nPI Resummation in 3D SU(N) Higgs Theory
PPrepared for submission to JHEP n PI Resummation in 3D SU( N ) Higgs Theory Mark C. Abraao York, Guy D. Moore
McGill University Department of Physics3600 Rue UniversityMontr´eal, QCH3A 2T8
Abstract:
We test the utility of the n PI formalism for solving nonperturbative dynamicsof gauge theories by applying it to study the phase diagram of SU( N ) Higgs theory in 3Euclidean spacetime dimensions. Solutions reveal standard signatures of a first order phasetransition with a critical endpoint leading to a crossover regime, in qualitative agreementwith lattice studies. The location of the critical endpoint, x ∼ .
14 for SU(2) with afundamental Higgs, is in rough but not tight quantitative agreement with the lattice. Weend by commenting on the overall effectiveness and limitations of an n PI effective actionbased study. In particular, we have been unable to find an n PI gauge-fixing procedurewhich can simultaneously display the right phase structure and correctly handle the large-VEV Higgs region. We explain why doing so appears to be a serious challenge. a r X i v : . [ h e p - ph ] J u l ontents ( N ) Yang-Mills + Higgs theory in the n PI formalism 5
Ans¨atze N ) Higgs theory 24B Self-energies computed in dimensional regularization 25 B.1 One-loop gluon self-energy 26B.2 One-loop Higgs self-energy 27B.3 One-loop ghost self-energy 27B.4 Two-loop topologies 27B.5 Two-loop gluon self-energy 29B.6 Two-loop Higgs self-energy 31
Thermal or off-equilibrium dynamics of hot nonabelian gauge theories have applicationsin heavy ion physics (see e.g. [1] and references therein) and in early Universe cosmology[2, 3]. An important feature of nonabelian gauge theory is asymptotic freedom, whichmakes the coupling smaller at shorter length (or higher energy) scales. Naively this meansthat perturbation theory should work better at higher temperatures, where the relevantenergy scale T is large. In fact this is only partly the case. As shown already in 1980[4, 5], the behavior on scales (cid:96) > /T is in fact that of a 3-dimensional theory, which goesrapidly to strong coupling at longer distances. Therefore the long-distance (cid:96) (cid:29) T behaviorof nonabelian gauge theory is strongly coupled at any temperature.– 1 –here is a dearth of tools for computing real-time dynamics of nonabelian gauge the-ories when nonperturbative physics is involved. This frustrates efforts to understand dy-namics, both at strong coupling and at weak coupling. A method which has shown muchpromise in scalar and Yukawa theories is the n -particle irreducible ( n PI) method [6–12].While the motivation for developing such methods lies largely in the hopes that they canbe applied to nonabelian gauge theories [13, 14], almost no work in that direction has oc-curred yet. There have been some results in abelian theories [15–18], and some argumentsas well as a computation demonstrating that a 3-particle-irreducible treatment of QCDwould automatically capture the leading perturbative effects relevant for transport andequilibriation [19]. But no one has made a concerted effort to apply the n PI approach tononabelian gauge dynamics.In a previous paper [20] we took a first step in this direction, by applying the 3-particleirreducible (3PI) method to the study of Yang-Mills theory in three Euclidean dimensions.The main motivation was to test out the methodology in a context where we do have othercomputational tools at our disposal, so we can appraise whether it is successful beforeundertaking the more challenging problem of applying 3PI methods to dynamics. Butif successful, the 3PI method could still have real utility as a potentially faster or moreefficient method of studying 3D theories, which are in fact of intrinsic interest. In particular,the 3D theory captures the large-distance nonperturbative physics of weakly-coupled hotgauge theories alluded to above, at least at the thermodynamical level.Unfortunately, in that work we were only able to solve the 3PI problem for pure gaugetheory in 3 dimensions. There are few long-distance sensitive observables in that theory, sowe lacked gauge-invariant measurements to match to (lattice) nonperturbative calculationsin 3D Yang-Mills theory and did not find effective ways to test whether the method “works.”In the present paper we intend to address this by extending our n PI treatment to 3-dimensional Yang-Mills Higgs theory, a theory which has a nontrivial phase structure. Wewill study whether the n PI approach can successfully reproduce the phase structure ofthe theory, a nontrivial and nonperturbative test of the technique. We emphasize thatour purpose is as a test of whether the n PI approach in gauge theory can reproducenonperturbative phenomena. The goal is not to improve our understanding of 3D Yang-Mills Higgs theory, which has been thoroughly studied using lattice gauge theory techniques[21–23]. In the remainder of the introduction, we will explain both 3D Yang-Mills Higgstheory and the n PI approach in a little more detail.The study of three dimensional nonabelian gauge theory is motivated by its relationshipto electroweak theory and QCD via dimensional reduction [24–28]. At high temperature T (cid:29) Λ QCD , QCD exhibits a natural separation of scales g T (cid:28) gT (cid:28) T so that non-zerobosonic and all fermionic Matsubara modes become heavy compared to the soft scalesof the theory. These modes can be integrated out to obtain an effective 3 dimensionaldescription of the soft physics, which is precisely SU(3) Yang Mills coupled to an adjointscalar with gauge coupling g = g T and mass m A = g ( N/ N f / T (identified withthe 0-mode of the A component of the 4D gauge field). If one is only interested in physicsat the supersoft scales, this can be taken one step further by integrating out the A fieldto yield pure 3D Yang-Mills. – 2 –ang-Mills theory, QCD and electroweak theory are known to undergo a phase transi-tion [29–32] over certain ranges of the model parameters. Naturally, for physical values ofthese parameters, one would ask whether we are in a first order, second order or crossoverregime. 3D effective models could potentially shed some light on this matter, except thatfor QCD, where the effective 3D description is an SU(3)+adjoint Higgs theory, T c ∼ Λ QCD .Thus, in the vicinity of the QCD phase transition (or crossover) the effective descriptionbreaks down, since the underlying assumption of weak 4D coupling and a separation ofscales is not valid. A 3D effective model is still useful for studying the nonperturbativeinfrared dynamics of hot QCD, just not at temperatures in the vicinity of the scale Λ
QCD .However, the situation is different for electroweak theory near its phase transition.The effective 3D description of electroweak theory resulting from dimensional reductionis an SU(2) × U(1) gauge theory coupled to both fundamental and adjoint scalars. Theadjoint scalars arise via the dimensional reduction, while the fundamental scalar is identifiedwith the 4D Higgs field. In practice, an accurate study of the 4D theory does not requiresuch elaborate field content; rather, quantitative predictions can be made by considering themuch simpler SU(2) + fundamental case [21–23, 33, 34]. Then, as a further refinement, onemay study the effects due to the inclusion of an adjoint field [35, 36], as well as a U(1) gaugefield [37]. Or, in the context of GUTs, the model with an SU(5) or SU(3) × SU(2) gaugegroup may be of interest [38, 39]. These models have received a fair amount of attentionin the past due to the significance of a phase transition on electroweak baryogenesis [2, 3].For the models considered, a first order phase transition at physical values of the Higgsmass has been ruled out.We show a cartoon of the phase diagram of SU( N ) fundamental-Higgs theory in Fig. 1.It is parametrized by two dimensionless variables x and y , describing the scalar self-couplingand renormalized mass respectively, normalized to the appropriate power of g . In termsof the 4D thermal theory, x is predominantly set by m H /m W and y is predominantly setby the temperature. Therefore an evolution in temperature in the 4D theory correspondsto a nearly vertical line on the phase diagram; moving horizontally is changing the vacuumparameters of the theory. The diagram has a first order line which ends at a second-order3D Ising universality [23] endpoint at ( x c , y c ); if y is varied holding x < x c fixed, thesystem encounters a first order phase transition. But the transition is not a symmetry-breaking phase transition, and no order parameter distinguishes one phase from the other,similar to the liquid-gas phase transition (which is in the same universality class). We havealso indicated upper and lower metastability lines, which show how deeply the system can“superheat” or “supercool” before encountering spinodal instability. The locations of theselines cannot be rigorously defined; but they will enter in our analysis so we indicate themfor completeness.At small x the phase transition can be studied perturbatively by computing the one-loop effective potential for the Higgs vacuum expectation value (VEV). Indeed one findsthat in this region, the phase transition is first order. However, the perturbative treatmentthen goes on to predict a first order phase transition for all values of x ! Higher-order com-putations [40, 41] show that the perturbative expansion parameter is actually x , indicatingthat perturbation theory fails for large values of x , which turns out to mean x > ∼ / xy ( x c , y c ) y c ( x ) y + ( x ) y − ( x ) Figure 1 . Phase diagram for SU( N ) Higgs theory, showing a first order line terminating at acritical point. The dashed lines indicate the appearance of metastable configurations in the effectivepotential. Therefore, the end point and crossover must be resolved nonperturbatively . Since thelattice has already provided us with a very accurate determination of the phase diagram,we are able to use these results to test the reliability and accuracy of an alternative non-perturbative approach to the lattice, namely that of n PI resummation in a gauge theorysetting. In this paper we will study the application of the n PI (specifically 2PI) formalismto SU( N ) Higgs theory.In the context of a hot gauge theory, the use of an n PI based resummation schemeis primarily motivated by the extremely poor convergence of a weak-coupling expansion[42], since it provides a systematic procedure for reorganizing a perturbation series. Ourapproach here is along a trajectory which differs from many of the previous works on thissubject mentioned earlier. That is, by applying the n PI formalism to SU( N ) Higgs theoryour goal is to directly solve the resulting self-consistent, Schwinger-Dyson (SD) resemblant integral equations in a manner reminiscent of [20], and then subsequently derive gauge-invariant quantities from the solutions.An n PI effective action Γ[ ¯ φ, G, ... ] generates equations of motion for n -point resummedvertices by variation with respect to these n -point functions. We will specifically considera three-loop truncation of the case n = 2, which in terms of diagrams can be interpretedas resumming one- and two-loop self-energy topologies. (In [20] we treated the pure-gaugetheory at the n = 3 level, that is, including as well a self-consistent one-loop resummationof three-point vertices. The result established that the corrections to these vertices aresmall, so we have avoided this technical complication.) By solving the resulting “SD”equations, we can subsequently compute the gauge-invariant scalar condensate (cid:104) φ † φ (cid:105) as afunction of the parameters x and y on the phase diagram. Then, at a specific value x , fromthe behavior of (cid:104) φ † φ (cid:105) over a range of y we can infer whether or not we are in the crossoveror first order phase transition region. This will allow us to bracket and locate the criticalend point.We will present the technical details of the computation for a single complex scalarfield in representation R of SU( N ). Results will be given for N = 2 (fundamental repre- The integral equations of motion obtained via the n PI formalism are not strictly speaking Schwinger-Dyson equations. However, they are qualitatively similar, so we will often refer to them as SD equationsthroughout the text. – 4 –entation) in Landau and Feynman gauges, however it should be noted that the methodstraightforwardly generalizes to the inclusion of additional scalar fields, higher representa-tions, and larger gauge groups.The text is organized as follows: in Section 2 we will present the three-loop truncated2PI effective action for SU( N ) Higgs theory, as well as the SD equations that it generates.Additionally, some details pertinent to regularization and renormalization will be reviewedhere. In Section 3 we will present certain extensions to the algorithm described in [20]which are needed to solve the 2PI equations of motion numerically. In Section 4 we willgive an overview of the results, as well as derived quantities such as the scalar condensateand the location of the critical end point. Finally, throughout Sections 3 and 4 we will alsodiscuss the properties of the effective action, and comment on the overall effectiveness ofthe method. ( N ) Yang-Mills + Higgs theory in the n PI formalism
It is useful to begin by reviewing a number of the basic conventions that are used through-out. It should be assumed that T aR is a generator of some representation R of SU( N ). Thefundamental and adjoint representations are denoted by F and A respectively, and d R isthe dimension of R , for instance d F = C A = N . We haveTr T aR T bR = T R δ ab (2.1) T sRim T sRmj = C R δ ij , (2.2)and additional group theory identities needed in this computation can be found in [43].Following gauge fixing, the Lagrangian can be divided into a Yang-Mills component and aHiggs component, L YM = 12 Tr F µν F µν + 12 ξ ( ∂ µ A aµ ) + ∂ µ ¯ c a ∂ µ c a − gf abc ∂ µ ¯ c a c b A cµ (2.3) L φ = ( D µ φ ) † ( D µ φ ) + ( m + δm ) φ † φ + λ ( φ † φ ) (2.4)so that L = L YM + L φ (in general covariant gauge as written). Note that we have not fixedto R ξ gauge; our gauge fixing only acts on the gauge degrees of freedom. We will discussthis more in Subsection 2.3.We define the dimensionless ratios x = λg y = m g , (2.5)which is the same as the definition introduced in Ref. [21] and commonly used throughoutthe literature. In Eq. (2.4) an additive counterterm has been explicitly included to cancelthe divergent two-loop self-energy graphs (its value is given in Appendix B). This leads toa scale dependence in m and y accordingly; for a fundamental SU(2) Higgs we have dyd log µ = − π (cid:16) x − x (cid:17) . (2.6)– 5 –he mass renormalization scale is fixed at µ = g throughout. For phenomenologicalapplications, the relation between x, y and 4D theory parameters are [21] x = − . . (cid:16) m H . (cid:17) (2.7) y = 0 . . (cid:16) m H . (cid:17) − . (cid:16) m H . (cid:17) − . m T (2.8)assuming a value of g = 2 / The 2PI effective effective action Γ[ G ij ] is formally defined as the Legendre transform ofthe generating function of connected diagrams W [ K ij ] with respect to a two particle source[7]. Using the generic label Φ i for fields, W [ K ij ] reads W [ K ij ] = − log (cid:90) D [Φ] e − S − Φ i K ij Φ j . (2.9)Even correlation functions can be obtained by differentiation with respect to K ij . Forinstance, δW [ K ij ] δK ij = 12 G ij (2.10)yields the two-point function G ij . For the two-point functions of SU( N ) Higgs theory, wecan assume that G ij is proportional to the color identity of the corresponding species, andhence so is K ij . Then, for a rotationally symmetric Lagrangian (cid:104) Φ i (cid:105) = (cid:82) D [Φ] Φ i e − S − Φ i K ij Φ j (cid:82) D [Φ] e − S − Φ i K ij Φ j = 0 , (2.11) i.e. , the presence of K ij does not alter the global rotational invariance of the originalLagrangian. So in fact, Eq. (2.10) generates the connected two-point function. The con-sequences of this statement in the context of a spontaneously broken gauge theory will bediscussed towards the end of this section, but for now we can proceed with the Legendretransform Γ[ G ij ] = K ij δW [ K ij ] δK ij − W [ K ij ] . (2.12)In setting K ij = 0, equations of motion for G ij are obtained from the stationarity condition δ Γ[ G ij ] δG ij = 0 . (2.13)The solutions we seek correspond to extrema of Γ[ G ij ]. Specializing now to the fieldcontent of SU( N ) Higgs theory, we can write Γ = Γ YM + Γ φ and explicitly state the loopexpansion, which we will truncate at three loops. Γ YM is defined so that it contains thosediagrams encountered in the pure Yang-Mills problem while Γ φ contains the additional– 6 –iagrams which arise when a single arbitrary representation Higgs field is included. Usingthe diagrammatic notation G µν = (2.14)∆ = (2.15) D = , (2.16)without loss of generality we write the two-point functions as G µν ( p ) = 1 p − Π T ( p ) T µν + ξp − ξ Π L ( p ) L µν , (2.17)∆( p ) = 1 p − Σ( p ) , (2.18) D ( p ) = 1 p + m − Π φ ( p ) . (2.19)All vertices appearing in the 2PI effective action are at tree level. These are drawn as p , a , µ p , a , µ p , a p , a = g V a a a a µ µ ( p , p , p , p ) (2.20)with the corresponding expressions given in Appendix A. Finally, the Higgs mass renormal-izes at two loops; it is necessary to subtract the divergence with an additive countertermof the form m = m φ + δm , with the corresponding vertex= − δm . (2.21)Explicitly factoring out minus signs due to ghost loops, we haveΓ YM = 12 Tr log G −
12 Tr G (0) µν G µν − Tr log ∆ + Tr[∆ (0) ] − ∆+ 112 + 18 −
12+ 148 + 124 + 18 − − . (2.22)– 7 –or an n -loop pure Yang-Mills planar diagram, the tracing over internal color indicesgenerically results in an overall color factor of ( N − N n − . Furthermore, since an n -loopvacuum bubble is also proportional to g n − , one finds as earlier that factors of g alwaysappear in the form of the ’t Hooft coupling g N . Hence, for the pure gauge problem, thenatural scale is not g , but rather g N . The Higgs contribution isΓ φ = Tr log D − Tr[ D (0) ] − D + 12 + 12 ++ 12 + 14 ++ 13 + 14 + . (2.23)These diagrams have a somewhat more complicated dependence on N (the associated colorfactors are stated in Table 1). We will nevertheless continue to use g N as the standardmass scale, but for clarity, units of g N will be explicitly stated throughout.The power of the 2PI formalism becomes apparent when we perform the variation ofΓ with respect to G T , G L , ∆ and D . For instance, from δ Γ /δD = 0, we have − D − ( p ) + D (0) − ( p ) = Π φ ( p ) (2.24)with (omitting charge arrows)Π φ ( p ) = + 2 + 12+ + + 2+ 2 + 12 + + . (2.25)Equations of the type Eq. (2.24) / Eq. (2.25) are generically referred to in this work as SDequations, and the topologies which appear in Eq. (2.25) correspond to the loop order of the– 8 – b cA B C D E ( a ) d A T R ( b ) 2 d A T R ( c ) d R (1 + d R )( A ) d A T R ( C R − C A )( B ) d A T R C A ( C ) d A T R (2 C R − C A )( D ) 2 d R (1 + d R )( E ) d A T R (4 C R − C A ) Table 1 . Color factors for the two and three-loop Higgs topologies. truncation of the effective action. By solving this equation self-consistently in a three-looptruncation, we fully resum one- and two-loop self-energy topologies to all orders.Eq. (2.25) contains terms that are linearly and logarithmically divergent; in dimen-sional regularization, only the logarithmic divergences appear explicitly as 1 /(cid:15) ’s, and theseare subtracted by the counterterm. This implies that the entire computation must be per-formed in MS, which requires the analytic continuation of these integrals to D dimensions.The regularization procedure which we adopt is described at length in [20]; to quickly recapthe key points, consider the tadpole graph A = − λ ( d R + 1) = 1¯ µ (cid:15) (cid:90) d D q (2 π ) D D ( q ) (2.26)with ¯ µ = µ e γ / π , and D = 3 + 2 (cid:15) . Since D ( q ) is an arbitrary function of q , this integralwould need to be performed numerically; in doing so we must set D →
3. To implementdimensional regularization, we adopt a procedure of “addition and subtraction,” as follows: A = 1¯ µ (cid:15) (cid:90) d D q (2 π ) D (cid:16) D ( q ) − q (cid:17) + 1¯ µ (cid:15) (cid:90) d D q (2 π ) D q . (2.27)The rightmost term is simple enough that it can be computed analytically (in MS itsvalue is zero), and the leftmost term is now only logarithmically divergent. Thus, we have– 9 –emoved the linear divergence by subtracting 1 /q , and now the next step is to remove thelogarithmic one. At large momenta, and near 3 dimensions, D ( q ) can be expanded as D ( q ) ∼ q + g C R (cid:0) (cid:15) (1 − ξ − log 4) (cid:1) µ (cid:15) q − (cid:15) + O (cid:16) q (cid:17) (2.28)where we have been careful to keep O ( (cid:15) ) corrections in the 1 /q term. Now, we can addand subtract the subleading term, A = 1¯ µ (cid:15) (cid:90) d D q (2 π ) D (cid:34) D ( q ) − q − g C R (cid:0) (cid:15) (1 − ξ + log 4) (cid:1) µ (cid:15) ( q + ω ) / − (cid:15) (cid:35) + 1¯ µ (cid:15) (cid:90) d D q (2 π ) D q + g C R (cid:0) (cid:15) (1 − ξ − log 4) (cid:1) µ (cid:15) µ (cid:15) (cid:90) d D q (2 π ) D q + ω ) / − (cid:15) . (2.29)The first line of Eq. (2.29) is finite, so we can set D = 3 and perform the integral numerically.What we have effectively done is shuffled all of the (cid:15) dependence into terms which can beintegrated analytically. Thus the regularized expression for A has the form A = (cid:90) d q (2 π ) (cid:34) D ( q ) − q − g C R q + ω ) / (cid:35) + g C R (cid:0) (cid:15) (1 − ξ − log 4) (cid:1) π ) / e (cid:15)γ Γ(3 / − (cid:15) ) Γ( − (cid:15) ) (cid:16) ωµ (cid:17) (cid:15) . (2.30)We can then subtract the 1 /(cid:15) divergence with the counterterm, and take the limit (cid:15) → ω . Its value is arbitrary, but itmust be included, otherwise one would introduce an IR divergence where originally therewas none. For simplicity, we can set ω = g N noting that the final results of the calculationare ω independent. Though it is certainly permitted, it is not a requirement that ω be setto the scalar mass m (and our reasoning for not doing so is explained in Appendix B).Other diagrams which appear in Γ are regularized in much the same fashion. Inthe end we need to compute all of the one- and two-loop gluon and Higgs self-energydiagrams which appear in perturbation theory (ensuring that IR divergent diagrams arenot introduced inadvertently); the results of this computation are contained in AppendixB. Since our approach is diagrammatic and is founded on trying to determine correlationfunctions of gauge dependent fields, we are obliged to perform some sort of gauge fixing.We have chosen covariant gauge with gauge-fixing functional ∂ µ A µ , that is, a gauge choicewhich does not make reference to the scalar field one-point function or vacuum value. Thischoice differs from what is usually done in perturbative treatments of the Higgs phase, andrequires some explanation. First we will argue that one can gauge fix as we do here; thenwe will explain why we believe it is preferable.That the gauge-fixing approach we have adopted is possible, has already been explainedclearly by Buchm¨uller, Fodor, and Hebecker [44] in the context of the electroweak phase– 10 –ransition. Perturbatively, in the broken symmetry phase we expect the typical contributionto the path integral to have a nonvanishing scalar VEV Φ i ; but since the gauge fixing doesnot eliminate the integration over the global gauge rotations, there are equal contributionsfrom every field direction choice, and the ensemble average of Φ i is zero. However theexistence of a VEV will still appear as a delta-function contribution to the scalar two-pointfunction, so the approach will still capture that physics. Nonperturbatively, while infraredgauge fields are suppressed, we do not expect them to vanish, and they can still destroyany infinite-range order in the scalar field. If this is the case then the scalar two-pointfunction will not in fact have a strict delta-function contribution. Instead it will have avery sharp structure near zero momentum, corresponding to long (but not infinite) distancecorrelations in the scalar field. Indeed, we expect this must be the correct behavior, sincethe symmetric and Higgs phases are analytically connected and are not distinct in thesense of being distinguished by a true order parameter. But the existence of infinite-rangeHiggs-field correlations in part but not all of the phase diagram would constitute an orderparameter and would forbid an analytic connection between the symmetric and Higgs sidesof the transition line.Now let us consider the alternative approach. It is to include explicitly a one-pointsource for the scalar field, W [ J i , K ij ] = − log (cid:90) D [ φ, A ] e − S − J i φ i − φ i K ij φ j . (2.31)The value of W [0 ,
0] is gauge-invariant [45], but the inclusion of nonzero J i explicitly breaksgauge invariance. Naturally we are then only interested in the J i → (cid:104) Φ i (cid:105) = δδJ i W [ J i , (cid:12)(cid:12)(cid:12) J i → ϑ ) (2.32)where 0( ϑ ) means “zero is approached along a direction ϑ on the manifold of SU( N )rotations.”Perturbatively, we expect the symmetric phase to display smooth behavior, so anapproach from any direction will yield the same result, consistent with a zero VEV. Inthe broken phase, W is expected to develop a conical singularity, so the direction has asignificance. Nonperturbatively, and still working in covariant gauge, we have just arguedthat we do not in fact expect infinite-range correlations in the scalar field when J i is takento zero; so the behavior of W near J i = 0 should always be differentiable, albeit thederivatives can become very large. Therefore, in covariant gauge we expect that thereshould be no VEV, even if J i is taken to zero along a particular direction. Therefore theintroduction of J i changes nothing; the J i → i – that is, we can use R ξ gauge. The gauge-fixing choice introduces– 11 –nto the action the gauge-fixing term L gf = 1 ξ (cid:0) ∂ µ A µa − ξ Φ i gT aij φ j (cid:1) , (2.33)where Φ i is the VEV and φ j is the field. This is balanced as usual by the appropriateFadeev-Popov determinant. Physically, the role of covariant gauge fixing can be understoodas using up the gauge freedom to force ∂ µ A µ to be as small as possible, which minimizes thetotal size of fluctuations in the gauge fields. R ξ gauge is instead a compromise, in whichthe gauge fixing is used to try to minimize ∂ µ A µ (gauge field fluctuations), but also tominimize fluctuations in the components of the scalar field not in the direction of the VEVΦ i (pseudo-Goldstone fluctuations). The limit ξ →
0, Landau gauge, is when all gaugefreedom is used to control gauge field fluctuations. The opposite limit, ξ → ∞ , Unitarygauge, is when all freedom is used to align the scalar in the direction of its VEV.One challenge with this approach is that in the current context the value of the VEVΦ i must be determined self-consistently as part of the procedure. In general the VEV willdepend on ξ [46], growing larger at large ξ as more fluctuations are forced into the VEV.Perturbatively this effect is suppressed by g in 4 dimensions. Here it will be suppressedby x . Since we are interested in a regime where perturbation theory requires resummation,the ξ dependence can be large.The first problem with this approach is that whether Φ i vanishes or not would con-stitute an order parameter, but we know that there should not be an order parameter forthis system. Second, it is possible that there are multiple self-consistent solutions for Φ i , inwhich case it is not clear which to use. Finally, the existence of a VEV Φ i for a given x, y value can and almost surely will depend on ξ , as large ξ biases the gauge fixing towardsthe development of a VEV. Therefore we anticipate that the details of the transition willhave no stability as a function of ξ , in other words, the methodology would not be reliable.(Similar issues are discussed in Appendix A of Ref. [47].)On the other hand, we emphasize that using covariant gauge will present considerablechallenges when the transition is strong or when the value of y places us deep in the Higgsphase. In this case we will be keeping track of very long-distance correlations in the φ fieldvia the corresponding very low-momentum structure in the two-point function. As we willexplain in the following, this proves numerically challenging, but it can also be a problemfrom the point of view of the convergence of the loop expansion and the stability of thesolutions we find within the space of possible G, D, ∆ choices. It is also not clear what ξ dependence the phase diagram will display in covariant gauge; if the dependence is strong,it indicates a problem with the method’s reliability. Ans¨atze
In [20], we extensively described an algorithm which can be used to extremize the effectiveaction when only gauge fields are present. Now we have to address the additional com-plications which arise due to the presence of a Higgs field. In the symmetric phase, the– 12 –resence of the Higgs does not really change much at the technical level, and obtainingself-consistent solutions for the gauge field and Higgs propagators proceeds much as earlier.To begin, we will review the details of the functions which enter into the problem. Sincewe have assumed a general covariant gauge, we are attempting to solve self-consistentlyfor the following 4 functions: G T ( p ), G L ( p ), ∆( p ) and D ( p ), which are respectively thetransverse and longitudinal gauge field propagators, the ghost propagator and the Higgspropagator. We can opt for the most part to simplify the problem further by working inLandau gauge, where G L falls out of the picture; however, computations in Feynman gaugedo require a treatment of G L .To realize the extremization, we will specify Ans¨atze for these functions in terms of afinite set C = { c i } of variational coefficients, such that the variational equations become δ Γ δ { G T /G L / ∆ /D } = 0 −→ δ Γ δc i = 0 . (3.1)Writing the functions in terms of a finite number of parameters in this way replaces theinfinite-dimensional functional space with a finite-dimensional subspace; and the problembecomes finding the extremum in this subspace. By increasing the size of C we enlargethe space of allowed functions, and the true extremum should be more closely approached.Our choice is to fit the self-energies as rational functions (Pad´e approximants), since thisgives a very flexible class of smooth functions. Specifically, for some { a i } ∪ { b j } ⊂ C , wewill define R i max − j max ( p, { a i } ∪ { b j } ) = a i max p i max + ... + a b j max p j max + g N . (3.2)Then the two-point functions Eq. (2.17), Eq. (2.18) and Eq. (2.19) are parametrized asfollows: Π T ( p ) = g (cid:16) N ( ξ + 2 ξ + 11)64 − T R (cid:17) p + R ( p, { c { Π T } i } ) (3.3)Π L ( p ) = R ( p, { c { Π L } i } ) (3.4)Σ( p ) = g N p p + g N + R ( p, { c { Σ } i } ) (3.5)Π φ ( p ) = g C R p p + g N + R ( p, { c { Π φ } i } ) . (3.6)Here we have incorporated the one-loop large- p behavior exactly and allowed the rest of theself-energy to be determined by extremization. Previous work [20] shows that third orderPad´e approximants are sufficient and we will use them here. The resulting SD equationshave the simple form Π Ansatz T ( p ) = Π T ( p ) (where Π T ( p ) is the gluonic analogue ofEq. (2.25)) and similarly for Π L , Σ and Π φ .However, we anticipate that the scalar propagator may display a very narrow structurenear zero momentum. Therefore we will add to the scalar propagator Ansatz an additional Using the labels “
Ansatz ” and “2PI” to distinguish between the value of the Pad´e approximant andself-energy functional constructed out of 2PI diagrams. – 13 –erm: D ( p ) = R ( p, { c { G } i } ) p ( p + g N ) + 1 p + m − Π φ ( p ) . (3.7)The added term is designed to allow for a sharp structure at small p ; its form has beenchosen phenomenologically. Technically the functional form allows for D ( p ) ∝ p − small- p behavior, whereas we expect that lim p → D ( p ) should be a constant. However, theextremization procedure is not sensitive to a turnover at very small p , so this functionalform near p = 0 is not very important. Generally, in the symmetric phase extremizationsetting R to zero produces essentially the same extremum as allowing the term to benonzero, whereas deep in the Higgs region the inclusion of R is essential to getting a goodsolution to the SD equation, which is modified to − D − ( p ) + D (0) − ( p ) = Π φ ( p ).We can also define the renormalized scalar “condensate” as D = (cid:90) d q (2 π ) (cid:34) D ( q ) − q − g C R q + g N ) / (cid:35) (3.8)= 1 N (cid:104) φ † φ (cid:105) ¯ µ = g N . The twice-subtracted integral is both UV and IR finite; as indicated it equals the expecta-tion value of the field squared when renormalized using ¯ µ = g N . This condensate will beuseful in distinguishing between coexisting phases.In Eq. (3.3), Eq. (3.5), and Eq. (3.6) we have fixed by hand the O ( p ) large-momentumbehavior of each propagator to match a one-loop perturbative calculation presented inAppendix B, leaving only the O ( p ) part to be determined variationally. This is actually arequirement; to see why this is the case, consider the UV expansion of G T resulting fromEq. (3.3), G T ( p (cid:29) g N ) = 1 p + g N ( ξ +2 ξ +11)64 − g T R p + O (cid:16) p (cid:17) , (3.9)as well as the variation of Γ with respect to c { Π T } i δ Γ δc { Π T } i = d A (cid:90) d p (2 π ) δG T ( p ) δc { Π T } i (cid:16) − Π Ansatz T ( p ) + Π T ( p ) (cid:17) . (3.10)By fixing the tree-level O (1 /p ) and one-loop O (1 /p ) behavior in Eq. (3.9), the term inparentheses in Eq. (3.10) is automatically O (1) at large momentum, while the derivativeof G T is O (1 /p ) (or milder, depending on which coefficient we are differentiating withrespect to). Hence, Eq. (3.10) is finite. Finally, it is worth noting that at one loop theinclusion of masses in bare diagrams is subleading in p relative to the massless diagrams;as we are not required to impose any constraints on the two-point functions at O (1 /p ), itsuffices to compute the one-loop corrections in the massless limit. Generally we expect self-energies to be nonzero at p = 0 and so propagators should go to constantvalues. The exception is the ghost propagator, where we showed in [20] that the self-energy must scale asΣ( p ) ∝ p at small p because a vertex always differentiates the external ghost line. – 14 –ince convergence to the perturbative limit is only necessary at large p , looking backat Eq. (3.5) and Eq. (3.6), we opted to include the one-loop contributions with an addi-tional IR suppression factor of p/ ( p + ω ). At sufficiently small momenta, a linear termin the denominator of a propagator can lead to the formation of a pole. The gauge fieldsdynamically generate a mass that is sufficiently large to prevent this sort of thing fromhappening so a suppression factor of this sort is not required. For the Higgs and ghost thisis not the case. In solving the problem we have set ω = g N . This arbitrary choice doesnot affect the final result, since a different choice of ω together with an appropriate shiftin the Ansatz parameters leaves the self-energy unchanged.
To extremize Γ[ G T , G L , ∆ , D ], we employ and algorithm based on conjugate gradient de-scent specialized to the problem at hand. We can visualize the root-finding algorithm as adynamical system where we choose an initial value for the coefficients c i and subsequentlyfollow a flow through the gradient field ∂ Γ /∂c i until we reach an attracting fixed point,which corresponds to a solution.For x below the critical end point, there is a region of metastability where two attractorscoexist. We will denote the solution with larger condensate D (and smaller small- p behaviorin G T ) as the Higgs solution and the solution with smaller condensate as the symmetricphase solution. We write the propagator (condensate) in the Higgs solution as D − ( D − )and that in the symmetric phase as D + ( D + ). In the space of initial guesses for the Ansatz parameters, each solution has a basin of attraction, which we write as γ + and γ − . Thesebasins of attraction do not cover the full space of initial guesses; because of the saddle-likenature of Γ, there is also a set of initial conditions c i ∈ γ which evolve towards divergentvalues of D . This is to be discussed in greater detail in Section 4.The number of iterations of gradient descent in the extremization procedure (denotedby N ) can be thought of as time evolution, and we are interested in the results at latetimes. We can observe convergence of the algorithm by plotting the evolution of the LHSand RHS of the SD equations with N ; this is shown generically in Fig. 2. From this figureit is apparent that convergence is attained, and that the variational Ansatz has captureda choice for the self-energy where the SD equations are quite accurately solved.To establish two distinct metastable solutions, it is also important not to be fooled byslow convergence to an extremal solution. We illustrate this idea in Fig. 3, which showsalgorithm convergence for two cases. On the left, we see convergence from two differentstarting configurations to two distinct final solutions. On the right, we see slow evolutionto a single solution. To distinguish these cases, it is important to fit the N dependence of D (or some other measure of the solution), to test convergence. We find that the fit form D fit ( N ) = A e −N /τ N N δ + c (3.11)gives a good description. – 15 – p/g N Π T /g N Π Ansatz T /g N Π φ /g N ( D − − D (0) − ) /g N -0.4-0.200.20.4 0 0.5 1 1.5 2 2.5 3 p/g N Π T /g N Π Ansatz T /g N Π φ /g N ( D − − D (0) − ) /g N Figure 2 . Evolution of the SD equations under gradient descent, which are solved when the pointsoverlap in the above figures (the ghost equation is not depicted, but it is qualitatively similar).The left panel corresponds to some initial choice of variational coefficients, and at the right we seeconvergence at late times. D / g N N [ x = 0 . , y = 0 . D fit ( N ); c = 0 . N → ∞
Symmetric phase -0.002-0.00100.0010.002 0 10 20 30 40 50 D / g N N [ x = 0 . , y = 0 . D fit ( N ); c = − . N → ∞
Symmetric phase
Figure 3 . Convergence of D (and hence D ) at late times; we show both the cases where D + (red)and D − (blue) are distinct (left panel) and equivalent (right panel). At generic values of x and y , D ( N ) will resemble one of these two graphs (when it converges). – 16 – .3 Issues of stability Is the extremization of Γ a minimization/maximization or a saddlepoint-seeking procedure?It is a saddlepoint-seeking procedure, as can be seen by considering the one-loop value,Γ [ G T , ∆ , D ] =Tr log D − DD (0) − + 12 Tr log G T − G T G T (0) − − Tr log ∆ + ∆∆ (0) − , (3.12)which is solved trivially by D = D , G = G , ∆ = ∆ . For the sign choice above, thisextremum is clearly a maximum for G , but a minimum for ∆, since the ghost enters withthe opposite sign. At least in the ultraviolet this property is not affected by additionaldiagrams, since the tree terms dominate in the UV.This does not cause a problem in practice, since we can alternately extremize withrespect to G, D holding ∆ fixed and with respect to ∆ holding
G, D fixed. The for-mer involves maximization, the latter involves minimization. This procedure shows rapidconvergence, and was used in our previous work [20]. The interpretation of this saddlebehavior is benign; it arises because of the peculiarities of gauge fixing and the presence ofthe “fermionic” ghost species which it introduces.But we are in trouble if the bosonic part of Γ at fixed ∆ is unbounded from aboveand below. When we move from one to three loops and we consider the possibility of largeinfrared contributions in D , we find that precisely this problem arises. The pure scalardiagrams, omitting group theoretic factors, are of the form − Γ scalar [ D ] = − Tr log D + DD (0) − + λ (cid:90) pk D ( p ) D ( k ) − λ (cid:90) pkq D ( p ) D ( k ) D ( q ) D ( p + k + q ) (3.13)where we have flipped the overall sign so the one-loop piece opens upwards. With thethree-loop term present, Γ is unbounded from above and below. This unbounded behaviorbecomes important whenever D becomes sufficiently large in some narrow momentumrange. For instance, when a small p range around zero supports a finite value of theintegral (cid:82) p D ( p ), then the small p, k, q contribution to the three-loop (basketball) termbecomes large, and it diverges as the phase space region supporting (cid:82) p D ∼ (cid:82) p D ( p )should receive a finite contribution from a very narrow momentum range near p = 0.Therefore, the extremum we seek is at best a local maximum as a function of G, D ; andin particular, we can expect trouble deep in the Higgs phase. The origin of this problemis that, when the field develops large long-distance correlations, the loopwise expansion ofthe 2PI functional breaks down. For instance, at the four-loop level we will encounter+ λ (cid:90) pkql D ( p ) D ( p + l ) D ( q ) D ( q + l ) D ( k ) D ( k + l ) (3.14)which diverges still more strongly. The sequence of such divergent graphs is resummed byincluding the one-point function in the procedure, and stripping the square of the one-point– 17 – p/g N [ x = 0 . p D + ( p ) [ y = 0 . p D + ( p ) [ y = 0 . p D + ( p ) [ y = 0 . p D + ( p ) [ y = 0 . p D + ( p ) [ y = 0 . p/g N [ x = 0 . p D + ( p ) [ y = 0 . p D + ( p ) [ y = 0 . p D − ( p ) [ y = 0 . p D − ( p ) [ y = 0 . Figure 4 . Evolution of the symmetric phase solution for Higgs two-point function with increasing y at fixed x = 0 .
125 (left), and coexistence of symmetric and Higgs phase solutions (right). function from the two-point correlator. However, as we have emphasized, any procedurefor including the one-point function appears to damage the properties which ensure thepossibility of a phase transition endpoint.Here we work in terms of the two-point function only, which will mean that we areunable to study cases where the solutions become strongly Higgs-like. We anticipate that,when we seek solutions which show strong Higgs-like behavior, we will instead find runawaybehavior in our extremization algorithm.Naively it appears that this problem is less severe at small x where the high-loop Higgsdiagrams are suppressed by more explicit powers of x . But the Higgs-phase value of thecondensate (cid:82) p D ( p ) grows as 1 /x , so in fact the problem is more severe, not milder, atsmall x . Therefore it will not be possible to make contact with the perturbative part ofthe phase diagram. We will concentrate on the analysis in Landau gauge (which eliminates the longitudinalgluon propagator), and set N = 2 with the scalar field in the fundamental representation.A comparison with the results in Feynman gauge appears towards the very end, in Section4.2. Solutions for the Higgs, gauge and ghost propagator are shown in Fig. 4 (Higgs) andFig. 5 (gauge/ghost). These plots are generated for the specific value of x = 0 .
125 (anda range of y ); however, at generic values of ( x, y ) solutions (when they exist) will take oneither of these forms. On the right panel in Fig. 4, we can distinguish between the peak-likeand massive behavior of Higgs ( D − ) and symmetric ( D + ) phase solutions. Furthermore, at x = 0 . y , which– 18 – p/g N [ x = 0 . p G T ( p ) [ y = 0 . p ∆( p ) [ y = 0 . Figure 5 . Transverse gauge field and ghost two-point functions (in the symmetric phase), showingnonperturbative massive behavior in G T . The corresponding broken phase solutions are not depictedsince for this particular ( x, y ) they would be nearly indistinguishable on this plot. However, it isworth noting that in general Π T becomes increasingly massive in the Higgs phase relative to thesymmetric phase (when these solutions coexist). is evidence of metastability. As shown and explained in Fig. 5, both solutions display gaugecorrelators with “massive” behavior (in the sense that lim p → G T ( p ) is finite; we are notclaiming that the propagator has a pole at imaginary p and we have not investigated thebehavior of spatial Wilson loops). In the symmetric phase this is due primarily to pure-glueloops; in the Higgs phase the mass is larger, due to additional Higgs-loop contributions.On the left panel of Fig. 4 we see that the symmetric solution D + terminates. Thisindicates that the symmetric phase has lost its metastability and become spinodally un-stable; so we identify the y value where this occurs as y − ( x ). The Higgs solution alsoterminates, and the possibility of metastability ceases to occur, at a larger y value whichwe interpret as y + ( x ). There is a third special value of y , where the Higgs solution becomesunstable to runaway behavior as described in Subsection 3.3. We will label this value y end .It does not have a physical interpretation in terms of the phase diagram; it is simply thepoint where the solution becomes so Higgs-like that our three-loop truncation encountersuncontrolled stability issues when we try to analyze the Higgs branch.We can map out the region of the ( x, y ) plane between the y + and y − curves by findingthose regions where two (meta)stable solutions for the propagators exist. The “symmetric”(small- D ) solution is obtained by seeding the gradient solver with a configuration found atlarger y , while the “broken” solution is found by starting at a smaller value of y with aninitial guess for the scalar propagator with strong small- p behavior. The critical value x c is the largest x such that metastability is observed.Plots of D ( y ) for x = 0 .
125 and x = 0 .
150 are shown in Fig. 6. In both cases wesee a branch of symmetric phase solutions which terminates at y − . But while x = 0 . D / g N y [ x = 0 . y c − y c + D − branch D + branch y end Higgs phaseSymmetric phase-0.00500.0050.010.015 0.1 0.11 0.12 0.13 0.14 0.15 D / g N y [ x = 0 . y c D + branch D − initialconditionsHiggs phaseSymmetric phase Figure 6 . Evolution of D with y at fixed x , showing the appearance of stable branch of Higgsphase solutions at x = 0 . supports a Higgs branch, x = 0 .
15 does not; so x c must occur between these two values.To determine where, we carry out a scan of the phase structure for several values of x , asshown in Fig. 7. The figure displays two-branch behavior at x = 0 .
14 but not x = 0 .
15, sowe conclude that 0 . < x c < .
15. – 20 – D / g N yD − branch y c + x = 0 . x = 0 . x = 0 . x = 0 . x = 0 . x = 0 . Figure 7 . Evolution of the stable branches with increasing x . The D − branch disappears by x = 0 . x c (cid:39) . Our original purpose in applying the 2PI formalism to SU( N ) Higgs theory was not specif-ically to determine the phase diagram (which is already known), but rather, to test theaccuracy with which n PI resummation is able to make predictions about the nonpertur-bative sector of a nonabelian gauge theory. The n PI method relies on approximating theeffective action by its truncation at a finite loop order, which results in a selective re-summation to all orders of a certain class of topologies. In a gauge theory, this inducesgauge-fixing dependence [48, 49], since at least perturbatively, one should include all di-agrams at every loop order. This effect could potentially be very mild, but a priori it isnot clear that accurate results can be obtained from this method anywhere on the phasediagram. The only way to test the reliability of the approximation is to directly computegauge-invariant observable quantities.Here we will attempt a direct comparison between lattice and 2PI determined valuesof x c , y − ( x ) and y + ( x ). An overview of many of the pertinent results from 3D latticestudies of SU(2) Higgs theory can be found in [50], which incorporates the original studies– 21 –21–23, 33]. The most relevant quantity to compare is the location of the critical endpoint.We find ( x c , y c ) (cid:39) (0 . , . x c , y c ) =(0 . ± . , − . ± . ∼
50% relative errors in x c (establishing relative errors in y c is harder since itdepended on an arbitrary renormalization point prescription).We could also try to compare the spacing y + − y − to the lattice, at a comparabledistance below x c . Unfortunately, the locations of the upper and lower metastability lineslack a clean nonperturbative definition. Technically, at any ( x, y ) value there is only onepossible phase, and the transition line is where the is an abrupt change in that phase’sproperties, such as D . In practice, for systems near the transition line there are very long-lived metastable states, and the transition from the metastable to the stable state involvesan extremely rare and spatially inhomogeneous configuration. The spatial inhomogeneityis the reason that our 2PI approach cannot explore such states, allowing us to explore thesupercooled or superheated phases. The lattice avoids this problem by sampling over allsuch states, typically using reweighting to make it more likely to sample the inhomoge-neous states which carry us between metastable and stable phases. Nevertheless, Ref. [21]presented a definition of the metastability limits. For x = . y + = . y − = − . y + − y − ) = . x = 0 .
10. So there is at least qualitative agreement here.We are not able to compare the discontinuity between condensates at y c as a function of x c − x to the lattice, because we have not implemented a procedure to find the Γ differencebetween the two phases and thereby determine the transition value y c . Up to now, we have argued diagrammatically that critical values of y are expected toexhibit dependence on the gauge parameter ξ . However, since it is difficult to quantify thiseffect without an explicit computation, we will now briefly present a comparison betweenLandau and Feynman gauges. The results in Feynman gauge are best summarized by a ξ = 1 analogue of Fig. 7, shown in Fig. 8.In setting ξ = 1 and resolving the SD equations (following the usual procedure), weobserve that qualitatively very little has changed. Feynman gauge solutions exhibit similarfeatures to those in Landau gauge, and once again we observe a disappearance of a stableHiggs branch somewhere between x = 0 .
125 and x = 0 . x enters primarily through diagrams without gauge fieldlines. The biggest change though is the observed shift in the critical range of y , fromaround y ∼ .
120 to y ∼ . G L propagators, which does not fully cancel between diagrams. For instance, the masscontributions (at vanishing external momentum) of G L in the two self-energy corrections+ 12 (4.1)– 22 – D / g N y [ ξ = 1] D − branch y c + x = 0 . x = 0 . x = 0 . Figure 8 . Feynman gauge analogue of Fig. 7, showing the evolution of the D + and D − brancheswith x . As in Landau gauge, the D − branch disappears by x = 0 . y has shifted. only cancel if the scalar propagator takes the free massless value 1 /p ; otherwise the tadpolecontribution is larger and leads to a positive mass contribution which is proportional tothe gauge parameter ξ . We directly solved the three-loop 2PI effective action for 3D SU( N ) Higgs theory andobtained resummed correlators which correspond to both the symmetric and Higgs phasesof the theory. We found that these solutions coexist over a region of the phase diagram,indicative of metastability and a first order phase transition. Subsequently, we have alsoobserved that there is a point x where the metastability ceases to be observed, which weidentified with the critical end point of the theory, x c .Concerning the numerical accuracy of the predictions made in Landau gauge, thelocation of the critical end point we inferred differs from the lattice value with a relative– 23 –rror of ∼ y c depends surprisingly strongly on thegauge parameter ξ .The most promising finding regarding the applicability of the n PI formalism to anonabelian gauge theory is the apparent qualitative evidence for a critical end point ( x c , y c )located relatively close to its known nonperturbative value. In this sense the 2PI approachhas successfully seen nonperturbative behavior in the phase diagram. However, the methodhas shown serious weaknesses as well. The quantitative level of agreement with the latticeis not very impressive, and the strong ξ dependence in y c is also worrying. More urgently,the method has failed completely to resolve the behavior of the Higgs phase when the scalarcondensate is large. The most straightforward way to fix this problem, via the introductionof a scalar one-point function and the use of R ξ gauge, would introduce new problems. Aswe have argued, the ξ dependence should be significant where the transition is weak, andthe gauge-fixing procedure may destroy the existence of a critical endpoint and analyticconnection between the two “phases.”Thus, the study of SU( N ) Higgs theory has therefore revealed several limitations to the n PI method in the context of a nonabelian gauge theory. In addition to the described ambi-guities in physical observables, the application of the formalism is difficult numerically, es-pecially if one wishes to consider higher-loop truncations or higher n -particle-irreducibility.If qualitative predictions can be made at best, then it may be hard to justify the numericalexpense. However, this work does not preclude the possibility that further refinements maybe possible with the goal of obtaining quantitatively accurate answers to nonperturbativeand gauge-invariant questions. This matter is left open for a future investigation. Acknowledgments
We would like to thank Meg Carrington and Marcus Tassler for useful comments. Thiswork was supported in part by the Natural Science and Engineering Research Council ofCanada (NSERC).
A Feynman rules for SU( N ) Higgs theory The Feynman rules for covariant-gauge perturbative calculations in SU( N ) Higgs theoryare derived from the Lagrangian L = 12 Tr F µν F µν + 12 ξ ( ∂ µ A aµ ) + ∂ µ ¯ c a ∂ µ c a − gf abc ∂ µ ¯ c a c b A cµ (A.1)+ ( D µ φ ) † ( D µ φ ) + ( m + δm ) φ † φ + λ φ † φ ) . (A.2)– 24 –auge field, scalar and ghost propagators are denoted by the symbols G, D and ∆. InEuclidean space at tree level these are G (0) µν ( p ) = 1 p (cid:16) T µν ( p ) + ξ L µν ( p ) (cid:17) (A.3) D (0) ( p ) = 1 p + m (A.4)∆ (0) ( p ) = 1 p (A.5)where the gauge-field propagator is specified by the transverse and longitudinal projectors T µν ( p ) = g µν − p µ p ν p (A.6) L µν ( p ) = p µ p ν p . (A.7)With all momenta assumed to be flowing outwards, the bare Yang-Mills vertices are gV (0) a a a µ µ µ = gF a a a (cid:0) ( p − p ) µ g µ µ + ( p − p ) µ g µ µ + ( p − p ) µ g µ µ (cid:1) (A.8) g V (0) a a a µ = gF a a a p µ (A.9) g V (0) a a a a µ µ µ µ = g (cid:0) F a a s F a a s ( g µ µ g µ µ − g µ µ g µ µ )+ F a a s F a a s ( g µ µ g µ µ − g µ µ g µ µ )+ F a a s F a a s ( g µ µ g µ µ − g µ µ g µ µ ) (cid:1) . (A.10)where ( a , p ) are the color indices and momentum of the outgoing ghost in V . The presenceof a complex scalar results in the following additional vertices, g V (0) a a a µ = gT a a a ( p − p ) µ (A.11) g V (0) a a a a µ µ = − g T { a a s T a } sa g µ µ (A.12) λ V (0) a a a a = − λ ( δ a a δ a a + δ a a δ a a ) . (A.13)where the outgoing scalar(s) are indexed by ( a , p ) (Eq. (A.11) and Eq. (A.12)) and a , a (Eq. (A.13)). B Self-energies computed in dimensional regularization
In regularizing the 2PI effective action, one makes use of one- and two-loop self-energycorrections computed in perturbation theory. In pure Yang-Mills, all one- and two-loopintegrals are massless from the onset. However, the inclusion of a Higgs field now inprinciple adds massive propagators to many of the diagrams. But, since we really onlyneed to know the UV limit of these diagrams, it actually suffices to compute them with amassless scalar field.In this Appendix, though some results are valid for arbitrary D, (cid:15) should be treatedas a small parameter, i.e. , it is assumed that we are working at or near 3 dimensions,– 25 – = D + 2 (cid:15) , with D = 3. Finally, since the Higgs mass renormalizes at the two-loop levelin three dimensions, it is useful to define the MS scale ¯ µ = µ e γ / π . The master one-looptopology is p p p = J (D)1 ( n , m ; n , m ) (cid:40) p = qp = q − p (B.1)with J (D)1 ( n , m ; n , m ) = (cid:16) µ (cid:17) D − D02 (cid:90) d D q (2 π ) D (cid:0) q + m (cid:1) n (cid:0) ( q − p ) + m (cid:1) n . (B.2) B.1 One-loop gluon self-energy
The presence of a scalar field adds two additional diagrams to the one-loop gluon self-energyrelative to the the pure Yang-Mills expression,Π (1 ,(cid:15) ) m ; µν = 12 + 12 − + + . (B.3)The result is strictly transverse; we will separate the Yang-Mills and Higgs contributionsas follows, Π (1 , m ; µν = g p (cid:16) π (1 , + π (1 , m (cid:17) T µν (B.4)Π (1 ,(cid:15) )0; µν = g (cid:18) p (cid:15) µ (cid:15) (cid:19) (cid:16) π (1 ,(cid:15) )YM + π (1 ,(cid:15) )0 (cid:17) T µν . (B.5)The terms which appear in the limit D → π (1 , = C A
64 ( ξ + 2 ξ + 11) (B.6) π (1 , m = − T R π (cid:18) − mp + 4 m + p p (cid:18) π − mp (cid:19)(cid:19) (B.7)and it is also useful to take the m → O ( (cid:15) ), π (1 ,(cid:15) )YM = C A (cid:0) ( ξ + 2 ξ + 11)(1 − (cid:15) log 2) + (cid:15) (12 − ξ − ξ ) (cid:1) (B.8) π (1 ,(cid:15) )0 = − T R
16 (1 − (cid:15) log 2 − (cid:15) ) . (B.9)– 26 – .2 One-loop Higgs self-energy The calculation of the one-loop correction of the Higgs self-energy proceeds forward inmuch the same manner,Π (1 ,(cid:15) ) φ ; m = + 2 + 12 (B.10)with Π (1 , φ ; m = g p π (1 , φ ; m (B.11)Π (1 ,(cid:15) ) φ ;0 = g (cid:18) p (cid:15) µ (cid:15) (cid:19) π (1 ,(cid:15) ) φ ;0 . (B.12)For the D → π (1 , φ ; m = (1 + d R ) x π mp + C R π (cid:18) (2 − ξ ) mp + 2( p − m ) p arctan pm (cid:19) (B.13) π (1 ,(cid:15) ) φ ;0 = C R − (cid:15) log 2 + (cid:15) (1 − ξ )) . (B.14)in terms of the dimensionless quartic coupling x = λ/g . B.3 One-loop ghost self-energy
The one-loop ghost self-energy is constructed out of a a single diagram,Σ (1 ,(cid:15) ) = (B.15)for which in D = 3 + 2 (cid:15) , ξ dependence only appears at O ( (cid:15) ),Σ (1 ,(cid:15) ) = g (cid:18) p (cid:15) µ (cid:15) (cid:19) σ (1 ,(cid:15) ) = g (cid:18) p (cid:15) µ (cid:15) (cid:19) C A
16 (1 − (cid:15) log 2 + (cid:15) (1 − ξ )) . (B.16) B.4 Two-loop topologies
The massless two-loop master topology is p p p p p p = J (D)2 ( n , n , n , n , n ) p = q p = q p = q − pp = q − pp = q − q (B.17)– 27 – (D)2 ( n , n , n , n , n ) = (cid:16) µ (cid:17) D − D (cid:90) d D q (2 π ) D d D q (2 π ) D (cid:0) q (cid:1) n (cid:0) q (cid:1) n (cid:0) ( q − p ) (cid:1) n (cid:0) ( q − p ) (cid:1) n (cid:0) ( q − q ) (cid:1) n . (B.18)The remaining two topologies are related to J (D)2 by shrinking one or more of the propa-gators to a point, for instance J (D)2 ( n , n , n , n ,
0) = J (D)1 ( n , n , J (D)1 ( n , n ,
0) (B.19) J (D)2 ( n , , , n , n ) = J (D)2 ( n , n , n ) (B.20) J (D)2 ( n , n , n , , n ) = J (D)2 ( n , n , n , n ) (B.21)where the number of propagators should be inferred from the arguments. In computingthe two-loop self-energies we encounter UV divergences arising from the integrals J (D)1 ( n , n , m ) = ( m ) D / − α − β Γ(D / − n )Γ( n + n − D / µ ) D − D02 (4 π ) D / Γ(D / n ) × F (cid:16) n , n + n − D2 ; D2 (cid:12)(cid:12)(cid:12) − p m (cid:17) (B.22) J (D)2 ( n , n , n ) = Γ(D / − n )Γ(D / − n )Γ(D / − n )(¯ µ ) D − D (4 π ) D Γ( n )Γ( n )Γ( n ) × Γ( n + n + n − D)Γ(3D / − n − n − n ) ( p ) D − n − n − n . (B.23)The massive one-loop scalar integral is needed since recursively one-loop diagrams (i.e. theone-loop diagrams with a self-energy insertion in one of the propagators) are IR divergentwhen they are massless. – 28 – .5 Two-loop gluon self-energy At two loops, a number of additional diagrams are present, π (UV2 ,(cid:15) )YM; µν ∝
16 + 12 + + 14 − − − π (IR2 ,(cid:15) )YM;0; µν ∝ Π + 12 Π (B.25) π (UV2 ,(cid:15) )0; µν ∝ + + 2 + 4 (B.26) π (IR2 ,(cid:15) )0; µν ∝ + Π (B.27)using a notation where the subscript zero refers to the mass of the scalars in the loops beingset to m = 0. As mentioned at the start of this appendix, for the purpose of regulariz-ing this calculation genuinely two-loop topologies can be computed in the massless limit.However, recursively one-loop diagrams (labeled with the superscript IR2) will exhibit IRdivergences without the inclusion of a regulator mass ω . We have (retaining the superscriptIR to indicate that the full expression involves the specifically IR regulated diagrams)Π (2 ,(cid:15) )0; µν = g (cid:18) p (cid:15) µ (cid:15) (cid:19) (cid:16) π (UV2 ,(cid:15) )YM; µν + π (IR2 ,(cid:15) )YM;0; µν + π (UV2 ,(cid:15) )0; µν + π (IR2 ,(cid:15) )0; µν (cid:17) (B.28)noting that it should not be interpreted that this expression is transverse. The IR regulatedgluon and scalar propagators are defined as G (IR1 ,(cid:15) ) µν ( q ) = g π (1 ,(cid:15) )YM + π (1 ,(cid:15) )0 µ (cid:15) ( q + ω ) − (cid:15) (cid:18) g µν − q µ q ν q + ω (cid:19) (B.29) D (IR1 ,(cid:15) ) ( q ) = g π (1 ,(cid:15) ) φ ;0 µ (cid:15) ( q + ω ) − (cid:15) . (B.30)Now, regarding the notation: at this point there are two quantities which can be regardedas masses, m and ω . m refers to the Higgs mass which enters the problem via the scalar– 29 –ropagator, which we have already set to zero. Whereas, ω is an unphysical regulatormass introduced to regulate IR divergences in some two-loop diagrams. So, for instance,the diagrams which comprise π (IR2 ,(cid:15) )YM;0; µν are calculated using finite ω , but setting m = 0.One may ask why we do not simply regulate the IR divergences by keeping the scalar fieldmassive from the onset? There are two reasons. First, a number of divergences arise froma 1 /p gauge field propagator, so this would not solve the problem entirely. Second, ingeneral these diagrams are introduced to regularize the UV divergences in the problem.To compute the leading order UV behavior, it is sufficient to set m = 0, which drasticallysimplifies the majority of the diagrams which must be calculated. Then, the IR divergenceswhich would arise in the bare perturbation theory are handled with ω , of which the finalresults will be independent regardless.Defining χ = p/m , the individual components are π (UV2 ,(cid:15) )YM; µν = C A π (cid:34) ( ξ + 2)( ξ + 2 ξ + 11)48 (cid:15) g µν − ξ + 75 ξ + 221 ξ + 233) + 18 ζ (2)( ξ + 3)( ξ + 2 ξ + 17)768 T µν − ξ + 32 ξ + 79 ξ + 4248 L µν (cid:35) (B.31) π (IR2 ,(cid:15) )YM;0; µν = C A π (cid:34) − ξ + 2) (cid:16) π (1 , + π (1 , φ ;0 (cid:17) (cid:15) g µν + (cid:20) (cid:16) π (1 , + π (1 , (cid:17) (cid:18) ξ + 2) log 4 χ − ξ + 2) χ + (20 ξ + 42) χ + 3(5 ξ + 11) χ + 4( ξ + 2) χ ( χ + 1) arcsinh( χ )+ (5 ξ + 16) χ + 5(2 ξ + 5) χ + 4( ξ + 2) χ (1 + χ ) (cid:19) + ( ξ + 2)( C A (cid:0) ξ + 6 ξ − (cid:1) − T R )24 (cid:21) T µν + (cid:20) (cid:16) π (1 , + π (1 , (cid:17) (cid:18) ξ + 2) log 4 χ − ξ + 2) χ + 2( ξ − χ − ξ + 2) χ ( χ + 1) arcsinh( χ )+ (5 ξ + 6) χ − ξ + 2) χ (cid:19) + ( ξ + 2) (cid:0) C A (cid:0) ξ + 6 ξ − (cid:1) − T R (cid:1) (cid:21) L µν (B.32) π (UV2 ,(cid:15) )0; µν = T R π (cid:34) C R − ( ξ + 2) C A (cid:15) g µν + 16 (18 ζ (2) − ξ − C R + (cid:0) ζ (2)( ξ −
5) + 80 ξ + 272 (cid:1) C A T µν − ξ + 5) C R − (3 ξ + 4) C A L µν (B.33) π (IR2 ,(cid:15) )0; µν = T R π (cid:34) − π (1 , φ ;0 (cid:15) g µν + (cid:20) π (1 , φ ;0 (cid:18) χ − χ + 1) χ arcsinh( χ )+ 22 χ + 16 χ (cid:19) + ( ξ − C R (cid:21) T µν + (cid:20) π (1 , φ ;0 (cid:18) χ − χ − χ − χ ( χ + 1) arcsinh( χ )+ 6 χ − χ (cid:19) + ( ξ − C R (cid:21) L µν . (B.34) – 30 – .6 Two-loop Higgs self-energy The two-loop Higgs self-energy is specified by the diagrams π (UV2 ,(cid:15) ) φ ;0 ∝ + + 2+ 2 + 12 + (B.35) π (IR2 ,(cid:15) ) φ ;0 ∝ + Π + 12 Π + 2 Π (B.36)where once again IR divergences are handled with a regulator mass ω . Including a counter-term, we have Π (2 ,(cid:15) ) φ ;0 = g (cid:18) p (cid:15) µ (cid:15) (cid:19) (cid:16) π (UV2 ,(cid:15) ) φ ;0 + π (IR2 ,(cid:15) ) φ ;0 (cid:17) − δm (B.37)with π (UV2 ,(cid:15) ) φ ;0 = 116 π (cid:34) C R (cid:0) C A ( ξ − ξ + 3) + 4 C R (2 ξ − (cid:1) − d R ) x (cid:15) + C R (cid:104) C A (cid:0) ζ (2) + 3 ξ + 22 ξ + 27 (cid:1) − C R (cid:0) ζ (2) + ξ + 6 ξ − (cid:1)(cid:105)
16+ 6(1 + d R ) x (cid:35) (B.38) π (IR2 ,(cid:15) ) φ ;0 = (1 + d R ) x π (cid:34) π (1 , φ ;0 (cid:15) − π (1 , φ ;0 (cid:0) log 4 χ − (cid:1) + (1 − ξ ) C R (cid:35) + C R π (cid:34) (cid:16) π (1 , + π (1 , (cid:17) − ξπ (1 , φ ;0 (cid:15) − (cid:20) (cid:16) π (1 , + π (1 , (cid:17) − ξπ (1 , φ ;0 (cid:21) log 4 χ + 8 (cid:20) (cid:16) π (1 , + π (1 , (cid:17) (cid:0) χ + 15 χ + 6 (cid:1) + 3 π (1 , φ ;0 ( χ + 1) (cid:0) ξ ( χ + 1) − (cid:1) (cid:21) χ ( χ + 1) arcsinh( χ ) − (cid:20) (cid:16) π (1 , + π (1 , (cid:17) χ + 3 χ + 1 + 96 π (1 , φ ;0 (7 ξ − C A ( ξ + 6 ξ − − T R − C R ξ ( ξ − (cid:21) (B.39) δm = 116 π (cid:15) (cid:20) C R (7 C A − C R − T R )8 g + C R ( d R + 1) g λ − ( d R + 1) λ (cid:21) (B.40) Due to the counter-term, the scale dependence of m is given by the RG equation dm d log µ = β m ( g , λ ) (B.41)– 31 –ith β m ( g , λ ) = − ∂δm ∂g dg d log µ − ∂δm ∂λ dλd log µ = − (cid:15)g ∂δm ∂g − (cid:15)λ ∂δm ∂λ . (B.42)For instance, with an SU(2) fundamental Higgs in Landau gauge, β m ( g , λ ) = − π (cid:20) g + 9 g λ − λ (cid:21) . (B.43) References [1]
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