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Topological charge on the lattice. The 2D CPN−1 model
Physics Letters B 306 (1993) 108-114 North-Holland
PHYSICS LETTERS B
Topological charge on the lattice. The 2D
C P N - 1 model
Federico Farehioni and Alessandro Papa Dipartimento di Fisica dell'Universiti~ and INFN, Piazza Torricelli 2, 1-56126 Pisa, Italy
Received 5 February 1993 Editor: R. Gatto We calculate the first two non-zero terms in the perturbative expansion of the additive and multiplicative renormalizations of the topological susceptibility, )~, in the 2D CP s - l model, when a density of topological charge is defined as a local operator on the lattice. We compare our results with some direct determinations, by Monte Carlo techniques.
1. Introduction
The topological charge of a spin field z (x) is defined by
Two-dimensional CP s - t models play a very important role in quantum field theory because of their similarities with non-abelian gauge theories. The models are asymptotically free and exhibit spontaneous mass generation. They also have a non-trivial topological structure. The action of the 2D CP N- 1 model, is [ 1 ] S=-~
2
f dExD--~z • D uz,
Du = Ou + iAu,
Au = ½i[-~(x).Ouz(x)-Ou-Z(x).z(x)],
(1)
(3)
and T is a coupling which plays the role of a temperature. At the quantum level the constraint (3) generates mass gap and asymptotic freedom. The gauge field A u is composite, but it behaves like a physical field after quantization. The theory is invariant under global S U ( N ) transformations of z. It is also invariant under local U ( 1 ) transformation z --~ ei~Cx)z,
A~ ~ A , - Oua.
(4)
The CP u - t model is equivalent to the 0 ( 3 ) amodel for N = 2. 108
,j
d x~u~Duz.D~z = ~
d x u A u.
(5)
Q is the number of times the gauge field A u winds the circle at infinity. The topological susceptibility X is a renormalization group invariant quantity, which measures the amount of topological excitations of the vacuum; X is defined as the correlation at zero momentum of two topological charge density operators, Q (x):
(2)
where z (x) is an N-component complex scalar field, constrained by the condition -Z(x) • z ( x ) = 1
Q = -~
x = fdax(oIT[Q(x)Q(O)]lO);
(6)
Q (x) is the divergence of a topological current Ku [2,3 ], Q (x) = OuKu ( x ) . The prescription defining the product of operators in eq. (6) is [4] ( O I T [ Q ( x ) Q ( O ) ll0) = Ou(OlT[Ku(x)Q(O)]10).
(7)
This prescription eliminates the contribution of possible contact terms (i.e. terms proportional to the 6 function or its derivatives) when x ~ O. The only known tool to compute X is simulation of the theory on the lattice. Many attempts to determine the topological susceptibility, X, were based on geometric definitions of the topological charge. The Elsevier SciencePublishers B.V.
Volume306, number 1,2
PHYSICSLETTERSB
so called geometrical method [ 5-9 ] uses an interpolation among spin variables to assign an integer topological charge to each lattice configuration. Since Monte Carlo data did not show the expected scaling behavior, the determination o f z was not possible. This failure was explained by the presence of exceptional configurations, called dislocations, whose unphysical contribution does not vanish in the continuum limit [ 10]. The geometrical approach meets similar difficulties in 4D non-abelian gauge theories [ 11 ]. Monte Carlo simulations performed on CP N-~ models [ 12 ] have recently shown that the contribution of dislocations is still effective for N = 4, while the geometrical approach seems to work for greater values of N. There alternative definitions of Z on the lattice have been considered: the field theoretical approach relying on a definition of topological charge density as a local operator, and the cooling method, which measures the topological charge on cooled configurations. The consistency of these two procedures was demonstrated for 4D SU (2) and for SU (3) non-abelian gauge theories and for 2D 0 ( 3 ) non-linear a-model or CP l model. For a review see refs. [ 13-15 ]. In the field theoretical method, a topological charge density operator is defined as having the appropriate classical continuum limit [ 16 ]: QL(x)a~0 ~ adQ(x) + O(ad+2),
(8)
where a is the lattice spacing and d the space-time dimension. As a consequence, Z L, the correlation at zero momentum of two QL (x) operators is given by
z L = (x~QL(x)QL(0) )
'(xZ
= V
(
QL(x))2
)
;
(9)
Z L is connected to Z by a non-trivial relation. The presence of irrelevant operators of higher dimension in QL (X) induces quantum corrections. Eq. (8) must be corrected by including a renormalization constant Z(fl) [17]: QL(x)a-.0 = adZ(fl)Q(x) + O(ad+2).
(10)
For S U ( N ) gauge theories, fl = 2N/g g, with go the coupling constant, for the 2D CP t¢-1 model, fl =
27 May 1993
2/NT; Z (fl) is a finite function offl, which will tend to 1 when the continuum limit is approached. If a perturbative expansion makes sense, then zl
z2
Z(fl) = 1 + ~ - + ~--~+....
(ll)
Furthermore, there are contributions of contact terms originating when x ~ 0 in eq. (9). A prescription equivalent to eq. (7) does not exist on the lattice, and therefore the contribution of the contact terms must be isolated and subtracted. These contact terms appear as mixings with the action density S (x) and with the unity operator I, which are the only available operators with equal dimension or lower. In formulae,
zL(fl) = adZ(fl)2 Z + adA(fl)(S(x)) + P(fl)(I)
+ O(ad+2).
(12)
In eq. (12) the quantity (S (x)) is intended to be the non-perturbative part of the expectation value of the action density, i.e. it is a signal of dimension d. In this paper we perform the calculation of the first two non-zero terms of the additive and multiplicative renormalizations of the topological susceptibility on the lattice, i.e. Z(fl) and P(fl), in the CP ~¢-1 model, for a general value of N. As we will see later, at the perturbative order of our calculation, the mixing with the action density can be neglected. Previous works have shown that the renormalization functions are well-defined quantities and that they are well approximated by the perturbative expansion (for a review, see refs. [ 13,14 ] for SU (2) non-abelian gauge theories and [15] for 2D 0 ( 3 ) a-model). A non-perturbative method has been recently developed, in which the renormalization functions are determined exploiting Monte Carlo techniques, through the observation of the topological charge and susceptibility on configurations obtained thermalizing an assigned instantonic configuration. This method, suggested by Teper [ 18] for the calculation of the multiplicative renormalization Z ( i l l has been extended to the additive renormalizations [ 19] and tested on the 0 ( 3 ) a-model [19,15] and, successively, on the CP t¢-1 models [12]. 109
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We compare the results of our calculations with the numerical data obtained in ref. [ 12 ] for the special values of N = 2 and N = 4.
The perturbation expansion suffers from infrared divergences, which can be cured by adding a magnetic term to the action
2. Renormalization and topological susceptibility
SM = -- -~
1/
2.1. The model
=
We regularize the theory on the lattice by taking the action [20]
sL= --ZlnIo(4]-fx+u'ZxO,
(13)
where I0 (x) is the modified Bessel function of order zero. For both continuum and lattice calculations we make use of Hikami parametrization of the fields z ( x ) [21]
zu =-zN =0";
zi=r/i
(i= 1,...,N-I)
(14)
_
d2x HO" (X)
71fd2xH~/1 S - ' ' , _
18)
Indeed S~ explicitly breaks the S U ( N ) invariance and acts as a mass term for the q field. The SU(N)invariant quantities (and the relations among them) are free of infrared divergences, and have a welldefined limit for H ~ 0.
2.2. Topological charge density renormalization Our regularized version of topological charge QL (X) is
QL (X) = -- ~
i
eUvTr (P (x)AuP (x)A~P ( x ) ) ,
and 0"2
(
(19)
1 + 0"' = --T--,ni
r/[ = x / 2 ( 1 + 0",)
(15)
where
Pij(x) = ~(x)zj(x)
with N-I
(a') 2 + E
r/[~"i = 1,
(16)
(20)
and Au is a symmetrized version of lattice derivative
i= 1
which allows us to eliminate a'. In the continuum calculations the gauge field A u is eliminated by use of the equation of motion. The quantization of the action introduces an additional ill-defined (infinite) determinant which allows to write the functional measure over the fields q' and r/'7 in an S U ( N ) invariant way. The effect of this determinant is to modify S L by an amount ~"~ In (a'(1 + a ' ) u - 2 ) .
(17)
x
In dimensional regularization, the measure term does not contribute, as a consequence of the rule f dak = 0~ 8a(0), where d is the space dimension. 110
Auf (x) = f (x + It) - f (x - lt) 2a
(21)
On the continuum, Q ( x ) is invariant under the renormalization group if a suitable renormalization scheme is chosen, for example the MS scheme. This can be directly checked by calculating the renorrnalized two-point proper function of the complex fields r/' and ~' with one insertion o f Q (x), F ~ s (in what follows the prime is suppressed). Since we are only interested in the renormalization of Q (x), in our calculations we putp + q = 0 after having factorized the treeorder two-point function .Q~,rtree= -(i/8~)Sijeu~p~,q~ (p and q are the external momenta). We define F(2~, by the relation FQ~. = /-,tree . Q~, × FQ~, and we compute FQ~ at p + q = O. By a calculation at two loops we
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PHYSICS LETTERSB
©
27 May 1993
Zq = 1 + N
1 2~e
3N2( 1 + - T - 2n~
Fig. 1. Diagrams contributing at one loop to the two-point proper function with one insertion of Q (x). White and black blobs indicate respectively the operator and the Lagrangian vertices; black squares include vertices coming from the measure term and from other terms vanishing in the continuum limit: the corresponding graphs are absent in the continuum.
1 ( l n 4 n - 7E)) Tr 4n 2
1 (ln4/r_ yE) ) Tr2 4n
+ O(T)),
(23)
while Z r is the renormalization of the coupling constant [21 ]
Z r = 1 + - ~N ( 2 ~1
4~1 ( l n 4 n _ y s ) ) T r
+
(24)
/t is an energy scale. Since the r.h.s, ofeq. (22) is finite when written in terms of renormalized quantities, it follows that Q ( x ) does not renormalize up to two loops. A finite multiplicative renormalization connects the matrix elements of QL (x) with those of Q (x), defined in the MS scheme. The antisymmetry of QL (x) forbids mixings with any other SU(N) invariant operator of dimension two. We write QL(x)
= Z(T)Q(x).
(25)
Since the anomalous dimension of Q (x), 7Q (Tr), is zero, Z (T) must satisfy the equation Fig. 2. Diagrams contributing at two loops to the two-point proper function with one insertion of Q (x).
d yQ(Tr) = l a - ~ l n Z ( T ) l r ,
a = 0
(26)
and therefore must be a finite function of T, which can be determined by imposing [22]
find (see fig. 1 and fig. 2)
-fig ( Tr, Hr, lZ; p, q ) = Z - [ ( T ) Z'ff'g ( T, Iz, a ) F~,~ Ft~-s~(Tr, Hr,/z; p, q )Ip+q=0
xF~-~,~(T,H,a;p,q),
(27)
= Z,s(T,e)F~,l(T,H,e;p,q)lp+¢=o iz 2 =l+N[lln(~rr)]Tr 3N2 l [ + T
In •Hr ).]
-r
+
O(Tr3),
(22)
where T = Tdz-~Zr, H = HrZrZ~-ln; here F h-g and F 8 are respectively the renormalized and the bare function; e = d - 2 ; Zq is the renormalization constant of the field r/ [21 ]
~/ 7MSTMS [/2 with T = TrZT-Ms, and H = aJrt.~ T ~,t/ ; F~Lq is defined by F~L~ = Ft~~ × Ft)L~, and FQ~,I is the twopoint proper function on the lattice, with one insertion of QL (x), and calculated in the limit a ~ 0. The functions Z ~ s (T,/1, a) and Z ~ s (T,/z, a) are respectively the field and the coupling renormalization constants that allow us to get, starting from the lattice regularization, the same Green functions as those of the MS renormalization scheme. In particular, Z (T) can be obtained by imposing eq. (27) at p + q = 0.
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In order to calculate Z ( T ) at two loops, we need to know Z ~ s (T, p, a ) to two loops and Z ~ s (T,/1, a ) to one loop. Following ref. [23] we determine them by imposing the relation Fig. 3. Diagram contributing at three loops to the mixing with the unity operator.
Zn (T, e ) - l G (x, ZrTr# -~ )
--
= z~s(T,p,a)
-l
L
G (X, ZT'~'gTr),
(28)
where G(x, ZrTd~-') and GL(x, ZrT,) are the S U ( N ) invariant two-point Green functions
(P(x)P(O))
1
N'
(29)
calculated respectively on the continuum and on the lattice (in the limit a ~ 0). The results are Fig. 4. Diagrams contributing at four loops to the mixing with the unity operator.
Z~-~'g = 1 -- NL(a2,u2)Tr -I- 3 N2[L(a2]z2) ]2T2,
+O(Tr 3) Z rMs = 1 - [½NL(a2l t2) +
Cl] rr
+ O (T2),
(30)
where L ( x ) = - ( 1 / 4 n ) l n x + ( 5 / 4 n ) l n 2 and cl =
1/8.
We have calculated F ~ q at p + q = 0 and at two loops. The diagrams at one loop and at two loops contributing to F ~ , are drawn respectively in fig. 1 and in fig. 2. We find
2.3. Topological susceptibility additive renormalization
F ~ , = 1 + N [L(aZU) + dl] T + ( N 2 [3(L(a2H)) 2 + dlL(a2H) + d2]
+ m [oL(a2H) + d 3 ] ) T 2 + O(T3),
(31)
with d l = - 0 . 4 0 9 1 4 , d2 = 0.05541 and d3 = -0.09826. Then by gathering the results in eqs. (27) and (30) and (31 ) and using the relation T = 2~Nil, we finally get
Z2 + O ( ~ 3 ) , Z ( f l ) = 1 + ~Zl- + ~-~ zl = 2da, 112
z2 = 4 d 2 +
4( d 3 + ~c,) •
~
That is, zl = - 0 . 8 1 8 2 8 and z2 = 0 . 2 2 1 6 4 0.35325/N. In table 1 we present a comparison between our results for Z ( f l ) and those obtained in ref. [12] for the cases of N = 2 and N = 4: there is a substantial agreement. The small difference observed is about the same, in size, as for the O (3) a - m o d e l [ 15 ]; it can be the result of either higher order or non-perturbative contribution, or both of them.
(32) (33)
As Z (fl), A (fl) and P (fl) can be calculated in perturbation theory following field theory prescriptions. The first contribution to A (fl) comes from a two-loop graph, that is O ( f l - 3 ) . So, consistently (in a perturbative sense) with Z (fl) calculated up to two loops, it can be neglected. With our symmetrized definition of QL (x), P(fl) starts from a fl-4 term (corresponding to three-loop graphs). We calculate the perturbative tail P(fl) = ~"~,~=4Pn/fln up to four loops (in figs. 3 and 4 we show the diagrams at three and four loops, respectively). The results for infinite volume are
P4 ---- 1.0178 × 10 -3 X ( NN3 - 1) ,
(34)
Volume 306, number 1,2
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Table 1 Z (p) versus ft. Z (fl) two loopsis the perturbative renormalization function calculated to two loops. Z (fl) MCis the renormalization function calculated by Monte Carlo techniques; L is the size of the lattice; Star is the statistic of the simulation. fl
N
Z
L
Stat
1.4
2
0.4385
0.40(1) 0.41(2)
48
2000 400
1.45
2
0.4571
0.43(1)
48
1000
1.5
2
0.4745
0.45(1)
48
500
1.15
4
0.3892
0.375(7) 0.375(8) 0.377(7)
48 60 60
1000 400 400
1.2
4
0.4107
0.413(5) 0.413(7)
48 60
2000 400
0.432(6)
48
1000
(fl)
two loops
Z (fl) MC 0.40(2)
500
Table 2 P(fl) versus ft. P(fl)four loopsis the perturbative tail calculated to four loops. P(fl) fit includes the higher order terms fitted from Monte Carlo data. P(fl)MC is the perturbative tail calculated by Monte Carlo techniques; L is the size of the lattice; Stat is the statistic of the simulation. fl
N
P(fl)four loopsx 105
P(fl)fit x 105
P(fl)MCX 105
L
Stat
1.15 1.20 1.25
4 4 4
2.484 2.103 1.792
2.94(6) 2.43(5) 2.03(4)
2.93(6) 2.44(4) 2.02(3)
36 36 36
5000 6000 6500
1.30
4
1.537
1.71(3)
1.70(7)
36
1000
P5 = ( - 0 . 3 6 2 7 + 1 " ~ 3 5 ) x 10-3 ( N - 1) N3
X -
(34 cont'd)
These numbers are extracted by performing numerical integrations at finite volume and extrapolating the results to infinite volume. In table 2 we compare our results for P ( f l ) with those obtained in ref. [ 12 ] for N = 4. Here the agreement is not as good as in the case o f Z (fl). We assume that the discrepancy is due to perturbative effects: the contribution of higher orders in the perturbative tail of topological susceptibility is expected to be still effective in the scaling region, to which the data in ref. [ 12 ] belong. On this basis, we have performed a fit of the Monte Carlo data and we have realized that
two further terms of the perturbative expansion are sufficient to well approximate P ( f l ) - the same situation took place in the O (3) a-model [ 15 ]. The results of the fit are P6 = ( - 1 . 0 1 + 0.06) x l0 -5 and P7 = (2.38 + 0.08) x l0 -5, (chi)2/d.o.f. = 0.1. In table 2 we also present the values of P (fl) including the terms fitted from Monte Carlo data.
3. Conclusions We have estimated by perturbation theory the renormalization functions entering the relation between Z and the corresponding quantity measured on the lattice. They can be used to extract X from 113
Volume 306, number 1,2
PHYSICS LETTERS B
a Monte Carlo simulation following the well tested procedures o f the field theoretical method. We have c o m p a r e d our results with those obtained by a direct estimate by Monte Carlo techniques o f the additive and multiplicative renormalizations, finding good agreement. This consistency is a further evidence o f the validity o f the new non-perturbative determination o f renormalization functions.
Acknowledgement We wish to thank Adriano Di G i a c o m o for having suggested the problem, Paolo Rossi and Ettore Vicari for many useful and stimulating conversations.
References I l l A. Actor, Fortschr. Phys. 33 (1985) 6, 333. [2] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. D 14 (1976) 3432. [ 3 ] A. D'Adda, M. Liischer and P. Di Vecchia, Nucl. Phys. B 146 (1978) 63. [4] R.J. Crewther, Nuovo Cimento, Rev. Ser. 3, 2 (1979) 8. [ 5 ] B. Berg and M. Liischer, Nucl. Phys. B 190 ( 1981 ) 412. [6] M. Liischer, Commun. Math. Phys. 85 (1982) 29. [7] A. Phillips and D. Stone, Commun. Math. Phys. 103 (1986) 599. [8] P. Woit, Phys. Rev. Lett. 51 (1982) 638; Nucl. Phys. B 262 (1985) 284.
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[9] M. G6ckeler, A.S. Kronfeld, A.L. Laursen, M.L. Schierholz and U.J. Wiese, Nucl. Phys. B 292 (1987) 349; A.S. Kronfeld, A.L. Laursen, M.L. Schierholz and U.J. Wiese, Nucl. Phys. B 292 (1987) 330. [10] M. Li~scher, Nucl. Phys. B 200 (1982) 61. [ 11 ] D.J.R. Pugh and M. Teper, Phys. Lett. B 218 (1989) 326. [ 12] M. Campostrini, P. Rossi and E. Vicari, Phys. Rev. D 46 (1992) 2647; Topological susceptibility and string tension in the lattice CP N-I models, Pisa preprint IFUP-TH 34/92, to appear in Phys. Rev. D. [ 13 ] M. Campostrini, A. Di Giacomo, H. Panagopoulos and E. Vicari, Nucl. Phys. B 329 (1990) 683. [14] A. Di Giacomo, Nucl. Phys. B 23 (1991) 191. [15] A. Di Giacomo, F. Farchioni, A. Papa and E. Vicari, Phys. Rev. D 46 (1992) 4630; Phys. Lett. B 276 (1992) 148. [16]P. Di Vecchia, K. Fabricius, G.C. Rossi and G. Veneziano, Nucl. Phys. B 192 (1981) 392; K. Ishikawa, G. Schierholz, H. Schneider and M. Teper, Phys. Lett. B 128 (1983) 309. [ 17 ] M. Campostrini, A. Di Giacomo and H. Panagopoulos, Phys. Lett. B 212 (1988) 206. [18] M. Teper, Phys. Lett. B 232 (1989) 227. [ 19 ] A. Di Giacomo and E. Vicari, Phys. Lett. B 275 (1992) 429. [20] P. Di Vecchia, A. Holtkamp, R. Musto, F. Nicodemi and R. Pettorino, Nucl. Phys. B 190 [FS3 ] ( 1981 ) 719. [21 ] S. Hikami, Progr. Theor. Phys. 62 (1979) 1,226. [22] B. Alles and E. Vicari, Phys. Lett. B 268 (1991) 241. [23] D.J. Amit and G.B. Kotliar, Nucl. Phys. B 170 [FSI ] (1980) 187.
PHYSICS LETTERS B
Topological charge on the lattice. The 2D
C P N - 1 model
Federico Farehioni and Alessandro Papa Dipartimento di Fisica dell'Universiti~ and INFN, Piazza Torricelli 2, 1-56126 Pisa, Italy
Received 5 February 1993 Editor: R. Gatto We calculate the first two non-zero terms in the perturbative expansion of the additive and multiplicative renormalizations of the topological susceptibility, )~, in the 2D CP s - l model, when a density of topological charge is defined as a local operator on the lattice. We compare our results with some direct determinations, by Monte Carlo techniques.
1. Introduction
The topological charge of a spin field z (x) is defined by
Two-dimensional CP s - t models play a very important role in quantum field theory because of their similarities with non-abelian gauge theories. The models are asymptotically free and exhibit spontaneous mass generation. They also have a non-trivial topological structure. The action of the 2D CP N- 1 model, is [ 1 ] S=-~
2
f dExD--~z • D uz,
Du = Ou + iAu,
Au = ½i[-~(x).Ouz(x)-Ou-Z(x).z(x)],
(1)
(3)
and T is a coupling which plays the role of a temperature. At the quantum level the constraint (3) generates mass gap and asymptotic freedom. The gauge field A u is composite, but it behaves like a physical field after quantization. The theory is invariant under global S U ( N ) transformations of z. It is also invariant under local U ( 1 ) transformation z --~ ei~Cx)z,
A~ ~ A , - Oua.
(4)
The CP u - t model is equivalent to the 0 ( 3 ) amodel for N = 2. 108
,j
d x~u~Duz.D~z = ~
d x u A u.
(5)
Q is the number of times the gauge field A u winds the circle at infinity. The topological susceptibility X is a renormalization group invariant quantity, which measures the amount of topological excitations of the vacuum; X is defined as the correlation at zero momentum of two topological charge density operators, Q (x):
(2)
where z (x) is an N-component complex scalar field, constrained by the condition -Z(x) • z ( x ) = 1
Q = -~
x = fdax(oIT[Q(x)Q(O)]lO);
(6)
Q (x) is the divergence of a topological current Ku [2,3 ], Q (x) = OuKu ( x ) . The prescription defining the product of operators in eq. (6) is [4] ( O I T [ Q ( x ) Q ( O ) ll0) = Ou(OlT[Ku(x)Q(O)]10).
(7)
This prescription eliminates the contribution of possible contact terms (i.e. terms proportional to the 6 function or its derivatives) when x ~ O. The only known tool to compute X is simulation of the theory on the lattice. Many attempts to determine the topological susceptibility, X, were based on geometric definitions of the topological charge. The Elsevier SciencePublishers B.V.
Volume306, number 1,2
PHYSICSLETTERSB
so called geometrical method [ 5-9 ] uses an interpolation among spin variables to assign an integer topological charge to each lattice configuration. Since Monte Carlo data did not show the expected scaling behavior, the determination o f z was not possible. This failure was explained by the presence of exceptional configurations, called dislocations, whose unphysical contribution does not vanish in the continuum limit [ 10]. The geometrical approach meets similar difficulties in 4D non-abelian gauge theories [ 11 ]. Monte Carlo simulations performed on CP N-~ models [ 12 ] have recently shown that the contribution of dislocations is still effective for N = 4, while the geometrical approach seems to work for greater values of N. There alternative definitions of Z on the lattice have been considered: the field theoretical approach relying on a definition of topological charge density as a local operator, and the cooling method, which measures the topological charge on cooled configurations. The consistency of these two procedures was demonstrated for 4D SU (2) and for SU (3) non-abelian gauge theories and for 2D 0 ( 3 ) non-linear a-model or CP l model. For a review see refs. [ 13-15 ]. In the field theoretical method, a topological charge density operator is defined as having the appropriate classical continuum limit [ 16 ]: QL(x)a~0 ~ adQ(x) + O(ad+2),
(8)
where a is the lattice spacing and d the space-time dimension. As a consequence, Z L, the correlation at zero momentum of two QL (x) operators is given by
z L = (x~QL(x)QL(0) )
'(xZ
= V
(
QL(x))2
)
;
(9)
Z L is connected to Z by a non-trivial relation. The presence of irrelevant operators of higher dimension in QL (X) induces quantum corrections. Eq. (8) must be corrected by including a renormalization constant Z(fl) [17]: QL(x)a-.0 = adZ(fl)Q(x) + O(ad+2).
(10)
For S U ( N ) gauge theories, fl = 2N/g g, with go the coupling constant, for the 2D CP t¢-1 model, fl =
27 May 1993
2/NT; Z (fl) is a finite function offl, which will tend to 1 when the continuum limit is approached. If a perturbative expansion makes sense, then zl
z2
Z(fl) = 1 + ~ - + ~--~+....
(ll)
Furthermore, there are contributions of contact terms originating when x ~ 0 in eq. (9). A prescription equivalent to eq. (7) does not exist on the lattice, and therefore the contribution of the contact terms must be isolated and subtracted. These contact terms appear as mixings with the action density S (x) and with the unity operator I, which are the only available operators with equal dimension or lower. In formulae,
zL(fl) = adZ(fl)2 Z + adA(fl)(S(x)) + P(fl)(I)
+ O(ad+2).
(12)
In eq. (12) the quantity (S (x)) is intended to be the non-perturbative part of the expectation value of the action density, i.e. it is a signal of dimension d. In this paper we perform the calculation of the first two non-zero terms of the additive and multiplicative renormalizations of the topological susceptibility on the lattice, i.e. Z(fl) and P(fl), in the CP ~¢-1 model, for a general value of N. As we will see later, at the perturbative order of our calculation, the mixing with the action density can be neglected. Previous works have shown that the renormalization functions are well-defined quantities and that they are well approximated by the perturbative expansion (for a review, see refs. [ 13,14 ] for SU (2) non-abelian gauge theories and [15] for 2D 0 ( 3 ) a-model). A non-perturbative method has been recently developed, in which the renormalization functions are determined exploiting Monte Carlo techniques, through the observation of the topological charge and susceptibility on configurations obtained thermalizing an assigned instantonic configuration. This method, suggested by Teper [ 18] for the calculation of the multiplicative renormalization Z ( i l l has been extended to the additive renormalizations [ 19] and tested on the 0 ( 3 ) a-model [19,15] and, successively, on the CP t¢-1 models [12]. 109
Volume 306, number 1,2
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27 May 1993
We compare the results of our calculations with the numerical data obtained in ref. [ 12 ] for the special values of N = 2 and N = 4.
The perturbation expansion suffers from infrared divergences, which can be cured by adding a magnetic term to the action
2. Renormalization and topological susceptibility
SM = -- -~
1/
2.1. The model
=
We regularize the theory on the lattice by taking the action [20]
sL= --ZlnIo(4]-fx+u'ZxO,
(13)
where I0 (x) is the modified Bessel function of order zero. For both continuum and lattice calculations we make use of Hikami parametrization of the fields z ( x ) [21]
zu =-zN =0";
zi=r/i
(i= 1,...,N-I)
(14)
_
d2x HO" (X)
71fd2xH~/1 S - ' ' , _
18)
Indeed S~ explicitly breaks the S U ( N ) invariance and acts as a mass term for the q field. The SU(N)invariant quantities (and the relations among them) are free of infrared divergences, and have a welldefined limit for H ~ 0.
2.2. Topological charge density renormalization Our regularized version of topological charge QL (X) is
QL (X) = -- ~
i
eUvTr (P (x)AuP (x)A~P ( x ) ) ,
and 0"2
(
(19)
1 + 0"' = --T--,ni
r/[ = x / 2 ( 1 + 0",)
(15)
where
Pij(x) = ~(x)zj(x)
with N-I
(a') 2 + E
r/[~"i = 1,
(16)
(20)
and Au is a symmetrized version of lattice derivative
i= 1
which allows us to eliminate a'. In the continuum calculations the gauge field A u is eliminated by use of the equation of motion. The quantization of the action introduces an additional ill-defined (infinite) determinant which allows to write the functional measure over the fields q' and r/'7 in an S U ( N ) invariant way. The effect of this determinant is to modify S L by an amount ~"~ In (a'(1 + a ' ) u - 2 ) .
(17)
x
In dimensional regularization, the measure term does not contribute, as a consequence of the rule f dak = 0~ 8a(0), where d is the space dimension. 110
Auf (x) = f (x + It) - f (x - lt) 2a
(21)
On the continuum, Q ( x ) is invariant under the renormalization group if a suitable renormalization scheme is chosen, for example the MS scheme. This can be directly checked by calculating the renorrnalized two-point proper function of the complex fields r/' and ~' with one insertion o f Q (x), F ~ s (in what follows the prime is suppressed). Since we are only interested in the renormalization of Q (x), in our calculations we putp + q = 0 after having factorized the treeorder two-point function .Q~,rtree= -(i/8~)Sijeu~p~,q~ (p and q are the external momenta). We define F(2~, by the relation FQ~. = /-,tree . Q~, × FQ~, and we compute FQ~ at p + q = O. By a calculation at two loops we
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Zq = 1 + N
1 2~e
3N2( 1 + - T - 2n~
Fig. 1. Diagrams contributing at one loop to the two-point proper function with one insertion of Q (x). White and black blobs indicate respectively the operator and the Lagrangian vertices; black squares include vertices coming from the measure term and from other terms vanishing in the continuum limit: the corresponding graphs are absent in the continuum.
1 ( l n 4 n - 7E)) Tr 4n 2
1 (ln4/r_ yE) ) Tr2 4n
+ O(T)),
(23)
while Z r is the renormalization of the coupling constant [21 ]
Z r = 1 + - ~N ( 2 ~1
4~1 ( l n 4 n _ y s ) ) T r
+
(24)
/t is an energy scale. Since the r.h.s, ofeq. (22) is finite when written in terms of renormalized quantities, it follows that Q ( x ) does not renormalize up to two loops. A finite multiplicative renormalization connects the matrix elements of QL (x) with those of Q (x), defined in the MS scheme. The antisymmetry of QL (x) forbids mixings with any other SU(N) invariant operator of dimension two. We write QL(x)
= Z(T)Q(x).
(25)
Since the anomalous dimension of Q (x), 7Q (Tr), is zero, Z (T) must satisfy the equation Fig. 2. Diagrams contributing at two loops to the two-point proper function with one insertion of Q (x).
d yQ(Tr) = l a - ~ l n Z ( T ) l r ,
a = 0
(26)
and therefore must be a finite function of T, which can be determined by imposing [22]
find (see fig. 1 and fig. 2)
-fig ( Tr, Hr, lZ; p, q ) = Z - [ ( T ) Z'ff'g ( T, Iz, a ) F~,~ Ft~-s~(Tr, Hr,/z; p, q )Ip+q=0
xF~-~,~(T,H,a;p,q),
(27)
= Z,s(T,e)F~,l(T,H,e;p,q)lp+¢=o iz 2 =l+N[lln(~rr)]Tr 3N2 l [ + T
In •Hr ).]
-r
+
O(Tr3),
(22)
where T = Tdz-~Zr, H = HrZrZ~-ln; here F h-g and F 8 are respectively the renormalized and the bare function; e = d - 2 ; Zq is the renormalization constant of the field r/ [21 ]
~/ 7MSTMS [/2 with T = TrZT-Ms, and H = aJrt.~ T ~,t/ ; F~Lq is defined by F~L~ = Ft~~ × Ft)L~, and FQ~,I is the twopoint proper function on the lattice, with one insertion of QL (x), and calculated in the limit a ~ 0. The functions Z ~ s (T,/1, a) and Z ~ s (T,/z, a) are respectively the field and the coupling renormalization constants that allow us to get, starting from the lattice regularization, the same Green functions as those of the MS renormalization scheme. In particular, Z (T) can be obtained by imposing eq. (27) at p + q = 0.
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In order to calculate Z ( T ) at two loops, we need to know Z ~ s (T, p, a ) to two loops and Z ~ s (T,/1, a ) to one loop. Following ref. [23] we determine them by imposing the relation Fig. 3. Diagram contributing at three loops to the mixing with the unity operator.
Zn (T, e ) - l G (x, ZrTr# -~ )
--
= z~s(T,p,a)
-l
L
G (X, ZT'~'gTr),
(28)
where G(x, ZrTd~-') and GL(x, ZrT,) are the S U ( N ) invariant two-point Green functions
(P(x)P(O))
1
N'
(29)
calculated respectively on the continuum and on the lattice (in the limit a ~ 0). The results are Fig. 4. Diagrams contributing at four loops to the mixing with the unity operator.
Z~-~'g = 1 -- NL(a2,u2)Tr -I- 3 N2[L(a2]z2) ]2T2,
+O(Tr 3) Z rMs = 1 - [½NL(a2l t2) +
Cl] rr
+ O (T2),
(30)
where L ( x ) = - ( 1 / 4 n ) l n x + ( 5 / 4 n ) l n 2 and cl =
1/8.
We have calculated F ~ q at p + q = 0 and at two loops. The diagrams at one loop and at two loops contributing to F ~ , are drawn respectively in fig. 1 and in fig. 2. We find
2.3. Topological susceptibility additive renormalization
F ~ , = 1 + N [L(aZU) + dl] T + ( N 2 [3(L(a2H)) 2 + dlL(a2H) + d2]
+ m [oL(a2H) + d 3 ] ) T 2 + O(T3),
(31)
with d l = - 0 . 4 0 9 1 4 , d2 = 0.05541 and d3 = -0.09826. Then by gathering the results in eqs. (27) and (30) and (31 ) and using the relation T = 2~Nil, we finally get
Z2 + O ( ~ 3 ) , Z ( f l ) = 1 + ~Zl- + ~-~ zl = 2da, 112
z2 = 4 d 2 +
4( d 3 + ~c,) •
~
That is, zl = - 0 . 8 1 8 2 8 and z2 = 0 . 2 2 1 6 4 0.35325/N. In table 1 we present a comparison between our results for Z ( f l ) and those obtained in ref. [12] for the cases of N = 2 and N = 4: there is a substantial agreement. The small difference observed is about the same, in size, as for the O (3) a - m o d e l [ 15 ]; it can be the result of either higher order or non-perturbative contribution, or both of them.
(32) (33)
As Z (fl), A (fl) and P (fl) can be calculated in perturbation theory following field theory prescriptions. The first contribution to A (fl) comes from a two-loop graph, that is O ( f l - 3 ) . So, consistently (in a perturbative sense) with Z (fl) calculated up to two loops, it can be neglected. With our symmetrized definition of QL (x), P(fl) starts from a fl-4 term (corresponding to three-loop graphs). We calculate the perturbative tail P(fl) = ~"~,~=4Pn/fln up to four loops (in figs. 3 and 4 we show the diagrams at three and four loops, respectively). The results for infinite volume are
P4 ---- 1.0178 × 10 -3 X ( NN3 - 1) ,
(34)
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Table 1 Z (p) versus ft. Z (fl) two loopsis the perturbative renormalization function calculated to two loops. Z (fl) MCis the renormalization function calculated by Monte Carlo techniques; L is the size of the lattice; Star is the statistic of the simulation. fl
N
Z
L
Stat
1.4
2
0.4385
0.40(1) 0.41(2)
48
2000 400
1.45
2
0.4571
0.43(1)
48
1000
1.5
2
0.4745
0.45(1)
48
500
1.15
4
0.3892
0.375(7) 0.375(8) 0.377(7)
48 60 60
1000 400 400
1.2
4
0.4107
0.413(5) 0.413(7)
48 60
2000 400
0.432(6)
48
1000
(fl)
two loops
Z (fl) MC 0.40(2)
500
Table 2 P(fl) versus ft. P(fl)four loopsis the perturbative tail calculated to four loops. P(fl) fit includes the higher order terms fitted from Monte Carlo data. P(fl)MC is the perturbative tail calculated by Monte Carlo techniques; L is the size of the lattice; Stat is the statistic of the simulation. fl
N
P(fl)four loopsx 105
P(fl)fit x 105
P(fl)MCX 105
L
Stat
1.15 1.20 1.25
4 4 4
2.484 2.103 1.792
2.94(6) 2.43(5) 2.03(4)
2.93(6) 2.44(4) 2.02(3)
36 36 36
5000 6000 6500
1.30
4
1.537
1.71(3)
1.70(7)
36
1000
P5 = ( - 0 . 3 6 2 7 + 1 " ~ 3 5 ) x 10-3 ( N - 1) N3
X -
(34 cont'd)
These numbers are extracted by performing numerical integrations at finite volume and extrapolating the results to infinite volume. In table 2 we compare our results for P ( f l ) with those obtained in ref. [ 12 ] for N = 4. Here the agreement is not as good as in the case o f Z (fl). We assume that the discrepancy is due to perturbative effects: the contribution of higher orders in the perturbative tail of topological susceptibility is expected to be still effective in the scaling region, to which the data in ref. [ 12 ] belong. On this basis, we have performed a fit of the Monte Carlo data and we have realized that
two further terms of the perturbative expansion are sufficient to well approximate P ( f l ) - the same situation took place in the O (3) a-model [ 15 ]. The results of the fit are P6 = ( - 1 . 0 1 + 0.06) x l0 -5 and P7 = (2.38 + 0.08) x l0 -5, (chi)2/d.o.f. = 0.1. In table 2 we also present the values of P (fl) including the terms fitted from Monte Carlo data.
3. Conclusions We have estimated by perturbation theory the renormalization functions entering the relation between Z and the corresponding quantity measured on the lattice. They can be used to extract X from 113
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a Monte Carlo simulation following the well tested procedures o f the field theoretical method. We have c o m p a r e d our results with those obtained by a direct estimate by Monte Carlo techniques o f the additive and multiplicative renormalizations, finding good agreement. This consistency is a further evidence o f the validity o f the new non-perturbative determination o f renormalization functions.
Acknowledgement We wish to thank Adriano Di G i a c o m o for having suggested the problem, Paolo Rossi and Ettore Vicari for many useful and stimulating conversations.
References I l l A. Actor, Fortschr. Phys. 33 (1985) 6, 333. [2] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. D 14 (1976) 3432. [ 3 ] A. D'Adda, M. Liischer and P. Di Vecchia, Nucl. Phys. B 146 (1978) 63. [4] R.J. Crewther, Nuovo Cimento, Rev. Ser. 3, 2 (1979) 8. [ 5 ] B. Berg and M. Liischer, Nucl. Phys. B 190 ( 1981 ) 412. [6] M. Liischer, Commun. Math. Phys. 85 (1982) 29. [7] A. Phillips and D. Stone, Commun. Math. Phys. 103 (1986) 599. [8] P. Woit, Phys. Rev. Lett. 51 (1982) 638; Nucl. Phys. B 262 (1985) 284.
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[9] M. G6ckeler, A.S. Kronfeld, A.L. Laursen, M.L. Schierholz and U.J. Wiese, Nucl. Phys. B 292 (1987) 349; A.S. Kronfeld, A.L. Laursen, M.L. Schierholz and U.J. Wiese, Nucl. Phys. B 292 (1987) 330. [10] M. Li~scher, Nucl. Phys. B 200 (1982) 61. [ 11 ] D.J.R. Pugh and M. Teper, Phys. Lett. B 218 (1989) 326. [ 12] M. Campostrini, P. Rossi and E. Vicari, Phys. Rev. D 46 (1992) 2647; Topological susceptibility and string tension in the lattice CP N-I models, Pisa preprint IFUP-TH 34/92, to appear in Phys. Rev. D. [ 13 ] M. Campostrini, A. Di Giacomo, H. Panagopoulos and E. Vicari, Nucl. Phys. B 329 (1990) 683. [14] A. Di Giacomo, Nucl. Phys. B 23 (1991) 191. [15] A. Di Giacomo, F. Farchioni, A. Papa and E. Vicari, Phys. Rev. D 46 (1992) 4630; Phys. Lett. B 276 (1992) 148. [16]P. Di Vecchia, K. Fabricius, G.C. Rossi and G. Veneziano, Nucl. Phys. B 192 (1981) 392; K. Ishikawa, G. Schierholz, H. Schneider and M. Teper, Phys. Lett. B 128 (1983) 309. [ 17 ] M. Campostrini, A. Di Giacomo and H. Panagopoulos, Phys. Lett. B 212 (1988) 206. [18] M. Teper, Phys. Lett. B 232 (1989) 227. [ 19 ] A. Di Giacomo and E. Vicari, Phys. Lett. B 275 (1992) 429. [20] P. Di Vecchia, A. Holtkamp, R. Musto, F. Nicodemi and R. Pettorino, Nucl. Phys. B 190 [FS3 ] ( 1981 ) 719. [21 ] S. Hikami, Progr. Theor. Phys. 62 (1979) 1,226. [22] B. Alles and E. Vicari, Phys. Lett. B 268 (1991) 241. [23] D.J. Amit and G.B. Kotliar, Nucl. Phys. B 170 [FSI ] (1980) 187.