- Email: [email protected]
Path integral formalism for a simple interacting nucleon model
I~Lgiqlll NP'_4Li mlN|~"i[6,1 |!
PROCEEDINGS SUPPLEMEHTS
Nuclear Physics B (Prec. Suppl.) 30 (1993) 944-948 North-Holland
Path integral formalism for a simple interacting nucleon model E. Mendel a aFB-8 Physik, Universit/it Oldenburg 2900 Oldenburg, Germany The early onset of the baryon density in QCD simulations can be explained by the high flavour degeneracy when using staggered fermions. A simple interacting nucleon gas model had already shown that the gas condenses at very low chemical potential as in the lattice simulations at four fiavours. In order to study more carefully the nucleon gas model in the condensation region we have developed the path integral formalism to treat the first quantization non perturbatively, describing the partition function for the interacting system of nucleons. First Monte Carlo results show good agreement with the lattice QCD simulations for the onset chemical potentials and saturation densities. The extrapolation to nature gives reasonable results.
1. I N T R O D U C T I O N In order to study hadronic matter at finite baryon density from first principles in QCD, a properly lattice-regularized [1, 2] chemical potential coupled to the fermion number operator had been introduced several years ago. For the free quark density one was able to find the correct continuum limit (corresponding to four flavours, as we all have used staggered fermions). In the interacting QCD case we expected the quarks to bind into nucleons in the normal confining phase while having a transition to a quarkgluon plasma phase at high densities. Furthermore, based on a free nucleon gas model, we expected at low temperatures that the chemical potential # at which the baryon density would have its onset would be close to raN~3. Even in the interacting case, in which nuclear matter is produced in nature, the binding energy is small and one expects only a slight decrease in the p needed for the onset. In contrast to these expectations, several groups working with staggered fermions have seen [3-5] that the onset of the baryon density as found on the lattice simulations is at much lower chemical potentials. Several proposals had been made as to the reason for this early onset, including the coarsness of the lattice [6-8] or the quenching of the complex fermion determinant [9, 7] with as yet no conclusive answer. Recently I presented a possible solution [10] to the early onset problem based on the consid-
eration of an interactive nucleon gas model. I had shown, perturbatively in quantum mechanics, that due to the use of staggered fermions which contain 4 degenerate flavours instead of the 2 in nature one obtains an onset at a much lower /l than naively expected and compatible with our quenched lattice results. There are two factors contributing to this early onset: the Kogut-Susskind quarks winding around the lattice can produce 40 lightest nucleons (from the 20 representation of SU(4)) instead of the usual 4, thus weakening the effective repulsion due to nuclear Fermi statistics. Furthermore, the attractive scalar Yukawa interaction is twice as strong as in nature due to the increase in light flavour channels. The grand canonical partition function for this nuclear model had been solved using just first order perturbation theory in quantum mechanics, which made it only valid in the gas phase up to nuclear condensation, giving a good estimate for the onset point. To further study the model at the condensation transition and above we need to treat the model nonperturbatively in quantum mechanics, for which I will use the path integral formalism. Our aim is then to solve the grand canonical partition function, Z(#, T), for a system of interacting nucleons (these come in D - degenerate classes of identical fermions). The simple interacting nucleon model consists of an attractive scalar Yukawa potential (controlled by mo as in the Nambu-Jona-Lasinio model) and a repulsive hard core potential (controlled by rn~).
0920-5632/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved.
E. Mendel ~Path integralformalism for a simple btteractbtgtmcleon model In the next section I want to describe the new features needed for the path integral formalism. In sec. 3 the Monte Carlo results for the model, as compared to the lattice QCD simulations, will be presented. In sec. 4 the extrapolation of the model to the parameters in nature will be discussed.
945
< E N > /N
3 mu+TT
2. N E W F E A T U R E S IN T H E P A T H INTEGRAL FORMALISM Our aim is to calculate the grand canonical partition function for a system of interacting nucleons. From it, one can obtain the desired expectation values for the particle number < N >Z,, or the energy < E >~,~ as:
=
=
~1
,lp 0 N
Figure i. Idealized energy per nucleon versus particle number in a fixed volume.
~ N / £ ) X 1. .DXN N e - S N + ~ N
1
~ E
N
< z > :1~
No
N Z N e z"N
-
<<~> Z N e z"~
10lnZ
¢~ 0 ~
(1)
(2)
N where SN and ZN are the action and canonical partition function fot a set of N particles. To evaluate directly < N > from the grand canonical partition function one would have to jump from one particle number to another while doing the Monte Carlo iterations. This is hard as we would have to add or substract full fermion world lines with a large change in the action. Furthermore, the relative normalization of the ZN is still not quite under control. As we are mainly interested in finding the onset chetnicM potential #o and the density < n >o to which the system jumps at the onset at relatively low temperatures, we can use as a first step a method which gives the exact result at zero T (as then Zjv --+ aN e x p ( - ~ E ° ) ) . It consists in finding the minimum for the curve < EN > / N as function of N (due to the attractive interaction the minimum can be at a nonzero density). For a chemical potential #o equal to that minimum the system has its onset by condensing to the density at the minimum < r) >o, as indicated in fig. (1).
We want to simulate then systems of N nucleons (in a fixed periodic volume) with a path integral over trajectories with the Monte Carlo method. As the fermions come in various degenerate classes of identical ones (D = 4 in nature and D = 40 for staggered fermions on the lattice) we have to consider the determinant, det(f~,(t ) ~At+,)), of all the propagators of identical fermions within the class k. In principle we would only need to take the det at one time slice but it is known [11] that the simulation converges faster when the det is taken at each time slice. A high fermion degeneracy actually makes the simulation easier as we have to consider only smaller determinants corresponding to the number of fermions in each class. To perform the Monte Carlo update at a position i (by shifting a point in fig. (2)), where that position corresponds to particles in a class k, we need only to consider the change in the action due
to: z(i,t)
o(
det(f.k (t - r ) , :(. )t ) det(f~,(t) k x/t+r))
e-r ~ , , , v(z/t)-z,(t))
(3)
where the potential V acting at the time t will be described later. One usually uses the nonrelativistic free propagator between two time slices, obtained by Feyn-
946
E. Mendel/Padz integralformalism for a simple interactingnucleon model
2
3
The nucleon interaction consists, in our simple model, of an attractive scalar Yukawa potential - U y e x p ( - m ~ r ) / r with mo assumed as in the Nambu - Jona-Lasinio to be -~mg and a hard core potential, with a hard core radius R = a/mw. This hard core potential can be smoothed in the future.
3
l
I
3. C O M P A R I S O N OF LATTICE RESULTS
periodic L Figure 2. Sample paths for four fermions, two of t h e m identical ones in class 3. man as the discretization of the K.E.. In our case the m o m e n t a involved can get relativistic so that we will assume that the nucleons propagate with the free relativistic dispersion relation between two time slices. The resulting propagator is: Ygt4T K 2 ( T T I 4 ( X i
L,(,)xj(,+r)
-
- - X j ) 2 -t- T 2) - x )2 + • (4)
This propagator does not allow creation of antiparticles. It obeys the composition rule for Green functions, f:cy = fdYf~:yfyz and has the correct N.R. limit. The propagator f and its time derivative are tabulated once by the program. The expectation value for the energy for a given N can be calculated then by the average over Monte Carlo path configurations of
o
!t
(5)
0Z
=6v=60=305.=5
Do
~.
LEGENO
o" .
< E >=
WITH
We want to compare now the model solved with the Monte Carlo method, with our earlier results [10] obtained on the lattice for quenched QCD with staggered fermions (as well as with the model solved perturbatively). The lattice simulations had been done at 3 values for the quark mass (mq = .01, .04 and .09) at gauge coupling 6.0 on a 164 lattice. The needed nucleon masses m g and m~ where obtained consistently on the same lattice, while for the scalar a channel we took -~mN. 2 The mass values used can be found in Table 1 of ref.[10].
o"
0 lnZ
MODEL
N
'" . . . . . . . . . . . . . . . . . . . . . . . .
,:.o
,,~.o
\ I
m m =.Og,m,=l.38
~
• ~'=:o1~,'.f3
k.P
o rn% 04 m =1 0
~itt,
zo.o
S$oothinQ Curves
3;.~
,&o
.~.o
~.o
~G.o
t6.~
,o.o
N
= _ 0
0~
ln(H
det(f~)) e - r E , E , < j v,j(t))
t,k
Figure 3. Nuclear model results for same parameters as lattice QCD simulations.
7"
=-
1 0 det(f~)+ ZE~J(t)" Zt,k det(f~)) 0fl 5 t i
The first t e r m in this last expression corresponds to the K . E . while the second to the P.E.
We have then two free parameters in our model, namely, the Yukawa coupling Uw and the scale parameter for the hard core radius a. Choosing
E. Mendel~Path hltegral fonnalism for a simple fllteracthlg nucleon model
these parameters as Uy = 30.5 and a = 1.5 we can fit the lattice onset chemical potential #o and condensation density no fairly well for the three mq presented in ref.[10]. The results presented in fig. (3) show, as in the perturbative solution of the model, a strong binding of the nucleons for this 4 flavour case in agreement with our staggered fermions lattice result. The factor 1.5 for is quite reasonable when considering that in nuclear models, as the Skyrmion one, the w couples to the the baryon density of the nucleon which is an extended object with a radius which scales again as one over the masses. 4. E X T R A P O L A T I O N
o d Extropolotion to noture
VT=60,V=20=,Ur=-15.25,a=1.5
I
o o ,% ~'.c s'.: ,I.o g.o ,'.o i.o
TO NATURE
It is important to check if the model gives reasonable results when extrapolated to parameters in nature. Now we have only two light flavour channels to produce the scalar interaction so that we have to take U~, = 30.5/2, which gives a plausible value (remember that for normal 7r, U~5 14). The nucleon statistics also changes as we have now only 4 degenerate nucleon species. Furthermore we extrapolate to lower m a s s e s (row = .57 in lattice units) as in nature. The temperature is also lowered significantly and the volume increased so that we get into the notorious minus sign problem for fermion paths. We have to consider then < O > = (~-'~n+ O+ -- ~ n - O - ) / A where O + - is the measured operator when the action is pos.(neg.) and A = (n + - n - ) . This quantity fluctuates strongly when A is small, giving large errors for < O >, but one can improve the measurement significantly by using the weighted average over results = E, < o / A;. The results for one temperature are shown in fig. (4). We notice that now the minimal energy per nucleon is only sligthly lower than m N indicating only a weak binding as found in nature. We are still computing for larger volumes and lower temperatures and we have seen a tendency for no to shift to lower densities, perhaps consistent in the thermodynamic limit with nuclear matter. The model seems then to give a consistent explanation for the results obtained with staggered fermions on the lattice. The ultimate check will be a calculation done with Wilson fermions. I
947
a'.o Lo ,Lo d.o ,~.o ,i.o d.o ,~.: N
Figure 4. Same model extapolated to nature.
would like to acknowledge helpful ideas from G. Nolte and B. Kleihaus and the usage of the GSI and Hannover supercomputers.
REFERENCES
1
J. Kogut, H. Matsuoka, M. Stone, H.W. Wyld, S. Shenker, J. Shigemitsu and D.K. Sinclair, Nucl. Phys. B225 (1983) 93. 2 P. Hasenfratz and F. Karsch, Phys. Lett. 125B (1983) 308. 3 I. Barbour, N. Behilil, E. Dagotto, F. Karsch, A. Moreo, M. Stone and H.W. Wyld, Nucl. Phys. B275 (1986) 296. 4 E. Mendel, Nucl. Phys. B4 (Proc. Suppl.) (1988) 308. 5 C. Davies and E. Klepfish, Phys. Lett. B256 (1991) 68. 6 N. BilK and K. Demeterfi, Phys. Lett. 212B (1988) 83. 7 E. Mendel, Nucl. Phys. B9 (Proc. Suppl.) (1989) 347. 8 J. Vink, Nucl. Phys. B323 (1989) 399. 9 I. Barbour, Nuel. Phys. B17 (Proc. Suppl.) (1990) 243. 10 E. Mendel, Nucl. Phys. B387 (1992) 485; Nucl. Phys. B20 (Proc. Suppl.) (1991) 343. l l M. Takahashi and M. Imada, Jour. of the Phys. Soc. of Japan 53 (1984) 963.
PROCEEDINGS SUPPLEMEHTS
Nuclear Physics B (Prec. Suppl.) 30 (1993) 944-948 North-Holland
Path integral formalism for a simple interacting nucleon model E. Mendel a aFB-8 Physik, Universit/it Oldenburg 2900 Oldenburg, Germany The early onset of the baryon density in QCD simulations can be explained by the high flavour degeneracy when using staggered fermions. A simple interacting nucleon gas model had already shown that the gas condenses at very low chemical potential as in the lattice simulations at four fiavours. In order to study more carefully the nucleon gas model in the condensation region we have developed the path integral formalism to treat the first quantization non perturbatively, describing the partition function for the interacting system of nucleons. First Monte Carlo results show good agreement with the lattice QCD simulations for the onset chemical potentials and saturation densities. The extrapolation to nature gives reasonable results.
1. I N T R O D U C T I O N In order to study hadronic matter at finite baryon density from first principles in QCD, a properly lattice-regularized [1, 2] chemical potential coupled to the fermion number operator had been introduced several years ago. For the free quark density one was able to find the correct continuum limit (corresponding to four flavours, as we all have used staggered fermions). In the interacting QCD case we expected the quarks to bind into nucleons in the normal confining phase while having a transition to a quarkgluon plasma phase at high densities. Furthermore, based on a free nucleon gas model, we expected at low temperatures that the chemical potential # at which the baryon density would have its onset would be close to raN~3. Even in the interacting case, in which nuclear matter is produced in nature, the binding energy is small and one expects only a slight decrease in the p needed for the onset. In contrast to these expectations, several groups working with staggered fermions have seen [3-5] that the onset of the baryon density as found on the lattice simulations is at much lower chemical potentials. Several proposals had been made as to the reason for this early onset, including the coarsness of the lattice [6-8] or the quenching of the complex fermion determinant [9, 7] with as yet no conclusive answer. Recently I presented a possible solution [10] to the early onset problem based on the consid-
eration of an interactive nucleon gas model. I had shown, perturbatively in quantum mechanics, that due to the use of staggered fermions which contain 4 degenerate flavours instead of the 2 in nature one obtains an onset at a much lower /l than naively expected and compatible with our quenched lattice results. There are two factors contributing to this early onset: the Kogut-Susskind quarks winding around the lattice can produce 40 lightest nucleons (from the 20 representation of SU(4)) instead of the usual 4, thus weakening the effective repulsion due to nuclear Fermi statistics. Furthermore, the attractive scalar Yukawa interaction is twice as strong as in nature due to the increase in light flavour channels. The grand canonical partition function for this nuclear model had been solved using just first order perturbation theory in quantum mechanics, which made it only valid in the gas phase up to nuclear condensation, giving a good estimate for the onset point. To further study the model at the condensation transition and above we need to treat the model nonperturbatively in quantum mechanics, for which I will use the path integral formalism. Our aim is then to solve the grand canonical partition function, Z(#, T), for a system of interacting nucleons (these come in D - degenerate classes of identical fermions). The simple interacting nucleon model consists of an attractive scalar Yukawa potential (controlled by mo as in the Nambu-Jona-Lasinio model) and a repulsive hard core potential (controlled by rn~).
0920-5632/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved.
E. Mendel ~Path integralformalism for a simple btteractbtgtmcleon model In the next section I want to describe the new features needed for the path integral formalism. In sec. 3 the Monte Carlo results for the model, as compared to the lattice QCD simulations, will be presented. In sec. 4 the extrapolation of the model to the parameters in nature will be discussed.
945
< E N > /N
3 mu+TT
2. N E W F E A T U R E S IN T H E P A T H INTEGRAL FORMALISM Our aim is to calculate the grand canonical partition function for a system of interacting nucleons. From it, one can obtain the desired expectation values for the particle number < N >Z,, or the energy < E >~,~ as:
=
=
~1
,lp 0 N
Figure i. Idealized energy per nucleon versus particle number in a fixed volume.
~ N / £ ) X 1. .DXN N e - S N + ~ N
1
~ E
N
< z > :1~
No
N Z N e z"N
-
<<~> Z N e z"~
10lnZ
¢~ 0 ~
(1)
(2)
N where SN and ZN are the action and canonical partition function fot a set of N particles. To evaluate directly < N > from the grand canonical partition function one would have to jump from one particle number to another while doing the Monte Carlo iterations. This is hard as we would have to add or substract full fermion world lines with a large change in the action. Furthermore, the relative normalization of the ZN is still not quite under control. As we are mainly interested in finding the onset chetnicM potential #o and the density < n >o to which the system jumps at the onset at relatively low temperatures, we can use as a first step a method which gives the exact result at zero T (as then Zjv --+ aN e x p ( - ~ E ° ) ) . It consists in finding the minimum for the curve < EN > / N as function of N (due to the attractive interaction the minimum can be at a nonzero density). For a chemical potential #o equal to that minimum the system has its onset by condensing to the density at the minimum < r) >o, as indicated in fig. (1).
We want to simulate then systems of N nucleons (in a fixed periodic volume) with a path integral over trajectories with the Monte Carlo method. As the fermions come in various degenerate classes of identical ones (D = 4 in nature and D = 40 for staggered fermions on the lattice) we have to consider the determinant, det(f~,(t ) ~At+,)), of all the propagators of identical fermions within the class k. In principle we would only need to take the det at one time slice but it is known [11] that the simulation converges faster when the det is taken at each time slice. A high fermion degeneracy actually makes the simulation easier as we have to consider only smaller determinants corresponding to the number of fermions in each class. To perform the Monte Carlo update at a position i (by shifting a point in fig. (2)), where that position corresponds to particles in a class k, we need only to consider the change in the action due
to: z(i,t)
o(
det(f.k (t - r ) , :(. )t ) det(f~,(t) k x/t+r))
e-r ~ , , , v(z/t)-z,(t))
(3)
where the potential V acting at the time t will be described later. One usually uses the nonrelativistic free propagator between two time slices, obtained by Feyn-
946
E. Mendel/Padz integralformalism for a simple interactingnucleon model
2
3
The nucleon interaction consists, in our simple model, of an attractive scalar Yukawa potential - U y e x p ( - m ~ r ) / r with mo assumed as in the Nambu - Jona-Lasinio to be -~mg and a hard core potential, with a hard core radius R = a/mw. This hard core potential can be smoothed in the future.
3
l
I
3. C O M P A R I S O N OF LATTICE RESULTS
periodic L Figure 2. Sample paths for four fermions, two of t h e m identical ones in class 3. man as the discretization of the K.E.. In our case the m o m e n t a involved can get relativistic so that we will assume that the nucleons propagate with the free relativistic dispersion relation between two time slices. The resulting propagator is: Ygt4T K 2 ( T T I 4 ( X i
L,(,)xj(,+r)
-
- - X j ) 2 -t- T 2) - x )2 + • (4)
This propagator does not allow creation of antiparticles. It obeys the composition rule for Green functions, f:cy = fdYf~:yfyz and has the correct N.R. limit. The propagator f and its time derivative are tabulated once by the program. The expectation value for the energy for a given N can be calculated then by the average over Monte Carlo path configurations of
o
!t
(5)
0Z
=6v=60=305.=5
Do
~.
LEGENO
o" .
< E >=
WITH
We want to compare now the model solved with the Monte Carlo method, with our earlier results [10] obtained on the lattice for quenched QCD with staggered fermions (as well as with the model solved perturbatively). The lattice simulations had been done at 3 values for the quark mass (mq = .01, .04 and .09) at gauge coupling 6.0 on a 164 lattice. The needed nucleon masses m g and m~ where obtained consistently on the same lattice, while for the scalar a channel we took -~mN. 2 The mass values used can be found in Table 1 of ref.[10].
o"
0 lnZ
MODEL
N
'" . . . . . . . . . . . . . . . . . . . . . . . .
,:.o
,,~.o
\ I
m m =.Og,m,=l.38
~
• ~'=:o1~,'.f3
k.P
o rn% 04 m =1 0
~itt,
zo.o
S$oothinQ Curves
3;.~
,&o
.~.o
~.o
~G.o
t6.~
,o.o
N
= _ 0
0~
ln(H
det(f~)) e - r E , E , < j v,j(t))
t,k
Figure 3. Nuclear model results for same parameters as lattice QCD simulations.
7"
=-
1 0 det(f~)+ ZE~J(t)" Zt,k det(f~)) 0fl 5 t i
The first t e r m in this last expression corresponds to the K . E . while the second to the P.E.
We have then two free parameters in our model, namely, the Yukawa coupling Uw and the scale parameter for the hard core radius a. Choosing
E. Mendel~Path hltegral fonnalism for a simple fllteracthlg nucleon model
these parameters as Uy = 30.5 and a = 1.5 we can fit the lattice onset chemical potential #o and condensation density no fairly well for the three mq presented in ref.[10]. The results presented in fig. (3) show, as in the perturbative solution of the model, a strong binding of the nucleons for this 4 flavour case in agreement with our staggered fermions lattice result. The factor 1.5 for is quite reasonable when considering that in nuclear models, as the Skyrmion one, the w couples to the the baryon density of the nucleon which is an extended object with a radius which scales again as one over the masses. 4. E X T R A P O L A T I O N
o d Extropolotion to noture
VT=60,V=20=,Ur=-15.25,a=1.5
I
o o ,% ~'.c s'.: ,I.o g.o ,'.o i.o
TO NATURE
It is important to check if the model gives reasonable results when extrapolated to parameters in nature. Now we have only two light flavour channels to produce the scalar interaction so that we have to take U~, = 30.5/2, which gives a plausible value (remember that for normal 7r, U~5 14). The nucleon statistics also changes as we have now only 4 degenerate nucleon species. Furthermore we extrapolate to lower m a s s e s (row = .57 in lattice units) as in nature. The temperature is also lowered significantly and the volume increased so that we get into the notorious minus sign problem for fermion paths. We have to consider then < O > = (~-'~n+ O+ -- ~ n - O - ) / A where O + - is the measured operator when the action is pos.(neg.) and A = (n + - n - ) . This quantity fluctuates strongly when A is small, giving large errors for < O >, but one can improve the measurement significantly by using the weighted average over results = E, < o / A;. The results for one temperature are shown in fig. (4). We notice that now the minimal energy per nucleon is only sligthly lower than m N indicating only a weak binding as found in nature. We are still computing for larger volumes and lower temperatures and we have seen a tendency for no to shift to lower densities, perhaps consistent in the thermodynamic limit with nuclear matter. The model seems then to give a consistent explanation for the results obtained with staggered fermions on the lattice. The ultimate check will be a calculation done with Wilson fermions. I
947
a'.o Lo ,Lo d.o ,~.o ,i.o d.o ,~.: N
Figure 4. Same model extapolated to nature.
would like to acknowledge helpful ideas from G. Nolte and B. Kleihaus and the usage of the GSI and Hannover supercomputers.
REFERENCES
1
J. Kogut, H. Matsuoka, M. Stone, H.W. Wyld, S. Shenker, J. Shigemitsu and D.K. Sinclair, Nucl. Phys. B225 (1983) 93. 2 P. Hasenfratz and F. Karsch, Phys. Lett. 125B (1983) 308. 3 I. Barbour, N. Behilil, E. Dagotto, F. Karsch, A. Moreo, M. Stone and H.W. Wyld, Nucl. Phys. B275 (1986) 296. 4 E. Mendel, Nucl. Phys. B4 (Proc. Suppl.) (1988) 308. 5 C. Davies and E. Klepfish, Phys. Lett. B256 (1991) 68. 6 N. BilK and K. Demeterfi, Phys. Lett. 212B (1988) 83. 7 E. Mendel, Nucl. Phys. B9 (Proc. Suppl.) (1989) 347. 8 J. Vink, Nucl. Phys. B323 (1989) 399. 9 I. Barbour, Nuel. Phys. B17 (Proc. Suppl.) (1990) 243. 10 E. Mendel, Nucl. Phys. B387 (1992) 485; Nucl. Phys. B20 (Proc. Suppl.) (1991) 343. l l M. Takahashi and M. Imada, Jour. of the Phys. Soc. of Japan 53 (1984) 963.