Lecture 6 The Super-Higgs Mechanism

Transcription

1 Lecture 6 The Super-Higgs Mechanism

2 Introduction: moduli space. Outline Explicit computation of moduli space for SUSY QCD with F < N and F N. The Higgs mechanism. The super-higgs mechanism. Reading: Terning

3 Moduli Space SUSY QFTs generally have a scalar potential that depends on the scalar fields in the theory. The minima of the scalar potential determine VEVs of the scalars. There is typically a manifold of such minima (characterized by some continuous and discrete parameters). Terminology: this space of vacua is known as moduli space. Moduli space is of central importance for the analysis of SUSY QFTs.

4 Flat directions F < N The scalar potential of SUSY QCD is: V (φ) = 1 2 Da D a where D a = g(φ in (T a ) m n φ mi φ in (T a ) m n φ mi) Define N N matrices so d n m φ in φ mi d n m φ in φ mi D a = g(t a ) m n (d n m d n m) Remark: the matrices d n m and d n m have rank F because they are formed from rectangular matrices φ mi, φ mi (with target space of dimension F ).

5 Flat directions F < N The Hermitean matrix d can be diagonalized by an SU(N) gauge transformation U d U Since d has rank F (or less), there will be at least N F zero eigenvalues. In the diagonal basis it therefore takes the form: d = v 2 1 v v 2 F... where v 2 i.

6 SUSY Minima for F < N The SUSY minima of V (φ) are precisely when for all a: D a = g(t a ) m n (d n m d n m) = The T a form a complete basis for traceless matrices, so the difference of the two matrices is proportional to the identity matrix: d n m d n m = αi In the basis where d is diagonal, d is also diagonal (because d d I), with at least N F zero eigenvalues (because its rank is F or less). Conclusion: α =, so d n m = d n m

7 Flat directions F < N d n m and d n m are invariant under SU(F ) SU(F ) transformations: φ mi φ mi Vj i, d n m V j i φ in φ mi Vj i = φ jn φ mj = d n m. Thus, up to a flavor transformation, we can write v 1... φ = φ = v F This is the canonical parametrization of moduli space for SUSY QCD with F < N.

8 Flat directions F N As before, define N N Hermitian matrices d n m φ in φ mi d n m φ in φ mi so the SUSY vacuum conditions are D a = g(φ in (T a ) m n φ mi φ in (T a ) m n φ mi) = g(t a ) m n (d n m d n m) = The generators T a form a complete set of traceless matrices so in the SUSY vacuum d n m d n m = ρi. Key remark: generally d n m and d n m have maximal rank N for F N.

9 Diagonalization In a SUSY vacuum : d n m can be diagonalized by an SU(N) gauge transformation: v 1 2 v 2 2 d =... vn 2 In the basis where d is diagonal, d must also be diagonal, since d d I. Notation: the eigenvalues of d are denoted v i 2. Eigenvalues are related as v i 2 = v i 2 + ρ.

10 Flat directions F N Since d n m and d n m are invariant under flavor transformations, we can use SU(F ) SU(F ) transformations to put φ and φ in the form v 1... v 1... φ =....., φ = v N.... v N This is the canonical parametrization of moduli space for SUSY QCD with F N.

11 Moduli Space: Physical Significance Moduli space: a space of degenerate vacua, all with vanishing potential V (φ) =. Coordinates on moduli space: vacua are distinguished by the VEVs of the scalar fields. Physical distinction between different vacua: the spectrum depends on the VEVs, as do couplings. As we move on moduli space, particle spectra change. Example: for F < N the VEVs of the scalars generically break gauge symmetry SU(N) SU(N F ). For F N the SU(N) gauge symmetry is completely broken at a generic point in moduli space. In either case, the masses acquired by the gauge particles of the broken symmetry depend on the position in moduli space.

12 The Super Higgs Mechanism Spontaneously broken gauge symmetry: a scalar field that develops a VEV breaks the gauge symmetry it is charged under. The pseudo-goldstone modes: the fluctuations of the scalar field along the broken symmetry direction correspond to massless modes (were it not that they are charged). The Higgs mechanism: the gauge fields corresponding to broken generators acquire a mass. The additional polarization needed for massive gauge particles arise from eating the would-be Goldstone components of the scalar field. Goal: realize the Higgs mechanism in a SUSY theory. Set-up: a scalar field develops a VEV that breaks gauge symmetry but not SUSY. The Super Higgs mechanism: a massless vector supermultiplet eats a chiral supermultiplet to form a massive vector supermultiplet.

13 Example in SUSY QCD Example: v 1 = v 1 = v and v i = v i =, for i > 1. Breaking of gauge symmetry: SU(N) SU(N 1). Breaking of global symmetry: SU(F ) SU(F ) SU(F 1) SU(F 1). Decompose the adjoint of SU(N) under SU(N 1): Ad N = Ad N 1 Basis of gauge generators G A = X, X α 1, X α 2, T a, with A = 1,..., N 2 1. Unbroken SU(N 1) generators T a with a = 1,..., (N 1) 2 1. Broken generators X, X α 1,2 with α = 1,..., N 1 span the coset of SU(N)/SU(N 1). Counting broken gauge generators: N 2 1 ((N 1) 2 1) = 2(N 1) + 1.

14 The Super Higgs mechanism The Xs are analogues of the Pauli matrices: X = 1 2N(N 1) N , X α 1 = , X α 2 = i.... i.

15 The Super Higgs Mechanism We can also define raising and lowering operators: X ±α = 1 2 (X α 1 ix α 2 ) so that X +α = , X α =

16 The Scalar Spectrum after SSB Expanding the squark field around its VEV φ : the auxiliary D A field becomes φ φ + φ. D A g = φ G A φ φg A φ φ G A φ φ G A φ + φ G A φ φ G A φ +φ G A φ φg A φ + φ G A φ φg A φ. We wish to evaluate the scalar potential V = 1 2 DA D A. For this, decompose the product of two SU(N) generators: G A G A = X X + X +α X α + X α X +α + T a T a.

17 The Scalar Spectrum after SSB Among the generators, G A = X, X α 1, X α 2, T a, the unbroken group T a annihilates φ so: A GA φ = X φ + α X α φ, φ A GA = φ X + φ α X+α. Fluctuations around the squark VEVs are decomposed as: ( ) ( h σi h H α ) φ =, φ =, H α φ mi σ i φ im where φ is a matrix with N 1 rows and F 1 columns.

18 The Scalar Spectrum after SSB Picking out the mass terms in the scalar potential V = 1 2 DA D A : V mass = g2 2 = g2 v 2 2 [ ( ) φ X φ + φ X φ φ X φ φx φ 2 ] +2( φ X +α φ φ X +α φ )(φ X α φ φx α φ ) [ ( h + h (h + h) ] +(H α H α )(H α H α ). (N 1) 2 2N(N 1) Diagonalize the mass matrix: H +α = 1 2 (H α H α ), π +α = 1 2 (H α + H α ), H α = 1 2 (H α H α ), π α = 1 2 (H α + H α ), h = Re(h h), π = Im(h h), Ω = 1 2 (h + h). ) 2

19 The Scalar Spectrum after SSB The mass terms reduce to The spectrum of scalars: V mass = g 2 v 2 [ N 1 N (h ) 2 + H +α H α]. A real scalar h with mass gv 2(N 1)/N. A complex scalar H ±α with mass gv. Massless complex scalars σ i, σ i, and Ω, neutral under the low energy SU(N 1). Massless pseudo Goldstone-modes π ±α, π. These become the longitudinal components of the massive gauge bosons (can be removed by going to unitary gauge). Massless complex squarks of the low energy theory: φ mi, φ im. Count real d.o.f. s : 1+2(N 1)+2(2F 1)+(2N 1)+4(N 1)(F 1) = 4NF.

20 Fermions The gauge sector couples to a single chiral multiplet by the Yukawa interaction L Yukawa = 2g(φ T a ψ)λ a + h.c. In the case of F fundamental chiral multiplets, F anti-fundamentals, and notation adapted to SSB: L Yukawa = 2g[(φ I GA Q I )λ A (Q I G A φ I)λ A ] + h.c. Fermion mass term arises when the scalars take their VEVs. In the case we analyze, the flavor index I reduces to I =. Decomposition of the gluino in basis adapted to SSB: G A λ A = X Λ + X +α Λ +α + X α Λ α + T a λ a. The scalar VEVs are such that φ X α =, φ T a = and also X +α φ =, T a φ =.

21 Fermion Spectrum after SSB The fermion mass terms generated by the Yukawa interactions: L F mass = 2g [( φ ( X Λ + φ X +α Λ +α) Q )] Q X Λ φ + X α Λ α φ + h.c. ( = gv ω Λ ω Λ ) ] + ω α Λ +α ω α Λ α [ N 1 N Decomposition of the quark field: ( ) ( ω ψ Q = i ω ω α, Q = ω α Q mi ψ i Q im ). + h.c. Comments: i is a flavor index (taking F 1 values). α and m are color indices (taking N 1 values). Q is a matrix with N 1 rows and F 1 columns. Q is a matrix with F 1 rows and N 1 columns.

22 Fermion Spectrum after SSB The fermion mass terms: L F mass = gv [ N 1 N ( ω Λ ω Λ ) + ω α Λ +α ω α Λ α ] + h.c. The fermion spectrum after SSB to SU(N 1) gauge theory: A Dirac fermion (Λ 1, 2 (ω ω 2(N 1) )) with mass gv N. Two sets of N 1 Dirac fermions (Λ +α, ω α ), (Λ α, ω α ) with mass gv. Massless Weyl fermions Q, Q, ψ, ψ, and 1 2 (ω + ω ).

23 The Super Higgs Mechanism Expand gauge fields in basis adapted to breaking: G B A B µ = X W µ + X +α W +α µ + X α W α µ + T a A a µ. The A 2 φ 2 terms leading to gauge boson masses are L A 2 φ 2 = g2 A A µ A B ν g µν φ G A G B φ = g 2 g µν φ (X WµX Wν + X +α W µ +α X α Wν α = g 2 v 2 g ( µν N 1 2N W µw ν W ) µ +α Wν α. There is an identical term arising from L A 2, giving a factor of 2. 2 φ The spectrum of gauge bosons is + X α W α µ X +α W +α ν ) A gauge boson W µ with mass gv 2(N 1)/N. It is neutral under the unbroken SU(N 1) gauge group. It acquired its third polarization by eating π. It is the superpartner of h and the neutral Dirac fermion.

24 Gauge bosons W +α µ and W α µ with mass gv. They are in the fundamental of the unbroken SU(N 1) gauge group. They acquired their third polarization by eating π ±α.these are superpartners of H ±α and the charged Dirac fermions. The massless gauge bosons A a µ of the unbroken SU(N 1) gauge group.

25 The super Higgs mechanism Reminder: the theory when the scalar VEV v = is a SUSY SU(N) gauge theory with the matter content: SU(N) SU(F ) SU(F ) b.d.o.f. Q 1 2NF Q 1 2NF Each massless chiral supermultiplet has 2 real bosonic degrees of freedom (b.d.o.f.) For v the gauge symmetry breaks SU(N) SU(N 1). The gauge particles are in massive supermultiplets : SU(N 1) SU(F 1) SU(F 1) b.d.o.f. W W (N 1) W 1 1 4(N 1)

26 Neutral massive vector supermultiplet: W =(W µ, h, Λ, 1 2 (ω ω )). The massive vector field W µ has 3 d.o.f. because it absorbed the scalar π (in unitary gauge). The mass of each field in the neutral massive super multiplet: 2(N 1) m W = gv N. Charged massive vector supermultiplets: W + =(W +α µ, H +α, Λ +α, ω α ) and W =(W α µ, H α, Λ α, ω α ). The massive vector fields W ±α µ have 3 d.o.f. because the absorbed the scalar π ±α (in unitary gauge). The mass of each field in the charged massive super multiplet: m W ± = gv.

27 The super Higgs mechanism For v also the massless chiral supermultiplets: SU(N 1) SU(F 1) SU(F 1) b.d.o.f. Q 1 2(N 1)(F 1) Q 1 2(N 1)(F 1) ψ 1 1 2(F 1) ψ 1 1 2(F 1) S Quark chiral supermultiplets Q = (φ, Q ) and Q = (φ, Q ). Gauge singlets ψ = (σ, ψ) and S = (Ω, 1 2 (ω + ω )). In both cases (v = and v ) a total of 2(N 2 1) + 4F N b.d.o.f. (and the same number of fermionic d.o.f.).

28 Summary The moduli space of a theory is its space of SUSY vacua. parametrized by the values of scalar VEVs. It is The gauge symmetry and other global symmetries are generally broken by the scalar VEVs, with breaking pattern dependent on the position in moduli space. The spectrum of the theory depends on the position in moduli space. It takes some effort to work out the details. The super Higgs mechanism: the gauge particles corresponding to broken generators acquire a mass. Scalars and fermions also acquire mass so that an entire massive vector supermultiplet emerges. The main example in this lecture: SUSY QCD with a single component of the squark acquiring a VEV (and the anti-quark acquiring the same VEV).