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--> --> -->Another interesting feature of the dimension-8 coefficient space is that, by exploiting its positive nature, we can either infer the existence of UV states and their quantum numbers [3] or exclude them. This originates from the positivity bounds that carve out a geometric object in the parameter space of the Wilson coefficients, namely a convex cone whose “edges” (to be more precisely defined later) are closely related to the properties of the UV states lying in specific irreducible representations (irreps) of the SM symmetry group. This relation, when combined with the positive nature of the dimension-8 coefficient space, can often provide striking information about the new physics states living in the UV.
As an extreme example, one could imagine that one succeeds in measuring the vector of dimension-8 Wilson coefficients with sufficient precision. If this vector coincides with one of the “edges” of the cone, the UV states generating the corresponding operators can be uniquely determined in the sense that they must all lie in a single irrep of the SM symmetries. If the UV completion is assumed to be tree-level and weakly coupled, one concludes that the underlying theory must be a “one-particle extension” of the SM. This provides an answer to the “inverse problem” [13-15] that can be summarized as follows. Given the measured values of the coefficients at the electroweak scale, how can we possibly determine the nature of the new physics beyond the SM? Similarly, if the measurement agrees with the SM value to a sufficient precision, one can simultaneously exclude the existence of any potential new physics state up to certain scales. This exclusion is guaranteed by the positive nature of the Wilson coefficients and cannot be removed by arranging the UV states in specific patterns that cancel each other's effects. Therefore, this would provide a model-independent confirmation of the SM, which is not possible when truncating the SMEFT at dimension-6.
Dimension-8 SMEFT operators [16, 17] have recently attracted increasing attention, in particular as the LHC accumulates more data. Various motivations for going beyond a truncation of the SMEFT Lagrangian at the dimension-6 level have correspondingly been presented, for example, in Refs. [4, 6, 18-24]. In addition, observables that can be used to disentangle the effects of dimension-8 operators from those of dimension-6 have been proposed and studied, as for example, in Refs. [25, 26]. However, positivity-related topics, including the possible tests of its violation and the option of inferring/excluding the existence of states in the UV, have not been discussed using realistic phenomenological examples.
This study aims to present some initial results in this direction in the context of future lepton colliders. In particular, we are interested in the following questions:
1. To what extent can we test any potential violation of the positivity bounds?
2. For realistic measurements (including the associated experimental errors), to what extent can we learn about the existence of UV states and their properties using the positive nature of the dimension-8 coefficient space?
The first point has been discussed in Ref. [12] but not in the SMEFT framework, and no realistic collider analysis has been presented. In contrast, the second point has not been discussed in literature.
Several proposals for a future electron-positron machine are currently discussed, including the CEPC [27], FCC-ee [28, 29], ILC [30, 31], and CLIC [32] projects. These colliders present an ideal means to perform high accuracy measurements, particularly as they are planned to be operated at various center-of-mass energies. Thus, they could allow us to distinguish the effects of dimension-8 operators from those of dimension-6 on a large set of observables, thus creating new opportunities to access to information on the SMEFT dimension-8 operators. Let us note that in addition to the energy dependence, the angular momentum can also be used as a discriminant [25].
For a first step in this direction, we consider the simplest
The remainder of this paper is organized as follows. In Section II, we list the
$ {\cal L}_{\rm{SMEFT}} = {\cal L}_{\rm{SM}}+\sum\limits_i\frac{C_i^{(6)}}{\Lambda^2}O_i +\sum\limits_i\frac{C_i^{(8)}}{\Lambda^4}O_i+\cdots\ , $ ![]() | (1) |
There are three four-fermion
$ \begin{aligned}[b] O_{ee} = &( \bar e \gamma^\mu e )\ ( \bar e \gamma_\mu e)\ ,\\ O_{el} =& ( \bar e \gamma^\mu e )\ ( \bar l \gamma_\mu l)\ ,\\ O_{ll} =& ( \bar l \gamma^\mu l )\ ( \bar l \gamma_\mu l)\ , \end{aligned} $ ![]() | (2) |
The full basis of dimension-8 operators has been presented recently [16, 17]. Three types of four-electron operators are relevant to our study and are of the forms
In this study, we are mainly interested in operators
$ \begin{aligned}[b] &O_{1} = \partial^\alpha(\bar e \gamma^\mu e) \partial_\alpha (\bar e \gamma_\mu e)\ ,\\ &O_{2} = \partial^\alpha(\bar e \gamma^\mu e) \partial_\alpha (\bar l \gamma_\mu l)\ ,\\ &O_{3} = D^\alpha(\bar e l)\ D_\alpha(\bar l e), \\ &O_{4} = \partial^\alpha(\bar l \gamma^\mu l)\ \partial_\alpha(\bar l \gamma_\mu l)\ ,\\ &O_{5} = D^\alpha(\bar l \gamma^\mu \tau^I l)\ D_\alpha(\bar l \gamma_\mu \tau^I l)\ , \end{aligned} $ ![]() | (3) |
Similar to a model-independent SMEFT framework, the other two classes of dimension-8 operators should be included as well. However, in the context of
In summary, our collider analysis only incorporates the effects of the dimension-8 operators of the first type
$\begin{aligned}[b] \vec C^{(6)} = (C_{ee},C_{el},C_{ll}), \quad \vec C^{(8)} = (C_1,C_2,C_3,C_4)\ .\end{aligned}$ ![]() | (4) |
$ \sigma = \sigma_{\rm{SM}} +\! \sum\limits_i\frac{C_i^{(6)}}{\Lambda^2}\sigma^{(6)}_i \!+\!\sum\limits_i\frac{[C_i^{(6)}]^2}{\Lambda^4}\sigma^{(6)}_{ii} \!+\!\sum\limits_i\frac{C_i^{(8)}}{\Lambda^4}\sigma^{(8)}_i \ , $ ![]() | (5) |
The conventional approach to derive positivity bounds utilizes forward and elastic scattering amplitudes (see, e.g., Ref. [1]). Briefly, it requires that the second order s-derivative of the elastic amplitudes (with poles subtracted) be positive. For instance, for the process considered in this study, it is given by
$ \frac{ {\rm{d}}^2}{ {\rm{d}} s^2}M(e^+e^-\to e^+e^-)\geqslant 0 . $ ![]() | (6) |
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However, the above list of bounds is not complete, as, by defining states through the superposition of different flavors and chiralities, one can consider extra elastic scattering processes [37]. According to this approach, the best bounds are derived in Appendix A from the scattering amplitudes presented in Appendix B,
$ C_1\leqslant 0 , $ ![]() | (7) |
$ C_4+C_5\leqslant 0, $ ![]() | (8) |
$ C_5\leqslant 0, $ ![]() | (9) |
$ C_3\geqslant 0, $ ![]() | (10) |
$ 2\sqrt{C_1(C_4+C_5)}\geqslant C_2, $ ![]() | (11) |
$2\sqrt{C_1(C_4+C_5)}\geqslant-(C_2+C_3). $ ![]() | (12) |
$ \begin{aligned}[b] | f_\pm \rangle \equiv &\frac{(C_4+C_5)^{1/4}} {\left[(C_4+C_5)^{1/2}+C_1^{1/2}\right]^{1/2}} | e_R\rangle \\& \pm \frac{C_1^{1/4}} {\left[(C_4+C_5)^{1/2}+C_1^{1/2}\right]^{1/2}} |\bar e_L\rangle , \end{aligned} $ ![]() | (13) |
The above approach is sufficient for the study of the four-fermion operators considered in this paper. However, in general, it is insufficient to obtain the best possible bounds. Accordingly, a new and better approach has been proposed recently [3]. The idea is to construct the allowed Wilson coefficient parameter space region directly as a convex cone, which is a convex hull of its extremal rays, and the latter can be identified using group theoretical considerations.
This new approach has at least two advantages. First, it always provides the tightest constraints available from the dispersion relation. They may be tighter than those that can be obtained by relying on the conventional elastic positivity approach, as for example, for the scattering of a pair of W-bosons. Second, more relevant to this study (see Section IV), it reveals a connection between the positivity bounds and the existence of new physics states in the UV. In order for this connection to be manifest, one needs to determine the exact shape of the parameter space allowed by the bounds, which cannot always be achieved with the conventional approach for complicated cases. We have verified that, for the operators under consideration, the two methods yield the same set of bounds of Eqs. (7)–(12).
We devote the remainder of this section to a discussion on the possible violation of positivity and its physical implications.
2
A.Quantifying positivity violation
A positivity violation would imply a breakdown of the fundamental principles of QFT. Hence, if such a violation is observed at a future collider, it would be mandatory to study the physics behind it. To this end, we first need a model-independent way to quantify the observed amount of violation, which can be connected later to possible physics scenarios.The physical quantity to consider for probing a potential positivity violation is the second order s-derivative of the studied amplitude with poles subtracted,
$ -\Delta^{-4}\equiv \min\left[ \min\limits_{\epsilon_1,\epsilon_2} \frac{1}{2} \frac{ {\rm{d}}^2M(s,t = 0)(\epsilon_1,\epsilon_2)}{ {\rm{d}} s^2},0\right]\ , $ ![]() | (14) |
While the physical interpretation of the
One often estimates
$ \Lambda_{\rm{BSM}}^{-4} = \max\limits_{\epsilon_1,\epsilon_2}\left| \frac{1}{2}\frac{ {\rm{d}}^2M(s,t = 0)(\epsilon_1,\epsilon_2)}{ {\rm{d}} s^2}\right|\ , $ ![]() | (15) |
To compute the positivity violation measure
$ \Delta^{-4} = \frac{\delta(\vec C^{8})}{\Lambda^4}\ , $ ![]() | (16) |
$ \begin{aligned}[b] \delta(\vec C^{(8)})\equiv & -\min\bigg[0, -4C_1,-4(C_4\!+\!C_5),C_3,-8C_5, \\ & \Theta[C_2 \!-\!2\max(C_1,C_4\!+\!C_5)] \frac{C_2^2\!-\!4C_1(C_4\!+\!C_5)}{C_1\!-\!C_2\!+\!C_4\!+\!C_5},\\ & \Theta[-C_2-C_3-2\max(C_1,C_4+C_5)]\\ & \times\frac{(C_2+C_3)^2-4C_1(C_4+C_5)}{C_1+C_2+C_3+C_4+C_5}\bigg] , \end{aligned} $ ![]() | (17) |
However, it is not necessarily sufficient to derive the
As an illustration, Fig. 1 shows the dependence of the quantity

2
B.Physical interpretations
What do possible violations of positivity bounds mean? It is known that the forward positivity bounds are derived by assuming that the scattering amplitudes computed in the UV-completed theory are unitary, Lorentz invariant, polynomially bounded in momenta and analytical in the complex s plane, apart from certain poles and branch cuts. The unitarity of the S-matrix indicates that the quantum mechanical probabilities of all possible scatterings add up to 1, which results in the optical theorem, or that the imaginary part of the UV amplitudes is positive in the physical region.Violations of unitarity, such as the existence of (bad) ghosts [38] in the UV, will lead to catastrophic instabilities in the theory (and thus should be avoided), unless the ghosts only appear in the effective field theory context and with a mass at or greater than the cutoff scale. Lorentz invariance has been tested at very high energy scales via various experiments, although certain properties are only weakly constrained [39]. Analyticity is implied by causality and can be proven to be valid at any perturbative order, although it has never been proven non-perturbatively. The polynomial boundedness of the amplitudes in the momentum space originates from locality, the lack of which would result in ill-defined Fourier transforms and non-locality in real space. In addition, polynomial boundedness, analyticity, and unitarity can be used to prove the Froissart bound that implies that forward UV amplitudes should grow slower than
In terms of the Wilson coefficient parameter space, the observation of small violations of the positivity bounds would indicate regions not too far away from the positive regime (e.g., the yellow/orange areas in Fig. 1) and would imply that some of the fundamental principles of QFT are violated at certain energy scales. A high experimental precision is required to detect those small violations. In contrast, the observation of a stronger violation of the positivity bounds would indicate regions further away from the positive regime (e.g., the purple/black areas in Fig. 1) and imply violations of the fundamental principles of QFT at more accessible energies.
On dimensional grounds, the quantity
$ \begin{aligned}[b] \frac12 \frac{ {\rm{d}}^2 M(s = 2m_e^2,t = 0)}{ {\rm{d}} s^2} =& \int^{\Lambda_*^2}_{4m_e^2} \frac{ {\rm{d}} s'}{2\pi} \frac{\Im[M(s',0)]}{(s'-2m_e^2)^3} \\& + \int_{{\cal C}'}\frac{ {\rm{d}} s'}{2\pi i} \frac{M(s',0)}{(s'-2m_e^2)^3} , \end{aligned} $ ![]() | (18) |
To estimate the integrand at
1.
$ \int_{{\cal C}'}\frac{ {\rm{d}} s'}{2\pi i} \frac{M(s',0)}{(s'-2m_e^2)^3} \simeq \frac{1}{\Lambda_*^4}\ . $ ![]() | (19) |
$ \int_{{\cal C}'}\frac{ {\rm{d}} s'}{2\pi i} \frac{M(s',0)}{(s'-2m_e^2)^3} \simeq \frac{1}{\Lambda_*^2 \Lambda_{\rm{BSM}}^2}\ . $ ![]() | (20) |
The fact that
First, for the UV completions of the SM that are intertwined with the massless graviton, the
If one assumes that the UV theory follows the Regge behavior,
$ \Im[M(s\to \infty, t\to 0)] = f(t)\left(\frac{\alpha^{\prime} s}{4}\right)^{2+j(t)}, $ ![]() | (21) |
$ \begin{aligned}[b] \frac{ {\rm{d}}^2 M}{ {\rm{d}}^2 s}>& -\frac{f \alpha^{\prime 2}}{4 \pi}\left[\frac{f^{\prime}}{f j^{\prime}}+\ln \left(\frac{\alpha^{\prime} M_{*}^{2}}{4}\right)\right]\\ & +\frac{f \alpha^{\prime 2}}{4 \pi}\left[\frac{j^{\prime \prime}}{2\left(j^{\prime}\right)^{2}}+O\left(\frac{1}{\alpha^{\prime} M_{*}^{2}}\right)\right] . \end{aligned} $ ![]() | (22) |
Second, one could devise a simple example demonstrating how unitarity violation leads to positivity violation in the Effective Field Theory (EFT). One such popular model having an interesting bearing for inflation consists of the so-called DBI model [47], whose effective Lagrangian is given by
$ {\cal L}_{\rm EFT} = \varepsilon \Lambda^{4}-\varepsilon \Lambda^{4} \sqrt{1-\frac{\varepsilon(\partial \phi)^{2}}{2\Lambda^{4}} } , $ ![]() | (23) |
$ {\cal L}_{\rm{pUV}} = \frac{ ( \partial \chi)^2}{2} + \frac{\varepsilon e^{\frac{\chi}{M}}( \partial \phi)^2}2 - \Lambda^4\left(\cosh \frac{\chi}{M}-1\right) ,$ ![]() | (24) |
Then, which of these QFT axiomatic principles is the weakest link? Arguably, it might be the polynomial boundedness/locality. Indeed, it is widely believed that gravity is non-local, and the UV completions of general relativity, such as string theories, violate polynomial boundedness [51]. This is intimately linked to the observation that black holes are formed in high-energy scatterings with gravity included, and their horizon radius increases with the scattering energy. Moreover, there are no local gauge invariant observables in gravity.
When contemplating UV completions for the SMEFT, one may consider that general relativity is also an EFT that needs to be UV completed and which might a priori be interconnected to the UV theory of the SMEFT. Hence, the SM and gravity may be (partially) UV completed together, potentially at an energy scale such as the TeV scale as in models with large extra dimensions. However, polynomial boundedness is also violated in some innocent looking (Minkowski space) field theories derived by taking certain low-energy limits of gravitational theories. For example, the galileon theory consists of a scalar EFT whose Lagrangian is given by
$ {\cal L} = \frac12 \partial^\mu\pi \partial_\mu \pi + \frac{ {\alpha}}{\Lambda^3} \partial^\mu\pi \partial_\mu \pi \partial^\rho \partial_\rho \pi + \cdots\ . $ ![]() | (25) |
Discussions on positivity bounds and their implications have recently restarted in the context of the galileon theory [7]. As the violation is marginal, adding a softly-breaking mass term for the galileon allows one to satisfy the forward positivity bounds [8, 11, 55, 56]. Moreover, at least in some parameter space region, generalized, t-derivative, positivity bounds [9, 57, 58] are fulfilled. However, further generalized positivity bounds exclude the entire parameter space [59].
An important feature of the galileon theory along with the DGP model and the dRGT model is that they embed the so-called Vainshtein mechanism (see Ref. [51] and the references therein). It has been argued that theories including the Vainshtein mechanism should not have standard UV completions whose low-energy EFT satisfies the positivity bounds. Instead, the high-energy behavior is characterized by a phenomenon called classicalization, where semi-classical contributions dominate [51, 60] (see ref. [61] for discussions on classicalization in the anti-DBI model). Examples of those non-standard UV completions also appear in gauge theories, in the context of chiral perturbation theories [62].
Lorentz invariance is also at risk of being violated at high energies. After all, our intuition of Lorentz invariance comes from low energy and weak gravity environments. In general relativity, Lorentz invariance still holds in local inertial frames, but this may just be a prejudice. Hence, Lorentz violating models are widely discussed in several contexts [63], including Horava-Lifshitz gravity [64]. In this case, gravity is Lorentz violating in the UV, so that the theory is potentially renormalizable and flows to Lorentz-invariant general relativity at low energies (so that it may be relevant for collider physics).
Finally, one last possibility that could justify a positive
In this study, we take an agnostic approach, leaving all these possibilities open, and, as a first step, focus on the phenomenological feasibility of probing positivity violation effects. If
● A convex cone (or cone) is a subset of a vector space that is closed under additions and positive scalar multiplications. A salient cone is a cone that contains no straight line. Thus, if
● An extremal ray of a convex cone
● The convex hull of a given set
● The conical hull of a given set
● The Krein-Milman theorem [65] states that a salient cone
We now begin with the second-order s-derivative of the forward elastic scattering
$ M^{ijkl} = \int_{(\epsilon\Lambda)^2}^{\infty} {\rm{d}}\mu {\sum\limits_{Z\;{\rm{ in }}\;{\bf{r}}}}' \frac{|\left\langle {{{Z}|{\cal M}|{\bf{r}}} |^2} \right\rangle } {\pi\left( \mu-\frac{1}{2}M^2 \right)^3} P_ {\bf r}^{i(j|k|l)}. $ ![]() | (26) |
The l.h.s. of Eq. (26) can be expanded in terms of the dimension-8 Wilson coefficients
$ \vec C^{(8)}\in{\rm{cone}}\left( \left\{ \vec c^{\ (8)}_{ {\bf r}} \right\} \right) \equiv {\cal{C}}\ . $ ![]() | (27) |
As a consequence of the Krein-Milman theorem, this cone
This new physics inference or exclusion would not be possible at the dimension-6 level due to the absence of any positive nature. For example, even if
For illustration purposes, a weakly coupled UV theory whose EFT manifestation is generated by integrating out some heavy states at the tree level is considered. Nevertheless, the conclusions obtained in the following are valid for loop-level and non-perturbative cases as well, as the positive nature of the dimension-8 parameter space originates from the dispersion relation of Eq. (26) that holds in general.
We thus extend the SM in the UV by several generations of the new states shown in Table 1, each of them being identified by a different set of quantum numbers specified by an irrep
Scalar | Vector | |||
![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() | ![]() ![]() |
Table1.New physics degrees of freedom in our UV setup aiming at illustrating the strength of the positivity bounds in inferring or excluding the existence of new states. The quantum numbers refer to
$ \begin{aligned}[b] {\cal L}_{\rm{int}} =& g_{Di} \bar L e D_i + g_{M_Li} \bar{L}^c \epsilon L M_{Li} + g_{M_Ri} \bar{e}^c e M_{Ri}\\ & +g_{Vi} \Big(\bar{L}\gamma^\mu L + \kappa_i \bar{e}\gamma^\mu e \Big) V_{i\mu} + g_{V'i} (\bar{e}^c\gamma^\mu L) V'^\dagger_i\\ &+ {\rm h.c.}, \end{aligned} $ ![]() | (28) |
By integrating these particles out, the resulting dimension-8 operator coefficients are obtained as follows, considering a specific particle species X at a time:
$ \vec{C}_{X}^{\,(8)}\equiv \sum\limits_i \vec{C}_{Xi}^{\, (8)} = \sum\limits_i w_{Xi} \, \vec c^{\ (8)}_X , $ ![]() | (29) |
$ w_{Xi} = \frac{g_{Xi}^2}{M_{Xi}^{4}}\geqslant 0\ , $ ![]() | (30) |
$ \begin{aligned}[b] \vec c^{\ (8)}_D = &\ (0,0,1,0),\\ \vec c^{\ (8)}_{M_L} = &\ (0,0,0,-1),\\ \vec c^{\ (8)}_{M_R} = &\ (-1,0,0,0),\\ \vec c^{\ (8)}_{V'} = &\ (0,0,-1,2),\\ \vec c^{\ (8)}_{V(\kappa)} = &\ (-\kappa^2/2,-\kappa,0,-1/2). \end{aligned} $ ![]() | (31) |
$ \vec C^{\,(8)} = \sum\limits_X w_X \vec c^{\ (8)}_X , $ ![]() | (32) |
$ \vec C^{(8)}\in {\cal{C}}_1\equiv {\rm{cone}}\left( \left\{ \vec c^{\ (8)}_X \right\} \right) , $ ![]() | (33) |
The set of vectors
In Fig. 2, we present three-dimensional cross sections for both the

An important observation is that, for tree level UV-completions, all
Let us now assume that the Wilson coefficients can all be determined experimentally, with the measurement being denoted
$ \vec C_{\exp} = \sum\limits_X w_X \vec c^{\ (8)}_X . $ ![]() | (34) |
We start by considering a measurement of
$ {w_X} = \left\{ {\begin{array}{*{20}{l}}{\lambda \quad }&{{\rm{if}}\;X = X'\;,}\\0&{{\rm{otherwise}}\;.}\end{array}} \right. $ ![]() | (35) |
In contrast, the above inference is not possible when one truncates the SMEFT expansion at the dimension-6 level, as there would always be an infinite number of positive solutions (with
More generally, if
$\vec C(\lambda)\equiv \vec C_{\exp}-\lambda \vec c_{X'} = \sum\limits_{X\neq X'} w_X\vec c_X+(w_{X'}-\lambda)\vec c_{X'} . $ ![]() | (36) |
Setting an upper limit on
$ 0\leqslant \frac{g_{Xi}^2}{M_{Xi}^4}\leqslant \sum\limits_i\frac{g_{Xi}^2}{M_{Xi}^4} = w_{X} , $ ![]() | (37) |
A similar reasoning can be achieved beyond the tree level by replacing the
In summary, we have shown so far that, in contrast to the dimension-6 case, a measurement of the dimension-8 Wilson coefficients would allow us to rule out or at least place a lower bound on the mass scale of each individual particle of a given type X without any model assumption. If a deviation from the SM is observed, these universal bounds narrow down the possible range of UV-complete BSM models that should be considered. On the contrary, if no deviation is observed, then model-independent exclusion limits on the BSM states can be set, at least up to certain scales depending on the precision of the measurement. This last point is crucial as a test of the SM. If no significant deviation from the SM is observed at future colliders, a global fit of the dimension-6 Wilson coefficients would only allow to set limits on the dimension-6 contributions without being able to further exclude the possibility that BSM exists in a way yielding the suppression of any dimension-6 effect (by virtue of cancellations or symmetry reasons). Hence, such a fit would not be sufficient to confirm the SM. In contrast, a global fit of the coefficients of operators ranging up to dimension-8 would allow for not only the extraction of limits on the coefficients, but, more importantly, also the exclusion of the existence of BSM states, thus confirming the SM.
The illustration presented in this section is based on the assumption of weakly-coupled UV completions. However, the conclusions hold in general, as positivity implies that any UV completion of the SM must lead to some non-vanishing dimension-8 effects. This should further motivate the study of dimension-8 operators through precision physics in the future.
On different grounds, we indicate that it is also possible to set some lower limits on a particular weight
$ {\cal H}_{X'} = {\rm{cone}}\left( \left\{ \vec c_{X\neq X'}, -\vec c_{X'}\right\}\right). $ ![]() | (38) |
We utilize FeynRules [34] to generate a UFO model [35] including all the operators introduced in Section II so that we could simulate
Given that
We have considered several future lepton collider pro-jects, mostly following the setups presented in Ref. [70]. However, we have omitted any operation run at the Z-pole, as the cross section is dominated by the Z-resonance contribution instead of any potential four-fermion operator effect. In addition, for the ILC case, we focus on a possible upgrade at a center-of-mass energy of 1 TeV [31]. We refer to Table 2 for details on the center-of-mass energies, luminosities, and beam polarization options of all collider configurations studied in this paper.
Scenario | Beam polarization ![]() ![]() | Runs (luminosity @ energy), [ab?1] @ [GeV] | |||
1 | 2 | 3 | 4 | ||
CEPC | None | 2.6@161 | 5.6@240 | ||
FCC-ee | None | 10@161 | 5@240 | 0.2@350 | 1.5@365 |
ILC-500 | (?80%, 30%) | 0.9@250 | 0.135@350 | 1.6@500 | |
(80%, ?30%) | 0.9@250 | 0.045@350 | 1.6@500 | ||
ILC-1000 | (?80%, 30%) | 0.9@250 | 0.135@350 | 1.6@500 | 1.25@1000 |
(80%, ?30%) | 0.9@250 | 0.045@350 | 1.6@500 | 1.25@1000 | |
CLIC | (?80%, 0%) | 0.5@380 | 2@1500 | 4@3000 | |
(80%, 0%) | 0.5@380 | 0.5@1500 | 1@3000 |
Table2.Different future collider operation runs considered in this study, presented together with the associated center-of-mass energy, expected luminosity, and beam polarization setup (if relevant).
2
A.Future lepton collider sensitivity to four-electron operators
The $ \vec C = (C_{ee},C_{el},C_{ll},C_1,C_2,C_3,C_4)\ , $ ![]() | (39) |
$ \chi^2\left(\vec C,\vec C_0\right)\ , $ ![]() | (40) |
In the case where the would-be observations would agree with the SM, limits on the four-electron Wilson coefficients can be set by enforcing
$\begin{aligned}[b]{{\rm{dimension - 6:}}\quad }&{{\Lambda _c} \equiv \dfrac{\Lambda }{{\sqrt {\dfrac{{{C_{{\rm{max}}}} - {C_{{\rm{min}}}}}}{2}} }}\;,}\\{{\rm{dimension - 8:}}\quad }&{{\Lambda _c} \equiv \dfrac{\Lambda }{{\sqrt[4]{{\dfrac{{{C_{{\rm{max}}}} - {C_{{\rm{min}}}}}}{2}}}}}\;,}\end{aligned}$ ![]() | (41) |
These scales are depicted in Fig. 3. For each collider and each specific coefficient (i.e., in each column), the lighter color represents the individual limit obtained when all other coefficients are enforced to vanish, whereas the darker color represents the marginalized limit obtained where all other coefficients are left floating. For an easy comparison of the strengths of the various machines, we show through the darkest color the largest collider center-of-mass energy reachable for each collider.

In the case of the dimension-6 operator coefficients, we observe that all future lepton colliders are sensitive to very large new physics scales (given in terms of
For the dimension-8 operators, the individual sensitivities range from
For all considered circular colliders, the marginalized limits for the operators
As our fit is at the quadratic level for the dimension-6 operator coefficients,
Finally, to estimate the error due to the SMEFT truncation at
2
B.Testing positivity at future lepton colliders
We now assume that some would-be observation at future colliders inWe start from the fact that, for any given
$ \Delta^{-1}\in \big[\Delta^{-1}_{\rm{low}},\Delta^{-1}_{\rm{high}}\big]\ , $ ![]() | (42) |
$ \begin{aligned}[b] \Delta^{-1}_{\rm{low}} = &\ \min\limits_{\chi^2({ \vec C,\vec C_0})\leqslant\chi^2_c} \left(\frac{\delta(\vec C)^\frac{1}{4}}{\Lambda}\right),\\ \Delta^{-1}_{\rm{high}} = &\ \max\limits_{\chi^2({ \vec C,\vec C_0})\leqslant\chi^2_c} \left(\frac{\delta(\vec C)^\frac{1}{4}}{\Lambda}\right). \end{aligned} $ ![]() | (43) |
We consider four benchmarks that differ by the
$ \begin{aligned}[b] B_1:\ & \vec{C}_0^{(8)} = (0,0,3,1.2),\\ B_2:\ & \vec{C}_0^{(8)} = (0,0.3,0.2,0),\\ B_3:\ & \vec{C}_0^{(8)} = (0,0.015,0.015,0),\\ B_4:\ & \vec{C}_0^{(8)} = (0,0,0.0006,0.00015). \end{aligned} $ ![]() | (44) |
![]() ![]() | ![]() ![]() | |||||
CEPC | FCC-ee | ILC-500 | ILC-1000 | CLIC | ||
![]() ![]() | 1.48 | 0 | 0.86 | 1.45 | 1.47 | 1.48 |
![]() ![]() | 0.74 | 0 | 0 | 0.66 | 0.73 | 0.74 |
![]() ![]() | 0.35 | 0 | 0 | 0 | 0.29 | 0.35 |
![]() ![]() | 0.16 | 0 | 0 | 0 | 0 | 0.10 |
Table3.Amount of positivity violation
The four points have been chosen to illustrate an increasing sensitivity to positivity violations at the five collider scenarios under consideration. The
To obtain a more intuitive picture, we present a slice of the dimension-8 Wilson coefficient parameter space for the two benchmarks

We observed differences between the solid and dashed contours, which indicates that there are correlations between the impacts of the dimension-6 and dimension-8 operators. Nevertheless, even after marginalizing the dimension-6 coefficients, a sensitivity to the dimension-8 operators remains, as illustrated in the case of the ILC-1000 collider (for the
It is evident that a large
A quantitative and accurate answer is difficult to provide, due to the quartic nature of our

Two particularly important features can be extracted. First, the smallest
Those possible evaluations of
Therefore, we conclude that future
2
C.Inferring/excluding states in the UV
In Section IV, we have argued that the positive nature of the dimension-8 parameter space can be used to infer or exclude the possible existence of new physics states in the UV, independent of the nature of the new physics model. In this section, this argument is demonstrated with realistic examples.We first consider an extension of the SM where the new physics sector of the theory solely includes a D-type scalar (see Table 1). For an illustrative benchmark scenario, we fix its mass to 2 TeV, its coupling
$\begin{aligned}[b] & \vec C_0^{(6)} = (0,-0.08, 0)\ , \\ &\vec C_0^{(8)} = (0,0,0.04,0)\ . \end{aligned}$ ![]() | (45) |
$ \begin{array}{*{20}{l}}{C_{ee}} = 0 \pm 0.0024,\qquad \;{C_{el}} = - 0.08 \pm 0.0035,\\{C_{ll}} = 0 \pm 0.0023,\\{C_1} = 0 \pm 0.0074,\qquad \;\;{C_2} = 0 \pm 0.0077,\\{C_3} = 0.04 \pm 0.020,\qquad {C_4} = 0 \pm 0.0071.\end{array}$ ![]() | (46) |
$ M_D/g_D\in [2.45,2.56]\ {\rm{TeV}} . $ ![]() | (47) |
$\frac{g_D^2}{2M_D^2}-\frac{g_{V'}^2}{M_{V'}^2} = 0.08\pm0.0035\ {\rm{TeV}}^{-2}. $ ![]() | (48) |
This shows that conclusions can only be drawn under very specific BSM assumptions, as we do not know a priori the exact particle content of the theory. Thus, this kind of interpretation is only practical for top-down studies of specific UV models, as for example the one carried out in Ref. [14].
One might naively believe that this lack of information is due to the fact that, at the dimension-6 level, we only measure three coefficients, whereas the number of possible UV states is infinite (as there can be several V particles with different
For example, for a particle species
$ \lambda_{\max} \equiv \max\limits_\lambda \Big[ \vec C_{\exp}^{(8)}- \lambda \vec c_{X'}^{\ (8)}\in {\cal{C}}\Big] , $ ![]() | (49) |
This has a simple interpretation at the tree level. If we remove from
However, the above statement assumes that
$ \lambda_{\max} \equiv \max\limits_\lambda \Big[ \vec C^{(8)}- \lambda \vec c_{X'}^{\ (8)}\in {\cal{C}};\ \chi^2\big(\vec C, \vec C_0\big)\leqslant \chi^2_c \Big] \ , $ ![]() | (50) |
Let us now apply this general procedure to the benchmark scenario defined in Eq. (45). As the coefficients
$ \frac{M_{V}}{\sqrt{g_{V}}}\geqslant 3.7\ {\rm{TeV}} .$ ![]() | (51) |
For an intuitive understanding of this universal bound, we investigate what happens when

The black, brown, and red dots lie on a straight line, which stems from the fact that
The same class of universal bounds can be set on all types of potential states extending the SM in the UV. The Wilson coefficients relevant for our benchmark assumption
X | ![]() ![]() | ![]() ![]() | ![]() ![]() |
![]() ![]() | ![]() ![]() | 0.0067 | ![]() ![]() |
![]() ![]() | ![]() ![]() | 0.0069 | ![]() ![]() |
V (with ![]() ![]() | ![]() ![]() | 0.0055 | ![]() ![]() |
V (with ![]() ![]() | ![]() ![]() | 0.0116 | ![]() ![]() |
![]() ![]() | ![]() ![]() | 0.0109 | ![]() ![]() |
Table4.Universal bounds imposed with positivity on the particle species X, and for a BSM setup in which the SM is extended by a D-type scalar. The bounds, expressed in terms of the
In contrast, for D-type scalars, we obtain a much larger
$ \lambda_{\max}\approx 0.049\ . $ ![]() | (52) |
$ \lambda_{\rm{min}}\approx 0.011\ . $ ![]() | (53) |
Perhaps a more practically useful example in the light of the current LHC results is the SM case itself, where
$ \frac{M_{D}}{g_{D}}\geqslant 16\;{\rm{TeV}}, \quad \frac{M_{M_L}}{g_{M_L}}\geqslant 17\;{\rm{TeV}} , \ \ldots $ ![]() | (54) |
Dimension-8 positivity instead provides a chance to exclude the potential existence of any class of UV states universally in a bottom-up way, hence without any specific model assumptions. Focusing on the ILC-1000 collider, we extract bounds by using the same approach as in the previous example and present the results in Table 5, together with the model-dependent one-particle dimension-6 bounds. As expected, the universal bounds extracted from the positive nature of the dimension-8 coefficient parameter space are weaker than the one-particle dimension-6 bounds. However, they are still much higher than the collider energy of 1 TeV. Assuming that no deviations from the SM would be observed at the ILC-1000, these bounds will allow for the exclusion of any UV model featuring all types of states up to certain scales. In contrast to the dimension-6 case, it will not be possible to remove those constraints by adding or arranging the properties of other states, unless the latter is done in a way that violates the positivity bounds.
X | ![]() ![]() | ![]() ![]() | ![]() ![]() |
D | ![]() ![]() | 0.0076 | ![]() ![]() |
![]() ![]() | ![]() ![]() | 0.0053 | ![]() ![]() |
![]() ![]() | ![]() ![]() | 0.0054 | ![]() ![]() |
![]() ![]() | ![]() ![]() | 0.0056 | ![]() ![]() |
V (with ![]() ![]() | ![]() ![]() | 0.0041 | ![]() ![]() |
V (with ![]() ![]() | ![]() ![]() | 0.0041 | ![]() ![]() |
Table5.Bounds imposed on the existence of a particle species X, assuming the absence of BSM physics. The bounds are provided as limits on model-dependent one-particle extensions of the SM extracted from a fit of the dimension-6 coefficients (second column), as well as universal and model-independent bounds on a given particle species X derived from positivity. The latter are expressed in terms of the
In summary, studying dimension-8 operators in the context of future high-precision machines will pave the way for universal exclusions of entire classes of BSM theories by forbidding the existence of specific types of new states in the UV. Moreover, null BSM search results will eventually allow for the confirmation of the SM at scales much higher than the collider energies, providing the dimension-8 operators a special role in the precision test of the SM.
The first is the possibility of probing positivity violations, which, if present, will revolutionize our understanding of the fundamental pillars of QFT or the S-matrix theory. Practically, it would provide very important information for model building. We have proposed a model-independent quantification of the amount of violation and discussed the implications of positivity violations in terms of breaking the axiomatic properties of QFT and paving the way for non-standard UV completions of the SM. We have observed that positivity violation at scales of order 1–10 TeV can potentially be probed by all future lepton collider projects currently discussed within the high-energy physics community. This includes the CEPC, FCC-ee, ILC (with a possible energy upgrade), and CLIC colliders. At each collider, positivity violation effects can be probed up to scales of a factor of a few higher than the highest expected collider center-of-mass energy, regardless of the presence of any four-electron dimension-6 operators. This suggests an important and novel avenue for testing the fundamental principles of QFT at future lepton colliders. If their violation is observed, a more tailored description for the violated effect may be designed, and results could be improved through further model-specific studies.
The second aspect of our study is that, if the fundamental principles of QFT hold in the UV, the positive nature of the dimension-8 coefficient space allows us to infer the existence of new states at the UV scales directly and characterize their quantum numbers from measurements. Conversely, it allows for the exclusion of certain particles up to scales depending on the future measurement precision. This originates from several concepts in convex geometry and can be achieved without any BSM model assumption, in contrast to the conventional SMEFT interpretation truncated at the dimension-6 level that always requires a specific model assumption. Therefore, while a dimension-6-level approach is useful mostly for a top-down investigation of any given model, the dimension-8-level one is as important, as it provides the possibility of setting model-independent bounds on certain types of particles in the UV, or in other words, solving the inverse problem. We have demonstrated this point with realistic examples, using projected measurements at the ILC with an energy upgrade at 1 TeV. In particular, if no deviation from the SM is observed, we have explicitly shown how the existence of any type of heavy particles up to scales much higher than ILC energies can be excluded, regardless of the UV model setup, thanks to the requirement that any UV completion of the SM has to satisfy positivity. This underlines a major difference between dimension-6 and dimension-8 operators (or the s and
Consequently, it is crucial to plan a comprehensive study of the dimension-8 operator effects, not only at the theoretical level but also at the experimental level.
Note added: After this paper appeared on the arXiv and was sent to Chin.Phys.C, Ref. [72] appeared on the arXiv. Its authors studied “explicit positivity bounds on dimension-six fermionic operators in the SMEFT.” Their results contrast with our observation in Section IV, where by explicitly integrating out all particle species in Table 1, we observed that any measured dimension-6 coefficient values could yield an infinite number of UV completions. Therefore, no bound exists. This difference from the results of Ref. [72] arises from neglecting the existence of new vector bosons in a UV completion.
$\tag{A1} f_{1,2}\equiv \epsilon_{1,2}^i F^i\ \ {\rm{with}}\ \ F^i\equiv \left( e_R,e_L,\nu_L,\bar e_R,\bar e_L,\bar \nu_L \right)\ , $ ![]() |
First, we show that mixings between fields with different chiralities do not provide any new bounds. To this end, we consider a forward scattering
$\tag{A2} f_{1,2}\equiv a_{1,2}^i f_L^i + b_{1,2}^i f_R^i\ . $ ![]() |
$\tag{A3} \begin{aligned}[b] M(f_1f_2\to f_1f_2) = & a^i_1a^{k\dagger}_1 a^j_2a^{l\dagger}_2 M(f_L^if_L^j \!\to\! f_L^kf_L^l) \!+\! b^i_1b^{k\dagger}_1 b^j_2b^{l\dagger}_2 M(f_R^if_R^j \!\to\! f_R^kf_R^l)\\ & + a^i_1a^{k\dagger}_1 b^j_2b^{l\dagger}_2 M(f_L^if_R^j \!\to\! f_L^kf_R^l) \!+\! b^i_1b^{k\dagger}_1 a^j_2a^{l\dagger}_2 M(f_R^if_L^j \!\to\! f_R^kf_L^l)\\ & + a^i_1b^{k\dagger}_1 a^j_2b^{l\dagger}_2 M(f_L^if_L^j \!\to\! f_R^kf_R^l) \!+\! b^i_1a^{k\dagger}_1 b^j_2a^{l\dagger}_2 M(f_R^if_R^j \!\to\! f_L^kf_L^l)\\ & + a^i_1b^{k\dagger}_1 b^j_2a^{l\dagger}_2 M(f_L^if_R^j \!\to\! f_R^kf_L^l) \!+\! b^i_1a^{k\dagger}_1 a^j_2b^{l\dagger}_2 M(f_R^if_L^j \!\to\! f_L^kf_R^l), \end{aligned} $ ![]() |
$\tag{A4} M(f_{1,R}f_{2,R} \to f_{1,R}f_{2,R}) , $ ![]() |
$\tag{A5} M(f_{1,L}f_{2,L} \to f_{1,L}f_{2,L}) , $ ![]() |
$\tag{A6} M(f_{1,R}f_{2,L} \to f_{1,R}f_{2,L}) , $ ![]() |
$\tag{A7} M(f_{1,L}f_{2,R} \to f_{1,L}f_{2,R}) , $ ![]() |
To investigate the amplitudes in Eqs. (A4)–(A7), we utilize the crossing symmetry that shows that Eqs. (A4) and (A5), as well as Eqs. (A6) and (A7), are equal. Furthermore, Eqs. (A4) and (A6) are related by an
$\tag{A8} M(f_{1,R}f_{2,R} \to f_{1,R}f_{2,R}) = C_{s}s^2 + C_{t} t^2 + C_{u} u^2, $ ![]() |
$\tag{A9} M(f_{1,R}f_{2,L} \to f_{1,R}f_{2,L}) = C_{s}u^2 + C_{t} t^2 + C_{u} s^2, $ ![]() |
Hence, we consider the
$ \tag{A10} f_{n,R} = a_n e_R+b_n \bar e_L+c_n \bar\nu_L \quad{\rm{for}}\quad n = 1,2 , $ ![]() |
$\tag{A11} \begin{aligned}[b] & \frac{1}{2}\frac{ {\rm{d}}^2}{ {\rm{d}} s^2}M(f_{1,R}f_{2,R} \to f_{1,R}f_{2,R}) = \\ &\qquad Af_A+Bf_B+Cf_C+Df_D+Ef_E , \end{aligned} $ ![]() |
$\tag{A12} \begin{aligned}[b] & f_A\! = \!-4C_1,\\& f_B\! = \!2C_2+C_3,\\&f_C\! = \!-4(C_4\!+\!C_5),\\ & f_D\! = \!-8C_5,\\& f_E\! = \!C_3, \end{aligned} $ ![]() |
$\tag{A13} \begin{aligned}[b] & A = |a_1a_2^\dagger|^2, \\ & B = 2 \Re (a_1a_2^\dagger)(\vec x^\dagger\!\cdot\!\vec y),\\ & C = \left| \vec x \cdot \vec y^\dagger \right|^2, \\ & D = \left| \vec x \times \vec y \right|^2, \\& E = |a_1|^2\left|\vec y\right|^2 + |a_2|^2\left|\vec x\right|^2\ . \end{aligned} $ ![]() |
$ \tag{A14}\begin{aligned}[b] & \vec x\equiv(b_1,c_1), \ \ \vec y\equiv(b_2,c_2),\\ & \vec u\cdot \vec v \equiv u_1v_1+u_2v_2,\ \ \vec u\times \vec v \equiv u_1v_2-u_2v_1\ . \end{aligned} $ ![]() |
$\tag{A15} A>0,\quad C>0,\quad D>0 . $ ![]() |
$\tag{A16} |B| \! = \! 2 \left|\Re (a_1a_2^\dagger)\left( \vec x^\dagger\!\cdot\!\vec y \right)\right| \!<\! 2\left|a_1a_2^\dagger\right| \left|\vec x^\dagger\!\cdot\!\vec y \right| \! = \! 2\sqrt{AC}, $ ![]() |
$\tag{A17} E\! = \!|a_1|^2\left|\vec y\right|^2 \!+\! |a_2|^2\left|\vec x\right|^2 \!>\! 2|a_1|\left|\vec y\right| |a_2|\left|\vec x\right| \! = \! 2\sqrt{A(C+D)}, $ ![]() |
We must then show that, for each set of parameters A, B, C, D, and E that satisfy the inequalities of Eqs. (A15)–(A17), there exists a corresponding set of coefficients
$\tag{A18} \begin{aligned}[b] a_1 =& rA^{1/4}{\rm e}^{{\rm i}\phi},\\ a_2 =& r^{-1}A^{1/4},\\ (b_1,c_1) =& \sqrt[4]{C+D}(1,0),\\ (b_2,c_2) =& \sqrt[4]{C+D}\left( \frac{C^{1/2}}{(C+D)^{1/2}},\frac{D^{1/2}}{(C+D)^{1/2}}\right) . \end{aligned} $ ![]() |
$ \tag{A19} f\equiv Af_A+Bf_B+Cf_C+Df_D+Ef_E >0 , $ ![]() |
$\tag{A20} \begin{aligned}[b]& A>0,\quad C>0, \quad D>0,\\ &-2\sqrt{AC}<B<2\sqrt{AC}, \\& 2\sqrt{A(C+D)}<E. \end{aligned} $ ![]() |
$\tag{A21} f_D>0,\quad f_E>0 . $ ![]() |
$\tag{A22} \begin{aligned}[b]{f > {f_{BE}} \equiv} & {A{f_A} + C{f_C} - 2|{f_B}|\sqrt {AC} + D{f_D}}\\&{ + 2\sqrt {A(C + D)} {f_E}.}\end{aligned} $ ![]() |
$ \tag{A23} f>f_{BED} = Af_A + 2(f_E-|f_B|)\sqrt{AC}+Cf_C . $ ![]() |
$\tag{A24} f_A>0,\; \; \; \; f_C>0, $ ![]() |
$ \tag{A25} f_E>|f_B|\quad{\rm{or}}\quad f_Af_C>(|f_B|-f_E)^2. $ ![]() |
$ \tag{A26} \sqrt{f_Af_C}>f_B-f_E\quad{\rm{and}}\quad \sqrt{f_Af_C}>-f_B-f_E . $ ![]() |
$\begin{aligned} &M(e_Re_R\to e_Re_R) = -4 s^2 C_1 \\ &M(e_Re_L\to e_Re_L) = C_3 (s+t)^2+2 t C_2 (s+t) \\ &M(e_Re_L\to e_Le_R) = C_3 t^2+2 (s+t) C_2 t \\ &M(e_R\nu _L\to e_R\nu _L) = C_3 (s+t)^2+2 t C_2 (s+t) \\ \end{aligned}$ ![]() |
$\begin{aligned}&M(e_R\nu _L\to \nu _Le_R) = C_3 t^2+2 (s+t) C_2 t \\ &M(e_R\bar{e}_R\to e_R\bar{e}_R) = -4 (s+t)^2 C_1 \\ &M(e_R\bar{e}_R\to e_L\bar{e}_L) = t^2 C_3-2 s t C_2 \\ &M(e_R\bar{e}_R\to \nu _L\bar{\nu }_L) = t^2 C_3-2 s t C_2 \\ &M(e_R\bar{e}_R\to \bar{e}_Re_R) = -4 t^2 C_1 \\ &M(e_R\bar{e}_R\to \bar{e}_Le_L) = C_3 (s+t)^2+2 s C_2 (s+t) \\ &M(e_R\bar{e}_R\to \bar{\nu }_L\nu _L) = C_3 (s+t)^2+2 s C_2 (s+t) \\ &M(e_R\bar{e}_L\to e_R\bar{e}_L) = s^2 C_3-2 s t C_2 \\ &M(e_R\bar{e}_L\to \bar{e}_Le_R) = C_3 s^2+2 (s+t) C_2 s \\ &M(e_R\bar{\nu }_L\to e_R\bar{\nu }_L) = s^2 C_3-2 s t C_2 \\ &M(e_R\bar{\nu }_L\to \bar{\nu }_Le_R) = C_3 s^2+2 (s+t) C_2 s \\ &M(e_Le_R\to e_Re_L) = C_3 t^2+2 (s+t) C_2 t \\ &M(e_Le_R\to e_Le_R) = C_3 (s+t)^2+2 t C_2 (s+t) \\ &M(e_Le_L\to e_Le_L) = -4 C_4 s^2-4 C_5 s^2 \\ &M(e_L\nu _L\to e_L\nu _L) = 4 s t C_4-4 s (2 s+3 t) C_5 \\ &M(e_L\nu _L\to \nu _Le_L) = 4 s (s+3 t) C_5-4 s (s+t) C_4 \\ &M(e_L\bar{e}_R\to e_L\bar{e}_R) = s^2 C_3-2 s t C_2 \\ &M(e_L\bar{e}_R\to \bar{e}_Re_L) = C_3 s^2+2 (s+t) C_2 s \\ &M(e_L\bar{e}_L\to e_R\bar{e}_R) = t^2 C_3-2 s t C_2 \\ &M(e_L\bar{e}_L\to e_L\bar{e}_L) = -4 C_4 (s+t)^2-4 C_5 (s+t)^2 \\ &M(e_L\bar{e}_L\to \nu _L\bar{\nu }_L) = 4 (s-2 t) (s+t) C_5-4 s (s+t) C_4 \\ &M(e_L\bar{e}_L\to \bar{e}_Re_R) = C_3 (s+t)^2+2 s C_2 (s+t) \\ &M(e_L\bar{e}_L\to \bar{e}_Le_L) = -4 C_4 t^2-4 C_5 t^2 \\ &M(e_L\bar{e}_L\to \bar{\nu }_L\nu _L) = 4 s t C_4-4 t (3 s+2 t) C_5 \\ &M(e_L\bar{\nu }_L\to e_L\bar{\nu }_L) = -4 t (s+t) C_4-4 (2 s-t) (s+t) C_5 \\ &M(e_L\bar{\nu }_L\to \bar{\nu }_Le_L) = 4 t (3 s+t) C_5-4 t (s+t) C_4 \\ &M(\nu _Le_R\to e_R\nu _L) = C_3 t^2+2 (s+t) C_2 t \\ &M(\nu _Le_R\to \nu _Le_R) = C_3 (s+t)^2+2 t C_2 (s+t) \\ &M(\nu _Le_L\to e_L\nu _L) = 4 s (s+3 t) C_5-4 s (s+t) C_4 \\ &M(\nu _Le_L\to \nu _Le_L) = 4 s t C_4-4 s (2 s+3 t) C_5 \\ &M(\nu _L\nu _L\to \nu _L\nu _L) = -4 C_4 s^2-4 C_5 s^2 \\ &M(\nu _L\bar{e}_R\to \nu _L\bar{e}_R) = s^2 C_3-2 s t C_2 \\ &M(\nu _L\bar{e}_R\to \bar{e}_R\nu _L) = C_3 s^2+2 (s+t) C_2 s \\ &M(\nu _L\bar{e}_L\to \nu _L\bar{e}_L) = -4 t (s+t) C_4-4 (2 s-t) (s+t) C_5 \\ &M(\nu _L\bar{e}_L\to \bar{e}_L\nu _L) = 4 t (3 s+t) C_5-4 t (s+t) C_4 \\ &M(\nu _L\bar{\nu }_L\to e_R\bar{e}_R) = t^2 C_3-2 s t C_2 \\ &M(\nu _L\bar{\nu }_L\to e_L\bar{e}_L) = 4 (s-2 t) (s+t) C_5-4 s (s+t) C_4 \\ &M(\nu _L\bar{\nu }_L\to \nu _L\bar{\nu }_L) = -4 C_4 (s+t)^2-4 C_5 (s+t)^2 \\ &M(\nu _L\bar{\nu }_L\to \bar{e}_Re_R) = C_3 (s+t)^2+2 s C_2 (s+t) \\ &M(\nu _L\bar{\nu }_L\to \bar{e}_Le_L) = 4 s t C_4-4 t (3 s+2 t) C_5 \\ &M(\nu _L\bar{\nu }_L\to \bar{\nu }_L\nu _L) = -4 C_4 t^2-4 C_5 t^2 \\ \end{aligned}$ ![]() |
$ \begin{aligned} &M(\bar{e}_Re_R\to e_R\bar{e}_R) = -4 t^2 C_1 \\ &M(\bar{e}_Re_R\to e_L\bar{e}_L) = C_3 (s+t)^2+2 s C_2 (s+t) \\ &M(\bar{e}_Re_R\to \nu _L\bar{\nu }_L) = C_3 (s+t)^2+2 s C_2 (s+t) \\ &M(\bar{e}_Re_R\to \bar{e}_Re_R) = -4 (s+t)^2 C_1 \\ &M(\bar{e}_Re_R\to \bar{e}_Le_L) = t^2 C_3-2 s t C_2 \\ &M(\bar{e}_Re_R\to \bar{\nu }_L\nu _L) = t^2 C_3-2 s t C_2 \\ &M(\bar{e}_Re_L\to e_L\bar{e}_R) = C_3 s^2+2 (s+t) C_2 s \\ &M(\bar{e}_Re_L\to \bar{e}_Re_L) = s^2 C_3-2 s t C_2 \\ &M(\bar{e}_R\nu _L\to \nu _L\bar{e}_R) = C_3 s^2+2 (s+t) C_2 s \\ &M(\bar{e}_R\nu _L\to \bar{e}_R\nu _L) = s^2 C_3-2 s t C_2 \\ &M(\bar{e}_R\bar{e}_R\to \bar{e}_R\bar{e}_R) = -4 s^2 C_1 \\ &M(\bar{e}_R\bar{e}_L\to \bar{e}_R\bar{e}_L) = C_3 (s+t)^2+2 t C_2 (s+t) \\ &M(\bar{e}_R\bar{e}_L\to \bar{e}_L\bar{e}_R) = C_3 t^2+2 (s+t) C_2 t \\ &M(\bar{e}_R\bar{\nu }_L\to \bar{e}_R\bar{\nu }_L) = C_3 (s+t)^2+2 t C_2 (s+t) \\ &M(\bar{e}_R\bar{\nu }_L\to \bar{\nu }_L\bar{e}_R) = C_3 t^2+2 (s+t) C_2 t \\ &M(\bar{e}_Le_R\to e_R\bar{e}_L) = C_3 s^2+2 (s+t) C_2 s \\ &M(\bar{e}_Le_R\to \bar{e}_Le_R) = s^2 C_3-2 s t C_2 \\ &M(\bar{e}_Le_L\to e_R\bar{e}_R) = C_3 (s+t)^2+2 s C_2 (s+t) \\ &M(\bar{e}_Le_L\to e_L\bar{e}_L) = -4 C_4 t^2-4 C_5 t^2 \\ &M(\bar{e}_Le_L\to \nu _L\bar{\nu }_L) = 4 s t C_4-4 t (3 s+2 t) C_5 \\ &M(\bar{e}_Le_L\to \bar{e}_Re_R) = t^2 C_3-2 s t C_2 \\ &M(\bar{e}_Le_L\to \bar{e}_Le_L) = -4 C_4 (s+t)^2-4 C_5 (s+t)^2 \\ \end{aligned}$ ![]() |
$\tag{B1} \begin{aligned} &M(\bar{e}_Le_L\to \bar{\nu }_L\nu _L) = 4 (s-2 t) (s+t) C_5-4 s (s+t) C_4 \\ &M(\bar{e}_L\nu _L\to \nu _L\bar{e}_L) = 4 t (3 s+t) C_5-4 t (s+t) C_4 \\ &M(\bar{e}_L\nu _L\to \bar{e}_L\nu _L) = -4 t (s+t) C_4-4 (2 s-t) (s+t) C_5 \\ &M(\bar{e}_L\bar{e}_R\to \bar{e}_R\bar{e}_L) = C_3 t^2+2 (s+t) C_2 t \\ &M(\bar{e}_L\bar{e}_R\to \bar{e}_L\bar{e}_R) = C_3 (s+t)^2+2 t C_2 (s+t) \\ &M(\bar{e}_L\bar{e}_L\to \bar{e}_L\bar{e}_L) = -4 C_4 s^2-4 C_5 s^2 \\ &M(\bar{e}_L\bar{\nu }_L\to \bar{e}_L\bar{\nu }_L) = 4 s t C_4-4 s (2 s+3 t) C_5 \\ &M(\bar{e}_L\bar{\nu }_L\to \bar{\nu }_L\bar{e}_L) = 4 s (s+3 t) C_5-4 s (s+t) C_4 \\ &M(\bar{\nu }_Le_R\to e_R\bar{\nu }_L) = C_3 s^2+2 (s+t) C_2 s \\ &M(\bar{\nu }_Le_R\to \bar{\nu }_Le_R) = s^2 C_3-2 s t C_2 \\ &M(\bar{\nu }_Le_L\to e_L\bar{\nu }_L) = 4 t (3 s+t) C_5-4 t (s+t) C_4 \\ &M(\bar{\nu }_Le_L\to \bar{\nu }_Le_L) = -4 t (s+t) C_4-4 (2 s-t) (s+t) C_5 \\ &M(\bar{\nu }_L\nu _L\to e_R\bar{e}_R) = C_3 (s+t)^2+2 s C_2 (s+t) \\ &M(\bar{\nu }_L\nu _L\to e_L\bar{e}_L) = 4 s t C_4-4 t (3 s+2 t) C_5 \\ &M(\bar{\nu }_L\nu _L\to \nu _L\bar{\nu }_L) = -4 C_4 t^2-4 C_5 t^2 \\ &M(\bar{\nu }_L\nu _L\to \bar{e}_Re_R) = t^2 C_3-2 s t C_2 \\ &M(\bar{\nu }_L\nu _L\to \bar{e}_Le_L) = 4 (s-2 t) (s+t) C_5-4 s (s+t) C_4 \\ &M(\bar{\nu }_L\nu _L\to \bar{\nu }_L\nu _L) = -4 C_4 (s+t)^2-4 C_5 (s+t)^2 \\ &M(\bar{\nu }_L\bar{e}_R\to \bar{e}_R\bar{\nu }_L) = C_3 t^2+2 (s+t) C_2 t \\ &M(\bar{\nu }_L\bar{e}_R\to \bar{\nu }_L\bar{e}_R) = C_3 (s+t)^2+2 t C_2 (s+t) \\ &M(\bar{\nu }_L\bar{e}_L\to \bar{e}_L\bar{\nu }_L) = 4 s (s+3 t) C_5-4 s (s+t) C_4 \\ &M(\bar{\nu }_L\bar{e}_L\to \bar{\nu }_L\bar{e}_L) = 4 s t C_4-4 s (2 s+3 t) C_5 \\ &M(\bar{\nu }_L\bar{\nu }_L\to \bar{\nu }_L\bar{\nu }_L) = -4 C_4 s^2-4 C_5 s^2 \end{aligned} $ ![]() |