introduced with the effect that even an effective field theory is created: thus the force between the observed particles is mediated by ”mesons”. Those particles are at their own also compound of quarks, but here they are regarded as being the ”elementary” force carrying entities. Thus one neglects the quarks and treats the hadron as being ”atomic” and described by an effective theory which gives them effective properties, without asking where they come from.
Another example is theBCS–theory which describes superconductivity in metals. The underlying theory in that case is the quantum mechanics of solid bodies, and especially of electrons and phonons3 Those interact with the effect of binding two
electrons in a Cooper–pair. This pair of electrons behaves as a bosonic particle, not obeying the rules for fermionic states.
3.2
Techniques
Despite the number of particles and that of gauge groups emergent from string theory, the latter also makes predictions which can be analyzed within the respective low energy effective action. The energy scale set in the framework of effective theories will also be the battle field where all the concepts introduced before, i.e. string theory, and effective actions will meet together supplied by some other mathematical ideas, to be presented in next chapters. As stated before, string theory has infinitely many vibrating modes, almost all of which are massive. So if would like to set up an effective theory and consider all those excitations, it would require an infinite number of differential equation coming from the effective action in order to describe them. The solution is just to neglect the majority of the string states. Since the string tension is in the region of the Planck mass ( 1019 GeV) the massive modes are extremely heavy such that we
can be sure that this simplification will not affect the effective theory, at least not at the scale we are looking at. It would require enormous energies to come in the regime where also the massive states would show up in the action with new effects. Thus, the heavy modes can simply be integrated out, such that in the effective action just the massless modes contribute. It is worth to notice that the number of the massless modes is finite and even rather small.
Several ways exist in which we can come to that effective theory beginning with string theory. One way is to formulate string theory from the beginning on in an given background. A background denotes just the fields describing the space–time in which the string evolves, thus the well known metric. Generally, this background is usually given in terms of an traceless metric Gµν, its antisymmetric part Bµν and the trace
Φ. These quantities are respectively called the graviton, antisymmetric tensor field and the dilaton. The formulation of string theory is, as already said, very restrictive, since we have to preserve all the symmetries from the classical action. This way also the background fields are constrained. When imposing conformal invariance on
3A phonon denotes a vibrating mode of a lattice in a solid body. It proves that such vibrations
can propagate and even stay localized showing particle properties, fact which animated physicists to describe it as a (quasi)particle.
32 3 The concept of effective theories
string theory in that background, also the background defined above has to fulfill some requirements. That consistency is nothing else than differential equations which are exactly the equations of motion for the named fields, see e.g. [68]. We shall though not follow that method because its difficulty but will approach another method, somehow more simpler to calculate the effective action. We will make use of the fact that correlation functions calculated in field theory are equal to the corresponding amplitudes in string theory. By ”corresponding amplitudes” it is meant that one takes the same topology of the interaction4, same number and sort of interacting particles
and of course the same background. This method we will use in the present work. As a next step we shall have some thoughts on the field theory. Since we have to compare the field theory with string theory (by computing the same scattering process) we could try to guess a Lagrangian for the field theory and then compare it with string theory and finally just adjust it. Thus one could write down the most general, non–redundant ansatz. The non–redundancy should be emphasized here, for a field theory Lagrangian is not unique, since field shifts and redefinitions don’t change the physics (S–matrix). (More specific field operations which let the physics unchanged will be presented in the Born–Infeld section). So when an ansatz is written down, great attention is to be paid in order not to count the same term twice or often, just because it is written different! Such an ansatz, which is also valid forDspace–time dimensions, could be of the following form
Sef f =α0 −D/2
X
n,m
0
α01/2(n+m)cnm∂nΦm, (3.1)
where the coefficients cnm are unknowns, to be determined from the corresponding
string scattering amplitudes. Actually those are the quantities which are wishful to be determined, for they exactly encode the ratio between the consecutive terms in the action. As the notation already suggests, cnm are ordered with respect to the number
of derivatives acting on the number of fields. Each term is a combination of derivatives and the respective fields under consideration. Thuscnm exactly determines how many
terms contain just powers of Φ or how many contain a number of derivatives acting on the specific number of fields Φ. As stated above, the prime on the sum indicates we have built the expression such that each term present is unique, i.e. every term is counted once and no field transformation or Lagrangian symmetry is able to relate two different terms in the expansion (3.1). Furthermore, the expression is organized as a power series in the string tension α0, which will necessary be encountered when
computing the amplitudes. Possible other constants may enter the series, like the string coupling constant gString. The latter organizes also the string loop expansion.
Last but not least, each term is in one to one correspondence with a specific Feynman– diagram.The correspondence can be established when considering solely the number
4Feynman graphs can be classified in tree–, one loop–, two loop–, etc, diagrams withN external
particles. Further, tree–diagrams can be reducible or not, depending wether some internal states are propagated in the process.