depend on the renormalization sheme) only near the ritial point at zero oupling while at intermediate and strong ouplings this statement is still ques

Transcription

1 ASYMMETRY PARAMETER ROLE IN DESCRIPTION OF PHASE STRUCTURE OF LATTICE GLUODYNAMICS AT FINITE TEMPERATURE. L. A. Averhenkova, K. V. Petrov, V. K. Petrov, G. M. Zinovjev Bogolyubov Institute for Theoretial Physis National Aademy of Sienes of Ukraine Kiev 43, UKRAINE June 0, 998 Abstrat The role of lattie asymmetry parameter in desription of the SU() gluodynamis phase struture at nite temperature is studied analytially. The fat that renormalization group relations whih permit to remove the lattie asymmetry parameter from the thermodynamial quantities in the "naive" limit don't do the same in the approximation SU(N) ' Z(N) keeping the "naive" limit results at the same time was point out. An additional ondition whih would x is needed for this dependene removal. INTRODUCTION. Following the heart point of the renormalization group theory after any regularization removal (in partiular, lattie one) observed quantities should go to their physial values and should not depend on the renormalization sheme. So, although going to the ontinuum from a symmetrial lattie and from asymmetrial one are two dierent renormalization proedures in the gauge elds theory it is natural to expet that in the ontinuum limit no lattie asymmetry eets will be revealed. At the same time, stritly speaking, the renormalization group trajetories are universal (i.e., they don't hedpitp@gluk.ap.org, gezin@hrz.uni-bielefeld.de, lily@knb.kiev.ua

2 depend on the renormalization sheme) only near the ritial point at zero oupling while at intermediate and strong ouplings this statement is still questionable. It is quite probable the trajetories will depend on the degree of lattie asymmetry []. Moreover, today's level of this problem understanding does not permit to exlude other possible revealings of the lattie asymmetry. In this paper we onsider the nite temperature SU(N)-gluodynamis on an asymmetrial lattie a 6= a ; = 0; ; ; 3; a 0 N 0 = ; a k N k = L k ; k = ; ; 3: a { the lattie spaing along the diretion. The most wide-spread example of suh asymmetry introduing is the Hamiltonian limit [, ] (where the lattie distane along the spae diretions is xed and the spaing along the time diretion is going to zero). The renormalization proedure independene atually means that renormalized ouplings of both theories (for example, Hamiltonian and Eulidean SU(N) lattie gauge theories) should be get equal. This naturally imposes the relation on the sales E H. Following hosen approah [, ] this is enough for two renormalized theories to give the "same physis". Indeed, if in the standard Wilson's ation S = x;< Re Tr( N U (x)u (x + )U + (x + )U + (x)) () we restrit ourselves to only "naive" limit (whih assumes the expansion of SU(N) matries around I-matrix, lose to I-matrix or any xed matrix) and we take U (x) =e ia g A (x) ' +ia g A (x) () = N g g a 0 a a a 3 a a then at least in the Hamiltonian (H) and the Eulidean (E) [a = a; g! g E ] limits the spetra of hamiltonians oinide. Moreover, not only in the lassial ase (g E = g H), but in the quantum ase (g E ' g H + O(g 4 H)) too. However, investigation of the nite temperature SU(N) lattie gauge theory in "naive" limit is arried out using SU(N) matries expansion around xed matrix thereby leaving the enter Z(N) out of the game. But the role of the gauge group enter in the nite temperature QCD phase struture is wellknown. Therefore, we will study the lattie asymmetry eets ontributed by the enter Z(N). To do this we will use the approah SU(N) ' Z(N), i.e., we will onsider the nite temperature lattie gauge theory in whih the (3)

3 link variables are the enter elements of the SU(N) gauge group (x) =e iq N ; q = 0; ; :::N (4) On the other hand, for theories with gauge groups whih have a trivial enter (for example, SU(N)=Z(N)) it may be quite reasonable to restrit ourselves to "naive" limit, whereas for the SU(N) gauge theories suh a restrition requires speial substantiation beause there is no naive limit for the enter Z(N) in ommon sense. Building the ontinuum quantum eld theory beomes possible only if Z(N) lattie gauge system undergoes the seond order phase transition at zero of the renormalization group funtion [3, 4, 5]. To ensure agreement with the "naive" limit results we hose (3) to guarantee the oinidene of naive limits for the models with SU(N) gauge group on the symmetrial lattie and asymmetrial one. In the future we will alulate the orretions to our approximation. As a justiation of our approah we should note that eet of the quantum utuations, whih are taken into aount besides the Z(N) topologial exitations, amounts to simple hange in the oupling onstant [, 6, 7] new = old (N )=4 (5) Additional term (5) depends neither on a nor on. Therefore, it does not depend on the lattie asymmetry. So, there is a reason to hope that inluding the quantum utuations besides of Z(N) solutions in our theory on asymmetrial lattie will not hange (5), i.e. will just result in the nite renormalization of ouplings. It will be enough for our purposes to onsider the simplest ase of lattie asymmetry, when 0n and nm, i.e., we allow the ouplings to be dierent for plaquettes lying in the temporal and in the spatial planes. Our results may be easily generalized to the ase (3). We will also limit ourselves to the analyses of four limiting ases: )! 0; )! 0; 3)! ; 4)! ; (6) (the estimations of phase struture in the whole plane we are planning to present in next paper). As known our Wilson's ation is splitting into two parts: S() = P N Re Tr n(~x) m (~x + n) + n (~x + m) + m(~x)! + 3

4 P! N Re Tr 0(~x;) n (~x; +0) 0(~x+n; + ) + n (~x;) (7) where = Na =g a and = Na =g a ; a (a ) is the spatial (temporal) spaing and the sum P (P ) runs over all purely spae-like (time-like) plaquettes. Let's rst onsider th and th independent temporary ignoring the dependene of g (g ) on a (a ) through the renormalization group relations, i.e., supposing that by varying a and a we an get to any point of the square (0 ; 0 ). We will investigate eah of the limiting ases separately, joining the results for the ouplings ritial values through the same parameter. THE AREA OF SMALL. In the ase ' 0 we may throw o the magneti part of ation thereby leaving only the spin ongurations with time-like plaquettes. In the stati gauge we an sum over the spae spin ongurations f n g, nally gaining the Ising model whih is muh known of. Now let us dwell upon that. Fixing the stati gauge 0 (~x;) =! ~x we onsider the following gauge transitions n (~x;)!! ~x n(~x;)! ~x+n + n(~x;)!! ~x+n + n(~x;)! ~x (8) Imposing the periodial boundary onditions in temporal diretion upon -matries: n (~x; = ) n (~x; = N ), all matries! ~x are grouped on the last links resulting in the Polyakov's loops We ome to the following ation = 4 N S P ~x;n = ~x = N Y =! ~x =! N ~x (9) N Re Tr n(~x;) + n(~x; +) + N Re Tr ~x n (~x;n ) + ~x+n + n (~x; ) i (0) 4

5 Summing over all ongurations f n g in Z() gauge system we get = ~ ~x? ~x+n () S P ~x;n th ~ = (th ) N = ~ As known, for the Ising model there exists the ritial value of the oupling ~ = separating two phases. Consequently, the partition funtion (0) will have a ritial point at = =N. THE AREA OF SMALL. In the other limiting ase ' 0 the funtional integral is highly peaked about ongurations with spae-like plaquettes and we may throw o the eletri part of ation. In this ase the partition funtion turns out to fall to N equivalent disonneted ontributions eah of whih is from a separate 3-dimensional layer. We get the set of standard 3-dimensional Wegner models S P ' N = x;n;m n m? n? m () There is no interation between the layers, so summing over n (x; ) may be done independently for every = onst layer. The lattie dual to original hyperubial lattie an be onstruted by shifting the lattie by half a lattie spaing in eah diretion (see, for example, [8]). Geometrial duality transforms q-dimensional manifolds into (d q)- dimensional ones. "Dual oupling onstant" ~ for the Z() gauge theory ~ = ln th is a monotonially dereasing funtion of the "original" oupling. The set of 3-dimensional Wegner models transforms into the set of 3-dimensional Ising models with spins in sites under duality transformations. These Ising models exhibit the transition at simultaneously. Therefore, for the partition funtion () there is a ritial point at = +. TWO ANOTHER LIMITING CASES:! AND!. 5

6 When onsidering the duality transformation in 4-dimensional spae-time it should be pointed out that just spae-like plaquettes transform into timelike ones. In the other words, 0 = ln th or 0 = + (3) and vie versa, 0 = ln th. This statement beomes lear from the following. Let us rewrite the partition funtion of Z() system in the form Z = P e x; x; (4) fg = fg Y x; = e Nf fg = e Nf fqg = e Nf fqg h ( + x; th ) Y x; q= Y x; Y x; qx; + ( x; th ) (e ln th qx; + ) (e ln th qx; + ) Y links = Y links ( () P 3 qx; + = 3;6= 3 = 3;6= q x; + ) where x; = (x) (x + )?(x + )? (x) and P f = N N 3 ln h. We introdued a new set of variables fqg - one for eah plaquette. The partition funtion is not equal to zero only if q x; satises the following ondition on the sum over six q x; (assoiated with six plaquettes whih adjoin the link x;, see g.): 3 = 3;6= (q x; +)=0 mod or 3 = 3;6= q x; = mod 4 (5) The solution of last equation an be found if we assoiate every q x; one of the ube plane and with q x; = s (x)s (x + )s? (x + )s? (x) 6= 6= 6= (6) where the dual link variable s (x) is the elementofthez() group. It beomes intuitively evident, if we onsider the starting ase when all s are equal. It ditates for P 6= q x; to be equal 6 mod 4 = mod 4. Every link enters the 6

7 solution twie (beause the plaquettes form a ube) and hanging the sign of a link to opposite results in hanging P 3 = 3;6= q x; only by 4. Consequently, in the plane there is the self-duality line (see g.). R. Balian, J.M. Droue, C. Itzykson [9] pointed out the possibility of the ritial behaviour at = 0:44 for the 4-dimensional Z() pure gauge theory on symmetrial lattie supposing this ritial point is single. So, now we realize that under duality transformation our original 4- dimensional theory (7) transforms into the same one but with new oupling onstants. S 0 = (7) For this dual representation we an onsider the two limiting ases in preisely the same way we have done for the original theory. The ase 0 = th 0 ' 0 for the original theory means ' in aording to (3). Throwing o time-like part of the ation we an go again from the set of 3-dimensional Wegner models to the set of standard 3-dimensional Ising models via the duality transformation. And after that we may ome bak to variables of the original theory. 0! ~ 0 s ~x s? ~x+n P ~xn th ~ 0 = ~ 0 = = ; + = = (8) Here tilde ~ (prime 0 ) means duality transformations in three (four) dimensions, respetively. The original theory undergoes the phase transition at ' and =. Similarly, let's onsider the limit 0 ' 0 (it orresponds ' for the original theory) as we have done already for the limit ' 0. We an nd the phase transition at ' ; = =N + =N. We may depit all these ritial points in the plane (g.3). To avoid arising innite onstant in the partition funtion at = and at =, we took and lose to but not equal exatly. 7

8 ANOTHER APPROACH TO THE CONSIDERATION OF LIMITS:! AND!. There is also another method to nd ritial points 3 and 4 (g.3), dierent from the above duality transformation approah. Let us build up the eetive ation for the ase!. It is obvious that spae-like plaquette variables in this ase nm =. The gauge is ompletely xed if we have a maximal tree, i.e. tree whih will have a losed loop after adding one more link to it. We may hoose the maximal tree in two dimensions like a snail as shown at the g.4. By analogy we may obtain suh a maximal tree for 3 dimensions. That hoie of the maximal tree provides that the equation = will have only single solution: n x =. Really, if we onsider the rst plaquette, whih have three link variables equal due to the gauge ondition and provided with =, the fourth link is to be equal also. Now we an apply these onsiderations to the seond plaquette in the snail and so on, getting nally x n =. As a result, only temporal links will survive in the ation. S = 0 (~x;) 0 (~x+n; ) (9) ~xn We've got the set of 3-dimensional independent Isings with ritial point at ' ; = for our original system. In the limit ' the plaquette variables 0n =. In the Hamiltonian gauge ( 0 = ) n (t) = n (t +)= n are time independent. We ome to the stati 3-dimensional Wegner system S = N n (~x) m (~x + n) n (~x + n + m) m (~x + m) (0) ~xn whih transforms into the Ising model under duality transformations. N = = = =N + =N ln 0 ln 0=N () So, this alternative method onrms the results obtained previously. 8

9 THE POTENTIAL BETWEEN PROBE SOURCES. We would like to estimate the onneted orrelation funtion h x x+r i for the Ising model with dierent ouplings in eah diretion within the spherial model. The ruial point is the following ondition: Then Z = fg Z + i N 3 d = () x x i en 3 P P x x + x;n 0 n x x+n (3) where 0 n = mk the onstant is hosen to ensure the legitimay of interhanging the integration and summation order. It means that is a line to the right of all -singularities. We an rewrite the partition funtion as: Z d Z = i en 3 x e xa x x 0 x 0 (4) where A x x 0 = x0 x 3 n= Z 0 n x+n x = ( 3 n= 0 n os n)e i(x x0) d 3 (5) The orrelation funtion h x x+r i an be alulated as the derivative of generation funtion over soures: h x x+r i = h 0 R Z d e N xa x x 0 x 0+ x x (6) R fg and after shifting integration's variables we have h x x+r R = A R = Z de 4 xa x x 0 x 0 0 e ir P 3n= 0 n os n d 3 (7) 0 is the sadle point whih is determined by the ondition Z d 3 0 P 3n= 0 n os n = h 0i = (8) 9

10 At R n! n! 0 and the pole will be determined by 3 n= 0 n n = Introduing "symmetrial" variables n= 0 n (9) n = p (30) 0 n vu u 3 = it( 0 0 ) n n= we obtain P h 0 R i = e 3 p rn n= 0 n an (3) where r = R n a n So, the potential between two probe soures depends on the hoie of n. THE PHASE STRUCTURE AND LIMIT a ;! 0. We will make suggestions about the phase struture in the whole area of oupling onstants and larify the nature of the phases previously obtained. In the ase ' 0 the 4-dimensional system transforms into the set of independent 3-dimensional subsystems with t = t j as mentioned already. The probe soures (the potential between them was alulated on dual lattie) orrespond to the magneti harges plaed inside ubes of the original lattie. The prodution over spae-like ube's plaquettes an be assoiated with magneti eld ux through the ube's surfae Y = expfonst ~B ~ng ube ube B k = kmnf mn (3) and is not equal to zero when the probe soure is plaed in orresponding dual site. In the other words, the probe soure of "eletri" harge in the site of the dual lattie orresponds to the monopol ("magneti" harge) of the original one. 0

11 As known, if for Wilson's loop C? in the plane [tx] of 4-dimensional dual lattie we'd x the time t = t 0 then the loop will piere the plane [zy] in two points (monopol - antimonopol). Dirak's string whih ties them together lies in the slie t = t 0 [0]. If the potential between probe soures in eah slie will inrease linearly with R (in the region of oupling < N ) then the average value of Wilson's loop hw i = Q T t=0 hs x0 s xr i (when the slies are independent) will derease exponentially aording to area law. Average value of the orresponding t'hooft's loop ht 0 Hi must behave in the same way in the region > +. ht 0 Y Hi original = T Y hm x0 ;m xr i=hwi dual = T hs x0 s xr i e T R (33) t= It is obvious that parameters area ( and ) falls to four setors depending on the behaviour of the average values of Wilson's and t'hooft's loops. t= I > < hw i e L C ht 0 Hi e 0 L C 0 II > > hwi e L C ht 0 Hi e 0 C 0 III < < hwi e C ht 0 Hi e 0 L C 0 IV < > hw i e C ht 0 Hi e 0 C 0 This piture overs all four types of possible behaviour of the averages under onsideration whih were found by G.t'Hooft [] from the ommutation relations analysis. It seems impossible to "see" all four phases on a lattie with xed asymmetry ( = onst ) inluding the symmetrial one. Our results are in good agreement with [0] in the areas they studied on the symmetrial lattie (line = ) Up to now we treated the ouplings ( ) freely enough onsidering them independent. However, it is obvious that underlying onstants g (g ) should depend on the lattie spaings through the renormalization group relations. This onnetion may make some areas of the square (0 ; 0 ) inaessible. To nd exat borders of the aessible area we should build the renormalization group relations on an asymmetrial lattie and put g(g ) into ( ). In "naive" limit g ' g. Introduing = a a we get =.

12 Investigating the phase struture of our theory we were dealing with effetive ouplings ;. Now we are interested in larifying whether ritial values ; are within aessible area of temperatures = (a N ) (i.e., 6= 0 and 6= ) or not. We should note that in the limit (a ;! 0) g ' g ' g << []. So, = 4N >> (shaded region at g.6). Say, at! 0;! as g 4 parameters area. Moreover, at N g 4 and = a a as ( = ; = 0) (see g.6). Taking into aount that in this area of parameters g. This narrows the aessible! the points and 4 move to point ~ ' expf N e g ' expf e a g (34) we may see the ritial point is aessible at nite temperature = (a N ) when g = N (35) ln a where = e a. As ; depend not only on a but also on a (diretly and via g ; too), nding funtional relations between ; and seems impossible if we don't x the onnetion between a ;a and N = N g a N = N g a N (36) In the "naive" limit g ' g and ' (a N ) (37) The investigation of the SU() gluodynamis phase struture on asymmetrial lattie in the plane g is beyond the frames of this paper and will be done later. The parameter = q usually is hosen arbitrarily ( Hamilt = and Eul = ). So, if this parameter is not restrited with additional ondition then hanging it arbitrarily wemay reahany point of the urve = onst at any small g, thereby rossing at least one line between phases (II and IV ). This says that thermodynamial quantities depend on and, moreover, the jump on this parameter is possible for some of them. It is ommonly believed that hanging the parameter should not result in any observable

13 eets. The renormalization group relations make possible exluding the dependene of observed quantities on in "naive" limit, thereby making lattie regularization with dierent equivalent [,, 3]. To save the independene of observed quantities from in "naive" limit, wehose ; in preisely the same way asin[,, 3]. In the approximation SU(N) ' Z(N) we failed to reate the renormalization group relations with parameter being a funtion of whih remove the lattie asymmetry parameter from the thermodynamial quantities keeping at the same time "naive" limit results. So, we suggest that lattie gauge theories need some additional ondition whih xes. Although all alulations were arried out for the Z() group, they an be fullled for the Z(3) group also, with small hanges. There are reasons to hope the results for SU() gauge group will be similar to those for the Z() gauge group, at least within approximations of [6]. The authors are indebted for fruitful disussions with Prof. Adriano Di Giaomo. They are grateful to O.A.Borisenko for the ritial notes. Referenes [] J. Shigemitsu, J.B. Kogut. Nul.Phys. B90 (98) 365 [] A. Hasenfratz, P. Hasenfratz. Nul.Phys. B93 (98) 0 [3] M. Creutz. Quarks, gluons and lattie. Cambridge University Press, 983, p.7 [4] E. Brezin and J.M. Droue. Nul.Phys. B00 (98) 93 [5] A. Pena, M. Soolovsky. DESY [6] T. Yoneya. Nul.Phys. B44 (978) 95 [7] M. Lusher, P. Weisz. MPI-PhT/95-7; DESY ; HEP-LAT [8] R. Savit. Rev.Mod.Phys.5, (980) 453 [9] R. Balian, J.M. Droue and C. Itzykson. Phys.Rev. D, 8 (975) 098 [0] A. Ukawa, P. Windey, A.H. Guth. Phys.Rev. D (980) 03 3

14 [] G.'t Hooft In High Energy Physis Proeedings of the European Physial Soiety International Conferene. Palermo [] F. Karsh. Nul.Phys. B05 (98) 85 [3] M. Billo, M. Caselle, A. D'Adda, S. Panzeri HEP-LAT/

15 * x Figure. 5

16 6 0:44 z = ln th = - 0:44 Figure. 6

17 6 =N =N + w 4 N w 3 w w - + Figure 3. Points -4 and -3 { dual symmetrial by pairs. 7

18 -st plaq Figure 4. 8

19 6 =N =N + w 4 I II N w V 3 w III w IV - + I II III IV V Figure 5. { deonnement of eletri and magneti harges { magneti onnement, eletri deonnement { eletri onnement, magneti deonnement { eletri onnement, magneti onnement { we annot investigate this region analytially 9

20 6 =N =N + y 6 N! y N py y - + Figure 6. 0