Decoupling of the right-handed neutrino contribution to the Higgs mass in supersymmetric models

Abstract

Recently, it has been argued that in the supersymmetric extension of the seesaw-extended Standard Model, heavy right-handed neutrinos and sneutrinos may give corrections as large as a few GeV to the mass of the lightest neutral CP-even Higgs boson, even if the soft supersymmetry-breaking parameters are of order the electroweak scale. The presence of such large corrections would render precise Higgs masses incalculable from measurable low-energy parameters. We show that this is not the case: decoupling is preserved in the appropriate sense and right-handed (s)neutrinos, if they exist, have negligible impact on the physical Higgs masses.

Appendix: Approximate renormalized self-energies and tadpoles

It is convenient to have analytic approximations for the self-energy functions and tadpoles in order to see how the terms sensitive to the seesaw scale explicitly cancel in the expressions for the Higgs masses [Eqs. (2.18) and (2.19)]. Following Ref. [27], we perform a series expansion in powers of $m_{D}^{2}$. At $\mathcal{O}(m_{D}^{0})$, the contributions are insensitive to the seesaw scale. At $\mathcal{O}(m_{D}^{4})$, each self-energy scales as $m_{M}^{-2}$, exhibiting decoupling independently, in agreement with Ref. [27]. In contrast, decoupling occurs in the $\mathcal{O}(m_{D}^{2})$ terms due to non-trivial cancellations among the various terms in Eqs. (2.18) and (2.19).

Below we give the $\mathcal{O}(m_{D}^{2})$ contributions to the real parts of the self-energy functions and tadpoles⁷ in d-spacetime dimensions using dimensional regularization, expanded to leading order with respect to the mass hierarchy

$$ \bigl\{m_Z^2,p^2,m_A^2,m_H^2 \bigr\}\ll m_S^2\ll m_M^2. $$

(A.1)

It is convenient to adopt the shorthand notation

$$ \log\widetilde{Q}^2\equiv\frac{1}{\epsilon}-\gamma+\log\bigl(4\pi Q^2\bigr), $$

(A.2)

where Q is the renormalization scale, $\epsilon\equiv2-\frac{1}{2} d$ and γ is Euler’s constant. The $\mathcal{O}(m_{D}^{2})$ contribution to Σ _hh (p ²) at leading order in the mass hierarchy [cf. Eq. (A.1)] is given by

$$\begin{aligned} \varSigma_{hh}\bigl(p^2\bigr)&= \frac{g^2 m_D^2}{64 \pi^2 c_W^2 m_Z^2} \biggl\{ \frac{2\cos^2\alpha }{\sin^2\beta} \biggl[2m_S^2 \log\frac{m_M^2}{\widetilde{Q}^2} \\ &\quad {}+ \bigl(m_Z^2-p^2\bigr) \biggl(1-\log\frac{m_M^2}{\widetilde{Q}^2} \biggr) \biggr] \\ &\quad {}+ m_Z^2 \biggl(1-\log\frac{m_M^2}{\widetilde{Q}^2} \biggr) \bigl[ \cos^2\alpha\bigl(4-3 \cot^2\beta\bigr) \\ &\quad {}+2 \sin2 \alpha \cot\beta-\sin^2\alpha\bigr] \biggr\}, \end{aligned}$$

(A.3)

where p is the incoming four momentum. Likewise, Σ _HH (p ²) is obtained by making the replacement $\alpha\to\alpha-\frac{1}{2}\pi$ in Eq. (A.3),

$$\begin{aligned} \varSigma_{HH}\bigl(p^2\bigr)&= \frac{g^2 m_D^2}{64 \pi^2 c_W^2 m_Z^2} \biggl\{ \frac{2\sin^2\alpha }{\sin^2\beta} \biggl[2m_S^2 \log\frac{m_M^2}{\widetilde{Q}^2} \\ &\quad {}+\bigl(m_Z^2-p^2\bigr) \biggl(1-\log\frac{m_M^2}{\widetilde{Q}^2} \biggr) \biggr] \\ &\quad {}+ m_Z^2 \biggl(1-\log\frac{m_M^2}{\widetilde{Q}^2} \biggr) \bigl[ \sin^2\alpha\bigl(4-3 \cot^2\beta\bigr) \\ &\quad {}-2 \sin2 \alpha \cot\beta-\cos^2\alpha\bigr] \biggr\}. \end{aligned}$$

(A.4)

For completeness, we provide the $\mathcal{O}(m_{D}^{2})$ contribution to the real parts of all the other relevant self-energy functions [at leading order in the mass hierarchy, Eq. (A.1)],

$$\begin{aligned} \begin{aligned} &\varSigma_{ZZ}\bigl(m_Z^2\bigr)= \frac{g^2 m_D^2}{64 \pi^2 c_W^2 m_Z^2}2m_Z^2 \biggl(1-\log \frac{m_M^2}{\widetilde{Q}^2} \biggr), \\ &\varSigma_{AA}\bigl(m_A^2\bigr)= \frac{g^2 m_D^2}{64 \pi^2 c_W^2 m_Z^2} \biggl\{\frac{\cos2 \beta}{\sin ^2\beta} \biggl[2 m_S^2 \log\frac{m_M^2}{\widetilde{Q}^2}\\ &\hphantom{\varSigma_{AA}\bigl(m_A^2\bigr)=}\quad {}- \bigl(m_A^2 +m_Z^2 \bigr) \biggl(1-\log\frac{m_M^2}{\widetilde{Q}^2} \biggr) \biggr] \\ &\hphantom{\varSigma_{AA}\bigl(m_A^2\bigr)=}\quad {}+\frac{1}{\sin^2\beta} \biggl[2 m_S^2 \log\frac{m_M^2}{\widetilde{Q}^2}-m_A^2\\ &\hphantom{\varSigma_{AA}\bigl(m_A^2\bigr)=}\quad {}\times \biggl(1-\log \frac{m_M^2}{\widetilde{Q}^2} \biggr) \biggr]\\ &\hphantom{\varSigma_{AA}\bigl(m_A^2\bigr)=}\quad {}+2 \cos2 \beta m_Z^2 \biggl(1-\log\frac{m_M^2}{\widetilde{Q}^2} \biggr) \biggr\} , \\ &\varSigma_{GG}(0)=\frac{g^2 m_D^2}{64 \pi^2 c_W^2 m_Z^2} \biggl\{ 4 m_S^2 \log\frac{m_M^2}{\widetilde{Q}^2}\\ &\hphantom{\varSigma_{GG}(0)=}\quad {}-2 \cos2 \beta m_Z^2 \biggl(1- \log\frac{m_M^2}{\widetilde{Q}^2} \biggr) \biggr\}, \\ &\frac{A_h}{\sqrt{2}v}= \frac{g^2 m_D^2}{64 \pi^2 c_W^2 m_Z^2} \biggl \{ \frac{\cos\alpha}{\sin\beta} \biggl[4 m_S^2 \log\frac{m_M^2}{\widetilde{Q}^2}\\ &\hphantom{\frac{A_h}{\sqrt{2}v}=}\quad {}-m_Z^2 \biggl(1-\log\frac{m_M^2}{\widetilde{Q}^2} \biggr) \biggr] \\ &\hphantom{\frac{A_h}{\sqrt{2}v}=}\quad {}+m_Z^2 (\sin\alpha\cos\beta+3 \cos\alpha\sin\beta)\\ &\hphantom{\frac{A_h}{\sqrt{2}v}=}\quad {}\times \biggl(1-\log\frac{m_M^2}{\widetilde{Q}^2} \biggr) \biggr\}, \\ &\frac{A_H}{\sqrt{2}v}= \frac{g^2 m_D^2}{64 \pi^2 c_W^2 m_Z^2} \biggl \{\frac{\sin\alpha}{\sin\beta} \biggl[4 m_S^2 \log\frac{m_M^2}{\widetilde{Q}^2}\\ &\hphantom{\frac{A_H}{\sqrt{2}v}=}\quad {}- m_Z^2 \biggl(1-\log\frac{m_M^2}{\widetilde{Q}^2} \biggr) \biggr] \\ &\hphantom{\frac{A_H}{\sqrt{2}v}=}\quad {}+m_Z^2 (3 \sin\alpha\sin\beta- \cos\alpha\cos\beta)\\ &\hphantom{\frac{A_H}{\sqrt{2}v}=}\quad {}\times \biggl(1-\log\frac{m_M^2}{\widetilde{Q}^2} \biggr) \biggr\}. \end{aligned} \end{aligned}$$

(A.5)

Next, we compute the $\mathcal{O}(m_{D}^{2})$ contributions [at leading order in the mass hierarchy, Eq. (A.1)] to the counterterm δtanβ in the various renormalization schemes. In the HM scheme, δtanβ is given by Eq. (2.26). Using the above expressions for the self-energy functions, along with Eq. (2.20) and the following tree-level relations (cf. Eq. (A.20) of Ref. [39]):

$$ \begin{aligned} &m_{ht}^2=-\frac{m_Z^2\cos2\beta\sin(\beta+\alpha)}{\sin(\beta-\alpha )},\\ &m_{Ht}^2=\frac{m_Z^2\cos2\beta\cos(\beta+\alpha)}{\cos(\beta-\alpha)}, \end{aligned} $$

(A.6)

we obtain after considerable simplification,

$$\begin{aligned} \delta\tan\beta_{{\rm HM}}&= \delta\tan\beta_{\overline{\rm DEC}} \\ &=\frac {g^2m_D^2}{32\pi^2c_W^2m_Z^2\sin2\beta} \\ &\quad {}\times \biggl({\frac{1}{\epsilon}-\gamma+\log{4\pi}-\log{\frac{m_M^2}{Q^2}}+1} \biggr) . \end{aligned}$$

(A.7)

Note that the $\mathcal{O}(m_{D}^{2})$ contributions to the counterterm δtanβ in the HM and DEC schemes are equivalent, in light of the absence of non-decoupling terms in Eq. (3.8). Indeed, the $\mathcal{O}(m_{D}^{2})$ contribution to δtanβ is independent of the tree-level Higgs mixing angle α. Although this result is obvious in the DEC scheme (which is defined via Higgs wave function counterterms that are evaluated at α=0), the cancellation of the α-dependence in the $\mathcal{O}(m_{D}^{2})$ contribution to $\delta\tan\beta_{\rm{HM}}$ [defined in Eq. (2.26)] is highly non-trivial.

In contrast, in the $\overline{\rm DR}$ scheme only the ϵ ⁻¹−γ+log4π is retained, so that the corresponding $\mathcal{O}(m_{D}^{2})$ contribution is simply

$$\begin{aligned} \delta\tan\beta_{\overline{\rm DR}}&=\frac{g^2m_D^2}{32\pi ^2c_W^2m_Z^2\sin2\beta} \biggl(\frac{1}{\epsilon}- \gamma+\log{4\pi} \biggr). \end{aligned}$$

(A.8)