Semileptonic
b;cto nucleon transitions in full QCD at light cone
K. Azizi,1,*M. Bayar,2,†Y. Sarac,3,‡and H. Sundu2,x1Physics Division, Faculty of Arts and Sciences, Dog˘us¸ University, Acbadem-Kadko¨y, 34722 Istanbul, Turkey 2Department of Physics, Kocaeli University, 41380 Izmit, Turkey
3Electrical and Electronics Engineering Department, Atilim University, 06836 Ankara, Turkey (Received 12 August 2009; published 30 November 2009)
The tree-level semileptonicb! pl and c! nl transitions are investigated using the light cone QCD sum rules approach in full theory. The spin1=2, Qbaryon with Q ¼ b or c, is considered by the most general form of its interpolating current. The time ordering product of the initial and transition currents is expanded in terms of the nucleon distribution amplitudes with different twists. Considering two sets of independent input parameters entering to the nucleon wave functions, namely, QCD sum rules and lattice QCD parameters, the related form factors and their heavy quark effective theory limits are calculated and compared with the existing predictions of other approaches. It is shown that our results satisfy the heavy quark symmetry relations for lattice input parameters and b case exactly and the maximum violation is for charm case and QCD sum rules input parameters. The obtained form factors are used to compute the transition rates both in full theory and heavy quark effective theory. A comparison of the results on decay rate ofb! pl with those predicted by other phenomenological methods or the same method in heavy quark effective theory with different interpolating current and distribution amplitudes of theb is also presented.
DOI:10.1103/PhysRevD.80.096007 PACS numbers: 11.55.Hx, 12.39.Hg, 13.30.a, 14.20.Mr I. INTRODUCTION
Motivated by the recent experimental progresses on the spectroscopy of the heavy baryons containing heavy b or c quark [1–8], theoretical studies on these baryons gain pace. Because of the heavy quark, these states are expected to be narrow, experimentally, hence their isolation and detection are easy compared to light systems. Theoretically, inves-tigation of the semileptonic decays of the heavy baryons, whose experimental testing may be in the future program of the large hadron collider (LHC), have attracted interests beside their mass and electromagnetic properties. For in-stance, the semileptonic b ! c and c! decays have been investigated in three points QCD sum rules and heavy quark effective theory (HQET) in [9]. Theb ! pl transition has also been studied in the same frame-works in [10] and using SU(3) symmetry and HQET in [11]. Constituent quark model have also been used to study thec! nl and b! pl form factors [12] and semi-leptonic decays of some heavy baryons containing single heavy quark in different quark models [12–14] are some other works in this respect.
In our recent work [15], we analyzed the semileptonic decay ofb, which has different interpolating current and structure thanQ, to proton in light cone QCD sum rules. In present study, we calculate the form factors related to the semileptonic decays of theb! pl and c ! nl also in light cone QCD in full theory and HQET limit. In full theory, these transitions are governed by six form factors,
but heavy quark effective theory limit reduces them to two. The vacuum to nucleon matrix element of the time-ordering product is expanded in terms of nucleon distribu-tion amplitudes (DAs) near light cone, x2 ’ 0. The nucleon wave functions contain eight independent parameters, which we consider two sets, namely, calculated using the QCD sum rules [16] and lattice QCD [17–19] approaches. In the calculations, the most general current ofQ general-izing the Ioffe current is used. The obtained form factors are used to compute the corresponding transition rates both their numerical values and in terms of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. Studying such type of transition provides a better understanding of the internal structure ofQ, information about the DAs and input parameters as well as determination of the CKM matrix elements. Note that, using different interpolating field, the semileptonic decay of bottom case, b! pl, has already been investigated in Refs. [20,21] in the same framework but HQET limit. In [20], the nucleon distribu-tion amplitudes are used only with QCD sum rules input parameters, while the distribution amplitudes ofb have been utilized to calculate the form factors in [21].
The layout of the paper is as follows: in Sec. II, the details of the calculation of the form factors in light cone QCD sum rules method are presented where the nucleon distribution amplitudes and the most general form of the interpolating currents for the Q baryon are used. The heavy quark limit of the form factors and the relations between the form factors in this limit is also discussed in this section. Section III comprises numerical analysis of the form factors and our predictions for the decay rate obtained in two different ways: first, using the DAs ob-tained from QCD sum rules and second, the DAs calcu-*kazizi@dogus.edu.tr
†melahat.bayar@kocaeli.edu.tr ‡ysoymak@atilim.edu.tr x
lated in lattice QCD. A comparison of our results on form factors and transition rates with the existing predictions of other approaches is also presented in this section.
II. THEORETICAL FRAMEWORK
In this section, following [15], we calculate the form factors of theb! p and c! n transitions in the frame-work of the light cone QCD sum rules and full theory. At quark level, these decays are governed by the tree-level Q ! q transition, where Q represents bðcÞ quark and q stands for uðdÞ quark for bðcÞ. The effective Hamiltonian responsible for these transitions at the quark level has the form
Heff ¼GFffiffiffi 2
p VqQqð1 5ÞQlð1 5Þ: (1) To calculate the amplitude, we need to sandwich the above equation between the initial and final states and compute the matrix element hNjqð1 5ÞQjQi, which is needed to study the Q! Nl decay. The starting point is to consider the following correlation function:
ðp; qÞ ¼ i Z
d4xeiqxhNðpÞ j TfJtr
ðxÞ JQð0Þg j 0i; (2)
where, JQ is interpolating currents of
bðcÞbaryon, Jtr ¼ qð1 5ÞQ is transition current and hNðpÞ j represents the nucleon state, where p denotes the proton (neutron) momentum and q ¼ ðp þ qÞ p is the transferred momentum.
One further step of the calculation is the saturation of the correlation function by a tower of hadronic states having the same quantum numbers as the interpolating currents. The obtained result from this procedure is called the phe-nomenological or physical side of the correlation function. From the general philosophy of the QCD sum rules ap-proach, this correlator is also calculated using the operator product expansion in deep Euclidean region. This part is called the theoretical or QCD side. Matching these two different representations of the same correlation function gives sum rules for form factors. To suppress the contribu-tion of the higher states and continuum, the Borel trans-formation is applied to both sides of the sum rules for physical quantities.
Let us first calculate the phenomenological part. After the insertion of the complete set of the initial hadronic state and performing the integral over x, we obtain the physical side as ðp; qÞ ¼ X s hNðpÞ j Jtr ðxÞ j Qðp þ q; sÞihQðp þ q; sÞ j JQð0Þ j 0i m2 Q ðp þ qÞ 2 þ . . . ; (3)
where, the ‘‘. . .’’ represents the contribution of the higher states and continuum. The matrix element hQðp þ q; sÞ j
JQð0Þ j 0i in (3) is given by
hQðp þ q; sÞ j JQð0Þ j 0i ¼ QuQðp þ q; sÞ; (4)
where Q is residue ofQbaryon. The transition matrix element, hNðpÞ j Jtrj Qðp þ q; sÞi can be written as hNðpÞ j Jtr ðxÞ j Qðp þ qÞi ¼ NðpÞ½f1ðQ2Þ þ iqf2ðQ2Þ þ qf3ðQ2Þ 5g1ðQ2Þ i5qg 2ðQ2Þ q 5g3ðQ2ÞuQðp þ qÞ; (5)
where Q2 ¼ q2. The fiand giare transition form factors in full theory and NðpÞ and uQðp þ qÞ are the spinors of nucleon and Q, respectively. Using Eqs. (3)–(5) and summing over spins of theQ baryon, i.e.,
X s
uQðp þ q; sÞuQðp þ q; sÞ ¼ p6 þ q6 þ mQ; (6) we attain the following expression:
ðp; qÞ ¼ Q m2 Q ðp þ qÞ 2 NðpÞ½f1ðQ2Þ þ iqf 2ðQ2þ qf3ðQ2Þ 5g1ðQ2Þ i5qg2ðQ2Þ q5g3ðQ2Þ ðp6 þ q6 þ mQÞ þ : (7) Using Nqu Q¼ i N½ðmNþ mQÞ ð2p þ qÞuQ; (8) in Eq. (7), the following final expression for the physical side of the correlation function is obtained:
ðp; qÞ ¼ Q m2 Q ðp þ qÞ 2 NðpÞ½2f1ðQ2Þpþ ff1ðQ2ÞðmN mQÞ þ f2ðQ 2Þðm2 N m2QÞg þ ff1ðQ2Þ f 2ðQ2ÞðmNþ mQÞg6 þ 2fq 2ðQ 2Þp q6 þ ff2ðQ2Þ þ f3ðQ2ÞgðmNþ mQÞq þ ff2ðQ2Þ þ f 3ðQ2Þgq6 þ 2gq 1ðQ2Þp5 fg1ðQ2ÞðmNþ mQÞ g2ðQ 2Þðm2 N m2QÞg5 þ fg1ðQ2Þ g 2ðQ2ÞðmN mQÞgq6 5þ 2g2ðQ 2Þp 6 q 5þ fg2ðQ2Þ þ g3ðQ2ÞgðmN mQÞq5 þ fg2ðQ2Þ þ g 3ðQ2Þgqq6 5 þ : (9)
In order to calculate the form factors f1, f2, f3, g1, g2, and g3, we will choose the independent structures p, pq6 , qq6 , p5, p6 q 5, and qq6 5from Eq. (7), respectively. For the theoretical side, to evaluate the correlation func-tion in deep Euclidean region where ðp þ qÞ2 0, the explicit expression of the interpolating field of the Q baryon is needed. Considering the quantum numbers, the most general form of interpolating current which can cre-ate theQ from the vacuum is given as
JQðxÞ ¼ 1ffiffiffi 6 p abcf2ðqaT 1 Cqb2Þ5Qcþ ðqaT1 C5qb2ÞQc þ ðqaT 1 CQbÞ5qc2þ ðqaT1 C5QbÞqc2 þ ðQaTCqb 2Þ5qc1þ ðQaTC5qb2Þqc1g; (10) where q1and q2are the u and d quarks, respectively, a, b, c are the color indices, and C is the charge conjugation operator and is an arbitrary parameter with ¼ 1 corresponding to the Ioffe current. Using the transition current, Jtr ¼ qð1 5ÞQ and JQ and contracting out all quark pairs by the help of the Wick’s theorem, we achieve ¼ i ffiffiffi 6 p abcZ d4xeiqxf½2ðCÞ ð5Þþ ðCÞðIÞ ð5Þ þ ðCÞ ð5Þ þ ½2ðC5Þ ðIÞ þ ðC5ÞðIÞ þ ðC5Þ ðIÞg½ð1 þ 5Þ SQðxÞhNðpÞjua ð0Þ ub ðxÞ dc ð0Þj0i; (11)
where, SQðxÞ is the heavy quark propagator which is given by [22] SQðxÞ ¼ SfreeQ ðxÞ igsZ d 4k ð2 Þ4eikx Z1 0 dv 6 þ mk Q ðm2 Q k2Þ2 GðvxÞ þ 1 m2Q k2vxG ; (12) where SfreeQ ¼ m 2 Q 4 2 K1ðmQpffiffiffiffiffiffiffiffiffix2Þ ffiffiffiffiffiffiffiffiffi x2 p i m2Q6x 4 2x2K2ðmQ ffiffiffiffiffiffiffiffiffi x2 p Þ; (13)
and Kiare the Bessel functions. Here, we neglect the terms proportional to the gluon field strength tensor since they can give contribution to four and five particle distribution functions and are expected to be small [23–25].
The matrix element hNðpÞ j abcua
ð0Þ ub ðxÞ dc ð0Þ j 0i appearing in Eq. (11), which is the nucleon wave function, is represented as [16,23–26]
4h0jabcua
ða1xÞubða2xÞdcða3xÞjNðpÞi ¼S1mNCð5NÞþ S2m2NCðx65NÞþP1mNð5CÞN þP2m2Nð5CÞðx6NÞþ V1þx 2m2 N 4 VM1 ðp6 CÞð5NÞ þV2mNðp6 CÞðx65NÞþV3mNðCÞð5NÞþV4m2Nðx6CÞð5NÞ þV5m2NðCÞðix5NÞþV6m3Nðx6CÞðx65NÞþ A1þx 2m2 N 4 AM1 ðp6 5CÞNþA2mNðp6 5CÞðx6NÞþA3mNð5CÞðNÞ
þA4m2Nðx65CÞNþA5m2Nð5CÞðixNÞþA6m3Nðx65CÞðx6NÞ þT1þx 2m2 N 4 TM1 ðpi CÞð5NÞþT2mNðxpiCÞð5NÞ þT3mNðCÞð5NÞþ T4mNðpCÞðx5NÞ þT5m2NðxiCÞð5NÞþT6m2NðxpiCÞðx65NÞ þT7m2NðCÞðx6 5NÞþT8m3NðxCÞðx5NÞ; (14) where, the calligraphic objects which have no definite twists are functions of the scalar product px and the parameters ai, i ¼1, 2, 3 and they are presented in terms of the nucleon distribution amplitudes (DAs) with definite and increasing twists. The scalar, pseudoscalar, vector, axial vector, and tensor DAs are explicitly shown in Tables I, II, III, IV, and V, respectively.
The distribution amplitudes FðaipxÞ ¼ Si, Pi, Vi, Ai, Ti can be written as FðaipxÞ ¼ Z dx1dx2dx3 ðx1þ x2þ x3 1Þ eipxixiaiFðx iÞ: (15)
where, xiwith i ¼1, 2, 3 corresponds to the longitudinal momentum fractions carried by the quarks.
In order to obtain the QCD or theoretical representation of the correlation function, the heavy quark propagator and nucleon distribution amplitudes are used in Eq. (11). Performing integral over x, equating the corresponding structures from both representations of the correlation function through the dispersion relations and applying Borel transformation with respect to ðp þ qÞ2 to suppress the contribution of the higher states and continuum, one can obtain sum rules for the form factors f1, f2, f3, g1, g2, and g3.
By means of the HQET, the number of independent form factors is reduced to two; F1 and F2. Hence, the transition matrix element can be parametrized in terms of these two form factors as [27,28]
hNðpÞ j ub j Qðp þ qÞi
¼ NðpÞ½F1ðQ2Þ þ v6 F2ðQ2ÞuQðp þ qÞ; (16) where, is any Dirac matrices and v6 ¼p6 þq6m
Q. One can
immediately obtain the following relations among the form factors in HQET limit comparing the Eq. (16) with the general definition of the form factors in Eq. (5) (see also [29,30]) g1¼ f1 ¼ F1þ mN mbF2 g2¼ f2 ¼ g3¼ f3 ¼ F2 mb: (17) Considering the above relations, one can obtain all form factors in terms of two form factors f1and f2. The explicit expressions for the form factors f1and f2 can be found in [31]. However, we will present the extrapolation of all form factors both in finite mass and HQET limit in terms of q2in the numerical analysis section.
In the following, some remarks about how the HQET limit of the form factors satisfy the above relations are in order. In HQET, all the ratios, f1
g1, f2 g2, f3 g3, f2 g3, f3 g2, f2 f3, and g2 g3
should be equal to one. The deviation of those ratios from unity are presented in TablesVIandVIIforb! p‘ and c! n‘, respectively. The bottom case and lattice QCD input parameters satisfy the HQET relations exactly, while the maximum violation of this symmetry is related to the
TABLE III. Relations between the calligraphic functions and nucleon vector DAs.
V1¼ V1 2pxV2¼ V1 V2 V3 2V3¼ V3 4pxV4¼ 2V1þ V3þ V4þ 2V5 4pxV5¼ V4 V3 4ðpxÞ2V 6¼ V1þ V2þ V3þ V4þ V5 V6
TABLE II. Relations between the calligraphic functions and nucleon pseudoscalar DAs.
P1¼ P1
2pxP2¼ P1 P2
TABLE IV. Relations between the calligraphic functions and nucleon axial vector DAs.
A1¼ A1 2pxA2¼ A1þ A2 A3 2A3¼ A3 4pxA4¼ 2A1 A3 A4þ 2A5 4pxA5¼ A3 A4 4ðpxÞ2A 6¼ A1 A2þ A3þ A4 A5þ A6
TABLE I. Relations between the calligraphic functions and nucleon scalar DAs.
S1¼ S1
2pxS2¼ S1 S2
TABLE V. Relations between the calligraphic functions and nucleon tensor DAs.
T1¼ T1 2pxT2¼ T1þ T2 2T3 2T3¼ T7 2pxT4¼ T1 T2 2T7 2pxT5¼ T1þ T5þ 2T8 4ðpxÞ2T 6¼ 2T2 2T3 2T4þ 2T5þ 2T7þ 2T8 4pxT7¼ T7 T8 4ðpxÞ2T 8¼ T1þ T2þ T5 T6þ 2T7þ 2T8
TABLE VI. Deviation of the ratio of the form factors from unity (violation of HQET symmetry relations) forb! p‘.
HQET QCD Sum Rules Input Parameters Lattice QCD Input Parameters f1 g1 0 0 f2 g2 20% 0 f3 g3 20% 0 f2 g3 20% 0 f3 g2 20% 0 f2 g3 0 0 g2 g3 0 0
charm case and QCD input parameters. When we consider all relations, we see that the violations for the charm case is larger than that of the bottom one.
The explicit expressions of the sum rules for form factors reveal that to get the numerical values of the form factors, the expression for residue Q is needed. This residue has been calculated in [32] using a two-point QCD sum rules method
2 Qe m2 Q=M2B ¼Zs0 m2Q eððsÞ=ðM2BÞÞðsÞds þ eððm2 QÞ=ðM2BÞÞ; (18) with ðsÞ ¼ ðh ddi þ huuiÞð 1Þ 192 2 m20 4mQ ½6ð1 þ Þc00 ð7 þ 11Þc02 6ð1 þ Þc11 þ ð1 þ 5ÞmQð2c10c11c12þ 2c21Þ þ m4Q 2048 4½5 þ ð2 þ 5Þ 12c10 6c20 þ 2c30 4c41þc42 12 ln s m2Q ; (19) ¼ð 1Þ 72 h ddihuui m2 Qm20 2M4 B ð13 þ 11Þ þ m20 4M2 B ð25 þ 23Þ ð13 þ 11Þ; (20) where, s0is the continuum threshold, M2Bis the Borel mass parameter, and cnm¼ ðsm2 QÞ n smðm2 QÞ
nm are some dimensionless
functions.
III. NUMERICAL RESULTS
The numerical analysis of the form factors and total decay rate for bðcÞ! pðnÞ‘ transition are presented in this section. Some input parameters used in the analysis of the sum rules for the form factors are huuið1 GeVÞ ¼ h ddið1 GeVÞ ¼ ð0:243Þ3 GeV3, m
n¼ 0:939 GeV, mp¼ 0:938 GeV, mb¼ 4:7 GeV, mc ¼ 1:23 GeV, mb ¼ 5:620 GeV, mc ¼ 2:286 GeV, and m20ð1 GeVÞ ¼ ð0:8 0:2Þ GeV2 [33]. The main inputs which are the nucleon DAs can be found in [16]. These DAs contain eight independent parameters fN, 1, 2, Vd
1, Au1, f1d, fu1, and f2d. These parameters have been calculated in the light cone QCD sum rules [16] and also most of these parame-ters have been computed in the framework of the lattice QCD [17–19]. For those parameters which have not calcu-lated in lattice, the data from QCD input parameters will be used. These parameters are given in TableVIII.
Three auxiliary parameters are encountered to the ex-pression of the sum rules for form factors, continuum threshold s0, Borel mass parameter M2B, and general pa-rameter entering to the most general form of the inter-polating current forQ baryon. A working region should be determined for these auxiliary and mathematical pa-rameters such that the form factors as physical quantities should be independent of them. The continuum threshold, s0is not completely arbitrary and it is related to the energy of the exited states. From our results, we observed that the form factors are weakly dependent on s0 in the interval, ðmQþ 0:5Þ2 s
0 ðmQþ 0:7Þ
2. To determine the working region for , we look at the variation of the
TABLE VII. Deviation of the ratio of the form factors from unity (violation of HQET symmetry relations) forc! n‘.
HQET QCD Sum Rules Input Parameters Lattice QCD Input Parameters f1 g1 0 0 f2 g2 45% 40% f3 g3 45% 40% f2 g3 45% 35% f3 g2 40% 35% f2 g3 0 0 g2 g3 0 0
TABLE VIII. The values of independent parameters entering to the nucleon DAs. The first errors in lattice values are statistical and the second errors represent the uncertainty due to the Chiral extrapolation and renormalization.
QCD Sum Rules [16] Lattice QCD [17–19]
fN ð5:0 0:5Þ 103GeV2 ð3:234 0:063 0:086Þ 103 GeV2 1 ð2:7 0:9Þ 102 GeV2 ð3:557 0:065 0:136Þ 102 GeV2 2 ð5:4 1:9Þ 102GeV2 ð7:002 0:128 0:268Þ 102 GeV2 Vd 1 0:23 0:03 0:3015 0:0032 0:0106 Au 1 0:38 0:15 0:1013 0:0081 0:0298 fd 1 0:40 0:05 -fu 1 0:07 0:05 -fd 2 0:22 0:05
form factors with respect to cos in the interval 1 cos 1 which corresponds to 1 1, where ¼tan . As a result, we attain a region at which the dependency is weak. The working region for is obtained to be 0:75 cos 0:25 for b and 0:25 cos 0:25 for c. The Ioffe current which corresponds to cos ¼ 0:71 is inside the working region for b but out of the region forc.
For further analysis, the upper and lower limits of M2B should be determined. To do that, we apply two conditions: The first one, which gives the upper limit, is that the series of the light cone expansion with increasing twist should be convergent, and the second one, which determines the lower limit, is that the contribution of higher states and continuum to the correlation function should be small enough; i.e., the contribution of the highest term with power 1=M2B is less than, say, 20%–25% of the highest power of M2B. In the present work, both conditions are satisfied in the region15 GeV2 MB2 30 GeV2 forb and4 GeV2 M2B 12 GeV2 forc, which we will use in numerical analysis. Taking into account the above
re-quirements, we obtained that the form factors obey the following extrapolations in terms of q2:
fiðq2Þ½giðq2Þ ¼ a ð1 q2 m2fitÞ þ b ð1 q2 m2fitÞ2 : (21)
The values of the parameters a, b, and mfitfor form factors and their HQET limit are given in TablesIX,X,XI, andXII
related to the QCD sum rules and lattice QCD input parameters. Because of the working near the light cone, x2 ’ 0 and concerning the considered correlation function, the results are not reliable at low q2; hence, to make the extension of our predictions to the full physical region, we need the above parametrization. From those Tables, we see that the pole of the form factors exist outside the physical region and the form factors are analytic in the whole physical interval. The values of form factors at q2 ¼ 0 obtained from fit functions are shown in Tables XIIIand
XIVfor b! p‘ and c ! n‘, respectively. A com-parison of the existing predictions from other approaches is also presented for bottom case. The Table XIII depicts a
TABLE XI. Parameters appearing in the fit function of the form factors at HQET limit for b ! p‘.
QCD Sum Rrules Lattice QCD
a b mfit a b mfit f1 0.041 0.040 4.82 0.0042 0.016 4.92 f2 0.033 0:097 4.83 0.013 0:030 5.92 f3 0.060 0:14 4.90 0.016 0:040 4.94 g1 0:0012 0.096 5.10 0:0022 0.029 5.30 g2 0:0094 0:018 5.36 0.0017 0:0043 5.36 g3 0:040 0.025 4.95 0:018 0.015 4.98
TABLE IX. Parameters appearing in fit function of the original form factors forb! p‘.
QCD Sum Rules Lattice QCD
a b mfit a b mfit f1 0.025 0.052 4.91 0.048 0.016 4.89 f2 0.007 0:050 4.92 0:003 0:006 4.92 f3 0.052 0:13 4.99 0.028 0:063 4.96 g1 0:059 0.13 5.29 0:17 0.32 5.32 g2 0.011 0:050 5.20 0.019 0:040 5.40 g3 0:009 0:017 4.90 0:015 0.012 4.98
TABLE X. Parameters appearing in fit function of the original form factors forc! n‘.
QCD Sum Rules Lattice QCD
a b mfit a b mfit f1 0:034 0.20 1.59 0:14 0.64 1.55 f2 0:015 0:77 1.57 0.018 0:32 1.60 f3 0:062 1:23 1.48 0.12 1:09 1.56 g1 0:015 0.54 1.53 0:20 0.71 1.59 g2 0:11 0:20 1.52 0:034 0:14 1.65 g3 0:088 0.085 1.48 0.009 0:41 1.50
good consistency on our result for f1ð0Þ HQET limit obtained from lattice QCD input parameters with the pre-diction of [21]; however, the f1ð0Þ HQET limit obtained from QCD sum rules parameters is almost 4 times larger than that of the [21] prediction. On the other hand, the similar comparison of our result on form factor f2ð0Þ at HQET and the prediction of [21] shows that the value presented in [21] is almost 2 times greater than our result obtained from lattice QCD input parameters and 1.5 times smaller than our result obtained from QCD input parameters.
In the next step, we calculate the total decay rate of Q! N‘ transition in the whole physical region, i.e., m2l q2 ðmQ mNÞ2. The decay width for such tran-sition is given by the following expression [34,35]: ðQ! NllÞ ¼ G2F 384 3m3 Q jVqQj2Z2 m2l dq2ð1 m2l=q2Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2 q2Þð2 q2ÞNðq2Þ (22) where Nðq2Þ ¼ F21ðq2Þð2ð4q2 m2lÞ þ 222ð1 þ 2m2l=q2Þ ð2þ 2q2Þð2q2þ m2lÞÞ þ F22ðq2Þð2 q2Þð22þ q2Þ ð2q2þ m2 lÞ=m2b þ 3F 2 3ðq2Þm2lð2 q2Þq2=m2bþ 6F1ðq 2ÞF 2ðq2Þð2 q2Þð2q2þ m2lÞ=mb 6F1ðq2ÞF3ðq2Þm2lð2 q2Þ=mb þ G 2 1ðq2Þð2ð4q2 m2lÞ þ 222ð1 þ 2m2l=q2Þ ð2þ 2q2Þð2q2þ m2 lÞÞ þ G22ðq2Þð2 q2Þð22þ q2Þð2q2þ m2lÞ=m2bþ 3G 2 3ðq2Þm2lð2 q2Þq2=m2b 6G1ðq2ÞG2ðq2Þð2 q2Þð2q2þ m2lÞ=mbþ 6G1ðq 2ÞG 3ðq2Þm2lð2 q2Þ=mb: (23)
TABLE XII. Parameters appearing in the fit function of the form factors at HQET limit for c! n‘.
QCD Sum Rules Lattice QCD
a b mfit a b mfit f1 0:066 1.14 1.51 0:039 0.37 1.55 f2 0.046 1:14 1.53 0.047 0:63 1.48 f3 0.071 1:33 1.50 0.039 0:52 1.53 g1 0:10 1.21 1.57 0:039 0.39 1.55 g2 0:070 0:11 1.56 0:027 0:046 1.60 g3 0:076 0:91 1.54 0:034 0:032 1.54
TABLE XIII. The values of the form factors at q2¼ 0 for b! p‘.
Original HQET
QCD Sum Rules Lattice QCD QCD Sum Rules Lattice QCD [21]
f1ð0Þ 0.077 0.064 0.081 0.021 0:023þ0:0060:005 f2ð0Þ 0:044 0:013 0:064 0:018 0:039þ0:0060:009 f3ð0Þ 0:079 0:036 - -g1ð0Þ 0.073 0.15 - -g2ð0Þ 0:039 0:021 - -g3ð0Þ 0:026 0:0035 -
-TABLE XIV. The values of the form factors at q2¼ 0 for c! n‘.
Original HQET
QCD Sum Rules Lattice QCD QCD Sum Rules Lattice QCD
f1ð0Þ 0.17 0.50 1.078 0.33 f2ð0Þ 0:78 0:31 1:09 0:58 f3ð0Þ 1.29 0.98 - -g1ð0Þ 0.52 0.51 - -g2ð0Þ 0:31 0:18 - -g3ð0Þ 0:0032 0:31 -
Here, F1ðq2Þ ¼ f1ðq2Þ, F2ðq2Þ ¼ mQf2ðq2Þ, F3ðq2Þ ¼ mQf3ðq2Þ, G1ðq2Þ ¼ g1ðq2Þ, G2ðq2Þ ¼ mQg2ðq2Þ, G3ðq2Þ ¼ mQg3ðq2Þ, ¼ mQþ mN, and ¼ mQ mN. GF¼ 1:17 105GeV2 is the Fermi cou-pling constant, and mlis the leptonic (electron, muon, or tau) mass. For the corresponding CKM matrix element Vub¼ ð4:31 0:30Þ103 and Vcd¼ ð0:230 0:011Þ are used [36].
Our final results for total decay rates are given in Table XV. As it can be seen from this Table, our results for e and and b cases are consistent for two sets of input parameters when the original form factors are used, especially when we consider the uncertainties. However, the QCD input parameters result is 1.5 times greater than that of the lattice input parameter for the decay rates of and bottom case. If we considerc, QCD sum rules input parameters gives the result 2 times greater than the lattice QCD input parameters. On the other hand, when we con-sider the uncertainties, results obtained using both sets of input parameters and original form factors coincide for all leptons. At HQET limit and QCD sum rules input parame-ters, our predictions for the decay rates are in the same order of magnitude with the original form factors and two sets for all leptons and both charm and bottom cases. In contrast, the results at HQET limit and lattice parameters are 2 orders of magnitude less than HQET limit and sum rules inputs as well as original form factors for bottom and
e and cases. For and bottom, and e and and charm cases, this difference is approximately 1 order of magni-tude. We also compare our results on decay rates in units of jVqQj2s1 with the predictions of Refs. [9–12,20,21] in TableXVI. From this Table, it is clear that our results for bottom case, lattice parameters and HQET limit are in the same order of magnitude with the predictions of [11,12] and HQET [20]. For all other cases the difference between our results with the existing predictions of the other ap-proaches presented in Table XVIis 1–2 order of magni-tudes. In Table XVI, HOSR refers to harmonic oscillator semi relativistic and HONR stands for harmonic oscillator nonrelativistic constituent quark models.
To summarize, using the most general form of the interpolating currents of Q and nucleon DAs with two sets of input parameters, namely, QCD sum rules and lattice QCD inputs, the transition form factors of the semi-leptonicQ! Nl have been calculated in the framework of the light cone QCD sum rules in full theory and HQET. The lattice input parameters satisfy the HQET relations exactly for the bottom case, while the maximum violation is for the charm case and QCD input parameters. The results of the form factors at HQET and q2¼ 0 have been compared with the existing predictions of the other approaches. These transition form factors have been used to estimate the corresponding tree-level semileptonic de-cay rates both in full theory and HQET limit. A comparison
TABLE XV. Values of theðQ! N‘Þ in GeV for different leptons and two sets of input parameters obtained from QCD sum rules and lattice QCD and also their HQET limit.
b! p b! pee b! p c! n c! nee c! n
For QCD Sum Rules Inputs ð3:07 1:05Þ 1015ð3:065 1:05Þ 1015ð3:82 1:35Þ 1015ð2:89 0:95Þ 1013ð2:86 0:95Þ 1013 -For Lattice QCD Inputs ð2:87 0:95Þ 1015 ð2:87 0:95Þ 1015 ð2:55 0:85Þ 1015ð1:35 0:45Þ 1013ð1:33 0:43Þ 1013 -HQET Limit for QCD
Sum Rules Inputs
ð5:84 1:81Þ 1015 ð5:83 1:81Þ 1015 ð7:90 2:45Þ 1015ð5:08 1:65Þ 1013ð5:01 1:60Þ 1013
-HQET Limit for Lattice QCD Inputs
ð4:70 1:60Þ 1017 ð4:60 1:55Þ 1017 ð2:36 0:85Þ 1016ð8:75 2:85Þ 1014ð8:74 2:83Þ 1014
-TABLE XVI. Values of the total decay rate (in jVqQj2 s1) of theQ! N‘ transition for different leptons and two sets of input parameters obtained from QCD sum rules and lattice QCD and also their HQET limit compared to the [9–12,20,21].
b! p b! pee b! p c! n c! nee c! n
For QCD Sum Rules ð2:5 0:85Þ 1014 ð2:5 0:85Þ 1014 ð3:12 1:05Þ 1014 ð8:3 2:85Þ 1012 ð8:21 2:80Þ 1012 For Lattice QCD ð2:35 0:85Þ 1014 ð2:35 0:85Þ 1014 ð2:08 0:70Þ 1014 ð3:88 1:25Þ 1012 ð3:82 1:20Þ 1012 HQET Limit for
QCD Rum Rules
ð4:78 1:75Þ 1014 ð4:77 1:75Þ 1014 ð6:46 2:15Þ 1014 ð1:46 0:55Þ 1013 ð1:44 0:55Þ 1013
HQET Limit for Lattice QCD ð3:84 1:25Þ 1012 ð3:76 1:20Þ 1012 ð1:93 0:70Þ 1012 ð2:51 0:85Þ 1012 ð2:51 0:85Þ 1012 [10] 2:05 1013 [9] 2:58 1013 [11] 6:48 1012 QCD Sum Rules [20] 3:65 1013 HQET [20] 5:62 1012
[12] 4:55 1012ðHONRÞ 4:01 1012ðHONRÞ 1:02 1010ðHONRÞ 7:55 1012ðHOSRÞ 6:55 1012ðHOSRÞ 1:35 1010ðHOSRÞ
of the obtained results and the existing predictions of the other approaches which all are at HQET limit, was also presented. The best consistency between our results and those predictions is related to the bottom case and lattice QCD input parameters at HQET. Our results can be checked in experiments held in the future such as the
LHC. Comparisons between the experimental data and our results could give essential information about the nature of the Q, nucleon distribution amplitudes, as well as determination of the CKM matrix elements, Vub and Vcd.
[1] D. Acosta et al. (CDF Collaboration), Phys. Rev. Lett. 96, 202001 (2006).
[2] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 97, 232001 (2006); 99, 062001 (2007); Phys. Rev. D 77, 012002 (2008).
[3] M. Mattson et al. (SELEX Collaboration), Phys. Rev. Lett. 89, 112001 (2002).
[4] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 99, 052002 (2007); 99, 202001 (2007).
[5] R. Chistov et al. (Belle Collaboration), Phys. Rev. Lett. 97, 162001 (2006).
[6] A. Ocherashvili et al. (SELEX Collaboration), Phys. Lett. B 628, 18 (2005).
[7] V. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 99, 052001 (2007); 101, 232002 (2008).
[8] E. Solovieva et al. (Belle Collaboration), Phys. Lett. B 672, 1 (2009).
[9] R. S. Marques de Carvalho, F. S. Navarral, M. Nielsen, E. Ferreira, and H. G. Dosch, Phys. Rev. D 60, 034009 (1999).
[10] C.-S. Huang, C.-F. Qiao, and H.-G. Yan, Phys. Lett. B 437, 403 (1998).
[11] A. Datta, arXiv:hep-ph/9504429.
[12] M. Pervin, W. Roberts, and S. Capstick, Phys. Rev. C 72, 035201 (2005); 74, 025205 (2006).
[13] D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Rev. D 73, 094002 (2006).
[14] C. Albertus, E. Hernandez, and J. Nieves, Phys. Rev. D 71, 014012 (2005).
[15] K. Azizi, M. Bayar, A. Ozpineci, and Y. Sarac, Phys. Rev. D 80, 036007 (2009).
[16] V. M. Braun, A. Lenz, and M. Wittmann, Phys. Rev. D 73, 094019 (2006).
[17] M. Gockeler et al. (QCDSF Collaboration), Proc. Sci., LAT2007 (2007) 147.
[18] M. Gockeler et al., Phys. Rev. Lett. 101, 112002 (2008).
[19] V. M. Braun et al. (QCDSF Collaboration), Phys. Rev. D 79, 034504 (2009).
[20] Ming-Qiu Huang and Dao-Wei Wang, Phys. Rev. D 69, 094003 (2004).
[21] Yu-Ming Wang, Yue-Long Shen, and Cai-Dian Lu, Phys. Rev. D 80, 074012 (2009).
[22] I. I. Balitsky and V. M. Braun, Nucl. Phys. B311, 541 (1989).
[23] V. M. Braun, A. Lenz, N. Mahnke, and E. Stein, Phys. Rev. D 65, 074011 (2002).
[24] V. M. Braun, A. Lenz, and M. Wittmann, Phys. Rev. D 73, 094019 (2006); A. Lenz, M. Wittmann, and E. Stein, Phys. Lett. B 581, 199 (2004).
[25] V. Braun, R. J. Fries, N. Mahnke, and E. Stein, Nucl. Phys. B589, 381 (2000).
[26] V. M. Braun, A. Lenz, G. Peters, and A. V. Radyushkin, Phys. Rev. D 73, 034020 (2006).
[27] T. Mannel, W. Roberts, and Z. Ryzak, Nucl. Phys. B355, 38 (1991).
[28] T. M. Aliev, A. Ozpineci, and M. Savci, Phys. Rev. D 65, 115002 (2002).
[29] C. H. Chen and C. Q. Geng, Phys. Rev. D 63, 054005 (2001); 63, 114024 (2001); 64, 074001 (2001).
[30] T. M. Aliev, A. Ozpineci, M. Savci, and C. Yuce, Phys. Lett. B 542, 229 (2002).
[31] K. Azizi, M. Bayar, Y. Sarac, and H. Sundu,
arXiv:0908.1758.
[32] K. Azizi, M. Bayar, and A. Ozpineci, Phys. Rev. D 79, 056002 (2009).
[33] V. M. Belyaev and B. L. Ioffe, JETP 56, 493 (1982). [34] A. Faessler, T. Gutsche, B. R. Holstein, M. A. Ivanov, J. G.
Korner, and V. E. Lyubovitskij, Phys. Rev. D 78, 094005 (2008).
[35] H. Pietschmann, Acta Phys. Austriaca Suppl. 12, 1 (1974). [36] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1
(2008).
![TABLE XVI. Values of the total decay rate (in jV qQ j 2 s 1 ) of the Q ! N‘ transition for different leptons and two sets of input parameters obtained from QCD sum rules and lattice QCD and also their HQET limit compared to the [ 9 – 12 , 20 , 21 ].](https://thumb-eu.123doks.com/thumbv2/9libnet/4076502.58205/8.918.71.840.831.1074/values-transition-different-leptons-parameters-obtained-lattice-compared.webp)
