1. F A C U L T Y O F S C I E N C E
U N I V E R S I T Y O F C O P E N H A G E N
Project outside of course scope
Anders Øhrberg Schreiber, wsl980@alumni.ku.dk
Many Leg Shifts for Computation of Amplitudes
N. Emil J. Bjerrum-Bohr
June 8, 2015
2. Abstract
In this project we explore methods for simplifying calculations of scattering amplitudes.
Specifically we focus on tree level amplitudes and develop recursions techniques, that vastly sim-
plify calculations of these amplitudes when compared to the traditional Feynman diagrammatic
method. We start by introducing the spinor-helicity formalism, which is used to write down
simple expression for amplitudes in terms of spinor products. Next up we introduce the Yang-
Mills lagrangian in the Gervais-Neveu gauge along with its Feynman rules. MHV classification
is then investigated, as specific helicity configurations, for pure gluon amplitudes in Yang-Mills
theory, make up vanishing amplitudes. Recursion techniques are then explored, where we start
by developing a general recursion procedure from complex momentum shifts of external mo-
menta resulting in an on-shell factorization of the amplitude. The BCFW recursion relations are
introduced and used to prove the famous Parke-Taylor formula as well as show an example of
the KLT relations for gauge theory and gravity. Finally we show an example of a multi leg shift
and how to incorporate a CSW prescription in an MHV vertex expansion.
4. 2 Spinor-helicity formalism Anders Ø. Schreiber
1 Introduction
The field of particle physics has expanded human knowledge about the Universe immensely. We
have gained insights of the inner most workings of atoms and their electron structure. We have
explored the depths of the atomic nuclei and its constituents. More importantly we have also
explored how these constituents interact with one and another, classifying three out of the assumed
four fundamental forces of nature, in terms of particle interactions in the quantum field theory
framework. These are the strong, the weak and the electromagnetic forces [1]. Gravity, the fourth
force, has yet to be written down as a consistent quantum field theory. However we can still model
gravity as an effective field theory and make predictions [2, 3].
To make predictions for particle physics processes, one has to calculate the cross section of the
specific process, as this is what we can measure in a modern collider experiments. To predict the
cross section for a process, one has to calculate the scattering amplitude (the S-matrix element)
for this process. It has however proven to be very difficult to do this with conventional methods
of Feynman diagrams [1]. Calculations with Feynman diagrams are realizations of a perturbation
expansion and when doing calculations with Feynman diagrams, one has to setup all topologically
different diagrams, for a given process, up to a given order of coupling in the theory. However as we
consider more complex processes, with more external particles (incoming and outgoing particles),
the number of topologically different diagrams we can write down goes more or less as n! with
n external particles. This very quickly makes calculations of scattering amplitudes a hot mess.
However new techniques have emerged in for doing these calcultations in a much simpler and elegant
way. These techniques can both be applied to phenomenological calculations, but might also hint
at an underlying, so far unknown, structure of quantum field theory [4].
We will start this project, by introducing the spinor helicity formulation in section 2, which has
deemed itself very useful to write down elegant and compact expressions for scattering amplitudes.
In section 3 we introduce the SU(N) Yang-Mills lagrangian and the Feynman rules for gluon vertices
and propagators in the Gervais-Neveu gauge. Furthermore we introduce the MHV classification on
the baggrund of vanishing amplitudes in certain helicity configurations for pure gluon amplitudes.
In section 4 we explore the main substance of this project, namely recursion techniques based on a
complex momentum shifts. We consider the BCFW recursion relations, but also multi leg shifts, the
MHV vertex expansion and the CSW prescription. Applications of these techniques are presented
as well.
2 Spinor-helicity formalism
In this section we introduce the the spinor helicity formalism, which is useful for writing down nicely
compact expressions for scattering amplitudes. The formalism comes from massless Dirac spinors,
which can be decomposed into two independent commuting spinors with opposite helicities (helicity,
together with momentum, being the only relevant quantum numbers). We start by considering the
massive Dirac equation [1]
(−i/∂ + m)Ψ(x) = 0 ⇒ Ψ(x) ∼ u(p)eipx
+ v(p)e−ipx
(2.1)
where /∂ = ∂µγµ, px = pµxµ and the momentum is on-shell p2 = −m2. If we do a Fourier transfor-
mation of the Dirac equation, we see that u and v solutions must satisfy
(/p + m)u(p) = 0, (−/p + m)v(p) = 0 (2.2)
Each of these equations have two independent solutions, which we label with a ± subscript.
Ψ(x) =
s=±
˜dp bs(p)us(p)eipx
+ d†
s(p)vs(p)e−ipx
, ˜dp =
d3p
(2π)32Ep
(2.3)
4
5. 2 Spinor-helicity formalism Anders Ø. Schreiber
and similarly for Ψ(x) with conjugated coefficients b†
s(p) and ds(p). We take the coefficients
(b±(p), d±(p)) and conjugates to be anticommuting fermionic creation and annihilation operators
when doing canonical quantization, so u±(p) and v±(p) are commuting 4-component spinors. Specif-
ically for u± and v± are eigenstates of the z-direction spin operators, so in the massless case the
± subscript denotes the helicity of the particle (projection of spin along the momentum). Also in
Feynman diagrams we take v± to describe an outgoing anti-fermion and u± to describe and outgo-
ing fermion. When calculating amplitudes, we will take all particles as outgoing, since we can use
crossing symmetry (which exchanges outgoing with incoming, fermion with antifermion and flips
helicity). Specifically this tells us that
u± = v , v± = u (2.4)
To get the specific form of the spinor helicity formalism, we notice that we can write the Dirac
equation for our outgoing spinors (which is actually the Weyl equation)
/pv±(p) = 0, u±(p)/p = 0, /p =
0 pa˙b
p˙ab 0
, pa˙b = pµ(σµ
)a˙b (2.5)
So this means that u± and v± will have the following solutions
v+(p) =
|p]a
0
, v−(p) =
0
|p ˙a
u−(p) = (0, p|˙a), u+(p) = ([p|a
, 0)
(2.6)
where angle and square spinors are 2-component commuting spinors. To lighten the notation a
little, we will now use the following shorthand (for several particles of momenta pi, i = 1, 2, . . . , n),
when we define spinor products
ij ≡ i|˙a |j ˙a
= u−(pi)v−(pj), [ij] ≡ [i|a
|j]a = u+(pi)v+(pj) (2.7)
Indeed we also see that products, where u and v spinors have opposite helicity, vanish. We can use
the spinor completeness relation for massless spinors to write [1]
u−(p)u−(p) + u+(p)u+(p) =
0 |p]a p|˙b
0 0
+
0 0
|p ˙a
[p|b 0
≡ |p] p| + |p [p| = −/p (2.8)
So
pa˙b = −|p]a p|˙b , p˙ab
= − |p ˙a
[p|b
(2.9)
Since the angle and square brakets have spinor indices, these are raised and lowered with Levi-
Civita symbols, so spinor products are antisymmetric pq = − qp and [pq] = −[qp]. Provided
that momenta are real, since Ψ is the Dirac conjugate of Ψ, we can change a [p|a = (|p ˙a
) and
|p]a = ( p|˙a) , so [pq] = qp . An important identity for reducing amplitudes to Mandelstam
variables is (we use conventions in the Appendix of [5])
pq [qp] = p|˙b |q
˙b
[q|a
|p]a = Tr(q
˙ba
pa˙b) = pµqν Tr[(σµ
)
˙ba
(σν
)a˙b] = −2p · q = spq (2.10)
A lot more identities exists [5] (see Appendix A for proofs)
Charge conjugation of current : [i|γµ
|j = j| γµ
|i]
Fierz rearrangement : i| γµ
|j] k| γµ|l] = 2 ik [jl]
Gordon identity : i| γµ
|i] = 2pµ
i
Momentum conservation :
n
i=1
|i [i| = 0 ⇒
n
i=1,i=j,k
[ji] ik = 0
Schouten identity : ij kl = ik jl + il kj
(2.11)
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6. 3 Color decomposition in SU(N) Gauge Theory Anders Ø. Schreiber
A final thing to note, regarding the spinor helicity formalism, is that polarization vectors for massless
spin-1 particles can be written as
µ
+(pi; q) = −
q| γµ|i]
√
2 qi
,
/+(pi; q) =
√
2
qi
(|i] q| + |q [i|),
µ
−(pi; q) = −
i| γµ|q]
√
2[qi]
,
/−(pi; q) =
√
2
[qi]
(|i [q| + |q] i|),
(2.12)
and pµ
µ
±(p) = 0 due to the massless Weyl equation.
3 Color decomposition in SU(N) Gauge Theory
We are interested in gluon amplitudes in this project, so we will investigate Yang-Mills theory and
specifically look into color decomposition, where we seperate the color structure and the kinematical
part of the gluon scattering amplitudes. We start by writing down the Yang-Mills lagrangian in the
Gervais-Neveu gauge [5]
L = −
1
4
Tr FµνFµν
= Tr −
1
2
∂µAν∂µ
Aµ
− i
√
2g∂µ
Aν
AνAµ +
g2
4
Aµ
Aν
AνAµ , (3.1)
where Aµ = Aa
µTa and Ta are gauge group generators, in this case for SU(N). Gluon fields are in
the adjoint representation, so color indices run over a, b, . . . = 1, 2, . . . , N2 − 1 and group generators
are normalized Tr(TaTb) = δab and satisfy [Ta, Tb] = i ˜fabcTc so i ˜fabc = Tr(TaTbTc) − Tr(TbTaTc).
The Feynman rule for the gluon propagator is δab ηµν
p2 , the 3-gluon vertex involve ˜fabc and the 4-gluon
vertex involve ˜fabx ˜fxcd + perm. and both include kinematic factors aswell. From calculating the
4-gluon tree level amplitude, we get an s-channel color factor
˜fa1a2b ˜fba3a4
∝ Tr(Ta1
Ta2
Ta3
Ta4
) − Tr(Ta1
Ta2
Ta4
Ta3
)
− Tr(Ta1
Ta3
Ta4
Ta1
) + Tr(Ta1
Ta4
Ta3
Ta2
),
(3.2)
where we have contracted two structure constants in terms of traces of group generators and used
the completeness relation [5, 6] (also known as Fierz rearrangement)
(Ta
) j
i (Ta
) l
k = δ l
i δ j
k −
1
N
δ j
i δ l
k . (3.3)
So we get an amplitude structure like
Afull tree
4 = g2
(A4[1234] Tr(Ta1
Ta2
Ta3
Ta4
) + permutations of (234)) , (3.4)
where objects like A4[1234] are color-ordered amplitudes, since we have seperated out the color
factor from the kinematical part of the amplitude. Indeed this is an example of a generalization of
the color structure to any n-point tree level amplitude involving only gluons (adjoint representations
of the gauge group) [6, 7]
Afull tree
n ({pi, hi, ai}) = gn−2
σ∈Sn/Zn
Tr(Taσ(1) · · · Taσ(n) )Atree
n (σ(1λ1
), . . . , σ(nλn
)), (3.5)
where pi are gluon momenta, hi are gluon helicities, and ai are color indices. Sn are all permutations
of n objects, while Zn is the subset of cyclic permutations, which preserves the trace; Sn/Zn means
all distinct cyclic orderings in the trace.
The smart thing about this decomposition is that we now only have to care about the color-ordered
amplitudes. These have specific Feynman rules in the Gervais-Neveu gauge
3-gluon vertex : V µ1µ2µ3
(p1, p2, p3) = −
√
2(ηµ1µ2
pµ3
1 + ηµ2µ3
pµ1
2 + ηµ3µ1
pµ2
3 ), (3.6)
4-gluon vertex : V µ1µ2µ3µ4
(p1, p2, p3, p4) = ηµ1µ3
ηµ2µ4
, (3.7)
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7. 3 Color decomposition in SU(N) Gauge Theory Anders Ø. Schreiber
and gluon propagator ηµν
p2 . A final thing to note is, that only planar diagrams contribute to color-
ordered amplitude (so e.g. no u-channel tree diagrams for two-to-two processes).
Now that we have the Feynman rules in the Gervais-Neveu gauge, we can look into calculating
specific color-ordered amplitudes. Let us start with the 3-gluon amplitude determined from the
3-gluon vertex
A3[123] = −
√
2(ηµ1µ2
pµ3
1 + ηµ2µ3
pµ1
2 + ηµ3µ1
pµ2
3 ) µ1
1 (p1) µ2
2 (p2) µ3
3 (p3)
= −
√
2(( 1 · 2)( 3 · p1) + ( 2 · 3)( 1 · p2) + ( 3 · 1)( 2 · p3))
(3.8)
If we pick gluons 1 and 2 to have negative helicity and 3 to have positive helicity, we have
A3[1−
2−
3+
] = −
12 [q1q2] q31 [13] + 2q3 [q23] 12 [2q1] + q31 [3q1] 23 [3q2]
[q11][q22] q33
(3.9)
where we have used the Fierz rearrangement. Now before we continue, we want to notice something
about three particle kinematics. We have momentum conservation so pµ
1 +pµ
2 +pµ
3 = 0, which means
that 12 [12] = 2(p1 · p2) = (p1 + p2)2 = p2
3 = 0. This means that either 12 or [12] must vanish.
Suppose now that [12] vanishes. Then consider 12 [23] = − 1| /p2
|3] = 1| (/p1
+ /p3
)|3] = 0. So
[23] = 0. Similarly, we can show that [13] vanishes and so we have [12] = [23] = [31] = 0, so the
three spinors are proportional
|1] ∝ |2] ∝ |3], (3.10)
and the same can be shown by letting 12 vanish
|1 ∝ |2 ∝ |3 . (3.11)
Therefore we can choose to eliminate a lot of spinor products by using this observation, which we
call 3-particle special kinematics. If we go back to (3.9), we can use special kinematics (3.11),
but this makes the amplitude vanish, so if we pick (3.10), use momentum conservation and the
Schouten identity, we end up with
A3[1−
2−
3+
] =
12 3
23 31
. (3.12)
This is an example of the simplicity of amplitudes, that we shall see to a greater extent with the
Parke-Taylor amplitude (4.11). We can also calculate the same 3-gluon color-ordered amplitude,
but with all the helicities reveresed, A3[1+2+3−], but here we simply switch all angle brakets with
square brakets and vice versa in the polarization vectors, so we get
A3[1+
2+
3−
] =
[12]3
[23][31]
. (3.13)
Likewise we can calculate the 4-gluon color-ordered amplitude with two minus helicity gluons and
two plus helicity gluons (see Appendix B) and get a similarly simple result
A4[1−
2−
3+
4+
] =
12 4
12 23 34 41
. (3.14)
This is an example of the Parke-Taylor amplitude (4.11), which we shall prove in section 4.2.
7
8. 3 Color decomposition in SU(N) Gauge Theory Anders Ø. Schreiber
3.1 MHV classification
When we calculate color-ordered amplitudes for only gluons, we can consider diagrams of n external
legs and some specific color-ordering of the external legs. However we also assign a specific helicity
configuration to all of the external gluons. In this section we will show that for certain helicity
configurations, n-gluon amplitudes vanish. We will start by considering the all plus helicity config-
uration with An[1+2+ · · · n+]. When we construct color-ordered amplitudes in the Gervais-Neveu
gauge and use Feynman rules (3.6) and (3.7), we get a bunch of resulting inner products between
either polarizations vector and polarization vector, polarization vector and momentum, or momen-
tum and momentum. And we also get products of internal momenta squared in the denominator,
so schematically we can construct an amplitude like
Atree
n ∼
diagrams
( ( i · j)) ( ( i · kj)) ( (ki · kj))
P2
I
(3.15)
where ki’s can be sums over external momenta pj. If we choose the reference momenta of all
polarization vectors to be the same, we get that
+
i · +
j ∼ q| γµ
|i] q| γµ|j] = 0 (3.16)
So the only way for an all plus helicity configuration to be non-zero is by having only terms of i ·kj
in the numerator of the amplitude. However since each vertex in an amplitude can contribute at
most one ki and we can have at most n − 2 vertices [6], we can at most have n − 2 polarization
vectors contracted with momenta (out of the n polarization vectors in total). So there must always
be atleast one +
i · +
j in each term in the numerator and thus the all plus helicity gluon amplitude
is zero.
Consider now the amplitude with one minus helicity gluon, Atree
n (1−2+ · · · n+). If we choose q2 =
q3 = · · · = qn = p1 then we have that +
i · +
j = 0 for i, j = 2, 3, · · · , n and also −
1 · +
i , i =
2, 3, · · · , n. Again any i · j contraction is zero, so we to contract all the polarization vectors
with momenta to get a non-zero amplitude, which we argued above is not possible. Therefore we
also have Atree
n (1−2+ · · · n+) = 0. We can also flip all the helicities which means angle and square
brackets exchange, but the charge conjugation rule allows us to get back to the original spinor
bracket configuration. So the flipped helicity amplitudes also vanish. So to sum up
Atree
n (1+
2+
· · · n+
) = Atree
n (1−
2+
· · · n+
) = 0,
Atree
n (1−
2−
· · · n−
) = Atree
n (1+
2−
· · · n−
) = 0.
(3.17)
We call these amplitudes helicity violating amplitudes. To see why this makes sense, remember first
that we are only considering outgoing particles, so an all plus helicity amplitude could translate into
g−
1 g−
2 → g+
3 · · · g+
n (3.18)
So we go from an all minus helicity incoming set of particles to an all plus set of outgoing particles,
which changes the helicity maximally (and too much, so it violates the helicity). For a one minus
amplitude we have something like
g+
1 g−
2 → g+
3 · · · g+
n (3.19)
so again we go from zero helicity to n − 2 plus helicity particles (and this is also too large of a
helicity change). So as it turns out the maximally helicity violating (MHV) process that has non-
zero amplitude is the two minus amplitude, Atree
n (1−2−3+ · · · n+) which has a process that goes
like
g+
1 g+
2 → g+
3 · · · g+
n (3.20)
8
9. 4 Recursion techniques Anders Ø. Schreiber
So here we only gain n − 4 extra plus helicities in the process (which turns out to be the maximal
change in helicity you can make). Furthermore, less helicity violating processes will be noted NMHV,
N2MHV, . . ., NkMHV and so on, where NMHV has three minus and rest plus gluons, N2MHV has
four minus and rest plus. We shall calculate the MHV amplitude (which is exactly the previously
mentioned Parke-Taylor amplitude, (4.11)) in the section 4.2 using the BCFW recursion relations
and an NMHV amplitude using a MHV vertex expansion in section 4.5.
4 Recursion techniques
In this section we will introduce methods of on-shell recursion relations, specifically BCFW recursion
relations and MHV vertex expansions (including the CSW prescription). Both of these methods are
based on the use of the complex plane, where we shift external particle momenta with a complex
parameter.
Let us start by considering an on-shell amplitude An with n external particles of momenta p2
i = 0,
i = 1, 2, . . . , n. We take all particles to be outgoing, so momentum conservation implies n
i=1 pµ
i = 0.
Now we want to shift the external momenta, because as we shall see, this will be a smart trick to
evaluate amplitudes recursively. We introduce exactly n complex-valued vectors rµ
i such that
(i) n
i=1 rµ
i = 0,
(ii) ri · rj = 0, ∀i, j = 1, 2, . . . , n and in particular r2
i = 0,
(iii) pi · ri = 0, ∀i = 1, 2, . . . , n.
Specifically we will use these n new vectors to shift momenta, introducing the hat notation for a
shifted momentum
ˆpµ
i ≡ pµ
i + zrµ
i , z ∈ C. (4.1)
The three conditions on rµ
i vectors now imply three conditions on ˆpµ
i
(a) n
i=1 ˆpµ
i = 0,
(b) ˆp2
i = 0, ∀i = 1, 2, . . . , n,
(c) For a non-trivial subset of generic momenta (a set of atleast two, and no more than n − 2
momenta) {pµ
i }i∈I, define Pµ
I = i∈I pµ
i and Rµ
I = i∈I rµ
i , then ˆP2
I is linear in z
ˆP2
I =
i∈I
ˆpi
2
= P2
I + z2PI · RI. (4.2)
Define now zI = −
P2
I
2PI ·RI
, then we can write
ˆP2
I = −
P2
I
zI
(z − zI). (4.3)
The reason why we want to consider leg-shifts for amplitudes is that, by doing these leg-shifts, we
have constructed a holomorphic function in the complex plane, ˆAn(z), where we reconstruct the
original amplitude by setting z = 0, An = ˆAn(z = 0). Specializing to tree-level amplitudes, the
structure of the function ˆAn(z) is pretty simple. Indeed they are rational functions of momenta
and there are no square roots, logarithms and so on as there are for loop diagrams. The analytic
structure of ˆAn(z) is captured by poles, which only appear from shifted propagators 1/ ˆP2
I , where
ˆPI is the sum of a non-trivial subset of momenta. By property (c) we then have that 1/ ˆP2
I has a
9
10. 4 Recursion techniques Anders Ø. Schreiber
simple pole at z = zI, where for generic momenta P2
I = 0 so zI = 0. So the structure is that ˆAn(z)
has simple poles, from propagators, away from the origin z = 0.
Now if we consider the function
ˆAn(z)
z and integrate around a contour, that surrounds the simple
pole at the origin, then by the residue theorem, the residue associated with the pole at the origin is
exactly the unshifted amplitude An = ˆAn(z = 0). Then we can deform the contour to capture all
other poles, and by the residue theorem we get
An = Resz=0
ˆAn(z)
z
= −
zI
Resz=zI
ˆAn(z)
z
+ Bn, (4.4)
where we note that the sign on the right-hand side is due to the orientation change of the contours
around the poles away from the origin. We also note that as we deform the contour out to infinity,
we get Bn, which is the possible pole of
ˆAn(z)
z at z → ∞. Indeed if we look at one of the zI-poles,
these are where the propagator 1/ ˆP2
I goes on-shell. Here a factorization happens near the pole
ˆAn(z) → ˆAL(zI)
1
ˆP2
I
ˆAR(zI) = −
zI
z − zI
ˆAL(zI)
1
P2
I
ˆAR(zI). (4.5)
So this means that
− Resz=zI
ˆAn(z)
z
= ˆAL(zI)
1
P2
I
ˆAR(zI) = L R
ˆPI
∧
∧
∧
∧
∧
∧
. (4.6)
Now why is this useful? Well now we can take an n-point amplitude and factorize it into ”less
than n”-point amplitudes and thus we actually have a way of recursively calculating amplitudes at
tree-level. Before we write down the full general recursion relation, we note that the Bn term does
not in general have an expression for it. However in most cases one proves (or assumes) that Bn is
zero, or even stronger, one shows that
ˆAn(z) → 0, z → ∞. (4.7)
If the statement of (4.7) is true, we say that the shift of the amplitude is valid. Now, the full
recursion relation is given by
An =
diagrams I
ˆAL(zI)
1
P2
I
ˆAR(zI) =
diagrams I
L R
ˆPI
∧
∧
∧
∧
∧
∧
. (4.8)
So we have to sum over all possible factorization channels I. Also we implicitly sum over possible on-
shell particle states for the internal line factorzation channel (for example sum over possible helcity
configuration for gluons). Equation (4.8) is the most general form of on-shell recursion relations we
can have for tree level amplitudes (by using complex leg-shifts).
4.1 BCFW recursion relations
Now that we have found a general recursion methods for doing on-shell tree level amplitudes, we
will now specialize to a case, that happens to be very useful for practical calculations, namely the
10
11. 4 Recursion techniques Anders Ø. Schreiber
BCFW recursion relations. For BCFW we shift only two of the legs, i and j and the rest of the legs
are unshifted (take rµ
k = 0, for k = i, j). We implement this shift on angle and square spinors by
|ˆi] = |i] + z|j], |ˆj] = |j], |ˆi = |i , |ˆj = |j − z |i , (4.9)
so spinor brakets [ˆij] and ˆji are linear in z and ˆiˆj = ij , [ˆiˆj] = [ij], ˆik = ik , and [ˆjk] = [jk]
are unshifted. We call this a [i, j shift. We can thus specialize the general recusion formula (4.8)
An =
diagrams I
ˆAL(zI)
1
P2
I
ˆAR(zI) =
diagrams I
L R
ˆPI
ˆjˆi
. (4.10)
where we have arranged exactly so that the shifted legs are on opposite sides of the factorization di-
agram (otherwise the internal momentum would not be shifted and there would be no corresponding
residue).
Let us consider different cases of shifts for the Parke-Taylor amplitude specifically, to see if we can
even make any valid shifts according to (4.7). The Parke-Taylor amplitude is given by [8]
An[1−
2−
3+
. . . n+
] =
12 4
12 23 · · · n1
. (4.11)
Consider a [−, − -shift (where we shift both of the minus helicity legs), then the amplitude gets
shifted according to (4.9)
ˆAn[ˆ1−ˆ2−
3+
. . . n+
] =
ˆ1ˆ2
4
ˆ1ˆ2 ˆ23 · · · nˆ1
=
12 4
12 ˆ23 · · · n1
∼
1
z
. (4.12)
Consider now a [−, + , where the − can be 1 or 2 and the + can be 3, . . . , n. For any non-adjacent
shift, we have ˆAn ∼ 1
z2 . However for an adjacent shift (a [2, 3 shift) then ˆAn ∼ 1
z . For [+, − -shifts,
we get the four powers of z in the numerator and one power in the denominator, so this goes as
z3. For a [+, + -shift we get something that goes as 1
z for shifting two adjacent legs, while we get
something that goes as 1
z2 for non-adjacent leg shifts. So all shifts, except the [+, − -shift, are valid
shifts.
4.2 An inductive proof of the Parke-Taylor formula
An important use of the BCFW relations (4.10), is to give an inductive proof of the Parke-Taylor
formula (4.11). To do this, we notice that the Parke-Taylor formula is indeed true for n = 3 and
n = 4, and we thus want to recursively build up to n external gluons. Suppose that for a given n,
(4.11) is true for n − 1 external gluons. Then consider a [1, 2 -shift. The BCFW relation (4.10) now
reads
An[1−
2−
3+
· · · n+
] =
n
k=4 hI =±
ˆAn−k+3[ˆ1−
, ˆPhI
I , k+
, . . . , n+
]
1
P2
I
ˆAk−1[− ˆP−hI
I , ˆ2−
, 3+
, . . . (k − 1)+
],
(4.13)
where we have PI = p2 + p3 + · · · + pk−1 and ˆPI = ˆp2 + p3 + · · · + pk−1. Since the An[− + · · · +]
amplitude vanish, except for n = 3, we can drastically reduce the number of BCFW diagrams
An[1−
2−
3+
· · · n+
] = ˆA3[ˆ1−
, − ˆP+
1n, n+
]
1
P2
1n
ˆAn−1[ ˆP−
1n, ˆ2−
, 3+
, · · · , (n − 1)+
]]
+ ˆAn−1[ˆ1−
, ˆP−
23, 4+
, . . . , n+
]
1
P2
23
ˆA3[− ˆP+
23, ˆ2−
, 3+
].
(4.14)
11
12. 4 Recursion techniques Anders Ø. Schreiber
where Pij = pi + pj and ˆPI is evaluated at the residue value where z = zI, such that the propagator
goes on-shell, ˆP2
I = 0. Consider now the 3-point anti-MHV amplitude in the first line of (4.14),
ˆA3[ˆ1−, − ˆP+
1n, n+]. We have shown in section 3, that it has the form
ˆA3[ˆ1−
, − ˆP+
1n, n+
] =
[ ˆP1nn]3
[nˆ1][ˆ1 ˆP1n]
, (4.15)
where we have chosen the convention [5] for analytic continuation of angle and square brackets
|−p = − |p and |−p] = |p]. If we use the on-shell condition for the internal line, it is possible to show
that [ˆ1n] = 0. The numerator in (4.15) also vanishes, as we can show that | ˆP1n [ ˆP1nn] = |1 [ˆ1n] = 0.
Similarly [ˆ1 ˆP1n] = 0. So all spinor products in (4.15) vanish, but since we have three powers of spinor
products in the numerator and two in the denominator, so we conclude that ˆA3[ˆ1−, − ˆP+
1n, n+] = 0.
We are now left with
An[1−
2−
3+
· · · n+
] = ˆAn−1[ˆ1−
, ˆP−
23, 4+
, . . . , n+
]
1
P2
23
ˆA3[− ˆP+
23, ˆ2−
, 3+
]. (4.16)
The 3-point anti-MHV amplitude we are left with does not vanish, since the leg 2 shift is on an
angle bracket, so we cannot use momentum conservation to make this amplitude vanish. Now we
can write down the explicit expression for the amplitudes on the righthand side of (4.16)
An[1−
2−
3+
· · · n+
] =
ˆ1 ˆP23
3
ˆP234 45 · · · nˆ1
×
1
23 [23]
×
[3 ˆP23]3
[ ˆP23ˆ2][ˆ23]
(4.17)
We wish to eliminate the spinor products including ˆP23, which is possible since 1 ˆP23 [3 ˆP23] =
− 12 [23] and ˆP234 [ ˆP232] = − 34 [23]. We then get our final result
An[1−
2−
3+
· · · n+
] =
12 4
12 23 34 45 · · · n1
, (4.18)
which by induction proves the validity of the Parke-Taylor formula.
This proof can also be generalized to non-adjacent legs, say legs i and j have negative helicity (with
an [i, j shift, which can also be checked to be valid), while the rest of the legs have positive helicity.
Then we have
An[1+
· · · i−
· · · j−
· · · n+
] = ˆA3[ˆi−
, − ˆP+
i−1,i, (i − 1)+
]
1
P2
i−1,i
ˆAn−1[ ˆP−
i−1,i, (i + 1)+
, . . . , ˆj−
, . . . , (i − 2)+
]
+ ˆAn−1[ˆi−
, ˆP−
j,j+1, (j + 2)+
, . . . , (i − 1)+
]
1
P2
j,j+1
ˆA3[− ˆP+
j,j+1, ˆj−
, (j + 1)+
].
(4.19)
The amplitude ˆA3[ˆi−, − ˆP+
i−1,i, (i − 1)+] will vanish, since P2
i−1,i = i − 1, i [i − 1,ˆi] = 0, and
ˆA3[− ˆP+
j,j+1, ˆj−, (j + 1)+] will not since only the j angle spinor is shifted. So
An[1+
· · · i−
· · · j−
· · · n+
] =
ij 4
12 · · · ij · · · n1
. (4.20)
which matches the form of the adjacent leg Parke-Taylor amplitude (4.11).
4.3 The 4-graviton amplitude and KLT relations
An interesting example of the use of the BCFW recursion relations is to calculate graviton ampli-
tudes and explore the so-called KLT relations [9]. To calculate graviton amplitudes, we first want
to consider the action of general relativity, the Einstein-Hilbert action
SEH =
1
2κ2
d4
x
√
−gR (4.21)
12
13. 4 Recursion techniques Anders Ø. Schreiber
where κ2 = 8πGN . However this action happens to inherit a huge problem from considering a
perturbation around the Minkowski metric. We introduce the graviton field as this perturbation
gµν = ηµν + κhµν [5]. Then the expansion of
√
−g contains an infinite number of vertices involving
two derivatives and n fields hµν. So we get very complicated Feynman rules for Einstein gravity.
However gravitons are spin-2 particles and in 4 spacetime dimensions they have helicity ±2 [5]. This
information makes us able to use a trick to calculate gravition amplitudes using BCFW relations,
because we can determine the 3-point graviton amplitude, for a given helicity configuration, from
little group scaling (see Appendix C).
We will now calculate the 4-point graviton amplitude, M4(1−2−3+4+). Specifically we shall use a
[1, 2 BCFW shifts (which is a valid shift [10, 11]). We write down the BCFW recursion relations
(note that gravity has non-planar diagrams contributing as opposed to Yang-Mills color-ordered
diagrams)
M4[1−
2−
3+
4+
] =
ˆ1− ˆ2−
3+4+
+
ˆ1− ˆ2−
4+3+
(4.22)
where have taken into account all possible helicity configuations for the internal line. Using Appendix
C, we can proceed to write down the exact expression for the amplitude (again using the convention
|−p = − |p and | − p] = |p])
M4[1−
2−
3+
4+
] =
1 ˆP14
6
14 2 ˆP144
2
1
14 [14]
[ ˆP143]6
[ ˆP142]2[23]2
+
1 ˆP13
6
13 2 ˆP133
2
1
13 [13]
[ ˆP134]6
[ ˆP13ˆ2]2[ˆ24]2
+
[ ˆP144]6
[ˆ1 ˆP14]2[ˆ14]2
1
14 [14]
ˆP14ˆ2
6
ˆP143
2
ˆ23
2
+
[ ˆP133]6
[ˆ1 ˆP13]2[ˆ13]2
1
13 [13]
ˆP13ˆ2
6
ˆP134
2
ˆ24
2
(4.23)
Consider now the first term
1 ˆP14
6
[ ˆP143]6
14 2 ˆP144
2
14 [14][ ˆP142]2[23]2
=
14 [43]6
[14][ˆ12]2[23]2
(4.24)
If we keep doing this, we get
M4[1−
2−
3+
4+
] =
14 [43]6
[14][12]2[23]2
+
13 [34]6
[13][12]2[24]2
+
[ˆ14]2 1ˆ2
6
14 [14] 43 2 ˆ23
2 +
[ˆ13]2 1ˆ2
6
13 [13] 34 2 ˆ24
2
(4.25)
We have some terms that depend on shifted spinors, so we must evaluate these expressions at the
appropriate poles. In the third term of (4.25), we have [ˆ14]. However this vanishes, as the internal
line for this BCFW diagram, has the on-shell condition ˆP2
14 = 14 [ˆ14] = 0. The last term in (4.25)
also vanishes, due to the on-shell condition ˆP2
13 = 13 [ˆ13] = 0. Then after some spinor helicity
manipulations, we are left with
M4[1−
2−
3+
4+
] =
12 7
[12]
13 14 23 24 34 2 (4.26)
The before mentioned KLT relation exists for 4-graviton amplitudes and it indeed reads
M4[1−
2−
3+
4+
] = s12A4[1−
2−
3+
4+
]A4[1−
2−
4+
3+
] =
12 7
[12]
13 14 23 24 34 2 (4.27)
13
14. 4 Recursion techniques Anders Ø. Schreiber
This is an example of an extensive set of relations between gauge theory and gravity, which have
been explored more comprehensively in [12]. This example also shows the simplicity of going through
the BCFW recursion relations as compared to using Feynman diagrams [13]. Indeed a very explicit
showcase of the power of recursion methods.
4.4 Multi leg shift
So far we have only worked with a special case of the general recursion relations (4.8), where only
two of the legs are shifted, the BCFW recursion relations (4.10). We now want to try to involve
shifts on more than just two legs and see if these can be useful. Consider the square spinor shift
|ˆi] = |i] + zci|X], |ˆi = |i (4.28)
where |X] is an arbitrary reference spinor and coefficient ci satisfy n
i=1 ci |i = 0. In general this is
known as a multi-leg shift and for all ci = 0 it is an all-leg shift. A specific realization of a multi-leg
shift is the Risager shift [14] where c1 = 23 , c2 = 31 , and c3 12 while ci = 0 for i = 4, . . . , n.
An all-leg shift is useful for calculating NkMHV gluon tree amplitudes, as it can be shown that, under
an all-leg shift, these go as 1/zk [15]. Therefore we can determine all gluon tree-level amplitudes
through all-leg shift recursion relations (except MHV amplitudes, but we have already determined
these with the Parke-Taylor formula).
4.5 MHV vertex expansion and the 6-gluon NMHV amplitude
Now that we have established the formalism for an all leg shift, we want to go ahead and use it in
an example calculation. It turns out that, using the all leg shift (4.28), we get MHV amplitudes as
building blocks for NkMHV amplitudes. Here we will take a look at the simplest example, namely
an NMHV amplitude. We are going to use the general recursion relation (4.8). We note that in
any case for gluon amplitudes, we must have atleast one negative helicity leg on each side of the
internal line in the recursion diagram (due to possible helicity assignments to the internal line) since
helicity violating diagrams vanish. For NMHV we then have one minus helicity leg on one side
and two minus helicity legs on the other side. The only two possibilities for nonvanishing helicity
assignments
−
+
+ −
−
−
+
or −
−
+
+ +
−
−
+
. (4.29)
So we have either anti-MHV3×NMHV or MHV×MHV. However during our proof of the Parke-
Taylor formula, we showed that the anti-MHV3 amplitude vanishes (since the all leg shift we are
using is a square spinor shift). Therefore NMHV amplitudes are constructed recursively from only
MHV amplitudes. We call this an MHV vertex expansion [16].
We will now show how this works in practice, by calculating the color-ordered NMHV 6-gluon
amplitude, ANMHV
6 [1−2−3−4+5+6+]. As we established, we will only have MHV×MHV diagrams,
14
15. 4 Recursion techniques Anders Ø. Schreiber
so we will get the following recursion expansion
ANMHV
6 [1−
2−
3−
4+
5+
6+
]
=
+ −
ˆ1−
ˆ2− ˆ3−
ˆ4+
ˆ5+ˆ6+
+
− +
ˆ6+
ˆ1− ˆ2−
ˆ3−
ˆ4+ˆ5+
+
− +
ˆ1− ˆ2−
ˆ5+ˆ6+
ˆ3−
ˆ4+
+
+ −
ˆ2− ˆ3−
ˆ6+ˆ1−
ˆ4+
ˆ5+
+
− +
ˆ1− ˆ2−
ˆ3−ˆ4+
ˆ5+
ˆ6+
+
+ −
ˆ2− ˆ3−
ˆ4+ˆ5+
ˆ6+
ˆ1−
.
(4.30)
Let us consider evaluating the first diagram in the expansion
+ −
ˆ1−
ˆ2− ˆ3−
ˆ4+
ˆ5+ˆ6+
=
12 3
2 ˆP126
ˆP1266 61
1
P2
126
ˆP1263
3
34 45 5 ˆP126
. (4.31)
To get rid of the hatted internal momentum spinor, we make the following rewriting
| ˆP126 = | ˆP126
[ ˆP126X]
[ ˆP126X]
= ˆP126|X]
1
[ ˆP126X]
= P126|X]
1
[ ˆP126X]
, (4.32)
where we have used that the shifted part of ˆP126 is proportional to [X|. Note also the P126 is not
equal to /p1
+ /p2
+ /p6
, but rather minus this. Diagrams must be invariant under little group scaling,
with respect to the internal line, which means that we should have an equal number of | ˆP126 in
the numerator and denominator. Therefore factors of 1
[ ˆP126X]
cancel out, and we are left with the
so-called CSW prescription (for a general internal line of momentum PI)
| ˆPI → PI|X]. (4.33)
This simplifies things greatly for our MHV expansion, as we can now remove any dependence on
shifted spinors. We can see that (4.31) changes under the CSW prescription as
+ −
ˆ1−
ˆ2− ˆ3−
ˆ4+
ˆ5+ˆ6+
=
12 3
34 45 61
×
3| P126|X]3
P2
126 2| P126|X] 6| P126|X] 5| P126|X]
.
(4.34)
15
16. 5 Summary and outlook Anders Ø. Schreiber
By evaluating the rest of the diagrams in (4.30) and make good choices of reference spinors, to
make as many of the diagrams as possible vanish, then the resulting amplitude can be shown to be
equivalent to
ANMHV
6 [1−
2−
3−
4+
5+
6+
] =
3| P12|6]3
P2
126[21][16] 34 45 5| P16|2]
+
1| P56|4]3
P2
156[23][34] 56 61 5| P16|2]
,
(4.35)
which is the expected result for this amplitude (see [5], eq. (3.34)).
We also want to note, that it might seem like the final amplitude depends on the reference spinor
|X], as this can be chosen freely, however we are safe as Cauchy’s theorem makes us sure, that
by using any recursion relation with a shifted leg, we always end up back at the real amplitude
evaluating diagrams at the appropriate poles. So any reference spinor will make us arrive at the
right amplitude.
5 Summary and outlook
In this project we have explored various aspects of scattering amplitudes and how to calculate them.
In section 2 we derived the spinor-helicity formalism, which is very useful for expressing scattering
amplitudes, in a very simple form, in terms of spinor products and Mandelstam invariants.
In section 3 we presented the SU(N) Yang-Mills lagrangian and showed how color decomposition
works, where we seperate amplitudes involving color into a color part and a color-ordered amplitude
(independent of color). We also derived how certain helicity configurations make color-ordered
amplitudes vanish and thus introduced the MHV classification of color-ordered amplitudes.
The main content of this project is in section 4, where we explored different recursion techniques.
Initially we introduced a general recursion relation based on complex momentum shifts of external
particle momenta. Then we specialized to the BCFW recursion relations with only two external
momenta being shifted. Then we showed the power of the BCFW relations by making an inductive
proof of the Parke-Taylor formula
Atree
n [· · · i−
· · · j−
· · · ] =
ij 4
12 · · · n1
, (5.1)
and by showing an example of the KLT relations between graviton and gluon amplitudes
M4[1−
2−
3+
4+
] = s12A4[1−
2−
3+
4+
]A4[1−
2−
4+
3+
]. (5.2)
Finally we considered multi leg shifts and the MHV vertex expansion, with the specific example of
expanding an 6-gluon NMHV amplitude in terms of MHV vertices only, as well as how the CSW
prescription works. We also saw that we can choose any reference spinor for the multi leg shift, as
the resulting amplitude is independent of this choice due to Cauchy’s theorem.
The techniques presented in this project could be extended to one loop or maybe higher loop
level amplitudes. This has been explored in [5]. One could also consider extending the recursion
techniques to other types of theories, say the Standard Model. For some specific models involving
the Higgs, and gluons or partons, this has been explored in [17, 18].
16
17. 6 Appendix A: Proof of spinor helicity identities Anders Ø. Schreiber
6 Appendix A: Proof of spinor helicity identities
Here we make a proof of the remaining spinor helicity identites showcased in the spinor helicity
section, equation (2.11). We start from the top
Charge conjugation of current : [i|γµ
|j = j| γµ
|i] (6.1)
This identity can easily be realized through the following manipulation of spinor indices
[i|γµ
|j = ([i|a
, 0)
0 (σµ)a˙b
(σµ)˙ab 0
0
|j
˙b =
˙b ˙d bc
j| ˙d (σµ
)a˙b|i]c = j| ˙d (σµ
)
˙dc
|i]c = j| γµ
|i] (6.2)
which verifies the identity.
Fierz rearrangement : i| γµ
|j] k| γµ|l] = 2 ik [jl] (6.3)
We will show this by manipulating spinor indices
i| γµ
|j] k| γµ|l] = i|˙a (σµ
)˙ab
|j]b k|˙c (σµ)˙cd
|l]d
= −2 ˙a˙c bd
i|˙a |j]b k|˙c |l]d
= 2 ˙a˙c db
i|˙a |j]b k|˙c |l]d
= 2 i|˙a |k ˙a
[j|d
|l]d
= 2 ik [jl]
(6.4)
which verifies the identity.
Gordon identity : i| γµ
|i] = 2pµ
i (6.5)
Consider again the spinor index manipulation
i| γµ
|i] = i|˙a (σµ
)˙ab
|i]b = (σµ
)˙ab
|i]b i|˙a = −(σµ
)˙ab
(pi)b˙a = −pν
i Tr((σµ
)(σν)) = 2pµ
i (6.6)
which verifies the identity.
Momentum conservation :
n
i=1
|i [i| = 0 ⇒
n
i=1,i=j,k
[ji] ik = 0 (6.7)
This identity holds from momentum conservation. If we have n momenta that sums to zero
n
i=1
pµ
i = 0 ⇔
n
i=1
pµ
i (σµ)a˙b = −
n
i=1
|i]a i|˙b = 0
We can also contract with σ to get
n
i=1
|i ˙a
[i|b
= 0 (6.8)
And indeed if j, k = i then
n
i=1
[ji] ik = 0 (6.9)
17
18. 6 Appendix A: Proof of spinor helicity identities Anders Ø. Schreiber
and when j, k = i then spinor products are trivially zero due to antisymmetry.
Schouten identity : ij kl = ik jl + il kj (6.10)
This identity can be shown from the fact that spinor brakets only have two components. So say we
have three two-component spinors, |i , |j , and |k , then we must be able to express one of them as
a linear combination of the two others
|k = a |i + b |j , (6.11)
then a = jk
ji and b = ik
ij , so
|k − a |i − b |j = |k +
jk
ij
|i +
ki
ij
|j = 0 ⇔ |k ij + |i jk + |j ki = 0 (6.12)
or with some fourth spinor l|
lk ij + li jk + lj ki = 0 (6.13)
which is equivalent to the given identity.
18
22. 8 Appendix C: Little group scaling and 3-particle amplitudes Anders Ø. Schreiber
8 Appendix C: Little group scaling and 3-particle amplitudes
Little group scaling is a property of amplitudes, and specifically it is realized through the following
transformation of angle and square spinors (simultaneously)
|p → t |p and |p] → t−1
|p], (8.1)
where t is the little group scaling parameter. Now to figure out the specific property for amplitudes
under little group scaling, we notice that amplitudes are always made up of propagators, vertices
and external lines. We are working with massless particles, so we can always write down the
ingredients of amplitudes in terms of angle and square spinors. Vertices and propagators do not scale
under little group transformations (specifically (2.9) is invariant under litte group transformation,
so propagators are invariant, and vertices are either numbers or matrices and are invariant too).
We are thus left with external lines. Scalar particles (zero helicity) are just 1, so these are invariant.
Weyl fermions scale with a factor t−2h (since they only have an angle or square spinor associated
with them). Vector bosons have an associated polarization vector and this also scales as t−2h (see
(2.12)). It so happens that for graviton polarization tensors, that we can write them as the product
of two vector polarization vectors [10], so since gravitons have helicity ±2, an external graviton
polarization tensor also scales as t−2h. Due to this scaling of external lines, we can write down the
following general amplitude relation
An({|1 , |1], h1}, . . . , {ti |i , t−1
i |i], hi}, . . .) = t−2hi
i An({|1 , |1], h1}, . . . , {|i , |i], hi}, . . .) (8.2)
As we saw in (3.10) and (3.11), we can write down a 3-point amplitude only in terms of either angle
or square spinors. Writing down a general ansatz for a 3-point amplitude (independent of the type
of particles)
A3(1h1
2h2
3h3
) = c 12 x12
13 x13
23 x23
(8.3)
where c is a constant, usually dependent on the specific theory we are considering (independent of
kinematics). Imploying (8.2) we find a relationship
−2h1 = x12 + x13, −2h2 = x12 + x23, −2h3 = x13 + x23 (8.4)
So we have completely fixed the general 3-point amplitude
A3(1h1
2h2
3h3
) = c 12 h3−h1−h2
13 h2−h1−h3
23 h1−h2−h3
(8.5)
Little group scaling is going to determine all the relevant 3-graviton amplitudes (up to a coupling
constant)
M3[1−
2−
3+
] =
12 6
13 2
23 2 , M3[1−
2+
3−
] =
13 6
12 2
23 2 , M3[1+
2−
3−
] =
23 6
12 2
13 2 (8.6)
M3[1+
2+
3−
] =
[12]6
[13]2[23]2
, M3[1+
2−
3+
] =
[13]6
[12]2[23]2
, M3[1−
2+
3+
] =
[23]6
[12]2[13]2
. (8.7)
22
23. References Anders Ø. Schreiber
References
[1] M. Srednicki, Quantum Field Theory, Cambridge University Press, 2007.
[2] N. Emil J. Bjerrum-Bohr, Quantum Gravity, Effective Fields and String Theory, PhD Thesis
(2004).
[3] N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. R. Holstein, L. Plante, and P. Vanhove, Bending of
Light in Quantum Gravity, Phys. Rev. Lett. 114, 061301 (2015).
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