Scale Setting Using the Extended Re normalization Group and the Principle of Maximal Conformality: the QCD Coupling at Four Loops Page: 3 of 7
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3
a(2# /B 11n(p/A'H), {0}). Here the word "associated"
means we are choosing the particular 't Hooft schemethat shares the same 't Hooft scale with the R-scheme.
Eq. (6) can be solved iteratively, and its solution can be
expanded as a power series of 1/L, i.e. up to four loops,a =+ L (C- In L) + L3 [C2 +C + c2 - (2C - In L +1) In L -1] +
C C2+C+3c2-2 - [C2+ 5C + 3c2 (3C -ln L + In Lln L + 0 (). (7)
L4 c 2C+32 ) 1 2 23 13As a cross-check, the above solution agrees with Ref.[13]
after proper parameter transformations and by identify-
ing the integration constant C* used there to be C*
C -n 0 ). When setting {c} {0} and C 0, we
recover the coupling constant under the 't Hooft scheme.
One can also obtain a relation between A'AH and AR, i.e.
ARH exp 1 CR1 AR (8)
As a special case, by choosing C- ln #o/ 1, we obtain
AffrAS (9)
AMS T2 AMB (A-
which agrees with the observation in Ref. [8]. The present
definition of AMS is the conventional one suggested by
Ref.[14, 15]; there are other choices for CM- [16], which
might be helpful in certain cases.
III. BLM SCALE SETTING UP TO NNLO
Generally, perturbative QCD prediction for a physical
observable p can be written as
p ro an(Q) + (A1 + A2nf)a7+1(Q)
+(B1 + B2nf + B3nf)a7+2(Q)
+(C1 + C2nf + Canj + C4n )a7+3(Q) + ] -(10)
where as(Q) ((Q)) and the overall tree-level param-
eter ro is scale-independent and is free of a8(Q). Here n f
stands for the quark flavor number and n(> 1) stands for
the initial a8 order at the tree level. After proper scale
setting, all nf-terms in the perturbative expansion can be
summed into the running coupling. Here, we shall con-
centrate on those processes in which all nf-terms are as-
sociated with the {/3, }-terms. In higher-order processes,
there may be nf-terms coming from the light-by-light
quark loops which are irrelevant to the ultra-violet cut-
off; they have no relation to the {3,}-terms [3]. Those
terms should be identified and kept separately after theBLM scale setting 3
The BLM scales can be set up in a general scheme-
independent way, and the generalization of the BLM pro-
cedure to higher order assigns a different renormalization
scale for each other in the perturbative series, which can
be done order by order. We can shift the renormaliza-
tion scale Q into effective ones until we fully absorb those
higher-order terms with nf-dependence into the running
coupling 4.
More explicitly, the first step of the BLM method is to
set the effective scale Q* at LO
p= ro as(Q*) + Aa7+1(Q*) + (B1 + B2nf)a+2(Q*)+(C1+C2nf+C3n)a+3(Q*)+.. .]
(11)
The second step is to set the effective scale Q** at NLO
p n ro [a (Q ) + Aia, (Q ) + Bia (Q )+(C1 + C2nf)a+3 (Q**) + ...]
(12)
and the final step is to set the effective scale Q*** at
NNLO
(P ro La(Q*) + Aa7+1(Q*) + Bla7+2(Q )(13)
When performing the scale shifts Q -> Q*, Q* -> Q**
and Q** -> Q***, we eliminate the nf-terms associated
with the {/,}-terms completely, but at the same time we
also have to modify the coefficients. To set the effective
scale for a7+3, one needs even higher order information
and here, a sensible choice is Q***, since this is the renor-
malization scale after shifting the scales in the final step
of the BLM procedure up to NNLO. Note that the ef-
fective scales should be a perturbative series of a, so as
3 Those nf-terms, coming from the light-quark loops connected to
at least four photon/gluon lines, are of higher twists and power
suppressed by hard scales, so they usually can be safely neglected.
4 Another way to set the BLM scale up to NNLO can be found
in Refs.[17, 18], where a unified effective scale Q* is used for all
orders.+~1n+3(Q***)+. .
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Brodsky, Stanley J. & Wu, Xing-Gang. Scale Setting Using the Extended Re normalization Group and the Principle of Maximal Conformality: the QCD Coupling at Four Loops, article, February 16, 2012; United States. (https://digital.library.unt.edu/ark:/67531/metadc836117/m1/3/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.