Excitations of elementary fermions in gauge Higgs theories
aa r X i v : . [ h e p - l a t ] J u l Excitations of elementary fermions in gauge Higgs theories
Jeff Greensite
Physics and Astronomy DepartmentSan Francisco State UniversitySan Francisco, CA 94132, USA (Dated: July 24, 2020)Static quark-antiquark states in QCD, at finite quark separation, have a spectrum of metastable states cor-responding to string-like excitations of the gauge field. In this article I suggest that there may also exist anexcitation spectrum of heavy fermions in some gauge Higgs theories deep in the Higgs phase. In this situationthere are no color electric flux tubes connecting quarks with antiquarks. There may, nonetheless, exist stableexcitations of the bosonic fields surrounding an isolated fermion, below the particle production threshold. Ipresent numerical evidence indicating the existence of such excitations in an SU(3) gauge Higgs theory, withthe scalar field in the fundamental representation of the gauge group.
I. INTRODUCTION
It has long been known that static quark-antiquark statesin QCD have a spectrum of string-like excitations of the colorelectric field joining the quarks. In a system with light dynam-ical quarks these excitations are of course only metastable,due to string breaking, and indeed the light quark-antiquarkstates themselves have a spectrum of metastable excitations,lying on linear Regge trajectories. We expect the same phe-nomena in the confinement phase of a gauge Higgs theory,with the scalar field in the fundamental representation of thegauge group.In the Higgs phase of a gauge Higgs theory, however, thereare no color electric flux tubes, and therefore no spectrum ofstring excitations associated with an isolated fermion. I willargue in this article, however, that there may still exist excitedstates of isolated fermions, corresponding to a spectrum ofexcitations of the surrounding gauge and Higgs fields. LatticeMonte Carlo evidence for such a possibility is presented be-low, in an SU(3) gauge Higgs theory with a unimodular scalarfield in the fundamental representation of the gauge group.
II. VARIETIES OF CONFINEMENT
Let us begin with ordinary QCD, and ask the question:what is the binding energy of a hadron, e.g. the J/ ψ ? Ofcourse it is impractical to address this question experimen-tally. Any attempt to “ionize” a quarkonium state will justresult in more hadrons, rather than a well separated pair ofcolor-charged particles. Nevertheless, there exist states in thephysical Hilbert space which correspond to precisely that lat-ter situation. For massive, static quarks, such states have theform Ψ V ( R ) = q a ( xxx ) V ab ( xxx , yyy ; U ) q b ( yyy ) Ψ , (1)where q , q are quark/antiquark operators transforming in thefundamental representation of the gauge group, a , b are colorindices, and Ψ is the vacuum state. The V operator trans-forms under a gauge transformation like a Wilson line running cloud of hadronsglue Ψ V = FIG. 1. Decay of a state with widely separated quark-antiquarkcolor charges and fractional electric charge into a set of color neutralhadrons of integer electric charge. The property of S c confinementis related to the energy of the color charge separated state Ψ V ( R ) , inthe limit of color charge separation R → ∞ . between points xxx and yyy , and depends only on the gauge field U i (we assume throughout a lattice regularization), and not onthe quark fields or any other matter fields. In QCD, a stateof this kind might represent a quark of electric charge + / − /
3, with no other electric charge in the region. Of coursesuch a state would not persist for long, and would soon decayinto a set of integer charged hadrons, as indicated in Fig. 1.But the point is that states with a large separation between aquark-antiquark pair, unscreened by any other matter fields,do exist in the Hilbert space. We are interested in how theenergy of this subclass of states varies with quark separation.Let E V ( R ) , with R = | xxx − yyy | , be the expectation value of theenergy of the state Ψ V above the vacuum energy. We say thatthe binding energy of a qq state is infinite, or that QCD has theproperty of “separation-of-charge” (S c ) confinement [1], ifflim R → ∞ E V ( R ) = ∞ (2)for any choice of the operator V ( xxx , yyy ; U ) , again with the im-portant restriction that V is a functional of the gauge field only.Of course it is always possible to choose a particular V suchthat this condition is satisfied; an example is a Wilson linerunning between xxx and yyy , in which case (2) holds even in anon-confining theory such as QED. The S c condition requiresthat the condition is satisfied for every V .Now suppose, instead of QCD, we consider a pure gaugetheory with only massive, static quarks at points xxx , yyy . Then atany R there is a spectrum of energy eigenstates Ψ n ( R ) = q a ( xxx ) V abn ( xxx , yyy ; U ) q b ( yyy ) Ψ , (3)which correspond to the ground and excited states of a colorelectric flux tube, running between points xxx , yyy . Such a spec-trum has in fact been observed in lattice Monte Carlo simula-tions [2]. Of course states whose excitation energy exceedsthe mass of a glueball cannot be energy eigenstates, since theycan emit a glueball and fall into a lower energy state. But be-low this limit the spectrum is stable. In QCD there are nostable flux tube states for separations R greater than some crit-ical distance, due to string breaking. Nevertheless, some ofthe Ψ n may still exist as metastable states in the theory. Infact, in QCD, this is exactly the case for the resonances whichlie on linear Regge trajectories. These are not stable states, ofcourse, but rather correspond to metastable excitations of thecolor electric flux tube.Let us now consider gauge Higgs theories, with the Higgsscalar in the fundamental representation of the gauge group.As was emphasized recently by Matsuyama and myself inref. [4], such theories have at least two distinct phases: aconfinement phase, with the property of S c confinement de-fined above, and a Higgs phase in which the S c confine-ment property is lost, and the global Z N center subgroup ofthe local SU(N) gauge group is spontaneously broken. TheHiggs phase turns out to be closely analogous to a spin glassphase, and there is a gauge invariant order parameter forthe confinement-to-Higgs transition which is a direct trans-lation, from condensed matter to a gauge theory context, ofthe Edwards-Anderson order parameter for spin glass transi-tions. The asymptotic spectrum of the Higgs phase still con-sists of color singlets, along the lines discussed by Fr¨ohlich,Morcio and Strocchi [5] and ‘t Hooft [6] (see also Maas et al.[7]). We refer to this property as “color” (C) confinement; itis much weaker than the S c confinement condition. Thus theHiggs phase is distinguished from the confinement phase bothby symmetry, and by type of confinement. The two phasesare not, however, necessarily separated by some line of non-analyticity in the free energy [8, 9]. In this sense the transitionmay be analogous to a Kertesz line [10]. III. PSEUDOMATTER FIELDS
In discussing the spectrum of elementary fermions in theHiggs phase, the concept of a “pseudomatter” operator willbe crucial. A pseudomatter operator ρ a ( xxx ; U ) is a non-localfunctional of the gauge field which transforms, at point xxx , likea matter field in the fundamental representation of the gaugegroup, with the important exception that, like the gauge field For a Nambu-Goto string stretched between two fixed points, the excitationspectrum was derived by Arvis [3]. itself, it is insensitive to transformations in the global centersubgroup of the gauge group. Hence a pseudomatter opera-tor transforms in the same way under gauge transformations g ( x ) and zg ( x ) , where z is an element of the global center sub-group. The simplest example of a pseudomatter operator, inan infinite volume, comes from the abelian theory ρ ( xxx ; A ) = exp (cid:20) − i e π Z d z A i ( zzz ) ∂∂ z i | xxx − zzz | (cid:21) . (4)Let us consider gauge transformations g ( x ) = e i θ ( x ) , and weseparate out the zero mode θ ( x ) = θ + e θ ( x ) . It is easy toverify that under such transformations ρ ( xxx ; g ◦ A ) = e i e θ ( x ) ρ ( xxx ; A ) . (5)Using this pseudomatter field, one may construct physicalstates corresponding to an isolated point charge Ψ ′ = ρ † ( xxx ; A ) ψ ( xxx ) Ψ (6)This construction is very well known, and was first introducedby Dirac [11]. Note that while the operator ρ † ( xxx ; A ) ψ ( xxx ) isinvariant under local gauge transformations, it still transformsunder global U(1) transformations. This is the hallmark of anoperator which can create a physical state associated with adefinite isolated charge, given that the ground state is itself aneigenstate of zero charge.Note also that the gauge transformation defined by g C ( xxx ; A ) = ρ † ( xxx ; A ) is precisely the transformation toCoulomb gauge, so in that gauge Ψ ′ = ψ ( xxx ) Ψ . The factthat a gauge choice defines a set of pseudomatter operatorsis quite general, and is not restricted to the abelian theory. Let g abF ( xxx ; U ) be the transformation to a physical gauge definedby some condition F ( U ) = N ) gauge theory. Then we may always express g † F ( xxx ; U ) at any point xxx in terms of its eigenvectors (enumerated by theindex n ) g † anF ( xxx ; U ) = u a ( n ) ( xxx ; U ) , u † a ( m ) ( xxx ; U ) u a ( n ) ( xxx ; U ) = δ mn . (7)Now let g be any infinitesimal gauge transformation. Then g F must have the property g † F ( xxx ; g ◦ U ) = g ( xxx ) g † F ( xxx ; U ) , (8)which means that u ( n ) ( xxx ; g ◦ U ) = g ( xxx ) u ( n ) ( xxx ; U ) , (9)from which we conclude that u ( n ) ( xxx ; U ) is a pseudomatterfield. This observation can be turned around: From a set of N orthogonal pseudomatter fields, with orthogonality definedby ∑ xxx ρ † an ( xxx ; U ) ρ am ( xxx ; U ) = δ nm , (10)it is possible to construct another set of pseudomatter fields u a ( n ) ( xxx ; U ) which define a gauge choice, i.e. a transformation g F to some physical gauge. This is the logic of the Laplaciangauge introduced by Vink and Wiese in [12], and the proce-dure for constructing the u ( n ) ( xxx ; U ) from a set of pseudomatteroperators ρ n ( xxx ; U ) is outlined in that reference.The eigenstates ζ n ( xxx ; U ) of the covariant lattice Laplacianoperator ( − D i D i ) abxxxyyy ζ bn ( yyy ; U ) = λ n ζ an ( xxx ; U ) , (11)where ( − D i D i ) abxxxyyy == ∑ k = h δ ab δ xxxyyy − U abk ( xxx ) δ yyy , xxx + ˆ k − U † abk ( xxx − ˆ k ) δ yyy , xxx − ˆ k i , (12)are all examples of pseudomatter fields, and will be espe-cially important here. Once again, these fields transform likematter in the fundamental representation of the gauge group,apart from their invariance under the global Z N subgroup ofthe SU(N) gauge group. The “Laplacian Landau gauge” in-troduced by Vink and Wiese [12] made use of the low-lyingeigenstates of the Laplacian operator in four Euclidean dimen-sions. In the next section we will be concerned with the low-lying eigenstates of the three dimensional lattice Laplacianoperator (12), defined at fixed time on a D = It was shown in [4] that in the spin glass (i.e. Higgs) phaseof the gauge Higgs theory, it is always possible to find a phys-ical gauge defined by F ( U ) = h φ i is non-zero, i.e. h g F ( xxx ; U ) φ ( xxx ) i 6 = . (13)A corollary is the loss of S c confinement in the Higgs phase.If g F ( xxx ; U ) is the gauge transformation to a gauge in which h φ i is non-zero (and g F , as just pointed out, can always bedecomposed into a set of pseudomatter fields), then one maychoose V ab ( xxx , yyy ; U ) = g † acF ( xxx ; U ) g cbF ( yyy ; U ) , (14)and show that E V ( R ) has a finite limit at R → ∞ . Conversely,in the phase of unbroken global Z N gauge symmetry, andassuming the absence of a massless phase, we must have E V ( R ) → ∞ , i.e. S c confinement, in the same limit. For de-tails, cf. [4]. Physical quark-antiquark states in the confinedphase, with finite energy in the R → ∞ limit, are created byoperators such as Q ( xxx , yyy ) = [ q a ( xxx ) φ a ( xxx )] × [ φ † b ( yyy ) q b ( yyy )] , (15) which can be thought of as creating two color neutral quark-scalar bound states. A. Pseudomatter in finite volumes
One cannot create a single electric charge in a finite vol-ume with periodic boundary conditions. The reason that theconstruction (4) doesn’t work in a finite volume, in ordinaryQED, is that the equation − ∇ D ( zzz ) = δ ( xxx − zzz ) is not solublein a finite periodic volume. Instead one must create ± chargesin pairs, as in (1), with V ( xxx , yyy ; A ) = exp (cid:20) − ie Z d z A i ( zzz ) ∂∂ z i D ( zzz ) (cid:21) − ∇ D ( zzz ) = δ ( zzz − xxx ) − δ ( zzz − yyy ) . (16)Likewise, in the non-abelian case, the eigenstates ζ n ( xxx ; U ) aredetermined only up to a global gauge-invariant phase. Un-less these operators occur in pairs such that the global phasescancel, i.e. ζ an ( xxx ; U ) ζ † bn ( yyy ; U ) , they will vanish in expectationvalues due to wild fluctuations in the global phase.In general, in a finite volume, we consider in the Higgsphase operators of the form V ab ( xxx , yyy ; U ) = ∑ n c n ρ an ( xxx ; U ) ρ † bn ( yyy ; U ) , (17)and consider taking the R = | xxx − yyy | → ∞ limit (along withthe infinite volume limit). Then instead of (13), the criterionfor spontaneous breaking of global Z N gauge symmetry is theexistence of a finite limit in the correlatorlim R → ∞ |h φ † a ( xxx ) V ab ( xxx , yyy ; U ) φ b ( yyy ) i| > , (18)for some V . In the Higgs phase, there will always exist a trans-formation g F ( xxx ; U ) to some F -gauge, such that this criterionis satisfied by the V operator in (14). IV. EXCITATIONS OF FERMIONS
I now put forward the conjecture that just as there is a setof metastable states (3) in the confined phase at fixed R , sothere is also a spectrum of excitations of a static fermion-antifermion system in the Higgs phase, at least for some gaugeHiggs theories, with a finite energy above the ground state There is an alternative approach to estimating the left hand side of (13) insome F -gauge, which is generally employed in computer simulations, viacomputation of the quantity1 V * L t ∑ t = (cid:12)(cid:12)(cid:12)(cid:12) ∑ xxx g F ( xxx ; U ) φ ( xxx , t ) (cid:12)(cid:12)(cid:12)(cid:12)+ on a V = L × L t lattice volume, and extrapolation to infinite spatial vol-ume. The modulus of the sum over xxx is used to eliminate the ambiguity withrespect to any remnant global symmetry transformations in the F -gauge. out to R → ∞ . The term “ground state” now refers not to thevacuum, but to the minimal energy state containing a staticfermion-antifermion pair. It is supposed that this gap in energyis too small to be explained simply by the presence of addi-tional vector or Higgs bosons. I support this conjecture withan example. The model is SU(3) lattice gauge theory witha standard Wilson action and a unimodular ( φ † ( x ) φ ( x ) = S = − β ∑ plaq ReTr [ U µ ( x ) U ν ( x + ˆ µ ) U † µ ( x + ˆ ν ) U † ν ( x )] − γ ∑ x , µ Re [ φ † ( x ) U µ ( x ) φ ( x + b µ )] . (19)The methods of ref. [4] can be used to determine the transitionbetween the S c confining and the spin glass (Higgs) phases. Inthis article I will work at the Wilson coupling β = . γ . At this β value the extrapolation method of [4]yields an estimate of γ = . ( ) at the transition.Now consider, at each R = | xxx − yyy | , the following set of four(in general non-orthogonal) states: Φ n ( R ) = Q n ( R ) Ψ , (20)where, for n = , , Q n ( R ) = [ q a ( xxx ) ζ an ( xxx ; U )] × [ ζ † bn ( yyy ; U ) q b ( yyy )] , (21)and Q ( R ) = [ q a ( xxx ) φ a ( xxx )] × [ φ † b ( yyy ) q b ( yyy )] , (22)where the ζ n ( xxx ; U ) are pseudomatter operators correspondingto eigenstates of the lattice Laplacian with the three largesteigenvalues λ n . Of course the Φ n states all have the form (1),with V abn ( xxx , yyy ; U ) = ζ an ( xxx ; U ) ζ † bn ( yyy ; U ) ( n = , , ) V ab ( xxx , yyy ; φ ) = φ a ( xxx ) φ † b ( yyy ) . (23)As already mentioned, the Higgs and confinement phases aredistinguished by the spontaneous breaking of the global Z center subgroup of SU(3) gauge symmetry in the Higgs phase,and what this implies is that in the unbroken, S c confinementphase, Φ ( R ) is orthogonal to the other three states in the R → ∞ limit. The reason is that the operator q a ( xxx ) φ a ( xxx ) is in-variant under all gauge transformations, while q a ( xxx ) ζ an ( xxx ; U ) transforms under the global Z N subgroup, since q transformsunder this symmetry, while ζ an ( xxx ; U ) does not. This impliesthe orthogonality just stated, providing the vacuum itself isinvariant under the global Z gauge symmetry.In order to compute energy expectation values E Φ n ( R ) cor-responding to the Φ n ( R ) states, we begin from the Euclideantime identity h Q † m ( R , t ) Q n ( R , ) i = h Φ m ( R ) | e − ( H − E ) t | Φ n ( R ) i , (24)where E is the vacuum energy, and Q ( R , t ) indicates that theoperator Q ( R ) is evaluated at time t . Then we see that the energy of state Φ n ( R ) above the vacuum energy is given by E Φ n ( R ) = − (cid:20) ddt log h Q † n ( R , t ) Q n ( R , ) i (cid:21) t = , (25)and the appropriately normalized overlap of states Φ m , Φ n is o mn ( R ) = h Φ m | Φ n i p h Φ m | Φ m ih Φ n | Φ n i = h Q † m ( R , ) Q n ( R , ) i{h Q † m ( R , ) Q m ( R , ) ih Q † n ( R , ) Q n ( R , ) i} / . (26)This may be generalized. We define E Φ n ( R , T ) = − (cid:20) ddt log h Q † n ( R , t ) Q n ( R , ) i (cid:21) t = T , (27)and o mn ( R , T ) = h Q † m ( R , T ) Q n ( R , ) i{h Q † m ( R , T ) Q m ( R , ) ih Q † n ( R , T ) Q n ( R , ) i} / . (28)These can be interpreted as the energies and the overlaps ofstates obtained by evolving the Φ n for a Euclidean time inter-val T /
2, i.e. Φ n ( R , T / ) = exp [ − HT / ] Φ n ( R ) , followed bynormalization.With discretized time on a hypercubic lattice, the logarith-mic time derivative must be replaced by the correspondinglattice expression E Φ n ( R , T ) = − log " h Q † n ( R , T ) Q n ( R , ) ih Q † n ( R , T − ) Q n ( R , ) i . (29)For T an odd integer, this is interpreted as the energy expec-tation value (minus the vacuum energy) of a state evolved for ( T − ) / Q † Q correlators arecomputed on the lattice as follows: Define a timelike Wilsonline P ( xxx , t , T ) = U ( xxx , t ) U ( xxx , t + ) ... U ( xxx , T − ) . (30)Then, after integrating out the static fermions, and discarding,since we are only interested in the energy due to the dynamicalfields, an irrelevant quark mass (hopping parameter) factor, h Q † ( R , T ) Q ( R , ) i = h Tr [ V † i ( xxx , yyy , U ( t + T )) P † ( xxx , t , T ) V j ( xxx , yyy ; U ( t )) P ( yyy , t , T )] i . (31)In the numerical calculation of this quantity we average overall xxx , yyy with fixed R = | xxx − yyy | .The overlap o ( R ) between the state Φ ( R ) constructedwith the Higgs field, and the state Φ ( R ) built with the ζ pseudomatter field, is displayed in Fig. 2(a) in the confinedphase, at β = . , γ = .
5. We see that this overlap tendsrapidly to zero as R → ∞ , as required by the invariance of thevacuum, in the confined phase, under global Z gauge trans- Φ - Φ o v e r l ap R β =5.5, γ =0.5 (a) Φ - Φ o v e r l ap R β =5.5, γ =3.5 (b) E Φ ( R ) R β =5.5, γ =0.5 Φ Φ (c) E Φ ( R ) R β =5.5, γ =3.5 Φ Φ (d) FIG. 2. Contrasting properties of pseudomatter states in the confinement and Higgs phases of an SU(3) gauge Higgs theory. (a) Overlap vs. R of normalized fermion-antifermion states using pseudomatter ( Φ ) and the Higgs field ( Φ ) states in the confined phase, at β = . , γ = . β = . , γ = .
5. (c) Energy expectation value E Φ ( R ) vs. separation R of the Φ and Φ states in the confined phase, β = . , γ = .
5. (d) Same as subfigure (c), but in the Higgs phase at β = . , γ = . formations. This global subgroup of the gauge symmetry isbroken in the Higgs phase, so that Φ ( R ) and Φ ( R ) are notnecessarily orthogonal in the R → ∞ limit. That is what wesee in Fig. 2(b), with data obtained in the Higgs phase, at β = . , γ = .
5, where the overlap between these states isquite large. It is also found in the confinement phase, in Fig.2(c), that the energy of the quark-pseudomatter state E Φ ( R ) rises linearly with R , consistent with S c confinement. The en-ergy of E Φ ( R ) is almost R independent at R >
1, which reflectsthe fact that Φ ( R ) consists of a non-interacting pair of colorsinglet (quark-Higgs) objects. In the Higgs phase the energiesof both the Φ and Φ states are nearly R -independent, as seenin Fig. 2(d). Now let τ = exp ( − H ) be the operator corresponding to It should be noted that the property of S c confinement in the confinementphase implies that the energies E Φ ( R ) of states Φ − ( R ) diverge to infinityas R → ∞ . But it is not necessarily true that the energies of these particularstates have a finite limit as R → ∞ everywhere in the Higgs phase, althoughthis finite limit is in fact seen for β = . γ > .
4. While there mustalways exist, everywhere in the Higgs phase, finite energy states corre-sponding to isolated (i.e. R → ∞ ) fermions, these need not be the n = , , the lattice transfer matrix, and we would like to calculatethe eigenstates and eigenvalues of this operator in the Higgsphase, in the four dimensional subspace of Hilbert spacespanned by the non-orthogonal set of states { Φ n } . We definethe 4 × [ T ] , [ O ] whose matrix elements are [ T ] mn = h Φ m | e − ( H − E ) | Φ n i = h Q † m ( R , ) Q n ( R , ) i [ O ] mn = o mn (32)respectively. The eigenstates and eigenvalues of T = τ e E inthe subspace are obtained by solving the generalized eigen-value problem [ T ] υυυ ( n ) = λ n [ O ] υυυ ( n ) , (33) states listed in (20), which correspond to a particular choice of the V oper-ator. For a further discussion of this point, cf. [4]. and we have energies above the vacuum energy E , given by E n ( R ) = − log ( λ n ) , (34)and ordered such that E n increases with n , corresponding toeigenstates in the subspace Ψ n ( R ) = ∑ i = υ ( n ) i Φ i ( R ) . (35)Likewise we consider evolving the states Ψ n in Euclidean time T nn ( R , T ) = h Ψ n | e − ( H − E ) T | Ψ n i = υ ∗ ( n ) i h Φ i | e − ( H − E ) T | Φ j i υ ( n ) j = υ ∗ ( n ) i h Q † i ( R , T ) Q j ( R , ) i υ ( n ) j (36)and compute E n ( R , T ) = − log (cid:20) T nn ( R , T ) T nn ( R , T − ) (cid:21) . (37)This can be regarded (for T an odd integer) as the en-ergy expectation value of state Ψ n ( R ) which has evolved for ( T − ) / T mn in an obvious way.There are several possibilities, for each Ψ n :1. Ψ n ( R ) is an exact eigenstate of the transfer matrix inthe full Hilbert space. Then E n ( R ) = E n ( R , T ) is timeindependent. This situation is rather unlikely.2. Ψ n ( R ) has a substantial overlap with the true groundstate, and therefore evolves steadily, in Euclidean time,towards that ground state. Then E n ( R , T ) drops rapidlyto the lowest possible energy of the static quark-antiquark system with increasing T .3. Ψ n ( R ) has very little overlap with the ground state,and rapidly evolves in Euclidean time to a stable ormetastable excited state. Then E n ( R , T ) converges toa value which is almost constant, over some range ofEuclidean time, above the ground state energy. This isthe interesting situation.Figure 3 displays energies E n ( R , T ) for n = , Ψ , , with T rangingfrom 4 to 12 for the n = n = R = | xxx − yyy | ≤
10, havingcomponents | x i − y i | ≤ β = . , γ = . ×
32. Data is obtained from 220lattices separated by 100 Monte Carlo update sweeps. Forboth n = , T to the ground ( n =
1) and an excited ( n =
2) stateenergy, respectively, separated by an energy gap of ≈ . n = E n ( R , T ) R β =5.5, γ =3.5 T=4,n=2T=6,n=2T=8,n=2T=10,n=2T=4,n=1T=6,n=1T=8,n=1T=10,n=1T=12,n=1 FIG. 3. Energies E n ( R , T ) , defined in (37), of states Ψ , Ψ afterevolution for a period of ( T − ) / β = . , γ = .
5. Note the energy gap, which persists outto the largest T values shown, of E ( R , T ) − E ( R , T ) ≈ . E n ( R , T ) for the n = , n =
2, thedata becomes rather noisy for T >
10; nevertheless the data at T =
12 simply fluctuates around the value obtained at lower T . In Figs. 4(a) and 4(b) we display separately the data for E ( R , T ) and E ( R , T ) , including the data at T = ,
2, and (for n =
2) the noisy data at T =
12. Note that while the Ψ , areclearly not energy eigenstates, they converge rapidly in Eu-clidean time to stable states already at T = Ψ , states are orthogonal by construction.But in principle this orthogonality need not persist underEuclidean time (as opposed to real time) evolution, beyond T =
1. However, the rapid convergence of Ψ , to states withdiffering energies implies the near-orthogonality of the twostates under Euclidean time evolution. In fact the overlap cor-responding to off-diagonal matrix elements O ( R , T ) = h Ψ | e − HT | Ψ i p h Ψ | e − HT | Ψ ih Ψ | e − HT | Ψ i = T ( R , T ) p T ( R , T ) T ( R , T ) (38)can be calculated for any R , T . This has the interpretation ofan overlap between states obtained from Ψ , evolved for T / T = , , ,
10 (again at β = . , γ = .
5) is shown in Fig.5. It is clear that the states obtained from evolving Ψ , Ψ inEuclidean time are very nearly orthogonal, as we had alreadydeduced.However, there is still the possibility that the energy gapseen in Fig. 3 is not really due to an excitation of the gaugefield surrounding the static fermions, but is rather due to somelow momentum particle excitation, e.g. a massive vector bo-son, or some other particle state. This would be the case inordinary QED, where any excited state of a static dipole sim- E ( R , T ) Rn=1, β =5.5, γ =3.5 T=1,n=1T=2,n=1T=4,n=1T=6,n=1T=8,n=1T=10,n=1T=12,n=1T=14,n=1 (a) E ( R , T ) Rn=2, β =5.5, γ =3.5 T=1T=2T=4T=6T=8T=10T=12 (b) FIG. 4. Energies (a) E ( R , T ) , and (b) E ( R , T ) at β = . , γ = . T = T =
14 ( n =
1) and T = n = Ψ nor Ψ is an energy eigenstate, but both appear torapidly converge towards different eigenstates after a short evolutionin Euclidean time. ply consists of the dipole field plus photons. In order to rulethis out, it is necessary to compare the excitation energy of anexcited state with the mass of the vector boson in the Higgsphase. In this connection it is important to observe that theparticle spectrum of an SU(3) gauge Higgs theory is not nec-essarily the spectrum that might be expected perturbatively,for reasons that have been discussed at length by Maas et al.[7]. Briefly, if one follows the approach of Fr¨ohlich, Mor-cio and Strocchi [5] and ’t Hooft [6], reasoning that particlesin the asymptotic spectrum are created by local gauge invari-ant operators or, more precisely, that they show up as polesin the correlation functions of such local operators, then thecorrespondence between the perturbative and the actual spec-trum in the electroweak theory is to some extent coincidental,a consequence of the approximate SU(2) custodial symmetry,and does not extend to higher gauge groups.The spectrum of an SU(3) gauge Higgs theory in the Higgs -1-0.5 0 0.5 1 1 2 3 4 5 6 7 8 9 10 O ( R , T ) R β =5.5, γ =3.5 T=2T=4T=8T=10 FIG. 5. The overlap O ( R , T ) between states Ψ ( R ) and Ψ ( R ) whichare evolved (and then normalized) for T / O ( R , T ) = T = ,
1, but the Euclidean time-evolved states are seen to remain approximately orthogonal for R > T . phase was determined by lattice simulations in [13], in a the-ory with somewhat different lattice couplings, and with a fluc-tuating (rather than unimodular) scalar field in the fundamen-tal representation. It was found that the lightest state was a1 −− vector meson, whose mass could also be determined an-alytically from correlators of the gauge-invariant operator φ † ( xxx ) U k ( xxx ) φ ( xxx + ˆ k ) . (39)Assuming that such an operator also creates the lightest statein the version of SU(3) gauge Higgs theory under considera-tion here, I have estimated the mass of this lightest state viathe standard procedure of projecting, at fixed time, to the zeromomentum component Q k ( t ) = ∑ xxx φ † ( xxx , t ) U k ( xxx , t ) φ ( xxx + ˆ k , t ) , (40)and then computing the Euclidean time correlation function ofthe zero momentum operators G ( T ) = h ∑ k = Q † k ( T ) Q k ( ) ih ∑ k = Q † k ( ) Q k ( ) i . (41)The data found on a 14 lattice volume, at β = . , γ = . G ( T ) Tvector boson correlator
FIG. 6. The Euclidean time correlator G ( T ) (see eq. (41)), shownon a log scale, associated with a zero-momentum vector meson state.The couplings are β = . , γ = .
5. The straight line is a best fitthrough the three data points at T >
0, and yields an estimate of thevector boson mass of 1.30(1) in lattice units. operator, i.e. Q ′ k ( t ) = ∑ xxx ζ †1 ( xxx , t ; U ) U k ( xxx , t ) ζ ( xxx + ˆ k , t ; U ) . (42)The corresponding time correlation function is almost indis-tinguishable from G ( T ) in (41), and the mass is in agreement,within error bars, with the vector boson mass extracted fromthe data in Fig. 6. Note that if one used different Laplacianeigenstates on the right and left side of U k in eq. (42), the timecorrelator would vanish, due to wild fluctuations in the globalphase of ζ n ( xxx ; U ) . One could construct gauge-invariant states containing two Laplacian eigen-states and two vector bosons, which would be independent of the globalphases. The energy calculation would then require computation of a timecorrelation function of four vector boson operators, which I have not at-tempted. Φ - Φ o v e r l ap Roverlap data γ =3.5 γ =2.5 γ =2.15 γ =1.8 γ =1.4 γ =1.3 FIG. 7. The overlap o (from eq. (26)) vs. R at fixed β = . γ values, where o is the overlap between the normal-ized Φ and Φ states, which use pseudomatter and the Higgs fieldrespectively to enforce gauge invariance. Note that γ = . γ values are inthe Higgs phase. It is natural to ask what happens as γ approaches theHiggs to confinement (spin glass to symmetric) transition at γ ≈ . , β = .
5. As one might expect, the overlap o be-tween the normalized Φ and Φ states, built from pseudo-matter and the Higgs field respectively, drops steadily as γ isreduced, as seen in Fig. 7, and is consistent with zero at large R at γ = .
3, which is either within or very near the confinedphase.As γ is reduced below γ = . β = .
5, both the ground and excited state energies graduallyrise, but an energy gap remains. The data, for the fixednumber of 220 lattices which were used at each γ , becomesnoisier for E ( R , T ) at small values of T as γ is reduced.These tendencies are illustrated in Fig. 8 for E n ( R , T ) , againat β = . γ = . , . , .
4. The data for E at T = γ = .
15, and at T = γ = .
8, has obviously quitea lot of statistical error, which presumably could be reducedby increasing statistics. Still, the existence of an energy gapbetween the n = n = γ values. At a still lower value of γ = . E ( R , T ) is very noisy beyond T =
4, and it is not possible to make any statement aboutconvergence to a stable excitation.
V. CONCLUSIONS
I have constructed four gauge-invariant states for a staticfermion-antifermion pair, at each fermion pair separation R ,by combining the fermion operators with Higgs and pseu-domatter operators. Working in the SU(3) gauge Higgs the-ory of eq. (19) at lattice couplings β = . γ = .
5, itis found that one of these states ( Ψ ) converges rapidly, un- E V ( R ) R β =5.5, γ =2.15 T=4,n=2T=6,n=2T=8,n=2T=10,n=2T=4,n=1T=6,n=1T=8,n=1T=10,n=1T=12,n=1 (a) E V ( R ) R β =5.5, γ =1.8 T=4,n=2T=6,n=2T=8,n=2T=4,n=1T=6,n=1T=8,n=1T=10,n=1T=12,n=1 (b) E n ( R , T ) R β =5.5, γ =1.4 T=4,n=2T=6,n=2T=4,n=1T=6,n=1T=8,n=1 (c) FIG. 8. Same as Fig. 3, but at smaller γ values, still at β = . γ = .
15; (b) γ = .
8; (c) γ = .
4, which is just above thetransition to the confined phase. Note that the data for n =
2, for datadrawn from a fixed number (220) of configurations, becomes noisyat smaller values of T , as γ approaches the transition. At γ = . T convergence of the n = der Euclidean time evolution, to the ground state, while the Ψ state also rapidly converges, but to a state with an energyabove the ground state. Of course Ψ and Ψ are orthogonalby construction, but it appears that they remain almost orthog-onal upon evolution in Euclidean time T . What is significantis that the energy gap between the n = n = T independent (for T ≥ R . The gap also seemsto be small compared to the mass of the lowest lying parti-cle excitation, assuming (based on the results of [13]) thatthe vector boson created by the operator (39) is the lightestparticle. This gap, below the vector meson threshold, indi-cates the existence of at least one gauge + Higgs field excita-tion of the fermion-antifermion system which cannot be read-ily interpreted as a fermion-antifermion ground state plus anadditional particle. It appears instead to be a stable excita-tion of the bosonic fields surrounding each of the elementaryfermions. Since the gap, judging from Fig. 3, appears to havea finite limit at R → ∞ , this is a physical excitation which isrelevant to fermion-antifermion pairs at infinite separation.Of course these are only some first results indicating ex-citations of elementary fermions, and there are many openquestions. First, it would be helpful to have a more system-atic examination of the particle spectrum of the action (19), toconfirm that the vector boson associated with the operator (39)is in fact the lightest particle in the spectrum, and to map outthe magnitude of the excitation gap throughout the Higgs/spinglass phase of the phase diagram. One would also like to gen-eralize the action beyond special case of a unimodular Higgsfield. Secondly, we would like to know whether there are ad-ditional excitations of the fermion-antifermion system beyondthe one found here; perhaps this could be studied with a largerbasis of Ψ states, and much improved statistics. Finally, itwould be very interesting to know whether any of this is rel-evant to the electroweak theory, or to phenomenology beyondthe Standard Model. We reserve these questions for later in-vestigation. ACKNOWLEDGMENTS
This research is supported by the U.S. Department of En-ergy under Grant No. de-sc0013682. [1] J. Greensite and K. Matsuyama, Phys. Rev.
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