Higher Derivative Corrections to Manifestly Supersymmetric Nonlinear Realizations
aa r X i v : . [ h e p - t h ] N ov August, 2014
Higher Derivative Corrections toManifestly Supersymmetric Nonlinear Realizations
Muneto Nitta † a and Shin Sasaki ‡ b † Department of Physics, and Research and Education Center for Natural Sciences,Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan ‡ Department of Physics, Kitasato UniversitySagamihara 252-0373, Japan
Abstract
When global symmetries are spontaneously broken in supersymmetric vacua, thereappear quasi-Nambu-Goldstone (NG) fermions as superpartners of NG bosons. In addi-tion to these, there can appear quasi-NG bosons in general. The quasi-NG bosons andfermions together with the NG bosons are organized into chiral multiplets. K¨ahler poten-tials of low-energy effective theories were constructed some years ago as supersymmetricnonlinear realizations. It is known that higher derivative terms in the superfield formalismoften encounter the auxiliary field problem; the auxiliary fields that accompanied withspace-time derivatives and it cannot be eliminated. In this paper, we construct higherderivative corrections to supersymmetric nonlinear realizations in the off-shell superfieldformalism free from the auxiliary field problem. As an example, we present the manifestlysupersymmetric chiral Lagrangian. a nitta(at)phys-h.keio.ac.jp b shin-s(at)kitasato-u.ac.jp Introduction
Low-energy field theories can be described by only light fields when one integrates out massiveparticles above the scale which one considers. In particular, when a global symmetry of La-grangian or Hamiltonian is spontaneously broken in the ground state or vacuum, there appearNambu-Goldstone (NG) bosons as massless scalar fields. The low-energy dynamics of these NGbosons is solely determined from the symmetry argument. When a symmetry group G is spon-taneously broken down to its subgroup H , the low-energy dynamics is governed by a nonlinearsigma model whose target space is the coset space G/H [1]. A prime example is the chiralLagrangian of pions which appear as NG bosons when the chiral symmetry of QCD is sponta-neously broken. Low-energy effective theories are usually expanded by the number of space-timederivatives, thereby they inevitably contain higher derivative corrections. It is known that thechiral perturbation theory includes derivative corrections to the chiral Lagrangian [2].On the other hand, supersymmetry plays important roles to control quantum corrections infield theories and determines the exact low-energy dynamics [3]. It is also a necessary ingredi-ent to define consistent string theories. It was also proposed as the most promising candidateto solve the naturalness problem in the Standard Model. Among other things, when a globalsymmetry is spontaneously broken in supersymmetric vacua, there appear quasi-NG fermions[4] in addition to the NG bosons. They are required to form chiral supermultiplets as super-partners of NG bosons. In model building of particle physics, quasi-NG fermions were identifiedas quarks in supersymmetric preon models [5]. The target spaces of supersymmetric nonlin-ear sigma models must be K¨ahler [6] because the lowest components of chiral superfields arecomplex scalar fields. When a coset space
G/H is eventually K¨ahler, there are no additionalmassless fields. However,
G/H is not K¨ahler in general, and in that case, there must appearquasi-NG bosons [7] in addition to the NG bosons, to parameterize a K¨ahler manifold. In thiscase, target spaces of low-energy effective theories are enlarged from
G/H . In general, the prob-lem to construct low-energy effective theories of massless fields reduces to finding G -invariantK¨ahler potentials. The most general framework to construct G -invariant K¨ahler potentials wasprovided as supersymmetric nonlinear realizations [8]. The authors of [8] classified NG super-multiplets into P-type, containing two NG boson, and M-type, containing one NG boson andone quasi-NG boson. In one extreme class called a pure realization, all supermultiplets are ofP-type and there are no quasi-NG bosons, which is possible only when G/H happens to beK¨ahler. In this case, the most general G -invariant K¨ahler potential up to K¨ahler transforma-1ions was constructed in Refs. [8, 9] (see Ref. [10] as a review), which is unique up to finitenumber of decay constants (K¨ahler class). This class was studied extensively in the literature(see, e.g., Refs. [11] and references in Ref. [10] ). In the other extreme class called a maximalrealization, all supermultiplets are of M-type so that there are the same number of quasi-NGbosons with NG bosons. The target manifold in this case is a cotangent bundle T ∗ ( G/H ),whose cotangent directions are parameterized by quasi-NG bosons. For instance, the chiralsymmetry breaking belongs to this class [12]. If there is at least one quasi-NG boson, theeffective K¨ahler potential is an arbitrary function of strict G -invariants [8]. Geometrically thisarbitrariness corresponds to a degree of freedom to deform non-compact directions of the targetspace, which cannot be controlled by the isometry G [12, 13, 14, 15]. These directions are asso-ciated with the quasi-NG bosons. It was proved that there must appear at least one quasi-NGboson in the absence of gauge interactions [16, 17, 18]. When there is a gauge symmetry onthe other hand, pure realizations without quasi-NG bosons are possible by absorbing M-typesuperfields by the supersymmetric Higgs mechanism [19].While the superfield formalism is one of the most powerful off-shell formulations to con-struct manifestly supersymmetric Lagrangians, it often encounters an auxiliary field problemwhen higher derivative terms exist in the Lagrangians. For example, chiral superfields withspace-time derivatives (e.g. ∂ m Φ) contain derivatives on the auxiliary fields F so that theycannot be eliminated by their equations of motion. This problem was recognized [20, 21] for asupersymmetric extension of Wess-Zumino-Witten (WZW) term [22] in the chiral Lagrangianof supersymmetric QCD. A supersymmetric WZW term proposed in Ref. [23] does not have thisproblem. Supersymmetric Lagrangians free from the auxiliary field problem were also knownbefore, such as supersymmetric Dirac-Born-Infeld action [24], supersymmetric higher derivative C P models [25, 26], supersymmetric baby Skyrme models [27, 28] and supersymmetric k-fieldtheories [29, 30]. The most general model of chiral superfields with higher derivative termswas recently presented in Ref. [31], where it was called a supersymmetric P ( X, ϕ ) model. Thehigher derivative interaction can be written by using a target space tensor with two holomor-phic and two anti-holomorphic indices which are both symmetric. This term was first found inRef. [32] as a quantum correction term in a chiral model, and the supersymmetric WZW termin Ref. [23] also contains it [33]. The model in Ref. [31] was extended by the introduction of asuperpotential [34] and coupling to supergravity [35, 36], and was applied to the supersymmet-ric Galileon inflation models [37] and the ghost condensation [38]. In our previous paper [39],2e have classified 1/2 and 1/4 Bogomol’nyi-Prasad-Sommerfield (BPS) equations for domainwalls, lumps, baby Skyrmions and domain wall junctions. See also Ref. [40] for further studyon baby Skyrmions.In this paper, we construct higher derivative corrections to supersymmetric nonlinear real-izations for spontaneous broken global symmetries with keeping supersymmetry. As the leadingtwo derivative terms for pure realizations without quasi-NG bosons, we find that the higherderivative terms are unique up to constants. On the other hand, higher derivative terms con-tain arbitrary functions in the presence of quasi-NG bosons. As one of the most importantexamples, we discuss chiral symmetry breaking in detail.This paper is organized as follows. In Sec. 2, we give a brief review on supersymmetricnonlinear realizations. In Sec. 3 we discuss higher derivative corrections to nonlinear realiza-tions. In Sec. 3.1, we introduce the supersymmetric higher derivative chiral model with foursupercharges. We write down the equation of motion for the auxiliary fields and analyze thestructure of the on-shell Lagrangians. In Sec. 3.2, we discuss higher derivative corrections topure realizations in the absence of quasi-NG bosons, for which each massless chiral superfieldcontains two NG bosons and there are no quasi-NG bosons. In Secs. 3.3 and 3.4, we discusshigher derivative corrections in the presence of quasi-NG bosons. In Sec. 4, we discuss higherderivative corrections for supersymmetric chiral symmetry breaking, which is a maximal real-ization where each massless chiral superfield contains one NG boson and one quasi-NG boson.Section 5 is devoted to conclusion and discussions. We use the notation of the textbook of Wessand Bagger [41].
In this section, we review supersymmetric nonlinear realizations formulated in Ref. [8].
When a global symmetry group G is spontaneously broken down to its subgroup H , thereappear massless Nambu-Goldstone (NG) bosons associated with broken generators of the cosetmanifold G/H . At low energies, interactions among these massless particles are described bythe so-called nonlinear sigma models, whose Lagrangians in the leading order of derivative ex-pansions are completely determined by the geometry of the target manifold
G/H parameterized3y NG bosons as was found by Callan, Coleman, Wess and Zumino [1].In four-dimensional N = 1 supersymmetric theories, scalar fields belong to chiral superfieldsΦ i ( i = 1 , · · · , N ) whose component expansion in the chiral base y m = x m + iθσ m ¯ θ isΦ i ( y, θ ) = ϕ i ( y ) + θψ i ( y ) + θ F i ( y ) , (2.1)where ϕ i is the complex scalar field, ψ i is the Weyl fermion and F i is the complex auxiliaryfield.When a global symmetry is spontaneously broken in supersymmetric vacua, there appearmassless fermions ψ i as supersymmetric partners of NG bosons [4]. These massless fermionstogether with NG bosons are described by chiral superfields. Since chiral superfields are com-plex, the supersymmetric nonlinear sigma models are closely related to the complex geometry;their target manifolds, where fields variables take their values, must be K¨ahler manifolds [6]. Ifthe coset manifold G/H itself happens to be a K¨ahler manifold, both real and imaginary partsof the scalar components of chiral superfields are NG bosons. If
G/H is not a K¨ahler manifold,on the other hand, there is at least one chiral superfield whose real or imaginary part is not aNG boson. This additional massless boson is called the quasi-NG boson [7].We explain how quasi-NG bosons appear. The spontaneous symmetry breaking of a globalsymmetry G in supersymmetric theories is caused by the superpotential W : the chiral su-perfields acquire the vacuum expectation values v = h ϕ i as a result of the F-term condition ∂W∂ϕ = 0. Since the superpotential W is holomorphic namely, it contains only chiral superfields,this condition is invariant under the complex extension of G , namely, G C . Hence, if we definethe complex isotropy group ˆ H ( ⊂ G C ) by ˆ Hv = v, ˆ H v = 0 , (2.2)the target space parameterized by NG and quasi-NG bosons can be written as a complex cosetspace: M ≃ G C / ˆ H. (2.3)In general, ˆ H is larger than H C , and it is decomposed asˆ H = H C ⊕ B , (2.4) We use the calligraphic font for a Lie algebra corresponding to a Lie group. B consists of non-hermitian generators E ∈ ˆ H and is called (the subalgebra of) the Borelsubalgebra in ˆ H [8]. • As an example, let us consider a doublet φ = ( φ , φ ) T of G = SU (2) and supposethat they acquire the vacuum expectation values v = (1 , T . Since the raising operator σ + = ( σ + iσ ) = satisfies σ + v = 0, it is the complex unbroken generator inˆ H . On the other hand, σ and the lowering operator σ − (= σ + † ) are the elements of thebroken generators in G C − ˆ H .The coset representative can be written as ξ (Φ) = exp( i Φ · Z ) ∈ G C / ˆ H, Z ∈ G C − ˆ H , (2.5)where Z are complex broken generators and Φ are NG chiral superfields generated by them.There are two kinds of broken generators: the hermitian broken generators X and the non-hermitian broken generators ¯ E : G C − ˆ H = { Z } = { X, ¯ E } . (2.6)The NG superfields Φ corresponding to non-hermitian and hermitian generators are called P-type (or non-doubled-type ) and
M-type (or doubled-type ) superfields, respectively [8, 16]. Notethat there are as many non-hermitian broken generators ¯ E as non-hermitian unbroken genera-tors E , since they are hermitian conjugate to each other. On a suitable basis, ¯ E and E can bewritten as off-diagonal lower and upper half matrices respectively. • In the previous example where the representative of G C / ˆ H is given by φ = exp i ( ϕ σ + ϕσ − ) · v , ϕ is a M-type and ϕ is a P-type superfield. The non-hermitian broken generator¯ E = σ − written as a lower half matrix is hermitian conjugate to the non-hermitian un-broken generator E = σ + written as a upper half matrix.The directions parameterized by quasi-NG bosons are non-compact, whereas those of NGbosons are compact. The scalar components of the M-type superfields consist of a quasi- In the group level, ˆ H can be written as a semi-direct product of H C and the Borel subgroup B : ˆ H = H C ∧ B .Here the symbol ∧ denotes a semi-direct product. If there are two elements of ˆ H , hb and h ′ b ′ , where h, h ′ ∈ H C and b, b ′ ∈ B , their product is defined as ( hb )( h ′ b ′ ) = hh ′ ( h ′− bh ′ ) b ′ = ( hh ′ )( b ′′ b ), where b ′′ = h ′− bh ′ ∈ B [8].It is, however, sufficient to consider only the Lie algebra in this paper. We use the word “compactness” in the sense of topology. The kinetic terms of quasi-NG bosons have thesame sign as those of NG bosons.
5G boson in addition to a NG boson, whereas those of the P-type superfields consist of twogenuine NG bosons. This can be understood as follows: note that, for each non-hermitianbroken generator ¯ E , there is a non-hermitian unbroken generator E . Since the vacuum isinvariant under ˆ H , we can multiply the representative of the coset manifold by an arbitraryelement of ˆ H from the right. Hence, for any P-type superfield Φ generated by a non-hermitiangenerator ¯ E , there exists an element exp( i Φ † E ) ∈ ˆ H such that ξv = exp i ( · · · + Φ ¯ E + · · · ) v = exp i ( · · · + Φ ¯ E + · · · ) exp( i Φ † E ) v = exp i ( · · · + ℜ Φ X + ℑ Φ X + O (Φ ) + · · · ) v, (2.7)where we have used the Baker-Campbell-Hausdorff formula and defined two hermitian brokengenerators X = ¯ E + E, X = i ( ¯ E − E ). Here ℜ and ℑ denote real and imaginary parts,respectively. Therefore two scalar components of the P-type superfield parameterize compactdirections, and hence are considered NG bosons. On the other hand, since any M-type superfieldis generated by an hermitian generator, there is no partner in ˆ H . Therefore its imaginary partof scalar component parameterizes a non-compact direction, and hence is considered to be aquasi-NG boson. • In our previous example, we can rewrite it as exp i ( ϕ σ + ℜ ϕσ + ℑ ϕσ ) · v by multiplyingan appropriate factor generated by σ + for sufficiently small | ϕ | and | ϕ | . The NG bosonsparameterizing S ≃ SU (2) are ℜ ϕ , ℜ ϕ, ℑ ϕ , whereas ℑ ϕ is the quasi-NG bosonparameterizing the radius of S .As a notation, we write the number of chiral superfields N Φ parameterizing the targetmanifold as N Φ = N M + N P , (2.8)where the numbers of the M-type and P-type superfields are denoted by N M and N P , respec-tively. The number of quasi-NG bosons is N Q = N M = 2 dim C ( G C / ˆ H ) − dim( G/H ) = dim(
G/H ) − dim B. (2.9) We use ‘dim C ’ for complex dimensions and ‘dim’ for real dimensions. B = dim( G/H ),there is no quasi-NG boson. This case is called the pure realization ( total pairing or non-doubling ). On the other hand, if there is no Borel subalgebra, dim B = 0, there appear as manyquasi-NG bosons as NG bosons. This case is called the maximal realization (or full-doubling ).It is known that a pure realization cannot occur in the model with a linear origin without gaugesymmetry [16, 17, 18]. It was shown in Ref. [16] that a maximal realization occurs when a fieldbelonging to a real representation obtains a vacuum expectation value or when NG boson part G/H brought by a vacuum expectation value is a symmetric space. In the presence of a gaugesymmetry, pure realizations without quasi-NG bosons are possible, since gauge fields absorbM-type superfields as a consequence of the supersymmetric Higgs mechanism [19]. G -invariant K¨ahler potentials The kinetic term in the effective Lagrangian is described by the K¨ahler potential K (Φ , Φ † ) ofNG chiral superfields L = Z d θd ¯ θ K (Φ , Φ † ) = − g i ¯ j ( ϕ, ¯ ϕ ) ∂ µ ϕ i ∂ µ ¯ ϕ ¯ j + (fermion terms) , (2.10)where we have eliminated the auxiliary fields F i by its equation of motion and g i ¯ j ≡ ∂∂ϕ i ∂∂ ¯ ϕ ¯ j K ( ϕ, ¯ ϕ )is the K¨ahler metric. Since the K¨ahler potential includes both chiral and anti-chiral superfields,the symmetry group of the effective theory is still G , but not its complexification. Hence ourgoal is to construct G -invariant K¨ahler potentials of complex coset spaces G C / ˆ H . Here the G -invariance means K (Φ , Φ † ) g → K (Φ ′ , Φ ′† ) = K (Φ , Φ † ) + F (Φ , g ) + F ∗ (Φ † , g ) , (2.11)where F ( F ∗ ) is a (anti-)holomorphic function of Φ (Φ † ) which depends on g ∈ G . The lattertwo terms in Eq. (2.11) disappear in the superspace integral R d θ . Since the redefinitionof the K¨ahler potential by adding holomorphic and anti-holomorphic functions is called the
K¨ahler transformation , we denote that it is G -invariant under a K¨ahler transformation or quasi G -invariant if F (Φ , g ) exists in Eq. (2.11).First of all, we note that the transformation law under G of the representative ξ of thecomplex coset G C / ˆ H is ξ g → ξ ′ = gξ ˆ h − ( g, ξ ) , (2.12) Here F (Φ , g ) is called the cocycle function , which satisfies the cocycle condition , F (Φ , g g ) = F ( g Φ , g ) + F (Φ , g ). h − ( g, ξ ) is a compensator to project gξ onto the coset representative, see Fig. 1.Figure 1: The G -transformation law for ξ .Bando et al. constructed the following three types of G -invariant K¨ahler potentials calledA-, B- and C-types [8]. A-type . We prepare a representation ( ρ, V ) of G in the representation space V . If thereare ˆ H invariant vectors v a , ρ ( ˆ H ) v a = v a , (2.13)the transformation law of the quantity ρ ( ξ ) v a under G is ρ ( ξ ) v a g → ρ ( ξ ′ ) v a = ρ ( g ) ρ ( ξ ) ρ (ˆ h − ) v a = ρ ( g ) ρ ( ξ ) v a . (2.14)Then, by using strict G -invariants X ab ≡ v † a ρ ( ξ † ξ ) v b , (2.15)we can construct a G -invariant K¨ahler potential K A (Φ , Φ † ) = f ( X ab ) , (2.16)where f is an arbitrary real function of all possible G -invariants X ab . B-type . It is sufficient to consider the fundamental representation [8], hence we do not write ρ for simplicity. We need the projection matrices, which project a fundamental representationspace onto an ˆ H invariant subspace. They satisfy the projection conditions, η † a = η a , η a ˆ Hη a = ˆ Hη a , η a = η a . (2.17)8efine the projected determinant asdet η A ≡ det( ηAη + − η ) , (2.18)where det η stands for the determinant in the projected space. By using these, if we construct K B (Φ , Φ † ) = X a c a log det η a ξ † ξ, (2.19)it is G -invariant up to a K¨ahler transformation:log det η ξ † ξ g → log det η ξ ′† ξ ′ = log det η ( ηξ ′† ξ ′ η )= log det η ( η ˆ h †− ξ † ξ ˆ h − η )= log det η ( η ˆ h †− ηξ † ξη ˆ h − η )= log det η ( η ˆ h †− ηηξ † ξηη ˆ h − η )= log det η ξ † ξ + log det η ˆ h − + log det η ˆ h †− , (2.20)where the last two terms include only chiral and anti-chiral superfields respectively, and disap-pear in the superspace integral R d θ . Here we have used Eq. (2.17).
C-type . Again, the fundamental representation is sufficient [8]. We define [ A ] − η ≡ [ ηAη + − η ] − , where the inverse is calculated in the projected space. The quantities defined by P a = ξη a [ ξ † ξ ] − η a η a ξ † (2.21)transform under G as P g → P ′ = ξ ′ η [ ξ ′† ξ ′ ] − η ηξ ′† = gξ ˆ h − η [ η ˆ h − † ξ † ξ ˆ h − η ] − η η ˆ h − † ξ † g † = gξη ( η ˆ h − η )[ η ˆ h − † ηηξ † ξηη ˆ h − η ] − η ( η ˆ h − † η ) ηξ † g † = gξη [ˆ h − ] η ([ˆ h − † ] η [ ξ † ξ ] η [ˆ h − ] η ) − η [ˆ h − † ] η ηξ † g † = gP g † . (2.22) This can be rewritten as [42], K B = P a log det ′ ( ξη a ξ † ), where det ′ is a determinant except zero eigenvalues. Here the cocycle function F (Φ , g ) = log det η ˆ h − ( g, ξ (Φ)) satisfies the cocycle condition. The meaning of P a can be understood as follows [42]. Since P a satisfies the properties P † a = P a , P a = P a , tr P a = tr η a , P a | Φ=0 = η a ,P a can be considered to be the transformation of η a from the origin Φ = 0 (or ξ = 1) to Φ = 0 in the manifold.
9y noting the relations P a = ξη a [ ξ † ξ ] − η a ( η a ξ † ξη a )[ ξ † ξ ] − η a η a ξ † = P a , (2.23)tr P a = tr ([ ξ † ξ ] − η a ( η a ξ † ξη a )) = tr η a = const , (2.24)a G invariant K¨ahler potential can be constructed as K C (Φ , Φ † ) = f (tr ( P a P b ) , tr ( P a P b P c ) , · · · ) , (2.25)where f is again an arbitrary real function and all the indices a, b, c, · · · are different. In this section we study higher derivative corrections to supersymmetric nonlinear realizations.In the first subsection, we present general higher derivative chiral models with multiple chiralsuperfields. In the second subsection, we consider pure realizations described by B-type K¨ahlerpotentials, for which each massless chiral superfield contains two NG bosons. In the third andfourth subsections, we consider A and C-type K¨ahler potentials, respectively, for which somechiral superfields are M-type superfields, consisting of one quasi-NG boson and one genuine NGboson.
We consider higher derivative terms generated by multiple chiral superfields Φ i in which nodynamical (propagating) auxiliary fields exist. The supersymmetric higher derivative term canbe given by [27, 28, 31, 39] L H . D . = 116 Z d θ Λ ik ¯ j ¯ l (Φ , Φ † ) D α Φ i D α Φ k ¯ D ˙ α Φ † ¯ j ¯ D ˙ α Φ † ¯ l . (3.1)Here the supercovariant derivatives are defined as D α = ∂∂θ α + i ( σ m ) α ˙ α ¯ θ ˙ α ∂ m , ¯ D ˙ α = − ∂∂ ¯ θ ˙ α − iθ α ( σ m ) α ˙ α ∂ m . (3.2)where the sigma matrices are σ m = ( , ~τ ) with the Pauli matrices ~τ = ( τ , τ , τ ). Since theterm D α Φ i behaves as a vector D α Φ ′ i = ∂ Φ ′ i ∂ Φ j D α Φ j (3.3)10nder field redefinition Φ i → Φ i ′ (Φ), Λ ik ¯ j ¯ l can be regarded as a (2 ,
2) K¨ahler tensor symmetricin holomorphic and anti-holomorphic indices, whose components are functions of Φ i and Φ † ¯ i (admitting space-time derivatives acting on them).We write down the bosonic components of the Lagrangian (3.1). The component expansionof the N = 1 chiral superfield in the x -basis isΦ i ( x, θ, ¯ θ ) = ϕ i + iθσ m ¯ θ∂ m ϕ i + 14 θ ¯ θ ✷ ϕ i + θ F i , (3.4)where only the bosonic components are presented. Then, the bosonic component of the super-covariant derivatives of Φ i can be calculated as D α Φ i D α Φ k ¯ D ˙ α Φ † ¯ j ¯ D ˙ α Φ † ¯ l = 16 θ ¯ θ h ( ∂ m ϕ i ∂ m ϕ k )( ∂ m ¯ ϕ ¯ j ∂ m ¯ ϕ ¯ l ) − (cid:0) ∂ m ϕ i F k + F i ∂ m ϕ k (cid:1) (cid:16) ∂ n ¯ ϕ ¯ j ¯ F ¯ l + ¯ F ¯ j ∂ n ¯ ϕ ¯ l (cid:17) + F i ¯ F ¯ j F k ¯ F ¯ l (cid:21) . (3.5)Since the bosonic part of the right hand side of (3.5) saturates the Grassmann coordinate θ ¯ θ ,only the lowest component of the tensor Λ ik ¯ j ¯ l contributes to the bosonic part of the Lagrangian.Therefore the bosonic part of the Lagrangian (2.10) with the higher derivative term (3.1) is L b = g i ¯ j ( − ∂ m ϕ i ∂ m ¯ ϕ ¯ j + F i ¯ F ¯ j ) + ∂W∂ϕ i F i + ∂ ¯ W∂ ¯ ϕ ¯ j ¯ F ¯ j + Λ ik ¯ j ¯ l ( ϕ, ¯ ϕ ) h ( ∂ m ϕ i ∂ m ϕ k )( ∂ n ¯ ϕ ¯ j ∂ n ¯ ϕ ¯ l ) − ∂ m ϕ i F k ∂ m ¯ ϕ ¯ j ¯ F ¯ l + F i ¯ F ¯ j F k ¯ F ¯ l i , (3.6)where we have introduced the superpotential W for generality. The model is manifestly (off-shell) supersymmetric and K¨ahler invariant provided that K and W are scalars and Λ ik ¯ j ¯ l isa tensor. The auxiliary fields F i do not have space-time derivatives and consequently can beeliminated by the following algebraic equation of motion, g i ¯ j F i − ∂ m ϕ i F k Λ ik ¯ j ¯ l ∂ m ¯ ϕ ¯ l + 2Λ ik ¯ j ¯ l F i F k ¯ F ¯ l + ∂ ¯ W∂ ¯ ϕ ¯ j = 0 . (3.7)Since NG fields are all massless, we consider the vanishing superpotential W = 0 . In thiscase, F i = 0 is a solution to this equation, and the on-shell Lagrangian becomes L b = − g i ¯ j ∂ m ϕ i ∂ m ¯ ϕ ¯ j + Λ ik ¯ j ¯ l ( ∂ m ϕ i ∂ m ϕ k )( ∂ n ¯ ϕ ¯ j ∂ n ¯ ϕ ¯ l ) . (3.8) If we consider the spontaneously breaking of approximate symmetries, a non-zero superpotential W thatprovides small mass to the pseudo-NG modes is possible.
11e call this canonical branch. We note that the second term in (3.8) contains more than theforth order of space-time derivatives for appropriate functions Λ ik ¯ j ¯ l . We will demonstrate anexample of sixth-derivative terms in Sec. 4.2.In general, there are more solution other than F i = 0, although an explicit solutions F i isnot easy to find except for one component field. Indeed, for single superfield models, we haveother on-shell branches associated with solutions F i = 0 [28, 39]. We call this non-canonicalbranch. In the non-canonical branch, the ordinary kinetic term with two space-time derivativesvanishes and the on-shell Lagrangian consists of only four-derivative terms. Although it isinteresting, we do not consider this branch because we are considering derivative expansions. When there are no quasi-NG modes, it is called a pure realization. This is possible only when
G/H is eventually K¨ahler. When there is a gauge symmetry, the pure realization withoutquasi-NG bosons is possible [19]. From the Borel’s theorem, compact K¨ahler coset spaces
G/H can be written as
G/H = G/ [ H s . s . × U (1) r ] (3.9)with H s . s . the semi-simple subgroup in H and r ≡ rank G − rank H s . s . [43]. In this case, thereexists the isomorphism G/H ≃ G C / ˆ H. (3.10)The most general G -invariant K¨ahler potential (up to K¨ahler transformations) was shown tobe written solely by B-type K¨ahler potentials and A and C-types were shown not to giveindependent K¨ahler potentials [8, 9, 10].Now we consider higher derivative terms. In this case, the problem is reduced to find G invariant (2 ,
2) tensors Λ ik ¯ j ¯ l on the target manifold G/H . The G -transformation on the fieldsare δ Φ iA = k iA , (3.11)where k iA (Φ) ( A = 1 , , · · · , dim G ) are holomorphic Killing vectors generated by the isometry G , preserving the metric L k g i ¯ j = 0. The (2 ,
2) tensors Λ ik ¯ j ¯ l for higher derivative term must be12reserved by the isometry G : L k Λ ik ¯ j ¯ l = 0. Then, G -invariant four derivative terms are givenby L (4) = 116 Z d θ Λ ik ¯ j ¯ l (Φ , Φ † ) D α Φ i D α Φ k ¯ D ˙ α Φ † ¯ j ¯ D ˙ α Φ † ¯ l , Λ ik ¯ j ¯ l = w g ( i ¯ j g k ¯ l ) + w R i ¯ jk ¯ l + w R ( i ¯ j R k ¯ l ) + w g ( i ¯ j R k ¯ l ) (3.12)where R i ¯ jk ¯ l and R i ¯ j are the Riemann curvature and Ricci-form, respectively, brackets ( ... ) implysymmetrization over holomorphic and anti-holomorphic indices, and w , , are real constants.The scalar curvature R is also invariant but it is just a constant for G/H . The explicit formof the curvature tensor can be found in Ref. [44]. In some cases, the terms in Eq. (3.12) arenot independent. For Einstein manifolds, R i ¯ j ∼ g i ¯ j holds. For instance, rank one cases ( r = 1)belong to this class.An important fact is that there are no strict G -invariant, unlike the case with quasi-NGbosons which we discuss in the next subsections. This is the reason why higher derivative termsare uniquely determined up to constants.As for derivative terms higher than four derivatives, one uses the covariant derivatives oftensors such as D g ¯ D ¯ h R i ¯ jk ¯ l . For instance, a six-derivative term can be constructed as L (6) = 116 Z d θ D g ¯ D ¯ h R i ¯ jk ¯ l ∂ m Φ g ∂ m Φ † ¯ h D α Φ i D α Φ k ¯ D ˙ α Φ † ¯ j ¯ D ˙ α Φ † ¯ l + · · · . (3.13) The K¨ahler potential of A-type is given in Eq. (2.16). There are two ways to construct G -invariant four-derivative terms using the A-type invariants. The first way is a geometricalmethod which is the same with pure realizations, and the second way is a group theoreticalmethod.In the first method, G -invariant four-derivative terms are given by L (4) = 116 Z d θ Λ ik ¯ j ¯ l (Φ , Φ † ) D α Φ i D α Φ k ¯ D ˙ α Φ † ¯ j ¯ D ˙ α Φ † ¯ l , Λ ik ¯ j ¯ l = w ( X ab ) g ( i ¯ j g k ¯ l ) + w ( X ab ) R i ¯ jk ¯ l + w ( X ab ) R ( i ¯ j R k ¯ l ) + w ( X ab ) g ( i ¯ j R k ¯ l ) . (3.14)Unlike the B-type case, w , , , are arbitrary functions of the strict G -invariants X ab . The scalarcurvature R is a function of X ab and is not included.Now we introduce the second method to construct G -invariant four-derivative terms. Herewe do not write the representation ρ for simplicity. First, the Maurer-Cartan one form on13 C / ˆ H is given by iξ − dξ = ( E Ii (Φ) X I + ω ai (Φ) H a ) d Φ i (3.15)with the holomorphic vielbein E Ii (Φ) and the holomorphic connection ω ai (Φ). By using thisexpression, we calculate D α ξ = D α Φ i ∂ i ξ = iξ ( E Ii (Φ) X I + ω ai (Φ) H a ) D α Φ i , (3.16) D α ξv a = i ( ξX I v a ) E Ii (Φ) D α Φ i . (3.17)Then, the supercovariant derivatives of the G -invariants X ab given in Eq. (2.15) can be calcu-lated to be D α X ab = ( v a ξ † ξX I v b ) E Ii (Φ) D α Φ i (3.18) D α D α X ab = ( v a ξ † ξX J X I v b ) E Ii (Φ) E Jj (Φ) D α Φ i D α Φ j (3.19)¯ D ˙ α D α X ab = ( v a X † J ξ † ξX I v b ) E Ii (Φ) E ∗ Jj (Φ † ) D α Φ i ¯ D ˙ α Φ † ¯ j , (3.20)¯ D ˙ α D α X ab ¯ D ˙ α D α X cd = ( v a X † J ξ † ξX I v b )( v c X † L ξ † ξX K v d ) E Ii (Φ) E Kk (Φ) E ∗ Jj (Φ † ) E ∗ Ll (Φ † ) × D α Φ i D α Φ k ¯ D ˙ α Φ † ¯ j ¯ D ˙ α Φ † ¯ l , (3.21)¯ D ˙ α ¯ D ˙ α D α D α X ab = ( v a X † J X † L ξ † ξX K X I v b ) E Ii (Φ) E Kk (Φ) E ∗ Jj (Φ † ) E ∗ Ll (Φ † ) × D α Φ i D α Φ k ¯ D ˙ α Φ † ¯ j ¯ D ˙ α Φ † ¯ l , (3.22) D α D α X ab ¯ D ˙ α ¯ D ˙ α X cd = ( v a ξ † ξX K X I v b )( v c X † J X † L ξ † ξv d ) E Ii (Φ) E Kk (Φ) E ∗ Jj (Φ † ) E ∗ Ll (Φ † ) × D α Φ i D α Φ k ¯ D ˙ α Φ † ¯ j ¯ D ˙ α Φ † ¯ l . (3.23)By using these relations, four-derivative terms can be given by L (4) = 116 Z d θ (cid:2) g ab ( X mn ) ¯ D ˙ α ¯ D ˙ α D α D α X ab + g abcd ( X mn ) ¯ D ˙ α D α X ab ¯ D ˙ α D α X cd + g abcd ( X mn ) D α D α X ab ¯ D ˙ α ¯ D ˙ α X cd + g abcdef ( X mn ) D α X ab D α X cd ¯ D ˙ α ¯ D ˙ α X ef + g abcdef ( X mn ) D α D α X ab ¯ D ˙ α X cd ¯ D ˙ α X ef + g abcdef ( X mn ) D α X ab D α ¯ D ˙ α X cd ¯ D ˙ α X ef + g abcdefgh ( X mn ) D α X ab D α X cd ¯ D ˙ α X ef ¯ D ˙ α X gh (cid:3) (3.24)with arbitrary real functions g ab ··· of the G -invariants X mn . From this equation, the components14f Λ ik ¯ j ¯ l can be read asΛ ik ¯ j ¯ l = h g ab ( X mn )( v a X † J X † L ξ † ξX K X I v b )+ g abcd ( X mn )( v a X † J ξ † ξX I v b )( v c X † L ξ † ξX K v d )+ g abcd ( X mn )( v a ξ † ξX K X I v b )( v c X † J X † L ξ † ξv d )+ g abcdef ( X mn )( v a ξ † ξX I v b )( v c ξ † ξX K v d )( v e X † J X † L ξ † ξv f )+ g abcdef ( X mn )( v a ξ † ξX K X I v b )( v c X † J ξ † ξv d )( v e X † L ξ † ξv f )+ g abcdef ( X mn )( v a ξ † ξX I v b )( v c X † J ξ † ξX K v d )( v e X † L ξ † ξv f )+ g abcdefgh ( X mn )( v a ξ † ξX I v b )( v c ξ † ξX K v d )( v e X † J ξ † ξv f )( v g X † L ξ † ξv h ) i × E Ii (Φ) E Kk (Φ) E ∗ Jj (Φ † ) E ∗ Ll (Φ † ) . (3.25)Note that Eq. (3.25) contains the multiple functions labeled by ab · · · , implying more generalthan Eq. (3.14).Derivative terms higher than four derivatives can be constructed by using space-time deriva-tive on X ab . For instance, six-derivative terms can be constructed as L (6) = 116 Z d θ X p =1 , Y p (cid:2) h ab ,p ( X mn ) ¯ D ˙ α ¯ D ˙ α D α D α X ab + h abcd ,p ( X mn ) ¯ D ˙ α D α X ab ¯ D ˙ α D α X cd + h abcd ,p ( X mn ) D α D α X ab ¯ D ˙ α ¯ D ˙ α X cd + h abcdef ,p ( X mn ) D α X ab D α X cd ¯ D ˙ α ¯ D ˙ α X ef + h abcdef ,p ( X mn ) D α D α X ab ¯ D ˙ α X cd ¯ D ˙ α X ef + h abcdef ,p ( X mn ) D α X ab D α ¯ D ˙ α X cd ¯ D ˙ α X ef (cid:3) (3.26)with arbitrary functions h ab ··· ,p of of the G -invariants X mn and the extra derivative terms Y p ( p = 1 ,
2) defined by Y = ∂ m ∂ m X a ′ b ′ , Y = ∂ m X a ′ b ′ ∂ m X c ′ d ′ . (3.27) Here, we discuss the construction of higher derivative terms from the C-type invariants. In thegeometrical method, G -invariant four-derivative terms are given by L (4) = 116 Z d θ Λ ik ¯ j ¯ l (Φ , Φ † ) D α Φ i D α Φ k ¯ D ˙ α Φ † ¯ j ¯ D ˙ α Φ † ¯ l , Λ ik ¯ j ¯ l = w (tr ( P a P b ) , · · · ) g ( i ¯ j g k ¯ l ) + w (tr ( P a P b ) , · · · ) R i ¯ jk ¯ l + w (tr ( P a P b ) , · · · ) R ( i ¯ j R k ¯ l ) + w (tr ( P a P b ) , · · · ) g ( i ¯ j R k ¯ l ) . (3.28)15 P \ D α D α P a ¯ D ˙ α ¯ D ˙ α P b ) non non3 tr ( P a D α D α P b ¯ D ˙ α ¯ D ˙ α P c ) non non4 tr ( D α P a D α P b ¯ D ˙ α P c ¯ D ˙ α P d ) tr ( D α P a D α P b )tr ( ¯ D ˙ α P c ¯ D ˙ α P d ) nontr ( P a D α D α P b · P c ¯ D ˙ α ¯ D ˙ α P d ) tr ( D α P a ¯ D ˙ α P b )tr ( D α P c ¯ D ˙ α P d )5 · · · · · · Table 1: Four-derivative terms X A ( D, ¯ D ; P a , P b , · · · ) constructed from the C-type invariants.The columns denote the number of traces, and the lows denote the number of P a . Each tracecontains more than two P a ’s with different a, b, c · · · .As the A-type case, w , , , are arbitrary functions of the strict G -invariants tr ( P a P b ), tr ( P a P b P c )and so on.In the group theoretical method, four-derivative terms can be constructed from the C-typeprojectors P a and the supercovariant derivatives D α and ¯ D ˙ α . All possible G -invariant terms X A ( D, ¯ D ; P a , P b , · · · ) including P a and two D ’s and two ¯ D ’s are summarized in Table 1. Theseterms are classified by the number of traces and the number of P a , where each trace shouldcontain more than two P a ’s with different a, b, c · · · . Then, the four-derivative term constructedfrom the C-type can be written as L (4) = 116 Z d θ X A ; a,b, ··· g Aab ··· (tr ( P c P d ) , · · · ) X A ( D, ¯ D ; P a , P b , · · · ) (3.29)where X A ( D, ¯ D ; P a , P b , · · · ) are the G -invariant four-derivative terms given in Table 1 and g Aab ··· are arbitrary functions of the C-type G -invariants tr ( P c P d ), tr ( P c P d P e ) and so on.This method can be generalized to derivative terms higher than four derivatives. It can beachieved by allowing g Aab ··· to contain linear terms including space-time derivatives or allowing16 A to contain space-time derivatives. For instance, six-derivative terms can be constructed as L (6) = 116 Z d θ h X p =1 , A ; a,b, ··· h ab ··· A,p (tr ( P e P f ) , · · · )tr ( Y p ) X A ( D, ¯ D ; P a , P b , · · · )+ X p =1 , a,b, ··· (cid:8) H ab ··· ,p (tr ( P e P f ) , · · · )tr ( Y p D α D α P a ¯ D ˙ α ¯ D ˙ α P b )+ H ab ··· ,p (tr ( P e P f ) , · · · )tr ( Y p P a D α D α P b ¯ D ˙ α ¯ D ˙ α P c )+ H ab ··· ,p (tr ( P e P f ) , · · · )tr ( Y p D α P a D α P b ¯ D ˙ α P c ¯ D ˙ α P d )+ H ab ··· ,p (tr ( P e P f ) , · · · )tr ( Y p P a D α D α P b · P c ¯ D ˙ α ¯ D ˙ α P d ) (cid:9) + · · · i (3.30)with arbitrary functions h ab ··· A,p and H ab ··· A,p of the C-type G -invariants, and the extra two-derivativeterms Y p ( p = 1 ,
2) given by Y = ∂ m ∂ m P a ′ , Y = ∂ m P a ′ ∂ m P b ′ . (3.31)The dots in Eq. (3.30) imply multi-trace terms such as tr ( ∂ m P a ′ D α D α P a )tr ( ∂ m P b ′ ¯ D ˙ α ¯ D ˙ α P b )and so on. In this section, we show an explicit example of higher derivative interactions of quasi-NG bosons.We consider higher derivative corrections for supersymmetric chiral symmetry breaking, whichis a maximal realization with each massless chiral superfield containing one NG boson and onequasi-NG boson.
Let us consider the chiral symmetry breaking G = SU ( N ) L × SU ( N ) R → H = SU ( N ) L+R . (4.1)The corresponding NG modes span the coset space G/H = SU ( N ) L × SU ( N ) R SU ( N ) L+R ≃ SU ( N ) . (4.2)We denote generators of the coset by T A ∈ SU ( N ). It was shown in Ref. [16] that a vacuumexpectation value belonging to a real representation gives rise to the same numbers of quasi-NG17osons and NG bosons, which is a maximal realization. The chiral symmetry breaking belongsto this class, and the total target space is G C / ˆ H ≃ SU ( N ) C = G C /H C ≃ SL ( N, C ) ≃ T ∗ SU ( N ) . (4.3)The coset representative is written as M = exp( i Φ A T A ) ∈ G C / ˆ H, (4.4)where the NG superfields are in the form ofΦ A ( y, θ ) = π A ( y ) + iσ A ( y ) + θψ A ( y ) + θθF A ( y ) , (4.5)with NG bosons π A , quasi-NG bosons σ A , and quasi-NG fermions ψ A .The nonlinear transformation law of the NG supermultiplets is M → M ′ = g L M g R , ( g L , g R ) ∈ SU ( N ) L × SU ( N ) R . (4.6)From the transformation M M † → g L M M † g † L , (4.7)the simplest K¨ahler potential is found to be K = f π tr ( M M † ) , (4.8)where f π is a constant. Therefore, the leading order of the bosonic part of the Lagrangian inthe derivative expansion reads L = − f π tr ( ∂ m M ∂ m M † ) , (4.9)where M is the lowest component of the NG superfield (4.4). However, the K¨ahler potential inEq. (4.8) is not general. In fact, the most general K¨ahler potential can be written as [12, 15] K = f (tr ( M M † ) , tr [( M M † ) ] , · · · , tr [( M M † ) N − ]) (4.10)with an arbitrary function of N − G but not G C , the target manifold is not homogeneous. One can deform the shape of thetarget manifold along the directions of the quasi-NG bosons, with keeping the isometry G . If we set all quasi-NG bosons to be zero [13, 12] U = M | σ A =0 ∈ SU ( N ) , (4.11)we have usual chiral Lagrangian L = − f π tr ( ∂ m U ∂ m U † ) = − f π tr ( U † ∂ m U ) (4.12)with the decay constant f π determined from f .One interesting feature of chiral symmetry breaking in supersymmetric vacua is that theunbroken group H = SU ( N ) L+R can be further broken to its subgroup due to the vacuumexpectation value of the quasi-NG bosons [12]. Some of quasi-NG bosons change to NG bosonsat less symmetric vacua [12, 15]
Let us discuss possible higher derivative terms for the supersymmetric chiral Lagrangian. Thesimplest candidate of a four-derivative term is L (4)0 = 116 Z d θ Λ ik ¯ j ¯ l (Φ , Φ † ) D α Φ i D α Φ k ¯ D ˙ α Φ † ¯ j ¯ D ˙ α Φ † ¯ l = Z d θ tr ( D α M ¯ D ˙ α M † D α M ¯ D ˙ α M † ) , (4.13)where components of Λ ik ¯ j ¯ l are determined from the right hand side. The bosonic part of thisterm is L (4)0 ,b = tr ( ∂ m M ∂ n M † ∂ m M ∂ n M † ) (4.14)in the canonical branch with F A = 0.However, Eq. (4.13) is not general. As in the leading term, we have a freedom to deformthe tensor along the directions of the quasi-NG bosons. The most general Lagrangian can be If one requires the Ricci-flat condition on the target manifold, the arbitrary function is fixed. That isknown as the Stenzel metric. This is not the scope of this paper. L (4) = 116 Z d θ Λ ik ¯ j ¯ l (Φ , Φ † ) D α Φ i D α Φ k ¯ D ˙ α Φ † ¯ j ¯ D ˙ α Φ † ¯ l = Z d θ h N − X k =1 g k (tr ( M M † ) , · · · , tr [( M M † ) N − ])tr ( D α M ¯ D ˙ α M † D α M ¯ D ˙ α M † ( M M † ) k )+ N − X k,l =1 g kl (tr ( M M † ) , · · · , tr [( M M † ) N − ]) × tr ( D α M ¯ D ˙ α M † ( M M † ) k )tr ( D α M ¯ D ˙ α M † ( M M † ) l ) i (4.15)with an arbitrary functions g k and g kl of N − G -invariants tr ( M M † ) , · · · , tr ( M M † ) N − . Thebosonic part of this term is L (4) b = N − X k =1 g k (tr ( M M † ) , · · · , tr [( M M † ) N − ]) tr ( ∂ m M ∂ n M † ∂ m M ∂ n M † ( M M † ) k )+ N − X k,l =1 g kl (tr ( M M † ) , · · · , tr [( M M † ) N − ]) tr ( ∂ m M ∂ n M † ( M M † ) k )tr ( ∂ m M ∂ n M † ( M M † ) l ) . (4.16)If we set all quasi-NG bosons to be zero as in Eq. (4.11), M M † | σ A =0 = U U † = N (tr [( M M † ) k ] | σ A =0 = N ) , (4.17)and the bosonic part in the canonical branch with F A = 0 becomes L (4) b | σ =0 = g , tr ( ∂ m U ∂ n U † ∂ m U ∂ n U † ) + g , tr ( ∂ m U ∂ n U † )tr ( ∂ m U ∂ n U † ) (4.18)with g , = P N − k =1 g k ( N, · · · , N ) and g , = P N − k,l =1 g k,l ( N, · · · , N ). One notes that the termtr ( ∂ m U ∂ m U † ∂ n U ∂ n U † ) or tr ( ∂ m U ∂ m U † )tr ( ∂ n U ∂ n U † ) is not allowed as a bosonic part of thesupersymmetric Lagrangian.Next, let us construct six-derivative terms. They can be written as L (6) = Z d θ h N − X k =1 h k (tr ( M M † ) , · · · )tr ( ∂ m M ∂ m M † D α M ¯ D ˙ α M † D α M ¯ D ˙ α M † ( M M † ) k )+ N − X k,l =1 h kl (tr ( M M † ) , · · · )tr ( ∂ m M ∂ m M † D α M ¯ D ˙ α M † ( M M † ) k ) × tr ( D α M ¯ D ˙ α M † ( M M † ) l )+ N − X k,l,j =1 h klj (tr ( M M † ) , · · · )tr ( ∂ m M ∂ m M † ( M M † ) k ) × tr ( D α M ¯ D ˙ α M † ( M M † ) l )tr ( D α M ¯ D ˙ α M † ( M M † ) j ) i (4.19)20ith arbitrary functions h k , h kl , h klj of N − G -invariants tr ( M M † ) , · · · , tr ( M M † ) N − . Thedots in Eq. (4.19) imply multi-trace terms such astr ( D α M ∂ m M † D α M ¯ D ˙ α M † ( M M † ) k )tr ( ∂ m M ¯ D ˙ α M † ( M M † ) l )and tr ( ∂ m M ¯ D ˙ α M † D α M ¯ D ˙ α M † ( M M † ) k )tr ( D α M ∂ m M † ( M M † ) l ) . The bosonic part of this term is L (6) b = N − X k =1 h h k (tr ( M M † ) , · · · )tr ( ∂ m M ∂ m M † ∂ n M ∂ o M † ∂ n M ∂ o M † ( M M † ) k )+ N − X k,l =1 h kl (tr ( M M † ) , · · · )tr ( ∂ m M ∂ m M † ∂ n M ∂ o M † ( M M † ) k )tr ( ∂ n M ∂ o M † ( M M † ) l )+ N − X k,l,j =1 h klj (tr ( M M † ) , · · · )tr ( ∂ m M ∂ m M † ( M M † ) k ) × tr ( ∂ n M ∂ o M † ( M M † ) l )tr ( ∂ n M ∂ o M † ( M M † ) j ) i + · · · . (4.20)If we set all quasi-NG bosons to be zero, these terms reduce to L (6) b | σ =0 = h , tr ( ∂ m U ∂ m U † ∂ n U ∂ o U † ∂ n U ∂ o U † )+ h , tr ( ∂ m U ∂ m U † ∂ n U ∂ o U † )tr ( ∂ n U ∂ o U † )+ h , tr ( ∂ m U ∂ m U † )tr ( ∂ n U ∂ o U † )tr ( ∂ n U ∂ o U † ) + · · · . (4.21)with h , = P N − k =1 h k ( N, · · · , N ), h , = P N − k,l =1 h k,l ( N, · · · , N ) and h , = P N − k,l,j =1 h k,l,j ( N, · · · , N ).We can construct the eight- or higher derivative terms in the same way. In this paper we have constructed higher derivative correction terms for massless NG and quasi-NG bosons and fermions in the manifestly supersymmetric off-shell formalism. In general, whena global symmetry is broken in supersymmetric vacua, massless quasi-NG bosons and fermionsappear. Low-energy effective theories are governed by supersymmetric nonlinear sigma modelsof the NG and quasi-NG fields. The number of the quasi-NG fields is determined by thestructure of the coset group G C / ˆ H . The G -invariant K¨ahler potentials of the nonlinear sigma21odels are classified into A-,B-, and C-types. We have shown the G -invariant quantities andexamples of K¨ahler potentials.In superfield formalism, the higher derivative term in the chiral model is given by a (2,2)K¨ahler tensor Λ ij ¯ k ¯ l symmetric in holomorphic and anti-holomorphic indices, whose compo-nents are functions of the chiral superfields Φ i . By using this formalism we have constructedhigher derivative corrections to supersymmetric nonlinear realizations. The tensors Λ ij ¯ k ¯ l areconstructed by the G -invariant K¨ahler metrics in the A-,B-,C-types. Remarkably, in the A-,C-types, the tensors Λ ij ¯ k ¯ l include degrees of freedom for the strict G -invariant quantities X ab andtr( P a P b · · · ). For the B-type, this is the pure realization, and there are no quasi-NG modes.We have found that the higher derivative terms are unique up to constants. For the A- andC-types, there are quasi-NG modes and higher derivative terms contain arbitrary functionswhich depends on the strict G -invariants. We have also constructed the higher derivative termsin purely group theoretical manners. As a practical example, we have further studied the caseof chiral symmetry breaking in more detail.Several discussions are addressed here.In this paper, we have studied spontaneous breaking of exact symmetry leading to exactlymassless NG bosons and quasi-NG bosons (fermions). For approximate symmetry, an explicitbreaking term should be introduced which give NG bosons masses. Consequently, they becomepseudo NG bosons, such as pions for the chiral symmetry breaking. In supersymmetric theory,a symmetry breaking potential term can be introduced by the superpotential W . The intro-duction of the superpotential can be treated perturbatively, which was done at least for singlecomponent cases [34, 39].As for another future work, the inclusion of the supersymmetric WZW term [23, 33] shouldbe discussed for supersymmetric chiral perturbation theory. For supersymmetric chiral pertur-bation theory with general target spaces, K¨ahler normal coordinates [45] should be useful as inRef. [33].A BPS Skyrme model was discovered some years back [46], which consists of only the sixth-order higher derivative term as well as appropriate potentials. Our result should be useful toinvestigate supersymmetric version of this model.In this paper, we have considered the canonical branch with F = 0 for solutions to theauxiliary field equations. It is known for the C P model that there is also a non-canonicalbranch with F = 0 [28, 39]. While the usual kinetic term disappears in this case, the theory22dmits a baby Skyrmion [28], which was shown to be a 1/4 BPS state [39]. Investigating non-canonical branches and 1/4 BPS baby Skyrmions for general K¨ahler G/H are one of interestingfuture directions.The supersymmetric C P N − model with four supercharges also appears as the world-volumeeffective action of a BPS non-Abelian vortex in N = 2 supersymmetric U ( N ) gauge theory with N hypermultiplets in the fundamental representation [47]. Higher derivative corrections to theeffective action were calculated in Ref. [48]. It was shown that 1/2 BPS lumps (sigma modelinstantons) are not modified in the presence of higher derivative terms [39, 48]. This should beso because a composite state of lumps inside a non-Abelian vortex is a 1/4 BPS state and it isnothing but a Yang-Mills instanton in the bulk point of view [49]. See Refs. [50, 51, 52] for areview of BPS composite solitons.As this regards, some other K¨ahler G/H manifolds are realized on a vortex in supersymmet-ric gauge theories with gauge groups G [53]. In particular, the cases of G = SO ( N ) , U Sp ( N )were studied in detail [54]. Therefore, 1/2 BPS lumps in sigma models on K¨ahler G/H withhigher derivative terms describe instantons in gauge theories with gauge group G . It should bechecked whether higher derivative corrections for lumps in these cases are canceled out.The supersymmetric chiral Lagrangian studied in Sec. 4 also appears as the effective the-ory on BPS non-Abelian domain walls in N = 2 supersymmetric U ( N ) gauge theories with2 N hypermultiplets in the fundamental representation with mass ± m [55]. A four-derivativecorrection was partly derived in Ref. [56].A general framework of a superfield formulation of the effective theories on BPS solitonworld-volumes with four supercharges was formulated in Ref. [57]. This should be generalizedto the case with higher derivative corrections. Acknowledgments
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