Extra Dimensions at Colliders

Extra Dimensions at Colliders Tilman Plehn SUPA, School of Physics, University of Edinburgh, Scotland You are looking at a rough script following my attempt to give an introduction into the phe- nomenology of extra dimensions to our SUPA graduate students. It covers the very basics of large and warped extra dimensions from an LHC phenomenologist's point of view. Moreover, I added a section on UV completions of models with large extra dimensions, because I got interested in this and even wrote a paper on it. Please notice that there are of course mistakes and typos all over the place, because I am also just learning the topic myself... Contents I. The Standard Model as an effective theory 2 A. Experimental hints 2 B. Theoretical hints 4 II. Fundamental Planck scale in (4+n) dimensions 7 A. Einstein{Hilbert action and proper time 7 B. Free fall, metric and Christoffel symbols 7 C. Simple invariant Lagrangian 8 D. Now we are ready! 9 III. Gravitons in flat (4+n) dimensions 11 A. Propagating an extra{dimensional graviton 11 B. Brane matter and bulk gravitons 13 C. Kaluza{Klein towers 14 D. Graviton Feynman rules 16 E. ADD gravitons at the LHC 17 IV. Warped extra dimensions 18 A. Newtonian gravity in a warped extra dimension 19 B. Gravitons in a warped dimension 20 V. Ultraviolet Completions 23 A. String theory 23 B. Fixed{point gravity 24 2 I. THE STANDARD MODEL AS AN EFFECTIVE THEORY Before we can even start talking about physics beyond the Standard Model, we have to define what we mean by the Standard Model. For those of you who attended Graham Kribs' journal club | remember the discussion between him and Thomas Binoth. There are different ways we can look at the Standard Model, regardless of it great success over by now many decades of measurements. Both approaches have basic structures in common: { a gauge theory with the group structure SU(3) SU(2) U(1) ⊗ ⊗ { massive electroweak gauge bosons masses through spontaneous symmetry breaking (Higgs mechanism with v = 246 GeV and mH unknown) { Dirac fermions in the usual doublets and with masses equal to their Yukawa couplings There are two philosophies behind writing down a Lagrangian for this model: 1. write a renormalizable Lagrangian with all dimension{4 operators consistent with the particle content and all symmetries 2. write a general effective{theory Lagrangian with these particles and all symmetries. Higher{dimension operators will appear and have to be suppressed by some scale Λ The difference between these two approaches are higher{dimensional operators, operators which have an explicit suppression by a large mass. If this scale Λ is large enough, we might never see the difference between the two approaches in high{energy experiments. If Λ is smaller, we might see the Standard Model break down as an effective theory, for example at the LHC, and we should be able to determine its ultraviolet completion. A. Experimental hints LEP and Tevatron LEP(2) and Tevatron experiments have for many years tested the Standard Model to energy scales of 100 500 GeV. All their results are in perfect agreement with the Standard Model, apart from that fact that we could· · ·have seen direct evidence for the Higgs boson: { electroweak gauge bosons discovered with masses mW 80 GeV, mZ 91 GeV no anomalous W; Z decays ∼ ∼ { 6 quarks found, mt 172 GeV typical decay t b∼W + observed, no anomalous decays ! { leptons, including τ as expected. { electroweak precision data with global fit: m 110 GeV best value H ∼ mH / 250 GeV 1 σ bound mH > 114 GeV from direct search at LEP2 e+ Z Z H + where ff are mostly H bb and Z l l−; νν. ! ! e− f f The possible problem with the electroweak precision data is the quality of the global fit. Its best χ2 value is poor. A reason might be that some for example b observables might be inconsistent, but we do not know not conclusive ) 3 Muon anomalous magnetic moment (g-2) The anomalous magnetic moment of the muon is one of the best{measured parameters in high{energy physics, even though most of the physics which goes into its determination we would call low{energy physics nowadays. Unfortunately, the Brookhaven experiment, which recently delivered the best available measurement, has been shut down. If you want to know more about this observable | Dominik Stöckinger recently wrote a great review on it. The measured value of the anomalous magnetic moment of µ is exp 1 exp 10 a = (g 2) = (11659208 6) 10− (1) µ 2 − µ · while the Standard Model predictions range between two different approaches exp SM 10 a a = (31:7 9:5) 10− : 3:3σ µ − µ · exp SM 10 a a = (20:2 9:0) 10− : 2:1σ (2) µ − µ · The general agreement in high{energy physics is: a 5 σ deviation from the background is called a discovery, everything else is either a rumor or a hint or a matter of taste not conclusive ) Atomic parity violation We know that electroweak gauge bosons have couplings which distinguish between the chirality of fermions in the ff 0W and ffZ vertices. In the interaction between the nucleus and the electrons in an atom, this interaction leads to parity violation, which can be measured: Cs γZ e e Beyond the Standard Model we can look for so{called leptoquarks, scalars or vectors which carry baryon and lepton number and occur in the crossed channel compared to the usual γ; Z exchange: S e e Again, this experiment in Boulder was terminated with around 2 σ discrepancy between the Standard Model prediction and the final measurement not conclusive ) Cosmology The last Nobel prize went to studies of the cosmic microwave background. The most recent WMAP data (combined with large{scale structure measurements) confirms conclusively the existence of cold dark matter in the universe: 2 ΩDMh = 0:094 ::: 0:129 (3) where Ω = 1 is critical density for flat universe and h = H =100 km/s/Mpc 0:7 is just a c{number connected 0 ∼ to the Hubble constant H0 73 km/s/Mpc. Different measurements determine the matter content of the universe (averaged over all distances)∼to: 4 { baryon density Ω h2 = 0:024 0:001 b { matter density Ω h2 = 0:14 0:02 m Error estimates are mixture of serious studies, chemistry and miracles. Such a dark matter particle is not part of the Standard Model. Most generally, it could be a stable particle with electroweak interactions and a mass around 200 GeV. Unfortunately, we have not observed such a particle in direct of indirect searches for dark matter ) conclusive! Flavor Physics Flavor physics has been a major effort over the last decade | in particular once we put together B and K physics, neutrino physics, and low{energy searches for example for proton decay or neutrinoless double{beta decay. Unfortu- nately, all these measurements have revealed little but the existence of a finite neutrino mass of which we still do not know the over{all scale: { proton decay not observed { flavor changing neutral current not observed { neutrinoless double{β decay not observed { no unexplained effects in B physics { no unexplained effects in K physics { .... If we want to phrase it positively, we conclude that we have not found anything and not gained any clues for physics beyond the Standard Model unfortunately conclusive ) B. Theoretical hints Let us start from assuming that the Standard Model is a renormalizable theory (it has no inverse powers of mass in the Lagrangian). This means that it does not have a built{in energy scale where it breaks down. An exception is gravity, because we know that at energies above the Planck scale 1019 GeV gravitational interactions become strong and our world should be described by some combination of the Standard Model and quantum gravity. All current observables probe scales E / 100 TeV = 105 GeV so we can ignore Planck{scale or quantum gravity effects for now. Standard Model beyond tree-level At next{to{leading order, the (bare) leading order Higgs mass gets corrected by loops involving Standard Model particles: H W-Z t + + + t + ... We can for example compute the 4{point Higgs loop with the coupling: H H 2 3 2 mH = 4ig 2 − mW The amplitude for this diagram is given in terms of the 4{momentum q in the loop and in terms of the cutoff scale Λ. Note that the Standard Model with only dimension{4 operators does not offer an interpretation for such a scale, so 5 at the end of the argument we have to perform the limit Λ . Introducing a cutoff scale and sending it to infinity is certainly a physical regularization scheme: ! 1 Λ d4q 3 1 2 = i::: MH (2π)4 −4 q2 m2 Z − H d4q 1 1 = Pauli{Villars regularization (2π)4 q2 m2 − q2 Λ2 Z − H − d4q 1 = (m H2 Λ2) − − (2π)4 (q2 m2 )(q2 Λ2) Z − H − d4q 1 1 2δ(1 x y) = (m2 Λ2) dx dy − − H 4 2 2 − (2π) 0 0 [(q2 m )x + (q2 Λ2)y] Z Z Z − H − 1 1 d4q 1 = 2(m2 Λ2) dx dy δ(1 x y) H 4 2 2 − 0 0 − − (2π) [q2 xm yΛ2] Z Z Z − H − 1 1 i 2(m2 Λ2) dx dy δ(1 x y) ∼ H − − − 16π2 Z0 Z0 2iΛ2 3 m2 = ig2 H now with couplings −16π2 −4 m2 W 2 3 2 mH 2 = 2 g 2 Λ (4) −32π mW where in the first line we have set ( 3=4i:::) = 1 for simplicity.

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