$S^2\times\mathbb{C}P^2$上的真空自旋-挠态势能密度与双模稳定性
Vacuum Spin-Torsion Energy Density and Two-Modulus Stability on $S^2\times\mathbb{C}P^2$
- University of Maryland, College Park（马里兰大学帕克分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究在$S^2\times\mathbb{C}P^2$的爱因斯坦-嘉当-卡鲁扎-克莱因紧致化中，推导真空自旋-挠态势能密度，分析双模势的稳定性，为更高层次综合提供输入。
AI中文摘要：
在$S^2\times\mathbb{C}P^2$上的爱因斯坦-嘉当-卡鲁扎-克莱因紧致化中，消除非传播挠率会产生四重自旋电流接触相互作用，即使在零费米子占据时其真空期望值也不为零。我们将十维$\Gamma_{ABC}$结构约化到零模 sector，发现该约化是通道分辨的：三个内部嘉当平面双线性项各自对外部（轴）通道贡献相反符号，因此完整迹为$R_{ch}=T_{\rm vac}/T_{\rm axial}=1+3\times(-1)=-2$。因此真空项是轴估计值的两倍且符号相反，结合通过显式克利福德计算验证的消除常数$c_{10}=-\kappa_{10}/32$，该项具有吸引力且与$N_{\rm fam}^2$成正比。我们得到分离点关联函数，其在连续介质中标度为$N_{\rm fam}^2 d_R/(V_6 r^6)$，并在物理量子边界规定的条件下，在最小支持分离处对其求值。同一关联函数的有限密度项符号相同且随化学势增大而增长。成核后结合加深；单粒子哈特里自收缩对手征零模消失。利用导出的带有爱因斯坦框架曲率和量子化通量贡献的$1/(a_1^2a_2^4)$真空项，我们分析耦合双模玩具势。在基准耦合下，$(m,n)=(3,3)$ sector 在两个 radion 方向上具有正黑塞矩阵的有限半径驻点，而四家族$(4,3)$最小值超出采用的量子曲率边界。三家族 sector $(1,5)$也局部稳定但结合较弱。三家族对热成核权重的主导地位进一步施加条件$\kappa_4\lesssim7.5(16/B)\ell_P^2$。这些结果作为我们更高层次综合的输入。
英文摘要：
In Einstein--Cartan--Kaluza--Klein compactification on $S^2\times\mathbb{C}P^2$, elimination of the non-propagating torsion generates a quartic spin-current contact interaction whose vacuum expectation value is nonzero even at zero fermion occupation. We reduce the ten-dimensional $Γ_{ABC}$ structure to the zero-mode sector and find that the reduction is channel-resolved: each of the three internal Cartan-plane bilinears contribute with the opposite sign to the external (axial) channel, so that the full trace is $R_{ch} = T_{\rm vac}/T_{\rm axial} = 1 + 3\,(-1) = -2$. The vacuum term is therefore twice the axial estimate and opposite in sign, and, combined with the elimination constant $c_{10} = -κ_{10}/32$, verified by explicit Clifford computation, it is attractive and proportional to $N_{\rm fam}^2$. We obtain the separated-point correlator, which in the continuum scales as $N_{\rm fam}^2 d_R/(V_6\,r^6)$, and, conditional on the physical quantum-boundary prescription, evaluate it at the minimum supported separation. The finite-density piece of the same correlator has the same sign and grows with chemical potential. Binding deepens after nucleation; the single-particle Hartree self-contraction vanishes for chiral zero modes. Using the derived $1/(a_1^2a_2^4)$ vacuum term with the Einstein-frame curvature and quantized-flux contributions, we analyze a coupled two-modulus toy potential. At fiducial couplings, the $(m,n)=(3,3)$ sector has a finite-radius stationary point with positive Hessian in both radion directions, while the four-family $(4,3)$ minimum lies beyond the adopted quantum-curvature boundary. The three-family sector $(1,5)$ is also locally stable but is less bound. 3-family dominance of the thermal nucleation weight further imposes the condition $κ_4 \lesssim 7.5\,(16/B)\,\ell_P^2$. These results serve as inputs to our higher-level synthesis.
