Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0197 Acta Polytechnica 62(1):197–207, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague COMPLEX TOPOLOGICAL SOLITON WITH REAL ENERGY IN PARTICLE PHYSICS Takanobu Taira City, University of London, Department of Mathematics, Northampton Square, London EC1V 0HB, UK correspondence: takanobu.taira@city.ac.uk Abstract. We summarise the procedure used to find the classical masses of Higgs particle, massive gauge boson and t’Hooft-Polyakov monopole in non-Hermitian gauge field theory. Their physical regions are explored, and the mechanism of the real value of the monopole solution is analysed in different physical regions. Keywords: t’Hooft-Polyakov monopole, quantum field theory, non-Hermitian quantum field theory. 1. Introduction Quantum field theory is a key tool to analyse particle physics. The most modern physical description of the fundamental particle interaction is described by the model called the Standard Model. However, the model possesses several problems such as incompatibil- ity with the general relativity, hierarchy problem, etc. Therefore, it is an active area to extend the standard model. Recently a growing number of research papers started exploring the non-Hermitian extension of the Standard Model [1–14]. We have contributed to this development by analysing the Goldstone theo- rem [8, 10], The Higgs mechanism [9] and t’Hooft- Polyakov monopoles [11]. The classical masses of Higgs particles, massive gauge boson and monopoles were analysed. However, a detailed analysis of their intersecting physical regions and the mechanism of the real value of the energy of monopole was not explored. The main aim of this contribution is to fill this gap. There are two separate mechanisms that guarantee the real value of the particle masses in question. First, the masses of Higgs particles are given by a non- Hermitian mass matrix M . Assume that the matrix possess anti-linear symmetry, which we refer to as PT symmetry, that satisfies [PT , M ] = 0, M v = λv, PT v = eiθ v, where {v, λ} are eigenvectors and eigenvalues of the mass matrix. From this, it is trivial to show that the eigenvalues are real PT M vi = PT λivi = λ∗i PT vi = λ ∗ i e iθi vi, PT M vi = M PT vi = M eiθi vi = λieiθi vi. It was shown in [8] that this PT symmetry is related to the CPT symmetry of the field-theoretic action. On the other hand, the classical energy of the soli- ton solution is found by inserting the solution into the Hamiltonian E = H[ϕ] = ∫ d3xH(ϕ). Therefore, the techniques from PT symmetric quantum mechan- ics shown above can not be applied. We will show below that the energy of the soliton solutions are real when the three conditions stated below holds. Therefore they are sufficient conditions to guarantee the real value of particles in the model. However, we do not claim that these are necessary conditions. Let {ϕ1, ϕ2} be a set of distinct (or identi- cal) solutions to the equations of motion δL/δϕ − ∂µ(δL/δ∂µϕ) = 0, where L(ϕ) is the field-theoretic Lagrangian density. The classical energies of the so- lution are given by inserting the solution into the Hamiltonian, Ei = H[ϕi] = ∫ d3xH(ϕi), for i ∈ {1, 2}. The classical mass of the solution ϕ1 and ϕ2 are real if there exist some anti-linear symmetry CPT (note that is it not the standard CPT symmetry in quantum field theory) such that three conditions are satisfied: (1.) CPT : H[ϕ(x)] → H[CPT ϕ(x)] = H†[ϕ(−x)]. (2.) CPT : ϕ1(x) → ϕ2(−x). (3.) H[ϕ1] = H[ϕ2]. If two solutions are identical ϕ1 = ϕ2, then the above condition reduces to the reality condition of the soliton solution already derived in [15]. Using the above three conditions, the real value of the classical mass can easily be shown by the following argument∫ d3xH[CPT ϕ(x)] (1) = ∫ d3xH†[ϕ(−x)] = M †1 , (2) = ∫ d3xH[ϕ2(−x)] = M2, =⇒ M †1 = M2 (3) =⇒ M †1 = M1, where numbers above the equal signs indicate the condition number. The above analysis can be performed directly on the complex model. However, the non-Hermitian theory is only well-defined once the inner-product is identified. The modern way of the well-defined non-Hermitian quantum mechanics was first realised by Frederik Scholtz, Hendrik Geyer, and Fritz Hahne in 1992, [16]. The authors used the mathematical condition on the operator called the quasi-Hermiticity (the term was first coined in [17], but the metric was 197 https://doi.org/10.14311/AP.2022.62.0197 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Takanobu Taira Acta Polytechnica not given) to define the positive definite inner prod- uct. The quasi-Hermiticity is defined as a condition on the bounded linear operator of the Hilbert space A : H → H, which satisfies (i)⟨v|ρv⟩ > 0 for all |v⟩ ∈ H and |v⟩ ≠ 0. (ii)ρA = A†ρ . Where the bounded Hermitian linear operator ρ : H → H is often called the metric operator because the inner product is defined by the operator ⟨·|·⟩ρ := ⟨·|ρ·⟩ restores the Hermiticity of the operator. This result can be shown by using the condition(ii) ⟨v|Aw⟩ ρ ≡ ⟨v|ρAw⟩ = ⟨v|A†ρw⟩ = ⟨Av|ρw⟩ = ⟨Av|w⟩ ρ , for all |v⟩ , |w⟩ ∈ H. Note that the quasi-Hermiticity alone does not guarantee the real energy spectrum of the Hamiltonian. In fact, one requires two extra conditions. (iii)The metric operator is invertible. (iv)ρ = η†η. The operator which satisfies only conditions (ii) and (iv) is referred to as the pseudo-Hermitian operator, which was first introduced in [18]. These extra con- ditions were considered in [16] to prove that, given a set of pseudo-Hermitian operators A = {Ai}, the metric operator ρA which satisfies conditions (i), (ii), (iii) and (iv) for all operators of set A is uniquely determined if and only if all operators of the set A are irreducible on the Hilbert space H. This procedure is analogous to the Dyson mapping first introduced by Freeman Dyson [19] used in the study of nuclear reaction [20–22], which maps the non-Hermitian oper- ator A to Hermitian operator η−1Aη via Dyson map η. The relation between the metric operator and the Dyson map is found by utilising the Hermiticity of the expression η−1Aη in the following way η −1 Aη = (η−1Aη)† =⇒ Aη†η = η†ηA† =⇒ η†η = ρ. (1) We will utilise this mapping to transform the non- Hermitian field-theoretic Hamiltonian to a Hermitian Hamiltonian. This procedure will resolve the issue of complex vacuum solution and Derrick’s scaling argu- ment, as we will see below. However, we note that the Dyson map used here introduces a negative kinetic sign in the kinetic term of one of the fields, indicat- ing the ghost field problem. This issue is removed if one further diagonalise the Hamiltonian. Such diago- nalisation can be realised via field-redefinition or via another Dyson map. A more detailed discussion of this is found in [23], and a Dyson map which diago- nalise the free part of the non-Hermitian Hamiltonian is found in [12]. 2. Methods In this section, we will summarise the method used in [8–11] to find the masses of the Higgs particles, mas- sive gauge particles and t’Hooft-Polyakov monopoles in non-Hermitian gauge field theory. We note that the explicit forms of the similarity transformation will not be discussed in this paper as non-Hermitian and Hermitian theories are isospectral as long as the CPT symmetry is preserved for Hamiltonian, Higgs particles and monopole solution. We begin with the non-Hermitian local SU (2) gauge theory with matter fields in the adjoint representation Lad2 = 1 4 Tr (Dϕ1) 2 + m21 4 Tr(ϕ21) (2) −i µ2 2 Tr(ϕ1ϕ2) − g 64 [ Tr(ϕ21) ]2 + 1 4 Tr (Dϕ2) 2 + m22 4 Tr(ϕ22) − 1 8 Tr ( F 2 ) . Here we take g, µ ∈ R, mi ∈ R and discrete values ci ∈ {−1, 1}. The two fields {ϕi}i=1,2 are Hermi- tian matrices ϕi(t, x⃗) ≡ ϕai (t, x⃗)T a, where ϕai (t, x⃗) is a real-valued field. The three generators {T a}a=1,2,3 of SU (2) in the adjoint representation are defined by three Hermitian matrices of the form (T a)bc = −iϵabc, satisfying the commutation relation [T a, T b] = iϵabcT c. One can check that T r(T aT b) = 2δab. The field strength tensor is defined as Fµν = ∂µAν − ∂ν Aµ − ie[Aµ, Aν ], where the gauge fields are Aµ = AaµT a. The partial derivative is replaced with the covariant derivative (Dµϕi)a := ∂µϕai + eεabcA b µϕ c i to compen- sate for the local symmetry group SU (2). This action is invariant under the local SU (2) transformation of the matter fields ϕi → eiα a(x)T a ϕie −iαa(x)T a and gauge fields Aµ → eiα a(x)T a Aµe −iαa(x)T a + 1 e ∂µα a(x)T a. It is also symmetric under modified CPT symmetry, which transforms two fields, ϕ1 and ϕ2 as CPT : ϕ1(t, x⃗) → ϕ1(−t, −x⃗) (3) : ϕ2(t, x⃗) → −ϕ2(−t, −x⃗) : i → −i. The equations of motion for the fields ϕi and Aµ of the Lagrangian (2) are (DµDµϕi) a + δV δϕa i = 0, (4) Dν F νµ a − eϵabcϕb1(Dµϕ)c + eϵabcϕb2(Dµϕ)c = 0, where repeated indices are summed over. We per- form the similarity transformation of the complex Lagrangian (2) by momentary resorting to a quantum theory where we assume an equal time commutation relation between the fields ϕai and their canonical momenta Πai = ∂0ϕ a i , satisfying the commutation re- lation [ϕai (t, x⃗), Π b j (t, y⃗)] = δ(x⃗ − y⃗)δij δab. Using this relation, we can transform the corresponding complex Hamiltonian of the Lagrangian (2) by H → eη± He−η± , (5) η± = ∏3 a=1 exp ( ± π2 ∫ d3xΠa2 ϕa2 ) , where H is the field-theoretic Hamiltonian of our model (2), obtained via Legendre transformation. This non-uniqueness of the metric is analogous to 198 vol. 62 no. 1/2022 Complex topological soliton with real energy in particle physics the non-uniqueness of the metric and its connection to the observables in the quantum mechanical set- ting discussed in [16, 24]. The adjoint action of η± maps the complex action in the equation (2) into the following real action s = ∫ d4x 1 4 T r (Dϕ1) 2 − 1 4 T r (Dϕ2) 2 (6) +c1 m21 4 T r(ϕ21) − c2 m22 4 T r(ϕ22) −c3 µ2 2 T r(ϕ1ϕ2) − g 64 ( T r(ϕ21) )2 − 1 8 T r(F 2) ≡ ∫ d4x 1 4 T r (Dϕ1) 2 − 1 4 T r (Dϕ2) 2 −V − 1 8 T r(F 2). The parameter c3 indicates the different similarity transformations by taking the values ±1 for η±, re- spectively. For convenience, let us rewrite the above real action in terms of each component of the fields ϕai as lad2 = 1 2 (Dµϕi)aIij (Dµϕj )a + 1 2 ϕai Hij ϕ a j (7) − g 16 ( ϕai Eij ϕ a j )2 − 1 4 F aµν F aµν , where the matrices H, I and E are defined as H := ( m21 −µ2 −µ2 −m22 ) , I := ( 1 0 0 −1 ) , (8) E := ( 1 0 0 0 ) . 2.1. Higgs and gauge masses Next, we define the trivial solution of the equa- tions of motion by solving δV = 0 and Dµϕi = 0. Such vacuum is often referred to as Higgs vac- uum. The first equation can be simplified by choos- ing an Ansatz (ϕ0i ) a(t, x⃗) = h0i r̂ a(x⃗) where r̂ = (x, y, z)/ √ x2 + y2 + z2 and {h0i } are some constants to be determined. Note that the vacuum solution has a rotational symmetry SO(3) since r̂ar̂a = 1. Insert- ing this Ansatz into the equation (7), we find V = − 1 2 hiHij hj + g 16 h41. (9) Then the vacuum equation δV = 0 is reduced to simple coupled third order algebraic equations g 4 (h01) 3 − c1m21h 0 1 + c3µ 2h02 = 0, (10) c2m 2 2h 0 2 + c3µ 2h01 = 0, Dµϕα = 0. The resulting vacuum solutions are h02 = − c2c3µ 2 m22 h01, (h01)2 = 4 c2µ 4+c1m21m 2 2 gm22 := R2, (11) (A0i ) a = − 1 er ϵiaj r̂j + r̂aAi, (A00)a = 0. The Ai are arbitrary functions of space-time. The Higgs particle can be identified with the fundamen- tal fields of the theory after spontaneous symmetry breaking of the continuous symmetry SU (2) by Taylor expanding around the vacuum solution. Performing a Taylor expansion around the Higgs vacuum and keeping a focus on the second-order terms with only matter fields ϕai , the real Lagrangian (7) contains the following term s = ∫ d4x 1 2 ϕai ( −∂µ∂µIij δab − H̃ abij ) ϕbj + . . . , where H̃ is a 6 × 6 block diagonal Hermitian matrix. After diagonalising the above term by redefining the fields with the eigenvectors of the non-Hermitian mass matrix M abij := Iikδ acH̃ cbkj , we find that the masses of the fundamental fields after the symmetry breaking to be equal to the eigenvalues of M abij given as m20 = c2 µ4 − m42 m22 , m2± = K ± √ K 2 + 2L, (12) where K = c1m21 − c2 m22 2 + 3µ4 2c2m22 and L = µ4 + c1c2m 2 1m 2 2. Notice that we only find three non-zero eigenvalues. The redefined fields with zero masses (eigenvalues) are called Goldstone fields, which can be absorbed into gauge fields Aaµ by defining the new massive gauge fields. This process of giving mass to the previously massless fields is called the Higgs mech- anism. The mass of the gauge fields can be found by expanding the kinetic term of ϕ around the Higgs vacuum ϕ0i = h 0 i r̂ a. Without loss of generality, we can choose a particular direction of the vacuum by taking r̂ = (0, 0, 1)T . This is possible due to the symmetry of the SO(3) vacuum as discussed above. Keeping the term only quadratic in the gauge field, we find 1 2 (Dµϕi + Dµϕ0i ) aIij (Dµϕi + Dµϕ0i ) a (13) = 1 2 ( eAµ × ϕ0i )a Iij ( eAµ × ϕ0j )a + . . . = 1 2 e2h0i Iij h 0 j ( A1µA 1µ + A2µA 2µ ) + . . . = 1 2 m2g ( A1µA 1µ + A2µA 2µ ) + . . . , where the mass of the gauge field is identified to be mg := √ h0i Iij h 0 j = e R √ m42−µ 4 m22 . 2.2. t’Hooft-Polyakov monopole To find the monopole solutions, let us consider the following Ansatz (ϕcli ) a(x⃗) = hi(r)r̂a , (Acli ) a = ϵiaj r̂j A(r) , (14) (Acl0 )a = 0, where the subscript cl denotes the classical solutions to the equations of motion (4). The difference be- tween this Ansatz (14) and the Higgs vacuum (11) 199 Takanobu Taira Acta Polytechnica is that the quantity hi now depends on the spatial radius hi = hi(r). Here we are only considering the static Ansatz to simplify our calculation, but one may, of course, also consider the time-dependent solution by utilising the Lorentz symmetry of the model and performing a Lorentz boost. According to Derrick’s scaling argument [25], for the monopole solution to have finite energy, we require the two matter fields of the equation (14) to approach the vacuum solutions in the equation (11) at spatial infinity lim r→∞ h1(r) = h0±1 = ±R , (15) lim r→∞ h2(r) = h0±2 = ∓ c2c3µ 2 m22 R. Also, notice that at some fixed value of the radius r, the vacuum solutions ϕ0α and monopole solutions ϕclα both belongs to the 2-sphere in the field configuration space. For example, ϕ01 belongs to the 2-sphere with radius R because (ϕ01)2 = R2. Therefore, solutions ϕcli can be seen as a mapping between 2-sphere in space- time (where the radius is given by the profile function hi) to 2-sphere in field configuration space. Such mapping has a topological number called the winding number n ∈ Z, which can be explicitly realised by redefining the unit vector r̂a as r̂an =   sin(θ) cos(nφ)sin(θ) sin(nφ) cos(θ)   . (16) Therefore different n represent topologically inequiva- lent solutions. Since we require the monopole and vacuum solu- tions to smoothly deformed into each other at spacial infinity, both solutions need to share the same wind- ing number. It is important to note that winding numbers of ϕ1 and ϕ2 need to be equal to satisfy Dϕ1 = Dϕ2 = 0, and therefore we will denote the winding numbers of ϕ1 and ϕ2 as n collectively. If they are not equal, we would have Dϕ1 = 0 but Dϕ2 ̸= 0. Next, let us insert our Ansatz equation (14) into the equations of motion equation (4). We will also redefine the Ansatz for the gauge fields to be Aai = ϵ aibr̂b ( 1−u(r) er ) , Aa0 = 0, which are more in line with the original Ansatz given in [26, 27], compared to equation (14). Inserting these expressions into the equations of motion equation (4), we find u ′′ (r) + u(r) [ 1 − u2(r) ] r2 (17) + e2u(r) 2 { h22(r) − h 2 1(r) } = 0, h ′′ 1 (r) + 2h ′ 1(r) r − 2h1(r)u 2(r) r2 (18) +g { −c1 m21 g h1(r) + c3 µ 2 g h2(r) + 14 h 3 1(r) } = 0, h ′′ 2 (r) + 2h ′ 2(r) r − 2h2(r)u2(r) r2 (19) +c2m22 { h2(r) + c3 µ2 m22 h1(r) } = 0. Notice that these differential equations are similar to the ones discussed in [26, 27], but with the extra field h2 and extra differential equation (19). In the Hermitian model, the exact solutions to the differential equations were found by taking the parameter limit called the BPS limit [26, 27], where parameters in the Hermitian model are taken to zero while keeping the vacuum solution finite. Here we will follow the same procedure and take the parameter limit where quantities in the curly brackets of equations (18) and (19) vanish but keep the vacuum solutions equation (11) finite. We will see in section 2.4 that we also find the approximate solutions in this limit. 2.3. The energy bound Surprisingly, by utilising Derrick’s scaling argument, one can find the lower bound of the monopole energy without the explicit form of the solution. The energy of the monopole can be found by in- serting the monopole solution into the corresponding Hamiltonian of equation (6). h = ∫ d3x T r ( E2 ) + T r ( B2 ) (20) +T r { (D0ϕ1)2 } + T r { (Diϕ1)2 } −T r { (D0ϕ2)2 } − T r { (Diϕ2)2 } + V, where E, B are Eia = Fa0i , Bia = − 12 ϵ ijkF jka , i, j, k ∈ {1, 2, 3}. The gauge is fixed to be the ra- diation gauge (i.e Aa0 = 0, ∂iAai = 0). Notice that our monopole Ansatz equation (10) is static with no electric charge Eai = 0 and therefore, the above Hamiltonian reduces to E = ∫ d3x T r ( B2 ) + T r { (Diϕ1)2 } (21) −T r { (Diϕ2)2 } + V = 2 ∫ d3x Bi aBi a + (Diϕ1)a(Diϕ1)a −(Diϕ2)a(Diϕ2)a + 1 2 V. Here, we simplified our expression by dropping the superscripts Acli → Ai , ϕ cl α → ϕα. We also keep in mind that these fields depend on the winding numbers n ∈ Z. In the Hermitian model (i.e. when ϕ2 = 0), one can rewrite the kinetic term as B2 + Dϕ2 = (B − Dϕ)2 + 2BDϕ and find the lower bound to be∫ 2BDϕ. Here we will follow a similar procedure but introduce some arbitrary constant α, β ∈ R such that B2 = α2B − β2B where α2 − β2 = 1. This will allow 200 vol. 62 no. 1/2022 Complex topological soliton with real energy in particle physics us to rewrite the above energy as E = 2 ∫ d3x α2 { Bi a + 1 α (Diϕ1)a }2 (22) −β2 { Bi a + 1 β (Diϕ2)a }2 +2 {−αBia(Diϕ1)a + βBia(Diϕ2)a} + 1 2 V. To proceed from here, we need to assume extra con- straints on α and β such that the following inequalities are true∫ d3x α2 { Bi a + 1 α (Diϕ1)a }2 (23) −β2 { Bi a + 1 β (Diϕ2)a }2 ≥ 0,∫ d3xV ≥ 0. With these constraints we can now write down the lower bound of the monopole as E ≥ 2 ∫ d 3 x {−αBia(Diϕ1)a + βBia(Diϕ2)a} (24) = 2 ∫ d 3 x − α { Bi a ∂iϕ a 1 + eBi a ϵ abc Ai b ϕ c 1 } +β { Bi a ∂iϕ a 2 + eBi a ϵ abc Ai b ϕ c 2 } = 2 ∫ d 3 x − α { Bi a ∂iϕ a 1 + ( −eϵabcAibBic ) ϕ a 1 } +β { Bi a ∂iϕ a 2 + ( −eϵabcAibBic ) ϕ a 1 ϕ c 2 } = 2 ∫ d 3 x − α {Bia∂iϕa1 + ∂iBi a ϕ a 1 } +β {Bia∂iϕa2 + ∂iBi a ϕ a 1 } = 2 ∫ d 3 x − α∂i (Biaϕ1a) + β∂i (Biaϕ2a) = lim r→∞ ( −2α ∫ Sr dSiBi a ϕ1 a + 2β ∫ Sr dSiBi a ϕ2 a ) , where in the fourth line, we used DiBai = 0, which can be shown from the Bianchi identity Dµϵµνρσ F aρσ = 0. The last line is obtained by using the Gauss theorem at some fixed value of the radius r. Since the ϕai in the integrand is only defined over the 2-sphere with a large radius, we can use the asymptotic conditions (15) and replace the monopole solutions {ϕaα, Bai } with the Higgs vacuum {(ϕ0α)a, (B0i ) a} E ≥ ( −2αϕ01 a + 2βϕ02 a) lim r→∞ ∫ Sr dSi(B0i ) a (25) = ( ∓2αRr̂an ∓ 2β c2c3µ 2 m22 Rr̂ a n ) lim r→∞ ∫ Sr dSi(B0i ) a , where the upper and lower signs of the above energy correspond to the upper and lower signs of the vacuum solutions in equation (11). The explicit value of Ba0i can be obtained by inserting the Higgs vacuum (11) into the definition of the magnetic field Bai = − 1 2 ϵi jk (∂j Ak − ∂kAj + eAj × Ak) a . (26) After a lengthy calculation, this expression can be simplified to Ba0i = ϕ̂0 a bi = r̂anbi, where ϕ̂0 a is a nor- malised solution ∑ a ϕ̂ 0 a ϕ̂0 a = 1. The bi is defined as bi ≡ − 1 2 ϵijk { ∂j Ak − ∂kAj + 1 e r̂n · ( ∂j r̂n × ∂kr̂n )} . (27) Where A was defined in equation (11). Notice that integrating the first term over the 2-sphere gives zero by Stoke’s theorem ∫ S ∂ × A = ∫ ∂S A = 0, where one can show that Stoke’s theorem on the closed surface gives zero by dividing the sphere into two open surfaces. The second term is a topological term which can be evaluated as∫ dSiBi = − 4πn e . (28) The explicit calculation is in [28]. This is the magnetic charge of the monopole solutions. Therefore integer n, which corresponds to the winding number of the solution, comes from the Ansatz Bai = ϕ̂0 a bi. In our case, there is an ambiguity of whether to choose Bai = ϕ̂01 a bi or Bai = ϕ̂02 a bi. Now we see explicitly the reason why we choose to keep the same integer values for solutions ϕ01 and ϕ02. If the integer values of r̂an in solutions ϕ01, ϕ02 are different, then the integration∫ Sr dSi(B0i ) a will be different, leading to inconsistent energy. Finally, we find our lower bound of the monopole energy E ≥ ∓2R ( α + β c2c3µ 2 m22 ) r̂anr̂ a n ( −4πn e ) (29) = ±8πnR e ( α + β c2c3µ 2 m22 ) . Notice that we have some freedom to choose α, β ∈ R as long as our initial assumptions (23) are satisfied. We will see in the next section that we can take a pa- rameter limit of our model, which saturates the above inequality and gives exact values to α and β. 2.4. The fourfold BPS scaling limit Our main goal is now to solve the coupled differential equations (17)-(19). Prasad, Sommerfield, and Bo- gomolny [26, 27] managed to find the exact solution by taking the parameter limit, which simplifies the differential equations. The multiple scaling limit is taken so that all the parameters of the model tend to zero with some combinations of the parameter re- maining finite. The combinations are taken so that the vacuum solutions stay finite in this limit. Inspired by this, we will take here a fourfold scaling limit g, m1, m2, µ → 0 , m21 g < ∞ , µ2 g < ∞ , µ2 m22 < ∞. (30) This will ensure that the vacuum solutions equation (11) stays finite, but crucially the curly bracket parts 201 Takanobu Taira Acta Polytechnica in equations (18) and (19) vanish. There is a physical motivation for this limit in which the mass ratio of the Higgs and gauge mass are taken to be zero (i.e. mHiggs << mg ) as described in [29]. We will see in the next section that the same type of behaviour is present in our model, hence justifying equation (30). The resulting set of differential equations, after taking the BPS limit, is similar to the ones considered in [26, 27] with the slightly different quadratic term in equation (17). It is natural to consider a similar Ansatz as given in [26, 27] u(r) = evr sinh (evr) , (31) h1(r) = −αf (r), (32) h2(r) = −βf (r), (33) where α, β ∈ R were introduced in section 2.3 and f (r) ≡ { v coth (evr) − 1 er } . One can check that this Ansatz indeed satisfies differential equations equation (17)-(19) in the BPS limit. We have decided to put a prefactor α and β in front of equations (32) and (33) to satisfy the differential equation (17). Note that if we take α = 1, we get exactly the same as given in [26, 27], which is known to satisfy the first-order differential equation called the Bogomolny equation Bi − Diϕ = 0. The Ansatz (31)-(33) only differs from the ones given in [26, 27] by the prefactors α and β, and therefore our Ansatz should satisfy the Bogomolny equation with the appropriate scaling to cancel the prefactor in equations (32) and (33) Bbi + 1 α (Diϕ1)b = 0, (34) Bbi + 1 β (Diϕ2)b = 0, (35) where ϕα ≡ hα(r)r̂n. If we compare these equations to the terms appearing in the energy of the monopole equation (22), then we can saturate the inequality in equation (29) by E[ϕ1, ϕ2] = ±8πnR e ( α + β c2c3µ 2 m22 ) , (36) where upper and lower signs correspond to the vacuum solutions equation (11), when taking the square root. We can calculate the explicit forms of α and β by comparing the asymptotic conditions in equation (15) lim r→∞ h±1 = h 0± 1 = ±R, (37) lim r→∞ h±2 = h 0± 2 = ∓ c2c3µ 2 m22 R, with the asymptotic values of equations (31)-(33) limr→∞ u(r) = 0 , limr→∞ h±1 (r) = −αv, (38) limr→∞ h±2 (r) = −βv. By Derrick’s scaling argument, the two asymptotic values (37) and (38) should match, resulting in al- gebraic equations for α and β. Using α2 − β2 = 1 and assuming m42 ≥ µ4, we find the four set of real solutions α = ∓(±) m22 l , v = (±) Rl m22 , β = ±(±) c2c3µ 2 l , (39) where l = √ m42 − µ4. The plus-minus signs in the brackets correspond to the two possible solutions to the algebraic equation α2 − β2 = 1. These need to be distinguished from the upper and lower signs of α and β, which correspond to the vacuums solutions (11). Inserting the explicit values of α and β to the energy equation (36) we find E[ϕ1, ϕ2] ≡ (±) 8πnR em22 ( −m42 + µ4 l ) (40) = (±) −8πnR em22 l, with corresponding solutions h±1 (r) = ±(±) m22 l [ Rl m22 coth ( eRl m22 r ) − 1 er ] , (41) h±2 (r) = ∓(±) c2c3µ 2 l [ Rl m22 coth ( eRl m22 r ) − 1 er ] . It is crucial to note that although it seems like there are two monopole solutions {h±1 , h ± 2 }, the two solutions are related non-trivially in their asymptotic limit by the constraint limr→∞ h±2 = (−c2c3µ2/m22) limr→∞ h ± 1 given in equation (10). For example, one can not choose {h+1 , h − 2 } as a solution as this will break the asymptotic constraint. The solution (41) can be constrained further by imposing that the energy (40) is real and positive. E[ϕ1, ϕ2] > 0 =⇒ −(±) 8πnR em22 l =⇒ −(±)n > 0. (42) Therefore we can ensure positive energy if (±) = sign(n). The final form of the monopole solution with positive energy are h ± 1 (r) = ±sign(n) m22 l [ Rl m22 coth ( eRl m22 r ) − 1 er ] , (43) h ± 2 (r) = ∓sign(n) c2c3µ 2 l [ Rl m22 coth ( eRl m22 r ) − 1 er ] . with energy E = 8|n|πlR/em22. We conclude this subsection by observing that the above solution de- pends on the parameter c3, which takes value {−1, 1} depending on the choice of the similarity transforma- tion. Choosing a different values of c3 also result in a different asymptotic values (37), meaning solutions for c3 = 1 and c3 = −1 are topologically different. Since the energy is independent of c3, two distinct solutions share the same energy. Respecting one of the main features of similarity transformation, which is to preserve the energy of the transformed Hamiltonian. In the next section, we will investigate in detail how the solution changes and a new CPT symmetry emerges by changing the parameter values. 202 vol. 62 no. 1/2022 Complex topological soliton with real energy in particle physics Figure 1. Monopole, gauge and Higgs masses plotted for m21/g = −0.44, µ/g = −0.14, e = 2, c1 = −c2 = −1. The solid line represents the real part, and the dotted line represents the imaginary part of the masses. The dotted vertical lines indicate the boundaries of the physical regions where all the masses acquire real positive values. 3. Results and discussion This section will investigate the behaviour of solution (43) in different regimes of the parameter spaces. We will compare the physical regions of gauge particles, Higgs particles and monopoles found in the previous section. We will see that the two regions coincide, but the solutions in different regions possess different CPT symmetries. Different symmetries of solutions in different regions are not coincident, but the conse- quence of the three reality conditions stated in the introduction. In fact, it is deeply related to the real value of energy, which will be discussed extensively in [30]. 3.1. Higgs mass and exceptional points Let us recall the masses of the particles and monopole m20 = c2 µ4−m42 m22 , m2± = K ± √ K 2 + 2L, (44) mg = e Rlm22 , Mmono = 8|n|πlR em22 . where K = c1m21 − c2 m22 2 + 3µ4 2c2m22 and L = µ4 + c1c2m 2 1m 2 2. Notice that the masses do not depend on c3, meaning they do not depend on the similarity transformation as expected. We also comment that in the BPS limit, we have m0 = m± = 0, but mg and M± stays finite, such that the ratios mHiggs/mg vanish in the BPS limit. This is in line with the Her- mitian case [29], providing the physical interpretation with mHiggs << mg for the BPS limit. One may notice that when c2 = 1, requiring positive mass m20 > 0, implies that µ4 − m42 > 0. This means the quantity l = √ m42 − µ4 is purely imaginary. One may then discard this region as unphysical. However, we will see in the next section that there is a discon- nected region beyond µ4 − m42 > 0, which admit real energy because R also becomes purely complex. This is not coincident, and in fact, we will see an emerging new CPT symmetry for the monopoles. In the rest of the section, we will exclusively fo- cus on the monopole and gauge masses. The main message of this section is the emerging symmetry re- sponsible for the real value of the monopole masses. The requirement to make the whole theory physical de- mands also to consider the intersection of the physical regions between monopole masses and Higgs masses. As an example, we plot all the masses of the theory in Figure 1. As one can see, intersection points of the physical re- gions of Higgs masses and monopole/gauge masses are non-trivial. In fact, they are bounded by two types of exceptional points. The first type is when two masses of Higgs particles coincide and form a complex con- jugate pair. Such a point is known as an exceptional point where the mass matrix is non-diagonalisable, and the corresponding eigenvectors coincide. The sec- ond type is when the gauge and the monopole masses vanishes. Interestingly, this is where one of the Higgs masses also vanish. Since the mass matrix already has a zero eigenvalue, as the result of the spontaneous symmetry breaking, it seems the number of massless fields is increased. However, at this point, the mass matrix is also non-diagonalisable. Therefore one can not diagonalise the Hamiltonian to identify the field which corresponds to the extra massless fundamen- tal field. Therefore this point is also an exceptional point. However, the eigenvalues do not become com- plex conjugate pairs beyond this point, and as one can see from Figure 1 that one of the mass square m20 become negative, and gauge and monopole masses become complex but with no conjugate pair. We dub such a point as zero exceptional point to distinguish from the standard exceptional point. 3.2. Change in CPT symmetry and complex monopole solution We begin by introducing the useful quantities m21/g ≡ X, µ2/g ≡ Y, µ2/m22 ≡ Z. The gauge mass, monopole mass and monopole solutions can be rewritten in terms of these quantities mg = eR √ 1 − Z 2, mmono = 8|n|πR e √ 1 − Z 2, (45) 203 Takanobu Taira Acta Polytechnica Figure 2. Monopole and gauge masses plotted for X = 1, Y = 0.8, e = 2, c1 = −c2 = 1. The solid line represents the real part, and the dotted line represents the imaginary part of the masses. Figure 3. Both panels are plotted for X = 1, Y = 0.8, n = 1, e = 2. The solid line represents the real part, and the dotted line represents the imaginary part of the masses. Panel (a) shows the monopole and gauge masses against Z ≥ 0, with vertical lines indicating the location of the boundaries of three regions. Panel (b) shows three profile function h1(r) defined on each region indicated in panel (a). h ± 1 (r) = ± sign(n) √ 1 − Z 2 [ R √ 1 − Z 2coth(r̂) − 1 er ] , (46) h ± 2 (r) = ∓ sign(n)c2c3Z√ 1 − Z 2 [ R √ 1 − Z 2coth(r̂) − 1 er ] , (47) where R2 = 4(c2ZY + c1X) and r̂ = eR √ 1 − Z 2r. The monopole masses are plotted against the gauge mass for fixed parameters with n ∈ {1, 2, 3, 4} in Figure 2 with weak and strong couplings e = 2, e = 10. Notice that the gauge mass is smaller than any of the monopole masses for weak coupling, but when e is large enough, some of the monopole masses can become smaller than the gauge mass. This is clear by inspecting the monopole, and gauge mass in equation (45) and two masses coincide when e = √ 8|n|π. Note that n = 0 is not a monopole mass as it corresponds to the solution with zero winding number, which is topologically equivalent to the trivial solution. From Figure 2, we also observe disconnected regions where both monopole and gauge masses become real to purely complex. A more detailed plot of this is shown in Figure 3. Region 2 is bounded by two points with lower bound µ2/m22 = 1 corresponding to the zero exceptional point where the vacuum manifolds stay finite (i.e. spontaneous symmetry breaking oc- cur). However, the Higgs mechanism fails because the Hamiltonian is non-diagonalisable, as discussed in the previous section. The upper bounds correspond to the point where the vacuum manifold vanishes. There- fore, the spontaneous symmetry breaking does not occur, implying that the gauge fields do not acquire a mass through the Higgs mechanism, resulting in a massless gauge field. Most crucially, an interesting region (denoted by region 3 in Figure 4) reappears as one increases the value of Z. The profile function in region 3 is purely complex, which signals that this may lead to complex energies. However, as one can see from Figure 3, the energy is real. The reason for the real energy is that the conditions stated in the introduction hold. We will specify below the CPT symmetry responsible for the real value of the energy. Note that the profile function h2 only differ from h1 by some factor in front. Therefore we omitted it from the plot. Another physical region is when c1 = −c2 = −1. The monopole and gauge masses for this case is plotted in Figure 4. We observe almost an identical plot from the Figure 3 but with real and imaginary parts swapped. The profile functions also respect these changes as regions 1 and 3 no longer have a definite asymptotic value. The boundaries are unchanged, as one can see from the Figure 4. Finally, there is an interesting parameter point X = Y where region 2 vanishes (see Figure 5). The two boundaries Z 2 = 1 and c2ZY + c1X = 0 coincide when X = Y and the zero exceptional point no longer exists because the spontaneous symmetry breaking does not occur in this case. Next, let us explain the real value of the energies in different regions. First, to realise the conditions 1-3, stated in the introduction, we require the following 204 vol. 62 no. 1/2022 Complex topological soliton with real energy in particle physics Figure 4. Both panels are plotted for X = 1, Y = 1, n = 1, e = 2. The solid line represents the real part, and the dotted line represents the imaginary part of the masses. Figure 5. Both panels are plotted for X = 1, Y = 1, n = 1, e = 2. The solid line represents the real part, and the dotted line represents the imaginary part of the masses. transformations h±2 (r) → −h ± 2 (r) , h ± 1 (r) → h ± 1 (r) in region 1 No symmetry in region 2 h±2 (r) → − ( h±2 (r) )∗ , h±1 (r) → ( h±1 (r) )∗ in region 3 . By using the explicit forms of the solutions (46) and (47). We can show that the above transformations sat- isfy condition 2 stated in the introduction, in regions 1 h±2 (r) → −h ± 2 (r) = h ∓ 2 (r) , h ± 1 (r) → h ± 1 (r), (48) and in region 3 h±2 (r) → − ( h±2 (r) )∗ = h±2 , (49) h±1 (r) → ( h±1 (r) )∗ = h∓1 . Notice that in regions 1 and 3, the CPT relates two distinct solutions in two different ways. For example, h±2 is mapped to h ∓ 2 in region 1, but it is mapped to itself in region 3. Finally, the condition 3 stated in the introduction is satisfied because the energy does not depend on the ± signs of the solutions. This explains the real energies of complex monopoles in region 3 and complex energy in region 2. Indeed, we observe the predicted behaviour in Figure 3. Region 2 is a hard barrier between two CPT symmetric regions where solutions are either real or purely imaginary. The same analysis can be carried out in the other physical region c1 = −c2 = −1 where the symmetry is now No symmetry in region 1 h ± 2 (r)→−(h±2 (r)) ∗ h ± 1 (r)→(h±1 (r)) ∗ in region 2 No symmetry in region 3 . (50) We have observed that one can find a well-defined monopole solution in two disconnected regions. How- ever, in the full theory where we include the Higgs particles, it is only one of the regions which are con- sidered physical. This is because the Higgs mass m20 is either positive or negative depending on which side of Z 2 = 1 it is defined. Because two disconnected re- gions are defined on either side of the zero exceptional point Z 2 = 1, the full physical region restricts one from moving region 1 to region 3 by changing Z. This is most clearly seen in Figure 1 where the plot of m20 (green line) becomes negative beyond the zero excep- tional point. This may imply that the purely complex monopole solution we observed is not a possible so- lution of the theory. However, the purely complex solution can exist in the full physical region. An exam- ple of this is shown in the Figure 6 where we observe that the profile function h1 (therefore h2) is purely complex, and the Higgs masses, gauge mass are all real and positive. 4. Conclusions We have found the t’Hooft-Polyakov monopole solu- tion (43) in the non-Hermitian theory by drawing an 205 Takanobu Taira Acta Polytechnica Figure 6. Both panels are plotted for X = −2, Y = −0.6, c1 = −c2 = 1, n = 1, e = 2. The solid line represents the real part, and the dotted line represents the imaginary part of the masses. analogue from the standard procedure in the Hermi- tian theory. 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Nuclear Physics B 971(2):115516, 2021. https://doi.org/10.1016/j.nuclphysb.2021.115516. 207 https://doi.org/10.1007/BF02724848 https://doi.org/10.1103/PhysRev.102.1230 https://doi.org/10.1143/PTP.31.1009 https://doi.org/10.1016/0029-5582(62)90416-9 https://doi.org/10.1016/0375-9474(71)90122-9 https://doi.org/10.1088/1742-6596/2038/1/012010 https://doi.org/10.1088/1751-8113/40/2/F03 https://doi.org/10.1063/1.1704233 https://doi.org/10.1103/PhysRevLett.35.760 https://doi.org/10.1063/1.522518 https://doi.org/10.1103/PhysRevD.24.999 https://doi.org/10.1016/j.nuclphysb.2021.115516 Acta Polytechnica 62(1):197–207, 2022 1 Introduction 2 Methods 2.1 Higgs and gauge masses 2.2 t'Hooft-Polyakov monopole 2.3 The energy bound 2.4 The fourfold BPS scaling limit 3 Results and discussion 3.1 Higgs mass and exceptional points 3.2 Change in CPT symmetry and complex monopole solution 4 Conclusions Acknowledgements References