Balkan Workshop BW2013 Beyond the Standard Models 25 – 29 April, 2013, Vrnjačka Banja, Serbia

1. 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, Vrnjaˇcka Banja, Serbia 1

2. 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
BW13, Vrnjaˇcka Banja, Serbia 2

3. 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
BW13, Vrnjaˇcka Banja, Serbia 3

4. Borut Bajc
RGEs and proton decay
From RGE’s and known exp values of αi(MZ):
mT ≈ 1015
GeV
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 ≈ 1015
GeV
msusy
1 TeV
5/6
In low-energy susy T too light, mediates too fast proton decay!
WT = Y ij
10 QiQj + uc
i ec
j T + Y ij
5 uc
i dc
j + QiLj
¯T
BW13, Vrnjaˇcka Banja, Serbia 4

5. Borut Bajc
τd=5
p ≈ 4.1033
yrs
4
tan β
2
˜m2
1,2
m ˜w 10 TeV
2
mT
1017 GeV
2
Possible solution split supersymmetry: m ˜w ˜m1,2
This same solution kills also all needed corrections to fermion
masses (will see later)
Increase msusy = ˜m1,2 ∼> m ˜w a bit (mT ∝ m
5/6
susy) !
For example this works for msusy ≈ 20 TeV ≈ 3 m ˜w
Notice that τd=5
∝ m
22/5
T ≈ m4
T similar to τd=6
∝ m4
V
BW13, Vrnjaˇcka Banja, Serbia 5

6. Borut Bajc
Fermion mass constraint
In minimal renormalizable susy SU(5) most general Yukawas
W
SU(5)
Y = Y ij
10 10i10j5H + Y ij
5
¯5i10j
¯5H
i, j = 1, . . . 3 (generation indices)
MSSM Yukawas parametrized by
WMSSM
Y = Y ij
U Qiuc
jHu + Y ij
D Qidc
jHd + Y ij
E Lec
Hd
BW13, Vrnjaˇcka Banja, Serbia 6

7. Borut Bajc
Easy to derive in our GUT that
MU = MT
U (∝ Y10)
BAD → MD = MT
E (∝ Y5)
3rd
generation (mb = mτ at GUT scale) OK
1st
and 2nd
generation bad after RGE’s from GUT scale to EW
scale
BW13, Vrnjaˇcka Banja, Serbia 7

8. Borut Bajc
More precisely at MZ (tan β ≈ 3):
δmd/md ≈ 2
δms/ms ≈ −0.75
δmb/mb ≈ 0.1
In this minimal renormalizable model
• no extra rep’s like for ex. 45H, extra vector like pairs
• no non-renormalizable terms 101,2
¯51,2
¯5H24H/Λ
the only way to improve → susy threshold corrections
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9. Borut Bajc
Gluino exchange dominates (maximized for common soft mass
˜m = m˜g ≈ ˜m1,2)
δmD
i = −
αs
3π
v
˜m
(AD
i cos β − µyD
i sin β)
Vacuum stability:
AD
i ≤ yD
i
√
3 m2
Hd
+ 2 ˜m2 1/2
Only possibility in mHd
>> ˜mi:
δmD
i
mD
i
≈
αs
3π
µ tan β − aD
i
√
3mHd
˜m
with |aD
i | ≤ 1
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10. Borut Bajc
µ term cannot dominate (different generations have opposite sign)
mHd
/ ˜m1,2 ≈ O(100) must overcome loop factor
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11. Borut Bajc
Higgs mass
m2
h = m2
tree + m2
log + m2
mix
m2
tree = m2
Z cos2
(2β)
m2
log =
3 sin2
βy2
t m2
t
4π2
log
m˜t1
m˜t2
m2
t
m2
mix =
3 sin2
βy2
t m2
t
4π2
f Xt, m˜t2
/m˜t1
For any choice of m˜t1,2 ∼> 1 TeV (both from ( ˜m10)3 at GUT scale)
the Higgs mass can always be mh ≈ 125 GeV for some
Xt ≡ At sin β − µyt cos β
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12. Borut Bajc
Very preliminary:
tan β ≈ 2, sign(µ) > 0, At = 0
0 200 400 600 800 1000
0
200
400
600
800
1000
m103
GUT
TeV
mH
GUT
TeV
mh mh
exp
mHu
2
MZ 0
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13. Borut Bajc
Electroweak symmetry breaking scale
To minimize the 1-loop effect one solves the RG improved tree level
Higgs potential at the scale
MEW SB = m˜t1
(MEW SB)m˜t2
(MEW SB)
At this scale (neglecting small MZ)
µ2
=
m2
Hd
− m2
Hu
tan2
β
tan2
β − 1
and similarly for Bµ.
In our case (large mHd
= mHu
≈ ( ˜m10)3)
→ µ ≈ O (few hundreds) TeV (modulo cancellations)
BW13, Vrnjaˇcka Banja, Serbia 13

14. Borut Bajc
An upper limit
At GUT scale
WH = ¯5H (η 24H + m5) 5H + m24 242
H + 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










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15. 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)
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16. Borut Bajc
From RGE:
mΣ
1016 GeV
3 mV
1016 GeV
6
=
103
GeV
msusy
2
Since mT ∼< mV and proton decay needs as large mT as possible
mT ≈ 1015
GeV
msusy
1 TeV
5/6
→ larger msusy allowed by smaller mΣ (i.e. small coupling λ)
Maximum msusy reached when mT ≈ mV ≈ MP lanck
→ maximum µmax
≈ mmax
susy ≈ 104
TeV
For higher msusy SU(5) becomes non-perturbative (η ∼> 1)
BW13, Vrnjaˇcka Banja, Serbia 16

17. Borut Bajc
More ambitious? Neutrino mass and dark
matter
Although not really necessary to explain (different sector), what
about 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 be
R-parity violating couplings
BW13, Vrnjaˇcka Banja, Serbia 17

18. 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
BW13, Vrnjaˇcka Banja, Serbia 18

19. Borut Bajc
Conclusions
The minimal renormalizable susy SU(5) (probably) still alive (or
almost alive) providing
• p decay → small tan β ≈ O(2 − 5)
• correct md,s → m˜g ≈ ˜md,s ∼< O(10) TeV at MZ
→ ( ˜m10,5)1,2 ≈ O(10) TeV at MGUT
• SU(5) relations at low scale → m ˜w ≈ m˜g/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
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