Qcd and Multiplicity Scaling

The notion of scaling is hardly new. One of the earliest scaling arguments dates back to 1638 when Galileo Galilei published his infamous masterpiece entitled “Dialogues Concerning Two New Sciences”. Among other fundamental observations he examined the principle of similitude, the elementary properties of similar physical/biological structures. Galileo realized that the strength S of a bone increases in direct proportion to its cross-sectional area (S ∼ l, if l is the linear size), whereas the weight of a bone increases in direct proportion to its volume (W ∼ l). Thus, there will be a characteristic point where a bone has insufficient strength to support its own weight: the intersection point of the quadratic and cubic curves denoting the strength and weight of a bone, respectively. This general engineering consideration implies that terrestrial bodies can not exceed a certain maximum size. The classical scaling argument of Galileo teaches us an important lesson: the physical laws are not invariant under a uniform change of the size of macroscopic objects. The gravitational force, governed by Newton’s constant GN with dimension of (mass), inevitably leads to the breakdown of dilatation symmetry. Classical scaling principles of the above sort are based on the key assumption that the physical bodies or processes are uniform, filling an interval in a smooth, continuous fashion. In the example given by Galileo, the strength of a bone was assumed to be uniformly distributed over the cross-sectional area with its weight having a similar uniformity. This is a major limitation of the principle of similitude because such assumptions are not necessarily accurate. In reality a vast number of biological and physical systems, the so-called fractals, exhibit highly irregular appearance as the result of their self-similar