Table Of ContentNote on helicity amplitudes in D V semileptonic 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 bythe recent extraction of thehelicity amplitudes for theD+ K¯∗0µνµ decay, done
by the FOCUS collaboration, we determine helicity amplitudes for the D+ → K¯∗0lν , D+ ρ0lν
l l
and D+ φlν semileptonic decays using the knowledge of the relevant for→m factors. The→vector
s → l
and axial form factors for D Vlν decays are parameterized byincluding contributions of charm
l
→
meson resonances and using the HQET and SCET limits. In the case that the vector form factor
receives contributions from two poles while axial form factors are dominated by a single pole for
D+ K¯∗0lν, we obtain better agreement with the experimental result then when all of them are
l
→
dominated by single poles.
6
0 PACSnumbers: 13.20.Fc,13.20.-v,12.39.Hg,12.39.Fe
0
2
Recently the FOCUS [1] collaboration has presented Next we follow the analysis of Ref. [2], where the F
n +
firstnon-parametricdeterminationofhelicityamplitudes formfactorinH P transitionsisgivenasasumoftwo
a
J inthesemileptonicdecayD+ K¯∗0µ+ν. Thismeasure- pole contribution→s,while theF0 formfactoris writtenas
4 mentallowsformoredetaileda→nalysisoftheD V form a single pole, based on form factor dispersion properties
→
factors, especially it enables the studying of the shapes as well as known HQET [6] and SCET [7, 8, 9] scaling
1 of the form factors. limits near zero and maximum recoilmomentum respec-
v We have recently proposed a generalization of the tively. Utilizing the same approach we have proposed
8
Be´cirevi´c-Kaidalov(BK)formfactorparameterization[2] a general parametrization of the heavy to light vector
2
0 forthesemileptonicH V formfactorsbasedonHQET formfactors,whichalsotakesintoaccountalltheknown
1 and SCET scaling pred→ictions [3]. Furthermore we have scalingandresonancepropertiesoftheformfactors. The
0 calculated the D P and D V form factors shapes detailsoftheanalysisareoutlinedinRef.[3]andweonly
6 within a model w→hich combine→s properties of the heavy givetheresultsforthe derivedformfactorparameteriza-
0 meson chiral Lagrangian by taking into account known tions:
h/ andpredictedcharmresonancesandutilizingthegeneral c′ (1 a)
V(q2) = H − ,
p form factor parameterizations [3, 4]. (1 x)(1 ax)
- − −
p In this note we determine helicity amplitudes for the c′ (1 a)
e D → V semileptonic decays and compare our model A1(q2) = ξ H1 −b′x ,
h predictions for the shapes of the form factors with the −
c′′(1 a′)
iv: nDe+w exKp¯e∗r0iµm+eνntdalecraeys.ults coming from FOCUS for the A0(q2) = (1−Hy)(1−−a′y),
X → c′′′
The standard decomposition of the current matrix el- A (q2) = H ,
r ements relevant to semileptonic decays between a heavy 2 (1 b′x)(1 b′′x)
a − −
pseudoscalar meson state H(pH) with momentum pH (1)
| i
and a light vector meson V(p ,ǫ ) with momentum
pV and polarization vector|ǫV Vis inV tierms of four form wa′h)]e/r(emc′H′′ m= )[(imsfiHxe+dbmyVt)hξec′Hre(l1ati−onab)e+twe2emnVthc′He′(fo1rm−
factors V, A0, A1 and A2, functions of the exchanged factors Hat−q2 =V 0 while ξ =m2 /(m +m )2 is the pro-
momentumsquaredq2 =(p p )2 [5]. HereV denotes H H V
the vector form factor andHis−expVected to be dominated portionality factor between A1 and V from the SCET
relation. Variables x = q2/m2 and y = q2/m2 en-
by vector meson resonance exchange, the axial A and H∗ H
1 sure, that the V and A form factors are dominated by
A form factors are expected to be dominated by ax- 0
2 the physical 1− and 0− resonance poles, while a and a′
ial resonances, while A denotes the pseudoscalar form
0 measure the contributions of higher states, parameter-
factor and is expected to be dominated by pseudoscalar
ized by additional effective poles. On the other hand
meson resonance exchange [5]. In order that the ma-
b′ in A and A measures the contribution of resonant
trix elements are finite at q2 = 0, the form factors must 1 2
states with spin-parity assignment 1+ which are param-
alsosatisfy the wellknownrelationA (0)+A (0)(m +
0 1 H eterized by the effective pole at m2 = m2 /b′ while
mV)/2mV A2(0)(mH mV)/2mV =0. He′∗ff H∗
− − the scaling properties and form factor relations require
an additional effective pole for the A form factor. At
2
the end we have parameterized the four H V vector
∗Electronicaddress:[email protected] form factors in terms of the six parameters c→′H, a, a′, b′,
†Electronicaddress:[email protected] c′′ and b′′.
H
2
We determine the above parameters via heavy meson compare them with the experimental results of FOCUS,
chiral theory (HMχT) calculation of the form factors scaledbyanoverallfactordeterminedbytheleastsquare
near q2 = (m m )2. We use the leading order fit of our model predictions, on FIGs. 1, 2 and 3. The
max H − V
heavy meson chiral Lagrangian in which we include ad- scale factor is common to all form factors.
ditional charm meson states. The details of this frame-
work are given in [3] and [4]. We first calculate values H+2Hq2L@GeV2D
of the form factors in the small recoil region. The pres- 4
enceofcharmmesonresonancesinourLagrangianaffects
the values of the form factors at q2 and induces sat-
max 3
uration of the second poles in the parameterizations of
the F (q2), V(q2) and A (q2) form factors by the next
+ 0
radial excitations of D∗ and D mesons respectively. 2
(s) (s)
UsingHQETparameterizationofthecurrentmatrixele-
ments[3],whichisespeciallysuitableforHMχT calcula-
1
tions of the form factors near zero recoil, we are able to
extract consistently the contributions of individual res-
onances from our Lagrangian to the various D V 0
form factors. We use physical pole masses of ex→cited 0.2 0.4 0.6 0.8
state charmed mesons in the extrapolation, giving for q2@GeV2D
the pole parameters a =m2 /m2 , a′ =m2 /m2 and
H∗ H′∗ H H′
b′ = m2 /m2 . Although in the general parameteriza- FIG.1: Ourmodelpredictions(doublepoleinsolidlineand
H∗ HA singlepoleindashedline)fortheq2dependenceofthehelicity
tion of the form factors the extra poles in V and A
0,1,2 amplitudeH2(q2)incomparisonwithscaledFOCUSdataon
parameterize all the neglected higher resonances beyond +
D+ K¯∗0 semileptonic decay.
thegroundstateheavymesonspindoublets(0−,1−),we →
are saturating those by a single nearest resonance. The
single pole q2 behavior of the A (q2) form factor is ex-
plained by the presence of a sing1le 1+ state relevant to H-2Hq2L@GeV2D
eachdecay,whileinA (q2)inadditiontothesestatesone
2 10
mightalsoaccountfortheirnextradialexcitations. How-
ever,duetothelackofdataontheirpresenceweassume
their masses being much higher than the first 1+ states 8
and we neglect their effects, setting effectively b′′ =0.
ThevaluesoftheunknownHMχTparametersappear-
6
ing in D Vlν decay amplitudes [3] are determined by
l
→
fitting the model predictions to knownexperimentalval-
ues of branching ratios and partial decay width ratios. 4
In order to compare our model predictions with re-
cent experimental analysis performedby FOCUS collab-
0.2 0.4 0.6 0.8
oration, following [10] we introduce helicity amplitudes
q2@GeV2D
H :
+,−,0
FIG.2: Ourmodelpredictions(doublepoleinsolidlineand
2m p~ (y)
H±(y) = +(mH +mV)A1(m2Hy)∓ mHH|+VmV |V(m2Hy) saimngplleitpuodleeHin2d(aqs2h)eidnlcinoem)pfoarritshoenqw2itdhepsecnaldeednFceOoCfUthSedhaetliaciotny
−
H (y) = + mH +mV [m2 (1 y) m2]A (m2 y) D+ K¯∗0 semileptonic decay.
0 2mHmV√y H − − V 1 H →
2mH|p~V(y)| A (m2 y) (2) In addition to the two pole contributions we calculate
−mV(mH +mV)√y 2 H helicity amplitudes in the case when all the form factors
exhibit single pole behavior. Putting contributions of
where y =q2/m2 and the three-momentum of the light higher charm resonances to be zero we fit the remaining
H
vector meson is given by: model parameters to existing branching ratios and par-
tial decay ratios. We obtain the values for the following
[m2 (1 y)+m2]2 parameter combinations as explained in [3]:
p~ (y)2 = H − V m2. (3)
| V | 4m2 − V
H
α˜µ˜ = 0
Becauseofthearbitrarynormalizationoftheformfac- α′ζ = 0.180 GeV3/2
tors in [1], we fit our model predictions for a common −
α′µ = 0.00273 GeV1/2
overallscaleinordertocomparetheresults. Weplotthe −
q2 dependence of the predicted helicity amplitudes and α = 0.203 GeV1/2 (4)
1
−
3
H02Hq2L@GeV2D H2Hq2L@GeV2D
i
60 H-
20 H-HsingleL
50 H+
H+HsingleL
15
40 H0
H HsingleL
0
30 10
20
5
10
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8
q2@GeV2D q2@GeV2D
FIG.3: Ourmodelpredictions(doublepoleinsolid lineand FIG. 4: Our model predictions for the q2 dependence of
singlepoleindashedline)fortheq2dependenceofthehelicity thehelicityamplitudesH2(q2)fortheD+ ρ0 semileptonic
amplitudeH2(q2)incomparisonwithscaledFOCUSdataon decay. Double pole predicitions are rendere→d in thick (black)
0
D+ K¯∗0 semileptonic decay. lines while single pole predictions are rendered in thin (red)
→
lines: H−(solidlines),H0(dashedlines)andH+(dot-dashed
lines).
AsshownonFIGs.1and2theexperimentaldataforH
±
do not favor such a parametrization, while in the case H2Hq2L@GeV2D
i
of H helicity amplitude there is almost no difference
0 H-
since the H0 helicity amplitude is defined via the A1,2 20 H-HsingleL
form factors, which are in our approach both effectively H+
dominated by a single pole. The agreement between the H+HsingleL
15
FOCUS results and our model predictions for the q2 de- H0
pendence of the helicity amplitudes is good, although as H0HsingleL
10
noted already in [1], the uncertainties of the data points
are still rather large. On FIGs. 4 and 5 we present he-
licity amplitudes for the D+ ρ0lν and D+ φlν 5
→ l s → l
decays. Both decay modes are most promissing for the
futureexperimentalstudies. Wemakepredictionsforthe
0.2 0.4 0.6 0.8
shapes of helicity amplitudes for both cases: where two q2@GeV2D
poles contribute to the vector form factor and a single
poleto the axialformfactors,andthesecondcasewhere
FIG. 5: Our model predictions for the q2 dependence of
all form factors exhibit single pole behavior.
the helicity amplitudes H2(q2) for the D+ φ semileptonic
In principle one can apply the above procedure to the i s →
decay. Double pole predictions are rendered in thick (black)
B ρlνl semileptonicdecays. However,duetothemuch lines while single pole predictions are rendered in thin (red)
→
broader leptons invariant mass dependence in this case, lines: H−(solidlines),H0(dashedlines)andH+(dot-dashed
our procedure is much more sensitive to the values of lines).
the form factors at q2 0. In addition, the semileptonic
≈
decayratesinourmodelfitarenumericallydominatedby
thelongitudinalhelicityamplitudeH whichhasabroad D+ φlν decays. In all three cases that we have con-
0 s → l
1/pq2 pole [11]. This is true especially for D V but sidered we used two approaches: one with a two poles
→
to minor extent also for B V transitions. Since our shape for the vector form factor and single pole for the
→
model parameters are determined at q2 , this gives a axialformfactors,andsecondlytheusuallyassumedsin-
max
poorhandleonthedominatingeffectsintheoveralldecay gle pole behavior of all three relevant form factors. Our
rate. Thus,accuratedeterminationofthemagnitudeand study indicates that the two pole shape for the V(q2)
shape of the H helicity amplitude near q2 = 0 would form factor in D+ K¯∗0 transition is favored over the
0 →
contribute much to clarifying this issue. single pole shape, when compared to the FOCUS result.
We can summarize: we have investigated the predic-
tions of the general H V form factor parametrization
combined with HMχT→calculation for the D+ K¯∗0 Acknowledgments
→
semileptonic helicity amplitudes, recently determined by
the FOCUS collaboration. In addition we have deter- We are thankful to D. Kim and J. Wiss from the FO-
mined the helicity amplitudes for the D+ ρ0lν and CUScollaborationforsendingustheirdataandforhelp-
l
→
4
ing us understand it. This work is supported in part by ogy of the Republic of Slovenia.
the Ministry of Higher Education, Science and Technol-
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