Instability of cosmic Yang-Mills fields
IInstability of cosmic Yang-Mills fields
Kaushlendra Kumar , Olaf Lechtenfeld and
Gabriel Pican¸co Costa
Institut f¨ur Theoretische Physik andRiemann Center for Geometry and PhysicsLeibniz Universit¨at HannoverAppelstraße 2, 30167 Hannover, Germany
Abstract
There exists a small family of analytic SO(4)-invariant but time-dependentSU(2) Yang–Mills solutions in any conformally flat four-dimensional spacetime.These might play a role in early-universe cosmology for stabilizing the symmet-ric Higgs vacuum. We analyze the linear stability of these “cosmic gauge fields”against general perturbations, by diagonalizing the (time-dependent) fluctua-tion operator around them and applying Floquet theory to its eigenfrequenciesand normal modes. Except for the exactly solvable SO(4) singlet perturbation,which is found to be marginally stable linearly but bounded nonlinearly, genericnormal modes often grow exponentially due to resonance effects. Even at veryhigh energies, all cosmic Yang–Mills backgrounds are rendered linearly unstable. a r X i v : . [ h e p - t h ] F e b A tale of three anharmonic oscillators
It is generally impossible to find analytic solutions to the coupled Einstein–Yang–Mills system of equa-tions, in part because they are coupled both ways. However, in a homogeneous and isotropic universe,where the metric is conformally flat, the Yang–Mills equations decouple due to their conformal invari-ance in four spacetime dimensions. Thus, if one can find isotropic Yang–Mills solutions on Minkowski,de Sitter, or anti de Sitter space, then their energy-momentum tensor will be compatible with anyFriedmann–Lemaˆıtre–Robertson–Walker (FLRW) metric (of the same topology) and allow for an ana-lytic computation of the scale factor from the Friedmann equation.Fortunately, for the de Sitter case and a gauge group SU(2), such Yang–Mills configurations with finiteenergy and action are available [1, 2, 3]. They are most easily constructed on the cylinder (0 , π ) × S ,which is related to de Sitter space by a purely temporal reparametrization and Weyl rescaling [4, 5, 6],d s = − d t + (cid:96) cosh t(cid:96) dΩ = (cid:96) sin τ (cid:0) − d τ + dΩ (cid:1) for t ∈ ( −∞ , + ∞ ) ⇔ τ ∈ (0 , π ) , (1.1)where dΩ is the round metric on S , and (cid:96) is the de Sitter radius. This provides an explicit relationbetween co-moving time t and conformal time τ , and it fixes the cosmological constant to Λ = 3 /(cid:96) . Oneemploys the identification SU(2) (cid:39) S to write an S -symmetric ansatz for the SU(2) gauge potential A µ ,which produces a solvable ODE for a parameter function ψ ( τ ) of conformal time. This ODE has theform of Newton’s equation for a mass point in a double-well potential V ( ψ ) = ( ψ − , (1.2)yielding a first anharmonic oscillator. A two-parameter family of (in general time-dependent) solutionscan be given in terms of Jacobi elliptic functions and describe SO(4) invariant Yang–Mills fields on theLorentzian cylinder over S .These Yang–Mills solutions then exist (at least locally) in any conformally related spacetime, but theconformal transformation will ruin isotropy unless we restrict ourselves to spatially closed FLRW metrics,d s = − d t + a ( t ) dΩ = a ( τ ) (cid:0) − d τ + dΩ (cid:1) for t ∈ (0 , t max ) ⇔ τ ∈ I ≡ (0 , T (cid:48) ) , (1.3)where we impose a big-bang initial condition a (0)=0, so thatd τ = d ta ( t ) with τ ( t =0) = 0 and τ ( t = t max ) =: T (cid:48) < ∞ . (1.4)The lifetime t max of the universe can be infinite (big rip, a ( t max )= ∞ ) or finite (big crunch, a ( t max )=0).Bouncing cosmologies as in (1.1) are also allowed but will not be pursued here. Since the energy-momentum tensor of our Yang–Mills configurations is SO(4) symmetric, their gravitational backreactionwill keep us inside the FLRW framework and merely modify the cosmic scale factor a ( τ ). The latter isfully determined by the Friedmann equation in the presence of the Yang–Mills energy-momentum and acosmological constant Λ, whose value may be dialed. It is well known that the Friedmann equation takesthe form of another Newton equation. Its (cosmological) potential for the case at hand reads W ( a ) = a − Λ6 a , (1.5)which is our second anharmonic oscillator (although inverted). Each pair ( ψ, a ) of solutions to thetwo systems (1.2) and (1.5) yields an exact classical Einstein–Yang–Mills configuration. One parameterin ψ is the conserved mechanical energy E in the potential V , which in turn determines the mechanicalenergy ˜ E for a in the potential W . This one-way coupling is the only relation between the two anharmonicoscillators.For a physical embodiment of the cosmological constant, we may add a third player, for instancea complex scalar Higgs field φ in the fundamental SU(2) representation. The standard-model Higgspotential U ( φ ) = λ (cid:0) φ † φ − v (cid:1) , (1.6)1here v/ √ λv is the Higgs mass, gives us a third anharmonic oscillator. Thedictate of SO(4) invariance, however, allows only the zero solution, φ ≡
0, which provides us with adefinite positive cosmological constant ofΛ = κ U (0) = κ λ v , (1.7)where κ is the gravitational coupling. The full Einstein–Yang–Mills–Higgs action (in standard notation), S = (cid:90) d x √− g (cid:110) κ R + g tr F µν F µν − D µ φ † D µ φ − U ( φ ) (cid:111) , (1.8)reduces in the SO(4)-invariant sector to S [ a, ψ, Λ] = 12 π (cid:90) T (cid:48) d τ (cid:110) κ (cid:0) − ˙ a + W ( a ) (cid:1) + g (cid:0) ˙ ψ − V ( ψ ) (cid:1)(cid:111) , (1.9)where g is the gauge coupling, and the overdot denotes a derivative with respect to conformal time.For large enough “gauge energy” E , the universe undergoes an eternal expansion, which is accom-panied by rapid fluctuations of the gauge field. The latter’s coupling to the Higgs field stabilizes thesymmetric vacuum φ ≡ U as a parametric resonance effect, as long as a is nottoo large. Eventually, when a exceeds a critical value a EW , the Higgs field will begin to roll down towardsa minimum of U , breaking the SO(4) symmetry. The corresponding time t EW signifies the electroweakphase transition in the early universe. This scenario was put forward recently by D. Friedan [6].The goal of the current paper is a stability analysis of these classical oscillating “cosmic” Yang–Mills fields. To begin with, Section 2 describes the geometry of S and reviews the classical configu-rations ( A µ , g µν ) in terms of Newtonian solutions ( ψ, a ) for the anharmonic oscillator pair ( V, W ). Toinvestigate arbitrary small perturbations of the gauge field departing from the time-dependent back-ground A µ parametrized by the “gauge energy” E , Section 3 linearizes the Yang–Mills equation aroundit and diagonalizes the fluctuation operator to obtain a spectrum of time-dependent natural frequencies.To decide about the linear stability of the cosmic Yang–Mills configurations we have to analyze the long-time behavior of the solutions to Hill’s equation for all these normal modes. In Section 4 we employFloquet theory to learn that their growth rate is determined by the stroboscopic map or monodromy,which is easily computed numerically for any given mode. We do so for a number of low-frequency normalmodes and find, when varying E , an alternating sequence of stable (bounded) and unstable (exponentiallygrowing) fluctuations. The unstable bands roughly correspond to the parametric resonance frequencies.With growing “gauge energy” the runaway perturbation modes become more prominent, and some ofthem persist in the infinite-energy limit, where we detect universal natural frequencies and monodromies.A special role is played by the SO(4)-invariant fluctuation, which merely shifts the parameter E of thebackground. We treat it exactly and beyond the linear regime in Section 5. This “singlet” mode turnsout to be marginally stable, i.e. it has a vanishing Lyapunov exponent. Its linear growth, however, getslimited by nonlinear effects of the full fluctuation equation, whose analytic solutions exhibit wave beatbehavior. Finally, some explicit data for the first few natural frequencies are collected in an Appendix.For a conclusion in a nutshell, the oscillating cosmic Yang–Mills fields are linearly unstable againstsmall perturbations, even for very high energy. In order to describe the classical Yang–Mills solutions we need to develop some elements of the spatial S geometry. Taking advantage of the fact that S (cid:39) SU(2) and so (4) (cid:39) su (2) L ⊕ su (2) R (2.1)we introduce a basis { L a } for a = 1 , , S generating the right multi-plication on SU(2) and forming the su (2) L algebra[ L a , L b ] = 2 ε cab L c . (2.2)2t is dual to a basis { e a } of left-invariant one-forms on S , i.e. e a ( L b ) = δ ab , subject tod e a + ε abc e b ∧ e c = 0 and e a e a = dΩ . (2.3)One may obtain this basis by expanding the left Cartan one-formΩ L ( g ) = g − d g = e a L a . (2.4)Here, the group element g provides the identification map g : S → SU(2) via ( α, β ) (cid:55)→ − i (cid:18) β α ∗ α − β ∗ (cid:19) with | α | + | β | = 1 , (2.5)which sends the S north pole (0 , i) to the group identity . We shall coordinatize S by an SU(2) groupelement g . The su (2) R half of the three-sphere’s so (4) isometry is provided by right-invariant vectorfields R a belonging to the left multiplication on the group manifold and obeying[ R a , R b ] = 2 ε cab R c . (2.6)The differential of a function f on I × S is then conveniently taken asd f = d τ ∂ τ f + e a L a f . (2.7)Functions on S can be expanded in a basis of harmonics Y j ( g ) with 2 j ∈ N , which are eigenfunctionsof the scalar Laplacian, − (cid:52) Y j = 2 j (2 j +2) Y j = 4 j ( j +1) Y j = − ( L + R ) Y j , (2.8)where L = L a L a and R = R a R a are (minus four times) the Casimirs of su (2) L and su (2) R , respectively, − L Y j = − R Y j = − (cid:52) Y j = j ( j +1) Y j . (2.9)The left-right (or toroidal) harmonics Y j ; m,n are eigenfunctions of L = R , L and R , i2 L Y j ; m,n = m Y j ; m,n and i2 R Y j ; m,n = n Y j ; m,n (2.10)for m, n = − j, − j +1 , . . . , + j , and hence the corresponding ladder operators L ± = ( L ± i L ) / √ R ± = ( R ± i R ) / √ i2 L ± Y j ; m,n = (cid:112) ( j ∓ m )( j ± m +1) / Y j ; m ± ,n and i2 R ± Y j ; m,n = (cid:112) ( j ∓ n )( j ± n +1) / Y j ; m,n ± . (2.12)The gauge potential is an su (2)-valued one-form on our spacetime. We use the I × S parametrizationand write A = A τ ( τ, g ) d τ + (cid:88) a =1 A a ( τ, g ) e a ( g ) with g ∈ SU(2) . (2.13)It has been shown [2, 6] that the requirement of SO(4) equivariance enforces the form A τ ( τ, g ) = 0 and A a ( τ, g ) = (cid:0) ψ ( τ ) (cid:1) T a (2.14)with some function ψ : I → R , where T a denotes the su (2) generators subject to[ T a , T b ] = 2 ε cab T c , (2.15) The SO(4) spin of these functions is actually 2 j , but we label them with half their spin, for reasons to be clear below.
3o that the adjoint representation produces tr( T a T b ) = − δ ab . The corresponding field strength reads F = d A + A ∧ A = ∂ τ A a d τ ∧ e a + (cid:0) R [ b A c ] − ε abc A a + [ A b , A c ] (cid:1) e b ∧ e c = ˙ ψ T a d τ ∧ e a + ε abc ( ψ − T a e b ∧ e c (2.16)where ˙ ψ ≡ ∂ τ ψ . The Yang–Mills action on this ansatz simplifies to S = − g (cid:90) I× S tr F ∧ ∗F = 6 π g (cid:90) I d τ (cid:2) ˙ ψ − V ( ψ ) (cid:3) with V ( ψ ) = ( ψ − , (2.17)where I = [0 , T (cid:48) ] and g here denotes the gauge coupling. Due to the principle of symmetric criticality [7],solutions to the mechanical problem ¨ ψ + V (cid:48) ( ψ ) = 0 (2.18)will, via (2.14), provide Yang–Mills configurations which extremize the action. Conservation of energyimplies that ˙ ψ + V ( ψ ) = E = constant , (2.19)and the generic solution in the double-well potential V is periodic in τ with a period T ( E ).Hence, fixing a value for E and employing time translation invariance to set ˙ ψ (0) = 0 uniquelydetermines the classical solution ψ ( τ ) up to half-period shifts. Its explicit form is ψ ( τ ) = k(cid:15) cn (cid:0) τ(cid:15) , k (cid:1) with T = 4 (cid:15) K ( k ) for < E < ∞ T = ∞ for E = ±√ (cid:0) √ τ (cid:1) with T = ∞ for E = ± k(cid:15) dn (cid:0) k τ(cid:15) , k (cid:1) with T = 2 (cid:15)k K ( k ) for 0 < E < ± T = π for E = 0 , (2.20)where cn and dn denote Jacobi elliptic functions, K is the complete elliptic integral of the first kind, and2 (cid:15) = 2 k − / √ E with k = √ , , ∞ ⇔ E = ∞ , , . (2.21)For E (cid:29) , we have k → , and the solution is well approximated by (cid:15) cos (cid:0) √ π Γ(1 / τ(cid:15) (cid:1) . At the critical valueof E = ( k =1), the unstable constant solution coexists with the celebrated bounce solution, and below itthe solution bifurcates into oscillations in the left or right well of the double-well potential, which halfensthe oscillation period. The two constant minima ψ = ± A = 0 and A = g − d g .Actually, the time translation freedom is broken by the finite range of I , so that time-shifted solutionsdiffer in their boundary values ψ (0) and ψ ( T (cid:48) ) and also in their value for the action.The corresponding color-electric and -magnetic field strengths read E a = F a = ˙ ψ T a and B a = ε bca F bc = ( ψ − T a , (2.22)which yields a finite total energy (on the cylinder) of 6 π E/g and a finite action [5, 8] g S [ ψ ] = 6 π (cid:90) I d τ (cid:2) E − ( ψ − (cid:3) = 6 π (cid:90) I d τ (cid:2) ˙ ψ − E (cid:3) ≥ − π T (cid:48) . (2.23)The energy-momentum tensor of our SO(4)-symmetric Yang–Mills solutions is readily found as T = 3 Eg a (cid:0) d τ + dΩ (cid:1) , (2.24)which is traceless as expected. 4 .001 0.002 0.003 0.004 0.005 0.006 0.007 τ - - - ψ ( τ ) τ - - - ψ ( τ ) τ ψ ( τ ) τ ψ ( τ ) Figure 1: Plots of ψ ( τ ) over one period, for different values of k :0 . . . (cid:40) − R + 4 Λ = 0 R ττ + R a − Λ a = κ T ττ (cid:41) ⇔ (cid:40) ¨ a + W (cid:48) ( a ) = 0 ˙ a + W ( a ) = κ g E =: E (cid:48) (cid:41) (2.25)with a gravitational coupling κ = 8 πG , a gravitational energy E (cid:48) and a cosmological potential W ( a ) = a − Λ6 a . (2.26)The two anharmonic oscillators, with potential V for the gauge field and potential W for gravity, arecoupled only via the balance of their conserved energies,1 κ (cid:2) ˙ a + W ( a ) (cid:3) = 12 g (cid:2) ˙ ψ + V ( ψ ) (cid:3) , (2.27)which is nothing but the Wheeler–DeWitt constraint H = 0.The Friedmann equation (2.25), being a mechanical system with an inverted anharmonic poten-tial (2.26), is again easily solved analytically, a ( τ ) = (cid:113)
3Λ 12 (cid:15) (cid:48) (cid:115) − cn (cid:0) τ(cid:15) (cid:48) ,k (cid:48) (cid:1) (cid:0) τ(cid:15) (cid:48) ,k (cid:48) (cid:1) with T (cid:48) = 2 (cid:15) (cid:48) K ( k (cid:48) ) for < E (cid:48) < ∞ (cid:113) tanh (cid:0) τ / √ (cid:1) with T (cid:48) = ∞ for E (cid:48) = (cid:113)
3Λ 12 (cid:15) (cid:48) (cid:115) − dn (cid:0) k (cid:48) τ(cid:15) (cid:48) , k (cid:48) (cid:1) (cid:0) k (cid:48) τ(cid:15) (cid:48) , k (cid:48) (cid:1) with T (cid:48) = 2 (cid:15) (cid:48) k (cid:48) K ( k (cid:48) ) for 0 < E (cid:48) < T (cid:48) = π for E (cid:48) = 0 , (2.28)5 - a - W ( a ) - - - ψ - V ( ψ ) Figure 2: Plots of the cosmological potential W ( a ) for Λ=1 and the double-well potential V .where we abbreviated (cid:15) (cid:48) = 2 k (cid:48) − / √ E (cid:48) so that k (cid:48) = √ , , ∞ ⇔ E (cid:48) = ∞ , , . (2.29)For E (cid:48) (cid:29) , we have k (cid:48) → , and the solution is well approximmated by (cid:113)
3Λ 12 (cid:15) (cid:48) tan (cid:0) √ π Γ(1 / τ(cid:15) (cid:48) (cid:1) . We τ a ( τ ) τ a ( τ ) τ a ( τ ) τ a ( τ ) Figure 3: Plots of a ( τ ) over one lifetime, for Λ=1 and different values of k (cid:48) :0 .
505 (top left), 0 . . . a (0) = 0 (big bang). There exist also (for E (cid:48) < ) bouncingsolutions, where the universe attains a minimal radius a min = (cid:2) (1 + √ − E (cid:48) ) (cid:3) / between infiniteextension in the far past ( t = −∞ ↔ τ =0) and the far future ( t =+ ∞ ↔ τ = T (cid:48) ). For E (cid:48) > Our k (cid:48) should not be confused with the dual modulus 1 − k , which is often denoted this way. → − dn in (2.28) above. The quantity T (cid:48) listed there is the (conformal) lifetimeof the universe, from the big bang until either the big rip (for E (cid:48) > ) or the big crunch of an oscillatinguniverse (for E (cid:48) < ). The solution relevant to our Einstein–Yang–Mills system is entirely determinedby the Newtonian energy E characterizing the cosmic Yang–Mills field: above the critical value of E crit = 2 g κ
38 Λ (2.30)the universe expands forever (until t max = ∞ ), while below this value it recollapses (at t max = (cid:82) T (cid:48) d τ a ( τ )).It demonstrates the necessity of a cosmological constant (whose role may be played by the Higgs expec-tation value) as well as the nonperturbative nature of the cosmic Yang–Mills field, whose contribution tothe energy-momentum tensor is of O ( g − ). Our main task in this paper is an investigation of the stability of the cosmic Yang–Mills solutions reviewedin the previous section. For this, we should distinguish between global and local stability. The former isdifficult to assess in a nonlinear dynamics but clear from the outset in case of a compact phase space. Thelatter refers to short-time behavior induced by linear perturbations around the reference configuration.We shall look at this firstly, in the present section and the following one. Here, we set out to diagonalizethe fluctuation operator for our time-dependent Yang–Mills backgrounds and find the natural frequencies.Even though our cosmic gauge-field configurations are SO(4)-invariant, we must allow for all kinds offluctuations on top of it, SO(4)-symmetric perturbations being a very special subclass of them. A genericgauge potential “nearby” a classical solution A on I × S can be expanded as A + Φ = A ( τ, g ) + (cid:88) p =1 Φ p ( τ, g ) T p d τ + (cid:88) a =1 3 (cid:88) p =1 Φ pa ( τ, g ) T p e a ( g ) (3.1)with, using ( µ ) = (0 , a ), Φ pµ ( τ, g ) = (cid:88) j,m,n Φ pµ | j ; m,n ( τ ) Y j ; m,n ( g ) , (3.2)on which we notice the following actions (supressing the τ and g arguments),( L a Φ pµ ) j ; m,n = Φ pµ | j ; m (cid:48) ,n (cid:0) L a ) m (cid:48) m , ( S a Φ) p = 0 , ( S a Φ) pb = − ε abc Φ pc , ( T a Φ) pµ = − ε apq Φ qµ , (3.3)where the L a matrix elements are determined from (2.10) and (2.12), and S a are the components of thespin operator. The (metric and gauge) background-covariant derivative reads D τ Φ = ∂ τ Φ and D a Φ = L a Φ + [ A a , Φ] with A a = (cid:0) ψ ( τ ) (cid:1) T a , (3.4)which is equivalent to D a Φ pb = L a Φ pb − ε abc Φ pc + [ A a , Φ b ] p since D a e b = L a e b − ε abc e c = ε abc e c . (3.5)The background A obeys the Coulomb gauge condition, A τ = 0 and L a A a = 0 , (3.6)but we cannot enforce these equations on the fluctuation Φ. However, we may impose the Lorenz gaugecondition, D µ Φ pµ = 0 ⇒ ∂ τ Φ p − L a Φ pa − (1+ ψ )( T a Φ a ) p = 0 , (3.7)which is seen to couple the temporal and spatial components of Φ in general. We then linearize theYang–Mills equations around A and obtain D ν D ν Φ µ − R µν Φ ν + 2[ F µν , Φ ν ] = 0 (3.8)7ith the Ricci tensor R µ = 0 and R ab = 2 δ ab . (3.9)After a careful evaluation, the µ =0 equation yields (cid:2) ∂ τ − L b L b + 2(1+ ψ ) (cid:3) Φ p − (1+ ψ ) L b ( T b Φ ) p − ˙ ψ ( T b Φ b ) p = 0 , (3.10)while the µ = a equations read (cid:2) ∂ τ − L b L b + 2(1+ ψ ) +4 (cid:3) Φ pa − (1+ ψ ) L b ( T b Φ) pa − L b ( S b Φ) pa − (1+ ψ )(2 − ψ )( S b T b Φ) pa − ˙ ψ ( T a Φ ) p = 0 . (3.11)It is convenient to package the orbital, spin, isospin, and fluctuation triplets into formal vectors, (cid:126)L = ( L a ) , (cid:126)S = ( S a ) , (cid:126)T = ( T a ) , (cid:126) Φ = (Φ a ) , (3.12)respectively, but they act in different spaces, hence on different indices, such that (cid:126)S = (cid:126)T = − p ) ∂ τ Φ − (cid:126)L · (cid:126) Φ − (1+ ψ ) (cid:126)T · (cid:126) Φ = 0 , (3.13) (cid:2) ∂ τ − (cid:126)L + 2(1+ ψ ) (cid:3) Φ − (1+ ψ ) (cid:126)L · (cid:126)T Φ − ˙ ψ (cid:126)T · (cid:126) Φ = 0 , (3.14) (cid:2) ∂ τ − (cid:126)L − (cid:126)S +2(1+ ψ ) (cid:3) Φ a − (1+ ψ ) (cid:126)L · (cid:126)T Φ a − (cid:126)L · ( (cid:126)S Φ) a − (1+ ψ )(2 − ψ ) (cid:126)T · ( (cid:126)S Φ) a − ˙ ψ T a Φ = 0 . (3.15)A few remarks are in order. First, except for the last term, (3.14) is obtained from (3.15) by setting (cid:126)S = 0, since Φ carries no spin index. Second, both equations can be recast as (cid:2) ∂ τ − − ψ (cid:126)L − ψ ( (cid:126)L + (cid:126)T ) − ψ )(1 − ψ ) (cid:3) Φ = ˙ ψ (cid:126)T · (cid:126) Φ , (3.16) (cid:2) ∂ τ − (1 − ψ )(2+ ψ )4 (cid:126)L − ψ (1+ ψ )4 ( (cid:126)L + (cid:126)T ) + ψ (1 − ψ )4 ( (cid:126)L + (cid:126)S ) − (1+ ψ )(2 − ψ )4 ( (cid:126)L + (cid:126)T + (cid:126)S ) − ψ )(1 − ψ ) (cid:3) (cid:126) Φ = ˙ ψ (cid:126)T Φ , (3.17)which reveals a problem of addition of three spins and a corresponding symmetry under ψ ↔ − ψ , (cid:126)L ↔ (cid:126)L + (cid:126)T + (cid:126)S and (cid:126)L + (cid:126)S ↔ (cid:126)L + (cid:126)T . (3.18)Third, for constant backgrounds ( ˙ ψ =0) the temporal fluctuation Φ decouples and may be gauged away.Still, the fluctuation operator in (3.17) is easily diagonalized only when the coefficient of one of the firstthree spin-squares vanishes, i.e. for (cid:126)L =0 ( j =0), for the two vacua ψ = ±
1, or for the “meron” ψ =0. Thelatter case has been analyzed by Hosotani [4].Let us decompose the fluctuation problem (3.13)–(3.15) into finite-dimensional blocks according to afixed value of the spin j ∈ N , (cid:126)L Φ pµ | j = − j ( j +1) Φ pµ | j (3.19)and suppress the j subscript. We employ the following coupling scheme, (cid:126)L + (cid:126)T =: (cid:126)U then (cid:126)U + (cid:126)S = ( (cid:126)L + (cid:126)T )+ (cid:126)S =: (cid:126)V . (3.20)Clearly, (cid:126)U and (cid:126)V act on (cid:126) Φ in su (2) representations j ⊗ j ⊗ ⊗
1, respectively. On Φ , we mustput (cid:126)S =0 and have just (cid:126)V = (cid:126)U act in a j ⊗ pµ ) = (Φ p , Φ pa ), we get a 12(2 j +1) × j +1) fluctuation matrix Ω j ) , (cid:2) δ pqµν ∂ τ + (Ω j ) ) pqµν (cid:3) Φ qν = 0 . (3.21)Actually, there is an additional overall (2 j +1)-fold degeneracy present due to the trivial action of the su (2) R generators R a , which plays no role here and will be suppressed. Roughly speaking, the 3(2 j +1) Another (less convenient) scheme couples (cid:126)L + (cid:126)S , then ( (cid:126)L + (cid:126)S )+ (cid:126)T =: (cid:126)V . are related to gauge modes, and we still must impose the gauge condition (3.13), which alsohas 3(2 j +1) components. Therefore, a subspace of dimension 6(2 j +1) inside the space of all fluctuationswill represent the physical gauge-equivalence classes in the end.Our goal is to diagonalize the fluctuation operator (3.21) for a given fixed value of j . It has a blockstructure, Ω j ) = (cid:32) ¯ N − ˙ ψ T (cid:62) − ˙ ψ T N (cid:33) , (3.22)where ¯ N and N are given by the left-hand sides of (3.16) and (3.17), respectively. We introduce a basiswhere (cid:126)U , (cid:126)V and V are diagonal, i.e. (cid:126)U | uvm (cid:105) = − u ( u +1) | uvm (cid:105) and (cid:126)V | uvm (cid:105) = − v ( v +1) | uvm (cid:105) with m = − v, . . . , v , (3.23)and denote the irreducible su (2) v representations with those quantum numbers as (cid:2) vu (cid:3) . On the Φ subspace, u is redundant since u = v as (cid:126)S =0. Working out the tensor products, we encounter the values (cid:2) vu (cid:3) = (cid:2) j − j − (cid:3) ; (cid:2) j − j − (cid:3) , (cid:2) j − j (cid:3) ; (cid:2) jj − (cid:3) , (cid:2) jj (cid:3) , (cid:2) jj +1 (cid:3) ; (cid:2) j +1 j (cid:3) , (cid:2) j +1 j +1 (cid:3) ; (cid:2) j +2 j +1 (cid:3) on (cid:126) Φ , (cid:2) vu (cid:3) = (cid:2) j − j − (cid:3) ; (cid:2) jj (cid:3) ; (cid:2) j +1 j +1 (cid:3) on Φ , (3.24)with some representations obviously missing for j< ψ T term in (3.22) as a perturbation and momentarily put it to zero, so that Ω j ) isblock-diagonal for the time being. Then, it is easy to see from (3.16) and (3.17) that [ (cid:126)V , ¯ N ] = [ (cid:126)U , ¯ N ] = 0and [ (cid:126)V , N ] = 0, even though [ (cid:126)U , N ] (cid:54) = 0 because ( (cid:126)L + (cid:126)S ) is not diagonal in our basis. Therefore, we havea degeneracy in m . Furthermore, both ¯ N and N decompose into at most three respectively five blockswith fixed values of v ranging from j − j +2 and separated by semicolons in (3.24). Moreover, the¯ N blocks are irreducible and trivially also carry a value of u = v . In contrast, N is not simply reducible; its (cid:126)V representations have multiplicity one, two or three. Only the N blocks with extremal v values in (3.24)are irreducible. The other ones are reducible and contain more than one (cid:126)U representation, hence the u -spin distinguishes between their (two or three) irreducible v subblocks. The only non-diagonal termin N is the ( (cid:126)L + (cid:126)S ) contribution, which couples different copies of the same v -spin to each other, butof course not to any u = v block of ¯ N , and does not lift the V = m degeneracy. As a consequence, theunperturbed fluctuation equations for Φ =Φ (¯ v ) and (cid:126) Φ=Φ ( v,α ) take the form (suppressing the m index) (¯ v ) (cid:2) ∂ τ + ¯ ω v ) (cid:3) Φ (¯ v ) = 0 and ( v ) (cid:2) ∂ τ + ω v,α ) (cid:3) Φ ( v,α ) = 0 for ˙ ψ = 0with ¯ v ∈ { j − , j, j +1 } and v ∈ { j − , j − , j, j +1 , j +2 } , (3.25)where ( v ) denotes a unit matrix of size 2 v +1, and α counts the multiplicity of the v -spin representationin N (between one and three). According to (3.16) the unperturbed frequency-squares for ¯ N are theeigenvalues ¯ ω v ) = 2(1 − ψ ) j ( j +1) + 2(1+ ψ ) ¯ v (¯ v +1) − ψ )(1 − ψ ) (3.26)with multiplicity 2¯ v +1, hence we get¯ ω j − = 2 ψ − j ψ + 2(2 j − , ¯ ω j ) = 2 ψ + 2(2 j +2 j − ω j +1) = 2 ψ + 4( j +1) ψ + 2(2 j +4 j +1) . (3.27)Considering N in (3.17), we can read off the eigenvalues at v = j ± (cid:126)L + (cid:126)S ) = (cid:126)U is already diagonal in the (cid:8) | uvm (cid:105) (cid:9) basis. For the other v -values we must diagonalize a Strictly, they are gauge modes only when ˙ ψ =0. Otherwise, the gauge modes are mixtures with the Φ a modes. × × ω j − = root of Q j − ( λ ) = − j − ψ + 2(2 j − j +1) ,ω j − ,α ) = two roots of Q j − ( λ ) ,ω j,α ) = three roots of Q j ( λ ) ,ω j +1 ,α ) = two roots of Q j +1 ( λ ) ,ω j +2) = root of Q j +2 ( λ ) = 2(2 j +3) ψ + 2(2 j +6 j +5) , (3.28)each with multiplicity 2 v +1, where Q v denotes a linear, quadratic or cubic polynomial. Let us now turn on the perturbation ˙ ψ T , which couples N with ¯ N , and consider the characteristicpolynomial P j ( λ ) of our fluctuation problem, P j ( λ ) := det (cid:16) ¯ N − λ − ˙ ψT (cid:62) − ˙ ψT N − λ (cid:17) = det( N − λ ) · det (cid:2) ( ¯ N − λ ) − ˙ ψ T (cid:62) ( N − λ ) − T (cid:3) = (cid:2)(cid:81) v det( N ( v ) − λ ) (cid:3) · det (cid:2) ( ¯ N − λ ) − ˙ ψ T (cid:62) { (cid:76) v ( N ( v ) − λ ) − } T (cid:3) , (3.29)where we made use of (cid:104) u v m | N | u (cid:48) v (cid:48) m (cid:48) > = (cid:0) N ( v ) (cid:1) uu (cid:48) δ vv (cid:48) δ mm (cid:48) . (3.30)Since T furnishes an su (2) representation (and not an intertwiner) it must be represented by squarematrices and thus cannot connect different v representations. Hence the perturbation does not coupledifferent v sectors but only links N and ¯ N in a common ¯ v = v sector. Therefore, it does not affect theextremal sectors v = j ±
2. Moreover, switching to a diagonal basis {| αvm (cid:105)} for N we can simplify to T (cid:62) { (cid:76) v ( N ( v ) − λ ) − } T (cid:3) = (cid:76) ¯ v { T (cid:62) ( N − λ ) − T } (¯ v ) = (cid:76) ¯ v (cid:110)(cid:80) α ( ω v,α ) − λ ) − (cid:0) T (cid:62) | α (cid:105)(cid:104) α | T (cid:1) (¯ v ) (cid:111) . (3.31)Observing that (cid:0) T (cid:62) | α (cid:105)(cid:104) α | T (cid:1) (¯ v ) = − t ¯ v,α (cid:0) (cid:126)T (cid:1) (¯ v ) = 8 t ¯ v,α (¯ v ) with some coefficient functions t ¯ v,α ( ψ ), with (cid:80) α t ¯ v,α = 1, we learn that the V degeneracy remains intact and arrive at (¯ v ∈ { j − , j, j +1 } ) P j ( λ ) = (cid:2)(cid:81) v Q v ( λ ) v +1 (cid:3) · (cid:81) ¯ v (cid:8) (¯ ω v ) − λ ) − ψ (cid:80) α t ¯ v,α ( ω v,α ) − λ ) − (cid:9) v +1 = ( ω j − − λ ) j − · ( ω j +2) − λ ) j +5 · (cid:81) ¯ v (cid:8) (¯ ω v ) − λ ) Q ¯ v ( λ ) − ψ P ¯ v ( λ ) (cid:9) v +1 = ( ω j − − λ ) j − · ( ω j +2) − λ ) j +5 · (cid:81) ¯ v R ¯ v ( λ ) v +1 , (3.32)where P ¯ v = Q ¯ v (cid:80) α t ¯ v,α ( ω v,α ) − λ ) − is a polynomial of degree one less than Q ¯ v since all poles cancel, and R ¯ v is a polynomial of one degree more. We list the polynomials Q v , P ¯ v and R ¯ v for j ≤ m (cid:48) = − j, . . . , j and m = − v, . . . , v ) (cid:8) | µ p m (cid:48) (cid:105) (cid:9) ⇒ (cid:8) | ¯ vm (cid:105) , | uvm (cid:105) (cid:9) ⇒ (cid:8) | ¯ vm (cid:105) , | αvm (cid:105) (cid:9) ⇒ (cid:8) | βvm (cid:105) (cid:9) (3.33)we have diagonalized (3.21) to (cid:2) ∂ τ − Ω j,v,β ) (cid:3) Φ ( v,β ) = 0 with v ∈ { j − , j − , j, j +1 , j +2 } , (3.34)where Ω j,v,β ) are the distinct roots of the characteristic polynomial P j in (3.32), and (for j ≥
2) themultiplicity label β takes 1 , , , , j,j ± = ω j ± , Ω j,j ± ,β ) = three roots of R j ± ( λ ) , Ω j,j,β ) = four roots of R j ( λ ) . (3.35)The reflection symmetry (3.18) implies that Ω v,j, · ) ( ψ ) = Ω j,v, · ) ( − ψ ). For j<
2, obvious modificationsoccur due to the absence of some v representations. For j< v<
10e still have to discuss the gauge condition (3.13), which can be cast into the form0 = ∂ τ Φ − (cid:2) (1 − ψ ) (cid:126)L + (1+ ψ ) (cid:126)U (cid:3) · (cid:126) Φ = ∂ τ Φ (¯ v, ¯ m ) − K v,m,α ¯ v, ¯ m ( ψ ) Φ ( v,m,α ) (3.36)with a 3(2 j +1) × j +1) linear (in ψ ) matrix function K . Here the v sum runs over ( j − , j, j +1) only,since the gauge condition (3.13) has components only in the middle three v sectors, like the gauge-modeequation (3.14). It does not restrict the extremal v sectors v = j ±
2, since these fluctuations do notcouple to the gauge sector Φ and are entirely physical. For the middle three v sectors (labelled by ¯ v ),the ˙ ψ T perturbation leads to a mixing of the N modes with the ¯ N gauge modes, so their levels will avoidcrossing. Performing the corresponding final basis change, the gauge condition takes the form (cid:2) L ¯ v (cid:48) , ¯ m (cid:48) ,β ¯ v, ¯ m ( ψ ) ∂ τ − M ¯ v (cid:48) , ¯ m (cid:48) ,β ¯ v, ¯ m ( ψ ) (cid:3) Φ (¯ v (cid:48) , ¯ m (cid:48) ,β ) = 0 (3.37)with certain 3(2 j +1) × j +1) matrix functions L and M . This linear equation represents conditions onthe normal mode functions Φ (¯ v, ¯ m,β ) and defines a 7(2 j +1)-dimensional subspace of physical fluctuations,which of course still contains a 3(2 j +1)-dimensional subspace of gauge modes. For j<
1, these numbersare systematically smaller. Together with the two extremal v sectors, we end up with (7 − j +1) =6(2 j +1) physical degrees of freedom for any given value of j ( ≥ j =0 backgrounds. For the vacuum background, say ψ = −
1, which is isospin degenerate, one gets (cid:0) ∂ τ − (cid:126)L − ( (cid:126)L + (cid:126)S ) (cid:1) (cid:126) Φ = 0 , (cid:126)L · (cid:126) Φ = 0 , Φ = 0 . (3.38)It yields the positive eigenfrequency-squares ω j,u (cid:48) ) = 2 j ( j +1) + 2 u (cid:48) ( u (cid:48) +1) = j at j ≥ u (cid:48) = j − j ( j +1) for u (cid:48) = j j +1) for u (cid:48) = j +1 (3.39)for j = 0 , , , . . . , but the (cid:126)L · (cid:126) Φ = 0 constraint removes the u (cid:48) = j modes. Clearly, all (constant)eigenfrequency-squares are positive, hence the vacuum is stable.For the “meron” background, ψ ≡
0, one has (cid:0) ∂ τ − (cid:126)L − ( (cid:126)L + (cid:126)T + (cid:126)S ) − (cid:1) (cid:126) Φ = 0 , (cid:0) (cid:126)L + (cid:126)T (cid:1) · (cid:126) Φ = 0 , Φ = 0 . (3.40)In this case, we read off ω j,v ) + 2 = 2 j ( j +1) + 2 v ( v +1) = j − j +1) for v = j − j for v = j − j ( j +1) for v = j (1 to 3 times)4( j +1) for v = j +1 (1 to 2 times)4( j +3 j +3) for v = j +2 (1 times) , (3.41)but the constraint removes one copy from each of the three middle cases (and less when j< { ω } = {− , , , , , . . . } with certain degeneracies [4]. The single non-degeneratenegative mode ω , = − pa = δ pa φ ( τ ), and it corresponds to rolling down the local maximumof the double-well potential. The meron is stable against all other perturbations.For a time-varying background, the natural frequencies Ω ( j,v,β ) inherit a τ dependence from thebackground ψ ( τ ). Direct diagonalization is still possible for j =0, where we should solve ∂ τ Φ − (1+ ψ ) (cid:126)T · (cid:126) Φ = 0 , (cid:2) ∂ τ + 2(1+ ψ ) (cid:3) Φ − ˙ ψ (cid:126)T · (cid:126) Φ = 0 , (cid:2) ∂ τ + 2(3 ψ − − (1+ ψ )(2 − ψ )( (cid:126)S + (cid:126)T ) (cid:3) (cid:126) Φ − ˙ ψ (cid:126)T Φ = 0 , (3.42) We have to bring back the m indices because the gauge condition is not diagonal in them. (cid:126)S + (cid:126)T ) = (cid:126)V = − v ( v +1) = 0 , − , −
24 for v = 0 , , . (3.43)It implies the unperturbed frequencies (suppressing the j index)¯ ω = 2( ψ +1) (3 × ) , ω = 2(3 ψ −
1) (1 × ) , ω = 2(2 ψ + ψ +1) (3 × ) , ω = 2(3 ψ +5) (5 × )(3.44)for (Φ ) p ≡ (cid:0) Φ (¯ v =1) (cid:1) p =: δ pb ¯ φ b , ( (cid:126) Φ) pa ≡ (cid:0) Φ (0) + Φ (1) + Φ (2) (cid:1) pa =: φ δ pa + (cid:15) pab φ b + ( φ ( ab ) − δ ab φ ) δ bp , (3.45)as long as ˙ ψ is ignored. There are no v -spin multiplicities (larger than one) here. Turning on ˙ ψ andobserving that ( (cid:126)T · (cid:126) Φ) p ∼ δ pb φ b , the characteristic polynomial of the coupled 12 ×
12 system in the | uvm (cid:105) basis reads P ( λ ) = det (¯ ω − λ ) − ˙ ψ T (cid:62) (1)
00 ( ω − λ ) − ˙ ψ T (1) ω − λ )
00 0 0 ( ω − λ ) . (3.46)Specializing the general discussion above to j =0, we find just t =1 so that P =1 and arrive at P ( λ ) = ( ω − λ ) ( ω − λ ) ( ω − λ ) (cid:2) (¯ ω − λ ) − ψ ( ω − λ ) − (cid:3) = ( ω − λ )( ω − λ ) (cid:8) (¯ ω − λ )( ω − λ ) − ψ (cid:9) . (3.47)We see that the frequencies Ω = ω and Ω = ω are unchanged and given by (3.44), while the gaugemode ¯ ω gets entangled with the (unphysical) v =1 mode to produce the pairΩ , ± ) = (¯ ω + ω ) ± (cid:113) (¯ ω + ω ) − ¯ ω ω + 8 ˙ ψ = 3 ψ +3 ψ +2 ± (cid:113) ψ ( ψ − + 8 ˙ ψ (3.48)with a triple degeneracy. There are avoided crossings at ψ =0 and ψ =1. Removing the unphysical andgauge modes in pairs, we remain with the singlet mode Ω , and the fivefold-degenerate Ω , . Forall higher spins j>
0, analytic expressions for the natural frequencies Ω ( j,v,β ) now require merely solvinga few polynomial equations of order four at worst. We have done so up to j =2 and list them in theAppendix but refrain from giving further explicit examples here. Below we display the cases of j =0 and j =2, with similar coloring for like v values, whose curves avoid crossing each other. One can see thatsome of the normal modes dip into the negative regime, i.e. their frequency-squares become negative,for a certain fraction of the time τ . Because of this and, quite generally, due to the τ variability of thenatural frequencies, it is not easy to predict the long-term evolution of the fluctuation modes. Clearly,the stability of the zero solution Φ ≡
0, equivalent to the linear stability of the background Yang–Millsconfiguration, is not simply decided by the sign of the τ -average of the corresponding frequency-square. The diagonalized linear fluctuation equation (3.34) represents a bunch of Hill’s equations, where thefrequency-squared is a root of a polynomial of order up to four with coefficients given by a polynomialof twice that order in Jacobi elliptic functions. A unique solution requires fixing two initial conditions,and so for each fluctuation Φ ( j,v,β ) there is a two-dimensional solution space. It is well known that Hill’sequation, e.g. in the limit of Mathieu’s equation, displays parametric resonance phenomena, which canstabilize otherwise unstable systems or destabilize otherwise stable ones.For oscillating dynamical systems with periodically varying frequency, there exist some general toolsto analyze linear stability. Switching to a Hamiltonian picture and to phase space, it is convenientto transform the second-order differential equation into a system of two coupled first-order equations(suppressing all quantum numbers), (cid:2) ∂ τ − Ω ( τ ) (cid:3) Φ( τ ) = 0 ⇔ ∂ τ (cid:32) Φ˙Φ (cid:33) = (cid:32) − Ω (cid:33) (cid:32) Φ˙Φ (cid:33) =: i (cid:98) Ω( τ ) (cid:32) Φ˙Φ (cid:33) , (4.1)12 .1 0.2 0.3 0.4 0.5 0.6 0.7 τ - v = = = = τ v = = = = τ v = = = = τ v = = = = Figure 4: Plots of Ω ,v,β ) ( τ ) over one period, for different values of k :0 .
51 (top left), 0 .
99 (top right), 1 .
01 (bottom left) and 5 (bottom right). τ - v = = = = = = = = = = = = τ - v = = = = = = = = = = = = τ v = = = = = = = = = = = = τ v = = = = = = = = = = = = Figure 5: Plots of Ω ,v,β ) ( τ ) over one period, for different values of k :0 .
505 (top left), 0 .
550 (top right), 0 .
999 (bottom left) and 1 .
001 (bottom right).where the frequency Ω( τ ) is T -periodic (sometimes T -periodic) in τ . The solution to this first-ordersystem is formally given by (cid:32) Φ˙Φ (cid:33) ( τ ) = T exp (cid:110)(cid:90) τ d τ (cid:48) i (cid:98) Ω( τ (cid:48) ) (cid:111) (cid:32) Φ˙Φ (cid:33) (0) , (4.2)where T denotes time ordering. Because of the time dependence of Ω, the time evolution operator aboveis not homogeneous thus does not constitute a one-parameter group, except when the propagation interval13s an integer multiple of the period T . For τ = T , one speaks of the stroboscopic map [9] M := T exp (cid:110)(cid:90) T d τ i (cid:98) Ω( τ ) (cid:111) ⇒ (cid:32) Φ˙Φ (cid:33) ( nT ) = M n (cid:32) Φ˙Φ (cid:33) (0) . (4.3)The linear map M is a functional of the chosen background solution ψ and hence depends on its param-eter E or k . This background is Lyapunov stable if the trivial solution Φ ≡ µ and µ of M . Since the system is Hamiltonian, det M =1, and we have three cases: | tr M | > ⇔ µ i ∈ R ⇔ hyperbolic/boost ⇔ strongly unstable , | tr M | = 2 ⇔ µ i = ± ⇔ parabolic/translation ⇔ marginally stable , | tr M | < ⇔ µ i ∈ U(1) ⇔ elliptic/rotation ⇔ strongly stable . (4.4)Clearly, | tr M | determines the linear stability of our classical solution.Let us thus try to evaluate the trace of the stroboscopic map M , making use of the special form ofthe matrix (cid:98) Ω,tr M = ∞ (cid:88) n =0 i n (cid:90) T d τ (cid:90) τ d τ . . . (cid:90) τ n − d τ n tr (cid:2)(cid:98) Ω( τ ) (cid:98) Ω( τ ) · · · (cid:98) Ω( τ n ) (cid:3) = 2 + ∞ (cid:88) n =1 ( − n (cid:90) T d τ (cid:90) τ d τ . . . (cid:90) τ n − d τ n H n ( τ , τ , . . . , τ n ) Ω ( τ ) Ω ( τ ) · · · Ω ( τ n )with H n ( τ , τ , . . . , τ n ) = ( τ − τ )( τ − τ ) · · · ( τ n − − τ n )( τ n − τ +1) and H ( τ ) = 1 . (4.5)It is convenient to scale the time variable such as to normalize the period to unity, τ = T x and Ω ( T x ) =: ω ( x ) , H ( { T x } ) =: h ( { x } ) , (4.6)hencetr M = 2 + ∞ (cid:88) n =1 (cid:0) − T (cid:1) n (cid:90) d x (cid:90) x d x . . . (cid:90) x n − d x n h n ( x , x , . . . , x n ) ω ( x ) ω ( x ) · · · ω ( x n )= ∞ (cid:88) n =0 n )! M n (cid:0) − T (cid:1) n =: 2 − M T + M T − M T + M T − . . . . (4.7)It is impossible to evaluate the integrals M n without explicit knowledge of ω ( x ). As a crude guess,we replace the weight function by its (constant) average value (cid:104) h n (cid:105) := 1 n ! (cid:90) d x (cid:90) x d x . . . (cid:90) x n − d x n h n ( x , x , . . . , x n ) = 2 n !(2 n )! (4.8)and obtain M n = (2 n )!2 (cid:104) h n (cid:105) (cid:90) d x (cid:90) x d x . . . (cid:90) x n − d x n n (cid:89) i =1 ω ( x i ) = (cid:16)(cid:90) d x ω ( x ) (cid:17) n =: (cid:104) ω (cid:105) n , (4.9)which yields tr M = 2 ∞ (cid:88) n =0 ( − n (2 n )! (cid:104) ω (cid:105) n T n = 2 cos (cid:0)(cid:112) (cid:104) ω (cid:105) T (cid:1) . (4.10)This expression indicates stability as long as (cid:104) ω (cid:105) >
0. However, the result for the j =0 singlet mode ω = Ω , in (4.11) already showed that the averaged frequency-squared may turn negative in certaindomains thus changing the cos into a cosh there. 14o do better, let us look at the individual terms M n in (4.7) for the simplest case of the SO(4) singletfluctuation, i.e. Ω , = 6 ψ − (cid:104) Ω , (cid:105) = 1 (cid:15) (cid:16) E ( k ) K ( k ) + 4 k − (cid:17) , (4.11)where E ( k ) and K ( k ) denote the second and first complete elliptic integrals, respectively. Plotting thisexpression as a function of the modulus k , we see that it becomes negative only in a very narrow rangearound k =1, namely for | k − | (cid:46) . k< k - < Ω ( ) > k - - - - < Ω ( ) > Figure 6: Plot of (cid:104) Ω , (cid:105) as a function of k , with detail on the right.simplicity) M = (cid:104) Ω , (cid:105) and M = (cid:104) Ω , (cid:105) − (cid:15) (cid:16) − k K ( k ) − E ( k ) K ( k ) + 9 π K ( k ) (cid:17) , (4.12)which does not suffice to rule out instability. Indeed, numerical studies show that M n as a function of k looses its positivity in a range around k =1 which increases with n , where the series (4.7) ceases to bealternating. Moreover, even in the limit of a very large background amplitude, k → , we find that (cid:104) Ω , (cid:105) → π (cid:15) Γ( ) ≈ . (cid:15) ⇒ (cid:112) (cid:104) Ω (cid:105) T → √ π ≈ . , (4.13)implying that we must push the series in (4.7) at least to O ( M T ), even though it turns out that M n < (cid:104) Ω , (cid:105) n at k = for n > (cid:98) Φ( τ ) = (cid:32) Φ Φ ˙Φ ˙Φ (cid:33) ( τ ) ⇒ ∂ τ (cid:98) Φ( τ ) = i (cid:98) Ω( τ ) (cid:98) Φ( τ ) (4.14)of our system (4.1) with some initial condition (cid:98) Φ(0) = (cid:98) Φ can be expressed in so-called Floquet normalform as (cid:98) Φ( τ ) = Q ( τ ) e τR with Q ( τ +2 T ) = Q ( τ ) , (4.15)where Q ( τ ) and R are real 2 × τ ) := Q ( τ ) − (cid:98) Φ( τ ) ⇒ ∂ τ Ψ( τ ) = R Ψ( τ ) . (4.16)Due to the identity (cid:98) Φ( τ + T ) = (cid:98) Φ( τ ) (cid:98) Φ(0) − (cid:98) Φ( T ) = (cid:98) Φ( T ) (cid:98) Φ(0) − (cid:98) Φ( τ ) = M (cid:98) Φ( τ ) (4.17)15e see that our stroboscopic map M is nothing but the monodromy, and M = (cid:98) Φ(2 T ) (cid:98) Φ(0) − = Q (0) (cid:98) Φ(0) − (cid:98) Φ(2 T ) Q (0) − = Q (0) e RT Q (0) − , (4.18)so that its eigenvalues (or characteristic multipliers) µ i = e ρ i T for i = 1 , ρ i whose real parts are the Lyapunov exponents. Since µ µ = 1 implies that ρ + ρ = 0, our system is linearly stable if and only if both eigenvalues ρ i of R arepurely imaginary (or zero).Generally it is impossible to find analytically the monodromy pertaining to a normal mode Φ ( j,v,β ) . However, we can evaluate it numerically for a number of examples. Before doing so, let us estimateat which energies E or, rather, moduli k , possible resonance frequencies might occur. To this end,we determine the period-average of the natural frequency Ω ( j,v,β ) and compare it to its modulationfrequency πT . If we modelΩ ( τ ) ≈ (cid:104) Ω (cid:105) (cid:0) h ( τ ) (cid:1) with (cid:104) Ω (cid:105) = T T ∫ d τ Ω ( τ ) and h ( τ ) ∝ cos(2 πτ /T ) , (4.20)where T = 4 (cid:15) K ( k ), then the resonance condition is met for (cid:112) (cid:104) Ω (cid:105) = (cid:96) πT ⇒ k = k (cid:96) ( j, v, β ) for (cid:96) = 1 , , , . . . . (4.21)Since this model reproduces only the rough features of Ω ( τ ), we expect potential instability due toparametric resonance effects in a band around or near the values k (cid:96) .Our expectation is confirmed by precise numerical evaluation of various monodromies as a functionof k . Below we display, together with the would-be resonant values k (cid:96) , the function tr M ( k ) for the samplecases of ( j, v ) = (2 ,
0) and (2 , E = (or k =1), k - - - - tr ( M ( ) ) k - - tr ( M ( ) ) Figure 7: Plot of tr M ( k ) for ( j, v ) = (2 , k (cid:96) values accumulate at the critical point. Butwhile for k> M ( k ) oscillates between values close to 2 in magnitudeand thus exponential growth is rare and mild, for k< M ( k ) comes with an amplitude exceeding 2 and growing with energy. Hence, in this latterregime stable and unstable bands alternate. This is supported by long-term numerical integration, as wedemonstrate in Figure 9 by plotting Φ( τ ) for ( j, v, β ) = (2 , ,
1) with initial values Φ(0)=1 and ˙Φ(0)=0on both sides of the first transition from instability to stability for tr M (2 , , shown in Figure 8 (at thehighest value of E or the lowest value of k ). An exception is the SO(4) singlet perturbation Φ (0 , , to be treated in the following section. .8 0.9 1.0 1.1 1.2 k - tr ( M ( ) ) k - - tr ( M ( ) ) k - - tr ( M ( ) ) k - - tr ( M ( ) ) Figure 8: Plots of tr M ( k ) for ( j, v ) = (2 ,
2) and β = 1 , , ,
4. Would-be resonances marked in red. τ Φ ( τ ) τ Φ ( τ ) Figure 9: Plot of Φ( τ ) for ( j, v, β ) = (2 , ,
1) and k =0 . k =0 . E → ∞ (or k → / √ T collapseswith (cid:15) = (cid:112) k − /
2, we rescale τ(cid:15) = z ∈ [0 , K ( )] , (cid:15) ψ = ˜ ψ , (cid:15) ˙ ψ = ∂ z ˜ ψ , (cid:15) Ω = ˜Ω , (cid:15) λ = ˜ λ (4.22)so that the tilded quantities remain finite in the limit, and find, with ¯ ω v ) → ψ , Q ( v ± ∼ ˜ λ ,Q ( v ± ∼ ˜ λ (˜ λ − ψ ) , R ( v ± ∼ ˜ λ (cid:2) (˜ λ − ψ )(˜ λ − ψ ) − ∂ z ˜ ψ ) (cid:3) ,Q ( v ) ∼ ˜ λ (˜ λ − ψ )(˜ λ − ψ ) , R ( v ) ∼ ˜ λ (cid:2) (˜ λ − ψ )(˜ λ − ψ ) − ∂ z ˜ ψ ) (cid:3) (˜ λ − ψ ) , (4.23)because all j -dependent terms in the polynomials are subleading and drop out in the limit. Factorizing For the cases ( j, v ) = (0 ,
1) and (1 , λ is missing; for ( j, v ) = (0 , R = Q ∼ (˜ λ − ψ ). R polynomials, we find the four universal natural frequency-squares˜Ω = 0 , ˜Ω = 3 ˜ ψ − (cid:113) ˜ ψ + 8( ∂ z ˜ ψ ) , ˜Ω = 3 ˜ ψ + (cid:113) ˜ ψ + 8( ∂ z ˜ ψ ) , ˜Ω = 6 ˜ ψ . (4.24)One must pay attention, however, to the fact that the avoided crossings disappear in the (cid:15) → (cid:2) ∂ z − ˜Ω j,v,β ) (cid:3) ˜Φ ( j,v,β ) = 0 (4.25)are ˜Ω j,j ± = 0 , ˜Ω j,j ± ,β ) ∈ (cid:8) min( ˜Ω , ˜Ω ) , max( ˜Ω , ˜Ω ) , ˜Ω (cid:9) , ˜Ω j,j,β ) ∈ (cid:8) min( ˜Ω , ˜Ω ) , max( ˜Ω , ˜Ω ) , min( ˜Ω , ˜Ω ) , max( ˜Ω , ˜Ω ) (cid:9) , (4.26)of which we show below the last list as a function of z . The monodromies are easily computed numerically, z - Ω ( j , j ) Figure 10: Plot of the universal limiting natural frequency-squares ˜Ω j,v,β ) for v = j and β = 1 , , , M ( j,j ± ( E →∞ ) = 2 , tr M ( j,j ± ,β ) ( E →∞ ) ∈ (cid:8) . , − . , − . (cid:9) , tr M ( j,j,β ) ( E →∞ ) ∈ (cid:8) . , − . , . , − . (cid:9) , (4.27)in agreement with the figures above. In particular, the extremal v -values become marginally stable, whilepart of the non-extremal cases are unstable for high energies.Of course, for each non-extremal value of v we still have to project out unphysical modes by imposingthe gauge condition (3.37). However, in the 12(2 j +1)-dimensional fluctuation space the gauge conditionhas rank 3(2 j +1) while we see that (for j ≥
2) in total 4(2 j +1) normal modes are unstable at high energy.Therefore, the projection to physical modes cannot remove all instabilities. We must conclude that,for sufficiently high energy E , some fluctuations grow exponentially, implying that the solution Φ ≡ Even though the Floquet representation helped to reduce the long-time behavior of the perturbationsto the analysis of a single period T , it normally does not give us an exact solution to Hill’s equation. For the cases ( j, v ) = (0 ,
1) and (1 ,
0) one gets { . , − . } ; for ( j, v ) = (0 ,
0) we have tr M = 2. ψ ( τ ), we can employ the fact that ˙ ψ trivially solvesthe fluctuation equation,( ˙ ψ ) ·· = ( ¨ ψ ) · = − (cid:0) V (cid:48) ( ψ ) (cid:1) · = − V (cid:48)(cid:48) ( ψ ) ˙ ψ = − (6 ψ −
2) ˙ ψ = − Ω , ( τ ) ˙ ψ , (5.1)with a frequency function which is T -periodic. This implies that all fluctuation modes are T -periodic.With the knowledge of an explicit solution to the fluctuation equation we can reduce the latter to afirst-order equation and solve that one to find a second solution. The normalizations are arbitrary, so wechooseΦ ( τ ) = − (cid:15) k ˙ ψ ( τ ) and Φ ( τ ) = Φ ( τ ) (cid:90) τ d σ Φ ( σ ) = − k(cid:15) ˙ ψ ( τ ) (cid:90) τ d σ ˙ ψ ( σ ) , (5.2)which are linearly independent since W (Φ , Φ ) ≡ Φ ˙Φ − Φ ˙Φ = 1 . (5.3)For simplicity, we restrict ourselves to the energy range
1) dn (cid:0) τ(cid:15) , k (cid:1) − k (cid:3) + sn (cid:0) τ(cid:15) , k (cid:1) dn (cid:0) τ(cid:15) , k (cid:1)(cid:2) τ(cid:15) + k − − k E (cid:0) am( τ(cid:15) , k ) , k (cid:1)(cid:3) , (5.4)where am( z, k ) denotes the Jacobi amplitude and E ( z, k ) is the elliptic integral of the second kind. As
10 20 30 40 τ - - - Φ ( τ )
10 20 30 40 τ - Φ ( τ ) Figure 11: Plot of the SO(4) singlet fluctuation modes Φ and Φ over eight periods for k =0 . (0) = 0 , ˙Φ (0) = 1 and Φ (0) = − , ˙Φ (0) = 0 , (5.5)which fixes the ambiguity of adding to Φ a piece proportional to Φ . Hence, (cid:98) Φ(0) = (cid:16) −
11 0 (cid:17) ⇒ M = (cid:98) Φ( T ) (cid:16) − (cid:17) . (5.6)We know that Φ ∼ ˙ ψ is T -periodic, and so is ˙Φ , but not the second solution,Φ ( τ + T ) = Φ ( τ ) + γ T Φ ( τ ) with γ = 1 T (cid:90) T d σ Φ ( σ ) (cid:12)(cid:12)(cid:12)(cid:12) reg =: k (cid:15) (cid:10) ˙ ψ − (cid:11) reg , (5.7)where the integral diverges at the turning points and must be regularized by subtracting the Weierstraß ℘ function with the appropriate half-periods. Since Φ has periodic zeros, Φ does return to − T . It follows that the Φ oscillation linearly grows in amplitude with a rate (per period) of γ = 1 (cid:15) (cid:104) k − − k E ( k ) K ( k ) (cid:105) , (5.8)19 .75 0.80 0.85 0.90 0.95 1.00 k γ ( k ) Figure 12: Plot of the linear growth rate γ as a function of k .which is always larger than 7.629, attained at k ≈ . M = (cid:32) − Φ ( T ) Φ ( T ) − ˙Φ ( T ) ˙Φ ( T ) (cid:33) = (cid:32) − γ T (cid:33) = exp (cid:110) − γ T (cid:0) (cid:1)(cid:111) (5.9)and thus easily obtain the Floquet representation, R = (cid:32) γ (cid:33) ⇒ e τR = (cid:32) γ τ (cid:33) and Q ( τ ) = (cid:32) Φ Φ − Φ γ τ ˙Φ ˙Φ − ˙Φ γ τ (cid:33) . (5.10)Obviously, we have encountered a marginally stable situation, since M is of parabolic type. There is noexponential growth, and Φ is periodic thus bounded, but Φ grows without bound as long as one staysin the linear regime. Note that we never made use of the form of our Newtonian potential. In fact, thisbehavior is typical for a conservative mechanical system with oscillatory motion.What to make of this linear growth? It can be (and actually is) easily overturned by nonlinear effects.Going beyond the linear regime, though, requires expanding the Yang–Mills equation to higher ordersabout our classical Yang–Mills solution (2.14). While this is a formidable task in general, it can actuallybe done to all orders for the singlet perturbation! The reason is that a singlet perturbation leaves usin the SO(4)-symmetric subsector, thus connecting only to a neighboring “cosmic background”, ψ → ˜ ψ .Since (2.20) gives us analytic control over all solutions ψ ( τ ), the full effect of such a shift can be computedexactly. Splitting an exact solution ˜ ψ into a background part and its (full) deviation,˜ ψ ( τ ) = ψ ( τ ) + η ( τ ) , (5.11)inserting ˜ ψ into the equation of motion (2.18) and remembering that V is of fourth order, we obtain0 = ¨ η + V (cid:48)(cid:48) ( ψ ) η + V (cid:48)(cid:48)(cid:48) ( ψ ) η + V (cid:48)(cid:48)(cid:48)(cid:48) ( ψ ) η = ¨ η + (6 ψ − η + 6 ψ η + 2 η , (5.12)extending the linear equation (5.1) by two nonlinear contributions. Perturbation theory introduces asmall parameter (cid:15) and formally expands η = (cid:15)η (1) + (cid:15) η (2) + (cid:15) η (3) + . . . , (5.13)20hich yields the infinite coupled system (cid:2) ∂ τ + (6 ψ − (cid:3) η (1) = 0 , (cid:2) ∂ τ + (6 ψ − (cid:3) η (2) = − ψ η , (cid:2) ∂ τ + (6 ψ − (cid:3) η (3) = − ψ η (1) η (2) − η ,. . . , (5.14)which could be iterated with a seed solution η (1) of the linear system.However, we know that the exact solutions to the full nonlinear equation (5.12) is simply given bythe difference η ( τ ) = ˜ ψ ( τ ) − ψ ( τ ) (5.15)of two analytically known backgrounds. The SO(4)-singlet background moduli space is parametrized bytwo coordinates, e.g. the energy E (or elliptic modulus k ) and the choice of an initial condition which fixesthe origin τ =0 of the time variable. In (2.20), we selected ˙ ψ (0) = 0, but relaxing this we can reintroducethis collective coordinate by allowing shifts in τ . We may then parametrize the SO(4)-invariant Yang–Mills solutions as ψ k,(cid:96) ( τ ) = ψ ( τ − (cid:96) ) with 2 E = 1 / (2 k − and (cid:96) ∈ R (5.16)where ψ is taken from (2.20). Note that ˙ ψ k,(cid:96) solves the background equation (5.1) with a frequency-squared ω k,(cid:96) = 6 ψ k,(cid:96) −
2. Without loss of generality we assign ψ = ψ k, and ˜ ψ = ψ k + δk,δ(cid:96) , hence η ( τ ) = δk ∂ k ψ ( τ ) − δ(cid:96) ˙ ψ ( τ ) + ( δk ) ∂ k ψ ( τ ) − δkδ(cid:96) ∂ k ˙ ψ ( τ ) + ( δ(cid:96) ) ¨ ψ ( τ ) + . . . = δk ∂ k ψ ( τ − δ(cid:96) ) + ( δk ) ∂ k ψ ( τ − δ(cid:96) ) + ( δk ) ∂ k ψ ( τ − δ(cid:96) ) + . . . , (5.17)because ∂ (cid:96) ψ = − ˙ ψ . Clearly, a shift in (cid:96) only shifts the time dependence of the frequency and does notalter the energy E , which is not very interesting. Its linear part corresponds to the mode Φ ∼ ˙ ψ of theprevious section. A change in k , in contract, will lead to a solution with an altered frequency and energy.Its linear part is given by Φ , which grows linearly in time. However, due to the boundedness of the fullmotion, the nonlinear corrections have to limit this growth and ultimately must bring the fluctuationback close to zero. This is the familiar wave beat phenomenon: the difference of two oscillating functions,˜ ψ and ψ , with slightly different frequencies, will display an amplitude oscillation with a beat frequencygiven by the difference. This is borne out in the following plots. As a result, we can assert a long-term τ - - - η ( τ )
50 100 150 200 τ - - - η ( τ ) Figure 13: Plots of the full perturbation η at k =0 .
95 for η (0)=0 .
02, ˙ η (0)=0, giving a beat ratio of ∼ Acknowledgments
K.K. is grateful to Deutscher Akademischer Austauschdienst (DAAD) for the doctoral research grant 57381412.21 ppendix j , v P( λ ) Q( λ ) R( λ )0 , − ψ ) − λ − (2 − ψ ) + λ , ψ + 4 ψ ) − λ (4 + 12 ψ + 20 ψ + 20 ψ + 8 ψ − ψ ) − (4 + 6 ψ +6 ψ ) λ + λ , ψ ) − λ N/A , − (1 + 6 ψ ) + λ − (1 + 2 ψ + 24 ψ ) + (2 +10 ψ ) λ − λ − (1 + 4 ψ + 28 ψ + 48 ψ − ψ − ψ ˙ ψ ) + (3 +16 ψ + 44 ψ − ψ ) λ − (3 + 12 ψ ) λ + λ , − (7 + 3 ψ ) + λ − (49 + 42 ψ + 32 ψ + 12 ψ ) +(14 + 6 ψ + 4 ψ ) λ − λ − (343 + 588 ψ + 574 ψ + 360 ψ + 136 ψ + 24 ψ −
56 ˙ ψ − ψ ˙ ψ ) + (147 + 168 ψ + 124 ψ + 48 ψ + 8 ψ − ψ ) λ − (21 + 12 ψ + 6 ψ ) λ + λ , N/A (17 + 8 ψ ) − λ N/A1 , − ψ + 4 ψ ) − λ (4 − ψ + 20 ψ − ψ + 8 ψ − ψ ) − (4 − ψ +6 ψ ) λ + λ , ψ ) − (12 + 6 ψ ) λ + λ (216 + 192 ψ + 104 ψ ) − (108 + 92 ψ + 24 ψ ) λ + (18 +10 ψ ) λ − λ (1296+1584 ψ +1008 ψ +208 ψ −
288 ˙ ψ − ψ ˙ ψ ) − (864 + 960 ψ + 432 ψ + 48 ψ −
96 ˙ ψ − ψ ˙ ψ ) λ +(216 + 188 ψ + 44 ψ − ψ ) λ − (24 + 12 ψ ) λ + λ , − (14 + 4 ψ ) + λ − (196+112 ψ +60 ψ +16 ψ )+(28 + 8 ψ + 4 ψ ) λ − λ − (2744 + 3136 ψ + 2128 ψ + 928 ψ + 248 ψ + 32 ψ −
112 ˙ ψ − ψ ˙ ψ )+(588+448 ψ +236 ψ +64 ψ +8 ψ − ψ ) λ − (42 + 16 ψ + 6 ψ ) λ + λ , ψ ) − λ N/A , − (7 − ψ ) + λ − (49 − ψ + 32 ψ − ψ ) +(14 − ψ + 4 ψ ) λ − λ − (343 − ψ + 574 ψ − ψ + 136 ψ − ψ −
56 ˙ ψ + 24 ψ ˙ ψ ) + (147 − ψ + 124 ψ − ψ + 8 ψ − ψ ) λ − (21 − ψ + 6 ψ ) λ + λ , (169 + 78 ψ ) − (26 + 6 ψ ) λ + λ (2197 + 962 ψ + 216 ψ ) − (507+204 ψ +24 ψ ) λ +(39+10 ψ ) λ − λ (28561 + 16900 ψ + 4732 ψ + 432 ψ − ψ − ψ ˙ ψ )+( − − ψ − ψ − ψ +208 ˙ ψ +48 ψ ˙ ψ ) λ + (1014 + 412 ψ + 44 ψ − ψ ) λ − (52 +12 ψ ) λ + λ , − (23 + 5 ψ ) + λ − (529+230 ψ +96 ψ +20 ψ )+(46 + 10 ψ + 4 ψ ) λ − λ − (12167+10580 ψ +5566 ψ +1880 ψ +392 ψ +40 ψ −
184 ˙ ψ + 40 ψ ˙ ψ ) + (1587 + 920 ψ + 380 ψ + 80 ψ +8 ψ − ψ ) λ + ( − − ψ − ψ ) λ + λ , N/A (37 + 12 ψ ) − λ N/A2 , − ψ ) − λ N/A2 , − (14 − ψ ) + λ − (196 − ψ +60 ψ − ψ )+(28 − ψ + 4 ψ ) λ − λ − (2744 − ψ + 2128 ψ − ψ + 248 ψ − ψ −
112 ˙ ψ +32 ψ ˙ ψ )+(588 − ψ +236 ψ − ψ +8 ψ − ψ ) λ − (42 − ψ + 6 ψ ) λ + λ , ψ ) − (44 + 6 ψ ) λ + λ (10648 + 2816 ψ + 360 ψ ) − (1452+348 ψ +24 ψ ) λ +(66+10 ψ ) λ − λ (234256 + 83248 ψ + 13552 ψ + 720 ψ − ψ − ψ ˙ ψ ) − (42592 + 13376 ψ + 1584 ψ + 48 ψ −
352 ˙ ψ − ψ ˙ ψ ) λ +(2904+700 ψ +44 ψ − ψ ) λ − (88 + 12 ψ ) λ + λ , − (34 + 6 ψ ) + λ − (1156 + 408 ψ + 140 ψ +24 ψ )+(68+12 ψ +4 ψ ) λ − λ − (39304 + 27744 ψ + 11968 ψ + 3312 ψ + 568 ψ +48 ψ −