QCD Axion, Moduli Stabilization and SUSY Breaking in String Theory - PowerPoint PPT Presentation

QCD Axion, Moduli Stabilization and SUSY Breaking in String Theory Kiwoon Choi (KAIST) ExDiP 2012, Superstring Cosmophysics, August (2012)

Tomorrow: TeV scale SUSY with the recent LHC results

Outline of Today’s Lecture QCD axion in string theory * Axion solution to the strong CP problem and its realization in string theory * Cosmological constraints on GUT scale QCD axion * Intermediate scale QCD axion with anomalous U ( 1 ) gauge symmetry Moduli stabilization and SUSY breaking with intermediate scale QCD axion * Effective theory for SUSY breaking by anti-brane in KKLT moduli stabilization * KKLT axiverse

Strong CP problem: Low energy QCD involves a CP-violating interaction 1 θ G a µν ˜ 32 π 2 ¯ G a µν which gives rise to the neutron EDM d n ∼ 10 − 16 ¯ θ e · cm and therefore is constrained as | d n | � 10 − 25 e · cm | ¯ θ | � 10 − 9 � � . On the other hand, ¯ SM : θ = θ QCD + ArgDet ( y u y d ) , MSSM : ¯ θ = θ QCD + ArgDet ( y u y d ) + 3Arg ( M ˜ g ) + 3Arg ( B µ ) , so it is quite unnatural that | ¯ θ | � 10 − 9 within the CKM paradigm which explains CP violations in the weak interactions through the complex Yukawa couplings y u , d .

Axion solution: Peccei and Quinn At scales below an appropriate energy scale f a , the theory is assumed to possess a non-linearly realized global U ( 1 ) symmetry: Axionic shift symmetry U ( 1 ) PQ : a → a + constant , which is explicitly broken dominantly by the QCD anomaly. � a � 1 2 ( ∂ µ a ) 2 + 1 1 ∂ µ a J µ + + ¯ G ˜ ⇒ L axion = θ G + ∆ L 32 π 2 f a f a � � ∆ L = nonderivative couplings of a other than aG ˜ G 1 ∂ ∂ µ J µ 32 π 2 G ˜ J µ � PQ = f a ∂ µ a + J µ � = G + f a ∂ a ∆ L PQ = QCD anomaly + other explicit U ( 1 ) PQ breakings V ( a ) = V QCD ( a ) + V UV ( a ) � a / f a + ¯ u + m 2 � � m 2 d + 2 m u m d cos θ − f 2 π m 2 = π ( m u + m d ) 2 α � = ¯ + ǫ M 4 � � � � UV cos a / f a + α θ

If explicit U ( 1 ) PQ breakings other than the QCD anomaly are highly suppressed, so that V UV ( a ) � 10 − 9 f 2 π m 2 π ∼ 10 − 78 M 4 GUT , then V QCD drives the axion VEV to cancel ¯ θ with an accuracy of O ( 10 − 9 ) , regardless of the values of the low energy parameter ¯ θ and the UV parameter α : � a � � � + ¯ � � � 10 − 9 . � θ � � f a � This is an elegant mechanism, but raises a question: Q1: What is the origin of such global symmetry which is explicitly broken in a quite peculiar way? Simply assuming such a global symmetry is not likely to be sensible as quantum gravity effects generically break global symmetries, so can generate V UV ( a ) ≫ f 2 π m 2 π .

Astrophysical and cosmological considerations lead to various constraints on the axion scale f a . * Axion emission from red giants, neutron stars, SN1987A: f a � 10 9 GeV . ⇒ * Relic axions produced by an initial misalignment δ a ≡ f a θ i : � 7 / 6 � f a Ω a h 2 ≃ 2 × 10 4 � θ 2 i � � 0 . 12 10 16 GeV (assuming no entropy production after the QCD phase transition) f a � 3 × 10 11 � θ 2 i � − 6 / 7 GeV . ⇒ So, if the misalignment angle takes a value in the natural range, i.e. θ i ∼ 1, the axion scale is required to be 10 9 GeV � f a � 3 × 10 11 GeV .

This raises another question: Q2: What is the dynamics to generate such an intermediate axion scale? In SUSY models, the axion scale f a is in fact a dynamical field, the saxion or modulus partner of axion, and then the axion scale is determined by the mechanism to stabilize saxion or moduli.

Higher-dim gauge symmetry as the origin of U ( 1 ) PQ Higher-dim theory with a p -form gauge field ( p = 1 , 2 , ... ), compactified on internal space involving a p -cycle S p : * Higher-dim gauge symmetry: � � G C : C p → C p + d Λ p − 1 Λ p − 1 = globally well-defined ( p − 1 ) -form * Axion fluctuation: C [ m 1 m 2 .. m p ] ( x , y ) = a ( x ) ω [ m 1 m 2 .. m p ] ( y ) = a ( x ) ∂ [ m 1 ˜ Λ m 2 .. m p ] x µ , y m � � � � = 4d Minkowski coordinates, internal coordinates S p ω p = 1 , so ˜ � Here ω p is a harmonic p -form with Λ p − 1 is only locally well-defined. Then the axionic shift symmetry U ( 1 ) PQ : a → a + constant is locally equivalent to the higher-dim gauge symmetry G C , but not globally: � G C : C p → C p + d Λ p − 1 for d Λ p − 1 = 0 , S p � U ( 1 ) PQ : C p → C p + constant × ω p for ω p � = 0 S p

U ( 1 ) PQ can be explicitly broken, but only through the effects associated with non-trivial global topology of the p -cycle S p , in particular associated with � ω p � = 0 S p * QCD anomaly: � � � aG ˜ C p ∧ G ∧ G → U ( 1 ) PQ -breaking ω p G C -invariant G 4D S p * UV instantons wrapping S p : � � V UV ( a ) = ǫ 0 e − S ins M 4 UV cos ( a / f a + α ) M UV ∼ M GUT or M string � n � � m 3 / 2 � ǫ 0 = model-dependent zero-mode factors possibly involving M Pl This suggests that if S p has a relatively large volume to have the instanton action S ins � O ( 100 ) , a good U ( 1 ) PQ can appear as a low energy remnant of higher-dim gauge symmetry. Obviously string theory is the best place to realize this scenario.

Axion scale ( = axion decay constant): Canonically normalized stringy axion typically has Planck-scale suppressed interactions, so a decay constant f a ∼ M GUT : a ˆ 1 ˆ a f a ∼ M Pl G ˜ G ˜ G ≡ → 32 π 2 ∼ M GUT G 32 π 2 M Pl f a This can be easily seen in supersymmetric compactification: √ a + θ 2 F T Axion Superfield: T = t + ia + 2 θ ˜ 1 � � t ∝ Vol ( S p ) , normalized as � t � ∼ g 2 GUT ∂ 2 K ∂ t 2 ∂ µ a ∂ µ a + 1 4 aG ˜ M 2 L axion = G + ... Pl � � ∂ 2 K ∂ 2 K M Pl ∂ t 2 × 10 16 GeV , ⇒ f a = 8 π 2 ∼ 3 ∂ t 2 For ∂ 2 K ≃ n t 2 ∼ g 4 K ≃ − n ln ( T + T ∗ ) → GUT , ∂ t 2 so the axion scale is indeed around 10 16 GeV. KC , Kim ( 1985 ); Svrcek , Witten ( 2006 )

� ∂ 2 K ∂ t 2 ≪ 1 , and therefore f a ≪ M GUT . It is in principle possible to have For instance, S p might be a relatively small cycle embedded in a much larger bulk volume (Large Volume Scenario), or it might be located at a highly warped region in the internal space (Warped Compactification): ∂ 2 K 1 small warp factor ∼ 10 − 10 , ∼ or ∂ t 2 large bulk volume which would give f a ∼ 3 × 10 11 GeV and � 7 / 6 � f a Ω a h 2 ∼ 0 . 1 θ 2 i ∼ 0 . 1 for θ i ∼ 1 . 3 × 10 11 GeV However in such scheme, the cutoff scale of 4D visible sector physics is red-shifted also, making it difficult to accommodate the unification scale M GUT ∼ 2 × 10 16 GeV within the scheme.

More on cosmology of GUT scale QCD axion: Fox , Pierce , Thomas ( 2004 ); Mack ( 2009 ) Even for GUT scale QCD axion, the relic axion mass density can have an acceptable value if (i) the misalignment angle θ i is small enough and/or (ii) there is a dilution of axions by late entropy production after the QCD phase transition: � 7 / 6 � f a Ω a h 2 2 × 10 4 � θ 2 � � ≃ i � No dilution 10 16 GeV � T RH � 2 � � f a Ω a h 2 � θ 2 ≃ i � 40 10 16 GeV 6 MeV � � Entropy production by late decaying massive particles with 6 MeV � T RH � Λ QCD In order for a GUT scale QCD axion to have Ω a h 2 � 0 . 1, � 6 MeV � � θ 2 i � � 5 × 10 − 6 2 . 5 × 10 − 3 or T RH

It is often argued that in inflationary cosmology with U ( 1 ) PQ non-linearly realized during inflation, such small � θ 2 i � can be explained by anthropic selection rule since galaxies in a Universe with Ω a h 2 > O ( 1 ) will have too large mass densities to accommodate life. Linde ( 1986 ); Tegmark , Rees , Wilczek ( 2009 ) On the other hand, in such scenario, the axion misalignment square receives an irreducible contribution from the axion fluctuation produced during inflation: � θ i � 2 + � ( θ i − � θ i � ) 2 � = � θ i � 2 + σ 2 � θ 2 i � = θ , H I � � f a σ θ ∼ = RMS axion fluctuation during inflation , 2 π There is no dynamical mechanism to determine � θ i � , so in principle it can have an arbitrarily small value, while for given inflation model, σ θ is predicted to be of the order of H I / 2 π . If anthropic selection is the correct explanation for small � θ 2 i � of GUT scale QCD axion, it implies that the actual value of Ω a h 2 should not be far below the anthropic upper bound ∼ O ( 1 ) , so it has a high probability to be around 0 . 1 .

Axion fluctuation produced during inflation is of isocurvature type, so leads to an isocurvature CMB fluctuation: Fox , Pierce , Thomas ( 2004 ) θ ( 2 � θ i � 2 + σ 2 � ( δ T / T ) 2 iso � (Ω a / Ω M ) 2 σ 2 θ ) 8 α ≡ tot � ≃ � ( δ T / T ) 2 � ( δ T / T ) 2 tot � ( � θ 2 i � ) 2 25 � 2 � Ω a h 2 2 � θ i � 2 + σ 2 1 . 5 × 10 11 σ 2 � � = . θ θ � θ 2 i � Imposing the observational bound on this isocurvature CMB fluctuation � 2 � H I � H I � 2 � � 2 � � Ω a h 2 2 � θ i � 2 + α ∼ 1 . 5 × 10 11 � 0 . 072 , � θ 2 2 π f a 2 π f a i � one finds a rather strong constraint on the inflation scale: For f a ∼ 10 16 GeV and Ω a h 2 ∼ 0 . 1: 6 × 10 8 GeV ( no dilution ) H I � 1 . 6 × 10 10 GeV / � � T RH / 6 MeV ( late entropy production )