1. Mixing of D s1 (2460) and D s1 (2536) Institute of High Energy Physics, CAS Xiao-Gang Wu Institute of High Energy Physics, CAS wuxiaogang@ihep.ac.cn In collaboration with Qiang Zhao QNP2012, April 16-20, Palaiseau, France

2. Outline  Introduction  Mixing through hadron loop  Coupling form factors in the chiral quark model  Result and discussion  Summary 2

3. 1.Introduction  Phys. Rev. D83, 072003 (2011). (BABAR) PDG2010  Constraints on the property and internal structure of D s1 (2536) and D s1 (2460)?  Mass, width and mixing angle Meson TetraquarkHadronic molecule 3

4. D s spectrum  Low mass: D s0 (2317) DK threshold D s1 (2460) D * K threshold  Narrow width : isospin violation D s0 (2317)-> D s π D s1 (2460)-> D s * π 4

5.  The mixing between 3 P 1 and 1 P 1 can shed light on our understanding of D s1 (2460) and D s1 (2536). 3 P 1 and 1 P 1  cs states are not charge conjugation eigenstates 3 P 1 and 1 P 1 have strong couplings to D * K through S wave Heavy quark limit: D s1 (2460) pure j=1/2, couple to D*K through S wave D s1 (2536) pure j=3/2, couple to D*K through D wave mass width mixing angle 3P13P1 1P11P1 D*D* K 5 a 0 -f 0 mixing, J.-J. Wu, Q. Zhao, and B. S. Zou, Phys. Rev. D 75, 114012(2007)

6. Hadron loop model The mass shift of charmonium E.Eichten etc., Phys.Rev., D17(1987)30900 。。。 Qian Wang etc, hep-ph/ 1202.3026 Xiao-Hai Liu etc, Phys. Rev., D81(2010)014017 Yuan-Jiang Zhang etc, Phys.Rev.Lett., 17(2009)172001 。。。 Gang Li and Qiang Zhao, Phys.Rev.D84(2011)074005 Feng-Kun Guo and Ulf-G Meissner, Phys.Rev.Lett., 108(2012)112002 Feng-Kun Guo etc., Phys.Rev., D83(2011)034013 。。。 See talk by Qian Wang 6

7. 2.Mixing through hadron loop Propagator matrix G ab Diagonalization  Mass and width: pole in complex s plane  Mixing angle and relative phase 7

8. Propagator matrix of two-state system  Scalar or pseudo-scalar e.g., a 0 -f 0 mixing  Vector or axial-vector e.g.,  and  (3770) ab a,μ b,ν 8

9. D s1 (2460) and D s1 (2536)  Mixing scheme  Mixing term: D * K, D s * η, DK * 9 Couplings and divergence?

10. 3.Couplings in the chiral quark model  Effective vertex in the hadron level  Couplings in the chiral quark model e.g., heavy-light meson decays by X.-h. Zhong and Q. Zhao, Phys. Rev. D 78, 014029(2008), Phys. Rev. D 81, 014031(2010) 10

11.  Exponential Form Factor from the NR chiral quark model Can remove UV divergence. Cutoff Λ is fixed and it characterizes the size of meson. No pole in the form factor and do not introduce unphysical freedom.  Couplings g s insensitive to the initial meson mass Strong coupling to D * K Couplings to DK * are vanishing in LO Isospin symmetry 11

12. 4.Result and discussion  The mixing term Real part and imaginary part Two kinks for charged and neutral D*K thresholds D s *η loop 1% Contribution from g μν is dominant Contribution from K μ k ν suffers an O(1/m 2 ) suppression … … 12 Xiao-Gang Wu and Qiang Zhao, Phys.Rev. D85 (2012) 034040

13. 1P11P1 3P13P1  Mass and width Bare masses are taken from GI model. Two poles in the propagator matrix. D s1 (2460) large mass shift, D s1 (2536) small mass shift. The higher pole is insensitive to cutoff Λ, while the lower pole is sensitive. width 13

14.  Mass shift procedure On shell state and virtual state Diagonal shift and off-diagonal shift D * K thresholds lower the mass spectrum Both have a larger 1 P 1 component 14

15.  Mixing parameters Heavy quark limit m a =m b D s1 (2460) deviate from pure j=1/2 state by D s1 (2536) deviate from pure j=3/2 state by relative phase Mixing angles are different for the two physical states! 15

16.  Experimental constraints Width puts the strongest constraint. Two solutions symmetric respect to the heavy quark limit Our analysis favors the bigger one. 16

17. 5.summary  When taking into the D * K loop corrections, we can explain the masses, widths and extract mixing angles of D s1 (2460) and D s1 (2536) with no additional free parameter.  Loop corrections can cause both large mass shifts from quark model and significant mixing angle shifts from the heavy quark limit.  The exponential form factor from the quark model can give a good estimate of the real part of the meson loop. Mass shiftMixing angle shiftD * K coupling D s1 (2460)115MeV12.3⁰S wave D s1 (2536)15MeV4.4⁰D wave 17