Higgs and gauge boson phenomenology of the 3-3-1 model with CKS mechanism

Higgs and gauge boson phenomenology of the 3-3-1 model with CKS mechanism

H. N. Long, Theoretical Particle Physics and Cosmology Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, VietnamFaculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, VietnamDepartment of Physics, Cantho University, 3/2 Street, Ninh Kieu, Cantho, VietnamFaculty of Physics, Hanoi Pedagogical University 2, Phuc Yen, Vinh Phuc, VietnamInstitute for Research and Development, Duy Tan University, Da Nang City, VietnamInstitute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, VietnamUniversidad Técnica Federico Santa María and Centro Científico-Tecnológico de Valparaíso,
Casilla 110-V, Valparaíso, Chile
N. V. Hop, Theoretical Particle Physics and Cosmology Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, VietnamFaculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, VietnamDepartment of Physics, Cantho University, 3/2 Street, Ninh Kieu, Cantho, VietnamFaculty of Physics, Hanoi Pedagogical University 2, Phuc Yen, Vinh Phuc, VietnamInstitute for Research and Development, Duy Tan University, Da Nang City, VietnamInstitute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, VietnamUniversidad Técnica Federico Santa María and Centro Científico-Tecnológico de Valparaíso,
Casilla 110-V, Valparaíso, Chile
L.T. Hue Theoretical Particle Physics and Cosmology Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, VietnamFaculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, VietnamDepartment of Physics, Cantho University, 3/2 Street, Ninh Kieu, Cantho, VietnamFaculty of Physics, Hanoi Pedagogical University 2, Phuc Yen, Vinh Phuc, VietnamInstitute for Research and Development, Duy Tan University, Da Nang City, VietnamInstitute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, VietnamUniversidad Técnica Federico Santa María and Centro Científico-Tecnológico de Valparaíso,
Casilla 110-V, Valparaíso, Chile
N. H. Thao Theoretical Particle Physics and Cosmology Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, VietnamFaculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, VietnamDepartment of Physics, Cantho University, 3/2 Street, Ninh Kieu, Cantho, VietnamFaculty of Physics, Hanoi Pedagogical University 2, Phuc Yen, Vinh Phuc, VietnamInstitute for Research and Development, Duy Tan University, Da Nang City, VietnamInstitute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, VietnamUniversidad Técnica Federico Santa María and Centro Científico-Tecnológico de Valparaíso,
Casilla 110-V, Valparaíso, Chile
A. E. Cárcamo Hernández Theoretical Particle Physics and Cosmology Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, VietnamFaculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, VietnamDepartment of Physics, Cantho University, 3/2 Street, Ninh Kieu, Cantho, VietnamFaculty of Physics, Hanoi Pedagogical University 2, Phuc Yen, Vinh Phuc, VietnamInstitute for Research and Development, Duy Tan University, Da Nang City, VietnamInstitute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, VietnamUniversidad Técnica Federico Santa María and Centro Científico-Tecnológico de Valparaíso,
Casilla 110-V, Valparaíso, Chile
Abstract

The gauge boson sector of the renormalizable 3-3-1 model for the SM fermion masses and mixings is explored. The experimental data of the parameter shows that the vacuum expectation value (VEV) associated with the first spontaneous symmetry breaking (SSB) chain ranges from 3.6 TeV up to 6.1 TeV. Therefore the mass of the new heavy neutral gauge boson ranges from 1.42 TeV up to 2.42 TeV, which is consistent with estimations done in other 3-3-1 models. In that region of masses, we find that the total cross section for the production of the heavy neutral gauge boson at the LHC via Drell-Yan mechanism ranges from pb up to pb. On the other hand, in a future TeV proton-proton collider the total cross section for the Drell-Yan production of a heavy neutral gauge boson gets significantly enhanced reaching values ranging from pb up to pb. By the way, the masses of the new bilepton gauge bosons and are around 800 GeV, which are quite good. The Higgs sector of the model is explored. The Higgs potential with lepton number conserving was considered in detail. The SM-like Higgs boson was identified and as expected, is mostly contained in the CP even part of . It has couplings very close to SM expectation with small deviations of the order of . For the total scalar potential including lepton number violating interactions, excepting the CP-even sector, the situation is similar. The potential consists of enough number of Goldstone bosons corresponding to the longitudinal components of the massive gauge bosons. The scalar potential contains a complex scalar candidate for Dark Matter, namely and a Majoron but it is harmless because it is a gauge singlet. The constraints arising from the estimation of the Dark Matter (DM) relic density, set the mass of the scalar dark matter candidate in the range GeV GeV, for a quartic scalar coupling in the window .

1 Introduction

Despite its great successes, the Standard Model (SM) still puzzles over the hierarchies and structure of the fermion sector, which remain without compelling explanation. It is known that in the SM, masses of the matter fields are determined through Yukawa interactions. In addition, the CKM matrix is also constructed from the same Yukawa couplings. To solve this puzzles, some mechanisms have been suggested. To the best of our knowledge, the first attempt to explain the huge differences in fermion masses in the SM is the Froggatt - Nielsen (FN) mechanism Froggatt:1978nt (). According to the FM mechanism, the mass differences between generations follow from suppression factors depending on FN charges of particles. It has been noticed that in order to implement the aforementioned mechanism, the effective Yukawa interactions were introduced, thus making this theory non-renormalizable. From this point of view, the recent mechanism proposed by Cárcamo, Kovalenko and Schmidt CarcamoHernandez:2016pdu () (called by CKS mechanism) based on sequential loop suppression mechanism, is more natural since its suppression factor is arisen from loop factor .

One of the main purposes of the models based on the gauge group (for short, 3-3-1 model) Valle:1983dk (); Pisano:1991ee (); Foot:1992rh (); Frampton:1992wt (); Foot:1994ym (); Hoang:1995vq (); Hoang:1996gi () is concerned with the search of an explanation for the number of generations of fermions. Combining with the QCD asymptotic freedom, the 3-3-1 models provide an explanation for the number of generations to be three. Some other advantages of the 3-3-1 models are: i) they solve the electric charge quantization deSousaPires:1998jc (); VanDong:2005ux (), ii) they contain several sources of CP violation Montero:1998yw (); Montero:2005yb (), and iii) they have a natural Peccei-Quinn symmetry, which solves the strong-CP problem Pal:1994ba (); Dias:2002gg (); Dias:2003zt (); Dias:2003iq ().

In the framework of the 3-3-1 models, most of research are focused on radiative seesaw mechanisms, and some but involving  nonrenormalizable interactions (see references in Ref.CarcamoHernandez:2017cwi ()). However, most researches on the 3-3-1 models are not concerned with vast different masses among the generations.

The FN mechanism was implemented in the 3-3-1 models in Ref.Huitu:2017ukq (). It is interesting that the FN mechanism does not produce new scale, i.e., the scale of the flavour breaking is the same as the breaking scale of the symmetry of the model.

The CKS mechanism has been included in the 3-3-1 model without exotic electric charges in Ref. CarcamoHernandez:2017cwi (). The implication of the CKS mechanism to the 3-3-1 model leads to the interesting 3-3-1 model in sense that the derived model is renormalizable, while it fits all current data on fermion masses and mixing CarcamoHernandez:2017cwi (). It is worth mentioning that there exists a residual discrete lepton number symmetry arising from the breaking of the global symmetry. Under this residual symmetry, the leptons are charged and other particles are neutral CarcamoHernandez:2017cwi ().

However, in the mentioned work, the authors have just focused on the data concerning fermions (both quarks and leptons including neutrino mass and mixing), but some questions are open for the future study.

The aim in this work is to consider, in more details, the phenomenology of the model such as gauge and Higgs sectors from which we can get a bound on the model scale as well as on the mass of the new heavy boson. Due to the implemented symmetries, the Higgs sector is rather simple and can be completely solved. All Goldstone bosons and the SM like Higgs boson are defined.

The further content of this paper is as follows. In Sect. 2, we briefly present particle content and SSB of the model. Sect.3 is devoted to gauge boson mass and mixing. Taking into account of data on the parameter, we get bounds on the VEV of the first step of the SSB and on the mass of the new heavy gauge boson. By the way, we also get a limit for masses of the bilepton charged/non-Hermitian bosons. The Higgs sector is considered in Sect. 4. The Higgs sector consists of two parts: the first part contains lepton number conserving terms and the second one is lepton number violating. We study in details the first part and show that the Higgs sector has all necessary ingredients. Sect. 5 is devoted to the production of the heavy and the heavy neutral scalar . In Sect. 6, we deal with the DM relic density. We make conclusions in Sect. 7.

2 Review of the model

To implement the CKS mechanism, only the heaviest particles such as the exotic fermions and the top quark get masses at tree level. The next - medium ones: bottom, charm quarks, tau and muon get masses at one-loop level. Finally, the lightest particles: up, down, strange quarks and the electron aquire masses at two-loop level. To forbid the usual Yukawa interactions, the discrete symmetries should be implemented. Hence, the full symmetry of the model under consideration is

 SU(3)C×SU(3)L×U(1)X×Z4×Z2×U(1)Lg, (1)

where is generalized lepton number defined in Refs. Chang:2006aa (); CarcamoHernandez:2017cwi (). It is interesting to note that, in this model, the light active neutrinos also get their masses by a combination of linear and inverse seesaw mechanisms at two-loop level.

As in the ordinary 3-3-1 model without exotic electric charges, the quark sector contains the following representations CarcamoHernandez:2017cwi ()

 QnL =(Dn,−Un,Jn)TL∼(3,3∗,0),Q3L=(U3,D3,T)TL∼(3,3,13),n=1,2, DiR ∼(3,1,−13),UiR∼(3,1,23),i=1,2,3, JnR ∼(3,1,−13),TR∼(3,1,23), ˜TL,R ∼(3,1,23),BL,R∼(3,1,−13), (2)

where denotes quantum numbers for the three above subgroups, respectively. Note that the singlet exotic up type quarks , down type quarks in the last line of Eq. (2) are newly introduced for implementation of the CKS mechanism.

In the leptonic sector, besides the usual lepton triplets, the model contains more three charged leptons () and four neutral leptons, i.e, and (). The leptonic fields have the following assignments:

 LiL (3) E1L E1R N1R (4)

where and () are the neutral and charged lepton families, respectively.

The Higgs sector contains three scalar triplets: , and and seven singlets ,, ,,, and . Hence, the content of the Higgs sector is

 \mathchar287 = ⟨\boldmath\mathchar287% ⟩+\boldmath\mathchar287′∼(1,3,−13), (5) ⟨\boldmath\mathchar287% ⟩ = (0,0,v%\boldmath$\mathchar287$√2)T,\boldmath\mathchar287′=(% \boldmath\mathchar28701,\boldmath\mathchar287−2,1√2(R\boldmath\mathchar28703−iI\boldmath\mathchar28703))T, \mathchar282 = (\boldmath\mathchar282+1,1√2(R\boldmath\mathchar282−iI\boldmath\mathchar282),% \boldmath\mathchar282+3)T∼(1,3,23), \mathchar273 = ⟨\boldmath\mathchar273% ⟩+\boldmath\mathchar273′∼(1,3,−13), ⟨\boldmath\mathchar273% ⟩ = (v\boldmath\mathchar273√2,0,0)T,% \boldmath\mathchar273′=(1√2(R\boldmath\mathchar27301−iI\boldmath\mathchar27301),\boldmath% \mathchar273−2,\boldmath\mathchar27303)T, \boldmath\mathchar29501 ∼ (1,1,0),\boldmath\mathchar29502∼(1,1,0), \boldmath\mathchar286+1 ∼ (1,1,1),\boldmath\mathchar286+2∼(1,1,1),% \boldmath\mathchar286+3∼(1,1,1),\boldmath\mathchar286+4∼(1,1,1), \boldmath\mathchar2800 = ⟨\boldmath\mathchar280% 0⟩+\boldmath\mathchar2800′,⟨\boldmath\mathchar2800⟩=v\boldmath\mathchar280√2,\boldmath\mathchar2800′=1√2(R\boldmath\mathchar2800−iI\boldmath\mathchar2800)∼(1,1,0). (6)

The assignments under of scalar fields is presented in Table 1

The fields with nonzero lepton number are presented in Table 2. Note that three singlets as well as the elements in bottom of the lepton triplets have lepton number equal .

In the model under consideration, the spontaneous symmetry breaking (SSB) occurs by two steps CarcamoHernandez:2017cwi (). The first step is triggered by the vacuum expectation values (VEVs) of the and scalar fields. At this step, all new extra fermions, non-SM gauge bosons as well as the gauge singlet lepton gain masses. In addition, the entries of the neutral lepton mass matrices with negative lepton number also get values proportional to . By this time, the initial group breaks down to that of the SM and . The second step is triggered by providing masses for the top quark as well as for the and gauge bosons and leaving the symmetry preserved. Here is residual symmetry where only leptons are charged. Thus the interactions having an odd number of leptons are forbidden. This is crucial to guarantee the proton stability CarcamoHernandez:2017cwi (). Thus

 SU(3)C×SU(3)L×U(1)X×Z4×Z2×U(1)Lg (7) v\boldmath\mathchar287,v\boldmath\mathchar280−−−−−−−−−−−−−−−−−−→ SU(3)C×SU(2)L×U(1)Y×Z4×Z(Lg)2 v\boldmath\mathchar273−−−−−−−−−→SU(3)C×U(1)Q×Z4×Z(Lg)2.

A consequence of the chain in (7) is

 v\boldmath\mathchar273=v=246GeV≪v\boldmath\mathchar287∼v\boldmath\mathchar280∼O(10)TeV. (8)

The corresponding Majoron is a gauge-singlet and, therefore, unobservable.

3 Gauge bosons

3.1 Gauge boson masses and mixing

After SSB, the gauge bosons get masses arising from the kinetic terms for the and scalar triplets, as follows:

 Lgaugemass=(D\boldmath\mathchar278⟨\boldmath\mathchar287⟩)†D\boldmath\mathchar278⟨% \boldmath\mathchar287⟩+(D%\boldmath$\mathchar278$⟨\boldmath\mathchar273⟩)†D\boldmath\mathchar278⟨\boldmath\mathchar273⟩, (9)

with the covariant derivative for triplet defined as

 D\boldmath\mathchar278=∂\boldmath\mathchar278−igA%\boldmath$\mathchar278$a\boldmath\mathchar277a2−igXX\boldmath\mathchar27792B\boldmath\mathchar278, (10)

where and are the gauge coupling constants of the and groups, respectively. Here, is defined such that , similarly as the usual Gell-Mann matrix . By matching gauge coupling constants at the breaking scale, the following relation is obtained Hoang:1995vq ()

 t≡gXg=3√2sin\boldmath\mathchar274W(MZ′)√3−4sin2\boldmath\mathchar274W(MZ′). (11)

Let us provide the definition of the Weinberg angle . As in the SM, ones put , where is gauge coupling of the subgroup satisfying the relation Hoang:1995vq ()

 g′=√3ggX√18g2−g2X. (12)

Thus

 tan\boldmath\mathchar274W=√3gX√18g2−g2X. (13)

Denoting

 W±\boldmath\mathchar278=1√2(A\boldmath\mathchar2781∓iA\boldmath\mathchar2782),Y±\boldmath\mathchar278=1√2(A\boldmath\mathchar2786±iA\boldmath\mathchar2787),X0\boldmath\mathchar278=1√2(A\boldmath\mathchar2784−iA\boldmath\mathchar2785), (14)

and substituting (10) and (14) into (9) ones get squared masses for charged/non-Hermitian gauge bosons as follows

 m2W=g24v2\boldmath\mathchar273,M2X0=g24(v2\boldmath\mathchar287+v2\boldmath\mathchar273),M2Y=g24v2\boldmath\mathchar287, (15)

and GeV, as expected.

From (15) it follows a splitting of gauge boson masses

 M2X0−M2Y=m2W. (16)

For neutral gauge bosons, the squared mass mixing matrix has the form

 Lngaugemass=12VTM2ngaugeV, (17)

where and

 M2ngauge=g24⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝v2\boldmath\mathchar273v2\boldmath\mathchar273√3−2t3√6v2\boldmath\mathchar27313(4v2\boldmath\mathchar287+v2\boldmath\mathchar273)2t9√2(2v2\boldmath\mathchar287−v2\boldmath\mathchar273)2t227(v2\boldmath\mathchar287+v2\boldmath\mathchar273)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (18)

The down-left entries in (18) are not written, due to the fact that the above matrix is symmetric.

The matrix in (18) has vanishing determinant, thus giving rise to massless gauge boson, which corresponds to the photon. Diagonalization of matrix in (18) separates into two steps. In the first step, the massive fields are identified as

 A\boldmath\mathchar278 = sWA\boldmath\mathchar2783+cW⎛⎝−tW√3A\boldmath\mathchar2788+√1−t2W3B%\boldmath$\mathchar278$⎞⎠, Z\boldmath\mathchar278 = cWA\boldmath\mathchar2783−sW⎛⎝−tW√3A\boldmath\mathchar2788+√1−t2W3B%\boldmath$\mathchar278$⎞⎠, (19) Z′\boldmath\mathchar278 = √1−t2W3A%\boldmath$\mathchar278$8+tW√3B%\boldmath$\mathchar278$,

where we have denoted . After the first step, matrix becomes block diagonal one, where the entry in the top is zero (due to the masslessness of the photon), while the matrix for in the bottom has the form

 M2(2×2)=(M2ZM2ZZ′M2ZZ′M2Z′). (20)

The matrix elements in (20) are given by

 M2Z = g2v2\boldmath\mathchar2734c2W=m2Wc2W, (21) M2ZZ′ = g24c2W√3−4s2Wv2\boldmath\mathchar273(1−2s2W), M2Z′ = g2c2W4(3−4s2W)⎡⎣4v2\boldmath\mathchar287\boldmath\mathchar287+v2\boldmath\mathchar273(1−2s2W)2c4W⎤⎦.

Note that our formula of is consistent with that given in Buras:2012dp ().

The last step of diagonalization is quite simple. The eigenstates are determined as

 Z1\boldmath\mathchar278 = Z\boldmath\mathchar278cos\boldmath\mathchar286−Z′\boldmath\mathchar278sin\boldmath\mathchar286, Z2\boldmath\mathchar278 = Z\boldmath\mathchar278sin\boldmath\mathchar286+Z′\boldmath\mathchar278cos\boldmath\mathchar286, (22)

where the mixing angle is given by

 tan2\boldmath\mathchar286=2M2ZZ′M2Z′−M2Z. (23)

It is very easy to prove that our definition of is consistent with that introduced in Ref. Hoang:1999yv (), which is needed to study the parameter.

The masses of physical neutral gauge bosons are determined as

 M2Z1 = 12{M2Z′+M2Z−[(M2Z′−M2Z)2+4(M2ZZ′)2]12}, M2Z2 = 12{M2Z′+M2Z+[(M2Z′−M2Z)2+4(M2ZZ′)2]12}. (24)

Ones approximate

 Δ = M4Z′(1−2M2ZM2Z′+M4Z+4(M2ZZ′)2M4Z′), ⇒√Δ ≃ M2Z′[1−M2ZM2Z′+2(M2ZZ′)2M4Z′+O(M6ZM6Z′)]. (25)

Therefore

 M2Z1 ≃ M2Z−(M2ZZ′)2M2Z′+M2Z×O⎛⎝v4%\boldmath$\mathchar273$v4\boldmath\mathchar287⎞⎠, (26) M2Z2 ≃ M2Z′+(M2ZZ′)2M2Z′+M2Z×O⎛⎝v4\boldmath\mathchar273v4%\boldmath$\mathchar287$⎞⎠≃M2Z′. (27)

In the limit , the mixing angle is

 tan\boldmath\mathchar286≃(1−2s2W)√3−4s2W4c4W⎛⎝v2\boldmath\mathchar273v2%\boldmath$\mathchar287$⎞⎠. (28)

Before turning to the next section, we remind the usual relation

 e=gsW. (29)

3.2 Limit on Z′ mass from the \mathchar282 parameter

The presence of the non SM particles modifies the oblique corrections of the SM, the values of which have been extracted from high precision experiments. Consequently, the validity of our model depends on the condition that the non SM particles do not contradict those experimental results. Let us note that one of the most important observables in the SM is the parameter defined as

 \boldmath\mathchar282=m2Wc2WM2Z. (30)

For the model under consideration, the oblique correction leads to the following form of the parameter Hoang:1999yv ()

 \boldmath\mathchar282−1 ≃ tan2\boldmath\mathchar286(M2Z′m2Z−1)+3√2GF16\boldmath\mathchar2812[M2++M20+2M2+M20M2+−M20lnM20M2+] (31) −\boldmath\mathchar267(mZ)4\boldmath\mathchar281 s2W[ t2WlnM20M2++\boldmath\mathchar2902(M+,M0)2+O(\boldmath\mathchar2903(M+,M0))],

where and .

Combining with Eq. (16), ones get

 \boldmath\mathchar282−1 ≃ tan2\boldmath\mathchar286(M2Z′m2Z−1)+3√2GF16\boldmath\mathchar2812[2M2Y++m2W−2M2Y+(M2Y++m2W)m2Wln(M2Y++m2W)M2Y+] (32) −\boldmath\mathchar267(mZ)4\boldmath\mathchar281 s2W[ t2Wln(M2Y++m2W)M2Y++m4W2(M2Y++m2W)2],

where  Tanabashi:2018oca ().

Taking into account Tanabashi:2018oca () and

 \mathchar282 = 1.00039±0.00019, (33)

we have plotted as a function of in Fig. 1 (the left-panel). From figure 1 (the left-panel), it follows

 3.57TeV≤v\boldmath\mathchar287≤6.09TeV. (34)

Substituting (34) into (27) and evaluating in figure 1(the right-panel) we get a bound on the mass as follows

 1.42TeV≤MZ2≤2.42TeV. (35)

It is worth mentioning that the second term in (32) is much smaller the first one. Consequently, the limit deduced from the tree level is slightly different from the one with the oblique correction.

From LHC searches, it follows that the lower bound on the boson mass in 3-3-1 models is around TeV Salazar:2015gxa (). Hence, the 3-3-1 scale is about TeV, while from the decays and CarcamoHernandez:2005ka (); Martinez:2008jj (); Buras:2013dea (); Buras:2014yna (); Buras:2012dp (), the lower limit on the boson mass ranges from TeV to TeV.

Then, the bilepton gauge boson mass is constrained to be in the range:

 465GeV≤MY≤960GeV. (36)

Here we have used Tanabashi:2018oca ()

 mW=80.379GeV.

Note that the above limit is stronger than the one obtained from the wrong muon decay Dong:2006mg ()

 MY≥230GeV.

For conventional notation, hereafter we will call and by and , respectively.

Now we turn into the main subject - the Higgs sector.

4 Higgs potential

The renormalizable potential contain three parts: the first one invariant under group in (1) is given by

 VLNC = \boldmath\mathchar2782\boldmath\mathchar287\boldmath\mathchar287†\boldmath\mathchar287+\boldmath\mathchar2782\boldmath\mathchar282\boldmath\mathchar282†\boldmath\mathchar282+\boldmath\mathchar2782\boldmath\mathchar273\boldmath\mathchar273†\boldmath\mathchar273+4∑i=1\boldmath\mathchar2782\boldmath\mathchar286+i% \boldmath\mathchar286+i\boldmath\mathchar286−i+2∑i=1\boldmath\mathchar2782\boldmath\mathchar295i\boldmath\mathchar2950i% \boldmath\mathchar2950∗i+\boldmath\mathchar2782\boldmath\mathchar280\boldmath\mathchar2800∗% \boldmath\mathchar2800 (37) +