Swampland, field distances andnaturalness
Irene Valenzuela
Utrecht University
Grimm,Palti,IV [arXiv:1802.08264 [hep-th]]
Ibanez,Martin-Lozano,IV [arXiv: 1707.05811 [hep-th]]
Ibanez,Martin-Lozano,IV [arXiv:1706.05392 [hep-th]]Strings 2018, Okinawa (Japan)

Swampland:
Apparently consistent (anomaly-free) quantum effective field
theories that cannot be UV embedded in quantum gravity(they cannot arise from string theory)

LandscapeLandscape

Swampland[Vafa’06][Ooguri-Vafa’06]

Consistent withquantum gravity

Not consistent withquantum gravity

Landscape

Swampland[Vafa’06][Ooguri-Vafa’06]

Not everything is possible in string
theory/quantum gravity!!!
Additional QG constraints = UV imprint at low energies= Quantum Gravity/String Theory predictions!

String Theory

Phenomenology

What are the constraints that an effective theory
must satisfy to be consistent with quantum gravity?

What distinguishes the landscape from the swampland?

Quantum Gravity Conjectures
Motivated by observing recurrent features of the string landscapeas well as black hole physics
Absence of global symmetries [Banks-Dixon’88][Horowitz,Strominger,Seiberg…]
Completeness hypothesis [Polchinski.’03]
Weak Gravity Conjecture [Arkani-Hamed et al.’06]
Swampland Distance Conjecture [Ooguri-Vafa’06]
No stable non-susy AdS vacua [Ooguri-Vafa’16]
No deSitter vacuum?

Quantum Gravity Conjectures
Motivated by observing recurrent features of the string landscapeas well as black hole physics
Absence of global symmetries [Banks-Dixon’88][Horowitz,Strominger,Seiberg…]
Completeness hypothesis [Polchinski.’03]
Weak Gravity Conjecture [Arkani-Hamed et al.’06]
Swampland Distance Conjecture [Ooguri-Vafa’06]
No stable non-susy AdS vacua [Ooguri-Vafa’16]
No deSitter vacuum?

Quantum Gravity Conjectures
Motivated by observing recurrent features of the string landscapeas well as black hole physics
Absence of global symmetries [Banks-Dixon’88][Horowitz,Strominger,Seiberg…]
Completeness hypothesis [Polchinski.’03]
Weak Gravity Conjecture [Arkani-Hamed et al.’06]
Swampland Distance Conjecture[Ooguri-Vafa’06]｝

No stable non-susy AdS vacua [Ooguri-Vafa’16]

Large field inflation

Particle physics and c.c.

No deSitter vacuum?

They can have significant implications in low energy physics!
-

UV sensitive effective theories
Naturalness issues

Quantum Gravity Conjectures
Motivated by observing recurrent features of the string landscapeas well as black hole physics
Absence of global symmetries [Banks-Dixon’88][Horowitz,Strominger,Seiberg…]
Completeness hypothesis [Polchinski.’03]
Weak Gravity Conjecture [Arkani-Hamed et al.’06]
Swampland Distance Conjecture[Ooguri-Vafa’06]｝

No stable non-susy AdS vacua [Ooguri-Vafa’16]

Large field inflation

Particle physics and c.c.

No deSitter vacuum?

They can have significant implications in low energy physics!
-

UV sensitive effective theories
Naturalness issues

Compactifications of the Standard Model
Ooguri-Vafa conjecture: non-susy stable AdS vacua are not consistent with QG

Problem: AdS vacua can arise when compactifying SM to lower dimensions(they might be stable if 4d bubbles are too big to be transferred to 3d)

Compactifications of the Standard Model
Ooguri-Vafa conjecture: non-susy stable AdS vacua are not consistent with QG

Problem: AdS vacua can arise when compactifying SM to lower dimensions(they might be stable if 4d bubbles are too big to be transferred to 3d)

Solution:

Avoid presenceof AdS vacua

Constraints on light
spectra of SM!
(Explain SM hierarchies!)

Upper bound on neutrino masses in terms of the cosmological constantXi

m4⌫i . O(⇤4 )(Upper bound on EW scale!) [Ibanez,Martin-Lozano,IV’17]

Constraints on BSM spectra [Ibanez,Martin-Lozano,IV’17]
SM by itself ruled out(see also [Hamada-Shiu’17])

MSSM survives [Gonzalo,Herraez,Ibanez’18]

Naturalness?
⇤1/4
Consistency with quantum gravity requires hHi . 1.6 Y⌫1[Ibanez,Martin-Lozano,IV’17](recall: sufficient but not necessary condition)

Parameters leading to a higher
EW scale do not yield theories
consistent with quantum gravity
No EW hierarchy problem

Y = 10 14

New approach to fine-tuning/hierarchy problems?
UV/IR mixing from quantum gravity?

Naturalness might not be a good principle, not everything goes!

Quantum Gravity Conjectures
Motivated by observing recurrent features of the string landscapeas well as black hole physics
Absence of global symmetries [Banks-Dixon’88][Horowitz,Strominger,Seiberg…]
Completeness hypothesis [Polchinski.’03]
Weak Gravity Conjecture [Arkani-Hamed et al.’06]
Swampland Distance Conjecture[Ooguri-Vafa’06]

No stable non-susy AdS vacua [Ooguri-Vafa’16]｝

Large field inflation

Particle physics and c.c.

No deSitter vacuum?

Very important to gather more evidence to prove (or disprove) themdoable task

Quantum Gravity Conjectures
Motivated by observing recurrent features of the string landscapeas well as black hole physics
Absence of global symmetries [Banks-Dixon’88][Horowitz,Strominger,Seiberg…]
Completeness hypothesis [Polchinski.’03]
Weak Gravity Conjecture [Arkani-Hamed et al.’06]
Swampland Distance Conjecture[Ooguri-Vafa’06]

No stable non-susy AdS vacua [Ooguri-Vafa’16]｝

Large field inflation

Particle physics and c.c.

No deSitter vacuum?

Very important to gather more evidence to prove (or disprove) themdoable task

Swampland Distance Conjecture
Definition and implications
Test in the complex structure moduli
space of Type IIB CY compactifications
General insights

Swampland Distance Conjecture
Definition and implications
Test in the complex structure moduli
space of Type IIB CY compactifications
General insights

Swampland Distance Conjecture[Ooguri-Vafa’06]

An effective theory is valid only for a finite scalar field variation
because an infinite tower of states become exponentially lightm ⇠ m0 e

when!1Consider the moduli space of an effective theory:
L = gij ( )@ i @ j

scalar manifold= geodesic distance
between P and Q

Q
P

1
2

m(P ) . m(Q)eSwampland Distance Conjecture[Ooguri-Vafa’06]

An effective theory is valid only for a finite scalar field variation
because an infinite tower of states become exponentially lightm ⇠ m0 e

when!1This signals the breakdown of the effective theory:⇤cut-o↵ ⇠ ⇤0 exp()

E
⇤QF T2⇤QF T⇤QG

Swampland Distance Conjecture
It gives an upper bound on the scalar field range
described by any effective field theory with finite cut-off
Phenomenological implications:.

Cosmological relaxation of the EW scale

string theory upon taking into account back-reaction on kinetic term)

Examples compatible with the Refined SDC:

Caveats!

log

Mp
E

◆

Large field inflation

It applies to any scalar (also axions realising axion monodromy in IIB

⇠1 !

1

✓

. O(1)Mp[Ooguri-Vafa’06][Klaewer,Palti'16]

non-geodesics?
mass hierarchies?

[Baume,Palti'16][I.V.,’16]

Swampland Distance Conjecture
Evidence: based on particular examples in string theory compactifications[Ooguri,Vafa’06] [Baume,Palti'16] [I.V.,’16] [Bielleman,Ibanez,Pedro,I.V.,Wieck’16] [Blumenhagen,I.V.,Wolf’17][Hebecker,Henkenjohann,Witkowski’17] [Cicoli,Ciupke,Mayhrofer,Shukla’18][Blumenhagen et al.’18]• Model-independent understanding missing…
• Very little is known about the tower of particles…
• What is the underlying QG obstruction?

Swampland Distance Conjecture
Evidence: based on particular examples in string theory compactifications[Ooguri,Vafa’06] [Baume,Palti'16] [I.V.,’16] [Bielleman,Ibanez,Pedro,I.V.,Wieck’16] [Blumenhagen,I.V.,Wolf’17][Hebecker,Henkenjohann,Witkowski’17] [Cicoli,Ciupke,Mayhrofer,Shukla’18][Blumenhagen et al.’18]• Model-independent understanding missing…
• Very little is known about the tower of particles…
• What is the underlying QG obstruction?

[Grimm,Palti,IV.’18]

Focus: Complex structure moduli space of IIB CY3 compactifications(4d N=2 string theory moduli space preserving special Kahler geometry)natural testing ground for QG

Aim: Prove the conjecture for any infinite distance path!

Swampland Distance Conjecture
Definition and implications
Test in the complex structure moduli
space of Type IIB CY compactifications
General insights

Complex structure moduli space of IIB CY compactifications
Prime example of a field space capturing information about ‘quantum gravity’Q
singular locus

T

P

Infinite geodesic distances can occur
only if approaching a singularity

Massless BPS states (wrapping D3-branes) arise at the singularities
Candidates for SDC tower!
Two types:- Infinite distance singularities: any trajectory approaching P has infinite length- Finite distance singularities: at least one trajectory approaching P has finite lengthAim: Identify infinite tower of exponentially massless BPS states at
any infinite distance singularity

Infinite distanceInfinite tower ofmassless states

Monodromy ofinfinite order• Nilpotent
orbit theorem• Schmidtheorem
globalsymmetry(wrapping D3-branes)• BPS stability

emergence from
integrating out[Grimm,Palti,IV.’18]

Focus on points belonging to a single singular divisor ✓Nilpotent orbit theorem• Distances given by:

Z pd (P, Q) =gIJ ẋI ẋJ ds

gI J¯ = @zI @z̄J K✓
Z
K = log
iD

YD

¯
⌦^⌦

◆

Q
singular locus

TP

Im(t) ! 1

t=

1
log z
2⇡i

Nilpotent orbit theorem• Distances given by:

Z pd (P, Q) =gIJ ẋI ẋJ ds• Periods of the (D,0)-form:⇧I =transform under monodromyZ⌦

gI J¯ = @zI @z̄J K✓
Z
K = logiDI⇧(e2⇡iz ) = T · ⇧(z)(remnant of higher dimensional gauge symmetries)

YD

¯
⌦^⌦

◆

Q
singular locus

TP

Im(t) ! 1

t=

1
log z
2⇡i

Nilpotent orbit theorem• Distances given by:

Z pd (P, Q) =gIJ ẋI ẋJ ds• Periods of the (D,0)-form:⇧I =transform under monodromyZ⌦I⇧(e2⇡iz ) = T · ⇧(z)(remnant of higher dimensional gauge symmetries)• Define nilpotent matrix N = log T

gI J¯ = @zI @z̄J K✓
Z
K = log
iD

YD

¯
⌦^⌦

◆

Q
singular locus

TP

Im(t) ! 1

t=

1
log z
2⇡i

(only non-zero if monodromy T is of infinite order)(no k s.t. T k = T )Nilpotent orbit theorem• Distances given by:

Z pd (P, Q) =gIJ ẋI ẋJ ds• Periods of the (D,0)-form:⇧I =transform under monodromyZ⌦

gI J¯ = @zI @z̄J K✓
Z
K = logiDQIsingular locus⇧(e2⇡iz ) = T · ⇧(z)(remnant of higher dimensional gauge symmetries)• Define nilpotent matrix N = log T

Nilpotent orbit theorem:

YD

¯
⌦^⌦

T

P

Im(t) ! 1

t=

1
log z
2⇡i

(only non-zero if monodromy T is of infinite order)(no k s.t. T k = T )[Schmid’73]⇧(t, ⌘) = exp (tN ) a0 (⌘) + O(e

2⇡it

◆

, ⌘)

d
gtt̄ =
+ ...
2
Im(t)

It gives local expression for the periods near singular locus!
⇧(t, ⌘) = (1 + tN + · · · + td N d )a0 (⌘) + O(e2⇡it,⌘ )Infinite distances - Infinite states
1) Infinite distances only if monodromy is of infinite order
Theorem:[Wang’97, Lee’16]

P is at infinite distanced (P, Q) =pd log(Im t)|P
Q !1

N a0 6= 0

Infinite distances - Infinite states
1) Infinite distances only if monodromy is of infinite order
Theorem:[Wang’97, Lee’16]

P is at infinite distanced (P, Q) =p

N a0 6= 0

d log(Im t)|P
Q !1

2) Monodromy can be used to populate an infinite orbit of BPS states
Mass given by central charge:
qm
..
.
q1
q0

K

Z =e q·⇧qm = T m q

If T is of infinite orderI mq = (qe , qI )m2Z

Starting with one state, we generate infinitely many!

Infinite distances - Infinite states
1) Infinite distances only if monodromy is of infinite order
Theorem:[Wang’97, Lee’16]

P is at infinite distanced (P, Q) =p

N a0 6= 0

d log(Im t)|P
Q !1

2) Monodromy can be used to populate an infinite orbit of BPS states
Mass given by central charge:
qm
..
.
q1
q0

Z =e q·⇧qm = T m q

If T is of infinite orderI mq = (qe , qI )K

m2Z

Starting with one state, we generate infinitely many!

3) Universal local form of the metric gives the exponential mass behaviour
Z=

d
1 X

td/2 j=1(q · N j a0 )tj + O(e2⇡it )Massless: q T N j a0 = 0 ,jd/2

Mq (P )' exp
Mq (Q)✓

1
p d (P, Q)
2d◆

Infinite distances - Infinite states
Infinite massless monodromy
orbit at the singularity

Infinite tower of states becoming
exponentially light
Swampland Distance Conjecture ✓Swampland Distance Conjecture (SDC) is reduced to prove the existence ofan infinite massless monodromy orbit at the singularity

9 q s.t. q T N j a0 = 0 ,N q 6= 0j

d/2(Massless)(Infinite orbit)

Tool: mathematical machinery of mixed hodge structure(introduce finer split of cohomology at the singularity adapted to N)[Deligne][Schmid][Cattani,Kaplan,Schmid][Kerr,Pearlstein,Robles’17](Subtleties regarding stability of BPS states: need to mod out states with q T N j a0 = 0 , 8j )

BPS states and stability
How many BPS states are when approaching singularity?
Do they cross a wall of marginal stability upon circling the monodromy locus?
Consider: qC = qB + qĀ

MqC  MqB + MqĀWall of marginal stability: ' (B)' (A) = 1 with ' (A) =Upon circling the monodromy locus n times:'I ! 'I

n
+O
⇡Im t

1
(Im t)

2!

1
Im log ZqA⇡..
unstableT n q0stableT q0..

n ⇠ Im t

q0

n
q
=
T
q0 with n ⌧ Im t are stable (grade does not change)
Charge states(states higher up in the tower are unstable)

Number of BPS states: n ⇠ Im(t) ⇠ ed (P,Q)(grows when approaching the
singularity and diverges there)Swampland Distance Conjecture
Definition and implications
Test in the complex structure moduli
space of Type IIB CY compactifications
General insights

Global symmetries
SDC as a quantum gravity obstruction to restore a global continuous
axionic shift symmetry at the singular point
K=

log[ pd (Im t) + O(e2⇡it ) ]At infinite distance singularities: Re t ! Re t + c, c 2 R

when Im t ! 1

Re t = axion with decay constant f 2 = gtt̄ ! 0(also, gauge coupling of dual 2-form gauge field goes to zero)
analogous to WGCGlobal symmetries
SDC = Magnetic Scalar WGC• Magnetic version:
WGC:⇤ < gMp

If g ! 0 global symmetry is restored

How small can the gauge coupling be?
SDC:⇤ ⇠ Mp exp()

If

! 1 global symmetry is restored

How large can the field variation be?

• Electric version:
chargem2｝｝

g ij (@i m) (@j m) Mp2

mass

satisfied for long distance if mass is
exponential in [Palti’17]

Emergence from integrating out the states
These moduli spaces are‘quantum in nature’geometry incorporates information
about integrating out BPS states

Conifold singularity: log-divergence of gauge coupling from integrating[Strominger'95]
out a single BPS D3-state

Similar computation at infinite distance singularities! [Grimm,Palti,IV.’18]

One-loop corrections from integrating out the tower of BPS statesmatches geometric result (generates the log. infinite distance)
At infinite distancesingularities:dgtt̄ =
+ ...
2
Im(t)
d (P, Q) =

Z P
Q

p

p

2
gtt̄ |dt| ⇠log(Im t)|PQ
2

Emergence from integrating out the states
Original theory:

light field

｝

1
1
2
2
2
(@ ) +(@hi ) + mi ( ) h2i
2
2
2
i=1

｝

L=

S 
X
1

integrate
them out!

tower of massive fields
mk = m0 + k m

We have to integrate out the tower of particles up to the UV cut-off of the original theory!
UV cut-off=

Species bound

⇤U V
S=
m( )
m( ) ⇠

｝

d/2

Mp
p
⇤U V =
S[Dvali’07]

number of fields below ⇤U V
⇤U V ( ) ⇠m( )1/3 ⇠d/6

Field dependent UV cut-off!

At the singularity: S ! 1 ) ⇤U V ! 0(growth of S matches withstability of BPS states!)

Emergence from integrating out the states
Original theory:

light field

Quantum correction to the field metric
g

/

S
X

k=1

2

(@ mk ) ⇠d2

tower of massive fieldsnmk = m0 + k m

integrate
them out!

matches geometric result✓ ◆p2d( 1 , 2 ) ⇠ d log

Quantum correction to the gauge kinetic function
gY2 M ⇠

｝

1
1
2
2
2
(@ ) +(@hi ) + mi ( ) h2i
2
2
2
i=1

｝

L=

S 
X
1

1

matches geometric result

⇠ m2n
0

UV cut-off decreases exponentially fast in the proper field distance ⇤ ⇠ Mp eeffective theory completely breaks down SDC!

✓

d( 1 , 2 )Emergence from integrating out the states
Infinite distance and weak coupling emerge from integrating out an infinite tower of states!
Can this be a general feature for any moduli space?

SDC
Infinite distanceInfinite states(specific structure)

Emergence?

Emergence from integrating out the states
Infinite distance and weak coupling emerge from integrating out an infinite tower of states!
Can this be a general feature for any moduli space?

SDC
Infinite distanceComments:

Infinite states(specific structure)

Emergence?

m( ) does not matter much as long as S( ) 6= const. is given by species✓ ◆bound of a tower of particles and @ m
1(see also [Heidenreich,Reece,Rudelius’18])

m

&O

when

!1

Emergence from integrating out the states
Infinite distance and weak coupling emerge from integrating out an infinite tower of states!
Can this be a general feature for any moduli space?
This limits correspond to restoring a continuous global symmetry, so global
symmetries would also be emergent from integrating out infinitely many states!
(emergence is continuous)
Mp⇤U V = pS0when S ! 1

unlessMp ! 1

Global symmetries only possible if gravity decouples

Summary
Swampland Distance Conjecture:
Upper bound on the scalar field range: Implications for inflation!

✓ Test in the complex structure moduli space of CY IIB compactifications• Infinite order monodromy as generator of the infinite tower• Emergence of infinite field distance and global symmetry
Generalizations:
Our results are valid for any CY (model-independent)(but only for infinite distance points that belong to a single singular divisor)

Other moduli spaces?

Summary
Consistency with quantum gravity implies constraints on low energy
physics.
Knowledge of Swampland is essential for UV sensitive
theories and might also be important for naturalness issues.

Very important to gather more evidence to prove or disprove theconjectures
doable and exciting task!

Summary
Consistency with quantum gravity implies constraints on low energy
physics.
Knowledge of Swampland is essential for UV sensitive
theories and might also be important for naturalness issues.

Very important to gather more evidence to prove or disprove theconjectures
doable and exciting task!

Thank you!

back-up slides

Infinite tower of states
Tool: mathematical machinery of mixed hodge structure
Problem: ‘Normal’ Hodge decomposition no longer useful whenapproaching a singularity H 3 (Y3 , C) = H 3,0 H 2,1

H 1,2

H 0,3

Idea: Introduce finer split of cohomology at the singularity adapted to thisX‘limiting’ Hodge decomposition
p,3 p
h
=
dim I p,qq

H 3,0 !

{I 3,3 , I 3,2 , I 3,1 , I 3,0 }[Deligne][Schmid][Cattani,Kaplan,Schmid][Kerr,Pearlstein,Robles’17]

The subspaces capture non-trivial information about the nilpotent monodromy
operator,
N I p,q ⇢ I p 1,q 1

For a CY:

a0 2 I 3,d ,d = 0, 1, 2, 3

Quantum Gravity Conjectures[Ooguri-Vafa’06][Baume,Klaewer,Palti’16][Arkani-Hamed et al.’06]

Weak Gravity
Conjecture

Swampland Distance
Conjecture
Applied toaxionsApplied toscalars

Applied tofluxes[Ooguri-Vafa’16][Freivogel-Kleban’16]

Scalar field range⇤cut-o↵ ⇠ ⇤0 exp()

Axionic
decay constantNon-susy AdS
vacua are unstablef < MpLarge field inflation
Cosmologicalrelaxation

No dual
CFT

Cosmologicalconstant and SM
neutrinosCasimir energy
Potential energy in 3d:2⇡r3 ⇤4 X
r3si
V (R) =
+
(2⇡R)
(
1)
ni ⇢i (R)
2
3
R
R
i

Casimir energy density:⇢(R) = ⌥1X
2m4 K2 (2⇡Rmn)

2 (2⇡Rmn)2
(2⇡)
n=1

For small mR:⇡2⇢(R) = ⌥90(2⇡R)4

2
⇡2
⇡
2
4
6
(mR)
+
(mR)
+
O(mR)
6(2⇡R)448(2⇡R)4

Compactifications of the Standard Model
Standard Model + Gravity on S 1:
2⇡⇤4
V (R) =
R2

tree-level[Arkani-Hamed et al.’07](also [Arnold-Fornal-Wise’10])+ Casimir energy
one-loop corrections

exponentially suppressed
for m 1/R

We can get AdS, Minkowski or dS vacua
in lower dimensions!
(depending on light spectra of SM and value of c.c.)Compactifications of the Standard Model

Problem:

AdS minimum appears if:• Neutrinos are MajoranaX4m⌫ & O(⇤4 )• Neutrinos have a Dirac mass:ii

For our c.c:m⌫1 > 7.7 meV (NH)m⌫1 > 2.1 meV (IH)

[Ibanez,Martin-Lozano,IV’17]

Assuming Ooguri-Vafa AdS conjecture + background independence:
We should not get stable non-susy AdS vacua from compactifying the SM !!!

Compactifications of the Standard Model
Solution:[Ibanez,Martin-Lozano,IV’17]

Make the minimum unstable

Impose absence of AdS vacua

Include Wilson Lines to generate
a runaway for small R [Hamada-Shiu’17]

Constraints on light spectra of SM!

not valid in orbifolds

Upper bound on neutrino masses interms of the cosmological constant

Assume existence of 4d bubble
instability which is transferred
to lower dimensions
not valid if Rbuble > lAdS3⇤4 & O(m4⌫ )(Upper bound on EW scale!)
SM by itself ruled out[Gonzalo,Herraez,Ibanez’18]

MSSM
survivesCompactification of the SM to 3d
1

Standard Model + Gravity on S :2⇡⇤4
V (R) =
R2

graviton, photon

massive particles:
neutrinos,…

X (2⇡R)4si+(
1)
ni ⇢i (R)
3
6
R
720⇡R
i

Dirac
neutrinos
Majorana
neutrinos

The more massive the neutrinos, the deeper the AdS vacuumLower bound on the cosmological constant

The bound for ⇤4 scales as m4⌫⇤4a(nf )30(⌃m2i )2 b(nf , mi )⌃m4i
384⇡ 2(as observed experimentally)witha(nf ) = 0.184(0.009)
b(nf , mi ) = 5.72(0.29)

for Majorana (Dirac)

First argument (not based on cosmology) to have ⇤4 6= 0

Adding BSM physics
Light fermions
Positive Casimir contribution

helps to avoid AdS vacuum

Majorana neutrinos are consistent if adding m . 2 meVexample. For m = 0.1 meV :Adding BSM physics
Axions1 axion: negative contribution

bounds get stronger

Multiple axions: can destabilise AdS vacuum

Bounds on the SM + light BSM physics
Model
SM (3D)
SM(2D)
SM+Weyl(3D)Majorana (IH)nonom⌫3  3 ⇥ 10 3
mf  4 ⇥ 10 3m⌫3  1 ⇥ 10 3
mf  2 ⇥ 10 3
mf  1 ⇥ 10 2
mf  0.9 ⇥ 10 2
noDirac (NH)m⌫1  7.7 ⇥ 10 3m⌫1  4.12 ⇥ 10 3m⌫1  1.5 ⇥ 10 2

Dirac (IH)m⌫3  2.56 ⇥ 10 3m⌫3  1.0 ⇥ 10 3m⌫3  1.2 ⇥ 10 2m⌫1  0.9 ⇥ 10 2m⌫3  0.7 ⇥ 10 2

SM+Dirac(3D)
SM+Dirac(2D)SM+1 axion(3D)Majorana (NI)nonom⌫1  0.9 ⇥ 10 2
mf  1.2 ⇥ 10 2m⌫1  0.5 ⇥ 10 2
mf  0.4 ⇥ 10 2
mf  2 ⇥ 10 2
mf  0.9 ⇥ 10 2
no

yes
yes
m⌫1  7.7 ⇥ 10 3

SM+1 axion(2D)nonom⌫1  4.0 ⇥ 10 3

2(10) axions

yes

yes

yes

yes
yes
m⌫3  2.5 ⇥ 10 3
ma 5 ⇥ 10 2m⌫3  1 ⇥ 10 3
ma 2 ⇥ 10 2
yes

SM+Weyl(2D)Compactifications of SM on T2(see also [Hamada-Shiu’17])

qualitatively similar,
but a bit stronger