Eur. Phys. J. C (2016) 76:98 DOI 10.1140/epjc/s10052-016-3944-9

Regular Article - Theoretical Physics

Spontaneous symmetry breaking, and strings defects in hypercomplex gauge field theories

R. Cartas-Fuentevilla1,a, O. Meza-Aldama2 1 Instituto de Física, Universidad Autónoma de Puebla, Apartado postal J-48, 72570 Puebla, Pue., Mexico 2 Facultad de Ciencias Físico-Matemáticas, Universidad Autónoma de Puebla, Apartado postal 1152, 72001 Puebla, Pue., Mexico

Received: 22 November 2015 / Accepted: 11 February 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract Inspired by the appearance of split-complex solutions representing the so-called BPS instantons and 3- structures in the dimensional reduction of string theory, and in branes. In the same context, the Lagrangian and the super- the theories emerging as byproducts, we study the hypercom- symmetric rules used in [7] for a description of the so-called plex formulation of Abelian gauge field theories by incorpo- D-instantons in terms of supergravity require by consistency rating a new complex unit to the usual complex one. The a substitution rule that changes the standard imaginary unit hypercomplex version of the traditional Mexican hat poten- i with i2 =−1, by a formally new imaginary unit j with tial associated with the U(1) gauge field theory, corresponds j2 = 1, which is different from ±1, and which corresponds to a hybrid potential with two real components, and with to an algebra of para-complex numbers. The formal descrip- U(1)×SO(1, 1) as symmetry group. Each component corre- tion of such a mysterious substitution rule is given in [8,9] sponds to a deformation of the hat potential, with the appear- in terms of para-complex manifolds endowed with a spe- ance of a new degenerate vacuum. Hypercomplex electro- cial para-Kähler geometry, with applications in the study dynamics will show novel properties, such as spontaneous of instantons, solitons, and cosmological solutions in super- symmetry breaking scenarios with running masses for the gravity and M-theory. vectorial and scalar Higgs fields, and such as Aharonov– In a different context, the para-complex numbers appear Bohm type strings defects as exact solutions; these topologi- as a hyperbolic unitary extension of the usual complex phase cal defects may be detected only by quantum interference of symmetry of electromagnetism in order to generalize it to charged particles through gauge invariant loop integrals. In a gravito-electromagnetic gauge symmetry [10]. Addition- a particular limit, the hyperbolic electrodynamics does not ally an alternative representation of relativistic field theo- admit topological defects associated with continuous sym- ries is given in [11–13] in terms of hyperbolic numbers; metries. in particular the Dirac equation and the Maxwell equations admit naturally such a representation, in which one has both the ordinary and the hyperbolic imaginary units; along the 1 Introduction same lines it is shown that the (1 + 1) string world-sheet possesses an inherent hyperbolic complex structure [13]. Explorations involving hypercomplex structures have Furthermore, in [14] the requirement of hermiticity on the appeared recently in the literature, for example, in the Poincaré mass operator defined on the commutative ring of dimensional reduction of M-theory over a Calabi–Yau man- the hyperbolic numbers H leads to a decomposition of the ifold, where a five-dimensional N = 2 supergravity theory corresponding hyperbolic Hilbert space into a direct prod- emerges. It turns out that the hyperbolic representation based uct of the Lorentz group related to the space-time symme- on para- or split-complex numbers is the most natural way tries and the hyperbolic unitary group SU(4, H), which is to parametrize the scalar fields of the five-dimensional uni- considered as an internal symmetry of the relativistic quan- versal multiplet, gaining insight in the understanding of the tum state; the hyperbolic unitary group is equivalent to the string theory landscape [1–6]. In this context, switching on group SU(4, C) × SU(4, C) of the Pati–Salam model [15]. the split-complex form of the theory solves automatically In [16] the hyperbolic Klein–Gordon equation for fermions the inconsistencies related with the finding of well-behaved and bosons is considered as a para-complex extension of groups and algebras formulated in terms of the product of a e-mail: [email protected] ordinary complex and hyperbolic units; this implies the exis- 123 98 Page 2 of 22 Eur. Phys. J. C (2016) 76:98 tence of hyperbolic complex gauge transformations and the bolic symmetry breaks down. Conveniently interpreted, these possibility of new interactions; however, although certainly results are consistent with the lack of strong and convincing there are no current experimental indications of them, either evidence of the existence of cosmic topological defects, and evidence against their existence. If these new interactions are consequently, with the possible presence of the hyperbolic effectively absent, then it is of interest to understand why the symmetry at some stage in the sequence of symmetry break- hyperbolic complex counterparts of the other interactions do downs in the early-universe phase transitions. Incidentally not exist in nature at presently known energies, in spite of the incorporation of non-compact gauge groups will allow the consistency of the hyperbolic extensions from the the- us to gain insight in certain aspects of gravity theories from oretical point of view. However, there exists the interesting the perspective of a deformed version of conventional gauge possibility of realizations of those hyperbolic counterparts theories. beyond presently known energies, including those close to Topologicalstructures have been object of intense research the Planck scale. due to their relevance in confinement and chiral symmetry On the other hand, the presence of hyperbolic phases breaking phenomena in quantum chromodynamics; it is well implies symmetries associated with non-compact gauge known the role that the monopoles may play as a possible groups; these symmetries can be realized as symmetries of source of confinement [21,22]; more recently the so-called the background space-time and as internal gauge symmetries; dyons have been considered as an alternative source for such for instance, within the context of gravity, it is well known a phenomenon [23]. In general the topological structure of the appearance of non-compact internal symmetries, related any gauge theory is conditioned by the existence of nontriv- to its invariance under diffeomorphisms as the fundamen- ial homotopy groups, and these are determined by the spe- tal gauge symmetry. Furthermore, although gauge theories cific gauge symmetry groups and their stability subgroups; are typically discussed for compact gauge groups, the inte- recently the Weyl symmetric structure of the classical QCD grable sectors of QDC, ghost-, and θ-sectors manifest the vacuum has been described by a second homotopy group presence of non-compact gauge groups [17], with surpris- constraint, which determines the monopole charge [24]. Sim- ing new features. For example, the spontaneous symmetry ilarly, the knot topology of QCD vacuum is determined by a breaking is possible in low dimensions provided that non- third homotopy group constraint [25]; this approach has been compact groups are present, evading the Mermin–Wagner useful in the construction of new analytic solutions. With theorem ([18], and references therein); at the quantum level, these motivations, in this work we determine the effects of the Hilbert space is non-separable. Similarly within the study the incorporation of hyperbolic rotations as part of the inter- of quantum non-compact σ models, as opposed to the case of nal gauge group that usually involves only compact gauge quantum electrodynamics, the theory can be correctly quan- groups, on the topological structures of the considered theo- tized only in a Hilbert space with indefinite metric [19]; in ries. the case of a positive-definite Hilbert space, the quantization In Sect. 3 we consider the hypercomplex numbers with requires an extended space that incorporates negative-energy the purpose of introducing the hyperbolic phases as part of modes [20]. In general the classical and quantum descrip- the gauge symmetry group, which in general will include the tions of non-compact σ models show problems related to the usual U(1) compact phases. Then the hypercomplex defor- non-unitarity of the S matrix and the spontaneous symmetry mation with global phases of the classical massive λφ4 model breaking realizations. is developed in Sect. 4; in particular in Sect. 4.1 the purely hyperbolic version of the traditional mexican hat potential is analyzed; hyperbolic version means the substitution of the 2 Motivations and an advance of results usual complex unit i by j. This theory does not allow topolog- ical defects associated with continuous symmetries. In Sects. One of the motivations of the present work is to explore 4.2 and 4.3 the hyperbolic deformations of the Mexican hat the realizations of the hyperbolic symmetries as an internal potential is developed; deformation will imply the incorpo- gauge symmetry in classical gauge field theories; we deter- ration of the new complex unit j, to the usual unit i. In Sect. mine the effects of the incorporation of those symmetries on 4.4 we describe geometrical and topologically the vacuum the geometry and topology of the vacuum manifolds, and the manifold, which will correspond to a two-dimensional non- subsequent effect on the formation of topological defects. compact space embedded in the four-dimensional hypercom- We shall find that by switching on the hyperbolic structures, plex space as ambient space; the homotopy constraints are the degenerate vacuum in gauge theories can correspond to analyzed. The polar description of the spontaneous symmetry non-compact manifolds; the non-compact character of the breakdown is developed in Sect. 4.5; the polar parametriza- vacuum will have a nontrivial effect on the possible forma- tion for the fields will allow us to describe circular, hyper- tion of topological defects associated to continuous symme- bolic, and radial oscillations, and we shall make a comparison tries through the Kibble mechanism, when the new hyper- with the usual treatments that involve only compact gauge 123 Eur. Phys. J. C (2016) 76:98 Page 3 of 22 98 groups. The formation of possible topological defects are or |trace| > 2, respectively. The respective subgroups are analyzed in Sect. 4.6; in this section Derrick’s theorem is con- obtained by incorporating ±I , with I the identity element; firmed. Furthermore, in the case of theories with U(1) gauge in particular, the elements of the hyperbolic subgroup are symmetries, the conventional vacuum manifold is defined identified with squeeze mappings, which geometrically cor- by the bottom of the Mexican hat potential, the circle; in the respond to preserving hyperbolas in the plane, with the hyper- formulation at hand that circle will be retained as a compact bolic angle playing the role of invariant measure of the sub- transversal section of the new non-compact vacuum manifold group. Since the image points of the squeeze mapping are that incorporates the hyperbolic phases. on the same hyperbola, such a mapping preserves the form Finally in Sect. 5 the circular and hyperbolic local rota- x · y, and can be identified with a hyperbolic rotation in tions are considered by the coupling to hypercomplex electro- analogy with circular rotations preserving circles. The hyper- dynamics; this theory leads to spontaneous symmetry break- bolic subgroup will play a central role in this paper, since the ing scenarios with hypercomplex scalar and vectorial fields invariance under its action will be revealed as a fundamen- with running masses which mimic the flows of the renormal- tal internal symmetry in field theory, due to the fact that the ization groups. In particular, in Sect. 5.3 a purely massive invariant form x · y appears recurrently in physics. electrodynamics, without the presence of scalar Higgs fields If h is a hyperbolic element, then |trace(h)| > 2, and is obtained. In Sect. 5.4 the local topological strings are stud- det(h) = 1, and can be parametrized as ied; these defects turn out to be of Aharonov–Bohm type,   detectable only by quantum interference of charged parti- ηe jχ 0 h = ,η=±1,χ∈ R −{0}; (3) cles, in consistency with previous studies on the subject; this 0 ηe− jχ issue is discussed in detail in Sect. 6.2 in our concluding remarks. We speculate at the end on possible cosmological the identity element can be incorporated by allowing χ = 0, implications of this formalism. and with the choice η = 1; similarly with η =−1 the ele- ment −I is added. Hence, Eq. (3) parametrizes the elements of the hyperbolic subgroup with χ ∈ R; the part connected 3 Incorporating the hyperbolic rotations to the identity will be denoted by SO+(1, 1), and repre- sents the subgroup of continuous transformations; the dis- As an extension of the conventional complex numbers, the crete transformation related to the element −I will act sepa- commutative ring of hypercomplex numbers, z ∈ H,is rately as a PT -like transformation. Note that the hyperbolic defined by [10], subgroup is an Abelian group, and SO+(1, 1) corresponds to an one-parameter Lie group, with a non-compact generator. z = x + iy + jv + ijw, z = x − iy − jv + ijw, Therefore, the quadratic form (2) is invariant under the full , ,v,w ∈ R θ χ x y (1) phase ei e j , corresponding to the group U(1) × SO(1, 1). where the hyperbolic unit j has the properties j2 = 1, and This symmetry largely ignored in the literature, will have a j =−j, and, as usual, i2 =−1, and i =−i. Hence, with direct impact in various directions, in particular in the vac- respect to the conjugation involving both complex units, the uum structure of theories with gauge symmetry, leading to square of the hypercomplex number is given by radical changes in its topology and geometry. The full non-compact group SL(2, R) already has been zz = x2 + y2 − v2 − w2 + 2ij(xw − yv), (2) considered previously as a structure group in a toy model; namely the kinematics of the SL(2, R) “Yang–Mills” theory which is not a real number, instead it is in general a Hermi- in 1 + 1 dimensions was studied in [26]; such an analysis tian number. Equation (2) is invariant under the usual circular was motivated in part by gaining insight in the formulation rotations eiθ represented by the Lie group U(1); similarly it is of gauge theories with non-compact structural groups, which invariant under hyperbolic rotations that can be represented may shed light in the quantization of any theory of gravity. by the connected component of the Lie group SO(1, 1) con- The analysis showed that the configuration space has a non- taining the group unit. A hyperbolic rotation is represented Hausdorff “network” topology, rather than a conventional by the hyperbolic versor e jχ ≡ cosh χ + j sinh χ, with the manifold, and that the emergent quantization ambiguity can- split-complex conjugate e− jχ = cosh χ − j sinh χ, and with not be resolved as opposed to the usual compact case. This   the operations e jχ · e jχ = e j(χ+χ ). toy model captures the relevant aspects of a four-dimensional The hyperbolic rotations correspond to a subgroup of non-compact Yang–Mills theory, which is physically closer the SL(2, R) group, which represents all linear transforma- to four-dimensional gravity. In that analysis the foliation of tions of the plane that preserve oriented area. The elements the SL(2, R) group by its conjugacy classes is used; the of the group are classified as elliptic, parabolic, or hyper- space of conjugacy classes associated with the elliptic and bolic, depending on whether the |trace| < 2, |trace|=2, hyperbolic subgroups has the non-compact topology of a 123 98 Page 4 of 22 Eur. Phys. J. C (2016) 76:98

γ − ij j − iγ two-sheet hyperboloid. A two-sheet hyperboloid has a cir- w = (z + z), v = (z − z), (6) cle and a hyperbola as factor spaces, with trivial homotopy 2(γ 2 + 1) 2(γ 2 + 1) groups, except π0 = 2; in the present work the vacuum man- ifold will have the circle and the hyperbola as factor spaces, where we have used the inverse expressions, but the homotopy groups will be nontrivial. − γ − ij − j − iγ We need to restrict ourselves to hypercomplex numbers (γ + ij) 1 = ,(iγ + j) 1 = , (7) γ 2 + 1 γ 2 + 1 that have only two degrees of freedom, i.e., that are defined only in terms of two real quantities, in order to obtain a mini- which are non-singular in spite of belonging to a ring. mal deformation of an ordinary complex number that encodes Just as any “not-null” hyperbolic number can be brought two real quantities. Such a deformed version will incorporate into polar form ρe jχ , where ρ ∈ R or ρ ∈ j ·R depending on ( ) the hyperbolic rotations to the usual U 1 -circular rotations whether the norm of the number is strictly positive or strictly in field theory; this requires the mutual identification of the negative, respectively, any number of the general form (1), four real variables in Eq. (1). To this aim, the appropriate with |z| = 0 can be written as identification is x = γwand y = γv, with γ a real parame- θ χ ter, which reduces Eq. (1)to z = ρei e j , (8) z = (γ + ij)w + (iγ + j)v, z = (γ + ij)w − (iγ + j)v, where θ ∈ (0, 2π], χ ∈ R and, in general, ρ must be a = (γ 2 − )(v2 + w2) + γ(w2 − v2); zz 1 2ij (4) Hermitian number, i.e., hence, the norm is invariant under the interchange of the field ρ = ρ + ijρ . (9) v ↔ w, and simultaneously the change γ →−γ . The effect R H of a combined circular and hyperbolic rotation is Equating the two expressions (Cartesian and polar) for z we θ χ ei e j z = (γ cos θ cosh χ − sin θ sinh χ)w obtain the relations +( θ χ − γ θ χ)v cos sinh sin cosh x = ρR cos θ cosh χ − ρH sin θ sinh χ, + [(γ θ χ + θ χ)v i cos cosh sin sinh y = ρR sin θ cosh χ − ρH cos θ sinh χ, +(γ θ χ + θ χ)w] sin cosh cos sinh v = ρR cos θ sinh χ − ρH sin θ cosh χ, + [( θ χ − γ θ χ)v j cos cosh sin sinh w = ρR sin θ sinh χ − ρH cos θ cosh χ; (10) +(γ cos θ sinh χ − sin θ cosh χ)w] which can in principle be inverted to obtain ρR, ρH , θ, and χ +ij[(sin θ cosh χ + γ cos θ sinh χ)v in terms of x, y, v, and w. The explicit expressions turn out +(cos θ cosh χ + γ sin θ sinh χ)w]; (5) to be relatively complicated, but we can straightforwardly arrive at the following implicit formulas: if γ = 0, then z = j(v+iw), which is essentially an ordinary ρ2 − ρ2 = 2 + 2 − v2 − w2,ρρ = w − v, complex number; however, if γ = 0, then, as opposed to the R H x y R H x y cases previously considered, the invariant norm in (4) con- 1(ρ2 + ρ2 ) χ = v + w, R H sinh 2 x y tains necessarily a ij-hybrid term, and it cannot be a purely 2 real quantity. However, the norm can be a purely hybrid quan- 1(ρ2 + ρ2 ) θ = + vw; R H sin 2 xy (11) tity by choosing γ 2 = 1; note that in this case the hypercom- 2 plex number in (4) can be reduced to a number proportional to that is, given the Cartesian components of z, one can use the a purely hyperbolic one, and the circular rotation is spurious. first two equations to obtain ρR and ρH in terms of them, and On the other hand, the case γ 2 = 1, represents a nontrivial then insert those values into the latter two to get the phases combined rotation (5), and thus a nontrivial algebraic defor- θ and χ. mation of the usual complex formalism; such a deformation In the hypercomplex formulation developed, the real will have a profound effect on the dynamical aspects of the objects such as Lagrangians, vector fields, masses, and cou- field theories considered. pling parameters will be generalized to Hermitian objects, Furthermore, note that the first term proportional to w in encoding two real objects. The four real components of a the hypercomplex number in Eq. (4) does not change under hypercomplex field have been identified to each other by conjugation, behaving as the real part of an ordinary com- using a real γ -parameter, leading to two real effective vari- plex number; similarly the term proportional to v will have a ables; hence, the new formulation is constructed as a γ - global change under conjugation, behaving as the imaginary deformation along a non-compact direction defined by the part of an ordinary complex number; thus, one can reverse new complex unit. As an effect of the γ -deformation, the the expression (4)as traditional Mexican hat potential will be hallowed out in two 123 Eur. Phys. J. C (2016) 76:98 Page 5 of 22 98    points in the valley that defines the degenerate vacuum; such d 1 i L(ψ, ψ) = dx ∂ ψ · ∂i ψ − V (ψ, ψ) , two points represent the new vacuum states. In a limit case, 2 a purely hyperbolic version of the Mexican hat potential will a λ 2 V (ψ, ψ) = m2ψψ + ψ2ψ , (14) be obtained. 2 4! which is Hermitian and a non-analytical function on ψ, and hence can attain relative minima and/or maxima; we consider λφ4 4 Hypercomplex version of the classical model : that the square mass is also Hermitian, with real and hybrid global symmetries 2 ≡ 2 + 2 =± parts m m R ijmH ; a 1, and similarly we consider that λ ≡ λ +ijλ , with (m2 , m2 ,λ ,λ ) real parameters. λφ4 R H R H R H The theory “ ” is not only a pedagogical model; for exam- The potential can be written explicitly in terms of its real and ple the potential λ(H † H)2 has been considered recently for hybrid parts as V = VR + ijVH , supporting the idea of a quantum origin of the Higgs poten-     tial and the electroweak scale [27]; specifically the renor- γ 2 − γ 2 − 1 2 2 2 1 2 2 2 VR = a m + γ m v + a m − γ m w malization group formalism for the corresponding Coleman– 2 R H 2 R H Weinberg potential obtained by radiative corrections is devel-   λ (γ 2 − 1)2 oped. Hence, the Coleman–Weinberg symmetry breaking + R (v2 + w2)2 − γ 2(v2 − w2)2 6 4 can be understood in terms of the running of the coupling λ constants. However, although the present formulation is far − H γ(γ2 − 1)(w4 − v4); (15) 6    in spirit from the Coleman–Weinberg dynamical symmetry γ 2 − γ 2 − 1 2 2 2 1 2 2 2 breaking scheme, the hypercomplex deformation of the U(1) VH = a m − γ m v + a m + γ m w 2 H R 2 H R λφ4 theory will be understood, in certain sense, in terms of γλR the running of the coupling constants, as functions of the + (γ 2 − 1)(w4 − v4) 6   parameter γ , which has up to this point, an algebraic origin. λ (γ 2 − )2 + H 1 (v2 + w2)2 − γ 2(v2 − w2)2 ; Now we shall show that, properly analytically continued, 6 4 λφ4 φ the d theory for a massive (real) scalar field ,inad- (16) dimensional flat background, can be re-formulated on the hypercomplex space; the conventional theory is described one can map the potentials VR and VH to each other, by the by discrete transformations    d 1 i L(φ) = dx ∂ φ · ∂i φ − V (φ) , 2 γ →−γ, (λR,λH ) → (λH ,λR), m R → m H . (17) 1 λ V (φ) = am2φ2 + φ4, (12) 2 4! The vacuum is defined as usual by the stationary points con- straint, where m2 > 0, and the self-interacting constant λ is assumed   to be positive in order to have the energy bounded from ∂ λ V 2 below; the action in invariant under the action of the cyclic = ψ0 am + ψ0ψ0 = 0; (18) ∂ψ0 6 group Z2 ={+1, −1}, manifested though the discrete sym- metry φ →−φ. The unbroken exact symmetry scenario which can be expressed explicitly is terms of real fields requires a = 1, and the vacuum manifold corresponds to a (v ,w ) and real parameters (m , m ); the zero-energy single point, without homotopy constraints, and hence with- 0 0 R H point for V and V is described by out possible topological defects. Furthermore, the sponta- R H =− neous symmetry breaking scenario requires a 1, and (v0 = 0,w0 = 0) ; (19) 0 ∼    the vacuum manifold corresponds to the 0-sphere, S Z2, γ 2 − 2 π = 1 4 2 4 with the nontrivial homotopy constraint 0 2, and hence VR = 0; (detH)( ) = 4 m − γ m ; VR 2 R H admitting the domain walls as possible topological defects.   The hypercomplex version is based on Eqs. (4) and (5), ∂2V γ 2 − 1 R = 2a m2 + γ m2 ; and hence the Lagrangian can be re-interpreted in terms of ∂v2 2 R H the two hypercomplexified field variables (ψ, ψ), with 0   ∂2 γ 2 − VR = 1 2 − γ 2 ; 2a m R m H (20) 2 2 2 2 2 ∂w2 ψψ = (γ − 1)(v + w ) + 2ijγ(w − v ), (13) 0 2    γ 2 − 2 1 4 2 4 iθ jχ VH = 0; det H( ) = 4 m − γ m ; which is invariant under the global phases e e , and under VH 2 H R the discrete transformation (v ↔ w, γ →−γ), 123 98 Page 6 of 22 Eur. Phys. J. C (2016) 76:98   ∂2 γ 2 − 2 ( − γ 2)λ + γλ VH 1 2 2 m H 1 H 2 R = 2a m − γ m , w0 = 0, → = , (28) ∂v2 2 H R m2 (1 − γ 2)λ − 2γλ 0   R R H 2 2 2 ∂ VH γ − 1 6am = 2a m2 + γ m2 ; (21) v2 = R ; (29) ∂w2 H R 0 ( − γ 2)λ − γλ 0 2 1 R 2 H fortunately, these conditions will induce two global minima where we have displayed the second order derivatives, and 2 for both potentials VR and VH in the case with γ = 1, and the determinant of the Hessian matrix; likewise, the other with a non-zero vacuum expectation value for the field v;this stationary points for VR and VH , related with the condition λ case will be analyzed in detail in Sect. 4.2. Alternatively one am2 + ψ ψ = 0, read 6 0 0 has the choice 2 ( − γ 2)λ − γλ ( − γ 2)(λ2 + λ2 )(v2 + w2) = (λ 2 + λ 2 ); m H 1 H 2 R 1 R H 0 0 6a Rm R H m H v0 = 0, → = , (30) m2 (1 − γ 2)λ + 2γλ (22) R R H 2 2 2 2 2 2 2 6am γ(λ + λ )(v − w ) = 3a(λRm − λH m ); (23) w2 = R ; R H 0 0 H R 0 2 (31) (1 − γ )λR + 2γλH which can be solved to favor of the fields (v0,w0): which can be obtained from (28) and (29) by the change γ →−γ ; this case will be developed in Sect. 4.3. Now we expand the theory around ψ0, using only the 2 3 1 1 v = λψ ψ =− am2 0 2a λ2 + λ2 γ(1 − γ 2) degenerate vacuum constraint 0 0 6 , R H × (γ 2 − )λ + γλ 2 + ( − γ 2)λ + γλ 2 , V (ψ + ψ , ψ + ψ ) 1 H 2 R m R 1 R 2 H m H 0 0 2 3 1 1 am w2 = = (ψ + ψ0)(ψ + ψ0) 0 2a λ2 + λ2 γ(1 − γ 2) 2 R H λ 2 2 × ( − γ 2)λ + γλ 2 − ( − γ 2)λ − γλ 2 , + (ψ + ψ0) (ψ + ψ0) 1 H 2 R m R 1 R 2 H m H 4! (24) 2 λ λ =−am ψψ + (ψψ)2 + (ψ2ψ2 + ψ2ψ2) 4 4 2 2 0 0 3 λR(m − m ) − 2λH m m 2 4! 4! V = H R H R ; (25) R 2 λ2 + λ2 λ R H + ψψ(ψ ψ + ψ ψ), (32) 12 0 0 ∂2V 1 R = (γ 4 − 6γ 2 + 1)λ + 4γ(γ2 − 1)λ v2; ∂v2 3 R H 0 note that we have not chosen yet a definite vacuum, but the 0 ∂2 expansion around a non-zero ground state value for the field VR 1 4 2 2 2 = (γ − 6γ + 1)λR − 4γ(γ − 1)λH w ; leads to a change of sign in the mass term. Furthermore, the ∂w2 3 0 0    first two terms in Eq. (32) have already the canonical form ∂2 2 λ 2 ψψ VR R 2 2 since depend on the norm given in (13). The third and = (γ + 1) v0w0 ; ∂v0∂w0 3 fourth terms correspond to the quadratic and cubic terms in   the fields (v, w); such terms do not have the canonical form γ(γ2 − ) 2 4 1 2 2 2 2 vw (detH)(V ) =− (λ + λ )v w ; (26) (due to the presence of mixed terms of the form ) and do R 3 R H 0 0 ψ ψ not depend on the modulus 0 0; explicitly we have, λ ( 4 − 4 ) − λ 2 2 3 H m R m H 2 Rm H m R 2 2 VH = . (27) ψ ψ2 + ψ2ψ = 2cosh2(χ − χ)cos 2(θ − θ) 2 λ2 + λ2 0  0 0 0 R H × (γ 4 + )(w2 − v2)(w2 − v2) − γ 2 1 0 0 2  Considering that VH is obtained from VR by means of the ×[(v2 + 3w2)w2 + (3v2 + w2)v2] transformations (17), the second order derivatives expres- 0 0 0 0 sions for V can be obtained directly from Eq. (26); in par- + γ (χ − χ) H 8 sinh 2 0 ticular (det H)V = (det H)V , which is strictly negative, H R × (γ 2 − 1) sin 2(θ − θ)(v2v2 − w2w2) according to the expression (26), and thus the points (24)are 0 0 0  in general saddle points for both potentials, unless det H = 0, −(γ 2 + ) (θ − θ)v w (v2 + w2) then anything is possible. For this purpose, one can fix to zero 1 cos 2 0 0 0 one of the vacuum expectation values, v0,orw0, which in + (γ 4 − ) (χ − χ) (θ − θ)v w (v2 − w2) 4 1 cosh 2 0  sin 2 0 0 0 fact will be required by spontaneous symmetry breaking; the +8(γ 2 + 1) cos 2(θ − θ) (γ 2 + 1)v w cosh 2(χ − χ) choice w0 = 0 leads to the following simplification: 0 0 0 0 123 Eur. Phys. J. C (2016) 76:98 Page 7 of 22 98   2 2 2 λ am2 +γ(v + w ) sinh 2(χ0 − χ) vw 1 2 1 6 0 0 E = 2T = |∂t ψ| + ∇ψ ·∇ψ + ψψ − . 00 2 2 4! λ − (γ 4 − )(v2 − w2) (χ − χ) (θ − θ)· vw 4 1 0 0 sinh 2 0 sin 2 0  (37)  +2ij 4γ(γ2 − 1) cosh 2(χ − χ)· cos 2(θ − θ)(w2w2 − v2v2) 0 0 0 0 Equations (35), (36), and (37) will allow us to study the pos-  sible formation of global strings with finite energy in Sect. + sinh 2(χ − χ)sin 2(θ − θ) (γ 4 + 1)(w2 − v2)(w2 − v2) 0 0 0 0 4.6.  − γ 2(v2 + w2)(v2 + w2) − γ 2(v2v2 + w2w2) 2 0 0 4 0 0 γ 2 = 2 = λ = 4.1 The case 1, m R 0, H 0: hyperbolic version + (γ 4 − ) (χ − χ) (θ − θ)v w (w2 − v2) of the Mexican hat 2 1 sinh 2 0 cos 2 0 0 0  − γ(γ2 + ) (χ − χ) (θ − θ)v w (w2 + v2) 4 1 cosh 2 0 sin 2 0 0 0 The case γ 2 = 1 is special, since the norm (13) reduces to  the hyperbolic part, a purely hybrid expression. In relation to + (γ 2 + ) (χ − χ) v w (θ − θ) 4ij 1 sinh 2 0 2 0 0 sin 2 0 Eq. (22) that defines the degenerate vacuum, the restriction  γ 2 = 1 implies more restrictions on the right-hand side; a 2 2 2 −(γ − 1)(w − v ) cos 2(θ0 − θ) vw 0 0  nontrivial choice is m R = 0, and λH = 0; then the hybrid + γ (γ 2 + ) (χ − χ) (θ − θ)(v2 + w2)vw; component of the potential VH (Eq. (16)) vanishes, and its 8 ij 1 cosh 2 0 sin 2 0 0 0  real part reduces to (33) λ   = = γ 2 (v2−w2)− (v2−w2)2, = ; ψψ(ψ ψ + ψ ψ) = 2 (γ 2 − 1)(v2 + w2) + 2γ ij(w2 − v2) · V VR a m H VH 0 (38)  0 0 6 × (γ 2 − ) (χ − χ) (θ − θ)(w w + v v) 1 cosh 0 cos 0 0 0 additionally the minima are defined by the hyperbola (23), irrespective of the signs of the parameters (a,γ,λR = λ); −2γ sinh(χ0 − χ)sin(θ0 − θ)(w0w − v0v) −(γ 4 + ) (χ − χ) (θ − θ)(v w − w v)  1 cosh 0 sin 0 0 0 3 aγ(v2 − w2) = m2 ; (39) 2 0 0 λ H +ij (γ − 1) sinh(χ0 − χ)sin(θ0 − θ)(w0w + v0v) + γ (χ − χ) (θ − θ)· (w w − v v) γ> 2 cosh 0 cos 0 0 0  the hyperbola with a 0 is the conjugate of that with γ< π v − w 2 a 0, and are related by a 2 -rotation in the plane . +(γ + 1) sinh(χ0 − χ)cos(θ0 − θ)(v0w − w0v) ; (34) In the appropriate field variables (1,2), the hyperbola   takes the form (  1 )2 − (  2 )2 = 1, and thus is rect- ψ = 2 2 where the fields have the general form (5), with 0 6m H 6m H | λ | | λ | ψ0(γ, v0,w0; θ0,χ0), and ψ = ψ(γ,v,w; θ,χ); note the √ presence of ij-hybrid terms in Eqs. (33), and (34). Further- angular with eccentricity 2, with the vertices localized at 2 vw 6m H more, the presence of mixed terms (both real and hybrid) ± | λ |, which coincide with the position of the two min- in the quadratic form (33) prevents us from determining the ima in the original Spontaneous Symmetry Breaking (SSB) masses of v and w; note also that in relation to the same scenario described in terms of the real field φ in Fig. 1. mixed terms, one must exploit the freedom of choosing the From the substitution of the expression (39) into Eq. (38), circular and hyperbolic parameters, and v.e.v. (v0,w0),in one obtains the energy of vacuum, order to obtain the simultaneous vanishing of the real and hybrid terms of the form vw. 3 m4 V (v ,w ) = H . (40) Finally the equations of motion are given by 0 0 2 λ    λ 6am2 For λ<0, the hyperbola corresponds to the global minima  + ψψ − ψ = 0,  = ∂2t −∇2, (35) 6 λ for the energy, and its value corresponds to the value for the two lowest energy states in the original SSB scenario with an energy-momentum tensor given by described in Fig. 1. For the stationary points described in Eq.   H ( , ) =− 4   (19) we have det (VR ) 0 0 4m H , and thus the zero- λ 2 2 1 κ 6am energy point is a saddle point. 2Tij = ∂i ψ·∂ j ψ−gij ∂ ψ · ∂κ ψ − ψ · ψ − ; 2 4! λ In Fig. 2, the potential is not the Mexican hat potential, par- (36) ticularly in relation to the existence of two connected regions for the possible vacuum states, and that each region is non- in particular, the energy density reads compact, as opposed to the compact circle associated to the 123 98 Page 8 of 22 Eur. Phys. J. C (2016) 76:98

In this case Eqs. (33), and (34) reduce to

2 2 ψ ψ2 + ψ2ψ = 4cosh2(χ − χ)cos 2(θ − θ) 0  0 0 0  3aγ × m2 (v2 − w2) − (v2 + w2)w2 − ( v2 + w2)v2 λ H 0 3 0 3 0 0 2 2 −16γ sinh 2(χ0 − χ)· cos 2(θ0 − θ)v0w0(v + w ) + (θ − θ) 16 cos 2 0  × v w (χ − χ)+ γ(v2 + w2) (χ − χ) vw 2 0 0 cosh 2 0 0 0 sinh 2 0  + (χ − χ) (θ − θ) 4ijsinh 2 0 sin 2 0  2  γ Fig. 1 The dashed potential shows the profile for the cases m 0, 3a 2 2 2 2 2 2 2 2 2 2 2 2 × m (v − w )−(v + w )(v +w ) − (v v + w w ) and the continuous potential shows the case m < 0; in the first case the λ H 0 0 2 0 0 vacuum manifold corresponds to a single point, thus π = 1,π  = 0 n 1 +16γ ijcosh 2(χ − χ)sin 2(θ − θ) {0}, and in the second case, it corresponds to two points (the 0-sphere),  0 0  × (v2 + w2)vw − v w (v2 + w2) with π0 = 2,πn1 ={0} 0 0 0 0 +16ijsinh 2(χ0 − χ)sin 2(θ0 − θ)v0w0 vw , (41) ψψ(ψ ψ + ψ ψ) = 8γ ij(w2 − v2)  0 0

× γ sinh 2(χ0 − χ)sin 2(θ0 − θ)(v0v − w0w)

+ cosh 2(χ0 − χ)sin 2(θ0 − θ)(w0v − v0w) + (χ − χ) (θ − θ)(v w − w v) ijsinh 2 0 cos 2 0 0 0 

+γ ijcosh 2(χ0 − χ)cos 2(θ0 − θ)(w0w − v0v) ; (42)

We can see that only some mixed terms vw (both real and hybrid), can be gauged away by fixing the hyperbolic param- eters χ = χ0; a particular choice, say χ0 = 0, leads to a break down of the symmetry SO+(1, 1), remembering that in the case γ 2 = 1, U(1) corresponds to a spurious rotation. Even λ< Fig. 2 The potential (38)for 0; the stable states are localized at so the remanent U(1) parameters must be fixed in Eq. (41)by the blue region, a hyperbola described by Eq. (39) demanding the vanishing of the remaining mixed term v · w, for example, by fixing γ conventional U(1)-symmetry. Let us see how the two scenar- θ = θ = ,v= ,w2 =−3a 2 ; 0 0 0 0 0 m H ios described in the conventional Lagrangian (12) are con- λ λ 2 2 tained in certain sense in this hypercomplex re-formulation; → (ψ ψ2 + ψ2ψ ) = aγ m2 (w2 + v2), ! 0 0 H (43) if the original Lagrangian corresponds to an Unbroken Exact 4 Symmetry (UES) scenario with L(φ, a = 1), then the poten- with aγ = 1, and λ<0; hence, taking into account that tial V (φ, a = 1) has a minimum (see Fig. 1), and it is shown − am2 ψψ = γ 2 (w2 − v2) 2 a m H , the mass matrix determined as a bold-face curve embedded in Fig. 2; the hypercom- by Eq. (32) for the Lagrangian (14) reads plex extension of φ leads to a new potential, transforming   00 the original stable minimum into a saddle point as the zero- , (44) − m2 w2 energy point, and with the appearance of other stable min- 0 2 H ima at the blue region. Similarly, the usual SSB scenario which corresponds to a massive ordinary scalar field w, and a with V (φ, a =−1) in Fig. 1 with two minima is shown massless field v. Conversely, if the field v develops a non-zero also as a bold-face curve in Fig. 2. Thus, the two minima v2 = 3aγ 2 γ =− vacuum expectation value with 0 λ m H , a 1, and have been extended to an infinite number of minima in two w0 = 0, then we shall have an ordinary massive term for the disconnected regions colored in blue, which correspond to field v, and the field w will be now massless; in this case the the hyperbola in Eq. (39); the original maximum point of mass matrix reads the potential V (φ, a =−1) is the saddle point for the new   −2m2 v2 0 potential. Hence, the Lagrangian (14) is build originally on a H . (45) 00 saddle point (the crossing point of the bold-face curves), and the choice of a stable vacuum in the blue region is mandatory, If the spurious U(1) rotations are broken for example with and we proceed now to the study of the realizations of the the choice (43), then the mixed terms vw in the real part of symmetry breakdown. Eq. (33) vanish trivially, and the complete mass term in Eq. 123 Eur. Phys. J. C (2016) 76:98 Page 9 of 22 98

(32) reduces to theory based on a hyperbolic symmetry may be a way of 2 evading the problem of the insignificant empirical evidence am λ 2 2 − ψψ + (ψ ψ2 + ψ2ψ ) of cosmic strings, contrary to the predictions of many field 2 4! 0 0  theory models. The possible cosmological implications of = 2 [ (χ −χ ) − ]v2 +[ + (χ −χ )]w2 m H cosh 2 0 1 1 cosh 2 0 these speculations are discussed in the concluding remarks.  Now we are going beyond the substitution of i by j, and the incorporation of j described previously in terms of a commu- −4γ sinh 2(χ − χ0)vw , (46) tative ring will allow us to study the hyperbolic deformation which is fully real, without ij-hybrid terms; similarly the of the Mexican hat potential; the parameter γ will govern cubic expression (42) is fully real under the choice (43). the competition between the circular and hyperbolic contri- Therefore, in relation to the expression (46), whereas the butions. hyperbolic rotation symmetry is not conveniently fixed, the 2 masses of the fields v and w cannot be determined due to the 4.2 The case γ = 1, (v0 = 0,w0 = 0), and λR = λH : presence of the mixed term vw. However, the choice χ0 = 0 hyperbolic deformation of the Mexican hat fixes a point on the hyperbola, and the condition χ = 0 leads 2 to the same mass matrix described previously in Eq. (44), The restrictions γ = 1 and γ = 0 imply that the norm and similarly for the mass matrix in Eq. (45). (13) will have nontrivial contributions from the circular and In the spontaneous symmetry breaking scenarios hyperbolic parts; the additional restriction λH = λR ≡ λ described above, the hyperbolic parameters are always cho- will allow us to simplify the analysis; hence Eqs. (28) and sen with finite values, in similarity with the compact circular (29) for the ratio between the squared masses and squared parameters; however, the hyperbolic parameters take in prin- expectation value, reduce to ciple values on a non-compact interval, with |χ| < ∞, and + m2 γ 2 − 2γ − 1 6m2 |χ0| < ∞. Hence, the remanent SO (1, 1) symmetry in H = ,v2 = R ; 0 (48) Eq. (46) can be spontaneously broken by considering that m2 γ 2 + 2γ − 1 aλ(1 − γ 2 − 2γ) χ R χ →+∞ χ ≈ χ ≈ e 0 0 , and hence sinh 0 cosh 0 2 , and for the hyperbolic functions in Eq. (46)wehave the first equation above defines a positive quotient, restricting the values of γ on the right-hand side; similarly positivity on 2χ0 e 2 the left-hand side of the second equation implies the inequal- cosh 2(χ − χ0) ≈ [sinh χ − cosh χ] , 2 ity χ e2 0 sinh 2(χ − χ ) ≈− [sinh χ − cosh χ]2, (47) 0 2 aλ(1 − γ 2 − 2γ)>0. (49) which are divergent, unless χ →+∞, and hence sinh χ − Other relevant quantities for inducing a stable vacuum are cosh χ ≈ 0; thus lim[cosh 2(χ−χ0)]=1, and lim[sinh 2(χ− det H for the potentials at the zero-energy point, χ0)]=0; therefore the mass matrix obtained from (46) coincides with that obtained previously with the choice 4 (γ 2 + )2 m R 1 Hv (v = ,w = ) = χ = χ = 0. Similarly, in the case of the other asymptotic det R 0 0 0 0 2 2 0 −χ (1 − γ − 2γ) limit χ →−∞we have − sinh χ ≈ cosh χ ≈ e 0 , and 0 0 0 2 ×[γ 4 + 4γ 3 − 6γ 2 − 4γ + 1], (50) the hyperbolic expressions depend now on sinh χ + cosh χ, 4 (γ 2 + )2 χ →−∞ m R 1 which goes to zero in the limit . det Hv (v = 0,w = 0) = H 0 0 ( − γ 2 − γ)2 The hyperbola has two connected regions, and hence π0 = 1 2 2; each connected region is topologically equivalent to R, ×[γ 4 − 4γ 3 − 6γ 2 + 4γ + 1]; (51) which has trivial homotopy groups, thus π = 0, for n ≥ 1; n H > there are no topological defects associated with continuous if det 0, the zero-energy point will be a minimum or H < symmetries, and only the domains walls are possible due to maximum; if det 0 it will be a saddle point. The poly- γ the nontriviality of π . nomials of in Eqs. (48), and (49), and those that determine 0 H Since the structure of this theory is basically hyperbolic, the signs of det in Eqs. (50) and (51), are shown in Fig. 3. it reduces in essence to the substitution of the ordinary imag- In Fig. 3, the vertical blue asymptote√ represents one root (γ 2 + γ − ) γ = − inary unit i, by the new hyperbolic unit j; as already men- of the polynomial 2 1 , 2 1; this singular tioned, this simple substitution generates nontrivial results point is out of the interval of interest. The arrow on the left- hand side points out a root of det H (0, 0), γ = 1 + in diverse scenarios [1,2,7–9]. In the hyperbolic “λφ4” the- √  √ VH H ory at hand, the topological defects such as strings, cannot 2 − 2(2 + 2) ≈−0.1989; similarly on the right-hand H ( , ) form; these defects are unavoidable in the breaking of an side an arrow points out a root of det VR 0 0 localized ( ) γ =−γ ≈ . Abelian U 1 symmetry. Hence, a phenomenological field at R H 0 1989; within this symmetric interval all 123 98 Page 10 of 22 Eur. Phys. J. C (2016) 76:98  λ (γ 2 − )2 = a v 2 v2 + a w 2 w2 + 1 (v2 +w2)2 VH PH m R PH m R 2 2 6 4  −γ 2(v2 − w2)2 + γ(γ2 − 1)(w4 − v4) ; (54)

where the polynomials P are defined as

γ 4 + 4γ 3 − 6γ 2 − 4γ + 1 (γ 2 + 1)2 Pv = , Pw = ; R γ 2 + 2γ − 1 R γ 2 + 2γ − 1 4 3 2 v γ − 4γ − 6γ + 4γ + 1 w w P = , P = P ; (55) H γ 2 + 2γ − 1 H R

the potentials (53), and (54) are shown in Fig. 4 as functions (v, w) γ (γ , −γ ) of , for a value of in the interval H H ; similarly ±γ the polynomials (55)areshowninFig.5. The values H H ( , ) Fig. 3 The continuous red curve represents essentially det VR 0 0 ; are critical, since Eqs. (50), and (51) vanish, and thus the H ( , ) the continuous black curve represents det VH 0 0 .Thecontinuous character of a local maximum for the zero-energy point and (γ 2 − γ − )/(γ 2 + γ 2 − ) blue curve represents the polynomial 2 1 2 1 ,the the form of the potentials with stable minima shown in Fig. ratio in Eq. (48); the dashed curve represents the polynomial (1 − γ 2 − 2γ),Eq.(49); the arrows point out the interval where all polynomials 4 are not guaranteed; therefore one must be careful on taking γ →±γ are positive the limit H . Furthermore, the vacuum energies are given by polynomials are positive, and a stable vacuum will be induced − 4 v(γ ) for the potentials. In this interval, the inequality (49) implies 3m R PR VR(±v , 0) = , that 0 2λ γ 2 + 2γ − 1 − 4 (γ 2 − )2 3m R 2 1 aλ>0. (52) VH (±v0, 0) = ; (56) 2λ (γ 2 + 2γ − 1)2

Under these simplifications the potentials will take the hence, the depth of the red regions in Fig. 4 depends on γ ;the following form: polynomials that deform the conventional vacuum energies in  λ (γ 2 − )2 the above expressions are shown in Fig. 6.Forγ = 0 one can = a v 2 v2 + a w 2 w2 + 1 (v2 + w2)2 VR PRm R PR m R recover from VR the vacuum energy for the usual U(1) field 2 2 6 4 Pv (γ )   R  = [γ , ] theory, since (γ 2+ γ − ) 1. In the subinterval H 0 −γ 2(v2 − w2)2 − γ(γ2 − )(w4 − v4) ; 2 1 γ =0 1 (53) the deformation is small with respect to the deformation in

/ 2 / 2 (γ , −γ ) (±v , ) Fig. 4 The potentials VR m R,andVH m R in the interval H H in the two red regions: 0 0 . The inequality (52) is satisfied with have essentially the same qualitative aspect, which we show here from a = 1andλ>0; the choice a =−1andλ<0 turns the potentials two different perspectives: the zero-energy point corresponds to the cen- upside down tral peak of the potential; the two minima are localized at the bottom

123 Eur. Phys. J. C (2016) 76:98 Page 11 of 22 98

Fig. 5 The mass polynomial coefficients: the blue curve represents Fig. 7 The continuous curve represents the mass polynomial coeffi- Pv ;theblack curve represents Pv ,andthered curve represents Pw. − v (γ ) R H R cient 2PH after SSB: the dashed curve represents the mass poly- In the shadowed interval (γ , −γ ) all polynomials are negative, v (γ ) γ = H H nomial coefficient PH before the SSB; the case 0 reproduces v (−γ ) = = v (γ ) v (γ ) ≈−. v (−γ ) ≈ ( ) w γ and PR H 0 PH H , PR H 1 1252 ,PH H the usual SSB of U 1 .Thefield is a fully massless field for any − . v ( ) = v ( ) = w( ) =− 2 7165, and all polynomials satisfy PR 0 PH 0 PR 0 1 in the interval

 2 2 am λ 2 2 1 − γ − ψψ + (ψ ψ2 + ψ2ψ ) = a m2 − γ m2 v2 2 4! 0 0 2 R H    1 − γ 2 1 − γ 2 +a m2 + γ m2 w2 + ija m2 + γ m2 v2 2 R H 2 H R   2 1 − γ a v + m2 − γ m2 w2 = ij (−2P m2 )v2; (58) 2 H R 2 H R

where the last equality follows from the mass relation (48); this expression must be compared to the mass terms in Eq. (53), and (54). Therefore, the field w is massless in both senses, real and hybrid. The field v has duplicated its hybrid v (γ )/(γ 2 + γ − ) Fig. 6 The blue curve represents PR 2 1 ;thered curve mass with a change of sign; note that the real mass of v has (γ 2− )2 represents 2 1 . In the interval (γ , −γ ) the polynomials take (γ 2+2γ −1)2 H H disappeared. The duplication of the mass with a change of positive values, and the vacuum energies are finite, even at the limits sign for any γ in the allowed interval is shown in Fig. 7. ±γ H The mass that arises from SSB in Eq. (58) is actually a run- v ning mass, since the polynomial PH takes values in the inter- val (0, −2.7165); hence the mass is running in the interval the subinterval [0, −γ ]; the polynomial is varying contin- H (0, 2.7165m2 ), from a nearly massless field to a “heavy” uously between the values [.828, 1] in the first subinterval, R field. Strict masslessness is prohibited, since the value −γ and between the values [1, 0] in the second subinterval, since H is critical in relation to the form of the potentials with local Pv(−γ ) = 0. Furthermore, from Fig. 6 it is evident that R H stable minima required for the spontaneous symmetry break- the deformation of V is strictly bigger than that of V ; thus, H R ing. the depth of the red regions in Fig. 4 is higher for V . Such H From Eq. (13) we can realize that in the limit γ → 0, the a difference in the deformation is maximum for −γ , and H usual norm of an ordinary U(1) complex field can be recu- minimum for γ . H perated; in this sense the case with γ = 0 can be understood The circular and hyperbolic rotations can be sponta- as a hyperbolic deformation around the usual formulation. neously broken by the choice Furthermore, according to Eq. (48), in such a limit there is no difference between m R and m H ; the vaccum expectation (θ − θ ) = ; (χ − χ ) = ; v2 ( ) sin 2 0 0 sinh 2 0 0 (57) value 0 will reduce to the usual U 1 expression. Similarly the inequality (49) will reduce to the inequality (52), which ( ) in addition to the vacuum expectation values (v = 0,w = corresponds to the usual constraint in the U 1 field the- 0 0 − 0); thus all mixed terms vw (both real and hybrid) can be ory. Likewise, all polynomials (55) reduce to 1, and conse- gauged away, reducing the quadratic terms in Eq. (32)tothe quently there is no difference between VR and VH . Therefore, canonical form, the usual Mexican hat potential can be recuperated from the 123 98 Page 12 of 22 Eur. Phys. J. C (2016) 76:98

w Fig. 9 Q R corresponds to the blue curve,withvaluesintheinterval (0, −1.1252); Qw is represented in red, with values in the interval γ → H Fig. 8 The mexican hat potential as the limit of VR and VH as 0 (0, −2.7165) deformed version described in Fig. 4; this process is shown in Fig. 8.

2 4.3 The case γ = 1, (v0 = 0,w0 = 0), and λR = λH

Along the same lines, the use of Eqs. (30), and (31) lead essentially to the same expressions (48), (49), (50), and (51), with the change γ →−γ ;Fig.3 is essentially the same; for example the red and black curves will convert to each other, without changing the interval (γH , −γH ); within this interval all new polynomials are negative. Furthermore, the inequality (52) remains valid, and the potentials can be described in Fig. 10 The description of the vacuum energies given in Fig. 6 is essen- terms of the same mass polynomials appearing in Eqs. (53) tially valid for this case, in relation to the behavior in the subintervals and (54); [γH , 0],and[0, −γH ], and the differences between the values of VR and V in the vacuum λ H a v 2 2 a w 2 2 VR = Q m v + Q m w + ··· ; (59) 2 R R 2 R R 6 λ ( , . 2 ) a v 2 2 a w 2 2 running in the interval 0 2 7165m R . Note that when the VH = Q m v + Q m w + ··· ; (60) 2 H R 2 H R 6 real mass goes to zero as γ → γH , the hybrid mass goes to γ →−γ where the dots represent exactly the same expressions for the its maximum value, and coversely in the limit H ; the masses coincide in the value m2 for γ = 0. λ-terms in Eqs. (53), and (54); the polynomials Q are defined R as In this case the vacuum energies are given by 4 v v w w v −3m P (−γ) Q = P (γ →−γ), Q = P (γ →−γ); V ( , ±w ) = R R , R R R R R 0 0 2 v v w v 2λ γ − 2γ − 1 Q = Q , Q = P (γ →−γ); (61) H R H H − 4 (γ 2 − )2 3m R 2 1 VH (0, ±w ) = ; (63) these potentials have the same form shown in Fig. 4.How- 0 2λ (γ 2 − 2γ − 1)2 ever, in spite of the similarities, there will be an important which can be obtained directly from Eq. (56) through the difference; although the field v will be massless as expected, change γ →−γ ; the polynomials in Eq. (63)areshownin the field w will develop a mass with both parts, real and Fig. 10. hybrid; explicitly we have 2 λ γ ∈ (−γ ,γ ) −am ψψ + (ψ2ψ2 + ψ2ψ2) 4.4 The vacuum manifold for H H 2 4! 0 0 = a 2 (− w)w2 + a 2 (− w )w2; In the usual formulation of the theory with a conventional m R 2Q R ijmR 2Q H (62) 2 2 complex field, the degenerate vacuum is identified with the which are shown in Fig. 9; the real mass in Eq. (62) is run- bottom of the Mexican hat potential, the S1 compact poten- ( , . 2 ) |v + w |2 = 6m2 ning in the interval 0 1 1252m R , and the hybrid mass is tial defined by the constraint 0 i 0 λ , and 123 Eur. Phys. J. C (2016) 76:98 Page 13 of 22 98   6m2 6m2 parametrized by v0 = λ cos θ, and w0 = λ sin θ; this case corresponds essentially to the choice γ = 0inthe formulation at hand. However, in the present case the hyper- complex field is undetermined additionally by a hyperbolic phase, leading to a two-dimensional manifold for the degen- erate vacuum; this case corresponds to the choice γ = 0 in Eq. (4). We shall see that the vacuum manifold will cor- respond to a non-compact space containing the hyperbola and the circle as factor spaces, and embedded in a four- dimensional ambient space. A parametrization for the vacuum manifold can be given by the expression (5), with γ ∈ (−γH ,γH ), and with λψ ψ =− 2 0 0 6am ; one can consider additionally the choice (29) with w0 = 0, and v0 = 0; 

ψ0 = v0 (cos θ0 sinh χ0 − γ sin θ0 cosh χ0) Fig. 11 The degenerate vacuum as a non-compact and not simply con- nected two-manifold embedded in the 3d space. Usually only the com- pact transversal sections related with U(1) have been considered as the +i(γ cos θ0 cosh χ0 + sin θ0 sinh χ0) vacuum; for this manifold of genus 1, we have π0 = 1,π1 = Z,and + j(cos θ cosh χ − γ sin θ sinh χ ) π = 1 0 0 0 0  n2 0, since it is homotopic to S

+ij(sin θ0 cosh χ0 + γ cos θ0 sinh χ0) . (64) 4.5 Polar parametrization for the fields One can embed the vacuum manifold as a two-dimensional subspace of the real four-dimensional space defined by the In the previous cases the hypercomplex field ψ is described in four components of the field ψ0 = x0 +iy0 + jz0 +ijw0 → terms of Cartesian components (v, w) as real variable fields; (x0, y0, z0,w0), and the result must be projected from 4D now we use a polar decomposition, considering that we have to 3D in order to be visualized, and to gain insight about its at hand only two real variable fields. Thus, following the geometrical and topological properties. The more practical (ρ + ideas at the end of Sect. 3, for an expression of the form R and direct method that can be used is the projection of the 2- ρ ) iξ jη ρ ρ ξ η ij H e e , where R , H , , and , are real field variables, manifold into the three-dimensional hyperplanes that define one must choose two of them as constants; similarly, the polar ξ η the four coordinate hyperplanes in the four-dimensional ϕ = (ρ R + ρ H ) i 0 j 0 form for the vacuum field reads 0 0 ij 0 e e . 2 ambient space, through (x0, y0, z0,w0) → (x0, y0, z0), and Case γ = 1: For this case studied previously in the Sect. similarly into the other three hyperplanes. Although in gen- 4.1, we see that the norm of the dynamical field reduces to eral such projections can be different depending of the orien- ψψ = 2ijγ(w2 − v2); additionally the norm of the corre- tation of the 2-manifold with respect to the coordinate hyper- sponding polar form will reduce to planes, in this case, the four projections coincide to each other, and are represented in Fig. 11. This projection is a sort ψψ = (ρ2 − ρ2 ) + ijρ ρ ; R H 2 R H (65) of product of a hyperbola and a circle; it can be visualized also as two-dimensional planes embedded in three dimen- ρ = γρ thus we have R H , and consequently sions and sharing a hole. The self-intersection is an effect  of the projection into a three-dimensional hyperplane; in the ψ = (γ + )ρ iξ jη,ρ≡ ρ = w2 − v2; ij e e H (66) original four-dimensional ambient space the two-manifold has no such self-intersections. This case is similar to the very therefore, in this case only one degree of freedom is encoded known Klein bottle, which cannot be realized in R3 without in the polar part, and the remaining degree of freedom will be encoded in one of the phases; similarly the vacuum field intersecting itself. ξ η ψ = (γ + )ρ i 0 j 0 We remark that the constraint defining the degener- takes the form 0 ij 0e e , with the constraint 6m2 λψ ψ =− 2 ρ2 = H ate vacuum 0 0 6am , retains the full symmetry 0 λ . + ξ( )/ρ η SO (1, 1) ×U(1), and the vacuum manifold is transformed Weconsider first the case ψ(x) = (γ +ij)ρ(x)ei x 0 e j , into itself by the action of these transformations, and we have with η constant, and excitations about the ground state with an infinite number of possible vacuum states with the same vanishing phases, ξ0 = 0 = η0, and hence the circular and 1 energy. The vacuum manifold is homotopic to S , and the hyperbolic rotations are broken. Now, we write the dynami- iξ(x) jη string defects can be formed as topological defects; we shall cal field as γ +ij)(ρ(x)+ρ0 e e , and the Lagrangian analyze in detail this topic in Sects. 4.6 and 5.4. becomes 123 98 Page 14 of 22 Eur. Phys. J. C (2016) 76:98   d 1 i 1 i terms in Eq. (70). Furthermore, since we are considering L = 2γ ij dx ∂ ρ∂i ρ + ∂ ξ∂i ξ 2  2 small oscillations around a point of the vacuum manifold, the dynamics of the circular and hyperbolic oscillations are −m2 ρ2 + higher terms , (67) H locally indistinguishable; in fact the expressions (68), (69), and (70) are invariant under the interchange iξ ↔ jη.As and, as expected, we have massive radial oscillations ρ and an option, one can distribute the global hybrid term ij out of circular angle massless oscillations ξ. There are no hyper- the integration in the Lagrangian (69), and hence the real and bolic angle oscillations, due to the fixing η = constant, since hybrid terms under integration will interchange their roles, they have disappeared completely from the Lagrangian. Sim- without changing the physical conclusions. ilarly, if we fix ξ = constant, and η → η(x)/ρ , then the 0 Case γ 2 = 1: In this case the comparison between the oscillations around the same ground state are described by a norm ψψ = (γ 2 − 1)(v2 + w2) + 2ijγ(w2 − v2), and Eq. Lagrangian of the form L = 1 (∂ρ)2 − 1 (∂η)2 −m2 ρ2,plus 2 2 H (65) leads to higher terms, where the massive radial oscillations remain ρ2 −ρ2 =(γ 2 − )(v2 +w2), ρ ρ =γ(w2 − v2); unchanged, but the hyperbolic angle oscillations appear now R H 1 R H (71) as massless modes with a global change of sign in the kinetic and similarly for the vacuum fields; these maps allow us to term. describe the fields with the pair (ρ ,ρ ) instead of the pair ρ = R H Now one can enforce the restriction constant, and (v, w) [ρ + ρ R + ; now, we rewrite the dynamical field as R 0 thus the two degrees of freedom will be encoded in the phases, (ρ + ρ H )] ij H 0 , leading to a Lagrangian of the form and all fluctuations will lie in the valley directions; such direc-   1 tions are defined by the circular and hyperbolic lines on the L(ρ ,ρ )= dxd ∂i ρ ∂ ρ − ∂i ρ ∂ ρ + 2ij∂i ρ ∂ ρ R H 2 R i R H i H R i H vacuum manifold, and the fluctuations will correspond to  purely massless modes, as expected. In this manner, if the two +2m2(ρ2 − ρ2 + 2ijρ ρ ) + higher terms ; (72) dynamical degrees of freedom are encoded in the phases, then R H R H ξ( )/ρ η( )/ρ ψ = (γ + )ρ i x 0 j x 0 with m2 = m2 + ijm2 ; hence, the two radial modes are ij 0e e   R H ξ( ) η( ) massive as expected. This expression can be rewritten in a = (γ + )ρ + x +··· + x +··· , ij 0 1 i 1 j compact form in terms of the Hermitian field  ≡ ρ +ijρ , ρ0 ρ0 R H = (γ + ij)[ρ0 + iξ(x) + jη(x) +···] (68) L(ρ + ijρ ) R  H ξ η 1 d i 2 2 where , , as well as their derivatives, are considered to = dx ∂ ∂i  + 2m  + higher terms ; (73) be small real fields, and the dots represent higher order 2 terms in the perturbations; therefore, the substitution into therefore, this Hermitian field  encodes two massive radial the Lagrangian yields the expression, oscillations that are orthogonal, in field space, to the two-  dimensional vacuum manifold described in Fig. 11; the cor- L(ξ, η) = γ ij dxd responding mass is also Hermitian. All these scalar fields, massless and massive bosons will i i i × ∂ ξ∂i ξ − ∂ η∂i η − 2ij∂ ξ∂i η + higher terms , (69) be completely eaten by a vector field through the Higgs mech- anism, once we consider the coupling to hypercomplex QED since the mass terms have disappeared completely, the fields and local rotations; this will lead to massive pure electrody- excitations in the valley directions are massless modes. Note, namics, without the presence of Higgs massive fields. however, that the kinetic terms have a nonconventional form, due to the presence of a hybrid term that mixes the gradients 4.6 Formation of global topological strings: confirming (ξ, η) of the field excitations ; however, one can interchange the Derrick’s theorem the canonical form of quadratic gradients and the mixed form through the invertible mapping Field configurations that define topological defects corre-      spond to domains where the symmetry is left unbroken, i.e. ξ 1 1 −1 ξ i i i ↔ ,∂ξ∂i η ↔ ∂ η∂i η − ∂ ξ∂i ξ. φ = η 2 11 η satisfy the constraint 0; we consider for example the case ( , , , ) (70) of a four-dimensional space-time x y z t as background; for time independent field configurations of the form (1), Alternatively, the kinetic terms in the Lagrangian (69) can be φ = φ1 + iφ2 + jφ3 + ijφ4, such a constraint implies that rewritten in terms of the derivatives of the full phase iξ + jη, φ1(x, y, z) = φ2(x, y, z) = φ3(x, y, z) = φ4(x, y, z) = 0, k by considering the identity [∂ (iξ + jη)][∂k(iξ + jη)]= which define monopoles as possible topological defects. In k i i −[∂ (iξ + jη)][∂k(iξ + jη)]=−[∂ ξ∂i ξ − ∂ η∂i η − the same background a conventional complex field of the i 2ij∂ ξ∂i η], which corresponds essentially to the kinetic form φ = φ1 + iφ2, leads to string defects. Therefore, a 123 Eur. Phys. J. C (2016) 76:98 Page 15 of 22 98

2 2 hypercomplex field of the form (1) leads to monopoles in  1  n l f + f − f − ( f 2 − 1) f + f = 0, (77) ρ (ρ )2 ρ2 a five-dimensional background, and to string defects in a 0r 0r 0 six-dimensional background. However, for a hypercomplex field of the form (4) defined in terms of two real functions, we with the exception of the last term f , which comes from the have again string defects as possible topological defects in new term f ∂2zejlz, all terms correspond to the usual ones a four-dimensional space-time, such as a conventional com- in the traditional scheme; hence, the approximate asymptotic plex field. These algebraic constraints complemented with solutions remain essentially the same, a vacuum manifold with nontrivial fundamental group will (ρ ) ≈ n +··· , → , yield the possibility of formation of string defects in the the- f 0r Cnr r 0 −2 ories considered in this paper. f (ρ0r) ≈ 1 − O(r ), r →∞. (78) In the usual description of U(1)-global strings in three Now the gradient of the field is spatial dimensions [28–30], the minima lie on a circle with 6m2 iα   ψ0 = λ e , where the phase α takes values in [0, 2π],  f θ and by continuity, within the circle the field must vanish ∇ψ = ρ f r + in θ + jlfz ein e jlz, (79) 0 r ψ = 0; the locus of such points lies on the core of the string defect. Thus, for the case at hand, one can relate each point with a new contribution in the z-direction; thus the gradient in the locus to each value of the hyperbolic parameter χ in energy density is the interval (−∞, +∞). Therefore, by enlarging the usual U(1) vacuum manifold to a cylindrical manifold, the string   f 2 defect will lie over the new non-compact transversal direc- |∇ψ|2 = ρ2 ( f )2 + n2 − l2 f 2 ; 0 2 (80) tion; the formation of this defect is consistent with the fact r that a cylindrical manifold is homotopic to S1; hence, topo- logically distinct string defects are labeled by the same ele- hence, the energy per unit length is given by 1 ments of the fundamental group of S , the usual winding   ∞  ∞  number n ∈ Z −{0}. πρ2 2 dr − 2 , 2 0 n l rdr (81) For the usual U(1)-strings with the core aligned with the r z-axis, the field asymptotically takes the form  where we have the usual logarithmical contribution to an infi- 2 nite energy, and additionally we have a new contribution with θ 6m lim ψ(r, z,θ)= ρ ein ,ρ= , (74) a quadratic divergence, which is worse than the logarithmical →+∞ 0 0 λ r contribution. Therefore, the energy is infinite, in fact tending to −∞ instead of +∞ as in the usual case; this result is con- in cylindrical polar coordinates (r, z,θ); and for the case sistent with the Derrick theorem [31], which establishes that at hand we have the extended version that incorporates a there are no finite-energy, time independent solutions, with hyperbolic phase that breaks the translational invariance in scalar fields only, that are localized in more than one dimen- the non-compact z-direction of Eq. (74), sion. However, in the present formulation such a divergence ψ( , ,θ)= ρ inθ jlz,ρ= ρ R + ρ H ; can be cured by considering compensating gauge fields (see lim r z 0e e 0 0 ij 0 (75) r→+∞ Sect. 5.4), such as in the usual U(1) global strings. where lz is adimensional in natural units; with this condition we identify the asymptotic form of the field with its ground state. 5 Hypercomplex electrodynamics: local symmetries Now, following the usual treatment for string defects, we look for a static exact solution for the equations of motion For the usual formulation that describes a charged scalar field (35) with the ansatz coupled to QED, we have the Lagrangian

inθ jlz 1 2 2 ψ = ρ0 f (ρ0r)e e , f (0) = 0, lim f (ρ0r) = 1; L =− Fμν +|(∂μ − ieAμ)ψ| − V (ψ, ψ), (82) r→+∞ 4 (76) with two coupling constants e and λ; the hyperbolic rotations where ρ0r, and lz are adimensional variables in natural units; can be incorporated as part of the local gauge symmetry of the asymptotic limit is required by consistency with the con- the Lagrangian (82), by considering the expression (4)for dition (75). With the substitution into the equations of motion the hypercomplex extension of the modulus ψ · ψ, and the (35), these reduce to a non-linear ordinary equation for f ; local gauge transformations 123 98 Page 16 of 22 Eur. Phys. J. C (2016) 76:98

μ iθ jχ 1 ∂ ψ · ∂ ψ ≈ (γ 2 − )(∂v2 + ∂w2) + γ(∂w2 − ∂v2), ψ → e e ψ, Aμ → Aμ + (∂μθ − ij∂μχ), μ 1 2ij e 2 ψ ∂μψ − ψ0∂μψ ≈ 2i(γ + 1)(w0∂μv − v0∂uw); (88) Fμν = ∂μ Aν − ∂ν Aμ → Fμν, (83) 0 Relevant terms in Eq. (32) that lead to quadratic terms in the where in general the arbitrary real functions depend on the Lagrangian (87) are given, to second order, by Eqs. (58), and background space-time coordinates, θ = θ(x), χ = χ(x). 1 (62), by approaching the circular and hyperbolic phases to Aμ → Aμ + (∂μθ − Furthermore, Eq. (83) imply that e first order, ij∂μχ), and thus, the condition Aμ − Aμ = 0 is preserved 2 λ under gauge transformations, in spite of the hypercomplex − am ψψ + (ψ2ψ2 + ψ2ψ2) ! 0 0 extension of the fields; hence the vector potential is “Hermi- 2 4 − v 2 v2 +··· ; (v = ,w = ) tian” in the hypercomplex sense. ≈ aij PH m R 0 0 0 0 ; − 2 ww2 − 2 w w2 +··· ; (v = ,w = ) The equations of motion and the energy-momentum tensor amR Q R aijmR Q H 0 0 0 0 for the action (82)are, (89)

μ ∂ Fμν = ie(ψ Dνψ − ψ Dνψ), (84) now, the vanishing requirement of interaction terms of the μ 2 form B · ∂μ(ϕ, ϕ,θ,χ) in Eq. (87) yields μ μ λ 6m (∂ − ieA )(∂μ − ieAμ)ψ + (ψψ − )ψ = , λ 0 12 ∂ γ(w2 − v2)θ − (γ 2 − )(w2 + v2)χ = , ij μ 2 0 0 1 0 0 0 (90) (85)  = ψ · ψ − 1 α − 1 L. ∂ (γ 2 − )(w2 + v2)θ + γ(w2 − v2)χ Tμν D(μ Dν) Fμ Fνα gμν (86) μ 1 0 0 2 0 0 2 2  Similarly in this case, the states with minimal energy are 2 −(γ + 1)(w0v − v0w) = 0; (91) ψ ψ = 6m2 given by 0 0 λ , and the vector potential is pure gauge, 0 = 1 (∂ θ − ∂ χ ) θ χ Aμ e μ 0 ij μ 0 , with 0 and 0 arbitrary space- Eq. (90)impliesthat time dependent functions; hence, for vacuum fields we must 0 0 0 0 0 2 2 have ∇ ψ ≡ ∂μψ − ieAμψ = 0, identically, which will 2γ w − v χ = 0 0 θ; (92) be used implicitly below. Hence, the degenerate vacuum is γ 2 − w2 + v2 1 0 0 essentially the manifold described in Fig. 11. First, we use the parametrization of the fields ψ and ψ0 and thus, Eq. (91) leads to given in Eq. (5); the expansions (32), (33), and (34), remain valid in the case at hand since they do not contain the (covari- (γ 4 − )(w2 + v2) 1 0 0 ant) derivatives of the fields; one only requires switching θ = (w0v − v0w); (γ 2 − 1)2(w2 + v2)2 + 4γ 2(w2 − v2)2 on the space-time dependence of the phases (χ, χ0; θ,θ0). 0 0 0 0 Additionally, one must develop the expansion of the first two (93) terms in Eq. (82); expanding around the vacuum requires 0 which corresponds to exploit the freedom of choosing the ψ → ψ + ψ0, and Aμ → Aμ + Aμ, and thus the covariant original gauge field Aμ. We require these expressions when derivative (∂μ − ieAμ)ψ → (∂μ − ieAμ)ψ − ie(Aμψ0 + 0 Aμψ); the Lagrangian (82) reads only one of the fields acquires a non-zero expectation value; ⎧ −γ 2 w 1 2 2 2 2 ⎨ 1 ; (v = ,w = ) L(ψ + ψ , + ) =− + |ψ | 2 v 0 0 0 0 0 A A0 Fμν e 0 Bμ θ = 1+γ 0 ; 4 2 (94)  ⎩ γ −1 v ; (v = ,w = ) γ 2+ w 0 0 0 0 +∂ ψ · ∂μψ + μ (ψ ∂ ψ − ψ ∂ ψ) 1 0 μ B ie 0 μ 0 μ  " 2γ 2 0 − θ; (v0 = 0,w0 = 0) + |ψ | (∂μθ − ∂μχ)+ (ψ ψ + ψ ψ) γ 2−1 2e 0 ij Aμ 0 0 χ = γ . (95) 2 θ; (v = 0,w = 0) γ 2−1 0 0 + (∂μθ − ∂μχ)(ψ ∂ ψ − ψ ∂ ψ) i ij 0 μ 0 μ μ Thus, using the general expressions (92), and (93), the +ieA (ψ∂μψ − ψ∂μψ) 0 quadratic terms in the Lagrangian reduce to 2 μ μ +|ψ0| (∂μθ − ij∂μχ)(∂ θ − ij∂ χ) 1 2 2 2 μ −V (ψ + ψ , ψ + ψ ) + higher terms, (87) L(ψ + ψ0, A + A0) =− Fμν + e |ψ0| Bμ B 0 0 4 = + 1 (∂ θ − ∂ χ) +(γ 2 − 1)(∂v2 + ∂w2) + 2ijγ(∂w2 − ∂v2) where Aμ has been replaced by Aμ Bμ e μ ij μ , 2 2 2 2 2 and Fμν is expressed now in terms of the new field Bμ. Addi- (1 − γ )(w + v ) + 2ijγ(w − v ) +(γ 2 + 1)2 0 0 0 0 tionally, we have (γ 2 − )2(w2 + v2)2 + γ 2(w2 − v2)2 1 0 0 4 0 0 123 Eur. Phys. J. C (2016) 76:98 Page 17 of 22 98

× w2∂v2 − v w ∂μv∂ w + v2∂w2 0 0 0 μ 0   2 am λ 2 2 − − ψψ + (ψ ψ2 + ψ2ψ ) + higher terms; 2 4! 0 0 (96) in this expression, we have not yet fixed one of the vacuum fields (v0,w0), and we have remanent mixed terms of the μ form ∂ v · ∂μw, and v · w, which turn out to be proportional to v0 · w0. Since only one v.e.v., v0 or w0 will acquire a non- zero value, such mixed terms will vanish at the end; now we shall study each case separately. B (γ ) B (γ ) Fig. 12 The blue curve represents MR ,andthered curve M H ; v (γ ) γ = the dashed curve represents PH in Eq. (97); the case 0repro- duces the usual SSB of U(1), with Pv (0) =−1 = M B (0),and 5.1 The case (v0 = 0,w0 = 0) H R B ( ) = MH 0 0 Using Eqs. (89), the Lagrangian (96) reduces to 2 6e B B 2 M (γH ) + ijM (γH ) m 1 2 2 2 μ aλ R H R L(ψ + ψ0, A + A0) =− Fμν + e |ψ0| Bμ B 4 6e2 +(γ 2 − − γ)∂v2 + v 2 v2 ≈ [−0.7071 + 0.2929ij] m2 . (99) 1 2ij aij PH m R aλ R + ; higher terms (97) Similarly in the limit γ →−γH , the figure shows that the fields will acquire the higher masses, the kinetic and the mass terms for the field w have disap- 6e2 peared, and then w corresponds to a Nambu–Goldstone field. M B(−γ ) + ijMB (−γ ) m2 λ R H H H R Additionally we have a residual massive field v with a non- a 6e2 zero vacuum expectation value given by Eq. (48); the v.e.v. ≈ [−1.7071 − .7071ij] m2 . (100) of this Higgs field determines the (Hermitian) mass of the aλ R longitudinal mode of the (Hermitian) vector field B, The limit γ → γH is the limit of light masses for the fields, and γ →−γH corresponds to the limit for massive fields; 2|ψ |2 = 2(γ 2 − − γ)v2 note that the difference between such limits is one mass unit e 0 e 1 2ij 0 for both, real and hybrid components. 6e2 = M B(γ ) + ijMB (γ ) m2 ; aλ R H R 5.2 The case (v0 = 0,w0 = 0) 1 − γ 2 2γ M B(γ ) ≡ , M B (γ ) ≡ ; (98) R γ 2 + γ − H γ 2 + γ − 2 1 2 1 In this case, the Lagrangian (96) reduces to

1 2 2 2 μ therefore, the Hermitian vector field B, which in general L(ψ + ψ , A + A ) =− Fμν + e |ψ | Bμ B 0 0 4 0 has the form B ≡ BR + ijBH , has acquired a Hermitian +(γ 2 − + γ)∂w2 + 2 ww2 + 2 w w2; mass through the Higgs mechanism; hence, one has two real 1 2ij amR Q R aijmR Q H masses for two real fields (BR, BH ). The flows of the masses +higher terms; (101) defined by these polynomials are shown in Fig. 12; this figure now the kinetic and the mass terms for the field v have disap- shows the global behavior, and the behavior in the interval peared, and the field v corresponds to a Nambu–Goldstone (γ , −γ ). The flows have the same asymptote, the root √H H field. The residual field w is massive in both senses, real and 2 − 1 of the polynomial (γ 2 + 2γ − 1). The real mass of hybrid; the non-zero vacuum expectation value for this field B vanishes at two roots of M B, γ 2 = 1, the purely hyper- R 6m2 is given by w2 = R (see Sect. 4.3), and deter- bolic limit for the theory; however, these roots are out of the 0 aλ(1−γ 2+2γ) interval (γH , −γH ). mines the masses for the vectorial field B shown in Fig. 13. The figure also shows that the Higgs boson field v may This figure is basically the mirrored image with respect to the have a mass as small as γ → γH , although strict massless- “y”-axis of Fig. 12; the qualitative and quantitative aspects ness is prohibited; hence, in this limit one could expect a of Eqs. (99), and (100) remain valid, and one only requires nearly massless Higgs boson. In this limit the masses for the interchanging the roles of the limits γH ↔−γH . The dif- vectorial field B will acquire the smallest values, ference with respect to the Lagrangian (97) is that the Higgs 123 98 Page 18 of 22 Eur. Phys. J. C (2016) 76:98

L(ψ + ψ0, A + A0) 1 2 2 2 2 2 =− F + e |ψ | B + (∂ρR + ij∂ρH ) 4 0 2 2 −|ψ0| (i∂ξ + i∂η) 2 μ +2e|ψ0| B ∂μ[θ − ξ + ij(η − χ)] 2 μ +|ψ0| ∂ (θ − ijχ)· ∂μ[θ − 2ξ + ij(2η − χ)] 2 2 +am (ρR + ijρH ) + higher terms. (104) The case γ 2 = 1: hyperbolic electrodynamics Along the lines followed in Sect. 4.5 for this case, we consider first Eq. (66), with η = constant; the vanishing of μ the interaction terms of the form B · ∂μ(ξ,η,θ,χ) in Eq. Fig. 13 The mirrored image of Fig. 12 (104) requires the identification θ(x) = ξ(x), and the fixing of the hyperbolic parameter χ = constant;

field w appears with real and hybrid masses, and their flows 1 6e2m2 L(ψ + ψ , A + A ) =− F2 − ij H B2 are shown as dashed curves; this behavior was discussed in 0 0 4 aλ Fig. 9, and hence, in the limit γ → γH , the Higgs field w will + γ [(∂ρ)2 + 2 ρ2]+ . 2 ij aijmH higher terms (105) have a light real mass and a hybrid mass with its maximum ξ( ) value, and conversely in the limit γ →−γH . Hence, the field x has disappeared and represents a Nambu–Goldstone boson in the two-dimensional valley. Therefore, as opposed to the global SSB scenario described 5.3 Polar parametrization for the fields: purely massive in Eq. (67), the field ξ has not survived. Furthermore, in electrodynamics, no massive Higgs bosons an orthogonal valley direction, we have a massive mode ρ, which has survived the gauging of the global SSB to the local Along the lines followed in Sect. 4.5, we consider the polar symmetry. decomposition for the fields with radial and circular modes; Similarly, in the case with a field of the form (66), with (ρ + ρ ) iξ jη in the expression of the form R ij H e e ,forthe ξ = constant, the quadratic terms in the Lagrangian have dynamical field we can consider for generality that the four essentially the same form shown in Eq. (105), where the inter- ρ ρ ξ η variables R , H , , and , are space-time coordinates depen- action terms, and the terms involving the fields (χ(x), η(x)) dent, and at the end we shall consider that two of them must vanish by fixing now θ = constant, and identifying χ(x) = be constants. Hence, one has the following expressions: η(x). Now, using the parametrization (68), which encodes all μ 2 2 2 θ( ) = ξ( ) ∂μψ · ∂ ψ = (∂ρR + ij∂ρH ) − (ρR + ijρH ) (i∂ξ+i∂η) , fluctuations in the phases, the identifications x x , 2 and χ(x) = η(x) lead to the vanishing of the interaction ψ∂μψ − ψ∂μψ = 2(ρR + ijρH ) (i∂ξ + i∂η), terms in Eq. (104), and all terms involving the scalar fields; (ψ + ψ , ψ + ψ ) = 2ρ2 + , V 0 0 m higher terms (102) the Lagrangian reduces to second order to a massive pure electrodynamics, and then 6e2m2 =−1 2 − H 2 + ; 2 2 2 L F ij B higher terms (106) |(∂μ − ieAμ)ψ| = (∂ρ + ij∂ρ ) + e(ρ + ijρ ) λ  R H R H 4 a μ μ × eB Bμ + 2B ∂μ[θ − ξ + ij(η − χ)] thus, the gauge vector boson has eaten the two Nambu– Goldstone bosons (ξ, η) and has acquired a mass; the 1 1 − (i∂ξ + i∂η)2 + ∂μ(θ − ijχ) field excitations in the valley directions described in the e e  Lagrangian (69) have not survived the gauging of a global ·∂μ[θ − 2ξ + ij(2η − χ)] , (103) SSB to a local symmetry, and there are no scalar Higgs fields. In the case of the conventional scalar electrodynamics one has the vector gauge field A, a complex field φ = φ1 + iφ2, = + 1 (∂ θ − ∂ χ) 1 where Aμ Bμ e μ ij μ ; fluctuations around with two real scalar fields, and the S vacuum manifold with the vacuum require one to rewrite the dynamical field as only one generator; thus, after the spontaneous symmetry ψ → (ρ + ρ0 + ij(ρ + ρ0 ))eiξ e jη R R H H , which inserted into breaking, only one scalar field can be eaten through the Higgs the above expressions leads to mechanism, leaving a massive vector field and additionally a 123 Eur. Phys. J. C (2016) 76:98 Page 19 of 22 98

The case with two scalar redundant modes can be devel- oped along the ideas behind Eq. (106), leading to the Lagrangian

1 L =− F2 + e2|ψ |2 B2 + higher terms; (109) 4 0

the only difference is the Hermitian mass for the vector field B; depending on the vacuum expectation values (v0,w0) such a running mass is described by Fig. 12, or the mirrored figure, Fig. 13; in both cases there will be no dashed curves, since the scalar Higgs fields have strictly disappeared; we Fig. 14 The constant real mass is represented by the horizontal dashed line; the running hybrid mass is represented by the dashed curve have again a purely massive electrodynamics, without any clues of scalar fields. massive scalar field, as opposed to the massive pure electro- dynamics at hand, with a hypercomplex field with two real 5.4 Local topological strings: Aharanov–Bohm-like defects fields, and a vacuum manifold with two generators. The case γ 2 = 1: hyperbolic deformation of QED The geometrical and topological description of the formation For this scenario of running parameters we shall consider of global string defects given in Sect. 4.6 is valid in essence two extremal cases; the first case corresponds to two massive for the local case, adding the asymptotic form for the gauge modes with oscillations orthogonal to the two-dimensional field, and the gradients of the fields, valley, and the second case with two redundant modes oscil- inθ jlz lim ψ(r, z,θ)= ρ0e e , lating on the valley. r→+∞ For the first case we consider the parametrization given in lim D ψ = 0, n ∈ Z −{0}; r→+∞ A Eq. (71), and the Lagrangian reads # $ 1 1 n lim A(r, z,θ)= ∇(nθ − ijlz) = θˆ − ijlzˆ , 1 2 2 2 2 2 →+∞ L =− F + e |ψ | B + (∂ρR + ij∂ρH ) r e e r 4 0 lim Fμν = 0; (110) 2 2 r→+∞ +am (ρR + ijρH ) + higher terms; (107) where these asymptotic expressions correspond to the boundary ⎧ # $ conditions for strings solutions of nontrivial windings of a ⎨ 2 + γ 2−2γ −1 ; (v = ,w = ) m R 1 ijγ 2+ γ − 0 0 0 0 circle onto the vacuum manifold. Therefore, with these con- m2 = # 2 1 $ ; (108) ⎩ 2 γ 2+2γ −1 ditions, the magnetic flux passing through a closed surface S m 1 + ij ; (v0 = 0,w0 = 0) R γ 2−2γ −1 is given by therefore, we have a Hermitian scalar field with a Hermitian  % 2πn mass, which has a constant real mass, and a running hybrid B · dS = A · dC =− , dC = rθˆdθ; (111) mass; for the first case of the above equation the mass of the S C e vector field B, is described in Sect. 5.1 andshowninFig. 12. In this case, the difference is the presence of a Hermitian where C is an infinitely large loop at spatial infinity, and we Higgs field with different masses; Fig. 14 shows again the have used the Stokes theorem; at this region, the vector θˆ masses of the field B shown in Fig. 12, but now with the points tangentially to the loop, and zˆ points orthogonally in Higgs masses given in Eq. (108). The figure shows that in the cylindrical direction. Therefore, the conventional quan- this case the limit γ → γH is not a limit for light Higgs tized magnetic flux for string defects remains intact. fields; however, the limit γ →−γH corresponds again to Now the ansatz includes Eq. (76) for the field ψ, and the a massive fields limit. The hybrid component of the mass following expression for the gauge field: 2 ∈ ( 2 (γ ), 2 (−γ )) ≈ is running in the interval m H m H H m H H (0.4142, 2.4142)m2 . i R A(r, z,θ) = f (r)einθ e jlz∇(e−inθ · e− jlz) Furthermore, the second case described in Eq. (108) can be A e # $ obtained from the first one by the transformation γ →−γ , 1 n = f (r) θ − ijlz , and the corresponding figure is basically the mirrored image e A r with respect to the “y”-axis of Fig. 14. f A(0) = 0, f A(∞) = 1; (112) 123 98 Page 20 of 22 Eur. Phys. J. C (2016) 76:98 in consistency with the asymptotic limits (110). This poten- topology, invokes immediately the Aharanov–Bohm effect tial results in a magnetic field [32]: a narrow, infinite length solenoid is added to the two- slit experiment for electrons; topologically the solenoid is  # $ f n a string-like defect, and thus the phase shift on the elec- B =∇×A = A z + ijlθ , (113) e r tron wave function is a topological effect determined by the magnetic flux inside the solenoid. Outside, there is no where we have a new hybrid contribution in the circular direc- magnetic field, with non-zero potential. Since the original = tion; note that B B. experimental confirmation [33], the Aharanov–Bohm effect Therefore, from Eqs. (76), (112), and (113), the equations has received considerable study, from theoretical generaliza- of motion (84) reduce to   tions, to a variety of experimental realizations; for example, n d f  ijl 1 d   the appearance of the electronic interference phenomenon in − A θ + rf z A carbon nanotubes suggests that the Aharanov–Bohm effect e dr r e r#dr $ n is relevant even at the microscopic scale [34]. With this = 2eρ2 f 2(1 − f ) · θ − ijlz , (114) 0 A r perspective, we establish now the analogy with the case at where the new contribution correspond to hybrid terms in the hand. z-direction on both sides; these equations reduce explicitly In the analogy, the infinite string core corresponds to the to Aharanov–Bohm solenoid; the azimuthal component of A in Eq. (120) falls off like 1/r, with the distance from the θ :  −  + 2ρ2 ( − ) 2 = , rfA f A 2e 0r 1 f A f 0 (115) core, such as in the Aharanov–Bohm effect. Additionally  :  +  + 2ρ2 ( − ) 2 = . ˆ z rfA f A 2e 0r 1 f A f 0 (116) the new contribution in the z-direction is constant every- where; this new contribution does not change the quantiza- Additionally, Eq. (85) reduces to tion condition of the magnetic flux in (111). Now an impor-   tant difference; in the core of the string defect the magnetic 2 2 1 d  2 n 2 m 2 field vanishes, as opposed to the Aharanov–Bohm solenoid, (rf )− f (1− f A) − l + (1− f ) f = 0, r dr r 2 2 inside which the B field is non-zero; the string defect at (117) hand is actually a ‘pure gauge” phenomenon. Furthermore, in the background, one has the same topological feature, where we have underbraced the new contribution coming namely, the existence of a non-simply connected space, and from the hyperbolic phase e jlz. Furthermore, Eq. (115)is thus the winding number around the loop is observable in exactly the same equation obtained in the usual formulation the Aharanov–Bohm effect; in the analogy, this fact repre- for the U(1) local strings; this equation together with the sents a phenomenological possibility for the string defects at corresponding equation of the form (117), have no closed- hand. form solutions. Hence the asymptotic analysis is used for Let us see how these “pure gauge” potentials are able large r, and close to the vortex core. However, the presence to make finite the energy of the defect, playing the role of of Eq. (116), distinctive of this hypercomplex formulation, compensating fields for the divergent effect of the scalar has a dramatic effect on the possible solutions, enforcing the fields. vacuum configuration for f A in full space, except for r = 0; The substitution of the solution (118) into Eq. (117) leads to a simplified equation, f A(0) = 0, f A(r) = 1, r ∈ (0, +∞), (118) 2 1 d  m (rf ) + (1 − f 2) f = 0; (121) thus, the potential is “pure gauge”, r dr 2 A(0) = 0; B(0) = 0, r = 0, in the vortex core, (119) # $ an immediate solution is f = 1, and thus, the field ψ settles 1 n A = θ − ijlz ; B(r) = 0, r ∈ (0, +∞), elsewhere; down to its vacuum configuration in the full space, except e r (120) for the core. Note that this solution for f does not solve Eq. (77) for global strings. In this case, all components of the this closed-form solution shows two representative features energy-momentum tensor (86) vanish trivially, and hence are of a topological defect, namely the field configurations where finite. Therefore, all classical Maxwell physical observables the symmetry is left unbroken, and the configurations at the vanish, and there is no classical experiment that allows us vacuum, where the symmetry will be spontaneously broken; to detect the string defect by its electrodynamical effects. note from Eq. (120) that the potential A cannot vanish at However, the loop integral (111) is a gauge invariant quantity the vortex core, due to the restriction n = 0. The pres- and, according to the Aharanov–Bohm effect, is detectable ence of “pure gauge” potentials in spaces with nontrivial by quantum interference. 123 Eur. Phys. J. C (2016) 76:98 Page 21 of 22 98

5.5 There are no other solutions for local strings 6 Concluding remarks

For the usual description of local string defects that involves 6.1 Cosmological implications only Eqs. (115) and (117), the asymptotic solutions are well known [30]; hence, close to the vortex core, one has The results obtained can be interpreted in various senses; in the inflationary cosmological context, the topological defects 2 n f A ∼ r , f ∼ r , r → 0; (122) are essential in the spontaneous symmetry breaking based phase transitions; if a phase transition occurs, then the topo- and for large r, logical defects may be generated provided that the vacuum manifold has a nontrivial topology. Since the dramatic effect of the substitution of the circular rotations by the hyper- λ − f ∼ e−mr , − f ∼ e−βr ,β= , r →∞; bolic rotations is the trivialization of the vacuum manifold 1 A 1 2 e from the homotopic point of view (Sect. 4.1), the insignifi- (123) cant observational support of the existence of cosmic string defects leads to the possibility that the vacuum can manifest physically, these expressions lead to an energy density more a non-compact topology in a certain phase of the early uni- localized that for the global strings, and show that the asymp- verse. This in its turn has various implications; for example totic behavior of f is controlled by the gauge field contribu- it suggests that the GUT philosophy must be extended by tion f A. These expressions are not valid in the present treat- incorporating non-compact gauge groups. Furthermore, the ment, since f A, has been relaxed to its vacuum configuration proliferation of cosmic defects is an essential feature of cer- ∈ ( , +∞) in the full interval r 0 due to Eq. (116); we have tain GUT’s; inflation was originally proposed as a form of only at hand Eq. (121), which is fully independent on f A. explaining the observational evidence against such a prolifer- Other closed-form solutions for Eqs. (121) are not known; ation. In the light of the present results, one can alternatively however, one can obtain asymptotic solutions for large r and postulate the hyperbolic symmetry as an essential symmetry close to the core using linearized versions; for large r the at some period of the early universe; thus, the inflationary = = background field is f 1, and close to the vortex f 0; scenarios can be modified drastically by the presence of the   new symmetry. More explorations are mandatory along these r f +f − m2r f = 0, r →∞, ideas.  f = AJ0(imr) + BN0(−imr), (124) 6.2 On Aharonov–Bohm strings and 2   m It has been shown that the dominant interaction between mat- r f +f + r f = 0, r → 0,  2    ter and cosmic strings is through an Aharonov–Bohm inter- m m action [35]; outside the tiny region of the inner core, the  f = AJ0 √ r + BN0 √ r , (125) 2 2 field strengths vanish, but one has non-vanishing potentials in the outer region, and hence a scattering mechanism of the where J0 is the zeroth-order Bessel function, N0 the zeroth- Aharonov–Bohm type will work at the outer region. The scat- order Neumann function, and A and B are adimensional con- tering cross sections and production rates do no go to zero as = stants. However, in Eq. (125), the condition A 0 is enforced the geometrical size of the string vanishes. One can realize = for ensuring a single-valued function at r 0; thus, the that the results obtained in the present approach are consistent energy density is given essentially by the expression with those results, but they suggest that the Aharonov–Bohm    interaction is the only interaction between matter and cosmic 2 2 2 1  2 B m m strings; the field configurations for the strings found in Sect. T00 =− ( f ) =− N1 √ r , (126) 2 4 2 5.4 do not distinguish between inner core and outer region, since the potentials are pure gauge as close to the core as one where N1 is the first-order Neumann function; thus, the wishes. energy diverges, similarly to the case of global strings. Since The Aharonov–Bohm strings have appeared previously local strings have their energy confined mainly close to the in the study of discrete gauge symmetries [36–39]; specifi- core, such a solution is not physically meaningful; in this cally in the effective Lagrangian description of Zk discrete case the gauge field f A has disappeared as a compensating gauge theory, this type of strings have confined magnetic field of the divergent effects of the scalar field. Therefore, the flux with a 1/k unit of fundamental magnetic charge; due Aharonov–Bohm like defects discussed early, are the only to this charge the Aharonov–Bohm string can interact with field configurations with finite energy. matter. Recently, certain cosmological constraints have been 123 98 Page 22 of 22 Eur. Phys. J. C (2016) 76:98 imposed on these theories, by studying the radiation of stan- 18. M. Niedermaier, E. Seiler, Nonamenability and spontaneous sym- dard model particles from these strings [40]. As opposed to metry breaking: the hyperbolic spin chain. Annales H. Poincaré 6, these discrete gauge symmetry approaches, the Aharonov– 1025 (2005). arXiv:hep-th/0312293 19. T. Morozumi, S. Nojiri, An analysis of noncompact nonlinear Bohm-like strings have been obtained in the approach at hand sigma models. Prog. Theor. Phys. 75, 677 (1986) by incorporating a (non-compact) continuous symmetry, the 20. J.W. van Holten, Quantum non-compact sigma models. J. Math. hyperbolic rotations; it may be interesting to study phe- Phys. 28, 1420 (1987) nomenological models that incorporate both, discrete, and 21. S. Mandelstam, Vortices and quark confinement in non-Abelian gauge theories. Phys. Rep. C 23, 245 (1976) non-compact gauge symmetries, following the ideas descri- 22. G. t Hooft, Topology of the gauge condition and new confine- bed in [40]. ment phases in non-Abelian gauge theories. Nucl. Phys. B 190, 455 (1981) Acknowledgments This work was supported by the Sistema Nacional 23. P. Faccioli, E. Shuryak, QCD topology at finite temperature: sta- de Investigadores (México), and the Vicerrectoria de Investigación y tistical mechanics of self-dual dyons. Phys. Rev. D 87, 074009 Estudios de Posgrado (BUAP). The authors would like to thank Dra. (2013). arXiv:1301.2523 [hep-ph] I. Rubalcava-Garcia, and Dr. A. Herrera-Aguilar for discussions. The 24. Y.M. Cho, D.G. Pak, P.M. Zhang, L.P. Zou, Weyl symmetric numerical analysis and graphics have been made using Mathematica. structure of QCD vacuum. Phys. Rev. D 86, 045025 (2012). arXiv:hep-th/12045970 Open Access This article is distributed under the terms of the Creative 25. L.P. Zou, P.M. Zhang, D.G. Pak, Knot topology in QCD. Phys. Commons Attribution 4.0 International License (http://creativecomm Lett. B 726, 436 (2013). arXiv:1302.3527 [hep-th] ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, 26. I. Bengtsson, J. Hallin, SL(2, R) Yang–Mills theory on a circle. and reproduction in any medium, provided you give appropriate credit Mod. Phys. Lett. A. 9, 3245 (1994). arXiv:hep-th/9407035 to the original author(s) and the source, provide a link to the Creative 27. C.T. Hill, Is the Higgs boson associated with Coleman–Weinberg Commons license, and indicate if changes were made. dynamical symmetry breaking? Phys. Rev. D 89, 073003 (2014). Funded by SCOAP3. arXiv:1401.4185 [hep-ph] 28. T.W.B. Kibble, Classification of topological defects and their rel- evance in cosmology and elsewhere, in Topological Defects and the Non-equilibrium Dynamics of Symmetry Breaking Phase Tran- sitions, ed. by Y.M. Bunkov, et al. (Kluwer Academic Publishers, References Berlin, 2000) 29. H. Nielsen, P. Olesen, Vortex-line models for dual strings. Nucl. 1. M.H. Emam, Five-dimensional 2-branes and the universal hyper- Phys. B 61, 45 (1973) multiplet. Nucl. Phys. B 817, 208 (2009). arXiv:hep-th/0701060 30. A. Vilenkin, E.P.S.Shellard, Cosmic Strings and Other Topological 2. M.H. Emam, BPS one-branes in five-dimensions. arXiv:1301.7338 Defects (Cambridge University Press, Cambridge, 1994) [hep-th] 31. G.H. Derrick, Comments on nonlinear wave equations as models 3. M.H. Emam, Phys. Rev. D 74, 125004 (2006) for elementary particles. J. Math. Phys. 5, 1252 (1964) 4. M.H. Emam, Phys. Rev. D 79, 085017 (2009) 32. Y. Aharonov, D. Bohm, Significance of electromagnetic potentials 5. M.H. Emam, Phys. Rev. D 71, 125020 (2005) in the quantum theory. Phys. Rev. 115, 485 (1959) 6. M.H. Emam, Phys. Rev. D 84, 045016 (2011) 33. R.G. Chambers, Shift of an electron interference pattern by 7. G.W. Gibbons, M.B. Green, M.J. Perry, Instantons and seven- enclosed magnetic flux. Phys. Rev. Lett. 5, 3 (1960) branes in type IIB superstring theory. Phys. Lett. B 370, 37 (1996). 34. J. Cao, Q. Wang, M. Rolandi, H. Dai, Aharonov–Bohm interfer- arXiv:hep-th/9511080 ence and beating in single-walled Carbon-nanotube interferome- 8. V.Cortes, C. Mayer, T. Mohaupt, F. Saueressing, Special geometry ters. Phys. Rev. Lett. 93, 216803 (2004). arXiv:cond-mat/0407185 of Euclidean supersymmetry. 1. Vector multiplets. JHEP 0403, 028 35. M.G. Alford, F. Wilczek, Aharonov–Bohm interaction of cosmic (2004). arXiv:hep-th/0312001 strings with matter. Phys. Rev. Lett. 62, 1071 (1989) 9. V.Cortes, C. Mayer, T. Mohaupt, F. Saueressing, Special geometry 36. L.M. Krauss, F. Wilczek, Discrete gauge symmetry in continuum of Euclidean supersymmetry II: hypermultiplets and the c-map. theories. Phys. Rev. Lett. 62, 1221 (1989) JHEP 0506, 025 (2005). arXiv:hep-th/0503094 37. M.G. Alford, J. March-Russell, F. Wilczek, Discrete quantum hair 10. S. Ulrych, Gravitoelectromagnetism in a complex Clifford algebra. on black holes and the nonAbelian Aharonov–Bohm effect. Nucl. Phys. Lett. B 633, 631 (2006). arXiv:gr-qc/0602018 Phys. B 337, 695 (1990) 11. S. Ulrych, Relativistic quantum physics with hyperbolic numbers. 38. J. Preskill, L.M. Krauss, Local discrete symmetry and quantum Phys. Lett. B 625, 313 (2005). arXiv:hep-th/9904170 mechanical hair. Nucl. Phys. B 341, 50 (1990) 12. S. Ulrych, The Poincaré mass operator in terms of a hyperbolic 39. T. Banks, N. Seiberg, Symmetries and strings in field theory and algebra. Phys. Lett. B 612, 89 (2005). arXiv:hep-th/0510198 gravity. Phys. Rev. D. 83, 084019 (2011). arXiv:1011.5120 [hep-th] 13. J. Hucks, Hyperbolic complex structures in physics. J. Math. Phys. 40. Y.Ookouchi, Discrete gauge symmetry and Aharonov–Bohm radi- 34, 5986 (1993) ation in string theory. JHEP 1401, 049 (2014). arXiv:1310.4026 14. S. Ulrych, Symmetries in the hyperbolic Hilbert space. Phys. Lett. [hep-th] B 618, 233 (2005). arXiv:hep-th/0510199 15. J. Pati, A. Salam, Unified lepton–hadron symmetry and a gauge theory of the basic interactions. Phys. Rev. D 8, 1240 (1973) 16. S. Ulrych, Considerations on the hyperbolic complex Klein–Gordon equation. J. Math. Phys. 51, 063510 (2010). arXiv:1006.5182 [math-ph] 17. L. Faddeev, G. Korchemsky, High energy QCD as a com- pletely integrable model. Phys. Lett. B 342, 311 (1995). arXiv:hep-th/9404173 123