Some comments about emission channels of non abelian vortices
aa r X i v : . [ h e p - t h ] F e b Some comments about emission channels of non abelian vortices
Osvaldo P. Santill´an ∗ Abstract
As is well established, several gauge theories admit vortices whose mean life time is very large. Insome cases, this stability is a consequence of the topology of the symmetry group of the underlyingtheory. The main focus of the present work is, given a putative vortex, to determine if it is nonabelian or not by analysis of its physical effects. The example considered here is the simplestone namely, a SU (2) gauge model whose internal orientational space is described by S . Axionand gravitational emission are mainly considered. It is found that the non abelian property isbasically reflected in a deviation of gravitational loop factor γ l found in [25]-[26]. The axion emissioninstead, is not very sensitive to non abelianity, at least for this simple model. Another importantdiscrepancy is that no point of the vortex reaches the speed of light when orientational modesare excited. In addition, the total power corresponding to each of these channels is compared,thus adapting the results of [1]-[3] to the non abelian context. The excitations considered hereare simple generalizations of rotating or spike string ansatz known in the literature [4]-[7]. It issuggested however, that for certain type of semi-local strings whose internal moduli space is noncompact, deviations due to non abelianity may be more pronounced.
1. Introduction
The dynamic of gauge vortices is a fascinating branch of physics whose role is not yet well understood.Historically, it was realised by Abrikosov that magnetic field lines play a fundamental role in phasetransitions in type low temperature II superconductors [8], and these objects were further studied in[9]. The dynamics of vortices in random environment is also of particular importance in the physics ofhigh temperature superconductors [10]-[12] . In addition, the phenomena of pinning of vortices mayalso have applications in the physics of neutron stars, as described for instance in [14] and referencestherein. Abelian vortices were intensively studied as sources of galaxy formation, some classic worksabout this topic are [15]-[30].One notable prediction related to such vortices is the phenomena of linear confinement of magnetsinside a low temperature type II superconductor [8]. Based on these phenomena, Mandelstam, Nambuand ’t Hooft suggested that a dual Meissner effect in which the field lines are chromo-electric, andthe electric and magnetic charge are interchanged, may explain the long standing question about howquarks are confined inside the hadrons [31]-[33]. The problem about this hypothesis is that it is notwell understood how to include objects like monopoles in ordinary QCD, with SU (3) gauge group, inorder to achieve this mechanism. ∗ Instituto de Matem´atica Luis Santal´o (IMAS), UBA CONICET, Buenos Aires, Argentina fi[email protected] [email protected]. See [13] and its references for an extensive review. SU (2) gauge theory with N= 2 supersymmetry. This theory admits very massive monopoles, whichbecomes massless in certain limit of the parameter space. The model posses a duality that interchangethe electric and magnetic field and charges E ↔ B, g → g , with g the abelian coupling constant of the model. The addition of certain term that breaks N= 2supersymmetry to N= 1 induce monopole condensation by Abrikosov lines. From the dual point ofview, these lines are magnetic. But in the original theory, they are electric. This means that Seibergand Witten found a realisation of a supersymmetric dual Meissner effect.One drawback of the Seiberg-Witten scenario is that is related to abelian vortices, however itmotivated a large amount of work about non abelian ones. In the context of supersymmetric theories,solutions of this type were found in [36]-[40]. Some of these models admit non abelian vortices whenthe s-quarks are the Higgs phase. These vortices induce monopole condensation at weak coupling, thusgeneralizing the Seiberg-Witten mechanism to the non abelian case. Another remarkable feature thatarise in this context is the presence of phases that are not identified neither with the Higgs, Coulombor confined one [41]. In particular, the ”instead of confinement” phase considered in [43], in whichthe quarks and gauge bosons of the model decay into monopole and anti-monopole pairs that formstringy mesons [44]. This phase is continuously connected to the fully Higgsed phase. This bears aresemblance with the Fradkin-Shenker scenario [42] generalized to the supersymmetric context, butwith the difference that the confined phase is replaced with the ’”instead of confinement” one.The vortices described above are related to N= 2 supersymmetric gauge theories. Since theirappearance, there have been investigations about these objects in theories with less supersymmetry[46]-[52]. In addition, some advances has been reported in the area of semilocal strings [53] appliedto supersymmetric theories [54]-[57]. In particular, an interesting link with critical superstrings waspointed out in [92] and further worked out in [93]-[99]. These works conjecture that in the strongcoupling regime, and in some specific thin limit, the resulting low energy theory can be identified witha IIA string over a target space which is the product of four dimensional space with a six dimensionalconifold. The Minkowski space represent the translational modes of the object, and the conifoldrepresents internal modes of the vortex. At classical level, the conifold is not represented by a Ricciflat metric, but the conjecture takes into account quantum corrections. After these corrections havebeen properly taken into account it is believed that the Ricci flat (Calabi-Yau) metric will emerge.The string theory techniques then may be applied in order to study the spectrum of the states of thetheory. More details can be found in [93]-[99].The physics of non abelian vortices is not only related to supersymmetric theories, and has in factapplications in ordinary QCD, even taking into account the drawback about monopoles mentionedabove. In particular, the study of non abelian vortices has proven to be fruitful in the so called2he colour-flavor locked phase of QCD [61]-[62]. This phase is supposed to appear for QCD at veryhigh densities, such as the ones in the core of a neutron star. In this phase the mean distancebetween two hadrons is much less than its mean radius r ∼ fm, and is expected for the quarkscomposing these composite particles to acquire a large mobility. The relevant excitations in such highdensity phase are sourced by quarks close to the Fermi surface. These low energy excitations thenhave a very large momentum, which implies that the system is asymptotically free and the confinedphase arguably does not take place [65]. The gluons are now part of the asymptotic spectrum of thetheory and induce an attractive interaction, giving rise to quark Cooper pairs which are not coloursinglets. A very rough estimation of the resulting gap is ∆ ∼ −
100 MeV, but there appear severalcorrections to this value due to the high chemical potential µ or the high temperature T of the neutronstar. The resulting state is symmetric under certain operation that interchange of color and flavoursimultaneously, a colour-flavour diagonal symmetry [63]. For this reason this phase sometimes isreferred as colour superconductivity or colour-flavor locked phase. Details of these affirmations maybe found in the reviews [64]-[65] and references therein. But is worthy to emphasize that the presenceof a gap may affect the transport properties of this regions and may influence the cooling rates or theirrotational properties of a neutron star [66]. This phase, as well as other hypothetical QCD phasesadmits non abelian vortices, as reviewed in [64]. Recent progress in the physics of these vortices havebeen reported in [67]-[91]. The colour-flavour diagonal symmetry, quotiented by a suitable subgroup,describe different inequivalent vortices. Therefore these objects acquire a moduli, which is non abelianin nature. Details of these affirmations can be found in [64] and references therein.The present work is focused on a simple and, at the moment, academic problem. This problem is,given an excited vortex, to understand if it is abelian or not abelian in nature by studying it emissionchannels. Particular attention is paid here on axion emission and also on gravitational waves. One ofthe main differences is that non abelian vortices may invest part of its energy in excitation of internalmoduli. This in particular implies that there are no points in the vortex reaching the speed of light,as all the velocities are slowed for sourcing these internal excitations. Another important difference isthe loop factor for gravitational radiation, whose value changes when internal orientations are excited,as will be discussed along the text.The organisation of this work is as follows. In section 2 some generalities about gauge theoriesadmitting non abelian vortices are stated. In section 3 the dynamics of these vortices is characterisedand some solutions are presented. In section 4, the coupling to axion particles is worked out and insection 5, formulas for the power radiated in axions are presented. The explicit power radiated forthe presented solutions is estimated in section 6. In section 7 the power radiated corresponding togravitational wave emission is discussed and, in particular, it is clarified how non abelian excitationsmay affect it. Section 8 also contains an axion radiation process, but in this case the internal modesexcitations play a more important role than in the examples of section 6. Section 9 contains thediscussion of the obtained results. 3 . Simple examples of non abelian vortices supersymmetric gauge models In this subsection, some basic features about supersymmetric models and about the colour-flavorlocked phase are briefly discussed, following [60] or [64]. The reader familiar with these subjects mayskip to the next subsection.A typical (but not unique) form of a bosonic lagrangian for N= 2 supersymmetric models, admittingnon abelian vortices as solutions, is the following [60] S = Z d x (cid:20) g F aµν F aµν + 14 g F µν F µν + 1 g |∇ µ a a | + 1 g | ∂ µ a | + Tr |∇ µ Φ | + Tr |∇ µ ¯˜Φ | + V (Φ , ˜Φ , a a , a ) (cid:21) . (2.1)Here the gauge group is generically SU ( N ) × U (1) and ∇ µ is the covariant derivative in the adjointrepresentation of the group ∇ µ = ∂ µ − i A µ − iA aµ T a . (2.2)The coupling constants g and g correspond to the U (1) and SU ( N ) sectors respectively, and a a andΦ kA are spin zero particles. The bosonic potential V (Φ A , ˜Φ A , a a , a ) is given by V (Φ A , ˜Φ A , a a , a ) = g (cid:18) g f abc ¯ a b a c + ¯Φ A T a Φ A − ˜Φ A T a ¯˜Φ A (cid:19) + g (cid:16) ¯Φ A Φ A − ˜Φ A ¯˜Φ A − N ξ (cid:17) + 12 N X A =1 (cid:26)(cid:12)(cid:12)(cid:12) ( a + √ m A + 2 T a a a )Φ A (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( a + √ m A + 2 T a a a ) ¯˜Φ A (cid:12)(cid:12)(cid:12) (cid:27) + 2 g (cid:12)(cid:12)(cid:12) ˜Φ A T a Φ A (cid:12)(cid:12)(cid:12) + g (cid:12)(cid:12)(cid:12)(cid:12) ˜Φ A Φ A − N ξ (cid:12)(cid:12)(cid:12)(cid:12) , (2.3)with f abc the structure constants of the Lie algebra SU ( N ). The parameters ξ i come from Fayet-Illopoulos terms. In the following, the choice ξ = 0, ξ = 0 and ξ = ξ will be employed. Byintroducing the field A = 12 a + T a a a , (2.4)the vacuum of the theory is parameterized as h A i = − √ m . . . . . . . . . . . . . . . m N . (2.5)For generic values of the parameter m N the subgroup SU( N ) is broken to U(1) N − . However, for thespecific choice of equal masses m = m = ... = m N , the classic group SU( N ) × U(1) is not broken.The presence of the Fayet-Illopoulos parameter induce the following non zero expectation values D Φ kA E = p ξ ...... ... ...... , D ¯˜Φ kA E = 0 , (2.6)4ith k = 1 , ..., N, and A = 1 , ..., N . The fact that the squarks Φ kA and the gauge field A acquireexpectation values proportional to the identity matrix implies that there exist the remanent symmetry SU ( N ) C + F Φ → U Φ U − , a a T a → U a a T a U − , M → U − M U, (2.7)with U an element of SU ( N ), leaving invariant the vacuum of the theory. Such type of situations werealready considered in the 70’s in another context by Bardacki-Halpern [59]. The symmetry (2.7) is offundamental importance in the presence of vortices. The reason is that vortices break the diagonalsymmetry, and different vortex solutions are then connected by a quotient Q = SU ( N ) C + F /I of theaction (2.7) with the group I leaving these vortex solutions invariant. This give rise to internal modulifor these objects, described by this quotient Q .A description similar to the one given above holds the colour-flavour locked phase [61]-[62], withthe Fayet-Illopoulos parameter ξ replaced by the scale ∆ cfl ∼ −
100 MeV. The details will not bemade explict here, they can be found for instance in the review [64].
The models described above admit vortex solutions in general. In the following, the simplest typeof non abelian type of vortices will be considered namely, vortices with moduli parameterized by thesphere Q = S . These vortices appear for instance in gauge scenarios with SU (2) × U (1) gauge group.The generic form for a vortex solution aligned along the ˆ z axis can be written as followsΦ kA = ∆ U (cid:18) e iθ φ ( r ) 00 φ ( r ) (cid:19) U − ,A i ( x ) = 12 ǫ ij x j r [1 − f ( r )] U τ U − , A z = A t = 0 . (2.8)Here the latin indices i = 1 , x and y components of the SU (2) gauge field A i . Theparameter ∆ is a characteristic energy scale of the system. It may be represent the gap ∆ ∼ − √ ξ . Thescalars Φ kA of the model compose a 2 × r and θ are the standardpolar coordinates on the plane defined by x = r cos θ , y = r sin θ . The SU (2) matrix U is a globalone, that is, it does not depend on the space time coordinates ( t , r , θ , z ). The following identity forthese matrices U τ U − = n a τ a , a = 1 , , , (2.9)is well known. The quantities n a represent a unit vector on S , that is, a vector satisfying n = 1.The matrices τ a are the standard Pauli matrices. The vortex solution can be expressed alternativelyas Φ kA = (cid:20) e iθ φ ( r ) + φ ( r )2 (cid:21) I + (cid:20) e iθ φ ( r ) − φ ( r )2 (cid:21) n a τ a ,A i ( x ) = 12 ǫ ij x j r [1 − f ( r )] n a τ a , i = 1 , . (2.10)5herefore, it is seen that the different vortices of the model are parameterized by the sphere S . Thissphere of course, is not representing any geometry in the space R or in the space-time M . Instead,it is a internal geometry describing different group elements characterizing all the possible vortexconfigurations. The function φ ( r ) describing the scalar field in (2.8) is slowly varying. The function φ ( r ) instead is not, and it is zero in the r = 0 line. In addition, φ ( r ) → r → ∞ . For thesupersymmetric case, the energy for unit length (tension) of the vortex [60] T = 2 π ξ, (2.11)is independent on the chosen orientation n a . For the colour-flavour locked phase, this tension isexpected to be proportional to the symmetry breaking scale ∆ cfl . It may roughly estimated as [64] T ∼ ζ (3)72 π µ ∆ cfl T c log Lm. (2.12)Here T c is the QCD critical temperature, of the order T c ∼ −
150 MeV and ζ ( x ) the Riemann zetafunction. The mass m is is related to an UV cutoff giving the vortex size core l c ∼ m ∼ − π T c ζ (3) log TT c . The cutoff L is an IR one, and is related to large but finite dimensions of the system. The chemicalpotential µ is this phase is assumed to be very high, of the order of µ ∼
300 MeV or even larger.The vortex solution described above is static. The region where φ ( r ) vanishes is a line, which isinterpreted as the string or vortex location. One of the tasks of the present work is to study the decaychannel of the vortex in axions. For this purpose, it is mandatory to describe the couplings betweenthe axion a and the gauge vector field A i . The axion is a Goldstone boson and it is coupled to thevortex by an interaction term S a = Z M a ( x µ ) f a Tr( F µν e F µν ) d x, (2.13)with e F µν the dual field strength corresponding to F µν . The axion is not usually coupled directly tothe gauge field, but this interaction is an effective one, induced by a triangle of heavy quarks in a ABJFeymann diagram [101]-[104].
3. Vortex excitations
Consider now a slightly excited vortex. The excitations arise by prompting the moduli n a of S described in (2.9) to a slowly varying field n a ( z, t ). Another type of excitation is obtained by deformingits shape. In this case the position of the vortex can fluctuate with a displacement δx µ ( z, t ) around thestatic position r = 0. For such excited vortex, the region of vanishing φ is a now a string x µ ( υ , υ )with time varying position in R . Here the coordinate υ is the temporal one while υ is the spatialone. The coordinates υ i swap a two dimensional surface, denoted by Σ, which is interpreted as the6orldsheet of the string. The equations of motions for the excited vortex, in the slow field or Mantonapproximation [105], were obtained in several references, see for instance [60] and references therein.In order to describe it, it is convenient to introduce six coordinates s µ = ( t, r, θ, φ, α, β ). The first fourcoordinates parameterize the Minkowski space M and describe the translation modes of the vortex.The last two coordinates describe the orientational S field n a by the relation n = sin α sin β, n = sin α cos β, n = cos α. (3.14)In these terms, the action describing the excitations of the vortex is [60] S = T Z p −| h | h ab g µν ∂ a s µ ∂ b s ν dτ dσ. (3.15)Here g µν is the canonical metric of M × S g = − dt + dr + r ( dθ + sin θdφ ) + R ( dα + sin αdβ ) . The physical interpretation of the radius R of the orientational sphere S deserve some comments.For the colour-flavour locked phase [64] it is given by R ∼ µ T − ∆ − cfl with T the vortex tension and µ the chemical potential of the environment where the vortices is located. It is estimated as R ∼ π ζ (3) T c ∆ cfl log Lm . (3.16)For the supersymmetric case instead, the radius is given by R ∼ ξg , (3.17)with the coupling g defined in the lagrangian (2.1). The moral of these expressions is that, the largerthe scale of broken symmetry is, the smaller the radius of the S results. In addition, h ab denotes isthe worldsheet metric of the string. It is an auxiliary field, as it does not contain any kinetic energy.In order to solve the equations of motion arising from (3.15), it is customary, although not manda-tory, to employ the conformal gauge p −| h | h ab = η ab = diag( − , υ = τ , υ = σ will be employed. The lagrangian corresponding to (3.15) in the conformal gauge isthen L = ∂ τ t ∂ τ t − ∂ τ r ∂ τ r − r ∂ τ θ ∂ τ θ − r sin θ∂ τ φ ∂ τ φ − R ∂ τ α ∂ τ α − R sin α∂ τ β ∂ τ β − ∂ σ t ∂ σ t + ∂ σ r ∂ σ r + r ∂ σ θ ∂ σ θ + r sin θ∂ σ φ ∂ σ φ + R ∂ σ α ∂ σ α + R sin α∂ σ β ∂ σ β. (3.18)On the other hand, the two conformal constraints of the model are − ∂ τ t ∂ τ t + ∂ τ r ∂ τ r + r ∂ τ θ ∂ τ θ + r sin θ∂ τ φ ∂ τ φ + R ∂ τ α ∂ τ α + R sin α∂ τ β ∂ τ β − ∂ σ t ∂ σ t + ∂ σ r ∂ σ r + r ∂ σ θ ∂ σ θ + r sin θ∂ σ φ ∂ σ φ + R ∂ σ α ∂ σ α + R sin α∂ σ β ∂ σ β = 0 , (3.19)and − ∂ τ t ∂ σ t + ∂ τ r ∂ σ r + r ∂ τ θ ∂ σ θ + r sin θ∂ τ φ ∂ σ φ + R ∂ τ α ∂ σ α + R sin α∂ τ β ∂ σ β = 0 . (3.20)The unperturbed string is given by τ = t , σ = z = r cos θ , ρ = r sin θ = 0, with α and β fixed. Thisin particular implies that θ = 0 or θ = π . 7 .2 Some simple excitations Consider now perturbed solutions. In the following the ansatz τ = t, σ = z, φ = ωt, β = νt, r = r ( z ) , θ = α = π , (3.21)will be considered. This ansatz bear some resemblance with classical string solutions considered forinstance in [4]-[6]. As now cos θ = 0, this perturbation describes z dependent oscillations in the radialcylindrical direction. The second conformal constraint is identically satisfied for this functional formof the excitation. Instead, the first one is not, and yields − ω r + R ν + r ′ = 0 . Here the ′ refers to a derivative with respect to σ . On the other hand, the equations of motion aresimply r ′′ + ω r = 0 . The last two equations are consistent, since the second arises by taking the derivative of the first withrespect to σ . In addition, R ν <
1, which means that the limit frequency is ν = R − . The integrationof the first equation throws the following result r = √ − R ν ω | sin ωz | . (3.22)For an infinitely large string there is no constraint in ω . In the following however, a large but finitestring will be considered, with size L much larger that its thickness and with fixed ends. The presenceof fixed endpoints require r ( z + L ) = r ( z ), and this implies that ω = 2 πm/L with m integer. Atthe end, it would be more desirable to consider closed loops, as these are likely the main objects toappear in physical applications. However, in order to deal with the complications of non abelianity,we will assume that these fixed end strings may approximate the excitations of a closed loop withradius R = L/ π . The task is now to understand the energy loss of this object by axion emission.
4. The coupling of the axion to the orientational and translationalmodes of the vortex
In order to study the string energy loss by axion emission, the couplings between the axion andthe orientational and translational modes of the vortex should be found. In order to figure out theorientational couplings, consider an excitation n a ( z, t ) of the unit vector defined in (2.9). The gaugefield components A i are assumed in this approximation to retain the same functional dependence onthe coordinates, except that now n a → n a ( z, t ). On the other hand, due to the non trivial dependenceof n a with respect to t and z , the gauge field components A α with α = 0 , A α = − iρ ( r )( ∂ α U ) U − . n a → n a ( z, t ) is neglected, then this expression vanishesidentically, and the solution (2.8) would be recovered. However, there are remanent U (1) symmetriesthat leave the vortex solution invariant [60]. By a proper quotient of this redundant action, it can beshown that gauge field components may be written as [60]2 A α = − ρ ( r ) ǫ abc n b ∂ α n c τ a , α = 0 , . (4.23)The field strength tensor is calculated by the convention F µν = ∂ µ A ν − ∂ ν A µ − i [ A µ , A ν ] . It is convenient to write the components i = 1 , A i = ǫ ij x j g ( r ) n a τ a , g ( r ) = 12 r [1 − f ( r )] . (4.24)Then, by use of the convention just introduced, it can be calculated from (4.24) that F = F xy = − [2 g ( r ) + g ′ ( r )] n a τ a . In addition, one has that F αi = (cid:20) ∂ α n a r ǫ ij x j f ( r )[1 − ρ ( r )] + x i r dρdr ǫ abc n b ∂ α n c (cid:21) τ a ,F = F zt = − ρ ( r ) ǫ abc [ ∂ z n b ∂ t n c − ∂ t n b ∂ z n c ] τ a + ρ ( r ) ǫ dbc [ ∂ z n b ∂ t n c − ∂ t n b ∂ z n c ] n d n a τ a . In these terms, a simple calculation throws the following resultTr( F µν e F µν ) = h ( r ) ǫ abc [ ∂ z n b ∂ t n c − ∂ t n b ∂ z n c ] n a , with h ( r ) a function of r whose explicit form is not very relevant for the following purposes, exceptthat is arguably non vanishing for a region of the size of the vortex δ . From this expression, by taking(2.13) into account, the following induced coupling between the axion and the orientational modes S effa = α a Z M a ( t, z, , ǫ abc n a ˙ n b n ′ c dzdt, is obtained. Here α a is a coupling arising due to the integration over the transversal coordinates x and y . It has dimensions [ α a ] =time=length in natural units. This effective action can be expressedalternatively as S effa = α a Z M a ( x µ ) ǫ abc n a ˙ n b n ′ c δ ( x ) δ ( y ) dx . (4.25)Consider now a dynamic string, such that the region of vanishing φ is a time varying line s µ ( τ, σ ),as the one described in (3.21) and (3.22). In this situation the Dirac delta δ ( x ) δ ( y ) is generalized to δ ( x ) δ ( y ) −→ Z Σ p −| γ | δ ( x µ − s µ ( τ, σ )) dτ dσ, (4.26)9eing γ αβ = η µν ∂ α s µ ∂ β s ν , α, β = τ, σ, (4.27)the world sheet metric. Then, with the help of (4.26), it is seen that effective action (4.25) is aparticular case of the following general functional form S effa = α a Z Σ Z M a ( x µ ) ǫ abc n a ˙ n b n ′ c p −| γ | δ ( x µ − s µ ( τ, z )) dτ dσdx , (4.28)or, after integrating in the spatial coordinates S effa = α a Z Σ a ( τ, z ) ǫ abc n a ˙ n b n ′ c p −| γ | dτ dσ. (4.29)Here the dots correspond to derivatives with respect to τ .The formula (4.29) gives the coupling of the axion to the orientational modes of the vortex. Inparticular, for the solution (3.21) and (3.22) found above, it is obtained that the determinant (4.27)is expressed as − γ = (cos ωz + R ν sin ωz )(1 − R ν ) cos ωz, (4.30)after identifying z = σ and t = τ . These formulas will be employed in the next sections.On the other hand, a typical coupling between the axion a and the scalar of the fields Φ ak of (2.8)is given by L aq = λa Tr(Φ A Φ A − I ∆ ) . The term a is needed, since the axion is a pseudo Goldstone boson and an odd power will make thelagrangian pseudoscalar. It is clear from (2.8) that this trace does not have any dependence on U or, what is the same, on the orientational modes n i . Thus, this coupling will solely induce a vertexbetween the axion and the translational modes. By taking into account (2.8) and the fact that φ isslowly varying with values near the unity while φ ( r ) tends to zero quickly near r ∼
0, the trace canbe approximated by Tr(Φ A Φ A − I ∆ ) = f ( r )∆ ≃ δ ∆ δ ( x ) δ ( y ) . In the previous expression δ denotes the thickness of the string, the function f ( r ) decays rapidly for r >> a and was approximated by the Dirac delta in the last step. A parametrization invariant formof the previous formula, for a non trivial loop, is the following S effaq = λ Z a Tr(Φ A Φ A − I ∆ ) d x ≃ λ ∆ δ Z p −| γ | a δ ( x µ − s µ ( τ, z )) dτ dσd x. (4.31)The last expression gives the coupling between the axion and the translational modes of the vortex.It is convenient to remark that (4.31) give rise to a single axion emission, while (4.29) representstwo axion emission. 10 . Some formulas for axion emitted power The coupllings found in the previous section are fundamental for deriving the power emitted by singleaxion and two axion emission. For single axion emission, the main object to be calculated is transitionamplitude < S ′ , a | S > from an initial string state | S > to a final one | S ′ , a > . This amplitude iscalculated by use of LSZ formulas, which shows as a result that < S ′ , a | S > = Z exp( ik · x ) < S ′ | ( (cid:3) + m a ) a ( x ) | S > d x. (5.32)In the following, the rough approximation that | S ′ > ∼ | S > will be employed, that is, the emission ofa two axions does not react back on the string. This is of course a simplifying assumption since, atthe end, the string excitations are expected to decay completely. The operator involved in (5.32) iscalculated by means of the following formula( (cid:3) + m a ) a ( x ) = ∂L∂a . The right hand has two type of contributions, one from the axion couplings to the translational modesand other due to the orientational ones. But the translational ones do not contribute to this process,as they involve two axion emission. The coupling to the orientational modes is obtained from (4.28),it is simply given by ∂L∂a (cid:12)(cid:12)(cid:12)(cid:12) o = α a Z Σ ǫ abc n a ˙ n b n ′ c p −| γ | δ ( x µ − s µ ( τ, z )) dτ dz. In these terms the amplitude (5.32) becomes < S ′ , a | S > = α a Z Z Σ e ik · x ǫ abc n a ˙ n b n ′ c p −| γ | δ ( x µ − s µ ( τ, z )) dτ dzd x. (5.33)The total radiated power by axion radiation in this case can be calculated from the expression P T = Z E | < S, a | S > | d k π ) E . (5.34)In the last formula, T ∼ δ (0) is the duration of the process, which is assumed to be infinitely large.Consider now the amplitude corresponding to two axion emission < S ′ , a , a | S > . In this case,the translational modes are the one contributing to the process. The corresponding amplitude is < S ′ , a , a | S > = Z exp( ik · x ) < S ′ , a | ( (cid:3) + m a ) a ( x ) | S > d x. (5.35)From the coupling (4.31) of the axion to the translational modes it is found that( (cid:3) + m a ) a ( x ) = ∂L∂a (cid:12)(cid:12)(cid:12)(cid:12) t = 2 λa Tr(Φ A Φ A − I ∆ ) ≃ δ ∆ Z p −| γ | aδ ( x µ − s µ ( τ, z )) dτ dz. The searched amplitude is then < S ′ , a , a | S > = λ ∆ δ Z Z Σ e i ( k + k ) · x p −| γ | δ ( x µ − s µ ( τ, z )) dτ dzd x. (5.36)The total radiated power in this case results P T = Z Z E | < S, a , a | S > | d k π ) k d k π ) k , (5.37)where E = k + k . 11 . Emitted power for the rotating string inspired ansatz It is of interest to apply the general formulas described above for the string solution (3.22). The stringlocation, in cartesian coordinates, is parameterized as follows s t = t, s x = √ − R ν ω | sin( ωz ) | cos( ωt ) , s y = √ − R ν ω | sin( ωz ) | sin( ωt ) , s z = z. (6.38)In addition, n ′ a = 0 for this ansatz which, together with (5.32), implies that only the translationalmodes contribute to the calculation of the decay. However, this does not mean that the orientationalmodes are irrelevant, as they contribute to the solution given above by the parameter R ν . Inparticular, for R ν = 0 one has that | ˙ s | < < S ′ , a , a | S > ≃ λ ∆ δ Z ∞−∞ Z πω e iEt e − ik x s x − ik y s y − ik z z × q ( R ν sin ωz + cos ωz )(1 − R ν ) cos ωz dzdt, with E = k + k and k = k + k the sum of the energy and the wave vector of the two axions,respectively. By making the redefinition k i → k i /ω and E → E/ω , together with the introduction ofthe new dimensionless variables η = ωz and ξ = ωt , the last expression may be written as < S ′ , a , a | S > ≃ λ ∆ δ ω Z ∞−∞ Z π e − ik x √ − R ν sin η cos ξ e − ik y √ − R ν sin η sin ξ ( e − ik z η − e ik z η ) q ( R ν sin η + cos η )(1 − R ν ) cos η e iEξ dηdξ, (6.39)where very elementary parity properties of the trigonometric functions were used to obtain this ex-pression. It is important to remark that the square root, which corresponds to the world sheet metricdeterminant, is independent on ξ . This simplifies the calculation done below.At first sight, the integral in η may be estimated by saddle point methods and the integration in ξ may be performed later on. The present author however, have found expressions that he could nothandle. For this reason, an alternative method will be employed. The use of the identity k x p − R ν sin η cos ξ + k y p − R ν sin η sin ξ = q ( k x + k y )(1 − R ν ) sin( η ) sin( ξ + δ ) , (6.40)sin δ = k x q k x + k y , cos δ = k y q k x + k y , together with the integral representation of the Bessel functions of the first kind [106] J n ( λ ) = 12 π Z π − π e iλ sin( u ) e − inu du, (6.41)12ields the following Fourier expansion e − ik x √ − R ν sin η cos ξ − ik y √ − R ν sin η sin ξ = ∞ X n = −∞ e in ( δ + π ) J n (cid:20)q ( k x + k y )(1 − R ν ) sin η (cid:21) e inξ . (6.42)The uniform convergence property of Fourier series imply that this equality can be integrated termby term. Thus, by inserting the last expression into (6.39), the searched amplitude becomes a sum ofthe form < S ′ , a , a | S > ≃ λ ∆ δ ω ∞ X n = −∞ e in ( δ + π ) δ ( E − n ) a n , with the coefficients a n given by a n = Z π ( e − ik z η − e ik z η ) J n (cid:20)q ( k x + k y )(1 − R ν ) sin η (cid:21)q ( R ν sin η + cos η )(1 − R ν ) cos η dη. (6.43)Note that this coefficients have purely imaginary values. In addition, the property J − n ( x ) = ( − n J n ( x )valid for integer n >
0, implies that a − n = ( − n a n . For the special case R ν = 1 the property that J n (0) = 0 for n = 0 implies that a n = 0 for n > a (0) is not necessarily zero, but it will be shown below that this term does not giveany contribution to the power radiated. In the terms given above, the squared amplitude can berewritten as | < S ′ , a , a | S > | ≃ λ ∆ δ Tω ∞ X n =0 δ ( E − n ) | a n | , with T = δ (0) /ω being the duration of the process. As the axion mass is assumed to be very tiny, themomentum can be parameterized by a set of pair of polar angles k xi = | k i | sin γ i sin ζ i , k yi = | k i | sin γ i cos ζ i , k zi = | k i | cos γ i , i = 1 , . From here it is calculated that P = 32 λ ∆ δ ω Z Z Ek dk d Ω k dk d Ω ∞ X n =0 δ ( E − n ) | a n | . Here the solid angle d Ω i = sin γ i dγ i dζ i was introduced, with i = 1 ,
2. Note that if R ν then a n = 0and a = 0, but the Dirac delta in the last expression forces P = 0. This is an important consistencytest, as the resulting static string should not radiate axions. By taking into account that E = k + k ,the last integral reduces to P = 32 λ ∆ δ ω ∞ X n =0 Z n nk ( n − k ) dk d Ω d Ω | a n | , (6.44)13here the coefficient a n is given by (6.43) with k x = k sin γ sin ζ + ( n − k ) sin γ sin ζ , k y = k sin γ cos ζ + ( n − k ) sin γ cos ζ ,k z = k cos γ + ( n − k ) cos γ . (6.45)Note that, if the coefficients a n were about to be approximated by a constant value, then afterintegration (6.44) would be a sum of terms proportional to n . However, a careful estimation of a n should be performed, as these coefficients may go to zero and change these powers into something ofthe form n l with l <
4. In addition, not only the coefficients should be estimated, but the integral of | a n | over d Ω d Ω as well. This analysis can be done by studying limits of the Bessel functions J n ( x )(6.43) involved in the problem, as shown below.For small arguments, the asymptotic behaviour of the Bessel functions of first kind is the following J n ( x ) ∼ n ! (cid:18) x (cid:19) n , x << , n ≥ . (6.46)This implies in particular that J n (0) = 0 for n = 0. For large arguments instead, the followingbehaviour holds J n ( x ) ∼ r x cos( x − nπ − π r x ( e ix − inπ − iπ + e − ix + inπ + iπ ) , x >> . (6.47)This formula is known to be valid for n < x , which is known as the Fraunhofer regime. However, thelimit that will be of interest is the Fresnel limit for which n ≤ x ≤ n . The use of the last asymptoticformula in this regime is dubious, and may introduce some considerable error. The Fresnel regime isless studied [108], but it will be described partially below.The coefficients a n introduced in (6.43) involve expressions of the form I ( E ) = Z ba f ( γ ) e iEφ ( γ ) dγ, with f ( γ ) and the phase φ ( γ ) continuous and differentiable functions of the integration variable γ and E = | k | . It is known that in this case, for E → ±∞ , the integral I ( E ) →
0. If the phase φ ( γ ) = 0 inthe interval of integration, then I ( E ) may be approximated as in page 258 of [107] I ( E ) ∼ sg φ ( γ ) iE (cid:20) f ( b ) φ ′ ( b ) e iEφ ( b ) − f ( a ) φ ′ ( a ) e iEφ ( a ) (cid:21) . (6.48)If instead there are N some points γ i in the integration interval for which φ ′ ( γ i ) = 0, then the saddlepoint approximation shows that [107] I ( E ) ∼ N X i =1 e iπ sg φ ′′ ( γ i ) f ( γ i ) e iEφ ( γ i ) s πE | φ ′′ ( γ i ) | . (6.49)Assume, by use of (6.45), that roughly k z ∼ n and q ( k x + k y )(1 − R ν ) ∼ n √ − R ν for n >> k , γ i and ζ i and which are small only for a narrow choice of these14arameters. Then, for the large region in the space described by k , γ i and ζ i one has that 1 << | k z | and 1 << q ( k x + k y )(1 − R ν ). The Bessel functions in (6.43) are in the Fresnel regime in thatregion. It may be assumed that they do not involve large oscillating phases. Thus, the only oscillatingphase is the one involving k z , which has no minima. By applying (6.48) it follows that a n ∼
0, asthe determinant is zero on η = π/ η = 0. For 1 << | k z | and0 < q ( k x + k y )(1 − R ν ) < < k z < n with n an integer with small or intermediate values. If k z is small and theenergy E = n is large, then energy conservation forces 1 << q ( k x + k y )(1 − R ν ). From (6.45)it is seen that roughly q ( k x + k y )(1 − R ν ) ∼ n √ − R ν . Thus, the expression (6.43) can beapproximated by a n ∼ Z π ( e − in η − e in η ) J n (cid:18)q ( k x + k y )(1 − R ν ) sin η (cid:19)q ( R ν sin η + cos η )(1 − R ν ) cos η dη, (6.50)and the Bessel function in the argument may be though as in entering into the Fresnel regime. Tothe knowledge of the author, there is no closed expression for this integral. However, there existintegration formulas [106] for the Bessel function of first kind such as Z π cos(2 µx ) J ν (2 a cos x ) dx = π J ν + µ ( a ) J ν − µ ( a ) . (6.51)The formula (6.50) is not exactly the same as (6.51), the difference is due to the square root factor.However, this factor does not induce a considerable deviation from the expression (6.51) for R ν < a n ∼ iJ n (cid:18)q ( k x + k y )(1 − R ν ) (cid:19) . In brief, the discussion given above together with (6.45) suggests that main contribution to a n isconcentrated in the directions defined by | k cos γ + ( n − k ) cos γ | ≤ , << ( k sin γ + ( n − k ) sin γ + 2 k ( n − k ) sin γ sin γ cos( ζ − ζ ))(1 − R ν ) . (6.52)The coefficients are then a n ∼ iJ n (cid:20)q ( k sin γ + ( n − k ) sin γ + 2 k ( n − k ) sin γ sin γ cos( ζ − ζ ))(1 − R ν ) (cid:21) . (6.53)From (6.52) it is seen that main contribution comes from angles cos γ i < n − , sin γ i ∼
1. The solidangle area subtended by these angles goes as d Ω i ∼ /n , therefore Z | a n | d Ω d Ω ∼ n J n (cid:18) n p − R ν (cid:19) . P = 32 λ ∆ δ ω lim N →∞ N X n =0 n J n (cid:18) n p − R ν (cid:19) , (6.54)The point is now to understand the behaviour for the Bessel functions in the last expression. Thereference [25] suggest that J ′ n ( an ) ∼ n . This formula of course, should not necessarily be integrated in n in order to find J n ( an ), as there maybe factors of n that do not correspond to the argument. In fact, integration gives a divergent resultfor large n , which is known not to be the case. On the other hand there are recurrence formulas suchas [106] J n +1 ( x ) + J n − ( x ) = 2 nx J n ( x ) , J n +1 ( x ) − J n − ( x ) = 2 J ′ n ( x ) . From the last formulas it may be reasonable to postulate that J n ( an ) ∼ c n n + another powers . In the following, it will be assumed that these extra powers are smaller or of almost the same orderas n − . This is of course not a rigorous result, but by playing with large numbers in Mathematica Ibelieve that it is a reasonable postulate. In these terms, the replacement of the sum in (6.54) by anintegral yields the following power radiated P = 32 λ c ∆ δ ω N c . (6.55)The constant c arise due to the c n factors, and it has controlled values. Its specific functional formis undetermined, except that c → R ν →
0. The cutoff N c holds because this descriptionmay not be valid for energies E > δ − . Its value is arguably of the order N c ∼ δ − ω − ∼ Lδ − , andtherefore P = 32 λ c ∆ δ L . (6.56)This result states that, for large objects, two axion radiation is suppressed. This is in qualitativeagreement with the results of [1]-[2], which suggest that for an extended object, with length of theorder of a Parsec, the axion radiation should be subdominant with respect to gravitational radiation.However, there is no real sensibility of this result with respect to the parameter R ν , except on c .The direction for which the power radiated by solid angle take relevant values is also not significantlydeformed. Thus, it is difficult to distinguish non abelianity by studying two axion emission, at leastfor the solution (6.38). 16 . Gravitational radiation The next task is to consider the gravitational radiation power emitted by the excited object. Thispower can be calculated with the help of formula [100] P = X n =0 P n , dP n d Ω = G n ω n π [ T µν ( ω n , k n ) T µν ( ω n , k n ) − | T µµ ( ω n , k n ) | ] , (7.57)with T µν ( k n , ω n ) the Fourier transform of the stress energy tensor T µν ( k n , ω n ) = 1 L Z L Z R e iω n t − k ni x i T µν ( t, x i ) d x i dt, k ni k in = ω n (7.58)The frequencies ω n = nπ/L . There are alternative formulas such as the quadrupole approximation,but they have additional assumptions such as that the emitted wavelength by the source is larger thanits size. Instead, the formula (7.57) employs a smaller amount of hypothesis, and for this reason it isthe one to be applied here.The formula (7.57) shows that the stress energy tensor T µν ( x i , t ) is the main quantity to be found.It can be obtained by varying the action (3.15) of the string, which it is written again below by furtherreference S = T Z p −| h | h ab g µν ∂ a s µ ∂ b s ν √− γδ ( x µ − s µ ( σ, τ )) d x. (7.59)In the conformal gauge, this action is S = T Z η ab (cid:20)e g µν ∂ a s µ ∂ b s ν + R ∂ a α∂ b α + R sin α∂ a β∂ b β (cid:21) √− γδ ( x µ − s µ ( σ, τ )) d x. In the last expression η ab = ( − , g µν → e g µν for the translational modestarget metric has been made. This change of notation is convenient, since the determinant √− γ = p − g µν ∂ a s µ ∂ b s ν , is the only quantity in the lagrangian related to the space time metric g µν . In the present situation, g µν and e g µν coincide with the Minkowski metric η µν . However, they have not to be identified, as oneis representing a sigma model arising from details of vortex interactions and the other is representingthe space time geometry the vortex is embedded in. In addition, it is g µν the one that has to beconsidered for calculating the stress energy tensor T µν . This tensor, in the conformal gauge, is givenby T µν = T η ab (cid:20)e g γδ ∂ a s γ ∂ b s δ + R ∂ a α∂ b α + R sin α∂ a β∂ b β (cid:21) √− γ δγδg µν δ ( x µ − s µ ( σ, τ )) . By use of the Jacobi formula for the determinant, it follows that δγδg µν = γγ ab δγ ab δg µν = γγ ab ∂ a s µ ∂ b s ν . By taking into account the last three formulas it is obtained the following expression for the stressenergy tensor of the configuration T µν = − T η ab (cid:20)e g γδ ∂ a s γ ∂ b s δ + R ∂ a α∂ b α + R sin α∂ a β∂ b β (cid:21) γ cd ∂ c s µ ∂ d s ν √− γδ ( x µ − s µ ( σ, τ )) .
17y use of the last expression and (6.38) it is calculated that T tt = − T (1 − R ν ) cos ωz cos ωz + R ν sin ωz √− γδ ( x µ − s µ ( σ, τ )) ,T tx = 2 T (1 − R ν ) cos ωz √ − R ν | sin ωz | sin ωt cos ωz + R ν sin ωz √− γδ ( x µ − s µ ( σ, τ )) ,T ty = − T (1 − R ν ) cos ωz √ − R ν | sin ωz | cos ωt cos ωz + R ν sin ωz √− γδ ( x µ − s µ ( σ, τ )) ,T xx = − T (1 − R ν ) cos ωz (cid:20) cos ωt − (1 − R ν ) sin ωz sin ωt cos ωz + R ν sin ωz (cid:21) √− γδ ( x µ − s µ ( σ, τ )) ,T yy = − T (1 − R ν ) cos ωz (cid:20) sin ωt − (1 − R ν ) sin ωz cos ωt cos ωz + R ν sin ωz (cid:21) √− γδ ( x µ − s µ ( σ, τ )) ,T xy = − T (1 − R ν ) cos ωz sin ωt cos ωt (cid:20) (1 − R ν ) sin ωz cos ωz + R ν sin ωz + 1 (cid:21) √− γδ ( x µ − s µ ( σ, τ )) . (7.60)From this point, the procedure of calculating the power radiated is quite analogous to the axion case.The energy momentum tensor is of the form T µν ( x i , t ) = T A µν ( x i , t ) √− γδ ( x µ − s µ ( σ, τ )) . The formula (7.58) is the analogous of (6.39), but the square of the metric determinant √− γ is replacedby A µν ( x i , t ) √− γ . In the same fashion than for the square of the world sheet metric determinant √− γ ,the factors A µν ( ωz, ωt ) are simple periodic functions whose values are not far from unity. The Fouriertransform of T µν is then T µν ( k n , ω n ) = TLω Z π Z π e − ik x √ − R ν sin η cos ξ e − ik y √ − R ν sin η sin ξ e − ik z η A µν ( η, ξ ) q ( R ν sin η + cos η )(1 − R ν ) cos η e iω n ξ dηdξ, (7.61)where the replacement E → E/ω and k → k/ω has been performed. The quantities A µν ( η, ξ ) arefunctions the two variable integrations. At first sight, this dependence makes the steepest descentmethod employed in previous sections more difficult to apply. However, from (7.60) it is seen thatthe dependence in ξ is very simple. It is given by linear combinations of the trigonometric functionscos ξ , 2 cos ξ = 1 + cos(2 ξ ), sin ξ , 2 sin ξ = 1 − cos(2 ξ ) and 2 sin ξ cos ξ = sin(2 ξ ). This makes thecalculation of T µν ( k n , ω n ) much easier than expected.Consider for example the Fourier component T xt ( k n , ω n ). From (7.60) and (7.61) it is seen that T xt ( k n , ω n ) = iT (1 − R ν ) Lω Z π Z π e − ik x √ − R ν sin η cos ξ e − ik y √ − R ν sin η sin ξ e − ik z η | sin η || cos η | p cos η + R ν sin η ( e i ( n +1) ξ − e i ( n − ξ ) dηdξ.
18y employing now (6.40)-(6.42) and by assuming that the integration order may be changed, thefollowing Fourier components are obtained T xt ( k n , ω n ) = iT (1 − R ν ) e inδ Lω Z π e − ik z η | sin η || cos η | p cos η + R ν sin η (cid:20) e − iδ J n +1 (cid:18)q ( k x + k y )(1 − R ν ) sin η (cid:19) − e iδ J n − (cid:18)q ( k x + k y )(1 − R ν ) sin η (cid:19)(cid:21) dη. This quantity can be estimated in terms of (6.48)-(6.49) by making a direct analogy with (6.43). Thisanalogy follows by replacing q ( R ν sin η + cos η )(1 − R ν ) cos η −→ | sin η || cos η | p cos η + R ν sin η , in (6.43) and by taking into account that now there are two Bessel functions involved, with label n + 1and n − n . In these terms, formulas analogous to the ones obtained in the previous sectionfor a n can be found for T xt ( k n , ω n ). The same type of formulas can be found for the other componentsof T µν ( k n , ω n ) as well. However, there is no need to go through all this calculation in order to estimatethe power emitted. By simply parameterizing the momentum as k x = n sin γ sin ζ, k y = n sin γ cos ζ, k z = n cos γ, it follows that the least decaying contributions to T µν ( k n , ω n ) are, as before, proportional to n − .Thus, the power radiated formula (7.57) then may be estimated as P ∼ G n cT πL ω (1 − R ν ) ∞ X n = n c n n , (7.62)up to the sum of the first terms, for which the steepest descent method for calculation does not apply,but which have moderate values as well. The sum in the last expression is convergent, thus P ∼ G n cT (1 − R ν ) , (7.63)with c an undetermined constant, which may depend on the internal parameter R ν .For abelian strings, this radiation was studied in several references [16]-[30]. In particular, theauthors of [25]-[26] postulate, for closed loops, that P = γ l G n T , (7.64)where γ l is a factor which depends on the shape of the loop but is independent on its perimeter L .Both (7.64) and (7.63) have the same dependence. At first sight, formula (7.63) does not exhibit asensible dependence on the internal rotations, unless R ν →
1. But in fact, this dependence may beimportant for detecting non abelianity. The point is that, in a Gedanken experiment, a loop with adefinite shape and tension T emits with a power of the form (7.64). However, if the factor γ l describesa loop with the wrong shape, it may be an indication that the vortex is in fact non abelian.19 . Internal rotations In the previous section, it was found that for the rotating string inspired ansatz (3.22) two axionemission power is not very sensitive to the excitations of the internal motion. The gravitationalemission instead may give a hint about non abelianity due to a deviation from the shape factor γ l .However, single axion emission (5.33)-(5.34) was not yet considered, as (3.22) does not allow thisprocess, at least at perturbative level. It may be of interest to find situations in which this emission ispresent. In fact, it is not the quantitative form of the power radiated for single axion emssion whichpoints out non abelianity. Instead, it is the fact that single axion emission is present that signals it, assingle axion emission (5.33)-(5.34) is only possible when orientational modes are present. The point isto determine if this effect is of the order or larger than the previously discussed, or if it is suppressedinstead.In the following, inspired by the spike string solution of [7], it will be assumed that the string isstraight and located at r = 0. The perturbations to be considered below are purely internal, thatis, the string position is always the same straight line r = 0. The perturbation is such that only thesphere unit vector n a is prompted to a varying function of z and time. The anzatz to be use for thissituation is [7] t = Rτ, α = α ( σ ) , β = ωτ + σ, z = λσ. (8.65)Here 2 πλ = L , with L the length of the vortex, and is not to be confused with the coupling present inthe interaction term (4.31). The parameter σ will be chosen proportional to the coordinate z of thestring, and τ is a time coordinate. Both τ and σ are dimensionless. Also ω is dimensionless, and isnot to be confused with the frequency ω = π/L of the previous sections.It should be recalled that (8.65) is not a true solution of the system composed by the equationsderived from (3.18) together with the conformal constraints (3.19)-(3.20). The equations of motionare in fact satisfied but the conformal constraint (3.20) is not . This is expected from the physicalpoint of view and is indicating that an straight vortex with varying orientational moduli tend tomove. The explicit form of the true solution may be in fact complicated. In the following this vortexmovement will be neglected and we conform ourselves to use (8.65), as we are intended to study theaxion emission due to pure orientational modes.The position of the string is static, but its moduli is varying with the height and time. The onlyunknown function in the previous anzatz is α ( σ ), and the solution to be employed is [7] α ′ = sin αC s ω sin α − C − ω sin α . (8.66)There are several cases to consider. In the situation ω < C < ω since otherwisethe square root would be imaginary. Thus C < ω <
1. The values of the angle are such that C ω < sin α < . It is a true solution for the reference [7] however, but these authors are studying strings in other curved backgrounds.
20n the other case ω > C < < ω , for which C ω < sin α < ω , < C < ω for which 1 ω < sin α < , or 1 < ω < C , which leads to the bound1 ω < sin α < C ω . The rotating vectors n a satisfying that n a n a = 1, and which define the internal orientation of thestring, are given by n = sin α cos( ωτ + σ ) , n = sin α sin( ωτ + σ ) , n = cos α. In these terms it follows that ǫ abc n a ˙ n b n ′ c = − ωα ′ sin α − ω cos α sin α sin( ωτ + σ ) cos( ωτ + σ ) . From the last expression is calculated that < S, a | S > = − α a Z ∞−∞ e iERτ (cid:18) Z π e − ik z λσ [ ωα ′ sin α + ω α sin α sin(2 ωτ + 2 σ )] dσ (cid:19) dτ By further parameterizing k z = E cos γ it follows that < S, a | S > = α a I δ ( ER ) + iα a δ ( ER − ω ) I − iα a δ ( ER + 2 ω ) I ∗ . with I = Z π e − iE cos γλσ ωα ′ sin αdσ, I = Z π ω α sin αe − iσ − iE cos γλσ dσ. (8.67)Then | < S, a | S > | = α a R I δ ( E ) δ (0) + α a R | I | [ δ ( E − ωR ) + δ ( E + 2 ωR )] δ (0) , and the power radiated P = dEdt T = 1(2 π ) Z d k E E | < S, a | S > | , with T = 2 πδ (0) is given explicitly by P = α a R π ) Z ∞ Z π Z π E dE sin γdγdζ I δ ( E ) + | I | [ δ ( E − ωR ) + δ ( E + 2 ωR )] . From the second (8.67) it follows that P = α a ω (2 π ) R Z π Z π | I | sin γdγdζ, I = Z π ω α sin αe − i (1 − ωλR cos γ ) σ dσ. If the parameter λ is large in comparison with R and cos γ > R ( ωλ ) − , then the formula (6.48) canbe employed for estimating I . The result is of the form I ∼ Rωλ .
The same follows for cos γ < R ( ωλ ) − , as the interval of integration has the small length l = R ( ωλ ) − .In these terms, the power is estimated as P ∼ α a R L , (8.68)up to numerical factors with controlled values. The power emitted seems to decay for large vorticesmore rapidly than for two axion emission, but of course this result is not rigorous as the employedsolution is just an approximation.
9. Discussion
In the present work it was argued that the main consequence of non abelianity, for certain typeof SU (2) gauge theories, is the modification of gravitational the loop factor γ l in (7.64). Anotherdifference is that the existence of single axion emission process. The task is now to understand inwhich regimes one or other process is dominant. Consider a phase similar to the CFL phase. It maybe assumed that the length α a in (8.68) and the thickness of the string δ are of the form α a = 1 √ T f (cid:18) µ ∆ (cid:19) , δ = 1 √ T g (cid:18) µ ∆ (cid:19) , with f ( x ) and g ( x ) unknown functions which takes moderate values for µ ∼ ∆ ∼ T c . By use of theformula (2.12) for the tension and (3.17) for the radius R , it follows that the one axion emission givenfor (8.68) is predominant over (6.56) for a size L given by L < ∆ µ (cid:18) µT c (cid:19) h (cid:18) µ ∆ (cid:19) , with h ( x ) also an unknown function. For values of the chemical of µ ∼ − MeV, T c ∼
100 MeVand ∆ ∼
300 MeV, under the assumption that h ( x ) takes moderate values, the resulting length is verysmall 10 − m < L < − m. Thus, it is likely that one axion emission is a suppressed process, exceptfor microscopically small objets, although the calculation presented here has not been rigourous. Onthe other hand, two axion emission (6.56) is predominant over gravitation (7.63) for L < (cid:18) T c µ (cid:19) (cid:18) M p ∆ (cid:19) k (cid:18) µ ∆ (cid:19) , with k ( x ) an unknown function with moderate values when the parameters take the values just men-tioned. A typical limit value is L ∼ m, which is 10 − d es with d es the distance between the earth22nd the sun. For an object of such size, or even larger, gravitational effects start to predominate. Fora parsec scale, gravitational radiation dominates completely over axion radiation. This is of coursefor energy scales characteristic of the CFL phase, for other models this calculation has to be repeated,but gravitational radiation will predominate at a very large scale.There may the case in which there is no coupling between the axion and the translational modes.In this case the typical length for which gravity dominates over single axion emission is L ≥
100 km.The picture above change for very small values of the symmetry breaking scale ∆. For ∆ ∼ m a ∼ − eV, single axion emission dominates over two axion emission until L ∼
10 m. For scales typicalof dark energy model, of the order ∆ ∼ − eV, single axion emission dominates until L ∼ m,a scale much larger than a parsec. A priori, detection of no abelianity is simpler in this case, but it ismore difficult to access to the physics at these small energies and large scales.The discussion given above is suggesting that single axion emission due to orientational modes maybe irrelevant, and the importance of these modes arise by modifying the gravitational loop factor γ l .However, there may be other situations, not considered here due to technical complications, for whichthis picture may change. The internal space considered here is S , but there are SU ( N ) × U (1) gaugetheories whose internal space is CP ( N − s i = a i ( x + t ) + b i ( x − t ), whichmay modify the analysis made in [25]-[26]. It is likely that for these hypothetical solutions (7.64) stillholds, but the correction of the loop factor may be more pronounced. Another interesting line of workwould be to consider semi-local strings [92]-[98], which have internal orientation space which is notcompact. The presence of non compact directions can generate a much more rich space of solutionsfor moving strings. It may be of interest to study these solutions and how the vortex position evolvesfor these models, together with axion and gravitational emission. This of course technically morecomplicated, as the couplings between axion and the string and the equations of motion in this caseare more involved. The study of emission channels for these largely non abelian objects deserves, inmy opinion, further attention. Acknowledgments
O. S is supported by CONICET, Argentina.