SUMMARY AND DISCUSSIONS

Our results of perturbative RG in the four dimensional real scalar mod l ar ummarized as follows.

(i) The present RG scheme based on the Bloch-Horowitz effective Hamiltonian is quite practical, because the scheme is free from the "vanishing energy d nominator" problem and the

Ei

dependence of the effective Hamiltonian causes no big problem in practic . Whil RG equations are easily derived in perturbation theory, RG flows d pend on eig n nergics

Ei

of

HAo,

since so does the effective Hamiltonian. The state dependence is negligibl for

A

much larger than

Mi.

(ii) Renormalization group (RG) equations for mass

r

and coupling

A

ar derived at on -loop order, where all irrelevant operators generated by RG transformation are remov d as a reasonable approximation. The invariant mass regularization [12,21] is adopted in this paper, but it breaks covariance and cluster property, so the running mass and coupling constant depend on momenta of spectators if they exist. The dependence is, however, very weak, as long as

_A

is at least several times larger than -/T ^and

Mi

which are assumed to be of order the physical mass scale

Aphys·

So it is neglected as a reasonable approximation in this paper. The regularization also excludes the zero-mode (a mode with zero longitudinal momentum) from the canonical Hamiltonian. The zero-mode is responsible for spontaneous symmetry breaking, that is, the order parameter (OI¢10) for the

Z2

symmetry never becomes nonzero without the mode. This means that the RG equations calculated with the initial cutoff Hamiltonian are correct only for the symmetric phase. In fact, a flow diagram drawn with the RG equations shows not only that two phases exist, but also that tachyons come out in the broken phase.

(iii) In light-front field theory, Hamiltonians are different between the two phases, while

their vacua are always trivial. Result (ii) indicates that for the symmetric phase the initial Hamiltonian HAo is obtained just by removing the zero-mode from th canonical Hamiltonian with an appropriate regularization. A problem is how to constru t another H Ao valid for the broken phase. Once the zero-mode is switched off, the syst m has to sit in th bottom of the effective potential. We then have to shift ¢as ¢(x) ⁼ cp(x)

+ v

in the initial Hamiltonian for the symmetric phase. Once a renormalization trajectory is found,

v

is d tcrmin d as a function of

A

on the trajectory.

(iv) The RG equations for

r, A

^and

v

calculated in the present framework arc compared with those in the covariant perturbation theory. For

A

>>

Aphys,

^{both agr} with each other, except for the following point. In the covariant theory a contribution of the tadpole (Fig.

1)

is explicitly calculated and then present in the RG equations. On th other hand, the contribution is removed in our normal ordered HAo, so it does not appear in our RG equations explicitly.

(v) At the one-loop level, we find by using the present RG equations that

6r(A)+A(A)v2(A)

^is

invariant for any

A

and by calculating the effective potential that the RG invariant quantity is zero when the system sits in the bottom of the potential. The resultant relation

6r(A) + A(A)v2(A)

⁼ 0 is thus a condition for the system to be in the bottom, and the renormalization trajectory should satisfy the relation. This is confirmed. A way of finding the relation without calculating the effective potential is furthermore presented.

Throughout all the results, we conclude that the present perturbative RG scheme is quite practical and valid at least near a Gaussian fixed point. This method is applicable for both the symmetric and broken phases. Application of the present method to QCD may not be straightforward, since it is known in the equal-time field theory that the QCD vacuum is much more complicated than that of the present model. Further study on how to construct HAo in the QCD case is highly expected.

According to perturbative analysis summarized above, we have shown how to renor­

malize the Hamiltonian nonperturbatively for the three-dimensional real scalar model and calculated the critical line and the critical surface. The results are as follows.

(i) There exist a region where the canonical Hamiltonian is unstable independently of per­

turbation. The mass spectrum is tachyonic in the left region of the critical lin . We have then introduced VEV to restore instability of the Hamiltonian and calculated th ritical surface. The asymmetric Hamiltonian on the surface describes physics for the brok n phase, if the phase has a massless mode. This is precisely true for SSB of any ontinuous ymm -try, and the present approach for finding the broken phas is applicable for the case. The bosonic calculation suggests that if we consider a fermionic theory and renormalize th mass term mif;'ljJ there seems to exist a region where the canonical Hamiltonian is unstable. It is reasonable to assume that the instability will be restored by introducing VEV of a campo ite bosonic field relevant to the symmetry.

(ii) The marginal coupling dependence of the critical line is weak. This r sult links to the

fact that Tamm-Dancoff approximation up to three-body state works well. The ES equation

-

-for 'ljJ1 has no marginal interaction, but the ES equation for 'ljJ3 has that. The ground state has marginal coupling dependence only through the three-body wavefunction {/;3 whi h is extremely small as a result of nonperturbative calculation. This is the reason why the marginal coupling dependence of the ground state is weak.

It is a problem whether all states are renormalized with same RG trajectory. This nonperturbative RG program seems to be energy independent. In order to check this more concretely, we need more than two bound states. For example, we can use spin singlet and degenerate triplet states in a fermionic theory such as Yukawa model. It is possible to draw RG flows by using a spectrum of one-body fermionic state in a similar way which has been done in this paper. We can say that the RG is energy independent if the spectra of the spin singlet and triplet states keep each own eigenmass on the same RG flow.

ACKNOWLEDGMENTS

I would like to acknowledge stimulating discussions with our colleagues in Kyushu Uni-versity, in particular K. Harada, K. Inoue , T. Kashiwa, Y. Masanobu, and S. Tominaga.

APPENDIX A: LOOP INTEGRALS

The loop integral A is easily performed by introducing th Jacobi coordinate

(x, s)

defined with

k1

=

(xP+, xP

1_ +

s), k2

=

((1- x)P+, (1- x)P

1_-

s).

The result is A ₌

JA(An)- IA(An+d

^with

IA(A)

=

2 j ^dxds B(x(1- x)A2- s2- r) x(1 - x )M2 - s2 - r

=27rln

₁ l -,fJJ0 -21rJa(M)In

₊

_,fJj0 ,fJ!0-;;Jji) _,fJj0

₊

ja(M) B(a(M))

+ 47r

V -a(M)

^arctan

{ _{J,fJJ0 a(-M)} } ^{e( -a(M)) (}

^A1

⁾

for

A� 2yr

and zero for

A< 2yr,

^where

a(z)

⁼

^{1- 4r/z2.}

The loop integral B is also given in a similar fashion. As an example let us consider the

diagram of Fig. 3, in which particles

1

and

2

interact with each other, while particl 3 is free. Each has a momentum

ki

in the initial state and

k�

in the intermediate state. For convenience we introduce the Jacobi coordinates

k1

=

(x(1- y)P+, x[(1- y)P

1_-

t]

⁺

s),

k2

=

( ( 1 -

X

) ( 1 - y) p+, ( 1 -

X

) [ ( 1 - y)

^p 1_

- t] -

S

)

,

(A2)

The momentum

( x, s)

represents a relative motion between interacting two particles in the initial state, and

( x', s')

the motion in the intermediate state. On the other hand,

(y, t)

is a conserved momentum describing a relative motion between the interacting pair and the third particle. The quantity B is obtained by making integrations over

x'

and s', so it is eventually given as a function of

y

and

t:

^B =

IB(An-1)- IB(An)

with

IB(A)

=

21r In l ₁ ^-

₊

Jb{i0 Jb{i0

^-

21rJb(M) I

ⁿ

Jb{i0 Jb{i0- Jb{ii5 B(b(M))

⁺

jb(M)

{ ^Jb(i0 }

+

47rJ -b(M)

^arctan

J -b(M) B( -b(M)) ⁽

^A3

⁾

for

A;:::: M

and zero for

A

<

M,

where

t2

=

ti + t�

and

_

(z2-M2)(1-y)

b(z)

⁼

(z2 -M2)(1 -y) + (2y'r)2' M 2

^{_ =}

1-y y y1-y -

^--

4r + - + ( r t2 )

^>

9r. (A4)

The invariant mass M of the initial state is smaller than

An_1,

sine the state is in the P space. The mass is related to

M

^as

r + s2 r t2

M

2

⁼

-(1-y)x(1-x) y y(1-y) + - + ;:::: M2 ;:::: 9r, (A5)

where use has been made of the inequality

( r+s2)/( x(1-x));:::: 4r.

Equation

(A5)

indicates that

M

is somewhere between

3y'r

^{and M.} TheM is conserved during the process Fig.

3,

since it is a function of the conserved momentum

(

^y,t

)

. In Fig.

4,

^B is drawn as a function of

M2

and

y (

instead of

t2

and

y)

and compared with A. For

An

⁼

100Aphys,

in Fig. 4

(

^a

)

^,

B agrees with A at all

M

and

y.

The agreement becomes poor gradually as

A

decreases to

2 y'r

. For example, for

An

⁼

10Aphys

^{in Fig.}

4(b),

^B is close to A at

M

^<

0.8An,

^but

undershoots A for

0.9An

^<

M

^<

An,

indicating that B can be approximated into A for any initial state with M less than

0.8An.

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