Abelian Higgs model with charge conjugate boundary conditions
aa r X i v : . [ h e p - l a t ] F e b Abelian Higgs model with charge conjugate boundary conditions
R.M. Woloshyn
TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, Canada V6T 2A3
The abelian Higgs model is studied on the lattice with charge conjugate boundary conditions. Alocally gauge invariant operator for the charged scalar field is constructed and the charged scalarparticle mass is calculated in the Coulomb phase of the lattice model. Agreement is found with themass calculated in Coulomb gauge. The gauge invariant scalar field operator is used to calculatethe Higgs boson mass in the Higgs region and to show that the charged particle disappears fromthe spectrum in the confined regime.
I. INTRODUCTION
The study of quantum chromodynamics using latticefield theory methods has progressed to the stage wheresmall effects due to electrodynamics have to be consid-ered. However, the description of charged particles on afinite lattice with periodic boundary conditions which aretypically used in lattice simulations poses some challengesdue to violation of Gauss’s law and the gauge dependenceof the charged particle propagator. See Refs. [1, 2] forearly work and [3, 4] for reviews of recent developments.Recently, Lucini et al . [5] have reconsidered the ideaof using charge conjugate boundary conditions [6, 7]. Inthis setup fields at positions differing by a distance equalto the lattice size are related by charge conjugation. Inthis way a charged particle on the lattice can have op-positely charged images in neighboring lattice volumesand Gauss’s law, which is an obstruction in the case ofperiodic boundary conditions, can be met. Furthermore,a gauge invariant form for the charged field can be ob-tained.In Ref. [5] the formalism for charged fields in a finitevolume with charge conjugate boundary conditions is setout and specific examples of operators for lattice QED areconstructed. In this work we apply the ideas discussedin [5] to a theory of electrodynamics with scalar fields,namely, the abelian Higgs model [8]. The primary pur-pose is to illustrate the calculation of the charged particlemass in a consistent gauge invariant way in the Coulombphase of the lattice model. In addition, using a gaugeinvariant definition of the charged scalar field the Higgsphenomenon and confinement, which are features of thelattice Higgs model in other regions of the phase diagram[9–11] , are demonstrated in a new way.The general formalism for scalar field electrodynam-ics with charge conjugate boundary conditions followsthe development of Ref. [5] and is given in Sec. II.The specific lattice model used in this work is given inSec. III. The results of lattice simulations are presentedin Sec. IV In Sec. IV A the scalar model results in the ab-sence of a gauge field for periodic and charge conjugateboundary conditions are compared to show that physicsis not affected by boundary conditions. In subsequentsubsections of Sec. IV some properties of the model inthe Coulomb, Higgs and confined regions are discussed.Sec. V gives a summary.
II. FORMALISMA. General
Consider the Euclidean space action for a complexscalar field φ with electrodynamics S = S G + S φ where S G = 14 ˆ d x F µν F µν (1)and S φ = ˆ d x [( D µ φ ( x )) ∗ D µ φ ( x ) + m c φ ∗ ( x ) φ ( x )+ λ c ( φ ∗ ( x ) φ ( x )) ] (2)with F µν = ∂ µ A ν ( x ) − ∂ ν A µ ( x ) and D µ = ∂ µ + iqA µ ( x ) . The action is invariant under the transformations A µ ( x ) → A µ ( x ) − ∂ µ α ( x ) , (3)and φ ( x ) → e iqα ( x ) φ ( x ) . (4)We consider the theory in a finite cubic spatial volumewith length L on a side. The commonly used boundaryconditions are periodic A µ ( x + L ˆ i ) = A µ ( x ) , (5) φ ( x + L ˆ i ) = φ ( x ) (6)for a shift L in the i th direction. However, as discussed in[5], it is advantageous to charge conjugate when makinga shift, that is, to apply the conditions A µ ( x + L ˆ i ) = − A µ ( x ) , (7) φ ( x + L ˆ i ) = φ ∗ ( x ) . (8)These charge conjugate boundary conditions are referredto as C ∗ boundary conditions in [5]. In order to preservethe charge conjugate boundary conditions the gaugetransformation must also have a particular form. Equa-tion (3) implies that α ( x ) = β ( x ) + constant , β ( x + L ˆ i ) = − β ( x ) . (9)Then (4) requires that the constant in (9) should be aninteger multiple of π/q. The most general gauge trans-formation is therefore a combination of a local spatiallyanti-periodic function and a global factor ± acting onthe scalar field. The global phase symmetry of the actionis broken by the boundary conditions from U (1) to Z . The charge conjugate boundary conditions also affectthe construction of momentum eigenstates in finite vol-ume [5, 12]. This has implications for the lattice simula-tions that we carry out. The real part of the scalar field isperiodic and a zero momentum field can be constructedby integrating over spatial positions. The mass of theparticle associated with the field can then be extracteddirectly from correlation function of the projected fieldoperator. On the other hand, the imaginary part of thefield is antiperiodic and in the lowest momentum eigen-state there is a half unit of momentum π/L associated toeach anti-periodic spatial direction. In the lattice simu-lation correlators of real and imaginary fields have to betreated separately. The correlation function of the imag-inary part of the field yields an energy which can be usedin a dispersion relation to determine the mass.
B. Charged field operator
The construction of the charge field operator followsRef. [5]. The charge q may be a multiple of some ele-mentary charge Q = q/q el . Consider the operator Φ J ( x ) = e − iq ´ d y A µ ( y ) J µ ( y − x ) φ ( x ) (10)where J µ ( x ) satisfies ∂ µ J µ ( x ) = δ ( x ) and J µ ( x + L ˆ i ) = − J µ ( x ) . Note that a sign is changed compared to Eq.(3.1) in [5] to be consistent with the gauge transformation(3). Under a global transformation A µ ( x ) is invariant but φ ( x ) → e iqα ( x ) φ ( x ) = ( − Q φ ( x ) (11)so Φ J ( x ) → ( − Q Φ J ( x ) . (12)Using the properties of J µ ( x ) and Eq. (4) it is easy toverify that Φ J ( x ) is invariant under a local (anti-periodic)gauge transformation.Lucini et al . [5] give specific examples of functions J µ ( x ) which yield operators that can be used in a cal-culation. We adopt two of them for this work. Firstconsider a solution for J µ ( x ) which has the form J ( x ) = 0 , J i ( x ) = δ ( x ) ∂ i Γ( x ) (13)where Γ( x ) is anti-periodic. Lucini et al . [5] give anexplicit representation for Γ but we do not need it here.Then the operator (10) takes the form Φ J ( x ) = e − iq ´ d y A µ ( x , y ) ∂ µ Γ( y − x ) φ ( x ) , = e iq ´ d y ∂ i A i ( x , y )Γ( y − x ) φ ( x ) . (14) In Coulomb gauge ∂ i A i ( x ) = Φ J ( x ) just becomes thegauge fixed scalar field which we will denoted as φ c ( x ) . The correlator of the scalar field in Coulomb gauge yieldsthe gauge invariant mass for the charged particle.Another solution is J µ ( x ) = 12 δ µ,i sgn ( x i ) Y ν = i δ ( x ν ) . (15)The operator (10) with this choice of J µ ( x ) , denoted as φ s ( x ) , takes the form φ s ( x ) = e i q ´ − xi dsA i ( x + s ˆ i ) φ ( x ) e − i q ´ L − xi dsA i ( x + s ˆ i ) . (16)The operator φ s ( x ) consists of the scalar field with stringsemanating in the positive and negative i th spatial direc-tions. The strings join at the boundary and due to theboundary conditions the operator is invariant under lo-cal gauge transformations. This operator is a very con-venient one for calculation since it can be constructedeasily without gauge fixing. III. LATTICE FORMULATION
The lattice version of the abelian Higgs model has beenextensively studied, for example, in the pioneering workof Refs. [10, 11, 13, 14]. With the compact form of thelattice gauge field in terms of links U x,µ = e iqA µ ( x ) thelattice action S = S G + S ϕ takes the form S G = − β X P ( U P + U ∗ P ) ,S ϕ = − κ X x,µ ( ϕ ∗ ( x ) U x,µ ϕ ( x + ˆ µ ) + h.c. ) (17) + X x ϕ ∗ ( x ) ϕ ( x ) + λ X x ( ϕ ∗ ( x ) ϕ ( x ) − where U P is the product of links around the elementaryplaquettes and β = 1 /q . This action is usually usedwith periodic boundary conditions in all directions. Thelattice field and parameters are related to the continuumquantities in (2) by φ ( x ) = ϕ ( x ) √ κ, λ c = λκ , m c = 1 − λ − κκ . (18)With charge conjugate boundary conditions one wouldlike to use the gauge invariant operator (16). As dis-cussed in [5] this is facilitated by introducing a latticeaction where the matter field carries two units of charge.Following [5] the scalar QED version of the action is S G = − β X P ( U P + U ∗ P ) ,S ϕ = − κ X x,µ ( ϕ ∗ ( x )( U x,µ ) ϕ ( x + ˆ µ ) + h.c. ) (19) + X x ϕ ∗ ( x ) ϕ ( x ) + λ X x ( ϕ ∗ ( x ) ϕ ( x ) − β κ ∞ HiggsCoulombconfined
Figure 1: Schematic phase diagram for the lattice abelianHiggs model at fixed λ . which will be implemented with charge conjugate bound-ary conditions in all spatial directions and periodic intime. This action is invariant under the local gauge trans-formations U x,u → Λ x U x,µ Λ ∗ x +ˆ µ , (20) ϕ ( x ) → Λ x ϕ ( x ) (21)where the transformation Λ satisfies Λ x + L ˆ i = Λ ∗ x . To investigate the properties of charged field the scalarfield after Coulomb gauge fixing ϕ c will be used as wellas the lattice version of (16) which takes the form ϕ s ( x ) = − Y s = − x i U x + s ˆ i,i ϕ ( x ) L − x i − Y s =0 U ∗ x + s ˆ i,i . (22) IV. RESULTS
The phase diagram for the lattice abelian Higgs model[10, 11] at a fixed λ is shown schematically in Fig. 1. Onecan identify three regions: confined, Higgs and Coulomb.However, the confined and Higgs regimes do not actuallycorrespond to distinct phases as they can be connectedby analytic continuation around the transition line thatseparates the confined and Higgs regions [9]. For λ & . the transition line ends at a value of β greater than 0 asshown in the figure. Free charges are expected to existonly in the Coulomb phase [9].The lattice simulations presented here were carried outon site lattices using a multi-hit Metropolis updatingalgorithm. The primary goal is to explore the calcula-tion of the charged particle mass in the Coulomb phase.Evertz et al . [13] studied charged particle mass in theabelian Higgs model long ago using an indirect method.To make some contact with this earlier work we choosethe same values β = 2.5, λ = 3 for most of the mass cal- κ 〈|ϕ|〉〈 Re ϕ x * ϕ x+ µ 〉〈ϕ x * ϕ x 〉 λ = 3 β = ∞ Figure 2: Observables calculated using the scalar field action S ϕ with periodic boundary conditions in the absence of agauge field. κ 〈|ϕ|〉〈 Re ϕ x * ϕ x+ µ 〉〈ϕ x * ϕ x 〉 λ = 3 β = ∞ Figure 3: Observables calculated using the scalar field action S ϕ with charge conjugate boundary conditions in the absenceof a gauge field. culations. To illustrate the confining feature of the modelsome calculations at other values of β were also done. A. β = ∞ In the absence of the gauge field, corresponding to β = ∞ , the model reduces to a ϕ theory. As a preliminarystep we compare calculations with periodic and chargeconjugate spatial boundary conditions (periodic in time)at β = ∞ . Ensembles of 32,000 scalar field configurationswere used in these calculations.When κ goes from small to large values there is atransition to a spontaneously broken symmetry phase.To calculate the vacuum expectation value of ϕ, whichwould serve as an order parameter, one should introduce κ M a ss [ l a tti ce un it s ] Charge conjugate b.c. Re ϕ Charge conjugate b.c. Im ϕ Periodic b.c. λ = 3 β = ∞ Figure 4: Scalar particle mass in lattice units as a function of κ calculated using the scalar field action in the absence of agauge field. a symmetry breaking term with an external field η , forexample, ηϕ into the action, calculate h ϕ i and take thethermodynamic and η → limits. However, there is asimpler procedure without an external field which pro-vides a reasonable estimator for h ϕ i (see Ref. [15]). Con-sider the field averaged over a single configuration withlattice volume V ϕ = 1 V X x ϕ ( x ) , (23)and the projection of ϕ in the direction of ϕ e ϕ ( x ) = ϕ ∗ ( x ) ϕ | ϕ | . (24)Then the expectation value h e ϕ i = h| ϕ |i (25)will be used as proxy for h ϕ i . The results for h| ϕ |i as afunction of κ at λ = 3 are shown in Fig. 2 and Fig. 3 forsimulations with periodic and charge conjugate spatialboundary conditions respectively. The transition in thevicinity of κ = 0.17 is seen clearly. The expectation valuesof the operators Re ( ϕ ∗ ( x ) ϕ ( x + ˆ µ )) and ϕ ∗ ( x ) ϕ ( x ) whichappear in the action are also shown. Although theseare not strictly speaking order parameters their behavioras function of κ can give an indication that the theoryundergoes a transition. Simulations with periodic andcharge conjugate boundary conditions yield compatibleresults on our lattice.Since the gauge field is absent correlation functions of ϕ can be used directly to calculate the scalar mass. The re-sults in the symmetric phase are shown in Fig. 4. Recallthat with charge conjugate boundary conditions Im ϕ isanti-periodic in space so Im ϕ is projected to momentum ( π/L ) (1,1,1). The energy extracted from the correlator κ 〈|ϕ s |〉〈|ϕ c |〉〈 Re ϕ x *U µ ϕ x+ µ 〉〈ϕ x * ϕ x 〉 λ = 3 β = 2.5 Figure 5: Observables as a function of κ calculated with theaction (19) using charge conjugate boundary conditions. of the momentum projected Im ϕ field is converted to amass using the lattice dispersion relation E ) = m + 8 − π/L ) . (26)The mass determined this way is consistent with the massextracted from the zero-momentum correlator of Re ϕ asit should be. B. Finite β Having seen that periodic and charge conjugate bound-ary conditions give compatible results in the absence of agauge field we turn in this section to the model at finite β . Figure 5 shows observables calculated at fixed valuesof β and λ (2.5 and 3 respectively) as a function of κ ona lattice with the action (19) using charge conjugateboundary conditions. The expectation values of | ϕ s | and | ϕ c | show clearly the transition from the Coulomb phaseto the Higg regime in the vicinity of κ equal to 0.177.This is very close to the transition point found in Ref.[14] using the action (17) with periodic boundary condi-tions and with the same values of β and λ . This givesconfidence that the physics of the lattice Higgs modelused in this work is the same as that of the action (17)used in earlier work.The primary objective here is to demonstrate the cal-culation of the charged scalar boson mass in the Coulombphase. This corresponds to the region that would be rele-vant for the use of a lattice U(1) gauge theory in more re-alistic applications such as electromagnetic corrections toQCD. Correlation functions of four different scalar fieldoperators Re ϕ s , Im ϕ s , Re ϕ c , Im ϕ c were analyzed. Recallthat the imaginary parts of the field are anti-periodic inspatial directions so for these fields projection to momen-tum ( π/L ) (1,1,1) is carried out and mass is determined κ m a ss [ l a tti ce un it s ] Re ϕ s Im ϕ s Re ϕ c Im ϕ c λ = 3 β = 2.5 Figure 6: Charged particle mass in lattice units as a functionof κ in the Coulomb phase calculated with different operators. from energy using (26). The real components of the fieldare projected to zero momentum in the usual way.For calculations with the gauge invariant operator anensemble of 32,000 field configurations was used. Thesewere constructed with a multi-hit Metropolis algorithmwith 30 sweeps between saved configurations. Sincegauge fixing is rather time consuming the Coulomb gaugefixed sample had only 8,000 configurations. The Eu-clidean time correlation functions were fit with two ex-ponential terms (symmetrized in time). Statistical errorswere calculated by a jackknife procedure. The masses inlattice units are plotted in Fig. 6. There is good con-sistency between different determinations over a range of κ values which provides some confidence that the formu-lation of the lattice theory presented in [5] can be usedeffectively to deal with charged particles. C. Higgs phase
Since the use of charge conjugate boundary conditionsallows for a gauge invariant operator for the scalar fieldwe have a new way to explore the Higgs region. Inthe standard semi-classical treatment of the Higgs phe-nomenon the Higgs boson is an elementary field. In con-trast, from the nonperturbative perspective of the latticeHiggs model the Higgs boson has been interpolated usinga gauge invariant composite operator [14]. For the action(19) the composite Higgs operator takes the form O H = Re X i ϕ ∗ ( x ) U x,i ϕ ( x + i ) (27)where the sum is over spatial directions. The construc-tion (22) provides a locally gauge invariant scalar fieldand it is natural to ask if it also describes the Higgs bo-son. Correlation functions of Re ϕ s and O H (with vac-uum expectation values subtracted) were analyzed in the κ m a ss [ l a tti ce un it s ] Re ϕ s O H λ = 3 β = 2.5 Figure 7: Scalar particle mass in lattice units as a function of κ in the Higgs region calculated with different operators. Higgs region above κ = 0.177. The mass in lattice unitsis shown in Fig. 7. The statistical errors are from ajackknife analysis. The masses extracted using the twodifferent fields are consistent over the range of κ valuesthat were investigated. At the upper end of this rangethe statistical uncertainties are growing so to go to evenlarger κ would require field ensembles much larger thanthose used in this study.The field ϕ s is composite but in way that is differentfrom O H . It consists of the elementary field ϕ with acloud of gauge field fluctuations. It gives a view of theHiggs phenomenon which has some similarity to the semi-classical treatment [8] but without the notion of sponta-neous local gauge symmetry breaking which, in the non-perturbative framework, would not be viable [16]. D. Confinement
At small β compact lattice QED is confining. We ex-plore the transition to the confined regime by calculatingat fixed κ and λ and decreasing β starting a point in theCoulomb phase. Figure 8 shows the values of some ob-servables as a function of β . The gauge field plaquettevariable Re U P shows the transition from the weak cou-pling to the strong coupling regime around β = 0.25. Thevacuum expectation values of observables involving the ϕ field are quite insensitive to the value of β and exhibitonly small changes in the transition from weak couplingto strong coupling. The mass of the charged scalar ex-tracted from the correlation function of Re ϕ s increasessteadily as β is decreased as shown in Fig. 9. Below β = 0.75 the correlation function falls very rapidly asfunction of time so even with an ensemble of 32,000 con-figurations it was not possible to make an accurate massdetermination. Figure 10 shows the correlation function h Re ϕ s ( t ) Re ϕ s (0) i at β = 0.25. In this region the correla- β 〈|ϕ s |〉〈 Re ϕ x *U µ ϕ x+ µ 〉〈ϕ x * ϕ x 〉〈 ReU P 〉κ = 0.173 λ = 3 Figure 8: Observables as a function of β calculated with theaction (19) with charge conjugate boundary conditions. β m a ss [ l a tti ce un it s ] κ = 0.173 λ = 3 Figure 9: Charged particle mass in lattice units as a functionof β calculated using the operator Re ϕ s . tion function is just noise. The charged scalar field doesnot propagate. It has disappeared from the spectrumwhich can be taken as a signature of confinement.In the strong coupling region the gauge field should alsobe confined. This can be demonstrated using the photonpropagator. For the photon interpolating operator onecan use O p = Im X i,j U P (28)which is the imaginary part of the gauge field plaquettesummed over spatial planes [14]. In the Coulomb phasethe photon is expected to be massless [9] so the correla-tion function should be calculated at a nonzero momen-tum. We use momentum ( π/L ) (1,1,1) consistent withour boundary conditions. The energy calculated from × -7 -2 × -7 × -7 × -7 〈 R e ϕ s ( t ) R e ϕ s ( ) 〉 κ = 1.73 λ = 3 β = 0.25 Figure 10: Correlation function h Re ϕ s ( t ) Re ϕ s (0) i at β =0.25. β E n e r gy [ l a tti ce un it s ] κ = 0.173 λ = 3 Figure 11: Ground state energy from the correlation functionof the operator O p (Eq. (28)) as a function of β calculated atmomentum ( π/L ) (1,1,1). The dashed line shows the energyof a zero mass particle at this momentum from the dispersionrelation (26). the momentum projected correlation function of O p isplotted in Fig. (11). The dashed line is the shows theenergy for a zero mass particle calculated using the dis-persion relation (26). In the Coulomb phase the gaugefield correlator is consistent with the presence of a zeromass photon. Around β = 0.25 the mass departs fromzero and at smaller values of β the correlator of O p is re-duced to noise similar to what is seen in Fig. 10 signalingthe confinement of the gauge field. V. SUMMARY
The use of charge conjugate boundary conditions, asdiscussed by Lucini et al . [5], provides an interestingoption for dealing with QED on the lattice. An attractivefeature of this formulation is that the mass of the chargedfield can be determined using a simple gauge invariantprocedure. In this paper we have implemented the ideasof [5] in a lattice theory of electrodynamics with scalarfields, the abelian Higgs model.In Sect. 4.1 the model in the absence of a gauge field( β = ∞ ) is compared for charge conjugate and peri-odic boundary conditions. The results for a variety ofobservables are compatible. At finite β and other pa-rameters within the pertubative region of the model thecharged scalar mass was calculated using both gauge in-variant (Eq. (22)) and Coulomb gauge fixed fields. Dueto the choice of boundary conditions the imaginary partsof the fields require projection to a non-zero momentumwith mass determined using the lattice dispersion rela-tion (26). As shown in Fig. (6) these technically varied procedures yield compatible charged particle masses.The gauge invariant field ϕ s is also useful for explor-ing the Higgs model in other regions of the phase dia-gram. In the Higgs regime the correlator of Re ϕ s givesmasses which are compatible with those extracted us-ing the composite scalar operator (27) which has beenused in the past to interpolate the Higgs boson. In thestrong coupling confining region we showed that the par-ticle associated with field ϕ s disappears from the physicalspectrum.In summary, this works demonstrates the efficacy ofthe formulation of [5] for numerical studies of lattice U(1)gauge theory and encourages further applications. Acknowledgments
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