An introduction to BCFW recursion relation with colour ... - BCFW.pdf · An introduction to BCFW...

An introduction to BCFW recursion relation withcolour ordered amplitudes

Simon Armstrong

Institute of Particle Physics Phenomenology, Durham University

October 21, 2013

Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation

IoP Three Minute Wonder Competition

Simon Armstrong IPPP

An introduction to BCFW recursion relation with colour ordered amplitudes


Outline

1 Introduction

2 Colour Ordered Amplitudes

3 Spinor Helicity Formalism

4 BCFW Recursion Relation




Outline

1 Introduction







Outline

1 Introduction







Outline

1 Introduction







Colour Ordered Amplitudes

ea

b

c

d

A = −g3f baz f czw f edy . . .





ea

b

c

d

A = −g3f baz f czw f edy . . .




Colour Rearrangements

f abc = − i√2

(tr[T aT bT c ]− tr[T aT cT b]

)T a

ijT a

ij = δi

jδij − 1

Ncδi

jδij (Fierz identity)

f baz f czy f edy = . . .





f abc = − i√2


)

T aijT a

ij = δi

jδij − 1

Ncδi


f baz f czy f edy =i

2√

2

(tr[T bT aT z ]− tr[T bT zT a]

)(tr[T cT zT y ]− tr[T cT yT z ])

(tr[T eT dT y ]− tr[T eT yT d ]

)





f abc = − i√2


)

T aijT a

ij = δi

jδij − 1

Ncδi



2√

2tr[T bT aT z ] tr[T cT zT y ] tr[T eT dT y ]+. . .





f abc = − i√2


)T a

ijT a

ij = δi

jδij − 1

Ncδi



2√

2tr[T bT aT yT c ] tr[T eT dT y ] + . . .





f abc = − i√2


)T a

ijT a

ij = δi

jδij − 1

Ncδi



2√

2tr[T eT dT cT bT a]± permutations




Colour Rearrangements - The Diagram Method

= −

= − 1

NC




Colour Rearrangement Example - The Diagram Method

= − = − + ...

= − + ... = ± permutations





A5 gluon,tree = g3∑

σ∈S5/Z5

tr[T aσ(1)T aσ(2)T aσ(3)T aσ(4)T aσ(5) ]

A5 gluon,tree(σ(1), σ(2), σ(3), σ(4), σ(5))




Colour Ordered Feynman Rules

k1

k2

k3

µ

ν

σ

− 1√2

(gµν(k1− k2)σ + gνσ(k2− k3)µ

+gσµ(k3− k1)ν)

kp

p′

µ i√2γµ

k

p

p′

µ − i√2γµ




Spinor Helicity Formalism

Definition (Helicity Projection Operator)

P± =1± γ5

2

Definition (Helicity Spinors)

u±(k) = P±u(k) =1± γ5

2u(k)

v∓(k) = P±v(k) =1± γ5

2v(k)





Definition (Helicity Projection Operator)

P± =1± γ5

2

Definition (Helicity Spinors)

u±(k) = P±u(k) =1± γ5

2u(k)

v∓(k) = P±v(k) =1± γ5

2v(k)





u(k)⊗ u(k) = v(k)⊗ v(k) = /k

u±(k) = v∓(k)





u(k)⊗ u(k) = v(k)⊗ v(k) = /k

u±(k) = v∓(k)





Definition

u+(ki ) = v−(ki ) ≡∣∣k+

i

⟩≡ |i〉

u−(ki ) = v+(ki ) ≡∣∣k−i ⟩ ≡ |i ]

u+(ki ) = v−(ki ) ≡⟨k+i

∣∣ ≡ [i |

u−(ki ) = v+(ki ) ≡⟨k−i∣∣ ≡ 〈i |




Spinor Helicity Formalism - An Example

k , a k ′, b

p −p′

k , a k ′, b

p −p′





k , a k ′, b

p −p′

k , a k ′, b

p −p′

A2 quark, 2 gluon = −iε∓µ (k ′)ε∓ν (k)

(−⟨p′∓∣∣γν (/p + /k

)γµ∣∣p±⟩

2spk

+〈p′∓|γρ|p±〉

2skk ′

[gµν(k − k ′)ρ − gνρ(2k + k ′)µ + gρµ(k + 2k ′)ν

])Simon Armstrong IPPP




k , a k ′, b

p −p′

k , a k ′, b

p −p′

A2,2(p+, p′−, k+, k ′−) = i〈kp〉2 〈kp′〉

〈kk ′〉〈k ′p′〉〈p′p〉

A2,2(p+, p′−, k−, k ′+) = i〈k ′p〉3

〈k ′k〉〈kp〉〈pp′〉




Some more simple amplitudes

A(g±1 , g

+2 , . . .

)= 0

A(g∓1 , g

−2 , . . .

)= 0

A(g−1 , g

+2 , . . . , g

+i−1, g

−i , g

+i+1, . . . , g

+n

)=

〈1, i〉4

〈1, 2〉 . . . 〈n − 1, n〉〈n, 1〉

A(g+

1 , g−2 , . . . , g

−i−1, g

−i , g

+i+1, . . . , g

−n

)= (−1)n

[1, i ]4

[1, 2] . . . [n − 1, n] [n, 1]




BCFW Recursion Relation

Definition

BCFW Recursion Relation for Gluons

An =∑r

Ahr+1

1

P2r

An−r+1

2

rr + 1

n − 1

n 1

=

r + 1

n − 1

n

1

P21,r

2

r

1




BCFW Derivation

λk → λk − z λn (1)

λn → λn + zλk (2)

Take the function A(z)z has poles in the complex plane.

One pole is at z = 0 and has A as it’s residue.

The others are due to internal propagators going onshell andhave residues of the form Ah

r+11P2rAn−r+1




BCFW Derivation

λk → λk − z λn (1)





r+11P2rAn−r+1




BCFW Derivation

λk → λk − z λn (1)





r+11P2rAn−r+1




BCFW Derivation

λk → λk − z λn (1)





r+11P2rAn−r+1




Take a complex contour integral on a circle at infinity, then byCauchy residue theorem A = I −

∑poles A

hr+1

1P2rAn−r+1 where I is

the integral.If it can be proved by power counting of the integrand (or othermethods) that I = 0 then we have the BCFW recursion relation.

For pure gluon amplitudes this corresponds to a condition that theshifted gluons can only have helicities (−,+), (+,+) or (−,−) butnot (+,−)




Take a complex contour integral on a circle at infinity, then byCauchy residue theorem A = I −

∑poles A

hr+1

1P2rAn−r+1 where I is

the integral.If it can be proved by power counting of the integrand (or othermethods) that I = 0 then we have the BCFW recursion relation.For pure gluon amplitudes this corresponds to a condition that theshifted gluons can only have helicities (−,+), (+,+) or (−,−) butnot (+,−)




L. J. Dixon, “Calculating scattering amplitudes efficiently,”arXiv:hep-ph/9601359 [hep-ph].

R. Britto, F. Cachazo, B. Feng, and E. Witten, “Direct proofof tree-level recursion relation in Yang-Mills theory,”Phys.Rev.Lett. 94 (2005) 181602, arXiv:hep-th/0501052[hep-th].



http://arxiv.org/abs/hep-ph/9601359

http://dx.doi.org/10.1103/PhysRevLett.94.181602

http://arxiv.org/abs/hep-th/0501052

http://arxiv.org/abs/hep-th/0501052

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