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Strongly Coupled Gauge Theory Strongly coupled gauge theory Yong-Hui Qi†,⋆ † Institute for High Energy Phenomenology, Newman Laboratory of Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA ⋆ Physics Department, Tsinghua University, Beijing 100084, China [email protected], [email protected] Abstract In this report, we discuss the physics of strongly coupled gauge theory, from ordinary space-time to super-space. Firstly we briefly review the topological objects of Yang-Mills theory in ordinary space-time, where we focus on the electric-magnetic duality. Later on we extend our discussion to Super Yang-Mills with matter added, where we discuss holomorphic gauge couplings and Seiberg duality. Finally we discuss Seiberg-Witten theory. The application to gauge-mediated dynamical SUSY breaking based on MSSM is straightforward. In above discussion, we will mainly focus on SUSY with N 2. [3][4] ≤ (Dated: Jan 22th, 2010) PACS numbers: I. TOPOLOGICAL FIELD THEORY IN PARTICLE PHYSICS A. 1+1 Dimension solitons We have the experience in 1+1 Dimensions for a real scalar field theories with a quartic or sine-Gordon potential. 1 1 = φ˙2 (∂ φ)2 V (φ) (1) L 2 − 2 x − where λ 2πφ V (φ)= (φ2 v2)2, or V (φ)= A(1 cos ) (2) 4! − − v The important physical results of these model is that the vacuum of the ground states are degenerate(two fold for the λφ4 potential and infinity for the sin-Gordon model). Consider small fluctuation around the vacuum, namely φ = v + φˆ (3) 2 ∂2V expands the potential of the scalar field in perturbative way, we find the mass(m = ∂φ2 >=0) of the correspond- ing scalar particle, due to fluctuation or excitation of the scalar field, as well as three and four point vertices, if the mass square m2 < 0, these particle are called tachyons, which are not physical and would make the vacuum unstable. The fact that the vacuum are degenerate, results in(by solving Euler-lagrange equation) these models posses highly non-trivial static solutions, that interpolate between the vacuum, which are called solitons or kinks, which are 1 2v m(x x ) φ(x)= v tanh m(x x ), φ(x)= arctan(e − 0 ) (4) 2 − 0 π for λφ4 and sin-Gordon potentials, separately. The difference is the first one is a renormalizable one while the second is non-renormalizable. Where x0 is position transition between two nearest vacuum take place. The solution of solitons(φ(x)) are basicly located at the vacuum value(v), and can be approximated by a Heaviside function φ(x)= vθ(x x ) for x x Γ 1/m (5) − 0 | − 0| ≫ ∝ which interpolate between two nearest vacuum, where the kinetic term and potential approach vanishes at spatial infinity( x ). Moreover the solution has finite positive energy and it turns out to be antipropotional to the self couplings.| | → ∞ m3 E (6) ∝ λ which means the solitons are very heavy and massive in the perturbative limit. In the perturbation language, we are dealing with a theory describing elementary particles with massive gap due to light fluctuation of vacuum. There ∂ is an ambiguity due the the sign in front of the φ(x), which leads to two kind solutions, solition( ∂x φ(x) > 0) ∂ ± and anti-soliton( ∂x φ(x) < 0). The picture of the theory can be sequences of solitons and anti-solitons, in the ”dilute 1 gas” approximation, means the solutions of them only valid for widely separated objects( x10 x20 m ). The anti-slotions are anti-particle of the solitions like in a complex scalar field, and the vacuum can| be− distinguished| ≫ by label it with the difference between the number of solitons and anti-solitions(∆n = n+ n ) for a potential with symmetry. − − If we consider the chiral fermions in a soliton background, namely we couple the fermions to the solition by the usual Yukawa coupling. = φ¯(γµ∂ + gφ(x))φ (7) L µ After spontaneous symmetry breaking of scalar field φ, it take the vacuum expectation value v every, the fermion also get a mass and become massive(mφ = gv) and the Hamiltonian of fermions come in pairs E with E m, ”electrons” and ”holenons” with positive and negative energy separately. The chiral symmetry of± the fermions| | ≥ are broken due to mass term of fermions. But the introducing of the opposite vacuum solution will restore the symmetry, in other words, making the Yukawa couplings term chiral invariant again. Thus the chiral invariant vacuum state of fermions are a Z2 degenerate vacuum, with φ and φ corresponding to the same energy E for electrons or E for the holenons. We are also interested in the zero modes(with− zero energy, and electron and holenons are the same− particle since E =0) of the fermions in a solition background. It turns out to be that the only normalisable(non-divergent) zero mode± state(wavefunctions) of fermions are spin up fermion (positive parityP = +1) localized at the transition position x0 of the soliton and spin down fermion(negative parity P = 1) localized on the transition position of x0 of the anti-solition. − For the concreteness, the wavefunctions of zero modes(Because they have eigen energy E = 0, they do not have effect on the energy whether the states are filled or not) of fermion are m m m 2 m 1 1 m 2 m 1 1 φ cosh (x x0)− ψ φ cosh (x x0)− ψ (8) ↑ ∼ 2 − √ 1 ↓ ∼ 2 − √ 1 2 2 − which are named Jackiw-Rebbi modes and they are both chiral eigen-state and related by a Z2 discrete symmetry. B. 1+ n(n ≥ 2) solitons - Domain Wall or Membrane or 2-brane The higher dimensional extension of solitons(kinks) are domain walls, which are transition regions between do- mains with different eigen-values ∆n to label the different(degenerate) vacuums, they are related by a discrete sym- metryx. The chiral fermions inside the domains are massive and on the domain walls they are massless, along the domain wall, we view them as lower dimensional Dirac fermions. The first (2+1) domain walls are vortices or strings. For a complex scalar field(or two component real, where φ = φ1 + iφ2), 2 µ λ v 2 = ∂ φ∗∂ φ (φ∗φ ) (9) L − µ − 2 − 2 The vacuum manifold(moduli) is a circle S1, with v φ eiϕ (10) → √2 , where ϕ is the polar angle in coordinate space. This complex scalar field theory in (2+1) dimension, with a U(1) 2 global invariance, characterize a vortex, and its energy E = dx (∂φ~ ∗∂φ~ + V (φ, φ∗)) is logarithmically divergent. The approach to cure the divergence is by making the U(1) globalR invariance a local one, namely it is necessary to introduce gauge field to cure the logarithmic divergent of the energy. ∂φ D φ = (∂ ieA )φ (11) → µ µ − µ The covariant derivativeDφ~ kinetic term will give a good converge than the ordinary derivative∂φ~ kinetic term. It is clear that the logarithmic divergent is absorbed in the gauge field A~. Intuitively speaking, the gauge field has to be along in the direction of the phase ϕ. It turns out that 1 x A ǫ j (12) i →− e ij r2 and as expected it has only a ϕ component, 1 A 0, A (13) r → ϕ → er and it is a pure gauge asymptotically and the field strength vanishes, 1 A~ ∂ϕ~ F 0 (14) → e ij → Thus the gauge filed A~ is a circular one and leads to a vortex with quantized magnetic flux, 2π 1 Φ= Bd~σ~ = Ad~x~ ngm = n = (ϕ(r, θ2π) ϕ(r, θ)) (15) ZS ZC=∂S ≡ e e − which is called topological conservation law for the original complex scalar field. The vortex carries a quantized magnetic flux number n, as the quantum number to label the different topological classes. The general total flux will 2π be of the form (n1 + n2 + n3) e . The choices for φ and A are solutions of the Euler-Lagrange equations asymptotically, the gauge field A runs into a singularity when we extend them naively towards the origin. Instead we can make an ansatz for φ and A and try to solve the problem. The model we have discussed so far is a 2+1 dimensional Abelian Higgs Model with gauge strength term added, 2 λ v 2 1 µν = D φ∗D φ (φ∗φ ) F F (16) L − µ µ − 2 − 2 − 4 µν then the energy of the model(static solution) is 2 1 2 λ 1 2 2 E = dx [D φ∗D φ + F + (φ∗φ v ) ] i i 2 12 2 − 2 2R 2 2 2 ~2 1 2 λ 1 2 2 = dx [(∂iφ) + e φ A + 2 B + 2 (φ∗φ 2 v ) ] 2− 2 = dx2[(∂ φ Reǫ A φ)2 eφ2B + 1 (B √λ(φ2 v ))2 √λ(φ2 v )B] (17) i ± ij j ± 2 ± − 2 ∓ − 2 R it turns out that before and after the spontaneously symmetry breaking, the massive scalar(Higgs) and the massive vector gauge boson have the mass v λ 2 mφ = √2e = ev, mA = v (18) √2 2 λ 2 2 2 v2 ~2 2 from the term 2 v φ and e 2 A separately. If we make a special choice, namely λ = e , the mass of the Higgs will be the same− as that| | of vector gauge boson. After SSB, the energy will be of the form 2 2 2 2 2 v 1 2 v 2 v 2 E = d x[(∂iφ eǫij Aj φ) e B + (B √λ(φ )) ] e Bd x (19) Z ± ± 2 2 ± − 2 ≥ 2 | Z | and has a lowest bound, which can be saturated with two equation of motions and it is v2 2π πm2 E e n = nπv2 = n (20) ≥ 2 e e2 m2 The energy will be typically proportional to g2 for a heavy topological objects in the perturbative viewpoint.
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