On capped Higgs positivity cone

1.   Introduction

Effective field theories (EFTs) serve as valuable tools for describing low-energy physics without explicit knowledge of the intricate high energy theory. The effectiveness of an EFT depends heavily on precisely determining its Wilson coefficients, which often poses a significant challenge because there can be numerous of them or they might be rather difficult to measure. Recent developments highlight that the general parameter space for the Wilson coefficients is mostly inconsistent with the fundamental principles of S-matrix theory, such as causality/analyticity and unitarity, except for a small subspace defined by the positivity bounds (see, for example, Refs. [1–43] and Ref. [44] for a review).

The positivity bounds have been used to constrain the standard model effective field theory (SMEFT)^{[10, 11, 45–69]}. The SMEFT parameterizes generic new physics beyond the standard model (SM) based on the SM field content and symmetries, and has been gaining popularity in both theoretical and experimental communities in the absence of new particle discoveries at the Large Hadron Collider (LHC). The SMEFT contains numerous Wilson coefficients particularly from dimension-8^{[70, 71]} and beyond, as the SM is a theory with many field degrees of freedom.

For a theory with multiple degrees of freedom, the positivity bounds significantly reduce the extensive parameter space. For instance, in vector boson scattering (VBS), the elastic positivity bounds can confine the physical dimension-8 (dim-8) Wilson coefficient space to approximately 2% of the total space^{[45, 46, 72]}. For the 10D dimension-8 VBS subspace involving only the transverse vector bosons, generalized elastic positivity bounds reduce the viable parameter space to about 0.7% of the total^[49]. Furthermore, for the $s^2$ amplitude coefficients ($s,t,u$ being the standard Mandelstam variables), the optimal positivity bounds can be obtained by a convex geometry approach^{[10, 12, 49, 59]}. When there are sufficient symmetries in the subsector we are interested in, one can use a group-theoretical method to compute the positivity cone, and the extremal rays of the cone in this case can be very useful in reverse engineering the UV theory in the event of an observation of nonzero Wilson coefficients^{[10, 12, 59]}. Generically, with fewer symmetries, one can employ a semidefinite programming (SDP) method to compute the the $s^2$ positivity cone^[11].

The preceding $s^2$ positivity cones are obtained by using only the positivity part of the UV unitarity conditions. Ref. [73] has initiated the use of the nonpositivity parts of the UV unitarity conditions to constrain the SMEFT coefficients, focusing on the scattering involving only the complex Higgs modes. Building upon the methods introduced in Refs. [6, 7, 27], the numerical bounds of Ref. [73] are obtained by discretizing the UV scales in the fixed-t dispersion relations and using the null constraints and linear programming to extract the constraints on the Wilson coefficients. Specifically, Ref. [73] derived a set of linear conditions from (nonlinear) partial wave unitarity, which allows the numerical optimizaiton to be easily carried out with some simple ${Mathematica}$ coding.

In this paper, we will revisit the capped positivity bounds on the SMEFT Higgs sector, making use of the nonlinear unitarity conditions on the imaginary part of the UV amplitudes. We will also more carefully take into account all available symmetries of the SMEFT Higgs Lagrangian. With these improvements, significantly better upper bounds are obtained. The paper is organized as follows. In Section 2, we will first derive the scattering amplitudes and dispersion relations from the SMEFT Lagrangian pertaining to the Higgs, and then introduce the null constraints and the nonlinear UV unitarity conditions we will use in this paper. In Section 3, we briefly set up the numerical optimization scheme for computing the two-sided, optimal numerical bounds. In Section 4, we present our results on the capped Higgs positivity cone, and compare with those obtained in Ref. [73]. We will see that, carefully taking into account the SMEFT Higgs symmetries, the linear unitarity bounds actually give rise to the same positivity bounds as those from the nonlinear unitarity conditions. We conclude in Section 5. In Appendix A.1, we will show that the linear unitarity conditions of Ref. [73] can be derived from the nonlinear unitarity conditions. In Appendix A.2, we will use a biscalar theory as an example to demonstrate that generically the nonlinear unitarity conditions are stronger than the linear ones.

2.   Model and setup

In this section, we derive the amplitudes for Higgs scattering in the SMEFT and the corresponding fixed-t dispersion relations that are used to extract positivity bounds on the dim-8 Wilson coefficients. Then, we proceed to obtain the so-called null constraints by imposing $st$ crossing symmetries on these dispersion relations, and present the nonlinear unitarity conditions we will use in this paper. Combining these ingredients together, we will derive two-sided bounds for the dim-8 Higgs coefficients in the following sections.

2.1   Amplitudes and dispersion relations

In the SMEFT, the Higgs retains the same symmetry as in the SM and is a ${\mathrm{SU}}(2)$ doublet. We shall parameterize the Higgs doublet with two complex fields,

and will use $\bar{\rm i}$ to denote the antiparticle of particle i. Thanks to the ${\mathrm{SU}}(2)$ internal symmetry, a generic 2-to-2 Higgs scattering amplitude can be parameterized by the invariant tensors of the ${\mathrm{SU}}(2)$ symmetry ${\cal{M}}^{\rm i \bar j k \bar l}(s,t) = \delta_{\rm i\bar j}\delta_{\rm k\bar l} {\alpha}(s,t) + \delta_{\rm i\bar l}\delta_{\rm \bar j k} {\beta}(s,t)$. Here we choose all the particles to be all-ingoing for the amplitudes. The rest amplitudes are related to ${\cal{M}}^{\rm i \bar j k \bar l}$ by crossing. The $su$ crossing symmetry implies that ${\cal{M}}^{\rm i \bar j k \bar l}(s,t)= {\cal{M}}^{\rm i \bar l k \bar j}(u,t)$, which means that we must have ${\alpha}(u,t)= {\beta}(s,t)$. Thus, we can express the Higgs amplitude as

Suppose that below the EFT cutoff the theory is weakly coupled so that we can take the tree level approximation. Then, at low energies, we can parameterized $f(s,t)$ as

which is the tree level approximation of $f(s,t)$ in the EFT region. We will be interested in constraining the Wilson coefficients of the dimension-8 SMEFT operators that will contribute to the 2-to-2 Higgs scattering. There are three of these operators, all of which contain four derivatives and are parameterized as follows:

where $D^\mu$ is the gauge covariant derivatives. Matching the $C_i$ coefficients with the amplitude coefficients in $f_{{\rm{EFT}}}(s,t)$, we find that

To make use of the null constraints and partial wave unitarity, we need to derive the dispersion relations where the UV amplitudes are expanded with partial waves. To that end, we shall perform the following partial wave expansion:

where $P_\ell(x)$ is the Legendre polynomial and we have defined

Note that $a_\ell^{s}$, $a_\ell^{t}$, and $a_\ell^{u}$ are related, as they are all expanded from the same function but with different arguments:

These relations will be taken into account by supplying the dispersion relations with null constraints. The expansions for the other amplitudes can be related to the above three via $a_\ell^{i j k l}(s)=(-1)^\ell a_\ell^{i j l k}(s)$. We adopt the convention that the indices i, j, k, l refer to particles, indices ${\rm \bar i, \bar j,\bar k, \bar l}$ refer to anti-particles, and indices $i,j,k,l$ refer to particles or anti-particles. More explicitly, we have

With the above ingredients as well as the Froissart–Martin bound^{[74, 75]}, a relative simple use of the residue theorem on the complex s plane for fixed t, plus some straightforward algebra, allows us to derive the twice subtracted dispersion relations for the amplitudes (for example, Ref. [2]):

where we have defined

and “EFT poles” denotes the poles of the amplitudes ${\cal{M}}^{i j k l}$ in the low energy EFT region, Λ is the EFT cutoff and ${\rho}^{i j k l}_\ell(\mu)={\rm{Im}}\,a^{i j k l}_\ell(\mu)$. $z_n(t)$ are some functions of t that we will not use in this paper, as we are constraining the coefficients in front of the terms $s^nt^m$ with $n\geq 2$. The fact that a fixed t dispersion relation naturally constrains the terms $s^nt^m$ with $n\geq 2$ is related to the Froissart–Martin bound. Specifically, the Froissart–Martin bound states that $\lim \limits _{s\rightarrow \infty} {\cal{M}}/s^2 = 0$ for fixed t. Therefore, a direct application of Cauchy’s integral formula on ${\cal{M}}$ leads to a dispersion relation with various diverging terms, which should cancel among themselves. The solution is to apply a twice subtraction, which directly allows us to constrain terms with $s^n t^m,\; n \ge 2$. In other words, on the right hand side of the dispersion relation, we have the $z_0(t)$ and $z_1(t) s$ term, where $z_0(t)$ and $z_1(t)$ contain some unknown information related to various contour integrals going like $\int {{\rm{d}}} \mu {\rm{Im}}\, {\cal{M}}/(\mu-s)$. Thus, the $s^0$ and $s^1$ EFT term on the left hand side can not be constrained. Of course, with crossing symmetries, coefficients in $z_n(t)$ can often be related to the $s^{n>2}t^m$ coefficients and thus also be bounded. Now, Eq. (14) is convergent on both sides of the equality, so we can taylor-expand both sides and match the coefficients in front of $s^n t^m$, which gives a set of sum rules that will be used to derive the positivity bounds. For example, let us consider the dispersion relation of amplitude ${\cal{M}}^{1 \bar{1} 2 \bar{2}}(s,t)$. Taylor-expanding both sides of the dispersion relation, we get

and matching the coefficients in front of the terms $s^n t^m$, we can get

These sum rules connect the unknown UV amplitudes with the low energy Wilson coefficients. The positivity bounds are the imprints of the UV information on the IR physics, passed down by these dispersion relations/sum rules.

2.2   Null constraints

The fixed t dispersion relations above or the sum rules extracted from them only include part of the full crossing symmetries of the amplitudes. To utilize the full crossing symmetries, we can simply impose the unrealized crossing symmetries, as extra conditions, on these fixed t dispersion relations or the sum rules. This gives rise to null constraints, which can significantly strengthen the positivity bounds, capable to bound the Wilson coefficients from the below and from the above^{[6, 7]}.

In the Higgs case, the null constraints can be obtained by equating different expressions of the same Wilson coefficient in various dispersion relations. For example, the coefficients in front of the terms $s^3$ and $s^2t$ in the dispersion relation of ${\cal{M}}^{1 \bar{1} 2 \bar{2}}(s,t)$ give rise to sum rules for $c_1$ and $c_2$, as shown in Eqs. (18) and (19). On the other hand, from the dispersion relation of ${\cal{M}}^{1 \bar{1} 2 \bar{2}}(t,s)$, the sum rule obtained from the $s^2t$ term is given by

Plugging the sum rules Eqs. (18) and (19) into the sum rule Eq. (20), we can get one null constraint:

To get independent null constraints, we only need to extract sum rules from the dispersion relations of ${\cal{M}}^{1 \bar{1} 2 \bar{2}}(s,t), \; {\cal{M}}^{1 \bar{1} 2 \bar{2}}(t,s)$, and ${\cal{M}}^{1 1 \bar{1} \bar{1}}(s,t)$. If a coefficient appears in multiple sum rules, we can obtain null constraints as illustrated above. As the order of the sum rules increases, the number of independent null constraints increases, but all of these can be easily handled by a symbolic algebra system.

2.3   Nonlinear unitarity

Ref. [73] derived a set of linearized unitarity conditions that can be used to obtain two-sided bounds on generic dim-8 Wilson coefficients. These linearized unitarity conditions are explicit, simple and, easy to use in a linear program. In fact, they can be easily implemented with simple ${Mathematica}$ coding to compute the numerical bounds. In this paper, we further use stronger, nonlinear unitarity conditions, which generally lead to stronger bounds; see Appendix A.2. For the Higgs case, however, due to the high degrees of the internal symmetries, the nonlinear unitarity conditions are actually equivalent to the linear ones, as we shall see in Section 4.

Recall that the full unitarity condition is $\boldsymbol{S} \boldsymbol{S}^\dagger=\boldsymbol{I}$, where $\boldsymbol{S}$ is the S-matrix and $\boldsymbol{I}$ is the corresponding identity matrix. If we restrict to a subspace of the space of all outgoing states, the reduced unitarity conditions can be written as $\hat{\boldsymbol{S}} \hat{\boldsymbol{S}}^\dagger \preceq \boldsymbol{I}$, where $\hat{\boldsymbol{S}}$ is the projection of ${\boldsymbol{S}}$ to the subspace. Splitting the projected S-matrix into an identity matrix plus a transfer matrix $\hat{\boldsymbol{T}}:\,\hat{\boldsymbol{S}} = \boldsymbol{I} + {\mathrm{i}} \hat{\boldsymbol{T}}$, we have $(\boldsymbol{I}-{\rm Im}\, \boldsymbol{T})^2 + ({\rm Re}\,\boldsymbol{T})^2 \preceq \boldsymbol{I}$. Since $({\rm Re}\,\boldsymbol{T})^2$ is semipositive, a weaker but simpler condition is $\boldsymbol{I}-(\boldsymbol{I}-{\rm Im}\, \boldsymbol{T})^2 \succeq 0$, which is equal to the following linear matrix inequalities

In the scattering, angular momenta are conserved, so the above inequalities also apply to each partial waves:

Note that in terms of partial wave amplitudes, these unitarity conditions are highly nonlinear. For the Higgs case we have in hand, for each partial wave, $\boldsymbol{T}_\ell$ is a $16\times16$ matrix and the partial wave amplitudes are related to it by

where the factor 2 comes from the bose symmetry.

These conditions are generically stronger than those linear unitarity conditions obtained in Ref. [73], as demonstrated in Appendix A.2 for the case of a simple biscalar theory. In Appendix A.1, we show how to rederive the linear conditions of Ref. [73] from the nonlinear conditions Eq. (23).

3.   Numerical implementation

In the last section, we have derived the sum rules which express the Wilson coefficients in terms of a sum of different UV spin contributions and each UV spin contribution is expressed as an integral over the UV energy scale. We do not know the exact values of the UV partial wave amplitudes, but they should satisfy the partial wave unitarity. Now, we shall numerically implement the nonlinear unitarity conditions Eq. (23) within SDPB. Additionally, the UV partial wave amplitudes should also satisfy the null constraints, which are also easy to implement with the SDPB package^[76]. In this section, we shall set up the numerical method to compute the optimal bounds on the Wilson coefficients.

Our strategy is to discretize the UV scale μ, after which we are left with a finite number of the imaginary part of the UV partial wave amplitudes ${\rho}^{s}_{\ell}(\mu)$, ${\rho}^{t}_{\ell}(\mu)$, and ${\rho}^{u}_{\ell}(\mu)$, the decision variables in the optimization problem. Specifically, in the numerical scheme, we choose $\ell=0,1,\dots,\ell_{{M}},\ell_{\infty}$ and $\varLambda^2/\mu= 1/N,\dots,1$, where $\ell_M$ and N are two sufficiently large integers for the numerics to converge and a larger partial wave $\ell_\infty\gg \ell_M$ is chosen to make the numerics converge faster. For example, under the discretization, Eq. (17) becomes

where we have defined ${\rho}^s_{\ell,n}= {\rho}^s_\ell(\varLambda^2 N/n)$ and so on. Note that now $b_1$ is a finite, linear combination of the decision variables. The same discretization is also applied to the null constraints. The null constraints are equality constraints, and in SDPB these equality constraints can be implemented by both imposing $(...)\geq 0$ and $(...)\leq 0$. With these setups, we can now propose our semidefinite program to get the bounds on the Wilson coefficients:

Decision variables

Maximize/Minimize

Subject to

Here, ${\alpha}_I$ are constants to be chosen by the user, which specifies the direction in the Wilson coefficient space $\{C_I\}$ that one wants to bound. In practice, since some of the unitarity conditions contain constants, we can introduce an extra decision variable and use the SDPB normalization to set this variable to 1.

In this paper, we shall only present 1D and 2D bounds. For the 1D bounds, we calculate the bounds on each of $C_{I}$. To get 2D bounds, we set one of ${\alpha}_I$ to zero and use the angular optimization method to compute the boundary of the bounds. For example, to obtain the bounds on $C_1$ and $C_2$, we set $({\alpha}_1, {\alpha}_2, {\alpha}_3)=(\cos\theta,\sin\theta,0)$. For each fixed θ, we use the SDPB package to obtain a lower and an upper bound on the objective $\cos\theta C_1 + \sin\theta C_2$, each upper or lower bound delineating a half plane in the $C_1$-$C_2$ space. Doing this for a number of θ, the many half-spaces carve out a 2D boundary in the $C_1$-$C_2$ space.

4.   Bounds on dim-8 Higgs operators

In this section, we present the numerical results on the SMEFT Higgs coefficients $C_{1}$, $C_2$, and $C_3$. With the Semidefinite Program (SDP) setup in the last section, we can find both upper and lower positivity bounds on them. We will compare our results with those from Ref. [73]. The positivity bounds obtained in Ref. [73] are already often stronger than the experiments bounds and the so-called partial wave unitarity bounds. As we see below, our bounds here are even stronger. Note that the partial wave unitarity bounds are not positivity bounds. The partial wave unitarity bounds reply only on partial wave unitarity within the low energy EFT, and dispersion relations are not used in their derivation. In comparison, the positivity bounds ¹ (sometimes also known as causality bounds) are built up on the dispersion relations, whose existence relies on causality of the S-matrix, and make use of the partial wave unitarity of the unknown UV theory.

In Table 1, we see that, comparing with the results in Ref. [73], labeled as “Linear”, our 1D “Nonlinear” bounds are much stronger, almost by a factor of 2. Comparing to the experiments bounds and the partial wave unitarity bounds that have also been obtained in Ref. [73], which we shall not repeat here, these new results will be more useful in helping the phenomenological analysis of the collider data. We also compute the 2D positivity bounds in the $C_1$-$C_2$, $C_1$-$C_3$, and $C_2$-$C_3$ plane respectively, which are shown in Fig. 1. From these plots, we consistently see an improvement by about a factor of 3 or 4, compared to the results of Ref. [73]. Clearly, in the total 3D parameter space spanned by $C_1$, $C_2$, and $C_3$, the improvement factor is even greater.

Table  1.  Comparison of positivity bounds on individual coefficients $C_I$ from the linear and nonlinear unitarity conditions.

	$\bar{C_1}=C_1 \varLambda^4 /(4\pi)^2$			$\bar{C_2}=C_2 \varLambda^4 /(4\pi)^2$			$\bar{C_3}=C_3 \varLambda^4 /(4\pi)^2$
	Lower	Upper		Lower	Upper		Lower	Upper
Linear	$-0.130$	$0.774$		0	$0.638$		$-0.508$	$0.408$
Linear2	$-0.086$	$0.467$		$0$	$0.378$		$-0.387$	$0.167$
Nonlinear	$-0.086$	$0.467$		$0$	$0.378$		$-0.387$	$0.167$
Here, the “Linear2” and “Nonlinear” results are our results in this paper obtained using linear and nonlinear unitarity conditions respectively, while the “Linear” results are from Ref. [73] using linear unitarity conditions but without using full Higgs symmetries. In this table, the numerical parameters are $\displaystyle N=10$ and $\displaystyle \ell_M=20$, and 42 null constraints are used.

 | Show Table

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Figure  1.  Positivity regions in the 2D subspaces of $C_1, C_2$, and $C_3$ by using linear and nonlinear unitarity conditions. Here, $\displaystyle \bar{C_i}=C_i \varLambda^4 /(4\pi)^2$. The orange and red regions are the results of the current paper using linear and nonlinear unitarity conditions respectively, while the blue region are from Ref. [73], which uses linear unitarity conditions but without using full Higgs symmetries. The orange and red regions are the same. We choose $\displaystyle N=10,\; \ell_M=20$ and use 42 null constraints.

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Note that, in Fig. 1, we are plotting 3 two-dimensional projections of the capped positivity cone. This is different from the positivity cone of Ref. [74] in that the capped positivity cone now additionally has upper bounds, much like an ice cream cone; see Fig. 1 of Ref. [73] for a cartoon explanation. That is, the rays of the cone of Ref. [10] go to infinity, while the current capped cone is a set of finite-length line segments. In the two-dimensional projections of the capped positivity cone, we can still see some parts of the Ref. [10] cone. For example, in the left plot of Fig. 1, the projection of the Ref. [10] cone is given by the lines of $C_1\ge0$ and $C_1+C_2\ge0$, which pass through the origin, and we see that this cone is now capped from the above by a smooth curve. The reasons that we can now bound the cone from above are: (ⅰ) we now have added the null constraints, which uses the information away from the forward limit; (ⅱ) we have made fuller use of the partial wave unitarity conditions in the UV. For more details about this, readers are referred to Ref. [73].

In Table 1 and Fig. 1, the “Linear2” bounds are the positivity bounds that can be obtained with the linear unitarity conditions of Ref. [73] but with all the symmetries of the SMEFT Higgs included. As it happens, these “Linear2” positivity bounds are numerically the same as our “Nonlinear” bounds. This is coincidental for the case of the SMEFT Higgs, due to the presence of strong internal symmetries. To see why this happens, let us compute the eigenvalues of the matrices ${\rm Im}\,\boldsymbol{T}_\ell$ and $2\boldsymbol{I}-{\rm Im}\,\boldsymbol{T}_\ell$, which are given respectively.

Distinct eigenvalues of ${\rm Im}\,\boldsymbol{T}_\ell$:

Distinct eigenvalues of $2\boldsymbol{I}-{\rm Im}\,\boldsymbol{T}_\ell$:

The semipositive definiteness of ${\rm Im}\,\boldsymbol{T}_\ell$ and $2\boldsymbol{I}-{\rm Im}\,\boldsymbol{T}_\ell$ are just the semipositivity of these eigenvalues. Remembering Eq. (9) and the relation $a_\ell^{i j k l}(s)=(-1)^\ell a_\ell^{i j l k}(s)$, it is easy to see that the positivity of these eigenvalues exactly give rise to the linear unitarity conditions for the SMEFT Higgs.

Nevertheless, the nonlinear unitarity conditions are in general stronger than the linear unitarity conditions derived in Appendix A.1. In Appendix A.2, as a simple example, we show that in a generic $\mathbb{Z}_2$ biscalar theory, the nonlinear bounds are indeed stronger than the linear bounds.

Finally, we would like to point out that the convergences of our numerically results are excellent. To see this, in Fig. 2, we plot how the 1D bounds varies with the number of null constraints used. In the above numerical results, we truncated the UV scales with $N=10$ and the UV spins with $\ell_M=20$, and we find that it is convenient to choose $\ell_{\infty}=100$. With this numerical setup, the computation of a single half-space bound uses about 110 CPU hours. As N increases, the positivity bounds become weaker, while the bounds becomes tighter as $\ell_M$ increases. In Fig. 3, we see that the results are quite stable against increasing the values of N and $\ell_M$.

Figure  2.  Convergence of positivity (upper and lower) bounds with the number of null constraint. We choose $\displaystyle N=10,\; \ell_M=20$.

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Figure  3.  Convergence of positivity (upper and lower) bounds with the numerical truncations $\ell_M$ and N. Here, $\displaystyle \bar{C_i}=C_i \varLambda^4 /(4\pi)^2$. 42 null constraints are used.

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5.   Conclusions

Positivity bounds are a set of highly restrictive conditions on the low-energy Wilson coefficients that have yet to be fully appreciated by the wider particle phenomenological and experimental communities. Although the formalism of positivity bounds itself is still under active development, highly constraining results are already available and straightforward to use. Here we take the Higgs scattering in the SMEFT as an example to illustrate how to numerically compute optimal, two-sided positivity bounds on ths dimension-8 Wilson coefficients. While the SMEFT formalism is generic, we assume that the SMEFT is weakly coupled below the EFT cutoff but may be strongly coupled in the UV. The formalism presented can be easily generalized to other sectors of the SMEFT, which is left for future work.

In this paper, we have improved the existing positivity bounds on the SMEFT Higgs by applying nonlinear unitarity conditions to the UV amplitude or spectral functions and by leveraging the full internal symmetries of the Higgs scattering. While the previous bounds can be obtained by simple ${Mathematica}$ coding with linear programming, our new results make use of the SDPB package, which can solve various field-theoretical semidefinite programs efficiently and highly accurately. We see that the new bounds are significantly stronger.

We have found that, in the Higgs case, the robust internal symmetries imply that the linear UV unitarity conditions used in Ref. [73] are actually tantamount to the nonlinear unitarity conditions. However, including the full internal symmetries does lead to tighter positivity bounds than the previous ones. As these new bounds are stronger than the current experimental bounds and the partial wave unitarity bounds, they will be useful in analyzing the current and upcoming phenomenological data for dimension-8 operators. These bounds may also be used to test the fundamental principles of quantum field theory or rule out UV particles from the collider data along the lines of Refs. [54, 63].

In general, the nonlinear unitarity conditions are of course more stringent. To demonstrate that the nonlinear unitarity conditions generally give rise to stronger bounds, we have calculated the two-sided positivity bounds for $\mathbb{Z}_2$ biscalar theory, a theory with two real scalar fields endowed with the reflection symmetry $\varphi_i \rightarrow -\varphi_i, \; i=1,2$. Nevertheless, we see that the linear unitarity conditions already give rise to bounds that are close to the bounds from the nonlinear conditions.