TRIUMF Summer Institute 2006 Collider and Energy Frontier Physics Beyond the Standard Model Lecture 4 Heather Logan Carleton University Plan Lecture 1 Monday July 17 - Why BSM? - Supersymmetry Lecture 2 Monday July 17 - Supersymmetry continued: phenomenology Lecture 3 Wednesday July 19 - Large extra dimensions: ADD - Universal extra dimensions; particle spins and UED vs. SUSY Lecture 4 Thursday July 20 - Deconstruction and the Little Higgs - T-parity Lecture 5 Friday July 21 - Warped extra dimensions: RS - RS and Technicolour Heather Logan Beyond the Standard Model – 4 TSI’06 The field theory defined by this diagram is both anomaly and asymptotically free for a Extra dimensional theories becamewide veryrange of popular,m and n. At dis buttances sh calculatingort compared to both 1/Λ and 1/Λs, the theory is well-described by (N copies of) four-dimensional weakly interacting massless fermions and some things in them was hard. gauge bosons. What does the theory look like at longer distances? In the limit where Λs ! Λ the long The gauge couplings run like a powerdist law,ance be insteadhavior is also ofsimp logarithmicallyle. At energy scales near Λs the SU(m) gauge coupling is quite weak, and may be treated perturbatively. At this scale each of the SU(n) groups become The theory requires a cutoff, where physicsstrong, causi becomesng the fermions stronglyto condense coupledin pairs: a non-zero expectation value forms for Physical results become dependent oneach detailspair of ferm ofions theconne cutoffcted to a g procedureiven strong gauge group: 3 "χi,i ψi,i+1# ∼ 4πfs Ui,i+1 i = 1, . , N (2.3) where fs ∼ Λs/(4π) and Ui,i+1 is an m × m unitary matrix parameterizing the direction of In the 4-D picture, problem was thethe co infinitendensate. The towerconfining s oftrong KKinteract modes.ions also produce a spectrum of “hadrons”, analogues of ordinary glueballs and baryons, all with masses on the order of Λs ∼ 4πfs. N Need a “toy theory” with onlyB theelow the firstscale Λs fewthe theo KKry can b modes.e described as a Π1 SU(m) gauge theory coupled to N → “Dimensional deconstructionno”!n-linear sigma model fields, each transforming as −1 Ui,i+1 → gi (x)Ui,i+1gi+1(x) . (2.4) Hill, Pokorski & Wang, hep-th/0104035; Arkani-Hamed, Cohen & Georgi, hep-th/0104005 We may use a diagram similar to the original moose to describe this “condensed” theory: - G G Discretize (or “latticize”) the 5th dimension. - - G G Introduce n copies of the extra-dimensional gauge group - “Connect” them with scalar fields G G Give the scalar fields a vev to break all but one of the - - gauge symmetries - G G The broken gauge generators are the n − 1 KK modes Figure 2: A condensed moose diagram Heather Logan Beyond the StandardThe low Model-energy e –ffec 4tive action for this non-line TSI’06ar sigma model is 1 N N S = d4x − tr F 2 + f 2 tr (D U )†DµU + · · · (2.5) 2g2 j s µ j,j+1 j,j+1 " j=1 j=1 & ! # # $ % j j+1 where the covariant derivative is DµUj,j+1 ≡ ∂µUj,j+1 − iAµUj,j+1 + iUj,j+1Aµ and the dots represent higher dimension operators that are irrelevant at low energies. The action for 3 2 SU(n) × SU(m) moose Our example field theories, all of which will be four dimensional, contain gauge fields and fermions, and are conveniently summarized in a pictorial representation, referred to variously as “moose” [3, for example] or “quiver” [4] diagrams. In such diagrams gauge groups are represented by open circles, and fermions by single directed lines attached to these circles. A line directed away from a circle corresponds to a set of Weyl fermions transforming as the fundamental representation of the gauge group, while a line directed toward a circle corre- sponds to a set of Weyl fermions transforming as the complex conjugate of the fundamental To avoid scalars, use trick from Technicolour:representation. The moose diagram we will consider is the N-sided polygon representing - G - Add a different gauge group “in between” each pair G s G - - G G of lattice sites s s - - G G Connect pairs of gauge groups with fermions - Send the “in between” gauge couplings strong Gs - Fermions condense and make a nonlinear sigma model G G - (just like the pions in QCD) - G G s s - All the extra Goldstone bosons get eaten by the mas- - - - G G G s sive gauge bosons Figure 1: A moose diagram. This diagram is called a moose N N a field theory with a G × Gs gauge group and fermions transforming bi-linearly under “nearest-neighbor” pairs of gauge transformations. For definiteness we will take G = SU(m) and Gs = SU(n). We will impose a cyclic In 4-dim, the moose gives a lightsym scalarmetry to ke fieldep all SU which(m) gauge c isoup thelings equ totallyal to a common value g, and all SU(n) symmetric linear combination ofg allauge c theouplings condensedequal to gs. By dimen scalars.sional transmutation we may equally well describe this theory by two corresponding dimensionful parameters, Λ and Λs. Each side of this polygon Classically it is massless: a (Nambu-)Goldstonedescribes two types of fermions boson.transforming under the three gauge groups associated with this side, SU (m) × SU (n) × SU (m): From quantum effects in the low-energyi theoryi i+1 it picks up a - - quadratically divergent mass. G Gs G χ But because it is a “nonlocal” object (stretchedi i around,i i ψi,i+1 thei+1 moose), the quadratic divergence is cut off by theχi,i energytransform scaleing as (m, ofn¯, 1) (2.1) ψi,i+1 transforming as (1, n, m¯ ) (2.2) the moose (would be 1/R in the continuum extra-dim version). where i = 1, . , N (and i = 0 is periodically identified with i = N). No longer massless: a pseudo-Nambu-Goldstone boson (PNGB). 2 Heather Logan Beyond the Standard Model – 4 TSI’06 6 top W W λt λt h g 2 _g 2 x x λt f h φ χL χR χR χL _ __λt 2f _ h h λ λ Figure 6. Canceling the top quark loop. Figure 7. Canceling the gauge and Higgs loops. constants required for the cancellations one has ticular also no mass. The “radial mode” r does only shifted the fine tuning from the Higgs mass obtain a mass from the potential and can be in- to the coupling constants of the new fields. How- tegrated out (Figure 7.). Note that this argu- Why is the “Goldstoneever, Little H boson”iggs theories hav massless?e such a symmetry built in. To discover it we will take a little detour. 3.1. Higgs as a Pseudo-Goldstone boson There is a globalI symmetry,n this section we identify whichthe symmetry sponta-which makes the identification of coupling constants neously breaks, givingof the prev aious Goldstonediscussion radiatively boson:stable and therefore “technically natural”. To do so we first r the flat directionb aroundriefly review Gol thedstone bo bottomsons. of the Massless Goldstone bosons always arise when potential. global symmetries are spontaneously broken by the vacuum. Consider for example the theory of a θ complex scalar field Φ with potential = ( Φ ). Why does the “pseudo-Nambu-GoldstoneThis potential preserves a global U(1)VsymVm|et|ry boson” get a mass? Φ ei!Φ. If the potential induces a vacuum The global symmetryexp→ectatio isn vaexplicitlylue for < Φ >= f tbroken.hen the U(1) Figure 8. The “Mexican hat” potential for Φ. The symmetry is spontaneously broken and a mass- black dot represents the vacuum expectation value There are some termsless Goldsto inne bo theson arises Lagrangian. To see this explicitly, (interactions)f, r is the radial mode and θ thatthe Goldst areone boson. it is convenient to parameterize Φ = (v + r)eiθ/f , not symmetric underwhere r a thend θ are globalreal fields. Bec symmetry.ause of the global U(1) symmetry θ can be removed from the poten- ment is based only on the existence of the U(1) Interactions (in loops!)tial by a spa communicatece-time dependent U(1) trans thatforma- breakingsymmetry and is tothere thefore sta PNGB.ble under radiative tion with " = θ/f. The resulting Lagrangian corrections. does not contai−n θ except in derivative interac- To summarize, we now have a simple mecha- tions. Therefore θ has no potential and in par- nism for generating massless scalar fields. Unfor- Why does it get only a small mass? Any one interaction preserves enough global symmetry to forbid the PNGB from getting a mass. Need multiple interactions acting together to generate the mass. To get multiple interactions, need to go to a higher loop order. PNGB mass is “protected” at 1-loop order! This idea is called “collective symmetry breaking”. Heather Logan Beyond the Standard Model – 4 TSI’06 Collective symmetry breaking offers a possible solution to the Little Hierarchy. What is the Little Hierarchy? The big hierarchy is the hierarchy between mW and MPl. There are strong constraints on Technicolour (and strongly- coupled theories in general) from precision electroweak measure- ments [at LEP]. A general analysis indicates that new strongly-coupled physics [without R- or KK-parity] shouldn’t be lighter than roughly 10 TeV. 2 2 2 2 ∆m ∼ (g /16π )Λ → mW ∼ (g/4π)Λ ∼ 0.1 Λ for naturalness. So we want Λ ∼ TeV. Because of EW precision constraints, this is difficult! But if ∆m2 appeared only at 2-loops, then: 2 2 2 2 2 2 ∆m ∼ (g /16π ) Λ → mW ∼ (g/4π) Λ ∼ 0.01 Λ So Λ ∼ 10 TeV is ok! Heather Logan Beyond the Standard Model – 4 TSI’06 Collective symmetry breaking offers a possible solution to the “Little Hierarchy problem”.