1 The ratio \(\rho _t=m_Z m_t/m_H\)

The problem of the mass of elementary particles has two independent aspects in particle physics. The first, how mass arises, it is answered in the SM by the Higgs mechanism.Footnote 1 The second aspect is why different elementary particles have their specific masses. Unless electromagnetic charge, there is not any, exact or approximate, known relation, structure or hierarchy among the masses of the SM elementary particles.

Evidence in favor of the existence of a boson with spin \(s=0\) and properties compatible with those of a SM Higgs particle (production cross sections, mass-proportional couplings) has been presented by the LHC experiments [2–8].

The aim of this work is to present some detailed phenomenological estimations of some ratios involving the experimental masses of the Higgs boson, the vector bosons and the top quark, the derivation from them of some model independent expressions and detailed study of them in the framework of the SM. In the light of the recent results from the LHC coming from the experiments ATLAS and CMS, the parameter defined by the relation

$$\begin{aligned} \rho _t&= \frac{m_Z m_t}{m_H^2} \end{aligned}$$
(2)

where \(m_Z,m_t\) are the masses of the \(Z^0\) gauge boson and the top quark and \(m_H\) is the Higgs mass has become experimentally measurable. We estimate its current value to be

$$\begin{aligned} \rho _t^{({\mathrm {exp}})}&= 0.9956\pm 0.0081 \end{aligned}$$
(3)

where we have used the current values for [1]

$$\begin{aligned} m_Z=91.1876\pm 0.0021~{{\mathrm {GeV}}}/\mathrm {c}^2, \end{aligned}$$
(4)
$$\begin{aligned} m_t=173.07\pm 0.52\pm 0.72~{{\mathrm {GeV}}}/\mathrm {c}^2 \end{aligned}$$
(5)

and the combined value of the boson masses presented by ATLAS and CMS [2–8],

$$\begin{aligned} m_H&= 125.9\pm 0.4 \pm 0.4~{{\mathrm {GeV}}}/\mathrm {c}^2. \end{aligned}$$
(6)

The combined value of the boson mass is obtained by standard statistic techniques, we neglect correlations among the systematic component of the errors. The value (3) is obtained by a MC simulation. First, a distribution of the quotient is obtained by generating Gaussian ensembles of the individual masses. Second, symmetric Gaussian fit, Fig. 1, is performed to the central part of this (close to symmetric) distribution avoiding the non-Gaussian tail. The central value and \(1\sigma \) errors appearing in (3) are extracted from this fitted Gaussian. If the individual values for each of the experiments are used instead (using a similar statistical procedure), we get (see Fig. 1 (up))

Fig. 1
figure 1

MC generated likelihood distributions for each of the quantities \(\rho _t\) (up) and \(\rho _{Wt}\) (down). The curves are in arbitrary units with normalized areas. The curves correspond to the global averaged Higgs mass (continuous) and individual ATLAS and CMS values (dashed and dotted lines). The shaded area is the \(1\sigma \) allowed region for each of the parameters

Full size image
$$\begin{aligned} \rho _t^{({\mathrm {exp}})} = 0.9940\pm 0.0102 \quad (m_{h,{\mathrm {ATLAS}}}), \end{aligned}$$
(7)
$$\begin{aligned} \rho _t^{({\mathrm {exp}})} = 0.9990\pm 0.0085 \quad (m_{h,{\mathrm {CMS}}}) \end{aligned}$$
(8)

for boson masses, respectively,

$$\begin{aligned} m_H=125.8\pm 0.4 \pm 0.4~{\mathrm {GeV}}/\mathrm {c}^2 \end{aligned}$$

and

$$\begin{aligned} m_H=126.0\pm 0.4 \pm 0.4~{\mathrm {GeV}}/\mathrm {c}^2. \end{aligned}$$

The conclusion is that the experimental value of the ratio \(\rho _t\) is close to one with a precision of the order or less than \(1~\%\). This precision is not far from the precision at which the well known ratio

$$\begin{aligned} \rho =m_W^2/m_Z^2\cos ^2\theta _W \end{aligned}$$

is presently measured, \(\rho =1.0008\pm 0.001\) [1] with \(\theta _W\) the Weinberg angle and \(m_W\) the charged electroweak gauge-boson mass. The closeness of this parameter \(\rho _t\) to one might be merely a coincidence which will disappear with any new measurement or might be not.

Note that the ratio would be exactly one for a boson mass (and nominal \(m_Z,m_t\) PDG masses) of

$$\begin{aligned} m_H (\rho _t=1)&\simeq 125.6~{\mathrm {GeV}}/ \mathrm {c}^2, \end{aligned}$$
(9)

a value somewhere in the 125–126 range of values currently measured by LHC and just on the borderline of the SM vacuum stability limits [29–34].

The ratio \(\rho _t\) would be still close to one, with a precision of \(5~\%\), if the Higgs mass appear finally anywhere in the range \(m_H=123\)–\(129~{\mathrm {GeV}}/\mathrm {c}^2\). If we vary the top mass in the range \(m_t\sim 170\)–\(175~\mathrm {GeV}/\mathrm {c}^2\) similar results are obtained.

Similar ratios involving other fermion masses instead of the top mass could be obviously defined, for example including all the fermions we could define \(\rho _\varSigma \) as

$$\begin{aligned} \rho _\varSigma&= \frac{m_Z m_\varSigma }{m_H^2}, \end{aligned}$$
(10)

with

$$\begin{aligned} m_\varSigma ^2&= \sum _f m_f^2 \end{aligned}$$
(11)

or including the third family quark doublet (\(m_Q^2=m_t^2+m_b^2\)) we could define the ratio

$$\begin{aligned} \rho _T&\equiv \frac{m_Z m_Q}{m_H^2},\end{aligned}$$
(12)
$$\begin{aligned}&\simeq \rho _t \biggl ( 1+ 2 \biggl (\frac{m_b}{m_t}\biggr )^2 \biggr ). \end{aligned}$$
(13)

In any case, any of these or similar ratios are deviated from \(\rho _t\) by a very moderate quantity \((m_b/m_t)^2\simeq 10^{-3}\).

It is also interesting to consider an alternative way to express the closeness of the ratio \(\rho _t\) to one. If we consider the individual mass rations \(m_Z/m_H, m_H/m_t\), their current experimental values are

$$\begin{aligned}&\frac{m_Z}{m_H} =0.725\pm 0.003, \end{aligned}$$
(14)
$$\begin{aligned}&\frac{m_H}{m_t} =0.727\pm 0.005 \end{aligned}$$
(15)

where we have taken the LHC combined value of \(m_H\). and PDG \(m_Z,m_t\) masses. Both ratios are the same at the level of 1 % (and totally compatible at even higher precision according to present error bars). Very similar results are obtained if we use any of the ATLAS or CMS individual measurements.

Similarly to \(\rho _t\) we define now another ratio of masses involving the Higgs, vector bosons and top quarks, whose experimental value is also seen to be close to one. Let us take

$$\begin{aligned} \rho _{Wt}&= \frac{m_W+m_t}{2 m_H} \end{aligned}$$
(16)

where \(m_W\) is the mass of the \(W\) boson. We estimate the current value of this ratio (using a similar MC technique as explained above, see Fig. 1 (down)) to be

$$\begin{aligned} \rho _{Wt}^{({\mathrm {exp}})}&= 1.0066\pm 0.0035 \end{aligned}$$
(17)

where we have used the current value for \(M_W\) [1]

$$\begin{aligned} M_W=80.385\pm 0.015~{\mathrm {GeV}}/\mathrm {c}^2 \end{aligned}$$

and the combined value for the Higgs mass, Eq. (6). If the individual values for each of the experiments are used instead, we get

$$\begin{aligned}&\rho _{Wt}^{({\mathrm {exp}})} = 1.0082\pm 0.0036 \quad (m_{h,{\mathrm {ATLAS}}}),\end{aligned}$$
(18)
$$\begin{aligned}&\rho _{Wt}^{({\mathrm {exp}})} = 1.0056\pm 0.0036 \quad (m_{h,{\mathrm {CMS}}}). \end{aligned}$$
(19)

The experimental value of the ratio \(\rho _{Wt}\) is close to one with a precision of the order of \(1~\%\). Nevertheless the favored value is around 2\(\sigma \) from being exactly one. The Higgs mass is numerically close to the arithmetic average of the charged boson and top masses \(m_H\sim (m_W+m_t)/2\). The ratios \(\rho _t\) and \(\rho _{Wt}\) are a priori independent.

The relations \(\rho _t\simeq \rho _{Wt}\simeq 1\) imply that any two of the quantities \(m_H,m_W,m_Z,m_t\) can be written in terms of the other two. Taking into account also the relation \(\rho \simeq 1\) we can write any two mass ratios as a function of, exclusively, the Weinberg angle (with a precision of the order of \(1~\%\) or better):

$$\begin{aligned} \frac{m_i}{m_j}&\simeq f_{ij}(\theta _W),\quad i,j=W,Z,H,t. \end{aligned}$$
(20)

Examples of these relations are

$$\begin{aligned}&\frac{m_W}{m_Z} \simeq \cos \theta _W ,\end{aligned}$$
(21)
$$\begin{aligned}&\frac{m_H}{m_Z} \simeq 1+\sqrt{2}\sin ^2 \frac{\theta _W}{2} ,\end{aligned}$$
(22)
$$\begin{aligned}&\frac{m_H}{m_t}\cos \theta _W \simeq 1-\sqrt{2}\sin ^2\frac{\theta _W}{2}. \end{aligned}$$
(23)

In the limit \(\cos \theta _W\rightarrow 1\) all the masses would become equal \(m_Z=m_W=m_t=m_H\).

2 In the SM

The latest LHC measurements [2–8], in particular the characteristics of the seen \(H\rightarrow \gamma \gamma \) channel, point to a preferred discovery of a neutral boson of spin 0. In a model independent way, thus the quantity \(\rho _t\) can be viewed as the ratio of the highest massive representatives of the spin \((0,1/2,1)\) particles of the Standard Model and, to a very good precision the experimental evidence tell us that

$$\begin{aligned} \rho _t^{({\mathrm {exp}})}\sim \frac{ m_{s=1} m_{s=1/2}}{m_{s=0}^2}&\simeq 1. \end{aligned}$$
(24)

Somehow the mass of the “lowest” scalar particle mass is numerically the geometric mean of the highest spin 1 and spin \(1/2\) masses.

Let us now assume that the new particle is a scalar Higgs boson. In the Standard Model (SM) with a Higgs sector consistent of one Higgs doublet \(\varPhi \) and scalar potential

$$\begin{aligned} V_{\mathrm {SM}}&= \mu ^2\varPhi ^\dagger \varPhi + \lambda \bigl (\varPhi ^\dagger \varPhi \bigr )^2, \end{aligned}$$
(25)

all masses are induced by spontaneous symmetry breaking and are proportional to the Higgs vacuum expectation \(\langle \phi _0\rangle =v/\sqrt{2}\) where

$$\begin{aligned} v^2&= -\frac{\mu ^2}{\lambda }=\frac{1}{\sqrt{2} G_F}= (246.218)^2\ \bigl ({\mathrm {GeV}}/\mathrm {c}^2 \bigr )^2. \end{aligned}$$
(26)

The tree level top, gauge and Higgs boson masses are given in terms of \(v\) and their respective Yukawa couplings

$$\begin{aligned}&m_W = g \frac{v}{2}, \qquad m_Z=\sqrt{g^2+g'^2} \frac{v}{2},\end{aligned}$$
(27)
$$\begin{aligned}&m_t = g_t \frac{v}{2},\end{aligned}$$
(28)
$$\begin{aligned}&m_H^2 = -2 \mu ^2= 2 \lambda v^2. \end{aligned}$$
(29)

Moreover we have \(g'=g \tan \theta _W\) or \(\sqrt{g^2+g'^2}=g/\cos \theta _W\), \(G_F m_W^2/\sqrt{2}=g^2/8\).

In terms of these quantities the tree level mass ratio \(\rho _t\) is simply given by

$$\begin{aligned} \rho _t^{0({\mathrm {SM}})}&= \frac{m_Z m_t}{m_H^2}= \sqrt{g^2+g'^2}\frac{v^{2}g_{t}}{4\sqrt{2} v^2 \lambda }\end{aligned}$$
(30)
$$\begin{aligned}&= \frac{1}{4\sqrt{2}}\frac{\sqrt{g^2+g'^2} g_t}{ \lambda } \end{aligned}$$
(31)
$$\begin{aligned}&= \frac{1}{4\sqrt{2}}\frac{g g_t}{\cos \theta _W \lambda }. \end{aligned}$$
(32)

Similarly, the tree level SM \(\rho _{Wt}\) ratio is given by

$$\begin{aligned} \rho _{Wt}^{0({\mathrm {SM}})}&= \frac{m_W +m_t}{2m_H}= \frac{g+ g_t}{4\sqrt{(2 \lambda )}}. \end{aligned}$$
(33)

Numerically, we estimate the top Yukawa, quartic coupling and other related ratios as (from measured masses, using tree level approximation, \(1\sigma \) errors):

$$\begin{aligned}&g_t^{0,({\mathrm {exp}})}= 1.409\pm 0.007,\end{aligned}$$
(34)
$$\begin{aligned}&\lambda ^{0,({\mathrm {exp}})} = 0.130\pm 0.001,\end{aligned}$$
(35)
$$\begin{aligned}&\biggl (\frac{\lambda }{g_t^2} \biggr )^{0,({\mathrm {exp}})} =\frac{1}{8} \biggl (\frac{m_H}{m_t} \biggr )^2=(6.6\pm 0.1)\times 10^{-2},\end{aligned}$$
(36)
$$\begin{aligned}&\biggl (\frac{g^2+g'{}^2}{\lambda } \biggr )^{0,({\mathrm {exp}})} =8 \biggl (\frac{m_Z}{m_H}\biggr )^2=4.21\pm 0.03. \end{aligned}$$
(37)

In the SM, the Higgs self-coupling \(\lambda \) is non-determined. However, assuming that both expressions \(\rho _t,\rho _{Wt}\simeq 1\) are not a coincidence, the relations (31) and (33) tell us that the scalar self-couplings, gauge couplings and Yukawa top couplings are related by two expressions. At tree level any two of the quantities \(\lambda ,g,g',g_t\) can be written in terms of the two others using the expressions

$$\begin{aligned} \lambda&\simeq c \sqrt{g^2+{g'} {}^2}g_t ,\end{aligned}$$
(38)
$$\begin{aligned} \lambda&\simeq c^2 (g+g_t)^2 \end{aligned}$$
(39)

where \(c\) is a numeric factor of order \({\sim } o(1)\). If we take into account only the first expression, the one related to the ratio \(\rho _t\), we arrive at the relation between the quartic, gauge and top couplings

$$\begin{aligned} \lambda&\sim g g_t. \end{aligned}$$
(40)

Let us take now into account both expressions. For \(g_t\gg g\) the second equation becomes \(\lambda \simeq c^2 g_t^2\), inserting it in the first one we arrive at

$$\begin{aligned} \lambda&\simeq \kappa \bigl (g^2+{g'} {}^2\bigr ) \end{aligned}$$
(41)

with \(\kappa \simeq 1+o(g/g_t)\).

The tree level relations (31, 33) are affected by SM quantum corrections. Including one-loop corrections, the three level relations above should be replaced, in particular by (where \(\mu _0\) the renormalization scale, \(\mu _0\sim m_Z-m_t\))

$$\begin{aligned}&g_t (\mu _0) = \frac{\sqrt{2 m_t}}{v} \bigl ( 1+\delta _t(\mu _0) \bigr ),\end{aligned}$$
(42)
$$\begin{aligned}&\lambda (\mu _0) = \frac{\sqrt{ m_H^2}}{2v^2} \bigl ( 1+\delta _\lambda (\mu _0) \bigr ), \end{aligned}$$
(43)

we consider negligible the running of the gauging couplings \(g_i(\mu _0)\). The first order corrected ratio \(\rho _t\) is then, using expressions (42, 43),

$$\begin{aligned} \rho _t^{\mathrm {SM}}&= \frac{m_Z m_t}{m_H^2}\end{aligned}$$
(44)
$$\begin{aligned}&= \frac{1}{4\sqrt{2}}\frac{g g_t}{\cos \theta _W \lambda } \frac{ 1+\delta _\lambda }{ 1+\delta _t}\end{aligned}$$
(45)
$$\begin{aligned}&\simeq \rho _t^0 ( 1+\delta _\lambda -\delta _t ). \end{aligned}$$
(46)

The top Yukawa \(\delta _t\) can be written as \(\delta _t=\delta _t^{\mathrm {QCD}}+\delta _t^w\). The corrections are ([9–11] and references therein), ignoring logarithm terms,

$$\begin{aligned}&\delta _\lambda = \frac{1}{16 \pi ^2} c_\lambda \lambda ,\end{aligned}$$
(47)
$$\begin{aligned}&\delta _t^w = \frac{1}{16 \pi ^2} \frac{c_t}{8}g_t^2,\end{aligned}$$
(48)
$$\begin{aligned}&\delta _t^{\mathrm {QCD}}= \bigl (-1/\bigl (3 \pi ^2\bigr )\bigr ) g_s^2, \end{aligned}$$
(49)

with the numerical coefficients \(c_\lambda \simeq 25/2-9 \pi /(2\sqrt{3})\simeq 4.3\), \(c_t \simeq 6.1\). Thus

$$\begin{aligned} \frac{\delta _\lambda }{\delta _t^w}&\simeq \frac{c_\lambda }{c_t} \biggl (\frac{m_H}{m_t}\biggr )^2\simeq 0.3. \end{aligned}$$
(50)

Then

$$\begin{aligned} \rho _t&= \rho _t^0 \bigl (1+c_1 \lambda -c_2 g_t^2-c_s g_s^2 \bigr ). \end{aligned}$$
(51)

The correction \(\delta _t^{\mathrm {QCD}}\sim 5~\%\) is the most important one, acting to diminish slightly the ratio. Both corrections, \(\delta _t^w,\delta _\lambda \), are of opposite sign and very small, of the order of \(1~\%\).

3 SM renormalization group equations

We explore here the behavior of the mass ratio (3, 31) at higher scales. We consider first a reduced system of one-loop renormalization group equations where only the Higgs self-coupling \(\lambda \) and the top Yukawa coupling \(g_t\) appear. All the other couplings are considered very small or not running at all. The RGE equations for the individual couplings take the form (see for example [12–24]) (with \(t=\log (\mu /\varLambda )\), an expression valid for high, but no so high, scales \(\mu \gg m_t,m_H\), or for \(\varLambda \rightarrow \infty \))

$$\begin{aligned}&\frac{d g_t^2}{d t} = \frac{9}{16 \pi ^2} g_t^4,\end{aligned}$$
(52)
$$\begin{aligned}&\frac{d \lambda }{d t} = \frac{6}{16 \pi ^2} \bigl (4\lambda ^2+2 \lambda g_t^2- g_t^4 \bigr ). \end{aligned}$$
(53)

If we introduce the variable

$$\begin{aligned} R&= \frac{\lambda }{g_t^2}, \end{aligned}$$
(54)

the RGE equations for \(g_t,R\) and \(\rho _t(t)\) become decoupled with nested solutions, \(g_t=g_t(\mu )\), \(R=R(g_t)\), \(\rho _t=\rho _t(R)\). In addition to Eq. (52), we have

$$\begin{aligned}&g_t^2\frac{d R}{d g_t^2} = \frac{1}{3} f(R),\end{aligned}$$
(55)
$$\begin{aligned}&\frac{d \rho _t}{d R} = -\frac{3\rho _t}{2 f(R)} \biggl (1+\frac{2f(R)}{3 R}\biggr ) \end{aligned}$$
(56)

with \(f(R)=8R^2+R-2\). Equations (52, 55, 56) can be solved explicitly, in particular for the ratio \(\rho _t\) we can write

$$\begin{aligned} \rho _t=k \biggl (\frac{R_0-R}{R_1+R} \biggr )^{R_0-R_1}R^2, \end{aligned}$$

where \(R_0,R_1\) are the fixed points of Eq. (55), \(f(R_{0,1})=0\). For a light Higgs and large top mass the ratio \(R\) is small, at low scales \(R^{{\mathrm {exp}}}\sim 10^{-1}\), Eq. (36). For such a small \(R\) the solution of the differential equations is approximately:

$$\begin{aligned} R(g_t)&= R_c-\frac{4}{3}\log g_t, \end{aligned}$$
(57)

and

$$\begin{aligned} \rho _t&\sim k R^2 \sim \biggl (R_c-\frac{4}{3}\log g_t\biggr )^2\sim k R_c^2 \sim \rho _t^0. \end{aligned}$$
(58)

At large energies (\(\mu \gg m_t\), as long as \(R>0\) or \(\lambda >0\)), the ratio \(\rho _t(\mu )\) keeps approximately constant, only slightly decreasing with the logarithm of \(g_t\).

If we consider a reduced Higgs–top–strong system where the \(\lambda ,g_t,g_s\) are non-vanishing and allowed to run together with the ratios \(R,\rho _t\). One ends with a similar system of equations where the evolution of \(\rho _t\) is of the type \( g_t^2d \rho _t/g_t^2\sim \rho _t h(R, g_t^2)\) and similar results are obtained.

At higher energies, and for more quantitative results, a full treatment is needed. Present state-of-the-art NLO and NNLo constraints on SM vacuum stability [29–34] seems to guarantee stability, and then a reasonably stable, positive, value for the quartic coupling, for a Higgs mass \(m_H\sim 126~\mathrm {GeV}/\mathrm {c}^2\) and to very high scales. If we assume a stable behavior for \(\lambda \) and ignoring the very modest variation of the coupling factor \(g^2+g'{}^2\),

$$\begin{aligned} \rho _t(\mu )\sim \rho _t^0\frac{g_t(\mu )}{g_t^0} \end{aligned}$$

the variation of the mass ratio \(\rho _t\) is governed by the top Yukawa up to scales where new physics is expected to emerge.

4 Conclusions and further discussion

We expect new physics that cuts off the divergent top, gauge and Higgs loop contributions to the Higgs mass at scales \(\lesssim 10\) TeV. Many different possibilities have been well explored, they usually include, more or less ad hoc, new particles with properties tightly associated to those of the SM. Some of these possibilities are for example (and any combinations among them) [25–28]: (a) The new particles are just the, softly broken, SUSY, superpartners with couplings and Yukawas strongly dictated by supersymmetry and the soft breaking itself. (b) The Higgs is a composite resonance, or (c) The “Little” Higgs is a pseudo-Nambu–Goldstone boson with respect a “softly” broken approximate global symmetry. This scalar sector is accompanied by some new particles belonging to enlarged multiplets together with the SM particles.

It is a general feature that, in all or most of these models, the quartic self-coupling, and then the Higgs mass, is related to the gauge coupling constants and to the top Yukawa in a more or less explicit way, reminding one of the relation (36) suggested by the experimental evidence \(\rho _t\simeq 1\). The reason is clear [25, 26], the new one loop which are proportional to the couplings of the SM gauge sector (or to a subsector of an enlarged gauge sector) have to match and cancel the top and the other quadratic loops.

We will briefly review the situation in the MSSM and Littlest Higgs scenarios. In the MSSM, the tree level top, gauge and lowest Higgs boson masses together their respective Yukawa couplings are given by the expressions

$$\begin{aligned}&v^2 = v_1^2+v_2^2,\qquad \tan \beta =v_2/v_1,\end{aligned}$$
(59)
$$\begin{aligned}&m_W = g \frac{v}{2}, \qquad m_Z=\sqrt{g^2+g'^2} \frac{v}{2},\end{aligned}$$
(60)
$$\begin{aligned}&m_t = g_t \frac{v}{2}\sin \beta , \end{aligned}$$
(61)

where the tree level Higgs quartic coupling is determined in terms of the gauge couplings

$$\begin{aligned} \lambda ^0&= \bigl (g^2+g'{}^2\bigr )\cos ^22\beta . \end{aligned}$$
(62)

Quantum corrections to the Higgs quartic coupling are very important. They lead for an expression for the lower neutral Higgs mass, of the form [35]

$$\begin{aligned} m_H^2&= m_Z^2\cos ^2 2\beta + \delta m_H^2\end{aligned}$$
(63)
$$\begin{aligned}&= m_Z^2 \cos ^2 2\beta + f\frac{3 G_F}{\sqrt{2}\pi ^2} m_t^4 \end{aligned}$$
(64)

where the factor \(f\) include logarithmic corrections, mainly associated to stops. From the expression (64) and from \(m_H^2= 2 \lambda (\mu ) v^2\) we can extract an improved value for the quartic effective coupling

$$\begin{aligned} \lambda (\mu )&= \frac{m_H^2}{2 v^2} \bigl (1+\delta _\lambda (\mu ) \bigr ). \end{aligned}$$
(65)

The effective quartic coupling is of the natural size \(\lambda \sim o(g^2, g_t^4)\). In terms of these quantities the mass ratio \(\rho _t\) is simply given by

$$\begin{aligned} \rho _t^{\mathrm {MSSM}}&= \frac{m_Z m_t}{m_H^2}\end{aligned}$$
(66)
$$\begin{aligned}&= \frac{\sqrt{g^2+g'^2} g_t \sin \beta }{(g^2+g'^2) \cos ^22\beta + g_t^4 \sin ^4\beta 3f/\pi ^2}. \end{aligned}$$
(67)

In the context of the MSSM, the experimental evidence \(\rho _t\simeq 1\) is a strong hint for the existence of a relation among the parameters of the expression above, principally top Yukawa and \(\tan \beta \) together with the gauge couplings.

As a second illustrative example, let us mention the “Littlest” Higgs scenario [25, 26]. Here the usual Higgs doublet, is the lightest of a set of pseudo Goldstone bosons in a non-linear sigma model including in its gauge group different \(SU(2)\times U(1)\) factors. The product group is broken to the diagonal, identified as the SM electroweak gauge group. The top Yukawa coupling generates a negative mass squared triggering electroweak symmetry breaking. New particles are added, in particular heavy top partners, which cancel the one-loop quadratically divergent corrections. The quartic self-coupling is related to the coupling constants of the gauge sector and to the top Yukawa with a natural size

$$\begin{aligned} \lambda \sim o\bigl (g^2,g_t^2\bigr ), \end{aligned}$$

reminding one, for example, of the expression (40) suggested by experimental evidence. Particular scenarios can be tuned so that either the gauge contributions or the top Yukawas dominate the Higgs quartic and \(m_H\sim m_Z\) or \(m_H\sim m_t\) as extreme cases. In fact we have seen, according to Eq. (3), that nature chooses, to a very high precision, just the geometric average of both extreme cases \(m_H=\sqrt{m_Z m_t}\). It seems plausible that a Little Higgs scenario can be found where this value appears naturally. Approximate accidental global symmetries related to the Little Higgs scenario could play a role in the understanding of the \(\rho _t\) ratio, as the global custodial \(SU(2)_c\) symmetry [36–38] plays for the \(\rho \) ratio.

In this short note we have presented some simple computations associated to the ratio of the product of \(Z^0\) and top masses to the squared Higgs mass, \(\rho _t\). We have shown how this ratio is surprisingly and robustly close to unity at the \(10^{-3}\) level. The Higgs mass seems to be just the geometrical mean of the \(m_Z\) and \(m_t\) masses.

We have briefly reviewed the theoretical predictions of this ratio in the SM and beyond. In the SM, the Higgs self-coupling \(\lambda \) is undetermined. However, assuming that the value \(\rho _t^{({\mathrm {exp}})}\simeq 1\) is not merely a coincidence, the relation (31) tell us that the scalar self-coupling and the spin 1 and spin \(1/2\) top couplings are subject to the tree level equality

$$\begin{aligned} \lambda&\simeq c \sqrt{g^2+{g'} {}^2} g_t \simeq c g g_t, \end{aligned}$$
(68)

where \(c\) is a numeric factor of order \({\sim } o(1)\). Such a relation is not very much affected by quantum effects at least up to scales \(\mu \sim m_t\) or slightly higher.

Moreover, the combined use of both the relations \(\rho _t^{({\mathrm {exp}})}\simeq 1\), \(\rho _{Wt}^{({\mathrm {exp}})}\simeq 1\) leads to the SM tree level relation (not very much affected by quantum effects)

$$\begin{aligned} \lambda&\simeq g^2+{g'} {}^2. \end{aligned}$$
(69)

In a model independent way, the relations \(\rho _t\simeq \rho _{Wt}\simeq 1\) imply that any two of the quantities \(m_H,m_W,m_Z,m_t\) can be written in terms of the other two. Taking into account also the relation \(\rho \simeq 1\) we can write any two mass ratios as a function of, exclusively, the Weinberg angle (with a precision of the order of \(1~\%\) or better) \(\frac{m_i}{m_j}\simeq f_{ij}(\theta _W)\), \( i,j=W,Z,H,t\). In the limit \(\cos \theta _W\rightarrow 1\) all the masses would become equal \(m_Z=m_W=m_t=m_H\). The existence of such relations could be interpreted as a hint for an instrumental role, together with some other unknown symmetry, of the \(SU(2)_c\) custodial symmetry in the explanation of the \(m_H/m_t\) ratio [36–38].

The closeness of the parameter \(\rho _t,\rho _{Wt}\) to 1, “the Higgs mass coincidence problem”, might be merely a coincidence or an artifact of the early status of the Higgs discovery, which will disappear with any new measurement. It is tempting to think that such a value of the ratios are, on naturalness grounds, a signal of a deeper mechanism or symmetry.