Inflation and long-range force from clockwork $D$-term

There exist a large number of well-motivated gauged extensions of the Standard Model (SM) containing an extra $U(1)$ group. These are proposed (a) on phenomenological grounds like explaining anomaly found in the muon anomalous magnetic moment Baek et al. (2001) (see also Lindner et al. (2018) for a review) or as explanation of the universality violation observed in the $B$ meson decays Altmannshofer et al. (2016), (b) on cosmological grounds such as need to explain the dark matter Holdom (1986); Hooper et al. (2012); Fabbrichesi et al. (2020), to provide secret interactions between sterile neutrinos of eV masses Hannestad et al. (2014); Dasgupta and Kopp (2014) to suppress their cosmological production in the early universe etc., (c) as a theoretical framework for the successful description of the inflation in the context of supersymmetric versions of the SM Stewart (1995); Binetruy and Dvali (1996); Halyo (1996), and (d) to provide a simple description of the long-range “fifth force” Fayet (1986, 1989) if it exists. Examples of such $U(1)$ are difference of any two of the leptonic charges $L_{e,\mu,\tau}$ He et al. (1991a, b); Foot et al. (1994) or an unbroken or mildly broken $B-L$ symmetry Heeck (2014).

Many extensions in category (a) and (b) need a very light gauge boson typically in the mass range eV-MeV. The models of the $D$-term inflation Stewart (1995); Binetruy and Dvali (1996); Halyo (1996) use the Fayet-Illiopoulos (FI) term Fayet and Iliopoulos (1974) which can be written when the gauge symmetry is $U(1)$. A large value for the FI parameter $\xi\sim 10^{32}$ GeV² leads to inflation in the early universe driven by an almost flat potential. Extensions in the category (d) correspond to an entirely different parameter range. If the first generation fermions are charged under the extra $U(1)$ then the induced long-range forces are constrained by the fifth force experiments Touboul et al. (2017) or by the precision tests of gravity Kapner et al. (2007); Schlamminger et al. (2008). These experiments constrain the couplings of electrons to the light gauge boson $M_{Z^{\prime}}$ and are not sensitive to the neutrino couplings. Constraints on the masses and couplings for the range of length $\>\raisebox{-4.73611pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>0.1\,{\rm eV}^{-1}$ follow from these experiments and restrict the coupling $g^{\prime}$ to be $\>\raisebox{-4.73611pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>10^{-25}$. If $U(1)$ group distinguishes between leptonic flavours then the long-range forces generated by electrons from the earth, Sun, Galaxies etc. induce the matter effects in neutrino oscillations Joshipura and Mohanty (2004); Grifols and Masso (2004). This effect can suppress the observed neutrino oscillations for a range in the gauge boson mass $M_{Z^{\prime}}$ and coupling $g^{\prime}$. Terrestrial experiments, as well as astrophysical and cosmological considerations, constrain the allowed $M_{Z^{\prime}}$-$g^{\prime}$ parameter space. It is found that there exists a region of parameters for which the $Z^{\prime}$ induced potential can be comparable to the Wolfenstein potential induced by the charged current interaction in the SM. This happens Smirnov and Xu (2019) for approximate ranges $M_{Z^{\prime}}\sim 10^{-17}$-$10^{-14}$ eV and $g^{\prime}\sim 10^{-27}$-$10^{-25}$. This implies a strong hierarchy $M_{Z^{\prime}}/\sqrt{\xi}\sim 10^{-40}$ between the allowed ultra-light mass and the inflation scale. Considering that the scales and parameters associated with the SM are much larger than $M_{Z^{\prime}}$ and $g^{\prime}$, it is natural to seek a theoretical explanation of their smallness.

It was pointed out by Fayet Fayet (1984, 1990) (see also Fayet (2017, 2018, 2019)) that the presence of a FI term allowed in case of the supersymmetric gauge theories can be used to relate the inflation scale $\sqrt{\xi}$ to a very small gauge coupling $g^{\prime}$. Consider a simple supersymmetric gauge theory based on a $U(1)$ group containing two oppositely charged superfields $\phi_{\pm}$. The scalar potential of this theory includes the following $D$-term contribution.

This simple potential is used to drive inflation when it is supplemented with a gauge singlet superfield $X$ - the inflaton and an $F$-term coming from a superpotential $\lambda X\phi_{+}\phi_{-}$. A large value of the inflaton field in the early universe leads to a supersymmetry breaking and $U(1)$ preserving minimum of $V_{D}$ with a value $\frac{1}{2}\xi^{2}$ at the minimum. For vanishing $X$ field which occurs after inflation, the $V_{D}$ has a supersymmetry preserving but the gauge symmetry breaking minimum with $g^{\prime}|\left\langle\phi_{+}\right\rangle|^{2}=\xi$. The above $D$-term leads to a scalar mass term $\mu^{2}=g^{\prime}\xi$¹¹1More precisely this will be a $D$-term contribution to the mass of the real part of $\phi_{+}$ when $g^{\prime}\left\langle\phi_{+}\right\rangle^{2}=\xi$.. Requiring that this mass parameter is less than the typical supersymmetry breaking scale $\sim$ TeV gives a small $g^{\prime}\leq{\rm TeV}^{2}/\xi\sim 10^{-26}$ Fayet (2018, 2019). Thus a large inflation scale relates to very small value of $g^{\prime}$. While small value of $g^{\prime}$ follows in this simple $U(1)$ example, it still cannot describe the long range forces. The $U(1)$ gauge boson in this case acquires a mass $M_{Z^{\prime}}^{2}=g^{\prime 2}|\left\langle\phi_{+}\right\rangle|^{2}=g^{\prime}\xi=\mu^{2}\sim{\rm TeV}^{2}$ which leads to a very short range potential. This is an artefact of the use of the SM singlet fields $\phi_{\pm}$ for breaking the $U(1)$ symmetry. As shown by Fayet Fayet (1984), it is possible to obtain ultra-light or even a massless $M_{Z^{\prime}}$ Fayet (1990), a small $g^{\prime}$ and the flat potential required for the inflation to start by using the SM non-singlet fields to break the $U(1)$ symmetry²²2A specific example based on the $SU(5)\times U(1)$ group proposed in Fayet (1984), leads to a mass relation $M_{Z^{\prime}}=\frac{g^{\prime}}{g}M_{W}$ leading to ultra-light gauge boson for very small $g^{\prime}$..

As an alternative to the above setup, we propose a series of $N+1$ gauged $U(1)_{i}$ groups ($i=0,1,...,N$) based on the clockwork (CW) mechanism Kaplan and Rattazzi (2016); Giudice and McCullough (2017). $N$ chiral superfields are introduced, each of which couples to only two adjacent $U(1)$ in the chain leading to characteristic nearest-neighbour interactions. FI term is introduced only for the $U(1)$ at the $N^{\rm th}$ site. This leads to inflation in a manner described above. The corresponding gauge coupling is of ${\cal O}(1)$. All the $U(1)$ symmetries, except a linear combination of them, get broken at the minimum, but the breaking scales are hierarchically related to the FI parameter $\xi$. Specifically, the $U(1)_{i}$ is broken at a scale $\sim q^{\frac{N-i}{2}}\sqrt{\xi}$, where $q$ being the $U(1)$ charge carried by chiral superfields which induce the symmetry breaking. The remaining $U(1)$ symmetry is broken by introducing another pair of chiral superfields which couple to one of the intermediate $U(1)$ in the chain. The localization of chiral superfields away from $U(1)_{0}$ leads to an explanation of a large hierarchy between the scales of inflation and the mass of the gauge boson mediating long-range force. The SM fields also interact with one of the intermediate $U(1)$ with ${\cal O}(1)$ gauge coupling. Exponentially small coupling with lightest gauge boson is then obtained in a manner used to describe the mini-charged particles within the standard CW frameworks Giudice and McCullough (2017); Lee (2018).

We introduce the basic framework of CW $D$-term in the next section. Inflation driven by the $D$-term along with the implications on inflationary observables is discussed in section III. In section IV, we collect various laboratory, astrophysical and cosmological constraints on the popular class of long-range forces and discuss their consequences on the parameters of the CW framework. We summarize the study in section V.

The framework of multiple $U(1)$ we discuss here is based on the clockwork constructions discussed in Giudice and McCullough (2017); Lee (2018); Ahmed and Dillon (2017). Consider a chain of $N+1$ supersymmetric $U(1)_{i}$, with $i=0,1,...,N$. A vector superfield ${\cal V}_{i}$ of $U(1)_{i}$ contains a vector filed $\hat{V}_{i,\mu}$, a pair of weyl fermions $\lambda_{i}$, $\lambda_{i}^{\dagger}$ and auxiliary field $D_{i}$. The supersymmetric Lagrangian involving gauge fields is given by

where $F_{i,\mu\nu}=\partial_{\mu}\hat{V}_{i,\nu}-\partial_{\nu}\hat{V}_{i,\mu}$. One can further include a gauge and supersymmetry invariant Fayet-Iliopoulos (FI) term for each $U(1)$. Here, we assume that only $U(1)$ at the $N^{\rm th}$ site possesses such a term.

The assumption of having only one vanishing FI term is technically natural Fischler et al. (1981) as the trace of each $U(1)_{i}$ factor is individually zero. We then consider $N$ pairs of chiral superfields $\Phi^{\pm}_{i}$ (with $i=1,...,N$) charged under $U(1)_{i-1}\times U(1)_{i}$ with charges $(\mp q_{i-1},\pm 1)$. These fields are chargeless under all the other $U(1)$ in the chain. The schematic presentation of the model is displayed in Fig. 1. We also consider a chiral superfield $X$ neutral under all the $U(1)$ groups. The relevant superpotential considered in the underlying framework is

The other terms in the potential may be forbidden by imposing additional symmetries³³3For example, an $R$-symmetry under which $W\to e^{i\alpha}W$ and $X\to e^{i\alpha}X$ along with a $Z_{2}$ symmetry under which only $X$ and $\Phi_{i}^{+}$ are odd can forbid all the other terms in $W$.. The gauge interaction between chiral and vector superfields contains the following interaction term between the scalars $\phi_{i}^{\pm}$ residing in $\Phi_{i}^{\pm}$ and $D_{i}$ in ${\cal V}_{i}$.

where $G_{i}=|\phi_{i}^{+}|^{2}-|\phi_{i}^{-}|^{2}$ and $g_{i}$ is the gauge coupling corresponding to $U(1)_{i}$.

Elimination of the auxiliary fields from Eqs. (2,3,5) using the equations of motion implies

where $j=0,1,...,N$ and $G_{0}=G_{N+1}=0$. Substituting this solution in Eqs. (2,3,5) leads to the following $D$-term scalar potential

This together with the $F$-term potential derived from Eq. (4),

gives the complete scalar potential of the underlying framework, $V=V_{F}+V_{D}$.

Inflation can proceed in a way analogous to the standard $D$-term inflation mechanism proposed in Stewart (1995); Binetruy and Dvali (1996); Halyo (1996) (see also Evans et al. (2017); Domcke and Schmitz (2017, 2018) for the recent versions). We identify the radial component $\sigma$ of

as the inflaton. As discussed earlier, for the inflaton field value $\sigma^{2}\geq 2g_{N}\xi/\lambda^{2}_{N}$ the potential has minimum at $|\phi^{\pm}_{i}|=0$ and it is given by

Non-zero $D$-term for the $N^{\rm th}$ $U(1)$, $D_{N}=\xi$, spontaneously breaks supersymmetry and splits the masses of fermions and bosons residing within $\Phi^{\pm}_{N}$. The fermion masses are given by $m_{f}=\lambda_{N}|X|$ while the masses of scalars, as can be read off from Eqs. (7,8), are given by $m_{\pm}^{2}=\lambda_{N}^{2}|X|^{2}\mp g_{N}\xi$. This splitting in turn generates Coleman-Weinberg correction Coleman and Weinberg (1973) to the tree level potential. The 1-loop correction to the potential can be estimated using

where $i$ runs over the scalars and fermions. $\Lambda$ is the ultraviolet cutoff of the theory which we identify with the reduced Planck scale, $M_{P}=1/(8\pi G)^{1/2}=2.4\times 10^{18}\,{\rm GeV}$. The 1-loop corrected effective potential is then given by

for $\lambda^{2}_{N}\sigma^{2}\gg 2g_{N}\xi$. The constant tree level contribution provides the vacuum energy density required to drive inflation and the slow roll is provided by the 1-loop correction.

The values of $g_{N}$ and $\xi$ can be estimated by fitting the potential in Eq. (32) with the observables from inflation models. These observables are minimum number of e-foldings $N$, amplitude of temperature anisotropy $A_{s}$, spectral index $n_{s}$ and tensor-scalar ratio $r$. They are given by

Here, $\epsilon$ and $\eta$ are the slow roll parameters which for the potential in Eq. (32) are obtained as

$N_{\rm CMB}$ is the number of e-foldings between the end of inflation and the time when the CMB modes are exiting the inflationary horizon and $\sigma_{\rm CMB}$ is the value of the inflaton field when the CMB modes are exiting the inflation horizon. $\sigma_{c}$ is the critical value of $\sigma$ when inflation ends. Inflation can end in two possible ways. If at some value of $\sigma$, $\epsilon\simeq 1$ then the slow roll phase ends. It is also possible that $\sigma$ reaches the critical value when the local supersymmetry breaking minimum becomes unstable and the fields roll along the $\phi^{+}_{N}$ direction. This critical value is given by $\sigma_{c}=\sqrt{2g_{N}\xi}/|\lambda_{N}|$.

In the following, we assume $\sigma_{\rm CMB}\simeq M_{P}$ at the start of inflation so that we get ${\cal O}(50)$ e-foldings consistent with the Lyth bound Lyth (1997). The different horizons in the universe can have different values of the inflaton field and the patch of the universe where $\sigma\gtrsim M_{P}$ will start inflating and will dominate the volume of the universe. The observables given in Eq. (33) are then functions of model parameters $g_{N}$, $\lambda_{N}$ and $\xi$. Among these only $A_{s}$ depends on $\xi$. We scan the values of $g_{N}$ and $\lambda_{N}$ to determine all the observables except $A_{s}$. Parameter $\xi$ is then determined using the value of $A_{s}$ as measured by Planck 2018, $A_{s}=(2.099\pm 0.101)\times 10^{-9}$ Akrami et al. (2018). The results are displayed in Fig. 3.

We find that $n_{s}\leq 0.9775$, the upper $3\sigma$ limit as measured by Planck 2018 Akrami et al. (2018), requires $g_{N}\geq 0.92$ almost independent of values of $\lambda_{N}$. For $g_{N}=0.92$, $N_{\rm CMB}$ can be as large as 46 for $\lambda_{N}\simeq\sqrt{4\pi}$. The greater value of $g_{N}$ decreases the number of e-foldings considerably. The scalar to tensor ratio remains well below the upper bound, $r<0.062$. For $g_{N}=0.92$, the value of $\xi$ fixed from $A_{s}$ is determined as $\sqrt{\xi}\simeq 6.3\times 10^{15}$ GeV.

The inflation parameters $A_{s}$ and $n_{s}$ measured by Planck 2018 Akrami et al. (2018) are reported for the scale $k=0.05\,{\rm Mpc^{-1}}$. The mode with co-moving wavenumber $k$ exits the inflation horizon when the physical length scale of the perturbation $a_{k}/k$ is the size of the horizon $H_{I}^{-1}$, i.e when $a_{k}=kH_{I}^{-1}$. Therefore the number of e-foldings $N(k)$ before the end of inflation when a given mode $k$ leaves the inflation horizon is given by Liddle and Lyth (1993); Liddle and Leach (2003); Dodelson and Hui (2003)

which implies

Here, we have assumed that the at the end of inflation the inflaton oscillates at the bottom of the potential and the energy density falls as $\rho\sim a^{-3}$ and then the universe reheats due to the coupling of inflaton with the SM fields. In Eq. (40), $H_{I}=(V/3M_{P}^{2})^{1/2}$ is the Hubble parameter during inflation where the potential $V\simeq\xi^{2}/2$. With $\xi=(6.3\times 10^{15}\,{\rm GeV})^{2}$ we obtain the value of $H_{I}=6.65\times 10^{12}\,{\rm GeV}$. The ratio $a_{0}/a_{eq}=3450$ and $T_{eq}=0.81$ eV. Using these parameters in Eq. (40) and taking $N(k=0.05\,{\rm Mpc^{-1}})\simeq 46$ the reheat temperature turns out to be $T_{reh}=10^{6}\,{\rm GeV}$. Reheating at the end of inflation takes place when the $X$ particles decay into $\phi^{\pm}_{N}$ and gauge bosons.

As discussed in the previous section, a viable inflation within this framework requires

Substitution of the above in Eqs. (25,27,28) determines the allowed values of $g^{\prime}$ and $M_{Z^{\prime}}$ as function of CW parameters $N$, $k$ and $q$. $M_{Z^{\prime}}$ also depends on $v_{\chi}\lesssim v_{k}$. To be more specific, we associate $v_{\chi}$ with the breaking scale of $U(1)_{k}$ by assuming

where the second equality results from Eq. (10). This, along with the above values of $g_{N}$ and $\xi$, leads to

from Eq. (25). Similarly, substitution of the values of $g_{N}$, $\xi$ and $M_{V_{M}}=1$ TeV in Eqs. (27,28) implies

Desired value of $g^{\prime}$ and $M_{Z^{\prime}}$ can therefore be obtained by choosing appropriate values of $N$ and $k$ for a given $q$. The ratio $k/N$ however does not depend on the value of $q$ and it can be constrained once the nature of the SM current $J^{\mu}$ is fixed. We do this by identifying $U(1)_{M}$ with $L_{\mu}-L_{\tau}$, $L_{e}-L_{\tau}$ and $B-L$ symmetries.