Little Group

Little group is a special case of the Lorentz group. This special transformation will leave the momentum of an particle invariant. To see the little group more concretely, we describe the one particle case as an example. For a massive particle, we can choose the rest frame where the particle momentum p^µ = (E, 0, 0, 0). When we rotate x-y plane, y-z plane and x-z plane of the frame, p^µ is invariant, so the little group is SO(3). For a massless particle, we can choose the reference frame where the particle momentum

p^µ = (E, 0, 0, E). When we rotate x-y plane of the frame, p^µ is invariant, so the little group is SO(2).

Little group scaling is a transformation which leave the momentum of on-shell particles invariant. As we know, for a massless particle momentum p_a˙b = |p]ahp|˙b (eq.(2.47)).

This relation is invariant under the scaling

|pi ! t|pi, |p] ! t ¹|p]. (2.68)

Equation(2.68) is just U(1) transformation. U(1) and SO(2) are isomorphic groups which means what little group scaling do is exactly rotate p^µ= (E, 0, 0, E) on x-y plane.

A crucial result for little group scaling: For an amplitude of massless particles, scaling the particle of this amplitude will transform the amplitude homogeneously like

An( 1, 2,· · · , ⁱ(ti|pⁱi, ti ¹|pⁱ], hi),· · · , ⁿ) = t_i ^2hiAn( 1, 2,· · · , ⁱ(|pⁱi, |pⁱ], hi),· · · , ⁿ).

(2.69) hi is the helicity of particle i. We can see eq.(2.69) from Feynman rules: (1). For scalar field theories, there are constant factor 1 on each fields and scalar field has zero helicity.

Eq.(2.69) is obviously true. (2). For spinor fields, there are one angle spinor for an left-handed Weyl field and one square spinor for an right-handed Weyl field. Left-handed and right-handed Weyl field have helicity ¹₂ and +¹₂ respectively. Eq.(2.69) works again!

(3). For spin-1 boson, the polarization vectors ✏^µ(p; q) = ^hp|p_2[pq]^µ|q] and ✏^µ₊(p; q) = ^hq|p₂^µ|p]

hqpi

have helicity -1 and +1 respectively. Polarization vectors ✏_± scale as t^±2.

This powerful result fix the massless three particles amplitude. For example, we consider the color order amplitude A3(1 2 3⁺) which is scattering of three gluons. From LSZ reduction, we know the tree point scattering amplitudes in Yang-Mills theory has mass dimension 1. Three particle special kinematics show a non-vanishing on-shell 3-particle amplitude can only depend on either angle brackets or square brackets. We choose the angle brackets to represent the amplitude, so that

A3(1 2 3⁺)/ |1i^a|2i^b|3i^c. The power a,b and c are fixed by little group

A3(1 2 3⁺)/ |1i²|2i²|3i ².

Non-vanishing A3(1 2 3⁺) which has correct mass-dimension must be A₃(1 2 3⁺) = h12i⁴

h12ih23ih31i which we have shown in eq.(2.65).

3 BCFW Recursion Relation

Recursion relations are a method for building higher point amplitudes from lower point amplitudes. Considering an on-shell amplitude, the key idea is to use complex analysis.

The most famous on-shell recursion relation is Britto, Cachazo, Feng and Witten (BCFW) recursion relation. We describe the general BCFW recursion relation in Section 3.1 and Section 3.2 shows how to do BCFW when we have boundary contribution in the complex plane.

3.1 BCFW

An n-point on shell amplitude An is a function of momenta. In general, we can write it as A_n(p₁, p₂,· · · , pn). Here we focus on massless particles so p²_i = 0 for i=1, 2, ..., n. However we can consider a more general functional form of amplitudes that An is a function not only of momenta but also a complex number z. To do this, we introduce n complex vectors v_i^µ which have the property that

Xn i=1

v_i^µ= 0 v^µ_i · vjµ = 0 p^µ_i · viµ= 0.

The last condition just contracts the index µ. ( Index i no sum. ) We introduce complex number z by shifting momenta like

ˆ

p^µ_i ⌘ p^µi + zv_i^µ.

Instead all of momenta of An to the shifted momenta. Now our amplitude depends on momenta pi and a complex number z. Obviously the amplitude An(p1, p2,· · · , pn, z) is equal to the unshifted amplitude when z is zero. So that we integrate the shifted amplitude

I

0

Aˆn(z)

z (3.1)

around z=0. Using Cauchy’s theorem we find Aˆn(z = 0) = X

zi

Resz=zi

Aˆn(z)

z + B. (3.2)

B is the residue of the pole at z =1. When we are doing good shift of momenta, we can drop the boundary term B out and find the contribution on finite z-plane. The problem is where are the poles of the amplitudes? How can we systematically find out all of the finite poles in the complex plane? To answer these questions, we need to use properties of vectors vi and pi. We note that the shifted momenta preserve momentum conservation Pn

i=1

ˆ

p^µ_i = 0 and the shifted momenta are also on-shell.

ˆ

p²_i = (p^µ_i + zv_i^µ)²

= p^µ_ipiµ+ 2zp^µ_iviµ+ z²r^µ_iriµ= 0.

The Feynman diagram tell us that amplitudes diverge while propagator approaches to zero. If we can find out all kind of shifted propagators, then we have already found all the poles on the complex plane. Here we focus on the discussion on tree amplitudes, so that we only need to consider single pole contributions.

By using momentum conservation, the generic shifted propagator looks like 1

Pˆ_I²

where ˆP_I^µ =P

Insert the result of eq.(3.3) into eq.(3.2). Because of the propagator go on-shell, the shifted amplitude factorize into two on-shell amplitudes which we call ˆAL and ˆAR.

X

AˆL and ˆAR are lower point amplitude than ˆAn(z). This is so called recursion relations.

In D=4 spacetime, we choose two particles and shift their momenta. Using spinor representation to demonstrate the shifted momentum:

Particle i:

We call this [i, ji-shift, and this is BCFW recursion relation. Note that hˆiˆji = hiji and [ˆiˆj] = [ij] remain unshifted because hiii and [jj] equal to zero.

Although we can use this powerful method to construct higher points tree amplitudes efficiently, but we may want to ask a question when does this method work? Can we use BCFW recursion relation on any theory we know? To answer this question, we should step back and study the Lagrangian more carefully. Then we would find the recursion relation workability rely on symmetry preservation on the Lagrangian. We will try to explain what this mean more concretely by using scalar-QED Lagrangian and ⁴ theory Lagrangian as examples. But before we investigate Lagrangians, we can answer this workability question in a simple way.

From the above derivation, BCFW recursion relation relies on vanish of boundary contribution after we do contour integration. This statement is discussed on eq.(3.2) before but we can show good shift and bad shift more concretely here. In pure Yang-Mills theory, [10] show the color-ordered gluon tree amplitudes under BCFW shift will

have large z behavior like

Minus and plus sign means -1 helicity gluon and +1 helicity gluon respectively. Here we choose two adjacent particles to shift. If we choose two shifted particles non-adjacent then we will get extra power 1/z in each case. Shifting like [ i, [ +i and [++i are called good shift and shifting like [+ i which would have boundary contribution is called bad shift. So a condition for BCFW recursion relation works is that the theory must have good shift. But this condition is so strong that we can only use BCFW recursion relation on some theories. This problem push us to extend recursion relation to theories which don’t have good shift. A systematic method called Multi-step BCFW [20] is one way to use recursion relation when we have boundary contribution. We will briefly introduce Multi-step BCFW in the next section.

Now we try to study the scalar-QED Lagrangian LQED = 1

4Fµ⌫² + (Dµ )^⇤(D^µ ) 1

4 | |⁴ (3.9)

where Dµ= @µ+ ieAµ is the covariance derivative. The interaction between scalar field and photon Aµ is encoded in (Dµ )^⇤(D^µ ). We can expand out this term

(D_µ )^⇤(D^µ ) =|@ |²+ ieA^µ[(@_µ )^{⇤ ⇤}@_µ ] e²A^µA_µ ^⇤ . (3.10) We find that e²A^µAµ ⇤ which will give us four-point vertex is not gauge invariant. But as we know four-point on-shell scattering amplitudes is a physical quantity which must be invariant under gauge transformation. We have same results on three-point vertex and three points on-shell scattering amplitudes. If we want to preserve gauge symmetry in Lagrangian, then we will have explicit relation between three points and four points amplitude. Actually when we write down the covariant derivative like eq.(3.10) the formulation between three-point vertex and four-point vertex are fixed. In this sense, we expect recursion relation work in scalar-QED. But we shouldn’t forget whether boundary contribution exist or not. We compute 4-point scalar amplitude for example. Using BCFW recursion relation, we will find

ABCF W( ^{⇤ ⇤}) = ˜e² h13i²h24i²

h12ih23ih34ih41i. (3.11)

We can compare this result with 4-point amplitude which we compute using Feynman rule

A_{F eynman}( ^{⇤ ⇤}) = + ˜e²+ ˜e² h13i²h24i²

h12ih23ih34ih41i. (3.12)

We meet a problem! First two terms on RHS ( and ˜e²) are the boundary terms when we are doing BCFW. As we claim BCFW recursion relation works when there are no boundary contributions, if + ˜e² = 0 then ABCF W should be the answer. This statement agrees with eq.(3.11) and eq.(3.12).

Now we try to study ⁴ theory Lagrangian L = 1

2(@_µ )² 1 2m^{2 2}

4!

4. (3.13)

We may think all of the information of n points amplitude(n > 4) as encoded in ⁴. So that we can use recursion relation construct higher points amplitude. From the above experience, we must be careful about the boundary contribution. On 6 points amplitudes we find that there is no way to shift the amplitude without boundary term. This means we can not use recursion relations.

Above two examples show when we can use BCFW recursion relation. We can do the same analysis to Yang-Mills theory color-ordered amplitudes, and this time we will find BCFW work in some good shifts.