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Physics Letters B

www.elsevier.com/locate/physletb

Confronting electroweak fine-tuning with No-Scale Supergravity

Tristan Leggetta, Tianjun Lib c, James A. Maxind'e'*, Joel W. Walkerh

Dimitri V. Nanopoulosa f g,

CrossMark

a George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA b State Key Laboratory of Theoretical Physics and Kavli Institute for Theoretical Physics China (K1TPC), Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, PR China

c School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, PR China d Department of Physics and Engineering Physics, The University of Tulsa, Tulsa, OK 74104, USA e Department of Physics and Astronomy, Ball State University, Muncie, IN 47306, USA

f Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, Woodlands, TX 77381, USA g Academy of Athens, Division of Natural Sciences, 28 Panepistimiou Avenue, Athens 10679, Greece h Department of Physics, Sam Houston State University, Huntsville, TX 77341, USA

A R T I C L E I N F 0

A B S T R A C T

Article history:

Received 23 August 2014

Received in revised form 11 November 2014

Accepted 13 November 2014

Available online 18 November 2014

Editor: M. Cvetic

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 F-SU(5), we express the Z-boson mass MZ as an explicit function of the boundary gaugino mass M\/2, M| = M2Z(My2), with implicit dependence upon a dimensionless ratio c of the supersymmetric Higgs mixing parameter // and Mi/2. Setting the top Yukawa coupling consistent with mt = 174.3 GeV at MZ = 91.2 GeV, the value of c naturally tends toward c ~ 1, which indirectly suggests underlying action of the Giudice-Masiero mechanism. Proportional dependence of all model scales upon the unified gaugino mass Mi/2 in the No-Scale F-SU(5) model suggests one possible mechanism of confronting the electroweak fine-tuning problem.

© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

1. 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. One possible mechanism for reconciliation of these considerations without an unnatural invocation of fine-tuning, vis-à-vis unmo-tivated cancellation of more than (say) a few parts per centum between contributions to physics at the electroweak (EW) scale, could be the providence of a unified framework wherein the entire physical spectrum (Standard Model + SUSY) may be expressed as functions of a single parameter.

* Corresponding author.

E-mail address: james-maxin@utulsa.edu (J.A. Maxin).

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 dimen-sionful mass-squared coefficients referencing the bi-linear Higgs mixing scale / 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 inclusion of soft mass terms mHu,d and the analog B/ of /, 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 M3/2. 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 "/ problem", suggesting that this SUSY-preserving coupling may likewise

http://dx.doi.org/10.1016/j.physletb.2014.11.023

0370-2693/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

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 Hu and Hd field directions yields two conditions on the pair of resulting vacuum expectation values (VEVs) (vu, vd ). The overall scale (vU + v2)1/2, in product with the gauge coefficients (g2 + gY2)1/2/2, is usually traded for the physical Z-boson mass MZ, whereas the relative VEV strengths are parameterized (tan fi = vu/vd) by an angle fi. This allows one to solve for ¡i and Bat the electroweak scale in terms of MZ, tan fi, and the soft masses mHu d. When addressing the question of fine-tuning, the solution for ¡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|.

— tan2 ßm

2 tan2 ß — 1

For moderately large tan ß, Eq. (1) reduces to

22 - —mHu — ß ■

At the outset, it must be noted that some level of cancellation should be here 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 HU field coefficient, provided by m2H + j2 — -M2Z/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 Veff ^ Vtree + VHoop to the effective scalar potential, and likewise via derivatives of VHoop to the minimization condition expressed in Eq. (1), which is recast with m2Hu ^ m2H + EU, and ml, ^m2Hd + .

'Hd ■

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 AEENz [1,2] first prescribed some 30 years ago by Ellis, En-qvist, Nanopoulos, and Zwirner (EENZ), consisting of the maximal logarithmic MZ derivative with respect to all fundamental parameters pi, evaluating at some high unification scale A as is fitting for gravity-mediated SUSY breaking. In this treatment, low fine-tuning mandates that heavy mass scales only weakly influence MZ , whereas strongly correlated scales should be light.

Aeenz = Max

d ln(MZ) Pi d Mz \

d ln(pz Mz dpi A

Lately, a prescription ÀEW emphasizing evaluation directly at the electroweak scale [3,4] has attracted attention, isolating contributions to the (loop-modified) right-hand side of Eq. (1) as a ratio with the left.

ÀEW = Max

2 mHd E d Ed m2Hu tan2 ß

tan2 ß — 1 tan2 ß — 1 tan2 ß — 1

EU tan2 ß

tan2 ß — 1

Mi 2 ■

More strictly [3], individual contributions to ^U'¡j may be compared to M2 /2 in isolation, without allowance for cancellation even within the internal summation.

A variation [4,5] of the prior, dubbed AHS, would further split the electroweak evaluation of the soft masses and ¡-term into the sum of their corresponding values at the high scale (HS) A plus a logarithmic correction from the renormalization group running, specifically m2H ^ m2H (A) + Sm2H , and j2 ^ J2 (A) + Sj2.

À Hs = Max

2 mHd SmHd m2Hu tan2 ß

tan2 ß — 1 tan2 ß — 1 tan2 ß — 1

tan2 ß

tan2 ß — 1

ß2, Sß2\ + MZ■

The scale A 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 d terms comprising Eq. (4). In any event, leading top-stop loops generate corrections SmHu ~ m2 x ln(m^i/A) 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 tri-linear soft term At, which may be generated radiatively at the intermediate scale, may engender cancellations in the light stop it1 tuning contribution while simultaneously lifting mh.

3. Impact of dynamics on tuning

In the SUGRA context, M| is generically bound to dimensionful inputs pi at the high scale A via a bi-linear functional, as shown following. The parameters pi may include scalar and gaugino soft SUSY breaking masses (whether universal or not), the bi- and tri-linear soft terms B j and Ai, as well as the ¡-term. The coefficients Ci and Cij are calculable, in principle, under the renormalization group dynamics.

MZ = £ Cip?(A) + £ Cijpi (A)pj (A). (6)

Applying Eq. (3) prescription, a typical contribution to the fine-tuning takes the subsequent form.

d ln(M2 ) d ln(Pi)

= MP2 x {2Q piCijpj} ■

Comparing with Eq. (6), each individual term in Eq. (7) sum is observed, modulo a possible factor of 2, to be simply the ratio of one contribution to the unified M| mass, divided by MZ/2. The structural similarity of the Aeenz and AHS 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 Pi may appear. The second is between each high-scale parameter and its running correction, for example between j2(A) and Sj2, as represented by, and absorbed into, the numerical coefficients Ci and Cij. AHS is therefore a harsher metric of tuning than Aeenz. It is sometimes argued [4,8] that AEW, 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 m2H and the SUSY-preserving ¡-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 г scale to have yet likewise been forged in the supersymmetry breaking event. If the former precept is granted, then the limit offered by AEW is potentially spurious. If the latter precept is additionally granted, then the very notion of electroweak fine-tuning may be moot.

4. 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 MV 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 M0 = A0 = B ¡¡, = 0 at the ultimate F-SU(5) gauge unification scale Mf — Mpi, and are enforced dynamically by invocation of a minimal Kahler potential [10,11]. Non-zero boundary values may be applied solely to the universal gaugino mass Mi/2, as necessarily implied by SUSY breaking, and to the ¡-parameter. In this perspective, the value of г is actually that evolved up from the scale dynamically established in EWSB; its retrospective similarity to (and proportional scaling with) M1 /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 Bgenerated in EWSB to its mandated vanishing at Mf ; in fact, this releases the constraint typically exhausted by the fixing of B^ to instead determine tan в — 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 MZ and top-quark mt masses) is an effectively one-parameter model, with all leading dynamics established by just the single degree of freedom allocated to M1/2. Inclusion of the flippon multiplets provides a vital modification to the в-function RGE coefficients, most notably resulting in nullification of the color-charge running (b3 = 0) at the first loop; however, dynamic dependence on the mass scale MV is quite weak, affecting gauge unification only via logarithmic feedback from a threshold correction term. Some vestigial freedom remains for the preservation of B^(Mf) = 0 by compensating adjustments to tan в and MV at fixed M1/2, 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 AEW 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 в-suppressed, and ЯЦ is computed to be rather small, possibly reflecting the presence of large, negative tri-linear

couplings. Dominant contributions to M|/2 are thus restricted to solely the pair of terms m2H and /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 Mi/2, with a ratio around 1.8 for Mi/2 ~ 400 GeV that drops to about 1.3 for M1/2 ~ 1500 GeV. The corresponding contributions to AEW therefore increase from about 140 to about 1000 over the same range of inputs. One interpretation of this circumstance, taking |mHu | and |/| 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 |mHu| and |/|, as dynamically induced by electroweak symmetry breaking, and preserved under projection onto the boundary scale Mf 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 /-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 M1/2 (and by extension its dependence in turn upon M3/2). Insofar as this may be demonstrated, the considered model can successfully attack the fine-tuning problem, i.e. no large cancellations enforced coincidentally, without a common dynamic origin. Insofar as the relation /(Mf) — M1/2 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 /-problem) while retaining consistency (even at the purely thermal level) with dark matter observations, limits on rare processes, and collider bounds.

5. Gaugino parameterization of F-SU(5)

The EW fine-tuning was numerically computed for No-Scale F-SU(5) according to Eq. (3) prescription in Ref. [13], yielding result of O(1). This absence of fine-tuning is equivalent to a statement that the Z-boson mass MZ can be predicted in F-SU(5) as a parameterized function of M1/2; 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 su-persymmetric Higgs mixing parameter / at the unification scale Mf with the gaugino mass M1/2:

m f ) Ml/2 '

This parameter c is a fixed constant if the ¡г 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 ¡ 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 MV is expected to

1 A parallel analytical treatment, making directly formal, albeit approximate, application of the No-Scale F-SU(5) renormalization group and boundary conditions

is underway, with results intended to follow in a separate publication.

develop an exponential compensation (for Bj = 0 and all else constant)

Mv - A{X]eMV2lR(X)

of the fundamental scale M1/2 due to its previously described appearance within a logarithmic threshold correction, where the undetermined dimensionful parameters A (A) and R (X) may be sensitive (among other things) to the top quark Yukawa coupling X. In the same vein, sensitivity to tan /) 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 M1/2.

We thus adopt an ansatz

M Z = fi + f2 Mi/2 + fs M2yi

consistent with Eq. (6), where the undetermined coefficients fi represent implicit functions of dimensionless quantities including c and X. Some evidence suggests that the dimensionful coefficients fi and f2 may additionally be sensitive to B, particularly to any potential deviations from the null No-Scale boundary value. If ( fi ^ M2/2) and ( f2 ^ M1/2), then a linearized approximation of the prior is applicable:

Mz = fa + fbMi/2.

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 < MZ < 500 GeV, and the top quark mass (equivalently its Yukawa coupling) within 125 < mt < 225 GeV. The region scanned for the gaugino mass boundary is within 100 < M1/2 < 1500 GeV. In order to truncate the scanning dimension, MV and tan /) are explicitly parameterized functions of M1/2 (consistent with the prior description) such that the physical region of the model space corresponding to MZ = 91.2 GeV and mt = 174.3, along with a valid thermal relic density, is continuously intersected2; this may be considered equivalent to fixing the top quark Yukawa coupling (and associated higher-order 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 Mf under the RGEs.

As an initial phase of the analysis, the only constraints applied are the No-Scale SUGRA boundary conditions M0 = A0 = 0, along with correct EWSB, a convergent j term, and no tachy-onic 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 < MZ < 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 MZ axis with c = 0 have no RGE solution for j at all. The red sliver within 83 < MZ < 93 GeV, narrowly bounding the physical region around MZ ~ 91 GeV, represents the sole surviving region of model space satisfying the boundary condition M0 = A0 = 0 with a convergent RGE solution. The gauge couplings being held fixed, the fluctuation of MZ 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 j, M1/2, and the top quark Yukawa X.

2 The No-Scale F-SU(5) model space favors a top quark mass of mt = 174.3 — 174.4 GeV in order to compute a Higgs boson mass of mh ~ 125 GeV [15-17]. The central world average top quark mass has recently ticked upward (along with an increase in precision) to mt = 174.34 GeV [18], affirming this preference.

Computed results for all points within 20 <MZ< 500 GeV Only region with correct RGE evolution within 20<MZ< 500 GeV

Mm = 1.2 TeV

M0=A„ = 0 No constraint on B

Fig. 1. Representation of the region of the model space for M1/2 = 1.2 TeV that cannot sustain correct RGE evolution (blue) versus the region that can compute a SUSY spectrum with no RGE errors, EWSB, convergent j term, and no tachyonic masses (red). All other M1/2 produce identical results. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

-Polynomial Fit ofMm = 1500 GeV

-Polynomial Fit of Mm = 1350 GeV

-Polynomial Fit of Mm = 1200 GeV

-Polynomial Fit of Mm = 1050 GeV

-Polynomial Fit of Mm = 900 GeV

-Polynomial Fit of Mm = 700 GeV

-Polynomial Fit of Mm = 500 GeV

M=A= B.= 0

Fig. 2. The Z-boson mass is shown as a function of the dimensionless parameter c for seven different values of My2. The black points are the results of the RGE calculations, while the curves are polynomial fits. The curves are only comprised of points with a vanishing B^ parameter at the Mf unification scale.

As a second phase of the analysis, the final No-Scale SUGRA constraint B^ = 0 must be applied at the Mf unification scale. The vanishing B^ = 0 requirement is enforced numerically with a width |B< 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 M1/2 in the model space.

The No-Scale SUGRA constraint on the B/x parameter naturally parameterizes all the particle and sparticle masses as a function of the dimensionless parameter c of Eq. (8). This is clearly shown in Fig. 3 for the Z-boson mass MZ, top quark mass mt, Higgs boson mass mh, and gluino mass mg. We use the gluino mass as an example, though the entire SUSY spectrum can also thusly be

Table 1

Results of RGE calculations for four different values of M1/2. These are only points with a vanishing B/ parameter at the Mf unification scale. The entries highlighted in bold are those that compute the observed experimental measurements for MZ , mt, and mh.

Af = 1.2 TeV

=500 GeV

c Mz mt mh mg

0.955 90.189 164.47 118.88 655

0.965 90.349 166.29 120.03 659

0.975 90.543 168.34 121.30 664

0.985 90.720 170.11 122. 48 669

1.005 91.181 174.31 125.28 681

1.015 91.444 176.49 126.84 687

1.025 91.741 178.81 128.45 694

1.035 92.082 181.31 130.21 702

M1/2 = 900 GeV

c Mz mt mh mg

0.933 90.496 167.69 119.46 1182

0.943 90.659 169.38 120.70 1190

0.953 90.818 170.95 121.85 1197

0.963 91.011 172.76 123.20 1205

0.973 91.189 174.35 124.39 1212

0.983 91.390 176.06 125.71 1220

0.993 91.652 178.17 127.28 1230

1.003 91.849 179.67 128.53 1238

M1/2 = 1200 GeV

c Mz mt mh mg

0.913 90.525 167.96 120.31 1574

0.923 90.672 169.49 121.48 1583

0.933 90.832 171.06 122.66 1591

0.943 90.999 172.63 123.88 1601

0.953 91.180 174.24 125.09 1610

0.963 91.371 175.87 126.41 1619

0.970 91.502 176.94 127.26 1626

0.983 91.780 179.11 129.04 1640

M1/2 = 1350 GeV

c Mz mt mh mg

0.905 90.537 168.06 120.76 1773

0.915 90.691 169.66 121.92 1782

0.925 90.850 171.21 123.12 1792

0.935 91.017 172.77 124.35 1802

0.945 91.188 174.29 125.55 1811

0.955 91.367 175.81 126.77 1822

0.969 91.644 178.03 128.59 1838

0.977 91.806 179.28 129.62 1847

parameterized as a function of c via the B/ = 0 condition. The point chosen in Fig. 3 to exhibit the correlation between the particle and sparticle masses is M1/2 = 1200 GeV. Table 1 itemizes certain numerical results from the RGE calculations for four of the Fig. 2 curves. The Higgs boson mass mh in Table 1 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 MZ .

The dimensionless parameter c is expected to be a fixed constant if the / term is generated by the GM mechanism. On any single-valued slice of the present parameterization with respect to c, particle and sparticle masses will residually be dependent upon just M1/2, as is visible for variation of the Z-boson mass in Fig. 2. This is made explicit for c = 0.80, 0.85, 0.90, 0.95, 1.00 in Fig. 4, where each curve is well fit by a quadratic in the form of Eq. (10). Fig. 5 demonstrates a fit against the linear approximation in Eq. (11). As the c parameter decreases, Fig. 5 illustrates that the linear fit approaches the precision of the quadratic fit. The dimensionful intercept fa is a function of c, but is observed gener-

— Z-boson mass Mz [GeV] —top quark mass mt [GeV]

— Higgs boson mass mh [GeV] = A = B.= 0

— gluino mass [GeV]

160 1.050

120.0S

H7.5gj]

1580 o

0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975 1.0001.025

Fig. 3. Depiction of the correlation between the Z-boson mass MZ, top quark mass mt, Higgs boson mass mh, and gluino mass mg, as a function of c, for Mi/2 = 1.2 TeV. All other SUSY particles can be expressed similarly. The curves are only comprised of points with a vanishing B^ parameter at the Mf unification scale. All other M1/2 produce comparable correlations.

Quadratic fit c= 1.00 -Quadratic fit c = 0.95 -Quadratic fit c = 0.90 -Quadratic fit c- 0.85 -Quadratic fit e-0.80

Ml =/1+/2Mi;2 + /3M12(

500 600 700

800 900 1000 1100 1200 1300 1400 1500 Gaugino mass M [GeV]

Fig. 4. Simple quadratic fits for M2Z as a function of Mj/2. Five different cases of c are shown. The curves are only comprised of points with a vanishing B/ parameter at the Mf unification scale. The black points are sampled from Fig. 2.

ically to take a value in the vicinity of 89 GeV. As seen in Fig. 2, larger values of M1/2 correlate with smaller values of c at fixed Z-boson mass; it is the region M1/2 > 900 GeV that remains viable for probing a prospective SUSY signal at the 13-14 TeV LHC in 2015-2016.

The relationship between the / term and M1/2 at the Mf unification scale is linear for fixed MZ , with a slope given by the ratio c from Eq. (8). This is expanded in Fig. 6 for MZ = 91.2. Parameterization of the flippon mass MV and tan/) as functions of M1/2 (with the top quark Yukawa and approximate relic density fixed) are illustrated in Fig. 7.

Having established a (family in c of) quadratic expression(s) for M2Z in Eq. (6) form, the Z-boson mass is extracted by reference only to M1/2 and c at the high scale A, and fine-tuning may be evaluated. Adopting the linear Eq. (11) form, we first consider tuning with respect to M1/2 at fixed c, as prescribed by Eq. (3).

> 91.5-

i91'0'

1 90.5-

N 89.5

89.088.5-

Lmcarfitc- 1.00 -Linear fit с = 0.95 -Linear fit с = 0.90 - Linear fit с = 0.85 -Linear fit с = 0.80

Mz=L+fbMu

500 600 700

800 900 1000 1100 1200 1300 1400 1500 Gaugino Mass M [GeV]

800 900 1000 1100 1200 1300 1400 1500 Gaugino Mass Myl [GeV]

Fig. 5. Linear fits for MZ as a function of M1/2. Five different cases of c are shown. The curves are only comprised of points with a vanishing Bj parameter at the Mf unification scale. The black points are sampled from Fig. 2.

500 600 700 800 900 1000 1100 1200 1300 1400 1500 Gaugino Mass Mm [GeV]

Fig. 6. Linear relationship between the j term at the Mf unification scale and M1/2 for MZ = 91.2 GeV.

Лл — ЛМ1/2 -

д ln(MZ) M1/2 д Mz

д ln(Mi/2) Mz дMi/2

Mz\ fb

Fig. 7. Vector-like flippon mass MV and tan p as functions of M1 /2 with Yukawa couplings fixed.

- MZ. Therefore,

using the numerical observation da + dibM1/2 stipulating the adopted high-scale context, we suggest that the more natural fine-tuning measure for No-Scale F-SU(5) may be Aeenz ~ 1.

6. Conclusions

We have shown here that by implementing only the No-Scale Supergravity boundary conditions of M0 = A0 = B j = 0 at the unification scale, the Z-boson mass MZ can be expressed as a simple quadratic function of the unified gaugino mass M1/2, i.e. M2Z = MZ(M2/2), in the supersymmetric GUT model No-Scale F-SU(5). A top-down string theoretic construction may be expected to fix the Yukawa couplings and a dimensionless boundary ratio c of the supersymmetric Higgs mixing parameter j with M1/2 at some heavy unification scale. The only degree of freedom left to influence MZ is M1/2. Setting the top Yukawa coupling consistent with mt = 174.3 GeV at MZ = 91.2 GeV, the value of c naturally tends toward c — 1, which suggests underlying action of the Giudice-Masiero mechanism. The regions of the model space in correspondence with the physical masses MZ = 91.2 GeV and mt = 174.3 GeV are further consistent with the correct Higgs boson mass mh — 125 GeV and dark matter observations, and possess overlap with the limits on rare processes and collider bounds. Proportional dependence of all model scales upon the unified gaug-ino mass M1/2 in the No-Scale F-SU(5) model could suggest one potential mechanism of confronting the electroweak fine-tuning problem.

Curiously, this expression evaluates very close to zero. It would appear this result is a consequence of the fact that the physical Z-boson mass can in fact be stably realized for a large continuum of M1/2 values, at the expense of variation in the ratio c. It may be better understood by attending in turn to the parallel functional dependence on the dimensionless c parameter itself. We have

Лл =

Лс —

д ln(MZ)

д ln(c)

c д MZ

mz д c

JL (f + sAmi/2 мД дc дc 1/2

-c ~ 1,

Acknowledgements

D.V.N. would like to thank Andriana Paraskevopoulou for inspiration and discussions during the writing of this paper. This research was supported in part by: the DOE grant DE-FG03-95-Er-40917 (D.V.N.), the Natural Science Foundation of China under grant numbers 10821504, 11075194, 11135003, and 11275246, the National Basic Research Program of China (973 Program) under grant number 2010CB833000 (T.L.), the Ball State University ASPiRE Research Grant Program under grant number I434-14 (J.A.M.), and the Sam Houston State University Enhancement Research Grant program (J.W.W.).

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