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Deformations along critical lines of field theories in two dimensions
Volume 257, number 1,2
PHYSICSLETTERSB
21 March 1991
Deformations along critical lines of field theories in two dimensions P. C h a s e l o n a, F. C o n s t a n t i n e s c u b and R. F l u m e a
a PhysikalischesInstitut, UniversitiitBonn, Nuflallee 12, W-5300Bonn 1, FRG b FachbereichMathematik, Johann WolfgangGoethe Universitiit, RobertMayerStrafle 10, W-6000Frankfurt, FRG Received 24 August 1990; revised manuscript received 26 October 1990
We investigatedeformations of two-dimensional quasirational conformal field theories through marginal deformations. A recursive representation of the perturbation expansion is found which allows to keep track of the variation of the data of the conformal field theory to arbitrary order. The emerginggeneralizedKadanoff-Browncriterion for true marginalityis discussed.
1. The explicitly known two-dimensional systems with critical lines or critical surfaces are all related to gaussian models and tensor products of gaussian models. The search for other (non-gaussian) theories with continuously varying critical exponents is interesting for the purpose of a general classification of two-dimensional critical phenomena and is secondly of practical importance for the attempts to construct superconformal quantum theories corresponding to classical a-models taking values in Calabi-Yau spaces (see ref. [ 1 ] for the review of the latter problem and references quoted therein). We consider in this paper perturbations around a quasirational #1 conformal field theory (QRCFT), call it C. The perturbation is given by an interaction of the form A S = g f d2x ~ ( x , .2) ,
(1)
where g is a (small) coupling constant and • is a primary scalar operator in C of scaling dimension two. We shall derive a representation of perturbation theory (PT) which allows for a recursive evaluation of necessary and sufficient conditions for "true" marginality of AS, that is, conditions for the vanishing of the (renormalization group) fl-function(s) in PT. The conditions we are looking for have been found to second order of PT by Kadanoff and Brown [ 2 ] a ~1 For an explanation of the terminologysee below.
long time ago. Partial results to third order have been obtained by Cardy [ 3 ]. Let X denote a bunch of local operators of C. The formal expression for the perturbative expansion of the correlation function of X is given by the GellMann-Low formula
( X ) s c + ~ = ( e x p (g f d2x ~(x, g) ) X I s c •
(2)
Sc denotes here the (fictitious) action characterizing the model C. We will partially follow the approach advocated in ref. [4] towards the analysis of short distance singularities of the Gell-Mann-Low series. It was shown in ref. [4] that the program of analytic regularization and renormalization can be implemented in a rather natural way through a successive transformation of two-dimensional integrals of the RHS of (2) into contour integrals. The formalism is however hampered by the fact that it is rather cumbersome to keep track of locality of the perturbing interaction in higher orders of PT. This defect will be cured in the present paper by establishing a completely recursive organization of PT. 2. To start with we collect some material from refs. [ 5-7 ] concerning the structure of QRCFT. For an alternative approach see refs. [ 8,9 ]. We assume for simplicity that all physical primary operators, denoted ~i, of our model C, which may be finite or (denumerably) infinite in number, are scalar with a
0370-2693/91/$ 03.50 © 1991 - ElsevierSciencePublishers B.V. ( North-Holland )
63
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decomposition into chiral and antichiral factors ~0,and ~, respectively, which we write formally as q)~= ~0~X ~i,
~=q~X~,
21 March 1991
~ ( ( z ) . , ..., (3), ) (-)
+(-)
=< {0 |inin[
(3)
(-)
(-)
Z n ) {O i n i n - l k n - 3 ( Z
n_l)...((7~)ilill((})l) >
(-)
where q~ is the above introduced interaction density. The operators q~ (and q~) are, to be precise, primary with respect to the chiral and antichiral symmetry algebras underlying the model C. Let these algebras be denoted by E and E respectively. We assume that E and F, are isomorphic to each other. The operators ~o, (@#) applied to the vacuum state 10) generate lowest weight vectors of representations of E (E). The corresponding representation space will be denoted by
(~i-): ~0,10) = [i), ~=Eli),
~10>=lf),
where the index 1 in ~0 ~xx and (~)xxl refers to the vacuum state. The sum in (5) embodies a finite number of terms due to quasirationality. In writing down eq. (5) we made the simplifying assumption that there are no degeneracies of the operator product algebra in the sense that the representation of E and F, do not occur with higher than unit multiplicity. The order of operators in (5) is ad hoc. Any other order leads to a new basis of block functions. Let P denote an element (a permutation ) of the symmetric group S,. We have (with self-explanatory notations) (-)k
F b( i)( (z)e(, ), ..., (Z)e( t))
~-=Eli).
(-)
The space H of physical states is given by
We assume that our model is quasirational what means that in the operator product expansion (OPE) of any pair of primaries only a finite number of different (primary) operators appear. Following Tsuchiya and Kanie [ 10] we introduce (anti)chiral vertex operators (-)
~o ok=P(7) ~ojP(~) ,
(4)
where P(~) denotes the projection onto the (anti)chiral subspace ~ %). The factorization of physical operators, eq. (3), together with the concept of vertex operators lead to the representation of physical correlation functions through products of holomorphic and antiholomorphic "block functions", < ~in (Zn, -~.
ert)'-'(J~il (Zl, Zl) >
...... ,,
C tn
× ~in...il ~'k,-3...k~(Z \ n,
( n-4 = ,_,,+=,,,}
- X F~C(k,i)ff~. #
C~%,
~pk,-3..*, (~,,
"", ZI ]~in...il
"",
f~ )
(-)
~o ~,,,~<~)~((z)p(1)) ) .
(6)
The functions (-)k Fb¢i) are naturally attached to the coordinate sector Re={(Zl,
...,
z°)l Ize.)-Ze(1)l
=
Izp(i),e(l)l
< Izeo),e(l ) [ for all i
(h
I
--dkJ-livo),kJ-2~ --2dim. ) ) Z P ( n ) , P ( 1 ) "'" ,
F~'(D;a = \j=2 Zp(j),p( 1 )
a~,b =da +db - d e
(7)
with dx denoting the scaling dimension of ~Ox (dx= l = 0). The omitted term on the RHS ofeq. (7) is given by a (in Re) convergent multidimensional power series in the difference (v)ariables Zet/),e(l ). The analytic continuation of F~(i) beyond Re is comprised in the linear relations between different bases of block functions. If suffices to specify the relation between bases with one pair of neighbouring operators in the fusion scheme being exchanged relative to each other: F...kj+ dgkj- ,... ...ip(j+ 1),iP(j)...
(5)
The C~x are here the coefficients ~f)the OPE. The (anti)holomorphic block functions F~ are given by 64
(-)
(-)
× ... ×
H= Z ~,®~,-. i
(-)
(-)
=
y. B± ( kj+ , iev+ I ) i p o ) k j - I k
J, kkj I~...kj+,kkj_,... ...iP(j),iP(j+ 1)... "
(8) The subscript + refers to the two homotopically in-
Volume257, number 1,2
PHYSICS LETTERSB
equivalent paths of analytic continuation. The matrices B+ constitute a representation of the braid group• They satisfy as such several consistency equations (guaranteeing that analytic continuation along homotopically equivalent paths gives rise to identical results): B+ (kj+ 1iPo+ l)ipo)kj_ 1)
XB_ (kj+, ipu.)iPu+,)k+_ , ) = 1 ,
21 March 1991
are satisfied. (B* denotes the complex conjugate of B. ) The coefficients Cg.c of the OPE are assumed to be fixed through relations (13) up to an overall normalization• 3. Let the bunch X o f fields in eq. (2) be a product of primary operators, x = ~o~.(z°) × ¢,.(2,)...q,~, (z,) × G (~,) •
(9)
E B+_ (kj+,ipo+l)ipo)kj_,) ~
Taking the factorization into chiral and antichiral block functions into account one achieves through the use of Stokes' theorem the following decomposition of the first order contribution of the Gell-Mann-Low series into contour integrals [ 4 ]:
Ol
XB+_ (otipo+t)ieu-l)kj-2)~_,
xB+_ (kj+, i~u~i~u-~)~)~ = Z B+-(kjieu)iPu-l)kj-2)~-, ot
f dZx (tp(x) × ~(X){o,.(z,,) ×~,.(2,,) × •••
XB+ (k/+, ipu+l)ipo_,)ot)~ j
XB++_(yipo+~)ipu)kj_z)~ ,
(10)
X~il (Zl) X ~i, (g[) >
1 (
B + ( kj+ l ipu+ l )ipu)kj- , ) Dj+ , -
= exp ( - -
•
Ol/2+O~ 1/2 I/2 C(k,!)~,.(ff~,) D
kj+l
(11)
with the notation
dk,+, il"o+ I } i l " o ) k j -
I =d,,u+,
Jr ,,rn>~32
~(F~,)e(1-m)C(k,i)~+(F~,)),
e(x) = 0 ,
(Dx)~,k. = 5k,k exp ( -- i2~Aki~Cx>kj_~) , ,
+di.%.) +dkj - , --dk++, •
=1
(12)
We shall use below the symbol Dx for a diagonal matrix of phase factors appearing at the xth intermediate state of the fusion scheme, irrespective of the particular basis of block functions we are dealing with. Correlation functions of the physical scalar operators q~ are required to be one-valued in the euclidean plane with respect to all their arguments• One-valuedness in the sector R, is built into the ansatz (5) through its diagonality. The phase factors being generated along closed paths in R~ cancel among the chiral and antichiral blocks (the latter functions being the complex conjugates of the former). The same will be true in other sectors if the relations kj
~ cg'(F~') i=,
127~A ieo+ , )ipo)kj_ 1 )
X D + B_ ( kj+ , ieu+ l )ieu)kj_t )
k
4 i ~.
k
C ij+ l.k C ij, k)- i
×B+ (kfij+, i/k/_, )~' B & (kjij+, ijkj_l )k., kj k' = (~k' ,k" Cii+l,k' Cij,kj-,
( 13
)
x=O, ,
x>O,
=--1
,
(14)
x
where we use the abbreviating notations ~(F~,)=
J" d t F ~ ( .... t , x , ) ,
(15)
te c~i k F 6 i ( . . . , t,
xi)=
~OkO+=PkgOP,,
(...~koi(t)Oiil
(Xi) ) ,
(16)
~ ( F ~ , ) = ~,.(Fk~) * .
The diagonal matrices D in the first term on the RHS of (14) are those introduced in eq. (12) referring in each case to the basis of block functions quoted in eq. (16). The contours (~i) encircling <2] with begin and end at infinity are displayed in fig. 1. The terms of the second sum on the RHS of (14) may be evaluated due to the invariance property, expressed through eq. (13) in an arbitrary basis• The expressions represented through the closed contours <~) (eq. ( 15 ) ) are free of singularities as 65
Volume 257, n u m b e r 1,2
PHYSICS LETTERS B
21 March 1991
in a different (but not unrelated) context. We define formally P
in
( zp,°, . .... zp,,, )
-'~ !fp <~lP(n)P(n)(ZP(n))'"~kj-IP~J)°l(ZP(J)) te ' (1) X
Fig. 1. A Contour ~,.
~O~o,_~ (t)~Okj_~eti_
PeS.,
1) " - 3 (_Tp(j_ 1) ) ' " > ,
(17)
~O.ok(t)=P~o(t)P~,
with ~feu) being the contour of fig. 2 attached to the point P(j). The functions ie¢P(t);a have to be interpreted as modified block functions. The subscript c~ has to be regarded as a degeneracy label. The singularity structure of P is easily inferred from the corresponding structure of the functions F (cf. eq. ( 7 ) ) ,
F_mo;, ~ * k = ,,j=2(h t Zp(j),P(1 --,Jki-tipuJ,ki-2"~ --2dim,o ) )Zp(n),P(l)... , Fig. 2. A Contour cCYeu).
functions of varying scaling dimensions ~2 of the involved field operators since the contours avoid the potential singularities of the integrands F and F respectively. The meromorphic dependence on the scaling dimensions is instead concentrated in the factor (Dl/2+D*l/2)/(Dl/2--D*l/2). This explicit extraction of the meromorphic structure is the main advantage of the present approach (which persists in higher orders o f P T (cf. ref. [4] ) ). It is however, as mentioned above, hard (within the formalism) to keep track of the consequences of locality in higher orders of PT. To cope with this problem we will rewrite eq. (14) in such a form that a complete recursive procedure becomes visible which can be iterated mechanically. We pass for this purpose from the up to now considered basis of contours to a new basis displayed in fig. 2. The choice of the new basis is inspired through Felder's work [ 11 ] where the usefulness of this type of contours has been demonstrated ~2 Such a variation m a y be achieved for example by tensoring with exponential functions of one or several free scalar fields.
66
(18)
where the omitted terms are again (in Re) convergent power series in the difference variables z,o~).e(~ ). One should note that the singularities o f f do not depend on the index c~labelling the fusion state appearing after the integrated operator ~a(t). The relations between functions /~p and Fe, attached to different coordinate sectors also follow straightforward (even though somewhat tedious to derive) from the corresponding relations, eq. (8), of the original block functions. The emerging modified braid matrices obey the same type of polynomial consistency relations as the original braid matrices. This follows from the fact that moves in the modified scheme can be resolved into (more complicated) moves in the original scheme and that homotopically equivalent paths in the modified scheme remain equivalent if translated back to the original scheme. The explicit verification of the consistency relations being of some interest for a better understanding of the nature of deformations along critical surfaces will be given in ref. [ 12 ]. Rewriting the result of first order PT, eq. (14), in terms of modified block functions/~ and ff we obtain after repeated use ofeqs. (9), ( 11 ) and (13)
Volume 257, number 1,2
PHYSICS LETTERSB
X~(X)rpi,(z,) X ~i,(g,,) X ...
~d2x (~(x)
x~,(z,) x~.(~) ) l
n--I
.
F L;~,jX~,j,,jC( k, fl, ~)F;:,~:,
j=2
( 19 )
of pole singularities (which have to be subtracted off). Coming back to the representation of the Born term through contours as given in eq. (14) we suppose that the ith of the n operators is equal to the interaction density,
"'
%aj,,o
U)# = B ++ ( kj_ , ~jOkj_ • # • X,~;. ~ ),~V
Let the OPE of two operators ~oX # give rise to an operator ~ , × ¢ with scaling dimension 2d~, near to marginality,
/ D ! n +Dy~/2]
d~,=l-y,
{D)': +D;":)
- ~,f,j ,~;p~,D)/2_DtW:]p p, C(k.,fl, i ) -- -w~¢ i n _
21 March 1991
l kn_ 2 ....
t"kJ-'~P ijfl ~ Okj_
(20) 2 ....
t'k'i 2 i I
"
(21)
The most important aspect of eq. (19) is that the chiral and antichiral modified block functions are again combined on the diagonal of the tensor product of the two (chiral and antichiral) fusion schemes. (We recall once more that the summation indices a j, a j in (19) are to be regarded as degeneracy labels of the new scheme. ) One-valuedness of the correlation functions is therewith manifest in the sector R~ = { (z~, .... z~)l Iz~.~l < Izj.~l for all i
~(~, c~', a, i) = Z x~,~,C(k., fl, i ) . ,a
Relations analogous to eq. ( 13 ) hold for the new braid matrices and the new coefficients C. The second order PT can now be organized as a first order contribution in the modified QRCFT. We have, in other words, completed the first recursion. 4. The breakdown of dilation invariance signalled through a nonvanishing fl function term in the renormalization group equations is in our approach of analytic regularization (i.e. analytic continuation in critical exponents) implied through the occurrence
y<
This coupling in the OPE leads to a contribution to the first sum on the RHS of (14) of the form 1 cos 3 ~ 4 sinai,
~(Fsik')C~i(F°~8')
[ (~k,=~,)
, ~,(___1 +O(yO))%(...)~(... = ~C~ ~Y
).
(22)
The holomorphic and antiholomorphic expressions ~g(F) and ~ ( F ) respectively are regular in y for y ~ 0, ~g ( Ft-)) = t-)
(-)°(t-))+O(y)ff F
(23)
where c~o and c~o are given by the residua (at y = 0 ) of the pole terms 1/(t-zi) and 1/ (F-~fi) o f F a n d F respectively times a factor 2n (from the integration around (27). The product of the residua is easily identified as emerging from a local counterterm supplying a contribution to the//-function of the induced interaction proportional to the coefficient C ~ of the OPE. Turning the last statement around one arrives at the Kadanoff-Brown criterion. There are no flfunctions generated in second order of PT if and only if the OPE of two interaction densities does not lead to operators of dimension two (the interaction itself in particular should not appear as a fused state). Let us now come to the second order of PT (in order to determine the fate of the fl-function in third order) assuming that the Kadanoff-Brown criterion is fulfilled. It follows from this assumption that the RHS of eq. (20) has an unambiguous meaning (that is, no subtractions are needed if the regularization is lifted). Looking on the second order of PT as first PT in the (to first order) deformed scheme we find analogously to the previous calculation, eqs. (22) and (23), as residue of a possible pole 67
Volume 257, number 1,2
PHYSICS LETTERS B
1 COSZl~, - ~ kl 4 sin 3,~ ,,~, ~ ~'(P6~'k~) C ( a ' a ' ) ~/(F~,~, )l (~k,=~,)
= - ,~yCg~,(2n)2(...~/×~) +O(y°) .
(24)
~ × ~ denotes here as before an operator with a scaling dimension near to marginality, d~,= 1 - y , y<< 1. The modified expansion coefficients C are given by
Z
or' ~ o t
" L OtOt'
cpCgo,
21 March 1991
before through a first order calculation with respect to a modified QRCFT (modified to ( n - 1 )th order) and will be given as such by an (n - 1 )-fold sum over intermediate fusion states combined through a product of braid matrices and an n-fold product of coefficients of the OPE. The vanishing condition for the residue is the generalization of the Kadanoff-Brown criterion. It is in our opinion an important problem to find conditions under which the terms in the sums over intermediate fusion states cancel each other.
otoL' f l
(-)
where the constants relations -
(X2)
d ~ are fixed through the
One of us (R.F.) thanks the members of the LPTHE and P.K. Mitter for hospitality extended to him.
(~aoo( t ) (POOl((X)l)[0)
(-) 6~, (-) ( _ ) )
~
References
= x~,~ °° a s ~<~)<~, (
68
[ 1 ] P. Candelas, P.S. Green and T. Hiibsch, in: Proc. String '88 (College Park, MD, 1988 ) p. 155. [ 2 ] L.P. Kadanoff and A.C. Brown, Ann. Phys. 121 (1979) 318. [3] J.U Cardy, J. Phys. A 20 (1987) L891. [4] F. Constantinescu and R. Flume, J. Phys. A 23 (1990) 2971. [5] G. Moore and N. Seiberg, Phys. Lett. B 212 (1988) 451. [6] G. Moore and N. Seiberg, Nucl. Phys. B 313 (1989) 16. [ 7 ] G. Felder, J. Fr6hlich and G. Keller, Commun. Math. Phys. 130 (1990) 1. [8] K.H. Rehren and B. Schroer, Phys. Lett. B 198 (1987) 480. [9] K. Fredenhagen, K.H. Rehren and B. Schroer, Commun. Math. Phys. 125 (1989) 201. [ 10] A. Tsuchiya and Y. Kanie, Lett. Math. Phys. 13 (1987) 303. [ 11 ] G. Felder, Nucl. Phys. B 317 ( 1989 ) 215. [ 12] P. Chaselon, in preparation.
PHYSICSLETTERSB
21 March 1991
Deformations along critical lines of field theories in two dimensions P. C h a s e l o n a, F. C o n s t a n t i n e s c u b and R. F l u m e a
a PhysikalischesInstitut, UniversitiitBonn, Nuflallee 12, W-5300Bonn 1, FRG b FachbereichMathematik, Johann WolfgangGoethe Universitiit, RobertMayerStrafle 10, W-6000Frankfurt, FRG Received 24 August 1990; revised manuscript received 26 October 1990
We investigatedeformations of two-dimensional quasirational conformal field theories through marginal deformations. A recursive representation of the perturbation expansion is found which allows to keep track of the variation of the data of the conformal field theory to arbitrary order. The emerginggeneralizedKadanoff-Browncriterion for true marginalityis discussed.
1. The explicitly known two-dimensional systems with critical lines or critical surfaces are all related to gaussian models and tensor products of gaussian models. The search for other (non-gaussian) theories with continuously varying critical exponents is interesting for the purpose of a general classification of two-dimensional critical phenomena and is secondly of practical importance for the attempts to construct superconformal quantum theories corresponding to classical a-models taking values in Calabi-Yau spaces (see ref. [ 1 ] for the review of the latter problem and references quoted therein). We consider in this paper perturbations around a quasirational #1 conformal field theory (QRCFT), call it C. The perturbation is given by an interaction of the form A S = g f d2x ~ ( x , .2) ,
(1)
where g is a (small) coupling constant and • is a primary scalar operator in C of scaling dimension two. We shall derive a representation of perturbation theory (PT) which allows for a recursive evaluation of necessary and sufficient conditions for "true" marginality of AS, that is, conditions for the vanishing of the (renormalization group) fl-function(s) in PT. The conditions we are looking for have been found to second order of PT by Kadanoff and Brown [ 2 ] a ~1 For an explanation of the terminologysee below.
long time ago. Partial results to third order have been obtained by Cardy [ 3 ]. Let X denote a bunch of local operators of C. The formal expression for the perturbative expansion of the correlation function of X is given by the GellMann-Low formula
( X ) s c + ~ = ( e x p (g f d2x ~(x, g) ) X I s c •
(2)
Sc denotes here the (fictitious) action characterizing the model C. We will partially follow the approach advocated in ref. [4] towards the analysis of short distance singularities of the Gell-Mann-Low series. It was shown in ref. [4] that the program of analytic regularization and renormalization can be implemented in a rather natural way through a successive transformation of two-dimensional integrals of the RHS of (2) into contour integrals. The formalism is however hampered by the fact that it is rather cumbersome to keep track of locality of the perturbing interaction in higher orders of PT. This defect will be cured in the present paper by establishing a completely recursive organization of PT. 2. To start with we collect some material from refs. [ 5-7 ] concerning the structure of QRCFT. For an alternative approach see refs. [ 8,9 ]. We assume for simplicity that all physical primary operators, denoted ~i, of our model C, which may be finite or (denumerably) infinite in number, are scalar with a
0370-2693/91/$ 03.50 © 1991 - ElsevierSciencePublishers B.V. ( North-Holland )
63
Volume 257, n u m b e r 1,2
PHYSICS LETTERS B
decomposition into chiral and antichiral factors ~0,and ~, respectively, which we write formally as q)~= ~0~X ~i,
~=q~X~,
21 March 1991
~ ( ( z ) . , ..., (3), ) (-)
+(-)
=< {0 |inin[
(3)
(-)
(-)
Z n ) {O i n i n - l k n - 3 ( Z
n_l)...((7~)ilill((})l) >
(-)
where q~ is the above introduced interaction density. The operators q~ (and q~) are, to be precise, primary with respect to the chiral and antichiral symmetry algebras underlying the model C. Let these algebras be denoted by E and E respectively. We assume that E and F, are isomorphic to each other. The operators ~o, (@#) applied to the vacuum state 10) generate lowest weight vectors of representations of E (E). The corresponding representation space will be denoted by
(~i-): ~0,10) = [i), ~=Eli),
~10>=lf),
where the index 1 in ~0 ~xx and (~)xxl refers to the vacuum state. The sum in (5) embodies a finite number of terms due to quasirationality. In writing down eq. (5) we made the simplifying assumption that there are no degeneracies of the operator product algebra in the sense that the representation of E and F, do not occur with higher than unit multiplicity. The order of operators in (5) is ad hoc. Any other order leads to a new basis of block functions. Let P denote an element (a permutation ) of the symmetric group S,. We have (with self-explanatory notations) (-)k
F b( i)( (z)e(, ), ..., (Z)e( t))
~-=Eli).
(-)
The space H of physical states is given by
We assume that our model is quasirational what means that in the operator product expansion (OPE) of any pair of primaries only a finite number of different (primary) operators appear. Following Tsuchiya and Kanie [ 10] we introduce (anti)chiral vertex operators (-)
~o ok=P(7) ~ojP(~) ,
(4)
where P(~) denotes the projection onto the (anti)chiral subspace ~ %). The factorization of physical operators, eq. (3), together with the concept of vertex operators lead to the representation of physical correlation functions through products of holomorphic and antiholomorphic "block functions", < ~in (Zn, -~.
ert)'-'(J~il (Zl, Zl) >
...... ,,
C tn
× ~in...il ~'k,-3...k~(Z \ n,
( n-4 = ,_,,+=,,,}
- X F~C(k,i)ff~. #
C~%,
~pk,-3..*, (~,,
"", ZI ]~in...il
"",
f~ )
(-)
~o ~,,,~<~)~((z)p(1)) ) .
(6)
The functions (-)k Fb¢i) are naturally attached to the coordinate sector Re={(Zl,
...,
z°)l Ize.)-Ze(1)l
=
Izp(i),e(l)l
< Izeo),e(l ) [ for all i
(h
I
--dkJ-livo),kJ-2~ --2dim. ) ) Z P ( n ) , P ( 1 ) "'" ,
F~'(D;a = \j=2 Zp(j),p( 1 )
a~,b =da +db - d e
(7)
with dx denoting the scaling dimension of ~Ox (dx= l = 0). The omitted term on the RHS ofeq. (7) is given by a (in Re) convergent multidimensional power series in the difference (v)ariables Zet/),e(l ). The analytic continuation of F~(i) beyond Re is comprised in the linear relations between different bases of block functions. If suffices to specify the relation between bases with one pair of neighbouring operators in the fusion scheme being exchanged relative to each other: F...kj+ dgkj- ,... ...ip(j+ 1),iP(j)...
(5)
The C~x are here the coefficients ~f)the OPE. The (anti)holomorphic block functions F~ are given by 64
(-)
(-)
× ... ×
H= Z ~,®~,-. i
(-)
(-)
=
y. B± ( kj+ , iev+ I ) i p o ) k j - I k
J, kkj I~...kj+,kkj_,... ...iP(j),iP(j+ 1)... "
(8) The subscript + refers to the two homotopically in-
Volume257, number 1,2
PHYSICS LETTERSB
equivalent paths of analytic continuation. The matrices B+ constitute a representation of the braid group• They satisfy as such several consistency equations (guaranteeing that analytic continuation along homotopically equivalent paths gives rise to identical results): B+ (kj+ 1iPo+ l)ipo)kj_ 1)
XB_ (kj+, ipu.)iPu+,)k+_ , ) = 1 ,
21 March 1991
are satisfied. (B* denotes the complex conjugate of B. ) The coefficients Cg.c of the OPE are assumed to be fixed through relations (13) up to an overall normalization• 3. Let the bunch X o f fields in eq. (2) be a product of primary operators, x = ~o~.(z°) × ¢,.(2,)...q,~, (z,) × G (~,) •
(9)
E B+_ (kj+,ipo+l)ipo)kj_,) ~
Taking the factorization into chiral and antichiral block functions into account one achieves through the use of Stokes' theorem the following decomposition of the first order contribution of the Gell-Mann-Low series into contour integrals [ 4 ]:
Ol
XB+_ (otipo+t)ieu-l)kj-2)~_,
xB+_ (kj+, i~u~i~u-~)~)~ = Z B+-(kjieu)iPu-l)kj-2)~-, ot
f dZx (tp(x) × ~(X){o,.(z,,) ×~,.(2,,) × •••
XB+ (k/+, ipu+l)ipo_,)ot)~ j
XB++_(yipo+~)ipu)kj_z)~ ,
(10)
X~il (Zl) X ~i, (g[) >
1 (
B + ( kj+ l ipu+ l )ipu)kj- , ) Dj+ , -
= exp ( - -
•
Ol/2+O~ 1/2 I/2 C(k,!)~,.(ff~,) D
kj+l
(11)
with the notation
dk,+, il"o+ I } i l " o ) k j -
I =d,,u+,
Jr ,,rn>~32
~(F~,)e(1-m)C(k,i)~+(F~,)),
e(x) = 0 ,
(Dx)~,k. = 5k,k exp ( -- i2~Aki~Cx>kj_~) , ,
+di.%.) +dkj - , --dk++, •
=1
(12)
We shall use below the symbol Dx for a diagonal matrix of phase factors appearing at the xth intermediate state of the fusion scheme, irrespective of the particular basis of block functions we are dealing with. Correlation functions of the physical scalar operators q~ are required to be one-valued in the euclidean plane with respect to all their arguments• One-valuedness in the sector R, is built into the ansatz (5) through its diagonality. The phase factors being generated along closed paths in R~ cancel among the chiral and antichiral blocks (the latter functions being the complex conjugates of the former). The same will be true in other sectors if the relations kj
~ cg'(F~') i=,
127~A ieo+ , )ipo)kj_ 1 )
X D + B_ ( kj+ , ieu+ l )ieu)kj_t )
k
4 i ~.
k
C ij+ l.k C ij, k)- i
×B+ (kfij+, i/k/_, )~' B & (kjij+, ijkj_l )k., kj k' = (~k' ,k" Cii+l,k' Cij,kj-,
( 13
)
x=O, ,
x>O,
=--1
,
(14)
x
where we use the abbreviating notations ~(F~,)=
J" d t F ~ ( .... t , x , ) ,
(15)
te c~i k F 6 i ( . . . , t,
xi)=
~OkO+=PkgOP,,
(...~koi(t)Oiil
(Xi) ) ,
(16)
~ ( F ~ , ) = ~,.(Fk~) * .
The diagonal matrices D in the first term on the RHS of (14) are those introduced in eq. (12) referring in each case to the basis of block functions quoted in eq. (16). The contours (~i) encircling <2] with begin and end at infinity are displayed in fig. 1. The terms of the second sum on the RHS of (14) may be evaluated due to the invariance property, expressed through eq. (13) in an arbitrary basis• The expressions represented through the closed contours <~) (eq. ( 15 ) ) are free of singularities as 65
Volume 257, n u m b e r 1,2
PHYSICS LETTERS B
21 March 1991
in a different (but not unrelated) context. We define formally P
in
( zp,°, . .... zp,,, )
-'~ !fp <~lP(n)P(n)(ZP(n))'"~kj-IP~J)°l(ZP(J)) te ' (1) X
Fig. 1. A Contour ~,.
~O~o,_~ (t)~Okj_~eti_
PeS.,
1) " - 3 (_Tp(j_ 1) ) ' " > ,
(17)
~O.ok(t)=P~o(t)P~,
with ~feu) being the contour of fig. 2 attached to the point P(j). The functions ie¢P(t);a have to be interpreted as modified block functions. The subscript c~ has to be regarded as a degeneracy label. The singularity structure of P is easily inferred from the corresponding structure of the functions F (cf. eq. ( 7 ) ) ,
F_mo;, ~ * k = ,,j=2(h t Zp(j),P(1 --,Jki-tipuJ,ki-2"~ --2dim,o ) )Zp(n),P(l)... , Fig. 2. A Contour cCYeu).
functions of varying scaling dimensions ~2 of the involved field operators since the contours avoid the potential singularities of the integrands F and F respectively. The meromorphic dependence on the scaling dimensions is instead concentrated in the factor (Dl/2+D*l/2)/(Dl/2--D*l/2). This explicit extraction of the meromorphic structure is the main advantage of the present approach (which persists in higher orders o f P T (cf. ref. [4] ) ). It is however, as mentioned above, hard (within the formalism) to keep track of the consequences of locality in higher orders of PT. To cope with this problem we will rewrite eq. (14) in such a form that a complete recursive procedure becomes visible which can be iterated mechanically. We pass for this purpose from the up to now considered basis of contours to a new basis displayed in fig. 2. The choice of the new basis is inspired through Felder's work [ 11 ] where the usefulness of this type of contours has been demonstrated ~2 Such a variation m a y be achieved for example by tensoring with exponential functions of one or several free scalar fields.
66
(18)
where the omitted terms are again (in Re) convergent power series in the difference variables z,o~).e(~ ). One should note that the singularities o f f do not depend on the index c~labelling the fusion state appearing after the integrated operator ~a(t). The relations between functions /~p and Fe, attached to different coordinate sectors also follow straightforward (even though somewhat tedious to derive) from the corresponding relations, eq. (8), of the original block functions. The emerging modified braid matrices obey the same type of polynomial consistency relations as the original braid matrices. This follows from the fact that moves in the modified scheme can be resolved into (more complicated) moves in the original scheme and that homotopically equivalent paths in the modified scheme remain equivalent if translated back to the original scheme. The explicit verification of the consistency relations being of some interest for a better understanding of the nature of deformations along critical surfaces will be given in ref. [ 12 ]. Rewriting the result of first order PT, eq. (14), in terms of modified block functions/~ and ff we obtain after repeated use ofeqs. (9), ( 11 ) and (13)
Volume 257, number 1,2
PHYSICS LETTERSB
X~(X)rpi,(z,) X ~i,(g,,) X ...
~d2x (~(x)
x~,(z,) x~.(~) ) l
n--I
.
F L;~,jX~,j,,jC( k, fl, ~)F;:,~:,
j=2
( 19 )
of pole singularities (which have to be subtracted off). Coming back to the representation of the Born term through contours as given in eq. (14) we suppose that the ith of the n operators is equal to the interaction density,
"'
%aj,,o
U)# = B ++ ( kj_ , ~jOkj_ • # • X,~;. ~ ),~V
Let the OPE of two operators ~oX # give rise to an operator ~ , × ¢ with scaling dimension 2d~, near to marginality,
/ D ! n +Dy~/2]
d~,=l-y,
{D)': +D;":)
- ~,f,j ,~;p~,D)/2_DtW:]p p, C(k.,fl, i ) -- -w~¢ i n _
21 March 1991
l kn_ 2 ....
t"kJ-'~P ijfl ~ Okj_
(20) 2 ....
t'k'i 2 i I
"
(21)
The most important aspect of eq. (19) is that the chiral and antichiral modified block functions are again combined on the diagonal of the tensor product of the two (chiral and antichiral) fusion schemes. (We recall once more that the summation indices a j, a j in (19) are to be regarded as degeneracy labels of the new scheme. ) One-valuedness of the correlation functions is therewith manifest in the sector R~ = { (z~, .... z~)l Iz~.~l < Izj.~l for all i
~(~, c~', a, i) = Z x~,~,C(k., fl, i ) . ,a
Relations analogous to eq. ( 13 ) hold for the new braid matrices and the new coefficients C. The second order PT can now be organized as a first order contribution in the modified QRCFT. We have, in other words, completed the first recursion. 4. The breakdown of dilation invariance signalled through a nonvanishing fl function term in the renormalization group equations is in our approach of analytic regularization (i.e. analytic continuation in critical exponents) implied through the occurrence
y<
This coupling in the OPE leads to a contribution to the first sum on the RHS of (14) of the form 1 cos 3 ~ 4 sinai,
~(Fsik')C~i(F°~8')
[ (~k,=~,)
, ~,(___1 +O(yO))%(...)~(... = ~C~ ~Y
).
(22)
The holomorphic and antiholomorphic expressions ~g(F) and ~ ( F ) respectively are regular in y for y ~ 0, ~g ( Ft-)) = t-)
(-)°(t-))+O(y)ff F
(23)
where c~o and c~o are given by the residua (at y = 0 ) of the pole terms 1/(t-zi) and 1/ (F-~fi) o f F a n d F respectively times a factor 2n (from the integration around (27). The product of the residua is easily identified as emerging from a local counterterm supplying a contribution to the//-function of the induced interaction proportional to the coefficient C ~ of the OPE. Turning the last statement around one arrives at the Kadanoff-Brown criterion. There are no flfunctions generated in second order of PT if and only if the OPE of two interaction densities does not lead to operators of dimension two (the interaction itself in particular should not appear as a fused state). Let us now come to the second order of PT (in order to determine the fate of the fl-function in third order) assuming that the Kadanoff-Brown criterion is fulfilled. It follows from this assumption that the RHS of eq. (20) has an unambiguous meaning (that is, no subtractions are needed if the regularization is lifted). Looking on the second order of PT as first PT in the (to first order) deformed scheme we find analogously to the previous calculation, eqs. (22) and (23), as residue of a possible pole 67
Volume 257, number 1,2
PHYSICS LETTERS B
1 COSZl~, - ~ kl 4 sin 3,~ ,,~, ~ ~'(P6~'k~) C ( a ' a ' ) ~/(F~,~, )l (~k,=~,)
= - ,~yCg~,(2n)2(...~/×~) +O(y°) .
(24)
~ × ~ denotes here as before an operator with a scaling dimension near to marginality, d~,= 1 - y , y<< 1. The modified expansion coefficients C are given by
Z
or' ~ o t
" L OtOt'
cpCgo,
21 March 1991
before through a first order calculation with respect to a modified QRCFT (modified to ( n - 1 )th order) and will be given as such by an (n - 1 )-fold sum over intermediate fusion states combined through a product of braid matrices and an n-fold product of coefficients of the OPE. The vanishing condition for the residue is the generalization of the Kadanoff-Brown criterion. It is in our opinion an important problem to find conditions under which the terms in the sums over intermediate fusion states cancel each other.
otoL' f l
(-)
where the constants relations -
(X2)
d ~ are fixed through the
One of us (R.F.) thanks the members of the LPTHE and P.K. Mitter for hospitality extended to him.
(~aoo( t ) (POOl((X)l)[0)
(-) 6~, (-) ( _ ) )
~
References
= x~,~ °° a s ~<~)<~, (
68
[ 1 ] P. Candelas, P.S. Green and T. Hiibsch, in: Proc. String '88 (College Park, MD, 1988 ) p. 155. [ 2 ] L.P. Kadanoff and A.C. Brown, Ann. Phys. 121 (1979) 318. [3] J.U Cardy, J. Phys. A 20 (1987) L891. [4] F. Constantinescu and R. Flume, J. Phys. A 23 (1990) 2971. [5] G. Moore and N. Seiberg, Phys. Lett. B 212 (1988) 451. [6] G. Moore and N. Seiberg, Nucl. Phys. B 313 (1989) 16. [ 7 ] G. Felder, J. Fr6hlich and G. Keller, Commun. Math. Phys. 130 (1990) 1. [8] K.H. Rehren and B. Schroer, Phys. Lett. B 198 (1987) 480. [9] K. Fredenhagen, K.H. Rehren and B. Schroer, Commun. Math. Phys. 125 (1989) 201. [ 10] A. Tsuchiya and Y. Kanie, Lett. Math. Phys. 13 (1987) 303. [ 11 ] G. Felder, Nucl. Phys. B 317 ( 1989 ) 215. [ 12] P. Chaselon, in preparation.