Spinor Helicity Amplitudes Method

Abstract

In this paper, we introduce the spinor helicity amplitude method. We provide the approaches to handle with massless fermions, massive spinors and polarization vectors respectively, then applied this method to some examples such as e ⁺ e ⁻→μ ⁺ μ ⁻, e ⁻ γ→e ⁻ γ, $\bar {q}q \to gg$ processes. Among the examples, $\bar {q}q\to gg$ process presented in the antecedent literatures, but we provide a more detailed calculation, and others are gave for the first time. We summarize all kinds of the situations, and give the needed formulas in appendix, which will be of use to collect these analysis formulas in one place.

Appendices

Appendix A: Polarization Vectors and Spinor Properties

We will use notations and conventions as in [14], and will summarize them here for ease of reference. Let ψ(p) be a massless Dirac spinor. We will denote its chiral projections as follows:

$$ |{p}\pm\rangle \,= \,{\psi_{\pm}(p)} \,= \frac{1}{2} (1\pm\gamma_5) \psi(p) \quad\quad \langle{p\pm}| \,= \overline{\psi_{\pm}(p)} $$

(A.1)

By convention, we will choose the spinor phases so as to satisfy the following identities:

$$ |{p}\pm\rangle \,=\,|{p}\mp\rangle^{c} \quad\quad \langle{p}\pm| \,=\,^{c}\! \langle{p}\mp|, $$

(A.2)

where the suffix c stands for the charge conjugation operation:

$$ |\,\rangle^{c} \,=\, C |\,\rangle^{*}, \quad\quad^{c}\langle\,| \,=\, -^{*}\langle\,|C^{-1} $$

(A.3)

$$ C \gamma_{\mu}^{*} C^{-1} \,=\, \gamma_{\mu}, $$

(A.4)

$$ C \,=\, C^{\dagger} \,=\, C^{-1} \,=\, C^{*} \,=\, C^{T}. $$

(A.5)

We will also introduce the following notation:

$$ \langle{pq}\rangle \,=\, \langle{p -}|{q +}\rangle \quad\quad [pq] \,=\,\langle{p +}|{q -}\rangle $$

(A.6)

The spinors are normalized as follows:

$$ \langle{p}|\gamma_{\mu} \vert p\rangle \,=\, 2 p_{\mu} $$

(A.7)

From the properties of the Dirac algebra, it is straightforward to prove the following useful identities:

$$ \langle{p +}|{q +}\rangle \,=\, \langle{p-}|{q -}\rangle \,=\,\langle{pp}\rangle \,=\, [{pp}] \,=\, 0 $$

(A.8)

$$ \langle{pq}\rangle \,=\, -\langle{qp}\rangle, \quad\quad [pq] \,=\, -[qp] $$

(A.9)

$$ 2\,|{p}\pm\rangle\,\langle{q}\pm| \,=\, \frac{1}{2} (1\pm\gamma_5) \gamma^{\mu} \langle{q}\pm|\gamma_{\mu} |{p}\pm\rangle , $$

(A.10)

$$ \langle{pq}\rangle^{*} \,=\, - sign(p\cdot q)[pq] \,=\, sign(p\cdot q) [qp] $$

(A.11)

$$ \vert\langle{pq}\vert^{2} \,= \,2(p \cdot q), $$

(A.12)

$$ \langle{p}\pm|\gamma_{\mu_{1}}\dots\gamma_{\mu_{2n+1}} |{q}\pm\rangle \,=\, \langle{q}\mp|\gamma_{\mu_{2n+1}}\dots\gamma_{\mu_{1}} |{p}\mp\rangle , $$

(A.13)

$$ \langle{p}\pm|\gamma_{\mu_{1}}\dots\gamma_{\mu_{2n}}|{q}\mp\rangle \,=\, - \langle{q}\pm |\gamma_{\mu_{2n}}\dots\gamma_{\mu_{1}} |{p}\mp\rangle , $$

(A.14)

$$ \langle{AB}\rangle \langle{CD}\rangle \,=\, \langle{AD}\rangle \langle{CB}\rangle + \langle{AC}\rangle \langle{BD}\rangle $$

(A.15)

$$ \langle{A +}|\gamma_{\mu}|B+\rangle \langle{C-}| \gamma^{\mu}|D -\rangle \,=\, 2[AD] \langle{CB}\rangle. $$

(A.16)

In the identity (A.11) the possibility of having spinors with energies of different sign is considered. This will be important in the following, since for simplicity we will always carry out the calculations of the matrix elements assuming all of the particles as being outgoing. Energy-momentum conservation will then force the energy of some of the particles to be negative.

Notice that the following equations hold for generic chiral spinors (not necessarily solutions of a Dirac equation) which satisfy (A.2): (A.9), (A.10), (A.13), (A.14), (A.15), (A.16).

The polarizations for vectors with momentum p, as defined in the text:

$$ {}\epsilon_{\mu}^{\pm}(p,k) \,=\, \pm \frac{{\langle{p}\pm}|\gamma_{\mu} |{k}\pm\rangle} {\sqrt{2}\langle{k}\mp|{p}\pm\rangle}, $$

(A.17)

$$ \epsilon^{\pm(p,k)} \cdot \gamma \,=\, \pm \frac{\sqrt 2}{\langle{k}\mp|{p}\pm\rangle} (\,|{p}\mp\rangle\,\langle{k}\mp|\, + \,|{k}\pm\rangle\,\langle{p}\pm|\,), $$

(A.18)

enjoy the following properties:

$$ \epsilon_{\mu}^{\pm}(p,k) \,=\, (\,\epsilon_{\mu}^{\mp}(p,k)\,)^{*} , $$

(A.19)

$$ \epsilon^{\pm}(p,k) \cdot p \,=\, \epsilon^{\pm}(p,k) \cdot k \,=\, 0 , $$

(A.20)

$$ \epsilon^{\pm}(p,k) \cdot \epsilon^{\pm}(p,k^{\prime}) \,=\, 0 , $$

(A.21)

$$ \epsilon^{\pm}(p,k) \cdot \epsilon^{\mp}(p,k^{\prime}) \,=\, -1 , $$

(A.22)

$$ \epsilon^{\pm}(p,k) \cdot \epsilon^{\pm}(p^{\prime},k) \,=\, 0 , $$

(A.23)

$$ \epsilon^{\pm}(p,k) \cdot \epsilon^{\mp}(k,k^{\prime}) \,=\, 0 , $$

(A.24)

$$ \epsilon_{\mu}^{+}(p,k) \, \epsilon_{\nu}^{-}(p,k) \,+\, \epsilon_{\mu}^{-}(p,k) \, \epsilon_{\nu}^{+}(p,k) \,=\, -g_{\mu\nu} \,+\, \frac{p_{\mu} k_{\nu}+p_{\nu} k_{\mu}}{p\cdot k}. $$

(A.25)

Appendix B: Notation and Conventions with Massive Spinors

We have used products of Dirac spinors

$$ [ij]\,=\,\bar{u}_{+}(i) u_{-}(j),\,\quad \langle{ij}\rangle \,=\, \bar{u}_{-}(i) u_{+}(j), \quad \, (ij)\,=\, \bar{u}_{\pm}(i) u_{\pm}(j) $$

(B.1)

with massive p _i ,p _j . To evaluate these we use two arbitrary four vectors k ₀ and k ₁, such that

$$ {k_{0}^{2}} \,=\, 0, \quad {k_{1}^{2}}\,=\, -1, \quad k_{0} \cdot k_{1} \,=\, 0. $$

(B.2)

Then

$$ [ij] \,=\, \frac{(p_{i} \cdot k_{0}) (p_{j} \cdot k_{1}) - (p_{j} \cdot k_{0})(p_{i} \cdot k_{1}) - i \epsilon_{\mu \nu \rho \sigma}k_{0}^{\mu} p_{i}^{\nu} p_{j}^{\rho} k_{1}^{\sigma}}{\sqrt{(p_{i}\cdot k_{0})(p_{j} \cdot k_{0})}} $$

(B.3)

$$ \langle{ij}\rangle \,=\, \frac{(p_{j} \cdot k_{0})(p_{i} \cdot k_{1}) -(p_{j} \cdot k_{1}) (p_{i} \cdot k_{0}) - i \epsilon_{\mu \nu \rho \sigma}k_{0}^{\mu} p_{i}^{\nu} p_{j}^{\rho} k_{1}^{\sigma}}{\sqrt{(p_{i}\cdot k_{0})(p_{j} \cdot k_{0})}} $$

(B.4)

$$ {}(ij) \,=\, m_{i} \left(\frac{p_{j} \cdot k_{0}}{p_{i} \cdot k_{0}}\right)^{\frac{1}{2}} \,+ \,i \leftrightarrow j $$

(B.5)

where m _i is negative if i is an antiparticle. Different choices of k ₀ correspond to different choices of the quantization axis of a massive fermion’s spin, as described in Section 2.

We use the notation

(B.6)

(B.7)

Whereas for massless vectors k _i ,k _j we have the familiar relation 2k _i ⋅k _j =〈i j〉 [j i], in the massive case this is extended to

$$ 2 p_{i} \cdot p_{j} \,=\, \langle{ij}\rangle[ji]\, + \,(ij)^{2}. $$

(B.8)

For any massive i,j and massless k,l we have

$$ \hspace{4pc}(ik)(jl) \,=\, (il)(jk), $$

(B.9)

$$ (ik)[li]\, + \,[ik](li) \,=\, m_{i} [lk], $$

(B.10)

(B.11)

The Schouten identity holds,

$$ \langle{a\,b}\rangle\, \langle{c\,d}\rangle\, + \, \langle{a\,c}\rangle\,\langle{d\,b}\rangle \,+\, \langle{a\,d}\rangle\, \langle{bc}\rangle\,=\,0. $$

(B.12)

For gluon polarization vectors we use

$$ \epsilon^{+}_{\mu}(p,k) \,=\, \frac{\bar{u}_{-}(k) \, \gamma_{\mu} \, u_{-}(p)}{{\sqrt{2}}\,\langle{kp}\rangle} , $$

(B.13)

$$ \epsilon^{-}_{\mu}(p,k) \,=\, \frac{\bar{u}_{+}(k) \, \gamma_{\mu} \, u_{+}(p)}{{\sqrt{2}} \,[pk]}, $$

(B.14)

which take the slashed form

(B.15)

(B.16)

We use a shorthand form for the amplitude in which we display only the helicities of the particles involved. So for example,

$$ M(\bar{q}^{+}_{1} \,,\, q^{-}_{2} \, ; \, 3^{+}\,,\, 4^{-}\,,\, 5^{+}) \sim M^{(+-+-+)}. $$

(B.17)