Low Energy Aspects of Heavy Meson Decays

TL;DR

This paper explores low energy phenomena in heavy meson decays involving at least one heavy meson, using heavy-light chiral perturbation theory and a chiral quark model to estimate non-factorizable effects.

Contribution

It provides a detailed analysis of heavy meson decay processes in the heavy quark limit, including estimates of coefficients for suppressed chiral Lagrangian terms beyond factorization.

Findings

Estimation of $1/N_c$ suppressed chiral Lagrangian coefficients

Application of heavy-light chiral perturbation theory to decay processes

Insights into non-factorizable contributions in heavy meson decays

Abstract

I discuss low energy aspects of heavy meson decays, where there is at least one heavy meson in the final state. Examples are $B -\bar{B}$ mixing, $B \to D \bar{D}$, $B \to D \eta'$, and $B \to D \gamma$. %and $B \to D W $ (Isgur-Wise function). The analysis is performed in the heavy quark limit within heavy-light chiral perturbation theory. Coefficients of $1/N_c$ suppressed chiral Lagrangian terms (beyond factorization) have been estimated by means of a heavy-light chiral quark model.

Figures

Equations

$${\mathcal{L}}_W=\sum _i{C}_i(\mu ){\hat{Q}}_i(\mu ),$$

$${\hat{Q}}_i=j_W^{\mu }(q_1\to q_2)j_{\mu }^W(q_3\to q_4),$$

$${\delta }_{ij}{\delta }_{ln}=\frac{1}{N_c}{\delta }_{in}{\delta }_{lj}+2t_{in}^a{t}_{lj}^a,$$

$${\hat{Q}}_i^F=\frac{1}{N_c}{j}_W^{\mu }(q_1\to q_4)j_{\mu }^W(q_3\to q_2)+2j_W^{\mu }(q_1\to q_4{)}^a{j}_{\mu }^W(q_3\to q_2{)}^a,$$

$${\mathcal{L}}_{Quark}=\pm \overline{Q_v^{(\pm )}}iv\cdot D{Q}_v^{(\pm )}+\mathcal{O}(m_Q^{-1})+\bar{q}i\gamma \cdot Dq+...$$

$${\mathcal{L}}_{\chi }(Bos)=\mp Tr\left[\overline{H_a^{(\pm )}}(iv\cdot {\mathcal{D}}_{fa})H_f^{(\pm )}\right]-g_{\mathcal{A}}Tr\left[\overline{H_a^{(\pm )}}{H}_f^{(\pm )}{\gamma }_{\mu }{\gamma }_5{\mathcal{A}}_{fa}^{\mu }\right]+...$$

$${\mathcal{V}}_{\mu }(\text{or}{\mathcal{A}}_{\mu })=\pm \frac{i}{2}({\xi }^{†}{\partial }_{\mu }\xi \pm \xi {\partial }_{\mu }{\xi }^{†}),$$

$$H^{(\pm )}=P_{\pm }(P_{\mu }^{(\pm )}{\gamma }^{\mu }-i{P}_5^{(\pm )}{\gamma }_5),P_{\pm }=(1\pm \gamma \cdot v)/2,$$

$$\overline{q_L}{\gamma }^{\mu }{Q}_{v_b}^{(+)}⟶\frac{{\alpha }_H}{2}Tr\left[{\xi }^{†}{\gamma }^{\mu }L{H}_b^{(+)}\right]\equiv J_b^{\mu },$$

$$\overline{Q_{v_b}^{(+)}}{\gamma }^{\mu }L{Q}_{v_c}^{(+)}⟶-\zeta (\omega )Tr\left[\overline{H_c^{(+)}}{\gamma }^{\mu }L{H}_b^{(+)}\right]\equiv J_{b\to c}^{\mu },$$

$${\mathcal{L}}_B=C_B{J}_b^{\mu }(J_{\bar{b}}{)}^{\mu },$$

$${\mathcal{L}}_{Fact}^{Spec}=(C_2+\frac{C_1}{N_c})J_{b\to c}^{\mu }(J_{\bar{c}}{)}_{\mu },$$

$${\mathcal{L}}_{Fact}^{Ann}=(C_1+\frac{C_2}{N_c})J_{c\bar{c}}^{\mu }(J_b{)}_{\mu }.$$

$$Tr\left[{\xi }^{†}{\sigma }^{\mu \alpha }L{H}_b^{(+)}\right]\cdot Tr\left[{\xi }^{†}{\sigma }_{\mu \alpha }R{H}_{\bar{b}}^{(-)}\right].$$

$$Tr\left[{\xi }^{†}{\sigma }^{\mu \alpha }L{H}_b^{(+)}\right]\cdot Tr\left[\overline{H_c^{(+)}}{\gamma }_{\alpha }L{H}_{\bar{c}}^{(-)}{\gamma }_{\mu }\right],$$

$$Tr\left[{\xi }^{†}{\sigma }^{\mu \alpha }L{H}_b^{(+)}\right]\cdot Tr\left[\overline{H_c^{(+)}}{\gamma }_{\alpha }L{H}_{\bar{c}}^{(-)}\right](\bar{v}-v_c{)}_{\mu }.$$

$$\chi (M)\equiv (\frac{g_{\mathcal{A}}{m}_M}{4\pi f}{)}^2\ln (\frac{{\Lambda }_{\chi }^2}{m_M^2}),$$

$${\mathcal{L}}_{HL\chi QM}={\mathcal{L}}_{HQET}+{\mathcal{L}}_{\chi QM}+{\mathcal{L}}_{Int},$$

$${\mathcal{L}}_{\chi QM}=\overline{\chi }\left[{\gamma }^{\mu }(i{D}_{\mu }+{\mathcal{V}}_{\mu }+{\gamma }_5{\mathcal{A}}_{\mu })-m\right]\chi .$$

$${\mathcal{L}}_{Int}=-G_H\left[{\overline{\chi }}_f\overline{H_v^f}{Q}_v+\overline{Q_v}{H}_v^f{\chi }_f\right].$$

$$-i{G}_H^2{N}_c(I_{3/2}+2m{I}_2+\frac{i(8-3\pi )}{384N_c{m}^3}h\frac{{\alpha }_s}{\pi }{G}^{2}i)=1,$$

$$g_s^2{G}_{\mu \nu }^a{G}_{\alpha \beta }^a\to 4{\pi }^{2}h\frac{{\alpha }_s}{\pi }{G}^{2}i\frac{1}{12}(g_{\mu \alpha }{g}_{\nu \beta }-g_{\mu \beta }{g}_{\nu \alpha }).$$

$${\left(\overline{q_L}{t}^a{\gamma }^{\alpha }{Q}_{v_b}^{(+)}\right)}_{1G}⟶-\frac{G_H{g}_s}{64\pi }{G}_{\mu \nu }^{a}Tr\left[{\xi }^{†}{\gamma }^{\alpha }L{H}_b^{(+)}{\Sigma }_{\mu \nu }\right],$$

$${\Sigma }^{\mu \nu }={\sigma }^{\mu \nu }-\frac{2\pi f^2}{m^2{N}_c}[{\sigma }^{\mu \nu },\gamma \cdot v_b{]}_{+}.$$

$$\frac{1}{m_c}Tr\left[\left(Z_0\overline{H_c^{(+)}}+Z_1{\gamma }^{\alpha }\overline{H_c^{(+)}}{\gamma }_{\alpha }+Z_2\overline{H_c^{(+)}}\gamma \cdot v_b\right){\gamma }^{\alpha }L{H}_b^{(+)}\right],$$

$${\hat{B}}_{B_q}=\frac{3}{4}\tilde{b}\left[1+\frac{1}{N_c}\left(1-{\delta }_G^B\right)+\frac{{\tau }_b}{m_b}+\frac{{\tau }_{\chi }}{32{\pi }^2{f}^2}\right],$$

$${\hat{B}}_{B_d}=1.51\pm 0.09{\hat{B}}_{B_s}=1.40\pm 0.16,$$

$$Br(B\to D{\eta }^{{}^{\prime }})\simeq 2\times 10^{-4}.$$

$$Br(\overline{B_d^0}\to \gamma D^{\ast 0}{)}_G\simeq 1\times 10^{-5};Br(\overline{B_s^0}\to \gamma D^{\ast 0}{)}_G\simeq 6\times 10^{-7}.$$

Full text

\PACS

PACS numbers 13.20.Hw , 12.39.St , 12.39.Fe , 12.39.Hg

1 Introduction

In this paper we consider non-leptonic “heavy meson to heavy meson(s)” transitions, for instance $B-\overline{B}$-mixing [1], $B\rightarrow D\bar{D}$ [2] and with only one $D$-meson in the final state, like $B\rightarrow D\eta^{\prime}$ [3] and $B\rightarrow\gamma\,D^{*}$ [4, 5, 6].

The methods [7] used to describe heavy to light tansitions like $B\rightarrow\pi\pi$ and $B\rightarrow K\pi$ are not suited for the decays we consider. We use heavy-light chiral perturbation theory (HL$\chi$PT). Lagrangian terms corresponding to factorization are then determined to zeroth order in $1/m_{Q}$, where $m_{Q}$ is the mass of the heavy quark ($b$ or $c$). For $B-\overline{B}$-mixing we have also calculated $1/m_{b}$ corrections [1].

Colour suppressed $1/N_{c}$ terms beyond factorization can be written down, but their coefficients are unknown. However, these coefficients can be calculated within a heavy-light chiral quark model (HL$\chi$QM) [8] based on the heavy quark effective theory (HQEFT) [9] and HL$\chi$PT [10]. The $1/N_{c}$ suppressed non-factorizable terms calculated in this way will typically be proportional to a model dependent gluon condensate [1, 2, 3, 6, 8, 11].

2 Quark Lagrangians for non-leptonic decays

The effective non-leptonic Lagrangian at quark level has the form [12]:

[TABLE]

where the Wilson coefficients $C_{i}$ contain $G_{F}$ and KM factors. Typically, the operators are four quark operators being the product of two currents:

[TABLE]

where $j_{W}^{\mu}(q_{i}\to q_{j})=\overline{(q_{j})_{L}}\,\gamma^{\mu}\,(q_{i})_{L}$, and some of the quarks $q_{i,j}$ are heavy. To leading order in $1/N_{c}$, matrix elements of $\hat{Q}_{i}$ factorize in products of matrix elements of currents. Non-factorizable $1/N_{c}$ suppressed terms are obtained from “coloured quark operators”. Using Fierz transformations and

[TABLE]

where $t^{a}$ are colour matrices, we may rewrite the operator $\hat{Q}_{i}$ as

[TABLE]

where $j_{W}^{\mu}(q_{i}\to q_{j})^{a}=\overline{(q_{j})_{L}}\,\gamma^{\mu}\,t^{a}\,(q_{i})_{L}$ is a left-handed coloured current. The quark operators in $\hat{Q}_{i}^{F}$ give $1/N_{c}$ suppressed terms.

3 Heavy-light chiral perturbation theory

The QCD Lagrangian involving light and heavy quarks is:

[TABLE]

where $Q^{(\pm)}_{v}$ are the quark fields for a heavy quark and a heavy anti-quark with velocity $v$, $q$ is the light quark triplet, and $iD_{\mu}=i\partial_{\mu}-e_{q}A_{\mu}-g_{s}t^{a}A_{\mu}^{a}$. The bosonized Lagrangian have the following form, consistent with the underlying symmetry [10]:

[TABLE]

where the covariant derivative is $i{\cal D}_{fa}^{\mu}\equiv\delta_{af}(i\partial^{\mu}-e_{H}A^{\mu})-{\cal V}_{fa}^{\mu}\;$ ; $a,f$ being SU(3) flavour indices. The axial coupling is $\;g_{{\cal A}}\simeq 0.6$. Furthermore,

[TABLE]

where $\xi=exp(i\Pi/f)$, and $\Pi$ is a 3 by 3 matrix containing the light mesons ($\pi,K\eta$), and the heavy $(1^{-},0^{-})$ doublet field $(P_{\mu},P_{5})$ is

[TABLE]

where superscripts $(\pm)$ means meson and anti-meson respectively. To bosonize the non-leptonic quark Lagrangian, we need to bosonize the currents. Then the $b$, $c$, and $\overline{c}$ quarks are treated within HQEFT, which means the replacements $b\rightarrow Q_{v_{b}}^{(+)}\;,c\rightarrow Q_{v_{c}}^{(+)}\;$, and $\overline{c}\rightarrow Q_{\bar{v}}^{(-)}$. Then the bosonization of currents within HQEFT for decay of a heavy $B$-meson will be:

[TABLE]

where $L$ is the left-handed projector in Dirac space, and $\alpha_{H}=f_{H}\sqrt{M_{H}}$ for $H=B,D$ before pQCD and chiral corrections are added. Here, $H_{b}^{(+)}$ represents the heavy meson (doublet) containing a $b$-quark. For creation of a heavy anti-meson $\overline{B}$ or $\overline{D}$, the corresponding currents $J_{\bar{b}}^{\mu}$ and $J_{\bar{c}}^{\mu}$ are given by (9) with $H_{b}^{(+)}$ replaced by $H_{b}^{(-)}$ and $H_{c}^{(-)}$, repectively. For the $B\rightarrow D$ transition we have

[TABLE]

where $\zeta(\omega)$ is the Isgur-Wise function, and $\;\omega=v_{b}\cdot v_{c}$. For creation of $D\overline{D}$ pair we have the same expression for the current $J_{c\bar{c}}^{\mu}$ with $H_{b}^{(+)}$ replaced by $H_{c}^{(-)}$, and $\zeta(\omega)$ replaced by $\zeta(-\lambda)$, where $\lambda=\bar{v}\cdot v_{c}\;$. In addition there are $1/m_{Q}$ corrections for $Q=b,c$. The low velocity limit is $\omega\rightarrow 1\,$. For $\,B\rightarrow D\overline{D}$ and $\,B\rightarrow D^{*}\gamma$ one has $\omega\simeq 1.3\,,$ and $\omega\simeq 1.6\,$, respectively.

3.1 Factorized lagrangians for non-leptonic processes

For $B-\overline{B}$ mixing, the factorized bosonized Lagrangian is

[TABLE]

where $C_{B}$ is a short distance Wilson coefficient (containing $(G_{F})^{2}$), which is taken at $\mu=\Lambda_{\chi}\simeq$ 1 GeV, and the currents are given by (9).

For processes obtained from two different four quark operators for $b\rightarrow c\bar{c}q\;\;(q=d,s)$, we find the factorized Lagrangian corresponding to Fig. 1:

[TABLE]

where $C_{i}=\frac{4}{\sqrt{2}}G_{F}V_{cb}V_{cq}^{*}\,a_{i}$, and [13] $a_{1}\simeq-0.35-0.07i$, $a_{2}\simeq 1.29+0.08i$. We have considered the process $\overline{B^{0}_{d}}\rightarrow D_{s}^{+}D_{s}^{-}\,$. Note that there is no factorized contribution to this process if both $D$-mesons in the final state are pseudoscalars! But the factorized contribution to $\overline{B^{0}_{d}}\rightarrow D^{+}D_{s}^{-}$ will be the starting point for chiral loop contributions to the process $\overline{B^{0}_{d}}\rightarrow D_{s}^{+}D_{s}^{-}$.

The factorizable term from annihilation is shown in Fig. 2, and is:

[TABLE]

Because $(C_{1}+C_{2}/N_{c})$ is a non-favourable combination of the Wilson coefficients, this term will give a small non-zero contribution if at least one of the mesons in the final state is a vector.

3.2 Possible $1/N_{c}$ suppressed tree level terms

For $B-\bar{B}$ mixing, we have for instance the $1/N_{c}$ suppressed term

[TABLE]

For $B\rightarrow D\bar{D}$, we have for instance the terms

[TABLE]

[TABLE]

One needs a framework to estimate the coefficients of such terms. We use the HL$\chi$QM, which will pick a certain linear combination of $1/N_{c}$ terms.

3.3 Chiral loops for non-leptonic processes

Within HL$\chi$PT, the leading chiral corrections are proportional to

[TABLE]

where $m_{M}$ is the appropriate light meson mass and $\Lambda_{\chi}$ is the chiral symmetry breaking scale, which is also the matching scale within our framework.

For $B-\overline{B}$ mixing there are chiral loops obtained from (6) and (11) shown in Fig. 3. These have to be added to the factorized contribution.

For the process $\overline{B^{0}_{d}}\rightarrow D_{s}^{+}D_{s}^{-}$ we obtain a chiral loop amplitude corresponding to Fig. 4. This amplitude is complex and depend on $\omega$ and $\lambda$ defined previously. It has been recently shown [5] that $(0^{+},1^{+})$ states in loops should also be added to the result.

4 The heavy-light chiral quark model

The Lagrangian for HL$\chi$QM [8] contains the Lagrangian (5):

[TABLE]

where ${\cal L}_{HQET}$ is the heavy quark part of (5), and the light quark part is

[TABLE]

Here $\chi_{L}=\xi^{\dagger}q_{L}\,$ and $\,\chi_{R}=\xi q_{R}$ are flavour rotated light quark fields, and $m$ is the light constituent mass.

The bosonization of the (heavy-light) quark sector is performed via the ansatz:

[TABLE]

The coupling $G_{H}$ is determined by bosonization through the loop diagrams in Fig 6. The bosonization lead to relations between the model dependent parameters $G_{H}$, $m$, and ${\cal h}\,\frac{\alpha_{s}}{\pi}G^{2}\,{\cal i}$, and the quadratic-, linear, and logarithmic- divergent integrals $I_{1},I_{3/2},I_{1}$, and the physical quantities $f_{\pi}$, ${\cal h}\,\overline{q}q\,{\cal i}$, $g_{{\cal A}}$ and $f_{H}$ ($H=B,D$).

For example, the relation obtained for identifying the kinetic term is:

[TABLE]

where we have used the prescription:

[TABLE]

The parameters are fitted in strong sector, with ${\cal h}\,\frac{\alpha_{s}}{\pi}G^{2}\,{\cal i}=[(0.315\pm 0.020)$ GeV]${}^{4}\;,$ and $\;{G_{H}}^{2}=\frac{2m}{f^{2}}\rho\,$, where $\rho\simeq 1$. For details , see [8].

5 $1/N_{c}$ terms from HL$\chi$QM

To obtain the $1/N_{c}$ terms for $B-\overline{B}$ mixing in Fig. 7 , we need the bosonization of colored current in the quark operators of eq. (4):

[TABLE]

[TABLE]

This coloured current is also used for $B\rightarrow D\overline{D}$ in Fig. 8, for $B\rightarrow D\,\eta^{\prime}$ in Fig. 9, and for $B\rightarrow\gamma D^{*}$ in Fig. 10 In addition there are more complicated bosonizations of coloured currents as indicated in Fig. 8.

For $B\rightarrow D\,\eta^{\prime}$ and $B\rightarrow\gamma D^{*}$ decays there are two different four quark operators, both for $b\rightarrow c\bar{u}q$ and $b\rightarrow\bar{c}uq$, respectively. At $\mu=1$ GeV they have Wilson coefficients $a_{2}\simeq 1.17\;,\;a_{1}\simeq-0.37$ (up to prefactors $G_{F}$ and KM-factors).

For $B\rightarrow D\,\eta^{\prime}$, we must also attach a propagating gluon to the $\eta^{\prime}gg^{*}$-vertex. Note that for $\overline{B_{s,d}^{0}}\rightarrow\gamma D^{0*}$, the $1/N_{c}$ suppressed mechanism in Fig. 10 dominates, unlike $\overline{B_{s,d}^{0}}\rightarrow\overline{\gamma D^{0*}}$. Factorized contributions are proportional to either the favourable contribution $a_{f}\;=\;a_{2}+a_{1}/N_{c}\simeq 1.05$ or the non-favourable contribution $a_{nf}\;=\;a_{1}+a_{2}/N_{c}\simeq 0.02$.

5.1 $1/m_{c}$ correction terms

For the $B\rightarrow D$ transition we have the $1/m_{c}$ suppressed terms:

[TABLE]

where the $Z_{i}$’s are calculable within HL$\chi$QM. The relative size of $1/m_{c}$ corrections are typically of order $20-30\%$.

6 Results

6.1 $B-\overline{B}$ mixing

The result for the B(ag) parameter in $B-\overline{B}$-mixing has the form [1]

[TABLE]

similar to the $K-\overline{K}$-mixing case [11]. From perturbative QCD we have $\widetilde{b}\simeq 1.56$ at $\mu=\Lambda_{\chi}$ = 1 GeV. From calculations within the HL$\chi$QM we obtain, $\delta_{G}^{B}=0.5\pm 0.1$ and $\tau_{b}=(0.26\pm 0.04)$GeV, and from chiral corrections $\tau_{\chi,s}=(-0.10\pm 0.04)$GeV2, and $\tau_{\chi,d}=(-0.02\pm 0.01)$GeV2. We obtained

[TABLE]

in agreement with lattice results.

6.2 $B\rightarrow D\,\overline{D}$ decays

Keeping the chiral logs and the $1/N_{c}$ terms from the gluon condensate, we find the branching ratios in the “leading approximation”. For decays of $\bar{B}_{d}^{0}$ ($\sim V_{cb}V^{*}_{cd}$) and $\bar{B}_{s}^{0}$ ($\sim V_{cb}V^{*}_{cs}$) we obtain branching ratios of order few $\times 10^{-4}$ and $\times 10^{-3}$, respectively Then we have to add counterterms $\sim m_{s}$ for chiral loops. These may be estimated in HL$\chi$QM.

6.3 $B\rightarrow D\,\eta^{\prime}$ and $B\rightarrow\gamma D^{*}$

decays

The result corresponding to Fig. 9 is:

[TABLE]

The partial branching ratios from the mechanism in Fig. 10 are [6]

[TABLE]

The corresponding factorizable contribibutions are roughly two orders of magnitude smaller. Note that the process $\overline{B^{0}_{d}}\rightarrow\gamma\,\overline{D^{*0}}$ has substantial meson exchanges (would be chiral loops for $\omega\rightarrow 1$), and is different.

7 Conclusions

Our low energy framework is well suited to $B-\overline{B}$ mixing, and to some extent to $B\rightarrow D\overline{D}$. Work continues to include $(0^{+},1^{+})$, states, counterterms, and $1/m_{c}$ terms. Note that the amplitude for $\overline{B^{0}_{d}}\rightarrow D_{s}^{+}D_{s}^{-}\,$ is zero in the factorized limit. For processes like $B\rightarrow D\eta^{\prime}$ and $B\rightarrow D\gamma\;$ we can give order of magnitude estimates when factorization give zero or small amplitudes.

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JOE is supported in part by the Norwegian research council and by the European Union RTN network, Contract No. HPRN-CT-2002-00311 (EURIDICE). He thanks his collaborators : A. Hiorth, S. Fajfer, A. Polosa, A. Prapotnik Brdnik, J.A. Macdonald Sørensen, and J. Zupan