- Email: [email protected]
Model study of nonleading mass singularities, I
ANNALS
OF PHYSICS
102, 156-169 (1976)
Model
Study of Nonleading
Mass Singularities,
I
B. SCHROER,T. T. TRUONG, AND P. WEISZ Institut fiiv Theoretische Physik, Freie Universitiit, Berlin, Germany
ReceivedJune 11, 1976
We investigatethe effectsof tinite masseson the short distancepropertiesin the context of the two-dimensional Federbnsh model. We comment in particular on the limiting procedure in deriving the equation of motion and on changesin Wilson expansionsof certain operators as compared to the masslesscase.
1. INTRODUCTION Little is known about superrenormalizable perturbations on scale invariant theories. The standard picture concerning this problem originates still from Wilson’s “spurion” analysis [l] supported by some nonrigorous considerations based on the conformal “bootstrap” [2] and, respectively, the Callan-Symanzik equation [3]. This analysis does not determine the nonleading behavior of the operator product’s uniquely but only modulo powers of logarithms. The presence of nonzero masses may lead to significant changes in the structure of a model for example in the Sine-Gordon equation for fl 3 4~ we encounter a mass effect which moves coherent states of the Sine-Gordon field out of the physical Hilbert space [4]. In this situation it seems worthwhile to study soluble models. In two-dimensional space-time, there exists a massive soluble and not completely trivial model: the Federbush model [5], consisting of a massive current-pseudocurrent coupling of two massive Dirac fields. At this point we should specify what we mean by “nontriviality” of a model. In high energy physics “on shell” triviality means that the S-matrix cannot be transformed into the identity matrix by using the conserved charges of the theory. An S-matrix of the form S = exp(2i?rhQ,Q,), where Q, and Q, are superselected charges, is considered therefore as the identity because all operators in a theory are only determined modulo function of charges. In contradiction to some discussions in the literature, the S-matrix of the 156 Copyright AU rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
NONLEADING
MASS SINGULARITIES,
157
I
Federbush model is less trivial. For the two-particle S-matrix of species I and II which we review in the next section, we have 1III; Ir)out = e--2inA1III; Ir)in, 1IIr, II)OUt = e2niA1IIr, II)‘“; i.e., the S-matrix is an energy independent phase factor which depends on whether species II enters on the left or right of species 1. This statement is not only in agreement with the basic invariance of the Federbusch Lagrangian under the combined action of X -+ --h and parity transformation, but also agrees with an explicit second-order perturbation computation and the application of the general Haag-Ruelle scattering theory. This will be shown in Appendix 1and, respectively, in the next section. In Section 3 we discuss certain operator products containing mass effects and the form of the field equations of motion in the sense of a space-time limit procedure. The more involved questions of the form of the n-point correlation functions and the form of the general operator-product expansions will be discussed in a subsequent paper.
2. THE FEDERBUSH
MODEL
S-MATRIX
AND PERTURBATION
THEORY
EFFECTS
The model was first proposed by Federbush [5] in 1960 before full understanding of the infrared problems in the massless Thirring model [6] had been achieved. Its interaction Lagrangian can be written as L = -2~rhJ,%,,J;~
(2.1)
where Jr, 1I are the conserved vector currents of the Dirac fields of species I and II. This model is in fact related to two noninteracting Thirring models in a formal massless limit. The remarkable property of the model is that it is exactly soluble, and indeed it is at present the only known example of soluble massive interacting fields. It is probably a simple member of a more general class of theories, to which the massive Thirring model, which we believe also to be explicitly soluble, belongs. The special feature of the coupling and of two dimensions which was exploited by Federbush is the property that the axial currents fi,:,, = E~‘J[,~~ can be expressed as gradients of local conserved vector currents [7]. To replace the rather formal quantities with which Federbush worked in his original formulation Wightman [8] proposed a
158
SCHROER,
TRUONG
AND
WEISZ
solution in which the Heisenberg fields are written entirely in terms of the free field quantities #I(x) = jexp -2hP2h~“(x)~ #b(x), (2.2)
$“(x) = iexp 2iP”X+‘(x)i
#f(x),
where the #A9”(x) are free Dirac fields of masses MI and Mu; the 41J1(x)0 are the corresponding pseudopotentials (1/7P2) aTp*‘(x)
= ~@‘y”y5api:
(x).
(2.3)
Clearly these fields satisfy equations closely related to the formal equations of motion (W,
- MI) #’ = 2~~Jth$‘,
(ifa,
(2.4)
- M,,) Ipi = -27rhJ,~‘y,~“.
The potentials +1J1 are not local with respect to the free Dirac fields; indeed, for spacelike separations we have for (x - y)” < 0 [Ifioi(x), P(y)]
= (&Z/2) &(x1
- yl) #oyx),
(2.5)
with i = I, II; j = I, II, and x1, y1 are the spatial components of x, y. Nevertheless it can be checked that the Heisenberg local fields 4’ and z/Y1are relatively local, e.g., for (x - y)” < 0 #I(x) #*(y)
= -[iexp(-i 27r1’2h+11(~))i ~&~(y)][~~(x) iexp(i 2~?‘~h+‘(y))i] = - [exp(irhr( y1 - xl)) #i’(y) i exp( - i 2~?‘~h4”(x)) i] x [jexp(2irX@( y))i zjol(x) exp(i&r(xl - y’))] = -WY)
$4x).
(2.6)
Similarly, {#w
~+YY>>= 0
for
(x - y)” < 0.
(2.7)
The validity of these simple relations requires the pseudoscalar potentials and not the Sine-Gordon potentials which have the B function in Eq. (2.5) instead of the function &(x1 - yl). Since the expressions for #rJ1 contain the +IJ1 they do not lie in the Borchers class [9] of the free fields $iil, and there is therefore a chance that the theory has a nontrivial S-matrix. This is indeed the case, unfortunately, however, the s-matrix elements are rather uninteresting ((‘nonobservable”) energy independent phases, as was pointed out by Federbush [5]. We would like to present the discussion of the S-matrix in a little more detail to correct minor errors in the present literature [8].
NONLEADING
MASS
SINGULARITIES,
Let fI , fir be two-dimensional two-component on the Mr,rr mass shell. Then smeared operators (&$)+
159
I
test functions with suppf,,,,
(t) = ei”“(&;~:,)+ emiHt
(2.8)
applied to the vacuum give one particle states with wave packets fr,rr . Suppose the wave packet fr moves to the right and fir to the left relative to each other as t --f -co. Then we have (2.9) where (G,,)’
(&,,’
(2.10)
I 0) = I 1151 9 IfI>.
The exponentials of @J1 applied to the vacuum give essentially 1 in the above expression since they are smeared with test functions with support on the oneparticle mass shell. In obvious notations we can then write 1III, Ir)Out = e--i2nA1III, Ir)‘*.
(2.11)
On the other hand, if the particles approach each other from the opposite direction the phase is reversed (2.12)
1IIr, I1)OUt = ei2rA 1IIr, Il)‘=. For particles of the same species we have simple 1ur, al)o”t = [ ar, alp
u = I, II.
(2. 13)
Thus, in summary, we have for the two-particle scattering matrix elements oUt<~l’P1’,
U2’P2’
I 4Pl
3 U2P2)
in =
~2,2(Wlt
~2P2)i%‘P,‘,
U2’P2’
I %Pl
9 a,p,P,
(2. 14)
with ai’ = I, II = u*’ and 5-2*2UP,
s2,2(I~l
,
2 IPJ
=
S,>,(IIP,
11~~)
=
exPG
3 IIP,)
=
1,
(2.15)
27W~~~wd).
The S-matrix in higher sectors has a product structure resulting from cluster properties applied to a phase factor, i.e., Out(u,‘p,‘, u2’p2’,. ..) dP9z’ =
&,&lPl
i* *
*** ***
I GPl
GPnYn
*-.
%P?J GL’Pn’
I alp1
...
%P?Y,
(2.16)
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SCHROER,
TRUONG
AND
WEISZ
with Lhm
,...,
add
=
I-I id
S2&ipi
, ajpj).
(2.17)
In terms of in-operators the S-matrix can be written as
where
mdk) = [@i(k) @h(k) - ai!fi(k) &dk>l.
(2.19)
One easily establishes the validity of these relations in second-order perturbation theory. The calculation is deferred to Appendix 1. Since we disagree with a conclusion in the literature [lo], some general remarks on using renormalized perturbation theory in two-dimensional models are in order. If one uses the systematic BPHZ [l l] procedure, then one should use, besides the obvious symmetries which hold for all parameters, the asymptotic symmetries which are only valid in the short distance limit but which are necessary for a complete characterization of the model. The corresponding Ward identities reinforce these typical model properties in the face of the BPHZ oversubtraction (e.g., absence of infinite coupling constant renormalization in this approach is not taken into account since it is not a graph by graph property). This is a somewhat difficult approach and certainly not the one discussed in the literature [lo]. In another more pedestrian approach, one deviates from the BPHZ graph by graph renormalization theory by using compensation between different graphs of the same order. An important question remains, however. What is then a minimal set of conservation laws, either exact or asymptotic, which completly specify the Federbush model ? We are dealing with a rather involved system of couplings in perturbation theory, for there are as many as eight four-fermion couplings between the two fermions preserving separate fermion number conservation. Five of these are parity conserving and three are parity violating. The type of classification we have in mind is illustrated by considerations of the SU(2) symmetry theory [12]. Among the five parity conserving couplings there are three SU(2) symmetric ones (meaningful if Mr = iMu). Thus in the three-dimensional space one may select a one-parametric “Thirring manifold” by demanding the absence of any hard breaking terms in the isoscalar and isovector axial currents [13]. This is an asymptotically scale invariant one-parametric subset. The question is whether it is the existence of analogous asymptotic symmetries, e.g., parity and reflection in isospin space, which characterize the one-dimensional “Federbush manifold” or whether it is the existence of infinitely many conservation laws. It is our contention that it is a combination of both.
NONLEADING
MASS SINGULARITIES,
161
I
The validity of an infinite set of conservation laws which essentially restrict the set of outgoing momenta to be equal to the set of incoming momenta is, of course, compatible with the S-matrix being a pure phase. The validity of such conservation laws does not seem sufficient to characterize the Federbush model because such conservation laws are perhaps also valid in the U(1) massive Thirring model. The validity of such conservation laws in the Thirring model was immediately speculated upon after the established Thirring model Sine-Gordon model connection [14] and the known validity of such conservation laws in the classical SineGordon theory [15]. Berg, Karowski, and Thun [16] have explicitly shown that in the Thirring model the S-matrix elements for n + n particle scattering vanish in the tree and one loop approximation unless the set of outgoing momenta is identical to the set of incoming momenta [17]. The interesting question of possible anomalies which could arise in higher orders remains to be answered [18]. 3. SHORT DISTANCE SINGULARITIES
AND EQUATIONS
OF MOTION
We now consider the structural properties of the operator solution @*“(x) = ~exp(~2~1’2iX#1~1(x))~ #>“(x). Let us first test the leading singularity structure of #I(x) by considering the two-point function, for (x2 < 0), G,(x), i.e., (here we write m = AI,,) (:(#)‘>z (I+$‘), (x): ~exp(-2~““iA~‘*(O))~) =- m sin 7rX 27r 2n ff coshd,f(ey:
0,) exp((x + 1/2)(ep - 6) + j(p + q) x).
(3.1) There exists a hierarchy of divergences with respect to X as x2 --f 0. For example:
O
C,(X) =
2m
X
n
$
s0
7Th KA"(m(-x2)l/2)
2m
$
??-A
m cosh(U - 1) j3 Ko(2m(-x2)1/2 cash /I) d/I; cash /3
4(x> = 2m $ nx [K,“(m(-x2)“‘) + **a (-1)” +
_
(--l)n+llm
- K,fe,(m(-x2)1’2)
KA-n(m(-x2)1/2) cosh[2(/\co;h$
-
x Ko(2m(-x2)1/2 cash p) d/31.
1lP
(3.2)
162
SCHROER,
TRUONG
AND
WEISZ
Clearly the leading singularity is (wz(-x~)~~~/~)-~” and the least nonleading is always Log m(-x2)‘12/2. If we assign the dimension h2 to jexp(-2i?r112h~‘,“)i, then the operator expansion iexp(2i~112~~r.xx(x)~ iexp(-2i.rr1/2h~1,Kx(0)i = F,,(x) jexp(-2i&/2(X - ~)u)yJ~(O))i + ...,
(3.3)
where FM,+(x) = ( iexp(2i?r1/2&*n(x)
i iexp( -2zW2Xq5’*“(0)i)
(3.4) would yield a leading scaling behavior of the type @z(-x~)~/~/~)-~*“. On the other hand, the operator product expansion :(ijf~“)~ (I$~“), (x)*, jexp(-2&P2qP1(0))~ = G,+(x) jexp(-2i~1i2h~1,11(0))~
+ ...
(3.5)
yields the scaling behavior previously shown, i.e., (m( -x2)1/2/2)-2$ since (t,b~“)+(z@l)+(x) h as canonical dimension 1, the consistency of the dimension assignment is checked, This holds if we test iexp(-27T112iX~1,11(0)~ with other A direct computation of the correlation functions of spinor polynomials. ~exp(-2i~1/2h~1.11(x)~ from which this assignment could be read off is beyond the scope of this paper. In Appendix 2 we only show how the two-point function can be represented as a determinant (exponential of the trace of a Log) of an integral transform defined by a kernel in Fourier transformed rapidity space. This kernel is of trace class for given x2 and formulas of this type are useful in the proof of the existence of our operator solution. Here we will be satisfied with the indication of how to prove this existence. From the explicit form of the triple dots ordered exponential in Appendix 2, it follows that in order to obtain an operator B,, whose leading dimension is always X2(even when h is an integer) we have to divide out a sin 7~hfactor from the triple dots ordered exponential, i.e. (dropping upper I, II indices for simplicity: B, = &
[iexp(-27WiX&O))i
-I]
(3.6)
According to Ref. [4], this operator at integer values, for example, at h = 1, is only different from a local function of the free fields z,@” by a nonleading term, for example, in the neutral sector,
v+T&
[ieM-- ~~1’2w(wi -11 = 1; :$(I + p) +: + -& $(O)/27ri. (3.7)
Therefore, up to nonleading terms, the solution for h an integer is in the Borchers
NONLEADING
MASS
SINGULARITIES,
163
I
[9] class of the free fields. The expected triviality at these integer values is consistent with our explicit computation of the S-matrix in the previous section. Now let us look at the form of the field equation. Clearly (iyQ,
- MI) #I = i2d/2hyu
ia,#~l* exp(-2W2X+r1)i
tiO’.
(3.8)
Consider the product 4(x) iexp(-2i&/2A$(0))i
= - &p J$ {iexp(--2 in1/2py5(x))i = &
$
F,,(x)
~exp(-2~~1~2(pd4
iexp(-22in1/2h~(0))i)u=0 + XyW)))~},=, (3.9)
= :4(x) exp(-2i+/2h4(0))i sin 7i-h 1 n2 At(x) +&j&n--
jexp(-2zW2X$(0))~,
where AlA is defined in Appendix 2. Consequently the limiting procedure
=$p
’
{a,+(x) :exp(-2’ ~d~~h~(O))~ + Iexp(-2ia1/2h+(0))i
8,$(-x)1)), (3.10)
which is well known from the massless Thirring model, works for this massive model. The occurrence of nonleading singularities for large X in AlA does not affect the algebraic form of the space-time limiting procedure. The main deviation from the zero mass case is the appearance of logarithmic modifications of the chiral symmetry breaking terms (chiral symmetry is already broken in the zero mass case). The simplest illustration is~ CO>= SW2 i7N2xp(x) - p(0)); F,,(x) x {(m/2~r) log(m(-x2)1/2) + nonsingular
operator *..I. (3.11)
The largest singularity is obtained by developing the bilocal exponent into a Taylor series and retaining only the lowest operator, i.e., the identity. Similar statements hold for integer values of A, however, they have to be derived from the X-limits of triple dots ordering by somewhat lengthier computations which we shall not present.
164
SCHROER, TRUONG AND WEISZ
The bilinear functions of the fermion fields in this model may be bosonized. The resulting field equation, in the zero charge sector, has the Sine-Gordon form .+*I1
= 2m2N[sin 27r15#9r1].
(3.12)
However, the local normal product is defined [4] by performing an additional additive renormalization on the multiplicative Wick ordering, which for dim &+& < 1 is absent. If we let the Federbush coupling act on two Thirring fields with dim #lt#Z < 1 instead of free Dirac fields then we are in a phase of mild chiral symmetry breaking, i.e., the zero mass algebra remains intact. All these considerations hold if we generalize the model by introducing more than two species of particles coupled by a current-pseudocurrent interaction.
APPENDIX
1: THE FOUR-POINT
VERTEX FUNCTIONS IN PERTURBATION THEORY
up TO SECOND ORDER
We define the N-point vertex functions by
(Al.l) where labels Ak: , Bz represent collective particle type and Dirac labels. Mi (i = I, II) are the physical masses, i.e., EL3(P,
Pba4,
= 0.
(A1.2)
The contribution to pi(l) Iol,~II~,~I~,,IIB,(p1p2 , qlq2) from the Federbush interaction Lagrangian L = --27~ti~%~ J$ to the first order (Fig. 1) is given by i2rXy$le,,y~as,.
FIGURE
1
NONLEADING
MASS
FIGURE
SINGULARITIES,
I
165
2
In second order, we have contributions ra, rb coming from diagrams a, b, of Fig. 2. The renormalized contributions are separately divergent but their sum is convergent:
where 1 J(u,
MI
, AI,,)
=
in-
I 0 dx [x% - x(u - &I$ + M12) + MI2 - jr]-‘.
(A1.4)
It is easily checked that (r” + rb) ,,nrenOrmauzedgives no contribution to the S-matrix element 0ut(p1p2 ] qlq2)in in the backward direction. In the forward direction one obtains a contribution to the real part of 0uf(p1p2 ] q1q2)in fixed by the first-order contribution and unitarity and zero contribution to the imaginary part. In the BPHZ formalism used by Tapper [lo] the vertex function is normalized such that He then obtains
Such an extra term which contributes to the imaginary part of the S-matrix element (also in the backward direction) can, of course, be cancelled by addition
166
SCHROER,
to the Zimmerman
TRUONG
AND
effective Lagrangian
WEISZ
of a scalar-scalar
interaction
term
L’ = gSr& with
(A1.7) If this is done one has, to second order,
out
I !mP,
G41.8)
which is consistent with the S-matrix element claimed by Federbush, OUYP~P~ I qlqzYn = exp ~2~Wqd%d19
iY~l~,
A similar situation occurs for the vertex function involving type, i.e.,
I qlqeY*.
(A1.9)
fermions of the same (A1.10)
~~~~,IIor~:IIB.IIB’(PP’, 44’).
The contribution to second order from the Federbush term in the effective Lagrangian is given by the two diagrams of Fig. 3. The unrenormalized integrals are already convergent and one obtains a contribution to the vertex f%ren(PP’;
44’) = 2M2~2rk4?3~ +
aA/A2)[Ja
{-g,,J(@ M
M , Md 9 MI,)
-
J(d2,
w
9
M31>
- 2M,“h~,rs”r~~~{-g,yJ(o, MI 9Md +
(QoQv/Q2)[J@,MI
3 WI)
-
J(Q2, MI
3
MJlh
(Al .11)
where A = p - q and Q = p’ - q.
and
FIGURE
We see, therefore, that we get zero element O~t(pp’ [ qq’)in as demanded imaginary part like a tree diagram of g’ = --2mh2 + 0(X3); i.e., we again Federbush out(pp’
1 qq’)in
p-p’
3
contribution to the real part of the S-matrix by unitary and we get a contribution to the a pure Thirring coupling -$ g’Jf; g,& with have consistency with the claimed form of =
in(pp’
1 qqf)in
if the counterterms are added, which will be required by the imposition Ward identities.
(A1.12) of correct
NONLEADING MASS SINGIJLARITIES, I
APPENDIX
167
2: GENERAL STRUCTURE OF THE TWO-POINT FUNCTION
F,,(x) = ( {exp(27G12+$(x))~ iexp( -2~?/~ih$(O))I> To calculate this function
one appeals to the Lehmann
jexp(-2rr1i2iX$(x))i
and Stehr’s result, i.e.,
= :exp((L,(x)):,
(A2.1)
where the kernel L,,(x) is fermion normal ordered: sin 7rX expG@, - 4)) LA(X) = r ff dep deq tcosh((0 9 - 19Q)/2)
x [exp(G + 4) 4 h+h+ + exp(--i(p + 4) 4 @GJ _ e&V, - 0) [exp(i(p - q) x - id) apta,. sinh((8, - Q/2)
+ exp(--i(p
- q) x + i~h) b,‘b,]\.
(A2.2)
Then making a power series expansion of :&A(*):, the function F,,(x) can be represented as I;l,(x)
=
exp
f
(-1)n+1
isinn
"","i"
Xx]
A",fQ),
(A2.3)
?l=l
where the coefficients A:+“(x) are defined by the integrals
*6‘11%+1 6929.+2 ***~an&%n~2n~~n+12)3 .**kl,,D)l -
(A2.4)
The S function products describe the contractions in the fermion operator product
I 0) arising from the nth term of the sum
. .* &,b&,,
= If0 -(jJ2
.
(A2.5)
We introduce now the Fourier transform
1
cosh((e, - e,y2) =
- eq>) d[, s--mmexp(-ii5(e9 cash rr.$
(A2.6)
168
SCHROER,
TRUONG
AND
WEISZ
and make use of the generalized Heine’s representation of Bessel functions to obtain a more compact form for At+‘(x). For the case of interest (spacelike x2 < 0) the modified Bessel function K, is of relevance. 2 (s)“”
Kv(z2 -
f2)lj2 = Jrn exp(-z -cc
cash t - t sinh t - vt) dt Re(z f =$)> 0.
(For x2 > 0, the Hankel
(A2.7)
functions would be used.) Hence, with u = h + p,
A reduction of this expression can be achieved by using the formula K”(z) K/(z’)
= 2 lo K,+,,(2.2 cash t) cosh[(v - v’) t] dt,
(A2.9)
so that
* BO(&,+l ,
&a+2
; x) a*- B”(E2ra-1
3 &, ; 4,
(A2.10)
where B”(& .$‘; x) = sp, exp(2utc-s;~
+ “)) Ki~r4~(2m(-x2)1~2
cash t) dt,
(A2.11)
which can be expressed as B”(t, 5’; 4 = m(-x”Y’“/i(5 - 6’) * (K-o+cl,2~-r(~(-~2)“2~~-~+~~,~~~~~~~-~2~1’2~ (A2.12) - K,-c,,,,-ut(~(-~~)~‘~)K-o-(l,2)+i~(~(-x~)~’~). We can then write A,“(x) as the trace of the integral operator [SC’@, E’; m( -x2)1/2>]” where we define WJ(~, f’; m(-x2)1/2)
= (l/r2)(Bu(& 5’; x)/(cosh & cash &‘)l/%), &Q(x) = trace(SP)n,
(A2.13) (A2.14)
and, consequently, F&)
= exp[Trace Log(l
+ sin rp sin &QP)],
(A2.15)
NONLEADING
MASS
SINGULARITIES,
I
169
which can be viewed as the value of the Fredholm determinant of XV at sin VP sin Z-X. This expression is pointwise convergent for large (-x2)l12. However, Fph vanishes if sin rp sin nX happens to be an eigenvalue of the kernel X0. We note that for h = + + T and p = 4 + p where T and p are both positive or zero, there exists the relation (a’ = p + T)
K~,+(1,2)--iE(m(-x2)l”) - P+p(f,
K,,+(,,,,-,,,(m(-~‘)l’~)
(cash nf cash &‘)li2 E’; m(-x2)1/2).
This splitting allows the estimate of the strongest short distance of F&) recursive way with respect to the parameters h and II.
(A2.16) in a
REFERENCES
1. K. WILSON, Phys. Rev. 179 (1969), 1499. 2. G. MACK, in “Strong Interaction Physics” (W. Ri.ihl and A. Vancura, Eds.), Lectures Notes in Physics, No. 17, Springer-Verlag, New York, 1972. 3. K. SYMANZIK, in “Particles, Quantum Fields and Statistical Mechanics,” Lectures Notes in Physics, No. 32, Springer-Verlag, New York, 1975. 4. B. SCHROER AND T. TRUONG, “Equivalence of Sine Gordon and Thirring Model and Cumulative Mass Effects,” FUB HEP 6. 5. P. FEDERBUSH, Phys. Rev. 121 (1960), 1247. 6. W. THIRRING, Ann. Phys. (N.Y.) 3 (1958), 91; K. JOHNSON,Nuovo Cimento 20 (1960), 773; B. KLAIBER, “Boulder Lectures in Theoretical Physics 1967,” Gordon and Breach, New York, 1968; G. F. DELL-ANTONIO, Y. FRISHMAN, AND D. ZWANZIGER, Phys. Rev. D 6 (1972), 988. 7. K. POHLMEYER, Commun. Math. Phys. 25 (1972), 73. 8. A. S. WIGHTMAN, in “High Energy Interactions and Field Theory” (M. Levy, Ed.), Cargese Lectures in Theoretical Physics 1964, Gordon and Breach, New York, 1966. 9. H. J. BORCHERS, Nuovo Cimento 24 (1962), 214. 10. M. L. TAPPER, Nuovo Cimento 25 A (1975), 165. 11. W. ZIMMERMAN, in “1970 Brandeis Lectures on Elementary Particles and Quantum Field Theory” (S. Deser et al., Eds.), M.I.T. Press, Cambridge, Mass., 1971; N. N. Booomov AND D. W. SHIRKOV, “Introduction to the Theory of Quantized Fields,” Interscience, New York, 1959; K. HEPP, Commun. Math. Phys. 2 (1966), 301. 12. P. K. MISTER AM) P. H. WEISZ, Phys. Rev. D 8 (1973), 4410. 13. B. SCHROER, Lectures Presented at the International School of Mathematical Physics Ettore Majorana 1975, FUB HEP 75/17. 14. S. COLEMAN, Phys. Rev. D 11 (1975), 2088. 15. A. C. Scorn, F. Y. F. Crru, AND D. W. MCLAUGHLIN, Proc IEEE 61(1973), 1443. 16. B. BERG, M. KAROWSKI, AND H. J. THUN, Phys. Lett., to appear; FUB HEP 76/5. 17. This is not a universal property of fermion models in two dimensions, e.g. the nontrivial three-particle scattering amplitudes in the SU(n) Thirring model are nonvanishing already in the tree approximation. 18. R. FLUME, “Anomalies of Currents in the Quantized Sine-Gordon Equation,” DESY Preprint 75/33. 19. H. LEHMANN AND K. STEHR, Hamburg Preprint DESY 76/29.
OF PHYSICS
102, 156-169 (1976)
Model
Study of Nonleading
Mass Singularities,
I
B. SCHROER,T. T. TRUONG, AND P. WEISZ Institut fiiv Theoretische Physik, Freie Universitiit, Berlin, Germany
ReceivedJune 11, 1976
We investigatethe effectsof tinite masseson the short distancepropertiesin the context of the two-dimensional Federbnsh model. We comment in particular on the limiting procedure in deriving the equation of motion and on changesin Wilson expansionsof certain operators as compared to the masslesscase.
1. INTRODUCTION Little is known about superrenormalizable perturbations on scale invariant theories. The standard picture concerning this problem originates still from Wilson’s “spurion” analysis [l] supported by some nonrigorous considerations based on the conformal “bootstrap” [2] and, respectively, the Callan-Symanzik equation [3]. This analysis does not determine the nonleading behavior of the operator product’s uniquely but only modulo powers of logarithms. The presence of nonzero masses may lead to significant changes in the structure of a model for example in the Sine-Gordon equation for fl 3 4~ we encounter a mass effect which moves coherent states of the Sine-Gordon field out of the physical Hilbert space [4]. In this situation it seems worthwhile to study soluble models. In two-dimensional space-time, there exists a massive soluble and not completely trivial model: the Federbush model [5], consisting of a massive current-pseudocurrent coupling of two massive Dirac fields. At this point we should specify what we mean by “nontriviality” of a model. In high energy physics “on shell” triviality means that the S-matrix cannot be transformed into the identity matrix by using the conserved charges of the theory. An S-matrix of the form S = exp(2i?rhQ,Q,), where Q, and Q, are superselected charges, is considered therefore as the identity because all operators in a theory are only determined modulo function of charges. In contradiction to some discussions in the literature, the S-matrix of the 156 Copyright AU rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
NONLEADING
MASS SINGULARITIES,
157
I
Federbush model is less trivial. For the two-particle S-matrix of species I and II which we review in the next section, we have 1III; Ir)out = e--2inA1III; Ir)in, 1IIr, II)OUt = e2niA1IIr, II)‘“; i.e., the S-matrix is an energy independent phase factor which depends on whether species II enters on the left or right of species 1. This statement is not only in agreement with the basic invariance of the Federbusch Lagrangian under the combined action of X -+ --h and parity transformation, but also agrees with an explicit second-order perturbation computation and the application of the general Haag-Ruelle scattering theory. This will be shown in Appendix 1and, respectively, in the next section. In Section 3 we discuss certain operator products containing mass effects and the form of the field equations of motion in the sense of a space-time limit procedure. The more involved questions of the form of the n-point correlation functions and the form of the general operator-product expansions will be discussed in a subsequent paper.
2. THE FEDERBUSH
MODEL
S-MATRIX
AND PERTURBATION
THEORY
EFFECTS
The model was first proposed by Federbush [5] in 1960 before full understanding of the infrared problems in the massless Thirring model [6] had been achieved. Its interaction Lagrangian can be written as L = -2~rhJ,%,,J;~
(2.1)
where Jr, 1I are the conserved vector currents of the Dirac fields of species I and II. This model is in fact related to two noninteracting Thirring models in a formal massless limit. The remarkable property of the model is that it is exactly soluble, and indeed it is at present the only known example of soluble massive interacting fields. It is probably a simple member of a more general class of theories, to which the massive Thirring model, which we believe also to be explicitly soluble, belongs. The special feature of the coupling and of two dimensions which was exploited by Federbush is the property that the axial currents fi,:,, = E~‘J[,~~ can be expressed as gradients of local conserved vector currents [7]. To replace the rather formal quantities with which Federbush worked in his original formulation Wightman [8] proposed a
158
SCHROER,
TRUONG
AND
WEISZ
solution in which the Heisenberg fields are written entirely in terms of the free field quantities #I(x) = jexp -2hP2h~“(x)~ #b(x), (2.2)
$“(x) = iexp 2iP”X+‘(x)i
#f(x),
where the #A9”(x) are free Dirac fields of masses MI and Mu; the 41J1(x)0 are the corresponding pseudopotentials (1/7P2) aTp*‘(x)
= ~@‘y”y5api:
(x).
(2.3)
Clearly these fields satisfy equations closely related to the formal equations of motion (W,
- MI) #’ = 2~~Jth$‘,
(ifa,
(2.4)
- M,,) Ipi = -27rhJ,~‘y,~“.
The potentials +1J1 are not local with respect to the free Dirac fields; indeed, for spacelike separations we have for (x - y)” < 0 [Ifioi(x), P(y)]
= (&Z/2) &(x1
- yl) #oyx),
(2.5)
with i = I, II; j = I, II, and x1, y1 are the spatial components of x, y. Nevertheless it can be checked that the Heisenberg local fields 4’ and z/Y1are relatively local, e.g., for (x - y)” < 0 #I(x) #*(y)
= -[iexp(-i 27r1’2h+11(~))i ~&~(y)][~~(x) iexp(i 2~?‘~h+‘(y))i] = - [exp(irhr( y1 - xl)) #i’(y) i exp( - i 2~?‘~h4”(x)) i] x [jexp(2irX@( y))i zjol(x) exp(i&r(xl - y’))] = -WY)
$4x).
(2.6)
Similarly, {#w
~+YY>>= 0
for
(x - y)” < 0.
(2.7)
The validity of these simple relations requires the pseudoscalar potentials and not the Sine-Gordon potentials which have the B function in Eq. (2.5) instead of the function &(x1 - yl). Since the expressions for #rJ1 contain the +IJ1 they do not lie in the Borchers class [9] of the free fields $iil, and there is therefore a chance that the theory has a nontrivial S-matrix. This is indeed the case, unfortunately, however, the s-matrix elements are rather uninteresting ((‘nonobservable”) energy independent phases, as was pointed out by Federbush [5]. We would like to present the discussion of the S-matrix in a little more detail to correct minor errors in the present literature [8].
NONLEADING
MASS
SINGULARITIES,
Let fI , fir be two-dimensional two-component on the Mr,rr mass shell. Then smeared operators (&$)+
159
I
test functions with suppf,,,,
(t) = ei”“(&;~:,)+ emiHt
(2.8)
applied to the vacuum give one particle states with wave packets fr,rr . Suppose the wave packet fr moves to the right and fir to the left relative to each other as t --f -co. Then we have (2.9) where (G,,)’
(&,,’
(2.10)
I 0) = I 1151 9 IfI>.
The exponentials of @J1 applied to the vacuum give essentially 1 in the above expression since they are smeared with test functions with support on the oneparticle mass shell. In obvious notations we can then write 1III, Ir)Out = e--i2nA1III, Ir)‘*.
(2.11)
On the other hand, if the particles approach each other from the opposite direction the phase is reversed (2.12)
1IIr, I1)OUt = ei2rA 1IIr, Il)‘=. For particles of the same species we have simple 1ur, al)o”t = [ ar, alp
u = I, II.
(2. 13)
Thus, in summary, we have for the two-particle scattering matrix elements oUt<~l’P1’,
U2’P2’
I 4Pl
3 U2P2)
in =
~2,2(Wlt
~2P2)i%‘P,‘,
U2’P2’
I %Pl
9 a,p,P,
(2. 14)
with ai’ = I, II = u*’ and 5-2*2UP,
s2,2(I~l
,
2 IPJ
=
S,>,(IIP,
11~~)
=
exPG
3 IIP,)
=
1,
(2.15)
27W~~~wd).
The S-matrix in higher sectors has a product structure resulting from cluster properties applied to a phase factor, i.e., Out(u,‘p,‘, u2’p2’,. ..) dP9z’ =
&,&lPl
i* *
*** ***
I GPl
GPnYn
*-.
%P?J GL’Pn’
I alp1
...
%P?Y,
(2.16)
160
SCHROER,
TRUONG
AND
WEISZ
with Lhm
,...,
add
=
I-I id
S2&ipi
, ajpj).
(2.17)
In terms of in-operators the S-matrix can be written as
where
mdk) = [@i(k) @h(k) - ai!fi(k) &dk>l.
(2.19)
One easily establishes the validity of these relations in second-order perturbation theory. The calculation is deferred to Appendix 1. Since we disagree with a conclusion in the literature [lo], some general remarks on using renormalized perturbation theory in two-dimensional models are in order. If one uses the systematic BPHZ [l l] procedure, then one should use, besides the obvious symmetries which hold for all parameters, the asymptotic symmetries which are only valid in the short distance limit but which are necessary for a complete characterization of the model. The corresponding Ward identities reinforce these typical model properties in the face of the BPHZ oversubtraction (e.g., absence of infinite coupling constant renormalization in this approach is not taken into account since it is not a graph by graph property). This is a somewhat difficult approach and certainly not the one discussed in the literature [lo]. In another more pedestrian approach, one deviates from the BPHZ graph by graph renormalization theory by using compensation between different graphs of the same order. An important question remains, however. What is then a minimal set of conservation laws, either exact or asymptotic, which completly specify the Federbush model ? We are dealing with a rather involved system of couplings in perturbation theory, for there are as many as eight four-fermion couplings between the two fermions preserving separate fermion number conservation. Five of these are parity conserving and three are parity violating. The type of classification we have in mind is illustrated by considerations of the SU(2) symmetry theory [12]. Among the five parity conserving couplings there are three SU(2) symmetric ones (meaningful if Mr = iMu). Thus in the three-dimensional space one may select a one-parametric “Thirring manifold” by demanding the absence of any hard breaking terms in the isoscalar and isovector axial currents [13]. This is an asymptotically scale invariant one-parametric subset. The question is whether it is the existence of analogous asymptotic symmetries, e.g., parity and reflection in isospin space, which characterize the one-dimensional “Federbush manifold” or whether it is the existence of infinitely many conservation laws. It is our contention that it is a combination of both.
NONLEADING
MASS SINGULARITIES,
161
I
The validity of an infinite set of conservation laws which essentially restrict the set of outgoing momenta to be equal to the set of incoming momenta is, of course, compatible with the S-matrix being a pure phase. The validity of such conservation laws does not seem sufficient to characterize the Federbush model because such conservation laws are perhaps also valid in the U(1) massive Thirring model. The validity of such conservation laws in the Thirring model was immediately speculated upon after the established Thirring model Sine-Gordon model connection [14] and the known validity of such conservation laws in the classical SineGordon theory [15]. Berg, Karowski, and Thun [16] have explicitly shown that in the Thirring model the S-matrix elements for n + n particle scattering vanish in the tree and one loop approximation unless the set of outgoing momenta is identical to the set of incoming momenta [17]. The interesting question of possible anomalies which could arise in higher orders remains to be answered [18]. 3. SHORT DISTANCE SINGULARITIES
AND EQUATIONS
OF MOTION
We now consider the structural properties of the operator solution @*“(x) = ~exp(~2~1’2iX#1~1(x))~ #>“(x). Let us first test the leading singularity structure of #I(x) by considering the two-point function, for (x2 < 0), G,(x), i.e., (here we write m = AI,,) (:(#)‘>z (I+$‘), (x): ~exp(-2~““iA~‘*(O))~) =- m sin 7rX 27r 2n ff coshd,f(ey:
0,) exp((x + 1/2)(ep - 6) + j(p + q) x).
(3.1) There exists a hierarchy of divergences with respect to X as x2 --f 0. For example:
O
C,(X) =
2m
X
n
$
s0
7Th KA"(m(-x2)l/2)
2m
$
??-A
m cosh(U - 1) j3 Ko(2m(-x2)1/2 cash /I) d/I; cash /3
4(x> = 2m $ nx [K,“(m(-x2)“‘) + **a (-1)” +
_
(--l)n+llm
- K,fe,(m(-x2)1’2)
KA-n(m(-x2)1/2) cosh[2(/\co;h$
-
x Ko(2m(-x2)1/2 cash p) d/31.
1lP
(3.2)
162
SCHROER,
TRUONG
AND
WEISZ
Clearly the leading singularity is (wz(-x~)~~~/~)-~” and the least nonleading is always Log m(-x2)‘12/2. If we assign the dimension h2 to jexp(-2i?r112h~‘,“)i, then the operator expansion iexp(2i~112~~r.xx(x)~ iexp(-2i.rr1/2h~1,Kx(0)i = F,,(x) jexp(-2i&/2(X - ~)u)yJ~(O))i + ...,
(3.3)
where FM,+(x) = ( iexp(2i?r1/2&*n(x)
i iexp( -2zW2Xq5’*“(0)i)
(3.4) would yield a leading scaling behavior of the type @z(-x~)~/~/~)-~*“. On the other hand, the operator product expansion :(ijf~“)~ (I$~“), (x)*, jexp(-2&P2qP1(0))~ = G,+(x) jexp(-2i~1i2h~1,11(0))~
+ ...
(3.5)
yields the scaling behavior previously shown, i.e., (m( -x2)1/2/2)-2$ since (t,b~“)+(z@l)+(x) h as canonical dimension 1, the consistency of the dimension assignment is checked, This holds if we test iexp(-27T112iX~1,11(0)~ with other A direct computation of the correlation functions of spinor polynomials. ~exp(-2i~1/2h~1.11(x)~ from which this assignment could be read off is beyond the scope of this paper. In Appendix 2 we only show how the two-point function can be represented as a determinant (exponential of the trace of a Log) of an integral transform defined by a kernel in Fourier transformed rapidity space. This kernel is of trace class for given x2 and formulas of this type are useful in the proof of the existence of our operator solution. Here we will be satisfied with the indication of how to prove this existence. From the explicit form of the triple dots ordered exponential in Appendix 2, it follows that in order to obtain an operator B,, whose leading dimension is always X2(even when h is an integer) we have to divide out a sin 7~hfactor from the triple dots ordered exponential, i.e. (dropping upper I, II indices for simplicity: B, = &
[iexp(-27WiX&O))i
-I]
(3.6)
According to Ref. [4], this operator at integer values, for example, at h = 1, is only different from a local function of the free fields z,@” by a nonleading term, for example, in the neutral sector,
v+T&
[ieM-- ~~1’2w(wi -11 = 1; :$(I + p) +: + -& $(O)/27ri. (3.7)
Therefore, up to nonleading terms, the solution for h an integer is in the Borchers
NONLEADING
MASS
SINGULARITIES,
163
I
[9] class of the free fields. The expected triviality at these integer values is consistent with our explicit computation of the S-matrix in the previous section. Now let us look at the form of the field equation. Clearly (iyQ,
- MI) #I = i2d/2hyu
ia,#~l* exp(-2W2X+r1)i
tiO’.
(3.8)
Consider the product 4(x) iexp(-2i&/2A$(0))i
= - &p J$ {iexp(--2 in1/2py5(x))i = &
$
F,,(x)
~exp(-2~~1~2(pd4
iexp(-22in1/2h~(0))i)u=0 + XyW)))~},=, (3.9)
= :4(x) exp(-2i+/2h4(0))i sin 7i-h 1 n2 At(x) +&j&n--
jexp(-2zW2X$(0))~,
where AlA is defined in Appendix 2. Consequently the limiting procedure
=$p
’
{a,+(x) :exp(-2’ ~d~~h~(O))~ + Iexp(-2ia1/2h+(0))i
8,$(-x)1)), (3.10)
which is well known from the massless Thirring model, works for this massive model. The occurrence of nonleading singularities for large X in AlA does not affect the algebraic form of the space-time limiting procedure. The main deviation from the zero mass case is the appearance of logarithmic modifications of the chiral symmetry breaking terms (chiral symmetry is already broken in the zero mass case). The simplest illustration is
operator *..I. (3.11)
The largest singularity is obtained by developing the bilocal exponent into a Taylor series and retaining only the lowest operator, i.e., the identity. Similar statements hold for integer values of A, however, they have to be derived from the X-limits of triple dots ordering by somewhat lengthier computations which we shall not present.
164
SCHROER, TRUONG AND WEISZ
The bilinear functions of the fermion fields in this model may be bosonized. The resulting field equation, in the zero charge sector, has the Sine-Gordon form .+*I1
= 2m2N[sin 27r15#9r1].
(3.12)
However, the local normal product is defined [4] by performing an additional additive renormalization on the multiplicative Wick ordering, which for dim &+& < 1 is absent. If we let the Federbush coupling act on two Thirring fields with dim #lt#Z < 1 instead of free Dirac fields then we are in a phase of mild chiral symmetry breaking, i.e., the zero mass algebra remains intact. All these considerations hold if we generalize the model by introducing more than two species of particles coupled by a current-pseudocurrent interaction.
APPENDIX
1: THE FOUR-POINT
VERTEX FUNCTIONS IN PERTURBATION THEORY
up TO SECOND ORDER
We define the N-point vertex functions by
(Al.l) where labels Ak: , Bz represent collective particle type and Dirac labels. Mi (i = I, II) are the physical masses, i.e., EL3(P,
Pba4,
= 0.
(A1.2)
The contribution to pi(l) Iol,~II~,~I~,,IIB,(p1p2 , qlq2) from the Federbush interaction Lagrangian L = --27~ti~%~ J$ to the first order (Fig. 1) is given by i2rXy$le,,y~as,.
FIGURE
1
NONLEADING
MASS
FIGURE
SINGULARITIES,
I
165
2
In second order, we have contributions ra, rb coming from diagrams a, b, of Fig. 2. The renormalized contributions are separately divergent but their sum is convergent:
where 1 J(u,
MI
, AI,,)
=
in-
I 0 dx [x% - x(u - &I$ + M12) + MI2 - jr]-‘.
(A1.4)
It is easily checked that (r” + rb) ,,nrenOrmauzedgives no contribution to the S-matrix element 0ut(p1p2 ] qlq2)in in the backward direction. In the forward direction one obtains a contribution to the real part of 0uf(p1p2 ] q1q2)in fixed by the first-order contribution and unitarity and zero contribution to the imaginary part. In the BPHZ formalism used by Tapper [lo] the vertex function is normalized such that He then obtains
Such an extra term which contributes to the imaginary part of the S-matrix element (also in the backward direction) can, of course, be cancelled by addition
166
SCHROER,
to the Zimmerman
TRUONG
AND
effective Lagrangian
WEISZ
of a scalar-scalar
interaction
term
L’ = gSr& with
(A1.7) If this is done one has, to second order,
out
I !mP,
G41.8)
which is consistent with the S-matrix element claimed by Federbush, OUYP~P~ I qlqzYn = exp ~2~Wqd%d19
iY~l~,
A similar situation occurs for the vertex function involving type, i.e.,
I qlqeY*.
(A1.9)
fermions of the same (A1.10)
~~~~,IIor~:IIB.IIB’(PP’, 44’).
The contribution to second order from the Federbush term in the effective Lagrangian is given by the two diagrams of Fig. 3. The unrenormalized integrals are already convergent and one obtains a contribution to the vertex f%ren(PP’;
44’) = 2M2~2rk4?3~ +
aA/A2)[Ja
{-g,,J(@ M
M , Md 9 MI,)
-
J(d2,
w
9
M31>
- 2M,“h~,rs”r~~~{-g,yJ(o, MI 9Md +
(QoQv/Q2)[J@,MI
3 WI)
-
J(Q2, MI
3
MJlh
(Al .11)
where A = p - q and Q = p’ - q.
and
FIGURE
We see, therefore, that we get zero element O~t(pp’ [ qq’)in as demanded imaginary part like a tree diagram of g’ = --2mh2 + 0(X3); i.e., we again Federbush out(pp’
1 qq’)in
p-p’
3
contribution to the real part of the S-matrix by unitary and we get a contribution to the a pure Thirring coupling -$ g’Jf; g,& with have consistency with the claimed form of =
in(pp’
1 qqf)in
if the counterterms are added, which will be required by the imposition Ward identities.
(A1.12) of correct
NONLEADING MASS SINGIJLARITIES, I
APPENDIX
167
2: GENERAL STRUCTURE OF THE TWO-POINT FUNCTION
F,,(x) = ( {exp(27G12+$(x))~ iexp( -2~?/~ih$(O))I> To calculate this function
one appeals to the Lehmann
jexp(-2rr1i2iX$(x))i
and Stehr’s result, i.e.,
= :exp((L,(x)):,
(A2.1)
where the kernel L,,(x) is fermion normal ordered: sin 7rX expG@, - 4)) LA(X) = r ff dep deq tcosh((0 9 - 19Q)/2)
x [exp(G + 4) 4 h+h+ + exp(--i(p + 4) 4 @GJ _ e&V, - 0) [exp(i(p - q) x - id) apta,. sinh((8, - Q/2)
+ exp(--i(p
- q) x + i~h) b,‘b,]\.
(A2.2)
Then making a power series expansion of :&A(*):, the function F,,(x) can be represented as I;l,(x)
=
exp
f
(-1)n+1
isinn
"","i"
Xx]
A",fQ),
(A2.3)
?l=l
where the coefficients A:+“(x) are defined by the integrals
*6‘11%+1 6929.+2 ***~an&%n~2n~~n+12)3 .**kl,,D)l -
(A2.4)
The S function products describe the contractions in the fermion operator product
I 0) arising from the nth term of the sum
. .* &,b&,,
= If0 -(jJ2
(A2.5)
We introduce now the Fourier transform
1
cosh((e, - e,y2) =
- eq>) d[, s--mmexp(-ii5(e9 cash rr.$
(A2.6)
168
SCHROER,
TRUONG
AND
WEISZ
and make use of the generalized Heine’s representation of Bessel functions to obtain a more compact form for At+‘(x). For the case of interest (spacelike x2 < 0) the modified Bessel function K, is of relevance. 2 (s)“”
Kv(z2 -
f2)lj2 = Jrn exp(-z -cc
cash t - t sinh t - vt) dt Re(z f =$)> 0.
(For x2 > 0, the Hankel
(A2.7)
functions would be used.) Hence, with u = h + p,
A reduction of this expression can be achieved by using the formula K”(z) K/(z’)
= 2 lo K,+,,(2.2 cash t) cosh[(v - v’) t] dt,
(A2.9)
so that
* BO(&,+l ,
&a+2
; x) a*- B”(E2ra-1
3 &, ; 4,
(A2.10)
where B”(& .$‘; x) = sp, exp(2utc-s;~
+ “)) Ki~r4~(2m(-x2)1~2
cash t) dt,
(A2.11)
which can be expressed as B”(t, 5’; 4 = m(-x”Y’“/i(5 - 6’) * (K-o+cl,2~-r(~(-~2)“2~~-~+~~,~~~~~~~-~2~1’2~ (A2.12) - K,-c,,,,-ut(~(-~~)~‘~)K-o-(l,2)+i~(~(-x~)~’~). We can then write A,“(x) as the trace of the integral operator [SC’@, E’; m( -x2)1/2>]” where we define WJ(~, f’; m(-x2)1/2)
= (l/r2)(Bu(& 5’; x)/(cosh & cash &‘)l/%), &Q(x) = trace(SP)n,
(A2.13) (A2.14)
and, consequently, F&)
= exp[Trace Log(l
+ sin rp sin &QP)],
(A2.15)
NONLEADING
MASS
SINGULARITIES,
I
169
which can be viewed as the value of the Fredholm determinant of XV at sin VP sin Z-X. This expression is pointwise convergent for large (-x2)l12. However, Fph vanishes if sin rp sin nX happens to be an eigenvalue of the kernel X0. We note that for h = + + T and p = 4 + p where T and p are both positive or zero, there exists the relation (a’ = p + T)
K~,+(1,2)--iE(m(-x2)l”) - P+p(f,
K,,+(,,,,-,,,(m(-~‘)l’~)
(cash nf cash &‘)li2 E’; m(-x2)1/2).
This splitting allows the estimate of the strongest short distance of F&) recursive way with respect to the parameters h and II.
(A2.16) in a
REFERENCES
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