Predictions for $\Xi_b^- \rightarrow \pi^- \left(D_s^- \right) \ \Xi_c^0 (2790) \left(\Xi_c^0 (2815) \right)$ and $\Xi_b^- \rightarrow \bar{\nu}_l l \ \Xi_c^0 (2790) \left(\Xi_c^0 (2815) \right)$

0.51 MB·

by  R. P. Pavao

(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
− − − 0 0 − 0 0
Predictions for Ξb → π Ds Ξc(2790) Ξc(2815) and Ξb → ν¯ll Ξc(2790) Ξc(2815)
R. P. Pavao,1,∗ W. H. Liang,2,† J. Nieves,1,‡ and E. Oset3,§
1Instituto de F´ısica Corpuscular (centro mixto CSIC-UV),
Institutos de Investigaci´on de Paterna, Apartado 22085, 46071, Valencia, Spain
2Department of Physics, Guangxi Normal University, Guilin 541004, China
3Departamento de F´ısica Teo´rica and IFIC, Centro Mixto Universidad de Valencia-CSIC
Institutos de Investigaci´on de Paterna, Aptdo. 22085, 46071 Valencia, Spain
(Dated: January 25, 2017)
We have performed the calculations for the nonleptonic Ξ− → π− Ξ0(2790)(cid:0)J = 1(cid:1) and Ξ− →
π− Ξ0(2815)(cid:0)J = 3(cid:1) and the same reactions replacing thebπ− by a Dc−. At the s2ame timeb we
c 2 s
evaluate the semileptonic rates for Ξ− → ν¯l Ξ0(2790) and Ξ− → ν¯l Ξ0(2815). We look at the
b l c b l c
reactionsfromtheperspectivethattheΞ0(2790)andΞ0(2815)resonancesaredynamicallygenerated
c c
7 fromthepseudoscalar-baryonandvector-baryoninteractions. Weevaluateratiosoftheratesofthese
1 reactionsandmakepredictionsthatcanbetestedinfutureexperiments. Wealsofindthattheresults
0 are rather sensitive to the coupling of the Ξ∗ resonances to the D∗Σ and D∗Λ components.
c
2

n PACSnumbers:
a
J
I. INTRODUCTION appears from the D∗N interaction in S-wave.
4
Support for the relevance of the vector-baryon compo-
2
h] meTsohne binatrryoodnucinttioenracotfiocnhsir[a1l,d2y]nhaams iaclslowinedthaersatpuiddydoe-f πnIne−nΛRtse(fi2.n6t2[h52e)1s]ewteshtreaetdestseucwdaayiessdrΛeacbnedn→tglyoπof−oduΛancgd(r2ei5ne9m5R)eenfastn.wd[2it1Λh,b2e2→x]-.
p velopment in this field. A qualitative step forward was c
perimentwasfoundfortheratioofthetwopartialdecay
- givenbyintroducingunitarityincoupledchannels,using
p widths. The role of the D∗N was found very impor-
thechiralLagrangiansasasourceoftheinteraction[3–7].
e tant, to the point that if the sign of the coupling of the
h In many cases the interaction is strong enough to gener- D∗N totheΛ (2595)waschanged,theratioofpartialde-
[ ate bound states in some channels, which decay into the c
cay widths was in sheer disagreement with experiment.
openstatesconsideredinthecoupledchannelformalism.
1 In Ref. [22] the semileptonic decay Λ → ν¯lΛ (2595)
The most renowned case is the one of the two Λ(1405) b l c
v and Λ → ν¯lΛ (2625) were studied and the ratio of the
4 states [5, 6, 8, 9]. The original works considered the in- b l c
partial decay widths was also found in agreement with
1 teraction of pseudoscalar mesons with baryons, but the
experiment. Once again, reversing the sign of the D∗N
9 extension to vector mesons with baryons was soon done
couplingtotheΛ (2595)ledtoresultsincompatiblewith
6 inRefs. [10,11]. Theextensiontovectormesonsfindsits c
0 natural framework in the use of the local hidden gauge experiment.
. In the present work we retake the ideas of Refs.
1 Lagrangians[12–14],whichextendthechiralLagrangians
[21, 22] and apply them to the study of the decays
0 and accommodate vector mesons.
7 Ξ− → π− Ξ0(2790)(1−), Ξ− → π− Ξ0(2815)(3−),
The mixing of pseudoscalar-baryon and vector-baryon b c 2 b c 2
1 Ξ− → D−Ξ0(2790), Ξ− → D−Ξ0(2815), Ξ− →
: channels in that framework was done in Ref. [15] in the b s c b s c b
Xiv lRigehfst.se[1c6to,r1,7a].ndAnwaaslteexrnteantidveedatpoprtohaechchtaormthissecmtoixrining Ξν¯l0l(Ξ280c1(257)(903)−,)Ξp−bla→yaνn¯llanΞa0cl(o2g8o1u5s).roTlehteoΞt0ch(e2Λ79(02)5(9215−))(a1n−d)
c 2 c 2
r has been done in Ref. [18], where the chiral Weinberg- and Λ (2625)(3−), substituting the u-quark by an s-
a Tomozawameson-baryoninteractionwasextendedtothe c 2
quark. In Ref. [18] the couplings of the Ξ0(2790) and
SU(8) spin-flavour symmetry group. c
Ξ0(2815)tothedifferentcoupledchannelswereevaluated
c
One case where the relevance of the mixing is found is
for both pseudoscalar-baryon and vector-baryon compo-
inthedescriptionoftheΛc(2595)(21−)andΛc(2625)(32−). nents, in particular the DΛ, D∗Λ, DΣ, D∗Σ which will
InearlyworksonthesubjecttheΛc(2595)appearedbasi- be those needed in the decays mentioned above. We will
callyasaDN molecule[19,20],butbothinRef. [18]and adapt the formalism developed in Refs. [21, 22] to the
Ref. [16] a coupling to the D∗N component was found present case and will make predictions for these partial
with similar strength. On the other hand the Λc(2625) decay modes, which are not yet measured.
II. FORMALISM
∗Electronicaddress: rpavao@ific.uv.es
†Electronicaddress: [email protected]
‡Electronicaddress: jmnieves@ific.uv.es We follow the steps of Ref. [23] for the weak decay
§Electronicaddress: oset@ific.uv.es of B mesons leading to the hadronic resonances in the
2
final state, generalized to the weak decay of Λ baryons parity, it is the c quark that must carry the nega-
b
into baryonic resonances in Ref. [24]. In Ref. [24] the tiveparityandhenceitwillbeproducedinp-wave
Λ → J/ψK−p and Λ → J/ψπΣ reaction in the re- (L=1) in the weak interaction of Fig. 1.
b b
gion of Λ(1405) for K−p(πΣ) were studied, and predic-
tions were made for the K−p invariant mass distribu- 3. The c quark will be incorporated into a final
tions, which were confirmed by experiment later in the D(D∗) meson and thus will go back to its ground
LHCb work disclosing pentaquark states [25]. The anal- state. Hence, the hadronization, introducing
ysis of Ref. [24] also predicted that the K−p and πΣ (cid:0)u¯u+d¯d+s¯s(cid:1) with the quantum numbers of the
would be produced with isospin I = 0, which was also vacuum, must involve the c quark.
confirmed in Ref. [25] since their partial wave analysis
only gave J/ψ and Λ∗ states. Work along the same lines With these constraints the hadronization proceeds as
as Ref. [24] was done in Ref. [26] in the decay of Λ shown in Fig. 2.
c
leading to Λ(1405) and Λ(1670), and in Ref. [27] in the
decay of Λ → J/ψKΞ. The idea of Ref. [24] to the
b
present case proceeds as depicted in Fig. 1.
FIG. 2: Hadronization after the weak process in Fig. 1
to produce a meson-baryon pair in the final state.
Technically the hadronization is implemented as fol-
low: The Ξ− state has a flavour function
b
FIG. 1: Diagrammatic representation of the weak decay
Ξ−b →π−Ξ∗c. (cid:12)(cid:12)Ξ−(cid:11)≡ √1 |b(ds−sd)(cid:105), (1)
b 2
The first point to take into account is that in the Ξ−
b and after the weak decay the b quark is substituted by a
the ds pair has spin S = 0. Symmetry of the wave
c quark and we have a state
function requires the flavour combination ds−sd, and
color provides the antisymmetry. The next step is the
1
hadronization of the final cds state into meson-baryon |H(cid:105)= √ |c(ds−sd)(cid:105). (2)
2
pairs.
We must consider some basic facts:
With the hadronization we have now
1. The ds quarks are spectators in the process. 3
1 (cid:88)
They have S = 0 and come in the combination |H(cid:48)(cid:105)= √ |P q (ds−sd)(cid:105), (3)
4i i
√1 (ds−sd). 2 i=1
2
2. The final Ξ∗ resonances have negative parity, and where P are the qq¯matrix elements.
c ij
they will be generated from the meson-baryon in- Next we write the qq¯matrix in terms of the physical
teraction in S-wave. Since the pair ds has positive mesons, P →φ, with φ given by
√1 π0+ √1 η+ √1 η(cid:48) π+ K+ D¯0
2 3 6
 π− −√1 π0+ √1 η+ √1 η(cid:48) K0 D−
φ≡ 2 3 6 (cid:113) . (4)
 K− K¯0 −√13η+ 23η(cid:48) Ds−
D0 D+ D+ η
s c
3
Then we can write
|H(cid:48)(cid:105)= √1 (cid:2)(cid:12)(cid:12)D0u(ds−sd)(cid:11)+(cid:12)(cid:12)D+d(ds−sd)(cid:11)+(cid:12)(cid:12)D+s(ds−sd)(cid:11)(cid:3). (5)
s
2
ThelaststateinEq. (5)containstwoextrasquarksand A. The weak vertex
corresponds to a more massive component that we omit
in our study. One must evaluate the weak transition matrix ele-
Nextweseethatwehaveamixedantisymmetriccom- ments. For this we follow the approach in Ref. [21].
ponent for the baryonic states of three quarks. If we The vertex W− →π− is of the type [29, 30]
evaluate the overlap with the mixed antisymmetric rep-
resentations of the Σ−, Σ0, Λ0 states [28], we find LWπ ∼Wµ∂µφ, (10)
while the bcW vertex is of the type
|H(cid:48)(cid:105)= √1 (cid:12)(cid:12)D0Σ0(cid:11)+(cid:12)(cid:12)D+Σ−(cid:11)− √1 (cid:12)(cid:12)D0Λ(cid:11). (6) L ∝q¯ W γµ(1−γ )q . (11)
2 6 qWq fin µ 5 in
Since we are dealing with heavy quarks, as in [21] we
Yet, we have to be careful here with the phase conven-
keep the dominant terms in a nonrelativistic expansion,
tions. By looking at the phase convention of Ref. [28] the γ0, γiγ5 (i=1,2,3) and, hence, combining the two
and the one inherent in the baryon octet matrix,
former Lagrangians we obtain a structure for the weak
transition at the quark level
√1 Σ0+ √1 Λ Σ+ p 
2 6 V ∼q0+(cid:126)σ·(cid:126)q, (12)
B = Σ− −√1 Σ0+ √1 n , (7) P
 2 6 
Ξ− Ξ0 −√2 Λ with qµ the four-momentum of the pion.
6
In Ref. [21] the operator in Eq. (12), which acts at
the quark level between the b and c quarks, was con-
which is used in the chiral Lagrangians, one can see that
one must change the phases of Σ+, Λ, Ξ0 from [28] to verted into an operator acting over the Λ∗c and Λb at the
agree with the chiral Lagrangians1. macroscopical level with the result
Withthisclarificationaboutthephases,thestatethat (cid:26)(cid:18) q0 (cid:19) q0√ (cid:27)
V ∼ i (cid:126)σ·(cid:126)q+iq δ −i 3 S(cid:126)+·(cid:126)q δ ME(q),
we obtain consistent with the chiral convention is: P q J,12 q J,32
(13)
|H(cid:48)(cid:105)= √1 (cid:12)(cid:12)D0Σ0(cid:11)+(cid:12)(cid:12)D+Σ−(cid:11)+ √1 (cid:12)(cid:12)D0Λ(cid:11). (8) where S(cid:126)+ is the spin transition operator from spin 1 to
2 6 2
spin 3 normalized such that
2
We also mention the phase convention for mesons in 1 3
terms of isospin states, where |π+(cid:105) = −|1,1(cid:105), |K−(cid:105) = (cid:104)M(cid:48)|Sµ+|M(cid:105)=C(2, 1, 2;M, µ, M(cid:48)), (14)
−−(cid:12)(cid:12)|112,,1−(cid:105),12Ξ(cid:11),−(cid:12)(cid:12)=D0−(cid:11)(cid:12)(cid:12)1=,−−1(cid:12)(cid:12)(cid:11)12.,−12(cid:11), and for baryons Σ+ = with µ in the spherical basis and C(cid:0)12, 1, 32;M, µ, M(cid:48)(cid:1)
2 2 theClebsch-Gordancoefficients. ME(q)isthequarkma-
In terms of isospin, |H(cid:48)(cid:105) can be written as
trix element for the radial wave functions
(cid:90)
|H(cid:48)(cid:105)=−(cid:114)3(cid:12)(cid:12)(cid:12)ΣD(J = 1)(cid:29)+ √1 (cid:12)(cid:12)(cid:12)ΛD(J = 1)(cid:29). (9) ME(q)= dr r2j1(qr)φin(r)φ∗fin(r). (15)
2(cid:12) 2 6(cid:12) 2
Here we do the same and the macroscopic states are Ξ∗
c
For D∗ production the flavour counting is the same and and Ξb respectively.
we would have the same combination substituting D by Since we require ratios of production rates, the matrix
D∗. element ME(q) cancels in the ratio and what matters to
differentiate the cases with spin 1 and 3 is the operator
2 2
in Eq. (13). One should note that the presence of the
factor j (qr) in Eq. (15) is due to the fact that the c
1
1 OnewaytoseethisistotakethesingletbaryonstateofRef. [28] quark is created with L=1 as we discussed previously.
withaminussign,introducethehadronizationwithu¯u+d¯d+s¯s
aswehavedonebeforeandseethemeson-baryoncontent. Then
wecomparethisresultwithTr(B·φ)obtainedwiththeoctetof B. The spin structure in the hadronization
mesonsEq. (4)forφ(takingonlythe3×3partofthematrix),
and Eq. (7) for B. The matrix φ contains also a singlet of
mesons, the octet matrix is the same putting in the diagonal The next issue is to see how the hadronization affects
(cid:16)√π0 + √η8,−√π0 + √η8,−2√η8(cid:17). the cases of DB or D∗B (with B = Σ,Λ) production in
2 6 2 6 6
4
spin J = 1 or 3. For this we follow again the approach sameastheangularmomentumofthecquarkafter
2 2
of Ref. [21]. The calculation proceeds as follows: the weak production.
1. The q¯q pair is created with J = 0+. Since the q¯
4. The angular momentum of the c quark and the q¯q
has negative intrinsic parity we need L = 1 in the
pairarerecombinedtogiveL(cid:48) =0,sinceallquarks
quarkstorestorethepositiveparityandthisforces
are in their ground state in the DΣ, D∗Σ, DΛ,
the q¯q pair to come with spin S =1 to give J =0.
and D∗Λ final states. The total angular momen-
This is the essence of the 3P model [28, 31].
0 tum of the c quark and that of the q¯ of the q¯q
pair are recombined to give j = 0,1, for the D or
2. Since what we want is to elaborate the spin de-
D∗ production. The total angular momentum of
pendenceofthematrixelements, weassumeazero
the q from the q¯q pair determines the spin of the
range interaction, as is also done in similar prob-
baryonΞ∗ sincethedsquarkscarryspinzero. The
lems like the study of pairing in nuclei [32, 33]. c
Clebsch-Gordan coefficient appearing in the differ-
3. Since the d, s quarks are spectators and carry J = ent combinations are recombined to give a Racah
0, the total angular momentum of the Ξ∗ is the coefficient [34] and the final result is
c
(cid:88)
|J, M;c(cid:105)|0, 0;q¯q(cid:105)|0, 0;cd(cid:105)= C(j,J)|J, M;meson-baryon(cid:105), (16)
j
where the coefficients C(j,J) are given in Table I. meson-baryon components.
The width for Ξ →π−Ξ∗ is given by
What we have done so far is to obtain the angular b c
structure of the mechanism for DB(D∗B) production,
dbnyuanntcawemseiΞcfia0cnl(la2yl7ly9g0ewn)eaarnnatdtetΞdo0c(rh2eas8vo1en5a)t.hnecTehpserioswddauyecpttiiooctnperdoofdinuthcFeeitgrhe.esso3e-. ΓΞb→π−Ξ∗c = 21πMMΞΞ∗cbq(cid:88)(cid:88)|t|2, (17)
It involves the amplitudes for Ξ → π−D(D∗)B pro-
b
duction studied before, together with the D(D∗)B loop with q the momentum of the pion in this decay.
functions and the couplings of the Ξ∗ resonance to these By combining Eqs. (9), (13), (16), we obtain
c
(cid:12) (cid:114)
J = 1 : (cid:88)(cid:88)|t|2 =C2(cid:0)q2+ω2(cid:1)(cid:12)(cid:12)1(− 3)G g + 1√1 G g
2 π (cid:12)(cid:12)2 2 ΣD R,ΣD 2 6 ΛD R,ΛD
(cid:114) (cid:12)2
1 3 1 1 (cid:12)
+ √ (− )G g + √ √ G g (cid:12) , (18)
2 3 2 ΣD∗ R,ΣD∗ 2 3 6 ΛD∗ R,ΛD∗(cid:12)(cid:12)
and
(cid:12) (cid:114) (cid:12)2
3 (cid:88)(cid:88) (cid:12) 1 3 1 1 (cid:12)
J = : |t|2 =C2 2ω2(cid:12)√ (− )G g + √ √ G g (cid:12) , (19)
2 π(cid:12)(cid:12) 3 2 ΣD∗ R,ΣD∗ 3 6 ΛD∗ R,ΛD∗(cid:12)(cid:12)
(cid:112)
whereω isthepionenergy m2 +q2,andG ,G contains the matrix element ME(q) and constants of the
π π BD BD∗
are the loop functions for the propagator of BD(BD∗) weak interaction. Since the mass of the two Ξ∗ that we
c
in the resonance formation mechanism of Fig. 3, and investigate are not very different, then we assume C to
g the coupling of the resonance Ξ∗ to any of beaconstantthatcancelsintheratiooftheratesforthe
R,BD(BD∗) c
thestatesBD(BD∗). C inEqs. (18)(19)isafactorthat
5
C(j,J) J = 1 J = 3
2 2
(pseudoscalar) j =0 1 1 0
4π2
(vector) j =1 1 √1 − 1 √1
4π2 3 4π 3
TABLE I: C(j,J) coefficients in Eq. (16).
(a) First step at quark level.
FIG. 3: Mechanism for the production of the Ξ∗
c
resonances by production of D(D∗)Σ(Λ) and coupling
of the meson-baryon components to Ξ∗.
c
production of the two resonances. In this case we find
Γ M p (1)(cid:80)(cid:80)|t|2(1)
R ≡ Ξb→π−Ξc(1) = Ξc(1) π , (20)
1 ΓΞb→π−Ξc(2) MΞc(2)pπ(2)(cid:80)(cid:80)|t|2(2)
where 1,2 refer to the Ξ (2790) and Ξ (2815) respec- (b) Hadronization to producte D(D∗)B.
c c
tively.
The case of D− production instead of π− is identical.
s
Instead of u¯d coupling to W we now have c¯s, which is
equally Cabbibo favoured and is proportional to cosθ
C
inbothcases. Theonlydifferenceinthiscaseisthatthe
momentum of the D− is smaller than in the case of pion
s
inproduction. ThemomentaofD−inthecasesΞ (2790)
s c
and Ξ (2815) are very similar and, hence, by analogy to
c
Eq. (20) we can write (c) Propagation of D(D∗)B and coupling to the Ξ∗.
c
FIG. 4: Different steps of Ξ∗ production in the
R ≡ ΓΞb→Ds−Ξc(1) = MΞc(1)pDs−(1)(cid:80)(cid:80)|t|2(1), (21) Ξb →ν¯llΞ∗c process. c
2 ΓΞb→Ds−Ξc(2) MΞc(2)pDs−(2)(cid:80)(cid:80)|t|2(2)
with p (1,2) evaluated for the Ξ (2790) and Ξ (2815)
respectDivs−ely, and (cid:80)(cid:80)|t|2(1,2) hacve to be reevacluated we have ν¯ll production. The semileptonic decays of BD
hadronsalongthelinesdescribedherehavebeenstudied
with the new momentum.
in Refs. [35, 36]. The weak decay of Λ →ν¯lΛ(1405) is
If we assume that ME(q) is not very different in the c l
adddressed in Ref. [37] and the Λ → ν¯lΛ (2595) and
case of π− or D− production we can also write b l c
s Λ → ν¯lΛ (2625) in Ref. [22]. The first step from the
b l c
Γ p (1)(cid:80)(cid:80)|t|2(1,D−) Ξb →ν¯llΞ∗c is shown in Fig. 4a
R3 ≡ ΓΞΞbb→→Dπ−s−ΞΞcc((11)) = pDπs−−(1)(cid:80)(cid:80)|t|2(1,π−s) . (22) iedThinetohnelyfodrimffeerrenseccetiwonitshitshtehneocnoluepptlionngicodfeWcaytostuν¯dl-.
l
Following Ref. [35] we have, for the combined Wν¯l and
We expect this equation to hold only at the qualitative l
Wcb vertices,
level since ME(q) is not necessarily the same for these
two different values of q.
V
t=−iG √bcLαQ , (23)
F α
2
III. SEMILEPTONIC DECAY
The semileptonic process, Ξ → ν¯lΞ0(2790), Ξ → with G the Fermi coupling constant, V the Cabbibo-
b l c b F bc
ν¯lΞ0(2815)proceedsinasimilarwaybutinsteadofaπ− Kobayashi-Maskawa matrix element for the b → c tran-
l c
6
sition, and Lα, Q the leptonic and quark currents: The rest of the work needed is identical to the one in
α
thenonleptoniccaseoftheformersections. Onecanalso
Lα =u¯lγα(1−γ5)uνl, (24a) do an angle integration analytically in the evaluation of
Q =u¯ γ (1−γ )u . (24b) Γ and one finally obtains
α c α 5 b
Once again we retain γ0 and γiγ from the quark
5
matrix elements, which are the leading terms in a
nonrelativistic reduction. Actually the ν¯ll pair comes dΓ = MΞ∗c2m 2m 1 p p˜(cid:88)(cid:88)|t|2, (26)
out with a large momentum [22] and the momenta of dMinv(ν¯ll) MΞb ν l(2π)3 Ξ∗c l
the baryons are small.
The first step in Fig. 4a produces a different structure
from Eq. (12) in the nonleptonic case, and one finds [22] where p is the Ξ∗ momentum in the Ξ rest frame
Ξ∗c c b
and p˜ the lepton momentum in the ν l rest frame, and
(cid:88) LαL† βQ Q† = 8 p p . (25) (cid:80)(cid:80)|lt|2 is given by [22] l
α β m m ν l
ν l
leptonpol.
(cid:88)(cid:88)|t|2 =C(cid:48)2 8 1 (cid:18)Minv(cid:19)2(cid:20)E˜2 − 1p(cid:126)˜2 (cid:21)A V (J), (27)
m m M2 2 Ξb 3 Ξb J had
ν l Ξb
with
(cid:12) (cid:32) (cid:114) (cid:33)
1 (cid:12)1 3 1 1
J = : A V (J)=(cid:12) − G g + √ G g +
2 J had (cid:12)(cid:12)2 2 ΣD R,ΣD 2 6 ΛD R,ΛD
(cid:32) (cid:114) (cid:33) (cid:12)2
1 3 1 1 (cid:12)
+ √ − G g + √ √ G g (cid:12) , (28)
2 3 2 ΣD∗ R,ΣD∗ 2 3 6 ΛD∗ R,ΛD∗(cid:12)(cid:12)
and,
(cid:12) (cid:32) (cid:114) (cid:33) (cid:12)2
3 (cid:12) 1 3 1 1 (cid:12)
J = : A V (J)=2(cid:12)√ − G g + √ √ G g (cid:12) , (29)
2 J had (cid:12)(cid:12) 3 2 ΣD∗ R,ΣD∗ 3 6 ΛD∗ R,ΛD∗(cid:12)(cid:12)
whereG , G andg , g arethesameasin
BD BD∗ R,BD R,BD∗
the nonleptonic decay and C(cid:48) is again a factor that con- Anapproximatevaluefortheratioofthesemileptonic
tains the matrix element ME(q) evaluated at the proper production for the two resonances is given by
value of q. A novelty here is that q is not constant when
oMnienvinpteeagkrasteasroduMndΓdinvthoevemraMxiimnvu.mHoawlloewveerd, itnhethfaecDt athliatzt R= ΓΓΞb→ν¯ll Ξc(2790) = AA12VVhad(cid:0)(cid:0)213(cid:1)(cid:1). (31)
plot[22],asweshowinFig. 5forthepresentcase,allows Ξb→ν¯ll Ξc(2815) 32 had 2
us to consider C(cid:48) constant over the while range of M .
inv
ThemagnitudesE˜ andp(cid:126)˜ inEq. (26)aretheener-
Ξb Ξb IV. RESULTS
gies of Ξ and its momentum in the rest frame of the ν l
b l
pair which are given by [35]
We use the values of g , g , g , g
R,ΣD R,ΣD∗ R,ΛD R,ΛD∗
M2 +M2 −M2 and of the G , G , G , G from Ref. [18]
E˜ = Ξb inv Ξ∗c, (30a) ΣD ΣD∗ ΛD ΛD∗
Ξb 2M which we have redone in order to evaluate the complex
inv
(cid:16) (cid:17) couplings and the G functions since only the modulus
p˜ = λ12 MΞ2b,Mi2nv,MΞ2∗c , (30b) of gR,i were given there and the values of Gi were not
Ξb 2M tabulated. We give all this information in Tables II and
inv
III.TheGfunctionsaretakenfromdimensionalregular-
with λ(x,y,z) the ordinary K¨allen function. ization subtracting the value of G at s=α(cid:0)M2 +m2 (cid:1)
th th
7
R R R
1 2 3
gR,ΣD∗ =0 0.882 0.626 0.686
gR,ΛD∗ =0 0.21 0.149 0.686
gR,ΣD∗ →−gR,ΣD∗ 0.512 0.363 0.686
gR,ΛD∗ →−gR,ΛD∗ 0.07 0.05 0.686
TABLE IV: Values of R , R , R by both changing the
1 2 3
sign of the g couplings and setting them to zero.
R,BD∗
R
gR,ΣD∗ =0 0.452
gR,ΛD∗ =0 0.108
gR,ΣD∗ →−gR,ΣD∗ 0.263
FIG. 5: The invariant mass distribution for ν¯l in the gR,ΛD∗ →−gR,ΛD∗ 0.036
l
Ξ →ν¯lΞ (2790). The one for the Ξ →ν¯lΞ (2815)
b l c b l c TABLE V: Values of R of the semileptonic decay,
decay is very similar.
obtained by both changing the sign of the g
R,BD∗
couplings and setting them to zero.
Σ D Σ D∗ Λ D Λ D∗
g 1.209+i0.107 -0.801-i0.286 1.438-i0.864 -0.602+i0.584
As we can see, the results shown above tell us the rel-
Gg -6.734-i0.270 3.483+i1.073 -8.570+i5.777 2.609-i2.777
evance of the D∗B components in the build up of these
TABLE II: Values of g and Gg for the different channels resonances.
for the resonance Ξ0(2790)(1−). As for the sector of the semileptonic decay rates cor-
c 2
responding to Eq. (31) we find that
Γ
with α=0.9698 and Mth+mth the mass of the lightest R= Ξb→ν¯ll Ξc(2790) =0.201, (35)
hadronic channel of all the coupled channels for a given ΓΞb→ν¯ll Ξc(2815)
quantum number [38].
Using the values in Tables II and III and Eq. (20) we and if we integrate Eq. (26) we find
obtain
R=0.207. (36)
Γ
R = Ξb→π−Ξ0c(2790) =0.404, (32)
1 Γ As we can see, the numbers are essentially the same.
Ξb→π−Ξ0c(2815) Once again, if the couplings to D∗B states are changes
and for Eq. (21) we obtain different results, shown in Table V.
Γ
R = Ξb→Ds−Ξ0c(2790) =0.287. (33)
2 Γ V. CONCLUSION
Ξb→Ds−Ξ0c(2815)
Similarly we can obtain for Eq. (22) We have studied the nonleptonic Ξ− →M +Ξ∗, with
b c
M = π−, D− and Ξ∗ = Ξ0(2790)(1−), Ξ0(2815)(3−).
Γ s c c 2 c 2
R3 = ΓΞΞbb→→Dπ−s−ΞΞ0c0c((22779900)) =0.686. (34) WcalelyhagveenearsastuemdefdrotmhatthethMe ΞB∗c arnesdonVaBnceisntaerreacdtiyonna,mais-
done in Ref. [18]. We saw that the present decays only
In order to see how sensitive these rates are to the involved the DΛ, DΣ, D∗Λ, D∗Σ channels and we took
values of the D∗B couplings we reevaluate them by first
theneededcouplingsfromthatwork. Giventhefactthat
settingthemtozerothenchangingtheirsign. Theresults
the momentum of the meson M is very similar for the
we obtain are shown in Table IV.
case of the production of the two resonances (since their
masses are very close) we could eliminate in the ratio of
Λ D∗ Σ D∗ widths the matrix element at the quark level involving
thewavefunctionsofthebandcquarks. Then,onlyfac-
g 2.344-i0.605 0.783+i0.483
torsrelatedtothespinstructure ofthechannelsandthe
Gg -12.286+i4.237 -4.108-i2.123
couplings of the hadronic model for the resonances were
relevant, which tells us that the measurement of these
TABLE III: Values of g and Gg for the different
channels for the resonance Ξ0(2815)(3−). partial decay widths are relevant to learn details on the
c 2 nature of the Ξ∗ resonances. With more uncertainty we
c
8
were able to also predict the ratio of Ξ− → π−Ξ∗ and evance of the mixing of pseudoscalar-baryon and vector-
b c
Ξ− →D−Ξ∗ for the same resonance. baryon components in the building up of the molecular
b s c
We also evaluated in the semileptonic rates. In this baryonic states, a subject which is catching up in the
case we can only evaluate one ratio, the semileptonic hadronic community [15–18, 39–41].
decay Ξ → ν¯lΞ∗ for the Ξ0(2790) and Ξ0(2815) res-
b l c c c
onances. Once again, the predictions will be valuable
when these partial decay width can be measured. We
Acknowledgments
should stress that both the nonleptonic and semilep-
tonic decay widths are measured for the case of Λ →
b
π−Λ (2595), Λ → π−Λ (2625) and Λ → ν¯lΛ (2595) R. P. Pavao wishes to thank the Generalitat Valen-
c b c b l c
and Λ → ν¯lΛ (2625) and the method used here gave ciana in the program Santiago Grisolia. This work is
b l c
results in agreement with experiment [21, 22], so we are partly supported by the National Natural Science Foun-
confident that the predictions done here are fair. In any dation of China under Grants No. 11565007, 11647309
case the experimental result could test the accuracy of and No. 11547307. This work is also partly sup-
the model of Ref. [18], which is one of the possible ways ported by the Spanish Ministerio de Economia y Com-
toaddressthemolecularstates,withaparticulardynam- petitividad and European FEDER funds under the con-
ics based on a particular symmetry. tract number FIS2011-28853-C02-01, FIS2011- 28853-
We also checked that the results were sensitive to the C02-02, FIS2014-57026-REDT, FIS2014-51948-C2- 1-P,
couplings of the D∗B components and confirmation of and FIS2014-51948-C2-2-P, and the Generalitat Valen-
this feature by experiment could give a boost to the rel- ciana in the program Prometeo II-2014/068.
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