Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Gauge Threshold Corrections for Local String Models Joseph Conlon (Oxford University) CERN String Seminar, March 30, 2009 Based on arXiv:0901.4350 (JC) Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Local vs Global There are many different proposals to realise Standard Model in string theory: ◮ Weakly coupled heterotic string / heterotic M-theory ◮ M-theory on G2 manifolds ◮ Intersecting/magnetised brane worlds in IIA/IIB string theory ◮ Branes at singularities ◮ F-theory GUTs Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Local vs Global These approaches are usefully classified as either local or global. Global models: ◮ Canonical example is weakly coupled heterotic string. ◮ Model specification requires global consistency conditions. ◮ Relies on geometry of entire compact space ◮ Limit V → ∞ also gives α SM → 0: cannot decouple string and Planck scales. ◮ Other examples: IIA/IIB intersecting brane worlds, M-theory on G2 manifolds Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Local vs Global Local models: ◮ Canonical example branes at singularity ◮ Model specification only requires knowledge of local geometry and local tadpole cancellation. ◮ Full consistency depends on existence of a compact embedding of the local geometry. ◮ Standard Model gauge and Yukawa couplings remain finite in the limit V → ∞ . It is possible to have M P ≫ M s by taking V → ∞ . ◮ Examples: branes at singularities, local F-theory GUTs. Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions BULK BLOW−UP U(2) Q L e L U(3) U(1) Q e R R U(1) Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Local vs Global Local models have various promising features: ◮ Easier to construct than fully global models. ◮ Typically have small numbers of families. ◮ Combine easily with moduli stabilisation, supersymmetry breaking and hierarchy generation (LARGE volume construction) ◮ Promising recent constructions of local stringy GUTs. Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Local vs Global One of the most phenomenologically important quantities in local models is the bulk volume. This determines ◮ String scale M s = M P √ V ◮ Gravitino mass through the flux superpotential � m 3 / 2 ∼ � G 3 ∧ Ω � V ◮ The unification scale in models where gauge couplings naturally unify. The purpose of this talk is to study this question precisely. Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Threshold Corrections ◮ If gauge coupling unification is non-accidental, it is important to understand the significance of M GUT ∼ 3 × 10 16 GeV. ◮ In particular, we want to understand the relationship of M GUT to the string scale M s and the Planck scale M P = 2 . 4 × 10 18 GeV. ◮ Is M GUT an actual scale or a mirage scale? ◮ I will discuss this first using supergravity arguments and subsequently directly in string theory. Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Threshold Corrections in Supergravity I In supergravity, physical and holomorphic gauge couplings are related by Kaplunovsky-Louis formula: phys (Φ , ¯ g − 2 Φ , µ ) = Re( f a (Φ)) (Holomorphic coupling) � M 2 � + b a 16 π 2 ln ( β -function running) P µ 2 + T ( G ) 8 π 2 ln g − 2 phys (Φ , ¯ Φ , µ ) (NSVZ term) + ( � r n r T a ( r ) − T ( G )) K (Φ , ¯ ˆ Φ) (K¨ ahler-Weyl anomaly) 16 π 2 − � T a ( r ) 8 π 2 ln det Z r (Φ , ¯ Φ , µ ) . (Konishi anomaly) r Relates measurable couplings and holomorphic couplings. Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions For local models in IIB ahler potential ˆ ◮ K¨ K is given by ˆ K = − 2 ln V + . . . ◮ Matter kinetic terms ˆ Z are given by Z = f ( τ s ) ˆ V 2 / 3 Why? When we decouple gravity the physical couplings Y αβγ ˆ ˆ K / 2 Y αβγ = e � Z α ˆ ˆ Z β ˆ Z γ should remain finite and be V -independent. Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Z = f ( τ s ) ˆ ˆ K = − 2 ln V , V 2 / 3 ◮ Local models require a LARGE bulk volume ( V ∼ 10 4 for M s ∼ M GUT , V ∼ 10 15 for M s ∼ 10 11 GeV). ◮ K¨ ahler and Konishi anomalies are formally one-loop suppressed. However if volume is LARGE, both anomalies are enhanced by ln V factors. ◮ This implies the existence of large anomalous contributions to physical gauge couplings! Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Plug in ˆ K = − 2 ln V and ˆ 1 Z = V 2 / 3 into Kaplunovsky-Louis formula. We restrict to terms enhanced by ln V and obtain: � M P Re( f a (Φ)) + ( � � r n r T a ( r ) − 3 T a ( G )) g − 2 phys (Φ , ¯ Φ , µ ) = ln 8 π 2 V 1 / 3 µ � ( RM s ) 2 � = Re( f a (Φ)) + β a ln . µ 2 ◮ Gauge couplings start running from an effective scale RM s rather than M s . ◮ Universal Re( f a (Φ)) implies unification occurs at a super-stringy scale RM s rather than M s . Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions ◮ Argument implies inferred low-energy unification scale is systematically above the string scale. ◮ Argument has only relied on model-independent V factors - result should hold for any local model (D3 at singularities, IIB GUTs, F-theory GUTs, local M-theory models) ◮ Unification scale is a mirage scale - new string states already occur at M s = M GUT / R ≪ M GUT . Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Threshold Corrections in Supergravity II Another argument: ◮ Consider gaugino condensation on a stack of branes wrapping a rigid collapsible cycle (del Pezzo) inside a large bulk. For definiteness assume an SU ( N c ) gauge group, b a = − 3 N c . ◮ Cycle size is measured by τ dP and classical gauge coupling is 4 π g 2 = τ dP ◮ Running gauge coupling is � Λ 2 � g 2 ( µ ) = τ dP 1 4 π − 3 N c UV 16 π 2 ln µ 2 Λ strong = Λ UV e − 2 π TdP 3 Nc Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Threshold Corrections in Supergravity II ◮ Gaugino condensation generates an effective holomorphic superpotential P e − 2 π TdP W = M 3 Nc ◮ This is identified with the strong coupling scale e K / 2 W � ¯ λλ � = � Λ strong � 3 = M 3 V e − 2 π TdP UV e − 2 π TdP Λ 3 P = . Nc Nc ◮ Consistency requires as before Λ UV = M P V 1 / 3 = RM s . Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Local and Global Models Threshold Corrections: Supergravity Threshold Corrections: String Theory Conclusions Threshold Corrections in String Theory ◮ We now want to investigate this directly in string theory. ◮ In string theory gauge couplings are � M 2 � 1 1 b a + ∆ a ( M , ¯ s a ( µ ) = + 16 π 2 ln M ) g 2 g 2 µ 2 0 , a ◮ ∆ a ( M , ¯ M ) are the threshold corrections induced by massive string/KK states. ◮ Study of threshold corrections pioneered by Kaplunovsky and Louis for weakly coupled heterotic string. ◮ For our calculations we use the background field method. Joseph Conlon (Oxford University) Gauge Threshold Corrections for Local String Models
Recommend
More recommend
