Baryon semileptonic decays: the Mexican contribution

0.1 MB·English

by  Ruben Flores-Mendieta

Baryon semileptonic decays: the Mexican
contribution
Rubén Flores-Mendieta and Alfonso Martínez†
∗
InstitutodeFísica,UASLP,ÁlvaroObregón64,SanLuisPotosí,S.L.P.78000,Mexico
∗
†ESFMdelIPN,ApartadoPostal75-702,México,D.F.07738,Mexico
7
0
0 Abstract. Wegiveadetailedaccountofthetechniquestocomputeradiativecorrectionsinbaryon
2
semileptonic decays developed over the years by Mexican collaborations. We explain how the

n method works by obtaining an expression for the Dalitz plot of semileptonic decays of polarized
a baryons including radiative corrections to order O(a q/p M ), where q is the four-momentum
1
J transfer and M is the mass of the decayingbaryon.From here we computethe totally integrated
1
9 spinangularasymmetrycoefficientoftheemittedbaryonandcompareitsvaluewithotherresults.

Keywords: Weakdecays,baryons,radiativecorrections
1
v PACS: 14.20.Lq,13.30.Ce,13.40.Ks
7
6
0 INTRODUCTION
1
0
7 RightafterPauliformulatedtheneutrinopostulate,Fermiintroducedthefieldtheoretical
0 treatmentoftheprocess n p+e +n intheearly 1930’s[1]. Thisisthefirst known
− e
h/ example of a V-theory. In→the ensuing years other forms of nuclear b decays were
p
observed, which prompted Gamow and Teller to formulate an A-theory. The concept
-
p ofa newclass ofinteractions,theweakinteractions,hadjustemerged.
e
Both theV and A theories were fused into theV A theory by Sudarshan and Mar-
h
−
: shak and also independently by Feynman and Gell-Mann in the late 1950’s, motivated
v
by the discoveryof parity non-conservation in weak interactions by Lee and Yang (the-
i
X
oretically)and Wuand Telegdi(experimentally).
r
a In this context the weak interactions are described by an effective Lagrangian
L (x) = (G /√2)J†(x)Jl (x)+h.c., where G is the Fermi coupling constant and
eff F l F
−
the weak current Jl (x) has the V A structure. Jl (x) can be separated into weak
leptonicJll (x) andweak hadronicJlh−(x)currents,namely,Jl (x)=Jll (x)+Jlh(x),where
the leptonic current can be written directly in terms of the lepton fields whereas the
hadronicone can be decomposed into parts having definite flavor SU(3) transformation
propertiesand can bewritteninterms ofquark fields [2].
The Lagrangian L however, faces many problems and cannot be taken as a self-
eff
consistent quantum field theory of weak interactions. Among other aspects, i) it is not
renormalizable; ii) at high energies it leads to a violation of unitarity, i.e., it brings in
probabilitynon-conservation;andiii)it hasno roomforneutral currents.
With the advent of gauge theories, the cornerstone of the theory of weak interactions
became the SU(2) U(1) Weinberg-Salam theory, which currently possesses a quite
×
impressiveexperimental evidence [3]. Further work, both theoretical and experimental,
has finally yielded to the standard model of elementary particles, which embodies our
knowledgeofthestrongandelectroweak interactions.
Thepurposeofthispaperistobrieflyreviewtheachievementsofthepastthirtyyears
of theoretical activity in baryon semileptonicdecays (BSD) from the Mexican perspec-
tive. We give a detailed account of BSD, focusing on techniques to the calculation of
radiative corrections to observables which have been the major contribution of local
research groups. The paper opens with a historical account of the development of the
theoreticalapproachandcontinueswithanapplicationtotheevaluationofradiativecor-
rections to the spin-asymmetry coefficient of the emitted baryon. Numerical results are
discussedafterwards. Thepaper closeswithabriefsummaryand conclusions.
RADIATIVE CORRECTIONS IN BSD: THE EARLY YEARS
In dealing with the radiative corrections (RC) to BSD it is very important to have an
organized systematic approach to deal with the complications that accompany them.
These RC depend on an ultraviolet cutoff, on strong interactions, and on details of
the weak interactions, other than the effectiveV A theory. In other words, they have
−
a model-dependent part [4, 5]. They also depend on the charge assignments of the
decayingandemittedbaryons.Theirfinalformdependsontheobservedkinematicaland
angular variables and on certain experimental conditions. Over the years, our approach
tothecalculationoftheseRChasbeentoadvanceresultswhichcanbeestablishedinas-
much-as possibleonce and for all. This task is considerably biased by the experimental
precisionattainedingivenexperimentsand bytheavailablephasespaceineach decay.
A systematic study of the calculation of RC in BSD was initiated back in 1980
[6] following an approach originally introduced by Sirlin [4] for the electron energy
spectrum in neutron decay and later extended to the decay of polarized neutrons [7, 8].
An expression for the electron energy spectrum including radiative corrections was
obtained in Ref. [6], which was accurate enough to allow experimental analyses to be
performed in a model-independent fashion, provided hard bremsstrahlung photons are
experimentally discriminated. At that time, those results were not directly applicable
to the experiments being performed, which had lower statistics and made no provision
to discriminateagainsthard photons.Later, in thesame spirit,theRC to thedifferential
decayrateofpolarizedneutralandchargedbaryonswerepresentedinRef.[9].Fromthat
result,obtainingtheRCto thedecay rates andangularcoefficients wasstraightforward.
Thiswasthefirst attempttoobtainRC tointegratedobservables.
A step further was taken in 1987, when in order to improve previous results on the
RCtotheelectron energy spectrumofbaryon b decay [9],anew theoreticalexpression
was obtained, this time including all terms of order a q/p M where q is the four-
1
momentum transfer and M is the mass of the decaying baryon [10]. The result is
1
suitable for model-independent analyses of very-high-statistics experiments, without
any experimental constraint on the detection of hard inner-bremsstrahlung photons. In
Ref. [10]itwasshownthatthebremsstrahlungcontributiontoBSD canbecomputedin
amodel-independentway upto termsoffirst orderinq, byusingtheLowtheorem[5].
In the meantime, new high-statisticsexperiments in BSD [11] made Dalitz plot (DP)
measurements feasibleand the application of RC was necessary. Much of the work had
already been advanced in early calculationsfor theenergy spectrum of thecharged lep-
tonand forthedecay distributionofpolarizedbaryons[6,8,9,10],sothenewtaskwas
to adapt the results for the DP. In this respect, an expression for the DP of semileptonic
decays of charged and neutral baryons including RC to order a and neglecting terms
of order a q was introduced in Ref. [12]. The virtual RC presented no new complica-
tions. In contrast, the bremsstrahlung RC became rather involved. The approximation
implemented in that work was that the real photons are not observed directly but indi-
rectly when the energies of the final baryon and charged lepton are found not to satisfy
thefinalthree-bodyoverallenergy-momentumconservation.Inconsequence, adetailed
kinematicalanalysisfordeterminingtheintegrationlimitsoverthephotonvariableswas
mandatory. Besides, the infrared divergence in the bremsstrahlung had to be identified
carefully along with the finite terms that accompany it. A proper choice of variables
yielded the integrations feasible. No doubt this paper marked an important path toward
newresults.TheimmediateapplicationwastothecomputationofRCtotheDPtoorder
a q/p M , bothforcharged [13]and neutral [14] decayingbaryons.
1
In summary, from the early years of research we learned that following the analysis
ofSirlinofthevirtualRCinneutronbetadecay[4]andarmedwiththetheoremofLow
for the bremsstrahlung RC [5] one can show that the model-dependent contributions
to both corrections (introducing new form factors) contribute to orders (a /p )(q/M )n
1
with n = 2,3,..., while orders n = 0,1 lead to model-independent final expressions,
because their model dependence is absorbed into the already existing form factors.
The RC to BSD obtained to these latter orders are then suitable for model-independent
experimentalanalyses and arevalidtoan acceptable degreeofprecisioninthenear and
intermediatefuture: theywillbeusefulin BSD involvingheavy quarks.
RADIATIVE CORRECTIONS TO BSD: THE RECENT YEARS
There are six different charge assignments in BSD, namely, A B0(l n ), A0
− − l
B+(l n ), A+ B0(l+n ), A0 B (l+n ), A++ B+(l+n ), and→A+ B++(l n →).
− l − l l − l
→ → → →
In Ref. [15] it was shown that it is not necessary to calculate each case separately. The
final results of the last four cases can be directly obtained from the final results of the
firsttwocases. Thissavesaconsiderableamountofeffort,sinceonlythefirsttwocases
need be calculated in detail. Such detail requires the choice of specific kinematic situa-
tions.It istheDP that isnormallystudiedexperimentally.However,energy-momentum
conservationmayallowtodiscriminateeventswherephotonsareemittedcarryingaway
energy such that the BSD is placed outside the so called three-body region (TBR) of
the DP of non-radiative BSD. The events with those photons belong to what we have
referred to as the four-body region (FBR). In addition, when the initial baryon is polar-
ized the angular correlations between that polarization and the emitted baryon and the
emitted charged lepton involve different RC, and it is not possible to obtain the final
resultsofonecorrelationfromthefinalresultsoftheothercorrelation.Backinthemid-
dle 1990’s the systematic study of RC to BSD was complete to both orders n = 0 and
n=1 for unpolarized decaying baryons so it was time to tackle new problems by con-
sidering the polarization of either the decaying or emitted baryons. After some work,
the analysis was finished to order n = 0 for polarized decaying baryons, covering the
spin-final baryon momentum and spin-final charged lepton momentum angular corre-
lations [16, 17]. Those results were indeed useful for obtaining theoretical expressions
for the angular spin asymmetry coefficients of the emitted baryon and charged lepton,
respectively.Within our approximationsand after producing some numbers, our results
agreed wellwithothers already published[18].
Nowadaysourgoalhasbeenextendedtocoverboththespin-finalbaryonmomentum
and the spin-final charged lepton angular correlations to order n = 1 of both neutral
and negatively charged decaying baryons, restricted to the TBR. The former problem
has been already solved [19] whereas the latter is in progress. A further analysis will
take into account the FBR contribution to the RC [20] and also the polarization of the
emittedbaryon.Thiswillbedoneinthenearfuture.Letusmentionthatuptothisorder
ourresultswillbeuseful inhigh-precisionexperimentsinvolvingheavyquarks.
In the next sections we will describe how to apply our methods to get the RC to the
TBR of the spin-final baryon momentum angular correlation to order n = 1. Basically
weborrowsomerecentresultsofRef. [19]andanalyzethecaseofanegativelycharged
decayingbaryon inorderto illustratetheprocedure.
VIRTUAL RADIATIVE CORRECTIONS
Fordefiniteness, letus considertheBSD
A B+l+n , (1)
l
→
and let A denote a negatively charged baryon and B a neutral one, so that l represents
a negatively charged lepton and n its accompanying antineutrino. The four momenta
l
and masses of the particles involved in process (1) will be denoted by p = (E ,p ),
1 1 1
p2 = (E2,p2), l = (E,l), and pn = (En0,pn ), and by M1, M2, m, and mn , respectively
[19]. Further notation and conventions can be found in those references. We organize
our results to explicitly exhibit the angular correlation sˆ pˆ , where sˆ is the spin of A.
1 2 1
·
This choice of the kinematical variables yields the angular spin-asymmetry coefficient
oftheemittedbaryon,denoted hereafter bya .
B
Theuncorrected transitionamplitudeM forprocess (1)isgivenby
0
G
M0 = V [uB(p2)Wm (p1,p2)uA(p1)][ul(l)Om vn (pn )], (2)
√2
whereuA, uB, ul, and vn are theDiracspinorsofthecorrespondingparticlesand
qn qm
Wm (p1,p2) = f1(q2)g m + f2(q2)s mn + f3(q2)
M M
1 1
qn qm
+ g1(q2)g m +g2(q2)s mn +g3(q2) g5. (3)
M M
1 1
(cid:20) (cid:21)
Here Om = g m (1+g5), q p1 p2 is the four-momentum transfer, and fi(q2) and
≡ −
g(q2) are theconventionalvectorand axial-vectorformfactors, respectively,which are
i
assumedtobereal in thiswork.
Theobservableeffects ofspinpolarizationareanalyzed by thereplacement
u (p ) S (s )u (p ) (4)
A 1 1 A 1
→
where S (s ) = (1 g s )/2 is the spin projection operator, and the polarization four-
1 5 1
− 6
vector s satisfies s s = 1 and s p = 0. In the center-of-mass frame of A, s
1 1 1 1 1 1
· − ·
becomesthepurelyspatialunitvectorsˆ whichpointsalongthespindirection.
1
The virtual RC can be separated into a model-independent part M which is finite
v
and calculable and into a model-dependent one which contains the effects of the strong
interactions and the intermediate vector boson [6, 13, 14]. This model-dependence can
be absorbed into M through the definition of effective form factors, hereafter referred
0
toas f andg . Thedecay amplitudewithvirtualradiativecorrections M isgivenby
i′ ′i V
M =M +M , (5)
V ′0 v
where
a
M = M F +M F . (6)
v 2p 0 p1 ′
The model-independent functions F (cid:2)and F contain(cid:3)the terms to order O(a q/p M )
′ 1
[13] and they reduce respectively to f and f of Ref. [12], in the limit of vanishing
′
a q/p M . Thesecondterm inEq. (6)can alsobefound inthisreference.
1
At this point we can construct the DP with virtual RC by leaving E and E as
2
the relevant variables in the differential decay rate for process (1). After making the
replacement(4)in (5), squaringit,summingoverfinal spinstates,wehave
(cid:229) M 2 = 1 (cid:229) M 2 1 (cid:229) M(s) 2. (7)
| V| 2 | V′ | −2 | V |
spins spins spins
The spin-independent contribution to M in Eq. (7), denoted here by M , was ob-
V V′
tained to order O(a /p ) in Ref. [12] and later improved to order O(a q/p M ) in
1
Ref. [13], so we will borrow the latter result. We now focus here on the spin-dependent
part M(s) to this order of approximation along the lines of Ref. [16], where only terms
V
oforder O(a /p )wereretained. Thedifferentialdecay ratecan bewrittenas
dG V = dE2d(E2pd)W 52dj lM2mmn 21 (cid:229) |MV′ |2−21 (cid:229) |MV(s)|2 =dG V′ −dG V(s), (8)
" spins spins #
where dG corresponds to the differential decay rate with virtual RC of unpolarized
V′
baryons given by Eq. (9) of Ref. [13], except that we have chosen here, without loss of
generality,acoordinateframeinthecenter-of-mass frameofAwiththezaxisalongthe
three-momentumoftheemittedbaryon,whereasinRef.[13]thezaxiswaschosenalong
the three-momentum of the emitted charged lepton. Similarly dG (s) can be obtained by
V
standardtechniques.Thusthedecay ratewithvirtualRC is
a a
dG =dW A + (B F +B F ) sˆ pˆ A + (B F +B F ) , (9)
V ′0 p ′1 ′1′ ′ 1 2 ′0′ p ′2 ′2′ ′
− ·
n h io
where dW is a phase space factor and A , B , and B are given in Ref. [13] whereas A
′0 ′1 ′1′ ′0′
canbefoundinRef.[16].Theyalldependonthekinematicalvariablesandarequadratic
functions of the form factors. Equation (9) is the differential decay rate with virtual RC
to order O(a q/p M ). It is model-independent and contains an infrared divergent term
1
inF whichat anyrate willbecanceled when thebremsstrahlungRC are added.
When including terms of order O(a q/p M ) the model dependence of the RC shows
1
up.Forthevirtualpart,onecanhandlethisbyintroducingeffectiveformfactors, f and
i′
g, in theuncorrected amplitudeM , insuch away that[6]
′i 0
a a
f (q2,p l)= f (q2)+ a (p l), g (q2,p l)=g (q2)+ b (p l),
1′ + 1 p 1 + ′1 + 1 p 1 +
· · · ·
a a
f (q2,p l)= f (q2)+ a , g (q2,p l)=g (q2)+ b , (k=2,3)
k′ + k p k ′k + k p k
· ·
i.e., f and g haveanew dependence in theelectron and emittedbaryon energies other
1′ ′1
thantheonesintheq2 dependenceoftheoriginalformfactors;thiscanbeseenthrough
a and b , which are functions of the product p l = (p +p ) l. For the remaining
1 1 + 1 2
· ·
formfactors, withinourapproximations,a and b (k=2,3)are constant.
k k
BREMSSTRAHLUNG RADIATIVE CORRECTIONS
Inthissectionwe turntotheemissionofreal photonsintheprocess
A B+ℓ+n +g , (10)
l
→
where A, B, ℓ, and n denote the same particles as in the virtual case and g represents a
l
real photonwithfour-momentumk=(w ,k).
The process (10) itself is strictly speaking a four-body decay whose kinematically
allowedregionisthejoinedareaA+BdepictedinFig.1.Thedistinctionbetweenthese
tworegionshasimportantphysicalimplications.RegionAis delimitedby
Emin E Emax, m E E (11)
2 2 2 m
≤ ≤ ≤ ≤
whereE =(M2 M2+m2)/2M whereas regionB isdelimitedby
m 1 2 1
−
M E Emin, m E E (12)
2 2 2 b
≤ ≤ ≤ ≤
with E = [(M M )2+m2]/2(M M ). Finding an event with energies E and E
b 1 2 1 2 2
in region B dem−ands the existence o−f a fourth particle which must carry away finite
energy and momentum. In contrast, in region A this fourth particle may or may not do
so. Consequently, region B is exclusively a four-body region whereas region A is both
a three-body and a four-body region. We will refer to them as the four-body and three-
bodyregions(FBR and TBR), respectively.OuranalysisofthebremsstrahlungRCwill
considerprocess (10) restrictedto theTBR.
Thestartingpointwillbetoobtaintheamplitudeofprocess(10)withthespineffects
included, retaining terms of order O(a q/p M ) by following the theorem of Low [5].
1
Theamplitudeforprocess (10), M , isgivenin Ref. [13]and willnotberepeated here.
B
E
2
A
B
E
FIGURE 1. Kinematical region (A+B) as a function of E and E of the four-bodydecay (10). The
2
regionAcorrespondstowhatisreferredtoasthethree-bodyregioninthiswork.
ThebremsstrahlungcontributiontotheDPisobtainedfromthedifferentialdecayrate
dG B = M(22mp )m8n dE3p22dE3l dE3pn n d23wksp(cid:229)ins,e |MB|2d 4(p1−p2−l−pn −k), (13)
wherethesumextendsoverthespinsofthefinal particles andthephotonpolarization.
In analogyto thevirtualRC, thesubstitution(4)in(cid:229) M 2 ofEq. (13)leadsto
B
| |
(cid:229) M 2 = 1 (cid:229) M 2 1 (cid:229) M(s) 2, (14)
| B| 2 | ′B| −2 | B |
spins,e spins,e spins,e
andtherefore thedifferentialdecay ratedG is
B
dG =dG dG (s). (15)
B ′B− B
Except for minor changes, the quantity dG in Eq. (15) corresponds to the
′B
bremsstrahlung differential decay rate for unpolarized baryons given by Eq. (42)
of Ref. [13]. As for the spin-dependent term, dG (s), one can find further details about it
B
inRef. [19]. Theresult can becastintotheform
a
dG (s) = dW sˆ pˆ [B I (E,E )+C ], (16)
B p 1· 2 ′2 0 2 A
where I (E,E ) contains the infrared divergent term [16] which cancels the one of
0 2
its virtual counterpart and C in infrared convergent. C is presented in two forms in
A A
Ref. [19]. Thefirst oneisgiveninterms oftripleintegralsoverkinematicalvariablesof
thephotonand thesecond oneisfullyanalytic.
ThefullbremsstrahlungdifferentialdecayratedG isnowconstructedbysubtracting
B
dG (s) fromdG . ThisdG isadded todG toobtain
B ′B B V
a
dG =dW A A sˆ pˆ + [q q sˆ pˆ ] , (17)
′0 ′0′ 1 2 p I II 1 2
− · − ·
h i
wherethefunctionsq and q can befoundin Ref. [19].
I II
Expression (17) is model-independent and strictly only well-defined in the TBR of
thekinematicallyallowed region. Althoughthis equation is composed of rather lengthy
expressions,ithas been organizedin suchamannerthatis easy touse.
a
SPIN-ASYMMETRY COEFFICIENT
B
TheDP (17)so organizedallowsthecalculationofa , whichis defined as
B
N+ N
a =2 − −. (18)
B
N++N
−
Here N+ [N ] denotes the number of emitted baryons with momenta in the forward
−
[backward] hemisphere with respect to thepolarization of thedecaying baryon. Appro-
priateintegrationofEq. (17)leadsto
B +(a /p )A
a = 2 2, (19)
B −B +(a /p )A
1 1
where
B = Em E2maxA dE dE, A = Em E2maxq dE dE,
2 ′0′ 2 2 II 2
m Emin m Emin
Z Z 2 Z Z 2
B = Em E2maxA dE dE, A = Em E2maxq dE dE,
1 ′0 2 1 I 2
m Emin m Emin
Z Z 2 Z Z 2
andA , q ,A , andq are defined inRef. [19].
′0′ II ′0 I
Equation (19) is a model-independent analytic expression for a , including radiative
B
correctionstoorderO(a q/p M ).Theuncorrectedasymmetrycoefficienta 0 isobtained
1 B
bydroppingtheterms proportionalto a /p from thisequation.
We can evaluatea in order to makea comparisonwith previousworks [18, 16]. We
B
use definite values of the form factors in order to compare under the same quotations,
but this does not mean that our calculation is compromised to any particular values of
them. Therefore we use f = 1.27, g = 0.89, and f = 1.20 for the decay L pen ,
1 1 2
f =1, g = 0.34, and f = 0.97 for the decay S nen , and f = 0, g →= 0.60,
1 1 2 − 1 1
and f =1.17−fortheprocess S − L en . →
2 −
First, we can evaluate a (E,E→) [the same Eq. (19) without performing the double
B 2
integrals] in several points of the DP. This evaluation is presented in Table 1 for the
processS nen ,wherethefirstpartcorrespondstoa (E,E )toorderO(a /p )from
− B 2
Ref.[16],th→esecondpartisa (E,E )toorderO(a q/p M )fromthiswork,andthelast
B 2 1
partwas computedinRef. [18]and reproduced hereforcomparison.
TABLE 1. Percentage da (E,E ) with RC over the TBR in S nen decay
B 2 −
(a) to order O(a /p ) of Ref. [16]; (b) to order O(a q/p M ) of this →work; and (c)
1
computedinRef.[18].
s (a)
0.8067 0.5 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1
0.8043 50.7 0.3 0.1 0.1 0.0 0.0 0.0 0.0 0.1 0.3
0.8020 1.2 0.3 0.2 0.1 0.1 0.1 0.0 0.1
0.7997 5.4 0.7 0.3 0.2 0.1 0.1 0.1 0.1
0.7974 1.6 0.6 0.3 0.2 0.1 0.1 0.1
0.7951 4.4 1.1 0.5 0.3 0.2 0.1 0.1
0.7928 19.8 2.1 0.9 0.4 0.2 0.1
0.7904 5.2 1.5 0.7 0.3
0.7881 3.2 1.1 0.3
0.7858 9.8 2.2 0.2
(b)
0.8067 0.6 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1
0.8043 51.2 0.3 0.1 0.0 0.0 0.0 0.0 0.0 0.1 0.2
0.8020 1.2 0.3 0.1 0.1 0.0 0.1 0.1 0.1
0.7997 5.5 0.6 0.2 0.1 0.1 0.1 0.1 0.2
0.7974 1.4 0.4 0.2 0.1 0.1 0.1 0.2
0.7951 4.1 0.8 0.3 0.2 0.1 0.1 0.1
0.7928 18.5 1.7 0.6 0.3 0.2 0.1
0.7904 4.4 1.1 0.4 0.2 0.1
0.7881 2.4 0.7 0.2
0.7858 8.3 1.4 0.1
(c)
0.8067 0.6 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1
0.8044 50.7 0.3 0.1 0.0 0.0 0.0 0.0 0.0 0.1 0.2
0.8020 1.2 0.2 0.1 0.1 0.0 0.0 0.1 0.1
0.7997 5.4 0.6 0.2 0.1 0.1 0.1 0.1 0.2
0.7974 1.4 0.4 0.2 0.1 0.1 0.1 0.2
0.7951 4.0 0.8 0.3 0.2 0.1 0.1 0.1
0.7928 18.4 1.7 0.6 0.3 0.2 0.1
0.7904 4.4 1.0 0.4 0.2 0.1
0.7881 11.1 2.4 0.7 0.2
0.7858 8.2 1.4 0.1
d 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Next, we integrate numerically Eq. (19) to obtain a . This evaluation is displayed in
B
Table2,wheretheentriesofthesecondcolumncorrespondtoa 0,thefollowingcolumn
B
is the percentage differences defined as da = a a 0, and the last two columns are
B B− B
reservedto comparisonswithRefs. [16,18]. Thereis averygoodoverallagreement.
TABLE2. Valuesofa andcomparisonwithotherworks.
B
Decay a da (thiswork) da Ref.[16] da Ref.[18]
0 B B B
L pen 58.6 0.09 0.2 0.1
S → nen −66.7 −0.05 −0.1 −0.0
−
S →L en 7.2 0.08
−
→
DISCUSSION
Wehavepresentedashortreviewofthesituation(pastandpresent)oftheachievements
in the computation of RC in BSD developed by Mexican research groups. We have
shownhowtoapplythemtotheparticularcaseofthespin-asymmetrycoefficientofthe
emittedbaryon and comparedwithotherapproaches.
We can claim that the advancement in this topic has been important over the past
years so that our understanding on the subject is now clear. Our approach to compute
RC can be used in model-independent analyses for charm-baryon semileptonic decays
to a high degree of precision. Even for semileptonic decays of baryons containing two
charm quarks, they provide a good first approximation. Of course the problem is still
open.Moreworkis needed buttheapproach can beappliedstraightforwardly.
Finantialsupportfrom CONACYT andCOFAA-IPN (Mexico)is acknowledged.
REFERENCES
1. For an interesting historical review see for example T.-P. Cheng and L.-F. Lee, Gauge theory of
elementaryparticlephysics(OxfordUniversityPress,Oxford,1984)andreferencestherein.
2. A. García and P. Kielanowski, The Beta Decay of Baryons, Lecture Notes in Physics Vol. 222
(Springer-Verlag,Berlin,1985).
3. S.Eidelmanetal.[ParticleDataGroup],Phys.Lett.B5921(2004).
4. A.Sirlin,Phys.Rev.164,1767(1967).
5. F.E.Low,Phys.Rev.110,974(1958).
6. A.GarciaandS.R.JuarezW.,Phys.Rev.D22,1132(1980);22,2923(E)(1980).
7. T.Shann,NuovoCimento5A,591(1971).
8. A.GarciaandM.Maya,Phys.Rev.D171376(1978).
9. A.Garcia,Phys.Rev.D25,1348(1982).
10. S.R.JuarezW.,A.MartinezandA.Garcia,Phys.Rev.D35232(1987).
11. M.Bourquinetal.Z.Phys.C21,1(1983);S.Y.Hsuehetal.Phys.Rev.D382056(1988).
12. D.M.Tun,S.R.JuarezW.andA.Garcia,Phys.Rev.D40,2967(1989).
13. D.M.Tun,S.R.JuarezW.,andA.Garcia,Phys.Rev.D44,3589(1991).
14. A.Martinez,A.GarciaandD.M.Tun,Phys.Rev.D47,3984(1993).
15. A.Martinez,J.J.Torres,A.GarciaandR.Flores-Mendieta,Phys.Rev.D66,074014(2002).
16. R.Flores-Mendieta,A.Garcia,A.MartinezandJ.J.Torres,Phys.Rev.D55,5702(1997).
17. A.Martinez,J.J.Torres,R.Flores-MendietaandA.Garcia,Phys.Rev.D63,014025(2001).
18. F.GluckandK.Toth,Phys.Rev.D46,2090(1992).
19. M. Neri, A. Martinez, R. Flores-Mendieta, J. J. Torres and A. Garcia, Phys. Rev. D 72, 057503
(2005);72,079901(E)(2005).
20. S.R.JuarezW.,Phys.Rev.D53,3746(1996);55(1997)2889.
21. R. Flores-Mendieta, J. J. Torres, M. Neri, A. Martinez and A. Garcia, Phys. Rev. D 71, 034023
(2005);70,093012(2004).