Notes on Supersymmetry Universidad de Santiago de Compostela 4 de diciembre de 2012 ´Indice general 1. Non perturbative aspects of N=1 Supersymmetric Theory 3 1.1. SU(Nc)-QCD with Nf flavours..........................3 1.1.1. Field Content . .3 1.1.2. Action and symmetries . .5 1.2. SQCD Classical Moduli Space: M0 .......................6 1.2.1. Classical Moduli Spaces . .6 1.2.2. M0(SQCD) ...............................6 1.3. SQCD, Quantum Effective Action . .9 1.3.1. SQCD: Renormalization Group Analysis . 12 1.3.2. Holomorphic β function . 15 1.3.3. Matching Conditions . 18 1.3.4. Konishi Anomaly . 19 1.4. Non Perturbative Superpotentials . 26 1.4.1. Nf < Nc massless: The Affleck-Dine-Seiberg Superpotential . 26 1.4.2. Nf < Nc mass deformation . 29 1.4.3. Integrating Out and In . 32 1.4.4. Nf ≥ Nc: The quantum moduli space . 36 1.5. Seiberg Duality . 39 1.5.1. Duality in the Conformal Window . 42 1.6. Conifold Field Theory (Klebanov-Witten) . 44 1.6.1. Free Theory . 44 1.6.2. Adding a Superpotential . 45 1.6.3. Conifold Connection . 46 1.6.4. Global Symmetries . 46 2. Selected Topics in Gauge Theories 49 2.1. Anomalies in Gauge Theories . 49 1 3. The Minimal Supersymmetric Standard Model 51 3.1. Introduction . 51 3.2. The MSSM . 52 3.2.1. Field content . 52 3.2.2. Couplings . 53 3.2.3. R-parity . 54 3.3. Electroweak symmetry breaking . 55 3.4. Supersymmetry Breaking . 57 2 Cap´ıtulo 1 Non perturbative aspects of N=1 Supersymmetric Theory 1.1. SU(Nc)-QCD with Nf flavours. Let us start by defining the class of microscopic models we will be dealing with in this course. They represent the minimal supersymmetric extension of QCD, and hence receive the generic name of SQCD. 1.1.1. Field Content Chiral Matter. i A single flavour is composed of a pair of chiral superfields (Q ; Q~i); i = 1; :::; Nc. With yµ = xµ − iθσµθ¯ p Qi(x; θ; θ¯) = φi(y) + 2θ i(y) + θ2F i(y) p 2 Q~i(x; θ; θ¯) = φ~i(y) + 2θ ~i(y) + θ F~i(y) The component fields receive the generic names of squark (φi), and quark fields i ~ ( α; α i). Gauge Fields a y In the Wess Zumino gauge V = TaW = V can be expanded as 1 V a(x; θ; θ¯) = θσµθA¯ a (x) + iθ2(θ¯λ¯a) − iθ¯2(θλ)a + θ2θ¯2D2(x) (1.1) µ 2 a ¯aα_ a in terms of a gaugino (λα; λ ), and a gauge boson Aµ. The gauge invariant field strength 1 W (x; θ; θ¯) = − D¯ 2e−2V D e2V (1.2) α 4 α a a µν a 2 µ ¯α_ a = −iλ (y) + θαD (y) − i(σ θ)αFµν(y) − θ σαα_ (Dµλ ) (y) Ta (1.3) 3 is a chiral superfield, i.e. DW¯ = 0. From here 2 1 µν a µ ¯ 2 i ∗ µν TrW 2 = − F Fµν + 2iλ σ Dµλa + D + F F (1.4) θ 2 2 µν hence 1 2 1 µν i µ ¯ 1 2 (TrW 2 + h:c:) = − F Fµν + λσ Dµλa + D 4g2 θ 4g2 g2 2g2 θ 2 θ ∗ µν −i (TrW 2 − h:c:) = F F (1.5) 32π2 θ 32π2 µν with 1 F ∗ = F αβ µν 2 µναβ Fµν = @µAν − @νAµ + i[Aµ;Aν] α_ α_ α_ Dµλ¯ = @µλ¯ + [Aµ; λ¯ ] (1.6) The gauge transformation of the different fields that enter is given in terms of a chiral i a i y ya y ya superfield Λ j = Λ (Ta) j and its antichiral hermitian conjugate Λ = Λ Ta = Λ Ta i - Q transforms in the Nc i −2iΛ i j y y 2iΛy i Q ! (e ) jQ ; Qj = Qi (e ) j (1.7) 1 - Q~i transforms in the N¯c, or 2iΛ i yi −2iΛy i y j Q~i ! Q~i(e ) j ; Q~ ! (e ) jQ~ (1.8) 2V - e transforms in the adjoint (Nc; N¯c) representation y y e2V ! e−2iΛ e2V e2iΛ ; e−2V ! e−2iΛe−2V e2iΛ (1.9) so that the combination Qye2V Q + Qe~ −2V Q~y is gauge invariant. Also for the chiral field α_ iΛ iΛ¯ strength, using [D¯ ; e ] = [Dα; e ] = 0 we find 1 W = − D¯ 2e−2V D e2V =) W ! e−2iΛW e2iΛ α 8 α α α 1 y y W¯ = − D2e2V D¯ e−2V =) W¯ ! e−2iΛ W e2iΛ (1.10) α_ 8 α_ α_ α 1Remember that given a representation D(g)D(g0) = D(gg0) of a group, automatically we have two new ones, D(g)∗ and D(g)−1t which in principle are inequivalent. The associated Lie algebra D(g) = i t −1t t i 1 + iαiD(L ) + ::: has correspondingly inequivalent representations −D , D(g) = 1 − iαiD (L ) + ::: and ∗ ∗ i i i t i ∗ D(g) ∼ 1 − iαiD (L ) + ::: . With D(L ) hermitian D(L ) = D(L ) , these are equivalent only for real parameters αi 2R. However, for general gauge transformations the parameter functions in (1.7) are chiral ∗ −1t superfields Λ 6= Λ . Hence by N¯c we must specify what we mean, and the choice is D , whence (1.8). 4 1.1.2. Action and symmetries The action can be written in superspace notation as follows τ Z 1 Z L = Im Tr d2θW 2 + d2θd2θ¯ Qy e2V Qf + Q~ e−2V Q~yf 8π 4 f f (1.11) Z 2 f + ( d θ W(Q ; Q~f ) + h:c:) fi where we have extended our chiral superfields to a set (Q ; Q~fi); i = 1; :::; Nc; f = 1; :::; Nf , 4πi θ τ = + (1.12) g2 2π hence the gauge-kinetic term reads as follows Im Z 1 Z θ Z τ Tr d2θW 2 = dθ2 TrW 2 + h:c: − i dθ2TrW 2 − h:c: 8π 4g2 32π2 1 i 1 θ = Tr − F F µν + λσµD λ¯ + D2 + F~ F µν(1.13) 4g2 µν g2 µ 2g2 32π2 µν The superpotential contains typically mass as well as higher interaction terms ~ g fi ~ fi gj hk ~fi ~gj ~hk W(Q; Q) = mf Q Qgi + afghijkQ Q Q +a ~fghijkQ Q Q + ::: (1.14) where afgh; a~fgh are SU(N) invariant tensors proportional to fgh [1] p.31. However these higher terms violate baryon number conservation. 1.1.2.1 Scalar Potential 2 Nc −1 X a 2 X 2 2 V = jD j + (jF f j + jF ~ j ) Q Qf a=1 f N 2−1 ! 1 Xc X @W @W¯ = jφ∗f T aφ − φ~ T aφ~∗f j2 + + (φ $ φ~) (1.15) 2 f f @φfi y a=1 f;i @φfi 1.1.2.2 Global symmetries In the absence of superpotential (W = 0) ! U(Nf ) × U(Nf ) × U(1)R ∼ SU(Nf ) × SU(Nf ) × U(1)B × U(1)R × U(1)A. SU(Nf )L SU(Nf )R UB(1) UA(1) UR(1) V 0 0 0 0 0 f Q Nf 0 1 1 0 Q~f 0 N¯f -1 1 0 5 U(1)R is a geometrical symmetry (θ; θ¯) ! (eiαθ; e−iαθ¯); (dθ; dθ¯) ! (e−iαdθ; eiαdθ¯) thus although the chiral superfields are neutral, the higher components inside are charged. In the presence of a superpotential of the form (1.14) and vanishing trilinear couplings, the global symmetry reduces to SU(N)V × U(1)B in the case of diagonal equal masses fg f f m = mδ g and to U(1)B for generic m g. 1.1.2.3 Theta term The gauge kinetic term (1.13) contains a θ-term. Remember that θ is a periodic variable. In other words θ ! θ+2π is a symmetry of the quantum theory. This is because spacetime integral of this term computes the\winding number" of the gauge field at infinity. Namely, there exist non-trivial gauge field configurations for which θ Z d4xFF~ = nθ (1.16) 32π2 with n an integer. Thus, although strictly speaking θ ! θ + 2π is not a symmetry of S, it shifts trivialy the phase factor in the path integral. 1.2. SQCD Classical Moduli Space: M0 1.2.1. Classical Moduli Spaces A common feature of supersymmetric models is the existence of a large amount of va- cua, spanning continuous manifolds which generically receive the name of \moduli space". Moreover, as we shall see, quantum corrections do not generically lift this degeneracy of vacuum states. This is unlike non-supersymmetric theories, for which vacua are usually dis- crete sets of points. The vacuum condition is simply V = 0. The space M0 is in many cases not a manifold, but rather a set of manifold or "branches", often joined along subspaces of lower dimension. As an example consider a model with 3 chiral multiplets X, Y and Z, and a superpotential W = λXY Z. The susy vacuum condition states that xy = xz = yz = 0 for the scalar component. So we find 3 branches, those are (y = z = 0 with x arbitrary) and cyclic permutations x ! y ! z ! x of the same statement. The three branches join at a single point x = y = z which is a singular point of W (all its gradients vanish). 1.2.2. M0(SQCD) Let us start by examining this manifold in the case of the lagrangian (1.11) with a vanishing superpotential to start with. Later on we will consider the quantum modification of this lagrangian, and we will show that nonperturbative effects allow for the appearance of a nontrivial W.