Soft thermal contributions to 3-loop gauge coupling

Journal of High Energy Physics, May 2018

Abstract We analyze 3-loop contributions to the gauge coupling felt by ultrasoft (“magnetostatic”) modes in hot Yang-Mills theory. So-called soft/hard terms, originating from dimension-six operators within the soft effective theory, are shown to cancel 1097/1098 of the IR divergence found in a recent determination of the hard 3-loop contribution to the soft gauge coupling. The remaining 1/1098 originates from ultrasoft/hard contributions, induced by dimension-six operators in the ultrasoft effective theory. Soft 3-loop contributions are likewise computed, and are found to be IR divergent, rendering the ultrasoft gauge coupling non-perturbative at relative order \( \mathcal{O}\left({\alpha}_{\mathrm{s}}^{3/2}\right) \). We elaborate on the implications of these findings for effective theory studies of physical observables in thermal QCD.

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Soft thermal contributions to 3-loop gauge coupling

HJE Soft thermal contributions to 3-loop gauge coupling M. Laine 0 1 3 P. Schicho 0 1 3 Y. Schr¨oder 0 1 2 0 Casilla 447 , Chilla ́n , Chile 1 Sidlerstrasse 5 , 3012 Bern , Switzerland 2 Grupo de Cosmolog ́ıa y Part ́ıculas Elementales, Universidad del B ́ıo-B ́ıo 3 AEC, Institute for Theoretical Physics, University of Bern We analyze 3-loop contributions to the gauge coupling felt by ultrasoft (“magnetostatic”) modes in hot Yang-Mills theory. So-called soft/hard terms, originating from dimension-six operators within the soft effective theory, are shown to cancel 1097/1098 of the IR divergence found in a recent determination of the hard 3-loop contribution to the soft gauge coupling. The remaining 1/1098 originates from ultrasoft/hard contributions, induced by dimension-six operators in the ultrasoft effective theory. Soft 3-loop contributions are likewise computed, and are found to be IR divergent, rendering the ultrasoft gauge coupling non-perturbative at relative order O(αs3/2). We elaborate on the implications of these findings for effective theory studies of physical observables in thermal QCD. Thermal Field Theory; Quark-Gluon Plasma; Resummation 1 Introduction 2 Form of EQCD 4 Details on the determination of dimension-six coefficients 3 Overlapping soft/hard and ultrasoft/hard contributions 1-loop results with dimension-six operators 2-loop results with dimension-six operators Contribution from dimension-six operators in MQCD Soft and overlapping ultrasoft/soft contributions Direct soft terms up to 3-loop level Contribution from dimension-six operators in MQCD Conclusions A Spacetime and colour tensors B Basic sum-integrals C Dimension-six vertices in the S/T basis D Basic vacuum integrals E Details concerning 2-loop and 3-loop results servables [6]. As examples, they can be used for estimating the so-called transverse collision kernel related to jet quenching in a hot QCD plasma [7, 8]; soft parts of the photon and dilepton production rates from a QCD plasma [9, 10]; and the interaction rate experienced by neutrinos in an electroweak plasma [11]. Following standard terminology, we refer to – 1 – the “soft” effective theory as EQCD, whereas the “ultrasoft” theory containing only the magnetostatic modes is called MQCD (cf. e.g. refs. [ 12–15 ]). The latter has been argued to give e.g. the leading non-perturbative contribution to jet quenching [ 16 ]. In the QCD context it is known, however, that EQCD fails to describe the full theory close to the phase transition or crossover temperature (Tc). This is obvious when light quarks are present: EQCD contains only gluonic degrees of freedom, and displays no remnant of the flavour symmetries that underlie the chiral transition. For pure-glue theory, the reason for the breakdown is more subtle. Even though the center symmetry that drives the transition in the imaginary-time formulation [17] is not explicit in EQCD, remnants of it are generated dynamically [18]. However the dynamical re-generation is incomplete, and a 3d lattice study in which soft EQCD dynamics was treated non-perturbatively did not achieve satisfactory agreement with thermodynamic functions obtained from full 4d lattice simulations [19]. One purpose of this paper is to demonstrate analytically that power-suppressed dimension-six operators, truncated from the super-renormalizable EQCD description, play an essential role in soft and ultrasoft observables, and are therefore a likely culprit for EQCD’s failure close to Tc. More concretely, we determine the MQCD gauge coupling in terms of the EQCD gauge coupling and mass parameter up to 3-loop level, including the 1- and 2-loop contributions of all dimension-six operators; the result is contained in eqs. (3.13), (3.14) and (4.4). Our presentation is organized as follows. After reviewing the form of EQCD and rederiving the coefficients of its dimension-six operators in section 2, we determine overlapping soft/hard and ultrasoft/hard contributions to the ultrasoft gauge coupling in section 3. In terms of four-dimensional Yang-Mills we go up to 3-loop level; this implies 2-loop level in effects originating from dimension-six operators, which are themselves generated by 1-loop diagrams. A 3-loop computation of soft effects, as well as of overlapping ultrasoft/soft contributions, is presented in section 4, whereas conclusions are collected in section 5. Spacetime and colour tensors, tensor-like 1-loop sum-integrals, Feynman rules related to dimension-six operators, d-dimensional vacuum integrals, and some lengthier results, are collected in five appendices, respectively. 2 2.1 Form of EQCD Super-renormalizable part The super-renormalizable truncation of the dimensionally reduced “electrostatic” QCD, called EQCD, is defined by the action SEQCD[A] ≡ Z X 41 Fiaj Fiaj + 12 DiabAb0 DiacAc0 + 2 m2E A0aA0a λ 4 κ 4 + E XabcdA0aAb0Ac0A0d + E A0aA0aAb0Ab0 . (2.1) Here RX ≡ T1 Rx, Fiaj ≡ ∂iAja − ∂j Aia + gEf abcAibAc, Dab ≡ δab∂i − gEf abcAic, A0a is an adjoint j i scalar, Xabcd is defined in eq. (A.6), Latin indices take values i, j ∈ {1, . . . , d}, we have – 2 – m2E = gB2Nc P PZ ′ (d − 1)2 P 2 + O g B4 , gE2 = g B2 1 + gB2Nc + O g 4 B , PZ ′ 25 − d P 6P 4 Z ′ 1 P λ E = gB4(d − 1)2(3 − d)P 3P 4 + O g B6 , κ E = O gB4Nf , where gB2 = g2µ 2ǫ(1+O(g2)) is the bare coupling of the original four-dimensional theory, µ is the scale parameter introduced in the context of dimensional regularization, and g2 ≡ 4παs is the renormalized coupling. By ΣR ′P we denote a sum-integral over P , with the prime indicating that the Matsubara zero mode is omitted. A 1-loop re-derivation of eqs. (2.2)– (2.4) can be found as a side product of section 2.3; 2-loop expressions were obtained in ref. [20]; the 3-loop level has been reached for m2E [21] and gE2 [22, 23]. For our higher-loop computations in section 3, it is helpful to express the dependence on λE and κE through the dimensionless combinations in mind d ≡ 3 − 2ǫ, and repeated indices are summed over. We employ a convention in which the fields Aia and A0a have the same dimensionality as in four-dimensional Yang-Mills theory. Then explicit factors of 1/T and T appear in configuration and momentum space integration measures, respectively, where T is the temperature. Focussing on pure SU(Nc) gauge theory,1 i.e. suppressing contributions proportional to the number of fermion flavours (Nf ), the parameters appearing in eq. (2.1) have the expressions 4gE4 κE(Nc2 + 1) 2gE2Nc , + + 10κE , gE2Nc 10λEκE + g4 E 2κ2E(Nc2 + 1) gE4Nc2 . We note in passing that fundamental representation couplings often used in the literature, viz. λ(E1)(Tr [A20])2 + λ(E2)Tr [A40], are given by λ(E1) = 3λE/2 + κE and λ(E2) = λENc/2. The theory can be renormalized through g E2 = gE2Rµ 2ǫ + δgE2 , m2E = m2ER + δm2E , and similarly for the scalar couplings. Within the super-renormalizable truncation, the counterterms take the forms [24, 25] δgE2 = 0 , δm2E = gE2RNcT 4π 2 κ2 − 4λ 4ǫ . 1We omit fermions for simplicity because they carry non-zero Matsubara freqencies and thus generate The starting point for our analysis is the 3-loop determination of gE2 from fourdimensional Yang-Mills theory [22, 23]. It is helpful to display the result in the form of a background field effective action [26]. After gauge coupling and wave function renormalization through vacuum counterterms, refs. [22, 23] found an expression containing a logarithmic (1/ǫ) divergence, 2 Bia(q) Bjb(r) δab δ(q + r) q2δij − qiqj ZB + δZB , Here ζn ≡ ζ(n) and µ¯2 ≡ 4πµ 2e−γE . The renormalized gauge coupling is given by gE2R = g2/ZB, and the corresponding counterterm by δgE2 = −g2µ 2ǫδZB + O(g10). We stress that eqs. (2.11) and (2.12) are gauge independent [27]. An essential technical goal of our investigation is to demonstrate how the divergence in eq. (2.12) is cancelled by overlapping soft/hard and ultrasoft/hard contributions, originating from dimension-six operators within EQCD and MQCD, respectively. At this point we would like to clarify why such logarithmic divergences (which are “universal”, i.e. present in any regularization scheme) originate first at 3-loop level. In three dimensions, 1-loop graphs may contain power divergences but no logarithmic divergences. Logarithmic divergences first originate at 2-loop level. However, within the super-renormalizable truncation of EQCD, they lead to the counterterms in eq. (2.9), i.e. the gauge coupling is finite. Divergences affecting the gauge coupling can only emerge when dimension-six operators are added to EQCD. Given that dimension-six operators are themselves generated by 1-loop diagrams, the divergences correspond to the 3-loop level in terms of the fundamental theory. In section 3, where effects originating from integrating out the hard scale are considered, 3-loop level corresponds to the relative accuracy O(g6), whereas in section 4, where effects originating from integrating out the soft scale are at focus, the expansion parameter is ∼ g, and the 3-loop effects are of relative magnitude O(g3). 2.2 Dimension-six operators The dimension-six operators that can be added to eq. (2.1) were determined in ref. [28]. We represent the operators as matrices in the adjoint representation. Letting Greek indices take the gauge coupling as the same gE as appears inside Fiaj and Diab, the dimension-six action – 4 – (2.10) (2.11) (2.12) −if abcDµcdFµνd . The value of the sum-integral over P evaluates to AabBbcCca, where (A0)ab ≡ −if abcAc , (Fµ 0)ab 0 ≡ −if abcFµ c0, and (Dµ Fµν )ab ≡ The colour trace refers to the adjoint representation: tr{AB} ≡ AabBba, tr{ABC} ≡ The values of ci were given for d = 3 in ref. [28]. We need to generalize the expressions to d dimensions, because some of the operators lead to divergent loop integrals at the second stage of our analysis (cf. section 3). Beyond leading order, the coefficients are also functions of g2, but these contributions are of higher order than the effects that we are interested in. As mentioned in section 2.1, we are also suppressing effects proportional to N . As a first step, it may be realized that the operator basis in eq. (2.13) is redundant: it can be verified that Z X tr igEhF0µ Fµν Fν0 + A0(Dµ Fµν )F0ν i + gE2 h− A02Fµν2 + A0Fµν A0Fµν i = 0 . (2.15) 2 Therefore a simultaneous change of the coefficients (cinew ≡ ci + δci, i = 4, . . . , 7) has no physical effect, provided that δc4 = δc5 = −2δc6 = 2δc7 . In particular, we could tune c7 to zero as was done in ref. [28],2 by choosing δc7 = −c7. Then eq. (2.16) implies that the other coefficients should appear in the combinations c(new) = c4 − 2c7 , 4 c(new) = c5 − 2c7 , 5 c(new) = c6 + c7 . 6 In the following we keep both c5 6= 0 and c7 6= 0 for generality; this offers for a good crosscheck in that only the combinations of eq. (2.17) appear in any physical expressions. In order to determine the values of the coefficients ci, we have computed 1-loop contributions to the 2-point, 3-point, 5-point and 6-point functions of the Matsubara zero modes in the background field Feynman gauge [26].3 Salient details from this computation are 2Tuning c5 to zero would yield eq. (2.13) more elegant and simplify a number of subsequent computations. 3In a general gauge, several of the coefficients depend on the gauge fixing parameter, but we have checked that the logarithmic divergences that we are ultimately interested in do not. – 5 – ζ ζ ′ 3 f (2.13) (2.14) (2.16) (2.17) presented in section 2.3. Matching the 2 and 3-point vertices yields c 1 = Adding the 5-point vertex permits for us to fix the combinations in eq. (2.17) as The coefficient c10 is also evanescent and can be determined from the 6-point vertex; we find c10 = (5 − d)(3 − d)(d − 1)2/180 but this does not contribute to any of our results. For d = 3 eqs. (2.18)–(2.20) agree with ref. [28]. (Expressions for a general d were derived in ref. [29], but unfortunately a rather different notation was employed.) 2.3 Details on the determination of dimension-six coefficients In this section we provide some details on the determination of the coefficients listed in eqs. (2.18)–(2.20). The derivation of eq. (2.13) is most conveniently formulated with the background field method [26], and as a reminder the gauge potentials are denoted by Bµa. The object computed is the background field effective action, ΓEQCD[B], whereby the vertices are automatically symmetrized in the appropriate way. After a field redefinition, viz. Aia = Bia(1 + O(gB2)) and A0a = B0a(1 + O(gB2)), the result is identified with SEQCD[A]. We choose to work directly in momentum space, with the background fields denoted by Bµa(q). The momenta q have spatial components only: qµ ≡ δµi qi . (2.21) Specific tensors are defined for showing the dependence of the vertices on spacetime and colour indices; these are summarized in appendix A. The structure naturally emerging from the computation is one in which there are Lorentz-invariant structures (δµν etc.) and additional terms that only appear for the zero components of the gauge potentials; the latter are identified through the tensors Tµν ≡ δµ 0δν0 etc. Results for various 1-loop sum-integrals in this basis are given in appendix B. Computing the 2-point and 3-point vertices in the background field gauge, we obtain the 1-loop correction Γ(E2Q+C3D)[B] = gB2Nc Bµa(q) Bνb (r) δab δ(q + r) γµν(2)(q) 2! 3! + igB3Nc Bµa(q) Bνb (r) Bρc(s) f abc δ(q + r + s) γµν(3ρ) (q, r, s) , (2.22) – 6 – 1/(πT )2, the 2-point vertex reads where summations and integrations are implied, and T Rq δ(q) ≡ 1. Expanding in 1/P 2 ∼ γ(2)(q) = P µν + O 1 (d − 1)(d − 3)q2 6P 4 , (2.23) where c1 and c2 have the values in eq. (2.18). The term proportional to ΣR ′P P12 yields the parameter m2E in eq. (2.2), whereas the terms proportional to ΣR ′P P14 yield wave function cor2 rections. The existence of a term ΣR ′P TνµP 4q indicates that temporal and spatial components of the gauge potentials need to be normalized differently. For the 3-point vertex a similar computation leads to γµν(3ρ) (q, r, s) = P P Z ′ (25 − d)qρδµν + (d − 1)(d − 3) qρTµν P 4 − − + 24c1 qµ qρrν + 12c3 qν (rµ qρ − qµ rρ) + O 1 , (2.24) where c3 and c4 − c5 have the values shown in eq. (2.18).4 The terms proportional to ΣR ′P P14 can be partly accounted for by wave function corrections; the remainder yields the effective gauge coupling of eq. (2.3). The same result for gE2 is obtained both from a purely spatial vertex (∼ qρδµi δνi) and from a vertex mixing two A0a’s and one Aia (∼ qρTµν ). The 4-point vertex can similarly be written as Γ(E4Q)CD[B] = g 4 4B! Bµa(q) Bνb (r) Bαc(s) Bβd(t) δ(q + r + s + t) γµν(4α)βabcd(q, r, s, t) , (2.25) where γµν(4α)βabcd(q, r, s, t) = P Z ′ P X{ab}{cd} 2(d − 1)2(3 − d)Tµναβ P 4 +X[ab][cd] 4(25 − d)δµα δνβ + 8(d − 1)(d − 3)Tµα δνβ + O 1 (2.26) The notations X{ab}{cd} and X[ab][cd] are defined in appendix A. The term proportional to to wave function corrections and gE2. The dimension-six part of the 4-point vertex is rather yields λE in eq. (2.4), whereas the other terms proportional to ΣR ′P P14 correspond complicated (it is shown in appendix C) and we have not used it for determining ci’s. 4This representation is not unique, cf. the comments below eq. (C.3). P 4 – 7 – denote gluons and dotted lines ghosts. The diagrams have been drawn with Axodraw [30]. Proceeding finally to the 5-point vertex, we find no contribution ∼ ΣR ′P P14 . The contribution of the dimension-six operators from eq. (2.13) can be written as Γ(E5Q)CD[B] = Bµa(q) Bνb (r) Bρc(s) Bαd (t) Bβe (u) δ(q + r + s + t + u) ×nX{ab}[cde] h−c1 δραδνβ + 4c1 δρβδνα − c1 δρν δαβ P Z ′ 8igE5sµ P −c2 Tραδνβ + 4c2 Tρβδνα − c2 Tρν δαβ −c2 δραTνβ + (c5 − 2c7) δρβTνα − c2 δρν Tαβ − c9Tρναβ i +X[ab]{cde} h(5c1 − 3c3) δραδνβ + (3c3 − 4c1) δρβδνα + 3c1 δρν δαβ +(c2 − c4 + c5) Tραδνβ + (c4 − c5) Tρβδνα + 3c2 Tρν δαβ +(c2 − c4 + c5) δραTνβ + (c4 − 4c2 − 2c7) δρβTνα + (c5 − c2 + 2c6) δρν Tαβ + (c8 − c9) Tρναβ i o . (2.27) We have computed the corresponding Feynman diagrams, shown in figure 1. Making use of momentum conservation and appropriate symmetrizations, and identifying g gB2(1 + O(gB2)), we obtain precisely the same structure from Feynman diagrams. There are 20 independent terms that permit for a crosscheck of eq. (2.18) and, most importantly, for E2 = a unique determination of the combinations appearing in eqs. (2.19) and (2.20). 3 Overlapping soft/hard and ultrasoft/hard contributions In EQCD, the gauge field components A0a have turned into massive adjoint scalar fields when the non-zero Matsubara modes were integrated out (cf. eq. (2.1)). Our goal now is to integrate out the massive A0a, and thereby construct the MQCD action. Its superrenormalizable part has the form of the spatial part of eq. (2.1). We denote it by SMQCD[A] ≡ Z X 14 Fiaj Fiaj , (3.1) even though Fiaj now contains a different gauge coupling than eq. (2.1): Fiaj = ∂iAja − ∂j Aia + gMf abcAibAjc. The main goal of this section is to determine the contributions to gM2 that originate from the dimension-six operators in eq. (2.13). These are termed soft/hard (sections 3.1 and 3.2) and ultrasoft/hard (section 3.3) contributions. We note that in analogy with eq. (2.13), SMQCD also has a dimension-six part, δSMQCD. It is given in eq. (3.16) and discussed in more detail in section 3.3. – 8 – from eq. (2.13), denoted by a filled blob. The adjoint scalar fields are denoted by solid lines. action, ΓMQCD[B]. In particular, we consider its quadratic part, 21 Bia(q) Bja(−q)(q2δij − qiqj ) Z B + δZB , (3.2) where δZB collects any possible divergences. In the background field gauge, Γ is gauge invariant in terms of B [26]. Consequently the 3-point and 4-point vertices are fully determined by eq. (3.2). After a subsequent field redefinition, this implies that Z B determines the gauge coupling of MQCD: g M2 = gE2R µ 2ǫ Z−1 − gE2R µ 2ǫ δZB + δgE2 + O(g10) . B (3.3) Here δgE2 is from eq. (2.8). The following discussion is carried out in terms of ZB and δZB. When the field A0a is integrated out and one vertex from eq. (2.13) is included, we expect to find terms of the types Z B + δZB = 1 + PZ ′ gE2Nc mERgE2RNcT 4π #(5) + (gE2RNcT )2 (4π)2 #(6) + . . . , (3.4) where #(6) may contain logarithms. The corresponding effects are of O(g5) and O(g6) in terms of the original QCD coupling. The latter effect is comparable to eq. (2.12). Before proceeding let us explain why we consider “2-loop soft × 1-loop hard” contributions, i.e. 2-loop graphs with one insertion of dimension-six operators, but not “1-loop soft × 2-loop hard” ones. In terms of Z B defined in eq. (3.2), “1-loop hard” gives a factor ∼ g2/T 2, “1-loop soft” gives a factor ∼ g2T mER ∼ g3T 2, and “2-loop soft” gives a factor ∼ (g2T )2 ∼ g4T 2. The overall effects of these orders are ∼ g5, g6, cf. eq. (3.4). In contrast “2-loop hard” would give dimension-six operators proportional to ∼ g4/T 2. The overall effect from “1-loop soft × 2-loop hard” would therefore be ∼ g7, i.e. of higher order than our computation. The same applies to dimension-eight operators, whose coefficients are ∼ g2/T 4 and who get a further suppression factor . g2T m3ER ∼ g5T 4 from soft effects. 3.1 1-loop results with dimension-six operators The 1-loop contribution to Z B from dimension-six operators originates from the graphs shown in figure 2. The vertices related to dimension-six operators have been indicated with a filled blob; we refer to them as “Chapman vertices”. In appendix C the vertices are written in a form convenient for computing these graphs. The 2-point vertex is parametrized 4-point vertex through ψ1, . . . , ψ44 and ω1, . . . , ω35, cf. eq. (C.5). q2/m2E, all of them can be related to a single 1-loop tadpole integral, denoted by (3.6) µ¯ 2mE ω26 d q 4 m2E I(mE) +O(ǫ2) . (3.5) Inserting the values of the coefficients in terms of the ci’s from appendix C, the terms proportional to m2Eδij and qiqj drop out as required by gauge invariance, and we are left with δΓ(M2Q)CD[B] = Bia(q) Bjb(r) δab δ(q + r) (q2δij − qiqj ) PZ ′ gE4Nc2 (c1 + c2) + 3c3 + (c4 − 2c7) + 4(c6 + c7) . (3.7) Inserting the coefficients c1, . . . , c7 from eqs. (2.18) and (2.19) and setting d → 3, the curly brackets evaluate to d→3 lim . . . = − lim d→3 d4 − 13 d3 + 312 d2 − 6404 d + 25424 1440 = − 875 144 . (3.8) The corresponding contribution to ZB is shown on the first row of eq. (3.13). 3.2 2-loop results with dimension-six operators At 2-loop level, the contributions of the 2-point, 3-point and 4-point Chapman vertices to Z B can be extracted from Feynman diagrams shown in figures 3–5. In addition the 5-point and 6-point Chapman vertex also contribute. The general expressions for these, eqs. (C.19) and (C.21), respectively, and the corresponding diagrams are shown in figure 6. +1 +2 +1 +1 +1 + +2 +2 + +1 +1 + 1 2 + 1 2 +2 1 2 + +2 + 1 2 +1 +1 + 1 4 + 1 2 +2 +1 +2 + 1 2 +2 +2 +1 +1 +2 +1 +2 +2 + 1 8 +1 +1 1 2 denoted by filled blobs. Adjoint scalars are denoted by solid lines. Graphs involving closed massless loops, which do not contribute to the matching, have been omitted. HJEP05(218)37 +1 +2 +1 + 1 2 + 1 4 + 1 4 (the notation is as in figure 3). (the notation is as in figure 3). man vertices (the notation is as in figure 3). In order to display the result, we introduce a 2-loop “sunset” integral, H (mE) ≡ Z T 2 p,q (p2 + m2E)(q2 + m2E)(p + q)2 = m2Ed−6Γ(1 − d2 )Γ(2 − d2 )T 2 (d − 3)(4π)d = 3−2ǫ T 2µ −4ǫ (4π)2 1 4ǫ µ ¯ 2mE 1 2 + ln + + O(ǫ) . (3.9) H(mE) × 4d m2E δij C1 + 4d q2δij − qiqj C2 + 4d qiqj C3 + O q 4 m2E where C1, C2, C3 are given in appendix E in terms of the coefficients η1, . . . , χ16. Inserting the values of the coefficients from appendix C, we find that C1 and C3 and terms proportional to α in C2 cancel. The remaining contribution reads δΓ(M2Q)CD[B] = −Bia(q) Bjb(r) δab δ(q + r) q2δij − qiqj 3d = −Bia(q) Bjb(r) δab δ(q + r) q2δij − qiqj × + + (d − 3)(d − 4)2(d3 − 10d2 + 23d − 44)(c1 + c2) 6d(d − 5)(d − 7) parameters, we obtain where in the last step we made use of eqs. (2.18)–(2.20). We note that the evanescent operators parametrized by c8 and c9 do not play a role for d ≈ 3, because the coefficients with which they contribute in eq. (3.11) themselves vanish for d → 3. Setting d = 3 − 2ǫ, inserting eqs. (2.14), (3.7) and (3.9), and going over to renormalized Remarkably, setting gE2R = g2 (1 + O(g2)), the divergence in eq. (3.14) cancels 1097/1098 of the coefficient of 1/ǫ in eq. (2.12). The remaining 1/1098 can be expressed as δZB + δZB = where in the round brackets we have isolated the master integral in eq. (2.14). , (3.10) (3.13) (3.15) As already alluded to below eq. (3.1), there are dimension-six operators also in MQCD. These originate from the purely spatial part of eq. (2.13), and also from 1-loop effects within EQCD, as will be discussed in section 4. The corresponding action can be written as6 Z X n δSMQCD[A] = 2gM2 tr C1 (DiFij )2 + igMC3 Fij FjkFki , o (3.16) where (recalling gM2 = gE2 (1 + O(g))) the hard contribution is δCi = ΣR ′P ci/P 6. The dimension-six operators in eq. (3.16) give a contribution to physical observables HJEP05(218)37 determined by MQCD, such as the spatial string tension or “magnetostatic” screening masses. Given that MQCD is a confining theory, these effects cannot be computed analytically. We would like to know, however, whether the MQCD dynamics can give an ultraviolet (UV) divergent contribution, compensating against the term in eq. (3.15). In order to determine the UV divergence, we employ a trick similar to that in ref. [ 31 ]. All infrared (IR) contributions are “shielded” by employing the propagators hAka(p)Alb(q)i ≡ δabδ(p + q) p2 + m2G δkl − α pkpl p2 + m2G , hca(p)c¯b(q)i ≡ δabδ(p − q) p2 + m2G , (3.17) where ca, c¯b are ghost fields, α is a gauge parameter, and mG ≡ gM2T /π is a fictitious mass. Once again, we compute a background field effective action, now denoted by ΓIR[B] given that the most IR fluctuations have been accounted for. We extract from it a 2-point function like in eq. (3.2). The technical implementation follows that in sections 3.1 and 3.2. Most contributions that we find are α-dependent and void of physical significance. For instance, the 1-loop result has a structure similar to eq. (3.6) but with mE → mG: δΓI(R2)[B] α=0 = 21 Bia(q) Bjb(r) δab δ(q + r) gM4Nc2 I(mG) × q2δij − qiqj 3 − 11C1 + 18 C3 + O(ǫ) q 4 m2G . (3.18) This result is finite and proportional to mG and vanishes when we send mG → 0. However, at 2-loop order a non-trivial and gauge-independent result emerges. Writing the contribution from Chapman vertices in a form reminiscent of eq. (3.10), we get 4d m2G δij D1 + 4d q2δij − qiqj D2 + 4d qiqj D3 + O q 4 m2G . (3.19) The function H3 is the three-mass variant of eq. (3.9), cf. eq. (D.10), and has the same UV divergence, viz. T 2µ −4ǫ/[(4π)24ǫ]. The coefficients Di contain a part ∝ H/H3 = 1 + O(ǫ). 6There are many alternative representations, for instance tr FijFjkFki 2(idN−c2) f abcǫijkFeiaFjblFkcl = iN2c f abcFeiaFejbFicj = 2(idN−c2) f abcǫijkFeiaFejbFekc, where we denoted the dual field strength by Feia ≡ ǫi2jk Fjak and defined ǫijkǫlmn ≡ δil(δjmδkn − δjnδkm) + δim(δjnδkl − δjlδkn) + δin(δjlδkm − = iN2c f abcFiajFjbkFkci = δjmδkl). For ǫ → 0, D1,3 are of O(ǫ) and yield no divergence, whereas D2 has a finite α-independent part: Substituting C3 → ΣR ′P c3/P 6, inserting c3 from eq. (2.18), and setting gM2 = g2µ 2ǫ (1 + O(g)), yields a gauge-independent UV divergence and logarithmic part: lished our main technical goal, demonstrating that the IR-divergence in eq. (2.12) is fully cancelled by soft/hard and ultrasoft/hard contributions from dimension-six operators. 4 Soft and overlapping ultrasoft/soft contributions In section 3 we considered the soft/hard contributions to the MQCD effective action, cf. eq. (3.4). However, there are other contributions to ZB, namely those associated with the purely “soft” contributions from the scale mE. In order to distinguish these from the effects considered in section 3, we denote them by ZeB. For this section, we can take the super-renormalizable truncation in eq. (2.1) as a starting point, and mE as the only scale being integrated out. 4.1 Direct soft terms up to 3-loop level couplings was added in ref. [20]):7 Up to 2-loop level, the value of ZeB was determined in ref. [32] (the dependence on scalar Z eB = 1 + gE2RNcT 48πmER + gE2RNcT 16πmER We now turn to the 3-loop contribution. The determination of ZeB is a rather straightforward exercise in computer-algebraic methods for loop integrals. The Feynman diagrams were generated with QGRAF [33]. After expanding in the external momentum and projecting onto the transverse and longitudinal polarizations, we have to deal with vacuum-like master integrals. The subsequent simplifications, making use of renamings of integration variables and integration-by-parts (IBP) identities [34, 35], have been programmed in FORM [36]. The values of the 3-loop master integrals can be found in refs. [ 31, 37 ] and are given in eqs. (D.12) and (D.13). As a crosscheck, we have carried out two independent computations, whose results coincide O(gE6Nc3), where the integrals are given in eq. (D.1). perfectly. Our final “bare” expression reads8 12 Bia(q) Bjb(r) δab δ(q + r) q2δij − qiqj have carried out the same computation by shielding all masses like in eq. (3.17), but with mG → mE. Then only the divergence proportional to 4(κ2 − 4λ) remains. This indicates that the divergence not containing scalar self-couplings is purely of IR origin. We can envisage two possible sources for the IR divergence. One is related to ultrasoft contributions of the same type as in section 3.3; these are analyzed in section 4.2. The other is related to the mass parameter m2E. It is well known that the physical Debye mass, defined as a screening mass related to a “heavy-light” state, is non-perturbative starting at next-to-leading order [38, 39]. Our m2E is not such a physical mass but rather a Lagrangian parameter. Nevertheless, m2E can still be considered IR sensitive at O(gE4RT 2). Indeed, if we compute the 2-point function of A0a at zero momentum, and shield all masses like in eq. (3.17), we find the UV divergence cancelled by the mass counterterm in eq. (2.9). In contrast, if we compute the 2-point function without IR-shielding, we find an additional 1/ǫ-divergence proportional to gE4RT 2, which depends on the gauge parameter α. This is an IR divergence, i.e. ∼ gE4RT 2/ǫIR. If we naively insert an ambiguity of this type into the 1-loop term in eq. (4.1) and re-expand up to 3-loop order, the result is − On the non-perturbative level, 1/ǫIR would turn into a multiple of ln(c mG/mER), where c is a non-perturbative constant and the scale mG was defined around eq. (3.17). Keeping in mind this expectation, we renormalize eq. (4.2) by employing the proper mass counterterm from eq. (2.9). The UV divergences proportional to κ2 − 4λ duly cancel, and we find the 3-loop result ZeB(3) + δZeB(3) = gE2RNcT 16πmER 3 + 1 + 1 6ǫ + + 8(κ2 − 4λ) 3 945 ln 9 µ¯ 2mER 2(23510 − 12600ζ2 − 1101 ln 2) 52λ + 24λ2 − κ1(5 − 8 ln 2) + κ2(19 − 24 ln 2) . (4.4) 8The full d-dimensional form is given in appendix E, cf. eqs. (E.4)–(E.12). ground field gauge. Wiggly lines denote ultrasoft gluons and solid lines adjoint scalars. Contribution from dimension-six operators in MQCD Parallelling section 3.3, let us finally consider contributions from ultrasoft effects to the gauge coupling, in the presence of dimension-six operators in MQCD. The action has the form in eq. (3.16), with the coefficients now completed to include the soft contribution: HJEP05(218)37 Ci = P Z ′ c P Pi6 + T Z c˜i p (p2 + m2E)3 , i = 1, 3 . (4.5) The spatial integral appearing is related to that in eq. (3.5) as shown by eq. (D.1), Z T p (p2 + m2E)3 = 2(4π) 2 d mdE−6Γ(3 − d2 )T 3−2ǫ T µ −2ǫ = 32πm3E µ¯ 2mE 1 + 2ǫ 1 + ln + O(ǫ2) . (4.6) Including the overall prefactor from eq. (3.16) and the integral from eq. (4.6), the new contributions to the coefficients of the dimension-six operators are ∼ gM2T /m3E at 1-loop level. Including a fictitious IR-regulator like in eq. (3.17), the 1-loop contribution from these operators to ZeB comes with a factor ∼ gM2T mG and vanishes for mG → 0, whereas the 2-loop contribution comes with a factor ∼ gM4T 2 and can yield a contribution ∼ gM6T 3/m3E ∼ O(g3) to ZeB. 2-loop contributions to the coefficients of dimension-six operators would be ∼ gM4T 2/m4E and therefore lead to effects suppressed by ∼ O(g4). Dimension-eight operators, whose coefficients are ∼ gM2T /m5E, lead to effects suppressed by ∼ gM10T 5/m5E ∼ O(g5). According to eq. (2.24), the value of c˜1 can be inferred from the 2-point and that of c˜3 from the 3-point vertex of the background field effective action. To be sure that no operators got overlooked, we have also determined them from the 5-point vertex, cf. the spatial part of eq. (2.27), which leads to several independent crosschecks (the diagrams are shown in figure 7). We find that the results are related in a curious way to the d-dependence of c1 and c3 in eq. (2.18):9,10 1 120 , c˜ 1 = − c˜ 3 = − 1 180 . (4.7) 9To our knowledge these values were first obtained for d = 3 by P. Giovannangeli (unpublished, 2005), along lines that have recently been documented in ref. [40]. 10We note in passing that even though the c˜i contribution in eq. (4.5) is parametrically larger by O(1/g3) than the ci contribution, the large value of c1 in eq. (2.18) implies that numerically c1 and c˜1 give similar contributions if g2 ∼ 2. If g2 ≫ 1, C1 becomes positive. Inserting these values into eq. (3.20), and substituting gM2 = gE2Rµ 2ǫ (1 + O(g)), we find a gauge-independent UV divergence and logarithmic part: µ¯ 3mG This implies that the counterterm needed in MQCD reads δZeB(3) = Obviously, eq. (4.8) does not match the divergence in eq. (4.4). In other words, if we subtract the part needed to serve as δZeB(3) from eq. (4.4), an IR divergence remains. In terms of the coefficient β introduced in eq. (4.3), it amounts to β = − 1135 . Let us stress that we have verified the gauge independence of this result. Therefore we are left to speculate that a nonperturbative mass ambiguity of the type discussed around eq. (4.3) prohibits a purely perturbative determination of ZeB(3), and thus of gM2 in terms of gE2R and mER at O(gE6RT 3/m3ER). gE2RNcT 3 415ǫ . 16πmER The main technical ingredient of this investigation was the analysis carried out in section 3. We considered dimension-six operators induced by integrating out the “hard” momenta ∼ πT from thermal QCD [28]. Specifically, we computed at 1-loop and 2-loop levels the influence of these operators on the gauge coupling felt by ultrasoft (magnetostatic) modes. Remarkably, including UV divergences originating both from “soft” loops at the Debye scale mE ∼ gT and “ultrasoft” loops at the non-perturbative scale ∼ g2T /π, we observed an exact cancellation of the IR divergence found in a 3-loop determination of the EQCD gauge coupling (cf. eq. (2.12)) [22, 23]. This represents a nice crosscheck of the effective theory setup as a whole. As a second technical ingredient, discussed in section 4, we considered the “soft” contributions to the ultrasoft gauge coupling. We determined direct 3-loop effects (cf. eq. (4.4)) and compared them with overlapping ultrasoft/soft contributions originating from dimension-six operators induced by integrating out the soft momenta ∼ mE (cf. eq. (4.8)). This time only a partial cancellation of soft IR divergences against ultrasoft/soft UV divergences was observed. As a culprit, we speculate that a non-perturbative ambiguity of the soft scale within EQCD sets an upper bound on the accuracy with which effects depending on mE can be determined within perturbation theory. This may be surprising insofar as no such problem was met in 3-loop or 4-loop studies of the EQCD vacuum energy density [ 15, 31 ]. However, the present quantity is different, being not directly a physical observable but rather an effective Lagrangian parameter (the MQCD gauge coupling gM2). On a more general level, the main conclusions that we draw are as follows: (i) Even if the colour-electric scale mE ∼ gT is formally larger than the colour-magnetic scale ∼ g2T /π, it does play an essential role in the IR dynamics. Concretely, in terms of the IR divergence found by integrating out the hard scale ∼ πT , the colour-electric scale is 1097 times more important than the colour-magnetic scale (cf. eq. (3.14)). (ii) Dimension-six operators need to be included in EQCD if good precision is required. Indeed, as we have demonstrated analytically (cf. point (i)), they do influence the IR dynamics of the system. This is a possible reason for why the super-renormalizable truncation of EQCD fails close to Tc even in pure Yang-Mills theory [19]. (iii) Apart from the indications in point (i) that the scale mE is important, we also find trouble if we try to integrate it out. The reason could be that EQCD is a confining theory, and that physics at the scale m2E should in general be affected by non-perturbative ambiguities of O(g4T 2/π2). Once mE is integrated out, some remnant of these ambiguities may remain, if the parameters of MQCD are determined up to the corresponding relative precision. It would be interesting to find a way to determine the leading non-perturbative contribution to gM2 through lattice methods, even if this requires the simultaneous inclusion of the 1/m3E-suppressed MQCD dimension-six operators in eq. (3.16). Acknowledgments This work was partly supported by the Swiss National Science Foundation (SNF) under grant 200020-168988, by the FONDECYT under project 1151281, and by the UBB under project GI-172309/C. A Spacetime and colour tensors Because the presence of a heat bath breaks Lorentz invariance, we need to introduce separate notation for spatial and zero spacetime indices. The full Kronecker symbol is denoted by δ µν ≡ T µν + Sµν , T µν ≡ δµ 0δν0 , S µν ≡ δµi δνi . We also introduce the totally symmetric tensors (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) For the colour indices, it is helpful to denote Tµνρσ ≡ δµ 0δν0δρ0δσ0 , Tµνρσαβ ≡ δµ 0δν0δρ0δσ0δα0δβ0 , δµνρσ ≡ δµν δρσ + 2 permutations , δµνρσαβ ≡ δµν δρσδαβ + 14 permutations . Xa1a2...an ≡ f mna1m1 f m1a2m2 · · · f mn−1anmn , as well as the symmetrized versions 1 2 X{a1...a2}... ≡ Xa1...a2... + Xa2...a1... , X[a1...a2]... ≡ Xa1...a2... − Xa2...a1... . (A.7) These objects satisfy Xanan−1...a2 a1 = (−1)nXa1a2...an−1an , Xa1a2...an−1an = Xa2...an−1ana1 . It follows that 1 2 X{a1a2}[a3a4] = X{a1a2}{a3a4a5} = X[a1a2][a3a4a5] = X{a1a2a3}[a4a5a6] = 0 . (A.8) Therefore we can write Xa1a2a3a4 = X{a1a2}{a3a4} + X[a1a2][a3a4] , Xa1a2a3a4a5 = X{a1a2}[a3a4a5] + X[a1a2]{a3a4a5} , Xa1a2a3a4a5a6 = X{a1a2a3}{a4a5a6} + X[a1a2a3][a4a5a6] . It may furthermore be noted that These are needed for the computations in section 2.3. P 8 P P P P 8 P 10 Z ′ Pµ Pν = PZ ′ (1 − d) Tµν + δµν , P Z ′ Pµ Pν = PZ ′ (3 − d) Tµν + δµν , P Z ′ Pµ Pν = PZ ′ (5 − d) Tµν + δµν , P Z ′ Pµ Pν PρPσ = P P Z ′ (3 − d)(1 − d)Tµνρσ Z ′ Pµ Pν PρPσ = P P Z ′ (5 − d)(3 − d) Tµνρσ P Z ′ Pµ Pν PρPσPαPβ = P P P 12 Z ′ (5 − d)(3 − d)(1 − d) Tµνρσαβ 480P 6 (3 − d) (Tµν δρσ + 5 permutations) + δµνρσ (5 − d) (Tµν δρσ + 5 permutations) + δµνρσ , 24P 4 48P 6 480P 6 480P 6 (5 − d)(3 − d) (Tµνρσ δαβ + 14 permutations) (5 − d) (Tµν δρσαβ + 14 permutations) + δµνρσαβ Xa1a2a3 = − Nc f a1a2a3 , Employing the notation defined in eqs. (A.1)–(A.5), the following relations can be estabP P P P P P 2 8 + + + + 2P 2 4P 4 6P 6 24P 4 48P 6 (A.9) (A.10) (A.11) (A.12) (A.13) (A.14) In section 2.3 we displayed (parts of) the vertices originating from eq. (2.13) in a basis in which spacetime indices appear in the form similar to appendix B. For the considerations of section 3, it is advantageous to employ a basis in which the spatial and temporal indices are strictly separated from each other. This can be implemented with the tensors Sµν ··· and Tµν ···, defined in eq. (A.1). In this section we display all the Chapman vertices originating from eq. (2.13) with such a notation. The 2-point Chapman vertex reads δSE(2Q)CD = Aµa (q) Aνa(−q) η1 q2 q2Sµν − qµ qν + η2 q4Tµν o , P PZ ′ gE2Nc n η2 = 2(c1 + c2) . where The 3-point Chapman vertex becomes δSE(3Q)CD = Aµa (q) Abν (r) Acρ(s) f abc δ(q + r + s) PZ ′ igE3Nc P 2 i ξ8 q2qρ + ξ9 q2rρ + ξ10 s qρ , (C.3) 2 i (C.1) (C.2) HJEP05(218)37 where qµ qν qρ and qµ qν qρ +qµ qν rρ −qµ rν qρ = −qµ (qν sρ +rν qρ) actually vanish as can be seen by the relabelling (r ↔ s, ν ↔ ρ, b ↔ c). Therefore any change δξ1 or any simultaneous change δξ2 = −δξ3 has no effect. It can be checked that eqs. (3.6) and (E.1)–(E.3) are invariant in these transformations. A representation of the coefficients can be chosen as ξ1 = 0 , ξ5 = −3c3 , ξ2 = 2c3 , ξ6 = 8c1 − 3c3 , ξ3 = −4c1 , ξ4 = −2c3 , ξ7 = 3c3 − 4c1 , ξ8 = −4c2 − 3c3 − c4 + c5 , ξ9 = 8c1 + 4c2 − 3c3 − c4 + c5 , ξ10 = 3c3 − 4c1 + c4 − c5 . (C.4) The 4-point vertex amounts to δSE(4Q)CD = Aµa (q) Abν (r) Acα(s) Adβ(t) δ(q + r + s + t) P Z ′ g4 E P X{ab}{cd} Sµα Sνβ ψ1 q2 + ψ3 q · r + Tµα Sνβ ψ4 q2 + ψ5 r2 + ψ6 q · r + Sµν Sαβ ψ10 q2 + ψ12 q · r + Sµν Tαβ ψ16 q2 + ψ18 q · r + T µν Sαβ ψ13 q2 + ψ15 q · r + Tµναβ ψ19 q2 + ψ21 q · r + Sµα + Tµα ψ22 qν qβ + ψ23 qν rβ + ψ24 rν qβ + ψ25 rν rβ ψ26 qν qβ + ψ27 qν rβ + ψ28 rν qβ + ψ29 rν rβ +Sαβ ψ38 qµ qν + ψ39 qµ rν + ψ40 rµ qν +Tαβ ψ42 qµ qν + ψ43 qµ rν + ψ44 rµ qν +X[ab][cd] ψi → ωi , where some coefficients have been dropped because they can be converted to the remaining ones through trivial renamings of indices and integration variables. The values are ψ1 = 0 , ψ3 = −8c1 , ψ13 = 0 , ψ15 = 0 , ψ16 = −8c1 − 2c5 + 4c7 , ψ18 = −8c1 − 2c5 − 4c6 , ψ22 = −8c1 , ψ23 = 12c1 , ψ24 = −4c1 , ψ25 = 4c1 , ψ28 = −4c1 + 12c2 − 4c5 + 8c7 , ψ29 = 4c1 + 4c2 , ψ30 = 4c1 , ψ31 = −4c1 , ψ34 = 4c1 + 4c2 , ψ35 = −4c1 − 4c2 , ψ38 = 4c1 , ψ39 = 0 , ψ40 = 8c1 , ψ42 = 4c1 − 4c2 + 2c5 − 4c7 , ψ43 = 8c2 − 2c5 + 4c7 , ψ44 = 8c1 − 8c2 + 4c5 + 4c6 − 4c7 , ω1 = −16c1 , ω3 = 8c1 − 12c3 , ω4 = −16c1 − 16c2 , ω5 = −16c1 − 4c5 + 8c7 , ω6 = 16c1 − 24c3 − 8c4 + 4c5 + 8c7 , ω22 = −24c1 , ω23 = −44c1 + 24c3 , ω24 = −12c1 , ω25 = 4c1 , ω26 = −24c1 − 24c2 , ω27 = −44c1 − 12c2 + 24c3 + 8c4 − 8c5 , ω28 = −12c1 − 28c2 + 4c5 − 8c7 , ω29 = 4c1 − 12c2 + 4c5 − 8c7 , ω30 = 0 , ω31 = 20c1 − 12c3 , ω34 = 0 , ω35 = 20c1 + 20c2 − 12c3 − 4c4 + 8c7 . In the case of ωi, all coefficients associated with operators containing Sαβ or Tαβ vanish, because of antisymmetry. The coefficients of the 4-point vertex listed above are not independent. Indeed momentum conservation leads to relations between the different structures defined in eq. (C.5), which implies that certain linear combinations of the coefficients couple to null operators. In the spirit of eq. (2.16), these ambiguities can be listed as transformations (Θ1 ...Θ12) whereby a simultaneous modification of the coefficients as indicated below has no physical (C.5) HJEP05(218)37 (C.6) where (C.7) (C.8) (C.9) (C.10) (C.11) (C.12) (C.13) (C.14) (C.15) (C.16) (C.17) (C.18) (C.19) (C.20) This list may not be complete. It can be checked that the expressions in eqs. (3.6) and (E.1)–(E.3) are invariant in these transformations. The 5-point Chapman vertex reads δSE(5Q)CD = Aµa(q)Abν(r)Acρ(s)Adα(t)Aeβ(u)δ(q + r + s + t + u) PZ′igE5sµ P P6 × X{ab}[cde] κ1 SραSνβ + κ2 SρβSνα + κ3 SρνSαβ +κ4 TραSνβ + κ5 TρβSνα + κ6 TρνSαβ +κ7 SραTνβ + κ8 SρβTνα + κ9 SρνTαβ + κ10 Tρναβ +X[ab]{cde} κi → λi , Θ12 : δψ34 = δψ35 = −δψ42 = −2δψ43 = −2δψ44 . κ1 = −8c1 , κ2 = 32c1 , κ3 = −8c1 , κ4 = −8c1 − 8c2 , κ5 = 32c1 + 32c2 , κ6 = −8c1 − 8c2 , κ7 = −8c1 − 8c2 , κ8 = 32c1 + 8c5 − 16c7 , κ9 = −8c1 − 8c2 , κ10 = 16c1 + 8c5 − 16c7 − 8c9 , λ1 = 40c1 − 24c3 , λ2 = −32c1 + 24c3 , λ3 = 24c1 , λ4 = 40c1 + 8c2 − 24c3 − 8c4 + 8c5 , λ5 = −32c1 + 24c3 + 8c4 − 8c5 , λ6 = 24c1 + 24c2 , λ7 = 40c1 + 8c2 − 24c3 − 8c4 + 8c5 , λ8 = −32c1 − 32c2 + 24c3 + 8c4 − 16c7 , λ9 = 24c1 − 8c2 + 8c5 + 16c6 , λ10 = 32c1 + 16c5 + 16c6 − 16c7 + 8c8 − 8c9 . Finally the 6-point vertex can be expressed as δSE(6Q)CD = Z X where χ1 = −4c1 + 2c3 , χ2 = 16c1 − 6c3 , χ3 = −4c1 , χ4 = −2c3 , χ5 = −8c1 + 6c3 , χ11 = −4c1 − 4c2 , χ12 = −6c3 − 2c4 + 4c7 , χ13 = −8c1 − 8c2 + 6c3 + 2c4 − 4c7 , χ16 = −12c1 − 6c5 − 4c6 + 8c7 − 2c8 + 4c9 , χ17 = −2c10 . (C.22) D Basic vacuum integrals For the computations of section 3 various d-dimensional vacuum integrals are needed. At 2-loop level their results can be expressed in terms of H defined in eq. (3.9), multiplied by rational functions of d. For notational simplicity we denote the mass by m, let Δp ≡ p2+m2, and omit the trivial factor T included in eq. (3.9). Making use of the integral Z 1 p Δpn = md−2nΓ(n − d2 ) , (4π) d2 Γ(n) m−2 = − 2(d − 3)H d − 2 , Z 1 p,q Δ2pΔq = (d − 3)H . factorized integrals can be expressed as Z 1 A sunset integral with a power of the massless propagator reads Z p,q ΔpΔq(p + q)2n = m2d−2n−4Γ( d2 − n)Γ(n + 2 − d)Γ2(n + 1 − d2 ) . (4π)dΓ( d2 )Γ(2n + 2 − d) Z 1 Z m2 p,q ΔpΔq(p + q)2 = H , p,q ΔpΔq(p + q)4 = − HJEP05(218)37 (D.1) (D.2) (D.3) (D.4) A sunset integral with a power of a massive propagator reads Z 1 p,q ΔpnΔq(p + q)2 = m2d−2n−4Γ(1 − d2 )Γ(n + 1 − d2 ) . (d − n − 2)(4π)dΓ(n) , 1 , + Tensor integrals can be reduced to scalar integrals through (Sµν Sαβ + Sµα Sνβ + Sµβ Sνα)hp4i d(d + 2) (Sµν Sαβ + Sµα Sνβ + Sµβ Sνα)hp2p · qi d(d + 2) (Sµα Sνβ + Sµβ Sνα)hd(p · q)2 − p2q2i S µν Sαβh(d + 1)p2q2 − 2(p · q)2i d(d − 1)(d + 2) d(d − 1)(d + 2) (D.5) . (D.6) (D.7) (D.8) , (D.9) (D.10) . (D.11) (D.12) (D.13) E Details concerning 2-loop and 3-loop results For completeness we report here technical results related to sections 3 and 4 that were too lengthy to fit the presentation in the main text. where h. . .i represents a generic rotationally invariant expectation value, and Sµν ≡ δµi δνi. In the considerations of section 3.3, another variant of the sunset integral was encountered, H3 ≡ Z p,q ΔpΔqΔp+q . It can be written in terms of the hypergeometric function 2F1 [41, 42], H3 = − 3(d − 2) 4(d − 3) 2F1 4 − d 2 , 1; 5 − d 3 2 d−5 2πΓ(5 − d) Γ( 4−2 d )Γ( 6−2 d ) Z m−2 At 3-loop level we need the values of two “basketball” integrals (cf. e.g. refs. [ 31, 37 ]): p,q,r ΔpΔq(p + r)2(q + r)2 B2 ≡ B4 ≡ Z Z = − mµ −6ǫ = − mµ −6ǫ (4π)3 2m 1 2m 1 2ǫ 1 6ǫ 6ǫ 1 ǫ p,q,r ΔpΔqΔp+rΔq+r + 4 + ǫ 26 + + O(ǫ2) , 25ζ2 4 + 8 − 4 ln 2 + ǫ 52 + 2 17ζ2 − 32 ln 2 + 4 ln2 2 + O(ǫ2) . Consider first the coefficients C1,C2 and C3, defined in eq. (3.10). Because of the general way in which we have parametrized the Chapman vertices (cf. appendix C), the expressions for these contain substantial “redundancies”, which we reproduce here in full. This permits for very strong crosschecks, as discussed e.g. in the context of eqs. (C.7)– (C.18) for the quartic Chapman vertex. The expressions read C1 = −8(d − 1) (2d + 3)η1 + 2d(d + 2)η2 + (d + 1)ξ5 − (d + 2)ξ6 − ξ7 + dξ10 −8(d − 1) (d + 1)(d + 2)ξ8 − (d2 + 3d + 1)ξ9 +2(d − 1) 4(ψ3 − ψ30 + ψ31) − 2(2d + 3)ψ10 + 4dψ12 − 3ψ22 + ω22 (d − 1) ψ6 − ω6 + ψ28 − ω28 + 2(5d + 1)(ψ34 − ψ35) − 2(d2 + 3)ω4 − (5d + 3)ω26 −10d(d − 3) κ10 − λ10 − 4χ14 − 2χ15 − 2χ16 + 4ψ19 − 2ψ21 , (E.1) d − 2 d − 2 d − 2 d − 2 d − 2 d − 2 3(d − 5) − 3(d − 5) 6(d − 5) 12 +(d − 2)(d − 3)(d − 7)(ψ4 + 3ω4 − 2ψ13) − (d3 − 8d2 + 51d − 84)ψ6 3(d − 5) +2(d2 − 8d + 9)ψ15 + d(23d − 21)ψ16 − 2(4d2 − 5d + 2)ψ18 3 3(d − 5) 6 12(d − 5) 2 −2 d6 − 13d5 + 49d4 − 83d3 + 208d2 − 114d − 156 η2 3(d − 5)(d − 7) − 4d5 − 55d4 + 226d3 − 335d2 + 484d − 336 ξ8 3(d − 5)(d − 7) + 4d5 − 55d4 + 226d3 − 323d2 + 388d − 252 ξ9 3(d − 5)(d − 7) −4 d4 − 10d3 + 25d2 − 51d + 51 η1 − 2 2d4 − 31d3 + 120d2 − 111d + 36 ξ10 (d − 1) (3d + 7)ψ1 − 4(ψ3 − ψ30 + ψ31) + 2(2d + 3)ψ10 − 4dψ12 + 3ψ22 d − 5 (d − 1) ψ28 − 2(d − 1)ω1 − 2ω22 − ω28 + d(37d − 39)ψ5 − d(3d − 1)ω5 C2 = 2 18(d − 1)ξ4 + (d + 1)(d2 − 9d + 12)(ξ6 − ξ5) + 12(d2 − 3)ξ7 4(d − 5) (d2 + 7d − 12)(ψ26 − 2ψ34 + 3ω26) + d(d + 1)ψ35 − 2(d − 2)ψ44 (d3 − 16d2 + 59d − 52)ω6 − (d − 2) (d2 − 33)ψ27 − (d2 − 24d + 87)ω27 6 12(d − 5) α(d − 1) ψ28 − ω28 − 2ω35 − 8(2η1 + ξ5 + ξ7) 4(d − 1) (d − 1)(η1 + ξ5) + 2(ξ2 + ξ3 + ξ4 + ξ6) + (d + 1)ξ7 6 d − 5 (d − 1) (3d + 7)(ψ1 + ψ25) − 4(ψ3 + ψ23 + ψ24 + ψ31) + 2(2d + 3)(ψ10 + ψ38) (d − 1) 4d(ψ12 + ψ39 + ψ40) − 10(ψ22 + ψ30) + (d − 1)(ω1 + ω25) d − 5 d − 5 +2d(d − 1) 3(ψ5 + ψ29) + 4(ψ16 − ψ18 + ψ42 − ψ43 − ψ44) − ω5 − ω29 . (E.3) After substituting the coefficients from appendix C, we get eq. (3.11). As a second ingredient, we report the full d-dimensional version of eq. (4.2). The result can be expressed as r1 + r˜1 (d) I3(mE) + r2(d) m2EB2(mE) + r3 + r˜3 (d) m2EB4(mE) , where the pure gauge contributions are parametrized by r1(d) = − r2(d) = r3(d) = (d − 2)p1(d) 384(d − 10)(d − 8)(d − 7)(d − 6)(d − 5)(d − 4)(d − 3)2(d − 1)d , (3d − 10)(3d − 8)p2(d) 128(d − 3)(d − 1)d(2d − 11)(2d − 9)(2d − 7) (3d − 10)(3d − 8)p3(d) 256(d − 10)(d − 8)(d − 6)(d − 4)(d − 1)d , , with the non-factorizable polynomials p1(d) = 12d 12 − 628d 11 + 14447d 10 − 193505d 9 + 1689420d 8 − 10234582d 7 +929595256d 2 − 791686464d + 314842752 , p2(d) = 12d 7 − 308d 6 + 3175d 5 − 17441d 4 + 57347d 3 where I, B2 and B4 are the master integrals from eqs. (3.5), (D.12) and (D.13), respectively. (E.2) (E.4) o (E.5) (E.6) (E.7) (E.8) (E.9) (E.10) In terms of the couplings from eqs. 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M. Laine, P. Schicho, Y. Schröder. Soft thermal contributions to 3-loop gauge coupling, Journal of High Energy Physics, 2018, 37, DOI: 10.1007/JHEP05(2018)037