TFPT scientific source excerpt
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000096 | \begin{corollary}[Determinant bounds for the infrared seam transfer]
000097 | \label{cor:ir-seam-transfer-bounds}
000098 | Let $\{\lambda_j(U_\Sigma)\}$ denote the admissible eigenvalues of the positive trace-class
000099 | operator $U_\Sigma$. Then
000100 | \[
000101 | \tr_{\mathrm{adm}}U_\Sigma
000102 | \le
000103 | -\log\det\nolimits_{\mathrm{adm}}(1-U_\Sigma)
000104 | \le
000105 | \frac{-\log(1-\rho(U_\Sigma))}{\rho(U_\Sigma)}\,\tr_{\mathrm{adm}}U_\Sigma.
000106 | \]
000107 | If, moreover, the infrared seam transfer is rank one,
000108 | \[
000109 | U_\Sigma=\rho(U_\Sigma)\,P_\Sigma^{(1)},
000110 | \]
000111 | then
000112 | \[
000113 | \frac{\Lambda_{\mathrm{IR}}}{\Mplunred^4}
000114 | =
000115 | -\log(1-\rho(U_\Sigma))
000116 | =
000117 | \rho(U_\Sigma)+O\!\bigl(\rho(U_\Sigma)^2\bigr).
000118 | \]
000119 | \end{corollary}
000120 | 
000121 | \begin{proof}
000122 | Because $U_\Sigma$ is positive trace class, its eigenvalues satisfy
000123 | \[
000124 | 0<\lambda_j(U_\Sigma)\le \rho(U_\Sigma)<1.
000125 | \]
000126 | For every $x\in[0,\rho(U_\Sigma)]$ one has the scalar bounds
000127 | \[
000128 | x\le -\log(1-x)\le \frac{-\log(1-\rho(U_\Sigma))}{\rho(U_\Sigma)}\,x.
000129 | \]
000130 | Applying this to each eigenvalue and summing gives the determinant bounds. If
000131 | $U_\Sigma=\rho(U_\Sigma)P_\Sigma^{(1)}$ is rank one, then
000132 | \[
000133 | \det\nolimits_{\mathrm{adm}}(1-U_\Sigma)=1-\rho(U_\Sigma),
000134 | \]
000135 | and the Taylor expansion of $-\log(1-\rho)$ yields the displayed asymptotic formula.
000136 | \end{proof}