Confronting Electroweak Fine-tuning with No-Scale Supergravity

Applying No-Scale Supergravity boundary conditions at a heavy unification scale to the Flipped $SU(5)$ grand unified theory with extra TeV-scale vector-like multiplets, $i.e.$ No-Scale ${\cal F}$-$SU(5)$, we express the $Z$-boson mass $M_Z$ as an explicit function of the boundary gaugino mass $M_{1/2}$, $M_Z^2 = M_Z^2 (M_{1/2}^2)$, with implicit dependence upon a dimensionless ratio $c$ of the supersymmetric Higgs mixing parameter $\mu$ and $M_{1/2}$. Setting the top Yukawa coupling consistent with $m_t = 174.3$ GeV at $M_Z = 91.2$ GeV, the value of $c$ naturally tends toward $c \simeq 1$, which indirectly suggests underlying action of the Giudice-Masiero mechanism. Proportional dependence of all model scales upon the unified gaugino mass $M_{1/2}$ in the No-Scale ${\cal F}$-$SU(5)$ model suggests one possible mechanism of confronting the electroweak fine tuning problem.


INTRODUCTION
Supersymmetry (SUSY) represents a solution to the big gauge hierarchy problem, logarithmically sequestering the reference to ultra-heavy (Grand Unification, Planck, String) scales of new physics. However, there is a residual little hierarchy problem, implicit in the gap separating TeV-scale collider bounds on (strong production of) yet elusive colored superpartner fields from the observation of a 125 GeV light CP-even Higgs; indeed, this same heaviness of the SUSY scale appears equally requisite to sufficient elevation of loop contributions to the physical Higgs mass itself. Reconciling these considerations without an unnatural invocation of fine tuning, vis-à-vis unmotivated cancellation of more than (say) a few parts per centum between contributions to physics at the electroweak (EW) scale, may require the providence of a unified framework wherein the entire physical spectrum (Standard Model + SUSY) may be expressed as functions of a single parameter.
The SUSY framework naturally provides for interplay between quartic and quadratic field strength terms in the scalar potential of the type essential to spontaneous destabilization of the null vacuum, the former emerging with dimensionless gauge-squared coupling coefficients from the D-term, and the latter with dimensionful masssquared coefficients referencing the bilinear Higgs mixing scale µ from the chiral F -term. Crucially though, this radiative electroweak symmetry breaking (EWSB) event, as driven by largeness of the top-quark Yukawa coupling, is not realizable without the supplementary in-clusion of soft mass terms m H u,d and the analog B µ of µ, which herald first the breaking of SUSY itself. In a supergravity (SUGRA) context, these terms may be expected to appear in proportion to the order parameter of SUSY breaking in the visible sector, as gravitationally suppressed from higher scale effects in an appropriately configured hidden sector, namely the gravitino mass M 3/2 . The gravitino mass may itself be exponentially radiatively suppressed relative to the high scale, plausibly and naturally taking a value in the TeV range. The Giudice-Masiero (GM) mechanism may be invoked to address the parallel "µ problem", suggesting that this SUSY-preserving coupling may likewise be of the same order, and likewise generated as a consequence of SUSY breaking, as evaluated at the high scale.
Minimization of the Higgs scalar potential with respect to the H u and H d field directions yields two conditions on the pair of resulting vacuum expectation values (VEVs) (v u , v d ). The overall scale (v 2 u + v 2 d ) 1/2 , in product with the gauge coefficients (g 2 L + g ′ 2 Y ) 1/2 /2, is usually traded for the physical Z-boson mass M Z , whereas the relative VEV strengths are parameterized (tan β ≡ v u /v d ) by an angle β. This allows one to solve for µ and B µ at the electroweak scale in terms of M Z , tan β, and the soft masses m H u,d . When addressing the question of fine tuning, the solution for µ 2 is typically inverted as follows in Eq. (1), and an argument is made regarding the permissible fraction of cancellation between terms on the right-hand side, whose individual scales may substantially exceed M 2 Z .
For moderately large tan β, Eq. (1) reduces to At the outset, it must be noted that some level of cancellation should here be permitted, and indeed expected, within Eq. (2) as an unavoidably natural consequence of the EWSB event itself. Specifically, destabilization of the symmetric vacuum occurs in conjunction with (the falsely tachyonic) negation of the quadratic H 2 u field coefficient, provided by m 2 Hu + µ 2 ≃ −M 2 Z /2, which is dynamically driven to vanish and then flow negative under the renormalization group. However, the proverbial devil may lurk in the details, an element of which are loop-level radiative corrections V eff ⇒ V tree +V 1−loop to the effective scalar potential, and likewise via derivatives of V 1−loop to the minimization condition expressed in Eq. (1), which is recast with m 2

VARIOUS MEASURES OF FINE-TUNING
Several approaches to quantifying the amount of fine tuning implicit in Eq. (1) have been suggested, one of the oldest being that ∆ EENZ [1,2] first prescribed some 30 years ago by Ellis, Enqvist, Nanopoulos, and Zwirner (EENZ), consisting of the maximal logarithmic M Z derivative with respect to all fundamental parameters ϕ i , evaluating at some high unification scale Λ as is fitting for gravity-mediated SUSY breaking. In this treatment, low fine-tuning mandates that heavy mass scales only weakly influence M Z , whereas strongly correlated scales should be light.
Lately, a prescription ∆ EW emphasizing evaluation directly at the electroweak scale [3,4] has attracted attention, isolating contributions to the (loop-modified) righthand side of Eq. (1) as a ratio with the left.
More strictly [3], individual contributions to Σ u,d u,d may be compared to M 2 Z /2 in isolation, without allowance for cancellation even within the internal summation.
A variation [4,5] of the prior, dubbed ∆ HS , would further split the electroweak evaluation of the soft masses and µ-term into the sum of their corresponding values at the high scale (HS) Λ plus a logarithmic correction from the renormalization group running, specifically The scale Λ referenced in Eq. (5) is not necessarily of GUT or Planck order, but may rather contextually refer to a SUSY breaking messenger as low as some tens of TeV [6], although this somewhat blurs distinction from the Σ u,d u,d terms comprising Eq. 4. In any event, leading top-stop loops generate corrections δm 2 Hu ∼ m 2 t1 × ln(mt 1 /Λ) that have been used [3,6] to infer naturalness bounds on the light stop mass (thereby also limiting vital parallel contributions to the Higgs mass), and similarly on the gluino mass. It has been suggested [3,6,7] that a large negative trilinear soft term A t , which may be generated radiatively at the intermediate scale, may engender cancellations in the light stop t 1 tuning contribution while simultaneously lifting m h .

IMPACT OF DYNAMICS ON TUNING
In the SUGRA context, M 2 Z is generically bound to dimensionful inputs ϕ i at the high scale Λ via a bilinear functional, as shown following. The parameters ϕ i may include scalar and gaugino soft SUSY breaking masses (whether universal or not), the bi-and tri-linear soft terms B µ and A i , as well as the µ-term. The coefficients C i and C ij are calculable, in principle, under the renormalization group dynamics.
Applying the Eq. (3) prescription, a typical contribution to the fine tuning takes the subsequent form.
Comparing with Eq. (6), each individual term in the Eq. (7) sum is observed, modulo a possible factor of 2, to be simply the ratio of one contribution to the unified M 2 Z mass, divided by M 2 Z /2. The structural similarity of the ∆ EENZ and ∆ HS prescriptions is therefore clear.
Differing conclusions drawn with respect to the supposed naturalness of a given SUSY model construction by the various fine tuning measures described are implicit within underlying assumptions that each makes regarding what may constitute a natural cancellation. For example, Eq. (3) permits two types of dynamic cancellation that are not recognized by Eq. (5). The first of these is between multiple Eq. (6) terms in which a single parameter ϕ i may appear. The second is between each high-scale parameter and its running correction, for example between µ 2 (Λ) and δµ 2 , as represented by, and absorbed into, the numerical coefficients C i and C ij . ∆ HS is therefore a harsher metric of tuning than ∆ EENZ . It is sometimes argued [4,8] that ∆ EW , which adopts a perfectly a priori view of only the low energy particle spectrum, sets an absolute lower bound on fine tuning via holistic recombination of high scale mass parameters and their potentially large running logarithms. Proponents of this opinion acknowledge [4] that the suggested bound is only as strong as an assumption that the SUSY breaking soft masses m 2 H u,d and the SUSY preserving µ-term harken from wholly disparate origins, such that cancellations amongst the two classes are inherently unnatural.
The positions of the authors in the current work are that (i) precisely the prior mode of cancellation is made natural by its essential role in promoting the electroweak symmetry breaking event, and that (ii) solid theoretical motivation exists (GM mechanism) for suspecting the SUSY preserving µ scale to have yet likewise been forged in the supersymmetry breaking event. If the former precept is granted, then the limit offered by ∆ EW is potentially spurious. If the latter precept is additionally granted, then the very notion of electroweak fine-tuning may be moot.

TUNING IN NO-SCALE F -SU (5)
Bottom up support for the prior opinions is provided here by consideration of a specific model of low energy physics named F -SU (5) (see Ref. [9] and references therein), which combines (i) field content of the Flipped SU (5) grand unified theory (GUT), with (ii) a pair of hypothetical TeV-scale vector-like supermultiplets ("flippons") of mass M V derivable within local F-Theory model building, and (iii) the boundary conditions of No-Scale Supergravity (SUGRA). The latter amount to vanishing of the scalar soft masses and the tri-/bi-linear soft couplings M 0 = A 0 = B µ = 0 at the ultimate F -SU (5) gauge unification scale M F ≃ M Pl , and are enforced dynamically by invocation of a minimal Kähler potential [10,11]. Non-zero boundary values may be applied solely to the universal gaugino mass M 1/2 , as necessarily implied by SUSY breaking, and to the µ-parameter. In this perspective, the value of µ is actually that evolved up from the scale dynamically established in EWSB; its retrospective similarity to (and proportional scaling with) M 1/2 is interpreted as a deeply suggestive accident. The model is highly constrained by the need to likewise dynamically tether (via the Renormalization Group Equations (RGEs)) the value of B µ generated in EWSB to its mandated vanishing at M F ; in fact, this releases the constraint typically exhausted by the fixing of B µ to instead determine tan β ≃ 20, which incidentally supports the approximation adopted by Eq. (2), while not being so large that impact of the bottom quark Yukawa coupling is substantially heightened.
The consequence (at fixed Z-boson M Z and top-quark m t masses) is an effectively one-parameter model, with all leading dynamics established by just the single degree of freedom allocated to M 1/2 . Inclusion of the flippon multiplets provides a vital modification to the βfunction RGE coefficients, most notably resulting in nullification of the color-charge running (b 3 = 0) at the first loop; however, dynamic dependence on the mass scale M V is quite weak, affecting gauge unification only via logarithmic feedback from a threshold correction term. Some vestigial freedom remains for the preservation of B µ (M F ) = 0 by compensating adjustments to tan β and M V at fixed M 1/2 , at the price of disrupting the natural tendency of this model to supply a suitable thermal dark matter (over 99% Bino) candidate; to be precise, there are associated fluctuations induced in the lightest neutralino (Bino) vs. the next to the LSP (stau) mass gap that synchronously affect the dark matter coannihilation rate.
Numerical analysis of the parameter interdependencies in No-Scale F -SU (5) is conducted with SuSpect 2.34 [12], utilizing a proprietary codebase modification that incorporates the flippon-enhanced RGEs. Applying the ∆ EW measure of Eq. (4) to the F -SU (5) model is thereby found to indicate a level of tuning that may indeed be considered large. Down-type contributions to Eq. (4) are tan β-suppressed, and Σ u u is computed to be rather small, possibly reflecting the presence of large, negative trilinear couplings. Dominant contributions to M 2 Z /2 are thus restricted to solely the pair of terms m 2 Hu and µ 2 appearing directly in Eq. (2). Each term, or its absolute square root evaluated at the EWSB scale, is larger than and roughly proportional to the boundary value of M 1/2 , with a ratio around 1.8 for M 1/2 ∼ 400 GeV that drops to about 1.3 for M 1/2 ∼ 1500 GeV. The corresponding contributions to ∆ EW therefore increase from about 140 to about 1000 over the same range of inputs. One interpretation of this circumstance, taking |m Hu | and |µ| to arise from disparate mechanisms, is that narrow tracking and cancellation of the two terms indicates fine tuning. However, we make the case here for a very different point of view: that it is precisely the close tracking of |m Hu | and |µ|, as dynamically induced by electroweak symmetry breaking, and preserved under projection onto the boundary scale M F by the renormalization group, in the context of a single parameter construction, that suggests a natural underlying interdependence.
It is not the purpose of this work to present a functioning hidden sector that is capable of producing at some high scale the requisite SUSY breaking, along with the associated soft term and µ-parameter boundary values. It is the purpose of this work to investigate, primarily by numerical means 1 , the dependency of all contextual low energy physics upon the single input scale M 1/2 (and by extension its dependence in turn upon M 3/2 ). Insofar as this may be demonstrated, the considered model possesses no fine tuning, i.e. no large cancellations enforced coincidentally, without a common dynamic origin. Insofar as the relation µ(M F ) ≃ M 1/2 may be validated, the providence of an underlying GM mechanism is supported through identification of the "fingerprints" which it has impressed upon the low energy phenomenology.
The most remarkable element of this construction may be the capacity to so severely constrain freedom of input parameterization (with all concomitant benefits to the interrelated questions of the little hierarchy and the µproblem) while retaining consistency (even at the purely thermal level) with dark matter observations, limits on rare processes, and collider bounds.

GAUGINO PARAMETERIZATION OF F -SU (5)
The EW fine-tuning was numerically computed for No-Scale F -SU (5) according to the Eq. (3) prescription in Ref. [13], yielding result of O(1). This absence of finetuning is equivalent to a statement that the Z-boson mass M Z can be predicted in F -SU (5) as a parameterized function of M 1/2 ; clarifying and rationalizing this intuition in a more quantitative manner is a key intention of the present section. First, we define a dimensionless ratio c of the supersymmetric Higgs mixing parameter µ at the unification scale M F with the gaugino mass M 1/2 .
This parameter c is a fixed constant if the µ term is generated via the Giudice-Masiero mechanism [14], which can, in principle, be computed from string theory. Its invocation implicitly addresses the need to otherwise explicitly refer to the µ parameter as an independent high scale input. We shall numerically scan over arbitrary values of this parameter, although the No-Scale F -SU (5) construction will be demonstrated to prefer a narrow range near c ≃ 1. The vector-like flippon mass parameter M V is expected to develop an exponential compensation (for B µ = 0 and all else constant) of the fundamental scale M 1/2 due to its previously described appearance within a logarithmic threshold correction, where the undetermined dimensionful parameters A(λ) and R(λ) may be sensitive (among other things) to the top quark Yukawa coupling λ. In the same vein, sensitivity to tan β is weak, and it is expected that any residual dependencies upon either parameter within the region of interest may be Taylor-expanded for absorption into a generic quadratic function of M 1/2 . We thus adopt an ansätz consistent with Eq. (6), where the undetermined coefficients f i represent implicit functions of dimensionless quantities including c and λ. Some evidence suggests that the dimensionful coefficients f 1 and f 2 may additionally be sensitive to B µ , particularly to any potential deviations from the null No-Scale boundary value. If (f 1 ≪ M 2 1/2 ) and (f 2 ≪ M 1/2 ), then a linearized approximation of the prior is applicable: The form of Eq. (10) must now be verified with explicit RGE calculations. This is accomplished via a numerical sampling, wherein the Z-boson mass is floated within 20 ≤ M Z ≤ 500 GeV, and the top quark mass (equivalently its Yukawa coupling) within 125 ≤ m t ≤ 225 GeV. The region scanned for the gaugino mass boundary is within 100 ≤ M 1/2 ≤ 1500 GeV. In order to truncate the scanning dimension, M V and tan β are explicitly parameterized functions of M 1/2 (consistent with the prior description) such that the physical region of the model space corresponding to M Z = 91.2 GeV and m t = 174.3, along with a valid thermal relic density, is continuously intersected 2 ; this may be considered equivalent to fixing the top quark Yukawa coupling (and associated higherorder feedback) within just this subordinate parameterization. The range of the ratio c from Eq. (8) is an output of this analysis, which is run from the EWSB scale up to M F under the RGEs.
As an initial phase of the analysis, the only constraints applied are the No-Scale SUGRA boundary conditions M 0 = A 0 = 0, along with correct EWSB, a convergent µ term, and no tachyonic sfermion or pseudoscalar Higgs boson masses, these latter conditions being the required minimum for proper RGE evolution. These constructionist elements carve out a narrow viable parameter space between 83 M Z 93 GeV, which is illustrated as a function of the dimensionless parameter c in FIG. 1. The blue region represents those points that cannot satisfy the minimum evolution requirements. The band of points clustered on the M Z axis with c = 0 have no RGE solution for µ at all. The red sliver within 83 M Z 93 GeV, narrowly bounding the physical region around M Z ∼ 91 GeV, represents the sole surviving region of model space satisfying the boundary condition M 0 = A 0 = 0 with a convergent RGE solution. The gauge couplings being held fixed, the fluctuation of M Z is attributable to fluctuation of the Higgs VEV magnitude, as compensated by differential contributions to the right-hand side of Eq. (1) induced by variation of µ, M 1/2 , and the top quark Yukawa λ. As a second phase of the analysis, the final No-Scale SUGRA constraint B µ = 0 must be applied at the M F unification scale. The vanishing B µ = 0 requirement is enforced numerically with a width |B µ | ≤ 1 GeV that is comparable to the scale of EW radiative corrections. The effect is to carve out a simply connected string of points from the narrow red region, depicted in FIG. 2 for seven values of M 1/2 in the model space.
The No-Scale SUGRA constraint on the B µ parameter naturally parameterizes all the particle and sparticle masses as a function of the dimensionless parameter c of Eq. curves. The Higgs boson mass m h in TABLE I includes both the tree level+1-loop+2-loop+3-loop+4-loop contributions [9] and the additional flippon contribution [19]. Sensitivity is observed to fluctuation of the VEV scale with M Z .  The dimensionless parameter c is expected to be a fixed constant if the µ term is generated by the GM mecha- nism. On any single-valued slice of the present parameterization with respect to c, particle and sparticle masses will residually be dependent upon just