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Distribution of the mean magnetization in O(N)-symmetric spin models and chiral perturbation theory
specifically explain, volume globally distribution purpose usually only from the derive O(N)-symmetry Physics indicated by dependence limit infinite-volume expansions in KFA, "measurements" consider Let (unnormalized) results these iswhat the Goldstone following, für Hence, O(N)-invariant GÖCKELER Theorie interested the Lsimulations In rather The the spontaneously designed Postfach large-volume Therefore BMonte denote Theoretische particular, formulae of procedures chiral (Proc method an finite from Euclidean THEORY the the small) which enables der N-component OF © modes we the Carlo in to perturbation mean chiral results observation box Suppl 1913, Elsevier Elementarteilchen, in (N one THE results of deal directly systems shall length distribution we in apply aexpansions one with chiral action that broken >_ Physik finite magnetization method needs perturbation lattice D-5170 obtain with MEAN from It17discuss 2) toScience referring whenever periodic of allows 4( have refer (1990) perturbation obtain box Inscalar S(4) aHowever, the of the theory finite-volume E,particular, are field reliable The are Jülich, the MAGNETIZATION to for W (see RWTH been finite-size aPublishers box 347-'349 expression to infinite-volume done to isfield boundary quantities spontaneously constants corresponding theory theories tells of aHöchstleistungsrechenzentrum derive also and supposed the devised Fcontinuous one method the fi(x) Aachen, on us theory we infiniteRef V effects for is calcuGermany about largefor finite = mean want B ulticonperinapconnected O(N)-symmetric for 2) L4 to toa1the Sommerfeldstrasse, IN distribution 0(N)-SYMMETRIC with thenow spin of mean spontaneously W to distribution with the M Therfore HLRZ, given the models in eD-5100 expansion modified Applications constant magnetization order o(N)-symmetry, perturbation some aby _ SPIN = lattice _ we into ~Jo~ Aachen, the function Jthe O(N)-invariant can Bessel (the J"0 broken Z(i)/i =find external Ae-s'4)b(M1 MODELS functional dMMN-1W(M spontaneously to JjD*e-(')'!f""" perform dMM`2v-1W(M)Yr so-called We the theory FMonte-Carlo M dNM function phase We can current corresponding ofGermany W AND integral W(1Îf)ey~âs define the the iswrite provides e-expansion) UV-regularization So aIangular system function _CHIRAL we broken of ~the calculations index want uspartition at expansion integrations with of phase, jM) vfinite to PERfor jM) M =solve aZ(j) =largeN12 isIoff! funcas(4) for in
Nuclear North-Balland
.
.)
347
DISTRIBUTION TURBATION Meinulf Institut and Gruppe c/o Using we volume. are
.R. .R.
.
.
All formed (and mately volume . extract lations . Among this is caused volume symmetry pearing quantities. volume results In broker to the We four-dimensional ditions . its . be The magnetization
is .
W(M)
jd44)
(Implicitly, sumed, .g. tion
.)
.
(2)
; :s S=
Z(~j . Due only.
.
: Z(a)
.
.
;YN(V
where
. .
with 1.
Chiral volume the for this
.
M 0920-5632/90/$3 .50 North-Holland
V
d4r,
f(Vj)
:r,) .V.
. . =
0
" (V
(4)
M. Gôckeler/ O(N)-symmetric spin models
348
and consider the argument of f as a complex variable. (Note that there is some additional V-dependence in f.) If W(M) decreases faster than exponentially for M -+ oo (as is plausible and will be confirmed by our calculation), f(z) is an entire function of z. In particular, one can put z = -i-r with T > 0: 2-vTv+1/2 f(-ZT)
00 dMMv+1 /2 W(M)(MT) 1 /2 jv(MT),
-.
0
(7)
i.e . 2-vTv +1 / 2 f(-ZT) is the Hankel transform 3 of M,,+1l2W(M) . Consequently W(M) = (2M)-'
J
(ZTT° +1f(-ZT) .I~(MT) .
(8)
Now we have to explain, what the E-expansion tells us about the large volume behaviour of Z(j) 4. The expansion parameter E is such that 1/L = O(e), j = O(E4) .
(9)
Therefore one gets expansions in powers of 1/L at a fixed value of Vj. For the function f introduced in (6) one finds f(t) = hYN(I,1Et)exp
(C2~-2 +O(E6 ), VF412
(10)
where 7a is a normalization constant and (N -1)01 _ (N - 1)(N - 8) P1 = 1 8-irF 2 L2 2F4 L4 647r x
10 i N-1
2-02--ln (
P2 = (1 + 02 + In 641r2 2
2
Inserting (10) in (8) one finds 5 W(M)
4 P1YF M 2P2 E VF4 x exp -(M2 -}- Pi E2) 4p2 E2 (
- 7tYF¢
(P1EM)- I~ 2wP2E2 '
With the help of the asymptotic expansion of I one gets the leading behaviour of W as V -+ oo : W(M) , const.(V/P2)1 /2 M v -1 /2 x exp
41r
-4 P2E2
(M - E) 2 .
e-vu(m) = CW(M)
,
A2 2 ))
4
(12)
The "shape coefficients" 01 , 02 depend only on the geometry of the box. For a hypercube one has 01 = -1 .765, 02 = -1 .175 . In addition, the above formulae contain four undetermined parameters : the field expectation value E, the "pion decay constant" F(specifying the transition matrix elements of the symmetry currents from the ground state to a state containing a single Goldstone boson) and the logarithmic scales Aj,r, A, . All these refer to the model in infinite volume and without external current.
(14)
To which extent can we trust (13) in view of the fact that terms of order e6 have been neglected? For L sufficiently large, W as given by (13) has a unique maximum close to M = E . The E-expansion amounts to a saddle point expansion of expectation values (moments, in particular) of the distribution W. Consequently, one probes only the neighbourhood of the maximum and all those features of W which influence the moments only at the order E6 have to be considered as artefacts of our procedure. It is illuminating to calculate W keeping only the lowest order in the E-expansion of Z(j) . One then obtains a b-function centered at E. Inclusion of the O(c2 )-correction leads to a shift of the b-function . Finally, at O(E4 ) one gets the above expression for W with a finite width. Now we want to discuss the expression for the constraint effective potential U (M) which follows from (13) . The constraint effective potential (CEP) is defined by 6
2~
A 47r
(13)
(15)
where r is a normalization constant and the Vdependence of U is not displayed. The CEP is not necessarily convex, but as V -+ oo it tends to the standard effective potential and hence becomes convex 7 . From (13) one finds U (M) =
4
E2M2 - Y In YN(rM) +const .,
4~ P1 VF4
(16)
2P2E
For V sufficiently large, this function has a unique minimum at M = Mo=EP1 -
N27, 1
+...
(17)
M. Göckeler/0(N)-symmetric spin models As V --* oo, it becomes indeed convex: The limit vanishes for all M >_ 0! This is not so unreasonable as it might seem at first, because the f-expansion probes U only near its minimum. Under plausible hypotheses on the V-dependence of the exact CEP one finds that the terms of the f-expansion given in (10) uniquely determine only the leading behaviour for V --> oo of the second derivative " F4 U (Mo) = 2p2E2 + . . .
(18)
In particular, one gets no information about the fourth derivative. What do these results teach us for the interpretation of Monte Carlo data obtained, e.g ., in the linear a-model? (See Ref. 8 for other applications of chiral perturbation theory in this field.) From (13) one can calculate the susceptibility (M2 ) - (M) 2 (a measure for the width of W) at j = 0. One finds (M2) - (M)2 -
2p2 = F2 + 0('66)* VF4
(19)
Given that E and F are known from other sources 10, one can determine the logarithmic sc4e~ A, contained in p2, which (via perturbation theory) can be related to the mass of the a-particle. Prelimintry results in this direction (for N = 4) are encouraging 9 . In a similar way one cdrj use the f-expansion results for two-point-functions to obtain a large-volume expansion for the two-point function of the "projected field" 4(x) = M . `i(x)/M 10 at j = 0 . Defining (20)
r' . One can subtract this expression from the measured r' and might try to fit the difference with a single cosh containing the a-mass . Let us conclude by remarking that the above calculations can be extended to three dimensions and that the successful use of chiral perturbation theory in the linear a-model should encourage applications in its original domain, QCD. ACKNOWLEDGEMENTS This work was done in collaboration with H. Leutwyler. It is a pleasure to thank him for his cooperation . REFERENCES 1 . J. Gasser and H. Leutwyler, Phys . Lett .188B (1987) 477; H. Leutwyler, Phys. Lett. 189B (1987) 197; J. Gasser and H . Leutwyler, Nucl . Phys . B307 (1988) 763. 2. H. Neuberger, Phys . Rev. Lett . 60 (1988) 889; Rutgers University preprint RU-42-87 1987). 3. A. Erdélyi et al ., Tables of integral transforms, vol. 2 (McGraw-Hill, N.Y ., 1954). 4. P.Hasenfratz and H.Leutwyler, in preparation ; P.Hasenfratz, this conference. 5. M . Göckeler and H. Leutwyler, in preparation. 6. R. Fukuda and E. Kyriakopoulos, Nucl . Phys . B85 (1975) 354. 7. L . O'Raifeartaigh, A. Wipf, and H. Yoneyama, Nucl . Phys . B271 (1986) 653 . 8. A . Hasenfratz, K . Jansen, J. Jersâk, C.B. Lang, H. Leutwyler, and T. Neuhaus, preprint FSU-SCRI-8942, BI-TP 89/08; T. Neuhaus, this conference . 9. K. Jansen, private communication.
one gets for IP(:r) - 1 id4,
(21)
the result 5 r'(a)
349
j,9 `2(N - 1)II(x.) + 2p2 lx) / +0(fr') (22)
where 11(aß), h'(x) are explicitly known functions of order .. This is the contribution of the Goldstone bosons to
10 . A.Hasenfratz, K . Jansen, C.B . Lang, T. Neuhaus, and H . Yoneyama, Phys. Lett . 199B (1987) 531 ; A. Hasenfratz, K . Jansen, J. Jersâk, C.B . Lang, T. Neuhaus, and H . Yoneyama, Nucl. Phys. B317 (1989) 81 ; K . Jansen, Nucl . Phys . B (Proc. Suppl.) 4 (1988) 422; C.B . Lan , Latt;ce Higgs Workshop, ed . B . Berg et al ., World Scientific, Singapore,1988) 158; T. Neuhaus, in 1988 Syrnposiran ora Lattice Field Theory in. Fcr»ailah. USA and preprint BI-TP 88/29.
Nuclear North-Balland
.
.)
347
DISTRIBUTION TURBATION Meinulf Institut and Gruppe c/o Using we volume. are
.R. .R.
.
.
All formed (and mately volume . extract lations . Among this is caused volume symmetry pearing quantities. volume results In broker to the We four-dimensional ditions . its . be The magnetization
is .
W(M)
jd44)
(Implicitly, sumed, .g. tion
.)
.
(2)
; :s S=
Z(~j . Due only.
.
: Z(a)
.
.
;YN(V
where
. .
with 1.
Chiral volume the for this
.
M 0920-5632/90/$3 .50 North-Holland
V
d4r,
f(Vj)
:r,) .V.
. . =
0
" (V
(4)
M. Gôckeler/ O(N)-symmetric spin models
348
and consider the argument of f as a complex variable. (Note that there is some additional V-dependence in f.) If W(M) decreases faster than exponentially for M -+ oo (as is plausible and will be confirmed by our calculation), f(z) is an entire function of z. In particular, one can put z = -i-r with T > 0: 2-vTv+1/2 f(-ZT)
00 dMMv+1 /2 W(M)(MT) 1 /2 jv(MT),
-.
0
(7)
i.e . 2-vTv +1 / 2 f(-ZT) is the Hankel transform 3 of M,,+1l2W(M) . Consequently W(M) = (2M)-'
J
(ZTT° +1f(-ZT) .I~(MT) .
(8)
Now we have to explain, what the E-expansion tells us about the large volume behaviour of Z(j) 4. The expansion parameter E is such that 1/L = O(e), j = O(E4) .
(9)
Therefore one gets expansions in powers of 1/L at a fixed value of Vj. For the function f introduced in (6) one finds f(t) = hYN(I,1Et)exp
(C2~-2 +O(E6 ), VF412
(10)
where 7a is a normalization constant and (N -1)01 _ (N - 1)(N - 8) P1 = 1 8-irF 2 L2 2F4 L4 647r x
10 i N-1
2-02--ln (
P2 = (1 + 02 + In 641r2 2
2
Inserting (10) in (8) one finds 5 W(M)
4 P1YF M 2P2 E VF4 x exp -(M2 -}- Pi E2) 4p2 E2 (
- 7tYF¢
(P1EM)- I~ 2wP2E2 '
With the help of the asymptotic expansion of I one gets the leading behaviour of W as V -+ oo : W(M) , const.(V/P2)1 /2 M v -1 /2 x exp
41r
-4 P2E2
(M - E) 2 .
e-vu(m) = CW(M)
,
A2 2 ))
4
(12)
The "shape coefficients" 01 , 02 depend only on the geometry of the box. For a hypercube one has 01 = -1 .765, 02 = -1 .175 . In addition, the above formulae contain four undetermined parameters : the field expectation value E, the "pion decay constant" F(specifying the transition matrix elements of the symmetry currents from the ground state to a state containing a single Goldstone boson) and the logarithmic scales Aj,r, A, . All these refer to the model in infinite volume and without external current.
(14)
To which extent can we trust (13) in view of the fact that terms of order e6 have been neglected? For L sufficiently large, W as given by (13) has a unique maximum close to M = E . The E-expansion amounts to a saddle point expansion of expectation values (moments, in particular) of the distribution W. Consequently, one probes only the neighbourhood of the maximum and all those features of W which influence the moments only at the order E6 have to be considered as artefacts of our procedure. It is illuminating to calculate W keeping only the lowest order in the E-expansion of Z(j) . One then obtains a b-function centered at E. Inclusion of the O(c2 )-correction leads to a shift of the b-function . Finally, at O(E4 ) one gets the above expression for W with a finite width. Now we want to discuss the expression for the constraint effective potential U (M) which follows from (13) . The constraint effective potential (CEP) is defined by 6
2~
A 47r
(13)
(15)
where r is a normalization constant and the Vdependence of U is not displayed. The CEP is not necessarily convex, but as V -+ oo it tends to the standard effective potential and hence becomes convex 7 . From (13) one finds U (M) =
4
E2M2 - Y In YN(rM) +const .,
4~ P1 VF4
(16)
2P2E
For V sufficiently large, this function has a unique minimum at M = Mo=EP1 -
N27, 1
+...
(17)
M. Göckeler/0(N)-symmetric spin models As V --* oo, it becomes indeed convex: The limit vanishes for all M >_ 0! This is not so unreasonable as it might seem at first, because the f-expansion probes U only near its minimum. Under plausible hypotheses on the V-dependence of the exact CEP one finds that the terms of the f-expansion given in (10) uniquely determine only the leading behaviour for V --> oo of the second derivative " F4 U (Mo) = 2p2E2 + . . .
(18)
In particular, one gets no information about the fourth derivative. What do these results teach us for the interpretation of Monte Carlo data obtained, e.g ., in the linear a-model? (See Ref. 8 for other applications of chiral perturbation theory in this field.) From (13) one can calculate the susceptibility (M2 ) - (M) 2 (a measure for the width of W) at j = 0. One finds (M2) - (M)2 -
2p2 = F2 + 0('66)* VF4
(19)
Given that E and F are known from other sources 10, one can determine the logarithmic sc4e~ A, contained in p2, which (via perturbation theory) can be related to the mass of the a-particle. Prelimintry results in this direction (for N = 4) are encouraging 9 . In a similar way one cdrj use the f-expansion results for two-point-functions to obtain a large-volume expansion for the two-point function of the "projected field" 4(x) = M . `i(x)/M 10 at j = 0 . Defining (20)
r' . One can subtract this expression from the measured r' and might try to fit the difference with a single cosh containing the a-mass . Let us conclude by remarking that the above calculations can be extended to three dimensions and that the successful use of chiral perturbation theory in the linear a-model should encourage applications in its original domain, QCD. ACKNOWLEDGEMENTS This work was done in collaboration with H. Leutwyler. It is a pleasure to thank him for his cooperation . REFERENCES 1 . J. Gasser and H. Leutwyler, Phys . Lett .188B (1987) 477; H. Leutwyler, Phys. Lett. 189B (1987) 197; J. Gasser and H . Leutwyler, Nucl . Phys . B307 (1988) 763. 2. H. Neuberger, Phys . Rev. Lett . 60 (1988) 889; Rutgers University preprint RU-42-87 1987). 3. A. Erdélyi et al ., Tables of integral transforms, vol. 2 (McGraw-Hill, N.Y ., 1954). 4. P.Hasenfratz and H.Leutwyler, in preparation ; P.Hasenfratz, this conference. 5. M . Göckeler and H. Leutwyler, in preparation. 6. R. Fukuda and E. Kyriakopoulos, Nucl . Phys . B85 (1975) 354. 7. L . O'Raifeartaigh, A. Wipf, and H. Yoneyama, Nucl . Phys . B271 (1986) 653 . 8. A . Hasenfratz, K . Jansen, J. Jersâk, C.B. Lang, H. Leutwyler, and T. Neuhaus, preprint FSU-SCRI-8942, BI-TP 89/08; T. Neuhaus, this conference . 9. K. Jansen, private communication.
one gets for IP(:r) - 1 id4,
(21)
the result 5 r'(a)
349
j,9 `2(N - 1)II(x.) + 2p2 lx) / +0(fr') (22)
where 11(aß), h'(x) are explicitly known functions of order .. This is the contribution of the Goldstone bosons to
10 . A.Hasenfratz, K . Jansen, C.B . Lang, T. Neuhaus, and H . Yoneyama, Phys. Lett . 199B (1987) 531 ; A. Hasenfratz, K . Jansen, J. Jersâk, C.B . Lang, T. Neuhaus, and H . Yoneyama, Nucl. Phys. B317 (1989) 81 ; K . Jansen, Nucl . Phys . B (Proc. Suppl.) 4 (1988) 422; C.B . Lan , Latt;ce Higgs Workshop, ed . B . Berg et al ., World Scientific, Singapore,1988) 158; T. Neuhaus, in 1988 Syrnposiran ora Lattice Field Theory in. Fcr»ailah. USA and preprint BI-TP 88/29.