Nathan’s talk and 1409.2510]

5

Outline gluon & gluino polariz. em, χα scattering amplitudes

6

Outline gluon & gluino polariz. em, χα 10 dim N = 1 SYM superfields scattering amplitudes Aα(x, θ) = eik·x

1 2 em (γmθ)α

−1

3(χγmθ)(γmθ)α + θ3ke

+θ4χk + θ5k2e + . . .

7

Outline gluon & gluino polariz. em, χα 10 dim N = 1 SYM superfields multiparticle superfields scattering amplitudes Aα(x, θ) = eik·x

1 2 em (γmθ)α

−1

3(χγmθ)(γmθ)α + θ3ke

+θ4χk + θ5k2e + . . .

• part I based on 1404.4986

8

Outline gluon & gluino polariz. em, χα 10 dim N = 1 SYM superfields multiparticle superfields Berends–Giele currents scattering amplitudes Aα(x, θ) = eik·x

1 2 em (γmθ)α

−1

3(χγmθ)(γmθ)α + θ3ke

+θ4χk + θ5k2e + . . .

• part I based on 1404.4986

part II based on 1404.4986

9

Outline gluon & gluino polariz. em, χα 10 dim N = 1 SYM superfields multiparticle superfields Berends–Giele currents BRST (pseudo-)invariants scattering amplitudes Aα(x, θ) = eik·x

1 2 em (γmθ)α

−1

3(χγmθ)(γmθ)α + θ3ke

+θ4χk + θ5k2e + . . .

• part I based on 1404.4986

part II based on 1404.4986 part III based on 1408.3605 part IV based on [to appear]

10

• I. Multiparticle superfields

Vertex operators for SYM states (unintegrated and integrated) V1 ≡ λαA1

α ,

U1 ≡ ∂θαA1

α + ΠmAm 1 + dαW α 1 + 1 2NmnF mn 1

superfields with known θ expansion ⊗ h = 1 fields as “bookkeeping var’s”

11

• I. Multiparticle superfields

Vertex operators for SYM states (unintegrated and integrated) V1 ≡ λαA1

α ,

U1 ≡ ∂θαA1

α + ΠmAm 1 + dαW α 1 + 1 2NmnF mn 1

superfields with known θ expansion ⊗ h = 1 fields as “bookkeeping var’s” BRST invariance QV1 = 0 and Q

• U1 =

∂

∂zV1 = 0 equivalent to

equations of motion (since Q = λαDα on superfields) [E. Witten 1986] 2D(αA1

β) = γm αβA1 m

DαA1

m = (γmW1)α + k1 mA1 α

DαW β

1 = 1 4(γmn)αβF mn 1

DαF mn

1

= 2k[m

1 (γn]W1)α

12

• I. Multiparticle superfields

Vertex operators for SYM states (unintegrated and integrated) V1 ≡ λαA1

α ,

U1 ≡ ∂θαA1

α + ΠmAm 1 + dαW α 1 + 1 2NmnF mn 1

superfields with known θ expansion ⊗ h = 1 fields as “bookkeeping var’s” BRST invariance QV1 = 0 and Q

• U1 =

∂

∂zV1 = 0 equivalent to

equations of motion (since Q = λαDα on superfields) [E. Witten 1986] 2D(αA1

β) = γm αβA1 m

components DαA1

m = (γmW1)α + k1 mA1 α

⇓ DαW β

1 = 1 4(γmn)αβF mn 1

kmem = 0 DαF mn

1

= 2k[m

1 (γn]W1)α

kαβχβ = 0

13

Define multiparticle superfields via OPE (where zij ≡ zi − zj) U1(z1)U2(z2) ∼ zα′k1·k2−1

12

• ∂θαA12

α +ΠmAm 12+dαW α 12+1 2NmnF mn 12

• + ∂

∂zi(. . .)

Same structure as U1 = ∂θαA1

α + ΠmAm 1 + dαW α 1 + 1 2NmnF mn 1

with A12

α = 1 2

• A2

α(k2 · A1) + A2 m(γmW 1)α − (1 ↔ 2)

14

Define multiparticle superfields via OPE (where zij ≡ zi − zj) U1(z1)U2(z2) ∼ zα′k1·k2−1

12

• ∂θαA12

α +ΠmAm 12+dαW α 12+1 2NmnF mn 12

• + ∂

∂zi(. . .)

Same structure as U1 = ∂θαA1

α + ΠmAm 1 + dαW α 1 + 1 2NmnF mn 1

with A12

α = 1 2

• A2

α(k2 · A1) + A2 m(γmW 1)α − (1 ↔ 2)

• A12

m = 1 2

• A1

pF 2 pm − A1 m(k1 · A2) + (W 1γmW 2) − (1 ↔ 2)

• W α

12 = 1 4(γmnW 2)αF 1 mn + W α 2 (k2 · A1) − (1 ↔ 2)

F 12

mn = F 2 mn(k2 · A1) + F 2 [m pF 1 n]p + 2k1 [m(W 1γn]W 2) − (1 ↔ 2)

15

Define multiparticle superfields via OPE (where zij ≡ zi − zj) U1(z1)U2(z2) ∼ zα′k1·k2−1

12

• ∂θαA12

α +ΠmAm 12+dαW α 12+1 2NmnF mn 12

• + ∂

∂zi(. . .)

Same structure as U1 = ∂θαA1

α + ΠmAm 1 + dαW α 1 + 1 2NmnF mn 1

with A12

α = 1 2

• A2

α(k2 · A1) + A2 m(γmW 1)α − (1 ↔ 2)

• A12

m = 1 2

• A1

pF 2 pm − A1 m(k1 · A2) + (W 1γmW 2) − (1 ↔ 2)

• W α

12 = 1 4(γmnW 2)αF 1 mn + W α 2 (k2 · A1) − (1 ↔ 2)

F 12

mn = F 2 mn(k2 · A1) + F 2 [m pF 1 n]p + 2k1 [m(W 1γn]W 2) − (1 ↔ 2)

⇓ Four superfield rep’s

• f the cubic vertex

2 1 . . . ↔ A12

α , Am 12, W α 12, F mn 12

16

V12 ≡ λαA12

α ,

U12 ≡ ∂θαA12

α + ΠmAm 12 + dαW α 12 + 1 2NmnF mn 12

Two-particle EOM ∼ = single-particle EOM ... 2D(αA1

β) = γm αβA1 m

DαA1

m = (γmW1 )α + k1 mA1 α

DαW β

1

= 1

4(γmn)αβF mn 1

DαF mn

1

= 2k[m

1 (γn]W1 )α

17

V12 ≡ λαA12

α ,

U12 ≡ ∂θαA12

α + ΠmAm 12 + dαW α 12 + 1 2NmnF mn 12

Two-particle EOM ∼ = single-particle EOM up to contact terms ∼ (k1 · k2) 2D(αA12

β) = γm αβA12 m + (k1 · k2)(A1 αA2 β − A2 αA1 β)

DαA12

m = (γmW12)α + k12 mA12 α + (k1 · k2)(A1 αA2 m − A2 αA1 m)

DαW β

12 = 1 4(γmn)αβF mn 12

+ (k1 · k2)(A1

αW β 2 − A2 αW β 1 )

DαF mn

12

= 2k[m

12 (γn]W12)α + (k1 · k2)(A1 αF mn 2

− A2

αF mn 1

) + 2(k1 · k2)(A[n

1 (γm]W2)α − A[n 2 (γm]W1)α)

where km

12 ≡ km 1 + km 2

18

V12 ≡ λαA12

α ,

U12 ≡ ∂θαA12

α + ΠmAm 12 + dαW α 12 + 1 2NmnF mn 12

Two-particle EOM ∼ = single-particle EOM up to contact terms ∼ (k1 · k2) 2D(αA12

β) = γm αβA12 m + (k1 · k2)(A1 αA2 β − A2 αA1 β)

DαA12

m = (γmW12)α + k12 mA12 α + (k1 · k2)(A1 αA2 m − A2 αA1 m)

DαW β

12 = 1 4(γmn)αβF mn 12

+ (k1 · k2)(A1

αW β 2 − A2 αW β 1 )

DαF mn

12

= 2k[m

12 (γn]W12)α + (k1 · k2)(A1 αF mn 2

− A2

αF mn 1

) + 2(k1 · k2)(A[n

1 (γm]W2)α − A[n 2 (γm]W1)α)

BRST invariance QV1 = 0 and Q

• U1 =

∂

∂zV1 replaced by covariance

QV12 = (k1 · k2)V1V2 , QU12 = ∂ ∂zV12 + (k1 · k2)(V1U2 − V2U1)

19

More particles by recursion V12 = 1

2

• V2(k2 · A1 ) + A2

m(λγmW1 ) − (1 ↔ 2)

• 2

1 . . . k12

20

More particles by recursion (replacing [1,2] by [12,3])

• V123 = 1

2

• V3(k3 · A12) + A3

m(λγmW12) − (12 ↔ 3)

• 2

1 3 . . . k12 k123 BRST variation cancels propagators ∼ k2

12, k2 123 of the cubic diagram

Q V123 = 1

2(k2 123 − k2 12)V12V3 + 1 2k2 12(V1V23 − V2V13)

21

More particles by recursion (replacing [1,2] by [12,3])

• V123 = 1

2

• V3(k3 · A12) + A3

m(λγmW12) − (12 ↔ 3)

• 2

1 3 . . . k12 k123 BRST variation cancels propagators ∼ k2

12, k2 123 of the cubic diagram

Q V123 = 1

2(k2 123 − k2 12)V12V3 + 1 2k2 12(V1V23 − V2V13)

Moreover – totally antisymmetric component is BRST closed and exact V123 = V123 + QH[123] = ⇒ V123 + V231 + V312 = 0 Reproduce Jacobi identity among color tensors f12afa3b +cyc(1, 2, 3) = 0 = ⇒ evidence for duality between color and kinematics

[Bern, Carrasco, Johansson 0805.3993] [Mafra, OS, Stieberger 1104.5224]

22

In multiparticle cases B = 12 . . . p ∃ 4 representatives of cubic tree diag’s 2 1 k12 3 k123 4 . . . p k12...p . . . ↔ AB

α , Am B, W α B, F mn B

with EOM of A1

α, Am 1 , W α 1 , F mn 1

up to contact terms ∼ k2

12...j

23

In multiparticle cases B = 12 . . . p ∃ 4 representatives of cubic tree diag’s 2 1 k12 3 k123 4 . . . p k12...p . . . ↔ AB

α , Am B, W α B, F mn B

with EOM of A1

α, Am 1 , W α 1 , F mn 1

up to contact terms ∼ k2

12...j

• iterate recursions, e.g. for

VB = λα AB

α

• V12...p = 1

2

• Vp(kp · A12...p−1) + Ap

m(λγmW12...p−1) − (12 . . . p − 1 ↔ p)

• discard BRST trivial components VB =

VB + Q(. . .) “by hand” = ⇒ symmetries of dual color tensor f12afa3bfb4c . . . fypz

24

Outline gluon & gluino polariz. em, χα 10 dim N = 1 SYM superfields multiparticle superfields Berends–Giele currents BRST (pseudo-)invariants scattering amplitudes Aα(x, θ) = eik·x

1 2 em (γmθ)α

−1

3(χγmθ)(γmθ)α + θ3ke

+θ4χk + θ5k2e + . . .

• AB

α , Am B, W α B, F mn B

• . . .

. . .

25

• II. Berends–Giele currents

Motivation: Package tree level subdiagrams and clean up BRST variation QV123 = (s123 − s12)V12V3 + s12(V1V23 − V2V13) with sij = (ki · kj)

26

• II. Berends–Giele currents

Motivation: Package tree level subdiagrams and clean up BRST variation QV123 = (s123 − s12)V12V3 + s12(V1V23 − V2V13) with sij = (ki · kj) Both can be achieved by assembling color ordered tree with off-shell leg M123 = k123 3 . . . 1 = 2 tree 2 1 k12 3 k123 . . . + 3 2 k23 1 k123 . . .

• s-channel:

V123 s12s123

• t-channel:

V321 s23s123

• 4pt tree

27

• II. Berends–Giele currents

Motivation: Package tree level subdiagrams and clean up BRST variation QV123 = (s123 − s12)V12V3 + s12(V1V23 − V2V13) with sij = (ki · kj) Both can be achieved by assembling color ordered tree with off-shell leg M123 = k123 3 . . . 1 = 2 tree 2 1 k12 3 k123 . . . + 3 2 k23 1 k123 . . .

• s-channel:

V123 s12s123

• t-channel:

V321 s23s123

• 4pt tree

Reward: Q action simply deconcatenates (with M12 ≡ V12

s12 and M1 = V1)

QM123 = M12M3 + M1M23

28

p-particle BG current M12...p ↔ (p + 1)-point tree (color ordered): M12...p ← → p 1 2 . . . tree k12...p =

• cubic diags

Vi1i2...ip (skl)p−1

• Mandelstams skl in QV12...p replaced by deconcatenation

QM12...p =

p−1

• j=1

M12...jMj+1...p =

• XY =12...p

MXMY

29

p-particle BG current M12...p ↔ (p + 1)-point tree (color ordered): M12...p ← → p 1 2 . . . tree k12...p =

• cubic diags

Vi1i2...ip (skl)p−1

• Mandelstams skl in QV12...p replaced by deconcatenation

QM12...p =

p−1

• j=1

M12...jMj+1...p =

• XY =12...p

MXMY Cancels overall propagator: M12...p ∼ 1 s12...p , QM12...p regular in s12...p

30

Application: color ordered SYM trees Atree(1, 2, . . . , N) =

N−2

• p=1

M12...p Mp+1...N−1 VN =

N−2

• p=1

p 1 2 M p VN p + 1 p + 2 N − 1 M N−p−1 fixed by pole structure and BRST invariance !

[Mafra, OS, Stieberger, Tsimpis 1012.3981]

31

Application: color ordered SYM trees Atree(1, 2, . . . , N) =

N−2

• p=1

M12...p Mp+1...N−1 VN =

N−2

• p=1

p 1 2 M p VN kN p + 1 p + 2 N − 1 M N−p−1 fixed by pole structure and BRST invariance !

[Mafra, OS, Stieberger, Tsimpis 1012.3981]

• divergent propagator M12...N−1 ∼

1 s12...N−1 ∼ 1 k2

N

→ ∞ avoids Q-exactness “Atree(1, 2, . . . , N) = Q(M12...N−1VN)”

• components fully explicit after λ3θ5 = 1

32

∃ further BG currents Am

B, Wα B, Fmn B

for Am

B, W α B, F mn B

with B = 12 . . . p Am

123 =

Am

123

s12s123 + Am

321

s23s123 “just a change of basis”

33

∃ further BG currents Am

B, Wα B, Fmn B

for Am

B, W α B, F mn B

with B = 12 . . . p Am

123 =

Am

123

s12s123 + Am

321

s23s123 “just a change of basis” ⇒ 4 representatives of tree subamplitudes with off-shell leg MB ← → p 1 2 tree . . .

M

Am

B ←

→ p 1 2 tree . . .

A

Wα

B ←

→ p 1 2 tree . . .

W

Fmn

B

← → p 1 2 tree . . .

F

Infer selection rules from zero mode saturation and BRST symmetry!

34

Q algebra ≡ universal single-particle EOM plus deconcatenation: QMB =

• XY =B

MXMY QAm

B = (λγmWB) + km B MB +

• XY =B

(MXAm

Y − MY Am X)

QWα

B = 1 4 (λγmn)α Fmn B

+

• XY =B

(MXWα

Y − MY Wα X)

QFmn

B

= 2 k[m

B (λγn]WB) +

• XY =B

(MXFmn

Y

− MY Fmn

X )

+ 2

• XY =B

(A[n

X(λγm]WY ) − A[n Y (λγm]WX))

35

Q algebra ≡ universal single-particle EOM plus deconcatenation: QMB =

• XY =B

MXMY QAm

B = (λγmWB) + km B MB +

• XY =B

(MXAm

Y − MY Am X)

QWα

B = 1 4 (λγmn)α Fmn B

+

• XY =B

(MXWα

Y − MY Wα X)

QFmn

B

= 2 k[m

B (λγn]WB) +

• XY =B

(MXFmn

Y

− MY Fmn

X )

+ 2

• XY =B

(A[n

X(λγm]WY ) − A[n Y (λγm]WX))

Simpler as compared to Q{Am

B, W α B, F mn B } ↔ (k12...j−1 · kj).

36

Outline gluon & gluino polariz. em, χα 10 dim N = 1 SYM superfields multiparticle superfields Berends–Giele currents BRST (pseudo-)invariants scattering amplitudes Aα(x, θ) = eik·x

1 2 em (γmθ)α

−1

3(χγmθ)(γmθ)α + θ3ke

+θ4χk + θ5k2e + . . .

• AB

α , Am B, W α B, F mn B

• . . .

. . . MB, Am

B, Wα B, Fmn B

• . . .

tree

37

• III. BRST (pseudo-)invariants at one-loop

Tree level example of string theory input: disk amplitude prescription Atree

string(α′)

=

• disk

V1 VN−1 VN U2 U3 . . . UN−2 Iterated OPEs (UAUB → UC) & (VAUB → VC) ⇒ kinematic pattern Atree

string(α′)

↔ MA MB MC ap a1 a2 tree

M M M

b1 bq tree cr c1 tree

38

• III. BRST (pseudo-)invariants at one-loop

Tree level example of string theory input: disk amplitude prescription Atree

string(α′)

=

• disk

V1 VN−1 VN U2 U3 . . . UN−2 Iterated OPEs (UAUB → UC) & (VAUB → VC) ⇒ kinematic pattern Atree

string(α′)

↔ MA MB MC ap a1 a2 tree

M M M

b1 bq tree cr c1 tree

39

At one-loop: Regulator N and b-ghost leave 3 zero modes unsaturated A1-loop

string (α′)

=

• cylinder

(Nb) × V1 U2 U3 U4 . . . UN−1 UN

• need zero modes dαdβNmn
• OPEs among Ui build up multiparticle vertices

UB ≡ ∂θαAB

α + ΠmAm B + dαWα B + 1 2NmnFmn B

40

At one-loop: Regulator N and b-ghost leave 3 zero modes unsaturated A1-loop

string (α′)

=

• cylinder

(Nb) × V1 U2 U3 U4 . . . UN−1 UN

• need zero modes dαdβNmn
• OPEs among Ui build up multiparticle vertices

UB ≡ ∂θαAB

α + ΠmAm B + dαWα B + 1 2NmnFmn B

Extraction of dαdβNmn ⇒ kinematic pattern A1-loop

string (α′)

↔ MA Wα

B Wβ C Fmn D

• ap

a1 tree d1 dt tree b1 bq tree cr c1 tree

M W W F

41

At one-loop: Regulator N and b-ghost leave 3 zero modes unsaturated A1-loop

string (α′)

=

• cylinder

(Nb) × V1 U2 U3 U4 . . . UN−1 UN

• need zero modes dαdβNmn
• OPEs among Ui build up multiparticle vertices

UB ≡ ∂θαAB

α + ΠmAm B + dαWα B + 1 2NmnFmn B

Extraction of dαdβNmn ⇒ kinematic pattern A1-loop

string (α′)

↔ MA Wα

B Wβ C Fmn D

• ap

a1 tree d1 dt tree b1 bq tree cr c1 tree

M W W F

42

Tensor structure of one-loop kinematic pattern is unique! A1-loop

string (α′)

↔ MA

• ∼λ

MB,C,D

need O(λ2)

• MB,C,D

≡

1 3 (λγmWB) (λγnWC) Fmn D

+ (D ↔ C, B) ap a1 tree d1 dt tree b1 bq tree cr c1 tree

W F M W F W F

43

Tensor structure of one-loop kinematic pattern is unique! A1-loop

string (α′)

↔ MA

• ∼λ

MB,C,D

need O(λ2)

• MB,C,D

≡

1 3 (λγmWB) (λγnWC) Fmn D

+ (D ↔ C, B) ap a1 tree d1 dt tree b1 bq tree cr c1 tree

W F M W F W F

By EOM for Wα

B and Fmn B : BRST covariance

QMB,C,D =

• XY =B

(MX MY,C,D − MY MX,C,D) + (B ↔ C, D)

44

BRST covariant building blocks suitable to construct scalar invariants QMA =

• XY =A

MX MY QMB,C,D =

• XY =B

(MX MY,C,D − MY MX,C,D) + (B ↔ C, D) e.g. at four- and five points, have QC1|B,C,D = 0 for C1|2,3,4 = M1M2,3,4 C1|23,4,5 = M1M23,4,5 + M12M3,4,5 − M13M2,4,5 + − 4 5 2 3 1 4 5 3 2 1 4 5 2 3 1

45

BRST covariant building blocks suitable to construct scalar invariants QMA =

• XY =A

MX MY QMB,C,D =

• XY =B

(MX MY,C,D − MY MX,C,D) + (B ↔ C, D) e.g. at four- and five points, have QC1|B,C,D = 0 for C1|2,3,4 = M1M2,3,4 C1|23,4,5 = M1M23,4,5 + M12M3,4,5

• M1⊗C2|3,4,5

− M13M2,4,5

• M1⊗C3|2,4,5

∃ recursion for C1|B,C,D = M1MB,C,D+ BRST-completion using MA ⊗ MB ≡ MAB concatenation “inverts” Q C1|B,C,D = M1MB,C,D +

• M1 ⊗ Cb1|b2...bp,C,D + 5 others

46

Also need vectors to contract loop momentum ℓm ↔ zero mode of Πm: Mm

A,B,C,D ≡

• MA,B,CAm

D + (D ↔ C, B, A)

• extra zero mode in UD=ΠmAm

D+...

• + Wα

A Wβ B Wγ C Wδ D tm αβγδ

• different b-ghost sector

47

Also need vectors to contract loop momentum ℓm ↔ zero mode of Πm: Mm

A,B,C,D ≡

• MA,B,CAm

D + (D ↔ C, B, A)

• + Wα

A Wβ B Wγ C Wδ D tm αβγδ

Tensor tm

αβγδ along with W4 achieves BRST-covariance

QMm

A,B,C,D =

• XY =A

(MX Mm

Y,B,C,D − MY Mm X,B,C,D)

+km

A MA MB,C,D + (A ↔ B, C, D)

MEMm

A,B,C,D

↔

treeE treeD treeA treeC treeB

ℓ

48

Also need vectors to contract loop momentum ℓm ↔ zero mode of Πm: Mm

A,B,C,D ≡

• MA,B,CAm

D + (D ↔ C, B, A)

• + Wα

A Wβ B Wγ C Wδ D tm αβγδ

Tensor tm

αβγδ along with W4 achieves BRST-covariance

QMm

A,B,C,D =

• XY =A

(MX Mm

Y,B,C,D − MY Mm X,B,C,D)

+km

A MA MB,C,D + (A ↔ B, C, D)

Suitable for vector invariants such as Cm

1|2,3,4,5

= M1Mm

2,3,4,5 +

• km

2 M12 M3,4,5 + (2 ↔ 3, 4, 5)

• 1

5 2 4 3 ℓ +

• + (2 ↔ 3, 4, 5)

4 5 3 2 1

49

Also need vectors to contract loop momentum ℓm ↔ zero mode of Πm: Mm

A,B,C,D ≡

• MA,B,CAm

D + (D ↔ C, B, A)

• + Wα

A Wβ B Wγ C Wδ D tm αβγδ

Tensor tm

αβγδ along with W4 achieves BRST-covariance

QMm

A,B,C,D =

• XY =A

(MX MY,B,C,D − MY MX,B,C,D) +km

A MA MB,C,D + (A ↔ B, C, D)

Suitable for vector invariants such as Cm

1|2,3,4,5

= M1Mm

2,3,4,5 +

• km

2 M12 M3,4,5

• km

2 M1⊗C2|3,4,5

+ (2 ↔ 3, 4, 5)

• ∃ recursion for Cm

1|A,B,C,D in terms of C1|B,C,D and lower point Cm 1|...

with concatenation MA ⊗ MB = MAB “inverting” Q.

50

Higher powers (ℓ)r of loop momentum from r zero modes of Πm ↔ Am

B:

Mm1m2...mr

B1,B2,...,Br+3 ≡ MB1,B2,B3(Ami Bi)r + W4 Bi(Ami Bi)r−1

BRST covariant up to anomalous tensor traces: QMm1m2...mr

B1,B2,...,Br+3 =

• XY =B1

(MX Mm1m2...mr

Y,B2,...,Br+3 − MY Mm1...mr X,B2,...,Br+3)

+ MB1k(m1

B1 Mm2...mr) B2,...,Br+3 + (B1 ↔ B2, . . . , Br+3)

• + δ(m1m2Ym3...mr)

B1,...,Br+3

51

Higher powers (ℓ)r of loop momentum from r zero modes of Πm ↔ Am

B:

Mm1m2...mr

B1,B2,...,Br+3 ≡ MB1,B2,B3(Ami Bi)r + W4 Bi(Ami Bi)r−1

BRST covariant up to anomalous tensor traces: QMm1m2...mr

B1,B2,...,Br+3 =

• XY =B1

(MX Mm1m2...mr

Y,B2,...,Br+3 − MY Mm1...mr X,B2,...,Br+3)

+ MB1k(m1

B1 Mm2...mr) B2,...,Br+3 + (B1 ↔ B2, . . . , Br+3)

• + δ(m1m2Ym3...mr)

B1,...,Br+3

At rank r = 2: supersymmetrization of ε10F 5 from hexagon anomaly

[Berkovits, Mafra 0607187]

YA,B,C,D,E ≡ (λγmWA) (λγnWB) (λγpWC) (WDγmnpWE)

52

Combine various MAMm1...mj

B1,...,Bj+3 to tensor superfields such as . . .

Cmn

1|2,3,4,5,6

= M1 Mmn

2,3,4,5,6 +

• M12 k(m

2

Mn)

3,4,5,6 + (2 ↔ 3, . . . , 6)

• −
• k(m

2 kn) 3 M213 M4,5,6 + (23 ↔ 24, 25, . . . , 56)

• . . . which is BRST invariant up to an anomalous trace

Q Cmn

1|2,3,4,5,6

= − δmn M1 Y2,3,4,5,6 “as invariant as you can get”

53

Combine various MAMm1...mj

B1,...,Bj+3 to tensor superfields such as . . .

Cmn

1|2,3,4,5,6

= M1 Mmn

2,3,4,5,6 +

• M12 k(m

2

Mn)

3,4,5,6 + (2 ↔ 3, . . . , 6)

• −
• k(m

2 kn) 3 M213 M4,5,6 + (23 ↔ 24, 25, . . . , 56)

• . . . which is BRST invariant up to an anomalous trace

Q Cmn

1|2,3,4,5,6

= − δmn M1 Y2,3,4,5,6 “as invariant as you can get” = ⇒ “Pseudo–invariants” at any tensor rank Cm1m2...mr

1|B1,B2,...,Br+3 ≡ M1 Mm1m2...mr B1,B2,...,Br+3 + recursively found completion

QCm1m2...mr

1|B1,B2,...,Br+3 = δmimj

• nly “anomalous” terms MA Ymk...ml

C1,C2,...

54

slots rank C1|A,B,C Cm

1|A,B,C,D

Cmn

1|A,B,C,D,E

Cmnp

1|A,B,C,D,E,F

Cmnpq

1|A,B,C,D,E,F,G single single single trace trace trace double trace

ր ր ր ր ր ր ր ր ր ց ց ց ց ց ց . . . . . . . . . = ⇒ “Pseudo–invariants” at any tensor rank

[C. Mafra, OS 1408.3605]

Cm1m2...mr

1|B1,B2,...,Br+3 ≡ M1 Mm1m2...mr B1,B2,...,Br+3 + recursively found completion

55

Outline gluon & gluino polariz. em, χα 10 dim N = 1 SYM superfields multiparticle superfields Berends–Giele currents BRST (pseudo-)invariants scattering amplitudes Aα(x, θ) = eik·x

1 2 em (γmθ)α

−1

3(χγmθ)(γmθ)α + θ3ke

+θ4χk + θ5k2e + . . .

• AB

α , Am B, W α B, F mn B

• . . .

. . . MB, Am

B, Wα B, Fmn B

• . . .

tree at one–loop: C1|A,B,C and Cm1...mr

1|B1,B2,...,Br+3

56

• IV. Amplitude examples

Four point one-loop amplitudes in SYM and SUGRA: A1–loop(1, 2, 3, 4) = Ibox C1|2,3,4 M1–loop

4

= Ibox C1|2,3,4 ˜ C1|2,3,4 Only the box graph with kinematic factor C1|2,3,4 = s12s23Atree(1, 2, 3, 4) Ibox = ℓ 3 4 2 1 =

• d10ℓ

ℓ2(ℓ − k1)2(ℓ − k12)2(ℓ − k123)2 Originally found from superstring theory.

[Brink, Green, Schwarz 1982]

57

For five point amplitude: have introduced 6 + 1 BRST invariants C1|23,4,5 and (23 ↔ 24, 25, 34, 35, 45)

• ; Cm

1|2,3,4,5

• for left- and right movers =

⇒ capture polarization dependence of 5pt superamplitudes of gauge

gravity

• multiplet of
• SYM & SUGRA

superstring

58

For five point amplitude: have introduced 6 + 1 BRST invariants C1|23,4,5 and (23 ↔ 24, 25, 34, 35, 45)

• ; Cm

1|2,3,4,5

• for left- and right movers =

⇒ capture polarization dependence of 5pt superamplitudes of gauge

gravity

• multiplet of
• SYM & SUGRA

superstring

• Color ordered SYM amplitude:

A1–loop(1, 2, 3, 4, 5) = 1 5 2 4 3 ℓ

• ℓmCm

1|2,3,4,5 + 1 2

• s23C1|23,4,5

+(23 ↔ 24, 25, 34, 35, 45)

• +

+ + C1|23,4,5 C1|2,34,5 C1|2,3,45 1 4 5 2 3 1 2 5 3 4 1 3 2 4 5

59

A1–loop(1, 2, 3, 4, 5) = 1 5 2 4 3 ℓ

• ℓmCm

1|2,3,4,5 + 1 2

• s23C1|23,4,5

+(23 ↔ 24, 25, 34, 35, 45)

• +

+ + C1|23,4,5 C1|2,34,5 C1|2,3,45 1 4 5 2 3 1 2 5 3 4 1 3 2 4 5 Supergravity amplitude follows by squaring pentagon- & box numerators: M1–loop

5

= 1 5 2 4 3 ℓ

• ℓmCm

1|2,3,4,5 + 1 2

• s23C1|23,4,5

+(23 ↔ 24, 25, 34, 35, 45)

• 2

+ +permutations in 2, 3, 4, 5 (not 1) s23C1|23,4,5 ˜ C1|23,4,5 1 4 5 2 3

60

SYM amplitude descends from string theory ancestor A1–loop

string (1, 2, 3, 4, 5)

= ∞ dt t

• dz2 . . . dz5
• i<j

eα′ki·kjGij ×

• ∂

∂z2G23s23 C1|23,4,5 + (23 ↔ 24, 25, 34, 35, 45)

• with worldsheet Green function Gij = G(zi − zj) & cylinder length t.

61

SYM amplitude descends from string theory ancestor A1–loop

string (1, 2, 3, 4, 5)

= ∞ dt t

• dz2 . . . dz5
• i<j

eα′ki·kjGij ×

• ∂

∂z2G23s23 C1|23,4,5 + (23 ↔ 24, 25, 34, 35, 45)

• ≡ Kopen

5

• with worldsheet Green function Gij = G(zi − zj) & cylinder length t.

Closed string is more than naive square of Kopen

5

[M. Green, C. Mafra, OS 1307.3534 & in progress]

M1–loop

string,5

=

• F

d2τ τ5

2

• Tτ

d2z2 . . . d2z5

• i<j

eα′ki·kjGij ×

• Kopen

5

˜ Kopen

5

+ π τ2 Cm

1|2,3,4,5 ˜

Cm

1|2,3,4,5

• with torus Tτ, Teichm¨

uller parameter τ and fundamental domain F.

62

Anomalous six point SYM amplitude

A1–loop(1, 2, . . . , 6) =

• ℓmℓnCmn

1|2,3,4,5,6 d10ℓ

ℓ2 (ℓ − k1)2 (ℓ − k12)2 (ℓ − k123)2 (ℓ − k1234)2 (ℓ − k12345)2 −

• P1|6|2,3,4,5 d10ℓ

(ℓ − k1)2 (ℓ − k12)2 (ℓ − k123)2 (ℓ − k1234)2 (ℓ − k12345)2 + BRST invariant

BRST variations in tensor–hexagon and pentagon ⇒ “super ε10F 5” QCmn

1|2,3,4,5,6 = −δmnV1Y2,3,4,5,6 ,

QP1|6|2,3,4,5 = −V1Y2,3,4,5,6

63

Anomalous six point SYM amplitude

A1–loop(1, 2, . . . , 6) =

• ℓmℓnCmn

1|2,3,4,5,6 d10ℓ

ℓ2 (ℓ − k1)2 (ℓ − k12)2 (ℓ − k123)2 (ℓ − k1234)2 (ℓ − k12345)2 −

• P1|6|2,3,4,5 d10ℓ

(ℓ − k1)2 (ℓ − k12)2 (ℓ − k123)2 (ℓ − k1234)2 (ℓ − k12345)2 + BRST invariant

BRST variations in tensor–hexagon and pentagon ⇒ “super ε10F 5” QCmn

1|2,3,4,5,6 = −δmnV1Y2,3,4,5,6 ,

QP1|6|2,3,4,5 = −V1Y2,3,4,5,6 QA1–loop(1, 2, . . . , 6) = V1Y2,3,4,5,6

• d10ℓ
• 1 − δmnℓmℓn

ℓ2

• ×

1 (ℓ − k1)2 (ℓ − k12)2 (ℓ − k123)2 (ℓ − k1234)2 (ℓ − k12345)2 Even though the integrand formally vanishes, get finite and rational anomaly QA1–loop(1, 2, . . . , 6) = − π5 5! V1Y2,3,4,5,6

64

• V. Conclusion & Outlook
• Identified LEGO–brickstones of tree subamplitudes

AB

α , Am B, W α B, F mn B

• . . .

. . . MB, Am

B, Wα B, Fmn B

• . . .

tree

• found systematic & unique BRST–completion @ tree level and 1–loop

Cm1...mr

1|B1,B2,...,Br+3

= M1 Mm1...mr

B1,B2,...,Br+3 + recursive completion

• BRST non–invariant traces reflect hexagon anomaly

QCm1...mr

1|B1,B2,...,Br+3

= −M1 δ(m1m2 Ym3...mr)

B1,...,Br+3 + recursive completion

65

Work in progress: extend superfield mappings to higher loops, e.g. treeB treeA treeC treeD ← → MA,B|C,D treeA treeD treeB treeC ← → M[A,B]|C|D + ℓm Mm

A,B|C|D

ℓ

66

Work in progress: extend superfield mappings to higher loops, e.g. treeB treeA treeC treeD ← → MA,B|C,D treeA treeD treeB treeC ← → M[A,B]|C|D + ℓm Mm

A,B|C|D

ℓ Thank you for your attention !