SUSY gauge theories

SUSY Yang–Mills under a gauge transformation gauge field, Aaµ, and gaugino field, λ

a, transform as:

δgaugeA a µ = −∂µΛa + gfabcAbµΛc

δgaugeλ a = gfabcλbΛc

where Λa is an infinitesimal gauge transformation parameter, g is the gauge coupling, and fabc are the antisymmetric structure constants of the gauge group which satisfy

[T ar , T b r ] = if

abcT cr

for the generators T a for any representation r. For the adjoint represen- tation :

(T bAd)ac = if abc

degrees of freedom Gauge invariance removes one degree of freedom from the gauge field, while the eqm projects out another, fermion eqm project out half the degrees of freedom, so

off-shell on-shell Aaµ 3 d.o.f. 2 d.o.f.

λaα, λ †a α̇ 4 d.o.f. 2 d.o.f.

for SUSY to be manifest off-shell add a real auxiliary boson field Da.

off-shell on-shell Da 1 d.o.f. 0 d.o.f.

SUSY Yang–Mills Lagrangian LSYM = − 14F

a µνF

µνa + iλ†aσµDµλ a + 12D

aDa

where the gauge field strength is given by

F aµν = ∂µA a ν − ∂νAaµ − gfabcAbµAcν

and the gauge covariant derivative of the gaugino is

Dµλ a = ∂µλ

a − gfabcAbµλc

auxiliary field has dimension [Da] = 2

infinitesimal SUSY transformations • should be linear in � and �†

• transform Aaµ and λa into each other

• keep Aaµ real

• maintain the correct dimensions of fields with [�] = − 12 • infinitesimal change in Da should vanish when the equations of

motion are satisfied

• infinitesimal change in the λa should involve the derivative of the Aaµ so that the infinitesimal changes in the two kinetic terms cancel

but gauge transformation of ∂µA a ν different from λ

a and F aµν

δAaµ = − 1√2 [ �†σµλ

a + λ†aσµ� ]

δλaα = − i2√2 (σ µσν�)α F

a µν +

1√ 2 �α D

a

δλ†aα̇ = i

2 √

2 (�†σνσµ)α̇ F

a µν +

1√ 2 �†α̇ D

a

δDa = −i√ 2

[ �†σµDµλ

a −Dµλ†aσµ� ]

SUSY gauge theories add a set of chiral supermultiplets

δgaugeXj = igΛ aT aXj

for Xj = φj , ψj ,Fj . gauge covariant derivatives are:

Dµφj = ∂µφj + igA a µ T

aφj Dµφ

∗j = ∂µφ ∗j − igAaµ φ∗jT a

Dµψj = ∂µψj + igA a µ T

aψj

new allowed renormalizable interactions:

(φ∗T aψ)λa, λ†a(ψ†T aφ) (φ∗T aφ)Da

all are required by SUSY with particular couplings. The first two are required to cancel pieces of the SUSY transformations of the gauge in- teractions of φ and ψ. The third is needed to cancel pieces of the SUSY transformations of the first two terms.

Lagrangian for a SUSY gauge theory

L = LSYM + LWZ − √

2g [ (φ∗T aψ)λa + λ†a(ψ†T aφ)

] + g(φ∗T aφ)Da.

LWZ is general Wess-Zumino with gauge-covariant derivatives. superpo- tential must be gauge invariant:

δgaugeW = igΛ a ∂W ∂φi

T aφi = 0.

infinitesimal SUSY transformations of φ and ψ are have derivatives promoted to gauge covariant derivatives, F has an additional term re- quired bythe gaugino interactions:

δφj = �ψj δψjα = −i(σµ�†)αDµφj + �αFj δFj = −i�†σµDµψj +

√ 2g(T aφ)j �

†λ†a

Integrating out auxiliary fields eqm for the auxiliary field Da:

Da = −gφ∗T aφ

scalar potential is given by “F-terms” and “D-terms”:

V (φ, φ∗) = F∗iFi + 12D aDa = W ∗i W

i + 12g 2(φ∗T aφ)2

as required by SUSY, is positive definite:

V (φ, φ∗) ≥ 0

for the vacuum to preserve SUSY V = 0 ⇒ Fi = 0 and Da = 0.

Feynman vertices

Figure 1:

Cubic and quartic Yang–Mills interactions; wavy lines denote gauge fields.

Figure 2:

Interactions required by gauge invariance. Solid lines denote fermions, dashed lines denote scalars, wavy lines denote gauge bosons, wavy/solid lines denote gauginos.

(a)

(c)

(b)

(d)

Figure 3:

Additional interactions required by gauge invariance and SUSY: (a) φ∗ψλ, (b) φ∗φD coupling, Note that these three vertices all have the same gauge index structure, being proportional to the gauge generator T a. Integrating out (c) the auxiliary field gives (d) the quartic scalar coupling proportional to T aT a.

(b)

(d)

(a)

(c)

Figure 4:

dimensionless non-gauge interaction vertices in a renormalizable su- persymmetric theory: (a) φiψjψk Yukawa interaction vertex −iyijk, (b) φiφjFk interaction vertex iyijk, (c) integrating out the auxiliary field yields, (d) the quartic scalar interaction −iyijny∗kln required for can- celling the Λ2 divergence in the Higgs mass

(e) (f)

(a)

(b)

(c) (d)

Figure 5:

dimensionful couplings: (a) ψψ mass insertion −iM ij , (b) φF mixing term insertion +iM ij , (c) integrating out F in cubic term, (d) integrating out F in mass term, (e) φ2φ∗interaction vertex −iM∗inyjkn, (f) φ∗φ mass insertion −iM∗ikMkj ensuring cancellation of the log Λ in the Higgs mass

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

Qi = (φi, Qi,Fi), i = 1, . . . , F ,

where φ is the squark and Q is the quark.

Qi = (φi, Qi,Fi) ,

in the antifundamental representation. Note the the bar ( ) is part of the name not a conjugation, the conjugate fields are

Q†i = (φ∗i , Q † i ,F∗i ), Q

† i = (φ

∗ i , Q

† i ,F∗i ).

SUSY QCD matter content is:

SU(N) SU(F ) SU(F ) U(1)B U(1)R

Q 1 1 F−NF Q 1 −1 F−NF

W = 0

R-charge [R,Qα] = −Qα.

chiral supermultiplet:

Rψ = Rφ − 1 ,

normalize the R-charge by

Rλa = λa ,

R-charge of the gluino is 1, and the R-charge of the gluon is 0.

Anomalies Since we can define an R-charge by taking arbitrary linear combina-

tions of the U(1)R and U(1)B charges we can choose Qi and Qi to have the same R-charge. For a U(1) not be to broken by instanton effects the SU(N)2U(1)R anomaly diagram vanishes

Figure 6:

fermion contributes itsR-charge times T (r). Sum over gluino, quarks:

1 · T (Ad) + (R− 1)T ( ) 2F = 0 ,

so R = F−NF

Renormalization group tree-level SUSY: Y =

√ 2g, λ = g2. 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

βg = µ dg dµ = −

g3

16π2

( 11 3 T (Ad)−

2 3T (F )−

1 3T (S)

) ≡ − g

3 b 16π2 ,

For SUSY QCD:

b = (3N − F )

Renormalization group the β function for the Yukawa coupling is :

(4π)2βjY = 1 2

[ Y †2 (F )Y

j + Y jY2(F ) ]

+ 2Y kY j†Y k

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

where

Y2(F ) ≡ Y j†Y j

Y †2 (F )Y j represents the scalar loop corrections to the fermion legs

2Y kY j†Y k contains the 1PI vertex corrections Y k TrY k†Y j represents fermion loop corrections to the scalar leg Cm2 (F ) is the quadratic Casimir of the fermion fields in the mth gauge group, and represents gauge loop corrections to the fermion legs

βY = √

2βg

so the relation between the Yukawa and gauge couplings is preserved under RG running

SUSY QCD Quartic RG SUSY also requires the D-term quartic coupling λ = g2. The auxiliary Da field is given by

Da = g(φ∗in(T a)mn φmi − φ in

(T a)mn φ ∗ mi)

and the D-term potential is

V = 12D aDa

The β function for a quartic scalar coupling at one-loop is

(4π)2βλ = Λ (2) − 4H + 3A+ ΛY − 3ΛS ,

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 ΛY to the Yukawa leg corrections ΛS corresponds to the gauge leg corrections

SUSY QCD Quartic RG (φ∗in(T a)mn φmi − φ

in (T a)mn φ

∗ mi)(φ

∗jq(T a)pqφpj − φ jq

(T a)pqφ ∗ pj) ,

(with flavor indices i 6= j, the case i = j is left as an exercise) we have

Λ(2) = ( 2F +N − 6N

) (T a)mn (T

a)pq + ( 1− 1N2

) δmn δ

p q ,

−4H = −8 ( N − 2N

) (T a)mn (T

a)pq − 4 ( 1− 1N2

) δmn δ

p q ,

3A = 3 ( N − 4N

) (T a)mn (T

a)pq + 3 ( 1− 1N2

) δmn δ

p q ,

ΛY = 4 ( N − 1N

) (T a)mn (T

a)pq , −3ΛS = −6

( N − 1N

) (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 operator vanishes and the D-term β function satisfies

βλ = 2gβg .

So SUSY is not anomalous at one-loop, and the β functio