[PPT] - On a hidden structure in the MHV Lagrangian Piotr Kotko IFJ PAN PowerPoint Presentation

SLIDE 1

On a hidden structure in the MHV Lagrangian

Piotr Kotko

IFJ PAN based on: P .K., A. Stasto, JHEP 1709 (2017) 047 supported by: DEC-2013/10/E/ST2/00656

THE HENRYK NIEWODNICZAŃSKI

INSTITUTE OF NUCLEAR PHYSICS POLISH ACADEMY OF SCIENCES

REF2017, Madrid

SLIDE 2

MOTIVATION

Amplitudes in QCD

crucial ingredients of factorization approaches (collinear and High Energy

(or kT) Factorization)

a lot of progress has been made, especially at tree-level
Feynman diagrams (not efficient in general)
recurrence methods
geometry (eg. amplituhedron in supersymmetric theory)
still a tip of the iceberg

1

SLIDE 3

MOTIVATION

Amplitudes in QCD

crucial ingredients of factorization approaches (collinear and High Energy

(or kT) Factorization)

a lot of progress has been made, especially at tree-level
Feynman diagrams (not efficient in general)
recurrence methods
geometry (eg. amplituhedron in supersymmetric theory)
still a tip of the iceberg

In this talk

1 We will forget about ordinary Feynman rules

(in favor of much more effective Cachazo-Svrcek-Witten (CSW) rules).

2 I’ll discuss what’s truly hidden in the Lagrangian generating the CSW rules

(the MHV Lagrangian – equivalent to standard Yang-Mills), and I’ll mention some consequences. 1

SLIDE 4

On-shell Amplitudes (1)

Collinear Factorization dσAB→n+X ∼ dxA xA dxB xB

a,b

fa/A (xA, µ) fb/B (xB, µ) dσab→n (xA, xB) dσab→n ∼ |M|2 dPS M – on-shell amplitude (on-shell limit of the amputated momentum space Green’s function) Gluon amplitudes: Ma1...an ελ1

1 , . . . , ελn n

=

. . .

k1 k2 a1, ελ1

1

a2, ελ2

2

ai – color index, ελi

i – polarization vector of gluon with momentum ki and helicity λi.

Ward identities: Ma1...an ελ1

1 , . . . , ki, . . . , ελn n

= 0

2

SLIDE 5

On-shell Amplitudes (2)

Selected methods of calculating on-shell amplitudes Tree-level techniques (some extended to NLO):

Berends-Giele recursion relations1
Britto-Cachazo-Feng-Witten (BCFW) recursion relations2
Cachazo-Svrcek-Witten (CSW) method3

Specific for loop corrections:

integrand reduction method4
generalized unitarity5

Automatization:

efficient computer codes for any tree-level amplitude
most amplitudes can be calculated automatically at NLO

1 F.A. Berends, W.T. Giele, Nucl.Phys. B306 (1988) 759-808 2 R. Britto, F. Cachazo, B. Feng, E. Witten, Phys.Rev.Lett. 94 (2005) 181602 3 F. Cachazo, P

. Svrcek, E. Witten, JHEP 0409 (2004) 006

4 G. Ossola, C.G. Papadopoulos, R. Pittau, Nucl.Phys. B763 (2007) 147-169 5 Z. Bern, L.J. Dixon, D.C. Dunbar, D.A. Kosower, Nucl.Phys. B425 (1994) 217; Nucl.Phys. B435 (1995) 59

3

SLIDE 6

Off-shell Amplitudes (1)

High Energy Factorization (HEF) dσAB→n+X ∼ dxA xA dxB xB d2kTAd2kTB Fg/A(xA, kTA) Fg/B(xB, kTB) dσg∗g∗→n (xA, xB, kTA, kTB)

+ . . . +

· · ·

. . . . . .

· · ·

pA pB kB kA

connected

k 2

A 0,

k 2

B 0

kA = xApA + kTA kB = xBpB + kTB dσg∗g∗→n ∼

˜

M

2 dPS

˜ M – off-shell gauge invariant amplitude. 4

SLIDE 7

Off-shell Amplitudes (2)

Methods of calculating off-shell amplitudes Tree-level techniques:

Lipatov’s effective action1
Analog of the Berends-Giele recursion relation (one off-shell leg)2
the “eikonalization” method3
Matrix Elements of straight infinite Wilson lines4
generalization of the BCFW recursion relations5

Specific for loop corrections:

generalization of the “eikonalization” method to one loop6

Automatization:

efficient computer code for any tree-level amplitude7

1 E. Antonov, L. Lipatov, E. Kuraev, I. Cherednikov, Nucl.Phys. B721 (2005) 111-135 2 A. van Hameren, P

.K., K. Kutak, JHEP 1212 (2012) 029

3 A. van Hameren, PK, K. Kutak, JHEP 1301 (2013) 078 4 P

.K., JHEP 1407 (2014) 128

5 A. van Hameren, JHEP 1407 (2014) 138, A. van Hameren, M. Serino, JHEP 1507 (2015) 010,

K. Kutak, A. van Hameren, M. Serino, JHEP 1702 (2017) 009

6 A. van Hameren, arXiv:1710.07609 7 A. van Hameren, arXiv:1611.00680

5

SLIDE 8

MHV amplitudes (1)

The helicity amplitudes with growing complexity

Vanishing amplitudes: (±, +, . . . , +)
Maximally Helicity Violating (MHV) amplitudes: (−, −, + . . . , +)
Next-to-MHV (NMHV) amplitudes (−, −, −, + . . . , +)
Next-to-next-to-MHV (NNMHV) amplitudes, and so on...

6

SLIDE 9

MHV amplitudes (1)

The helicity amplitudes with growing complexity

Vanishing amplitudes: (±, +, . . . , +)
Maximally Helicity Violating (MHV) amplitudes: (−, −, + . . . , +)
Next-to-MHV (NMHV) amplitudes (−, −, −, + . . . , +)
Next-to-next-to-MHV (NNMHV) amplitudes, and so on...

Color decomposition1 Ma1...an ελ1

1 , . . . , ελn n

=
noncyclic

permutations

Tr (ta1 . . . tan) M

1λ1, . . . , nλn

For example for four-point MHV amplitude Ma1a2a3a4 ε−

1, ε− 2, ε+ 3 , ε+ 4

k1

− − k2 k3 k4 + +

M(1−, 2−, 3+, 4+) =

k1 − + k3 k2 k4 − +

M(1−, 3+, 2−, 4+) = and so on.

1 M.L. Mangano, S.J. Parke, Phys.Rep. 200 301-367

6

SLIDE 10

MHV amplitudes (3)

Spinor algebra

Spinor products

ij = u− (ki) u+ (kj) ≡ ǫαβλα

i λβ j ,

[ij] = u+ (ki) u− (kj) ≡ ǫ ˙

α˙ β˜

λ ˙

α i ˜

λ

˙ β j

where u± (ki) = 1

2 (1 ± γ5) u (ki) and λα i ≡ u+ (ki), ˜

λ ˙

α i ≡ u− (ki).

Momenta ki are light-like. One can express (ki)α ˙

α = λα i ˜

λ ˙

α i .

7

SLIDE 11

MHV amplitudes (3)

Spinor algebra

Spinor products

ij = u− (ki) u+ (kj) ≡ ǫαβλα

i λβ j ,

[ij] = u+ (ki) u− (kj) ≡ ǫ ˙

α˙ β˜

λ ˙

α i ˜

λ

˙ β j

where u± (ki) = 1

2 (1 ± γ5) u (ki) and λα i ≡ u+ (ki), ˜

λ ˙

α i ≡ u− (ki).

Momenta ki are light-like. One can express (ki)α ˙

α = λα i ˜

λ ˙

α i .

Polarization vectors
ε+

i

α ˙

α =

√ 2 λα

q˜

λ ˙

α i

qi , (ε−

i )α ˙ α =

√ 2 ˜ λ ˙

α qλα i

[iq] where q is a null reference momentum. 7

SLIDE 12

MHV amplitudes (3)

Spinor algebra

Spinor products

ij = u− (ki) u+ (kj) ≡ ǫαβλα

i λβ j ,

[ij] = u+ (ki) u− (kj) ≡ ǫ ˙

α˙ β˜

λ ˙

α i ˜

λ

˙ β j

where u± (ki) = 1

2 (1 ± γ5) u (ki) and λα i ≡ u+ (ki), ˜

λ ˙

α i ≡ u− (ki).

Momenta ki are light-like. One can express (ki)α ˙

α = λα i ˜

λ ˙

α i .

Polarization vectors
ε+

i

α ˙

α =

√ 2 λα

q˜

λ ˙

α i

qi , (ε−

i )α ˙ α =

√ 2 ˜ λ ˙

α qλα i

[iq] where q is a null reference momentum. Parke-Taylor amplitudes1 M

1−, 2−, 3+, . . . , n+

= 124 12 23 . . . n1

1 S.J. Parke, T.R. Taylor, Phys.Rev.Lett. 56, 2459 (1986)

7

SLIDE 13

Off-shell gauge invariant MHV amplitudes

Suprising result One off-shell leg pocess g∗g → g . . . g1

+

. . .

N − m

. . .

m m = 1 M − 1

+ . . .

. . .

N − m − k

. . .

k

+

m = 1 M − 2 k = 1 N − m − 1

. . .

m

. . .

M

Mg∗g→g...g

1∗, 2−, 3+, . . . , n+

∼ 1∗24 1∗22334 . . . n − 1nn1∗ Spinor products for off-shell states involve only longitudinal component of the

ff-shell momentum 1∗i = p1i, where k1 = p1 + kT1, k 2

1 0, p2 1 = 0.

Similar formula holds for g∗g∗ → g . . . g 2. Note: it is essential that the amplitude is gauge invariant, otherwise we get a mess!

1 A. van Hameren, P

.K., K. Kutak, JHEP 1212 (2012) 029

2 A. van Hameren, JHEP 1407 (2014) 138

8

SLIDE 14

Cachazo-Svrcek-Witten (CSW) Method (1)

General idea Glue any amplitude from the MHV amplitudes continued off-shell. 9

SLIDE 15

Cachazo-Svrcek-Witten (CSW) Method (1)

General idea Glue any amplitude from the MHV amplitudes continued off-shell. Off-shell continuation of spinors If k is light-like, we have kα ˙

α = λkα˜

λk ˙

α

=⇒ λkα = kα ˙

α˜

λ ˙

α q/ [kq]

where q is auxiliary light-like momentum. If k is off-shell we define the off-shell continuation of spinor in the same way: λ(∗)

kα = kα ˙ α˜

λ ˙

α q

9

SLIDE 16

Cachazo-Svrcek-Witten (CSW) Method (1)

General idea Glue any amplitude from the MHV amplitudes continued off-shell. Off-shell continuation of spinors If k is light-like, we have kα ˙

α = λkα˜

λk ˙

α

=⇒ λkα = kα ˙

α˜

λ ˙

α q/ [kq]

where q is auxiliary light-like momentum. If k is off-shell we define the off-shell continuation of spinor in the same way: λ(∗)

kα = kα ˙ α˜

λ ˙

α q

MHV vertices ij4 12 23 . . . n1 ≡

. . . . . . . . . . . . i− j− + + + +

The spinor products are made from off-shell spinors ij = ǫαβλ(∗)α

i

λ(∗)β

j

. 9

SLIDE 17

Cachazo-Svrcek-Witten (CSW) Method (2)

Example: NMHV amplitude M(1−, 2−, 3−, 4+, 5+)

1− 2− 3− 4+ 5+ = 3− 4+ 5+ 1− 2− + − 5+ 1− 2− + − 3− 4+ 3− 4+ 5+ 1− 2− + − 5+ 1− 2− − + 3− 4+

10

SLIDE 18

Cachazo-Svrcek-Witten (CSW) Method (2)

Example: NMHV amplitude M(1−, 2−, 3−, 4+, 5+)

1− 2− 3− 4+ 5+ = 3− 4+ 5+ 1− 2− + − 5+ 1− 2− + − 3− 4+ 3− 4+ 5+ 1− 2− + − 5+ 1− 2− − + 3− 4+

The result: M

1−, 2−, 3−, 4+, 5+

= [45]4 [12] [23] [34] [45] [51] 10

SLIDE 19

Yang-Mills action on the light-front (1)

Yang-Mills action SY−M = −1 4

d4x Tr FµνFµν

where: Fµν =

i g′ [Dµ, Dν]

Dµ = ∂µ − ig′ ˆ A µ ˆ A µ = A µ

a ta

ta, tb

= i √ 2f abctc 11

SLIDE 20

Yang-Mills action on the light-front (1)

Yang-Mills action SY−M = −1 4

d4x Tr FµνFµν

where: Fµν =

i g′ [Dµ, Dν]

Dµ = ∂µ − ig′ ˆ A µ ˆ A µ = A µ

a ta

ta, tb

= i √ 2f abctc Light-cone coordinates Basis vectors: η = 1 √ 2 (1, 0, 0, −1) , ˜ η = 1 √ 2 (1, 0, 0, 1) , ε±

⊥ =

1 √ 2 (0, 1, ±i, 0) Contravariant coordinates: v+ = v · η , v− = v · ˜ η , v• = v · ε+

⊥ ,

v⋆ = v · ε−

⊥

Scalar product: u · v = u+w− + u−w+ − u•w⋆ − u⋆w• Three-vectors: x ≡ (x−, x•, x⋆) , p ≡

p+, p•, p⋆

11

SLIDE 21

Yang-Mills action on the light-front (2)

Yang-Mills action in transverse fields only

Light cone gauge:

A · η = A + = 0

Integration of A − fields out of the action

12

SLIDE 22

Yang-Mills action on the light-front (2)

Yang-Mills action in transverse fields only

Light cone gauge:

A · η = A + = 0

Integration of A − fields out of the action

S(LC)

Y−M [A •, A ⋆] =

dx+

L(LC)

+− + L(LC) ++− + L(LC) +−− + L(LC) ++−−

12

SLIDE 23

Yang-Mills action on the light-front (2)

Yang-Mills action in transverse fields only

Light cone gauge:

A · η = A + = 0

Integration of A − fields out of the action

S(LC)

Y−M [A •, A ⋆] =

dx+

L(LC)

+− + L(LC) ++− + L(LC) +−− + L(LC) ++−−

L(LC)

+− [A •, A ⋆] = −

d3x Tr ˆ

A • (x) ˆ A ⋆ (x) L(LC)

++− [A •, A ⋆] = −2ig′

d3x Tr γx ˆ

A • ∂− ˆ A ⋆, ˆ A • L(LC)

−−+ [A •, A ⋆] = −2ig′

d3x Tr γx ˆ

A ⋆ ∂− ˆ A •, ˆ A ⋆ L(LC)

++−− [A •, A ⋆] = −g2

d3x Tr
∂− ˆ

A •, ˆ A ⋆ ∂−2

−

∂− ˆ

A ⋆, ˆ A • where γx = ∂−1

− ∂•,

γx = ∂−1

− ∂⋆,

g′ = g/ √ 2. 12

SLIDE 24

The MHV action (1)

Transformation of fields1 (A •, A ⋆) → (B•, B⋆)

1 P

. Mansfield, JHEP 03 (2006) 037

13

SLIDE 25

The MHV action (1)

Transformation of fields1 (A •, A ⋆) → (B•, B⋆)

1 Transformation is canonical such that B• = B• [A •]

∂−A ⋆

a (x) =

d3y δB•

c (y)

δA •

a (x)∂−B⋆ c (y)

1 P

. Mansfield, JHEP 03 (2006) 037

13

SLIDE 26

The MHV action (1)

Transformation of fields1 (A •, A ⋆) → (B•, B⋆)

1 Transformation is canonical such that B• = B• [A •]

∂−A ⋆

a (x) =

d3y δB•

c (y)

δA •

a (x)∂−B⋆ c (y)

2 The vertex (+ + −) is removed

L(LC)

+− [A •, A ⋆] + L(LC) ++− [A •, A ⋆] = L(LC) +− [B•, B⋆]

d3y Tr

      

D⋆, γy ˆ

A • (y)

tc

       δB•

a (x)

δA •

c (y) = ωxB• a (x)

where ωx = ∂•∂⋆∂−1

− .

1 P

. Mansfield, JHEP 03 (2006) 037

13

SLIDE 27

The MHV action (2)

Solution to the transformations in momentum space ˜ A •

a = ˜

B•

a + ∞

n=2

˜ Ψa{b1...bn}

n

⊗ ˜ B•

b1 . . . ˜

B•

bn

˜ A ⋆

a = ˜

B⋆

a + ∞

n=2

˜ Ωab1{b2...bn}

n

⊗ ˜ B⋆

b1 ˜

B•

b2 . . . ˜

B•

bn

14

SLIDE 28

The MHV action (2)

Solution to the transformations in momentum space ˜ A •

a = ˜

B•

a + ∞

n=2

˜ Ψa{b1...bn}

n

⊗ ˜ B•

b1 . . . ˜

B•

bn

˜ A ⋆

a = ˜

B⋆

a + ∞

n=2

˜ Ωab1{b2...bn}

n

⊗ ˜ B⋆

b1 ˜

B•

b2 . . . ˜

B•

bn

The MHV action S(LC)

Y−M

˜ B•, ˜ B⋆ =

dx+

L(LC)

+− + L(LC) −−+ + · · · + L(LC) −−+···+ + . . .

14

SLIDE 29

The MHV action (2)

Solution to the transformations in momentum space ˜ A •

a = ˜

B•

a + ∞

n=2

˜ Ψa{b1...bn}

n

⊗ ˜ B•

b1 . . . ˜

B•

bn

˜ A ⋆

a = ˜

B⋆

a + ∞

n=2

˜ Ωab1{b2...bn}

n

⊗ ˜ B⋆

b1 ˜

B•

b2 . . . ˜

B•

bn

The MHV action S(LC)

Y−M

˜ B•, ˜ B⋆ =

dx+

L(LC)

+− + L(LC) −−+ + · · · + L(LC) −−+···+ + . . .

where the MHV vertex is:

L(LC)

−−+···+ = ˜

Vb1...bn

−−+···+ ⊗ ˜

B⋆

b1 ˜

B⋆

b2 ˜

B•

b3 . . . ˜

B•

bn

˜ V−−+···+ (p1, . . . , pn) = 1 n! (g′)n−1 p+

1

p+

2

2 ˜ v∗4

21

˜ v∗

1n˜

v∗

n(n−1)˜

v∗

(n−1)(n−2) . . . ˜

v∗

21

with ˜ v∗

(i)(j) = −p• i + p+ i

p•

j /p+ j

– this is a particular off-shell continuation of ij.

14

SLIDE 30

The new solution to the field transformation

The solution B•[A •] Introduce a vector: ε+

α = ε+ ⊥ − αη, =⇒ (−α, −1, 0) ≡ eα

If α = p•/p+ it is a polarization vector for momentum p. 15

SLIDE 31

The new solution to the field transformation

The solution B•[A •] Introduce a vector: ε+

α = ε+ ⊥ − αη, =⇒ (−α, −1, 0) ≡ eα

If α = p•/p+ it is a polarization vector for momentum p. The solution can be expressed as the the Wilson line in A + = 0 gauge: B•

a (x) =

∞

−∞

dα Tr

1

2πig′ ta∂−P exp

ig′

∞

−∞

ds ˆ A • (x + seα)

15

SLIDE 32

The new solution to the field transformation

The solution B•[A •] Introduce a vector: ε+

α = ε+ ⊥ − αη, =⇒ (−α, −1, 0) ≡ eα

If α = p•/p+ it is a polarization vector for momentum p. The solution can be expressed as the the Wilson line in A + = 0 gauge: B•

a (x) =

∞

−∞

dα Tr

1

2πig′ ta∂−P exp

ig′

∞

−∞

ds ˆ A • (x + seα)

Diagrammatically:

ig′ ˆ ˜ B

=

+ + + . . .

where

p

=

i eα·p+iǫ = ig′ ˆ A•

b(p)

15

SLIDE 33

Gauge invariant off-shell currents

Partially reduced Green’s function

. . . + − − + +

ff−shell
n−shell

16

SLIDE 34

Gauge invariant off-shell currents

Partially reduced Green’s function

. . . + − − + +

ff−shell
n−shell

Matrix element of the Wilson line At tree-level the on-shell fields B can be replaced by A. Jn (p1...n) =

d4x eix·p1...n 0| B•

a [A •] (x) |p1, +; p2, −; . . . ; pn, −

where |pi, ± is on-shell gluon state. 16

SLIDE 35

Gauge invariant off-shell currents

Partially reduced Green’s function

. . . + − − + +

ff−shell
n−shell

Matrix element of the Wilson line At tree-level the on-shell fields B can be replaced by A. Jn (p1...n) =

d4x eix·p1...n 0| B•

a [A •] (x) |p1, +; p2, −; . . . ; pn, −

where |pi, ± is on-shell gluon state.

n

. . .

=

. . .

n

igJ =

p1 pn p1...n

+

. . .

n − m

. . .

m m = 1 n − 1

+ . . .

. . .

n − m − k

. . .

k

. . .

m

+

m = 1 n − 2 k = 1 n − m − 1

J satisfies the Ward identities. 16

SLIDE 36

Relation to High Energy Factorization

There is a striking analogy between MHV vertexes and off-shell amplitudes

The form of the vertex and the off-shell amplitude is the same, but different

continuations of spinor products appear (note: a different gauge is used in both approaches)

Topologically, the diagrammatic content is the same

Further questions

What if we take the high energy limit of the MHV action (ala Lipatov’s

effective action)?

Is there an analog of the CSW contruction for off-shell amplitudes?

17

SLIDE 37

Summary & Outlook

Summary

The MHV Lagrangian is equivalent to the Yang Mills action but contains

vertices which are off-shell continuations of the MHV amplitudes.

Most of the fields in this new form of the action are specific Wilson lines.
There seems to be a relation between amplitudes used in High Energy

Factorization and the MHV vertices.

Not discussed:
Interpretation in terms of light-front diagrams and related applications.
Inverse of the Wilson line (inverse field transformation) and its

geometric interpretation. 18

SLIDE 38

Summary & Outlook

Summary

The MHV Lagrangian is equivalent to the Yang Mills action but contains

vertices which are off-shell continuations of the MHV amplitudes.

Most of the fields in this new form of the action are specific Wilson lines.
There seems to be a relation between amplitudes used in High Energy

Factorization and the MHV vertices.

Not discussed:
Interpretation in terms of light-front diagrams and related applications.
Inverse of the Wilson line (inverse field transformation) and its

geometric interpretation. Further directions

Self-dual Yang Mills
2D integrable models
Loop corrections – can the Wilson lines help?

18

SLIDE 39

BACKUP

SLIDE 40

The diagrammatic content of transformations (1)

Solution B•[A •]

+ 1

s2

+

1 s2s3

+

1 s2s3

B• = +

1 s2s3

+ . . .

Vertical dashed lines – energy denominators:

D1...i = 2        Einitial −

j ∈ intermediate

Ej         , Ep = p⋆p• p+

Triple gluon vertices – helicity (− + +).

˜ Γn (P; p1, . . . , pn) = 1 n! (−g′)n−1 1 ˜ v∗

1(1...n)˜

v∗

(12)(1...n) . . . ˜

v∗

(1...n−1)(1...n)

δ3 (p1...n − P) where p1...i ≡ p1 + · · · + pi. ˜ Γn has an interpretation of the gluon wave function1.

1 L. Motyka, A. Stasto, Phys.Rev.D 79 (2009) 08016

20

SLIDE 41

The diagrammatic content of transformations (2)

Inverse solution A •[B•]

+ 1

s2

+

1 s2s3

+

1 s2s3

A• = +

1 s2s3

+ . . .

Vertical double-dashed lines – redefined energy denominators:

˜ D1...i = 2        

i ∈ final

Ei −

j ∈ intermediate

Ej         , Ep = p⋆p• p+

Triple gluon vertices – helicity (− + +), same as before.

˜ Ψn (P; p1, . . . , pn) = − 1 n! (−g′)n−1 ˜ v∗

(1...n)1

˜ v∗

1(1...n)

1 ˜ v∗

n(n−1) . . . ˜

v∗

32 ˜

v∗

21

δ3 (p1...n − P) ˜ Ψn has an interpretation of the gluon fragmentation amplitude1.

1 L. Motyka, A. Stasto, Phys.Rev.D 79 (2009) 08016

21

SLIDE 42

Gauge invariant off-shell currents

Light-front recurrence relation for off-shell MHV current1,2,3

k(1...j) k(j+1...N) + k(1...N) + +

j = 2 N − 1

− + + k1 k2 kj + + + kj+1 kj+2 kN k(1...j) k(j+1...N) + k(1...N) − + − + + k1 k2 kj + + + kj+1 kj+2 kN

j = 1 N − 1

k(1...i) k(j+1...N) − +

j = 2 N − 1 i = 1 j − 1

k(i+1...j) + + k(1...N) − + + k1 k2 ki + + + ki+1 ki+2 kj + + + kj+1 kj+2 kN k(1...i) − k(i+1...j) + k(j+1...N) +

j = 2 N − 1 i = 1 j − 1

+ k(1...N) − + + k1 k2 ki + + + ki+1 ki+2 kj + + + kj+1 kj+2 kN 1 C. Cruz-Santiago and A. Stasto, Nucl.Phys.B 875 (2013) 368-387 2 C. Cruz-Santiago, P

. Kotko, A. Stasto, Nucl.Phys. B895 (2015) 132-160

3 P

. Kotko, M. Serino, A. Stasto, JHEP 1608 (2016) 026

22

SLIDE 43

Gauge invariant off-shell currents

Light-front recurrence relation for off-shell MHV current1,2,3

k(1...j) k(j+1...N) + k(1...N) + +

j = 2 N − 1

− + + k1 k2 kj + + + kj+1 kj+2 kN k(1...j) k(j+1...N) + k(1...N) − + − + + k1 k2 kj + + + kj+1 kj+2 kN

j = 1 N − 1

k(1...i) k(j+1...N) − +

j = 2 N − 1 i = 1 j − 1

k(i+1...j) + + k(1...N) − + + k1 k2 ki + + + ki+1 ki+2 kj + + + kj+1 kj+2 kN k(1...i) − k(i+1...j) + k(j+1...N) +

j = 2 N − 1 i = 1 j − 1

+ k(1...N) − + + k1 k2 ki + + + ki+1 ki+2 kj + + + kj+1 kj+2 kN

J (−−+···+)

n

(p1...n) = Jn (p1...n) − ig′

n−1

j=2

Jj (p1...j) p+

1...n

p+

j+1...n˜

v∗

(1...j)(j+1)

J (−+···+)

n−j

(pj+1...n)

1 C. Cruz-Santiago and A. Stasto, Nucl.Phys.B 875 (2013) 368-387 2 C. Cruz-Santiago, P

. Kotko, A. Stasto, Nucl.Phys. B895 (2015) 132-160

3 P

. Kotko, M. Serino, A. Stasto, JHEP 1608 (2016) 026

22

SLIDE 44

Inverse transformation in position space (1)

Inverse to the path-ordered exponential A •

a = Tr

i g′ ta∂−U g′ 2π ˆ B•

U

ˆ φ

=

∞

n=1
ds1dα1 ˆ

φ (x+s1eα1)

n

i=2
dsidαi

−∞

dτi−1∂−ˆ φ (x+τi−1eαi−1 +sieαi) where eα = (−α, −1, 0) [recall x ≡ (x−, x•, x⋆)]. The n-th term in the expansion: U ˆ φ (n) =

ds1 . . . dsn
dα1 . . . dαn

−∞

dτ1 . . . dτn−1 ˆ φ (x+s1eα1) ∂−ˆ φ (x+τ1eα1 +s2eα2) ∂−ˆ φ

x+τ2eα2 +s3eα3
. . .

23

SLIDE 45

Geometric interpretation (1)

Define a new object pα (τ, α′) = ∂− +∞

−∞

ds φ (x + τeα′ + seα)

The Wilson line B• [A •] can be expressed in terms of pα.

Set φ = ˆ A •. The n-th term in expansion:

dα
dα1 . . . dαn

+∞

−∞

dτ1 τ1

−∞

dτ2 . . . τn−1

−∞

dτn ˆ pα1 (τ1, α) ˆ pα2 (τ2, α) . . . ˆ pαn (τn, α)

ℓ(n)

α

The inverse functional A • [B•]

Set φ = ˆ B•. The n-th term in expansion:

dα
dα1 . . . dαn

+∞

−∞

dτ1

−∞

dτ2 . . .

−∞

dτn ˆ pα1 (τ1, α) ˆ pα2 (τ2, α1) ˆ pα3 (τ3, α2) . . .

L(n)

α

24

SLIDE 46

Geometric interpretation (2)

Vector field in the 2D space Consider 2D space made of points (a⋆, a−) , a⋆, a− ∈ R. In that space eα → (1, α)

a− a∗ (1, α1) 1 α1 α2 (1, α2)

The object pα (τ, α′) can be thought of as a vector attached in a point (τ, τα′) and having a direction given by α.

a− a∗ τ τα′ α′ pα(τ, α′)

25

SLIDE 47

Geometric interpretation (3)

Objects ℓ(n)

α

and L(n)

α

in 2D space

a− a∗ (1, α) pα1(τ1, α) pα2(τ2, α) pα3(τ3, α) pα4(τ4, α) pα5(τ5, α) pα6(τ6, α) pαn(τn, α)

ℓ(n)

α

26

SLIDE 48

Geometric interpretation (3)

Objects ℓ(n)

α

and L(n)

α

in 2D space

a− a∗ pα1(τ1, α) α1 α α2 pα2(τ2, α1) pα3(τ3, α2) α3 pα4(τ4, α3)

L(n)

α

26

SLIDE 49

Geometric interpretation (3)

Objects ℓ(n)

α

and L(n)

α

in 2D space

10
5

5 10

5

5

10
8
6
4
2
5

5

26