Exponentially Suppressed Cosmological Constant with Gauge Enhanced - - PowerPoint PPT Presentation

Exponentially Suppressed Cosmological Constant with Gauge Enhanced Symmetry in Heterotic Interpolating Models

Sota Nakajima (Osaka City University) with Hiroshi Itoyama (OCU, NITEP)

@ YITP, 8/2, 2019

Based on arXiv: 1905.10745

When a top-down approach from string theory is considered, there are two choices depending on where SUSY breaking scale is ;

1.SUSY is broken at low energy in supersymmetric EFT ; 2.SUSY is already broken at high energy like string/Planck scale. In this talk, the second one is focused on, and non-supersymmetric string models are considered. In particular, the 𝑻𝑷 𝟐𝟕 × 𝑻𝑷 𝟐𝟕 model is a unique tachyon-free non-supersymmetric string model in ten-dimensions.

[Dixon, Hervey, (1986)]

Int ntroduc uction

Int ntroduc uction

Considering non-supersymmetric string models, however, we face with the problem of vacuum instability arising from nonzero dilaton tadpoles;

𝑊(𝜚) ∝ Λ

𝑊 𝜚 : dilaton tadpole Λ: cosmological constant (vacuum energy)

𝜚

∝

At 1-loop level, The desired model is a non-supersymmetric one whose cosmolosical constant is vanishing or as small as possible. Interpolating models have the possibility of such properties.

[Itoyama, Taylor, (1987)]

Ou Outlin ine

• 1. Introduction
• 2. Heterotic Strings
• 3. 9D Interpolating models
• 4. 9D Interpolating models with Wilson line
• 5. Summary

• 1. Introduction
• 2. Heterotic Strings
• 3. 9D Interpolating models
• 4. 9D Interpolating models with Wilson line
• 5. Summary

Ou Outlin ine

Ide dea a of

• f Het

eter erot

• tic

ic S Strin ings

Left mover: 26d bosonic string out of which internal 16d realize rank 16 current algebra Adopting the lightcone coordinates, the worldsheet contents are Right mover: 10d superstring

L R

10=(8+2)d 16d

• n torus

10=(8+2)d Heterotic strings are hybrid closed strings of bosonic string in 26D and superstrings in 10D.

[Gross, Hervey, Martinec, Rohm, (1985)]

The he o

• ne

ne-loo

• op

p pa partit itio ion n fu func

• nc. &

& St Stat ate e Co Coun untin ing

 The one-loop partition function is the trace over string Fock space:  𝑎 𝜐 counts #(states) at each mass level as coeff. in 𝑟 ത 𝑟 expansion.

𝒃𝒏𝒐 denotes #(bosons) minus #(fermions) at mass levels (𝒏, 𝒐)

In the string model with spacetime SUSY, 𝑏𝑛𝑜 = 0 for all (𝑛, 𝑜) because of fermion-boson degeneracy. for supersymmetric string models.  In order for the string model to be consistent, 𝑎(𝜐) has to be invariant under modular transformation:

 𝑎(𝜐) is written in terms of 𝑇𝑃 2𝑜 characters 𝑃2𝑜, 𝑊

2𝑜, 𝑇2𝑜, 𝐷2𝑜 and the

Dedekind eta function 𝜃(𝜐), e.g,

Cha Charac acter ers

𝑇𝑃 32 hetero: 𝑇𝑃 16 × 𝑇𝑃(16) hetero: 𝐹8 × 𝐹8 hetero: , , the Jacobi’s abstruse identity:

SU SUSY SY br brea eaki king ng by by Co Comp mpac actif ific icat atio ion

 Compactification on a circle  Compactification on a twisted circle 𝑌9

2𝜌𝑆 The translation operator for 𝑌9 satifies

identify

identify with 𝟑𝝆 rot.

• n the 7-8 plane

𝑌9

2𝜌𝑆

𝑌7 𝑌8

The translation operator for 𝑌9 satifies This comp. affects bosonic and fermionic states in the same way. SUSY is NOT broken. This comp. affects bosonic and fermionic states in the different way. It induces the mass splitting between bosonic and fermionic states.

SUSY is broken

[Rohm, (1984)]

Ou Outlin ine

• 1. Introduction
• 2. Heterotic Strings
• 3. 9D Interpolating models
• 4. 9D Interpolating models with WL
• 5. Summary

Interp rpolat ation betwe ween SU SUSY SY an and non-SU SUSY Y mo models

Model 𝑵𝟐 Model 𝑵𝟑

• Comp. on a

twisted circle T-dual

Interpolating model

Radius 𝑆

∞

10 dim. 9 dim. SUSY breaking SUSY non-SUSY non-SUSY

In the large 𝑆 (small 𝑏) region, the cosmological constant is

[Itoyama, Taylor, (1987)]

If 𝒐𝑮 = 𝒐𝑪, the cosmological constant is exponentially suppressed.

: # of massless fermions, bosons

An interpolating model is a lower dimensional string model relating two different higher dimensional string models continuously.

Interp rpolat ation betwe ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕)

 The one-loop partition function where the sum is taken over

• 𝑆 → ∞: contribution from the zero

winding # only

• 𝑆 → 0: contribution from the zero

momentum only

 The limiting case: 𝑺 → ∞

the one-loop partition function of SUSY 𝑇𝑃(32) heterotic model, which is vanishing

SUSY is restored in 𝑺 → ∞ (𝒃 → 𝟏)

Interp rpolat ation betwe ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕)

 The limiting case: 𝑺 → 𝟏

the one-loop partition function of 𝑇𝑃(16) × 𝑇𝑃(16) heterotic model realizes SUSY 𝑇𝑃(32) model in 𝑆 → ∞ 𝑇𝑃(16) × 𝑇𝑃(16) model in 𝑆 → 0

Interp rpolat ation betwe ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕)

𝑻𝑷(𝟒𝟑) 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟕) Radius 𝑆

∞

SUSY non-SUSY

non-SUSY

9D Int. model

 Massless spectrum at generic R, massless states come from n=w=0 part

Massless bosons Massless fermions

• 9-dim. graviton, anti-symmetric tensor, dilaton:
• Fermions
• Gauge bosons in adj rep of 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟕) × 𝑽𝑯,𝑪

𝟑 (𝟐)

𝑕9𝜈, 𝐶9𝜈

Interp rpolat ation betwe ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕)

• 1. Introduction
• 2. Heterotic Strings
• 3. 9D Interpolating models
• 4. 9D Interpolating models with Wilson line
• 5. Summary

Ou Outlin ine

• The 𝑒-dimensional compactifications are classified by the transformation

whose DOF agree with that of 𝐷

𝐵𝑏.

Bo Boost st on mo mome mentum um lat attice

𝑇𝑃(16+𝑒,𝑒) 𝑇𝑃 16+𝑒 ×𝑇𝑃(𝑒),

• Considering 𝑒-dimensional compactification, the boost in the momentum

lattice corresponds to putting massless constant backgrounds, that is, adding the following term to the worldsheet action 𝐷𝑐𝑏: metric and antisymmetric tensor, 𝐷𝐽𝑏: 𝑉(1)16 gauge fields (WL)

[Narain, Sarmadi, Witten, (1986)] 𝑏 = 10 − 𝑒, ⋯ , 9 𝐵 = 𝑏, 𝐽 = 10 − 𝑒, ⋯ , 26

• In this work, we will consider one-dimensional compactification and put a single

WL background 𝐵 = 𝐷𝐽=1,𝑏=9 for simplicity.

Bo Boost st on mo mome mentum um lat attice

boost and rotation

The effective change in the 1-loop partition function is

𝑚𝑀 is the left-moving momentum of 𝑌𝑀

𝐽=1

After turning on WL, the momenta of 𝑌𝑀

𝐽=1, 𝑌𝑀 𝑏=9 and 𝑌𝑆 𝑏=9 are changed as

introduction

• f WL

Th The fu fundam amental al re region of f mo modul uli sp spac ace

Do all the points in moduli space correspond to different models? It is convenient to introduce a modular parameter ǁ 𝜐 as NO! Momentum lattice Λ(𝛿,𝜀)

(𝛽,𝛾) is invariant under the shift

The fundamental region of moduli space is ǁ 𝜐1 ǁ 𝜐2

− 2 2

 The one-loop partition function

• 𝑆 → ∞:
• 𝑆 → 0:

Interp rpolat atio ion betwe ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

 The one-loop partition function

Interp rpolat atio ion betwe ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

 The limiting cases

For any WL 𝐵, realizes SUSY 𝑇𝑃 32 model in 𝑆 → ∞ 𝑇𝑃(16) × 𝑇𝑃(16) model in 𝑆 → 0

• 𝑆 → ∞: the 1st and 2nd lines survive
• 𝑆 → 0: the 1st and 3rd lines survive

𝑻𝑷(𝟒𝟑) 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟕) Radius 𝑆

∞

SUSY non-SUSY

non-SUSY

+ WL 𝑩 9D Int. model

 Massless spectrum

Massless bosons Massless fermions

• 9-dim. graviton, anti-symmetric tensor, dilaton:
• Gauge bosons in adj rep of 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟓) × 𝑽(𝟐) × 𝑽𝑯,𝑪

𝟑 (𝟐)

• 8

at generic R, massless states come from n=w=0 part

Interp rpolat atio ion betwe ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

 Massless spectrum

• two
• two

new massless states：

𝑻𝑷 𝟐𝟕 × 𝑻𝑷 𝟐𝟓 × 𝑽(𝟐) 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕)

condition ①

a few conditions under which the additional massless states appear

∃

Interp rpolat atio ion betwe ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

 Massless spectrum

• two
• two

new massless states：

𝑻𝑷 𝟐𝟕 × 𝑻𝑷 𝟐𝟓 × 𝑽(𝟐) 𝑻𝑷 𝟐𝟗 × 𝑻𝑷(𝟐𝟓)

condition ②

a few conditions under which the additional massless states appear

∃

Interp rpolat atio ion betwe ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

Actually, there are only four inequivalent orbits in the fundamental region: We have found the two conditions under which the additional massless states appear:

 Summary of the conditions

Interp rpolat atio ion betwe ween 𝑻𝑷 𝟒𝟑 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

condition ① condition ② Condition

𝒐1 = 𝟏 and 𝟑 (or −𝟑) 𝒐𝟑 = −𝟐 and 𝟐

Gauge gp

𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟕) 𝑻𝑷(𝟐𝟗) × 𝑻𝑷(𝟐𝟓) 𝒐𝑮 > 𝒐𝑪 𝒐𝑮 = 𝒐𝑪

 The one-loop partition function

Interp rpolat atio ion betwe ween 𝑭𝟗 × 𝑭𝟗 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

• 𝑆 → ∞:
• 𝑆 → 0:

 The one-loop partition function

Interp rpolat atio ion betwe ween 𝑭𝟗 × 𝑭𝟗 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

 The limiting cases

For any WL 𝐵, realizes SUSY 𝐹8 × 𝐹8 model in 𝑆 → ∞ 𝑇𝑃(16) × 𝑇𝑃(16) model in 𝑆 → 0

• 𝑆 → ∞: the 1st and 2nd lines survive
• 𝑆 → 0: the 1st and 3rd lines survive

𝑭𝟗 × 𝑭𝟗 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟕) Radius 𝑆

∞

SUSY non-SUSY

non-SUSY

+ WL 𝑩 9D Int. model

 Massless spectrum at generic R, massless states come from n=w=0 part

Massless bosons Massless fermions

• 9-dim. graviton, anti-symmetric tensor, dilaton:
• Gauge bosons in adj rep of 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟓) × 𝑽(𝟐) × 𝑽𝑯,𝑪

𝟑 (𝟐)

• 8

Interp rpolat atio ion betwe ween 𝑭𝟗 × 𝑭𝟗 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

 Massless spectrum

condition ①

• two

new massless states：

𝑻𝑷 𝟐𝟕 × 𝑻𝑷 𝟐𝟓 × 𝑽(𝟐) 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) Furthermore, the different additional massless states appear depending on whether 𝒐𝟐/𝟑 𝐣𝐭 𝐟𝐰𝐟𝐨 𝐩𝐬 𝐩𝐞𝐞.

a few conditions under which the additional massless states appear

∃

Interp rpolat atio ion betwe ween 𝑭𝟗 × 𝑭𝟗 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

 Massless spectrum

• two

new massless states：

condition ①-1

• two

In the fundamental region, this condition is ෤ 𝝊𝟐 = 𝟏, which corresponds to the no WL case.

a few conditions under which the additional massless states appear

∃

Interp rpolat atio ion betwe ween 𝑭𝟗 × 𝑭𝟗 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

 Massless spectrum

• two

new massless states：

condition ①-2

• two

In the fundamental region, this condition is ෤ 𝝊𝟐 = 𝟑 (or ෤ 𝝊𝟐 = − 𝟑). 𝑻𝑷 𝟐𝟕 × 𝑻𝑷 𝟐𝟓 × 𝑽(𝟐) 𝑻𝑷 𝟐𝟕 × 𝑭𝟗

a few conditions under which the additional massless states appear

∃

Interp rpolat atio ion betwe ween 𝑭𝟗 × 𝑭𝟗 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

 Massless spectrum new massless states：

condition ②

a few conditions under which the additional massless states appear

∃

• two

Gauge group is not enhanced

Interp rpolat atio ion betwe ween 𝑭𝟗 × 𝑭𝟗 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

In the fundamental region, this condition is ෤ 𝝊𝟐 = 𝟑/𝟑 and ෤ 𝝊𝟐 = − 𝟑/𝟑.

There is no condition such that 𝒐𝑮 = 𝒐𝑪 in this example.

condition ①-2 condition ①-1 condition ②

Actually, there are only four inequivalent orbits in the fundamental region:

Interp rpolat atio ion betwe ween 𝑭𝟗 × 𝑭𝟗 an and 𝑻𝑷 𝟐𝟕 × 𝑻𝑷(𝟐𝟕) wi with WL

Condition 𝒐𝟐 = 𝟏 𝒐𝟐 = 𝟑 (or −𝟑) 𝒐𝟑 = 𝟐 and −𝟐 Gauge gp 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟕) 𝑻𝑷(𝟐𝟕) × 𝑭𝟗 𝑻𝑷(𝟐𝟕) × 𝑻𝑷(𝟐𝟓) × 𝑽(𝟐) 𝒐𝑮 > 𝒐𝑪 𝒐𝑮 < 𝒐𝑪 𝒐𝑮 < 𝒐𝑪

We have found the three conditions under which the additional massless states appear:

 Summary of the conditions

Th The lead ading term rms s of f the he cosm smological al const stan ant

The cosmological constant is written as Up to exponentially suppressed terms, the results are  𝑇𝑃 32 - 𝑇𝑃(16) × 𝑇𝑃(16) interpolation  𝐹8 × 𝐹8 - 𝑇𝑃(16) × 𝑇𝑃(16) interpolation These results reflect the shift symmetry ǁ 𝜐 → ǁ 𝜐 + 2 2 and the conditions under which the additional massless states appear.

• 1. Introduction
• 2. Heterotic Strings
• 3. 9D Interpolating models
• 4. 9D Interpolating models with Wilson line
• 5. Summary

Ou Outlin ine

Co Conclus usion

 We have constructed 9D interpolating models with two parameters, radius 𝑆 and WL 𝐵, and studied the massless spectra.  We have found some conditions for (𝑆, 𝐵) under which the additional massless states appear.  We have found that an example under which the cosmological const. is exponentially suppressed simultaneously with the gauge group enhancement to 𝑇𝑃 18 × 𝑇𝑃 14 .

Ou Outlook

How are SM-like or GUT-like 4D models with 𝑜𝐺 = 𝑜𝐶 constructed ? Are the conditions found in this work preferred ? Where are stable points in moduli space ? We can generalize by putting more WL and the other backgrounds. In fact, compactifying 𝑒-dimensions, the compactifications are classified by

𝑇𝑃(16+𝑒,𝑒) 𝑇𝑃(16+𝑒)×𝑇𝑃(𝑒), whose DOF is 𝑒(16 + 𝑒).

Th Than ank k yo you! u!