SUPPLEMENTS
ELSEVIER Nuclear Physics B (Proc. Suppl.) 96 (2001) 497-505
www.elsevier.nUlocate/npe
Predicting the inter-monopole potential in gluo.dynamics.
M.N. Chernodub, F.V. Gubarev, M.I. Polikarpo? and V.I. Zakharov”
a Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya, 25, 117259 MOSCOW b Max-Planck Institut fiir Physik,
Fiihringer Ring 6, 80805 Miinchen, Germany.
We discuss predictions for the interaction energy of the nmdamental monopoles in gluodynamics. The inter- action energy is defined, as usual, in terms of the ‘t Hooft loop. At short distances, the potential is calculable starting from first principles. At larger distances, we apply the Abelian dominance models. Non-zero tempera- tures are also discussed. The predictions turn to be in reasonable agreement with the existing data, in cases when the comparison is possible.
Monopole-antimonopole potential.
Fundamental monopoles can be introduced on the lattice via the ‘t Hooft loop [l]. In a simpli- fied way, the monopoles are visualized as the end- point of the corresponding Dirac strings which in turn are defined as piercing negative plaque- ttes. Proceeding to more detailed definitions, the one-plaquette action of SU(2) lattice gauge the- ory (LGT) is given by:
P
where p = 4/y” and 9 is the bare coupling, the sum is taken over all elementary plaquettes p and U, is the ordered product of link variables Ul along the boundary of p. The ‘t Hooft loop is formulated then (see [2] and references therein) in terms of the action
S(A--P)=P C *Up-P c nUpp
p4M PEM (2)
where M is a manifold which is dual to the sur- face spanned on the monopole world-line j. In- troducing the corresponding partition function, Z(p, -,0) and considering a planar rectangular T x R, T >> R contour j one can define V,,(R) 3 1 Z(P, -P)
- ?; In Z(/?,P) . (3)
By the analogy with expectation value of the Wil- son loop the quantity Vmm(R) is referred to as
monopole-antimonopole, or inter-monopole po- tential. The potential l&e(R), in a way, is the same fundamental quantity as the heavy-quark potential V&-J and its understanding within the fundamental QCD would be of great importance.
Motivated by the first recent measurements of VmrR(R) on the lattice [2,3], we tried [4,5] to ex- plore the predicting power with respect to the monopole potential of both the fundamental glu- odynamics and of the Abelian dominance mod- els. We have found that quite detailed predictions concerning VmtFL(R) can in fact be made. In this talk we discuss the following predictions for V,,r, [4,51:
(i) At short distances the potential is Coulomb- like:
V,,(R) = r ’
-g”(R)3 (4)
where g(R) is the running coupling of the gluo- dynamics;
(ii) As for the power-like corrections to (4), there is no linear in R term at small R;
(iii) At larger distances the potential is of the Yukawa type:
Vmm( R) = e--mvR
- const . -
R ’ (5)
where the mass rnv can be obtained, at least in principle, from independent fits. Moreover, the fJ920-5632/01/$ - see liont malter CL3 2001 l%evier Science I%\! ,411 rights reserved.
I’ll SO920-5632COl)OI 172-O
49x MN. Cherr~oduh et trl /Ndeur Physics B (Proc. Suppl.) 96 (2001) 497-505
coordinates Z(o). The star symbol denotes the duality operation. The unit three-dimensional vector field nn (a), 6” = 1 is defined on the world- sheet:
nQ(g) ZZ (t. *,a) [Q. *Fyy (9)
(t. P) = &/(o-) F,n,(;c) 1
45 (10)
Y(C) = Det[ d,S+ dpZc, 1.
Note that n”(u) is not in fact an independent variable but is completely determined in terms of the components of the field strength tensor Fiv.
Table 1
The screening mass p (see Eq. (5)) at different temperatures from Ref. [2] and our predictions for mv obtained with different A, Eqs. (39,40).
potential (5) is due to an exchange of a vector particle;
(iv) Temperature dependence of the screening mass, Eq. (5). The predictions are given in the Table 1.
In the subsequent sections we will outline the derivations of (i)-( iv) and discuss the assumptions made. Here let us only mention that the points (i) and (ii) are based on the fundamental gluody- namics while the points (iii) and (iv) are conse- quences of the Abelian dominance models. Fur- ther details and references can be found in [4,5].
Let us also mention that the basic facts about the monopoles in non-Abelian theories can be found in [6].
The ‘t Hooft loop in the continuum limit.
Our starting point is the continuum analog [4]
of the ‘t Hooft loop:
H(Cj) = exp{ S(F) - S(F + 27r*Cj) } , (6)
S(F) = $, J d4x (F,:,)” ,
qp” =
J c12Q” e(u) 6@)(2 - 2(u)) ) (8)
where the surface C4 is assumed to be non- intersecting and is described by the world-sheet
It can be shown [4] that the Eqs. (6)-(10) cor- rectly define the ‘t Hooft loop operator. In partic- ular, the expectation value of (6) does not depend on the position of the surface Cj. If one considers the rectangular T x R time-like contours j with
T >> R then:
(H(Cj) = (H(T, R)) N e--TVmm(R), (11)
where V,, is the continuum limit of the lattice monopole potential (3).
Although the definitions given above may look somewhat formal, the physics involved is simple.
Namely, the surface Cj is swept by a Dirac string.
The field flowing through the string is character- ized, in particular, by its color orientation, that is by the vector n”. Using the freedom to rotate the field in the color space, it can be oriented, for example, along the third axis, ,n3, everywhere.
The energy of the Dirac string is infinite in the continuum limit and the fundamental monopoles, therefore, can be introduced only as external ob- jects.
The Coulomb-like potential.
Because of the asymptotic freedom, we expect that at short distances the potential V,,(R) can be evaluated classically. Consider, therefore, the corresponding equations of motion:
I~;~(F;,,(A)+ 27r*C;,,) = 0, (12)
which should be supplemented by the Bianchi identities:
D;"Fj,, = 0. (13)
M.N. ClwrnoduD et al. /Nuclear Plysirs B (Pmt. Suppl.) 96 (2001) 497-505 499
Note that (12) is consistent with the covariant conservation of the electric current:
D&F,, = - 2nD,D,‘C,, = (14)
=
/ d%,/F” PV ,n”(cr) 5(4)(z - 5) = 0.
Next, let us choose the gauge such that CzV has a constant color orientation characterized by the vector Q^. Then n” N Q” and Eq (12) becomes:
a, (81,&l) = - 27rd” *c@” . (15) The solution of this equation in the Landau gauge,
(16)
corresponds to the gauge potential of an Abelian monopoles current dC embedded into the SU(2) group. The corresponding potential energy of a monopole-antimonopole pair is given by (4) (with g” not running in the classical approximation).
Dual gluon as an Abelian particle.
There is a long standing interest in construct- ing the dual gluodynamics, for review and further references see [7]. The dual gluon, by definition, interacts with monopoles. The motivation is to realize in the field theoretical language the dual superconductor model of the quark confinement [S] according to which the quarks are connected at large distances by an Abrikosov-type vortex.
The key element is then the construction of non- Abelian monopoles. Usually they are modeled after the Polyakov-‘t Hooft monopoles. Namely, one introduces first non-Abelian dual gluons in- teracting with Higgs fields and then assumes con- densation of the Higgs fields which mimics the condensation of the monopoles. In the realistic case of the SU(3) gauge group one needs an octet of dual gluons and three octets of the Higgs fields, all of them understood in terms of an effective field theory valid in the infrared region.
While such a construction might be viable as an effective theory, we need in fact tools to describe interaction of non-Abelian monopoles at arbi- trary short distances as well [4,5,9]. Indeed, in the lattice version of the theory external monopoles
can be introduced via the ‘t Hooft loop operator (6) and in the continuum limit these monopoles are point like. Thus, we are encouraged to con- sider the dual gluodynamics at short distances, or at the findamental level.
Moreover, since the expectation value of the
‘t Hooft loop depends only on its boundary, j,, it seems natural to introduce a dual gluon which interacts directly with the fundamental point-like monopoles, or with j,. In the context of electro- dynamics, the idea is of course very old and goes back to the papers in Ref. [lo]. The Zwanziger Lagrangian which describes interaction of a gauge field with point-like monopoles is [lo]:
Lz,.,(4, B) = ;(m . [a A A])2 + ;(m . [a A B])“+
+;(m . [a A A])(,m + *[a A B])- (17)
-;(m . [a A B])(m . *[aAA])+ij,.A-tij,.B,
where j,, j, are electric and magnetic currents, respectively, m, is a unit space-like vector, m2 = 1 and
[A A &v = A,& - A&,, (me [A A B]), = m,(A A B)pv.
At first sight, two different vector fields, A, B have been introduced to describe interactions with electric and magnetic charges, respectively.
If it were so, however, we would have solved a wrong problem because we need to have a single photon interacting both with electric and mag- netic charges. And this is what is achieved by the construct (17). Indeed, the fields A, B are not diagonal and the bilinear in A, B interference terms in (17) are such that the field strength ten- sors constructed on the potentials A and B are in fact related to each other:
F@“(A) = F;,(B). (13)
Which means in turn that there are only two physical degrees of freedom corresponding to the transverse photons which can be described either in terms of the potential A or B. Topological ex- citations, however, can be different for the fields A and B.
500 M.N. Chernodub el al /Nuclear Physics B (Proc. Suppl.) 96 (2001) 497-50s
The physical content of (17) is revealed by the propagators for the fields A, B. In the Feynman-
‘t Hooft gauge, one can derive:
(A,&) = (BP&) = +’ , (19)
(A,&) = -(&A,) = &*[m A k],, . The propagators should reproduce, as usual the classical solutions. And indeed, one can readily see that the (AA), (BB) propagators describe the Coulomb-like interaction of two charges and mag- netic monopoles, respectively. While the (AB) propagator reproduces the interaction of the mag- netic field of a monopole with a moving electric charge.The appearance of the poles in (Icm) is a manifestation of the Dirac strings.
To summarize, the Zwanziger Lagrangian in electrodynamics [lo] reproduces the classical in- teraction of monopoles and charges. Upon the quantization, it describes the correct number of the degrees of freedom associated with the pho- ton.
Now, if we would introduce a Zwanziger-type Lagrangian for the dual gluodynamics, we im- mediately come to a paradoxical conclusion that the dual field, if any, is Abelian [5,9]. Indeed, monopoles associated with, say, SU(N) gauge group are classified according to U(l)N-’ sub- groups [ll]. Thus, there is no place for a non- Abelian dual gluon because the monopoles do not constitute representations of the non-Abelian group.
The potential (16) can be obtained by postu- lating the following Lagrangian for the dual glu- odynamics:
&al(An, B) = $F;J’+ (20)
+;[rn. (a A B - ~“TPF~)]~ + i&B, where CL is the color index and for simplicity we concentrate on the SU(2) case so that CL = 1,2,3, nn is an arbitrary constant vector while the j, is the magnetic current, and F& is the non-Abelian field strength tensor.
Since the Lagrangian (20) is apparently violat- ing both the global and local SU(2) symmetry let
us add a few comments on the meaning and rules of using the Lagrangian (20).
(a) First, if the magnetic current is vanishing, j7n = 0 then the integration over the field B re- produces the standard Lagrangian of the gluody- namics.
(b) As far as the quantization is concerned, the Lagrangian (20) reproduces the correct degrees of freedom of the free gluons. Indeed, the procedure at this point is essentially the same as in case of a single photon.
(c) At first sight, the most serious problem with the Lagrangian (20) is the emergence of the vector nn in (20), which breaks the SU(2) invariance of the gluodynamics. Thus, one should specify the rules of using the Lagrangian (20).
It is irnportant to realize that the arbitrari- ness in choosing the vector nn in (20) reflects the freedom in choosing the color orientation of the monopoles. Namely, picking up a particular nn is nothing else but using the gauge fixing freedom to fix the vector no(a) which lives on the surface Cj, see Eq. (8). To restore the apparent SU(2) symmetry, we can either average over the direc- tions of nn or fix nn but evaluate only color singlet quantities, like the Wilson loop (note somewhat similar remarks in the paper in Ref. [12]).
To summarize this subsection, if one tries a Lagrangian approach to describe the interaction of the fundamental monopoles in gluodynamics, there is no much choice but to introduce an Abelian dual gluon. This does not imply any vi- olation of the SU(2), however.
Running of the coupling.
To test the Lagrangian approach, we will con- sider now radiative corrections to the Coulomb- like interaction at short distances. Obviously enough, one would expect that the radiative cor- rections result in the standard, non-Abelian run- ning of the coupling y”. Which is indeed our main conclusion. Moreover, since for a given vec- tor nn the non-Abelian monopoles essentially co- incide with the Dirac rnonopoles, the derivation has much in common with the QED case, see [6,13]. There are some points specific for the non Abelian case as well, however.
M.N. Chernodub et al. /Nuclear Plysirs B (Proc. Suppl.) 96 (2001) 497-jO5 501
It is worth emphasizing that evaluation of the radiative corrections addresses in fact two dif- ferent, although closely related problems. That is, running of the coupling and stability of the classical solutions. Both aspects are unified, of course, into evaluation of a single loop in the clas- sical background. However, the running of the coupling can be clarified by keeping track of the ultraviolet logs, In AIJV alone and are universal since in the ultraviolet all the external fields can be neglected. Therefore, the coefficient in front of In Rr~v can be found by evaluating the loop graph with two external legs, i.e. the standard pertur- bative polarization operator. This is true despite of the fact that the monopole field is strong (i.e.
the product of the magnetic and electric coupling is of order unity). On the other hand, the stability of the classical solution is decided by the physics in the infrared. Here, the fact that ge + gn N 1 could be crucial.
In this talk we will consider only the running of the coupling. Our first step is to reproduce the results known in QED in a simple way which can be generalized then to the monopoles in the non- Abelian theories. Moreover, for the sake of defi- niteness we concentrate on the Dirac monopole with the minimal magnetic charge interacting with electrons and in one-loop approximation, for a review and further references see [13]. Then, the evaluation of, say, first radiative correction to the propagator (BB) in the Zwanziger formalism (19) is straightforward and reduces to taking a product of two (AB) propagators and inserting in between the standard polarization operator of two electromagnetic currents. The result is [13]:
(BJ3,) = 6,” $ (1 - L)+
+$-&‘” - ?m)L~
L= 7 In A&/k2
(21)
and we neglect the electron masses so that the in- frared cut-off is provided, in the logarithmic ap- proximation, by the momentum k.
At first sight, there is nothing disturbing about the result (21). Indeed, we have a renormaliza- tion of the original propagator which is to be
absorbed into the running coupling, and a new structure with the factor (k vr~)-~ which is non:
vanishing, however, only on the Dirac string. The latter term would correspond to the renormaliza- tion of the Dirac-string self-energy which we do not follow in any case (since it is included into self-energy of the external monopoles). What is, actually, disturbing is that according (21) the magnetic coupling would run exactly the same as the electric charge,
(A,&) = 6,” F l (1 -L),
violating the Dirac quantization condition eg = 1.
The origin of the trouble is not difficult to fig- ure out. Indeed, using the propagator (AB) in the Feynman graphs is equivalent, of course, to using the full potential corresponding to the Dirac monopole A$. Then, switching on the interaction with electrons would bring terms like A$ $774~.
Since AD includes the potential of the string, elec- trons do interact with the Dirac string and we are violating the Dirac “veto” which forbids any di- rect interaction with the string.
Let us demonstrate that, indeed, it is the incor- rect treatment of the Dirac string that changes the sign of the radiative correction. This can be done in fact in an amusingly simple way.
First, let us note that it is much simpler to re- move the string if one works in terms of the field strength tensor, not the potential. Indeed, we have H = Hrnd + &ring while in terms of the potential A any separation of the string would be ambiguous.
Thus, we start with relating the potential V,, to the interference term in the magnetic field squared:
V mm =
s H, .H2d3r. (22)
Now, it is not absolutely trivial, how we should understand the product Hi . HZ. Consider for example the magnetic field of a single monopole:
H = Hrad + &ring , div H = 0 , (23) H rcrd =
502 M.N Chernodub ef al. /Nuclear Physics B (Proc. Suppl.) 96 (2001) 497-505
div H string = S(r - q) (25)
Then, by the analogy with the the case of two electric charges, we would like to have the follow- ing expression for the interaction energy:
Vme = s Hi,rad H2,,,dd3r = --- 1 1
477 ]rl - r2) . (26) Note, however, that if we substitute the sum of the radial and string fields for Hl,s , then we would have an extra term in the interaction en- ergy:
%a = J’ (Hl,mJ-b,string+ (27) +H 1 ,string Hz,rnrl)d3T = +2 & ,rI t _
In other words, the account of the string field would flip the sign of the interaction energy! This contribution, although looks absolutely finite, is of course a manifestation of the singular nature of the string magnetic field. Note that the integral in (27) does not depend on the shape of the string.
Thus, in the zero, or classical approximation we should write:
Hi . H2 = Hl,rod . H2,rod. (28)
However, in the above example the electrons in- teract with full potential A$ and therefore the first radiative correction would bring the prod- uct of the total Hi e Hz which includes also the string contribution’. Indeed, the result in the log approximation would be as follows:
@Hi . Hz) = L(Hl,string + Hl,ran 1. (29)
W2,string + H2,rnd) = - LHl,rnd . H2,rnd >
where at the last step we have used the observa- tion (27).
Now, it is clear how we could ameliorate the situation. Namely, to keep the Dirac string un- physical we should remove the string field from the expression (29) which arises automatically if we use the propagators (19) following from the
‘At this point we assume in fact that A~TV is larger than the inverse size of the string, which is convenient for our purposes here. Other limiting procedures could be consid- ered as well, however.
Zwanziger Lagrangian. Removing the string re- verses then the sign of the radiative correction and
(39) as a result of the radiative corrections.
One might still wonder, how it happens that the couplings in the electric and magnetic poten- tial run in opposite ways. Indeed, now we reduced the product Hi . Hz to exactly the same form as the product Ei . E2 in case of two electric charges (since the radial magnetic and electric fields are the same, up to a change of the overall constants).
The resolution of the paradox is that the renor- malization of the electric and magnetic fields are indeed similar in the language of the Lagrangian.
However, the small corrections to the Lagrangian and Hamiltonian are related as:
6L = - 6H. (31)
Since E” and H” enter with the same sign into the expression for the Hamiltonian and with the op- posite signs into the Lagrangian, Eq (31) implies that the running of the couplings in the electric and magnetic potentials V, and V,, respectively, are opposite in sign. Which is, of course, in full agreement with expectations since V, m y” and v, N y-2.
So far, we simply assumed that the effect of the Dirac string should be removed from the renor- malization effects. This seems to be a reasonable assumption but a crucial point is that the lattice regularization indeed results in this renormaliza- tion procedure. For the demonstration of this we refer the reader to [5,9].
So far we considered the Abelian case. It is clear, however, that the non-Abelian case can be treated in exactly the same way as far as we stick to the background-field gauge. It is worth empha- sizing that the use of this gauge is not a matter of convenience now but a matter of principle. The point is that only in this case we can still assume that the fundamental monopoles introduced via the ‘t Hooft loop are oriented along the same di- rection in the color space. Use of other gauges would result in a fluctuating vector nn (see Eq.
(20)). Which, in turn, would require for a gener- alization of the framework presented here.
M.N. Chernodub ei al. /Nuclear Plz_vsics B (Proc. Suppl.) 96 (,7001) 497-505 503
The Yukawa potential
To describe the interaction of the fnndamen- tal monopoles at larger distances we accept the standard assumption on the condensation of the monopoles in the QCD vacuum, for review and further references see [14]. It is worth empha- sizing that the monopoles which condense are of course not the fundamental monopoles which are introduced via the ‘t Hooft loop as external objects. Instead, the monopoles “living” in the QCD vacuum have a double magnetic charge.
The corresponding U(1) monopoles are known [15] to be unstable as classical solutions and they cannot be obtained via the quasiclassical approx- imation for this reason, for further details and references see [4,5].
To describe the monopole condensation phe- nomenologically we modify the Lagrangian (20) by adding the effective Higgs interaction where the role of the Higgs field is played by the monopole field (Pm:
S eff = hial (A’, B) + S~iggs (B, hn> . (32) S,qiggs is the standard action of the Abelian Higgs model. The vacuum expectation value of the Higgs, or monopole field is, of course, of order
&CD*
Despite its apparent simplicity, Eq. (32) is highly speculative. Namely, it unifies so to say fundamental gluons, A”, B and & which is an effective field. One may justify the use of (32) by assuming that the effective size of the monopoles with Qm = 2 is in fact numerically small, al- though generically it is of order AQ~D. While in our presentation here we follow mostly the lines of Refs. [4,5,9,16], let us note that the approx- imation of the numerically small monopole size was formulated also in [17].
What is actually specific about the Lagrangian (32) is that the dual gluon is a U(1) field so that the color symmetry is maintained by averaging over all possible embeddings of the (dual) U(1) into the SU(2). We emphasized already this point in connection with the Lagrangian for the dual gluodynamics (20).
Without going into a full investigation of the phenomenological consequences of (32) we note
here only that the dual gluon becomes massive and the Coulomb-like potential is replaced by a Yukawa potential at larger distances, see Eq. (5).
The appearance of a screening (Yukawa) mass is confirmed by the existing data [2]. It is worth emphasizing, however, two less trivial points about the Yukawa potential (5):
(a) Within the model considered, the Yukawa potential is predicted to be mediated by a an ex- change of a vector particle. On the other hand, in the strong-coupling limit one can argue that at large distances the exchange is by a scalar glue- ball [18]. Thus, there is a conflict between the two predictions and direct measurements of the spin structure of the V,,(R) would be of great interest.
(b) An amusing point is that the Yukawa po- tential at short distances comes in contradiction with the theory of the power corrections in &CD, for a recent review a further references see [19].
Namely, at short distances there are no linear cor- rections to the Coulomb-like potential:
;l$vmffr = - --&; + O(R"). (33)
In a somewhat simplified form, the argument is that that in the fundamental QCD there is no (gauge invariant) parameter of the dimension d = 2. In more details, the argumentation is very similar to that used to demonstrate the absence of the linear in R corrections to the heavy-quark potential [20].
Thus, according to the standard picture the po- tential should be of a more complicated form than a mere Yukawa potential. Which looks a some- what bizarre conjecture. If further studies of the potential would reveal no anomalies and confirm the simple Yukawa form (5), then it would a new evidence in favor of validity of the effective La- grangian (32) at the distances currently available to direct measurements. The crucial difference between the effective Lagrangian (32) and than of the fundamental gluodynamics is that the La- grangian (32) does have d = 2 quantities entering through its Higgs part [17,19).
504 M. N. Chernoduh e/ al. /N~tclenr Pl~ysics B (Proc. S~cppl.) 96 (2001) 497-505
Non-vanishing temperatures.
In Ref. [2] the numerical calculations of the
‘t Hooft loop have also been performed at finite temperature and first results on the temperature dependence of the screening (Yukawa) mass p have been obtained. Thus, theoretical predictions of this dependence are worth developing.
To this end, we can use the idea of the Abelian dominance and estimate the mass p using the fact that the Abelian model which corresponds to the high temperature gluodynamics is the 3D compact U(1) theory. Therefore, at high temper- atures the screening mass p coincides with the corresponding Debye mass:
rn& = 16nv
ef ’ (34)
where p is the density of monopoles and es is the corresponding three-dimensional coupling constant.
To estimate the temperature dependence of mD we use the numerical results of Ref. [21], where the density of Abelian monopoles was obtained”:
p = 2-7(1 & 0.02) eg , (35) where es is the 30 electric charge. Therefore
mD = 1.11(2) ei (36)
At high temperatures we can use the dimensional reduction formalism and express 3D coupling con- stant es in terms of 4D Yang-Mills coupling y. At the tree level approximation one has
e;(T) = &A, T) T, (37)
where y(A,T) is the running coupling calculated at the scale T,
y-?(h,T) = &log($)+
+&1og[2log(~)] >
“Note that the original result of Ref. [21] for the lattice monopole density is: piat, = 0.50(l) /J&, where pg is a three dimensional coupling constant which is expressed in terms of the 30 electric charge es and lattice spacing n as 0:; = 4/(aeg). The physical density p of monopoles is given by p = piat. a-’ which can easily be transformed into Eq. (35).
and h is a dimensional constant which can be determined from lattice simulations.
At present the lattice measurements of the A parameter are ambiguous and depend on the quantity which was used to determine it. We use two “extreme” values of A. In Ref. [22] the lattice data for the gluon propagator have been used to determine the so-called “magnetic mass”
in high temperature SU(2) gluodynamics. These measurements indirectly give the following value:
A = 0.262(18) T, = 0.197(14) fi, (39) where T, is the temperature of the deconfinement phase transition, T, z 0.75J-6. In Ref. [23] the spatial string tension has been calculated and the three times smaller value of A has been found:
A = 0.076(13) T, = 0.057(10) J;;. (49) Collecting eqs. (36)-(40) we get the predictions for the Debye mass which are shown Table 1 together with the values of mass p obtained in Ref. [2].
One can see that the prediction and the numerical results are in the quantitative agreement.
Note apparent sources of uncertainties in our analysis. First, the value of A is not determined precisely as we already noted. Second, we have used the dimensional reduction arguments which are supposed to work well only at asymptotically large temperatures while only one temperature T = 3.676 fi M 5T, in the above table may be considered as large enough. Also, the experimen- tal fits [2] do not always satisfy the renormgroup constraints yet. Thus, at the moment we would conclude only that the predictions on the tem- perature dependence of the screening mass are in qualitative agreement with the data.
1. Conclusions.
We conclude that, on the theoretical side, there exist rather detailed predictions for the potential V,,. Their tests on the lattice are desirable and, in particular, would be crucial for the Abelian dominance models.
2. Acknowledgments
One of the authors (V.Z.) would like to ac- knowledge gratefully the hospitality of the Cen-
M.N. Chernodub et al. /Nuclear Physics B (Proc. Suppl.) 96 (2001) 497-505 505
tre National de la Recherche Scientifique (CNRS), and especially of S. Narison, during his stay at the University of Montpellier where part of this work was done.
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