Scattering of charged tensor bosons in gauge and superstring theories
aa r X i v : . [ h e p - t h ] J u l NRCPS-HE-13-09CERN-PH-TH/2009-126
July, 2009
Scattering of Charged Tensor BosonsinGauge and Superstring Theories
Ignatios Antoniadis ∗ and George Savvidy Department of Physics, CERN Theory Division CH-1211 Geneva 23, Switzerland Demokritos National Research Center, Ag. Paraskevi, GR-15310, Athens
Abstract
We calculate the leading-order scattering amplitude of one vector and two tensor gaugebosons in a recently proposed non-Abelian tensor gauge field theory and open superstringtheory. The linear in momenta part of the superstring amplitude has identical Lorentzstructure with the gauge theory, while its cubic in momenta part can be identified withan effective Lagrangian which is constructed using generalized non-Abelian field strengthtensors. ∗† On leave of absence from CPHT ´Ecole Polytechnique, F-91128, Palaiseau Cedex,France.
Introduction
An infinite tower of particles of high spin naturally appears in the spectrum of differentstring theories. In the zero slope limit massless states of open and closed strings can beidentified with vector - Yang-Mills and tensor - graviton gauge quanta [1, 2, 3, 5, 4, 6, 7].Massive string states can be described by string field theory developed in [8, 9, 10, 12, 11,13, 14]. Nevertheless to represent the Lagrangian and equations in terms of componentsof tensor fields still remains a challenge [10]. In this respect higher spin field theories havereceived large attention [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] together with therecent development of interacting field theories of high spins [29, 30, 31, 32, 33, 34, 35, 36].As we mentioned the massless states of open superstring theory with Chan-Patoncharges [37] have been identified with the Yang-Mills gauge quanta [1, 6, 7, 38, 39, 40, 41]and it is of great importance to identify also massive higher spin string states with statesof some Lagrangian quantum field theory.One can imagine that massive states of open string may be described by some extensionof Yang-Mills theory to non-Abelian tensor gauge field theory. Such extension of Yang-Millstheory which includes charged tensor gauge fields was suggested recently in [42, 43, 44, 45].Not much is known about physical properties of this gauge field theory and our intension isto compare tree-level scattering amplitudes of tensor gauge bosons in non-Abelian tensorgauge field theory and open superstring theory.Non-Abelian tensor gauge fields are defined as rank-(s+1) tensor potentials A aµλ ...λ s . † The gauge invariant Lagrangian describing dynamical tensor gauge bosons of all ranks hasthe form [42, 43, 44, 45] L = L Y M + L + L + ...., (1.1)where L Y M is the Yang-Mills Lagrangian and defines cubic and quartic interactions with dimensionless coupling constant . ‡ For the lower-rank tensors, the Lagrangian has the fol-lowing form [42, 43, 44]: L = − G aµν,λ G aµν,λ − G aµν G aµν,λλ (1.2)+ 14 G aµν,λ G aµλ,ν + 14 G aµν,ν G aµλ,λ + 12 G aµν G aµλ,νλ , † Tensor gauge fields A aµλ ...λ s ( x ) , s = 0 , , , ... are totally symmetric with respect to the indices λ ...λ s ,but with no a priori symmetry with respect to the first index µ . In particular, we have A aµλ = A aλµ and A aµλρ = A aµρλ = A aλµρ . The adjoint group index a = 1 , ..., N − SU ( N ) gauge group. ‡ In D-dimensions the coupling constant has dimension (4 − D ) / G aµν = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν ,G aµν,λ = ∂ µ A aνλ − ∂ ν A aµλ + gf abc ( A bµ A cνλ + A bµλ A cν ) , (1.3) G aµν,λρ = ∂ µ A aνλρ − ∂ ν A aµλρ + gf abc ( A bµ A cνλρ + A bµλ A cνρ + A bµρ A cνλ + A bµλρ A cν ) . The Lagrangian forms L s for higher-rank fields can be found in Refs.[42, 43, 44].Here we shall focus our attention on the lower-rank tensor gauge field A aµλ , which de-composes into a symmetric tensor T S of spin two and an antisymmetric tensor T A , Poincar´edual of spin zero, charged gauge bosons [44]. The Feynman rules for these propagatingmodes and their interaction vertices can be extracted from the Lagrangian (1.2) and allowto calculate tree-level scattering amplitudes for processes involving vector and tensor gaugebosons [44].In the spectrum of open superstring theory with Chan-Paton charges there is also amassless vector gauge boson V on the first excited level and a rank-two massive tensorboson T S at the second level. The emission vertices for these states are defined as follows(in the zero and -1 ghost picture for V and T S , respectively) [6, 7, 38]: V : e µ ( k )( ˙ X µ − iα ′ k · ψψ µ ) e ikX α ′ k = 0 T S : ε µν ( k ) ψ { µ ( ˙ X ν − iα ′ k · ψψ ν } ) e ikX α ′ k = − . (1.4)and allow to calculate different tree level scattering amplitudes involving vector and tensorbosons V and T S . Our intension is to compare tree-level scattering amplitudes of tensorgauge bosons in the above non-Abelian tensor gauge field theory and in open superstringtheory.
We have found that the linear in momenta part of the 3-point superstring amplitudehas similar Lorentz structure with the one in gauge theory defined by the Lagrangian L (1.2) and that the cubic in momenta part of the superstring amplitude can be identi-fied with an effective Lagrangian L ∂ (6.23) and L ′ ∂ (6.24) constructed using non-Abelianfield strength tensors (1.3). This result suggests that most probably non-Abelian tensorgauge field theory describes a sub-sector of excited states of open superstring theory withhigher helicities, similar to a Yang-Mills theory describing the first excited state. Morecomplicated amplitudes should be analyzed in order to solidify this proposal.2 VTT Amplitude in Tensor Gauge Theory In massless tensor gauge field theory, the on-shell tensor-vector-tensor amplitude VTT is[46] M gauge theory = ε α ´ α ( k ) e β ( k ) ε γγ ′ ( k ) F α ´ αβγ ´ γ ( k , k , k ) δ ( k + k + k ) (2.5)where F α ´ αβγ ´ γ ( k , k , k ) = + k β ( η αγ η α ′ γ ′ + η αγ ′ η α ′ γ ) (2.6)+ 14 k α ( η βγ η α ′ γ ′ + η βγ ′ η α ′ γ )+ 14 k α ′ ( η βγ η αγ ′ + η βγ ′ η αγ )+ 14 k γ ( η αβ η α ′ γ ′ + η α ′ β η αγ ′ )+ 14 k γ ′ ( η αβ η α ′ γ + η α ′ β η αγ ) , where k i = 0 i = 1 , , k ij = k i − k j . It is important that in this massless theory themomentum conservation δ ( k + k + k ) has a nontrivial solution k = ( ω, , , r ) , k = ( ω, , , r ) , k = ( − ω, , , − r )( ω = r ) that can be deformed by a complex parameter z [47, 48, 49, 50, 51, 52, 53], k = ( ω, z, iz, r ) , k = ( ω, − z, − iz, r ) , k = ( − ω, , , − r ) . (2.7)Thus, the above expression for VTT F α ´ αβγ ´ γ ( k , k , k ) has a nonzero phase space of validity which is parameterized by the complex parameter z (2.7). Depending on polarizationsof scattered particles one can see that there are only four non-zero helicity amplitudes M (+2 , +1 , − M ( − , +1 , +2) , M (+2 , − , −
2) and M ( − , − , +2).Contracting Lorentz indices one can see that in tensor gauge theory the amplitude (2.5),(2.6) can explicitly be written in the form M gauge theory = ε α ´ α ( k ) e β ( p ) ε γγ ′ ( q ) F α ´ αβγ ´ γ ( k, p, q ) δ ( k + p + q ) == 4( k · e p ) ( ε k · ε q ) − p · ε k · ε q · e p ) + 2( e p · ε k · ε q · p ) . (2.8)Furthermore, one should take into account that all particles are massless k = p = q = 0and that the momentum conservation k + p + q = 0 gives k · p = p · q = q · k = 0 . Thus the3TT amplitude is nontrivial only if one considers complex momenta (2.7) or the space-timesignature η µν = ( − + − +) . The polarization vectors and tensors we shall take are then inthe form e + k = 1 √ zω , , − i, − zr ) , e + p = 1 √ − zω , , − i, zr ) , e − q = 1 √ , , i, ε α ´ α ( k ) = e α ( k ) e ´ α ( k ) and ε γ ´ γ ( q ) = e γ ( q ) e ´ γ ( q ). We can calculate now the amplitude(2.8) using the relations e k · e q = e p · e q = 1, k · e p = − p · e k = 2 √ z and e k · e p = 0 M (+2 , +1 , − gauge theory = (2.9)4( k · e p ) ( ε k · ε q ) − p · ε k · ε q · e p ) + 2( e p · ε k · ε q · p ) = 12 √ z, so that the VTT amplitude has non-trivial analytical continuation and is proportional to thedeformation parameter z . In the same way one can compute other polarization amplitudes. Mass-shell gauge invariance of VTT in tensor gauge theory
The expression for the VTT amplitude (2.8) is on mass-shell gauge invariant e β ( p ) → e β ( p ) + ξ p β ε αα ′ ( k ) → ε αα ′ ( k ) + k α ξ α ′ + k α ′ ξ α , k = 0 , k · ξ = 0 (3.10)where ξ and ξ α are gauge parameters. Indeed the gauge variation of (2.8) under (3.10) δe p ∼ p is δM gauge theory = 4( k · p ) ( ε k · ε q ) − p · ε k · ε q · p ) + 2( p · ε k · ε q · p ) = 0 (3.11)and under (3.10) δε k ∼ k ⊗ ξ + ξ ⊗ k is δM gauge theory = 8( k · e p ) ( k · ε q · ξ ) − (3.12) − p · k ) ( ξ · ε q · e p ) − p · ξ ) ( k · ε q · e p ) + 2( e p · k ) ( ξ · ε q · p ) + 2( e p · ξ ) ( k · ε q · p ) = 0 , because p · k = 0 , k · e q = p · e q = 0. Thus, in tensor gauge field theory, the VTT amplitudegets non-trivial values in four cases M (+2 , +1 , − M ( − , +1 , +2), M (+2 , − , − M ( − , − , +2) and is explicitly gauge invariant quantity. This completes the analysis ofVTT scattering amplitude in tensor gauge field theory.4 VTT Amplitude in Superstring Theory
In open superstring theory, the linear in momenta part (7.37), (7.40) of the full VTTamplitude (7.35), (7.36) for massless vector and massive tensors in ten dimensions is givenby the expression § M string theory = ε α ´ α ( k ) e β ( k ) ε γγ ′ ( k ) F αα ′ βγγ ′ ( k , k , k ) δ ( k + k + k ) , (4.13)where F αα ′ βγγ ′ ( k , k , k ) = + k β ( η αγ η α ′ γ ′ + η αγ ′ η α ′ γ )+ k α ( η βγ η α ′ γ ′ + η βγ ′ η α ′ γ )+ k α ′ ( η βγ η αγ ′ + η βγ ′ η αγ )+ k γ ( η αβ η α ′ γ ′ + η α ′ β η αγ ′ )+ k γ ′ ( η αβ η α ′ γ + η α ′ β η αγ ) . (4.14)Formally comparing these amplitudes in tensor gauge theory (2.6) and in string theory(4.14) one can see that they have identical Lorentz structure and are linear in momenta.But there is a difference in coefficients between these two expressions in last four terms:1 / m T = 1 /α ′ and the vector boson ismassless m V = 0, therefore the momentum conservation δ ( k + k + k ) has no solutions atall . The expression for the amplitude (4.13) has therefore a formal character because it ismultiplied by a delta function which vanishes identically. The idea is to find a reasonableextension of the string scattering amplitude considering some non-trivial limit that willallow to define it away from the zeros of the delta function. Let us first consider the wave § The details of the calculation are given in section 7 and in Appendix. e αα ′ = − , , ǫ = r + m T (4.15) − r/m T − r/m T ǫ/m T ǫ/m T , − r/m T
00 0 0 0 − r/m T ǫ/m T ǫ/m T , r /m T − rǫ/m T − / − / − rǫ/m T r /m T , where the first two wave functions correspond to helicities ±
2, the next two correspond tohelicities ± m T → ∞ and r → ± ¶ k = ( m T , , , , ... ) , k = ( ω, , , r, ... ) , k = ( − q r + m T , , , − r, ... ) , and its complex deformation k = ( m T , z, iz, , ... ) , k = ( ω, − z, − iz, r, ... ) , k = ( − q r + m T , , , − r, ... )one can see that the equation fulfills if 2 m T r = 0. Therefore, if we take the limit m T → ∞ , r →
0, so that m T r →
0, then the momentum conservation indeed can be fulfilled.Thus it seems possible to define this amplitude in superstring model. Physically, this limitcorresponds to the interaction between infinitely heavy tensor bosons ( m T → ∞ ) andmassless vector bosons which are in the deep infrared region of the spectrum ( r → superstring theory the amplitude (4.13), (4.14) can be written in the form M string theory = ε α ´ α ( k ) e β ( p ) ε γγ ′ ( q ) F α ´ αβγ ´ γ ( k, p, q ) == 4( k · e p ) ( ε k · ε q ) − p · ε k · ε q · e p ) + 8( e p · ε k · ε q · p ) . (4.16) ¶ Without loss of generality, all momenta and components of the polarization tensors in the additionalsix space dimensions will be taken equal to zero. Note also that discussing only the properties of masslessand massive modes of strings without loop corrections we could restrict ourselves to lower dimensions. k = ( m T , z, iz, , ... ) , p = ( ω, − z, − iz, r, ... ) , q = ( − q r + m T , , , − r, ... ) , (4.17)( ω = r ) and the polarization tensors can be chosen as follows e k = 1 √ zm T , , − i, − zm T , ... ) , e p = 1 √ − zω , , − i, zr , ... ) , e q = 1 √ , , i, , ... )where ε α ´ α ( k ) = e α ( k ) e ´ α ( k ), ε γ ´ γ ( q ) = e γ ( q ) e ´ γ ( q ). We can calculate now the amplitude(4.16) using the relations e k · e p = 0 , e k · e q = 1 , e p · e q = 1, k · e p = z (2 + m T /r ) / √ p · e k = − z (2 + 4 r/m T ) / √
2. Thus we shall get nontrivial analytical continuation of theVTT amplitude in string theory M (+2 , +1 , − string theory = 4( k · e p ) ( ε k · ε q ) − p · ε k · ε q · e p ) ++ 8( e p · ε k · ε q · p ) = 12 √ z + 2 √ z ( m T r + 8 rm T ) , (4.18)which has a part which is identical with the massless tensor gauge theory 12 √ z in (2.9)and an additional part which depends on the mass of the tensor particle m T . One shouldtake now the limit m T → ∞ , r → m T r →
0, keeping z m T r = Z fixed, so that M (+2 , +1 , − string = 2 √ Z . Mass-shell gauge invariance in superstring theory
This expression is also on mass-shell gauge invariant e β ( p ) → e β ( p ) + ξ p β ε αα ′ ( k ) → ε αα ′ ( k ) + k α ξ α ′ + k α ′ ξ α , k = − m T , k · ξ = 0 (5.19)where ξ and ξ α are gauge parameters. The gauge variation of (4.16) under (5.19) δe p ∼ p is δM string theory = 4( k · p ) ( ε k · ε q ) − p · ε k · ε q · p ) + 8( p · ε k · ε q · p ) = 4 rm T → δε k ∼ k ⊗ ξ + ξ ⊗ k is δM string theory = 8( k · e p ) ( k · ε q · ξ ) − (5.21) − p · k ) ( ξ · ε q · e p ) − p · ξ ) ( k · ε q · e p ) + 8( e p · k ) ( ξ · ε q · p ) + 8( e p · ξ ) ( k · ε q · p ) == − rm T ( ξ · ε q · e p ) → , because p · k = rm T → , k · e q = 0 , p · e q = 0. Thus, this expression is also gauge invarianton mass-shell for gauge variations ξ α in any direction of the ten dimensional space-time.7 Effective Action in Terms of Tensor Gauge Fields
In the full superstring amplitude (7.35), (7.36) together with the linear part (7.37) we havealso an additional term which is cubic k in momenta α ′ { − η α ′ γ ′ k α k γ k β + η βγ ′ k α k α ′ k γ − η βα ′ k α k γ k γ ′ ++ ( k · k ) k α ′ ( η αγ ′ η βγ − η αγ η βγ ′ ) + ( k · k ) k γ ′ ( η αβ η α ′ γ − η αγ η α ′ β ) ++ ( k · k )[ ( η βγ η α ′ γ ′ − η α ′ γ η βγ ′ ) k α + ( η αγ η βγ ′ − η γβ η αγ ′ ) k α ′ + ( η αγ ′ η βα ′ − η αβ η α ′ γ ′ ) k γ + ( η αβ η α ′ γ − η αγ η α ′ β ) k γ ′ + ( η αγ ′ η α ′ γ − η αγ η α ′ γ ′ ) k β ] } . (6.22)In particular it contains scalar products k · k , k · k and k · k . In the three particlescattering amplitude, which we are considering here, they can take only fixed values k · k = k · k = 0 , α ′ k · k = − α ′ ( k + k ) = 2 , therefore they do not appear in the final expression(7.36). Nevertheless let us keep them all, as they are, in order to examine if they can bereproduced by an effective Lagrangian which is constructed using generalized field strengthtensors (1.3).Naturally we should try to associate these cubic terms with a gauge invariant effectiveLagrangian which has higher derivatives. Indeed, there are two independent gauge invariantforms which can be constructed in tensor gauge field theory using the field strengths (1.3)and are cubic in derivatives L ∂ = α ′ [ T r ( G µν,λ G νρ G ρµ,λ ) + 12 T r ( G µν G νρ,λλ G ρµ ) ] (6.23)and L ′ ∂ = α ′ [ − T r ( G µν,λ G νρ G ρµ,λ ) + T r ( G µλ,λ G µν G νρ,ρ ) + T r ( G µν,λ G µρ G ρλ,ν ) ++ T r ( G µν G µρ,ν G ρλ,λ ) + T r ( G µν G µρ,λ G ρλ,ν ) + T r ( G µν G µρ,λ G ρν,λ ) ++ T r ( G µλ,λ G µν,ρ G νρ ) + T r ( G µν,λ G µρ,λ G ρν ) + T r ( G µν,λ G µρ,ν G ρλ ) ++ 2 T r ( G µν G νλ,ρµ G ρλ ) − T r ( G µν G νρ,λλ G ρµ ) ] (6.24)It is interesting that reproducing the higher derivative part (6.22) of the VTT vertex, thereare no “traces” of any higher derivative string (gravity) vertex VVT between two vectorsand a tensor (two photons and a graviton). What is also striking is that one reproduces8ll terms with scalar products of momenta k · k , k · k and k · k in (6.22). In ouron-mass-shell scattering amplitude they have fixed values and did not “show up”, but theywill certainly contribute to other more complicated amplitudes. Therefore it is importantthat they are present in the effective Lagrangian.In the next section we shall present the actual calculation of the superstring scatter-ing amplitude (4.13), (7.35). As our calculation shows, the ( α ′ ) k terms are absent insuperstring theory. Open Type I Superstring Tree-Level Amplitudes
To set up notation let us begin with the simplest example of the tree-level scatteringamplitude for three on-shell massless vector bosons. The vertex operator has the followingform k [6, 7]: V = e α ( k )( ˙ X α − iα ′ k · ψψ α ) e ikX ( y ) V − = e α ( k ) e − φ ψ α e ikX ( y ) (7.25)and we shall represent the disk as the upper half-plane so that the boundary coordinate y is real y ∈ [ −∞ , + ∞ ]. The tree amplitude can take the form V µ µ µ a a a ( k , k , k ) = F µ µ µ ( k , k , k ) tr ( λ a λ a λ a ) + F µ µ µ ( k , k , k ) tr ( λ a λ a λ a ) , where λ a are isotopic matrices and the matrix element F is given below F µ µ µ ( k , k , k ) = < c V − ( y ) c V − ( y ) c V ( y ) == < ce − φ ψ µ e ik X ( y ) ce − φ ψ µ e ik X ( y ) c ( ˙ X µ − iα ′ k · ψψ µ ) e ik X ( y ) > = y y y y − { F µ y η µ µ y − + 2 iα ′ k µ y − η µ µ y − − iα ′ k µ y − η µ µ y − } . Here y ij = y i − y j , y < y < y and we have to sum over two orderings of the vertexoperators on the disk. The vector function F µ y is given by the expression F µ y = − iα ′ ( k µ y − y + k µ y − y ) = − iα ′ k µ y y y . k In eq. (7.25) and below, the superscript -1 stands for the ( − < c ( y ) c ( y ) c ( y ) > = y y y , < e − φ ( y ) e − φ ( y ) > = y − , while the contraction of world-sheet fermions is < ψ µ ( y ) ψ ν ( y ) > = η µν y − . All bosons are on mass-shell α ′ k = α ′ k = α ′ k = 0 and k + k + k = 0. Their wavefunctions are e µ ( k ) , e µ ( k ) , e µ ( k ) and are transversal to the corresponding momenta k i · e ( k i ) = 0 , i = 1 , ,
3. One sees that the matrix element F µ µ µ is linear in momentum[ F µ µ µ ] ∼ α ′ k . (7.26)Unlike the bosonic open string amplitude, there is no k term and so no G term in thelow energy effective action. Thus for the F µ µ µ ( k , k , k ) tr ( λ a λ a λ a ) we have2 iα ′ [ k µ η µ µ − k µ η µ µ − k µ η µ µ ] tr ( λ a λ a λ a ) . (7.27)Adding the equal term ∗∗ iα ′ [ − k µ η µ µ + k µ η µ µ + k µ η µ µ ] tr ( λ a λ a λ a ) and the re-versed cyclic orientation amplitude a , µ , k ↔ a , µ , k , we can get the total matrixelement: iα ′ [ ( k − k ) µ η µ µ + ( k − k ) µ η µ µ + ( k − k ) µ η µ µ ] tr ([ λ a , λ a ] λ a ) . (7.28)This expression coincides with the Yang-Mills vertex projected to the mass-shell. Tree-Level Amplitude for Two Symmetric Tensors and a Vector
The vertex operator for the symmetric rank-2 tensor boson T S on the second level is V − = ε αα ′ ( k ) e − φ ψ α ( ˙ X α ′ − iα ′ k · ψψ α ′ ) e ikX ( y ) (7.29)and together with the vertex (7.25) can be used to calculate now the scattering amplitudebetween a vector and two tensor bosons: V αα ′ βγγ ′ abc ( k, p, q ) = F αα ′ βγγ ′ ( k, p, q ) tr ( λ a λ b λ c ) + F γγ ′ βαα ′ ( q, p, k ) tr ( λ c λ b λ a ) (7.30) ∗∗ One should use momentum conservation and transversality of the wave functions. ε αα ′ ( k ) , e β ( p ) , ε γγ ′ ( q ) . (7.31)We shall define for convenience k ≡ k, k ≡ q, k ≡ p, and k + k + k = 0. The mass-shellconditions are α ′ k = α ′ k = − , α ′ k = 0 (7.32)and therefore it follows that k · k = k · k = 0 , α ′ k · k = − α ′ ( k + k ) = 2 . (7.33)We have to calculate the correlation function: < : ce − φ ψ α ( ˙ X α ′ − iα ′ k · ψψ α ′ ) e ikX ( y ) : : ce − φ ψ γ ( ˙ X γ ′ − iα ′ q · ψψ γ ′ ) e iqX ( y ) :: c ( ˙ X β − iα ′ p · ψψ β ) e ipX ( y ) : > (7.34)We shall split it into four terms. The first one gives (the details of the calculation are givenin the Appendix) < ce − φ ψ α ˙ X α ′ e ik X ( y ) ce − φ ψ γ ˙ X γ ′ e ik X ( y ) c ( ˙ X β − iα ′ k · ψψ β ) e ik X ( y ) > == i (2 α ′ ) [ η αγ ( η βγ ′ k α ′ + η α ′ β k γ ′ + η α ′ γ ′ k β ) + η α ′ γ ′ ( η αβ k γ + η βγ k α )]( − iα ′ ) [ η αγ k β + η αβ k γ + η γβ k α ] k α ′ k γ ′ and contains linear as well as cubic in momentum expressions . The other three remainingterms have only cubic in momentum expressions. Indeed the second one gives < ce − φ ψ α ˙ X α ′ e ik X ( y ) ce − φ ψ γ ( − iα ′ ) k · ψψ γ ′ e ik X ( y ) c ( ˙ X β − iα ′ k · ψψ β ) e ik X ( y ) > == ( − iα ′ ) k α ′ [ η αγ k β k γ ′ + η βγ ′ k α k γ − η βγ k α k γ ′ − η αγ ′ k β k γ + k · k ( η αγ ′ η βγ − η αγ η βγ ′ )]and contains only cubic in momentum terms . A new feature of this expression is that itcontains a scalar product ( k · k ). The third one gives < ce − φ ψ α ( − iα ′ ) k · ψψ α ′ e ik X ( y ) ce − φ ψ γ ˙ X γ ′ e ik X ( y ) c ( ˙ X β − iα ′ k · ψψ β ) e ik X ( y ) > == ( − iα ′ ) k γ ′ [ η αγ k β k α ′ − η αβ k α ′ k γ + η βα ′ k α k γ − η α ′ γ k β k α + k · k ( η αβ η α ′ γ − η αγ η α ′ β )]11nd also contains only cubic in momentum expressions as well as a scalar product( k · k ).Finally, the last term gives < c e − φ ψ α ( − iα ′ ) k · ψψ α ′ e ik X ( y ) ce − φ ψ γ ( − iα ′ ) k · ψψ γ ′ c ( ˙ X β − iα ′ k · ψψ β ) e ik X ( y ) > = ( − iα ′ ) { + k β [ η αγ ( − k · k η α ′ γ ′ + k γ ′ k α ′ ) − k α ( − k γ η α ′ γ ′ + k γ ′ η α ′ γ ) + η αγ ′ ( − k γ k α ′ + k · k η α ′ γ )] − k α [ η βγ ( − k · k η α ′ γ ′ + k γ ′ k α ′ ) − k β ( − k γ η α ′ γ ′ + k γ ′ η α ′ γ ) + η βγ ′ ( − k γ k α ′ + k · k η α ′ γ )] − k α ′ [ η βγ ( k · k η αγ ′ − k γ ′ k α ) − k β ( − k γ ′ η αγ + k γ η αγ ′ ) + η βγ ′ ( k γ k α − k · k η αγ )]+ k γ [ η αβ ( − k · k η α ′ γ ′ + k γ ′ k α ′ ) − k β ( − k α η α ′ γ ′ + k α ′ η αγ ′ ) + η βα ′ ( − k γ ′ k α + k · k η αγ ′ )]+ k γ ′ [ η αβ ( k · k η α ′ γ − k γ k α ′ ) − k β ( k α η α ′ γ − k α ′ η αγ ) + η βα ′ ( k γ k α − k · k η αγ )]+ k · k [ η βγ ( k α ′ η αγ ′ − k α η α ′ γ ′ ) − k β ( − η αγ η α ′ γ ′ + η α ′ γ η αγ ′ ) + η βγ ′ ( k α η α ′ γ − k α ′ η αγ )] − k · k [ η αβ ( k γ ′ η α ′ γ − k γ η α ′ γ ′ ) − k β ( − η αγ η α ′ γ ′ + η α ′ γ η αγ ′ ) + η βα ′ ( k γ η αγ ′ − k γ ′ η αγ )] } and again contains only cubic in momentum expressions as well as a scalar product ( k · k ).The scalar products k · k and k · k can be dropped because of (7.33). Summing allremaining terms together one can see that many terms which are cubic in momentumcancel each other so that we are left with the expression i (2 α ′ ) [ η αγ ( η βγ ′ k α ′ + η α ′ β k γ ′ + η α ′ γ ′ k β ) + η α ′ γ ′ ( η αβ k γ + η βγ k α ) ] +( − iα ′ ) { − η α ′ γ ′ k α k γ k β + η βγ ′ k α k α ′ k γ − η βα ′ k α k γ k γ ′ ] +( k · k )[ + ( η βγ η α ′ γ ′ − η α ′ γ η βγ ′ ) k α + ( η αγ η βγ ′ − η γβ η αγ ′ ) k α ′ + ( η αγ ′ η βα ′ − η αβ η α ′ γ ′ ) k γ + ( η αβ η α ′ γ − η αγ η α ′ β ) k γ ′ + ( η αγ ′ η α ′ γ − η αγ η α ′ γ ′ ) k β ] } . (7.35)Note that the terms proportional to the nonzero product 2 α ′ k · k = 2 (7.33) can also bedropped because they are antisymmetric with respect to the indices αα ′ and γγ ′ while thewave functions ε αα ′ and ε γγ ′ are symmetric.Thus, we arrive to the following expression i (2 α ′ ) [ η αγ ( η βγ ′ k α ′ + η α ′ β k γ ′ + η α ′ γ ′ k β ) + η α ′ γ ′ ( η αβ k γ + η βγ k α ) ] +( − iα ′ ) [ − η α ′ γ ′ k α k γ k β + η βγ ′ k α k α ′ k γ − η βα ′ k α k γ k γ ′ ] , (7.36)which is linear and cubic in momenta . Its linear in momenta part is4[ η αγ ( η βγ ′ q α ′ + η α ′ β p γ ′ + η α ′ γ ′ k β ) + η α ′ γ ′ ( η αβ p γ + η βγ q α )] . (7.37)12t should be symmetrized over αα ′ and γγ ′ + ( η αγ η α ′ γ ′ + η αγ ′ η α ′ γ ) k β + ( η βγ η α ′ γ ′ + η βγ ′ η α ′ γ ) q α + ( η βγ η αγ ′ + η βγ ′ η αγ ) q α ′ + ( η αβ η α ′ γ ′ + η α ′ β η αγ ′ ) p γ + ( η αβ η α ′ γ + η α ′ β η αγ ) p γ ′ (7.38)and we can add to it an equal term − ( η αγ η α ′ γ ′ + η αγ ′ η α ′ γ ) q β − ( η βγ η α ′ γ ′ + η βγ ′ η α ′ γ ) p α − ( η βγ η αγ ′ + η βγ ′ η αγ ) p α ′ − ( η αβ η α ′ γ ′ + η α ′ β η αγ ′ ) k γ − ( η αβ η α ′ γ + η α ′ β η αγ ) k γ ′ (7.39)in order to get a symmetric expression: (1) F αα ′ βγγ ′ ( k, p, q ) = + ( k − q ) β ( η αγ η α ′ γ ′ + η αγ ′ η α ′ γ )+ ( q − p ) α ( η βγ η α ′ γ ′ + η βγ ′ η α ′ γ )+ ( q − p ) α ′ ( η βγ η αγ ′ + η βγ ′ η αγ )+ ( p − k ) γ ( η αβ η α ′ γ ′ + η α ′ β η αγ ′ )+ ( p − k ) γ ′ ( η αβ η α ′ γ + η α ′ β η αγ ) . (7.40)Substituting this result into the expression (7.30) with the terms in the reversed cyclicorientation a, ( α, α ′ ) , k ↔ c, ( γ, γ ′ ) , q , we get: V αα ′ βγγ ′ abc ( k, p, q ) = tr ([ λ a , λ b ] λ c ) F αα ′ βγγ ′ ( k, p, q ) . (7.41)The remaining part of the vertex VTT (7.36), which is cubic in momenta, (3) F is (3) F αα ′ βγγ ′ ( k, p, q ) = 8 α ′ [ − η α ′ γ ′ k α k γ k β + η βγ ′ k α k α ′ k γ − η βα ′ k α k γ k γ ′ ] (7.42)and can be written in the form (3) M string = ε α ´ α ( k ) e β ( k ) ε γγ ′ ( k ) (3) F αα ′ βγγ ′ ( k, p, q ) = (7.43)8 α ′ [ 12 ( p · ε k · ε q · p ) (( q − k ) · e p ) − ( p · ε k · p )( k · ε q · e p ) + ( q · ε k · e p )( p · ε q · p ) ] .
13e can calculate its value in the limit considered in section 4, that gives (3) M (+2 , +1 , − string = ε α ´ α ( k ) e β ( k ) ε γγ ′ ( k ) (3) F αα ′ βγγ ′ ( k, p, q ) = 0 , (7.44)because k · e q = 0 , p · e q = 0. Its gauge variation under (3.10) δe p ∼ p vanishes δ (3) M string = 0and under (5.19) δε k ∼ k ⊗ ξ + ξ ⊗ k vanishes as well δ (3) M string = 0 , because of the same relations k · e q = 0 , p · e q = 0.The cubic in momenta part of the VTT vertex can be associated with an effective actionwhich have higher derivative terms constructed by generalized field strength tensors (1.3).The gauge invariant effective action which is cubic in field strength tensors is L ∂ = G µν,λ G νρ G ρµ,λ + 12 G µν G νρ,λλ G ρµ . (7.45)Indeed, as one can easily check, its gauge variation vanishes δ L ∂ = ([ G µν,λ ξ ] + [ G µν ξ λ ]) G νρ G ρµ,λ + G µν,λ [ G νρ ξ ] G ρµ,λ + G µν,λ G νρ ([ G ρµ,λ ξ ] + [ G ρµ ξ λ ]) +12 { [ G µν ξ ] G νρ,λλ G ρµ + G µν ([ G νρ,λλ ξ ] + 2[ G νρ,λ ξ λ ] + [ G νρ ξ λλ ]) G ρµ + G µν G νρ,λλ [ G ρµ ξ ] } = 0On the other hand, the second invariant we have found has the form L ′ ∂ = − T r ( G µν,λ G νρ G ρµ,λ ) + T r ( G µλ,λ G µν G νρ,ρ ) + T r ( G µν,λ G µρ G ρλ,ν ) ++ T r ( G µν G µρ,ν G ρλ,λ ) + T r ( G µν G µρ,λ G ρλ,ν ) + T r ( G µν G µρ,λ G ρν,λ ) ++ T r ( G µλ,λ G µν,ρ G νρ ) + T r ( G µν,λ G µρ,λ G ρν ) + T r ( G µν,λ G µρ,ν G ρλ ) ++ 2 T r ( G µν G νλ,ρµ G ρλ ) − T r ( G µν G νρ,λλ G ρµ ) . (7.46) Acknowledgements
The work of (I.A.) was supported in part by the European Commission under the ERCAdvanced Grant 226371 and in part by the CNRS grant GRC APIC PICS 3747. The workof (G.S.) was partially supported by ENRAGE (European Network on Random Geometry),a Marie Curie Research Training Network, contract MRTN-CT-2004- 005616.14 ppendix
Let us consider four terms of the interaction vertex VTT. We have Y i
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