Helicity amplitudes in light-cone and Feynman-diagram gauges

Abstract

Recently proposed Feynman-diagram (FD) gauge propagator for massless and massive gauge bosons is obtained from a light-cone (LC) gauge propagator, by choosing the gauge vector along the opposite direction of the gauge boson three-momentum. We implement a general LC gauge propagator for all the gauge bosons of the standard model (SM) in the helicity amplitude subroutines (HELAS) codes, such that all the SM helicity amplitudes can be evaluated at the tree level in the LC gauge by using MadGraph. We confirm that our numerical codes produce helicity amplitudes which agree among all gauge choices. We then study interference patterns among Feynman amplitudes, for a few $2\rightarrow 3$ scattering processes in QED and QCD, and the process $\gamma \gamma \rightarrow W^+W^-$ followed by the $W^{\pm}$ decays. We find that the particular choice of the FD gauge vector has advantages over generic LC gauge, not only because all the terms which grow with energy of off-shell and on-shell currents are removed systematically from all the diagrams, but also because no artificial gauge vector direction dependence of individual amplitudes appears.

Appendices

Appendix 1: Comments on the LC gauge singularities

In this Appendix, we report the study on the LC gauge results with other gauge vectors, Eqs. (54b) and (54c).

Fig. 12

Distribution of the scattering angle for $\gamma \gamma \rightarrow W^-W^+$ at $\sqrt{s}=2$ TeV in the LC gauge with different LC vectors, when both $W^+$ and $W^-$ are exactly on the mass shell

We first note that all the LC gauge amplitudes give exactly the same cross section when both W’s are on the mass shell:

$$\begin{aligned} m(e^- \bar{\nu }_e) = m(\mu ^+ \nu _{\mu} ) = m_W. \end{aligned}$$

(56)

The three plots of Fig. 12 show the differential cross section with respect to $\cos \theta _{W^-}$ when the final state is constrained to satisfy the double on-shell conditions (56).

The black solid histograms are obtained from the physical amplitudes where the three Feynman diagrams of Fig. 10 are summed before squaring. The gauge invariance of the physical amplitudes are clearly observed from the exact agreement of the three black histograms in Fig. 12.

The individual Feynman diagram contributions, however, are very different among the three LC gauges.

The gauge vector (54a) gives distributions which are consistent with the behaviour of the corresponding Feynman propagator in the t- and u-channels, and its absence in the contact term, just as in the FD gauge.

For the gauge vector (54b), each Feynman amplitude gives approximately the same magnitude at $\left| \cos \theta _{W^-}\right| \gtrsim 0.6$ as that of the LC gauge (54a).

All the three Feynman amplitudes, however, have large magnitude at around $\cos \theta _{W^-}\approx 0$, where destructive interference among the three amplitudes give the physical cross section.

In case of the gauge vector (54c), the enhancement of the magnitude of all three individual Feynman amplitudes is observed in the wide region of the scattering angle besides near the two edges around $\left| \cos \theta _{W^-}\right| \approx 1$.

We study in detail the LC gauge vector dependence of each amplitude in order to understand their behaviours shown in these figures.

Fig. 13

Feynman diagrams for $\gamma \gamma \rightarrow e^-\bar{\nu }_e\mu ^+\nu _{\mu}$, generated by Madgraph

We first note that the enhancement found for the contact interaction diagram, Fig. 10c, which is given by red curves in Fig. 12, can only come from the two weak boson propagators attached to the final lepton pairs.

The lepton current,

$$\begin{aligned} J^{\mu} = u_L(p_e)^{\dagger} \sigma _-^{\mu} v_L(p_{\bar{\nu }}) \end{aligned}$$

(57)

, can be expressed in general as a summation over three W boson helicity components as follows:

$$\begin{aligned} J^{\mu}= & {} \left( g^{\mu }_{}{}_{\nu } -\frac{q^{\mu} q_{\nu} }{q^2}\right) J^{\nu} \nonumber \\= & {} \left( -\sum _{h=\pm 1,0} \epsilon ^{\mu} (q,h)^*~~ \epsilon _{\nu} (q,h)\right) J^{\nu} \nonumber \\= & {} -\sum _{h=\pm 1,0} \epsilon ^{\mu} (q,h)^* ~~ \epsilon (q,h)\cdot J, \end{aligned}$$

(58)

where we assume zero lepton masses so that the current is conserved and has no Goldstone boson coupling.

In the above expression, the term

$$\begin{aligned} \epsilon (q,h)\cdot J \end{aligned}$$

(59)

gives Wigner’s d-functions [35] in the rest frame of the decaying weak boson or the splitting amplitudes [3] in boosted frames.

Since the transverse polarization components $h=\pm 1$ do not grow with the weak boson energy, it is only the $h=0$ component which grows with energy.

In the unitary gauge, or in an arbitrary covariant $R_{\xi}$ gauge, it is this $h=0$ component which grows with the weak boson energy.

In the FD gauge, this $h=0$ component is replaced by the $\tilde{\epsilon }^{\mu} (q,0)$ term, whose magnitude decreases with energy.

Let us examine what happens to the $h=0$ component of the leptonic current in the general LC gauge:

$$\begin{aligned} P_{\textrm{LC}}{}^{\mu} {}_{\nu} \epsilon ^{\nu} (q,0) = P_{\textrm{LC}}{}^{\mu} {}_{\nu} \left( \frac{q^{\mu} }{Q}+\tilde{\epsilon }^{\nu} (q,0)\right) . \end{aligned}$$

(60)

For our example of $E_W=1~\textrm{TeV}$, the magnitude of $\tilde{\epsilon }^{\nu} (q,0)$ decreases by a factor of $e^{-\eta }\approx 0.04$ for $\cosh \eta =E_W/m_W$, and we can safely neglect its contribution against $q^{\mu} /Q\sim {{\mathcal {O}}}(E_W/m_W)$. Therefore, it is only the behaviour of the scalar current term $q^{\mu} /Q$ we should examine.

We first note that the general LC gauge polarization tensor can be decomposed as

$$\begin{aligned} P_{\textrm{LC}}^{}{}^{\mu \nu } = P_T^{\mu \nu } + q^2 \frac{n^{\mu} n^{\nu} }{(n \cdot q)^2}, \end{aligned}$$

(61)

where the so-called transverse polarization tensor satisfies

$$\begin{aligned} P_T^{\mu \nu } q_{\mu} = P_T^{\mu \nu } n_{\nu} = 0. \end{aligned}$$

(62)

Because of the property (62), the scalar current component of the leptonic current reduces to

$$\begin{aligned} P_{\textrm{LC}}{}^{\mu} _{\ \nu }~ \frac{q^{\nu} }{Q} = Q ~\frac{n^{\mu} }{n \cdot q}. \end{aligned}$$

(63)

Since $Q \approx m_W$ around the resonance peak, and since $n^{\mu}$ is a constant gauge vector, it is the gauge term $n\cdot q$ in the denominator which we should examine.

Fig. 14

a Differential cross section $d\sigma /d\cos \theta _{W^-}$ for the process $\gamma \gamma \rightarrow e^-\bar{\nu }_e\mu ^+\nu _{\mu}$ at $\sqrt{ s}=2$ TeV in the LC gauge with $n^{\mu} =(1,\sin \theta _{W^-},0,\cos \theta _{W^-})= (1, \overrightarrow{p}_{W^-}/|\overrightarrow{p}_{W^-}|)$, where $\theta _{W^-}$ is the polar angle of the $e^-\bar{\nu }_e$ pair. The invariant masses of the lepton pair are fixed at $m(\mu ^+\nu _{\mu} )=69.3$ GeV and $m(e^-\bar{\nu }_e)=m_W$. The red dashed curve shows the contribution of the three double-resonant diagrams (${{\mathcal {M}}}_{\textrm{DR}}$), while the blue dotted curve shows the contribution of all the other diagrams, single resonant (${{\mathcal {M}}}_{\textrm{SR}}$) or non-resonant (${{\mathcal {M}}}_{\textrm{NR}}$). The black solid curve shows the sum of all 13 Feynman diagrams. b Imaginary part of the amplitudes when the photon helicities are $\lambda _1=\lambda _2=-1$ and $e^-$ and $\mu ^+$ momenta are along the $W^+(\mu ^+\nu _{\mu} )$ momentum direction in the $\gamma \gamma$ rest frame. We set $\mathrm{\Gamma }_W=0$ in the off-shell $W^-$ propagator

Let us write down the four momentum of $W^{\pm}$ in the colliding photon rest frame.

We find

$$\begin{aligned} p^{\mu} _{W^-}&= (E_{W^-},p^*\sin \theta _{W^-},0,p^*\cos \theta _{W^-}) \end{aligned}$$

(64a)

$$\begin{aligned} p^{\mu} _{W^+}&= (E_{W^+},-p^*\sin \theta _{W^-},0,-p^*\cos \theta _{W^-}) \end{aligned}$$

(64b)

where

$$\begin{aligned}{} & {} E_{W^{\pm} } = \frac{\sqrt{s}}{2} \left( 1+\frac{p_{W^{\pm} }^2-p_{W^{\mp} }^2}{s}\right) \end{aligned}$$

(65)

and

$$\begin{aligned} p^* = \frac{\sqrt{s}}{2} {\bar{\beta }}\left( \frac{p_{W^-}^2}{s},\frac{p_{W^+}^2}{s}\right) \end{aligned}$$

(66)

with

$$\begin{aligned} \bar{\beta }(a,b)=\sqrt{1-2(a+b)+(a-b)^2} \end{aligned}$$

(67)

is the common momentum of the W pair in the c.m. frame.

For the gauge vector (54a), the product is

$$\begin{aligned} n\cdot p_{W^{\pm} } = E_{W^{\pm} } \pm p^*\cos \theta _{W^-} \end{aligned}$$

(68)

which can become small around $\cos \theta _{W^-}=\pm 1$, but after contracting all the four currents in the contact-term amplitudes Fig. 10c, the gauge terms proportional to $n^{\mu}$ cancel out: They vanish in the product of the $W^-$ and $W^+$ currents because of $n\cdot n=0$, whereas the product of $W^{\pm}$ and the initial photon polarization vectors vanish, since

$$\begin{aligned} n \cdot \epsilon (p_{\gamma} ,\pm 1) = 0 \end{aligned}$$

(69)

for both photons with momenta along the z-axis.

In case of the gauge vector (54b), we find

$$\begin{aligned} n\cdot p_{W^{\pm} } = E_{W^{\pm} } \pm p^*\sin \theta _{W^-} \end{aligned}$$

(70)

and one of which can be small at $\cos \theta _{W^-}\approx 0$. In the contact interaction amplitude, the enhanced current survives when contracted with the initial photon polarization vectors, because

$$\begin{aligned} \left| n \cdot \epsilon (p_{\gamma} ,\pm 1)\right| = \left| -\epsilon (p_{\gamma} ,\pm 1)^1\right| = \frac{1}{2} \end{aligned}$$

(71)

for both photons. The enhancement in the red histogram around $\cos \theta _{W^-}\approx 0$ observed in Fig. 12b is hence due to the LC gauge term.

Finally, for the gauge vector of (54c), we find

$$\begin{aligned} n\cdot p_{W^{\pm} } = E_{W^{\pm} } \pm p^* \end{aligned}$$

(72)

which is independent of $\cos \theta _{W^-}$. In fact, the above term grows with energy for the $W^+$ propagator, because the three-vector direction $\overrightarrow{n} = \overrightarrow{p}_{W^-}/|\overrightarrow{p}_{W^-}|$ is opposite of the $W^+$ momentum direction in the c.m. frame, which is the prescription for the FD gauge. It is hence only the $W^-$ propagator whose gauge term can grow with energy. Because the enhancement factor in the $W^-$ propagator is independent of the scattering angle, we observe broad enhancement in Fig. 12c. The enhancement effect diminishes near $\cos \theta _{W^-} = \pm 1$, because the gauge vector becomes orthogonal to one of the incoming photon polarization vectors in the limits.

Although we understand the gauge artefacts observed for the contact interaction term as above, we find that the most serious problem arises in the t- and u-channel exchange amplitudes, from their gauge terms. The four momentum of the weak boson exchanged in the t-channel can be expressed as follows:

$$\begin{aligned} q^{\mu}= & {} p^{\mu} (\gamma _1) -p^{\mu} (W^-) \nonumber \\= & {} (E-E_{W^-}, -p^*\sin \theta _{W^-}, 0, E-p^*\cos \theta _{W^-}) \end{aligned}$$

(73)

with $E=\sqrt{s}/2$. Let us show the gauge term for our three gauge vectors. For (54a), we find

$$\begin{aligned} n\cdot q = -( E_{W^-} -p^*\cos \theta _{W^-}). \end{aligned}$$

(74)

This term can never vanish as long as both $W^{\pm}$ invariant mass is in the resonance region. For (54b), we find

$$\begin{aligned} n\cdot q = E-E_{W^-} +p^*\sin \theta _{W^-}. \end{aligned}$$

(75)

When both W’s are on-shell, $E_{W^-}=E$, and hence the gauge term vanishes at $\sin \theta _{W^-}=0$, or at $\cos \theta _{W^-}=\pm 1$. Once the invariant mass of $W^-$ is larger than that of $W^+$, the zero appears in the physical region. Although the LC gauge pole singularity can be integrated out in loop calculation e.g. by taking the principal value integral [34], once the pole appears in the physical region of the phase space integral, the MC integral fails instantly, since the singular terms appear in the absolute value square of the amplitudes.

Let us study how the LC gauge pole singularity appears in the resonant amplitudes, and how they are cancelled in the physical amplitudes, by using the gauge vector (54c). The four momentum of the W boson exchanged in the t-channel is given by Eq. (73) and the gauge term for (54c) becomes

$$\begin{aligned} n\cdot q = E-E_{W^-}+p^*-E\cos \theta _{W^-}. \end{aligned}$$

(76)

The $n\cdot q=0$ pole can appear at

$$\begin{aligned} \cos \theta _{W^-} = (E-E_{W^-}+p^*)/E. \end{aligned}$$

(77)

When both W’s are on-shell, this happens at

$$\begin{aligned} \cos \theta _{W^-} = \sqrt{1-(m_W/E)^2}, \end{aligned}$$

(78)

which is 0.9968 at $\sqrt{s}=2$ TeV. We find, however, that this pole does not show up in the amplitudes because each on-shell amplitude vanishes exactly at the corresponding pole.

We find that the above vanishing of individual Feynman amplitude at the pole of the LC gauge term does not hold when one or both of the weak bosons is off-shell. There appear nonzero amplitudes which are proportional to the off-shellness of decaying W’s, $\left( m(e^- \bar{\nu }_e\right) ^2 -m_W^2)$ and/or $\left( m(\mu ^+ \nu _{\mu }\right) ^2 -m_W^2)$, as residues of the pole term. We further find that the singular terms do not cancel even after summing up the three resonant diagrams of Fig. 10a–c.

We therefore evaluate all the 13 diagrams contributing to the process

$$\begin{aligned} \gamma \gamma \rightarrow e^- {\bar{\nu }}_e \mu ^+ \nu _{\mu} \end{aligned}$$

(79)

as shown in Fig. 13, generated by MadGraph [17,18,19]. We note that the three double-resonant (DR) diagrams of Fig. 10 are now labelled as 11, 12, 13 at the bottom of Fig. 13. The diagrams 2, 3, 7, 8 have $W^+$ propagator, while the diagrams 5, 6, 9, 10 have $W^-$ propagator only, and we call them single resonant (SR) diagrams. The diagrams 1 and 4 have no $W^{\pm }$ propagator, and are labeled as non-resonant (NR).

In Fig. 14a, we show the $\cos \theta _{W^-}$ distribution when

$$m(e^{ - } \bar{\nu }_{e} ) = 69.3~GeV \approx m_{W} - 5.3~\Gamma _{W} {\text{ }}$$

(80a)

$$\begin{aligned} m(\mu ^+ \nu _{\mu} )= & {} m_W=80.4~\textrm{GeV}, \end{aligned}$$

(80b)

at $\sqrt{s}=2$ TeV. The red dashed curve shows the contribution of the sum of the three double-resonant amplitudes,

$$\begin{aligned} \left| {{\mathcal {M}}}_{\textrm{DR}}\right| ^2=\left| \sum _{k=11}^{13} {{\mathcal {M}}}_k\right| ^2. \end{aligned}$$

(81)

Shown by the blue dashed curve is for the sum of all the other amplitudes

$$\begin{aligned} \left| {{\mathcal {M}}}_{\textrm{SR}}+{{\mathcal {M}}}_{\textrm{NR}}\right| ^2 = \left| \sum _{k=1}^{10} {{\mathcal {M}}}_k\right| ^2, \end{aligned}$$

(82)

which sum over the eight single resonant and two non-resonant amplitudes. We can tell from the red and blue dashed curves that they have the same singular behaviour at $\cos \theta _{W^-}$

$\approx 0.9976$, which is the pole position of Eq. (77) for the invariant masses of Eq. (80). Although the red and blue dashed curves in Fig. 14a confirm that the sum of the three double-resonant amplitudes (${{\mathcal {M}}}_{\textrm{DR}}$) and that of the single and non-resonant amplitudes (${{\mathcal {M}}}_{\textrm{SR}}+{{\mathcal {M}}}_{\textrm{NR}}$) have the same singular behaviour, we find that the total sum of all the 13 diagrams in Fig. 13,

$$\begin{aligned} \left| {{\mathcal {M}}}_{\textrm{DR}}+{{\mathcal {M}}}_{\textrm{SR}}+{{\mathcal {M}}}_{\textrm{NR}}\right| ^2 = \left| \sum _{k=1}^{13} {{\mathcal {M}}}_k\right| ^2 \end{aligned}$$

(83)

still gives the singular behaviour, as shown by the black solid curve in the same plot. The magnitude of the singular term is significantly reduced, and hence there is a hint of cancellation among the 13 amplitudes of Fig. 13 .

We find that this non-cancellation of the LC gauge pole singularity is due to the width term in the W boson propagator, unitarized by the Breit-Wigner prescription [36]. At the kinematical point on top of the pole, $n\cdot p_W=0$, the non-vanishing off-shell amplitudes are proportional to $p_W^2-m_W^2$. This term cancels the W propagator factor exactly, only if the width is set to be zero. In the presence of the finite width term, the cancellation is not exact, and the pole singularity in the resonant amplitudes survive even after summing over all the non-resonant amplitudes.

In order to demonstrate cancellation of the LC gauge pole singularity between the resonant and non-resonant amplitudes, we show in Fig. 14 the amplitude of the double-resonant diagrams, and that of the single $W^+$ resonant diagrams for the lepton-pair invariant masses of Eq. (76). We show only the amplitude when both incoming photons are left-handed $\left( \lambda _{\gamma _1}=\lambda _{\gamma _2}=-1\right)$, and when the final $e^-$ and $\mu ^+$ momenta are along the $W^+$ $(\mu ^+\nu _{\mu} )$ momentum direction. With this set-up, the double-resonant amplitude is proportional to the helicity amplitude for

$$\begin{aligned} \gamma _1\left( -1\right) +\gamma _2\left( -1\right) \rightarrow W^+\left( +1\right) + W^-\left( +1\right) , \end{aligned}$$

(84)

where $\pm 1$ inside parentheses denote helicities. We keep the width factor of the $W^+$ propagator so that the amplitude is finite and pure imaginary on top of the $W^+$ boson mass shell, (80b). We set the width of off-shell $W^-$ propagator to zero, in order to demonstrate the cancellation of the LC gauge pole singularities between the double-resonant amplitudes (${{\mathcal {M}}}_{\textrm{DR}}={{\mathcal {M}}}_{11}+{{\mathcal {M}}}_{12}+{{\mathcal {M}}}_{13}$) and the single $W^+$ resonant amplitudes (${{\mathcal {M}}}_{\textrm{SR}}={{\mathcal {M}}}_2+{{\mathcal {M}}}_3+{{\mathcal {M}}}_7+{{\mathcal {M}}}_8$). These amplitudes are pure imaginary (due to on-shell $W^+$), and we show the imaginary part of $M_{\textrm{DR}}$ by the red dashed line, that of ${{\mathcal {M}}}_{\textrm{SR}}$ by the blue dashed line, and their sum by the black line. We can clearly observe exact cancellation of the LC gauge pole singularity between the ${{\mathcal {M}}}_{\textrm{DR}}$ and the ${{\mathcal {M}}}_{\textrm{SR}}$ amplitudes. Similar cancellation takes place in the real part of the amplitudes between the $W^-$ SR amplitudes and NR amplitudes, with significantly smaller magnitudes.

The apparent LC gauge vector dependence of the amplitudes is hence the artefact of the Breit-Wigner unitarization  [36] of the W boson propagator, which violates the order-by-order gauge invariance of the perturbative amplitudes.

Since it is not practical to sum up all the non-resonant amplitudes before unitarizing the resonant W amplitudes, we propose that the LC gauge vector should be carefully chosen such that the LC gauge term does not have a pole in the physical region of all the contributing gauge boson propagators. Among our examples, the choice (54a) should be safe, as long as the final-state particles are massive or has large $p_T$.

Appendix 2: Comments on event generators with the LC gauge

From the perspective of high energy computation tools development, our LC gauge quantization approach has an absolute advantage over the previously proposed method [2] of modifying the amplitudes generated in the unitary or $R_{\xi}$ gauges. If we start from the unitary gauge, as in ref. [2], although the number of Feynman diagrams can be reduced due to the absence of the Goldstone boson propagation, some of the fundamental vertices like the four-point neutral Goldstone boson coupling are absent from the Feynman rule. As a consequence, the authors of ref. [2] added a few vertices to the SM interactions in order to generate appropriate Feynman diagrams and the corresponding amplitudes. If we start from the $R_{\xi}$ gauge, things become even more complicated. It is because all the Goldstone boson propagators should first be cancelled against the corresponding $R_{\xi}$ gauge boson propagators which share the same four momentum, by using the BRST identities. At this stage, all the amplitudes with the Goldstone boson propagators are removed, and the vertices which are missing in the unitary gauge should be re-introduced. Although there are only four such vertices in the SM of the EW theory [2], we anticipate more missing vertices, in models beyond the SM or even in the SM effective field theory (SMEFT).

On the other hand, in the LC gauge quantization of the weak bosons, no unphysical particles propagates because all the Goldstone bosons are the ${5}^{\hbox {th}}$ component of the corresponding physical weak bosons. The Feynman rules are then straightforward, since we do not distinguish the first 4 and the ${5}^{\hbox {th}}$ components of the weak bosons when generating Feynman diagrams. All the vertices are straightforward to obtain once the Higgs sector of the model is specified. The method can be applied for all SMEFT operators, and for all BSM models with spontaneously broken gauge symmetries. We therefore believe that the next generation of scattering amplitude generators should adopt the LC gauge form of the massless and massive gauge boson propagators. The FD gauge amplitudes are then obtained by choosing the gauge vector along the opposite direction of each off-shell gauge boson momentum.