[Paper Review] ERRATA for "One-Loop Threshold Effects in String Unification"

Vadim Kaplunovsky|ArXiv.org|May 19, 1992

Nuclear reactor physics and engineering16 citations

TL;DR

This errata corrects critical errors in the original 1988 paper on one-loop threshold effects in string unification, primarily identifying a missing factor of 2 in key formulas related to gauge coupling unification and super-trace calculations. The corrections, especially the reinstatement of the factor of 2 in the effective GUT scale and beta-function coefficients, restore consistency between field-theory and string-theory formulations, resolving long-standing confusion in phenomenological string unification studies.

ABSTRACT

The original paper, as published in Nuclear Physics B in 1988, had a few factor-of-two errors. Some people got confused by those errors. The purpose of these errata is to make things clear. The revised version of the complete article is also posted to hep-th.

Motivation & Objective

• To correct a series of persistent errors in the original 1988 paper on one-loop threshold effects in string unification that had led to widespread confusion in the literature.
• To clarify that the $¯{MS}$ scheme used was actually $¯{DR}$, the modified minimal subtraction scheme for dimensional reduction, not standard dimensional regularization.
• To correct the missing factor of 2 in the effective GUT scale formula (eq. 26), which affects the numerical value of the unification scale.
• To fix erroneous coefficients in field-theory and string-theory formulas, including $16\pi^2$ instead of $4\pi^2$, and $1/16\pi^2$ instead of $1/4\pi^2$.
• To correct the $b_a$ coefficients and associated super-trace functions, which were too small by a factor of 2, and to restore consistency across field-theory and string-theory expressions.

Proposed method

• Identifies and corrects the mislabeling of the $¯{MS}$ scheme as $¯{DR}$, clarifying the renormalization scheme used in the original paper.
• Revises the formula for $\xi^{\prime\prime}$ on page 154, correcting the sign and logarithmic term to $\xi^{\prime\prime} = 1 + \log(2/\sqrt{27}\pi) - \gamma \approx -1.6767$.
• Restores the missing factor of 2 in equation (26), which defines the effective GUT scale $M_{\rm GUT}$, now correctly given as $M_{\rm GUT} = \frac{2e^{(1-\gamma)/2}3^{-3/4}}{\sqrt{2\pi\alpha^\prime}} \approx g_{\rm GUT} \times 5.27 \times 10^{17}\ \text{GeV}$.
• Corrects the coefficients in equations (1), (2), (7), (24) from $4\pi^2$ to $16\pi^2$, and adjusts inverse coefficients in (5) and (21) from $1/4\pi^2$ to $1/16\pi^2$.
• Reinstates the missing factor of 2 in the super-trace expressions for $\mathbf{B}_a(t)$ and $\mathcal{B}_a(\tau,\bar{\tau})$, including in equations (5), (8), (22), and (23), ensuring consistency between field-theory and string-theory formulations.
• Resolves inconsistent trace conventions by specifying that traces over massless states count each CPT-conjugate pair only once, and corrects the infrared limits of $\mathbf{B}_a(t)$ and $\mathcal{B}_a(\tau,\bar{\tau})$ to match the corrected $b_a$ coefficients.

Experimental results

Research questions

• RQ1 What is the correct value of $\xi^{\prime\prime}$ in the context of one-loop threshold corrections, and how does it differ from the original publication?
• RQ2 Why did the original paper's formula for the effective GUT scale $M_{\rm GUT}$ lack a factor of 2, and what is the corrected expression?
• RQ3 How do the corrected coefficients in the field-theory and string-theory formulas—specifically $16\pi^2$ instead of $4\pi^2$—affect the computation of threshold corrections?
• RQ4 What is the impact of the missing factor of 2 in the $b_a$ coefficients and super-trace functions on the consistency between field-theory and string-theory unification calculations?
• RQ5 How do differing normalization conventions for gauge generators ($\tr(Q_a^2) = \frac{1}{2}$ vs. $\tr(Q_a^2) = 2$) affect the interpretation of the coupling constants and the final unification scale?

Key findings

• The formula for $\xi^{\prime\prime}$ is corrected to $\xi^{\prime\prime} = 1 + \log(2/\sqrt{27}\pi) - \gamma \approx -1.6767$, resolving a sign and logarithmic error in the original paper.
• The effective GUT scale is corrected to include a missing factor of 2, resulting in $M_{\rm GUT} \approx g_{\rm GUT} \times 5.27 \times 10^{17}\ \text{GeV}$, which is consistent with phenomenological normalization conventions.
• The coefficients in equations (1), (2), (7), (24) are corrected from $4\pi^2$ to $16\pi^2$, and the inverse coefficients in (5) and (21) are corrected from $1/4\pi^2$ to $1/16\pi^2$, restoring consistency with standard field-theory normalization.
• The $b_a$ coefficients and their associated super-trace functions $\mathbf{B}_a(t)$ and $\mathcal{B}_a(\tau,\bar{\tau})$ are corrected to include a missing factor of 2, which doubles the values of the threshold corrections $\Delta_a$.
• The corrected super-trace formula for one-loop threshold corrections is $\Delta_a = 2\,\mathrm{str}_{M\sim M_{\rm GUT}}\left(Q_a^2\left(\frac{1}{12} - \chi^2\right)\log\frac{M^2_{\rm GUT}}{M^2}\right)$, now properly accounting for the full contribution.
• The infrared limits of $\mathbf{B}_a(t)$ and $\mathcal{B}_a(\tau,\bar{\tau})$ are corrected to match the $b_a$ coefficients, now defined as $b_a = -\frac{11}{3}\tr_V(Q_a^2) + \frac{2}{3}\tr_F(Q_a^2) + \frac{1}{3}\tr_S(Q_a^2)$, with traces counting CPT-conjugate pairs once.