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On the short distance behavior of the critical Ising model perturbed by a magnetic field
ELSEVIER
Nuclear Physics B 483 [FS] (1997) 563-579
On the short distance behavior of the critical Ising model perturbed by a magnetic field Riccardo Guida a,l, Nicodemo Magnoli b,c,2 a CEA-Saclay, Service de Physique Th(orique, F-91191 Gif-sur-Yvette Cedex, France h Dipartimento di Fisica, Universitgt di Genova, Via Dodecaneso, 33, 16146 Genova, Italy c lstituto Nazionale di Fisica Nucleare, Sez. Genova, Via Dodecaneso, 33, 16146 Genova, Italy
Received 19 June 1996; accepted 10 October 1996
Abstract We apply here a recently developed approach to compute the short distance corrections to scaling for the correlators of all primary operators of the critical two-dimensional lsing model in a magnetic field. The essence of the method is the fact that if one deals with OPE Wilson coefficients instead of correlators, all-order IR safe formulas can be obtained for the perturbative expansion with respect to the magnetic field. This approach yields in a natural way the expected fractional powers of the magnetic field, which are clearly absent in the naive perturbative expression for correlators. The technique of the Mellin transform has been used to compute the IR behavior of the regularized integrals. As a corollary of our results, by comparing the existing numerical data for the lattice model we give an estimate of the vacuum expectation value of the energy operator of the continuous theory. PACS: ll.25.Hf; ll.15.Bt; ll.10.-z; ll.10.Gh Keywords: Ising model; Conformal field theories; Perturbationtheory; IR divergences;Operator product
expansion
1. Introduction It is well known that ( D = 2) conformai field theories [ 1 ] can be used to describe statistical systems at the critical point [2]. The second step from this point of view is to obtain information on the behavior of the system near criticality, that is to study 1E-mail: [email protected]. 2 E-mail: [email protected]. 0550-3213/97/$17.00 Copyright C) 1997 Elsevier Science B.V. All rights reserved Pll S0550-32 13 (96)00585-8
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a conformal field theory perturbed by a relevant operator (i.e. with scale dimension 0
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computed by use of form factor techniques [ 16,17]. The plan of the paper is as follows: in Section 2 we will summarize the OPE approach, while in Section 2.1 we describe a known mathematical technique that appears to be useful to regularize and disentangle IR divergences of integrals, the Mellin transform. In Section 3 we apply the approach to the Ising model (all the bulk computations are confined to Appendix A); in Section 3.1 we use non-perturbative as well as numerical inputs to estimate the parameters left unfixed by the OPE method; the obtained results are discussed in Section 3.2. Conclusions are given in Section 4.
2. All-order IR finite formulas
The goal of the method presented in Ref. [7] (see also Refs. [8-10] and Refs. [ 1113] for many preexisting ideas) is to obtain information on the short distance behavior of a conformal field theory perturbed by relevant operators, i.e. when to the conformal action ScFr is added a perturbation of the form
AS = - I dx ~ ais OiB (X), d
( 2.1 )
i
where A} (0i8) are bare couplings (operators) and 0 < xi < 2 (xi = xo,, xo being the scale dimension of operator 0 ) . This is achieved by expanding in powers of the corresponding renormalized couplings a i (Taylor expansion) the so-called Wilson coefficients Cab(r, ~l) that enter in the operator product expansion
CCb(r, A)(q0c (0)) a
(qOa (r)qOb(0)) a ,~ ~
(2.2)
¢
for the complete theory (4'a are deformations of the conformal theory operators, the suffix A refers to the complete theory correlators; the dependence on a i of the Wilson coefficients will be omitted in the following). Assuming the regularity of the Wilson coefficients in terms of the renormalized couplings Ai (minimality of the renormalization scheme), the validity of an action principle for the derivative with respect to A i and the asymptotic convergence of OPE, all reasonably satisfied by the complete theory, an IR finite representation has been given for the nth derivative of C~b with respect to the couplings evaluated at ai = 0, involving integrals of (eventually renormalized) conformal correlators: ~
Oil . . .ogi,,Cbla2 <[(I)bXoo ]} b
= lim { f d X l . . . f X<[:
Oin : ...
:
Oil :
(~a,~02 --
b ~-~Col02(~)b)Xoo]> b
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566
b
x ( [ : Oi,, : . . . --
*
: 0i2 : q~bX~]) + p . + . . .
Oil...Oi,,_lCala2
dXnOR,,(Xn)([:
Oi,, : q ~ b X ~ ] ) + p .
),
(2.3)
b
where c~i = a--~, O R ( X ) is any generic IR cutoff function (such that l i m g ~ O R ( x ) = 1), p. means all non-trivial permutations over labels of indices i, XR, is any arbitrary operator localized outside the sphere of radius R t >> R --~ e~, chosen to give adequate boundary conditions (and possibly including powers of Rt), and ~ is restricted to operators C/'b such that xb <<. ( D - xik) . . . + ( D - xi,,) - x x , . See Ref. [7] for more details. While each term in the sum in the right-hand side is IR divergent (and consequently requires a regularization), the complete right-hand side is IR finite and independent on the choice of the cutoff function (what we presented above is an "IR renormalization" more than a simple regularization). The structure of the expression is simple: the first term on the right-hand side is a naive (generalized) perturbative one, while the others, containing lower derivatives of Wilson coefficients, play the role of non-local IR counterterms, naturally induced by the theory itself. (A conjecture for the existence of this mechanism can be found in Ref. [ 18], in the general context of quantum field theories. Also a rigorous proof of the conjecture within the MS renormalization scheme for perturbative superrenormalizable quantum field theories can be found in Ref. [ 13 ] .) If the cutoff function is rotationally invariant (OR (r) = OR ( ]rl ) ) only scalar operators contribute in all expressions above. Also, if by dimensional analysis one knows a priori that no powers R° can be obtained from the IR counterterms available in that model, then it suffices to compute the first (naive) term of the right-hand side in some chosen IR regularization, and to keep only the regular term of its asymptotic expansion when R i --+ cxz, the singularity of the naive term being automatically killed by the IR counterterms 3 (which, in this case, cannot give additional finite contributions). Roughly speaking, one expects that this picture is realized when the IR counterterms do not need UV renormalization (i.e. are locally integrable), because in this case the renormalization scale ~ is absent and there is no other possibility to obtain an adimensional quantity from the log R terms that are expected in general from the R° IR counterterms (apart from miraculous cancellations of log R divergences). This shortcut can be applied in particular to the case we will consider here. We conclude by observing that the complete correlators are then obtained by combining the derived expressions for the Wilson coefficients with those for the required operator vacuum expectation values (eventually parameterized by unknown but fixed and universal constants, see Section 3.1 and the discussion in Ref. [7] ).
3This idea was already present in the first-ordercomputationsof Ref. [8].
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2.1. Mellin transform and IR divergences Having in mind that all we need are asymptotic expansions of (multiple) integrals in 1 ---+0 above, it is worth introducing here a natural method to deal the limit of mi ~ RTi with these expansions: the Mellin transform 4 (see Ref. [31 ] for a nice introduction). Given a function l(m) locally integrable on (0, ~ ) , of order m '~ when m ~ 0 and m - ~ when m ~ ~ , one can introduce in the complex strip - a < s < /3 its Mellin transform
[(s) =_~ dmmsl(m). J m o
(2.4)
If [(s) has a meromorphic extension in the complex plane on the left of Ires = - a one can derive from the inverse Mellin transform the asymptotic expansion
l(m) ,,~ ~ ' ~ R e s m-S[(s)).~=_,~i
(2.5)
i
with al = cr < ce2 < . . the powers of m in the asymptotic expansion of l(m) (notice that multiple poles give logarithms of m). In other words: singularities of the Mellin transform drive the asymptotic expansion. In our formulas we have always (possibly nested) integrals of the form
l(m) = / d 2 z O(mlzl)g(z )
(2.6)
(we refer from here on to the rotation invariant cutoff; notice also the slight and obvious change of notations). It is easy to prove the following convolution theorem:
[(s) = O(s)G(l - s),
(2.7)
where
~(1 - s) = f d2z Izl-Sg(z)
(2.8)
is essentially the Mellin transform of g (up to angular integrals). Thus in the cases in which (~ is known the integrals are easily done. Notice that if the Mellin transform of g strictly speaking does not exist but the Mellin transform of I(m) is well defined, one can try to bypass the problem by extending the integral [(s) to a more general one, [(s; 8) (in which g is now extended to some adequate g6), such that it reduces to the original one for some value 6 = 60, and such that the convolution theorem can be applied: if [ ( s ; 8) can be analytically continued to 8 = 80, the Mellin transform of I is obtained accordingly. 4 This technique was essential for the analog analysis in perturbative superrenormalizable quantum field theories, Ref. [ 13t (and references therein).
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In particular if the IR counterterms give no finite contributions (see Section 2), then the derivatives of the Wilson coefficients can be unambiguously obtained simply by taking the residue of the first, "naive perturbative", term in (2.3). In Appendix A we give the analytical expression of [ ( s ) for a general case that allows us to compute all the integrals involved in the first-order derivatives of the Wilson coefficients of primary fields of the conformal field theory underlying the critical Ising model.
3. Wilson coefficients for the perturbed Ising model The scaling limit of the lattice (D = 2) Ising model (see Ref. [20] and references therein) at the critical point, T = Tc, H = 0, is described by a continuous unitary conformal field theory, technically 34(3/4) [1]. The operator space of this conformal theory is generated by the primary operators 1, o-,g of dimension x = 0, 1/8, 1. The related fusion rules are y[o-] [o-] = [1] + [C],
[C][E] = [11,
[o-] [C] = [o-],
(3.1)
implying ( [ E l k [ l l " [ o - ] ~) = 0
(3.2)
for I odd. Also the Kramers-Wannier duality implies ( [ g l " [ l ] t) = ( - ) n ( [ g l " [ 1 ] t ) .
(3.3)
The object of our study will be the extended theory in the presence of a magnetic field, S = Slsing --}-- I ho'd2z"
(3.4)
J
Notice that with this notation each derivative Oh will be realized by the insertion of the operator - f dZz ~r( z ) (action principle). By application of the general formula (2.3) and of the selection rules (3.2) and (3.3) we obtain the following (non-trivial) first-order relations: -
c9I ,r,,~ ~ , ~ "r)(o'otrR,)
--
c9h
Cee(r)(o-oo-R, ) tr
=
=
- OhC~e(r) = - OhC,~ee(r){£OgR,) =
/' /' /' /'
d2z(trzO.R,[OrrO.O
d2z(O'zO'R,[£r£O
-
__
C ~1 r~(r)
- C~e~(r)Eo]),
C~e(r)] ),
(3.5) (3.6)
d2z(o'z [o'rgo - C~e (r)o'0]),
(3.7)
d2z(O'zgR,[~rgo -- C ~ (r)o-o - C~"e(r)~rll)
(3.8)
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569
(where the prime on the integrals denotes a rotationally invariant IR regularization, all limits have been omitted and we defined o-l = L_1L_1~r). It is easily realized by dimensional analysis that in all expressions above no finite contribution can arise from IR counterterms: we are thus in the case described in Section 2.1, where computations are easier. To reconstruct the short distance corrections of complete correlators we must have the expressions of the VEV of the lowest dimensional operators, o- and g. As is well known in perturbative quantum field theories and from direct renormalization group considerations, see e.g. Ref. [ 19], composite operators VEVs evolve by RG as d([q~a])
F~([q~b]),
(3.9)
in which 1 is the logarithm of the scale and the matrix F~ contains only positive integer powers of the renormalized couplings ,~i and forbids the mixing with higher dimensional operators (it is lower diagonal if operators are ordered with increasing dimensions). From these properties and from the fact that h has dimension 15/8 and that it is not renormalized it follows by simple dimensional considerations that o" and g do not mix with other operators, that F~, F~ reduce to the fixed point scale dimensions and that no logarithms appear in their VEVs. We can thus parameterize the VEVs of the complete continuous theory as
(~)h = A~lhll/15Sign( h),
(C)h = AEIhl 8/j5
(3.10)
(where it has also been kept into account that (o')h ((g)h) has to be odd (even) in h as is easily seen directly from the corresponding lattice expressions). Defining C - CIr] aimc, FA,B =- (AB)hlrl °iman and the scaling variable t =_ Ih[[r[15/8 we have the following expression for the short distance (small t) behavior: ~-'-~,8/15 -k- A~rSS~t 16/15+ O(t2), Fo.o.=C~o, + A~,~,~,
(3.11)
FCC = cl-~g + Acrffh~ggt 16/15 + O(t2),
(3.12)
KF~E = a o , ~ t 1/15 + OhC~---'~ct+ a E ~ t
23/'5 + O( t31/15),
(3.13)
where K = Sign(h) and
I-----_
t~hCff°" --
l (F(-1/2)F(7/4)) 3-2
2
F(5/4)
'
(3.14)
OhC~e =0,
(3.15)
OhC~-----~e= 1 sin(3~/8) F2(11/8) sin(15~r/8) F2(15/8) '
&,-'-~c=sin(Tr/8) s i n ( 3 ~ / 8 ) ( ~ r ( - l l / 8 ) F ( 7 / 8 ) ) r(-1/2)
(3.16)
2
(3.17)
have been computed in Appendix A (see also Appendix B for zeroth-order expressions) and the only unknown constants, up to now, are Ae and A,~. While in general the OPE
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approach leaves some universal unfixed constants, in this case the (approximate) value of A,~ (Ac) can be obtained from existing knowledge, see next section.
3.1. The missing VEVs The coefficient A,~ in (3.10) can be easily extracted by use of the results of Ref. [21 ] (see also references therein), where the integrability of 3,4 (3/4) + 412 (~ O') was used essentially (in particular the thermodynamic Bethe ansatz and the method of external fields have been used). We report here Eqs. (2.9) and (3.15) of Ref. [21 ],
47r2
[
]4/15 4sin(Tr/5)F(1/5)hS/, 5
M= [y2(3/16)y(l/4 )
r(2/3)r(8/~)
= 4.404908579981566037... h 8/15, ( •=-
(3.18)
M )2 sin(~'/5) 2 sin(7"r/5) 8 sin(Tr/3) sin(87r/15)
= -0.061728589822368... M 2,
(3.19)
where M is the minimum mass of the particle of the theory, y(x) =_ l'(x)/I'(1 - x), h > 0 in this section and the bulk energy • is defined by the partition function of the theory IR regulated in a cylinder of dimensions L, R,
Z(h) = (e- f h~) ,,~ e -'IcL,
L,R--* cxD.
(3.20)
Noticing then that (o-)/, = (o-e- f h,r)/(e- f h,~) ~ lim( --0/, log L,R
Z(h)/LR),
(3.21)
it follows that
(cr)h = Ohe = A~h 1/15,
(3.22)
where 8(
A,~= -
47r2
)8/,S
\ r2 (3/1--~-~,(1/4)
= - 1.27758227605119295...
sin~ ( F(1/5 ) ,~2 ~ 8 . /'(273-)T(ff/15)J sin 3 sm 73
(3.23) (3.24)
On the other hand, no non-perturbative information is available (to our knowledge) for the VEV of operators different from the perturbation, o-. In principle, numerical estimates of the constant Ag can be obtained by comparison with Monte Carlo simulations on the lattice for the normalized (connected) correlator (S(R)S(O))c/{S) 2 in the regime in which sc, R >> 1 and t 8/15 o( R/( is not too big, i.e. ( e( H -8/15, in a small field (S, R, sc, H being lattice spin, position vector, correlation
R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
571
length and magnetic field and the lattice spacing is assumed to be one everywhere). 5 In practice with the available data (Ref. [22], cubic lattice), one has only a small window of applicability ( ( ~ 15.5 for H = 0.001). To improve convergence and enlarge this window of applicability we fitted, together with the powers given in Eq. (3.11) also the next to leading terms t 2, t 32/15 (notice that logarithmic terms are absent for the same considerations given in Section 3). From the known first two coefficients in the small t expansion for the continuous expression of (o-(r)o-(0))c/(O-) 2 we can fix the overall normalization (i.e. the lattice magnetization (S)) and the correlation length (, related to continuous variables by R / ( = Mr (M being the mass in Eq. (3.18)). Notice that the predictions (3.11)-(3.13) refer to the complete correlators, and that the normalization factor between the continuous spin VEV and the lattice one (physical magnetization), is negative in our conventions, see also Ref. [22]. At this point all is fixed and we get a prediction for the coefficient of t 8/15 (Ac) and for the coefficient of t 16/15 (which actually is known exactly). In practice we selected only those fits that give a good prediction for the known coefficient, fixing in this way the above-mentioned (but unknown) window of applicability, a good representative of which is (for H = 0.001) the R = 5-54 range that corresponds to Ac ~ 0.321
(3.25)
(see Eq. (B.1) for our normalization) with an error of the order of a few percent (estimated roughly from the variation of this quantity among different "good" fits). As a byproduct we also get the estimate sc ~ 15.4,
(S) ,-~ 0.634,
(3.26)
in agreement with the results sc = 0 . 3 8 ( 1 ) H -s/15 of Ref. [22] and (S) = 1.003(2)H 1/15 of Ref. [23] (with a larger lattice). As an independent check that our estimate is correct we can use the known exact sum rule [ 16,17,24] /I,~ = _ 27rh(2 - 2A '~) f dZr(o.(r)o.(O))h,c" 4rr(o')
(3.27)
We split the integral into two pieces, Mr < A and Mr > A. We estimated the first term by use of our expression (3.11 ) and the second one by use of the first three terms of the long distance expansion 3
(o'(r)o-(0))h,c -~ Z
1
o- 2
~(F,~ ) Ko(c,mr),
(3.28)
i=1
where coefficients ci and form factors F/'~ can be found in Ref. [ 16] (and references therein). By use of (3.27) we can give an expression of AE in terms of A. Clearly, for 5 Notice that the relation between lattice spin and continuous theory operators has the general form S o- + . . . in w h i c h . . , means higher descendants of o- that are irrelevant in the continuous limit because they have higher scale dimensions: it is easy to check that their relative contribution in a cubic lattice is suppressed at least by a factor 1/~"2 (not bigger than the relative errors of our estimations, Eqs. (3.25) and (3.26)).
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A small we have a big error from the approximated long distance expansion, while for small A we have a big error from the approximated small distance expansion. If (by a minimal sensitivity criterion) we choose the value of A that minimizes AE (A ~ 1.5) we obtain the value Ac ,'~ 0.322, with an error of the order of 2% (estimated by repeating the procedure with only one mass term) which is compatible with Eq. (3.25). 3.2. Comments
A first glance at our results shows that a coefficient of the expansion for (CE) (3.15) vanishes non-trivially (unexpectedly from the point of view of the selection rules of the conformal field theory). This might sound strange, but one must bear in mind that the complete theory is an integrable model with powerful symmetries: we can thus interpret the zero found as a (short distance) signal of these symmetries and possibly of the existence of non-trivial differential equations satisfied by the correlator (analog of Painlev6 equations for spin-spin correlators in thermal perturbations of the Ising model [20] ). Another important feature of our prediction is the presence of fractional powers of the coupling (or equivalently of the scaling variable t = IhllrlS/~5): in the OPE approach these powers come out naturally from the VEV of operators (times some integer power of t coming from the Wilson coefficients, analytic in t). In particular the fractional correction t 8/j5 that is found in (3.11) for the (o-o-) correlator is expected if one introduces as infrared cutoff in the perturbative expansion for the correlators the correlation length Rc ~ Ih1-8/~5 (see Refs. [14,6]). The existence of this fractional contribution to the spin-spin correlator can be recovered from the results of Ref. [ 12] (based also on OPE and Callan-Symanzik equations in the 4 - e framework) by inserting the exact values of the critical indices given by the conformal theory. Also it is interesting to see that the scaling of the (o-E) correlator (vanishing at the critical point) is dominated by a fractional power, t I/i5, coming from the VEV (o'). We must remark here that our results are in disagreement with an existing expression of the short distance behavior of spin-spin correlator in the same context [25], coming from the effort to apply in this case the perturbative expansion for correlators developed in Ref. [6] and to obtain the expected fractional powers of h as well. The main difference in Ref. [25] with respect to our (3.11) is the presence of powers of logt in the first terms that, in our mind, are not motivated due to the absence of renormalization of the lowest dimensional operators o-,E (and of L - 2 L - z l as well, which is the next scalar operator that can have a non-zero VEV in this theory, as shown in Refs. [8,19]), see the discussion near Eq. (3.9). Also the power t 24/15 present there is not motivated from the OPE point of view (there is no corresponding VEV). In Fig. 1 we give a comparison of the numerical data for (SS), see Ref. [22], with our prediction Eq. (3.11 ) (without the higher-order terms added in previous section to improve the fit) where the estimated values (3.25) as well as (3.26) are used. What is noticed immediately is that the convergence of the expansion is quite good (see also Ref. [8] ): with the first-order corrections to Wilson coefficients, we can achieve a
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573
0.3
0.25
0.2
[3.15
0.i
0.05
. . . . . . . . . . . . . . . . . . . . . . 5
i0
20
15
77.'7rr~ 25
30
Fig. 1. Values of (S(R)S(O)) from lattice simulation (dots) and as predicted from the OPE approach (solid line) versus R (lattice position) for H = 0.001 (ge ~ 15.5).
2.4 ! 2.2
. . . .
0. . . .
i
. . . .
l'.s . . . .
1.8
1.6
1.4
1.2
Fig. 2. Estimates of {o'(r)g(O))/(tr)(g) versus Mr: comparisonbetween the OPE approach (solid line) and the form factor method with one (dashed line) three (dot-dashed line) and eight (dots) terms. relative error of order 2% at R / ( = M r ~ 1. No numerical results exist to our knowledge for (o-g) and (g,f). Nevertheless in Fig. 2 we show a comparison between our (short distance) prediction and the (long distance) form factor result (up to the first eight terms) [ 17] in the case of {o'g). It appears from Fig. 2 that there is a good agreement of the two methods in the intermediate region 1 < M r < 1.5 and a reasonable evidence of convergence of the long distance expansion towards our result in the region M r < 1, as is expected. This agreement could be regarded as an independent test of the form factor clustering property that has been assumed in Ref. [ 171 to get the form factors for (o-g) and ( g g ) . (The origin of this property will be explained in Ref. [24].)
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4. Conclusions We computed the short distance behavior of all the correlators of primary fields of the critical two-dimensional Ising model perturbed by a magnetic field up to O(t 2) excluded, t = IhLIr IJs/s being the scaling variable. The IR finite OPE approach, extensively developed in Ref. [7] has been used. This method does not give information about VEVs by itself. An unfixed constant for (o-) has been obtained by use of non-perturbative results coming from the integrability of the model (TBA), while by comparison with the existing numerical data we got the non-trivial estimate (3.25) of (g) in the continuous theory. We point out once more that with the OPE approach the expected [14] fractional powers of the coupling h are naturally obtained from the operators' VEVs, while in alternative perturbative direct estimates of the correlators (IR regularized in some way) these non-integer powers are absent. In particular we immediately found the expected t 8/15 correction to the spin-spin correlator [ 14], and we predicted that the dominant term of (o-g) is the fractional term t 1/15 (which should be observable in adequate lattice simulations). The result for (o'o-) when compared with the available data signals a good convergence of the approach: this encourages one to search for some resummation technique that could further enlarge the convergence radius. Moreover the use of exact sum rules to extract information from both short and long distance approximations, which has been done in Section 3.1, should stimulate the research in this direction, with the final goal of connecting the two regimes. We conclude by emphasizing that the OPE approach, being very general (integrability of the theory is not necessary to compute Wilson coefficients) and fast convergent, can always be seen as a bridge towards the study of non-integrable perturbations and as a test for the ansatz behind integrable theories predictions (as done in Section 3.2). It is furthermore an open question if the powerful Coulomb gas technique of Refs, [26,27] and the IR regularization of Ref. [6] could be married with the OPE approach to reach higher orders of the expansion in this and other statistical models.
Acknowledgements One of the authors (R.G.) is indebted to D. Bernard and J. Zinn-Justin for useful discussions and careful reading of the manuscript. The authors acknowledge J. Bros, M. Caselle, G. Delfino, P. Di Francesco, G. Mussardo, S. Nonnemacher, A1.B. Zamolodchikov, J.B. Zuber for interesting information and discussions. The authors also thank G. Delfino and P. Simonetti for sending a table of the (eight terms) form factors prediction for the spin energy correlator. The work of R.G. is supported by a TMR EC grant, contract No. ERB-FMBI-CT-95.0130. R.G. also thanks the INFN group of Genova for the kind hospitality.
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575
Appendix A. Computation of the integrals First of all we give the general expression for the Mellin transform (see Eq. (2.4)) with respect to m (IR cutoff) of the integral
l~./z,~( m; x )
= fd wO mlwl)lwl °lw-
xJ2 lw-
(A.1)
in which m > 0, x E (0, 1), a, f l , 7 > - 1 (to have local integrability) and we will fix the IR cutoff function O(t) = e -t (such that its Mellin transform is O(s) = F ( s ) ) . By use of the convolution theorem (2.7) it easy to obtain
[,,~,r(s;x) = F ( s ) D ( a - s/2,fl, y , x ) ,
(A.2)
where the integral
O ( a , b , c , x ) =_
f d:wlwl2alw-xl2Clw-II
(a.3)
has been computed in Ref. [27] by use of contour deformation in the variable Im w and was found to give
S(a)S(c) S(b)S(a + b + c) D ( a , b , c , x ) = S ( a + c ) ]l°xl2 + S(a+c) 111°°12'
(A.4)
where
S(a) - sin 7"ra, lox =- f d u u a ( x - u)C(1 - u) b o
= xl+a+cF(a + 1)F(c + 1) F(-b,a+ F ( a + c + 2)
l ; a + c + 2;x),
O0
fduua(. -- x)C(u -1
F(-a-b-c1)F(b+l) F(-a -c)
F(-a-
b- c-
l,-c;-a-
c;x)
and F _= 2F1 is the hypergeometric function (see e.g. Ref. [29]). From the Mellin transform (A.2) we can obtain the asymptotic expansion of the original integral (A.1) when m -~ 0, by use of (2.5). All the required integrals in (3.5)-(3.8) can be then computed by use of the general expression (A.2). See Appendix B for the conformal correlators. In our particular case we know that the IR counterterms cannot give m ° corrections, see Section 3, so that the derivative of the Wilson coefficient is obtained by extracting the m ° contribution of the naive perturbative term (the first of right-hand side) in (3.5)-(3.8), i.e. by computing the residue of its Mellin transform in s = 0. Notice that due to the absence of m ° log m
R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
576
IR counterterms, there cannot be log m factors in the naive term and correspondingly its Mellin transform will have only simple poles. While (3.7) and (3.8) are trivially reduced to (A.2), some few words must be spent for (3.5), i.e. for the Mellin transform of the four-spin correlator (B.6). In the limit R'm --~ c~ (R' being the argument of the R'-z _-- I boundary operator o'R,) we have in the correlator the simplification x = (R'-~)r (r = 1) So that it is better to change variable and obtain (in the notation of the convolution theorem (2.4))
O(l_s)= f daxlxl,,,_4ll_x,_26(l+ 2 lv/-(=7-x+
1- lVq-S-~-x) 2 .
(A.5)
As explained in Section 2.1 we have introduced an additional parameter 6 to justify the use of the convolution theorem. The analytic continuation to 8 = 1/8 will give the wanted Mellin transform of the four-spin correlator with exponential cutoff. The final step to reduce (A.5) to (A.2) is the simple substitution w = x/1 - x and the observation that the integrand is invariant for w --+ - w . By use of these considerations Eqs. (3.14)-(3.17) follow. As a general check of the regularization independence our approach, we report the alternative derivation of OhC~% obtained regularizing the integrals by restricting them to Izl < R. We will not use the Mellin transform technique and we will keep explicitly the IR counterterms to show explicitly how cancellations work. In the limit R ' / R --~ oo (being x = l / z ) we can rewrite (3.5) as
-ohc,,%=
~~ Ig+(x)lR+lg-(x)12-1- Ixl,
(A.6)
I.~L>I/R where we omitted the remaining overall limit R ~ oo in the left-hand side and defined
g + ( x ) --
W/1 + v / 1 - x 2~ ----x~"
(A.7)
Unfortunately, we do not have a closed expression for the integral (A.6). However, the calculation can be performed by series, splitting the integral in two regions, 1 < ix I < 1 and Ix I > 1. The contribution of the first region is then obtained by expanding OO
g+(x)
Z g i x "+ i .i--o
OO
g - ( x ) = ~--~gTx.i+l/2 j---o ~
-
(A.8)
(note also that ( g + ( x ) ) * = g ~ ( x * ) ) , exchanging the series with the integral and performing simple integrals of powers of x x*. At this stage the role of IR counterterms is essential to give a finite R ~ oo limit. The contribution of the second region is obtained after making the change of variable x = - 3 ' from which we obtain
2 I Iwl
d2w]wl3/Zlh(w)12'
(A.9)
R. Guida,N. Magnoli/NuclearPhysicsB 483 [FS] (1997)563-579 h(w) =
577
(A.10)
1 + w2) 1/4
for the sum of g± contributions (integrals of IR counterterms are easily performed). Again expanding
h(w) = ~ h.iwi .j_--o
(A. 11)
the integral can be easily performed by series. The final result is
(1 "--".,.Ig~122jlg;IZ)--
=
+
1
+
43
4
+ 7 +
.j>~2
\ 2j2[hj12 +7/2 ]
1 +
/
'
(A.12) where the coefficients g~:, hj satisfy the recursive relations
(j+l)
(j+3)
gi-+l-2(J+2)(j+~)gf+(jz-~j+5)g~_l=O,
(j+l)(j+2)hj+2+j
( ~ )
2j-
hi+
(
7 45) j2-2j+T~ hi_z=0 (A.13)
due to the existence of the following differential equations satisfied by g±, h:
(x3-- 2x2 + x)g
+',
+
(~
7
x2---~x +
~)
'
3
±
g± ---~xg =0,
(l + 2w2 + w4) h- + ~(w 3 3 + w)h ' - ~-~w3 2h=0.
(A.14)
As they stand the series are very slowly convergent (g:~, hj ~ j-7/8), so to increase the convergence of the series we added and subtracted to the jth term its asymptotic powerlike behavior 6 that can be resummed and gives the well known Lerch functions (q0(z, s, a) = ~-']~,+,,+0 z"/( n+a)s, n = 0, 1. . . . ) which are obtained with high precision from Mathematica. After numerical resummation we obtain then
OhCff,~ = 0,403745631213448123 ....
(A.15)
which agrees up to the obtained precision with the analytic result (3.14). 6 The first two terms are obtainedfrom the contribution of the singularities nearestto zero (Darbouxtheorem, see Refs. [ 30,31 ] ) while the others can be convenientlyextracted by expanding in j--7/8+k/2the recursive relations (A.13) up to the desired order k.
578
R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
Appendix B. Conformal correlators
To fix notations we give here the expressions for the fixed point (conformal theory) quantities involved in our computations. See Ref. [28,10] for their derivation. Wilson coefficients: 1
C~E (r) - 47r2tr[~,
cl,~(r) -
! Irll/4'
(B.1)
C ~ ( r ) = 7fir[ 3/4,
(B.2)
1
C ~ ( r ) = 4~rtrl
(B.3)
Useful correlators: 1 Izl2l3/4 (o-(z,)o(z2)£(z3)}- 47r Iz131Iz231 '
(B.4)
Iz12(z32 + z42) 2Z32Z4212 (O'( Zl ) O-(Z2) E (Z3) E (Z4) ) = 16~ [---z42--z32-z41 "2 z----3~111z43121 1 - - - -z12 - ~ /I4 -
(O"(Zt)O'(Z2)O'(Z3)O'(Z4)) =
1(1
-
'
(B.5)
- x)z12z34[ - I / 4
(B.6) where
x = z12z34/(z13z24), Z~j = zi - zj.
References I 1] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. 121 C. Itzykson, H. Saleur and J.-B. Zuber, eds., Conformal lnvariance and Applications to Statistical Mechanics (World Scientific, Singapore, 1988); J.L. Cardy, in Fields, Strings and Critical Phenomena, Les Houches Session XLIX, ed. E. Br6zin and J. Zinn-Justin (Elsevier, Amsterdam, 1989); A.B. Zamolodchikov, Rev. Math. Phys. 1 (1990) 197. [3] A.B. Zamolodchikov, Adv. Stud. Pure Math. 19 (1989) 641; Int. J. Mod. Phys. A 3 (1988) 743; A.B. Zamolodchikov and AI B. Zamolodchikov, Ann. Phys. 120 (1979) 253; see also G. Mussardo, Phys. Rep. 218 (1992) 215. 141 EA. Smirnov, Form Factors in Completely lntegrable Models of Quantum Field Theory (World Scientific, Singapore, 1992); J.L. Cardy and G. Mussardo, Nucl. Phys. B 410 [FS] (1993) 451. 151 H. Saleur and C. Itzykson, J. Stat. Phys. 48 (1987) 449; A. Cappelli and J.l. Latorre, Nucl. Phys. B 340 (1990) 659; A.W.W. Ludwig and J.L. Cardy, Nucl. Phys. B 285 [FS19] (1987) 687. 161 VI.S. Dotsenko, Nucl. Phys. B 314 (1989) 687. [7] R. Guida and N. Magnoli, preprint University Genova Gef-Th-12/1995, hep-th/9511209, to be published in Nucl. Phys. B. 18] AI.B. Zamolodchikov, Nucl. Phys. B 348 (1991) 619. [9] H. Sonoda, Nucl. Phys. B 383 (1992) 172; Nucl. Phys. B 394 (1993) 302.
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[ 10l B. Mikhak and A.M. Zarkesh, Nucl. Phys. B 430 [FSI (1994) 656. [ 11 ] K.G. Wilson, Phys. Rev. 179 (1969) 1499. [ 12] E. Brrzin, J.C. Le Guillou and J. Zinn-Justin, Phys. Rev. Lett. 32 (1974) 473; see also E. Brrzin, J.C. Le Guillou and J. Zinn-Justin, in Phase Transitions and Critical Phenomena, Vol. 6, ed. C. Domb and M.S. Green (Academic Press, London, 1976). [13] M.C. Berg6re and E David, Ann. Phys. (NY) 142 (1982) 416. 1141 A.M. Polyakov, Zh. Eksp. Teor. Fiz. 57 (1969) 271 I Sov. Phys. JETP 30 (1970) 151]. [151 B.M. McCoy and T.T. Wu, Phys. Rev, D 18 (1978) 1259; B.M. McCoy and Mu-Lin Yan, Nucl. Phys. B 257 [FSI41 (1985) 303. 116] G. Delfino and G. Mussardo, Nucl. Phys. B 455 (1995) 724. 1171 G. Delfino and P. Simonetti, preprint OUTP-96-198, SWAT~95-96~ 110, hep-th 9605065. I18t G. Parisi, Nucl. Phys. B 150 (1979) 163. 1191 H. Sonoda, Nucl. Phys. B 352 (1991) 585. [201 B.M. McCoy and T.T. Wu, The Two-Dimensional Ising Model (Harvard University Press, Cambridge, 1973); B.M. McCoy, hep-th 9403084. 1211 V.A. Fateev, Phys. Lett. B 324 (1994) 45. 1221 P.G. Lauwers and V. Rittenberg, Phys. Lett. B 233 (1989) 197; preprint Bonn University BONN-HE89-11 (unpublished). [23] C. Destri, F. Di Renzo, E. Onofri, E Rossi and G.P. Tecchiolli, Phys. Lett. B 278 (1992) 311. [24] J. Cardy, G. Delfino and P. Simonetti, in preparation. [251 VI.S. Dotsenko, Int. J. Mod. Phys. B 5 (1990) 1039. [261 V1.S. Dotsenko and V.A. Fateev, Nucl. Phys. B 240 [FSI21 (1984) 312; B 251 (1985) 691. [27] V1.S. Dotsenko, Adv. Stud. Pure Math. 16 (1987) 123. [281 P. Di Francesco, H. Saleur and J.B. Zuber, Nucl. Phys. B 290 [FS20] (1987) 527. [29] A.E Prudnikov, Yu.A. Brychov, O.I. Marichev, Integrals and Series, Vol. 3 (Gordon and Breach, London, 1990). 1301 J.G. Darboux, J. Math. 3 (1878) 377. ]311 R. Wong, Asymptotic Approximations of Integrals (Academic Press, New York, 1989) p. 122.
Nuclear Physics B 483 [FS] (1997) 563-579
On the short distance behavior of the critical Ising model perturbed by a magnetic field Riccardo Guida a,l, Nicodemo Magnoli b,c,2 a CEA-Saclay, Service de Physique Th(orique, F-91191 Gif-sur-Yvette Cedex, France h Dipartimento di Fisica, Universitgt di Genova, Via Dodecaneso, 33, 16146 Genova, Italy c lstituto Nazionale di Fisica Nucleare, Sez. Genova, Via Dodecaneso, 33, 16146 Genova, Italy
Received 19 June 1996; accepted 10 October 1996
Abstract We apply here a recently developed approach to compute the short distance corrections to scaling for the correlators of all primary operators of the critical two-dimensional lsing model in a magnetic field. The essence of the method is the fact that if one deals with OPE Wilson coefficients instead of correlators, all-order IR safe formulas can be obtained for the perturbative expansion with respect to the magnetic field. This approach yields in a natural way the expected fractional powers of the magnetic field, which are clearly absent in the naive perturbative expression for correlators. The technique of the Mellin transform has been used to compute the IR behavior of the regularized integrals. As a corollary of our results, by comparing the existing numerical data for the lattice model we give an estimate of the vacuum expectation value of the energy operator of the continuous theory. PACS: ll.25.Hf; ll.15.Bt; ll.10.-z; ll.10.Gh Keywords: Ising model; Conformal field theories; Perturbationtheory; IR divergences;Operator product
expansion
1. Introduction It is well known that ( D = 2) conformai field theories [ 1 ] can be used to describe statistical systems at the critical point [2]. The second step from this point of view is to obtain information on the behavior of the system near criticality, that is to study 1E-mail: [email protected]. 2 E-mail: [email protected]. 0550-3213/97/$17.00 Copyright C) 1997 Elsevier Science B.V. All rights reserved Pll S0550-32 13 (96)00585-8
564
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a conformal field theory perturbed by a relevant operator (i.e. with scale dimension 0
R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
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computed by use of form factor techniques [ 16,17]. The plan of the paper is as follows: in Section 2 we will summarize the OPE approach, while in Section 2.1 we describe a known mathematical technique that appears to be useful to regularize and disentangle IR divergences of integrals, the Mellin transform. In Section 3 we apply the approach to the Ising model (all the bulk computations are confined to Appendix A); in Section 3.1 we use non-perturbative as well as numerical inputs to estimate the parameters left unfixed by the OPE method; the obtained results are discussed in Section 3.2. Conclusions are given in Section 4.
2. All-order IR finite formulas
The goal of the method presented in Ref. [7] (see also Refs. [8-10] and Refs. [ 1113] for many preexisting ideas) is to obtain information on the short distance behavior of a conformal field theory perturbed by relevant operators, i.e. when to the conformal action ScFr is added a perturbation of the form
AS = - I dx ~ ais OiB (X), d
( 2.1 )
i
where A} (0i8) are bare couplings (operators) and 0 < xi < 2 (xi = xo,, xo being the scale dimension of operator 0 ) . This is achieved by expanding in powers of the corresponding renormalized couplings a i (Taylor expansion) the so-called Wilson coefficients Cab(r, ~l) that enter in the operator product expansion
CCb(r, A)(q0c (0)) a
(qOa (r)qOb(0)) a ,~ ~
(2.2)
¢
for the complete theory (4'a are deformations of the conformal theory operators, the suffix A refers to the complete theory correlators; the dependence on a i of the Wilson coefficients will be omitted in the following). Assuming the regularity of the Wilson coefficients in terms of the renormalized couplings Ai (minimality of the renormalization scheme), the validity of an action principle for the derivative with respect to A i and the asymptotic convergence of OPE, all reasonably satisfied by the complete theory, an IR finite representation has been given for the nth derivative of C~b with respect to the couplings evaluated at ai = 0, involving integrals of (eventually renormalized) conformal correlators: ~
Oil . . .ogi,,Cbla2 <[(I)bXoo ]} b
= lim { f d X l . . . f X<[:
Oin : ...
:
Oil :
(~a,~02 --
b ~-~Col02(~)b)Xoo]> b
R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
566
b
x ( [ : Oi,, : . . . --
*
: 0i2 : q~bX~]) + p . + . . .
Oil...Oi,,_lCala2
dXnOR,,(Xn)([:
Oi,, : q ~ b X ~ ] ) + p .
),
(2.3)
b
where c~i = a--~, O R ( X ) is any generic IR cutoff function (such that l i m g ~ O R ( x ) = 1), p. means all non-trivial permutations over labels of indices i, XR, is any arbitrary operator localized outside the sphere of radius R t >> R --~ e~, chosen to give adequate boundary conditions (and possibly including powers of Rt), and ~ is restricted to operators C/'b such that xb <<. ( D - xik) . . . + ( D - xi,,) - x x , . See Ref. [7] for more details. While each term in the sum in the right-hand side is IR divergent (and consequently requires a regularization), the complete right-hand side is IR finite and independent on the choice of the cutoff function (what we presented above is an "IR renormalization" more than a simple regularization). The structure of the expression is simple: the first term on the right-hand side is a naive (generalized) perturbative one, while the others, containing lower derivatives of Wilson coefficients, play the role of non-local IR counterterms, naturally induced by the theory itself. (A conjecture for the existence of this mechanism can be found in Ref. [ 18], in the general context of quantum field theories. Also a rigorous proof of the conjecture within the MS renormalization scheme for perturbative superrenormalizable quantum field theories can be found in Ref. [ 13 ] .) If the cutoff function is rotationally invariant (OR (r) = OR ( ]rl ) ) only scalar operators contribute in all expressions above. Also, if by dimensional analysis one knows a priori that no powers R° can be obtained from the IR counterterms available in that model, then it suffices to compute the first (naive) term of the right-hand side in some chosen IR regularization, and to keep only the regular term of its asymptotic expansion when R i --+ cxz, the singularity of the naive term being automatically killed by the IR counterterms 3 (which, in this case, cannot give additional finite contributions). Roughly speaking, one expects that this picture is realized when the IR counterterms do not need UV renormalization (i.e. are locally integrable), because in this case the renormalization scale ~ is absent and there is no other possibility to obtain an adimensional quantity from the log R terms that are expected in general from the R° IR counterterms (apart from miraculous cancellations of log R divergences). This shortcut can be applied in particular to the case we will consider here. We conclude by observing that the complete correlators are then obtained by combining the derived expressions for the Wilson coefficients with those for the required operator vacuum expectation values (eventually parameterized by unknown but fixed and universal constants, see Section 3.1 and the discussion in Ref. [7] ).
3This idea was already present in the first-ordercomputationsof Ref. [8].
R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
567
2.1. Mellin transform and IR divergences Having in mind that all we need are asymptotic expansions of (multiple) integrals in 1 ---+0 above, it is worth introducing here a natural method to deal the limit of mi ~ RTi with these expansions: the Mellin transform 4 (see Ref. [31 ] for a nice introduction). Given a function l(m) locally integrable on (0, ~ ) , of order m '~ when m ~ 0 and m - ~ when m ~ ~ , one can introduce in the complex strip - a < s < /3 its Mellin transform
[(s) =_~ dmmsl(m). J m o
(2.4)
If [(s) has a meromorphic extension in the complex plane on the left of Ires = - a one can derive from the inverse Mellin transform the asymptotic expansion
l(m) ,,~ ~ ' ~ R e s m-S[(s)).~=_,~i
(2.5)
i
with al = cr < ce2 < . . the powers of m in the asymptotic expansion of l(m) (notice that multiple poles give logarithms of m). In other words: singularities of the Mellin transform drive the asymptotic expansion. In our formulas we have always (possibly nested) integrals of the form
l(m) = / d 2 z O(mlzl)g(z )
(2.6)
(we refer from here on to the rotation invariant cutoff; notice also the slight and obvious change of notations). It is easy to prove the following convolution theorem:
[(s) = O(s)G(l - s),
(2.7)
where
~(1 - s) = f d2z Izl-Sg(z)
(2.8)
is essentially the Mellin transform of g (up to angular integrals). Thus in the cases in which (~ is known the integrals are easily done. Notice that if the Mellin transform of g strictly speaking does not exist but the Mellin transform of I(m) is well defined, one can try to bypass the problem by extending the integral [(s) to a more general one, [(s; 8) (in which g is now extended to some adequate g6), such that it reduces to the original one for some value 6 = 60, and such that the convolution theorem can be applied: if [ ( s ; 8) can be analytically continued to 8 = 80, the Mellin transform of I is obtained accordingly. 4 This technique was essential for the analog analysis in perturbative superrenormalizable quantum field theories, Ref. [ 13t (and references therein).
R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
568
In particular if the IR counterterms give no finite contributions (see Section 2), then the derivatives of the Wilson coefficients can be unambiguously obtained simply by taking the residue of the first, "naive perturbative", term in (2.3). In Appendix A we give the analytical expression of [ ( s ) for a general case that allows us to compute all the integrals involved in the first-order derivatives of the Wilson coefficients of primary fields of the conformal field theory underlying the critical Ising model.
3. Wilson coefficients for the perturbed Ising model The scaling limit of the lattice (D = 2) Ising model (see Ref. [20] and references therein) at the critical point, T = Tc, H = 0, is described by a continuous unitary conformal field theory, technically 34(3/4) [1]. The operator space of this conformal theory is generated by the primary operators 1, o-,g of dimension x = 0, 1/8, 1. The related fusion rules are y[o-] [o-] = [1] + [C],
[C][E] = [11,
[o-] [C] = [o-],
(3.1)
implying ( [ E l k [ l l " [ o - ] ~) = 0
(3.2)
for I odd. Also the Kramers-Wannier duality implies ( [ g l " [ l ] t) = ( - ) n ( [ g l " [ 1 ] t ) .
(3.3)
The object of our study will be the extended theory in the presence of a magnetic field, S = Slsing --}-- I ho'd2z"
(3.4)
J
Notice that with this notation each derivative Oh will be realized by the insertion of the operator - f dZz ~r( z ) (action principle). By application of the general formula (2.3) and of the selection rules (3.2) and (3.3) we obtain the following (non-trivial) first-order relations: -
c9I ,r,,~ ~ , ~ "r)(o'otrR,)
--
c9h
Cee(r)(o-oo-R, ) tr
=
=
- OhC~e(r) = - OhC,~ee(r){£OgR,) =
/' /' /' /'
d2z(trzO.R,[OrrO.O
d2z(O'zO'R,[£r£O
-
__
C ~1 r~(r)
- C~e~(r)Eo]),
C~e(r)] ),
(3.5) (3.6)
d2z(o'z [o'rgo - C~e (r)o'0]),
(3.7)
d2z(O'zgR,[~rgo -- C ~ (r)o-o - C~"e(r)~rll)
(3.8)
R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
569
(where the prime on the integrals denotes a rotationally invariant IR regularization, all limits have been omitted and we defined o-l = L_1L_1~r). It is easily realized by dimensional analysis that in all expressions above no finite contribution can arise from IR counterterms: we are thus in the case described in Section 2.1, where computations are easier. To reconstruct the short distance corrections of complete correlators we must have the expressions of the VEV of the lowest dimensional operators, o- and g. As is well known in perturbative quantum field theories and from direct renormalization group considerations, see e.g. Ref. [ 19], composite operators VEVs evolve by RG as d([q~a])
F~([q~b]),
(3.9)
in which 1 is the logarithm of the scale and the matrix F~ contains only positive integer powers of the renormalized couplings ,~i and forbids the mixing with higher dimensional operators (it is lower diagonal if operators are ordered with increasing dimensions). From these properties and from the fact that h has dimension 15/8 and that it is not renormalized it follows by simple dimensional considerations that o" and g do not mix with other operators, that F~, F~ reduce to the fixed point scale dimensions and that no logarithms appear in their VEVs. We can thus parameterize the VEVs of the complete continuous theory as
(~)h = A~lhll/15Sign( h),
(C)h = AEIhl 8/j5
(3.10)
(where it has also been kept into account that (o')h ((g)h) has to be odd (even) in h as is easily seen directly from the corresponding lattice expressions). Defining C - CIr] aimc, FA,B =- (AB)hlrl °iman and the scaling variable t =_ Ih[[r[15/8 we have the following expression for the short distance (small t) behavior: ~-'-~,8/15 -k- A~rSS~t 16/15+ O(t2), Fo.o.=C~o, + A~,~,~,
(3.11)
FCC = cl-~g + Acrffh~ggt 16/15 + O(t2),
(3.12)
KF~E = a o , ~ t 1/15 + OhC~---'~ct+ a E ~ t
23/'5 + O( t31/15),
(3.13)
where K = Sign(h) and
I-----_
t~hCff°" --
l (F(-1/2)F(7/4)) 3-2
2
F(5/4)
'
(3.14)
OhC~e =0,
(3.15)
OhC~-----~e= 1 sin(3~/8) F2(11/8) sin(15~r/8) F2(15/8) '
&,-'-~c=sin(Tr/8) s i n ( 3 ~ / 8 ) ( ~ r ( - l l / 8 ) F ( 7 / 8 ) ) r(-1/2)
(3.16)
2
(3.17)
have been computed in Appendix A (see also Appendix B for zeroth-order expressions) and the only unknown constants, up to now, are Ae and A,~. While in general the OPE
570
R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
approach leaves some universal unfixed constants, in this case the (approximate) value of A,~ (Ac) can be obtained from existing knowledge, see next section.
3.1. The missing VEVs The coefficient A,~ in (3.10) can be easily extracted by use of the results of Ref. [21 ] (see also references therein), where the integrability of 3,4 (3/4) + 412 (~ O') was used essentially (in particular the thermodynamic Bethe ansatz and the method of external fields have been used). We report here Eqs. (2.9) and (3.15) of Ref. [21 ],
47r2
[
]4/15 4sin(Tr/5)F(1/5)hS/, 5
M= [y2(3/16)y(l/4 )
r(2/3)r(8/~)
= 4.404908579981566037... h 8/15, ( •=-
(3.18)
M )2 sin(~'/5) 2 sin(7"r/5) 8 sin(Tr/3) sin(87r/15)
= -0.061728589822368... M 2,
(3.19)
where M is the minimum mass of the particle of the theory, y(x) =_ l'(x)/I'(1 - x), h > 0 in this section and the bulk energy • is defined by the partition function of the theory IR regulated in a cylinder of dimensions L, R,
Z(h) = (e- f h~) ,,~ e -'IcL,
L,R--* cxD.
(3.20)
Noticing then that (o-)/, = (o-e- f h,r)/(e- f h,~) ~ lim( --0/, log L,R
Z(h)/LR),
(3.21)
it follows that
(cr)h = Ohe = A~h 1/15,
(3.22)
where 8(
A,~= -
47r2
)8/,S
\ r2 (3/1--~-~,(1/4)
= - 1.27758227605119295...
sin~ ( F(1/5 ) ,~2 ~ 8 . /'(273-)T(ff/15)J sin 3 sm 73
(3.23) (3.24)
On the other hand, no non-perturbative information is available (to our knowledge) for the VEV of operators different from the perturbation, o-. In principle, numerical estimates of the constant Ag can be obtained by comparison with Monte Carlo simulations on the lattice for the normalized (connected) correlator (S(R)S(O))c/{S) 2 in the regime in which sc, R >> 1 and t 8/15 o( R/( is not too big, i.e. ( e( H -8/15, in a small field (S, R, sc, H being lattice spin, position vector, correlation
R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
571
length and magnetic field and the lattice spacing is assumed to be one everywhere). 5 In practice with the available data (Ref. [22], cubic lattice), one has only a small window of applicability ( ( ~ 15.5 for H = 0.001). To improve convergence and enlarge this window of applicability we fitted, together with the powers given in Eq. (3.11) also the next to leading terms t 2, t 32/15 (notice that logarithmic terms are absent for the same considerations given in Section 3). From the known first two coefficients in the small t expansion for the continuous expression of (o-(r)o-(0))c/(O-) 2 we can fix the overall normalization (i.e. the lattice magnetization (S)) and the correlation length (, related to continuous variables by R / ( = Mr (M being the mass in Eq. (3.18)). Notice that the predictions (3.11)-(3.13) refer to the complete correlators, and that the normalization factor between the continuous spin VEV and the lattice one (physical magnetization), is negative in our conventions, see also Ref. [22]. At this point all is fixed and we get a prediction for the coefficient of t 8/15 (Ac) and for the coefficient of t 16/15 (which actually is known exactly). In practice we selected only those fits that give a good prediction for the known coefficient, fixing in this way the above-mentioned (but unknown) window of applicability, a good representative of which is (for H = 0.001) the R = 5-54 range that corresponds to Ac ~ 0.321
(3.25)
(see Eq. (B.1) for our normalization) with an error of the order of a few percent (estimated roughly from the variation of this quantity among different "good" fits). As a byproduct we also get the estimate sc ~ 15.4,
(S) ,-~ 0.634,
(3.26)
in agreement with the results sc = 0 . 3 8 ( 1 ) H -s/15 of Ref. [22] and (S) = 1.003(2)H 1/15 of Ref. [23] (with a larger lattice). As an independent check that our estimate is correct we can use the known exact sum rule [ 16,17,24] /I,~ = _ 27rh(2 - 2A '~) f dZr(o.(r)o.(O))h,c" 4rr(o')
(3.27)
We split the integral into two pieces, Mr < A and Mr > A. We estimated the first term by use of our expression (3.11 ) and the second one by use of the first three terms of the long distance expansion 3
(o'(r)o-(0))h,c -~ Z
1
o- 2
~(F,~ ) Ko(c,mr),
(3.28)
i=1
where coefficients ci and form factors F/'~ can be found in Ref. [ 16] (and references therein). By use of (3.27) we can give an expression of AE in terms of A. Clearly, for 5 Notice that the relation between lattice spin and continuous theory operators has the general form S o- + . . . in w h i c h . . , means higher descendants of o- that are irrelevant in the continuous limit because they have higher scale dimensions: it is easy to check that their relative contribution in a cubic lattice is suppressed at least by a factor 1/~"2 (not bigger than the relative errors of our estimations, Eqs. (3.25) and (3.26)).
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A small we have a big error from the approximated long distance expansion, while for small A we have a big error from the approximated small distance expansion. If (by a minimal sensitivity criterion) we choose the value of A that minimizes AE (A ~ 1.5) we obtain the value Ac ,'~ 0.322, with an error of the order of 2% (estimated by repeating the procedure with only one mass term) which is compatible with Eq. (3.25). 3.2. Comments
A first glance at our results shows that a coefficient of the expansion for (CE) (3.15) vanishes non-trivially (unexpectedly from the point of view of the selection rules of the conformal field theory). This might sound strange, but one must bear in mind that the complete theory is an integrable model with powerful symmetries: we can thus interpret the zero found as a (short distance) signal of these symmetries and possibly of the existence of non-trivial differential equations satisfied by the correlator (analog of Painlev6 equations for spin-spin correlators in thermal perturbations of the Ising model [20] ). Another important feature of our prediction is the presence of fractional powers of the coupling (or equivalently of the scaling variable t = IhllrlS/~5): in the OPE approach these powers come out naturally from the VEV of operators (times some integer power of t coming from the Wilson coefficients, analytic in t). In particular the fractional correction t 8/j5 that is found in (3.11) for the (o-o-) correlator is expected if one introduces as infrared cutoff in the perturbative expansion for the correlators the correlation length Rc ~ Ih1-8/~5 (see Refs. [14,6]). The existence of this fractional contribution to the spin-spin correlator can be recovered from the results of Ref. [ 12] (based also on OPE and Callan-Symanzik equations in the 4 - e framework) by inserting the exact values of the critical indices given by the conformal theory. Also it is interesting to see that the scaling of the (o-E) correlator (vanishing at the critical point) is dominated by a fractional power, t I/i5, coming from the VEV (o'). We must remark here that our results are in disagreement with an existing expression of the short distance behavior of spin-spin correlator in the same context [25], coming from the effort to apply in this case the perturbative expansion for correlators developed in Ref. [6] and to obtain the expected fractional powers of h as well. The main difference in Ref. [25] with respect to our (3.11) is the presence of powers of logt in the first terms that, in our mind, are not motivated due to the absence of renormalization of the lowest dimensional operators o-,E (and of L - 2 L - z l as well, which is the next scalar operator that can have a non-zero VEV in this theory, as shown in Refs. [8,19]), see the discussion near Eq. (3.9). Also the power t 24/15 present there is not motivated from the OPE point of view (there is no corresponding VEV). In Fig. 1 we give a comparison of the numerical data for (SS), see Ref. [22], with our prediction Eq. (3.11 ) (without the higher-order terms added in previous section to improve the fit) where the estimated values (3.25) as well as (3.26) are used. What is noticed immediately is that the convergence of the expansion is quite good (see also Ref. [8] ): with the first-order corrections to Wilson coefficients, we can achieve a
R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
573
0.3
0.25
0.2
[3.15
0.i
0.05
. . . . . . . . . . . . . . . . . . . . . . 5
i0
20
15
77.'7rr~ 25
30
Fig. 1. Values of (S(R)S(O)) from lattice simulation (dots) and as predicted from the OPE approach (solid line) versus R (lattice position) for H = 0.001 (ge ~ 15.5).
2.4 ! 2.2
. . . .
0. . . .
i
. . . .
l'.s . . . .
1.8
1.6
1.4
1.2
Fig. 2. Estimates of {o'(r)g(O))/(tr)(g) versus Mr: comparisonbetween the OPE approach (solid line) and the form factor method with one (dashed line) three (dot-dashed line) and eight (dots) terms. relative error of order 2% at R / ( = M r ~ 1. No numerical results exist to our knowledge for (o-g) and (g,f). Nevertheless in Fig. 2 we show a comparison between our (short distance) prediction and the (long distance) form factor result (up to the first eight terms) [ 17] in the case of {o'g). It appears from Fig. 2 that there is a good agreement of the two methods in the intermediate region 1 < M r < 1.5 and a reasonable evidence of convergence of the long distance expansion towards our result in the region M r < 1, as is expected. This agreement could be regarded as an independent test of the form factor clustering property that has been assumed in Ref. [ 171 to get the form factors for (o-g) and ( g g ) . (The origin of this property will be explained in Ref. [24].)
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4. Conclusions We computed the short distance behavior of all the correlators of primary fields of the critical two-dimensional Ising model perturbed by a magnetic field up to O(t 2) excluded, t = IhLIr IJs/s being the scaling variable. The IR finite OPE approach, extensively developed in Ref. [7] has been used. This method does not give information about VEVs by itself. An unfixed constant for (o-) has been obtained by use of non-perturbative results coming from the integrability of the model (TBA), while by comparison with the existing numerical data we got the non-trivial estimate (3.25) of (g) in the continuous theory. We point out once more that with the OPE approach the expected [14] fractional powers of the coupling h are naturally obtained from the operators' VEVs, while in alternative perturbative direct estimates of the correlators (IR regularized in some way) these non-integer powers are absent. In particular we immediately found the expected t 8/15 correction to the spin-spin correlator [ 14], and we predicted that the dominant term of (o-g) is the fractional term t 1/15 (which should be observable in adequate lattice simulations). The result for (o'o-) when compared with the available data signals a good convergence of the approach: this encourages one to search for some resummation technique that could further enlarge the convergence radius. Moreover the use of exact sum rules to extract information from both short and long distance approximations, which has been done in Section 3.1, should stimulate the research in this direction, with the final goal of connecting the two regimes. We conclude by emphasizing that the OPE approach, being very general (integrability of the theory is not necessary to compute Wilson coefficients) and fast convergent, can always be seen as a bridge towards the study of non-integrable perturbations and as a test for the ansatz behind integrable theories predictions (as done in Section 3.2). It is furthermore an open question if the powerful Coulomb gas technique of Refs, [26,27] and the IR regularization of Ref. [6] could be married with the OPE approach to reach higher orders of the expansion in this and other statistical models.
Acknowledgements One of the authors (R.G.) is indebted to D. Bernard and J. Zinn-Justin for useful discussions and careful reading of the manuscript. The authors acknowledge J. Bros, M. Caselle, G. Delfino, P. Di Francesco, G. Mussardo, S. Nonnemacher, A1.B. Zamolodchikov, J.B. Zuber for interesting information and discussions. The authors also thank G. Delfino and P. Simonetti for sending a table of the (eight terms) form factors prediction for the spin energy correlator. The work of R.G. is supported by a TMR EC grant, contract No. ERB-FMBI-CT-95.0130. R.G. also thanks the INFN group of Genova for the kind hospitality.
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Appendix A. Computation of the integrals First of all we give the general expression for the Mellin transform (see Eq. (2.4)) with respect to m (IR cutoff) of the integral
l~./z,~( m; x )
= fd wO mlwl)lwl °lw-
xJ2 lw-
(A.1)
in which m > 0, x E (0, 1), a, f l , 7 > - 1 (to have local integrability) and we will fix the IR cutoff function O(t) = e -t (such that its Mellin transform is O(s) = F ( s ) ) . By use of the convolution theorem (2.7) it easy to obtain
[,,~,r(s;x) = F ( s ) D ( a - s/2,fl, y , x ) ,
(A.2)
where the integral
O ( a , b , c , x ) =_
f d:wlwl2alw-xl2Clw-II
(a.3)
has been computed in Ref. [27] by use of contour deformation in the variable Im w and was found to give
S(a)S(c) S(b)S(a + b + c) D ( a , b , c , x ) = S ( a + c ) ]l°xl2 + S(a+c) 111°°12'
(A.4)
where
S(a) - sin 7"ra, lox =- f d u u a ( x - u)C(1 - u) b o
= xl+a+cF(a + 1)F(c + 1) F(-b,a+ F ( a + c + 2)
l ; a + c + 2;x),
O0
fduua(. -- x)C(u -1
F(-a-b-c1)F(b+l) F(-a -c)
F(-a-
b- c-
l,-c;-a-
c;x)
and F _= 2F1 is the hypergeometric function (see e.g. Ref. [29]). From the Mellin transform (A.2) we can obtain the asymptotic expansion of the original integral (A.1) when m -~ 0, by use of (2.5). All the required integrals in (3.5)-(3.8) can be then computed by use of the general expression (A.2). See Appendix B for the conformal correlators. In our particular case we know that the IR counterterms cannot give m ° corrections, see Section 3, so that the derivative of the Wilson coefficient is obtained by extracting the m ° contribution of the naive perturbative term (the first of right-hand side) in (3.5)-(3.8), i.e. by computing the residue of its Mellin transform in s = 0. Notice that due to the absence of m ° log m
R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
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IR counterterms, there cannot be log m factors in the naive term and correspondingly its Mellin transform will have only simple poles. While (3.7) and (3.8) are trivially reduced to (A.2), some few words must be spent for (3.5), i.e. for the Mellin transform of the four-spin correlator (B.6). In the limit R'm --~ c~ (R' being the argument of the R'-z _-- I boundary operator o'R,) we have in the correlator the simplification x = (R'-~)r (r = 1) So that it is better to change variable and obtain (in the notation of the convolution theorem (2.4))
O(l_s)= f daxlxl,,,_4ll_x,_26(l+ 2 lv/-(=7-x+
1- lVq-S-~-x) 2 .
(A.5)
As explained in Section 2.1 we have introduced an additional parameter 6 to justify the use of the convolution theorem. The analytic continuation to 8 = 1/8 will give the wanted Mellin transform of the four-spin correlator with exponential cutoff. The final step to reduce (A.5) to (A.2) is the simple substitution w = x/1 - x and the observation that the integrand is invariant for w --+ - w . By use of these considerations Eqs. (3.14)-(3.17) follow. As a general check of the regularization independence our approach, we report the alternative derivation of OhC~% obtained regularizing the integrals by restricting them to Izl < R. We will not use the Mellin transform technique and we will keep explicitly the IR counterterms to show explicitly how cancellations work. In the limit R ' / R --~ oo (being x = l / z ) we can rewrite (3.5) as
-ohc,,%=
~~ Ig+(x)lR+lg-(x)12-1- Ixl,
(A.6)
I.~L>I/R where we omitted the remaining overall limit R ~ oo in the left-hand side and defined
g + ( x ) --
W/1 + v / 1 - x 2~ ----x~"
(A.7)
Unfortunately, we do not have a closed expression for the integral (A.6). However, the calculation can be performed by series, splitting the integral in two regions, 1 < ix I < 1 and Ix I > 1. The contribution of the first region is then obtained by expanding OO
g+(x)
Z g i x "+ i .i--o
OO
g - ( x ) = ~--~gTx.i+l/2 j---o ~
-
(A.8)
(note also that ( g + ( x ) ) * = g ~ ( x * ) ) , exchanging the series with the integral and performing simple integrals of powers of x x*. At this stage the role of IR counterterms is essential to give a finite R ~ oo limit. The contribution of the second region is obtained after making the change of variable x = - 3 ' from which we obtain
2 I Iwl
d2w]wl3/Zlh(w)12'
(A.9)
R. Guida,N. Magnoli/NuclearPhysicsB 483 [FS] (1997)563-579 h(w) =
577
(A.10)
1 + w2) 1/4
for the sum of g± contributions (integrals of IR counterterms are easily performed). Again expanding
h(w) = ~ h.iwi .j_--o
(A. 11)
the integral can be easily performed by series. The final result is
(1 "--".,.Ig~122jlg;IZ)--
=
+
1
+
43
4
+ 7 +
.j>~2
\ 2j2[hj12 +7/2 ]
1 +
/
'
(A.12) where the coefficients g~:, hj satisfy the recursive relations
(j+l)
(j+3)
gi-+l-2(J+2)(j+~)gf+(jz-~j+5)g~_l=O,
(j+l)(j+2)hj+2+j
( ~ )
2j-
hi+
(
7 45) j2-2j+T~ hi_z=0 (A.13)
due to the existence of the following differential equations satisfied by g±, h:
(x3-- 2x2 + x)g
+',
+
(~
7
x2---~x +
~)
'
3
±
g± ---~xg =0,
(l + 2w2 + w4) h- + ~(w 3 3 + w)h ' - ~-~w3 2h=0.
(A.14)
As they stand the series are very slowly convergent (g:~, hj ~ j-7/8), so to increase the convergence of the series we added and subtracted to the jth term its asymptotic powerlike behavior 6 that can be resummed and gives the well known Lerch functions (q0(z, s, a) = ~-']~,+,,+0 z"/( n+a)s, n = 0, 1. . . . ) which are obtained with high precision from Mathematica. After numerical resummation we obtain then
OhCff,~ = 0,403745631213448123 ....
(A.15)
which agrees up to the obtained precision with the analytic result (3.14). 6 The first two terms are obtainedfrom the contribution of the singularities nearestto zero (Darbouxtheorem, see Refs. [ 30,31 ] ) while the others can be convenientlyextracted by expanding in j--7/8+k/2the recursive relations (A.13) up to the desired order k.
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R. Guida, N. Magnoli/Nuclear Physics B 483 [FS] (1997) 563-579
Appendix B. Conformal correlators
To fix notations we give here the expressions for the fixed point (conformal theory) quantities involved in our computations. See Ref. [28,10] for their derivation. Wilson coefficients: 1
C~E (r) - 47r2tr[~,
cl,~(r) -
! Irll/4'
(B.1)
C ~ ( r ) = 7fir[ 3/4,
(B.2)
1
C ~ ( r ) = 4~rtrl
(B.3)
Useful correlators: 1 Izl2l3/4 (o-(z,)o(z2)£(z3)}- 47r Iz131Iz231 '
(B.4)
Iz12(z32 + z42) 2Z32Z4212 (O'( Zl ) O-(Z2) E (Z3) E (Z4) ) = 16~ [---z42--z32-z41 "2 z----3~111z43121 1 - - - -z12 - ~ /I4 -
(O"(Zt)O'(Z2)O'(Z3)O'(Z4)) =
1(1
-
'
(B.5)
- x)z12z34[ - I / 4
(B.6) where
x = z12z34/(z13z24), Z~j = zi - zj.
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