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We can relate the 4-momentum: (P0 , P ) and (y, PT ) as below. From the deﬁnition of rapidity, we have ey = P 0 + Pz and e−y = P 0 − Pz Adding these equations we get P0 = mT cosh y where mT is the transverse mass of the particle m2T = m2 + PT2 P0 − Pz P0 + Pz 46 1. Quark-Gluon Plasma: An Overview Subtracting the above two equations gives Pz = mT sinh y Thus, the information contained in (P0 , P ) is all contained in (y, PT ). We saw that the rapidity of a particle in a moving frame is equal to the rapidity in the laboratory frame minus the rapidity of the frame.
So g increases with increasing t and is driven towards g0 . Similarly, if g > g0 , then β < 0 and dg dt < 0, so g decreases towards g0 with increasing t. Thus, g0 is an ultraviolet (large t) stable ﬁxed point and g(∞) = g0 . Note that g0 is an infrared unstable ﬁxed point. Because for g < g0 , β > 0 so g decreases away from g0 with decreasing t. Similarly, for g > g0 , β < 0, so decreasing t takes g away from g0 . By the same arguments, g = 0 is an infrared stable ﬁxed point. 2. Now consider the other possibility.
This implies n (−n/2−1) ∂Zφ (n) ∂ ∂gR ∂ ∂mR ∂ (−n/2) (n) − Zφ μ Γ =0 μ +μ +μ Γ + Zφ 2 ∂μ R ∂μ ∂μ ∂gR ∂μ ∂mR R n/2 Multiplying the above with Zφ −nμ ∂ ln ∂μ gives Zφ + μ ∂ (n) + ... ΓR = 0 ∂μ Deﬁne μ ∂ ln ∂μ Zφ = γ(g) β(g) = μ mγm (g) = ∂g ∂μ ∂m μ ∂μ We then get the renormalization group (RG) equation: μ ∂ ∂ ∂ Γ(n) = 0 + β(g) − nγ(g) + mγm (g) ∂μ ∂g ∂m β(g) is called the β function of the theory. The renormalization group equation expresses how the renormalized vertex functions change when we change the arbitrary scale μ.