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Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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test

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test

The Standard Model

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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
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Last modified: January 26, 2018

The Gauge Hierarchy Problem --------------------------- The Guahe hierarchy problem is perhaps the main motivation for supersymmetry. To understand what the problem is, it is necessary to look more closely at the mass of the Higgs boson. Consider the following 1-loop Feynman diagrams: (a) Represents the bare Higgs mass. (b) Represents the Higgs SELF INTERACTION. (c) Represents the Yukawa interaction. The corresponding Lagrangian is:    _ L = ∂μφ∂μφ - μ2φ2/2 + λφ4/4 - gYψψφ Case (b). Applying Feynman rules we get: I = λ∫d4p/(2π)4 1/(p2 - mH2) Where i/(p2 - mH2) is the massive bosonic propagator. From the discussion on Regularization and Renormalization we had (using dimensional regularization): I = (iλΔ/16π2)[2/ε - ln(μ2/Δ) + ln(4π) - γE + 1] __ = (iλmH2/16π2)(1 + ln(μ2/mH2) after MS Which is quadratically divergent. Likewise, case (c) is also calculated in the notes on Regularization and Renormalization. The 1 loop Yukawa correction to the Higgs mass is given by: Π(k) = -4g2(-i)[1/8π2(k2/6 + m2) - 3/16π2(ln(μ2/Δ))] Which, again, is quadratically divergent. Aside: We can schematically see this as follows: Π = -gY2∫ d4k/(2π)4μk + m)2/(k2 - m2)2μk + m)2 = k2 + 2km + m2 Therefore, I = -gY2∫ d4k/(2π)4 (k2 + 2km + m2)/(k2 - m2)2 = -gY2∫ d4k/(2π)4 (k2 + m2)/(k2 - m2)2 Write the numerator as k2 - m2 + 2m2 and the integral becomes: I = -gY2∫ d4k/(2π)4 1/(k2 - m2) + 2m/(k2 - m2)2 Note: There is a -ve in front of the g2 term because it represents a fermionic loop. The sign difference between fermionic and bosonic loops can be seen in the following diagram. Bosons Fermions ---------- ---------- A B AB A B AB - - -- - - -- + + + + - - - - + - + - Therefore, bosonic loops make a positive contribution to the vacuum energy while fermionic loops make a negative contribution. The diagram also illustrates the SPIN-STATISTICS THEOREM. If A and B swap positions there is no change in sign for the bosons but there is a change for the fermions. Thus, φ(x1)φ(x2) = φ(x2)φ(x1) While, ψ(x1)ψ(x2) = -ψ(x2)ψ(x1) The Problem ----------- We can write the physical Higgs mass with loop corrections (neglecting all factors of π etc.) as: mH2/2 = m02/2 + λmH2(1 + log terms) - gY2(p2 + 2mH2 + log terms) The log terms are of lesser importance because they are very nuch smaller than the squared terms. Again, we can write the above schematically as: mH2/2 ~ m02/2 + λmH2 - gY2mH2 Unfortunately, these quadratically divergent terms cannot be removed by renormalization. They are considered to be unphysical or 'unnatural'. As a result, one would expect that these large quantum corrections to mH2 would make the Higgs mass huge as mH approaches the scale at which new physics appears (the Planck mass). In fact, unless there is an incredible amount of fine tuning cancellation between the quadratic terms and the bare mass, the 1-loop correction to mH2 would be more than 30 orders of magnitude larger than the measured mH of a few hundred GeV (~(1019)2/(100)2). To compound issues, there are many, many combinations of possible Feynman diagrams leading to additional corrections to the mass, some positive and some negative with different powers of the coupling constant. For example, every fermion in the SM has a Yukawa coupling to the Higgs field and will quadratically diverge. In addition there are contributions from the gauge bosons. All of these terms would need to collectively cancel each other to achieve the physical mass! The Mass of the Higgs --------------------- The Higgs field is described by: V(φ) = -μ2φ2/2 + λφ4/4 This has a minimum (VEV) at dV(φ)/dφ = 0. Therefore, -μ2φ = λφ3 or φMIN = f = μ/√λ We know that f ~ 246 GeV and μH ~ 125 GeV. Therefore, λ < 1. Supersymmetry to the Rescue --------------------------- What if the Higgs had a fermionic superpartner, the Higsino, ~ H, that couples to the field with a coupling constant, λS. This would be represented by (b) in the following diagram: The cancellations would be exact if the mass and charge of the partner/superpartner pair were the same and λS = λY2. This is exactly the idea behind supersymmetric theories. However, because they have yet to be observed in the laboratory, superpartners must be heavier than their partners. The fact that there is not a perfect symmetry implies that supersymmetry is not an exact, unbroken symmetry. The implication is that the true symmetry is somehow spontaneously broken. The difference in masses jeopardizes the solution to the hierarchy problem because exact cancellation is spoiled - but not totally. The prediction is that the mass difference is small enough to still provide a high degree of cancellation. R-Parity -------- R-parity is a set of rule that define possible partner/ superpartner interactions. If we define 'normal' SM particles, N as + and supersymmetric particles, S as -1, the rules are: N -> NN N -> SS S -> NS For 2-body particle transitions the rules are: NN -> NN NN -> SS SS -> SS NS -> NS Therefore, R parity is a multiplicative quantum number. For example ~ ~~ H -> H -> H is not allowed while H -> FF -> H or ~~ H -> HH -> is allowed. (a), (b), (c) and (d) are valid. However, a diagram like the one below would not be. Dark Matter ----------- As we said before, superpartners can only be pair produced from SM particles and a superpartner can only produce a another superpartner and a normal partner. Thus, S / / N ------ \ \ S and, S / / S ------ \ \ N Because of their large mass, superpartners decay and are not freely floating around in the Universe. Superpartners will decay in a chain until the lightest superpartner (LSP) is produced (cascade decay). Thus, LSP / / ------ N / . . S / / ------ N S / / S ------ \ \ N The resulting LSP) is thought to be stable and electrically neutral. If it were charged the theory would be cosmologically disfavored. In addition, LSPs must interact weakly with ordinary matter and are not seen in detectors since they interact only by exchange of heavy virtual particles. These are exactly the characteristics required for Dark Matter, thought to make up most of the matter in the universe and to hold galaxies together. Fermionic and Gauge Boson Mass ------------------------------ The Hierachy problem does not impact fermions or the gauge bosons. In both cases the self energy varies logarithmically and are not quadratically divergent. (see the discussion on regularization and renormalizationfor the fermionic self energy calculation). More fundamentally, for fermions it is the chiral symmetry that protects the mass. The Lagarangian has a symmetry as the fermionic masses goes to 0. This guarantees that all corrections to the mass must also vanish as the mass goes to 0. We can see this from the discussion on regularization and renormalization. There we showed that the fermionic self energy correction to the mass is given by (schematically): δm ~ αmln(μ2/Δ) for 1 loop. δm is very small. For example, for μ2/m2 = 1030 the log = 69 Now, α = 1/137 ∴ δm ~ (1/137)* 69 ~ 0.5*m and as m -> 0 δm -> 0 For the gauge bosons, it is the gauge invariance that protects the mass. A mass term for a gauge field is m2AμAμ. However, such terms are not invariant and break the local symmetry. Therefore, m2 must be zero. The Vacuum Catastrphe --------------------- While Supersymmetry offers a possible solution to the Gauge Hierarchy problem. It does not explain the vacuum catastrophe problem that is discussed in another note. The Fine Tuning Problem ----------------------- The problem with the Higgs mass being so finely tuned is part of a wider question as why the Universe in generai is so finely tuned. The current standard model of particle physics has 25 freely adjustable parameters. In addition gravity has the cosmological constant. A small change in several of the dimensionless physical constants would make the Universe radically different from what we observe today. The famous scientist Stephen Hawking has stated, "The laws of science, as we know them at present, contain many fundamental numbers, like the size of the electric charge of the electron and the ratio of the masses of the proton and the electron. The remarkable fact is that the values of these numbers seem to have been very finely adjusted to make possible the development of life." If any of several fundamental constants were only slightly different, the Universe would be unlikely to be conducive to the establishment and development of matter, astronomical structures, elemental diversity, or life as we know it. For example, if the coupling constant for the strong force was 2% larger, while the other constants were unchanged, the physics of stars would be drastically different. Possible explanations for fine-tuning are the subject of vigorous discussions among philosophers, scientists, theologians, and proponents of creationism.