CP asymmetries of ${\overline{B}}\to X_{s}/X_{d}\gamma$ in models with
three Higgs doublets

In the fully inclusive approach only a photon from the signal $\overline{B}$ (or $B$) meson in the $B\overline{B}$ event, which decays via $b\to s/d\gamma$, is selected. Consequently, this method cannot distinguish between hadronic states $X_{s}$ and $X_{d}$, and what is measured is actually the sum of ${\overline{B}}\to X_{s}\gamma$ and ${\overline{B}}\to X_{d}\gamma$. From the other $\overline{B}$ (or $B$) meson (”tag $B$ meson”) either a lepton ($e$ or $\mu$) can be selected or full reconstruction (either hadronic or semi-leptonic) can be carried out. The former method has a higher signal efficiency, but the latter method has greater background suppression. Measurements of ${\overline{B}}\to X_{s+d}\gamma$ using the fully inclusive method with leptonic tagging have been carried out by the CLEO collaboration Chen:2001fja , the BABAR collaboration Lees:2012ufa and the BELLE collaboration Belle:2016ufb . A measurement of ${\overline{B}}\to X_{s+d}\gamma$ using the fully inclusive method with full (hadronic) reconstruction of the tag ${\overline{B}}$ meson has so far only been carried out by the BABAR collaboration Aubert:2007my . At the current integrated luminosities (0.5 to 1 ab^-1) the errors associated with measurements that involve full reconstruction are significantly larger than the errors from measurements with leptonic tagging. However, with the larger integrated luminosity at BELLE II (50 ab^-1) it is expected that both approaches will provide roughly similar errors. To obtain a measurement of ${\overline{B}}\to X_{s}\gamma$ alone, the contribution of ${\overline{B}}\to X_{d}\gamma$ (which is smaller by roughly $|V_{td}/V_{ts}|^{2}\approx 1/20$ in the SM, which has also been confirmed experimentally) is subtracted.

In the sum-of-exclusives approach the selection criteria are sensitive to as many exclusive decays as possible in the hadronic final states $X_{s}$ and $X_{d}$ of the signal ${\overline{B}}$, as well as requiring a photon from $b\to s/d\gamma$. In contrast to the fully inclusive approach, no selection is made on the other $B$ meson in the $B\overline{B}$ event. The sum-of-exclusives method is sensitive to whether the decay $b\to s\gamma$ or $b\to d\gamma$ occurred and so this approach measures ${\overline{B}}\to X_{s}\gamma$ or ${\overline{B}}\to X_{d}\gamma$. It has different systematic uncertainties to that of the fully inclusive approach. Measurements of ${\overline{B}}\to X_{s}\gamma$ have been carried out by the BABAR collaboration Lees:2012wg and the BELLE collaboration Saito:2014das . Currently, 38 exclusive decays in ${\overline{B}}\to X_{s}\gamma$ (about 70% of the total BR) and 7 exclusive decays in ${\overline{B}}\to X_{d}\gamma$ delAmoSanchez:2010ae are included. At current integrated luminosities the error in the measurement of ${\overline{B}}\to X_{s}\gamma$ is about twice that of the fully inclusive approach, whereas at BELLE II integrated luminosities the latter is still expected to give the more precise measurement.

Measurements in both the above approaches are made with a lower cut-off on the photon energy $E_{\gamma}$ in the range 1.7 GeV to 2.0 GeV, and then an extrapolation is made to $E_{\gamma}>1.6$ GeV using theoretical models. The current world average for the above six measurements of ${\overline{B}}\to X_{s}\gamma$ is Amhis:2019ckw :

$$\mathcal{B}^{\text{exp}}_{s\gamma}=(3.32\pm 0.15)\times 10^{-4}\quad{\rm with}\quad E_{\gamma}>1.6\,\text{GeV}\,.\\$$

(1)

The error is currently $4.5\%$, and is expected to be reduced to around $2.6\%$ with the final integrated luminosity at the BELLE II experiment Kou:2018nap .

The theoretical prediction including corrections to order $\alpha^{2}_{s}$ (i.e. Next-to-Next-to leading order, NNLO) is Misiak:2020vlo :

$$\mathcal{B}^{SM}_{s\gamma}=(3.40\pm 0.17)\times 10^{-4}\quad{\rm with}\quad E_{\gamma}>1.6\,\text{GeV}\,.\\$$

There is excellent agreement between the world average and the NNLO prediction in the SM. Consequently, $\mathcal{B}^{\text{exp}}_{s\gamma}$ allows stringent lower limits to be derived on the mass of new particles, most notably the mass of the charged scalar ($m_{H^{\pm}}>480$ GeV Misiak:2015xwa , as mentioned earlier) in the 2HDM (Type II).

The parameters $\tilde{\Lambda}^{u}_{17},\tilde{\Lambda}^{c}_{17},\tilde{\Lambda}_{78}$ are hadronic parameters that determine the magnitude of the long-distance contribution. Their allowed ranges were updated in Ref. Gunawardana:2019gep to be as follows:

	$\displaystyle-660\;\text{MeV}<\tilde{\Lambda}^{u}_{17}<+660\;\text{MeV}\,,$
	$\displaystyle-7\;\text{MeV}<\tilde{\Lambda}^{c}_{17}<+10\;\text{MeV}\,,$
	$\displaystyle 17\;\text{MeV}<\tilde{\Lambda}_{78}<190\;\text{MeV}\,.$		(4)

The short-distance contributions to $\mathcal{A}_{X_{s(d)}\gamma}$ are the terms that are independent of $\Lambda_{ij}$, and $\mathcal{A}^{0}_{X_{s(d)}\gamma}=\mathcal{A}^{\pm}_{X_{s(d)}\gamma}$ if long-distance terms are neglected. Other parameters are as follows: $\Lambda_{c}=0.38\;\text{GeV}$, $\epsilon_{s}=(V_{ub}V^{*}_{us})/(V_{tb}V^{*}_{ts})=\lambda^{2}(i\bar{\eta}-\bar{\rho})/[1-\lambda^{2}(1-\bar{\rho}+i\bar{\eta})]$ (in terms of Wolfenstein parameters), $\epsilon_{d}=(V_{ub}V^{*}_{ud})/(V_{tb}V^{*}_{td})=(\bar{\rho}-i\bar{\eta})/(1-\bar{\rho}+i\bar{\eta})$. The $C_{i}$’s are Wilson coefficients of relevant operators that are listed in Ref. Kagan:1998bh . In the SM the Wilson coefficients are real and the only term in $\mathcal{A}_{X_{s(d)}\gamma}$ that is non-zero is the term with $\epsilon_{s(d)}$. Due to $\epsilon_{s}$ being of order $\lambda^{2}$ while $\epsilon_{d}$ is of order 1, for the imaginary parts one has ${\rm Im}(\epsilon_{d})/{\rm Im}(\epsilon_{s})\approx-22$. For the short-distance contribution only (i.e. neglecting the term with $(\Lambda^{u}_{17}-\Lambda^{c}_{17})/{m_{b}}$ in eq. (3)) one has $\mathcal{A}_{X_{s}\gamma}\approx 0.5\%$ and $\mathcal{A}_{X_{d}\gamma}\approx 10\%$. The small value of $\mathcal{A}_{X_{s}\gamma}$ in the SM suggests that this observable could probe models of physics beyond the SM that contain Wilson coefficients with an imaginary part.

After the publication of Ref. Kagan:1998bh , several works calculated $\mathcal{A}_{X_{s}\gamma}$ (for the short-distance contribution only) in the context of specific models of physics beyond the SM Chua:1998dx , usually in supersymmetric extensions of it. Values of $\mathcal{A}_{X_{s}\gamma}$ of up to $\pm 16\%$ were shown to be possible in specific models, while complying with stringent constraints from electric dipole moments (EDMs). Including the long-distance contributions, it was shown in Ref. Benzke:2010tq that the the SM prediction using eq. (3) is enlarged to the range $-0.6\%<\mathcal{A}_{X_{s}\gamma}<2.8\%$, and (using updated estimates of the $\Lambda_{ij}$ parameters) is further increased to $-1.9\%<\mathcal{A}_{X_{s}\gamma}<3.3\%$ in Ref. Gunawardana:2019gep . This revised SM prediction has decreased the effectiveness of $\mathcal{A}_{X_{s}\gamma}$ as a probe of physics beyond the SM. Consequently, in Ref. Benzke:2010tq the difference of CP asymmetries for the charged and neutral $B$ mesons $\Delta\mathcal{A}_{X_{s}\gamma}=\mathcal{A}^{\pm}_{X_{s}\gamma}-\mathcal{A}^{0}_{X_{s}\gamma}$ was proposed, which is given by:

$\displaystyle\Delta\mathcal{A}_{X_{s}\gamma}\approx 4\pi^{2}\alpha_{s}\frac{\tilde{\Lambda}_{78}}{m_{b}}\text{Im}\frac{C_{8g}}{C_{7\gamma}}\,.$

(5)

This formula is obtained from eq. (3) in which only the terms with $e_{\text{spec}}$ do not cancel out. The SM prediction is $\Delta\mathcal{A}_{X_{s}\gamma}=0$ (due to the the Wilson coefficients being real) and hence this observable is potentially a more effective probe of new physics than $\mathcal{A}_{X_{s}\gamma}$. Note that $\Delta\mathcal{A}_{X_{s}\gamma}$ depends on the product of a long-distance term (${\tilde{\Lambda}_{78}}$, whose value is only known to within an order of magnitude) and two short-distance terms ($C_{8}$ and $C_{7}$).

An alternative observable is the untagged (fully inclusive) asymmetry given by

$\displaystyle A_{X_{s+d}\gamma}=$

$\displaystyle\ \frac{(A_{X_{s}\gamma}^{0}+r_{0\pm}A_{X_{s}\gamma}^{\pm})+R_{ds}(A_{X_{d}\gamma}^{0}+r_{0\pm}A_{X_{d}\gamma}^{\pm})}{(1+r_{0\pm})(1+R_{ds})}\,.$

(6)

Here $R_{ds}$ is the ratio ${\rm BR}({\overline{B}}\to d\gamma)/{\rm BR}({\overline{B}}\to s\gamma)\approx|V_{td}/V_{ts}|^{2}$. The parameter $r_{0\pm}$ is defined as the following ratio:

$$r_{0\pm}\equiv\frac{N_{X_{s}}^{+}+N_{X_{s}}^{-}}{N_{X_{s}}^{\bar{0}}+N_{X_{s}}^{0}},\,$$

(7)

where $N_{X_{s}}^{+}$ is the number of $B^{+}$ mesons that decay to $X_{s}\gamma$ etc. Experimentally, $r_{0\pm}\approx 1.03$ Kou:2018nap and in our numerical analysis we take $r_{0\pm}=1$. In the fully inclusive measurement of BR($b\to s/d\gamma$) the asymmetry $\mathcal{A}_{CP}({\overline{B}}\to X_{s+d}\gamma)$ is measured by counting the difference in the number of positively and negatively charged leptons from the tagged (not signal) $B$ meson. The SM prediction of $\mathcal{A}_{CP}({\overline{B}}\to X_{s+d}\gamma)$ is essentially 0 Soares:1991te ; Kagan:1998bh (up to tiny $m^{2}_{s}/m^{2}_{b}$ corrections), even with the long-distance contribution included. Hence this observable is a cleaner test of new physics than $\mathcal{A}_{X_{s}\gamma}$. The first studies of the magnitude of the untagged asymmetry in the context of physics beyond the SM were in Ref. Akeroyd:2001cy , and the importance of this observable was emphasised in Ref. Hurth:2003dk . In this work we will consider the above three direct CP asymmetries in the context of 3HDMs: i) $\mathcal{A}_{X_{s}\gamma}$, ii) $\mathcal{A}_{CP}({\overline{B}}\to X_{s+d}\gamma)$, iii) $\Delta\mathcal{A}_{X_{s}\gamma}$.

Measurements of all three asymmetries have been carried out, and the most recent BELLE and BABAR measurements are summarised in Tab. 2. In Tab. 2 the CP asymmetry ${\mathcal{A}}^{\rm tot}_{X_{s}\gamma}$ would have the same magnitude as the average $\overline{\mathcal{A}}=(\mathcal{A}^{0}_{X_{s}\gamma}+\mathcal{A}^{\pm}_{X_{s}\gamma})/2$ if the production cross-sections of $B^{+}B^{-}$ and $B^{0}\overline{B^{0}}$ were the same. The BELLE measurement Watanuki:2018xxg of $\overline{\mathcal{A}}=(0.91\pm 1.21\pm 0.13)\%$ differs only slightly from the BELLE measurement of ${\mathcal{A}}^{\rm tot}_{X_{s}\gamma}$ in Tab. 2. The world averages are taken from Ref. Tanabashi:2018oca . The given averages for ${\mathcal{A}}^{\rm tot}_{X_{s}\gamma}$ and $\Delta\mathcal{A}_{X_{s}\gamma}$ are obtained from the two displayed measurements in Tab. 2, while the average for $\mathcal{A}_{CP}({\overline{B}}\to X_{s+d}\gamma)$ also includes two earlier BABAR measurements and the CLEO measurement $(-7.9\pm 10.8\pm 2.2)\%$ Coan:2000pu .

At BELLE II all three asymmetries will be measured with greater precision Kou:2018nap . At present around 74 fb^-1 of integrated luminosity have been accumulated, which is about one tenth of the integrated luminosity at the BELLE experiment, and about one sixth that at the BABAR experiment. By the end of the year 2021 about 1 ab^-1 is expected, and thus measurements of $b\to s\gamma$ at BELLE II will then match (or better) in precision those at BELLE and BABAR. For an integrated luminosity of 50 ab^-1 (which is expected to be obtained by the end of the BELLE II experiment in around the year 2030), the estimated precision for $\mathcal{A}^{\rm tot}_{X_{s}\gamma}$ is 0.19%, for $\mathcal{A}_{CP}({\overline{B}}\to X_{s+d}\gamma)$ is $0.48\%$ (leptonic tag) and $0.7\%$ (hadronic tag), and for $\Delta\mathcal{A}_{X_{s}\gamma}$ is 0.3% (sum-of-exclusives) and 1.3% (fully inclusive with hadronic tag, and so it measures a sum of $b\to s\gamma$ and $b\to d\gamma$). These numbers are summarised in Tab. 3, together with the SM predictions. Due to the SM prediction of $\mathcal{A}_{CP}({\overline{B}}\to X_{s+d}\gamma)$ being essentially zero, a central value of 2.5% with 0.5% error would constitute a $5\sigma$ signal of physics beyond the SM. For $\Delta\mathcal{A}^{\rm tot}_{X_{s}\gamma}$, whose prediction in the SM is also essentially zero, a central value of 1.5% with 0.3% error would constitute a $5\sigma$ signal. Note that the current $2\sigma$ allowed range of $\mathcal{A}^{\rm tot}_{X_{s}\gamma}$ is $-0.7\%<\mathcal{A}^{\rm tot}_{X_{s}\gamma}<3.7\%$ ($-1.8\%<\mathcal{A}^{\rm tot}_{X_{s}\gamma}<4.8\%$ at $3\sigma$). Comparing this range with the SM prediction of $-1.9\%<\mathcal{A}^{\rm tot}_{X_{s}\gamma}<3.3\%$ shows that it is less likely that the observable $\mathcal{A}^{\rm tot}_{X_{s}\gamma}$ alone could provide a clear signal of physics beyond the SM, e.g. a future central value of above $4.3\%$ (which is outside the current $2\sigma$ range) with the expected of error 0.19% would be required to give a $5\sigma$ discrepancy from the upper SM prediction of 3.3%.

The values of $d$, $u$, and $\ell$ in these matrix elements are given in Tab. 4 and depend on which of the five distinct 3HDMs is being considered. The choice of $d=1$, $u=2$, and $\ell=3$ indicates that the down-type quarks receive their mass from $v_{1}$, the up-type quarks from $v_{2}$, and the charged leptons from $v_{3}$ (and is called the “Democratic 3HDM”). The other possible choices of $d$, $u$, and $\ell$ in a 3HDM are given the same names as the four types of 2HDM.

The elements of the matrix $U$ can be parametrised by four parameters $\tan\beta$, $\tan\gamma$, $\theta$, and $\delta$, where

$$\tan\beta=v_{2}/v_{1},\qquad\tan\gamma=\sqrt{v_{1}^{2}+v_{2}^{2}}/v_{3}\,.$$

(12)

The angle $\theta$ and phase $\delta$ can be written explicitly as functions of several parameters in the scalar potential Cree:2011uy . The explicit form of $U$ is:

	$\displaystyle U$	$\displaystyle=$	$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\ 0&e^{-i\delta}&0\\ 0&0&1\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\ 0&c_{\theta}&s_{\theta}e^{i\delta}\\ 0&-s_{\theta}e^{-i\delta}&c_{\theta}\end{array}\right)\left(\begin{array}[]{ccc}s_{\gamma}&0&c_{\gamma}\\ 0&1&0\\ -c_{\gamma}&0&s_{\gamma}\end{array}\right)\left(\begin{array}[]{ccc}c_{\beta}&s_{\beta}&0\\ -s_{\beta}&c_{\beta}&0\\ 0&0&1\end{array}\right)$		(25)
		$\displaystyle=$	$\displaystyle\left(\begin{array}[]{ccc}s_{\gamma}c_{\beta}&s_{\gamma}s_{\beta}&c_{\gamma}\\ -c_{\theta}s_{\beta}e^{-i\delta}-s_{\theta}c_{\gamma}c_{\beta}&c_{\theta}c_{\beta}e^{-i\delta}-s_{\theta}c_{\gamma}s_{\beta}&s_{\theta}s_{\gamma}\\ s_{\theta}s_{\beta}e^{-i\delta}-c_{\theta}c_{\gamma}c_{\beta}&-s_{\theta}c_{\beta}e^{-i\delta}-c_{\theta}c_{\gamma}s_{\beta}&c_{\theta}s_{\gamma}\end{array}\right).$		(29)

Here $s$ and $c$ denote the sine or cosine of the respective angle. Hence the functional forms of $X_{i}$, $Y_{i}$, and $Z_{i}$ in a 3HDM depend on four parameters. As mentioned earlier, this is in contrast to the analogous couplings in the 2HDM for which $\tan\beta$ is the only free coupling parameter.

The parameters $X_{i}$, $Y_{i}$ and $Z_{i}$ are constrained (for a specific value of $m_{H_{i}^{\pm}}$) by direct searches for $H^{\pm}_{i}$ (e.g. at the LHC) and by their effect on low-energy observables in flavour physics. A summary of these constraints can be found in Ref. Cree:2011uy , in which the lightest charged scalar is assumed to give the dominant contribution to the observable being considered. A full study in the context of the 3HDM with both charged scalars contributing significantly has not been performed, and is beyond the scope of this work. The coupling $Y_{i}$ is most strongly constrained from the process $Z\to b\overline{b}$ from LEP data. For $m_{H^{\pm}}$ around 100 GeV the constraint is roughly $|Y_{i}|<0.8$ (assuming $|X_{i}|\leq 50$, so that the dominant contribution is from the $Y_{i}$ coupling), and weakens with increasing mass of the charged scalar. Constraints on the $X_{i}$ and $Z_{i}$ are weaker and we take $|X_{i}|<50$ and $|Z_{i}|<50$ as being representative of these constraints for $m_{H_{i}^{\pm}}$ around 100 GeV.

The couplings $Z_{i}$ do not enter the partial width of $b\to s\gamma$, and only the couplings to quarks are relevant ($X_{i}$ and $Y_{i}$). Consequently, the partial width for $b\to s\gamma$ in Type I and the lepton-specific structures (which have identical functional forms for $X_{i}$ and $Y_{i}$ due to $u=d$ in Tab. 4) has the same dependence on the parameters of $U$. Likewise, the partial width for $b\to s\gamma$ in Type II, flipped and democratic structures ($u\neq d$ in Tab. 4) is the same. The contribution of $H^{\pm}_{1}$ and $H^{\pm}_{2}$ to BR(${\overline{B}}\to X_{s}\gamma$) has been studied in the 3HDM at the leading order (LO) in Ref. Hewett:1994bd (no $\alpha_{s}$ corrections arising from diagrams with charged scalars) and at next-to-leading order (NLO) in Ref. Akeroyd:2016ssd ($\alpha_{s}$ corrections arising from diagrams with charged scalars). In Ref. Akeroyd:2016ssd the effect of a non-zero phase $\delta$ was not studied, and direct CP asymmetries were not considered. Previous studies of $\mathcal{A}_{X_{s}\gamma}$ (and $\mathcal{A}_{X_{d}\gamma}$), but not $\mathcal{A}_{CP}({\overline{B}}\to X_{s+d}\gamma)$ and $\Delta\mathcal{A}_{X_{s}\gamma}$, in models with one charged scalar (e.g. 2HDM, or the lightest $H^{\pm}$ of a 3HDM or multi-Higgs doublet model) have been carried out in Refs. Wolfenstein:1994jw ; Asatrian:1996as ; Borzumati:1998tg ; Kiers:2000xy ; Jung:2010ab ; Jung:2012vu .

The NLO Wilson coefficients at the matching scale are as follows:

	$\displaystyle C^{1,\text{eff}}_{1}(\mu_{W})=$	$\displaystyle\ 15+6\hskip 5.69046pt\text{ln}\frac{\mu^{2}_{W}}{M^{2}_{W}},$		(34)
	$\displaystyle C^{1,\text{eff}}_{4}(\mu_{W})=$	$\displaystyle\ E_{0}+\frac{2}{3}\hskip 5.69046pt\text{ln}\frac{\mu^{2}_{W}}{M^{2}_{W}}+|Y_{1}|^{2}E_{H_{2}}+|Y_{2}|^{2}E_{H_{3}}$		(35)
	$\displaystyle C^{1,\text{eff}}_{i}(\mu_{W})=$	$\displaystyle\ 0\quad(i=2,3,5,6)$		(36)
	$\displaystyle C^{1,\text{eff}}_{7\gamma}(\mu_{W})=$	$\displaystyle\ C^{1,\text{eff}}_{7,SM}(\mu_{W})+|Y_{1}|^{2}C^{1,\text{eff}}_{7,Y_{1}Y_{1}}(\mu_{W})+|Y_{2}|^{2}C^{1,\text{eff}}_{7,Y_{2}Y_{2}}(\mu_{W})$
		$\displaystyle\ +(X_{1}Y_{1}^{*})C^{1,\text{eff}}_{7,X_{1}Y_{1}}(\mu_{W})+(X_{2}Y_{2}^{*})C^{1,\text{eff}}_{7,X_{2}Y_{2}}(\mu_{W})$		(37)
	$\displaystyle C^{1,\text{eff}}_{8g}(\mu_{W})=$	$\displaystyle\ C^{1,\text{eff}}_{8,SM}(\mu_{W})+|Y_{1}|^{2}C^{1,\text{eff}}_{8,Y_{1}Y_{1}}(\mu_{W})+|Y_{2}|^{2}C^{1,\text{eff}}_{8,Y_{2}Y_{2}}(\mu_{W})$
		$\displaystyle\ +(X_{1}Y_{1}^{*})C^{1,\text{eff}}_{8,X_{1}Y_{1}}(\mu_{W})+(X_{2}Y_{2}^{*})C^{1,\text{eff}}_{8,X_{2}Y_{2}}(\mu_{W})\,.$		(38)

Explicit forms for all functions are given in Ref. Borzumati:1998tg . Renormalisation group running is used to evaluate the Wilson coefficients at the scale $\mu=m_{b}$.

The partial width for ${\overline{B}}\to X_{s}\gamma$ has four distinct parts: i) Short-distance contribution from the $b\to s\gamma$ partonic process (to a given order in perturbation theory); ii) Short-distance contribution from the $b\to s\gamma g$ partonic process; iii) and iv) Non-perturbative corrections that scale as $1/m_{b}^{2}$ and $1/m_{c}^{2}$, respectively. The partial width of $\bar{B}\to X_{s}\gamma$ is as follows:

	$\displaystyle\Gamma({\overline{B}}\to X_{s}\gamma)$	$\displaystyle=$	$\displaystyle\frac{G^{2}_{F}}{32\pi^{4}}|V^{*}_{ts}V_{tb}|^{2}\alpha_{em}m^{5}_{b}$		(39)
		$\displaystyle\times$	$\displaystyle\Bigg{\{}|\bar{D}|^{2}+A+\frac{\delta^{NP}_{\gamma}}{m^{2}_{b}}|\text{C}^{0,\text{eff}}_{7}(\mu_{b})|^{2}$
		$\displaystyle+$	$\displaystyle\frac{\delta^{NP}_{c}}{m^{2}_{c}}{\rm Re}\Bigg{[}[\text{C}^{0,\text{eff}}_{7}(\mu_{b})]^{*}\times\bigg{(}\text{C}^{0,\text{eff}}_{2}(\mu_{b})-\frac{1}{6}\text{C}^{0,\text{eff}}_{1}(\mu_{b})\bigg{)}\Bigg{]}\Bigg{\}}\,.$

The short-distance contribution is contained in $|\bar{D}|^{2}$, with $\bar{D}$ given by:

$$\bar{D}=\text{C}^{0,\text{eff}}_{7}(\mu_{b})+\frac{\alpha_{s}(\mu_{b})}{4\pi}[\text{C}^{1,\text{eff}}_{7}(\mu_{b})+V(\mu_{b})]\,.$$

(40)

The Wilson coefficient $C^{0,\text{eff}}_{7}(\mu_{b})$ is a linear combination of $C^{0,\text{eff}}_{7}(\mu_{W})$, $C^{0,\text{eff}}_{8}(\mu_{W})$ and $C^{0,\text{eff}}_{2}(\mu_{W})$, while $C^{1,\text{eff}}_{7}(\mu_{b})$ is a linear combination of these three LO coefficients as well as the NLO coefficients $C^{1,\text{eff}}_{7}(\mu_{W})$, $C^{1,\text{eff}}_{8}(\mu_{W})$, $C^{1,\text{eff}}_{4}(\mu_{W})$, and $C^{1,\text{eff}}_{1}(\mu_{W})$. The parameter $V(\mu_{b})$ is a summation over all the LO Wilson coefficients which are evaluated at the scale $\mu_{b}=m_{b}$. The contribution from $b\to s\gamma g$ is contained in $A$, and the remaining two terms are the non-perturbative contributions. In $|\bar{D}|^{2}$ there are terms of order $\alpha_{s}^{2}$, but to only keep terms to the NLO order for a consistent calculation (to $\alpha_{s}$) the following form is used in Ref. Borzumati:1998tg :

$\displaystyle|\bar{D}|^{2}=|C^{0,\text{eff}}_{7}(\mu_{b})|^{2}\{1+2{\rm Re}(\Delta\bar{D})\}\,,$

(41)

$\displaystyle\Delta\bar{D}=\frac{\bar{D}-C^{0,\text{eff}}_{7}(\mu_{b})}{C^{0,\text{eff}}_{7}(\mu_{b})}=\frac{\alpha_{s}(\mu_{b})}{4\pi}\frac{C^{1,\text{eff}}_{7}(\mu_{b})+V(\mu_{b})}{C^{0,\text{eff}}_{7}(\mu_{b})}\,.$

(42)

The $m^{5}_{b}$ dependence is removed by using the measured value of the semi-leptonic branching ratio ${\rm BR}_{SL}\approx 0.1$ and its partial width $\Gamma_{SL}$ (which also depends on $m^{5}_{b}$), and BR$({\overline{B}}\to X_{s}\gamma)$ can be written as follows:

$${\rm BR}({\overline{B}}\to X_{s}\gamma)=\frac{\Gamma({\overline{B}}\to X_{s}\gamma)}{\Gamma_{SL}}{\rm BR}_{SL}\,.$$

(43)