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Supersymmetry to the rescue?

   One troubling aspect of spontaneously broken gauge field theories based on the Higgs mechanism of giving mass to gauge bosons is that it's not only the coupling constants, but also the masses, that get renormalized by quantum corrections from taking into account all possible virtual processes at all possible momentum scales.
    Suppose there is new physics at some scale L, so that the Standard Model of particle physics is no longer adequate to describe physics at higher momentum scales. The quantum corrections to fermion masses would depend on that cutoff scale L only logarithmically

Fremion mass correction

whereas the scalar Higgs particles would exhibit a quadratic dependence on the cutoff scale

This means that the masses of Higgs particles are very sensitive to the scale at which new physics emerges.
   This sensitivity is called the gauge hierarchy problem, because the Higgs mass is related to the masses of the gauge bosons in the spontaneously broken gauge theory. The original question "How do the gauge bosons get mass without spoiling gauge invariance?" was only partially answered by the Higgs mechanism. In a way, question wasn't answered by the Higgs mechanism, it was just transferred up to a new level, to the question: "Why does the Higgs mass remain stable against large quantum corrections from high energy scales?"
   The interesting thing about scalar mass divergences from virtual particle loops is that virtual fermions and virtual bosons contribute with opposite signs and could cancel each other completely if for every boson, there were a fermion of the same mass and charge.
   At the level of quantum mechanics, this type of Fermi-Bose symmetry would entail some quantum operator, let's call it Q, whose action would be to transform bosons into fermions, and vice versa. In operator language this would be written

Supercharge operator

And since this is a symmetry, this operator must commute with the Hamiltonian

Supercharge commutes with H

Such a theory is called a supersymmetric theory, and the operator Q is called the supercharge. Since the supercharge corresponds to an operator that changes a particle with spin one half to a particle with spin one or zero, the supercharge itself must be a spinor that carries one half unit of spin of its own.
   Supersymmetry is such a powerful idea because it is a symmetry under the exchange of classical and quantum physics. Bosons are particles that obey Bose statistics, meaning that any number of them can occupy the same quantum state at the same time. Fermions obey Fermi statistics and only one fermion can occupy any given quantum state at one time. But the classical limit of quantum physics is approached when the occupation numbers of available states are very high. For example, in this limit, the quantum photon field behaves like the classical electromagnetic field as described by Maxwell's equations. But then the conclusion for fermions is that there is no classical limit for fermions. Fermionic fields are inherently quantum relativistic phenomena.
   Therefore, any symmetry that exchanges fermions and bosons is a symmetry that exchanges physics that has a classical limit with physics that has no classical limit. So such a symmetry should have very powerful consequences indeed.
   One big problem with supersymmetry: in the particle physics that is observed in today's accelerators, every boson most definitely does NOT have a matching fermion with the same mass and charge. So if supersymmetry is a symmetry of Nature, it must somehow be broken. It's easy enough for an expert to construct a supersymmetric theory. It's breaking the symmetry, without destroying the beneficial effects of that symmetry, that has been the hardest part of the program to fulfill.
   But would a broken supersymmetric theory still be able to solve the gauge hierarchy problem? That depends on the scale at which the supersymmetry is broken, and the method by which it is broken. In other words, it's still an open question. Stay tuned.

How was supersymmetry developed?

   Supersymmetry was not developed originally as a means of solving the gauge hierarchy problem. Supersymmetry was first developed independently by two different groups of theorists separated by the Cold War back in the 1970s. One group in the USSR was exploring the mathematics of space-time symmetry, and the other group in the West was trying to add fermions to bosonic string theory.
   In the USSR, mathematicians Gol'fand and Likhtman wanted to do something exotic with the group theory of spacetime symmetries. The usual group of spacetime symmetries in relativistic quantum field theory is called the Poincaré group. This group includes symmetries under spatial rotations, spacetime boosts and translations in space and time.    
   The action of the group can be described by the algebra of the group, which is defined by a set of commutation relations between the generators of infinitesimal group transformations. The algebra of the Poincaré group looks like:

Poincare algebra

The momentum generator Pm generates space and time translations. The Lorentz matrices Jmn generate rotations in space and Lorentz boosts in spacetime. These are all bosonic symmetries, which ought to be true because momentum conservation and Lorentz invariance are present in classical physics.
   But the Poincaré group also has representations that describe fermions. Since spin 1/2 particles arise as solutions to a relativistically invariant equation -- the Dirac equation -- this is to be expected. If there are spin 1/2 particles, could there be spin 1/2 symmetry generators in a spacetime symmetry algebra? Yes! One way to add them is shown below:

Super-Poincare algebra

What are the new symmetry generators labeled by Q? These are the supercharges mentioned above.
    What Gol'fand and Likhtman ended up with was the group theory of supersymmetric transformations in four spacetime dimensions, and using this new type of symmetry, they constructed the first supersymmetric quantum field theory.
   Unfortunately for them, their work was ignored, both in the Soviet Union and in the West, until years later when supersymmetry finally mushroomed into a major topic of investigation in particle physics. In 1972, Gol'fand was judged one of the least important researchers in his group at FIAN in Moscow, and so he was let go in a cost reduction drive in 1973. He remained unemployed for seven years, until pressure from the world physics community led to his rehiring in 1980.

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Introduction to Supersymmetry by Adel Bilal
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Introduction to Supersymmetry: Astrophysical and Phenomenological Constraints by Keith A. Olive

String Theory Origins of Supersymmetry by John H. Schwarz
Introduction to the Yuri Golfand Memorial Volume "Many Faces of the Superworld" by M. Shifman


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