Magnetic dipole moment of neutral particles from quantum corrections at two-loop order

The tentative gamma-ray line in the Fermi data at $\sim 135\mathrm{GeV}$ motivates a dark matter candidate that couples to photons through loops of charged messengers. It was recently shown that this model can explain the observed line, but achieving the correct phenomenology requires a fairly sizable coupling between the weakly interacting massive particle (WIMP) and the charged messengers. While strong coupling by itself is not a problem, it is natural to wonder whether the phenomenological success is not spoiled by higher order quantum corrections. In this work we compute the dominant two-loop contributions to the electromagnetic form factors of the WIMP and show that over a large portion of the relevant parameter space these corrections are under control, and the phenomenology is not adversely affected. We also discuss more generally the effects of these form factors on signals in direct-detection experiments as well as on the production of the WIMP candidate in colliders. In particular, for low masses of the charged messengers, the production rate at the LHC enjoys an enhancement from the threshold singularity associated with these charged states.

Figure 1

Magnetic dipole operator generated at one loop.

Figure 2

All diagrams at two-loop order contributing to the magnetic dipole moment involving only the Yukawa interactions.

Figure 3

All diagrams at two-loop orders contributing to the magnetic dipole moment involving the scalar quartic vertex.

Figure 4

One- and two-loop contributions to the form factors $F_1$ (top pane) and $F_2$ (bottom pane) as a function of the messenger mass, for different values of the RG scale. Here we set the WIMP mass $m_{\chi }=135\mathrm{GeV}$, the messenger masses $M_s=M_f=M_{\mathrm{mess}}$, the momentum transfer $q^2=4m_{\chi }^2$, and the couplings ${\alpha }_{\lambda }=1$, $\kappa =0$.

Figure 5

Two-loop contributions to the form factors $F_1$ and $F_2$ as a function of the quartic coupling $\kappa$. The form factors are normalized to their value at $\kappa =0$. Here we set the WIMP mass $m_{\chi }=2\mu =135\mathrm{GeV}$, the messanger mass $M_f=M_s=160\mathrm{GeV}$, and the Yukawa coupling ${\alpha }_{\lambda }=1$.

Figure 6

One- and two-loop contributions to the form factors $f_1$ (up) and $F_2$ (down) in the low momentum exchange limit $q^2=0$. Here $m_{\chi }=135\mathrm{GeV}$, ${\alpha }_{\lambda }=1$, $\kappa =0$, in terms of the messenger mass, with one set of messenger fields, for different values of the renormalization scale $\mu$.

Figure 7

The Yukawa coupling vs the messengers’ mass fixing the freeze-out rate at $\sigma v=6\times {10}^{-26}{\mathrm{cm}}^3/\mathrm{s}$, at one- and two-loop order, for different values of the RG scale $\mu$, in the case of one family of messengers (top pane) and two families (bottom pane). Note that the two-loop corrections in the latter case are relatively smaller, as discussed in the text.

Figure 8

Ratios between the one- and two-loop contributions to the form factors $F_1$ and $F_2$, for $m_{\chi }=2\mu =135\mathrm{GeV}$, $q^2=4m_{\chi }^2$, $\kappa =0$, and ${\alpha }_{\lambda }$ fixed by the relic abundance as in Fig. 7, for one messenger family. The jagedness of the solid curve is due to numerical precision issues coming from the fact that the overall form factor $F_1$ is close to zero (see Fig. 4). In the case of $F_2$, the ratio of one-to two-loop contributions blows up as the two-loop correction approaches zero while the one-loop result stays finite, as can be seen from Fig. 4.

Figure 9

The magnetic dipole moment normalized to the nuclear magneton plotted against the messengers’ mass fixing the freeze-out rate at $\sigma v=6\times {10}^{-26}{\mathrm{cm}}^3/\mathrm{s}$, for different values of the RG scale $\mu$ and for the case of one family of messengers (top pane) and two families (bottom pane).

Figure 10

Real and imaginary parts of $F_1$ and $F_2$ at one loop as a function of $q^2$, for $m_{\chi }=135\mathrm{GeV}$, $\alpha =2$, $\kappa =0$, and $M_{\mathrm{mess}}=200\mathrm{GeV}$. Notice the threshold singularity at the messenger mass.

Figure 11

LHC production cross section in terms of the messenger mass, in the case of unresolved form factors ($F_1=0$, $F_2=1$), at one-loop and also including two-loop effects from the determination of ${\alpha }_{\lambda }$ and from field renormalization effects for $\mu =1/2m_{\chi }$. The couplings were fixed so as to get the correct relic abundance. Note that the $q$-dependent form factors tend to enhance the cross section; the dropping of the two-loop curve at large messenger masses is related with two-loop corrections demanding lower values of ${\alpha }_{\lambda }$ for high messenger masses (see Fig. 7). We also show bounds from CMS monojet searches.

Figure 12

Diagram contributing to field and mass renormalization.