Vacuum loops in light-front field theory

TL;DR

This paper shows that vacuum diagrams in light-front field theory are non-zero and explores their behavior, revealing subtle roles of zero modes and the connection to traditional equal-time field theory results.

Contribution

It demonstrates that vacuum diagrams in genuine light-front field theory are non-zero and analyzes their behavior using Hamiltonian perturbation theory and DLCQ, clarifying the role of zero modes.

Findings

Vacuum diagrams are non-zero despite simple kinematical arguments.

Vacuum amplitudes behave as $C\,\lambda^2\mu^{-2}$ in D=2 and diverge in D=4.

DLCQ analysis confirms vacuum amplitudes converge to continuum values.

Abstract

We demonstrate that vacuum diagrams in the genuine light front (LF) field theory are non-zero, in spite of simple kinematical counter-arguments (positivity and conservation of the LF momentum $p^+$, absence of Fourier zero mode). Using the light-front Hamiltonian (time-ordered) perturbation theory, the vacuum amplitudes in self-interacting scalar $\lambda\phi^3$ and $\lambda\phi^4$ models are obtained as $p=0$ limit of the associated self-energy diagrams, where $p$ is the external momentum. They behave as $C\lambda^2\mu^{-2}$ in D=2, with $\mu$ being the scalar-field mass, or diverge in D=4, in agreement with the usual "equal-time" form of field theory, and with the same value of the constant $C$. The simplest case of the vacuum bubble with two internal lines is analyzed in detail, displaying the subtle role of the small $k^+$ region and its connection to the $p=0$ limit. However, the vacuum bubbles in the genuine light-front field theory are non-vanishing not due to the Fourier mode carrying LF momentum $k^+=0$ (as is the case in the LF evaluation of the covariant Feynman diagrams), in full accord with the observation that the LF perturbation theory formula breaks down in the exact zero-mode case. This is made explicit using the DLCQ method - the discretized (finite-volume) version of the theory, where the light-front zero modes are manifestly absent, but the vacuum amplitudes still converge to their continuum-theory values with the increasing "harmonic resolution" K.

Tables

Table 1. Table 1: Smooth approach of the one-loop self-energy amplitude Σ 3 ​ ( p ) subscript Σ 3 𝑝 \Sigma_{3}(p) of the λ ​ ϕ 3 𝜆 superscript italic-ϕ 3 \lambda\phi^{3} model from Eq.( 22 ) to its (rescaled) value S = 1 − K − 1 𝑆 1 superscript 𝐾 1 S=1-K^{-1} at p = 0 𝑝 0 p=0 ( K 𝐾 K =512).

$p^2/{\mu }^2$	${10}^{-2}$	${10}^{-3}$	${10}^{-4}$	${10}^{-5}$	${10}^{-6}$	0
$S$	1.50902	1.09320	1.00667	0.99890	0.99813	0.99805

Table 2. Table 2: Convergence of the vacuum-loop constant C 𝐶 C from Eq.( 11 ) represented by the double sum of Eq.( 27 ) to its continuum value C = 2.343908 ​ … 𝐶 2.343908 … C=2.343908... with the resolution K 𝐾 K .

$K$	32	64	128	256	512	1024	2048
$C$	1.921	2.099	2.205	2.266	2.301	2.320	2.331

Equations

$$V_3(\mu )=N_4{\lambda }^2\int d^2{k}_1\int d^2{k}_{2}G(k_1)G(k_2)G(k_1+k_2),$$

$$G(k)=\frac{i}{k^2-{\mu }^2+iϵ},N_4=\frac{1}{3!}\frac{1}{i(2\pi {)}^4}.$$

$$V_3(\mu )=-\frac{i{\lambda }^2}{{\mu }^2}{\pi }^2{N}_{4}C,C=\mathrm{2.343908...},$$

$$C=\underset{0}{\overset{1}{\int }}d{x}_1\underset{0}{\overset{1-x_1}{\int }}d{x}_2\frac{1}{D(x_1,x_2)},$$

$$D(x_1,x_2)=x_1(1-x_1)+x_2(1-x_2)-x_1{x}_2.$$

$${\Sigma }_4(p^2)=N_4\underset{0}{\overset{1}{\int }}d{x}_1\underset{0}{\overset{1-x_1}{\int }}d{x}_2\frac{{\lambda }^2}{A(x_1,x_2)p^2-D(x_1,x_2){\mu }^2},$$

$$A(x_1,x_2)=x_1{x}_2(1-x_1-x_2).$$

$$\tilde{V}\sim \underset{0}{\overset{\infty }{\int }}\frac{d{k}_1^{+}}{k_1^{+}}\underset{0}{\overset{\infty }{\int }}\frac{d{k}_2^{+}}{k_2^{+}}\underset{0}{\overset{\infty }{\int }}\frac{d{k}_3^{+}}{k_3^{+}}\frac{\delta (k_1^{+}+k_2^{+}+k_3^{+})}{(-{\mu }^2)[\frac{1}{k_1^{+}}+\frac{1}{k_2^{+}}+\frac{1}{k_3^{+}}]}.$$

$${\Sigma }_4(p^2)={\tilde{N}}_4{\lambda }^2\underset{0}{\overset{p^{+}}{\int }}\frac{d{k}_1^{+}}{k_1^{+}}\underset{0}{\overset{p^{+}-k_1^{+}}{\int }}\frac{d{k}_2^{+}}{k_2^{+}(p^{+}-k_1^{+}-k_2^{+})}$$

$$\times \frac{1}{p^{-}-\frac{{\mu }^2}{k_1^{+}}-\frac{{\mu }^2}{k_2^{+}}-\frac{{\mu }^2}{p^{+}-k_1^{+}-k_2^{+}}+iϵ},{\tilde{N}}_4=\frac{1}{3!}\frac{1}{(4\pi {)}^2}.$$

$${\Sigma }_4(p^2)=\underset{0}{\overset{1}{\int }}\frac{dx}{x}\underset{0}{\overset{1-x}{\int }}\frac{dy}{y(1-x-y)}\frac{{\tilde{N}}_4{\lambda }^2}{p^2-\frac{{\mu }^2}{x}-\frac{{\mu }^2}{y}-\frac{{\mu }^2}{1-x-y}}.$$

$$F(x)=\frac{1}{{\mu }^2}\frac{4x}{\sqrt{3x^2-2x-1}}\arctan \frac{1-x}{\sqrt{3x^2-2x-1}}.$$

$${\Sigma }_4(0)={\lambda }^2{\tilde{N}}_4\underset{0}{\overset{1}{\int }}\frac{dx}{x}F(x)$$

$${\Sigma }_4(0)\equiv V_3(\mu )=-\frac{{\lambda }^2}{{\mu }^2}{\tilde{N}}_{4}C,C=2.343908\dots ,$$

$$-i{\Sigma }_3(p^2)=\frac{1}{2}\frac{(-i\lambda {)}^2}{(2\pi {)}^2}\int d^{2}kG(k)G(p-k),$$

$$V_2(\mu )=\frac{i}{16{\pi }^2}\underset{-\infty }{\overset{+\infty }{\int }}d{k}^{+}\underset{-\infty }{\overset{+\infty }{\int }}d{k}^{-}\frac{1}{(k^{+}{k}^{-}-{\mu }^2+iϵ{)}^2}.$$

$$V_{2}(\mu)=\tilde{N}_{3}\int\limits_{-\infty}^{+\infty}\frac{dk^{+}}{k^{+}}\Big{[}\frac{1}{\Lambda k^{+}-\mu^{2}+i\epsilon}-\frac{1}{-\Lambda k^{+}-\mu^{2}+i\epsilon}\Big{]}.$$

$$V_{2}(\mu)=\tilde{N}_{3}\frac{\Lambda}{\mu^{2}}\int\limits_{-\infty}^{+\infty}dk^{+}\Big{[}\frac{1}{\Lambda k^{+}-\mu^{2}+i\epsilon}-\frac{1}{\Lambda k^{+}+\mu^{2}-i\epsilon}\Big{]}.$$

$$\displaystyle V_{2}(\mu)=\frac{\tilde{N}_{3}}{\mu^{2}}\int\limits_{-\infty}^{+\infty}dk^{+}\Big{[}\frac{1}{k^{+}+i\epsilon}-\frac{1}{k^{+}-i\epsilon}\Big{]}=$$

$$=\frac{{\tilde{N}}_3}{{\mu }^2}(-2\pi i)\underset{-\infty }{\overset{+\infty }{\int }}d{k}^{+}\delta (k^{+})=-\frac{1}{8\pi }\frac{1}{{\mu }^2}.$$

$${\Sigma }_3(p)=\frac{1}{8\pi }\underset{0}{\overset{p^{+}}{\int }}\frac{d{k}^{+}}{k^{+}(p^{+}-k^{+})}\frac{1}{p^{-}-\frac{{\mu }^2}{k^{+}}-\frac{{\mu }^2}{p^{+}-k^{+}}+iϵ}.$$

$${\Sigma }_3(p)=\frac{1}{8\pi }\underset{0}{\overset{1}{\int }}\frac{dx}{p^{2}x(1-x)-{\mu }^2+iϵ}.$$

$${\Sigma }_3(\eta )=\frac{1}{8\pi }\underset{0}{\overset{\eta }{\int }}\frac{d{k}^{+}}{k^{+}(\eta -k^{+})}\frac{1}{\eta -\frac{{\mu }^2}{k^{+}}-\frac{{\mu }^2}{\eta -k^{+}}+iϵ}.$$

$${\Sigma }_3(\eta )$$

$$G(k)$$

$$\Sigma_{3}(\eta)=-\frac{1}{8\pi}\frac{1}{\mu^{2}}\Big{[}1+\frac{\eta^{2}}{4\mu^{2}}+{\cal O}(\eta^{4})\Big{]}.$$

$${\Sigma }_3(p^2)=\frac{i}{16{\pi }^2}\underset{-\infty }{\overset{+\infty }{\int }}d{k}^{+}\underset{-\infty }{\overset{+\infty }{\int }}d{k}^{-}\frac{1}{(k^{+}{k}^{-}-{\mu }^2+iϵ)}$$

$$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\frac{1}{\big{(}(p^{+}-k^{+})(p^{-}-k^{-})-\mu^{2}+i\epsilon\big{)}},$$

$$\phi(0,x^{-})=\frac{1}{\sqrt{2L}}\sum_{n}^{\infty}\frac{1}{\sqrt{k^{+}_{n}}}\big{[}a_{n}e^{-ik^{+}_{n}x^{-}}+a^{\dagger}_{n}e^{ik^{+}_{n}x^{-}}\big{]},$$

$$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\Sigma_{3}(p)={\cal N}_{3}\sum_{k^{+}}^{p^{+}}\frac{1}{k^{+}(p^{+}-k^{+})}\frac{1}{\big{[}p^{-}-\frac{\mu^{2}}{k^{+}}-\frac{\mu^{2}}{p^{+}-k^{+}}\big{]}},$$

$$p^{+}=\frac{2\pi }{L}K,k^{+}\equiv k_n^{+}=\frac{2\pi }{L}n,n=1,2,\dots ,K-1.$$

$$V_2(\mu )=-\frac{1}{8\pi {\mu }^2}S,$$

$$\displaystyle S=\sum_{n=1}^{K-1}\frac{1}{n(K-n)}\frac{1}{\big{[}\frac{1}{n}+\frac{1}{K-n}\big{]}}=\frac{K-1}{K}.$$

$${\Sigma }_4(p)={\lambda }^2{\mathcal{N}}_4\sum _{q^{+}}^{p^{+}}\sum _{k^{+}}^{p^{+}-q^{+}}\frac{1}{k^{+}{q}^{+}(p^{+}-k^{+}-q^{+})}$$

$$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\frac{1}{\big{[}p^{-}-\frac{\mu^{2}}{k^{+}}-\frac{\mu^{2}}{q^{+}}-\frac{\mu^{2}}{p^{+}-k^{+}-q^{+}}\big{]}},$$

Full text

• the discretized (finite-volume) version of the theory, where the light-front zero modes are manifestly absent, but the vacuum amplitudes still converge to their continuum-theory values with the increasing ”harmonic resolution” K.

I Introduction

Quantum field theory (QFT) formulated in terms of light-front (LF) variables Dir ; KS ; ChMa ; LKS ; BPP has a few unusual features which are sometimes interpreted as indicating its inconsistency or incompleteness McCR ; Burk . One problem appeared to be paradoxically related to the most celebrated property of the LF quantization, namely vacuum simplicity. As is well known, positivity of the LF momentum $p^{+}$ together with its conservation implies that the ground state of any interacting model cannot contain quanta carrying $p^{+}\neq 0$. Only a tiny subset of all field modes, namely those carrying $p^{+}=0$ - the dynamical LF zero modes (ZMs) - can contribute. In addition, some modes (like the scalar-field zero mode) which appear as dynamical ones in the conventional (”space-like”, SL for short) theory become constrained, that is, non-dynamical, in the LF form of the interacting theory (or vanish in the free theory), and hence cannot contribute to vacuum processes directly. The subtle role of the constrained zero modes has been known since the work of Maskawa and Yamawaki MY , who analyzed the self-interacting scalar models using the discretized light-cone quantization (DLCQ), that is a finite-volume formulation with fields satisfying (anti)periodic boundary conditions in space variables. A natural question is if the LF formalism with its ”trivial” vacuum is then capable to describe phenomena, related to nontrivial vacuum structure, like spontaneous symmetry breaking. More generally, is the LF form of QFT equivalent to the usual SL one, can it reproduce the known results or even predict some truly new insights?

The equivalence issue has been realized and studied by Chang and Ma and by T.-M. Yan already in the pioneering LF papers ChMa ; Yan , including the vacuum problem at the level of the perturbation theory (i.e., the vacuum bubbles). Here one should distinguish two methods. In the first method, one starts from the covariant Feynman amplitudes and rewrites the corresponding integrals in terms of LF variables $k^{\pm}=k^{0}\pm k^{3}$. The delicate step is to integrate over the $k^{-}$ variable (using the contour integration and the residue theorem) since the propagators (in two dimensions) behave as $(k^{+}k^{-}-m^{2}+i\epsilon)^{-1}$ instead of $(k_{0}^{2}-k_{1}^{2}-m^{2}+i\epsilon)^{-1}$, so there arises the convergence issue. Moreover, vacuum loop diagrams (diagrams with no external lines) seemed to vanish altogether when calculated in terms of LF variables. However, when the naive evaluation was replaced by a careful mathematical treatment ChMa ; Yan , these vacuum amplitudes were found to be nonzero. The missing contribution was identified as coming from the $k^{+}=0$ field mode due to the delta function $\delta(k^{+})$ resulting from the $k^{-}$ integration ChMa ; Yan . Later on, this kind of zero modes turned out to be equally important in the case of non-vacuum LF Feynman amplitudes of various precesses related to LF wavefuctions, electroweak currents and formfactors melost ; brohw ; bakji .

The $k^{+}=0$ modes have been also found to give non-negligible contributions in model studies of pair creation tomrs ; ild , in the LF analysis of the effective potential hein1 ; hein2 and in the LF path integral formulation hein2 . It was shown in tanigy1 ; tanigy2 that when the ZM was missed in the discretized evaluation of the Feynman integrals expressed in terms of the LF variables (summation in $k^{+}$, integration in $k^{-}$), the forward scattering amplitude was not correctly reproduced in the continuum limit. This violation of Lorentz invariance is actually an artifact of the method used, however. In the genuine (non-manifestly covariant) DLCQ perturbation theory, where $k^{-}$ integration is not present (as is not the $k^{+}=0$ Fourier mode MY ), the limit of the forward-scattering amplitude is reached smoothly to an arbitrary precision with the increased value of the resolution parameter $K$ (i.e., in the continuum limit) lmhava .

The recent study mannh of the ”vacuum-sector diagrams”, concentrating on the tadpole graph of the four-dimensional scalar field and claiming failure of the LF Hamiltonian method due to the intrinsic impossibility to capture the contribution of an arch at infinity, deserves a separate discussion lmannh .

So far we have mostly mentioned the applications of the first method of computation of LF amplitudes, based on Feynman diagram technique. The second method is the LF perturbation theory (LFPT), which is not manifestly covariant and uses energy denominators instead of Feynman propagators. The integration over the $k^{-}$ variable is not present by construction. The same simple kinematical argument (positivity and conservation of $p^{+}$) suggests that vacuum bubbles are zero in LFPT. A vacuum diagram can be non-vanishing only if all internal lines carry $k^{+}=0$, but the rules of LFPT give an ill-defined result in this case Yan (see the formula (6) below.) Thus, it has not been clear for a long time how the vacuum amplitudes can be correctly evaluated in the LF perturbation theory. They have often been considered as vanishing. One implication was that the cosmological constant problem could be solved in this way bros . If true, this would mean that the conventional SL form and the LF form of the relativistic dynamics are not equivalent, since the vacuum loops diverge (logarithmically or quadratically) in the SL theory in D=3+1.

Recently, J. Collins drew attention to this contradiction coll . He used an example with the simplest vacuum diagram (two propagators) to identify the mathematical inconsistency present in the LF computation of the corresponding integrals. He also indicated an alternative approach, based on analyticity, to calculate the LF vacuum amplitudes as the limit of vanishing external momentum of the associated self-energy diagrams.

In the present letter, we extend this argument to the vacuum bubbles of two-dimensional $\phi^{3}$ and $\phi^{4}$ models. We show that the self-energy diagrams, when represented in the LF perturbation theory, reduce for external momentum $p=0$ to nonzero limits equal to the values of vacuum bubbles known from the conventional Feynman diagrams. For example, the value of the vacuum loop of the $\lambda\phi^{3}$ model is given by the $p=0$ limit of the two-loop self-energy of the $\lambda\phi^{4}$ model. Moreover, the non-vanishing of LF vacuum loops is not due to the Fourier mode carrying $k^{+}=0$: same results as in the continuum theory are obtained in the DLCQ framework, namely in the finite-volume formulation with (anti)periodic boundary conditions, where the zero mode is manifestly absent. In this approach, the spectrum of field modes is discrete, enumerated by non-negative integers $n,1\leq n\leq K-1$, where $K$ is called the ”harmonic resolution”. We show that although the Fourier modes with $n=0$ are not present, the corresponding summations (which replace integrals of the continuum theory) still converge to the same values of the vacuum amplitudes for increasing values of $K$. This adds further evidence to the attitude that the LF form of QFT is fully equivalent to the usual SL one. However, the mechanisms and mathematical appearance may be different, and the LF theory requires fresh and independent thinking.

II Vacuum amplitudes in the conventional and light-front theory

In the language of perturbation theory, the non-trivial structure of the ground state in relativistic QFT manifests itself by vacuum-polarization amplitudes (diagrams). In the case of the two-dimensional self-interacting $\lambda\phi^{3}$ and $\lambda\phi^{4}$ theories, these diagrams are shown in Fig. 1. Such a process of creation of two or more particles from the vacuum does not violate the momentum conservation in the SL theory since $p^{1}$ can acquire both the positive and negative values which add to zero (momenum conservation). Once again, this seems impossible in the LF case where $p^{+}$ values carried by the two quanta are strictly positive (or vanishing if there exist dynamical LF zero modes in the considered model).

Let us remind briefly how the vacuum amplitudes are calculated in the SL form with the example of the $\phi^{3}$ bubble (see Fig. 1(b)). The corresponding Feynman rules lead to the double two-dimensional integral expression

[TABLE]

The coefficient $1/3!$ is the symmetry factor. The double integral (1) can be evaluated in a few ways: by using the Feynman parameters fey , by means of $\alpha$-representation smi or via more sophisticated mathematical methods (Mellin-Barnes representation for powers of massive propagators datau ). All of them yield the same result

[TABLE]

where the constant $C$ has a particular representation in each of the computational method. For example, the first method requires to combine the propagators into one denominator by means of the auxiliary integrals in terms of the Feynman parameters $x_{i}$, then to go over to Euclidean space and calculate the integrals in $k_{1}$ and $k_{2}$ variables. The result is the double-integral representation

[TABLE]

The first integral can be calculated analytically in terms of $arctan$ function and square roots of polynomials, the numerical evaluation of the second integral then yields the above value of C.

If we consider the self-energy diagram instead of the vacuum bubble ($G(k_{1}+k_{2})$ replaced by $G(p-k_{1}-k_{2})$ in Eq.(1)), the analogous calculation yields

[TABLE]

Our goal now is to demonstrate that the result (3) can be obtained also in the LF perturbation theory, contrary to the general belief. Let us recall that in the LFPT Weinb ; ChMa ; KS ; LB , no $T$-product is introduced. The integrals over the LF time variables are explicitly performed in the iterative solution of the equation for the $S$-matrix in the interaction representation. As the result, the particles in the intermediate states are on mass-shell with the usual Feynman propagators replaced by the LF energy denominators. The loss of manifest covariance is of no harm and the advantage is that one does not need to perform extra integration over the LF energy in the perturbative amplitudes. However, when the LFPT rules are applied to vacuum diagrams, the result is ill-defined Yan , as the delta function requires all three momenta to be equal to zero:

[TABLE]

It is actually not very difficult to resolve this inconsistency. One simply has to start from the graph with nonvanishing external momentum $p$ and to compute the $p=0$ limit in the corresponding integrals. The assumed analyticity in $p$ guarantees existence of this limit which should be equal to the vacuum amplitude. In this way, the expression (6) is first replaced by the self-energy foot

[TABLE]

Introducing the dimensionless variables $x=\frac{k_{1}^{+}}{p^{+}}$, $y=\frac{k_{2}^{+}}{p^{+}}$, $\Sigma_{4}(p)$ becomes

[TABLE]

Now we can set $p=0$ foota . The integral over the variable $y$ can be performed explicitly, yielding

[TABLE]

The numerical computation of the second integral

[TABLE]

gives

[TABLE]

in the complete agreement with the space-like result (3). The overall situation is in fact very simple. Multiplying out the terms in the denominator of Eq.(8) for $p=0$, we find that the corresponding double integral is precisely equal to the representation of the constant $C$ in Eq.(4). The LF and SL schemes match at this point. The only difference is that in the SL theory one can start directly with the vacuum diagram while in the LF case one has to consider the associated self-energy diagram first. Its $p=0$ value can be taken after changing the LF momenta to relative variables which makes the integrand manifestly covariant, that is depending symmetrically on both $p^{+}$ and $p^{-}$. Obviously, $\Sigma_{4}(p^{2})$ is also the same in the both schemes, the difference being that the LF scheme needs for that just two steps while the conventional Feynman procedure is considerably lengthier.

III The simplest diagram with two internal lines

The above result can be understood in more detail in a simpler case of the one-loop self-energy diagram in the $\lambda\phi^{3}$ theory (see Fig. 1(a) with two external lines attached). The corresponding Feynman amplitude is

[TABLE]

The vacuum bubble ($p=0,\lambda=1$) rewritten in terms of the LF variables takes the form

[TABLE]

To correctly evaluate the integral over $k^{-}$, one has to impose a cutoff $\Lambda$ Yan , leading to ($\tilde{N}_{3}=-i/16\pi^{2}$)

[TABLE]

This integral can be equivalently rewritten as Yan

[TABLE]

For $\Lambda\rightarrow\infty$ we get

[TABLE]

This result coincides with the one obtained in the usual SL calculation as well as with the direct LF computation coll ; lmh . The same diagram, being the simplest one, sheds light upon the mechanism at work in the genuine LF case. The LFPT formula for the self-energy $\Sigma_{3}$ (12) is ($\lambda=1$)

[TABLE]

Going over to the variable $x=k^{+}/p^{+}$, one finds

[TABLE]

For $p=0$ one indeed easily reproduces $V_{2}(\mu)$ of (14) (see Fig. 1(a)). Alternatively, we can work directly with the form (15). Taking $p^{+}=p^{-}=\eta$ for simplicity, we have

[TABLE]

The integral can be evaluated exactly with the result

[TABLE]

The expansion for infinitesimal $\eta$ gives

[TABLE]

One can see that the correct result is recovered for $\eta=0$. The technical reason is simple: the integrand in (17) is $\eta^{-1}[k^{+}(\eta-k^{+})-\mu^{2}]^{-1}$. For very small $\eta$ the expression in the brackets has almost a constant value very close to $(-\mu^{2})$ at the interval $(0,\eta)$, while the diverging $\eta^{-1}$ factor is canceled by the length $\eta$ of the integration domain. However, setting $\eta=0$ from very beginning as in the formula (6) yields the wrong (ill-defined) result. Note also that the change of variables $x=k^{+}/\eta$ transforms the integral (17) to the form (16) with $p^{2}$ replaced by $\eta^{2}$, displaying clearly the relation between the small $k^{+}$ region and the $p\rightarrow 0$ limit.

The above calculation can be confirmed in yet another way, using the limiting approach instead of Yan’s cutoff in the Feynman amplitude (13). In the self-energy (12) rewritten in terms of the LF variables,

[TABLE]

integration over $k^{-}$ can be performed using the method of residue, because with $p\neq 0$ the poles are at a finite distance from the origin. The result exactly matches the LFPT amplitude (15) and hence its value at $p=0$ obviously yields the correct value of the vacuum amplitude.

IV Calculations in a finite volume: the DLCQ method

It is remarkable that the same result for the LF vacuum bubble is obtained in the discretized (finite-volume) treatment with (anti)periodic boundary conditions (BC). In both cases, the field mode carrying $k^{+}=0$ is manifestly absent. The field expansion at $x^{+}=0$ is

[TABLE]

where $k^{+}_{n}=2\pi n/L$ and $L$ is the box lenght. The index $n$ runs over half-integers for antiperiodic BC, so there is no zero mode by construction, and over integers for periodic BC. In this case the term with $n=0$ is excluded because the field equation in the ZM sector $\mu^{2}\phi_{0}=0$ requires the field mode $\phi_{0}\equiv\phi(k^{+}=0)$ to vanish for $\mu\neq 0$.

The DLCQ analog of the $\Sigma_{3}(p)$ amplitude is

[TABLE]

${\cal N}_{3}$ is the normalization constant. For $p=0$, choosing periodic BC, $\Sigma_{3}(0)=V_{2}(\mu)$ becomes

[TABLE]

For $K\rightarrow\infty$, $S$ obviously converges to the continuum value $1$. The same result is obtained for the antiperiodic BC lmd . In Table I, the smooth approach of the self-energy value to the vacuum-loop value as $p\rightarrow 0$ is shown.

The vacuum bubble with three internal lines can be computed in the same way. The discrete version of the self-energy (8) is

[TABLE]

${\cal N}_{4}$ is a normalization constant. For $p=0$, choosing periodic BC, $\Sigma_{4}$ becomes

[TABLE]

Numerical values of the double sum in $\Sigma_{4}(0)$ for increasing values of $K$ are shown in Table II. They readily approach the continuum value (11) of the constant $C$.

The same pattern is valid also for the LF vacuum loops with four and more internal lines lmd . This may be of interest for the LF solution of the sine-Gordon model Grif ; Burk .

V Conclusions

We have demonstrated that the vacuum amplitudes are non-zero in LF perturbation theory and match the known values. The calculations were performed within the self-interacting scalar models in two dimensions. Generalization of these results to higher dimensions where the vacuum loops diverge, is straightforward. One can also consider vacuum diagrams with internal lines corresponding to fields with different masses (the Yukawa model, e.g.).

Our results have been obtained first in the usual continuum form of field theory. Switching to the discretized (finite-volume) formulation, the values of vacuum amplitudes were shown to be reproduced for the resolution parameter $K$ tending to infinity (i.e., in the infinite-volume limit). As the scalar zero modes are manifestly absent in the finite-volume treatment with (anti)periodic boundary conditions, the non-vanishing of vacuum bubbles is solely due to non-zero field modes. However, to see this, one has to start from the associated diagram with non-zero incoming momentum $p$ and take the limit $p\rightarrow 0$ at the end of the calculation. Starting with $p=0$ leads to mathematically ill-defined amplitudes like Eq.(6). Let us mention here that the limiting approach of the present work was used previously in the context of the LF scattering amplitudes lmhava computed in DLCQ. We note also that the LF momentum is conserved in the diagram vertices for arbitrarily small value of $p$ and hence in the limiting sense also for $p=0$. This observation is reinforced by the fact that the LF self-energy diagrams when re-expressed in terms of relative variables for $p=0$ exactly match the covariant vacuum amplitudes expressed in terms of Feynman parameters. In view of this it is not too surprising that the DLCQ evaluation gives the correct answers, as it is just a discrete approximation to the corresponding integrals.

On the other hand, as emphasized in coll , non-vanishing of vacuum bubbles in the genuine LF theory does not spoil its fundamental property, namely the possibility to construct a consistent Fock expansion for composite systems with well-defined coefficients - the LF wave functions. The crucial feature that distinguishes the LF formulation from the conventional SL one is that in the former, creation and annihilation operators can be consistently defined even in the interaction case, so that the vacuum-annihilation condition exists for an interacting theory in the Heisenberg picture LKS ; yam98 . The ultimate reason for this is the positivity of the spectrum of a kinematical quantity - the LF momentum $P^{+}$, in addition to positivity of the energy $P^{-}$.

In summary, while in the evaluation of the covariant vacuum Feynman diagrams in terms of the LF variables for the self-interacting scalar models the non-zero values of the vacuum amplitudes are due to the sharp $k^{+}=0$ contribution (with $k^{+}$ being the loop variable), the situation in the genuine LF theory is somewhat different. The Fourier mode carrying $k^{+}=0$ is not present in the LF perturbation theory but the correct results are nevertheless obtained if one uses a regularized approach, which adequatly treats the integration region close to $k^{+}=0$. Passing to the relative LF variables (which coincide with the Feynman parameters) maps this picture to the $p=0$ limit of the corresponding self-energy Feynman diagrams.

VI Acknowledgements

The authors are grateful to V. Karmanov for many useful discussions. LM thanks the Slovak Committee for the collaboration with JINR for a long-term support.