The Super-Higgs Mechanism in non-linear Supersymmetry

Transcription

1 Università degli Studi di Padova DIPARTIMENTO DI FISICA E ASTRONOMIA GALILEO GALILEI Corso di Laurea Magistrale in Fisica Tesi di Laurea Magistrale The Super-Higgs Mechanism in non-linear Supersymmetry Laureando: Elia Zanoni Matricola Relatore: Prof. Gianguido Dall Agata Anno Accademico

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3 Contents Introduction 5 1 Linear Supersymmetry Realizations Supermultiplets N=1 Supersymmetry Supermultiplets Superfields as Supermultiplets Superfields Chiral Superfields Vector Superfields Invariant Actions Matter actions Vector Actions Vector-Matter Interactions Kähler Chiral Models Supersymmetry breaking Non-Linear SUSY and Constrained Superfields 31.1 Why Non-Linear Realizations? An Historical approach Low Energy Lagrangians One Chiral Field Two Chiral Superfields Constrained Superfields One Chiral Constrained Superfield Two Chiral Constrained Superfields

4 4 CONTENTS.5 Constrained Superfields Theory Constrained Superfields in Supergravity Supergravity as Local Supersymmetry Constrained Matter Scalar-less models Models Constraining Fermions Constrained Supergravity The Super-Higgs Mechanism Super-Higgs in Minkowski Super-Higgs with a Cosmological Constant Super-Higgs with Non-Linear Supersymmetry Constrained Superfields in a Non-Minkowski Vacuum Four Fermion Interaction Summary and Outlook 83 Bibliography 85

5 Introduction The Standard Model is, so far, the most successful theory that describes elementary particles and their interactions. It fits perfectly with the experimental data but it leaves some problems unsolved as the hierarchy problem in the Higgs mass, the gap between the running coupling constants in Grand Unification theories and the presence of dark matter. Supersymmetry had a great impact when introduced because it could give an answer to these problems. It provided a dark matter candidate, the neutralino. It removed the gap in Grand Unifications theories, the three running coupling constant of the standard model intersect perfectly at GeV. Also, if supersymmetry were linearly realized at the T ev scale then it would solve also the Higgs hierarchy problem. Every boson s,interacting with the Higgs field, would have a fermionic superpartner f, which also interacts with the Higgs and at energies above the supersymmetry breaking scale the one loop contributions from these interactions would cancel. The Higgs mass would then be compatible with the standard model expectations for a supersymmetry breaking scale around the T ev. Supersymmetry can also be seen as a low energy theory of the String Theory and so the enthusiasm about supersymmetry was well motivated by phenomenological and theoretical reasons. It is now clear that the supersymmetry breaking scale is not around the T ev scale but it is much higher. However, even if supersymmetry is broken at a very high energy scale, it can still be used to constrain effective lagrangians, in fact, when a symmetry is broken, we still have a non-linear realization of such symmetry on the effective degrees of freedom [1]. A generic consequence of supersymmetry breaking is a mass splittings in the spectrum, where the heavy states can be close to or higher than the supersymmetry breaking scale and therefore might be integrated out. The effective theory for the remaining states is then constrained 5

6 by a non-linearly realized supersymmetry. If global non linear supersymmetry is exact then the fermionic goldstone modes are massless. This is the most common scenario, but, starting from Volkok-Akulov work [], non-linear supersymmetry were also used to study light fermions as pseudo-goldstone modes of an approximate supersymmetry. More recently inflationary theories in supergravity, [3 13], and brane supersymmetry breaking scenarios, [14 17], were described using non-linearly realized supersymmetry. Non linear realizations of supersymmetry were initially described using the component fields formulation and then they were implemented in superspace. In superspace methods various properties of supersymmetry are manifest even when the spectrum is not supersymmetric anymore. Among these methods, an interesting approach is to describe non-linear supersymmetry trough constrained superfields [18 1]. This thesis is a review about constrained superfields and eventually it focuses on a open question about the unitarity bounds in inflationary models. In [13] the authors stated that there is no problem with unitarity thanks to the effective cutoff Λ = (V + 3m 3/ )1/4 that they believed universal. The authors of [6] instead found that the energy range of validity in inflationary models is constrained by unitarity. In this work we will show that, for a simple model with constrained superfields in supergravity, Λ is not the only relevant scale and so the absence of problems linked to unitarity bound was a feature only of models analogous to those studied in [13]. This work starts with a discussion about supersymmetry and its linear realizations. The superfields formalism is introduced and a general way to construct invariant lagrangians under supersymmetry transformations is developed. Our interest in effective lagrangians in which supersymmetry is spontaneously broken leads us to the description of non-linear realizations and the constrained superfields formalism allows us to obtain them in a efficient way. This formalism is extended also to supergravity and this is useful in order to have a set-up compatible with inflationary models. The breaking of local supersymmetry is well described by the super-higgs mechanism. This mechanism is analyzed first in linear supersymmetry and then in a simple model with constrained superfields. Eventually the interaction terms of this model are computed and it is shown that Λ is not the 6

7 only relevant scale. 7

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9 Chapter 1 Linear Supersymmetry Realizations This chapter is a brief review about supersymmetry and its linear realizations based on []. After a brief discussion on supermultiplets there is a focus on superfields and on their properties. The aim of this chapter is to highlight how the superfields formalism can simplify the construction of supersymmetric lagrangians and to derive them for vector and chiral superfields. 1.1 Supermultiplets Coleman and Mandula showed that the most general symmetry, compatible with the Poincaré symmetry group, that can be realized in a local QFT is the direct product of the Poincaré group and an an internal symmetry group [3]. It was then natural to define particles as unitary irreducible representations of the Poincaré algebra and to label them with the possible values of the two Poincaré Casimirs, P and W. The first one is the square-mass operator while W is related to the the spin (helicity) operator. Mass and spin (helicity) values describe completely a particle and they are invariant under the action of both the Poincaré group and the internal symmetry group. Coleman and Mandula considered only bosonic symmetries. Haag, Łopuszański and Sohnius investigated also fermionic symmetries and they found that most general symmetry compatible with the Poincaré group is given by the product of the Super-Poincaré group, an extension of the Poincaré group that contains the supersymmetry generators Q I α, and an internal symmetry 9

10 CHAPTER 1. LINEAR SUPERSYMMETRY REALIZATIONS group [4]. The irreducible representations of the Super-Poincaré algebra can be written as a collection of irreducible representations of the Poincaré algebra and they are called supermultiplets. Supermultiplets play in a supersymmetry-invariant theory the same role that particles play in a Lorentz-invariant one. It is of great importance then to highlight some fundamental properties: Particles in the same supermultiplet have the same mass but different spin. W can not be a Casimir of the Super-Poincaré algebra because particles in the same supermultiplet are related by the action of the supersymmetry generators that change the spin by half a unit. P instead is still a Casimir because P µ commutes with the supersymmetry generators. In every supermultiplet the bosonic and fermionic d.o.f. are the same. The operator ( 1) s, where s is the spin, acts on bosonic and fermionic states as ( 1) s B = B, ( 1) s F = F. (1.1) This operator anti-commutes with Q α and so 0 =Tr ( Q α ( 1) s B β + ( 1) s Q βq α ) =Tr ( ( 1) s {Q α, Q β} ) = σ µ α β Tr[ ( 1) s] P µ. (1.) Choosing P µ 0 it follows that Tr( 1) s = 0, namely that n B = n F. Every state has positive energy. From the positive metric assumption [4] 0 α φ {Q α, Q α } φ = Tr(σ µ ) φ P µ φ = 4 φ P 0 φ. α N =1 Supersymmetry Supermultiplets In this work we are interested in N = 1, or minimal, supersymmetry. This means that the supersymmetry generators are given by a complex Weyl fermion Q α. As seen before, in a supermultiplet there are particles related by the action of this generator. Unitary irreducible representations od the Poincaré group were derived 10

11 1.1. SUPERMULTIPLETS acting on a Clifford vacuum with creation operators. A similar procedure can be developed in order to create unitary irreducible representations of the super- Poincaré group. Massless Supermultiplets The first step for building massless supermultiplets is finding creation and annihilation operators. By evaluating {Q, Q} = σ µ P µ in the rest frame, it follows that only Q is non trivial. From this generator it is possible to define a 1 4E Q, a 1 4E Q, (1.3) such that they satisfy the anti-commutator relation for creation and annihilation operators: {a, a } = 1. (1.4) When acting on some state, the operators a and a respectively lower and rise the helicity of 1. The second step is to define a Clifford vacuum E, λ 0, where λ 0 is the helicity, such that a E, λ 0 = 0. (1.5) The full supermultiplet is obtained acting with a on E, λ 0 λ 0 : a λ 0 = λ (1.6) The last step is imposing CPT-invariance by doubling the supermultiplet, namely by adding its CPT conjugate. The useful massless supermultiplets for this work are: Matter or chiral multiplet λ 0 = 0 ( 0, + 1 ) CP T ( 1 ), 0. (1.7) There are two bosonic degrees of freedom from a complex scalar and two fermionic from a Weyl fermion. This multiplet is also known as Wess-Zumino 11

12 CHAPTER 1. LINEAR SUPERSYMMETRY REALIZATIONS multiplet. Gauge or vector multiplet λ 0 = 1 ( + 1, +1 ) CP T ( 1, 1 ). (1.8) There are two bosonic degrees of freedom from a massless vector and two fermionic from a Weyl fermion. Graviton multiplet λ 0 = 3 ( + 3, + ) CP T This multiplet contains a graviton and a gravitino. (, 3 ). (1.9) Massive Supermultiplets The procedure for constructing massive supermultiplets is similar to the previous one. The main differences are that there are not vanishing generators and that now the spin is taken in account rather than the helicity. The creation and annihilation operators a 1, 1 m Q 1,, a 1, 1 m Q 1,, (1.10) respectively lower and raise the spin j by half unit. Starting from different Clifford vacua j 0 it is possible to construct: Matter multiplet: ( j 0 = 0 1, 0, 0, + 1 ) ; (1.11) this multiplet is made of a massive complex scalar and a massive Majorana fermion; Gauge or vector multiplet: j 0 = 1 ( 1, 1, 0, +1, 1 ), (1.1) 1

13 1.1. SUPERMULTIPLETS the degrees of freedom are those of one massive vector, one massive Dirac fermion and one massive real scalar Superfields as Supermultiplets Until now the focus was on unitary irreducible representations. For the Poincaré group there are also finite dimensional irreducible representations. They are labelled by a couple of numbers (m, n) and their dimension is d = (m + 1)(n + 1). They are very useful because in a relativistic quantum field theory all the fields belong to one of these representations. It is possible to develop a similar formalism also in Super-Poincaré? The next section will provide a detailed answer but here a useful first attempt in this direction is made using the procedure illustrated above. For simplicity let s start from a complex scalar field φ(x) such that [ Q α, φ(x)] = 0. (1.13) Thanks to this constraint if the field were real it would be a constant. The action of Q α on φ gives: [Q α, φ(x)] ψ α (x). (1.14) A new field ψ α is defined by the action of Q α on φ. In the same multiplet now there are a complex scalar and a Weyl fermion. Acting again with the generators on ψ α : {Q α, ψ β (x)} = F αβ (x) ; (1.15) { Q α, ψ β } = X αβ (x). (1.16) After some calculations one gets: X αβ µ φ ; (1.17) F αβ (x) = ɛ αβ F (x). (1.18) F (x) is a new scalar field that must be added to the field multiplet. No new fields are introduced with a further step and so all the fields that appear in the 13

14 CHAPTER 1. LINEAR SUPERSYMMETRY REALIZATIONS supermultiplet constructed starting from φ are (φ, ψ, F ). (1.19) This object is called chiral or Wess-Zumino multiplet. This multiplet starts with a complex scalar, whose associated state can be represented by 0. The action of Q α gave a Weyl fermion, ψ α. This operations is analogous to the action of a on 0 that creates 1/. The first two fields correspond exactly to the particle states of a chiral supermultiplet. A problem arises when a third field, F, is generated. The degrees of freedom of the collection of fields generated with this procedure are four bosonic, two from φ, and two from F and four fermonic from ψ α. The equivalence of the degrees of freedom is still valid but they are not the ones of the chiral supermultiplet. That multiplet was on-shell. The Weyl fermion looses two d.o.f. thanks to the Dirac equation. Also the bosonic number is diminished of two units, as will be shown later, because F is an auxiliary field. The on-shell numbers are then n F = n B =. (1.0) A collection of fields related by supersymmetry is called superfield. When on-shell superfields are considered there is a perfect correspondence with the superparticle states. 1. Superfields Superfields are defined as functions of superspace coordinates (x µ, θ α, θ α ). The Grassmann variables θ and θ have been introduced in order to transform the graded supersymmetry algebra in a Lie algebra with generators: Q α θq, (1.1) Q α θ Q. (1.) 14

15 1.. SUPERFIELDS Since θ and θ are Grassmann variables, the most general superfield Y(x, θ, θ) has the following expansion Y(x, θ, θ) =f(x) + θψ(x) + θ χ(x) + θθm(x) + θ θn(x) + θσ µ θvµ (x) + θθ θ λ(x) + θ θθρ(x) + θθ θ θd(x). (1.3) The Y superfield has this name because it is a collection of ordinary fields. A supersymmetry transformation on Y with parameters (ɛ α, ɛ α ) is defined as Y(x + δx, θ + δθ, θ + δ θ) = e i(ɛq+ ɛ Q) Y(x, θ, θ)e i(ɛq+ ɛ Q). (1.4) In this notation Q is the abstract operator. From the superfield variation is possible to derive the explicit expression for the coordinate variations: δx µ = iθσ µ ɛ iɛσ µ θ ; (1.5) δθ α = ɛ α ; (1.6) δ θ α = ɛ α. (1.7) The supersymmetry variation of Y can be written as δ ɛ, ɛ Y = (iɛq + i ɛ Q)Y, (1.8) with Q the differential supersymmetry operator: Q α = i α σ µ α β θ β µ, (1.9) Q α = +i α + θ β σ µ β α µ. (1.30) In (1.3) there is the general expression for a superfield. The supermultiplets derived above had less component than Y. They can be obtained by imposing some supersymmetric constraint. The most common superfields are the chiral and the vector ones. 15

16 CHAPTER 1. LINEAR SUPERSYMMETRY REALIZATIONS 1..1 Chiral Superfields In N=1 supersymmetry covariant derivatives D α and D α can be constructed in order to anti-commute with Q α and Q β defined above. D α = α + iσ µ α β θ β µ ; (1.31) D α = α iθ α σ µ α α µ. (1.3) Since D and D anti-commute with Q and Q they also commute with the variation δ ɛ, ɛ : δ ɛ, ɛ (D α Y ) = D α (δ ɛ, ɛ Y ). (1.33) This implies that if Y is a superfield, then also D α Y is a superfield. The constraints D α Φ = 0, (1.34) D α Φ = 0, (1.35) are supersymmetric invariant. A field Φ that satisfies the first constraint is called chiral while a field Φ that satisfies the second anti-chiral. Finding a general expression for Φ is quite simple with the following coordinates redefinition y µ = x µ + iθσ µ θ. (1.36) With this coordinates D α y µ = D α θ = 0. (1.37) The general expression for a chiral superfield is Φ = A(y) + θψ(y) + θθf (y). (1.38) In the x µ coordinates it becomes Φ =A(x) + iθσ µ θ µ A(x) θθ θ θ A(x) + θθf (x) + θψ(x) i θθ µ ψ(x)σ µ θ. (1.39) 16

17 1.. SUPERFIELDS Analogously for an anti-chiral superfield, defining ȳ µ = x µ iθσ µ θ, Φ =A (ȳ) + θ ψ(ȳ) + θ θf (ȳ) (1.40) =A (x) oθσ µ θ µ A (x) θθ θ θ A (x) + θ θf (x) + θ ψ(x) + i θ θθσ µ µ ψ(x). (1.41) This superfield is equivalent to the field multiplet in (1.19). A chiral superfield, under supersymmetry transformations, transforms as: δ ɛ, ɛ Φ(y, θ) = (iɛq + i ɛ Q)Φ(y, θ). (1.4) Q and Q have to be expressed in term of the new coordinates: Q = i α, (1.43) Q = i α + θ α σ µ α α y. (1.44) µ Plugging these definitions in (1.4) the transformation becomes δ ɛ, ɛ Φ(y, θ) = ɛψ + ( θ + ɛf + iσ µ ɛ ) ( y A +θθ i ɛ σ µ ) µ y ψ. (1.45) µ The supersymmetry transformations for each component of the chiral superfield multiplet are: δ ɛ A = ɛψ, (1.46) δ ɛ ψ = i σ µ ɛ µ A + ɛf, (1.47) δ ɛ F = i ɛ σ µ µ ψ. (1.48) The fermionic nature of supersymmetry is easily seen in these transformations. The variations of the scalars A and F are proportional to the spinor ψ while the variation of ψ is proportional to the scalars. From these transformation it is clear why superfields are linear realizations of the supersymmetry. Every field variation depends linearly on the other fields. 17

18 CHAPTER 1. LINEAR SUPERSYMMETRY REALIZATIONS 1.. Vector Superfields Vector superfields are defined imposing the condition: V = V. (1.49) Their power series expansion in θ and θ is: V(x, θ, θ) =C(x) + iθχ(x) i θ χ + i θθ[m(x) + in(x)] i [ θ θ[m(x) in(x)] θσ µ θvµ (x) + iθθ θ λ(x) + i ] σµ µ χ(x) [ i θ θ λ(x) + i ] σµ µ χ(x) + 1 [D(x) θθ θ θ + 1 ] C(x). (1.50) The fields C, D, M, N and v µ are real. generalization of a gauge transformation, Under the following supersymmetric V V + Φ + Φ, (1.51) where Φ and Φ are respectively a chiral and an anti-chiral superfield, the component fields transform as C C + A + A, (1.5) χ χ i ψ, (1.53) M + in M + in if, (1.54) v µ v µ i µ (A A ), (1.55) λ λ, (1.56) D D. (1.57) (1.58) It is possible to choose a gauge, often called the Wess-Zumino gauge, in which C, χ, M and N are all zero. Only a vector field, with a usual gauge transformation, 18

19 1.3. INVARIANT ACTIONS a spinor and a scalar remain: V = θσ µ θvµ (x) + iθθ θ λ(x) i θ θθλ(x) + 1 θθ θ θd(x). (1.59) The supersymmetry transformations for the component fields are: δ ɛ v µ = i λ σ µ ɛ + i ɛ σ µ λ, (1.60) δ ɛ λ = σ µν ɛ( µ v ν ν v µ ) + iɛd, (1.61) δ ɛ D = ɛσ µ µ λ λσµ ɛ. (1.6) Also these transformations, as the chiral ones, are linear in the fields. 1.3 Invariant Actions The real strength of the superfields formalism is linked to the possibility of building supersymmetric actions in a simple way. If only a set of fields, transforming as in (1.46)-(1.48), were taken in account, it would be problematic to construct an invariant lagrangian. Every time a new term is added at least another one must be taken in account in order to compensate its variation. With superfields it becomes quite easy because if Y is a superfield, then S = d 4 xd θd θy(x, θ, θ), (1.63) is supersymmetric invariant: δ ɛ, ɛ S = d 4 xd θd θδɛ, ɛ Y = d 4 xd θd θ[ɛ α α Y + ɛ α α Y + µ [ i(ɛσ µ θ θσµ ɛ)y]] = 0. (1.64) The first equality holds because the integral in the Grassmann variables is translational invariant by construction while the last holds because the terms with α and α don t have enough θs and µ [... ] is a total derivative. The integral in the full superspace of a superfield always gives a supersymmetric action, if it is also suitably defined then, by integrating it over the Grassmann coordinates, it is pos- 19

20 CHAPTER 1. LINEAR SUPERSYMMETRY REALIZATIONS sible to get a Lagrangian density of dimension four, which is real and transforms as a scalars. S = d 4 xd θd θa(x, θ, θ) = d 4 xl(φ(x), ψ(x), A µ (x),... ). (1.65) Matter actions Starting from a set of chiral superfields Φ i, the combination Φ i Φ i has the right dimension for giving a four-dimension lagrangian L. The only contribution comes from the θθ θ θ-component of the integrated superfield. Φ i Φ i = F i F i A i A i A i A i 1 µa i µ A i + i ψ i σ µ µ ψ i i ψ i σ µ µ ψ i. (1.66) Up to total derivatives, L is L = i µ ψi σ µ ψ i + A i A i + F i F i. (1.67) This lagrangian contains the canonical kinetic term for a complex scalar and a Weyl spinor. may come from Another kind of contribution, when considering chiral superfields d 4 xd θσ(x, θ, θ) = d 4 yd θσ(y, θ). (1.68) Σ is chiral and the equation holds because x µ = y µ up to total derivatives. These contributions were not included in the integral in the full superspace because, even if d 4 xd θd θy = d 4 xd θ D Y, (1.69) it is not true that all the integrals in d θ can be written as integrals in d θd θ. Every holomorphic function P of a chiral superfield Φ, namely a function that satisfies P = 0, is chiral Φ D α P (Φ) = P Φ D α Φ + P Φ D α Φ = 0. (1.70) 0

21 1.3. INVARIANT ACTIONS The contribution to the lagrangian is L int = d 4 xd θp (Φ i ) + d 4 xd θ P ( Φi ) = + P Φ F i 1 P i Φ i Φ j ψi ψ j + h.c.. (1.71) The derivatives are evaluated at Φ i = A i. The superpotential P has to satisfy some simple properties. First, as said before it has to be holomorphic, then it can not contain covariant derivatives since D α Φ is not chiral. Finally, P has to have dimension three in order to have a lagrangian of dimension four. This means that, for having renormalizable theories, P can be at most cubic in Φ i. The lagrangian for matter fields is given by L = d 4 xd θd θ Φi Φ i + d 4 xd θp (Φ i ) + d 4 xd θ P ( Φi ) (1.7) ( P =i µ ψi σ µ ψ i + A i A i + F i F i + A F i 1 ) P i A i A j ψi ψ j + h.c.. (1.73) This lagrangian is invariant, up to total derivatives, under the transformations (1.46)-(1.48). Furthermore there are not derivatives of the F fields so they are auxiliary fields and they can be integrated out: F i = P A i, F i = P A i. (1.74) The on-shell Lagrangian obtained thanks to the F i equations of motion is ( ) 1 L = i µ ψi σ µ ψ i + A i P A i A i A j ψi ψ j + h.c. V (A i, A i ). (1.75) The scalar potential V (A i, A i ) is V = P A i. (1.76) It is important to underline that the interactions that are present in the on-shell lagrangian are due to the equation of motion for F. With a vanishing F -term there would be no interactions and masses but only the kinetic terms. 1

22 CHAPTER 1. LINEAR SUPERSYMMETRY REALIZATIONS 1.3. Vector Actions The vector superfield V can be seen as a generalization of the Yang-Mills potential. The generalization of the field strength can be defined as: W α = 1 4 D DD α V, W α = 1 4 DD D α V. (1.77) These superfields are chiral and gauge invariant: D α W α =0, D α W α = 0, (1.78) W α 1 4 D DD α (V + Φ + Φ) =W α 1 4 D{ D, D α }Φ = W α. (1.79) In the Wess-Zumino gauge they have a simple expression: W α = iλ α (y) + [ δα β D(y) i ] (σµ σ ν ) α β ( µ v ν (y) ν v µ (y)) θ β + θθσ µ α α µ λ α (y) ; (1.80) [ W α = i λ α (ȳ) + ɛ βd(ȳ) α + i ] ɛ α γ( σ µ σ ν ) γ β( µ v ν (ȳ) ν v µ (ȳ)) θ β m ɛ α β θ θ σ βα µ λ α (ȳ). (1.81) In the chiral superfields W α and W α there are only the gauge invariant fields, λ α and D, and the gauge invariant field strength v µν = µ v ν ν v µ. Since W α is chiral an invariant action can be obtained from the integration in d θ of W α W α : S = d 4 xl = 1 4 = ( d 4 x d θw α W α + d θw ) α W α ; (1.8) d 4 x 1 D 1 4 vµν v µν iλσ µ µ λ. (1.83) In this lagrangian there are the kinetic terms for a gauge vector, v µ, and for a spinor, λ α. The lagrangian above can be written as an integral in d θd θ thanks to the definition and the chirality of W α : L = d θd θ 1 4 (W α D α V + W α D α V ). (1.84)

23 1.3. INVARIANT ACTIONS Mass term can be added but they aren t gauge invariant and so they ca not be expressed in the Wess-Zumino gauge. L m = d θd θv. (1.85) Vector-Matter Interactions In the standard model, when the matter lagrangian is globally invariant under some symmetry group, a set of vectors must be introduced in order to have local symmetry and these vectors interacts with the matter fields. The procedure for gauging chiral superfields is analogous. If a chiral lagrangian is invariant under the global action of a group with generators {T a } then introducing a vector superfields V a this invariance can be preserved also locally. The gauge invariant lagrangian is: L = 1 (H 16kg ab + ξ A ) d θw a W b + h.c. + d θd θ Φe V Φ d 4 xd θp (Φ i ) + d 4 xd θ P ( Φi ). (1.86) d θ d θv A + The contribution proportional to ξ A is the Fayet-Iliopulos term and it is present only for the abelian factors, H ab is a holomorphic function of Φ and obviously P has to be gauge invariant.the explicit off-shell lagrangian, with H ab = δ ab, is: L = 1 4 F µνf a aµν i λ a σ µ D µ λ a + 1 Da D a D µ A D µ A i ψ σ µ D µ ψ + F F + i g(a T a ψλ a λ a T a A ψ) + gd a A T a A + gξ A D A ( P + A F i 1 ) P i A i A j ψi ψ j + h.c.. (1.87) where D µ A = µ A + igv a µt a A, (1.88) D µ ψ = µ ψ + igv (a) µ T (a) ψ, (1.89) Dλ (a) = µ λ (a) gt abc v (b) µ λ (c), (1.90) F µν (a) = µ v ν (a) ν v µ (a) gt abc v µ (b) v ν (c). (1.91) 3

24 CHAPTER 1. LINEAR SUPERSYMMETRY REALIZATIONS The transformation laws for the components of the superfield multiplets are linear and their expressions are: δ ɛ A = ɛψ, (1.9) δ ɛ ψ = i σ µ ɛd µ A + ɛf, (1.93) δ ɛ F = i ɛ σ µ D µ ψ + igt (a) A ɛ λ (a), (1.94) δ ɛ v (a) µ = i λ (a) σ µ ɛ + i ɛ σ µ λ (a), (1.95) δ ɛ λ (a) = σ µν ɛv (a) µν + iɛd (a), (1.96) δ ɛ D (a) = ɛσ µ D µ λ(a) Dλ (a) σ µ ɛ. (1.97) The equations of motion for the auxiliary fields are: F = P A, (1.98) F = P A, (1.99) D a = ga T a A gξ a. (1.100) The ξ a contribution to D a is present only for the Abelian factors. The on-shell lagrangian is L = 1 4 F µνf a aµν i λ a σ µ D µ λ a D µ A D µ A i ψ σ µ D µ ψ + i ( ) g(a T a ψλ a λ a T a A ψ) 1 P A i A j ψi ψ j + h.c. V, (1.101) where the scalar potential V is: V = P A + g A T a A + ξ a. (1.10) The scalar potential is always positive in a supersymmetry invariant gauge theory. 4

25 1.3. INVARIANT ACTIONS Kähler Chiral Models In the previous description only Φ i Φ i appeared in the full superspace integral. This can be generalized integrating an analytic functions of the superfields K(Φ i, Φ j ). This function has to satisfy all the condition necessary to give a meaningful lagrangian and it has an important property: d 4 xd θd θk(φ i, Φ j ) = d 4 xd θd θ[k(φ i, Φ j ) + Λ(Φ) + Λ( Φ)], (1.103) where Λ is a chiral superfield that depends only on Φ. This relation is true because the θ θ term of the variation is a total derivative. This kind of transformation is called Kähler transformation and K(a i, a i ) is called Kähler potential. It describes a manifold with metric: g ij = K. (1.104) a i aj Obviously the metric is hermitian, positive defined and invariant under the Kähler transformations: K(a i, a i ) K(a i, k i ) + F (a i ) + F (a i ). (1.105) The only non vanishing Christoffel symbols in a Kähler geometry are: Γ k ij = g kl a g i jl, Γk i j = glk g lj a i. (1.106) The covariant derivative on this manifold is defined as i V j = i V j Γ k ijv k. (1.107) The curvature of a Kähler metric is defined as [ i, j ]V k = R l ij kv l. (1.108) The explicit expression for the curvature is R ij kl = g ml a j Γm ik. (1.109) 5

26 CHAPTER 1. LINEAR SUPERSYMMETRY REALIZATIONS By using this formalism for integrating K(Φ i, Φ j ) in d θd θ, the resulting lagrangian is: L =g ij F i F j + 1 { 1 4 g ij,kl χi χ k χ j χ l F i g im Γm j k χj χ k P } A { i 1 F i g mi Γm jkχ j χ k P } g ij µ A i µ A j ig ij χ j σ µ D µ χ i A i 1 P A i A j χi χ j 1 P A i A j χi χ j. (1.110) Here D µ χ i = µ χ i + Γ i jk µa j χ k. The equations of motion for F i are The on-shell lagrangian becomes g ij F i 1 g kj Γk mlχ m χ l + P = 0. (1.111) A j L = g ij µ A i µ A j ig ij χ j σ µ D µ χ i R ij kl χi χ k χ j χ l 1 D id j P χ i χ j 1 D i D j P χ i χ j g ij D i P D j P, (1.11) where D i P = D i P D j P = A P, i (1.113) A i A P j Γk ij A P. k (1.114) Kähler isometries Since we are dealing with a Kähler manifold there can be analytic isometries that have to be gauged including vector fields. These isometries are generated by holomorphic Killing vectors, X (b) = X i(b) (a j ) a i, (1.115) X (b) = X i(b) (a j ). (1.116) a i 6

27 1.3. INVARIANT ACTIONS Here (b) = 1,..., d, where d is the dimension of the isometry group. The Killing equations, for a Kähler manifold, imply the existence of d real scalar function D (a) (a, a ) such that g ij X j(a) = i a i D(a), (1.117) g ij X i(a) = i a j D(a). (1.118) The killing potentials D (a) are defined modulo a constant c (a). The Killing vectors are a representation of the isometry group: [X (a), X (b) ] = f abc X (c) ; (1.119) [X (a), X (b) ] = f abc X (c) ; (1.10) [X (a), X (b) ] =0. (1.11) Also D (a) can be chosen to transform in the adjoint representation of the isometry group and this fixes the c (a) for non-abelian group. For abelian U(1) factors the constants c (a) are undetermined. These constants are related to the Fayet-Iliopulos terms. The variations of the Kähler potential and of the superpotential are: δk =[ɛ (a) X (a) + ɛ (a) X (a) ]K ; (1.1) δp =ɛ (a) X (a) P. (1.13) Since the action must be invariant the variation of P has to vanish. The function F (a) = X (a) K + id (a) satisfy j F (a) = 0 and so δk = ɛ (a) F (a) + ɛ (a) F (a) i(ɛ (a) ɛ (a) )D (a) (1.14) is a Kähler transformation for real parameters ɛ (a). There is no need to make this term vanish because the action is invariant under Kähler transformations. By promoting the global symmetry to a local symmetry the parameters ɛ (a) become chiral fields Λ (a) that are complex. The K variation is not a Kähler transformation anymore because of the term proportional to D (a) and so a new terms must be added for canceling it. The counterterms involve a vector superfield V = V (a) T (a), 7

28 CHAPTER 1. LINEAR SUPERSYMMETRY REALIZATIONS where the T (a) are the generators of the isometry group. The D (a) components of the V (a) superfields are exactly the Killing potential defined above. After some calculation the explicit expression for a Kähler gauge invariant model is L = g ij D µ A i D µ A j iλ (a) σ µ D µ λ(a) 1 g D ig ij χ i σ µ D µ χ j 1 F (a) µν F µν(a) + g g ij [X i(a) χ j λ(a) + X j(a) χ i λ ( a)] 1 D id j P χ i χ j 1 D i D j P g ij D i P D j P R ij kl χi χ k χ j χ l. (1.15) where The transformation laws are: D µ A i = µ A i gv (a) µ X i(a), (1.16) D µ χ i = µ χ i + Γ i jkd µ A j χ k gv µ (a) X i(a) χ j, A j (1.17) Dλ (a) = µ λ (a) gf abc v µ (b) λ (c), (1.18) D i P = A P, i (1.19) D i P D j P = A i A P j Γk ij A P. k (1.130) δa i =ɛ (a) X i(a), (1.131) δχ i (a) Xi(a) =ɛ χ j, A j (1.13) δλ (a) =f abc ɛ (b) λ (c), (1.133) δv (a) µ =g 1 µ ɛ (a) + f abc ɛ (b) v (c) m. (1.134) 1.4 Supersymmetry breaking In order to be compatible with experimental data supersymmetry must be broken at least at the T ev scale. Spontaneous symmetry breaking is a scenario in which the theory is supersymmetric but the scalar potential admits a supersymmetry breaking vacuum. On a vacuum, respecting Lorentz invariance, all the fields but 8

29 1.4. SUPERSYMMETRY BREAKING the scalars have vanishing VEV and VEV s derivatives and so the transformation laws are δ φ i = 0, δ F i = 0, δ ψ i ɛ F i, δ Fµν a = 0, δ D a = 0, δ λ a ɛ D a. (1.135) If the F and D expectation values vanish, the vacuum is supersymmetric, otherwise it breaks supersymmetry. Since the scalar potential is positive defined V = F F + 1 D, F i = W φ i, D a = g( φ i (T a ) i jφ j + ξ a ), (1.136) then supersymmetric vacua are global minima of the potential and V vanishes on them. In a supersymmetry breaking vacuum V 0 and the potential VEV is related to the supersymmetry breaking scale. The contributions from the F and D terms to the masses of the particles of the theory are: vector mass matrix [(M 1 ) ] ab = g A T a T b A = D a i D bi, (1.137) where D a i = D a / A i ; fermionic mass matrix ( ) F ij i D b M 1/ = i i D a j 0, (1.138) where F ij = F/ A i A j ; scalar mass matrix (M 0 ) = ( ) V V A i A j A i A l V V A j A k A j A l. (1.139) These matrices satisfy the supertrace mass formula STrM = g D a TrT a. (1.140) 9

30 CHAPTER 1. LINEAR SUPERSYMMETRY REALIZATIONS Every time supersymmetry is broken the fermionic mass matrix has a zero eigenvalue: ( ) ( ) F i F ij F j + Di b D b M 1/ = = i D a Dj a F j The goldstino ψ G can be defined as the massless fermion: ( ψ i λ a Its explicit form is ψ G F i + i D a. ) = ( F i i D a ) ( ) V A i = 0. (1.141) δw a ψ G + ψ G. (1.14) The Goldstone theorem implies the existence of a massless scalar in the spectrum every time a symmetry generator is broken. Since supersymmetry generators are fermionic, the extension of the Goldstone theorem to supersymmetry breaking implies the existence of a massless spinor, the goldstino. 30

31 Chapter Non-Linear SUSY and Constrained Superfields In this chapter we describe three different approaches to construct models at energies much below the supersymmetry breaking scale. Some properties of linearly realized supersymmetry, such as the equivalence of the bosonic and fermionic numbers, the mass degeneracy for the fields in the same multiplet and the Kähler geometry of the scalar σ-model, are lost in these models but the superfields formalism is still present in two of them and it will be clear that constraining superfields is the easiest way to obtain non-linear representations..1 Why Non-Linear Realizations? There is no experimental evidence of linearly realized supersymmetry and so the main focus is on effective supersymmetric lagrangians. In order to have a better comprehension of what happens with supersymmetry it is useful to discuss briefly non linear realizations of a simple bosonic global symmetry. Let us consider a model with four real scalars φ i with SO(4) symmetry: L = 1 µφ i µ φ i µ φ iφ i λ 4 (φ iφ i ). (.1) 31

32 CHAPTER. NON-LINEAR SUSY AND CONSTRAINED SUPERFIELDS If µ < 0 the vacuum choice breaks the symmetry SO(4) SO(3). The choice of the vacuum is arbitrary and we consider the following vacuum: φ 1,,3 = 0, φ 4 = With the following parametrization φ i = Π i (v + ρ(x)), Π i Π i = 1 Π 4 = µ λ = v. (.) 1 Π I, (.3) with I = 1,, 3. Under the initial SO(4) symmetry the scalar fields Π I transform as Π I = Λ J I Π J + Λ 4 I 1 Π K. (.4) Their transformation laws are non-linear. The lagrangian, after the redefinitions, becomes L = 1 [(v + ρ) g IJ µ Π I µ Π J + µ ρ µ ρ] λv ρ λvρ 3 λ 4 ρ4. (.5) The expression for g IJ is g IJ = δ IJ + Π IΠ J 1 Π k. (.6) The only massive scalar is ρ, m ρ = λv, and if v, then ρ can be integrated out. The equations of motion for ρ in the vacuum are satisfied for ρ = 0. After the substitution Π I Π I /v the effective lagrangian becomes L = 1 [ δ IJ 1 v Π I Π J 1 Π K v ] µ Π I µ Π J. (.7) This lagrangian can be expanded in a series is 1/v for v. The first terms are: L = 1 µπ I µ Π I + 1 v (Π I µ Π I ) (.8) This lagrangian can have infinite contributions from the expansion as a power series of 1/v. All the contributions are SO(3) invariant. The rescaled Π I transform 3

33 .. AN HISTORICAL APPROACH under the original SO(4) symmetry group as: Π I = Λ J I Π J + vλ 4 I 1 Π K v. (.9) After the symmetry breaking the action of SO(4) is not linear on the remaining fields and this is a general feature when a symmetry is broken. The lagrangian (.7) is invariant under the non-linear SO(4) transformations in (.9). The approach in this section starts from a symmetry that is broken in a vacuum and it shows that integrating out a massive field gives an effective lagrangian with infinite contributions from the massless fields. A different approach is to consider the non-linear transformation laws and to build an invariant lagrangian starting from them. In the model considered in this section the effective lagrangian has to be invariant under linear SO(3) and non-linear SO(4). All the terms that satisfy this constraints can be inserted in the effective lagrangian.. An Historical approach The first attempt to write an effective lagrangian in which supersymmetry was not-linearly realized was done following the general method illustrated by Callan, Coleman, Wess and Zumino in [1]. This approach is different from the one in the previous section because the effective lagrangian is built starting from non-linear supersymmetry transformations. This procedure starts considering the supersymmetry coordinates transformations: x = x + i(θσ ɛ ɛσ θ), (.10) θ = θ + ɛ, (.11) θ = θ + ɛ. (.1) Introducing an arbitrary spinor field λ(x) analogous to θ such as θ = kλ the transformations above become: λ (x ) = λ(x) + 1 k ɛ, (.13) 33

34 CHAPTER. NON-LINEAR SUSY AND CONSTRAINED SUPERFIELDS λ (x ) = λ(x) + 1 ɛ. (.14) k The variation of λ at the same point is δ ɛ λ α = λ α (x) λ α (x) = 1 k ɛα ik(λσ µ ɛ ɛσ µ λ) µ λ α. (.15) Since (δ η δ ɛ δ ɛ δ η )λ α = i(ησ µ ɛ ɛσ µ η) µ λ α, (.16) then the transformation law above realizes non-linearly the supersymmetry algebra. In (.15) there are the non-linear transformation for λ we were interested in. Now we have to construct an effective lagrangian for λ that is invariant under that transformation. By using differential forms the coordinates transformations can be written as: dx µ = dx µ + idθσ µ ɛ iɛσ µ d θ ; (.17) dθ α = dθ α ; (.18) d θ α = d θ α. (.19) The following combinations of differentials are invariant under the above transformations e µ = dx µ idθσ µ θ + iθσ µ d θ, (.0) e α = dθ α, (.1) e α = d θ α. (.) In terms of λ, e µ becomes e a dx µ [δ a µ ik µ λσ a λ + ik λσ a µ λ] = dx µ A a µ. (.3) 34

35 .3. LOW ENERGY LAGRANGIANS By considering the expression for the invariant quantity e a, an invariant lagrangian under the non-linear transformation for λ may be: L = 1 deta. (.4) k This lagrangian, known as Volkov-Akulov lagrangian [], describes a massless spinor: L = 1 k i (λσµ µ λ µ λσ µ λ) + [interactions]. (.5) The constant term is related to a non vanishing scalar potential in a linear realization of supersymmetry and so supersymmetry is spontaneously broken for non-linear realization..3 Low Energy Lagrangians In this section the most common way for obtaining effective lagrangian for supersymmetric theories is described trough some simple examples. The procedure is analogous to the one introduced for the SO(4) SO(3) symmetry breaking discussed above. Expanding around a non-supersymmetric vacua some fields acquire mass and they can be integrated out leading to an effective lagrangian. The remaining fields transform non-linearly under the original supersymmetry action..3.1 One Chiral Field The first model describes one chiral field X with the following Kähler potential and superpotential: K = XX 1 Λ ( XX), W = fx. (.6) For simplicity Λ, f R. Λ is a very high energy scale and f is related to the supersymmetry breaking energy scale as will be shown later and so 1TeV< f Λ. The metric and the Christoffel symbols at the first order in 1/Λ are: g x x = 1 4 Λ xx, Γx xx = 4 Λ x, Γ x x x = 4 x. (.7) Λ 35

36 CHAPTER. NON-LINEAR SUSY AND CONSTRAINED SUPERFIELDS The lagrangian of this model is L = (1 4Λ )F xx x F x 1Λ ( χχ χ χ F x Λ ) ( x χ χ f F x Λ ) xχχ f (1 4Λ ) xx µ x µ x i (1 4Λ ) xx χ σ µ µ χ + 4i Λ x µx χ σ µ χ. (.8) This lagrangian is invariant under the following supersymmetry transformations: δ ξ x = ξχ, (.9) δ ξ χ = i σ µ ξ µ x + ξf x, (.30) δ ξ F x = i ξ σ µ µ χ. (.31) Since F x is an auxiliary field, it can be integrated out in order to give the on-shell lagrangian L = (1 4Λ ) xx µ x µ x i (1 4Λ ) xx χ σ µ µ χ + i 4 Λ x µx χ σ µ χ f f xχχ Λ Λ x χ χ 1 χχ χ χ V, (.3) Λ where V is the following scalar potential: V = f ( Λ xx ). (.33) The scalar potential never vanishes and it has a minimum for x = 0 in which V = f. Supersymmetry is spontaneously broken and the breaking scale is f. As expected there is a massless fermion, χ, namely the goldstino, while the scalar acquires a mass, m x = f/λ. If the energy scale given by m φ is much higher than the scale we are interested in, than x can be integrated out. Considering only zero-momenta contributions, the equation of motion for x is x = χχ f. (.34) 36

37 .3. LOW ENERGY LAGRANGIANS Lagrangian (5.14), considering the substitution and taking the limit Λ becomes: L = + ψ ψ (ψψ) 4f i χ σ µ µ χ f. (.35) This lagrangian is equivalent to (.5), derived following the historical approach. The supersymmetry transformation laws now are non-linear and equivalent to (.15). δ ξ χ = i σ µ ξ µ (χχ) fξ. (.36) f.3. Two Chiral Superfields The second model describes two chiral superfields, A and B, with the following Kähler potential K and prepotential W : K = XX + ȲY 1 Λ ( XX) 1 XXȲY, W = fx. (.37) Λ The lagrangian is given by L = d θd θk(x, ( X, Y, Ȳ) + ) d θw (X, Y) + h.c.. (.38) This lagrangian is invariant under the usual linear supersymmetry transformations. The metric g i j of the Kähler manifold is ( g x x g y x g xȳ g yȳ ) = ( 1 4 Λ x x 1 Λ yȳ 1 Λ xy 1 Λ xȳ 1 1 Λ xx ). (.39) The inverse metric g i j is ( g x x g y x g xȳ g yȳ ) = ( Λ x x 1 Λ yȳ + 1 Λ xȳ + 1 Λ xy Λ xx ). (.40) The only non vanishing, at first order in 1/Λ, Christoffel symbols are Γ x xx = 4 Λ x, Γx xy = 1 Λ ȳ, Γy xy = 1 x (.41) Λ 37

38 CHAPTER. NON-LINEAR SUSY AND CONSTRAINED SUPERFIELDS and their complex conjugates. The curvature R i jk l is R x xx x = 4 Λ, R x xyȳ = 1 Λ = R xȳy x. (.4) The scalar potential is: V = f (1 + 4 Λ x x + 1 Λ yȳ B ). (.43) There aren t supersymmetric vacua. There is a minimum for x = y = 0. The masses of the the two scalars are m x = f/λ and m y = f/λ while the fermions are massless. As seen before the scalars can be integrated out. The zero-momenta solution of the equation of motion for x and y are. The effective lagrangian, in the limit Λ, is: L = + χ χ χχ 4f 4 x = χχ f, (.44) y = χψ f. (.45) + χ ψ χψ f The transformation law for the spinors are non-linear: i χ σ µ µ χ i ψ σ µ µ ψ f. (.46) δ ξ χ = i ( ) χχ σ µ ξ µ fξ, (.47) f δ ξ ψ = i ( ) χψ σ µ ξ µ. (.48) f Also this lagrangian has the form of lagrangian [] and the spinors carry a nonlinear realization of supersymmetry. Both fields are massless but the goldstino is χ as can be seen by the f contribution to the χ transformation laws. 38

39 .4 Constrained Superfields.4. CONSTRAINED SUPERFIELDS The third method for obtaining effective lagrangians is constraining superfields. It is better than the historical one because it uses the superfields formalism that is the most efficient. It also is more convenient than the second because it eliminates the unwanted fields in a easier way. As usual a first description in given trough some simple examples..4.1 One Chiral Constrained Superfield The most simple model of supersymmetry with constrained superfield describes a superfield X that satisfies the constraint X = 0: 0 = x + (θχ)(θχ) + xθχ + xθθf x = x + xθχ + θθ(xf x χχ). (.49) The only non trivial solution is x = χχ F x. (.50) The most simple Kähler potential and superpotential that break supersymmetry are The lagrangian for X(x, χ, F x ) is K = XX, W = fx. (.51) L = F x F x µ x µ x i χ σ µ µ χ + ff x + f F x. (.5) This lagrangian is invariant under the following supersymmetry transformations: δ ξ x = ξχ, (.53) δ ξ χ = i σ µ ξ µ x + ξf x, (.54) δ ξ F x = i ξ σ µ µ χ. (.55) 39

40 CHAPTER. NON-LINEAR SUSY AND CONSTRAINED SUPERFIELDS By adding the constraint (.50), lagrangian (.5) becomes L = F x F x χ χ F x ( χχ F x ) i χ σ µ µ χ + ff x + f F x. (.56) The transformation laws now are not linear: δ ξ χ = i ( ) χχ σ µ ξ µ + ξf x, (.57) F x δ ξ F x = i ξ σ µ µ χ. (.58) The equation of motion for F x is F x = f χ χ F x ( χχ F x ). (.59) This equation can be solved iteratively. This is possible because only finite combination of χ, χ and their derivatives contributes to F x. The first step is F x = f. The second is The computation of F 1 x The third step is F x = f χ χ χχ 4f 3. (.60) = A + B χ χ(χχ) + C(... ) is done imposing F x F 1 x = 1: F 1 x = 1 f + χ χ (χχ) 4f 5. (.61) F x = f + 1 ( 1 4 χ χ f χχ ( χ χ) ) [ ( χχ 1 f 6 f + χ χ (χχ) )]. (.6) 4f 5 The solution is F x = f χ χ (χχ) 4f χ χχχ ( χ χ) (χχ) 16f 7. (.63) A fourth step is not necessary because the last contribution of (.63) vanishes when inserted in (.59). 40

41 .4. CONSTRAINED SUPERFIELDS With this expression for F x the lagrangian (5.18) becomes L = f + χ χ (χχ) 4f This lagrangian is invariant under δ ξ χ = + i σ µ ξ µ [ χχ + ξ χ χχχ ( χ χ) (χχ) 16f 6 i χ σ µ µ χ. (.64) ( 1 f + χ χ (χχ) )] 4f 5 ( f χ χ (χχ) + 3 χ χχχ ( χ χ) (χχ) 4f 3 16f 7 ). (.65) The two lagrangians (.64) and (.35) differ by a term with eight fermions. This contribution is suppressed by a factor k 4 /f 6, where k is the momentum carried by the fermions. For momenta well below the supersymmetry breaking scale the two lagrangians coincide..4. Two Chiral Constrained Superfields The second model describes the two chiral fields introduced in the previous section but constrained by: X = 0, XY = 0. (.66) The first constraint gives The second is equivalent to x = χχ F x. (.67) The θ -term vanishes for: 0 = χχ F x y + θ(ψ χχ F x + χy) + θ (F y χχ F x + F x y χψ) (.68) y = χψ F yχχ. (.69) F x This makes the scalar term to vanish trivially while the θ contribution becomes ((θψ) (χχ) F x 41 F x + (θχ) (χψ) ) F x (.70)

42 CHAPTER. NON-LINEAR SUSY AND CONSTRAINED SUPERFIELDS Since (θχ)(χψ) = (χχ)(θψ)/ also this term vanish. The two expressions (.67) and (.69) satisfy the constraints imposed and the lagrangian without the scalar fields is. ( ) χχ F x F x ( χψ F y ) F ) yχχ A F x Fx L = i χ σ µ µ χ i ψ σ µ µ ψ + χ χ + ( χ ψ F x χ χ F x + F x F x + F y F y + ff x + f F x. (.71) This lagrangian is invariant under the following non linear supersymmetry transformations δ ξ χ = i ( ) χχ σ µ ξ µ + ξf x, (.7) F x δ ξ ψ = i ( χψ σ µ ξ µ χχ F F x Fx y )+ ξf y, (.73) δ ξ F x = i ξ σ µ µ χ, (.74) δ ξ F y = i ξ σ µ µ ψ. (.75) The equations of motion for the fields F x and F y are F x = f + χ χ ( ) χχa F x F x ( χ ψ + χ χ ( χψ F F x F x 3 y ) F ) yχ A χ, (.76) F x Fx F y = + χ χ ( χψ F F ) yχχ. (.77) x F x Fx These equations can be solved iteratively. The first step is trivial F x = f, F y = 0. (.78) The second step gives F x = f χ χ χχ 4f 3 χ ψ χψ f 3 ; (.79) 4

43 .4. CONSTRAINED SUPERFIELDS F y = χ χ χψ f 3. (.80) The expression for F x after the third step is F x = f χ χ χχ 4f 3 + χ χχχ χ ψ χ ψ 4f 7 χ ψ χψ f 3 + χ χχχ χ ψ χχ f χ χχχ χ χ χχ + χ ψ χ ψ χψ χψ 16f 7 f 7 χψ χ ψ χ χ χχ χ ψχψ χ ψ χψ f 7 f 7 χ ψχχ χ χ χψ χ ψχχ χ χ χψ + + f 7 4f 7 χ χχχ χ ψ χψ + + χ ψχψ χ ψ χψ f 7 f 7 + ( more than two ). (.81) We stress that if we switch off the superfield Y we recover (.63) as we expect. With the third step F y becomes F y = χ χ χψ f χ χχψ χ χ χχ 8f 7 χ χχψ χ ψ χψ f 7 + χ χχχ χ χ χχ A 4f 7 + χ χχχ χ χ χψ 8f 7 + ( more than two ). (.8) By substituting the expressions for F x and F y, the lagrangian (.71) becomes L = i χ σ µ µ χ f + χ χ χχ 4f i ψ σ µ µ ψ + χ χχχ χ ψ χψ 4f 6 χ ψ χψ f χχ χ χ χχ χ χ 16f 6 χψ χ ψ χψ χ ψ f 6 χ ψ χ ψ χψ χψ f 6 43

44 CHAPTER. NON-LINEAR SUSY AND CONSTRAINED SUPERFIELDS χ ψχψ χ ψ χψ f 6 χ ψχψ χ χ χχ 4f 6 χ ψχχ χ χ χχ 4f 6 + ( more than two ). (.83) Considering ψ = 0, this lagrangian is identical to the lagrangian (.64) found in the case of a single chiral field. By taking the low energy limit, or in other words by forgetting the two contribution, it is equivalent to the lagrangian (.46) computed from supersymmetry breaking. Most General Model The two constraints seen until now can be used for constructing the most general non linear representation with two fermions. The constraints X = 0 = XY imply Y 3 = 0: Y 3 = y y θψ + 6y(θψ) + 3y F y θ. (.84) y 3 and y ψ vanish trivially. The non trivial part is the θ coefficient y F y yψ = (χψ) F x F y + χχψψ F Fx y = 0. (.85) The most general Kähler potential with the constraints above is K = XX + ȲY + a( XY + XȲ ) + b(ȳy + YȲ ) + c(ȳy). (.86) All the terms f(x) or g(y) are swept away by a Kähler transformation while d( XY + ȲX) is absorbed using a linear combination of X and Y. The most general superpotential is W = fx + gy + hy. (.87) From the Kähler potential the metric g i j, with i, j = {X, Y}, can be derived ( ) 1 aȳ g i j =. (.88) ay 1 + b(y + ȳ) + 4c(yȳ) 44

45 .4. CONSTRAINED SUPERFIELDS The inverse is g i j = b(y + ȳ) + 4(c a )yȳ ( ) 1 + b(y + ȳ) + 4cyȳ ay aȳ 1. (.89) The non vanishing Christoffel symbols are Γ x yy = Γ y yy = The lagrangian of this model is a(1 + bȳ) 1 + b(y + ȳ) + 4(c a )yȳ, (.90) b + 4(c a )ȳ 1 + b(ȳ + y) + 4(c a )yȳ. (.91) L =F x Fx + aȳf x Fy + ay F x F y + (1 + b(y + ȳ))f y Fy i χ σ µ µ χ i ψ σ µ µ ψ ( F x (aψψ f)+ F y (bψψ g hy) hψψ + h.c.) + ( 4 fermions). (.9) The equations of motion for the auxiliary fields give F x = f + F x + ( 4 fermions), (.93) F y = g + F y + ( 4 fermions). (.94) where F i indicates the terms with two fermions. In order to compute the masses of the fermions let s calculate L =f f(f x + F x ) + afg(y + ȳ) + g (1 + b(y + ȳ)) Eventually it becomes g(f y + F y ) + (faψψ f + f F x + h.c.) (+bgψψ g hyg + g F y + h.c.) hψψ h ψ ψ i χ σ µ µ χ i ψ σ µ µ ψ. (.95) L = f g i χ σ µ µ χ i ψ σ µ µ ψ 1 [ ψψ(af + bg h) 45