La dignidad humana como limite a las penas indeterminadas o desproporcionadas

Weyl anomaly coefficients

In the previous section we have mostly been interested in the classical behavior of conformal field theories. When considering quantum field theories with conformal invariance, the situation becomes more involved. To make sense out of a quantum field theory, its expressions need to be regularized and renormalized, thereby auto- matically introducing a scale into the classically scale invariant theory [83]. For this reason, it might be that a classically conformally invariant theory loses its confor- mality as a quantum field theory. The departure from conformal invariance, can be expressed in terms of a violation of the hallmark of a conformal field theory, i.e. the trace of the stress-energy tensor is no longer vanishing,

Θ =−β(u)δL

δu. (3.13)

The notation of the coefficients,β(u), is no accident, as they are closely related to the renormalization groupβfunctions (3.2) [103–106]. Conceptually this is easy to understand. Only a quantum field theory with a vanishingβfunction will remain a fixed point for the renormalization group flow, without the introduction of any new scale into the theory. All relevant and irrelevant operators induce a renormalization of their couplings and therefore a scale dependence.

Technically the relation may be seen by the effect of an explicit scale transforma- tionx0α =eωxαon the couplingδu=ωβ(u), whereβ(u) now really is the renormal- ization groupβfunction [72]. As a result the action changes as

δS = Z ddxδL= Z ddxωβ(u)δL δu.

Compared to the definition (3.8) of the stress-energy tensor in terms of a scale trans- formation of the metrich0_αβ(x0)=e−2ωhαβ(x),

δS =−1 2 Z ddx Tαβδhαβ=− Z ddxΘω, (3.14) the renormalization groupβfunctions appear as coefficients in

Z

ddxΘ =−

Z

ddxβ(u)δL

δu.

Hence, the renormalization groupβfunctions and the Weyl anomaly coefficientsβ appearing in (3.13) are related in the same way as global and local scale invariance

are related. The requirementsR Θ =0 andΘ =0 for global and local scale invariant theories respectively are directly transferred to theβfunctions. Although this relation can and is used [107–109] to simplify the computation of Weyl anomaly coefficients, we will not emphasize the distinction.

In this section we have specifically restricted ourselves to flat metrics only. As we will see when we turn our attention to string theory, conformal symmetry on a curved space introduces another source for Weyl anomaly, due to the curvature scale that is introduced.

Computation of theβfunctions

A conformal field theory, with vanishingβfunctions and traceless stress-energy ten- sor, is a fixed point for the renormalization group flow. To study the flow perturba- tively, we consider a perturbation of the conformal field theoryS0by operators Oa

with couplinguaand dimension∆0,a,

Su =S0+

Z

ddx uaO_a(x). (3.15) The trace of the stress-energy tensor is

Θ =−βa_(u)O

a. (3.16)

The coefficients βa(u) can be calculated perturbatively by considering the Callan- Symanzik equation (3.1). The anomalous dimension γ appearing in the Callan- Symanzik equation actually becomes a matrix of anomalous dimensions γa

b for the

multi-operator case under consideration. It can be related to theβfunctions [110] via3 γa b(u)= ∂βa ∂ub −(∆0,b−d)δ a b. (3.17)

Writingβa(u)=Aa+Ba_bub+. . .and remembering thatuahas mass dimensionµd−∆0,a_,

applying the Callan-Symanzik equation to the partition function of the perturbed the- ory, Z =he− R dd_{x u}a_O a_i 0=h1− Z ddx uaOa+. . .i0,

enables us to computeβ(u) recursively as a perturbation series inu[86, 110–112]. To compute the higher order coefficients ofβ(u), an operator product expansion is necessary, whose singularities have to be regularized. The regularization scheme

dependence is carefully explained in [111]. In the limit where the operators are nearly marginal,|∆0,a−d| 1, the result in the Zamolodchikov scheme [110, 111] is

βa_(u)=(∆

0,a−d)ua+2πCabcu b

uc+. . . (3.18)

to second order inu. In the first term, there is no summation over thea-index. As we see, conformal perturbation theory enables us to study the renormalization group flow around a conformal fixed point. The flow is determined by theβfunctions, which can be expressed in terms of the scaling dimension∆0,aof the operatorOain

the unperturbed conformal field theory, as long as the deviation from the fixed point is small,ua_{1 and the operators under consideration are nearly marginal|∆}

a−d| 1.