On the heavy monopole potential in gluodynamics

9 August 2001

Physics Letters B 514 (2001) 88–96 www.elsevier.com/locate/npe

On the heavy monopole potential in gluodynamics M.N. Chernodub a , F.V. Gubarev a,b , M.I. Polikarpov a , V.I. Zakharov b a Institute of Theoretical and Experimental Physics, B.Cheremushkinskaya 25, Moscow, 117259, Russia b Max-Planck Institut für Physik, Föhringer Ring 6, 80805 München, Germany

Received 9 January 2000; received in revised form 26 March 2001; accepted 8 May 2001 Editor: P.V. Landshoff

Abstract We consider predictions for the interaction energy of the fundamental monopoles in gluodynamics introduced via the ’t Hooft loop. At short distances, the heavy monopole potential is calculable from first principles. At larger distances, we apply the Abelian dominance models. We discuss measurements which could be crucial to distinguish between various models. Non-zero temperatures are also considered.  2001 Elsevier Science B.V. All rights reserved.

1. Introduction

references therein) in terms of a modified action S(β, −β):

In this Letter we consider the interaction of the fundamental monopoles in gluodynamics. The fundamental monopoles can be introduced via the ’t Hooft loop [1], and are best understood on the lattice. The monopoles are visualized as end-points of the Dirac strings which in turn are defined as piercing negative plaquettes. In more detail, consider the standard Wilson action of SU(2) lattice gauge theory (LGT):

S(β, −β) = −β

Slat (U ) = −β


1 p

2

Tr Up ,

(1)

where β = 4/g 2 , g is the bare coupling, the sum is taken over all elementary plaquettes p and Up is the ordered product of link matrices in the fundamental representation along the boundary of p. Then the ’t Hooft loop is formulated (see, e.g., [2,3] and

E-mail address: [email protected] (F.V. Gubarev).


1
1 Tr Up + β Tr Up , (2) 2 2 ∗ ∗

p∈ / Σj

p∈ Σj

∗Σ j

where is a manifold which is dual to a surface spanned on the monopole world-line j . Introducing the corresponding partition function, Z(β, −β) and considering a time-like planar rectangular T × R, T  R contour j one can define Vm m (R) ≡ −

1 Z(β, −β) ln . T Z(β, β)

(3)

Since the external monopoles become point-like particles in the continuum limit the potential Vm m (R) is the same fundamental quantity as, say, the heavyquark potential VQQ  (R) related to the expectation value of the Wilson loop. By analogy, we will call the quantity Vm m (R) the heavy monopole potential. At large distances the potentials VQQ m (R)  (R) and Vm are expected to behave in a complementary way [4]. Namely, if the color electric charges are confined and VQ Q (R) → σ R at large R, then the magnetic charges

0370-2693/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 6 0 0 - 1

M.N. Chernodub et al. / Physics Letters B 514 (2001) 88–96

are not confined and the Vm m (R) is of a Yukawa form at large R. This fundamental prediction of the theory is confirmed by the first measurements of the Vm m (R) [2,3]. In anticipation of further measurements of the heavy monopole potential, we try to clarify in this note further predictions for Vm m (R) which follow both from the fundamental gluodynamics and from various models. It turns out that various approaches result in a wide spectrum of predictions. In particular, we will argue that within the fundamental gluodynamics one cannot expect that Vm m (R) fits the Yukawa form at all the distances. On the other hand, the Yukawa potential is a trade mark of the Abelian dominance models. Thus, if the Yukawa potential indeed fits Vm m (R) at all the distances this would imply that the distances available for the measurements are in fact not small enough to allow for the use of fundamental QCD and the physics is rather described in terms of effective theories. Which would be, in turn, an important confirmation of the theoretical speculations on the existence of a numerically large scale of non-perturbative effects [5]. Another crucial test of the applicability of the fundamental theory could be provided by measurements of the coefficient in front of the Coulombic potential at small R. Within the fundamental theory the coefficient turns to be fixed with great accuracy while the effective theories could accommodate other values as well. So far we assumed that the Vm m (R) is measured via the lattice simulations within the full SU(2) theory. In fact, the first measurements of the screening mass in the Vm m (R) were reported within the Abelian projected theory [6]. Moreover, the error bars of those measurements are still the smallest and it would be desirable to include these results into the theoretical analysis. Within this approach, one evaluates the ’t Hooft loop knowing the monopole trajectories in the vacuum. Similar calculations of the Wilson loop are well known since long [7]. We will argue that in case of the ’t Hooft loop this procedure can be justified only in the limit of either vanishing or very high temperatures. This remark allows us to conclude that there exist indications that the screening mass at the vanishing temperature is lower than the mass of the lightest glueball. If confirmed, this observation will be a first example of the situation when the “masses” entering

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an effective Lagrangian in the Euclidean space do not match any physical masses, i.e., masses measured in the Minkowski space. As we emphasize later, there is no contradiction with spectral representations. 2. Short distances Because of the asymptotic freedom, one would expect that at short distances the potential Vm m (R) is predictable from first principles. Some preliminary work is needed, however, since originally Vm m (R) is formulated in the lattice terms. The continuum version of the ’t Hooft loop was worked out in [8]. Moreover, it turns possible [9] to reformulate the problem in the Lagrangian approach a la Zwanziger [10]. As a result, the heavy monopole potential was found on the classical level and in one-loop approximation. Here we add comments on higher loops and consider the power corrections. 2.1. Coulomb-like potential On short distances, the heavy monopoles potential is Coulomb-like (see [8,9] and references therein):   2π 2 1 , Vm (4) m (R) = − g(R) 4πR where g(R) is the running coupling of the gluodynamics. 1 Eq. (4) makes manifest that monopoles in gluodynamics unify Abelian and non-Abelian features. Namely, the magnetic charge, 2π/g is the same as in the Abelian theory with fundamental electric charge e = g while the running of the coupling g 2 is governed by the non-Abelian interactions. Eq. (4) can be found within a Lagrangian approach. In case of QED a well known example of a Lagrangian which allows for interaction of the photon with both electric and magnetic charges can be found in Ref. [10]. In case of gluodynamics the Zwanzigertype Lagrangian was found in [9]:   1  a 2 Ldual Aa , B = Fµν 4 2 1  2π + m · ∂ ∧ B − i ∗na F a + i j B, (5) 2 g 1 For simplicity we consider only the SU(2) gauge group.

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a is the nonwhere j is the magnetic current, Fµν Abelian field strength tensor and mµ is an arbitrary constant vector, m2 = 1. Note that the vector m enters also the original Zwanziger Lagrangian, that is in the QED case. The vector field B can be called dual gluon, although because the Aa and B fields are mixed the number of the physical degrees of freedom corresponding to (5) is the same as in the gluodynamics itself. A salient feature of the Eq. (5) is that the dual gluon B is a U (1) particle despite of the fact that the “ordinary” gluon are in an adjoint representation of the non-Abelian group. There is no violation, however, of the color symmetry through the mixing of the B and Aa fields since the choice of na is a matter of gauge fixing, see for details [8,9]. In the simplest form, the argument is that an averaging over all possible embeddings of the U (1) into the non-Abelian group is understood. To define the theory (5) on the quantum level one should supplement, as usual, the Lagrangian by specific rules of the ultraviolet regularization. The rules reflect existence of the Dirac string connecting the monopoles, see [8,9]. The message so far is that at one-loop level the interaction of the heavy monopoles is understood theoretically no worse than the interaction of the heavy quarks. The most important conclusion is that the Dirac quantization condition, Qm · Qe = 2π holds not only for the bare couplings but for the one-loop running couplings as well. As for the higher loops, a straightforward calculation seems very difficult. The point is that the monopole field is proportional to g −1 . As a result, any number of interaction of virtual gluons with the external monopole field is equally important. The leading log approximation is an exception to the rule [9] and is given by the simple graphs with insertion of two external fields. However, it seems most plausible that the Dirac quantization condition will survive further quantum corrections as well. Then if one defines the running coupling gV2 (R) in terms of the heavy quark potential:

lim VQQ  (R) = −

R→0

3gV2 (R) , 4 · 4πR

(6)

then the same gV2 (R) enters in fact (4). Note that the definition (6) incorporates higher loops as well, 2

for further discussions and explicit two-loop results see [11]. 2.2. Power corrections As far as the physics of short distances is concerned, the next step is to consider power corrections to (4). Theoretically, the prediction is that there is no linear in R term at small R: Vm m (R) ≈ −

π g 2 (R)

1 + a0 ΛQCD R

+ a2 Λ3QCD R 2 + · · · ,

(7)

where a0,2 are constants sensitive to the physics in the infrared. To the contrary, the coefficients in front of the odd powers of R are sensitive to the short distances. The same is true in case of the heavy quark potential and we refer the reader to [12] for the reasoning and further references. The absence of the linear term follows then from the absence of a dimension d = 2 quantity in the gluodynamics. 3 Note that in case of the heavy quark potential this logic can in fact be challenged [5]. The picture that we have in mind is that monopoles condense in the vacuum (see below). Then the external quarks are connected by a Dirac string and since the string is infinitely thin, the physics in the ultraviolet can 2 Let us note that g 2 (R) could be determined in terms of the V potential VQQ  (R) with the quarks Q having arbitrary isospin T as well. Thus, for the Dirac quantization condition to be consistent at higher loop level the Coulomb-like potential should be proportional to T (T + 1) even if all the logs are included. This property of the heavy quark potential can indeed be established directly and is called perturbative Casimir scaling [11]. 3 One of the standard means to clarify what kind of the power corrections are allowed is provided by the renormalons. This is one of the ways to derive (7). Renormalons relate non-perturbative contributions to the divergences of the perturbative expansions. It is amusing to notice that in case of the heavy monopole potential there seems to exist a power correction which defies this logic. Namely, the SU(2) monopoles are actually Z2 monopoles although perturbatively they behave as U (1) monopoles, (for a discussion of the interplay between the two classifications see, e.g., [9,13]). As a result, two monopoles with the total U (1) charge zero can undergo a transition into a state with total magnetic charge Qm = 2 according to the U (1) classification. One can easily see that the corresponding correction to the heavy monopole potential is of order αs2 R. Such a correction is interesting of course only on theoretical grounds and is negligible for any practical purpose.

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change [5]. The case that we are considering now, that is, magnetically charged external probes embedded into the vacuum with monopoles condensed, is much simpler. Indeed, in the dual language we do not need then Dirac strings at all. To summarize, the absence of the linear in R term at short distances follows also from the first principles and eventually is related to the fact that the monopoles are not confined. It is a common point nowadays that at large distances Vm m (R) is of the Yukawa type, e−µR . (8) R Now, we see that such a simple form, if valid on all the distances, is in variance with the theory. Indeed, Eq. (8) incorporates a linear correction at short distances. To summarize, the Coulomb-like potential can be evaluated from first principles with high accuracy. Moreover, one cannot expect that the Coulomb like potential at small distances continues smoothly into the Yukawa-like potential at larger distances. Vm m (R) = −C

3. Abelian dominance 3.1. Yukawa potential The Yukawa potential (8) arises naturally within effective field theories with monopole condensation. 4 To describe the condensation in the field theoretical language, one introduces a new effective field φ which is interacting minimally with the dual gluon (for review and classical references see, e.g., [15]). Consider first the Lagrangian proposed recently in [9]:   Leff = Ldual Aa , B + SHiggs(B, φm ), (9) where Ldual is given by Eq. (5) and LHiggs is the standard Higgs part of the Abelian Higgs model action. The vacuum expectation value of the Higgs, or monopole field is, of course, of order ΛQCD . Within 4 Note that the monopoles which condense are of course not the

fundamental monopoles which are introduced via the ’t Hooft loop as external probes. Instead, the monopoles “living” in the vacuum have a double magnetic charge.

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this model, µ = mV ,

C=

π , g2

(10)

where mV is the mass of the vector field B acquired through the Higgs mechanism. The effective theory applies at distances much larger than the size of the monopoles. Respectively, the constant g 2 is rather frozen coupling than the running one. An attractive feature of the model (9) is that it incorporates the Casimir scaling [9]. Note that historically the first prediction for the screening mass µ was obtained in [14]. Namely, it was shown that to all orders in the strong coupling expansion the mass µ coincides with the mass of 0++ glueball: µ = mG .

(11)

As for the constant C, there is again no reason for it to be the same for the Yukawa and Coulomb like potentials so that, generally speaking, C = π/g 2 . It is worth emphasizing, therefore, that the predictions (10) in fact differ substantially from (11). First, mV may not coincide with any glueball mass. This is not in contradiction with the general principles since there is no spectral representation for the correlator of two magnetic currents j (which are sources of the field B). Indeed, although the currents do not carry any color index explicitly they are not color singlets either since their color orientation is a matter of a gauge fixing. Moreover, the monopole trajectory j is to be understood as the boundary of the Dirac-string world sheet. The Dirac string is infinitely heavy in the continuum limit so that all the intermediate states are in fact infinitely heavy. As for the magnitude of mV , the estimates usually give mV ≈ 1 GeV, see, e.g., [16]. More generally, if it turns out that µ is indeed smaller than the lowest glueball mass, it would be a serious argument in favor of the Abelian dominance models. If the value of µ does not distinguish between the models, then it would be crucial to check that the potential is vector like, as is predicted by the model (9). At very short distances the potential (8) should yield to the potential (7) obtained within the fundamental gluodynamics. However, there exist various pieces of evidence that the effective theory (9) is valid down to such small scales that in practice it covers all

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the distances available for the lattice measurements nowadays [5]. Clearly, a careful study of the Vm m (R) could be crucial to confirm or reject these speculations. It is worth noticing that the value C = π/g 2 is specific for the model (9) and is not true in a generic Abelian dominance model. To use an analogy, consider the world with spontaneously broken chiral symmetry of strong interactions and with weakly interacting vector bosons added. Then the model (9) would be analogous to assuming that the massless pions are coupled to the W bosons but do not interact directly with the nucleons. Thus, we could have added to (9) interaction of the Higgs bosons with the fundamental monopoles. In other words, since the monopoles are condensed the monopole charge of particles is not well defined and the constant C is not constrained, generally speaking. 3.2. Non-vanishing temperatures The screening mechanism at high temperatures is the Debye screening. As is noted above the classical limit of the state created by ’t Hooft loop is an Abelian monopole pair. Thus it is natural to evaluate the Debye mass within the Abelian dominance models. We have already considered the corresponding predictions in [8] and here just summarize the results. The estimates of the mass µ(T ) [8] at high T are based on the observation that the Abelian model which corresponds to the high temperature gluodynamics is the 3D compact U (1) theory. Therefore, at high temperatures the screening mass µ coincides with the corresponding Debye mass: µ2 = m2D = 16π 2

ρ , e32

(12)

where ρ is the density of monopoles and e3 is the corresponding three-dimensional coupling constant. To estimate the temperature dependence of mD we use the numerical results of Ref. [17]. Moreover, at high temperatures we can use the dimensional reduction formalism and express 3D coupling constant e3 in terms of 4D Yang–Mills coupling g. At the tree level one has e32 (T ) = g 2 (Λ, T )T ,

(13)

where g(Λ, T ) is the running coupling calculated at the scale T , 11 log(T /Λ) 12π 2   17 + (14) log 2 log(T /Λ) , 2 44π and Λ is a dimensional constant which can be determined from lattice simulations. At present, the lattice measurements of the Λ parameter are ambiguous. We will use Λ varying within the following limits: g −2 (Λ, T ) =

Λ = (0.0760–0.262)Tc ,

(15)

where the extreme values of Λ here correspond to the data of Ref. [19] and of Ref. [18], respectively, for details see [8]. Another source of uncertainty is the use of the dimensional reduction which is supposed to work well only at asymptotically high temperatures. 3.3. The heavy monopole potential in the Abelian projection So far we assumed that the heavy monopole potential is measured via the ’t Hooft loop in full, non-Abelian lattice simulations. However, within the Abelian dominance model all the interactions are described by QED-like interactions of the magnetic and electric charges. Moreover, the magnetic currents in the vacuum can be measured directly. Then the potential energy of external sources can, in principle, be evaluated using the ensembles of the magnetic currents. In case of external color charges, or Wilson loop such an approach is well known. Also, the intermonopole potential has been studied on the lattice both at zero [20] and non-zero [6] temperatures. In this section we address this issue anew and emphasize that in fact calculation of the heavy monopole potential involves extra parameters. However, the situation is simplified greatly in the limits of very low and very high temperatures. The Abelian monopole action in the Maximal Abelian projection depends on many parameters. The action contains the term S (Coul) [∗j ] = const(∗j, +−1 ∗j ), responsible for the Coulomb exchange between monopoles as well as various n-point, n
2, local monopole interaction terms, Ref. [21]. Here +−1 is the lattice inverse Laplacian, ∗j is the monopole current on

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the dual lattice. Throughout this section we are using the differential form notations on the lattice [22]. Let us consider the following simple monopole action with 2-point (self-)interaction as an example:     S (mon) ∗j = 4π 2 β ∗j, +−1 ∗j + 4π 2 γ ∗j 2 . (16) This action corresponds to the London limit of the (dual) lattice Abelian Higgs model [23] in which the role of the Higgs field is played by the monopole field. Another representation of model (16) is the compact gauge field representation [21]: π Z=

Dθ e−S

(comp) (dθ)

,

−π

e

−S (comp) (dθ)

=




exp − βdθ + 2πn2  

n(c2 )∈Z − γ dθ + 2πn, +(dθ + 2πn) ,

(17) where θ is the compact gauge field and n is the integer valued auxiliary plaquette field. The heavy monopole potential can be studied then with the help of the Abelian – ’t Hooft loop, HΣabJ ,q = exp S (comp) (dθ )

− S (comp) (dθ + 2πq ∗ΣJ ) , (18) where the surface ΣJ ends on the trajectory of an external monopole with charge q.

(a)

93

Similarly to Eq. (3) the quantum average of the ’t Hooft operator gives the static monopole potential: ab Vm m,q (R) = −

Zqmon =




1 Zqmon ln , T Z0mon e−S

(mon) [∗j +q ∗J ]

,

(19)

∗j (∗c )∈Z 1 δ ∗j =0

where the summation is taken over all closed monopole trajectories. The 2-point interaction term provides the local interaction between external J and dynamical j currents, Sint (∗j, ∗J ) = 8qπ 2 γ (∗j , ∗J ). In our case the external current consists of two disconnected pieces, ∗J = ∗Jm + ∗J∗m , separated by the distance R. The term Sint (j, J ) is non-zero if the dynamical monopole current touches the external monopole current J , see Fig. 1(a). Thus, the 2-point interaction terms affect the heavy monopole potential provided the dynamical monopoles overlap locally with the external sources. Let us discuss the role of the local n-point interaction terms at finite temperature. At sufficiently small temperatures the system is in the confinement phase so that the entropy of the monopole currents prevails over their energy. Thus there exist arbitrarily long dynamical monopole currents j connecting the external (anti-)monopole trajectories Jm (Jm  ), Fig. 1(a). In other words, the monopole currents in the confinement phase of gluodynamics are percolating [24]: the probability for two distinct points to be connected by a

(b)

Fig. 1. The static external currents Jm and Jm  and the dynamical monopole currents j at confinement, (a), and the deconfinement, (b), phases.

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Fig. 2. Temperature dependence of the screening mass in the heavy monopole potential (8). Shaded region denotes the theoretical prediction, Eqs. (10), (12)–(14). Symbols denote: × — the dual gluon mass mV ≈ 1 GeV as reported in Ref. [16]; * — the mass of the lightest 0++ glueball;  — magnetic screening mass measured in Maximal Abelian gauge, Ref. [6]; ✸ and ✷ — the screening masses measured at various temperatures in Ref. [2] and Ref. [3], respectively.

monopole trajectory does not depend on the distances between them at sufficiently large separations. Since the dynamical currents are not getting suppressed as their length is increased the distance dependent part of the inter-monopole potential is not affected by the n-point interaction terms. As temperature increases the system goes into the deconfinement phase. Now the n-point terms become important since the monopole currents acquire non-zero free energy per unit monopole trajectory length. However at sufficiently high temperatures the dynamical monopole trajectories become static (i.e., wrapped on the compactified direction with a minimal length), see Fig. 1(b), and n-point terms do not affect the distance dependence of the correlators for well separated external monopoles. To summarize, we expect that the magnetic screening mass obtained using only the long range part of the action is correct at asymptotically small and large tem-

peratures while at the intermediate temperatures the result is modified by the local current interactions.

4. Status of the data. Conclusions Data on Vm m (R) are only beginning to accumulate and we will summarize briefly the comparison of the lattice measurements with predictions above. Short distances The Coulomb-like potential (4) is confirmed in the numerical simulations in the classical approximation [2,8]. There is no running of g 2 on this level of course. Within the full quantum simulations, the fitting coefficient of the Yukawa potential is smaller than predicted but grows towards this value at higher 1/g 2 [3].

M.N. Chernodub et al. / Physics Letters B 514 (2001) 88–96

As for the power corrections, all the data so far [2, 3] are fitted smoothly with a Yukawa potential (8). Thus, there is no evidence whatsoever that the linear term at short distances is to be subtracted. However, no explicit bound on the linear term at short distances has been obtained either. The data on the screening mass µ as function of the temperature are reproduced on Fig. 2. It would be most interesting to get µ at T = 0 with smaller error bars. The point is that the extrapolation to T = 0 of the data obtained within the Abelian projections [6] seem to fall below the lightest glueball mass. And, as noticed in the previous section, the method used [6] is justified in the limit of the vanishing temperatures. As for the full non-Abelian simulations they do not allow to resolve µ = mG from µ = mV at T = 0. As for the higher temperatures, there are two comments which we would like to make. First, at higher temperatures the screening mass obtained in Ref. [6] falls definitely lower than is indicated by the measurements in the full gluodynamics [2,3]. It is worth emphasizing therefore that the calculations [6] took into account only the Coulomb-like interaction between the monopoles. As is explained in the preceding section, this is justified at very low and very high temperatures, but not at the intermediate temperatures. This observation allows to understand why the results of [2,3] and [6] tend to agree at T = 0 and disagree at intermediate temperatures. Second, the numerical simulations in full SU(2) do indicate the growth of the screening mass with the temperature. However, it is early to judge on the the level of the quantitative agreement with the predictions outlined in Section 3.2. To summarize, the heavy monopole potential at short distances is fixed within the fundamental theory. It is Coulomb-like vector potential with known overall coefficient. There is no linear in R term at short distances and it is difficult to expect therefore a smooth matching of the Coulomb like potential at short distances with the Yukawa potential at larger distances. Within the Abelian dominance models, the vector type of the potential is maintained at larger distances as well. The screening mass does not coincide, generally speaking, with any glueball mass. Experimental confirmation of this would be most interesting since would demonstrate existence of a new kind of particles which are neither confined nor allow for analytical continuation from the Euclidean to

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Minkowski space–time. On the other hand, within the strong coupling expansion (considered to all orders) the potential at large distances is of the scalar type with the screening mass coinciding with the mass of the 0++ glueball. At present, there are some indications that the screening mass at T = 0 is actually lower than the glueball mass. However, such “eclectic fits” as unifying the overall coefficient obtained within the fundamental theory with the screening mass equal to the mass of the scalar glueball are not ruled out now either. Further data would be of great interest.

Acknowledgements Authors are thankful to Ph. de Forcrand for providing us with lattice data. M.N.Ch. and M.I.P. acknowledge the kind hospitality of the staff of the Max-Planck Institut für Physik (München), where the work was initiated. Work of M.N.C. and M.I.P. was partially supported by RFBR 99-01230a and Monbusho grants, and CRDF award RP1-2103.

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