Baryon vector and axial-vector currents in the 1/N_c expansion

0.17 MB·English

by  Raul Acosta

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0
0
2 BARYON VECTOR AND AXIAL-VECTOR CURRENTS IN
n THE 1/NC EXPANSION
a
J

5
2 RAU´LACOSTA ANDRUBE´N FLORES-MENDIETA
Instituto de F´ısica, Universidad Aut´onoma de San Luis Potos´ı,

1 A´lvaro Obreg´on 64, Zona Centro,
v
San Luis Potos´ı, S.L.P. 78000, Mexico
1
3
2
Thederivationofthe1/Nc expansionsforthebaryonvectorandaxial-vectorcur-
1
rents, in the SU(3) symmetry limit and including perturbative SU(3) breaking,
0
is reviewed. Symmetry breaking effects in the hyperon semileptonic decay form
5
factors arereanalyzed. Fits to experimental data yieldcorrections to the leading
0
vector and axial-vector form factors consistent with expectations. The values of
/
h theCabibbo-Kobayashi-MaskawamatrixelementsVudandVusalsoextractedfrom
p theanalysisarecomparabletotheonesrecommendedbytheParticleDataGroup.
-
p
e
h 1. Introduction
:
v
A powerful method which has been decisive in the understanding of low-
i
X energy QCD hadron dynamics is the 1/N expansion.1,2 This method pro-
c
r motesQCDto anSU(Nc)non-Abeliangaugetheory,whereNc isthe num-
a
berofcolors. InthelimitN ithasbeenshownthatthebaryonsector
c
→∞
possessesacontractedSU(2F)spin-favorsymmetry,whereF isthenumber
oflightquarkflavors.3,4 Large-N baryonsformirreduciblerepresentations
c
of the spin-flavor algebra5 and their static properties can be computed in
a systematic expansion in 1/N . Outstanding evidence of the predictions
c
of the 1/N expansion can be found in the analysis of baryon masses,5,6
c
magneticmoments,5,7,8 andaxial-vectorandvectorcurrents,5,7,9 whichare
in good agreement with the experimental data.
An interesting topic that can be tackled in the context of the 1/N
c
expansionistheevaluationofflavorSU(3)symmetrybreaking(SB)correc-
tions in the hyperon β-decay form factors. A deep understanding of these
correctionsisindeedimportantforaprecisedeterminationofthe Cabibbo-
Kobayashi-Maskawa (CKM) matrix elements V and V from hyperon
ud us
semileptonic decays (HSD). Currently, V can be precisely obtained from
ud
superallowed 0+ 0+ β decays whereas V is presumably more reliably
us
→
1
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extracted from K rather than HSD10 because of the larger theoretical
e3
uncertainties due to first-order SU(3) SB in the axial-vector form factors.
Here we come back to the study of HSD form factors in a combined
expansion in 1/N and SU(3) flavor SB by following the lines of previous
c
works,5,7,9,11 with the introduction of some variants. First (Sec. 2) we re-
view the constructionofthe baryonvectorandaxial-vectorcurrentswhose
matrix elements between SU(6) symmetric states yield the actual values
of the vector and axial-vector form factors at q2 = 0, namely f and g ,
i i
with i = 1,2. We also briefly review (Sec. 3) the existent information in
HSD. We then proceed (Sec. 4) to perform a comparison of the theoret-
ical expressions with the available experimental data for the decay rates,
angular correlations and angular spin-asymmetry coefficients of the octet
baryons, and for the widths (converted to axial-vector couplings by using
the Goldberger-Treiman relation) of the decuplet baryons. Finally we dis-
cuss our findings in a closing section (Sec. 5).
2. Operator analysis
Thetheoreticalgroundworkonthespin-flavorstructureoflarge-N baryons
c
has been established,5 so we willonly providea brief summaryofthe main
results here. For the analysis of HSD data, however, we follow the lines of
previous works.7,9,11
The 1/N expansion using quark operators as the operator basis12,13,5
c
providesaframeworkforstudyingthe spin-flavorstructureofbaryons. For
F =3, the lowestlying baryonstates fall into a representationof the spin-
flavor group SU(6). When N = 3, this corresponds to the familiar 56
c
dimensional representation of SU(6).
A complete set of operators can be constructed using the zero-body
operator I and the one-body operators5
σi
Ji = q† I q (1,1),
2 ⊗
(cid:18) (cid:19)
λa
Ta = q† I q (0,8), (1)
⊗ 2
(cid:18) (cid:19)
σi λa
Gia = q† q (1,8),
2 ⊗ 2
(cid:18) (cid:19)
where Ji are the baryon spin generators, Ta are the baryon flavor gener-
ators, and Gia are the baryon spin-flavor generators. The transformation
properties of these operators under SU(2) SU(3) are indicated as (j,d) in
×
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Eq. (1), where j is the spin and d is the dimension of the SU(3) flavor
representation.
Any QCD one-body operator transforming according to a given
SU(2) SU(3) representation has a 1/N expansion of the form5
c
×
Nc 1
= c , (2)
OQCD nNn−1On
n=0 c
X
where c are unknown coefficients which have power series expansions in
n
1/N beginningatorderunity. ThesuminEq.(2)extendsoverallpossible
c
independent n-body operators with the same spin and flavor quantum
n
O
numbers as . By using operator identities5 it is always possible to
QCD
O
reduce the operator basis to a set of independent operators.
We nowproceedto derivethe 1/N expansionsfor the HSD amplitudes
c
tofirstorderinflavorSB,andtoleadingorderin1/N formostoftheform
c
factors, except for f which is protected by the Ademollo-Gatto theorem
1
againstSU(3)breakingcorrectionstolowestorderin(m mˆ);thereforewe
s
−
needtoinclude second-orderflavorSBcorrectionsinf . Thecontributions
1
of f and g to the decay amplitudes are suppressed by the momentum
2 2
transfer. Inthesymmetrylimitthehyperonmassesaredegenerateandthen
such contributions vanish. Thus, the first-order SB to f and g actually
2 2
contribute to second order in the decay amplitude and can be neglected.
The axialformfactor g is computedto firstorderin SB. Finally, the form
1
factors f and g , for electron or positron emission, have contributions
3 3
proportional to the electron mass squared so we can safely neglect them.
2.1. Vector form factor f1
To begin with, let us write downthe 1/N expansionfor the baryonvector
c
currentin the SU(3) flavorsymmetry limit. Atq2 =0, the hyperonmatrix
elements for the vector current are given by the matrix elements of the
associatedSU(3)generator. LetV0a denotetheflavoroctetbaryoncharge9
λa
V0a = B′ qγ0 q B , (3)
* (cid:12)(cid:12)(cid:18) 2 (cid:19)QCD(cid:12)(cid:12) +
(cid:12) (cid:12)
whose matrix elements between(cid:12) SU(6) symmet(cid:12)ric states yield f1. In the
(cid:12) (cid:12)
above expression the subscript QCD emphasizes that q and q are QCD
quarkfields,notthequarkcreationandannihilationoperatorsofthequark
representation. V0a is spin-0 and a flavor octet, so it transforms as (0,8)
under SU(2) SU(3).
×
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The1/N expansionfora(0,8)operatorhasbeenalreadyobtained.6 By
c
using operator reduction rules it is found that only n-body operators with
a single factor of either Ta or Gia appear. The allowed one- and two-body
operators are given by
a =Ta, a = Ji,Gia ,
O1 O2 { }
and the remaining operators (n 3) are obtained as = J2, .
n+2 n
≥ O { O }
The 1/N expansion of V0a is then
c
Nc 1
V0a = c a. (4)
nNn−1On
n=1 c
X
At q2 =0 V0a is the generator of SU(3) symmetry transformations so
c =1, c =0, n>1.
1 n
Therefore, in the limit of exact SU(3) flavor symmetry one has
V0a =Ta, (5)
to all orders in the 1/N expansion.
c
2.2. Vector form factor with perturbative SU(3) breaking
Flavor SU(3) SB is due to the strange quark mass in QCD and transforms
asaflavoroctet. Constructingthemostgeneral1/N expansionforV0a up
c
to second-order in symmetry breaking requires to consider all the possible
spin-0 SU(2) SU(3) representations of the quark operators contained in
×
the SU(6) representations 1, 35, 405 and 2695 allowed by time reversal
invariance,i.e.(0,1),(0,8),(0,27),(0,64),and(0,10+10),sincethebaryon
1/N expansionextendsonlytothree-bodyoperatorsifwerestrictourselves
c
to physical baryon states. This problem has been already solved6 and the
results can be summarized as follows.
For a (0,1) operator the 1/N expansion starts with the zero-body
c
operator = I and the remaining operators are obtained as =
0 2m
O O
J2, , for m 1.
2m−2
{ O } ≥
The 1/N expansion for a (0,8) operator has the same form as Eq. (4)
c
whereas for a (0,27) operator the 1/N expansion contains the two- and
c
three-body operators
ab = Ta,Tb , ab = Ta, Ji,Gib + Tb, Ji,Gia ,
O2 { } O3 { { }} { { }}
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where the flavor singlet and octet components of the above operators are
subtracted off. For a (0,64) operator, the 1/N expansion starts with a
c
single three-body operator
abc = Ta, Tb,Tc ,
O3 { { }}
wherethe singlet,octetand27componentsaresubtractedofftoleaveonly
the 64 component. Finally, for a (0,10+10) operator one obtains
ab = Ta, Ji,Gib Tb, Ji,Gia .
O3 { { }}−{ { }}
First-order SB terms in V0a are given by setting one free flavor index
equal to 8 in the appropriate operators. At second-order in SB, two free
flavor indices are set equal to 8. One thus obtains9
1 1
V0a+δV0a =(1+ǫa )Ta+ǫa Ji,Gia +ǫa J2,Ta
1 2N { } 3N2{ }
c c
1
+ǫb dab8Tb+ǫb dab8 Ji,Gib
1 2
N { }
c
1 1
+ǫb dab8 J2,Tb +ǫa Ta,T8
3N2 { } 4N { }
c c
1
+ǫa Ta, Ji,Gi8 + T8, Ji,Gia
5N2 { { }} { { }}
c
1 (cid:0) (cid:1)
+ǫa Ta, Ji,Gi8 T8, Ji,Gia
6N2 { { }}−{ { }}
c
1 (cid:0) 1 (cid:1)
+ǫ2b dab8 Tb,T8 +ǫ2a Ta, T8,T8
4N { } 7N2{ { }}
c c
1
+ǫ2b dab8 Tb, Ji,Gi8 + T8, Ji,Gib
5N2 { { }} { { }}
c
1 (cid:0) (cid:1)
+ǫ2b dab8 Tb, Ji,Gi8 T8, Ji,Gib . (6)
6N2 { { }}−{ { }}
c
(cid:0) (cid:1)
Here ǫ is a measure of SU(3) breaking. Equation (6) has been written in
such a way that there is no SB for the ∆S = 0 weak decays, since isospin
symmetry is not broken by the strange quark mass.
Ifweintroducethenumberofstrangequarks,N ,andthestrangequark
s
spin, Ji, which are defined through5
s
1 1
T8 = (N 3N ), Gi8 = (Ji 3Ji),
2√3 c− s 2√3 − s
then Eq.(6) canbe given,after rearrangingterms andabsorbingfactorsof
N−1 and N−2, as
c c
V0a =Ta, (7)
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for ∆S =0 HSD, and
V0a =(1+v )Ta+v Ta,N +v Ta, I2+J2 , (8)
1 2{ s} 3{ − s}
for ∆S =1HSD.HereI istheisospin. ThebaryonsareeigenstatesofJ2,
| |
I2, J2, and N , so the matrix elements of Eq. (8) can be computed rather
s s
easily.9 Thus, for any process, the matrix elements of V0a are given as the
sumofthematrixelementsoftheoperatorsinvolvedintheexpansiontimes
their respective coefficient.
2.3. Axial-vector form factor g1
The1/N expansionfortheaxial-vectorcurrentAia,whosematrixelements
c
yield g , has been already obtained.5 It reads
1
1
Aia = aGia+bJiTa+∆a(c Gia+c JiTa)+c Gia,N +c Ta,Ji
2 1 2 3{ s} 4{ s}
1 d 1
+ δa8Wi ( J2,Gia Ji, Jj,Gja ), (9)
√3 − 2 { }− 2{ { }}
where
Wi =(c 2c )Ji+(c 2c )N Ji 3(c +c )N Ji,
4− 1 s 3− 2 s − 3 4 s s
and ∆a = 1 for a = 4,5,6, or 7 and vanishes otherwise. In order to
avoidthemixingbetweenSBeffectsand1/N correctionsinthesymmetric
c
couplingsatermproportionaltodisadded.7 ThiswillalsoallowtheSU(3)
symmetric parameters D, F, and to have arbitrary values.
C
For definiteness, in the present analysis Eq. (9) can be split into two
contributions, according to the physical processes we are concerned with.
Thus one has,
1 d 1
Aia =aGia+bJiTa ( J2,Gia Ji, Jj,Gja ), (10)
2 − 2 { }− 2{ { }}
for ∆S =0 processes, and
1
Aia =a′Gia+b′JiTa+c Gia,N +c Ta,Ji , (11)
2 3{ s} 4{ s}
for ∆S =1 processes.
| |
Inthefirstcasethecouplingshavebeenparametrizedinsuchawaythat
only the parameters a, b, and d contribute to strangeness-zero processes,
namely the ∆S =0 HSD andthe strong decays∆ Nπ, Ξ∗ Λπ, Σ∗
→ → →
Σπ,andΞ∗ Ξπ. Inthesecondcaseweignorethecontributionoftheterm
→
proportional to d, which results in redefinitions of the parameters a and b
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of Eq. (9) into a′ and b′, which absorbthe terms c and c , respectively, of
1 2
the original expansion. Equations (10) and (11) are the ones used in the
fits to the experimental data.
2.4. Weak magnetism form factor f2
InthelimitofexactSU(3)flavorsymmetrytheformfactorf isdetermined
2
by twoinvariants,m andm , whichcanbe extractedfromthe anomalous
1 2
magneticmomentsofthenucleons.7Themagneticmomentisaspin-1octet
operator so it has a 1/N expansion identical in structure to Aia. For
c
convenience m and m are defined through5,7
1 2
Mi =m GiQ+m JiTQ, (12)
1 2
where Q represents the SU(3) generator (the electric charge), so GiQ
≡
Gi3+Gi8/√3, and TQ T3+T8/√3.
≡
Previous works (see e.g. Ref. [9] and references therein) have shown
that reasonableshifts from the SU(3) predictions of f haveno perceptible
2
effects upon χ2 or g in a global fit to experimental data. We therefore
1
follow these works and determine f with the best fit values7 m = 2.87
2 1
and m = 0.77.
2
−
2.5. Weak electricity form factor g2
In the SU(3) flavor symmetry limit, the form factor g vanishes, so that it
2
is proportional to SU(3) SB at leading order. g transforms oppositely to
2
g and f under time-reversal, and therefore has a different 1/N operator
1 2 c
expansion.
Let Wia be the operatorwhose matrix elements yield g . At firstorder
2
in SU(3) SB, the contribution to g transforms as (1,8) and (1,10 10)
2
−
under spin and flavor. The (1,8) expansion reads9
JiTb
δWia ib fab8Gib+ib fab8 , (13)
8 ∝ 1 2 N
c
whereas the i(10 10) expansion is
−
δWia if8cgdach Gig,Th Tg,Gih . (14)
10−10 ∝ { }−{ }
For any process, the matrix elem(cid:0)ents of Wia can be gi(cid:1)ven as a sum of
the three parameters b times the operator matrix elements involved
1−3
in Eqs. (13) and (14). Previous analyses,9 however, have concluded that
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the HSD data are not accurate enough for an extraction of the small g -
2
dependenceofthedecayamplitudes. Inconsequence,wealsotakethevalue
g =0 in this work in order to have a consistent analysis.
2
3. HSD data
3.1. Integrated observables
The totaldecay rate R andangularcorrelationandasymmetrycoefficients
are some integrated observables in HSD which can be used when experi-
ments have low statistics and no analysis of the differential decay rate is
possible. These observables are defined using only kinematics and no par-
ticular theoretical approach is assumed in their definitions. The charged
lepton-neutrino angular correlation coefficient, for instance, is defined as
N(Θ <π/2) N(Θ >π/2)
ℓν ℓν
α =2 − , (15)
ℓν
N(Θ <π/2)+N(Θ >π/2)
ℓν ℓν
where N(Θ < π/2) [N(Θ > π/2)] is the number of charged lepton-
ℓν ℓν
neutrino pairs emitted in directions that make an angle between them
smaller[greater]thanπ/2. Similarexpressionsareobtainedforthecharged
lepton α , neutrino α , and emitted hyperon α asymmetry coefficients,
ℓ ν B
where Θ , Θ , and Θ are this time the angles between the ℓ, ν, and B
ℓ ν B 2
directions and the polarization of B , respectively. If the polarization of
1
the emitted hyperon is observed, two more asymmetry coefficients, A and
B, can be defined.
The unpolarizedtotaldecayrateR0 is a quadraticfunction ofthe form
factors and can be written in the most general form asa
6 6
R0 = aRf f + bR(f λ +f λ ). (16)
ij i j ij i fj j fi
i≤j=1 i≤j=1
X X
Here, for the sake of shortening Eq. (16), we have momentarily redefined
g = f , g =f , g = f , λ =λ , λ = λ , and λ =λ . Each sum
1 4 2 5 3 6 g1 f4 g2 f5 g3 f6
in Eq. (16) contains 21 terms due to the restriction i j. Although the
≤
formfactorshavebeenassumedtobeconstant,theirq2-dependencecannot
always be neglected because they may contribute to the observables. We
havealreadypointedoutthattheq2-dependenceoff andg canbeignored
2 2
becausetheyalreadycontributetoorder (q)tothedecayrate. Forf (q2)
1
O
aAsuperscript0onagivenobservableindicatesthatnoradiativecorrectionshavebeen
incorporatedintoit.
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and g (q2), however, non-negligible contributions can be obtained with a
1
linear expansion in q2, namely,
q2 q2
f (q2)=f (0)+ λf, g (q2)=g (0)+ λg,
1 1 M2 1 1 1 M2 1
1 1
where the slope parameters λf and λg are both of order unity.14
1 1
Similar expressions to Eq. (16) also hold for the products R0α0, where
α0 is any of the angular coefficients defined above. Once R0 and R0α0
are determined, α0 is obtained straightforwardly. Updated values of the
coefficients aR and aRα can be found elsewhere.11,15
ij ij
3.2. Experimental data in HSD
The measured quantities in HSD are the total decay rate R, angular cor-
relation coefficients α , and angular spin-asymmetry coefficients α , α ,
eν e ν
α , A, and B. Often the measured g /f ratios are also presented. This
B 1 1
latter set, however, is not as rich as the former and will not be used in
the currentanalysisunless noted otherwise. These experimentaldata10 for
HSD are listed in Tables 1 and 2. Besides, the experimentally measured
quantity forthe decupletbaryonsis the decaywidth convertedto anaxial-
vector coupling g for each decay using the Goldberger-Treiman relation.
Thisinformationcanbe foundinRefs.[7, 9]andwillnotbe repeatedhere.
Table1. Experimentaldataonthree∆S=0HSD.Theunits of
Rare10−3s−1forneutrondecayand106s−1fortheotherdecays.
np Σ+Λ Σ−Λ
R 1.1291 ±0.0010 0.249±0.062 0.387±0.018
αeν −0.0766 ±0.0036 −0.35 ±0.15 −0.404±0.044
αe −0.08559±0.00086
αν 0.983 ±0.004
A 0.07 ±0.07
B 0.85 ±0.07
g1/f1 1.2695 ±0.0029
4. Fits to experimental data
At this point we can proceed to perform detailed comparisons with the
experimental data of Tables 1 and 2 through a number of fits. The exper-
imental data which are used are the decay rates and the spin and angular
correlation coefficients of the HSD listed in these tables. The value of
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Table2. Experimentaldataonfive|∆S|=1HSD.TheunitsofRare106s−1.
Λp Σ−n Ξ−Λ Ξ−Σ0 Ξ0Σ+
R 3.161±0.058 6.88 ±0.24 3.44±0.19 0.53 ±0.10 0.93±0.14
αeν −0.019±0.013 0.347±0.024 0.53±0.10
αe 0.125±0.066 −0.519±0.104
αν 0.821±0.060 −0.230±0.061
αB −0.508±0.065 0.509±0.102
A 0.62±0.10
g1/f1 0.718±0.015 −0.340±0.017 0.25±0.05 1.287±0.158 1.32±0.22
the ratio g /f is not used since it is not an independent measurement.
1 1
For the processes Ξ− Σ0e−ν and Ξ0 Σ+e−ν , however, we do use
e e
→ →
g /f because no information on the angular coefficients is available yet.
1 1
The theoretical expressions for the integrated observables can be found in
Refs. [11, 15]. In orderto havea reliably determination ofV and V , we
ud us
systematically incorporateradiative correctionsinto the various integrated
observables and include the momentum-transfer contributions of the form
factors. As for the decuplet baryons,we use the axial coupling g.
Theparameterstobefittedarethosearisingoutofthe1/N expansions
c
of the baryon operators, namely, v for f and a, b, d, c for g . We
1−3 1 1−4 1
use the values of f and g in the limit of exact SU(3) flavor symmetry.
2 2
The matrix elements V and V are allowed to be free parameters too.
ud us
Hereafter, the quoted errors of the best fit parameters will be from the χ2
fit only, and will not include any theoretical uncertainties.
4.1. ∆S = 0 fit and the extraction of Vud
As stated above, we can proceed to make an investigation of the ∆S = 0
processes only, namely, n p and Σ± Λ semileptonic decays and the
→ →
strong decays ∆ Nπ, Ξ∗ Λπ, Σ∗ Σπ, and Ξ∗ Ξπ. We proceed
→ → → →
right away to evaluate first-order SB effects in g , which depends on the
1
parameters a, b, d, c and c , as introduced in Eq. (10). The fit produces
3 4
a = 0.89 0.02, b = 0.21 0.04, d = 0.06 0.01, c = 0.08 0.01,
3
± − ± − ± − ±
c =0.01 0.01, and
4
±
V =0.9741 0.0005, (17)
ud
±
with χ2 = 8.62 for 5 degrees of freedom. The results are consistent with
expectations. The leading parameter a is of order unity, b 1/N is small
c
∼
compared to a, d 1/N2 is rather small and c ,c ǫ/N are consistent
∼ c 3 4 ∼ c
with first-order breaking ǫ divided by N .
c