高位无味介子的共形对称破缺和简并
Conformal symmetry breaking and degeneracy of high-lying unflavored mesons
摘要：
我们表明，虽然共形对称性可以打破的电子，这样可以发生在不打破共形简并模式的光谱。我们从AdS_5的R^1XS^3切片出发，注意到S^3的半径倒数R与解除禁闭相变的温度有关，并且必须满足\hbar c/R>> Λ_{QCD}。然后我们集中讨论S^3共形拉普拉斯算子的特征值问题，由1/R^2（K^2 + 1）给出，其中K^2代表so（4）代数的卡西米尔不变量。这样一个谱的特征在于其能级的（K +1）^2倍简并度，K为\in [0，\infty）。然后我们将共形S^3度规破缺为d \tilde{s}^2 = e ^{-b x}（（1 + b^2/4）d χ ^2 + \sin^2 χ（d θ ^2 + \sin^2 θ d φ ^2）），并将对称破缺标度b \hbar^2 c^2/R^2归因于介子。我们证明了这样一种度量变形等价于S^3的共形曲率被一个与b \cot χ成比例的项破坏，并且扰动的共形拉普拉斯算子等价于（\tilde{K}^2 + c_K），其中c_K是表示常数，并且\tilde{K}^2也是so（4）Casimir不变量，但这次是在一个酉不等价于4D旋转的表示中。实际上，对称性破缺前后的谱都是由一个so（4）代数的卡西米尔不变量的本征值决定的，这就是为什么尽管共形群对称性在其代数表示的水平上破缺，但简并性保持不变的原因。我们用S^3半径和\hbar^2 c^2 b/R^2标度拟合了纯介子谱中的高激发态，并观察到\hbar c/R = 373 MeV值明显超过Λ_{QCD}的正确趋势.对称性破缺标度的大小计算为\hbar c \sqrt{b}/R = 673.7 MeV。
英文摘要：
We show that though conformal symmetry can be broken by the dilaton, such can happen without breaking the conformal degeneracy patterns in the spectra. We departure from R^1XS^3 slicing of AdS_5 noticing that the inverse radius, R, of S^3 relates to the temperature of the deconfinement phase transition and has to satisfy, \hbar c/R >> Λ_{QCD}. We then focus on the eigenvalue problem of the S^3 conformal Laplacian, given by 1/R^2 (K^2+1), with K^2 standing for the Casimir invariant of the so(4) algebra. Such a spectrum is characterized by a (K+1)^2 fold degeneracy of its levels, with K\in [0,\infty). We then break the conformal S^3 metric as, d\tilde{s}^2=e^{-bχ} ((1+b^2/4) dχ^2 +\sin^2χ(dθ^2 +\sin^2θdφ^2)), and attribute the symmetry breaking scale, b\hbar^2c^2/R^2, to the dilaton. We show that such a metric deformation is equivalent to a breaking of the conformal curvature of S^3 by a term proportional to b\cot χ, and that the perturbed conformal Laplacian is equivalent to (\tilde{K}^2 +c_K), with c_K a representation constant, and \tilde{K}^2 being again an so(4) Casimir invariant, but this time in a representation unitarily inequivalent to the 4D rotational. In effect, the spectra before and after the symmetry breaking are determined each by eigenvalues of a Casimir invariant of an so(4) algebra, a reason for which the degeneracies remain unaltered though the conformal group symmetry breaks at the level of the representation of its algebra. We fit the S^3 radius and the \hbar^2c^2b/R^2 scale to the high-lying excitations in the spectra of the unflavored mesons, and observe the correct tendency of the \hbar c /R=373 MeV value to notably exceed Λ_{QCD}. The size of the symmetry breaking scale is calculated as \hbar c \sqrt{b}/R=673.7 MeV.
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