^a^ainstitutetext: Department of Physics and State Key Laboratory of Nuclear Physics and Technology,
Peking University,
Beijing 100871, China^b^binstitutetext: Collaborative Innovation Center of Quantum Matter,
Beijing 100871, China^c^cinstitutetext: Center for High Energy Physics, Peking University,
Beijing 100871, China

Massive On-shell Recursion Relations for $n$ -point Amplitudes

Chao Wu a,b,c and Shou-Hua Zhu wuch7@pku.edu.cn shzhu@pku.edu.cn

Abstract

We construct two and three-line shifts for tree-level amplitude with massless and/or massive particles, and provide a method to construct general multi-line shifts for all masses. We choose the massless-massive BCFW shift from these shifts and examine its validity in renormalizable theories. Using such a shift, we find that amplitudes with at least one massless vector boson are constructible. This reveals the importance of gauge theory in the construction of amplitudes with massive particles. We also find that this kind of amplitudes have a cancellation related to group structure among different channels, which is essential for constructibility. Furthermore, we show that in the limit of large shift parameter $z$ , the amplitude with four massive vector bosons, which can include transverse massive vector particles, have structures proportional to the amplitude with shifted vector particles replaced by Goldstone bosons in the leading order. This is responsible for the failure of massive-massive BCFW recursion relations in the amplitudes with four massive vector bosons.

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1 Introduction

In comparison with Feynman diagrams, on-shell recursion relations provide a more efficient approach to construct higher-point tree-level amplitudes from lower-point amplitudes. It was first motivated by Britto-Cachazo-Feng-Witten (BCFW) recursion relations Britto:2004ap; Britto:2005fq in the calculation of gluon scattering, and other versions of recursion relations were proposed to study the amplitudes in gauge theories Badger:2005zh; Risager:2005vk, gravity theories Cachazo:2005ca, supersymmetric theories Brandhuber:2008pf; Arkani-Hamed:2008owk, scalar effective field theories Cheung:2015ota and more general theories Cohen:2010mi; Cheung:2015cba. They based on the same idea, namely using complex deformation of the external momenta and calculating the residues of deformed amplitudes in the complex plane, to collect the information of factorized lower-point amplitudes.

Contrary to massless particles, the momenta of massive particles cannot be written as a direct product of two spinors. To analyze amplitudes with massive particles, the method of decomposing the massive momenta into two light-like vectors was developed Schwinn:2005pi; Schwinn:2006ca; Schwinn:2007ee; Craig:2011ws; Boels:2011zz. However, this formalism is not the most convenient for specific calculation of amplitudes, since it’s not little-group covariant. Recently, Arkani-Hamed, Huang and Huang Arkani-Hamed:2017jhn introduced a new method by regarding the massive particle as a representation of its little group. In this notation, both amplitudes and complex shifts are simplified into a little-group covariant form. There have been some efforts in constructing massive BCFW shift in the massless-massive case Aoude:2019tzn; Ballav:2020ese and massive-massive case Herderschee:2019dmc; Franken:2019wqr afterwards.

After considering little-group invariance, the on-shell constructibility of amplitudes with massive particles was investigated Franken:2019wqr. Various multi-line shifts were used to estimate the large- $z$ behavior of amplitudes with all particles massive, where $z$ is the shift parameter. Although people prefer using three and more line shifts to investigate the constructibility, two-line shifts are more convenient in the computation of amplitudes.

In this work, we take the little-group covariant spinor helicity formalism and regard different spin states from one massive external leg as a whole, so these states should be deformed by a same shift. In the case that all particles are massless, some researches ArkaniHamed:2008yf; Cheung:2008dn proved that in a gauge theory coupled to scalars and fermions, any massless amplitude with at least one gluon is two-line constructible, which is a strong conclusion. We want to examine whether the amplitude with both massless and massive particles is also two-line constructible.

The present paper is organized as follows. In section 2 we review the basic idea of recursion relations and construct all possible two and three-line shifts for both massive and massless particles. Assuming the coupling is dimensionless, we evaluate the Feynman rule in a diagrammatic way in section 3 and discuss why gauge-fixing is not enough to improve the behavior of amplitude in the large- $z$ limit. In section 4 we evaluate $n$ -point massive amplitudes with at least one massless vector boson in the large- $z$ limit. We find that these amplitudes vanish in the large- $z$ limit except for the all vector amplitudes, which can have a cancellation among different channels. In section LABEL:sec5, we give the relation between such a cancellation and the group structure of massive vectors, and explore why this cancellation fails in the amplitudes without massless particles. Finally, section LABEL:sec6 presents our conclusion and discussion. Appendix LABEL:AppA gives our conventions. Appendix LABEL:AppB explicitly calculates the large- $z$ behavior of polarization vectors in the center-of-mass frame. Appendix LABEL:AppC gives an example of evaluating the diagrammatic expressions.

2 Recursion relations for all masses

For completeness, we review the general complex shift for $A_{n}$ , a tree-level amplitude with $n$ massless particles. For each external particle, we shift their momentum vector $p_{i}$ by complex-valued vector $r_{i}$ ,

\hat{p}_{i}^{\mu}=p_{i}^{\mu}+zr_{i}^{\mu},

(1)

where $i=1,2,\dots,n$ . Now we restrict these shift vectors $r_{i}$ by three conditions,

	$\displaystyle\sum_{i}r_{i}^{\mu}=0,$		(2)
	$\displaystyle r_{i}\cdot r_{j}=0,$		(3)
	$\displaystyle p_{i}\cdot r_{i}=0.$		(4)

These three conditions (2–4) respectively guarantee that (a) momentum conservation holds for shifted momenta, (b) shifted momenta are still on-shell, (c) shifted propagators are linear in $z$ . We can construct a complex function $\hat{A}_{n}(z)/z$ , whose residue at $z=0$ is the unshifted amplitude $A_{n}$ . Then Cauchy’s theorem tells us,

A_{n}=-\sum_{z_{I}}\mathrm{Res}_{z=z_{I}}\frac{\hat{A}_{n}(z)}{z}+B_{n},

(5)

where the boundary term $B_{n}$ is the residue at infinity. The first term on the right-hand side can factorize into two on-shell subamplitudes when the momentum $P_{I}$ of the internal line goes on-shell,

-\mathrm{Res}_{z=z_{I}}\frac{\hat{A}_{n}(z)}{z}=\hat{A}_{L}(z_{I})\frac{1}{P_{I}^{2}}\hat{A}_{R}(z_{I}).

(6)

Now we consider massive amplitudes. The three conditions (2–4) still keep the shifted momenta on-shell $\hat{p}_{i}^{2}=p_{i}^{2}=m^{2}$ . To generalize formula (5) into the massive case, we add a new term which corresponds to massive on-shell propagators,

A_{n}=-\sum_{z_{I}}\mathrm{Res}_{z=z_{I}}\frac{\hat{A}_{n}(z)}{z}-\sum_{z_{J}}\mathrm{Res}_{z=z_{J}}\frac{\hat{A}_{n}(z)}{z}+B_{n}.

(7)

At a $z_{J}$ -pole one of massive internal particles goes on-shell. We use little-group covariant spinors Arkani-Hamed:2017jhn to describe massive particles, so the two subamplitudes $\hat{A}_{L}$ and $\hat{A}_{R}$ have little-group indices and should be contracted,

-\mathrm{Res}_{z=z_{J}}\frac{\hat{A}_{n}(z)}{z}=\epsilon^{J_{1}}_{(J^{\prime}_{1}}\epsilon^{J_{2}}_{J^{\prime}_{2}}\cdots\epsilon^{J_{n}}_{J^{\prime}_{n})}\hat{A}_{L,J_{1}J_{2}\cdots J_{n}}(z_{J})\frac{1}{P_{J}^{2}}\hat{A}_{R}^{J^{\prime}_{1}J^{\prime}_{2}\cdots J^{\prime}_{n}}(z_{J}),

(8)

where the lower indices in the parenthesis means symmetrization of these indices. Here we only write the little-group indices related to the internal momentum $P_{J}$ and neglect other little-group indices. If the boundary term $B_{n}=0$ , the $n$ -point on-shell amplitude $A_{n}$ will be completely determined in lower-point on-shell amplitudes and this recursive formula (7) becomes an on-shell recursion relation under a valid shift.

Before we discuss whether amplitudes vanish in the large- $z$ limit, we should construct complex shifts for the composition of all masses. In this section, we will solve equations (2–4) to give all possible shift vectors $r$ in two and three-line shifts. Since massless shifts have been well studied, their generalizations in the massive case will be based on these massless shifts.

2.1 Shifting massless particles

In spinor-helicity formalism, which is briefly reviewed in appendix LABEL:AppA, we write massless amplitudes in terms of two kinds of Weyl spinor $|p]_{\alpha}$ and $|p\rangle^{\dot{\alpha}}$ . They are two inequivalent fundamental representations of $SL(2,\mathbb{C})$ . Both of them can be shifted, we refer to the former as holomorphic shift and the latter as anti-holomorphic shift.

Since the massive particles takes little-group representation into account, the three conditions (2–4) may not be valid in the massive case. We review some specific massless amplitude recursion relations to translate the three conditions (2–4) into massless spinor-helicity variables. Although all shift-vectors $r_{i}$ could be non-trivial $r_{i}\neq 0$ , two or three-line shifts are enough to construct amplitudes in many applications. Let’s start from these few-line shifts.

1) Two-line shift

Since the shifted momentum is linear in $z$ , we can’t use holomorphic and anti-holomorphic shifts simultaneously for one particle. Otherwise, the momentum conservation condition would be violated. When we shift two external lines $i$ and $j$ , there are two choices. We choose holomorphic shift for particle $i$ and anti-holomorphic shift for particle $j$ ,

|\hat{i}]=|i]+z|j],\quad|\hat{j}\rangle=|j\rangle-z|i\rangle.

(9)

We call this a $[i,j\rangle$ -shift. The shift-vector $r_{i}=-r_{j}=r$ , so momentum conservation condition (2) is automatically satisfied. The shift vector is

2r^{\mu}=\langle i|\gamma^{\mu}|j],

(10)

so conditions (3) and (4) lead to

2r^{2}=\langle ii\rangle[jj],\quad 2r\cdot p_{i}=\langle i|p_{i}|j],\quad 2r\cdot p_{j}=\langle i|p_{j}|j].

(11)

We find that $\langle ii\rangle=[jj]=0$ is responsible for the condition (3). Weyl equations $p_{j}|j]=p_{i}|i\rangle=0$ are responsible for condition (4).

2) Risager-type three-line shift

In Risager-type, all the shifted external lines are holomorphic shifts. The shifted spinors are

|\hat{i}]=|i]+z\langle jk\rangle|X],\quad|\hat{j}]=|j]+z\langle ki\rangle|X],\quad|\hat{k}]=|k]+z\langle ij\rangle|X],

(12)

where $|X]$ is an arbitrary reference spinor. Here we ignore the dimension analysis for convenience¹¹1Actually The mass dimension of $\langle jk\rangle$ , $\langle ki\rangle$ and $\langle ij\rangle$ is $1$ . In order to ensure the dimensionless $z$ , we should write the shifts as $|\hat{i}]=|i]+zc\langle jk\rangle|X]$ , $|\hat{j}]=|j]+zc\langle ki\rangle|X]$ and $|\hat{k}]=|k]+zc\langle ij\rangle|X]$ , where the mass dimension of constant $c$ is $-1$ . Since we only want to discuss the large- $z$ behavior, we ignore such constants, which will not change the result of our following analysis.. The shift vectors are

\displaystyle 2r_{i}^{\mu}=\langle jk\rangle\langle i|\gamma^{\mu}|X],\quad 2r_{j}^{\mu}=\langle ki\rangle\langle j|\gamma^{\mu}|X],\quad 2r_{k}^{\mu}=\langle ij\rangle\langle k|\gamma^{\mu}|X].

(13)

Using Schouten identity $\langle ki\rangle\langle j|+\langle ki\rangle\langle j|+\langle ij\rangle\langle k|=0$ , we easily verify condition (2) $r_{i}+r_{j}+r_{k}=0$ . We find that $[XX]=0$ ensures condition (3), and that Weyl equations $p_{i}|i\rangle=p_{j}|j\rangle=p_{k}|k\rangle=0$ are responsible for condition (4).

3) BCFW-type three-line shift

In BCFW-type, only one shifted external line is anti-holomorphic shift, the other shifted external lines are holomorphic shifts. The shifted spinors are,

|\hat{i}]=|i]+z\langle jX\rangle|k],\quad|\hat{j}]=|j]+z\langle Xi\rangle|k],\quad|\hat{k}\rangle=|k\rangle+z\langle ij\rangle|X\rangle,

(14)

where $|X\rangle$ is an arbitrary reference spinor. The shift vectors are

\displaystyle 2r_{i}^{\mu}=\langle jX\rangle\langle i|\gamma^{\mu}|k],\quad 2r_{j}^{\mu}=\langle Xi\rangle\langle j|\gamma^{\mu}|k],\quad 2r_{k}^{\mu}=\langle ij\rangle\langle X|\gamma^{\mu}|k].

(15)

We use Schouten identity to verify condition (2), $\sum_{i}r_{i}=0$ . We find that $[kk]=0$ ensures condition (3) and Weyl equations $p_{i}|i\rangle=p_{j}|j\rangle=p_{k}|k]=0$ are responsible for condition (4).

So far all possible two and three-line shifts in the massless case are presented. Since Risager and BCFW-type shifts are the only two distinct classes of recursion relations Cheung:2015cba, the analysis of four and more-line shifts are just generalizations of three-line shifts. They are sufficient for analyzing mixed or massive recursion relations. We find that the antisymmetry of brackets $\langle\cdots\rangle$ and $[\cdots]$ ensure condition (3), and the equation of motion for massless particles is responsible for condition (4).

2.2 Shifting mixed particles

Now let’s think about how to satisfy the three conditions when massive particles are taken into account. The massive spinor-helicity variables $|\mathbf{p}^{I}]_{\alpha}$ and $|\mathbf{p}^{I}\rangle^{\dot{\alpha}}$ are not only fundamental representations of Lorentz group $SO(3,1)$ , but also fundamental representations of little group $SU(2)_{LG}$ . We write them as $(\frac{1}{2};\frac{1}{2},0)$ and $(\frac{1}{2};0,\frac{1}{2})$ , which are the representations of $SU(2)_{LG}\times SO(3,1)$ . In the massive case, the antisymmetric brackets

[\mathbf{i}^{I}\mathbf{i}^{I^{\prime}}]=-\langle\mathbf{i}^{I}\mathbf{i}^{I^{\prime}}\rangle=m_{i}^{2}\epsilon^{II^{\prime}}\neq 0

(16)

and massive Dirac equation

p|\mathbf{p}^{I}]=-m|\mathbf{p}^{I}\rangle\neq 0

(17)

are different from the massless case. Therefore, conditions (3) and (4) wouldn’t be satisfied automatically in the massive case. For example, we can give up condition (4) for massive particles and construct a shift, where masses are no longer invariants. The validity of this kind of shifts was examined numerically for the amplitudes with graviton and scalar bosons Britto:2021pud. Here we try to satisfy all conditions.

The key point is to use some variables to contract the little-group index or Weyl-spinor index of massive spinors. For example, we can introduce an unknown variable $\eta^{I}$ , which is the representation $(\frac{1}{2};0,0)$ . We use $\eta^{I}$ to contract the little-group index of $|\mathbf{i}^{I}]$ , so the inner product $\eta_{I}|\mathbf{i}^{I}]$ gives the antisymmetric bracket

\eta_{I}[\mathbf{i}^{I}\mathbf{i}^{I^{\prime}}]\eta_{I^{\prime}}=m_{i}^{2}\eta_{I}\epsilon^{II^{\prime}}\eta_{I^{\prime}}=0,

(18)

which ensures condition (3). Another way is to introduce $|Y]$ and contract the Weyl-spinor index of $|\mathbf{i}^{I}]$ , which has been used in ref. Franken:2019wqr.

Our strategy is the former one. First, we take massless shifts in Section 2.1 and replace the massless variables $|p]_{\alpha}$ and $|p\rangle^{\dot{\alpha}}$ with the massive spinor-helicity variables $|\mathbf{p}^{I}]_{\alpha}$ and $|\mathbf{p}^{I}\rangle^{\dot{\alpha}}$ for massive particles, while the spinor part of the shifts remain as massless shifts. Then we introduce some unknown variables with little-group indices ( $\eta^{I}$ , $\zeta^{J}$ , $\xi^{L}$ , etc.), whose number equals the number of massive shifted external legs. With the shifts which have been replaced multiplied by or contracted with these unknown variables, condition (3) must be satisfied. Since the above manipulation is based on massless shifts, condition (2) is still satisfied. Finally, we use the last condition (4) to determine these unknown variables.

2.2.1 Two-line shifts for mixed particles

Let’s consider two-line shifts for massive particle $i$ and massless particle $j$ . There are two ways to shift them. We can choose the $[i,\mathbf{j}\rangle$ -shift: the massless line $i$ is shifted holomorphically, while the massive line $j$ is shifted anti-holomorphically. We introduce one unknown $\zeta^{J}$ , so the shifted spinors are

|\hat{i}]=|i]+z|\mathbf{j}^{J}]\zeta_{J},\quad|\mathbf{\hat{j}}^{J}\rangle=|\mathbf{j}^{J}\rangle-z|i\rangle\zeta^{J}.

(19)

The shift vector is $2r^{\mu}=\langle i|\gamma^{\mu}|\mathbf{j}^{J}]\zeta_{J}$ . It is orthogonal to the massless momentum $p_{i}$ because of the Weyl equation $\langle i|p_{i}=0$ . Condition (4) leads to

2p_{j}\cdot r=\langle i|p_{j}|\mathbf{j}^{J}]\zeta_{J}=m_{j}\langle i\mathbf{j}^{J}\rangle\zeta_{J}=0.

(20)

The solution is $\zeta^{J}=\langle i\mathbf{j}^{J}\rangle$ . We substitute the solution into eq. (19), so the explicit form of $[i,\mathbf{j}\rangle$ -shift is

|\hat{i}]=|i]+z|\mathbf{j}^{J}]\langle i\mathbf{j}_{J}\rangle,\quad|\mathbf{\hat{j}}^{J}\rangle=|\mathbf{j}^{J}\rangle-z|i\rangle\langle i\mathbf{j}^{J}\rangle.

(21)

Another choice is the $[\mathbf{j},i\rangle$ -shift: the massless line $i$ uses holomorphic shift, the massive line $j$ uses anti-holomorphic shift. For real-valued momenta, angle and square spinors are not independent. For massless particles, we have $(|i\rangle)^{*}=[i|$ and $(|i])^{*}=\langle i|$ . For the massive particle $j$ , the complex conjugation of massive spinors lowers the little indices: $(|\mathbf{j}^{J}\rangle)^{*}=[\mathbf{j}_{J}|$ and $(|\mathbf{j}^{J}])^{*}=\langle\mathbf{j}_{J}|$ . Therefore, the $[\mathbf{j},i\rangle$ -shift can be implemented from the complex conjugate of $[i,\mathbf{j}\rangle$ -shift,

|\mathbf{\hat{j}}^{J}]=|\mathbf{j}^{J}]+z|i][i\mathbf{j}^{J}],\quad|\hat{i}\rangle=|i\rangle-z|\mathbf{j}^{J}\rangle[i\mathbf{j}_{J}].

(22)

2.2.2 Three-line Risager type shifts for mixed Particles

We take massless shift (12) and choose one or two shifted particles to be massive. Since all shifts are holomorphic shifts in Risager type, each shift vector must be $r^{\mu}\propto\langle\cdots|\gamma^{\mu}|X]$ . Condition (3) is satisfied, because $r_{i}\cdot r_{j}\propto[XX]=0$ .

1) One massive and two massless

Let particle $i$ be massive particle and particles $j$ and $k$ be massless particles. We introduce one unknown $\eta^{I}$ , so the shifted spinors are

|\mathbf{\hat{i}}^{I}]=|\mathbf{i}^{I}]+z\eta^{I}\langle jk\rangle|X],\quad|\hat{j}]=|j]+z\eta^{I}\langle k\mathbf{i}_{I}\rangle|X],\quad|\hat{k}]=|k]+z\eta^{I}\langle\mathbf{i}_{I}j\rangle|X].

(23)

The shift vectors are

2r_{i}^{\mu}=\eta^{I}\langle jk\rangle\langle\mathbf{i}_{I}|\gamma^{\mu}|X],\quad 2r_{j}^{\mu}=\eta^{I}\langle k\mathbf{i}_{I}\rangle\langle j|\gamma^{\mu}|X],\quad 2r_{k}^{\mu}=\eta^{I}\langle\mathbf{i}_{I}j\rangle\langle k|\gamma^{\mu}|X],

(24)

so condition (4) leads to

2p_{i}\cdot r_{i}=\eta^{I}\langle jk\rangle\langle\mathbf{i}_{I}|p_{i}|X]=m_{i}\eta^{I}\langle jk\rangle[\mathbf{i}_{I}X]=0.

(25)

The solution is $\eta^{I}=[\mathbf{i}^{I}X]$ .

2) Two massive and one massless

Let particles $i$ and $j$ be massive particles and particle $k$ be massless particle. We introduce two unknowns $\eta^{I}$ and $\zeta^{J}$ , so the shifted spinors are

$\displaystyle\|\mathbf{\hat{i}}^{I}]$	$\displaystyle=\|\mathbf{i}^{I}]+z\eta^{I}\zeta^{J}\langle\mathbf{j}_{J}k\rangle\|X],$	(26)
$\displaystyle\|\mathbf{\hat{j}}^{J}]$	$\displaystyle=\|\mathbf{j}^{J}]+z\eta^{I}\zeta^{J}\langle k\mathbf{i}_{I}\rangle\|X],$
$\displaystyle\|\hat{k}]$	$\displaystyle=\|k]+z\eta^{I}\zeta^{J}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\|X].$

The shift vectors are

$\displaystyle 2r_{i}^{\mu}$	$\displaystyle=\eta^{I}\zeta^{J}\langle\mathbf{j}_{J}k\rangle\langle\mathbf{i}_{I}\|\gamma^{\mu}\|X],$	(27)
$\displaystyle 2r_{j}^{\mu}$	$\displaystyle=\eta^{I}\zeta^{J}\langle k\mathbf{i}_{I}\rangle\langle\mathbf{j}_{J}\|\gamma^{\mu}\|X],$
$\displaystyle 2r_{k}^{\mu}$	$\displaystyle=\eta^{I}\zeta^{J}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\langle k\|\gamma^{\mu}\|X],$

so condition (4) leads to

	$\displaystyle 2p_{i}\cdot r_{i}$	$\displaystyle=\eta^{I}\zeta^{J}\langle\mathbf{j}_{J}k\rangle\langle\mathbf{i}_{I}\|p_{i}\|X]=m_{i}\eta^{I}\zeta^{J}\langle\mathbf{j}_{J}k\rangle[\mathbf{i}_{I}X]=0,$		(28)
	$\displaystyle 2p_{j}\cdot r_{j}$	$\displaystyle=\eta^{I}\zeta^{J}\langle k\mathbf{i}_{I}\rangle\langle\mathbf{j}_{J}\|p_{j}\|X]=m_{j}\eta^{I}\zeta^{J}\langle k\mathbf{i}_{I}\rangle[\mathbf{j}_{J}X]=0.$		(28)

The shift vectors should be non-trivial, so $\zeta^{J}\langle\mathbf{j}_{J}k\rangle\neq 0$ , $\eta^{I}\zeta^{J}\langle k\mathbf{i}_{I}\rangle\neq 0$ . The only solutions are $\eta^{I}=[\mathbf{i}^{I}X]$ , $\zeta^{J}=[\mathbf{j}^{J}X]$ .

2.2.3 Three-line BCFW type shifts for mixed particles

We take massless shift (14) and choose one or two shifted particles to be massive. If particle $k$ is massive, each shift vector in this type of shifts must be $r^{\mu}\propto\langle\cdots|\gamma^{\mu}|\mathbf{k}^{K}]\xi_{K}$ . Condition (3) is satisfied, because $r_{i}\cdot r_{j}\propto[\mathbf{k}^{K}\mathbf{k}^{K^{\prime}}]\xi_{K}\xi_{K^{\prime}}=\epsilon^{KK^{\prime}}\xi_{K}\xi_{K^{\prime}}=0$ . If particle $k$ is massless, condition (3) is also satisfied for the same reason as the case of massless three-line BCFW type shifts.

1) One massive and two massless

Since BCFW type recursion relations use holomorphic and anti-holomorphic shift, there are two kinds of compositions. The first kind is that the massive particle uses holomorphic shift. Let particle $i$ be massive particle and particles $j$ and $k$ be massless particle. We introduce one unknown $\eta^{I}$ , so the shifted spinors are

|\mathbf{\hat{i}}^{I}]=|\mathbf{i}^{I}]+z\eta^{I}\langle jX\rangle|k],\quad|\hat{j}]=|j]+z\eta^{I}\langle X\mathbf{i}_{I}\rangle|k],\quad|\hat{k}\rangle=|k\rangle+z\eta^{I}\langle\mathbf{i}_{I}j\rangle|X\rangle.

(29)

The shift vectors are

2r_{i}^{\mu}=\eta^{I}\langle jX\rangle\langle\mathbf{i}_{I}|\gamma^{\mu}|k],\quad 2r_{j}^{\mu}=\eta^{I}\langle X\mathbf{i}_{I}\rangle\langle j|\gamma^{\mu}|k],\quad 2r_{k}^{\mu}=\eta^{I}\langle\mathbf{i}_{I}j\rangle\langle X|\gamma^{\mu}|k],

(30)

so condition (4) leads to

2p_{i}\cdot r_{i}=\eta^{I}\langle Xj\rangle\langle\mathbf{i}_{I}|p_{i}|k]=m_{i}\eta^{I}\langle Xj\rangle[\mathbf{i}_{I}k]=0.

(31)

The solution is $\eta^{I}=[\mathbf{i}^{I}k]$ .

The second kind is that the massive particle uses anti-holomorphic shift. Let particle $k$ be massive particle and particles $i$ and $j$ be massless particles. We introduce one unknown $\xi^{K}$ , so the shifted spinors are

|\hat{i}]=|i]+z\xi_{K}\langle jX\rangle|\mathbf{k}^{K}],\quad|\hat{j}]=|j]+z\xi_{K}\langle Xi\rangle|\mathbf{k}^{K}],\quad|\mathbf{\hat{k}}^{K}\rangle=|\mathbf{k}^{K}\rangle-z\xi^{K}\langle ij\rangle|X\rangle.

(32)

The shift vectors are

2r_{i}^{\mu}=\xi_{K}\langle jX\rangle\langle i|\gamma^{\mu}|\mathbf{k}^{K}],\quad 2r_{j}^{\mu}=\xi_{K}\langle Xi\rangle\langle j|\gamma^{\mu}|\mathbf{k}^{K}],\quad 2r_{k}^{\mu}=\xi_{K}\langle ij\rangle\langle X|\gamma^{\mu}|\mathbf{k}^{K}],

(33)

so condition (4) leads to

2p_{k}\cdot r_{k}=\xi_{K}\langle ij\rangle\langle X|p_{k}|\mathbf{k}^{K}]=-m_{k}\xi_{K}\langle ij\rangle\langle X\mathbf{k}^{K}\rangle=0.

(34)

The solution is $\xi^{K}=\langle X\mathbf{k}^{K}\rangle$ .

2) Two massive and one massless

Let particles $i$ and $j$ be massive particles and particle $k$ be massless particle. We introduce two unknowns $\eta^{I}$ and $\zeta^{J}$ , so the shifted spinors are

$\displaystyle\|\mathbf{\hat{i}}^{I}]$	$\displaystyle=\|\mathbf{i}^{I}]+z\eta^{I}\zeta^{J}\langle\mathbf{j}_{J}X\rangle\|k]$	(35)
$\displaystyle\|\mathbf{\hat{j}}^{J}]$	$\displaystyle=\|\mathbf{j}^{J}]+z\eta^{I}\zeta^{J}\langle X\mathbf{i}_{I}\rangle\|k]$
$\displaystyle\|\hat{k}\rangle$	$\displaystyle=\|k\rangle+z\eta^{I}\zeta^{J}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\|X\rangle.$

The shift vectors are

	$\displaystyle 2r_{i}^{\mu}=\eta^{I}\zeta^{J}\langle\mathbf{j}_{J}X\rangle\langle\mathbf{i}_{I}\|\gamma^{\mu}\|k],$		(36)
	$\displaystyle 2r_{j}^{\mu}=\eta^{I}\zeta^{J}\langle X\mathbf{i}_{I}\rangle\langle\mathbf{j}_{J}\|\gamma^{\mu}\|k],$
	$\displaystyle 2r_{k}^{\mu}=\eta^{I}\zeta^{J}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\langle X\|\gamma^{\mu}\|k],$

so condition (4) leads to

	$\displaystyle 2p_{i}\cdot r_{i}$	$\displaystyle=\eta^{I}\zeta^{J}\langle\mathbf{j}_{J}X\rangle\langle\mathbf{i}_{I}\|p_{i}\|k]=m_{i}\eta^{I}\zeta^{J}\langle\mathbf{j}_{J}X\rangle[\mathbf{i}_{I}k]=0,$		(37)
	$\displaystyle 2p_{j}\cdot r_{j}$	$\displaystyle=\eta^{I}\zeta^{J}\langle X\mathbf{i}_{I}\rangle\langle\mathbf{j}_{J}\|p_{j}\|k]=m_{j}\eta^{I}\zeta^{J}\langle X\mathbf{i}_{I}\rangle[\mathbf{j}_{J}k]=0.$		(37)

The solutions are $\eta^{I}=[\mathbf{i}^{I}k]$ , $\zeta^{J}=[\mathbf{j}^{J}k]$ .

Let particles $i$ and $k$ be massive particles and particle $j$ be massless particle. We introduce two unknowns $\eta^{I}$ and $\xi^{K}$ , so the shifted spinors are

$\displaystyle\|\mathbf{\hat{i}}^{I}]$	$\displaystyle=\|\mathbf{i}^{I}]+z\eta^{I}\xi^{K}\langle jX\rangle\|\mathbf{k}_{K}],$	(38)
$\displaystyle\|\hat{j}]$	$\displaystyle=\|j]+z\eta^{I}\xi^{K}\langle X\mathbf{i}_{I}\rangle\|\mathbf{k}_{K}],$
$\displaystyle\|\mathbf{\hat{k}}^{K}\rangle$	$\displaystyle=\|\mathbf{k}^{K}\rangle-z\eta^{I}\xi^{K}\langle\mathbf{i}_{I}j\rangle\|X\rangle.$

The shift vectors are

$\displaystyle 2r_{i}^{\mu}$	$\displaystyle=\eta^{I}\xi^{K}\langle jX\rangle\langle\mathbf{i}_{I}\|\gamma^{\mu}\|\mathbf{k}_{K}],$	(39)
$\displaystyle 2r_{j}^{\mu}$	$\displaystyle=\eta^{I}\xi^{K}\langle X\mathbf{i}_{I}\rangle\langle j\|\gamma^{\mu}\|\mathbf{k}_{K}],$
$\displaystyle 2r_{k}^{\mu}$	$\displaystyle=\eta^{I}\xi^{K}\langle\mathbf{i}_{I}j\rangle\langle X\|\gamma^{\mu}\|\mathbf{k}_{K}],$

so condition (4) leads to

	$\displaystyle 2p_{i}\cdot r_{i}$	$\displaystyle=\eta^{I}\xi^{K}\langle jX\rangle\langle\mathbf{i}_{I}\|p_{i}\|\mathbf{k}_{K}]=m_{i}\eta^{I}\xi^{K}\langle jX\rangle[\mathbf{i}_{I}\mathbf{k}_{K}]=0,$		(40)
	$\displaystyle 2p_{k}\cdot r_{k}$	$\displaystyle=\eta^{I}\xi^{K}\langle\mathbf{i}_{I}j\rangle\langle X\|p_{k}\|\mathbf{k}_{K}]=m_{j}\eta^{I}\xi^{K}\langle\mathbf{i}_{I}j\rangle\langle X\mathbf{k}_{K}\rangle=0.$		(40)

The solutions are $\eta^{I}=[\mathbf{i}^{I}|p_{k}|X\rangle$ , $\xi^{K}=[\mathbf{k}^{K}X]$ . The solutions become more complicated.

2.3 Shifting massive particles

The all-massive recursion relations have been worked out to study the constructibility of all-massive amplitudes in spontaneously broken gauge theories Franken:2019wqr. Now we want to reproduce these massive shifts with our method. Similarly, we introduce $n$ unknown variables for $n$ -line shifts and then solve eq. (4) for these unknown variables.

2.3.1 Two-line shift for massive particles

Unfortunately, we can’t construct consistent massive BCFW shift for two massive lines as simply as what we do in last subsection. We don’t have a natural choice of massless spinors to contract with Weyl-spinor index, since two external lines are both massive particles. It means that the general form of two-line shift doesn’t exist. We must choose a specific spinor or reference frame to write down a particular shift. For example, we can only shift one of the helicity states instead of both Herderschee:2019dmc; Franken:2019wqr in a special frame.

Now we still introduce two unknowns $\eta^{I}$ and $\zeta^{J}$ in $[\mathbf{i},\mathbf{j}\rangle$ -shift, so the shifted spinors are

|\mathbf{\hat{i}}^{I}]=|\mathbf{i}^{I}]+z\eta^{I}\zeta^{J}|\mathbf{j}_{J}],\quad|\mathbf{\hat{j}}^{J}\rangle=|\mathbf{j}^{J}\rangle+z\eta^{I}\zeta^{J}|\mathbf{i}_{I}\rangle.

(41)

The shift vectors are $2r_{i}^{\mu}=-2r_{j}^{\mu}=\eta^{I}\zeta^{J}\langle\mathbf{i}_{I}|\gamma^{\mu}|\mathbf{j}_{J}]$ , so condition (4) leads to

	$\displaystyle 2p_{i}\cdot r_{i}$	$\displaystyle=\eta^{I}\zeta^{J}\langle\mathbf{i}_{I}\|p_{i}\|\mathbf{j}_{J}]=m_{i}\eta^{I}\zeta^{J}[\mathbf{i}_{I}\mathbf{j}_{J}]=0,$		(42)
	$\displaystyle 2p_{j}\cdot r_{j}$	$\displaystyle=-\eta^{I}\zeta^{J}\langle\mathbf{i}_{I}\|p_{j}\|\mathbf{j}_{J}]=m_{j}\eta^{I}\zeta^{J}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle=0.$		(42)

The tensor $\eta^{I}\zeta^{J}$ has three degrees of freedom, since its determinant is zero. Two equations are not enough to determine all degrees of freedom, so we should choose $\eta^{I}$ and $\eta^{J}$ from other information. The massive momenta of two particles can be written as different linear combinations of two null vectors $l_{i}$ and $l_{j}$ :

	$\displaystyle p_{i}$	$\displaystyle=l_{i}+\alpha_{j}l_{j},$		(43)
	$\displaystyle p_{j}$	$\displaystyle=l_{j}+\alpha_{i}l_{i},$		(43)

where $\alpha_{i}$ and $\alpha_{j}$ are coefficient. We choose $\eta^{I}=[l_{j}\mathbf{i}^{I}]/[l_{j}l_{i}]$ , $\zeta^{J}=\langle\mathbf{j}^{J}l_{i}\rangle/\langle l_{j}l_{i}\rangle$ . We can verify condition 4,

	$\displaystyle\eta^{I}\zeta^{J}[\mathbf{i}_{I}\mathbf{j}_{J}]$	$\displaystyle=-\frac{[l_{j}\mathbf{i}^{I}][\mathbf{i}_{I}\|p_{j}\|l_{i}\rangle}{[l_{j}l_{i}]\langle l_{j}l_{i}\rangle}=-\frac{[l_{j}\mathbf{i}^{I}][\mathbf{i}_{I}l_{j}]}{[l_{j}l_{i}]}=0,$		(44)
	$\displaystyle\eta^{I}\zeta^{J}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle$	$\displaystyle=\frac{[l_{j}\|p_{i}\|\mathbf{j}_{J}\rangle\langle\mathbf{j}^{J}l_{i}\rangle}{[l_{j}l_{i}]\langle l_{j}l_{i}\rangle}=\frac{\langle l_{i}\mathbf{j}_{J}\rangle\langle\mathbf{j}^{J}l_{i}\rangle}{\langle l_{j}l_{i}\rangle}=0.$		(44)

Now the shift can be written as

\displaystyle|\mathbf{\hat{i}}^{I}]=|\mathbf{i}^{I}]-z\frac{[l_{j}\mathbf{i}^{I}]}{[l_{j}l_{i}]}|l_{j}],\quad|\mathbf{\hat{j}}^{J}\rangle=|\mathbf{j}^{J}\rangle+z\frac{\langle\mathbf{j}^{J}l_{i}\rangle}{\langle l_{j}l_{i}\rangle}|l_{i}\rangle.

(45)

2.3.2 Three-line Risage-type shifts for massive particles

We introduce three unknowns $\eta^{I}$ , $\zeta^{J}$ and $\xi^{K}$ . Since they are all massive, there is a permutation symmetry between these shifted lines. The shifted spinors are

$\displaystyle\|\mathbf{\hat{i}}^{I}]$	$\displaystyle=\|\mathbf{i}^{I}]+z\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{j}_{J}\mathbf{k}_{K}\rangle\|X],$	(46)
$\displaystyle\|\mathbf{\hat{j}}^{J}]$	$\displaystyle=\|\mathbf{j}^{J}]+z\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{k}_{K}\mathbf{i}_{I}\rangle\|X],$
$\displaystyle\|\mathbf{\hat{k}}^{K}]$	$\displaystyle=\|\mathbf{k}^{K}]+z\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\|X].$

The shift vectors are

$\displaystyle 2r_{i}^{\mu}$	$\displaystyle=\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{j}_{J}\mathbf{k}_{K}\rangle\langle\mathbf{i}_{I}\|\gamma^{\mu}\|X],$	(47)
$\displaystyle 2r_{j}^{\mu}$	$\displaystyle=\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{k}_{K}\mathbf{i}_{I}\rangle\langle\mathbf{j}_{J}\|\gamma^{\mu}\|X],$
$\displaystyle 2r_{k}^{\mu}$	$\displaystyle=\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\langle\mathbf{k}_{K}\|\gamma^{\mu}\|X],$

so condition (4) leads to

$\displaystyle 2p_{i}\cdot r_{i}$	$\displaystyle=\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{j}_{J}\mathbf{k}_{K}\rangle\langle\mathbf{i}_{I}\|p_{i}\|X]=m_{i}\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{j}_{J}\mathbf{k}_{K}\rangle[\mathbf{i}_{I}X]=0,$	(48)
$\displaystyle 2p_{j}\cdot r_{j}$	$\displaystyle=\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{k}_{K}\mathbf{i}_{I}\rangle\langle\mathbf{j}_{J}\|p_{j}\|X]=m_{j}\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{k}_{K}\mathbf{i}_{I}\rangle[\mathbf{j}_{J}X]=0,$
$\displaystyle 2p_{k}\cdot r_{k}$	$\displaystyle=\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\langle\mathbf{k}_{K}\|p_{k}\|X]=m_{k}\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle[\mathbf{k}_{K}X]=0.$

The solutions are simple, $\eta^{I}=[\mathbf{i}^{I}X]$ , $\zeta^{J}=[\mathbf{j}^{J}X]$ , $\xi^{K}=[\mathbf{k}^{K}X]$ .

2.3.3 Three-line BCFW-type shifts for massive particles

We introduce three unknowns $\eta^{I}$ , $\zeta^{J}$ and $\xi^{K}$ , so the shifted spinors are

$\displaystyle\|\mathbf{\hat{i}}^{I}]$	$\displaystyle=\|\mathbf{i}^{I}]+z\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{j}_{J}X\rangle\|\mathbf{k}_{K}],$	(49)
$\displaystyle\|\mathbf{\hat{j}}^{J}]$	$\displaystyle=\|\mathbf{j}^{J}]+z\eta^{I}\zeta^{J}\xi^{K}\langle X\mathbf{i}_{I}\rangle\|\mathbf{k}_{K}],$
$\displaystyle\|\mathbf{\hat{k}}^{K}\rangle$	$\displaystyle=\|\mathbf{k}^{K}\rangle-z\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\|X\rangle.$

The shift vectors are

$\displaystyle 2r_{i}^{\mu}$	$\displaystyle=\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{j}_{J}X\rangle\langle\mathbf{i}_{I}\|\gamma^{\mu}\|\mathbf{k}_{K}],$	(50)
$\displaystyle 2r_{j}^{\mu}$	$\displaystyle=\eta^{I}\zeta^{J}\xi^{K}\langle X\mathbf{i}_{I}\rangle\langle\mathbf{j}_{J}\|\gamma^{\mu}\|\mathbf{k}_{K}],$
$\displaystyle 2r_{k}^{\mu}$	$\displaystyle=\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\langle X\|\gamma^{\mu}\|\mathbf{k}_{K}],$

so condition (4) leads to

$\displaystyle 2p_{i}\cdot r_{i}$	$\displaystyle=\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{j}_{J}X\rangle\langle\mathbf{i}_{I}\|p_{i}\|\mathbf{k}_{K}]=m_{i}\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{j}_{J}X\rangle[\mathbf{i}_{I}\mathbf{k}_{K}]=0,$	(51)
$\displaystyle 2p_{j}\cdot r_{j}$	$\displaystyle=\eta^{I}\zeta^{J}\xi^{K}\langle X\mathbf{i}_{I}\rangle\langle\mathbf{j}_{J}\|p_{j}\|\mathbf{k}_{K}]=m_{j}\eta^{I}\zeta^{J}\xi^{K}\langle X\mathbf{i}_{I}\rangle[\mathbf{j}_{J}\mathbf{k}_{K}]=0,$
$\displaystyle 2p_{k}\cdot r_{k}$	$\displaystyle=\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\langle X\|p_{k}\|\mathbf{k}_{K}]=-m_{k}\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\langle X\mathbf{k}_{K}\rangle=0.$

We can choose $\eta^{I}=[\mathbf{i}^{I}\mathbf{k}^{K}]\xi_{K},\zeta^{J}=[\mathbf{j}^{J}\mathbf{k}^{K}]\xi_{K},\xi^{K}=\langle X\mathbf{k}^{K}\rangle$ . However, it isn’t the final solution. After substituting $\xi^{K}=\langle X\mathbf{k}^{K}\rangle$ , we get the final result: $\eta^{I}=[\mathbf{i}^{I}|p_{k}|X\rangle,\zeta^{J}=[\mathbf{j}^{J}|p_{k}|X\rangle$ .

2.4 Explicit form of three-line shifts

In previous discussions, we figured out solutions for all two and three-line shifts. To simplify the expressions of these shifts, we define some new Weyl spinors,

$\displaystyle\|k_{m}\rangle$	$\displaystyle=p_{m}\|k],$	(52)
$\displaystyle\|X_{m}\rangle$	$\displaystyle=p_{m}\|X],$
$\displaystyle\|X_{m,n}\rangle$	$\displaystyle=p_{m}\|X_{n}]=p_{m}p_{n}\|X\rangle,$

where $p_{m}$ and $p_{n}$ correspond to massive particles. The simplified expressions of three-line shifts are shown in table 1, in which the massive spinor helicity variables are denoted in BOLD notation. Now we can write down massless three-line shifts (12) and (14) and use replacements listed in table 1 to rederive three-line shifts for all masses.

External Legs Type Shifted Spinors Shift-Vectors Replacement 1 massive 2 massless Risager $|\mathbf{\hat{i}}]=|\mathbf{i}]+z\langle jk\rangle[\mathbf{i}X]|X]$ $|\hat{j}]=|j]+z\langle kX_{i}\rangle|X]$ $|\hat{k}]=|k]+z\langle X_{i}j\rangle|X]$ $r_{i}^{\mu}=\langle jk\rangle\langle X_{i}|\gamma^{\mu}|X]$ $r_{j}^{\mu}=\langle kX_{i}\rangle\langle j|\gamma^{\mu}|X]$ $r_{k}^{\mu}=\langle X_{i}j\rangle\langle k|\gamma^{\mu}|X]$ $|i\rangle\rightarrow|X_{i}\rangle$ BCFW $|\mathbf{\hat{i}}]=|\mathbf{i}]+z\langle Xj\rangle[\mathbf{i}k]|k]$ $|\hat{j}]=|j]+z\langle k_{i}X\rangle|k]$ $|\hat{k}\rangle=|k\rangle+z\langle jk_{i}\rangle|X\rangle$ $r_{i}^{\mu}=\langle Xj\rangle\langle k_{i}|\gamma^{\mu}|k]$ $r_{j}^{\mu}=\langle k_{i}X\rangle\langle j|\gamma^{\mu}|k]$ $r_{k}^{\mu}=\langle jk_{i}\rangle\langle X|\gamma^{\mu}|k]$ $|i\rangle\rightarrow|k_{i}\rangle$ $|\hat{i}]=|i]+z\langle Xj\rangle|X_{k}]$ $|\hat{j}]=|j]+z\langle iX\rangle|X_{k}]$ $|\mathbf{\hat{k}}\rangle=|\mathbf{k}\rangle+z\langle ji\rangle\langle\mathbf{k}X\rangle|X\rangle$ $r_{i}^{\mu}=\langle Xj\rangle\langle i|\gamma^{\mu}|X_{k}]$ $r_{j}^{\mu}=\langle iX\rangle\langle j|\gamma^{\mu}|X_{k}]$ $r_{k}^{\mu}=\langle ji\rangle\langle X|\gamma^{\mu}|X_{k}]$ $|k]\rightarrow|X_{k}]$ 2 massive 1 massless Risager $|\mathbf{\hat{i}}]=|\mathbf{i}]+z\langle X_{j}k\rangle[\mathbf{i}X]|X]$ $|\mathbf{\hat{j}}]=|\mathbf{j}]+z\langle kX_{i}\rangle[\mathbf{j}X]|X]$ $|\hat{k}]=|k]+z\langle X_{i}X_{j}\rangle|X]$ $r_{i}^{\mu}=\langle X_{j}k\rangle\langle X_{i}|\gamma^{\mu}|X]$ $r_{j}^{\mu}=\langle kX_{i}\rangle\langle X_{j}|\gamma^{\mu}|X]$ $r_{k}^{\mu}=\langle X_{i}X_{j}\rangle\langle k|\gamma^{\mu}|X]$ $|i\rangle\rightarrow|X_{i}\rangle$ $|j\rangle\rightarrow|X_{j}\rangle$ BCFW $|\mathbf{\hat{i}}]=|\mathbf{i}]+z\langle Xk_{j}\rangle[\mathbf{i}k]|k]$ $|\mathbf{\hat{j}}]=|\mathbf{j}]+z\langle k_{i}X\rangle[\mathbf{j}k]|k]$ $|\hat{k}\rangle=|k\rangle+z\langle k_{j}k_{i}\rangle|X\rangle$ $r_{i}^{\mu}=\langle Xk_{j}\rangle\langle k_{i}|\gamma^{\mu}|k]$ $r_{j}^{\mu}=\langle k_{i}X\rangle\langle k_{j}|\gamma^{\mu}|k]$ $r_{k}^{\mu}=\langle k_{j}k_{i}\rangle\langle X|\gamma^{\mu}|k]$ $|i\rangle\rightarrow|k_{i}\rangle$ $|j\rangle\rightarrow|k_{j}\rangle$ $|\mathbf{\hat{i}}]=|\mathbf{i}]+z\langle Xj\rangle[\mathbf{i}X_{k}]|X_{k}]$ $|\hat{j}]=|j]+z\langle X_{k,i}X\rangle|X_{k}]$ $|\mathbf{\hat{k}}\rangle=|\mathbf{k}\rangle+z\langle jX_{k,i}\rangle\langle\mathbf{k}X\rangle|X\rangle$ $r_{i}^{\mu}=\langle Xj\rangle\langle X_{k,i}|\gamma^{\mu}|X_{k}]$ $r_{j}^{\mu}=\langle X_{k,i}X\rangle\langle j|\gamma^{\mu}|X_{k}]$ $r_{k}^{\mu}=\langle jX_{k,i}\rangle\langle X|\gamma^{\mu}|X_{k}]$ $|i\rangle\rightarrow|X_{k,i}\rangle$ $|k]\rightarrow|X_{k}]$ 3 massive Risager $|\mathbf{\hat{i}}]=|\mathbf{i}]+z\langle X_{j}X_{k}\rangle[\mathbf{i}X]|X]$ $|\mathbf{\hat{j}}]=|\mathbf{j}]+z\langle X_{k}X_{i}\rangle[\mathbf{j}X]|X]$ $|\mathbf{\hat{k}}]=|\mathbf{k}]+z\langle X_{i}X_{j}\rangle[\mathbf{k}X]|X]$ $r_{i}^{\mu}=\langle X_{j}X_{k}\rangle\langle X_{i}|\gamma^{\mu}|X]$ $r_{j}^{\mu}=\langle X_{k}X_{i}\rangle\langle X_{j}|\gamma^{\mu}|X]$ $r_{k}^{\mu}=\langle X_{i}X_{j}\rangle\langle X_{k}|\gamma^{\mu}|X]$ $|i\rangle\rightarrow|X_{i}\rangle$ $|j\rangle\rightarrow|X_{j}\rangle$ $|k\rangle\rightarrow|X_{k}\rangle$ BCFW $|\mathbf{\hat{i}}]=|\mathbf{i}]+z\langle XX_{k,j}\rangle[\mathbf{i}X_{k}]|X_{k}]$ $|\mathbf{\hat{j}}]=|\mathbf{j}]+z\langle X_{k,i}X\rangle[\mathbf{j}X_{k}]|X_{k}]$ $|\mathbf{\hat{k}}\rangle=|\mathbf{k}\rangle+z\langle X_{k,j}X_{k,i}\rangle\langle\mathbf{k}X\rangle|X\rangle$ $r_{i}^{\mu}=\langle XX_{k,j}\rangle\langle X_{k,i}|\gamma^{\mu}|X_{k}]$ $r_{j}^{\mu}=\langle X_{k,i}X\rangle\langle X_{k,j}|\gamma^{\mu}|X_{k}]$ $r_{k}^{\mu}=\langle X_{k,j}X_{k,i}\rangle\langle X|\gamma^{\mu}|X_{k}]$ $|i\rangle\rightarrow|X_{k,i}\rangle$ $|j\rangle\rightarrow|X_{k,j}\rangle$ $|k]\rightarrow|X_{k}]$

Table 1: Three-line shifts for all masses, where the little-group indices are suppressed.

Furthermore, the notation $|X_{m,n}\rangle$ is not necessary in the expressions. We can rewrite $|X_{m,n}\rangle$ and $|X\rangle$ in terms of $|Y]=|X_{n}]$ ,

	$\displaystyle\|X_{m,n}\rangle$	$\displaystyle=p_{m}\|X_{n}]=p_{m}\|Y\rangle=\|Y_{m}\rangle,$		(53)
	$\displaystyle\|X\rangle$	$\displaystyle=\frac{p_{n}p_{n}\|X\rangle}{m^{2}_{n}}=\frac{p_{n}\|Y]}{m^{2}_{n}}=\frac{\|Y_{n}\rangle}{m^{2}_{n}}.$		(53)

For example, we set $|Y]=|X_{k}]$ . The all-massive BCFW-type three-line shifts reduce to

$\displaystyle\|\mathbf{\hat{i}}]$	$\displaystyle=\|\mathbf{i}]+z\langle Y_{k}Y_{j}\rangle[\mathbf{i}Y]\|Y],$	(54)
$\displaystyle\|\mathbf{\hat{j}}]$	$\displaystyle=\|\mathbf{j}]+z\langle Y_{i}Y_{k}\rangle[\mathbf{j}Y]\|Y],$
$\displaystyle\|\mathbf{\hat{k}}\rangle$	$\displaystyle=\|\mathbf{k}\rangle+z\langle Y_{j}Y_{i}\rangle\langle\mathbf{k}X\rangle\|Y_{k}\rangle.$

This expression coincides with Franken:2019wqr.

3 Feynman rules in the large- $z$ limit

There is no general constructive method to give an expression of the contribution $B_{n}$ in eq. (7), so the recursion relations hold as long as $A(z\rightarrow\infty)=0$ . To investigate the validity of recursion relations, there are many works on the large- $z$ behavior of tree-level massless amplitudes in various shifts ArkaniHamed:2008yf; Cheung:2008dn; Cohen:2010mi; Cheung:2015cba. In refs. ArkaniHamed:2008yf; Cheung:2008dn the authors focused on the BCFW recursion relations and used background field method to show that, in a theory of spin $\leq 1$ , any massless amplitudes with at least one gluon is constructible. We want to examine whether this argument is applicable to the massive case, so the steps in their proof should be carefully reconsidered.

We label two external particles of massless amplitudes $A_{n}$ by 1 and 2. Their momenta are chosen to be deformed,

\hat{p}_{1}=p_{1}+zr,\quad\hat{p}_{2}=p_{2}-zr,

(55)

which corresponds to eq. (9). In the background field method, the large- $z$ behavior of amplitudes $A_{n}(z)$ have a nice physical interpretation. We take particles 1 and 2 to be incoming and outgoing, so this process can be interpreted as a hard particle shooting through a soft background. In the hard limit $z\rightarrow\infty$ , the $z$ -independent soft physics is treated as a classical background, while the large- $z$ behavior of amplitudes is completely determined by the hard fluctuations.

Now, let’s discuss the $z$ -dependent propagators, vertices and external legs separately. We will see differences in the case when hard fluctuations correspond to massive particles.

3.1 Hard propagators

The first problem is how massive propagators scale at large- $z$ . Both massive fermions and scalar propagators scale as the same as massless propagators, while a massive vector propagator goes as $\mathcal{O}(z)$ :

\Pi^{\mu\nu}=\frac{g^{\mu\nu}-\frac{\hat{p}^{\mu}\hat{p}^{\nu}}{m^{2}}}{\hat{p}^{2}-m^{2}}=\frac{g^{\mu\nu}-\frac{(p^{\mu}+zr^{\mu})(p^{\nu}+zr^{\nu})}{m^{2}}}{p^{2}+2zp\cdot r-m^{2}}\overset{z\rightarrow\infty}{=}-z\frac{r^{\mu}r^{\nu}}{2m^{2}p\cdot r}.

(56)

Notice that a massless vector propagator goes as $\mathcal{O}(1/z)$ , so we should distinguish massless and massive vectors in the following discussion. As in table 2, we use single and double wavy lines to make a distinction between massless and massive vectors propagators.

particle	massless	massive
scalar	{fmfgraph*}(50,10) \fmflefti1 \fmfrighto1 \fmfdashesi1,o1
fermion	{fmfgraph*}(50,10) \fmflefti1 \fmfrighto1 \fmfplaini1,o1
vector boson	{fmfgraph*}(50,10) \fmflefti1 \fmfrighto1 \fmfwigglyi1,o1	{fmfgraph*}(50,10) \fmflefti1 \fmfrighto1 \fmfdbl_wigglyi1,o1

Table 2: Propagators

Here we introduce a diagrammatic expression to represent the numerator in propagator (56):

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(57)

where the line segment and the letters attached with it, which are either indices or momentum $\hat{p}$ , compose a representation of Lorentz group. The first diagram represents a symmetric tensor $g^{\mu\nu}$ , in which the line segment connects two Lorentz indices. The second diagram represents two Lorentz vectors, where the line segments connect the shifted momentum $\hat{p}$ to Lorentz indices.

Furthermore, Einstein summation also has a diagrammatic representation,

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}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{23.9166pt}{0.0pt}\pgfsys@curveto{23.9166pt}{2.20903pt}{22.12587pt}{3.99976pt}{19.91684pt}{3.99976pt}\pgfsys@curveto{17.70781pt}{3.99976pt}{15.91708pt}{2.20903pt}{15.91708pt}{0.0pt}\pgfsys@curveto{15.91708pt}{-2.20903pt}{17.70781pt}{-3.99976pt}{19.91684pt}{-3.99976pt}\pgfsys@curveto{22.12587pt}{-3.99976pt}{23.9166pt}{-2.20903pt}{23.9166pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.40121pt}{-1.18056pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=p^{2}=m^{2},

(58)

where two same indices are equivalent to a line segment. Therefore, the first diagram reduces to a line segment with two momenta $p$ attached to it, which represents Lorentz scalar $p^{2}$ .

3.2 External polarizations

External spinors are constructed in the little-group notation,

\bar{u}^{I}(p)=\begin{pmatrix}[\mathbf{p}^{I}|&\langle\mathbf{p}^{I}|\end{pmatrix},\quad v^{I}(p)=\begin{pmatrix}|\mathbf{p}^{I}]\\ |\mathbf{p}^{I}\rangle\end{pmatrix},

(59)

where the little-group indices $I=1,2$ characterize two different solutions of the Dirac equation.

For massive vector bosons, the polarization vectors transform under the three-dimensional tensor representation of the little group,

\epsilon_{\mu}^{I_{1}I_{2}}(p)=-\frac{1}{\sqrt{2}m}\left[\mathbf{p}^{I_{1}}|\gamma_{\mu}|\mathbf{p}^{I_{2}}\right\rangle.

(60)

However, this little-group covariant expression is not convenient when we are talking about amplitudes in the large- $z$ limit. Usually we don’t choose the rest frame of massive particles as a reference system, since any shift should include more than one external leg. It implies that we can choose the spin axis along the 3-momentum direction and write down the spin state for massive vector particles in terms of $\epsilon^{I_{1}I_{2}}$ :

\epsilon_{i}^{+}=\epsilon^{11}(p_{i}),\quad\epsilon_{i}^{0}=\frac{1}{2}\left(\epsilon^{12}(p_{i})+\epsilon^{21}(p_{i})\right),\quad\epsilon_{i}^{-}=-\epsilon^{22}(p_{i}),

(61)

where we ignore Lorentz indices.

3.2.1 $[1,\mathbf{2}\rangle$ -shift

To consider two-line shifts, we take particles 1 and 2 to be massless and massive respectively. Particle 1 is always a gauge boson, while particle 2 can be any massive particle whose spin $\leq 1$ . Since the large- $z$ behavior is independent of the reference system, we can choose the center-of-mass frame of the particles for simplicity. The momenta of particle 1 and 2 and shift vectors become

\displaystyle p_{1}^{\mu}=(1,0,0,1),\quad p_{2}^{\mu}=(\sqrt{m^{2}+1},0,0,-1),\quad r^{\mu}=(0,-1,-i,0),

(62)

where the 3-momenta are normalized. The shift vector $r$ is basically the same as the polarization vectors for the real momentum $p_{1}$ .

In pure Yang-Mills theory, the shift vector $r$ is enough to construct the polarization vectors. We add a new vector $\bar{p}_{2}$ to give the longitudinal polarization, whose spatial components point opposite to the direction of spatial components of $p_{2}$ . Similarly, we construct a new null vector $\bar{p}_{1}$ . The expressions of $\bar{p}_{1}$ and $\bar{p}_{2}$ are,

\bar{p}_{1}^{\mu}=(1,0,0,-1),\quad\bar{p}_{2}^{\mu}=(1,0,0,-\sqrt{m^{2}+1}).

(63)

We choose $[1,\mathbf{2}\rangle$ -shift (21). The shifted polarization vectors of particle 1 are

\displaystyle\hat{\epsilon}_{1}^{+}=r^{*}+z\bar{p}_{1},\quad\hat{\epsilon}_{1}^{-}=r,

(64)

which are the same as in the all-massless case. They should stay orthogonal to momentum $\hat{p}_{1}$ and their product $(\epsilon_{1}^{+}\epsilon_{1}^{-})$ is maintained. The Ward identity is still valid for complexified amplitudes:

\hat{p}_{1}^{\mu}\hat{A}_{\mu}(z)=(p_{1}^{\mu}+zr^{\mu})\hat{A}_{\mu}(z)=0.

(65)

Therefore, we can use it to replace the negative polarization,

\hat{\epsilon}_{1}^{-}\rightarrow-\frac{1}{z}p_{1}.

(66)

If particle 2 is a vector boson, we choose the $z$ axis as the spin direction, and the shifted external polarizations are

\hat{\epsilon}_{2}^{+}=r,\quad\hat{\epsilon}_{2}^{-}=r^{*}-zCp_{1},\quad\hat{\epsilon}_{2}^{0}=\frac{\bar{p}_{2}}{m}-z\frac{r}{m},\\

(67)

where $C=2/(\sqrt{m^{2}+1}+1)$ . Since we focus on the large- $z$ behavior of amplitudes, the overall factor $1/m$ in the expression of $\hat{\epsilon}_{2}^{0}$ can be ignored. The polarizations are modified appropriately to remain normalized to unity and orthogonal to $\hat{p}_{2}$ . What’s more, the longitudinal polarization should be orthogonal to other two transverse polarizations. A more detailed discussion is performed in appendix LABEL:AppB. Using Goldstone boson equivalence theorem, the scaling behaviour of massive vector bosons can also be improved Franken:2019wqr. However, this improvement changes type of particles, so we don’t apply it in the following analysis.

If particle 2 is a fermion, we need the shifted Dirac spinors,

\hat{\bar{u}}^{J}_{2}=\bar{u}^{J}_{2}-z\bar{u}^{-}_{1}\langle 1\mathbf{2}^{J}\rangle,\quad\hat{v}^{J}_{2}=v^{J}_{2}-zv^{-}_{1}\langle 1\mathbf{2}^{J}\rangle.\\

(68)

Although particle 1 is a gauge boson, we still use $\bar{u}^{-}_{1}=\langle 1|$ and $v^{-}_{1}=|1\rangle$ for consistency.

3.2.2 $[\mathbf{1},\mathbf{2}\rangle$ -shift

Particles 1 and 2 have equal mass. The momenta of them become

\displaystyle p_{1}^{\mu}=(\sqrt{m^{2}+1},0,0,1),\quad p_{2}^{\mu}=(\sqrt{m^{2}+1},0,0,-1).

(69)

We set the shift vector $r$ to be the same as in eq. (62). Now both $p_{1}$ and $p_{2}$ are time-like vectors, we need two vectors $\bar{p}_{1}$ and $\bar{p}_{2}$ to give the longitudinal polarizations,

\displaystyle\bar{p}_{1}^{\mu}=(1,0,0,\sqrt{m^{2}+1}),\quad\bar{p}_{2}^{\mu}=(1,0,0,-\sqrt{m^{2}+1}).

(70)

Here we only discuss the case that both particles 1 and 2 are massive vector bosons. The shifted external legs of massive vectors are

	$\displaystyle\hat{\epsilon}_{1}^{+}=r^{*}+zC^{2}\frac{p_{2}+\bar{p}_{2}}{2},\quad\hat{\epsilon}_{1}^{-}=r,\quad\hat{\epsilon}_{1}^{0}=\frac{\bar{p}_{1}}{m}+z\frac{r}{m},$		(71)
	$\displaystyle\hat{\epsilon}_{2}^{+}=r,\quad\hat{\epsilon}_{2}^{-}=r^{*}-zC^{2}\frac{p_{1}+\bar{p}_{1}}{2},\quad\hat{\epsilon}_{2}^{0}=\frac{\bar{p}_{2}}{m}-z\frac{r}{m},$		(71)

where $C=2/(\sqrt{m^{2}+1}+1)$ . In the high energy limit, $C\rightarrow 1$ , $\bar{p}_{1}\rightarrow p_{1}$ and $\bar{p}_{2}\rightarrow p_{2}$ . Therefore, eqs. (64) and (67) become the high energy limit of eq. (71).

3.3 $\mathcal{O}(z)$ vertices

Along the hard fluctuation, the spin of the hard particle may be changed by $z$ -dependent vertices which involve soft background fields. Since we are discussing renormalizable field theory, the $z$ -dependent vertices from derivative interactions must be linear in $z$ . In massless gauge theory, they are eliminated by choosing appropriate light-cone and $R_{\xi}$ gauges ArkaniHamed:2008yf; Elvang:2013cua. However, the massive vector bosons don’t have such degrees of freedom to eliminate these $z$ -dependence. In renormalizable field theory, there are two classes of $\mathcal{O}(z)$ vertices that cannot be eliminated by gauge fixing in the massive case: triple vector coupling ( $VVV$ ) and Vector-Vector-Scalar ( $VSS$ ) interaction. For simplicity, we only consider massive vector bosons that have equal mass.

Since all on-shell 3-point amplitudes in the Standard Model have been figured outChristensen:2018zcq, we can translate them into the Feynman rules to find out the vertices. Then we deform these vertices to give their $z$ -dependence explicitly.

{fmfgraph*}

(60,60) \fmflefti1,i2 \fmfrighto1 \fmfdbl_wigglyi1,v1 \fmfphotoni2,v1 \fmfdbl_wigglyv1,o1

(a)

{fmfgraph*}

(60,60) \fmflefti1,i2 \fmfrighto1 \fmfdbl_wigglyi1,v1 \fmfdbl_wigglyi2,v1 \fmfdbl_wigglyv1,o1

(b)

Figure 1:

VVV

vertices with massive vectors

There are two kinds of possible $VVV$ amplitudes (see figure 1). Since $VVV$ amplitude has three vector external legs, the vertex should be a order-3 Lorentz tensor $V_{k,p,q}^{\mu\nu\lambda}$ , which refers to $V$ attached with three line segments in the diagrammatic expression. One kind of vertex includes two massive vectors and one massless vector. Its diagrammatic representations are

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.4cm}}=\frac{\sqrt{2}}{m}x\langle\mathbf{i}\mathbf{j}\rangle^{2},\qquad\parbox[b]{79.6678pt}{ \leavevmode\hbox to64.11pt{\vbox to31.54pt{\pgfpicture\makeatletter\hbox{\hskip 5.41147pt\lower-6.06064pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}} {}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{25.6073pt}{0.0pt}\pgfsys@lineto{52.63777pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ 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}{}\pgfsys@moveto{30.8949pt}{0.0pt}\pgfsys@curveto{30.8949pt}{2.92029pt}{28.52759pt}{5.2876pt}{25.6073pt}{5.2876pt}\pgfsys@curveto{22.68701pt}{5.2876pt}{20.3197pt}{2.92029pt}{20.3197pt}{0.0pt}\pgfsys@curveto{20.3197pt}{-2.92029pt}{22.68701pt}{-5.2876pt}{25.6073pt}{-5.2876pt}\pgfsys@curveto{28.52759pt}{-5.2876pt}{30.8949pt}{-2.92029pt}{30.8949pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.57953pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$V$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ 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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{49.29492pt}{-1.64166pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\epsilon_{j}^{J_{1}J_{1}}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@lineto{25.6073pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ 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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{30.7655pt}{19.91684pt}\pgfsys@curveto{30.7655pt}{22.76567pt}{28.45613pt}{25.07504pt}{25.6073pt}{25.07504pt}\pgfsys@curveto{22.75847pt}{25.07504pt}{20.4491pt}{22.76567pt}{20.4491pt}{19.91684pt}\pgfsys@curveto{20.4491pt}{17.06801pt}{22.75847pt}{14.75864pt}{25.6073pt}{14.75864pt}\pgfsys@curveto{28.45613pt}{14.75864pt}{30.7655pt}{17.06801pt}{30.7655pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{25.6073pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.03195pt}{18.13351pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\epsilon_{k}^{-}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.4cm}}=\frac{\sqrt{2}}{m}\frac{1}{x}[\mathbf{i}\mathbf{j}]^{2},

(72)

where the $x$ factor is introduced by Arkani-Hamed:2017jhn, which carries $+1$ helicity. The vertex $\gamma W^{+}W^{-}$ in the Standard Model belongs to this kind of vertex.

Another kind of amplitude includes three massive vectors. Its diagrammatic representation is

		$\displaystyle=\frac{[\mathbf{i}\mathbf{j}]\langle\mathbf{j}\mathbf{k}\rangle[\mathbf{k}\mathbf{i}]+\langle\mathbf{i}\mathbf{j}\rangle[\mathbf{j}\mathbf{k}]\langle\mathbf{k}\mathbf{i}\rangle}{\sqrt{2}m^{2}}+\frac{[\mathbf{i}\mathbf{j}][\mathbf{j}\mathbf{k}]\langle\mathbf{k}\mathbf{i}\rangle+\langle\mathbf{i}\mathbf{j}\rangle\langle\mathbf{j}\mathbf{k}\rangle[\mathbf{k}\mathbf{i}]}{\sqrt{2}m^{2}}$		(73)
		$\displaystyle\;\quad+\frac{\langle\mathbf{i}\mathbf{j}\rangle[\mathbf{j}\mathbf{k}][\mathbf{k}\mathbf{i}]+[\mathbf{i}\mathbf{j}]\langle\mathbf{j}\mathbf{k}\rangle\langle\mathbf{k}\mathbf{i}\rangle}{\sqrt{2}m^{2}}.$		(73)

There is no such vertex in the Standard Model, because $W$ and $Z$ bosons have different masses.

Now we give the expression for the $VVV$ vertex $V_{k,p,q}^{\mu\nu\lambda}$ ,

\displaystyle V_{k,p,q}^{\mu\nu\lambda}\equiv\parbox[b]{51.21504pt}{ \leavevmode\hbox to49.49pt{\vbox to29.28pt{\pgfpicture\makeatletter\hbox{\hskip 4.72421pt\lower-5.68759pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}} {}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{19.91684pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ 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}\hbox{{$\mu$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ 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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{36.917pt}{-3.47221pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\lambda$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@lineto{19.91684pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ 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}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{23.19296pt}{19.91684pt}\pgfsys@curveto{23.19296pt}{21.72621pt}{21.72621pt}{23.19296pt}{19.91684pt}{23.19296pt}\pgfsys@curveto{18.10747pt}{23.19296pt}{16.64072pt}{21.72621pt}{16.64072pt}{19.91684pt}\pgfsys@curveto{16.64072pt}{18.10747pt}{18.10747pt}{16.64072pt}{19.91684pt}{16.64072pt}\pgfsys@curveto{21.72621pt}{16.64072pt}{23.19296pt}{18.10747pt}{23.19296pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.44693pt}{17.76407pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\nu$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.15cm}}=g^{\mu\nu}(p-k)^{\lambda}+g^{\nu\lambda}(q-p)^{\mu}+g^{\lambda\mu}(k-q)^{\nu},

(74)

where $k+p+q=0$ . It is easy to check this expression by dotting it into vector boson polarizations. This manipulation will give the amplitudes in eqs. (72) and (73) again. We can use shifted momenta $k+zr$ and $q-zr$ instead of $k$ and $q$ to deform this vertex. Diagrammatically, shifted vertex is represented as

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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.44693pt}{17.76407pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\nu$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.17cm}}=\hat{V}_{k+zr,p,q-zr}^{\mu\nu\lambda}=V_{k,p,q}^{\mu\nu\lambda}+zR^{\mu\nu\lambda}=\parbox[b]{51.21504pt}{ \leavevmode\hbox to49.49pt{\vbox to29.28pt{\pgfpicture\makeatletter\hbox{\hskip 4.72421pt\lower-5.68759pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ 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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.44693pt}{17.76407pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\nu$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.13cm}},

(75)

where $R^{\mu\nu\lambda}=(-g^{\mu\nu}r^{\lambda}-g^{\nu\lambda}r^{\mu}+2g^{\lambda\mu}r^{\nu})$ . In the diagram, the three lines correspond to three Lorentz indices of the vertex. Two horizontal lines represent hard fluctuations, so the momenta they carry should be shifted (e.g. $k+zr$ and $q-zr$ in eq. (75)).

{fmfgraph*}

(60,60) \fmflefti1,i2 \fmfrighto1 \fmfdashesi1,v1 \fmfdashesi2,v1 \fmfdbl_wigglyv1,o1

Figure 2:

VSS

vertex with massive vector

Next, $VSS$ amplitude (see figure 2) will give a simpler vertex. Since the only one external vector boson contributes one Lorentz index, the vertex must be a Lorentz vector. Suppose the momenta of scalar bosons are $p$ and $q$ , the vertex $V_{p,q}^{\mu}$ will be

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(76)

This vertex can be realized in various BSM models, such as 2HDM, MSSM and the simplest Little Higgs model Gunion:1989we; He:2017jjx. In the last case, there is a Higgs-Goldstone mixing term in the non-linear Lagrangian.

Notice that the $VSS$ vertex seems to be a substructure of $VVV$ vertex. Diagrammatically, this means

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(77)

Therefore, the large- $z$ behavior of $VSS$ vertex is contained in $VVV$ vertex.

4 Large- $z$ behavior of $n$ -point amplitudes

For massless amplitudes, appropriate gauges eliminate all large- $z$ contributions from derivative interactions except in the so-called “unique diagrams". Therefore, the large- $z$ behavior of amplitudes depends on the number of hard propagators. If there are more hard boson propagators, the amplitudes will be suppressed by higher powers of $z$ . As for hard fermion propagators, the contribution can be simplified by using anti-commuted gamma matrices and $\not{r}\not{r}=r^{2}=0$ . At last, the hard polarizations are dotted into the sum of all contributions.

In the massive case, we have seen that the $\mathcal{O}(z)$ vertices are not eliminated completely, so the diagrams with hard vector propagators cannot be ignored. Thus, the large- $z$ behavior in the massive case should be evaluated directly instead of estimated.

One may wonder if the Goldstone boson equivalence theorem Lee:1977eg; Chanowitz:1985hj is useful in the complex deformation. In the high energy limit, this theorem treats the longitudinal modes of vector bosons as Goldstone bosons to simplify calculations. However, the large- $z$ limit is not exactly the same as the high energy limit. Actually, it has been used in amplitudes under the all-line shift Cohen:2010mi, but there are also many cases where it doesn’t work. Consider the following amplitude in the large- $z$ limit with particles 1 and 2 shifted:

\begin{gathered}\fmfgraph*(90,50)\fmfbottom{i1,o1}\fmfv{l=1,l.a=70,l.d=.06w}{i1}\fmf{photon}{i1,v1}\fmf{dbl_{w}iggly}{v1,o1}\fmfv{l=2,l.a=110,l.d=.05w}{o1}\fmffreeze\fmftop{i2}\fmfv{l=3,l.a=-30,l.d=.08w}{i2}\fmf{dbl_{w}iggly}{v1,i2}\end{gathered}\quad=\quad\begin{gathered}\fmfgraph*(90,50)\fmfbottom{i1,o1}\fmf{photon}{i1,v1}\fmfv{l=1,l.a=70,l.d=.06w}{i1}\fmf{dashes}{v1,o1}\fmfv{l=2,l.a=120,l.d=.06w}{o1}\fmffreeze\fmftop{i2}\fmfv{l=3,l.a=-30,l.d=.08w}{i2}\fmf{dashes}{v1,i2}\end{gathered}\times\left(1+\mathcal{O}\left(\frac{m_{\Phi}^{2}}{E^{2}}\right)\right).\\

(78)

where $m_{\Phi}$ is the mass of scalar bosons and $E$ is the energy of particle 3. Particle 3 is unshifted, so $\frac{m_{\Phi}^{2}}{E^{2}}$ is no longer a negligible quantity. If we insist on applying this expansion, we should sum over the contributions from infinite terms. Thus, we won’t use the Goldstone boson equivalence theorem in our analysis.

4.1 $n$ -point Vector Boson Scattering Amplitudes

In the massive case, the large- $z$ behavior of amplitudes depends on massive vector bosons, so we can first consider vector boson scattering. Besides particle 1, we set all particles to be massive. The massive vector propagator (56) has two terms. The first term connects vertices while the second term splits the diagram in two. Therefore, the amplitudes can be split into two parts,

\displaystyle\hat{A}_{n}=\hat{A}_{n}^{C}+\hat{A}_{n}^{D},

(79)

where $\hat{A}_{n}^{C}$ and $\hat{A}_{n}^{D}$ correspond to the connected and disconnected diagrammatic expressions respectively.

When a hard particle shoots through a soft background, it will interact with the classical field more than once. The soft physics is parameterized by currents $B^{\mu_{j}}$ , so the amplitude $\hat{A}_{n}^{C}$ becomes

\displaystyle\hat{A}_{n}^{C}=

\displaystyle N_{\mu_{3}}B^{\mu_{3}}_{3}+\sum_{i=4}^{n}\sum_{\sigma\in S_{i-2}}\frac{N_{\sigma(\mu_{3}\mu_{4}\cdots\mu_{i})}}{\prod_{j=3}^{i-1}D_{\sigma(j)}}\prod_{j=3}^{i}B^{\mu_{j}}_{i},

(80)

where the second sum is over all permutations of the labels $(3,4,\dots,i)$ . Here $D_{j}$ is the denominator of shifted propagators. After permutation, it becomes

\displaystyle D_{\sigma(j)}

\displaystyle=\left(\hat{p}_{1}+\sum_{k=3}^{j}p_{\sigma(k)}\right)^{2}-m^{2}=2z\sum_{k=3}^{j}p_{\sigma}(k)\cdot r+\mathcal{O}(1).

(81)

{fmfgraph*}

(100,70) \fmfbottomi1,o1\fmfvl=1,l.a=70,l.d=.06wi1 \fmfphotoni1,v1 \fmfdbl_wigglyv1,o1\fmfvl=2,l.a=110,l.d=.05wo1 \fmffreeze\fmftopi2 \fmfphantomi2,v3,v2 \fmfdbl_wigglyv3,v2,v1 \fmfblob.15wv3

(a)

Figure 3: The diagram without hard propagators. The blob represents the soft background.

The first term in eq. (80) corresponds to the diagram without hard propagators (see figure 3). Since the hard particle interacts with the classical field only once, the large- $z$ behavior of the numerator should be given by a Lorentz vector $N_{\mu_{3}}$ . Inserting polarizations (67) and (66), it becomes

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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{35.6559pt}{-3.52556pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{\epsilon}_{2}^{+}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@lineto{19.91684pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ 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}{}\pgfsys@moveto{24.24106pt}{19.91684pt}\pgfsys@curveto{24.24106pt}{22.30507pt}{22.30507pt}{24.24106pt}{19.91684pt}{24.24106pt}\pgfsys@curveto{17.52861pt}{24.24106pt}{15.59262pt}{22.30507pt}{15.59262pt}{19.91684pt}\pgfsys@curveto{15.59262pt}{17.52861pt}{17.52861pt}{15.59262pt}{19.91684pt}{15.59262pt}\pgfsys@curveto{22.30507pt}{15.59262pt}{24.24106pt}{17.52861pt}{24.24106pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.9041pt}{18.73628pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.2cm}}=-\parbox[b]{51.21504pt}{ \leavevmode\hbox to48.86pt{\vbox to30.18pt{\pgfpicture\makeatletter\hbox{\hskip 5.41196pt\lower-5.54013pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}} {}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{19.91684pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ 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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{37.439pt}{-2.15277pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$r$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@lineto{19.91684pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ 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}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{24.24106pt}{19.91684pt}\pgfsys@curveto{24.24106pt}{22.30507pt}{22.30507pt}{24.24106pt}{19.91684pt}{24.24106pt}\pgfsys@curveto{17.52861pt}{24.24106pt}{15.59262pt}{22.30507pt}{15.59262pt}{19.91684pt}\pgfsys@curveto{15.59262pt}{17.52861pt}{17.52861pt}{15.59262pt}{19.91684pt}{15.59262pt}\pgfsys@curveto{22.30507pt}{15.59262pt}{24.24106pt}{17.52861pt}{24.24106pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.9041pt}{18.73628pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.15cm}}+\mathcal{O}\left(\frac{1}{z}\right)\rightarrow\mathcal{O}\left(\frac{1}{z}\right),$	(82)
$\displaystyle N^{-,-}_{\mu_{3}}$	$\displaystyle=\parbox[b]{59.75095pt}{ \leavevmode\hbox to53.8pt{\vbox to31.62pt{\pgfpicture\makeatletter\hbox{\hskip 6.9813pt\lower-6.9813pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}} {}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{19.91684pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ 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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{35.6559pt}{-3.31166pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{\epsilon}_{2}^{-}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@lineto{19.91684pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ 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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.2cm}}$
	$\displaystyle=zC\parbox[b]{54.06006pt}{ \leavevmode\hbox to50.66pt{\vbox to30.18pt{\pgfpicture\makeatletter\hbox{\hskip 5.41196pt\lower-5.54013pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}} {}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{19.91684pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ 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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.9041pt}{18.73628pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.15cm}}\parbox[b]{8.5359pt}{\Bigg{)}\vspace{-0.2cm}}+\mathcal{O}\left(\frac{1}{z}\right)$
	$\displaystyle=[(r^{*}r)+C(p_{1}p_{2})]p_{1}^{\mu}-\mathcal{O}\left(\frac{1}{z}\right)\rightarrow\mathcal{O}\left(\frac{1}{z}\right),$
$\displaystyle N^{-,0}_{\mu_{3}}$	$\displaystyle=\parbox[b]{59.75095pt}{ \leavevmode\hbox to54.04pt{\vbox to31.87pt{\pgfpicture\makeatletter\hbox{\hskip 6.9813pt\lower-7.22934pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}} {}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{19.91684pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ 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}\hbox{{$\hat{\epsilon}_{1}^{-}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ 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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{35.6559pt}{-3.61111pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{\epsilon}_{2}^{0}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@lineto{19.91684pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{24.47055pt}{0.0pt}\pgfsys@curveto{24.47055pt}{2.51497pt}{22.43181pt}{4.55371pt}{19.91684pt}{4.55371pt}\pgfsys@curveto{17.40187pt}{4.55371pt}{15.36313pt}{2.51497pt}{15.36313pt}{0.0pt}\pgfsys@curveto{15.36313pt}{-2.51497pt}{17.40187pt}{-4.55371pt}{19.91684pt}{-4.55371pt}\pgfsys@curveto{22.43181pt}{-4.55371pt}{24.47055pt}{-2.51497pt}{24.47055pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.13905pt}{-3.61111pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{V}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{24.24106pt}{19.91684pt}\pgfsys@curveto{24.24106pt}{22.30507pt}{22.30507pt}{24.24106pt}{19.91684pt}{24.24106pt}\pgfsys@curveto{17.52861pt}{24.24106pt}{15.59262pt}{22.30507pt}{15.59262pt}{19.91684pt}\pgfsys@curveto{15.59262pt}{17.52861pt}{17.52861pt}{15.59262pt}{19.91684pt}{15.59262pt}\pgfsys@curveto{22.30507pt}{15.59262pt}{24.24106pt}{17.52861pt}{24.24106pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.9041pt}{18.73628pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.2cm}}$
	$\displaystyle=z\parbox[b]{54.06006pt}{ \leavevmode\hbox to48.86pt{\vbox to30.18pt{\pgfpicture\makeatletter\hbox{\hskip 5.41196pt\lower-5.54013pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}} {}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{19.91684pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{5.01196pt}{0.0pt}\pgfsys@curveto{5.01196pt}{2.76805pt}{2.76805pt}{5.01196pt}{0.0pt}{5.01196pt}\pgfsys@curveto{-2.76805pt}{5.01196pt}{-5.01196pt}{2.76805pt}{-5.01196pt}{0.0pt}\pgfsys@curveto{-5.01196pt}{-2.76805pt}{-2.76805pt}{-5.01196pt}{0.0pt}{-5.01196pt}\pgfsys@curveto{2.76805pt}{-5.01196pt}{5.01196pt}{-2.76805pt}{5.01196pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.91562pt}{-1.18056pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{25.05698pt}{0.0pt}\pgfsys@curveto{25.05698pt}{2.83885pt}{22.75569pt}{5.14014pt}{19.91684pt}{5.14014pt}\pgfsys@curveto{17.07799pt}{5.14014pt}{14.7767pt}{2.83885pt}{14.7767pt}{0.0pt}\pgfsys@curveto{14.7767pt}{-2.83885pt}{17.07799pt}{-5.14014pt}{19.91684pt}{-5.14014pt}\pgfsys@curveto{22.75569pt}{-5.14014pt}{25.05698pt}{-2.83885pt}{25.05698pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } 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}{}\pgfsys@moveto{43.04315pt}{0.0pt}\pgfsys@curveto{43.04315pt}{1.77255pt}{41.60623pt}{3.20947pt}{39.83368pt}{3.20947pt}\pgfsys@curveto{38.06113pt}{3.20947pt}{36.6242pt}{1.77255pt}{36.6242pt}{0.0pt}\pgfsys@curveto{36.6242pt}{-1.77255pt}{38.06113pt}{-3.20947pt}{39.83368pt}{-3.20947pt}\pgfsys@curveto{41.60623pt}{-3.20947pt}{43.04315pt}{-1.77255pt}{43.04315pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{37.439pt}{-2.15277pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$r$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ 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}{}\pgfsys@moveto{25.20444pt}{0.0pt}\pgfsys@curveto{25.20444pt}{2.92029pt}{22.83713pt}{5.2876pt}{19.91684pt}{5.2876pt}\pgfsys@curveto{16.99655pt}{5.2876pt}{14.62924pt}{2.92029pt}{14.62924pt}{0.0pt}\pgfsys@curveto{14.62924pt}{-2.92029pt}{16.99655pt}{-5.2876pt}{19.91684pt}{-5.2876pt}\pgfsys@curveto{22.83713pt}{-5.2876pt}{25.20444pt}{-2.92029pt}{25.20444pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.88907pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$V$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{24.24106pt}{19.91684pt}\pgfsys@curveto{24.24106pt}{22.30507pt}{22.30507pt}{24.24106pt}{19.91684pt}{24.24106pt}\pgfsys@curveto{17.52861pt}{24.24106pt}{15.59262pt}{22.30507pt}{15.59262pt}{19.91684pt}\pgfsys@curveto{15.59262pt}{17.52861pt}{17.52861pt}{15.59262pt}{19.91684pt}{15.59262pt}\pgfsys@curveto{22.30507pt}{15.59262pt}{24.24106pt}{17.52861pt}{24.24106pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.9041pt}{18.73628pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.15cm}}\parbox[b]{8.5359pt}{\Bigg{)}\vspace{-0.2cm}}+\mathcal{O}\left(\frac{1}{z}\right)$
	$\displaystyle=-2[(p_{1}\bar{p}_{2})-(p_{1}p_{2})]r^{\mu}+\mathcal{O}\left(\frac{1}{z}\right)\rightarrow\mathcal{O}\left(\frac{1}{z}\right).$

The leading diagrams are equal to zero. We used $(r^{*}r)+C(p_{1}p_{2})=0$ and $(p_{1}\bar{p}_{2})=(p_{1}p_{2})$ to cancel the subleading diagrams in $N^{-,+}_{\mu_{3}}$ and $N^{-,0}_{\mu_{3}}$ . It shows that the first term in eq. (80) vanishes in the large- $z$ limit.

{fmfgraph*}

(150,50) \fmfbottomb1,b5\fmfvl=1,l.a=70,l.d=.06wb1 \fmfphotonb1,b2 \fmfdbl_wigglyb2,b3,b4,b5 \fmfvl=2,l.a=110,l.d=.05wb5 \fmffreeze\fmftopns6 \fmfdbl_wigglys2,b2 \fmfdbl_wigglys3,m1,b3 \fmfdbl_wigglys5,m2,b4 \fmfblob.15ws2,s3,s5 \fmffreeze\fmfphantomm1,m3,m4,m5,m2 \fmfvd.shape=circle,d.f=full,d.size=1.5thickm3,m4,m5

(a)

{fmfgraph*}

(150,50) \fmfbottomb1,b5\fmfvl=1,l.a=70,l.d=.06wb1 \fmfphotonb1,b2 \fmfdbl_wigglyb2,b3,b4,b5 \fmfvl=2,l.a=110,l.d=.05wb5 \fmffreeze\fmftopns6 \fmfdbl_wigglys2,m1,b2 \fmfdbl_wigglys3,m2,b3 \fmfdbl_wigglys4,m3,b3 \fmfdbl_wigglys5,m4,b4 \fmfblob.15ws2,s3,s4,s5 \fmffreeze\fmfphantomm1,ma1,ma2,ma3,m2 \fmfphantomm3,mb1,mb2,mb3,m4 \fmfvd.shape=circle,d.f=full,d.size=1.5thickma1,ma2,ma3 \fmfvd.shape=circle,d.f=full,d.size=1.5thickmb1,mb2,mb3

(b)

Figure 4: The diagrams that give the leading contributions in the large-

z

limit. The blobs correspond to soft backgrounds.

As for the second term in eq. (80), the large- $z$ behavior is given by a Lorentz tensor $N_{\sigma(\mu_{3}\mu_{4}\cdots\mu_{i})}$ . Since this term includes more than three external particles, 4-vertices should be considered. We use a tensor $V_{4}^{\mu\nu\sigma\rho}$ to represent a 4-vertex. Since there are three ways to contract four polarization vectors, the diagrammatic expression of a 4-vertex should be

\displaystyle V_{4}^{\mu\nu\sigma\rho}\equiv\parbox[b]{51.21504pt}{ \leavevmode\hbox to48.23pt{\vbox to27.2pt{\pgfpicture\makeatletter\hbox{\hskip 4.72421pt\lower-5.68759pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}} {}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{19.91684pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ 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}\hbox{{$\mu$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ 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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{33.91576pt}{17.07182pt}\pgfsys@curveto{33.91576pt}{19.30324pt}{32.10689pt}{21.1121pt}{29.87547pt}{21.1121pt}\pgfsys@curveto{27.64406pt}{21.1121pt}{25.83519pt}{19.30324pt}{25.83519pt}{17.07182pt}\pgfsys@curveto{25.83519pt}{14.84041pt}{27.64406pt}{13.03154pt}{29.87547pt}{13.03154pt}\pgfsys@curveto{32.10689pt}{13.03154pt}{33.91576pt}{14.84041pt}{33.91576pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{29.87547pt}{17.07182pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{27.2904pt}{15.89127pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\rho$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.13cm}}=c_{\alpha}g^{\mu\rho}g^{\nu\sigma}+c_{\beta}g^{\mu\sigma}g^{\nu\rho}+c_{\gamma}g^{\mu\nu}g^{\sigma\rho},

(83)

where $c_{\alpha}$ , $c_{\beta}$ and $c_{\gamma}$ are arbitrary coefficients. If we use one $\mathcal{O}(1)$ 4-vertex instead of two $\mathcal{O}(z)$ 3-vertices, one $\mathcal{O}(z^{-1})$ propagator will decrease. Basically, the more 4-vertices diagrammatic expressions have, the lower order they are. The Lorentz tensor $N_{\sigma(\mu_{3}\mu_{4}\cdots\mu_{i})}$ can be expanded as

\displaystyle N_{\sigma(\mu_{3}\mu_{4}\cdots\mu_{i})}=N^{(0)}_{\sigma(\mu_{3}\mu_{4}\cdots\mu_{i})}+\sum_{k=3}^{i-1}D_{\sigma(k)}N^{(1),\sigma(k)}_{\sigma(\mu_{3}\cdots\mu_{i})}+\cdots,

(84)

where the superscript $(n)$ is the number of 4-vertices. Only the first two terms $N_{0}$ and $N_{1}$ (see figure 4) give the same contributions in $\hat{A}_{n}^{C}$ . Their diagrammatic expressions are

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(85)

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}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{95.60728pt}{18.73628pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu_{i}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.2cm}\vspace{-0.2cm}},

(86)

where the script $k$ denotes the position of the 4-vertex. Now we also evaluate these connected diagrammatic expressions in specific helicity and spin states.

\displaystyle N^{(0),-,+}_{\mu_{3}\mu_{4}\cdots\mu_{i}}=

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{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@lineto{19.91684pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{25.05698pt}{0.0pt}\pgfsys@curveto{25.05698pt}{2.83885pt}{22.75569pt}{5.14014pt}{19.91684pt}{5.14014pt}\pgfsys@curveto{17.07799pt}{5.14014pt}{14.7767pt}{2.83885pt}{14.7767pt}{0.0pt}\pgfsys@curveto{14.7767pt}{-2.83885pt}{17.07799pt}{-5.14014pt}{19.91684pt}{-5.14014pt}\pgfsys@curveto{22.75569pt}{-5.14014pt}{25.05698pt}{-2.83885pt}{25.05698pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.08177pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$R$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{25.32187pt}{19.91684pt}\pgfsys@curveto{25.32187pt}{22.902pt}{22.902pt}{25.32187pt}{19.91684pt}{25.32187pt}\pgfsys@curveto{16.93169pt}{25.32187pt}{14.51181pt}{22.902pt}{14.51181pt}{19.91684pt}\pgfsys@curveto{14.51181pt}{16.93169pt}{16.93169pt}{14.51181pt}{19.91684pt}{14.51181pt}\pgfsys@curveto{22.902pt}{14.51181pt}{25.32187pt}{16.93169pt}{25.32187pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.5041pt}{18.73628pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu_{3}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@lineto{39.83368pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{44.97382pt}{0.0pt}\pgfsys@curveto{44.97382pt}{2.83885pt}{42.67253pt}{5.14014pt}{39.83368pt}{5.14014pt}\pgfsys@curveto{36.99483pt}{5.14014pt}{34.69354pt}{2.83885pt}{34.69354pt}{0.0pt}\pgfsys@curveto{34.69354pt}{-2.83885pt}{36.99483pt}{-5.14014pt}{39.83368pt}{-5.14014pt}\pgfsys@curveto{42.67253pt}{-5.14014pt}{44.97382pt}{-2.83885pt}{44.97382pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{35.99861pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$R$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{45.23871pt}{19.91684pt}\pgfsys@curveto{45.23871pt}{22.902pt}{42.81883pt}{25.32187pt}{39.83368pt}{25.32187pt}\pgfsys@curveto{36.84853pt}{25.32187pt}{34.42865pt}{22.902pt}{34.42865pt}{19.91684pt}\pgfsys@curveto{34.42865pt}{16.93169pt}{36.84853pt}{14.51181pt}{39.83368pt}{14.51181pt}\pgfsys@curveto{42.81883pt}{14.51181pt}{45.23871pt}{16.93169pt}{45.23871pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{39.83368pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{35.42094pt}{18.73628pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu_{4}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{79.6678pt}{0.0pt}\pgfsys@lineto{79.6678pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{84.80794pt}{0.0pt}\pgfsys@curveto{84.80794pt}{2.83885pt}{82.50665pt}{5.14014pt}{79.6678pt}{5.14014pt}\pgfsys@curveto{76.82895pt}{5.14014pt}{74.52766pt}{2.83885pt}{74.52766pt}{0.0pt}\pgfsys@curveto{74.52766pt}{-2.83885pt}{76.82895pt}{-5.14014pt}{79.6678pt}{-5.14014pt}\pgfsys@curveto{82.50665pt}{-5.14014pt}{84.80794pt}{-2.83885pt}{84.80794pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{79.6678pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{75.83273pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$R$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{84.70833pt}{19.91684pt}\pgfsys@curveto{84.70833pt}{22.70067pt}{82.45163pt}{24.95737pt}{79.6678pt}{24.95737pt}\pgfsys@curveto{76.88397pt}{24.95737pt}{74.62727pt}{22.70067pt}{74.62727pt}{19.91684pt}\pgfsys@curveto{74.62727pt}{17.13301pt}{76.88397pt}{14.87631pt}{79.6678pt}{14.87631pt}\pgfsys@curveto{82.45163pt}{14.87631pt}{84.70833pt}{17.13301pt}{84.70833pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{79.6678pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{75.69043pt}{18.73628pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu_{i}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.2cm}}+\mathcal{O}(z^{i-4}),

(87)

	$\displaystyle N^{(0),-,-}_{\mu_{3}\mu_{4}\cdots\mu_{i}}=$	$\displaystyle z^{i-2}C\parbox[b]{113.81102pt}{ \leavevmode\hbox to110.41pt{\vbox to33.38pt{\pgfpicture\makeatletter\hbox{\hskip 5.41196pt\lower-7.65756pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}} {}{}{{}} {}{}{{}} {}{}{{}} {}{}{{}} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{19.91684pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@lineto{59.75096pt}{0.0pt}\pgfsys@lineto{79.6678pt}{0.0pt}\pgfsys@lineto{99.58466pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{5.01196pt}{0.0pt}\pgfsys@curveto{5.01196pt}{2.76805pt}{2.76805pt}{5.01196pt}{0.0pt}{5.01196pt}\pgfsys@curveto{-2.76805pt}{5.01196pt}{-5.01196pt}{2.76805pt}{-5.01196pt}{0.0pt}\pgfsys@curveto{-5.01196pt}{-2.76805pt}{-2.76805pt}{-5.01196pt}{0.0pt}{-5.01196pt}\pgfsys@curveto{2.76805pt}{-5.01196pt}{5.01196pt}{-2.76805pt}{5.01196pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.91562pt}{-1.18056pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{25.05698pt}{0.0pt}\pgfsys@curveto{25.05698pt}{2.83885pt}{22.75569pt}{5.14014pt}{19.91684pt}{5.14014pt}\pgfsys@curveto{17.07799pt}{5.14014pt}{14.7767pt}{2.83885pt}{14.7767pt}{0.0pt}\pgfsys@curveto{14.7767pt}{-2.83885pt}{17.07799pt}{-5.14014pt}{19.91684pt}{-5.14014pt}\pgfsys@curveto{22.75569pt}{-5.14014pt}{25.05698pt}{-2.83885pt}{25.05698pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.08177pt}{-3.41666pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$R$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ 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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ 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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\vspace{-0.2cm}}+\cdots$		(88)

$\displaystyle\|\mathbf{\hat{i}}^{I}]$	$\displaystyle=\|\mathbf{i}^{I}]+z\eta^{I}\zeta^{J}\langle\mathbf{j}_{J}k\rangle\|X],$	(26)
$\displaystyle\|\mathbf{\hat{j}}^{J}]$	$\displaystyle=\|\mathbf{j}^{J}]+z\eta^{I}\zeta^{J}\langle k\mathbf{i}_{I}\rangle\|X],$
$\displaystyle\|\hat{k}]$	$\displaystyle=\|k]+z\eta^{I}\zeta^{J}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\|X].$

$\displaystyle\|\mathbf{\hat{i}}^{I}]$	$\displaystyle=\|\mathbf{i}^{I}]+z\eta^{I}\zeta^{J}\langle\mathbf{j}_{J}X\rangle\|k]$	(35)
$\displaystyle\|\mathbf{\hat{j}}^{J}]$	$\displaystyle=\|\mathbf{j}^{J}]+z\eta^{I}\zeta^{J}\langle X\mathbf{i}_{I}\rangle\|k]$
$\displaystyle\|\hat{k}\rangle$	$\displaystyle=\|k\rangle+z\eta^{I}\zeta^{J}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\|X\rangle.$

	$\displaystyle 2r_{i}^{\mu}=\eta^{I}\zeta^{J}\langle\mathbf{j}_{J}X\rangle\langle\mathbf{i}_{I}\|\gamma^{\mu}\|k],$		(36)
	$\displaystyle 2r_{j}^{\mu}=\eta^{I}\zeta^{J}\langle X\mathbf{i}_{I}\rangle\langle\mathbf{j}_{J}\|\gamma^{\mu}\|k],$
	$\displaystyle 2r_{k}^{\mu}=\eta^{I}\zeta^{J}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\langle X\|\gamma^{\mu}\|k],$

$\displaystyle\|\mathbf{\hat{i}}^{I}]$	$\displaystyle=\|\mathbf{i}^{I}]+z\eta^{I}\xi^{K}\langle jX\rangle\|\mathbf{k}_{K}],$	(38)
$\displaystyle\|\hat{j}]$	$\displaystyle=\|j]+z\eta^{I}\xi^{K}\langle X\mathbf{i}_{I}\rangle\|\mathbf{k}_{K}],$
$\displaystyle\|\mathbf{\hat{k}}^{K}\rangle$	$\displaystyle=\|\mathbf{k}^{K}\rangle-z\eta^{I}\xi^{K}\langle\mathbf{i}_{I}j\rangle\|X\rangle.$

$\displaystyle\|\mathbf{\hat{i}}^{I}]$	$\displaystyle=\|\mathbf{i}^{I}]+z\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{j}_{J}\mathbf{k}_{K}\rangle\|X],$	(46)
$\displaystyle\|\mathbf{\hat{j}}^{J}]$	$\displaystyle=\|\mathbf{j}^{J}]+z\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{k}_{K}\mathbf{i}_{I}\rangle\|X],$
$\displaystyle\|\mathbf{\hat{k}}^{K}]$	$\displaystyle=\|\mathbf{k}^{K}]+z\eta^{I}\zeta^{J}\xi^{K}\langle\mathbf{i}_{I}\mathbf{j}_{J}\rangle\|X].$

Massive On-shell Recursion Relations for 𝒏n-point Amplitudes

Abstract

1 Introduction

2 Recursion relations for all masses

2.1 Shifting massless particles

2.2 Shifting mixed particles

2.2.1 Two-line shifts for mixed particles

2.2.2 Three-line Risager type shifts for mixed Particles

2.2.3 Three-line BCFW type shifts for mixed particles

2.3 Shifting massive particles

2.3.1 Two-line shift for massive particles

2.3.2 Three-line Risage-type shifts for massive particles

2.3.3 Three-line BCFW-type shifts for massive particles

2.4 Explicit form of three-line shifts

3 Feynman rules in the large-zz limit

3.1 Hard propagators

3.2 External polarizations

3.2.1 [1,𝟐⟩[1,\mathbf{2}\rangle-shift

3.2.2 [𝟏,𝟐⟩[\mathbf{1},\mathbf{2}\rangle-shift

3.3 𝒪(z)\mathcal{O}(z) vertices

4 Large-zz behavior of nn-point amplitudes

4.1 nn-point Vector Boson Scattering Amplitudes

Massive On-shell Recursion Relations for $n$ -point Amplitudes

3 Feynman rules in the large- $z$ limit

3.2.1 $[1,\mathbf{2}\rangle$ -shift

3.2.2 $[\mathbf{1},\mathbf{2}\rangle$ -shift

3.3 $\mathcal{O}(z)$ vertices

4 Large- $z$ behavior of $n$ -point amplitudes

4.1 $n$ -point Vector Boson Scattering Amplitudes