Dark matter candidate with well-defined mass and couplings

TL;DR

This paper introduces a novel spin 1/2 dark matter candidate with Higgs-like mass and unique couplings, predicting new charged particles at higher energies, advancing dark matter model understanding.

Contribution

It proposes a new Higgs-related spin 1/2 dark matter candidate with distinctive couplings, differentiating it from neutralinos and predicting new charged particles.

Findings

Identifies a Higgs-like mass for the dark matter candidate.

Predicts unconventional couplings to W and Z bosons.

Forecasts higher-energy charged spin 1/2 particles.

Abstract

We propose a Higgs-related but spin $1/2$ dark matter candidate with a mass that is comparable to that of the Higgs. This particle is a WIMP with an R-parity of $-1$, but it can be distinguished from a neutralino by its unconventional couplings to W and Z bosons. Charged spin 1/2 particles of a new kind are also predicted at higher energy.

Equations

$$S_{\Phi }=S_1+S_2$$

$$S_1=\int d^{4}x\left(\frac{1}{2}{\left(i{\overline{\sigma }}^{\mu }{D}_{\mu }\Phi \left(x\right)\right)}^{†}\left(i{\sigma }^{\nu }{D}_{\nu }\Phi \left(x\right)\right)+h.c.\right)$$

$$\displaystyle\Phi=\left(\begin{array}[]{c}\Phi_{\uparrow}\\ \Phi_{\downarrow}\end{array}\right)$$

$$\displaystyle\gamma^{\mu}=\left(\begin{array}[]{cc}0&\sigma^{\mu}\\ \overline{\sigma}^{\mu}&0\end{array}\right)\;,$$

$$S_1$$

$$=$$

$$/D^2=-D^{\mu }{D}_{\mu }+S^{\mu \nu }{F}_{\mu \nu }$$

$${\mathcal{S}}_1=\int d^{4}x\left(\frac{1}{2}{\Phi }^{†}\left(x\right)D^{\mu }{D}_{\mu }\Phi \left(x\right)-\frac{1}{2}{\Phi }^{†}\left(x\right)S^{\mu \nu }{F}_{\mu \nu }\Phi \left(x\right)\right)+h.c.$$

$$S^{\mu \nu }=\frac{1}{2}{\sigma }^{\mu \nu }$$

$$\displaystyle S^{kk^{\prime}}=\frac{1}{2}\varepsilon_{kk^{\prime}k^{\prime\prime}}\left(\begin{array}[]{cc}\sigma^{k^{\prime\prime}}&0\\ 0&\sigma^{k^{\prime\prime}}\end{array}\right)\quad,\quad S^{0k}=-\frac{i}{2}\left(\begin{array}[]{cc}\sigma^{k}&0\\ 0&-\sigma^{k}\end{array}\right)\;\;.$$

$$F_{k{k}^{\prime }}=-{\epsilon }_{k{k}^{\prime }{k}^{\prime \prime }}{B}_{k^{\prime \prime }},F_{0k}=E_k$$

$$\displaystyle-S^{\mu\nu}F_{\mu\nu}=\left(\begin{array}[]{cc}\left(\overrightarrow{B}+i\overrightarrow{E}\right)\cdot\overrightarrow{\sigma}&0\\ 0&\left(\overrightarrow{B}-i\overrightarrow{E}\right)\cdot\overrightarrow{\sigma}\end{array}\right)$$

$${\mathcal{S}}_1=\int d^{4}x\left({\Phi }^{†}\left(x\right)D^{\mu }{D}_{\mu }\Phi \left(x\right)+{\Phi }^{†}\left(x\right)\overrightarrow{B}\cdot \overrightarrow{\sigma }\Phi \left(x\right)\right).$$

$$\displaystyle\Phi^{r}=\left(\begin{array}[]{c}\Phi_{\uparrow}^{r}\\ \Phi_{\downarrow}^{r}\end{array}\right)\;,\;r=0\text{ or }+$$

$${\Phi }^r={\varphi }^r{\chi }^r\text{[no sum on }r\text{]}$$

$${\chi }_{\downarrow }^{r†}\overrightarrow{\sigma }{\chi }_{\downarrow }^r=-{\chi }_{\uparrow }^{r†}\overrightarrow{\sigma }{\chi }_{\uparrow }^r\text{[no sum on }r\text{]}$$

$${\chi }^{r†}\overrightarrow{\sigma }{\chi }^r=0.$$

$${\Phi }^{†}\left(x\right)\overrightarrow{B}\cdot \overrightarrow{\sigma }\Phi \left(x\right)=0.$$

$${\chi }^{r†}{\chi }^r=1\text{[no sum on }r\text{]},$$

$${\mathcal{S}}_1=\int d^{4}x{\varphi }^{†}\left(x\right)D^{\mu }{D}_{\mu }\varphi \left(x\right)$$

$$-{\mathcal{L}}_2=\sum _i{m}_i^2{\varphi }_i^{\ast }{\varphi }_i$$

$$-{\mathcal{L}}_2=\sum _i{m}_i^2{\Phi }_i^{†}{\Phi }_i$$

$$\displaystyle\Phi_{i}=\left(\begin{array}[]{c}H_{i,\uparrow}\\ H_{i,\downarrow}\end{array}\right)\;,$$

$$-{\mathcal{L}}_2=\sum _i{m}_i^2\left(H_{i,\uparrow }^{†}{H}_{i,\uparrow }+H_{i,\downarrow }^{†}{H}_{i,\downarrow }\right).$$

Full text

Although there is as yet no confirmed and statistically significant evidence for direct, indirect, or collider-based detection of dark matter drees , all of these experiments are entering regimes where there is now a reasonable possibility of success bertone ; snowmass1 ; snowmass2 ; snowmass3 ; freese ; lisanti ; cirelli ; klasen ; undagoitia ; fermi ; ams ; kahlhoefer ; bergstrom . Weakly interacting massive particles (WIMPs) are among the leading dark matter candidates, largely because they would have been created in about the right abundance as thermal relics if their mass is $\sim 100$ GeV.

A favorite hypothetical WIMP is the lowest-mass neutralino, a linear combination of the neutral fermionic superpartners predicted by supersymmetry (susy). However, the tension that currently exists between experiment and simple supersymmetric models may indicate that it is desirable to consider alternative scenarios for how WIMPs can naturally arise.

Here we propose a new candidate which resembles a neutralino, in that it has spin 1/2 and is made stable by having an R-parity of -1, but it is distinguished by various unusual features, including unconventional couplings to the W and Z bosons and a well-defined mass that is simply related to that of the recently discovered Higgs boson. One of the motivations for this paper is in fact the Higgs discovery ATLAS ; CMS . In the present theory, the Higgs boson of fundamental physics is somewhat analogous to the Higgs mode observed for superconductors in condensed matter physics varma . Namely, the scalar Higgs field is interpreted as an amplitude in a more complex structure described below.

We begin with an action that follows from fundamental arguments given elsewhere statistical , but is here simply postulated as a phenomenological model:

[TABLE]

where

[TABLE]

and $S_{2}$ consists of mass terms, discussed below, which result from potentially very complicated field interactions in the complete potential for Higgs scalars. Here $h.c.$ means Hermitian conjugate, the $\sigma$ matrices have their usual definitions (and are always implicitly multiplied by an appropriate identity matrix), and $D_{\mu}$ is the usual covariant derivative for the electroweak gauge fields.

Stripped of its fundamental derivation, this action still has a strong a posteriori motivation: It leads to a dark matter candidate with about the right abundance, which has a well-defined mass and well-defined couplings, which is in a desirable mass range for direct detection, which is in the accessible energy range at the LHC, and whose annihilation products are well-defined for indirect detection.

As described below, $\Phi$ consists of two Higgs-like doublets $\Phi_{\uparrow}$ and $\Phi_{\downarrow}$, with each doublet having a neutral and a charged component. This is reminiscent of the simplest description in general models with multiple electroweak Higgs doublets 2-doublets . But here each of these four components is itself a 2-component spinor, and scalar Higgs bosons are interpreted as amplitudes of combinations of these spinors. This very unusual picture is discussed below. The essential idea is that the underlying structure involves spin 1/2 bosons, which are permitted because there is a violation of Lorentz invariance, in this high-energy and previously unexplored sector of the theory, which is fully consistent with existing tests of Lorentz invariance liberati ; kostelecky . Rotational invariance is still exact, but invariance under a boost is not.

With

[TABLE]

it is convenient to use the same Weyl representation as for Dirac fields, where

[TABLE]

so that (after integration by parts with neglect of boundary terms)

[TABLE]

According to a result schwartz that can easily be extended to the nonabelian case, we have

[TABLE]

with a $(-+++)$ convention for the metric tensor. The second term gives an addition to standard physics, involving the field strength tensor $F_{\mu\nu}$ for the electroweak gauge fields and the Lorentz generators $S^{\mu\nu}$ which act on Dirac spinors:

[TABLE]

where

[TABLE]

or schwartz

[TABLE]

This can be rewritten in terms of the “magnetic” and “electric” fields $B_{k}$ and $E_{k}$ defined by

[TABLE]

since schwartz

[TABLE]

where $a\cdot b=a_{k}b^{k}$ (with $\mu=0,1,2,3$ and $k=1,2,3$). We then obtain

[TABLE]

The second term (which is analogous to the interaction of an electron spin with a magnetic field) is invariant under a rotation, but not under a boost, making it the only aspect of the theory that does not have complete Lorentz invariance. As discussed below, this term will have observable effects only at high energy (or in extremely weak radiative corrections), and in conjunction with the new spin 1/2 particles predicted here.

Let us regroup the components according to charge:

[TABLE]

and then write

[TABLE]

where $\phi^{r}$ is a complex scalar and $\chi^{r}$ is a 4-component spinor. We can achieve a scalar condensate and scalar excitations by requiring that

[TABLE]

or

[TABLE]

As a bonus the unconventional term in (24) also vanishes:

[TABLE]

With the normalization

[TABLE]

(24) is then reduced to

[TABLE]

where $\phi$ has the scalar components $\phi^{r}$. Each amplitude mode $\phi^{r}$ thus has only its standard coupling to the gauge fields through the covariant derivative, and we have returned to standard physics with scalar Higgs bosons.

In the present theory, however, there can also be spin $1/2$ excitations, analogous to quasiparticle excitations in a superconductor. They will be harder to create than scalar Higgs bosons, because angular momentum conservation requires that they be created in pairs. They will also be harder to observe, because there are no apparent decay modes for the lowest-mass of these particles: They have an R-parity of $-1$, with spin $1/2$ and with no lepton or baryon number.

The masses of these particles depend on the potentially very complicated set of parameters determining $S_{2}$, made even more nontrivial by the requirements of susy baer-tata ; kane-susy , which doubles the number of Higgs fields yet again. But if the action is extremalized, by setting the first derivatives of the full Higgs potential $V$ equal to zero, and the scalar mass eigenstates $\phi_{i}$ are then determined, by diagonalizing the mass matrix obtained from the second derivatives of $V$, the resulting lowest-order Lagrangian has the form

[TABLE]

which is equivalent to

[TABLE]

if we can still write $\Phi_{i}=\phi_{i}\chi_{i}$ for the mass eigenstates, with $\chi_{i}$ constant. More generally, with

[TABLE]

we then have

[TABLE]

The masses for the 2-component spinors $H_{i}$ are then the same as the masses for the Higgs scalars $\phi_{i}$ in this simplest description, and the couplings are also the same.

In the simplest version of susy, with two Higgs doublets $\phi_{u}$ and $\phi_{d}$ – which correspond to $\Phi_{u}$ and $\Phi_{d}$ in the present picture – there are 3 would-be Goldstone bosons, one charged Higgs, and 3 neutral Higgses, which then account for the 4 charged and 4 neutral degrees of freedom. The same basic approach is needed here, but there must be 3 charged and 7 neutral scalar Higgs bosons (resulting from the various scalar combinations of fields in $\Phi_{u}$ and $\Phi_{d}$), and the same number of types of spin 1/2 particles, to account for the 8 charged and 8 neutral degrees of freedom. Their masses must arise from the field interactions, including the variety of quartic terms that couple the fields to one another.

According to the spin-statistics theorem, spin $1/2$ bosonic excitations are impossible, but the requirements of this theorem are not satisfied in this one specific context, since the second term of (24) is not fully Lorentz invariant: It is invariant under a rotation, but not a Lorentz boost with respect to the original (cosmological) coordinate system.

The present theory is, however, fully Lorentz invariant if the internal (spinor) degrees of freedom in $\Phi$ are not excited – as can be seen in (33) and (34) – and these excitations can be observed only at the high energies that are now becoming available. Furthermore, the extremely weak virtual effects of these excitations are irrelevant to the many existing sensitive tests of Lorentz invariance, which probe only rotational invariance (rather than boosts) or else those phenomena in various areas of physics and astrophysics where the present theory is fully Lorentz invariant liberati ; kostelecky . In particular, it is hard to imagine how any of the many experiments listed in Ref. kostelecky could have any measurable sensitivity to virtual processes involving pairs of 125 GeV spin 1/2 particles which only participate in the weak interaction, exactly obey rotational invariance, and have free-particle propagators (resulting from (24) and (35)) that are fully Lorentz invariant.

Two historical precedents may be relevant: After the electron was discovered in 1897, and the photon was introduced by Einstein in 1905, the richness of behavior associated with spin 1/2 fermions and spin 1 gauge bosons emerged slowly during the following decades. (J.J. Thomson in 1897 did not picture the electron as described by a 4-component Dirac field.) More than a century later, the third kind of Standard Model particle, with spin 0, has finally been discovered, and one should not be completely surprised if some of its implications are yet to be determined. Similarly, it should not be completely surprising if Lorentz invariance, like previously well-established principles such as P and CP invariance, is ultimately found to have exceptions in a more nearly complete theory.

We obtain no direct interaction of these new particles with fermions in the present formulation, but they can be produced by quarks in colliding protons, via virtual W or Z bosons.

Details of the phenomenology for the various kinds of experiments are of great interest, but beyond the scope of the present paper. We only note that once a lowest-mass particle of this kind has left the region where it was created, it is unable to decay without violating lepton number or baryon number conservation, since the net decay products must have angular momentum $1/2$. This implies that these (weakly-interacting) particles are dark matter candidates, roughly similar to neutralinos, but distinguished by both their quite different couplings and the fact that their mass is simply related to that of the recently discovered Higgs boson.