---

The sine-Gordon model and the small k+ region

arXiv:hep-th/9207082v1 24 Jul 1992UFIFT-HEP-17May 1992The sine-Gordon model and the small k+ regionof light-cone perturbation theory†Paul A. Griffin††Department of Physics, University of FloridaGainesville, FL 32611AbstractThe non-perturbative ultraviolet divergence of the sine-Gordon model is used to studythe k+ = 0 region of light-cone perturbation theory. The light-cone vacuum is shown tobe unstable at the non-perturbative β2 = 8π critical point by a light-cone version ofColeman’s variational method.

Vacuum bubbles, which are k+ = 0 diagrams in light-conefield theory and are individually finite and non-vanishing for all β, conspire to generateultraviolet divergences of the light-cone energy density. The k+ = 0 region of momentumalso contributes to connected Green’s functions; the connected two point function will notdiverge, as it should, at the critical point unless diagrams which contribute only at k+ = 0are properly included.

This analysis shows in a simple way how the k+ = 0 region cannotbe ignored even for connected diagrams. This phenomenon is expected to occur in higherdimensional gauge theories starting at two loop order in light-cone perturbation theory.†Supported in part by the U.S. Department of Energy, under grant DE-FG05-86ER-40272†† Internet: pgriffin@ufhepa.phys.ufl.edu

1. IntroductionRecently, much attention has been given to the issue of regulating the non-covariantdivergences which occur in canonical light-cone field theory[1][2][3][4].

In the naive appli-cation of this perturbation theory, the k+ = 0 region of light-cone Feynman diagrams isregulated by applying a cutoffto k+ ∼ǫ to momentum integrals. It is then assumed thatfor physical (gauge invariant) processes the ǫ →0 limit can be taken at the end of the fullcalculation, i.e.

the k+ = 0 region does not contribute to physical processes. This assump-tion is analogous to the (in this case correct) expectation that infared (IR) divergencescancel in inclusive processes for theories with massless particles.

In fact, a straight cutoffof the k+ = 0 region does give the right renormalization structure at one loop for QED[4]and the right beta function β(g) for QCD[5]. Furthermore, it is standard lore that thelight-cone vacuum is trivial; i.e.

that the interacting vacuum is the free-field vacuum. Inlight-cone perturbation theory this implies that all bubble diagrams, which have supportonly in the k+ = 0 region, actually vanish; certainly with a sharp k+ cutoffthey can notcontribute.

However, that the light-cone vacuum actually must be non-trivial has beendiscussed recently in refs. [6][7][8], and in fact was actually realized many years ago[9].In this letter, the sine-Gordon model in two dimensions will be used to explore thesequestions.

We will see that vacuum bubbles do not vanish, and (more importantly) thatthe k+ = 0 region of light-cone perturbation theory contributes to connected Green’sfunctions at second order in the light-cone Hamiltonian perturbation theory; these factsare intimately related. For the more physical gauge theories in four dimensions, it will beclear that the k+ = 0 region can contribute to connected Green’s functions at two-looporder and beyond.The analysis of this paper is in the framework of canonical quantization on the light-cone (null plane) x+ = (t+x)/√2 = 0, and Dyson’s (old fashioned) Hamiltonian perturba-tion theory for the light-cone[10][11][12].

While the issue addressed here is not the same asthe k+ = 0 divergence in the covariant Minkowski-space Feynman perturbation theory ofgauge theories in the light-cone gauge, which is treated with the Mandelstam-Leibbrandtprescription[13][14], they are probably deeply related.Note that no new properties will discovered about the sine-Gordon model per se, andonly sine-Gordon perturbation theory[15][16][17][18] will be discussed. (The exact sine-Gordon solution via the inverse scattering method in given in ref.

[19].) The reason forusing this model is that the k+ = 0 issue appears to lowest non-vanishing order in sine-Gordon perturbation theory, and is easy to calculate because the sine-Gordon model is1

ultraviolet (UV) finite diagram by diagram[15]. In addition, we will use existence of thephase transition at β2 = 8π as a check on the validity of light-cone perturbation theory.2.

The instability at β2 = 8πTo warm up, let us calculate the critical point of the sine-Gordon model using Cole-man’s variation method[15] and light-cone quantization. This will lead to insight on thenature (IR verses ultraviolet UV ) of the k+ = 0 singularity.

In particular, we want toshow that for β2 ≥8π, the Hamiltonian density H is unbounded from below.For canonical light-cone quantization on the null-plane, the momentum operator isP + =Rdx−(H + P)/√2 and the Hamiltonian operator is P −=Rdx−(H −P)/√2. Forthe sine-Gordon model, they are given byP + =Zdx−∂−φ∂−φ ,P −=Zdx−α0β2 (1 −cos βφ) ,(2.1)where α0 and β are taken to be positive[15].

If we expand about the configuration ofminimum P −, (φ = 0), then 1 −cos βφ is an even power series in φ; the quadratic termhas coefficient 12α0 = 12m2, where m is mass associated with perturbation theory aboutφ = 0. The canonical boson field has mode expansion at x+ = 0φ =1√4πZ ∞0dk+k+ha(k+)e−ik+x−+ a†(k+)e+ik+x−i,(2.2)where the 1/k+ term in the integrand of eqn.

(2.2) comes from the covariant measure forfree particles. The canonical commutation relations are[a(k+), a†(k′+)] = k+δ(k+ −k′+) .

(2.3)In terms of the creation a† and annihilation a operators, the light-cone momentum densityoperator isNm[∂−φ∂−φ] + 14πZ ∞0dk+k+ ,(2.4)where normal ordering Nm[ ] is with respect to the free-field vacuum with mass m. Inorder to perform a variational calculation of the Hamiltonian density with respect a newvacuum state |µ⟩that corresponds to different free-field mass µ, it necessary to calculatethe divergent part of eqn. (2.4) with respect to a UV momentum cutoff[15].2

To this end, introduce “IR” and “UV” cutoffs δ+ and Λ+,Z ∞0dk+ →Z Λ+δ+dk+ . (2.5)These cutoffs are actually related by parity.

Consider the solution to the free-field equationof motion,φ =Z Λ+δ+dk+k+hb(k+)e+ik+x−e+m2x+/2k+ + c.c.i. (2.6)Under parity x →−x, the modes b transform as b(k+) →b(m2/2k+), and the cutoffstransform as Λ+ →m2/2δ+ and δ+ →m2/2Λ+.Hence to maintain parity with ourregularization of the divergent part of P +, letδ+ = m22Λ+ .

(2.7)This relation introduces the mass m into the divergent part of P +. The regulated free-fieldlight-cone vacuum is therefore sensitive to the free-field mass.

And note that the k+ = 0region is clearly not just IR, since parity interchanges the IR region with the UV regionof k+. The UV cutoffΛ+ is by Lorentz invariance a function of the free-field mass and amass-independent momentum cutoffΛ,Λ+ = Λ +√Λ2 + m2√2.

(2.8)Using relations (2.7) and (2.8), the regulated light-cone momentum isP + = Nm[P +] + 18π2Λ2 + m2 + O(m/Λ). (2.9)To similarly regularize the light-cone energy density, relate the exponentials in the cos βφterm to their normal ordered forms:e±iβφ = e±iβφ+e±iβφ−eβ2[φ+,φ−]/2 = Nm[e±iβφ] m24Λ2 β28π(1 + O(m/Λ)) ,(2.10)where φ+ and φ−contain only raising and lowering operators respectively.With eqns.

(2.9) and (2.10), we can easily reproduce the variational estimate of Cole-man. In the quantum perturbation theory with respect to the mass m vacuum state, thecorrect expression for the Hamiltonian density isH =1√2Nm[∂−φ∂−φ + αβ2 (1 −cos βφ)] .

(2.11)3

Normal ordering the exponential eliminates divergent tadpole terms, and renormalizes thecoupling constant α0 →α. The resulting perturbation theory is UV finite, diagram bydiagram[15].

Now consider the variational estimate of the energy density with respect tothe perturbative ground states corresponding to free-field mass µ. Using eqns.

(2.9) and(2.10),⟨µ|H|µ⟩=1√218π (µ2 −m2) −αβ2 µ2m2 β28π,(2.12)where the cutoffΛ →∞. For finite α and β > 8π the energy density of the µ vacuum stateis unbounded from below as µ becomes large, and the theory as defined has no groundstate for these values of β.The underlying reason that the sine-Gordon perturbation theory is sick for β2 > 8πis based on the fact that that the anomalous dimension of the cosine term is β2/4π.

Forβ2 < 8π the anomalous dimension of the interaction Hamiltonian is less than two andthe theory is super-renormalizable, at β2 = 8π the cosine term is a marginal operatorand the theory is renormalizable[18][20], and for β2 > 8π the anomalous dimension isgreater than 2, the dimension of the renormalized coupling α is negative, and the theoryis non-renormalizable.3. The light-cone vacuum is non-trivialWe will now try to understand the UV divergence for β2 ≥8π diagrammaticallyin light-cone perturbation theory.

We proved in the previous section that the light-conevacuum is unstable in this regime.This is possible only if vacuum bubbles are non-vanishing, and the interacting light-cone vacuum is not the free-field light-cone vacuum.As we will explicitly see, vacuum bubbles in light-cone field theory are k+ = 0 diagrams,and the sharp cutoffof the k+ = 0 region used in variational estimate of the previoussection is not a suitable regulator for these diagrams.The light-cone Hamiltonian density is broken up into free and interacting parts,P −free = 12m2Zdx−N[φ2] ,P −int = αβ2Zdx−N[(1 −cos βφ + 12β2φ2)] ,(3.1)where m2 = α, and the normal ordering is with respect to m.The Dyson perturba-tion expansion is defined for operators in the interaction representation, O(x−, x+) =4

eiP −freex+O(x−)e−iP −freex+.In particular, the free-field Green’s function is G(x, x′) =⟨0|T[φ(x−, x+)φ(x′−, x′+)]|0⟩, where the time ordering T[ ] is with respect to the light-cone time x+:G(x) =Θ(+x+)4πZ ∞0dk+k+ e−i[k+x−+m2x+/2k+]+Θ(−x+)4πZ ∞0dk+k+ e+i[k+x−+m2x+/2k+] . (3.2)The integrals over k+ are well defined because the singularity at k+ = 0 is cancelled bythe k+ = ∞region.

The result is just the covariant propagator for both time-like andspace-like separations. The one explicit property of G(x) that is required in the followinganalysis is that for |x|m ≪1, where |x| is the invariant distance, the Green’s function isG(x) = −ln[m2x2]/4π.

A standard tool from light-cone perturbation theory that will beapplied is Wick’s theorem for x+-ordered exponentials,T[eiRd2xj(x)φ(x)] = e−12Rd2xd2yj(x)j(y)G(x−y)N[eiRd2xj(x)φ(x)] . (3.3)The interacting vacuum light-cone energy density E−is given by a straightforwardrewriting of the Gell-Mann Low equal time formula[21] for the light-cone case,E−= ⟨0|P−int(0) Texp−iZ 0−∞dx+P−int|0⟩conn ,(3.4)where ⟨0| · · · |0⟩conn is the connected (to the light-cone Hamiltonian density P−int(0)) free-field vacuum expectation value.

This expression is a perturbation theory in α. To firstorder in α, E−1 = ⟨0|P −int(0)|0⟩= 0; but to order α2, we get the non-vanishing resultE−2 = −iα2β4Z 0−∞dx+Z ∞−∞dx−cosh (β2G(x)) −β4G2(x) −1.

(3.5)To recover the light-cone perturbation theory expression for E−2 , expand eqn. (3.5) ina power series in β, and integrate over coordinates x+ and x−.

Using eqn. (3.2), one easilyfindsE−2 = −α2β4∞Xn=2β2n(2n)!Z ∞0nYp=1dk+pk+pδ(Pnq=1 k+q )Pnr=1m22k+r −iǫ.

(3.6)This power series expansion in β has the diagrammatic interpretation shown in figure 1.It is an infinite sum of two-point connected vacuum bubbles, which are non-vanishingonly in the k+ →0 limit. Because of the ratio δ(P k+)/ Q k+, this limit is ill-defined;regulating the k+ = 0 region by introducing a cutoffk+ ≥δ+ would miss this contribution5

altogether. How one should properly evaluate these integrals is no mystery however, sincethe coordinate space representation given by eqn.

(3.5) is perfectly well defined.To see that eqns. (3.5) and (3.6) must be non-vanishing, consider the non-perturbativeUV divergence of the sine-Gordon model.

The divergence occurs when the point separation|x| is very small, i.e. close to the light-cone, where the most singular term in the light-cone operator product expansion of the interacting light-cone Hamiltonian densities willcontribute.The divergence is regulated by introducing a spatial UV cutoffa2, wherema ≪1, and letting G(m2x2) →G(m2(x2+a2)).

Then the most singular part of eqn. (3.5),comes from the exponential exp β2G(x) ≈exp[−lnm2(x2 + a2)/4π].

The singular term isisolated by restricting the d2x integral to the region x2 < l2, where ml ≪1 and l > a,−i α22β4Z 0−∞dx+Zx2

The space-like region for x+ < 0 is x−> 0,and it can be parametrized as x−= −re+θ and x+ = −re−θ.The measure is justR l/√20rdrR ∞−∞dθ. Similarly for the time-like region x−< 0, the parametrization is x−=−re+θ and x+ = −re−θ, and the measure is the exactly the same as the space-like region.The Green’s functions are independent of θ, and the space-like and time-like contributionsadd.

The light-cone result is the same as the equal-time result; isolating the a dependenceof eqn. (3.7),E−2 ∼−iα2(1−β24π )β4Zdθa2(1−β24π )(1 −β2/4π) .

(3.8)This result is valid for all β2 ̸= 4π. As a →0, the expression vanishes for β2 < 4π, and hasa power-law divergence for β2 > 4π.

At β2 = 4π, the a dependence is really ln a, i.e. E−2diverges logarithmically as a →0.

Note that this logarithmic divergence does not occur atthe phase transition point β2 = 8π. And according to ref.

[17], the 2p point contributionin the equal time-case diverges logarithmically at β2/8π = 1 −(2p)−1 and the odd p-pointfunctions are UV finite for all β2 < 8π. For β2 < 8π, the divergences are less severe asp increases; for β2 ≥8π they become worse, and in this sense perturbation theory breaksdown.It is clear from the above analysis that vacuum bubbles in light-cone field theory,which correspond to the k+ = 0 region of momentum space, are in general non-vanishing.And one proper way of evaluating them is to use the coordinate space diagram approach,which follows directly from the Dyson expansion in perturbation theory.6

4. The k+ = 0 contribution to the Feynman propagatorThe vacuum energy density is not a physical observable for the sine-Gordon model(no gravity) so its singularity properties are a relatively mild concern.What is moreimportant are singularities in physical connected Green’s functions, which from the equal-time analysis[17][18], occur only at β2 = 8π.

To study this we will calculate to secondorder the in light-cone perturbation theory the connected two-point Green’s function (theFeynman propagator) Γ(2)(k),Γ(2)(k) =Zd2xeik·x⟨0|Tφ(x)φ(0) exp−iZdx+P −int|0⟩conn . (4.1)The leading order contribution Γ(2)0isRd2xeik·xG(x) = G0(k),G0(k) =iΘ(k+)2k+ [k−−m2/(2k+) + iǫ] +iΘ(−k+)2k+ [k−−m2/(2k+) −iǫ]=i[k+k−−m2/2 + iǫ] .

(4.2)The next non-vanishing term is order α2. The simplest way to calculate it is to temporarilyexponentiate φ(x)φ(0) to exp (aφ(x)) exp (bφ(x)), use eqn.

(3.3), and then pick out theorder ab terms;Γ(2)2 (k) = G20(k)4α2β4Zd2x cosh (β2G(x)) −β4G2(x) −1−e−ik·x sinh (β2G(x)) −β2G(x). (4.3)This equation is manifestly covariant and equivalent to the equal-time result[17].Now we can interpret it in terms of light-cone diagrams.

After integrating over x−,the [cosh · · ·] term of eqn. (4.3) is given byG20(k) α22β4∞Xn=2β2n(2n)!Z ∞0nYp=1dk+pk+pδ(nXq=1k+q )Z ∞−∞dx+Θ(−x+)eix+ Pnr=1m22k+r + Θ(x+)e−ix+ Pnr=1m22k+r.

(4.4)These are “connected bubble” diagrams as shown in figure 2a-b.Integrating over x+will generate 1/(P −0 −iǫ) denominators. Like the true vacuum bubble diagrams of eqn.7

(3.6) and fig. 1, they receive support only in the k+ = 0 region of light-cone momentum.Similarly, the [sinh · · ·] term of eqn.

(4.3) has the light-cone momentum space expansionG20(k) α22β4∞Xn=1β2n+1(2n + 1)!Z ∞0nYp=1dk+pk+pZ ∞−∞dx+(Θ(−x+)δ(k+ −nXq=1k+q )e−ix+k−−Pnr=1m22k+r+ Θ(x+)δ(k+ +nXq=1k+q )e−ix+k−+Pnr=1m22k+r). (4.5)The Θ(x+) · · · term of this expression are the “Z”-diagrams of fig.

3b. They vanish for allphysical (i.e.

particles moving forward in time) momenta k+ > 0, because of the constraintδ(k+ + Pp k+p ). The regular light-cone diagrams of fig.

3a contain a contribution fromthe k+ = 0 region of each internal momentum, since like the bubble diagrams, theirdenominators are singular when Qp k+p (Pr m2/2k+r −k−) vanishes. The singularity occurswhen two internal light-cone momenta simultaneously vanish while the constraint Pq k+q =k+ is still preserved.

(Note that this is impossible for φ3 theory[9]. ) Therefore both theconnected bubble diagrams of fig.

2a-b and the regular light-cone diagrams of fig. 3a havecontributions from the k+ = 0 region.

These contributions do not cancel between the twotypes of diagrams, because they occur to different orders in β perturbation theory. Theconnected bubble diagrams are even order in β, while the regular diagrams are odd order.We can verify the assertion that the connected bubble diagrams of fig.

2a-b contributeto the connected Green’s function Γ(2)2by considering the non-perturbative UV divergenceof the sine-Gordon model that arises from integrating over the small |x| region of eqn. (4.3).Separately, the sum of connected bubble diagrams of fig.

2 and the sum of the diagrams offig. 3a diverge at β2 = 4π.

This is easy to see; the sources of the divergence are the eβ2G(x)terms in eqn. (4.3), and the result follows from the analysis for the true bubble diagramsof eqn.

(3.6). However, when the terms are combined and the net divergence of eqn.

(4.3)is considered, the result for the regulated contribution from the small |x| region isΓ(2)2 (k) = G20(k) ∼α2(1−β24π )β4|x|2Zdθa2(1−β28π )(1 −β2/8π) . (4.6)This is valid for all β2 ̸= 8π; for β2 = 8π the net contribution diverges logarithmically asa →0.

The important point is that without the k+ = 0 contribution from the connectedbubble diagrams one gets the false, and unphysical, result that the phase transition is atβ2 = 4π. The interplay between regular light-cone diagrams and the connected bubblediagrams that produces the correct non-perturbative (in β) UV divergence of the sine-Gordon model is expected to occur order by order in α perturbation theory[18].8

5. DiscussionThe light-cone vacuum of the sine-Gordon model is unstable at β2 ≥8π, by a sim-ple application of Coleman’s variational technique to light-cone field theory.

The crucialelement in the light-cone analysis is that the small and large k+ cutoffs required for thevariational calculation are related by parity. This introduces free-field mass dependenceinto the regulated expressions of the vacuum expectation values of light-cone energy andmomentum.This momentum cutoffprescription is not a sufficient regulator of light-cone perturba-tion theory; the k+ = 0 region of light-cone perturbation theory contributes to connectedGreen’s functions in sine-Gordon model perturbation theory, and incorrect results will oc-cur if this region of momentum space is discarded.

This region contributes to connecteddiagrams at the two-loop order. (The one loop contributions are tadpoles, which are elim-inated by normal ordering.) The existence of the four and higher point vertices in thetheory is crucial to have non-vanishing connected bubble diagrams and k+ = 0 singulari-ties in the regular connected diagrams; φ3 theory does not have any contribution from thek+ = 0 region to connected diagrams[9].While in this paper, we focused on the effect of k+ = 0 digrams on the propagator, allof the vertices that occur in the sine-Gordon model also receive contributions from k+ = 0type diagrams.

Therefore, it is clear that the naive process of implementing a sharp cutoffof the k+ = 0 region without making any corrections to the interaction light-cone Hamil-tonian in perturbation theory is doomed to failure. With the sharp cutoff, corrections tothe propagator and vertices will have to be made order by order in perturbation theory;neglecting the k+ = 0 region will in general violate gauge or Lorentz symmetries, andthe counterterms are necessary to restore them.

It is not clear, however, that symmetryrestoration provides strong enough constraints to recover a unique Hamiltonian. There-fore, the light-cone Hamiltonian with cutoffof the k+ = 0 region is a phenomenologicalconstruction that needs to be tuned order by order in perturbation theory.Symmetry restoration in “naive” light-cone perturbation theory has already been con-sidered[2][3] by Burkardt and Langnau.

In particular, in ref. [3], the non-covariant two loopterm that is added to light-cone perturbation theory is a k+ = 0 contribution.

In theiranalysis, they recovered this term by starting with the covariant momentum space ap-proach and correctly integrating over k−. I have argued here that light-cone field theoryis completely valid at the level of the Dyson perturbative expansion, and that neglectingk+ = 0 diagrams when integrating over x± is a source of similar problems.9

One may ask how these results can be applied to more physical models. Gauge theoriesquantized on the light-cone in light-cone gauge A−= 0 have an effective four point Fermicoupling which arises from the constraint equation for A+[10].

Because of its non-localnature, the vertex carries an extra 1/(k+)2. Therefore for QED as well as QCD, a potentialcontribution from the k+ = 0 region via the connected bubble type diagrams exists startingat two loop order.We have noted that the coordinate space expression for the light-cone vacuum en-ergy density that comes directly from the Dyson perturbative expansion is perfectly welldefined diagram by diagram, while the “equivalent” momentum space expression suffersfrom k+ = 0 singularities.

The mathematical source of this disparity is the integral overx−that generates the constraint δ(P k+). It might be worthwhile to try to regulate thisdistribution, and therefore alter the light-cone vertices, so as to make bubble diagrams welldefined in momentum space light-cone field theory.In principle, it might also be possible to build a sharp k+ cutoffinto the theory at theLangrangian level.

This is the approach suggested in ref. [1] in the context of discrete lightcone quantization[22].

It would be interesting to study how this approach can recover thek+ = 0 region of regular light-cone perturbation theory.Acknowledgements: This analysis was partially motivated by discussions with M. Burkardton Lorentz non-covariant counterterms in light-cone perturbation theory. I would also liketo thank D. Robertson for explaining the results of ref.

[1] to me, and C. Thorn for manyuseful comments regarding light-cone field theory.Note added: K. Hornbostel has brought to my attention ref. [23], in which the variationaltechnique of sec.

2 is applied to φ4 theory in two dimensions. Their analysis uses a light-cone momentum regularization scheme which, in the limit of infinite cutoffΛ, reduces tothe parity symmetric prescription advocated here.10

References[1]G. McCartor and D. Robertson, Z. Phys. C53 (1992)679.[2]M.

Burkardt and A. Langnau, Phys. Rev.

D44 (1991)1187.[3]M. Burkardt and A. Langnau, Phys.

Rev. D44 (1991)3857.[4]D.

Mustaki, S. Pinsky, J. Shigemitsu,K. Wilson, Phys.

Rev. D43 (1991)3411.[5]C.

Thorn, Phys. Rev.

D20 (1979)1934.[6]K. Hornbostel, Phys.

Rev. D45 (1992)3781.[7]F.

Lenz, S. Levit, M. Thies, and K. Yazaki, Ann. Phys.

208 (1990)1.[8]E. Prokvatilov and V. Franke, Yad.

Fiz. 49 (1989)1109.[9]S.

Chang and S. Ma, Phys. Rev.

180 (1969)1506.[10]J. Kogut and D. Soper, Phys.

Rev. D1 (1970)2901.[11]J.D.

Bjorken,J. Kogut, and D. Soper, Phys.

Rev. D3 (1971)1382.[12]S.

Chang, R. Root, and T. Yan, Phys. Rev.

D7 (1973)1133,1147.[13]S. Mandelstam, Nucl.

Phys. B213 (1983)149.[14]G.

Leibbrandt, Phys. Rev.

D29 (1984)1699.[15]S. Coleman, Phys.

Rev. D11 (1975)2088.[16]S.

Samuel, Phys. Rev.

D18 (1978)1916.[17]P. Minnhagen, A. Rosengren, and G. Grinstein, Phys.

Rev. B18(1978)1356.[18]D.

Amit, Y. Goldschmidt, and G. Grinstein, J. Phys. A13 (1980)585.[19]E.

Sklyanin, L. Takhtadzhyan, and L. Faddeev, Teor. Mat.

Fiz. 40 (1979)194.[20]V.A.

Miransky, Nuovo Cimento 90A (1985)149.[21]M. Gell-Mann, and F. Low, Phys.

Rev. 84(1951)350.[22]S.

Brodsky and H.C. Pauli, Phys. Rev.

D32 (1985)1993.[23]A. Harindranath and J.

Vary, Phys. Rev.

D37 (1988)3010.11

---

출처: arXiv:9207.082 • 원문 보기