Transition Form Factors of P Wave
Bottomonium
χ
b0
(1P) into B
c
Meson
J. Y. Süngü
∗, K. Azizi
†and H. Sundu
∗∗Department of Physics, Kocaeli University, 41380 Izmit, Turkey
†Department of Physics, Do˘gu¸s University, Acıbadem-Kadıköy, 34722 Istanbul, Turkey
Abstract. Taking into account the two-gluon condensate contributions, the transition form factors enrolled to the low energy effective Hamiltonian describing the semileptonic χb0 → Bcν,( = (e,μ,τ)) decay channel are calculated within three-point QCD sum rules.
Keywords: QCD sum rules, Semileptonic transition, Form factor, Decay width PACS: 11.55.Hx, 14.40.Pq, 13.20.-v, 13.20.Gd
The quarkonia, especially the bottomonium states, are approximately non-relativistic systems since they do not contain intrinsically relativistic light quarks. Hence, these states are the best candidates to examine the hadronic dynamics and investigate both perturbative and non-perturbative characteristic of QCD. The QCD sum rule approach [1] is one of the most powerful and applicable tools for hadron physics. In the present work, we apply this method to calculate the transition form factors of theχb0→ Bcν.
The effective Hamiltonian responsible for theχb0→ Bcν can be written as:
He f f =
GF
√
2Vcbν γμ(1 −γ5)l cγμ(1 −γ5)b, (1) where GF is the Fermi weak coupling constant and Vcb is an element of the
Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix. The transition amplitude is obtained via
M = Bc(p) | He f f |χb0(p). (2)
To proceed, we need to know the transition matrix element Bc(p) | cγμ(1 −γ5)b |
χb0(p) whose vector part do not contribute due to parity considerations. The
axial-vector part of transition matrix element can be parameterized in terms of form factors:
Bc(p) | cγμγ5b|χb0(p) = f1(q2)Pμ+ f2(q2)qμ, (3)
where f1(q2) and f2(q2) are transition form factors; and Pμ = (p + p)μ and qμ =
(p− p)
μ. In order to calculate the transition form factors, the starting point is to consider
the following tree-point correlation function: Πμ= i2 d4x d4ye−ipxeipy0 | T JBc(y)J A;V μ (0)Jχ†b0(x) | 0 (4)
where T is the time ordering product, JBc(y) = cγ5b and Jχb0(x) = bUb are the
U γ5 γμ(1 − γ5) l ¯ν U γ5 γμ(1 − γ5) l ¯ν U γ5 γμ(1 − γ5) l ¯ν U γ5 γμ(1 − γ5) l ¯ν W W W W l ¯ν W U γ5 γμ(1 − γ5) l ¯ν W γμ(1 − γ5) U γ5 l ¯ν W γμ(1 − γ5) U γ5 (a) (b) (c) (d) (e) (f) (g) b b c b b c b b c bb c b c b bb c bbc
FIGURE 1. Feynman diagrams contributing to the correlation function for theχb0→ Bcν decay: (a)
the bare loop and (b, c, d, e, f, g) two-gluon condensate diagrams.
JμA(0) = cγμγ5b are the vector and axial-vector parts of the transition current. The
phe-nomenological or physical side of the correlation function is obtained inserting two com-plete sets of intermediate states with the same quantum numbers as the interpolating currents JBc and Jχb0. As a result, we obtain
ΠPHY S μ = 0 | JBc(0) | Bc(p )B c(p) | JμA(0) |χb0(p)χb0(p) | Jχ†b0(0) | 0 (p2− m2 Bc)(p 2− m2 χb0) + ··· (5) where··· represents the contributions coming from higher states and continuum. The matrix elements of interpolating current between the vacuum and hadronic states are parameterized in terms of the leptonic decay constants, i.e.,
0 | JBc(0) | Bc(p ) = i fBcm 2 Bc mb+ mc, χb0(p) | J † χb0(0) | 0 = −imχb0fχb0. (6)
Putting all expressions together, the final version of the phenomenological side of the correlation function is obtained as
ΠPHY S μ (p2, p2) = (p2− mfχ2b0mχb0 Bc)(p 2− m2 χb0) fBcm 2 Bc mb+ mc f1(q2)Pμ+ f2(q2)qμ + .... (7) In QCD side, we can write the coefficient of each structure in correlation function as a sum of a perturbative (diagram a in Fig. 1) and a non-perturbative (diagrams. b, c, d, e, f and g in Fig. 1) parts as follows:
ΠQCD μ = (Π1pert+ Π nonpert 1 )Pμ+ (Π pert 2 + Π nonpert 2 )qμ (8)
where, theΠipert functions are written in terms of double dispersion integrals as: Πpert i = − 1 (2π)2 ds ds ρi(s,s ,q2) (s − p2)(s− p2)+ subtraction terms, (9)
124
here,ρi(s,s,q2) are the spectral densities with i = 1 or 2. Applying the usual Feynman
integral technique to the bare loop, the spectral densities are obtained as follows:
ρ1(s,s,q2) = 2NcI0(s,s,q2) mb(mc− 3mb) − A(h + s) − B(h + s) , ρ2(s,s,q2) = 2NcI0(s,s,q2) mb(mb+ mc) − A(h − s) + B(h − s) , (10) where I0(s,s,q2) = 1 4λ1/2(s,s,q2), A = 1 λ(s,s,q2) (m2 b− m2c)u+ us , B= 1 λ(s,s,q2) 2(m2b− m2c)s + su, h = 2mb(mb− mc), (11) here alsoλ(s,s,q2) = s2+s2+q4−2ss−2sq2−2sq2, u= q2+s−s, u= q2−s+s,
u= q2−s−sand Nc= 3 is the number of colors. For the non-perturbative part Πnonpert1,2 ,
we take into account the two-gluon condensate diagrams (b, c, d, e, f, g) in Figure 1. They have very lengthy expressions and we do not present them here. For their explicit expressions see [2]. The next step is to apply double Borel transformations with respect to the variables p2and p2(p2→ M2, p2→ M2). Then, the QCD sum rules for the form factors are obtained as:
f1,2(q2) = (mb+ mc) fBcm 2 Bc 1 fχb0mχb0 em2χb0/M2em2Bc/M2 −(2π1)2 s0 4m2bds s 0 (mb+mc)2 ds × ρ1,2(s,s,q2)θ[1 − f2(s,s)]e−s/M 2 e−s/M2+ ˆBΠnonpert1,2 , (12)
where s0 and s0 are continuum thresholds in initial and final meson channels,
re-spectively. In our calculation, we choose the intervals of continuum thresholds as
s0= (97.7 − 99.2) GeV2and s0= (40 − 41) GeV2slightly higher than the mass of pole
squared of the initial and final mesonic channels for the continuum thresholds. Our cal-culations show that the form factors are truncated at q2 9 GeV2. Therefore, to extend our results to the full physical region, the following fit parametrization is used:
fi(q2) = a (1 − q2 m2f it) + b (1 − q2 m2f it) 2 (13)
where, the values of the parameters a, b and mf it obtained using M2 = 25 GeV2 and
M2= 15 GeV2for theχb0→ Bcνchannel are given in Table 1. We present the results
TABLE 1. Parameters appearing in the fit function of the form factors.
a b m2f it(GeV2)
f1(χb0→ Bcν) -0.055 0.062 21.86
f2(χb0→ Bcν) 0.225 -0.254 19.79
TABLE 2. Numerical results for decay rate for different lepton chan-nels. Γ(GeV) χb0→ Bceνe 1.46 × 10−14 χb0→ Bcμνμ 1.45 × 10−14 χb0→ Bcτντ 0.91 × 10−14 Fit Total Perturbative NonPerturbative 0 2 4 6 8 10 12 0.00 0.05 0.10 0.15 q2GeV2 f1 Fit Total Perturbative NonPerturbative 0 2 4 6 8 10 12 1.0 0.8 0.6 0.4 0.2 0.0 q2GeV2 f2
FIGURE 2. Dependence of the form factors f1and f2on q2at M2= 25 GeV2and M2= 15 GeV2.
Figure 2. Also, the numerical values of the decay width at different lepton channels are obtained as shown in Table 2.
Any measurement on the form factors as well as decay rate of the channel under consideration and comparison of the obtained results with theoretical predictions in the present study can give valuable information about the internal structures of the participating mesons specially nature of the scalarχb0(1P) state.
ACKNOWLEDGMENTS
This work has been supported in part by Scientific and Technological Research Council of Turkey (TUBITAK) under project No: 110T284 and in part by Kocaeli University Scientific Research Center (BAP) under project No: 2011/52.
REFERENCES
1. M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979). 2. K. Azizi, H. Sundu and J.Y. Süngü, arXiv:1207.5922v1[hep.ph].
