Chandana Jayalath Department of Physics Hampton University

1. Strong and Electromagnetic Mass Splitting in Heavy Mesons Chandana Jayalath Department of Physics Hampton University

2. Chandana Jayalath Hampton University Outline • Motivation • Calculation Tools and Analysis • Results • Summary

3. Chandana Jayalath Hampton University Motivation • Mesons containing a heavy quark (c or b) and light anti-quark (u, d, s) are • called Heavy Mesons. • D-meson : contains charm quark and light anti-quark • B-meson : contains bottom quark and light anti-quark • Heavy meson studies are important for understanding both weak • and strong interactions. • D-meson and B-meson masses and mass differences are experimentally • determined with high precision.

4. q Q Q • These masses are determined by : • Heavy Quark mass (mQ) • Light Quark mass (mq) • Heavy and light quark electric charges Chandana Jayalath Hampton University • How we determine heavy meson mass? • Consider H atom but in QCD level. • Important to quantify these contributions to the total mass.

5. Pseudo-scalar heavy mesons • Vector heavy mesons Where, mQ – Heavy quark pole mass mq – Light quark mass α – Fine structure constant QQ – Heavy quark charge Qq – Light quark charge 1st term : Contribution in the limit and 2nd term :SU(3) breaking effects by the Light quark masses to first order 3rd term : SU(3) breaking effects by the Electromagnetic interaction 1st part :Electromagnetic self-energy of light anti-quark 2nd part :Electromagnetic interaction between heavy & light quarks Chandana Jayalath Hampton University Mass Formulas

6. Heavy quark effective lagrangian in 1/mQ expansion: Spin independent terms Spin dependent term • CM(μ) satisfies the Renormalization Group Equation : G.Amorós,M.Beneke,M.Neubert Phys. Lett. B 401 (1997) 81-90 Matching to full QCD; E.Eichten, B.Hill Phys. Lett. B 243 (1990) 427 CA = 3, CF = 4/3, β0 = 11 – (2/3)Nf αs : Strong coupling constant Chandana Jayalath Hampton University Heavy quark effective theory

7. Chandana Jayalath Hampton University Effective Parameters • Strong hyperfine interaction • mQ is the heavy quark pole mass (charm or bottom quark mass). • mρ is the ρ-meson mass, used as the reference QCD mass scale, for dimensional purpose. • h1 can be expressed in the renormalization group invariant form; • Neglecting the running of parameter h2 is a good approximation, • since the terms O(1/mQ2) play a minor role in the B-mesons.

8. mq(μ) is defined in Scheme. • κ0 & κ1 are spin-independent, thus independent of mQ. • Their dependence on μ is given by the running of mq(μ) ; • κ2 has an extra running factor similar to that of h1 ; Chandana Jayalath Hampton University • Light quark mass effects (i = 0, 1)

9. has the same running factor like h1 ; Chandana Jayalath Hampton University • Electromagnetic effects • Self-energy of light quark

10. has two general contributions. (i.) Coupling of the photon to the heavy quark spin. [ No mQ dependent] (ii.) Coupling of photon plus gluon to the heavy quark spin. [ Proportional to CM ] Chandana Jayalath Hampton University • EM interaction between the light & heavy quarks

11. Consider five different mass splittings possible in each multiplet. D meson: B meson: • The mass formulae leave one parameter independent mass relation: Where, • Neglecting the h2 term ( O(1/mQ2) terms in strong hyperfine), • there is an additional relation: Chandana Jayalath Hampton University Analysis • Deviation from this relation indicates the importance of the 1/mQ2 term in Strong Hyperfine.

12. Therefore, three linearly dependent terms in mass formulas. • and can be absorbed into and • into and • Eliminate sub-leading terms. Chandana Jayalath Hampton University • There are 12 parameters in mass difference formulas with one • parameter free mass relation.

13. Light quark mass ratios; 2 ms / (mu + md) and ms / (md – mu) • Heavy quark pole mass ratio; ( mc / mb ) • Experimental mass differences for D-meson system; • Experimental mass differences for B-meson system; Chandana Jayalath Hampton University Inputs H.Leutwyler, Phys. Lett. B378 (1996) 313 A.Pineda & F.J.Yndurain, Phys. Rev. D58 (1998) 094022 Particle Data Group, W.M.Yao et al., J.Phys. G33(2006) 1 • ΛQCD = 200 MeV

14. Chandana Jayalath Hampton University Results • Mass Contributions by Strong HF, Light quark masses & Electromagnetism in MeV

15. Predicted experimentally unavailable data points. Examples: • Difference between D and B mesons of the O(1/mQ) spin independent • part of the isospin breaking by the quark masses; Chandana Jayalath Hampton University Summary • With the current accuracy of the heavy meson masses, analyzed the different • contributions to the mass splittings, with significant precision, • in heavy ground state mesons.

16. Compare results with light mesons: • Electromagnetic shift of pseudo-scalar mesons (D+ - D0) & (B- - B0) are similar • within the errors to (K- - K0)EM • Electromagnetic shift of vector mesons (D*+ - D*0) & (B*- - B*0) are • comparable to (ρ+ - ρ0) Chandana Jayalath Hampton University • Obtained results can be useful for constraining models of heavy mesons, • and for lattice QCD calculations of heavy meson masses.