Effective potential and dynamical symmetry breaking up to five loops in a massless abelian Higgs model

The Abelian Higgs (AH) model is one of the most fundamental field theories in both condensed matter and particle physics. As an example, it is the prime textbook example for the superconducting transition and the Anderson-Higgs mechanism (ZinnJustin1996, ; Peskin2018, ; Altland2010a, ; Herbut2007, ). The AH model features a complex scalar field coupled to an $U(1)$ gauge field, and it displays two distinct phases separated by a sharp transition: the symmetric phase and the phase with spontaneously broken symmetry. In the context of superconductors, the symmetric phase is related to the normal metallic state and the broken one to the superconducting Meissner state. The transition between spontaneously broken and symmetric phases is characterized by a dimensionful parameter that serves as an order parameter.

Starting from a classical scale invariant Lagrangian, Coleman and Weinberg (CW) demonstrated in (Coleman1973, ) how the order parameter could be generated by radiative corrections, i.e., the spontaneous symmetry breaking can occur as a dynamical mechanism, the radiative corrections being entirely responsible for the appearance of the nontrivial minima of the effective potential. This Dynamical Symmetry Breaking (DSB) is a key concept that has many applications in particle physics (Elias2003, ; Chishtie2006, ; Chishtie2011, ; Steele2013, ) and condensate matter systems (Liu2013, ; Uchino2014, ; Burmistrov2020, ; Chodos1994, ; A.G.Quinto2021, ).

In order to study the CW mechanism, we need to calculate the effective potential, a powerful tool to explore many aspects of the low-energy sector of a quantum field theory. In many cases, the one-loop approximation is good enough, but it can be improved by adding higher order contributions to the loop expansion. A standard tool for improving a perturbative calculation is the Renormalization Group Equation (RGE), which, together with a reorganization of the perturbative series in terms of leading logs, have been shown to be very effective in several instances (AHMADY2003221, ; PhysRevD.72.037902, ; Souza2020, ; Quinto2016, ; Dias2014, ; CHISHTIE2007, ; A.G.Quinto2021, ). We refer the reader to section 3 in (Quinto2016, ) for a short review of the method, and (Elias2003, ; Chishtie2006, ; Chishtie2011, ; Steele2013, ) for some of the interesting results that have been reported with the use of the RG improvement, in the context of a scale-invariant approximation of the Standard Model.

In this work we study the behavior of effective potential in a massless AH model with $N$-component complex scalar field in $(3+1)$ dimensional space-time, which is scale invariant at the classical level. We observed that the effective potential, computed up to five loops, leads to an interesting phase structure arising from DSB. This result was achieved by the use of RGE with the help of renormalization group functions, $\widetilde{\beta}$ and $\widetilde{\gamma}$, which were calculated up to four loops in the minimal subtraction (MS) scheme in (Ihrig2019, ). From these renormalization group functions, we need to obtain the corresponding functions in a different renormalization scheme, which we call CW scheme, using the multi-scales techniques reported in (Chishtie2008, ).

This paper is organized as follows: in Sec. II, we present our model, together with the renormalization group functions found in the literature. In Section III we obtain the corresponding functions in the CW scheme and we use them in Section IV for the calculation of the effective potential using the RGE approach. This effective potential is used in Section V to study different aspects of the DSB in our model. Section VI presents our conclusions and perspectives.

We start with the $N$-component massless Abelian Higgs (AH) model defined in $d$-dimensional Euclidean space-time by the Lagrangian

$\displaystyle\mathcal{L}$

$\displaystyle=\left|D_{\mu}\phi\right|^{2}+\frac{1}{4}F_{\mu v}^{2}+\lambda\left(|\phi|^{2}\right)^{2}+\mathcal{L}_{\mathrm{gf}},$

(1)

where $\phi=\left(\phi_{1},\ldots,\phi_{N}\right)$ describes the $N$-component complex scalar field with quartic self-interaction $\lambda$. This scalar is minimally coupled to an $U(1)$ gauge field $A_{\mu}$ via covariant derivative $D_{\mu}=\partial_{\mu}-ieA_{\mu}$, $e$ being the analogous to the “electric charge”. The field strength tensor is defined as $F_{\mu v}=\partial_{\mu}A_{v}-\partial_{\nu}A_{\mu}$ and the gauge-fixing Lagrangian is $\mathcal{L}_{\mathrm{gf}}=-\frac{1}{2\xi}\left(\partial_{\mu}A^{\mu}\right)^{2}$, $\xi$ being the gauge-fixing parameter. For the case of a single complex scalar field, $N=1$, and in three spatial dimensions, this model is used to describe transitions on superconductors (Ginzburg1950, ) and liquid crystals (Halperin1974, ).

The renormalized Lagrangian of the massless AH model is

$\displaystyle\mathcal{L}^{\prime}$

$\displaystyle=Z_{\phi}\left|D_{\mu}\phi\right|^{2}+Z_{\phi^{4}}\lambda\widetilde{\mu}^{\epsilon}\left(|\phi|^{2}\right)^{2}+\frac{Z_{A}}{4}F_{\mu v}^{2}-\frac{1}{2\xi}\left(\partial_{\mu}A^{\mu}\right)^{2},$

(2)

where $D_{\mu}\phi=\left(\partial_{\mu}-ie\widetilde{\mu}^{\epsilon/2}A_{\mu}\right)\phi$ and $\widetilde{\mu}$ is a mass scale introduced by the (dimensional) regularization scheme (Collins1984, ; Peskin2018, ; Altland2010, ). The wave-function renormalization constants $Z_{\phi}$ and $Z_{A}$ relate the bare and the renormalized fields in the Lagrangian through $\phi_{0}=Z_{\phi}^{1/2}\phi$ and $A_{0,\mu}=Z_{A}^{1/2}A_{\mu}$. Also, we obtain the relations between bare and renormalized coupling constants as

	$\displaystyle\alpha$	$\displaystyle\equiv e^{2}=e_{0}^{2}\widetilde{\mu}^{-\epsilon}Z_{A},$		(3)
	$\displaystyle\lambda$	$\displaystyle=\lambda_{0}\widetilde{\mu}^{-\epsilon}Z_{\phi}^{2}Z_{\phi^{4}}^{-1},$		(4)
	$\displaystyle\xi$	$\displaystyle=\xi_{0}Z_{A}^{-1}.$		(5)

It is interesting to note that the gauge-fixing parameter is also renormalized (Collins1984, ) in this case, meaning it will have a corresponding $\beta$ function.

In the MS scheme, the beta functions of the model are defined by

$\displaystyle\widetilde{\beta}_{\alpha}=-\widetilde{\mu}\frac{d\alpha}{d\widetilde{\mu}},\;$

$\displaystyle\widetilde{\beta}_{\lambda}=-\widetilde{\mu}\frac{d\lambda}{d\widetilde{\mu}},\;\widetilde{\beta}_{\xi}=-\widetilde{\mu}\frac{d\xi}{d\widetilde{\mu}}.$

(6)

These functions were computed in the Ref. (Ihrig2019, ) up to four loops, from which we quote the expressions below. The contributions to $\widetilde{\beta}_{\alpha},\widetilde{\beta}_{\lambda}$ and $\widetilde{\beta}_{\xi}$, can be cast as, for the gauge coupling constant,

$\displaystyle\widetilde{\beta}_{\alpha}$

$\displaystyle=\widetilde{\beta}_{\alpha}^{\left(2\right)}+\widetilde{\beta}_{\alpha}^{\left(3\right)}+\widetilde{\beta}_{\alpha}^{\left(4\right)}+\widetilde{\beta}_{\alpha}^{\left(5\right)},$

(7)

where

	$\displaystyle\widetilde{\beta}_{\alpha}^{\left(2\right)}$	$\displaystyle=-\frac{N}{3}\alpha^{2},$		(8a)
	$\displaystyle\widetilde{\beta}_{\alpha}^{\left(3\right)}$	$\displaystyle=-2N\alpha^{3},$		(8b)
	$\displaystyle\widetilde{\beta}_{\alpha}^{\left(4\right)}$	$\displaystyle=\left(\frac{49N^{2}}{72}-\frac{29N}{8}\right)\alpha^{4}-\frac{1}{2}\left(N^{2}+N\right)\alpha^{3}\lambda+\frac{1}{8}\left(N^{2}+N\right)\alpha^{2}\lambda^{2},$		(8c)
	$\displaystyle\widetilde{\beta}_{\alpha}^{\left(5\right)}$	$\displaystyle=\left(\frac{3N}{16}-\frac{323N^{3}}{3888}+\left(\frac{451}{54}-\frac{38\zeta_{3}}{9}\right)N^{2}\right)\alpha^{5}+\left(\frac{5N^{3}}{72}-\frac{41N^{2}}{9}-\frac{37N}{8}\right)\alpha^{4}\lambda$
		$\displaystyle+\left(\frac{N^{3}}{24}+\frac{139N^{2}}{48}+\frac{137N}{48}\right)\alpha^{3}\lambda^{2}-\left(\frac{5N^{3}}{48}+\frac{7N^{2}}{16}+\frac{N}{3}\right)\alpha^{2}\lambda^{3},$		(8d)

for the scalar self-interaction,

$\displaystyle\widetilde{\beta}_{\lambda}$

$\displaystyle=\widetilde{\beta}_{\lambda}^{\left(2\right)}+\widetilde{\beta}_{\lambda}^{\left(3\right)}+\widetilde{\beta}_{\lambda}^{\left(4\right)}+\widetilde{\beta}_{\lambda}^{\left(5\right)},$

(9)

where

	$\displaystyle\widetilde{\beta}_{\lambda}^{\left(2\right)}$	$\displaystyle=-6\alpha^{2}+6\alpha\lambda-\left(4+N\right)\lambda^{2},$		(10a)
	$\displaystyle\widetilde{\beta}_{\lambda}^{\left(3\right)}$	$\displaystyle=\left(\frac{14N}{3}+30\right)\alpha^{3}-\left(\frac{71N}{6}+\frac{29}{2}\right)\alpha^{2}\lambda-\left(4N+10\right)\alpha\lambda^{2}+\left(\frac{9N}{2}+\frac{21}{2}\right)\lambda^{3},$		(10b)
	$\displaystyle\widetilde{\beta}_{\lambda}^{\left(4\right)}$	$\displaystyle=\left(-45\zeta_{3}-\frac{7N^{2}}{18}+\left(\frac{203}{8}-27\zeta_{3}\right)N+\frac{367}{8}\right)\alpha^{4}+\left(\left(18\zeta_{3}-\frac{989}{8}\right)N-54\zeta_{3}-\frac{5N^{2}}{216}-\frac{889}{4}\right)\alpha^{3}\lambda,$
		$\displaystyle+\left(126\zeta_{3}+\frac{43N^{2}}{16}+\left(18\zeta_{3}+\frac{1749}{16}\right)N+\frac{1093}{8}\right)\alpha^{2}\lambda^{2}+\left(6\zeta_{3}+\left(\frac{25}{2}-6\zeta_{3}\right)N+\frac{29}{2}\right)\alpha\lambda^{3}$
		$\displaystyle+\left(-33\zeta_{3}-\frac{33N^{2}}{16}-\left(15\zeta_{3}+\frac{461}{16}\right)N-\frac{185}{4}\right)\lambda^{4},$		(10c)
	$\displaystyle\widetilde{\beta}_{\lambda}^{\left(5\right)}$	$\displaystyle=\left(504\zeta_{3}-390\zeta_{5}+\left(\frac{1}{81}-\frac{2\zeta_{3}}{9}\right)N^{3}+\left(28\zeta_{3}-\frac{\pi^{4}}{10}-\frac{55709}{1296}\right)N^{2}+\left(310\zeta_{5}-\frac{578\zeta_{3}}{3}-\frac{19\pi^{4}}{60}-\frac{13987}{36}\right)N\right.$
		$\displaystyle\left.-\frac{33\pi^{4}}{20}-\frac{12751}{16}\right)\alpha^{5}+\left(2095\zeta_{5}-\frac{1191\zeta_{3}}{2}+\left(\frac{19\zeta_{3}}{18}+\frac{67}{2592}\right)N^{3}+\left(20\zeta_{5}-\frac{77\zeta_{3}}{2}+\frac{\pi^{4}}{5}+\frac{12779}{162}\right)N^{2}\right.$
		$\displaystyle+\left.\left(\frac{431\zeta_{3}}{2}+305\zeta_{5}+\frac{13\pi^{4}}{10}+\frac{209}{12}\right)N+\frac{26\pi^{4}}{5}-\frac{19127}{96}\right)\alpha^{4}\lambda-\left(725\zeta_{3}-1520\zeta_{5}+\left(\frac{7\zeta_{3}}{6}+\frac{139}{1944}\right)N^{3}\right.$
		$\displaystyle\left.-\left(40\zeta_{5}-\frac{157\zeta_{3}}{3}-\frac{\pi^{4}}{60}+\frac{289817}{7776}\right)N^{2}-\left(1080\zeta_{5}-\frac{1813\zeta_{3}}{2}-\frac{5\pi^{4}}{12}+\frac{88871}{96}\right)N+\frac{6\pi^{4}}{5}-\frac{66851}{48}\right)\alpha^{3}\lambda^{2}$
		$\displaystyle+\left(\left(-\frac{27\zeta_{3}}{2}+40\zeta_{5}-\frac{\pi^{4}}{180}-\frac{6145}{48}\right)N^{2}-768\zeta_{3}-960\zeta_{5}-\left(\frac{3157\zeta_{3}}{6}+80\zeta_{5}+\frac{143\pi^{4}}{180}+\frac{25123}{24}\right)N\right.$
		$\displaystyle\left.-\frac{12\pi^{4}}{5}-\frac{18503}{16}\right)\alpha^{2}\lambda^{3}+\left(150\zeta_{5}-435\zeta_{3}+\left(\frac{29\zeta_{3}}{2}-\frac{\pi^{4}}{60}-\frac{377}{96}\right)N^{2}+\frac{103\pi^{4}}{60}-\frac{185}{12}\right.$
		$\displaystyle\left.+\left(50\zeta_{5}-\frac{269\zeta_{3}}{2}+\frac{7\pi^{4}}{10}-\frac{1403}{32}\right)N\right)\alpha\lambda^{4}+\left(\frac{583\zeta_{3}}{2}+465\zeta_{5}-\frac{5N^{3}}{96}+\left(\frac{63\zeta_{3}}{2}+20\zeta_{5}-\frac{\pi^{4}}{12}+\frac{395}{12}\right)N^{2}\right.$
		$\displaystyle\left.+\left(191\zeta_{3}+275\zeta_{5}-\frac{31\pi^{4}}{60}+\frac{10057}{48}\right)N-\frac{11\pi^{4}}{15}+\frac{24581}{96}\right)\lambda^{5},$		(10d)

and finally, for the gauge parameter,

$\displaystyle\widetilde{\beta}_{\xi}$

$\displaystyle=\widetilde{\beta}_{\xi}^{\left(2\right)}+\widetilde{\beta}_{\xi}^{\left(3\right)}+\widetilde{\beta}_{\xi}^{\left(4\right)}+\widetilde{\beta}_{\xi}^{\left(5\right)},$

(11)

where

	$\displaystyle\widetilde{\beta}_{\xi}^{\left(2\right)}$	$\displaystyle=\frac{8N}{3}\alpha\xi,$		(12a)
	$\displaystyle\widetilde{\beta}_{\xi}^{\left(3\right)}$	$\displaystyle=16N\alpha^{2}\xi,$		(12b)
	$\displaystyle\widetilde{\beta}_{\xi}^{\left(4\right)}$	$\displaystyle=\left(29-\frac{49N}{9}\right)N\alpha^{3}\xi+\frac{17}{6}N\left(N+1\right)\alpha^{2}\lambda\xi-\frac{5}{12}N\left(N+1\right)\alpha\lambda^{2}\xi,$		(12c)
	$\displaystyle\widetilde{\beta}_{\xi}^{\left(5\right)}$	$\displaystyle=\left(\frac{1}{486}N\left(323N^{2}+72\left(228\zeta_{3}-451\right)N-729\right)\right)\alpha^{4}\xi-\frac{25}{288}N\left(5N^{2}-328N-333\right)\alpha^{3}\lambda\xi$
		$\displaystyle-\frac{3}{32}N\left(2N^{2}+139N+137\right)\alpha^{2}\lambda^{2}\xi+\frac{11}{192}N\left(5N^{2}+21N+16\right)\alpha\lambda^{3}\xi.$		(12d)

The anomalous dimensions are defined through the relation

$\displaystyle\widetilde{\gamma}_{\phi}$

$\displaystyle\equiv-\frac{\widetilde{\mu}}{\phi}\frac{d\phi}{d\widetilde{\mu}}=-\frac{\widetilde{\mu}}{Z_{\phi}}\frac{d}{d\widetilde{\mu}}Z_{\phi},$

(13)

and, up to four loops, the contributions to $\gamma_{\phi}$ read as

$\displaystyle\widetilde{\gamma}_{\phi}$

$\displaystyle=\widetilde{\gamma}_{\phi}^{\left(1\right)}+\widetilde{\gamma}_{\phi}^{\left(2\right)}+\widetilde{\gamma}_{\phi}^{\left(3\right)}+\widetilde{\gamma}_{\phi}^{\left(4\right)},$

(14)

where

	$\displaystyle\widetilde{\gamma}_{\phi}^{\left(1\right)}$	$\displaystyle=-2\alpha,$		(15a)
	$\displaystyle\widetilde{\gamma}_{\phi}^{\left(2\right)}$	$\displaystyle=-\alpha\xi+\frac{1}{12}\left(11N+9\right)\alpha^{2}+\frac{1}{4}\left(N+1\right)\lambda^{2},$		(15b)
	$\displaystyle\widetilde{\gamma}_{\phi}^{\left(3\right)}$	$\displaystyle=\frac{5}{4}\left(N+1\right)\alpha\lambda^{2}-\frac{1}{16}\left(N+1\right)\left(N+4\right)\lambda^{3}-\frac{1}{8}\left(24\text{$\zeta_{3}$}-13\right)\left(N+1\right)\alpha^{2}\lambda,$
		$\displaystyle+\left(\frac{1}{8}-3\text{$\zeta_{3}$}\left(N-1\right)+\frac{1}{432}N\left(5N+3267\right)\right)\alpha^{3}$		(15c)
	$\displaystyle\widetilde{\gamma}_{\phi}^{\left(4\right)}$	$\displaystyle=\left(\frac{125}{64}-\frac{5}{64}N^{3}+\frac{5}{8}N^{2}+\frac{85}{32}N\right)\lambda^{4}+\left(\frac{5}{6}-\zeta_{3}+\left(\frac{\zeta_{3}}{2}-\frac{19}{96}\right)N^{2}+\left(\frac{61}{96}-\frac{\zeta_{3}}{2}\right)N\right)\alpha\lambda^{3}$
		$\displaystyle+\left(\frac{63\zeta_{3}}{2}-45\zeta_{5}+\left(\frac{13}{5184}-\frac{\zeta_{3}}{36}\right)N^{3}+\left(\frac{9\zeta_{3}}{4}-\frac{\pi^{4}}{120}-\frac{1505}{432}\right)N^{2}+\left(\frac{231}{16}-\frac{87\zeta_{3}}{2}-\frac{\pi^{4}}{24}\right)N-\frac{\pi^{4}}{20}+\frac{133}{64}\right)\alpha^{4}$
		$\displaystyle+\left(\frac{345}{16}-4\zeta_{3}-20\zeta_{5}+\left(\frac{5\zeta_{3}}{3}-\frac{\pi^{4}}{180}+\frac{199}{144}\right)N^{2}+\left(-\frac{7\zeta_{3}}{3}-20\zeta_{5}+\frac{2\pi^{4}}{45}+\frac{413}{18}\right)N+\frac{\pi^{4}}{20}\right)\alpha^{3}\lambda$
		$\displaystyle+\left(\frac{11\zeta_{3}}{2}+5\zeta_{5}+\left(\frac{19\zeta_{3}}{12}-\frac{\pi^{4}}{120}-\frac{641}{288}\right)N^{2}+\left(\frac{85\zeta_{3}}{12}+5\zeta_{5}-\frac{\pi^{4}}{24}-\frac{179}{18}\right)N-\frac{\pi^{4}}{30}-\frac{247}{32}\right)\alpha^{2}\lambda^{2}.$		(15d)

The superscript present in the previous expressions denotes the aggregate power of coupling constants. So, for instance, $\widetilde{\beta}_{\alpha}^{\left(2\right)}$ means the terms in $\widetilde{\beta}_{\alpha}$ which contain exactly two powers of coupling constants.

The renormalization group functions presented in this section were calculated in the MS scheme by (Ihrig2019, ). In order to use the RGE improvement method, we need to convert these functions to a different renormalization scheme (Quinto2016, ; AHMADY2003221, ; Dias2010, ; PhysRevD.72.037902, ; Dias2014, ; CHISHTIE2007, ). This will be done in the next section.

In this section we will use the renormalization group function obtained in section II to calculate the $\beta$ and $\gamma$ function in the CW scheme. We know the effective potential will involve terms with logarithms, of the general form

	$\displaystyle L$	$\displaystyle=\ln\left(\frac{\sigma^{2}}{\mu^{2}}\right)\,\,\text{for CW scheme },$		(16)
	$\displaystyle\widetilde{L}$	$\displaystyle=\ln\left(x\frac{\sigma^{2}}{\widetilde{\mu}^{2}}\right)\,\,\text{for MS scheme },$		(17)

where $x$ is associated to some coupling constant present in the model, and $\sigma$ is the classical value of one of the components of $\phi$, the one which is shifted as $\phi_{i}\rightarrow\phi_{i}+\sigma$ in order to study the symmetry breaking (see details in Section V).

We can obtain the relation between the renormalization group function in the CW scheme from the knowledge of the corresponding function in the MS scheme. This procedure is not straightforward because we have multiple coupling constants and then we have to use the multi-scale procedure described in (Chishtie2008, ). In order to do that, we start with Eq. (17) applying to our model, i.e., $x=\lambda,\alpha,\xi$, then we get

	$\displaystyle\widetilde{L}_{1}$	$\displaystyle=\ln\left(\lambda\frac{\sigma^{2}}{k_{1}^{2}}\right),$		(18a)
	$\displaystyle\widetilde{L}_{2}$	$\displaystyle=\ln\left(\alpha\frac{\sigma^{2}}{k_{2}^{2}}\right),$		(18b)
	$\displaystyle\widetilde{L}_{3}$	$\displaystyle=\ln\left(\xi\frac{\sigma^{2}}{k_{3}^{2}}\right),$		(18c)

where $k_{i}$ with $i=1,2,3$ are different scales. Notice the gauge parameter appears here as if it were a coupling constant, being dimensionless and having its own $\beta$ function as $\lambda$ and $\alpha$. If we compare $\widetilde{L}_{i}$ with Eq. (16) we obtain the following relations:

	$\displaystyle k_{1}$	$\displaystyle=\lambda^{1/2}\mu,$
	$\displaystyle k_{2}$	$\displaystyle=\alpha^{1/2}\mu,$		(19)
	$\displaystyle k_{3}$	$\displaystyle=\xi^{1/2}\mu.$

The renormalization group function in the CW scheme can be defined as

	$\displaystyle\beta_{\lambda}=$	$\displaystyle\mu\frac{d}{d\mu}\lambda=\mu\frac{\partial\lambda}{\partial k_{1}}\frac{dk_{1}}{d\mu}+\mu\frac{\partial\lambda}{\partial k_{2}}\frac{dk_{2}}{d\mu}+\mu\frac{\partial\lambda}{\partial k_{3}}\frac{dk_{3}}{d\mu},$		(20)
	$\displaystyle\beta_{\alpha}=$	$\displaystyle\mu\frac{d}{d\mu}\alpha=\mu\frac{\partial\alpha}{\partial k_{1}}\frac{dk_{1}}{d\mu}+\mu\frac{\partial\alpha}{\partial k_{2}}\frac{dk_{2}}{d\mu}+\mu\frac{\partial\alpha}{\partial k_{3}}\frac{dk_{3}}{d\mu},$		(21)
	$\displaystyle\beta_{\xi}=$	$\displaystyle\mu\frac{d}{d\mu}\xi=\mu\frac{\partial\xi}{\partial k_{1}}\frac{dk_{1}}{d\mu}+\mu\frac{\partial\xi}{\partial k_{2}}\frac{dk_{2}}{d\mu}+\mu\frac{\partial\xi}{\partial k_{3}}\frac{dk_{3}}{d\mu},$		(22)
	$\displaystyle\gamma_{\phi}=$	$\displaystyle\frac{\mu}{\phi}\frac{d}{d\mu}\phi=\frac{\mu}{\phi}\frac{\partial\phi}{\partial k_{1}}\frac{dk_{1}}{d\mu}+\frac{\mu}{\phi}\frac{\partial\phi}{\partial k_{2}}\frac{dk_{2}}{d\mu}+\frac{\mu}{\phi}\frac{\partial\phi}{\partial k_{3}}\frac{dk_{3}}{d\mu}.$		(23)

Now, if we use the relations Eq. (19) in the last set of equations with the condition $k_{1}=k_{2}=k_{3}=\widetilde{\mu}$, we get the final relation between the renormalization group functions in the CW computed from MS scheme,

	$\displaystyle\beta_{\lambda}$	$\displaystyle=-\widetilde{\beta}_{\lambda}\left(3+\frac{\beta_{\text{$\lambda$}}}{2\lambda}+\frac{\beta_{\text{$\alpha$}}}{2\alpha}+\frac{\beta_{\text{$\xi$}}}{2\xi}\right),$		(24a)
	$\displaystyle\beta_{\alpha}$	$\displaystyle=-\widetilde{\beta}_{\alpha}\left(3+\frac{\beta_{\text{$\lambda$}}}{2\lambda}+\frac{\beta_{\text{$\alpha$}}}{2\alpha}+\frac{\beta_{\text{$\xi$}}}{2\xi}\right),$		(24b)
	$\displaystyle\beta_{\xi}$	$\displaystyle=-\widetilde{\beta}_{\xi}\left(3+\frac{\beta_{\text{$\lambda$}}}{2\lambda}+\frac{\beta_{\text{$\alpha$}}}{2\alpha}+\frac{\beta_{\text{$\xi$}}}{2\xi}\right),$		(24c)
	$\displaystyle\gamma_{\phi}$	$\displaystyle=-\widetilde{\gamma}_{\phi}\left(3+\frac{\beta_{\text{$\lambda$}}}{2\lambda}+\frac{\beta_{\text{$\alpha$}}}{2\alpha}+\frac{\beta_{\text{$\xi$}}}{2\xi}\right).$		(24d)

Notice that the minus sign come from the definition of the renormalization group function in the MS scheme (see Eqs. (6) and (13)).

We obtain the CW RG functions through an order by order comparison of the previous expression. For example, for the lowest order, we have

	$\displaystyle\gamma_{\phi}^{\left(1\right)}$	$\displaystyle=-3\widetilde{\gamma}_{\phi}^{\left(1\right)},$		(25a)
	$\displaystyle\beta_{\lambda}^{\left(2\right)}$	$\displaystyle=-3\widetilde{\beta}_{\lambda}^{\left(2\right)},$		(25b)
	$\displaystyle\beta_{\alpha}^{\left(2\right)}$	$\displaystyle=-3\widetilde{\beta}_{\alpha}^{\left(2\right)},$		(25c)
	$\displaystyle\beta_{\xi}^{\left(2\right)}$	$\displaystyle=-3\widetilde{\beta}_{\xi}^{\left(2\right)},$		(25d)

where these relations are expected because at this order the renormalization group function in the CW and MS are easily related (see for example, the section 3 of the Ref. (Quinto2016, ) for more details).

For the next order, i.e, $\beta_{i}^{\left(3\right)}$ and $\gamma_{\phi}^{\left(2\right)}$, the relation between the two schemes is more complex,

	$\displaystyle\beta_{\lambda}^{\left(3\right)}$	$\displaystyle=-3\widetilde{\beta}_{\lambda}^{\left(3\right)}+3\widetilde{\beta}_{\lambda}^{\left(2\right)}\left(\frac{\widetilde{\beta}_{\lambda}^{\left(2\right)}}{2\lambda}+\frac{\widetilde{\beta}_{\alpha}^{\left(2\right)}}{2\alpha}+\frac{\widetilde{\beta}_{\xi}^{\left(2\right)}}{2\xi}\right),$		(26a)
	$\displaystyle\beta_{\alpha}^{\left(3\right)}$	$\displaystyle=-3\widetilde{\beta}_{\alpha}^{\left(3\right)}+3\widetilde{\beta}_{\alpha}^{\left(2\right)}\left(\frac{\widetilde{\beta}_{\lambda}^{\left(2\right)}}{2\lambda}+\frac{\widetilde{\beta}_{\alpha}^{\left(2\right)}}{2\alpha}+\frac{\widetilde{\beta}_{\xi}^{\left(2\right)}}{2\xi}\right),$		(26b)
	$\displaystyle\beta_{\xi}^{\left(3\right)}$	$\displaystyle=-3\widetilde{\beta}_{\xi}^{\left(3\right)}+3\widetilde{\beta}_{\xi}^{\left(2\right)}\left(\frac{\widetilde{\beta}_{\lambda}^{\left(2\right)}}{2\lambda}+\frac{\widetilde{\beta}_{\alpha}^{\left(2\right)}}{2\alpha}+\frac{\widetilde{\beta}_{\xi}^{\left(2\right)}}{2\xi}\right),$		(26c)
	$\displaystyle\gamma_{\phi}^{\left(2\right)}$	$\displaystyle=-3\widetilde{\gamma}_{\phi}^{\left(2\right)}+3\widetilde{\gamma}_{\phi}^{\left(1\right)}\left(\frac{\widetilde{\beta}_{\lambda}^{\left(2\right)}}{2\lambda}+\frac{\widetilde{\beta}_{\alpha}^{\left(2\right)}}{2\alpha}+\frac{\widetilde{\beta}_{\xi}^{\left(2\right)}}{2\xi}\right).$		(26d)

For the order $\beta_{i}^{\left(4\right)}$ and $\gamma_{\phi}^{\left(3\right)}$ we get

	$\displaystyle\beta_{\lambda}^{\left(4\right)}$	$\displaystyle=-3\widetilde{\beta}_{\lambda}^{\left(4\right)}+3\widetilde{\beta}_{\lambda}^{\left(3\right)}\left(\frac{\widetilde{\beta}_{\lambda}^{\left(2\right)}}{2\lambda}+\frac{\widetilde{\beta}_{\alpha}^{\left(2\right)}}{2\alpha}+\frac{\widetilde{\beta}_{\xi}^{\left(2\right)}}{2\xi}\right)-\widetilde{\beta}_{\lambda}^{\left(2\right)}\left(\frac{\beta_{\lambda}^{\left(3\right)}}{2\lambda}+\frac{\beta_{\alpha}^{\left(3\right)}}{2\alpha}+\frac{\beta_{\xi}^{\left(3\right)}}{2\xi}\right),$		(27a)
	$\displaystyle\beta_{\alpha}^{\left(4\right)}$	$\displaystyle=-3\widetilde{\beta}_{\alpha}^{\left(4\right)}+3\widetilde{\beta}_{\alpha}^{\left(3\right)}\left(\frac{\widetilde{\beta}_{\lambda}^{\left(2\right)}}{2\lambda}+\frac{\widetilde{\beta}_{\alpha}^{\left(2\right)}}{2\alpha}+\frac{\widetilde{\beta}_{\xi}^{\left(2\right)}}{2\xi}\right)-\widetilde{\beta}_{\alpha}^{\left(2\right)}\left(\frac{\beta_{\lambda}^{\left(3\right)}}{2\lambda}+\frac{\beta_{\alpha}^{\left(3\right)}}{2\alpha}+\frac{\beta_{\xi}^{\left(3\right)}}{2\xi}\right),$		(27b)
	$\displaystyle\beta_{\xi}^{\left(4\right)}$	$\displaystyle=-3\widetilde{\beta}_{\xi}^{\left(4\right)}+3\widetilde{\beta}_{\xi}^{\left(3\right)}\left(\frac{\widetilde{\beta}_{\lambda}^{\left(2\right)}}{2\lambda}+\frac{\widetilde{\beta}_{\alpha}^{\left(2\right)}}{2\alpha}+\frac{\widetilde{\beta}_{\xi}^{\left(2\right)}}{2\xi}\right)-\widetilde{\beta}_{\xi}^{\left(2\right)}\left(\frac{\beta_{\lambda}^{\left(3\right)}}{2\lambda}+\frac{\beta_{\alpha}^{\left(3\right)}}{2\alpha}+\frac{\beta_{\xi}^{\left(3\right)}}{2\xi}\right),$		(27c)
	$\displaystyle\gamma_{\phi}^{\left(3\right)}$	$\displaystyle=-3\widetilde{\gamma}_{\phi}^{\left(3\right)}+3\widetilde{\gamma}_{\phi}^{\left(2\right)}\left(\frac{\widetilde{\beta}_{\lambda}^{\left(2\right)}}{2\lambda}+\frac{\widetilde{\beta}_{\alpha}^{\left(2\right)}}{2\alpha}+\frac{\widetilde{\beta}_{\xi}^{\left(2\right)}}{2\xi}\right)-\widetilde{\gamma}_{\alpha}^{\left(1\right)}\left(\frac{\beta_{\lambda}^{\left(3\right)}}{2\lambda}+\frac{\beta_{\alpha}^{\left(3\right)}}{2\alpha}+\frac{\beta_{\xi}^{\left(3\right)}}{2\xi}\right).$		(27d)

And finally, for the order $\beta_{i}^{\left(5\right)}$and $\gamma_{\phi}^{\left(4\right)}$,

	$\displaystyle\beta_{\lambda}^{\left(5\right)}$	$\displaystyle=-3\widetilde{\beta}_{\lambda}^{\left(5\right)}+3\widetilde{\beta}_{\lambda}^{\left(4\right)}\left(\frac{\widetilde{\beta}_{\lambda}^{\left(2\right)}}{2\lambda}+\frac{\widetilde{\beta}_{\alpha}^{\left(2\right)}}{2\alpha}+\frac{\widetilde{\beta}_{\xi}^{\left(2\right)}}{2\xi}\right)-\widetilde{\beta}_{\lambda}^{\left(3\right)}\left(\frac{\beta_{\lambda}^{\left(3\right)}}{2\lambda}+\frac{\beta_{\alpha}^{\left(3\right)}}{2\alpha}+\frac{\beta_{\xi}^{\left(3\right)}}{2\xi}\right)$
		$\displaystyle-\widetilde{\beta}_{\lambda}^{\left(2\right)}\left(\frac{\beta_{\lambda}^{\left(4\right)}}{2\lambda}+\frac{\beta_{\alpha}^{\left(4\right)}}{2\alpha}+\frac{\beta_{\xi}^{\left(4\right)}}{2\xi}\right),$		(28a)
	$\displaystyle\beta_{\alpha}^{\left(5\right)}$	$\displaystyle=-3\widetilde{\beta}_{\alpha}^{\left(5\right)}+3\widetilde{\beta}_{\alpha}^{\left(4\right)}\left(\frac{\widetilde{\beta}_{\lambda}^{\left(2\right)}}{2\lambda}+\frac{\widetilde{\beta}_{\alpha}^{\left(2\right)}}{2\alpha}+\frac{\widetilde{\beta}_{\xi}^{\left(2\right)}}{2\xi}\right)-\widetilde{\beta}_{\alpha}^{\left(3\right)}\left(\frac{\beta_{\lambda}^{\left(3\right)}}{2\lambda}+\frac{\beta_{\alpha}^{\left(3\right)}}{2\alpha}+\frac{\beta_{\xi}^{\left(3\right)}}{2\xi}\right)$
		$\displaystyle-\widetilde{\beta}_{\alpha}^{\left(2\right)}\left(\frac{\beta_{\lambda}^{\left(4\right)}}{2\lambda}+\frac{\beta_{\alpha}^{\left(4\right)}}{2\alpha}+\frac{\beta_{\xi}^{\left(4\right)}}{2\xi}\right),$		(28b)
	$\displaystyle\beta_{\xi}^{\left(5\right)}$	$\displaystyle=-3\widetilde{\beta}_{\xi}^{\left(5\right)}+3\widetilde{\beta}_{\xi}^{\left(4\right)}\left(\frac{\widetilde{\beta}_{\lambda}^{\left(2\right)}}{2\lambda}+\frac{\widetilde{\beta}_{\alpha}^{\left(2\right)}}{2\alpha}+\frac{\widetilde{\beta}_{\xi}^{\left(2\right)}}{2\xi}\right)-\widetilde{\beta}_{\xi}^{\left(3\right)}\left(\frac{\beta_{\lambda}^{\left(3\right)}}{2\lambda}+\frac{\beta_{\alpha}^{\left(3\right)}}{2\alpha}+\frac{\beta_{\xi}^{\left(3\right)}}{2\xi}\right)$
		$\displaystyle-\widetilde{\beta}_{\xi}^{\left(2\right)}\left(\frac{\beta_{\lambda}^{\left(4\right)}}{2\lambda}+\frac{\beta_{\alpha}^{\left(4\right)}}{2\alpha}+\frac{\beta_{\xi}^{\left(4\right)}}{2\xi}\right),$		(28c)
	$\displaystyle\gamma_{\phi}^{\left(4\right)}$	$\displaystyle=-3\widetilde{\gamma}_{\phi}^{\left(4\right)}+3\widetilde{\gamma}_{\phi}^{\left(3\right)}\left(\frac{\widetilde{\beta}_{\lambda}^{\left(2\right)}}{2\lambda}+\frac{\widetilde{\beta}_{\alpha}^{\left(2\right)}}{2\alpha}+\frac{\widetilde{\beta}_{\xi}^{\left(2\right)}}{2\xi}\right)-\widetilde{\gamma}_{\alpha}^{\left(2\right)}\left(\frac{\beta_{\lambda}^{\left(3\right)}}{2\lambda}+\frac{\beta_{\alpha}^{\left(3\right)}}{2\alpha}+\frac{\beta_{\xi}^{\left(3\right)}}{2\xi}\right)$
		$\displaystyle-\widetilde{\gamma}_{\alpha}^{\left(1\right)}\left(\frac{\beta_{\lambda}^{\left(4\right)}}{2\lambda}+\frac{\beta_{\alpha}^{\left(4\right)}}{2\alpha}+\frac{\beta_{\xi}^{\left(4\right)}}{2\xi}\right).$		(28d)

Now, with the four-loop renormalization group functions written in the CW scheme, in the next section we will compute the effective potential and study the CW mechanism by using the RGE in the leading log approximation up to five loops.

The main object we shall be interested in studying is the effective potential. In order to compute this object, we consider a shift in the $N$-th component of $\phi$ in (1),

$\displaystyle\phi_{N}$

$\displaystyle=\phi_{N}^{q}+\sigma,$

(29)

where $\phi_{N}^{q}=\frac{1}{\sqrt{2}}\left(\phi_{1N}+i\phi_{2N}\right)$ with $\phi_{1N}$ and $\phi_{2N}$ being two real scalar fields and $\sigma$ is a constant expectation value of scalar field, called background field. This scalar field has the same properties of $\phi_{N}$. If we substitute (29) into (1) we can find the effective potential in the classical approximation

$\displaystyle V_{\mathrm{eff}}^{\left(0\ell\right)}\left(\sigma\right)$

$\displaystyle=\frac{1}{4}\lambda\sigma^{4}.$

(30)

It is easy to see that $\sigma=0$ is the minimum of $V_{\mathrm{eff}}^{(0\ell)}$, so there is no spontaneous symmetry breaking at classical level. Our aim is to compute loop corrections to $V_{\mathrm{eff}}\left(\sigma\right)$ in order to understand if these corrections are capable to induce a spontaneous symmetry breaking and the corresponding generation of mass as given below

	$\displaystyle m_{\phi_{1N}}^{2}$	$\displaystyle=\frac{3}{2}\lambda\sigma^{2},$		(31)
	$\displaystyle m_{\phi_{2N}}^{2}$	$\displaystyle=\frac{1}{2}\lambda\sigma^{2}-\xi m_{A}^{2},$		(32)
	$\displaystyle m_{A}^{2}$	$\displaystyle=\frac{1}{2}e^{2}\sigma^{2}.$		(33)

Notice that the gauge dependence on the mass of $\phi_{2N}$ is a consequence of a $R_{\xi}$-gauge, $\mathcal{L}_{\mathrm{gf}}=-\frac{1}{2\xi}\left(\partial_{\mu}A^{\mu}-\xi e\sigma\phi_{2N}\right)^{2}$. Actually, in the spontaneously symmetry broken phase, the $\phi_{2N}$ degree of freedom is absorbed by the photon, which becomes massive.

As discussed in (Quinto2016, ; A.G.Quinto2021, ), the knowledge of $V_{\mathrm{eff}}\left(\sigma\right)$ is sufficient for investigating the dynamical breaking of gauge symmetry. We will be able to calculate it by using an ansatz for the RGE, motivated by dimensional analysis, together with the renormalization group functions for the model found in section III. Specifically, we shall use for $V_{\mathrm{eff}}\left(\sigma\right)$ the ansatz

$\displaystyle V_{\mathrm{eff}}\left(\sigma_{cl}\right)$

$\displaystyle=\sigma^{4}S_{\mathrm{eff}}\left(\sigma,\lambda,\alpha,\xi,L\right),$

(34)

where

$\displaystyle S_{\mathrm{eff}}\left(\sigma;\lambda,\alpha,\xi,L\right)$

$\displaystyle=A\left(\lambda,\alpha,\xi\right)+B\left(\lambda,\alpha,\xi\right)L+C\left(\lambda,\alpha,\xi\right)L^{2}+D\left(\lambda,\alpha,\xi\right)L^{3}+\cdots,$

(35)

and $A,B,C,$ and $D$ are defined as power series in the coupling constants $\lambda,\alpha,\xi$, and $L$ is defined in (16). This ansatz follows from the conformal invariance at the tree-level, leading to the fact that we can have only one type of logarithm appearing in the quantum corrections. Comparison with (30) show us that

$\displaystyle A\left(\lambda,\alpha,\xi\right)$

$\displaystyle=A^{\left(1\right)}=\frac{1}{4}\lambda.$

(36)