Origin of Neutrino Masses on the Convex Cone of Positivity Bounds

We exhibit the geometric structure of the convex cone in the linear space of the Wilson coefficients for the dimension-8 operators involving the left-handed lepton doublet $L$ and the Higgs doublet $H$ in the Standard Model effective field theory (SMEFT). The boundary of the convex cone gives rise to the positivity bounds on the Wilson coefficients, while the extremal ray corresponds to the unique particle state in the theory of ultra-violet completion. Among three types of canonical seesaw models for neutrino masses, we discover that only right-handed neutrinos in the type-I seesaw model show up as one of extremal rays, whereas the heavy particles in the type-II and type-III seesaw models live inside the cone. The experimental determination of the relevant Wilson coefficients close to the extremal ray of type-I seesaw model will unambiguously pin down or rule out the latter as the origin of neutrino masses. This discovery offers a novel way to distinguish the most popular seesaw model from others, and also strengthens the SMEFT as an especially powerful tool to probe new physics beyond the Standard Model.

Solving the inverse problem provides us with a novel way to identify the seesaw model and understand the nature of neutrino masses.
In this letter, we study the seesaw models of neutrino masses and establish for the first time the cone structure for the WC space of dim-8 operators containing L and H. New positivity bounds on the LLHH dim-8 operators are obtained, which are universal and accessible to future lepton colliders. We also solve the inverse problem to extract the UV information in the positivity region with the convex optimization algorithm.
Framework.-We focus on the second derivative of the forward 2-to-2 amplitudes M ij→kl (s, t → 0) with respect to s (where s, t are the ordinary Mandelstam variables, and the indices i, j, k, l refer to the low-energy particles, including particle species, polarizations and quantum numbers), and introduce the tensor In the consideration of analyticity and unitarity, as well as the generalized optical theorem at the tree level, the dispersion relation in Eq. (1) can be recast into [27,39] where the summation is over all the intermediate UV states X and the crossing channel is taken into account. Defining m ij ≡ M ij→X , one can regard the result on the right-hand side of Eq. (2) as a positive linear combination of m ij m * kl + (j ↔ l), since the integration can be understood as a limit of summation. Consequently, M ijkl belongs to the convex cone formed by m ij m * kl + (j ↔ l), i.e., M ijkl = cone m ij m * kl + m il m * kj 2 wherej andl denote the antiparticle states. For an EFT with n low-energy states i, j = 1, ..., n, m ij by construction must be an n-dimensional matrix. The extremal ray (ER) of the convex cone is defined as the element that cannot be decomposed into any nontrivial positive sum of other elements in the cone. One observation from Eq. (2) is that the UV state residing in the irrep of the symmetry group corresponds to the ER of the cone [31]. We choose X to be the irrep r of the SU(3) ⊗ SU(2) L ⊗ U(1) Y gauge group, the particles i and j belong to the irrep r i and r j , respectively. By the decomposition rule r i ⊗ r j = α C i,j r,α r, where C i,j r,α are the Clebsch-Gordan (CG) coefficients and the summation over all the states α's in r is implied, we can rewrite Eq. (2) as below with G ijkl r ≡ α C i,j r,α C k,l r,α * + (j ↔ l) being defined as the "generator", which plays the role of an ER, and M ijkl can be generated by positive combinations of G ijkl r 's from different irrep's. Extending the amplitudes m ij to include the CP-conjugate process, one can construct the generator as [38] G ijkl =m ij m * kl + m il m * kj + mk j m * īl + mklm * īj and thus the cone is defined by C = cone G ijkl . Now the primary goal is to examine the positions of seesaw models in the convex cone, and to identify them with the help of dim-8 operators. We notice the UV states of heavy particles in three types of seesaw models are in the irrep's of 1, 3, 3 of SU(2) L gauge symmetry, respectively, so they naturally fit into this framework. Given all known symmetries in the theory, it is straightforward to find all UV states that lead to dim-8 operators.
The UV States.-As mentioned before, nonzero neutrino masses may hint at the existence of the Weinberg operator O (5) ≡ LHH T L c and thus new physics associated with lepton and Higgs doublets. To gain more information about possible UV states, we examine the minimal space of WC's, including all four-particle operators O ijkl with i, j, k, l = H or L. For simplicity, only one lepton flavor is considered, but the extension to three flavors is straightforward. The dim-8 operators that contribute to the forward scattering amplitudes with the s 2 -dependence can be classified into three types of subspaces [52,53]: • LLHH: • LLLL: • HHHH: where σ I (for I = 1, 2, 3) stand for the Pauli matrices and ← → D ν ≡ D µ − ← − D µ with D µ being the covariant derivative in the SM has been defined. Note that there are another two dim-8 operators in the subspace of LLHH, namely, . For the forward scattering H(p 1 )L(p 2 ) → H(p 1 )L(p 2 ), where the four-momenta are specified, the d'Alembert operators will produce p 2 1 = 0, leading to a vanishing amplitude. On the other hand, for the k ↔ l exchanged process H(p 1 )L(p 2 ) → L(p 1 )H(p 2 ), the covariant derivative D µ acting on the Higgs field H provides a p µ 1 or p µ 2 factor, which will be contracted with γ µ in the fermion current and renders the amplitude to vanish. Due to the crossing symmetry, other amplitudes induced by these two dim-8 operators also vanish. Therefore, they don't contribute to the amplitudes of our interest.
The positivity bounds in two subspaces of HHHH and LLLL have been studied in the literature [28,31,32,38]. In the present letter, we enlarge the space of WC's by further combining those two operators O 1 and O 2 in Eq. (6). As we shall show later, the results obtained in the previous works can be reproduced when restricted to the subspace HHHH or LLLL.
Since both L and H are assigned as the irrep 2 of the SU(2) L group in the complex-field basis, the CG coefficients of direct product decomposition 2 × 2 = 1 + 3 and 2 ×2 = 1 + 3, labeled by C ab 1/3,c andC ab 1/3,c , read where ≡ iσ 2 and the subscript "c" is trivial for 1 but c = I for 3. With those CG coefficients, the m ij matrix can be found immediately, i.e.
where x is an arbitrary real parameter, representing the relative size of the coupling constant between X and HH  (or H † H) to that between X and LL (orLL). The generator G ijkl r can be derived from Eq. (5) for each irrep r, and will be matched into the WC space by identifying . In fact, it can be effectively viewed as a vector c r in the WC space. On the other hand, all the generators can be interpreted as the tree-level exchange of a single heavy state X in the irrep r. Therefore, one can evaluate M ijkl in the UV theory and integrate X out to match M ijkl into the WC space. In this way, another vector will be obtained, but it must be identical to c r up to an overall positive factor.
In Table I, we list all possible scenarios of tree-level UV completion. In each scenario, the single heavy state is specified with the spin, the quantum number r Y under the SU(2) L ⊗ U(1) Y group, the interactions with light SM particles, and the corresponding vector c in the WC space. Once all those vectors in the WC space are obtained, the ER's will be identified the subset of c's so that other vectors can be positively decomposed into the ER's. We also explicitly indicate which vector is ER in one column. In particular, three types of seesaw models for neutrino masses are also marked.
Some comments on Table I are in order. First, the subspace of LLHH formed by the WC's C 1 and C 2 is orthogonal to two other types of subspaces, i.e., LLLL and HHHH. In the LLHH subspace, there are four scenarios of UV completion, for which all the UV states are fermions. In other scenarios, the UV states are bosons. This is reasonable since one fermion cannot mediate interactions between two L's or two H's in the s-channel. Meanwhile, for the forward scattering, one boson cannot be tree-level UV completion of the effective vertices in O 1 and O 2 . Otherwise, the resultant amplitude takes the form of p 2 2v (p 1 ) / p 1 u(p 2 ) with v(p 1 ) and u(p 2 ) being the wave functions of external leptons (with four-momenta p 1 and p 2 ), which would vanish after applying equations of motion. This observation allows us to discuss the LLHH subspace without worrying about the other WC's in the next section.
Second, if we restrict ourselves into the HHHH subspace, corresponding to the last three components of c, the results in Refs. [28,31] can be reproduced with the positivity bounds C 6 ≥ 0, C 5 + C 6 ≥ 0, C 5 + C 6 + C 7 ≥ 0. Similarly, when reduced to the LLLL subspace, the same conclusions of the positivity bounds C 3 + C 4 ≤ 0, C 4 ≤ 0 in Refs. [32,38] are reached .
Finally, it is worthwhile to notice that the c's in the subspaces LLLL and HHHH don't contain the components linear in x. Since the generator G is the product of two m's, while m ij is the linear function of x, it is expected that c's have the components both linearly and quadratically dependent on x, and independent of x. The reason is simply that the components linear in x can only appear in the LLHH subspace, but the one-boson realization of the operators O 1 and O 2 is impossible. This feature guarantees that the ER's in the subspace LLLL and HHHH, respectively, remain to be so in the enlarged space, keeping the previously derived positivity bounds intact.
The LLHH Subspace.-In the two-dimensional LLHH subspace, the x-and y-axis actually refer to C 1 and C 2 , respectively. This subspace is orthogonal to the rest and thus we simply set C 3 , · · · , C 7 = 0 for the moment. The first four rows in Table I  vectors of the two edges, i.e.
which are first derived here. Interestingly, we find that the type-I and type-III seesaw models belong to this subspace, but only the type-I seesaw lives on one of two edges. On the other hand, the type-II seesaw lives in the five-dimensional subspace of LLLL and HHHH and appears in the convex cone, as shown in its threedimensional projection in Fig. 2. To better visualize the convex cone and the UV models, we have chosen a particular two-dimensional direction as explained in the caption of Fig. 2. Another practical application of the convex geometry is to solve the inverse problem [32,38], as any UV completion must have net dim-8 effects that cannot be completely lifted by the contributions from other possible UV completions. Then we proceed to explain how to infer the information about the UV physics. Once the collider experiments observe the benchmark point that fixes the vector C = (−3/2, 0), it should be a positive combination of the generator vectors, i.e., C = i ω i c i with i = E, N, Σ, Σ 1 and c i being the vector corresponding to each UV state in the C 1 -C 2 plane. The coefficients ω i = g 2 i /M 4 i are positive, and they carry the very information about the UV theory, namely, the relevant couplings and masses.
For instance, if the measured data point is located exactly on the edge represented by c = (−1/2, 1/2), then one can pin down the existence of N , i.e., the UV state in the type-I seesaw. At the same time, the existence of other UV states E, Σ, Σ 1 can be excluded. These conclusions are guaranteed by the salient feature of the ER of the convex cone. If the data point lies on the edge, the associated vector cannot be decomposed into any other vectors. Therefore, the only possible UV state X should be the one in the irrep r corresponding to that edge.
Generally, the measured data point may be not on the edge but inside the cone. In reality, the experimental results of the WC's are usually reported as a region bounded by the multidimensional ellipsoid, which is determined by the ∆χ 2 -value. Then the question is how to extract the constraints on ω i from experimental data. The solution has been provided in Refs. [32,38]. If the experimental result is represented by a point C 0 in the C 1 -C 2 plane, then the upper bound on ω i can be derived by finding the maximal value λ max of λ such that the following vector breaks the positivity condition The value of λ can be stated as the maximum possibility for the UV state i to exist and explain the experimental data. Quantitatively, the upper bound on ω i is given by λ max , i.e., λ max ≥ g 2 i /M 4 i . Unlike the numerical solution in Refs. [32,38], we find that this can be identified as a conic optimization problem, thanks to the convex nature of the WC space.
Given the uncertainty as a multidimensional ellipsoid, the upper bound on ω i can be determined since the conic optimization reduces to the second-order cone program maximize λ where A is the covariant matrix from the χ 2 -analysis, C 0 is the best-fit point, and ∆ is determined by the desired confidence level and by the number of free parameters. If the ∆ constraints are absent, the problem automatically reduces to the linear optimization program. Both these two optimization problems can be solved by the wellestablished computer algorithms.
For illustration, we take the best-fit point C 0 = (−3/2, 0) and the constraint as the disc (C 1 + 3/2) 2 + C 2 2 ≤ 0.1, whose boundary has been plotted as the dashed circle in Fig. 1. In Table II, we summarize the results by solving Eq. (13) in such a simple setup. The bounds on ω i for the benchmark point C 0 = (−3/2, 0) and that for the point on the edge C 0 = (−1/2, 1/2) have been derived and then converted into the bounds on M i / √ g i in units of TeV for each UV state. In the former case, it is difficult to solve the inverse problem, i.e., all the UV models fit the measurement equally well. But, in the latter case, the type-I seesaw model is singled out even if the experimental uncertainty is taken into account. In contrast, if the experimental results point to C 0 = (0, 0), all the UV models will unambiguously be ruled out up to a certain mass scale.
Summary.-Motivated by nonzero neutrino masses observed in neutrino oscillation experiments, we stress that the Weinberg operator O (5) ≡ LHH T L c for tiny Majorana neutrino masses may naturally exist in the SMEFT and new physics beyond the SM is very likely connected to the lepton and Higgs doublets. Therefore, we examine three classes of dim-8 operators involving lepton and Higgs doublets and reveal the geometric structure of the convex cone of positivity bounds in the subspace of the relevant WC's at the tree level. The discus-sions about positivity bounds at the one-loop level can be found in Ref. [37].
In the subspace of the WC's for two LLHH operators, we discover that the type-I seesaw model resides on the edge of the convex cone, indicating that the measurement of the WC's close to the edge will unambiguously confirm or rule out the type-I seesaw model as the true theory of neutrino masses. However, type-II and type-III seesaw models live inside the convex cone. This discovery provides a new and highly nontrivial way to distinguish between the type-I seesaw model and its analogues. We also explain how to extract the constraints on the UV theories once the experimental measurements of the WC's of dim-8 operators are available.
Obviously the key point is to experimentally measure the relevant WC's of dim-8 operators in the LLHH class. More and more data will be accumulated at the CERN Large Hadron Collider and future lepton or hadron colliders, offering the possibility to probe dim-8 operators [26,29,[52][53][54][55][56][57][58][59][60]. For example, one can probe the pair production of the Higgs bosons via e + e − → hh in future electron-positron colliders [61], to determine the values of C 1 and C 2 . As tree-level SM contributions will be highly suppressed by the electron Yukawa coupling and the Higgs self-coupling, this channel may be an ideal place to test the effects of dim-8 operators. Based on the studies in Ref. [61], observing the Higgs pair production at a lepton collider requires the center-of-mass energy √ s to be higher than 400 GeV, which can be reached at the linear colliders, e.g., the Compact Linear Collider (CLIC) [62]. Given the nominal setup √ s = 1.5 TeV and the integrated luminosity L = 1.5 ab −1 at the CLIC, one can roughly estimate the sensitivity as |C dim-8 /Λ 4 | ≤ O(10 −2 ) TeV −4 at the 95% confidence level. Further detailed studies in this direction are interesting and desirable.