SUSY QCD. Consider a SUSY SU(N) with F flavors of quarks and squarks

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1 SUSY gauge theories

2 SUSY QCD Consider a SUSY SU(N) with F flavors of quarks and squarks Q i = (φ i, Q i, F i ), i = 1,..., F, where φ is the squark and Q is the quark. Q i = (φ i, Q i, F i ), in the antifundamental representation. Note the the bar ( the name not a conjugation, the conjugate fields are ) is part of Q i = (φ i, Q i, F i ), Q i = (φ i, Q i, F i ).

3 SUSY QCD matter content is: SU(N) SU(F ) SU(F ) U(1) B U(1) R Q 1 1 F N F Q 1 1 F N F W =

4 R-charge [R, Q α ] = Q α. chiral supermultiplet: R ψ = R φ 1, normalize the R-charge by Rλ a = λ a, R-charge of the gluino is 1, and the R-charge of the gluon is.

5 Group Theory: Bird Tracks Identify the group generator with a vertex as in Fig.??. a m r ) n (T = n m Figure 1: Bird-track notation for the group generator T a.

6 Bird Tracks quadratic Casimir C 2 (r) and the indext (r) of the representation r, are given diagrammaticly as (T a r ) m l (T a r ) l n = C 2 (r)δ m n, (T a r ) m n (T b r ) n m = T (r)δ ab, m C (r) δ = 2 n n m T(r) δ ab = a b Figure 2:

7 Bird Tracks Contracting the external legs: In the first diagram setting m equal = Figure 3: to n and summing over n yields a factor of d(r). In the second diagram setting a equal to b and summing yields a factor d(ad). d(r)c 2 (r) = d(ad)t (r).

8 Casimirs d( ) = N, d(ad) = N 2 1 T ( ) = 1 2, T (Ad) = N so C 2 ( ) = N 2 1 2N, C 2(Ad) = N.

9 Sum over Generators For the fundamental representation : (T a ) l p(t a ) m n = 1 2 (δl nδ m p 1 N δl pδ m n ). = 1 2 2Ν 1 Figure 4: We can reduce the sums over multiple generators to an essentially topological exercise

10 Anomalies Since we can define an R-charge by taking arbitrary linear combinations of the U(1) R and U(1) B charges we can choose Q i and Q i to have the same R-charge. For a U(1) not be to broken by instanton effects the SU(N) 2 U(1) R anomaly diagram vanishes Figure 5: fermion contributes its R-charge times T (r). Sum over gluino, quarks: 1 T (Ad) + (R 1)T ( ) 2F =, so R = F N F

11 Renormalization group tree-level SUSY: Y = 2g, λ = g 2. For SUSY to be a consistent quantum symmetry these relations must be preserved under RG running. the β function for the gauge coupling at one-loop is β g = µ dg dµ = g3 16π 2 ( 11 3 T (Ad) 2 3 T (F ) 1 3 T (S)) g3 b 16π 2, For SUSY QCD: b = (3N F )

12 where Renormalization group the β function for the Yukawa coupling is : [ ] (4π) 2 β j Y = 1 2 Y 2 (F )Y j + Y j Y 2 (F ) + 2Y k Y j Y k +Y k Tr Y k Y j 3gm{C 2 2 m (F ), Y j }, Y 2 (F ) Y j Y j Y 2 (F )Y j represents the scalar loop corrections to the fermion legs 2Y k Y j Y k contains the 1PI vertex corrections Y k Tr Y k Y j represents fermion loop corrections to the scalar leg C m 2 (F ) is the quadratic Casimir of the fermion fields in the mth gauge group, and represents gauge loop corrections to the fermion legs

13 SUSY QCD RG For SUSY QCD the Yukawa coupling of quark i with color index m, gluino a, and antisquark j with color index n is given by Y jn im,a = 2g(T a ) n mδ j i. Q φ λ Q = Y (Q) 2 λ Q φ λ = Y ( λ) 2 φ Q λ φ = Tr Y Y Figure 6: Feynman diagrams and associated bird-track diagrams. Y 2 (Q) = 2g 2 C 2 ( ), Y 2 (λ) = 2g 2 2F T ( )

14 SUSY QCD RG no scalar corrections corresponding to Y k Y j Y k. As for the fermion loop correction it always has a quark (antisquark) and gluino for the internal lines so we have Y k Tr Y k Y j = Y kq kq im,a (Yfp,b ) Y jn fp,b = 2g2 C 2 ( )(T a ) n mδ j i, gauge loop corrections are {C m 2 (F ), Y j } = (C 2 ( ) + C 2 (Ad))Y j. all the terms in β j Y proportional to C 2( ) cancel: (4π) 2 β j Y = 2g 3 (C 2 ( ) + F + 2C 2 ( ) 3C 2 ( ) 3N) = 2g 3 (3N F ) = 2(4π) 2 β g so the relation between the Yukawa and gauge couplings is preserved under RG running

15 SUSY QCD Quartic RG SUSY also requires the D-term quartic coupling λ = g 2. The auxiliary D a field is given by and the D-term potential is D a = g(φ in (T a ) m n φ mi φ in (T a ) m n φ mi) V = 1 2 Da D a The β function for a quartic scalar coupling at one-loop is (4π) 2 β λ = Λ (2) 4H + 3A + Λ Y 3Λ S, Λ (2) corresponds to the 1PI contribution from the quartic interactions H corresponds to the fermion box graphs A to the two gauge boson exchange graphs Λ Y to the Yukawa leg corrections Λ S corresponds to the gauge leg corrections

16 SUSY QCD Quartic RG (a) (b) (c) (d) (e) + Figure 7:

17 SUSY QCD Quartic RG = 1 4 4Ν 1 4Ν Ν 2 2 Ν 2 = + 2Ν 2 Ν 1 4Ν 2 Figure 8: The bird-track diagram for the sum over four generators quickly reduces to the sum over two generators and a product of identity matrices.

18 SUSY QCD Quartic RG (φ in (T a ) m n φ mi φ in (T a ) m n φ mi)(φ jq (T a ) p qφ pj φ jq (T a ) p qφ pj), (with flavor indices i j, the case i = j is left as an exercise) we have Λ (2) = ( ) 2F + N 6 N (T a ) m n (T a ) p q + ( ) 1 1 N δ m 2 n δq p, 4H = 8 ( ) N 2 N (T a ) m n (T a ) p q 4 ( ) 1 1 N δ m 2 n δq p, 3A = 3 ( ) N 4 N (T a ) m n (T a ) p q + 3 ( ) 1 1 N δ m 2 n δq p, Λ Y = 4 ( ) N 1 N (T a ) m n (T a ) p q, 3Λ S = 6 ( ) N 1 N (T a ) m n (T a ) p q. individual diagrams that renormalize the gauge invariant, SUSY breaking, operator (φ mi φ mi )(φ pj φ pj ) but the full β function for this operator vanishes and the D-term β function satisfies β λ = β g 2T a T a, β g 2 = 2gβ g. So SUSY is not anomalous at one-loop, and the β functions preserve the relations between couplings at all scales.

19 SUSY RG Figure 9: The couplings remain equal as we run below the SUSY threshold M, but split apart below the non-susy threshold m. If we had added dimension 4 SUSY breaking terms to the theory then the couplings would have run differently at all scales

20 one-loop squark mass Figure 1: The squark loop correction to the squark mass. Σ squark () = ig 2 (T a ) l n(t a ) m l = ig2 16π 2 C 2 ( )δ m n Λ 2 d 4 k (2π) 4 dk 2. i k 2

21 one-loop squark mass Figure 11: The quark gluino loop correction to the squark mass. Σ quark gluino () = ( i 2g) 2 (T a ) l n(t a ) m l ( 1) d 4 k = 2g 2 C 2 ( )δn m d 4 k = 4ig2 16π 2 C 2 ( )δ m n 2k 2 (2π) 4 k 4 Λ 2 dk 2. (2π) Tr ik σ 4 k 2 ik σ k 2

22 one-loop squark mass (a) (b) Figure 12: (a) The squark gluon loop and (b) the gluon loop. Σ quark gluino () = (ig) 2 (T a ) l n(t a ) m l = ξig2 16π 2 C 2 ( )δ m n Λ 2 d 4 k (2π) 4 dk 2, i k k µ ( i) (g µν+(ξ 1) 2 k 2 kµkν k 2 ) k ν Σ gluon () = 1 2 ig2 { (T a ) l n, (T b ) m l = (3+ξ)ig2 16π 2 C 2 ( )δ m n } δ ab g µν d 4 k Λ 2 dk 2. (2π) 4 i k ( i) (g µν+(ξ 1) 2 k 2 kµkν k 2 )

23 one-loop squark mass Adding all the terms together we have Σ() = ( ξ (3 + ξ)) ig2 16π 2 C 2 ( )δ m n Λ 2 dk 2 =. The quadratic divergence in the squark mass cancels! In fact for a massless squark all the mass corrections cancel. This means that in a SUSY theory with a Higgs the Higgs mass is protected from quadratic divergences from gauge interactions as well as from Yukawa interactions

24 Flat directions F < N and the scalar potential is: D a = g(φ in (T a ) m n φ mi φ in (T a ) m n φ mi) V = 1 2 Da D a maximal rank F. In a SUSY vacua: define d n m φ in φ mi d n m = φ in φ mi D a = T am n (d n m d n m) = Since T a is a complete basis for traceless matrices, we must therefore have that the difference of the two matrices is proportional to the identity matrix: d n m d n m = αi

25 Flat directions F < N d n m can be diagonalized by an SU(N) gauge transformation U d U In this diagonal basis there will be at least N F zero eigenvalues d = v 2 1 v v 2 F... where v 2 i. In this basis dn m must also be diagonal, and it must also have N F zero eigenvalues. This tells us that α =, and hence that d n m = d n m

26 Flat directions F < N d n m and d n m are invariant under SU(F ) SU(F ) transformations since φ mi φ mi Vj i, d n m V j i φ in φ mi V i φ jn φ mj = d n m. j, Thus, up to a flavor transformation, we can write v 1... φ = φ = v F D-term potential has flat directions, as we change the VEVs, we move between different vacua with different particle spectra, generically SU(N F ) gauge symmetry.

27 Flat directions F N d n m and d n m are N N positive semi-definite Hermitian matrices of maximal rank N in a SUSY vacuum : d n m d n m = ρi. d n m can be diagonalized by an SU(N) gauge transformation: v 1 2 v 2 2 d =... vn 2 In this basis, d n m must also be diagonal, with eigenvalues v i 2, so v i 2 = v i 2 + ρ.

28 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 Again we have a space of degenerate vacua. At a generic point in the moduli space the SU(N) gauge symmetry is completely broken.

29 The super Higgs mechanism a massless vector supermultiplet eats a chiral supermultiplet to form a massive vector supermultiplet Fayet

30 The super Higgs mechanism Consider the case when v 1 = v 1 = v and v i = v i =, for i > 1 SU(N) SU(N 1) and SU(F ) SU(F ) SU(F 1) SU(F 1). The number of broken gauge generators is N 2 1 ((N 1) 2 1) = 2(N 1) + 1, decompose the adjoint of SU(N) under SU(N 1), we have Ad N = Ad N 1 convenient basis of gauge generators is G A = X, X α 1, X α 2, T a where A = 1,..., N 2 1, α = 1,..., N 1, and a = 1,..., (N 1) 2 1. Xs are the broken generators (span the coset of SU(N)/SU(N 1)), T s are the unbroken SU(N 1) generators

31 The super Higgs mechanism The Xs are analogs of the Pauli matrices: X = 1 2(N 2 N) N , X α 1 = , X α 2 = i.... i.

32 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 =

33 The super Higgs mechanism We can then write the sum of the product of two generators as: G A G A = X X + X +α X α + X α X +α + T a T a Expanding the squark field around its VEV φ φ φ + φ, we have A GA φ = X φ + α X α φ, φ A GA = φ X + φ α X+α, since T a annihilates φ. label the components of the gluino field as G A λ A = X Λ + X +α Λ +α + X α Λ α + T a λ a,

34 The super Higgs mechanism write the quark field as ( ω ψ Q = i ω α Q mi ), Q = ( ω ω α ψ i Q im ), where i is a flavor index, α and m are color indices, Q is a matrix with N 1 rows and F 1 columns, and Q is a matrix with F 1 rows and N 1 columns. 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 + h.c.]. [ N 1 N So we have a Dirac fermion (Λ, (1/ 2)(ω ω )) with mass gv 2(N 1)/N, two sets of N 1 Dirac fermions (Λ +α, ω α ), (Λ α, ω α )) with mass gv, and massless Weyl fermions Q, Q, ψ, ψ, and (1/ 2)(ω + ω )).

35 The super Higgs mechanism decompose the squark field as φ = ( ) h σi, φ = H α φ mi ( h σ i H α φ im ), where φ is a matrix with N 1 rows and F 1 columns. Shifting the scalar field by its VEV so that φ φ + φ we have that the auxiliary D A field is given by D A g = 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.

36 The super Higgs mechanism 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 2(N 2 N) 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

37 The super Higgs mechanism mass terms reduce to V mass = g 2 v 2 [ N 1 N (h ) 2 + H +α H α]. real scalar h with mass gv 2(N 1)/N, a complex scalar H +α (and its conjugate H α ) with mass gv, massless complex scalars σ i, σ i, and Ω. πs become the longitudinal components of the massive gauge bosons, can be removed by going to Unitary gauge

38 The super Higgs mechanism We can write the gauge fields as: G B A B µ = X W µ + X +α W +α µ + X α W α µ + T a A a µ. Then the A 2 φ 2 terms which lead 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. + X α W α µ X +α W identical term arising from L A2 φ 2 gauge boson Wµ with mass gv 2(N 1)/N, gauge bosons W µ +α and Wµ α with mass gv, the massless gauge bosons A a µ of the unbroken SU(N 1) gauge group. all the particles fall into supermultiplets

39 The super Higgs mechanism v= SU(N) SU(F ) SU(F ) b.d.o.f. Q 1 2NF Q 1 2NF for v we have massive states (in Unitary gauge): SU(N 1) SU(F 1) SU(F 1) b.d.o.f. W W (N 1) W 1 1 4(N 1) massive vector supermultiplet (Wµ, h, Λ, (1/ 2)(ω ω )) 2(N 1) m W = gv N, massive vector supermultiplets (W +α µ, H +α, Λ +α, ω α ) and (W α µ, H α, Λ α, ω α m W ± = gv.

40 The super Higgs mechanism for v also have the massless states: 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 supermultiplet Q = (φ, Q ) gauge singlets ψ = (σ, ψ) and S = (1/ 2)(h + h), (1/ 2)(ω + ω ) In both cases (v = and v ) a total of 2(N 2 1) + 4F N b.d.o.f. (and, of course, the same number of fermionic degrees of freedom).