Charm meson resonances in $D \to P \ell ν$ decays

0.22 MB·

by  S. Fajfer

Charm meson resonances in D Pℓν decays
→
Svjetlana Fajfer1,2, and Jernej Kamenik1,
∗ †
1J. Stefan Institute, Jamova 39, P. O. Box 3000, 1001 Ljubljana, Slovenia
2Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
(Dated: February 2, 2008)
Motivated by recent experimental results we reconsider semileptonic D Pℓνℓ decays within a
→
model which combines heavy quarksymmetry and properties of the chiral Lagrangian. We include
excitedcharmmesonstates,someofthemrecentlyobserved,inourLagrangiananddeterminetheir
impact on thecharm meson semileptonic form factors. Wefindthat theinclusion of excited charm
mesonstatesinthemodelleadstoarathergood agreementwiththeexperimentalresultsontheq2
shape of theF+(q2) form factor. Wealso calculate branchingratios for all D Pℓνℓ decays.
→
PACSnumbers: 13.20.Fc,12.39.Hg,12.39.Fe,14.40.Lb
5
0
0
I. INTRODUCTION nificance [4].
2
Recently new experimental studies of charm meson

n resonances have provided a lot of new information on
a The knowledge of the form factors which describe the
the charm sector. Several experiments on open charm
J weak heavy light semileptonic transitions is very im-
→ hadrons have reported very interesting discoveries: First
5 portant for the accurate determination of the CKM pa-
BaBar[20]announcedanew,narrowmesonD (2317)+.
2 rametersfromtheexperimentallymeasuredexclusivede- sJ
ThiswasconfirmedbyFocus[21]andCLEO[22]whoalso
3 cayrates. Alackofpreciseinformationabouttheshapes observedanothernarrowstate,DsJ(2463)+. Bothstates
of various form factors is thus still the main source of
v were confirmed by Belle [23] who also provided first evi-
uncertainties especially in these processes.
0 dencefortwonew,broadstatesD (2308)andD (2427),
0∗ 1′
4 In addition to many studies of exclusive B-meson both ca. 350 400 MeV higher above the usual D0, D
∗
1 −
semileptonic decays there are new interesting results on states and with opposite parity. Finally, Selex [24] has
2
D-meson semileptonic decays. The CLEO and FOCUS announced a new, surprisingly narrow state D+ (2632)
1 sJ
h/04 ciπno−flolℓar+bmνoaratanitodinoDnosn0h→thaveKeD−s0tℓu→+dνiπe[d1−,ℓs2+e].mνTialnehpdetirDond0iac→tadpeKcrao−yvℓsi+dνDesf0onre→mw wabneitrdhsDotfhsJeth(s2ep4i6(n03+)p+,a1r+hita)yvseapsiasnlirgednaomdueybnltebte1ec−nh.irpaBrlooppthoasreDtdnseJar(ss2o3mf1e7tm)h+e-
factors. Usually in D semileptonic decays a simple pole heavy-light pseudoscalar and vector D mesons [25, 26],
p s
- parametrization has been used in the past. The results while the states D0∗(2308) and D1′(2427) have also been
p of Refs. [1, 2] for the single pole parameters required by proposedas chiralpartners of the D-meson doublet [26].
e
the fitoftheirdata,however,suggestpolemasses,which On the other hand, a proposal has been put forward
h
are inconsistent with the physical masses of the lowest for the D+ (2632) state as the first radial excitation of
: sJ
v lyingcharmmesonresonances. Intheiranlysesthey also the D (2112)[27, 28]. While strongandelectromagnetic
s∗
Xi utilized a modified pole fit as suggested in [3] and their transitions of these new states have already been stud-
results indeed suggest the existence of contributions be- ied[25],theyhavenotyetbeenappliedtothedescription
r
a yond the lowest lying charm meson resonances [1]. ofweakdecaysofcharmedmesons. Specificallyinthede-
scription of heavy to light weak transition form factors
There exist many theoretical calculations describing
they are yet to be taken into account as possible transi-
semileptonicdecaysofheavytolightmeson: quarkmod-
tion resonances.
els (QM) [4, 5, 6, 7, 8, 9], QCD sum rules (SR) [10, 11,
The purpose of this paper is to investigate contribu-
12, 13, 14, 15], lattice QCD [16, 17] and a few attempts
tions ofthe newly discoveredandtheoretically predicted
tousecombinedheavymesonandchiralLagrangianthe-
charmmesonswithinaneffectivemodelbasedonHMχT
ory (HMχT) [18, 19]. Most of the above methods have
byincorporatingthenewlydiscoveredheavymesonfields
limited range of applicability. For example, QCD sum
rules are suitable only for describing the low q2 region into the HMχT Lagrangian. In this paper we only fo-
cus onto the D Pℓν transitions, since available ex-
while lattice QCD and HMχT give good results only for ℓ
→
the high q2 region. However, the quark models, which perimental data can be used in determination of all the
do provide the full q2 range of the form factors, can- required parameters. We restrain our discussion to the
leading chiral and 1/m terms in the expansion, but we
noteasilybe relatedtotheQCDLagrangianandrequire H
hope to capture the main physical features about the
input parameters,which may not be of fundamental sig-
impact of the nearest poles in the t-channel to the q2-
dependence of the form factors.
Within the context of heavy to light transitions the
∗ElectronicAddress:[email protected] HMχT is only valid at small recoil momentum (large
†ElectronicAddress:[email protected] q2) and in order to predict the whole q2 region depen-
2
dence, the form factors have to be modeled with a pre- odd parity fields:
sumed pole ansatz. After including excited charm vec-
tor meson states we notice that their presence implies a L′int = ihhGbγµγ5AµbaH¯ai+h.c.. (3)
natural appearance of the double pole shape for the F
+
Finally we include the radially excited states into our
formfactor. Assuming suchform factor shape we obtain
good agreement with the observed q2 dependence of F discussion by introducing another odd parity heavy me-
in D0 K/π transitions as well as with the measure+d sonmultipletfieldH′ =1/2(1+/v)[Pµ′∗γµ−P′γ5]contain-
→ ing the radial excitations of ground state pseudoscalar
branching ratios for these decays.
We compare our results for the form factor q2 depen- and vector mesons. Such excited states were predicted
in Ref. [31]. The strong interactions between these fields
dence with existing experimental [1, 2, 29] and lattice
and ground state meson fields H can again be described
QCD [17] results, as well as results of other theoreti-
by the lowest order interaction Lagrangian analogous
cal studies [4, 9]. We complete our study by calculating
to (3)
branching ratios for all D Pℓν (P = π, K, η, η )
ℓ ′
→
transitions.
˜ = ig˜ H γ γ µ H¯ +h.c.. (4)
Sec. II describes the framework we use in our calcu- Lint h b′ µ 5Aba ai
lations: first we write down the HMχT Lagrangian for
heavyandlightmesonsandextendittoincorporatenew
B. Weak interactions
heavy meson fields. In Sec. III we study q2 behavior of
the form factors in D Pℓν decays. Finally, a short
ℓ
summary of the results→and comparisonwith experimen- For the semileptonic decays the weak Lagrangian can
tal data is given in Sec. IV. be given by the effective current-current Fermi interac-
tion
G
II. THE FRAMEWORK = F ℓ¯γµ(1 γ5)ν , (5)
eff ℓ µ
L −√2 − J
(cid:2) (cid:3)
A. Strong interactions where G is the Fermi constant and is the effec-
F µ
J
tive hadronic current. In heavy to light meson decays
Interactions, relevant for our study, between odd par- it can be written as = K Ja, where constants Ka
Jµ a µ
ity heavy mesons and light pseudoscalar mesons are parametrize the SU(3) flavor mixing, while the leading
described by the leading order interaction Lagrangian, order weak current Ja in 1/m (where m is the heavy
µ H H
namely (see e.g. [18, 19]) meson mass) and chiral expansion can be written as
= ig H γ γ µ H¯ , (1) 1
Lint h b µ 5Aba ai Jaµ = 2iαhγµ(1−γ5)Hbξb†ai (6)
withH =1/2(1+/v)[P γµ Pγ ],thematrixrepresenta-
µ∗ − 5
tion of the heavy meson fields, where Pµ∗ and P are cre- for the 0− and 1− heavy mesons, while similarly for the
ation operators for heavy-light vector and pseudo scalar 0+ and 1+ states we write [19]
mesons respectively. Light pseudoscalarmeson fields are
egnocldosdteodneinflaΣvo=r meaxtpr(ix2iM/f) where M is the pseudo- Ja′µ = 12iα′hγµ(1−γ5)Gbξb†ai. (7)
1 (η +π0) π+ K+ Finally for the radially excited pseudoscalar and vector
√2 q
= π 1 (η π0) K0 . (2) fields Ja can be written as:
M  − √2 q− 
 K− K¯0 ηs  J˜aµ = 12iα˜hγµ(1−γ5)Hb′ξb†ai. (8)
To describe η η mixing we follow the work of Ref. [30],
′
where η = η cosφ η sinφ and η = η sinφ+
q s ′ q
| i | i −| i | i | i
ηs cosφ, and φ is the mixing angle between the flavor III. FORM FACTOR CALCULATION
| i
andmasseigenstates. Thelightpseudoscalarmesonaxial
current is defined as = 1/2(ξ ∂ ξ ξ∂ ξ ), where
ξ2 = Σ so that Aµ(i/f)∂ †. µFu−rtherµm†ore, ... Nowweturntothediscussionoftheformfactorq2 dis-
Aµ ≈ µM h i tribution. The H P currentmatrix elementis usually
indicate a trace over spin matrices and summation over →
parametrized as
light quark flavor indices.
In order to incorporate positive parity heavy meson P(p )(V A)µ H(p )
P H
statesintothemodel,weintroducethescalar-axialvector h | − | i
m2 m2
fields doublet G = 1/2(1+/v)[Sµ∗γµγ5−S] representing =F+(q2) (pH +pP)µ− Hq−2 Pqµ
axial-vector (Sµ∗) and scalar (S) mesons and incorporate (cid:18) (cid:19)
it into the interaction Lagrangian by adding an addi- m2 m2
+F (q2) H − Pqµ, (9)
tional leading order interaction term between even and 0 q2
3
where (V A) is the weaklefthanded quark currentand thus get
−
q =(p p )istheexchangedmomentum. HereF de-
notes tHhe−vePctor form factor and it is dominated by+vec- P(p )Jµ H(v) = αvµ
P
h | | i −f
tor meson resonances, while F denotes the scalar form
0
factor and is expected to be dominated by scalar meson +αgvµv·pP −pµP
resonance exchange [5, 32]. In order that these matrix f v pP +∆H∗
elements are finite at q2 = 0, the form factors must also α˜ vµ·v p pµ α vµv p
satisfy the relation + g˜ · P − P + ′h · P . (15)
f v·pP +∆H′∗ f v·pP +∆HS
F (0)=F (0) . (10) Weapplytheprojectorsv andv v p p oneq. (15)
+ 0 µ µ P Pµ
· −
and extract the form factors F (q2) and F (q2) at q2
+ 0 max
using Eqs. (12), (14a) and (14b):
In Ref. [33] it was pointed out that in the limit of a
static heavy meson α m
F (q2 ) = g H
1 + max −2√mHf mP +∆H∗
H(v) = lim H(p ) (11) α˜ m
| iHQET mH→∞√mH| H i −2√mHfg˜mP +H∆H′∗, (16)
one can use the following decomposition:
and
hP=(pP[p)µ|(V (−vAp)µ|)Hvµ(]vf)iH(vQEpT ) F0(qm2ax) = −√mαHf + √mα′HfhmPm+P∆HS. (17)
P − · P p · P
+vµfv(v pP), (12) If one uses directly relation (9) instead of this extrac-
·
tionofformfactorsatlargeq2 [33]oneendsupwiththe
max
where the form factors f are functions of the variable mixed leading √m terms and the subleading 1/√m
p,v H H
terms. Furthermore, the scalar meson contribution ap-
m2 +m2 q2 pears in the F+ form factor. The extractionof form fac-
v·pP = H 2mP − . (13) torswe followhere[33] givesa correct1/mH behaviorof
H
the form factors and the contributions of 1 resonances
−
enterinF ,while0+ resonancescontributetoF asthey
The form factors F given in (9) and the form fac- + 0
+,0 must[32]. InprevioustheoreticalstudiesoftheD meson
tors f (v p ) are related to each other by matching
P,v · P form factors [19] a single pole parametrization has been
QCD to Heavy quark effective theory (HQET) at the
used when extrapolating from F (q2 ). In such calcu-
scaleµ m (seeEq. (14)ofRef.[33]). As in[33]wefix + max
the mat≃chincg constants to their tree level values. This lations only the D∗ resonance contributed, and a lower
valueofthegstrongcouplingwasused. Atthattimeonly
approach immediately accounts for the F behavior at
+,0 few decay rates were measured. In comparison with the
q2 . Attheleadingorderinheavyquarkexpansion,the
max presentexperimentaldatathepredictedbranchingratios
two definitions are then related near zero recoil momen-
were too large. The approach of Ref. [34] was developed
tum (q2 q2 =(m m )2 or equivalently p~ 0)
≃ max H− P | P|≃ to treat D meson semileptonic decay within heavy light
as
meson symmetries in the allowed kinematical region by
1 using the full propagators. We find that this approach
F0(q2)|q2≈qm2ax = √mHfv(v·pP) (14a) fcaacntnoorst.reproduce the observed q2 shape of the F+ form
F (q2) = √mHf (v p ), (14b) Inourstudyofformfactors’q2 distributions,wefollow
+ |q2≈qm2ax 2 p · P theanalysisofRef.[3],wheretheF+ formfactorisgiven
as a sum of two pole contributions, while the F form
0
Note, however,that the F+ andF0 formfactorsmight factor is written as a single pole. This parametrization
contain 1/mc corrections [33], which we do not consider includesallknownpropertiesofformfactorsatlargemH
within present approach. andtheQCDsumrulesforlowq2 region. Theirproposal
We use Feynman rules for HMχT coming from the for the form factor parametrization is
strong and weak Lagrangians described in the previ-
1 a
ous section. For the heavy meson propagators we use F (q2) = c (18a)
+ B
iδ /2(v k ∆) and iδ (g v v )/2(v k ∆) 1 x − 1 x/γ
ab · − − ab µν − µ ν · − (cid:18) − − (cid:19)
for the pseudoscalar(scalar) and vector (axial) mesons c (1 a)
respectively, where kµ = qµ m vµ and ∆ = ∆ is F0(q2) = B − , (18b)
H R 1 bx
−
themasssplittingbetweentheheavyresonancemesonR −
and the groundstate heavypseudoscalarmeson(see e.g. where x = q2/m2 . Using the relation which connects
H∗
Ref. [19]). For the hadronic current matrix element we theformfactorswithinlargeenergyreleaseapproach [35]
4
the authors in Ref. [3] then find a=1/γ and so obtain a mD′ =2.3 GeV from Ref. [31].
simplified double pole function for the F form factor In our calculations we use for the heavy meson weak
+
current coupling α = f √m [33], which we calculate
H H
F (q2)= F+(0) (19) from the lattice QCD value of fD = 0.225 GeV [38] and
+
(1 x)(1 ax) experimental D meson mass m = 1.87 GeV [39] yield-
− − D
ing α = 0.33 GeV3/2. For light pseudoscalar mesons we
Although the D mesons might not be considered heavy
use f = 130 MeV, while for the η η mixing angle we
′
enough,weemploytheseformulaswiththemodelmatch- −
use the value of φ 40 [30]. For the g-coupling we
◦
ingconditionatq2 . Atthesametimewefixtheparam- ≃
max use the experimentally determined value of g =0.59 [40]
etersaandbinEqs.(18b)and(19)bythenext-to-nearest
while for the h coupling we use the value of h = 0.6
resonances. We use physicalpole masses of excited state −
from the global estimate of Ref. [41]. We can also de-
charmed mesons in this extrapolation, giving for the ra-
rive a constraint on the absolute value of g˜ from the
tiosa=m2H′∗/m2H∗ andb=m2HS/m2H∗. Althoughinthe Selex bound on the DsJ(2632) D0K+ decay rate of
originalidea[3]theextrapoleinF+ parametrizedallthe Γ(DsJ(2632) D0K+) < 17 M→eV [24]. In HMχT this
neglectedhigherresonances,weareheresaturatingthose decay rate ca→n be derived directly from the interaction
by a single nearest resonance. Lagrangian (4) and reads Γ = 1/6π[g˜/f]2 p~ 3 which
K
In our numerical analyses we use available experimen- gives for the coupling constant g˜ <0.21. | |
talinformationandtheoreticalpredictionsoncharmme- Eqs. (18a) (18b) and (19) can| |be combined to obtain
son resonances. Particularly for the scalar resonance a theoretical estimate for the value of g˜α˜ in the limit of
we use the DsJ(2317)+ state with mass mDsJ(2317)+ = infinite heavy meson mass. By equating both terms in
2.317 GeV. For the radially excited vector resonance Eqs. (16) and (18a) at q2 and then imposing a = 1/γ
we then have the Selex Ds+J(2632) state with mass one obtains max
m = 2.632 GeV. It is important to note, how-
evDers+,J(t2h6a32t)so-far the Selex discovery has not been con- α˜g˜ αgmP +∆H′∗ (mH −mP)2−m2H∗ . (20)
firmed by any other searches. In D decays, the situa′- ∼− mP +∆H∗ × (mH −mP)2−m2H′∗
tion is similarly ambiguous. Although the vector D
∗
resonance was discovered by Delphi [36] with a mass We apply this formula to the D π transitions where
→
of mD′∗ = 2.637 GeV and spin-parity 1−, its existence the chiral symmetry breaking corrections are smallest
was not confirmed by other searches [31]. On the other and thus the results most reliable. This yields α˜g˜
∼
hand recent theoretical studies [31, 37] indicate that 0.15 GeV3/2. On the other hand the 1/m and chi-
D
−
both radially excited states of D as well as Ds should ralcorrectionsmightstillmodify thisresultsignificantly.
have slightly larger masses of mD′∗ ≃ 2.7 GeV and Such corrections were explicitely written out in Ref. [42]
mDs′∗ ≃ 2.8 GeV [31]. We use these theoretically pre- but they include additional parameters which cannot be
dicted values in our analysis. Even less clear is the sit- fixed within this context.
uation with the scalar resonance (D ), which was pre- Similarly, in this limit we can infer on the value of
′
dicted in Ref. [31] but not yet confirmed by experiment. α. By applying equalities (10) and (20) to Eqs. (17)
′
Here we use the theoretically predicted mass value of and (18b) we obtain
α α mP +∆HS 1 g mH 1 m2H∗ 1− (mHm−2Hm∗P)2 , (21)
′ ∼ h × mP  − 2 × mP +∆H∗ − m2H′∗!1− (mHm−2HmSP)2
 
which gives, when applied to the D π transitions a does not contribute. For the decay width we get [34]:
→
value of α 0.69 GeV3/2.
′
∼−
G2m2 K 2 ym
Γ= F H| HP| dy F (m2 y)2 p~ (y)3, (22)
24π3 | + H | | P |
Alternatively the values of the new model parameters Z0
can be determined by fitting the model predictions to
known experimental values of branching ratios (D0 where y =q2/m2 , so that
K ℓ+ν), (D+ K¯0ℓ+ν), (D0 π ℓ+ν), B(D+ → H
− −
π0ℓ+ν), B(D+ →ηℓ+ν) andB (D+→ η ℓ+ν)B[39]. →In
our decayBwidsth→calculations wBe shsall→negl′ect the lepton mP 2
y = 1 (23)
mass, so the form factor F0, which is proportionalto qµ, m (cid:18) − mH(cid:19)
5
goodagreementwith experimentalresults. This discrep-
TABLE I: The pole mesons and the flavor mixing constants
ancy further increases if only the first resonance contri-
KHP for theD→P semileptonic decays. bution is kept in the F+(qm2ax) calculation for the single
H P H∗ H′∗ HS KHP poleextrapolationwithinHMχTasdoneinpreviouscal-
D0 K− Ds∗+ Ds′∗+ DsJ(2317)+ Vcs culations [19]. Note also that the experimental fits on
D+ K¯0 Ds∗+ Ds′∗+ DsJ(2317)+ Vcs tDh0e singπle p(Dol0e paraKme)tritzraatnisointioonfsthdeoFne+ifnorRmeffsa.ct[o1r, i2n]
Ds+ η Ds∗+ Ds′∗+ DsJ(2317)+ Vcssinφ yield→ed e−ffective→pole−masses which are somewhat lower
Ds+ η′ Ds∗+ Ds′∗+ DsJ(2317)+ Vcscosφ thanthephysicalmassesoftheD (D )mesonresonances
D0 π− D∗+ D′∗+ D′+ Vcd used in this analysis. ∗ s∗
D+ π0 D∗+ D′∗+ D′+ Vcd/√2 WealsocompareourpredictionsfortheF scalarform
0
D+ η D∗+ D′∗+ D′+ Vcdcosφ/√2 factor q2 dependence for the D0 K and D0 π
− −
D+ η′ D∗+ D′∗+ D′+ Vcdsinφ/√2 transitionswiththoseofquarkmo→delinRef.[4]and→with
Ds+ K0 D∗+ D′∗+ D′+ Vcd lattice QCDpole fitanalysisofRef.[17]. The resultsare
depicted in FIG. 3 and FIG. 4. Note that without the
scalar resonance, one only gets a contribution from the
α/√m f term from Eq. (17). This gives for the q2 de-
and H
pendence of F a constant value F (q2) = 1.81 for both
0 0
p~ (y)2 = [m2H(1−y)+m2P]2 m2. (24) D0 → π− and D0 → K− transitions, which largely dis-
| P | 4m2 − P agreeswithlatticeQCDresultsaswellasheavilyviolates
H relation (10).
The constants K parametrize the flavor mixing rele- Finally, using numericalvalues as explained above,we
HP
vant to a particular transition, and are given in Table I calculatethe branchingratiosforallthe relevantD P
→
together with the pole mesons. semileptonic decays and compare the predictions of our
Wecalculatetheresultforg˜α˜ byaweightedaverageof model with various other model predictions found in the
valuesobtainedfromallthemeasureddecayratestaking literature, and with experimental data from PDG. The
account for the experimental uncertainties. The calcula- results are summarized in Table II. For comparison we
tion yields α˜g˜ = 0.0050 GeV3/2, which is rather small also include the results for the rates obtained with our
compared to estim−ation given by (20). This discrepancy approach for F (q2 ) (Eq. 16) but using a single pole
+ max
can be attributed to the presence of 1/m and chiral fit. It is very interesting that our model extrapolated
D
corrections which are not included systematically into with a double pole gives branching ratios for D Pℓν
ℓ
→
consideration here due too many new parameters [42] in rather good agreement with experimental results for
which cannot be fixed within this approach. However, the already measured decay rates, while the predictions
we estimate the influence of such corrections on the fit- for the unmeasured decay rates agree with results of ex-
ted value of α˜g˜ by varying the value of αg/f in Eq. (16) isting approaches. It is also obvious that the single pole
by 10% [43] and inspecting the fit results. We obtain a fit gives the rates up to a factor of two larger than the
range of α˜g˜ [ 0.05,0.04]GeV3/2. experimentalresults. Onlyfordecaystoηandη′asgiven
Knowing α∈˜g˜−we can further infer on the value of α. in Table II, an agreementwith experiment of the double
′
The equality(10),whenappliedto the samedecayswith pole version of the model is not better but worse than
α˜g˜ = 0.0050 GeV3/2, gives an average value of α = for the single pole case.
′
0.47−GeV3/2, which is close to the estimate from (21). We also calculate branching ratios within our model,
−
We use these values for α˜g˜ and α in all our subsequent when the value of α˜g˜ is estimated from Eq. (20) rather
′
analyses. then from the branching ratio fit. While the D π
We next draw the q2 dependence of the F form fac- decay rates remain in reasonableagreementwith ex→peri-
+
tors for the D0 K and D0 π transitions and mentalvalues,predictionsforotherdecayratescomeout
− −
→ →
compare it with results of quark models in Refs. [4, 9], upto30%lower. Thisagainindicateslargecontributions
lattice QCD double pole fit analysis [17], as well as the from 1/m and especially chiral corrections in these de-
D
experimental results of a double pole fit from CLEO [1] cayamplitudes. We alsoattempt using experimentalfits
and FOCUS [2] with F (0) values taken from Ref. [29]. for the parameter a in Eq.(19) instead of saturating the
+
The results are depicted in FIG. 1 and FIG. 2. For second pole by physicalpole masses a=m2 /m2 . By
H∗ H′∗
comparison we also plot results when single pole fit is using the experimentally fitted values of a = 0.36(0.37)
used. Also in this case we calculate F (q2 ) within for D K(D π) decays from Ref. [1] we notice that,
+ max → →
HMχTandtakeintoaccountbothresonances(Eq.(16)). while the fitted values of α˜g˜ tend to shift towards larger
From the plots it becomes apparent, that our model’s negative values, the overall goodness of fit on the ex-
predictions for both D0 K and D0 π tran- perimentalbranchingratiosandconsequentlydecayrate
− −
→ →
sition F form factors are in good agreement with ex- predictions of our model remain almost the same. This
+
perimental and lattice data when extrapolated with a indicatesthattheactualpositionsofthepoles(massesof
double pole, while single pole extrapolations are not in excited charmed resonances) are not very volatile input
6
parameters of our model. On the other hand corrections obtained q2 dependence of the form factors is in good
of the order 1/m , as well as chiral corrections in the agreement with recent experimental results and the lat-
D
case ofD Kℓν andD η(η )ℓν mightimprovethe tice calculation for D0 K (π )ℓν . The calculated
ℓ s ′ ℓ − − ℓ
→ → →
agreement with the experimental data, but due to the branching ratios are close to the experimental ones.
presence of a large number of new couplings it is impos- Our study shows that the single physical pole
sible to include them into the calculation within present parametrizationcannotexplaintheq2 distributionofthe
framework. We expect that the errors in the predicted semileptonic form factors. The calculated rates for this
decays rates stemming from the uncertainties in the in- form factor parametrization are not in good agreement
put parameters we used can be 30%. with the experimental results.
IV. SUMMARY Acknowledgments
We have investigated semileptonic form factors for We thank D. Be´cirevi´c for many very fruitful discus-
D P decayswithinanapproachwhichcombinesheavy sions and his comments on the manuscript. We would
→
mesonandchiralsymmetry. Thecontributionsofexcited also like to thank D. Y. Kim and A. Khodjamirian for
charmmesonstates areincluded into analysis. The dou- their comments. This work is supported in part by the
blepolebehavioroftheF formfactorsisaresultofthe MinistryofEducation,ScienceandSportoftheRepublic
+
presence of the two 1 charm meson resonances. The of Slovenia.
−
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8
F+Hq2L
1.6
ThisworkHdoublepoleL
ThisworkHsinglepoleL
QM@4D
1.4 QM@9D
Latt.@17DHdoublepoleL
Expt.@1DHdoublepoleL
1.2 Expt.@2DHdoublepoleL
1
0.8
0 0.25 0.5 0.75 1 1.25 1.5 1.75
q2@GeVD
FIG. 1: Comparison of the D0 K− transition F+ form
factor q2 dependence of our mo→del double pole extrapola-
tion(thicksolid(black)line),singlepoleextrapolation(thick
dashed (black) line), quark model of Ref. [4] (thick dotted
(magenta)line),quarkmodelofRef.[9](thindotted(purple)
line), lattice QCD fitted to a double pole [17] (dot-dashed
(blue) line) and experimental double pole fits [1, 2] (thin
(green) solid and dashed lines).
F+Hq2L
ThisworkHdoublepoleL
ThisworkHsinglepoleL
QM@4D
3
QM@9D
Latt.@17DHdoublepoleL
Expt.@1DHdoublepoleL
2
1
0
0 0.5 1 1.5 2 2.5 3
q2@GeVD
FIG. 2: Comparison of D0 π− transition F+ form factor
q2 dependenceof our model→doublepole extrapolation (thick
solid (black) line), single pole extrapolation (thick dashed
(black)line),quarkmodelofRef.[4](thickdotted(magenta)
line), quark model of Ref. [9] (thin dotted (purple) line), lat-
ticeQCDfittedtoadoublepole[17](dot-dashed(blue)line)
and experimental double pole fit [1] (thin solid (green) line).
9
F Hq2L
0
ThisworkHpoleL
1 QM@4D
Latt.@17D
0.9
0.8
0.7
0 0.25 0.5 0.75 1 1.25 1.5 1.75
q2@GeVD
FIG. 3: Comparison of the D0 K− transition F0 form
factor q2 dependenceof our mode→l (solid (black) line), quark
model of Ref. [4] (dotted (magenta) line) and lattice QCD
fitted to a pole [17] (dot-dashed (blue) line).
F Hq2L
0
1.4 Thiswork
QM@4D
1.2
Latt.@17D
1
0.8
0 0.5 1 1.5 2 2.5 3
q2@GeVD
FIG.4: ComparisonoftheD0 π−transitionF0formfactor
q2 dependenceof ourmodel(s→olid (black) line), quarkmodel
of Ref. [4] (dotted (magenta) line) and lattice QCD fitted to
a pole [17] (dot-dashed (blue) line).
10
TABLE II: The branching ratios for the D P semilep-
→
tonicdecays. Comparison ofdifferentmodelpredictionswith
experiment as explained in thetext.
Decay [%] Reference
B
D0 K− 3.4 Thiswork(doublepole)
→
4.9 Thiswork(singlepole)
3.75 1.16 QM[8]
±
4.0 QM[4]
3.9 1.2 QM[9]
±
2.7 0.6 SR[10]
3.4±−+11..20 SR[12]
3.7 1.4 SR[15]
±
3.43 0.14 Expt.
±
D0 π− 0.27 Thiswork(doublepole)
→
0.56 Thiswork(singlepole)
0.236 0.034 QM[8]
±
0.39 QM[4]
0.30 0.09 QM[9]
±
0.16 0.3 SR[11]
0.28−+±00..0098 SR[12]
0.27 0.10 SR[15]
±
0.36 0.06 Expt.
±
Ds+→η 1.7 Thiswork(doublepole)
2.5 Thiswork(singlepole)
1.8 0.6 QM[8]
±
2.45 QM[4]
2.5 0.7 Expt.
±
Ds+→η′ 0.61 Thiswork(doublepole)
0.74 Thiswork(singlepole)
0.93 0.29 QM[8]
±
0.95 QM[4]
0.89 0.33 Expt.
±
D+ K¯0 8.4 Thiswork(doublepole)
→
12.4 Thiswork(singlepole)
6.8 0.8 Expt.
±
D+ π0 0.33 Thiswork(doublepole)
→
0.70 Thiswork(singlepole)
0.31 0.15 Expt.
±
D+ η 0.10 Thiswork(doublepole)
→
0.15 Thiswork(singlepole)
<0.5 Expt.
D+ η′ 0.016 Thiswork(doublepole)
→
0.019 Thiswork(singlepole)
<1.1 Expt.
Ds+→K0 0.20 Thiswork(doublepole)
0.32 Thiswork(singlepole)
0.3 QM[4]