Perturbative expansion in DLCQ of field and string theories

ELSEVIER

Perturbative expansion in DLCQ of field and string theories *

Satoshi Yamada a

Nuclear Physics B (Proc. Suppl.) 108 (2002) 118-122

PROCEEDINGS SUPPLEMENTS www.elsevier.com/locale/npe

aDepartment of Physics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan

We explore the perturbative aspects in discrete light-cone quantization (DLCQ) of field and string theories. In field theory, we develop a systematic DLCQ perturbation and establish the Feynman rules. From the Feynman rules, we show that DLCQ S-matrix does not have a covariant continuum limit for processes with p+ = 0 exchange. Furthermore, we study the relationship between Feynman rules in (conventional) DLCQ and those in DLCQ as a light-like limit (L3) and show that the difference between DLCQ and L3 is only the existence of the divergences by the zero-mode loop diagrams in L3, which are absent in DLCQ. In string theory, we study the scattering amplitudes in L3 and discuss that at any order in perturbation, the zero-mode loop divergences are absent even in L3. This result suggests that the scattering amplitudes in L3 string theory are well-defined and coincides with those in DLCQ string theory.

1. Introduction

Discrete light-cone quantization (DLCQ) was proposed [4] to solve the zero-mode problem

in light-front quantization. Later on a non-

perturbative calculation was carried out in some field theory [5]. Recently the idea played an important role in non-perturbative formulations of superstring or M-theory [6,7].

In DLCQ, the compactified coordinate is the light-like coordinate z-(2: Z- + 2L)2 and the corresponding momentum is quantized as p+ = (Nn)/L. The meaning of “DLCQ” is sometimes confusing. Here we use “DLCQ” as a quantization on the exact light-front surface (i.e., conventional DLCQ in Refs.[4] and [5]) and “Light-Like Limit (L3)” as a procedure to quantize on an al-

most light-front surface and then to take a limit, which makes the surface the exact light-front one

[8-101. DLCQ has a long history. In the original paper

[4], the light-like coordinate x- was compactified in order to isolate the zero modes on the light- front and it was shown that the zero modes are non-dynamical degrees of freedom, i.e., they are

*This talk is baaed on the papers [l-3] 21n this paper, we use a convention of the light-cone coor- dinates as, x@ = (z+,z-, zl), where x* = &(z” h CC~),

x~=(z~,~~~,~~~~). Andwetreatz+astimeandz-as the light-like space.

written in terms of the non-zero modes due to a nonlinear operator equation (zero-mode con-

straint), and hence they can be removed from the physical Fock space. Because of that, DLCQ

proves the popular statement that the light-cone vacuum is trivial.

From the very beginning of DLCQ, it was known that Poincarb invariance in DLCQ is not recovered as the operator algebra in the continuum limit (L + co) [4], and it was further shown

[ll] that Wightman function does not agree with the Lorentz-invariant result in the continuum limit of DLCQ (or any other regularization fixed

to an exact light-front), since the dynamical infor- mation of the zero mode is lost: Namely, DLCQ has no Lorentz-invariant continuum limit as field theory itself (operator, Hilbert space, etc.) [12]. However, it has been expected that the Lorentz- invariant continuum limit may be realized on the S-matrix even though the theory itself has no covariant limit [12]. Actually it was argued [13,14] that DLCQ S-matrix has a covariant continuum limit in the scalar field theory through perturbation. Unfortunately, scattering diagrams having exchange of p+ = 0 3 are not treated in [13], and

3The kinematics, where the light-cone momentum p+ is not exchanged, is very important in the context of Matrix theory (6,7]. In fact, as a first step, the tree level processes in this kinematics have been considered intensively,

0920-5632/02/$ - see front matter 0 2002 Elsevier Science B.V. All rights reserved.

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S. Yamada /Nuclear Physics B (Proc. Suppl.) 108 (2002) 118-122 119

they are missing in [14] and hence the statement

is not correct. In this paper, we study whether or not DLCQ S-matrix really has a covariant continuum limit and we find that DLCQ S-matrix elements for processes with p+ = 0 exchange do not have covariant continuum limits [2].

For this purpose, we develop a systematic

DLCQ perturbation theory in the operator formalism. It is well-known that by solving the zero- mode constraint perturbatively, we can see that the zero modes propagate schematically in internal lines in the Feynman diagrams [15,13,14]. We perform the perturbative expansion (Dyson expansion) including such diagrams precisely and establish the Feynman rules of DLCQ in scalar field theory.

L3 has been the subject for intensive research in the context of superstring or M-theory [1,17]. Since the zero modes are dynamical degrees of freedom in L3, it is obvious that the perturbative expansion in DLCQ is different from that in

L3. In particular, in scalar field theory, the diagrams with zero-mode loop(s) diverge pathologically in L3 [lo]. Here, ‘Lzero-mode loop” means a loop which consists of only the zero-mode propagators. Thus, the relationship between two sorts of DLCQ is not clear. However, in the context of Matrix theory [6,7], the relationship is very important. In fact, although finite N Matrix theory was originally proposed [7] as a model to describe DLCQ of M-theory, what was formally proved [9] is that finite N Matrix theory describes L3 of M- theory. Hence, in order to make clear whether finite N Matrix theory conjecture is correct or not, we need to give the precise relationship between DLCQ and L3. In this paper, we consider field theory first. In particular, we study the precise

relationship between the Feynman rules in DLCQ and those in L3 and show that the difference between DLCQ and L3 is only the existence of the pathological divergences by the zero-mode loop diagrams in L3, which are absent in DLCQ [3]. Next, we consider string theory. In that case, we study the scattering amplitudes in L3 and discuss that at any order in perturbation, the zero-mode

and the remarkable agreements between the potentials in finite N Matrix theory and those in (linear and non-linear) DLCQ supergravity have been reported [16].

loop divergences are absent even in L3 [l]. This

result suggests that the scattering amplitudes in

L3 string theory are well-defined and agree with those in DLCQ string theory.

2. Field theory

2.1. DLCQ perturbation theory Let us begin with formulating the perturbative

expansion (Dyson expansion) of scalar field theory in the operator formalism. We consider a d- dimensional X44 scalar field theory with the mass m and put the system in a light-like box (-L 5 x- < L) with periodic boundary conditions [4]. We decompose the scalar field 4(z) into the zero

mode 40@+, 21) (E (1/2L) J_“, qb(s)dz-) and

the non-zero mode parts q(z). Then the Hamil- tonian is given by

+&o(m’ - @_Mo + $(v + 401'1 . (1)

Furthermore, from the equation of motion for 40, we obtain the following zero-mode constraint [4],

L

J [ dx- Cm2 - a340 + g(p + 40)3] = 0. (2) -L

It is obvious that the zero mode is nothing but an auxiliary field having no kinetic term. Solv-

ing this zero-mode constraint perturbatively (i.e., the zero mode is written in terms of the non- zero modes perturbatively) and plugging the result into the Hamiltonian (l), we obtain the effective interaction terms for cp. Then using the effective Hamiltonian, we can perform the per-

turbative expansion of the vacuum and scattering amplitudes [15,14,2,3]. In the perturbative expansion, we treat the first term in the r.h.s. of

eq.(l) as the free part of the Hamiltonian and the rests as interactions.

First, classically or putting aside the operator ordering first, we can straightforwardly solve the

zero-mode constraint (2) perturbatively and we obtain

&J(x+,x*) = -ix J 3(Y) ddy Azy y

120 S. Yamada/Nuclear Physics B (Proc. Suppl.) 108 (2002) 118-122

+ ( -CQ2 J ddy ddz AEy !!?-- 7Y) yr V”(Z) 21 As - 3!

+ 0(X3), (3)

where AzV is the Feynman propagator for the zero

mode,

1 1 Ax9 - - 0 2iL m2 - 8,2,

x 6(x+ - y+)6(d-2)(x1 - y1).

Plugging eq.(3) into the Hamiltonian (l), we obtain the effective interaction Hamiltonian,

-i J

dx+Hint = -iA J

ddx(p4(x)

+k(-ix)’ J ddxddy (!?f$i;$&$)

The first three terms on the r.h.s. of eq.(4) cor- respond to the diagrams in Fig.1, respectively.

Figure 1. The effective interactions to the third order in X. The dotted lines represent the zero-

mode propagators and the solid lines represent the non-zero modes.

Now we comment on the operator ordering. In the ordinary quantization, the equal-time

commutator [$(z),$(y)],o+ is zero due to the micro-causality, while in the light-cone quantization, the equal light-cone time commutator

kf+;, cp(Y)lz+=y+ 7 in general, has a non-zero

. Thus we must fix the operator ordering in eq.(4). Here we choose the Weyl ordering for the equal light-cone time operators.

Having fixed the ordering of the equal light-cone time operators, we can perform

the perturbative expansion (Dyson expansion) with the effective interaction Hamilto-

nian (4) systematically. From the Schrijdinger equation of the light-cone time evolution,

i(a/&+)ls+) = Hint(&)(b) in the interaction picture, we obtain the S-operator, S =

2” exp [-i J-“, d&Hint (&)I , where T-symbol

means the light-cone time ordered product. Now that we have given the systematic DLCQ

perturbation theory, we can calculate the scattering amplitudes straightforwardly.

Figure 2. Examples of the scattering diagram of order X2.

From the effective interactions in Fig.1, it is easily seen that two diagrams in Fig.2 contribute to the one-loop scattering, where external momenta are given by pi = (pi, ngr/L,pil), (ni > 0, i = l;.- ,4), respectively. Actually we can

calculate each amplitude of these diagrams, and combine them to obtain the following scattering amplitude A,

A = kc’ J d;;;j::lF m2 _ ,,-; + (k1)2 1

1 X

mi _ 2k- Q--nz;ns)r + (j42 ' (5)

where

i-=k--p,+p,, in = kl -pzl +pu,

rn: = m2 - ic and Ci stands for the summation over 1 without the zero-mode loop contribution

(I = 0 and 1 - ns + ns = 0). Notice that although the zero modes can propagate schematically as internal lines due to the effective interactions in (4), we never have zero-mode loop diagrams.

2.2. DLCQ Feynman rules From a concrete example in the previous sub-

section, it is obvious that using the effective interaction Hamiltonian (l), which is obtained by

S. Yamada/Nuclear Physics B (Proc. Suppb) 108 (2002) 118-122 121

solving the zero-mode constraint (2), we can get

all the Feynman diagrams where only the zero-

mode loops are absent. Hence, we can deduce DLCQ Feynman rules as follows [3]:

(1)

(2)

(3)

(4)

2.3.

Write down the Feynman rules in the ordinary covariant quantization.

Replace the momentum variables k” and /cl

by k+ and k- ( k”,l + k* = w).

Put the system in a light-like box (-L 5

2- 5 L) with periodic boundary conditions, i.e., discretize the variable k+ (k+ + %) and change the integrals over k+ to the

summations over I (& J dk+ + T& Cl).

Remove the Feynman diagrams with zero-

mode loop(s).

Does DLCQ S-matrix have a covariant continuum limit?

We consider the continuum limit (L + cm) of

eq.(5). One may think that eq.(5) would agree with the covariant answer in the continuum limit

[14]. However, in DLCQ, the zero-mode loop contribution is absent in the summation, and it is not so straightforward. First we consider the n2 # n3 case in eq.(5). In this case, the zero- mode loop contribution is kinematically absent and XI’ = C1 from the outset. Hence, it is obvi-

ous that eq.(5) h as a covariant continuum limit. Next we consider the n2 = n3 case in eq.(5).

For clarity we consider in two dimensions (d = 2)

where the external momenta pi (i = 1, . . . ,4) sat- ,isfy pi = p4 and pz = ps in Fig.P(a), i.e., the forward scattering. Note that there is no p+- momentum exchange in this process and hence

the diagram of Fig.2(b) does not exist. From eq.(5) the amplitude AF is calculated as

since J dx (x 3~ if)- 2 = 0. Thus the scattering amplitude of this process is also zero in the continuum limit.

On the contrary, the covariant amplitude of the

same process AFV is given by

Ay = /~(m;_(&+(kl)2)2

This is non-zero and finite! Thus the continuum limit of the DLCQ S-matrix does not coincide with the covariant one in the specific kinematics where there is no p+ exchange. That is, the informations on the zero-mode loop diagram are lost in DLCQ and hence DLCQ S-matrix does not have a covariant continuum limit.

2.4. L3 Feynman rules In L3, it is obvious that the zero modes are

dynamical degrees of freedom and the zero-mode loop diagrams exist. Hence, the Feynman rules are summarized as follows [3]:

(1)

(2)

(3)

Write down the Feynman rules in the ordi-

nary covariant quantization.

Replace the momentum variables k” and k’

by k+ and k- (k”ll + k* = w).

Put the system in a light-like box (-L 5 X- 5 L) with periodic boundary conditions, i.e., discretize the variable k+ (k+ + g) and change the integrals over k+ to the summations over I (& J dk+ + & Cl).

Thus only the difference between Feynman rules in DLCQ and those in L3 is the zero-mode loop

diagrams in L3, which are absent in DLCQ. Now, in L3, we consider the same forward scat-

tering process in the previous subsection. Then

the zero-mode loop contribution of the forward scattering amplitude is given by

The integrand of eq.(8) is independent of k- and the k--integral volume is trivially divergent. This is in fact the divergence in Ref.[lO]. Thus L3 of field theory is pathological.

122 S. Yamada/Nuclear Physics B (Proc. Suppl.) 108 (2002) 118-122

3. String theory

We discuss DLCQ in string theory. Though the

continuum light-front formulation of string field theory is well-known [18], it is very difficult to for- mulate DLCQ string field theory, and we do not have a satisfactory formulation of DLCQ string theory. Hence, we cannot consider the precise relationship between DLCQ and L3 in string theory. However, from the above results in field theory,

we expect that also in string theory, the differ- ences between DLCQ and L3 are only the zero- mode loop divergences in L3. Hence, if the zero- mode loop divergences are absent in S-matrix of L3 string theory, we can expect that DLCQ S- matrix agrees with L3 S-matrix completely in

string theory and it has a covariant continuum limit. Actually, it was shown in Ref.[l] that in L3 the zero-mode loop divergences are absent in multi-loop string scattering amplitudes due to the

striking stringy nature. In field theory, we must add all possible Feynman diagrams in a certain order of perturbation and the zero-mode loop diagrams among them diverge pathologically because the integrands in zero-mode loop amplitudes are independent of the integral variables Ic- (See es.(s)). On the other hand, in string

theory, due to the s-t channel duality, a scattering amplitude in a certain order of perturbation contains the resonances in all the possible field theoretical Feynman diagrams in the same order of perturbation. Hence the integrands in string scattering amplitudes inevitably depend on the integral variables Ic-. From this reason, in string

theory, the zero-mode loop divergences are absent even in L3.

Acknowledgments

I am thankful to organizers for a stimulating

meeting. I also thank T. Sugihara, M. Taniguchi, S. Uehara and K. Yamawaki for useful discus-

sions .

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Perturbative expansion in DLCQ of field and string theories