Explaining the cosmological dark matter coincidence in asymmetric dark QCD

A fixed point of the coupled RG evolution of the gauge couplings of visible and dark QCD, denoted $(\alpha_{s}^{\star},\alpha_{d}^{\star})$, is given by

Rewriting Eqn. II as

and with $\beta_{d}(\alpha_{s},\alpha_{d})$ given by exchanging $\alpha_{s}$ and $\alpha_{d}$ and interchanging the indices $v\leftrightarrow d$, there are four solutions to Eqn. 4:

Note that these are not necessarily infrared fixed points; they may be UV fixed points, or they may be saddle points of the coupled evolution of the gauge couplings. Indeed, these nominal solutions are not necessarily physical, let alone perturbative.

In our previous work, we considered models in which the last of these solutions – Eqn. 9 – is an infrared fixed point, with both $\alpha_{s}^{\star}$ and $\alpha_{d}^{\star}$ taking perturbative values⁴⁴4In this work, we consider gauge couplings $\alpha_{s},\alpha_{d}<0.3$ to be perturbative. This ensures the pertubative validity of our two-loop $\beta$-function calculations: see Sec. V A in our previous work [11].. This is not the only case one could consider, and so we refer to this case specifically as the ‘finite-coupling fixed point’ approach. This contrasts it with the case we consider in the next section, the ‘zero-coupling fixed point’ approach, where the first of these solutions – Eqn. 6 – is an infrared fixed point.

In Eqn. 9 we immediately see one of the shortcomings that this approach faces; that is, the values of $\alpha_{s}^{\star}$ and $\alpha_{d}^{\star}$ have a highly non-trivial dependence on the various $\beta$-function coefficients. Firstly, this prevents any simple links being drawn between the field content of a given model and the value of the couplings at the fixed point, especially since a given field contributes to multiple $\beta$-function coefficients. Secondly, as stated already, for a generic model these values may not be perturbative, or even physical. To obtain a model that has a perturbative FCFP requires a delicate interplay between the $\beta$-function coefficients, and we find that this occurs in only a small fraction of models. For example, in our previous work we initially analysed a set of 12,288 models; of these models, 1354 ($\sim$10%) have an infrared fixed point, and only 155 ($\sim$1%) of the models have a perturbative fixed point.

The last observation we make here is that for the FCFP of Eqn. 9 to be an infrared fixed point, $b_{v}^{(0)}$ and $b_{d}^{(0)}$ must both be negative. The argument is as follows: if the theory contains an infrared FCFP, then initially small couplings $\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}}\ll 1$ must increase towards the IRFP with decreasing energy scale (that is, the zero-coupling fixed point of Eqn. 6 must be a UV fixed point). So, for these small couplings, the $\beta$-functions must be negative; also, for small couplings the $\beta$-functions are dominated by the one-loop terms. Thus, the one-loop coefficients $b_{v}^{(0)}$ and $b_{d}^{(0)}$ must be negative.

We search for models with the largest possible value for the viability fraction. In our previous work, the largest value we found for $\epsilon_{v}$ was around 0.4, and we considered models with $\epsilon_{v}>0.3$ to be viable. These comprised a smaller fraction again of the total models. For example, in the set of 12,288 models mentioned previously, of the 155 models with perturbative infrared FCFPs, only 19 had $\epsilon_{v}>0.3$, comprising $\sim$0.1% of the total models.

This restriction is due to the fact that the models with the largest viability fractions all have ‘small’ IRFPs with $\alpha_{s}^{\star},\alpha_{d}^{\star}\lesssim 0.1$. The reason for this is in two parts. Firstly, at the decoupling scale of the new physics, we need $\alpha_{s}<\alpha_{s}(M_{Z})=0.11729$ in order to be able to match the strong coupling to its measured value at the $Z$ mass, and thus to calculate $\Lambda_{\mathrm{dQCD}}$. So, for this to be achieved for a wide range of initial coupling values in the UV, $\alpha_{s}$ must be evolving towards a fixed point value $\alpha_{s}^{\star}<0.11729$. Secondly, to have $\Lambda_{\mathrm{dQCD}}\sim\Lambda_{\mathrm{QCD}}$ for a large area of the UV coupling parameter space, we need $\alpha_{s}$ and $\alpha_{d}$ to be similar in size at the decoupling scale. This indicates that the couplings should be evolving towards fixed points with similar values. Since $\alpha_{s}^{\star}$ must be small, $\alpha_{d}^{\star}$ must also be small.

We note that this second point assumes that similar gauge couplings at the decoupling scale imply similar confinement scales in the two sectors. This is only true if the gauge coupling evolution is similar in the two sectors after decoupling. If $N_{d}\neq 3$, or if the number of light dark quarks differs substantially from the number of light SM quarks, then this is not necessarily true. Indeed, we find in general that the value of $\epsilon_{v}$ depends strongly on the number of light dark quarks; this will be discussed further in Section V.

We also note that the intuitive concept of the success of a viable model now differs from the original idea of Bai and Schwaller. The relationship between the couplings in the infrared results from their evolution towards some fixed point whose couplings take small values; that is, they are ‘focused’ toward a point near the origin of the $(\alpha_{s},\alpha_{d})$ coupling plane, and thus brought close to each other in value. This motivates us to consider the zero-coupling fixed point approach that we introduce in the next section, where the origin itself, $\alpha_{s}=\alpha_{d}=0$, serves as the infrared fixed point that draws the couplings together from their values in the UV.

An issue that arises generically for these viable models is that in much of the $(\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}})$ parameter space, the required mass scale of new physics $M$ is below a TeV. These new coloured particles with $\mathcal{O}(100)$ GeV masses would be copiously produced at colliders, leading to the exclusion of these models. In our previous work, we thus searched for models in which $M$ is TeV-scale or higher.

The general idea is that to obtain a high mass scale, the running of the gauge couplings in the full theory must be ‘fast’, so that $\alpha_{s}$ evolving from the UV matches to the required coupling to generate $\alpha_{s}(M_{Z})$ at a sufficiently high scale. Strong running requires large values for the $\beta$-function coefficients. In the FCFP approach, the one-loop $\beta$-function coefficients $b_{v}^{(0)}$ and $b_{d}^{(0)}$ must be negative, but there is no restriction on the sign of the two-loop terms. So, the strongest running occurs when $b_{v}^{(0)}$ and $b_{d}^{(0)}$ are negative and close to zero. This, however, requires a fine cancellation between the gauge and matter contributions to these coefficients, and so again only occurs for select sets of field content, further reducing the proportion of total models that are viable and phenomenologically allowed.

In our first paper, we achieved strong running by considering models with high multiplicities for the jointly-charged $(\mathbf{3},\mathbf{3})$ scalars, while finding that successful models also had large multiplicities for the other heavy new scalars. Each additional $(\mathbf{3},\mathbf{3})$ scalar contributes to all $\beta$-function coefficients, and so strengthens the running more than scalars charged under only one gauge group.

We note that this means that we are considering models in which the one-loop coefficients are much smaller in magnitude than the two-loop coefficients, thanks to the precise cancellation between the gauge and matter contributions to the one-loop coefficient. This could lead to questions about the perturbative validity of this analysis. Our answer is to consider the magnitude of each term that contributes to a given coefficient, not their sum (as this ignores the artificial smallness that occurs due to cancellations between terms), and to ensure that individual two-loop terms are smaller in magnitude than individual one-loop terms.

The issue now is that not only do these successful models comprise a very small proportion of the parameter space, they also require specific and unwieldly selections of field content, with ten or more joint scalars, and up to nineteen dark $(\mathbf{1},\mathbf{3})$ scalars⁵⁵5See Fig. 10 of Ref. [11]..

To summarise the conclusions of our previous work regarding the FCFP approach:

• •

  For a model to have a perturbative FCFP, there must be a precise cancellation between combinations of different $\beta$-function coefficients. In addition, to obtain a decently large value for $\epsilon_{v}$, the couplings at the fixed point must be small. This means that only a very small proportion of models ($\lesssim 1\%$) even have such a fixed point.

• •

  To then have $M>$ TeV in most of the UV-coupling parameter space, we need the couplings to run strongly in the full theory while keeping the one-loop $\beta$-function coefficients negative to ensure the existence of an infrared FCFP. This requires large multiplicities for certain fields, e.g. jointly-charged $(\mathbf{3},\mathbf{3})$ scalars.

The zero-coupling point $\alpha_{s}^{\star}=\alpha_{d}^{\star}=0$ is trivially always a fixed point of the RG evolution. For it to be an infrared fixed point, the $\beta$-functions must be positive for gauge couplings in the vicinity of the fixed point: that is, for small couplings $\alpha_{s},\alpha_{d}\ll 1$. In this regime, the one-loop terms dominate, and so the one-loop $\beta$-function coefficients must be positive:

This is a much less restrictive requirement than the delicate cancellations that we needed to obtain an infrared FCFP. Since matter contributions to the $\beta$-function coefficients are positive, all a model needs in order to have an infrared ZCFP is a sufficient amount of new field content such that the one-loop terms become positive.

In addition, infrared ZCFPs can be achieved generically with low multiplicities for the new field content, thanks to the presence of fields in higher-dimensional representations of both visible and dark QCD. These fields contribute more strongly to the $\beta$-function coefficients in Eqn. II, due to their high dimension and larger group theory factors. So, for example, if $N_{d}=3$, a model with 6 quarks, 1 dark quark, and just a single $(\mathbf{6},\mathbf{6})$ fermion exhibits an infrared ZCFP.

To see which higher-dimensional representations have this effect of obtaining infrared ZCFPs with low multiplicities, we consider models containing the 6 SM quarks, 1 dark quark, and some number of copies of an $(\mathbf{R}_{v},\mathbf{R}_{d})$ fermion or scalar. In Table 1 we give the minimum number of copies of this field that must be included for a model to exhibit an infrared ZCFP: that is, for the one-loop $\beta$-function coefficients $b_{v}^{(0)}$ and $b_{d}^{(0)}$ to be positive. Fermions of a given representation require smaller multiplicities than the corresponding scalar, since the fermions make a larger contribution to each $\beta$-function coefficient; this is due in part to Dirac fermions having more degrees of freedom than complex scalars. We also note that $N_{d}=4$ models require more higher-dimensional representations than $N_{d}=2,3$, which correlates with a larger negative gluonic contribution to the $\beta$-function coefficients.

One may be perturbed by the fact that positive one-loop $\beta$-function coefficients seem to imply the loss of asymptotic freedom. Indeed, since the gauge couplings grow with increasing energy scale, they diverge at some Landau pole, which we must ensure does not fall below the Planck scale.

We first note that in this framework, $b_{v}^{(0)}$ and $b_{d}^{(0)}$ are only positive above the mass scale of new physics. At energies below this scale, when the new heavy fields have decoupled, the running of $\alpha_{s}$ is SM-like, and the running of $\alpha_{d}$ is similar so long as $n_{dq}$ is not too large; so, the theory is asymptotically free at low energies.

The values of the gauge couplings at high energies, and thus the location of the Landau pole, depend upon the mass scale $M$ at which the RG evolution changes. In our calculations, we begin by specifying a pair of (perturbative) gauge couplings at the UV scale, $(\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}})$, and then solve for $M$ by matching $\alpha_{s}$ to its measured value at $M_{Z}$. This ensures that we only ever consider mass scales $M$ for which each coupling is perturbative from its associated confinement scale all the way up to the Planck scale.

Indeed, the philosophy behind our idea of a model that ‘naturally’ explains the coincidence problem is that we are agnostic to physics above the UV scale. We thus treat $\alpha_{s}^{\mathrm{UV}}$ and $\alpha_{d}^{\mathrm{UV}}$ as input parameters, and when defining the viability fraction $\epsilon_{v}$ we consider them to take perturbative values that are randomly, uniformly and independently chosen. This ‘flat prior’ may well be invalid, as the ‘initial’ gauge coupling values in the UV are in fact generated by the unknown high-scale physics. This physics above the Planck scale may also change the RG evolution, avoiding a Landau pole in the theory by either regaining asymptotic freedom, or by the couplings approaching an UV fixed point as in the ‘asymptotic safety’ idea [42, 43, 44]. But, given our ignorance of specific Planck-scale scenarios for generating the coupling constant values, a flat prior is justified.

Models with an infrared ZCFP generally require a high mass scale for the new physics. In these models, the positive $\beta$-function coefficients in the full theory lead to strong running of the gauge couplings from their initial values in the UV, unless there is a precise cancellation such that the $\beta$-function coefficients happen to be small. As a result of this ‘fast’ RG evolution, to generate the correct value for $\alpha_{s}$ at $M_{Z}$, the new physics must generically decouple at a high scale.

We illustrate this analytically at one-loop level. Ignoring threshold corrections, for a given value of the strong coupling in the UV, $\alpha_{s}^{\mathrm{UV}}$, the mass scale $M$ for the new physics is found by running $\alpha_{s}$ up from $\alpha_{s}(M_{Z})$ using the SM RGE, and down from $\alpha_{s}^{\mathrm{UV}}$ using the RGE of the full theory until they meet. At one-loop level, this is given by

where $b_{n}=(2n/3-11)/2\pi$ is the one-loop $\beta$-function coefficient for QCD with $n$ quark flavours, and $b\equiv b_{v}^{(0)}/2\pi$ is the one-loop coefficient for the full theory. Taking the maximum value of $\alpha_{s}^{\mathrm{UV}}=0.3$ gives the minimum value for the mass scale as

While the resultant mass scale can be low, this requires $b$ to be small. As will be discussed shortly, $M$ cannot be more than 100s of TeV without introducing a severe naturalness issue. To obtain such a mass scale,

For this to occur, the matter contribution to $b$ must cancel closely with the gluonic contribution. We encountered a similar restriction in the FCFP approach when searching for models with a sufficiently large value of $M$, and in that case we required specific and unwieldy selections of field content. We find similar behaviour in this case, which we explore in Section V.1.

We identified earlier that we would be interested in models containing fields in higher-dimensional representations, as these models generically obtain an infrared ZCFP with low multiplicities for the new fields. However, it is the inclusion of such representations that strengthens the running to the point that the mass scale is very high. For example, for $N_{d}=3$, models including a $(\mathbf{6},\mathbf{6})$ fermion have $b\gtrsim 2$, leading to $M\gtrsim 10^{11}$ GeV.

In Table 2 we show the minimum mass scales of new physics for models including a single fermion in the representation $(\mathbf{R}_{v},\mathbf{R}_{d})$. We only show the representations for which a model with the 6 SM quarks, 1 dark quark, and 1 $(\mathbf{R}_{v},\mathbf{R}_{d})$ fermion has an infrared ZCFP: that is, the representations in Table 1 with minimum multiplicity 1. We reiterate that these are only the minimum values for $M$, calculated at one-loop; using the two-loop RGEs, working with any UV coupling below $\alpha_{s}^{\mathrm{UV}}=0.3$, and including any additional field content in these models only increases $M$.

We now come to the naturalness issue that arises due to the presence of heavy new physics. As we have seen, the ZCFP approach to this framework generally necessitates such heavy fields.

Dirac fermions and complex scalars contribute to the running of the Higgs mass parameter. If these fields are sufficiently heavy, then these contributions are sizeable, making its low-energy value $\mu^{2}(M_{Z})$ very sensitive to physics at a high scale. This is true even for coloured fermions, which only contribute to the Higgs mass parameter at three-loop level.

To assess the scale of this issue, we adopt the approach of Ref. [41], where a model is unnatural if, given a flat prior for the Higgs mass parameter in the UV, it is very unlikely for it to obtain its measured value at $M_{Z}$. This is similar in spirit to our perspective on naturalness, but is implemented quantitatively using a Bayesian sensitivity measure that can be roughly interpreted as a level of fine-tuning.

In Ref. [41], limits are placed on the masses of new gauge multiplets added to the SM. These limits are calculated for threshold values of the sensitivity measure; we look at the results corresponding to a 10% fine-tuning, where for colour-triplet, electroweak-singlet Dirac fermions, the limits are in the 10—100 TeV mass range, and for colour-triplet, electroweak-singlet complex scalars, the limits are in the 1—10 TeV mass range.

We note that these limits on $M$ may be optimistic for the models we consider. Firstly, they only apply to colour triplets; any field in a higher-dimensional representation of $SU(3)_{\mathrm{QCD}}$ will be subject to a stronger constraint. Secondly, these limits are calculated for models that only have a single additional gauge multiplet. In models where multiple new fermions and scalars are introduced, $M$ must be smaller again to avoid excessive fine-tuning. For these reasons it is possible that even for models with $M\lesssim 1$ TeV the naturalness issue could remain problematic. This suggests that it could even be impossible to find models in this framework that avoid a naturalness problem, an observation which also applies to the FCFP approach.

To consider models in which the mass scale $M$ is much greater than 100s of TeV, we introduce supersymmetry, which prevents the new heavy fields from contributing to the running of the Higgs mass parameter. We always take the scale of supersymmetry breaking to be $M_{\mathrm{SUSY}}=5$ TeV. Below $M_{\mathrm{SUSY}}$ the running is as before, with the running of $\alpha_{d}$ due to $n_{dq}$ light dark quarks. Between $M_{\mathrm{SUSY}}$ and $M$ the gauge couplings run separately; the RG evolution of $\alpha_{s}$ is MSSM-like, and the RG evolution of $\alpha_{d}$ is due to $n_{dq}$ light dark quarks and $n_{dq}$ light dark squarks.

The heavy field content in this case consists of chiral supermultiplets. In the non-SUSY case, we introduce Dirac fermions and complex scalars to ensure anomaly cancellation and write down renormalisable mass terms. For the same reasons, in the SUSY case when we say we introduce a ‘field’ in the representation $(\mathbf{R}_{v},\mathbf{R}_{d})$ of visible and dark QCD, we are adding two chiral supermultiplets $\Phi\sim(\mathbf{R}_{v},\mathbf{R}_{d})_{y}$ and $\bar{\Phi}\sim(\mathbf{\bar{R}}_{v},\mathbf{\bar{R}}_{d})_{-y}$ with superpotential mass term $M\bar{\Phi}\Phi$.

Introducing SUSY slightly alters the conclusions we made previously in this section about the existence of a zero-coupling IRFP and the size of the mass scale $M$ in these models.

In Table 3, we replicate Table 1 in the supersymmetric case. We find that with SUSY, even fewer copies of each new field are required for a model to exhibit a zero-coupling infrared fixed point. This is expected, as each new field we introduce is a pair of chiral supermultiplets containing the same field content as a Dirac fermion and two complex scalars.

We also find that the required mass scale $M$ is larger when SUSY is included. For a given $\alpha_{s}^{\mathrm{UV}}$, at one-loop level and ignoring threshold corrections, $M$ is found by

where the the one-loop $\beta$-function coefficient $b_{\mathrm{SUSY}}=-3/2\pi$. Since the MSSM running is weaker than the SM running, to match the measured strong coupling $\alpha_{s}(M_{Z})$, the heavy fields must decouple at a higher scale. With $M_{\mathrm{SUSY}}=5$ TeV and again taking $\alpha_{s}^{\mathrm{UV}}=0.3$, the minimum value for the mass scale in the SUSY case is

In Table 4 we replicate Table 2 for the SUSY case, showing the minimum mass scales of new physics for models including a single field in the representation $(\mathbf{R}_{v},\mathbf{R}_{d})$. As before, we only show the values of $M$ for representations where a model with the 6 SM (s)quarks, 1 dark (s)quark, and 1 $(\mathbf{R}_{v},\mathbf{R}_{d})$ field has an infrared ZCFP. Since this occurs for lower-dimensional representations in the SUSY case, more values of $M$ are shown than in Table 2. There are also two representations marked with crosses; these models do happen to have a small value of $b$, and so in fact have $M<M_{Z}$. This is not possible, and indicates that one cannot solve for $M$ in such a model for $\alpha_{s}^{\mathrm{UV}}=0.3$.

As a comparison to the point singled out in the non-SUSY case, for $N_{d}=3$, models including a $(\mathbf{6},\mathbf{6})$ field have $b\gtrsim 4$, leading to $M\gtrsim 10^{15}$ GeV.

Finally, we note that although these heavy superfields do not introduce a naturalness issue, we still include their mass scales as they are relevant for some of the discussion in Section VI.

To find models with a low mass scale of new physics $M$, we first must specify the upper limit on $M$ such that we avoid a naturalness issue from the contribution of these new fields to the running of the Higgs mass parameter. As mentioned in the previous section, we are guided by the results of Ref. [41] which provides limits on the mass of new Dirac fermion and complex scalar gauge multiplets. Conservatively, to introduce less than 10% fine-tuning requires:

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  $M\lesssim 100$ TeV for colour-triplet, electroweak-singlet Dirac fermions.

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  $M\lesssim 10$ TeV for colour-triplet, electroweak-singlet complex scalars.

We use these limits as rough criteria for determining whether a model avoids a naturalness problem. For models containing only new fermions we look at the viable proportion of $(\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}})$ parameter space for which $M<100$ TeV: we call this parameter $\epsilon_{v}^{100}$. For models also containing new scalars we look at the viable proportion of UV coupling parameter space for which $M<10$ TeV: we call this parameter $\epsilon_{v}^{10}$.

So, what conditions do we need to place on a model to have $M\lesssim 10\text{---}100$ TeV in a sizeable proportion of the UV coupling parameter space?

In the previous section, to one-loop level we found that a one-loop $\beta$-function coefficient $b_{v}^{(0)}/2\pi\sim 0.4\text{---}0.6$ results in a minimum value of $M$ between 1 and 100 TeV. This changes when we include the two-loop RG evolution. At two-loop level the running is stronger, and so to obtain the same value of $M$ requires a smaller value for $b_{v}^{(0)}/2\pi$. The value of $M$ now also depends on $\alpha_{d}^{\mathrm{UV}}$, and so for a given model the minimum mass scale $M_{\mathrm{min}}$ occurs for $(\alpha_{s}^{\mathrm{UV}},\alpha_{d}^{\mathrm{UV}})=(0.3,0)$. We now find that

In models incorporating SUSY, we face no naturalness issue when adding in new fields. The mass scale of new physics $M$ can now be as high as the Planck scale, and there are no restrictions on the size of the $\beta$-function coefficients as there were in the previous section. This allows us to generically obtain infrared ZCFPs with minimal new field content; in particular we can now include fields of higher-dimensional representations which immediately ensure the existence of an infrared ZCFP.

We are interested in the values of $\epsilon_{v}$ that these models can obtain. The hope is that models in this framework will generically be viable; however we find that this is not always the case. This desire is primarily confronted by the fact that the value of $\epsilon_{v}$ strongly depends on the number of light dark quarks, $n_{dq}$. This is because changing $n_{dq}$ greatly alters the running of $\alpha_{d}$ following the decoupling of the new field content, which now occurs at a high mass scale.

In this section we consider sets of models with a given heavy field content; for each selection of heavy field content, we choose the number of light dark quarks $n_{dq}$ such that $\epsilon_{v}$ is a maximum. This is done first for $N_{d}=3$, and then for $N_{d}=2$ and 4. A summary of the general findings is as follows:

• •

  For $N_{d}=3$, the best value of $\epsilon_{v}$ is $\sim$0.35, which is decently large. However, there is a fairly wide spread of maximum values for $\epsilon_{v}$, and only a small proportion of models have $\epsilon_{v}>0.3$. While this may challenge our desire to generically obtain viable models, $\sim$40% of models have $\epsilon_{v}>0.25$, which constitutes a decent proportion of models with a reasonable viability fraction.

• •

  For $N_{d}=4$, the situation improves. The best value of $\epsilon_{v}$ is $\sim$0.42, and $\sim$65% of models have $\epsilon_{v}>0.3$.

• •

  For $N_{d}=2$, the situation is worse. There are no models with $\epsilon_{v}>0.3$, and less than 1% even have $\epsilon_{v}>0.25$.

• •

  The required number of light dark quarks peaks at $n_{dq}=6$ for $N_{d}=3$, $n_{dq}=11$ for $N_{d}=4$, and $n_{dq}=1$ for $N_{d}=2$. For a given $N_{d}$, the maximum possible value of $\epsilon_{v}$ increases with $n_{dq}$.