Rare baryon decays
b! ‘
þ‘
(‘ ¼ e, , ) and
b! : Differential and total rates,
lepton- and hadron-side forward-backward asymmetries
Thomas Gutsche,1Mikhail A. Ivanov,2Ju¨rgen G. Ko¨rner,3Valery E. Lyubovitskij,1,*and Pietro Santorelli4,5
1Institut fu¨r Theoretische Physik, Universita¨t Tu¨bingen, Kepler Center for Astro and Particle Physics,
Auf der Morgenstelle 14, D-72076 Tu¨bingen, Germany
2Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia 3PRISMA Cluster of Excellence, Institut fu¨r Physik, Johannes Gutenberg-Universita¨t, D-55099 Mainz, Germany 4Dipartimento di Fisica, Universita` di Napoli Federico II, Complesso Universitario di Monte Sant’ Angelo, Via Cintia,
Edificio 6, 80126 Napoli, Italy
5Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, 80126 Napoli, Italy
(Received 17 January 2013; published 23 April 2013)
Using the covariant constituent quark model previously developed by us, we calculate the differential rate and the forward-backward asymmetries on the lepton and hadron side for the rare baryon decays b! ‘þ‘ (‘ ¼ e, , ) and b ! . We use helicity methods to write down a threefold joint
angular decay distribution for the cascade decay b! ð! pÞ þ Jeffð! ‘þ‘Þ. Through appropriate
angular integrations, we obtain expressions for the rates, the lepton-side forward-backward asymmetry, and the polarization of the daughter baryon leading to a hadron-side forward-backward asymmetry. We present numerical results on these observables using the covariant quark model and compare our results to the results of other calculations that have appeared in the literature.
DOI:10.1103/PhysRevD.87.074031 PACS numbers: 12.39.Ki, 13.20.He, 14.20.Jn, 14.20.Mr
I. INTRODUCTION
In a recent paper, the CDF Collaboration has reported on 24 events of the rare baryon decay b! þ þ[1].
The collaboration measures the total branching ratio and gives first results on the q2 spectrum. This experimental
result is quite remarkable since, given the measured branching ratio of 1:73 106, the CDF Collaboration must have had a data sample of at least 42 million b’s.
The physics of heavy baryon decays appears to have entered a new era with this experimental result. With the LHC running so well, many more b’s will be recorded by,
e.g., LHCb in the near future, which makes the study of rare b decays a worthwhile enterprise. The decay b !
‘þ‘can be considered to be a welcome complement to the well-analyzed rare meson decays B ! KðÞ‘þ‘, Bs! ‘þ‘, etc., to study the short- and long-distance dynam-ics of rare decays induced by the transition b ! s‘þ‘.
There has been a number of theoretical papers on the rare b! baryon decays involving the one-photon
mode b! and the dilepton modes b! ‘þ‘
(‘ ¼ e, , ). They use the same set of (penguin) operators or their non–Standard Model extensions to describe the short-distance dynamics but differ in their use of theoreti-cal models to theoreti-calculate the nonperturbative transition ma-trix element hjOijbi. Among the phenomenological
models used are the bag model [2], a pole model [3,4], the covariant oscillator quark model [5], nonrelativistic quark models [6,7], and a perturbative QCD approach
using light-cone distribution amplitudes calculated from QCD sum rules [8]. The authors of Refs. [3,4,9–13] have made use of the heavy quark mass limit to write their form factors in terms of two independent heavy quark effective theory (HQET) form factors [14–16] for which they pro-vide some estimates. Mannel and Recksiegel [3] and Colangelo et al. [4] use experimental input for this esti-mate, while Refs. [9–13] use QCD sum rules to determine the two HQET form factors. Recently, the brare decays
were studied in lattice QCD [17] and used an improved version of the QCD sum rules [18].
As has been emphasized by Refs. [9,19,20] heavy quark symmetry should not be relied on at small q2, i.e., close to maximal recoil, in particular for heavy-to-light transitions. Heavy quark symmetry is expected to be reliable close to zero recoil where not much momentum is transferred to the spectator system. As one moves away from the zero recoil point and more momentum gets transferred to the spectator system, hard gluon exchange including spin flip interac-tions become more important, and the heavy quark sym-metry can be expected to break down [19,20]. One should, therefore, not rely on a form factor parametrization in terms of only two heavy quark symmetry form factors for the whole q2 range as has been done in Refs. [3,9–13].
Furthermore, the results of QCD sum rules for heavy-to-light transitions have been shown to be unreliable close to maximum recoil [21]. One should rather rely on light-cone sum rules for the near maximum recoil region for the evaluation of the hadronic form factors. It is for these reasons that most recent calculations have employed light-cone sum rules to calculate the whole set of hadronic form factors [22–32]. The form factors are then extrapo-lated to the low recoil region by using some pole-type
*On leave of absence from Department of Physics, Tomsk State University, 634050 Tomsk, Russia.
parametrizations. The authors of Refs. [26–28] have de-rived some very interesting form factor relations in the large recoil region using the soft collinear effective theory. Apart from rates, q2-spectra, lepton-side
forward-backward asymmetries treated in most of the papers, some of the authors have also discussed polarization effects of the final state particles. For example, in Refs. [3,10,11,26] the polarization of the daughter baryon was calculated, which can be measured experimentally by analyzing the decay ! p. The polarization components of a single lepton have been considered in Refs. [11,12], while the authors of Ref. [13] have even studied double-lepton polarization asymmetries. Polarization effects of the decaying b have
been investigated in Ref. [29]. Non-Standard Model effects for various observables have been examined in Refs. [4,9,12,13,23,24,30–32].
There are large discrepancies in the predictions of the different models for the various observables, in particular, for the photonic decay b! and for the
near-zero-recoil -mode b! þ. It will be interesting to
compare the results of future experiments on rare baryon decays with the various model predictions, in particular, for the above two decay modes.
In the present paper, we use the covariant constituent quark model (for short, covariant quark model) as dynami-cal input to dynami-calculate the nonperturbative transition matrix elements. In the covariant quark model, the current-induced transitions between baryons are calculated from two-loop Feynman diagrams with free quark propagators in which the divergent high-energy behavior of the loop integrations is tempered by Gaussian vertex functions [33–43]. An attractive new feature has recently been added to the covariant quark model in as much as quark confine-ment has now been incorporated in an effective way, i.e., there are no quark thresholds and thus no free quarks in the relevant Feynman diagrams [44–46]. We emphasize that the covariant quark model described here is a truly frame-independent field-theoretic quark model in contrast to other constituent quark models that are basically quantum mechanical with built-in relativistic elements. One of the advantages of the covariant quark model is that it allows one to calculate the transition form factors in the full accessible range of q2 values.
We somehow deviate from the traditional order when presenting our results. We first discuss the model-independent aspects of the problem involving spin physics. This is done by the use of the helicity amplitude method, which leads to comprehensive, compact, and clearly or-ganized formulas for the joint angular decay distributions of the decay products and the spin density matrix elements
of the final state particles. Lepton mass effects are auto-matically included. At a later stage, we present the details of the dynamics for which we give numerical results toward the end of the paper. We believe that, in this paper and in a forthcoming paper, we provide the first complete and comprehensive discussion of all the spin physics as-pects of the decay b! ‘þ‘.
We mention that our helicity formulas are ideally suited as input in an event generator. This has been previously demonstrated for the charged current decay 0! þ‘‘ (‘ ¼ e, ) followed by the decay þ ! p0
including even polarization effects for the 0 [47]. We
have written a Monte Carlo (MC) event generator for the above process based on decay distribution formulas de-rived from a corresponding helicity analysis. The event generator is based on the genbod routine from the CERNLIB library and has been used by the NA48 experi-ment to analyze its data on the above process.
Our paper is structured as follows. In Sec.II, we present a detailed discussion of the helicity formalism that allows one to write down the joint angular distribution of the cascade decay b! ð! pÞ þ jeffð! ‘þ‘Þ. In Sec. III, we
review the basic notions of our dynamical approach—the covariant quark model for baryons. In particular, we i) derive the phenomenological Lagrangians describing the interaction of baryons with their constituent quarks, ii) introduce the corresponding interpolating 3-quark cur-rents with the quantum numbers of the respective baryon, iii) discuss the idea and implementation of quark confine-ment, and iv) present the calculational loop integration techniques. In Sec. IV, we consider the application of our approach to the rare one-photon decay b! and the
dilepton decay b! ð! pÞ þ jeffð! ‘þ‘Þ. Finally,
in Sec.V, we summarize our results. Some technical mate-rial has been relegated to AppendicesA,B,C,D, andE.
II. JOINT ANGULAR DECAY DISTRIBUTION As in the case of the rare meson decays B ! KðÞ‘þ‘ (‘ ¼ e, , ) treated in Ref. [48], one can exploit the cascade nature of the decay b!ð!pÞþjeffð!‘þ‘Þ to
write down a joint angular decay distribution involving the polar angles , Band the azimuthal angles defined by the
decay products in their respective center-of-mass (CM) systems as shown in Fig.1. The angular decay distribution involves the helicity amplitudes hm
1 2 for the decay jeff !
‘þ‘, Hm
jfor the decay b ! þ jeff, and h
B
p0for the
decay ! p þ . The joint angular decay distribution for the decay of an unpolarized breads
Wð; B; Þ / X 1; 2; j; 0j;J;J 0;m;m0; ; 0; p hm 1 2ðJÞh m0 1 2ðJ 0Þeið j 0jÞ j ; 0j 0 JJ 0dJ j; 1 2ðÞd J0 0j; 1 2ðÞH m jðJÞ Hm0y 0 0 jðJ 0Þd1=2 pðBÞd 1=2 0 pðBÞh B p0h By p0: (1)
The Kronecker delta in j ; 0j 0 in Eq. (1) expresses
the fact that we are considering the decay of an unpo-larized b. In Eq. (1) one has to observe that j j
j ¼ j 0j 0j ¼ 1=2 due to the spin 1=2 nature of the decaying b. The corresponding Kronecker delta
has not been included in Eq. (1) and will also be omitted in the subsequent formulas. The djmm0 in Eq. (1) with
(j ¼ 0, 1=2, 1) are Wigner’s d functions where d000¼ 1.
In AppendixAwe provide an explicit expression for the threefold angular decay distribution by expanding out the rhs of Eq. (1). We mention that it is not difficult to generalize Eq. (1) to the case of a decaying polarized b
as has been done in Ref. [29] by transcribing the results of the corresponding semileptonic charged current de-cays [47,49].
Let us discuss the helicity amplitudes appearing in Eq. (1) in turn. The lepton-side helicity amplitudes hm
1 2
for the process jeff! ‘þ‘ are defined by ( ^ j¼
1 2), m ¼ 1ðVÞ h1 ^j; 1 2ðJÞ ¼ u2 ð 2Þv1ð 1Þ ð ^ jÞ; m ¼ 2ðAÞ h2 ^ j; 1 2ðJÞ ¼ u2 ð 2Þ5v1ð 1Þ ð ^ jÞ: (2)
We have put a hat on the helicity label ^ j to emphasize that ^ j is not the jappearing in Eq. (1). We have also
included the label ( ^ j¼ 1 2) in Eq. (2) for clarity
even if the notation is redundant. We evaluate the helicity amplitudes in the (‘þ‘) CM system with ‘þdefining the z direction. The label (J) takes the values (J ¼ 0) with j¼ 0 and (J ¼ 1) with j¼ 1, 0 for the scalar and
vector parts of the effective current jeff, respectively. In order to distinguish between the two j¼ 0 cases, we
write j¼ t for the (J ¼ 0) scalar case and j¼ 0 for the (J ¼ 1) vector case. The lepton-side helicity amplitudes can be calculated as h1 t;1 212ðJ ¼ 0Þ ¼ 0; h 1 0;1 212ðJ ¼ 1Þ ¼ 2m‘ ; h1 þ1;1 212ðJ ¼ 1Þ ¼ ffiffiffiffiffiffiffiffi 2q2 q ; h2 t;1 212ðJ ¼ 0Þ ¼ 2m‘ ; h2 0;1 212ðJ ¼ 1Þ ¼ 0; h 2 þ1;1 212ðJ ¼ 1Þ ¼ ffiffiffiffiffiffiffiffi 2q2 q v; (3) where v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4m2‘=q2 q
is the lepton velocity in the (‘þ‘) CM frame and m‘ is the leptonic mass.
From parity one has h1 j; 1 2 ¼ h 1 j; 1 2; h 2 j; 1 2 ¼ h 2 j; 1 2: (4)
In Eq. (1) we are summing over the lepton helicities. A closer look at the relations (3) and (4) shows that there are no (J ¼ 0; J ¼ 1) interference effects in the joint angular decay distribution (1). This has been annotated by the Kronecker delta JJ0 in Eq. (1). (J ¼ 0; J ¼ 1)
interfer-ence effects come into play when one leaves the lepton helicities unsummed, i.e., when one considers lepton polarization effects. In this case one has to replace JJ0
by ð1ÞJþJ0 in Eq. (1) where the extra minus sign for the (J ¼ 0; J ¼ 1) interference contribution results from the Minkowskian form of the metric tensor [47,49,50]. We mention that (J ¼ 0; J ¼ 1) interference effects occur in charged current transitions already in the unpolarized lep-ton case [47,49,50].
From Eq. (3) it is clear that the scalar contributions enter the game only for nonzero lepton masses m‘ 0.
The scalar contributions will thus only be important for the mode.
The hadronic helicity amplitudes Hm
jðJÞ in Eq. (1)
describe the full dynamics of the current-induced transi-tions b! þ jeffincluding the structure and the values
of the short-distance coefficients of the pertinent penguin operators. The helicity labels on the helicity amplitudes Hm
jðJÞ take the values ¼
1
2, j¼ t for (J ¼ 0)
and j¼ 1, 0 for (J ¼ 1) as described above. The
superscript m on Hm
jðJÞ defines whether the hadronic
helicity amplitude multiplies the lepton vector current uð‘Þvð‘þÞ (m ¼ 1) or the axial vector current
uð‘Þ5vð‘þÞ (m ¼ 2). More details on the definitions
of the hadronic helicity amplitudes and their calculation in the covariant quark model can be found in Sec.III.
If desired, one can switch from the helicity amplitudes used here to the transversity amplitudes used, e.g., in Ref. [51] by use of the relations
Am ?;k¼ ðH m þ1H m 1Þ= ffiffiffi 2 p ; Am 0¼ H m 0; Am t¼ H mt: (5) The advantage of the transversity amplitudes is that they have definite transformation properties under parity.
The helicity amplitudes hB p ¼0, finally, describe the decay ! p þ . We shall use experimental input for
FIG. 1 (color online). Definition of angles , B, and in the
the relevant bilinear forms of the helicity amplitudes. The helicity labels on the helicity amplitudes hB
p ¼0are
self-explanatory.
As mentioned in the introduction, the joint angular decay distribution Eq. (1) is ideally suited as input to an event generator using sequential boosts to the respective rest systems of the secondary particles as exemplified by the cascade-type decay distribution (1).
We mention that the decay distribution Eq. (1) reprodu-ces the results in Ref. [48] for B ! KðÞð! KÞ‘þ‘ when one replaces the helicity amplitudes for ! p þ by the corresponding helicity amplitude for K! K. The necessary replacement is
Hm jH m0y 0 0jd 1=2 pðBÞd 1=2 0 pðBÞh B p0h By p0 ! Hm K jH m0y 0K 0jd10 jð Þd1 0 0j ðÞhK 00h Ky 00 ; (6)
where, in the mesonic case, K ¼ j and 0
K¼ 0 j
since the decaying B has spin zero. In Eq. (6) we have omitted the label (J) on the hadronic helicity amplitudes for brevity. Note that there is only one helicity amplitude for K ! K compared to the two helicity amplitudes for ! p þ . A new feature of the baryonic case is that the decay ! p þ is parity nonconserving, which leads to a forward-backward asymmetry on the hadron side as is the case on the lepton side.
In this paper we will not further consider the azimuthal dependence of the joint angular decay distribution Eq. (1), but we shall rather integrate Eq. (1) over the azimuthal angle . This leads to j¼ 0j via
R2
0 d exp ½ið j
0jÞ ¼ 2 j 0
j. Since we are considering the decay of
an unpolarized b such that b ¼
0
b, one also has the
equality ¼ 0 after azimuthal integration due to the fact that b ¼ j and 0b ¼ j
0
. One then
obtains the twofold angular decay distribution Wð; BÞ / 2 X 1; 2; j;J;m;m0; ; p hm 1 2ðJÞh m0 1 2ðJÞ dJ j; 1 2ðÞd J j; 1 2ðÞH m jðJÞH m0y jðJÞ d1=2 pðBÞd 1=2 pðBÞh B p0h By p0: (7)
The fact that ¼ 0 in Eq. (7) implies that the spin density matrix of the daughter baryon appearing in Eq. (7) is purely diagonal implying that, in a polar angle analysis such as the one in Eq. (7), one can only probe the longitudinal polarization P
z of the .
In Eq. (1) we have summed over the helicity labels of the leptons, i.e., we have taken the trace of the respective spin density matrices. By leaving the respective helicity labels unsummed, one can then obtain single lepton and double lepton spin asymmetries as have been discussed in Refs. [11,13], respectively. As mentioned before one can treat the decay of a polarized b in a similar vein.
In the following subsections, we will consider various integrated forms of Eq. (7).
A. Differential rate
The differential rate is obtained from the twofold angular decay distribution Eq. (7) by integrations over ðcos B; cos Þ. For the cos B integration, one uses
R1
1d cos Bd1=2 pðBÞd
1=2
pðBÞ ¼ 1. The cos
integra-tion, finally, can be done by usingR11dcosdJ j; 1 2ðÞ
dJ
j; 1 2ðÞ ¼ 2=3 for J ¼ 1 and 2 for J ¼ 0. One obtains
W /2 3 2 X 1; 2; j;J;m; hm 1 2ðJÞh m 1 2ðJÞð3J0þ J1Þ Hm jðJÞH ym jðJÞ X p hB p0h By p0: (8)
Note that the differential rate obtains only parity-conserving diagonal contributions such that m ¼ m0.
Let us define rate functions mmX 0 (X ¼ U, L, S) through
dmm0 X dq2 ¼ 12 G2 F ð2Þ3 j tj 2 2jp 2jq2v 12M21 HXmm0; (9) where ¼ 1=137:036 is the fine structure constant, GF ¼ 1:16637 105 GeV2 is the Fermi coupling constant, t¼ V
y
tsVtb¼ 0:041 is the product of
corre-sponding Kabayashi-Maskawa matrix elements, and jp2j ¼ 1=2ðM12; M22; q2Þ=2M1 is the momentum of the
hyperon in the b rest frame. Note that we have included
the statistical factor 1=ð2Sbþ 1Þ ¼ 1=2 in the definition
of the rate functions.
The bilinear expressions Hmm0
X (X ¼ U, L, S) are defined by Hmm0 U ¼ ReðHm1 21 Hym1 0 21 Þ þ ReðHm 1 21 Hym10 21 Þ unpolarize transverse; Hmm0 L ¼ ReðHm1 20 Hym1 0 20 Þ þ ReðH m 1 20 Hym10 20 Þ longitudinal; Hmm0 S ¼ ReðHm1 2t H1ym0 2t Þ þ ReðH m 1 2t Hym10 2t Þ scalar: (10) Note that, compared to Ref. [48], we have redefined the scalar structure function HmmS 0 by omitting a factor of 3 in the definition of the scalar structure function.
Putting in the correct normalization factors, one obtains the differential rate d=dq2, which reads
dðb ! ‘þ‘Þ dq2 ¼ v2 2 ðU 11þ22þ L11þ22Þ þ 2m2‘ q2 32 ðU 11þ L11þ S22Þ; (11)
where we have adopted the notations dmmX 0=dq2¼ Xmm
0
an importance sampling of our rate expressions by sorting the contributions according to powers of the threshold factor v. When one wants to compare our results to the corresponding results for the mesonic case written down in Ref. [48], one has to rearrange the contributions in Ref. [48] accordingly. And one has to take into account the factor of 3 difference in the definition of the scalar structure function. We mention that the authors of Ref. [51] have also written their mesonic decay distributions in terms of powers of the threshold factor v. The second term proportional to 2m2
‘=q2 in Eq. (11) can be seen to arise
from the s-wave contributions to ‘þ‘ production: (J ¼ 1) in U11and L11associated with the vector current m ¼ 1 and (J ¼ 0) in S22associated with the axial vector current
m ¼ 2 [see Eq. (3)].
The total rate, finally, is obtained by q2 integration
in the range 4m2
‘ q2 ðM1 M2Þ2: (12)
For the lower q2 limit, one has 4m2
‘¼ ð1:04106;
0:045;12:6284Þ GeV2 for ‘ ¼ ðe; ; Þ. The upper limit of the q2 integration is given by ðMb MÞ2 ¼ 20:29 GeV2. For ‘ ¼ ðe; Þ one is practically probing
the whole q2 region, while for ‘ ¼ the q2 range is
restricted to the low-recoil half of the phase space starting at pffiffiffiffiffiq2¼ 3:55 GeV just below the position of the ð2SÞ
vector meson resonance.
B. Lepton-side decay distribution
Integrating the twofold decay distribution (7) over cos B, one obtains
WðÞ / 2 X 1; 2; j;J;m;m0; hm 1 2ðJÞh m0 1 2ðJÞd J j; 1 2ðÞ dJ j; 1 2ðÞH m jðJÞH ym0 jðJÞ X p hB p0h By p0: (13)
The lepton-side decay distribution involves one more parity-odd structure function, which is defined by
Hmm0 P ¼ ReðH1m 21 H1ym0 21 Þ ReðH m 1 21 Hym10 21 Þ parity odd: (14) Putting in the correct normalization, one obtains dðb! ‘þ‘Þ dq2dcos ¼ v2 3 8ð1þcos 2Þ1 2U 11þ22 þ3 4sin 2 1 2L 11þ22 v3 4cos P 12 þ2m2‘ q2 34½U11þL11þS22: (15)
One can define a lepton-side forward-backward asymmetry A‘
FB by A‘FB ¼ ðF BÞ=ðF þ BÞ, where F
and B denote the rates in the forward and backward hemispheres: A‘FBðq2Þ ¼ 3 2 v P12 v2 ðU11þ22þ L11þ22Þ þ2m2‘ q2 3 ðU11þ L11þ S22Þ : (16)
Note that the lepton-side forward-backward asymmetry vanishes at threshold q2! 4m2
‘. The integrated forward-backward
asymmetry is defined as A‘FB ¼ 3 2 RðM1M2Þ2 4m2 ‘ dq2ðv P12Þ RðM1M2Þ2 4m2 ‘ dq2ðv2 ðU11þ22þ L11þ22Þ þ2m2‘ q2 3 ðU11þ L11þ S22ÞÞ : (17)
C. polarization and hadron-side decay distribution Integrating the twofold decay distribution (7) over cos , one obtains WðBÞ/2 32 X 1; 2; j;J;m;m0; ; p hm 1 2ðJÞh m0 1 2ðJÞðJ1þ3J0Þ Hm jðJÞH ym0 jðJÞd 1=2 pðBÞd 1=2 pðBÞh B p0h By p0: (18) In fact, one finds from the structure of the lepton helicity amplitudes, Eq. (3) and (4), that only (m ¼ m0) contribu-tions survive in Eq. (18).
By chopping off the ! p decay structure d1=2 pðBÞd 1=2 pðBÞh B p0h By
p0 and leaving the helicity
of the unsummed, one obtains the diagonal spin density matrix of the given by
W / 23 2 X 1; 2; j;J;m hm 1 2ðJÞh m 1 2ðJÞðJ1þ 3J0Þ Hm jðJÞH ym jðJÞ: (19)
The z component of the polarization of the daughter baryon can then be calculated from the diagonal spin density matrix elements according to
P z ¼ W1 212 W1212 W1 212þ W1212 ; (20) which gives Pz ¼ v2 ðP11þ22þ L11þ22 P Þ þ 2m2 ‘ q2 3 ðP11þ L11P þ S22PÞ v2 ðU11þ22þ L11þ22Þ þ2m2‘ q2 3 ðU11þ L11þ S22Þ : (21) One has to keep in mind that Xmm0 stands for the
differen-tial expression dmm0
X =dq2(X ¼ U, L, S, P, LP, SP). When
averaging the polarization P
z over q2, one has to
remem-ber to separately average the numerator and denominator in Eq. (21).
In Eq. (21) we have defined two new parity-violating structure functions according to
Hmm0 LP ¼ ReðH m 1 20 H1ym0 20 Hm 1 20 Hym10 20 Þ longitudinal polarized; Hmm0 SP ¼ ReðH m 1 2t Hym1 0 2t Hm 1 2t Hym10 2t Þ scalar polarized: (22) Returning to Eq. (18) we write down the correctly normalized single-angle decay distribution, which reads
dðb ! ð! pÞ‘þ‘Þ dq2d cos B ¼ Brð ! pÞ 1 2 v2 2 ½U 11þ22þ L11þ22 þ ðP11þ22þ L11þ22 P Þ Bcos B þ 2 m2‘ q2 32 ½U11þ L11þ S22þ ðP11þ L22 P þ S22PÞ Bcos B ; (23)
which, using the rate, Eq. (15), and the polarization, Eq. (21), can be rewritten as
dðb! ð! pÞ‘þ‘Þ dq2d cos B ¼ Brð ! pÞ 1 2 dðb! ‘þ‘Þ dq2 ð1 þ BPz cos BÞ: (24)
In Eqs. (23) and (24), we have made use of the asymmetry parameter B in the decay ! p þ , which is
defined by B¼ jhB 1 20 j2 jhB 1 20 j2 jhB 1 20 j2þ jhB 1 20 j2: (25)
The asymmetry parameter has been measured to be B¼ 0:642 0:013 [52]. As a check on Eq. (23), one recovers the differential rate expression, Eq. (8), by integrating over cos Band removing the factor Brð ! pÞ.
Corresponding to the lepton-side forward-backward asymmetry A‘
FB, one can define a hadron-side
forward-backward asymmetry Ah
FB defined by AhFB¼ ðF BÞ=
ðF þ BÞ. Contrary to the mesonic case B ! Kð! KÞ þ
‘þ‘, the hadron-side forward-backward asymmetry in b! ‘þ‘ is nonzero since the decay ! p þ
is parity nonconserving, i.e., one has hB
1 20 h B 1 20 with 64% analyzing power. Using the form Eq. (24), one finds that the forward-backward asymmetry is simply related to the polarization P z via Ah FBðq2Þ ¼ B 2 P zðq2Þ: (26)
The integrated forward-backward asymmetry is defined as
Ah FB¼ B 2 RðM1M2Þ2 4m2 ‘ dq2ðv2 ðP11þ22þ L11þ22P Þ þ2m2‘ q2 3 ðP11þ L11P þ S22PÞÞ RðM1M2Þ2 4m2 ‘ dq2ðv2 ðU11þ22þ L11þ22Þ þ2m2‘ q2 3 ðU11þ L11þ S22ÞÞ : (27)
For the sake of completeness, let us also list the Bdependence of the polarization PzðBÞ, which is obtained by chopping
off the ! p decay structure in Eq. (7) and proceeding as before when calculating the cos B—independent
polarization (21). One obtains
P
zðq2; BÞ ¼
v2ð3
8ð1 þ cos2BÞ P11þ22þ32sin2B L11þ22P Þ v32 cos B U12þ 2m2
‘
q2 32 ðP11þ L11P þ S22PÞ
v2ð38ð1 þ cos2BÞ U11þ22þ32sin2B L11þ22Þ v32 cos B P12þ 2m2
‘
q2 32 ðU11þ L11þ S22Þ
: (28)
In this section we have only discussed the z component of the polarization of the daughter baryon . As mentioned before the contribution of the transverse polarization component P
x (as, e.g., calculated in Ref. [9]) averages out after
integration since it enters the angular decay distribution with an angular factor / P
x Bcos sin B (see Eq. (1) or
III. -TYPE BARYONS IN THE COVARIANT QUARK MODEL
In this section we discuss the basic ingredients of the covariant quark model, which will be used for the calcu-lation of the rare decays of the b baryon. A detailed
description of baryons as bound states of three quarks can be found in Refs. [39,41–43,45]. This includes a description of the structure of the Gaussian vertex factor, the choice of interpolating baryon currents, as well as the compositeness condition for baryons.
The new features introduced to the meson sector in Refs. [44,46] and applied to the baryon sector in Ref. [45] are both technical and conceptual. Instead of using Feynman parameters for the evaluation of the two-loop baryonic quark model Feynman diagram, we now use Schwinger parameters. The technical advantage is that this leads to a simplification of the tensor loop integrations in as much as the loop momenta occurring in the quark propa-gators can be written as derivative operators. Furthermore, the use of Schwinger parameters allows one to incorporate quark confinement in an effective way. Details of these two new features of the covariant quark model have been described in Refs. [44–46].
In the following we consider ¼ ðQ½udÞ-type baryons needed in the present application. They consist of a heavy quark and two light quarks in a1S0 spin 0 configuration. The coupling of a -type baryon to its constituent quarks is described by the Lagrangian
L
intðxÞ ¼ gðxÞ J ðxÞ þ gJðxÞ ðxÞ; (29)
where we make use of the same Jð JÞ interpolating three quark currents for all three -type baryons.
In general, for the -type baryons, one can construct three types of currents without derivatives—pseudoscalar JP, scalar JS, and axial-vector JA(see Refs. [34,38,39,43]):
JP Q½ud ¼ a1a2a3Qa1ua2C 5da3; JS Q½ud ¼ a1a2a35Qa1ua2Cda3; JA Q½ud ¼ a1a2a3Qa1ua2C 5da3: (30)
The symbol ½ud in the suffixes of the currents denote antisymmetrization of flavor and spin indices with respect to the light quarks u and d. We will consider the three flavor types of the baryons: 0sðsudÞ, þcðcudÞ, and
0bðbudÞ. In Ref. [42] we have shown that, in the
non-relativistic limit, the JPand JAinterpolating currents of the
Q½ud baryons become degenerate and attain the correct
nonrelativistic limit (in the case of single-heavy baryons, this limit coincides with the heavy quark limit), while the JScurrent vanishes in the nonrelativistic limit. On the other
hand, the JP and JA interpolating currents of the -type
baryons become degenerate with SUðNfÞ-symmetric
cur-rents. In Ref. [34] we have shown that, in the case of the heavy-to-light baryon transition þc ! 0eþe, the use of
a SU(3) symmetric 0 hyperon is essential in order to
describe data on ðþc ! 0eþeÞ (see also the
discus-sion in Refs. [6,53]). In AppendixBwe explicitly demon-strate that our JP and JA are degenerate with the
SUðNfÞ-symmetric currents. Therefore, in the following
we restrict ourselves to the simplest pseudoscalar JPcurrent.
The nonlocal interpolating three-quark current is written as JðxÞ ¼Z dx1Z dx2Z dx3Fðx; x1; x2; x3Þ JðÞ3qðx1; x2; x3Þ; J3qðÞðx1; x2; x3Þ ¼ 1 2 a1a2a3Qa1ðx 1Þðua2ðx2ÞC5da3ðx3Þ da2ðx 3ÞC5ua3ðx2ÞÞ ¼ a1a2a3Qa1ðx 1Þua2ðx2ÞC5da3ðx3Þ; (31) JðxÞ ¼Z dx1Z dx2Z dx3Fðx; x1; x2; x3Þ JðÞ3qðx1; x2; x3Þ; JðÞ3qðx1; x2; x3Þ ¼ a1a2a3da3ðx 3Þ5Cua2ðx2Þ Qa1ðx1Þ;
where Q ¼ s, c, b. Here the matrix C ¼ 02is the usual charge conjugation matrix, and the ai (i ¼ 1, 2, 3) are
color indices.
The vertex function F characterizes the finite size of the -type baryon. We assume that the vertex function is real. To satisfy translational invariance, the function FN has to fulfill the identity
Fðxþa;x1þa;x2þa;x3þaÞ ¼ Fðx;x1;x2;x3Þ (32) for any given four-vector a. In the following we use a particular form for the vertex function
Fðx; x1; x2; x3Þ ¼ ð4Þ x X 3 i¼1 wixi X i<j ðxi xjÞ2 ; (33) where is the correlation function of the three
constitu-ent quarks with the coordinates x1, x2, x3 and masses m1, m2, m3, respectively. The variable wiis defined by wi¼ mi=ðm1þ m2þ m3Þ such that P3
i¼1wi¼ 1.
We shall make use of the Jacobi coordinates 1;2and the CM coordinate x, which are defined by
x1 ¼ x þ 1ffiffiffi 2 p w3 1 1ffiffiffi 6 p ð2w2þ w3Þ 2; x2 ¼ x þ 1ffiffiffi 2 p w3 1þ 1ffiffiffi 6 p ð2w1þ w3Þ 2; x3 ¼ x 1ffiffiffi 2 p ðw1þ w2Þ 1þ 1ffiffiffi 6 p ðw1 w2Þ 2: (34)
The CM coordinate is given by x ¼P3
i¼1wixi. In terms of
X
i<j
ðxi xjÞ2 ¼ 21þ 22: (35)
Note that the choice of Jacobi coordinates is not unique. By using the particular choice of Jacobi coordinates given by Eq. (34), one obtains the following representation for the correlation function in Eq. (33):
X i<j ðxixjÞ2 ¼Z d4p1 ð2Þ4 Z d4p 2 ð2Þ4e ip1ðx1x3Þip2ðx2x3Þ ðP21P22Þ; ðP21P22Þ ¼ 19 Z d4 1 Z d4 2eiP1 1þiP2 2 ð 21þ 22Þ; P1¼ 1ffiffiffi 2 p ðp1þp2Þ; P2¼ 1ffiffiffi 6 p ðp1p2Þ: (36)
This representation is valid for any choice of the set of Jacobi coordinates. The particular choice (34) is a pre-ferred choice since it leads to the specific form of the argument P2
1 P22 ¼ 23ðp12þ p22þ p1p2Þ. Since this
expression is invariant under the transformations p1 $ p2, p2 ! p2 p1 and p1 ! p1 p2, the rhs in Eq. (36) is invariant under permutations of all xias it should be.
In the next step, we have to specify the function
ðP21 P22Þ ðP2Þ, which characterizes the
fi-nite size of the baryons. We will choose a simple Gaussian form for the function :
ðP2Þ ¼ exp ðP2=2Þ; (37)
where is a size parameter parametrizing the distribution
of quarks inside a -type baryon. We use different values of the parameter for different types of the -type baryon:
s, c, and b for the , c, and bbaryons,
respec-tively. One has to note that we have used another definition of the in our previous papers: ¼ old =ð3
ffiffiffi 2 p
Þ. Since P2 turns into P2E in Euclidean space, the form (37) has the appropriate falloff behavior in the Euclidean region. We emphasize that any choice for is appropriate
as long as it falls off sufficiently fast in the ultraviolet region of Euclidean space to render the corresponding Feynman diagrams ultraviolet finite. The choice of a Gaussian form for has obvious calculational advantages.
FIG. 2 (color online). q1½q2q3 baryon mass operator.
FIG. 3 (color online). Electromagnetic vertex function of the q1½q2q3 baryon: (a) triangle diagram with the (off-shell) photon
attached to the quark q1; (b) triangle diagram with the (off-shell) photon attached to quarks q2or q3; (c) bubble diagram with the (off-shell) photon attached to the vertex of the ingoing baryon; (d) bubble diagram with the (off-(off-shell) photon attached to the vertex of the outgoing baryon.
The coupling constants g are determined by the com-positeness condition suggested by Weinberg [54] and Salam [55] (for review, see Ref. [56]) and extensively used in our approach (for details, see Ref. [57]). The compositeness condition in the case of baryons implies that the renormalization constant of the baryon wave func-tion is set equal to zero:
Z ¼ 1 0ðmÞ ¼ 0; (38) where 0 is the on-shell derivative of the -type baryon mass function , i.e., 0¼ @=@6p, at 6p ¼ m. The
compositeness condition is the central equation of our covariant quark model. The physical meaning, the impli-cations, and corollaries of the compositeness condition have been discussed in some detail in our previous papers (see, e.g., Ref. [44]).
The calculation of the -type mass function (Fig.2) and the electromagnetic vertex (Fig. 3) proceed in the same way as shown in the nucleon case in Ref. [45]. The matrix elements in momentum space read
ðpÞ ¼ 6g2hh 2ðz0ÞSQðk1þ w1pÞ tr½Suðk2 w2pÞ5
Sdðk2 k1þ w3pÞ5ii; (39)
where we use the same shorthand notation for the double loop-momentum integration. The variable z0 is defined as z0¼ 1 2ðk1 k2Þ 2þ 1 6ðk1þ k2Þ 2: (40)
The various contributions to the electromagnetic vertex are given by Qðp; p0Þ ¼ 6eQg2 ðz0Þ 1 2ðk1 k2þ w3qÞ 2 1 6ðk1þ k2þ ð2w2þ w3ÞqÞ 2 SQðk1þ w1p0ÞSQðk1þ w1p0þ qÞ tr½Suðk2 w2p0Þ5Sdðk2 k1þ w3p0Þ5 ; uðp; p0Þ ¼ 6eug2 ðz0Þ 1 2ðk1 k2þ w3qÞ 2 1 6ðk1þ k2 ð2w1þ w3ÞqÞ 2 SQðk1þ w1p0Þ tr½Suðk2 w2p0 qÞSuðk2 w2p0Þ5Sdðk2 k1þ w3p0Þ5 ; dðp; p0Þ ¼ 6edg2 ðz0Þ 1 2ðk1 k2 ðw1þ w2ÞqÞ 2 1 6ðk1þ k2 ðw1 w2ÞqÞ 2 SQðk1þ w1p0Þ tr½Suðk2 w2p0Þ5S dðk2 k1þ w3p0ÞSdðk2 k1þ w3p0þ qÞ5 ; ðaÞðp; p0Þ ¼ 6g2 ðz0Þ ~E ðk1þ w1p0; k2þ w2p0; k2 k1þ w3p0; qÞ SQðk1þ w1p0Þ tr½Suðk2 w2p0Þ5S dðk2 k1þ w3p0Þ5 ; ðbÞðp; p0Þ ¼ 6g2 ðz0Þ ~E ðk1þ w1p; k2þ w2p; k2 k1þ w3p; qÞ SQðk1þ w1pÞ tr½Suðk2 w2pÞ5Sdðk2 k1þ w3pÞ5 : (41)
The free quark propagator in momentum space is given by SfðkÞ ¼
1
mf 6k; (42)
where f ¼ u, d, s, c, b denotes the flavor of the freely propagating quark. We restrict ourselves to isospin invariance mu¼ md. The function ~E ðr1; r2:r3; rÞ is defined as ~ E ðp1; p2; p3; rÞ ¼ X3 i¼1 eqiZ1 0 dfwi1ðwi1r þ 2q 1Þ 0 ðz1Þ wi2ðwi2r þ 2q 2Þ 0 ðz2Þg: (43) The variables q1 ¼ P3
i¼1wi1riand q2¼ P3i¼1wi2riin ~Eðr1; r2:r3; rÞ can be seen as related to the loop momenta by
q1¼ 1ffiffiffi 2
p ðk1 k2Þ; q2¼ 1ffiffiffi 6
for both bubble diagrams. By using Eq. (44), one finds the q ¼ 0 relations ðaÞðp; pÞ þ ðbÞðp; pÞ ¼ 8g2 hhðQ1k 1 þ Q2k 2Þ 0 ðz0Þ ðz0ÞSQðk1þ w1pÞ tr½Suðk2 w2pÞ5 Sdðk2 k1þ w3pÞ5ii; Q1¼ e1ðw2þ 2w3Þ e2ðw1 w3Þ e3ð2w1þ w2Þ; Q2¼ e1ðw2 w3Þ e2ðw1þ 2w3Þ þ e3ðw1þ 2w2Þ; (45)
where the subscripts on the charges eirefer to the flavors of the three quarks: ‘‘i ¼ 1” ! s; c; b,’’ ‘‘i ¼ 2” ! u,’’ and ‘‘i ¼ 3” ! d.’’ Next we will use an integration-by-parts identity to write @ @ki f 2 ðz0ÞSQðk1þw1pÞtr½Suðk2w2pÞ5 Sdðk2k1þw3pÞ5g 0; ði ¼ 1;2Þ: (46) One finds hhk 1A0ii ¼ 1 4hhð2A 1 þ A 2 A 3Þii; hhk 2A0ii ¼ 1 4hhðA 1 þ 2A 2 þ A 3Þii; (47) where A0 ¼ 0ðz0Þ ðz0ÞSQðk1þ w1pÞ tr½Suðk2 w2pÞ5 Sdðk2 k1þ w3pÞ5; A1 ¼ 2ðz0ÞSQðk1þ w1pÞSQðk1þ w1pÞ tr½Suðk2 w2pÞ5Sdðk2 k1þ w3pÞ5 A2 ¼ 2 ðz0ÞSQðk1þ w1pÞ tr½Suðk2 w2pÞ Suðk2 w2pÞ5Sdðk2 k1þ w3pÞ5; A3 ¼ 2ðz0ÞSQðk1þ w1pÞ tr½Suðk2 w2pÞ5 Sdðk2 k1þ w3pÞS dðk2 k1þ w3pÞ5: (48) Using these identities and collecting all the pieces together, one obtains
ðp; pÞ ¼ ðeQþ euþ edÞ
@ðpÞ
@p ; p ¼ m: (49)
As was discussed above, this Ward identity allows one to use the compositeness condition Z¼ 0 written in the form ðp; pÞ ¼ ; p ¼ m; (50) where we take eQ ¼ ec for the present discussion. Again
we have checked analytically that, on the -type baryon mass shell, the triangle diagrams are gauge invariant by
themselves, and the non-gauge invariant parts coming from the bubble diagrams, Figs.3(c)and3(d), cancel each other. The standard definition of the electromagnetic form factors is ðp; p0Þ ¼ F1ðq2Þ iq 2m F2ðq2Þ; (51) where q¼i
2ð Þq. The magnetic moment
of the -type baryon is defined by ¼ ðF1ð0Þ þ F2ð0ÞÞ e
2m
: (52)
In terms of the nuclear magneton2me
pthe -hyperon
mag-netic moment is given by
¼ ðF1ð0Þ þ F2ð0ÞÞmp
m; (53)
where mpis the proton mass.
The magnetic moment has been measured for the s
only and is given by [52]
s ¼ 0:613 0:004: (54) Since we want to fit the size parameters s, c, and
balso to semileptonic b ! c and c ! s charged current
transitions, we need to briefly set up the formalism for the description of these transitions, i.e., for the transitions MðQ½ud! Q0½ud‘‘Þ
¼GffiffiffiF
2
p VQQ0hQ0½udj Q0OQjQ½udið‘O‘Þ; (55)
where O¼ð1
5Þ. These processes are described
in our model by the triangle diagram shown in Fig. 4. The hadronic matrix element in Eq. (55) is expanded in terms of the dimensionless form factors fJ
i (i¼1, 2, 3
and J ¼ V, A), viz.
hQ0½udj Q0QjQ½udi ¼ u2ðp2Þ½fV1ðq2Þ f2Vðq2Þiq=M1 þ fV 3ðq2Þq=M1u1ðp1Þ; hQ0½udj Q05QjQ½udi ¼ u2ðp2Þ½fA1ðq2Þ fA2ðq2Þiq=M1 þ fA 3ðq2Þq=M15u1ðp1Þ: (56)
The calculation of the form factors in our approach is automated by the use of FORM [58] and FORTRAN pack-ages written for this purpose. To be able to compare with our earlier calculations that did not contain confinement, the packages exist for the confined and the unconfined versions of the covariant quark model.
The results of our numerical calculations are well rep-resented by the double-pole parametrization
fðsÞ ¼ fð0Þ
1 as þ bs2; (57)
where s ¼ q2=M2
i and Mi is the mass of the initial
baryon. Using such a parametrization facilitates further integrations. The values of fð0Þ, a, and b are listed in Tables I,II, and III. We plot the form factors in the full kinematical regions (0 s smax) in Figs. 5 (c ! s)
and 6 (b ! c): solid and dotted lines correspond to approximated and exact results, respectively. The agree-ment between the approximate and numerically calcu-lated form factors is excellent except for the form factors A2ðsÞ and TA1ðsÞ for which the agreement is not so good. This is due to the steep ascent of A2 and descent of TA1 at the high end of the q2 spectrum. A better fit would
require the addition of a linear s term in the numerator of Eq. (57) for these two form factors.
As in the case of the rare meson decays B ! KðKÞ ‘‘ and Bc ! DðDÞ ‘‘ treated in Ref. [48], all physical
observables (the rate ðQ½ud! Q0½udþ ‘‘Þ and
asymmetry parameter , etc.) are conveniently written down in terms of helicity amplitudes H 2; j. Note that the
corresponding helicity amplitudes do not carry any super-scripts as they are needed in the description of the corre-sponding rare decays. The relations of these helicity amplitudes to the invariant form factors fJ
i is given in
Appendix C. The rate for the charged current transitions can be written as ðQ½ud! Q0½udþ ‘‘Þ ¼G2FjVQQ0j2 1923M2 1 ZðM1M2Þ2 m2 ‘ dq2 q2 ðq2 m2‘Þ2jp2jH : (58)
For the asymmetry parameter in these decays (see Ref. [34] for the definition of the asymmetry parameter), one obtains ¼ RðM1M2Þ2 m2 ‘ dq2 q2 ðq2 m2‘Þ2jp2jG RðM1M2Þ2 m2 ‘ dq2 q2 ðq2 m2‘Þ2jp2jH ; (59) where H ¼ HUþ HLþ m2 ‘ 2q2ð3HSþ HUþ HLÞ; G ¼ HPþ HLPþ m2 ‘ 2q2ð3HSPþ HPþ HLPÞ; (60)
TABLE I. Parameters for the approximated form factors fðsÞ¼fð0Þ=ð1asþbs2Þ, s¼q2=m2 bin the c! transition. fV 1 fV2 f3V f1A f2A fA3 fð0Þ 0.468 0.204 0.059 0.431 0:078 0:256 a 1.017 1.148 0.698 0.939 0.870 1.208 b 0.249 0.337 0.221 0.211 0.195 0.377
TABLE II. Parameters for the approximated form factors fðsÞ¼fð0Þ=ð1asþbs2Þ, s¼q2=m2 bin the b!ctransition. fV 1 fV2 fV3 fA1 fA2 fA3 fð0Þ 0.600 0.098 0.042 0.594 0.038 0:107 a 0.961 1.127 1.008 0.951 0.971 1.148 b 0.233 0.344 0.267 0.228 0.254 0.357
TABLE III. Parameters for the approximated form factors fðsÞ ¼ fð0Þ=ð1 as þ bs2Þ, s ¼ q2=m2
b in the b! transition. fV 1 fV2 fV3 fA1 fA2 f3A fTV1 f2TV f1TA fTA2 fð0Þ 0.107 0.043 0.003 0.104 0.003 0:052 0:043 0:105 0.003 0:105 a 2.271 2.411 2.815 2.232 2.955 2.437 2.411 0.072 2.955 2.233 b 1.367 1.531 2.041 1.328 3.620 1.559 1.531 0.001 3.620 1.328
FIG. 4 (color online). Diagrams contributing to the flavor-changing transition Q½ud! Q0½udþ X, where X ¼ ‘‘,
and the HX are the following combinations of the helicity amplitudes: HU¼jH1 21j 2þjH 1 21j 2; H L¼jH1 20j 2þjH 1 20j 2; HP¼jH1 21j 2jH 1 21j 2; H LP¼jH120j 2jH 1 20j 2; HS¼jH1 2tj 2þjH 1 2tj 2; H SP¼jH12tj 2jH 1 2tj 2: (61)
We determine the set of size parameters s, c, and
b by fitting data on the magnetic moment of the
hyperon (54) and the nominal branching ratios of the semileptonic decays c ! ‘þ‘ and b ! c‘‘ by
a one-parameter fit to these values. Using the results of Table II for the b! c case, one finds the zero
recoil values fV1 ¼ 0:87 and f1A¼ 0:86. These form
factor values are somewhat lower than the values fV
1 ¼ fA1 ¼ 1 predicted by HQET. This can be interpreted
as an indication that the nominal value for the b!
c‘‘ branching ratio listed by the Particle Data Group
[52] and used by us in our fit is underestimated. With the choice of dimensional parameters s ¼ 0:490 GeV,
c ¼ 0:864 GeV, and b ¼ 0:569 GeV, we get a
rea-sonable agreement with current data on exclusive Cabibbo-allowed decays of c and b (see Tables IV
and V. For the magnetic moments, we get the follow-ing results:
s¼ 0:73; c¼ 0:39; b¼ 0:06; (62) which compares well with data for the s and theoreti-cal estimates for the c and b (see the detailed
0.00 0.05 0.10 0.15 0.20 0.25 0.50 0.55 0.60 s V1 s 0.00 0.05 0.10 0.15 0.20 0.25 0.44 0.46 0.48 0.50 0.52 0.54 0.56 s A1 s 0.00 0.05 0.10 0.15 0.20 0.25 0.22 0.24 0.26 0.28 s V2 s 0.00 0.05 0.10 0.15 0.20 0.25 0.095 0.090 0.085 0.080 s A2 s 0.00 0.05 0.10 0.15 0.20 0.25 0.060 0.062 0.064 0.066 0.068 0.070 s V3 s 0.00 0.05 0.10 0.15 0.20 0.25 0.36 0.34 0.32 0.30 0.28 0.26 s A3 s
discussion in Ref. [40]). In particular, our present results for the magnetic moments of heavy hyperons are very close to our predictions done before in the model without taking account of the mechanism of quark con-finement: c¼ 0:42 and b ¼ 0:06 [40]. Note, the
other model parameters mq and are taken from the fit done in Ref. [46]: mu ms mc mb 0:235 0:424 2:16 5:09 0:181 GeV: (63) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.60 0.65 0.70 0.75 0.80 0.85 s V1 s 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.60 0.65 0.70 0.75 0.80 0.85 s A1 s 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.10 0.11 0.12 0.13 0.14 0.15 s V2 s 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.040 0.045 0.050 0.055 s A2 s 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.060 0.055 0.050 0.045 s V3 s 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.16 0.15 0.14 0.13 0.12 0.11 s A3 s
FIG. 6 (color online). Form factors defining the transition b! c: approximated results (solid line), exact result (dotted line).
TABLE IV. Branching ratios of semileptonic decays of heavy baryons (in %).
Mode Our results Data [52]
c! eþe 2.0 2:1 0:6
c! þ 2.0 2:0 0:7
b! cee 6.6 6:5þ3:22:5
b! c 6.6
b! c 1.8
TABLE V. Asymmetry parameter in the semileptonic decays of heavy baryons.
Mode Our results Data [52]
c! eþe 0.828 0:86 0:04
c! þ 0.825
b ! cee 0.831
b ! c 0.831
IV. THE RARE BARYON DECAYS b! þ ‘þ‘AND b! þ
The effective Hamiltonian [59] leads to the quark decay amplitudes b ! slþl and b ! s: Mðb ! s‘þ‘Þ ¼GFffiffiffi 2 p t 2 Ceff 9 ðsObÞð ‘‘Þ þ C10ðsObÞð ‘5‘Þ 2 q2Ceff7 ½mbðsiqð1 þ 5ÞbÞ þ msðsiqð1 5ÞbÞð ‘‘Þ : (64) and Mðb ! sÞ ¼ GFffiffiffi 2 p e t 42C eff 7 ½mbðsiqð1 þ 5ÞbÞ þ msðsiqð1 5ÞbÞ ; (65) where q¼i 2ð Þq , O ¼ ð1 5Þ, and t V y
tsVtb. The Wilson coefficient Ceff9 effectively
takes into account, first, the contributions from the four-quark operators Qiði ¼ 1; . . . ; 6Þ and, second, the
nonperturbative effects (long-distance contributions) coming from the cc-resonance contributions that are, as
usual, parametrized by a Breit-Wigner ansatz [60] (see details in AppendixD).
The Feynman diagrams contributing to the exclusive transitions b! ‘‘ and b! are shown in Fig.6.
The corresponding matrix elements of the exclusive tran-sitions b ! ‘‘ and b! are defined by
Mðb! ‘‘Þ ¼ GFffiffiffi 2 p t 2
Ceff9 hjsObjbi ‘‘ þ C10hjsObjbi ‘5‘
2mb q2 C eff 7 hjsiqð1þ5Þbjbi ‘‘ (66) and Mðb! Þ ¼ GFffiffiffi 2 p e t 42mbCeff7 hjsiqð1 þ 5Þbjbi : (67)
The hadronic matrix elements in Eqs. (66) and (67) are expanded in terms of dimensionless form factors fJ
i (i ¼ 1, 2, 3
and J ¼ V, A, TV, TA), viz. hB2jsbjB 1i¼ u2ðp2Þ½f1Vðq2ÞfV2ðq2Þiq=M1þfV3ðq2Þq=M1u1ðp1Þ; hB2js5bjB 1i¼ u2ðp2Þ½f1Aðq2Þf2Aðq2Þiq=M1þfA3ðq2Þq=M15u1ðp1Þ; hB2jsiq=M 1bjB1i¼ u2ðp2Þ½f1TVðq2Þðq2q6qÞ=M21f2TVðq2Þiq=M1u1ðp1Þ; hB2jsiq5=M 1bjB1i¼ u2ðp2Þ½f1TAðq2Þðq2q6qÞ=M21fTA2 ðq2Þiq=M15u1ðp1Þ: (68)
One can see that, in comparison with the Cabibbo-allowed b ! c and c ! s transitions, one has four more form factors fTV;TA1;2 . As was mentioned before, the nu-merical results for the invariant form factors are well represented by the double-pole parametrization (57). The values of fð0Þ, a, and b for the approximated form factors describing the b ! s flavor transitions are listed in Table III. The plots of the form factors in the full kine-matical regions (0 s smax) are shown in Fig. 7: the
solid and dotted lines correspond to approximated and exact results, respectively. One can see that both curves are in close agreement with each other. There is only a small disagreement for the suppressed form factors fA 2
and fTA 2 .
Note that a form factor approximation similar to the form (57) was successfully used by us in Ref. [48] in the
analysis of rare decays of bottom mesons. The relations of the helicity amplitudes and invariant form factors are given in AppendixC.
Similar to Eq. (24), the angular decay distribution for the cascade decay b ! ð! pÞ can be written as
dðb ! ð! pÞÞ
d cos B
¼ Brð ! pÞ 1
2ðb! Þ ð1 þ BPz cos BÞ; (69)
where B is the asymmetry parameter in the decay !
p þ for which we take the experimental value B¼
0:642 0:013 [52]. The b! decay rate is calculated
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.2 0.4 0.6 0.8 1.0 s V1 s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.2 0.4 0.6 0.8 s A1 s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 s V2 s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.008 0.007 0.006 0.005 0.004 0.003 s A2 s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.10 0.08 0.06 0.04 0.02 0.00 s V3 s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.6 0.5 0.4 0.3 0.2 0.1 0.0 s A3 s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.5 0.4 0.3 0.2 0.1 0.0 s TV 1 s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.003 0.004 0.005 0.006 0.007 0.008 s TA 1 s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 1.0 0.8 0.6 0.4 0.2 0.0 s TV 2 s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.6 0.4 0.2 0.0 s TA 2 s
ðb! Þ ¼ 2 GFM21j tj 42pffiffiffi2 2 jp 2j 2M21 h jHV 1 21 j2þ jHV 1 21 j2 þ jHA 1 21 j2þ jHA 1 21 j2i ¼ 2 GFmbj tjCeff7 42pffiffiffi2 2ðM2 1 M22Þ3 M3 1 ½ðfTV 2 ð0ÞÞ2þ ðfTA2 ð0ÞÞ2: (70)
As before, the expressions of helicity amplitudes in terms of invariant form factors are given in AppendixC. The z component of the polarization of the appearing in Eq. (69) is given by ~ P z ¼ ~ W1 212 ~W1212 ~ W1 212þ ~W1212 ; (71) where ~ W / H ; j¼2 H y ; j¼2 : (72)
We have used a tilde notation in ~P
z and ~W in order
to distinguish these quantities from the corresponding quantities in the dilepton modes. One has
~ P z ¼ 2 fTV 2 ð0Þf2TAð0Þ ðfTV 2 ð0ÞÞ2þ ðfTA2 ð0ÞÞ2 : (73) Note that fTV
2 ð0Þ fTA2 ð0Þ (see the proof in AppendixE,
which is in agreement with statement of Ref. [61]). Therefore, ~Pz 1, and, finally,
1 tot dðb! ð! pÞÞ dcosB ¼ Brð ! pÞ1 2Brðb! Þð1 BcosBÞ: (74) V. NUMERICAL RESULTS
In this section we present a detailed numerical analysis of the rare decays b! ‘þ‘ and b !
. In Figs. 8–26 we present two-dimensional and
three-dimensional polar angle and polarization distribu-tions. Our predictions for differential rates are shown in the two-dimensional plots in Figs. 8–10, and lepton-side and hadron-side forward-backward asymmetries are dis-played in Figs.14–16and in Figs.23–25, respectively. In
NLD LD 0.40 0.45 0.50 0.55 0.60 0 5 10 15 s 1 tot d ds
FIG. 10 (color online). Differential rate 1 tot dðb!þÞ ds in units of 107 GeV2. NLD LD 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 10 20 30 40 50 60 s 1 tot d ds
FIG. 9 (color online). Differential rate 1 tot dðb!þÞ ds in units of 107 GeV2. NLD LD 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 10 20 30 40 50 60 70 s 1 tot d ds
FIG. 8 (color online). Differential rate 1 tot
dðb!eþeÞ ds in units
of 107GeV2.
FIG. 11 (color online). Lepton-side angular decay distribution
1 tot
dðb!eþeÞ
all the three cases, we plot two respective results that are labelled by ‘‘LD’’ (including long-distance contribu-tions) and ‘‘NLD’’ (no long-distance contribucontribu-tions). In Figs. 11–13 we provide three-dimensional plots of the s
dependence of the lepton-side polar angle decay distribu-tions for each of the e, , and cases. In Figs.17–19we do the same for the hadron-side decay distribution. One clearly sees the long-distance contributions of the
FIG. 13 (color online). Lepton-side angular decay distribution
1 tot
dðb!þÞ
dsd cos in units of 107 GeV2.
NLD LD 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.2 0.0 0.2 0.4 s A eFB
FIG. 14 (color online). Lepton-side forward-backward asym-metry Al
FBin the decay b! eþe.
FIG. 12 (color online). Lepton-side angular decay distribution
1 tot
dðb!þÞ
dsd cos in units of 107 GeV2.
NLD LD 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.2 0.1 0.0 0.1 0.2 s A FB
FIG. 15 (color online). Lepton-side forward-backward asym-metry Al FB in the decay b! þ. LD NLD 0.40 0.45 0.50 0.55 0.60 0.001 0.000 0.001 0.002 s A FB
FIG. 16 (color online). Lepton-side forward-backward asym-metry Al
FB in the decay b! þ.
FIG. 17 (color online). Hadron-side angular decay distribution
1 tot
dðb!eþeÞ
dsd cos B in units of 10
charmonium resonances. In Figs.20–22we show plots of of the cos and s dependence of the longitudinal polar-ization Pz of the daughter baryon , again for the e, ,
and cases. The polarization is large and negative in all
cases. Finally, in Fig.26we show the (hadron-side) polar angle distribution of the radiative decay b! ð!
pÞ. As expected from Eq. (74) and from the discus-sion in Sec. IV, the cos B dependence is given by a FIG. 19 (color online). Hadron-side angular decay distribution
1 tot
dðb!þÞ
dsd cos B in units of 10
7 GeV2.
FIG. 20 (color online). Polarization Pzðs; cos BÞ for the decay
b! eþe.
FIG. 21 (color online). Polarization P
zðs; cos BÞ in the decay
b ! þ.
FIG. 22 (color online). Polarization P
zðs; cos BÞ in the decay
b ! þ.
FIG. 18 (color online). Hadron-side angular decay distribution
1 tot dðb!þÞ dsd cos B in units of 10 7 GeV2. NLD LD 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.32 0.31 0.30 0.29 0.28 s A e FB
FIG. 23 (color online). Hadron-side forward-backward asym-metry Ah
straight-line plot with a slope proportional to the asym-metry parameter B.
In Table VI we present our results for the branching ratios of the rare dileptonic decay b ! ‘þ‘. The
results without long-distance effects are shown in brackets. Our predictions for the radiative decay b! are
shown in Table VII. Here, we also present the results of other approaches using the compilation of Ref. [18]. The results for the integrated lepton-side and hadron-side forward-backward asymmetries are shown in TableVIII.
In our calculations we do not include the regions around the two charmonium resonances Rcc¼ J=c,
ð2SÞ. We exclude the regions MJ= 0:20 GeV to
MJ=þ 0:04 GeV and Mð2SÞ 0:10 GeV to Mð2SÞþ 0:02 GeV. As stressed in Ref. [7], these regions are ex-perimentally vetoed because the rates of nonleptonic de-cays b! þ Rcc, followed by the dileptonic decays of
the charmonium, are much larger than rates of the b ! s-induced rare decays b! ‘þ‘. Vetoing the regions
near the charmonium resonances leads to physically ac-ceptable results—the predictions with and without the inclusion of long-distance effects are comparable with each other. Otherwise (without such a vetoing), the results
NLD LD 0.40 0.45 0.50 0.55 0.60 0.30 0.28 0.26 0.24 s A FB
FIG. 25 (color online). Hadron-side forward-backward asym-metry Ah FBin the decay b! þ. 1.0 0.5 0.0 0.5 1.0 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 cos B 1 tot d d co s B
FIG. 26 (color online). Angular decay distribution for the decay b! ð! pÞ in units of 105. NLD LD 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.32 0.31 0.30 0.29 0.28 s A FB
FIG. 24 (color online). Hadron-side forward-backward asym-metry Ah
FBin the decay b! þ.
TABLE VI. Branching ratios of semileptonic decays b! ‘þ‘ with (without) long-distance contributions (in units of 106).
Mode Our results Theoretical predictions Data
b! eþe 1.0 (1.0) 2:79 0:56 [12]; 4:6 1:6 [25]; 53 (2.3) [10] b! þ 1.0 (1.0) 26:5 5:5 (0:8 0:2) [7]; 1:73 0:42 0:55 [1] 53 (2.1) [10]; 2:64 0:56 [12]; 46 (6.1) [22]; 39 (5.9) [23]; 4:0 1:2 [25]; 3:96þ0:380:08[18]; 2:03þ0:260:09[18] b! þ 0.2 (0.3) 0:63 0:13 (0:30 0:08) [7]; 0:23 0:05 [12]; 4.3 (2.4) [22]; 4.0 (2.1) [23]; 0:8 0:3 [25]; 11 (0.18) [10]
TABLE VII. Branching ratio of the radiative decay b!
(in units of 105).
Our result Theoretical predictions Data [52] 0.4 2:75 1:75 [3]; 3:7 0:5 [9]; <130
3:1 0:6 [10]; 0:68 0:05 [24]; 1:99þ0:340:31[18]; 0:61þ0:140:13[18];
1:55 0:35 [2]; 0.6 [2]; 0.23 [5]; 5:55 1:25 [8]
TABLE VIII. Asymmetries Al
FB and AhFB with (without)
long-distance contributions. Mode Al FB AhFB b ! eþe 3:2 1010ð1:2 108Þ 0:321ð0:321Þ b ! þ 1:7 104ð8:0 104Þ 0:300ð0:294Þ b ! þ 5:9 104ð9:6 104Þ 0:265ð0:259Þ
with long-distance effects are dramatically enhanced (as shown in different theoretical calculations; see also results in TableVI).
VI. SUMMARY AND CONCLUSIONS
(i) We have used the helicity formalism to express a number of observables in the rare baryon decay b ! ð! pÞ‘þ‘ in terms of a basic set of
hadronic helicity structure functions. In the helicity method, one provides complete information on the spin density matrix of each particle in the cascade decay chain that can be conveniently read out by considering angular decay distributions in the rest frame of that particular particle. We hope that we have demonstrated the advantages of the helicity method over a traditional covariant calculation. Every conceivable observable can be written in terms of bilinear forms of the basic hadronic helicity amplitudes calculated in this paper, while a covariant evaluation requires an ab initio calculation for every new observable. We have provided some examples of such observables in this paper.
(ii) There is a multitude of observables to be explored experimentally and theoretically. These include the polarization of the decaying baryon and single-lepton and double-single-lepton polarization asymmetries that have not been discussed in this paper. The advantage of the helicity method is that it is straight-forward to define any of the observables of the problem and to express them in terms of bilinear forms of the hadronic helicity matrix elements de-fined and calculated in this paper. There is no need to restart a covariant calculation for every new spin observable. We mention that it is well-known that hadronically produced hyperons are found to be partially polarized perpendicular to the production plane. Similar polarization effects are expected to occur for hadronically produced b’s. Also b’s
from Z ! bb are expected to be highly
polar-ized. It would be important to take into account such polarization effects in the angular decay distribution of the b.
(iii) We have provided results with and without taking the so-called long-distance effects into account, for which the long-distance effects are calculated by the contributions of the J= and ð2SÞ resonances.
(iv) We have described from a unified point of view exclusive Cabibbo-allowed semileptonic decays b! c‘‘, c ! ‘þ‘ and rare decays
b! ‘þ‘, b! with the use of only three
model parameters: the size parameters s, c,
and b defining the distribution of quarks in the
, c, and bbaryons.
(v) The helicity formulas introduced in this paper can be used as input in a MC event generator patterned after the existing event generator for 0ð"Þ ! þð! p0Þ‘‘ ‘ ¼ ðe; Þ, which is described
and put to use in Ref. [47] and which has been used by the NA48 Collaboration to analyze its data on the above decay [62]. Such a MC event generator would require a viable parametrization of the hadronic transition helicity amplitudes for the whole range of q2, which we provide in this
paper.
(vi) In a future work, we plan to discuss further rare baryonic (b ! s) and (b ! d) decays such as b !
‘þ‘,
b !
‘þ‘, etc., and
b! n‘þ‘. We shall then also compare our
form factor results with the results of other model calculations.
(vii) In this future work, we shall also discuss the full threefold joint angular decay distribution including a treatment of b polarization effects as well as
single-lepton polarization effects. ACKNOWLEDGMENTS
This work was supported by the DFG under Contract No. LY 114/2-1, by Federal Targeted Program ‘‘Scientific and scientific-pedagogical personnel of innovative Russia’’ Contract No. 02.740.11.0238. The work is done partially under the Project No. 2.3684.2011 of Tomsk State University. M. A. I. acknowledges the support of the Forschungszentrum of the Johannes Gutenberg– Universita¨t Mainz ‘‘Elementarkra¨fte und Mathematische Grundlagen (EMG)’’ and the Heisenberg-Landau Grant. M. A. I. and V. E. L. would like to thank Dipartimento di Fisica, Universita` di Napoli Federico II, and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, for warm hospitality.
APPENDIX A: JOINT FOURFOLD ANGULAR DECAY DISTRIBUTION FOR THE DECAY OF
AN UNPOLARIZED b
We write out the threefold angular decay distribution Eq. (1), where we collect together terms with the threshold behavior v0, v1, and v2. Including the q2dependence, one
obtains a fourfold joint angular decay distribution for the decay of an unpolarized b. One has
Wð; B; Þ / 32q2 9 ðjh B 1 20 j2þ jhB 1 20 j2Þ Av2þ Bv þ C 2m2‘ q2 ; (A1) where the coefficients A, B, and C are given by
A ¼ 9 64ð1 þ cos 2ÞðU11þ U22Þ þ 9 32sin 2ðL11þ L22Þ þ 9 32 Bcos B sin2ðL11 P þ L22PÞ þ 1 2ð1 þ cos 2ÞðP11þ P22Þ þ 9
16p ffiffiffi2 Bsin 2 sin B½cos ðI1
11 P þ I122PÞ sin ðI211P þ I222PÞ; B ¼ 9 16cos ½P 12þ Bcos BU12 9
4p ffiffiffi2 Bsin sin B½cos I3
12 P sin I412P; C ¼ 9 16ðU 11þ L11þ S22Þ þ 9 16 Bcos BðP 11þ L11 P þ S22PÞ:
The bilinear expressions Hmm0
X (X ¼ U, L, S, P, LP, SP, I1P, I2P, I3P, I4P) are defined by Hmm0 U ¼ ReðH1m 21 Hym1 0 21 Þ þ ReðH m 1 21 Hym10 21 Þ transverse unpolarized; Hmm0 L ¼ ReðH1m 20 H1ym0 20 Þ þ ReðH m 1 20 Hym10 20 Þ longitudinal unpolarized; HSmm0 ¼ ReðH1m 2t Hym1 0 2t Þ þ ReðH m 1 2t Hym10 2t Þ scalar unpolarized; Hmm0 P ¼ ReðH1m 21 H1ym0 21 Þ ReðH m 1 21 Hym10 21 Þ transverse parity-oddpolarized; Hmm0 LP ¼ ReðH m 1=20H ym0 1=20 H m 1=20Hym 0 1=20Þ longitudinal polarized;
HSmmP 0 ¼ ReðH1=2tm H1=2tym0 Hm1=2tHym1=2t0 Þ scalar polarized;
(A2) HI1mmP0 ¼ 1 4ReðH m 1=21H ym0 1=20þ Hm1=20Hym 0 1=21 H m 1=21Hym 0 1=20 H m 1=20H ym0 1=21Þ longitudinal-transverse interferenceð1Þ; Hmm0 I2P ¼ 14ImðH m 1=21H ym0 1=20 Hm1=20Hym 0 1=21þ H m 1=21Hym 0 1=20 H m 1=20H ym0 1=21Þ longitudinal-transverse interferenceð2Þ; HI3mm0 P ¼ 14ReðH m 1=21H ym0 1=20þ Hm1=20Hym 0 1=21þ H m 1=21Hym 0 1=20þ H m 1=20H ym0 1=21Þ longitudinal-transverse interferenceð3Þ; Hmm0 I4P ¼ 14ImðH m 1=21H ym0 1=20 Hm1=20Hym 0 1=21 H m 1=21Hym 0 1=20þ H m 1=20H ym0 1=21Þ longitudinal-transverse interferenceð4Þ:
Note the threefold joint angular decay distribution for the decay of an unpolarized b is factorized in terms of
fully transverse (unpolarized and parity-odd polarized), longitudinal (upolarized and polarized), and scalar (unpolarized and polarized) bilinear helicity combina-tions and four combinacombina-tions of longitudinal-transverse interference.
Another important property of the threefold joint angular decay distribution is its invariance with respect to the choice of coordinate systems. For example, using the completeness relation for the polarization vectors of the effective current, one can explicitly show that the angular decay distribution Eq. (A1) is the same for two specific choices of coordinate systems: system (i) ~Jeff is directed along the z axis as in this paper, and system (ii) ~Jeff is antiparallel to the direction of the z axis as used, e.g., in Refs. [42,43,47]). In particular, only the transverse helicity amplitudes H1=21m change sign when going from system i to system ii, while the other helicity amplitudes remain invariant. The change of sign for the transverse amplitudes
Hm1=21 can be seen to be compensated by the effects of rotating the coordinate system i by 180 around the x axis when going from system i to system ii.
APPENDIX B: INTERPOLATING CURRENTS OF -HYPERONS
When constructing interpolating baryon currents, it is convenient to use Fierz transformations and corresponding identities in order to interchange the quark fields. First, we specify five possible spin structures J ; ¼
1
ðC2Þ defining the Fierz transformation of the baryon
currents: P ¼ I C5; S ¼ 5 C; A ¼ C 5; V ¼ 5 C ; T ¼ 1 2 5 C : (B1)
The Fierz transformation of the structures J ¼ fP; S; A; V; Tg read
