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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

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

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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

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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

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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)

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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

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

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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

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

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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

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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

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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

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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

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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

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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

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

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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

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