Update docs for radiative correction
Added summaries for Sirlin's articles and spec file based on them.
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# Accurate Verification of the Conserved-Vector-Current and Standard-Model Predictions
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A. Sirlin and R. Zucchini
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*Physical Review Letters*, 20 October 1986
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Department of Physics, New York University, New York, New York 10003
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(Received 11 June 1986)
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PACS numbers: 23.40.Bw, 12.15.Ji, 13.40.Ks
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---
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## Abstract
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An approximate analytic calculation of \(O(Z\alpha^{2})\) corrections to Fermi decays is presented. When the analysis of Koslowsky et al. is modified to take into account the new results, it is found that each of the eight accurately studied \(\mathcal{F}t\) values differs from the average by \(\leq 1\sigma\), thus significantly improving the comparison of experiments with conserved-vector-current predictions. The new \(\mathcal{F}t\) values are lower than before, which also brings experiments into very good agreement with the three-generation standard model, at the level of its quantum corrections.
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---
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## 1. Introduction and Motivation
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The analysis of superallowed Fermi transitions has long been a cornerstone in the study of weak interactions and plays a basic role in verification of the standard model (SM) at the level of its quantum corrections.\(^{1,2}\) In recent years, experiments have attained great precision (\(\approx 0.1\%\)), and at this level a sharp discrepancy has been found between the four low-\(Z\) and the four high-\(Z\) decays.\(^{3,4}\)
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Motivated by this experimental situation and compelling theoretical arguments, the authors have reexamined the \(O(Z\alpha^{2})\) corrections to Fermi decays. These are defined as the residual \(O(Z\alpha^{2})\) corrections not contained in the product \(F(Z,E)(1 + \delta_{1})\), where \(F(Z,E)\) is the Fermi function and \(\delta_{1}\) the \(O(\alpha)\) correction.
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The problem of evaluating \(O(Z\alpha^{2})\) and \(O(Z^{2}\alpha^{3})\) corrections was tackled in the influential papers of Jaus and Rasche\(^{5}\) (called I) and Jaus\(^{6}\) (called II). These authors developed a systematic perturbative method in the framework of the independent-particle model. However, the results obtained in these pioneering papers face severe difficulties when confronted with theoretical expectations based on general theorems of perturbative quantum field theory.
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---
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## 2. General Theoretical Considerations
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### 2.1 Diagram Classification
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The crucial diagrams are depicted in two figures:
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- **Figure 1**: Leading corrections of \(O(Z\alpha^{2})\). Here \(q\) stands for the momentum of the Coulombic photon interacting with the daughter nucleus, while \(k\) represents the momentum of the usual four-dimensional photon. In diagram (a), one subtracts from the diagram on the left the same diagram with \(q\) set equal to zero in the indicated propagator.
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- **Figure 2**: Nonleading corrections of \(O(Z\alpha^{2})\).
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The combination of graphs in Fig. 1 is finite in both the ultraviolet and infrared domains. The same is true of the diagrams of Fig. 2 after charge renormalization [which affects Fig. 2(a)] and mass renormalization [which must be included in Fig. 2(b)]. Because all relevant mass scales are \(<< m_{W}\), these \(O(Z\alpha^{2})\) contributions may be treated in the local Fermi theory.
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### 2.2 Logarithmic Structure and Mass Singularities
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According to I, when the daughter nucleus is regarded as a point particle, the graphs in Fig. 1 lead to a large contribution to the decay rate, namely \(Z\alpha^{2}\ln(M/m)\), where \(M\) and \(m\) are the proton and positron masses, respectively. After the finite nuclear size is taken into account (II), \(M\) is essentially replaced by \(\Lambda \equiv \sqrt{6}/a\), where \(a\) is the rms radius of the charge distribution of the daughter nucleus.
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A key theoretical question is posed: Knowing the coefficient of \(\ln M\) in the corrections to the total decay rate, can one determine on the basis of general principles the coefficients of the other possible logarithms, \(\ln E_m\) and \(\ln m\), in the limit \(m/E_m \to 0\)? (Here \(E\) and \(E_m\) are the relativistic energy of the positron and its maximum value, respectively.)
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The answer is yes, by invoking the theorems on cancellation of mass singularities for total decay rates.\(^{7}\) If \(F(Z,E)\) is expanded in powers of the bare charge \(e_0\), there are no \(m\) singularities in the corrections to the total decay rate. Thus, the only \(m\) singularities are induced by renormalization. To obtain the coefficient of \(\ln m\), one expands: \(F(Z,E) = 1 - \pi Z a_0 / \beta + \ldots\) (with \(a_0 = e_0^2/4\pi\)) and inserts \(a_0 = a[1 + (2a/3\pi)\ln(1/m) + \ldots]\). The \(m\) singularity is associated with the renormalization of the vacuum-polarization graph of Fig. 2(a).
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With knowledge of the coefficients of \(\ln M\) and \(\ln m\), the coefficient of \(\ln E_m\) is determined, leading to the conclusion that the corrections to the total decay rate must be of the form:
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\[\frac{\overline{\Delta P}}{\overline{P}_0} = Z a^2 \left[ \ln\left(\frac{M}{2E_m}\right) - \ln\left(\frac{2E_m}{m}\right) + c + \ldots \right] \quad (1)\]
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for a point daughter nucleus, and a similar expression with \(M \to \Lambda\) when finite nuclear size is considered. Here the bars denote integration over the positron spectrum, \(c\) is a constant, and the ellipses stand for terms that vanish as \(m/E_m \to 0\) as well as very small energy-dependent contributions of \(O(E/M, E_m/\Lambda)\).
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Neglecting terms that vanish as \(m/E_m \to 0\) corresponds to treating the positrons in the extreme relativistic approximation (ERA). Because in all accurately measured superallowed decays the positron is relativistic throughout most of the spectrum, this is expected to be a good approximation, especially for high-\(Z\) nuclei.
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### 2.3 Discrepancy with Previous Results
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It is very difficult to reconcile the results of I and II with Eq. (1). The contribution from Fig. 1 was expressed in those papers as \(Z\alpha^2[\ln(M/m) + \Delta^0(E_0)]\), where \(E_0 = E_m/m\). The term \(\Delta^0(E_0)\) was evaluated numerically and found to be a positive and monotonically increasing function of \(E_0\). Instead, from Eq. (1) one expects contributions from Fig. 1 to be of the form:
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\[Z\alpha^2[\ln(M/2E_m) + c' + \ldots] = Z\alpha^2[\ln(M/m) - \ln(2E_0) + c' + \ldots]\]
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so that \(\Delta^0(E_0)\) should be a monotonically decreasing function of \(E_0\). Moreover, unless \(c'\) is a large positive constant, one expects \(\Delta^0(E_0)\) to be negative, especially for the high-\(Z\) cases.
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---
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## 3. Detailed Calculation
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### 3.1 Point Nucleus Contribution from Figure 1
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The starting point is the consideration of Fig. 1 for a point daughter nucleus. Only terms of leading order in \(M\) are retained because other contributions involve additional powers of \(k\) and \(q\) and, as explained in II, are suppressed by small factors of \(O(\Lambda/M)\) when finite nuclear size is considered.
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After evaluating the two-loop integrals over \(k\) and \(q\) and performing the appropriate traces, the contribution of Fig. 1 at the level of the positron spectrum is:
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\[\frac{\Delta P_1}{P_0} = \frac{2Z\alpha^2 M}{\pi} \int_0^1 \frac{u du}{(1-u)^{1/2}} \int_0^1 dv \int_0^1 dz \frac{1}{A^2} \left[ Lv \tan^{-1}\left(\frac{Lv}{X}\right) + \frac{9(L^2 v^2 - A^2)}{X} \frac{\pi}{2} + \ldots \right], \quad (2)\]
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where \(L\) is the three-momentum of the positron, \(A^2 \equiv a_0^2 - L^2(1-uv)\), \(X \equiv (A^2 - L^2 v^2)^{1/2}\), \(a_0 = Muv(1-z) + E(1-uv)\), and the ellipsis represents subdominant contributions.
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In the ERA, including subdominant contributions:
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\[\Delta P_1 = P_0 Z\alpha^2[\ln(M/2E) + 5/2 - \pi^2/6]. \quad (3)\]
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In the opposite (nonrelativistic) limit, setting \(L = 0\) in Eq. (2):
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\[\Delta P_1 = P_0 Z\alpha^2[\ln(M/m) + 4 - 6\ln 2].\]
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Writing \(\Delta P_1/P_0 = Z\alpha^2[\ln(M/E) + f(E)]\), one finds that \(f(E)\) varies from a small positive constant (\(\sim -0.16\)) in the extreme relativistic domain to a small negative one (\(\sim -0.16\)) in the NR limit, indicating that \(\ln(M/E)\) is by far the dominant contribution.
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### 3.2 Figure 2 Contributions
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The diagrams from Fig. 2 are much less important numerically. In the ERA:
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\[\Delta P_2 = P_0 Z\alpha^2\left[-\frac{5}{12}\ln(2E/m) + \frac{1}{9}\pi^2 - \frac{7}{18}\right]. \quad (4)\]
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### 3.3 Combined Result
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Combining Eqs. (3) and (4):
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\[\frac{\Delta P}{P_0} = Z\alpha^2\left[\ln\left(\frac{M}{2E}\right) - \frac{5}{12}\ln\left(\frac{2E}{m}\right) + \frac{43}{18}\right]. \quad (5)\]
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Averaging over the positron spectrum\(^{8}\):
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\[\frac{\overline{\Delta P}}{\overline{P}_0} = Z\alpha^2\left[\ln\left(\frac{M}{2E_m}\right) - \frac{5}{12}\ln\left(\frac{2E_m}{m}\right) + \frac{133}{36}\right], \quad (6)\]
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or, recalling \(E_0 = E_m/m\):
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\[\frac{\overline{\Delta P}}{\overline{P}_0} = Z\alpha^2\left[\ln\left(\frac{M}{m}\right) - \frac{17}{12}\ln(2E_0) + \frac{133}{36}\right]. \quad (7)\]
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Equation (6) is in agreement with the theoretically expected Eq. (1) and determines \(c = 133/36\). The constant \(c'\) is obtained by averaging Eq. (3): \(c' = \frac{5}{2} - \pi^2/6 + \frac{47}{60} \approx 1.64\). Thus \(-\ln(2E_0) + c'\) is negative for the eight decays, confirming the expectation that \(\Delta^0(E_0)\) should be negative.
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---
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## 4. Finite Nuclear Size Effects
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Neglecting terms of \(O(\Lambda/M)\), the fractional correction is:
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\[\Delta = Z\alpha^2\left[1 - \gamma - \ln(Ma) - 4\pi \int_0^\infty \rho(r)r^2 \ln(r/a)dr\right],\]
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where \(\gamma = 0.5772\) and \(\rho(r)\) is the nuclear charge density of the daughter nucleus.
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For the modified Gaussian distribution\(^{9,10}\): \(\rho(r) = \mathrm{const}\times (1 + \alpha k^2 r^2/a^2)e^{-r^2 k^2/a^2}\) with \(k^2 = 3(2+5\alpha)/2(2+3\alpha)\) and \(\alpha = (Z-2)/3\):
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\[\Delta = -Z\alpha^2[\ln(M/\Lambda) + \kappa_1(Z)], \quad (8)\]
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where \(\kappa_1(Z) \equiv \frac{1}{2}[\gamma + \ln(3/2k^2) + 2\alpha/(2+3\alpha)]\) is a small constant varying from 0.33 for \(Z=7\) to 0.36 for \(Z=26\).
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For a uniformly charged sphere of radius \(R\) [\(a = (\frac{3}{5})^{1/2}R\)], \(\Delta\) is analogous with \(\kappa_1(Z)\) replaced by \(\kappa_2(Z) \equiv \gamma - \frac{4}{3} + \frac{1}{2}\ln 10 = 0.40\).
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The variation in \(\kappa\) is negligible; the authors use Eq. (8) with \(\kappa_1(Z) = 0.35\) and \(a = (\frac{3}{5})^{1/2}r_0 A^{1/3}\) fm (equivalently, \(M/\Lambda = r_0 A^{1/3}/0.665\)), with \(r_0\) given by Wilkinson.\(^{11}\)
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The sum \(\langle\delta_2(E)\rangle\) of Eqs. (7) and (8) is the final expression for the \(O(Z\alpha^2)\) corrections.\(^{12}\)
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---
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## 5. Estimate of \(O(Z^2\alpha^3)\) Corrections
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An estimate of \(O(Z^2\alpha^3)\) corrections (residual contributions not contained in \(F(Z,E)(1+\delta_1+\delta_2)\)) is given. Jaus\(^{6}\) evaluated the coefficient of the ultraviolet logarithm from diagrams analogous to Fig. 1 (with one additional Coulombic photon attached to the positron). Based on the earlier discussion, this is identified as \(\ln(\Lambda/E)\). A heuristic estimate that includes \(m\) singularities is:\(^{14}\)
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\[\langle\delta_1^{\mathrm{HE}}(E)\rangle = Z^2 a^3 \left[ a\ln(\Lambda/E_m) + b\ln(2E_m/m) + d f(E)\ln(2E_m/m) + \ldots \right]\]
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where \(\alpha = 1.232\), \(b = (4/3\pi)(\frac{11}{4} - \gamma - \pi^2/6)\), \(d = 4/3\pi\), \(f(E) \equiv \ln(2E_m/m) - 5/6\), \(g(E) \equiv \frac{1}{2}[\ln^2(Rm) - \ln^2(2E/m)] + (\frac{5}{4})\ln(2RE)\), \(R = (\frac{5}{4})^{1/2}a\), and \(h = -0.649\) is a constant adjusted to cancel spurious \(m\) singularities. This will be used as an estimate of the error made in stopping the calculation at \(O(Z\alpha^2)\).
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---
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## 6. Results: \(\mathcal{F}t\) Values and CVC Test
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**Table I** presents fractional radiative corrections (in percent) and \(\mathcal{F}t\) values:
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| Decay | \(\langle\delta_1^{\mathrm{qu}}\rangle\) | \(\langle\delta_2\rangle\) | \(\langle\delta_1^{\mathrm{HE}}\rangle\) | \(\mathcal{F}t\) (old)\(^a\) (s) | \(\mathcal{F}t\) (new)\(^b\) (s) |
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|-------|---------------------|----------------|------------------------|---------------------------|---------------------------|
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| \(^{14}\mathrm{O}\) | 1.29 | 0.21 | 0.02 | 3075.5 ± 3.9 | 3073.4 ± 3.9 |
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| \(^{26}\mathrm{Al}^{m}\) | 1.11 | 0.30 | 0.04 | 3072.9 ± 3.7 | 3066.9 ± 3.9 |
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| \(^{34}\mathrm{Cl}\) | 1.00 | 0.35 | 0.06 | 3076.9 ± 4.7 | 3066.9 ± 5.0 |
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| \(^{38}\mathrm{K}\) | 0.96 | 0.37 | 0.07 | 3076.6 ± 4.6 | 3064.2 ± 5.1 |
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| \(^{42}\mathrm{Sc}\) | 0.94 | 0.40 | 0.08 | 3089.3 ± 7.5 | 3074.7 ± 7.9 |
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| \(^{46}\mathrm{V}\) | 0.90 | 0.41 | 0.10 | 3088.6 ± 4.3 | 3071.3 ± 5.2 |
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| \(^{50}\mathrm{Mn}\) | 0.87 | 0.43 | 0.11 | 3085.9 ± 5.7 | 3066.2 ± 6.5 |
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| \(^{54}\mathrm{Co}\) | 0.84 | 0.44 | 0.12 | 3087.5 ± 4.4 | 3065.4 ± 5.7 |
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| **Average** | | | | **3080.1 ± 2.4** | **3068.6 ± 1.8** |
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| **CL** | | | | **3%** | **80%** |
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\(^a\) Taken from Ref. 4.
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\(^b\) Modified values obtained in this paper on the basis of \((\delta_1^{\mathrm{qu}} + \delta_2)\); \(\langle\delta_1^{\mathrm{HE}}\rangle\) has been included as a theoretical error.
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The \(\langle\delta_2(E)\rangle\) correction ranges from 0.21% for \(^{14}\mathrm{O}\) to 0.44% for \(^{54}\mathrm{Co}\), while the corresponding correction reported in II varies from 0.26% to 0.93%. The main difference can be traced to the evaluation of \(\Delta^0(E_0)\) arising from Fig. 1.\(^{13}\) A more physical characterization: while papers I and II state that the leading contribution is \(Z\alpha^2\ln(M/m)\), this analysis indicates the dominant contribution to the spectrum is roughly \(Z\alpha^2\ln(\Lambda/E)\), a significantly different logarithm.
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The overall pattern is much better than before. Seven of the \(\mathcal{F}t\) values differ from the new weighted average by \(< 1\sigma\), while \(^{14}\mathrm{O}\) differs by 1.2\(\sigma\). The \(\chi^2/\nu\) for the old and new fits are 2.2 and 0.55, respectively (\(\nu = 7\)), corresponding to confidence levels of 3% and 80%. If \(\langle\delta_1^{\mathrm{HE}}\rangle\) is not included in the error, the CL is slightly reduced to 72%.
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The weighted averages of the four low-\(Z\) and four high-\(Z\) values are now \(3068.4\pm 2.2\) s and \(3069.0\pm 3.1\) s, respectively. Thus, the sharp discrepancy pointed out in Refs. 3 and 4 has disappeared.
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---
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## 7. Standard Model Test and \(V_{ud}\) Determination
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**Table II** lists values of \(V_{ud}\) and \(V_{ud}^2 + V_{us}^2 + V_{ub}^2\) based on various combinations of the new \(\mathcal{F}t\) values and \(V_{us} = 0.220\pm 0.002\):\(^{17}\)
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| Combination | \(V_{ud}\) | \(V_{ud}^2 + V_{us}^2 + V_{ub}^2\) |
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|-------------|--------|-------------------------------|
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| (a) \(^{14}\mathrm{O}\) | 0.9740 ± 0.0015 | 0.9971 ± 0.0031 |
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| (b) Average of \(^{14}\mathrm{O}\) and \(^{26}\mathrm{Al}^{m}\) | 0.9745 ± 0.0013 | 0.9981 ± 0.0027 |
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| (c) Average of four low-\(Z\) decays | 0.9748 ± 0.0011 | 0.9986 ± 0.0023 |
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| (d) Average of 8 decays | 0.9747 ± 0.0011 | 0.9984 ± 0.0023 |
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The treatment of inner corrections and calculation of errors is the same as in Ref. 2.\(^{16}\) There is now very good agreement between a large number of experiments and the three-generation SM, at the level of its quantum corrections. If the latter were not included, the unitarity bound would be exceeded by \(\approx 3.7\%\) and the SM would not be tenable.
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In spite of small errors, the possibility of a fourth generation with substantial mixing is still open. Using the last entry in Table II, one finds \(|V_{ub}| \leq 0.064\) (90% CL). These calculations can put sharp constraints on various types of new physics, as competing theories must match the SM at a high level of precision. On the experimental side, accurate measurements of \(^{10}\mathrm{C}\) (Ref. 18) and \(\pi\beta\) decay would be important to verify the consistent trend.
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---
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## References
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1. A. Sirlin, Nucl. Phys. B71, 29 (1974); Rev. Mod. Phys. 50, 573 (1978); Ann. N.Y. Acad. Sci. 461, 521 (1986).
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2. W. J. Marciano and A. Sirlin, Phys. Rev. Lett. 56, 22 (1986).
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3. I. S. Towner and J. C. Hardy, in *Proceedings of the Seventh International Conference on Atomic Masses and Fundamental Constants*, edited by O. Klepper (Gesellschaft fur Schwerionenforschung, Darmstadt, 1984), p. 564.
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4. V. T. Koslowsky et al., in Ref. 3, p. 572.
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5. W. Jaus and G. Rasche, Nucl. Phys. A143, 202 (1970).
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6. W. Jaus, Phys. Lett. 40, 616 (1972).
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7. T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959); T. Kinoshita, J. Math. Phys. (N.Y.) 3, 650 (1962); T. D. Lee and M. Nauenberg, Phys. Rev. B 133, 1549 (1964).
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8. The positron spectrum \(S(E)\) in the ERA: \(S(E)dE = E_0^2 x^2(1-x)^2 dx\), \(x \equiv E/E_m\), which leads to \(\langle\ln x\rangle = -\frac{1}{20}\).
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9. R. Hofstadter, Annu. Rev. Nucl. Sci. 7, 231 (1957).
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10. W. Jaus, Nucl. Phys. A177, 70 (1971).
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11. D. H. Wilkinson, in *Nuclear Physics with Heavy Ions and Mesons*, edited by R. Balian, M. Rho, and G. Ripka (North-Holland, Amsterdam, 1978), p. 877.
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12. All reference to the nucleon (in contraposition to the nucleus) has disappeared in \(\delta_1(E)\), which strongly suggests that the leading contributions do not depend on the independent-particle model.
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13. Note that an additional contribution of the form \(\mathrm{const}/E_0\) cannot explain the difference for all the nuclei.
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14. A. Sirlin, unpublished.
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15. A. Sirlin, Phys. Rev. 164, 1767 (1967).
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16. The \(\mathcal{F}t\) values from Ref. 4 include the Coulomb correction evaluated by I. S. Towner, J. C. Hardy, and M. Harvey, Nucl. Phys. A284, 296 (1977), with an allowance for theoretical uncertainties. In extracting \(V_{ud}\), the total errors in the \(\mathcal{F}t\) values have been further doubled (Ref. 2).
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17. H. Leutwyler and M. Roos, Z. Phys. C 25, 91 (1984); J. Donoghue, B. Holstein, and S. Klimt, to be published.
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18. The ERA is less reliable for low-\(Z\) decays such as \(^{10}\mathrm{C}\) but the \(O(Z\alpha^{2})\) corrections are less important for such cases. It would be desirable to perform new computer-based calculations that retain the small terms neglected in the ERA as well as those of \(O(\Lambda/M)\).
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