Thomas-Fermi approximation to functional determinants in external gauge potentials

Author(s): Spence, CD; Bander, M | Abstract: The functional determinant and vacuum expectation value of a fermion field in an external gauge potential are evaluated in a Thomas-Fermi approximation. This approximation should be valid for slowly varying potentials. The expressions are readily transcribable to a lattice. We do not encounter the problem of doubling of fermionic degrees of freedom. © 1986 The American Physical Society.


I. INTRODUCTION
The inclusion of fermions in lattice calculations has run into two major problems.The first one is calculational in that Monte Carlo integrations over Grassmann variables cannot be performed and one is forced into approximate numerical methods' for the evaluation of inverses and determinants of functional matrices; an extreme approxi- mation to the determinant consists of setting it identically equal to one2 (quenched approximation).The other diffi- culty is that the naive continuum limit of fermions on a lattice has too many degrees of freedom.This problem has been dealt with through the introduction of a quadra- tic term into the Lagrangian for the fermions (Wilson fermions) or by appropriately staggering them on the lat- tice~ (Kogut-Susskind fermions).The first method yields an action that is not invariant under chiral transforma- tions, even in the massless fermion limit.The second technique reduces the number of fermionic degrees of freedom but does not completely eliminate the multiplication of fermionic species.These two problems could be avoided should we be able to perform the integrations over the fermions in the continuum resulting in a func- tional of the gauge potentials.These expressions could then be discretized and placed on a lattice.In this article we shall present an approximate evaluation of the func- tional determinant for fermions moving in an external gauge potential as well as the expectation value of (Pg).
The approximation we shall use is a variant of the Thomas-Fermi method in that we treat the external gauge potentials as slowly varying.Confinement and chiral- symmetry breaking are long-distance problems and thus rapid short-distance variations of the potentials are not expected to play a dominant role.For technical reasons we shall, for the present, limit ourselves to the gauge group SU(2).
The properties of functional determinants for particles of various spins and isospins moving in constant external potentials were previously studied by Brown and Weis- berger.
Although they did discuss the situation of a spin--, ' color--, ' particle moving in such a potential, these authors did not obtain an expression for the functional determinant in a form suitable for the purposes discussed previously.For both the effective action and for (~) we will obtain renormalized expressions.
In a subsequent publication we hope to return to a Monte Carlo calcula-tion using these results.
In Sec.II the Thomas-Fermi approximation for particles moving in external gauge potentials is discussed, with special attention paid to the gauge-invariance properties of this approximation.Conclusions will be given in Sec.III along with some observations on configurations of gauge potentials that may possibly be responsible for chiral-symmetry breakdown.
The details of the evalua- tion of the effective action are presented in the Appendix.

II. THOMAS-FERMI APPROXIMATION
The contribution of a single Dirac fermion of mass m moving in an external gauge potential A to the effective action for the gauge potential is -F= x x trlni -m x +const.

2.1
We are working in Euclidean space and the coupling constant is absorbed into the definition of the potential.The gauge potentials themselves are matrix valued.The trace is over all spin and color variables.The other quantity we shall be interested in is the expectation value of Pg.This (2.2) Both of these may be obtainmi from Z(x;ri)=(x i trexp[ sl(iQ g --m)] ix-) by using the relations (2.3) F= 4x " Z x;g -x e-&~x , 2.4a 0 f d x(f(x)g(x)) = f d x f driZ(x;g) .The diagonal matrix element of the integrand of Eq.
(2.3) is the object we shall try to evaluate in a Thomas- Fermi approximation.
Namely, we approximate a diagonal matrix element of a function of the operators P and Had we been dealing with the problem of a particle moving in an ordinary potential, this approximation would have consisted of treating the potential-energy term as a constant c number ~hose value is the potential energy 33 1113 1986 The American Physical Society at x. Gauge invariance prevents us from treating the po- tential as a constant; applying a gauge transformation changes it.Before developing a variant of this approxi- mation suitable for this problem we will write a different representation for the function Z(x;ri) of Eq. (2.3).A standard path-integral representation for this function is where we integrate over all paths that begin and end at x.
The integrand is path ordered both in the color and Dirac matrices.An equivalent expression is The first path ordering refers to the Dirac matrices only, while the second refers to the color ones.
Again, because of the choice of gauge freedom we can- not set the potential A equal to its value at the point x.
The lowest-order Thomas-Fermi approximation will con- sist of choosing, for each x, a constant potential that reproduces T Pexp i A ~x for small paths beginning and ending at x.As discussed in Ref. 5, one can find a constant gauge transformation and Lorentz transformation that brings an SU(2) potential to the form where only three out of the twelve com- ponents are nonzero.Thus there are only three combina- tions of contours that we can reproduce.With M""=Pexpi IIIC""A.dx -1, where C"" is the square contour of side a as shown in Fig. (2.8) The types of infinitesimal loops around the point x which are combined in these expressions are shown in Fig. 1.
For ordinary potential problems higher-order corrections to the Thon1as-Fermi approximation retain higher-order derivatives of the potential at each point; adding terms linear, quadratic, etc. , in x to the gauge potential would permit us to reproduce the path-ordered integrals for a larger class of loops.The results for the loops presented in Fig. 1 may also be given in terms of the field strengths Ep~e (2.9) G4 --tr(FI, ~"~~~2 FI i.F"i.F~~-) .
For non-Abelian theories the field strengths do not deter- mine the field configurations; again, only larger loops will have the property of not being expressible in terms of the F""'s.For the present case the Thomas-Fermi ap- proxlHlatlon becomes Z(x;g)= I d p trexp[ ri(iI g -m)] .
-- Answers will be presented using 8 F/(Bm ), A term behaving as E"+""lm must be subtracted prior to the m' integration and then added on as a logarithmi- cally infinite fermion contribution to the coupling-constant renormalization.
Details of the evaluation are presented in the Appendix.Using the notation ' 1/2 C(~)= r +4r -+m -HGz 4-rm G3 ) 64 2 4 Gi (2.12) These are the main results of this work.We may immediately transcribe these results to a lattice versian of this problem by regarding the curves of Fig. 1 as going around elementary plaquettes.

III. CONCLUSIONS
Equations (2.11) -(2.13) provide us with an expression for the fermion-loop correction to the free effective action af an SU(2) gauge field theory and for the fermion prapagator.These expressions may be put into a version suitable for a lat- tice calculation without encountering any problems of doubling of fermion degrees af freedom.Equation (2.13) is singular for vanishing masses and for potential configurations in which G& is small while Gz and G4 remain finite.Although we cannot present any rigarous results we speculate that these are the configurations that are, in this approximation, responsible for chiral-symmetry breakdown.These matters should be settled by numerical calculations.

ACKNO%'LEDG MENT
This work was supported in part by the National Science Foundation.APPENDIX We wish to further evaluate the approximate expression for F given by (2.4a) and (2.10).The g integration can be The determinant was worked out in Ref. 5 in a gauge and Lorentz frame in which Ao --0, A =A;5;, i =1 -3 is the spatial index and a is color.Defining 4X X and using the integral representation for the logarithm gives f(»)= f 4 f (expI -ri[H +(po +m )Ai Az A& ]I e" ), - Perturbatively (A 1) has infinite zeroth-, second-, and fourth-order terms.They can be subtracted from our expressions for f, and we will do so.For simplicity we will not include them in our expressions until the end, but we will invoke them when convenient.In particular, we choose to drop the constant term e " in (A2).
Using the formula Xexpq + . +i(y+l'ei) gp A + -(y+iel) gA A ' i (z +i e2)(p + , ' A 2-+i e2) +(p p +m )A i A 2 A q (AS) For the p integral to converge, we therefore need eq)eiAl2 for all i .
The p integrations can now be done, giving (A6) The e's were introduced to make the p integration man- ifestly finite.They will also determine how the integra- tion contours pass around branch paints.
The spatial part of the momentum appears in the ex- ponential as 3 g p [iq(z+ie, ) i ri(y+ie', )A, '] .
In the z plane, R has branch points at Z= -rA1 A2 A3 -tE'2 and Z=JAj +l61Aj -l6'2, Which are both in the lower half-plane because of (A6).Since R goes as z for large z, we can do the z integration of the first term in (Al 1), giving zero.
The z integral of RP does not converge.The branch points af RP are again in the lower half-plane, so we can integrate to a cutoff radius r, and close the contour with a semicircle of radius r, in the upper half-plane giving zero, or The g integration diverges at the lower bound.Cutting it off at gp and rescaling g = cpu gives I f dzR(z)P(z)= ir, f dee' R(r, e' -)P(r, e'e) .j Expanding this in powers of r, gives r, e" 4 2+A1'A2'A3' . 1 2 2 r, 4E2+A1'A2'A3 d8 - . + +i( -, 'A +mi) = -.im-' +w( -, ' A'+m') .0 4(elly) 8(eily) 2(eiiy) 8(e,iy) The first term is constant in the A s, so it is removed by the subtractian of the zeroth-order term.The y integration of the second term gives zero.The third is removed by the zeroth-and second-order perturbative subtractions.
gp can now be taken to zero, and we are left with (A12) Using the formula g(x)=g(y)+(xy) (y)+ f dx'(x' -x) (x') dg J', , Q g dx X for RPln(P/M ) as a function of m gives AP P &2 f = ~f,&2 f dz RPln s +RP AR-Pi+ f, dm' (m'2 -mi), RP(m'z)ln 64m (eiiy)'r2 (A13) where P =Po+m Pi, A' has been assumed large and only terms which do not vailish as p goes to infinity have been kept.
Th«erma RP and -A RPi have essentially been dealt with above and are removed by the subtractions.The z in- teg«of RP»(& Pi/M ) can be done like that of RP since the branch points are again in the lower half-plane.

Pg
As before, most of the terms are either subtracted off or are finite as r, goes to infinity and zero when the y in- tegration is performed.However, the term gives, after y is integrated to a radius r"using the same method used for the z integral, z, with roots at zz --2(y+iei){ , A +-m )+iC{iy), where C(iy)=i 4(y+iei) ( , A +-m ) +iy (y+iei) -(y +i ei) g A; AJ This is ill defined.
The terms in (A13) outside of the m' integral cancel the Az-dependent part of this integral.We will show, however, that perturbatively this integral is finite beyond the fourth order.Hence, (A14} must be finite.To have the right units, the sixth-order term must be proportional to m in F. (A14) is independent of m, so it must be zero.
Letting A go to infinity, we are left with f= f, f dz f, dm'(m' -m ) (&i iy)'r'- By separating the real and imaginary parts of -C2, it can be shown this quantity goes clockwise around the origin as y goes fromao to 00, crossing the real axis once.Thus, if the phaM of -iC is defined to have a positive imaginary part at some real y, it is always positive for real y and so zi is always in the upper half z plane.
From (A16) onward we have dropped e2 since it no longer determines how the integration contours pass around branch points.The z integration contour had to be deformed into the upper half-plane first.f, dm' (m' m) f dy(y+-ie'i)' (Ai Az Ai -izi) ff [i(y+iei}AJ izi] 'r C(iy)- 16' m j=1 There is a branch point at y = -ie1.The others are all a finite distance away from the origin, on the imaginary axis, so for simplicity we will deform the contour to pass above the origin and let e1 go to zero.
FIG. 1.Types of contours which contribute to 62 63 and 64.62 involves contours of type (a).63 involves type {b).64 involves types (c) -(f).A variety of other contours are involved in the definitions of the 6's which cancel parts that are infinite as a goes to zero.
of i are introduced to make the final form more convenient. 2 The phase of C(iy) on the negative imaginary y axis.