( ) Brane Vector (Proca) Fields p p 2. Brane Vector Production - PowerPoint PPT Presentation

2008 Phenomenology Symposium: LHC Turn On University of Wisconsin-Madison 28-30 April 2008 Brane Oscillations At The TeVatron and LHC • T.E. Clark*, S.T. Love, C. Xiong--Purdue University • Muneto Nitta--Keio University Χ • T. ter Veldhuis--Macalester College Outline: Χ 1. Brane-Standard Model Effective Action: Massive ( ) Brane Vector (Proca) Fields p p 2. Brane Vector Production → γ + → γ + q q XX E p 3. TeVatron Bounds on Brane Vector Parameter Space γ 4. TeVatron Reach In Parameter Space 5. LHC Reach In Parameter Space *Speaker

1. Flexible Brane Effective Action Poincaré ⊗ Brane Co-Volume Bulk ⊗ ISO(1,D-1) ISO(1,3) SO(N) Symmetries = μ φ = M Bulk Coordinates: x ( x , ( )) ( x N 1 Domain Wall Case) Nonlinear Realization: μ μν μ = = M MN Symmetry Generators: P ( P , Z ), M ( M , K ) μ μ φ = ix P iv ( ) x K i ( ) x Z μ μ U x ( ) e e e Coset Element: C o s e t C o o rd in a te s : { } μ φ μ x , ( x ) , v ( x ) Maurer-Cartan Form= Covariant Building Blocks: Global Symmetries M − ∂ 1 ( ) U x U x ( ) mn Spin Connection μ P m ω = ∂ − ∂ + � mn m n n m v v v v Z K μ μ μ m Induced Vierbein Covariant Extrinsic Curvature μ φ m ( , ) e v Derivative = = ∂ + � n K e K v μν ν μ μ ν n ∇ μ φ

Covariant Constraint = Eliminate Redundant ∇ φ = ⇒ ∂ φ = + � 0 v Coordinate v μ Field Equation: μ μ μ = η − ∂ φ ∂ φ Invariant Nambu-Goto Action = Induced Gravity: g ( ) x ( ) x μν μν μ ν ∫ ⎡ ⎤ Γ = − − + L 4 4 d x det g f ( ) g ⎣ ⎦ SM Expand in powers of a rescaled Branon field φ and add a mass for the branons as a curved bulk requires energy to deform the brane ⎡ ⎤ ∫ Γ = η + ∂ φ ∂ μ φ − φ + ∂ μ φ ∂ ν φ − φ η μν + L 2 � 4 2 2 SM 2 SM m 1 1 1 d x ( ) m T T ⎣ ⎦ μ μν μν Effective SM 2 2 4 4 2 f 8 f Standard Model fields couple to the branon fields Alcaraz, Cembranos, Dobado and Maroto: SM ( ) Phys. Rev. D 67 , 075010 (2003), etc. via the SM energy-momentum tensor T x μν Creminelli and Strumia: Nucl. Phys. B596 , 125 (2001)

General Coordinate and Local Lorentz Transformations Dynamic Gravitational Fields require locally covariant Maurer-Cartan Form: = + + + γ m m mn Gravitational Fields: E ( ) x E ( ) x P X ( ) x Z V ( ) x K ( ) x M μ μ μ μ μ m m mn Locally Covariant Maurer-Cartan Form= Covariant Building Blocks: − ∂ + ⋅ − ⋅ 1 ( )[ ix P ix P U x ie E e ] U x ( ) μ μ Use Broken Local Lorentz Transformation to go to Partial Unitary Gauge: v μ = 0 = δ + + φ m m m m e E 2 V μ μ μ μ ∇ φ = ∂ φ + X μ μ μ = m m K V μ μ ω = γ mn mn μ μ

Covariant Building Blocks: = ∇ ⎡ ∇ ⎤ φ = ∂ − ∂ F , X X ⎣ ⎦ Field Strength Tensor: μν μ ν μ ν ν μ { } = − ≡ ∇ ∇ ν φ 1 m 1 K e V , Extrinsic Curvature Constraint: μν ν μ μ m 2 Invariant Action is obtained (normal field dimensions): τ ∫ Γ = Λ + − μν + + ∇ μ φ ∇ φ + ∇ μ φ ∇ ν φ L 4 SM 1 1 1 d x [ R F F ( ) e T μν μ μν κ 2 4 SM 2 4 2 F X 2 ( ) M � + + μρ ν + � X K B K B F K ] μν μν ρ 1 2 4 2 F X Use broken bulk general coordinate transformations to go to Full Unitary Gauge: φ=0 (gravi-photon X μ eats φ to get Mass M X ) = δ + ∇ φ = m m m e E ; M X μ μ μ μ μ X ( ) = = ∂ + ∂ ω = γ mn mn 1 K V X X ; μν μν μ ν ν μ μ μ 2

i X μ Ignore gravity and expand in powers of , where the index i=1,2,…,N , the number of additional space dimensions, to obtain the Brane Vector Effective Action τ 2 M ∫ Γ = η − μν + μ + μ ν L 4 i i 2 i i i SM i 1 1 X d x [ ( ) F F M X X X T X μν μ μν SM X 4 2 4 2 F X 2 ( ) M � μρ ν + + i i X K B K B F K ] μν μν ρ 1 2 4 2 F X with phenomenologically determined mass M X and effective brane tension F X as well as = ∂ − ∂ couplings τ, K 1 , K 2 and where i i i F X X are the N extra dimension-Abelian field μν μ ν ν μ strength tensors for the brane vector (Proca) gauge fields . The covolume SO(N) symmetry is envisioned to be spontaneously broken, hence their gauge fields are massive and not considered here. Although the SO(N) symmetry amongst the brane vectors is now broken, for simplicity the covolume is taken to be isotropic, thus the brane vectors have a common mass M X and effective brane tension F X . Similarly, the bilinear X coupling can be to any SU(3), SU(2), U(1) invariant. These have been chosen to be equal for simplicity, hence the Standard Model energy-momentum tensor appears.

2. Brane Vector Missing Energy: → γ + → γ + LEP-II has searched for and we determined an + - e e XX E excluded/allowed region of F X , M X , K 1 , K 2 parameter space based on the agreement with the Standard Model. Brane Vector Creminelli and Strumia: Nucl. Phys. B596 , 125 (2001); Branon Alcaraz, Cembranos, Dobado and Maroto: Phys. Rev. D 67 , 075010 (2003); L3 Collaboration, P. Achard et al.: Phys. Lett. B 597 (2004) 145; μν T S. Mele, Search for Branons at LEP, Int. Europhys. Conf. = = on High Energy Phys., PoS(HEP2005)153. K 0 K 1 2 → γ + → γ + pp XX pp XX at the TeVatron and at the LHC where the 2 X particles escape the detector as missing energy have also been used to bound parameter space. Likewise, the TeVatron Ib and II data exclude regions of brane parameter space. The TeVatronII reach based on an integrated luminosity of 6000 pb -1 and the LHC reach can be used to delineate accessible/inaccessible regions of parameter space.

E + γ → XX + i γ i X X γ → i X γ i i i The Feynman Diagrams for Brane Vector Production: q q X X γ X , Z γ q q q q q q i γ i X X γ i i X i X X γ i X , Z γ γ q q q q q q

− q q The differential cross-section for spin averaged collisions producing a photon and 2 X particles with summed over polarizations and the X species, i=1,2,…,N = # of extra dimensions σ ⎡ ⎤ − α 2 2 2 d k 4 M 1 N 1 γ ⎡ ⎤ = + + × XX X 2 2 2 ⎢ ⎥ ˆ 2 sk u t ⎣ ⎦ π π 2 8 3 ⎣ ⎦ ˆ dk dt 4 15,360 F s ut 2 k X { ( ) ( ) ( ) 2 ⎡ ⎤ × τ + − + + 2 2 2 2 2 2 2 ˆ sk 4 ut k 4 M 20 M k 2 M ⎣ ⎦ X X X ) } ( ) ( ( ⎡ ⎤ ) ⎡ ⎤ 2 + + − 2 2 2 2 2 2 2 ˆ K k K k 4 M S M 8 0 M s k ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ 1 2 X X with p p , the antiquark and quark momenta, the photon momenta and q k k , the brane vector momenta. The 1 2 1 2 = + = − = − = + 2 2 2 2 2 ˆ Mandelstam variables s ( p p ) , t ( p q ) , u ( p q ) and k ( k k ) . Neglecting the quark mas ses 1 2 1 2 1 2 + + = 2 ˆ implies s t u k . The Standard Model coupling and propagator factor ( ) ⎡ ⎤ ⎛ ⎞ − ⎛ ⎞⎛ θ 2 2 ⎞ ⎛ ⎞ k M 2 cos 1 1 1 1 1 ⎢ ⎜ ⎟ ⎥ ≡ πα + + θ − + θ θ − Z 2 2 2 2 W ⎜ ⎟⎜ (sin ) ⎟ 2cos ⎜ sin ⎟ . SM ⎜ ⎟ − + Γ θ ⎢ ⎝ W ⎠ W ⎝ W ⎠ ⎥ 2 2 2 2 2 2 2 2 2 ⎝ ⎠ ( k ) ( k M ) M c os 16 4 4 k ⎝ ⎠ ⎣ ⎦ Z Z Z W No Interference—Energy-Momentum Tensor and Extrinsic Curvature Terms Add

3. TeVatron Bounds on Parameter Space: → γ + → γ + Total Cross Section for p p XX E The single photon cross section with missing energy of the brane vectors is found by integrating over the parton distribution functions using CTEQ-6.5M distributi ons: 2 σ k t 1 1 = ∫ d m ax max ∫ ∫ ∫ γ σ 2 X X ˆ dx dy f x y s ( , ; ) dk dt dk dt γ XX 2 2 x y t k min min min min where the quark distribution function is = + 2 ˆ 1 2 ˆ ˆ ˆ ˆ f x y s ( , ; ) ( ) [ u ( , ) x s u ( , ) y s u ( , ) x s u ( , )] y s 3 3 p p p p + − + 2 1 1 ˆ ˆ ˆ ˆ ( ) [ d ( , ) x s d ( , ) y s d ( , ) x s d ( , )]. y s 3 3 p p p p = − 2 E 2 2 ˆ The kinematic constraints are given by k (4 M , (1 s )), T X ˆ s − η max 1 tanh = − η max 2 ˆ min t ( k s x y ) . The pseudo-rapidity is denoted and + + − η min max ( x ) ( y x )tanh min E is the minimum transverse energy of the photon while and denote the fr x y action T 1 of proton CM energy s that the quark and antiquark have, respectively. 2 η = ± max 1.0 is taken for the TeVatron and LHC plots while the transverse min energy is 45-50 GeV for the TeVatron and sca led to 350 GeV for the LHC.