QCD Scales in Finite Nuclei

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QCD Scales in Finite Nuclei

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QCD Scales in Finite Nuclei
J. L. Friar and D. G. Madland
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545
B. W. Lynn
Clarendon Laboratories, Oxford University, Oxford OX1 3PU, Great Britain
Abstract
The role of QCD scales and chiral symmetry in nite nuclei is examined. The Dirac-Hartree mean- eld coupling constants of Nikolaus, Hoch, and Madland (NHM) are scaled in accordance with the QCD-based prescription of Manohar and Georgi. Whereas the nine empirically-based coupling constants of NHM span thirteen orders of magnitude, the scaled coupling constants are almost all natural, being dimensionless numbers of order one. We speculate that this result provides good evidence that QCD and chiral symmetry apply to nite nuclei.
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Although QCD is widely believed to be the underlying theory of the strong interaction, a direct description of nuclear properties in terms of the natural degrees of freedom of that theory, quarks and gluons, has proven elusive. The problem is that at suciently low energy, the physical degrees of freedom of nuclei are nucleons and (intranuclear) pions. Nevertheless, QCD can be mapped onto the latter Hilbert space and the resulting e ective eld theory is capable in principle of providing a dynamical framework for nuclear calculations. This framework is usually called chiral perturbation theory (PT).
Two organizing principles govern this PT: (1) (broken) chiral symmetry (which is manifest in QCD) and (2) an expansion in powers of (Q=), where Q is a general intranuclear
momentum or pion mass, and  is the generic QCD large-mass scale 1 GeV, which in a
loose sense indicates the transition region between the two alternative sets of degrees of freedom indicated above (that is, quark-gluon versus nucleon-pion). Typically, one constructs Lagrangians (that is, interactions) that display (broken) chiral symmetry and retains only those terms with exponents less than or equal to some xed power of (1/). The chiral
symmetry itself provides a crucial constraint: a general term has the structure  (Q=)N and N  0 is mandated. This guarantees that higher-order constructions in perturbation
theory (viz., loops) will have even higher (not lower) powers of (Q=). The price one pays for this mapping from natural to e ective degrees of freedom is an in nite series of interaction terms, where coecients are unknown and must be determined from experiment.
To date only a few nuclear calculations have been performed within this framework. The seminal work of Weinberg [1] highlighted the role of power counting and chiral symmetry in weakening N-body forces. That is, two-nucleon forces are stronger than three-nucleon
forces, which are stronger than four-nucleon forces,    . This chain makes nuclear physics
tractable. Van Kolck and collaborators [2] developed a nuclear potential model, including one-loop (two-pion exchange) contributions. Friar and Coon [3] developed non-adiabatic two-pion-exchange forces, while van Kolck, Friar and Goldman [4] examined isospin violation in the nuclear force. Rho, Park, and Min [5] were the rst to treat external electromagnetic and weak interactions with nuclei. Essentially all of this work was focused on few-nucleon
3

systems, where computational techniques are sophisticated. Only the work of Lynn [6] on

(nuclear) chiral liquids was speci cally directed at heavier nuclei and, more recently, Gelmini

and Ritzi [7] have calculated nuclear matter properties using lowest order nonlinear chiral

e ective Lagrangians.

Is there any evidence for chiral symmetry or QCD scales in nite nuclei? The tractability and astonishing success of the recent few-nucleon calculations of 2H, 3H, 3He, 4He, 5He, 6He, 6Li, and 6Be with only a weak three-nucleon force and no four-nucleon force con rms

Weinberg's power-counting prediction [1] and yields strong but indirect evidence for chiral

symmetry. The work of Lynn [6] established a procedure for going beyond few-nucleon

systems. Nuclear (N-body) forces either have zero range or are generated by pion exchange.

Following Manohar and Georgi [8] we can scale a generic Lagrangian component as

L

clmn



f

2



l



~ f

m



@



;m 





n

f2

2

(1)



where and ~ are nucleon and pion elds, respectively, f and m are the pion decay
constant, 92.5 MeV, and pion mass, 139.6 MeV, respectively,   1 GeV has been discussed
above, and (@, m) signi es either a derivative or a power of the pion mass. Dirac matrices and isospin operators (we use ~t here rather than ~) have been ignored. Chiral symmetry

demands [9]

=l+n 20:

(2)

Thus the series contains only positive powers of (1/). If the theory is natural [6,8], the dimensionless coecients clmn are of order (1). Thus, all information on scales ultimately resides in the clmn. If they are natural, scaling works. Our limited experience with nuclearforce models suggests that natural coecients are the rule.
Unfortunately, zero-range nuclear-force models are not widely used. However, a recent calculation has been performed using zero-range forces for an extended range of mass number A and this work provides signi cant new information on QCD and chiral symmetry in nuclei. Nikolaus, Hoch, and Madland (NHM) [10] used a series of zero-range interactions to perform
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Dirac-Hartree calculations in mean- eld approximation for a total of fty-seven nuclei. Their Lagrangian [using their notation] is given by

L = Lfree + L4f + Lhot + Lder + Lem ;

(3)

where Lfree and Lem are the kinetic and electromagnetic terms, respectively, and

L4f = 12 S(  )(  ) 21 V (   )(   )

12 TS( ~ )( ~ ) 21 TV ( ~  )( ~  ) ;

(4)

Lhot =

13 S(  )3

1 4

S( 

)4

1 4

V [( 



)( 



)]2 ;

and

(5)

Lder =

1 2

S

(@



)(@ 

)

21V (@   )(@   ) :

(6)

In these equations, is the nucleon eld, the subscripts S and V refer to the isoscalar-scalar and isoscalar-vector densities, respectively, and the subscripts T S and T V refer to the isovector-scalar and isovector-vector densities, respectively, containing the nucleon isospin operator ~. The nine coupling constants of the NHM Lagrangian were determined in a self-consistent procedure that solved the model equations for several nuclei simultaneously

in a nonlinear least-squares adjustment algorithm with respect to measured ground-state observables (Table IV of Ref. [10]). The predictive power of the extracted coupling constants

is quite good both for other nite nuclei and for the properties of saturated nuclear matter

(see Tables VIII, IX, and XI of Ref. [10]).
L4f contains four two{nucleon{force terms corresponding to  = 0, the rst term of Lhot
is a three{nucleon{force term corresponding to  = 1, whereas the remaining two terms are
four{nucleon{force terms corresponding to  = 2. Finally, Lder contains two nonlocal two{
nucleon{force terms, also corresponding to  = 2. The derivative terms act on  , rather
than on just one of the elds, because the latter generate a factor E = M, the nucleon mass,
whereas the former generate an energy di erence that is considerably smaller. The latter

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terms would spoil the series in Eq. (1) since M = . However, either by a transformation
or by rearranging the series, this problem could in principle be eliminated [6]. The construction of the NHM Lagrangian was motivated by empirically-based improve-
ments to a Walecka type scalar-vector model [11,12], but using contact (zero-range) interactions to allow treatment of the Fock (exchange) terms. It was not motivated either by power counting or by chiral symmetry. The pion degrees of freedom are ignored and the Lagrangian is not complete; additional operators in each order of (1/) are possible. Speci cally, the NHM Lagrangian, Eqs. (4){(6), has four operators in order (1/)0, one operator in order (1/)1, and four operators in order (1/)2, constituting an incomplete mix of three di erent orders in (1/).
Nevertheless, a meaningful comparison can be made of the generic chiral Lagrangian given by Eqs.(1) and (2) and the NHM Lagrangian given by Eqs.(4){(6), precisely because our test of naturalness does not care whether a speci c clmn coecient is 0.5 or 2.0. Changing (re ning) the model by adding terms would change all of the clmn, but the same test of naturalness still applies. Adding new terms would simply change a speci c coecient by an
amount  1 (or less).
The nine coupling constants of the NHM Lagrangian are shown in Table 1, both in dimensional and dimensionless form [the latter obtained by equating Eqs.(1) and (4){(6), with  = 1 GeV, using isospin operators ~t in Eq.(1), and solving for clmn in terms of , , , and ]. In the former form they span more than thirteen orders of magnitude, while in the latter form six of the nine coupling constants can be regarded as natural. Only the very small TS and large S and V are unnatural. However, the sum of the latter appears to be natural, and we speculate that the di erence may not be well determined in the leastsquares adjustments to the measured observables. The unnaturally small TS, if correct, would presuppose a symmetry to preserve its small value.
Although these results were not obtained as a test of chiral symmetry and QCD scales (NHM at that time were unaware of these developments) and hence are imperfect, they are conversely completely unbiased. This result is very indicative of the role of chiral symmetry
6

and QCD in nite nuclei. A systematic study of this approach is clearly indicated. 7

REFERENCES [1] S. Weinberg, Phys. Lett. 251B, 288 (1990). [2] C. Ordon~ez and U. van Kolck, Phys. Lett. B291, 459 (1992); C. Ordon~ez, L. Ray, and
U. van Kolck, Phys. Rev. Lett. 72, 1982 (1994). [3] J. L. Friar and S. A. Coon, Phys. Rev. C 49, 1272 (1994); S. A. Coon and J. L. Friar,
Phys. Rev. C 34, 1060 (1986).
[4] U. van Kolck, J. L. Friar, and T. Goldman, Phys. Lett. (submitted).
[5] T.-S. Park, D.-P. Min, and M. Rho, Phys. Rev. Lett. 74, 4153 (1995); M. Rho, Phys. Rev. Lett. 66, 1275 (1991); T.-S. Park, D.-P. Min, and M. Rho, Phys. Rep. 233, 341
(1993).
[6] B. W. Lynn, Nucl. Phys. B402, 281 (1993). [7] G. R. Gelmini and B. Ritzi, Phys. Lett. B 357, 431 (1995). [8] A. Manohar and H. Georgi, Nucl. Phys. B234, 189 (1984). [9] S. Weinberg, Physica 96A, 327 (1979). [10] B. A. Nikolaus, T. Hoch, and D. G. Madland, Phys. Rev. C 46, 1757 (1992). [11] B. D. Serot and J. D. Walecka, Phys. Lett. 87B, 172 (1979). [12] C. J. Horowitz and B. D. Serot, Nucl. Phys. A368, 503 (1981).
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TABLES TABLE I. Optimized Coupling Constants for the NHM Lagrangian and Corresponding Dimen-

sional Power Counting Coecients and Chiral Expansion Order

Coup. Const.

Magnitude

Dimension

clmn

Order

S

-4.50810 4

MeV 2

-1.93

0

T S

7.40310 7

MeV 2

0.013

0

V

3.42710 4

MeV 2

1.47

0

T V

3.25710 5

MeV 2

0.56

0

S

1.11010 11

MeV 5

0.27

1

S

5.73510 17

MeV 8

8.98

2

V

-4.38910 17

MeV 8

-6.87

2

S

-4.23910 10

MeV 4

-1.81

2

V

-1.14410 10

MeV 4

-0.49

2

9
PhysChiral SymmetryLettConstantsTerms