chiral particle/photon emission from heavy-light mesons

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Chiral Particle/Photon emissionChiral Particle/Photon emissionfrom heavy-light mesonsfrom heavy-light mesons

Koichi SEO Gifu City Womens’ Col.

Takayuki MATSUKI Tokyo Kasei Univ.

HNP13(07/20/2013)

7/20/2013

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1.1. IntroductionIntroduction2.2. Spectroscopy of heavy-light systemSpectroscopy of heavy-light system3.3. Fundamental Formulation of one-particle Fundamental Formulation of one-particle

decaydecay4.4. Relativistic Formulation of decay widthsRelativistic Formulation of decay widths5.5. Numerical Results for one chiral particle/one Numerical Results for one chiral particle/one

photon emission from heavy-light mesonphoton emission from heavy-light meson6.6. SummarySummary

OutlineOutline

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Mass spectrum of heavy-light mesons has been explained by our group successfully in a semi-relativistic potential model （ Matsuki & Morii , Phys. Rev. D 56, 5646 (1997) 　Matuski et. al. , Prog. Theor. Phys. 117, 1077 (2007) ; Eur. Phys. J. A31, 701(2007) ）→　 Semi-Leptonic Decay Form Factor

　（ Matsuki & Seo, Prog. Theor. Phys. 118,　 1087 (2007)　）→　 π or K emitting Hadronic Decay Width

　（ Matsuki & Seo, Phys. Rev. D85,　 014036 (2012)　）

§1.§1.　　 IntroductionIntroduction

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Atom(Hydrogen)

B / D

Heavy particle 940 4500/1500

Light particle 0.5 10～ 100

ΔE (=Ei - Ef) 10-6 300～ 500

in units of MeV

Non-relativistic method used in the atomic transition is not appropriate for B and D decays.

Relativistic calculation is necessary !

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OutlineOutline

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• Successful prediction/reproduction of Ds mass spectra using our semi-relativistic potential model

– Lowering 0+ and 1+ of Ds0*(2317) and Ds1

’(2460) compared with other potential models

prediction by conventional potentialmodel (Godfrey & Kokski, PRD43, 1679 (1991))

prediction by our semi-relativisiticpotential model (Prog. Theor. Phys.117 (2007) 1077)

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Belowthreshold

§2.§2.　　 Spectroscopy of heavy-light systemSpectroscopy of heavy-light system 　　 ((Mass Spectra of DMass Spectra of DsJsJ))

Other Mass Spectra of Our ModelOther Mass Spectra of Our Model

D

Successful reproduction of the following spectraD0*(2318) and D1’(2427) by BelleDs0(2860) and Ds

*(2715) by BaBar & Belle (n=2; 0+ and 1- states of Ds)B1(5720) and B2*(5745) by D0 (1+ and 2+ states of B)Bs2*(5839) by D0 (2+ state of Bs)

B Bs

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Belowthreshold

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Our Numerical Values/Present Exp. StatusOur Numerical Values/Present Exp. Status

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(0 ) (1 ) (0 ) (1 ) (1 ) (2 )

observed 186 18 2421 2427 24607 2008

predic 2283 2421ted 1869 2 2425 24011 68

PJ D D D D D D− − + + + +

(0 ) (1 ) (0 ) (1 ) (1 ) (2 )

observed 1969 21 2317 2460 2535 2512

predicted 196 27 21 325 2467 25

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2 25 2 80 56

2

− − + + + +Ps s s s s sJ D D D D D D

(0 ) (1 ) (0 ) (1 ) (1 ) (2 )

observed 5279 5325

predicted 5 5720 5737270 5329 5621 56

5723 5745

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PJ B B B B B B− − + + + +

− −

(0 ) (1 ) (0 ) (1 ) (1 ) (2 )

observed 536 58407 5415

predicte 5617 5682 5831 58d 5378 5440

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47

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Ps s s s s sJ B B B B B B− − + + + +

− −

Below BK/B*K threshold

CDF data

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0-th order wave function in 1/mQ expansion

Fermi-Yang EquationFermi-Yang Equation　　

…　 Angular & spin wf

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Input parametersInput parameters

Radial wave functionRadial wave function

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1-st order corrections to wave function can be included as




OutlineOutline

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§ 3. Fundamental § 3. Fundamental Formulation of one particle decayFormulation of one particle decay

Following “Excited heavy-light systems and hadronic transitions” by Di Perro and Eichten, PRD 64, 114004 (2001); 　　Goity and Roberts PRD 60, 034001 (1999).Georgi-Manohar interaction between quarks and π, K,… （ chiral multiplets) 　　　　　　　

“Chiral multiplets of heavy-light mesons”　　 by Bardeen, Eichten, and Hill, PRD 68, 054024 (2003) 　 effective Lagrangian among heavy meson and π, K,…, γ 　　　　　　

Intermultipletpionic transition

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light quark current

Calculation of Hadronic-Decay like Goity & Roberts (potential model)

Assuming the infinitely heavy mQ, people used to use the static

w.f. for the meson.

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In this talkCalculate decay widths by taking into account the recoil effects of mesonsDecay processes are

1-pion decay radiative decay

1 2H H π→ +

1 2H H γ→ +7/20/2013

Transition amplitude based on the field theory (1)Transition amplitude based on the field theory (1) 　　

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ξ＝ 0　 pos. of HQ (X=y) ξ＝ 1　 pos. of LQ (X=x)

(Internal coordinate)

(External coordinate)

Wave function in the moving frame

Wave function in the rest frame

(Confining linear potential)

(Color Coulomb potential)7/20/2013

Transition amplitude based on the field theory (2)Transition amplitude based on the field theory (2)

Fermi-Yang Equat ion (Eigen value prob lem)

Trans i t ion Ampl i tude

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phaseno phase factor

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Inser t 　　　　　　　　　　　　　　 and use val ence quark approximat ion.Wave func t ion in moving frame (equal t ime of two quarks)

Time di f f erence in the re s t - f rame

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Velocity of the meson

Boost operator

Boost matrix

　

Est imate of T. A. i f a paren t moves with +V in the Bre i t f rame

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　 Comparing with the methods which have been u sed Replace the pion wave funct ion as

There are correc t ion s to perpendicular d irec t ion s which vani sh i f contrac t ing with pion momentum

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Relativistic formula for the matrix elements of the EM currentin the Breit frame

independent of ξ !

In the case of EM currentIn the case of EM current

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Relativistic formula for the matrix elements of Relativistic formula for the matrix elements of the EM current in the Breit framethe EM current in the Breit frame

Breit frame: Parent meson is moving with +V. Daughter meson is moving with –V.

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q ~ k

(photon momentum)

“effective mass” of the light quark

Radiative decay widths of heavy-light mesonsRadiative decay widths of heavy-light mesons in a in a nonnon-relativistic potential model-relativistic potential model

Bardeen et al., PRD 68, 054024 (2003) …　　 E1 and M1

Close and Swanson, PRD 72, 094004 (2005) 　…　 E1 and M1

Godfrey, PRD 72, 054029 (2005)　…　 E1 only

　

wave function in the rest frame

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OutlineOutline

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§4.§4.　　 Relativistic Formulation of Decay WidthRelativistic Formulation of Decay Width (Tensor structures of Transition Amplitude)(Tensor structures of Transition Amplitude)　

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Couples to pion

Tensor structures of T. A. for photonTensor structures of T. A. for photon

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Estimate of transition amplitude (1)Estimate of transition amplitude (1)　

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Wave Function at Rest

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Estimate of T. A. (2)Estimate of T. A. (2)　

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Valuse of parametersｋ＝-1

ｋ＝1

ｋ＝-2

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（ Ex.）　 0+ 　 (3P0： k=+1) → 0- 　

(1S0:k=-1)

Estimate of transition amplitude (3)Estimate of transition amplitude (3)　

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Relation between the boosted wave function and Relation between the boosted wave function and the static wave functionthe static wave function　　

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Wave function in the moving frame (equal time of two quarks)

(different time of two quarks)

Boost operator

Boost matrix

Velocity of the meson

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1/m1/mQQ corrections corrections in the in the relationrelation between the between the boosted boosted wfwf && the the static static wfwf

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OutlineOutline

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§5§5　　 Numerical resultsNumerical results π/γπ/γ emission from excited D meson states emission from excited D meson states

Values in the parentheses are 0-th order results in 1/mQ expansion7/20/2013 31

Xiao-Hai Lin

π/γπ/γ emission from excited B meson statesemission from excited B meson states

Values in the parentheses are 0-th order results in 1/mQ expansion7/20/2013 32

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π/γπ/γ emission from excited Ds / Bs meson statesemission from excited Ds / Bs meson states

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Comparison with non-relativistic calculationsComparison with non-relativistic calculationsin units of keV

Ref.1…Bardeen et al., PRD 68, 054024 (2003)

Ref.2…Close and Swanson, PRD 72, 094004 (2005)

Ref.3…Godfrey, PRD 72, 054029 (2005)　　7/20/2013

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OutlineOutline

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§6§6　　 SummarySummary

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Formulas for the one pion decay/radiative decay widths were given including the 1st order corrections of wave functions(wf)& the relation of the moving wf to the static wf in 1/mQ expansion

For charged D*or Ds*, sizable decay widths were obtained by including the 1st order corrections in 1/mQ expansion.

For DsJ, large decay widths were obtained compared with non-relativistic works.

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§7§7　　 non-relativistic limitnon-relativistic limit　　

　 cf）　 Result of Bardeen et al.

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non-relativistic limit (2)non-relativistic limit (2)　　

　 cf）　 Result of Bardeen et al.

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chiral particle/photon emission from heavy-light mesons

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