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Supersymmetry and the Origin of the Weak Scale

James WellsMichigan Center for Theoretical Physics University of

Michigan

APCTP Summer Institute August 2006

1

Why Study Supersymmetry

Several good reasons to study supersymmetry,

• stabilizes the hierarchy (MW/MP ≪ 1)

• gauge coupling unification

• natural dark matter candidate

• string theory seems to like it

• electroweak symmetry breaking occurs radiatively

• excellent laboratory for exact field theory results

Supersymmetry also has some challenges, or is at least neutralto several fundamental questions

• suppression of Flavor Changing Neutral Currents

• suppression of CP violation

• proton stability

• cosmological constant

2

Outline

1. Rules for building SUSY lagrangians

2. Minimal Supersymmetric Standard Model

3. Generating weak-scale µ term

4. Unification studies

3

Part 1.

Rules for building SUSY lagrangians

4

What is supersymmetry?

Supersymmetry is a symmetry that transforms bosons andfermions into each other.

Why introduce such a symmetry?Answer: Possible =⇒ Good Implications

We will minimize formalism here, and focus more on practi-cal aspects, such as how to build invariant lagrangians andinvestigate consequences.

Formalism interests? See Wess and Bagger, or Bailin andLove.

5

Chiral Superfields

One representation of the supersymmetry is “chiral super-field”:

Φ = φ +√ψθ + Fθθ

θ is a Grassman variable and φ, ψ and F are scalar, fermionand auxiliary fields, respectively.

Under a supersymmetry transformation, F transforms likea total derivative.

δξF = ∂µ

(i√2ψσµξ

)

SUSY-invariant lagrangians built from F terms since∫d4xδξF = 0

6

Multiplication of chiral superfields

The product of two chiral superfields, is a chiral superfield.

Gauge invariant and supersymmetric lagrangians can be con-structed from this rule.

For example, Φ+ has charge Q = +1 and Φ− has chargeQ = −1. The gauge invariant and SUSY invariant lagrangianis ∫

d2θΦ+Φ− = [Φ+Φ−]F = FΦ+Φ−

7

The Superpotential

The superpotentialW is the collection of gauge invariant poly-nomials of chiral superfields that describe the theory. The Fterm of W is gauge invariant and SUSY invariant.

∫d2θW = [W ]F

For example,∫d2θΦ2Φ

2−1 = FΦ2Φ

2−1 + 2Φ2Φ−1FΦ−1

Only reason why W is an invented concept, is because it isuseful to do so.

8

Vector Superfields

ΦΦ† is not a chiral superfield, but rather a vector superfieldor real superfield.

In Wess-Zumino gauge, vector superfield can be written as

V (x, θ, θ) = θσµθVµ(x) + iθθθλ(x) − iθθθλ(x) +1

2θθθθD(x)

The D term of a vector superfield transforms as a totalderivative and thus is susy invariant.

SUSY-invariant lagrangian from∫d2θd2θ V = [V ]D

9

Gauge interactions

For the gauge fields, a field strength tensor can be formedfrom Wα = D2DαV , which yields a “spinor chiral multiplet”

Wα = 4iλα − [4δβαD + 2i(σµσν)βαVµν]θβ + 4θ2σµαα∂µλα

= 4iλα + θβφαβ + θθ Fα

Because it is a chiral superfield, the F term ofWαWα is SUSY

invariant and contains F µνFµν

1

32

∫d2θW αWα = −1

4V µνVµν + iλσµ∂µλ +

1

2D2

10

Example Supersymmetric Lagrangian

Start with two oppositely charged superfields S and T andone vector superfield V associated with U (1) gauge symmetry(see Bailin and Love). The gauge invariant superpotential isW = mST .

The supersymmetric Lagrangian is compactly written as

L =1

32

∫d2θW αWα +

∫d4θ (S†e2qVS + T †e−2qV T )

+

∫d2θ (mST + h.c.)

This translates into the following lagrangian in componentnotation

L = (DµφS)†(DµφS) + (DµφT )†(DµφT ) + iψSσ

µD†µψS

iψTσµDµψT + F †

SFS + F †TFT + i

√2q(φ†SψS − φ†TψT )λ

i√

2q(φT ψT − φSψS)λ + q(φ†SφS − φ†TφT )D

m(φsFT + φTFS + φ†SF†T + φ†TF

†T − ψSψT − ψSψT )

−1

4VµνV

µν + iλσµ∂µλ +1

2D2.

11

Note that φS, ψS and FS are scalar, fermionic and auxiliarycomponents of superfield S. The notation is similar for Tsuperfield.

λ is the gaugino (fermionic superpartner of the gauge boson),and D is the auxiliary field in the gauge vector multiplet. Asusual, the gauge bosons are in the covariant derivativeDµ andfield tensors Vµν .

12

Auxiliary Fields and the Scalar Potential

We call F and D do not propagate (no derivatives). They arenot physical and can be eliminated by imposing their equa-tions of motion:

FS,T = −mφ†S,TD = −q(φ†SφS − φ†TφT )

These can be obtained by the more general solution

F †i = −dW

dφiDa = gφ†iT

aijφj

The lagrangian can now be written as

L = Lspin−1,1/2 + |DµφS|2 + |DµφT |2 − F †TFT − F †

SFS −1

2D2

where FS,T and D are given above in terms of φS,T .

The scalar potential famously can be written as

V (φS, φT ) = F †TFT + F †

SFS +1

2D2

Note that it is positive definite. Any vevs of FS,T or D nec-essarily yield V > 0. (This is a general property of globalsupersymmetry.)

13

Supersymmetry breaking

SUSY transformation properties of component fields of chiralmultiplet:

δφ =√

2ξψ

δψ =√

2ξF + i√

2∂µφσµξ

δF =i√2∂µ(ψσ

µξ)

Only Lorentz invariant possibility to break the symmetry is〈δψ〉 6= 0, which implies that 〈F 〉 6= 0. This is called F termbreaking. An analagous argument indicates 〈D〉 6= 0 alsobreaks supersymmetry (D term breaking).

Thus,

V = |F |2 +1

2D2 > 0

implies susy breaking and vice versa. V = 0 is the global susyminimum.

If it exists, SUSY preserving vacua is always the global mini-mum. “A SUSY theory will break only when forced to.”

14

Spurions of hidden-sector SUSY Breaking

Convenient to represent susy breaking as a spurion superfieldX , which has a nonzero FX component:

X = FXθθ, X† = F †X θθ

Treating X as a gauge singlet and defining m ≡ 〈FX〉/MP ,we can write the soft susy breaking terms of the lagrangian:

Soft Scalar Masses:∫d4θ

X†

MP

X

MPΦ†Φ =

F †X

MP

FXMP

φ†φ = m†mφ†φ.

A-terms:∫d2θ

X

MPΦ2Φ

2−2 =

FXMP

φ2φ2−1 = mφ2φ

2−1.

Gaugino Masses:∫d2θ

X

MPWαW

α =FXMP

λλ = mλλ

15

Part 2.

The Minimal Supersymmetry Standard Model

16

Standard Model Superfields

To supersymmetrize the Standard Model, promote all fieldsto superfields and construct gauge invariant operators out ofthe superfields.

First, here are the superfields:

Names spin 0 spin 1/2 SU(3)C , SU(2)L, U(1)Y

squarks, quarks Q (uL dL) (uL dL) ( 3, 2 , 1

6)

(×3 families) u u∗R u†R ( 3, 1, −2

3)

d d∗R d†R ( 3, 1, 1

3)

sleptons, leptons L (ν eL) (ν eL) ( 1, 2 , −1

2)

(×3 families) e e∗R e†R ( 1, 1, 1)

Higgs, higgsinos Hu (H+u H0

u) (H+u H0

u) ( 1, 2 , +1

2)

Hd (H0d H−

d ) (H0d H−

d ) ( 1, 2 , −1

2)

Names spin 1/2 spin 1 SU(3)C , SU(2)L, U(1)Y

gluino, gluon g g ( 8, 1 , 0)

winos, W bosons W± W 0 W± W 0 ( 1, 3 , 0)

bino, B boson B0 B0 ( 1, 1 , 0)

S.Martin, Primer

17

Superpotential and SUSY breaking terms

The most general gauge-invariant superpotential is

W = λuQHuuc + λdQHdd

c + λeQHdec − µHuHd

λQLdc + λ′LLec − µ′LHu + λ′′ucdcdc

(The flavor indices have been suppressed.)

The supersymmetric part of the Lagrangian arises from theF term of W .

The soft susy breaking A terms come from, for example,

d2θX

MPλuQHuu

c −→ AuλuQHuuc

and the Bµ term comes from∫d2θ

X

MPHuHd −→ BµHuHd.

The other soft terms, for the scalar masses and gaugino masses,arise in the same way as discussed earlier.

18

Mass Eigenstates: Fermionic Superpartners

Charginos in the χ±i =

{W±, H±

}basis,

U †XV −1 = diag(mχ±1, mχ±2

), where

X =

M2

√2sβmW

√2cβmW µ

Neutralinos in the χ0i = {B, W 0, H0

d , H0u} basis,

N ∗Y N−1 = diag(mχ01,mχ0

2, mχ0

3,mχ0

4), where

Y =

M1 0 −cβsWmZ sβsWmZ

0 M2 cβcWmZ −sβcWmZ

−cβsWmZ cβcWmZ 0 −µsβsWmZ −sβcWmZ −µ 0

19

Mass Eigenstates: Scalar Superpartners

For sfermions in the fi = {fL, fR} basis,

m2fi

=

m2

fL+m2

f + ∆fLmf(Af − µηf)

mf(Af − µηf) m2fR

+m2f + ∆fR

where

ηf =

1/ tan β, for up type fermions

tanβ, for down type fermions

and

∆f = (T f3 −Qfem sin2 θW ) cos 2βm2

Z.

The mixing of fL and fR is defined such that f1

f2

=

cos θf sin θf

− sin θf cos θf

fL

fR

20

MSSM Higgs Sector

For SUSY invariance, two Higgs doublets are required to givemass to all the fermions:

L = ytQHutR + ybQHdbR + yτLHdτR

When Hu and Hd obtain vevs (more on this later)

mt = yt〈Hu〉 = ytv√2

sinβ

mb = yb〈Hd〉 = ybv√2

cos β

mτ = yτ〈Hd〉 = yτv√2

cosβ

where tanβ = 〈Hu〉/〈Hd〉 and v ≃ 246 GeV.

Any doublet that gets a vev contributes to the vector bosonmasses

m2W =

1

2g2(〈Hu〉2 + 〈Hd〉2

)=

1

4g2v2

Two Higgs doublets have eight degrees of freedom. Three areeaten by W±

L and ZL, leaving five real physical scalars (Higgsboson) in the spectrum (h,H,H±, A).

21

Higgs mechanism and supersymmetry

Recall in the Standard Model

V (Φ) = m2|Φ|2 + λ|Φ|4

Symmetry breaking (〈Φ〉 6= 0) occurs when m2 < 0.

Mysteries: Why is m2 < 0 and why should it be near mW

and not MP?

In SUSY, RG flow induces negative mass2 from top Yukawacoupling:

m2Hu

(mW ) ≃ m2Hu

(MX) − Cy2tm

2t log

MX

mW< 0.

No other scalar mass2 goes negative

m2t > 0

m2e > 0

•••

(If mt < 0 were to happen, the photon would get mass, etc.)

22

RGE of MSSM mass parameters

KKRW, 93

m0 = common scalar massesm1/2 = common gaugino masses

23

Light Higgs boson in SUSY

SUSY invariance requires H4 couplings be gauge couplings inthe minimal model. (Compare with SM which has an arbi-trary coupling λ.)

As m→ high, a complete doublet of Higgs bosons decouples(H,A,H±), and one physical Higgs boson (h) remains light.

Above m theory is susy; below m theory is SM with matchingboundary condition

λ(m) =1

4(g2Y + g2

2)(m) cos2 2β

The light Higgs eigenvalue is approximated by

m2h = m2

Z cos2 2β +3

4π2y4t v

2 logm2

m2h

where the first term is the matching boundary condition andthe second term is the leading RG contribution

βλ = − 3

4π2y4t + · · · .

For m <∼ 1 TeV, one can predict mh<∼ 135 GeV.

24

R-parity conservation and proton lifetime

The most general gauge invariant superpotential is Wgood +Wbad:

Wgood = yeLHdec + ydQHdd

c + yuQHuuc − µHuHd

Wbad = λBucdcdc + λLdQLd

c + λLeLLec + µLHd

If all couplings are non-zero in Wbad, proton decay rapid.

Banish Wbad with R-parity conservation:Rp = +1 for all matter fields (Q, uc, dc, L, ec)Rp = −1 for all Higgs fields (Hd, Hu)

25

R-parity conservation perhaps not as ad hoc as it appears.

All interactions in Wbad violate B − L by one unit. GaugedU (1)B−L forbids Wbad.

But U (1)B−L must be broken (don’t have a massless B − Lgauge boson).

Let’s say that φ breaks U (1)B−L when it condenses. If it hasthe right charge (qφ = 1), then for example

1

Mφucdcdc

is gauge invariant and so

1

M〈φ〉ucdcdc ⇒ λBu

cdcdc

and R-parity is not conserved.

However, if qφ = 2, for example, then φucdcdc is not allowed,and R-parity survives.

SO(10) contains U (1)B−L and so it is possible to automat-ically get R-parity in the low energy theory if SO(10) werebroken down to the SM with irreps that carry the right B−Lcharge (e.g., 126).

26

Phenomenological Consequences of R-parity

With R-parity conservation, the LSP (lightest supersymmet-ric partner) is stable.

You will learn from other lectures that cosmologically, oneprefers the stable LSP to be neutral (neutralino, usually) soas to be the cold dark matter. In model building, this is ageneric implication from RG evolution.

27

Electric dipole moments

The EDM of fermion ψ is defined as dψ in

Hedm = idψ2ψσµνγ5ψFµν

Hedm is a T -odd operator =⇒ CP violating observable.

For the neutron, the experimental bound is

dexptn<∼ 10−25 e · cm.

The SM has a small neutron edm (if the QCD angle θ is zero):

dsmn ≃ 10−31 e · cm.

28

In SUSY, there are several sources of CP violation if couplingshave an imaginary part:

mdvdAddLd

cLHd triscalar soft term

12mλλλ gaugino mass

BµHdHu bi − scalar higgs interactionmdvdµdLd

cHu

Some of these phases (but not all!) can be absorbed by re-definitions of the fields. (Minimal CMSSM has two physicalphases left.)

CP violating interactions ultimately feed into EDMs:

××↓

Im (Ad − µ tanβ)γ

dL g dR

dL dR

dn ∝ md Im (Ad − µ tan β)

29

Doing this calculation yields

dsusyn

e≃ 8αs

27πmdmg

Im (Ad − µ tan β)

m4q

I(m2g/m

2q)

where

I(x) =1

(1 − x)2

[1

2+x

2+x ln x

1 − x

].

We can simply by assuming

mq = mg = m

Ad − µ tan β = eiφm

This leads to the simplfied expression

dsusyn

e=

4αs81π

md

m2sinφ

For m <∼ 1 TeV, one requires then φ <∼ 1/20.

In order for low-scale susy not to have a problem with EDMs,we usually insist on the phases being small.

30

Flavor Changing Neutral Currents

Sleptons and squarks can mediated large flavor-violating in-teractions.

×

bR tL sL

H−d

W−γ

↑Flavor angle Vts

There are three main ways to suppress these effects: (1) Alignsusy phases just right, (2) impose universal scalar masses, and(3) decouple the squark and sleptons by making them veryheavy (unqualified safety at the 100 TeV level).

31

Part 3.

Generating weak-scale µ term

32

Two Problems in the MSSM

1. Hard-pressed to generate mh > 114 GeV (experimentallimit).

2. The µ-term has no right to be µ ∼ v ∼ msusy

WMSSM = yuQHuuc + ydQHdd

c + yeLHdec + µHuHd

{Hu, Hd} are a vector-like pair that requires no symmetrybreaking to obtain mass. No reason to suspect it should beweak scale.

33

Higgs mass in MSSM

Prediction of Higgs mass (at leading order) is

m2h = m2

Z cos2 2β + η3GFm

4t√

2π2ln

∆2S

m2t

where

η = 1 − 2αsπ

(lnm2t

m2t

− 2

3

)

and

∆2S = m2

t ≡1

2(m2

t1+m2

t2)

Going beyond leading order (especially for large tanβ), ∆2S

becomes much more complicated.

34

Plot of mh vs. ∆S in MSSM

0.2 0.5 1.0 2.0 105.0

60

80

100

120

140

∆S [TeV]

m[G

eV]

MSSM

experimental limit

tantantantan

β = 2

βββ

= 3

= 5= 30

h

Figure 1: Tobe, Wells, hep-ph/0204196

Typically, need ∆S>∼ 1 TeV. Finetuning issue?

Radical naturalists ⇒ NMSSM (add a singlet)

35

NMSSM and the Higgs Mass

Introduce a SM singlet and a Z3 symmetry to the superpo-tential:

W = λSHuHd + λ′S3 + · · ·φ4-like contributions to Higgs mass arise from F †

SFS contri-butions to the scalar potential.

The Higgs mass bound then becomes

m2h = m2

Z cos2 2β + λ2v2 sin2 2β + δm2h,rad

Assuming λ perturbative up to unification scale (MU ≃ 2 ×1016 GeV), gives predictions for the “reasonable upper limit”of lightest Higgs in NMSSM.

36

Plot of mh vs. ∆S in NMSSM

0.2 0.5 1.0 2.090

100

110

130

120

140

∆ S [TeV]

m[G

eV]

experimental limit

NMSSM

Λ = 2 1016

GeV (sin2θW(Λ)= 3/8)

tantantantan

β = 2

βββ

= 3= 5= 30

h

Figure 2: Tobe, Wells, hep-ph/0204196

tan β = 2 example: ∆S went from ∼ 5 TeV (MSSM) to ∼400 GeV (NMSSM)

37

Problems with NMSSM

Maybe NMSSM is end of this story....However, there are well-known challenges.

Add a singlet S and impose a Z3 to bypass new µ′ problem:

W = λSHuHd + κS3

However, domain wall problem in breaking the Z3 duringEWPT.

⇒ Must break Z3 at higher order.

38

Problems with NMSSM (cont.)

Some terms that break the Z3 symmetry:

Wbreak = λ1S4

MPl+ λ2

S2(HuHd)

MPl+ λ3

(HuHd)2

MPl+ · · ·

This leads to tadpole divergences, and the return of the µproblem:

S S S

S

S S S

S

S

Hu

Hu

Hu Hd

Hd

Hd

Vind ∼ λ1κ

(16π2)2SMPlm

23/2 +

λ2λ

(16π2)2SMPlm

23/2 +

λ3λ2κ

(16π2)3SMPlm

23/2 + h.c.

Tadpole divergence of “corrected NMSSM” destabilizes hier-archy (Abel, Sarkar, White).

39

Putting a gauge charge into the µ-term

Issues ameliorated by giving S a gauge charge.

W = λSHuHd + · · ·We investigate the case where all fields, including S, are chargedunder a U (1)X symmetry, that is spontanously broken by〈S〉 6= 0.

Of course, if s = 0, then hu 6= −hd (charges are denoted bylower-case letters).

Hu and Hd no longer form a vector-like pair, and explicit µterm is forbidden.

λ〈S〉 → µeff

For shorthand, we’ll say “µ is charged” in this scenario.

40

Higgs mass and charged µ term

Higgs mass is similar to NMSSM except there is an additionalterm from

V =1

2D2X + · · ·

where,

DX = gX∑

i

qiQ∗iQi = gXhu|Hu|2 + gXhd|Hd|2 + · · ·

Thus,

m2h = m2

Z cos2 2β + λ2v2 sin2 2β

+2g2Xv

2(hu cos2 β + hd sin2 β)2 + δm2h,rad

Higgs boson is even heavier than the NMSSM case.

A better solution to original problems 1 and 2.

41

So what’s the problem?

A new problem arises. Gauge coupling unification is nowsuspect.

We will investigate this issue.

My final view will be (i.e., there is no “set up” intended):It is possible to have gauge coupling unification, and even“automatic gauge coupling unification” (as defined later), butthere is an aesthetic price and a model-building complexityprice for doing it.

42

Important assumptions

1. All the terms present in the MSSM superpotential appearin the superpotential of the extended model. (Ensuresthe model reproduces correct low-energy physics)

2. The U (1)X charges of the MSSM matter fields are family-universal. (Prevents FCNC problems)

3. The exotic matter needed to cancel the U (1)X anoma-lies consists either of GSM singlets, or of complete SU (5)multiplets (with the usual GSM ⊂ SU (5) embedding).(Preserves “automatic gauge coupling unification”)

4. The full set of exotic matter is vector-like in its GSM rep-resentation. (Ensures SM anomalies are ok, and preci-sion electroweak corrections controlled)

Any one of the above could be dismissed or relaxed, but notwithout model-building or phenomenological viability price.

Another way to look at it: The above assumptions are a wish-list applied to the problem.

43

Superpotential Invariance Conditions

Q = (3, 2, 1/6, q) U c = (3,1,−2/3, u)

Dc = (3, 1,−1/3, d) L = (1,2,−1/2, l)

Ec = (1, 1, 1, e) S = (1,1, 0, s)

Hu = (1,2, 1/2, hu) Hd = (1,2,−1/2, hd)

Gauge invariance of the superpotential requires

q + u + hu = 0 q + d + hd = 0

l + e + hd = 0 s + hu + hd = 0.

Four equations and eight variables ⇒ four “basis U (1)’s”, un-der which each field charge can be expressed: U (1)Y , U (1)B+L,U (1)ψ and U (1)χ, where last two are E6 subgroups.

(We’ll come to anomalies later.)

44

U (1)X charges that preserve W

27 GSM 2√

6U(1)ψ 2√

10U(1)χ

Q (3, 2, 1/6) 1 -1

L (1, 2,−1/2) 1 3

U c (3, 1,−2/3) 1 -1

Dc (3, 1, 1/3) 1 3

Ec (1, 1, 1) 1 -1

H (1, 2,−1/2) -2 -2

Hc (1, 2, 1/2) -2 2

N c (1, 1, 0) 1 -5

P c (3, 1, 1/3) -2 -2

P (3, 1,−1/3) -2 2

S (1, 1, 0) 4 0

The charge of any field i that preserves gauge invariance ofthe superpotential is

QiX =

2

5(−3d + e + 2hu − 3hd)Q

iY

+1

2(−3d− e + hu − 3hd)Q

iB+L

−1

4(hu + hd)2

√6Qi

ψ

+1

20(6d− 2e + hu + hd)2

√10Qi

χ,

45

Anomaly Conditions

We will demand that all gauge symmetries are anomaly free,including U (1)X be anomaly free.

Three classes:

• Pure SM anomalies: SU (3)U (1)2Y , U (1)Y -gravity, U (1)3Y ,etc.

• Mixed anomalies: SU (3)2U (1)X, SU (2)2U (1)X, U (1)2YU (1)X,U (1)YU (1)2X.

• Pure Singlet anomalies: U (1)3X and U (1)X-gravity.

Suppose s 6= 0 (charge µ term), and add only GSM singlets.Can all conditions be met?

46

Standard Model Exotics Required

Adding only GSM implies all “pure SM anomalies” are satis-fied.

Let’s look at only one mixed anomaly:

SU (3)2U (1)Y ⇒ 2q + u + d = 0

However, from W gauge invariance we know that

(q + u + hu = 0) + (q + d + hd = 0)

⇒ (2q + u + d) + (hu + hd) = 0

⇒ hu + hd = 0

Thus, the mixed anomaly equation requires that s(= −hu −hd) = 0 and µ cannot be charged.

The way out of this restriction is to add SM exotics.

47

Exotics and gauge coupling unification

By our assumptions, if we add states that can be organizedunder vector-like SU (5) multiplets, unification is still “auto-matic” and precision electroweak is ok. What can we add?

dαidt

= − bi2π

α2i , where t = ln(Q/MZ)

Gauge coupling works well (i.e., exact unification can occurfor reasonable high-scale threshold corrections).

48

Exotics and gauge coupling unification (cont.)

The vector-like additions to the matter content shift bi equallyfor each coupling:

∆bi = −N5+5 − 3N10+10 − 7N15+15 − 10N24 + . . .

Perturbative unification puts a limit on how many of thesereps are allowed.

α−1G,MSSM ≃ 1

24.1→ α−1

G,MSSM + (5.3)∆b

Thus, −∆b <∼ 4.5, or

N5+5 ≤ 4 or N10+10 ≤ 1

N15+15 = N24 = . . . = 0

For GUT perturbativity of gX,

∑di(gXQi)

2 <∼8π2

ln(MGUT/MZ)≃ 2.4.

49

Split X-charge Claim

Claim: It is impossible to arrange the collection of GSM ex-otics into SU (5) multiplets, given the above requirements andassumptions, such that each field within the SU (5) multiplethas the same U (1)X charge. Thus, the charges within each“GUT multiplet” are split.

Implication: It is not possible to interpret our theory easilyas an SU (5) × U (1)X (or G× U (1)X where SU (5) ⊂ G).

Epicyclic response: For example, take many 5 and 5 repswith various U (1)X charges, split them apart, project out/givemass to unwanted components, and piece the remaining partsback together again in complete 5 + 5 multiplets, from GSM

point of view, with split X charges.

50

Split X-charge Exotics

Assume our GSM exotics spectrum is one copy of 5 + 5 (withD1 6= L1):

5 = Dc1 ⊕ L1 = (3, 1, 1/3, D1) ⊕ (1,2,−1/2, L1),

5 = D1 ⊕ Lc1 = (3,1,−1/3, D1) ⊕ (1, 2, 1/2, L1),

Also add GSM singlets:

S = (1,1, 0, s), A = (1, 1, 0, a),

B = (1, 1, 0, b) Zm = (1,1, 0, zm).

S generates µ term, while A and B are to help give mass tothe 5+ 5 exotics. Zm are extra singlets that we need for otherreasons.

Claim: If there are no Zm fields in spectrum, some U(1)X charges are

necessarily relatively irrational (i.e., ratios of charges are irrational).

51

Singlet Engineering Theorems

Claim 1: There is always a solution to the anomaly equationsonce the GSM -dependent anomalies and conditions are met.

Proof: Equivalently, we can always solve

N∑

m=0

zm = α,N∑

m=0

z3m = β. (α, β ∈ Integers)

Choose z0 = α, then we must solve

N∑

m=1

zm = 0,N∑

m=1

z3m = γ. (γ = β − α3 ∈ Integers)

The cubic sum of any integer partition of zero is a multipleof 6. Utilize this factor of 6. Solutions:

36γ sets of zm =

{2

6,−1

6,−1

6

}

6γ sets of zm =

{4

6,−3

6,−1

6

}

γ sets of zm =

{7

6,−5

6,−1

6,−1

6

}

52

Singlet Engineering Theorems (cont.)

Claim 2: An unpaired, massless singlet field can always begiven mass with more singlets, while simultaneously not gen-erating any gauge/anomalies problems, and not introducingother massless field, nor introducing another µ problem in thesinglet sector.

Proof: “Partitions of zero” are important in this considerationas well....

Simplest, Guaranteed, All-Purpose Solution:

Field A of charge a is unpaired. Add following reps

Xa = {a, 5a,−6a} , Yb = {−2a,−3a, 5a}No singular mass matrix and no anomalies.

53

Model Building Features

Here are some generalized results/implications when pursuinga charged µ term (see paper for complete example model):

1. The most general U(1)X symmetry, when acting on the fields of the

MSSM, is a linear combination of U(1)Y , U(1)B+L, and the U(1)ψand U(1)χ subgroups of E6. Of these basis U(1)’s, only U(1)ψ is able

to forbid the µ term and protect the singlet S that replaces it.

2. Anomaly cancellation requires the introduction of exotics charged un-

der GSM to cancel all mixed anomalies in the theory. Such SM exotics

can only be dismissed if the effective µ-term is not charged under the

U(1)X .

3. Adding complete multiplets of SU(5) according to their SM charges,

which is required for “automatic gauge coupling unification”, neces-

sitates assigning different U(1)X charges to the SM-like component

states within each exotic multiplet (SM GUT multiplets are ok).

4. Solutions with rational U(1)X charges for all SM-charged states gen-

erally require a large set of Zm singlet states; nevertheless, a solution

to the singlet-dependent anomaly equations from these states Zm is

guaranteed, and we have shown an algorithm to obtain that solution.

54

Some E6 Comments

Last week, King, Moretti, Nevzorov.

Low scale gauge group is SM×U (1)N , where

U (1)N =1

4U (1)χ +

√15

4U (1)ψ

such that νc has no charge (for seesaw).

Three generations of 27 reps survive to low scale, and anadditional vector-like

{H ′, H ′

}doublet pair.

Challenges: Lots of Higgses to induce FCNC. Splitting R-parity to allow decay of exotic quarks. And... a µ problemfor the

{H ′, H ′} doublet pair.

This is a special case of all we’ve talked about today. Equiv-alent to three copies of (split) 5 + 5 added, and 3 copies ofE6-specific singlets in addition to the νci .

55

Getting the µ-term from NROs

So far, we have considered a single charged S field.

Let’s consider µeff arrising from multiple fields.

Cvetic, Langacker, and Martin, and others, have consideredvector-like states

W =SS

MPlHu ·Hd +

(SS)2

M 2Pl

where S and S are vector complements with vevs such that

〈S〉 ∼ 〈S〉 ∼√mWMPl ∼ 1012 GeV

W = µHuHd, µ′SS is forbidden by a selection rule.

Related approach:

W ∼ Hu ·HdS

(S

MPl

)P+ S3

(S

MPl

)K

The right relations between P and K (i.e., P = K) can yieldµeff → mW . Discrete symmetries must play a role. Exploitthis with multiple fields.

56

Background: Discrete Gauge Symmetries

Consider: U (1) with fields {φi} and φH of integer charges Qi

and QH .

If 〈φH〉 6= 0 the symmetry of the theory breaks down to

U (1) −→ ZQH

Operators that have charge 0 modQH are allowed.

(Krauss, Wilczek, etc.)

57

Supersymmetric discrete symmetry

Consider: Supersymmetric U (1) with fields {φi} and φH ofinteger charges Qi and QH , and FI-term ξ.

Supersymmetric minimum occurs where 〈D〉 = 0

D = QH|φH|2 − ξ + · · · (assuming QH, ξ > 0)

Thus 〈φH〉 =√ξ/QH breaks the gauge symmetry but pre-

serves susy:

U (1) −→ ZholQH

where ZholQH

≃ ZQH except ZholQH

is intended to relate moreinformation about holomorphic restrictions of superpotential.

58

Resulting Weff operators (1 of 3)

The operators allowed in the effective theory superpotentialcan arise from three sources:

1. Replace φH with its vev in the superpotential of the fulltheory. For example,

∫d2θ

1

M 3−a−b∗

φaHφbi →

∫d2θ

〈φH〉aM 3−a−b

∗φbi

This mechanisms creates effective superpotential operatorswith QH charges in the numerator in the form of 〈φH〉/M∗insertions. Note, no insertions of the form 〈φ†H〉/M∗ are al-lowed.

59


2. Integrate out fields whose masses are ∼ 〈φH〉. For example,consider the theory

W = λ1φ4φ2−2 + λφ−2φ

21

where the field φq has charge q. When φ4 gets a vev, φ4 andφ−2 get large masses ∼ 〈φ4〉, and upon integrating them outone gets

Weff ⊂ − λ22φ

41

4λ1〈φ4〉This mechanism creates effective superpotential operators withQH in the denominator of the form 1/〈φH〉 insertions.

60


3. supersymmetry breaking terms that transfer Kahler poten-tial terms to the effective superpotential. This last mechanismcan be schematically represented by

∫d4θ

X

MPlP(φ†H, φH, {φi}) →

∫d2θWK,

where WK =FXMPl

P(〈φ†H〉, 〈φH〉, {φi}),

This mechanism always involves supersymmetry breaking ofthe scale m = FX/MPl ∼ mW , and thus is usually a sub-dominant contribution to the effective theory. Nevertheless,insertions of mvav† b/M c

∗ are allowed, where the power of M∗is dictated by dimensionality requirements.

61

Effective Theory discussion (1 of 2)

Consider: U (1) theory with cutoffM∗. Break the U (1) theorywith φH of charge QH .

The low-scale effective superpotential is

Weff = µO(2) +∞∑

∆=0

c∆1

v∆O(3+∆)

where O(d) is operator of dimension d.

Properties of µO(2):

• O(2) should be charged otherwise µ ∼ M∗, which is notacceptable to effective theory.

• µ must have value below v to be part of effective theory.

• Being dimensionful, µ should be constructed out of inser-tions, such as

µ ∼ 〈φH〉aMa−1

∗,m〈φ†H〉bM b

∗, etc.

62

Effective theory discussion (2 of 2)

Consider the c∆ coefficients of

Weff = µO(2) +∞∑

∆=0

c∆1

v∆O(3+∆)

Properties of c∆:

• The only c∆ ∼ 1 are those for operators O(3+∆) withchargeQH∆. All others are suppressed by factors of v/M∗or m/v, etc.

• Chargeless NROs are suppressed by M∗ (c∆ ∼ (v/M∗)∆).

• c∆ ∼ 1 for operator of chargeQH∆, whereas c∆ ∼ (v/M∗)2∆

for operator of charge −QH∆.

• Holomorphy puts a strong connection between dimension-ality of operator and size of coefficient. Thus, Zhol

QHis more

informative than ZQH .

63

Discrete symmetry along flat direction

Consider two fields φa and φ−b, with charges a and −b re-spectively

D = a|φa|2 − b|φb|2 + · · ·The D-term cancels if we assume that φa and φ−b obtain vevswith the relation

|〈φa〉|2 =a

b|〈φ−b〉|2

The D-term potential is completely flat along this direction.

Gauge symmetry breaking is

U (1)〈φa,−b〉−→ Zhol

(a,−b)

If a and b are co-prime, (a,−b) = 1 and Z(a,−b) = Z1 =“nothing.”

In ordinary field theory, we’d say symmetry completely bro-ken, and all operators allowed. Conclusion not full story insusy.

64

Now to the µ term

Consider MSSM augmented by U (1) theory at high-scale withφa and φ−b (a, b > 0 and coprime) breaking symmetry alongD-flat direction. The charge of (Hu ·Hd) is +1.

W = λ1

φqaa φqb−b

Mqa+qb−1∗

(Hu ·Hd) + λ2

φbaφa−b

Ma+b−3∗

where qa and qb are whatever needed for gauge invariance.

Second term lifts potential. Negative soft masses m2a ∼ m2

−b ∼−m2 push solution away from origin.

Definining T = φbaφa−b ∼ τ a+b (polynomial characterizing flat

direction), the effective potential for τ (vevs of φa and φ−b) is

Vτ = −m2τ 2 + λ2M 4∗

(τ

M∗

)2(a+b)−2

65

µ term solution

The solution for τ is

τ ∼M∗

(m2

M 2∗

)1/(2a+2b−4)

Not all choices of (a,−b) will generate µeff ∼ m. In termsof the parameters a, b, qa and qb,

µeff ∼M∗

(m

M∗

) qa+qba+b−2

If we wish µeff ∼ m, we require

(qa + qb)/(a + b− 2) = 1

A general solution that guarantees this is obtained by thechoices

(a,−b) = (n,−n + 1) implying

qa = n− 2, qb = n− 1 ⇒ qa + qba + b− 2

= 1

where n is a positive integer.

66

µ term solution – some numbers

Some numbers: choose (a,−b) = (4,−3), then (qa, qb) =(2, 3) and

τ ∼M∗

(m

M∗

)1/5

∼ 1015 GeV

µeff ∼M∗

(m

M∗

)1

∼ m ∼ mW

A different choice of coprime charges could break the symme-try completely and leave a hierarchy for the µ term comparedto m (split susy relevance).

Choose (a,−b) = (3,−1), then

µeff ∼M∗

(m

M∗

)5/2

∼ m

(m

M∗

)3/2

≪ m

Note, Z(a,−b) = Z1 masks the various suppressions and asym-metries between +Q and −Q operators.

67

µ term thoughts

Evidence could be accrued at colliders for the right theory ofthe µ term: Find Z ′ boson and its couplings to particles, andfind SM/Exotic Higgs mixing effects. Separate and indepen-dently interesting phenomenology.

Significant tension exists between gauge coupling unificationand a low-scale charged µ term, when we wish to keep all otheraspects of supersymmetry simply in check (FCNC, etc.).

Present thought: Charging the µ term by a single field vev isnot obviously the right explanation. High-scale origin of thethe µ term may be more palatable.

Future possible thought: If we find evidence for low-scaleorigin of µ term, the standard notions of unification may needrethinking.

68

Part 4.

Unification Studies

69

Gauge coupling unification

Gauge coupling unification possible in weak-scale susy

2 4 6 8 10 12 14 16 18Log10(Q/1 GeV)

0

10

20

30

40

50

60

α−1

α1

−1

α2

−1

α3

−1

Martin, 97

70

Third generation Yukawa unification

Gauge coupling unification implies that our low scale SU (3)×SU (2)×U (1)Y theory is unified somehow – either string unifi-cation and/or grand unification into a simple group like SU (5)or SO(10).

SO(10) is especially powerful because all known SM states ofeach generation fit nicely into a 16 rep. E.g., all 3rd generationfermions in one rep.

The simplest model has third family Yukawa unification atthe GUT scale: yb = yτ = yt from W = y16310H163.

71

How rigorous is Yukawa unification?

So why isn’t Yukawa unification on the “Supersymmetry Suc-cesses” list?

Lazy answer: Because we’re not sure if it works.

Better answer: Because, unlike gauge coupling unification,Yukawa unification is extremely sensitive to low-scale super-partner masses and parameters which haven’t been measuredyet.

Bad news: We presently cannot have confidence in Yukawaunification for model building like we do for gauge couplingunification.

Good news: the extreme sensitivity is a great opportunityto test the idea. If all possible patterns of unification theo-ries were insensitive to low-scale superpartner parameters we’dnever make progress.

72

Goals of our Analysis

Two choices: (1) Wait for superpartner measurements andthen revisit the question of Yukawa unification, or (2) Askwhat the well-motivated hypothesis of Yukawa unification doesto the requirements on the superpartner spectrum.

We did (2), and we will happily do (1) also when the timecomes.

Why do (2)?

• Warns of possibly difficult superpartner spectrum patternsto measure at collider ...

• Determining the requirements, we can then apply our aes-thetic judgement of whether the idea is likely to work ornot ...

• Sets priorities on model building ...

“We Report, You Decide” [Kazu Tobe, JW, 2004]

73

Technical Remarks

Here is how we compute Yukawa couplings at high scale:

Find low-scale DR gauge couplings from experimental mea-surements

ComputeDRYukawa couplings from SMMS fermion masses:

yt(mZ) =

√2mMSSM

t (mZ)

v(mZ) sin β=

√2mSM

t (mZ)

v(mZ) sin β(1 + δt(mZ)),

yb(mZ) =

√2mMSSM

b (mZ)

v(mZ) cos β=

√2mSM

b (mZ)

v(mZ) cos β(1 + δb(mZ)),

yτ(mZ) =

√2mMSSM

τ (mZ)

v(mZ) cos β=

√2mSM

τ (mZ)

v(mZ) cos β(1 + δτ(mZ)).

where δf(mZ) are the weak-scale corrections (both finite andlog) due to SUSY particle loops.

Yukawa couplings and gauge couplings are then run up tohigh-scale using 2-loop RGEs.

Depending on what we do, sometimes δi are left unknownand fit to, so that a superpartner spectrum later must givethose values. And sometimes δi are directly computed givena superpartner spectrum.

74

Demonstration of Gauge Coupling Unification

At a high scale M ≃ MG0 (MG0 is where g1 = g2, the gaugecouplings are

g1(M) ≃ 0.734 (1 + 3δg1− 0.007δg2

+ 0.02δg3− 0.02δyt

− 0.005δyb

−0.002δyτ− 0.007δtanβ + 0.02 log

M

MG0

+ δGUTg1

+ O(δ2)

),

g2(M) ≃ 0.734 (1 − 0.003δg1+ δg2

+ 0.03δg3− 0.02δyt

− 0.008δyb

−0.001δyτ− 0.01δtanβ + 0.004 log

M

MG0

+ δGUTg2

+ O(δ2)

),

g3(M) ≃ 0.722 (1 − 0.001δg1− 0.002δg2

+ 0.4δg3− 0.01δyt

− 0.005δyb

−0.0002δyτ− 0.005δtanβ − 0.01 log

M

MG0

+ δGUTg3

+O(δ2)

),

The success is very insensitive to even large weak-scale susycorrections from δgi or δyi.

We call susy gauge coupling unification a success because un-naturally large δGUT

giare not needed.

75

Yukawa unification – basic results

We can do a similar exercise for Yukawa couplings at a high-scale M near MG0 (tan β0 = 50):

yt(M) ≃ 0.63(1 + 0.9δg1 + 3δg2 − 3δg3 + 7δyt + 0.7δyb

+0.02δyτ + 0.7δtanβ − 0.01 logM

MG0

+ δGUTt +O(δ2)

),

yb(M) ≃ 0.44(1 + 0.7δg1 + 2δg2 − 2δg3 + δyt + 3δyb + 0.2δyτ

+3δtanβ − 0.02 logM

MG0

+ δGUTb + O(δ2)

),

yτ(M) ≃ 0.52(1 + 0.1δg1 + δg2 − 0.6δg3 + 0.2δyt + δyb + 2δyτ

+3δtanβ − 0.005 logM

MG0

+ δGUTτ + O(δ2)

),

Note extreme sensitivity to δg3 and δyt,b.

Unification gets better when δt < 0 (check! log corrections)and δb > 0 but not very big (check! finite corrections).

Must do things numerically, since above expansion is not pre-cise.

76

Needed corrections for Yukawa unification

0.04−0.06 −0.02−0.04 0.02 0.06 0.08 0.1−0.12

−0.08−0.06−0.04−0.02

0

0

τ

δ

δt

b

0.140.120.1

0.080.06

0.040.02−0.02 0−0.1

δ(a)

−0.06−0.04−0.02 0 0.02 0.04 0.060.08 0.1

−0.020

−0.04

−0.08−0.06

−0.1−0.12

52 50 48 46 44

δτ

δt

βtan

(b)

77

Typical top and tau corrections

The size of corrections for δt and δτ in typical weak scalesupersymmetric theories with tanβ ∼ 50 are roughly

−δt ≃ g23

6π2log

(MSUSY

mZ

)<∼ 10%, (negative correction)

δτ ∼ g22

32π2

M2µ tanβ

M 2SUSY

<∼ ± few%.

Therefore, from previous graph of relations needed between δi,only relatively small corrections are tolerated for the b-quark:δb <∼ few%.

78

Expectation for δb corrections

−0.1 −0.05 0.050 0.1 0.15 0.240

45

50

55

60

tanβ

δb

δτ = 0.02 δt = −0.08

79

Typical size for b quark mass corrections

At high tanβ, if all supersymmetry masses are roughly equal,the finite b-quark mass corrections are about:

δb ∼ ± g23

12π2tanβ

Therefore, for b-τ -t unification, which is at large tanβ, weexpect that typical b mass corrections are

δb ∼ ±50%

50% corrections are too big to allow Yukawa unification.

Therefore, basic conclusion: b mass corrections must besmaller than naively is expected to allow 3rd generationYukawa unification.

(Conclusion survives a more detailed/numerical analysis.)

80

Things don’t shift too much with g3 shift

−0.12 −0.1 −0.08−0.06 −0.04

0−0.02−0.04

−0.08−0.06

−0.12−0.10

δt

δb

0.120.14

0.080.1

0.060.04

δτ = 0.02

(a)

0.02−0.02 0

δg3

0−0.02

−0.06−0.04

−0.08−0.10−0.12

βtan

−0.12 −0.1 −0.08−0.06−0.04

tδ

44464850

52

(b)δg3

= 0.02τδ

81

What does 3Y-unification tells us about susy masses?

A more detailed expression for finite b quark mass correctionsis

δfiniteb ≃ − g2

3

12π2

µMg tan β

m2b

+y2t

32π2

µAt tan β

m2t

+ . . .

This suggests several ways to get the small but non-zero δbneeded for 3Y unification.

• suppress by making R-charged masses (Mi and Ai) verysmall compared to scalar superpartners

• suppress by making PQ-charged masses (µ) very smallcompared to the scalar superpartners

• manufacture a cancellation between terms

• some combination of the above

82

The correlation with b→ sγ

Before going further, must point out corrrelation with b→ sγ

Even for super-KM=KM, this observable can be very large athigh tanβ. The SUSY coefficient to the operator

O7 = mbsLσµνbRFµν

is tanβ enhanced, since chirality flips in supersymmetry canbe done with λb ∝ mb/ cos β ∼ mb tan β.

In general, we expect wildly HUGE contributions to b → sγfor low-scale susy with tanβ ∼ 50.

×

bR tL sL

H−d

W−γ

↑yb ∝ mb/ cos β

83

The Blazek-Dermisek-Raby solution (BDR)

BDR’s analysis is based on GUT SO(10) with separate massesfor m16 and m10. They get 3Y unification by the followingmeans:

1. tanβ ∼ 50 for the third generation Yukawa couplings to approach unification.

2. m1/2 ∼ µ ≪ m16 to suppress, but not to zero, the gluino contributions to δband b→ sγ.

3. Large A0 such that the weak-scale At is larger than Mg and the positivechargino-stop contributions to δb cancels, and slightly overcomes, the largenegative contributions due to gluino-sbottoms finite and logarithmic correc-

tions. (Note, we are expressing this criteria in our sign convention for b-quarkcorrections which is opposite to BDR sign convention: δb ∝ −∆mBDR

b ).

4. µ > 0 so that the large chargino-stop corrections to b → sγ can be oppositein sign to SM (and charged Higgs) contributions. This is necessary to be

consistent with the large choice of A0 term above, which when combined withthe right sign of µ gives the chargino-stop loops a sufficiently large canceling

contribution to change the sign of the b → sγ amplitude. This enables largetanβ supersymmetry to be consistent with the B(b → sγ) measurementsdespite the SUSY contributions being much larger than the SM contributions.

84

Changing sign of b→ sγ?

Define ∆C7 to be the susy contributions to the O7 operator.

The susy prediction can be approximated as

B(b→ sγ)susy = B(b→ sγ)SM

∣∣∣∣1 + 0.45∆C7

CSM7

∣∣∣∣2

The experiment is nicely consistent with SM theory (within∼ 10%). Therefore, either

∆C7 ≃ 0, or

1 + 0.45∆C7

CSM7

≃ −1

thereby flipping the sign of the amplitude. But amplitude getssquared for b→ sγ, so satisfies experimental constraint.

Because ∆C7 is expected to be so large, the latter possibilityis real and is exploited by BDR.

85

Disquieting features of BDR solution

Main worry ... “finetuned cloaking of large tanβ effects”

δb has to be suppressed by a combination of susy hierarchiesand cancellation between gluino-sbottom and chargino-stopcontributions.

b→ sγ has to be cloaked by a just-so flipping of amplitude.

Is there a better way?

86

The partially-decoupled solution

We can suppress both δb, as is needed for 3Y unification, andb → sγ, as is needed for consistency with experiment, byconsidering very massive scalars.

There are pluses and minuses to considering very massivescalars:

Main minus: EWSB breaking more tuned – heavy massscales in potential have to eject a low-mass mZ.

Main plus: FCNC and CP violating observables are ok nomatter what super-KM angles are.

So...

Our framework: mi ≫ Mλ, Ai, µ such that no cancellationsor cloaking of large tanβ effects is necessary.

Roughtly speaking m >∼√

tan βMλ, etc.

87

Example model approach to spectrum

Anomaly mediated supersymmetry breaking (AMSB) workswell for our needs.

Ordinarily, mass of gauginos come from∫d2θ

S

MplWW → FS

Mplλλ

If there is no singlet in theory carrying susy breaking thengaugino mass very supressed. However, AMSB (via conformalanomaly) introduces susy breaking contributions at one loop:

Mλ =βgλgλ

m3/2

16π2

where m3/2 is the gravitino mass.

Scalar masses generally arise at order F †F/M 2pl and so are

naturally of orderm3/2. They can be supressed (e.g., no scale)all the way to AMSB levels, but probably should not rely onthat. We assume only a mild suppression at most

m2 = η m23/2

where η is maybe one-loop suppressed.

88

Illustration of 3Y-unification in pAMSB

Choose AMSB spectrum normalized such thatM2 = 150 GeV(M1 = 500 GeV andM3 = 1300 GeV). Gravitino mass is near600 GeV in this case.

For given value of µ andMSUSY (defined to be all scalar massesexcept Higgs), compute the value of tanβ (usually near 50)such that ǫ is minimized.

ǫ =

√(yb − yτ

yb

)2

+

(yt − yτ

yt

)2

+

(yt − yb

yt

)2

(computed at GUT scale)

Plot contours of constant ǫ – values less than about 5% (ǫ <∼ 0.05)are reasonable requirements on the unification.

89

Contours of constant ǫ

2 4 6 8 10 12 14

M SUSY

20

10

30

40µ

0.050.10.15

= 150 GeVM 2

ε

[TeV]

[TeV]

Contours of ǫ = 0.05 (0.1, 0.15) corresponds to about 5% (10%, 15%)GUT threshold correction needed to achieve Yukawa couplingunification. GUT-scale Yukawa corrections are expected tobe less than about 1%. MSUSY is the low-energy mass forall scalar superpartners. The gaugino and A-term massesare equal to their anomaly-mediated values normalized toM2 = 150 GeV.

90

Is b→ sγ ok?

200

600

1000

1400

1800

500 1500 2000 2500 3000

µ

2

1.51.25

1.15

= 150 GeV2

M

1000

SUSY

M

1.1 1.05=0.15ε

0.15

0.1

0.05

0.05

0.1

sγbB(γsbB(

R =))SM

(a)[GeV]

[GeV]

Kagan and Neubert say that 0.37 < R < 1.25 where

R =B(b→ sγ)

B(b→ sγ)SM.

Msusy over a few TeV seems necessary. This graph was madeunder the most favorable FCNC assumptions (super-KM=KM),otherwise limit would be higher.

91

g − 2 of the muon

23

5

10

30

500 1000 1500 2000 2500 3000

1800

1400

1000

200

600

M SUSY

µ

M2 = 150 GeVaSUSY

µ

1

[GeV]

[GeV]

= 0.15ε0.1

0.10.15

0.050.05

(b)

[10−10 ]

A conservative view of the g − 2 experimental uncertaintiesand theoretical uncertainties implies that asusy

µ /10−10 shouldbe between about -37 and 90 (Martin and JW).

A less conservative interpretation of the experiment and the-ory implies (Hoecker) that

asusyµ /10−10 = 25 ± 9 (2.7σ effect).

Positive aµ is mild preference for this model?

92

Comments on high-scale threshold corrections

Gravity effects yield order 1% corrections to gauge and Yukawacouplings:

δy, δg ∼ MG

Mpl∼ 1%

Heavy GUT-scale particles in loops generally do not con-tribute much to Yukawa corrections (but can contribute muchto gauge coupling corrections).

If neutrino Yukawa yν is unified at the high scale than itsYukawa coupling affects the evolution of the other Yukawas.The correction can be expressed as a GUT-scale thresholdcorrection:

δGUTt ≃ δGUTτ<∼

y2ν

16π2log

MG

1013 GeV∼ 4%, (1)

δGUTb ≃ 0. (2)

This correction is in the positive direction δt ≃ δτ directionwhich has almost no impact on needed δb. (Contours of con-stant “needed δb” are in δt ≃ δτ direction.)

93

Prediction of top mass

Old (and not so old) Claims: “S0(10) 3Y unification predictsa top mass of 175 GeV, just like experiment shows.”

That was derived assuming (either consciously or, usually,subconsciously) that δb = 0 and doing a not-so-super-preciseanalysis.

Actually, any mt would have been fine. In fact, perhaps alower value of mt would have been better since we naturallyexpect δb ∼ tens of percent.

0 0.1 0.15 0.2 0.25100

120

140

160

180

bδ0.05

mtop

[GeV]

δτ = 0.02 tδ = −0.08

94

tan β dependence on top mass in 3Y unification

20 40 5025 3530 45 55100

120

140

160

180

tanβ

mtop

[GeV]

δτ = 0.02 δt = −0.08

95

Yukawa unification discussion

• judging viability of b− τ − t unification is highly sensitiveto the low-scale superpartner spectrum.

• bmass finite corrections must be much smaller than naivelywould be expected to make unification work out.

• BDR approach: cancellations to cloak unwanted large tanβeffects

• Our approach: deep suppressions of unwanted large tanβeffects at the possible expense of natural EWSB

• Experiment will tell! (The benefits of IR sensitivity)

96

Exact gauge coupling unification

We saw earlier that gauge couplings do not unify exactly, whenapplying only IR considerations.

Nor do we expect it! Rather, we expect high-scale thresholdcorrections to have an effect.

From minimal SU (5) point of view, we can illustrate howimportant non-renormalizable operators (NROs) are to unifi-cation.

97

Minimal SU (5) is dead...

Consider minimal SU (5):

{10i, 5i, 1i} matter sector

24 gauge sector

{24H, 5H, 5H}

High scale threshold corrections come from the massive com-ponents of the these reps: MV , MΣ, MHc.

98

Gauge coupling running

The relationships between the GUT scale gauge coupling gGand the low-scale gauge couplings gi(Q) of the MSSM effectivetheory are

1

g2i (Q)

=1

g2G(Q)

+ ∆Gi (Q) + ciǫ

The ∆Gi (Q) functions are the threshold corrections due to

heavy GUT states; ∆Gi (Q) = 1/(8π2)

∑a bai ln(Q/Ma) where

bai and Ma are β function coefficient of a heavy particle andits mass, respectively. They are explicitly written by

∆G1 (Q) =

1

8π2

(−10 ln

Q

MV+

2

5ln

Q

MHc

)

∆G2 (Q) =

1

8π2

(−6 ln

Q

MV+ 2 ln

Q

MΣ

)

∆G3 (Q) =

1

8π2

(−4 ln

Q

MV+ ln

Q

MHc

+ 3 lnQ

MΣ

).

99

Importance of Higgs triplet

One linear combination isolates MHc (Hisano et al.):

− 1

g21(Q)

+3

g22(Q)

− 2

g23(Q)

=3

5π2lnMHc

Q.

Evaluate at unification scale ΛU , which we define to be theplace where g1(ΛU) = g2(ΛU) = gU ,

1

g2U

− 1

g23(ΛU)

=3

10π2lnMHc

ΛU.

ΛU depends mildly on the low-scale superpartner masses, butit is always within the range

1 × 1016 GeV <∼ ΛU<∼ 2 × 1016 GeV

for superpartner masses at the TeV scale and below.

100

Conflict between unification and proton stability?

1

g2U

− 1

g23(ΛU)

=3

10π2lnMHc

ΛU.

g3(ΛU) < gU , albeit by less than 1%.

This implies that the LHS of is necessarily negative. We seethat

MHc < ΛU ≃ 1016 GeV (gauge unification)

is required for the RHS to be negative and successful gaugecoupling unification to occur.

But this is in conflict with the proton decay requirement that

MHc > 1017 GeV(> ΛU) (proton decay)

p

K+

u

d

u

u

d

H3

s

νµ

u

101

NRO effects

We expect gravity to induce MP operators of the type∫d2θ

[S

8MPlWW +

yΣ

MPlWW

]

where Σ = 24H and 〈S〉 = MPl/g2G + θ2FS contains the

effective singlet supersymmetry breaking. The SU (5) gaugecoupling is gG and the universal contribution to the masses ofall gauginos is M1/2 = −g2

GFS/(2MPl).

Indeed, such interactions are necessary for gaugino massesfrom susy breaking.

The GUT symmetry breaking is accomplished by

〈Σ〉 = vΣ diag

(2

3,2

3,2

3,−1,−1

)

to break SU (5) to SU (3) × SU (2)L × U (1)Y at the GUTscale. The numerical value of vΣ depends on details of thecouplings but should be around the GUT scale of 1016 GeV.

102

Shifted gauge couplings

The NRO operator involving the 24 affects the SU (3) ×SU (2) × U (1) couplings. They are now

1

g2i (Q)

=1

g2G(Q)

+ ∆Gi (Q) + ciǫ

where

ǫ = 8yvΣ

MPand ci = {−1/3,−1, 2/3}

for groups i = {U (1)Y , SU (2)L, SU (3)} respectively.

We can recompute the combination that isolated Hc:

1

g2U

− 1

g23(ΛU)

=3

10π2lnMHc

ΛU− 2ǫ

=3

10π2lnM eff

Hc

ΛU.

where

M effHc

= MHc exp(−20π2ǫ/3)

Thus, with ǫ ∼ few percent, we can have MHc > 1017 GeVand MHc < 1016 GeV.

103

Triplet Higgs mass for exact unification

��

��

��

��

��

��

m0 M1/2= =µ=mA = 3 TeV

m0=M1/2=µ=mA = 500 GeV

m0=M1/2=µ=mA = 200 GeV

1 2 3 4 5 6 7

100 ε (%)

010

10

10

14

15

1610

10

1010

10

MH

c(G

eV)

21

20

19

18

17

Tobe, JW, 04

104

SUSY breaking effects

VEVs usually develop in superspace

〈Σ〉 ≃ (vΣ + FΣθ2) diag

(2

3,2

3,2

3,−1,−1

)

The superpotential and soft lagrangian terms we assume are

W =1

2MΣTrΣ2 +

f

3TrΣ3 +M55H5H + λ5HΣ5H + · · ·

−Lsoft =1

2BΣMΣTrΣ2 +

f

3AΣTrΣ3 + B5M55H5H + Aλλ5HΣ5H + h.c. + · · ·

where upon minimizing the full potential we find

FΣ ≃ vΣ(AΣ − BΣ) =ǫMPl

8y(AΣ −BΣ)

which generates a correction to gaugino masses via the NRO.∫d2θ

[S

8MPlWW +

yΣ

MPlWW

]

105

Gaugino mass spectrum

M1(ΛU) = g2UM + g2

U

[1

6ǫ(AΣ − BΣ) − 1

16π2

(10g2

UM + 10{AΣ −BΣ} +2

5B5

)]

M2(ΛU) = g2UM + g2

U

[1

2ǫ(AΣ − BΣ) − 1

16π2

(6g2

UM + 6AΣ − 4BΣ

)]

M3(ΛU) = g23(ΛU)M + g2

U

[−1

3ǫ(AΣ − BΣ) − 1

16π2

(4g2

UM + 4AΣ − BΣ + B5

)]

where M = −FS/(2MPl) ∼ O(mz) is the supersymmetry mass scale from the

singlet field F -term.

There are four parameters of the GUT theory that are affecting the ratios of the

gaugino mass values at ΛU ,

ǫ, AΣ/M, BΣ/M, B5/M.

δ1−2 =M1(ΛU) −M2(ΛU)

M2(ΛU)and δ3−2 =

M3(ΛU) −M2(ΛU)

M2(ΛU).

The δ’s are defined at the g1 = g2 unification scale ΛU .

106

Relative shifts in gaugino mass boundary conditions

−40 −20 0 20 40

−40

−20

0

20

40

100

100

δ 3−

2(%

)

= 0%ε

= 3%ε= 5%ε

= 10%ε

δ1−2(%)

A/M <3 B/M, <3

Tobe, JW, 04

107

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