- Email: [email protected]
Renormalization group flows in models with exponential interactions
Volume 234, number 1,2
PHYSICS LETTERS B
4 January 1990
RENORMALIZATION GROUP FLOWS IN MODELS WITH EXPONENTIAL INTERACTIONS M.T. G R I S A R U ,,l, A. L E R D A
b,2,
S. P E N A T I ,,3 and D. Z A N O N c,1
" Department of Physics, Brandeis University, Waltham, MA 02254, USA ~' Center jbr Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA c Dipartimento di Fisica dell' Universitdl di Milano and INFN, Sezione di Milano, 1-20133 Milan, Italy
Received 26 September 1989
We study the renormalization group properties of sine-Gordon type models in the presence of a background charge. We exhibit RG trajectories connecting UV and IR fixed points where the central charge equals that of the minimal models Mp and Mp_ j.
Recently, the study o f critical statistical systems by means o f conformal field theory techniques has been extended to off-critical systems, described as perturbed conformal field theories. In particular, it has been shown that under suitable p e r t u r b a t i o n s minimal models [ 1 ] Mp flow in the infrared to models Mp_ ~ [ 2,3 ], with the central charge decreasing in accordance with the c-theorem [4]. It has also been shown that for these p e r t u r b e d systems conserved charges, constructed out of p o l y n o m i a l s in the stress tensor and its derivatives, exist, at least to lowest order of p e r t u r b a t i o n theory [ 5 ]. The existence of an i n f i n i t e n u m b e r o f exactly conserved charges would make these systems solvable, Indeed in a C o u l o m b gas description o f the m i n i m a l m o d e l s [6 ], perturbations by ( e x p o n e n t i a l ) vertex operators lead to s i n e - G o r d o n or T o d a h a m i l t o n i a n s [7]. It is well known that such h a m i l t o n i a n s a d m i t an infinite set o f conserved charges at the classical and q u a n t u m level and are integrable [8,91. W h a t is not clear is whether these hamiltonians describe other features o f p e r t u r b e d m i n i m a l models, in particular the Mp-+Mp_ 1 flows between fixed points. Work partially supported by the National Science Foundation, under grant PHY-88-18853. z Work partially supported by the US Department of Energy, under grant DE-AC-02-76ER03069. 3 Work partially supported by the National Science Foundation, under grant PHY-88-18853 and by INFN, Sezione di Milano. 88
R e n o r m a l i z a t i o n group studies of the conventional sine-Gordon system reveal the presence of only one fixed point, where the theory becomes free and c = 1 [ 10,1 1 ]. A d d i t i o n a l interaction terms can produce a second fixed point, b u t with c > 1 [ 12 ]. In this letter we study the r e n o r m a l i z a t i o n group properties o f s i n e - G o r d o n type models in the presence o f a b a c k g r o u n d charge and d e m o n s t r a t e the existence o f trajectories with two fixed points and c < 1. In particular, we exhibit trajectories for which the c-function of the system at the ultraviolet ( U V ) and infrared ( I R ) fixed points approaches the value of the central charge of the m i n i m a l models Mp and Mp_ ~, respectively. In this fashion we p r o v i d e a field theory analogue o f the work in refs. [ 2,3 ]. The m o d e l we consider is described by the action 1
+ ? + exp(i~@) + 7 _ e x p ( - i 2 ~ 0 ) + i q R ~ ] ,
(1)
where gu~ and R are the two-dimensional metric and scalar curvature respectively, The coupling to R gives rise to a stress-energy tensor with an i m p r o v e m e n t term
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V, ( North-Holland )
Volume 234, number 1,2
PHYSICS LETTERS B 2~0=2R~0R.
T ~ = 2 ~ g g 8 8 ~~
- ~h.,[7+ exp(i2{o)+7- exp(-i2q}) ] +2iq(r/~. 0~'0.- 0u0.) q ,
(2)
so that q can be interpreted as a background charge. Using complex coordinates 1
_~-x/~(a,+ia2),
1
g=~(a,-ia2),
we define T=- 7T., 7~-- Tze with T = O:q~8=q}- 2iq 02 ~o.
(3)
The classical trace O -= 7~e is given by
+ ( 1 - q 2 ) 7- exp(-i)~p) ] .
q2-=Z;'q~, (4)
We have used the flat-space equations of motion 2 0:0.~o=i2[p+ exp(i2~p)-7_ exp(-i2~0) l
(5)
which also ensure the conservation equation (6)
At the classical level, in complete analogy with the sine-Gordon [8] or Toda systems [9], this model possesses higher grade conserved charges when the condition q2=
l
)~
(8)
where the subscript R denotes renormalized quantities. As shown in ref. [ 1 1 ], the sine-Gordon model exhibits a fixed point in the IR (UV) for 22> 4 ( < 4 ) , where it becomes a free theory. The quantization of the more general action in ( 1 ) follows a very similar pattern. We find again that in the neighborhood of 22=4 coupling constant and wave-function renormalizations remove all divergences, but, in contradistinction to the sine-Gordon case, the presence of a background charge affects the scaling of the various couplings in a nontrivial manner. We define renormalized quantities at an arbitrary mass # by ~0-71/2,- - L"~o WR ,
69= - [ ( 1 +q).) 7+ exp (i2~0)
O._T+O~O=O .
4 January 1990
) 2=Z~-1~.2
7±=122Z±7R.
(9)
We emphasize that 2~0=2R~0R and q~0= qR~R, since the exponential interactions are simply renormalized by normal ordering (which corresponds to a renormalization ofT± and the curvature term in ( 1 ) is not renormalized at all. The dependence of the couplings on the renormalization mass is given by the conventional/?-functions d22R /~-=fla,
dq2R /l~-=flu,
dyR+ l,-~g =-2y~+fl±, (10)
where, to lowest order, we find [ 12,14 ] (7) fla=222y+y_,
is satisfied [ 8,13 ]. The quantization of models such as ( 1 ) was originally discussed for q = 0 , and 7 + = p _ (the sineGordon model). For small values of 22 < 4 short-distance divergences arise only from tadpoles and can be removed by a renormalization of the coupling constant 7 [ 10 ]. For 22 ~ 4 additional ultraviolet divergences appear from the infinite summation of graphs with an arbitrary number of lines joining two vertices [ 1 1,12 ]. These divergences can be removed by a wave-function renormalization (and subtraction of an infinite vacuum energy). Moreover, since exp( _+iA~0) is not renormalized, the wave-function renormalization must be accompanied by a renorrealization of 2 such that
flq=2q27+7_, fl_+=½22y±.
(ll)
Here, and henceforth, 2, 7_+ and q denote renormalized quantities. On the other hand, the scaling behavior of the theory is governed by the trace of the stress-energy tensor, which can be obtained by performing a Weyl transformation
g~,v-~-Q2(a)g,,v on the effective action. The vacuum expectation value of the local trace is then given by
(T~(a)) = - 2 z r I 2 ( a ) 8 W [ J = 0 ] •=. 8£2(a)
,
(12)
89
Volume 234, number 1,2
PHYSICS LETTERSB
4 January 1990
where W[J] is the generating functional for the theory in ( 1 ). Standard a-model techniques give [ 15 ]
dy_ _ 27_ [22 ( ~ + r) - 1 ] . dt
,/g
Straightforward algebra gives dfl 2
= 27r /?~ ~
+ flo
+ {/?+ - 27+ - 2q27+ ) 07-2-
+(fl
+2q27_)0-~---
°)
dt
17 cont'd)
-- ½(2 4 -- 82 2 ) "31-200
( I8)
where all the terms involving the background charge arise when taking the Weyl variation of the two-dimensional curvature R in ( 1 ). The expression in ( 13 ) determines the behavior of the renormalized couplings under a scale transformation a ~ - , a ' e -t
where the integration constant 2o determines different renormalization group trajectories. In the range b 2 = - 2 ( 8 - 2 o ) > 0 , fi~ has two zeros at 2 2 = 4 ¥ b so that for small b the theory has the potential of exhibiting two nearby fixed points. Therefore we restrict the following analysis to this situation. Again for definiteness we choose b> 0 and, in order to keep 2 z positive, b < 4 . From (17), (18), with - b ~ Z 2 - 4 < ~ b , we find the solutions
/ ~ = - -d22 ~=2227+P-,
t - t o = b In \ 2 2 _ 2 2 + j ,
-27
W[0]
+ ~ ( 1 - 2 4 q 2) v @ R ,
(13)
(x2_x2
/~-~ dq2 dt =2q2y+7- '
fi_+=_ dT+ _27_+(~Z 2-I-T-qZ). dt
(14)
The fixed points of the model correspond to the vanishing of the generalized/?-functions which appear on the right-hand side of (14). Therefore at a fixed point, where the background charge has critical value q., (7"~) = ~ 2 ( 1 - 2 4 q ,2) R ,
(15)
and this leads to the identification of the central charge c = 1 - 2 4 q . 2.
(16)
In order to study the flows described by (14), we observe first that r = q / 2 is a renormalization group invariant. For definiteness we choose r > 0 ( r < 0 simply corresponds to interchanging Y+ ~ Y- ). Since the divergence which induces the wave-function renormalization and, correspondingly, a renormalization of Z and q arises for values of 22~ 4, it is consistent with the order of the perturbative calculation to set 22= 4 on the RHS of the first equation in ( 14 ), since this term is already a second order expression in the couplings. Therefore we have to solve d22 dt - 8 7 + 7 _ , d;,+ d-~- =27+ [ 2 2 ( ~ - r ) 90
1] ,
(17)
q2=r222
y2_~.T~Y0 (22__22)1+ (4r/b))t 2 ( 2 2 __22) I--{4r/b)). 2_ l.~ (22__22+)l_(4r/b)22+(22 __22)1+{4r/b).~2
)12 --
1670
(19)
where to, 7o are integration constants. We can study now the behavior of the couplings m the UV (t-~ + ~ ) and IR (t-, - ~ ) limits. We obtain
22-+22+,
q2-~r222+
t~+oo,
72+ --+ ( 2 2 - - 2 2 ) l+(4r/b)22+ ,
7-2 --+ (22--2+2)l-(4r/b).~2+ ,
(20) and ~2 ~,~2
q2_.+r2~2 t--+--O0,
72~_;,()2__22)1--(4r/0) 22- '
y2__.(,~2__22)l+(4r/b)22
(21) Therefore the model has both an IR and an UV fixed point [i.e. vanishing & t h e right-hand side in (17) ], whenever the constants r and b satisfy 4r 1 - ~- ( 4 + b ) >~0.
(22)
In particular, for the case of a strict inequality 7+ = 7 - =0, both in the UV and in the IR, while 2 and q flow as indicated in (20) and (21). Choosing instead in (22) the equality condition, i.e.
V o l u m e 234, n u m b e r 1,2
b=
PHYSICS LETTERS B
16r
(23)
- 1 -4r'
we obtain the fixed points 1 -8r
,~2+ = 4 ~ _ 4 r 2+=0,
,
'
1-8r
qs- = 4 r 2 l - 4 r '
7_=0
(24)
4 J a n u a r y 1990
has in the IR limit the same central charge as the Mp_ minimal model, while in the UV limit it approaches for large p the value corresponding to the Mp conformal theory. Also, since the anomalous dimension o f the exponentials exp ( +_i2~0) is A+ = J22T-2q, we find in the IR limit A+-~l
- '~(p-I) --~(-1,1)
,
A --
p+l__ = A ( p(3.1) _l ) p-- 1
(29)
in the ultraviolet limit, and 4
22 _
q2_ _
1 --4r'
7+ = c o n s t a n t ,
and in the UV limit to O( 1/p 2)
4r 2 1 --4r
'
7- = 0
(25)
in the infrared limit. From (16), we also obtain the values of the central charge at the fixed points, c+_=1-24q 2, in t h e U V ( + ) and in the IR ( - ) limits respectively. We notice that c+ > c_, i.e. the central charge decreases when flowing towards the IR region. The consistency o f this result with Zamolodchikov's c-theorem [4 ] indicates that, in spite of the lack o f manifest hermiticity of the action in ( l ) , the theory may be unitary. Finally we show that for an appropriate choice o f the ratio q/Z, our model shares at the fixed points c o m m o n features with the minimal models, so that it behaves like an interpolating theory between Mp ( U V ) and Mp_l ( I R ) [2]. To simplify the analysis we consider the case in (23) [the more general condition in (22) leads to similar results but the algebra is more cumbersome]. We choose q
4p'
q2 _
1
q2+ _
4p(p- 1) '
p-2 4p2(p - 1) '
(27)
and the corresponding values of the central charge --I
c+=l-
p-l=j{~5), p+l 1 =A/~!_~),
(30)
where AI~,,, ) denote the conformal weights of the pris) mary fields (O/m,~) of the M, minimal model. Hence, the two exponential interactions have the same anomalous dimensions as the fields ~01~!3) , ~01~)_ ~) in the UV limit and g,{P_q~)), ~0(3.~)(P-l)in the IR limit. It is natural to identify in this way the relevant perturbations that in Zamolodchikov's approach [ 2 ] drive the RG flows. To conclude, we observe that, according to our perturbative calculations, a generic exponential operator e x p ( i a ¢ ) renormalizes in such a way that a / 2 is a renormalization group invariant. In particular, at the fixed points oz /2 = a + / 2 + . Choosing t h e i R value .... ) = ( l - m )
P
1
(26)
where p is an integer and, substituting r = 1/4p in (24) and (25), we obtain
c
2 p
3_--* (1 + p ) ( p - 2 ) p(p-l)
a_=a(
1
2 -
~+~l
6 p(p--1) ' 6
p--2 --1 p2(/-~_ 1 )
6 +O(~2)" p ( p + 1)
(28)
We thus find that with the choice in (26), the system
(31) so that the operator ¢/$7~1)) =exp(ic~l,PnTnl))tp) represents a primary field in the Mp_ 1 minimal model, and 2~=p/(p-l) [cf. (25) with r = l / 4 p ] , we determine a + . To O ( l / p 2) we find that the 1R value d (t~Ln) (p- ~) becomes 3 (n,2n-(p) rn) in the UV limit. We stress that although these results are in complete agreement with those in refs. [2,3], our analysis is only at the level ofconformal weights, not of conformal primary fields. A complete identification of the theory at the fixed points with the minimal models requires a detailed study of the Hilbert space, suitably restricted by a projection of the null states [ 16]. 91
Volume 234, number 1,2
PHYSICS LETTERS B
Further work, including the calculation of the stress tensor two-point correlation functions will be reported elsewhere [ 14 ].
Daniela Zanon thanks Brandeis and Harvard Universities for their hospitality.
References [1] A.A. Balavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl, Phys. B 241 (1984) 333; D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575. [2] A.B. Zamolodchikov, Soy. J. Nucl. Phys. 46 (1987) 1090. [ 3 ] A.W.W. Ludwig and J.L. Cardy, Nucl. Phys. B 285 [ FS 19 ] (1987) 687. [4] A.B. Zamolodchikov, JETP Lett. 43 (1986) 730. 15 ] A.B, Zamolodchikov, Kyoto preprint (1988), Adv. Slud. Pure Math,, to appear; Rutherford-Appleton preprint RAL89-001 (1989). [ 6 ] B.L. Feigin and D.B. Fuchs, Representations of the Virasoro algebra, in: Representations of infinite-dimensional Lie groups and Lie algebras (Gordon and Breach, New York, 1986);
92
4 January 1990
VI.S. Dotsenko and V.A. Fateev, Nucl. Phys. B 240 [ FS 12 ] (1984) 312;B25l [FS13] (1985) 691. [7 ] T. Eguchi and S.K. Yang, Phys. Len. B 224 (1989) 373. [81R. Sasaki and I. Yamanaka, Adv. Stud. Pure Math. 16 ( 1988 ) 27 l, and references therein. [9] V.G. Drinfeld and V.V. Sokolov, J. Soy. Math. 30 (1985) 1977; D. Olive and N. Turok, Nucl. Phys. B 257 [FS14] (1985) 277. [10] S. Coleman, Phys. Rev. D 11 (1975) 2088. [ 11 ] D.J. Amit, Y.Y. Goldschmidt and G. Grinstein, J. Phys. A 13 (1980) 585; T. Ohta, Prog. Theor. Phys. 60 (1978) 968; P.B. Weigmann, J. Phys. C 11 (1978) 1585; C. Lovelace, Nucl. Phys. B 273 (1986) 413. [ 12] D. Boyanovsky, J. Phys. A 22 (1989) 2601; D. Boyanovsky and R. Holman, Pittsburgh preprint PITT89-07 (1989). [ 131 T.L. Curtright and C.B. Thorn, Phys. Rev. Lett. 48 ( 1982 ) 1309; H. Yoshii, Hiroshima preprint RRK 89-5 (1989). [14]M.T. Grisaru, A. Lerda, S. Penati and D. Zanon, m preparation. [ 15 ] A.A. Tseytlin, Phys. Len. B 178 ( 1986 ) 34; G.M. Shore, Nucl. Phys. B 286 (1987) 349; G. Curci and G. Paffuti, Nucl. Phys. B 286 (1987) 399; H. Osborn, Nucl. Phys. B 294 ( 1987 ) 595. [ 16 ] F.A. Smirnov, Leningrad preprint LOMI-E-4-89 (1989); A. LeClair, Princeton preprint PUPT-1124 (1989).
PHYSICS LETTERS B
4 January 1990
RENORMALIZATION GROUP FLOWS IN MODELS WITH EXPONENTIAL INTERACTIONS M.T. G R I S A R U ,,l, A. L E R D A
b,2,
S. P E N A T I ,,3 and D. Z A N O N c,1
" Department of Physics, Brandeis University, Waltham, MA 02254, USA ~' Center jbr Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA c Dipartimento di Fisica dell' Universitdl di Milano and INFN, Sezione di Milano, 1-20133 Milan, Italy
Received 26 September 1989
We study the renormalization group properties of sine-Gordon type models in the presence of a background charge. We exhibit RG trajectories connecting UV and IR fixed points where the central charge equals that of the minimal models Mp and Mp_ j.
Recently, the study o f critical statistical systems by means o f conformal field theory techniques has been extended to off-critical systems, described as perturbed conformal field theories. In particular, it has been shown that under suitable p e r t u r b a t i o n s minimal models [ 1 ] Mp flow in the infrared to models Mp_ ~ [ 2,3 ], with the central charge decreasing in accordance with the c-theorem [4]. It has also been shown that for these p e r t u r b e d systems conserved charges, constructed out of p o l y n o m i a l s in the stress tensor and its derivatives, exist, at least to lowest order of p e r t u r b a t i o n theory [ 5 ]. The existence of an i n f i n i t e n u m b e r o f exactly conserved charges would make these systems solvable, Indeed in a C o u l o m b gas description o f the m i n i m a l m o d e l s [6 ], perturbations by ( e x p o n e n t i a l ) vertex operators lead to s i n e - G o r d o n or T o d a h a m i l t o n i a n s [7]. It is well known that such h a m i l t o n i a n s a d m i t an infinite set o f conserved charges at the classical and q u a n t u m level and are integrable [8,91. W h a t is not clear is whether these hamiltonians describe other features o f p e r t u r b e d m i n i m a l models, in particular the Mp-+Mp_ 1 flows between fixed points. Work partially supported by the National Science Foundation, under grant PHY-88-18853. z Work partially supported by the US Department of Energy, under grant DE-AC-02-76ER03069. 3 Work partially supported by the National Science Foundation, under grant PHY-88-18853 and by INFN, Sezione di Milano. 88
R e n o r m a l i z a t i o n group studies of the conventional sine-Gordon system reveal the presence of only one fixed point, where the theory becomes free and c = 1 [ 10,1 1 ]. A d d i t i o n a l interaction terms can produce a second fixed point, b u t with c > 1 [ 12 ]. In this letter we study the r e n o r m a l i z a t i o n group properties o f s i n e - G o r d o n type models in the presence o f a b a c k g r o u n d charge and d e m o n s t r a t e the existence o f trajectories with two fixed points and c < 1. In particular, we exhibit trajectories for which the c-function of the system at the ultraviolet ( U V ) and infrared ( I R ) fixed points approaches the value of the central charge of the m i n i m a l models Mp and Mp_ ~, respectively. In this fashion we p r o v i d e a field theory analogue o f the work in refs. [ 2,3 ]. The m o d e l we consider is described by the action 1
+ ? + exp(i~@) + 7 _ e x p ( - i 2 ~ 0 ) + i q R ~ ] ,
(1)
where gu~ and R are the two-dimensional metric and scalar curvature respectively, The coupling to R gives rise to a stress-energy tensor with an i m p r o v e m e n t term
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V, ( North-Holland )
Volume 234, number 1,2
PHYSICS LETTERS B 2~0=2R~0R.
T ~ = 2 ~ g g 8 8 ~~
- ~h.,[7+ exp(i2{o)+7- exp(-i2q}) ] +2iq(r/~. 0~'0.- 0u0.) q ,
(2)
so that q can be interpreted as a background charge. Using complex coordinates 1
_~-x/~(a,+ia2),
1
g=~(a,-ia2),
we define T=- 7T., 7~-- Tze with T = O:q~8=q}- 2iq 02 ~o.
(3)
The classical trace O -= 7~e is given by
+ ( 1 - q 2 ) 7- exp(-i)~p) ] .
q2-=Z;'q~, (4)
We have used the flat-space equations of motion 2 0:0.~o=i2[p+ exp(i2~p)-7_ exp(-i2~0) l
(5)
which also ensure the conservation equation (6)
At the classical level, in complete analogy with the sine-Gordon [8] or Toda systems [9], this model possesses higher grade conserved charges when the condition q2=
l
)~
(8)
where the subscript R denotes renormalized quantities. As shown in ref. [ 1 1 ], the sine-Gordon model exhibits a fixed point in the IR (UV) for 22> 4 ( < 4 ) , where it becomes a free theory. The quantization of the more general action in ( 1 ) follows a very similar pattern. We find again that in the neighborhood of 22=4 coupling constant and wave-function renormalizations remove all divergences, but, in contradistinction to the sine-Gordon case, the presence of a background charge affects the scaling of the various couplings in a nontrivial manner. We define renormalized quantities at an arbitrary mass # by ~0-71/2,- - L"~o WR ,
69= - [ ( 1 +q).) 7+ exp (i2~0)
O._T+O~O=O .
4 January 1990
) 2=Z~-1~.2
7±=122Z±7R.
(9)
We emphasize that 2~0=2R~0R and q~0= qR~R, since the exponential interactions are simply renormalized by normal ordering (which corresponds to a renormalization ofT± and the curvature term in ( 1 ) is not renormalized at all. The dependence of the couplings on the renormalization mass is given by the conventional/?-functions d22R /~-=fla,
dq2R /l~-=flu,
dyR+ l,-~g =-2y~+fl±, (10)
where, to lowest order, we find [ 12,14 ] (7) fla=222y+y_,
is satisfied [ 8,13 ]. The quantization of models such as ( 1 ) was originally discussed for q = 0 , and 7 + = p _ (the sineGordon model). For small values of 22 < 4 short-distance divergences arise only from tadpoles and can be removed by a renormalization of the coupling constant 7 [ 10 ]. For 22 ~ 4 additional ultraviolet divergences appear from the infinite summation of graphs with an arbitrary number of lines joining two vertices [ 1 1,12 ]. These divergences can be removed by a wave-function renormalization (and subtraction of an infinite vacuum energy). Moreover, since exp( _+iA~0) is not renormalized, the wave-function renormalization must be accompanied by a renorrealization of 2 such that
flq=2q27+7_, fl_+=½22y±.
(ll)
Here, and henceforth, 2, 7_+ and q denote renormalized quantities. On the other hand, the scaling behavior of the theory is governed by the trace of the stress-energy tensor, which can be obtained by performing a Weyl transformation
g~,v-~-Q2(a)g,,v on the effective action. The vacuum expectation value of the local trace is then given by
(T~(a)) = - 2 z r I 2 ( a ) 8 W [ J = 0 ] •=. 8£2(a)
,
(12)
89
Volume 234, number 1,2
PHYSICS LETTERSB
4 January 1990
where W[J] is the generating functional for the theory in ( 1 ). Standard a-model techniques give [ 15 ]
dy_ _ 27_ [22 ( ~ + r) - 1 ] . dt
,/g
Straightforward algebra gives dfl 2
= 27r /?~ ~
+ flo
+ {/?+ - 27+ - 2q27+ ) 07-2-
+(fl
+2q27_)0-~---
°)
dt
17 cont'd)
-- ½(2 4 -- 82 2 ) "31-200
( I8)
where all the terms involving the background charge arise when taking the Weyl variation of the two-dimensional curvature R in ( 1 ). The expression in ( 13 ) determines the behavior of the renormalized couplings under a scale transformation a ~ - , a ' e -t
where the integration constant 2o determines different renormalization group trajectories. In the range b 2 = - 2 ( 8 - 2 o ) > 0 , fi~ has two zeros at 2 2 = 4 ¥ b so that for small b the theory has the potential of exhibiting two nearby fixed points. Therefore we restrict the following analysis to this situation. Again for definiteness we choose b> 0 and, in order to keep 2 z positive, b < 4 . From (17), (18), with - b ~ Z 2 - 4 < ~ b , we find the solutions
/ ~ = - -d22 ~=2227+P-,
t - t o = b In \ 2 2 _ 2 2 + j ,
-27
W[0]
+ ~ ( 1 - 2 4 q 2) v @ R ,
(13)
(x2_x2
/~-~ dq2 dt =2q2y+7- '
fi_+=_ dT+ _27_+(~Z 2-I-T-qZ). dt
(14)
The fixed points of the model correspond to the vanishing of the generalized/?-functions which appear on the right-hand side of (14). Therefore at a fixed point, where the background charge has critical value q., (7"~) = ~ 2 ( 1 - 2 4 q ,2) R ,
(15)
and this leads to the identification of the central charge c = 1 - 2 4 q . 2.
(16)
In order to study the flows described by (14), we observe first that r = q / 2 is a renormalization group invariant. For definiteness we choose r > 0 ( r < 0 simply corresponds to interchanging Y+ ~ Y- ). Since the divergence which induces the wave-function renormalization and, correspondingly, a renormalization of Z and q arises for values of 22~ 4, it is consistent with the order of the perturbative calculation to set 22= 4 on the RHS of the first equation in ( 14 ), since this term is already a second order expression in the couplings. Therefore we have to solve d22 dt - 8 7 + 7 _ , d;,+ d-~- =27+ [ 2 2 ( ~ - r ) 90
1] ,
(17)
q2=r222
y2_~.T~Y0 (22__22)1+ (4r/b))t 2 ( 2 2 __22) I--{4r/b)). 2_ l.~ (22__22+)l_(4r/b)22+(22 __22)1+{4r/b).~2
)12 --
1670
(19)
where to, 7o are integration constants. We can study now the behavior of the couplings m the UV (t-~ + ~ ) and IR (t-, - ~ ) limits. We obtain
22-+22+,
q2-~r222+
t~+oo,
72+ --+ ( 2 2 - - 2 2 ) l+(4r/b)22+ ,
7-2 --+ (22--2+2)l-(4r/b).~2+ ,
(20) and ~2 ~,~2
q2_.+r2~2 t--+--O0,
72~_;,()2__22)1--(4r/0) 22- '
y2__.(,~2__22)l+(4r/b)22
(21) Therefore the model has both an IR and an UV fixed point [i.e. vanishing & t h e right-hand side in (17) ], whenever the constants r and b satisfy 4r 1 - ~- ( 4 + b ) >~0.
(22)
In particular, for the case of a strict inequality 7+ = 7 - =0, both in the UV and in the IR, while 2 and q flow as indicated in (20) and (21). Choosing instead in (22) the equality condition, i.e.
V o l u m e 234, n u m b e r 1,2
b=
PHYSICS LETTERS B
16r
(23)
- 1 -4r'
we obtain the fixed points 1 -8r
,~2+ = 4 ~ _ 4 r 2+=0,
,
'
1-8r
qs- = 4 r 2 l - 4 r '
7_=0
(24)
4 J a n u a r y 1990
has in the IR limit the same central charge as the Mp_ minimal model, while in the UV limit it approaches for large p the value corresponding to the Mp conformal theory. Also, since the anomalous dimension o f the exponentials exp ( +_i2~0) is A+ = J22T-2q, we find in the IR limit A+-~l
- '~(p-I) --~(-1,1)
,
A --
p+l__ = A ( p(3.1) _l ) p-- 1
(29)
in the ultraviolet limit, and 4
22 _
q2_ _
1 --4r'
7+ = c o n s t a n t ,
and in the UV limit to O( 1/p 2)
4r 2 1 --4r
'
7- = 0
(25)
in the infrared limit. From (16), we also obtain the values of the central charge at the fixed points, c+_=1-24q 2, in t h e U V ( + ) and in the IR ( - ) limits respectively. We notice that c+ > c_, i.e. the central charge decreases when flowing towards the IR region. The consistency o f this result with Zamolodchikov's c-theorem [4 ] indicates that, in spite of the lack o f manifest hermiticity of the action in ( l ) , the theory may be unitary. Finally we show that for an appropriate choice o f the ratio q/Z, our model shares at the fixed points c o m m o n features with the minimal models, so that it behaves like an interpolating theory between Mp ( U V ) and Mp_l ( I R ) [2]. To simplify the analysis we consider the case in (23) [the more general condition in (22) leads to similar results but the algebra is more cumbersome]. We choose q
4p'
q2 _
1
q2+ _
4p(p- 1) '
p-2 4p2(p - 1) '
(27)
and the corresponding values of the central charge --I
c+=l-
p-l=j{~5), p+l 1 =A/~!_~),
(30)
where AI~,,, ) denote the conformal weights of the pris) mary fields (O/m,~) of the M, minimal model. Hence, the two exponential interactions have the same anomalous dimensions as the fields ~01~!3) , ~01~)_ ~) in the UV limit and g,{P_q~)), ~0(3.~)(P-l)in the IR limit. It is natural to identify in this way the relevant perturbations that in Zamolodchikov's approach [ 2 ] drive the RG flows. To conclude, we observe that, according to our perturbative calculations, a generic exponential operator e x p ( i a ¢ ) renormalizes in such a way that a / 2 is a renormalization group invariant. In particular, at the fixed points oz /2 = a + / 2 + . Choosing t h e i R value .... ) = ( l - m )
P
1
(26)
where p is an integer and, substituting r = 1/4p in (24) and (25), we obtain
c
2 p
3_--* (1 + p ) ( p - 2 ) p(p-l)
a_=a(
1
2 -
~+~l
6 p(p--1) ' 6
p--2 --1 p2(/-~_ 1 )
6 +O(~2)" p ( p + 1)
(28)
We thus find that with the choice in (26), the system
(31) so that the operator ¢/$7~1)) =exp(ic~l,PnTnl))tp) represents a primary field in the Mp_ 1 minimal model, and 2~=p/(p-l) [cf. (25) with r = l / 4 p ] , we determine a + . To O ( l / p 2) we find that the 1R value d (t~Ln) (p- ~) becomes 3 (n,2n-(p) rn) in the UV limit. We stress that although these results are in complete agreement with those in refs. [2,3], our analysis is only at the level ofconformal weights, not of conformal primary fields. A complete identification of the theory at the fixed points with the minimal models requires a detailed study of the Hilbert space, suitably restricted by a projection of the null states [ 16]. 91
Volume 234, number 1,2
PHYSICS LETTERS B
Further work, including the calculation of the stress tensor two-point correlation functions will be reported elsewhere [ 14 ].
Daniela Zanon thanks Brandeis and Harvard Universities for their hospitality.
References [1] A.A. Balavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl, Phys. B 241 (1984) 333; D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575. [2] A.B. Zamolodchikov, Soy. J. Nucl. Phys. 46 (1987) 1090. [ 3 ] A.W.W. Ludwig and J.L. Cardy, Nucl. Phys. B 285 [ FS 19 ] (1987) 687. [4] A.B. Zamolodchikov, JETP Lett. 43 (1986) 730. 15 ] A.B, Zamolodchikov, Kyoto preprint (1988), Adv. Slud. Pure Math,, to appear; Rutherford-Appleton preprint RAL89-001 (1989). [ 6 ] B.L. Feigin and D.B. Fuchs, Representations of the Virasoro algebra, in: Representations of infinite-dimensional Lie groups and Lie algebras (Gordon and Breach, New York, 1986);
92
4 January 1990
VI.S. Dotsenko and V.A. Fateev, Nucl. Phys. B 240 [ FS 12 ] (1984) 312;B25l [FS13] (1985) 691. [7 ] T. Eguchi and S.K. Yang, Phys. Len. B 224 (1989) 373. [81R. Sasaki and I. Yamanaka, Adv. Stud. Pure Math. 16 ( 1988 ) 27 l, and references therein. [9] V.G. Drinfeld and V.V. Sokolov, J. Soy. Math. 30 (1985) 1977; D. Olive and N. Turok, Nucl. Phys. B 257 [FS14] (1985) 277. [10] S. Coleman, Phys. Rev. D 11 (1975) 2088. [ 11 ] D.J. Amit, Y.Y. Goldschmidt and G. Grinstein, J. Phys. A 13 (1980) 585; T. Ohta, Prog. Theor. Phys. 60 (1978) 968; P.B. Weigmann, J. Phys. C 11 (1978) 1585; C. Lovelace, Nucl. Phys. B 273 (1986) 413. [ 12] D. Boyanovsky, J. Phys. A 22 (1989) 2601; D. Boyanovsky and R. Holman, Pittsburgh preprint PITT89-07 (1989). [ 131 T.L. Curtright and C.B. Thorn, Phys. Rev. Lett. 48 ( 1982 ) 1309; H. Yoshii, Hiroshima preprint RRK 89-5 (1989). [14]M.T. Grisaru, A. Lerda, S. Penati and D. Zanon, m preparation. [ 15 ] A.A. Tseytlin, Phys. Len. B 178 ( 1986 ) 34; G.M. Shore, Nucl. Phys. B 286 (1987) 349; G. Curci and G. Paffuti, Nucl. Phys. B 286 (1987) 399; H. Osborn, Nucl. Phys. B 294 ( 1987 ) 595. [ 16 ] F.A. Smirnov, Leningrad preprint LOMI-E-4-89 (1989); A. LeClair, Princeton preprint PUPT-1124 (1989).