Fixing the EW scale in supersymmetric models after the Higgs discovery

TeV-scale supersymmetry was originally introduced to solve the hierarchy problem and therefore fix the electroweak (EW) scale in the presence of quantum corrections. Numerical methods testing the SUSY models often report a good likelihood L (or chi^2=-2ln L) to fit the data {\it including} the EW scale itself (m_Z^0) with a {\it simultaneously} large fine-tuning i.e. a large variation of this scale under a small variation of the SUSY parameters. We argue that this is inconsistent and we identify the origin of this problem. Our claim is that the likelihood (or chi^2) to fit the data that is usually reported in such models does not account for the chi^2 cost of fixing the EW scale. When this constraint is implemented, the likelihood (or chi^2) receives a significant correction (delta_chi^2) that worsens the current data fits of SUSY models. We estimate this correction for the models: constrained MSSM (CMSSM), models with non-universal gaugino masses (NUGM) or higgs soft masses (NUHM1, NUHM2), the NMSSM and the general NMSSM (GNMSSM). For a higgs mass m_h\approx 126 GeV, one finds that in these models (delta_chi^2)/ndf> 1.5 (approx 1 for GNMSSM), which violates the usual condition of a good fit (total chi^2/ndf approx 1) already before fitting observables other than the EW scale itself (ndf=number of degrees of freedom). This has (negative) implications for SUSY models and it is suggested that future data fits properly account for this effect, if one remains true to the original goal of SUSY. Since the expression of delta_chi^2 that emerges from our calculation depends on a familiar measure of fine-tuning, one concludes that EW fine-tuning is an intrinsic part of the likelihood to fit the data that includes the EW scale (m_Z^0).Comment: 18 pages; (v4: added text in Conclusions

One-loop potential with scale invariance and effective operators

We study quantum corrections to the scalar potential in classically scale invariant theories, using a manifestly scale invariant regularization. To this purpose, the subtraction scale $\mu$ of the dimensional regularization is generated after spontaneous scale symmetry breaking, from a subtraction function of the fields, $\mu(\phi,\sigma)$. This function is then uniquely determined from general principles showing that it depends on the dilaton only, with $\mu(\sigma)\sim \sigma$. The result is a scale invariant one-loop potential $U$ for a higgs field $\phi$ and dilaton $\sigma$ that contains an additional {\it finite} quantum correction $\Delta U(\phi,\sigma)$, beyond the Coleman Weinberg term. $\Delta U$ contains new, non-polynomial effective operators like $\phi^6/\sigma^2$ whose quantum origin is explained. A flat direction is maintained at the quantum level, the model has vanishing vacuum energy and the one-loop correction to the mass of $\phi$ remains small without tuning (of its self-coupling, etc) beyond the initial, classical tuning (of the dilaton coupling) that enforces a hierarchy $\langle\sigma\rangle\gg \langle\phi\rangle$. The approach is useful to models that investigate scale symmetry at the quantum level.Comment: 10 pages; Contribution to the Proceedings of the Corfu Summer Institute 2015, Sep 2015, Corfu, Greec

Quantum implications of a scale invariant regularisation

We study scale invariance at the quantum level (three loops) in a perturbative approach. For a scale-invariant classical theory the scalar potential is computed at three-loop level while keeping manifest this symmetry. Spontaneous scale symmetry breaking is transmitted at quantum level to the visible sector (of $\phi$) by the associated Goldstone mode (dilaton $\sigma$) which enables a scale-invariant regularisation and whose vev $\langle\sigma\rangle$ generates the subtraction scale ($\mu$). While the hidden ($\sigma$) and visible sector ($\phi$) are classically decoupled in $d=4$ due to an enhanced Poincar\'e symmetry, they interact through (a series of) evanescent couplings $\propto\epsilon^k$, ($k\geq 1$), dictated by the scale invariance of the action in $d=4-2\epsilon$. At the quantum level these couplings generate new corrections to the potential, such as scale-invariant non-polynomial effective operators $\phi^{2n+4}/\sigma^{2n}$ and also log-like terms ($\propto \ln^k \sigma$) restoring the scale-invariance of known quantum corrections. The former are comparable in size to "standard" loop corrections and important for values of $\phi$ close to $\langle\sigma\rangle$. For $n=1,2$ the beta functions of their coefficient are computed at three-loops. In the infrared (IR) limit the dilaton fluctuations decouple, the effective operators are suppressed by large $\langle\sigma\rangle$ and the effective potential becomes that of a renormalizable theory with explicit scale symmetry breaking by the "usual" DR scheme (of $\mu=$constant).Comment: 18 pages; v3: minor clarification

SUSY naturalness without prejudice

Unlike the Standard Model (SM), supersymmetric models stabilize the electroweak (EW) scale $v$ at the quantum level and {\it predict} that $v$ is a function of the TeV-valued SUSY parameters ($\gamma_\alpha$) of the UV Lagrangian. We show that the (inverse of the) covariance matrix of the model in the basis of these parameters and the usual deviation $\delta\chi^2$ (from $\chi^2_{min}$ of a model) automatically encode information about the "traditional" EW fine-tuning measuring this stability, {\it provided that} the EW scale $v\sim m_Z$ is indeed regarded as a function $v=v(\gamma)$. It is known that large EW fine-tuning may signal an incomplete theory of soft terms and can be reduced when relations among $\gamma_\alpha$ exist (due to GUT symmetries, etc). The global correlation coefficient of this matrix can help one investigate if such relations are present. An upper bound on the usual EW fine-tuning measure ("in quadrature") emerges from the analysis of the $\delta\chi^2$ and the s-standard deviation confidence interval by using $v=v(\gamma)$ and the theoretical approximation (loop order) considered for the calculation of the observables. This upper bound avoids subjective criteria for the "acceptable" level of EW fine-tuning for which the model is still "natural".Comment: 13 pages. LaTeX, (v4: minor corrections

Manifestly scale-invariant regularization and quantum effective operators

Scale invariant theories are often used to address the hierarchy problem, however the regularization of their quantum corrections introduces a dimensionful coupling (dimensional regularization) or scale (Pauli-Villars, etc) which break this symmetry explicitly. We show how to avoid this problem and study the implications of a manifestly scale invariant regularization in (classical) scale invariant theories. We use a dilaton-dependent subtraction function $\mu(\sigma)$ which after spontaneous breaking of scale symmetry generates the usual DR subtraction scale $\mu(\langle\sigma\rangle)$. One consequence is that "evanescent" interactions generated by scale invariance of the action in $d=4-2\epsilon$ (but vanishing in $d=4$), give rise to new, finite quantum corrections. We find a (finite) correction $\Delta U(\phi,\sigma)$ to the one-loop scalar potential for $\phi$ and $\sigma$, beyond the Coleman-Weinberg term. $\Delta U$ is due to an evanescent correction ($\propto\epsilon$) to the field-dependent masses (of the states in the loop) which multiplies the pole ($\propto 1/\epsilon$) of the momentum integral, to give a finite quantum result. $\Delta U$ contains a non-polynomial operator $\sim \phi^6/\sigma^2$ of known coefficient and is independent of the subtraction dimensionless parameter. A more general $\mu(\phi,\sigma)$ is ruled out since, in their classical decoupling limit, the visible sector (of the higgs $\phi$) and hidden sector (dilaton $\sigma$) still interact at the quantum level, thus the subtraction function must depend on the dilaton only. The method is useful in models where preserving scale symmetry at quantum level is important.Comment: 16 pages (added references; published version

Decoupling of Heavy Kaluza-Klein Modes In Models With Five-Dimensional Scalar Fields

We investigate the decoupling of heavy Kaluza-Klein modes in $\phi^{4}$ theory and scalar QED with space-time topology $\mathbb{R}^{3,1} \times S^{1}$. We calculate the effective action due to integrating out heavy KK modes. We construct generalized RGE's for the couplings with respect to the compactification scale $M$. With the solutions to the RGE's we find the $M$-scale dependence of the effective theory due to higher dimensional quantum effects. We find that the heavy modes decouple in $\phi^{4}$ theory, but do not decouple in scalar QED. This is due to the zero mode of the 5-th component $A_{5}$ of the 5-d gauge field. Because $A_{5}$ is a scalar under 4-d Lorentz transformations, there is no gauge symmetry protecting it from getting mass and $A_{5}^{4}$ interaction terms after loop corrections. In light of these unpleasant features, we explore $S^{1}/\mathbb{Z}_{2}$ compactifications, which eliminate $A_{5}$, allowing for the heavy modes to decouple at low energies. We also explore the possibility of decoupling by including higher dimensional operators. It is found that this is possible, but a high degree of fine tuning is required.Comment: 9 pages, no figures; sign error on equations 20, 36, 37; Added additional reference

A review of naturalness and dark matter prediction for the Higgs mass in MSSM and beyond

Within a two-loop leading-log approximation, we review the prediction for the lightest Higgs mass (m_h) in the framework of constrained MSSM (CMSSM), derived from the naturalness requirement of minimal fine-tuning (Delta) of the electroweak scale and dark matter consistency. As a result, the Higgs mass is predicted to be just above the LEP2 bound, m_h=115.9\pm 2 GeV, corresponding to a minimal Delta=17.8, value obtained from consistency with electroweak and WMAP (3\sigma) constraints, but without the LEP2 bound. Due to quantum corrections (largely QCD ones for m_h above LEP2 bound), Delta grows \approx exponentially on either side of the above value of m_h, which stresses the relevance of this prediction. A value m_h>121 (126) GeV cannot be accommodated within the CMSSM unless one accepts a fine-tuning cost worse than Delta>100 (1000), respectively. We review how the above prediction for m_h and Delta changes under the addition of new physics beyond the MSSM Higgs sector, parametrized by effective operators of dimensions d=5 and d=6. For d=5 operators, one can obtain values m_h\leq 130 GeV for Delta<10. The size of the supersymmetric correction that each individual operator of d=6 brings to the value of m_h for points with Delta<100, is found to be small, of few (<4) GeV for M=8 TeV, where M is the scale of new physics. This value decreases (increases) by approximately 1 GeV for a 1 TeV increase (decrease) of the scale M. The relation of these results to the Atlas/CMS supersymmetry exclusion limits is presented together with their impact for the CMSSM regions of lowest fine-tuning.Comment: 27 pages, 19 figures; (new figures and references added; improved presentation

Palatini quadratic gravity: spontaneous breaking of gauged scale symmetry and inflation

We study quadratic gravity $R^2+R_{[\mu\nu]}^2$ in the Palatini formalism where the connection and the metric are independent. This action has a {\it gauged} scale symmetry (also known as Weyl gauge symmetry) of Weyl gauge field $v_\mu= (\tilde\Gamma_\mu-\Gamma_\mu)/2$, with $\tilde\Gamma_\mu$ ($\Gamma_\mu$) the trace of the Palatini (Levi-Civita) connection, respectively. The underlying geometry is non-metric due to the $R_{[\mu\nu]}^2$ term acting as a gauge kinetic term for $v_\mu$. We show that this theory has an elegant spontaneous breaking of gauged scale symmetry and mass generation in the absence of matter, where the necessary scalar field ($\phi$) is not added ad-hoc to this purpose but is "extracted" from the $R^2$ term. The gauge field becomes massive by absorbing the derivative term $\partial_\mu\ln\phi$ of the Stueckelberg field ("dilaton"). In the broken phase one finds the Einstein-Proca action of $v_\mu$ of mass proportional to the Planck scale $M\sim \langle\phi\rangle$, and a positive cosmological constant. Below this scale $v_\mu$ decouples, the connection becomes Levi-Civita and metricity and Einstein gravity are recovered. These results remain valid in the presence of non-minimally coupled scalar field (Higgs-like) with Palatini connection and the potential is computed. In this case the theory gives successful inflation and a specific prediction for the tensor-to-scalar ratio $0.007\leq r \leq 0.01$ for current spectral index $n_s$ (at $95\%$CL) and N=60 efolds. This value of $r$ is mildly larger than in inflation in Weyl quadratic gravity of similar symmetry, due to different non-metricity. This establishes a connection between non-metricity and inflation predictions and enables us to test such theories by future CMB experiments.Comment: 22 pages, 2 figures, LaTe