- Email: [email protected]
Universality in the mixed SU(2) lattice gauge theory: Nonperturbative approach to the ratio of Λ-parameters
Volume 126B, number 1,2
PHYSICS LETTERS
23 June 1983
UNIVERSALITY IN THE MIXED SU(2) LATTICE GAUGE THEORY: NONPERTURBATIVE APPROACH TO THE RATIO OF A - PARAMETERS
Yu.M. MAKEENKO, M.I. POLIKARPOV and A.V. ZHELONKIN Institute for Theoretical and Experimental Physics, 11 7259 Moscow, USSR Received 8 March 1983
The mixed SU(2) lattice gauge theory (LGT) is approximately represented as an effective SU(2) LGT with Wilson's action. This approach is applied to the nonperturbative calculation of the ratio of A-parameters in the mixed SU(2) LGT. It is shown that our formulas describe the Monte Carlo data fairly so that universality holds in the mixed SU(2) LGT.
Recent numerical Monte Carlo (MC) studies of lattice gauge theories (LGT) performed i n t h e original paper by Creutz [1] as well as in subsequent papers [2] have shown that confinement occurs in LGT's with non-abelian gauge groups [SU(2) and SU(3)] for any values of the coupling constant,/3. For/3 > /3c , 1 (the scaling region) the string tension, ¢r(/3), depends on/3 as prescribed by asymptotic freedom, so that in the scaling region the continuum limit sets in. Moreover Lang and Rebbi [3] showed by straightforward MC calculations that rotational symmetry does restore for/3 >/3e" To further test that for/3 >/3 c one probes the continuum limit, in refs. [ 4 - 7 ] MC calculations were performed for LGT's with variant actions which differ from the original Wilson one. In the region, where the continuum limit has set in, physical results should be universal, i.e. independent of the choice of lattice action. Universality has been shown for Manton's and Villain's actions. String tension calculated in ref. [4] using these actions depends on/3 as predicted by asymptotic freedom. Universality holds in the following sense. The ratio of A-parameters computed to the leading order of the weak coupling expansion [8,9] is in agreement with that obtained from the condition that MC calculations with Wilson's, Manton's and ,1 The region near ~3c is called the crossover region; ~3c ~ 2.2 for the SU(2) group. 82
Villain's actions yield the same values of string tension [4] or of deconfinement temperature [7]. Moreover corrections of order O(1//3) to the weak coupling results improve [ 10] the agreement with the MC data. However there is no such agreement for the mixed fundamental-adjoint LGT whose action [for the SU(N)] group reads * 2
S = ~ [(~3IN) Re tr Up + ([3A/N2)ltr U p [ 2 ] • p
(1)
The leading order of the weak coupling expansion for the A-ratio in the mixed SU(2) LGT yields [9,6] AmixedlAWilson 1s L t--L = exp[--~/r2/3A/(fl + 2/3A)] .
(2)
As was shown by Bhanot and Dashen [6], this formula does not describe the A-ratio evaluated from the MC calculations. At/3A = 1.21 the discrepancy is of a factor 4, when/3 is varied near the endpoint of the phasetransition line. Having assumed that corrections to eq. (2) are small for/3A > 0, Bhanot and Dashen concluded that universality is violated in the mixed SU(2) LGT. In fact many authors have shown corrections to eq. (2) to be quite large near the endpoint, so that one cannot use the weak coupling expansion to calculate the A-ratio, As was shown by Gavai et al. [7] the dominant next order correction to eq. (2) estimated by Sharatchandra and Weisz [ 10] reduces the discre• 2 Our normalization of 3's reproduces for N = 2 that of ref. [6]. 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
Volume 126B, number 1,2
PHYSICS LETTERS
23 June 1983
pancy by a factor 2. So one may hope that effects of higher orders in 1/3 will remove the remaining discrepancy. In the present paper we propose an approximate nonperturbative approach to calculating the ratio of A-parameters in the mixed LGT. The obtained formulas fairly describe the MC data of refs. [5-7] so that universality holds for the mixed LGT as well. We study the mixed SU(2) LGT by representing it as the Wilson SU(2) LGT with some effective coupling constant. Our approach extends that of refs. [11,12], employed for proving universality of SU(N) lattice gauge theories in the large-N limit, to the SU(2) group. In the large-N limit, universality immediately follows from the fact that any physical quantity, R, in the mixed LGT with the couplings fl,/3A can be expressed via that in the Wilson LGT by
To determine the dependence of~- on/3 and 3A, let us consider the simplest quantity, the plaquette average
n m~ed (/3, t~A) -- n WUso~(~),
PO,/3A) = ((~ tr Up)2) mixed - wZ(fl,flA)-
(3)
where the effective coupling fl is defmed by ~ =/3 + 2/3a co(~-) .
(4)
Here co(3) is the plaquette average in the Wilson LGT with the coupling 36o(3) = ( N - 1 tr Up) Wilson .
(5)
The formulas (3), (4) are proven in refs. [11,12] for large N. For finite N, those can be justified by factorization. In other words, if the factorization property approximately holds, the mixed LGT reduces to the Wilson LGT, guaranteeing universality of the continuum limit. For this reason the property (3) is called in ref. [11] universality of lattice gauge theories. For finite N the property (3) is not valid in a sense that lines of constant R in the 3,/3A plane (called in ref. [5] lines of "constant physics") may be different for various R, being coincided in the continuum limit only. However our experience of MC simulations in LGT's says that lines of "constant physics" approximately coincide for various R. For example, the loop average for rectangles of different size have cross-over at the same values of 3,/3AIn what follows we accept the hypothesis that the property (3) is valid in the SU(2) case with/3- being independent oUR (hypothesis of lattice universality). We shall show that.consequences of this hypothesis are confirmed by MC data.
W(/3, 3A) = (51 tr Up)Mixed . _
(6)
Then eq. (3) reads
W(3,/3A) = to(3),
(7)
where a~(3-) is defined by eq. (5). One can obtain W(3,/3A) in the whole/3, 3A plane by solving the equation [11]
W~A = 2WW¢ + p# ,
(8)
with the boundary condition at/3A = 0 W(/3, (3) = co(/3).
(9)
By def'mition, P(3,/3A) is the irreducible correlator of two plaquettes in the mixed LGT (10)
Eq. (8) possesses an exact solution of the form (7) with/3 defined by ~A ~=/3+2/316o(~)+ 1 ~ f dT?pff(~,r/). (ll) ~'(~) 0 This solution differs from that of ref. [I1] by a redef'lnition of the 3-parameter. It is essential for our purpose that W(/3,/3A) remains constant in the/3, 3A plane along the lines of constant fl defined by eq. (11) [see eq. (7)]. Due to our hypothesis of lattice universality, we can say that lines of constant ~ represent lines of "constant physics". Let us first consider the case of small/3 A when one can expand tile integral in eq. (11) in powers of/3A: P(3,/3A) = P(~) + O(3A).
(12)
Here p(3) -= 0(3, 0) is the irreducible correlator in the Wilson LGT. Substitution of (12) into eq. (11) yields the following equation for lines of "constant physics"
= 3 +/3a [2~(~-) + p'(~)/~o'(~)] .
(13)
Note that only quantities calculable in the Wilson LGT enter this equation. Using the weak coupling expansion ,3 ,3 We have obtained this expression for p in a standard way. Note that each term of the difference p(3-) = ~c~/4) (c~2/2)2 ((cos c~ = ((1/2) tr Up)Wilson) can be calculated to given order without corrections to the saddle-point value. 83
Volume 126B, number 1,2
PHYSICS LETTERS
expansion is applicable in this region. However in the region ~ ~ 2.2 that we are interested in, 6o(~) and p(]~-) are strongly dependent on ~ so that one cannot use the weak coupling expansion resulting in eq. (1 5). Analogously, the description of lines of "constant physics" based on the strong coupling expansion [ 1 5] cannot possess an accuracy better than that of describing co and p by the strong coupling expansion in this region. Let us now calculate the ratio A mixed L [A~Wilson within our approach. The following scale parameter
! 4.O
I
l l 3.O
%% 2.O
1.0
A
-I.0
°I
-2.0
A~ixed I
I
I
I
I
2
I
0.6g
I
5
× e x p [ - - ~ 7r2(/3 + 2/3A) ]
6o(~) = 1 - 3/4~- + O(1/~-2), p(~) = 3/8~ 2 + O(1/~-3) ,
(14)
(15)
This formula was first advocated by Grossman and Samuel [13] who showed that it describes the MC data for lines of "constant physics" [6] for /3A < 0. However our exact formula (13) is quite different from the corresponding formula of ref. [13]. The lines of "constant physics" given by eq. (13) are depicted in fig. I. We used the MC data of Berg and Stehr [2] for 6o03) and 0(/3). The derivative, p'03), is found by approximating the inverse function,/3(P), with a quartic polynomial and then by analytically differentiating. 6o'03) is taken from the MC data of ref. [14] for the specific heat C03) =/326o'03). One can see from fig. 1 that eq. (13) fairly describes the MC data of Bhanot and Dashen [6] until /3A is not loo large (positive or negative). For large /3A the MC data deviate from straight lines while eq. (13) always leads to the straight lines of "constant physics". We hope that the exact eq. (1 1) could describe the curving. The difference between eqs. (13) and (1 5) becomes insignificant for ~- >t 3. This is why the weak coupling
(16)
is used in refs. [ 5 - 7 ] for extracting physical numbers from the MC data in the mixed LGT. As shown in refs. [ 13,16], it is more convenient to use the improved scale parameter defined by the lhs of the equation a-l(~l/r2~)
one gets -~=/3+ 2/3A -- }/3A/~- •
(/3A) = a - 1[~¢r203 + 2/3A)] 51/121
I
Fig. 1. Lines of "constant physics" in the ~, ~3A plane. The dots are the MC data of ref. [6]. The solid lines are obtained from eq. (13).
84
23 June 1983
51/121 exp(--~ rrZff) = AWils°n "'L
(17)
Eq. (1 7) represents the fact that this improved scale parameter should be equal to the scale parameter in the Wilson LGT due to our hypothesis of lattice universality. Now the A-ratio can be evaluated to be mixed Wilson 5 1/ 1 2 1 ?/L /AL = [(/3 + 2/3A)/~-] X exp[-l~irr 2 ( ~ - / 3 - 2 / 3 A ) ] .
(18)
For not too large/3A, the substitution of eq. (13) into eq. (18) gives A~ixed/AWils°n = [1 + C(~)/3A [~] S 1/1 2 1 /* Lt
× exp [--/3A ~ 7r2C(~)] ,
(1 9)
where C(~) = 2 - 26o(~) - p'(-~)16o'(~).
(20)
When ~ is large, we reproduce eq. (2) by substituting the expansion (14). However eq. (2) is not applicable near the endpoint of the phase-transition line corresponding to ~ ~ 2.2. Therefore we calculated C(~) as described above using the MC data for co and p in the Wilson LGT. We compare in fig. 2 the A-ratio given by eq. (19) with the MC data of Bhanot and Creutz [5]. The small deviations from the straight lines are due to
V o l u m e 126B, n u m b e r 1,2
PHYSICS LETTERS
I
23 J u n e 1983
0.,~ 0.7, 0.6 0.5 0.4
I
tO,O
i
i
•
i
.
i
f5.4 = 1.21
y
•
•
" . . .
0.$ O,2 ,~.
1,0
x
1
x~ Nx\
Ao
O.tO
\\
Ao (OJ
0.06
N\ \~
0.1
0.06 \\\\\
0.0#
,,
0.0!
x(21
+ "X.C~)
"-.
Xf,*) o 'X.CS)
"
o
0.02 O.OOI -2
- -
I
-1
I
0
I
1
__
I
g
:A
ROt
• m i x e d , . Wilson Fig. 2. A L /A L versus ~A. The d o t s are t h e MC d a t a o f ref, [5]. The lines are o b t a i n e d f r o m eq. (19). The solid line as well as t h e f u r circles c o r r e s p o n d to ff ~ 2.4. The d a s h e d line a n d the o p e n circles c o r r e s p o n d to 3 ~ 2.8.
the pre-exponential factor in eq. (19). The solid line corresponds to 3- ~ 2.4 and well describes the MC data for the same value of ~- (the full circles). Analogously the dashed line as well as the open circles correspond to 3 ~ 2.8. As one can see from fig. 2, the slope of the solid line is slightly different from that of the dashed line, which qualitatively agrees with the MC data. Eq. (19) for the A-ratio is valid at small/3 A . It is interesting to consider the case/3A = 1.21 for which MC calculations are performed in refs. [6,7]. The expansion (12) in/3 A is no longer correct at/3 A = 1.21 so that, in describing the MC data, eq. (13) should be replaced with the exact eq. (11). However, since 0(/3,/3A) is unknown except for/3 A = 0, we straightforwardly used eq. (7) to Irmd 3(/3,/3 A = 1.21) substituting the MC data of Bhanot and Dashen [6] on the lhs and those of Berg and Stehr [2] on the rhs. Having obtained/3 versus/3 at/3A = 1.21 in this way, we can apply the improved scaling formula V'o = (0.011 + 0.002)- l a - I(T~I rr23)51/121 X e x p ( - ~/r2~-) .
(21)
1,36
I
I
I
I
t
I
1,4
L44
~48
1,52
1.56
~60
Fig. 3. x ( I ) versus 3 at 3A = 1.21. The MC d a t a as well as the d a s h e d lines, o b t a i n e d f r o m eq. ( 2 2 ) , are t a k e n f r o m ref. [ 6 ] . The solid curves are f r o m eq. (21).
to describe the MC data for X(I) [6] in the mixed SU(2) LGT at/3 A = 1.21. The value of the ratio, AWils°n/v'o = 0.011 -+ 0.002, is taken from the MC data [2] for the Wilson LGT. If we were not able to describe the MC data using eq. (11), this would mean violation of universality. In fig. 3 we plot the MC data of Bhanot and Dashen [6] for X(I) at/3 A = 1.21 as well as the curve given by eq. (21). As one can see from fig. 3, this curve is indeed an envelope of the MC data for X(1) so that universality holds. The discrepancy, seen in rel. [6], is associated with the fact that the weak coupling scaling formula [6] V-o-= [(3.0 + 0.3)X 10-3] -1 a - I [~x n2(/3+ 2/3A)] 51/121 × exp[-r]rr2(/3+ 2/3A)+ ~-rtZ/3A/(/3+ 2/3A) ] (22) was used instead of (21). As shown above, eq. (22) is not applicable near the endpoint and leads to a wrong functional behavior for/3 ~ 1.5, ~A = 1.21 [compare the slop of the solid curve of fig. 3 given by eq. (21) with that of the dashed line given by eq. (22)]. 85
Volume 126B, number 1,2
PHYSICS LETTERS
This is also the reason why a different behavior of the Wilson line, (ILI), as a function of T/A wils°n is obtained in ref. [7]. We have verified that if the MC data of Gavai et al. [7] for (ILl) in the mixed LGT are with A wilton given by eq. plotted versus T/Awils°n, L L (17), the functional behavior is the same, so that universality holds in the mixed LGT at f'mite temperature as well. Thus, in the present paper we have obtained the quantitative formulas relating the mixed SU(2) LGT to the Wilson SU(2) LGT with the effective coupling ~ . For large ~- our formulas reproduce the weak coupling results, but in the crossover region, where MC calculations are performed, the weak coupling formulas are not applicable. Nonetheless we show that our formulas fairly describe the MC data of refs. [ 5 - 7 ] so that universality holds in the mixed SU(2) LGT. We expect that universality will hold for the SU(N) gauge group at N~> 3 as well because the larger N the better is the accuracy of our approach.
[ 12]
One of the authors (Yu.M) thanks L. Jacobs for bringing ref. [13] to his attention.
[13]
References
[3] [4] [5] [6] [7] [8] [9]
[10] [11]
[14] [ 15]
[1] M. Creutz, Phys. Rev. D21 (1980) 2308; Phys. Rev. Lett. 45 (1980) 343. [ 2] D. Petcher and D.H. Weingarten, Phys. Rev. D22 (1980) 2465 ; G. Bhanot and C. Rebbi, Nucl. Phys. B180 [FS2] (1981) 469;
86
[16]
23 June 1983
B. Berg and J. Stehr, Z. Phys. C9 (1981) 333; E. Pietarinen, Nuel. Phys. B190 [FS3] (1981) 349; M. Creutz and K.J.M. Moriarty, Phys. Rev. D26 (1982) 2166; E.-M. Ilgenfritz and M. Mueller-Preussker,JINR preprint E2-82-473 (1982). C.B. Lang and C. Rebbi, Phys. Lett. 115B (1982) 137. C.B. Lang, C. Rebbi, P. Solomonson and R.-S. Skagerstam, Phys. Lett. 101B (1981) 173. G. Bhanot and M. Creutz, Phys. Rev. D24 (1981) 3212. G. Bhanot and R. Dashen, Phys. Lett. II3B (1982) 299. R. Gavai, F. Karsch and H. Satz, Bielefeld preprint BITP 82/26 (1982). C.B. Lang, C. Rebbi, P. Solomonson and B.-S. Skagerstam, Phys. Rev. D26 (1982) 2028. A. Gonzales-Arroyo and C.P. Korthals Altes, Nucl. Phys. B205 [FSS] (1982)46; A. di Giacomo and G. Paffuti, Nucl. Phys. B205 [FS5] (1982) 313. H.S. Sharatchandra and P.H. Weisz, preprint DESY 81083 (1981). Yu.M. Makeenko and M.I. Polikarpov, Nucl. Phys. B205 [FS5] (1982) 386. T.-L. Chen, C.-I. Tan and Z.-T. Zheng, Phys. Lett. 109B (1982) 383; S. Samuel, Phys. Lett. 112B (1982) 237. B. Grossman and S. Samuel, Phys. Lett. 120B (1983) 387. B. Lautrup and M. Nauenberg, Phys. Rev. Lett. 45 (1980) 755. A. Gonzales-Arroyo, C.P. Korthals Altes, J. Peiro and M. Perrottet, Phys. Lett. l16B (1982) 414. Yu.M. Makeenko, Lectures Arctic School of Physics (1982).
PHYSICS LETTERS
23 June 1983
UNIVERSALITY IN THE MIXED SU(2) LATTICE GAUGE THEORY: NONPERTURBATIVE APPROACH TO THE RATIO OF A - PARAMETERS
Yu.M. MAKEENKO, M.I. POLIKARPOV and A.V. ZHELONKIN Institute for Theoretical and Experimental Physics, 11 7259 Moscow, USSR Received 8 March 1983
The mixed SU(2) lattice gauge theory (LGT) is approximately represented as an effective SU(2) LGT with Wilson's action. This approach is applied to the nonperturbative calculation of the ratio of A-parameters in the mixed SU(2) LGT. It is shown that our formulas describe the Monte Carlo data fairly so that universality holds in the mixed SU(2) LGT.
Recent numerical Monte Carlo (MC) studies of lattice gauge theories (LGT) performed i n t h e original paper by Creutz [1] as well as in subsequent papers [2] have shown that confinement occurs in LGT's with non-abelian gauge groups [SU(2) and SU(3)] for any values of the coupling constant,/3. For/3 > /3c , 1 (the scaling region) the string tension, ¢r(/3), depends on/3 as prescribed by asymptotic freedom, so that in the scaling region the continuum limit sets in. Moreover Lang and Rebbi [3] showed by straightforward MC calculations that rotational symmetry does restore for/3 >/3e" To further test that for/3 >/3 c one probes the continuum limit, in refs. [ 4 - 7 ] MC calculations were performed for LGT's with variant actions which differ from the original Wilson one. In the region, where the continuum limit has set in, physical results should be universal, i.e. independent of the choice of lattice action. Universality has been shown for Manton's and Villain's actions. String tension calculated in ref. [4] using these actions depends on/3 as predicted by asymptotic freedom. Universality holds in the following sense. The ratio of A-parameters computed to the leading order of the weak coupling expansion [8,9] is in agreement with that obtained from the condition that MC calculations with Wilson's, Manton's and ,1 The region near ~3c is called the crossover region; ~3c ~ 2.2 for the SU(2) group. 82
Villain's actions yield the same values of string tension [4] or of deconfinement temperature [7]. Moreover corrections of order O(1//3) to the weak coupling results improve [ 10] the agreement with the MC data. However there is no such agreement for the mixed fundamental-adjoint LGT whose action [for the SU(N)] group reads * 2
S = ~ [(~3IN) Re tr Up + ([3A/N2)ltr U p [ 2 ] • p
(1)
The leading order of the weak coupling expansion for the A-ratio in the mixed SU(2) LGT yields [9,6] AmixedlAWilson 1s L t--L = exp[--~/r2/3A/(fl + 2/3A)] .
(2)
As was shown by Bhanot and Dashen [6], this formula does not describe the A-ratio evaluated from the MC calculations. At/3A = 1.21 the discrepancy is of a factor 4, when/3 is varied near the endpoint of the phasetransition line. Having assumed that corrections to eq. (2) are small for/3A > 0, Bhanot and Dashen concluded that universality is violated in the mixed SU(2) LGT. In fact many authors have shown corrections to eq. (2) to be quite large near the endpoint, so that one cannot use the weak coupling expansion to calculate the A-ratio, As was shown by Gavai et al. [7] the dominant next order correction to eq. (2) estimated by Sharatchandra and Weisz [ 10] reduces the discre• 2 Our normalization of 3's reproduces for N = 2 that of ref. [6]. 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
Volume 126B, number 1,2
PHYSICS LETTERS
23 June 1983
pancy by a factor 2. So one may hope that effects of higher orders in 1/3 will remove the remaining discrepancy. In the present paper we propose an approximate nonperturbative approach to calculating the ratio of A-parameters in the mixed LGT. The obtained formulas fairly describe the MC data of refs. [5-7] so that universality holds for the mixed LGT as well. We study the mixed SU(2) LGT by representing it as the Wilson SU(2) LGT with some effective coupling constant. Our approach extends that of refs. [11,12], employed for proving universality of SU(N) lattice gauge theories in the large-N limit, to the SU(2) group. In the large-N limit, universality immediately follows from the fact that any physical quantity, R, in the mixed LGT with the couplings fl,/3A can be expressed via that in the Wilson LGT by
To determine the dependence of~- on/3 and 3A, let us consider the simplest quantity, the plaquette average
n m~ed (/3, t~A) -- n WUso~(~),
PO,/3A) = ((~ tr Up)2) mixed - wZ(fl,flA)-
(3)
where the effective coupling fl is defmed by ~ =/3 + 2/3a co(~-) .
(4)
Here co(3) is the plaquette average in the Wilson LGT with the coupling 36o(3) = ( N - 1 tr Up) Wilson .
(5)
The formulas (3), (4) are proven in refs. [11,12] for large N. For finite N, those can be justified by factorization. In other words, if the factorization property approximately holds, the mixed LGT reduces to the Wilson LGT, guaranteeing universality of the continuum limit. For this reason the property (3) is called in ref. [11] universality of lattice gauge theories. For finite N the property (3) is not valid in a sense that lines of constant R in the 3,/3A plane (called in ref. [5] lines of "constant physics") may be different for various R, being coincided in the continuum limit only. However our experience of MC simulations in LGT's says that lines of "constant physics" approximately coincide for various R. For example, the loop average for rectangles of different size have cross-over at the same values of 3,/3AIn what follows we accept the hypothesis that the property (3) is valid in the SU(2) case with/3- being independent oUR (hypothesis of lattice universality). We shall show that.consequences of this hypothesis are confirmed by MC data.
W(/3, 3A) = (51 tr Up)Mixed . _
(6)
Then eq. (3) reads
W(3,/3A) = to(3),
(7)
where a~(3-) is defined by eq. (5). One can obtain W(3,/3A) in the whole/3, 3A plane by solving the equation [11]
W~A = 2WW¢ + p# ,
(8)
with the boundary condition at/3A = 0 W(/3, (3) = co(/3).
(9)
By def'mition, P(3,/3A) is the irreducible correlator of two plaquettes in the mixed LGT (10)
Eq. (8) possesses an exact solution of the form (7) with/3 defined by ~A ~=/3+2/316o(~)+ 1 ~ f dT?pff(~,r/). (ll) ~'(~) 0 This solution differs from that of ref. [I1] by a redef'lnition of the 3-parameter. It is essential for our purpose that W(/3,/3A) remains constant in the/3, 3A plane along the lines of constant fl defined by eq. (11) [see eq. (7)]. Due to our hypothesis of lattice universality, we can say that lines of constant ~ represent lines of "constant physics". Let us first consider the case of small/3 A when one can expand tile integral in eq. (11) in powers of/3A: P(3,/3A) = P(~) + O(3A).
(12)
Here p(3) -= 0(3, 0) is the irreducible correlator in the Wilson LGT. Substitution of (12) into eq. (11) yields the following equation for lines of "constant physics"
= 3 +/3a [2~(~-) + p'(~)/~o'(~)] .
(13)
Note that only quantities calculable in the Wilson LGT enter this equation. Using the weak coupling expansion ,3 ,3 We have obtained this expression for p in a standard way. Note that each term of the difference p(3-) = ~c~/4) (c~2/2)2 ((cos c~ = ((1/2) tr Up)Wilson) can be calculated to given order without corrections to the saddle-point value. 83
Volume 126B, number 1,2
PHYSICS LETTERS
expansion is applicable in this region. However in the region ~ ~ 2.2 that we are interested in, 6o(~) and p(]~-) are strongly dependent on ~ so that one cannot use the weak coupling expansion resulting in eq. (1 5). Analogously, the description of lines of "constant physics" based on the strong coupling expansion [ 1 5] cannot possess an accuracy better than that of describing co and p by the strong coupling expansion in this region. Let us now calculate the ratio A mixed L [A~Wilson within our approach. The following scale parameter
! 4.O
I
l l 3.O
%% 2.O
1.0
A
-I.0
°I
-2.0
A~ixed I
I
I
I
I
2
I
0.6g
I
5
× e x p [ - - ~ 7r2(/3 + 2/3A) ]
6o(~) = 1 - 3/4~- + O(1/~-2), p(~) = 3/8~ 2 + O(1/~-3) ,
(14)
(15)
This formula was first advocated by Grossman and Samuel [13] who showed that it describes the MC data for lines of "constant physics" [6] for /3A < 0. However our exact formula (13) is quite different from the corresponding formula of ref. [13]. The lines of "constant physics" given by eq. (13) are depicted in fig. I. We used the MC data of Berg and Stehr [2] for 6o03) and 0(/3). The derivative, p'03), is found by approximating the inverse function,/3(P), with a quartic polynomial and then by analytically differentiating. 6o'03) is taken from the MC data of ref. [14] for the specific heat C03) =/326o'03). One can see from fig. 1 that eq. (13) fairly describes the MC data of Bhanot and Dashen [6] until /3A is not loo large (positive or negative). For large /3A the MC data deviate from straight lines while eq. (13) always leads to the straight lines of "constant physics". We hope that the exact eq. (1 1) could describe the curving. The difference between eqs. (13) and (1 5) becomes insignificant for ~- >t 3. This is why the weak coupling
(16)
is used in refs. [ 5 - 7 ] for extracting physical numbers from the MC data in the mixed LGT. As shown in refs. [ 13,16], it is more convenient to use the improved scale parameter defined by the lhs of the equation a-l(~l/r2~)
one gets -~=/3+ 2/3A -- }/3A/~- •
(/3A) = a - 1[~¢r203 + 2/3A)] 51/121
I
Fig. 1. Lines of "constant physics" in the ~, ~3A plane. The dots are the MC data of ref. [6]. The solid lines are obtained from eq. (13).
84
23 June 1983
51/121 exp(--~ rrZff) = AWils°n "'L
(17)
Eq. (1 7) represents the fact that this improved scale parameter should be equal to the scale parameter in the Wilson LGT due to our hypothesis of lattice universality. Now the A-ratio can be evaluated to be mixed Wilson 5 1/ 1 2 1 ?/L /AL = [(/3 + 2/3A)/~-] X exp[-l~irr 2 ( ~ - / 3 - 2 / 3 A ) ] .
(18)
For not too large/3A, the substitution of eq. (13) into eq. (18) gives A~ixed/AWils°n = [1 + C(~)/3A [~] S 1/1 2 1 /* Lt
× exp [--/3A ~ 7r2C(~)] ,
(1 9)
where C(~) = 2 - 26o(~) - p'(-~)16o'(~).
(20)
When ~ is large, we reproduce eq. (2) by substituting the expansion (14). However eq. (2) is not applicable near the endpoint of the phase-transition line corresponding to ~ ~ 2.2. Therefore we calculated C(~) as described above using the MC data for co and p in the Wilson LGT. We compare in fig. 2 the A-ratio given by eq. (19) with the MC data of Bhanot and Creutz [5]. The small deviations from the straight lines are due to
V o l u m e 126B, n u m b e r 1,2
PHYSICS LETTERS
I
23 J u n e 1983
0.,~ 0.7, 0.6 0.5 0.4
I
tO,O
i
i
•
i
.
i
f5.4 = 1.21
y
•
•
" . . .
0.$ O,2 ,~.
1,0
x
1
x~ Nx\
Ao
O.tO
\\
Ao (OJ
0.06
N\ \~
0.1
0.06 \\\\\
0.0#
,,
0.0!
x(21
+ "X.C~)
"-.
Xf,*) o 'X.CS)
"
o
0.02 O.OOI -2
- -
I
-1
I
0
I
1
__
I
g
:A
ROt
• m i x e d , . Wilson Fig. 2. A L /A L versus ~A. The d o t s are t h e MC d a t a o f ref, [5]. The lines are o b t a i n e d f r o m eq. (19). The solid line as well as t h e f u r circles c o r r e s p o n d to ff ~ 2.4. The d a s h e d line a n d the o p e n circles c o r r e s p o n d to 3 ~ 2.8.
the pre-exponential factor in eq. (19). The solid line corresponds to 3- ~ 2.4 and well describes the MC data for the same value of ~- (the full circles). Analogously the dashed line as well as the open circles correspond to 3 ~ 2.8. As one can see from fig. 2, the slope of the solid line is slightly different from that of the dashed line, which qualitatively agrees with the MC data. Eq. (19) for the A-ratio is valid at small/3 A . It is interesting to consider the case/3A = 1.21 for which MC calculations are performed in refs. [6,7]. The expansion (12) in/3 A is no longer correct at/3 A = 1.21 so that, in describing the MC data, eq. (13) should be replaced with the exact eq. (11). However, since 0(/3,/3A) is unknown except for/3 A = 0, we straightforwardly used eq. (7) to Irmd 3(/3,/3 A = 1.21) substituting the MC data of Bhanot and Dashen [6] on the lhs and those of Berg and Stehr [2] on the rhs. Having obtained/3 versus/3 at/3A = 1.21 in this way, we can apply the improved scaling formula V'o = (0.011 + 0.002)- l a - I(T~I rr23)51/121 X e x p ( - ~/r2~-) .
(21)
1,36
I
I
I
I
t
I
1,4
L44
~48
1,52
1.56
~60
Fig. 3. x ( I ) versus 3 at 3A = 1.21. The MC d a t a as well as the d a s h e d lines, o b t a i n e d f r o m eq. ( 2 2 ) , are t a k e n f r o m ref. [ 6 ] . The solid curves are f r o m eq. (21).
to describe the MC data for X(I) [6] in the mixed SU(2) LGT at/3 A = 1.21. The value of the ratio, AWils°n/v'o = 0.011 -+ 0.002, is taken from the MC data [2] for the Wilson LGT. If we were not able to describe the MC data using eq. (11), this would mean violation of universality. In fig. 3 we plot the MC data of Bhanot and Dashen [6] for X(I) at/3 A = 1.21 as well as the curve given by eq. (21). As one can see from fig. 3, this curve is indeed an envelope of the MC data for X(1) so that universality holds. The discrepancy, seen in rel. [6], is associated with the fact that the weak coupling scaling formula [6] V-o-= [(3.0 + 0.3)X 10-3] -1 a - I [~x n2(/3+ 2/3A)] 51/121 × exp[-r]rr2(/3+ 2/3A)+ ~-rtZ/3A/(/3+ 2/3A) ] (22) was used instead of (21). As shown above, eq. (22) is not applicable near the endpoint and leads to a wrong functional behavior for/3 ~ 1.5, ~A = 1.21 [compare the slop of the solid curve of fig. 3 given by eq. (21) with that of the dashed line given by eq. (22)]. 85
Volume 126B, number 1,2
PHYSICS LETTERS
This is also the reason why a different behavior of the Wilson line, (ILI), as a function of T/A wils°n is obtained in ref. [7]. We have verified that if the MC data of Gavai et al. [7] for (ILl) in the mixed LGT are with A wilton given by eq. plotted versus T/Awils°n, L L (17), the functional behavior is the same, so that universality holds in the mixed LGT at f'mite temperature as well. Thus, in the present paper we have obtained the quantitative formulas relating the mixed SU(2) LGT to the Wilson SU(2) LGT with the effective coupling ~ . For large ~- our formulas reproduce the weak coupling results, but in the crossover region, where MC calculations are performed, the weak coupling formulas are not applicable. Nonetheless we show that our formulas fairly describe the MC data of refs. [ 5 - 7 ] so that universality holds in the mixed SU(2) LGT. We expect that universality will hold for the SU(N) gauge group at N~> 3 as well because the larger N the better is the accuracy of our approach.
[ 12]
One of the authors (Yu.M) thanks L. Jacobs for bringing ref. [13] to his attention.
[13]
References
[3] [4] [5] [6] [7] [8] [9]
[10] [11]
[14] [ 15]
[1] M. Creutz, Phys. Rev. D21 (1980) 2308; Phys. Rev. Lett. 45 (1980) 343. [ 2] D. Petcher and D.H. Weingarten, Phys. Rev. D22 (1980) 2465 ; G. Bhanot and C. Rebbi, Nucl. Phys. B180 [FS2] (1981) 469;
86
[16]
23 June 1983
B. Berg and J. Stehr, Z. Phys. C9 (1981) 333; E. Pietarinen, Nuel. Phys. B190 [FS3] (1981) 349; M. Creutz and K.J.M. Moriarty, Phys. Rev. D26 (1982) 2166; E.-M. Ilgenfritz and M. Mueller-Preussker,JINR preprint E2-82-473 (1982). C.B. Lang and C. Rebbi, Phys. Lett. 115B (1982) 137. C.B. Lang, C. Rebbi, P. Solomonson and R.-S. Skagerstam, Phys. Lett. 101B (1981) 173. G. Bhanot and M. Creutz, Phys. Rev. D24 (1981) 3212. G. Bhanot and R. Dashen, Phys. Lett. II3B (1982) 299. R. Gavai, F. Karsch and H. Satz, Bielefeld preprint BITP 82/26 (1982). C.B. Lang, C. Rebbi, P. Solomonson and B.-S. Skagerstam, Phys. Rev. D26 (1982) 2028. A. Gonzales-Arroyo and C.P. Korthals Altes, Nucl. Phys. B205 [FSS] (1982)46; A. di Giacomo and G. Paffuti, Nucl. Phys. B205 [FS5] (1982) 313. H.S. Sharatchandra and P.H. Weisz, preprint DESY 81083 (1981). Yu.M. Makeenko and M.I. Polikarpov, Nucl. Phys. B205 [FS5] (1982) 386. T.-L. Chen, C.-I. Tan and Z.-T. Zheng, Phys. Lett. 109B (1982) 383; S. Samuel, Phys. Lett. 112B (1982) 237. B. Grossman and S. Samuel, Phys. Lett. 120B (1983) 387. B. Lautrup and M. Nauenberg, Phys. Rev. Lett. 45 (1980) 755. A. Gonzales-Arroyo, C.P. Korthals Altes, J. Peiro and M. Perrottet, Phys. Lett. l16B (1982) 414. Yu.M. Makeenko, Lectures Arctic School of Physics (1982).