Reduction of Quark Mass Scheme Dependence in B bar -> Xs gamma at the NNLL Level Page: 2 of 14
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The branching ratio of B -+ Xa7 is a very sensitive probe for new degrees of freedom be-
yond the standard model (SM) (for a review, see ). Within supersymmetric extensions
of the SM for example, one can derive stringent bounds on the parameter space of these
models [2-8]. Clearly, such bounds will be most valuable when the general nature of the
new physics beyond the SM will be identified at the forthcoming LHC experiments.
Because of the heavy mass expansion that is valid for inclusive decay modes, the
decay rate of B - Xa7 is dominated by the perturbatively calculable partonic decay rate
F(b - Xy). QCD corrections to the latter, due to hard-gluon exchange, are the most
important perturbative contributions; they were calculated in the past up to the next-to-
leading logarithmic (NLL) level [9-18]. Subsequently, also electroweak corrections were
calculated [19-22]. After completion of these computations, it was generally believed that
the theoretical uncertainty of the branching ratio is below 10%.
However, as first pointed out in 2001 in , there is an additional uncertainty in the
NLL results for F(b - X,8y) which is related to the definition (renormalization scheme)
of the charm quark mass. Technically, the charm quark mass depencence enters through
the matrix elements (s2 01,2b) which in the context of a NLL have to be calculated up
to O(as). As these matrix elements vanish at the lowest order, the charm quark rmc only
enters (through the ratio nc/mb) at O(as). As a consequence, the charm quark mass
does not get renormalized in a NLL calculation, which means that the symbol mc can
be identified with n,,pole or with the MS mass Hic(pc) at some scale pc or with some
other definition of m,. Formally, all these assignements are equivalent, as they lead to
differences which are of order a.
Note that in contrast to the c-quark mass the b-quark mass does get renormalized in
a NLL calculation and we choose to express all the following results in terms of nb,pole.
In this respect we do not follow ref. , where the mb,1S mass was used. Unless stated
otherwise, the symbol mb stands for nb,pole in all the formulas in this paper. Numerically,
we use mb = 4.8 GeV throughout.
Numerically, it turns out that the NLL result for F(b - Xy) strongly depends on
which mass definition of the charm quark mass is used in the NLL expressions. To
illustrate this, we first identify mc with mc,,pole as it was done in all analyses before the
paper of Gambino and Misiak . Numerically, we use mc,,pole/Mb,pole = 0.29 which is
based on the mass difference nb,pole - 1c,pole = 3.4 GeV fixed through the heavy mass
expansion of nB and HID and nb,pole = 4.8 GeV. The corresponding branching ratio then
BR[B - XS2]E,>m6/20 = 3.35 x 10-4. (1)
As the charm quarks which are propagating in a loop have a typical virtuality of mb/2,
the authors of Ref.  suggested to use ihc(pc) with pc E [m,, mb] instead of nc,pole. A
typical value for the corresponding ratio is ntcpc)/mb,pole = 0.22. Using this value, the
branching ratio gets increased w.r.t. (1) by about 11% :
BR[B - XS2]E,>m6/20 = 3.73 x 10-4. (2)
In a recent theoretical update of the NLL prediction of this branching ratio, the uncer-
tainty related to the definition of m, was taken into account by varying m/mb in the
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Asatrian, H.M.; Inst., /Yerevan Phys.; Greub, C.; U., /Bern; Hovhannisyan, A.; Inst., /Yerevan Phys. et al. Reduction of Quark Mass Scheme Dependence in B bar -> Xs gamma at the NNLL Level, article, June 20, 2005; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc874863/m1/2/: accessed January 19, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.