Chromoelectric flux tubes and coherence length in QCD

Chromoelectric flux tubes and coherence length in QCD

Paolo Cea paolo.cea@ba.infn.it Dipartimento di Fisica dell’Università di Bari, I-70126 Bari, Italy
and INFN - Sezione di Bari, I-70126 Bari, Italy
   Leonardo Cosmai leonardo.cosmai@ba.infn.it INFN - Sezione di Bari, I-70126 Bari, Italy    Alessandro Papa papa@cs.infn.it Dipartimento di Fisica dell’Università della Calabria, I-87036 Arcavacata di Rende, Cosenza, Italy
and INFN - Gruppo collegato di Cosenza, I-87036 Arcavacata di Rende, Cosenza, Italy
July 1, 2019
Abstract

The transverse profile of the chromoelectric flux tubes in SU(2) and SU(3) pure gauge theories is analyzed by a simple variational ansatz using a strict analogy with ordinary superconductivity. Our method allows to extract the penetration length and the coherence length of the flux tube.

pacs:
11.15.Ha, 12.38.Aw

I Introduction

The presence of chromoelectric flux tubes in QCD vacuum is a clear signal of color confinement Bander (1981); Greensite (2003). Monte Carlo simulations of lattice QCD can produce a sample of vacuum configurations, thus allowing a thorough nonperturbative study of tube-like structures that emerge by analyzing the chromoelectric fields between static quarks Fukugita and Niuya (1983); Kiskis and Sparks (1984); Flower and Otto (1985); Wosiek and Haymaker (1987); Di Giacomo et al. (1990a, b); Singh et al. (1993a); Cea and Cosmai (1993); Matsubara et al. (1994a); Cea and Cosmai (1994a, b, 1995a, 1995b, 1995c); Bali et al. (1995); Haymaker and Matsuki (2007); D’Alessandro et al. (2007). A direct consequence of the tube-like structure of the chromoelectric fields between static quarks is the linear potential and hence the color confinement.

A striking physical analogy exists between the QCD vacuum and an electric superconductor. As conjectured long time ago by ’t Hooft ’t Hooft (1975) and Mandelstam Mandelstam (1976), the vacuum of QCD could be modeled as a coherent state of color magnetic monopoles, what is well known as dual superconductor Ripka (2004). In the dual superconductor model of QCD vacuum the condensation of color magnetic monopoles is analogous to the formation of Cooper pairs in the BCS theory of superconductivity. Even if the dynamical formation of color magnetic monopoles is not explained by the ’t Hooft construction, there is a lot of lattice evidences Shiba and Suzuki (1995); Arasaki et al. (1997); Cea and Cosmai (2000, 2001); Di Giacomo et al. (2000a, b); Carmona et al. (2001); Cea et al. (2004); D’Alessandro et al. (2010) for the color magnetic condensation in QCD vacuum. It should be recognized ’t Hooft (2004) that the color magnetic monopole condensation in the confinement mode of QCD could be a consequence rather than the origin of the mechanism of color confinement, that actually could be originated from additional dynamical causes. Notwithstanding the dual superconductivity picture of the QCD vacuum remains at least a very useful phenomenological frame to interpret the vacuum dynamics.

In the usual electric superconductivity tube-like structures arise Abrikosov (1957) as a solution of the Ginzburg-Landau equations. Similar solutions were found by Nielsen and Olesen Nielsen and Olesen (1973) in the case of the Abelian Higgs model, where they showed that a vortex solution exists independently of the type I or type II superconductor behavior of the vacuum. In previous studies Cea and Cosmai (1994a, b, 1995a, 1995b, 1995c); Cardaci et al. (2011) performed by some of the present authors, color flux tubes made up of chromoelectric field directed along the line joining a static quark-antiquark pair has been investigated, in the cases of SU(2) and SU(3).

In the present work we would like to push forward the analogy with electric superconductivity and exploit some results Clem (1975) in the superconductivity to further extract information from flux tube configurations in SU(2) and SU(3) vacuum. The method and the numerical results for both SU(2) and SU(3) are reported in Section II. In Section III we check the scaling of the penetration and coherence lengths for both SU(2) and SU(3), and compare with previous studies. In Section IV we critically discuss the contribution of the longitudinal chromoelectric field to the string tension. Finally, in Section V we summarize our results and present our conclusions.

Ii Chromoelectric flux tubes on the lattice

To explore on the lattice the field configurations produced by a static quark-antiquark pair we exploit the following connected correlation function Di Giacomo et al. (1990a, b); Kuzmenko and Simonov (2000); Di Giacomo et al. (2002)

(1)

where is the plaquette in the plane, connected to the Wilson loop by a Schwinger line , is the number of colors (see Fig. 1 in Refs. Cea and Cosmai (1995c); Cardaci et al. (2011)). The correlation function defined in Eq. (1) measures the field strength. Indeed, in the naive continuum limit Di Giacomo et al. (1990b)

(2)

where denotes the average in the presence of a static pair and is the vacuum average. According to Eq. (2), we define the color field strength tensor as

(3)

By varying the distance and the orientation of the plaquette with respect to the Wilson loop , one can probe the color field distribution of the flux tube. In particular, the case of plaquette parallel to the Wilson loop corresponds to the component of the chromoelectric field longitudinal to the axis defined by the static quarks. In previous studies the formation of chromoelectric flux tubes was investigated in SU(2) lattice gauge theory Cea and Cosmai (1993, 1994a, 1994b, 1995a, 1995b, 1995c) and in SU(3) lattice gauge theory Cardaci et al. (2011) by exploiting the connected correlation function Eq. (1). It was found that the flux tube is almost completely formed by the longitudinal chromoelectric field, , which is constant along the flux axis and decreases rapidly in the transverse direction . By interpreting the formation of chromoelectric flux tubes as dual Meissner effect in the context of the dual superconductor model of confinement, the proposal was advanced Cea and Cosmai (1993, 1994a, 1994b, 1995a, 1995b, 1995c) to fit the transverse shape of the longitudinal chromoelectric field according to

(4)

Here, is the modified Bessel function of order zero, is the external flux, and is the London penetration length. Equation (4) is valid in the region , being the coherence length which measures the coherence of the magnetic monopole condensate (the dual version of the Cooper condensate). In fact, we expect that Eq. (4) gives an adequate description of the transverse structure of the flux tube if . This means that Eq. (4) should be valid for (type II superconductor), where is the Ginzburg-Landau parameter,

(5)

However, several numerical studies Suzuki (1988); Maedan et al. (1990); Singh et al. (1993b, a); Matsubara et al. (1994b); Schlichter et al. (1998); Bali et al. (1998); Schilling et al. (1999); Gubarev et al. (1999); Koma et al. (2001, 2003) in both SU(2) and SU(3) lattice gauge theories indicated that the vacuum behaves like an effective dual superconductor which belongs to the borderline between a type I and type II superconductor with . Thus, we see that Eq. (4) is no longer adequate to account for the transverse structure of the longitudinal chromoelectric field. Remarkably, it turns out that we may re-analyze our lattice data for chromoelectric flux tubes by exploiting the results presented in Ref. Clem (1975) where, from the assumption of a simple variational model for the magnitude of the normalized order parameter of an isolated vortex, a simple analytic expression is derived for the magnetic field and supercurrent density that solve Ampere’s law and the Ginzburg-Landau equation. In particular, the transverse distribution of the magnetic filed reduces to the London model results outside the vortex core, but has the added advantage of yielding realistic values in the vortex core vicinity. Accordingly, from Eq. (4) of Ref. Clem (1975) we derive

(6)

with

(7)

where is a variational core radius parameter found to be Clem (1975) of the order of . Equation (6) can be written as

(8)

with

(9)

By fitting Eq. (8) to our flux tubes data, we may obtain both the penetration length and the ratio of the penetration length to the variational core radius parameter . It is worth to recall that, by means of Eq. (8), we can extend our fit up to . Moreover by using Eq. (16) of Ref. Clem (1975), we may also obtain the Ginzburg-Landau parameter,

(10)

with the modified Bessel function of order 1. The coherence length is obtained from Eqs. (5) and (10). Our data for chromoelectric fields between static quark-antiquark sources have been obtained through the connected correlation function Eq. (1). In order to reduce the quantum fluctuations we adopted the controlled cooling algorithm. It is known Campostrini et al. (1989) that by cooling in a smooth way equilibrium configurations, quantum fluctuations are reduced by a few order of magnitude, while the string tension survives and shows a plateau. We shall show below that the penetration length behaves in a similar way. The details of the cooling procedure are described in Ref. Cea and Cosmai (1995c) for the case of SU(2). Here we adapted the procedure to the case of SU(3), by applying successively this algorithm to various SU(2) subgroups. The control parameter was fixed at the value 0.0354, as in Ref. Cea and Cosmai (1995c). As described in Ref. Cardaci et al. (2011), in the construction of the lattice operator given in Eq. (1) we have considered also noninteger distances to check the restoration of the rotational symmetry on our lattices.

ii.1 SU(2) data

We analyzed our lattice SU(2) data collected for three different values of , namely (for further details we refer to Ref. Cardaci et al. (2011)). Indeed, we find that Eq. (8) is able to reproduce the transverse distribution of the longitudinal chromoelectric field in the whole region .

Figure 1: SU(2): versus for after 8 cooling steps.

An example of the of effectiveness of Eq. (8) to fit all data for the transverse distribution of the chromoelectric field down to is given in Fig. 1, where we also display the points calculated at noninteger distances, which were not included in the fit. We see that there are slight deviations from the fit curve due to the failure of rotational invariance on a discrete lattice. In fact, fitting all the available data to Eq. (8) results in an increase of the reduced chi-squared without affecting appreciably the fit parameters.

SU(2)   
cooling
5 0.886( 89) 0.829(267) 0.590(452) 0.512(400) 0.3
6 1.214( 66) 0.782(145) 0.567(270) 0.485(365) 0.2
7 1.590( 53) 0.749( 82) 0.538(162) 0.453(325) 0.2
8 1.998( 49) 0.711( 51) 0.530(110) 0.444(314) 0.1
9 2.423( 49) 0.685( 39) 0.508( 82) 0.420(285) 0.1
10 2.680( 23) 0.652( 22) 0.530( 44) 0.444(313) 0.3
Table 1: Summary of the fit values for SU(2) at .

In Table I the results of our fit of Eq. (8) to the SU(2) data at are reported. The Ginzburg-Landau parameter has been obtained through Eq. (10).

Figure 2: SU(2): versus cooling.
Figure 3: SU(2): versus cooling. Here and in other figures below data have been slightly shifted along the horizontal axis for the sake of readability.
Figure 4: SU(2): versus cooling.
Figure 5: SU(2): versus cooling.

In Figs. 2, 3, 4, 5 we display the fitted parameters versus the cooling steps. As regards the parameters , , and , a short plateau is always visible. This corroborates our expectation that the long range physics is unaffected by the cooling procedure. On the other hand, Fig. 2 shows that the overall normalization of the transverse distribution of the longitudinal chromoelectric field is more affected by the cooling. In fact, the parameter seems to displays an approximate short plateau after 9 - 10 cooling steps in accordance with previous studies Cea and Cosmai (1995c).

ii.2 SU(3) data

Figure 6: SU(3): at after 10 cooling steps.

We re-analyzed the SU(3) lattice data presented in Ref. Cardaci et al. (2011). We have fitted the longitudinal chromoelectric field transverse distribution to Eq. (8) for and up to 16 cooling steps. Again, we find that Eq. (8) accounts for the transverse distribution of the longitudinal chromoelectric field in the whole region . In Fig. 6 we display our data for the transverse shape of the longitudinal chromoelectric field between static quark-antiquark sources after 10 cooling steps at together with the fit to Eq. (8). As for the SU(2) case we also display the points calculated at noninteger distances and checked that the fit to all the available data of Eq. (8) does not change the values of the fit parameters.

SU(3)    
cooling
5 4.564( 14) 0.726( 8) 0.516( 14) 0.428(295) 2.8
6 6.617( 9) 0.683( 4) 0.449( 7) 0.355(216) 0.3
7 7.644( 9) 0.652( 4) 0.412( 6) 0.316(177) 2.2
8 8.107( 8) 0.630( 3) 0.384( 5) 0.287(150) 4.6
9 8.254( 8) 0.612( 3) 0.360( 4) 0.263(130) 5.5
10 8.227( 7) 0.598( 3) 0.341( 4) 0.244(114) 5.2
11 8.108( 7) 0.585( 3) 0.325( 4) 0.229(103) 4.5
12 7.943( 7) 0.574( 3) 0.313( 4) 0.217( 94) 3.6
13 7.759( 6) 0.563( 3) 0.304( 4) 0.208( 88) 2.8
14 7.570( 6) 0.553( 3) 0.296( 4) 0.201( 83) 2.1
15 7.383( 6) 0.544( 3) 0.291( 4) 0.196( 79) 1.5
16 7.204( 6) 0.534( 3) 0.287( 4) 0.192( 77) 1.1
Table 2: Summary of the fit values for SU(3) at .
Figure 7: SU(3): versus cooling.
Figure 8: SU(3): versus cooling.
Figure 9: SU(3): versus cooling.
Figure 10: SU(3): versus cooling.

In Table II we collect the results of our fit of Eq. (8) to the SU(3) data at for cooling steps ranging from 5 up to 16. In Figs. 7, 8, 9, 10 we show the fitted parameters versus the cooling steps.
In fact, we see that the parameters , , and , display a short plateau during the controlled cooling procedure as in the SU(2) case. Moreover, Fig. 7 shows that, at variance with the previous case, even the overall normalization of the transverse distribution of the longitudinal chromoelectric field seems to displays an approximate plateau after 7–9 cooling steps in agreement with the results of Ref. Cardaci et al. (2011).

Iii Penetration and coherence lengths

Figure 11: SU(2): versus . Open circles corresponds to the fit with Eq. (4) after 8 cooling steps, full squares correspond to the fit with Eq. (8) after 10 cooling steps.
Figure 12: SU(2): versus for 10 cooling steps.

In Refs. Cea and Cosmai (1995c); Cardaci et al. (2011) it was found that the inverse penetration length exhibits approximate scaling with the string tension . To check the scaling of our new determination of with the string tension, we use a parameterization for the SU(2) string tension obtained by means of a Chebyshev polynomial interpolation to the string tension data collected in Table 10 of Ref. Teper (1998).
In Fig. 11 we display our determination of the ratio for three different values of . For comparison we also report , where the inverse of the penetration length is obtained by fitting the transverse profile of the longitudinal chromoelectric field to Eq. (4) after 8 cooling steps (for details, see Ref. Cardaci et al. (2011)). We see that our new determination of is in satisfying agreement with the results of Ref. Cardaci et al. (2011). Thus, we confirm that displays an approximate scaling with the string tension . Fitting our data for with a constant, we estimate

(11)

where the quoted error take care also of the systematic errors due to the scaling violations displayed by our data. Assuming  MeV, Eq. (11) gives for the penetration length

(12)

Moreover, we have also checked the scaling of the Ginzburg-Landau parameter obtained through Eq. (10). In fact, Fig. 12 shows that is almost insensitive to . By fitting the data with a constant we get

(13)

We would like to stress that our continuum extrapolation for the penetration length and the Ginzburg-Landau parameter are in reasonable agreement with the results obtained in Refs. Suzuki et al. (2008); Schlichter et al. (1998); Bali et al. (1998). In particular, we may confirm that the Ginzburg-Landau parameter is consistent with the critical value , i.e. the SU(2) vacuum behaves as a dual superconductor which lies at the borderline between the type I - type II superconductor regions.
Also for the SU(3) gauge theory we studied the scaling of the “plateau” values of with the string tension. For this purpose, we have expressed our values of in units of , using the parameterization

(15)

given in Ref. Edwards et al. (1998). Figure 13 suggests that the ratio displays a nice plateau in , as soon as is larger than 6. Accordingly, fitting the ratio to a constant we get:

(16)

which, assuming again the standard value for the string tension  MeV, corresponds to:

(17)

The quoted error takes into account our estimation of systematic effects due to the small scaling violations present in our lattice data. It is interesting to compare the present determination of the SU(3) penetration length with the one obtained previously Cardaci et al. (2011) by fitting the lattice data to Eq. (4). In fact in Ref. Cardaci et al. (2011) we obtained which, unlike to the SU(2) gauge theory, seems not to agree with Eq. (16). This discrepancy must be ascribed to the fact that, as discussed below, in the present case the coherence length exceeds the penetration length. In other words, the SU(3) vacuum behaves like a type I superconductor and, as we already discussed, the London equation Eq. (4) is not good enough to describe the transverse distribution of the longitudinal chromoelectric field.
As concerns the Ginzburg-Landau parameter, in Fig. 14 we present our lattice data for different values of . Indeed, is almost insensitive to . By fitting the data with a constant we get

(18)

which confirms that (type I superconductor). It is worthwhile to note that our Eqs. (17), (18) are not in agreement with the recent determinations in Ref. Cardoso et al. (2010), where it is reported fm and . We believe that the origin of the discrepancies resides in the use of different lattice operators to extract the longitudinal chromoelectric field. In fact, the authors of Ref. Cardoso et al. (2010) use a lattice operator which is sensitive to the square of the chromoelectric field instead of our correlation function, Eq. (1), which measures the chromoelectric field strength. In any case, we believe that these discrepancies deserve further studies.

Figure 13: SU(3): versus . Full squares correspond to the fit with Eq. (8) after 10 cooling steps.
Figure 14: SU(3): versus for 10 cooling steps.

To summarize, we have found that the transverse behavior of the longitudinal chromoelectric field can be fitted according to Eq. (6) for both SU(2) and SU(3) gauge theories. This allows us determine the coherence and penetration lengths. In Ref. Cardaci et al. (2011) it was stressed that the ratio between the penetration lengths respectively for SU(2) and SU(3) gauge theories recalls the analogous behavior seen in a different study of SU(2) and SU(3) vacuum in a constant external chromomagnetic background field Cea and Cosmai (2005). In fact, in Ref. Cea and Cosmai (2005) numerical evidence was presented that the deconfinement temperature for SU(2) and SU(3) gauge systems in a constant Abelian chromomagnetic field decreases when the strength of the applied field increases. Moreover, as discussed in Refs. Cea and Cosmai (2001, 2005); Cea et al. (2007), above a critical strength of the chromomagnetic external background field the deconfined phase extends to very low temperatures. It was found Cea and Cosmai (2005) that the ratio between the critical field strengths for SU(2) and SU(3) gauge theories was

(19)

It is interesting to compare the ratio between the critical field strengths Eq. (19) with the analogous ratio between penetration and coherence lengths. Combining Eqs. (5), (11), (13), (16), and (18), we readily obtain

(20)
(21)

It is remarkable that the ratio between the penetration lengths, respectively for SU(3) and SU(2) gauge theories, agrees with the analogous ratio between the coherence lengths, albeit within the rather large statistical uncertainty. Moreover, both ratios are in fair agreement with the ratio between the critical field strengths, Eq. (19).
As stressed in the Conclusions of Ref. Cea and Cosmai (2005), the peculiar dependence of the deconfinement temperature on the strength of the Abelian chromomagnetic field could be naturally explained if the vacuum behaved as a disordered chromomagnetic condensate which confines color charges due both to the presence of a mass gap and the absence of color long range order, such as in the Feynman picture for Yang-Mills theory in (2+1) dimensions Feynman (1981). The circumstance that the ratio between the SU(2) and SU(3) penetration and coherence lengths agrees within errors with the above discussed ratio of the critical chromomagnetic fields, suggests us that the Feynman picture of the Yang-Mills vacuum could be a useful guide to understand the dynamics of color confinement.

Iv Chromoelectric flux tube and string tension

Figure 15: SU(2): versus . Full points correspond to Eq. (24); open points refer to the lattice string tension (see the discussion in the text).
Figure 16: SU(3): versus . Full points correspond to Eq. (24); open points refer to the lattice string tension (see the discussion in the text).

We have shown Cea and Cosmai (1995c); Cardaci et al. (2011) that the color fields of a static quark-antiquark pair are almost completely described by the longitudinal chromoelectric field, which in turn is approximately constant along the flux tube. This means that the long-distance potential acting on the color charges is linear. Using our data and the parameterization Eq. (8) for the chromoelectric flux tube, we are able to compute the string tension given as the energy stored into the flux tube per unit length:

(22)

where, to avoid confusion, we have denoted the flux-tube string tension as , while will indicate the lattice string tension. It is worth to note that the string tension defined by Eq. (22) does not depend on as long as the longitudinal chromoelectric field is constant along the flux tube. Obviously this last condition is not strictly fulfilled on a finite lattice. From Eqs. (22) and (8) we obtain an explicit relation between the string tension and the parameters , , and of the fit Eq. (8) to the chromoelectric flux tube profile:

(23)

After performing the integration, we get:

(24)

Naively, one expects that the string tension defined as the energy per unit length stored into the flux tube chromoelectric field Eq. (24) should agree, at least approximatively, with the string tension measured on the lattice. To check this, in Figs. (15) and (16) we compare the flux-tube string tension in Eq. (24) with the lattice string tension (obtained as detailed in the previous Section) for SU(2) and SU(3) gauge theories, respectively. It is evident from Figs. (15) and (16) that

(25)

for both SU(2) and SU(3) gauge theories. At first sight this result looks quite surprising. In fact, the lattice string tension should contain the total energy per unit length stored into the flux tube. As a consequence we can write

(26)

where takes into account the contribution due to the order parameter condensate. We may, in turn, obtain an estimate of this contribution to the total string tension as follows. Since within the vortex core the order parameter condensate vanishes, we have

(27)

where is the condensation energy density and is approximately the vortex core size. Note that the minus sign is due to the loss of condensation energy in the normal region with respect to the confining vacuum where the order parameter is nonzero. Now, it is usually assumed that in the confining vacuum it is energetically favored to have a condensation of the order parameter as in ordinary BCS superconductors where the superconducting transition is energetically driven by the coherent condensation of Cooper pairs. Therefore, one is led naturally to suppose that . Thus we see from Eqs. (26) and (27) that one should obtain . On the contrary our numerical results clearly indicate that exceeds . The only possible conclusion we can derive is that the order parameter condensation energy is positive. Thus, the confining transition must be driven by disordering the gauge system. In other words, even though the condensation of the confining order parameter costs energy there is a huge number of degenerate physical configurations such that the configurational entropy easily overcomes the energy cost. This means that the deconfining transition is an order-disorder transition, much like the Berezinskii-Kosterlitz-Thouless transition than the BCS superconducting transition. It is remarkable that this conclusion reinforces our previous picture of the confining vacuum that behaves like a disordered chromomagnetic condensate which confines color charges due both to the presence of a mass gap and the absence of color long range order, such as in the Feynman qualitative picture Feynman (1981).

V Conclusions

In the present paper we studied the chromoelectric field distribution between a static quark-antiquark pair in SU(2) and SU(3) pure gauge theories. By means of the connected correlator given in Eq. (1) we were able to compute the chromoelectric field that fills the flux tube along the line joining a quark-antiquark pair. Since our connected correlator is sensitive to the field strengths instead of the squared field strength, we were able to follow the transverse shape of the color fields up to sizable distances. Using some dated results in ordinary superconductivity based on a simple variational model for the magnitude of the normalized order parameter of an isolated vortex, we proposed that the transverse behavior of the longitudinal chromoelectric field can be fitted according to Eq. (8), which allowed us to get informations on the penetration and coherence lengths. In fact we found that our Eq. (8) is able to reproduce the transverse distribution of the longitudinal chromoelectric field in the whole available region. In the case of the SU(2) gauge theory we argued that the confining vacuum behaves as a dual superconductor which lies at the borderline between the superconductor type I - type II regions. On the other hand, we found that the SU(3) vacuum belongs to the superconductor type I region. We found that the ratio between the penetration lengths respectively for SU(3) and SU(2) gauge theories agrees with the analogous ratio between the coherence lengths, albeit within the rather large statistical uncertainty, and both ratios are in fair agreement with the ratio between the critical chromomagnetic fields. Finally, we suggested that the deconfining transition resembles the order-disorder Berezinskii-Kosterlitz-Thouless transition and that the confining vacuum behaves like a disordered chromomagnetic condensate in agreement with the Feynman qualitative picture of the Yang-Mills vacuum.

References

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