Lecture 6

Chasse_neige

Four Particle Test

From 3 pt amplitude, using consistency to guess the result of 4 pt.

Tension: non-zero spin will not be compatible with locality.

We assume perturbativity in the 4 pt test and amplitudes are doinated by the on-shell pole terms. (+ analytic terms required by consistency), this means that $M$s are rational functions of all kinetic variables. We call this “tree-level approximation”.

DoF Counting

$$4\times 4-4-10=2$$$$\begin{array}{c}s\equiv -(p_1+p_2{)}^2\\ t\equiv -(p_1+p_4{)}^2\\ u\equiv -(p_1+p_3{)}^2\end{array}$$

And we have

$$s+u+t=0$$

The 4 pt amplitude can be expressed as

$$M_4=M_4(s,t,u)$$

We assume that $M$ is a rational function.

Most General Form of $M_4$

Self-Interact Massless Scalar

On-shell Factorization Theorem gives that in the $s$ channel

$$\underset{s\to 0}{lim}i{M}_4(1,2,3,4)=i{M}_3(p_1,p_2,-p_s)\times i{M}_3(p_s,p_3,p_4)\times \frac{-i}{-s-iϵ}$$

Doing the same thing for $t$ and $u$ channels, we can derive that

$$M_4=-{\mu }^2(\frac{1}{s}+\frac{1}{t}+\frac{1}{u})+\text{regular terms}$$

The regular term will not contain any 1-order terms of stu

$$\text{regular term}={\lambda }_0+{\lambda }_1(s^2+t^2+u^2)+{\lambda }_{2}stu+\cdots$$

From the field theory we can claim that

$$\frac{1}{s}+\frac{1}{t}+\frac{1}{u}\to \mu {\varphi }^3\times \mu {\varphi }^3$$$$Const\to {\lambda }_0{\varphi }^4$$$$s^2+t^2+u^2\to {\lambda }_1({\partial }_{\mu }{\varphi }^4)$$$$stu\to {\lambda }_2({\partial }_{\mu }\varphi {)}^2◻({\partial }_{\nu }\varphi {)}^2$$

Yang-Mills Theory

Massless spin-1 particles ($n\ge 3$), we call these particles gluons, and the species colors. The amplitude

$$M_4(1_a^{+},2_b^{+},3_c^{-},4_d^{-})=[12{]}^2⟨34{⟩}^2\cdot f(s,t,u)$$

s-channel ($s\to 0$)

$$p_s\equiv p_1+p_2=-|s⟩[s|$$

Under this representation

$$\begin{array}{c}\underset{s\to 0}{lim}i{M}_4=\frac{i}{s}\sum _{e}i{M}_3(1_a^{+},2_b^{+},-s_e^{-})\times i{M}_3(s_e^{+},3_c^{-},4_d^{-})\\ =-\frac{i}{s}\sum _e{\kappa }_{abe}{\kappa }_{cde}\frac{[12{]}^3}{[1s][s2]}\frac{⟨34{⟩}^3}{⟨3s⟩⟨s4⟩}\end{array}$$

Simplify

$$[1s]⟨s4⟩=[11]⟨14⟩+[12]⟨24⟩=[12]⟨24⟩$$$$⟨3s⟩[s2]=⟨34⟩[42]$$

Define

$$\alpha =\sum _e{\kappa }_{abe}{\kappa }_{ecd}$$

Then the amplitude can be expressed as

$$\underset{s\to 0}{lim}{M}_4=[12{]}^2⟨34{⟩}^2\frac{\alpha }{st}$$

t-channel ($t\to 0$)

$$p_t=-|t⟩[t|$$

Amplitude is

$$\begin{array}{c}\underset{t\to 0}{lim}i{M}_4=\frac{i}{t}\sum _{e}i{M}_3(1_a^{+},4_d^{-},-t_e^{+})\times i{M}_3(t_e^{-},2_b^{+},3_c^{-})+i{M}_3(1_a^{+},4_d^{-},-t_e^{-})\times i{M}_3(t_e^{+},2_b^{+},3_c^{-})\\ =-\frac{i}{t}\sum _e{\kappa }_{dae}{\kappa }_{bce}[\frac{[1t{]}^3}{[14][4t]}\frac{⟨3t{⟩}^3}{⟨32⟩⟨2t⟩}+\frac{[2t{]}^3}{[23][3t]}\frac{⟨4t{⟩}^3}{⟨41⟩⟨1t⟩}]\end{array}$$

while $t\to 0$, we have

$$\begin{array}{c}[14]⟨41⟩\approx O(t)\\ [23]⟨32⟩\approx O(t)\end{array}$$

We suppose that $⟨14⟩\approx O(t),[14]\approx O(1)$, so we can derive that

$$⟨1t⟩\approx ⟨4t⟩\approx O(t)$$

$[23]\approx O(t),⟨23⟩\approx O(1)$

$$[23]\approx [2t]\approx [3t]\approx O(t)$$

Under this assumption, the amplitude has the form

$$\underset{t\to 0}{lim}{M}_4=[12{]}^2⟨34{⟩}^2\frac{\beta }{tu}$$

Similarly, in the u-channel, we have

$$\underset{u\to 0}{lim}{M}_4=[12{]}^2⟨34{⟩}^2\frac{\gamma }{us}$$

From these limit forms, we can guess the rational function $f(s,t,u)$

$$f(s,t,u)=\frac{A}{st}+\frac{B}{tu}+\frac{C}{us}$$

where

$$\begin{array}{c}A-C=\alpha \\ B-A=\beta \\ C-B=\gamma \end{array}$$

So this form is only possible if

$$\sum _e{\kappa }_{abe}{\kappa }_{cde}+{\kappa }_{ade}{\kappa }_{bce}+{\kappa }_{ace}{\kappa }_{dbe}=0$$

called Jacobi’s Identity.

Consistant interaction among $n$ ($n\ge 3$) massless spin-1 particles (by minimal coupling) must have a Lie-algebra structure.

Self-Interaction of Spin-j (j > 1)

$$\underset{s\to 0}{lim}{M}_4=\frac{\alpha }{s}{\left(\frac{[12{]}^2⟨34{⟩}^2}{t}\right)}^j$$

$u$ and $t$ channels are similar.

Gravity (j = 2)

$$\begin{gathered} M_4(1_a^{+2},2_b^{+2},3_c^{-2},4_c^{-2})=[12{]}^4⟨34{⟩}^4\frac{A}{stu} \\ \to \{\begin{array}{l}-\frac{A}{s{t}^2}(s\to 0)\\ -\frac{A}{t{u}^2}(t\to 0)\\ -\frac{A}{u{s}^2}(u\to 0)\end{array} \end{gathered}$$

This gives out

$$\alpha =\beta =\gamma$$$$\sum _e{\kappa }_{abe}{\kappa }_{cde}=\sum _e{\kappa }_{ade}{\kappa }_{bce}=\sum _e{\kappa }_{ace}{\kappa }_{bde}$$

With a proper choice of basis, we can give out a simple form of $\kappa$

$${\kappa }_{abc}={\kappa }_a{\delta }_{ab}{\delta }_{ac}$$

There can not be more than one massless spin-2 particles interaction by each other through minimal coupling.

Amplitude:

$$M_4=[12{]}^4⟨34{⟩}^4\frac{{\kappa }^2}{stu}$$

Consider spin-j (j>2)

$$M_4(1_a^{+j},2_b^{+j},3_c^{-j},4_d^{-j})=[12{]}^{2j}[34{]}^{2j}f(s,t,u)$$$$[M_4]=0\Rightarrow [f]=-4j$$

Do consistent factorization similarly, which requires $f\propto {\kappa }^2,[\kappa ]=1-j$

$$f={\kappa }^2\tilde{f}(s,t,u)$$

where $\tilde{f}$ is a rational function of $s,t,u$

$$[\tilde{f}]=-2-2j$$

If $j>2$, we can say that $[\tilde{f}]<-6$, which is impossible because this will cause double-pole singularity in any channel.

So massless spin-j particles cannot have minimal self-interaction.

Remarks

1. No underlying theory assumed in our derivation. Tension: spin $\leftrightarrow$ locality
2. $j=1,2$ 4pt amplitudes’s form is nearly “uniquely” fixed.
3. “constructable”: 3pt $\to$ n-pt ($n>3$) “BCFW”

Gravitational Couplings of all Massless Particles

Simplify: we only consider how 1 species of spin-j particles coulple with gravitons.

$$M(1^{+j},2^{+j},3^{-j},4^{-j})$$

is impossible if $j>2$ (same reason as analyzed above, they can’t minimal couple with gravitons), so we only consider $j=0,1,2$

Gravitional Compton Scattering

Gravitional coupling ${\kappa }_j$ of a spin-j particles equals the graviton cubic self-coupling $\kappa$.

If we consider more than 1 massless spin-2 particles, then any other massless can only couple gravitationally to one of them.

Remarks

1. Minimal couplings are assumed in all our derivation.
2. Massive particles can also be derived using the same trick, but the spinor-helicity formalism will be more complicated (need to spinors to construct one momentum).