Can a Vanishingly Small Cosmological Constant Be Natural ?

Introduction Basic Idea The Large Volume Scenario in Type IIB String Theory Summary

Can a Vanishingly Small Cosmological Constant be Natural ?

Henry Tye 戴自海

Hong Kong University of Science and Technology Cornell University

February 20, 2014 KEK Theory Workshop, KEK, Japan

Henry Tye A Vanishingly Small Λ 1/41 Introduction Basic Idea The Large Volume Scenario in Type IIB String Theory Summary Hong Kong University of Science and Technology

Henry Tye A Vanishingly Small Λ 2/41 Introduction Basic Idea The Large Volume Scenario in Type IIB String Theory Summary Insitute for Advanced Study Building

Henry Tye A Vanishingly Small Λ 3/41 Introduction Basic Idea The Large Volume Scenario in Type IIB String Theory Summary

Coming Workshop:

New Perspectives on Cosmology

May 19-23, 2014

Henry Tye A Vanishingly Small Λ 4/41 Introduction Basic Idea The Large Volume Scenario in Type IIB String Theory Summary

This talk is based on work done with Yoske Sumitomo :

arXiv:1204.5177, arXiv:1209.5086, arXiv:1211.6858 and arXiv:1305.0753 (also with student Sam Wong)

Under preparation (with Sam Wong)

Henry Tye A Vanishingly Small Λ 5/41 Introduction Basic Idea The Large Volume Scenario in Type IIB String Theory Summary

Some of the works relevant to us : Bousso and Polchinski, hep-th/0004134 Kachru, Kallosh, Linde and Trivedi, hep-th/0301240 Balasubramanian, Berglund, Conlon and Quevedo, hep-th/0502058 Westphal, hep-th/0611332 Denef and Douglas, hep-th/0404116 Douglas and Kachru, hep-th/0610102 Becker, Becker, Haack and Louis, hep-th/0204254 Lust, Reﬀert, Schulgin, Stieberger, hep-th/0506090 Rummel and Westphal, arXiv:1107.2115 [hep-th] de Alwis and Givens, arXiv:1106.0759 [hep-th] Aazami and Easther, hep-th/051205 Chen, Shiu, Sumitomo and Tye, arXiv:1112.3338 [hep-th] Bachlechner, Marsh, McAllister and Wrase, arXiv:1207.2763 [hep-th] Blanco-Pillado, Gomez-Reino and Metallinos, arXiv:1209.0796 [hep-th] Martinez-Pedrera, Mehta, Rummel and Westphal, arXiv:1212.4530 [hep-th] Danielsson and Dibitetto, arXiv:1212.4984 [hep-th] Rummel and Sumitomo, arXiv:1310.4202 [hep-th]

Henry Tye A Vanishingly Small Λ 6/41 Introduction 10500 possible solutions with diﬀerent Λ values. Basic Idea Pressing Question The Large Volume Scenario in Type IIB String Theory The Stringy Mechanism Summary Puzzle

I There is very strong evidence that we are living in a de-Sitter vacuum with a positive cosmological constant Λ. I With c = 1 and ~ = 1, we can express Newton’s constant in 2 terms of Planck mass : GN = MP

122 4 Λ +10− M ∼ P I This vanishingly small Λ value poses a puzzle in physics.

Henry Tye A Vanishingly Small Λ 7/41 I Since Λ is calculable in string theory, string theory is the place to search for an explanation why it is so small.

MS MP , Λ →

I Our universe has probably gone through an inﬂationary period, when the vacuum energy is much higher than today’s value. So our universe has to evolve dynamically to this small value.

Introduction 10500 possible solutions with diﬀerent Λ values. Basic Idea Pressing Question The Large Volume Scenario in Type IIB String Theory The Stringy Mechanism Summary

I Since we can always introduce an arbitrary Λ into Einstein’s relativity theory, this very small value of Λ can either be obtained by ﬁne-tuning there, or explained by ”the anthropic principle”.

Henry Tye A Vanishingly Small Λ 8/41 I Our universe has probably gone through an inﬂationary period, when the vacuum energy is much higher than today’s value. So our universe has to evolve dynamically to this small value.

Introduction 10500 possible solutions with diﬀerent Λ values. Basic Idea Pressing Question The Large Volume Scenario in Type IIB String Theory The Stringy Mechanism Summary

I Since Λ is calculable in string theory, string theory is the place to search for an explanation why it is so small.

MS MP , Λ →

I Since Λ is calculable in string theory, string theory is the place to search for an explanation why it is so small.

MS MP , Λ →

I Our universe has probably gone through an inﬂationary period, when the vacuum energy is much higher than today’s value. So our universe has to evolve dynamically to this small value.

Henry Tye A Vanishingly Small Λ 8/41 Introduction 10500 possible solutions with different Λ values. Basic Idea Pressing Question The Large Volume Scenario in Type IIB String Theory The Stringy Mechanism Summary Bousso and Polchinski observed that fluxes in string theory are i quantized. E.g., J types of quantized 4-form fluxes Fµνρσ contribute to the Λ. 2.4 MultipleStabilization: Before four-forms After General compactifications actually give rise to several four-form fluxes, and this can solve the gap problem. Let there be J such fluxes, with

CC=0 CC = 0 1 J λ = λ + n2q2 . (2.21) bare 2 i i i=1 ! The question is whether there exists a set of ni such that [Bousso, Polchinski, 00] J 2 2 37 2 λ Henry Tye< A Vanishinglyn q Small Λ< 2(9/41 λ + ∆λ) , (2.22) | bare| i i | bare| i=1 ! 120 where ∆λSunday,corresponds 8 January, 12 to the observational bound, roughly 10− in Planck units. This can be visualized in terms of a J-dimensional grid of points, spaced by qi and labeled by ni (see Fig. 1). Consider a sphere of radius r = 2λ 1/2 centered at n =0.Ifoneofthepoints(n ,n ,...,n )is | bare| i 1 2 J sufficiently close to the sphere, the field configuration corresponding to this point will lead to an acceptable value of the cosmological constant. More precisely, one should think of a thin shell, whose width encodes the width of the observational range,

1/2 ∆r = 2λ − ∆λ . (2.23) | bare| We need at least one point to lie within the shell. As we will discuss, there may be large degeneracies — let the typical degeneracy be D.Thevolume J 1 per D grid points must then be less than the volume of the shell, ωJ 1r − ∆r, J/2 − where the area of a unit sphere is ωJ 1 =2π /Γ(J/2). Thus − J ωJ 1 J 1 q − 2λ 2 − ∆λ , (2.24) i ! D | bare| i=1 " or J D qi ∆λ 1 ! . (2.25) ω 2 2λ J 1 i=1 2λbare bare − " | | | | In other words, the typical spacing of the spectrum of the cosmological constant in a model with given J, ei,andλbare will be given by J D i=1 qi ∆λmin = . (2.26) J 1 ωJ 1 2λbare 2 − − |# | 9 Introduction 10500 possible solutions with diﬀerent Λ values. Basic Idea Pressing Question The Large Volume Scenario in Type IIB String Theory The Stringy Mechanism Summary Flux number can change via tunneling through a brane-ﬂux annihilation process.

Henry Tye A Vanishingly Small Λ 10/41 Why nature picks such a very small positive Λ ?

We argue that there may be a statistical preference for a very small (either positive or negative) Λ. We’ll illustrate with some examples in Type IIB string theory.

Introduction 10500 possible solutions with diﬀerent Λ values. Basic Idea Pressing Question The Large Volume Scenario in Type IIB String Theory The Stringy Mechanism Summary Pressing Question and our Proposal

String theory may have 10500 possible solutions. They live in the so called string theory (or cosmic) landscape. Surely some will have a Λ at about the right value, as proposed by Bousso and Polchinski.

Henry Tye A Vanishingly Small Λ 11/41 We argue that there may be a statistical preference for a very small (either positive or negative) Λ. We’ll illustrate with some examples in Type IIB string theory.

Why nature picks such a very small positive Λ ?

Henry Tye A Vanishingly Small Λ 11/41 Introduction 10500 possible solutions with diﬀerent Λ values. Basic Idea Pressing Question The Large Volume Scenario in Type IIB String Theory The Stringy Mechanism Summary Pressing Question and our Proposal

Why nature picks such a very small positive Λ ?

We argue that there may be a statistical preference for a very small (either positive or negative) Λ. We’ll illustrate with some examples in Type IIB string theory.

Henry Tye A Vanishingly Small Λ 11/41 I Solve V (nj , ui ) for all the meta-stable vacua. For every meta-stable vacuum with a given set nj , each ui is { } determined in terms of nj : ui min(nj ). { } , So Λ(nj ) = Vmin(nj , ui,min).

I Treat each nj as a parameter with some (typically smooth or { } uniform) probability distribution Pj (nj ). Find the probability distribution P(Λ) for Λ(nj ) as we sweep through allowed nj . { }

Introduction 10500 possible solutions with diﬀerent Λ values. Basic Idea Pressing Question The Large Volume Scenario in Type IIB String Theory The Stringy Mechanism Summary Approach in IIB

I Consider a string model with a set of moduli ui and 2-form { } fields C2 and B2. The 3-form fluxes F3 = dC2 and H3 = dB2 wrap cycles in a Calabi-Yau like manifold. The quantized fluxes lead to a set of discrete values labelled as nj , yielding { } V (B2, C2, ui ) V (nj , ui ), → where each nj can take a discretum of values.

Henry Tye A Vanishingly Small Λ 12/41 I Treat each nj as a parameter with some (typically smooth or { } uniform) probability distribution Pj (nj ). Find the probability distribution P(Λ) for Λ(nj ) as we sweep through allowed nj . { }

Introduction 10500 possible solutions with diﬀerent Λ values. Basic Idea Pressing Question The Large Volume Scenario in Type IIB String Theory The Stringy Mechanism Summary Approach in IIB

I Solve V (nj , ui ) for all the meta-stable vacua. For every meta-stable vacuum with a given set nj , each ui is { } determined in terms of nj : ui min(nj ). { } , So Λ(nj ) = Vmin(nj , ui,min).

Henry Tye A Vanishingly Small Λ 12/41 Introduction 10500 possible solutions with diﬀerent Λ values. Basic Idea Pressing Question The Large Volume Scenario in Type IIB String Theory The Stringy Mechanism Summary Approach in IIB

I Solve V (nj , ui ) for all the meta-stable vacua. For every meta-stable vacuum with a given set nj , each ui is { } determined in terms of nj : ui min(nj ). { } , So Λ(nj ) = Vmin(nj , ui,min).

I Treat each nj as a parameter with some (typically smooth or { } uniform) probability distribution Pj (nj ). Find the probability distribution P(Λ) for Λ(nj ) as we sweep through allowed nj . { } Henry Tye A Vanishingly Small Λ 12/41 The Basic Idea is very simple :

It is based on the properties of the probability distribution of functions of random variables.

Does Λ(nj ) has the right functional form ? Do the parameters nj have the right distribution ?

An example :

Consider a set of random variables xi (i = 1, 2, ..., n). Let the probability distribution of each xi be uniform in the range [ L, +L]. What is the probability distribution of their product z ? −

Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary Peaking behavior of P(Λ) at Λ = 0

As we shall see, P(Λ) tends to peak at Λ = 0.

Henry Tye A Vanishingly Small Λ 13/41 An example :

Consider a set of random variables xi (i = 1, 2, ..., n). Let the probability distribution of each xi be uniform in the range [ L, +L]. What is the probability distribution of their product z ? −

Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary Peaking behavior of P(Λ) at Λ = 0

As we shall see, P(Λ) tends to peak at Λ = 0.

The Basic Idea is very simple :

It is based on the properties of the probability distribution of functions of random variables.

Does Λ(nj ) has the right functional form ? Do the parameters nj have the right distribution ?

Henry Tye A Vanishingly Small Λ 13/41 Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary Peaking behavior of P(Λ) at Λ = 0

As we shall see, P(Λ) tends to peak at Λ = 0.

The Basic Idea is very simple :

It is based on the properties of the probability distribution of functions of random variables.

Does Λ(nj ) has the right functional form ? Do the parameters nj have the right distribution ?

An example :

Consider a set of random variables xi (i = 1, 2, ..., n). Let the probability distribution of each xi be uniform in the range [ L, +L]. What is the probability distribution of their product z ? −

Henry Tye A Vanishingly Small Λ 13/41 Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary

Probability distribution of z = x1x2 and z = x1x2x3

z=x1…xn, Uniform P z

1.4 H L 1.2

1.0

0.8

0.6

0.4

0.2

z -1.0 -0.5 0.0 0.5 1.0

n 1 1 1 − P(z) = ln 2(n 1)! z − | | Henry Tye A Vanishingly Small Λ 14/41 Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary

Suppose we have n random variables xi (i = 1, 2, , n), each ··· with probability distribution Pi (xi ), where dxi Pi (xi ) = 1. Let R z = f (x1, x2, , xn) ··· Then the probability distribution P(z) of z is given by

P(z) = dx1P1(x1) dx2P2(x2) dxnPn(xn) δ (f (xi ) z) ··· − Z P(z)dz = 1 Z so the probability distribution P(z) of z can always be properly normalized, even when P(z) diverges at z = 0 and/or elsewhere.

E.g. : P(z) = dx1 dx2 δ (x1x2 z) = dx1/x1 − z R R Henry Tye A Vanishingly Small Λ 15/41 Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary

z=x1…xn, Normal P z

1.4 H L 1.2

1.0

0.8

0.6

0.4

0.2

z -2 -1 0 1 2

Figure: The product distribution P(z) is for z = x1 (solid brown curve for normal distribution), z = x1x2 (red dashed curve), and z = x1x2x3 (blue dotted curve), respectively. In general, the curves are given by the Meijer-G function.

Henry Tye A Vanishingly Small Λ 16/41 Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary n Probability distribution P(z) for z = x1

n z=x1 , Uniform P z

1.4 H L 1.2

1.0

0.8

0.6

0.4

0.2

z 1+1/n -1.0 -0.5 0.0 0.5 1.0 P(z) z ∼ −

Henry Tye A Vanishingly Small Λ 17/41 Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary

Henry Tye A Vanishingly Small Λ 18/41 Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary

Henry Tye A Vanishingly Small Λ 19/41 Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary

Henry Tye A Vanishingly Small Λ 20/41 Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary Probability distribution P(z)

z Asymptote of P(z) at z = 0 n 1 x1 xn (ln(1/ z )) − ···n 1+1| | /n x1 z− n n 1+1/n m 1 x1 xm z− (ln(1/ z )) − ···m n 1+1/m 1+1|/n| x1 x2 (z− z− )/(m n) − m 1 − x1 xm / y1 yn (ln(1/ z )) − ··· m n ··· 1+1| | /m x1 /y1 z− xn1 + + xnm z 1+1/n1+ 1/nm 1 ··· m − ··· x1x2, 0 < c = x1/x2 < smooth ∞ x1x2, 0 c = x1/x2 or c ln(1/ z ) ≤ ≤ ∞ | |

Henry Tye A Vanishingly Small Λ 21/41 6 20 Suppose the 10 solutions are now at Λ = 10− and the one is still at Λ = 1. Then Λ = 10 20 while Λ does not change. 50% − h i Introduce ΛY % : Y % of the solutions have a value below ΛY %. 20 In this special case, Λ10% = Λ90% = 10− also.

Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary Median as a useful measure for the expected values

Λ 50% For our purpose, | |Λ is a good measure of the preference for a small Λ. h| |i 6 9 Example : 10 solutions at Λ = 10− and one solution at Λ = 1. Then Λ = 10 6 while Λ = 10 9. h i − 50% −

Henry Tye A Vanishingly Small Λ 22/41 Introduce ΛY % : Y % of the solutions have a value below ΛY %. 20 In this special case, Λ10% = Λ90% = 10− also.

Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary Median as a useful measure for the expected values

Λ 50% For our purpose, | |Λ is a good measure of the preference for a small Λ. h| |i 6 9 Example : 10 solutions at Λ = 10− and one solution at Λ = 1. Then Λ = 10 6 while Λ = 10 9. h i − 50% − 6 20 Suppose the 10 solutions are now at Λ = 10− and the one is still at Λ = 1. Then Λ = 10 20 while Λ does not change. 50% − h i

Henry Tye A Vanishingly Small Λ 22/41 Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary Median as a useful measure for the expected values

Henry Tye A Vanishingly Small Λ 22/41 The peaking of P(Λ) at Λ = 0 is necessary for the preference of a small Λ. Fortunately, gravity couples to all sectors, so this decoupling should not happen. But it may happen in over-simpliﬁed models, as in standard perturbative calculations : Λ = Λ0 + Λ1 + Λ2 + .....

Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary

In general, if V = V1(nj , ui ) + V2(mk , vl ) where the 2 terms in V do not couple, then

Λ = Λ1(nj ) + Λ2(mk )

Even if P1(Λ1) and P2(Λ2) are peaked at zero, P(Λ) has no or only weak peaking at Λ = 0.

Henry Tye A Vanishingly Small Λ 23/41 Fortunately, gravity couples to all sectors, so this decoupling should not happen. But it may happen in over-simpliﬁed models, as in standard perturbative calculations : Λ = Λ0 + Λ1 + Λ2 + .....

Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary

In general, if V = V1(nj , ui ) + V2(mk , vl ) where the 2 terms in V do not couple, then

Λ = Λ1(nj ) + Λ2(mk )

Even if P1(Λ1) and P2(Λ2) are peaked at zero, P(Λ) has no or only weak peaking at Λ = 0.

The peaking of P(Λ) at Λ = 0 is necessary for the preference of a small Λ.

Henry Tye A Vanishingly Small Λ 23/41 Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms Summary

In general, if V = V1(nj , ui ) + V2(mk , vl ) where the 2 terms in V do not couple, then

Λ = Λ1(nj ) + Λ2(mk )

Even if P1(Λ1) and P2(Λ2) are peaked at zero, P(Λ) has no or only weak peaking at Λ = 0.

The peaking of P(Λ) at Λ = 0 is necessary for the preference of a small Λ. Fortunately, gravity couples to all sectors, so this decoupling should not happen. But it may happen in over-simpliﬁed models, as in standard perturbative calculations : Λ = Λ0 + Λ1 + Λ2 + .....

Henry Tye A Vanishingly Small Λ 23/41 32 14/8/12

Bousso-Polchinski Introduction Basic features Basic Idea P(z) The Large Volume Scenario in Type IIB String Theory Non-interacting case: e.g., Sum of terms 4-form quantization Summary No peaking behavior for P(Λ)

In Denef and Douglas : W , Di W and Di D32j W are random. 14/8/12 Another example : Bousso-Polchinski Model Assume randomness in Bousso-Polchinski; Bousso-Polchinski 4-form quantization

But… Moduli fields couple each term Assume randomness in Bousso-Polchinski;

But… Moduli fields couple each term correlation generated via stabilization

Henry Tye A Vanishingly Small Λ 24/41 correlation generated via stabilization Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1

Type IIB String Theory (MP = 1)

Consider the superpotential W0 (Gukov-Vafa-Witten)

W0(Ui , S) = G3 Ω = (F3 iSH3) Π(Ui ) ∧ − · cycles X Z = (f3j iSh3j ) j (Ui ) − F c1 + bj Uj S(c2 + dj Uj ) ' − j j X X where f3j and h3j take discrete ﬂux values.

E.g., Only linear terms in Uj in orientifolded toroidal orbifolds (Font, ..., Lust, Reﬀert, Schulgin, Stieberger, ... ).

Henry Tye A Vanishingly Small Λ 25/41 Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1

K J¯I 2 V =e K DJ WD¯W¯ 3 W , I − | | K = 2 ln( + ξ/2) ln(S + S¯) ln(Uj + U¯j ) − V − − j X 3 3/2 3/2 =Vol/α0 = γ1(T1 + T¯1) γi (Ti + T¯i ) , V − i=2 X NK ai Ti W =W0(Uj , S) + Ai e− , i=1 X W0(Uj , S) =c1 + bj Uj S(c2 + dj Uj ) − j j X X

where ξ is the α0 correction (Becker, Becker, Haack and Louis: Pedro, Rummel and Westphal) that can provide the K¨ahleruplift to de Sitter solutions (Rummel and Westphal, deAlwis and Givens.)

Henry Tye A Vanishingly Small Λ 26/41 Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1

We shall illustrate the statistical preference for a small Λ with 2 types of examples : aT A single K¨ahler modulus T with K¨ahleruplift : W = W0 + Ae− :

B = 0 V W0A ( 1 + ξ/ ) → ∝ − V

Henry Tye A Vanishingly Small Λ 27/41 Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1

A single K¨ahlermodulus T In a racetrack with K¨ahleruplift :

aT bT W = W0 + Ae− + Be−

a3AW e x β e βx Cˆ V 0 − cos y − cos(βy) + ' − 2 − x2 − z x2 x9/2 !

3a3/2W ξ aT = x + iy, z = A/B, β = b/a, Cˆ = 0 − 32√2A

a = 2π/N1 b = 2π/N2 β = N1/N2 & 1

Henry Tye A Vanishingly Small Λ 28/41 β = 26/25 = 1.04

7 19 Λ = 6.36 10− , Λ = 5.47 10− h i × 50% × 54 Λ = 2.83 10− 10% × This shows how the functional form of Λ(z), with a uniformly distributed variable z, can lead to a sharp peaking of P(Λ) at Λ = 0.

Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1

3/2 2 4 62√2a B β 2β 5 2 3 β−1 3 / Λ β− z ln( β− z) ∼ 243ξ√β 1 − − − − When β & 1 and simultaneously z is small, we have an exponential suppression in the cosmological| | constant.

Henry Tye A Vanishingly Small Λ 29/41 Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1

β = 26/25 = 1.04

Henry Tye A Vanishingly Small Λ 29/41 Introduction Examples Basic Idea Multi-Complex Structure Moduli case ! "#$%&'%()*+%,-)+.The/, Large&. Volume0$+ Scenario%&*. in Type IIB String Theory12343,Probability567 Distribution P(Λ) Summary P(Λ) as a function of h2,1 Typical Calabi-Yau Manifold with many warped throats.

Wednesday, 19 February, 14 ! 8.%&-)+./'0$#9,+..&(&$+%&*.,1:+;()#<,=/+)>*.<,?/)$&.@/<,5A7Henry Tye A Vanishingly Small Λ 30/41 ! B/./)+%/,-)+./>,+>,;+.@&@+%/,&.0$+%*.> ! C9&%,D'%()*+%<,)*$$,%()*#E(,-#$F<,>/%%$/,@*G.,&.,+.*%(/),%()*+% ! C.*#E(,G+)H&.EI,JDK,&.0$+%&*.L, M$+%,H*%/.%&+$I,>$*G')*$$,&.0$+%&*.3 Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1 Typical Manifolds Studied

χ(M) = 2(h1,1 h2,1) −

Manifold h1,1 h2,1 χ 4 2 272 540 P[1,1,1,6,9] − 11 3 111 216 F − 18 5 89 168 F − 4 1 (100) ( 200) CP[1,1,1,1,1] O O −

A manifold has h1,1 number of K¨ahlermoduli and h2,1 number of complex structure moduli.

Henry Tye A Vanishingly Small Λ 31/41 I All ﬂux parameters bi , ci and di are treated as real random variables with some uniform probability distributions.

I Find the supersymmetric solution w0 = W0 of W0 for the |min complex structure moduli and the dilaton and insert this w0 into V to stabilize the K¨ahlermodulus.

I The functional form of Λ = Vmin in terms of the parameters allow us to ﬁnd P(Λ).

Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1 Approach for the Multi-Complex Structure Moduli case

I Consider the above model W0(Ui , S) = c1 + bj Uj S(c2 + dj Uj ) j − j with the dilation S and h2,1 number of complex structure P P moduli Ui .

Henry Tye A Vanishingly Small Λ 32/41 I Find the supersymmetric solution w0 = W0 of W0 for the |min complex structure moduli and the dilaton and insert this w0 into V to stabilize the K¨ahlermodulus.

I The functional form of Λ = Vmin in terms of the parameters allow us to ﬁnd P(Λ).

I Consider the above model W0(Ui , S) = c1 + bj Uj S(c2 + dj Uj ) j − j with the dilation S and h2,1 number of complex structure P P moduli Ui .

I All ﬂux parameters bi , ci and di are treated as real random variables with some uniform probability distributions.

Henry Tye A Vanishingly Small Λ 32/41 Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1 Approach for the Multi-Complex Structure Moduli case

I Consider the above model W0(Ui , S) = c1 + bj Uj S(c2 + dj Uj ) j − j with the dilation S and h2,1 number of complex structure P P moduli Ui .

I All ﬂux parameters bi , ci and di are treated as real random variables with some uniform probability distributions.

I Find the supersymmetric solution w0 = W0 of W0 for the |min complex structure moduli and the dilaton and insert this w0 into V to stabilize the K¨ahlermodulus.

I The functional form of Λ = Vmin in terms of the parameters allow us to ﬁnd P(Λ).

DS W0 = ∂S W0 + KS W0 = 0, Di W0 = 0

W0(ui , s) = c1 + bj uj s(c2 + dj uj ) − j j X X 2,1 Solution : ui = (c1 sc2)/(h 2)(bi sdi ) − − − − 2,1 c + sc h b + sd (h2,1 2) 1 2 = i i − c1 sc2 bi sdi i=1 − X − 2(c sc ) 2(c + sc )Π (b sd ) w = W = 1 2 = 1 2 i i i 0 0 min 2,1− − | − h 2 i (bi + sdi )Πj=i (bj sdj ) − 6 − P Then insert w0 into the V for the K¨ahlermodulus and ﬁnd the solution : 5/2 2 3 e− 2 w0a A 5 Λ = − xm 9 5 γ2 − 2 Henry Tye A Vanishingly Small Λ 33/41 Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1

Figure: The probability distribution P(Λ) of Λ at meta-stable vacua as a function of h2,1 = 2, 5, 8 number of complex structure moduli and a single 1,1 K¨ahlermodulus (h = 1). Although the range is 0 ≤ Λ . 1, the probability distributions for only 0 ≤ Λ ≤ 10−3 are shown. P(Λ) becomes more peaked at Λ = 0 as h2,1 increases.

Henry Tye A Vanishingly Small Λ 34/41 Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1

L æ æ æ à æà à æ ì ì à - æ æ 10 6 ì à ì à æ à æ æ æ æ æ ì à ì à -13 ì 10 ì ì à

- à 10 20 ì à

-27 10 ì à

- 10 34 ì

10-41 ì 2,1 0 5 10 15 20 25 30 h

Figure: The ﬁgure shows hΛi (red circles), Λ80% (blue squares) and Λ10% 2,1 (green diamonds) as a function of h . Here, the bi parameters are ﬁxed or have limited ranges. At h2,1 = 30 : Λ10% ' 1.5 × 10−41 (green diamonds) while hΛi ' 10−8 (red circles).

Λ 10 1.1h2,1 while Λ 10 8 50% ∼ − h i ' − Henry Tye A Vanishingly Small Λ 35/41 Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1 The Supersymmetric KKLT Case

L æà ìæà ì ò æ æ æ æ æ à 0.001 ò ì È à È -9 ò à 10 ì à à ò ì 10-15 ì -21 ò 10 ì

10-27 ò

ò h2,1 0 5 10 15 20 25 30

0.82h2,1+2.7 3 Λ median = Λ 10 while Λ 10 . | | | |50% ∼ − h| |i ∼ −

Henry Tye A Vanishingly Small Λ 36/41 If we set Λ 10 122M4 , the gravitino mass ∼ − P 61 33 m 10− MP = 10− eV 3/2 ∼

Note in Swiss-Cheese type Large Volume Scenario (LVS) :

2 2 2 2 2 Di W 3 W m Di W / , Λ | | − | | 3/2 ∼ | | V ∼ 3 V where we can take volume to 1030. V ∼

Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1 m3/2 In the 3 supergravity cases described above, we ﬁnd that

m2 Λ 3/2 ∼ | | for both positive and negative Λ, with broken supersymmetry.

Henry Tye A Vanishingly Small Λ 37/41 Note in Swiss-Cheese type Large Volume Scenario (LVS) :

2 2 2 2 2 Di W 3 W m Di W / , Λ | | − | | 3/2 ∼ | | V ∼ 3 V where we can take volume to 1030. V ∼

Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1 m3/2 In the 3 supergravity cases described above, we ﬁnd that

m2 Λ 3/2 ∼ | | for both positive and negative Λ, with broken supersymmetry.

If we set Λ 10 122M4 , the gravitino mass ∼ − P 61 33 m 10− MP = 10− eV 3/2 ∼

Henry Tye A Vanishingly Small Λ 37/41 Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1 m3/2 In the 3 supergravity cases described above, we ﬁnd that

m2 Λ 3/2 ∼ | | for both positive and negative Λ, with broken supersymmetry.

If we set Λ 10 122M4 , the gravitino mass ∼ − P 61 33 m 10− MP = 10− eV 3/2 ∼

Note in Swiss-Cheese type Large Volume Scenario (LVS) :

2 2 2 2 2 Di W 3 W m Di W / , Λ | | − | | 3/2 ∼ | | V ∼ 3 V where we can take volume to 1030. Henry TyeV ∼ A Vanishingly Small Λ 37/41 What about a quantum loop correction δΛ?

Λ0 Λ = Λ0 + δΛ → δΛ will typically shift Λ0 by a large amount and destroy the preference for a very small Λ.

Quantum or string corrections must be re-expressed in terms of nj and ui into δV (nj , ui ).

V (nj , ui ) = V0(nj , ui ) + δV (nj , ui ) Λ(nj ) P(Λ) → →

V0(nj , ui ) Λ0(nj ) → What happens if we choose a more complete V (nj , ui )? We expect/guess that the peaking behavior of P(Λ) will remain, but may be modiﬁed or sharpened.

Henry Tye A Vanishingly Small Λ 38/41 Quantum or string corrections must be re-expressed in terms of nj and ui into δV (nj , ui ).

V (nj , ui ) = V0(nj , ui ) + δV (nj , ui ) Λ(nj ) P(Λ) → →

V0(nj , ui ) Λ0(nj ) → What happens if we choose a more complete V (nj , ui )? We expect/guess that the peaking behavior of P(Λ) will remain, but may be modiﬁed or sharpened.

What about a quantum loop correction δΛ?

Λ0 Λ = Λ0 + δΛ → δΛ will typically shift Λ0 by a large amount and destroy the preference for a very small Λ.

Henry Tye A Vanishingly Small Λ 38/41 Introduction Examples Basic Idea Multi-Complex Structure Moduli case The Large Volume Scenario in Type IIB String Theory Probability Distribution P(Λ) Summary P(Λ) as a function of h2,1 The potential we study is an approximate one :

V0(nj , ui ) Λ0(nj ) → What happens if we choose a more complete V (nj , ui )? We expect/guess that the peaking behavior of P(Λ) will remain, but may be modiﬁed or sharpened.

What about a quantum loop correction δΛ?

Λ0 Λ = Λ0 + δΛ → δΛ will typically shift Λ0 by a large amount and destroy the preference for a very small Λ.

Quantum or string corrections must be re-expressed in terms of nj and ui into δV (nj , ui ).

V (nj , ui ) = V0(nj , ui ) + δV (nj , ui ) Λ(nj ) P(Λ) → →

Henry Tye A Vanishingly Small Λ 38/41 I At high vacuum energies, hardly any meta-stable vacua exist. Most vacua accumulate around Λ = 0.

Introduction Basic Idea New Picture The Large Volume Scenario in Type IIB String Theory Summary The Picture

+ I Peaking of P(Λ) at Λ = 0 happens for both Λ and Λ−. I Introducing ”multi-complex structure moduli” into the racetrack potential for a single K¨ahler modulus can yield a vanishingly small Λ.

Henry Tye A Vanishingly Small Λ 39/41 Introduction Basic Idea New Picture The Large Volume Scenario in Type IIB String Theory Summary The Picture

I At high vacuum energies, hardly any meta-stable vacua exist. Most vacua accumulate around Λ = 0.

Henry Tye A Vanishingly Small Λ 39/41 Introduction Basic Idea New Picture The Large Volume Scenario in Type IIB String Theory Summary Rolling down after the inﬂationary epoch, our universe reaches the small positive Λ region before the small negative Λ region.

Stabilization: Before After

CC = 0

[Bousso, Polchinski, 00] [Sumitomo, Tye]

1 Henry Tye A Vanishingly Small Λ 40/41 Technical challenge : When we simplify the model too much, the moduli are not coupled to each other so P(Λ) does not peak at Λ = 0. On the other hand, when we include more couplings, the meta-stable vacua can be found only numerically; so it is difficult to find Λ(nj ) and P(Λ) for a large number of flux values nj . { }

THANKS

Introduction Basic Idea New Picture The Large Volume Scenario in Type IIB String Theory Summary Summary : There may be a dynamical stringy alternative to the ”Anthropic Principle” in explaining the very small positive Λ.

Henry Tye A Vanishingly Small Λ 41/41 Introduction Basic Idea New Picture The Large Volume Scenario in Type IIB String Theory Summary Summary : There may be a dynamical stringy alternative to the ”Anthropic Principle” in explaining the very small positive Λ.

Technical challenge : When we simplify the model too much, the moduli are not coupled to each other so P(Λ) does not peak at Λ = 0. On the other hand, when we include more couplings, the meta-stable vacua can be found only numerically; so it is difficult to find Λ(nj ) and P(Λ) for a large number of flux values nj . { }

THANKS

Henry Tye A Vanishingly Small Λ 41/41