Lecture 5 The Renormalization Group

Lecture 5 The Renormalization Group

Outline The canonical theory: SUSY QCD. Assignment of R-symmetry charges. Group theory factors: bird track diagrams. Review: the renormalization group. Explicit Feynman diagram computations I: running of coupling constants (with emphasis on group theory factors). Explicit Feynman diagram computations II: quadratic divergences in the squark mass. Reading: Terning 3.1-3.3, App. B1-2

SUSY QCD The theory: SUSY gauge theory with SU(N) gauge group. The matter is F flavors of quarks and squarks. Each flavor is a chiral multiplet (φ is the squark and Q is the quark) in the fundamental representation of the gauge group Q i = (φ i, Q i, F i ), i = 1,..., F, and a chiral multiplet in the antifundamental representation: Q i = (φ i, Q i, F i ), i = 1,..., F. ) is part of the name, not a conjugation. The con- Remark: the bar ( jugate fields are Q i = (φ i, Q i, F i ), Q i = (φ i, Q i, F i ).

SUSY QCD: Symmetries Flavor symmetry: there is a global SU(F ) acting on the Q i and another global SU(F ) acting on the Q i. (A mass term would violate this separation). The overall U(1) flavor symmetries (enhancing the SU(F ) s to U(F ) s): an axial U(1) A (acting identically on Q i,q i ) that is broken by a quantum anomaly and the baryon number symmetry U(1) B. Summary of symmetries: SU(N) SU(F ) SU(F ) U(1) B U(1) R Q 1 1 F N F Q 1 1 F N F Comment 1): the R-charge is discussed below. Comment 2): there is no superpotential W = 0.

R-charge The R-symmetry acts on the superspace coordinate θ (which by convention has charge one). In commutator form: In a chiral supermultiplet: In a vector supermultiplet: [R, Q α ] = Q α. R ψ = R φ 1. Rλ a = λ a. The R-charge of the gluino is 1 (the lowest component of W α ) and the R-charge of the gluon is 0 (next component and θ has charge one).

Group Theory: Bird Tracks Computation of group theory factors: bird track diagrams. Key identification: represent the group generator as a vertex: Figure 1: Bird-track notation for the group generator T a. Remark: this diagram is not a Feynman diagram. It is a representation of the group theory.

Basic Bird Tracks The quadratic Casimir C 2 (r) and the index T (r) of the representation r, are given diagrammaticly as (T a T a ) m n = (T a r ) m l (T a r ) l n = C 2 (r)δ m n, TrT a T b = (T a r ) m n (T b r ) n m = T (r)δ ab,

Contracting the external legs: Bird Tracks Figure 2: Identity in diagram form In the first diagram setting m equal 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). Result: d(r)c 2 (r) = d(ad)t (r).

Quadratic Casimirs of SU(N) Dimensions of the simplest representations: d( ) = N, d(ad) = N 2 1. Indices of the simplest representations: T ( ) = 1 2, T (Ad) = N. Quadratic Casimirs of the simplest representations: C 2 ( ) = N 2 1 2N, C 2 (Ad) = N.

Sum over Generators Reduction of the fundamental representation, using explicit generators: Diagrammatic form of reduction: (T a ) l p(t a ) m n = 1 2 (δl nδ m p 1 N δl pδ m n ). Figure 3: Application: the sums over multiple generators reduce to an essentially topological exercise.

Anomaly Cancellation For U(1) R to be conserved, the SU(N) 2 U(1) R anomaly must vanish. The condition (more details in later lecture) is that the triangle diagram cancels when summed over all particles with R-charge: Figure 4: Triangle diagram giving the quantum anomaly to the chiral U(1) current. The color factor: Tr(T a T b ) = T (r)δ ab where a, b are the colors of the currents to the right. So: the triangle diagram is proportional to the index of the representation under the gauge group.

The triangle diagram is also proportional to the fermion R-charge, through the vertex at the left. The R-charge is a priori ambiguous: we can add a conserved charge (that θ is neutral under). Taking linear combinations of the U(1) R and U(1) B charges we choose Q i and Q i to have the same R-charge. The R-charge R of a chiral multiplet is the charge of its scalar component (by convention); so the R-charge of the fermion component is R 1. The R-charge of the gaugino is 1. Requirement that the anomalies of the R-current cancel: Conclusion: 1 T (Ad) + (R 1)T ( ) 2F = 0, R = F N F

Running Couplings SUSY relates the couplings of the SUSY QCD Lagrangian. Denoting the gauge coupling g, the φ ψλ Yukawa coupling must be Y = 2g. The quartic D term coupling must be λ = g 2. These are relations in the classical Lagrangian. For SUSY to be a consistent quantum symmetry these relations must be preserved under the running of coupling due to quantum effects. It is an instructive exercise to verify the predictions explicitly at one-loop level.

Renormalization group The running of the coupling constants in QFT are determined by the β-function, which in turns depends on the matter content in the theory. The running of the gauge coupling, at one-loop order: 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, the fermions are one gaugino and 2F quarks: T (F ) = T (Ad) + 2F T ( ) For SUSY QCD, the scalars are 2F squarks: T (S) = 2F T ( ) Collecting contributions from matter: b = 3N F

RG for Yukawa The β-function for a general Yukawa coupling Y j ik of a boson (type j) to two fermions (of type i, k): [ ] (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 }. Explanations: 1PI vertex corrections: 2Y k Y j Y k. Scalar/fermion loop corrections to the fermion leg: Y 2 (F )Y j +h.c.. Notation: Y 2 (F ) Y j Y j. Fermion loop corrections to the scalar leg: Y k Tr Y k Y j. Gauge loop corrections to the fermion legs: C m 2 (F ) (the quadratic Casimir of the fermion fields in the mth gauge group).

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. Bird-track diagrams for external leg corrections (of Y 2 type): Figure 5: Feynman diagrams and associated bird-track diagrams.

The Running of Yukawa Correction of quark leg, due to gaugino/scalar loop: Y 2 (Q) = 2g 2 C 2 ( ). Correction of gaugino leg, due to quark/scalar loop: Y 2 (λ) = 2g 2 2F T ( ). Correction of squark leg, due to quark (antisquark)/gluino fermion loop: Y k Tr Y k Y j = 2g 2 C 2 ( )Y j. Corrections of quark/gluino legs due to gauge loop: 3g 2 {C2 m (F ), Y j } = 3g 2 (C 2 ( ) + C 2 (Ad))Y j. To do: add contributions.

Running SUSY Collecting contributions to the running of the Yukawa coupling: Remarks: (4π) 2 β j Y = 2g 3 (C 2 ( ) + F + 2C 2 ( ) 3C 2 ( ) 3N) = 2g 3 (3N F ) = 2(4π) 2 β g The 1PI vertex corrections (proportional to Y k Y j Y k ) were not considered because they cancel by themselves. The terms in β j Y proportional to C 2( ) cancel each other. The remaining terms combine to the β-function for the gauge coupling. The overall numerical coefficient is such that the relation Y = 2g is maintained by the running. Conclusion: the (one-loop) runnings of the gauge coupling and the Yukawa coupling in SUSY QCD are consistent with SUSY.

SUSY QCD Quartic RG SUSY also requires the D-term quartic coupling λ = g 2. The auxiliary D a field is given by D a = g(φ in (T a ) m n φ mi φ in (T a ) m n φ mi) The D-term potential is V = 1 2 Da D a Goal: check the relation λ = g 2 at one-loop order in perturbation theory.

The quartic coupling at one loop. The β function for a quartic scalar coupling at one-loop: (4π) 2 β λ = Λ (2) 4H + 3A + Λ Y 3Λ S. Contributions: The 1PI contribution from the quartic interactions: Λ (2). The fermion box graphs: H. The two gauge boson exchange graphs: A. Corrections to external legs due to Yukawa couplings: Λ Y. Corrections to external legs due to gauge couplings: Λ S.

Diagrams for the quartic coupling Figure 6: Reduction from Feynman to Bird-track

SUSY QCD Quartic RG Figure 7: The bird-track diagram for the sum over four generators reduces to the sum over two generators and a product of identity matrices.

SUSY QCD Quartic RG The quartic coupling (with flavor indices i j): (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), Explicit contributions: Adding: 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. (4π) 2 β λ = 2(3N F )(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.

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.

SUSY RG Figure 8: 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

One-loop squark mass The mass of a light scalar generally receives quantum corrections that are quadratically divergent. (See appetizer in Lecture 1 for the case of Higgs). Suppose that the light scalar is the scalar component of a chiral supermultiplet (the squark). Compute the quantum corrections to the scalar mass due to a SUSY coupling between the chiral supermultiplet and a non-abelian gauge field. Result: quadratic divergences cancel. The details are instructive so we carry them out.

Scalar Self-Coupling The D-term potential: a quartic coupling between the scalars, with coupling constant g 2 T a T a. Bird-track diagram: Figure 9: The squark loop correction to the squark mass. Loop correction to the mass: Σ squark (0) = 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 0 dk 2. i k 2

The Quark/Gluino Loop Figure 10: The quark gluino loop correction to the squark mass. Loop correction due to the Yukawa coupling between the scalar and the quark/gluino: Σ quark gluino (0) = ( 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 0 dk 2. (2π) Tr ik σ 4 k 2 ik σ k 2

The Gluon Loops Figure 11: (a) The squark gluon loop and (b) the gluon loop. Σ quark gluino (0) = (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 0 dk 2, i k k µ ( i) (g µν+(ξ 1) 2 k 2 kµkν k 2 ) k ν Σ gluon (0) = 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 0 dk 2. (2π) 4 i k ( i) (g µν+(ξ 1) 2 k 2 kµkν k 2 )

One-loop Squark Mass Adding all the quadratically divergent contributions to the squark mass: Σ(0) = ( 1 + 4 + ξ (3 + ξ)) ig2 16π 2 C 2 ( )δ m n Λ 2 0 dk 2 = 0. Result: the quadratic divergence in the squark mass cancels! More detailed result: for a massless squark all the mass corrections cancel. Application in model building: in a SUSY theory, the Higgs mass is protected from quadratic divergences from gauge interactions as well as from Yukawa interactions.