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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
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Outline
1 Introduction
2 Colour Ordered Amplitudes
3 Spinor Helicity Formalism
4 BCFW Recursion Relation
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Outline
1 Introduction
2 Colour Ordered Amplitudes
3 Spinor Helicity Formalism
4 BCFW Recursion Relation
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Outline
1 Introduction
2 Colour Ordered Amplitudes
3 Spinor Helicity Formalism
4 BCFW Recursion Relation
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Outline
1 Introduction
2 Colour Ordered Amplitudes
3 Spinor Helicity Formalism
4 BCFW Recursion Relation
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Colour Ordered Amplitudes
ea
b
c
d
A = −g3f baz f czw f edy . . .
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Colour Ordered Amplitudes
ea
b
c
d
A = −g3f baz f czw f edy . . .
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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 = . . .
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Colour Rearrangements
f abc = − i√2
(tr[T aT bT c ]− tr[T aT cT b]
)
T aijT a
ij = δi
jδij − 1
Ncδi
jδij (Fierz identity)
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 ]
)
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Colour Rearrangements
f abc = − i√2
(tr[T aT bT c ]− tr[T aT cT b]
)
T aijT a
ij = δi
jδij − 1
Ncδi
jδij (Fierz identity)
f baz f czy f edy =i
2√
2tr[T bT aT z ] tr[T cT zT y ] tr[T eT dT y ]+. . .
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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 =i
2√
2tr[T bT aT yT c ] tr[T eT dT y ] + . . .
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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 =i
2√
2tr[T eT dT cT bT a]± permutations
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Colour Rearrangements - The Diagram Method
= −
= − 1
NC
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Colour Rearrangement Example - The Diagram Method
= − = − + ...
= − + ... = ± permutations
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Colour Ordered Amplitudes
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))
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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)
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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)
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Spinor Helicity Formalism
u(k)⊗ u(k) = v(k)⊗ v(k) = /k
u±(k) = v∓(k)
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Spinor Helicity Formalism
u(k)⊗ u(k) = v(k)⊗ v(k) = /k
u±(k) = v∓(k)
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Spinor Helicity Formalism
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 |
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Spinor Helicity Formalism - An Example
k , a k ′, b
p −p′
k , a k ′, b
p −p′
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Spinor Helicity Formalism - An Example
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
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
Spinor Helicity Formalism - An Example
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′〉
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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]
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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 (+,−)
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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 (+,−)
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes
Introduction Colour Ordered Amplitudes Spinor Helicity Formalism BCFW Recursion Relation
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].
Simon Armstrong IPPP
An introduction to BCFW recursion relation with colour ordered amplitudes