8.324 Relativistic Quantum Field Theory II

Transcription

1 Lecture Relativistic Quantum Field Theory II Fall 8.34 Relativistic Quantum Field Theory II MIT OpenCourseWare Lecture Notes Hong Liu, Fall α(μ) Lecture : Beta-Functions of Quantum Electrodynamics In the case of quantum electrodynamics, we have the Lagrangian L = F B ν ( ( ν F B iψ B γ ) ) ie B A B m B 4 ψ B, () B e with ψ B = Z ψ, A = Z3 A, m or α B = Z 3 α ϵ B = m + δm, and e B =, where α = 4π. From our earlier result that Z 3 = α ϵ, we have Z 3 e ϵ [ ] α α B = ϵ α , () ϵ and α 4α α β α = + = + O(α 3 ). (3) Hence, the running of α() is given by α() = α( ) log, (4) or, equivalently, α α() = α. (5) log In quantum electrodynamics, α() increases as is increased, and α() decreases as decreases. In particular, Figure : We find a linear relationship between α () and log() with a negative gradient for quantum electrodynamics. the Landau pole at α() is given by log(μ) = Λ e α, (6) independently of the choice of. Quantum electrodynamics becomes strongly coupled near the scale Λ. It is convenient to express α() in terms of the physical α phys 37 which we measure. Consider e() k e() k e () Π(k ). (7) At the one-loop level, Π(k ) = e () π ˆ ( dx x( x) ϵ ( )) D log (Z 3 ), (8)

2 Lecture Relativistic Quantum Field Theory II Fall where 4π + x( x)k and m = m e γ, D m e + O(α), where m e is the physical electron mass. It is convenient to introduce α() α B αˆ(k) Π(k ) = Π B (k ). (9) Note that this quantity is finite, although the numerator and denominator of the last term are divergent. αˆ(k) is an effective k-dependent coupling. In particular, Now, and so We therefore have We can compare this to and we find Therefore, in the MS scheme, (k ) = α(m) π (k = ) = e () 6π α(k) ^ α e = ˆα(k = ) 37. () ˆ dx x( x) log D, () α e = α() α() = α(). (). (3) α( α() ) =, (4) α() log α e = α( = m e ). (5) α() = and the Landau pole occurs at Λ = m e e αe m e e We now consider ˆα(k) =. For k m e, D x( x)k, and so Therefore, α e (k ) = α( ) ˆα(k) = α( ) α e log m e, (6) α( ) α( ) Π M S (k ) : log k (7) log k +..., (8) and so αˆ(k) α() for k. Note that this is scheme-independent: in any scheme for m e, α() αˆ().. When k m e, we have ˆα(k) α e, but α() as m. For < m e e, α() differs qualitatively from the physical coupling. Physically, m e becomes important, but that is not tracked by the MS or 37 -log(m ) e -log(k) Figure : ˆα(k) as a function of log k. At the scale of k m e, we have ˆα(m e ) 37.

3 Lecture Relativistic Quantum Field Theory II Fall MS schemes: α() is no different for a theory with m e =. It is more transparent to understand the behaviour of ˆα(k) using the Wilsonian approach. The coupling in the Wilsonian action, by definition, should track ˆα(k) closely. Below the scale of m e, the electron becomes heavy, and we can then integrate it out, leaving a pure Maxwell theory, in which the coupling constant does not run. 3. For massless quantum electrodynamics, with m e =, α() is qualitatively correct for. We then find that α eff (k) as k : The theory is marginally irrelevant : Beta-Function of Quantum Chromodynamics The Lagrangian of quantum chromodynamics, with an SU(N c ) gauge group and N f quarks, where N c = 3 and N f = 6, is given by N f L = 4 TrF ν F ν i ψ j (γ D m j ) ψ j, (9) with j= F ν = A ν ν A ig [A, A ν ], A = A a t a, F ν = F a derivative is given by ν t a, where t a are the generators of the fundamental representation of SU(N c ). The covariant We redefine A A, and so the Lagrangian becomes g D ψ j = ψ j iga ψ j. () with N f L = 4 g TrF ν F ν i ψ j (γ D m j ) ψ j, () j= D ψ j = ψ j ia ψ j. () We will outline the computation of the β function for the pure gauge theory using the language of the Wilsonian approach, in the Euclidean case. Consider that the theory is provided with a cut off Λ, and bare coupling g = g(λ). We now write = A + A, (3) (Λ ) where A is the part below the scale Λ, and A is the high-energy part, above the scale Λ. Then we have [ ] [ ] ] = S Y M S + S A (Λ ) Y M [A, Ã, (4) and ˆ ˆ [ ] ˆ [ ] DA e S[A ] = DA (Λ ) e S DÃ e S,Ã ˆ [ ] [ ] = D e S S. We take the derivative expansion of S, S c log Λ ˆ d 4 x F +..., (5) Λ Λ giving Λ = + c log, (6) g (Λ ) g (Λ) Λ 3

4 Lecture Relativistic Quantum Field Theory II Fall where c is a pure g independent number. A precise calculation gives c = (4π) 3 N C. So, for the β function, we find ( ) g 3 β g = N C N F. (7) (4π) 3 3 Considering the fermionic sector, we have ˆ ( ) Dψ Dψ e dd x ψ (D/ Λ m)ψ+... det D/ Λ m log det(d/ Λ m) = e. If we define α s = g, we find 4π ( ) β α α NF = bα, (8) = π 3 N C 3 as we found in the gϕ 3 theory. So, we have α s () α s ( ) = b log, (9) and hence The Landau pole occurs at α s ( ) α s () =. (3) + α s ( )b log α(μ) QED QCD Figure 3: α scales linearly with log() with a positive gradient in quantum chromodynamics, as with the gϕ 3 theory. and so log(μ) Λ QCD α s ( )b log =, (3) bαs( ) Λ QCD = e 5MeV, (3) independently of our choice of. Finally, we put our coupling constant in the form α s () =. (33) b log ΛQCD Near Λ QCD, quantum chromodynamics becomes strongly coupled. The form of this coupling leads to many interesting phenomena, including confinement, and chiral symmetry breaking: U L U R =. 4

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