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Non-Tachyonic Semi-Realistic
Non-Supersymmetric Heterotic String Vacua
Johar Muhammad Ashfaque
University of Liverpool
based on 1506.03114
in collaboration with
P. Athanasopoulos, A. E. Faraggi, and H. Sonmez
Johar Muhammad Ashfaque String Phenomenology

Motivation
Seek string models which closely resemble the MSSM with
4D N = 1 theory

Motivation
4D N = 1 theory
with SM gauge group embedding

Motivation
4D N = 1 theory
3 chiral generations of quarks and leptons

Motivation
4D N = 1 theory
minimum of 1 Higgs doublet pair

Motivation
4D N = 1 theory
N = 1 → N = 0

Motivation
4D N = 1 theory
N = 1 → N = 0
See also talks by S. Groot Nibbelink, E. Mavroudi

All roads lead to Rome. (Spanish Proverb)
Conformally Invariant

4D, N = 1 Theory

4D, N = 1 Theory
SM gauge group embedding

4D, N = 1 Theory
3 generations

4D, N = 1 Theory
3 generations
ROME: N = 1 → N = 0

The Free Fermionic Construction
A general boundary condition basis vector is of the form
α = ψ1,2
, χi
, yi
, ωi
|yi
, ωi
, ψ
1,...,5
, η1,2,3
, φ
1,...,8
where i = 1, ..., 6
ψ
1,...,5
- SO(10) gauge group
φ
1,...,8
- SO(16) gauge group

The Very Simple Rules
The ABK Rules
[Antoniadis, Bachas, Kounnas, 1987]
One-Loop Phases
C
bi
bj
= ±1 or ±i
Virasoro Level-Matching Condition
M2
L = ?1
2 +
α2
L
8 + vL = ?1 +
α2
R
8 + vR = M2
R

Non-Supersymmetric Tachyon-Free String Vacua
SO(16) × SO(16)
tachyonic state(s) produced
????????????????→ untwisted sector
[Alvarez-Gaume, Ginsparg, Moore, Vafa, 1986]
The SUSY generator S
Guarantees The Untwisted Tachyon Is Projected OUT.

However...
Problem: There is an abundance of sectors in the 3 generation
semi-realistic models that can potentially give rise to tachyonic
states.

An Explicit Non-Supersymmetric Tachyon-Free Model
1 = {ψ?
, χ1,..,6
, y1,...,6
, ω1,...,6
|y1,...,6
, ω1,...,6
, ψ
1,...,5
, η1,2,3
, φ
1,...,8
}
S = {ψ?
, χ1,..,6
}
b1 = {ψ?
, χ1,2
, y3,...,6
|y3,...,6
, ψ
1,...,5
, η1
}
b2 = {ψ?
, χ3,4
, y1,2
, ω5,6
|y1,2
, ω5,6
, ψ
1,...,5
, η2
}
b3 = {ψ?
, χ5,6
, ω1,...,4
|ω1,...,4
, ψ
1,...,5
, η3
}
α = {y1,...,6
, ω1,...,6
|ω1
, y2
, ω3
, y4,5
, ω6
, ψ
1,2,3
, φ
1,...,4
}
β = {y2
, ω2
, y4
, ω4
|y1,...,4
, ω5
, y6
, ψ
1,2,3
, φ
1,...,4
}
γ = {y1
, ω1
, y5
, ω5
|ω1,2
, y3
, ω4
, y5,6
, ψ
1,2,3
= 1
2 ,
η1,2,3
= 1
2 , φ
2,...,7
= 1
2 }

The Non-Supersymmetric Choice of GGSO Phases
?
?
?
?
?
?
?
?
?
?
?
?
1 S b1 b2 b3 α β γ
1 1 1 ?1 ?1 ?1 1 1 i
S 1 1 1 1 1 1 1 ?1
b1 ?1 ?1 ?1 ?1 ?1 ?1 ?1 i
b2 ?1 ?1 ?1 ?1 ?1 ?1 ?1 i
b3 ?1 ?1 ?1 ?1 ?1 ?1 1 i
α 1 1 1 1 1 1 1 1
β 1 1 ?1 ?1 ?1 ?1 ?1 ?1
γ 1 ?1 1 ?1 1 ?1 ?1 1
?
?
?
?
?
?
?
?
?
?
?
?

The Observable Sector
SO(10)
α, β

SO(6) × SO(4)
γ

SU(3)C × U(1)C × SU(2)L × SU(2)R

The Hidden Sector
SO(16)
α, β

SO(8) × SO(8)
γ

U(1) × SU(3)H1 × U(1)H1 × SU(3)H2 × U(1)H2 × U(1)

No More SUSY
3 Types of Sectors
Sectors that respect SUSY
Sectors that exhibit SUSY breaking explicitly
For a given sector
ρ ∈ Ξ
SUSY partners would be obtained from
S + ρ
But
(ρ)2
L = 4, (S + ρ)2
L = 8
Sectors with equal number of bosons and equal number of
fermions but with di?erent gauge charges

The Fermi-Bose Degeneracy The Mismatch
nb = nf ? Λ is exponentially suppressed.
Talk by E. Mavroudi
[Abel, Dienes, Mavroudi, 2015]
However, in our non-tachyonic, non-supersymmetric model there is
a mismatch.
Having said that, we believe we can engineer our way to tackle this
problem.

Summary
Generically models without SUSY have tachyons
We found a non-supersymmetric semi-realistic model
Furthermore
This model is tachyon-free (all tachyons are projected out)
Various Types of Sectors

Future Work
Searching For Semi-Realistic Models With Exponentially
Suppressed Cosmological Constant
Cancellation of Non-Vanishing Vacuum Energy Tadpole
Against U(1)A Tadpole

GRACIAS POR TU ATENCI?ON

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JM ASHFAQUE

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