Landau Gauge Yang-Mills Correlation Functions

Landau gauge Yang-Mills correlation functions

Anton K. Cyrol,1 Leonard Fister,2 Mario Mitter,1 Jan M. Pawlowski,1, 3 and Nils Strodthoff1, 4 1Institut fürTheoretische Physik, UniversitätHeidelberg, Philosophenweg 16, 69120 Heidelberg, Germany 2Institut de Physique Théorique,CEA Saclay, F-91191 Gif-sur-Yvette, France 3ExtreMe Matter Institute EMMI, GSI, Planckstr. 1, D-64291 Darmstadt, Germany 4Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA We investigate Landau gauge SU(3) Yang-Mills theory in a systematic vertex expansion scheme for the effective action with the functional renormalisation group. Particular focus is put on the dynamical creation of the gluon mass gap at non-perturbative momenta and the consistent treatment of quadratic divergences. The non-perturbative ghost and transverse gluon propagators as well as the momentum-dependent ghost-gluon, three-gluon and four-gluon vertices are calculated self- consistently with the classical action as only input. The apparent convergence of the expansion scheme is discussed and within the errors, our numerical results are in quantitative agreement with available lattice results.

PACS numbers: 12.38.Aw, 12.38.Gc

I. INTRODUCTION tex expansion scheme. Therefore, the results [9, 33] for the YM correlations functions give no access to system- The past two decades have seen tremendous progress in atic error estimates. In general, many applications of the description of QCD with functional approaches such functional methods to bound states and the QCD phase as the functional renormalisation group (FRG), Dyson- diagram use such mixed approaches, where part of the Schwinger equations (DSE), and n-particle irreducible correlation functions is deduced from phenomenological methods (nPI). These approaches constitute ab initio constraints or other external input. Despite the huge descriptions of QCD in terms of quark and gluon cor- success of mixed approaches, a full ab initio method is relation functions. The full correlation functions satisfy wanted for some of the most pressing open questions of a hierarchy of loop equations that are derived from the strongly-interacting matter. The phase structure of QCD functional FRG, DSE and nPI relations for the respec- at large density is dominated by fluctuations and even a tive generating functionals. By now, systematic compu- partial phenomenological parameter fixing at vanishing tational schemes are available, which can be controlled by density is bound to lead to large systematic errors [34]. apparent convergence. In the present work on pure Yang- The same applies to the details of the hadron spectrum, Mills (YM) theory we complement the work in quenched in particular with regard to the physics of the higher QCD [1], where such a systematic expansion scheme has resonances, which requires knowledge about correlation been put forward within the FRG. Equipped with such a functions deep in the complex plane. controlled expansion, functional approaches to QCD are In the present work we perform a systematic vertex specifically interesting at finite temperatures and large expansion of the effective action of Landau gauge YM density, where reliable ab initio theoretical predictions theory within the functional renormalisation group ap- and experimental results are missing at present. proach, discussed in Sec. II. The current approximation Most progress with functional approaches has been is summarised in Sec. II B, which also includes a compari- made in Landau gauge QCD, which has many convenient son to approximations used in other works. This ab initio properties for non-perturbative numerical computations. approach starts from the classical action. Therefore the Applications of functional methods include the first-ever only parameter is the strong coupling constant αs at a arXiv:1605.01856v2 [hep-ph] 9 Sep 2016 calculation of qualitative non-perturbative Landau gauge large, perturbative momentum scale. The most distinct propagators as well as investigations of the phase struc- feature of YM theory is confinement, which is reflected ture of QCD. For reviews see [2–13], for applications to by the creation of a gluon mass gap in Landau gauge. Yang-Mills theory see e.g. [14–27], and e.g. [28] for re- We discuss the necessity of consistent infrared irregular- lated studies in the Coulomb gauge. The formal, alge- ities as well as mechanisms for the generation of a mass braic, and numerical progress of the past decades sets gap in Sec. III. Numerical results from a parameter-free, the stage for a systematic vertex expansion scheme of self-consistent calculation of propagators and vertices are Landau gauge QCD. Quantitative reliability is then ob- presented in Sec. IV. Particular focus is put on the im- tained with apparent convergence [1] as well as further portance of an accurate renormalisation of the relevant systematic error controls inherent to the method, see e.g. vertices. We compare with corresponding DSE and lat- [5, 29–32]. In the aforementioned quenched QCD in- tice results and discuss the apparent convergence of the vestigation [1], the gluon propagator was taken from a vertex expansion. Finally, we present numerical evidence separate FRG calculation in [9, 33]. This gluon propa- for the dynamic mass gap generation in our calculation. gator shows quantitative agreement with the lattice re- Further details, including a thorough discussion of the sults, but has been obtained within an incomplete ver- necessary irregularities, can be found in the appendices. 2

II. FRG FLOWS FOR YANG-MILLS THEORY IN A VERTEX EXPANSION

Functional approaches to QCD and Yang-Mills theory are based on the classical gauge fixed action of SU(3) Yang-Mills theory. In general covariant gauges in four full. mom. dep. full. mom. dep. sym. point and dimensions it is given by tadpole config. Z Z Z 1 a a 1 a 2 a ab b FIG. 1. Approximation for the effective action. Only clas- Scl = Fµν Fµν + (∂µAµ) c¯ ∂µDµ c . x 4 2ξ x − x sical tensor structures are included. See Fig. 2 for diagrams (1) that contribute to the individual propagators and vertices.

Here, ξ denotes the gauge fixing parameter, which is R R 4 where R = R d4p/(2π)4 . Here we have introduced the taken to zero in the Landau gauge and x = d x. The p field strength tensor and covariant derivative are given RG-time t = ln(k/Λ) with a reference scale Λ , which is by typically chosen as the initial UV cutoff Λ . Although a a a abc b c this flow equation comes in a simple one-loop form, it F = ∂µA ∂ν A + gf A A , µν ν − µ µ ν provides an exact relation due to the presence of the full field-dependent propagator, ab ab abc c D = δ ∂µ gf A , (2) µ − µ 1 G [φ](p, q) = (p, q) , (7) using the fundamental generators T a, defined by k (2) Γk [φ] + Rk 1 T a,T b = if abcT c , tr T aT b = δab . (3) on its right-hand side. Furthermore, the flow is infrared 2 and ultraviolet finite by construction. Via the integration In general, our notation follows the one used in the works of momentum shells in the Wilsonian sense, it connects [1, 35, 36] of the fQCD collaboration [37]. the ultraviolet, bare action Scl = Γk→Λ→∞ with the full quantum effective action Γ = Γk→0 . The flow equations for propagators and vertices are A. Functional Renormalisation Group obtained by taking functional derivatives of (6). At the vacuum expectation values, these derivatives give equa- (n) n n We use the functional renormalisation group approach tions for the 1PI correlation functions Γk = δ Γk/δφ , as a non-perturbative tool to investigate Yang-Mills the- which inherit the one-loop structure of (6). As the cutoff- ory. The FRG is built on a flow equation for the one- derivative of the regulator functions, ∂tRk, decays suffi- particle irreducible (1PI) effective action or free energy ciently fast for large momenta, the momentum integra- of the theory, the Wetterich equation [38]. It is based tion in (6) effectively receives only contributions for mo- 2 2 on Wilson’s idea of introducing an infrared momentum menta p . k . Furthermore, the flow depends solely cutoff scale k. Here, this infrared regularisation of the on dressed vertices and propagators, leading to a consis- gluon and ghost fluctuations is achieved by modifying tent RG and momentum scaling for each diagram result- the action Scl Scl + ∆Sk with ing from derivatives of (6). Despite its simple structure, → the resulting system of equations does not close at a fi- Z Z 1 a ab b a ab b nite number of correlation functions. In general, higher ∆Sk = A R A + c¯ R c . (4) 2 µ k,µν ν k (n+2) x x derivatives up to the order Γk of the effective action appear on the right hand side of the functional relations The regulator functions Rk are momentum-dependent (n) masses that suppress the corresponding fluctuations be- for the correlation functions Γk . low momentum scales p2 k2 and vanish in the ultraviolet for momenta p2 ≈k2. See App. E for details on B. Vertex expansion of the effective action the regulators used in the present work. Consequently, the effective action, Γk[φ] , is infrared regularised, where φ denotes the superfield The structural form of the functional equations discussed in the previous section necessitates the use of ap- φ = (A, c, c¯) . (5) proximations in most practical application. One systematic expansion scheme is the vertex expansion, i.e. an The fluctuations of the theory are then successively taken expansion of the effective action in terms of 1PI Green’s into account by integrating the flow equation for the ef- functions. This yields an infinite tower of coupled equa- fective action, see e.g. [39, 40], tions for the correlation functions that has to be trun- Z Z cated at a finite order. This expansion scheme allows a 1 µν ba ab ba ∂tΓk[φ] = Gk,ab[φ] ∂tRk,µν Gk [φ] ∂tRk . (6) systematic error estimate in terms of apparent conver- p 2 − p gence upon increasing the expansion order or improving 3

vertices has been summarised as 3 and , which 1 A ,cl Acc,¯ cl ∂t − = + are listed explicitly in App. D. InT the case ofT the transversally projected ghost-gluon vertex, Acc,¯ cl represents already a full basis whereas a full basisT for the transversally 1 1 ∂t − = 2 − − 2 projected three-gluon vertex consists of four elements. However, the eﬀect of non-classical tensor structures has been found to be subleading in this case [41]. ∂t = + perm. − − The most important four-point function is given by the four-gluon vertex, which appears already on the classical level. Similarly to the three-gluon vertex, we approxi- ∂t = +2 + + perm. − mate it with its classical tensor structure

(4) abcd abcd [ΓA4 ]µνρσ(p, q, r) = ZA4,⊥(¯p)[ A4,cl]µνρσ(p, q, r) , (10) ∂t = + + 2 + perm. − − T see App. D for details. The dressing function of the four- gluon vertex is approximated from its momentum depen- FIG. 2. Diagrams that contribute to the truncated flow of dence at the symmetric point via the average momen- p propagators and vertices. Wiggly (dotted) lines correspond tump ¯ p2 + q2 + r2 + (p + q + r)2/2 , which has been to dressed gluon (ghost) propagators, filled circles denote shown≡ to be a good approximation of the full momen- dressed (1PI) vertices and regulator insertions are represented by crossed circles. Distinct permutations include not only tum dependence [42, 43]. To improve this approximation (anti-)symmetric permutations of external legs but also per- further, we additionally calculate the momentum depen- mutations of the regulator insertions. dence of the four-gluon dressing function ZA4,⊥( p , q , t) on the special configuration (p, q, r) = (p, q, p) .| We| | | use this special configuration exclusively in the− tadpole di- further approximations for example in the momentum agram of the gluon propagator equation, cf. Sec. IV B. resolution or tensor structures of the included correla- We show the difference between the special configuration functions. We discuss the convergence of the vertex tion and the symmetric point approximation in the ap- expansion in Sec. IV B. pendix in Fig. 11. Although the four-gluon vertex has Here we calculate the effective action of SU(3) Yang- been the subject of several studies [42–46], no fully con- Mills theory in Landau gauge within a vertex expansion, clusive statements about the importance of additional see Fig. 1 for a pictorial representation. The diagrams non-classical tensors structures are available. contributing to the resulting equations of the constituents In summary we have taken into account the propa- of our vertex expansion are summarised graphically in gators and the fully momentum-dependent classical ten- Fig. 2. The lowest order contributions in this expan- sor structures of the three-point functions, as well as se- sion are the inverse gluon and ghost propagators param- lected momentum-configurations of the gluon four-point eterised via function, see the paragraph above, and App. D. For a comparison of the current approximation with that used (2) ab 2 ab ⊥ 1 ab [ΓAA]µν (p) = ZA(p) p δ Πµν (p) + δ pµpν , in other functional works one has to keep in mind that ξ FRG, Dyson-Schwinger or nPI equations implement dif-

(2) ab 2 ab ferent resummation schemes. Thus, even on an identical [Γcc¯ ] (p) = Zc(p) p δ , (8) approximation level of a systematic vertex expansion, the included resummations diﬀer. with dimensionless scalar dressing functions 1/ZA and ⊥ 2 In the present work we solve the coupled system of 1/Zc. Here, Πµν (p) = δµν pµpν /p denotes the corresponding projection operator.− We use this splitting in all momentum-dependent classical vertex structures and tensor structures with canonical momentum dimension propagators. In former works with functional methods, and dimensionless dressings also for the higher order ver- see e.g. [14–23, 41–53], only subsets of these correlation tices. functions have been coupled back. A notable exception On the three-point level we include the full transverse is [26], where a similar self-consistent approximation has ghost-gluon vertex and the classical tensor structure of been used for three-dimensional Yang-Mills theory. the three-gluon vertex

(3) abc abc C. Modified Slavnov-Taylor identities and [ΓAcc¯ ]µ (p, q) = ZAcc,¯ ⊥( p , q , t)[ Acc,¯ cl]µ (p, q) , | | | | T transversality in Landau gauge (3) abc abc [Γ 3 ]µνρ(p, q) = ZA3,⊥( p , q , t)[ A3,cl]µνρ(p, q) . (9) A | | | | T In Landau gauge, the dynamical system of correlation Here, the momentum p (q) corresponds to the indices a functions consists only of the transversally projected cor- (b) and t denotes the cosine of the angle between the relators [9]. Those with at least one longitudinal gluon momenta p and q . The classical tensor structure of the leg do not feed back into the dynamics. To make these 4 statements precise, it is useful to split correlation func- Consequences of the STIs & mSTIs tions into purely transverse components and their complement with at least one longitudinal gluon leg. The For the purposes of this work, the most important ef- (n) purely transverse vertices Γ⊥ , are defined by attach- fect of the modification of the STIs due to the regulator ing transverse projection operators to the corresponding term is that it leads to a non-vanishing gluon mass pa- gluon legs, rameter [54],

h (n)i h i h iab Γ Π⊥ Π⊥ Γ(n) , (11) (2) ab 2 ⊥ µ1ν1 µnA νnA ∆mSTI ΓAA ∝ δ δµν α(k) k . (16) µ1···µnA ≡ ··· ν1···νnA µν where nA is the number of gluon legs and group indices At k = 0, where the regulators vanish, this modification and momentum arguments have been suppressed for the disappears, as the mSTIs reduce to the standard STIs. In sake of brevity. This defines a unique decomposition of particular, this entails that, at k = 0, the inverse longi- (2) n-point functions into tudinal gluon propagator, ΓAA,L, reduces to the classical one, solely determined by the gauge fixing term (n) (n) (n) Γ = Γ + ΓL , (12) ⊥ (2) ab (2) ab pµ [ΓAA,L]µν (p) [SAA,L]µν (p) = 0 . (17) (n) − where the longitudinal vertices ΓL , have at least one longitudinal gluon leg. Consequently, they are always This provides a unique condition for determining the projected to zero by the purely transverse projection op- value of the gluon mass parameter (16) at the ultraviolet erators of (11). initial scale Λ. However, it can only serve its purpose, if Functional equations for the transverse correlation the longitudinal system is additionally solved. functions close in the Landau gauge, leading to the struc- One further conclusion from (15) is that the mSTIs do ture [9], not constrain the transverse correlation functions without further input. This fact is not in tension with one of the main applications of STIs in perturbation theory, i.e. Γ(n) = Diag[ Γ(n) ] . (13) ⊥ { ⊥ } relating the running of the relevant vertices of Yang-Mills theory that require renormalisation. As Yang-Mills the- In (13) Diag stands for diagrammatic expressions of ei- ory is renormalisable, only the classical vertex structures ther integrated FRG, Dyson-Schwinger or nPI equations. are renormalised and hence the renormalisation functions Equation (13) follows from the fact that all internal legs of their transverse and longitudinal parts have to be iden- are transversally projected by the Landau gauge gluon tical. propagator. Hence, by using transverse projections for As an instructive example we consider the ghost-gluon the external legs one obtains (13). In contradistinction vertex. For this example and the following discussions to this, the functional equations for the vertices with at (n) we evaluate the STIs within the approximation used in least one longitudinal gluon leg, ΓL , are of the form the present work: on the right hand side of the STIs we only consider contributions from the primitively diver- (n) (n) (n) ΓL = Diag[ ΓL , Γ⊥ ] . (14) gent vertices. In particular, this excludes contributions { } { } from the two-ghost–two-gluon vertex. The ghost-gluon In other words, the solution of the functional equations vertex can be parameterised with two tensor structures, (14) for Γ(n) requires also the solution of the transverse L (3) abc abch i set of equations (13). [ΓAcc¯ ]µ (p, q) = if qµZAcc,¯ cl(p, q) + pµZAcc,¯ ncl(p, q) . In the present setting, gauge invariance is encoded in (18) modified Slavnov-Taylor identities (mSTIs) and Ward- Takahashi identities (mWTIs). They are derived from In (18) we have introduced two dressing functions ZAcc,¯ cl the standard Slavnov-Taylor identities (STIs) by includ- and ZAcc,¯ ncl as functions of the gluon momentum p and ing the gauge or BRST variations of the regulator terms, anti-ghost momentum q . In a general covariant gauge see [5, 14, 54–57] for details. The mSTIs are of the only ZAcc,¯ cl requires renormalisation. Similar splittings schematic form into a classical tensor structure and the rest can be used in other vertices. Trivially, this property relates the per- (n) (n) (n) ΓL = mSTI[ ΓL , Γ⊥ ,Rk] , (15) turbative RG-running of the transverse and longitudi- { } { } nal projections of the classical tensor structures. Then, which reduce to the standard STIs in the limit of van- the STIs can be used to determine the perturbative RG- ishing regulator, Rk 0. The STIs and mSTIs have a running of the classical tensor structures, leading to the similar structure as (14)≡ and can be used to obtain infor- well-known perturbative relations mation about the longitudinal part of the correlators. Al- 2 2 ternatively, they provide a non-trivial consistency check ZAcc,¯ cl ZA3,cl ZA4,cl 2 = 3 = 2 , (19) for approximate solutions of (14). Zc ZA ZA ZA 5 at the renormalisation scale µ. Consequently, (19) al- variant gauges. Furthermore, in Yang-Mills theory for- lows for the definition of a unique renormalised two-loop mulated in covariant gauges, the gapping of the gluon coupling αs(µ) from the vertices. relative to the ghost is necessary and sufficient for pro- The momentum dependent STIs can also be used to ducing a confining potential for the corresponding order make the relation (19) momentum-dependent. Keeping parameter, the Polyakov loop. Hence, understanding the only the classical tensor structures, we are led to the details of the dynamical generation of a gluon mass gap momentum dependent running couplings gives insight into the confinement mechanism. This relation holds for all potential infrared closures of Z2 (p) 1 Acc,¯ ⊥ the perturbative Landau gauge. The standard infrared αAcc¯ (p) = 2 , 4π ZA(p) Zc (p) closure corresponds to a full average over all Gribov regions. This leads to the standard Zinn-Justin equation 2 1 ZA3,⊥(p) as used in the literature, e.g. [4]. In turn, the restriction αA3 (p) = 3 , to the first Gribov regime can be implemented within the 4π ZA(p) refined Gribov-Zwanziger formalism, e.g. [63–67], that

1 ZA4,⊥(p) leads to infrared modifications of the STIs. In the follow- α 4 (p) = , (20) A 4π Z2 (p) ing we discuss the consequences of the standard STIs, a A discussion of the refined Gribov-Zwanziger formalism is where the used transverse projection is indicated by the deferred to future work. subscript , for details see App. D. Additionally, the vertices appearing⊥ in (20) are evaluated at the symmetric point, see Sec. IV B for the precise definition. The A. Gluon mass gap and irregularities STIs and two-loop universality demand that these running couplings become degenerate at large perturbative In order to study the dynamical generation of the mass momentum scales, where the longitudinal and transverse gap, we first discuss the consequences of the STI for the parts of the vertices agree. longitudinal gluon two point function (17). It states that In Landau gauge, the ghost-gluon vertex is not renor- no quantum fluctuations contribute to the inverse lon- malised on specific momentum configurations, and we gitudinal gluon propagator, i.e. the longitudinal gluon can alternatively define a running coupling from the wave propagator is defined by the gauge fixing term. There- function renormalisation of ghost and gluon [15, 58], fore, the dynamical creation of a gluon mass gap requires different diagrammatic contributions to the longitudinal 1 g2 αs(p) = 2 . (21) and transverse gluon mass parameter. The discussion 4π ZA(p)Zc (p) of the prerequisites for meeting this condition is qualita- Note that the momentum-dependence of the running cou- tively different for the scaling and the decoupling solu- pling (21) does not coincide with that of the correspond- tions. Hence, these two cases are discussed separately. ing running couplings obtained from other vertices, i.e. The scaling solution is characterised by the infrared (20). This is best seen in the ratio αAcc¯ (p)/αs(p) = behaviour [15, 18, 69–75] 2 2 ZAcc,¯ ⊥(p)/g . In this context we also report on an im- 2 2 κ lim Zc(p ) (p ) , portant result for the quark-gluon vertex coupling, p→0 ∝ 2 1 ZAqq,¯ ⊥(p) 2 2 −2 κ αAqq¯ (p) = 2 , (22) lim ZA(p ) ∝ (p ) , (23) 4π ZA(p)Zq(p) p→0 with the dressing function of the classical tensor struc- 2 with the scaling coefficient 1/2 < κ < 1. A simple calcu- ture of the quark-gluon vertex ZAqq,¯ ⊥(p) and the quark lation presented in App. A shows that the ghost loop with dressing function 1/Zq(p) [1]. The solution of the cor- an infrared constant ghost-gluon vertex and scaling ghost responding STI reveals that the quark-gluon vertex cou- propagator is already capable of inducing a splitting in pling αAqq¯ agrees perturbatively with αs(p) in (21), and the longitudinal and transverse gluon mass parameter. hence it differs from the other vertex couplings in (20). Next we investigate the decoupling solution, e.g. [21, Note that the present truncation only considers contri- 22], which scales with butions from primitively divergent vertices. Accordingly, 2 the two-quark–two-ghost vertex contribution in the STI lim Zc(p ) ∝ 1 , for the quark-gluon vertex, see e.g. [4], has been dropped. p→0

2 2 −1 lim ZA(p ) (p ) , (24) p→0 ∝ III. CONFINEMENT, GLUON MASS GAP, AND IRREGULARITIES at small momenta. Assuming vertices that are regular in the limit of one vanishing gluon momentum, one ﬁnds It has been shown in [40, 59–62] that a mass gap in that all diagrammatic contributions to the longitudinal the gluon propagator signals conﬁnement in QCD in co- and transverse gluon mass parameter are identical. For 6

3 FRG, scaling FRG, scaling FRG, decoupling FRG, decoupling 10 Sternbeck et al. Sternbeck et al. 2

1 ghost propagator dressing gluon propagator dressing

0 0.1 1 10 0.1 1 10 p [GeV] p [GeV]

FIG. 3. Gluon dressing 1/ZA (left) and ghost dressing 1/Zc (right) in comparison to the lattice results from [68]. The scale setting and normalisation procedures are described in App. F. example, if the ghost loop were to yield a non-vanishing pearing in the gluon propagator equation. Consequently, contribution to the gluon mass gap, the ghost-gluon ver- if all vertices were regular, no gluon mass gap would be tex would have to be a function of the angle θ = arccos(t) created. In particular, regular vertices would entail the between the gluon and anti-ghost momenta p and q, absence of conﬁnement. The necessity of irregularities for the creation of a gluon mass gap was already realised (3) abc (3) abc lim [ΓAcc¯ ]µ ( p , q , t) = [ΓAcc¯ ]µ (0, q , t) , (25) by Cornwall [76]. |p|→0 | | | | | | even in the limit of vanishing gluon momentum p 0 . In the light of these ﬁndings it is interesting to in- Since the above limit depends on the angle, the vertex| | → is vestigate the consistency of irregularities with further irregular. See App. A for more details on this particular Slavnov-Taylor identities. Therefore, we consider the case. Similar conclusions can be drawn for all vertices ap- Slavnov-Taylor identity of the three-gluon vertex, e.g. [4],

(3) abc abc 1 ˜ 2 2 ⊥ ˜ 2 2 ⊥ irρ[ΓA3 ]µνρ(p, q, r) f 2 Gµσ(p, q)q ZA(q )Πσν (q) Gνσ(q, p)p ZA(p )Πσµ(p) , (26) ∝ Zc(r ) −

where G˜µν relates to the ghost-gluon vertex via three-gluon vertex projected with one non-zero longitudinal leg rρ . Although this momentum configuration does (3) abc abc ˜ [ΓAcc¯ ]µ (p, q) = igf qν Gµν (p, q) . (27) not enter the gluon mass gap directly, crossing symmetry implies the necessary irregularity. In summary, these ˜ For a regular Gµν in the limit p 0 in (26), the scaling arguments illustrate that also the three-gluon vertex STI → solution leads to a singular contribution of the type is consistent with the necessity of irregularities for both 2 1−2κ ˜ ⊥ types of solutions. lim(p ) Gνσ(q, 0) Πσµ(p) + regular , (28) p→0 We close the discussion of vertex irregularities with the remark that the infrared modification of the propagator- where κ is the scaling coefficient from (23). This is con- STI in the refined Gribov-Zwanziger formalism may re- sistent with the expected scaling exponent of the three- move the necessity for irregularities in the vertices. gluon vertex in this limit [74]. In the same limit, the decoupling solution leads to a singular contribution of the form B. Origin of irregularities ˜ ⊥ lim Gνσ(q, 0) Πσµ(p) + regular . (29) p→0 As discussed in the previous section, self-consistency ⊥ Since the transverse projector Πσµ(p) introduces an an- in terms of the Slavnov-Taylor identities entails a corre- gular dependence in the limit p 0 , the STI again spondence between the dynamical generation of a gluon demands an irregularity in limit of→ one vanishing mo- mass gap and the presence of irregularities. But the STIs mentum. Note that this is just a statement about the do not provide a mechanism for the creation of irregular- 7

8 ] FRG, scaling 0 -2 10 FRG, decoupling 6 Sternbeck et al. -2 2 10 1.5 α 4 Acc -4 α 3 10 1 A α 4 running couplings A 2

gluon propagator [GeV -6 10 0.5 0 0.1 1 10 0.1 1 10 100 p [GeV] p [GeV]

FIG. 4. Left: Gluon propagator in comparison to the lattice results from [68]. Right: Effective running couplings defined in (20) as obtained from different Yang-Mills vertices as function of the momentum. ities, the gluon mass gap, and in turn confinement. sufficient for creating a physical mass gap in the gluon. In the scaling solution, (23), the irregularities arise nat- This scenario provides a direct relation of confinement urally from the non-trivial scaling. Hence they are tightly and spontaneous symmetry breaking. Therefore it is pos- linked to the original Kugo-Ojima confinement scenario sibly connected to the presence of resonances that are [77], that requires the non-trivial scaling. Note, however, triggered in the longitudinal sector of the theory, where that this simply links different signatures of confinement they do not spoil the gapping of the completely trans- but does not reveal the mechanism at work. verse sector. A purely longitudinal massless mode, as a For the decoupling solution (24), we want to discuss source for irregularities in the gluonic vertices, has been two possible scenarios. In the first scenario, the irregu- worked out in [80, 81], for a short summary see [82]. As larities are generated in the far infrared. A second possi- a consequence, an irregularity appears in the purely lon- bility is that they are triggered via a condensate and/or a gitudinal three-gluon vertex in a way that preserves the resonance, providing a direct connection of confinement corresponding Slavnov-Taylor identity. The creation of and spontaneous symmetry breaking. a purely transverse background and the presence of longitudinal massless mode would then be two sides of the In the first scenario it is sufficient to focus on ghost same coin. Furthermore, the longitudinal resonance has loops as possible sources of such irregularities, since the to occur at about the same scale as the gluon conden- gluonic diagrams decouple from the infrared dynamics sate, in order to trigger the correct gluon mass gap. A due to the gluon mass gap. This is a seemingly appealing more detailed discussion and computation of this scenario scenario as it is the dynamical ghost that distinguishes cannot be assessed in the purely transverse system and confining Yang-Mills theory from e.g. QED. However, in is therefore deferred to future work. the decoupling solution (24) both, the ghost-gluon vertex as well as the ghost propagator, have infrared finite quantum corrections: no ghost-loops contribute to their equation and (infrared) constant dressing functions can C. The purely transverse system be assumed for both. As a consequence the ghost loop contributions to correlation functions have the same in- In this work we restrict ourselves to a solution of the frared structure as perturbative ghost-loop contributions. purely transverse system (13), which is closed. The only However, none of these perturbative ghost loops yields relevant UV parameters in this system are the strong the necessary irregularities, see App. B for an explicit coupling and the transverse gluon mass parameter. In calculation. the UV the transverse mass parameter agrees with the In the second scenario, the generation of irregulari- longitudinal one. The latter is fixed by the mSTI for ties can be based on the dynamical generation of a non- the longitudinal gluon propagator. Hence, the only in- a a vanishing transverse background, Fµν Fµν = 0 , in the formation needed from the longitudinal system is the ini- infrared. This gluon condensate is the Savvidi6 vacuum tial value for the transverse gluon mass parameter (16). [78], and its generation in the present approach has been Note also that there is at least one value for the initial a a 4 discussed in [79] with Fµν Fµν 1 GeV . Then, a ver- gluon mass parameter that yields a valid confining solu- tex expansion about this non-trivial≈ IR-solution of the tion. In the following we vary the gluon mass parameter equation of motion introduces an IR-splitting of trans- and discuss the properties of the ensuing solutions. We verse and longitudinal vertices due to the transversality find a confining branch with both scaling and decoupling of the background field. This IR-splitting automatically solutions. In addition, we observe a transition to the implies irregularities as discussed in Sec. III A, and is deconfined Higgs-type branch. No Coloumb branch is 8

3.5 4 FRG, scaling FRG, scaling FRG, decoupling FRG, decoupling 3

2 3 1

0 three-gluon vertex dressing ghost-gluon vertex dressing

2.5 -1 0.1 1 10 0.1 1 10 p [GeV] p [GeV]

1 1 FIG. 5. Ghost-gluon vertex (left) and three-gluon (right) vertex dressing functions ZAcc,¯ ⊥ p, p, − 2 and ZA3,⊥ p, p, − 2 in the symmetric point conﬁguration. More momentum conﬁgurations and comparisons to Dyson-Schwinger and lattice results can be found in Figs. 11-13. In contrast to Fig. 3, the decoupling dressings are normalised to the scaling solution in the UV.

2 found. The unique scaling solution satisfies the original Slavnov-Taylor identities. The initial value mΛ can be Kugo-Ojima confinement criterion with ZC (p = 0) = 0 . uniquely fixed by demanding that the resulting propaga- We emphasise that the existence of the scaling solution is tors and vertices are of the scaling type. Consequently, dynamically generated in a highly non-trivial way. The the only parameter in this calculation is the value of the details are discussed in Sec. IV D. strong running coupling at the renormalisation scale, as initially stated. We also produce decoupling solutions by varying the gluon mass parameter towards slightly larger IV. NUMERICAL RESULTS values. Our reasoning for their validity as confining solutions is presented in Sec. IV D. We calculate Yang-Mills correlation functions by integrating the self-consistent system of flow equations ob- A. Correlation functions and running couplings tained from functional derivatives of (6), see Fig. 2 for diagrammatic representations. Technical details on the numerical procedure are given in App. C. We use constant We show our results for the Yang-Mills correlation dressing functions as initial values for the 1PI correlators functions as well as the momentum-dependent transverse at the ultraviolet initial scale Λ . Consequently, the ini- running couplings in Figs. 3-6, see also Figs. 11-13 in tial action ΓΛ is given by the bare action of QCD and the the appendices for a comparison of the vertices to re- Slavnov-Taylor identities enforce relations between these cent lattice and DSE results. A discussion of truncation constant initial correlation functions. As is well-known, effects is deferred to Sec. IV B. In order to be able to and also discussed in Sec. II C, the Landau gauge STIs compare to results from lattice simulations, we set the leave only three of the renormalisation constants inde- scale and normalise the dressings as described in App. F. pendent, namely the value of the strong running coupling At all momenta, where the difference between the scaling and two trivial renormalisations of the fields that drop (solid line) and decoupling (band bounded by dashed-dot out of any observable. To eliminate cutoff effects, we line) solutions is negligible, our results for the correlations choose the constant initial values for the vertex dressings functions agree very well with the corresponding lattice such that the momentum-dependent running couplings, results. In the case of the scaling solution we find the (20) are degenerate at perturbative momentum scales p consistent scaling exponents with ΛQCD p Λ i.e. the STIs (19) are only fulfilled κghost = 0.579 0.005 , on scales considerably below the UV cutoff scale. The ± modification of the Slavnov-Taylor identity, caused by κgluon = 0.573 0.002 , (31) the regulator term, requires a non-physical gluonic mass ± 2 term mΛ at the cutoff Λ. The initial value for the inverse where the uncertainties stem from a least square fit with gluon propagator is therefore taken as the ansatz

2 κghost (2) ab 2 2 ab ⊥ Zc(p) ∝ (p ) , [ΓAA,Λ]µν (p) = ZA,Λ p + mΛ δ Πµν (p) . (30)

2 −2 κgluon 2 ZA(p) ∝ (p ) . (32) The non-physical contribution mk to the gluon propagator vanishes only as the renormalisation group scale, k , As discussed in Sec. III C, the scaling solution is a self- is lowered to zero, where the mSTIs reduce to the usual consistent solution of the purely transverse system in the 9

3 RG scale dep. 100 FRG, scaling 1D mom. dep. FRG, decoupling 3D mom. dep. 2 DSE, scaling DSE, decoupling

10 1 gluon propagator dressing four-gluon vertex dressing

0 0.1 1 10 0.1 1 10 p [GeV] p [GeV]

FIG. 6. Left: Four-gluon vertex dressing function as defined in (10) at the symmetric point in comparison to Dyson-Schwinger computations [42]. We normalised all curves to match the scaling result at p = 2 GeV. Right: Gluon propagator dressings obtained with different momentum approximations, see Sec. IV B for details. used approach, and has no systematic error related to running couplings have been obtained within an approx- the lack of solving the longitudinal system. In turn, the imation that takes only one momentum variable into ac- presented decoupling solutions suffer from the missing so- count in the vertices, see Sec. IV B. At large perturbative lution of the longitudinal system, leading to a small addi- momentum scales, we find them to be perfectly degen- tional systematic error. This argument already suggests erate, as is demanded by the Slavnov-Taylor identities. that it is the presented scaling solution that should agree The degeneracy of the running couplings is lifted at a best with the lattice results in the regime p & 1 GeV , scale of roughly 2 GeV , which coincides with the gapping where the solutions show no sensitivity to the Gribov scale of the gluon. Furthermore, the three-gluon vertex problem. This is confirmed by the results, see in partic- shows a zero crossing at scales of 0.1 GeV to 0.33 GeV, ular Fig. 3. which is the reason for the spike in the corresponding In the infrared regime, p . 1 GeV , the different solu- running coupling. This zero crossing, which is caused by tions approach their infrared asymptotics. In Fig. 3 and the infrared-dominant ghost-loop, is well-known in the Fig. 4 we compare the FRG solutions with the lattice literature [41, 49, 50, 90]. Even though we are look- data from [68]. In agreement with other lattice results ing at the scaling solution, we find that the running [83–85] in four dimensions, these propagators show a de- couplings defined from the purely gluonic vertices are coupling behaviour, for a review see [12]. Taking the IR still strongly suppressed in the infrared. In particular behaviour of all correlators into account, cf. also Fig. 13, the three-gluon vertex running coupling becomes more the lattice solution [68] is very close to the decoupling so- strongly suppressed than the four-gluon vertex running lution (dot-dashed line) that is furthest from the scaling coupling. However, as demanded by scaling, they seem solution (solid line). Note however, that the systematic to settle at tiny but finite fixed point values, which has errors of both approaches, FRG computations and lattice also been seen in Dyson-Schwinger studies [41, 42, 44]. simulations increase towards the IR. While the FRG computations lack apparent convergence in this regime, the lattice data are affected by the non-perturbative gauge B. Quality of the approximation fixing procedure, i.e. the choice of Gribov copies [86–88] and discretisation artefacts [89]. Consequently, compar- In Fig. 6 (right panel), we show the scaling solution for ing the FRG IR band to the lattice propagators has to the propagators in different truncations. In all cases, the be taken with a grain of salt. In the case of the vertices, full momentum dependence of the propagators is taken we compare also to results obtained within the Dyson- into account whereas different approximations are used Schwinger equation approach [42, 47, 50], see Fig. 7 and for the vertices. Including only RG-scale-dependent con- 13. A comparison of the different running vertex cou- stant vertex dressing functions is the minimal approxima- plings is given in Sec. IV C. tion that can produce a scaling solution with a physical We find that it is crucial to ensure the degeneracy gluon mass gap. The dot-dashed (magenta) line in Fig. 6 in the different running vertex couplings at perturba- (right panel) corresponds to an approximation with con- tive momentum scales in order to achieve quantitative stant vertex dressing functions evaluated at the symmet- accuracy, see also Sec. IV B. The transverse effective run- ric configuration with momentum (250 MeV) . Hence ning couplings, as defined in (20), are shown in the right the vertices are only RG-scale-dependentO vertices. For panel of Fig. 4. To be able to cover a larger range of the dashed blue results the dressing functions for the momenta with manageable numerical effort, the shown transversally projected classical tensor structures have 10

1 been approximated with a single momentum variable α [47, 92] 2 1 Pn 2 Acc p¯ n i=1 pi . Reducing the momentum dependence α [91, 93] to a≡ single variable requires the definition of a momen- Acc 0.8 α 3 [91, 93] tum configuration to evaluate the flow. Here, we use the A α 3 2 A [50] symmetric point configuration, defined by pi pi = p and · α 4 [42] pi pj = 1/(n 1) for i = j , where n = 3 (4) for the 0.6 A three(four)-gluon· − − vertex. Finally,6 the solid red line corre- 0.3 sponds to our best truncation. As described in Sec. II B, 0.2 it takes into account the full momentum dependence of 0.4 0.1 the classical tensors structures of the three-point func- rescaled running couplings tions as well as the four-gluon vertex in a symmetric point 0.2 approximation. Additionally, all (three-dimensional) mo- 2 3 4 5 6 7 8 9 10 mentum configurations of the four-gluon vertex that are p [GeV] needed in the tadpole diagram of the gluon propagator equation have been calculated and coupled back in this FIG. 7. Running couplings (20) in comparison with DSE run- diagram. The reliability of our approximation can be as- ning. The grey band gives the spread of vertex couplings from sessed by comparing the two simpler truncations to the the FRG in the present work. The DSE results are shown result obtained in our best truncation scheme. We ob- rescaled to fit our ghost-gluon vertex running coupling at serve that our results apparently converge towards the 10 GeV to facilitate the comparison. The inlay shows the un- lattice result, as we improve the momentum approxima- scaled couplings. Note that the FRG running couplings nat- urally lie on top of each other and are not depicted rescaled. tion for the vertices. The effects of non-classical tensor structures and vertices are beyond the scope of the current work and have to be checked in future investigations, see however [41] running couplings to the FRG result at large momen- for an investigation of non-classical tensor structures of tum scales in Fig. 7. For the sake of visibility, we only the three-gluon vertex. Within the present work, the al- have provided a band for the spread of the FRG cou- ready very good agreement with lattice results suggests, plings as obtained from different vertices. The shown that their influence on the propagators is small. DSE running couplings are based on a series of works The final gluon propagator is sensitive to the correct [42, 47, 50, 53, 91], where the explicitly shown results are renormalisation of the vertices. For example, a one per- taken from [42, 50, 92, 93]. Additionally, we provide the cent change of the three-gluon vertex dressing at an UV raw DSE running couplings that have not been rescaled scale of 20 GeV magnifies by up to a factor 10 in the final by a constant factor in the inlay. gluon propagator. Therefore, small errors in the perturbative running of the vertices propagate, via renormalisation, into the two-point functions. We expect a five D. Mass gap, mSTIs and types of solutions percent uncertainty in our results due to this. Despite these uncertainties, we interpret the behaviour As discussed in Sec. II C, the introduction of the regu- in Fig. 6 (right panel) as an indication for apparent con- lator in the FRG leads to a modification of the Slavnov- vergence. Taylor identities. In turn the inverse gluon propagator (2) 2 obtains a contribution proportional to ∆ΓAA ∝ k α(k) for all k > 0 . Disentangling the physical mass gap contri- C. Comparison to other results bution from this mSTI contribution to the gluon mass parameter is intricate, both conceptually and numerically. In Fig. 13, numerical results for the ghost-gluon and The resulting numerical challenge is illustrated in the ap- three-gluon vertices are shown in comparison to other pendix in Fig. 10, where we show the k-running of the functional methods as well as lattice results. In sum- gluon mass parameter. This is the analogue of the prob- mary, the results from various functional approaches and lem of quadratic divergences in Dyson-Schwinger equa- the lattice agree to a good degree. But these correlation tions with a hard momentum cutoff, see e.g. [94]. How- functions are not renormalisation group invariant, and ever, there has to exist at least one choice for the gluon 2 a fully meaningful comparison can only be made with mass parameter mΛ that yields a valid confining solution, RG invariant quantities. Therefore, we compare our re- see Sec. II C. To resolve the issue of finding this value, we sults for the RG invariant running couplings with the first recall that a fully regular solution has no confine- respective results from DSE computations. To be more ment and necessarily shows a Higgs- or Coulomb-type precise, it is actually the β functions of the different ver- behaviour. Although we do not expect these branches to tices that are tied together by two-loop universality in be consistent solutions, we can trigger them by an appro- the sense that they should agree in the regime where priate choice of the gluon mass parameter in the UV. The three-loop effects are negligible. Since constant factors confinement branch then lies between the Coulomb and drop out of the β functions, we have normalised the DSE the Higgs branch. We need, however, a criterion for dis- 11

confined branch ] confined branch 10 -2 Higgs branch 6 Higgs branch

1 gluon propagator [GeV ghost propagator dressing

0 0.1 1 10 0.1 1 10 p [GeV] p [GeV]

FIG. 8. Ghost dressing functions 1/Zc (left) and gluon propagators (right) for different values of the ultraviolet gluon mass parameter. Blue results correspond to the Higgs-type branch and red results to the confined branch. The solutions in all branches have been normalised to the scaling solution in the UV. tinguishing between the confinement and the Higgs-type fining and Higgs solutions, we use the presence of a max- branch. imum at non-vanishing momenta in the gluon propaga- To investigate the possible solutions in a controlled tor, which signals positivity violation [4]. In the right way, we start deep in the Higgs-type branch: an asymp- panel of Fig. 9, we show the location of the maximum in 2 totically large initial gluon mass parameter mΛ triggers the gluon propagator, again as a function of the gluon an explicit mass term of the gluon at k = 0 . If we could mass parameter, m2 m2 . We clearly see a re- Λ − Λ,scaling trigger this consistently in the present SU(3) theory, it gion of confining solutions that show a back-bending of would constitute a Higgs solution. Note that in the cur- the gluon propagator at small momenta, see Fig. 8. The rent approximation it cannot be distinguished from mas- dashed line, separating the shaded from the white region sive Yang-Mills theory, which has e.g. been considered in in the right panel of Fig. 9, indicates the smallest mo- [23, 95]. Starting from this Higgs-type branch, we can mentum value at which the gluon propagator has been then explore the limit of smaller initial mass parame- calculated. With this restriction in mind, the fit in the ters. This finally leads us to the scaling solution, which inlay demonstrates that the location of the maximum of forms the boundary towards an unphysical region char- the propagator scales to zero as one approaches the crit- 2 acterised by Landau-pole-like singularities. It is left to ical value mc . We fit with distinguish between the remaining confining and Higgs- 2 2 α type solutions, shown in Fig. 8, without any information 2 mΛ mc pmax(mΛ) −2 , (33) from the longitudinal set of equations. For that purpose ∝ mc we use two criteria: which yields the critical exponent In the left panel of Fig. 9, we show the mass gap of 2 (2) α = 1.95 0.6 , (34) the gluon, m = ΓAA,k=0(p = 0) , as a function of the ± 2 chosen initial value for the gluon mass parameter mΛ in the 1D approximation. Within the numerical accuracy, 2 subtracted by the corresponding value for the scaling so- this boundary value mc is equivalent to the minimal value 2 2 lution mΛ,scaling . The latter solution corresponds to zero mmin of our first confinement criterion. Hence, the value on the x-axis in Fig. 9. As mentioned before, going be- of the UV mass parameter that results in the minimal 2 2 yond the scaling solution, mΛ < mΛ,scaling , leads to sin- gluon mass gap, is also the one that shows minimal back- gularities. We interpret their presence as a signal for the bending. Note that the lattice simulations show a gluon invalidity of the Coulomb branch as a possibly realisation propagator that is at least very close to this minimal mass of non-Abelian Yang-Mills theory. The decisive feature gap. of the left panel of Fig. 9 is the presence of a minimum at As discussed in detail in Sec. III and appendices A 2 mmin . If there were no dynamical mass gap generation, and B, a gluon mass gap necessitates irregularities. The 2 2 m would have to go to zero as we lower mΛ . In con- scaling solution by definition contains these irregularities trast to this, we find that the resulting gluon mass gap is already in the propagators, cf. (23). For the decoupling- 2 2 always larger than the value it takes at mΛ = mmin . In type solutions, we excluded infrared irregularities of dia- particular, this entails that all solutions to the left of the grammatic origin, see App. B. Thus, for the decoupling- 2 2 minimal value, mΛ < mmin , are characterised by a large type solutions our arguments for the validity of the so- dynamical contribution to the gluon mass gap, which we lutions are weaker and remain to be investigated in a interpret as confinement. solution including at least parts of the longitudinal sys- As a second criterion for differentiating between con- tem, see the discussion in Sec. III. Additionally, it might 12

0.3 0.1 confined branch Higgs branch ] 2 1 0.2 0.01 [GeV

2 confined 0.001 0.005

m Higgs 2 2 2 0.1 (m Λ-m c)/m c 2 fit points 2 m min m c fit 0 0.1 position of gluon prop. max. [GeV] 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 2 2 2 2 2 2 m Λ - m Λ,scaling [GeV ] m Λ - m Λ,scaling [GeV ]

2 2 2 FIG. 9. Left: Gluon mass gap as a function of the gluon mass parameter mΛ − mΛ, scaling , where mΛ, scaling denotes the gluon mass parameter that yields the scaling solution. Right: Momentum value at which the gluon propagator assumes its maximum, 2 2 as a function of the gluon mass parameter mΛ −mΛ, scaling . The inlay exposes the power law behaviour of the gluon propagator maximum in the vicinity of the transition region, see (33). Both plots were obtained from our numerically less-demanding 1D approximation. We have repeated this analysis in the transition regime from Higgs-type to confinement branch also with the best approximation and find the same behaviour. The shaded area marks momentum scales that are not numerically resolved in the present work. The points in this region rely on a generic extrapolation. be necessary to expand about the solution of the equation ture also explains and supports the semi-quantitative na- of motion, see [79]. ture of the results in low-order approximations. We summarise the findings of the present section. In This implies that self-consistent calculations of most or the right panel of Fig. 9 we can distinguish a confining all vertices have to reproduce this universality, in partic- branch with positivity violation and a Higgs-type branch ular for momenta 2 GeV . p . 10 GeV . When starting with a massive gluon propagator. A Coulomb-type so- from the value of the strong running coupling at pertur- lution, on the other hand, can never be produced with bative momenta, we find that a violation of the degen- the functional renormalisation group since any attempt eracy of the running couplings, (20), in this regime goes to do so leads to Landau-pole-like singularities. The non- hand in hand with the loss of even qualitative properties existence of the Coulomb branch is tightly linked to the of the non-perturbative results in self-consistent approx- non-monotonous dependence of the mass gap on the ini- imations. This surprising sensitivity to even small devi- tial gluon mass parameter, see left panel of Fig. 9. This ations of the couplings from their universal running ex- behaviour is of a dynamical origin that is also responsible tends to the fully dynamical system with quarks, see e.g. for the existence of the scaling solution for the smallest [1, 96]. Note in this context that the quark-gluon cou- possible UV gluon mass parameter. pling αAqq¯ , (22), agrees with the ghost-gluon coupling αs defined in (21), and not the vertex coupling αAcc¯ , see Sec. II C. It can be shown in the full QCD system, that E. Discussion deviations from universality on the percent level have a qualitative impact on chiral symmetry breaking. The ori- As has been discussed already in Sec. IV A, one non- gin of this is the sensitivity of chiral symmetry breaking trivial feature of the different vertex couplings is their to the correct adjustment of physical scales, i.e. ΛQCD , quantitative equivalence for momenta down to p in all subsystems. These observations underline the rel- 2 GeV , see Fig. 4 (right panel). This property ex-≈ evance of the present results for the quantitative grip on tends the universal running of the couplings into the chiral symmetry breaking. A full analysis will be pre- semi-perturbative regime. On the other hand, the cou- sented in a forthcoming work, [96]. plings violate universality in the non-perturbative regime We close this discussion with the remark that univer- for p . 2 GeV . The universality down to the semi- sality in the semi-perturbative regime is tightly linked perturbative regime is a very welcome feature of Landau with the consistent renormalisation of all primitively di- gauge QCD, as it reduces the size of the non-perturbative vergent correlation functions. We find it crucial to de- regime and hence the potential systematic errors. In par- mand the validity of the STIs (19) only on momentum ticular, one running coupling is sufficient to describe Lan- scales considerably below the ultraviolet cutoff Λ . On dau gauge Yang-Mills theory down to momentum scales the other hand, the relations (19) are violated close to of the order of the gluon mass gap. This suggests to use the ultraviolet cutoff, due to the BPHZ-type subtraction the propagators together with the ghost-gluon vertex for schemes. This constitutes no restriction to any prac- simple semi-quantitative calculations. The above struc- tical applications, since the cutoff can always be cho- 13 sen large enough, such that no violations effects can be Nuclear Physics in the US Department of Energy’s Office found at momenta p Λ . One particular consequence of Science under Contract No. DE-AC02-05CH11231. of BPHZ-type subtraction schemes is then that the calculated renormalisation constants necessarily have to violate (19), since they contain contributions from momen- Appendix A: Gluon mass gap and irregularities tum regions close to the ultraviolet cutoff. In this section we illustrate the arguments from Sec. III. We restrict ourselves to the case of vanishing V. CONCLUSION background fields. We first show that the infrared behaviour of the scaling propagators generically induce a mass gap. We then demonstrate that the decoupling so- In this work we investigate correlation functions in lution necessitates irregular vertices for a mass gap gen- Landau gauge SU(3) Yang-Mills theory. This analysis eration due to the infrared finiteness of the decoupling is performed in a vertex expansion scheme for the effec- propagators. In App. B we show that the vertex irregu- tive action within the functional renormalisation group larities required for a decoupling mass gap cannot be of approach. Besides the gluon and ghost propagators, our diagrammatic origin. approximation for the effective action includes the self- A rather general comment is in place here: When one consistent calculation of momentum-dependent dressings is dealing with the gluon mass gap, it is crucial to care- of the transverse ghost-gluon, three-gluon and four-gluon fully take the vanishing momentum limit. In the FRG vertices. Starting from the gauge fixed tree-level pertur- approach this also means that one must first take the bative action of Yang-Mills theory, we obtain results for limit k 0 and then p 0 . the correlators that are in very good agreement with cor- → → responding lattice QCD simulations. Furthermore, the comparison of different vertex truncations indicates the Scaling solution apparent convergence of the expansion scheme. Special emphasis is put on the analysis of the dynami- The infrared-relevant part of the self-energy contribu- cal creation of the gluon mass gap at non-perturbative tion of the ghost loop to the inverse gluon propagator is momenta. Self-consistency in terms of the Slavnov- given by Taylor identities directly links this property to the re- h i Z Λ Z 1 quirement of IR irregularities in the correlation functions. (2),gh-loop 3 p 2 ΓAA (p) dq dt q 1 t The source of these irregularities is easily traced back µν ∝ −1 − to the IR-divergent ghost propagator for the scaling solution. In the decoupling-type solutions, the source of q (q + p) µ ν , (A1) these irregularities is harder to identify, where the cre- · (q2)1+κ ((q + p)2)1+κ ation of diagrammatic infrared irregularities is ruled out by general arguments. Within our truncation, we can where we inserted the infrared ghost propagator exclude irregularities of non-diagrammatic origin in the from (23) and a classical ghost-gluon vertex, i.e. (3) abc abc purely transverse subsystem. Hence it is necessary to [ΓAcc¯ ]µ (p, q) = if qµ . Ignoring the angular integra- solve the longitudinal system to answer whether the re- tion in (A1) for the moment and setting p = 0 , we find quired irregularities are generated for decoupling-type so- Z Λ h (2),gh-loopi −1−4κ lutions, which is not done in this work. Nevertheless, we ΓAA (p) dq qµqν q , (A2) are able to produce decoupling-type solutions by invok- µν ∝ ing two consistent criteria, which allow for the differen- which is infrared-divergent with 2−4κ if κ > 0.5 . This tiation between confining and Higgs-like solutions. The has to be the case in order to obtain a divergent gluon decoupling-type solutions are bound by the solution that mass gap consistent with (23). To investigate the mass shows the minimal mass gap, which is also the solution 1 ⊥ L gap, we project (A1) with 3 Πµν (p) Πµν (p) , where the with minimal back-bending of the gluon propagator. 1 − factor 3 accounts for the three modes of the transverse projection operator. We obtain h i Γ(2),gh-loop Γ(2),gh-loop (p) ACKNOWLEDGMENTS AA,⊥ − AA,L ∝

|p| Z Λ Z 1 5 2 We thank Markus Q. Huber, Axel Maas, Fabian Ren- q p 1 4t |q| t dq dt 1 t2 − − . necke, Andre Sternbeck and Richard Williams for discus- 2 1+κ 2 1+κ 0 −1 3 − (q ) ((q + p) ) sions as well as providing unpublished data. This work · (A3) is supported by EMMI, the grant ERC-AdG-290623, the FWF through Erwin-Schr¨odinger-Stipendium No. J3507- One can easily show numerically that the above integral N27, the Studienstiftung des deutschen Volkes, the DFG does not vanish in the limit p 0, but diverges with 1−2κ → through grant STR 1462/1-1, and in part by the Oﬃce of p2 . 14

0 scaling 1 decoupling -1 ] 2 -2 0.3 0.5 [GeV 0.2 k

2 -3

m 0.1 -4 0 -5 -0.1 different gluon dressings 0 0.1 1 -6 0 5 10 15 0.1 1 10 k [GeV] k [GeV]

2 (2) FIG. 10. Left: Gluon mass parameter mk = ΓAA,k(p = 0) over k . Right: Possible choices for the scaling prefactor in the gluon regulator: ZA,k(k) (black, solid), Z¯A,k(k) (red, dashed), ZÂ,k(k) (blue, dot-dashed) and ZÃ,k (green, dotted) as defined in (E3) and (E5). Independence of the results from the above choice has been checked explicitly.

Decoupling solution diagram:

Z 1 2 2 p 2 Using again the ghost-loop diagram as an example, we k∂k mgh-loop,⊥ mgh-loop,L dt 1 t show that a decoupling gluon mass gap requires irregu- − ∝ −1 − lar vertices. We choose the ghost-loop diagram since the 2 1 4t ZAcc,¯ cl(0, q , t) ZAcc,¯ cl(0, q , t) , (A6) ghost-gluon vertex has the smallest tensor space of all · − | | | | − vertices, which makes the example easy to comprehend. We checked explicitly that a similar analysis can be car- where θ = arccos(t) is the angular variable between the ried out for all diagrams and vertices contributing to the loop momentum and the gluon momentum that is taken inverse gluon propagator. This can be most easily done to zero. The dressing ZAcc,¯ cl(0, q , t) is independent of | | by assuming regular vertices (which allows to set p = 0) the angular variable t if the ghost-gluon vertex is regular. and then showing that the mass gap is zero. Thus, the mass gap contribution evaluates to zero: To make this point absolutely clear, we demonstrate Z 1 p this argument for two different tensor bases, for the basis dt 1 t2 1 4t2 = 0 . from Sec. III and for one with an explicit splitting into −1 − − transverse and longitudinal tensors. The former basis, given in (18), reads Hence, a gluon mass gap requires requires irregular vertices in the case of the decoupling solution. Since we consider differences between the vanishing (3) abc abc [ΓAcc¯ ]µ (p, q) = if qµZAcc,¯ cl(p, q) + pµZAcc,¯ ncl(p, q) , longitudinal and the transverse mass, it might seem more (A4) appropriate to split the tensor basis of the ghost-gluon vertex into a purely longitudinal and a purely transverse part. We show now that this leads to the same con- where p is the gluon and q the anti-ghost momentum. We clusion. Transverse and longitudinal projection of the assume that the ghost-gluon vertex is regular. Therefore classical tensor structure already gives us a complete or- the second tensor structure has to be less divergent than thogonal basis: 1/ p in the limit of vanishing gluon momentum, i.e., | | " (3) abc abc ⊥ [ΓAcc¯ ]µ (p, q) =if Πµν (p) qν ZAcc,¯ ⊥(p, q) lim p ZAcc,¯ ncl(q, p) = 0 . (A5) |p|→0 | | # L Note that logarithmic divergences, which for example oc- + Πµν (p) qν ZAcc,¯ L(p, q) , (A7) cur in the classical tensor structure of the three-gluon vertex and the non-classical tensor structures of the four- gluon vertex, do not suffice to violate their respective where p is the gluon and q the anti-ghost momentum. L 2 equivalents of (A5). Utilising the finiteness of the ghost The projection operators are given by Πµν (p) = pµpν /p ⊥ L dressing function and (A5), we can take the limit p 0 and Πµν (p) = 1lµν Πµν (p). Note that the basis (A7) to obtain the mass gap contribution of the ghost| | loop → contains a discontinuity− at p = 0 due to the projection 15

FIG. 11. Left: Four-gluon vertex dressing function ZA4,⊥(p, q, 0) in the tadpole configuration. The angular dependence is small compared to the momentum dependence. Right: Four-gluon tadpole configuration evaluated in the symmetric point approximation, showing a quantitative and qualitative deviation from the full calculation. operators. The mass gap contribution of the ghost dia- zero while the others are non-vanishing. Those vertex gram with this ghost-gluon vertex basis evaluates to irregularities can be generated either by back-coupling of momentum dependence or by diagrammatic infrared Z 1 2 2 p 2 singularities. We cannot observe the former in our com- k∂k mgh-loop,⊥ mgh-loop,L dt 1 t − ∝ −1 − putation of the purely transverse system. To investigate the latter case we use that the gluonic diagrams decouple 1 t2 from the infrared dynamics due to the gluon mass gap. − ZAcc,¯ ⊥(0, q , t) ZAcc,¯ ⊥(0, q , t) · 3 | | | | − Therefore we can focus on the ghost loops as possible sources of diagrammatic IR irregularities without loss of 2 generality. The ghost-gluon vertex as well as the ghost t ZAcc,¯ L(0, q , t) ZAcc,¯ L(0, q , t) . (A8) − | | | | − propagator are constant and finite in the infrared. In the Regularity (A5), implies a degenerate tensor space in the following we show explicitly that the three-gluon vertex limit of vanishing gluon momentum. The ghost-gluon does not obtain an irregular contribution from the ghost triangle. Its relevant part is given by vertex can then be fully described by ZAcc,¯ cl(0, q ) ⊥| | ≡ ZAcc,¯ L(0, q , t). Using the identity 1lµν = Πµν (p) + Z d L | | (3),gh-loop d l (l + p)µ lν (l q)ρ Π (p), we find [Γ 3 ]µνρ(p, q, r) − . µν A ∝ (2π)d (l + p)2 l2 (l q)2 − (B1) ZAcc,¯ cl(0, q ) = ZAcc,¯ ⊥(0, q ) = ZAcc,¯ L(0, q ) . (A9) | | | | | | Using (A9) we can perform the angular integration in To confirm that (B1) does not generate an irregularity |p| (A8) and find that the mass gap contribution vanishes. in the limit 0, we consider the low and high mo- |q| → We want to stress that this statement is general and mentum integration regions separately. If the loop mo- holds for any diagrammatic method. For example, the mentum l is of the order of q , then p l and the same conclusion can be drawn for the ghost-loop diagram p dependence| | in (B1) is suppressed.| | Thus| | no | | irregular of the gluon propagator Dyson-Schwinger equation that structure can be generated from this integration region. is also proportional to (A6) or (A8). Consequently, for For small loop momenta l p we have l q and | | ≈ | | | | | | the decoupling solution there can be no mass gap with the contribution to the integral in the limit |p| 0 is |l| → regular vertices. given by Z d qρ d l (l + p)µ lν Appendix B: Ghost-triangle . (B2) q2 (2π)d (l + p)2 l2 In Sec. A it is shown that the decoupling solution re- This integral can easily be solved analytically for d = quires irregular vertices. In the three gluon vertex, this 4 2 to show that it has no irregularities, which one irregularity has to occur if one momentum is sent to also− expects from a dimensional analysis of (B2). Hence, 16

FIG. 12. Left: Ghost-gluon vertex dressing function ZAcc,¯ ⊥(p, q, 0) . Right: Three-gluon vertex dressing function ZA3,⊥(p, q, 0) . we conclude, that the decoupling ghost triangle cannot YM three-point vertices are defined by generate the irregularity necessary for the dynamical gen- abc abcn o eration of a gluon mass gap. Note that the ghost triangle 3 A ,cl µνρ (p, q) = if (p q)ρδµν + perm. , develops a non-trivial pole structure in the case of the T − scaling solution, see [97]. We have verified these findings abc abc [ Acc,¯ cl] (p, q) = if qµ , (D1) numerically, and since they are in accordance with per- T µ turbation theory, we expect similar arguments to hold for and by the ghost loops contributing to higher n-point functions. abcd abn cdn 4 (p, q, r) = f f δµρδνσ + perm. , (D2) TA ,cl µνρσ

Appendix C: Numerical implementation for the four-point function. For the transversally projected ghost-gluon vertex this single tensor constitutes already a full basis and the projection is uniquely defined. The algebraic flow equations are derived using Do- However, additional allowed tensors exist in the case of Fun [98]. The projected flow equations are then traced the three-gluon and four-gluon vertices. We obtained the using the FormTracer [99], a Mathematica package that dressing functions by contracting the equations with uses FORM [100, 101]. The output is exported as opti- mized C code with FORMs optimisation algorithm [102]. ⊥ ⊥ ⊥ abc Πµµ¯(p)Πνν¯(q)Πρρ¯(p + q) A3,cl (p, q) , (D3) The calculation is performed with the frgsolver, a flexi- T µ¯ν¯ρ¯ ble, object-orientated, parallelised C++ framework de- and veloped by the fQCD collaboration [37], whose devel- ⊥ ⊥ ⊥ ⊥ abcd opment was initiated in [1]. The framework uses the Π (p)Π (q)Π (r)Π (p + q + r) 4 (p, q, r) , µµ¯ νν¯ ρρ¯ σσ¯ TA ,cl µ¯ν¯ρ¯σ¯ adaptive ordinary differential equation solver from the (D4) BOOST libraries [103], the Eigen linear algebra library [104] and an adaptive multidimensional integration rou- respectively. tine from [105] which implements [106, 107] to solve the integro-differential equations. Appendix E: Regulators

In the functional renormalisation group, the choice of Appendix D: Tensor structures of YM-vertices the regulator, together with the choice of the cutoff- independent parts of the initial effective action corre- In this section we define our conventions for the tensor sponds to defining a renormalisation scheme, for a more structures in which we expanded our vertices and the detailed discussion see [5]. Moreover, to any given order used projections. The tensor structures for the classical of a given approximation scheme there exist optimised 17

4 FRG, scaling FRG, scaling 4 FRG, decoupling FRG, decoupling DSE, scaling 3 DSE, scaling DSE, decoupling DSE, decoupling 3.5 2

1 3

0 three-gluon vertex dressing ghost-gluon vertex dressing 2.5 -1 0.1 1 10 0.1 1 10 p [GeV] p [GeV] 1 1 (a) ZAcc,¯ ⊥(p, p, ). (b) Z 3 (p, p, ). − 2 A ,⊥ − 2 4 FRG, scaling FRG, scaling 4 FRG, decoupling FRG, decoupling DSE, scaling 3 DSE, scaling DSE, decoupling DSE, decoupling 3.5 2

1 3

0 three-gluon vertex dressing ghost-gluon vertex dressing 2.5 -1 0.1 1 10 0.1 1 10 p [GeV] p [GeV]

4 FRG, scaling FRG, scaling 4 FRG, decoupling FRG, decoupling DSE, scaling 3 DSE, scaling DSE, decoupling DSE, decoupling 3.5 2

1 3

0 three-gluon vertex dressing ghost-gluon vertex dressing 2.5 -1 0.1 1 10 0.1 1 10 p [GeV] p [GeV]

(e) ZAcc,¯ ⊥(p, 0, 0) . (f) ZA3,⊥(p, 0, 0) .

FIG. 13. Left: Ghost-gluon vertex dressing function ZAcc,¯ ⊥(p, q, cos ^(p, q)) in comparison to SU(2) lattice [108–110] and DSE results [47, 92]. The lattice results are obtained from N = 324 lattices. The magenta/orange/green points (colour online) correspond to β ∈ {2.13, 2.39, 2.60} and lattice spacing a−1 ∈ {0.8 GeV, 1.6 GeV, 3.2 GeV} , respectively. Right: Three-gluon vertex dressing function ZA3,⊥(p, q, cos ^(p, q)) compared with SU(2) lattice [108–110] and Dyson- Schwinger [50] results. The coloured lattice points are taken from [108, 109] and correspond to β ∈ {2.2, 2.5} and different lattice sizes 1.4 fm < L < 4.7 fm. The lattice results shown in black are based on [108, 109] but stem from [110]. These are gained from N ∈ {244, 324} lattices with β ∈ {2.13, 2.39, 2.60} and lattice spacing a−1 ∈ {0.8 GeV, 1.6 GeV, 3.2 GeV} . The comparison with SU(2) lattice simulations is justified since the propagators of SU(2) and SU(3) Yang-Mills theory agree well for a large range of momenta [111, 112] after a respective normalisation in this regime. We rescaled all DSE results such that they match the scaling solution in the symmetric point configuration at p = 2 GeV . Note that the scaling and decoupling solutions differ in the UV just due to the different field renormalisations, cf. Fig. 3. The physically relevant couplings, given by (20), agree in the UV. 18

(2) regulators that lead to the most rapid convergence of we parameterise ΓAA,k(p) by the results, hence minimising the systematic error, see [5, 29, 30]. For recent extensions and applications rel- (2) 2 Γ (p) =: ZA,k(p) p evant for the present work see [31, 32]. In the present AA,k · work we use 2 2 =: Z¯A,k(p) p + m · k 2 2 =: ZÂ,k(p) (p + m ) , (E5) ab ˜ 2 2 2 ab ⊥ · k Rk,µν (p) = ZA,k r(p /k ) p δ Πµν (p) , 2 (2) where we define mk := Γ (0) to guarantee the unique- ab 2 2 2 ab AA,k R (p) = Z˜c,k r(p /k ) p δ , (E1) k ness of ZÂ,k. We see that these choices differ considerably below 1 GeV. For more details see Fig. 10 (right panel). In particular the naive choice ZA,k diverges since it car- for the gluon and the ghost fields, respectively. For the ries the gluon mass gap. Consequently, we freeze ZA,k ¯ shape function we choose a smooth version of the Litim at a scale k close to 1 GeV. We have checked explicitly ¯ or flat regulator [29]: that varying the value of k and n has no influence on our results.

1 1 Appendix F: Scale setting and normalisation r(x) = 1 x−1 , (E2) x − · 1 + e a When comparing to lattice results, the momentum scales as well as the global normalisations of the fields where we set a = 0.02 . It has been argued in [5] that have to be fixed. We set the scale by smooth versions of the flat regulator satisfy the functional pGeV = c pinternal , optimisation criterion put forward there. · where we choose c such that the scale of the maximum of In (E1) we multiply the regulators with scaling factors ˜ the gluon dressing 1/ZA(p) agrees with the lattice scale Z , related to the corresponding wave function renormal- from [68], which lies at p 0.955 GeV . isations of the gluon and ghost fields ≈ −1 We then rescale the gluon dressing by ZA (p) −1 → a ZA (p) with a chosen such that it minimises

2 ˜ : n ¯n 1/n X ∆xi h −1 L,−1 ZA,k = ZA,k((k + k ) ) , NZ (a) = a Z (pi) Z (pi) A ∆E2 · A − A i i ˜ 2 Zc,k := Zc,k(k) , (E3) −1 L,−1 i + a ∂pZ (pi) ∂pZ (pi) , (F1) A − A where we sum over all lattice points that fulfil 0.8 GeV ¯ ≤ where we choose n 6 and k 1 GeV . The cut- pi 4 GeV . We do not include points with smaller mo- ≈ ≈ ≤ off scale running of ZÃ is held constant below scales of menta since they can be affected by the global gauge fix- about 1 GeV as the gluon wave function renormalisation ing procedure. Points with momentum larger than 4 GeV ZA,k(p k) diverges for k 0. Separating the tensor are also not included since they might contain finite vol- ≈ → structure by ume effects. In (F1), we weight the lattice points with 2 ∆xi/∆Ei , where ∆xi denotes the distance to the next point and ∆Ei is the statistical error of the point. The (2) ab (2) ab ⊥ superscript L in (F1) marks lattice dressing functions. [ΓAA]µν (p) =: ΓAA,k(p) δ Πµν (p) , (E4) The ghost dressing is rescaled analogously.

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