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Quantum Kinks: Solitons at Strong Coupling

arXiv:hep-th/9207074v1 22 Jul 1992JHU-TIPAC-920014July, 1992Quantum Kinks: Solitons at Strong CouplingStephen G. Naculich∗Department of Physics and AstronomyThe Johns Hopkins UniversityBaltimore, MD 21218ABSTRACTWe examine solitons in theories with heavy fermions. These “quantum” soli-tons differ dramatically from semi-classical (perturbative) solitons because fermionloop effects are important when the Yukawa coupling is strong.

We focus on kinksin a (1 + 1)–dimensional φ4 theory coupled to fermions; a large-N expansion isemployed to treat the Yukawa coupling g nonperturbatively. A local expression forthe fermion vacuum energy is derived using the WKB approximation for the Diraceigenvalues.

We find that fermion loop corrections increase the energy of the kinkand (for large g) decrease its size. For large g, the energy of the quantum kinkis proportional to g, and its size scales as 1/g, unlike the classical kink; we arguethat these features are generic to quantum solitons in theories with strong Yukawacouplings.

We also discuss the possible instability of fermions to solitons.∗NACULICH @ CASA.PHA.JHU.EDU

1. IntroductionTopological solitons, despite their inherently nonperturbative character, aretypically studied semi-classically, that is, in a perturbative expansion in the cou-pling constants [1].The first term in this expansion, the classical soliton, isthe solution to a nonlinear classical field equation.

This solution is nonpertur-bative because its energy diverges as the coupling constants—which parametrizethe nonlinearity—vanish. Perturbative corrections to the soliton are important:they split the degeneracies of the classical solution resulting from Poincar´e and in-ternal symmetries, and project the solitons onto eigenstates of momentum, angularmomentum, and charge.

If the coupling constants are small, however, correctionsto the shape and energy of the soliton are small, and the classical description ofthe soliton is essentially accurate.If the couplings are large, on the other hand, there is no reason to expect thequantum soliton states to resemble the classical solitons, at least quantitatively. Ingeneral, the strong coupling behavior of solitons in a quantum field theory is notwell known.

One notable exception is the sine-Gordon kink in (1+1) dimensions;because of the equivalence of the sine-Gordon theory to the massive Thirring model[2], the sine-Gordon kink at strong coupling becomes a weakly-coupled fermion inthe Thirring model, which is well described by perturbation theory.In this paper we study strongly-coupled solitons more generally, when such afortuitous equivalence does not arise. We focus in particular on solitons in the-ories with large Yukawa couplings.

One motivation for doing so is the following.Fermions can acquire mass through a Yukawa coupling to a scalar field with non-vanishing vacuum expectation value. Solitons in such theories often carry (possi-bly fractional) fermion number.

It has recurrently been suggested that when theYukawa coupling is large such a soliton may have less energy than a fermion in aconstant scalar field background; consequently, fermions may be unstable to theformation of solitons [3–11]. To determine whether this is so, however, one mustknow the form and energy of solitons in a strongly-coupled theory, which may differ2

appreciably from classical solitons. Indeed, we expect fermion loop corrections tosignificantly affect the solitons when the Yukawa coupling is large.One means of studying a strongly-coupled Yukawa theory is through a large-N expansion [5,10–13].

To leading order in 1/N, the theory can be solved forarbitrary values of the Yukawa coupling.This expansion captures some of thestrong-coupling behavior of the theory, which one hopes is representative evenwhen N is not large. To carry out this expansion, we introduce N fermion flavorsand choose the N-dependence of the couplings so that the theory has a sensibleN →∞limit, with only fermion loops contributing to Green functions to leadingorder in 1/N.

The total contribution of the fermion loops can be summed in closedform to give the exact large-N effective actionSeff[φ] = S [φ] −iN log det (i /D) ,(1.1)where S [φ] is the classical scalar field action and /D is the Dirac operator in thepresence of the field φ.Solitons in this large-N theory are c-number configurations of the scalar fields;scalar field fluctuations are suppressed because scalar loops do not contribute tothe effective action to leading order in 1/N. The shape of the large-N solitondiffers from the classical soliton, however, since it extremizes not the classicalaction but the effective action (1.1).

The fermion loop contribution significantlyalters the form of the soliton when the Yukawa coupling is large. In this regime,where quantum effects are so important, the large-N soliton is truly a “quantumsoliton.”To determine the form of the quantum soliton, we need to know −iN log det (i /D)explicitly for an arbitrary scalar field configuration.One generally resorts tosome local approximation, such as the gradient expansion [5, 14], accurate forslowly-varying configurations.

The gradient expansion, however, breaks down fortopological solitons in the theories that we are considering.Another approach3

to computing the fermion loop contribution relies on the fact that for static soli-tons (iN/T) log det (i /D) is just the energy of the “Dirac sea,” the sum of negativeeigenvalues of the Dirac equation in the soliton background [15]. Unfortunately, theDirac eigenvalues must be numerically computed [16] for each separate backgroundconsidered, rendering this approach inconvenient for a variational problem.In this paper, we propose a hybrid of the gradient expansion and eigenvaluesum methods.

Following an idea of Wasson and Koonin [17], we use the WKBapproximation to estimate the Dirac sea eigenvalues for an arbitrary static scalarfield background. We then sum these to obtain a local expression for the fermionvacuum energy.

Unlike the gradient expansion, this expression is finite for topo-logically nontrivial configurations. Using this WKB approximation, we extremizethe effective action to find the form of the quantum soliton in the large-N theory.We illustrate this method on a well-known example, the kink of the (1+1)–dimensional φ4 theory coupled to fermions.

The classical kink is reviewed in sect. 2.In sect.

3, we derive the WKB approximation for the large-N effective action inthis theory. This result is used in sect.

4 to find the form of the quantum kink,which is contrasted to the classical kink. The question of fermion stability is alsodiscussed.

In sect. 5, we present our conclusions and discuss the features of themodel that we expect are generic to strongly-coupled solitons.2.

Classical KinksWe begin by recalling the form and quantum numbers of the classical kink[1]. The (1+1)–dimensional φ4 theory coupled to N flavors of fermion has theLagrangianL = 12 (∂µφ)2 −λ4Nφ2 −Nv22 +NXi=1ψi i/∂−g√Nφψi.

(2.1)The N-dependence of the parameters has been chosen so that this theory has asensible N →∞limit. If we rewrite φ as√Nϕ, the parameter N becomes an4

overall scale,L = Nh12 (∂µϕ)2 −14λϕ2 −v22i+NXi=1ψi (i/∂−gϕ) ψi. (2.2)In the vacuum state |ϕ| = v, the scalar field has mass√2λv and the fermionfield mass gv.

In two dimensions, v is dimensionless, the scalar self-coupling λ hasdimension 2, and the Yukawa coupling g dimension 1. It is convenient to substitutefor λ and g the parametersxcl =r2λv2,y = gr2λ.

(2.3)The parameter xcl is proportional to the scalar field Compton wavelength (and,as we will see, the size of the classical kink), and will serve as the overall scale oflength and energy in the theory. There are two dimensionless parameters, v andy, the latter being proportional to the ratio of fermion and scalar masses.The Lagrangian (2.2) gives rise to the field equations∂2ϕ + λϕ3 −λv2ϕ = −g 1NNXi=1ψiψi,(2.4)(i/∂−gϕ) ψi = 0.

(2.5)The topologically nontrivial solutions of these equations give a “classical” descrip-tion of the soliton states in the Hilbert space, which is accurate when the quantumcorrections are small. If we neglect the fermion source term, the scalar field equa-tion (2.4) has the well-known static kink solutionϕcl(x) = v tanh xxcl,(2.6)which is the lowest energy state with topological charge [ϕ(∞) −ϕ(−∞)]/2v = 1.There is also an anti-kink solution which interpolates from v to −v, with topological5

charge −1. The Dirac equation (2.5)in the kink background (2.6)has a self-conjugate zero mode solutionψ0(x) =[sech(x/xcl)]y0,γ0 = σ1,γ1 = iσ3.

(2.7)The state with the zero mode occupied has the same energy as that with the zeromode unoccupied. Since there is a zero mode for each flavor i, the kink is 2N-folddegenerate.

If n of the zero modes are occupied, the kink has fermion numbern −12N, which ranges from −12N to 12N [18]. The anti-kink too has degeneracy2N, and fermion number ranging from −12N to 12N.

Although the fermion zeromodes increase the degeneracy of the kink, their contribution to the source term inthe scalar field equation (2.4) vanishes, so the kink (2.6) remains a solution evenin the presence of fermions. The energy of the classical kinkE [ϕcl] = 2√23 N√λv3 = 43v2 Nxcl(2.8)has no dependence on the Yukawa coupling g, even though the kink carries fermionnumber due to the zero modes.This classical picture leads to the fascinating possibility that, even if fermionnumber is conserved, “ordinary” fermions may be unstable to the formation ofsolitons carrying fermion number.

A configuration consisting of a widely-separatedkink and anti-kink, each carrying fermion number 12N, has zero topological charge,fermion number N, and energy 83v2(N/xcl). On the other hand, a set of N widely-separated fermions in the vacuum background ϕ = v, a state which has the samequantum numbers, has energy Ngv = y(N/xcl).

Thus, when y > 83v2, it is energet-ically favorable for a state of N fermions to coalesce onto a spontaneously createdkink/anti-kink pair. Each kink acts as a kind of bound state of 12N fermions.

Evenmore surprising, in a theory with one flavor of fermion (N = 1), a single fermioncould split into a kink/anti-kink pair, each with fermion number 12.6

This putative instability occurs only when y is large, however, where the quan-tum corrections from fermion loops are important and the semi-classical approxi-mation breaks down. To determine whether fermions are truly unstable, one mustcompare their energy not with that of a classical kink, but of a “quantum kink,”which includes the effects of quantum corrections.

The quantum kink extremizesnot the action but rather the effective action. In the next section we will derive alocal expression for the effective action suitable for finding the quantum kink.3.

Effective Action for KinksQuantum solitons are field configurations that extremize the effective action,which includes quantum corrections. To find the form of quantum solitons, oneneeds an explicit local expression for the effective action.

The familiar gradientexpansion, however, diverges for topologically nontrivial configurations in (1+1)–dimensional φ4 theory. In this section, we derive an alternative local approximationfor the effective action that is finite for kinks.Since we are interested in the properties of solitons for large Yukawa couplingg, the effective action must be calculated nonperturbatively in g. This can be doneby taking the number N of fermion flavors to be large, holding λ, v, and g fixed,and calculating to leading order in 1/N.

Scalar field fluctuations are subleading in1/N, so only fermion loops contribute to the large-N effective actionSeff[ϕ] =Zd2x Leff(ϕ)= NZd2xh12 (∂µϕ)2 −14λϕ2 −v22i−iN log det (i/∂−gϕ)+Zd2x δL(ϕ)+iN log det (i/∂−gv) . (3.1)We have added the countertermδL(ϕ) = AN(ϕ2 −v2)(3.2)to tame the divergent contributions of the fermion determinant to the one- and7

two-point functions, and the overall constant iN log det (i/∂−gv) to ensure thatSeff[ϕ = v] = 0. The coefficient A is fixed by requiring the one-point function tovanish at ϕ = v,0 = 2Av + igZd2p(2π)2 tri/p −gv,(3.3)so that v remains the minimum of the effective potential.

With a cutoffΛ on thespatial momentum p1, eq. (3.3) givesδL(ϕ) = −Ng22π (ϕ2 −v2)ΛZ0dp1qp21 + g2v2.

(3.4)This counterterm also renders finite the two-point functionΓ(2)σσ(p)p=0 = −2λv2 −g2π ,(3.5)where σ = ϕ −v. Fermion loop contributions to all other Green functions arefinite.We must write the effective action (3.1) in a more tractable form if we are tofind the quantum kink explicitly.

The gradient expansion [5, 14]Leff(ϕ) = −Veff(ϕ) + L(2)eff(ϕ) + · · ·(3.6)is a useful approximation for slowly-varying fields. The first term in this expansionis minus the effective potentialVeff(ϕ)N= λ4ϕ2 −v22 + g24πϕ2 lnϕ2v2−g24πϕ2 −v2.

(3.7)The term with two derivatives isL(2)eff(ϕ)N= 121 +112πϕ2(∂µϕ)2 . (3.8)At this point, we discover that the gradient expansion fails for topological soli-tons in this theory; any configuration ϕ(x) with unit topological charge must pass8

through ϕ = 0 somewhere, at which point L(2)eff(ϕ), as well as higher order terms,diverges. This failure is quite general.

For the gradient expansion to converge, fieldgradients must be small relative to gϕ, the “local fermion mass.” Since the lattervanishes at the core of solitons with fermion zero modes, the gradient expansionnecessarily breaks down there, no matter how slowly varying the field.An alternative approach for a static scalar field background such as the kinkis to express the effective action in terms of Dirac equation eigenvalues [15,4]. Fortime-independent ϕ(x), the effective action equals −Eeff[ϕ] T, where T =Rdt andEeff[ϕ] is the energy of the configuration,Eeff[ϕ] = Ecl [ϕ] + Q [ϕ] ,(3.9)a sum of the classical energyEcl [ϕ] = Nxcl∞Z−∞dz"12dϕdz2+ 12v2ϕ2 −v22#,z = xxcl,(3.10)and the quantum correction, the fermion vacuum energy,Q [ϕ] = iNT log det (i/∂−gϕ) −iNT log det (i/∂−gv) + δE [ϕ] .

(3.11)The first term in eq. (3.11) can be interpreted as the energy of the Dirac sea in thebackground ϕ(x).To write eq.

(3.11) more explicitly, we observe that the Dirac equation (2.5) im-plies that the spinor components ψi =ψi+ψi−obey the Schr¨odinger-type equations d2dz2 −y2V±(z) −y2 + x2clǫ2±ψ± = 0(3.12)in a static background ϕ(z), whereV±(z) =ϕ2v2 −1∓1yvdϕdz . (3.13)We restrict ϕ(z) to configurations of unit topological charge that obey ϕ(−z) =−ϕ(z); the Schr¨odinger potentials Vσ(z) are then even, and the solutions ψσ(z)9

can be taken to be parity eigenstates. (Here σ = ± labels the upper and lowerspinor components.

)Sinceφ(±∞) = v, the potentials Vσ(z) vanish at ±∞,so eq. (3.12) has a continuous spectrum of states classified by their asymptoticmomentum, k =qx2clǫ2 −y2, and their parity.The asymptotic forms of thecontinuum wavefunctionsψσ,even(k, z) −→z→±∞cos(kz ± 12δσ,even(k)),ψσ,odd(k, z) −→z→±∞sin(kz ± 12δσ,odd(k)),(3.14)serve to define the phase shifts δσ,even(k)δσ,odd(k)for the even (odd) paritystates.

If we put the system into a box, |z| ≤12L, with periodic boundary con-ditions, eq. (3.14) implies that the allowed momenta satisfy kσnL + δσ(k) = 2πn.Eq.

(3.12) may also have a series of discrete bound states with eigenvalues ǫ2σi

Thus, any configurationwith unit topological charge can carry fermion quantum numbers.The difference of fermion loop contributions can be written as the shift of theDirac sea energy [15]iNT log det (i/∂−gϕ) −iNT log det (i/∂−gv) = −12NXσXλǫσλ −ǫ(0)σλ, (3.15)where ǫσλ denotes the positive root of ǫ2σλ, and ǫ(0)σλ are the Dirac eigenvalues in theconstant configuration ϕ(x) = v. Eq. (3.15) may be separated into the sum overdiscrete eigenvaluesEdisc [ϕ] = −12NXσXiǫσi −yxcl(3.16)and the sum over continuum eigenvaluesEcont [ϕ] = −12NXσXn>0Xparityhǫ(kσn) −ǫ(k(0)σn)i,ǫ(k) =pk2 + y2xcl.

(3.17)Using kσnL + δσ(k) = k(0)σnL = 2πn, and letting L →∞, we can write eq. (3.17) as10

[15]Econt [ϕ] = NXσΛZ0dk4πdǫdkδσ,even(k) + δσ,odd(k)= NXσΛZ0dk2πdǫdkδσ(k), (3.18)where δ = 12 (δeven + δodd). The integral over k diverges as the momentum cutoffΛ is removed, but this divergence is cancelled by the counterterm energyδE [ϕ] = −∞Z−∞dz δL(ϕ) = y22π NxclΛZ0dkpk2 + y2∞Z−∞dzϕ2v2 −1.

(3.19)The sum of eqs. (3.16), (3.18), and (3.19),Q [ϕ] = Edisc [ϕ] + Econt [ϕ] + δE [ϕ](3.20)is precisely the fermion vacuum energy (3.11).The expression (3.20) for the fermion vacuum energy is much more explicitthan eq.

(3.1), and can even be computed analytically for certain scalar field con-figurations [19].For an arbitrary background, however, ǫσi and δσ(k) must becomputed numerically [16]. Wasson and Koonin [17] showed how to speed up theconvergence of these “brute force” numerical calculations by employing the WKBapproximation for the high momentum phase shifts, but the discrete eigenvaluesand low momentum phase shifts must still be computed numerically for each sepa-rate field configuration.

Thus, eq. (3.20) is still not very convenient for extremizingthe effective action.⋆Taking our cue from ref.

[17], we adopt the WKB approximation for all theDirac eigenvalues, both continuous and discrete, and use them in eq. (3.20) toobtain a local expression for the energy of an arbitrary scalar field configuration.⋆Campbell and Liao [4] were able to extremize(3.9) using powerful inverse scatteringmethods, but only for the special case y = 1.11

The resulting expression will be accurate for field configurations slowly varying onthe scale of the fermion Compton wavelength, but unlike the gradient expansion,does not diverge for solitons. We will then use this approximate expression to findthe form of the quantum kink in sect.

4.In the WKB approximation, the continuum eigenfunctions of eq. (3.12) areψWKBσ,even(k, z) =1pkσ(z)cosR z0 kσ(z′)dz′,ψWKBσ,odd(k, z) =1pkσ(z)sinR z0 kσ(z′)dz′,kσ(z) =qk2 −y2Vσ(z),(3.21)whence the phase shift defined through eq.

(3.14) is given byδWKBσ(k) =∞Z−∞dzkσ(z) −k,(3.22)independent of parity. (We assume Vσ(z) ≤0 everywhere; this will be true if ϕ(z)does not vary too rapidly.) Using the WKB phase shifts (3.22) in the integral(3.18) and adding the counterterm energy (3.19), we findEWKBcont [ϕ] + δE [ϕ] = y24π Nxcl∞Z−∞dz 1 −ϕ2v2−p−V+ +p−V−+ (1 + V+) log1 +p−V++ (1 + V−) log1 +p−V− .

(3.23)We also need to approximate the sum over discrete eigenvalues (3.16).In theWKB approximation, the Schr¨odinger equation (3.12) has discrete eigenvalues ǫwhenever wσ(ǫ), defined bywσ(ǫ) = 1π∞Z−∞dz kσ(z)Θ(k2σ(z)),kσ(z) =qx2clǫ2 −y2 −y2Vσ(z),(3.24)equals half an odd integer, w ∈Z + 12. The number of discrete eigenstates is givenby the integer closest to wσ(y/xcl).

We define ǫσ(w) by inverting eq. (3.24) and12

setting ǫσ(w) = 0 for 0 ≤w ≤wσ(0). The sum over discrete eigenvalues (3.16) inthe WKB approximation is then writtenEWKBdisc [ϕ] = −12XσXw∈Z+ 120

(3.25)We separate this into two termsEWKBdisc [ϕ] = E(1)disc [ϕ] + E(2)disc [ϕ] ,(3.26)where E(1)disc [ϕ] is the integral approximation of the sum (3.25)E(1)disc [ϕ] = −12Xσwσ(y/xcl)Z0dwǫσ(w) −yxcl,(3.27)and E(2)disc [ϕ] is the remainder. The integral (3.27) may be rewrittenE(1)disc [ϕ] = −12Xσy/xclZ0dǫ wσ(ǫ)= −12π∞Z−∞dzXσy/xclZ0dǫ kσ(z)Θ(k2σ(z))= y24π Nxcl∞Z−∞dz p−V+ +p−V−+ (1 + V+) log p|1 + V+|1 + √−V+!+ (1 + V−) log p|1 + V−|1 + √−V−!

. (3.28)Adding the contributions from the continuum (3.23) and discrete (3.26) states, we13

obtainQWKB [ϕ] = y24π Nxcl∞Z−∞dz 1 −ϕ2v2+ 12ϕ2v2 + 1yvdϕdzlogϕ2v2 + 1yvdϕdz+ 12ϕ2v2 −1yvdϕdzlogϕ2v2 −1yvdϕdz+ E(2)disc [ϕ](3.29)for the fermion vacuum energy in the WKB approximation.When ϕ(z) is slowly varying on the scale of the fermion Compton wavelength,the number of discrete states wσ(y/xcl) is large, the sum (3.25) is well approximatedby the integral (3.27), and E(2)disc [ϕ] is much smaller than E(1)disc [ϕ] . If we thereforeneglect E(2)disc [ϕ] , eq.

(3.29) provides a completely explicit local expression for theenergy of a static configurationEWKBeff[ϕ] = Nxcl∞Z−∞dz"12dϕdz2+ 12v2ϕ2 −v22#+ y24π Nxcl∞Z−∞dz 1 −ϕ2v2+ 12ϕ2v2 + 1yvdϕdzlogϕ2v2 + 1yvdϕdz+ 12ϕ2v2 −1yvdϕdzlogϕ2v2 −1yvdϕdz. (3.30)For ϕ(z) constant, eq.

(3.30) reduces to the effective potential (3.7). When ϕ(z) isnot constant, eq.

(3.30) yields a correction to the effective potential which, unlikethe gradient expansion, does not diverge for configurations going through ϕ = 0.We conclude this section by comparing the WKB approximation of the fermionvacuum energy of the classical kink, ϕcl(z) = v tanh(z), with the known exact re-sult.The WKB approximation should be accurate for y ≫1, when ϕcl(z) isslowly-varying relative to the fermion Compton wavelength. The Dirac equationcan be solved analytically in the classical kink background.Using the result-ing eigenvalues, Chang and Yan [19] computed the exact fermion loop correction14

(3.20) to the energy of the classical kinkQ [ϕcl] = Nxcl∆(y). (3.31)The function ∆(y) is given by a complicated integral, but for integer y it simplifiesto [19]∆(y) = y2π +y−1Xn=1 −p2yn −n2 + 2πpy2 −n2 arctanry2n2 −1!,y ∈Z.

(3.32)Using the Euler-Maclaurin formula, we obtain the large y behavior of eq. (3.32)∆(y) = 32π −π8y2+2√2β +13√2 √y+O(1),β =∞Xk=1(4k −5)!!22k(2k)!

B2k ≈.0206 . .

. (3.33)where B2k are the Bernoulli numbers.

The series defining β is asymptotic, so weonly keep 4 or 5 terms in the sum.The WKB approximation is obtained by substituting ϕcl(z) into eq. (3.29) andexpanding for large yQWKB [ϕcl] = Nxcl 32π −π8y2 + 16√y + O(1)+ E(2)disc [ϕcl] .

(3.34)Using the WKB approximation for the discrete eigenvalues ǫσi together with theEuler-Maclaurin formula, we find the leading behavior of the remainder termE(2)disc [ϕcl] = Nxcl2√2β +13√2 −16 √y + O(1). (3.35)Thus the WKB approximation of the fermion vacuum energy isQWKB [ϕcl] = Nxcl 32π −π8y2 +2√2β +13√2 √y + O(1),(3.36)in agreement with eq.

(3.33) to this accuracy. The y2 term, of course, is just thecontribution from the effective potential (3.7).

The non-analytic subleading √y15

dependence cannot be seen in the gradient expansion, but is correctly given by theWKB approximation (3.29).Obviously, the coefficient obtained for the subleading √y dependence would beincorrect if we made the further approximation of dropping E(2)disc [ϕ] , as was donein obtaining the local expression (3.30). Nonetheless, eq.

(3.30) correctly gives theorder of the subleading dependence. In general, it provides a useful estimate of thecorrection to the effective potential for a spatially-varying field.4.

Quantum KinksThe quantum kink extremizes the effective action of the (1+1)–dimensional φ4theory. For small Yukawa coupling y, the effective action (3.1) differs only slightlyfrom the classical action, so the quantum kink nearly coincides with the classicalkink.

When y is large, however, fermion loop corrections are important, and thequantum kink differs significantly from the classical kink.To find the explicit form of the quantum kink, we use the local approximation(3.30) for the energy Eeff[ϕ] of a static scalar field configuration derived in sect. 3.The equation of motion for the quantum kink follows from extremizing eq.

(3.30),1 +14πv4ϕ2f(ϕ) d2ϕdz2 = 2ϕϕ2v2 −1+y24πv2ϕ logf(ϕ) +12πv4ϕf(ϕ)dϕdz2,f(ϕ) = ϕ4v4 −1y2v2dϕdz2. (4.1)Using the program COLSYS [20], we have solved this equation numerically forvarious values of the parameters subject to the boundary condition ϕ(±∞) =±v.The solutions obtained interpolate smoothly between −v and v.Indeed,their profiles are almost indistinguishable from the hyperbolic tangent shape ofthe classical kink (see fig.

1). The slope of the quantum kink differs from that ofthe classical kink, however, being much steeper for certain values of the parameters.16

We can more easily see how the slope of the quantum kink depends on the pa-rameters of the theory by restricting ϕ(x) to the one-parameter family of functionsϕx0(x) = v tanh xx0,(4.2)where x0 is the “size” of the ansatz. We write the energy of the ansatzEeff(z0) = Ecl(z0) + Q(z0),z0 = x0xcl,(4.3)where z0 is the ratio of the size of the ansatz to that of the classical kink.

Theclassical contributionEcl(z0) = Nxcl23v2z0 + 1z0(4.4)has a minimum at z0 = 1, of course. The quantum contribution is obtained bysubstituting the ansatz (4.2) into the WKB approximation (3.29) and retainingthe leading power of yQWKB(z0) = Nxcl 32π −π8y2z0 + Or yz0.

(4.5)The WKB approximation is accurate when the neglected terms are small, whichrequires z0 ≫1/y. That is, the size of the ansatz must be much larger than thefermion Compton wavelength (x0 ≫1/gv).In the following discussion, we assume large Yukawa coupling y ≫1.

The sizeof the quantum kink is found by minimizing (4.3),z0 =1 + 94π −3π16 y2v2−1/2,(4.6)and depends on the values of both dimensionless parameters y and v. When v ≫y,the kink size z0 ≈1 and the quantum kink reduces to the classical kink, because the17

classical contribution to the energy is dominant in this regime. On the other hand,when v ≪y (but v ≫1), the kink size z0 ≈ 94π −3π16−12 (v/y) ≈2.8(v/y); thequantum kink is much smaller than the classical kink.

The energy of the quantumkink in this regime, Eeff≈ 4π −π3 12 vy(N/xcl) ≈.48 vy(N/xcl), is larger than theclassical kink energy (2.8) due to the positive fermion vacuum energy (4.5).When v <∼1, the WKB approximation (4.5) breaks down because the kink sizeis no longer much larger than the fermion Compton wavelength. By using the exactDirac eigenvalues for the background (4.2) rather than the WKB eigenvalues, how-ever, we can calculate the fermion vacuum energy Q(z0) without approximation,just as for the classical kink.

We findQ(z0) = Nxcly U(yz0),U(t) = ∆(t)t,(4.7)where ∆(y) is the fermion vacuum energy of the classical kink defined in sect. 3.The function U(t) is shown in fig.

2, and equals 1/π at its minimum t = 1. (Thatits minimum is at t = 1 can be seen from eq.

(3.32) and fromd∆dt= 12δt0 + tπ +t−1Xn=1 −rn2t −n + 2πrt −nt + n arctanrt2n2 −1!,t ∈Z,(4.8)obtained by a calculation similar to that in ref. [19].) When v ≪1 (and y ≫1), thefermion vacuum contribution (4.7) dominates the energy, so the size of the quantumkink is determined by the minimum of Q(z0), that is, z0 ≈1/y.

The quantum kinkenergy, Eeff≈(y/π)(N/xcl), is much larger than that of the classical kink (2.8).The classical contribution to the energy Ecl(z0) is minimized for z0 = 1, whenthe ansatz size equals the scalar field Compton wavelength, x0 = xcl. The quantumcontribution Q(z0) is minimized for z0 = 1/y, when the ansatz size equals thefermion Compton wavelength, x0 = xcl/y = 1/gv.

The size of the quantum kinkalways lies somewhere between these two values. The three limits we considered18

above1 ≪y ≪v⇒z0 ≈1,Eeff≈4v23 Nxcl,1 ≪v ≪y⇒z0 ≈ 94π −3π16−1/2 vy,Eeff≈4π −π31/2vy Nxcl,v ≪1 ≪y⇒z0 ≈1y,Eeff≈yπ Nxcl. (4.9)correspond to the regime in which the classical energy is dominant (v ≫y), theregime in which the fermion vacuum energy is dominant (v ≪1), and the regime inwhich both contributions are important (1 ≪v ≪y).

For v >∼y, the classical andquantum kink nearly coincide, while for v <∼y, the quantum kink is smaller andhas greater energy than the classical kink. Note that due to the fermion vacuumcontribution (4.7), the energy of the kink is bounded below by (y/π)(N/xcl) =Ngv/π, that is, 1/π times the mass of N fermions.We now turn to the question of fermion stability.In sect.

2, we saw thatfor sufficiently strong Yukawa coupling, y > 83v2, a state of N widely-separatedfermions has greater energy than a kink/anti-kink pair, computed in the classicalapproximation, so one might expect a kink/anti-kink pair to appear spontaneously,with the fermions coalescing to occupy the zero modes. Since the zero modes donot increase the kink energy, the energy of the fermions on the kinks is independentof y in the classical approximation, and would be much less than the energy of thefermions in a constant scalar field background for y ≫v2.

The kink binding energycould approach 100% for very large Yukawa coupling.Instead we have found that, for large y, quantum corrections significantly in-crease the energy of the kink.For v >∼1 (but v ≪y), a kink/anti-kink pairhas energy ∼yv(N/xcl), greater than the energy of N fermions, so the fermionsare stable.For v <∼1, the energy of a kink/anti-kink pair may be less thany(N/xcl) = Ngv, in which case a state of N fermions may be unstable to the for-mation of a kink and anti-kink, each carrying fermion number 12N. Since the kink19

energy is never less than (y/π)(N/xcl), however, the energy of a widely-separatedkink/anti-kink pair is not significantly less than that of the original fermions; thebinding energy per fermion cannot exceed 1 −2π ∼36%.Up to this point, we have been chiefly concerned with large Yukawa coupling,y ≫1; we conclude this section by briefly considering y <∼1. When y is not large,the WKB approximation is no longer useful, but we can use the exact solution(4.7)for the ansatz (4.2).

The case y = 1 is interesting, because then z0 = 1minimizes both the classical and quantum contributions to the energy; the classicalkink is an extremum of the effective action restricted to the subspace of functions(4.2). One might suspect from this that the classical kink extremizes the effectiveaction over the space of all functions.

Campbell and Liao [4] proved this to be thecase by using inverse scattering methods (which were tractable only when y = 1).⋆Thus, the quantum kink exactly coincides with the classical kink (for all values ofv) when y = 1.† As we have seen, they differ when y ̸= 1.The energy of the kink when y = 1 isEeff=43v2 + 1π Nxcl,y = 1,(4.10)so a kink/anti-kink pair will have less energy than N widely-separated fermionswhen v q14π is a bag [4].

(See also ref. [9].

)Finally, for small Yukawa coupling, y < 1, the fermion Compton wavelength islarger than the scalar field Compton wavelength, so quantum corrections tend toincrease the size of the kink. When v2 ≪y < 1, the quantum contribution domi-nates the energy, and the kink has size z0 ≈1/y and energy Eeff≈(y/π)(N/xcl).⋆The stationary phase approximation of ref.

[4] is equivalent to our large-N approximation.†Interestingly, the theory is supersymmetric precisely when y = 1 [4, 21].20

5. ConclusionsWe have examined the effects of quantum corrections on solitons in a (1+1)–dimensional φ4 theory with a large Yukawa coupling y to fermions.

To treat theYukawa coupling nonperturbatively, we have solved the theory in the large-N limit,where N is the number of flavors. The solitons in this theory are kinks which carryfermion number ranging from −12N to 12N.

In the classical approximation, theenergy of the kink is independent of y, and its size is proportional to the scalarfield Compton wavelength. We have found that fermion loop corrections increasethe energy of the kink and (when y > 1) reduce its size.

As a result of the fermionvacuum contribution, the kink energy is bounded below by (y/π)(N/xcl) = Ngv/π,and its size can be as small as the fermion Compton wavelength.When y is large, a state of N fermions is expected on classical grounds to beunstable to the formation of a kink and anti-kink, each carrying fermion number12N. Quantum corrections eliminate this instability for v >∼1 by increasing thekink/anti-kink energy.

The instability persists for v <∼1, but the difference inenergy between the N fermions and the kink/anti-kink pair is only about 36%because the kink energy is proportional to the Yukawa coupling in the large ylimit.In the large-N limit, scalar loops are suppressed. The energy of scalar field fluc-tuations is of order 1/xcl, small compared to the classical kink energy ∼Nv2/xcl.What happens when N is not large?

Will a single fermion decay into a kink/anti-kink pair when N = 1? Scalar field fluctuations are still relatively unimportant aslong as v is large; 1/v2 is the usual semi-classical expansion parameter.

We foundfermions to be stable in this regime. Scalar corrections become more importantfor small v, but on the other hand 1/xcl is still small relative to the quantum kinkenergy y/πxcl when y is large.

It is difficult to say whether a single fermion isunstable when v <∼1.Some aspects of the model discussed in this paper are peculiar to two dimen-sions. Presumably only in two dimensions can a fermion split into a pair of solitons,21

each carrying fermion number 12. We expect other features of the quantum kink tobe more universal, however.

First, its energy acquires a linear dependence on theYukawa coupling in the strong coupling limit through the fermion vacuum energy.Second, for large Yukawa coupling, fermion loop corrections tend to reduce the sizeof the soliton in the direction of the fermion Compton wavelength. Both of thesefeatures apply not only to the (1+1)–dimensional solitons described in this paper,but also to (3+1)–dimensional (large-N) nontopological solitons [10].General arguments can be adduced to suggest that these are generic featuresof quantum solitons in any (large-N) strongly-coupled Yukawa theory in (3+1) di-mensions.

The fermion loop contribution to the effective action is −iN log det (i /D).After renormalization, its contributions to the effective potential (of order g4) andto the two-derivative term (of order g2) overwhelm the tree-level contributionswhen g is large.⋆For large Yukawa coupling, therefore, quantum solitons are deter-mined by the fermion vacuum energy (iN/T) log det (i /D). If this has a minimumfor given boundary conditions, the resulting configuration must have size R ∼1/gv(the only scale present) and energy ∼(gv)4R3−d, where d is the dimension of thesoliton.

For point-like solitons (d = 0), the energy is proportional to the Yukawacoupling. Assuming that the large-N restriction is only a technical one to facili-tate calculations at strong coupling, we conjecture that these properties hold forquantum solitons in any strongly-coupled Yukawa theory.ACKNOWLEDGEMENTSI would like to thank R. Perry for a useful conversation, and for drawing myattention to ref.

17. I am also grateful to J. Bagger, E. Poppitz, and S. Mrennafor helpful discussions.

This work has been supported by the NSF under grantPHY-90-96198.⋆Skyrmions, in which the fermion loop contribution to the two-derivative term vanishes afterrenormalization, apparently present an exception [5,6,14,16].22

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FIGURE CAPTIONS1) A representative quantum kink. The solid line shows the solution of eq.

(4.1) forthe parameters y = 20 and v = 4.The dashed line shows the ansatzϕ(z) = v tanh(z/z0), with z0 given by eq. (4.6).The dotted line showsthe classical kink, z0 = 1.2) The function U(t).

The exact fermion vacuum energy for the ansatz ϕ(z) =v tanh(z/z0) is given by (N/xcl) y U(yz0).25

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