On the perturbative aspects of deformed Yang-Mills theory

Centre-stabilised $SU(N)$ Yang-Mills theories on $\mathbb{R}^3 \times S^1$ are QCD-like theories that can be engineered to remain weakly-coupled at all energy scales by taking the $S^1$ circle length $L$ to be sufficiently small. In this regime, these theories admit effective long-distance descriptions as Abelian $U(1)^{N-1}$ gauge theories on $\mathbb{R}^3$, and semiclassics can be reliably employed to study non-perturbative phenomena such as colour confinement and the generation of mass gaps in an analytical setting. At the perturbative tree level, the long-distance effective theory contains $(N-1)$ free photons with identical gauge couplings $g^2_3 \equiv g^2/L$. Vacuum polarisation effects, from integrating out heavy charged fields, lift this degeneracy to give $\floor{\frac{N}{2}}$ distinct values: $g^2(\frac{2}{L})\lesssim g_{3,\ell}^2 L \lesssim g^2(\frac{2\pi}{NL}) $. In this work, we calculate these corrections to one-loop order in theories where the centre-symmetric vacuum is stabilised by $2\leq n_f \leq 5$ massive adjoint Weyl fermions with masses of order $m_\lambda \sim \frac{2\pi}{NL}$, (also known as"deformed Yang-Mills,") and show that our results agree with those found in previous studies in the $m_\lambda \to 0$ limit. Then, we show that our result has an intuitive interpretation as the running of the coupling in a"lattice momentum"in the context of the non-perturbative"emergent latticised fourth dimension"in the $N\to \infty$, fixed-$NL$ limit.

2 distinct values: g 2 ( 2 L ) g 2 3, L g 2 ( 2π N L ). In this work, we calculate these corrections to one-loop order in theories where the centre-symmetric vacuum is stabilised by 2 ≤ n f ≤ 5 massive adjoint Weyl fermions with masses of order m λ ∼ 2π N L , (also known as "deformed Yang-Mills,") and show that our results agree with those found in previous studies in the m λ → 0 limit. Then, we show that our result has an intuitive interpretation as the running of the coupling in a "lattice momentum" in the context of the non-perturbative "emergent latticised fourth dimension" in the N → ∞, fixed-N L limit.

Contents 1 Introduction
Analytical methods to study the long-distance properties of four-dimensional asymptotically free non-Abelian gauge theories are few and far between; broadly speaking, it is a difficult problem to handle because the flow to strong coupling causes theoretical control over the system to be lost at lo-energy scales. While there are known models that are well-behaved enough to be studied analytically, (e.g., Seiberg-Witten theory [1],) these typically require special structures such as supersymmetry, or otherwise make use of gauge-gravity duality arguments and string-inspired tools (such as in Ref. [2]).
Over the past years, studies performed on "centre-stabilised" gauge theories on R 3 × S 1 have been remarkably fruitful for providing insight into the non-perturbative dynamics of four-dimensional gauge theories. These models are distinguished from the few known analytically-calculable models in four dimensions by the fact that they can be engineered to remain weakly-coupled at all energy scales, so that a semiclassical expansion in terms of objects defined in the UV theory is reliable and self-consistent.
The basic idea behind these models is as follows: by compactifying R 4 to R 3 × S 1 , and "deforming" the pure Yang-Mills (YM) theory by adding a non-local and non-renormalisable potential to the Lagrangian, the well-known deconfining phase transition (cf. thermal Yang-Mills [3],) at small circle lengths L can be circumvented, and the theory remains in the colour-confining phase for all values of L. Adiabatic continuity to the full R 4 theory of ultimate interest can therefore be argued on grounds that the theories share identical (non-spacetime) global symmetries for all L ∈ [0, ∞]. That is, they belong in the same "universality class" [4,5].
To be certain, the non-renormalisable "deformed" theory that we are describing can be viewed as a lattice theory with a fixed finite lattice spacing [6]. On the other hand, it is also possible to define a UV-complete continuum theory with the same desired properties by introducing n f S 1 -periodic adjoint-representation fermion fields to the pure YM Lagrangian: The desired deformation potential is realised as the fermionic contribution to the dynamically-generated Gross-Pisarski-Yaffe (GPY) effective potential at energy scales below ∼ 1 L [4,5,7,8]. If the fermions are massless, 1 this class of theories is referred to as QCD(adj) if 2 ≤ n f ≤ 5, and super Yang-Mills (SYM) if n f = 1. It is called "deformed Yang-Mills" (dYM), when the 2 ≤ n f ≤ 5 fermions are massive, or if the deformation potential is added "by hand," as in the lattice formulation.
From the theorist's perspective, one of the most alluring features of these admittedly artificial setups is that they admit a "weak-coupling regime" at ΛN L 2π, (where Λ is the strong-coupling scale,) in which the gauge coupling g 2 (and more pertinently, g 2 N ,) remains small at all energy scales. Thus, in this regime, the semiclassical expansion over high-energy monopole-instanton configurations is trustworthy, and can be reliably employed to study the effects of the non-perturbative physics on the low energy theory. The result is a theoretical laboratory in which a wide variety of non-perturbative low-energy phenomena can be studied analytically: For example, colour confinement, the generation of a nonperturbative mass gap [4,8], a deconfining phase transition [9,10], and certain aspects of chiral symmetry breaking [6]. For this reason, ΛN L 2π is sometimes also called the "calculable regime" in the context of centre-stabilised R 3 × S 1 theories. For comprehensive reviews, see Refs. [11][12][13].
At leading perturbative order and finite N , the IR effective theory of SU (N ) dYM and QCD(adj) in the calculable regime is sometimes described as being "rather boring" [11], because the gauge sector describes (N − 1) free, massless photons in R 3 . When vacuum polarisation effects are accounted for, the photons acquire N 2 . These corrections have been calculated to one-loop order by various methods when the fermions are assumed to be massless: for N = 2, 3 QCD(adj) in Ref. [14], for SYM (i.e., n f = 1) with arbitrary N in Ref. [10], and in QCD(adj) with arbitrary N in Ref. [15].
In this study, we derive a more general expression for these corrections that in particular covers the massive fermion case, for generic masses m 1 , ... m n f such that the theory remains in the centre-symmetric and weak-coupling regime. The final result is contained in Equation (2.13), and the bulk of our exposition explains how we arrive at this result. Our motivating aim is to confirm that the perturbative corrections to finite-N SU (N ) dYM theory yield no unpleasant surprises even when the stabiliser fermions are assumed to be heavy. This is a very reasonable assumption to make, since ultimately we are interested in obtaining insight on pure Yang-Mills on R 4 , and a continuum QCD-like theory that continues smoothly to pure YM should not contain light adjoint fermions in its IR spectrum.
Nevertheless, our results are not entirely devoid of novelty: Ref. [16] showed that in the N → ∞, L → 0, fixed-N L limit of SYM, an emergent latticised fourth dimension appears, emerging out of the space of fields -even though we should expect that taking L → 0 ought to result in a 3d theory. In particular, this emergent dimension exhibits z = 2 Lifschitz scaling invariance in SYM. In other words, the action is quartic, rather than quadratic, in the momentum: ∼ |∂ 2 y Φ| 2 , where ∂ y is the partial derivative in the emergent latticised dimension. Simply put, this is because in SYM, there is a discrete Z N chiral symmetry (not to be confused with the Z N centre symmetry,) that forbids monopoleinstantons from contributing a bosonic potential of the form ∼ |∂ y Φ| 2 in the semiclassical expansion. Such a potential is permitted, however, when the chiral symmetry is explicitly broken by a non-zero fermion mass, as is in the case we study here. We find a satisfying and intuitive interpretation of our massive correction in this emergent dimension as the flow towards strong coupling for large values of the "lattice momentum." The rest of this paper is structured as follows: Section 2 contains an overview of the essentials of dYM theory in an effort to make this paper more self-contained. For the benefit of the impatient reader, we have placed our main result, Equation (2.13) and its accompanying discussion, in Section 2.1.1. A discussion of this result in the context of the emergent latticised dimension of Ref. [16] is contained in Section 2.2.1.
Section 3 covers the derivation of Equation (2.13) in detail, starting from the very beginning with the UV dYM Lagrangian. Since this calculation is fairly long and convoluted, we briefly summarise what we have done at the end of subsections 3.1 and 3.2 to help the reader keep track of our progress. The main "meat" of the calculation, and therefore of this paper, is mostly contained in Section 3.3, especially Section 3.3.2.
In our calculation, we use the Mellin transform to rewrite certain infinite sums in a form that allows their asymptotic behaviour to be more easily seen. The details of this manipulation, which is mostly just complex analysis in one variable, is given in Appendix A.
2 Background, results and discussion 2.1 Review of dYM: I. Perturbative aspects Consider pure SU (N ) Yang-Mills theory on compactified R 3 × S 1 , where the S 1 is a circle of circumference L: This theory enjoys a global Z N = Z(SU (N )) centre symmetry, as it only contains fields transforming in the adjoint representation of the gauge group. The action of this symmetry may be thought of as a "gauge" 2 transformation g(x µ , x 4 ) : R 3 × S 1 → SU (N ) that is periodic over the S 1 modulo a Z N factor: This acts on the fundamental representation Polyakov loop Ω, the gauge holonomy along the S 1 , where A 4 is the S 1 part of the gauge field A. At large L, the centre symmetry is unbroken. That is to say, trΩ n = 0 in the ground state for all n = 0 mod N . On the other hand, it is a well-known fact [3] that in the small-L limit, the theory undergoes a deconfining phase transition associated with the breaking of centre symmetry: In this regime, where perturbative analyses can be trusted because of asymptotic freedom, Ref. [3] showed that the theory produces a (GPY) effective potential, V pert. [Ω]: This result can be found by integrating out the Kaluza-Klein modes at one-loop order. This potential is minimised by Ω of the form Ω = ω k 1 N for any integer k, suggesting that the theory has N degenerate vacua related by the Z N symmetry and describes a gluon plasma phase. The basic idea behind R 3 × S 1 theories such as dYM is to re-enforce the stability of the Z N at small L by "flipping" the shape of the GPY potential, so to speak. This can be done in the most direct way by simply adding a "double-trace" term to the YM action: But such a term is manifestly non-local, being defined in terms of a non-local operator. It is also non-renormalisable, as it contains infinitely many irrelevant operators which blow up uncontrollably in the UV. As such, such a deformation of the theory may be considered problematic to those with a philosophical preference for continuum theories. We will return 2 Though, of course, the g(x µ , x 4 ) so defined is not a true gauge transformation by any means. That is, it is not a transition function between local trivialisations of the principle bundle. 3 According to the modern viewpoint, this ZN belongs to a class of "generalised" global symmetries, which act on operators with non-trivial spatial extent. In this context, Equation (2.4) defines the symmetry [17] to address this objection later, and focus on the effects of the double-trace deformation potential on the IR theory for now. The Z N symmetry is said to be preserved if and only if the vacuum state of the theory satisfies trΩ n = 0 for all n = 0 mod N , so the coefficients a n > 0 in Equation (2.6b) must each be chosen so as to dominate the dynamically generated V pert.
[Ω]. With the centre symmetry stabilised, we can remove the gauge redundancy of Ω by choosing a diagonal representative from the class of physically equivalent minima: This choice is in fact unique, up to permutations of the coefficients corresponding to Weyl reflections that can also be gauged-fixed away by working in the (affine) Weyl chamber. This allows us to write Ω as a physically meaningful expectation value despite its uncontracted matrix indices. 4 This vev precipitates a simulacrum of the Higgs mechanism in which the A 4 field plays the role of an adjoint Higgs field. The gauge group generators left unbroken by Ω form a Cartan subalgebra t ⊂ su(N ), generating the maximal torus U (1) N −1 ⊂ SU (N ). The Higgs mechanism endows fields in t ⊥ (i.e., fields that carry charge under the U (1) N −1 ) with an effective mass ≥ 2π N L ≡ m W , the so-called Abelianisation scale. We can now perform the path integral around the centre symmetric vacuum: Working perturbatively, (the treatment of the non-perturbative physics is left to Section 2.2,) weakcoupling ensures that the A 4 fluctuations around Ω , of mass g 2 N m W , can only effect small corrections to the effective action. Weak-coupling, in turn, is guaranteed by the weak-coupling assumption m W Λ -meaning that all dynamic charged fields can be safely integrated out before the onset of strong coupling as we carry the theory towards the infrared. When the dust settles, we are left with a weakly-coupled U (1) N −1 gauge theory containing no light charged fields in its spectrum. In fewer words: everything works out fine.
In settings where it is desirable to have a theory that respects both locality and UVcompleteness, and yet preserve centre symmetry at all scales, we can opt to have V deformed [Ω] generated dynamically as well, by adding sufficiently light, or massless, S 1 -periodic adjoint fermions λ I to the theory [8], rather than inserting the deformation potential "by hand." In such a setting, the periodicity requirement λ I (x µ , x 4 ) = +λ I (x µ , x 4 + L) prohibits a thermal interpretation for the S 1 , which must therefore be taken to be a spatial circle.
For the theory with 1 ≤ n f ≤ 5 Weyl fields 5 of masses m I indexed by 1 ≤ I ≤ n f , the dynamically generated (GPY) effective potential is [5,10,18] (2.9) 4 To be certain, the action of the centre ZN in this gauge is, with Ωi denoting the i'th diagonal of Ω [13], (2.8) The cyclic permutation is necessitated by gauge-fixing to the Weyl chamber. 5 The theory loses asymptotic freedom for n f > 5.
where K 2 is the modified Bessel function of order 2. It is not hard to find constraints on the m I for each value n f that stabilise the centre-symmetric Ω in Equation (2.7); see e.g., Ref. [18]. We also note in passing that the n f = 1 potential vanishes (in fact, to all perturbative orders,) in the massless case, and is centre-unstable otherwise. This particular case is known as super Yang-Mills (SYM), in which the UV theory enjoys an exact N = 1 supersymmetry, which allows many aspects of its rich non-perturbative physics to be calculated exactly. But SYM is outside of the scope of this study, along with the massless QCD(adj), and we henceforth only consider 2 ≤ n f ≤ 5 and m I > 0.
Assuming the fermion masses to be roughly equal, it turns out that centre stability requires m I m W . In particular, this means that we can assume that the m I are O(m W ) so that the fermions disappear from the low-energy theory, and the effective action can be written on R 3 as: κ ab F a µν F b µν + (A 4 and higher order terms) (2.10a) for Abelian field strengths F a µν and R 3 indices µ, ν ∈ {1, 2, 3} and Lie algebra indices a, b ∈ {1 ... N }. There is also a a neutral scalar field A a 4 in the IR theory, which descends from the Abelian part of A 4 and corresponds to the oscillations of the eigenvalues of Ω around the centre-symmetric vev. But this field receives a (mass) 2 ∼ g 2 N m 2 W correction from the GPY potential and can be integrated out by moving the theory to still lower energies.
The quantity κ ab in Equation (2.10a) is the quantum-corrected photon coupling matrix: where in Equation (2.10b), the gauge coupling is normalised with respect to the L → ∞, m I → 0 limit: and b 0 is the one-loop coefficient of the beta function of (g 2 N ) −1 in that limit. As stated before, these corrections have been calculated in previous studies for arbitrary N and 1 ≤ n f ≤ 5 in the limit m I = 0. Our calculation generalises to the massive case, and is a new result. We also believe it to be a non-trivial problem in terms of significance (as we will argue in this Section,) as well difficulty (which we will demonstrate in the next).

The one-loop corrections to κ ab
Compared to the writing out the matrix entries of κ ab explicitly in the Cartan-Weyl basis, (given in Equation (3.46),) it is more enlightening to present its eigenvalues, κ : N a,b=1 where again ω = e i 2π N , so Equation (2.12) are really just discrete Fourier transforms in the indices a, b. Then, assuming centre-stabilising fermion masses m I , 6 (2.13) pure function of the masses m I in units of m W , which enjoys the following properties: A plot of W as a function of N for a few select values of m is given in Figure 1. Equation (2.13) is written so that all the information about the one-loop corrections due to the fermion masses is encoded in the pure function W . In particular, Equation (2.14d) implies that the = 1 mode receives a vanishingly small mass correction in the N → ∞ limit; conversely, Equation (2.14e) implies the N/2 mode receives the largest mass correction. Equations (2.14a) and (2.14b) together imply that κ 1 ≥ κ ≥ κ N/2 for all m I and . Note that in order for our results to make sense, we must require κ N/2 > 0; we will discuss the conditions that fulfill this requirement later.
We present two analytic expressions for W with different convergence properties: one expression holds for m < m W , and the other, for m m W . The former of these is: for (m/m W ) < 1, where ζ is the Riemann zeta function, and Li s is the polylogarithm function of order s. This expression fails to converge when m > m W 7 ; it is in this regime where our second 6 Please note that the = N mode must be excluded from the spectrum as it corresponds to the trace of  expression is more useful: K 0 is the modified Bessel function of order 0. Equation (2.15b) is one-loop exact for all m, but it is more useful at large m m W , where it may be very well approximated by the first term of the series, as K 0 (t) ∼ e −t at large t. Conversely, Equation (2.15b) is less useful at small m/m W as K 0 (t) ∼ log t at small t.
Since W ≤ 0, the massive correction competes against the massless-limit corrections encoded in the log-sine term. Indeed, by taking m I m W , the I'th fermion decouples from the theory 8 , n f → (n f − 1), up to an overall renormalisation of g 2 N , or, equivalently, a re-definition of the strong-coupling scale Λ.
We can also use Equation (2.11) to define a "lattice-renormalised" 't Hooft coupling λ : Observing that the terms in the square brackets of (2.15a) are absolutely bounded for all n ≥ 1, and that (2n)! (n!) 2 the root test gives a radius of convergence of (m/mW ) < 1. 8 Assuming, of course, that the theory still remains in the centre-symmetric regime.
The dependence of κ on N is illustrated in Fig. 2 for a few sample values of m λ , for fixed nf = 4 and m W = e 3 Λ. Given a fixed value of m W /Λ = 2π ΛN L 1, it is a straightforward exercise in numerical analysis to find constraints on m λ so that κ N/2 > 0 in order for our result to make sense. Conversely, Z N -stability requires that e.g. m λ 1.08m W for n f = 4, and this bound can in particular be saturated by taking m W /Λ e 3 . On the other hand, Z N stability requires m λ 1.2m W for n f = 5, and saturation of that bound would require m W /Λ e 9 , which is substantially larger.
Equations (2.13), (2.15a), and (2.15b) comprise the main results of this paper; they are derived in detail in Section 3, with reference to some results from Appendix A. The rest of this Section discusses how to interpret these results in the context of the non-perturbative physics of dYM theory, particularly with regards to the "emergent latticised dimension" of Ref. [16].

Review of dYM: II. Non-perturbative aspects
Let us now very quickly summarise the derivation of the low-energy effective Lagrangian in dYM theory at leading order in the semiclassical expansion. The basic idea is essentially the same as Polyakov's version of confinement in the Georgi-Glashow (GG) model in 2 + 1 dimensions [19], although there are crucial differences due to the intrinsically fourdimensional nature of dYM theory. The reader interested in a more detailed exposition is referred to Refs. [4,20,21].
The contribution of the non-perturbative physics to the path integral in a weaklycoupled Euclidean QFT can be approximated to first exponential order by summing over classical field configurations that are inundated by a "gas" of weakly interacting minimalaction instantons. This is the so-called dilute instanton gas approximation, and it is applicable in dYM because weak coupling can be reliably assumed to hold at all scales provided that N LΛ 2π. In addition to a topological charge Q ∼ tr F ∧ F = 1 N , the instantons of dYM theory carry a magnetic charge ( ∼ F ) under the U (1) N −1 -they are essentially 't Hooft-Polyakov monopoles, with A 4 again standing in for the adjoint Higgs field. In particular, we call them monopole-instantons.
Among these, there are (N − 1) "BPS" 9 monopoles, each carrying a magnetic charge corresponding to a simple root α i of the gauge group. In distinction to the 3-dimensional Polyakov model, in R 3 × S 1 theories there is also an N 'th "twisted," or Kaluza-Klein, (KK) monopole, which carries charge N −1 i=1 (−α i ) ≡ α N , the affine root. In addition to these, there are also the antiparticles carrying charge −α i . In a sense, the (N − 1) BPS+KK monopole-instantons can be thought of as the "dissociation" of the BPST instanton in 4-dimensional SU (N ) Yang-Mills into N sub-constituents [22,23].
We unfortunately do not have exact expressions for the monopole-instantons outside of the supersymmetric n f = 1, m = 0 case. But as it turns out, they will not be required as far as our presentation is concerned: we need only know that these charged objects interact with a long-range Coulombic interaction, and have a non-linear "A 4 /Higgs condensate" core of size ∼ m −1 W . In addition, there is also a "medium-range" (∼ 1/g √ N m W ) Yukawa interaction arising from A 4 /Higgs exchange.
Every insertion of a monopole-instanton in the path integral comes with three translation zero modes and a Boltzmann suppression factor is the one-monopole action. This means the typical monopole-instanton separation d ∼ e S 0 /3 is much greater than the monopole diameter ∼ m −1 W , allowing us to ignore the contribution from paths with overlapping monopole-instanton cores. It also means that we can ignore the effects of A 4 exchange. The proliferation of magnetic charges in the vacuum gives rise to a potential for the photon. This potential which is most conveniently described in terms of the dual photon σ a , defined as Written in terms of σ a , the IR behaviour of dYM theory is described to first order in the semiclassical expansion, by the 3d Lagrangian L 3d,dual : where σ N +1 ≡ σ 1 , and κ −1 ab is the inverse 10 of κ ab , and ζ is the monopole fugacity: This is a common abuse of terminology: outside of the supersymmetric case, the BPS bound cannot be saturated because the "Higgs" potential V [Ω] cannot be set to zero. So strictly speaking we are expanding around "almost-BPS" configurations. 10 Actually, it should be the pseudoinverse since the eigenvalue corresponding to the = N mode diverges, but the difference is immaterial since the = N mode is unphysical.
pre-exponential factor. 11 In the Fourier basis, the 3d dual photon Lagrangian is, to quadratic accuracy in the fields, whereσ is the discrete Fourier transform of σ a : From this expression we can read off the dual photon masses-squared: Let us take ΛN L to be sufficiently small so that 2 3 n f λ W 1 can be treated as a small correction for all . In that case, we can write a mass-corrected expression for the scaling behaviour of the k-wall thicknesses. Recalling (2.16a), The multiplicative sine factor is the expected tree-level scaling behaviour; the factor of (λ 1 /λ k ) 1/2 is due to the one-loop corrections in the massless limit, and the factor in the square brackets gives the massive correction. The dependence of m 2 σ, on in units of m 2 σ,N/2 is graphically depicted in Fig. 3. 11 As an aside, let us note that calculating the pre-factor A({mI }, n f ) is a highly non-trivial open calculation, and has only been performed in the SYM case, first in [24], and later corrected in [9,10]. This is because it involves matrix determinants in a monopole-instanton background, for which we do not even have an exact analytic expression, as mentioned. We make no attempt to calculate A here.

Emergent dimension at large N : a 4d interpretation of the mass correction
Let us now consider the large-N limit. To do this, we simultaneously take N → ∞ and L → 0 whilst keeping N L constant so as to stay inside the weak-coupling regime. 12 This is known as the "Abelian large-N limit" [11]. In this setup, we can treat N ∈ [0, 1] as though it were on a continuum, and the potential in Equation (2.20) has an interpretation as the kinetic energy on a latticised and compact fourth dimension, with a quadratic (as opposed to quartic, as is the case in SYM,) dependence on a lattice momentum p y . But what is the scale of this momentum? Since the mass gap for the dual photon m 2 σ, 1 vanishes in the large-N limit, the only remaining mass scale to characterise the low-energy theory is m 2 σ, N/2 ≡ m N/2 , the (Debye) mass of the heaviest dual photon: up to small corrections. This allows us to define p y, as an honest-to-goodness lattice momentum: p y, ≡ m N/2 sin π N . (2.25) We can also read off the two-point function directly from (2.20): Defining x M ≡ ( x, y), p M ≡ ( p, p y ), and momentarily disregarding the massive correction, We observe that there is a restored Lorentz symmetry which is broken by an anomalous scaling dimension ∆ = b 0 λ as λ ∼ p b 0 λ y . Put another way, the dual photon coupling λ exhibits logarithmic running in the lattice momentum p y ∼ sin(π N ): In particular, the scaling behaviour is opposite to that of the R 4 theory (cf. Equation (2.11)): We can also show how this analogy can be extended to encompass the mass-correction terms ∼ W : The one-loop correction to the coupling due to a single adjoint fermion with mass m in an SU (N ) theory on R 4 , renormalised at some scale µ in the M S scheme is, (cf. Equation (3.38)): where x is a Feynman parameter. We can compare this with our result of the contribution in the R 3 × S 1 theory, which can be read off from (2.13): This result is consistent with our interpretation of p y as a momentum, with the mass correction behaving as we should expect in the R 4 theory, albeit with opposite momentumscaling behaviour in p y .

Perturbative analysis: theory and practice
The remainder of this paper mainly focuses on deriving and calculating loop integrals and Matsubara sums. Our approach is extremely straightforward -essentially identical to the analysis of a thermal gauge theory at temperatures T = 1/L, but for the fact that our S 1 is spacelike rather than timelike. This means, in particular for the fermions, that the S 1 momenta ω n assume integer values ω n = 2πn L , rather than half-integer ω n = 2π L (n + 1 2 ). As the calculation is rather involved, our presentation will try to go into as much detail as we can without being overly cumbersome. For the convenience of the reader, we will summarise the contents of Sections 3.1 and 3.2 at the end of their respective Sections.
We also define the (over-complete) Cartan-Weyl basis on su(N ): which span the Cartan subalgebra t. These are accompanied by the raising and lowering operators spanning t ⊥ , the orthogonal complement of t,  Perhaps a bit idiosyncratically, we say that the subscripts on β ij are a set of antisymmetric indices labelling the roots of su(N ): β ij = −β ji , and the superscript a denotes its a'th vector component. E β ij , E −β ij are respectively raising and lowering operators for the su(2) subalgebra associated with the root β ij : (no sums over i, j). We also have These generators are normalised as: In the interest of brevity, we will frequently abuse notation and treat β as though it were the index on the root space and omit the subscripts ij, as we have just done above. To avoid confusion, there will be no implicit sum over su(N ) indices unless otherwise specified,.
As a matter of convenience, we normalise the components of su(N )-valued fields ψ as: obeying hermiticity conditions:

3b)
and constrained by a trace-free condition: so that the expansion (3.3a) is unique although it is written in terms of an over-complete basis.

Formal setup: beginnings
Let us start with a 4-dimensional Euclidean SU (N ) gauge theory with non-Abelian field strength F M N and n f two-component massive adjoint fermions λ I . As we are performing a perturbative calculation, the vacuum angle is "invisible" to us, so we might as well set the fermion masses to be real and the topological angle θ = 0:

(3.4)
∇ M is the covariant derivative on adjoint-representation fields: andσ M = ( i σ , 1 2 ) are the Euclidean sigma matrices. Formally integrating out the high-energy ( m W ) degrees of freedom around the centresymmetric Ω gives us the effective 3d Lagrangian, (2.10a).
We want to explicitly integrate out the high-energy ( m W ) degrees of freedom to obtain the effective 3d Lagrangian, (2.10a) to find the one-loop corrections to κ ab , the photon coupling matrix. The methods we use can also be applied almost verbatim to find ρ ab , the corrected scalar couplings, as well as M ab the scalar masses. Since these are not as interesting to us, we simply quote their Fourier-transformed results in Equations(3.55) and (3.56).
Following Abbott's approach, (e.g., Ref. [25],) we use an adapted background field gauge method to calculate vacuum polarisation. This is fairly standard textbook material, but to review: first, we treat the su(N )-valued gauge field A M as the sum of a "classical" background field and a "quantum" high-frequency field: (3.6) The normalisation is for convenience. We say that these fields have two complementary expressions of gauge symmetry: for U,Ũ : Anticipating a 3d and Abelian theory, we take A M to be Abelian and trivial over x 4 , and call its field strength F M N : We want to fix the gauge underŨ in order to integrate out a M , which, as we will see, are basically W -bosons. To do this, we would impose the condition where D M is the "covariant derivative with connection A M " : and a Lagrangian L c for scalar-yet-Grassmannian su(N )-valued ghost fields c,c.
Since A 4 has a non-zero vev, we must write and A 0 4 represents the fluctuations around the vev. But we can mostly ignore A 0 4 as it is U (1) N −1 -neutral and therefore not involved in the corrections to κ ab at one-loop. Gaugefixing the centre-symmetric Ω as in Equation (2.7), φ has vector components: where the sum in Equation (3.12) is over the positive roots. In particular, this means All together, the gauge-fixed Lagrangian has the form: classical fields where L a contains the ∼ Aaa, AAaa terms upon expanding L 4d in terms of A M and a M , and similarly for L c and L λ . We observe that by choosing A M to be Abelian, the Abelian parts of each of the quantum fields a, λ, c,c cannot contribute to κ ab at one-loop order, so we may forget about them altogether for the rest of this analysis. Let us take any su(N )-valued field ψ and simultaneously expand in the KK modes and the Cartan-Weyl basis, recalling our convention as in Equation (3.3a), so asymptotically, the derivative operator iD 4 diagonalises with eigenvalues so that fields in t ⊥ with charge β and circle momentum 2πz/L of a field with mass m obtains an effective 3d (mass) 2 : Thus only U (1) N −1 -neutral and x 4 -trivial fields survive in the IR theory at scales m W , consistent with our hypotheses on A M .

Summary of Section 3.1:
We outlined the background field approach to perturbation theory: with an eye toward the infrared theory, we set the background A M to be x 4 -trivial and Abelian, and showed that only "quantum fields" proportional to the broken gauge generators may contribute to the corrections of κ ab at one-loop order. We further showed that this assumption is selfconsistent, because all fields with non-vanishing x 4 momentum or carrying charge under the U (1) N −1 acquire an effective mass ≥ m W through the Higgs mechanism.

The 1-loop Wilsonian action
We can write an expression for the Wilsonian effective action Γ[A] by formally integrating out the quantum fields under the path integral sign. Setting the vacuum energy to zero,  There is a lot of notation to define in Equation (3.18), but it will make life easier by formatting the problem so that the entire non-trivial part of the calculation is contained in the single expression "Tr log(−D 2 (s) + m 2 fs,s )," which we will only have to evaluate once to cover all the relevant cases, rather than having to work with massive or massless, spinor, scalar, and vector integrals separately.
"Tr" refers to the trace over the respective Hilbert spaces, and −D 2 (s) is a differential operator defined in Equation (3.21). The terms on the third row of Equation (3.18) are due to the W -bosons a, (s = 1,) the gauge ghosts c,c, (s = 0,) and the fermions λ I (s = 1/2). The s = 1/2 term is obtained by doubling then halving the trace-log of the massive Weyl operator: fs is a sum over flavour indices I when s = 1/2; m 2 fs,s = 0 for s = 1/2. χ(s) is a pre-factor determined by the statistics of each field:  To define −D 2 (s) , let A, B denote indices in the spin-s irrep of the (Euclidean) Lorentz group, then    There is no ∼ Σ F Σ 1 cross-term because the trace of σ

(3.25)
We note in passing that the GPY potential V [Ω] still appears in Equation (3.25) through M 2 ab , its second derivative. Now we are ready to draw some Feynman diagrams. Let p M = (p µ , 0) denote the external momentum of A M , and for convenience, define an effective loop momentum K M (β,z) .
These integrals are pictorially represented by the Feynman diagrams in Fig. 4. In each of the integrals above we have employed a Feynman parameter x, and shifted our loop momentum K M → K M − xp M . We have also defined an effective (mass) 2 , ∆ s (not to be confused with the 3d effective (mass) 2 in Equation (3.17),) where the traces are over the spin indices, which we have omitted. The rest of our report will be largely concerned with evaluating these three integrals.

Summary of Section 3.2:
We introduced some formal notation to write down the one-loop effective action in a more compact form, Equation (3.18). This allowed us to write the integrals of each of a, λ, c in terms of the loop integrals Π  3.27c)). As we will see, the evaluation of these integrals are by no means a trivial task, but we will make them much more tractable with a handful of clever manipulations.

Outline of the calculation
We have written the integrals in (3.27) to superficially respect the Euclidean Lorentz group SO(4). But to evaluate them, we must rewrite (3.27) to reflect the broken rotational symmetry SO(4) → SO (3). Symmetry considerations tell us that averaging K M K N must give: where P µν M N and P 44 M N are projection operators to R 3 and S 1 respectively: Integrating over the angular coordinates and summing the three graphs in Equation (3.27), we get (3.31a) We also write out the (44) part, which are needed to renormalise: (3.31b) where (Π (s) ) ab µν and (Π (s) ) ab 44 are defined in the obvious way: (Π (s) ) ab M N ≡ (Π (s) ) ab µν P µν M N + (Π (s) ) ab 44 P 44 M N , (3.31c) and we have also defined: and dimensionless sums over the KK modes, S 0,1,2 : The third sum, S 0 , is a standard result. It can be evaluated exactly by e.g., Matsubara summation: Re Where we have defined a function I vac.
0 ≡ 1 2ωL that falls off as a negative power in ωL, and another, δI 0 ≡ 1 2ωL Re( 1 e Lω+ib −1 ), that falls off exponentially. Since the summand of S 0 is monotone decreasing in |n|, differentiation commutes with summation, so S 1,2 can be trivially evaluated by taking derivatives of both sides of Equation (3.31f): where I vac. 1,2 and δI 1,2 are defined in terms of derivatives of I vac.
The basic idea is this: we can see by the asymptotics that the (Π (s),vac. ) ab M N integrals remain unchanged in the L → ∞ limit. This means we can evaluate those integrals in terms of the familiar loop integrals in R 4 , in a way we show explicitly. Obviously these integrals are UV divergent, but they can be renormalised in the M S scheme in the usual way. On the other hand, the SO(4)-breaking, L-dependent parts of (Π (s) ) ab M N are contained entirely within (δΠ (s) ) ab M N : the exponential decay of the δI 0,1,2 means that the integrands of (δΠ (s) ) ab µν are uniformly convergent in kL. Then we may use the identities (3.32a), (3.32b) to integrate by parts in kL and obtain a much more tractable expression.

The vacuum integrals
We can evaluate the loop integrals in Π (s),vac. by "undoing" an integral over an auxiliary continuous variable k 4 . For example, (defining ω ≡ √ k 2 + ∆ for positive ∆,) thus mapping the integral over k ∈ R 3 to one overk ∈ R 4 . Then we regulate the expression by taking the analytic continuation to d ≡ 4 − ε dimensions. In summary: . (3.34c) The expressions on the LHS are the relevant R 3 integrals, and µ is the M S scale of the theory. "−→" means "analytically continues to." On the other hand, we also have the following series of relations under the integral sign: (3.36) Expanding in powers of 1 ε > 0, it is easy to regulate thek integral to get a convergent result. The (Abelian part of the) UV counterterm δZ s trF M N F M N contributes, diagrammatically, and the sum of the three regulated vacuum integrals is therefore: (3.38)

The pseudo-thermal integrals
Now we consider the pseudo-thermal integrals. Using Equations (3.32a), (3.32b), we can simplify the loop integrals immensely by integrating by parts by changing variables ∂ ∂(ωL) 2 = 1 2kL ∂ ∂(kL) . We find that all boundary terms vanish, and the results are, in summary, Plugging into Equations (3.31a), (3.31b), the pseudo-thermal integrals may be written: Where we have defined 14 where K 0 is the modified Bessel function of order 0. This represents the only remaining non-trivial sum, as far as the corrections to κ ab are concerned. We have not given an expression for (δΠ (s) ) ab 44 as it is not needed to find κ ab . Summing the result with the vacuum contribution, (Π (s),vac. ) ab µν + (δΠ (s) ) ab µν + (counterterms) where we have defined (3.43) In Appendix A.1 we explicitly show that whereR b 0 (t) is a pure function that has a power series expansion around t = 0 for fixed b ∈ (0, 2π). We know that Equation (3.44) must be true because the running of the coupling g 2 must freeze out at scales below m W . Equation and 14 The integral in the second line can be carried out by expanding in series in |e − √ (kL) 2 +∆L 2 −ib | < 1.

The sums over β: linear algebra on the root lattice
Let us consider the sums over the root vectors β. It is not hard to show by standard Fourier analysis that, for any integer n, The relation "≡" is to be understood here as equality in the mod N sense (we instead use ":=" to denote "is defined to be" for this subsection). The matrix in Equation (3.47) is diagonalised by the (trace-free) eigenvectors u with vector components: and have eigenvalues indexed by : (3.51) When m m W , this series is very well-approximated by the p = 1 term. However, some extra work is needed to extract information about the m m W case. In Appendix A.2, we perform the sum over p by taking the Mellin transform and find (cf. Equation (A.24)): where, as mentioned before, W is an O(1) function such that W (0) = 0, and has a power series expansion for τ ≡ (m/m W ) < 1: Note that although heretofore the fermion masses only appeared in the combination mL, Equations (3.52a), (3.53) suggest that they are in fact more naturally measured in units of m W , as we should expect.
On the other hand, the 44 parts of the integrals also give us M 2 ab , the scalar (mass) 2 matrix. Omitting the intermediate steps, where K 2 is the modified Bessel function of order 2; this matches the result from taking the second derivative of the GPY potential, (2.9), which serves as a "sanity check" on our calculations. For completeness, we present M 2 , the physical scalar (masses) 2 : We also present ρ , the eigenvalues of ρ ab , where X is an O(1) function defined in terms of W : Equations (3.55) and (3.56) are only presented for completeness, although they may be found without too much difficulty using the methods described in this paper.

Future directions
In this study we have derived an explicit one-loop expression for the eigenvalues of κ ab , the polarisation operator of the SU (N ) dYM theory with massive fermions, and provisionally surveyed some properties of the emergent fourth dimension. It would be interesting to numerically examine the effect of these one-loop corrections on the k-string tensions, (as was done for SYM in Ref. [26],) but to do so would require us to compute the matrix determinants in the monopole measure, ζ -a daunting task (see the discussion in footnote (11)). Additionally, the topological angle θ-dependence in Yang-Mills theory has been the subject of much attention [27][28][29][30][31]: we should also like to examine the dependence of the kstring tensions on the topological angle θ as well as on the circle length L, at the tree-order level, to compare against results on the lattice.
Finally, we would also like to further study the confining properties of dYM outside of the calculable regime, N LΛ 1 and its conjectured continuity with the small N LΛ regime, on the lattice.

A The Mellin transform, and some results
In this Appendix, we explicitly evaluate the sums over p in (3.51) to obtain an expression for κ in terms of analytic functions. To this end, we introduce the Mellin transform, an integral transform on real-valued functions.
Definition A.1. The Mellin transform M is an integral transform defined on the space of real integrable functions f : R + → R as: (A.1a) In particular, for each λ > 0, The inverse transform M −1 is, formally, Now consider the inverse transform. Since Γ(s) has poles at s = 0, −1, −2..., we evaluate the integral by limiting the integration contour c → 0 + and closing the contour over the Re(s) < 0 half-plane. The integral over the arc goes to zero at large radius, so where ψ (0) (z) ≡ d dz log(Γ(z)) is the polygamma function (of order 0).

A.1 Proof of Equation (3.44)
We are now prepared to prove Equation (3.44) and derive a series expression forR b 0 . The idea is to perform the sums over n in "Mellin space," then transform back to "mass space" to obtain a series expansion in t. Like in Example A.1, the inverse transform involves evaluating the residue of a chain of poles on the real axis.
To begin, we observe the Mellin transform of the modified Bessel function of order ν, K ν , is known to be [33, 03.04.22.0004.01]: Plugging this into Equation (3.41), Li s e ib + Li s e −ib , where Li s is the polylogarithm function of order s: Changing back to the original variable t, Li s e ib + Li s e −ib . (A.8) Note that the integral in Equation (A.8) is over the order s of the polylogarithm, rather than its argument. As in Example (A.1), we can evaluate this integral by letting the integration contour approach the imaginary axis from the right, c → 0 + , and close the contour over the half plane Re(s) ≤ 0. The polylogarithm terms are regular for all s for real 0 < b < 2π, so we are left with the residues from the chain of poles at s = 0, −2, −4... where the gamma function diverges. Unfortunately, the poles of Γ(s/2) 2 are of order 2, so evaluating the residues with the integrand of (A.8), as is, would involve the expression d ds Li s e ib , which produces a result that is even more opaque than our original expression.
Observing that which concludes the proofs for the statements we made about W in Equation (2.14).