Quantum field theory of K-mouflage

We consider K-mouflage models, which are K-essence theories coupled to matter. We analyze their quantum properties and in particular the quantum corrections to the classical Lagrangian. We setup the renormalization program for these models and show that, contrary to renormalizable field theories where renormalization by infinite counterterms can be performed in one step, K-mouflage theories involve a recursive construction whereby each set of counterterms introduces new divergent quantum contributions which in turn must be subtracted by new counterterms. This tower of counterterms can be in principle constructed step by step by recursion and allows one to calculate the finite renormalized action of the model. In particular, it can be checked that the classical action is not renormalized and that the finite corrections to the renormalized action contain only higher-derivative operators. We concentrate then on the regime where calculability is ensured, i.e., when the corrections to the classical action are negligible. We establish an operational criterion for classicality and show that this is satisfied in cosmological and astrophysical situations for (healthy) K-mouflage models which pass the solar system tests. These results rely on perturbation theory around a background and are only valid when the background configuration is quantum stable. We analyze the quantum stability of astrophysical and cosmological backgrounds and find that models that pass the solar system tests are quantum stable. We then consider the possible embedding of the K-mouflage models in an UV completion. We find that the healthy models which pass the solar system tests all violate the positivity constraint which would follow from the unitarity of the putative UV completion, implying that these healthy K-mouflage theories have no UV completion. We then analyze their behavior at high energy, and we find that the classicality criterion is satisfied in the vicinity of a high-energy collision, implying that the classical K-mouflage theory can be applied in this context. Moreover, the classical description becomes more accurate as the energy increases, in a way compatible with the classicalization concept.

Figure 1

Derivative of the kinetic function, $K^{\prime }(\chi )$, for the cubic, arctan, and square root (sqrt) models.

Figure 2

Squared-mass $R_K^2{\overline{m}}^2(r)$, as a function of $r/R_K$, and potential $V(x)$, as a function of $x$, for the cubic model.

Figure 3

Squared-mass $R_K^2{\overline{m}}^2(r)$, as a function of $r/R_K$, and potential $V(x)$, as a function of $x$, for the arctan and sqrt models.

Figure 4

Ground-state energy $E_0$ as a function of the parameter $\nu$ that multiplies the potential $V(x)$, for the arctan and sqrt models. The physical case corresponds to $\nu =1$, where we find $E_0>0$.

Figure 5

Squared mass $m_v^2(a)$ from Eq. (145) as a function of the scale factor, for the cubic, arctan, and sqrt models. We show the combination $-(2a/H_0^2{\mathrm{\Omega }}_{\mathrm{m}0})m_v^2$, which is positive (i.e., $m_v^2<0$). We also plot the case $m_v^2=-a^{\prime \prime }/a$ (lower line).

Figure 6

Growing mode $v_k^{+}(\tau )$, as a function of the scale factor $a$, for the arctan and sqrt models at $k=0$. We show the ratio $v_0^{+}/a$, which is constant at both high and low redshifts for these models.