**4. NLO**

Now consider the NLO contributions that can be parameterized as:

$$
\Sigma^{\text{(NLO)}}(p) = \left(\frac{8}{N}\right)^2 B \frac{(p^2)^{-a}}{(4\pi)^3} \left(\Sigma\_A + \Sigma\_1 + 2\Sigma\_2 + \Sigma\_3\right),
\tag{13}
$$

where each contribution to the linearized gap equation is represented graphically in Figure 2. When these contributions are added to the LO result, Equation (7), the gap equation has the following general form:

$$1 = \frac{(2+\xi)\beta}{L} + \frac{\overline{\Sigma}\_A(\xi) + \overline{\Sigma}\_1(\xi) + 2\overline{\Sigma}\_2(\xi) + \overline{\Sigma}\_3(\xi)}{L^2},\tag{14}$$

where Σ*i* = *π* Σ*i*, (*i* = 1, 2, 3. *<sup>A</sup>*).

Contribution Σ*A*, see (A) in Figure 2, comes from the LO value of *<sup>A</sup>*(*p*) and is singular. Using dimensional regularization for an arbitrary parameter *ξ*, it takes the form:

$$\overline{\Sigma}\_{A}(\underline{\boldsymbol{x}}) = 4 \frac{\overline{\mu}^{2\varepsilon}}{p^{2\varepsilon}} \beta \left[ \left( \frac{4}{3} (1 - \underline{\boldsymbol{x}}) - \underline{\boldsymbol{\xi}}^{2} \right) \left[ \frac{1}{\varepsilon} + \underline{\boldsymbol{\Psi}}\_{1} - \frac{\underline{\boldsymbol{\mathcal{E}}}}{4} \right] + \left( \frac{16}{9} - \frac{4}{9} \underline{\boldsymbol{\xi}} - 2 \underline{\boldsymbol{\xi}}^{2} \right) \right], \tag{15}$$

where

$$\Psi\_1 = \Psi(a) + \Psi(1/2 - a) - 2\Psi(1) + \frac{3}{1/2 - a} - 2\ln 2,\tag{16}$$

and Ψ is the digamma function.

The contribution of diagram (1) in Figure 2 is finite (the shaded blob contains the diagrams shown in Figure 3) and reads:

$$
\overline{\Sigma}\_1(\xi) = -2(2 + \xi) \,\beta \,\Omega, \qquad \widehat{\Pi} = \frac{92}{9} - \pi^2, \tag{17}
$$

where the gauge dependence comes from the fact that we are working in a nonlocal gauge, and Πˆ arises from the two-loop polarization operator in the dimension *D* = 3 [27,28,48–51].

The contribution of diagram (2) in Figure 2 is again singular. Dimensionally regularizing it gives:

$$\begin{split} \Sigma\_{2}(\xi) &= -2\frac{\overline{\mu}^{2\varepsilon}}{p^{2\varepsilon}}\beta \left[ \frac{(2+\xi)(2-3\xi)}{3} \left( \frac{1}{\varepsilon} + \Psi\_{1} - \frac{\beta}{4} \right) + \frac{\beta}{4} \left( \frac{14}{3} \left( 1-\xi \right) + \xi^{2} \right) \right. \\ &\left. + \frac{28}{9} + \frac{8}{9}\xi - 4\xi^{2} \right] + (1-\xi)\,\Sigma\_{2} \, , \\ \Sigma\_{2}(a) &= -(4a-1)\beta \left[ \Psi'(a) - \Psi'(1/2-a) \right] + \frac{\pi}{2a} \, I\_{1}(a) + \frac{\pi}{2(1/2-a)} I\_{1}(a+1) \, , \end{split} \tag{18}$$

where Ψ is the trigamma function and ˜*<sup>I</sup>*1(*α*) is a dimensionless integral that was defined in [1]. *Particles* **2020**, *3*

The singularities in <sup>Σ</sup>*A*(*ξ*) and <sup>Σ</sup>2(*ξ*) cancel each other, so their sum is finite. Defining: <sup>Σ</sup>2*A*(*ξ*) = <sup>Σ</sup>*A*(*ξ*) + <sup>2</sup>Σ2(*ξ*), the latter reads:

$$
\nabla\_{2A}(\frac{\pi}{2}) = 2(1-\xi)\mathfrak{L}\_2(a) - \left(\frac{14}{3}(1-\xi) + \xi^2\right)\mathfrak{f}^2 - 8\mathfrak{f}\left(\frac{2}{3}(1+\xi) - \xi^2\right).
\tag{19}
$$

Finally, the contribution of diagram (3) in Figure 2 is finite and reads:

$$\begin{aligned} \Sigma\_3(\xi) &= \quad \hat{\Sigma}\_3(a, \xi) + \Big( 3 + 4\xi - 2\xi^2 \big) \beta^2, \ \hat{\Sigma}\_3(a, \xi) = \frac{1}{4} \Big( 1 + 8\xi + \xi^2 + 2a(1 - \xi^2) \big) \pi I\_2(a) \\ &+ \frac{1}{2} \Big( 1 + 4\xi - a(1 - \xi^2) \big) \pi I\_2(1 + a) + \frac{1}{4} \Big( -7 - 16\xi + 3\xi^2 \big) \pi I\_3(a) \end{aligned} \tag{20}$$

where the dimensionless integrals ˜ *<sup>I</sup>*2(*α*) and ˜*<sup>I</sup>*3(*α*) were defined in [1].

**Figure 2.** NLO diagrams for dynamically generated mass <sup>Σ</sup>(*p*). The symbol (A) shows the contribution of the LO fermion wave function and symbols (1), (2) and (3) correspond to the different topologies of the NLO corrections themselves. The shaded blob contains the sum of the diagrams shown in Figure 3.

**Figure 3.** The diagrams contributing to the shaded blob is shown in Figure 2. Symbols (**a**) and (**b**) correspond to the different topologies of the corrections to the polarization operator.

Combining all the above results, the gap Equation (14) can be written explicitly as:

$$1 = \frac{(2+\tilde{\zeta})\beta}{L} + \frac{1}{L^2} \left[ 8S(a,\tilde{\zeta}) - 2(2+\tilde{\zeta})\hat{\Pi}\beta + \left( -\frac{5}{3} + \frac{26}{3}\tilde{\zeta} - 3\tilde{\zeta}^2 \right) \beta^2 - 8\beta \left( \frac{2}{3}(1-\tilde{\zeta}) - \tilde{\zeta}^2 \right) \right], \quad \text{(21)}$$

where *<sup>S</sup>*(*<sup>α</sup>*, *ξ*) = Σˆ <sup>3</sup>(*<sup>α</sup>*, *ξ*) + 2(1 − *ξ*)Σ<sup>ˆ</sup> 2(*α*) /8.

#### *4.1. Extraction of the Most "Important" Terms*

Following Ref. [26], we would like to resum the term LO along with some of the NLO contributions containing terms ∼*β*2. To do this, we will now rewrite the gap Equation (21) in a more suitable form.This is equivalent to extracting the terms ∼*β* and ∼*β*<sup>2</sup> from the complex parts of the fermion self-energy, Equations (18) and (20). Such calculations give:

$$
\hat{\Sigma}\_2(\mathfrak{a}) = \beta(3\mathfrak{\beta} - 8) + \hat{\Sigma}\_2(\mathfrak{a}) \,, \ \hat{\Sigma}\_3(\mathfrak{\zeta}) = -4\mathfrak{z}(4 + \mathfrak{\zeta})\mathfrak{\beta} + \hat{\Sigma}\_3(\mathfrak{a}, \mathfrak{\zeta}) \,. \tag{22}
$$

Then, using the results (22), the gap Equation (21) can be written as:

$$1 = \frac{(2+\tilde{\varsigma})\beta}{L} + \frac{1}{L^2} \left[ 8\tilde{s}(a,\tilde{\varsigma}) - 2(2+\tilde{\varsigma})\acute{\Omega}\beta + \left(\frac{2}{3} - \tilde{\varsigma}\right)(2+\tilde{\varsigma})\left\beta^2 + 4\beta\left(\zeta^2 - \frac{4}{3}\xi - \frac{16}{3}\right) \right],\tag{23}$$

where *<sup>S</sup>*˜(*<sup>α</sup>*, *ξ*) = Σ˜ <sup>3</sup>(*<sup>α</sup>*, *ξ*) + 2(1 − *ξ*)Σ˜ 2(*α*) /8. At this point, Equations (21) and (23) are strictly equivalent to each other and give the same values for *Nc*(*ξ*).

## *4.2. Gap Equation*

Following the Addendum to [2], we now proceed to the calculation of the NLO correction to the parameter *β*−<sup>1</sup> of the solution of the SD equation. From (23), we have:

$$\beta^{-1} = \frac{2+\xi}{L} + \frac{1}{L^2} \left[ \frac{8}{\beta} \mathbb{S}(\boldsymbol{\beta}, \boldsymbol{\xi}) - 2(2+\xi)\boldsymbol{\Omega} + \left(\frac{2}{3} - \boldsymbol{\xi}\right) \left(2 + \boldsymbol{\xi}\right) \beta + 4 \left(\boldsymbol{\xi}^2 - \frac{4}{3}\boldsymbol{\xi} - \frac{16}{3}\right) \right] + O(L^{-3}). \tag{24}$$

It is clear from this equation that the first term in brackets is of the order ∼1/*L* (as can be seen from the iterative solution of the Equation (24)) and, therefore, its contribution is of the order ∼1/*L*<sup>3</sup> and should be neglected in the present study. So, with NLO accuracy, we ge<sup>t</sup> that:

$$\boldsymbol{\beta}^{-1} = \frac{2 + \frac{\boldsymbol{\gamma}}{3}}{L} + \frac{1}{L^2} \left[ \left( \frac{2}{3} - \boldsymbol{\xi} \right) \left( 2 + \boldsymbol{\xi} \right) \boldsymbol{\beta} - 2(2 + \boldsymbol{\xi}) \boldsymbol{\hat{\Pi}} + 4 \left( \boldsymbol{\xi}^2 - \frac{4}{3} \boldsymbol{\xi} - \frac{16}{3} \right) \right] + O(L^{-3}).\tag{25}$$

Now we are able to calculate *β*−<sup>1</sup> from Equation (25) as a combination of the terms ∼1/*L* and ∼1/*L*2. This, however, is not so important in this analysis. Since we are interested in the critical mode, we can obtain *Lc* in a simple way from (25) (or equally from the Equation (21) using the condition *<sup>S</sup>*˜(*β*, *ξ*) = 0) by setting *β* = 16 and preserving the conditions *<sup>O</sup>*(1/*L*<sup>2</sup>). This gives:

$$L\_c^2 - 16(2+\xi)L\_c + 32\left[ (2+\xi)\hat{\Pi} + 2\xi\left(\frac{20}{3} + 3\xi\right) \right] = 0. \tag{26}$$

Solving this equation, we have two standard solutions:

$$L\_{\mathfrak{c},\pm} = 8\left(2 + \mathfrak{z} \pm \sqrt{d\_1(\mathfrak{z})}\right), \; d\_1(\mathfrak{z}) = 4 - \frac{8}{3}\mathfrak{z} - 2\mathfrak{z}^2 - \frac{2 + \mathfrak{z}}{2}\hat{\Pi}.\tag{27}$$

Combining these values with the one of Πˆ in the Equation (17), we obtain:

$$N\_{\mathbb{C}}(\xi=0) = \text{3.17}, \quad N\_{\mathbb{C}}(\xi=2/3) = \text{2.91},\tag{28}$$

where the " −" solution is unphysical, and there is no solution in the Feynman gauge (*ξ* = 1). The range of *ξ*-values for which a solution exists corresponds to *ξ*− ≤ *ξ* ≤ *ξ*<sup>+</sup>, where *ξ*+ = 0.82 and *ξ*− = −2.24.
