Lecture 1 the SUSY Algebra Outline of Course • Introduction to Supersymmetry: Algebra, field Theory Representa- Tions, Superspace Etc
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Lecture 1 The SUSY Algebra Outline of Course • Introduction to Supersymmetry: algebra, field theory representa- tions, superspace etc. Covered in the first few lectures, but not in great depth. Students are encouraged to work through additional material, as needed. • SUSY phenomenology: the MSSM and SUSY breaking. Central to approach in textbook, but we cover it minimally. • Modern developments: nonperturbative methods. This is empha- sis. Effective Lagrangians, symmetry breaking, anomalis, ... : key con- cepts of advanced QFT that are well illustrated by SUSY examples. • Some (relatively) recent applications: SUSY breaking in Beyond the Standard Model scenarios. Textbook: "Modern Supersymmetry" by John Terning Outline of Lecture 1 • Appetizer: radiative corrections to the Higgs mass. The quadratic divergence in the SM and its cancellation in SUSY theory. • The SUSY Algebra. An anti-commutator mixing spacetime and internal symmetry. • Spontaneous SUSY breaking: a first look. The vaccum energy: to vanish or not to vanish, that is the question. • Representations of SUSY. SUSY generalizations of the simplest QFTs. Reading: Terning 1.1-1.3 Unreasonable effectiveness of the SM • Key ingredient in Standard Model: the Higgs particle. • Required properties: a vacuum expectation value (VEV) of 246 GeV and mass broadly of the same order. • Radiative corrections: there are quadratically divergent contribu- tions to the mass. • Interpretation in effective QFT: the Higgs mass is naturally of the same order as unknown high energy physics so the low energy de- scription is inconsistent (it is not a systematic approximation). • Proposed resolution: a symmetry (SUSY) requires couplings such that quadratic divergences cancel. Thus the low energy description is consistent, if the high energy theory is supersymmetric. The Higgs Mechanism Yukawa coupling between the Higgs field (H0) and a top quark: L = − pyt H0t t + h:c: Yukawa 2 L R The Higgs field acquires a VEV (somehow: details are unimportant) : H0 = hH0i + h0 = v + h0 so that the top acquires a mass (this is the Higgs mechanism): m = ypt v t 2 Corrollary: other couplings between the Higgs excitation h0 and the top follows from the same Yukawa coupling. The Quadratic Divergence Figure 1: The top loop contribution to the Higgs mass term. Correction to h0 mass due to a top loop: 4 h −iy∗ i 2 R d k −piyt i p t i −iδm jtop = (−1)Nc 4 Tr h (2π) 2 k6 −mt 2 k6 −mt 4 2 2 2 R d k k +mt = −2Ncjytj 4 2 2 2 (2π) (k −mt ) 2 2 2 2 2 iNcjytj R Λ 2 kE (kE −mt ) = 2 dkE 2 2 2 8π 0 (kE +mt ) Details: 2 2 2 • Traces over γ-matrices: Tr(6k + mt) = 4(k + mt ) 2 2 • Euclidean continuation: k0 ! ik4, k ! −kE 4 2 3 2 2 2 • d kE = 2π kEdkE = π kEdkE 2 2 Evaluation of integral (using x = kE + mt ): 2 2 2 4 2 Ncjytj R Λ 3mt 2mt δm jtop = − 2 2 dx 1 − + 2 h 8π mt x x 2 h 2 2 i Ncjytj 2 2 Λ +mt = − 2 Λ − 3 mt ln 2 + ::: 8π mt Notation: ... are terms that are finite as Λ ! 1. Lesson from computation: virtual top quarks give quadratically and log- aritmically divergent contributions to the Higgs mass. Interpretation (reminder): we consider Λ the large (but finite!) scale where high energy physics omitted in the low energy description becomes important. Thus: the quadratic \divergence" shows that the unknown high energy physics is the natural scale for the Higgs mass. UV Completion: a Toy Model Hypothesis: there are also N light scalar particles φL; φR and they couple to the Higgs as: λ 0 2 2 2 0 2 2 Lscalar = − 2 (h ) (jφLj + jφRj ) − h (µLjφLj + µRjφRj ) 2 2 2 2 −mLjφLj − mRjφRj Figure 2: Scalar boson contribution to the Higgs mass term via the quartic coupling. Contribution to the Higgs mass from the quartic coupling to virtual new scalars: 2 R d4k h i i i −iδmhj2 = −iλN 4 2 2 + 2 2 (2π) k −mL k −mR h 2 2 2 2 i 2 λN 2 2 Λ +mL 2 Λ +mR ) δmhj2 = 2 2Λ − mL ln 2 − mR ln 2 + ::: : 16π mL mR Cancellation: the new contribution cancels the Λ2 divergence from top 2 quarks, if it happens that N = Nc and λ = jytj . Figure 3: Scalar boson contribution to the Higgs mass term via the trilinear coupling. Contribution to the Higgs mass from the cubic coupling to virtual new scalars: 2 2 2 R d4k i i −iδmhj3 = N 4 −iµL 2 2 + −iµR 2 2 (2π) k −mL k −mR h 2 2 2 2 i 2 N 2 Λ +mL 2 Λ +mR ) δmhj3 = − 2 µL ln 2 + µR ln 2 + ::: : 16π mL mR 2 2 2 Cancellation: If mt = mL = mR and µL = µR = 2λmt , then log Λ are canceled as well. Conclusion: the toy model of the high energy completion of the SM cancels all the divergences from the top quark, for specific values of the couplings between the Higgs and the new scalars. Status • The standard model requires a Higgs particle, and it contributes quadratic divergences the Higgs mass. • Interpretation: new physics at the same scale as the standard model is important. So it is mandatory that we model such new physics and include it. • A minimal model of the additional physics: two new scalars in- cluded in the low energy theory, with specific couplings to Higgs. This model is simple and acceptable, in that no further new physics must be included in the low energy theory. The model is ad hoc in that masses and couplings of the new par- ticles are postulated to have specific values. • The point: SUSY guarantees these relations and so offers a princi- pled (based on symmetry) model of the new physics. SUSY algebra The novelty: the SUSY algebra has generators that are spinors under Lorentz invariance. Notation for SUSY generators: complex, anticommuting Weyl spinors y Qα and their complex conjugates Qα_ . Convention: Weyl spinors are just left-handed (α = 1; 2), but their com- plex conjugates are right-handed. Fundamental SUSY anti-commutator (with Pµ the four-momentum): y µ fQα;Qα_ g = 2σαα_ Pµ; Pauli-matrices: µ i µαα_ i σαα_ = (1; σ ) ; σ = (1; −σ ) 0 1 0 −i 1 0 σ1 = σ2 = σ3 = 1 0 i 0 0 −1 SUSY is an internal symmetry in that it is independent on spacetime position. In Fourier space, this is expressed by the commutator y [Pµ;Qα] = [Pµ;Qα_ ] = 0 It is useful to keep track of \SUSY-ness" by the R-symmetry generator R: y y [Qα;R] = Qα [Qα_ ;R] = −Qα_ y In short: Qα decreases the R-quantum number by 1, while Qα_ increases it by 1. The SUSY generator is a spacetime spinor so due to the spin-statistics theorem its action turns fermions into bosons, and vice versa. Formally: F f(−1) ;Qαg = 0 where (−1)F jbosoni = +1 jbosoni (−1)F jfermioni = −1 jfermioni Fermion-Boson degeneracy All particles in a representation of SUSY (=a supermultiplet) have the µ 2 same mass (because PµP = M commutes with Qα). Gauge generators also commute with supercharges, so all particles in a supermultiplet have the same gauge charges. The Hamiltonian is part of the SUSY algebra: 0 1 y y y y H = P = 4 (Q1Q1 + Q2Q2 + Q1Q1 + Q2Q2) P Consequence for spectrum (derived using completeness i jiihij = 1 in any given supermultiplet): P F 0 1 P F y P F y ihij(−1) P jii = 4 ihij(−1) QQ jii + ihij(−1) Q Qjii 1 P F y P F y = 4 ihij(−1) QQ jii + ijhij(−1) Q jjihjjQjii 1 P F y P F y = 4 ihij(−1) QQ jii + ijhjjQjiihij(−1) Q jji 1 P F y P F y = 4 ihij(−1) QQ jii + jhjjQ(−1) Q jji 1 P F y P F y = 4 ihij(−1) QQ jii − jhjj(−1) QQ jji = 0: Conclusion: in each supermultiplet, there is the same number of fermions and bosons. NF = NB The Vacuum The vacuum is the ground state of the theory. It is denoted j0i. In a SUSY theory the vacuum energy (density) is non-negative 1 y y y y h0jHj0i = 4 h0j(Q1Q1 + Q2Q2 + Q1Q1 + Q2Q2)j0i ≥ 0 The vacuum energy vanishes exactly when the ground state is invariant under SUSY y h0jHj0i , Qαj0i = Qα_ j0i = 0 Conversely, SUSY is broken exactly when the ground state energy is positive y h0jHj0i= 6 0 , Qαj0i 6= 0 or Qα_ j0i= 6 0 SUSY representations Goal: determine the particle spectrum in simple supermultiplets. Reminder: representations of Relativisitic QFT are labeled by mass, spin, and a component of spin: jm; s; s3i. The fundamental anti-commutator for a massive particle in the rest frame: pµ = (m;~0): y fQα;Qα_ g = 2 m δαα_ fQα;Qβg = 0 fQy ;Qy g = 0 α_ β_ Intuition: Qα are fermionic annihilation oscillators (for α = 1; 2), and y Qα_ are the corresponding creation operators. Remark: there is no absolute distinction so the opposite assignment of creation/annihilation works as well. 0 0 Implementation: starting from any state jm; s ; s3i, define the Clifford vacuum: 0 0 jΩsi = Q1Q2jm; s ; s3i By construction, the Clifford vacuum is annihilated by the annihilation operators Q1jΩsi = Q2jΩsi = 0 Starting from the Clifford vacuum, we construct the general massive supermultiplet: jΩsi y y Q1jΩsi;Q2jΩsi y y Q1Q2jΩsi There are several relevant values for s, the spin of the Clifford vacuum jΩsi. The massive \chiral" multiplet( s = 0): state s3 jΩ0i 0 y y 1 Q1jΩ0i;Q2jΩ0i ± 2 y y Q1Q2jΩ0i 0 Spectrum: a complex boson and a massive two component fermion (=a Majorana fermion).