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Lecture 5The 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 couplingconstants (with emphasis on group theory factors).

• Explicit Feynman diagram computations II: quadratic divergencesin the squark mass.

Reading: Terning 3.1-3.3, App. B1-2

SUSY QCDThe theory: SUSY gauge theory with SU(N) gauge group. The matteris F “flavors” of “quarks” and squarks.

Each flavor is a chiral multiplet (φ is the squark and Q is the quark) inthe fundamental representation of the gauge group

Qi = (φi, Qi,Fi) , i = 1, . . . , F ,

and a chiral multiplet in the antifundamental representation:

Qi = (φi, Qi,Fi) , i = 1, . . . , F .

Remark: the bar ( ) is part of the name, not a conjugation. The con-jugate fields are

Q†i = (φ∗i , Q†i ,F∗i ) , Q†i = (φ

∗i , Q

†i ,F∗i ).

SUSY QCD: SymmetriesFlavor symmetry: there is a global SU(F ) acting on the Qi and an-other global SU(F ) acting on the Qi. (A mass term would violate thisseparation).

The overall U(1) flavor symmetries (enhancing the SU(F )’s to U(F )’s):an axial U(1)A (acting identically on Qi,Qi) that is broken by a quantumanomaly 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−NF

Q 1 −1 F−NF

Comment 1): the R-charge is discussed below.

Comment 2): there is no superpotential W = 0.

R-chargeThe R-symmetry acts on the superspace coordinate θ (which by conven-tion has charge one).

In commutator form:

[R,Qα] = −Qα.

In a chiral supermultiplet:

Rψ = Rφ − 1 .

In a vector supermultiplet:

Rλa = λa .

The R-charge of the gluino is 1 (the lowest component of Wα) and theR-charge of the gluon is 0 (next component and θ has charge one).

Group Theory: Bird TracksComputation 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 representationof the group theory.

Basic Bird TracksThe quadratic Casimir C2(r) and the index T (r) of the representation r,

(T aT a)mn = (T ar )ml (T ar )ln = C2(r)δmn ,TrT aT b = (T ar )mn (T br )nm = T (r)δab ,

are given diagrammaticly as

Bird TracksContracting the external legs:

Figure 2: Identity in diagram form

In the first diagram setting m equal to n and summing over n yields afactor of d(r). In the second diagram setting a equal to b and summingyields a factor d(Ad).

Result:

d(r)C2(r) = d(Ad)T (r) .

Quadratic Casimirs of SU(N)Dimensions of the simplest representations:

d( ) = N ,d(Ad) = N2 − 1 .

Indices of the simplest representations:

T ( ) = 12 ,

T (Ad) = N .

Quadratic Casimirs of the simplest representations:

C2( ) = N2−12N ,

C2(Ad) = N .

Sum over GeneratorsReduction of the fundamental representation , using explicit generators:

(T a)lp(Ta)mn = 1

2 (δlnδmp − 1

N δlpδmn ) .

Diagrammatic form of reduction:

Figure 3:

Application: the sums over multiple generators reduce to an essentiallytopological exercise.

Anomaly CancellationFor U(1)R to be conserved, the SU(N)2U(1)R anomaly must vanish.The condition (more details in later lecture) is that the triangle diagramcancels when summed over all particles with R-charge:

Figure 4: Triangle diagram giving the quantum anomaly to the chiralU(1) current.

The color factor: Tr(T aT b) = T (r)δab where a, b are the colors of thecurrents to the right. So: the triangle diagram is proportional to theindex of the representation under the gauge group.

The triangle diagram is also proportional to the fermionR-charge, throughthe 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 chooseQi and Qi 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:

1 · T (Ad) + (R− 1)T ( ) 2F = 0 ,

Conclusion:

R = F−NF

Running CouplingsSUSY 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 = g2.

These are relations in the classical Lagrangian. For SUSY to be a con-sistent quantum symmetry these relations must be preserved under therunning of coupling due to quantum effects.

It is an instructive exercise to verify the predictions explicitly at one-looplevel.

Renormalization groupThe running of the coupling constants in QFT are determined by theβ-function, which in turns depends on the matter content in the theory.

The running of the gauge coupling, at one-loop order:

βg = µ dgdµ = − g3

16π2

(113 T (Ad)− 2

3T (F )− 13T (S)

)≡ − g3 b

16π2 .

For SUSY QCD, the fermions are one gaugino and 2F quarks:

T (F ) = T (Ad) + 2FT (�)

For SUSY QCD, the scalars are 2F squarks:

T (S) = 2FT (�)

Collecting contributions from matter:

b = 3N − F

RG for YukawaThe β-function for a general Yukawa coupling Y jik of a boson (type j) totwo fermions (of type i, k):

(4π)2βjY = 12

[Y †2 (F )Y j + Y jY2(F )

]+ 2Y kY j†Y k

+Y k TrY k†Y j − 3g2m{Cm2 (F ), Y j} .

Explanations:

• 1PI vertex corrections: 2Y kY j†Y k.

• Scalar/fermion loop corrections to the fermion leg: Y †2 (F )Y j+h.c..

• Notation: Y2(F ) ≡ Y j†Y j .

• Fermion loop corrections to the scalar leg: Y k TrY k†Y j .

• Gauge loop corrections to the fermion legs: Cm2 (F ) (the quadraticCasimir of the fermion fields in the mth gauge group).

SUSY QCD RGFor 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 jnim,a =√

2g(T a)nmδji .

Bird-track diagrams for external leg corrections (of Y 2 type):

Figure 5: Feynman diagrams and associated bird-track diagrams.

The Running of YukawaCorrection of quark leg, due to gaugino/scalar loop:

Y2(Q) = 2g2C2( ) .

Correction of gaugino leg, due to quark/scalar loop:

Y2(λ) = 2g2 2F T ( ) .

Correction of squark leg, due to quark (antisquark)/gluino fermion loop:

Y k TrY k†Y j = 2g2C2( )Y j .

Corrections of quark/gluino legs due to gauge loop:

−3g2{Cm2 (F ), Y j} = −3g2(C2( ) + C2(Ad))Y j .

To do: add contributions.

Running SUSYCollecting contributions to the running of the Yukawa coupling:

(4π)2βjY =√

2g3(C2( ) + F + 2C2( )− 3C2( )− 3N)= −

√2g3(3N − F )

=√

2(4π)2βg

Remarks:

• The 1PI vertex corrections (proportional to Y kY †jY k) were notconsidered because they cancel by themselves.

• The terms in βjY proportional to C2( ) cancel each other.

• The remaining terms combine to the β-function for the gauge cou-pling.

• The overall numerical coefficient is such that the relation Y =√

2gis maintained by the running.

Conclusion: the (one-loop) runnings of the gauge coupling and the Yukawacoupling in SUSY QCD are consistent with SUSY.

SUSY QCD Quartic RGSUSY also requires the D-term quartic coupling λ = g2.

The auxiliary Da field is given by

Da = g(φ∗in(T a)mn φmi − φin

(T a)mn φ∗mi)

The D-term potential is

V = 12D

aDa

Goal: check the relation λ = g2 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)2βλ = Λ(2) − 4H + 3A+ ΛY − 3ΛS .

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: ΛY .

• Corrections to external legs due to gauge couplings: ΛS .

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 re-duces to the sum over two generators and a product of identity matrices.

SUSY QCD Quartic RGThe quartic coupling (with flavor indices i 6= j):

(φ∗in(T a)mn φmi − φin

(T a)mn φ∗mi)(φ

∗jq(T a)pqφpj − φjq

(T a)pqφ∗pj) ,

Explicit contributions:

Λ(2) =(2F +N − 6

N

)(T a)mn (T a)pq +

(1− 1

N2

)δmn δ

pq ,

−4H = −8(N − 2

N

)(T a)mn (T a)pq − 4

(1− 1

N2

)δmn δ

pq ,

3A = 3(N − 4

N

)(T a)mn (T a)pq + 3

(1− 1

N2

)δmn δ

pq ,

ΛY = 4(N − 1

N

)(T a)mn (T a)pq ,

−3ΛS = −6(N − 1

N

)(T a)mn (T a)pq .

Adding:

(4π)2βλ = −2(3N − F )(T a)mn (T a)pq

Individual diagrams that renormalize the gauge invariant, SUSY break-ing, operator (φ∗miφmi)(φ∗pjφpj) but the full β function for this operatorvanishes.

The D-term β function satisfies

βλ = βg2TaT a ,

βg2 = 2gβg .

So SUSY is not anomalous at one-loop, and the β functions preserve therelations between couplings at all scales.

SUSY RG

Figure 8: The couplings remain equal as we run below the SUSY thresh-old M , but split apart below the non-SUSY threshold m.

If we had added dimension 4 SUSY breaking terms to the theory thenthe couplings would have run differently at all scales

One-loop squark massThe mass of a light scalar generally receives quantum corrections thatare quadratically divergent. (See “appetizer” in Lecture 1 for the caseof Higgs).

Suppose that the light scalar is the scalar component of a chiral super-multiplet (the squark).

Compute the quantum corrections to the scalar mass due to a SUSYcoupling 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-CouplingThe D-term potential: a quartic coupling between the scalars, with cou-pling constant g2T aT a.

Bird-track diagram:

Figure 9: The squark loop correction to the squark mass.

Loop correction to the mass:

Σsquark(0) = −ig2(T a)ln(T a)ml∫

d4k(2π)4

ik2

= −ig216π2C2( )δmn

∫ Λ2

0dk2 .

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 thequark/gluino:

Σquark−gluino(0) = (−i√

2g)2(T a)ln(T a)ml (−1)∫

d4k(2π)4 Tr ik·σk2

ik·σk2

= −2g2C2( )δmn∫

d4k(2π)4

2k2

k4

= 4ig2

16π2C2( )δmn∫ Λ2

0dk2 .

The Gluon Loops

Figure 11: (a) The squark–gluon loop and (b) the gluon loop.

Σquark−gluino(0) = (ig)2(T a)ln(T a)ml∫

d4k(2π)4

ik2 k

µ(−i) (gµν+(ξ−1)kµkν

k2)

k2 kν

= ξig2

16π2C2( )δmn∫ Λ2

0dk2 ,

Σgluon(0) = 12 ig

2{

(T a)ln, (Tb)ml}δabgµν

∫d4k

(2π)4ik2 (−i) (gµν+(ξ−1)

kµkν

k2)

k2

= −(3+ξ)ig2

16π2 C2( )δmn∫ Λ2

0dk2 .

One-loop Squark MassAdding all the quadratically divergent contributions to the squark mass:

Σ(0) = (−1 + 4 + ξ − (3 + ξ)) ig2

16π2C2( )δmn∫ Λ2

0dk2 = 0 .

Result: the quadratic divergence in the squark mass cancels!

More detailed result: for a massless squark all the mass corrections can-cel.

Application in model building: in a SUSY theory, the Higgs mass isprotected from quadratic divergences from gauge interactions as well asfrom Yukawa interactions.