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Quark confinement and number of flavors
Nuclear Physics B (Proc. Suppl.) 26 (1992) 311-313 North-Holland
QUARK CONFINEMENT AND NUMBER OF FLAVORS Y. Iwasaki, K . Kanaya, S. Sakar, T. Yoshi6 Institute of Physics, University of Tsukuba, Ibaraki 305, Japan Constraint on the number of flavors for quark confinement and spontaneous breakdown of chiral symmetry to take place in QCD is investigated with Wilson fermions . It is shown that even in the strong coupling limit, when the number of flavors exceeds 7, quarks are not confined and chiral symmetry is not spontaneously broken for light quarks . We make a conjecture for what is "light quark" at finite Q and give numerical results which support the conjecture. Theses results imply that quarks are not confined for NJ > 7 in the continuum limit.
We report the results of a numerical investigation on the problem whether there is constraint on the number of flavors for quark confinement and spontaneous breakdown of chiral symmetry using Wilson quark action . We employ the same method as in our previous paper[1] to discriminate the phases of QCD with a various number of flavors: With the quark mass defined through the axial-vector current Ward identity, the value of the pion mass at zero quark mass determines whether chiral symmetry is spontaneously broken or not. It will turn out that confinement is closely related with the spontaneous breakdown of chiral symmetry. We generate gauge configurations using the hybrid-molecular-dynamics R algorithm[2] . The inversion of the quark matrix (x = D-1 b) is made by the minimal residual method or the conjugate gradient (CG) method. The lattice size is 82X 10 x T (T = 4, 6, or 8) and 182 x 24 x T (T=18) . When the hadron spectrum is calculated in the former case, the lattice is duplicated in the direction of the lattice size 10, which we call the z direction. We use an anti-periodic boundary condition for quarks in the t direction *
PRESENTED BY Y. IWASAKI
0920-5632/924$05 .00 0 1942- Elsevier Science Publishers B.V
and periodic boundary conditions otherwise. Errors of physical quantities are estimated by the jackknife method. More detailed description will be given elsewhere[3] Let us begin with the investigation of confinement in the strong coupling limit ,Q = 0. We note that there is no proof for confinement at Q = 0 in full QCD. We have found an evidence for a phase transition around K = 0.20 for Nf = 18 on a T = 4 lattice in the behavior of the plaquette W(1 x 1) and the Polyakov loop. The jump of the Polyakov loop suggests that the transition from the confining phase to the deconfining phase takes place around K = 0.20. The data of the pion mass and the quark mess indicate that for K >_ 0.20 chiral symmetry is not spontaneously broken. Thus the deconfining transition and the chiral symmetry transition occur at an identical hopping parameter . Decreasing the number of flavors from 18 to 6 with K = 0.25 fixed on the T = 4 lattice, we find that the number of iteration of CG needed for the quark matrix inversion (r., several hundreds) does not change so much down to Nf = 7 . However, it suddenly increases when Nf becomes 6 and exceeds 10,000 with Ar = 0.01 . This implies that All rights reserved.
Y Iwasaki et d/Quark confu:ement andnumber offlavors
312 03
2.5
N;=7 --0- W(1x1) (T-4)
02
15
46 W(ix1) (T-6) 0 W(ixt) (T-8)
O
W(1x1) (T-18)
01
00 1
38
4
4.2
44
46
Fig. 1. W(1 x 1) for
I
48
5
52
Nf = 7.
at K = 0.25 the phase for Nf = 16 - 7 is identical with that of Nf = 18, but is different from that of Nf = 6. Therefore we study the Nf = 7 and 6 cases in detail. The results of W(1 x 1) for Nf = 7 and those of the quark mass m q and the pion mass m,, on the T = 4, 6, 8 and 18 lattices are given in fig.1 and 2. W(1 x 1) and m,, for T = 4 have jumps between K = 0.24 and K = 0.245. W(l x 1) at K = 0.25 for T = 6, 8 and 18 are consistent with that for T = 4. These results strongly imply that the transition is not a finite temperature transition. From K = 0.20 to K = 0.24, m.., is consistent with the strong coupling result without quark loops. However, m,r 's for K > 0.245 are completely different from the strong coupling result. The m,r 's at K = 0.25 are 1.251(1),1 .136(3), 1 .08(6) and 1.12(2) for T = 4, 6, 8 and 18, respectively. The m,'s for T = 6 N 18 lattices are larger than the lowest Matsubara frequencies. The Polyakov loop also has a jump between K = 0.24 and K = 0.245. Therefore we interpret the state for K > 0.245 as a deconfined state. Thus we conclude that when the quark mass is small, quarks are not confined and chiral symmetry is not spontaneously broken for Nf = 7
Fig . 2. mr and 2 mq for Nf = 7: Circles (T = 4), triangles (T = 6), boxes (T = 8) and diamonds(T = 18) . The solid lines are the strong coupling results .
at ,Q = 0. We believe that this conclusion should also apply to Nf = 8 - 18, since the Nf = 7 case is the critical case. Indeed we have confirmed the conclusion for the Nf = 12 and 18 cases. As we mentioned already, at K = 0.25, the Nf = 6 case is completely different from the case of Nf = 18 - 7. We find that the point K = 0.25 at ,Q = 0 for Nf = 6 cannot be reached from the following three paths in that the number of iterar tions exceeds 10,000 for quark matrix inversions with OT = 0.01: (1) Change Nf from 7 to 6 with ,Q = 0 and K = 0.25 fixed. (2) Increase the hopping parameter K toward 0.25 with Nf = h and ,6 = 0 fixed. (3) Decrease 6 on the chiral-limit line K,,(,6) with Nf = 6 and T = 4 fixed. This fact indicates that the point K = 0.25 at ,Q = 0 for Nf = 6 belongs to the confining phase. Now we state the major conclusion of this article: For Nf > 7 quarks are not confined and chiral symmetry is not spontaneously broken for light quarks at ,Q = 0. (Although we have not investigated the case for Nf > 18, we conjecture that these features hold for this case as well.) Combining all the results obtained, we can draw the line of the hopping parameter Kd for the de-
Y. Iwasaki et al. IQuark confinement and number offlavors
0.25
1
deconfining phase K
0.2
confining phase 0.15
0
5
10
N
f
15
20
Fig. 3. Kd versus Nf
confining transition as a function of the number of flavor ( see fig.3) . The fundamental problem which remains is what will happen in the continuum limit. A problem which should be clarified in this connection is what is "light quark" at finite # . Writing the critical quark mass at the deconfining phase transi tion as mgMti~t (,Q) = c"i'at/a (a is the lattice spacing), we conjecture that ccritica! is of order unity for Nf >_ 7, because the effects of quark loops become essential when mq N 1/a. To substantiate the conjecture we performed simulations for Nf = 12 at Q = 0.0, 2.0, 4.0 and 4.5 . We found that the twice critical quark mass 2mgritiaat are roughly 0.5, 0.7,1 .1, and 1.2 in unit of the 1/a for ,Q = 0.0, 2.0, 4.0 and 4.5, respectively. Thus the critical quark mass is indeed of order unity as conjectured. Note that it increases with increasing Q. We have also made the calculation for Nf = 7 at Q = 4.5. The result is that 2mqrittcal is roughly 1.0 as in accord with the conjecture . If our conjecture is correct, the coefficient ccritical limit (Q) is not zero in the continuum (a -+ 0). This means that the critical quark mass is oo. Then in the continuum limit quarks are not confined for any finite mass for Nf >_ 7. We have investigated the case where all the quarks have the same masses. The problem to
313
be considered is what happens in the case where each quark has a different mass. As discussed above the essential ingredient which determines the phase of a quark system is whether the quark mass is small compared with the 1/a. Thus it can be safely assumed that when each quark mass is different, crucial is the number of flavor ßíf which is light compared with the 1/a. In the continuum limit all the finite quark masses are much smaller than 1/a and therefore 1Kf equals to the number of flavors. Thus we conclude, assuming that only QCD has the responsibility for quark confinement, that even in the general case where each quark has a different mass, the critical flavor number is 6 above which quarks are not confined. In nature quarks are confined and therefore it implies that the number of flavors does not exceed 6. It should be noted that the decoupling theorem for heavy particles cannot be applied in the usual sense for Nf >_ 7, because the theory for Nf >_ 7 itself is completely different from that for Nf < 6. The numerical calculations have been performed with HITAC 5820/80 at KEK and with QCDPAX . We would like to thank members of KEK for their warm hospitality and strong support and other members of QCDPAX collaboration for their help. We also would like to thank Sinya Aoki and Akira Ukawa for valuable discussions. This project is in part supported by the Grant-in-Aid of Ministry of Education, Science and Culture (No .62060001 and No .02402003) .
References [1] Y. Iwasaki, K. Kanaya, S. Sakai and T. Yoshié. Phys. Rev. Lett. 67 (1991) 1494. [2] S. Gottlieb et at. . Phys. Rev. D35 (1987) 2531. [3] Y. Iwasaki, K. Kanaya, S. Sakai and T. Yoshié . Preprint of Tsukuba UTHEP-226 and in preparation .
QUARK CONFINEMENT AND NUMBER OF FLAVORS Y. Iwasaki, K . Kanaya, S. Sakar, T. Yoshi6 Institute of Physics, University of Tsukuba, Ibaraki 305, Japan Constraint on the number of flavors for quark confinement and spontaneous breakdown of chiral symmetry to take place in QCD is investigated with Wilson fermions . It is shown that even in the strong coupling limit, when the number of flavors exceeds 7, quarks are not confined and chiral symmetry is not spontaneously broken for light quarks . We make a conjecture for what is "light quark" at finite Q and give numerical results which support the conjecture. Theses results imply that quarks are not confined for NJ > 7 in the continuum limit.
We report the results of a numerical investigation on the problem whether there is constraint on the number of flavors for quark confinement and spontaneous breakdown of chiral symmetry using Wilson quark action . We employ the same method as in our previous paper[1] to discriminate the phases of QCD with a various number of flavors: With the quark mass defined through the axial-vector current Ward identity, the value of the pion mass at zero quark mass determines whether chiral symmetry is spontaneously broken or not. It will turn out that confinement is closely related with the spontaneous breakdown of chiral symmetry. We generate gauge configurations using the hybrid-molecular-dynamics R algorithm[2] . The inversion of the quark matrix (x = D-1 b) is made by the minimal residual method or the conjugate gradient (CG) method. The lattice size is 82X 10 x T (T = 4, 6, or 8) and 182 x 24 x T (T=18) . When the hadron spectrum is calculated in the former case, the lattice is duplicated in the direction of the lattice size 10, which we call the z direction. We use an anti-periodic boundary condition for quarks in the t direction *
PRESENTED BY Y. IWASAKI
0920-5632/924$05 .00 0 1942- Elsevier Science Publishers B.V
and periodic boundary conditions otherwise. Errors of physical quantities are estimated by the jackknife method. More detailed description will be given elsewhere[3] Let us begin with the investigation of confinement in the strong coupling limit ,Q = 0. We note that there is no proof for confinement at Q = 0 in full QCD. We have found an evidence for a phase transition around K = 0.20 for Nf = 18 on a T = 4 lattice in the behavior of the plaquette W(1 x 1) and the Polyakov loop. The jump of the Polyakov loop suggests that the transition from the confining phase to the deconfining phase takes place around K = 0.20. The data of the pion mass and the quark mess indicate that for K >_ 0.20 chiral symmetry is not spontaneously broken. Thus the deconfining transition and the chiral symmetry transition occur at an identical hopping parameter . Decreasing the number of flavors from 18 to 6 with K = 0.25 fixed on the T = 4 lattice, we find that the number of iteration of CG needed for the quark matrix inversion (r., several hundreds) does not change so much down to Nf = 7 . However, it suddenly increases when Nf becomes 6 and exceeds 10,000 with Ar = 0.01 . This implies that All rights reserved.
Y Iwasaki et d/Quark confu:ement andnumber offlavors
312 03
2.5
N;=7 --0- W(1x1) (T-4)
02
15
46 W(ix1) (T-6) 0 W(ixt) (T-8)
O
W(1x1) (T-18)
01
00 1
38
4
4.2
44
46
Fig. 1. W(1 x 1) for
I
48
5
52
Nf = 7.
at K = 0.25 the phase for Nf = 16 - 7 is identical with that of Nf = 18, but is different from that of Nf = 6. Therefore we study the Nf = 7 and 6 cases in detail. The results of W(1 x 1) for Nf = 7 and those of the quark mass m q and the pion mass m,, on the T = 4, 6, 8 and 18 lattices are given in fig.1 and 2. W(1 x 1) and m,, for T = 4 have jumps between K = 0.24 and K = 0.245. W(l x 1) at K = 0.25 for T = 6, 8 and 18 are consistent with that for T = 4. These results strongly imply that the transition is not a finite temperature transition. From K = 0.20 to K = 0.24, m.., is consistent with the strong coupling result without quark loops. However, m,r 's for K > 0.245 are completely different from the strong coupling result. The m,r 's at K = 0.25 are 1.251(1),1 .136(3), 1 .08(6) and 1.12(2) for T = 4, 6, 8 and 18, respectively. The m,'s for T = 6 N 18 lattices are larger than the lowest Matsubara frequencies. The Polyakov loop also has a jump between K = 0.24 and K = 0.245. Therefore we interpret the state for K > 0.245 as a deconfined state. Thus we conclude that when the quark mass is small, quarks are not confined and chiral symmetry is not spontaneously broken for Nf = 7
Fig . 2. mr and 2 mq for Nf = 7: Circles (T = 4), triangles (T = 6), boxes (T = 8) and diamonds(T = 18) . The solid lines are the strong coupling results .
at ,Q = 0. We believe that this conclusion should also apply to Nf = 8 - 18, since the Nf = 7 case is the critical case. Indeed we have confirmed the conclusion for the Nf = 12 and 18 cases. As we mentioned already, at K = 0.25, the Nf = 6 case is completely different from the case of Nf = 18 - 7. We find that the point K = 0.25 at ,Q = 0 for Nf = 6 cannot be reached from the following three paths in that the number of iterar tions exceeds 10,000 for quark matrix inversions with OT = 0.01: (1) Change Nf from 7 to 6 with ,Q = 0 and K = 0.25 fixed. (2) Increase the hopping parameter K toward 0.25 with Nf = h and ,6 = 0 fixed. (3) Decrease 6 on the chiral-limit line K,,(,6) with Nf = 6 and T = 4 fixed. This fact indicates that the point K = 0.25 at ,Q = 0 for Nf = 6 belongs to the confining phase. Now we state the major conclusion of this article: For Nf > 7 quarks are not confined and chiral symmetry is not spontaneously broken for light quarks at ,Q = 0. (Although we have not investigated the case for Nf > 18, we conjecture that these features hold for this case as well.) Combining all the results obtained, we can draw the line of the hopping parameter Kd for the de-
Y. Iwasaki et al. IQuark confinement and number offlavors
0.25
1
deconfining phase K
0.2
confining phase 0.15
0
5
10
N
f
15
20
Fig. 3. Kd versus Nf
confining transition as a function of the number of flavor ( see fig.3) . The fundamental problem which remains is what will happen in the continuum limit. A problem which should be clarified in this connection is what is "light quark" at finite # . Writing the critical quark mass at the deconfining phase transi tion as mgMti~t (,Q) = c"i'at/a (a is the lattice spacing), we conjecture that ccritica! is of order unity for Nf >_ 7, because the effects of quark loops become essential when mq N 1/a. To substantiate the conjecture we performed simulations for Nf = 12 at Q = 0.0, 2.0, 4.0 and 4.5 . We found that the twice critical quark mass 2mgritiaat are roughly 0.5, 0.7,1 .1, and 1.2 in unit of the 1/a for ,Q = 0.0, 2.0, 4.0 and 4.5, respectively. Thus the critical quark mass is indeed of order unity as conjectured. Note that it increases with increasing Q. We have also made the calculation for Nf = 7 at Q = 4.5. The result is that 2mqrittcal is roughly 1.0 as in accord with the conjecture . If our conjecture is correct, the coefficient ccritical limit (Q) is not zero in the continuum (a -+ 0). This means that the critical quark mass is oo. Then in the continuum limit quarks are not confined for any finite mass for Nf >_ 7. We have investigated the case where all the quarks have the same masses. The problem to
313
be considered is what happens in the case where each quark has a different mass. As discussed above the essential ingredient which determines the phase of a quark system is whether the quark mass is small compared with the 1/a. Thus it can be safely assumed that when each quark mass is different, crucial is the number of flavor ßíf which is light compared with the 1/a. In the continuum limit all the finite quark masses are much smaller than 1/a and therefore 1Kf equals to the number of flavors. Thus we conclude, assuming that only QCD has the responsibility for quark confinement, that even in the general case where each quark has a different mass, the critical flavor number is 6 above which quarks are not confined. In nature quarks are confined and therefore it implies that the number of flavors does not exceed 6. It should be noted that the decoupling theorem for heavy particles cannot be applied in the usual sense for Nf >_ 7, because the theory for Nf >_ 7 itself is completely different from that for Nf < 6. The numerical calculations have been performed with HITAC 5820/80 at KEK and with QCDPAX . We would like to thank members of KEK for their warm hospitality and strong support and other members of QCDPAX collaboration for their help. We also would like to thank Sinya Aoki and Akira Ukawa for valuable discussions. This project is in part supported by the Grant-in-Aid of Ministry of Education, Science and Culture (No .62060001 and No .02402003) .
References [1] Y. Iwasaki, K. Kanaya, S. Sakai and T. Yoshié. Phys. Rev. Lett. 67 (1991) 1494. [2] S. Gottlieb et at. . Phys. Rev. D35 (1987) 2531. [3] Y. Iwasaki, K. Kanaya, S. Sakai and T. Yoshié . Preprint of Tsukuba UTHEP-226 and in preparation .