1 Introduction

An alternative version of the standard model (SM), constructed using an ultraviolet finite quantum field theory with nonlocal field operators, was investigated in previous work [1, 2]. In place of Dirac delta functions, \(\delta (x)\), the theory uses distributions \({\mathcal {E}}(x)\) based on finite-width Gaussians. The Poincaré and gauge-invariant model adapts perturbative quantum field theory (QFT), with a finite renormalization, to yield finite quantum loops. For the weak interactions, \(SU(2)\times U(1)\) is treated as an ab initio broken symmetry group with nonzero masses for the W and Z intermediate vector bosons and for left and right quarks and leptons. The model guarantees the stability of the vacuum. Two energy scales, \(\Lambda _M\) and \(\Lambda _H\), were introduced; the rate of asymptotic vanishing of all coupling strengths at vertices not involving the Higgs boson is controlled by \(\Lambda _M\), while \(\Lambda _H\) controls the vanishing of couplings to the Higgs. Experimental tests of the model, using future linear or circular colliders, were proposed. The present observations are consistent with \(\Lambda _M \ge 10\) TeV. The Higgs boson mass hierarchy problem will be solved if future experiments confirm the prediction \(\Lambda _H\lesssim 1\) TeV.

In the following, we will investigate the consequences of an application of renormalization group (RG) methods for the perturbative finite renormalizable model. We will concentrate on a nonlocal spin 0 scalar field \(\phi =\phi _H\) Lagrangian model which is perturbatively formulated in Euclidean momentum space and might describe the Higgs boson field if nonlocality were fundamental. The ultraviolet finite theory resolves the Higgs mass hierarchy problem, the scalar field model triviality problem and removes the Landau pole singularity for the Higgs field.

2 Scalar field theory

The Lagrangian we consider for a real scalar field \(\phi \equiv \phi _H\) describing the Higgs boson in Euclidean space is

$$\begin{aligned} {\mathcal {L}}_H=\frac{1}{2}(-\phi \Box \phi +m_0^2\phi ^2)+\frac{1}{4!}\lambda _0\phi ^4. \end{aligned}$$
(1)

Using the formalism of [3], we assume that the vacuum expectation of the bare field \(\phi \) vanishes and write \(\phi =Z^{1/2}\phi _r\), where \(\phi _r\) is the renormalized field. Expressed as series expansions in powers of the physical coupling \(\lambda \), mass m and energy scale \(\Lambda _H\), the field strength renormalization constant Z and the bare parameters \(m_0\) and \(\lambda _0\) are given by:

$$\begin{aligned}&Z=1+\delta Z(\lambda , m, \Lambda _H^2), \end{aligned}$$
(2)
$$\begin{aligned}&Zm_0^2=m^2+\delta m^2(\lambda , m,\Lambda _H^2),\end{aligned}$$
(3)
$$\begin{aligned}&Z^2\lambda _0=\lambda +\delta \lambda (\lambda , m, \Lambda _H^2). \end{aligned}$$
(4)

The propagator in Euclidean momentum space is given by

$$\begin{aligned} i\Delta _H(p)\equiv \frac{i{\mathcal {E}^2(p)}}{p^2+m^2}, \end{aligned}$$
(5)

where \({\mathcal {E}}(p)\) is the entire function:

$$\begin{aligned} {\mathcal {E}}(p)=\exp \biggl [-\biggl (\frac{p^2+m^2}{2\Lambda _H^2}\biggr )\biggr ]. \end{aligned}$$
(6)

Evaluating the one-loop self-energy graph gives a constant shift to the Higgs boson bare self-energy [3]:

$$\begin{aligned} -i\Sigma _0=\frac{-iZ^{-2}\lambda }{32\pi ^2}m^2\,\Gamma \biggl (-1,\frac{m^2}{\Lambda _H^2}\biggr ), \end{aligned}$$
(7)

where \(\Gamma (n,z)\) is the incomplete gamma function:

$$\begin{aligned} \Gamma (n,z)=\int _z^\infty dt\,t^{n-1}\exp (-t)=(n-1)\Gamma (n-1,z)+z^{n-1}\exp (-z). \end{aligned}$$
(8)

Setting \(n=0\) in (8) gives:

$$\begin{aligned} \Gamma (0,z)= & {} E_1(z)=\int _z^\infty dt\frac{\exp (-t)}{t} = -\ln (z)-\gamma -\sum ^{\infty }_{n=1}\frac{(-z)^n}{nn!}, \end{aligned}$$
(9)
$$\begin{aligned} \Gamma (-1,z)= & {} -\Gamma (0,z)+\frac{\exp (-z)}{z}. \end{aligned}$$
(10)

The renormalized one-loop self-energy \(\Sigma _R(p^2)\) can then be written in the form:

$$\begin{aligned} \Sigma _R(p^2)=\delta Z(p^2+m^2)+\delta m^2 +\frac{Z^{-1}\lambda }{32\pi ^2}m^2\Gamma \biggl (-1,\frac{m^2}{\Lambda _H^2}\biggr )+{\mathcal {O}}(\lambda ^2). \end{aligned}$$
(11)

The renormalized mass and field strength are given by

$$\begin{aligned} \delta m^2= & {} -\frac{\lambda }{32\pi ^2}m^2\Gamma \biggl (-1,\frac{m^2}{\Lambda _H}\biggr ) +{\mathcal {O}}(\lambda ^2), \end{aligned}$$
(12)
$$\begin{aligned} \delta Z= & {} {\mathcal {O}}(\lambda ^2). \end{aligned}$$
(13)

The expansion of the one-loop Higgs boson self-energy mass correction for \(m \ll \Lambda _H\) is

$$\begin{aligned} \delta m^2=\frac{\lambda }{32\pi ^2} \biggl [-\Lambda _H^2+m^2\ln \biggl (\frac{\Lambda _H^2}{m^2}\biggr ) +m^2(1-\gamma )+{\mathcal {O}}\biggl (\frac{m^2}{\Lambda _H^2}\biggr )\biggr ]+{\mathcal {O}}(\lambda ^2). \end{aligned}$$
(14)

The one-loop vertex correction is given by

$$\begin{aligned} \delta \lambda =\frac{3\lambda ^2}{16\pi ^2}\int _0^{1/2}dx\, \Gamma \biggl (0,\frac{1}{1-x}\frac{m^2}{\Lambda _H^2}\biggr ) +{\mathcal {O}}(\lambda ^3). \end{aligned}$$
(15)

For \(m \ll \Lambda _H\), this can be expanded for the Higgs boson to give

$$\begin{aligned} \delta \lambda =\frac{3\lambda ^2}{16\pi ^2} \biggl [\frac{1}{2}\ln \biggl (\frac{\Lambda _H^2}{m^2}\biggr ) +\frac{1}{2}(\ln (2)-1-\gamma )+{\mathcal {O}} \biggl (\frac{m^2}{\Lambda _H^2}\biggr )\biggr ]+{\mathcal {O}}(\lambda ^3). \end{aligned}$$
(16)

3 Callan–Symanzik equation and running of \(\lambda \)

Let us consider the Callan–Symanzik equations [4,5,6,7] satisfied with our energy (length) scales \(\Lambda _i\) playing the roles of finite renormalization scales. In finite QFT theory, the equations for the regularized amplitudes \(\Gamma ^{(n)}(x-x')\) are

$$\begin{aligned} \biggl [\Lambda _i\frac{\partial }{\partial \Lambda _i}+\beta (g_i)\frac{\partial }{\partial g_i}-2\gamma (g_i)\biggr ]\Gamma ^{(n)}=0, \end{aligned}$$
(17)

where \(g_i\) are the running coupling constants associated with diagram vertices. The correlation functions will satisfy this equation for the nth-order \(\Gamma ^{(n)}\) for the Gell-Mann–Low functions \(\beta (g_i)\) and the anomalous dimensions in nth-loop order.

For the Higgs field, the RG equation is given by

$$\begin{aligned} \biggl [\Lambda _H\frac{\partial }{\partial \Lambda _H}+\beta (\lambda )\frac{\partial }{\partial \lambda }-2\gamma (\lambda )\biggr ]\Gamma ^H=0. \end{aligned}$$
(18)

where the coupling \(\lambda \) runs with \(\Lambda _H\). Neglecting the anomalous dimension term \(\gamma (\lambda )\) and replacing the measured Higgs mass m by the RG scaling mass \(\mu \) yields the equation:

$$\begin{aligned} \beta (\lambda )= -\frac{d\lambda }{d\ln \left( \frac{\Lambda _H}{\mu }\right) }. \end{aligned}$$
(19)

We obtain from (15) the Higgs field \(\beta \) function:

$$\begin{aligned} \beta (\lambda )=\frac{3\lambda ^2}{16\pi ^2}I(\mu ^2/\Lambda _H^2)+{\mathcal {O}}(\lambda ^3), \end{aligned}$$
(20)

where

$$\begin{aligned} I(\mu ^2/\Lambda _H^2)=\int _0^{1/2}dx\,\Gamma \biggl (0,\frac{1}{1-x}\frac{\mu ^2}{\Lambda ^2_H}\biggr ). \end{aligned}$$
(21)

Using the identities \(\Gamma (0,y)=E_1(y)=-\mathrm{Ei}(-y)\) yields:

$$\begin{aligned}&I(\mu ^2/\Lambda _H^2)=-\int _0^{1/2}dx\, \mathrm{Ei}\biggl (-\frac{1}{1-x} \frac{\mu ^2}{\Lambda ^2_H}\biggr )\nonumber \\&\quad = \frac{1}{2}\left( \exp \left( \frac{-2 \mu ^2}{\Lambda _H^2}\right) + \left( 1 + \frac{2 \mu ^2}{\Lambda _H^2}\right) \mathrm{Ei}\left( \frac{-2 \mu ^2}{\Lambda _H^2}\right) \right) \nonumber \\&\quad -\exp \left( \frac{-\mu ^2}{\Lambda _H^2}\right) - \left( 1 + \frac{\mu ^2}{\Lambda _H^2}\right) \mathrm{Ei}\left( \frac{-\mu ^2}{\Lambda _H^2}\right) . \end{aligned}$$
(22)

We have \(\lambda =\lambda _0+\delta \lambda \) and

$$\begin{aligned} \frac{d\lambda }{d\left( \frac{\Lambda _H}{\mu }\right) } =\frac{d\delta \lambda }{d\ln \left( \frac{\Lambda _H}{\mu }\right) }=-\beta (\lambda ). \end{aligned}$$
(23)

From (20), we obtain

$$\begin{aligned} \frac{d\lambda }{\lambda ^2}=-\frac{3}{16\pi ^2}dI(\mu ^2/\Lambda _H^2). \end{aligned}$$
(24)

Integrating this equation, we get

$$\begin{aligned} \frac{1}{\lambda }=\frac{1}{\lambda _0}+J(\mu ^2/\Lambda _H^2), \end{aligned}$$
(25)

where

$$\begin{aligned} J(\mu ^2/\Lambda _H^2)=\frac{3}{16\pi ^2}\int \frac{d\Lambda _H}{\Lambda _H}I(\mu ^2/\Lambda _H^2). \end{aligned}$$
(26)

Evaluating the integral for \(J(\mu ^2/\Lambda _H^2)\), using \(x=\frac{\mu ^2}{\Lambda _H^2}\), gives

$$\begin{aligned} J(x)= & {} \frac{3}{128\pi ^2}\left( -2\exp (-2 x) + 4\exp (-x) + \pi ^2 - (2+4 x)\mathrm{Ei}(-2 x) + (4+4 x)\mathrm{Ei}(-x)\right. \nonumber \\&+4 x \,_3\mathrm{F}_3(1,1,1;2,2,2;-2 x) - 4 x \,_3\mathrm{F}_3(1,1,1;2,2,2;-x) \nonumber \\&- \left. \ln (2)^2 - \ln (4)\gamma + 2( \gamma -\ln (2))\ln (x) + \ln (x)^2\right) , \end{aligned}$$
(27)

where \(_p\mathrm{F}_q(a_1,\ldots ,a_p;b_1,\ldots ,b_q;z)\) is a generalized hypergeometric function.

From (25), we obtain:

$$\begin{aligned} \lambda =\frac{\lambda _0}{1+\lambda _0J(\mu ^2/\Lambda _H^2)}, \end{aligned}$$
(28)

or

$$\begin{aligned} \lambda _0=\frac{\lambda }{1-\lambda J(\mu ^2/\Lambda _H^2)}. \end{aligned}$$
(29)

We can compare (25) with the equation obtained in SM:

$$\begin{aligned} \frac{1}{\lambda }=\frac{1}{\lambda _0}+\frac{3}{16\pi ^2}\ln \biggl (\frac{\Lambda _C}{\mu }\biggr ), \end{aligned}$$
(30)

or

$$\begin{aligned} \lambda =\frac{\lambda _0}{1+\frac{3\lambda _0}{16\pi ^2}\ln \left( \frac{\Lambda _C}{\mu }\right) }, \end{aligned}$$
(31)

and

$$\begin{aligned} \lambda _0=\frac{\lambda }{1-\frac{3\lambda }{16\pi ^2}\ln \left( \frac{\Lambda _C}{\mu }\right) }. \end{aligned}$$
(32)

In the SM, the \(\lambda \phi ^4\) model is renormalizable and produces finite scattering amplitudes and cross sections, but renormalization theory demands that the cutoff \(\Lambda _C\) must be taken to infinity, \(\Lambda _C\rightarrow \infty \) [8, 9]. Then, from (30), the renormalized coupling constant \(\lambda =0\). This is known as the triviality problem [10,11,12,13,14,15]. This result holds even in the limit \(\lambda _0\rightarrow \infty \):

$$\begin{aligned} \frac{1}{\lambda }\sim \frac{3}{16\pi ^2}\ln \biggl (\frac{\Lambda _C}{\mu }\biggr ). \end{aligned}$$
(33)

In the earlier paper [16], it was demonstrated that the triviality problem for the scalar field field could be resolved in the finite QFT theory. Because \(\Lambda _H=1/{\ell _H}\) is a fundamental constant to be measured, it cannot be taken to infinity as in the case of infinite renormalization theory. Thus, we cannot take the limit \(\ell _H\rightarrow 0\) corresponding to the \(\delta \)-function limit. From Fig. 1, we observe that when we choose \(\Lambda _H\lesssim 1\) TeV, the Higgs mass hierarchy problem is resolved, for we have \(\delta m^2/m^2\sim {\mathcal {O}}(1)\) where \(m=125\) GeV. From Fig. 1, we observe that for \(\Lambda _H > \frac{1}{2}\mu \), we avoid a Landau pole and, in particular, for \(700< \Lambda _H < 1\) TeV, we resolve the triviality problem for the scalar Higgs field and the Higgs mass fine-tuning hierarchy problem.

Fig. 1
figure 1

Running of \(\lambda \) versus \(\Lambda _H/\mu \) for finite QFT

Full size image

Choosing an energy \(\mu _0\) above \(\Lambda _H\) as a measurement probe of the running of \(\lambda \) is attempting to make a measurement within the finite Gaussian distribution length size \(\ell _H\) [1] and is prohibited within the perturbation approximations we have assumed. The results obtained for the running of \(\lambda \) are for a single Higgs particle interacting with another Higgs particle. This cannot describe a fully realistic situation, for the Higgs coupling to other particles such as the top quark (the top quark-Higgs coupling \(\lambda _t\sim {\mathcal {O}}(1)\)) may play an important role.

The above one-loop calculations have employed a perturbative formulation in Euclidean momentum space and rely on analytic continuation to obtain corresponding Lorentzian results. At tree level, the theory is completely equivalent to the classical field theory with the same Lagrangian, provided that spacetime Fourier transforms exist. Although loop diagrams should be finite at all levels, convergence of the quantum perturbative formalism has not been demonstrated. Because the Fourier transform to momentum space is generally not well-defined in curved spacetime, no claims can be made about whether or how the theory might be applied in the context of an expanding universe; the energy scales \(\Lambda _M\), \(\Lambda _H\) may well-depend on emergent and evolving properties of the classical universe (e.g., entropy density), thus bridging the gap between quantum and classical. Whether a nonperturbative quantum formulation can be developed remains to be determined.