b. bajc, the still available parameter space of the minimal supersymmetric su(5)

Borut Bajc

The Still Available Parameter

Space of the Minimal

Supersymmetric SU(5)

Borut Bajc

J. Stefan Institute, Ljubljana, Slovenia

BB, Stephane Lavignac and Timon Mede, work in progress

BW13, Vrnjacka Banja, Serbia 1

Borut Bajc

Outline

• The minimal susy SU(5)

• RGEs and proton decay

• Fermion mass constraint

• Higgs mass

• Electroweak symmetry breaking scale

• An upper limit

• More ambitious? Neutrino mass and dark matter

• Conclusions


Borut Bajc

The minimal susy SU(5)

It is made of

• 3 generations of matter in 5i + 10i

• Higgses in 24H and 5H + 5H

• gauge superfield in 24V

Furthermore we will assume

• renormalizability

• susy broken above MGUT , soft terms SU(5) symmetric

• vacuum (global) stability


Borut Bajc

RGEs and proton decay

From RGE’s and known exp values of αi(MZ):

mT ≈ 1015GeV(m3

m8

)5/2 (msusy

1 TeV

)5/6

m3,8 . . . masses of weak triplet and color octet in 24H .

In minimal renormalizable susy SU(5):

m3 = m8 → mT ≈ 1015GeV(msusy

1 TeV

)5/6

In low-energy susy T too light, mediates too fast proton decay!

WT = Y ij10

(QiQj + ucie

cj

)T + Y ij5

(ucid

cj +QiLj

)T


Borut Bajc

τd=5p ≈ 4.1033 yrs

(4

tanβ

)2(

m21,2

mw 10 TeV

)2 ( mT

1017 GeV

)2

Possible solution split supersymmetry: mw � m1,2

This same solution kills also all needed corrections to fermionmasses (will see later)

Increase msusy = m1,2 ∼> mw a bit (mT ∝ m5/6susy) !

For example this works for msusy ≈ 20 TeV ≈ 3 mw

Notice that τd=5 ∝ m22/5T ≈ m4

T similar to τd=6 ∝ m4V


Borut Bajc

Fermion mass constraint

In minimal renormalizable susy SU(5) most general Yukawas

WSU(5)Y = Y ij10 10i10j5H + Y ij5 5i10j 5H

i, j = 1, . . . 3 (generation indices)

MSSM Yukawas parametrized by

WMSSMY = Y ijU Qiu

cjHu + Y ijD Qid

cjHd + Y ijE Le

cHd


Borut Bajc

Easy to derive in our GUT that

MU = MTU (∝ Y10)

BAD → MD = MTE (∝ Y5)

3rd generation (mb = mτ at GUT scale) OK

1st and 2nd generation bad after RGE’s from GUT scale to EWscale


Borut Bajc

More precisely at MZ (tanβ ≈ 3):

δmd/md ≈ 2δms/ms ≈ −0.75δmb/mb ≈ 0.1

In this minimal renormalizable model

• no extra rep’s like for ex. 45H , extra vector like pairs

• no non-renormalizable terms 101,251,25H24H/Λ

the only way to improve → susy threshold corrections


Borut Bajc

Gluino exchange dominates (maximized for common soft massm = mg ≈ m1,2)

δmDi = −αs

3πv

m(ADi cosβ − µyDi sinβ)

Vacuum stability:

∣∣ADi ∣∣ ≤ yDi √3(m2Hd

+ 2m2)1/2

Only possibility in mHd>> mi:

δmDi

mDi

≈ αs3π

µ tanβ − aDi√

3mHd

m

with |aDi | ≤ 1


Borut Bajc

µ term cannot dominate (different generations have opposite sign)

mHd/m1,2 ≈ O(100) must overcome loop factor


Borut Bajc

Higgs mass

m2h = m2

tree +m2log +m2

mix

m2tree = m2

Z cos2 (2β)

m2log =

3 sin2 βy2tm

2t

4π2log(mt1

mt2

m2t

)m2mix =

3 sin2 βy2tm

2t

4π2f(Xt,mt2

/mt1

)For any choice of mt1,2 ∼> 1 TeV (both from (m10)3 at GUT scale)the Higgs mass can always be mh ≈ 125 GeV for some

Xt ≡ At sinβ − µyt cosβ


Borut Bajc

Very preliminary:

tanβ ≈ 2, sign(µ) > 0, At = 0

0 200 400 600 800 10000

200

400

600

800

1000

m�

103

GUTHTeVL

m�

HGUTHTeVL

mh ~ mhexp

mHu2HMZL > 0


Borut Bajc

Electroweak symmetry breaking scale

To minimize the 1-loop effect one solves the RG improved tree levelHiggs potential at the scale

MEWSB =√mt1

(MEWSB)mt2(MEWSB)

At this scale (neglecting small MZ)

µ2 =m2Hd−m2

Hutan2 β

tan2 β − 1

and similarly for Bµ.

In our case (large mHd= mHu

≈ (m10)3)

→ µ ≈ O (few hundreds) TeV (modulo cancellations)


Borut Bajc

An upper limit

At GUT scale

WH = 5H (η 24H +m5) 5H +m24 242H + λ 243

H

SU(5)→ SU(3)C × SU(2)L × U(1)Y

〈24H〉 = v24

2 0 0 0 0

0 2 0 0 0

0 0 2 0 0

0 0 0 −3 0

0 0 0 0 −3


Borut Bajc

Doublet-triplet splitting fine-tuning:

m5 = η v24

Mass spectrum:

mT = η v24

mΣ = m3 = m8 = λ v24

mX = g5 v24

It follows:

mT ∼< mV (perturbativity)

mΣ could be also much smaller in principle (if λ� 1)


Borut Bajc

From RGE:

( mΣ

1016 GeV

)3 ( mV

1016 GeV

)6

=(

103 GeVmsusy

)2

Since mT ∼< mV and proton decay needs as large mT as possible

mT ≈ 1015GeV(msusy

1 TeV

)5/6

→ larger msusy allowed by smaller mΣ (i.e. small coupling λ)

Maximum msusy reached when mT ≈ mV ≈MPlanck

→ maximum µmax ≈ mmaxsusy ≈ 104 TeV

For higher msusy SU(5) becomes non-perturbative (η ∼> 1)


Borut Bajc

More ambitious? Neutrino mass and dark

matter

Although not really necessary to explain (different sector), whatabout neutrinos and dark matter?

In this minimal renormalizable model

• no extra representations 1F , 15H , 24F (type I, II, III resp.)

Without adding anything the only source of ν mass could beR-parity violating couplings


Borut Bajc

ν mass

SU(5) relations→ λ ≈ λ′ ≈ λ′′ →

• either in conflict with d = 4 p-decay τ ≈ 1/(λ′λ′′)2

• or not enough for neutrino mass mν ∝ λ2, λ′2

Only possibility bilinear R-parity violation µ′LHu

dark matter

neutralino decays too fast → only dark matter candidate: gravitino

due to diffuse photon background constraints

mgravitino ∼< O(1) GeV


Borut Bajc

Conclusions

The minimal renormalizable susy SU(5) (probably) still alive (oralmost alive) providing

• p decay → small tanβ ≈ O(2− 5)

• correct md,s → mg ≈ md,s ∼< O(10) TeV at MZ

→ (m10,5)1,2 ≈ O(10) TeV at MGUT

• SU(5) relations at low scale → mw ≈ mg/3 at MZ

• vacuum stability + Higgs mass →mHd

= mHu≈ (m10)3 ∼> O(100) TeV at MGUT

• (m5)3(MGUT ) constrained only by stability (not too small)

• Higgsino mass µ ≈ O(10− 100) TeV


b. bajc, the still available parameter space of the minimal supersymmetric su(5)

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