Positivity from J-Basis operators in the standard model effective Field Theory

JHEP05(2024)221 Published for SISSA by Springer Received: December 14, 2023 Revised: March 23, 2024 Accepted: March 29, 2024 Published: May 17, 2024 Positivity from J-Basis operators in the standard model effective Field Theory Chengjie Yang,a,c Zhe Ren b,c,d,∗and Jiang-Hao Yub,c,e,f a Theory Division, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100190, China bCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China cSchool of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China dDepartamento de Física Teórica y del Cosmos, Universidad de Granada, Campus de Fuentenueva, E–18071 Granada, Spain e School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China fInternational Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China E-mail: [email protected],[email protected] Abstract: In the effective field theory (EFT), the positivity bound on dim-8 effective operators tells us that the s2 contribution in the scattering amplitude of 2-to-2 process geometrically corresponds to the convex cone composed of the ultraviolet (UV) states as the extremal rays. The J-Basis method can provide a complete group theory decomposition of the scattering amplitude on the direct product of the gauge group and the Lorentz group, thus to search for all UV states. Compared to previous methods, which can only perform direct product decomposition on the gauge groups, the J-Basis method greatly improves the strictness of the restrictions and also provides a systematic scheme for calculating the positivity bounds of the dim-8 operators. Keywords: Effective Field Theories, SMEFT, Other Weak Scale BSM Models ArXiv ePrint: 2312.04663 ∗Corresponding author. Open Access,©The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP05(2024)221 JHEP05(2024)221 Contents 1 Introduction 1 2 Positivity bounds based on extremal rays 3 2.1 Dispersion relation 3 2.2 Cone construction 6 2.3 Cone calculation and obtaining bounds 8 3 J-Basis theory framework 9 3.1 Poincare Casimir and partial wave basis 9 3.2 Gauge eigen-basis and SU(N)Casimir 10 3.3 Lorentz eigen-basis construction 11 3.4 Gauge J-Basis from gauge Casimir 13 4 The updated extremal ray positivity bounds 14 4.1 4 SM Higgs scattering 14 4.2 4Wscattering 17 4.3 4 lepton scattering 22 4.4 2-to-2 scattering involving Wand Bin the CP-conservation case 27 4.5 2-to-2 scattering involving Wand Higgs 28 4.6 2-to-2 scattering involving Wand quark 30 5 Summary and discussion 31 A Matching results 33 A.1 SM Higgs 33 A.2 4Wboson 36 A.3 Fermions with the multi-generation 37 1 Introduction The Standard Model Effective Field Theory (SMEFT) framework provides a systematic approach to parameterize new physics (NP) effects at high energy by using low energy degrees of freedom. As a non-renormalizable theory, SMEFT Lagrangian contains many operators with higher mass dimension written as L=LSM + ∞ X n=5 1 Λ(n−4) X i C(n) iO(n),(1.1) where C(n) and O(n) are Wilson Coefficients (WCs) and effective operators respectively of mass dimension n . These effective operators are written based on the standard model field building blocks, following the Lorentz and gauge symmetries [ 1 – 3 ]. They are enumerated order by order via the canonical mass dimension and form the complete and independent basis up to dimension 8 and higher in refs. [ 4 – 11 ], with generalization to any mass dimension – 1 – JHEP05(2024)221 in refs. [ 12 , 13 ]. The WCs parameterize ultraviolet (UV) information from NP theory. In the top-down approach, once the heavy states of a UV theory are integrated out, effective operators at the low energy scale can be obtained, called the matching procedure. Since the WCs comprise the information from UV theory, if the experimental data shows deviation from Standard Model (SM) prediction, the WCs can be determined. Given the null signal of NP, the WCs can only be restricted by current data or bounded theoretically. Using various processes, it is possible to restrict the WCs by experimental data via global fits [ 14 – 22 ]. On the other hand, positivity bound was proposed [ 23 – 28 ] to constrain WCs based on the unitarity, analyticity, locality properties of quantum field theory. There are many works to discuss positivity restriction of SMEFT operator coefficients. The earliest work of the positivity bound can be traced back to ref. [ 23 ], which established a positivity bound in the forward scattering limit of 2-to-2 elastic scattering (see also [24–28] for earlier discussions and applications in strong dynamics). The main idea of the elastic positivity bound is using unitary and analyticity characters to point out that 2-to-2 elastic forward scattering amplitude is non-negative. Recent literatures use the mathematical concept Arc to give positivity bound as a semi-positive Hankel matrix filled by the WCs linked to involving effective operators at different mass dimensions [ 29 , 30 ]. The partial wave analysis and unitary are also used to restrict the dim-6 operators’ WCs [ 31 ] and various motivation for going beyond dim-6 have been discussed in the refs. [ 32 – 38 ]. Since WCs contain UV information, it’s possible to enumerate possible NP particles based on effective operators, which is called the inverse problem [ 39 , 40 ]. The top-down approach is a well-studied and systematized procedure via matching and running [ 41 – 50 ]. The bottom-up inverse problem, however, has been rarely discussed in literature. The main difficulty is that each effective operator can be mapped to infinitely many UV theories. This case is referred to as “degeneracy”. Some articles propose to search for the possible UV states based on group representation decomposition [ 23 , 51 – 58 ]. The positivity can also be used to find possible UV states in the bottom-up way by combining theoretical bounds in the SMEFT and its UV states. The theoretical framework of positivity is that from a geometric perspective, the s2 contribution of SMEFT amplitude exists in a salient cone formed by the extremal rays linked to the corresponding UV completion with different quantum numbers [ 52 , 59 – 61 ]. Thus, the whole procedure only relies on principles of quantum field theory: i.e. unitarity and UV’s locality, thus the positivity framework is quite universal. In this work, a local UV quantum field theory (QFT) is assumed in order to link the s2 -order contribution of the scattering amplitude to the convex geometry, and thus the positivity bound is linked to the cone space shaped by the UV particles, as discussed in refs. [ 52 , 59 , 60 , 62 – 64 ]. Starting with the analyticity behavior of the forward scattering amplitude Mij→kl ( s )and the generalized optical theorem, the dispersion relation can be derived as Mijkl =Z∞ (ϵΛ)2 dµDisc Mij→kl(µ) 2iπ µ−1 2M23+ (j↔l) + c.c. = ′ X XZ∞ (ϵΛ)2X K=R,I dµmX,ij KmX,kl K πµ−1 2M23+ (j↔l). (1.2) – 2 – JHEP05(2024)221 Here i, j, k, l means the color and polarization of 4 outer legs while X stands for the heavy states, M is the mass of the external particles. By applying the convex hull theory, one shows that the salient cone which contains s2 contribution of amplitude has the form C = cone nmi(jm|k|l)|mij ∈Rn2o which sum over all the possible UV amplitude products mX,ij KmX,kl K ( i, j means extremal particles while X means the heavy state, K = R, I means the real and the imaginary part of the amplitude), while every UV state stands for the possible extremal ray of the cone which provides geometric perspective on the UV physics of the SMEFT operators. From the geometry perspective, it is essential to find a complete list of the UV states in a systematic way. In previous works [ 59 , 60 , 65 , 66 ], the gauge group projectors formed by Clebsch-Gordan (CG) coefficients is utilized and UV states are enumerated to form the cone to obtain bounds for scattering processes in the SMEFT. This is called the projection method. However, this method can not guarantee finding all the possible UV states without a systematic program on the UV completion searching. In recent work [ 7 , 9 , 12 , 53 , 57 , 67 , 68 ], the Pauli-Lubanski operator W2 and Casimir operator are introduced to decompose contact scattering amplitude to different eigenstates with specific quantum numbers. By identifying these eigenstates as the UV particles with corresponding quantum numbers, our work provides a systematic method to exhaust all the possible UV states for the effective operators in the SMEFT, which is called the J-Basis method [ 12 , 53 , 57 ]. In this work, both the convex geometry and the J-Basis method are applied in the dispersion relation to derive the positivity bounds in the SMEFT. After utilizing the J-Basis method to find the complete UV completion, the previous positivity bound based on the complete UV states according to the salient cones formed by extremal rays are updated. By comparing our results with the previous projection method [ 69 ], we point out that the previous method of searching UV states ignores some Lorentz structures in the group decomposition, so that it exists defects. From the comparison of the results, a more complete UV completion for the specific 2-to-2 scattering process at the Lagrangian level can be obtained so that our bounds are more precise than before. The paper is organized as follows. In section 2, we derive the dispersion relation for 2-to-2 forward scattering amplitude and show how to use the dispersion relation to give a geometry perspective of amplitudes. In section 3, we introduce relevant the Pauli-Lubanski operator for the momentum and the Casimir operator for the gauge structure. Then we show how to build a set of amplitudes representing possible UV states with definite angular momenta J and gauge quantum number R , that is, the J-Basis method. In section 4, for some typical scattering processes discussed in previous works, we show our bounds by using the J-Basis method and the UV selection to search for more complete UV completion at tree level and compare ours with previous bounds to show the rigour of the J-Basis. 2 Positivity bounds based on extremal rays 2.1 Dispersion relation Any 2-to-2 forward scattering amplitude Mij→kl ( s, t )for the full UV theory can be expanded in the low-energy EFT to get Mij→kl(s, t) = c0+c2s2+c2,1s2t+. . . +cn,msntm.(2.1) – 3 – JHEP05(2024)221 By taking derivatives on the amplitude, applying analyticity of amplitudes and considering the contour integral as shown in figure 1, the dispersion relation can be obtained (e.g. ref. [ 70 ] by replacing 0,4 m2 in the contour Γwith m2 − and m2 + ) by defining m±≡m1±m2 , c2=d2 ds2Mij→kl(s, t = 0)s=µ2 =1 2πi IΓ′dsMij→kl(s, 0) (s−µ2)3 =1 2πi Z0 −∞ +Z∞ m2 +!dsDisc Mij→kl(s, 0) (s−µ2)3. (2.2) Here Disc M ( s, 0) = M ( s + iϵ, 0) −M ( s−iϵ ). After setting t = 0 and applying the variable replacement u = m2 +−s , we obtain c2=1 2πi Z∞ m2 + duDisc Mij→kl m2 +−u, 0 m2 +−u−µ23+1 2πi Z∞ m2 + dsDisc Mij→kl(s, 0) (s−µ2)3 =1 2πi Z∞ m2 + ds"1 (s−µ2)3+1 s+µ2−m2 +3#Disc Mij→kl(s, 0) =1 πZ∞ m2 + ds"1 (s−µ2)3+1 s+µ2−m2 +3#Im Mij→kl(s, 0). (2.3) The above discussion is quite general: the second derivative of the low energy scattering amplitude is related to the imaginary part of the high energy scattering amplitude in the forward limit. This statement applies to both the elastic and the inelastic scatterings. For the elastic scattering ij →ij , by further applying the optical theorem, the positivity dispersion relation eq. (2.4) can be obtained, ImMij→ij(s, 0) = X XZdΠX|Mij→X(s, 0)|2(2π)4δ4(ij −X) = [(s−m2 −s−m2 +i1/2σtot >0, (2.4) where σt is total scattering cross section of process ij →X . Further, taking m±< ϵ Λ < Λ to subtract the SM contribution, the general expression on elastic positivity bound has the form c2> 0[ 71 , 72 ]. For the inelastic scattering ij →kl , to utilize the more general optic theorem, we need to do a little more work. By adding conjugate term on Mij→kl ( s, t ), Mij→kl s=m2 +/2 is defined as the real part of the derivative of the forward amplitude Mij→kl ( s, t )for scattering ij →kl process. By applying M∗ kl→ij ( s + iε ) = Mij→kl ( s−iε )to connect the time reversal ij →kl and its conjugate terms, eq. (2.3) becomes Mij→kl m2 + 2!≡1 2 d2 ds2Mij→kl s=m2 +/2,0+c.c. =Z∞ m2 + ds 2iπ Disc Mij→kl(s, 0) s−m2 + 23+Z∞ m2 + ds 2iπ Disc Mil→kj(s, 0) s−m2 + 23+c.c. , (2.5) From the above equation, we note that in the forward limit, a twice-subtracted dispersion relation can be derived for Mij→kl ( s, t ), assuming that a UV completion exists and is consistent with the fundamental unitary principles of the QFT. – 4 – JHEP05(2024)221 Figure 1. Diagram of the analytic structure of the forward amplitude in the complex s plane in the case m1 = m2 = m . The simple poles at s = m2 and 3 m2 and the branch cuts starting at s = 4 m2 and 0 correspond to resonances and multi-particle thresholds in the sand u-channels, respectively. In above eq. (2.5), the contributions of the kinematic poles are subtracted out [ 73 – 76 ]. Furthermore, by assuming the Λis the scale of the UV theory, we can compute the amplitude in the IR to a desired accuracy within the EFT in the energy scale within − ( ϵ Λ) 2≤s≤ ( ϵ Λ) 2 ( ϵ≤ 1). Then we choose the lower limit of the integral of eq. (2.5) turn to the value ϵ Λlarger than m+ , so that we can subtract out the low energy parts of the dispersion relation integrals corresponding to the EFT theory and keep the denominator of the integrands positive. Besides, the SM contribution of the eq. (2.6) will be suppressed by inverse powers of ϵ Λin ref. [ 77 ]. The above dispersion relation can be much simplified to Mij→kl (ϵΛ)2=Z∞ (ϵΛ)2 dµDisc Mij→kl(s) 2iπs−m2 + 23+ (j↔l).(2.6) This equation can be traced back to the improved positivity bounds discussed in refs. [ 52 , 78 ], and can also be regarded as the Arc defined in refs. [ 29 , 30 ], with a radius ( ϵ Λ) 2 . Now by applying the more general form of the optical theorem, Mij→kl −˜ M∗ kl→ij =iX X Mij→XM∗ kl→X,(2.7) the dispersion relation can be written as Mij→kl((ϵΛ)2) = 1 2πZ∞ (ϵΛ)2 ds s−m2 + 23X X [Mij→XM∗ kl→X+ (j↔l)] .(2.8) The power of analyticity is that the EFT and UV amplitude can be connected [ 79 ]. Considering the s2 contribution corresponding to the dim-8 effective operators O(8) i , we – 5 – JHEP05(2024)221 obtain that EFT: L=CiO(8) i, Mij→kl(ϵΛ) = ∂2 sMEFT ij→kl(ϵΛ) = Ci ∂ ∂sO(8) i|s=ϵΛ (2.9) which means we establish the link between the dispersion relation of the full theory and the EFT theory to obtain the convex geometry of the EFT. Several comments are in order. First, if choosing ij =kl, it recovers the elastic bounds. Second, the sum in the integrand on the r.h.s. is over all the intermediate states, denoted by X , which might contain infinite states. Thus it provides a geometric perspective that the UV physical amplitudes PXMij→X→kl exist in a cone Cspanned by many rays in which each ray represents contributions from UV particles X with certain quantum numbers. Taking the shorthand notation Mij→X→mij , all the UV amplitudes constitute the cone with Cn4≡cone nmijm∗kl +mi¯ lm∗k¯ jo.(2.10) To find the boundary of the cone, it is necessary to find all the possible immediate states with certain quantum numbers. So the problem becomes how to find all the possible UV states for a scattering process. 2.2 Cone construction From above, we notice that the s2 contribution of the 2-to-2 amplitude should stay in the cone formed by the UV states. Now the problem becomes how to find all possible UV states: one way is the projection method by using the Irrep’s (irreducible representation) projectors formed by CG coefficients and another one is the J-Basis method which is discussed in section 3. Here we focus on introducing the projection method and show its incompleteness in searching for UV completions. If we don’t know all possible UV states, naturally, we can use the CG coefficients to establish projectors that can expand the EFT operators [ 59 , 64 , 66 , 71 , 80 ], for the dimn Irrep X which comes from the direct product of the two basic representation, the projectors can be written as follow, PX ijkl = n XmX,ij nmX,kl n+j↔l . (2.11) Here mX,ij n is the CG coefficient where X represents Irrep with different quantum number, n represents the dimension of the Irrep X , the indices i, j represent the component of the two basic representation, and j↔l represents that the crossing symmetry [ 81 , 82 ] is imposed to the projectors. Taking the 4 H scattering as an example to show concrete steps to search for all projectors, The H is a complex field with the SU (2) w symmetry, which can be written as H = ( H2 + iH1, H4−iH3 ). Thus, by considering the direct product of HHX :2⊗2=1⊕3, HH†X:2⊗¯ 2=1⊕3, H†HX :¯ 2⊗2=1⊕3, H†H†X:¯ 2⊗¯ 2=1⊕3. (2.12) – 6 – JHEP05(2024)221 Particles Irrep(SU(2)w) CG coefficients matrix Projector HH +H†H†1C,Cγ4Ci(jC|k|l)+ (Cγ4)i(j(Cγ4)|k|l) 3CγI,Cγ4γI(CγI)i(j(CγI)|k|l)+ (Cγ4γI)i(j(Cγ4γI)|k|l) HH†or H†H 1S11i(j1|k|l) 3Sγ4γI(γ4γI)i(j(γ4γI)|k|l) 1Aγ4γi(j 4γ|k|l) 4 3AγIγi(j Iγ|k|l) I Table 1. Projectors represent 4Hscattering process. Particle Spin SU(2)w/U(1)yInteraction c(M)c(p) H1231gM−1HµνI† 1∂µHTϵτI∂νH+h.c. (3,−2,3) (1,6,6) Ξ1031gMΞI† 1HTϵτIH+h.c. (0,1,0) (1,0,0) B1111gBµ† 1HTϵi ↔ DµH+h.c. (1,0,−1) (1,0,2) H0230(S)gM−1HµνI 0∂µH†τI∂νH(−7,3,8) (−4,1,−14) W130(A)gWµI 0H†τIi↔ DµH(1,1,−2) (2,−1,2) Ξ0030(S)gMΞI 0H†τIH(2,0,−1) (2,1,4) G210(S)gM−1Gµν ∂µH†∂νH(3,3,−2) (6,1,6) B0110(A)gBµ 0H†i↔ DµH(−1,1,0) (0,−1,−2) S010(S)gMSH†H(0,0,1) (0,1,0) Table 2. Tree level UV completion in 4 H scattering process. The ( A )and ( S )after SU (2) w/Y means anti-symmetry and symmetry for the amplitude ij →X under the ij exchanges. In this paper, c ( M ) is the UV-EFT matching results in the basis defined in ref. [ 8 ], while c ( p )is the UV-EFT matching results in the Partial Wave (P-)Basis defined in ref. [7]. Here X is the heavy state while the indices of the Lorentz and the gauge group are omitted for the simplification of marking. We can obtain the projectors listed in table 1for expanding the 4 H scattering amplitudes. However, in ref. [ 8 ], there are only six projectors. Once hypercharge is considered in, HHX and H†H†X ’s same dimension Irreps should be merged, so the number of the UV states standing for HHX and H†H†X is only 2, so the number of projectors reduces to 6. MX,n kl→X is the matrix formed by CG coefficients for Irrep X and its component n , k , l , while i ( j|k|l )means that crossing symmetry in QFT is imposed to the projectors. However, by using the J-Basis method and the UV selection, nine UV states can be found in table 2. This shows that finding UV states by decomposing the gauge group direct product miss the spin-2 UV states in that case. Except the spin-2 states, the rest UV states can be checked in ref. [ 69 ]. Similarly, for 4 W scattering, we obtain projectors as follow, P1 αβγσ =1 Nδαβδγσ, P2 αβγσ =1 2(δαγδβσ −δασδβγ), – 7 – JHEP05(2024)221 P3 αβγσ =1 2(δαγδβσ +δασδβγ)−1 Nδαβδγσ .(2.13) With N = 3, these above projectors represent SU (2) adjoint representation decompositions, while N = 2 stands for decompositions in SO (2) or spin space. After imposing the crossing symmetry on these projectors, as what we did in section 2.1. We reach the conclusion that for tree-level UV completion of 4 W scattering, there are 9 possible UV states. However, in the tree level, we point out that the old framework of searching UV completion may cause a mistake. By applying the UV selection analysis in the vector boson scattering (VBS) case, we find that some UV states in the tree level completion corresponding to projectors couldn’t exist because their Lagrangian is zero or they are eliminated by the equation of motion (EOM), i.e. UV state corresponding to such projector doesn’t exist. Besides, the construction of the projectors for 4 fermions scattering amplitudes is a little more complicated [ 65 ]. The crossing symmetry j↔l changes to ik ↔¯ k¯ i into consideration in this case so that the projectors of the 4 fermion scattering can be written as, PX ijkl =1 2X αmX,ij αmX,kl α+mX,i¯ l αmX,k¯ j α+mαX,¯ kjmX,¯ il α+mαX,¯ k¯ lmX,¯ i¯ j α.(2.14) Easily, the cone for the 4 fermions scattering can be defined as follows, C= conenmX,ij αmX,kl α+mX,i¯ l αmX,k¯ j α+mαX,¯ kjmX¯ il α+mαX,¯ k¯ lmX,¯ i¯ j α+(i↔j,k ↔l)|m∈C2n×2no. (2.15) 2.3 Cone calculation and obtaining bounds Now we know how to construct projectors which represents UV states. Then the projectors can used to expand corresponding EFT amplitudes, and we can calculate positivity bounds. First, we need to determine the dimension of projectors, then choose a set of basis BY ijkl to expand projectors and EFT amplitudes to acquire a group of vectors {cXY } for different UV states X in the basis space by applying eq. (2.16). PX ijkl =cXY BY ijkl .(2.16) For example, in table 2, the corresponding P-Basis EFT operators On,ijkl is chosen as the basis BY ijkl to obtain the {c(p)} . On,ijkl =cnY BY ijkl .(2.17) If other BY ijkl rather than the operators On,ijkl are chosen as basis, these can be linked according to the basis transformation relationship eq. (2.17). Mijkl =CnOn,ijkl =CncnY BY ijkl .(2.18) Then, the amplitude Mijkl is expanded by applying eq. (2.18) to obtain the corresponding vector CncnY . Nm Y·(CncnY )≥0,(2.19) – 8 – JHEP05(2024)221 channel (Spin,SU(3)c,SU(2)w,U(1)y) P-Basis {H1, H2},nH† 3, H† 4oOi (2,1,3,1) −8Of 1−48Of 2−48Of 3 (0,1,3,1) 8Of 1 (1,1,1,1) 8Of 1+ 16Of 3 nH1, H† 3o,nH2, H† 4oOj (2,1,3,0) 16Of 1−4Of 2+ 56Of 3 (1,1,3,0) 8Of 1−4Of 2+ 8Of 3 (0,1,3,0) 8Of 1+ 4Of 2+ 16Of 3 (2,1,1,0) −24Of 1−4Of 2−24Of 3 (1,1,1,0) −4Of 2−8Of 3 Table 3. J-Basis analysis results for the 4Hscattering. Q(2) H4=DµH†DνHDµH†DνH, Q(3) H4=DµH†DµHDνH†DνH.(4.1) In the extremal ray method, first in ref. [ 59 ], the gauge group SU (2) w CG-coefficients of the SU (2) w gauge group are used to form projectors in table 1. Projectors in table 1match the UV states B1,S,B,W, Ξ 0, Ξ 1 in table 2. After utilizing the J-Basis method, we find extra new spin-2 UV states G,H0,H1 . Here we present the details of the J-Basis method applying to the 4 H scattering. First, we list the 6 P-Basis operators for the type D4H4 involved in the 4 H scattering, Of 1=1 4Y[p r ]Y[s t ]HpiHrj DµDνH†i sDµDνH†j t, Of 2=1 4Y[p r ]Y[s t ]H†i pHri (DµDνHjs)DµDνH†j t, Of 3=1 4Y[p r ]Y[s t ]Hpi (DµHrj)DνH†i sDµDνH†j t, Of 4=1 4Y[p r]Y[s t ]HpiHrj DµDνH†i sDµDνH†j t, Of 5=1 4Y[p r]Y[s t]H†i pHri (DµDνHjs)DµDνH†j t, Of 6=1 4Y[p r]Y[s t]Hpi (DµHrj)DνH†i sDµDνH†j t. (4.2) Acting the Poincare Casimir operator W2 on these P-Basis operators, we obtain the eigenstates and the eigenvalues of J-Basis in table 3. In detail, we process these steps by using the program ABC4EFT in ref. [ 12 ]. Then we transform the P-Basis to the basis in ref. [ 8 ]. By applying the UV selection, all the possible UV states that match nine eigenstates are written out. So we can obtain the table 2in section 2.2 corresponding to table 3. After obtaining all the UV states, we can apply eq. (2.17), eq. (2.18) and eq. (2.19) to obtain positivity bounds. More detailed, we choose the EFT operators On,ijkl as basis Bijkl to expand the UV – 15 – JHEP05(2024)221 Figure 4. The positivity cone for the 4-Higgs operators, with the corresponding generators. The x-axes represents ( C1 + C3 ) / (2 C1 + 3 C2 + C3 ), the y axes represents ( C1−C3 ) / (2 C1 + 3 C2 + C3 ), the dashed line presents the cone we obtained before in ref. [59], and the new cone is the solid line. amplitude. So we just need to search for all the normal vectors of the cone constructed by all the matching results from the fifth column of table 2directly. The number of rank-2 subsets of { c ( p )} is C2 9 = 36. Thus we could obtain 36 normal vectors corresponding to every rank-2 subset which represents the corresponding possible facet of the cone. To select the correct facets of the cone, we need to select the normal vectors which satisfy the positivity argument eq. (2.19). However only the following 4 normal vectors n ( p )in these 36 normal vectors which are listed eq. (4.4) satisfying that for every c ( p ) i , c(p)i·n(p)≥0.(4.3) The normal vectors that satisfies the positivity argument eq. (4.4) are (1,1,1),(1,1,1 2),(5,9,1),(1,3,2) .(4.4) The EFT amplitude ( C1, C2, C3 )should exist in the cone, so we obtain new positivity bounds, C1+C2+C3≥0, C1+C2+1 2C3≥0, 5C1+ 9C2+C3≥0, C1+ 3C2+C3≥0. (4.5) Thus extremal rays are changed to H1,H0,B1,B0 . In the perspective of the cone’s bottom, we obtain figure . 4. Based on figure 4, the Monte Carlo Sampling shows that the allowed area of the WC space is larger than the one obtained by the projection method, and the cone is a quadrangular pyramid actually. By applying the J-Basis method in the SM Higgs sector we find that in ref. [ 59 ] the projection to the UV states representing potential extremal ray bounds provides tighter bounds. – 16 – JHEP05(2024)221 group: (Spin, SU(3)c, SU(2)w,U(1)y) O(p) 1=WI LµνWIνρ LWJ LλρWJλµ L O(p) 2=WI LλρWI LµνWJλρ LWJµν L {WL1, WL2},{WL3, WL4} O(m) jO(p) j (2,1,5,0) (144,−12,0,−96,8,0,144,−12,0) 8(44,3) (1,1,5,0) (48,−12,24,−32,8,−16,48,−12,24) 0 (0,1,5,0) (0,12,0,0,−8,0,0,12,0) 8(4,3) (2,1,1,0) (0,0,0,48,−4,0,0,0,0) 8(−4,−3) (1,1,1,0) (0,0,0,16,−4,8,0,0,0) 0 (0,1,1,0) (0,0,0,0,4,0,0,0,0) 8(−2,0) (2,1,3,0) (−48,4,0,0,0,0,48,−4,0) 0 (1,1,3,0) (−16,4,−8,0,0,0,16,−4,8) 8(4,1) (0,1,3,0) (0,−4,0,0,0,0,0,4,0) 0 O(p) 1=WI LµνWJ LλρWIνρ RWJλµ R O(p) 2=WI LλρWI LµνWKλµ RWKνρ R {WL1, WL2},{WR3, WR4} O(m) jO(p) j (0,1,5,0) (−48,32,−48) 32(−3,1) (0,1,3,0) (16,0,−16) 0 (0,1,1,0) (0,−16,0) 16(0,−1) {WL1, WR3},{WL2, WR4} O(m) jO(p) j (2,1,5,0) (32,−48,−48) 16(−1,−3) (2,1,3,0) (0,16,−16) 16(−1,1) (2,1,1,0) (−16,0,0) 16(−1,0) Table 4. J-Basis analysis results for the 4 W scattering. Column of O(m) j represents the m-Basis results, and the column of O(p) j represents the P-Basis results. The combination of groups is defined as (Spin, SU(3)c, SU(2)w, Y ). 4.2 4Wscattering 4.2.1 Amplitude analysis and redundancy By applying the J-Basis method in 4 W scattering, we obtain table 4listing all the possible UV states. In table 4, the 6 involved operators in the P-Basis are listed as follows, O(p) W4 L,1 =WI LµνWJ LλρWIνρ LWJλµ L,O(p) W4 L,2=WI LµνWJµν LWI LλρWJλρ L, O(p) W2 LW2 R,1=WI LµνWJ LλρWIνρ RWJλµ R,O(p) W2 LW2 R,2=WI LµνWI LλρWJνρ RWJλµ R, O(p) W4 R,1=WI RµνWJ RλρWIνρ RWJλµ R,O(p) W4 R,2=WI RµνWJµν RWI RλρWJλρ R. (4.6) The WC space of the 4 W operators can be defined as (C(p) W4 L,1 ,C(p) W4 L,2,C(p) W2 LW2 R,1,C(p) W2 LW2 R,2,C(p) W4 R,1,C(p) W4 R,2)≡(C1, C2, C3, C4, C5, C6).(4.7) – 17 – JHEP05(2024)221 The m-basis for the operator type W4 L are O(m) W4 L,1=WI L1µνWJ L2λρWIνρ L3 WJλµ L4 ,O(m) W4 L,2=WI L1µνWJµν L2 WI L3λρWJλρ L4 , O(m) W4 L,3=WI L1µνWJ L2λρWIµν L3 WJλρ L4 ,O(m) W4 L,4=WI L1µνWI L2λρWJνρ L3 WJλµ L4 , O(m) W4 L,5=WI L1µνWIµν L2 WJ L3λρWJλρ L4 ,O(m) W4 L,6=WI L1µνWI L2λρWJµν L3 WJλρ L4 , O(m) W4 L,7=WI L1µνWJ L2λρWJνρ L3 WIλµ L4 ,O(m) W4 L,8=WI L1µνWJµν L2 WJ L3λρWIλρ L4 , O(m) W4 L,9=WI L1µνWJ L2λρWJµν L3 WIλρ L4 . (4.8) In m-basis the W boson in the operators are marked by the number. It means we don’t consider identical particle so as to cause the redundance. Similarly, the WC space of 4 W operators in m-basis for the type W4 L can be defined as (C(m) W4 L,1,C(m) W4 L,2,C(m) W4 L,3,C(m) W4 L,4,C(m) W4 L,5,C(m) W4 L,6,C(m) W4 L,7,C(m) W4 L,8,C(m) W4 L,9)≡ (C1, C2, C3, C4, C5, C6, C7, C8, C9). (4.9) As for the operator W2 LW2 R , the involving m-basis operators are O(m) W2 LW2 R,1=WI L.1µνWJ L2λρWIνρ R3 WJλµ R4 ,O(m) W2 LW2 R,2=WI L1µνWI L2λρWJνρ R3 WJλµ R4 , O(m) W2 LW2 R,3=WI L1µνWJ L2λρWJνρ R3 WIλµ R4 . (4.10) By applying the UV selection to table 4, we can check whether some UV states are ruled out or not. 1. Let’s us write out such UV Lagrangian WWV where W is the W boson, and V represents the heavy vector. In the term WWV , the indices of the Lorentz and the gauge groups has been omitted for simplification of marking. The first leading contribution would match to D2W4 which corresponds to dim-10. So WWV couplings can be excluded. 2. Meanwhile, table 3gives the possibility of existing spin-2 UV couplings as WLWLX . However, if you calculate the matching of UV state Wµν I LWI LνρGρµ to the P-Basis, Wµν I LWI LνρGρµWαβ J LWJ LβγGγα = Wµν I LWI LνρWαβ J LWJ Lβγ ∗1 M2 Sgργgµα +gραgµγ −2 3gρµgγα∝ WI LµνWJ LλρWIνρ LWJλµ L=O(p) W4 L,1 . (4.11) Thus, the matching result for this UV state Wµν I LWI LνρGρµ exists in the ray (1 , 0 , 0 , 0 , 0 , 0) of the WC space. The result violates the J-Basis analysis result ( − 4 ,− 3 , 0 , 0 , 0 , 0) for the UV state (2 , 1 , 1 , 0) in the channel ( WL, WL, WL, WL ). Besides, ref. [ 59 ] provides another character of the dispersion relation in eq. (2.8) that the amplitude cone is a salient cone. This means there shouldn’t exist any other UV state in the negative – 18 – JHEP05(2024)221 direction of the UV state with quantum number (0 , 1 , 5 , 0) for the 4 W scattering case. In table 4, it shows that in the opposite direction of (0 , 1 , 5 , 0), there exists the UV state with the quantum number (2 , 1 , 1 , 0). The three result from the J-Basis method, from the UV matching, and from geometry perspective seem incongruous in that case. However, there is no conflict among the three results because the UV states of tensor particle with the form WWG can be eliminated by the EOM. For the tensor coupling WI LµνWI LρνGµρ, the interaction Lagrangian can be rewritten as follow WIαβ LWI γ LαGβγ ˙α˙α= (σµν)αβ(σmn)γ α(σu1)˙α β(σv1)γ˙αWIµν LWImn LGµ1ν1.(4.12) By applying the characters of the σmatrix, eq. (4.12) can be expanded as WIαβ LWI γ LαGβγ ˙α˙α=−2gµµ1σν t˙α+ 2gνµ1σµ t˙α+ 2iεµνµ1λσλt ˙αεtα∗ −2gmν1σn α˙ β+ 2gnν1σm α˙ β+ 2iεmnν1λσλ˙ βε˙α˙ βWIµν LWImn LGµ1ν1. (4.13) There are many kinds of terms in the expansion of eq. (4.13), but all the terms can be transformed to the form WI LµνWI LρνGµρ by using Tr (σλ¯σρ)=2gλρ, gµµ1σν t˙αgmν1σn α˙ βWIµν LWImn LGµ1ν1=gµµ1gmν1Tr [σt¯σn)] WIµν LWImn LGµ1ν1 =WI LµνWI LρνGµρ .(4.14) Likewise, εµνν1λεmnνρσλtασρ˙α˙ βεtαεα˙ βWµν LWmn LGµ1ν1 =εµνµ1λεmnνρ Tr (σλ¯σρ)Wµν LWmn LGµν1 =−4Wµν LWLνρGρ µ. (4.15) Finally, eq. (4.16), the transformation relationship can be obtained WI LµνWI LρνGµρ ∝WIαβ LWI γ LαGβγ ˙α˙α =WIαβ LWI γ LαGµν −gµνϵβγ −iσµν βγ.(4.16) 3. By applying the EOM of the massive spin-2 particles, we can show that the eq. (4.16) equals zero. The free Lagrangian of the massive spin-2 quantum theory [90,91] is SFP hh;m2 1i≡SLG[h] + Smhh;m2 1,−m2 1i.(4.17) The expressions SLG (kinetic term) and Sm(mass term) are SLG[h] = Zd4x1 2h∂2h−hµν∂µ∂νh−1 2hµν∂2hµν +hµν∂ν∂ρhµρ, Smhh;m2 1, m2 2i=1 2Zd4xm2 1hµνhµν +m2 2h2. (4.18) By applying the Euler-Lagrange equation, we can obtain the EOMs and find that the hµν is traceless. ∂2−m2 1hµν(x)=0, ∂µhµν(x)=0, hµ µ=h(x)=0. (4.19) – 19 – JHEP05(2024)221 (Spin,SU(3),SU(2),U(1)y) Interaction Lagrangian c(p) (0,1,5,0) WI LµνWJµν LTAIJ SA+ x1WI RµνWJµν RTAIJ SA(4,3,−12x1,4x1,4x2 1,3x2 1) (0,1,1,0) WI LµνWIµν LS+x2WI RµνWIµν RS(−2,0,0,−4x2,−2x2 2,0) (2,1,5,0) WIµν LWJ RνρTIJ KGKρµ (0,0,−1,−3,0,0) (2,1,3,0) WIµν LWJ RνρεIJKGKρµ (0,0,−1,1,0,0) (2,1,1,0) WIµν LWI RνρGρµ (0,0,−1,0,0,0) Table 5. Matching Results for the 4Wscattering. By using the relation Gβγ ˙α˙α = Gµνϵ˙α˙ βσµ β˙ασν γ˙ β = Gµν −gµνϵβγ −iσµν βγ , the contraction between symmetry tensor Gµν and anti-symmetry tensor σµν βγ is 0. Then according to eq. (4.19), the Gµνgµν is 0. So the couplings with the form WLWLG and WRWRG are eliminated by the EOMs. The above discussions show that not all the amplitude decompositions correspond to the determined UV states in any case. The results of amplitude decomposition require the UV selection by the EOMs, the repeat field and other identities. Finally, we can write out all possible UV states for the 4Wscattering in table 5. Now according to eq. (2.17), eq. (2.18) and eq. (2.19), the normal vectors of the 4 W scattering amplitude cone can be calculated to obtain bounds. The cone has three categories of normal vectors which have the form in the WC space as              (0,1,0,0,0,0) , (1,−4 3,0,0,0,0) , (−x2,4(x1+x2) 3,1,1,−1 x2 ,4(x1+x2) 3x1x2 ) (x1≤0, x2≤0)) . (4.20) The EFT amplitude ( C1, C2, C3, C4, C5, C6 )should exist in the cone. So the product between the EFT WCs and the normal vector above should be positive, which represents the positivity bounds. Then the positive argument of the vectors in the eq. (4.20) in the WCs space can be obtained by solving such a system of binary quadratic inequalities:              C2≥0, −C1+4C2 3≥0, C5−4C6 3x1−4C6x2 3−(C3+C4)x1x2−4 3C2x1−C1x2+4 3C2x2x1x2≥0 (x1,x2≤0) . (4.21) To solve the third inequality, there are some tricks, i.e. we can regard x1 as a known number so as to calculate the single quadratic inequality. Then we obtain quadratic inequality of – 20 – JHEP05(2024)221 x2 from b2≥ 4 ac . Finally, we obtain the bounds as                                C6≥0, C2≥0, C1−4C2 3≤0, C5−4C6 3≤0, −C3−C4−8√C2C6 3≤2sC1−4C2 3C5−4C6 3. (4.22) The volume of the allowed WC space is 0.435% by the Monte Carlo Sampling. Despite that the cone is described by more than 6 WCs, we can still show the structure of the cone in 3D space as in figure 5by choosing the specific slicing in the dim-6 WC space. In the scheme of the slice in figure 5, the UV state (2 , 1 , 3 , 0) is projected to origin while the UV states (2,1,5,0) and (2,1,1,0) are projected to the yaxes. More than that, the circle corresponds to the UV state (0 , 1 , 5 , 0), and the (0 , 1 , 1 , 0) is degenerated to a linear ray y = 4 x . All of them are in the inner or surface of the slice. In the previous result in ref. [ 59 ], the projectors formed by SO (2) and SU (2) w CG coefficients were used to represent UV states. the previous work considered the CP-conserving case and reached the results of the 9 possible extremal rays (UV states) presented by Em,n where m, n are different Irreps of SU (3) c and SU (2) w . However we reach the conclusion that there are only 5 possible UV states in the tree level completion. For example, E1,2 means (0 , 1 , 3 , 0) in table 4whose contribution is zero after the decomposition of the Lorentz and the gauge group. In conclusion, we find that not all irrep projectors can be realized with the UV completion. 4.2.2 Comment with the 4 gluon scattering According to the detailed discussions about the UV completion of the 4 W scattering in the section 4.2.1, we find that the number of UV states in the tree level completion to restrict vector boson cones is less than previous results obtained by projection in ref. [ 69 ]. Hence, the 4 gluon scattering is similar. More specifically, color group direct product decompositions (projectors) are listed as follows, while the ¯ 10 representations in 8 ⊗ 8 = 1 + 8 + ¯ 8 + 10 + ¯ 10 + 27 from ref. [ 66 ] is eliminated for it doesn’t correspond to the inverted symmetry ( ij →ji , kl →lk ). The projectors corresponding the group decompositions of the direct product of the two SU (3) c adjoint representations are listed in eq. (4.23). The SO (2) group decompositions are the same as eq. (2.13). Finally, in ref. [ 69 ], it reaches the conclusion that there are 15 possible UV states for the 4 Gluon scattering case, Pab,cd 1=δabδcd N2−1, Pab,cd D=N N2−4dabedcde, Pab,cd F=fabefcde N, – 21 – JHEP05(2024)221 (a) (b) Figure 5. The WC space of the 4 W scattering process. Figure 5(b) is the slice of 4W scattering cone with different value of C1 C2 and C5 C6 . While the red semiconical represents C1 C2 = 4 3 , C5 C6 = 4 3 point in figure 5(a); the green semiconical represents curve 9 16 ( C1 C2−4 3 )( C5 C6−4 3 ) = 1 4 in figure 5(a); the pink semiconical represents curve 9 16 (C1 C2−4 3)(C5 C6−4 3)=1in figure 5(a). Pab,cd T=N2−4 4N2δacδbd −δadδbc−1 2Ndacedbde −dadedbce−1 4dbcefade +dadefbcc, Pab,cd X=N+ 2 4Nδacδbd +δadδbc−N+ 2 2N(N+ 1)δabδcd +1 4daccedbde +dadedbce −N+ 4 4(N+ 2)dabcdcde .(4.23) However, according to discussions in section 4.2.2, five spin-1 UV states couldn’t exist for their leading contribution correspond to the dim-10 EFT operators. Besides, the UV state Gµν iGjνρfijkS corresponding to the quantum number ( Spin = 0, SU (3) c = 1) obviously equals to zero in Lagrangian. This means that based on the J-Basis method, searching UV states by applying the UV selection can obtain the more reasonable bounds. 4.3 4 lepton scattering In this case, the involved P-Basis operators can be divided into four categories based on the their symmetry of the corresponding Young-Tableau in eq. (4.24). O(p) D2L2L†2,1=1 4Y[1 2 ,3 4 ]pδi1 i3δi2 i4(Lp1i1Lp2i2)DµL†i3 p3DµL†i4 p4, O(p) D2L2L†2,2=1 4Y[12,3 4 ]pδi1 i3δi2 i4(Lp1i1σµνLp2i2)DµL†i3 p3DνL†i4 p4, O(p) D2L2L†2,3=1 4Y[1 2,3 4]pδi1 i3δi2 i4(Lp1i1Lp2i2)DµL†i3 p3DµL†i4 p4, O(p) D2L2L†2,4=1 4Y[1 2,3 4]pδi1 i3δi2 i4(Lp1i1σµνLp2i2)DµL†i3 p3DνL†i4 p4.(4.24) – 22 – JHEP05(2024)221 State Spin SU(2)w/U(1)yInteraction c(p) W1131gp1p2ϵi1mτIi2 mWµI 1Lp1i1i↔ DµLp2i2+h.c.(0,0,0,−4) Ξ031gp1p2ϵi1mτIi2 mΞI(Lp1i1Lp2i2) + h.c.(−4,0,0,0) B1111gp1p2ϵi1i2Bµ 1Lp1i1i↔ DµLp2i2+h.c.(0,−4,0,0) S011gp1p2ϵi1i2S(Lp1i1Lp2i2) + h.c.(0,0,−4,0) H230gp1p3τIi1 i3HµνI Lp1i1iσµ ↔ DνL†i3 p3(−5,9,15,−3) W0130gp1p3τIi1 i3WµI 0Lp1i1σµL†i3 p3(−1,−3,3,1) G210gp1p3δi1 i3Gµν Lp1i1iσµ ↔ DνL†i3 p3(−5,−3,−5,−3) B0110gp1p3δi1 i3Bµ 0Lp1i1σµL†i3 p3(−1,1,−1,1) Table 6. UV completion for the 4 lepton scattering. Here the gpigpj means coupling constant of fermions between different generations pi, pj. Here pi represents the generation of the particle i . Thus, the corresponding Young-Tableau gives the corresponding tensor structure of the generation of operators. So the WC space can be defined as (C(p) D2L2L†2,1,C(p) D2L2L†2,2,C(p) D2L2L†2,3,C(p) D2L2L†2,4).(4.25) By applying the J-Basis method in amplitude decomposition, we can obtain table 6as a possible list of the UV completion. 4.3.1 One generation In this case, the involved operators become degenerate O1=∂µ¯ lγνl∂µ¯ lγνl,O2=∂µ¯ lγντIl∂µ¯ lγντIl,(4.26) because the last two types of operators in eq. (4.24) are eliminated for the Young Diagram’s anti-symmetry character of O(p) D2L2L†2,3,O(p) D2L2L†2,4 . So we can obtain the positivity bounds as C1≤0,9C1+ 5C2≤0.(4.27) Eq. (4.27) gives a cone marked by purple with extremal rays respectively representing the UV states H and Ξ 1 in figure 6. However in ref. [ 69 ], it only obtained UV states B1,B, Ξ 1,W . Hence, the bounds in ref. [ 69 ] are C1≤ 0 , C1 + C2≤ 0which shows a looser bounds marked by purple in figure 6than this results in eq. (6). 4.3.2 How to deal with the multi-generation case We need to expand the generation indices of the operators in eq. (4.25), because when we choose different generations ( p1p2p3p4 )in the same type of operators, the coefficients gp1,p2, g∗ p3,p4 – 23 – JHEP05(2024)221 Figure 6. The 2-D cone of the 4 Lepton scattering amplitude in one-generation case. Here the green area is previous result while the purple are is now result. are different. We use the UV state Bµ 0 as an example to show how to expand generation indices. For simplification we only consider the lepton coupling with two-generation like L=g12δi1 i3Bµ 0L1i1σµL†i3 2+g21δi1 i3Bµ 0L2i1σµL†i3 1.(4.28) Next, we use the combination ( pipjpkpl )where the index pi represents the generation of the particle i in the operator to refer to the operators with different generation combinations. Then based on the permutation group, combination of generation indices ( p1p2p3p4 )can take (1212) , (1221) , (2112) , (2121). For the operator with the type O(p) D2L2L†2,1 or O(p) D2L2L†2,2 , we can obtain that (1212) = (1221) = (2121) = (2112). As for the operators with the form O(p) D2L2L†2,3 or O(p) D2L2L†2,4 ,(1212) = − (1221) = (2121) = − (2112). Let’s try to write the matching vectors with components of generations tensor ( p1, p2, p3, p4 ) in the WC space as (C(p)(1111) D2L2L†2,1,C(p)(1111) D2L2L†2,2,C(p)(2222) D2L2L†2,1,C(p)(2222) D2L2L†2,2,C(p)(1122) D2L2L†2,1,C(p)(1122) D2L2L†2,2, C(p)(2211) D2L2L†2,1,C(p)(2211) D2L2L†2,2,C(p)(1212) D2L2L†2,1,C(p)(1212) D2L2L†2,2,C(p)(1212) D2L2L†2,3,C(p)(1212) D2L2L†2,4) ≡(C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, C12). After expanding the generation indices, we could obtain the matching results in table 7. We can obtain positivity bounds as                    C1+5C2 9<= 0 , C3+5C4 9<= 0 , 5C10 9+ 2C5+10C7 9+C9≤2sC1+5C2 9C3+5C4 9. (4.29) 4.3.3 The full flavor case Considering two-generation of fermion, the UV Lagrangian can be written as the form L= (g12L1L† 2+g21L2L† 1+g11L1L† 1+g22L2L† 2)X . – 24 – JHEP05(2024)221 group: (Spin, SU(3)c, SU(2)L,U(1)y) nQ1, Q† 3o,nQ2, Q† 4oO(m) jO(p) j (2,1,3,0) (5,3,−10,−6,0,0,0,0) (-10, 6, 5, -3) (1,1,3,0) (−1,1,2,−2,0,0,0,0) (2, 2, -1, -1) (2.1.1.0) (−5,−3,0,0,0,0,0,0) (0, 0, -5, 3) (1,1,1,0) (1,−1,0,0,0,0,0,0) (0, 0, 1, 1) Table 14. J-Basis analysis results for the 4 Q scattering. Here the P-Basis O(p) j are (O(p) Q4 1 ,O(p) Q4 2 ,O(p) Q4 3 ,O(p) Q4 4 ). (Spin,SU(3)c,SU(2)w,U(1)y)Matching result in P-Basis (0,1,5,0) (4,3,−12x1,4x1,4x2 1,3x2 1,0,0,0,0,0,0) (0,1,1,0) (−2,0,0,−4x2,−2x2 2,0,0,0,0,0,0,0) (2,1,5,0) (0, 0, -1, -3, 0, 0, 0, 0, 0, 0, 0, 0) (2,1,3,0) (0,0,−1,1,0,0,−10x2 3,6x2 3,5x2 3,−3x2 3,−4x3,0) (2,1,1,0) (0,0,−1,0,0,0,0,0,−5x2 4,3x2 4,0,−4x4) (1,1,3,0) (0, 0, 0, 0, 0, 0, 2, 2, -1, -1, 0, 0) (1,1,1,0) (0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0) Table 15. Matching results for the 2-to-2 scattering involving Wand Q. marking. We could only discuss the weak sector. As the UV vector boson coupling term in the {Q†Q}{WLWL} channel cannot exist because the WWV coupling’s contribution starts at the dim-10 operators at the tree level which are discussed in section 4.2.1. So we can obtain the conclusion that the degeneracy of ¯ QQ and WW exists in the Irreps with quantum numbers (2 , 1 , 3 , 0) and (2 , 1 , 1 , 0). The matching results are listed in table 15, where we consider that for decompositions with different quantum numbers, the coupling constants xi of degeneracy between WWX and QQX are different. In the table 15, the matching results show that the corresponding cone has curved surface parametrized by four parameters x1, x2, x3, x4 which match to the degeneracy between W and quark. Hence, obtaining the positivity bounds equals solving a hard quaternion quadratic polynomials problem so only numerical solutions can be obtained. 5 Summary and discussion Positivity bounds. Positivity bounds for the EFT operators involving 2-to-2 scattering can be transformed into geometry problems: every UV state contributing the EFT operators corresponds to the possible extremal ray that form the cone in the WC space of the EFT operators. It means that the more complete UV states we find, the more accurate shape of the cone we can acquire so as to obtain the exact bounds for the WCs. Previously, using the projection method based on the CG coefficients to represent UV or enumerating all possible UV states either provide redundant UV states or omit some UV states so as to – 31 – JHEP05(2024)221 obtain a not so strict constraint. Among the results obtained previously, the bounds of the 4 W scattering show a significant difference. the J-Basis method and the UV selection. We introduce the J-Basis method in section 2. In fact, the J-Basis takes the Lorentz structure into consideration to provide direct product decompositions of the spin structure and uses the Casimir Operators to give decompositions of gauge structure. Then according to a quantum number of decompositions, all possible UV Lagrangian in tree-level can be written. After that, we need to process the UV selection to check whether its contribution to tree-level matching is eliminated by the EOMs, the repeat field and other redundancy or not, to give an accurate UV completion. We apply the J-Basis method and the UV selection to calculate the bounds of some typical processes, such as the 4 H ,4 W and 4lepton scattering, and present the results in section 4. Despite that the J-Basis can give a systematic scheme to find all the UV states, it’s hard to obtain the analytical bounds in some cases. Especially for the 4 fermion scattering with multi-generation we cannot obtain fully analytical solutions due to too many parameters represents couplings between different generations. However, by imposing limitations such as the MFV case, the numerical solution can be obtained. In summary, the J-Basis idea and the UV selection provide a systematic framework to find all the UV states and gives more rigorous limitations in positivity-bound problems. Discussion. The positivity bounds based on extremal rays, by itself, is a powerful tool to determine the exact boundary of the UV-completable EFTs and supersedes bounds from the elastic scattering, and has a better physical interpretation of the relationship between the UV and the SMEFT. Many typical 2-to-2 scattering involving the SM particles are calculated in previous work have been updated in our works by the J-Basis method and the UV selection. However, obtaining the full set of bounds for all the SMEFT operators seems impossible because the degeneracy of two states with the same quantum number turns to obtain bounds to solve corresponding complex multivariate quadratic inequalities. So we should be able to obtain numerical bounds for all the SMEFT operators. Acknowledgments Thanks to Hao-Lin Li and Yu-Han Ni’s passionate instruction in the J-Basis framework and ABC4EFT Code. Besides, thanks for Hao Sun’s help in the basis for the MFV case. Thanks to Shuang-Yong Zhou for his valuable comments on the manuscript. The work is supported in part by the National Science Foundation of China under Grants No. 12022514, No. 12375099, No. 12047503, No.11725520, No.11675002, No.12075257, No.12235001, and National Key Research and Development Program of China Grant No. 2020YFC2201501, and No.2021YFA0718304. Zhe Ren is support by the grant CNS2022-136024 funded by the European Union NextGenerationEU/PRTR. – 32 – JHEP05(2024)221 A Matching results In fact, the J-Basis method only provides the possibility of UV particles’ existence and the further UV selection step give all the UV completion. For cross-checking, we need to calculate the UV-EFT matching results to process crosscheck. In this appendix, we list all calculations of tree-level matching for the UV states involved in the 4 H , the 4 W , and the 4lepton scattering processes in the P-Basis. The transformation matrices between the different basis can be acquired in refs. [ 7 , 9 , 53 ]. Here we introduce some notations that would be used later in this appendix. The Proca Lagrangian for massive spin-1 particle: L=−1 2AµGµνAν,(A.1) where Gµν =−(□+M2)gµν +∂µ∂ν.(A.2) A.1 SM Higgs The channel: {H1, H2},{H† 3, H† 4}. All the possible UV resonances and the matching results to the SMEFT operators with the form H2H†2D4 are listed as follows, The UV states ( Spin, SU (2) w, U(1) y ) = (2 , 3 , 1) : The UV amplitude =ig2 M4δi kδj l+δi lδj ks2 13 +s2 14 −2 3s2 12+··· (A.3) =i8g2 3M43−2 3     M(1) D4H2H†2 M(2) D4H2H†2 M(3) D4H2H†2    +··· .(A.4) The UV states ( Spin, SU (2) w, U(1) y ) = (1 , 3 , 1): Lint =gWµI 1(HTϵτIi↔ DµH) + h.c. . (A.5) The UV amplitude =i2g2 M4ejm τIi mτIn kϵnl s2 13 −s2 14+ϵjm τIi mτIn lϵnk s2 14 −s2 13+··· (A.6) = 0 .(A.7) The UV states ( Spin, SU (2) w, U(1) y ) = (0 , 3 , 1): Lint =gMΞI† 1(HTϵτIH) + h.c.. Considering that, Hjϵjk τIi kHi† =−H†iτIk iϵkjH†j, – 33 – JHEP05(2024)221 we have the UV amplitude =−i2g2 M4ϵjm τIi mτIn lϵnk +ϵjm τIi mτIn kϵnls2 12 +··· (A.8) =i4g2 M4δi kδj l+δi lδj ks2 12 +··· (A.9) =i32g2 M40 1 0     M(1) D4H2H†2 M(2) D4H2H†2 M(3) D4H2H†2    +··· .(A.10) The UV states (Spin, SU(2)w,U(1)y) = (1 , 1 , 1): Lint =gBµ† 1HTϵi ↔ DµH+h.c. . Considering the conjugate relationship Hjϵjii↔ DµHi† =iDµH†iϵijH†j−H†iϵijDµH†j=−H†iϵiji↔ DµH†j,(A.11) we have the UV amplitude =i4g2 M4−δi kδj l+δi lδj ks2 13 −s2 14+··· (A.12) =i32g2 M41 0 −1    M(1) D4H2H†2 M(2) D4H2H†2 M(3) D4H2H†2    +··· .(A.13) Now we discuss about the channel nH1, H† 3o,nH2, H† 4o . The UV states ( Spin, SU (2) w, U(1) y ) = (2 , 3 , 0): Lint =gM−1HµνI 0∂µH†τI∂νH.(A.14) The UV amplitude =ig2 4M4δi kδj ls2 12 −7 3s2 14 +8 3s2 13+δi lδj ks2 12 −7 3s2 13 +8 3s2 14+··· (A.15) =i2g2 3M4−7 3 8     M(1) D4H2H†2 M(2) D4H2H†2 M(3) D4H2H†2    +··· .(A.16) The UV states ( Spin, SU (2) w, U(1) y ) = (1 , 3 , 0): LUV =gWµI 0H†τIi↔ DµH,(A.17) The UV amplitude =ig2 M4δi kδj ls2 12 +s2 14 −2s2 13+δi lδj ks2 12 +s2 13 −2s2 14+··· (A.18) =i8g2 M41 1 −2    M(1) D4H2H†2 M(2) D4H2H†2 M(3) D4H2H†2    +··· .(A.19) – 34 – JHEP05(2024)221 The UV states ( Spin, SU (2) w, U(1) y ) = (0 , 3 , 0): Lint =gMΞI 0H†τIH. The UV amplitude =ig2 M4δi kδj l2s2 14 −s2 13+δi lδj k2s2 13 −s2 14+··· (A.20) =i8g2 M42 0 −1    M(1) D4H2H†2 M(2) D4H2H†2 M(3) D4H2H†2    +··· .(A.21) The UV states ( Spin, SU (2) w, U(1) y ) = (2 , 1 , 0): Lint =gM−1Gµν ∂µH†∂νH. The UV amplitude =ig2 8M4δi kδj ls2 12 +s2 14 −2 3s2 13+δi lδj ks2 12 +s2 13 −2 3s2 14+··· (A.22) =ig2 3M43 3 −2    M(1) D4H2H†2 M(2) D4H2H†2 M(3) D4H2H†2    +··· .(A.23) The UV states (Spin, SU(2)w,U(1)y) = (1 , 1 , 0): Lint =gBµ 0H†i↔ DµH. The UV amplitude =ig2 M4δi kδj ls2 12 −s2 14+δi lδj ks2 12 −s2 13+··· (A.24) =i8g2 M4−1 1 0     M(1) D4H2H†2 M(2) D4H2H†2 M(3) D4H2H†2    +··· .(A.25) The UV states( Spin, S U(2) w, U(1) y ) = (0 , 1 , 0): Lint =gMSH†H. The UV amplitude =ig2 M4δi kδj ls2 13 +δi lδj ks2 14+··· (A.26) =i8g2 M40 0 1     M(1) D4H2H†2 M(2) D4H2H†2 M(3) D4H2H†2    +··· .(A.27) – 35 – JHEP05(2024)221 A.2 4Wboson A.2.1 Scalar couplings The UV states ( Spin, SU (3) c,SU (2) w, U(1) y ) = (0 , 1 , 1 , 0): LUV =−1 2S□+M2S+gWI LµνWIµν LS+g∗WI RµνWIµν RS . •W2 LW2 R— The EOM of S: (□+M2)S=gWI LµνWIµν L+g∗WI RµνWIµν R, LEFT =gg∗ M2WI LµνWIµν LWJ RρλWJρλ R =4gg∗ M20−1 O(p) W2 LW2 R,1 O(p) W2 LW2 R,2 . (A.28) •W4 L— The EOM of S: (□+M2)S=gWI LµνWIµν L, LEFT =g2 2M2WI LµνWIµν LWJ LρλWJρλ L =2g2 M2−1 0  O(p) W4 L,1 O(p) W4 L,2 . (A.29) The UV states (Spin, SU(3)c,SU(2)w,U(1)y) = (0 , 1 , 5 , 0): LUV =−1 2SΛ□+M2SA+gWI LµνWJµν LTAIJ SA+g∗WI RµνWJµν RTAIJ SA. •W2 LW2 R— The EOM of SA: (□+M2)SA=gWI LµνWJµν LTAIJ +g∗WI RµνWJµν RTAIJ , LEFT =gg∗ M2WI LµνWJµν LWK RρλWLρλ RTAIJ TAKL =4gg∗ 3M2−3 1  O(p) W2 LW2 R,1 O(p) W2 LW2 R,2 . (A.30) •W4 L— The EOM of SΛ: (□+M2)SA=gWI LµνWJµν LTAIJ , LEFT =g2 2M2WI LµνWJµν LWK LρλWLρλ LTAIJ TAKL =g2 3M24 3  O(p) W4 L,1 O(p) W4 L,2 . (A.31) – 36 – JHEP05(2024)221 A.2.2 Massive Spin-2 couplings We have already discussed, there are only W2 LW2 R terms. The UV states ( Spin, SU (3) c,SU (2) w, U(1) y ) = (2 , 1 , 5 , 0): Lint =WI LµνWJ RρνTAIJ HAµρ 5 LEFT =g2 2M2gµλgνρ +gνλgµρ −2 3gµνgλρTAIJ TAKL WI LµξWJ RνξWK LλσWL Rρ =g2 3M2(−1−3)  O(p) W2 L,W2 R,1 O(p) W2 LW2 R,2 .(A.32) The UV states ( Spin, SU (3) c,SU (2) w, U(1) y ) = (2 , 1 , 3 , 0): Lint =ϵIJKWI LµνWJ RρνHKµρ 3. LEFT =g2 2M2gµλgνρ +gνλgµρ −2 3gµνgλρϵIJM ϵKLM WI LµξWJ RνξWK LλσWL Rρ =g2 M21−1 O(p) W2 LW2 R,1 O(p) W2 LW2 R,2 .(A.33) The UV states ( Spin, S U(3) c, S U(2) w, U(1) y ) = (2 , 1 , 1 , 0): Lint =WI LµνWI RρGµρ , LUV =−1 2gµλgνρ +gνλgµρ −2 3gµνgλρ−1 Gµν □+M2Gλρ +gWI LλσWI RρGλρ , LEFT =g2 2M2gµλgνρ +gνλgµρ −2 3gµνgλρWI LµξWI RνξWJ LλσWJ Rρσ =g2 M2−1 0  O(p) W2 LW2 R,1 O(p) W2 LW2 R,2 .(A.34) A.3 Fermions with the multi-generation The UV states ( Spin, S U(3) c, S U(2) w, U(1) y ) = (1 , 1 , 3 , 1): Lint =gp1p2ϵi1mτIi2 mWµI 1Lp1i1i↔ DµLp2i2+h.c. , LUV =W†I 1µ□+M2WµI 1+gp1p2ϵi1mτIi2 mWµI 1Lp1i1i↔ DµLp2i2 −g∗ p1p2τIm i2 ϵmi1W†µ 1L†i2 p2i↔ DµL†i1 p1.(A.35) •The EOM of WµI: □+M2WI 1µ=g∗ p1p2τIm i2 ϵmi1L†2 p2i↔ DµL†i1 p1. – 37 – JHEP05(2024)221 •The EOM of W†µI 1: (□+M2)W†I 1µ=−gp1p2ϵi1mτIi2 mLp1i1i↔ DµLp2i2. LEFT =gp1p2g∗ p3p4 M2ϵi1mτIi2 mτIn i4 ϵni3Lp1i1i↔ DµLp2i2L†4 p4i↔ DµL†i3 p3 =−gp1p2g∗ p3p4 M2ϵi1mτIi2 mτIn 4 ϵni3(⟨12⟩⟨14⟩[14][34] −⟨12⟩⟨13⟩[13][34]) =gp1p2g∗ p3p4 M20 0 0 −4       O(p) L2L†2D2,1 O(p) L2L†2D2,2 O(p) L2L†2D2,3 O(p) L2L†2D2,4        .(A.36) The UV states ( Spin, S U(3) c, S U(2) w, U(1) y ) = (0 , 1 , 3 , 1): Lint =gp1p2ϵi1m(τI)i2 mΞI(Lp1i1Lp2i2)+h.c., LUV =−Ξ†I□+M2ΞI+gp1p2ϵi1mτIi2 mΞI(Lp1i1Lp2i2)−g∗ p1p2τIm i2 ϵmi1Ξ†IL†i2 p2L†i1 p1. •The EOM of ΞI: −□+M2ΞI=g∗ p1p2τIm i2 ϵmi1L†i2 p2L†i1 p1. •The EOM of Ξ†I: (□+M2)Ξ†I=gp1p2ϵi1mτIi2 m(Lp1i1Lp2i2). •After integrating out ΞI, =−gp1p2g∗ p3p4 M44 0 0 0        O(p) L2L†2D2,1 O(p) L2L†2D2,2 O(p) L2L†2D2,3 O(p) L2L†2D2,4        .(A.37) The UV states ( Spin, S U(3) c, S U(2) w, U(1) y ) = (1 , 1 , 1 , 1): Lint =gp1p2ϵi1i2Bµ 1(Lp1i1i↔ DµLp2i2) + h.c. , LEFT =gp1p2g∗ p3p4 M2ϵi1i2ϵi4i3Lp1i1i↔ DµLp2i2Li4 p4i↔ DµL†i3 p3 =−gp1p2g∗ p3p4 M2ϵi1i2ϵi4i3(⟨12⟩⟨14⟩[14][34] −⟨12⟩⟨13⟩[13][34]) .(A.38) – 38 – JHEP05(2024)221 The UV states ( Spin, S U(3) c, S U(2) w, U(1) y ) = (0 , 1 , 1 , 1): Lint =gp1p2ϵi1i2S(Lp1i1Lp2i2) + h.c. , LEFT =gp1p2g∗ p3p4 M4ϵi1i2ϵi4i3□(Lp1i1Lp2i2)L†i4 p4L†i3 p3 =gp1p2g∗ p3p4 M4ϵi1i2ϵi4i3(⟨12⟩⟨12⟩[12][34]) =gp1p2g∗ p3p4 M40 0 −4 0        O(p) L2L+2D2,1 O(p) L2L+2D2,2 O(p) L2⊥+2D2,3 O(p) L2L+2D2,4        .(A.39) The UV states ( Spin, S U(3) c, S U(2) w, U(1) y ) = (2 , 1 , 3 , 0): Lint =gp1p3τIi1 i3HµνI Lp1i1iσµ ↔ DνLi3 p3, LUV =−1 2gµλgνρ +gνλgµρ −2 3gµνgλρ−1 HµνI □+M2HλρI (A.40) +gp1p3τIi1 i3HµνI Lp1i1iσµ ↔ DνL†i3 p3. •The EOM of HµνI: (□+M2)HµνI =τIi1 i3gµλgνρ +gνλgµρ −2 3gµνgλρLp1i1σλ ↔ DρL†i3 p3, LEFT =gp1p3gp2p4 2M2τIi1 i3τIi2 i4gµλgνρ +gνλgµρ −2 3gµνgλρ∗ Lp1i1iσµ ↔ DνLi3 p3Lp2i2iσλ ↔ DρL†i4 p4 =gp1p3gp2p4 2M2τIi1 i3τIi2 i4 [2⟨12⟩[34](−⟨12⟩[12]+⟨14⟩[14]) −(⟨14⟩[43]⟨23⟩[34]−⟨14⟩[43]⟨21⟩[14]−⟨12⟩[23]⟨23⟩[34]+⟨12⟩[23]⟨21⟩[14])] =gp1p3gp2p4 M2−5 9 15 −3      O(p) L2L†+D2,1 O(p) L2L†2D2,2 O(p) L2L†2D2,3 O(p) L2L†D2,4      . (A.41) The UV states ( Spin, S U(3) c, S U(2) w, U(1) y ) = (1 , 1 , 3 , 0) : Lint =gp1p3(τI)i1 i3WµI 0(Lp1i1σµL†i3 p3), LUV =1 2WI 0µ□+M2WµI 0+gp1p3τIi1 i3WµI 0Lp1i1σµL†i3 p3.(A.42) •The EOM of WI 0µ: (□+M2)WI 0µ=−gp1p3τIi1 i3Lp1i1σµL†i3 p3, – 39 – JHEP05(2024)221 LEFT =gp1p3gp2p4 2M4τIi1 i3τIi2 i4Lp1i1σµL†i3 p3□Lp2i2σµL†i4 p4 =gp1p3gp2p4 M4τIi1 i3τIi2 i4⟨12⟩⟨24⟩[24][34] =−gp1p2g∗ p3p4 M21 3 −3−1       O(p) L2L†D2,1 O(p) L2L†D2,2 O(p) L2L†2D2,3 O(p) L2L†D2,4        .(A.43) The UV states ( Spin, S U(3) c, S U(2) w, U(1) y ) = (2 , 1 , 1 , 0): Lint =gp1p3δi1 i3Gµν(Lp1i1iσµ ↔ DνL†3 p3), LEFT =gp1p3gp2p4 2M2δi1 i3δi2 i4gµλgνρ +gνλgµρ −2 3gµνgλρLp1i1iσµ ↔ DνL†i3 p3Lp2i2iσλ ↔ DρL†i4 p4 =gp1p3gp2p4 2M2δi1 i3δi2 i4−8⟨12⟩⟨34⟩[34]2+6⟨13⟩⟨24⟩[34]2 =gp1p3gp2p4 M2−5−3−5−3       O(p) L2L†2D2,1 O(p) L2L†2D2,2 O(p) L2L†2D2,3 O(p) L2L†D2,4        .(A.44) The UV states ( Spin, SU (3) c, S U(2) w, U(1) y ) = (1 , 1 , 1 , 0): Lint =gp1p3δi1 i3Bµ 0(Lp1i1σµLi3 p3), LEFT =gp1p3gp2p4 2M4δi1 i3δi2 i4Lp1i1σµL†i3 p3□Lp2i2σµL†i4 p4 =gp1p3gp2p4 M4δi1 i3δi2 i4⟨12⟩⟨24⟩[24][34] =−gp1p3gp2p4 M21−1 1 −1       O(p) L2L†+2D2,1 O(p) L2L†2D2,2 O(p) L2L†2D2,3 O(p) L2L+2D2,4        .(A.45) Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] S. Weinberg, Baryon and Lepton Nonconserving Processes,Phys. Rev. Lett. 43 (1979) 1566 [INSPIRE]. [2] W. Buchmuller and D. Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation,Nucl. Phys. B 268 (1986) 621 [INSPIRE]. – 40 –