Worldline master formulas for the dressed electron propagator, part 2: On-shell amplitudes

The starting point in the second-order formalism is the following factorisation of the $x$-space Dirac propagator $S^{x^{\prime}x}[A]$ in a Maxwell background (we suppress spin indices for brevity),

$\displaystyle S^{x^{\prime}x}[A]$

$\displaystyle=$

$\displaystyle\bigl{[}m+i\not{D}^{\prime}\bigr{]}K^{x^{\prime}x}[A]\,,$

(1)

where $\not{D}^{\prime}=\gamma^{\mu}D^{\prime}_{\mu}$, $D_{\mu}^{\prime}=\partial_{\mu}^{\prime}+ieA_{\mu}(x^{\prime})$ is the covariant derivative and we introduced the matrix elements¹¹1See appendix A for our conventions.

$\displaystyle K^{x^{\prime}x}[A]$

$\displaystyle\equiv$

$\displaystyle\langle x^{\prime}\big{|}\bigl{[}m^{2}+\hskip 3.00003pt\widehat{\pi}\hskip 3.99994pt^{\mu}\hskip 3.00003pt\widehat{\pi}\hskip 3.99994pt_{\mu}+{i\over 2}\,e\gamma^{\mu}\gamma^{\nu}F_{\mu\nu}(\hskip 3.00003pt\widehat{x}\hskip 3.99994pt)\bigr{]}^{-1}\big{|}x\rangle$

(2)

where $\hskip 3.00003pt\widehat{\pi}\hskip 3.99994pt_{\mu}=\hskip 3.00003pt\widehat{p}\hskip 3.99994pt_{\mu}+eA_{\mu}(\hskip 3.00003pt\widehat{x}\hskip 3.99994pt)$. For this “kernel” function, following fragit , we then derived the following path integral representation:

	$\displaystyle K^{x^{\prime}x}[A]$	$\displaystyle=$	$\displaystyle\int_{0}^{\infty}dT\,\,{\rm e}^{-m^{2}T}\,{\rm e}^{-{1\over 4}\frac{(x-x^{\prime})^{2}}{T}}\int_{q(0)=0}^{q(T)=0}Dq\,{\rm e}^{-\int_{0}^{T}d\tau\bigl{[}{1\over 4}\dot{q}^{2}+ie\,\dot{q}\cdot A(x_{0}+q)+ie\frac{x^{\prime}-x}{T}\cdot A(x_{0}+q)\bigr{]}}$		(3)
			$\displaystyle\times 2^{-\frac{D}{2}}{\rm symb}^{-1}\int_{\psi(0)+\psi(T)=0}D\psi\,\,{\rm e}^{-\int_{0}^{T}d\tau\,\bigl{[}{1\over 2}\psi_{\mu}\dot{\psi}^{\mu}-ie(\psi+\eta)^{\mu}F_{\mu\nu}(x_{0}+q)(\psi+\eta)^{\nu}\bigl{]}}\,.$

Here $x_{0}=x+\frac{\tau}{T}(x^{\prime}-x)$ is the straight-line path between the endpoints, $\eta^{\mu}$ is an external Grassmann Lorentz vector, and the inverse of the “symbol map” symb converts products of $\eta$’s into fully antisymmetrised products of Dirac matrices,

$\displaystyle{\rm symb}\bigl{(}\gamma^{[\alpha_{1}\alpha_{2}\cdots\alpha_{n}]}\bigr{)}\equiv(-i\sqrt{2})^{n}\eta^{\alpha_{1}}\eta^{\alpha_{2}}\ldots\eta^{\alpha_{n}}\,.$

(4)

When working in four dimensions, this reduces to the three cases

	$\displaystyle{\rm symb}^{-1}(1)$	$\displaystyle=$	$\displaystyle{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}\,,$		(5)
	$\displaystyle{\rm symb}^{-1}(\eta^{\alpha_{1}}\eta^{\alpha_{2}})$	$\displaystyle=$	$\displaystyle-\frac{1}{4}[\gamma^{\alpha_{1}},\gamma^{\alpha_{2}}]\,,$		(6)
	$\displaystyle{\rm symb}^{-1}(\eta^{\alpha_{1}}\eta^{\alpha_{2}}\eta^{\alpha_{3}}\eta^{\alpha_{4}})$	$\displaystyle=$	$\displaystyle-\frac{i}{4}\varepsilon^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}\gamma_{5}\,.$		(7)

We then performed the usual projection onto an $N$-photon background of fixed momenta and polarisations, and used Gaussian integration on both the orbital path integral, $\int Dq(\tau)$, and spin path integral, $\int D\psi(\tau)$, to derive Bern-Kosower type master formulas for the $N$-photon kernel $K$ both in configuration and in momentum space. We also showed a convenient way to decompose these amplitudes into contributions with a fixed number of orbital and spin interactions, which we refer to as a spin-orbit decomposition.

As in I, in our applications here we will focus entirely on momentum-space amplitudes. Fourier transforming the starting identity (1) to momentum space, and projecting it onto the $N$-photon sector, it turns into

	$\displaystyle S_{N}^{p^{\prime}p}[k_{1},\varepsilon_{1};\ldots;k_{N},\varepsilon_{N}]$	$\displaystyle=$	$\displaystyle({\not{p}^{\prime}}+m)K_{N}^{p^{\prime}p}[k_{1},\varepsilon_{1};\ldots;k_{N},\varepsilon_{N}]$		(8)
			$\displaystyle-e\sum_{i=1}^{N}{\not{\varepsilon}_{i}}K_{(N-1)}^{p^{\prime}+k_{i},p}[k_{1},\varepsilon_{1};\ldots;\hat{k}_{i},\hat{\varepsilon}_{i};\ldots;k_{N},\varepsilon_{N}]\,.$

Here in the second term the ‘hat’ on $\varepsilon_{i}$ and $k_{i}$ means omission. The terms involving $K_{(N-1)}$ are called “subleading” and arise because one of the $N$ photons could be taken from the $A_{\mu}$ appearing in the covariant derivative in the factor $\bigl{[}m+i\not{D}^{\prime}\bigr{]}$ in (1). An important advantage of the “worldline representation” is that it automatically takes care of the permutations over the $N$ photons, and thus avoids the break-up of the amplitude into individual Feynman diagrams. This may not seem very relevant at the tree-level, but becomes an important issue when tree-level amplitudes are used for the construction of multiloop amplitudes by sewing 15 ; 41 ; 100 .

In I our focus was on the use of the off-shell amplitudes as a building block for loop amplitudes, which we exemplified by a recalculation of the one-loop electron self-energy in an arbitrary gauge and dimension. Now, our objective is to explore the simplifications which can be achieved in the on-shell case, both for the amplitudes themselves as well as for the linear ($N=2$) and non-linear ($N>2$) Compton scattering cross sections that can be constructed out of them. Thus our principal object of interest is the on-shell matrix element corresponding to the dressed electron propagator with fixed spins $s,s^{\prime}$. In its construction we must remember that the second-order representation of the dressed electron propagator still contains the external propagators, which must be removed before going on-shell. Thus the matrix element has to be written as

$\displaystyle{\cal M}_{Ns^{\prime}s}^{p^{\prime}p}=\bar{u}_{s^{\prime}}(-p^{\prime})(-{\not{p}^{\prime}}+m)S_{N}^{p^{\prime}p}(\not{p}+m)u_{s}(p)\,,$

(9)

where the spinors satisfy the on-shell relations

$\displaystyle\bar{u}_{s^{\prime}}(-p^{\prime})(-{\not{p}^{\prime}}+m)=0=(\not{p}+m)u_{s}(p)\,.$

(10)

Thus these zeroes must be cancelled by corresponding poles to get a non-vanishing matrix element, which is the fermionic version of the LSZ theorem. Looking at the decomposition (8), we can see how this works for the leading term: as explained in part I, $K_{N}^{p^{\prime}p}$ is, in the second-order formalism, built from untruncated Feynman diagrams involving only scalar propagators. Thus it contains a factor $\frac{1}{\left(p^{\prime 2}+m^{2}\right)\left(p^{2}+m^{2}\right)}$, which led us to the redefinition (I. 5.11),

$\displaystyle K_{N}^{p^{\prime}p}$

$\displaystyle=$

$\displaystyle(-ie)^{N}\frac{\mathfrak{K}_{N}^{p^{\prime}p}}{\left(p^{\prime 2}+m^{2}\right)\left(p^{2}+m^{2}\right)}\,.$

(11)

Since $m^{2}+p^{2}=(\not{p}+m)(-\not{p}+m)$, the cancellation of the poles can then be made explicit:

	$\displaystyle\bar{u}_{s^{\prime}}(-p^{\prime})(-{\not{p}^{\prime}}+m)(\not{p}^{\prime}+m)\frac{\mathfrak{K}_{N}^{p^{\prime}p}}{\left(p^{\prime 2}+m^{2}\right)\left(p^{2}+m^{2}\right)}(\not{p}+m)u_{s}(p)$
	$\displaystyle=\bar{u}_{s^{\prime}}(-p^{\prime})\frac{\mathfrak{K}_{N}^{p^{\prime}p}}{-\not{p}+m}u_{s}(p)=\bar{u}_{s^{\prime}}(-p^{\prime})\frac{\mathfrak{K}_{N}^{p^{\prime}p}}{2m}u_{s}(p)\,.$		(12)

The same does not work for the subleading terms, $K_{(N-1)}^{p^{\prime}+k_{i},p}$, since in those the pole $\frac{1}{p^{\prime 2}+m^{2}}$ has been shifted to $\frac{1}{(p^{\prime}+k_{i})^{2}+m^{2}}$. Thus they can be omitted in the calculation of on-shell matrix elements, and our final formula for the matrix elements of the $N$-photon dressed electron propagator becomes

$\displaystyle{\cal M}_{Ns^{\prime}s}^{p^{\prime}p}$

$\displaystyle=$

$\displaystyle\frac{(-ie)^{N}}{2m}\bar{u}_{s^{\prime}}(-p^{\prime})\mathfrak{K}_{N}^{p^{\prime}p}u_{s}(p)\,.$

(13)

As is clear from (7), in $D=4$ the Dirac matrix structure of $\mathfrak{K}_{N}^{p^{\prime}p}$ is (I 5.11),

$\displaystyle\mathfrak{K}_{N}^{p^{\prime}p}$

$\displaystyle=$

$\displaystyle A_{N}1\!\!1+B_{N{\alpha\beta}}\sigma^{\alpha\beta}-iC_{N}\gamma_{5}\,,$

(14)

with $\sigma^{\alpha\beta}=\frac{1}{2}[\gamma^{\alpha},\gamma^{\beta}]$. Moreover, from the spin-orbit decomposition it is clear that $A_{N}$ can be split as

$\displaystyle A_{N}=A_{N}^{\rm scal}+A_{N}^{\psi}\,,$

(15)

where $A_{N}^{\rm scal}$ is, up to the coupling constant factor, the truncated scalar propagator (see (I.1.2) for its path integral representation),

$\displaystyle(-ie)^{N}A_{N}^{\rm scal}=\hskip 3.00003pt\widehat{D}\hskip 3.99994pt_{N}^{p^{\prime}p}\,,$

(16)

while terms in $A_{N}^{\psi}$ involve the spin interaction at least once.

A less obvious simplification concerning the coefficients $A_{N},B_{N{\mu\nu}},C_{N}$ is that there are useful relations between them whenever both $p$ and $p^{\prime}$ are on-shell. We will see that, for $D=4$ ($\widetilde{B}$ is the dual to $B$),

	$\displaystyle A_{N}(m^{2}-p\cdot p^{\prime})$	$\displaystyle=$	$\displaystyle 2p^{\mu}B_{N\mu\nu}p^{\prime\nu}$		(18)
	$\displaystyle C_{N}(m^{2}+p\cdot p^{\prime})$	$\displaystyle=$	$\displaystyle 2p^{\mu}\widetilde{B}_{N\mu\nu}p^{\prime\nu}\,,$		(19)

so that in principle the full information on the matrix element is already contained in $B_{N\mu\nu}$.

After summing over electron spins, which we are still able to do for arbitrary $N$ and photon helicity assignments, further simplification is possible, leading to the following very compact representation for the spin-averaged cross sections:

$\displaystyle\big{\langle}\big{|}{\cal M}_{N}^{p^{\prime}p}\big{|}^{2}\big{\rangle}$

$\displaystyle=$

$\displaystyle e^{2N}\Bigl{[}\left|A_{N}\right|^{2}+2B_{N}^{\alpha\beta}B_{N{\alpha\beta}}^{\ast}-\left|C_{N}\right|^{2}\Bigr{]}.$

(20)

We then apply all this machinery to the $N=2$ case, performing a complete recalculation of the fully polarised matrix elements for Compton scattering, as well as the spin-averaged and the fully unpolarised cross sections, and recover the known results with relatively little effort.

There are four appendices. In appendix A we summarise our conventions; contrary to I, where Euclidean conventions were used throughout, for the computation of on-shell quantities it is, as usual, preferable to Wick rotate from Euclidean to Minkowski space. Apart from changing the metric from $(++++)$ to $(-+++)$, this will also induce a factor of $(-i)$ for each propagator and a factor of $i$ for each vertex. This amounts to a global factor of $(-i)$ for the dressed scalar or spinor propagators $D_{N}^{p^{\prime}p}$ or $S_{N}^{p^{\prime}p}$, but a factor $i$ for their truncated versions $\hskip 3.00003pt\widehat{D}\hskip 3.99994pt_{N}^{p^{\prime}p}$ and $\hskip 3.00003pt\widehat{S}\hskip 3.99994pt_{N}^{p^{\prime}p}$, defined by

	$\displaystyle\hskip 3.00003pt\widehat{D}\hskip 3.99994pt_{N}^{p^{\prime}p}$	$\displaystyle\equiv$	$\displaystyle\left(p^{\prime 2}+m^{2}\right)D_{N}^{p^{\prime}p}\left(p^{2}+m^{2}\right)\,,$		(21)
	$\displaystyle\hskip 3.00003pt\widehat{S}\hskip 3.99994pt_{N}^{p^{\prime}p}$	$\displaystyle\equiv$	$\displaystyle(-{\not{p}^{\prime}}+m)S_{N}^{p^{\prime}p}(\not{p}+m)\,.$		(22)

Thus in Minkowski space we have $i{\cal M}_{{\rm scal},N}^{p^{\prime}p}=i\hskip 3.00003pt\widehat{D}\hskip 3.99994pt_{N}^{p^{\prime}p}$ and $i{\cal M}_{{\rm spin},N}^{p^{\prime}p}=i\hskip 3.00003pt\widehat{S}\hskip 3.99994pt_{N}^{p^{\prime}p}$.

The basic idea of the R-representation has been explained already in section 2.3 of I. The electromagnetic potential is specialised to a sum of plane waves, $A^{\mu}(x)=\sum_{i=1}^{N}\varepsilon_{i}^{\mu}\,{\rm e}^{ik_{i}\cdot x}$, and we project onto the contribution to the kernel that is multi-linear in the $\varepsilon_{i}$. The photons inserted along the line are then represented by vertex operators,

$\displaystyle V_{\rm scal}[k,\varepsilon]=\int_{0}^{T}d\tau\,\varepsilon\cdot\dot{x}(\tau)\,{\rm e}^{ik\cdot x(\tau)}\,.$

(23)

The simplest way of achieving manifest transversality of photonic amplitudes is to rewrite each vertex operator from the beginning in terms of the photon field-strength tensor. In scalar QED one can do this by adding to the vertex operator a total-derivative term supplying the “missing half” of that tensor:

$\displaystyle V_{\textrm{scal}}[k,\varepsilon;r]:=V_{\rm scal}[k,\varepsilon]+i\frac{\varepsilon\cdot r}{k\cdot r}\int_{0}^{T}d\tau\frac{d}{d\tau}\,{\rm e}^{ik\cdot x(\tau)}=\int_{0}^{T}d\tau\frac{r\cdot f\cdot{\dot{x}}}{r\cdot k}\,{\rm e}^{ik\cdot x(\tau)}\,.$

(24)

Here $r$ is a “reference vector” that is arbitrary except for the constraint $k\cdot r\neq 0$. In the open-line case, adding this term causes non-vanishing boundary terms, but these do not have both of the poles necessary for contributing to the on-shell matrix element ${\cal M}_{{\rm scal},N}^{p^{\prime}p}$, defined by the on-shell limit of $\left(p^{\prime 2}+m^{2}\right)D_{N}^{p^{\prime}p}\left(p^{2}+m^{2}\right)$. Thus, as far as the calculation of ${\cal M}_{{\rm scal},N}^{p^{\prime}p}$ is concerned, the $N$-photon dressed scalar propagator $D_{N}^{p^{\prime}p}$ could as well be used with the replacement

$\displaystyle\varepsilon_{i}\rightarrow\frac{r_{i}\cdot f_{i}}{r_{i}\cdot k_{i}},\quad i=1,\ldots,N$

(25)

throughout. Thus, in particular, the master formula (I. 2.25) (originally due to dashsu and ahmbassch-16 ),

	$\displaystyle D_{N}^{p^{\prime}p}(k_{1},\varepsilon_{1};\cdots;k_{N},\varepsilon_{N})$	$\displaystyle=$	$\displaystyle(-ie)^{N}\int_{0}^{\infty}dT\,{\rm e}^{-m^{2}T}$
		$\displaystyle\times$	$\displaystyle\int_{0}^{T}\prod_{i=1}^{N}d\tau_{i}\,{\rm e}^{\sum_{i,j=0}^{N+1}\big{[}\frac{1}{2}|\tau_{i}-\tau_{j}|k_{i}\cdot k_{j}-i{\rm sgn}(\tau_{i}-\tau_{j})\varepsilon_{i}\cdot k_{j}+\delta(\tau_{i}-\tau_{j})\varepsilon_{i}\cdot\varepsilon_{j}\big{]}}\Big{|}_{\varepsilon_{1}\varepsilon_{2}\cdots\varepsilon_{N}}\,,$

$(k_{0}=p^{\prime}\,,k_{N+1}=p\,,\tau_{0}=T\,,\tau_{N+1}=\varepsilon_{0}=\varepsilon_{N+1}=0)$ can then as well be written purely in terms of the field strength tensors of the $N$ photons in the form

	$\displaystyle D_{N}^{p^{\prime}p}(k_{1},\varepsilon_{1},r_{1};\cdots;k_{N},\varepsilon_{N},r_{N})$	$\displaystyle=(-ie)^{N}\int_{0}^{\infty}dT\,{\rm e}^{-m^{2}T}$
		$\displaystyle\times\int_{0}^{T}\prod_{i=1}^{N}d\tau_{i}\,{\rm e}^{\sum_{i,j=0}^{N+1}\big{[}|\tau_{i}-\tau_{j}|{1\over 2}k_{i}\cdot k_{j}-i{\rm sgn}(\tau_{i}-\tau_{j}){r_{i}\cdot f_{i}\cdot k_{j}\over r_{i}\cdot k_{i}}-\delta(\tau_{i}-\tau_{j}){r_{i}\cdot f_{i}\cdot f_{j}\cdot r_{j}\over r_{i}\cdot k_{i}\,r_{j}\cdot k_{j}}\big{]}}\Bigg{|}_{f_{1}f_{2}\ldots f_{N}}$

even though the right-hand sides of (LABEL:master-dss) and (LABEL:linemastercov) are not actually equal.

The result of expanding out the exponential factor in (LABEL:linemastercov) will be named $(-i)^{N}\bar{R}_{N}\,{\rm e}^{(\cdot)}$, where ${\rm e}^{(\cdot)}$ was already introduced in part I,

	$\displaystyle{\rm e}^{(\cdot)}$	$\displaystyle\equiv$	$\displaystyle{\rm e}^{{1\over 2}\sum_{i,j=0}^{N+1}|\tau_{i}-\tau_{i}|k_{i}\cdot k_{j}}$		(28)
		$\displaystyle=$	$\displaystyle{\rm e}^{-Tp^{\prime 2}+\frac{1}{2}\sum_{i,j=1}^{N}|\tau_{i}-\tau_{i}|k_{i}\cdot k_{j}+(p-p^{\prime})\cdot\sum_{i=1}^{N}k_{i}\tau_{i}}\,.$

The on-shell master formula (LABEL:linemastercov) can then be rewritten as

$\displaystyle D_{N}^{p^{\prime}p}(k_{1},\varepsilon_{1},r_{1};\cdots;k_{N},\varepsilon_{N},r_{N})$

$\displaystyle=$

$\displaystyle(-e)^{N}\int_{0}^{\infty}dT\,{\rm e}^{-m^{2}T}\int_{0}^{T}\prod_{i=1}^{N}d\tau_{i}\ \bar{R}_{N}\,{\rm e}^{(\cdot)}.$

(29)

For use below let us also write down the first two polynomials $\bar{R}_{N}$:

$\displaystyle\begin{split}\bar{R}_{1}&=\sum_{i=0}^{2}{\rm sgn}(\tau_{1}-\tau_{i})\frac{r_{1}\cdot f_{1}\cdot k_{i}}{r_{1}\cdot k_{1}}=\frac{r_{1}\cdot f_{1}\cdot(p-p^{\prime})}{r_{1}\cdot k_{1}}\,,\\ \bar{R}_{2}&=\sum_{i,j=0}^{3}{\rm sgn}(\tau_{1}-\tau_{i})\frac{r_{1}\cdot f_{1}\cdot k_{i}}{r_{1}\cdot k_{1}}{\rm sgn}(\tau_{2}-\tau_{j})\frac{r_{2}\cdot f_{2}\cdot k_{j}}{r_{2}\cdot k_{2}}+2\delta(\tau_{1}-\tau_{2}){r_{1}\cdot f_{1}\cdot f_{2}\cdot r_{2}\over r_{1}\cdot k_{1}\,r_{2}\cdot k_{2}}\,.\end{split}$

(30)

To obtain additional simplifications, we observe that the most complicated term in $\bar{R}_{N}$ is always given by the product

$\displaystyle\prod\limits_{j=1}^{N}\Bigl{(}\sum_{i_{j}=0}^{N+1}{\rm sgn}(\tau_{j}-\tau_{i_{j}})\frac{r_{j}\cdot f_{j}\cdot k_{i_{j}}}{r_{j}\cdot k_{j}}\Bigr{)}$

(31)

which, upon choosing²²2Excluding the case when $N=1$, since here $p^{\prime}\cdot k_{1}=0$ on-shell. $r_{j}=p^{\prime}$ for $j=1,\ldots,N$, turns into

$\displaystyle\prod\limits_{j=1}^{N}\frac{p^{\prime}\cdot f_{j}\cdot\bigl{(}p+\sum_{i_{j}=1}^{N}{\rm sgn}(\tau_{j}-\tau_{i_{j}})k_{i_{j}}\bigr{)}}{p^{\prime}\cdot k_{j}}\,.$

(32)

For every ordering of the variables $\tau_{1},\ldots,\tau_{N}$, there is a $j$ such that the $j$th term in the product is given by

$\displaystyle p^{\prime}\cdot f_{j}\cdot\Bigl{(}p+\sum_{i_{j}=1}^{N}{\rm sgn}(\tau_{j}-\tau_{i_{j}})k_{i_{j}}\Bigr{)}=p^{\prime}\cdot f_{j}\cdot\Bigl{(}p+\sum_{i_{j}=1,i_{j}\neq j}^{N}k_{i_{j}}\Bigr{)}=-p^{\prime}\cdot f_{j}\cdot(p^{\prime}+k_{j})=0\,.$

(33)

Thus the term in question can be made to vanish by this choice of reference momenta, leaving only those contributions in $\bar{R}_{N}$ that involve at least one of the $\delta(\tau_{i}-\tau_{j})$ (these $\delta$-functions generate the seagull vertex of scalar QED).

Proceeding to the spinor-line case, in section 6 of part I we had seen that the $N$-photon kernel $K_{N}^{p^{\prime}p}$ can be decomposed naturally as

$\displaystyle K_{N}$

$\displaystyle=$

$\displaystyle\sum_{S=0}^{N}K_{NS}\,,$

(34)

since in the representation (3) after specialising to an $N$-photon background a certain number $S$ of photons have to be taken out of the spin path integral $\int D\psi$, and the remaining $N-S$ ones out of the coordinate (orbital) path integral $\int Dq$. The former ones are by convention associated to the variables $\tau_{1},\ldots,\tau_{S}$, and since the spin path integral couples to the background field only through the field-strength tensor $F_{\mu\nu}$, manifest transversality in those indices is automatic, as is also borne out by our explicit formula for these terms, written in terms of functions $W_{\eta}$, (6.8) of part I (some of these are given below). Thus only the transversality of the orbital part has to be taken care of, and here it is obvious that the only change necessary in the final formula for $K_{NS}$, (I. 6.10), is to replace the orbital prefactor polynomials $\bar{P}_{NS}^{\{i_{1}i_{2}\ldots i_{S}\}}$ by the corresponding objects of the R representation, defined by (compare (LABEL:linemastercov) and (I.6.11))

$\displaystyle{\rm e}^{\sum_{i,j=0}^{N+1}\big{[}-i{\rm sgn}(\tau_{i}-\tau_{j}){r_{i}\cdot f_{i}\cdot k_{j}\over r_{i}\cdot k_{i}}-\delta(\tau_{i}-\tau_{j}){r_{i}\cdot f_{i}\cdot f_{j}\cdot r_{j}\over r_{i}\cdot k_{i}\,r_{j}\cdot k_{j}}\big{]}}\big{|}_{f_{i_{1}}=\cdots=f_{i_{S}}=0}\big{|}_{f_{i_{S+1}}\cdots f{i_{N}}}$

$\displaystyle\equiv$

$\displaystyle(-i)^{N-S}\bar{R}_{NS}^{\{i_{1}i_{2}\ldots i_{S}\}}\,.$

(35)

Thus we arrive at the following manifestly transversal version of the spin-orbit decomposition (compare with (I 6.10)):

	$\displaystyle K_{NS}$	$\displaystyle=$	$\displaystyle\sum_{\{i_{1}i_{2}\ldots i_{S}\}}K_{NS}^{\{i_{1}i_{2}\ldots i_{S}\}}\,,$
	$\displaystyle K_{NS}^{\{i_{1}i_{2}\ldots i_{S}\}}$	$\displaystyle=$	$\displaystyle(-e)^{N}\textrm{symb}^{-1}\int_{0}^{\infty}dT\,{\rm e}^{-m^{2}T}\prod_{i=1}^{N}\int_{0}^{T}d\tau_{i}\,W_{\eta}(k_{i_{1}},\varepsilon_{i_{1}};\ldots;k_{i_{S}},\varepsilon_{i_{S}})\bar{R}_{NS}^{\{i_{1}i_{2}\ldots i_{S}\}}{\rm e}^{(\cdot)}\,,$

where the sum runs over all choices of $S$ out of the $N$ variables, and ${\rm e}^{(\cdot)}$ was given in (28).

This fact we can see most directly from the scalar master formula in its “first alternative version,” equation (I.2.23):

	$\displaystyle D_{N}^{p^{\prime}p}(k_{1},\varepsilon_{1};\cdots;k_{N},\varepsilon_{N})=(-ie)^{N}\int_{0}^{\infty}dT\,{\rm e}^{-T(m^{2}+p^{\prime 2})}\int_{0}^{T}\prod_{i=1}^{N}d\tau_{i}$
	$\displaystyle\times{\rm e}^{\sum_{i=1}^{N}(-2k_{i}\cdot p^{\prime}\tau_{i}+2i\varepsilon_{i}\cdot p^{\prime})+\sum_{i,j=1}^{N}\big{[}\bigl{(}\frac{|\tau_{i}-\tau_{j}|}{2}-\frac{\tau_{i}+\tau_{j}}{2}\bigr{)}k_{i}\cdot k_{j}-i({\rm sgn}(\tau_{i}-\tau_{j})-1)\varepsilon_{i}\cdot k_{j}+\delta(\tau_{i}-\tau_{j})\varepsilon_{i}\cdot\varepsilon_{j}\big{]}}\Big{|}_{\varepsilon_{1}\varepsilon_{2}\cdots\varepsilon_{N}}\,.$		(58)

If the scalar is massless, $p^{\prime}$ is a null vector on-shell, so we can take it as the reference vector for all the photons. This removes the terms proportional to $\varepsilon_{i}\cdot p^{\prime}$ in the exponent. Then if all the photons have the same helicity, the identity

$\displaystyle\varepsilon^{\pm}(k_{i};q)\cdot\varepsilon^{\pm}(k_{j};q)=0$

(59)

removes all the terms in the exponent involving $\varepsilon_{i}\cdot\varepsilon_{j}$. The only terms left are of the form

$\displaystyle\prod_{i=1}^{N}\Bigl{(}{\rm sgn}(\tau_{i}-\tau_{j_{i}})-1\Bigr{)}\varepsilon_{i}\cdot k_{j_{i}}.$

(60)

Now we apply a similar argument to the one given below (30): for any given ordering of the $\tau_{i}$ there will be one $\tau_{i_{0}}$ with the largest value, so that ${\rm sgn}(\tau_{i_{0}}-\tau_{j_{i_{0}}})=1$, which makes the whole term vanish (taking also into account that on-shell we cannot have $j_{i_{0}}=i_{0}$ since $\varepsilon_{i_{0}}\cdot k_{i_{0}}=0$). This completes the proof of the vanishing of this amplitude.