The Basics of String Theory
I will also give the basics of internal symmetry groups and symmetry breaking in quantum field theory, in particular in the Stan- dard Model of weak interaction, before going to the usage of supersymmetry in todays particle physics, viz. I also include something about the most common models for media- tion of supersymmetry breaking and a short survey of the state of affairs in the search for supersymmetric partners and the Higgs bosons.
For the reader who is interested in the phenomenology of supersymmetry but who does not wish to delve too deep into the mathematics of the subject I suggest skipping chapter 2, except for section 2. The summary should be enough for the reader to be able to follow the subsequent development of the subject in this thesis. There are sev- eral reasons for this.
First, it seems appealing in its own right to have a theory which unites the two different groups of particles found in nature, bosons and fermions, into a common representation.
Second, supersymme- try is necessary in superstring theory, by many seen as the best candidate of a TOE theory of everything ; in particular, local supersymmetry gives rise to supergravity, which could possibly be an answer to the attempts to unite general relativity with quantum field theory1.
Third, in the Standard Model there is the problem with quadratic divergences the mass hierarchy prob- lem, to be discussed below which can be solved by supersymmetry. In the scalar case, though, masses are quadratically divergent. This holds in particular for the Higgs boson, the only elementary scalar particle in the Standard Model.
This problem can be solved by fine-tuning the bare mass mH so that it contains a large negative term which almost exactly cancels the 1-loop correction leaving the small electroweak scale mass. Such a series of cancellations might be technically feasible, but seems ad hoc without a deeper explanation of why the cancellation terms would have such fine-tuned values. An alternative, and much simpler, solution is obtained if we postulate the existence of new particles with similar masses but spin that differs by one-half.
But thus we need a theory which in- cludes bosons and fermions within the same framework, i. Which are then the largest mass differences that can be allowed? Also, an understanding of the concepts presented here should give the reader enough background to be able to go further on her own reading about e. My intention is that I shall have given enough information for any reader fulfilling the requirements of section 1.
Elements of a basis of the Lie algebra are thus generators of the corresponding Lie group. A familiar example is the isospin group of transformations taking up-quarks into down-quarks. This means that such transformations are by necessity bosonic, since they cannot transform like spinors under Lorentz transfor- mations see below. Thus we need to introduce some extension of the Lie algebra in order to create fermionic operators. Since the supersymmetric algebra is a Z2 graded Lie algebra I here restrict my attention to these.
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If we further impose the Majorana condition eq. This applies in all cases we will discuss here. Another representation of the Dirac equation is the Weyl, or Chiral, rep- resentation, appropriate for taking the ultra-relativistic limit. We consider the subgroup of transformations with the homoge- neous Lorentz transformations factored out, i. For reasons that will be apparent later, when we want to construct the chiral and vec- tor superfields in section 2.
We now demonstrate how eq. Also note that, by construction, the differential repre- sentations of the spinor charges satisfy the supersymmetry algebra 2. The covariant derivatives will be used in section 2. In this section the first task lies before us, while the latter will have to wait until section 2.
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If we thus construct an action using dx d x , this action will be invariant under supersymmetry transformations provided the fields fall off sufficiently fast at infinity a condition that is always assumed to be fulfilled. Chiral fields are used to represent matter fields, and vector fields are used to represent gauge fields, as will be seen later. A y and F y describe complex scalar fields. The equality of the number of bosonic and fermionic degrees of freedom is a feature common to all supersymmetric multiplets see section 2.
Products of scalar fields are again scalar fields i. Making the correct identifications in the expansion of a superfield, eq. Thus any action using this component will be invariant under supersymmetric transformations, just as an action using the d-component field in eq. But apparently the spinor cannot come alone. The field F x is called an auxiliary field since it can be eliminated using the equations of motion of the fields obtained through the Lagrangian more of auxiliary fields in section 2.
In order to construct Lagrangians for chiral superfields we use, just as for chiral spinor fields, the product of a left-handed and a right-handed field. Since this term transforms into a total derivative it is the proper term to use as the Lagrangian. But this will be further explored in section 2. The vector supermultiplet has a considerably larger number of component fields than the chiral superfields. In order to find this we need to go into some theory of generalized gauge transformations, in order to choose a specific gauge in which most of the component fields disappear.
This is done in the next section. We adjust the expansion 2. This leads to the following transformations of the component fields using eqs.
This particular gauge is called Wess-Zumino WZ gauge. D x is an auxiliary field. The Wess-Zumino gauge is not covariant though; under a supersymmetric transformation the variation of the elimi- nated fields does not vanish. Hence the gauge must be forced onto the field for each choice of supersymmetric frame.
From the proper- ties of integration of Grassman numbers see section 2. In order to get a renormalizable theory scalar superfields cannot enter the Lagrangian to an order higher than 3. The product of two or more left-handed chiral superfields is again a chiral superfield, and similarly for right-handed chiral superfields see section 2. The reason why it is necessary to have an auxiliary field in the off-shell action is that it is required in order for the number of fermionic and bosonic degrees of freedom to be equal here 4N each.
This is an important property of the scalar potential of supersymmetric theories.
Equations 2. Equation 2. The Dirac equation gives a coupling between the first and second el- ement and the third and fourth element of the Majorana spinor, re- ducing the number of fermionic degrees of freedom to 2, while the Klein-Gordon equation does not give any coupling between the bosonic degrees of freedom. Thus we still, after taking the equations of motion into account, have equal numbers of degrees of freedom for bosons and fermions 2 each.
Thus we must, if supersymmetry is to be inherent in nature, find some mechanism that differentiates the masses of the fermions and their scalar supersymmetric partners. This will be explored in section 4 on supersymmetric extensions of the Standard Model. Furthermore, they are both invariant under supersymmetric gauge transformations of eq.
The real scalar field D is an auxiliary field and can be eliminated using the equations of motion just like the field F in eq. The invariance of eq. By means of the superpotential it is possible to write actions in a more concise manner. The general scalar superfield Lagrangian 2. We now get the following expression equivalent to eq. In conventional gauge theory the electromagnetic interaction can be defined through a local U 1 gauge transformation see section 3. For the vector field V we use the Wess-Zumino gauge of eq.
There are no terms in the Lagrangian 2. This gives us the full gauge invariant action cf. Using the general ideas leading to this action, the reader should now be able to construct any supersymmetric gauge theories such as SU 2 and SU 3 gauge theories. The complexities arising are the same as in the non- supersymmetric case. The superspace formalism, including the superfields and the construction of invariant Lagrangians, is indeed powerful!
The generators Qa are defined to form a Majorana spinor. F x is an auxiliary field which can be eliminated from the action integral using the equations of motion. Products of left-handed chiral superfields are again left-handed chiral su- perfields and analogously for right-handed chiral superfields. Then comes the particle content of the unbroken gauge theory, how SU 2 symmetry breaking is accomplished using the Higgs mechanism and the resulting particle spectrum, and finally something about families and the difference between gauge and mass eigenstates This allows for choosing different conditions on the potentials, thus simplifying the calculation at hand.
This, however, is only the simplest of its consequences. As will be seen in the following, gauge theory has so many important properties that it has actually become the most important tool available for the development of theoretical physics. An understanding of gauge theory is fundamental for the understanding of the subjects to come. Gauge theories originated with the electromagnetic theory of Maxwell. The electromagnetic theory is an Abelian symmetric gauge theory, in con- trast to the non-Abelian Yang-Mills theories.
Since the SU 2 symmetry of the Standard Model is a non-Abelian gauge theory, we will first take a look at general Yang-Mills theories before continuing to the Standard Model gauge symmetries. This section might be skipped in a first reading, although it is important to understand the origin of the formulas in the rest of this chapter.
Introductory supersymmetry literature for the perplexed | An American Physics Student in England
A local U 1 symmetry means that the field equations involved of the fields involved, i. This is just the kinetic energy Lagrangian Maxwell proposed for the electromagnetic field. You should be familiar with words like superpotential, Kahler potential, super covariant derivative, etc. Once again Bailin and Love is my first choice here, while Wess and Bagger have some very useful insights if at times cryptically terse.