Study of Rare Semileptonic ]] Decay in the Light-Cone ...downloads.hindawi.com/journals/ahep/2018/2943406.pdf · shownintheFigure. e yreceivecontributionsfromthe penguin and box diagrams

Research ArticleStudy of Rare Semileptonic 119861+119888 rarr 119863+]] Decay in theLight-Cone Quark Model

Nisha Dhiman and Harleen Dahiya

Department of Physics Dr B R Ambedkar National Institute of Technology Jalandhar 144011 India

Correspondence should be addressed to Harleen Dahiya dahiyahnitjacin

Received 7 November 2017 Accepted 4 March 2018 Published 3 April 2018

Academic Editor Luca Stanco

Copyright copy 2018 Nisha Dhiman and Harleen Dahiya This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited The publication of this article was funded by SCOAP3

We study the exclusive semileptonic rare 119861+푐 rarr 119863+]] decay in the framework of light-cone quark model The transition formfactors 119891+(1199022) and 119891푇(1199022) are evaluated in the timelike region using the analytic continuation method in 119902+ = 0 frameThe analyticsolutions of these form factors are compared with the results obtained from the double pole parametric form The branching ratiofor 119861+푐 rarr 119863+]] decay is calculated and compared with the other theoretical model predictions The predicted results in this modelcan be tested at the LHCb experiments in near future which will help in testing the unitarity of CKM quark mixing matrix thusproviding an insight into the phenomenon of CP violation

1 Introduction

In the past few years great progress has been made inunderstanding the semileptonic decays in the 119861 sector asthese are among the cleanest probes of the flavor sector of theStandard Model (SM) which not only provide valuable infor-mation to explore the SM but also are powerful means forprobing different new physics (NP) scenarios beyond the SM(BSM) [1ndash3] Due to the Glashow-Iliopoulos-Maiani (GIM)mechanism [4] flavor changing neutral current (FCNC)induced semileptonic 119861 decays are rare in the SM becausethese decays are forbidden at tree level and can proceedat the lowest order only via electroweak penguin and boxdiagrams [5 6] Therefore these decay processes providesensitive probes to look into physics BSM [7] They also playa significant role in providing a new framework to study themixing between different generations of quarks by extractingthe most accurate values of Cabibbo-Kobayashi-Maskawa(CKM) matrix elements which help us to test the charge-parity (CP) violation in the SM and to dig out the status ofNP [8 9]

The theoretical analysis of CP violating effects in raresemileptonic 119861 decays requires knowledge of the transitionform factors that are model dependent quantities and are

scalar functions of the square of momentum transfer [10]These form factors also interrelate to the decay rates andbranching ratios of all the observed decay modes of 119861mesons and their calculation requires a nonperturbativetreatment Various theoretical approaches such as relativisticconstituent quark model [11ndash15] QCD sum rules [16ndash20]lattice QCD calculations [21ndash23] chiral perturbation theory[24 25] and the light-front quark model (LFQM) [26ndash34] have been applied to the calculations of hadronic formfactors for rare semileptonic 119861 decays Experimentally asignificant effort has been made for the advancement of ourknowledge of the flavor structure of the SM through thestudies of inclusive [35] as well as exclusive [36] rare119861 decaysThe violation of CP symmetry in 119861 meson decays was firstobserved in 2001 (other than in neutral 119870 meson decays)by two experiments the Belle experiment at KEK and theBabar experiment at SLAC [37] Both these experiments wereconstructed and operated on similar time scales andwere ableto take flavor physics into a new realm of discovery [38] TheBabar and Belle experiments completed taking data in 2008and 2010 respectively Recently numerous measurements of119861 decays have been performed by the LHC experiments atCERN in particular the dedicated 119861 physics experimentLHCbmakes a valuable contribution in the understanding of

HindawiAdvances in High Energy PhysicsVolume 2018 Article ID 2943406 7 pageshttpsdoiorg10115520182943406

2 Advances in High Energy Physics

CP violation through the precise determination of the flavorparameters of the SM [39ndash41]

In particular there has been an enormous interest instudying the decay properties of the 119861푐 meson due to itsoutstanding properties [42] Unlike the symmetric heavyquark bound states 119887119887 (bottomonium) and 119888119888 (charmonium)119861푐 meson is the lowest bound state of two heavy quarks (119887and 119888) with different flavors and charge Due to the explicitflavor numbers 119861푐 mesons can decay only through weakinteraction and are stable against strong and electromagneticinteractions thereby providing us an opportunity to test theunitarity of CKM quark mixing matrix The study of anexclusive semileptonic rare 119861+푐 rarr 119863+]] decay is prominentamong all the 119861푐 meson decay modes as it plays a significantrole for precision tests of the flavor sector in the SM and itspossible NP extensions At quark level the decay 119861+푐 rarr 119863+]]proceeds via 119887 rarr 119889 FCNC transition with the intermediate119906 119888 and 119905 quarks and most of the contribution comesfrom the intermediate 119905 quark Also due to the neutral andmassless final states (]]) it provides an unique opportunityto study the 119885 penguin effects [10] As a theoretical inputhadronic matrix elements of quark currents will be requiredto calculate the transition form factors [43] in order to studythe decay rates and branching ratios of the above-mentioneddecay

The semileptonic rare 119861+푐 rarr 119863+]] decay has beenstudied by various theoretical approaches such as constituentquark model (CQM) [44] and QCD sum rules [45] In thiswork we choose the framework of light-cone quark model(LCQM) [46] for the analysis of this decay process LCQMdeals with thewave function defined on the four-dimensionalspace-time plane given by the equation 119909+ = 1199090 + 1199093 andincludes the important relativistic effects that are neglectedin the traditional CQM [47 48] The kinematic subgroupof the light-cone formalism has the maximum number ofinteraction-free generators in comparison with the pointform and instant form [49] The most phenomenal featureof this formalism is the apparent simplicity of the light-conevaccum because the vaccum state of the free Hamiltonian isan exact eigen state of the total light-cone Hamiltonian [50]The light-cone Fock space expansion constructed on this vac-uum state provides a complete relativisticmany-particle basisfor a hadron [51] The light-cone wave functions providing adescription about the hadron in terms of their fundamentalquark and gluon degrees of freedom are independent of thehadronmomentummaking them explicitly Lorentz invariant[52]

The paper is organized as follows In Section 2 we discussthe formalism of light-cone framework and calculate thetransition form factors for 119861+푐 rarr 119863+]] decay process in 119902+ =0 frame In Section 3 we present our numerical results forthe form factors and branching ratios and compare themwithother theoretical results Finally we conclude in Section 4

2 Light-Cone Framework

In the light-cone framework we can write the bound state ofa meson119872 consisting of a quark 1199021 and an antiquark 119902 withtotal momentum 119875 and spin 119878 as [53]

1003816100381610038161003816119872 (119875 119878 119878푧)⟩ = int 119889119901+푞11198892p푞1perp161205873119889119901+푞1198892p푞perp161205873

sdot 1612058731205753 ( minus 119901푞1 minus 119901푞)times sum휆1199021 휆119902

Ψ푆푆119911 (119901푞1 119901푞 120582푞1 120582푞)sdot 100381610038161003816100381610038161199021 (119901푞1 120582푞1) 119902 (119901푞 120582푞)⟩

(1)

where 119901푞1 and 119901푞 denote the on-mass shell light-frontmomenta of the constituent quarks The four-momentum 119901is defined as

119901 = (119901+ pperp) pperp = (1199011 1199012) 119901minus = 1198982 + p2perp119901+ 100381610038161003816100381610038161199021 (119901푞1 120582푞1) 119902 (119901푞 120582푞)⟩

= 119887dagger (119901푞1 120582푞1) 119889dagger (119901푞 120582푞) |0⟩ 119887 (119901耠 120582耠) 119887dagger (119901 120582) = 119889 (119901耠 120582耠) 119889dagger (119901 120582)

= 2 (2120587)3 1205753 (119901耠 minus 119901) 120575휆1015840휆

(2)

The momenta 119901푞1 and 119901푞 in terms of light-cone variables are

119901+푞1 = 1199091119875+119901+푞 = 1199092119875+

p푞1perp = 1199091Pperp + kperpp푞perp = 1199092Pperp minus kperp

(3)

where 119909푖 (119894 = 1 2) represent the light-cone momentumfractions satisfying1199091+1199092 = 1 and kperp is the relative transversemomentum of the constituent

The momentum-space light-cone wave function Ψ푆푆119911 in(1) can be expressed as

Ψ푆푆119911 (119901푞1 119901푞 120582푞1 120582푞) = 119877푆푆119911휆1199021휆119902

(119909 kperp) 120601 (119909 kperp) (4)

where 120601(119909 kperp) describes the momentum distribution of theconstituents in the bound state and 119877푆푆119911

휆1199021휆119902constructs a state

of definite spin (119878 119878푧) out of the light-cone helicity (120582푞1 120582푞)eigenstates For convenience we use the covariant form of119877푆푆119911휆1199021휆119902

for pseudoscalar mesons which is given by

119877푆푆119911휆1199021휆119902

(119909 kperp)

= radic119901+푞1119901+푞radic2radic11987220 minus (119898푞1 minus 119898푞)2

119906 (119901푞1 120582푞1) 1205745V (119901푞 120582푞) (5)

Advances in High Energy Physics 3

where

11987220 = 1198982푞1 + 997888rarr119896 2perp1199091 + 1198982푞 + 997888rarr119896 2perp1199092 (6)

The meson state can be normalized as

⟨119872(119875耠 119878耠 119878耠푧) | 119872 (119875 119878 119878푧)⟩= 2 (2120587)3 119875+1205753 (耠 minus ) 120575푆1015840푆120575푆1015840119911푆119911

(7)

so that

int 1198891199091198892kperp2 (2120587)3 1003816100381610038161003816120601 (119909 kperp)10038161003816100381610038162 = 1 (8)

We choose the Gaussian-type wave function to describe theradial wave function 120601

120601 (119909 kperp) = radic 1120587321205733 exp(minusk2

21205732) (9)

where 120573 is a scale parameter and k2 = k2perp + 1198962푧 denotes theinternal momentum of meson The longitudinal component119896푧 is defined as

119896푧 = (119909 minus 12)1198720 +1198982푞1 minus 1198982푞21198720 (10)

21 Form Factors for the Semileptonic 119861+푐 rarr 119863+]] Decay inLCQM The form factors 119891+(1199022) and 119891푇(1199022) can be obtainedin 119902+ = 0 frame with the ldquogoodrdquo component of current thatis 120583 = + from the hadronic matrix elements given by [53]

⟨119863+ 10038161003816100381610038161003816119889120574휇11988710038161003816100381610038161003816 119861+푐 ⟩ = 119891+ (1199022) 119875휇 + 119891minus (1199022) 119902휇 (11)

⟨119863+ 10038161003816100381610038161003816119889119894120590휇]119902]11988710038161003816100381610038161003816 119861+푐 ⟩= 119891푇 (1199022)(119872퐵+119888 +119872퐷+) [119902

2119875휇 minus (1198722퐵+119888 minus1198722퐷+) 119902휇] (12)

It is more convenient to express the matrix element definedby (11) in terms of 119891+(1199022) and 1198910(1199022) as

⟨119863+ 10038161003816100381610038161003816119889120574휇11988710038161003816100381610038161003816 119861+푐 ⟩ = 119865+ (1199022) [119875휇 minus 1198722퐵+119888 minus1198722퐷+1199022 119902휇]

+ 1198910 (1199022) 1198722퐵+119888minus1198722퐷+1199022 119902휇

(13)

with

119865+ (1199022) = 119891+ (1199022) 1198910 (1199022) = 119891+ (1199022) + 11990221198722

퐵+119888minus1198722퐷+

119891minus (1199022) (14)

Here 119875 = 119875퐵+119888 + 119875퐷+ and 119902 = 119875퐵+119888 minus 119875퐷+ and 0 le 1199022 le (119872퐵+119888 minus119872퐷+)2Using the parameters of 119887 and 119889 quarks the form factors119891+(1199022) and119891푇(1199022) can be respectively expressed in the quark

explicit forms as follows [46]

119891+ (1199022) = int10119889119909int1198892kperpradic120597119896耠푧120597119909 radic120597119896푧120597119909 120601푑 (119909 k耠perp) 120601푏 (119909 kperp)

sdot 119860푏119860푑 + kperp sdot k耠perpradic1198602푏+ k2perpradic1198602푑 + k耠2perp

119891푇 (1199022)

= minusint10119889119909int1198892kperpradic120597119896耠푧120597119909 radic120597119896푧120597119909 120601푑 (119909 k耠perp) 120601푏 (119909 kperp)

times 119909 (119872퐵+119888 +119872퐷+) [(119898푑 minus 119898푏) ((kperp sdot qperp) q2perp) + 119860푏]radic1198602푏+ k2perpradic1198602푑 + k耠2perp

(15)

where k耠perp = kperp minus 119909qperp represents the final state transversemomentum 119860푏 = 119909119898푏 + (1 minus 119909)119898푞 and 119860푑 = 119909119898푑 + (1 minus119909)119898푞 The term 120597119896푧120597119909 denotes the Jacobian of the variabletransformation 119909 kperp rarr k = (119896푧 kperp)

The LCQM calculations of form factors have been per-formed in the 119902+ = 0 frame [54 55] where 1199022 = 119902+119902minus minus q2perp =minusq2perp lt 0 (spacelike region) The calculations are analyticallycontinued to the 1199022 gt 0 (timelike) region by replacing qperp to119894qperp in the form factors To obtain the numerical results of theform factors we use the change of variables as follows

kperp = ℓperp + 1199091205732퐵+1198881205732퐵+119888+ 1205732퐷+

qperp

k耠perp = ℓperp minus 1199091205732퐷+1205732퐵+119888+ 1205732퐷+

qperp(16)

Thedetailed procedure of analytic solutions for theweak formfactors in timelike regionhas been discussed in literature [56]

For the sake of completeness and to compare our analyticsolutions we use a double pole parametric form of formfactors expressed as follows [44]

119891 (1199022) = 119891 (0)1 +A119904 +B1199042 (17)

where 119904 = 11990221198722퐵+119888 119891(1199022) denotes any of the form factorsand 119891(0) denotes the form factors at 1199022 = 0 Here A Bare the parameters to be fitted from (17) While performingcalculations we first compute the values of 119891+(1199022) and 119891푇(1199022)from (15) in 0 le 1199022 le (119872퐵+119888 minus 119872퐷+)2 followed by extractionof the parametersA andB using the values of119872퐵+119888 and119891(0)and then finally fit the data in terms of parametric form

22 Decay Rate and Branching Ratio for 119861+푐 rarr 119863+]] DecayAt the quark level the rare semileptonic 119861+푐 rarr 119863+]] decay


d

c

c

W

b

b

d

W

W

B+c

D+

u c t

u c t

u c t

Z()

]

]

]

]

Figure 1 Loop diagrams for 119861+푐 rarr 119863+]] decay process

is described by the 119887 rarr 119889 FCNC transition As mentionedearlier these kinds of transitions are forbidden at the treelevel in the SM and occur only through loop diagrams asshown in the Figure 1 They receive contributions from thepenguin and box diagrams [44] Theoretical investigationof these rare transitions usually depends on the effectiveHamiltonian density The effective interacting Hamiltoniandensity responsible for 119887 rarr 119889 transition is given by [57]

Heff (119887 997888rarr 119889]])= 119866퐹radic2

120572119881푡푏119881lowast푡푑2120587 sin2120579푊119883(119909푡) 119889120574휇 (1 minus 1205745) 119887]120574휇 (1 minus 1205745) ] (18)

where 119866퐹 is the Fermi constant 120572 is the electromagnetic finestructure constant 120579푊 is the Weinberg angle 119881푖푗 (119894 = 119905 119895 =119887 and 119889) are the CKMmatrix elements and 119909푡 = 1198982푡 1198722푊

The function 119883(119909푡) denotes the top quark loop functionwhich is given by

119883(119909푡) = 119909푡8 (2 + 119909푡119909푡 minus 1 +3119909푡 minus 6(119909푡 minus 1)2 ln119909푡) (19)

The differential decay rate for 119861+푐 rarr 119863+]] can beexpressed in terms of the form factors as [46]

119889Γ119889119904 = 1198725퐵+119888 1198662퐹281205875sin4120579푊1205722 1003816100381610038161003816119881푡푏119881lowast푡푑10038161003816100381610038162 1003816100381610038161003816119883 (119909푡)10038161003816100381610038162 12060132퐷+ 1003816100381610038161003816119891+10038161003816100381610038162 (20)

where 120601퐷+ = (1 minus 119903퐷+)2 minus 2119904(1 + 119903퐷+) + 1199042 with 119904 = 11990221198722퐵+119888and 119903퐷+ = 1198722퐷+1198722퐵+119888

The differential branching ratio (119889BR119889119904) can be obtainedby dividing the differential decay rate (119889Γ119889119904) by the totalwidth (Γtotal) of the 119861+푐 meson and then by integrating thedifferential branching ratio over 119904 = 11990221198722퐵+119888 we can obtainthe branching ratio (BR) for 119861+푐 rarr 119863+]] decay

00 01 02 03 04 0500

05

10

15

20

s

AnalyticParametric

f+

Figure 2 Analytic solutions of119891+ (thick solid curve) comparedwiththe parametric results (dashed curve) with definition 119904 = 11990221198722퐵+119888

3 Numerical Results

Before obtaining the numerical results of the form factors forthe semileptonic 119861+푐 rarr 119863+]] decay we first specify theparameters appearing in the wave functions of the hadronsWe have used the constituent quark masses as [53 58]

119898푏 = 48GeV119898푑 = 025GeV119898푐 = 14GeV

(21)

The parameter 120573 that describes the momenta distribution ofconstituent quarks can be fixed by the meson decay constants119891퐵+119888 and119891퐷+ respectivelyThe 120573 parameters that we have usedin our work are given as [44]

120573퐵+119888 = 081GeV120573퐷+ = 046GeV (22)

Using the above parameters we present the analytic solutionsof the form factors 119891+ and 119891푇 (thick solid curve) for 0 le1199022 le (119872퐵+119888 minus119872퐷+)2 in Figures 2 and 3 respectively We havealso shown the results obtained from the parametric formula(dashed curve) given by (17) We would like to mention herethat the point 1199022 = 0 represents the maximum recoil pointand the point 1199022 = 1199022max = (119872퐵+119888 minus 119872퐷+)2 represents thezero recoil point where the produced meson is at rest Aswe can see from Figures 2 and 3 the form factors 119891+ and119891푇 increase and decrease exponentially with respect to 1199022The analytic solutions of form factors given by (15) are wellapproximated by the parametric form in the physical decayregion 0 le 1199022 le (119872퐵+119888 minus 119872퐷+)2 For a deeper understandingof the results we have listed the numerical results for the formfactors 119891+ and 119891푇 at 1199022 = 0 and the parameters A and B ofthe double pole form in Table 1 For the sake of comparisonwe have also presented the results of other theoreticalmodels


Table 1 Form factors for 119861+푐 rarr 119863+]] decay process at 1199022 = 0 and the parametersA andB defined by (17) and their comparison with othertheoretical model predictions

Model 119891+(0) A B 119891푇(0) A B

This work 0140 minus3263 2846 minus0234 minus3430 3174CQM [44] 0123 minus335 303 minus0186 minus352 338SR [45] 022 minus110 minus248 minus027 minus072 minus324Linear [46] 0086 minus350 330 minus0120 minus335 306HO [46] 0079 minus320 281 minus0108 minus318 277

00 01 02 03 04 05

minus30

minus25

minus20

minus15

minus10

minus05

00

s

AnalyticParametric

fT

Figure 3 Analytic solutions of119891푇 (thick solid curve) comparedwiththe parametric results (dashed curve) with definition 119904 = 11990221198722퐵+119888

Table 2 Branching ratio for 119861+푐 rarr 119863+]] decay in LCQM and itscomparison with the other models

Model Branching ratios (in units of 10minus8)This work 333CQM [44] 274QCD sum rules [45] 338Linear [46] 131HO [46] 081

It can be seen from the table that the values of form factors119891+and119891푇 at 1199022 = 0 in our model agree quite well with the CQMThe difference in the values with respect to other modelsmight be due to the different assumptions of the models ordifferent choices of parameters

To estimate the numerical value of the branching ratiofor 119861+푐 rarr 119863+]] decay (defined in (20)) the various inputparameters used are [46] 120572minus1 = 129 |119881푡푏119881lowast푡푑| = 0008119872푊 = 8043GeV119898푡 = 1713GeV and sin2120579푊 = 02233 Thelifetime of 119861+푐 (120591퐵+119888 = 0507 ps) is taken from the Particle DataGroup [59] Our results for the differential branching ratio asa function of 119904 is shown in Figure 4

Our prediction for the decay branching ratio of 119861+푐 rarr119863+]] decay is listed in Table 2 and compared with the othertheoretical predictions As we can see from Table 2 the resultpredicted by LCQMapproximately agrees with the prediction

00 01 02 03 04 0500

02

04

06

08

10

s

(dBR

ds)

times10

7

Figure 4 Differential branching ratios as a function of 119904 for 119861+푐 rarr119863+]] decay

given by QCD sum rules whereas it is slightly larger whencompared with the results of CQM At present we do nothave any deep understanding of these values however theydo indicate that these results may be important even ina more rigorous model The measurements can perhapsbe substantiated by measurement of the decay width of 119861mesons Several experiments at LHCb are contemplating thepossibility of searching for more 119861meson decays

4 Conclusions

We have studied the exclusive semileptonic rare 119861+푐 rarr119863+]] decay within the framework of LCQM In our analysiswe have evaluated the transition form factors 119891+(1199022) and119891푇(1199022) in the 119902+ = 0 frame and then extended themfrom the spacelike region (1199022 lt 0) to the timelike region(1199022 gt 0) through the method of analytical continuationusing the constituent quark masses (119898푏 119898푑 and 119898푐) andthe parameters describing the momentum distribution of theconstituent quarks (120573퐵+119888 and120573퐷+) respectivelyThe numericalvalues of 120573퐵+119888 and 120573퐷+ have been fixed from the meson decayconstants 119891퐵+119888 and 119891퐷+ respectively We have also comparedthe analytic solutions of transition form factors with theresults obtained for the form factors using the double poleparametric form Using the numerical results of transitionform factors we have calculated the decay branching ratioand compared our result with the other theoretical modelpredictions The LCQM result for the decay branching ratioof 119861+푐 rarr 119863+]] decay comes out to be 333 times 10minus8 whichapproximately agrees with the prediction given by QCD


sum rules [45] This result can also be tested at the LHCbexperiments in near future

To conclude new experiments aimed at measuring thedecay branching ratios are not only needed for the profoundunderstanding of 119861 decays but also to restrict the modelparameters for getting better knowledge on testing the uni-tarity of CKM quark mixing matrixThis will provide us witha useful insight into the phenomenon of CP violation

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to acknowledge Chueng-Ryong Ji(North Carolina State University Raleigh NC) for the helpfuldiscussions and Department of Science and Technology (Refno SBS2HEP-0042013) Government of India for financialsupport

References

[1] J Dingfelder and TMannel ldquoLeptonic and semileptonic decaysof 119861 mesonsrdquo Reviews of Modern Physics vol 88 no 3 ArticleID 035008 2016

[2] H-M Choi and C-R Ji ldquoSemileptonic and radiative decays ofthe 119861푐 meson in the light-front quark modelrdquo Physical ReviewD vol 80 Article ID 054016 2009

[3] W Jaus ldquoSemileptonic decays of 119861 and 119863 mesons in thelight-front formalismrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 41 no 11 Article ID 3394 1990

[4] S L Glashow J Iliopoulos and L Maiani ldquoWeak interactionswith lepton-hadron symmetryrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 2 no 7 pp 1285ndash12921970

[5] A Ali ldquo119861 decays in the standard model mdash status and perspec-tivesrdquo Acta Physica Polonica B vol 27 p 3529 1996

[6] G Buchalla and A Buras ldquoQCD corrections to rare 119870- and 119861-decays for arbitrary top quarkmassrdquoNuclear Physics B vol 400pp 225ndash239 1993

[7] T Blake G Lanfranchi and R Khosravi ldquoRare 119861 decays as testsof the StandardModelrdquo Progress in Particle and Nuclear Physicsvol 92 pp 50ndash91 2017

[8] C S Kim T Morozumi and A I Sanda ldquo119861 rarr 119883푞119897+119897minus(119902 = 119889 119904)and determination of |119881td119881ts|rdquo Physical Review D vol 56 pp7240ndash7246 1997

[9] T Aliev C Kim and M Savcı ldquoExclusive 119861 rarr 119872ℓ+ℓminus(119872 =120587119870 120588 119870) decays and determinations of 119881ts (and 119881td119881ts)rdquoPhysics Letters B vol 441 no 1-4 pp 410ndash418 1998

[10] M Wick PhD thesis Technical University of Munich 2010[11] D Melikhov ldquoForm factors of meson decays in the relativistic

constituent quark modelrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 53 no 5 pp 2460ndash2479 1996

[12] D Melikhov N Nikitin and S Simula ldquo Rare exclusivesemileptonic rdquo Physical Review D Particles Fields Gravitationand Cosmology vol 57 no 11 pp 6814ndash6828 1998

[13] D Melikhov and B Stech ldquoWeak form factors for heavymeson decays an updaterdquo Physical Review D Particles FieldsGravitation and Cosmology vol 62 no 1 Article ID 0140062000

[14] M Wirbel B Stech and M Bauer ldquoExclusive semileptonicdecays of heavy mesonsrdquo Zeitschrift fur Physik C vol 29 no4 pp 637ndash642 1985

[15] W Jaus and D Wyler ldquoRare decaysrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 41 no 11 pp3405ndash3413 1990

[16] P Ball V M Braun and H G Dosch ldquoForm factors ofsemileptonic D decays from QCD sum rulesrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 44 no 11pp 3567ndash3581 1991

[17] P Ball ldquo119861 rarr 120587 and 119861 rarr 119870 and 119861 rarr 119870 transitions from QCDsum rules on the light-conerdquo Journal of High Energy Physics vol9 p 5 1998

[18] P Ball and V M Braun ldquoExclusive semileptonic and rarerdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 58 no 9 Article ID 094016 1998

[19] P Colangelo F De Fazio P Santorelli and E Scrimieri ldquoQCDsum rule analysis of the decays 119861 rarr 119870ℓ+ℓminus and 119861 rarr 119870ℓ+ℓminusrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 53 no 7 pp 3672ndash3686 1996

[20] V V Kiselev A E Kovalsky andA K Likhoded ldquo119861푐 decays andlifetime in QCD sum rulesrdquo Nuclear Physics B vol 585 no 1-2pp 353ndash382 2000

[21] J M Flynn and C T Sachrajda ldquoHeavy quark physics fromlattice QCDrdquo Journal of High Energy Physics vol 15 pp 402ndash452 1998

[22] A Abada D Becirevic P Boucaud et al ldquoDecays of heavymesonsrdquo Nuclear Physics BmdashProceedings Supplements vol 83-84 no 1-3 pp 268ndash270 2000

[23] K C Bowler et al ldquoImproved 119861 rarr 120587119897]푙 form factors from thelatticerdquo Physics Letters B vol 486 pp 111ndash117 2000

[24] R Casalbuoni A Deandrea N Di Bartolomeo R Gatto FFeruglio and G Nardulli ldquoPhenomenology of heavy mesonchiral lagrangiansrdquo Physics Reports vol 281 no 3 pp 145ndash2381997

[25] D Du C Liu and D Zhang ldquoThe rare decay 119861 rarr 119870푇+푇minus inheavy meson chiral perturbation theoryrdquo Physics Letters B vol317 pp 179ndash182 1993

[26] H Choi and C Ji ldquoNonleptonic two-body decays of the 119861푐meson in the light-front quark model and the QCD factoriza-tion approachrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 80 Article ID 114003 2009

[27] H-M Choi and C-R Ji ldquoKaon electroweak form factors in thelight-front quark modelrdquo Physics Letters D vol 59 Article ID034001 1999

[28] H-M Choi ldquoDecay constants and radiative decays of heavymesons in light-front quark modelrdquo Physical Review D vol 75Article ID 073016 2007

[29] H-M Choi and C-R Ji ldquoLight-front quark model analysis ofexclusive 0minus rarr 0minus semileptonic heavy meson decaysrdquo PhysicsLetters B vol 460 pp 461ndash466 1999

[30] H Cheng C Chua and C Hwang ldquoCovariant light-frontapproach for s-wave and p-wave mesons Its application todecay constants and form factorsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 69 no 7 2004

[31] CQGeng CWHwang C C Lih andWMZhang ldquoMesonictensor form factors with the light front quark modelrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 64no 11 Article ID 114024 2001

[32] C Y Cheung C W Hwang and W M Zhang ldquo119861 rarr 120587119897119873form factors calculated on the light-frontrdquo Zeitschrift fur PhysikC Particles and Fields vol 75 pp 657ndash664 1997


[33] C Q Geng C C Lih and W Zhang ldquo Radiative leptonic rdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 57 no 9 pp 5697ndash5702 1998

[34] C C Lih C Q Geng and W-M Zhang ldquoStudy of 119861+푐 rarr 119897+]푙120574decays in the light front modelrdquo Physical Review D vol 59Article ID 114002 1999

[35] M S Alam et al ldquoFirstmeasurement of the rate for the inclusiveradiative penguin decay 119887 rarr 119904120574rdquo Physical Review Letters vol74 Article ID 2885 1995

[36] R Ammar et al ldquoEvidence for penguin-diagram decays firstobservation of 119861 rarr 119870(892)120574rdquo Physical Review Letters vol 71p 674 1993

[37] A J Bevan et al ldquoThe physics of the B factoriesrdquoThe Physics ofthe B Factories vol 74 p 3026 2014

[38] A J Bevan ldquoB factoriesrdquo Comptes Rendus Physique vol 13 no2 pp 145ndash151 2012

[39] C Langenbruch LHCb Collaboration et al ldquoContribution tothe proceedings of the 51st Rencontres de Moriondrdquo QCDSession 2016

[40] B Adeva LHCbCollaboration et al ldquoFlavour physics at LHCbrdquoin Proceedings of the 4th International Conference on NewFrontiers in Physics vol 126 2016

[41] J He et al ldquoElectroweak penguins at LHCbrdquo Nuclear andParticle Physics Proceedings vol 273 pp 1370ndash1375 2016

[42] S S Gershtein V V Kiselev A K Likhoded and AV Tkabladze ldquoReviews of topical problems physics of 119861푐-mesonsrdquo Physics-Uspekhi vol 38 no 1 pp 1ndash37 1995

[43] H Cheng C Cheung and C Hwang ldquoMesonic form factorsand the Isgur-Wise function on the light frontrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 55 no 3 pp1559ndash1577 1997

[44] C Q Geng C W Hwang and C C Liu ldquoStudy of rare 119861+푐 rarr119863()+푑푠

119897119897 decaysrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 65 Article ID 094037 2002

[45] K Azizi and R Khosravi ldquoAnalysis of the rare semileptonicrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 78 no 3 2008

[46] H-M Choi ldquoLight-front quark model analysis of the exclusiverare 119861푐 rarr 119863(푠)(ℓ+ℓminus ]ℓ]ℓ) decaysrdquo Physical Review D vol 81Article ID 054003 2010

[47] S J Brodsky H-C Pauli and S S Pinsky ldquoQuantum chromo-dynamics and other field theories on the light conerdquo PhysicsReports vol 301 no 4-6 pp 299ndash486 1998

[48] G P Lepage and S J Brodsky ldquoExclusive processes in pertur-bative quantum chromodynamicsrdquo Physical ReviewD ParticlesFields Gravitation and Cosmology vol 22 article 2157 1980

[49] P A Dirac ldquoForms of relativistic dynamicsrdquo Reviews of ModernPhysics vol 21 pp 392ndash399 1949

[50] S J Brodsky and H C Pauli ldquoLight-cone quantization ofquantum chromodynamicsrdquo Lecture Notes in Physics vol 396pp 51ndash121 1991

[51] S J Brodsky ldquoSLAC-PUB-8627 Presented at VII HadronPhysics 2000rdquo Caraguatatuba Sao Paulo Brazil April 10-152000

[52] S J Brodsky ldquoQCD phenomenology and light-front wavefunc-tionsrdquo Acta Physica Polonica B vol 32 pp 4013ndash4068 2001

[53] C-D LuWWang and Z-T Wei ldquoHeavy-to-light form factorson the light conerdquo Physical Review D vol 76 Article ID 0140132007

[54] S D Drell and T-M Yan ldquoConnection of elastic electro-magnetic nucleon form factors at large 1198762 and deep inelasticstructure functions near thresholdrdquo Physical Review Letters vol24 p 181 1970

[55] G B West ldquoPhenomenological model for the electromagneticstructure of the protonrdquo Physical Review Letters vol 24 no 21pp 1206ndash1209 1970

[56] H Choi C Ji and L S Kisslinger ldquo Light-front quark modelanalysis of rare rdquo Physical ReviewD Particles Fields Gravitationand Cosmology vol 65 no 7 2002

[57] B Grinstein M B Wise and M J Savage ldquo119861 rarr 119883푠119890+119890minus in thesix-quark modelrdquoNuclear Physics B vol 319 no 2 pp 271ndash2901989

[58] T Wang T Liu D Zhang and B Ma ldquo119861푐 meson rare decays inthe light-cone quark modelrdquo The European Physical Journal Cvol 71 no 9 p 1758 2011

[59] C Patrignani Particle Data Group et al ldquoReview of particlephysicsrdquo Chinese Physics C vol 40 no 10 Article ID 1000012016

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components


Shock and Vibration


High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

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Advances inPhysical Chemistry


RotatingMachinery



Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom


CP violation through the precise determination of the flavorparameters of the SM [39ndash41]

In particular there has been an enormous interest instudying the decay properties of the 119861푐 meson due to itsoutstanding properties [42] Unlike the symmetric heavyquark bound states 119887119887 (bottomonium) and 119888119888 (charmonium)119861푐 meson is the lowest bound state of two heavy quarks (119887and 119888) with different flavors and charge Due to the explicitflavor numbers 119861푐 mesons can decay only through weakinteraction and are stable against strong and electromagneticinteractions thereby providing us an opportunity to test theunitarity of CKM quark mixing matrix The study of anexclusive semileptonic rare 119861+푐 rarr 119863+]] decay is prominentamong all the 119861푐 meson decay modes as it plays a significantrole for precision tests of the flavor sector in the SM and itspossible NP extensions At quark level the decay 119861+푐 rarr 119863+]]proceeds via 119887 rarr 119889 FCNC transition with the intermediate119906 119888 and 119905 quarks and most of the contribution comesfrom the intermediate 119905 quark Also due to the neutral andmassless final states (]]) it provides an unique opportunityto study the 119885 penguin effects [10] As a theoretical inputhadronic matrix elements of quark currents will be requiredto calculate the transition form factors [43] in order to studythe decay rates and branching ratios of the above-mentioneddecay

The semileptonic rare 119861+푐 rarr 119863+]] decay has beenstudied by various theoretical approaches such as constituentquark model (CQM) [44] and QCD sum rules [45] In thiswork we choose the framework of light-cone quark model(LCQM) [46] for the analysis of this decay process LCQMdeals with thewave function defined on the four-dimensionalspace-time plane given by the equation 119909+ = 1199090 + 1199093 andincludes the important relativistic effects that are neglectedin the traditional CQM [47 48] The kinematic subgroupof the light-cone formalism has the maximum number ofinteraction-free generators in comparison with the pointform and instant form [49] The most phenomenal featureof this formalism is the apparent simplicity of the light-conevaccum because the vaccum state of the free Hamiltonian isan exact eigen state of the total light-cone Hamiltonian [50]The light-cone Fock space expansion constructed on this vac-uum state provides a complete relativisticmany-particle basisfor a hadron [51] The light-cone wave functions providing adescription about the hadron in terms of their fundamentalquark and gluon degrees of freedom are independent of thehadronmomentummaking them explicitly Lorentz invariant[52]

The paper is organized as follows In Section 2 we discussthe formalism of light-cone framework and calculate thetransition form factors for 119861+푐 rarr 119863+]] decay process in 119902+ =0 frame In Section 3 we present our numerical results forthe form factors and branching ratios and compare themwithother theoretical results Finally we conclude in Section 4

2 Light-Cone Framework

In the light-cone framework we can write the bound state ofa meson119872 consisting of a quark 1199021 and an antiquark 119902 withtotal momentum 119875 and spin 119878 as [53]

1003816100381610038161003816119872 (119875 119878 119878푧)⟩ = int 119889119901+푞11198892p푞1perp161205873119889119901+푞1198892p푞perp161205873

sdot 1612058731205753 ( minus 119901푞1 minus 119901푞)times sum휆1199021 휆119902

Ψ푆푆119911 (119901푞1 119901푞 120582푞1 120582푞)sdot 100381610038161003816100381610038161199021 (119901푞1 120582푞1) 119902 (119901푞 120582푞)⟩

(1)

where 119901푞1 and 119901푞 denote the on-mass shell light-frontmomenta of the constituent quarks The four-momentum 119901is defined as

119901 = (119901+ pperp) pperp = (1199011 1199012) 119901minus = 1198982 + p2perp119901+ 100381610038161003816100381610038161199021 (119901푞1 120582푞1) 119902 (119901푞 120582푞)⟩

= 119887dagger (119901푞1 120582푞1) 119889dagger (119901푞 120582푞) |0⟩ 119887 (119901耠 120582耠) 119887dagger (119901 120582) = 119889 (119901耠 120582耠) 119889dagger (119901 120582)

= 2 (2120587)3 1205753 (119901耠 minus 119901) 120575휆1015840휆

(2)

The momenta 119901푞1 and 119901푞 in terms of light-cone variables are

119901+푞1 = 1199091119875+119901+푞 = 1199092119875+

p푞1perp = 1199091Pperp + kperpp푞perp = 1199092Pperp minus kperp

(3)

where 119909푖 (119894 = 1 2) represent the light-cone momentumfractions satisfying1199091+1199092 = 1 and kperp is the relative transversemomentum of the constituent

The momentum-space light-cone wave function Ψ푆푆119911 in(1) can be expressed as

Ψ푆푆119911 (119901푞1 119901푞 120582푞1 120582푞) = 119877푆푆119911휆1199021휆119902

(119909 kperp) 120601 (119909 kperp) (4)

where 120601(119909 kperp) describes the momentum distribution of theconstituents in the bound state and 119877푆푆119911

휆1199021휆119902constructs a state

of definite spin (119878 119878푧) out of the light-cone helicity (120582푞1 120582푞)eigenstates For convenience we use the covariant form of119877푆푆119911휆1199021휆119902

for pseudoscalar mesons which is given by

119877푆푆119911휆1199021휆119902

(119909 kperp)

= radic119901+푞1119901+푞radic2radic11987220 minus (119898푞1 minus 119898푞)2

119906 (119901푞1 120582푞1) 1205745V (119901푞 120582푞) (5)


where




(7)

so that

int 1198891199091198892kperp2 (2120587)3 1003816100381610038161003816120601 (119909 kperp)10038161003816100381610038162 = 1 (8)



21205732) (9)


119896푧 = (119909 minus 12)1198720 +1198982푞1 minus 1198982푞21198720 (10)


⟨119863+ 10038161003816100381610038161003816119889120574휇11988710038161003816100381610038161003816 119861+푐 ⟩ = 119891+ (1199022) 119875휇 + 119891minus (1199022) 119902휇 (11)

⟨119863+ 10038161003816100381610038161003816119889119894120590휇]119902]11988710038161003816100381610038161003816 119861+푐 ⟩= 119891푇 (1199022)(119872퐵+119888 +119872퐷+) [119902

2119875휇 minus (1198722퐵+119888 minus1198722퐷+) 119902휇] (12)



+ 1198910 (1199022) 1198722퐵+119888minus1198722퐷+1199022 119902휇

(13)

with

119865+ (1199022) = 119891+ (1199022) 1198910 (1199022) = 119891+ (1199022) + 11990221198722

퐵+119888minus1198722퐷+

119891minus (1199022) (14)





119891푇 (1199022)



(15)



kperp = ℓperp + 1199091205732퐵+1198881205732퐵+119888+ 1205732퐷+

qperp


qperp(16)



119891 (1199022) = 119891 (0)1 +A119904 +B1199042 (17)




d

c

c

W

b

b

d

W

W

B+c

D+

u c t

u c t

u c t

Z()

]

]

]

]



Heff (119887 997888rarr 119889]])= 119866퐹radic2









00 01 02 03 04 0500

05

10

15

20

s

AnalyticParametric

f+


3 Numerical Results


119898푏 = 48GeV119898푑 = 025GeV119898푐 = 14GeV

(21)


120573퐵+119888 = 081GeV120573퐷+ = 046GeV (22)




Model 119891+(0) A B 119891푇(0) A B


00 01 02 03 04 05

minus30

minus25

minus20

minus15

minus10

minus05

00

s

AnalyticParametric

fT







00 01 02 03 04 0500

02

04

06

08

10

s

(dBR

ds)

times10

7



4 Conclusions







Acknowledgments


References

































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






where




(7)

so that

int 1198891199091198892kperp2 (2120587)3 1003816100381610038161003816120601 (119909 kperp)10038161003816100381610038162 = 1 (8)



21205732) (9)


119896푧 = (119909 minus 12)1198720 +1198982푞1 minus 1198982푞21198720 (10)


⟨119863+ 10038161003816100381610038161003816119889120574휇11988710038161003816100381610038161003816 119861+푐 ⟩ = 119891+ (1199022) 119875휇 + 119891minus (1199022) 119902휇 (11)

⟨119863+ 10038161003816100381610038161003816119889119894120590휇]119902]11988710038161003816100381610038161003816 119861+푐 ⟩= 119891푇 (1199022)(119872퐵+119888 +119872퐷+) [119902

2119875휇 minus (1198722퐵+119888 minus1198722퐷+) 119902휇] (12)



+ 1198910 (1199022) 1198722퐵+119888minus1198722퐷+1199022 119902휇

(13)

with

119865+ (1199022) = 119891+ (1199022) 1198910 (1199022) = 119891+ (1199022) + 11990221198722

퐵+119888minus1198722퐷+

119891minus (1199022) (14)





119891푇 (1199022)



(15)



kperp = ℓperp + 1199091205732퐵+1198881205732퐵+119888+ 1205732퐷+

qperp


qperp(16)



119891 (1199022) = 119891 (0)1 +A119904 +B1199042 (17)




d

c

c

W

b

b

d

W

W

B+c

D+

u c t

u c t

u c t

Z()

]

]

]

]



Heff (119887 997888rarr 119889]])= 119866퐹radic2









00 01 02 03 04 0500

05

10

15

20

s

AnalyticParametric

f+


3 Numerical Results


119898푏 = 48GeV119898푑 = 025GeV119898푐 = 14GeV

(21)


120573퐵+119888 = 081GeV120573퐷+ = 046GeV (22)




Model 119891+(0) A B 119891푇(0) A B


00 01 02 03 04 05

minus30

minus25

minus20

minus15

minus10

minus05

00

s

AnalyticParametric

fT







00 01 02 03 04 0500

02

04

06

08

10

s

(dBR

ds)

times10

7



4 Conclusions







Acknowledgments


References

































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






d

c

c

W

b

b

d

W

W

B+c

D+

u c t

u c t

u c t

Z()

]

]

]

]



Heff (119887 997888rarr 119889]])= 119866퐹radic2









00 01 02 03 04 0500

05

10

15

20

s

AnalyticParametric

f+


3 Numerical Results


119898푏 = 48GeV119898푑 = 025GeV119898푐 = 14GeV

(21)


120573퐵+119888 = 081GeV120573퐷+ = 046GeV (22)




Model 119891+(0) A B 119891푇(0) A B


00 01 02 03 04 05

minus30

minus25

minus20

minus15

minus10

minus05

00

s

AnalyticParametric

fT







00 01 02 03 04 0500

02

04

06

08

10

s

(dBR

ds)

times10

7



4 Conclusions







Acknowledgments


References

































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







Model 119891+(0) A B 119891푇(0) A B


00 01 02 03 04 05

minus30

minus25

minus20

minus15

minus10

minus05

00

s

AnalyticParametric

fT







00 01 02 03 04 0500

02

04

06

08

10

s

(dBR

ds)

times10

7



4 Conclusions







Acknowledgments


References

































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery










Acknowledgments


References

































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





Documents

Study of Rare Semileptonic ]] Decay in the Light-Cone ...downloads.hindawi.com/journals/ahep/2018/2943406.pdf · shownintheFigure. e yreceivecontributionsfromthe penguin and box diagrams