Setting the Renormalization Scale in QCD: The Principle of Maximum Conformality Page: 2 of 12
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sums all vacuum polarization contributions to the dressed photon propagator, both proper and improper. (Here
H(t) = H(t, 0) is the sum of proper vacuum polarization insertions, subtracted at t = 0). Formally, one can choose
any initial renormalization scale pg = to, since the final result when summed to all orders will be independent
of to. This is the invariance principle used to derive renormalization group results such as the Callan-Symanzik
equations [4, 5]. However, the formal invariance of physical results under changes in to does not imply that there is no
optimal scale. In fact, as seen in QED, the scale choice p2 q2, the photon virtuality, immediately sums all vacuum
polarization contributions to all orders exactly. With any other choice of scale, one will recover the same result, but
only after summing an infinite number of vacuum polarization corrections.
Thus, although the initial choice of renormalization scale to is arbitrary, the final scale t which sums the vacuum
polarization corrections is unique and unambiguous. The resulting perturbative series is identical to the conformal
series with zero -function. In the case of muonic atoms, the modified muon-nucleus Coulomb potential is precisely
-Za(-q&)/&; i.e., p2 -2. Again, the renormalization scale is unique.
One can employ other renormalization schemes in QED, such as the MS scheme, but the physical result will be
the same once one allows for the relative displacement of the scales of each scheme. For example, one can compute
the standard one-loop charged lepton pair vacuum polarization contribution to the photon propagator at photon
virtuality q2 using dimensional regularization. The result in MS scheme for spacelike argument q2 _Q2 is
log M % 6 j x(1 -x)log m2 + Q.(1 x) (3)
me o me
At large Q2 this is
log MS= log - /; 4
m2 m
i.e., M = Q2e-5/3. Thus if Q2 >> 4m, we can identify
a__(e-5/3 2) aGM-L(q2). (5)
The e-5/3 displacement of renormalization scales between the MS and Gell-Mann Low schemes is a result of the
convention [6] which was chosen to define the minimal dimensional regularization scheme. One can use another
definition of the renormalization scheme, but the final physical prediction cannot depend on the convention. This
invariance under choice of scheme is a consequence of the transitivity property of the renormalization group [3, 7 9].
The same principle underlying renormalization scale-setting in QED must also hold in QCD since the nF terms
in the QCD ) function have the same role as the lepton N vacuum polarization contributions in QED. QCD and
QED share the same Yang-Mills Lagrangian. In fact, one can show [10] that QCD analytically continues as a
function of NC to Abelian theory when NC -> 0 at fixed a = CFas with CF N -1. For example, at lowest
2 2c
order )oGD = QN - inF -- -nF at NC 0. Thus the same scale-setting procedure must be applicable to all
renormalizable gauge theories.
Thus there is a close correspondence between the QCD renormalization scale and that of the analogous QED process.
For example, in the case of ee- annihilation to three jets, the PMC/BLM scale is set by the gluon jet virtuality, just
as in the corresponding QED reaction. The specific argument of the running coupling depends on the renormalization
scheme because of their intrinsic definitions; however, the actual numerical prediction is scheme-independent.
The basic procedure for PMC/BLM scale setting is to shift the renormalization scale so that all terms involving
the ) function are absorbed into the running coupling. The remaining series is then identical with a conformal theory
with ) = 0. Thus, an important feature of the PMC is that its QCD predictions are independent of the choice of
renormalization scheme. The PMC procedure also agrees with QED in the NC -> 0 limit.
The determination of the PMC-scale for exclusive processes is often straightforward. For example, consider the
process +e - cc -> ccg* -> ccbb, where all the flavors and momenta of the final-state quarks are identified. The nff
terms at NLO come from the quark loop in the gluon propagator. Thus the PMC scale for the differential cross section
in the MS scheme is given simply by the MS scheme displacement of the gluon virtuality: pP4Mc = e-5/3 (Pb+pb)2
In practice, one can identify the PMC/BLM scale for QCD by varying the initial renormalization scale pg to identify
all of the /-dependent nonconformal contributions. At lowest order )o = 11/3Nc - 2/3nF. Thus at NLO one can
simply use the dependence on the number of flavors nf which arises from the quark loops associated with ultraviolet
renormalization as a marker for )o. Of course in QCD, the nF terms arise from the renormalization if the three-gluon
and four-gluon vertices as well as from gluon wavefunction renormalization.
It is often stated that the argument of the coupling in a renormalization scheme based on dimensional regularization
has no physical meaning since the scale p was originally introduced as a mass parameter in extended space-time
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Brodsky, Stanley J.; /SLAC /Southern Denmark U., CP3-Origins & Di Giustino, Leonardo. Setting the Renormalization Scale in QCD: The Principle of Maximum Conformality, article, August 19, 2011; United States. (https://digital.library.unt.edu/ark:/67531/metadc836522/m1/2/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.