Low Energy Aspects of Heavy Meson Decays
Low Energy Aspects of Heavy Meson Decays .∗
Jan O. Eeg
Department of Physics, University of Oslo,
P.O.Box 1048 Blindern, N-0316 Oslo, Norway
I discuss low energy aspects of heavy meson decays, where there is at
least one heavy meson in the final state. Examples are B − B mixing,
B → DD, B → Dη′, and B → Dγ. The analysis is performed in the heavy
quark limit within heavy-light chiral perturbation theory. Coefficients of
1/Nc suppressed chiral Lagrangian terms (beyond factorization) have been
estimated by means of a heavy-light chiral quark model.
PACS numbers: 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 − B-mixing [1], B → DD̄ [2] and with only one
D-meson in the final state, like B → Dη′ [3] and B → γ D∗ [4, 5, 6].
The methods [7] used to describe heavy to light tansitions like B → ππ
and B → Kπ are not suited for the decays we consider. We use heavy-light
chiral perturbation theory (HLχPT). Lagrangian terms corresponding to
factorization are then determined to zeroth order in 1/mQ, where mQ is the
mass of the heavy quark (b or c). For B−B-mixing we have also calculated
1/mb corrections [1].
Colour suppressed 1/Nc terms beyond factorization can be written down,
but their coefficients are unknown. However, these coefficients can be cal-
culated within a heavy-light chiral quark model (HLχQM) [8] based on the
heavy quark effective theory (HQEFT) [9] and HLχPT [10]. The 1/Nc
suppressed non-factorizable terms calculated in this way will typically be
proportional to a model dependent gluon condensate [1, 2, 3, 6, 8, 11].
Presented at the Euridice meeting in Kazimierz, Poland, 24-27th of august 2006
http://arxiv.org/abs/0704.0167v1
2 KazProc printed on November 4, 2018
2. Quark Lagrangians for non-leptonic decays
The effective non-leptonic Lagrangian at quark level has the form [12]:
Ci(µ) Q̂i(µ) , (1)
where the Wilson coefficients Ci contain GF and KM factors. Typically, the
operators are four quark operators being the product of two currents:
Q̂i = j
W (q1 → q2) j
µ (q3 → q4) , (2)
where j
W (qi → qj) = (qj)L γ
µ (qi)L, and some of the quarks qi,j are heavy.
To leading order in 1/Nc, matrix elements of Q̂i factorize in products of
matrix elements of currents. Non-factorizable 1/Nc suppressed terms are
obtained from “coloured quark operators”. Using Fierz transformations
δijδln =
δinδlj + 2 t
lj , (3)
where ta are colour matrices, we may rewrite the operator Q̂i as
Q̂Fi =
W (q1 → q4) j
µ (q3 → q2) + 2 j
W (q1 → q4)
a jWµ (q3 → q2)a , (4)
where j
W (qi → qj)a = (qj)L γµ ta (qi)L is a left-handed coloured current.
The quark operators in Q̂Fi give 1/Nc suppressed terms.
3. Heavy-light chiral perturbation theory
The QCD Lagrangian involving light and heavy quarks is:
LQuark = ±Q
v iv ·DQ(±)v +O(m−1Q ) + q̄iγ ·Dq + ... (5)
where Q
v are the quark fields for a heavy quark and a heavy anti-quark
with velocity v, q is the light quark triplet, and iDµ = i∂µ− eqAµ− gstaAaµ.
The bosonized Lagrangian have the following form, consistent with the un-
derlying symmetry [10]:
Lχ(Bos) = ∓Tr
a (iv · Dfa)H
− gATr
f γµγ5A
+ ...(6)
where the covariant derivative is iDµ
≡ δaf (i∂µ− eHAµ)−Vµfa ; a, f being
SU(3) flavour indices. The axial coupling is gA ≃ 0.6. Furthermore,
Vµ(orAµ) = ±
(ξ†∂µξ ± ξ∂µξ†) , (7)
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where ξ = exp(iΠ/f), and Π is a 3 by 3 matrix containing the light mesons
(π,Kη), and the heavy (1−, 0−) doublet field (Pµ, P5) is
H(±) = P±(P
µ − iP (±)5 γ5) , P± = (1± γ · v)/2 , (8)
where superscripts (±) means meson and anti-meson respectively. To bosonize
the non-leptonic quark Lagrangian, we need to bosonize the currents. Then
the b, c, and c quarks are treated within HQEFT, which means the replace-
ments b → Q(+)vb , c → Q
vc , and c → Q
v̄ . Then the bosonization of
currents within HQEFT for decay of a heavy B-meson will be:
µQ(+)vb −→
ξ†γµLH
≡ Jµb , (9)
where L is the left-handed projector in Dirac space, and αH = fH
for H = B,D before pQCD and chiral corrections are added. Here, H
represents the heavy meson (doublet) containing a b-quark. For creation of
a heavy anti-meson B or D, the corresponding currents J
and J
c̄ are given
by (9) with H
b replaced by H
b and H
c , repectively. For the B → D
transition we have
µ LQ(+)vc −→ −ζ(ω)Tr
≡ Jµb→c , (10)
where ζ(ω) is the Isgur-Wise function, and ω = vb · vc. For creation of DD
pair we have the same expression for the current J
cc̄ with H
replaced
c , and ζ(ω) replaced by ζ(−λ), where λ = v̄ · vc . In addition there
are 1/mQ corrections for Q = b, c. The low velocity limit is ω → 1 . For
B → DD and B → D∗γ one has ω ≃ 1.3 , and ω ≃ 1.6 , respectively.
3.1. Factorized lagrangians for non-leptonic processes
For B −B mixing, the factorized bosonized Lagrangian is
LB = CB J
b (Jb̄)
µ , (11)
where CB is a short distance Wilson coefficient (containing (GF )
2), which
is taken at µ = Λχ ≃ 1 GeV, and the currents are given by (9).
For processes obtained from two different four quark operators for b →
cc̄q (q = d, s), we find the factorized Lagrangian corresponding to Fig. 1:
LSpecFact = (C2 +
b→c (Jc̄)µ , (12)
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B0 D+
Fig. 1. Factorized contribution for B0
→ D+D−s through the spectator mechanism,
which does not exist for decay mode B0
→ D+s D−s .
Fig. 2. Factorized contribution for B0
→ D+s D−s through the annihilation mecha-
nism, which give zero contributions if both D+s and D
s are pseudoscalars.
where Ci =
GFVcbV
cq ai, and [13] a1 ≃ −0.35 − 0.07i, a2 ≃ 1.29 +
0.08i. We have considered the process B0d → 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 B0
→ D+D−s will be
the starting point for chiral loop contributions to the process B0
→ D+s D−s .
The factorizable term from annihilation is shown in Fig. 2, and is:
LAnnFact = (C1 +
cc̄ (Jb)µ . (13)
Because (C1 + C2/Nc) is a non-favourable combination of the Wilson coef-
ficients, 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/Nc suppressed tree level terms
For B − B̄ mixing, we have for instance the 1/Nc suppressed term
ξ†σµαLH
ξ†σµαRH
. (14)
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π0, η8
π0, η8
π0, η8
π0, η8
Fig. 3. Chiral corrections to B −B mixing, i.e the bag parameter BBq for q = d, s.
The black boxes are weak vertices.
B0 B∗0
Fig. 4. Two classes of non-factorizable chiral loops for B0
→ D+s D−s based on the
factorizable amplitude proportional to the IW function ∼ ζ(ω).
For B → DD̄, we have for instance the terms
ξ†σµαLH
c γαLH
c̄ γµ
, (15)
ξ†σµαLH
c γαLH
(v̄ − vc)µ . (16)
One needs a framework to estimate the coefficients of such terms. We use
the HLχQM, which will pick a certain linear combination of 1/Nc terms.
3.3. Chiral loops for non-leptonic processes
Within HLχPT, the leading chiral corrections are proportional to
χ(M) ≡ (
)2 ln(
) , (17)
where mM is the appropriate light meson mass and Λχ is the chiral symme-
try breaking scale, which is also the matching scale within our framework.
For B − B mixing there are chiral loops obtained from (6) and (11)
shown in Fig. 3. These have to be added to the factorized contribution.
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Fig. 5. The HLχQM ansatz: Vertex for quark meson interaction
For the process B0d → D
s we obtain a chiral loop amplitude cor-
responding to Fig. 4. This amplitude is complex and depend on ω and λ
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 HLχQM [8] contains the Lagrangian (5):
LHLχQM = LHQET + LχQM + LInt , (18)
where LHQET is the heavy quark part of (5), and the light quark part is
LχQM = χ [γµ(iDµ + Vµ + γ5Aµ)−m]χ . (19)
Here χL = ξ
†qL and χR = ξqR 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:
LInt = −GH
v Qv +Qv H
. (20)
The coupling GH is determined by bosonization through the loop diagrams
in Fig 6. The bosonization lead to relations between the model depen-
dent parameters GH , m, and 〈 αsπ G
2 〉, and the quadratic-, linear, and
logarithmic- divergent integrals I1, I3/2, I1, and the physical quantities fπ,
〈 qq 〉, gA and fH (H = B,D). For example, the relation obtained for iden-
tifying the kinetic term is:
− iG2HNc (I3/2 + 2mI2 +
i(8 − 3π)
384Ncm3
G2 〉) = 1 , (21)
where we have used the prescription:
αβ → 4π2〈
G2 〉 1
(gµαgνβ − gµβgνα) . (22)
The parameters are fitted in strong sector, with 〈 αs
G2 〉 = [(0.315 ± 0.020)
GeV]4 , and GH
2 = 2m
ρ , where ρ ≃ 1. For details , see [8].
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Fig. 6. Diagrams generating the strong chiral lagrangian at mesonic level. The
kinetic term and and the axial vector term ∼ gA.
Fig. 7. Non-factorizable contribution to B −B mixing; Γ ≡ ta γµ L
5. 1/N
terms from HLχQM
To obtain the 1/Nc terms for B − B mixing in Fig. 7 , we need the
bosonization of colored current in the quark operators of eq. (4):
a γαQ(+)vb
GH gs
GaµνTr
ξ†γαLH
b Σµν
, (23)
Σµν = σµν − 2πf
[σµν , γ · vb]+ . (24)
This coloured current is also used for B → DD in Fig. 8, for B → Dη′ in
Fig. 9, and for B → γD∗ in Fig. 10 In addition there are more complicated
bosonizations of coloured currents as indicated in Fig. 8.
For B → Dη′ and B → γD∗ decays there are two different four quark
operators, both for b → cūq and b → c̄uq, respectively. At µ = 1 GeV they
have Wilson coefficients a2 ≃ 1.17 , a1 ≃ −0.37 (up to prefactors GF and
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Fig. 8. Non-factorizable 1/Nc contribution for B0 → D+s D−s through the annihila-
tion mechanism with additional soft gluon emision.
Fig. 9. Diagram for B → Dη′ within HLχQM . Γ = γµ(1− γ5)
KM-factors). For B → Dη′, we must also attach a propagating gluon to the
η′gg∗-vertex. Note that for B0
→ γD0∗, the 1/Nc suppressed mechanism
in Fig. 10 dominates, unlike B0
→ γD0∗. Factorized contributions are
proportional to either the favourable contribution af = a2 + a1/Nc ≃ 1.05
or the non-favourable contribution anf = a1 + a2/Nc ≃ 0.02.
5.1. 1/mc correction terms
For the B → D transition we have the 1/mc suppressed terms:
c + Z1γ
c γα + Z2H
c γ · vb
, (25)
where the Zi’s are calculable within HLχQM. The relative size of 1/mc
corrections are typically of order 20− 30%.
6. Results
6.1. B −B mixing
The result for the B(ag) parameter in B −B-mixing has the form [1]
B̂Bq =
1− δBG
32π2f2
, (26)
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B D B D
Fig. 10. Non-factorizable contributions to B → γD∗ from the coloured operators
similar to the K − K-mixing case [11]. From perturbative QCD we have
b̃ ≃ 1.56 at µ = Λχ = 1 GeV. From calculations within the HLχQM we
obtain, δBG = 0.5±0.1 and τb = (0.26±0.04)GeV, and from chiral corrections
τχ,s = (−0.10 ± 0.04)GeV2, and τχ,d = (−0.02 ± 0.01)GeV2 . We obtained
B̂Bd = 1.51± 0.09 B̂Bs = 1.40 ± 0.16 , (27)
in agreement with lattice results.
6.2. B → DD decays
Keeping the chiral logs and the 1/Nc terms from the gluon condensate,
we find the branching ratios in the “leading approximation”. For decays of
B̄0d (∼ VcbV ∗cd) and B̄0s (∼ VcbV ∗cs) we obtain branching ratios of order few
×10−4 and ×10−3, respectively Then we have to add counterterms ∼ ms
for chiral loops. These may be estimated in HLχQM.
6.3. B → Dη′ and B → γD∗ decays
The result corresponding to Fig. 9 is:
Br(B → Dη′) ≃ 2× 10−4 . (28)
The partial branching ratios from the mechanism in Fig. 10 are [6]
Br(B0d → γ D
∗0)G ≃ 1× 10−5 ; Br(B0s → γ D∗0)G ≃ 6× 10−7 . (29)
The corresponding factorizable contribibutions are roughly two orders of
magnitude smaller. Note that the process B0d → γ D∗0 has substantial
meson exchanges (would be chiral loops for ω → 1), and is different.
7. Conclusions
Our low energy framework is well suited to B −B mixing, and to some
extent to B → DD. Work continues to include (0+, 1+), states, countert-
erms, and 1/mc terms. Note that the amplitude for B
→ D+s D−s is zero
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in the factorized limit. For processes like B → Dη′ and B → Dγ we
can give order of magnitude estimates when factorization give zero or small
amplitudes.
* * *
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
REFERENCES
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J. O. Eeg, S. Fajfer, and A. Prapotnik Eur. Phys. J. C42, 29-36 (2005).
See also: J.O. Eeg, S. Fajfer, J. Zupan, Phys. Rev. D 64, 034010 (2001).
[3] J. O. Eeg, A. Hiorth, A. D. Polosa, Phys. Rev. D 65, 054030 (2002).
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http://arxiv.org/abs/hep-ph/0606065
http://arxiv.org/abs/hep-ph/0605078
	Introduction
	Quark Lagrangians for non-leptonic decays
	Heavy-light chiral perturbation theory
	Factorized lagrangians for non-leptonic processes
	Possible 1/Nc suppressed tree level terms 
	Chiral loops for non-leptonic processes
	The heavy-light chiral quark model
	1/Nc terms from HLQM
	1/mc correction terms 
	Results
	B- B mixing
	B D  D decays
	B D  ' and B D* decays
	Conclusions
