Dual versions of N = 2 supergravity and spontaneous supersymmetry breaking

Transcription

1 Dual versions of N = supergravity and spontaneous supersymmetry breaking arxiv:hep-th/ v1 1 Oct 1994 Yu. M. Zinoviev Institute for High Energy Physics Protvino, Moscow Region, 1484, Russia Abstract In this paper, using a model of N = supergravity - vector multiplets interaction with the scalar field geometry SU(1,m)/SU(m) U(1) as an example, we show that even when the geometry is fixed one can have a whole family of the Lagrangians that differ by the vector field duality transformations. As a byproduct, for this geometry we have constructed a model of (m 1) vector multiplets interacting with the hidden sector admitting spontaneous supersymmetry breaking with two arbitrary scales and without a cosmological term. address: ZINOVIEV@MX.IHEP.SU

2 1 Introduction Recently there was essential progress in understanding of general coupling of N = supergravity to vector multiplets (as well as hypermultiplets) and of geometrical aspects in such models [1]. As is well known (e.g. []) not all of the isometries of the manifold spanned by scalar fields turn out to be the symmetry of the Lagrangian, part of them (containing vector fields duality transformations) being a symmetry of the equations of motion only. First of all this leads to the existence of different (we will call them dual) versions of supergravity - vector multiplets interaction with the same geometry in which part of the vector fields are replaced by axial vector ones. For example, there exist two different models for one vector multiplets with the scalar field geometry SU(1, 1)/U(1). One of them is so called minimal model [3] when the Lagrangian is invariant under non-compact subgroup O(1, 1). Another one appeared as a part of the hidden sector in our search [4] for mechanism of spontaneous supersymmetry breaking with two arbitrary scales (including partial super-higgs effect) without a cosmological term. In this model two vector fields (graviphoton and the matter one) enter the Lagrangian and the supertransformation laws through the complex combinations A µ ±γ 5 B µ, the Lagrangian being invariant under the compact U(1)-subgroup. The aim of this paper is to show that the invariance of the equations of motion under duality transformations leads to the existence of the whole family of the Lagrangians with the same scalar field geometry. At first we have constructed the general coupling of one vector multiplet to N = supergravity with the geometry SU(1, 1)/U(1) which can smoothly interpolates between two models mentioned above. Then we have managed to generalize this model to the case of arbitrary number of vector multiplets, the scalar field geometry being SU(1, m)/su(m) U(1). As a partial but physically a most interesting case we obtained the model for (m 1) vector multiplets interacting with our hidden sector. It turns out that in this model one really can have spontaneous supersymmetry breaking without a cosmological term. Dual versions of N = supergravity LetususeasanexamplethewellknownmodeloftheN = supergravity vectormultiplets interaction with the scalar field geometry SU(1, m)/su(m) U(1). The most simple way to describe this model is to introduce, apart from the graviton e µr and gravitini Ψ µi, i = 1,, m+1 vector multiplets {A µ A,Ω ia,z A }, A = 0,1,...m. The following constraints correspond to the model with required geometry: z A z A = z A Ω ia = 0 (1) Note that we use the γ-matrix representation in which majorana spinors are real, so in all expressions with spinors the matrix γ 5 will play a role of the imaginary unit i, e.g., z A Ω ia = (x A + γ 5 y A ) Ω ia. In this, the theory has local axial U(1) invariance with composite gauge field U µ = ( z A µ z A ). The corresponding covariant derivatives look like, e.g., D µ z A = µ z A + 1 ( z µz)z A, z A D µ z A = 0 1

3 D µ η i = D µ η i 1 4 ( z µz)η i () Here and further we drop, wherever possible, the repeated indices. In this notations the Lagrangian has the form: L = 1 R+ i εµνρσ Ψµ i γ 5 γ ν D ρ Ψ σi + Ω 1 i ˆDΩi + 1 D µ zd µ z 1 4 A µν + 1 { } 1 4 z (za µν)(z(a µν +iãµν))+h.c. 1 εij Ψµi z A γ z (Aµν 5 Ã µν ) A Ψ νj 1 εij Ωi A γ µ γ ν D ν z A Ψ µj Ω i [ ] A i γ µ (σa) A za (z(σa)) Ψ z µi (3) where now and further on we omit all four fermion terms which will not be essential for our consideration. This Lagrangian is invariant under the following local supertransformations: δe µr = i( Ψ i µ γ r η i ) δψ µi = D µ η i i ε (z(σa)) ij γ µ η j z δa A µ = ε ij ( Ψ µi z A η j )+i( Ω ia γ µ η i ) (4) δω ia = 1 [ ] (σa) A za (z(σa)) η z i iε ij ˆDz A η j δx A = ε ij ( Ω ia η j ) δy A = ε ij ( Ω ia γ 5 η j ) Now let us discard for a moment vector multiplets with a =,3...m and consider the most general interaction of one vector multiplet with N = supergravity keeping the scalar field geometry fixed (it is just SU(1,1)/U(1) in this case). The requirement of axial U(1) invariance and the constraints (1) lead to the following ansatz for the supertransformations: δψ µi = D µ η i i ε ije α (σa) α γ µ η j δa µ α = ε ij ( Ψ µi z a K a α η j ) i( λ ia γ µ K a α η i ) (5) δλ ia = 1 ε abz b (M α (σa) α )η i iε ij ˆDz a η j δx a = ε ij ( λ ia η j ) δy a = ε ij ( λ ia γ 5 η j ) where K a α is a constant two by two matrix, and E α and M α are functions of z a with axial charge +. The requirement of the closure of the superalgebra leads to the following equation for them: K α E β +N α M β = δ αβ (6) where K α = z a K a α, N α = z a N aα and N aα = ε ab Kb α. This equation allows one to express the functions E α and M α in terms of K α, namely E α = ε αβn β, M α = ε αβk β, = εαβ K α N β (7)

4 The corresponding Lagrangian has the form: L = 1 R+ i εµνρσ Ψµi γ 5 γ ν D α Ψ β + i λ i γ µ D µ λ i + 1 Dµ z a D µ z a 1 4 [f αβa µν α A µν β g αβ A µν α Ã µν β ] 1 εij λia γ µ γ ν D ν z a Ψ µj 1 εij Ψµi (A µν γ 5 Ã µν ) α Ē α Ψ νj i 4 λ i aε ab z b γ µ M α (σa) α Ψ µi (8) where f αβ and g αβ are real and symmetric functions to be determined. The requirement of the Lagrangian to be invariant under the supertransformations given above leads to the following equations: f αβ = (E α Ē β +M α Mβ ) δ(f αβ +γ 5 g αβ ) δz a = 4ε ab z b M (α E β) (9) The first one determines the function f αβ and, in order this function to be symmetric, leads to the (only) constraint on the matrix K a α : ε αβ Ka α K aβ = 0 (10) As for the second one, it could be solved by the following ansatz: f αβ +γ 5 g αβ = H α E β +G α M β (11) where H α = z a H aα, G α = z a G aα and G aα = ε ab Hb α. Now both equations (9) are satisfied if H a αk aβ +H aα Ka β = δ αβ (1) Note, that this equation determines the matrix H aα only up to the arbitrary constant due to the possibility to make a shift H aα H aα +ωγ 5 ε αβ K β a. This, in turn, implies that the function g αβ is determined up to the arbitrary constant, but that leads to the Lagrangians that differ only by the total divergency. Thus, we have managed to construct a whole class of Lagrangians with the same scalar field geometry and different dual forms for vector fields, which are determined by the constant matrix K α a with the only constraint (10) on ( it. As) partial cases we have usual minimal 1 1 case, corresponding to the choice K α a =, and the dual version that we mentioned above, with K α γ5 1 1 ( ) 1 a = (we will give corresponding Lagrangian and the 1 γ 5 supertransformations in the next section). At thirst sight it seems that it is not easy to generalize the solution obtained to the case of arbitrary number of vector multiplets. In order to do that note, first of all, that the part of the supertransformations for gravitini containing vector fields could be rewritten in a very suggestive form: δψ µi = i ε (z a Ẽ aα (σa) α ) ij γ z a g ab z b µ η j (13) 3

5 where we introduced: Ẽ aα = ε ab ε αβ Kbβ g ab = ε αβ K aα N bβ (14) Moreover, inthecasewhendet(g ab ) 0thecorrespondingpartoftheλ-supertransformations could be rewritten as: δλ ia = 1 [ E aα (σa) α g abz b (z c E cα (σa) α ] ) η i (15) (zgz) Now we are ready to consider the generalization of this formulas on the case of arbitrary number of vector multiplets. For that we make the following ansatz for the supertransformations: and for the Lagrangian: δe µr = i( Ψ i µ γ r η i ) δψ µi = D µ η i i ε z A E Aα (σa) α ij γ µ η j (zgz) δa α m = ε ij ( Ψ µi z A K α A η j )+i( Ω ia γ µ K α A η i ) (16) δω ia = 1 [ E Aα (σa) α g ABz B (z C E Cα (σa) α ] ) η i iε ij ˆDz A η j (zgz) δx A = ε ij ( Ω ia η j ) δy A = ε ij ( Ω ia γ 5 η j ) L = 1 R+ i εµνρσ Ψµ i γ 5 γ ν D ρ Ψ σi + Ω i ia ˆDΩiA + 1 D µ z A D µ z A 1 4 A µν α E Aα Ē A βa β µν + 1 [ ] (zeaµν )(ze(a µν +γ 5 Ã µν )) +h.c. 4 (zgz) 1 εij Ψµi z A Ē A α(a µν γ 5 Ã µν ) α Ψ νj 1 ( zḡ z) εij ΩiA γ µ γ ν D ν z A Ψ µj Ω i [ ia γ µ E Aα (σa) α g ABz B ] (ze(σa)) Ψ µi (17) (zgz) In this, the requirements of the closure of the superalgebra and the invariance of the Lagrangian give: K Aα E Aβ = δ α β g AB = K A α E Bα E A[α Ē A β] = 0 (18) The first two equations allow one to express K A α and g AB in terms of the E Aα while the last one turns out to be the only constraint on E Aα. Note that in such model vector fields A µ α and spinor Ω ia and scalar z A fields carry different kind of indices exactly as in the general construction of [1]. 3 Spontaneous supersymmetry breaking In this section we would like to explain how the results obtained above are connected with the problem of spontaneous supersymmetry breaking. As we have already mentioned there 4

6 exist the hidden sector (N = supergravity, vector and hypermultiplet) admitting spontaneous supersymmetry breaking with two arbitrary scales and without a cosmological term. The two vector fields (graviphoton and the matter one) enter through complex combinations A µ ± γ 5 B µ and it seemed that the only natural generalization of this model to the case of arbitrary number of vector multiplets is the model with the scalar field geometry SO(,m)/SO(m) SO(). Indeed it has been shown [5, 6] that for this case one can have spontaneous supersymmetry breaking for arbitrary number of vector and hypermultiplets and the corresponding soft breaking terms have been calculated. Now using the general formulas given in the previous section we can obtain as a partial case the generalization of our hidden sector to the case of scalar field geometry being SU(1, m)/su(m) U(1). But the case of interest turns to be the singular one corresponding to the degenerate matrix E aα and hence g ab. Let us stress however that our general solution for one vector multiplets had no singularities in this case. It is in rewriting it in a covariant form that one faces the requirement the matrix g ab to be nondegenerate. So let us split the vector fields onto the hidden sector (graviphoton and one vector fields) and matter ones. Moreover, in order to be able to compare the results with the previously obtained, it turns out to be useful to write all the formulas in term of the physical scalar and spinor fields. In order to solve the constraints (1) we introduce a kind of light cone variables, namely, we split z A = {x+y,x y,z a }, a =,3...m. Then the scalar field constrains takes the form: xy +xȳ = 1+ z az a Now we introduce a new variable π = i( x x ) (a choice of such combination will become ȳ y clear later) so that we can exclude the field x and reexpress all formulas in term of y, π and z a. It turns out that in order to have canonical form of the scalar field kinetic terms one has to make a change z a yz a. Then the only scalar field having nonzero axial charge is y and we can use local axial U(1) invariance to make it real. Denoting this real field by Φ we obtain: 1 D µ z A D µ z A = ( µ Φ Φ (19) ) + 1 Φ µ z µ z 1 4 U µ (0) where U µ = ( z A µ z A ) = Φ (i µ π + 1 ( z z)). Analogously, let us introduce Ω ia = { (λ i + χ i ),(χ i λ i ),Ω ia }. Then the spinor field constraint xλ i +yχ i za Ω ia = 0 (1) allows one to exclude spinor χ i. Again in order to have canonical kinetic terms one has to make two changes Ω ia Ω ia + z a λ i and λ i ȳ λ i. We have i Ω ia ˆDΩiA = λ i i γ µ [D µ 3 4 U µ]λ i + Ω i ia γ µ [D µ 1 4 U µ]ω ia i λi γ µ Φ µ z a Ω ia () It is not hard to rewrite supertransformations in term of the new fields (here we give the terms without vector fields only): δe µr = i( Ψ µ i γ r η i ) δψ µi = D µ η i 1 U µη i 5

7 [ δλ i = iε ij γ µ µ Φ Φ + 1 U µ ]η j δω ia = iε ij γ µ Φ µ z a η j δφ = Φ ε ij ( λ i η j ) δπ = Φ εij ( λ i γ 5 η j ) 1 Φ εij ( Ω ia γ 5 z a η j ) (3) δx a = 1 Φ εij ( Ω ia η j ) δy a = 1 Φ εij ( Ω ia γ 5 η j ) The corresponding part of the Lagrangian looks like: L = 1 R+ i εµνρσ Ψµ i γ 5 γ ν [D ρ 1 4 U ρ]ψ σi + i λ i γ µ [D µ 3 4 U µ]λ i + + i Ω ia γ µ [D µ 1 4 U µ]ω ia + ( µ Φ Φ ) 1 4 U µ + 1 Φ µ z µ z 1 εij λi γ µ γ ν [ νφ Φ + 1 U ν ]Ψ µj 1 εij Ωi a γ µ γ ν Φ ν z a Ψ µj i λi γ µ Φ µ z a a Ω i (4) One can see that the π field enter through the divergency µ π only, one of the isometries acting as a translation π π + Λ. That explains our special choice for π. Let us stress, that the last two formulas are universal in a sense that they are determined by the choice of scalar field geometry and do not depend on the choice of dual version. Now it is not hard to get from the formulas (5), (8) our hidden sector in terms of the physical fields(note that our current notations differ a little from ones used in[4]). Additional terms for the supertransformations looks like: δψ µi = i 4Φ ε ijσ(a γ 5 B)γ µ η j [ δa µ = Φ ε ij ( Ψ µi η j )+ i ] ( λ i γ µ η i ) [ δb µ = Φ ε ij ( Φ µi γ 5 η j )+ i ] ( λ i γ µ γ 5 η i ) (5) δλ i = 1 Φ σ(a+γ 5B)η i and whose for the Lagrangian have the form: L = 1 4Φ (A µν +B µν ) π 4 (A µνãµν +B µν Bµν ) 1 4Φ εij Ψµi (A µν γ 5 Ã µν +γ 5 B µν + B µν )Ψ νj + + i 4 λ i γ µ σ(a+γ 5 B)Ψ µi (6) Φ Note here that for this model (just because the matrix g ab is degenerate) the global symmetry of the Lagrangian is enhanced. At first we have axial U(1) transformations acting 6

8 on the spinor fields as well as on the complex vector field A µ + γ 5 B µ. Then we have scale transformationssuchasφ e Λ Φ, (A µ,b µ ) e Λ (A µ,b µ ), π e Λ π. Andatlastthewhole Lagrangian including terms with vector fields is invariant under translations π π + Λ. Asforthemattervectorfields, therearetwoways. Onecanusegeneralformulas(16),(17) and, makingsomekindofregularizationformatrixk a α inordertoavoidsingularities, rewrite them in term of the physical fields (keeping only terms which survive when regularization parameter goes to zero and making field rescaling if necessary). On the other hand one can start directly from the hidden sector and add to it the additional vector multiplets. We have explicitly checked that both ways lead to the same answer, namely, that δc µ a = i( Ω ia γ µ η i )+ε ij ( Ψ µi Φz a η j )+ 1 ( λ i γ µ Φz a η i ) δω ia = 1 [(σc)a x a (σa) y a (σb)]η i (7) are the only additional terms for the supertransformations, while the Lagrangian has to be completed with: L = 1 4 (C µν a ) + 1 C µν a [(x a A µν +y a B µν ) (y a à µν x a Bµν )] 1 4 (xa A µν +y a B µν )[(x a A µν +y a B µν ) (y a à µν x a Bµν )]+ + i 4 Ω ia γ µ [(σc) a x a (σa) y a (σb)]ψ µi (8) As we have already mentioned the hidden sector [4] includes also the hypermultiplet in a formulation with a singlet Φ and a triplet π i j of scalar fields as well as a doublet χ i of spinor fields. It is not hard to introduce such a multiplet in our model by adding to the Lagrangian L = 1 ( ) µ Φ and to the supertransformations: Φ + Φ ( µ π) 1 εij χ i γ µ γ ν [δ j k ν Φ Φ + Φ ν π j k ]Ψ µk + + i χi γ µ [D µ U µ]+ 1 8Φ εij χ i σ(a γ 5 B)χ j i 4 Φ[ε µνρσ Ψµ i γ 5 γ ν ρ π i j Ψ σj + λ i γ µ µ π i j λ j χ i γ µ µ π i j χ j + Ω ia γ µ µ π i j Ω i a ] (9) δ Ψ µi = Φ µ π i j η j δχ i = iε ij γ µ [δ j k µ Φ Φ + Φ µ π j k ]η k (30) δ Φ = Φε ij ( χ i η j ) δ π = 1 Φε ij ( χ i ( τ) j k η k ) Here ( τ) i j - is the usual Pauli matrices and we use π i j = π( τ) i j. The peculiar feature of this formulation for the hypermultiplets is the invariance of the Lagrangian under translations π π + Λ. This allows one to introduce masses for two 7

9 vector fields A µ and B µ using e.g. fields π 1 and π as the Goldstone ones. So let us make in the Lagrangian and in the supertransformations the replacement: µ π i j µ π i j A µ (t 1 ) i j B µ (t ) i j (31) where(t 1, ) j i = m 1, (τ 1, ) j i. Asusualinsupergravitysuchareplacement spoilstheinvariance of the Lagrangian under the supertransformations but it can easily be restored by adding to the lagrangian { L = Φ Φ 1 µi σ 4 Ψ µν ε ij (M + ) k j Ψ νk i Ψ i µ γ µ (M ) j i [λ j + χ j ] 1 ε ij (M ) k j χ k 1 } λi 4 χ iε ij (M ) k j χ k (3) where (M ± ) i j = (t 1 ) i j ±γ 5 (t ) i j and to the supertransformations: δ Ψ µi = i Φ Φγ µ ε ij (M + ) j k η k δ λ i = 1 Φ Φ(M + ) i j η j (33) δ χ i = Φ Φ(M + ) i j η j From this formulas one can see that in this model we can really have spontaneous supersymmetry breaking with two different scales, gravitini masses being connected with the vector field masses through the relations µ 1 = m 1 m and µ = m 1+m. There are no any scalar field potential in this model so the cosmological term is automatically equals to zero. Note that one could introduce a gauge interaction for vector multiplets for some nonabelian gauge group G as well. In this case a nonzero scalar field potential for the z a fields will be generated but the cosmological term remains to be zero. At the same time one can see that no soft breaking terms are generated in this model in great contrast with the case of the geometry SO(, m)/so(m) SO(). 4 Conclusion Thus we have shown that for the system of N = supergravity - vector multiplets with fixed scalar field geometry one can have a whole family of the Lagrangians interpolating between previously known dual versions for such models. We have seen that it is the choice of dual version that determines the possibility to have spontaneous supersymmetry breaking without a cosmological term while the scalar field geometry determines the pattern of soft breaking terms that are generated (or not generated) after symmetry breaking takes place. Acknowledgments Work supported by International Science Foundation grant RMP000 and by Russian Foundation for Fundamental Research grant

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