Lecture 9Soft SUSY Terms and Spurions

Outline• Soft SUSY breaking: a general discussion.

• Spurions.

• SUSY breaking scenario: gravity mediation.

• SUSY breaking scenario: gauge mediation.

• Computation of soft SUSY breaking terms: an example.

Reading: Terning 2.8, 5.2, 6.1, 6.2.

Chiral Symmetry Protects MassesQuantum corrections to fermion masses are logarithmically divergent (atmost). For cutoff Λ,

mferm = m0 + cfα

16π2m0 ln(

Λm0

).

The reason: chiral symmetry for m0 = 0 is preserved by quantum effectsso corrections vanish for m0 = 0 ⇒ linearly divergent corrections areabsent even though they are allowed by dimensional analysis.

In SUSY QFT: the scalar mass is given by the same formula. In otherwords, quadratic corrections (to m2

bos) are absent even though they areallowed by dimensional analysis.

The challenge: break SUSY such that quadratic corrections to scalarmasses are absent also after SUSY breaking.

Soft Breaking of SUSYReminder: Higgs receives quadratically divergent corrections from theYukawa coupling to the top.

The quadratic divergence cancels if the top has a scalar partner (a stop)with quartic coupling to the Higgs taking the SUSY value λ = |yt|2:

δm2h ∝ (λ− |yt|2)Λ2

Analysis: we want to break SUSY such that SUSY relations betweendimensionless coupling are protected.

Conclusion: SUSY breaking couplings with explicit mass dependencepreserve SUSY cancellations of quadratic divergences. Those are relevantterms, i.e. terms with dimension< 4.

Terminology: relevant terms in the Lagrangian are called soft SUSYbreaking terms.

Effective Theory of Soft BreakingThe effective theory of softly broken SUSY allows all relevant termsconsistent with symmetries.

General soft terms in a SUSY gauge theory coupled to chiral matterfields:

Lsoft = − 12 (Mλλ

aλa + h.c.)− (m2)ijφ∗jφi

−( 12bijφiφj + 1

6aijkφiφjφk + h.c.)

− 12cjki φ

i∗φjφk + eiφi + h.c.

Comments:

• eiφi is only allowed if φi is a gauge singlet.

• The cjki term may introduce quadratic divergences if there is agauge singlet multiplet in the model.

• The ei and cjki couplings are not allowed in the MSSM.

(a) (b)

(d) (e)

(c)

(f)

Figure 1:

Soft SUSY breaking interactions:(a) gaugino mass Mλ.(b) non-holomorphic mass m2.(c) holomorphic mass bij .(d) holomorphic trilinear coupling aijk.(e) non-holomorphic trilinear coupling cjki .(f) tadpole ei.

SpurionsSpurions are convenient book-keeping devices for broken symmetries.

Spurions are fictitious background fields that transform under a symme-try group ⇒ spurions break the symmetry “spontaneously”.

Example: a chiral superfield Φ with a real superfield Z as wavefunctionrenormalization factor that breaks SUSY (but not Lorentz):

Z = 1 + b θ2 + b∗θ2 + c θ2θ2

Thus (recalling Φ(y, θ) = φ(y) +√

2θψ(y) + θ2F(y)):∫d4θ ZΦ†Φ = Lfree + bF∗φ+ b∗φ∗F + cφ∗φ

Integrate out the auxiliary field F :∫d4θ ZΦ†Φ = ∂µφ∗∂µφ+ iψ†σµ∂µψ + (c− |b|2)φ∗φ

Conclusion: the spurion field Z encodes the non-holomorphic soft SUSYbreaking mass: m2 = |b|2 − c.

Superpotential SpurionsHolomorphic spurions are θ2 components in:a) Yukawa couplings.b) Masses.c) The coefficient of WαW

α.These generate a, b, and Mλ soft SUSY breaking terms.

The c term requires a term like∫d4θ Cjki Φi†ΦjΦk + h.c.

where Cjki has a nonzero θ2θ2 component.

Thus the spurion interpretation applies to all soft SUSY breaking terms:

Lsoft = − 12 (Mλλ

aλa + h.c.)− (m2)ijφ∗jφi

−( 12bijφiφj + 1

6aijkφiφjφk + h.c.)

− 12cjki φ

i∗φjφk + eiφi + h.c.

Gravity-Mediated ScenarioScenario:

• A hidden sector field X acquires a nonzero 〈FX〉 and SUSY break-ing is transmitted by universal gravitational interaction.

• The MSSM soft terms are estimated as

msoft ∼ 〈FX〉MPl.

• Scale: msoft around the weak scale requires√〈FX〉 ∼ 1010–1011

GeV.

• Example: gaugino condensation in the hidden sector 〈0|λaλb|0〉 =δabΛ3 6= 0 implies

msoft ∼ Λ3

M2Pl,

which requires Λ ∼ 1013 GeV. (Equivalently, 〈FX〉 = Λ3/MPl.)

Effective LagrangianMSSM is decoupled from the hidden sector for MPl → ∞ so no softterms in this limit. For finite MPl there may be couplings:

Leff = −∫d4θ X∗

MPlb′ijψiψj + XX∗

M2Pl

(mijψiψ

j∗ + bijψiψj

)+ h.c.

−∫d2θ X

2MPl

(M3G

αGα + M2WαWα + M1B

αBα

)+ h.c.

−∫d2θ X

MPlaijkψiψjψk + h.c.

Notation: Gα, Wα, Bα, ψi are the chiral superfields of the MSSM; thehatted symbols are dimensionless.

SUSY breaking in the hidden sector gives 〈X〉 = 〈FX〉 so

Leff = − 〈FX〉2MPl

(M3GG+ M2WW + M1BB

)+ h.c.

− 〈FX〉〈F∗X〉

M2Pl

(mijψiψ

j∗ + bijψiψj

)+ h.c.

− 〈FX〉MPlaijkψiψjψk − 〈F

∗X〉

MPl

∫d2θ b′ijψiψj + h.c.

Assumptions in Gravity MediationThe messenger field X does not have MSSM quantum numbers so uni-versality of gravity motivates assumptions of its couplings:

• Universal gaugino masses: Mi = M .

• Universal masses for scalars mij = mδij .

• Diagonal Yukawa couplings: aijk = aY ijk.

• The coefficients b′ in the µ-term µij = b′δiHuδjHd〈F∗X〉/MPl and b in

the b = Bµ term bij = bδiHuδjHd

are of the same order of magnitude.

Simplifications in Minimal SUGRAAssumptions give soft parameters a universal form (when renormalizedat MPl):

• Gaugino masses are equal:

Mi = m1/2 = M 〈FX〉MPl

.

• The scalar masses are universal:

m2f = m2

Hu= m2

Hd= m2

0 = m |〈FX〉|2

M2Pl

.

• The A terms are given by:

Af = AYf = a 〈FX〉MPlYf .

• The SUSY conserving µ2 and soft breaking b = Bµ = bb′〈FX〉MPl

µ arenaturally of the same order of magnitude.

DiscussionDesirable consequence: the assumptions avoid problems with FCNCs.

Motivation: gravity is flavor-blind, as is hidden sector.

Problem: the motivation does not hold up. Specifically, there are non-universal terms like a Kahler function of the form

Kbad = f(X†, X)ij ψ†jψi ,

which leads directly to off-diagonal terms in the matrix mij .

Terminology: the minimal supergravity scenario assumes just µ and thefour universal SUSY breaking parameters at unification scale (ratherthan the Planck scale).

Gauge-Mediated ScenarioScenario:

• SUSY is broken in the hidden sector by a gauge group becomingstrong.

• There are “messenger” chiral supermultiplets that are charged un-der the hidden gauge group. Their fermions and bosons masses aretherefore split.

• The messenger fields also couple to the MSSM so the superpartnersacquire masses through loops:

msoft ∼ αi4π〈F〉Mmess

• If Mmess ∼√〈F〉, then the SUSY breaking scale can be as low as√

〈F〉 ∼ 104–105 GeV.

Messengers of SUSY breakingNotation: goldstino multiplet X with an expectation value:

〈X〉 = M + θ2F

Assume Nf messengers φi, φi coupling to hidden sector as

W = Xφiφi

Constraint: for gauge unification, φi and φi should form complete GUTmultiplets. Messengers shift the coupling at the GUT scale

δα−1GUT = −Nm2π ln

(µGUTM

)where

Nm =∑Nfi=1 2T (ri)

For the unification to remain perturbative we need

Nm < 150ln(µGUT/M)

Soft MassesVEV of goldstino multiplet:

〈X〉 = M + θ2F

⇒ messenger fermion mass = M⇒ messenger scalars mass2 = M2 ±F

λ λ

One-loop gaugino mass:

Mλi ∼ αi4πNm

FM

Scalar Soft MassesMessenger couple to MSSM gauge fields (by assumption) but not toMSSM matter (there are no superpotential couplings).

Soft mass for matter is therefore a two-loop effect: messenger loop cor-rections in the one-loop sfermion mass diagrams spoils the cancellation.

Two-loop mass2 for squarks and sleptons

M2s ∼

∑i

(αi4πFM

)2 ∼M2λ

Direct computation at two loop is laborious but there is a short cut.

RG Calculation of Soft MassesEffective Lagrangian below the messenger mass:

LG = − i16π

∫d2θ τ(X,µ)WαWα + h.c.

where

τ = θ2π + 4πi

g2 .

Recall

WαWα = λaαλaα + 2iλa†σµDµλ θ

2 + . . . .

so

Mλ = − 12τ

∂τ∂X

∣∣∣X=M

F = − 12∂ ln τ∂ lnX

∣∣∣X=M

FM

The standard running of the gauge coupling can be recast as:

µ ddµτ = ib

2π .

The β-function coefficients: b′ at high scale (includes the messengers) butb in the MSSM at lower scale (where messengers are inert). Matching atthe messenger scale µ0:

τ(X,µ) = τ(µ0) + i b′

2π ln(Xµ0

)+ i b2π ln

(µX

).

The shift as messengers freeze out:

b′ = b−Nm .

The gaugino mass is simply given by:

Mλ = − 12∂ ln τ∂ lnX

∣∣∣X=M

FM = α(µ)

4π NmFM .

Gaugino MassesThe gaugino masses in the gauge mediated model:

Mλ = α(µ)4π Nm

FM .

Remark: the ratio of the gaugino mass to the gauge coupling is universalMλ1α1

= Mλ2α2

= Mλ3α3

= NmFM .

Historical note: this was once thought to be a signature of gravity me-diation models, but here it appears rather simply in gauge mediation.

Sfermion MassesConsider wavefunction renormalization for the matter fields of the MSSM:

L =∫d4θ Z(X,X†)Q′†Q′ .

Notation: wave function renormalization factor Z is real and the primeindicates that the fields are not yet canonically normalized.

Taylor expand in the superspace coordinates

L =∫d4θ

(Z + ∂Z

∂XFθ2 + ∂Z

∂X†F†θ2

+ ∂2Z∂X∂X†

Fθ2F†θ2) ∣∣∣

X=MQ′†Q′

Introduce canonical normalizations of the matter fields:

Q = Z1/2(1 + ∂ lnZ

∂X Fθ2) ∣∣∣X=M

Q′

so

L =∫d4θ

[1−

(∂ lnZ∂X

∂ lnZ∂X†

− 1Z

∂2Z∂X∂X†

)Fθ2F†θ2

] ∣∣∣X=M

Q†Q

Sfermion MassesRead off the soft sfermion masses

m2Q = − ∂2 lnZ

∂ lnX∂ lnX†

∣∣∣X=M

FF†MM†

.

Wave function renormalization also introduces an A term in the effectivepotential. The rescaled superpotential:

W (Q′) = W(QZ−1/2

(1− ∂ lnZ

∂X Fθ2) ∣∣∣X=M

),

gives the A term (∼ aφ3) in the Lagrangian

Z−1/2 ∂ lnZ∂X

∣∣∣X=M

FQ ∂W∂(Z−1/2Q)

.

Remark: it is proportional to the Yukawa coupling in the superpotential.

RG CalculationSo far: expressions for soft breaking terms from spurion.Next: evaluate these expressions using the RG.

First, compute the wave function renormalization factor Z in the con-ventional manner. At one-loop

µd lnZdµ = C2(r)

π α(µ)

Replace the RG-scale M by√XX†, so that Z(X,X†) is invariant under

X → eiβX.

RG CalculationIntegrate, using the β-function equation for the running of the cou-

pling:

Z(µ) = Z0

(α(µ0)α(X)

)2C2(r)/b′ (α(X)α(µ)

)2C2(r)/b

,

where

α−1(X) = α−1(µ0) + b′

4π ln(XX†

µ20

),

α−1(µ) = α−1(X) + b4π ln

(µ2

XX†

).

Insert in expression for the soft mass and obtain

m2Q = 2C2(r)α(µ)2

16π2 Nm(ξ2 + Nm

b (1− ξ2)) ( F

M

)2,

whereξ = 1

1+ b2πα(µ) ln(M/µ)

.

Remark: two-loop scalar masses are determined by a one-loop RG equa-tion!

Spurions: the Big PictureReminder: in SUSY gauge theory, there is an extended gauge symmetry,where the gauge parameter Λ is a chiral superfield.

Characterization of SUSY breaking: some chiral field has non-vanishingauxiliary component F . (We do not consider D-breaking in this discus-sion). This chiral field could be composite, the point is just its transfor-mation property.

Computation of soft SUSY breaking terms: the renormalization scaleis a chiral superfield. All counterterms (couplings and wave functionrenormalization factors) depend on that renormalization scale.

The big picture: in SUSY QFT the SUSY transformation properties areencoded in superfields so everything is a superfield.

Examples: masses, coupling constants, renormalization scale, gauge pa-rameter...

These quantities typically have definite values, but treating them as fieldsgive extended symmetries.