Dyonic non-Abelian vortex strings in supersymmetric and non-supersymmetric theories: tensions and higher derivative corrections
YYGHP-14-03
Dyonic non-Abelian vortex stringsinsupersymmetric and non-supersymmetric theories — tensions and higher derivative corrections —
Minoru Eto a and Yoshihide Murakami a a Department of Physics, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata,Yamagata 990-8560, Japan
Abstract
Dyonic non-Abelian local/semi-global vortex strings are studied in detail in supersymmetric/non-supersymmetric Yang-Mills-Higgs theories. While the BPS tension formula is known to bethe same as that for the BPS dyonic instanton, we find that the non-BPS tension formula isapproximated very well by the well-known tension formula of the BPS dyon. We show thatthis mysterious tension formula for the dyonic non-BPS vortex stings can be understood fromthe perspective of a low energy effective field theory. Furthermore, we propose an efficientmethod to obtain an effective theory of a single vortex string, which includes not only lowerderivative terms but also all order derivative corrections by making use of the tension formula.We also find a novel dyonic vortex string whose internal orientation vectors rotate in timeand spiral along the string axis. e-mail address: meto(at)sci.kj.yamagata-u.ac.jp e-mail address: cosmoboe(at)gmail.com a r X i v : . [ h e p - t h ] D ec Introduction
When a non-trivial symmetry is spontaneously broken, it is possible that topological solitons,which are non-perturbative objects, appear in spectra. They are non-dissipative lumps ofenergy and behave as if they are isolated particles. They play important roles in quantum fieldtheories. For instance, instantons are necessary for understanding non-perturbative structureof non-Abelian gauge theories, and magnetic monopoles may solve the long standing problemof color confinement in QCD via the so-called dual Meissner effect [1, 2, 3], and so on.There exist topological solitons, the so-called Bogomol’nyi-Prasad-Sommerfield (BPS) [4,5] solitons, which have several special features and are deeply related to supersymmetry.Although an inhomogeneous field configuration generally breaks supersymmetry, the BPSsolitons preserve a part of it. Furthermore, thanks to the supersymmetry, net interactionsamong multiple BPS solitons vanish, so that their masses are proportional to topologicalwinding numbers. While many kinds of BPS solitons are characterized by only topologicalcharges, there also exist extended BPS solitons which are also characterized by conservedcharges for some local/global symmetries. They are called dyonic solitons or Q-solitons in theliterature. Several examples are in order: dyonic instantons in 1 + 4 dimensions [6], dyons in1 + 3 dimensions [7], Q-lumps in 1 + 2 dimensions [8], and Q-kinks in 1 + 1 dimensions [9, 10].The charged semi-local Abelian vortices in 1+2 dimensions [11] are also one of the dyonic BPSsolitons. As explained in Appendix, these dyonic BPS solitons have special mass formulae,and they can be classified into two types: the dyonic-instanton type and the dyon type. Theformer has the BPS mass formula M d-inst = | T | + | Q | , (1.1)while the latter has another formula M dyon = (cid:112) T + Q , (1.2)where T and Q stand for a topological charge and an electric (Noether) charge, respectively.The dyonic instantons, the Q-lumps, and the charged semi-local Abelian vortices belong to thedyonic-instanton type while the dyons and the Q-kinks are of the dyon type. Common featuresof these solitons are that firstly they are all BPS states, and secondly spatial co-dimensionsof the solitons of the dyonic-instanton type are even while those of the dyon-type solitons areodd. Thus, we naively guess that all dyonic solitons belonging to the dyonic-instanton (dyon)type are BPS and their spatial co-dimensions are even (odd). Let us examine this conjecture by another dyonic soliton which was found relatively re-cently compared to the other dyonic solitons listed above. It is called dyonic non-Abelian1ocal vortex (DNALV) in 2 + 1 dimensions [12]. It is a dyonic extension of the so-called non-Abelian local vortex (NALV). A non-Abelian local vortex string (NALVS) [13, 14, 15, 16] in3 + 1 dimensional Yang-Mills-Higgs theories is a natural extension of the Nielsen-Olsen vortexstring in Abelian-Higgs theories, which is also called the Abelian local vortex string (ALVS).NALVS carries a genuine non-Abelian magnetic flux. In other words, they have non-Abelianmoduli parameters that are points on a compact manifold, typically C P N C − [13, 14, 15, 16].So far, lots of works have been done for revealing many roles by the non-Abelian vortex stringsin supersymmetric theories, see the review articles [17, 18, 19, 20] and references therein, andsee also recent studies [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. While much of them havebeen focused on static BPS configurations, few studies concentrated on a time-dependent(stationary) configuration, namely dyonic non-Abelian local vortex string (DNALVS). As willbe explained in Sec. 2, in a certain condition it is a BPS object of co-dimensions two. TheBPS mass formula was found to be of the dyonic-instanton type [12]. Hence, DNALVS makesus further believe that our conjecture is correct.The aim of this paper is to study dyonic non-Abelian vortex strings in more detail. Firstly,we will focus on DNALVS in supersymmetric U ( N C ) Yang-Mills-Higgs model in 3 + 1 dimen-sions. Among the known BPS dyonic solitons listed above, DNALVSs have a distinct featurethat they can continuously change from BPS to non-BPS by increasing Noether charges [12].While the BPS DNALV in 2 + 1 dimensions was studied in Ref. [12], non-BPS one with alarge Noether charge has not been studied in detail. Especially, a tension (mass per unitlength) formula for the non-BPS state is not clarified so far. Generally speaking, there areno reasons for expecting any simple mass formulae like Eqs. (1.1) and (1.2) for non-BPSsolitons. Nevertheless, we find an interesting result that tensions of the non-BPS DNALVSsare reproduced pretty well by the dyon-type tension formula. However, unlike BPS cases,it is not so easy to derive some exact results for non-BPS states. Actually, we numericallyverify that DNALVS appears to obey the dyon-type formula, so that it is hard for us to con-clude that the dyon-type formula is an exact result. It is a little bit pity that we will proofthat the dyon-type tension formula cannot be exact. However, the fact that the dyon-typeformula is just an approximation but accurately reproduces the tensions is still surprising.Thus, DNALVS changes the type according to the magnitude of the Noether charge. For asmall Noether charge, it remains BPS and belongs to the dyonic-instanton type. On the otherhand, for a large Noether charge, it becomes non-BPS and changes to the approximate dyontype. To the best of our knowledge, no such solitons having this feature have been known inthe literature.In order to investigate such novel solitons further, we will next study a dyonic extension2f a similar non-Abelian vortex string in a non-supersymmetric SU ( N C ) Yang-Mills-Higgsmodel. Ungauging the overall U (1) gauge symmetry of U ( N C ) makes the non-Abelian vortexstring a sort of global vortex string whose tension is logarithmically divergent. They werefirst found in the color-flavor locked phase of color superconductor in the high density QCD,and are sometimes called the semi-superfluid vortex strings [35, 36, 37, 38, 39, 40, 41, 23]. Inthis work, superfluidity is not relevant, so we christen them non-Abelian semi-global vortexstrings (NASGVSs) in this paper. It is an important feature that NASGVS has a well-squeezed flux tube of non-Abelian magnetic fields inside its logarithmically expanded core ofthe scalar fields. This implies that they have normalizable non-Abelian zero modes, typically C P N C − . Furthermore, they are non-BPS solitons even when they carry no Noether charges.We numerically find a non-BPS dyonic extension of NASGVS by taking a rotation insidethe internal moduli space C P N C − into account. While its tension is still logarithmicallydivergent, we numerically find that a difference in tensions of the dyonic one and non-dyonicone remains finite. Remarkably, the tension formula again is of the approximate dyon type.Furthermore, we will try to go beyond the numerical and accordingly approximate tensionformula for the non-BPS DNALVS. For that purpose, we will consider DNALVS in 2 + 1dimensions, and will make use of the low energy effective theory on the vortex string worldvolume. Apart from the center of mass of DNALVS, the low energy effective theory is 0 + 1dimensional C P N C − non-linear sigma model [17, 18, 19, 20]. So far, much of works havestudied the effective theory with quadratic derivative terms, which is a lowest approximationtaking only massless degrees of freedom into account. In general, finding higher derivativecorrections is not a very easy task. Indeed only the next leading order, namely a quarticderivative correction, was obtained [21]. For our purpose, this quartic derivative term willplay an important role. We will combine it with the numerical result for the tension ofDNALVS which is obtained by solving the equations of motion for not the effective theorybut the original 2 + 1 dimensional theory. As a result, we will find a plausible low energyeffective Lagrangian including higher derivative corrections to all order. This Lagrangian hasonly one parameter which can be determined by the next leading order effective Lagrangian.From the effective Lagrangian, we can derive analytic expressions of the tension and theNoether charge. Hence, we will find an implicit relation between them, which reproducesthe numerical results very well. The method used here for deriving the low energy effectivetheory including all order higher derivative corrections may be possible to apply for othertopological solitons. As mentioned above, derivation of higher derivative corrections is usuallyvery involved. Therefore, our method may offer an efficient way.This paper is organized as follows. In Sec. 2, we will mainly explain in detail the BPS3D)NALVS in the supersymmetric U ( N C ) Yang-Mills-Higgs theory. The most of this sectionwill be a review but the results in Sec. 2.6 are new. The main results of this work will beshown in subsequent Secs. 3 – 5. Sec. 3 will be devoted to the non-BPS DNALVSs. We willnumerically solve the full equations of motion and find the tension formula of the approximatedyon type. In Sec. 4, we will derive a low energy effective Lagrangian of DNALVS includinghigher derivative corrections to all order. Making use of it, we will find an implicit relationbetween the tension and the Noether charge, which improves the approximate dyon-typetension formula. In Sec. 5, we will study the non-BPS DNASGVSs in non-supersymmetric SU ( N C ) Yang-Mills-Higgs theory. We will conclude this work in Sec. 6. Several BPS dyonicsolitons in various models are reviewed in the Appendix. In this section we briefly review on the BPS NALVSs [17, 18, 19, 20] and their dyonic exten-sions [12] in N = 2 supersymmetric U ( N C ) gauge theories. While Secs. 2.1 – 2.5 are almostreviews, Sec. 2.6 includes new results. A bosonic part of N = 2 SUSY Lagrangian in 3 + 1 dimensions is given by L N =2 = Tr (cid:20) − g F µν F µν + 1 g D µ Σ( D µ Σ) † + D µ H i ( D µ H i ) † − g (cid:16) ( σ a ) ij H i H † j − c a N C (cid:17) − (Σ H i − H i M )(Σ H i − H i M ) † (cid:21) . (2.1)Here, W µ and Σ belong to a U ( N C ) vector multiplet. An N C by N F matrix field H i ( i = 1 , SU (2) R index), the lowest component of hypermultiplet, is in the fundamental representationof U ( N C ) gauge group. Covariant derivatives are given by F µν = ∂ µ W ν − ∂ ν W µ + i [ W µ , W ν ] , (2.2) D µ Σ = ∂ µ Σ + i [ W µ , Σ] , (2.3) D µ H i = ∂ µ H i + iW µ H i . (2.4)We will use a matrix notation for W µ = W aµ T a and Σ = Σ a T a (Tr[ T a T b ] = δ ab / g stands for a U ( N C ) gauge coupling constant, c a ( a = 1 , ,
3) is the SU (2) R triplet of the Fayet-Iliopoulos parameter (in what follows, we set c a = (0 , , v ), ( v > SU (2) R transformation), and M is a diagonal mass matrixas M = diag( m , m , · · · , m N F ). In general, the elements m i are complex variables but we4mpose them to be real valued, and we order them as m i ≥ m i +1 without loss of generality.Furthermore, the mass matrix M can be always set to be traceless ( (cid:80) i m i = 0) by shifting Σby a constant. Thus, the potential becomes V = Tr (cid:20) g (cid:16) H H † + H H † − v N C (cid:17) + g ( H H † )( H H † ) † + (Σ H i − H i M )(Σ H i − H i M ) † (cid:21) . (2.5)In the massless case M = 0, the model has not only U ( N C ) gauge symmetry but also SU ( N F )flavor symmetry, which act on the fields as H i → U C H i U F , Σ → U C Σ U † C , U C ∈ U ( N C ) , U F ∈ SU ( N F ) . (2.6)In general case with M (cid:54) = 0, the flavor symmetry SU ( N F ) reduces to a subgroup whichcommutes with the mass matrix M . The minimal flavor symmetry is the maximum torus U (1) N F − in the case that all the elements of the mass matrix are not degenerate, namely m i > m i +1 .It is well known that the above model possesses rich topological excitations, for example,magnetic monopoles, vortex strings and domain walls as half and/or quarter BPS states [42,43, 44, 45]. In Sec. 2 and 3, we will put our focus onto NALVSs which appear in the caseof N C = N F with M = 0. So, we will concentrate on the case N C = N F hereafter. Beforegoing to the vortex strings, we explain the vacuum of the model. A supersymmetric vacuumis uniquely determined up to gauge transformation as H = v N C , H = 0 , Σ = 0 . (2.7)The vacuum is invariant under the color-flavor locked (CFL) global symmetry( H , H , Σ) → U C+F ( v N C , , U † C+F , U
C+F ∈ SU ( N ) C+F . (2.8)Thus, this vacuum is in the so-called CFL phase. Note that the vacuum is unique and all themassive fields have the same mass thanks to the N = 2 supersymmetry µ = g v . (2.9)Next, we consider the BPS NALVSs. In the following, we will set H = 0 because H willbe irrelevant to the subsequent discussions. This is, of course, consistent with the equationsof motion. Furthermore, we will use simple notation H → H in the rest of the paper. Then,the equations of motion for the non-trivial fields H , W µ and Σ are given by − g D µ F µν = ig (cid:8)(cid:2) Σ , ( D ν Σ) † (cid:3) − (cid:2) D ν Σ , Σ † (cid:3)(cid:9) + i (cid:8) H ( D ν H ) † − ( D ν H ) H † (cid:9) , (2.10)5 g D µ D µ Σ = (Σ H − HM ) H † , (2.11) D µ D µ H = − g (cid:0) HH † − v N (cid:1) H + (Σ H − HM ) M − Σ † (Σ H − HM ) . (2.12)Here we retain the generic mass matrix M . These are complicated second order partialdifferential equations, so that it is not easy to solve them in practice. However, for BPSstates, one does not need to solve the equations of motion. Instead, we solve BPS equations.In this work, we are interested in a BPS vortex string. For simplicity, we assume that it isa straight string perpendicular to the x x plane, so we will set W α = 0 and ∂ α = 0 with α = 0 ,
3. Furthermore, we will set Σ = 0 which solves Eq. (2.11) when M = 0. Now weare ready to derive the BPS equations which can be derived through a standard Bogomol’nyitechnique for the Hamiltonian. Namely, the Hamiltonian per unit length can be cast into thefollowing perfect square form H = (cid:90) dx dx Tr (cid:20) g ( F ) + D m H ( D m H ) † + g (cid:0) HH † − v N C (cid:1) (cid:21) = (cid:90) dx dx Tr (cid:20) g (cid:18) F − g (cid:0) HH † − v N C (cid:1)(cid:19) + ( D H + i D H )( D H + i D H ) † − v F + ∂ m J m (cid:21) ≥ − v (cid:90) dx dx Tr[ F ] , (2.13)where m = 1 , J m ≡ i(cid:15) mn H ( D n H ) † which rapidly goes to zero at the boundaryof x x plane since D m H = 0 in the vacuum. The Bogomol’nyi bound is saturated when( D + i D ) H = 0 , F = g (cid:0) HH † − v N C (cid:1) , (2.14)are satisfied, and the tension of the BPS NALVS reads T = − πv (cid:90) d x Tr[ F ] = 2 πv k, k ∈ Z > . (2.15)One can easily find anti-BPS solitons for a negative integer k by appropriately adjustingsigns in the BPS equations. It is obvious that any solutions of the BPS equations solve theequations of motion (2.10) and (2.11) because the BPS states saturate the energy bound frombelow.The BPS equations are the first order partial differential equations, so solving them ismuch easier compared to the equations of motion. In order to further simplify the BPS6quations, let us use the symmetry. A straight string is axially symmetric, so one can makea natural Ansatz for minimally winding ( k = 1) NALVS H ( x , x ) = vh ( r ) e iθ diag . (1 , , · · · , , (2.16) W ( x , x ) + iW ( x , x ) = − i w ( r ) e iθ r diag . (1 , , · · · , , (2.17)with x + ix = re iθ . The magnetic field F is expressed by F = − w (cid:48) ( r ) r diag . (1 , , · · · , . (2.18)Plugging these into Eq. (2.14), we are lead to first order ordinary differential equations for h ( ρ ) and w ( ρ ) with respect to a dimensionless coordinate ρ ≡ µr, (2.19)as h (cid:48) ( ρ ) + h ( ρ ) ρ ( w ( ρ ) −
1) = 0 , − w (cid:48) ( ρ ) ρ = 12 ( h ( ρ ) − , (2.20)where the prime stands for a derivative by ρ . These should be solved with the boundaryconditions h (0) = 0 , w (0) = 0 , h ( ∞ ) = 1 , w ( ∞ ) = 1 , (2.21)which guarantees a non-trivial topological charge ( k = 1) − π (cid:90) d x Tr[ F ] = − (cid:90) ∞ dr r (cid:18) − w (cid:48) r (cid:19) = 1 . (2.22)Unfortunately, no analytic solutions for Eq. (2.20) have been known, so we numerically solvethem. A numerical solution is shown in Fig. 1.Note that the BPS equations in Eq. (2.20) are exactly identical to those of the Nielsen-Olesen vortex string in the BPS limit of the Abelian-Higgs model, see for example Ref. [46].In this sense, the k = 1 BPS NALVS can be thought of as a merely embedding solution ofthe BPS Nielsen-Olesen vortex string into the N C × N C matrices in the U ( N C ) non-Abeliangauge theory. This realization immediately leads us to non-Abelian moduli parameters cor-responding to degrees of freedom for the embedding. A key ingredient is the CFL symmetry SU ( N C ) C+F of the vacuum state. The single NALVS given in Eqs. (2.16) and (2.17) sponta-neously breaks it down to U (1) C+F × SU ( N C − C+F . Thus there are infinitely degeneratestates whose moduli space is given by M k =1 = C × SU ( N C ) C+F U (1) C+F × SU ( N C − C+F (cid:39) C × C P N C − . (2.23)7 Ρ p r o f il e f un c ti on s hw Figure 1: The profile functions h ( ρ ) and w ( ρ ) for the minimally winding NALVS.The first factor C corresponds to the position of the vortex string on the x x plane, whilethe second factor C P N C − is genuine non-Abelian moduli which is the so-called orientationalmoduli, see for example Ref. [17, 18, 19, 20]. Let us now extend the BPS NALVSs in the previous subsection, which are static configura-tions, to dyonic solutions [12] by allowing time dependence. In order to remain solutions to beBPS states, we now need to include the non-trivial adjoint scalar field Σ and the generic massmatrix M . So we restore Σ and W while set H to zero as before. We still keep N C = N F ,and the flavor symmetry is the maximum torus U (1) N F − . We begin with the Bogomol’nyicompletion of the Hamiltonian H = (cid:90) dx dx Tr (cid:20) g (cid:8) ( F ) + ( F ) + ( F ) + ( F ) + ( F ) + ( F ) (cid:9) + 1 g (cid:8) D Σ( D Σ) † + D Σ( D Σ) † + D m Σ( D m Σ) † (cid:9) + D H ( D H ) † + D H ( D H ) † + D m H ( D m H ) † + g (cid:0) HH † − v N C (cid:1) + (Σ H − HM )(Σ H − HM ) † (cid:21) = (cid:90) dx dx Tr (cid:20) g (cid:18) F − g (cid:0) HH † − v N C (cid:1)(cid:19) + ( D H + i D H )( D H + i D H ) † + 1 g ( F m ∓ D m Σ)( F m ∓ D m Σ) † + ( D H ± i (Σ H − HM )) ( D H ± i (Σ H − HM )) † g (cid:8) ( D Σ)( D Σ) † + ( D Σ)( D Σ) † + ( F ) + ( F ) + ( F ) + D H ( D H ) † (cid:9) − v Tr[ F ] + ∂ m J m ± g ∂ m (cid:0) (Σ + Σ † ) F m (cid:1) ∓ i (cid:8) ( D H ) M H † − HM ( D H ) † (cid:9) ± Σ (cid:18) − g D m F m − iH ( D H ) † (cid:19) ± Σ † (cid:18) − g D m F m + i ( D H ) H † (cid:19) (cid:21) ≥ (cid:90) dx dx Tr (cid:2) − v F (cid:3) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) dx dx Tr F (cid:2) M J (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) , (2.24)where vanishing of the last line of the second equality of Eq. (2.24) will be explained at theend of this subsection. We have defined the SU ( N C ) F flavor current by J µ F ≡ − i (cid:0) H † D µ H − ( D µ H ) † H (cid:1) . (2.25)Note that among elements of J µ F only the elements which commute with the mass matrix M are conserved. The conserved charge per unit length is defined by Q a = 2 (cid:90) dx dx Tr (cid:2) J T a (cid:3) , for [ T a , M ] = 0 . (2.26)Thus, the tension of the BPS DNALVSs reads T = 2 πv k + | Tr[
M Q ] | , (2.27)with Q = Q a T a . As is mentioned in the Introduction, this is the dyonic-instanton type. Notethat there is a maximum value for the second term | Tr[
M Q ] | , namely it cannot be arbitrarylarge. A reason for that will be given in the subsequent section 2.3.Eq. (2.24) also includes the electric charge density Q e = 1 g ∂ m Tr (cid:2) (Σ + Σ † ) F m (cid:3) . (2.28)Hence, the dyonic vortex is electrically charged object but the net electrical charge is zero, (cid:82) dx dx Q e = 0, since it is screened in the Higgs phase (a superconductor).The energy bound is saturated when the following first order equations are satisfied( D + i D ) H = 0 , F = g (cid:0) HH † − v N C (cid:1) , (2.29) F = F = F = 0 , D Σ = 0 , D H = 0 , (2.30) F m ∓ D m Σ = 0 , D H ± i (Σ H − HM ) = 0 , D Σ = 0 . (2.31)These first order equations can be also derived from a 1/4 BPS condition in N = 2 SUSYYang-Mills-Higgs theories [34]. Eq. (2.29) is exactly the same as Eq. (2.14). Therefore, x x dependence of the solution for H and W , are identical to those obtained by solvingEq. (2.20). Eq. (2.30) is trivially satisfied by ∂ = 0 and W = 0. In order to solve the firstequation in Eq. (2.31), we fix the gauge byΣ( x , x ) = ∓ W ( x , x ) . (2.32)Note that this also solves D Σ = 0. Then the second equation of Eq. (2.31) becomes ∂ H = ± iHM. (2.33)This can be easily solved by H ( t, x , x ) = H (cid:63) ( x , x ) e ± iMt , (2.34)where H (cid:63) ( x , x ) can be any of static BPS solutions for Eq. (2.29). Finally, we should deter-mine W ( x , x ) by solving the Gauss law − g D m F m − ig (cid:8)(cid:2) Σ , ( D Σ) † (cid:3) − (cid:2) D Σ , Σ † (cid:3)(cid:9) − i (cid:8) H ( D H ) † − ( D H ) H † (cid:9) = 0 . (2.35)By using Eq. (2.31), this can be rewritten as2 g D m D m Σ + H (cid:63) (cid:0) H † (cid:63) Σ − M H † (cid:63) (cid:1) + (Σ H (cid:63) − H (cid:63) M ) H † (cid:63) = 0 (2.36)Note that this is consistent with the equation of motion for Σ given in Eq. (2.11) with thecondition Σ = Σ † . Note also that Σ does not depend on t and x because neither does H (cid:63) . Insummary, for a given solution H and W m , this equation determines Σ( x , x ) = ∓ W ( x , x ).A final comment is on the last line of the second equality of Eq. (2.24). Since Σ = Σ † , itcan be written as ± Σ (cid:26) − g D m F m − i (cid:0) H ( D H ) † − D HH † (cid:1)(cid:27) . (2.37)This is precisely zero due to the Gauss law (2.35), so that there are no contributions to theenergy density. There is a clear-cut view point for understanding the BPS DNALVs [12]. It is a low energyeffective theory on the vortex world volume. As explained in Sec. 2.1, the single NALVS hasthe massless moduli C P N C − in the case of M = 0. In this case, the low energy effectivetheory is 1 + 1 dimensional non-linear sigma model whose target space is C P N C − .10e will study the massless case M = 0 in this subsection. Let us see the simplest caseof N C = 2 as a concrete example. First of all, we have to specify the orientational moduli inthe solution. It turns out that the singular gauge is simpler. So we transform H and W , given in Eqs. (2.16) and (2.17) to the following ones by the gauge transformation H → V H and W µ → V W µ V † + i ( ∂ µ V ) V † with V = diag . ( e − iθ , H = v (cid:18) h ( r ) 00 1 (cid:19) , W + iW = (cid:18) − i w ( r ) − r e iθ
00 0 (cid:19) . (2.38)In order to specify the orientational moduli, we transform these by an SU (2) C+F transforma-tion as H = U (cid:20) v (cid:18) h
00 1 (cid:19)(cid:21) U − , W + iW = U (cid:18) − i w − r e iθ
00 0 (cid:19) U − . (2.39)Here U is an SU (2) C+F matrix which we parametrize by a complex parameter φ ∈ C as U = √ | φ | φ √ | φ | − φ ∗ √ | φ | √ | φ | ∈ SU (2) C+F . (2.40)Note that the phase of the left-upper component of U can be always fixed to be real andpositive by using exp( iασ / ∈ U (1) C+F transformation: U → U exp( iασ / U (1) C+F does not change H and W , given in Eq. (2.38). In other words, this is an isotropyof the moduli space. Hence, the moduli space is isomorphic to SU (2) C+F /U (1) C+F (cid:39) C P and φ is an inhomogeneous coordinate of C P .In order to derive a low energy effective theory for the orientational moduli in the masslesscase M = 0, we promote the complex parameter φ to be a field φ ( x , x ) on the vortex world-volume [47], which gives us x and x dependence in H and W , . We also need to clarify x , x dependence of W , while we set Σ to zero. To this end, we make an Ansatz as follows [15] W α ( x , x ) = − iλ ( r ) (cid:16) U σ U − (cid:17) ∂ α (cid:16) U σ U − (cid:17) , ( α = 0 , , (2.41)where x , dependence enters through U , and λ ( r ) is an unknown function of r . Finally,plugging H ( x , x ; φ ( x , t )), W , ( x , x ; φ ( x , t )) and W , ( x , x ; φ ( x , t )) into the Lagrangian(2.1), and integrating the quadratic derivative terms in x and t over x and x , we get L (2)eff = (cid:90) dx dx Tr (cid:20) − g F mα F mα + D α H ( D α H ) † (cid:21) = β ∂ α φ∂ α φ ∗ (1 + | φ | ) , (2.42)with m = 1 , α = 0 , β = 4 πg (cid:90) dρ ρ (cid:20) λ (cid:48) + (1 − λ ) (1 − w ) ρ + λ (cid:0) h + 1 (cid:1) + (1 − λ )(1 − h ) (cid:21) . (2.43)11e regard this as a potential for λ ( ρ ) which is minimized by λ (cid:48)(cid:48) + λ (cid:48) ρ + (1 − λ )(1 − w ) ρ + 12 (cid:0) (1 + h )(1 − λ ) − h (cid:1) = 0 . (2.44)Note that the same equation can also be derived by plugging H and W µ given above into theGauss law (2.35) with Σ = 0. By using the BPS equation (2.20), this can be solved by [15] λ ( ρ ) = 1 − h ( ρ ) , (2.45)where h ( ρ ) is the background profile function shown in Fig. 1. Plugging this back intoEq. (2.43), and using the BPS equations (2.20), we get an analytic result [15] β = 4 πg (cid:90) dρ d ( w − dρ = 4 πg . (2.46)In the expression (2.42), the U (1) global symmetry φ → e iη φ , which is a subgroup ofisometry of C P (cid:39) S is manifest. The corresponding Noether current is given by j α = iβ φ∂ α φ ∗ − φ ∗ ∂ α φ (1 + | φ | ) . (2.47)Thus the corresponding Noether charge per unit length is given by q = iβ φ ˙ φ ∗ − φ ∗ ˙ φ (1 + | φ | ) . (2.48)Note that an origin of this e iη ∈ U (1) symmetry in the effective theory can be traced toexp( − iησ / ∈ SU (2) C+F symmetry of the original theory in 3 + 1 dimensions. This can beseen by transforming U given in Eq. (2.40) with the U (1) F transformation associated withthe U (1) C as U → e iη σ U e − iη σ ⇔ φ → e iη φ. (2.49)Therefore, q should be identified with Q / Q from Eqs. (2.26), (2.39), (2.41), and (2.45) as Q = i (cid:16) φ ∗ ˙ φ − φ ˙ φ ∗ (cid:17) (1 + | φ | ) (cid:90) dx dx v (cid:0) hλ + ( h − (cid:1) = 2 q , (2.50)where we have also used Eq. (2.20). 12 .4 A low energy effective theory in the massive case Next, we turn on non-zero mass matrix M = m σ , ( m > . (2.51)Then, we cannot set Σ to zero anymore. The mass term explicitly breaks the flavor symmetry SU (2) F down to U (1) F subgroup. However, as long as the masses are kept sufficiently small( m (cid:28) µ ), SU (2) F can be dealt with as an approximate symmetry, and the low energy effectivetheory remains C P non-linear sigma model. Instead, the small symmetry breaking termgenerates a small effective potential. Namely the effective theory becomes massive C P sigmamodel. In order to derive the effective potential, we make the following Ansatz for Σ [15]Σ( t, x ) = m (cid:110) (1 − σ ( r )) σ + 4 σ ( r )Tr (cid:104)(cid:16) U σ U − (cid:17) σ (cid:105) (cid:16) U σ U − (cid:17)(cid:111) , (2.52)with an unknown function σ ( r ). We plug this into the Lagrangian (2.1). Then the quadraticterm in m yields the desired effective potential V eff = − (cid:90) dx dx Tr (cid:20) g (cid:0) D Σ( D Σ) † + D Σ( D Σ) † (cid:1) − (Σ H − HM ) (Σ H − HM ) † (cid:21) = χ m | φ | (1 + | φ | ) . (2.53)Here the coefficient χ is given by the following integral χ = 4 πg (cid:90) dρ ρ (cid:20) σ (cid:48) + (1 − σ ) (1 − w ) ρ + σ (cid:0) h + 1 (cid:1) + (1 − σ )(1 − h ) (cid:21) . (2.54)Note that this is exactly the same form as β given in Eq. (2.43), so that the coefficient χ isalso minimized by σ ( r ) = λ ( r ) = 1 − h ( r ) [15]. Hence, we find χ = β = 4 πg . (2.55)In summary, the effective Lagrangian is sum of the kinetic term given in Eq. (2.42) and thepotential term given in Eq. (2.53) L (2)eff = β (cid:20) ∂ α φ∂ α φ ∗ (1 + | φ | ) − m | φ | (1 + | φ | ) (cid:21) . (2.56)There are two vacua: the one for φ = 0 and the other for φ = ∞ . The former corresponds to U = given in Eq. (2.40), so it gives NALVS living in the left-upper corner, see Eq. (2.39).On the other hand, φ = ∞ corresponds to U = iσ , so it gives NALVS living in the right-bottom corner. We identify the former and the latter to the north and the south poles of C P , respectively. All the other configurations are lifted by the non-zero mass M .13t is also useful to rewrite everything in terms of spherical coordinates φ = e i Φ tan Θ2 , ≤ Θ ≤ π, ≤ Φ < π. (2.57)Then the massive C P non-linear sigma model is expressed by L (2)eff = β (cid:2) ∂ α Θ ∂ α Θ + ∂ α Φ ∂ α Φ sin Θ − m sin Θ (cid:3) . (2.58)In terms of the spherical coordinate, the two vacua are Θ = 0 and Θ = π . We are now ready to reconstruct the BPS DNALVS in the effective theory. As can be seen fromEq. (2.34), DNALVS can be generated by time-dependent flavor rotation e imtσ / ∈ U (1) F .Combining this and Eq. (2.49), we see that φ should be transformed as φ → φe imt . Namely,we should have a time dependence of Φ as Φ( t ) = mt . Actually, the BPS nature of the dyonicconfiguration tells us that this is the case H (2)eff = β (cid:2) ( ∂ Θ) + ( ∂ Θ) + (cid:8) ( ∂ Φ) + ( ∂ Φ) (cid:9) sin Θ + m sin Θ (cid:3) = β (cid:2) ( ∂ Θ) + ( ∂ Θ) + ( ∂ Φ) sin Θ + ( ∂ Φ ∓ m ) sin Θ + 2 m∂ Φ sin Θ (cid:3) ≥ m | q | , (2.59)with the conserved Noether charge per unit length in the spherical coordinate q = β ∂ Φ sin Θ . (2.60)The Bogomol’nyi bound is saturated when ∂ Θ = ∂ Θ = ∂ Φ = 0 , ∂ Φ = ± m, (2.61)which gives the desired time dependenceΦ = ± mt. (2.62)Thus, the BPS tension of this state in the low energy effective theory reads m | q | . Thenon-dyonic NALVS gains this as an additional tension, so that the total tension of the singleDNALVS becomes T (BPS) = 2 πv + m | q | . (2.63)14et us compare this with the BPS tension formula (2.27) in the original theory. Since wehave M = mσ /
2, Eq. (2.27) with k = 1 gives T = 2 πv + m | Q | /
2. Since | Q | = 2 | q | fromEq. (2.50), we see that Eqs. (2.27) and (2.63) are indeed identical.Assuming that Θ and Φ are function of t only, we can eliminate Φ from the effectiveLagrangian by using the equation of motion, L (2)eff = β (cid:20) ∂ Θ − m sin Θ − q ) β sin Θ (cid:21) . (2.64)There are two minima of the effective potential V eff = m sin Θ + 4( q ) /β sin Θ atsin Θ = 2 q mβ → Θ = ± sin − (cid:115) q mβ . (2.65)Thus, for a given q , there exist two BPS dyonic solutions [12]. ⇡v m | q | O BPS non-BPS
Figure 2: Tension of the dyonic non-Abelian vortex as function of the angular momentum q .The BPS solutions with | q | ≤ mβ/ | q | > mβ/ (cid:12)(cid:12) q (cid:12)(cid:12) BPS = (cid:12)(cid:12)(cid:12)(cid:12) β ∂ Φ sin Θ (cid:12)(cid:12)(cid:12)(cid:12) BPS ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂ Φ β (cid:12)(cid:12)(cid:12)(cid:12) BPS = mβ ≡ q . (2.66)Therefore, any configurations with the conserved charge greater than the critical charge q cannot be BPS state. For such solutions, a centrifugal force in the internal space forces Θ to15e π/
2. Namely, while there are two degenerate configurations for BPS states with | q | ≤ q ,there is a unique non-BPS state for | q | > q . From Eq. (2.60) with Θ = π/
2, the non-BPSsolution is given by Φ non-BPS = ωt, q = βω . (2.67)Thus, gain in the tension of the solution above q c is δT = β (cid:34)(cid:18) q β (cid:19) + m (cid:35) = ( q ) β + βm , (2.68)and the total tension of the non-BPS DNALVS reads T (non-BPS) = 2 πv + ( q ) β + βm . (2.69)In summary, DNALVS continuously changes from BPS to non-BPS at | q | = mβ/
2, see Fig. 2.While the BPS tension formula (2.63) is of the dyonic-instanton type, Eq. (2.69) for thenon-BPS state is neither of the dyon nor of the dyonic-instanton type. However, note that theresults obtained in this section are valid only for small ω (cid:46) m (cid:28) µ since they are all derivedfrom the low energy effective action. In Sec. 3, we will focus on the non-BPS states above | q | = mβ/
2, especially on the case with m = 0 where all the dyonic solutions are non-BPS. DNALVS considered so far has no x dependence, and it trivially extends along the x axis.Therefore, all the results in the previous sections also hold in 2 + 1 dimensions without anychanges. Now let us generate new solutions which depend on x . Namely, we are going to studygenuine 3+1 dimensional configurations. An easy way for finding such solutions is boosting the x -independent solution along the x direction. We begin with the x -independent solution,namely DNALVS; H ( t, x , x ) = H (cid:63) ( x , x ) e ± iMt given in Eq. (2.34), the solution W , ( x , x )of Eq. (2.29) which is independent of both t and x , Σ( x , x ) = ∓ W ( x , x ) are also t -and x -independent which are determined by Eq. (2.36), and W = 0. Now we boost theseconfigurations. This can be done by just replacing t by ( t − ux ) / √ − u with 0 ≤ u < H = H (cid:63) ( x , x ) exp (cid:18) iM t − ux √ − u (cid:19) , (2.70) W , = W , ( x , x ) , (2.71) W = ∓ √ − u Σ( x , x ) , (2.72)16 = ∓ − u √ − u Σ( x , x ) , (2.73)where W , ( x , x ) and Σ( x , x ) in the right hand side are the solutions of Eqs. (2.29) and(2.36). The Noether charge per unit length is transformed as Q → √ − u Q . (2.74)Plugging this into the BPS tension formula (2.27), we get the following formula for the boostedvortex string T = 2 πv k + 1 √ − u | Tr[
M Q ] | . (2.75)Note that only the second term is transformed while the first term remains unchanged becauseit corresponds to the vortex tension, which is nothing to do with t and x . In order tounderstand this configuration better, let us look at t - and x -dependence in the phase of H ,see Eq. (2.70). The phase can be identified with t - and x -dependent flavor transformation,so that θ = t − ux √ − u is nothing but rotating angle on σ -axis in the internal orientation modulispace. Therefore, if u = 0, the internal orientation uniformly rotates with an angular velocity m , see Fig. 3 (a). On the other hand, once u (cid:54) = 0, the rotating angle depends on x , so theorientation spirals along x -axis with a phase velocity u , see Fig. 3 (b).The same can be seen in the low energy effective theory on the vortex string world volume.The BPS solution corresponding to DNALVS is given by Φ = mt and Θ = Θ . This istransformed by a Lorentz boost asΦ( t, x ) = m t − ux √ − u , Θ = Θ . (2.76)Of course, this solves the equations of motion ∂ α ∂ α Θ − (cid:0) ∂ α Φ ∂ α Φ − m (cid:1) sin 2Θ = 0 , ∂ α (cid:0) ∂ α Φ sin Θ (cid:1) = 0 . (2.77)The conserved charge is given by q = β ∂ Φ sin Θ = 1 √ − u mβ Θ , (2.78)and an increment in the tension reads δT = β (cid:0) ( ∂ Φ) + ( ∂ Φ) + m (cid:1) sin Θ = 1 √ − u m | q | . (2.79)This is consistent with the above result in Eq. (2.75).17 x x (a)(b) C P u Figure 3: (a) The trivial DNALVS and (b) the spiral DNAVLS. The green thick line standsfor the non-Abelian vortex string extending along the x axis in the real space while the redarrows show the internal orientation Φ at each x of the strings.A comment on the spinning non-Abelian vortex strings: In Ref. [48], a sort of non-Abelianvortex strings whose profile functions depend not only on x , x but also t, x were found inthe U ( N C ) Yang-Mills-Higgs model with N F flavors. This is a kind of spinning soliton andsuch string exists only in the case N F > N C . Namely, it is the spinning non-Abelian semi-local vortex strings. The spinning non-Abelian semi-local vortex strings is always non-BPS, andthe internal orientation does not rotate in neither t nor x unlike the spiral DNALVS studiedin this subsection. In the previous section, we have seen that the dyonic extension makes the BPS NALVS benon-BPS, if the Noether charge exceeds the critical value | Q | = mβ = 4 πm/g . This factindicates, if m = 0, that any DNALVSs are non-BPS states. We will investigate the non-BPSDNALVSs in the massless case M = 0 in this section, which have not been studied in the18iterature. Let us first see what happens if we set M = 0 in the Bogomol’nyi completion (2.24). Sincethe Noether charge cannot contribute to the tension, we just return to the tension formulafor the non-dyonic BPS NALVS H ≥ (cid:90) dx dx Tr (cid:2) − v F (cid:3) . (3.1)The BPS equations (2.29) – (2.31) with M = 0 allow only x - and x -independent configura-tions. Namely, they are nothing but the non-dyonic BPS NALVSs explained in Sec. 2.1. Thisindicates that any time-dependent vortex strings in the massless case are non-BPS states,which is consistent with the previous observation from the view point of 1 + 1 dimensionaleffective theory in Sec. 2.5.Therefore, instead of solving the BPS equations, we have to solve the full equations ofmotion given in Eqs. (2.10) – (2.12) with M = 0 for time-dependent configurations. Sincewe are interested in x -independent configurations, we will set W = 0 and ∂ = 0 hereafter.Furthermore, we impose Σ = 0 which solves Eq. (2.11) for M = 0. Hence, we are left withthe following equations D µ D µ H + g HH † − v ) H = 0 , (3.2)1 g D µ F µν + i (cid:0) H ( D ν H ) † − ( D ν H ) H † (cid:1) = 0 . (3.3)In order to solve these differential equations, we need to make an appropriate Ansatz for H and W , , . To this end, we first note that the flavor symmetry SU (2) F is manifest becausewe have M = 0. Hence, we have three conserved flavor charges Q a ( a = 1 , ,
3) associatedwith the generators of SU (2) F . Without loss of generality, we will consider configurationswith Q a = Qδ a . Since the non-Abelian vortex has the orientational moduli C P (cid:39) S forthe case of N C = 2, it may be instructive to imagine a free particle confined on a sphere. Afree particle on a sphere always moves along a great circle, and the motion is specified byangular momentum. In particular, a particle with Q a = Qδ a rotates on the equator. Interms of the spherical coordinate defined in Eq. (2.57), the motion is expressed as Θ = π/ ωt . With these observation in mind, let us now make an appropriate Ansatz. Webegin with the BPS non-dyonic NALVS given in Eqs. (2.16) and (2.17) for N C = 2. Since theconfiguration is diagonal, any flavor transformation generated by T are absorbed by U (2) C gauge transformation. In other words, it corresponds to a particle on the north pole of S .19herefore, we first need to transform it by π/ T = σ /
2, by which we send a particlefrom the north pole to a point on the equator. Now we are ready to make an Ansatz as [21] H ( x , x , t ) = ˜ U † ( t ) (cid:20) v (cid:18) h ( r ) e iθ h ( r ) (cid:19)(cid:21) ˜ U ( t ) , (3.4) W ( x , x , t ) + iW ( x , x , t ) = ˜ U † ( t ) (cid:20) − ie iθ r (cid:18) w ( r ) 00 w ( r ) (cid:19)(cid:21) ˜ U ( t ) , (3.5)with ˜ U ( t ) = exp (cid:16) i π T (cid:17) exp ( iωtT ) = 1 √ (cid:32) e iωt − e − iωt e iωt e − iωt (cid:33) . (3.6)Remember that the right-bottom elements of H and W , are trivial ( h = 1 , w = 0)for the BPS non-dyonic NALVS with ω = 0. On the contrary, for time-dependent dyonicconfigurations, it will turn out that they are non-trivial, so we leave the right-bottom elementsto be unknown profile functions. We also need to make an Ansatz for W [21] W ( x , x , t ) = ˜ U † ( t ) (cid:20) ω (cid:18) − e iθ f ( r )1 − e − iθ f ( r ) 0 (cid:19)(cid:21) ˜ U ( t ) . (3.7)Plugging H and W , , into the equations of motion (3.8) and (3.9), we have the followingordinary differential equationseq ≡ h (cid:48)(cid:48) + h (cid:48) ρ − (cid:18) h −
12 + (1 − w ) ρ (cid:19) h + ˜ ω (cid:20) (1 − f ) h + ( h − h ) f (cid:21) = 0 , (3.8)eq ≡ h (cid:48)(cid:48) + h (cid:48) ρ − (cid:18) h −
12 + w ρ (cid:19) h + ˜ ω (cid:20) (1 − f ) h − ( h − h ) f (cid:21) = 0 , (3.9)eq ≡ w (cid:48)(cid:48) − w (cid:48) ρ + h (1 − w ) − ˜ ω f (1 − w + w ) = 0 , (3.10)eq ≡ w (cid:48)(cid:48) − w (cid:48) ρ − h w + ˜ ω f (1 − w + w ) = 0 , (3.11)eq ≡ f (cid:48)(cid:48) + f (cid:48) ρ − (1 − w + w ) ρ f − (cid:18) f h − h ) − (1 − f ) h h (cid:19) = 0 , (3.12)with a dimensionless parameter ˜ ω = ωµ . (3.13)As we will see below, an absolute value of ˜ ω should be less than one. We solve these differentialequations for h , ( ρ ), w , ( ρ ) and f ( ρ ) with the following boundary conditions h (0) = 0 , h (cid:48) (0) = 0 , w (0) = 0 , w (0) = 0 , f (0) = 0 , (3.14)20 fh h w w Figure 4: Profile functions for the non-BPS DNALVS with ˜ ω = 0 .
8. Those for BPS NALVSwith ˜ ω = 0 are given in Fig. 1.and h ( ∞ ) = 1 , h ( ∞ ) = 1 , w ( ∞ ) = 1 , w ( ∞ ) = 0 , f ( ∞ ) = 1 . (3.15)The energy density is expressed as H = g v ( K s + K t + V ) , (3.16)with K s = 1 g v Tr (cid:20) g F + D m H ( D m H ) † (cid:21) = w (cid:48) + w (cid:48) ρ + (cid:0) h (cid:48) + h (cid:48) (cid:1) + h (1 − w ) + h w ρ , (3.17) K t = 1 g v Tr (cid:20) g F m + D H ( D H ) † (cid:21) = ˜ ω (cid:20) f (cid:48) + f ( − w + w + 1) ρ + 12 (cid:0) (1 + f )( h + h ) − f h h (cid:1) (cid:21) , (3.18) V = 1 g v Tr (cid:20) g (cid:0) HH † − v (cid:1) (cid:21) = 14 (cid:16)(cid:0) h − (cid:1) + (cid:0) h − (cid:1) (cid:17) , (3.19)where the prime stands for a derivative in terms of ρ . Furthermore, the conserved Noether21harge density is given by Q a = 2 i Tr (cid:2)(cid:0) ( D H ) † H − H † D H (cid:1) T a (cid:3) = ωv (cid:0) h + h − f h h (cid:1) δ a . (3.20)Thus, the tension and the Noether charge per unit length are given by T (˜ ω ) = 2 πv (cid:90) ∞ dρ ρ [ K t + K s + V ] , (3.21)and Q a (˜ ω ) = (cid:90) dx dx Q a = 2 πv µ ˜ ωδ a (cid:90) dρ ρ (cid:0) h + h − f h h (cid:1) . (3.22)Before solving the equations of motion, let us first study stability of asymptotic state at r → ∞ . From Eq. (3.15), we perturb the fields as h , = 1 − δh , , w = 1 − δw , w = δw and f = 1 − δf . Then the linearized equations for δh and δh read ∇ (cid:18) δh δh (cid:19) = M (cid:18) δh δh (cid:19) , M = (cid:18) µ − ω ω ω µ − ω (cid:19) . (3.23)The eigenvalues of the mass square matrix are µ and µ − ω . Thus, the fluctuations aroundconfigurations with ω > µ become tachyonic, so that we have to set | ω | < µ , namely | ˜ ω | should be less than 1 as mentioned before.We are now ready to solve the equations of motion. Since no analytic solutions can beobtained, we solve them numerically. A numerical solution is shown in Fig. 4. As can be seenfrom Figs. 1 and 4, DNALVS with non-zero ω becomes fatter than the non-dyonic NALVS.This is consistent with Eq. (3.23) that leads to an asymptotic behavior e − √ µ − ω r in theprofile functions.In Sec. 2.6, we have considered the spiral extension of the BPS DNALVS by boosting thesolution along the string axis. Obviously, similar spiral solution for the non-BPS DNALVScan be easily constructed. Let us next study the tension of DNALVS. In Fig. 5, we show ˜ ω -dependence in T and Q for˜ ω = [0 , .
94] with a step δ ˜ ω = 0 .
01. For small ˜ ω , we find that the numerical results are wellfitted by the following functions, see Fig. 5 T πv = 1 + 12 ˜ ω + O (˜ ω ) , µQ πv = ˜ ω + O (˜ ω ) . (3.24)22 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) Ω(cid:142) Q Μ (cid:144) Π v (cid:72) (cid:98) Ω(cid:142) (cid:76) (cid:230) Q Μ (cid:144) Π v (cid:72) Ω(cid:142) (cid:98) (cid:76)
Ω(cid:142) T (cid:144) Π v (cid:72) (cid:98) Ω(cid:142) (cid:76) (cid:230) T (cid:144) Π v (cid:72) Ω(cid:142) (cid:98) (cid:76) (cid:43) Ω(cid:142) (cid:72) (cid:45)Η (cid:76) Ω(cid:142) (cid:43) Η (cid:45) Ω(cid:142) (cid:43) (cid:45) Η Figure 5: The ˜ ω dependence of T and Q for k = 1 DNALVS are shown. The dots arenumerical results for 0 ≤ ˜ ω ≤ . . ≤ ˜ ω ≤ .
94. Theblue dashed line corresponds to T πv = 1 + ˜ ω / µQ πv = ˜ ω . The blue solid curve stands for T (˜ ω ) given in Eq. (3.26).Note that, since we have the identification | q | = | Q | /
2, the second equation is consistentwith Eq. (2.67) obtained from the low energy effective action with m = 0. From these, wefind a direct relation of T and Q for small | Q | , T = 2 πv + 14 β ( Q ) + · · · . (3.25)This is also consistent with the observation Eq. (2.69).Remember that the effective theory description is valid only for sufficiently small ˜ ω . Onthe other hand, since we have solved the full equations of motion, we are now able to gobeyond the effective theory. Namely, it may be possible to find an appropriate function whichcan reproduce the numerical data not only for a small ˜ ω but also for a large ˜ ω . Such functionshould be a function of ˜ ω and singular at ˜ ω = 1. Indeed, we find that the following functionfits the numerical data quite well, see Fig. 5, T πv = 1 + (cid:18) − η (cid:19) ˜ ω + η ˜ ω − ˜ ω , η = 0 . . (3.26)This is somewhat surprising because the almost all numerical data are fitted by the functionwith only one parameter η . Two comments are in order. Firstly, the expression (3.26) can be23xpanded in terms of ˜ ω as T πv = 1 + ˜ ω η (cid:0) ˜ ω + ˜ ω + · · · (cid:1) . (3.27)The first two terms are independent of η , and they are consistent with Eq. (3.24). Secondly,as can be seen in Fig. 5, while Eq. (3.26) reproduces the numerical data below ˜ ω = 0 .
90 quitewell, the data in 0 . ≤ ˜ ω ≤ .
94 are not fitted very well. As to the solutions for ˜ ω > . { h , , w , , f } as functions of ρ andan artificial time τ and add dissipative terms to the right hand side Eqs. (3.8) – (3.12) aseq α = ∂X α ( ρ, τ ) ∂τ , X α = ( h , h , w , w , f ) . (3.28)We integrate these in τ with a suitable initial configuration X α ( ρ, τ = 0) that satisfies theboundary conditions given in Eqs. (3.14) and (3.15). After a while in τ , the profile functionsconverge. Since we have ∂X α /∂τ = 0, the converged profile functions X α ( ρ, ∞ ) solve thegenuine equations of motion. In this way, we get the numerical solutions of k = 1 for 0 ≤| ˜ ω | ≤ .
94. Unfortunately, we failed to get numerical solutions above | ˜ ω | > .
94. In thisregion, what we get is not a configuration localized around the vortex center but expandingtoward spatial infinity. Indeed, when we increase the computational domain, the numericalsolution becomes just fatter. Namely, they are sensitive to the box size, and we cannot acceptthem as genuine vortex solutions. Relatedly, we also observe that a numerical error for alarger | ˜ ω | tend to be larger than that for a smaller | ˜ ω | , see Fig. 6 where the values of eq α in Eqs. (3.8) – (3.12) at the end of numerical integration in τ are shown. For the numericalsolutions with | ˜ ω | ≥ .
91, the numerical errors are not so small. Therefore, we guess thatdiscrepancy between the formula (3.26) and the numerical results for | ˜ ω | ≥ .
91 is caused bythe numerical errors.Furthermore, we show a direct relation between the tension T and the Noether charge Q by regarding ˜ ω as a parameter, see Fig. 7. Surprisingly, we find that the dyon-type tensionformula can reproduce the numerical result quite well, T k =1 ∼ = (cid:114) (2 πv ) + (cid:16) µ Q (cid:17) . (3.29)Here we have used the symbol ∼ =, in order not to forget the fact that the equality is notanalytically proven but is just verified by the numerical calculation. Anyway, the dyon-type formula works pretty well. Actually, if we expand Eq. (3.29) to the quadratic order in24 .0 0.2 0.4 0.6 0.8 1.0 (cid:45) (cid:180) (cid:45) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) Ρ v a l u e o f e q Α Ω(cid:142) (cid:61) (cid:45) (cid:45) Ρ v a l u e o f e q Α Ω(cid:142) (cid:61)
Figure 6: Numerical errors, namely values of eq α at the end of integration are shown. Theleft panel is for ˜ ω = 0 .
80 and the right one is for ˜ ω = 0 . µQ / / (2 πv ), we reproduce the result in Eq. (3.25). We now encounter an unexpectedresult: Remember, as is explained in Sec. 2, that the tension formula (2.27) of the BPS
DNALVS is of the dyonic-instanton type. This matches the conjecture that any solitonswhose mass formulae are of the dyonic-instanton (dyon) type are BPS and their spatial co-dimensions are always even (odd). Nevertheless, we have found that the tension of the non-BPS
DNALVS is approximated pretty well by the dyon-type tension formula (3.29). Tothe best of our knowledge, no such non-BPS solitons have been known in the literature. Ifthe relation (3.29) was really exact, this would have been the first counter-example to theconjecture. However, as mentioned above, Eq. (3.29) is verified only by numerical calculations,so we cannot conclude that the conjecture fails at this stage. Indeed, in Sec. 4.2, we will showthat Eq. (3.29) cannot be exact. Therefore, the conjecture survives. We will propose anothertension formula in Sec. 4.2.In addition to the minimally winding solution, we also examine axially symmetric solutionswith higher winding numbers k = 2 and 3. Since the dyonic solutions are non-BPS and havethe Noether charge of the same sign, axially symmetric solution is very likely to be unstable.Despite of these, we assume an axial symmetry and solve the equations of motion (3.8) –(3.12) with modifying them for k = 2 ,
3. Unexpectedly again, we find that the dyon-typeformula approximately holds T k ≥ ≈ (cid:114) (2 πv k ) + (cid:16) µ Q (cid:17) . (3.30)For small Q , this formula works well. However, increasing Q , the numerical data is off thedyon-type formula. The coincidence between the numerical results and the formula (3.30) is25 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) Μ Q (cid:144) Π v T (cid:144) Π v eff th (cid:72) (cid:76) eff th (cid:72) (cid:76) Ω(cid:142) (cid:62) (cid:230)
Ω(cid:142) (cid:98)
Figure 7: Comparison with the dyon-type formula (green line) given in Eq. (3.29) and resultsby the effective field theories including terms up to quadratic (red-dashed line) and quartic(red line) derivative terms in Eqs. (4.3) and (4.4). The numerical data are the blue dots andthe blue circles.not as good as k = 1 case especially for a large ˜ ω , so we have used the symbol ≈ in Eq. (3.30). Let us return to Eq. (3.27). The first term is the tension of the static non-Abelian vortex andthe second term is the correction of order O ( ∂ t ) which can be explained from the view pointof the low energy effective theory in Sec. 2.3. The third term is of order O ( ∂ t ), and so itcorresponds to a higher derivative correction to the effective Lagrangian given in Eq. (2.42).The higher derivative correction of the quartic order was obtained in Ref. [21] L (2+4)eff = β (cid:20) | ∂ α φ | (1 + | φ | ) + γµ | ∂ α φ∂ α φ | (1 + | φ | ) (cid:21) , (4.1)where γ is a numerical constant defined by γ ≡ (cid:90) dρ ρ (cid:0) − h ( ρ ) (cid:1) ≈ . . (4.2)From this, the conserved charge and energy are given by q = iβ (cid:20) φ∂ φ ∗ − ∂ φφ ∗ (1 + | φ | ) + 2 γµ φ ( ∂ φ )( ∂ φ ∗ ) − φ ∗ ( ∂ φ ∗ )( ∂ φ ) (1 + | φ | ) (cid:21) , (4.3)26 T (2+4) = β (cid:20) | ∂ φ | (1 + | φ | ) + 3 γµ | ∂ φ | (1 + | φ | ) (cid:21) . (4.4)Note that, under the presence of the four derivative term, φ = e iωt remains a solution ofthe equations of motion. Plugging φ = e iωt into these, the increment in the tension and theNoether charge per unit length are obtained as δT (2+4) πv = 2 µ (cid:20) ω γµ ω (cid:21) = 12 ˜ ω + 3 γ ω , (4.5) µq πv = 2 µ (cid:20) ω γµ ω (cid:21) = ˜ ω + γ ω . (4.6)Thus the coefficient of the quartic order reads3 γ . . (4.7)Comparing this with the coefficient of the third term of Eq. (3.27), we should identify η = 3 γ , (4.8)as is indeed taken in Eq. (3.26). In other words, we succeeded in computing the coefficientof the quartic derivative term only from the tension formula, avoiding any complicated com-putations [21]. In Fig. 7, we show the numerical solutions, the dyonic-type tension formula(3.29), and the results from effective action including the quadratic and quartic derivativecorrections. Encouraged by success in finding the coefficient of the four derivative term from the tensionformula (3.26), we entertain hope to figure out higher derivative corrections of all order tothe effective theory for a single non-Abelian vortex. To the best of our knowledge, higherderivative corrections only up to the quartic order have been obtained in the literature. Inorder to simplify the problem, we consider DNALV in 2 + 1 dimensions in this subsection. Anadvantage of this is that the effective theory is 0 + 1 dimensional theory, so that there are lessvarieties for the higher derivative terms. For example, the 2 n -th order is simply proportionalto | ∂ φ | n . On the contrary, there are several choices for higher derivative terms in higherdimensions. For instance, there are two possibilities for four derivative terms, ( ∂ α φ∂ α ¯ φ ) and | ∂ α φ∂ α φ | , although only the latter appears in Eq. (4.1).Firstly, we expand Eq. (3.26) and rewrite it in the following form T k =1 = 2 πv + β (cid:32) ω η ∞ (cid:88) n =2 ω n µ n − (cid:33) . (4.9)27he second term should be compared with the effective theory with higher derivative correc-tions. To this end, let us assume the effective Lagrangian to be the following form L ( ∞ )eff = β (cid:32) | ∂ φ | (1 + | φ | ) + ∞ (cid:88) n =2 a n µ n − | ∂ φ | n (1 + | φ | ) n (cid:33) , (4.10)with a n being real constants. Reflecting the symmetry of the original Lagrangian, thiseffective Lagrangian respects the SU (2) F symmetry ( | ∂ φ | / (1+ | φ | ) is an SU (2) F invariant).Note that, regardless of the coefficient a n , φ = e iωt solves the equation of motion for thisLagrangian. The corresponding Hamiltonian reads H ( ∞ )eff = β (cid:32) | ∂ φ | (1 + | φ | ) + ∞ (cid:88) n =2 (2 n − a n µ n − | ∂ φ | n (1 + | φ | ) n (cid:33) . (4.11)Plugging φ = e iωt into the Hamiltonian, we get δT ( ∞ ) = β (cid:32) ω ∞ (cid:88) n =2 (2 n − a n n ω n µ n − (cid:33) (4.12)Equating this with T k =1 − πv in Eq. (4.9), we find the expansion coefficients a n as a n = 2 n − n − η. (4.13)Plugging this back into Eq. (4.10), we end up a prediction to the effective Lagrangian includinghigher derivative corrections of all order as L ( ∞ )eff = 2 πv (cid:18) − η X + η √ X tanh − √ X (cid:19) , (4.14)where we have used (cid:80) ∞ n =1 X n / (2 n −
1) = √ X tanh − √ X and X ≡ | ∂ φ | µ (1 + | φ | ) . (4.15)Our prediction fully depends on the global tension formula (3.26) which we have verifiedonly by numerical data. Since numerical errors are unavoidable in any numerical results,Eq. (3.26) could not be exact but just an approximation. If it is the case, the expressionof the effective action given in Eq. (4.14) would be modified (Eq. (3.26) is correct up to thequartic order). However, this is not a big matter for us. What we would like to stress isthat, once somehow one gets the tension formula of DNALVS, it is always possible to derivean effective action including all order higher derivative corrections. Usually, obtaining the28ension formula is much easier than a straightforward but very complicated calculation forthe higher derivative corrections. So, the tension formula helps us very much.Regardless of the fact that the effective Lagrangian (4.14) is exact or approximate, theeffective Lagrangian (4.14) has an important property which the true effective action has tohave: the true effective action should be singular at ˜ ω = 1 since DNALVS with ˜ ω = 1 doesnot exist. The effective theory with the quartic derivatives given in Eq. (4.1) does not havethis property because ˜ ω = 1 is beyond the scope. On the other hand, the effective Lagrangian(4.14) is actually singular at X = 1. Note that this occurs at ˜ ω = 1 because of X (cid:12)(cid:12) φ = e iωt = ˜ ω . (4.16)Remember the fact that, when we derive the effective action (4.14), we used only thetension formula (3.26) but we have not used any informations about the Noether charge.Note that, since the Lagrangian (4.14) is invariant under the U (1) transformation φ → e iα φ ,the Lagrangian (4.14) has its own Noether charge. For the solution φ = e iωt , the tension andthe Noether charge per unit length for the Lagrangian (4.14) are given by δT ( ∞ ) πv = (cid:18) − η (cid:19) ˜ ω + η ˜ ω − ˜ ω , (4.17) µq ∞ ) πv = 2 (cid:18) − η (cid:19) ˜ ω + η ˜ ω − ˜ ω + η tanh − ˜ ω. (4.18)By definition, Eq. (4.17) coincides with Eq. (3.26). Of course, if we expand Eq. (4.17) in ˜ ω ,it also reproduces the result (4.5) from the effective action including the quartic derivatives.Similarly, if q ∞ ) is expanded in ˜ ω , we have µq ∞ ) πv = ˜ ω + 4 η ω + 6 η ω + · · · , (4.19)which is again consistent with (4.6). From Eqs. (4.17) and (4.18), one can in principle derivea direct relation beween T ( ∞ ) / πv = 1 + δT ( ∞ ) / πv and µq ∞ ) / πv . But in practice itis not easy to express T ( ∞ ) as a simple function of µq ∞ ) . At least it is obviously differentfrom the dyon-type tension formula (3.29). Now a question is which is more plausible, thedyon-tension formula (3.29) or the relation from Eqs. (4.17) and (4.18).In order to verify validity of the former, let us assume that the global tension formula(3.26) and the dyon-type tension formula (3.29) are correct. Then, the ˜ ω dependence in q reads µq πv = (cid:115)(cid:18) (cid:18) − η (cid:19) ˜ ω + η ˜ ω − ˜ ω (cid:19) − (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) q (cid:144) Π v effective theory (cid:72) (cid:165) (cid:76) T (cid:144) Π v (cid:72) (cid:98) Ω(cid:142) (cid:76) (cid:230) T (cid:144) Π v (cid:72) Ω(cid:142) (cid:98) (cid:76) dyon type formula
Figure 8: Comparison with the dyon-type formula (green line) given in Eq. (3.29) and therelation between T and Q derived from Eqs. (4.17) and (4.18). The dots are the numericalresults. = ˜ ω + (cid:18)
18 + η (cid:19) ˜ ω + 1128 (cid:0) − η + 176 η − (cid:1) ˜ ω + · · · . (4.20)The difference from Eq. (4.19) appears at the order ˜ ω . Since this is inconsistent with theexpression (4.6) obtained from the effective Lagrangian, the dyon-type tension formula (3.29)cannot be exact but just an appropriate expression. Nevertheless, it is still surprising that thedyon-type tension formula (3.29) approximates the numerical data pretty well, see Fig. 7. Thiscoincidence can be also seen in the coefficients of Eqs. (4.19) and (4.20); 4 η/ . / η = 0 . η/ . − η + 176 η − /
128 = 0 . d ( T ( ∞ ) / πv ) d ( µq ∞ ) / πv ) = dT ( ∞ ) d ˜ ω d ˜ ωd ( µq ∞ ) ) = ˜ ω → µq πv for ˜ ω →
01 for ˜ ω → . (4.21)This asymptotic behavior is consistent with that for the dyon-type tension formula (3.29).Therefore, Eqs. (4.17) and (4.18) gives us an inplicit function T ( Q ) which is almost degeneratewith the dyon-type tension formula T ( Q ) = (cid:112) (2 πv ) + ( µQ / . In Fig. 8, we show thenumerical data, the dyon-type tension formula (3.29), and the relation from Eqs. (4.17) and(4.18). Indeed, it is hard to find differences among these three below ˜ ω ≤ . In this section, we are going to study the dyonic non-Abelin vortex strings not in the U ( N C )Yang-Mills-Higgs model but in the SU ( N C ) Yang-Mills-Higgs model. A difference is justungauging the overall U (1) symmetry of the model in the previous sections. However, thisdrastically changes properties of the non-Abelian vortex strings. The distinction stands outeven in the model with N C = 1. In the U (1) gauge theory (the Abelian-Higgs theory), vorticesare the so-called local vortex strings (or the Nielsen-Olesen vortex strings) whose tensions arefinite. On the other hand, strings in a global model (no gauge symmetries) are the so-calledglobal vortex strings. In contrast, tensions of the global vortices are logarithmically divergent.In addition, no global vortices as BPS states have been found in the literature. For N C > SU ( N C ) is gauge symmetry and the overall U (1) is global symmetry, thesame can be said. Their tensions are logarithmically divergent and they are always non-BPS states. They are sometimes called the semi-superfluid vortex strings especially in thecontext of the high density QCD in which they were firstly found [35]. In this way, thenon-Abelian strings in U ( N C ) gauge theory and SU ( N C ) gauge theory are quite different.Nevertheless, they share an important property that they have normalizable non-Abelianorientational moduli. Hence, as in the U ( N C ) case, the non-Abelian vortex strings in SU ( N C )gauge theories can be extended to dyonic ones. As explained in the Introduction, we preferthe non-Abelian semi-global vortex string (NASGVS) rather than the semi-superfluid string.Please do not confuse it with the other non-Abelian vortex strings: the non-Abelian localvortex string [13, 14, 15, 16], the non-Abelian semi-local vortex string [13, 49, 50, 51], andthe non-Abelian global vortex string [52, 53, 54, 55].In order to study a dyonic NASGVS (DNASGVS), we slightly change the bosonic La-grangian of N = 2 supersymmetric Yang-Mills-Higgs Lagrangian given in Eq. (2.1). Firstly,we ungauge the overall U (1) symmetry. Then, we discard unimportant fields Σ and H i =2 from the beginning, and set M = 0. As result, we start with the following simple Lagrangian L SU ( N C ) = Tr (cid:20) − g ˆ F µν ˆ F µν + ˆ D µ H ( ˆ D µ H ) † − g (cid:0) HH † − v N C (cid:1) (cid:21) , (5.1)where ˆ F µν and ˆ D µ are the field strength and covariant derivative for the SU ( N C ) gaugefield. Note that this Lagrangian cannot be embedded into any supersymmetric models. Thisis because the overall U (1) symmetry is not gauge symmetry, so that it is impossible tointroduce the term proportional to unit matrix [ D U (1) = (Tr[ HH † ] /N C − v ) N C ] in the scalar31otential. The symmetry of the model is G = SU ( N C ) C × U (1) B × SU ( N C ) F . (5.2)The first SU ( N C ) C is the gauge symmetry while the rest symmetry U (1) B × SU ( N C ) F is theglobal symmetry. The vacuum moduli space is S which is parametrized by H = ve iϕ N C . (5.3)The phase e iϕ corresponds to the spontaneously broken U (1) B . At any points in the vacuummoduli space, the symmetry G is spontaneously broken as SU ( N C ) C × U (1) B × SU ( N C ) F → SU ( N C ) C+F . (5.4)The spectra in the Higgs vacuum split into two masses; the one is µ = g v for the SU ( N C )gauge field ( N − H ( N degrees of freedom),and the other is a Nambu-Goldstone zero mode associated with the spontaneously broken U (1) B symmetry (the phase of the trace part of H ). The massless mode exists alone becauseof the absence of the U (1) gauge field.Since neither NASGVS nor DNASGVS is a BPS state, we have to solve the equations ofmotion, ˆ D µ ˆ D µ H + g HH † − v ) H = 0 , (5.5)1 g ˆ D µ ˆ F µν + i (cid:28) H (cid:16) ˆ D ν H (cid:17) † − (cid:16) ˆ D ν H (cid:17) H † (cid:29) = 0 , (5.6)where (cid:104) X (cid:105) stands for traceless part of an N C × N C matrix X . In what follows, we will considerthe case of N C = 2 for simplicity. For a static (non-dyonic) configuration, we set ∂ , = 0 and W , = 0, and make the following Ansatz [38] H = v (cid:18) h ( r ) e iθ h ( r ) (cid:19) , (5.7) W + iW = − ie iθ r (cid:18) w ( r ) 00 − w ( r ) (cid:19) . (5.8)Plugging these into Eqs. (5.5) and (5.6), we get the three 2nd order differential equations for h , and w . In terms of the dimensionless coordinate ρ = µr , they are expressed as h (cid:48)(cid:48) + h (cid:48) ρ − (1 − w ) ρ h − (cid:0) h − (cid:1) h = 0 , (5.9)32 h h h h f Figure 9: Profile functions for a static semi-global non-Abelian vortex with ˜ ω = 0 (the upperpanel) and for a dyonic semi-global non-Abelian vortex with ˜ ω = 0 . h (cid:48)(cid:48) + h (cid:48) ρ − w ρ h −
12 ( h − h = 0 , (5.10) w (cid:48)(cid:48) − w (cid:48) ρ + 12 (cid:0) h (1 − w ) − h w (cid:1) = 0 . (5.11)We solve these with the boundary conditions h → , h → , w → , as ρ → ∞ , (5.12) h → , h (cid:48) → , w → , as ρ → . (5.13)It is hard to analytically solve the above differential equations but is easy to numerically solvethem. A numerical solution is shown in the upper panel of Fig. 9. Note that h looks like33niformly 1 in Fig. 9, but it is slightly different from 1 near the origin. Indeed, h = 1 is nota solution of Eq. (5.10), if w (cid:54) = 0.Let us study asymptotic behaviors of the non-dyonic NASGVS by perturbing the fieldsfar from the origin as h = 1 − δh , h = 1 − δh , w = 12 − δw . (5.14)The linearized equations at r → ∞ for trace and traceless parts δF ≡ δh + δh , δG ≡ δh − δh (5.15)are given by ( (cid:52) − δF = − ρ , ( (cid:52) − δG = 0 , (5.16)and that for δw is given by ( (cid:52) (cid:48) − δw = − δG, (5.17)where we defined (cid:52) = d dρ + ρ ddρ and (cid:52) (cid:48) = d dρ − ρ ddρ . Eq. (5.17) can be cast into a standardform ( (cid:52) − δw ρ = 0 . (5.18)Solving these equations, we find δF = 12 ρ + O (cid:0) ρ − (cid:1) , (5.19) δG = q s K ( ρ ) + O (cid:0) ( e − ρ ) (cid:1) , (5.20) δw = q g ρK ( ρ ) + O (cid:0) ( e − ρ ) (cid:1) , (5.21)where K n ( ρ ) is the modified Bessel functions of the second kind and q s and q g are numericalconstants. While the Higgs mechanism for the SU ( N C ) part ensures the exponentially smalltails for δG and δw , the trace part δF has a long tail of power behavior due to the mass-less Nambu-Goldstone mode [38]. Note that this long tail in the trace part makes tensionlogarithmically divergent. This can be seen by looking at the kinetic term of H (cid:90) d x Tr[ ˆ D i H ˆ D i H † ] ∼ (cid:90) d x v ρ = πv (cid:90) dρ ρ = πv log L, (5.22)where L is a IR cut off in unit of µ − . 34ext we extend NASGVS to dyonic ones. As in Sec. 3, we rotate the static Ansatz inEqs. (5.7) and (5.8) by the time-dependent SU (2) C+F matrix defined by Eq. (3.6) H = ˜ U † ( t ) (cid:20) v (cid:18) h ( r ) e iθ h ( r ) (cid:19)(cid:21) ˜ U ( t ) , (5.23)¯ W = ˜ U † ( t ) (cid:20) − ie iθ r (cid:18) w ( r ) 00 − w ( r ) (cid:19)(cid:21) ˜ U ( t ) , (5.24) W = ˜ U † ( t ) (cid:20) ω (cid:18) − e iθ f ( r )1 − e − iθ f ( r ) 0 (cid:19)(cid:21) ˜ U ( t ) . (5.25)Plugging these into the equations of motion (5.5) and (5.6), we find h (cid:48)(cid:48) + h (cid:48) ρ − (1 − w ) ρ h − (cid:0) h − (cid:1) h = − ˜ ω (cid:0)(cid:0) f + 1 (cid:1) h − f h (cid:1) , (5.26) h (cid:48)(cid:48) + h (cid:48) ρ − w ρ h −
12 ( h − h = − ˜ ω (cid:0) − f h + (1 + f ) h (cid:1) , (5.27) w (cid:48)(cid:48) − w (cid:48) ρ + 12 (cid:0) h (1 − w ) − h w (cid:1) = ˜ ω − w ) f , (5.28) f (cid:48)(cid:48) + f (cid:48) ρ − (1 − w ) ρ f − (cid:0) ( h + h ) f − h h (cid:1) = 0 , (5.29)where we have again used ρ = µr and ˜ ω = ω/µ . The boundary conditions for h , h and f are the same as those given in Eqs. (3.14) and (3.15). The one for w is given by w (0) = 0 , w ( ∞ ) = 12 . (5.30)A numerical solution is shown in the lower panel in Fig. 9. As in the case of the U (2) gaugetheory, the dyonic configuration becomes fatter than the static configuration. This fact alsoreflects in the asymptotic behavior. To see this, let us perturb the fields as Eq. (5.14) with f = 1 − δf . Then we have the following linearized equations( (cid:52) − δF = − ρ , (5.31) (cid:0) (cid:52) − (1 − ˜ ω ) (cid:1) δG = 0 , (5.32) (cid:0) (cid:52) (cid:48) − (1 − ˜ ω ) (cid:1) δw = − δG, (5.33)( (cid:52) − δf = 0 . (5.34)The equation for δF is unchanged from Eq. (5.15), so the asymptotic behavior is the sameas one given in Eq. (5.19). The equations for δG and δw are the same as those given inEqs. (5.15) and (5.17) except for the mass square being replaced by 1 → − ˜ ω . Thus theasymptotic behaviors are δG = q (cid:48) s K (cid:16) √ − ˜ ω ρ (cid:17) , δw = q (cid:48) g (cid:16) √ − ˜ ω ρ (cid:17) K (cid:16) √ − ˜ ω ρ (cid:17) , δf = q K ( ρ ) , (5.35)35able 1: The masses (in the unit of 2 πv ) of a semi-global vortex as function of the frequency˜ ω (= 0 , . , . , . , . , . , .
6) and the IR cutoff scale L (= 50 , , µ − . L T (0; L ) δT (0 . δT (0 . δT (0 . δT (0 . δT (0 . δT (0 . q (cid:48) s , q (cid:48) g and q are numerical constants. The quantity (cid:112) µ − ω plays a role of an“effective” mass for the dyonic configurations, and so it should be positive definite. When itbecomes negative, the configuration becomes unstable since the effective mass is tachyonic.Thus, for stable configurations, we should restrict | ω | < µ, (5.36)as in the U (2) case in Sec. 3.The Noether charge density is formally the same as that in the U (2) case. Q a ≡ i Tr (cid:104)(cid:16) ( ˆ D H ) † H − H † ˆ D H (cid:17) T a (cid:105) = ωv (cid:0) h + h − f h h (cid:1) δ a . (5.37)Since the tension of NASGVS is logarithmically divergent, one may anticipate that theNoether charge also diverges. But this is not the case. Indeed, it is exponentially smallat spatial inifnity Q a → ωv δf δ a = 2 ωv q K ( ρ ) δ a , ( ρ → ∞ ) , (5.38)so that the Noether charge per unit length is finite, Q a = (cid:90) dx dx Q a . (5.39)Hence, nevertheless the tension itself diverges, we expect that excess in the tension of DNAS-GVS from NASGVS remains finite. To see this, let T (˜ ω ; L ) be a logarithmically divergentmass with L being a IR cut off scale in the unit of µ − , T (˜ ω ; L )2 πv = (cid:90) L dρ ρ (cid:20) (cid:18) h (cid:48) + h (cid:48) + ( h (1 − w ) + h w ) ρ (cid:19) + 2 w (cid:48) ρ + 14 (cid:110)(cid:0) − h (cid:1) + (cid:0) − h (cid:1) (cid:111) + ˜ ω (cid:26) f (cid:48) + f (1 − w ) ρ + 12 (cid:0)(cid:0) f + 1 (cid:1) (cid:0) h + h (cid:1) − f h h (cid:1)(cid:27) (cid:21) . (5.40)36 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) Ω(cid:142) (cid:230) Q Μ (cid:144) Π v (cid:230) ∆ T (cid:144) Π v Figure 10: Numerical results for Q (red dots) and δT (blue dots) for 0 . ≤ ˜ ω ≤ .
9. Theasymptotic behaviors given in Eqs. (5.42) and (5.43) for small ˜ ω are shown in dashed blueand red lines, respectively.Then our guess is that δT (˜ ω ) defined by δT (˜ ω ) = T (˜ ω ; L ) − T (0; L ) , (5.41)is finite. We numerically compute δT by changing the size of computational domain, and wefind that δT is indeed independent of size of computational box L and is finite. The resultsare summarized in Table 1.Behaviors of Q and δT as functions of ˜ ω (0 ≤ ˜ ω ≤ .
90) are shown in Fig. 10. For small˜ ω , as can be seen from Fig. 10, they are well approximated by linear and quadratic functions,respectively. We find numerical fitting curves δT (˜ ω )2 πv = ξ ˜ ω + O (cid:0) ˜ ω (cid:1) , ξ = 0 . , (5.42) µQ (˜ ω )2 πv = 2 ξ ˜ ω + O (cid:0) ˜ ω (cid:1) , ξ = 1 . . (5.43)The coefficients ξ and ξ are slightly different from those in Eq. (3.24) for the supersymmetricmodel. Nevertheless, a similar relation to the supersymmetric case holds2 ξ = ξ . (5.44)The reason why this special relation holds can be clearly explained from the view pointof a low energy effective theory on the vortex world volume. The orientational zero modes37re associated with the spontaneous symmetry breaking SU (2) C+F → U (1) C+F , so that thecorresponding effective Lagrangian should be a non-linear sigma model whose target space is C P (cid:39) SU (2) C+F /U (1) C+F L (2)eff; SU (2) = ˜ β ∂ α φ∂ α φ ∗ (1 + | φ | ) . (5.45)Unlike the case of supersymmetric case, the overall coefficient ˜ β cannot be analytically ob-tained but can be determined by numerical computations [39]. But here, for a while, we leaveit an unknown constant. As before, we are interested in a time-dependent solution φ = e iωt .Then, we find excess in the tension and the Noether charge per unit length δT eff = ˜ β | ˙ φ | (1 + | φ | ) = ˜ β ω , q = i ˜ β φ∂ φ ∗ − φ ∗ ∂ φ (1 + | φ | ) = ˜ β ω. (5.46)Comparing this with Eqs. (5.42) and (5.43) and remembering the relation Q = 2 q , irre-spective of value of ˜ β , we reach the relation 2 ξ = ξ . Furthermore, as a byproduct, we candetermine the unknown constant ˜ β by comparing Eqs. (5.42) and (5.46)˜ β = 8 πv ξ µ = 2 ξ × πg = 1 . × πg . (5.47)In order to check validity of the above indirect computation for ˜ β , let us calculate ˜ β in adirect way [39]. To this end, we mimic the derivation for β done in Sec. 2.3. First, we prepare x α ( α = 0 ,
3) dependent background configurations H = U (cid:20) v (cid:18) h h (cid:19)(cid:21) U − , (5.48) W + iW = U (cid:20) − i e iθ r (cid:18) w − − w + 1 (cid:19)(cid:21) U − , (5.49)with U given in Eq. (2.40). Here h , h and w are a static solution of Eqs. (5.9) – (5.11),see Fig. 9. For W , , we use the same Ansatz given in Eq.(2.41). Plugging these into theLagrangian (5.1) and picking up the terms quadratic in ∂ α , we find the effective Lagrangiangiven in Eq. (5.45) as L (2)eff; SU (2) = (cid:90) dx dx Tr (cid:20) − g ˆ F iα ˆ F iα + ˆ D α H ( ˆ D α H ) † (cid:21) = ˜ β ∂ α φ∂ α φ ∗ (1 + | φ | ) . (5.50)with ˜ β = 4 πg (cid:90) dρ ρ (cid:20) λ (cid:48) + (1 − w ) ρ (1 − λ ) + λ (cid:0) h + h (cid:1) + (1 − λ )( h − h ) (cid:21) (5.51)38 Ρ Λ Figure 11: A numerical solution of λ for Eq. (5.52).This is slightly different from β given in Eq. (2.43). If we replace h → h , h → w → w , β and ˜ β are formally identical. However, since h , h and w do not solve the BPSequation but solve the equations of motion (5.9) – (5.11), the integral in Eq. (5.51) is notidentical to β = 4 π/g . In order to find an appropriate λ , we need to numerically solve theequation of motion λ (cid:48)(cid:48) + λ (cid:48) ρ + (1 − w ) ρ (1 − λ ) + 12 (cid:0) ( h + h )(1 − λ ) − h h (cid:1) = 0 . (5.52)The boundary conditions are of the form λ (0) = 1 , λ ( ∞ ) = 0 . (5.53)A numerical solution is shown in Fig. 11. Plugging the numerical configurations of h , h , w and λ into Eq. (5.51) and integrating it over ρ , we get the following result˜ β = 1 . × πg . (5.54)The coefficient is in good agreement with the one in Eq. (5.47), which ensures that the lowenergy effective theory gives a correct view point to understand DNASGVSs at least for small˜ ω . Finally, we turn to a global aspect including not only small ˜ ω but also large ˜ ω (cid:46) δT as a function of Q with ˜ ω being aparameter in Fig. 12. The result is again surprising. Remember that the tension of NASGVS39 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) Μ Q (cid:144) Π v ∆ T (cid:144) Π v (cid:230) numerical data (cid:72) Ω(cid:142) (cid:163) (cid:76) dyon type formulaeff th (cid:72) (cid:76) Figure 12: Relation between Q and δT for a dyonic semi-global non-Abelian vortex. Thedots shows the numerical results, the red-dashed line corresponds to Eq. (5.46), and the greensolid line stands for the square root formula (5.55).logarithmically diverges and it is not BPS state even when ˜ ω = 0. Nevertheless, the excess δT obeys the square root formula quite well, which is similar to the supersymmetric model δT (cid:12)(cid:12) k =1 ∼ = (cid:115) (2 πv × . + 0 . (cid:18) µQ (cid:19) − πv × . . (5.55)Here the second term in the left hand side is an artifact which is added for tuning δT to bezero at ˜ ω = 0. Similarly, we find the following square root formulae for higher winding axiallysymmetric configurations δT (cid:12)(cid:12) k =2 ≈ (cid:115) (2 πv × . + 0 . (cid:18) µQ (cid:19) − πv × . , (5.56) δT (cid:12)(cid:12) k =3 ≈ (cid:115) (2 πv × . + 0 . (cid:18) µQ (cid:19) − πv × . . (5.57)Comparing these formulae with that for the supersymmetric model given in Eq. (3.29), thecoefficients of ( Q ) in the square root are not unique for different k .For k = 1 DNASGVS, we also numerically find the following global relation between δT and ω δT (cid:12)(cid:12) k =1 ∼ = 2 πv (cid:18) .
063 ˜ ω + 0 . ω . − ˜ ω (cid:19) . (5.58)Although this fitting curve reproduces the numerical data quite well, it has a pole at ˜ ω = √ . (cid:39) . ω (cid:38) . ω < .
9. Then,expanding this in ˜ ω , we have δT (cid:12)(cid:12) k =1 ∼ = 2 πv (cid:0) ξ ˜ ω + 0 .
585 ˜ ω + · · · (cid:1) = ˜ β (cid:18) ω + ω µ + · · · (cid:19) , (5.59)where we used the relation ˜ β = 2 ξ β and replaced 0 . /ξ by 1 because of ξ ≈ . L (2+4)eff = ˜ β (cid:20) | ∂ α φ | (1 + | φ | ) + ˜ γµ | ∂ α φ∂ α φ | (1 + | φ | ) (cid:21) . (5.60)Then, increment in the tension for the configuration φ = e iωt reads δT (2+4)eff = ˜ β (cid:18) ω + 3˜ γ µ ω (cid:19) . (5.61)Comparing this with Eq. (5.59), we are able to make a prediction for the coefficient of thefour derivative term ˜ γ ≈ . (5.62)We leave verification of this prediction as a future problem. Furthermore, as is done inSec. 4.2, we may pursue deriving all order higher derivative corrections to the effective theoryfor NASGVS. We began this paper with making the conjecture that all dyonic solitons of the dyonic-instanton (dyon) type are always BPS and their spatial co-dimensions are always even (odd).41he conjecture is true for all known dyonic solitons so far in the literature. In order to exam-ine further the conjecture, we have focused on the dyonic extension of the non-Abelian vortexstrings both in the supersymmetric and non-supersymmetric Yang-Mills-Higgs theories.In the supersymmetric U ( N C ) Yang-Mills-Higgs model, the non-dyonic NALVSs are theBPS states while their dyonic extensions, DNALVSs, can be either BPS or non-BPS accordingto amount of the dyonic charges. This is a distinctive feature of DNALVS. As expected fromthe conjecture, its BPS tension formula is of the dyonic-instanton type. How about thenon-BPS tension formula? The conjecture does not say anything about non-BPS states.So we numerically solved the equations of motion and found the non-BPS tension formula.Surprisingly, the non-BPS tension formula is approximately of the dyon-type. For BPS states,the simple mass formulae can be understood from the perspective of central charges of superalgebra. In contrast, there are no reasons that such simple formulae like the dyonic-instantonor dyon type hold for non-BPS states. Thus, it still remains a question why the dyon-typeformula for the non-BPS DNALVS holds, although it is the approximate relation.Furthermore, we have studied a dyonic extension of another kind of non-Abelian vortexstring, NASGVS, in the non-supersymmetric Yang-Mills-Higgs model with SU ( N C ) gaugesymmetry. Regardless that NASGVS has a dyonic charge or not, it is always non-BPS, andits tension is logarithmically dirvegent as a global vortex. Nevertheless, we found that anincrement in the tension due to an additional dyonic charge not only remains finite but alsoapproximately obeys the dyon-type tension formula.Thus, we have found the common property for the different non-BPS dyonic vortex stringsin the SUSY and non-SUSY Yang-Mills-Higgs theories. Since the vortex strings are non-BPSand the property holds among SUSY and non-SUSY theories, a reason for the property to holdhas nothing to do with either BPS-ness or SUSY. Or rather, we should make our attention onthe common feature that the vortex strings have the normalizable non-Abelian zero modes.The dyonic extension can be understood as the rotation of the internal orientational modes.For the BPS case, the profile functions of the scalar fields and the gauge fields are not affectedby the additional motion of the internal orientations. Therefore, the tension formula is justthe dyonic-instatnton type that the topological term and the Noether charge are merelyadded. On the contrary, for the non-BPS cases, the profile functions are deformed. Thus, itis natural that the non-BPS tension formula changes from the dyonic-instanton type. Whatis not natural is that the formula is approximated very well by the dyon-type formula.In order to have a deeper insight for understanding this, we have studied the low energyeffective theory for the internal orientation zero modes. While there are many works on theeffective theory in the literature, much of them are on the lowest order theory. Namely, it42ncludes only the quadratic derivative terms. There are also very few works for the higherderivative corrections of the quartic order [21]. However, such effective theories with lowerorder terms are not enough to understand the global property of the non-BPS tension formula.In order to have a breakthrough, we have made the prediction for the low energy effectivetheory (4.14) including the higher derivative corrections to the all order. This has been doneby combining the global tension formula (3.26) which has been obtained by the numericalcomputation and the Ansatz (4.10). We have shown that the effective action has correctproperties that the true effective theory has to have. Furthermore, we have derived theimplicit relation between the tension and the Noether charge from Eqs. (4.17) and (4.18),which reproduces the numerical results and explain a reason why the mysterious dyon-typetension formula approximately holds.Understanding the tension formula for the non-BPS dyonic strings is a practical purpose ofthis paper. On the way to carry out the mission, we have found the efficient method to derivethe low energy effective theory with the all order higher derivative corrections. We expect thatour indirect method can be applied also to other solitons like magnetic monopoles, instantonsand so on.Apart from things on the tension formula, we have also found the novel dyonic soliton, thespiral DNALVS, which is generated by boosting DNALVS along the string. The orientationvector spins with angular frequency m/ √ − u while it twists around the string axis withwavenumber mu/ √ − u (the phase velocity is u ). We have confirmed that existence of thespiral DNALVS from the view points of the original Yang-Mill-Higgs theory in 3+1 dimensionsand the low energy effective theory in 1 + 1 dimensions.Before closing this section, let us make comments on future directions. Firstly, we wouldlike to mention about possibility of detecting dyonic solitons in nature. Although importanceof topological solitons in high energy physics is widely accepted, non of them have beendetected in laboratories thus far. No non-dyonic solitons have been found, much less dyonicsolitons. Nevertheless, we entertain hope that DNASGVS studied in Sec. 5 would be foundinside a compact star such as a quark star or optimistically in a core of a neutron star. Itis expected that QCD enters color superconducting phase at a high baryon density region.Especially, at asymptotically high density region, the phase goes into so-called the CFL phasewhere QCD is weakly coupled theory. The effective non-Abelian Ginzburg-Landau theorywas obtained [56, 57], which is similar to the Lagrangian dealt with in Sec. 5. A possibilityof finding non-dyonic NASGVSs in the CFL phase has been already pointed out [35, 36, 37,38, 39, 40, 41, 23]. So it would be natural to expect that not only DNASGVS but also spiralDNASGVS exist. We will study DNASGV in the context of high density QCD elsewhere.43econdly, the low energy effective Lagrangian including the all order derivative correctionsgiven in Eq. (4.14) is bosonic. Since NALV is a half BPS state, the effective Lagrangian shouldbe generalized to a supersymmetric form. For that purpose, it would be useful to make useof a manifestly supersymmetric way for constructing higher derivative corrections which wasrecently found [58]. Acknowledgements
This work is supported by Grant-in Aid for Scientific Research No.26800119 (M. E.). M. E.thanks to M. Arai, F. Blaschke and M. Nitta for useful comments.
A BPS dyonic solitons
A.1 Dyonic instantons
Dyonic instantons in 1 + 4 dimensional N = 1 supersymmetric SU (2) Yang-Mills theory canbe found through the following Bogomol’nyi completion of Hamiltonian [6] M d-inst = (cid:90) d x Tr (cid:20) F I + 12 F IJ + ( D Σ) + ( D I Σ) (cid:21) = (cid:90) d x Tr (cid:20) ( F I − D I Σ) + 14 (cid:16) F IJ − ˜ F IJ (cid:17) + ( D Σ) + 12 F IJ ˜ F IJ + 2 F I D I Σ (cid:21) ≥ (cid:90) d x Tr (cid:20) F IJ ˜ F IJ (cid:21) + (cid:90) d x Tr [2 F I D I Σ] , (A.1)with I, J = 1 , , , F µν = ∂ µ W ν − ∂ ν W µ + ig [ W µ , W ν ], D µ Σ = ∂ µ + ig [ W µ , Σ], and Tr[ T a T b ] = δ ab /
2. The energy bound is saturated when the BPS equations F IJ = ˜ F IJ , F I = D I Σ, D Σ = 0 and Gauss’s law D I F I = i [Σ , D Σ] = 0 are satisfied. The self-dual equations F IJ = ˜ F IJ are identical to those for static instantons. A famous one-instanton solution insingular gauge is given by W I = − g ¯ η IJ x J ρ x ( x + ρ ) , where ¯ η IJ is the anti self-dual ’t Hooft tensor and ρ corresponds to size modulus. Plugging this into the first term in the last line of Eq. (A.1),we have Q i = g (cid:82) d x Tr (cid:104) F IJ ˜ F IJ (cid:105) = 8 π (the gauge coupling has mass dimension − / F I = D I Σ and D Σ = 0 are solved by W = − Σ which weassume time independent functions. Finally, we determine Σ by solving the Gauss’s law D I D I Σ = 0. A solution is known to be Σ = v x x + ρ σ . Here, v is vacuum expectation value atspatial infinity. When v (cid:54) = 0, the gauge symmetry is spontaneously broken to U (1) subgroup.This gives rise to a non-zero electric charge which stabilizes the size of instanton by therelation Q e = (cid:82) d x ∂ I Tr (cid:2) F I Σ v (cid:3) = (cid:82) d x Tr (cid:2) F I D I Σ v (cid:3) = 4 π ρ v. In summary, the one44yonic instanton with the electric charge Q e has the BPS mass M d-inst = Q i g + Q e v. (A.2)Note that the spatial gauge field configurations W I are independent of Q e and the BPS massis summation of the topological term 8 π /g and electric contributions Q e v by W and Σ. A.2 Dyons
Let us next recall the BPS dyons in 1 + 3 dimensional N = 2 supersymmetric SU (2) Yang-Mills theory. The Bogomol’nyi completion for the Hamiltonian is performed in the followingway M dyon = (cid:90) d x Tr (cid:20) E i + B i + |D Σ | + |D i Σ | + g (cid:2) Σ , Σ † (cid:3) (cid:21) = (cid:90) d x Tr (cid:20) ( E i − D i Σ sin α ) + ( B i − D i Σ cos α ) + ( D Σ) +2 B i D i Σ cos α + 2 E i D i Σ sin α (cid:21) ≥ (cid:18)(cid:90) d x Tr [2 B i D i Σ] (cid:19) cos α + (cid:18)(cid:90) d x Tr [2 E i D i Σ] (cid:19) sin α, (A.3)with B i = (cid:15) ijk F jk , E i = F i , i, j, k = 1 , , α being an arbitrary constant. We setΣ = Σ † for the second equality. With the Bianchi identity D i B i = 0 and Guass’s low D i E i = ig [Σ , D Σ] = 0, we have Q m = v (cid:82) d x ∂ i Tr [ B i Σ] = v (cid:82) d x Tr [ B i D i Σ] and Q e = v (cid:82) d x ∂ i Tr [ E i Σ] = v (cid:82) d x Tr [ E i D i Σ], where v stands for vacuum expectation value ofΣ at spatial infinity. Plugging these into Eq. (A.3), the energy bound reduces to M dyon ≥ v ( Q m cos α + Q e sin α ). Although α is arbitrary, the most stringent bound is obtained whentan α = Q e /Q m . Thus BPS mass formula for the BPS dyon becomes M dyon = v (cid:112) Q + Q . (A.4)Note that the BPS equation B i = Q m √ Q + Q D i Σ obviously depends on the electric charge Q e ,so that the spatial gauge field configurations (the magnetic fields) change from those for themagnetic monopole without electric charge. Furthermore, the BPS mass formula is not meresuperposition of the topological charge and the electric charge unlike the case of the dyonicinstants. 45 .3 Q-lumps There are low dimensional analogue of the dyonic instantons and dyons. They are Q-lumpsin 1 + 2 dimensions and Q-kinks in 1 + 1 dimensions [8]. One of the simplest model for theQ-lumps is 1 + 2 dimensional massive non-linear sigma model whose target space is C P manifold. Let φ be an inhomogeneous complex coordinate of C P , then the Lagrangian isgiven by L = | ∂ µ φ | − m | φ | (1+ | φ | ) . The Hamiltonian can be cast into the following perfect square form M Q-lump = (cid:90) d x | φ | ) (cid:104) | ˙ φ | + | ∂ i φ | + m | φ | (cid:105) = (cid:90) d x | ¯ ∂φ | + | ˙ φ ∓ imφ | + i ( ∂ φ∂ φ ∗ − ∂ φ∂ φ ∗ ) ∓ im (cid:16) ˙ φφ ∗ − φ ˙ φ ∗ (cid:17) (1 + | φ | ) ≥ (cid:90) d x i ( ∂ φ∂ φ ∗ − ∂ φ∂ φ ∗ )(1 + | φ | ) ∓ m (cid:90) d x i (cid:16) ˙ φφ ∗ − φ ˙ φ ∗ (cid:17) (1 + | φ | ) , (A.5)with ¯ ∂ = ∂ + i∂ . The BPS energy bound is saturated for the solutions of the BPS equations¯ ∂φ = 0 and ˙ φ = ∓ imφ . Generic solution is φ = e ∓ imt f ( z ) for an arbitrary rational function f ( z ) of the complex coordinate z = x + iy . The first term in the last line of Eq. (A.5)is topological term Q L = (cid:82) d x i ( ∂ φ∂ φ ∗ − ∂ φ∂ φ ∗ )(1+ | φ | ) = 2 πk with k being positive integer. Thesecond term is nothing but the Noether charge for the U (1) global transformation φ → e iα φ , Q N = (cid:82) d x i ( ˙ φφ ∗ − φ ˙ φ ∗ ) (1+ | φ | ) . Therefore, the BPS mass for the Q-lump is similar to the one for thedyonic instantons M Q-lump = Q L + m | Q N | . (A.6)Note that one of the BPS equation ¯ ∂φ = 0 is same as that for the static BPS lumps for themassless case with m = 0. Therefore, the energy density contributed from spatial derivativesare unaffected. Reflecting this fact, the BPS mass is mere summation of the topological massand Noether charge, which is parallel to the BPS dyonic instantons. A.4 Q-kinks
The last example is BPS Q-kinks [9, 10]. The simplest model is again the massive C P non-linear sigma model in 1 + 1 dimensions. The Bogomol’nyi completion of Hamiltonianreads M Q-kink = (cid:90) dx | φ | ) (cid:20) | ˙ φ ∓ im (sin α ) φ | + | φ (cid:48) − m (cos α ) φ | The Noether charge Q N is finite only for k ≥ im sin α ( ˙ φφ ∗ − ˙ φ ∗ φ ) + m cos α ( φ (cid:48) φ ∗ + φ ∗(cid:48) φ ) (cid:21) ≥ m cos α (cid:90) dx φ (cid:48) φ ∗ + φ ∗(cid:48) φ (1 + | φ | ) ∓ m sin α (cid:90) dx i ( ˙ φφ ∗ − ˙ φ ∗ φ )(1 + | φ | ) , (A.7)where α is an arbitrary constant, and dot and prime stand for derivatives by t and x ,respectively. Inequality is saturated by solutions of the BPS equations φ (cid:48) = m (cos α ) φ and ˙ φ = ± im (sin α ) φ . The first integral in the second line of Eq. 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