Non-perturbative methods in non-linear field theories and their supersymmetric extensions

Departamento de F´ısica de Part´ıculas NON-PERTURBATIVE METHODS IN NON-LINEAR FIELD THEORIES AND THEIR SUPERSYMMETRIC EXTENSIONS Jos´e Manuel Fern´andez Queiruga Santiago de Compostela, Xu˜no de 2013. UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Departamento de F´ısica de Part´ıculas NON-PERTURBATIVE METHODS IN NON-LINEAR FIELD THEORIES AND THEIR SUPERSYMMETRIC EXTENSIONS Jos´e Manuel Fern´andez Queiruga Santiago de Compostela, Xu˜no de 2013. UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Departamento de F´ısica de Part´ıculas NON-PERTURBATIVE METHODS IN NON-LINEAR FIELD THEORIES AND THEIR SUPERSYMMETRIC EXTENSIONS Tese presentada para optar ao grao de Doutor en F´ısica por: Jos´e Manuel Fern´andez Queiruga Compostela, Xu˜no de 2013 UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Departamento de F´ısica de Part´ıculas Joaquin S´anchez Guill´en, Catedr´atico de F´ısica Te´orica e Christoph Adam, Profesor Contratado Doutor da Universidade de Santiago de Compostela, CERTIFICAN: que a memoria titulada Non-perturbative methods in non-linear field theories and their supersymmetric extensions foi realizada, baixo a nosa direcci´on, por Jos´e Manuel Fern´andez Queiruga, no departamento de F´ısica de Part´ıculas desta Universidade e constit´ue o traballo de Tese que presenta para optar ao grao de Doutor en F´ısica. Asinado: Joaqu´ın S´anchez Guill´en Christoph Adam Compostela, Xu˜no de 2013. Chapter 1 Introduction 1.1 Motivation Despite its non-renormalizability, field theories with kinetic terms with powers higher than two (usually called K field theories) arise naturally in many areas in theoretical physics as effective field theories. Moreover, if we focus on a specific non-linear phenomenon of special importance, the topological solitons, and we want to ensure its stability in higher dimensions, we have two natural ways. One possibility consists of the inclusion of gauge fields. Known examples are the Abelian Higgs or the Chern-Simons Higgs models in 2 + 1 dimensions, the BPS monopole in 3 + 1... These models have a rich topological structure which can be exploited via their supersymmetric extensions (remember the relation between topological charges and central extensions of the supersymmetric algebra). The other possibility consists of the addition of higher derivative terms, leading us again to K field theories (therefore, it is natural to think on supersymmetric extensions of these models because of the intimate relation between supersymmetry and topology). The study of K field theories is interesting in itself (from a formal point of view), but, in addition, it has multiple applications. The canonical example of a K field theory is the Skyrme model (SkM). Other K field models have direct applications in cosmology, for example, they are used to describe phenomena like K-inflation [2] or K-essence [3]. 15 CHAPTER 1. INTRODUCTION 16 Moreover, these theories include new features studied in for example [4], [5], [7], [33] and [51]. Of course, in the context of cosmology (in the inflationary epoch) the supersymmetric extension of such theories becomes relevant and arises naturally. Relevant phenomena associated to the supersymmetric versions of K field theories (Galileons, ghost condensates, DBI inflation) have been studied in [135] and in lower dimensions in [132]. More applications of these models can be found in [31], [60] and [59]. The existence of topological defects with compact support (compactons), a usual feature of this kind of theories, have been discovered in [36]. In general, topological defects resulting from K field theories are quite different from the corresponding ones of the standard theories [5], [74] and [12]. However, under certain conditions both defects can share the same energy density and profile (Doppelg¨anger effect [51]), in this case we say that the theories are twin-like. This feature makes them interesting in a wide range of applications of K field theories. In the framework of K field theories, the SkM is singled out especially due to its rich structure and applications. The most popular application of the SkM is found in strong interaction and nuclear physics ([102], [103], [150], [106]), for which it was formulated. In this field, the SkM is interpreted as a low energy effective model for QCD when the number of colors becomes large, [109] and [108]. Its supersymmetric extensions in 3 + 1 dimensions have been studied in [16] and [15] (in its CP1restriction). On the other hand, the baby Skyrme model (bSkM) is a non-linear theory with topological solitons in 2 + 1 dimensions and S2as target space [156], [81],and [82]. Topologically speaking, this model is similar to the SkM (for example, solitons in this model are labeled by a winding number), moreover, the fact that it is almost impossible to obtain analytic solutions directly from SkM justifies the study of its restrictions to simpler models, in our case the CHAPTER 1. INTRODUCTION 17 bSkM. This model is interesting in itself and has its specific applications for example to Hall ferromagnets [85] or spin textures [86] and [87]. Moreover, the analysis of supersymmetric extensions of the bSkM can be useful as a model for the study of general properties of supersymmetric topological solitons, an issue widely discussed throughout this thesis. 1.2 Content of this thesis It this Ph.D. thesis, supersymmetric extensions of non-linear field theories are investigated, in particular, supersymmetric extensions of the so-called K field theories. Moreover, features of the specific solutions of this models inherited from supersymmetry are analyzed. For example, BPS solutions, energy bounds or central charges of the SUSY models are directly related to the topological charges. We will see that supersymmetry provides enough structure to determine systematically first-order BPS equations. We will pay special attention to the SkM in lower dimensions (the so-called bSkM) which is a paradigmatic example of a theory with higher derivatives. We will show how supersymmetry constrains in 2 + 1 dimensions the coexistence of quadratic, quartic and potential terms. It is even more interesting to see what happens if you try to reconcile the bSkM with N= 2 extended supersymmentry. In this case, the quadratic term is absent due to supersymmetry, but we can add a potential which depends on the metric of the target manifold. It will be shown how to obtain systematically the BPS equations for gauged and ungauged models, from the corresponding SUSY transformations. These results are interesting by themselves, but moreover, taking into account the dimensional reduction from N= 1 in d= 3 + 1 to N= 2 in d= 2 + 1, we can extend our lower dimension results with extended supersymmetry to three dimensional space with ordinary supersymmetry. In a slightly different way of investigation with K field theories we will find the conditions that ensure the existence of twin models, even up to equivalence between fluctuation spectra. These results provide a kind of dictionary CHAPTER 1. INTRODUCTION 18 of a correspondence between K and standard theories. This correspondence allows us to investigate, in a specific range, more complex theories (in this case, K field theories) in terms of standard theories, which are generally simpler. This construction is then extended to supersymmetric models. Finally, symmetries and solutions of the BPS Skyrme model are analyzed. The BPS Skyrme model is a Skyrme type model in 3 + 1 dimensions which includes a sextic term in derivatives. The BPS bound for the energy as well as different solutions preserving symmetries of subgroups of the group of Volume Preserving Diffeomorphisms are calculated. 1.3 Structure of this thesis This Ph.D. thesis is organized as follows: •Chapters 2 to 4 are dedicated to present general features of supersymmetry which will be necessary in the following chapters. In chapter 5 a brief presentation of classical results about complex geometry and its relation with supersymmetry is given. •Chapter 6 includes a basic introduction to the SkM and some relevant properties. •In chapter 7 a first example of a supersymmetric extension of K field theories is presented. We will see in this section why supersymmetric extensions of K field theories are not at all trivial. •In chapter 8 we present a supersymmetric extension of K field theories of the form L=PiαiXi+V(φ) where X=∂µφ∂µφand V(φ) is a potential. General properties and solutions are analyzed. •In chapter 9 domain wall solutions and BPS bounds of K field models of the form L=PiαiXi+V(φ) are calculated. Concretly, we demonstrate that all the domain wall solutions which exist for this family of theories are BPS solutions and that the corresponding BPS energy reappears as a central charge in the SUSY algebra. CHAPTER 1. INTRODUCTION 19 •In chapter 10 algebraic conditions which imply the existence of the so-called twin-like models are found. •In chapter 11 the algebraic conditions which imply the existence of the twin models are generalized to imply the equivalence of the linear fluctuation spectra between the corresponding solutions. •In chapter 12 an explicit N= 1 supersymmetric extension of the baby Skyrme model is presented, and different consequences of the supersymmetrization are analyzed. •In chapter 13, in a first step an N= 1 SUSY extension of a the gauge bSkM is presented. Then we analyze a N= 2 SUSY extensions of both gauged and ungauged bSkM. Moreover, the relation between Bogomol’nyi equations and extended supersymmetry is studied more generally. A general scheme that generates Bogomolny equations in 2 + 1 dimensions is found both for general gauged and ungauged theories with N= 2 supersymmetry. •Chapter 14 is devoted to the study of the BPS Skyrme model (A Skyrme type model consisting of a sextic term in derivatives in 3+1 dimensions). The symmetries of Volume Preserving Diffeomorphisms symmetry are used to calculated solutions. •Chapter 15 contains a brief summary of the results obtained along this thesis. •Finally, chapter 16 contains the main conclusions of this work. Chapter 2 SUSY N=1 d=1+1 and d=2+1 For this chapter we will follow the conventions of [18]. N= 1 supersymmetry in 2+1-dimensional (and also in 1+1-dimensional) space is specially simple, because in this case the Lorentz group is SL(2,R) (instead of SL(2,C)) and the fundamental representation acts on real (Majorana) 2-component spinors. We can lower or rise spinor indices with the totally antisymmetric symbol, Cαβ =−iαβ, with 12 = 1, i.e.: ψα=ψβCβα , ψα=Cαβψβ(2.1) where the Majorana spinor ψα= (ψ+, ψ−). To denote a general coordinate on superspace we use the short notation z= (xµ, θα), where the first components correspond to usual space-time coordinates and the second ones to the anticommutative part of the superspace: {θα, θβ}= 0 →(θi)2= 0 (2.2) 2.1 Berezin integration At the end of the day we will need to integrate over anticommuting variables (Grassmann variables) in order to obtain the supersymmetric invariant actions. In this section we introduce the concept of integration over anticommuting objects (Berezin integration). The main two ingredients are 21 CHAPTER 2. SUSY N=1 D=1+1 AND D=2+1 22 the linearity and the invariance under translations in the Grassman variable. Suppose that our superspace has only one anticommuting variable, then, the dimension of the Grassmann part of the space is exactly one. If we are working in 3-dimensional Minkowski space we denote the corresponding superspace as R3|1. The most general superfunction that we can construct is: Φ(θ) = a+θb (2.3) where a:M3→N, and bis an anticommuting field, being M3the 3-dimensional Minkowski space and Nwhatever manifold. Imposing translation invariance we have ZdθΦ(θ) = ZdθΦ(θ+η) (2.4) or equivalently: Zdθbη = 0 (2.5) by linearity: Zdθ1 = 0 (2.6) Now integrating again Φ(θ): ZdθΦ(θ) = bZdθθ (2.7) and what we need now is to fix the normalization of the integral, for example we can take: Zdθθ = 1 (2.8) finally the result for the total inegral is: ZdθΦ(θ) = b(2.9) Note that: CHAPTER 2. SUSY N=1 D=1+1 AND D=2+1 23 Zdθ ≡d dθ (2.10) Now it is straightforward to generalize this integration to Grassmann algebras of arbitrary dimension. Let {θ1, ..., θN}be anticommuting variables: {θα, θβ}= 0 (2.11) From this we can construct a Grassmann algebra of dimension 2N. A generic basis has the following form: f(θ1, ..., θN) = f0+Xfiθi+Xfijθiθj+... +f1,2,..,N θ1θ2...θN(2.12) We must be careful with the order of θ0sin the integration, because a minus sign appears if the integration variable appears in an odd position w.r.t. the number of anticommuting variables, i.e.: Zdθiθ1...θi...θn= (−1)i+1θ1...ˆ θi...θn(2.13) (in this case the superindex labelled the position of the variable in the product). One interesting and fundamental feature of the Berezin integration is that after integration over all Grassmann space only the component with the highest order in θ0ssurvives, for the previous expression: ZdθN...dθ1(f0+Xfiθi+Xfijθiθj+...+f1,2,..,N θ1θ2...θN) = f1,2,..,N (2.14) 2.2 Superfields In order to construct the correct algebra we need to grade the Poincar`e algebra by introducing the generators of supersymmetry (Qα). The commutation relations involving translations and Qαare (assuming no central extension): [Pµν, Pρσ] = 0 {Qµ, Qν}= 2Pµν (2.15) [Qµ, Pνρ] = 0 CHAPTER 2. SUSY N=1 D=1+1 AND D=2+1 24 where a single index is interpreted as a spinor index and a double index means: Aαβ = (γµ)αβAµ. The algebra (2.15) can be realized on superfields (functions depending on both space-time coordinates and Grassmann coordinates) in terms of: Pµν =i∂µν , Qµ=i(∂µ−iθν∂µν) (2.16) If we have N= 1 supersymmetry in either 1 + 1 or 2 + 1 dimensions the supersymmetric generators are two-component spinors. This implies that our superspace will be Rd|2, i.e. the most general superfield we can costruct in these dimensions is: Φ(x, θ) = φ(x) + θαψα(x)−θ2F(x) (2.17) (with θ2=iθ+θ−). φ(x) is a real scalar field, ψα(x) a real two-component Majorana spinor and F(x) a real auxiliary field. As we will see, usually Fis non-dynamical and we can eliminate it using its (usually algebraic) equations of motion, but it is necessary to have it in the superfield formulations to compensate the bosonic and fermionic degrees of freedom. The supersymmetric derivative is defined to be: Dα=∂α+iθν∂µν.(2.18) From a general supersymmetric transformation it is easy to obtain the corresponding transformations on the components: δΦ(x, θ) = iαQαΦ(x, θ) = −α(∂α−iθβ∂αβ)Φ(x, θ) (2.19) or δΦ(x, θ) = δφ(x) + θαδψα(x)−θ2δF(x).(2.20) Now equating powers of θ: CHAPTER 2. SUSY N=1 D=1+1 AND D=2+1 31 but (Q±−Q−)2≥0, so P++P−≥T. If we think in a particle at rest of mass Mthen P+=P−=M, then eqns. (2.65) implies: M≥1 2|T|(2.66) And now we can ask: When is (2.66) saturated? To answer this question we can analyze first the energy of the model: E=1 2Z[(∂φ ∂t )2+ (∂φ ∂x)2+V(φ)2]dx = (2.67) =1 2Z[(∂φ ∂t )2+ (∂φ ∂x ∓V(φ))2]dx ±ZV(φ)∂φ ∂xdx ≥(2.68) ≥ |ZV(φ)∂φ ∂xdx|(2.69) the inequality for Eturns into an equality iff: ∂φ ∂t = 0 ,∂φ ∂x =±V(φ) (2.70) Now if we go back to supersymmetry, the condition for a state |αito saturate (2.66) is (Q++Q−)|αi= 0 or (Q+−Q−)|αi= 0. But this condition is automatically satisfied (taking into account (2.54)) if: ∂φ ∂t = 0 ,∂φ ∂x =±V(φ) (2.71) the same condition we obtained before! Generalizing this result we can conclude that supersymmetry provides a systematic way to obtain Bogomol’nyi solutions. The strategy in principle seems to be simple: 1. Supersymmetric extension of the corresponding bosonic model. 2. Calculation of supercharges. 3. Try to find a solution which annihilates certain combination of supercharges. Chapter 3 SUSY N=2 d=1+1 and d=2+1 3.1 Introduction In this section we will study extended supersymmetry in 2−dimensional space-time following the conventions of [174]. Of course it is interesting by itself, but, moreover, it constitutes the dimensional reduction from N= 1, d= 3+ 1 to N= 2, d= 1+ 1 and therefore we can say that we can translate the results in this dimension to 3 + 1 space-time with one supersymmetry. In this chapter we present N= 2 SUSY in 1 + 1 dimensions, but, due tot he similarity of the spinor representation, it is formally equivalent to N= 2 in 2 + 1 dimensions. In chapter 13 we will fix the notation for this dimension. Note that, in this case, the Grassmann space has twice the number of Grassmann variables (4 in this case): θ+, θ−,¯ θ+,¯ θ−(3.1) satisfying the usual anti-commuting algebra. A general superfield is a function defined in superspace: G(x0, x1, θ+, θ−,¯ θ+,¯ θ−) = g0(x0, x1) + θ+g+(x0, x1) + (3.2) +θ−g−(x0, x1) + ¯ θ+¯g+(x0, x1) + ... (3.3) +θ+θ−g+−(x0, x1) + ... (3.4) 33 CHAPTER 3. SUSY N=2 D=1+1 AND D=2+1 34 Note that a general superfield has 16 terms. In analogy with N= 1 supersymmetry we introduce the the supersymmetry generators: Q±=∂ ∂θ±+i¯ θ±∂±(3.5) ¯ Q±=−∂ ∂¯ θ±−iθ±∂±(3.6) Where ∂±are the derivatives in light-cone coordinates: ∂±=1 2∂ ∂x0±∂ ∂x1(3.7) As usual we introduce the set of superderivatives: D±=∂ ∂θ±−i¯ θ±∂±(3.8) ¯ D±=−∂ ∂¯ θ±+iθ±∂±(3.9) satisfying the following anticommutation relations: {Q±,¯ Q±}=−2i∂±(3.10) {D±,¯ D±}= 2i∂±.(3.11) For centrally extended N= 2 superalgebras we have: {QL α, QM β}=−2iδLM ∂αβ +TLM αβ (3.12) (with L, M = 1,2). Due to the dimension of the Grassmann space, it will be a little bit more difficult to construct supersymmetric actions. First of all, we need to constrain our general superfields, so we define a superfield Φ satisfying the following equation, ¯ D±Φ = 0 (3.13) which is called chiral superfield. The complex conjugate of the previous equation is CHAPTER 3. SUSY N=2 D=1+1 AND D=2+1 35 D±¯ Φ = 0 (3.14) and this superfield ¯ Φ is called anti-chiral superfield. We can also define superfields Σ with twisted conditions, for example: ¯ D+Σ = D−Σ = 0 (3.15) called twisted chiral superfield, but we will not use it. 3.2 Supersymmetric actions We will repeat the same formalism as before but taking into account the new Grassmann space. From the form of supercharges it is obvious that any action constructed in terms of general superfields and superderivatives and integrated over all the Grassmann space is supersymmetric, i.e.: S=Zd2xd4θH(Φ,¯ Φ, D±Φ...) = Zd2xdθ+dθ−d¯ θ+¯ θ−H(Φ,¯ Φ, D±Φ...). (3.16) We see that δS =Zd2x∂µFµ(3.17) where δis the supersymmetric transformation δ=+Q−−−Q+−¯+¯ Q−+ ¯−¯ Q+.(3.18) Taking into account that the first part of Q0sis a derivative w.r.t. θ, the contribution of this terms after integration over θ0sgives zero and the second term of Q0sis directly a derivative, this shows (3.17). As we will see later, this kind of integrals (D-terms) give us kinetic terms, so we need another integration in order to generate potentials. Suppose that we only integrate in half the Grassmann space, for example: SP=Zd2xd2θW(Φ) = Zd2xdθ−dθ+W(Φ)|¯ θ±=0 (3.19) CHAPTER 3. SUSY N=2 D=1+1 AND D=2+1 36 where Φ is a chiral superfield. This kind of terms are called F-terms. Let’s check that this action is supersymmetric. First of all, we will restrict to the component +of δ(the calculation is the same for −): δ|+SP=Zd2xdθ−dθ+∂ ∂θ−+i¯ θ−∂−W(Φ) (3.20) The second term vanishes because we put ¯ θ±= 0 and the first vanishes after θ-integration. But if we look now at the term involving ¯+: δ|¯+SP=Zd2xdθ−dθ+−∂ ∂¯ θ−−iθ−∂−W(Φ) (3.21) in principle we can not guarantee that it is a total derivative, but using the following relation ¯ Q±=D±−2iθ±∂±,(3.22) we obtain δ|¯+SP=Zd2xdθ−dθ+¯+¯ D−−2iθ−∂−W(Φ).(3.23) Now it is clear that the second term is a total derivative in xµand the first term is zero because Φ is chiral, ¯ D−W(Φ) = W0(Φ) ¯ D−Φ = 0 (3.24) since ¯ D−Φ = 0. We have seen that, in order to construct susy F-terms (which generate the potentials) we have to use chiral superfields integrating over chiral anti-commuting variables (θ±) or anti-chiral superfields integrating over anti-chiral anti-commuting coordinates (¯ θ±). The way to generate supersymmetric actions is different from the one with N= 1, for example, to generate a standard action we need a D-term of the form: SD=Zd2xd4θ¯ ΦΦ (3.25) being Φ and ¯ Φ chiral and antichiral superfields. After the expansion of the product we need only to integrate in θ0s: CHAPTER 3. SUSY N=2 D=1+1 AND D=2+1 37 SD=Zd2x∂µ¯ φ∂µφ+i¯ ψ−(∂0+∂1)ψ−+i¯ ψ+(∂0−∂1)ψ++¯ FF(3.26) Performing the same calculation for the F-term with superpotential W we get SF=Zd2xW0F+¯ W0¯ F−W00ψ+ψ−−¯ W00 ¯ ψ+¯ ψ−.(3.27) We can add these actions, and after the elimination of the auxiliary field we obtain SD+SF=Zd2x(∂µ¯ φ∂µφ+i¯ ψ−(∂0+∂1)ψ−+ +¯ ψ+(∂0−∂1)ψ+−W00ψ+ψ−−¯ W00 ¯ ψ+¯ ψ−−|W0|2).(3.28) We observe in this actions small but fundamental differences w.r.t the N= 1 action. First of all, the kinetic part has been built in terms of a real combination of complex superfields (this is a K¨ahler potential) and this condition constrains the possible N= 2 supersymmetric actions. The second part is that the superpotential is coming from the F-term which is a sum of holomorphic plus antiholomorphic functions of the superfields, and this fact constrains again the possibilities. It is always possible to reduce an N= 2 model to N= 1 by restriction of the superspace. Obviously the other way is not true in general, but only under certain conditions: if we accomodate Majorana spinors of a N= 1 scalar multiple into complex spinors and there exists a U(1) symmetry under the fermion rotation ψ→eiαψit is possible to accomodate N= 1 supermultiples in N= 2 supermultiplets. 3.3 Gauge invariant N= 2 actions We can think on the lagrangian L=Zd4θ¯ ΦΦ.(3.29) CHAPTER 3. SUSY N=2 D=1+1 AND D=2+1 38 If we generalize the usual phase rotation φ→eiαφwhich leads to gauge invariance, in terms of superfields we have the natural generalization Φ → eiAΦ, being Aa chiral superfield, which sends chiral fields to chiral fields. The integrand of (3.29) is not invariant under such a transformation: ¯ ΦΦ →¯ Φe−i¯ A+iAΦ,(3.30) but if we introduce a real superfield Vthat transforms as V→V+i(¯ A−A) (3.31) when Φ→eiAΦ (3.32) then a gauge invariant lagragian under transformations (3.31) and (3.32) can be written as L=Zd4θ¯ ΦeVΦ.(3.33) The real superfield in the Wess-Zumino gauge (equivalent to V3= 0) is expessed in the form: V=θ−¯ θ−(v0−v1) + θ+¯ θ+(v0+v1) + iθ−θ+(¯ θ−¯ λ−+¯ θ+¯ λ+) (3.34) +iθ−θ+(θ−λ−+θ+λ+)−θ−¯ θ+σ−θ+¯ θ−¯σ+θ−θ+¯ θ+¯ θ−D where vµis the gauge field, σis a complex field, λis a Dirac fermions and Dis a real auxiliary field. If we have chiral superfields coupled to vector superfields, the supersymmetric transformations of the chiral fields are modified because of the gauge symmetry, in the present case, the supersymmetric transformation for Φ = (φ, ψ, F) and V= (vµ, λ, σ, D) are: CHAPTER 3. SUSY N=2 D=1+1 AND D=2+1 39 δφ =i¯ε±λ±+iε±¯ λ±(3.35) δψ+=i¯ε−(D0+D1)φ+ε+Fε+¯σφ (3.36) δψ−=−iε +−(D0−D1)φ+ε−F+ ¯ε−σφ (3.37) δF =−iε +−(D0−D1)ψ+−i¯ε−(D0+D1)ψ−+ (3.38) +ε+¯σψ−+ ¯ε−σψ++i(¯ε−¯ λ+−¯ε+¯ λ−)φ δv±=i¯ε±λ±+iε±¯ λ±(3.39) δσ =i¯ε+λ−−iε−¯ λ+(3.40) δD =−¯ε+∂−λ+−¯ε−∂+λ−+ε+∂−¯ λ++ε−∂+¯ λ−(3.41) δλ+=iε+(D+ivµν)+2ε−∂+¯σ(3.42) δλ−=iε−(D−ivµν)+2ε+∂−σ(3.43) with vµν =∂µvν−∂νvµ. This supersymmetric transformation will lead us to BPS equations of the gauged bSkM in chapter 13. Again the N= 2 and gauge invariant lagrangians are constructed like always, for example, if we gauge the lagrangian (3.29) we obtain: Lk,gauged =Zd4θ¯ ΦeVΦ = (3.44) =−Dµ¯ φDµφ+i¯ ψ−(D0+D1)ψ−+i¯ ψ+(D0−D1)ψ++ +D|φ|2+|F|2−|σ|2|φ|2−¯ ψ−σψ+−¯ ψ+¯σψ−−i¯ φλ−ψ++ +i¯ φλ+ψ−+i¯ ψ+¯ λ−φ−i¯ ψ−¯ λ+φ The super Yang-Mills lagragian in 1+1 dimensions is constructed in terms of the superfield strength Σ := ¯ D+D−Vas: LY M =−1 2e2Zd4θ¯ ΣΣ = (3.45) =1 2e2−∂µ¯σ∂µσ+i¯ λ−(∂0+∂1)λ−+i¯ λ+(∂0−∂1)λ++v2 01 +D2 (e2is the gauge coupling). In chapter 13 we will use the generalization of (3.44) with a general K¨ahler potential which will allow us to construct the N= 2 bSkM. Chapter 5 Geometry and Supersymmetry 5.1 Introduction In this chapter we will present briefly the relation between supersymmetry and geometry. We use the results of ´ Alvarez-Gaum´e and Freedman [68] and also [69] and [70], to discuss the relation between the complex structure of the target space manifold for bosonic non-linear σ-model and the number of supersymmetries which this model can allow. We start with 2-dimensional σ-models. 5.2 Relation between complex geometry and SUSY Given a n-dimensional Riemannian manifold with metric gij(Φk) one can define a supersymmetric σ-model with N= 1 supersymmetry. The superfield action is S[Φ] = 1 4iZd2xd2θgij(Φk)¯ DΦiDΦj,(5.1) where Φkis a real scalar superfield, Φk(x, θ) = φk(x) + ¯ θψk(x) + 1 2¯ θθFk(x).(5.2) 47 CHAPTER 5. GEOMETRY AND SUPERSYMMETRY 48 After integration in θand the elimination of the auxiliary field, we obtain, in terms of physical fields, S[φ, ψ] = 1 2Zd2x{gij(φ)∂µφi∂µφj+igij(φ)¯ ψiγµDµψj+1 6Riklj(¯ ψiψl)( ¯ ψkψj)} (5.3) with the covariant derivative Dµψk=∂µψk+ Γk ji∂µφjψk. The action is invariant under the following supersymmetric transformations, δφk= ¯εψk(5.4) δψk=−i/ ∂φkε−Γk ji(¯εψj)ψi(5.5) We want to study the possibility of additional supersymmetric invariances of the action (5.3). First of all, we know that the action is invariant under (5.5) and also under reparametrizations of the target manifold M: φ0k=φ0k(φ) (5.6) ψ0k=∂φ0k ∂φ0jψj(5.7) It can be checked that the most general Ansatz for SUSY transformation rules which is consistent with dimensional arguments, and Lorentz and parity invariance is δφk=fk j¯εψj(5.8) δψk=−ihk j/ ∂ψjε−Sk ji(¯εψj)ψi−(5.9) −Vk ji(¯εγµψj)γµψi−Pk ji(¯εγ5ψj)γ5ψi(5.10) Commutations with diffeomorphisms implies that f,h,Vand Pare tensors. We require that the action (5.3) be stationary under the variations (5.10). Absence of linear term in ψof δS requires the conditions gikfk j=gjkhk i(5.11) ∇kfi j= 0,(5.12) CHAPTER 5. GEOMETRY AND SUPERSYMMETRY 49 then fk jis covariantly constant. Let us suppose now the general rules (5.10) obey the supersymmetric algebra {Qa,¯ Qb}= 2δab / P, (5.13) which implies fi jhj k=δi k,(5.14) and these relations plus the previous one between hand fallow us to write gijfi kfj l=gkl.(5.15) We now assume that there are several supersymmetries with covariantly constant tensors f(a)i j. Then (5.13) implies f(a)f(b)−1+f(b)f(a)−1= 2δab.(5.16) Assuming that one of these transformations is the original (f(0)i j=δi j) and b= 0 we have f(a)i kf(a)k j=−δi j.(5.17) We collect these properties for the tensor f, ∇kfi j= 0 (5.18) gijfi kfj l=gkl (5.19) fi kfk j=−δi j.(5.20) The third one implies that the dimension of Mis even. From the three relations it follows that Mcan be covered smoothly with complex coordinate charts (zα, z¯α) such that transition functions in overlapping coordinate patches are holomorphic. In complex coordinates the line element ds2can be written as: ds2= 2gα¯ βdzαdz¯ β; (5.21) CHAPTER 5. GEOMETRY AND SUPERSYMMETRY 50 the two form F=igα¯ βdzα∧dz¯ β(5.22) is closed which implies locally gα¯ β=∂2 ∂zα∂z¯ βK(z, ¯z),(5.23) where K(z, ¯z) is the K¨ahler potential. Thus a supersymmetric σ-model on a Riemann manifold Madmits a second supersymmetry if and only if M is K¨ahler. It is possible to extend these results to more dimensions or more supersymmetry. In the following table we summarize some of these results: dimension SUSY manifold d=2 N=1 no restriction on M N=2 M is K¨ahler N=4 M is hyper-K¨ahler d=4 N=1 M is K¨ahler N=2 M is hyper-K¨ahler N=4 No extension exists (bosonic sector with spin 1) . We emphasize that these geometric constraints hold for SUSY extensions of σ-models, the inclusion of gauge fields changes the geometry in general. This relationship between supersymmetry and geometry will be important in Chapter 13. Chapter 6 The Skyrme model Let us analyze first the sine-Gordon model in order to suggest a kind of generalization to the Skyrme model. To do this, we need two components φ0 and φ1which are functions on the (1 + 1)-dimensional space-time with the constraint φ2 0+φ2 1= 1,(6.1) then it is possible to fulfill this constraint by setting φ0= cos α(x, t), φ1= sin α(x, t).(6.2) Now substituting this explicit form for the fields in the original Lagrangian proposed by Skyrme to describe nucleon fields interacting with pseudo scalar meson field, L=1 2X ρ [(∂µφρ)2+1 2k2φ4 ρ] + ¯ ψ[iγµ∂µ+g(φ0+iγ5τ·φ)]ψ(6.3) we obtain the lagrangian of the sine-Gordon model, LSG =1 2[(∂tα)2−(∂xα)2]−k2(1 −cos α) (6.4) with the corresponding Euler-Lagrange equation ∂2 xα−∂2 t−k2sin α= 0 (6.5) 51 CHAPTER 6. THE SKYRME MODEL 52 with solutions of the form (2π-solitons) α(x) = 4 tan−1[exp{±k(x−x0)}] (6.6) with the following profile -4 -2 2 4 x 1 2 3 4 5 6 ΑHxL There are solutions interpolating all different neighboring values of the vacuum. 6.0.1 The topological charge All solutions of the model satisfy the following boundary condition α(x)→0(mod2π)as |x| → ∞.(6.7) In this family of solutions satisfying these boundary conditions we can distinguish between solutions where α(x) takes the zero value at both boudaries x=±∞, on the other hand we have α(−∞) = 0 and α(∞) = 2π. Those solutions are not transformable into each other by any continuous transformation, hence the space of solutions with the boundary condition (6.7) is split into distinct connected components. Since a continuous deformation could be regarded as an evolution of the classical system, we can assign to these solutions a‘characteristic” which does not change its value under time evolution. Now taking into account the existence of the conserved current Jµ=1 2πεµν∂να(6.8) CHAPTER 6. THE SKYRME MODEL 53 such that ∂µJµ= 0 independent of the equations of motion, we can define the topological charge Q=Z∞ −∞ dxJ0=1 2πZ∞ −∞ dx∂α ∂x =1 2π]α(+∞)−α(−∞)] (6.9) which is exactly the quantity (winding number, Chern-Pontryagin number...) which labels the different sectors where solutions live. If we write our fields in the form φ=φ0+iφ1or φ(x) = exp(iα(x)), the constraint (6.1) can be written as φ(x)φ(x)?= 1 (6.10) and the integrand of the second term in (6.9) as ∂α ∂x =−iφ?∂φ ∂x (6.11) such that φ?=φ−1. We can try to construct an analog of this kind of models in (3+1) dimensions. The boundary condition at infinity compactifies R3onto the three sphere S3, which is topologycally isomorphic to SU(2). Hence we can use a quarternionic representation of SU(2). Our fields now are: U(x, t) = φ0(x, t) + iτ ·φ(x, t) (6.12) where x∈R3and τare the Pauli matrices. The constraint (6.1) in terms of Uis rewritten as U·U†=1.(6.13) To ensure the compactification R2∪∞ wS3the field satisfies U(x)→1as |x|→∞.(6.14) We can write now a straightforward generalization on the condition (6.11) in terms of the quaternionic field Rµ=U−1∂µU=iτaRa µ,(6.15) CHAPTER 6. THE SKYRME MODEL 54 this field must satisfy the Maurer-Cartan structural equations of the zero curvature conditions, ∂µRν−∂νRµ+ [Rµ, Rν] = 0.(6.16) These conditions are necessary and sufficient conditions for the reconstruction of the field Uin terms of BRµ. At this point we can write down the lagrangian proposed by Skyrme as a low energy effective theory for QCD, becoming exact as the number of quark colors becomes large, L=ε 4π2{κ2Ra µRa µ−1 2[(Ra µRa µ)2−(Ra µRa ν)2]},(6.17) or in terms of the quaternionic field, L=F2 π 16 Tr(∂µU∂µU†) + 1 32e2Tr[∂µUU†, ∂νUU†]2.(6.18) This is the lagrangian corresponding to the Skyrme model (with λ= 2/Fπand ε= (√2e)−1). We will analyze different features of this model in this chapter. After scaling the parameters the Euler-Lagrange equations of motion can be written as ∂µRµ+1 4[Rν,[Rν, Rµ]]= 0.(6.19) If one restricts to static fields, then the Skyrme energy functional derived from the Lagrangian is E=1 12π2Zd3x{−1 2Tr(/RiRi)−1 16Tr([Ri, Rj][Ri, Rj])}.(6.20) Now the boundary condition at infinity implies a one-point compactification of space, so that U:S37→ S3, where the domain S3is to be identified with R3∪ {∞}.The homotopy group π3(S3) is Z, which implies that maps between 3-spheres fall into homotopy classes indexed by an integer. This integer Bis also the degree of the map Uand has the explicit representation B=−1 24π2Zd3xijkTr(RiRjRk).(6.21) CHAPTER 6. THE SKYRME MODEL 55 As Bis a topological invariant, it is conserved under continuous deformations of the field, including time evolution. It is this conserved topological charge which Skyrme identified with baryon number. But the existence of this invariant is not enough to ensure the existence of stable topological solitons. Note that the energy decomposes into two contributions, quadratic and quartic in derivatives E=E2+E4. Under a rescaling of the spatial coordinates x→µx. the energy becomes E(µ) = 1 µE2+µE4.(6.22) We see that these two terms scale in an opposite way, leading to a minimal value of E(µ) for a finite µ6= 0. From this discussion is now clear why the σ-model (consisting only on the quadratic term) does not support stable solitons. Any term with 4 or more derivatives can cure this problem but the Skyrme term is the unique expression of minimal degree (4 in this dimension) which is Lorentz invariant and for which the resulting equations of motion are second order in the time derivative. However, notice that the antisymmetric sextic contribution, of topological origin, also satisfies this consistency requirement, this fact will be important in our work. 6.1 The Baby Skyrme model Faddeev suggested [173] that stable closed strings may exist as topological solitons of an O(3) σ-model modified by terms with higher derivatives. The following model realizes this idea. If we restrict the Skyrme field to the 2- sphere, S2, the usual SU(2) target space, the field corresponding to the SkM is a real three-component vector of unit length, φa, with φaφa= 1. This field is related to the original Skyrme field via U=iφaτa(where τaare the Pauli matrices). Subtsituting this into the Skyrme lagrangian results in L=∂µφa∂µφa−1 2(εabc∂µφb∂νφc)2.(6.23) The above lagrangial is known as Skyrme-Faddev lagrangian. We can generalize this lagrangian by adding an additional higher derivative term following [169], [168] CHAPTER 6. THE SKYRME MODEL 56 L=∂µφa∂µφa−1 2(εabc∂µφb∂νφc)2+1 2(∂µφa∂µφa)2.(6.24) Now if we go to 2 + 1 dimensions, the basic field of the reduced model maps the three-dimensional Minkowski space M3onto S2, φ:M37→ S2.(6.25) Now, after the addition of a potential term depending on the third component of the field, we obtain the class of baby Skyrme models we shall consider along this thesis, L=λ2 2∂µφa∂µφa−λ4 4(εabc∂µφb∂νφc)2+ (6.26) +˜ λ4 4(∂µφa∂µφa)2−λ0V(φ3) with φaφa= 1. The relevant homotopy group is π2(S2) = Zwhich implies that maps between 2-spheres fall into homotopy classes indexed by an integer (like for the original Skyrme model). The degree of such maps Qcan be written explicitly as: Q=1 4πZdxdyabcφa∂xφb∂yφc(6.27) After the stereographic projection φa= (u+u?,−i(u−u?),(|u|2−1)/(1 + |u|2)) (6.28) the above lagrangian can be written as L= 2λ2 ∂µu∂µu? (1 + |u|2)2+ 2λ4{(∂µu)2(∂νu?)2 (1 + |u|2)4+ (6.29) + (2˜ λ4 λ4−1)(∂µu∂µu?)2 (1 + |u|2)4+V((|u|2−1)/(1 + |u|2))}. We will see that the N= 1 supersymmetrization of this model if possible for any ˜ λ4, because each term can be extended separately. However, N= 2 extended SUSY constrains the parameters of the model enforcing the condition ˜ λ4= 0. CHAPTER 7. FIRST TRY: N=1 SUSY K FIELD MODELS 63 and is, therefore, identical to Eq. (7.18). The Euler–Lagrange equation for the spinor field is ∂µ(f00(X)∂νφ∂νψα∂µφ+f0(X)∂µψα) + V00(φ)ψα= 0.(7.26) We remark that again the auxiliary field Fdoes not couple to either the scalar or the spinor field and may be treated as auxiliary or unphysical in this sense. Finally, let us demonstrate that the fermionic zero mode in a kink background continues to be the derivative of the kink. For a static, one-dimensional scalar field φ(x) the Euler–Lagrange equation (7.25) reads −∂x(f0(X)φx) + V0(φ) = 0 (7.27) or, with the help of Xx=−φxφxx, f00φ2 xφxx −f0φxx +V0= 0.(7.28) on the other hand, the Euler–Lagrange equation (7.26) for a static spinor ψα(x) in a kink background φ(x) reads ∂x(f00φ2 xψα,x −f0ψα,x) + V00ψα= 0 (7.29) and is identically satisfied for a spinor ψα=αφxwhere αis a constant spinor. Indeed, inserting this spinor in the above equation we get α∂x(f00φ2 xφxx −f0φxx +V0) = 0 (7.30) i.e., just the xderivative of the kink equation (7.28). 7.4 Supercurrent and SUSY algebra It is a well-known fact that a standard supersymmetric scalar field theory in 1+1 dimensions has a centrally extended SUSY algebra if it supports topological soliton solutions (kinks) [22], where the central charges are related to the topological charges of the solitons. Here we want to investigate whether this phenomenon continues to hold in the case of the supersymmetric extensions CHAPTER 7. FIRST TRY: N=1 SUSY K FIELD MODELS 64 of K field theories introduced in the last section. The SUSY transformations of the fields are δφ = ¯ψ , δψ =−iγµ∂µφ−F δF =i¯γµ∂µψ , δ ¯ ψ=i¯γµ∂µφ−¯F. (7.31) The supersymmetric K field Lagrangian related to the action (7.24) transforms under the SUSY transformations by the following total derivative δLK,SUSY =i¯∂µ[f0(X)∂νφγµ∂νψ−V0(φ)γµψ]≡∂µJµ 2(7.32) where the following relations are useful for the calculation, ¯ψ =¯ ψ , ¯γµψ=−¯ ψγµ , (7.33) ¯γµγνψ= ¯1 2{γµ, γν}+1 2[γµ, γν]ψ=¯ ψ1 2{γµ, γν}− 1 2[γµ, γν] . (7.34) The part of the SUSY Noether current related directly to the field variations is Jµ 1≡δφ ∂ ∂(∂µφ)+δF ∂ ∂(∂µF)+δψ ∂ ∂(∂µψ)+δ¯ ψ∂ ∂(∂µ¯ ψ)LK,SUSY = ¯f0(X)[∂µFψ −F∂µψ+i∂µφ/∂ψ +i/∂φ∂µψ] + ¯f00(X)1 2(∂µ¯ ψ∂νφ∂νψ+∂ν¯ ψ∂νφ∂µψ)ψ+ ∂µφ[(∂νφ∂νF+1 2∂ν¯ ψ∂νψ)ψ+i∂νφ/∂φ∂νψ−F∂νφ∂νψ] +1 2¯f000(X)∂µφ(∂λφ∂λ¯ ψ∂νφ∂νψ)ψ(7.35) (here in the first line it is understood that the field insertions should be made exactly at the positions where the corresponding field derivatives act), and the full SUSY Noether current is Jµ SUSY =Jµ 1−Jµ 2≡¯Jµ≡¯αJµ α(7.36) where we introduced some notation at the r.h.s. It may be checked by a lengthy but straight forward calculation that this current is conserved onshell. CHAPTER 7. FIRST TRY: N=1 SUSY K FIELD MODELS 65 For an evaluation of the SUSY algebra it is useful to study the simpler case f(X) = X(the model of Section 2.C) first. The current in this case is Jµ α=∂µFψα−F∂µψα+i∂µφ(/∂ψ)α+i(/∂φ∂µψ)α−i∂νφ(γµ∂νψ)α+iV 0(γµψ)α (7.37) and the correct field equal time (anti) commutators are [φ(x),˙ F(y)] = iδ(x−y),[F(x),˙ φ(y)] = iδ(x−y) (7.38) {ψα(x),˙ ¯ ψβ(y)}=iδαβδ(x−y),{˙ ψα(x),¯ ψβ(y)}=−iδαβδ(x−y).(7.39) The bosonic commutators are obvious from the action (7.13), whereas the anticommutators are obvious up to an overall sign. An easy way to check that our sign choice is right is to observe that with this sign choice the correct SUSY transformations of the fields are produced, i.e., [iQ, φn] = δφnφn= (φ, ψ, ¯ ψ, F) (7.40) where Qα=ZdxJ0 α.(7.41) For the SUSY anticommutator {J0 α(x),¯ Qβ}we find after another lengthy calculation {J0 α(x),¯ Qβ}= 2T0ν(¯γν)αβ + 2i(¯γ5)αβV0φ0(7.42) (remember (γµ)α≡¯γµαββ≡γµαββin the barred spinor and spinor metric notations, respectively, where is an arbitrary spinor; further, γ5=γ0γ1). The corresponding energy momentum tensor is Tµν =∂µφ∂νF+∂νφ∂µF+1 2∂µ¯ ψ∂νψ+∂ν¯ ψ∂µψ− gµν ∂λφ∂λF+1 2∂λ¯ ψ∂λψ−1 2V00 ¯ ψψ −V0F.(7.43) It is interesting to contrast this result with the corresponding one for a standard theory like the one in Section 2.B (where the energy-momentum tensor is different, of course), {J0 α(x),¯ Qβ}= 2T0ν(γν)αβ + 2i(γ5)αβP0φ0.(7.44) CHAPTER 7. FIRST TRY: N=1 SUSY K FIELD MODELS 66 The result looks formally almost identical, with the only difference that in the second term at the r.h.s. the prepotential Pappears instead of the potential Vitself. This difference is, however, important. Indeed, in the standard case a further integration Rdx leads to the SUSY algebra with central extension, {Qα,¯ Qβ}= 2Pν(γν)αβ + 2i(γ5)αβ(P(φ+)−P(φ−)) (7.45) where φ±=φ(x=±∞), and Pνis the momentum operator. For a kink, φ+6=φ−, and also P(φ+) and P(φ−) are different, so a central extension appears in the SUSY algebra in a kink background. For the anticommutator (7.42), on the other hand, a further integral leads to {Qα,¯ Qβ}= 2Pν(γν)αβ + 2i(γ5)αβ(V(φ+)−V(φ−)) = 2Pν(γν)αβ (7.46) because φ±must take vacuum values, and V(φ) is zero by definition for a vacuum value. Therefore, for the theory of Section 2.C there is no central extension in the SUSY algebra in a kink background. It remains to calculate the SUSY algebra for the supersymmetric K field theories of Section 3.B. For this purpose it is useful to introduce the canonical momenta from the variations of the Lagrangian ∂LK,SUSY ∂(∂µφ)=f000(X)∂µφ∂λφ∂λ¯ ψ∂νφ∂νψ+ +f00(X)∂µ¯ ψ∂νφ∂νψ+∂νφ∂ν¯ ψ∂µψ+∂µφ∂νφ∂νF+ +f0(X)∂µF(7.47) ∂LK,SUSY ∂(∂µF)=f0(X)∂µφ(7.48) ∂LK,SUSY ∂(∂µ¯ ψα)=1 2(f00(X)∂µφ∂νφ∂νψα+f0(X)∂µψα) (7.49) ∂LK,SUSY ∂(∂µψα)=1 2f00(X)∂µφ∂νφ∂ν¯ ψα+f0(X)∂µ¯ ψα.(7.50) For the bosonic fields we have directly Πφ≡∂LK,SUSY ∂(∂0φ),ΠF≡∂LK,SUSY ∂(∂0F),(7.51) CHAPTER 7. FIRST TRY: N=1 SUSY K FIELD MODELS 67 whereas for the fermi fields we have to take into account that ψand ¯ ψare not independent, i.e, ¯α(Πψ)α≡(Π¯ ψ)αα= ¯α ∂LK,SUSY ∂0¯ ψα +∂LK,SUSY ∂0ψα α(7.52) for an arbitrary spinor . It follows that e.g. (Πψ)α=f00(X)˙ φ∂νφ∂ν¯ ψα+f0(X)˙ ¯ ψα,(7.53) and the SUSY charge density is J0 α=ψαΠφ+i(/∂ψ)αΠf+i(/∂φΠψ)α−F(Πψ)α−i∂νφ(γ0∂νψ)α−iV 0(φ)(γ0ψ)α. (7.54) Finally, the equal time (anti) commutators are [φ(x),Πφ(y)] = iδ(x−y),[F(x),ΠF(y)] = iδ(x−y) (7.55) {ψα(x),(Πψ)β(y)}=iδαβδ(x−y),{¯ ψα(x),(Π¯ ψ)β(y)}=−iδαβδ(x−y) (7.56) (the anticommutators for ψand ¯ ψare of course not independent). For the SUSY charge and charge density algebra we find again Eq. (7.42). The SUSY algebra in a kink background, therefore, again contains no central extension. The energy-momentum tensor is, of course, different from the one in Eq. (7.43). Its explicit expression is rather long and not particularly illuminating, therefore we do not display it here. 7.5 Problems of the extension Remember that, in fact, all these constructions are explicitly supersymmetric, because, first of all, we are promoting bosonic fields to superfields. We leave space-time derivatives unchanged and do not promote them to superderivates as is usually done. Space-time derivatives, however, are anticommutators of superderivatives and, therefore, map superfields into superfields, {Dα, Dβ}= 2i∂αβ.(7.57) Then our scheme for this supersymmetrization is the following, CHAPTER 7. FIRST TRY: N=1 SUSY K FIELD MODELS 68 Lbos =L(φ, ∂µφ, ...)−→ LSUSY =L(Φ, ∂µΦ, ...).(7.58) Although the bosonic sector in the SUSY version is not the same we had in the original bosonic model, the variation of the action w.r.t the auxiliary field Fgenerates the e.o.m. of the bosonic field, remember the trivial example Lbos =1 2∂µφ∂µφ−1 2(P0(φ))2(7.59) with e.o.m. ∂µ∂µφ+P0(φ)P00(φ) = 0 (7.60) and the corresponding extension LSUSY =∂µφ∂µF+1 2∂µψα∂µψα−1 2V00(φ)ψαψα−V0(φ)F. (7.61) Variation w.r.t. Fimplies: ∂µ∂µφ+V0(φ) = 0.(7.62) Now, making the right choice for the superpotential, i.e., such that V0(φ) = P0(φ)P00(φ), we have the same equation. Up to here nothing new. Another interesting observation is that this scheme is absolutely general (at least for scalar field theories), we always obtain the equation of motion of the original model from the variation of the susy model w.r.t. the auxiliary field. Then, where are the problems? If we have a look at (7.61), we see that Fbecomes dynamical, and if we change the field like φ=A+B(7.63) F=A−B, (7.64) we can rewrite the lagrangian as LSUSY,ghost =∂µA∂µA−∂µB∂µB+1 2∂µψα∂µψα−(7.65) −1 2V00(φ)ψαψα−V0(φ)(A−B), CHAPTER 7. FIRST TRY: N=1 SUSY K FIELD MODELS 69 and the field Bconstitutes a ghost which allows to have infinitely negative energy. Still, the study of these models has been instructive in understanding the new structures and difficulties in SUSY extensions of K field theories, because we were able to go rather far in the explicit calculation and even determine the complete SUSY algebra with its central extensions. In the following chapters we propose different extensions to avoid the problem mentioned above with the ghost field. Chapter 8 N=1 SUSY extension of K field theories After having displayed the inherent complications related with the supersymmetrization of K field theories, we propose in this chapter a possible SUSY extension of these models with a detailed analysis of solitonic solutions and exact calulations of energies. This chapter consists of a paper published in [95]. Supersymmetric K field theories and defect structures C. Adam 1, J.M. Queiruga 1, J. Sanchez-Guillen 1, A. Wereszczynski 2 1Departamento de F´ısica de Part´ıculas, Universidad de Santiago de Compostela and Instituto Galego de F´ısica de Altas Enerxias (IGFAE) E-15782 Santiago de Compostela, Spain 2Institute of Physics, Jagiellonian University, Reymonta 4, Krak´ow, Poland Abstract: We construct supersymmetric K field theories (i.e., theories with a non-standard kinetic term) in 1+1 and 2+1 dimensions such that the bosonic sector just consists of a nonstandard kinetic term plus a potential. Further, we study the possibility of topological defect formation in these supersymmetric models. Finally, we consider more general supersymmetric K field theories where, again, topological defects exist in some cases. 71 CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 72 8.1 Introduction Topological defects are of fundamental importance in a wide range of physical theories. Both in particle theory and in condensed matter physics, topological defects may exist as stable, particle-like excitations above the ground state of a theory. In some cases, states containing topological defects are even energetically preferred over the homogeneous state, such that the true ground state of the system is a condensate or lattice of topological defects. Another field where topological defects are deemed relevant is cosmology. On the one hand, topological defects are crucial in inflationary scenarios, where they may form domain walls separating different vacua of some primordial fields in the symmetry-breaking phase. As a consequence, it is widely believed that a pattern of these topological defects might be responsible for the structure formation in the very early universe, see e.g. [23], [24], [25]. On the other hand, topological defects also play an important role in the so-called brane-world scenario, where it is assumed that the visible universe is a 3+1 dimensional subspace “brane”) in a higher-dimensional bulk universe. The brane may be either strictly 3+1 dimensional “thin brane”) or have a small but nonzero extension also in the additional dimensions “thick brane”). In the latter, thick brane case, these branes are normally topological defects in the higher-dimensional bulk space [26], [27], [28], [29], [30] [9]. In all these cosmological applications, the relevant topological defects are usually solutions of some effective field theories of one or several scalar fields. The scalar field theories may either consist of the standard kinetic term of the scalar fields plus a potential, in which case the specific properties of the defects are related to the properties of the potential. Or one may relax the condition on the kinetic term and allow for more general field theories with a Lagrangian depending both on the fields and their first derivatives. These so-called K field theories have been increasing in importance during the last years, beginning with the observation about a decade ago that they might be relevant for the solution of some problems in cosmology, like K-inflation [2] and K- essence [3]. K field theories have found their applications in cosmology [4], [5], [7], [33], [51], and they introduce some qualitatively new phenomena, like the formation of solitons with compact support, so-called compactons [35] - CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 79 condition therefore reads (L(α,F) b),X = N X k=1 2kαk(∂µφ∂µφ)k−1≥0.(8.23) Again, this condition is automatically satisfied if only αkfor odd kare nonzero, or if the αkfor even kobey certain restrictions. Remark: in [50] a class of models based on the superfield (8.82) of Section 4.2 were introduced. These models satisfy neither energy positivity nor the null energy condition. They support, nevertheless, topological kink solutions, and their energy densities can be expressed as the squares of the corresponding supercharges. A more complete analysis of these models which would resolve the issue of stability is, therefore, an open problem at the moment which requires further investigation. 8.3 Solutions The Euler-Lagrange equation for the Lagrangian density (8.19) is N X k=1 2kαk∂µ[(∂νφ∂νφ)k−1∂µφ]+(−1)k−1(2k−1)F2k−1F,φ= 0 (8.24) which, in 1+1 dimensions and for static (time-independent) fields simplifies to N X k=1 2k(−1)k−1αk−∂x(φ2k−1 ,x ) + (2k−1)F2k−1F,φ= 0.(8.25) 8.3.1 Generic static solutions First of all, we want to demonstrate that, due to the restrictions imposed by supersymmetry, this equation has a class of static, one-dimensional solutions which are completely independent of the coefficients αk, which we shall call the ”generic” solutions. Indeed, if we impose equation (8.25) for each k(i.e., for each term in the sum) independently, the resulting equation is ∂x(φ2k−1 ,x ) = (2k−1)F2k−1F,φ (8.26) CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 80 or, after multiplying by φ,x and dividing by (2k−1), φ2k−1 ,x φ,xx =F2k−1F,φφ,x (8.27) which may be integrated to φ2k ,x =F2kand, therefore, to the kindependent solution φ,x ≡φ0=±F. (8.28) That is to say, these solutions only depend on the choice of F=F(φ), but do not depend on the αkand, therefore, exist for an infinite number of theories defined by different values of the αk. Depending on the choice for F(φ), the static solutions may be topological solitons. E.g. for the simple choice F= 1 −φ2, the solution of (8.28) is just the well-known φ4kink solution φ(x) = tanh(x−x0) where x0is an integration constant (the position of the kink). As another example, for F=p|1−φ2|, we get the compacton solution φ(x) =      −1x−x0≤ −π 2 sin(x−x0)−π 2≤x−x0≤π 2 1x−x0≥π 2 (8.29) where, again, x0is an integration constant. The energy of a generic supersymmetric kink solution may be calculated with the help of the first order formalism, which has the advantage that an explicit knowledge of the kink solution is not needed for the determination of its energy (for details on the first order formalism we refer to [32]). All that is needed is the field equation of a generic solution φ0=±F(we shall choose the plus sign corresponding to the kink, for concreteness). The idea now is to separate a factor φ0in the energy density with the help of the field equation, because this allows to rewrite the base space integral of the energy functional as a target space integral with the help of the relation φ0dx ≡dφ. Concretely, we get for the energy density of the generic kink solution E= N X k=1 (−1)k−1αk(φ02k+ (2k−1)F2k) = N X k=1 (−1)k−1αk2kF2k =φ0 N X k=1 (−1)k−1αk2kF2k−1≡φ0W,φ (8.30) CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 81 where W,φ and its φintegral W(φ) are understood as functions of φ. For the energy this leads to E=Z∞ −∞ dxφ0W,φ =Zφ(∞) φ(−∞) dφW,φ =W(φ(∞)) −W(φ(−∞)).(8.31) As indicated, all that is needed for the evaluation of this energy is the root φ0=F(φ) and the asymptotic behaviour φ(±∞) of the kink. We remark that the integrating function W(φ) of the first order formalism is identical to the prepotential P(φ), W(φ) = P(φ) (8.32) as is obvious from Eq. (8.18). This is exactly as in the case of the standard supersymmetric scalar field theory with the standard, quadratic kinetic term. It also remains true for the class of models introduced and studied in [17], as we shall discuss in some more detail in Section 8.4.2 Both for the standard supersymmetric scalar field theories and for the models introduced in [17] it is, in fact, possible to rewrite the energy functional for static field configurations in a BPS form, such that both the first order field equations for static fields and the simple, topological expressions E=P(φ(∞)) −P(φ(−∞)) for the resulting energies are a consequence of the BPS property of the energy functional (for the models introduced in [17] we briefly recapitulate the BPS property of static kink solutions in Section 8.4.2). on the contrary, for the models introduced in the previous section there is no obvious way to rewrite them in a BPS form, despite the applicability of the first order formalism, because the energy functional contains, in general, many more than two terms (just two terms are needed to complete a square and arrive at the BPS form). on the other hand, for the additional, specific solutions of the theories of Section 8.2 to be discussed in the following two subsections, the relation W=Pis no longer true, although it is still possible to calculate the energies of the specific solutions with the help of the first order formalism. 8.3.2 Specific solutions: an example Next, we want to study whether in addition to the solutions φ2 ,x =F2, which do not depend on the specific Lagrangian (i.e., on the coefficients αk), there CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 82 exist further (static) solutions which do depend on the Lagrangian. Both the existence of such additional solutions and their properties (e.g., being topological solitons) will depend on the Lagrangian, therefore the results will be less general and have to be discussed separately for each model. So, let us select a specific Lagrangian (specific values for the αk) as an example. Concretely, we want to study the simplest case which gives rise to a potential with several vacua and obeys certain additional restrictions (positivity of the energy). Positivity of the energy requires that both the highest and the lowest nonzero αkare for odd k, so we choose nonzero α3and α1for the simplest case. Further, we want that the potential factorizes and gives rise to several vacua, so we choose the concrete example α3=1 5,α2=2 3, and α1= 1, which gives rise to the Lagrangian density Lex b=1 5(∂µφ∂µφ)3+2 3(∂µφ∂µφ)2+∂µφ∂µφ−F6+ 2F4−F2(8.33) where, indeed, the potential in terms of Ffactorizes, F6−2F4+F2= F2(1 −F2)2. Next, we want to assume the simplest relation between Fand φ, namely F2(φ) = φ2. The resulting Lagrangian is Lex b=1 5(∂µφ∂µφ)3+2 3(∂µφ∂µφ)2+∂µφ∂µφ−φ2(1 −φ2)2.(8.34) We already know that it gives rise to the static solutions (φ,x)2=φ2⇒φ(x) = exp ±(x−x0).(8.35) These solutions have infinite energy and are not solitons. We want to investigate whether there exist additonal solutions and, specifically, whether there exist topological solitons. The potential has the three vacua φ= (0,1,−1), therefore topological solitons (static solutions which interpolate between these vacua) are not excluded. We shall find that these solitons exist in the space C1of continuous functions with a continuous first derivative, but not in the spaces Cn(with continuous first nderivatives) for n > 1. The once-integrated field equation for static solutions (with the integration constant set equal to zero, as required by the finiteness of the energy) reads (φ0≡φ,x) φ06−2φ04+φ02=φ6−2φ4+φ2(8.36) CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 83 and obviously has the solutions (8.35). For a better understanding of further solutions the following observations are useful. Firstly, for a fixed value x= ˜x of the independent variable x, the field φand its derivative φ0have to obey the equation V(φ) = V(φ0) = c(8.37) where cis a real, positive constant (or zero) and V(λ)≡λ6−2λ4+λ2=λ2(1 −λ2)2(8.38) is the potential (see Figure (8.1)). In general, the equation V(λ) = chas six solutions λ=±λi(c), i = 1 . . . 3. In other words, if we choose the initial condition φ(˜x) = ˜ φ, then φ0(˜x) is not uniquely determined (as would be the case for a linear first order equation) and may take any of the six values ±λi(c) such that V(±λi(c)) = V(˜ φ) = c. obviously, the choice φ0(˜x) = ±˜ φ leads to the exponential solutions (8.35), whereas other choices will lead to additional solutions. -1.0 -0.5 0.5 1.0 Φ 0.05 0.10 0.15 0.20 0.25 VHΦL Figure 8.1: The potential V(φ) = φ6−2φ4+φ2. Secondly, the field equation (8.36) leads to the following equation for the second derivative φ00 =(3φ4−4φ2+ 1)φ 3φ04−4φ02+ 1 (8.39) where the numerator is zero for all critical points (minima and maxima) φ= (0,±1,±1 √3) of the potential V(φ), whereas the denominator is zero for the critical points φ0= (±1,±1 √3). For later convenience we also remark that CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 84 at the two local maxima φ=±1 √3the potential takes the value V(±1 √3) = 4 27 and that the equation V(λ) = 4 27 has the two further solutions λ=±2 √3which are not critical points. We observe that the equation V(φ0) = V(φ) can in fact be solved algebraically for φ0and leads to the solutions φ0=±φ(8.40) (that is, the exponential solutions (8.35)) and to the four further solutions φ0=±1 2φ±p4−3φ2.(8.41) For this last expression, reality of φ0requires that |φ| ≤ 2 √3. The resulting integral for φ Zφ 0 d˜ φ ±(φ±p4−3φ2)=1 2(x−x0) (8.42) may be resolved explicitly, providing an implicit solution x−x0=H(φ) where, for each choice of signs, H(φ) is a combination of logaritms and inverse trigonometric functions. The explicit expressions for Hare, however, rather lenghty and not particularly illuminating, therefore we prefer to continue our discussion with a combination of qualitative arguments and numerical calculations. We want to remark, however, that the graphs of the numerical solutions shown in the figures below agree exactly with the graphs of the analytic solutions (8.42) (we remind the reader that for the graph of a function the implicit solution is sufficient). For the qualitative discussion, we now assume that we choose an ”initial value” at a given point ˜x. Due to translational invariance we may choose this point at zero ˜x= 0, i.e. φ(0) = φ0, without loss of generality. For 0<|φ0|<1 √3,1 √3<|φ0|<1 and 1 <|φ0|<2 √3,φ0(0) may take any of the six real solutions of the equation V(φ0) = V(φ0). Further, φ00(0) as well as all higher order derivatives at x= 0 are uniquely determined by linear equations, as we shall see in a moment. Therefore, for these ”initial conditions” φ0, there exist indeed the six solutions (8.35) and (8.42). For |φ0|>2 √3, only the two solutions φ0(0) = ±φ0of the equation V(φ0) = V(φ0) are real, therefore only the two exponential solutions (8.35) exist. At the CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 85 critical points φ0= 0,±1 and φ0=±1 √3,±2 √3the situation is slightly more complicated (strictly speaking φ0=±2 √3are not critical points, because V0(±2 √3)6= 0; however, φ0=±2 √3provides the same level height of the potential like the critical points φ0=±1 √3, that is, V(±1 √3) = V(±2 √3) = 4 27, therefore these points play a special role in the analysis, too). In order to understand what happens it is useful to insert the Taylor expansion about x= 0, φ(x) = ∞ X k=0 fkxk(8.43) into the field equation V(φ)−V(φ0) = 0, which, up to second order, reads 0 = f2 0(1 −f2 0)2−f2 1(1 −f2 1)2+ (8.44) [2f0(1 −4f2 0+ 3f4 0)f1−4f1(1 −4f2 1+ 3f4 1)f2]x+ (8.45) [(1 −12f2 0+ 15f4 0)f2 1+ 2f0(1 −4f2 0+ 3f4 0)f2− −4(1 −12f2 1+ 15f4 1)f2 2−6f1(1 −4f2 1+ 3f4 1)f3]x2+. . . (8.46) It can be inferred easily that for generic values of f0and f1(values which are not critical points), f2is determined uniquely by a linear equation from the term of order x1. on the other hand, if f1takes a critical value, then the coefficient multiplying f2in the order x1term is zero, and f2is determined, instead, by a quadratic equation coming from the term of order x2. These points will be important in the following, because precisely at these points we may join different solutions such that the resulting solution belongs to the class C1of continuous functions with a continuous first derivative. Specifically, we find the following possible values for f1and f2for a given, critical f0 (we only consider the cases f0≥0 because of the obvious symmetry φ→ −φ of the theory). For f0= 0 (f0, f1, f2) = (0,0,0) or (0,±1,±1 4) (8.47) where the first case corresponds to the trivial vacuum solution φ≡0, and the second case corresponds to the four solutions (8.42). The exponential solutions (8.35) are obviously incompatible with the ”initial condition” φ(0) = 0 (the vacuum solution φ(x) = 0 can be understood as a limiting case of the two CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 86 exponential solutions for infinite integration constant x0). Next, for f0= 1 (f0, f1, f2) = (1,0,0) or (1,±1,±1 2) (8.48) where the first case corresponds to the trivial vacuum solution φ≡1. The second case consists of the exponential solutions (two solutions) and of two of the four solutions (8.42). The other two are incompatible with the ”initial conditions”. For f0=1 √3we get (f0, f1, f2) = ( 1 √3,±1 √3,±1 2√3) or ( 1 √3,±2 √3,0) (8.49) where the first case contains both the two exponential solutions and two of the four solutions (8.42), and the second case corresponds to the other two solutions (8.42). Finally, for f0=2 √3we find (f0, f1, f2) = ( 2 √3,±2 √3,1 √3) or ( 2 √3,±1 √3,∞) (8.50) where the first case provides the two exponentail solutions (8.35), whereas the second case shows that the solutions (8.42) run into a singularity when |f0|=2 √3. Now let us study some of these cases in more detail. Concretely, we investigate the case (f0, f1, f2) = (0,1,1 4). Firstly, for negative x,φ(x) diminishes from φ(0) = 0 towards −1, and φ0(x) diminishes from φ0(0) = 1 towards 0, such that for a fixed value of x φ and φ0have the same height on the graph of V, see Fig. (8.1). If xis sufficiently negative such that φ(x) is close to its vacuum value −1 and φ0(x) is close to zero, the field equation may be linearized about the vacuum −1, and it follows easily that the vacuum is approached exponentially, like φ(x)∼ −1 + exp(4x) (remember that xis negative). In other words, for negative xthe solution behaves like a nice kink or topological soliton and does not reach the vacuum value −1 for finite x. For positive x, in a first instant both φ(x) and φ0(x) grow till they reach the values φ(x1) = 1 √3and φ0(x1) = 2 √3for some x1>0. At this point φ00(x1) = 0 therefore φ0may change direction in a smooth way. For x>x1,φ continues to grow while φ0shrinks till they reach the values φ(x2) = 2 √3and φ0(x2) = 1 √3for some x=x2. At this point φ00(x2) = ∞, and the solution CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 87 -4 -2 2 4 x -1.0 -0.5 0.5 1.0 ΦHxL -4 -2 2 4 x -1.0 -0.5 0.5 1.0 Φ¢HxL Figure 8.2: For the ”initial condition” φ(0) = 0 all the five solutions (including the trivial solution φ≡0) φ(x) (left figure) and the first derivatives φ0(x) (right figure). The singularity at φ(x2) = ±2 √3,φ0(x2) = ±1 √3for some x2, where the integration breaks down, is clearly visible. hits a singularity. A numerical integration confirms these findings, see Figure (8.2). There exists, however, the possibility to form a topological soliton or kink solution in the class C1of continuous functions with continuous first derivatives by simply joining the solution (f0, f1, f2) = (0,1,1 4) for negative xwith the solution (f0, f1, f2) = (0,1,−1 4) for positive x. Indeed, both φ(0) and φ0(0) agree, so the resulting solution is C1. Further, φ0in the second case diminishes for positive xbecause φ00(0) is negative. Therefore, φ(x) approaches 1 and φ0approaches 0 for large positive x, and a linearized analysis reveals that in that region φ(x)∼1−exp(−4x). As a consequence, the solution obtained by the joining procedure behaves exactly like a kink interpolating between the vacuum −1 at x=−∞ and the vacuum +1 at x=∞. For the corresponding result of a numerical integration, see Figure (8.3). Finally, let us discuss the possibility to form a kink in the class of C1 functions which interpolates, e.g., between the vacuum 0 and the vacuum 1. For this purpose, we should join the solution (f0, f1, f2) = ( 1 √3,1 √3,1 2√3) for x < 0 with the solution (f0, f1, f2)=(1 √3,1 √3,−1 2√3) for x > 0. Indeed, the solution (f0, f1, f2)=(1 √3,1 √3,1 2√3) is just the exponential solution exp xand behaves well (approaches 0 exponentially) for negative x. For the solution (f0, f1, f2) = ( 1 √3,1 √3,−1 2√3), on the other hand, both φ(0) and φ0(0) are equal CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 88 -4 -2 2 4 x -1.0 -0.5 0.5 1.0 ΦHxL -4 -2 2 4 x 0.2 0.4 0.6 0.8 1.0 Φ¢HxL Figure 8.3: For the ”initial condition” φ(0) = 0, the kink solution interpolating between φ=−1 and φ= 1 (left figure) and its first derivative (right figure). to 1 √3at x= 0. For increasing x,φ(x) increases and φ0(x) decreases until they get close to 1 and 0, respectively. But near these values, again, a linearized analysis applies and tells us that φbehaves like φ(x)∼1−exp(−4x). Therefore, the solution produced by the joining procedure describes a kink which interpolates between the vacuum φ= 0 at x=−∞ and the vacuum φ= 1 at x=∞. The general solution for the initial condition φ(0) = 1 √3is displayed in Figure (8.4), and the kink solution is shown in Figure (8.5). -4 -2 2 4 x -1.0 -0.5 0.5 1.0 1.5 ΦHxL -4 -2 2 4 x -1.5 -1.0 -0.5 0.5 1.0 1.5 Φ¢HxL Figure 8.4: For the ”initial condition” φ(0) = 1 √3all six solutions φ(x) (left figure) and the first derivatives φ0(x) (right figure). Again, the singularities at φ(x2) = ±2 √3,φ0(x2) = ±1 √3for some x2for the non-exponential solutions are clearly visible. The remaining kink and antikink solutions which we have not discussed explicitly may be easily found with the help of the obvious symmetries x→ −xand φ→ −φ. We remark that from the point of view of the CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 95 We remark that this kink has the interesting feature that its first derivative (and, therefore, also the energy density) has a local minimum at the position of the kink center, whereas the two local maxima are slightly displaced to the right and left of the center. We further remark that this example demonstrated explicitly that C∞kinks may exist not only among the generic solutions but also among the specific solutions of our supersymmetric K field theories (which was not obvious in the other two examples studied so far). 8.4 Further models 8.4.1 Field-dependent αk Here we want to construct further supersymmetric K field theories based on the observation that the models introduced in the last section remain supersymmetric when the factors αkmultiplying each power of the kinetic term depend on φinstead of being constants. Indeed, a superfield which is just an arbitrary function α(Φ) of the basic superfield Φ has the superspace expansion in the bosonic sector ψ= 0 (α(Φ))ψ=0 =α(φ)−θ2α0(φ)F(8.69) (the prime α0(φ) denotes the derivative w.r.t. the argument φ). If this superfield is multiplied by the superfield (DαΦDαΦ)ψ=0 which only has a θ2 component in the bosonic sector, then in the product only the θ-independent component of α(Φ) contributes, (α(Φ))ψ=0|=α(φ) (8.70) and the multiplication with the superfield α(Φ) corresponds to a multiplication with the ordinary field α(φ) of the Lagrangian densities (8.15), i.e., to the new building blocks −(L(k,n))ψ=0 = (8.71) =D2[α(k,n)(Φ)(1 2DαΦDαΦ)(1 2DβDαΦDβDαΦ)k−1(D2ΦD2Φ)n]|ψ=0 = =−α(k,n)(φ)(F2+∂µφ∂µφ)kF2n(8.72) CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 96 where again k= 1,2, . . . and n= 0,1,2, . . .. The cancellation of the mixed terms (∂µφ∂µφ)iF2jmay again be achieved by calculating the sum analogous to (8.17) provided all the α(k,n)in the sum are equal. Linear combinations of these Lagrangians are therefore L(α,P) b= N X k=1 αk(φ)[(∂µφ∂µφ)k+ (−1)k−1F2k] + P0(φ)F(8.73) just like in Section 8.2, but now with φdependent coefficients αk(φ). Also, the field equation for the auxiliary field Fis the same and leads to the Lagrangian L(α,F) b= N X k=1 αk(φ)[(∂µφ∂µφ)k−(−1)k−1(2k−1)F2k] (8.74) where F=F(φ) is an arbitrary function of φ, like in (8.19), but now with field dependent coefficients αk(φ). This result provides us with a new class of supersymmetric K field theories where now different powers of the kinetic term may be multiplied by functions of the scalar field. Now we shall discuss an explicit example, where we choose the functions αk(φ) and F(φ) such that the resulting model possesses a simple defect solution. Concretely we choose the nonzero αk α3=1 24 , α2=1 4(1 −φ2)2, α1=1 2[1 + (1 −φ2)4].(8.75) For non-constant αk(φ) positivity of the energy and the null energy condition become slightly more involved. For the moment we only consider the null energy condition, which is satisfied for this specific model. Indeed, the resulting Lagrangian is L=1 24[(∂µφ∂µφ)3−5F6] + 1 4(1 −φ2)2[(∂µφ∂µφ)2+ 3F4] + (8.76) +1 2[1 + (1 −φ2)4](∂µφ∂µφ−F) and for LXwe get LX=X2+ 2(1 −φ2)2X+ (1 −φ2)4+ 1 = (X+ (1 −φ2)2)2+ 1 >0 (8.77) CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 97 (remember X≡1 2∂µφ∂µφ), so the null energy condition holds. In order to have simple defect solutions we now choose for F F2= (1 −φ2)2(8.78) such that the Lagrangian becomes L=1 24[8X3−5(1 −φ2)6] + 1 4(1 −φ2)2[4X2+ 3(1 −φ2)4] + (8.79) +1 2[1 + (1 −φ2)4][2X−(1 −φ2)2]. The once integrated field equation for static solutions is equivalent to the condition that the one-one component of the energy momentum tensor is constant (see, e.g. [32, 17]). Further, for finite energy solutions this constant must be zero, T11 =L−2XLX= 0 (8.80) where now X=−1 2φ02because φis a static configuration. For the concrete example this equation reads −5 3X3−3(1−φ2)2X2−[1+(1−φ2)4]X+1 24(1−φ2)6−1 2(1−φ2)2= 0.(8.81) It may be checked easily that this equation is solved by X=−1 2(1 −φ2)2, i.e., φ02= (1 −φ2)2which is just the field equation of the φ4kink with the solution φ(x) = ±tanh(x−x0). Therefore, our concrete example has the standard φ4kink as a defect solution (it was specifically chosen to have this solution). Due to the nonlinear character of the above field equation, the model probably has more solutions (like the ones of Section 8.3), but this issue is beyond the scope of the present paper. Finally, let us remark that, although for the above model (i.e., the choice (8.75) for the αk) the energy density is not positive semi-definite, it is easy to find a small variation of the model with positive semi-definite energy density. All one has to do is to increase the relative size of α3and (or) α1as compared to α2. Choosing, for example, α1and α2as in (8.75), and α3=1 3, the resulting energy density is positive semi-definite for arbitrary F(φ), as may be shown easily. If we further choose F2∼(1−φ2)2then the resulting potential will have at least the two vacua φ=±1, and a linearization of the model CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 98 near these two vacua shows that the vacua are approached exponentially, like in the case of the standard kink. Therefore, the model most likely supports kinks which interpolate between the two vacua, although the explicit kink solutions will be more complicated. 8.4.2 The models of Bazeia, Menezes and Petrov The supersymmetric K field theories of Bazeia, Menezes and Petrov (BMP) [17] are based on the superfield ∂µΦ∂µΦ = ∂µφ∂µφ+ 2θα∂µφ∂µψα−2θ2∂µφ∂µF−θ2∂µψα∂µψα.(8.82) Indeed, the bosonic component of the superfield DαΦDαΦ only consists of a term proportional to θ2, therefore multiplying this superfield by an arbitrary function of the above superfield (8.82), f(∂µΦ∂µΦ), only the theta independent term f(∂µφ∂µφ) will contribute, leading to the Lagrangian LBMP =−D2[f(∂µΦ∂µΦ)1 2DαΦDαΦ]|ψ=0 =f(∂µφ∂µφ)(F2+∂µφ∂µφ). (8.83) obviously, these Lagrangians produce a coupling of the auxiliary field Fwith the kinetic term ∂µφ∂µφ. on the other hand, the auxilliary field only appears quadratically, implying a linear (algebraic) field equation for F. First of all, we want to remark that for functions f(∂µφ∂µφ) which have a Taylor expansion about zero, the same bosonic Lagrangians may be constructed from the building blocks (8.15) of Section 8.2 by taking a different linear combination (the fermionic parts of the corresponding Lagrangians will in general not coincide) L(k) BMP ≡(L(k,0))ψ=0 −k−1 1(L(k−1,1))ψ=0 +k−1 2(L(k−2,2))ψ=0 +. . . . . . +(−1)k−1k−1 k−1(L(1,k−1))ψ=0 = (8.84) = (∂µφ∂µφ+F2)(∂µφ∂µφ)k−1. We may easily recover the Lagrangian (8.83) by taking linear combinations CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 99 of these, LBMP =∞ X k=1 βkL(k) BMP = (F2+∂µφ∂µφ)X k βk(∂µφ∂µφ)k−1≡(8.85) ≡(F2+∂µφ∂µφ)f(∂µφ∂µφ). In order to have more interesting solutions, BMP added a potential term, as we did in Section 8.2. The resulting theories can, in fact, be analyzed with methods very similar to the ones employed in the previous sections. Concretely, they studied the Lagrangians L(P) BMP =f(∂µφ∂µφ)(F2+∂µφ∂µφ) + P0(φ)F(8.86) which after eliminating the auxiliary field Fusing its algebraic field equation F=−P0 2f(8.87) becomes L(P) BMP =f(P02 4f2+∂µφ∂µφ)−P02 2f=∂µφ∂µφf −P02 4f(8.88) where we suppressed the arguments of Pand fin the last expression to improve readability. The Xderivative of this Lagrangian is (L(P) BMP),X =f,X P02 4f2+ 2X+ 2f(8.89) (please remember that X≡1 2∂µφ∂µφand f=f(2X) such that f,X = 2f0). The null energy condition already imposes rather nontrivial restrictions on the function f. A sufficient condition is f≥0, f,X ≥0 and f≥ |Xf,X|as may be checked easily. Finally, the once integrated field equation (8.80) for static fields, after a simple calculation, leads to 1 4f2(f+ 2Xf,X)(8Xf2+P02) = 0 (8.90) where now X=−1 2φ02and f=f(−φ02) and, therefore, to the two equations 2φ0(x)f(−φ0(x)2) = ±P0(φ) (8.91) CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 100 where we reinserted the arguments in the last expression for the sake of clarity. For some choices of fand Pthese equations lead to defect solutions. Finally, in the models of BMP the energy of a kink may be calculated with the help of the first order formalism first introduced in [32], in close analogy to the calculations presented in Section 8.3.1. It also remains true that, like in Section 8.3.1, the prepotential P(φ) is equal to the integrating function W(φ). The energy functional for static configurations (but without the use of the field equation) may, in fact, be re-written in a BPS form (exactly like for the standard supersymmetric scalar field theory), from which both the first order equations and the equality P=Wfollow immediately. Indeed, the energy functional may be written like E(P) BMP =Zdx φ02f+P02 4f=Zdx 1 4f(2φ0f∓P0)2±φ0P0(8.92) and for a solution to the first order (or BPS) equation (8.91) (we take the plus sign for definiteness) the resulting energy is therefore E(P) BMP =Z∞ −∞ dxφ0P0=Zφ(∞) φ(−∞) dφP0=P(φ(∞)) −P(φ(−∞)) (8.93) which proves the above statement. For a more detailed discussion we refer to [17] (we remark that BMP use a slightly different notation in [17]: they use the notation hinstead of Pfor the prepotential, and their Xis defined like X=∂µφ∂µφinstead of the definition X=1 2∂µφ∂µφused in the present paper and in [32]). 8.5 Discussion We developed and described a method to construct general supersymmetric scalar K field theories in 1+1 and 2+1 dimensions. Among these theories, we found a large class of models which, in the purely bosonic sector, consist of a generalized kinetic term plus a potential, where the vacuum structure of the potential is crucial for the determination of the topological defect solutions, similarly to the standard case (i.e., kinetic term ∼∂µφ∂µφ). Due to the enhanced nonlinearity of these supersymmetric K field models there CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 101 are, nevertheless, some significant differences, like different roots of the first order field equations leading to a larger number of kink solutions, or the possibility to join different solutions forming additional kinks in the space C1 of continuous functions with a continuous first derivative 1. These results are new and are by themselves interesting, broadening the range of applicability of supersymmetry to a new class of field theories and, at the same time, enhancing our understanding of these field theories. As far as possible applications are concerned, the natural arena seems to be the area of cosmology, as already briefly mentioned in the introduction. Indeed, if these scalar field theories are interpreted as effective theories which derive from a more fundamental theory with supersymmetry (like string theory), then it is natural to study the supersymmetric versions of the effective models. If, in addition, the defect formation and phase transition (e.g. from a symmetry breaking phase to a symmetric phase) relevant for cosmological considerations occur at time or energy scales where supersymmetry is still assumed unbroken, then also the defect solutions of the supersymmetric effective field theories are the relevant ones. At this point, several questions show up. The first one is the inclusion of fermions. It is, e.g., expected that, as a consequence of supersymmetry and translational invariance, there should exist a fermionic zero mode for each kink background where, in addition, the fermionic zero mode is equal to the spatial derivative of the kink. This fact has already been confirmed explicitly in some supersymmetric K field theories [17], [50]. The inclusion of fermions in the Lagrangians studied in the present article does not present any difficulty on a fundamental level, the only practical obstacle being that, for purely combinatorical reasons, the resulting expressions will be rather lengthy. A second question is whether the SUSY algebra in a kink background contains central extensions related to the topological charge of the kink, as happens for the standard supersymmetric kink [44]. Again, this prob- 1Whether such C1solutions are physically relevant depends, of course, on the concrete physical system under consideration, but we want to emphasize that if the supersymmetric scalar field theory is interpreted as an effective theory then the C1solutions should be taken into account. In this case, the spike in the field derivative (and in the energy density) of aC1solution will, in any case, be resolved by the true UV degrees of freedom. CHAPTER 8. N=1 SUSY EXTENSION OF K FIELD THEORIES 102 lem has already been studied for some supersymmetric K field theories [50]. A further question concerns the issue of quantization. If the SUSY K field theories are interpreted as effective theories, as would be appropiate, e.g., in a cosmological context, then already the classical model contains some relevant information of the underlying quantum theory, like spontaneous symmetry breaking or the existence of topological defects. In this context, the quantization of quadratic fluctuations about the topological defect is the correct procedure to obtain further information about the underlying theory. Independently, one may, nevertheless, pose the problem of a full quantization of the supersymmetric K field theory, where the enhanced degreee of nonlinearity certainly implies further complications. one may ask, e.g.,. whether the additional, non-quadratic kinetic terms may be treated perturbatively, like the non-quadratic terms of the potential in the standard case. The answer will certainly depend on the space-time dimension. A related question is whether supersymmetry simplifies the task by taming possible divergences, as happens in the standard case. Here it is interesting to note that, even at the classical level, supersymmetry implies some restrictions on possible Lagrangians which are visible already in the bosonic sector. Indeed, as is obvious e.g. from Eq. (8.19), there exists a relation between the kinetic and the potential terms (this relation is responsible for the existence of the so-called generic solutions). one wonders what this relation implies for the quantum theory, e.g., in the form of Ward-like identities. These and related questions will be investigated in future publications. Finally, we think that the supersymmetric models we found present some independent mathematical interest of their own, given their high degree of non-linearity, on the one hand, and the possibility to obtain rather precise information on their solutions (e.g. all kink solutions together with their exact energies), on the other hand. Further investigations in this direction (e.g., time-dependent solutions, or the stability of topological solitons) will be pursued, as well. Chapter 9 BPS bounds in N=1 K field theories Continuing the line of analysis of supersymmetric K field theories presented in the previous chapter, we demonstrate that all domain wall solutions of such models are, in fact, BPS solutions. Moreover, in this chapter a first analysis of the supersymmetric algebra is done, finding that the central charge of the SUSY algebra coincides with the one for standard models. This chapter consists of a paper published in [96]. Supersymmetric K field theories and defect structures C. Adam 1, J.M. Queiruga 1, J. Sanchez-Guillen 1, A. Wereszczynski 2 1Departamento de F´ısica de Part´ıculas, Universidad de Santiago de Compostela and Instituto Galego de F´ısica de Altas Enerxias (IGFAE) E-15782 Santiago de Compostela, Spain 2Institute of Physics, Jagiellonian University, Reymonta 4, Krak´ow, Poland Abstract: We demonstrate that in the supersymmetric extensions of a class of generalized (or K) field theories introduced recently, the static energy satisfies a BPS bound in each topological sector. Further, the corresponding soliton solutions saturate the bound. We also find strong indications that the BPS bound shows up in the SUSY algebra as a central extension, as is the 103 CHAPTER 9. BPS BOUNDS IN N=1 K FIELD THEORIES 104 case in the well-known supersymmetric field theories with standard kinetic terms. 9.1 Introduction If a quantum field theory is assumed to be applicable to physical processes at arbitrary energy scales, then both its field contents and possible terms contributing to the Lagrangian are quite constrained, mainly by the requirement of renormalizability. Recently, however, a different point of view has gained support, where the field theory under consideration is interpreted as a low-energy effective field theory which, at sufficiently high energies, is superseded by a more fundamental theory (string theory being the most prominent proposal). In this effective field-theory interpretation, the presence of non-renormalizable terms in the lagrangian just indicates the existence of a natural cutoff in the effective field theory, beyond which calculations within the effective field theory framework are no longer trustworthy, and effects of the fundamental theory have to be taken into account. The effective field theory point of view, therefore, allows to consider a much broader class of Lagrangians, which may, in a first instance, be rather general functions of the fields and their space-time derivatives. Allowing for higher than first derivatives in the Lagrangian, however, may introduce some further problems like, e.g., the necessity to introduce ghosts, so it is natural to consider a class of generalized field theories given by Lagrangians which depend in a Poincare-invariant way on the fields and on their first derivatives. Specifically, a broader class of kinetic terms, generalizing the standard quadratic kinetic terms, may be considered. These theories with generalized kinetic terms (termed K field theories) have been studied with increasing effort in the last years, especially in the context of cosmology, where they might resolve some problems like inflation or late time acceleration (K-inflation [2] or K-essence [3]). Another relevant issue in cosmology is the formation of (topological or non-topological) defects [23], [27], [28], [30], [29], [26], [9] where, again, K field theories allow for a much richer phenomenology [4], [33], [51], [36], [10], [11], [39], [40]. Specifically, the formation of domain walls is described by effectively 1+1 dimensional theories [5], [7], [31], [32], [12], with CHAPTER 9. BPS BOUNDS IN N=1 K FIELD THEORIES 111 Let us point out that if we require kinks to be solutions of the corresponding variational problem, then solutions in the space C1are perfectly valid. They lead to well-defined energy densities and, therefore, provide well-defined critical points of the corresponding energy functional. For more details and some explicit examples, we refer to [56]. 9.2.3 Kink energies and BPS bounds In a next step, we want to study the energies of kinks. The energy density for the Lagrangian (9.11) is E(α,F) b= N X k=1 αk(φ)(˙ φ2−φ02)k−1((2k−1) ˙ φ2+φ02)+(−1)k−1(2k−1)F2k (9.21) and, for static configurations, E= N X k=1 (−1)k−1αk(φ)φ02k+ (2k−1)F2k.(9.22) With the help of Eq. (9.17), for kink solutions this may be expressed like E= N X k=1 (−1)k−12kαk(φ)φ02k=φ0 N X k=1 (−1)k−12kαk(φ)φ02k−1≡φ0w(φ, φ0) (9.23) where the last expression is especially useful for the calculation of the corresponding energies. Indeed, for the energy calculation we should now replace φ0in w(φ, φ0) by the root Riwhich corresponds to the kink solution, and interpret the resulting function of φas the φderivative of another function. That is to say, we define an integrating function Wi(φ) for each root Rivia Wi,φ ≡w(φ, Ri(φ)) = N X k=1 (−1)k−12kαk(φ)R2k−1 i,(9.24) then the kink energy is E=Zdxφ0Wi,φ =ZdφWi,φ =Wi(φ+)−Wi(φ−).(9.25) CHAPTER 9. BPS BOUNDS IN N=1 K FIELD THEORIES 112 For the calculation of the kink energy we, therefore, do not have to know the kink solution. We just need the root and the two vacuum values φ±of the kink. For the C1kinks described above which are constructed by joining local solutions for two different roots Riand Rj, we need the two corresponding integrating functions and the joining point φs. The energy then results in E=Wj(φ+)−Wj(φs) + Wi(φs)−Wi(φ−).(9.26) Until now, the energy considerations have been for arbitrary roots, but now we shall see that the generic root R1≡Fapparently plays a particular role. Firstly, the integrating function of the generic root is just the superpotential, W1=P. Indeed, we find W1,φ = N X k=1 (−1)k−12kαk(φ)F2k−1≡P0(φ) (9.27) see Eq. (9.10). Secondly, if the generic root has a kink solution, then this solution is, in fact, a BPS solution and saturates a BPS bound, as we want to demonstrate now. In general, an energy density has a BPS bound if it may be written off-shell (i.e. without using the static Euler-Lagrange equation) as E= (PSD)(φ, φ0) + t(x) (9.28) where (PSD) is a positive semi-definite function of φand φ0, and t(x) is a topological density, i.e., a total derivative whose integral only depends on the boundary values φ±. Further, a soliton solution (a kink φk) is of the BPS type, i.e., saturates the BPS bound if the positive semi-definite function is zero when evaluated for the kink, (PSD)(φk, φ0 k) = 0. In our case, the possible topological terms are the expressions φ0Wi,φ for the different roots. In any case, a possible topological term must be linear in φ0in order to be a total derivative (we emphasize, again, that the BPS form (9.28) must be valid off-shell, i.e., it is not legitimate to replace φ0by a root Rior vice versa). Let us now demonstrate that the energy density may be expressed in BPS form (9.28) for the generic topological term t=φ0W1,φ ≡φ0P,φ, and that the corresponding positive semi-definite function is zero precisely for the generic kink, i.e., for φ0=F. Indeed, we find for the difference E −tfor the generic CHAPTER 9. BPS BOUNDS IN N=1 K FIELD THEORIES 113 topological term E −φ0P,φ = N X k=1 (−1)k−1αk(φ)φ02k+ (2k−1)F2k−2kφ0F2k−1= = (φ0−F)2S(φ0, F)≡ ≡(φ0−F)2 N X k=1 (−1)k−1αk(φ)Hk(φ0, F) (9.29) where Hk(φ0, F)≡ 2k−1 X i=1 iφ02k−1−iFi−1.(9.30) Before proving this algebraic identity, we want to make some comments. The above result implies a genuine BPS soliton provided that the positive semi-definite function is zero only iff φobeys the corresponding generic kink equation φ0=F. This implies that S(φ0, F) must be strictly positive for any nontrivial field configuration (for the trivial vacuum φ0= 0 and F= 0 it holds that S(0,0) = 0), i.e., S(a, b)>0 unless a= 0 and b= 0. This inequality, indeed, holds for each individual term Hk(a, b), i.e., Hk(a, b)>0 unless a= 0 and b= 0 (the proof requires two complete inductions, therefore we relegate it to appendix A). The inequality S(a, b)>0 for the complete function S, therefore, implies some restrictions on the functions αk(φ) (one possible choice is that the αkare zero for even kand positive semi-definite for odd k, but there are less restrictive choices). This is similar to the conditions of positivity of the energy density, or the NEC, which, too, imply some restrictions on the αk, (again, αkzero for even kand positive semi-definite for odd kis a possible choice), and we shall assume in the sequel that the αkobey these restrictions (i.e., the restrictions resulting from the condition S > 0, and either positivity of the energy density or the NEC; these restrictions are probably related, but we shall not investigate this problem further and assume the two restrictions independently). Now let us prove the algebraic identity between Eq. (9.29) and Eq. (9.29). This follows from the following CHAPTER 9. BPS BOUNDS IN N=1 K FIELD THEORIES 114 identities (we set φ0=a,F=b) a2k+ (2k−1)b2k−2kab2k−1(9.31) = (a−b)a2k−1+a2k−2b+a2k−3b2+. . . +ab2k−2−(2k−1)b2k−1 = (a−b)2a2k−2+ 2a2k−3b+ 3a2k−4b2+. . . + (2k−1)b2k−2 ≡(a−b)2Hk(a, b) (9.32) where the equality of adjacent lines may be checked easily. So we found, indeed, that generic kinks (if they exist) saturate a BPS bound, whereas up to now we could not make a comparable statement about additional ”specific” kinks. This special role played by the generic kink solution is not surprising from the point of view of the supersymmetric extension, because only the generic kink obeys the simple equation φ0=F, and only the generic kink has a topological charge which may be expressed in terms of the superpotential. on the other hand, the special character of the generic kink is surprising from the point of view of the purely bosonic theory Lb= N X k=1 αk(φ)(∂µφ∂µφ)k−V(φ) (9.33) (with given αkand a given potential V), whose once-integrated static field equation just leads to the 2Nroots φ0=±Ri(φ), i = 1, . . . , N (9.34) without distinguishing them in terms of an auxiliary field or a superpotential. The resolution of the puzzle may be understood if we express the onceintegrated static field equation both in terms of the potential and in terms of the on-shell auxiliary field, N X k=1 (2k−1)(−1)k−1αk(φ)φ02k−F2k= N X k=1 (2k−1)(−1)k−1αk(φ)φ02k−V= 0. (9.35) Up to now we assumed a given F(φ) which lead to the two generic roots φ0=Fand the remaining, specific roots. But now we may interpret this equation in a different way. We may treat only Vand the αkas given and CHAPTER 9. BPS BOUNDS IN N=1 K FIELD THEORIES 115 try to find all the solutions for Fof the equation N X k=1 (2k−1)(−1)k−1αk(φ)F2k=V. (9.36) obviously, the solutions are just the roots Fi=Ri(φ) (see Eq. (9.17)), and the corresponding first order equations now just read φ0=±Fi. We remark that different on-shell choices Fifor the auxiliary field Flead to different superpotentials and, therefore, to different supersymmetric extensions. As a result, the resolution of the puzzle is that one given bosonic theory allows for Ndifferent supersymmetric extensions such that each kink solution is the generic solution of its corresponding supersymmetric extension. As a consequence, the energy density allows for BPS bounds for all kink solutions. The existence of several BPS bounds for one and the same energy density may seem surprising, but the different bounds exist, of course, in different topological sectors (i.e., for different boundary values), so there is no contradiction. Finally, all topological charges (i.e., all BPS energies) are now given in terms of the corresponding superpotentials. Indeed, we calculate (see Eqs. (9.10), (9.23) and (9.24)) Wi,φ(φ) = w(φ, Ri(φ)) = w(φ, Fi) = P0(Fi(φ)) ≡P0 i(φ).(9.37) We remark that from a practical point of view it is still useful to choose a specific on-shell F(φ), because in this way we may choose simple functions with simple kink solutions. For generic αkand a generic V, on the other hand, the resulting roots will usually be quite complicated. 9.3 SUSY algebra and central extensions From now on, we will, again, restrict to a fixed supersymmetric extension, i.e., to fixed, given αk, a fixed, given on-shell auxiliary field F(φ) and the corresponding superpotential given by Eq. (9.10). The SUSY transformations of the fields read δφ =αψα, δψα=−i(γµ)αββ∂µφ−αF , δF =iα(γµ)αβ∂µψβ(9.38) CHAPTER 9. BPS BOUNDS IN N=1 K FIELD THEORIES 116 (where α= (1, 2) are the Grassmann-valued SUSY transformation parameters, and α= (i2,−i1)), or more explicitly δφ =i(2ψ1−1ψ2) δF =i2(ψ0 1−˙ ψ2)−1(˙ ψ1−ψ0 2) δψ1=1(φ0−F)−2˙ φ δψ2=1˙ φ−2(φ0+F).(9.39) obviously, for a generic kink solution ( ˙ φ= 0, φ0=F, ψα= 0) the SUSY transformation restricted to 2= 0 is zero, whereas for a generic antikink the restriction 1= 0 gives zero. On the other hand, the SUSY transformations of the fields are generated by the SUSY generators Q=αQαvia the commutators δφ = [iQ, φ], etc., where Qshould be determined from the Noether current of the SUSY transformations, and the commutators are evaluated with the help of the canonical (anti-)commutation relations of the fields. The supercharges Qα are known to obey the algebra {Qα, Qβ}= 2Πν(γν)αβ+ 2iZ(γ5)αβ(9.40) or, explicitly, Q2 1= Π0+Z Q2 2= Π0−Z {Q1, Q2}= 2Π1(9.41) where the curly bracket is the anti-commutator, Πν= (Π0,Π1) are the energy and momentum operators, and Zis a possible central extension which the SUSY algebra may contain. An explicit calculation of the operators which appear in the SUSY algebra requires the knowledge of the Noether current and the canonical momenta and, therefore, of the complete SUSY Lagrangian, including the fermionic terms, which, in general, is quite complicated. If we only want to determine the central charge, however, it is enough to evaluate the SUSY algebra for a specific field configuration, because the central charge is essentially a number (it commutes with all operators) and, therefore, must CHAPTER 9. BPS BOUNDS IN N=1 K FIELD THEORIES 117 take the same value for all field configurations within a given topological sector. We now evaluate the SUSY algebra for a generic kink solution and make the reasonable assumption that not only the restricted SUSY transformation (i.e., the action of the corresponding SUSY charge on the fields) for a generic kink is zero, but that the corresponding SUSY charge itself is zero when evaluated for the generic kink. As we know the energy of the kink, this allows then to determine the central charge. Concretely, for the kink the corresponding charge is Q2, and we get Q2 2= 0 = Ek−Z=P(φ+)−P(φ−)−Z⇒Z=P(φ+)−P(φ−),(9.42) where Pis the superpotential, and φ±are the asymtopic values of the kink. For the antikink, Q1is zero, and we find Z=P(φ−)−P(φ+). We remark that for positive semi-definite energy densities the resulting restrictions on the functions αkimply that the central extension Zis always positive, because P0≥0 for the kink, and P0≤0 for the antikink, as follows from the energy density (9.22) and the defining equation for P0, Eq. (9.10). We, therefore, found exactly the same result for the central extension as in the case of the SUSY extension of a standard scalar field theory with a quadratic kinetic term for the boson field. 9.3.1 Central extensions for the models of Bazeia, Menezes and Petrov Here we want to demonstrate that the same central extensions of the SUSY algebra in terms of the superpotential may be found for another class of supersymmetric K field theories, originally introduced by Bazeia, Menezes and Petrov (BMP) [17]. They are based on the superfield SBMP =f(∂µΦ∂µΦ)1 2DαΦDαΦ (9.43) and lead to the bosonic Lagrangian LBMP =f(∂µφ∂µφ)(F2+∂µφ∂µφ).(9.44) Here, the Lagrangian produces a coupling of the auxiliary field Fwith the kinetic term ∂µφ∂µφ, but, on the other hand, the auxiliary field only appears CHAPTER 9. BPS BOUNDS IN N=1 K FIELD THEORIES 118 quadratically, implying a linear (algebraic) field equation for F. The same bosonic Lagrangians may, in fact, be constructed from the building blocks (9.5) of Section 9.2 by taking a different linear combination (the fermionic parts of the corresponding Lagrangians will in general not coincide) S(k) BMP ≡ k−1 X n=0 (−1)nk−1 nS(k−n,n)(9.45) leading to the bosonic Lagrangians L(k) BMP = (F2+∂µφ∂µφ)(∂µφ∂µφ)k−1.(9.46) We may easily recover the Lagrangian (9.44) by taking linear combinations of these, LBMP =∞ X k=1 βkL(k) BMP = (F2+∂µφ∂µφ)X k βk(∂µφ∂µφ)k−1≡(9.47) ≡(F2+∂µφ∂µφ)f(∂µφ∂µφ). Adding a superpotential, the resulting bosonic Lagrangians are L(P) BMP =f(∂µφ∂µφ)(F2+∂µφ∂µφ)−P0(φ)F, (9.48) or, after eliminating the auxiliary field Fusing its algebraic field equation F=P0 2f,(9.49) L(P) BMP =f·(P02 4f2+∂µφ∂µφ)−P02 2f=f·∂µφ∂µφ−P02 4f.(9.50) The energy functional for static configurations may be written in a BPS form. Indeed, E(P) BMP =Zdx φ02f+P02 4f=Zdx 1 4f(2φ0f∓P0)2±φ0P0(9.51) and for a solution to the first order (or BPS) equation 2φ0(x)f(−φ02) = P0(9.52) CHAPTER 9. BPS BOUNDS IN N=1 K FIELD THEORIES 119 (we take the plus sign for a kink) the resulting energy is E(P) BMP =Z∞ −∞ dxφ0P0=Zφ(∞) φ(−∞) dφP0=P(φ+)−P(φ−).(9.53) Finally, from Eq. (9.49) for Fand the BPS equation (9.52) it follows that the equation φ0=Fstill holds for a kink solution and, therefore, the restricted SUSY transformation with only 1nonzero is, again, zero when evaluated for the kink. We conclude that the central charge in the SUSY algebra is, again, given by the topological term Z=|P(φ+)−P(φ−)|(9.54) for this class of models. 9.4 Conclusions In this paper we carried further the investigation of a class of SUSY K field theories originally introduced in [56]. Concretely, we demonstrated that all the domain wall solutions which exist for this class of field theories are, in fact, BPS solutions. Further, these BPS solutions are invariant under part of the SUSY transformations. We also found strong indications (based on a very reasonable assumption) that the topological charges carried by the domain wall solutions show up in the SUSY algebra as central extensions. That is to say, the situation we found is exactly equivalent to the case of standard SUSY theories with BPS solitons, despite the much more complicated structure of the SUSY K field theories investigated here. Let us emphasize, again, that from an effective field theory point of view, K field theories are as valid as field theories with a standard kinetic term, and there exists no reason not to consider them. Even one and the same topological defect with some given, well-known physical properties may result either from a theory with a canonical kinetic term, or from a certain related class of K field theories (so-called noncanonical twins of the standard, canonical theory), [51], [57]. K field theories should, therefore, be considered on a par with standard field theories in all situations where they cannot be excluded a priori. This implies that also the study of their possible SUSY extensions is a valid and relevant CHAPTER 9. BPS BOUNDS IN N=1 K FIELD THEORIES 120 subject. Structural investigations of the type employed in the present paper are, then, important steps towards a better understanding of these supersymmetric generalized field theories with nonstandard kinetic terms. Appendix A We want to prove that a2k−2+ 2a2k−3b+. . . + (2k−1)b2k−2>0∀k(9.55) unless a= 0 and b= 0. For a= 0, b6= 0, and for a6= 0, b= 0 this is obvious, so we may restrict to the case a6= 0 and b6= 0. In this case, we may divide by b2k−2, so that we have to prove (x≡a/b) fk(x)≡x2k−2+ 2x2k−3+. . . + (2k−1) >0 (9.56) which we do by complete induction. obviously, the statement is true for k= 1: f1(x) = x2+ 2x+ 3 = (x+ 1)2+ 2 >0. Now we assume that it holds for fkand calculate fk+1. We get fk+1(x) = x2k+ 2(x2k−1+x2k−2+. . . + 1) + fk(x)≡gk(x) + fk(x) (9.57) and the statement is true if gk(x)≥0∀k. This, again, we prove by induction. obviously, it is true for k= 1: g1(x) = x2+2x+ 2 ≥0. For gk+1 we calculate gk+1(x) = x2k(x+ 1)2+gk(x) (9.58) and it is obviously true that gk(x)≥0⇒gk+1(x)≥0 and, therefore, fk(x)>0⇒fk+1(x)>0, which is what we wanted to prove. CHAPTER 10. TWIN-LIKE MODELS AND SUSY 127 For theories with a standard kinetic term Xand a potential V(φ), Ls=X−V, (10.10) the integrated static field equation simply is −X−V= 0 ⇒φ02= 2V(10.11) with the two roots φ0=±√2V. If the potential Vhas at least two vacua (which we assume from now on), then there will exist, in general, finite energy solutions of Eq. (10.11) which interpolate between different vacua (kinks), and the two roots correspond to kink and antikink, respectively. The static energy density for the standard theory is Es=−X+V=1 2φ02+V(10.12) and for a kink solution it may be written as Es|φk= (−X+V)|φk= 2V(φk) = −2X|φk(10.13) where φk(x) is the kink solution under consideration, and the notation |φk means that the expression (in general, a function of φand φ0), is evaluated at the kink solution φ=φk(x). Finally, the energy of a kink in this standard case simply is E=Z∞ −∞ dxφ02=Zφ(∞) φ(−∞) dφφ0=Zφ(∞) φ(−∞) dφ(±√2V) = (10.14) =±[WV(φ(∞)) −WV(φ(−∞))] where WV,φ =√2V(10.15) and the explicit expression for WVdepends, of course, on V. The two signs correspond to kink and antikink, respectively. 10.2.2 Twin or doppelgaenger defects In [51] the authors observed the possibility of twin-like models within the class of generalized K field theories, that is, of field theories which share CHAPTER 10. TWIN-LIKE MODELS AND SUSY 128 the same kink solution with the same energy density with a given standard field theory. They discussed a Dirac-Born-Infeld (DBI) like example in some detail where, however, the DBI term is multiplied by a target space geometric factor, because a pure DBI theory cannot be the twin of a standard field theory. Then they derived a necessary and sufficient geometrical condition which a second field theory L2has to obey in order to be the twin of a given field theory L1. From their geometric description they already concluded that there exist, in principle, infinitely many twin theories for a given standard scalar field theory. We shall review this geometric construction in a first step, because we will find that combining it with the first order formalism provides us with a simple and purely algebraic method to explicitly calculate an infinite number of twin models for any given field theory. The authors of [51] demonstrated that if the theory L1has a kink solution φk(x) with energy density Ek(x), then a necessary and sufficient condition for a second theory L2to have the same kink solution with the same energy density is that both Land L,X agree when evaluated for the kink solution, that is, L1|φk=L2|φk(10.16) L1,X|φk=L2,X |φk.(10.17) obviously, the first condition implies that the energy densities are equal, see Eq. (10.3). Further, the first order equation Eq. (10.7) holds for L1 by assumption, then the two conditions (10.16) and (10.17) imply that Eq. (10.7) is an identity for L2. It follows that the two conditions (10.16) and (10.17) are sufficient for L2to be a twin of L1. That the two conditions are necessary follows easily from the fact that the two equations (10.3) and (10.7) are linear in Land LX. From what has been said above, it might appear that for the explicit construction of a twin model L2for a given theory L1it is necessary to know an explicit kink solution φkof the theory L1, and to use this kink in the evaluation of possible twin models L2, which would render calculations rather cumbersome. But this is, in fact, not true. The important point is that the lagrangian densities are functions of the target space variables φand φ0 only, therefore it is sufficient to implement the root φ0=±fi(φ) which leads to the kink (or antikink) solution under consideration. Further, we shall use CHAPTER 10. TWIN-LIKE MODELS AND SUSY 129 the fact that all lagrangians we consider depend on φ0only via X=−1 2φ02 (for static configurations), so that the above conditions transform into L1|2X=−f2 i=L2|2X=−f2 i(10.18) L1,X|2X=−f2 i=L2,X|2X=−f2 i(10.19) where fi(φ) is a known root (10.8) of the theory L1leading to a kink solution. The above conditions are purely algebraic conditions in the target space variables φand Xand do not involve the base space variable xor explicit knowledge of a kink solution φk(x) at all. Up to now we allowed for completely general lagrangians L1and L2to emphasize the general character of the procedure. Now, however, we will concentrate on the case of a standard lagrangian L1=Ls=X−Vfor concreteness, so the problem consists in finding possible twins to standard scalar field theories. Here V(φ) is a positive semi-definite potential with at least two vacua (zeros) such that kink solutions exist. The two roots for kink and antikink may be combined into X=−V, and the above conditions read (we write Lfor L2) L|X=−V=−2V(10.20) L,X|X=−V= 1.(10.21) Again, these two conditions are purely algebraic and allow an easy calculation of twin models, as we shall see in the next section. 10.2.3 Examples of twin models As a first class of twin models let us consider the class of Lagrangians L= K X k=1 fk(φ)Xk−U(φ) (10.22) where the kinetic terms Xkare multiplied by functions of φin a sigma-model like fashion. Here, the condition L,X|X=−V= K X k=1 kfk(φ)(−V)k−1≡1 (10.23) CHAPTER 10. TWIN-LIKE MODELS AND SUSY 130 imposes one condition on the functions fk(φ). one may, for instance, choose arbitrary fkfor k≥2, then the above condition determines f1in terms of the remaining fkand V. We remark that it is not possible to choose all fk constant, but if at least one fkhas a nontrivial φdependence then the above condition can always be fulfilled. The second condition L|X=−V= K X k=1 fk(φ)(−V)k−U(φ)≡ −2V(φ),(10.24) in turn, determines U(φ) in terms of the fkand V. One question to ask is whether the resulting twin models constitute viable field theories on their own, that is, whether they obey certain stability requirements like energy positivity or the null energy condition (NEC). Here we shall mainly be concerned with the NEC, because i) it is deemed sufficient for stability, ii) it is weaker than the condition of positivity of the energy density and iii) it is easier to implement for the class of models we study in this paper. The NEC in general is the condition that nµnνTµν ≥0 (10.25) where Tµν is the energy-momentum tensor and nµis an arbitrary null vector. For the class of models L(X, φ) the NEC simply reads L,X ≥0.(10.26) It is, in general, not completely trivial to reconcile the NEC with the two twin conditions (10.23) and (10.24), but it is easy to find certain special classes of models where the NEC holds by construction. A first class of models which obeys both the NEC and the condition (10.23) by construction is given by field theories which obey L,X =Kf(φ)(X+V)K−1+ 1 (10.27) where fis an arbitrary, positive semi-definite function f(φ)≥0 and Kis an odd integer. The resulting Lagrangian (i.e., Xintegral) is L=f(φ)(X+V)K+X−˜ U(φ) (10.28) CHAPTER 10. TWIN-LIKE MODELS AND SUSY 131 (where the integration ”constant” ˜ U(φ) is an arbitrary function of φ), and the second twin condition (10.24) requires ˜ U=Vsuch that the class of twin Lagrangians reads L=f(φ)(X+V)K+X−V , K = 3,5, . . . (10.29) As a concrete example, we may e.g. choose f= 1 and K= 3 which results in the Lagrangian L=1 3X3+V X2+ (V2+ 1)X+1 3V3−V(10.30) which shares both the kink solution φ0=±√2Vand the corresponding energy density with the standard scalar model Ls=X−V. A second class of twin models obeying the NEC may be constructed from the equation L,X =f1−K(X+V+f)K−1(10.31) (where f=f(φ)≥0, and Kis an odd integer) with Lagrangian L=f1−K K(X+V+f)K−˜ U. (10.32) Here the second twin condition leads to ˜ U= 2V+ (f/K) and, therefore, to the Lagrangian L=f1−K K(X+V+f)K−2V−f K.(10.33) Next, let us describe another class of examples of twin models, different from the power expansion in Xof Eq. (10.22). We start from the ansatz L=f(φ)g(X)−U(φ) (10.34) and calculate L,X =f(φ)g0(X) (10.35) and the NEC leads to the conditions f≥0, g0≥0.(10.36) CHAPTER 10. TWIN-LIKE MODELS AND SUSY 132 Further, the two twin conditions lead to f(φ) = (g0(−V))−1and U= 2V+ (g(−V)/g0(−V)) and, therefore, to the Lagrangian L=g(X) g0(−V)−g(−V) g0(−V)−2V. (10.37) Among this class we may easily recover the DBI type example originally presented and discussed in [51]. Indeed, choosing for the kinetic function g(X) the DBI type expression g(X) = −√1−2X(10.38) we calculate g0(X) = 1 √1−2X, f(φ) = √1+2V , U = 2V−(1+2V) = −1 (10.39) and the resulting Lagrangian is L=−√1+2V√1−2X+ 1.(10.40) It is obvious from the derivation that the nontrivial target space geometry factor f(φ) = √1+2Vis necessary for this DBI type action to be the twin of a standard scalar field theory, as announced above. 10.3 Supersymmetric twin models To begin with, let us remind that a standard scalar field theory Ls=X−V with a positive semi-definite potential V≥0 may always be viewed as the purely bosonic sector of a supersymmetric scalar field theory. Indeed, before the elimination of the auxiliary field Fthe bosonic sector of the supersymmetric standard scalar field theory reads Ls=1 2(∂µφ∂µφ+F2)−FP0 s(φ) (10.41) where Ps(φ) is the prepotential (also sometimes called superpotential) of the standard SUSY scalar field theory. Elimination of the auxiliary field Fwith the help of its algebraic field equation F=P0 sleads to the lagrangian Ls=1 2∂µφ∂µφ−1 2P02 s(10.42) CHAPTER 10. TWIN-LIKE MODELS AND SUSY 133 which is just the standard scalar field lagrangian with the identification V=1 2P02 s≥0.(10.43) This observation leads to the obvious question whether there exist supersymmetric K field theory twins for the supersymmetric standard field theories. For this purpose, in a first instance we have to know whether there exist supersymmetric scalar K field theories at all. The answer is that these supersymmetric K theories do exist. Some classes of examples have been introduced and studied in [63], [17] (these theories exist both in 1+1 and in 2+1 dimensional Minkowski space, due to the similar spin structure in the two spaces), and we shall use some of these examples for the construction of our supersymmetric K field twins. In [63] a class of supersymmetric models was introduced such that their purely bosonic sector before the elimination of the auxiliary field reads L(α,P)= N X k=1 αk(φ)[(∂µφ∂µφ)k+ (−1)k−1F2k]−P0(φ)F. (10.44) Next, the auxiliary field Fshould be eliminated via its algebraic field equation N X k=1 (−1)k−12kαkF2k−1−P0(φ) = 0 (10.45) which in general is, however, a rather complicated algebraic equation for F. As no assumption was made yet about the functional dependence of P, this equation may be understood in a second, equivalent way: one assumes that Fis an arbitrary given function of φ, which in turn determines the prepotential P(φ). This second way of interpreting Eq. (10.45) is more useful for our purposes. Eliminating the resulting P0(φ) we arrive at the Lagrangian density L(α,F)= N X k=1 αk(φ)[(∂µφ∂µφ)k−(−1)k−1(2k−1)F2k] (10.46) where now F=F(φ) is a given function of φwhich may be chosen freely depending on the theory or physical problem under consideration. This class CHAPTER 10. TWIN-LIKE MODELS AND SUSY 134 of lagrangians is exactly of the type (10.22), therefore the conditions for being the twin of a standard theory are exactly analogous to the conditions (10.24) and (10.23). The restrictions implied by supersymmetry (i.e., the requirement to express the ”potential function ” U(φ) in Eq. (10.22) in terms of F(φ)), nevertheless, impose some additional restrictions, as we want to show now. Indeed, the second twin condition L,X |2X=−F2 s= 1 leads to X2kαk(−F2 s)k−1= 1,(10.47) where we introduced the function Fs(φ) of the standard SUSY theory, i.e., the auxiliary field Fof the standard theory evaluated at its field equation via 2V(φ)≡Ps02(φ)≡F2 s(φ) (10.48) for convenience. The first twin condition L|2X=−F2 s=−F2 sthen leads to L|2X=−F2 s=X k αk((−F2 s)k−(−1)k−1(2k−1)F2k) =X k αk((−F2 s)k−(−F2)k−2kF2(−F2)k−1) ≡ −F2 s where we used (10.47) in the last step. This condition is solved by F=±Fs.(10.49) As we shall see in a concrete example below, this is typically the only acceptable solution, therefore supersymmetry seems to imply the additional relation F=Fsfor the algebraic solutions of the auxiliary fields of standard and K field twin theories. Again, the NEC is not automatic in these theories, but a more specific class of examples which obeys the NEC by construction may be given, analogous to the last subsection. Concretely, we give some examples starting from the Xderivative L,X = 2K(2X+ 2V)K−1+ 1 (10.50) (we write 2Xinstead of Xin order to be as close as possible to the notation used in Ref. [63] and in Eq. (10.46); Kis an odd integer). The resulting Lagrangian is L= (2X+ 2V)K+X−˜ U(φ) (10.51) CHAPTER 10. TWIN-LIKE MODELS AND SUSY 135 and obeys the NEC and the twin condition L,X |X=−V= 1 by construction. For a more concrete example, let us now assume K= 3 which leads to the Lagrangian L= (2X)3+ 6V(2X)2+ (12V2+1 2)(2X)+8V3−˜ U(10.52) and therefore to α3= 1 , α2= 6V , α1= 12V2+1 2(10.53) and to the Lagrangian L= (2X)3−5F6+ 6V((2X)2+ 3F4) + (12V2+1 2)(2X−F2) (10.54) which explicitly is of the form (10.46) (we replaced the arbitrary integration“constant” ˜ U(φ) by the required Fterms). Now the second twin condition L|X=−V=−2Vleads to 5F6−18V F4+ (12V2+1 2)F2+ 8V3−V= 0 (10.55) which may be viewed as a third order algebraic equation for F2. The only acceptable (i.e., real and positive) solution is F2= 2V≡F2 s(10.56) and leads to the Lagrangian L= (2X)3−40V3+ 6V((2X)2+ 12V2) + (12V2+1 2)(2X−2V) (10.57) which is the desired supersymmetric twin of the standard Lagrangian. As already remarked for the more general class of examples above, it holds that also the (algebraic) field equations for the auxiliary fields coincide, see Eq. (10.49). This equality is not a further condition, but a consequence of the twin conditions and supersymmetry. Another class of supersymmetric theories has the following purely bosonic sector (before the elimination of the auxiliary field F) [17] L=g(φ)f(X)(F2+ 2X2)−P0(φ)F(10.58) CHAPTER 10. TWIN-LIKE MODELS AND SUSY 136 where fand gare arbitrary, fixed functions of their arguments and for the moment we just assume g≥0. The algebraic field equation for the auxiliary field Fhas the solution F=P0 2gf (10.59) and leads to the Lagrangian L= 2Xgf −P02 4gf ≡2gXf −h f(10.60) where h(φ)≡P02 8g2.(10.61) The Xderivative of this lagrangian is L,X = 2gf+Xf,X +hf,X f2.(10.62) A sufficient condition for the NEC consists in the following inequalities f+Xf,X ≥0, f,X ≥0 (10.63) but we have not been able to find a function fwhich obeys these inequalities. There exists, however, another possiblity to obey the NEC, and for this possibility we found solutions. Concretely, assume that f+Xf,X ≥1 (10.64) and that further  f,X f2≤1 (10.65) and h≤1 (10.66) then the NEC holds. A specific function fobeying these conditions is f= 1 + X2(10.67) which indeed leads to f+Xf,X = 1 + 3X2≥1, f,X f2 =2|X| (1 + X2)2≤1.(10.68) CHAPTER 10. TWIN-LIKE MODELS AND SUSY 143 10.5 Discussion In this article, we have derived a simple and purely algebraic method for the construction of K field twins of a standard scalar field theory, that is, of K field models which share the same topological defect with the same energy density with a given standard scalar field theory. This method may be derived for the cases of non-supersymmetric field theories, supersymmetric field theories and for self-gravitating fields. Further, we gave several examples for all these cases. The interest of these twin models lies in the fact that the field profile together with the energy density are the most relevant physical data of a defect which makes the twins almost indistinguishable from their standard counterparts in many situations. A pattern of defects in the very early universe will look the same irrespective of whether it is formed by defects of a standard theory or of a K field twin. The spectrum of linear fluctuations, on the other hand, is in general different between the standard theory and its twins [51], [58], so small differences will set in once dynamics (i.e., time dependence) is taken into account. A similar question is related to the behaviour of additional matter fields (e.g., fermion fields) coupled to twin defects. In the non-supersymmetric case there exist different possibilities to couple fermions to each field theory, therefore general statements cannot be made. The situation is different, however, for supersymmetric (SUSY) K field twins of standard SUSY scalar field theories. Here, the first important piece of information is, of course, the existence of SUSY K field theories [63], [17], [49], [136]. We want to point out again that defects of supersymmetric theories are the relevant ones to study if the symmetry breaking and the subsequent defect formation in the early universe occur at an energy scale where supersymmetry is still intact (e.g., at the end of inflation). For supersymmetric twins of the standard supersymmetric scalar field theories, it is interesting to observe that these SUSY twin models not only share the defect solution and its energy density with the standard theory. Also the (algebraic) field equation for the auxiliary field in the kink background is identical for the standard theory and the twin. Another interesting problem of these SUSY theories concerns, of course, the inclusion of fermions which we have set equal to zero in our discussion. In general, the fermionic sectors CHAPTER 10. TWIN-LIKE MODELS AND SUSY 144 of the standard theory and the twin will be different, like the bosonic sectors. Standard and twin theory will, however, share some common features in the fermionic sector, too. They will, e.g., share the same fermionic zero mode in the background of the same kink solution. This is a consequence of translational invariance, on the one hand, which implies that both theories in the kink background have the same bosonic zero mode (or Goldstone mode) equal to the derivative of the kink. The second ingredient is, of course, supersymmetry, which guarantees that each bosonic Goldstone mode is paired by a fermionic zero mode which is, again, equal to the derivative of the kink field. A final issue is the existence of twin models when the gravitational backreaction is taken into account, i.e., of twin defects sharing the same field profile, energy density and induced metric. Already for defect structures in cosmology (i.e., in the early universe) the full self-gravitating case should, in principle, be taken into account, although in many circumstances a Minkowski space calculation is sufficient. In the brane world scenario taking into account the full self-gravitating solution is mandatory. We found that, again, there exists a simple algebraic method to calculate infinitely many K field twins of a standard self-gravitating scalar field theory and gave several examples. We emphasize that for the 4+1 dimensional case relevant for the brane world scenario an example of a self-gravitating twin has already been given in [58] with the help of the first order formalism. It was the main aim of the present article to shed more light on the existence of K field twin defects and the mathematical structures behind them, on the one hand, and to provide a simple calculational tool for the construction and study of twin-like models, on the other hand. We want to point out that, whenever K field theories cannot be excluded on purely theoretical grounds, they have to be considered on a par with the standard field theories as an immediate consequence of the existence of twin defects, because for twin-like models their most relevant physical manifestations are completely indistinguishable. This is the case, e.g., for effective field theories resulting from the integration of UV degrees of freedom, where higher kinetic terms are naturally induced. Chapter 11 More on twin-like models After the general framework which provides the algebraic conditions that twin-like models must verify, in this chapter we show that it is possible to add more purely algebraic constraints to the lagranian of the twin-like model to ensure that both linear fluctuation spectra coincide. The interesting result is that a semiclassical quantization about the topological defect provides the same results for the standard field theory and its K field twins. This chapter consists of a paper published in [98]. Twinlike models with identical linear fluctuation spectra C. Adam 1, J.M. Queiruga 1, 1Departamento de F´ısica de Part´ıculas, Universidad de Santiago de Compostela and Instituto Galego de F´ısica de Altas Enerxias (IGFAE) E-15782 Santiago de Compostela, Spain Abstract: Recently, the possibility of so-called twinlike field theories has been demonstrated, that is, of different field theories which share the same topological defect solution with the same energy density. Further, purely algebraic conditions have been derived which the corresponding Lagrangians have to obey in order that the field theories be twins of each other. A further diagnostical tool which, in general, allows to distinguish the topological defects of a given theory from the corresponding defects of its twins is the spectrum of linear fluctuations about these defects. Very recently, however, 145 CHAPTER 11. MORE ON TWIN-LIKE MODELS 146 explicit examples of twin theories have been constructed such that not only their shapes and energy densities coincide, but also their linear fluctuation spectra are the same. Here we show that, again, there exist purely algebraic conditions for the Lagrangian densities which imply that the corresponding field theories are twins and that the fluctuation spectra about their defects coincide. These algebraic conditions allow to construct an infinite number of twins with coinciding fluctuation spectra for a given theory, and we provide some explicit examples. The importance of this result is related to the fact that coinciding defects with coinciding energy densities and identical fluctuation spectra are almost indistinguishable physically, that is, indistinguishable in a linear or semiclassical approximation. This implies that the measurable physical properties of a kink, in general, do not allow to determine the theory which provides the kink uniquely. Instead, in principle an infinite number of possible theories has to be considered. 11.1 Introduction One of the most fertile concepts in theoretical physics in the last decades has been the concept of topological defects or topological solitons (see e.g. [162]). They are ubiquitous in condensed matter systems and, besides this, are deemed relevant for the cosmology of the early universe. Topological defects may, for instance, contribute to the structure formation in the very early universe (e.g., during or at the end of inflation) [23]-[25]. A topological soliton is, in general, a static solution of the Euler–Lagrange equations of the given field theory with finite energy which obeys nontrivial boundary conditions. Further, the stability of the topological soliton against transitions to the vacuum is guaranteed by the fact that a deformation to the vacuum configuration with trivial boundary conditions would require to change the field in an infinite volume and, therefore, cost an infinite amount of energy. The relevant data characterizing the physical properties of a soliton are, first of all, its shape or profile (i.e., the soliton solution itself), and its energy density. Additional important information is contained in the so-called spectrum of linear fluctuations about the topological defect. In order to determine this spectrum, one calculates the fluctuations about the soliton up to second or- CHAPTER 11. MORE ON TWIN-LIKE MODELS 147 der in the action (or up to first order in the Euler–Lagrange equations). For the fluctuation field then one introduces a temporal Fourier decomposition, which results in a stationary second order equation of the Schr¨odinger type. The (in general, infinitely many) solutions of this equation together with the allowed frequencies constitute the spectrum of linear fluctuations. The first relevant information contained in the spectrum of linear fluctuations is linear stability. For a stable soliton, the spectrum should contain no negative mode (i.e., no imaginary frequency). Another aspect where the fluctuation spectrum is important is the issue of semiclassical quantization in the presence of solitons [72] (for an easy to follow discussion see [73]). Concretely, the discrete part of the fluctuation spectrum describes some excited states of the soliton or, equivalently, soliton-meson bound states. Here by ”meson” we mean a fluctuation field which is Gaussian in the leading approximation and obeys the boundary conditions of the vacuum configuration. Further, the continuous part of the spectrum describes soliton-meson scattering. The discussion so far has been for general soliton models, but now we want to restrict to the case of a real scalar field in 1+1 dimensions. The standard scalar field theory in 1+1 dimensions is Ls=X−U(φ), X ≡1 2∂µφ∂µφ(11.1) and we shall require that Uis nonnegative, U(φ)≥0∀φ(11.2) This theory may support topological solitons (kinks) provided that the potential Uhas at least two vacua, i.e., there exist at least two (constant) values φ=φisuch that U(φi) = 0. A kink is a static solution φk(x) which, in general, interpolates between two adjacent vacua, i.e., φk(−∞) = φi, φk(∞) = φi+1. The corresponding static kink equation is (φ0≡∂xφ) 1 2φ02≡ −X=U(11.3) with the two roots (for kink and antikink) φ0=±√2U. (11.4) CHAPTER 11. MORE ON TWIN-LIKE MODELS 148 The kink equation (11.3) results from the static second order Euler–Lagrange equation by performing one integration, where the integration constant must be set equal to zero in order to satisfy the kink boundary conditions. Finally, the linear fluctuation equation in the kink background may be derived by inserting the decomposition φ(t, x) = φk(x)+η(t, x) and the temporal Fourier decomposition η(t, x) = cos(ωt)η(x) into the Euler–Lagrange equation and keeping terms linear in η. Explicitly, the linear fluctuation equation reads (U,φ ≡∂φU, etc.) −η00 = (ω2−U,φφ|φk)η(11.5) where the notation |φkmeans that the expression has to be evaluated for the kink solution. The solutions of this Schr¨odinger type equation together with the allowed frequencies ωdetermine the spectrum of linear fluctuations in this case. Up to now the logical line of reasoning has been to begin with a field theory and to derive from this starting point the topological defect (kink) and its properties. Now we want to see whether and how far this logical arrow can be reversed. That is to say, we start with a kink solution together with its properties, like energy density and linear fluctuation spectrum, and we want to know whether or to which degree we may recover the theory which gives rise to this defect solution with its properties. The answer depends on the class of Lagrangians we are willing to admit. For a standard scalar field theory (11.1), the kink solution itself is already sufficient to recover the Lagrangian, i.e., the potential, by inverting the solution φ=φk(x)⇒x= xk(φ) and by inserting the resulting expression into the kink equation, φ02(x) = φ02(xk(φ)) ≡2U(φ),(11.6) which determines U(φ). on the other hand, the situation will be different if we allow for a more general class of Lagrangians. Concretely, we want to admit Lagrangians which are general functions of both φand X≡(1/2)∂µφ∂µφ. There are several reasons which make these theories with a generalized kinetic term (the so-called K field theories) worth considering. First of all, K field theories have been applied already to some problems in cosmology, like inflation (so-called K-inflation [2]), late time acceleration (so-called K-essence [3]), or in the brane world scenario [31], [7], [74]. Secondly, generalized kinetic CHAPTER 11. MORE ON TWIN-LIKE MODELS 149 terms may serve to stabilize static field configurations, evading thereby the Derrick theorem and allowing the existence of soliton solutions. The third and probably strongest case in favor of K field theories is related to the fact that in many circumstances scalar field theories are interpreted as effective field theories which result from the integration of UV degrees of freedom of some more fundamental theory. In this case of an effective field theory, higher powers of derivatives are induced naturally, and therefore they have to be taken into account. In this paper we are specifically interested in K field theories whose topological defects coincide with the standard ones, but let us mention, nevertheless, that K field theories in general give rise to a much richer phenomenology of possible topological defects [5], [61], like, e.g. solitons with compact support (so-called compactons) [35] - [40]. other more mathematical aspects of K field theories have been discussed, e.g., in [4] and in [62]. For the generalized dynamics of K field theories (i.e., for general Lagrangians L(X, φ)) it was found recently [51] that different field theories may exist which share the same topological defect with the same energy density. The coinciding kinks with their coinciding energy densities were dubbed twin or Doppelg¨anger defects in [51], and the models which give rise to these identical kink solutions are called twinlike models. The investigation of twinlike models was carried further in [75] and in [76]. Specifically, in [76] it was demonstrated that there exist purely algebraic necessary and sufficient conditions for a Lagrangian L(X, φ) to be the twin of a standard theory Ls=X−U. As these conditions are algebraic, they do not require the knowledge of the topological defect solution and, therefore, allow the simple construction of an infinite number of twins for any given standard field theory supporting topological defects. Very recently, in [77] explicit examples of K field theories were found which not only are twin models of standard field theories, but where also the fluctuation spectra of the standard defect and its K field twins coincide, making the standard defect and its twins almost completely indistingushable physically. This implies that the measurable physical properties of a kink, in general, do not allow to determine the theory which provides the kink uniquely. Instead, in principle an infinite number of possible theories has to be considered. CHAPTER 11. MORE ON TWIN-LIKE MODELS 150 It is the purpose of the present paper to show that, again, there exist purely algebraic conditions for a Lagrangian density which imply that the corresponding field theory is the twin of a standard scalar field theory and that the fluctuation spectra about their defects coincide. Further, these algebraic conditions allow to explicitly construct an infinite number of twins with coinciding fluctuation spectra for any given standard field theory. Concretely, in Sec. 11.2 we briefly review some known facts about twinlike models which we need. In Sec. 11.3, we derive the algebraic conditions for coinciding fluctuation spectra and provide some explicit examples. Further we discuss the relation of our results with the examples of Ref. [77]. Finally, Sec. 11.4 contains our conclusions. 11.2 Twinlike models The algebraic twin conditions require the first order form of the static field equations, so let us briefly review this issue (for more details see, e.g., [76], [32]). For a general Lagrangian L(X, φ) where X≡1 2∂µφ∂µφ=1 2(˙ φ2−φ02), the Euler–Lagrange equation reads ∂µ(L,X∂µφ)−L,φ = 0.(11.7) Further, the energy momentum tensor is Tµν =L,X∂µφ∂νφ−gµνL(11.8) which, for static configurations φ=φ(x), φ0≡∂xφ, simplifies to T00 =E=−L (11.9) T11 =P=L,Xφ02+L(11.10) where Eis the energy density and Pis the pressure. The static Euler– Lagrange equation may be integrated once to give −2XL,X +L≡P= 0.(11.11) The general first integral allows for a nonzero constant on the r.h.s. (nonzero pressure), but the boundary conditions for finite energy field configurations CHAPTER 11. MORE ON TWIN-LIKE MODELS 151 require this constant to be zero (zero pressure condition). For a standard field theory Ls=X−U, the energy density and pressure read Es=−X+U=1 2φ02+U(11.12) Ps=−X−U=1 2φ02−U, (11.13) and for a kink solution φkobeying φ02 k= 2Uthese simplify to Es|φk=−2X|φk= 2U|φk(11.14) −Ps=X+U≡0.(11.15) obviously, a K field theory will be the twin of a standard theory (i.e., have the same kink solution φkwith the same energy density) if both Eand P ≡ 0 agree when evaluated for the kink solution. A necessary and sufficient condition for the K field Lagrangian is [51] L|φk=−2U(11.16) L,X|φk= 1,(11.17) as may be checked easily. Now the important point is that the first order form φ02=−2X= 2Uof the static kink equation may be interpreted as an algebraic equation involving the variables Xand φon which the K field Lagrangian depends. As a consequence, the evaluation condition |φkmay be replaced by the purely algebraic condition |X=−U, leading to the so-called algebraic twin conditions [76] L|X=−U≡ L| =−2U(11.18) L,X|X=−U≡ L,X |= 1 (11.19) (here and below the evaluation of an expression at X≡ −(1/2)φ02=−U (and its prolongations, when required) will always be denoted by the vertical line |, and will be called on-shell condition or on-shell evaluation frequently). CHAPTER 11. MORE ON TWIN-LIKE MODELS 152 11.3 The algebraic conditions 11.3.1 The fluctuation equation We start from the Euler–Lagrange equation (11.7) and insert the decomposition φ(t, x) = φk(x) + η(t, x) (11.20) where φkis the kink solution and ηis the fluctuation field. In first order in ηwe find ∂µ(L,X∂µη+L,XX∂νφk∂νη∂µφk+L,Xφη∂µφk)− − L,φφη−L,Xφ∂µφk∂µη= 0.(11.21) Now we use the fact that φkonly depends on x, and the ansatz for the fluctuation field η(t, x) = cos(ωt)η(x) (11.22) and get −L,Xη0+L,XX(φ0 k)2η0−L,Xφφ0 kη0− − L,φφη+L,Xφφ0 kη0−ω2L,Xη= 0 (11.23) or, more explicitly −(L,X + 2XL,XX )η00 − −(L,Xφ + 2XL,XXφ −φ00 k(3L,XX + 2XL,XXX ))φ0 kη0= =ω2L,X +L,φφ −2XL,Xφφ +φ00 k(L,Xφ + 2XL,XXφ)η. (11.24) This expression should now be evaluated for the defect solution φk, i.e., implementing the on-shell condition X|=−Uand its first prolongation (that is, the original second order static field equation) φ00| ≡ φ00 k=U,φ. Inserting these on-shell expressions above produces an expression containing Uand its derivative, whereas the variables of Land its derivatives are X(= −U) and φ. 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