Nontriviality of the scattering matrix for weakly coupled Φ34 models

ANNALS

OF PHYSICS

Nontriviality

108,

37-48 (1977)

of the Scattering FLORIN

Lyman

Laboratory

of Physics,

Matrix

for Weakly

Coupled

@,4 Models

CONSTANTINESCU * ’

Harrard

University,

Cambridge,

Massachusetts

02138

ReceivedMarch 7, 1977

It is shownthat the weakly coupled,three space-timedimensions,bosonquantumfield-theorymodelwith D4self-interaction hasa scatteringmatrix whichis differentfrom 1.

1. INTR~D~CTION We consider the three space-time dimensions, boson quantum field model with a weak q4 self-interaction. Using techniques developed by Glimm and Jaffe [I], both Feldman and Osterwalder and Magnen and SCnCor [2, 31 proved that this model exists and satisfies all Wightman axioms, including a strictly positive mass gap. Burnap [4] proved the existence of an isolated one-particle hyperboloid; hence by Haag-Ruelle theory [5, 61, it has a well-defined scattering matrix. S-matrix elements

can be obtained from the LSZ reduction formulas in the form given by Hepp [7], at least for nonoverlapping in-going and out-going velocities, by amputating the time-ordered Green’s functions and restricting them to the massshell. We show that the scattering matrix of this model (for small coupling) is nontrivial, i.e., describes a nontrivial scattering process. A similar result iti two space-time dimensions (weakly coupled boson quantum field theory with polynomial self-interaction) has been proved recently both by Osterwalder and S6nCor [S] and by Eckmann et al. [9]. In [9] it is shown that perturbation theory in the coupling constant is asymptotic to the S-matrix elements. Dimock [lo] has proved differentiability of the Green’s

functions which is related to the nontriviality of the S-matrix (two dimensions). We follow here the idea in [S], i.e., we use the reduction formulas. The (generalized) time-ordered Green’s functions are recovered from the (generalized) Euclidean Green’s functions (Schwinger functions) mainly by the Osterwalder-Schrader analytic continuation [l] (see also [S, 121). In Section 2 we present the necessary notation and definitions and give an expansion for the truncated Schwinger functions. In Section- 3 we prove that truncated (generalized) Schwinger functions are bounded by locally integrable functions; * Supportedin part by the Deutsche Forschungsgemeinschaft andtheNationalScience Foundation underGrant PHY 75-21212. t Permanentaddress:Institute for Applied Mathematics, University of Frankfurt, 6 Frankfurt,‘M., West Germany. Copyright .\I1 rights

Q 1977 by Academic Press, Inc. of reproduction in any form reserved.

37 rsss

0003-4916

38

FLORIN CONSTANTINESCU

a result which is needed for the sake of analytic continuation to imaginary times. In Section 4 it is shown that the scattering matrix is nontrivial, and we obtain the asymptotic expansion for the scattering matrix up to the first order.

2. THE TRUNCATED

~-POINT

SCHWINCER

FUNCTION

Let #(t, x) be the Gaussian process with mean zero and covariance

where t > 0; x, y E R3; and mo2 is a positive constant. The momentum is taken to be [2]

cutoff field (2)

and the propagator

c(x,Y)= G,(x, v>= (@‘(x> L>@(.Y, &rJ)= 1,” GO,x, Y)dtFor t, = 0, C(x, y) is the free propagator representing the kernel of the operator C = (-A + m,2)-1. t, plays the role of a momentum cutoff (momenta larger than t;;Ll” are suppressed). Let dpG be the Gaussian measure with mean zero and covariance G(t, x, y). The cutoff interacting measure is e-Av’tm*g)dp, , where V(t, , g) is the cutoff Euclidean action and is given by vtm , g) = VI + vc 9

(4)

where V, =

.r

d3x g(x) :Q4(x, t,J:,

Vc = U/~)WI~)

- (1/6KV,3) + v,, ,

(5)

l’,,, = (h/2) 1 d3x g(x)’ 8m2(x, t,,) :Q2(x, t,,J:, srn2 = 42 . 6 j- d3y [j-I dt G(t, x, P,]“.

Here <*) means J . dp. We will also use the notation (.)t,,s for (.e-Y(tm,g)). In (4) and (5), h is the coupling constant and g(x) is the volumic cutoff which is supposed to be sufficiently regular, with compact support, and 0 < g(x) < 1. The cutoff Euclidean Green’s functions are now defined as s’“‘(t,

2 s;fi ,...,fn) = <@CA> .** Wn)>t,,,

. z-*3

(6)

SCATTERING

MATRIX

FOR

OS4

39

MODELS

where z = at,,

5 g) = <1;,,,,.,

is the partition function andf, E 5$(R3), i = 1, 2,..., II (YR(R3) is the real Schwartz space). We now consider the no cutoff limits syfi

,...),

f,,)

=

,,kn.,

S(?‘)(t,,i

)

g:

j;

. . . . .

I,,.

tt

q

I. 2 ,...,

(7)

which we know to exist for weak coupling [2], and to satisfy the following inequalities:

where a: is a constant independent of n and 1f’l is the L?-norm offE YQR3). Following [8] we will change the constant m, in the cutoff theory to the new value trr, which is the renormalized mass without changing the Euclidean Green’s functions. 1~1~ is a constant used to define the interaction; it is not the usual bare mass,which is infinite and requires renormalization. The Wick ordering in (4) has to be changed from .. ..)l*oto : :1,1. This will generate an extra :@:, term in V with a coefficient which depends on h and nz, and thus on A, and stays bounded as h varies in a small interval [0, A,]. The new free propagator will now be C(x, J) := C,JX, ii). We consider the truncated n-point Euclidean function T,,.?and apply the contraction formula in each @(‘Xi), i = 1. 2,..., II (see for instance [13]). We get for 173 2 [8]

where the sum runs over all partitions r of (1, 2..... II] into ; 7r nonempty mutually disjoint sets u, with ) T j taking all values I,,... IZ. / (J 1 is the number of elements in ; 0 1 and Yl”l(.) is the / G 1 derivative of Y(.). There is no term in the sum (9) if the partition 7r contains a u with 1(I / 3 4 (the case / u 1 z= 4 is excluded by truncation). On the other hand, again because of truncation, we can neglect the constant term in V”( .) = 12 :@( -): + h &H (provided n :> 2). For II =~4 we get from (9)

40

FLORIN CONSTANTINESCU

We prove in the next section that the no cutoff limits,

in (9) exist as locally integrable functions, and we will find bounds on the local singularities. We discuss in detail the (generalized) products in (9’) but the same proof applies in general.

3. LOCAL SINGULARITIES OF(GENERALIZED)TRUNCATED

SCHWINGER FUNCTIONS

We now prove that the no cutoff limits,

exist. We use the method in [2, 141 (see also [15]) based on the phase cell expansion [II]. We take the family of momentum cutoffs t, = co, 0 < tl < 1, tm = til+v)m-l, m = 2, 3,.... Let

(11)

C&G v) = j, dt G(t,x, u>dt and introduce the interpolated ct,(o)k

propagator

u> = 4,(x

- u) + (1 - 4 CL1

= Ctmml + u 11-l G(t, x, y) dt,

O,(U
We have for a graph G: n

((3 t,,,.g= c ((G)t,,g - (G>t,,-,,,I+ (Gh,,, m=z

SCATTERING

MATRIX

FOR

$*

41

MODELS

where t,(u) indicates the use of the covariance Ct,(,) . For G we take the graph in (10). (G),m(0),B will be a function F[(a, 0) depending on u and @(a). For the evaluation of (4/&)(G)t,(0),s we use the change of covariance formula [ 161

is the Gaussian measure with covariance C+) . By taking F(a, .) := G(o, .) e-r’(!p.) and performing cancellations by using Wick ordering as in [2, 151

where

&,

,cj

fields -

4X(++)

X fields and unit cubes

+72

T-2-342A2~(~)T

A

“‘,I$$~

)T-

-2(12)*x3

I: A,K’,A

-6(12)*X3

A *$;i”:

tx

w:

4 1z3.x4

c
-+fy$

f

jr

pgg:)‘-

-“‘“‘““:,‘,“‘im

: +&,:)r I

,

Here x represents contributions arising from the explicit -+- is the propagator (d/do) Cr,,L(0)= Ctm - Ctm-,) and

A’K

The vertices .r‘i f,

*

:-f&:

A-

:

could generate logarithmic singularities. Yd discussion in [ 151 shows that the only diagrams which could lead to trouble are

where i stays for ri ,

and

= :

u dependence of G,

05)

The

42

FLORIN

CONSTANTINESCU

The first two diagrams are combined to give further cancellations -3.4”

>&

+ 3.6.42

=-3*4’:&$

@--

: -3.42.6:-+&

-3.4’.6

ic7

+ 3,4’*6

:

(17)

F

The last two diagrams in (17) can be combined to give a mass cancellation. diagram in (16) can be written as }m{

But the diagram

=

:)m(:

1

i

f

4 :7Q7:+4i@j

is eliminated

The last

(W

by the truncation.

Finally

we will

contract (by using the contractions formula [13]) all free legs yi , i = 1,2, 3,4. We will be left with a finite number of graphs Cd for each of the graphs G in (10). We now have

We now have to estimate ( Gi)t,(0) ,g . This can be done as in [2, 151 by using the /I . j)6,a norms extracted from the phase cell expansion. We use the inequality -f- < > 0) and then separate the ( yi, vj) propagators from (Gi). In each e&-l *(/I
1 G

5(l)

/yi

_

l,j

-(m~l4)l~i-!Jjl llf2B

e

for

@> 0.

cm

SCATTERING

MATRIX

FOR

@::

43

JIODELS

We may have some other diagrams which give rise to singularities ( j’I . ~3~)lines. A typical example is

but do not have

From [4 (Thesis), Lemmas 5.4.2 and 5.4.3~1 we find that it has singularities of the form l/i J‘, - ~7~/1+~2a.There are also diagrams which have .I’ vertices but give continuous contributions, like for instance

as we will see below. Denoting for the moment by B( ~7) a bound on the product in ,
of ( yi , yj) singularities

i(Gi ‘f&),g I ‘., NJ.1 f,,,‘, for a positive E. From (12), (19), and (21) follows

the existence of ultraviolet

(21) limit:

The volumic cutoff can now be removed B la cluster expansion [2] and we get ‘G ,\ < (I?!)” B(y).

(23)

Here
(34)

(25)

44

FLORIN

CONSTANTINESCU

Diagrams like

can be split into the product of

because the y vertices are not integrated. The first diagram has already been discussed; it has a bound l/[ y, - yz [1+28.The second one can be easily shown to be continuous in y, , y3. The third one is also continuous in y3. Indeed we have (26)

and the last graph gives a finite contribution because the graph obtained by identifying the two y3 vertices and integrating on y3 E d, is logarithmically divergent and therefor controlled by a positive power oft. We have a similar discussion of the diagram

which can be split into the product of two diagrams

The last diagram has a l/(yz - y3 singularity. type

Finally, we can have diagrams of the

which can be split into the procut of

This type of diagram has been already discussed. In (26) HS means the HilbertSchmidt norm and the last graph in (26) was estimated by identifying the y3 vertices and integrating over d, . The integration in y3 EL& is legitimate because of translational invariance. All other graphs in the expansion of (V”(yJ V’(yJ V’(Y&)~ are either continuous in yi , i = 1,2, 3 or can be split in a product of a continuous

SCATTERING

and a singular part with singularities obtain finally

MATRIX

FOR

which

Qs4

4.5

MODELS

are less severe than those of (24). We

: : V”( 4’1) V’( y2) V’( y3))7‘ / < Of 1) c-4--=.

(27)

where E = mini1 yi - yj 1

for

1 :’ i - : j -< 3).

(28)

In (27) we have not indicated in C” dependence in X. Exactly in the same way we get

(29)

where E has a similar meaning as in (28), and generally (see also [2]) (30) where a: is a constant and 1 an integer both independent

of I rr 1and h E [0, h,], and

E = min(i yi - J’~ 1for I < i ,< j < : 7r !>. In (30) the constant multiplying E- (3n/2)-2B comes from the cluster expansion (removing the cutoff g). The integer I comes from the fact that the cluster expansion is applied on (IO) after performing some cancellations, thus increasing the number of external legs. Remarks 1. The singularities (27), (29), and (30) are locally integrable. Indeed, because of truncation there are no nonconnected diagrams in (Gi) which could lead to trouble. For instance in ( V’(yI) V’(y2) V’( y3) V’(y.))’ the diagram

is eliminated nonintegrable 2.

by truncation. singularity.

The factors

Again because of truncation

in the general products.

Without

l/i yi - yj ~3would

generate in this case a

we never looked at graphs like ,. truncation

or

a

i for tl = 2 the corresponding

46

FLORIN CONSTANTINESCU

expressions do not exist as distributions. contraction formula gives

In the case of the two point function the

<@(Xl) @(x2)) = C(Xl - x2) + 1 C(Xl - Yl) C(x2 - Y2)

x KV’(Yl) WY2D - %Yl - VZ)(~YYl))l @I gY, * Here ( v’(yl) V’(y&) do not exist as distribution, expression (V’(y,) V’(y2)> - 6(y, - JJJ(V”(~,)),

(31)

but if we consider the whole then the divergent diagram

2 will be cancelled by the mass counterterm, giving sense to (@(x1) Ia @(x2)) as locally integrable function (see also [4, 151). 3. We have not been concerned with the problem of getting the best possible bounds on local singularities of the [generalized) products. Actually it can be shown that we can put /3 = 0 in (27), (29, and (30) provided we do the (ui, vj) contractions before going to the ultraviolet limit by the methods of this section. This corresponds to the fact that in superrenormalizable theories the singularities of the (generalized) products of the interacting field coincide with those of the free field. 4. The results of this section show that the Schwinger functions and the generalized truncated Schwinger functions can be analytically continued to their time variables by the methods in [II] (see also [8, 121). In particular the time-ordered products exist as distributions for the weakly coupled Q3* quantum field theory. The proof goes on the lines of [S, 121. Instead of the logarithmic singularities of the P(@)z model we have now some algebraic singularities which do not influence the validity of the proof in [8, 121. 4. NONTRIVIALITY

OF THE SCATTERING

MATRIX

The results in Section 3 and the methods in [8, 11, 121allow us to state the following: THEOREM. The time ordered products (T(n~=, @(xi))) as well as the truncated VI”I(yo))T (see Section 3) are well (generalized) time ordered products (T(&, defined as distributions for the weakly coupled @is4model.

We now apply the LSZ reduction formula in the form given by Hepp [7] for in-going and out-going particles having all different velocities. We reproduce first some notations from [7] (see also [S]). Let Y(G) be the space of test functions jg Y’(R3) with suppf”~ (p” > 0 10 < p” < m’2}, m’ > m. p is the Fourier transform off. For PE Y(G) and w = w(p) = (p2 + m2)1/2, one defines f(x, t) = (2n)-l 1 ei”y(p) ei(“‘-“jt d%, (32) m

= jl(% at

f(x, t) = Z(x,

t).

SCATTERING

MATRIX

47

FOR @s4 MODELS

For the connected part of the S-matrix element we have [S]
I A%%m?eted

Here (& E P’(G) are supposed to be nonoverlapping ({h} E Y(G) are called nonoverlapping if for all pi E suppfi , wi = w,(pi) we have w-lpi # w-lpi for all i -fj). Then from (9) (32), [SJ we get PROPOSITION.

For nonoverlapping ( A} E Y(G), i = 1,2, 3,4 we hue

x 6(w1 + oJ2 -

co3 -

w4)

%iil

+

a,

-

p3

-

fi4)

+

O(h2).

(34)

From (33) follows that in X is sufficiently small the Vatrix is nontrivial. As remarked in [8] the proof of this proposition makes use of the fact that the finite mass renormalization has been done correctly (we have worked on the mass shell). O(h2) in (34) collects apart h2 functions of h, which are bounded in h for h E [0, X,]. ACKNOWLEDGMENTS I would like to thank Professors A. Jaffe, K. Osterwalder, and R. Seneor, and Dr. J. Feldman for many discussions. I would also especially like to thank Professor A. Jaffe for very kind hospitality at Harvard University.

REFERENCES 1. J. GLIMM AND A. JAFFE,Fortschr. Phys. 21 (1973), 327. 2. J. FELDMAN AND K. OSTERWALDER,Ann. Phys. 97 (19761, 80. 3. J. MAGNEN AND R. SBtio~, The Infinite Volume Limit of the (‘2~~)~Model, Ann. Inst. H. Poincark, to appear. 4. C. BURNAP, thesis, Harvard Univ. 1976. 5. R. HAAG, Phys. Rev. 112 (1958), 669. 6. D. RUELLE, Heh. Phys. Acta 35 (1962), 147. 7. K. HEPP, Commun. Math. Phys. 1 (1965), also see i/r “Axiomatic Field Theory” (M. Chretien and S. Deser, Eds.), 1965 Brandeis Lectures, Vol. I, Gordon and Breach, New York, 1968. 8. K. OSTERWALDER AND R. SENSOR, The Scattering Matrix is Nontrivial for Weakly Coupled Pi Models, Helv. Phys. Acta, to appear. 9. J.-P. ECKMANN, H. EPSTEIN, AND J. FRBHLICH, Asymptotic Perturbation Expansion for the S-Matrix and Definition of Time Ordered Functions in Relativistic Quantum Field Models, Ann. Inst. H. Poincarg, to appear. 595/108/I-.t

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FLORIN

CONSTANTINESCU

10. J. DIMOCK, The P(y)* Green’s Functions: Smoothness in the Coupling Constant, Adu. Phys., to appear; Asymptotic Perturbation Expansion, preprint. 11. K. OSTERWALDER AND R. SCHRADER, Commun. Math. Phys. 31 (1973), 83; 42 (1975), 281, 12. K. OSTERWDER, Time Ordered Operator Products and the Scattering Matrix in Pi Models, in Proceedings of the 1975 Bielefeld Conference. 13. J. Gm AND A. JAFFE, Commun. Math. Phys. 44 (1975), 293. 14. J. FELDMAN, Commun. Math. Phys. 37 (1974), 93. 15. J. FELDMAN AND R. RACZKA, The Relativistic Field Equation of the A@43Quantum Field Theory, preprint, 1976. 16. J. Gm, A. JAFFE, AND T. SPENCER, The Particle Structure of the Weakly Coupled P(@)2 Model and Other Applications of High Temperature Expansions, Parts I and II, in “Constructive Field Theory,” Lecture Notes in Physics, Vol. 25, Springer-Verlag, Berlin, 1973.