aa r X i v : . [ h e p - l a t ] J un A composite massless vector boson
Vincenzo Aﬀerrante, Axel Maas, and Pascal T¨orekInstitute of Physics, NAWI Graz, University of Graz, Universit¨atsplatz 5, A-8010 Graz,AustriaJune 22, 2020
In a non-perturbative gauge-invariant formulation of grand-uniﬁed theories all low energy vectorstates need to be composite with respect to the high-scale gauge group, including the photon. Weinvestigate this by using lattice methods to spectroscopically analyze the vector channel in a toygrand-uniﬁed theory, an SU(2) adjoint Higgs model. Our results support indeed the existence ofa massless composite vector particle.
Observable particles need to be described bymanifestly gauge-invariant operators. Beyondperturbation theory, BRST symmetry is insuf-ﬁcient to ensure this in non-Abelian gauge the-ories. Instead composite operators are needed,irrespective of the actual value of any couplingconstants [1–3]. Thus, in electroweak physics anidentiﬁcation of the observed particles and theelementary, gauge-dependent degrees of freedomof the Lagrangian is not directly possible. How-ever, due to a combination of the Brout-Englert-Higgs (BEH) eﬀect together with the Fr¨ohlich-Morchio-Strocchi (FMS) mechanism [1, 2] thishappens eﬀectively, up to corrections suppressed by powers of the Higgs vacuum expectationvalue. This has been conﬁrmed in lattice calcu-lations [3–5], including subleading contributions[6–8], and has potentially experimentally observ-able consequences [6, 8–10]. For a review see .However, this can potentially change for the-ories with a diﬀerent structure than the stan-dard model, in particular in scenarios for newphysics [11–13]. Especially, the physical observ-able spectrum of particles can diﬀer qualitativelyfrom the one of the elementary particles, andthus from those in perturbation theory. Thishas also been supported in lattice calculations[14, 15]. Though this does not invalidate newphysics scenarios as such, it does require to takemanifest gauge invariance in their construction1nto account, by augmenting perturbation the-ory with the FMS mechanism . This yieldedin all cases tested so far on the lattice [14, 15] cor-rect predictions [12, 14] even when conventionalperturbation theory did not.In the context of grand-uniﬁed theories(GUTs) [16, 17], this program faces a particu-lar challenge when it comes to model-building[12, 13]. In GUTs all low-energy interactions arecreated from a single non-Abelian gauge group,including QED . This requires the presence ofa massless, uncharged vector particle, which iscomposite with respect to the GUT gauge group,to play the role of the low-energy photon . FMS-augmented perturbation theory indeed predictsthat such states can arise when adjoint Higgsﬁelds are present . And, in fact, early ex-ploratory lattice investigations seem to supportthe presence of such a composite massless vectorparticle . Our aim is to substantiate these re-sults. In addition, also the massive vector statesare predicted  to diﬀer from those of pertur-bation theory. Thus, we also test this.To this end, we will simulate the simplest the-ory which is expected to show this behavior,SU(2) Yang-Mills theory with a single Higgs inthe adjoint representation. We will discuss thistheory and the relevant predictions, both of per-turbation theory and the FMS mechanism, inSection 2. Our lattice implementation will be Gauge-invariance is already non-trivial in QED,where a Dirac string is needed to make the photon gauge-invariant with respect to the electromagnetic gauge group[3, 18]. This has been conﬁrmed in lattice simulations,see e. g. [19, 20]. This can be included straight-forwardlyinto the FMS description of the electroweak sector of thestandard model [3, 21], which is again conﬁrmed by latticeinvestigations [22, 23]. Note that composite massless photons appear also innon-GUT contexts, e. g. as bound states of new fermions[24, 25]. given in Section 3, with some details relegated toappendix A. In particular, we found that earlyinvestigations of the phase diagram of this theory[27–31] likely underestimated systematic eﬀectsdue to the ﬁnite volume and length of MonteCarlo trajectories, similar to what has happenedin the fundamental case . These eﬀects arequite severe, and thus also in our case we can-not yet oﬀer a full systematic analysis in termsof discretization artifacts, though volume eﬀectswill be investigated in great detail.Since perturbation theory uses gauge-ﬁxedcalculations, we need to replicate this on thelattice to provide the corresponding results forcomparison. For this, we use the minimal ’tHooft-Landau gauge [3, 14]. This also allows usto determine the running gauge coupling in theminiMOM scheme , and to compare gauge-ﬁxed correlation functions to their perturbativepredictions. By this we verify that we indeedwork at weak coupling. This is also a necessarystep to obtain the FMS predictions [3, 12]. Thisis discussed in Section 4.Finally, the central result is the spectroscop-ical analysis of the vector channel in Section 6,which is obtained with the methods described inSection 5. The spectrum is found to be compat-ible with the results from the FMS mechanism. Especially, we ﬁnd the massless compositevector state, which would act as the photon ina GUT scenario. We do not ﬁnd evidence forfurther massive states. These ﬁndings are sum-marized and put into perspective in Section 7.Some preliminary results can be found in .2
SU(2) theory cou-pled to an adjoint scalar
The theory we investigate is described by theLagrangian L = − W aµν W aµν + tr (cid:2) ( D µ Φ) † ( D µ Φ) (cid:3) − V (Φ) . The scalar ﬁeld can be expanded as Φ( x ) =Φ a ( x ) T a , where T a are the generators of the Liealgebra of the group. The components Φ a form athree dimensional real-valued vector. The scalarﬁeld transforms under a gauge transformation G as Φ( x ) → G ( x )Φ( x ) G ( x ) † . The covariantderivative acts as D µ Φ = ∂ µ Φ + ig [ W µ , Φ] . Thegauge ﬁelds W µ = W aµ T a and their ﬁeld strengthtensor W µν = W aµν T a are the usual ones of Yang-Mills theory.The potential is taken to be the most generalone renormalizable by power-counting and con-serving the Z transformation Φ → − Φ, V (Φ) = − µ tr (cid:2) Φ (cid:3) + λ (cid:2) Φ (cid:3) + ˜ λ tr (cid:2) Φ (cid:3) . (1)However, in the case of the gauge group SU(2)trΦ is the only nontrivial invariant Casimir, andwe can therefore combine the last two terms intoone with a single coupling constant λ . In addi-tion, because the ﬁeld is in the not-faithful ad-joint representation of the pseudo-real SU(2), the Z symmetry is not an independent ﬁeld trans-formation when the theory is gauged. Hence,there is no global (custodial) symmetry, andthere are no global quantum numbers in this the-ory except for spin and parity. To test the FMS mechanism and compare tousual perturbative treatments it is necessary to consider the gauge-ﬁxed theory. Since our inter-est is the BEH domain, only this case will beconsidered. For the present theory, the (only)breaking pattern is SU(2) → U(1) , i.e., anunbroken U(1) subgroup is left.It is then possible to choose a suitable gauge,here minimal ’t Hooft-Landau gauge , wherethe scalar ﬁeld can be split into a constant anda ﬂuctuating part, i.e.,Φ( x ) = h Φ i + φ ( x ) ≡ w Φ + φ ( x ) . (2)Φ is the direction of the vacuum expectationvalue obeying Φ a Φ a = 1, and w is its magni-tude. Φ can always be chosen inside the Car-tan . Gauge transformations in the unbro-ken U(1) subgroup leave Φ invariant. The ﬁeld φ = φ a T a is the ﬂuctuation ﬁeld.Inserting the split (2) into the Lagrangianyields the tree-level mass matrix( M A ) ab = − gw ) tr (cid:16) [ T a , Φ ][ T b , Φ ] (cid:17) , for the gauge bosons. This leads to a masslessgauge ﬁeld for the unbroken U(1) subgroup. Themasses of the two SU(2) coset gauge bosons are m A = gw . In addition, one degree of freedom ofthe scalar Higgs ﬁeld remains with mass m H = √ λw . As discussed in the introduction, the observ-able spectrum needs to be manifestly and non-perturbatively gauge-invariant [1, 2]. For thepresent theory this spectrum has been predictedin  for the 0 + and 1 − channels, implying thepresence of non-scattering states in both. Forcompleteness, we will rehearse here the predic-tions of  for these two channels.3onsider ﬁrst the 0 + channel. The simplestcomposite gauge-invariant operator with thesequantum numbers is O + ( x ) = tr (cid:2) Φ (cid:3) ( x ) . To obtain the leading-order prediction for the as-sociated mass spectrum for this operator, FMS-augmented perturbation theory requires to ex-pand this operator in the vacuum expectationvalue w to leading non-constant order , yield-ing O + ( x ) = w w H ( x ) + O ( w ) , (3)with the Higgs ﬁeld H ( x ) = 2tr(Φ φ ( x )). Thus,at this order, the operator is, up to an irrelevantconstant, identical to the Higgs. States createdby this operator should thus have the same massspectrum as the elementary Higgs. Especially, attree-level the scalar singlet should have the massof the Higgs at tree-level, i.e., m H .The situation is somewhat more involved forthe vector channel 1 − . Because of the special fea-tures of the present theory, the simplest opera-tor, generalized from the fundamental case [1, 2],is  O µ − = ∂ ν ∂ tr (cid:2) Φ F µν (cid:3) . (4)Performing the same expansion yields  O µ − = − w tr (cid:2) Φ W µ ⊥ (cid:3) ( x ) + O ( w ) , (5)with W µ ⊥ = W µ ⊥ + g ∂ ν ∂ [ W µ , W ν ] , (6)the ﬁeld-strength tensor with one index trans-versely contracted and W µ ⊥ = (cid:18) δ µν − ∂ µ ∂ ν ∂ (cid:19) W ν , the transverse part of the gauge ﬁeld.At tree-level (5) reduces to O µ − = − w tr (cid:2) Φ W µ ⊥ (cid:3) ( x ) + O ( w , g , λ ) . (7)The trace with Φ projects precisely to the trans-verse gauge boson of the unbroken U(1) sub-group. Thus, the state created by this opera-tor should contain a massless pole. Hence, thispredicts  a massless, composite vector boson.This gauge-invariant state could potentially playthe role of an eﬀective low-energy photon in aGUT setup.At leading order in w , but next-to-leading or-der in g , this changes. While the ﬁrst term in(6) will give rise only to a scattering threshold,this is no longer obvious for the second term. Adetailed analysis in a constituent-like evaluation yields that a second pole at 2 m A could arise,and thus a second, massive vector particle. Ofcourse, such a particle, like the scalar, will notbe stable against decay into the massless vectors,but the level can still show up in the spectrumas a resonance, if it is present and decays weaklyenough.Unfortunately, it turns out that the scalar isfar too noisy to obtain reliable results with aboutﬁve million core hours of computing time avail-able to us in this project. The reason is that ithas vacuum quantum numbers, and thus suﬀersfrom the presence of disconnected contributions.This substantially enlarges the noise. Thoughwe saw a signal in the lattice simulations pre-sented here at short times, the signal drownedto quickly in noise to determine spectral infor-mation. We estimate that at least an order ofmagnitude more statistics, and probably furtherimproved operators, will be necessary for a resultof similar quality as in the vector channel.Thus, we will concentrate here only on the pre-dictions in the vector channel. In principle, there4ould also be non-scattering states in other chan-nels. But because of the lack of elementary par-ticles with other spin-parity quantum numbersno one-to-one mapping in the sense of the FMSmechanism, e. g as in (3) for the 0 + channel andthe Higgs, is possible. They would therefore benon-trivial bound states, and could be searchedfor along the lines of [36, 37] in the fundamen-tal case. Based on the experience with thesecases, this will likely require substantially morestatistics than even for the 0 + , and we will leavethese others channels therefore to future investi-gations. The lattice action can be obtained by discretiza-tion of the action as  S [Φ , U ] = S W [ U ]+ X x (cid:16) (cid:2) Φ( x )Φ( x ) (cid:3) + λ (cid:0) (cid:2) Φ( x )Φ( x ) (cid:3) − (cid:1) − κ X µ =1 tr (cid:2) Φ( x ) U µ ( x ) Φ( x + ˆ µ ) U µ ( x ) † (cid:3)(cid:17) , with S W the standard Wilson action and U µ ( x )are the usual links. The action can be rewrittenin component form: S [Φ , U ] = S W [ U ]+ X x " X a =1 (cid:16) Φ a ( x )Φ a ( x )+ λ (cid:0) Φ a ( x )Φ a ( x ) − (cid:1) (cid:17) − κ X µ =1 3 X a,b =1 Φ a ( x ) V abµ ( x ) Φ b ( x + ˆ µ ) , with V abµ ( x ) = tr (cid:2) T a U µ ( x ) T b U µ ( x ) † (cid:3) , which are the links in the adjoint representation.In fact, the latter form of the action has beenused for our simulations.Lattice of sizes L = 8 , 12 , 16 , 20 , 24 ,and 32 have been used. For the simulation, amulti-hit Metropolis Monte-Carlo algorithm hasproven to be eﬀective for the purpose of gener-ating the conﬁgurations, like in [12, 14], see alsoappendix A. For every update of the scalar ﬁeldﬁve updates of the gauge ﬁeld have been em-ployed, and ﬁve hits have been used for everyupdate. This created a new conﬁguration. We have scanned, similarly to [14, 15], a widerange of lattice parameters within the ( β, κ, λ )volume. However, we encountered severe criti-cal slowing down. This is discussed in detail inappendix A. Especially, we found that with bet-ter thermalization properties the results on thephase diagram from exploratory investigations[27–31] changed, and especially the phase tran-sition shifted to larger values of κ for larger vol-umes. The reason for this is likely the presence ofthe massless gauge-invariant vector particle, andthus slow decorrelation and large ﬁnite-volumeeﬀects.However, these results, together with our own,suggest a transition from a QCD-like phase toa BEH phase at any ﬁxed values of β and λ when increasing κ suﬃciently. Based on thescan, and since we do aim at a proof-of-principle,we thus decided to ﬁx β = 4 and λ = 1, andperform a scan in κ from κ = 1 /
8, i.e., a tree-level massless scalar, to κ = 2. As will be seen,5e ﬁnd a transition at about κ ≈ . κ ∈ [0 . , . κ = 0 .
55 and κ = 0 . κ & . κ ≈ .
5. However,the susceptibility suggests either a cross-over orat least a very small critical region for a phasetransition, due to the absence of volume scaling.Although being close to an actual second-ordertransition point , if it exists, would be preferablefor a better approach to large correlation lengths,for the purpose at hand it will be suﬃcient tohave suﬃciently large correlation lengths. Aswill be seen, our choice of large-statistics sim-ulation points, κ = 0 .
55 and κ = 0 .
65, indeedprovide suitable conditions.In total, we have simulated then 12 lattice se-tups in detail: For each κ = 0 .
55 and κ = 0 . , 12 , 16 , 20 ,24 , and 32 . For the gauge-invariant states, weused (1 − × conﬁgurations for the smallervolumes, 8 and 12 , and (1 − × for thelarger volumes, while for the gauge-ﬁxed calcu-lations an order of magnitude less conﬁgurationswas used. This was necessary to compensatefor the substantially increased computing timefor gauge ﬁxing, which increases with volume by Even if no genuine second-order phase transition ex-ists, we expect  that low-energy observables are suf-ﬁciently reliable, just as is the case with the standardmodel Higgs sector . one to two orders of magnitude in comparisonwith the generation of not gauge-ﬁxed conﬁgu-rations. However, as the elementary gauge-ﬁxedobservables contain less ﬁeld operators than thecomposite gauge-invariant ones, a similar level ofstatistical accuracy was nonetheless achieved, asit is expected from results on gauge dependantobservables in Yang Mills theories [40, 41].
As Section 2.2 shows, testing the FMS mech-anism requires information from the gauge-dependent spectrum. We therefore ﬁx a sub-set of the conﬁgurations to minimal Landau-’tHooft gauge. This is done like in [14, 15], byﬁrst ﬁxing minimal Landau gauge, and then per-forming a global gauge transformation to satisfythe ’t Hooft gauge condition by rotating the ex-pectation value of the Higgs ﬁeld into the Car-tan. In a ﬁnite volume this is always possible,even in a QCD phase, where the vacuum expec-tation value in any gauge vanishes in the inﬁnite-volume limit.Once ﬁxed, we calculate separately the gaugeboson propagators in the Cartan direction andin the remainder direction, as in . Further-more, we calculate the ghost propagator to de-termine the running gauge coupling in the min-iMOM scheme , again as in . This allowsus to verify that we are indeed in a weak couplingregime. Finally, we also investigated the scalarboson propagator to conﬁrm the existence of theGoldstone boson, as in .The results for the gauge boson propagatorsfor both simulation points are shown in ﬁgure 2.In addition tree-level ﬁts based on Section 2.1 D ( p ) = Z ( ap ) + ( am ) , (8)6 .25 0.50 0.75 1.00 1.25 1.50 1.75 2.00κ0.800.810.820.830.840.850.860.87 P l a q u e tt e Plaquette ∂ | P |/ ∂ κ Plaquette susceptibility Figure 1: The plaquette as a function of κ (left panel) for various volumes, as well as its derivativewith respect to κ (right panel). The scatter of the susceptibility at large values of κ is an artifactof the critical slowing down discussed in appendix A. L/a κ am A
32 0.55 0.338(1)24 0.55 0.261(1)16 0.55 0.207(2)32 0.65 0.54(2)24 0.65 0.623(3)16 0.65 0.585(9)Table 1: The ﬁt parameter for the ﬁt form (8)of the gauge-ﬁxed gauge boson propagator fordiﬀerent lattice sizes
L/a . For the 8 lattice nostable ﬁt was possible. In ﬁgure 2 the values forthe 16 lattices have been used.for the propagators are shown, with p the stan-dard improved momentum p µ = 2 sin(2 πn µ /L ).This ﬁt describes the data quite well, except forthe two lowest momentum points. However, thecomparison of diﬀerent volumes show that thesepoints are strongly aﬀected by ﬁnite-volume ef-fects, and can thus be dismissed from the ﬁts. The ﬁt values for the masses of the massive prop-agator are an important ingredient in Section6 and we list them therefore in Table 1. Thisyields that the gauge boson in the unbroken sec-tor is indeed compatible with a massless particle,while the ones in the broken sector are compati-ble with tree-level massive ones. However, we ob-serve qualitatively diﬀerent, and strong, volume-dependencies for the diﬀerent κ values. This isactually consistent with the predictions from theFMS mechanism and the fact that the physicalstates cross various decay thresholds as a func-tion of volume, as will be discussed in detail inSection 6, and can be seen in ﬁgure 7.The running gauge coupling in the miniMOMscheme is shown in ﬁgure 3. The picture is quitesimilar to the case with a fundamental Higgs .At large momenta the running coupling of thebroken sector and the unbroken sector uniﬁes.The momenta where they split depends on thelattice parameter, and is larger the larger m A , as7 .00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00ap10 a D ( a p ) Propagator, κ=0.55 massless fitmassive fit8 broken8 unbroken16 broken16 unbroken24 broken24 unbroken32 broken32 unbroken 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00ap0.81.01.21.188.8.131.52 ( p + m ) D ( a p ) Gauge boson dressing function,k=0.55 broken16 unbroken24 broken24 unbroken32 broken32 unbroken0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00ap10 a D ( a p ) Propagator, κ=0.65 massless fitmassive fit8 broken8 unbroken16 broken16 unbroken24 broken24 unbroken32 broken32 unbroken 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00ap0.81.01.21.184.108.40.206 ( p + m ) D ( a p ) Gauge boson dressing function,k=0.65 unbroken24 unbroken32 unbroken16 broken24 broken32 broken Figure 2: The gauge boson propagator (left panels) and dressing function (right panels) for κ = 0 . κ = 0 .
65 (bottom panels) against tree-level ﬁts for the 16 case in lattice units.Momenta are along an edge of the lattice. The masses used to calculate the dressing functions arezero for the Cartan propagator and the ﬁtted mass am A in Table 1 for the broken sector in theright panels. Momenta are along a lattice edge.expected. For the lower scale with its larger vol-ume and lower maximal physical momenta thisis at ap split ≈ .
6, while for the ﬁner lattice it isat ap split ≈ .
1. The lowest momenta are visiblyaﬀected by ﬁnite volume eﬀects. Ignoring them,the coupling in the broken sector is typical for a theory with BEH eﬀect , and never exceedsabout 0.1. For the unbroken sector, the couplingis almost momentum-independent, but also is atmost 0 .
12, even at the smallest momenta. Thus,both lattice settings are indeed weakly coupled,at least for the gauge interaction.8 .25 0.50 0.75 1.00 1.25 1.50 1.75 2.00ap0.040.060.080.100.120.140.160.18 α ( a p ) Running coupling, κ=0.55 broken8 unbroken16 broken16 unbroken24 broken24 unbroken32 broken32 unbroken 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00ap0.040.060.080.100.120.140.160.180.20 α Running coupling, κ=0.65 broken8 unbroken16 broken16 unbroken24 broken24 unbroken32 broken32 unbroken Figure 3: The running gauge coupling in the miniMOM scheme. The left panel shows the result for κ = 0 .
55 and the right panel for κ = 0 .
65. Note that, the lowest momentum point is very stronglyaﬀected by ﬁnite volume eﬀects, and thus often outside the plotting range. Momenta are along anedge of the lattice. D H ( a p ) / D t l Higgs dressing function, κ=0.55 Goldstone8 Fluctuation16 Goldstone16 Fluctuation24 Goldstone24 Fluctuation32 Goldstone32 Fluctuation 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00ap0.000.250.500.751.001.251.501.752.00 D H ( a p ) / D t l Higgs dressing function, κ=0.65 Goldstone8 Fluctuation16 Goldstone16 Fluctuation24 broken24 unbroken32 broken32 unbroken Figure 4: The renormalized Higgs dressing function normalized to the tree-level propagator in thewould-be pole scheme of . The left panel shows the result for κ = 0 .
55 and the right panel for κ = 0 .
65. Note that the lowest momentum point is very strongly aﬀected by ﬁnite volume eﬀects,and thus often outside the plotting range. Momenta are along an edge of the lattice.Finally, we show the Higgs dressing function in ﬁgure 4. Because the Higgs propagator requires9lso a mass renormalization, we needed to choosea scheme. For this purpose, we used the one of, which shows almost no volume-dependence, and appears to be a suitable approxima-tion to the pole scheme on an Euclidean lattice, at least in a BEH phase. However, as wedo not know the corresponding pole mass with-out access to the gauge-invariant scalar , wecould only choose arbitrary masses for the renor-malization condition, except for the Goldstonemasses. These are massless in ’t Hooft-Landaugauge. For the ﬂuctuation propagator, we set therenormalized masses to 0.5 and 1.2 for κ = 0 . κ = 0 .
65, respectively. They provided rea-sonably stable results for all volumes, though es-pecially the ﬂuctuation mode on κ = 0 .
65 turnedout to be quite ﬁckle. The resulting dressingfunction do not deviate substantially from thetree-level form 1 / (( ap ) + ( am ) ), and especiallythe Goldstone modes are well compatible withbeing massless. However, strong volume depen-dencies are also seen here at small momenta. As the scalar channel is too strongly dominatedby noise from the disconnected contributions, weconcentrate here on the vector channel. To de-termine the spectrum in this channel, we employa standard variational analysis, solving a gener-alized eigenvalue problem .The following operators have been employedin this variational analysis for the study of the J P = 1 − channel. All operators are averaged over time slices to reduce noise. The ﬁrst oper-ator is the simplest discretization of the contin-uum operator (4), see : B i − ( x ) = Im tr h Φ( x ) U jk ( x ) iq (cid:2) Φ( x ) (cid:3) , (9)where U jk is the usual plaquette, and the indices( ijk ) are even permutations of the spatial indices(123). We enlarge the basis by adding two moreoperators B Φ ,i − ( x ) = 2 tr (cid:2) Φ( x ) (cid:3) B i − ( x ) , (10) B ,i − ( x ) = X j =1 B j − ( x ) B j − ( x ) ! B i − ( x ) . (11)These represent scattering states in this channel.The ﬁrst one has an insertion of another opera-tor with quantum numbers 0 + constructed fromthe scalar ﬁeld. The second one also has an in-sertion of a 0 + operator, but this one has beenconstructed using a product of the vector opera-tor. Both insertions are multiplied with the op-erator described in (9) to provide the spin-parity.The additional two operators therefore describea scattering state of a scalar and a vector, andof three vectors, respectively, with zero relativemomenta.In addition, we performed APE smear-ing, like in the fundamental case [45, 46],up to n = 5 levels. The smearing pro-cedure for the ﬁelds reads as follows: U ( n ) µ ( x ) = 1 q det R ( n ) µ ( x ) R ( n ) µ ( x ) , (12) R ( n ) µ ( x ) = α U ( n − µ ( x ) + 1 − α X ν = µ h U ( n − ν ( x + ˆ µ ) U ( n − µ ( x + ˆ ν ) † U ( n − ν ( x ) † + U ( n − † ν ( x + ˆ µ − ˆ ν ) U ( n − µ ( x − ˆ ν ) † U ( n − ν ( x − ˆ ν ) i , Φ a ( n ) ( x ) = 17 " Φ a ( n − ( x ) + X µ (cid:16) V abµ ( x ) Φ b ( n − ( x + ˆ µ ) + V baµ ( x − ˆ µ ) Φ b ( n − ( x − ˆ µ ) (cid:17) , U (0) = R (0) and Φ (0) describe the un-smeared ﬁelds. We select the tuning parameter α = 0 .
55, as in the fundamental case . Thiscreated in total a maximal basis of four opera-tors per smearing level, and 24 in total. Fromthese we chose a subset of up to six operators,which provided for every lattice setting the leastnoisy results for the lowest energy levels.One particular problem is that, even in Eu-clidean space-time and on a ﬁnite lattice, mass-less vector particles cannot have a ﬁnite mass.Otherwise, a third degree of freedom would benecessary. This additional degree of freedomcannot be provided by the ﬁnite volume. Thus,to study a massless vector particle requires towork in a boosted frame .Thus we boosted our operators to a non-zeromomentum via O j ( ~p, t ) = 1 √ L Re X ~x O j ( ~x, t ) e i~p · ~x , (13)with the operators O j being (9)-(11), and it isfound that also the boosted operators remainreal. We chose the momentum in z -direction ~p = (cid:18) , , p z = 2 πL n z (cid:19) , and consider n z = 1 for all operators. In ad-dition, we enlarge the operator basis further byusing the operator (9) also with n z = 2. Thisturned out to be necessary to capture all rele-vant trivial scattering states for the analysis inSection 6.The correlators are divided in a transversecomponent C ⊥ and a longitudinal component C k An alternative may be to use twisted boundary con-ditions . Note in this context also [20, 26]. deﬁned as C ⊥ ( t ) = 1 L L − X t ′ =0 2 X j =1 (cid:10) O j ( ~p z , t ′ ) O j ( ~p z , t + t ′ ) † (cid:11) , (14) C k ( t ) = 1 L L − X t ′ =0 (cid:10) O ( ~p z , t ′ ) O ( ~p z , t + t ′ ) † (cid:11) , (15)where time-slice averaging is performed over thepoints in the 4-direction of the lattice. We ﬁndthat the longitudinal component is zero for theground state within statistical uncertainties, seealso . This is shown for an example in ﬁgure5, and required for a massless vector particle.Hence, this is already a strong hint for the exis-tence of a massless state in this channel. As formassive states the longitudinal component canbe at most constant, we will concentrate in thefollowing on the transverse part only.Because we work with boosted states, we needto take the kinetic energy into account whensearching for the energy levels. For this, we em-ploy the lattice dispersion relation cosh( aE ) = cosh( am )+ X i =1 (cid:0) − cos( ap i ) (cid:1) . (16)Especially, in the case of massless states with anon zero momentum component p z only in thethird direction the behavior should becosh( aE ) = 2 − cos( ap z ) . (17)In addition, there can be massless states withhigher momenta. Furthermore, because of theperturbative and FMS predictions, we also testfor other energy levels with once or twice themass of the elementary gauge boson. In this casewe can use equation (16) and the results in Table11 −1 C ⟂ ( t ) Transverse correlator
Massless tree-level particleData 0 1 2 3 4 5 6 7 8t/a−0.015−0.010−0.0050.0000.0050.0100.015 C ∥ ( t ) Longitudinal∥correlator
Figure 5: Examples for the correlator decomposition (15), showing the transverse part (left panel)and longitudinal part (right panel) of the gauge-invariant vector correlator (9) in a boosted frameon a 16 lattice. The simulation has been performed at κ = 3 /
4. We also indicate the expectedbehavior for a massless vector particle (solid lines).1 for the lattice energy prediction with either m = m A or m = 2 m A .We demonstrate the resulting ﬁts in ﬁgure 6for a particular lattice setup. Shown are theeﬀective masses from the lowest eigenvalues ofthe variational analysis. They are compared tothe expected lowest levels for a massless particle.While in this case only a single cosh was neces-sary for the ﬁts, sometimes at short times theﬁts deviate from the expected levels due to con-tamination from higher levels. In these cases weincluded a second cosh in our ﬁts. The resultingﬁts then agree very well with the expected levelsat large times. Thus, our operator basis is notsuﬃcient to disentangle very heavy states, but issuitable to identify the lowest levels quite well.Higher eigenvalues turned out to be too noisy onall but the smallest volumes, and thus we couldusually only identify two levels for each volume. Before studying the ﬁnal results, it is worthwhileto list the expectations. On the one hand, thereshould be a massless state. In our boosted framewe expect it to have energies corresponding toone or more units of kinetic energy, which be-have like aE n z ≈ πn z /L . In addition, thereare two diﬀerent predictions for massive states.The one from perturbation theory should havea mass m A , while the one from the FMS mech-anism should have 2 m A . In the boosted frame,both will have at least one unit of kinetic en-ergy E as well. In addition, any massive stateof mass am in this frame can only decay intoat least three massless ones. Thus, this is onlypossible if3 + cosh( am ) − cos (cid:18) πL (cid:19) < − cos (cid:18) πL (cid:19) a E ( t ) κ=0.55,V=12 κ=0.55,V=12 Fi st levelG ound state (exact)Single cosh fit, aE=0.53(1)Second levelScatte ing state (exact)Single cosh fit, aE=0.98(4) 1 2 3 4 5 6 7t/a−1.0−0.50.00.51.0 a E ( t ) κ=0.65,V=12 κ=0.65,V=12 Fir t levelGround tate (exact)Single co h fit, aE=0.52(1)Second levelScattering tate (exact)Single co h fit, aE=0.97(4)
Figure 6: The plots show the eﬀective energy obtained at κ = 0 .
55 (left panel) and κ = 0 .
65 (rightpanel) and a volume of 12 . They have been obtained in a basis with four operators smeared ﬁvetimes. Besides single-cosh ﬁts to the data (dotted lines) also the expected behavior for a masslessparticle (17) with one unit of kinetic energy is shown (dashed lines).is satisﬁed. Though the masses show somevolume-dependence, this eﬀect is dominated forour lattice setups by the volume-dependence ofthe kinetic energy. As a function of volume,both predicted massive states eventually crossthe elastic decay threshold when increasing thevolume, though at diﬀerent ones.Note that adding the perturbative state di-rectly does not make full sense, as it has diﬀerentquantum numbers: It is charged under the resid-ual gauge U(1). Thus, it cannot be observable atall. However, it could be argued that it shouldstill be manifest in the spectrum, by dominatingsome other state. Its absence is again a predic-tion of the FMS mechanism , which warrantschecking.The ﬁnal results are shown in ﬁgure 7, com-pared to these expectations. While we were notable to extract more than the two lowest-lying states on all volumes, we see a rather clear pic-ture emerging.First, the ground state is throughout consis-tent with the expected massless state. Hence,the ground state in the vector channel in thistheory is pretty likely a massless, composite par-ticle. Thus, this basic prediction of a compositemassless vector from the FMS mechanism is con-ﬁrmed. We also see very clearly and consistentlya state which is compatible with a massless statewith two units of kinetic energy. Thus, the exis-tence of a massless, composite vector particle inthis theory is well supported.We do, however, not see any indications of ei-ther of the massive states. Especially, we do notsee any hints of these states even on volumeswere we would expect them to be stable, as theyare below the corresponding decay threshold, thethird massless level, which is also indicated in ﬁg-13 .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 .
10 0 .
12 0 .
14 0 . a/L . . . . . . . . a E κ = 0 . First levelGround state (exact)Second levelScattering state (exact): p z Third levelScattering state (exact): p z Massive PT prediction: m A Massive FMS prediction: m A .
00 0 .
02 0 .
04 0 .
06 0 .
08 0 .
10 0 .
12 0 .
14 0 . a/L . . . . . . . . . a E κ = 0 . First levelGround state (exact)Second levelScattering state (exact): p z Third levelScattering state (exact): p z Massive PT prediction: m A Massive FMS prediction: m A Figure 7: The plots show the volume-dependent low-lying spectrum for κ = 0 .
55 (left) and κ = 0 . am A = 0 .
25 and am A = 0 .
6, respectively, have been used, as reasonable proxies to themasses in Table 1. For the massive predictions eﬀects from avoided level crossing have not beenincluded in this plot.ure 7. We also see no deformation indicative ofavoided level crossing or additional states. Thus,at the moment, neither of the additional massivestates is seen.The reason for this may, of course, be the op-erator basis, which always included the primi-tive operator (9). Other operators  may beneeded, e.g., like those employed in . Un-fortunately, for massless (vector) particles noL¨uscher analysis is (yet) available to check forpossible resonances. There is, of course, also thepossibility of further discretization artifacts, ﬁ-nite volume eﬀects, or too little statistics for onlysmall admixtures. Such improvements would bestraightforward, but would require substantiallymore computing time. If even such extensions would not detect the-ses states, this would have diﬀerent implications.In perturbation theory, the (unstable) massivevector state is unambiguously predicted. Its ab-sence would therefore be in direct contradictionto perturbation theory. In the FMS approach,this would invalidate the simpliﬁed constituentmodel in , but may be understood in a moreadvanced analysis  yet to be performed.
Summarizing, we have obtained substantial ev-idence for a massless, composite vector statein the Brout-Englert-Higgs regime of the SU(2)theory with a Higgs in the adjoint represen-tation. This conﬁrms the exploratory study1426]. Moreover, we ﬁnd no indications for addi-tional massive states. The latter would, however,necessarily be resonances in the inﬁnite-volumelimit.We have thus provided evidence that sucha theory can create, in a manifestly gauge-invariant way, a particle which could be regardedas a low-energy eﬀective photon in a grand-uniﬁed-theory setting. This is needed to ob-tain a non-perturbatively gauge-invariant con-struction of a GUT [12, 13]. In addition, thisis also a proof-of-principle that massless non-scalar bound states can emerge without a broken(global) symmetry, and thus not as a Goldstoneboson. This may be an interesting option alsoin other extensions of the standard model, andmay also be relevant to quantum gravity .In addition, by comparison to the gauge-ﬁxedvector particles, we support the analytic predic-tion for the bound state spectrum in the vectorchannel by the FMS mechanism for this theoryfor the ground state . That the ground-statecomes out correctly in such calculations is bynow familiar from other theories [4, 5, 14, 15].However, we do not see additional massive stateswith non-trivial internal structure, which havebeen argued for [12, 13]. Only for trivial inter-nal structure this has so far happened, experi-mentally conﬁrmed, in the standard model forthe photon and the Z -boson .In total, these results are therefore a vi-tal step towards a fully gauge-invariant con-struction of a GUT, and another example thatFMS-mechanism augmented perturbation the-ory is the best method to deal with (non-Abelian) gauge theories involving the Brout-Englert-Higgs eﬀect.Nonetheless, a full determination of the spec-trum in other channels remains desirable for theoutlined gauge-invariant description of GUTs. A logical next step is therefore to focus on thescalar channel in the future. Understandingthe scalar channel would potentially also helpto shed more light on the results in the vectorchannel, and is a necessary input for further an-alytic calculations in FMS-augmented perturba-tion theory. Acknowledgments
We are grateful to C. B. Lang and R. Son-denheimer for useful discussions, to R. Son-denheimer also for a critical reading of themanuscript, and to C. Pica for providing us withthe HiRep code in early stages of this work. V.A. is supported by the FWF doctoral schoolW1203-N16. The computations have been per-formed on the HPC clusters at the University ofGraz and the Vienna Scientiﬁc Cluster (VSC).
A Thermalization properties ofthe algorithm
As noted in Section 3.2, we found that the theoryis very hard to thermalize, especially when beingdeep in the BEH region. Given the observationof the massless mode, this does not come as asigniﬁcant surprise, as light modes usually yieldlong correlations.Originally, we started this project using amodiﬁed  variant of the HiRep code .This code is based on a hybrid Monte Carlo.We have augmented it to deal with the adjointHiggs. For this purpose, we used various de-compositions of the Higgs ﬁeld. Especially weexplicitly attempted to decouple the radial andthe angular mode. We found that this algorithmsuﬀered from a lack of thermalization for valuesof κ larger than 0 .
2. Especially, for all practi-cal purposes even volumes as small as 24 eﬀec-15ively no longer thermalized. The algorithm re-quired an extensive amount of time for updatesand in general a really low acceptance rate forthe new proposed conﬁgurations in the regimewith κ larger than 0.2. This applied both tothe vacuum expectation value of the Higgs, buteven to local quantities like the plaquette. Weare not sure what precisely created this behav-ior, but we suspect that the attempted globalupdate in the hybrid Monte-Carlo yielded onlytoo small steps inside the potential trough of theHiggs, and could therefore not move eﬃciently.We thus reverted to a local algorithm, a multi-hit Metropolis algorithm, as was already success-fully used previously for Yang-Mills-Higgs sys-tems [14, 15]. This proved successfull also in ourcase, allowing us to perform simlations with setof parameters which were practically inaccess-bile with the previous algorithm. However, wefound that even in this case thermalization be-came problematic at too large values of κ & . κ the acceptance rate decreased signiﬁ-cantly, while this eﬀect was much less harsh withlocal updates.When investigating Monte-Carlo trajectories,we ﬁnd that the reason for the jumping behav-ior comes from excursions to conﬁgurations withvastly diﬀerent values of the Higgs vacuum ex-pectation value and the Polyakov loop norm.Occasionally, it also happens that the algorithm | P | Polyakov loop norm | ϕ | Vacuum expectation value P l a q u e tt e Plaquette Figure 8: The norm of the Polyakov loop(top panel), the Higgs vacuum expectationvalue (middle panel), and the plaquette (bot-tom panel) for various volumes as a function of κ for β = 4 and λ = 1. For the Higgs vacuumexpectation value the statistical error has beenenlarged by a factor of ten to demonstrate thatthe observed eﬀect is deﬁnitely not a statisticalproblem.16ets stuck. Since the plaquette seems to changediscontinuously, this could be due to a two-statesystem. However, neither of the phases shows avanishing Higgs vacuum expectation value, as isalso visible in ﬁgure 8. Also, it would usually notbe expected that this becomes a stronger prob-lem further away from the phase transition. Wetherefore expect that this is still a sign of slowthermalization, which allows for large excursionsin conﬁguration space. This is also consistentwith the observation that the values of the ob-servables in the various trajectories seems to berather random. We therefore conclude that alsoour multi-hit Metropolis algorithm is not ableto thermalize quickly enough for κ & .
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