pQCD running couplings finite and monotonic in the infrared: when do they reflect the holomorphic properties of spacelike observables?

According to general principles of Quantum Field Theories, the physical spacelike observables ${ \mathcal D }({Q}^{2})$ (such as the quark current correlators) and even unphysical amplitudes (such as the dressing functions of quark and transverse gluon propagators in QCD) are holomorphic (analytic) functions in the complex Q²-plane (where ${Q}^{2}\equiv -{q}^{2}\,=-{({q}^{0})}^{2}+{\vec{q}}^{2}$) except on the negative Q² semiaxis [1, 2]. On the other hand, QCD running coupling a(Q²) ≡ α_s (Q²)/π can be defined, in a specific renormalization scheme, as a product of the Landau gauge gluon dressing function and the square of the ghost dressing function [3]. Further, the leading-twist part of the spacelike physical QCD amplitudes ${ \mathcal D }({Q}^{2})$ is a function of the running coupling, ${ \mathcal D }{({Q}^{2})}^{({\rm{l}}.{\rm{t}}.)}={ \mathcal F }(a(\kappa {Q}^{2});\kappa )$ (where κ ∼ 1 is a positive renormalization scale parameter). Therefore, a natural consequence of the holomorphic behaviour of QCD amplitudes ${ \mathcal D }({Q}^{2})$ would be that QCD running coupling a(Q²) reflected these properties, i.e., that a(Q²) were a holomorphic function in the complex Q²-plane with the exception of a negative semiaxis, ${Q}^{2}\in {\mathbb{C}}\setminus (-\infty ,-{M}_{\mathrm{thr}}^{2}]$, where M_thr is a threshold mass, $0\leqslant {M}_{\mathrm{thr}}^{2}\lesssim 0.1\ {\mathrm{GeV}}^{2}$.

However, the QCD coupling a(Q²) is evaluated often in such renormalization schemes in which it is not an observable, and consequently a(Q²) is not a holomorphic [on ${\mathbb{C}}\setminus (-\infty ,0)]$ function, but it may have singularities in the mentioned region, called Landau singularities. These singularities are a serious problem especially in evaluations of low-energy QCD observables, where the coupling often has to be evaluated in the regimes of the complex Q²-plane which are close to those singularities and thus the obtained values lose predictability. Therefore, it is important to have a reliable method to find whether such Landau singularities exist, and if they exist, where in the complex Q²-plane they are situated and what is their nature.

Specifically, if the considered observable ${ \mathcal D }({Q}^{2})$ is spacelike and the spacelike momentum Q² is positive, the leading-twist part of ${ \mathcal D }({Q}^{2})$ is evaluated as a perturbation series in powers of a(κ Q²) where κ is a positive renormalization scale parameter (κ ∼ 1); if the coupling $a({Q}^{{\prime} 2})$ has Landau singularities in the complex ${Q}^{{\prime} 2}$-plane at values on or close to the positive semiaxis, then the evaluation of ${ \mathcal D }({Q}^{2})$ becomes unreliable for Q² close to such Landau singularities.

Further, if the considered QCD observable ${ \mathcal R }(s)$ is timelike (s = − Q² > 0), then it is usually evaluated as a contour integral involving the corresponding spacelike quantity Π(Q²) in the complex ${Q}^{{\prime} 2}$-plane, with a contour of radius $| {Q}^{{\prime} 2}| \sim s$. In such a case, there are at least two problems appearing when $a({Q}^{{\prime} 2})$ has Landau singularities in the complex ${Q}^{{\prime} 2}$-plane. The first is the following: the quantity ${ \mathcal R }(s)$ is originally expressed as an integral involving the corresponding physical (measured) spectral function $\omega (\sigma )=\mathrm{Im}{\rm{\Pi }}(-\sigma -i\epsilon )$ along the physical cut 0 < σ < s; this integral cannot be evaluated directly in pQCD; it is transformed via the Cauchy theorem into a contour integral involving ${\rm{\Pi }}({Q}^{{\prime} 2})$ along a circle of radius $| {Q}^{{\prime} 2}| \sim s$ (a form of sum rules). In pQCD, the leading-twist part of the spacelike quantity ${\rm{\Pi }}({Q}^{{\prime} 2})$ in this contour integral is usually expressed as a perturbation series in powers of $a(\kappa {Q}^{{\prime} 2})$ where κ is a positive renormalization scale parameter, κ ∼ 1. If the pQCD coupling has Landau singularities, the evaluated ${\rm{\Pi }}({Q}^{{\prime} 2})={ \mathcal F }(a(\kappa {Q}^{{\prime} 2}))$ function does not possess the holomorphic properties in the complex ${Q}^{{\prime} 2}$-plane (outside the negative axis) which it was assumed to possess when applying the Cauchy theorem. The mentioned sum rule relation is thus inconsistent in the case of pQCD with Landau singularities. The second problem that can appear here is more of a practical nature: if there are Landau singularities ${Q}_{* }^{2}$ in the complex ${Q}^{{\prime} 2}$-plane such that $| {Q}_{* }^{2}| \sim s$, then the contour integral may come close to such singularities and the evaluation may turn numerically unstable.

The perturbative QCD (pQCD) frameworks usually used in the literature are the $\overline{\mathrm{MS}}$-type mass independent renormalization schemes (such as $\overline{\mathrm{MS}}$, 't Hooft, MiniMOM, Lambert schemes), which give the running coupling a(Q²) which is not holomorphic in the mentioned sense, but has a (Landau) branching point at ${Q}_{* }^{2}\gt 0$ (∼0.1-1 GeV²) for the cut, i.e., the cut reaches beyond the negative semiaxis to the positive IR regime, i.e., there is a Landau ghost cut $(0,{Q}_{* }^{2})$. Further, the coupling often diverges at the branching point, $a({Q}_{* }^{2})=\infty$ (Landau pole). These properties are mathematical consequences of the form of the beta function β(a) which appears in the RGE determining the flow of a(Q²) with the squared momentum Q². These properties contradict the earlier mentioned holomorphic properties for a(Q²) which are motivated physically. If the Landau branching point ${Q}_{* }^{2}$ is on the (positive) real axis, it is relatively straightforward to encounter it in practice, for example by one-dimensional numerical integration of the RGE along the positive Q²-axis. On the other hand, if there are no Landau singularities on the positive real semiaxis, they could still appear within the complex plane ${Q}^{2}\in {\mathbb{C}}\setminus {\mathbb{R}};$ in such a case, it may be practically more difficult to find whether such singularities exist, and if they do exist, where they are and what is their nature. In this work we will concentrate on this problem, in the case of pQCD couplings in large classes of mass-independent renormalization schemes.

In our work, the considered QCD coupling a(Q²) ≡ α_s (Q²)/π will be such that it has so called freezing in the infrared regime, i.e., a(0) = a₀ is finite positive. This behaviour is suggested by the scaling solutions for the gluon and ghost propagators in the Landau gauge in the Dyson-Schwinger equations (DSE) approach [3–7], in the functional renormalization group (FRG) approach [8–10], stochastic quantization [11], and by Gribov-Zwanziger approach [12, 13]. Further, 0 < a(0) ≡ a₀ < + ∞ is also obtained in various physically motivated models for the running QCD coupling, among them the minimal analyticity dispersive approach [14–35] and its modifications or extensions [36–55], ³ and the AdS/CFT correspondence modified by a dilaton backgound [56, 57]. For reviews, we refer to [58, 59]. Such a behaviour has also been suggested in [60], where the running coupling definition involves explicitly the dynamical gluon mass and thus gives positive (nonzero) a(0) even in the case of so called decoupling solution of DSE [61–65] for gluon and ghost propagator in the Landau gauge. ⁴ All these approaches lead to nonperturbative (NP) running coupling ${ \mathcal A }({Q}^{2})$, which in general differs from the underlying perturbative coupling a(Q²) (i.e., the pQCD coupling in the same renormalization scheme) by power terms $\sim 1/{({Q}^{2})}^{N}$, i.e., terms of the type $\exp (-C/a)$ which cannot be Taylor-expanded around the pQCD point a = 0.

However, there are also pure pQCD frameworks (beta functions) in which the running coupling achieves a finite positive value in the infrared limit a(0) ≡ a₀ < ∞. Among such couplings are those where the coupling is a physical observable, such as in the effective (physical) charge approach [102–105] (cf.also [106]) where the coupling is a (spacelike or timelike) observable; such an observable can have variable and even very low momentum scales ∣Q²∣ < 1 GeV² [107], and such physical charges can even be related at the perturbative level to each other analytically [108–110]. Application of the principle of minimal sensitivity [111] also leads to schemes which give finite positive value of a(0). There exist yet other renormalization schemes with a(0) > 0, namely such that the resulting pQCD coupling a(Q²) is holomorphic in ${Q}^{2}\in {\mathbb{C}}\setminus (-\infty ,-{M}_{\mathrm{thr}}^{2}]$ (with $0\lt {M}_{\mathrm{thr}}^{2}\sim 0.1\ {\mathrm{GeV}}^{2}$) and reproduces the correct high- and low-energy QCD phenomenology [112–114].

In section 2 we define the class of considered pQCD beta functions β(a) (i.e., renormalization schemes), which are meromorphic functions leading to a finite positive a(0), present the implicit solution of the RGE in the complex Q²-plane, and discuss the renormalization scheme parameters β_j (j ≥ 2) that such beta functions generate. In section 3 we then present a practical algebraic procedure which allows us to find for a chosen beta function (in the considered class) the Landau singularities in the complex Q²-plane, i.e., the points where the behaviour of the running coupling a(Q²) does not reflect the holomorphic properties of the spacelike Green functions ${ \mathcal D }({Q}^{2})$ as required by the general principles of Quantum Field Theories. In section 4 we present some practical examples, and check with (2-dimensional) numerical integration of the RGE in the complex Q²-plane that the algebraic procedure gives us the correct answer. In section 5 we summarize our results.

The renormalization group equation (RGE) for the coupling parameter F(z) ≡ a(Q²) ≡ α_s (Q²)/π, where $z\equiv \mathrm{ln}({Q}^{2}/{Q}_{\mathrm{in}}^{2})$ is in general complex (and the initial scale is ${Q}_{\mathrm{in}}^{2}\gt 0$), can be written in the following way:

$$\begin{eqnarray}&&\displaystyle \frac{{dF}(z)}{{dz}}=\beta (F(z)),\end{eqnarray} \tag{ 1 }$$

where the beta function β(F) characterizes a mass independent renormalization scheme in perturbative QCD (pQCD), i.e., it has a well defined expansion around F = 0

$$\begin{eqnarray}&&\beta {\left(F\right)}_{\exp }=-{\beta }_{0}{F}^{2}-{\beta }_{1}{F}^{3}-{\beta }_{2}{F}^{4}-\ldots .\end{eqnarray} \tag{ 2 }$$

Here, β₀ and β₁ are universal constants, β₀ = (11 − 2n_f /3)/4 and β₁ = (102 − 38n_f /3)/16, where n_f is the number of active quark flavours. In the low-momentum regime (∣Q²∣ ≲ 10¹ GeV²), this number is usually taken to be n_f = 3, corresponding to the three lightest, almost massless, active quarks u, d and s. The coefficients β_j (j ≥ 2) characterize the pQCD renormalization scheme [111].

As mentioned in the Introduction, there exist several theoretical arguments which suggest that the running coupling F(z) ≡ a(Q²) is a finite function for all positive Q² and that it possibly acquires a finite positive value in the infared limit, 0 < a(0) ≡ a₀ < + ∞. In this case, it turns out that β(F) for positive couplings F ≡ a has a root at F = a₀ [and double root at F = 0 according to equation (2)], and for 0 < F < a₀ it has no roots. In view of this, we will consider the following class of beta functions:

$$\begin{eqnarray}&&{\left.\beta (F)\left(\equiv \beta {\left(F\right)}_{\left[M/N\right]}\right)=-{\beta }_{0}{F}^{2}(1-Y)\displaystyle \frac{{T}_{M}(Y)}{{U}_{N}(Y)}\right|}_{Y\equiv F/{a}_{0}},\end{eqnarray} \tag{ 3 }$$

where T_M (Y) and U_N (Y) are polynomials of degree M and N, respectively, both normalized in such a way that T_M (0) = 1 = U_N (0). Specifically, we denote as 1/t_j the roots of T_M (Y), and 1/u_j the roots of U_N (Y)

$$\begin{eqnarray}&&{T}_{M}(Y)=(1-{t}_{1}Y)\cdots (1-{t}_{M}Y),\end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray}&&{U}_{N}(Y)=(1-{u}_{1}Y)\cdots (1-{u}_{N}Y).\end{eqnarray} \tag{ 4b }$$

The parameters t_j and u_k are such that the polynomials T_M (Y) and U_N (Y) have real coefficients; this means that some of these parameters t_j and u_k can be real, and others complex conjugate pairs. We will restrict ourselves, for physical reasons, to such beta functions of the form (3) in which those t_j and u_k which are real and positive are all below unity: 0 < t_j < 1 and 0 < u_k < 1. This means that:

• a = a₀ is the smallest positive root of the beta function;
• and that all those poles of the beta function which are positive are larger than a₀.

If the latter conditions were not fulfilled, the running coupling a(Q²) would obviously have (Landau) singularities on the positive Q²-axis, contradicting the theoretical arguments mentioned in the Introduction. The former condition only means that we define a₀ as the smallest positive root of the beta function, and demand that at least one such positive root exist. An important practical consequence of these restrictions will be highlighted in section 3 (the first paragraph).

The first universal coefficient β₀ in the expansion of the beta function (2) is reproduced automatically by our construction. The second universal coefficient β₁ in equation (2) imposes the following restriction on the polynomials T_M (Y) and U_N (Y):

$$\begin{eqnarray}&&-\displaystyle \sum _{j=1}^{M}{t}_{j}+\displaystyle \sum _{k=1}^{N}{u}_{k}=1+{\beta }_{1}{a}_{0}{/\beta }_{0},\end{eqnarray} \tag{ 5 }$$

In addition, we will restrict the considered class of meromorphic beta functions to M + 1 ≥ N. In such a case, it turns out that the RGE (1) can be integrated algebraically and leads to an implicit solution of the form z = G(F) [for F ≡ F(z)]. Namely, the integration of the RGE (1) gives

$$\begin{eqnarray}&&z={\int }_{{a}_{\mathrm{in}}}^{F}\displaystyle \frac{{dF}^{\prime} }{\beta (F^{\prime} )},\end{eqnarray} \tag{ 6 }$$

and if we introduce a new integration variable t ≡ a₀/F, this can be written as

$$\begin{eqnarray}&&z=\displaystyle \frac{1}{{\beta }_{0}{a}_{0}}{\int }_{{a}_{0}/{a}_{\mathrm{in}}}^{{a}_{0}/F}{dt}\displaystyle \frac{t\ {U}_{N}(1/t)}{(t-1){T}_{M}(1/t)},\end{eqnarray} \tag{ 7 }$$

where ${a}_{\mathrm{in}}=a({Q}_{\mathrm{in}}^{2})=F(z=0)$ has a real positive value, 0 < a_in < a₀. When M + 1 ≥ N, the integrand can be written as a sum of simple partial fractions 1/(t − t_j ), where t₀ = 1 and t_j (j = 1, ..., M) are the roots of the (M-degree) polynomial t^M T_M (1/t)

$$\begin{eqnarray}&&{t}^{M}{T}_{M}(1/t)\equiv {t}^{M}(1-{t}_{1}/t)\cdots (1-{t}_{M}/t)=(t-{t}_{1})\cdots (t-{t}_{M}).\end{eqnarray} \tag{ 8 }$$

Namely, we have

$$\begin{eqnarray}&&\displaystyle \frac{{{tU}}_{N}(1/t)}{(t-1){T}_{M}(1/t)}=\displaystyle \frac{{t}^{M+1}{U}_{N}(1/t)}{(t-{t}_{0})(t-{t}_{1})\cdots (t-{t}_{M})}\end{eqnarray} \tag{ 9a }$$

$$\begin{eqnarray}&&=\ \left[1+\displaystyle \sum _{j=0}^{M}{B}_{j}\displaystyle \frac{1}{(t-{t}_{j})}\right],\end{eqnarray} \tag{ 9b }$$

where the M + 1 constants B_j are

$$\begin{eqnarray}&&{B}_{j}=\displaystyle \frac{{t}_{j}^{M+1}{U}_{N}(1/{t}_{j})}{({t}_{j}-{t}_{0})\cdots ({t}_{j}-{t}_{j-1})({t}_{j}-{t}_{j+1})\cdots ({t}_{j}-{t}_{M})}.\end{eqnarray} \tag{ 10 }$$

As a special case, we see that

$$\begin{eqnarray}&&{B}_{0}={U}_{N}(1)/{T}_{M}(1),\end{eqnarray} \tag{ 11 }$$

which is a real number. Using this, and the expression (3), we also obtain the following relation:

$$\begin{eqnarray}&&{\beta }^{{\prime} }(a){| }_{a={a}_{0}}={\beta }_{0}{a}_{0}\displaystyle \frac{{T}_{M}(1)}{{U}_{N}(1)}=\displaystyle \frac{{\beta }_{0}{a}_{0}}{{B}_{0}}.\end{eqnarray} \tag{ 12 }$$

Incidentally, in the limit of large t the relations (9a ) imply the following sum rule:

$$\begin{eqnarray}&&\displaystyle \sum _{j=0}^{M}{B}_{j}=1-{t}_{1}^{(M)}+{u}_{1}^{(N)}=-\displaystyle \frac{{\beta }_{1}{a}_{0}}{{\beta }_{0}},\end{eqnarray} \tag{ 13 }$$

where the last equality is obtained by using the relation (5). Using the form (9b) for the integrand in equation (7) leads us immediately to the implicit solution of the RGE

$$\begin{eqnarray}&&z=\displaystyle \frac{1}{{\beta }_{0}{a}_{0}}\left[\left(\displaystyle \frac{{a}_{0}}{F(z)}-\displaystyle \frac{{a}_{0}}{{a}_{\mathrm{in}}}\right)+\displaystyle \sum _{j=0}^{M}{B}_{j}\mathrm{ln}\left(\displaystyle \frac{{a}_{0}/F(z)-{t}_{j}}{{a}_{0}/{a}_{\mathrm{in}}-{t}_{j}}\right)\right].\end{eqnarray} \tag{ 14 }$$

Each logarithm has an ambiguity (winding number) because

$$\begin{eqnarray}&&\mathrm{ln}A={\mathrm{ln}}_{(\mathrm{pb})}A+i2\pi {n}_{A}=\mathrm{ln}| A| +i\mathrm{Arg}(A)+i2\pi {n}_{A},\quad ({n}_{A}=0,\pm 1,\pm 2,\ldots ),\end{eqnarray} \tag{ 15 }$$

where ${\mathrm{ln}}_{(\mathrm{pb})}$ is the principal branch: $-\pi \lt {\mathrm{Imln}}_{(\mathrm{pb})}A=\mathrm{Arg}(A)\leqslant +\pi ;$ further, n_A is the winding number representing the ambiguity. When A is positive, we consider that $\mathrm{ln}A$ is automatically the principal branch. This would then suggest that the right-hand side of equation (14) has M + 1 independent winding numbers n_j , correspondig to each logarithm there. The physically acceptable winding numbers of the logarithms on the right-hand side of equation (14) are such that they give for the expression (14) a number $z=\mathrm{ln}({Q}^{2}/{Q}_{\mathrm{in}}^{2})$ corresponding to the squared impulse Q² on the first Riemann sheet, i.e., $| \mathrm{Im}z| \leqslant \pi$ (cf. also the discussion in the beginning of section 3.1).

However, in general some of the roots t_j of the polynomial T_M (Y), equation (4a), are not real, but form complex conjugate pairs. For example, if the first complex conjugate pair is $({t}_{1},{t}_{2}={t}_{1}^{* })$, then it is straightforward to check that the corresponding coefficients B₁ and B₂ are mutually complex conjugate, and the corresponding two terms in the sum (9b) are

$$\begin{eqnarray}&&{B}_{1}\displaystyle \frac{1}{(t-{t}_{1})}+{B}_{2}\displaystyle \frac{1}{(t-{t}_{2})}=2\,\displaystyle \frac{t\mathrm{Re}({B}_{1})-\mathrm{Re}({t}_{1}^{* }{B}_{1})}{{t}^{2}-2t\mathrm{Re}({t}_{1})+| {t}_{1}{| }^{2}}\end{eqnarray} \tag{ 16 }$$

and the coresponding contribution to the integral (7) is

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{{\beta }_{0}{a}_{0}}{\displaystyle \int }_{{a}_{0}/{a}_{\mathrm{in}}}^{{a}_{0}/F}{dt}\,2\,\displaystyle \frac{t\mathrm{Re}({B}_{1})-\mathrm{Re}({t}_{1}^{* }{B}_{1})}{{t}^{2}-2t\mathrm{Re}({t}_{1})+| {t}_{1}{| }^{2}}\\ \ =\ \displaystyle \frac{1}{{\beta }_{0}{a}_{0}}\left\{\displaystyle \frac{2\left((\mathrm{Re}{B}_{1})(\mathrm{Re}{t}_{1})-\mathrm{Re}({t}_{1}^{* }{B}_{1})\right)}{| \mathrm{Im}{t}_{1}| }\left[\mathrm{ArcTan}\left(\displaystyle \frac{{a}_{0}/F(z)-\mathrm{Re}{t}_{1}}{| \mathrm{Im}{t}_{1}| }\right)-\mathrm{ArcTan}\left(\displaystyle \frac{{a}_{0}/{a}_{\mathrm{in}}-\mathrm{Re}{t}_{1}}{| \mathrm{Im}{t}_{1}| }\right)\right]\right.\\ \ \left.+(\mathrm{Re}{B}_{1})\left[\mathrm{ln}\left({\left(\displaystyle \frac{{a}_{0}}{F(z)}\right)}^{2}-2(\mathrm{Re}{t}_{1})\displaystyle \frac{{a}_{0}}{F(z)}+| {t}_{1}{| }^{2}\right)-\mathrm{ln}\left({\left(\displaystyle \frac{{a}_{0}}{{a}_{\mathrm{in}}}\right)}^{2}-2(\mathrm{Re}{t}_{1})\displaystyle \frac{{a}_{0}}{{a}_{\mathrm{in}}}+| {t}_{1}{| }^{2}\right)\right]\right\}.\end{array}\end{eqnarray} \tag{ 17 }$$

Therefore, the expression on the right-hand side of equation (14) can be rewritten more explicitly, for the case when t_j (j = 1,...,2P) are P complex conjugate pairs and t_j (j = 2P + 1,...,M) are real

$$\begin{eqnarray}&&\begin{array}{l}z=\displaystyle \frac{1}{{\beta }_{0}{a}_{0}}\left\{\left(\displaystyle \frac{{a}_{0}}{F(z)}-\displaystyle \frac{{a}_{0}}{{a}_{\mathrm{in}}}\right)+{B}_{0}\mathrm{ln}\left(\displaystyle \frac{{a}_{0}/F(z)-1}{{a}_{0}/{a}_{\mathrm{in}}-1}\right)+\displaystyle \sum _{j=2P+1}^{M}{B}_{j}\mathrm{ln}\left(\displaystyle \frac{{a}_{0}/F(z)-{t}_{j}}{{a}_{0}/{a}_{\mathrm{in}}-{t}_{j}}\right)\right.\\ \ +\displaystyle \sum _{k=0}^{P-1}\displaystyle \frac{2\left((\mathrm{Re}{B}_{2k+1})(\mathrm{Re}{t}_{2k+1})-\mathrm{Re}({t}_{2k+1}^{* }{B}_{2k+1})\right)}{| \mathrm{Im}{t}_{2k+1}| }\\ \quad \times \ \left[\mathrm{ArcTan}\left(\displaystyle \frac{{a}_{0}/F(z)-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)-\mathrm{ArcTan}\left(\displaystyle \frac{{a}_{0}/{a}_{\mathrm{in}}-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)\right]\\ \ +\displaystyle \sum _{k=0}^{P-1}(\mathrm{Re}{B}_{2k+1})\left[\mathrm{ln}\left({\left(\displaystyle \frac{{a}_{0}}{F(z)}\right)}^{2}-2(\mathrm{Re}{t}_{2k+1})\displaystyle \frac{{a}_{0}}{F(z)}+| {t}_{2k+1}{| }^{2}\right)\right.\\ \ \left.\left.-\mathrm{ln}\left({\left(\displaystyle \frac{{a}_{0}}{{a}_{\mathrm{in}}}\right)}^{2}-2(\mathrm{Re}{t}_{2k+1})\displaystyle \frac{{a}_{0}}{{a}_{\mathrm{in}}}+| {t}_{2k+1}{| }^{2}\right)\right]\right\}.\end{array}\end{eqnarray} \tag{ 18 }$$

We note that among the P z-dependent $\mathrm{ArcTan}$ terms, which are in general complex, each has one winding number because for $A=| A| \exp (i\theta )$ (∣θ∣ ≤ π) ⁵

$$\begin{eqnarray}&&\mathrm{ArcTan}(A)={\mathrm{ArcTan}}_{(\mathrm{pb})}(A)+\pi {n}_{A},\quad ({n}_{A}=0,\pm 1,\pm 2,\ldots ),\end{eqnarray} \tag{ 19 }$$

where we regard as the principal branch ${\mathrm{ArcTan}}_{(\mathrm{pb})}(A)$ the one which fulfills the inequality $-\pi /2\,\lt {\mathrm{ReArcTan}}_{(\mathrm{pb})}(A)\leqslant +\pi /2$. When A is real, we consider that $\mathrm{ArcTan}$ is automatically the principal branch.

Further, each of the M − P + 1 z-dependent logarithms appearing in equation (18) has a winding number according to the relation (15). This means that we have in general in total M + 1 winding numbers. This realization will play a role in the next section 3.

We recall that the considered β(a) functions are such that a(Q²) is a holomorphic function in and around any positive point Q² > 0. However, at Q² = 0, where a = a₀ < ∞, the function a(Q²) could be nonholomorphic (nonanalytic), i.e., certain (high enough) derivative ${(d/{{dQ}}^{2})}^{n}a({Q}^{2})$ at Q² = 0 could be infinite. In our considered cases we have for the Taylor expansion around Q² = 0

$$\begin{eqnarray}&&a({Q}^{2})={a}_{0}+{C}_{0}{\left(\displaystyle \frac{{Q}^{2}}{{{\rm{\Lambda }}}^{2}}\right)}^{\kappa }+{C}_{1}{\left(\displaystyle \frac{{Q}^{2}}{{{\rm{\Lambda }}}^{2}}\right)}^{2\kappa }+\ldots \end{eqnarray} \tag{ 20 }$$

This implies

$$\begin{eqnarray}&&\beta (a({Q}^{2}))=\kappa (a({Q}^{2})-{a}_{0})+{ \mathcal O }\left({\left(a\left({Q}^{2}\right)-{a}_{0}\right)}^{2}\right),\end{eqnarray} \tag{ 21a }$$

$$\begin{eqnarray}&&\Rightarrow \,{\beta }^{{\prime} }(a){| }_{a={a}_{0}}=\kappa .\end{eqnarray} \tag{ 21b }$$

The use of relation (12) then gives the power index κ in terms of the parameters contained in the considered beta function equation (3)

$$\begin{eqnarray}&&\kappa ={\beta }_{0}{a}_{0}\displaystyle \frac{{T}_{M}(1)}{{U}_{N}(1)}=\displaystyle \frac{{\beta }_{0}{a}_{0}}{{B}_{0}}.\end{eqnarray} \tag{ 22 }$$

In general, κ is noninteger, and consequently the coupling is in general not analytic at Q² = 0 (z = − ∞).

We wish to point out that the class of the β-functions considered here, equation (3), in addition to having a Padé form P[M + 3/N](a), have specific restrictions which result in a finite positive and monotonically decreasing running coupling a(Q²) < a₀ on the entire nonnegative Q²-axis Q² ≥ 0 [with a(Q²) → a₀ when Q² → 0]. This is reflected in the formal requirement that those of the parameters t_j and u_k of equations (4a ) which are not complex and are positive must fulfill the restrictions 0 < u_k < 1 and 0 < t_j < 1.

On the other hand, there are specific classes of Padé-type QCD β-functions which do not fulfill the above restrictions [i.e., they do not give finite a(Q²) on the entire positive Q²-axis], ⁶ but give explicit solutions a(Q²) of the RGE where a(Q²) involves the Lambert function W. Specifically, when β(a) is of a Padé-form P[2/1](a) such that it reproduces the correct β_j -coefficients up to two-loop (β₀, β₁) [115–117] ⁷ ; when β(a) is of a Padé-form P[3/1](a) such that it reproduces the chosen β_j -coefficients up to three-loop (β_j , j = 0, 1, 2) [115]; when β(a) is of a Padé-form P[4/4](a) reproducing the chosen β_j -coefficients up to four-loop (β_j , j = 0, 1, 2, 3) [118]; ⁸ and even up to five-loop [118] [in that case β(a) is of a Padé-form P[5/6](a)].

In the considered class of β-functions (3)–(4a ), with the mentioned restrictions t_j < 1 and u_k < 1 when t_j or u_k are real, the following question may arise: when expanding the β-function in powers of F, equation (2), which values of the renormalization scheme parameters c_n ≡ β_n /β₀ (n ≥ 2) can be generated? Direct expansion gives the relations

$$\begin{eqnarray}&&{c}_{n}^{(r)}\equiv {a}_{0}^{n}{c}_{n}=\displaystyle \sum _{s=0}^{n}{\left(-1\right)}^{n-s}\sum {t}_{{j}_{1}}\cdots {t}_{{j}_{n-s}}\sum {u}_{{k}_{1}}\cdots {u}_{{k}_{s}},\end{eqnarray} \tag{ 23 }$$

where the sum is over 0 ≤ j₁ < ... < j_n−s ≤ M (taking t₀ = 1) and 1 ≤ k₁ ≤ ... ≤ k_s ≤ N. For n = 1 this relation reduces to the condition (5), where c₁ ≡ β₁/β₀ is a universal coefficient (c₁ = 51/22 when n_f = 0; c₁ = 16/9 when n_f  = 3). In a considered β-function form (3), for chosen M and N and a chosen value of a₀ ≡ a(0) > 0, the relation (5) relates the (M + N) parameters t_j (1 ≤ j ≤ M) and u_k (1 ≤ k ≤ N), and consequently we have (M + N − 1) degrees of freedom (d.o.f.). These (M + N − 1) d.o.f. then give us the first (M + N − 1) independent scheme parameters c₂,...,c_M+N . It turns out that in general the values of the latter scheme parameters cover the entire real axis (i.e., all the values) once the (M + N − 1) independent coefficients t_j and u_k are varied across all the allowed range; the only exception may be the last scheme coefficient c_M+N which may vary only over a part of the real axis.

For example, if M + N = 2, only the first scheme parameter c₂ ≡ β₂/β₀ is independent. More specifically, there are three cases

$$\begin{eqnarray}&&M=N=1\Rightarrow {c}_{2}\lt \displaystyle \frac{3{c}_{1}}{{a}_{0}},\end{eqnarray} \tag{ 24a }$$

$$\begin{eqnarray}M=2\,\&\,N=0\Rightarrow -\displaystyle \frac{1}{{a}_{0}}\left(\displaystyle \frac{3}{{a}_{0}}+2{c}_{1}\right)\lt {c}_{2}.\end{eqnarray} \tag{ 24b }$$

$$\begin{eqnarray}M=0\,\&\,N=2\Rightarrow {c}_{2}\lt \mathrm{Max}\left[\displaystyle \frac{1}{4}\left(\displaystyle \frac{1}{{a}_{0}}+{c}_{1}\right)\left(-\displaystyle \frac{1}{{a}_{0}}+3{c}_{1}\right),\,{c}_{1}^{2}\right]\end{eqnarray} \tag{ 24c }$$

When M = N = 1, the coefficients t₁ and u₁ must be real and are thus both below unity (t₁, u₁ < 1), which gives us the restriction (24a). When M = 2 and N = 0, and t₁ and t₂ are mutually complex conjugate, the restriction on c₂ is (1/4)(1/a₀ + c₁)(− 3/a₀ + c₁) ≤ c₂; and when t₁ and t₂ are real (and thus below unity), the restriction is − (1/a₀)(+3/a₀ + 2c₁) < c₂ < (1/4)(1/a₀ + c₁)(−3/a₀ + c₁); combining these two restrictions gives us the restriction (24b ). When M = 0 and N = 2, a similar analysis leads to the restriction (24c).

The infrared limit a(0) ≡ a₀ ( > 0) can be, in principle any number. Nonetheless, the QCD phenomenology requires in practice that ${a}_{0}\gt {({a}_{0})}_{\min }$. In such case, in all the restrictions (24a ) we must replace a₀ by ${({a}_{0})}_{\min }$.

We can see from the restrictions (24a ) that in the case M + N = 2 the first (two-loop) scheme parameter c₂ covers all possible (real) values once we allow, for example, in addition to the form M = N = 1 also the form M = 2 and N = 0.

Alternatively, if enlarge the M = N = 1 form to the form M = 2 and N = 1, we can also see that this generates all possible values of c₂ (and a restricted range of values of c₃).

In general, for the first (n − 1) scheme parameters c₂,...,c_n (where n ≥ 2 is fixed), all their values can be generated if we consider a sufficiently large set of β-functions of the type (3), i.e., with various choices for the values of the indices M and N and with full variation of the parameters t_j and u_k under the mentioned restrictions.

We note that, by restrictions on the beta function mentioned in the previous section, the running coupling a(Q²) ≡ F(z) has no singularities on the real positive Q²-axis, i.e., there are no positive-Q² Landau singularities. This is so because a(Q²) is constrained there to run between the value a(0) = a₀ (>0) and a(+∞ ) = 0, the latter equality being valid by the asymptotic freedom of QCD reflected in the form (2) of beta function when F → 0. Namely, by our restrictions on the class of considered β(a) functions, when a(Q²) is RGE-running with increasing positive Q², beta function β(a(Q²)) will be negative finite all the time, since no new roots or poles of the beta function are encountered. Therefore, by the mentioned restrictions on the roots and poles of the beta function (3) we ensured in advance that the positive-Q² Landau singularities (poles and/or cuts) do not exist. ⁹

3.1. Landau poles

We will now construct an algebraic algorithm which allows us to verify whether in the (first Riemann sheet of the) complex Q²-plane the solution (14) has poles outside the real Q²-axis (Landau poles). We assume that only the first sheet of the complex Q²-plane has physical meaning, ¹⁰ i.e., ${Q}^{2}=| {Q}^{2}| \exp (i\phi )$ where − π ≤ ϕ < + π. This corresponds for the $z\equiv \mathrm{ln}({Q}^{2}/{Q}_{\mathrm{in}}^{2})$ variable to be a band in the complex z-plane with $-\pi \leqslant \mathrm{Im}z\lt +\pi$, cf. figures 1 (a), (b). As argued in the Introduction, if a(Q²) is to reflect the holomorphic properties of spacelike Green functions and observables, such as current correlators or structure functions, then a(Q²) can have singularities (cut) only along the negative Q²-axis: $-\infty \lt {Q}^{2}\leqslant -{M}_{\mathrm{thr}}^{2}$, where the threshold mass ${M}_{\mathrm{thr}}^{2}$ is either positive ( ∼ 0.1 GeV²) or zero. This cut corresponds in the z-stripe to the cut along the $\mathrm{Im}z=-\pi$ border line.

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As explained, the possible Landau singularities in the considered pQCD renormalization schemes are within the Q²-complex plane outside the real Q²-axis. In the z-plane this corresponds to the possible Landau singularities within the interior of the z-stripe, − π < z < + π, but not along the real axis, $z\rlap{/}{\in }{\mathbb{R}}$.

Landau pole z_* = x_* + iy_* $[\iff {Q}_{* }^{2}={Q}_{\mathrm{in}}^{2}\exp ({x}_{* })\exp ({{iy}}_{* })]$ is usually a branching point of a cut singularity of F(z), such that F(z_*) = ∞ , and it is situated on the first Riemann sheet outside the timelike semiaxis ($| \mathrm{Im}{z}_{* }| \lt \pi$). Let us denote a_* ≡ F(x_*) (0 < a_* < a₀). We then apply the implicit solution equation (18) to the points z₁ = x_* and z₂ = x_* + iy_*, and subtract the two equations; this then gives us the equation

$$\begin{eqnarray}&&{y}_{* }={{ \mathcal G }}_{\vec{n}}({a}_{* })\end{eqnarray} \tag{ 25 }$$

where

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal G }}_{\vec{n}}({a}_{* })\equiv \displaystyle \frac{(-i)}{{\beta }_{0}{a}_{0}}\left\{-\displaystyle \frac{{a}_{0}}{{a}_{* }}+{B}_{0}\left[i\pi -\mathrm{ln}\left(\displaystyle \frac{{a}_{0}}{{a}_{* }}-1\right)\right]+\displaystyle \sum _{j=2P+1}^{M}{B}_{j}\left[{\mathrm{ln}}_{(\mathrm{pb})}(-{t}_{j})-\mathrm{ln}\left(\displaystyle \frac{{a}_{0}}{{a}_{* }}-{t}_{j}\right)\right]\right.\\ \ +\displaystyle \sum _{k=0}^{P-1}\displaystyle \frac{2\left((\mathrm{Re}{B}_{2k+1})(\mathrm{Re}{t}_{2k+1})-\mathrm{Re}({t}_{2k+1}^{* }{B}_{2k+1})\right)}{| \mathrm{Im}{t}_{2k+1}| }\left[\mathrm{ArcTan}\left(\displaystyle \frac{-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)-\mathrm{ArcTan}\left(\displaystyle \frac{{a}_{0}/{a}_{* }-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)\right]\\ \left.+\displaystyle \sum _{k=0}^{P-1}(\mathrm{Re}{B}_{2k+1})\left[\mathrm{ln}| {t}_{2k+1}{| }^{2}-\mathrm{ln}\left({\left(\displaystyle \frac{{a}_{0}}{{a}_{* }}\right)}^{2}-2(\mathrm{Re}{t}_{2k+1})\displaystyle \frac{{a}_{0}}{{a}_{* }}+| {t}_{2k+1}{| }^{2}\right)\right]\right\}\\ +\displaystyle \frac{2\pi }{{\beta }_{0}{a}_{0}}\left({B}_{0}{n}_{0}+\displaystyle \sum _{j=2P+1}^{M}{B}_{j}{n}_{j}\right)\\ +\displaystyle \frac{2\pi }{{\beta }_{0}{a}_{0}}\displaystyle \sum _{k=0}^{P-1}(\mathrm{Re}{B}_{2k+1}){N}_{k}+i\displaystyle \frac{2\pi }{{\beta }_{0}{a}_{0}}\displaystyle \sum _{k=0}^{P-1}\displaystyle \frac{\left(\mathrm{Re}({t}_{2k+1}^{* }{B}_{2k+1})-(\mathrm{Re}{B}_{2k+1})(\mathrm{Re}{t}_{2k+1})\right)}{| \mathrm{Im}{t}_{2k+1}| }{{ \mathcal N }}_{k}.\end{array}\end{eqnarray} \tag{ 26 }$$

Here, we accounted for the nonuniqueness of the (z-dependent) logarithms equation (15) and ArcTan equation (19)

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{y\to {y}_{* }}\mathrm{ln}\left(\displaystyle \frac{{a}_{0}}{F({x}_{* }+{iy})}-{t}_{j}\right)={\mathrm{ln}}_{(\mathrm{pb})}(-{t}_{j})+i2\pi {n}_{j}\,(j=0;2P+1,\ldots ,M),\end{eqnarray} \tag{ 27a }$$

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{y\to {y}_{* }}\mathrm{ln}\left[{\left(\displaystyle \frac{{a}_{0}}{F({x}_{* }+{iy})}\right)}^{2}-2(\mathrm{Re}{t}_{2k+1})\displaystyle \frac{{a}_{0}}{F({x}_{* }+{iy})}+| {t}_{2k+1}{| }^{2}\right]=\mathrm{ln}| {t}_{2k+1}{| }^{2}+i2\pi {N}_{k}\quad (k=0,\ldots ,P-1),\end{eqnarray} \tag{ 27b }$$

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{y\to {y}_{* }}\mathrm{ArcTan}\left(\displaystyle \frac{\tfrac{{a}_{0}}{F({x}_{* }+{iy})}-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)=\mathrm{ArcTan}\left(\displaystyle \frac{-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)+\pi {{ \mathcal N }}_{k},\end{eqnarray} \tag{ 27c }$$

where (k = 0,...,P − 1), and we denoted the M + 1 winding numbers

$$\begin{eqnarray}&&\vec{n}\equiv \{{n}_{0},{n}_{2P+1},\ldots ,{n}_{M};{N}_{0},\ldots ,{N}_{P-1};{{ \mathcal N }}_{0},\ldots ,{{ \mathcal N }}_{P-1}\},\end{eqnarray} \tag{ 28 }$$

where ${n}_{j},{N}_{k},{{ \mathcal N }}_{k}=0,\pm 1,\pm 2,\ldots$. We note that the terms ${\mathrm{ln}}_{(\mathrm{pb})}(-{t}_{j})$ may have t_j either negative or positive, and therefore

$$\begin{eqnarray}&&{\mathrm{ln}}_{(\mathrm{pb})}(-{t}_{j})=\mathrm{ln}| {t}_{j}| +{\rm{\Theta }}({t}_{j})i\pi .\end{eqnarray} \tag{ 29 }$$

As a special case, we used in equation (26): $\mathrm{ln}(-{t}_{0})=\mathrm{ln}(-1)=i\pi$. We note that, since the real roots t_j fulfill the inequality t_j ≤ 1 [our initial physical restrictions on β-function, cf. comments after equations (4a )], the logarithm $\mathrm{ln}({a}_{0}/{a}_{* }-{t}_{j})$ in equation (26) is a real number because it has positive argument. For the same reason, also the P logarithms of the trinomials in (a₀/a_*) in equation (26) are real. We point out that that winding numbers $\vec{n}$ appear when integrating the RGE (1) from z₁ = x_* to z₂ = x_* + iy*.

equation (25) for the poles represents two equations, one for the imaginary and one for the real parts

$$\begin{eqnarray}&&\mathrm{Im}{{ \mathcal G }}_{\vec{n}}({a}_{* })=0,\quad \mathrm{Re}{{ \mathcal G }}_{\vec{n}}({a}_{* })={y}_{* },\end{eqnarray} \tag{ 30 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{Im}{{ \mathcal G }}_{\vec{n}}({a}_{* }) & \equiv & \displaystyle \frac{1}{{\beta }_{0}{a}_{0}}\left\{\displaystyle \frac{{a}_{0}}{{a}_{* }}+{B}_{0}\mathrm{ln}\left(\displaystyle \frac{{a}_{0}}{{a}_{* }}-1\right)+\displaystyle \sum _{j=2P+1}^{M}{B}_{j}\left[-{\mathrm{ln}}_{(\mathrm{pb})}| {t}_{j}| +\mathrm{ln}\left(\displaystyle \frac{{a}_{0}}{{a}_{* }}-{t}_{j}\right)\right]\right.\\ & & +\,\displaystyle \sum _{k=0}^{P-1}\displaystyle \frac{2\left(\mathrm{Re}({t}_{2k+1}^{* }{B}_{2k+1})-(\mathrm{Re}{B}_{2k+1})(\mathrm{Re}{t}_{2k+1})\right)}{| \mathrm{Im}{t}_{2k+1}| }\\ & & \times \,\left[-\mathrm{ArcTan}\left(\displaystyle \frac{-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)+\mathrm{ArcTan}\left(\displaystyle \frac{{a}_{0}/{a}_{* }-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)\right]\\ & & \left.+\,\displaystyle \sum _{k=0}^{P-1}(\mathrm{Re}{B}_{2k+1})\left[-\mathrm{ln}| {t}_{2k+1}{| }^{2}+\mathrm{ln}\left({\left(\displaystyle \frac{{a}_{0}}{{a}_{* }}\right)}^{2}-2(\mathrm{Re}{t}_{2k+1})\displaystyle \frac{{a}_{0}}{{a}_{* }}+| {t}_{2k+1}{| }^{2}\right)\right]\right\}\\ & & +\,\displaystyle \frac{2\pi }{{\beta }_{0}{a}_{0}}\displaystyle \sum _{k=0}^{P-1}\displaystyle \frac{\left(\mathrm{Re}({t}_{2k+1}^{* }{B}_{2k+1})-(\mathrm{Re}{B}_{2k+1})(\mathrm{Re}{t}_{2k+1})\right)}{| \mathrm{Im}{t}_{2k+1}| }{{ \mathcal N }}_{k},\end{array}\end{eqnarray} \tag{ 31a }$$

$$\begin{eqnarray}&&\mathrm{Re}{{ \mathcal G }}_{\vec{n}}({a}_{* })\equiv \displaystyle \frac{2\pi }{{\beta }_{0}{a}_{0}}\left\{{B}_{0}\left({n}_{0}+\displaystyle \frac{1}{2}\right)+\displaystyle \sum _{j=2P+1}^{M}{B}_{j}\left({n}_{j}+\displaystyle \frac{1}{2}{\rm{\Theta }}({t}_{j})\right)+\displaystyle \sum _{k=0}^{P-1}(\mathrm{Re}{B}_{2k+1}){N}_{k}\right\}.\end{eqnarray} \tag{ 31b }$$

We note that $\mathrm{Im}{{ \mathcal G }}_{\vec{n}}({a}_{* })$ depends only on the winding numbers ${{ \mathcal N }}_{k}$ (k = 0,...,P − 1) coming from $\mathrm{ArcTan};$ and $\mathrm{Re}{{ \mathcal G }}_{\vec{n}}({a}_{* })$ depends on the winding numbers n_j (j = 0, 2P + 1, 2P + 2,...,M) and N_k ( = 0,...,P − 1), both coming from logarithms.

The necessary conditions for the existence of a Landau pole are

• 1.  
  $\mathrm{Im}{{ \mathcal G }}_{\vec{n}}({a}_{* })=0$ for a chosen set ${{ \mathcal N }}_{k}$ (k = 0,...,P − 1), and the value a_* lies between 0 and a₀ (0 < a_* < a₀);
• 2.  
  and simultaneously, $\mathrm{Re}{{ \mathcal G }}_{\vec{n}}({a}_{* })$ ( = y_*) is within the interior of the first Riemann sheet, i.e., inside the first stripe of z, $| \mathrm{Re}{{ \mathcal G }}_{\vec{n}}({a}_{* })| \lt \pi$, for certain choices of n_j (j = 0, 2P + 1, 2P + 2,...,M) and N_k ( = 0,...,P − 1).

If, for example, all B_j coefficients are real, then $\mathrm{Im}\,{{ \mathcal G }}_{\vec{n}}(a)=\mathrm{Im}\,{{ \mathcal G }}_{\vec{0}}(a);$ if in such a case $\mathrm{Im}\,{{ \mathcal G }}_{\vec{0}}({a}_{* })$ has no zero in the positive interval 0 < a_* < a₀, then one necessary condition for the existence of Landau poles is not fulfilled, i.e., there are no Landau poles.

3.2. Landau branching points

In the previous Subsection we presented an algorithm which allows us to find, inside the complex z-stripe, the (Landau) poles where the coupling is infinite F(z_*) = ∞. However, the complex function F(z) can have also another type of Landau singularities, namely a cut with a finite-valued branching point z_*.

One illustrative mathematical example is $F{(z)=(z-{z}_{* })}^{1/2}$, where z_* = x_* + iy_* is such a branching point, F(z_*) = 0 and $F^{\prime} (z)=\infty$. The cut in this case is usually defined along the semiaxis to the left of z_*: x + iy_* (x ≤ x_*).

However, we may worry at first that other, even more 'finite,' type of Landau branching points ${z}_{* }\,\rlap{/}{\in }\,{\mathbb{R}}$ may appear, such as $F{(z)=(z-{z}_{* })}^{3/2}$, for which $F^{\prime} ({z}_{* })\lt \infty$ and F''(z_*) = ∞. We show that this does not occur for the considered class of meromorphic beta functions (3)–(4a ). Namely,

$$\begin{eqnarray}&&F^{\prime\prime} (z)=\displaystyle \frac{d}{{dz}}\beta (F(z))=\beta (F(z))\displaystyle \frac{\partial }{\partial F}\beta (F(z)).\end{eqnarray} \tag{ 32 }$$

The poles of the right-hand side are at the same values F = a₀/u_s (s = 1, ..., N) as in the beta function β(F) itself, cf. equations (3)–(4a ). This means that, if F''(z_*) = ∞ , then $F^{\prime} ({z}_{* })=\infty$. We can continue this argumentation, by applying further derivatives (d/dz)ⁿ to equation (32). E.g., if F⁽³⁾(z_*) = ∞, then $F^{\prime} ({z}_{* })=\infty$.

Therefore, the only relevant situation of finite-valued Landau branching points z_* for the considered beta functions is: $F^{\prime} ({z}_{* })=\infty$ and F(z_*) < ∞. Since $F^{\prime} ({z}_{* })=\beta (F({z}_{* }))$, such a branching point is one of the poles of the beta function, ${z}_{* }^{(s)}={x}_{* }^{(s)}+{{iy}}_{* }^{(s)}$ such that $F({z}_{* }^{(s)})={a}_{0}/{u}_{s}$ (s = 1, ..., N), cf. equations (3)–(4a ). This means, in analogy with equations (25)–(26) and using the notations (28), that we have the relation

$$\begin{eqnarray}&&{y}_{* }^{(s)}={{ \mathcal K }}_{\vec{n}}({a}_{* }^{(s)};{u}_{s}),\end{eqnarray} \tag{ 33 }$$

where ${a}_{* }^{(s)}\equiv F({x}_{* }^{(s)})$ ($0\lt {a}_{* }^{(s)}\lt {a}_{0}$), and

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal K }}_{\vec{n}}({a}_{* }^{(s)};{u}_{s})\equiv \displaystyle \frac{(-i)}{{\beta }_{0}{a}_{0}}\left\{\left({u}_{s}-\displaystyle \frac{{a}_{0}}{{a}_{* }^{(s)}}\right)+{B}_{0}\left[{\mathrm{ln}}_{(\mathrm{pb})}({u}_{s}-1)-\mathrm{ln}\left(\displaystyle \frac{{a}_{0}}{{a}_{* }^{(s)}}-1\right)\right]\right.\\ \ +\displaystyle \sum _{j=2P+1}^{M}{B}_{j}\left[{\mathrm{ln}}_{(\mathrm{pb})}({u}_{s}-{t}_{j})-\mathrm{ln}\left(\displaystyle \frac{{a}_{0}}{{a}_{* }^{(s)}}-{t}_{j}\right)\right]\\ \ +\displaystyle \sum _{k=0}^{P-1}\displaystyle \frac{2\left((\mathrm{Re}{B}_{2k+1})(\mathrm{Re}{t}_{2k+1})-\mathrm{Re}({t}_{2k+1}^{* }{B}_{2k+1})\right)}{| \mathrm{Im}{t}_{2k+1}| }\\ \ \times \,\left[\mathrm{ArcTan}\left(\displaystyle \frac{{u}_{s}-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)-\mathrm{ArcTan}\left(\displaystyle \frac{{a}_{0}/{a}_{* }^{(s)}-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)\right]\\ \ +\left.\displaystyle \sum _{k=0}^{P-1}(\mathrm{Re}{B}_{2k+1})\left[\mathrm{ln}\left({u}_{s}^{2}-2(\mathrm{Re}{t}_{2k+1}){u}_{s}+| {t}_{2k+1}{| }^{2}\right)-\mathrm{ln}\left({\left(\displaystyle \frac{{a}_{0}}{{a}_{* }^{(s)}}\right)}^{2}-2(\mathrm{Re}{t}_{2k+1})\displaystyle \frac{{a}_{0}}{{a}_{* }^{(s)}}+| {t}_{2k+1}{| }^{2}\right)\right]\right\}\\ \ +\displaystyle \frac{2\pi }{{\beta }_{0}{a}_{0}}\left({B}_{0}{n}_{0}^{(s)}+\displaystyle \sum _{j=2P\,+\,1}^{M}{B}_{j}{n}_{j}^{(s)}\right)\\ \ +\displaystyle \frac{2\pi }{{\beta }_{0}{a}_{0}}\displaystyle \sum _{k=0}^{P-1}(\mathrm{Re}{B}_{2k+1}){N}_{k}^{(s)}+i\displaystyle \frac{2\pi }{{\beta }_{0}{a}_{0}}\displaystyle \sum _{k=0}^{P-1}\displaystyle \frac{\left(\mathrm{Re}({t}_{2k+1}^{* }{B}_{2k+1})-(\mathrm{Re}{B}_{2k+1})(\mathrm{Re}{t}_{2k+1})\right)}{| \mathrm{Im}{t}_{2k+1}| }{{ \mathcal N }}_{k}^{(s)}.\end{array}\end{eqnarray} \tag{ 34 }$$

The winding numbers are generated in a limiting process analogous to that in equations (27a )

$$\begin{eqnarray}&&\begin{array}{l}\mathop{\mathrm{lim}}\limits_{y\to {y}_{* }^{(s)}}\mathrm{ln}\left(\displaystyle \frac{{a}_{0}}{F({x}_{* }^{(s)}+{iy})}-{t}_{j}\right)=\mathrm{ln}({u}_{s}-{t}_{j})\\ =\ {\mathrm{ln}}_{(\mathrm{pb})}({u}_{s}-{t}_{j})+i2\pi {n}_{j}^{(s)}\quad (j=0;2P+1,2P+2,\ldots ,M),\end{array}\end{eqnarray} \tag{ 35a }$$

$$\begin{eqnarray}&&\begin{array}{l}\mathop{\mathrm{lim}}\limits_{y\to {y}_{* }^{(s)}}\mathrm{ln}\left[{\left(\displaystyle \frac{{a}_{0}}{F({x}_{* }^{(s)}+{iy})}\right)}^{2}-2(\mathrm{Re}{t}_{2k+1})\displaystyle \frac{{a}_{0}}{F({x}_{* }^{(s)}+{iy})}+| {t}_{2k+1}{| }^{2}\right]\\ \ \ =\ \mathrm{ln}\left({u}_{s}^{2}-2(\mathrm{Re}{t}_{2k+1}){u}_{s}+| {t}_{2k+1}{| }^{2}\right)+i2\pi {N}_{k}^{(s)}\quad (k=0,\ldots ,P-1),\end{array}\end{eqnarray} \tag{ 35b }$$

$$\begin{eqnarray}&&\begin{array}{l}\mathop{\mathrm{lim}}\limits_{y\to {y}_{* }^{(s)}}\mathrm{ArcTan}\left(\displaystyle \frac{\tfrac{{a}_{0}}{F({x}_{* }^{(s)}+{iy})}-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)=\mathrm{ArcTan}\left(\displaystyle \frac{{u}_{s}-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)\\ =\ {\mathrm{ArcTan}}_{(\mathrm{pb})}\left(\displaystyle \frac{{u}_{s}-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)+\pi {{ \mathcal N }}_{k}^{(s)}\quad (k=0,\ldots ,P-1).\end{array}\end{eqnarray} \tag{ 35c }$$

As in equations (27a )–(28), the M + 1 winding numbers appear

$$\begin{eqnarray}&&\vec{n}\equiv \{{n}_{0}^{(s)},{n}_{2P+1}^{(s)},\ldots ,{n}_{M}^{(s)};{N}_{0}^{(s)},\ldots ,{N}_{P-1}^{(s)};{{ \mathcal N }}_{0}^{(s)},\ldots ,{{ \mathcal N }}_{P-1}^{(s)}\},\end{eqnarray} \tag{ 36 }$$

when integrating the RGE (1) in the z-plane along the vertical direction, from ${z}_{1}={x}_{* }^{(s)}$ toward ${z}_{2}={x}_{* }^{(s)}+{{iy}}_{* }^{(s)}$.

We note that one of the physically motivated restrictions on the β-function, from the outset, was that those roots u_s which are real satisfy u_s < 1 [cf. the comments after equations (4a )]. This means that for such u_s , the branching point ${z}_{* }^{(s)}$ where F(z_*) = a₀/u_s ( > a₀) cannot be achieved at real ${z}_{* }^{(s)}={x}_{* }^{(s)}$, i.e., also in such cases ${z}_{* }^{(s)}$ must have ${y}_{* }^{(s)}\ne 0$, and thus we can have also in such a case nonzero winding numbers $\vec{n}\ne \vec{0}$.

Here, the procedure described in the previous section 3.1 for $\mathrm{Im}\,{{ \mathcal G }}_{\vec{n}}({a}_{* })$ and $\mathrm{Re}\,{{ \mathcal G }}_{\vec{n}}({a}_{* })$ [for a_* ≡ F(x_*) in the interval 0 < a_* < a₀, and for ${n}_{j},{N}_{k},{{ \mathcal N }}_{k}=0,\pm 1,\ldots ,$], is now performed for $\mathrm{Im}\,{{ \mathcal K }}_{\vec{n}}({a}_{* }^{(s)};{u}_{s})$ and $\mathrm{Re}\,{{ \mathcal K }}_{\vec{n}}({a}_{* }^{(s)};{u}_{s})$, again with ${a}_{* }^{(s)}=F({x}_{* }^{(s)})$ in the interval $0\lt {a}_{* }^{(s)}\lt {a}_{0}$ and for ${n}_{j}^{(s)},{N}_{k}^{(s)},{{ \mathcal N }}_{k}^{(s)}=0,\pm 1,\ldots$, but now also for each u_s (s = 1, ..., N). This means that equation (33) for the branching points represents two real equations, in analogy with equations (30)–(31a ).

$$\begin{eqnarray}&&\mathrm{Im}{{ \mathcal K }}_{\vec{n}}({a}_{* }^{(s)};{u}_{s})=0,\quad \mathrm{Re}{{ \mathcal K }}_{\vec{n}}({a}_{* }^{(s)};{u}_{s})={y}_{* }^{(s)},\end{eqnarray} \tag{ 37 }$$

where

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{Im}{{ \mathcal K }}_{\vec{n}}({a}_{* }^{(s)};{u}_{s})\equiv \displaystyle \frac{1}{{\beta }_{0}{a}_{0}}\left\{\left(-{u}_{s}+\displaystyle \frac{{a}_{0}}{{a}_{* }^{(s)}}\right)+{B}_{0}\left[-{\mathrm{ln}}_{(\mathrm{pb})}({u}_{s}-1)+\mathrm{ln}\left(\displaystyle \frac{{a}_{0}}{{a}_{* }^{(s)}}-1\right)\right]\right.\\ \ +\,\displaystyle \sum _{j=2P\,+\,1}^{M}{B}_{j}\left[-{\mathrm{ln}}_{(\mathrm{pb})}({u}_{s}-{t}_{j})+\mathrm{ln}\left(\displaystyle \frac{{a}_{0}}{{a}_{* }^{(s)}}-{t}_{j}\right)\right]\\ \ +\,\displaystyle \sum _{k=0}^{P-1}\displaystyle \frac{2\left((\mathrm{Re}{B}_{2k+1})(\mathrm{Re}{t}_{2k+1})-\mathrm{Re}({t}_{2k+1}^{* }{B}_{2k+1})\right)}{| \mathrm{Im}{t}_{2k+1}| }\\ \ \times \,\left[-\mathrm{ArcTan}\left(\displaystyle \frac{{u}_{s}-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)+\mathrm{ArcTan}\left(\displaystyle \frac{{a}_{0}/{a}_{* }^{(s)}-\mathrm{Re}{t}_{2k+1}}{| \mathrm{Im}{t}_{2k+1}| }\right)\right]\\ \ +\,\displaystyle \sum _{k=0}^{P-1}(\mathrm{Re}{B}_{2k+1})\left[-\mathrm{ln}\left({u}_{s}^{2}-2(\mathrm{Re}{t}_{2k+1}){u}_{s}+| {t}_{2k+1}{| }^{2}\right)\right.\\ \ +\,\left.\left.\mathrm{ln}\left({\left(\displaystyle \frac{{a}_{0}}{{a}_{* }^{(s)}}\right)}^{2}-2(\mathrm{Re}{t}_{2k+1})\displaystyle \frac{{a}_{0}}{{a}_{* }^{(s)}}+| {t}_{2k+1}{| }^{2}\right)\right]\right\}\\ \ +\,\displaystyle \frac{2\pi }{{\beta }_{0}{a}_{0}}\displaystyle \sum _{k=0}^{P-1}\displaystyle \frac{\left(\mathrm{Re}({t}_{2k+1}^{* }{B}_{2k+1})-(\mathrm{Re}{B}_{2k+1})(\mathrm{Re}{t}_{2k+1})\right)}{| \mathrm{Im}{t}_{2k+1}| }{{ \mathcal N }}_{k}^{(s)},\end{array}\end{eqnarray} \tag{ 38 }$$

and

$$\begin{eqnarray}&&\mathrm{Re}{{ \mathcal K }}_{\vec{n}}({a}_{* }^{(s)};{u}_{s})\equiv \displaystyle \frac{2\pi }{{\beta }_{0}{a}_{0}}\left\{\left({B}_{0}{n}_{0}^{(s)}+\displaystyle \sum _{j=2P+1}^{M}{B}_{j}{n}_{j}^{(s)}\right)+\displaystyle \sum _{k=0}^{P-1}(\mathrm{Re}{B}_{2k+1}){N}_{k}^{(s)}\right\}.\end{eqnarray} \tag{ 39 }$$

We notice that $\mathrm{Im}{{ \mathcal K }}_{\vec{n}}({a}_{* }^{(s)};{u}_{s})$ depends only on the winding numbers ${{ \mathcal N }}_{k}^{(s)}$, cf. equation (34). The existence of a Landau branching point means that equations (37) have a solution, for an s, and ${a}_{* }^{(s)}$ and ${y}_{* }^{(s)}$ such that: $0\lt {a}_{* }^{(s)}\lt {a}_{0}$ and $| {y}_{* }^{(s)}| \lt \pi$.

The procedures described in this section 3 for $\mathrm{Im}\,{{ \mathcal G }}_{\vec{n}}({a}_{* })$ and $\mathrm{Re}\,{{ \mathcal G }}_{\vec{n}}({a}_{* })$, and for $\mathrm{Im}\,{{ \mathcal K }}_{\vec{n}}({a}_{* }^{(s)};{u}_{s})$ and $\mathrm{Re}\,{{ \mathcal K }}_{\vec{n}}({a}_{* }^{(s)};{u}_{s})$, represent a relatively simple algebraic instrument for practical verification of whether the pQCD scheme with a given beta function of the form (3) described in section 2 has Landau singularities or has no such singularities, and where these singularities are.

We will consider three specific cases of application of the above algebraic formalism: (a) when the β(F) function (3) has (cubic) polynomial structure and only real roots: M = 2, N = 0; ${t}_{1},{t}_{2}\in {\mathbb{R}};$ (b) β(F) has (cubic) polynomial structure and complex roots: M = 2, N = 0; ${t}_{2}={t}_{1}^{* }\,\rlap{/}{\in }\,{\mathbb{R}};$ (c) β(F) has a Padé structure with (one) pole: M = N = 1. Although these are cases with low indices M and N, we believe that they are representative to a certain degree, and show in practice how the presented formalism works. The cases (a) and (b) are specific low-index cases belonging to the set of beta-functions discussed in section 3.1 where Landau poles are expected to appear in the complex Q²-plane. The case (c) is a specific low-index case belonging to the set of beta-functions discussed in section 3.2 where a cut structure of Landau singularities is expected.

4.1. Polynomial β with real roots

Here we consider the case of (M = 2, N = 0)

$$\begin{eqnarray}\begin{array}{rcl}\beta (F) & = & -{\left.{\beta }_{0}{F}^{2}(1-Y)\times P[2/0](Y)\right|}_{Y\equiv F/{a}_{0}}\\ & = & -{\left.{\beta }_{0}{F}^{2}(1-Y)(1-{t}_{1}Y)(1-{t}_{2}Y)\right|}_{Y\equiv F/{a}_{0}},\end{array}\end{eqnarray} \tag{ 40 }$$

where t₁ and t₂ are real [and t₁, t₂ < 1 by physical requirements, cf. the text after equations (4a )]. In order to present numerical results, we choose specific numerical input values for a₀ (>0) and t₁ (which we choose to be positive)

$$\begin{eqnarray}&&{a}_{0}=0.4;\qquad {t}_{1}=+0.3.\end{eqnarray} \tag{ 41 }$$

The condition (5) then gives

$$\begin{eqnarray}&&{t}_{2}=-1-({\beta }_{1}/{\beta }_{0}){a}_{0}-{t}_{1}\,(\approx -2.0111),\end{eqnarray} \tag{ 42 }$$

where the numerical value is obtained by using in the universal β-coefficients β₀ and β₁ the number of active quark flavours N_f = 3 (β₀ = 9/4; β₁ = 4). The resulting renormalization scheme parameters c_j ≡ β_j /β₀ (j ≥ 2) are then c₂ = −14.4653, c₃ = 9.4271 and c₄ = 0 (in $\overline{\mathrm{MS}}$ scheme, for N_f = 3, they are: c₂ = 4.4711, c₃ = 20.990, c₄ = 56.588). ¹¹ The κ coefficient of equations (20)–(22) then has the value

$$\begin{eqnarray}&&\kappa =\displaystyle \frac{{\beta }_{0}{a}_{0}}{{B}_{0}}\approx 1.897.\end{eqnarray} \tag{ 43 }$$

Since t₁ and t₂ are real (hence: M = 2; P = 0), the only winding numbers (28) are $\vec{n}=({n}_{0},{n}_{1},{n}_{2})$, and thus $\mathrm{Im}{{ \mathcal G }}_{\vec{n}}({a}_{* })$ is independent of $\vec{n}$. The first condition of equation (30) then immediately gives for a_* (we recall: 0 < a_* < a₀)

$$\begin{eqnarray}&&\mathrm{Im}{{ \mathcal G }}_{\vec{n}}({a}_{* })=0\,\Rightarrow {a}_{* }\approx 0.320\,816;\end{eqnarray} \tag{ 44 }$$

and the corresponding x_* is ¹²

$$\begin{eqnarray}&&{x}_{* }=-5.03423,\end{eqnarray} \tag{ 45 }$$

as can be easily checked by the implicit solution (18) when using there for F(z) the value of a_* equation (44). When we now numerically integrate the RGE (1) along the line $\mathrm{Re}(z)={x}_{* }$ in the z-plane, ¹³ we obtain for the real and imaginary part of the coupling F(z = x_* + iy) the values presented in figures 2, which clearly show that there are singularities (poles) of the running coupling a(Q²) ≡ F(z) at ${z}_{* ,\pm }^{(j)}={x}_{* }\pm {{iy}}_{* }^{(j)}$

$$\begin{eqnarray}&&{y}_{* }^{(1)}=1.59783;\qquad {y}_{* }^{(2)}=1.71434.\end{eqnarray} \tag{ 46 }$$

The obtained points ${z}_{* ,\pm }^{(j)}$ are the Landau poles. We can cross-check that these points are really the Landau poles by evaluating the algebraic expression ${{ \mathcal G }}_{\vec{n}}({a}_{* })$, equation (26), for various winding numbers $\vec{n}=({n}_{0},{n}_{1},{n}_{2})$, and we find that

$$\begin{eqnarray}&&{{ \mathcal G }}_{\vec{n}}({a}_{* })=+{y}_{* }^{(1)}(=+1.59783)\qquad \mathrm{for}\,\vec{n}=(0,0,0);\end{eqnarray} \tag{ 47a }$$

$$\begin{eqnarray}&&{{ \mathcal G }}_{\vec{n}}({a}_{* })=-{y}_{* }^{(1)}(=-1.59783)\qquad \mathrm{for}\,\vec{n}=(-1,-1,0);\end{eqnarray} \tag{ 47b }$$

$$\begin{eqnarray}&&{{ \mathcal G }}_{\vec{n}}({a}_{* })=+{y}_{* }^{(2)}(=+1.71434)\qquad \mathrm{for}\,\vec{n}=(0,-1,0);\end{eqnarray} \tag{ 47c }$$

$$\begin{eqnarray}&&{{ \mathcal G }}_{\vec{n}}({a}_{* })=-{y}_{* }^{(2)}(=-1.71434)\qquad \mathrm{for}\,\vec{n}=(-1,0,0);\end{eqnarray} \tag{ 47d }$$

On the other hand, without the algebraic seminumeric approach described above, it would be difficult to find the (four) Landau poles on the first Riemann sheet of Q². In figures 3(a), (b) we present ∣β(F(z))∣ for the considered couplings, obtained by the 2-dimensional numerical integration of the RGE in the complex z-stripe. In figure 3(a) it is difficult to see that there are two Landau poles close to each other, at positive (and negative) values of $\mathrm{Im}(z)=y;$ only the strongly 'zoomed' figure 3(b) suggests that there are two poles near to each other, at ${z}_{* ,+}^{(j)}={x}_{* }+{{iy}}_{* }^{(j)}$ (j = 1, 2), as clearly obtained in equations (47a ) by our algebraic seminumeric analysis.

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4.2. Polynomial β with complex roots

Here we consider the case of (M = 2, N = 0)

$$\begin{eqnarray}\begin{array}{rcl}\beta (F) & = & -{\beta }_{0}{F}^{2}(1-Y)\times P[2/0](Y){| }_{Y\equiv F/{a}_{0}}\\ & = & -{\beta }_{0}{F}^{2}(1-Y)(1-{t}_{1}Y)(1-{t}_{2}Y){| }_{Y\equiv F/{a}_{0}},\end{array}\end{eqnarray} \tag{ 48 }$$

where t₁ and t₂ are complex nonreal and thus mutually complex conjugate $[{t}_{2}={({t}_{1})}^{* }$]. Since we want to present numerical results, we choose as an example the following specific input values:

$$\begin{eqnarray}&&{a}_{0}=0.5;\qquad \mathrm{Im}{t}_{1}=+0.60.\end{eqnarray} \tag{ 49 }$$

The condition (5) then gives

$$\begin{eqnarray}&&\mathrm{Re}{t}_{1}=-\displaystyle \frac{1}{2}\left(1+({\beta }_{1}/{\beta }_{0}){a}_{0}\right)\,(\approx -0.9444),\end{eqnarray} \tag{ 50 }$$

where, as in section 4.1, the numerical value is obtained by using in the universal β-coefficients β₀ and β₁ with N_f = 3 The renormalization scheme parameters c_j ≡ β_j /β₀ (j ≥ 2) are in this case c₂ = − 2.5477, c₃ = − 10.0158 and c₄ = 0. ¹⁴ The coefficient κ of equations (20)–(22), has now the value

$$\begin{eqnarray}&&\kappa =\displaystyle \frac{{\beta }_{0}{a}_{0}}{{B}_{0}}\approx 4.6585\end{eqnarray} \tag{ 51 }$$

Since t₁ and t₂ are complex nonreal (hence: M = 2; P = 1), the winding numbers (28) are $\vec{n}=({n}_{0},{N}_{0},{{ \mathcal N }}_{0})$, and thus $\mathrm{Im}{{ \mathcal G }}_{\vec{n}}({a}_{* })$ depends on ${{ \mathcal N }}_{0}$ and $\mathrm{Re}{{ \mathcal G }}_{\vec{n}}({a}_{* })$ depends on n₀ and N₀.

The first condition of equation (30) then gives for a_* the acceptable solution (i.e., in the interval 0 < a_* < a₀) only when ${{ \mathcal N }}_{0}\geqslant -1$

$$\begin{eqnarray}&&\mathrm{Im}{{ \mathcal G }}_{\vec{n}}({a}_{* })=0\,\Rightarrow \end{eqnarray} \tag{ 52a }$$

$$\begin{eqnarray}&&{a}_{* }\approx 0.492\,229({{ \mathcal N }}_{0}=-1);\,0.433\,899({{ \mathcal N }}_{0}=0);\,0.305\,849({{ \mathcal N }}_{0}=1);\,0.215\,326({{ \mathcal N }}_{0}=2);\,\mathrm{etc}.\end{eqnarray} \tag{ 52b }$$

and the corresponding x_* (we use ${\alpha }_{s}({M}_{Z}^{2};\overline{\mathrm{MS}})=0.1179$ as described in section 4.1) is obtained from the implicit solution (18) with F(z) = a_*

$$\begin{eqnarray}&&{x}_{* }=-4.73364({{ \mathcal N }}_{0}=-1);\,-4.18465({{ \mathcal N }}_{0}=0);\,-3.63565({{ \mathcal N }}_{0}=1);\,-3.08666({{ \mathcal N }}_{0}=2);\,\mathrm{etc}.\end{eqnarray} \tag{ 53 }$$

When we now perform the simple (1-dimensional) numerical integration of the RGE (1) along the line $\mathrm{Re}(z)={x}_{* }$ in the z-plane, we obtain for the real and imaginary part of the coupling F(z = x_* + iy) on the first Riemann sheet (∣y∣ ≤ π) singular structure only when ${{ \mathcal N }}_{0}=0$ (x_* = − 4.18465), with the values presented in figures 4. These figures clearly show that there are singularities (poles) of the running coupling a(Q²) ≡ F(z) at ${z}_{* ,\pm }^{(j)}={x}_{* }\pm {{iy}}_{* }^{(j)}$

$$\begin{eqnarray}&&{y}_{* }^{(1)}=0.67438;\qquad {y}_{* }^{(2)}=2.02315.\end{eqnarray} \tag{ 54 }$$

As in section 4.1, we conclude that the obtained points ${z}_{* ,\pm }^{(j)}$ are the Landau poles. We cross-check that these points are really the Landau poles by evaluating the algebraic expression ${{ \mathcal G }}_{\vec{n}}({a}_{* })$, equation (26), for various values of the winding numbers $\vec{n}=({n}_{0},{N}_{0},{{ \mathcal N }}_{0})$, and we find

$$\begin{eqnarray}&&{{ \mathcal G }}_{\vec{n}}({a}_{* })=+{y}_{* }^{(1)}(=+0.67438)\qquad \mathrm{for}\,\vec{n}=(0,0,0);\end{eqnarray} \tag{ 55a }$$

$$\begin{eqnarray}&&{{ \mathcal G }}_{\vec{n}}({a}_{* })=-{y}_{* }^{(1)}(=-0.67438)\qquad \mathrm{for}\,\vec{n}=(-1,0,0);\end{eqnarray} \tag{ 55b }$$

$$\begin{eqnarray}&&{{ \mathcal G }}_{\vec{n}}({a}_{* })=+{y}_{* }^{(2)}(=+2.02315)\qquad \mathrm{for}\,\vec{n}=(+1,0,0);\end{eqnarray} \tag{ 55c }$$

$$\begin{eqnarray}&&{{ \mathcal G }}_{\vec{n}}({a}_{* })=-{y}_{* }^{(2)}(=-2.02315)\qquad \mathrm{for}\,\vec{n}=(-2,0,0);\end{eqnarray} \tag{ 55d }$$

On the other hand, the fully numerical (2-dimensional) integration of the RGE (1) in the first Riemann sheet of the complex squared momenta Q² (i.e., in the complex z-stripe with $| \mathrm{Im}z| \leqslant \pi$) gives us the results in figures 5(a), (b) where we present ∣β(F(z))∣ for the considered couplings. In figure 5(a) it is hard to see two of the four mentioned Landau poles, namely those with $\mathrm{Im}(z)=\pm 2.023\,15$. Only the strongly 'zoomed' figure 5(b) suggests that there are Landau poles also at z = x_* ± i 2.023 15.

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4.3. Padé β with a real pole

Here we consider a numerical example for the types of β-function of section 3.2 where a finite-valued Landau branching point is realized. We will take the simplest case M = 1 and N = 1 (and P = 0) where β-function equation (3) has a Padé form with one real pole

$$\begin{eqnarray}&&{\left.{\left.\beta (F)=-{\beta }_{0}{F}^{2}(1-Y)\times P[1/1](Y)\right|}_{Y\equiv F/{a}_{0}}=-{\beta }_{0}{F}^{2}(1-Y)\displaystyle \frac{(1-{t}_{1}Y)}{(1-{u}_{1}Y)}\right|}_{Y\equiv F/{a}_{0}}\end{eqnarray} \tag{ 56 }$$

Here, both u₁ and t₁ are real and related via the relation (5). The β-function has a pole at the coupling value F(z_*) = a₀/u₁. We will present numerical results, so we choose as a representative example the following specific input values:

$$\begin{eqnarray}&&{a}_{0}=0.3;\qquad {u}_{1}=+0.50.\end{eqnarray} \tag{ 57 }$$

The condition (5) then gives

$$\begin{eqnarray}&&{t}_{1}={u}_{1}-\left(1+({\beta }_{1}/{\beta }_{0}){a}_{0}\right)=0.5-\left(1+({\beta }_{1}/{\beta }_{0}){a}_{0}\right)\,(\approx -1.0333),\end{eqnarray} \tag{ 58 }$$

where, as in the previous examples, we use the values of β₀ and β₁ with N_f = 3. The resulting renormalization scheme parameters c_j ≡ β_j /β₀ (j ≥ 2) are then c₂ = − 8.5185, c₃ = − 14.1975, c₄ = − 23.6626, etc. ¹⁵ The κ coefficient of equations (20)–(22) has in this case the value

$$\begin{eqnarray}&&\kappa =\displaystyle \frac{{\beta }_{0}{a}_{0}}{{B}_{0}}=2.7450.\end{eqnarray} \tag{ 59 }$$

Since in the considered case we have M = 1 and P = 0 (and N = 1), the only winding numbers are $\vec{n}=\{{n}_{0}^{(1)},{n}_{1}^{(1)}\}$. The (real) value of the coupling $F({x}_{* }^{(1)})={a}_{* }^{(1)}$ ($0\lt {a}_{* }^{(1)}\lt {a}_{0}$) is then obtained by the condition $\mathrm{Im}{{ \mathcal K }}_{\vec{n}}({a}_{* }^{(1)},{u}_{1})=0$, cf. equation (37), where $\mathrm{Im}{{ \mathcal K }}_{\vec{n}}({a}_{* }^{(1)},{u}_{1})$ is independent of the winding numbers $\vec{n}\equiv \{{n}_{0}^{(1)},{n}_{1}^{(1)}\}$. This then immediately gives us

$$\begin{eqnarray}&&\mathrm{Im}{ \mathcal K }({a}_{* }^{(1)},{u}_{1})=0\,\Rightarrow {a}_{* }^{(1)}=0.268253.\end{eqnarray} \tag{ 60 }$$

The corresponding value of ${x}_{* }^{(1)}$ is [we use ${\alpha }_{s}({M}_{Z}^{2};\overline{\mathrm{MS}})=0.1179$ as in section 4.1] is obtained from the implicit solution (18) with F(z) = a_* at $z={x}_{* }^{(1)}$

$$\begin{eqnarray}&&{x}_{* }^{(1)}=-4.38168.\end{eqnarray} \tag{ 61 }$$

Now performing the simple (1-dimensional) numerical integration of the RGE (1) along the line $\mathrm{Re}(z)={x}_{* }^{(1)}$ in the z-plane, gives us the real and imaginary part of the coupling $F(z={x}_{* }^{(1)}+{iy})$ on the first Riemann sheet (∣y∣ ≤ π) with the values presented in figures 6. These figures clearly show that, for $\mathrm{Re}(z)={x}_{* }^{(1)}$ ( = − 4.38168), there is a singular behaviour of the running coupling a(Q²) ≡ F(z) in the Riemann sheet only at the points ${z}_{* ,\pm }^{(j)}={x}_{* }^{(1)}\pm {{iy}}_{* }^{(1)}$ where

$$\begin{eqnarray}&&{y}_{* }^{(1)}\approx 1.144\,48.\end{eqnarray} \tag{ 62 }$$

On the other hand, the second condition in equation (37) should give us in this case the same values $\pm {y}_{* }^{(1)}=\pm 1.144\,48$. Indeed, the evaluation of the algebraic expression ${{ \mathcal K }}_{\vec{n}}({a}_{* }^{(1)},{u}_{1})$, equation (34), gives

$$\begin{eqnarray}&&{{ \mathcal K }}_{\{\mathrm{0,0}\}}({a}_{* }^{(1)},{u}_{1})\approx 1.144\,48,\qquad {{ \mathcal K }}_{\{-\mathrm{1,0}\}}({a}_{* }^{(1)},{u}_{1})\approx -1.14448.\end{eqnarray} \tag{ 63 }$$

This is consistent with the results (62), and clearly shows that in the considered case the Landau branching point is achieved in the first Riemann sheet only at the two complex conjugate points ${z}_{* }^{(1)}={x}_{* }^{(1)}\pm {{iy}}_{* }^{(1)}$ with ${x}_{* }^{(1)}=-4.38168$ and ${y}_{* }^{(1)}=1.14448$, and with the corresponding winding numbers $\{\vec{n}\}\equiv \{{n}_{0}^{(1)},{n}_{1}^{(1)}\}$ equal to {0, 0,} and {− 1, 0}, respectively. Further, figures (6) indicate that the coupling $F({z}_{* }^{(1)})$ at this point achieves the (real) value 0.60 which coincides with the value a₀/u₁, i.e., the value where β-function diverges (but not the coupling).

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The fully numerical (two-dimensional) integration of the RGE (1) in the first Riemann sheet of the complex squared momenta gives us the results in figures 7. Figure 7(a) shows ∣β(F(z))∣ and indicates the Landau singularities at ${z}_{* ,\pm }^{(1)}={x}_{* }^{(1)}\pm {{iy}}_{* }^{(1)}$. Figure 7(b) shows $\mathrm{Im}F(z)$ and indicates that the previously mentioned singularities are indeed branching points, with the cut in the complex-z stripe extending from ${z}_{* ,+}^{(1)}={x}_{* }^{(1)}+{{iy}}_{* }^{(1)}$ along the line $z={x}_{* }^{(1)}+{iy}$ with $y\geqslant {y}_{* }^{(1)}$, and the complex-conjugate cut from ${z}_{* ,-}^{(1)}={x}_{* }^{(1)}-{{iy}}_{* }^{(1)}$ along the line $z={x}_{* }^{(1)}+{iy}$ with $y\leqslant -{y}_{* }^{(1)};$ the same indication can be obtained when evaluating $\mathrm{Re}F(z)$ in the z-complex stripe. ¹⁶ For example, at $z={x}_{* }^{(1)}+i1.5$ we have numerically: $F({x}_{* }^{(1)}+\epsilon +i1.5)-F({x}_{* }^{(1)}+i1.5)\approx -0.189-i\,0.327$ (when ≈ 10⁻⁵–10⁻³).

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On the other hand, the algebraic seminumeric analysis above, equations (60)–(63) and figures 6, shows that the Landau singularities ${z}_{* ,\pm }^{(1)}={x}_{* }^{(1)}\pm {{iy}}_{* }^{(1)}$ are indeed branching points (with cuts) and correspond to specific winding numbers, and that no other branching points exist in the first Riemann sheet.

We present in figure 8 the behaviour of the couplings a(Q²) for positive Q² in all three cases considered in this section. This figure confirms that the considered class of running couplings has qualitatively similar behaviour in the regime Q² > 0, i.e., a(Q²) is a continuous and monotonically decreasing function of Q², with finite values in the IR limit at Q² = 0.

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Finally, we present in figures 9, for the case of the coupling of section 4.2, the discontinuity (spectral) function ${\rho }_{1}(\sigma )=\mathrm{Im}\,a({Q}^{2}=-\sigma -i\epsilon )$ and the corresponding timelike coupling ${ \mathcal H }(s)$ (s = q² ≡ − Q² > 0). The timelike coupling is defined in the usual form [127] (cf. also [19, 128, 129])

$$\begin{eqnarray}&&{ \mathcal H }(s)=\displaystyle \frac{1}{\pi }{\int }_{s}^{+\infty }\displaystyle \frac{d\sigma }{\sigma }{\rho }_{1}(\sigma ),\end{eqnarray} \tag{ 64 }$$

and fulfills the relation $\pi {sd}{ \mathcal H }(s)/{ds}=-{\rho }_{1}(s)$. We notice in figure 9(b) that ${ \mathcal H }(0)\approx 0.168$ which is less than a(0)( ≡ a₀) = 0.5; this is a consequence of the Landau singularities of the coupling a(Q²). Only if a(Q²) had no Landau singularities, would we obtain ${ \mathcal H }(0)=a(0)$ [19].

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In this work we presented an algebraic algorithm for finding possible Landau singularities of the pQCD running coupling a(Q²) in the complex plane of the squared momenta Q² (first Riemann sheet). We considered a large class of β-functions, representative of the scenarios where the running coupling a(Q²) is a monotonic function of Q² at positive Q² and 'freezes' in the IR sector, a(Q²) → a₀ for Q² → 0, where the IR freezing value a₀ is considered positive finite. The consideration of the running coupling a(Q²) ≡ F(z) was performed on the corresponding complex z-stripe, $-\pi \leqslant \mathrm{Im}(z)\lt \pi$, where $z=\mathrm{ln}({Q}^{2}/{Q}_{\mathrm{in}}^{2})$ and ${Q}_{\mathrm{in}}^{2}\gt 0$ was an initial scale for the integration of the RGE. The analysis was performed by explicit integration of the RGE which led to the implicit (inverted) solution of the form $z={ \mathcal H }(F)$. An analysis of this implicit solution than led us to an algebraic procedure for the search of the Landau singularities of F(z) on the z-stripe. We considered two types of such singularities, the poles F(z) = ∞ and the branching points (for cuts) β(F(z)) = ∞. For illustration, we then presented the mentioned algebraic (and seminumeric) analysis for three specific representative cases of the β-function, and compared the found Landau singularities with those seen directly by the numerical 2-dimensional integration of the RGE in the entire complex z-stripe, the latter approach being numerically demanding. The presented specific cases suggest that our algebraic seminumeric approach is reliable and has high precision in finding the Landau singularities, while the 2-dimensional integration of the RGE gives these singularities with less precision and sometimes we may miss some of the singular points with this purely numerical method, especially if the numerical scanning over the entire z-stripe is made with limited density. Therefore, the presented algebraic seminumeric formalism appears to be useful when we want to find out whether the pQCD running coupling has Landau singularities, and if there are any, to find the location of these singularities with high precision.

This work was supported in part by the FONDECYT (Chile) Grants No. 1191434 (C.C.), 1180344 (G.C.) and 1181414 (O.O.).

The data that support the findings of this study are available upon reasonable request from the authors.