Reduction of Charm Quark Mass Scheme Dependence in ¯ B → X s γ at the NNLL Level

The uncertainty of the theoretical prediction of the ¯ B → X s γ branching ratio at NLL level is dominated by the charm mass renormalization scheme ambiguity. In this paper we calculate those NNLL terms which are related to the renormalization of m c in order to get an estimate of the corresponding uncertainty at the NNLL level. We ﬁnd that these terms signiﬁcantly reduce (by typically a factor of two) the error on BR( ¯ B → X s γ ) induced by the deﬁnition of m c . Taking into account the experimental accuracy of around 10% and the future prospects of the B factories, we conclude that a NNLL calculation would increase the sensitivity of the observable ¯ B → X s γ to possible new degrees of freedom beyond the SM signiﬁcantly.


Introduction
The branching ratio ofB → X s γ is a very sensitive probe for new degrees of freedom beyond the standard model (SM) (for a review, see [1]). Within supersymmetric extensions of the SM for example, one can derive stringent bounds on the parameter space of these models [2][3][4][5][6][7][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 ofB → X s γ is dominated by the perturbatively calculable partonic decay rate Γ(b → X s γ). 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-toleading logarithmic (NLL) level [9][10][11][12][13][14][15][16][17][18]. Subsequently, also electroweak corrections were calculated [19][20][21][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 [23], there is an additional uncertainty in the NLL results for Γ(b → X s γ) which is related to the definition (renormalization scheme) of the charm quark mass. Technically, the charm quark mass depencence enters through the matrix elements sγ|O 1,2 |b which in the context of a NLL have to be calculated up to O(α s ). As these matrix elements vanish at the lowest order, the charm quark m c only enters (through the ratio m c /m b ) at O(α s ). As a consequence, the charm quark mass does not get renormalized in a NLL calculation, which means that the symbol m c can be identified with m c,pole or with the MS massm c (µ c ) at some scale µ c or with some other definition of m c . Formally, all these assignements are equivalent, as they lead to differences which are of order α 2 s . 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 m b,pole . In this respect we do not follow ref. [23], where the m b,1S mass was used. Unless stated otherwise, the symbol m b stands for m b,pole in all the formulas in this paper. Numerically, we use m b = 4.8 GeV throughout.
Numerically, it turns out that the NLL result for Γ(b → X s γ) strongly depends on which mass definition of the charm quark mass is used in the NLL expressions. To illustrate this, we first identify m c with m c,pole as it was done in all analyses before the paper of Gambino and Misiak [23]. Numerically, we use m c,pole /m b,pole = 0.29 which is based on the mass difference m b,pole − m c,pole = 3.4 GeV fixed through the heavy mass expansion of m B and m D and m b,pole = 4.8 GeV. The corresponding branching ratio then reads [23] BR[B → X s γ] Eγ>m b /20 = 3.35 × 10 −4 . (1) As the charm quarks which are propagating in a loop have a typical virtuality of m b /2, the authors of Ref. [23] suggested to usem c (µ c ) with µ c ∈ [m c , m b ] instead of m c,pole . A typical value for the corresponding ratio ism c (µ c )/m b,pole = 0.22. Using this value, the branching ratio gets increased w.r.t. (1) by about 11% [23]: In a recent theoretical update of the NLL prediction of this branching ratio, the uncertainty related to the definition of m c was taken into account by varying m c /m b in the conservative range 0.18 ≤ m c /m b ≤ 0.31 which covers both, the pole mass (with its numerical error) value and the running massm c (µ c ) value with µ c ∈ [m c , m b ] [24]: There exists a large number of measurements of the inclusive decayB → X s γ [25][26][27][28][29][30] and the present experimental accuracy has reached the 10% level [31]: In the near future, more precise data on this mode are expected from the B factories. Thus, it is mandatory to reduce the present theoretical uncertainty accordingly. A systematic improvement certainly consists in performing a complete NNLL calculation . This is, however, a very complicated task (for discussion and some results see [32][33][34][35]) and a certain motivation is needed to enter such an enterprise. In the present paper we try to give such a motivation: By calculating those NNLL terms which are induced by renormalizing the charm quark mass in the NLL expressions, i.e. those terms which are sensitive to the definition of the charm quark mass, we show that the large error at the NLL level related to the m c definition gets significantly reduced. As this error is the dominant one at the NLL level (see eq. (3)), we conclude that a complete NNLL calculation will drastically improve the theoretical prediction of the branching ratio. We stress here that in the present paper we only make a statement about the reduction of the error at the NNLL level, and not about the central value of the branching ratio; this remains the topic of a complete NNLL calculation! The remainder of this paper is organized as follows. In section 2 we discuss in some detail how to calculate the NNLL terms induced by renormalizing m c in the NLL results. In order to make the paper self-contained, we first list in section 3 the structure of the NNL results and then we present the analytical results for the new terms discussed in section 2. Finally, in section 4, we numerically investigate by how much the error related to the definition of m c gets reduced at the NNLL level.

NNLL terms related to m c renormalization
As already explained in the introduction, the matrix elements M virt 1,2 (m c ) = sγ|O 1,2 (µ b )|b only start at order O(α 1 s ), or, in other words at the NLL order 2 . As a consequence, the definition of m c is not fixed at this order, because m c does not get renormalized. This is also true for the bremsstrahlung contributions M (m c ), followed by expanding in δm c up to linear order: As δm c is ultraviolet divergent, the matrix elements M virt(ǫ) 1,2 (m c ) are needed in our application up to order ǫ 1 , as indicated by the notation in eq. (5).
The explicit shift δm c depends of course on the renormalization scheme. When aiming at expressing the results for M On the other hand, when the result is expressed in terms of m c,pole , the shift reads The infinities induced by the 1/ǫ terms in δm c get cancelled in a full NNLL calculation, in particular by self-energy diagrams depicted in the right frame of Fig. 1. As we do not perform a full NNLL calculation, we suggest to consider self-energy insertions, where the self-energy Σ(p 2 ) is replaced by Σ 1 (p 2 = m 2 c ). The Σ 1 -part of the self-energy Σ 1 (p 2 ) is defined through the decomposition of the full unrenormalized self-energy Σ(p 2 ) as At the one-loop level, the corresponding pieces Σ R 1 and Σ R 2 of the renormalized self-energy are where Z c = 1 + δZ c denotes the wave function renormalization constant of the charm quark. Eq. (7) implies that the sigularities in δM virt(ǫ) 1,2 (m c ) · δm c cancel when combined with the diagrams with Σ 1 (p 2 ) insertions. However, for general p 2 , the function Σ 1 (p 2 ) depends on the gauge parameter ξ: These momentum independent Σ 1 (p 2 = m 2 c ) insertions can be straightforwardly aborbed into δm eff c insertions: Finally, if we wish to express the matrix elements M

Analytical results
Before turning to the contributions induced through the renormalization of the charm quark mass, which are NNLL terms, we first summarize the structure of the NLL result for the branching ratio for b → X s γ. We write the decay width for b → X s γ using a photon energy cut where the two parts are defined as follows: The expressions for the Wilson coefficients C i (µ b ) can be found in [36]. Their numerical values we take from table 5.1 in ref. [37]. Writing the results in this specific form, the functions f ij (δ) and r i are understood to be taken from [11] and not from the original paper [10] where the results were parametrized differently.
Following common practice, we write the branching ratio (without taking into account non-perturbative corrections) as where the semileptonic decay rate is given by g(z) = 1 − 8 z + 8 z 3 − z 4 − 12 z 2 ln(z) is the phase-space factor and the function accounts for O(α s ) QCD corrections. We note that m c is understood to be the pole mass in eq. (14).
We now turn to that part of NNLL corrections which is responsible for the reduction of the charm quark mass renormalization scheme dependence, as explained in section 2. We first turn to terms δM up to oder ǫ 1 . In [10] have calculated these matrix elements up to terms ǫ 0 , using Mellin-Barnes representations for generalized propogator to obtain analytic results in the form of the series in z = (m c /m b ) 2 and L = ln(z). As in these calculations the expansion in ǫ was the last step, it is straightforward to calculate M virt(ǫ) 1,2 up to order ǫ 1 .
In order to get finite results for these matrix elements, we add counterterms related to operator mixing as in ref. [10], adapted however, to the operator basis defined in ref. [11]. This step leads to M virt,ren 1,2 , which we decompose as in ref. [10]: We obtain for r 2 = r (−67 + 66 L + 9 L 2 + 12 π 2 )z 3 In these formulas we retained all terms up to order z 3 , as higher order terms contribute much less than 1%. Nevertheless, in the numerical evaluations in section 4 all terms up to z 6 were included.
At the level of the decay width, the implementation of the contribution coming from renormalization of the c-quark mass in the virtual contributions is (according to eq. (5)) most easily achieved by replacing r 1,2 in eq. (11) by r (0) 1,2 + ∆r 1,2 , where At the NLL order, the bremsstrahlung corrections to the decay width are encoded in the quantities f ij (δ) (see eq. (12)), which correspond to the interference terms (O i , O j ). In the following, we calculate the shifts ∆f ij to these quantities induced by the renormalization of the charm quark mass. In principle, we calculate the decay width using a photon energy cut δ = 0.9 (see eq. (10)). However, as all bremsstrahlung contributions which contain charm quark loops are finite for δ → 1, we can approximate these terms by putting δ = 1. Numerically the relative error is of order 10 −4 .
We first calculate the shift ∆f 27 . To this end, we shift the charm quark mass in the matrix element of sγg|O 2 |b as in eq. (5) and then work out the interference with sγg|O 7 |b . Because of the 1/ǫ term in δm c , the result is ultraviolet singular. In a full NNLL calculation this singularity gets cancelled when combined with self-energy insertions in the charm quark lines in the matrix element of O 2 . We therefore do the phase space integrations involved in the derivation of f 27 (or ∆f 27 ) in d=4 dimensions. As only the matrix element of O 2 depends on m c , the shift ∆f 27 can be constructed by first considering the quantity f 27 itself. Using the integral representation for the building block for photon and gluon emission from the c-quark loop [10], one obtains after integration over all but one of the phase space parameters Here x, y are Feynman parameters and u is the remaining phase space parameter, 0 ≤ x, y, u ≤ 1. To solve the integrals, we use the Mellin-Barnes representation for the generalized propagator s appearing in eq. (19). γ denotes the integration path parallel to imaginary axes which hits the real axes somewhere between (−1 − ǫ) and 0. Closing the integration path in the right s-half plane, one gets an expansion for f 27 in z = (m c /m b ) 2 and L = ln(z).
The shift ∆f 27 is then obained as To summarize, the NNLL contributions due to renormalization of m c in the (O 2 , O 7 ) interference are taken into account by replacing f 27 → f (0) 27 + ∆f 27 in eq. (12). Explicitly, we find: Note that f 0 27 in eq. (21) is an expanded version in z of the integral expression for f 27 in ref. [11]. We further note that f 28 = − 1 3 f 27 , f 17 = − 1 6 f 27 , f 18 = 1 18 f 27 ; the same relations also hold for the respective ∆f ij (see for instance, [38]).  We decided to give the expansion coefficients in these equations in numerical form, because the exact results are somewhat lenghty. We note that f 0 22 in eq. (25) is an expanded version in z of the integral expression for f 22 in ref. [11]. We further note that f 11 = 1 36 f 22 and f 12 = − 1 3 f 22 ; the same relations also hold for the respective ∆f ij . These analytical results are defined parts of the complete NNLL contribution which can be used within a future NNLL calculation.

Numerical results
In the following analysis we show that the NNLL terms, induced through the renormalization of m c , drastically reduce the error related to the definition of the charm quark mass in BR(b → X s γ). To illustrate this feature as clearly as possible, we take the fixed values shown in Table 1 for the input parameters. In particular, we use the fixed ratio  Table 1: Input parameters used in the numerical analysis. m c,pole /m b,pole = 0.29. Furthermore, we always leave the semileptonic decay width, which enters the branching ratio for b → X s γ through eq. (13), expressed in terms of m c,pole as given in eq. (14). In this way the m c definition dependence of the BR(b → X s γ) only comes from the numerator in eq. (13). For our studies, we neglect electroweak corrections and non-perturbative effects. As already mentioned, in the bremsstrahlung contribution we use δ = 0.9 for the lower cut in the photon energy (see eq. (10)).
Starting from m c,pole = 0.29 · 4.8 GeV = 1.392 GeV, we first calculatem c (m c,pole ), using the one-loop expression To getm c (µ c ) for an arbitrary scale (typically between 1.25 GeV and 5 GeV), we use two-loop running (with 5 flavours) according tō with µ 0 = m c,pole . Numerically, we get the values shown in table 2. In Figure 2   For each µ b the left string shows the value of the branching ratio at the NLL level, while the right string shows the corresponding value where in addition δm c mass insertions and Σ 1 (p 2 = m 2 c ) insertions were taken into account, as explained in detail in section 2. Because the combination of these insertions is zero by construction for the pole scheme (see eq. (8)), the solid dots are at the same place in the left and the right string for a given value of µ b .
From Figure 2 we see that the error related to the charm quark mass definition gets significantly reduced when taking into account NNLL terms connected with mass insertions. Taking as an example the results for µ b = 5 GeV, we find that at the NLL level the branching ratio evaluated form c (2.5 GeV) is 12.6% higher than the one based on m c,pole , in agreement with ref. [23]. Including the new contributions, these 12.6% get reduced to 5.1%.
A remark concerning the remaining NNLL terms is in order: As these terms give contributions to the branching ratio which (up to terms of order α 3 s ) do not depend on charm quark mass definition, the error related to m c in the full NNLL result is expected to stay essentially the same as estimated in the present paper.
However, to obtain a NNLL prediction for the central value of the branching ratio, it is of course necessary to calculate all NNLL terms.
Summing up, we have shown that the relatively large error related to the definition of the charm quark mass in the NLL result for BR(b → X s γ) gets significantly reduced (typically by a factor of 2) at the NNLL level. Taking into account the present experimental accuracy of around 10% and the future prospects of the B factories and also of possible Super-B factories [40,41], we conclude that a future NNLL QCD calculation of the b → X s γ branching ratio will significantly increase the sensitivity of this observable to possible new physics.