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A reformulation of the perturbative treatment of a system of fermions in a deformed basis
Volume 148B, number 6
PHYSICS LETTERS
6 December 1984
A REFORMULATION OF THE PERTURBATIVE TREATMENT OF A SYSTEM OF FERMIONS IN A DEFORMED BASIS V. ALESSANDRINI a, D.R. BES
b,1,
O. CIVITARESE c,1 and M.T. MEHR b
a LPTHE, 91504 Orsay, France b Departamento de Fisica, CNEA, 1429 Buenos Aires, Argentina c Departamento de Fisica, UNLP, 1900 La Plata, Argentina Received 15 August 1984 The constraints that eliminate infrared divergences wttich are characteristic of a deformed basis are perturbatively taken into account through the coupling to an extra phonon, in analogy to QED in the Lorentz gauge. The method is free from cumbersome limiting procedures which are present in previous treatments of the same constraints. It is exemplified in the case of a two-dimensional, exactly soluble model, for which it yields exact results.
In a previous publication [1] we have applied collective coordinate methods to set a consistent perturbation theory free o f infrared divergencies for systems o f fermions which, at lowest order, break a symmetry of the problem such as, for example, rotational invariance. The symmetry is of course restored by quantum fluctuations, as it is the case o f translational invariance in the quantization of kinks and instantons. We followed the path-integral formulation o f Gervais, Jevicki and Sakita [2]. The method was used for two- [3] and three- [4] dimensional rotations. We briefly review the main points in what follows in the case of two-dimensional rotations [3]. The vacuum-to-vacuum transition amplitude for a system of fermions is given by the path integral over the fermion fields b +, b m
z=f~m
D[b~n]D[bm]exp(ifZ?(b+,b) dt),
£ = i ~ b+bm - H(b +, b ) .
(1)
m
A rotation to the intrinsic system is induced by the transformation a+(t)= exp [ - i 0 ( t ) L ] b+(t), where L = L(a +, a) is the microscopic angular momentum. By introducing, as in ref. [2], the collective angular 1 Fellow of the CONICET, Argentina.
m o m e n t u m I ( t ) and its conjugate collective angle O(t) as formal variables, the amplitude Z can be written as [1,2] Z = f I-I D [ a + l D[am] m
D[II D[0I
8 [ I - L]
.12= i ~ a+ a m + I0 - H(a +, a) ,
(2)
rn
where ~b= ¢(a +, a) is the microscopic angle * 1 conjugate to L and C(t) is an arbitrary function of time. Next, the integral over I(t) and O(t) can be explicitly carried out [2], and the result of this integration is to replace 6 [I - L] by 6 [L 0 - L], where L 0 is the eigenvalue of the angular momentum operator. We are therefore left with a fermion system with a constraint L = L 0 and a "gauge condition" ¢ = C(t). By using the fact that Z in eq. (2) is independent of the arbitrary function C(t), the 8 function related to the gauge condition can easily be exponentiated [5]. In order to exponentiate the constraint, we have chosen in ref. [1] to represent 8(L - L0) as the limit of a gaussian. This gives the effective interaction
,1 We may also use another ~ and introduce a jacobian. 395
Volume 148B, number 6
PHYSICS LETTERS
H e f t = lim [ H + ½ D - I ( L - L o ) D~0
2+½A-1521,
(3)
where A is an arbitrary gauge parameter [1,5]. The limit D ~ 0 has to be taken in the final expressions for the matrix elements o f physical operators, which must be independent o f the gauge parameter A. However, application o f the method to realistic cases proved to be difficult [6] since it requires exact cancellations between all the contributions in terms carrying negative powers of D. We adopt in this paper an alternative procedure which is free o f these difficulties. By expressing the angular m o m e n t u m constraint as a functional Fourier transform, we add a term f p ( t ) ( L - LO) dt to the action. The Lagrange multiplier p(t) becomes thus a new dynamical variable in the problem. In order to give this new degree o f freedom a kinetic energy term, we replace C(t) by C(t) + F[~ in the gauge condition, where F is an arbitrary constant. The new effective lagrangian and hamiltonian are "/~eff = i ~
ama.
- H + p(L - LO) - ½ A - I ( $ -
F[9) 2
m
=
6 December 1984
bation theory. The same will be true in our case: the Lagrange multiplier/9 - analogous to A 0 - will cancel the contribution o f the spurious state. Moreover, our new gauge condition is the analogous of the Lorentz gauge condition in the sense that now ~bis centered around FIS. In eq. (4) n e f f contains two coupled phonons without restoring forces. The first one corresponds to the new degree of freedom (P, p). The second one is included in H, since we are interested in systems such that the (small) fluctuations around the g.s. include a zero frequency mode. Quite generally [8], within RPA, H = [(L - L 0 ) / 2 ~ ] RPA + finite frequency modes. Therefore, we have a coupled boson hamiltonian * 3 H b = (L - L0)RPA/2 9 + ( A / 2 F 2 ) p 2 + P(~RPA/F -
(o
-
Oo)(L
-
L0)RPA •
(5)
It is convenient to introduce boson creation and annihilation operators through the equations (L - L0)RP A = - ( ½ QE')1/2(7+ + 7 ) , q~RPA = i( 2Q E ' ) - I/2(~t + - 7 ) ,
+
i ~ama
•
m
+
*
Pp
--
P = 2 - t / 2 i ( j 3 + - 13),
t
Heft,
m
(P - P0) = 2-1/2(/3+ +/3). H e ft = H -
( A / 2 F 2 ) p 2 + P(o/F - p(L - L O ) ,
where P = 0Z?eff/0tS.The quantization condition Lo, P] = i is assumed ,2 The two alternative approaches have their counterpart in the different gauge f'Lxing procedures used in the quantization of gauge theories [7]. The physical gauge where there are no spurious states is the Coulomb gauge d i v A = 0. In our case, it follows from eq. (3) that (~b) = 0 in the intrinsic frame and that the spurious state has a frequency co = ( A D ) - 1 / 2 which goes to infinity and disappears from the spectrum ofH'ef f as the constraint is enforced by l e t t i n g D ~ 0. In the Lorentz gauge, where div A + .zi0 = 0, the Lagrange multiplier A 0 that enforces the constraint (Gauss law) is introduced as a new dynamical variable which will ultimately cancel the longitudinal photons in pertur,2 The constant parameters A, F are arbitrary, and no physical result should depend on them (as we later verify). Thus the fact that the kinetic energy in (4) is only positive for A < 0 does not have any physical consequence. 396
(6)
(4) Here the parameter E ' represents an average p a r t i c l e hole energy. The transformation to uncoupled pho+ + + nons F v = Xv~3, + Xv~13 -/av~/7 - ~vt3~ (v = 1, 2) yields the (doubly degenerate) roots w = +l/x/ft. As usual in the RPA, we use the two positive roots. The fact that they are finite ensures the ehmination o f the infrared divergences associated with the zero frequency modes. In addition, one obtains the amplitudes k l . r = ½ [(colE') 1/2 + (E'/co)I/2] , •113 = l ( Q / ~ ) l / 2 ' lal,y = 21[(~IE')l/2 - (E'/6o)1/2] , //1~ =
-½(O/w) 1/2 ,
;k2.Y = ~(co/E') 112 ,
(7)
,3 A part -oo(L - Lo) of the last term in (4) will be incorporated to H. The constant Po is to be chosen such that the g.s. average (L) is fixed at the value Lo. The remaining contribution, -(O - Po)( L - Lo) is included in (5).
Volume 148B, number 6
PHYSICS LETTERS
where the subindex n = f denotes the last filled level. The angular momentum and the angle 0 are
~'2~ = ½[(9/6o) 1/2 - (6°/9)1/21, /.127 = I(6O/E')112 , U2~ = _ 1 [(~/6o)1/2 + (6o/9)1/21 ,
6 December 1984
(7 cont'd)
L = L 0 + l i B - 2(A + + A ) , 0 = (i/2EZ) [A + - A - (A +2 - A 2)II/ZE
where use has been made of the orthonormalization relations
[Pl, P~] = [r'+2, r21 = 1,
[ r 2, 71] = IF2,
vii = 0,
(8) There are no solutions if we impose usual normalization conditions for both phonons. This is reasonable, since spurious states (due to the presence of extra degrees of freedom) can only be eliminated through a non-canonical transformation. According to (8) the g.s. contractions have the value (PlP~ -) = - ( P 2 P ~ ) = 1. The zeroth order hamiltonian consists of an independent-particle, deformed hamiltonian, the finitefrequency RPA bosons and the two spurious bosons [eqs. (6)-(8)]. The residual interaction may be constructed according to the NFT formalism. We exemplify the procedure for the case of particles moving in a two-dimensional h.o. shell N and coupled through quadruj~ole forces H 0 = - 1 9 - 1 ( Q 2 + $2), Q = IMW X £ i ( x ~ - y 2 ) and S = M W ~i xi Yi" This case was already studied in ref. [3]. Exact results and the construction of a deformed basis are given, respectively, in sections 2 and 3.1 o f ref. [3]. The "Nilsson" singleparticle states are labelled by the quantum number n (n = 0, 1, ..., At) plus additional numbers] (spin, isospin, etc.) which distinguish between £2-degenerate states. The levels are equidistant, and their distance has the value E = 2 Z / 9 , where G is the largest possible value of Q. Use is made of the operators A +=~rlnAn,
+
,4 A mistake which is present in eq. (3.9) of ref. [3] has been corrected in (10).
Table 1 Vertices used in the calculation. In addition the pure fermion vertices ofeq. (3.8) in ref. [3] have to be used. Vertices denoted by A are particle-vibration (fig. 3, ref. [3]) or pure boson (fig. 2, ref. [3]) vertices. IfLo is increased by l the pure phonon vertex is replaced by Aqv + Aqv , Hll by Hll + h l l B and there appears a term h2o(A + + A). The Q+-vare the pbonon vertices corresponding to the quadrupole operator (c.f. fig. 6, ref. [3]). Ab 1 = (E + w)(E~- 2 - to)/2~21/2"~(wE) 1/2
Asl = _ [ I I ( t o E ) l l 2 / 2 Z
B = ~ onB n n>f
~ OnBn, n<~f
an,.ian, ] ,
on = N - 2 n ,
n <,f,
(9)
1/2 ] (+ On )
As2 = 0
A q l = _(IRo/2~l12-.2)(tolE)l/2
Aq2 --- - A q 1
Aql = l(toE)l/2/2Z1/2
Aq2 --" 0
n >f,
1
a n j a n , ],
1/2
Af2 = [ ( E - e0)12$21/2~1 ( t o / E ) 1/2
1
1
where 4) is given ,4 as an expansion in powers of Z -1. Here 11 = L O / E and li2 + 22 = 1. The average energy E ' in (6) and (7) results to be equal to E ~-2. The operator A ; creates particle-hole excitations. The particle-phonon and the phonon-creation vertices are constructed through the replacement of A ; by its phonon expression x/~3 '+ = (~.171"'~ +/./i3,Pl -- X2vP ~ +/a2vP2)X/-~ , as usual in the NFT. The resultant vertices are given in table 1. In addition, a boson contraction in the interaction term P~)/F between the P operator and the A ; operator in the third term of ~ (eq. (10)), gives rise to a singleparticle contribution - w B / 2 Y , which should be added
Afl = - ( E - to)(E~ 2 + t o ) / 2 1 2 1 / 2 ~ ( ~ E )
An+ = . an+l,yan,],
=
(10)
Ab2 = [(E + w)/2121/2,.~] (to/E) 1/2
r/n= [ ( N - n ) ( n + l ) ] l / 2 ,
n
Bn =
- (A+B - B A ) / Z + ...],
hll
= -IFIE/2z
Q+I
= (]~I/2/N)(FI
Q+2 = Q - 2
h2o = IE--/2E -T- E~2Iuo)(oo/E)
1/2
= - ( g 1 5 2 1 / 2 / ~ , ) ( c o l E ) 1/2
397
Volume 148B, number 6
PHYSICS LETTERS
6 December 1984 p
Table 2 Diagrammatic calculations of the particle-hole excitation energy, of the moment of inertia and of the quadrupole transition rate between two consecutive rotational states. The notation Zn for a given contribution indicates that the calculation corresponds to graph Z in fig. n of ref. [3]). A4 + C4 + D4 = -EH2/E B 4 + K 4 + L 4 = -~E--2/r. + co/~, E4 + F4 + G 4 + H 4
= 1 E ~ 2 / E - (to/~) (1 + H2) + (EH2/E)(E~-2/2w - II 2)
I4 + J4 + M4 + N4 = -(II2/Y_,)IE(E~-~/2to - n 2 - I)+ ton2/x 2 ] + (n21~)to/-2
0 6 + X6 = ( I H / ~ ) ( 1 E - co/2.-2) P6 = - 1 2 E [ 4~ + (IH/~)t°/2~'2 A7 = -(~./2to)(E~ 2 - 2w) + I'I2(E~ 2 - to)/E ~-
The hamiltonian corresponding to the case L 0 = L 0 + l differs from the one corresponding to L 0 through the value of some vertices as indicated in the caption to table 1. We take the difference between the g.s. energies corresponding to L~ and L 0. The sum of diagrams 0 6 , P6, X6 plus the constant A 0 equals l L o / 9 + 1 2 / 2 9 , which yields the exact moment of inertia 9. Finally the transition rate between two consecutive members of the rotational band differs from the unperturbed value --.E by the quantity _--/(1 + II), which is obtained adding together the third group of contributions (A 7 ..... D7) of table 2. In all cases the exclusion of all diagrams with spurious phonons yields a wrong result. Inclusion of only the zero frequency boson leads to infrared divergences. The present method for carrying out perturbation procedures in deformed bases appears to be conceptually equivalent to the one previously developed, but very much simpler from the practical point of view. Therefore it is hoped that applications to realistic cases will be much more feasible.
D7 = -E~_/2co B7 = _yI2(E,~2 _ to)/E ~_
Illuminating discussions with Professor H. Fanchiotti are greatly appreciated.
C7 = -(l/_--)(rl - E ~ 2 / w ) References
to the corresponding term from HQ - p0(Z - L 0) in eq. (3.8) of ref. [3]. In the following we calculate particle-hole excitation energies, the m o m e n t of inertia and quadrupole transition rates. The corresponding diagrams are to be found in figs. 4, 6 and 7 of ref. [3]. It is understood that a phonon line indicates summation over the phonons 1 and 2. The unperturbed energy E gives the exact p a r t i c l e hole excitation energy. The sum of contributions appearing in the first part of table 2 (A4, ..., N4) vanishes. This corresponds to a non-trivial cancellation of the E / E corrections.
398
[1] V. Alessandrini, D.R. Bes and B. Machet, Nucl. Phys. B142 (1978) 489. [2] J.L. Gervais, A. Jevickiand B. Sakita, Phys. Rep. 23C (1976) 281. [3] D.R. Bes, G.G. Dussel and R.P.J. Perazzo, Nucl. Phys. A340 (1980) 157. [4] D.R. Bes, O. Civitarese and H.M. Sofia, Nucl. Phys. A370 (1981) 99. [5] G. 't Hooft, Nucl. Phys. B33 (1971) 173. [6] D.R. Bes, O. Civitarese, R.J. Liotta and M.T. Mehr, to be published. [7] E.S. Abers and B.W. Lee, Phys. Rep. 9C (1973) 1; J. Bernstein, Rev. Mod. Phys. 46 (1974) 1. [8] E.R. Marshalek and J. Weneser, Phys. Rev. C2 (1970) 1682.
PHYSICS LETTERS
6 December 1984
A REFORMULATION OF THE PERTURBATIVE TREATMENT OF A SYSTEM OF FERMIONS IN A DEFORMED BASIS V. ALESSANDRINI a, D.R. BES
b,1,
O. CIVITARESE c,1 and M.T. MEHR b
a LPTHE, 91504 Orsay, France b Departamento de Fisica, CNEA, 1429 Buenos Aires, Argentina c Departamento de Fisica, UNLP, 1900 La Plata, Argentina Received 15 August 1984 The constraints that eliminate infrared divergences wttich are characteristic of a deformed basis are perturbatively taken into account through the coupling to an extra phonon, in analogy to QED in the Lorentz gauge. The method is free from cumbersome limiting procedures which are present in previous treatments of the same constraints. It is exemplified in the case of a two-dimensional, exactly soluble model, for which it yields exact results.
In a previous publication [1] we have applied collective coordinate methods to set a consistent perturbation theory free o f infrared divergencies for systems o f fermions which, at lowest order, break a symmetry of the problem such as, for example, rotational invariance. The symmetry is of course restored by quantum fluctuations, as it is the case o f translational invariance in the quantization of kinks and instantons. We followed the path-integral formulation o f Gervais, Jevicki and Sakita [2]. The method was used for two- [3] and three- [4] dimensional rotations. We briefly review the main points in what follows in the case of two-dimensional rotations [3]. The vacuum-to-vacuum transition amplitude for a system of fermions is given by the path integral over the fermion fields b +, b m
z=f~m
D[b~n]D[bm]exp(ifZ?(b+,b) dt),
£ = i ~ b+bm - H(b +, b ) .
(1)
m
A rotation to the intrinsic system is induced by the transformation a+(t)= exp [ - i 0 ( t ) L ] b+(t), where L = L(a +, a) is the microscopic angular momentum. By introducing, as in ref. [2], the collective angular 1 Fellow of the CONICET, Argentina.
m o m e n t u m I ( t ) and its conjugate collective angle O(t) as formal variables, the amplitude Z can be written as [1,2] Z = f I-I D [ a + l D[am] m
D[II D[0I
8 [ I - L]
.12= i ~ a+ a m + I0 - H(a +, a) ,
(2)
rn
where ~b= ¢(a +, a) is the microscopic angle * 1 conjugate to L and C(t) is an arbitrary function of time. Next, the integral over I(t) and O(t) can be explicitly carried out [2], and the result of this integration is to replace 6 [I - L] by 6 [L 0 - L], where L 0 is the eigenvalue of the angular momentum operator. We are therefore left with a fermion system with a constraint L = L 0 and a "gauge condition" ¢ = C(t). By using the fact that Z in eq. (2) is independent of the arbitrary function C(t), the 8 function related to the gauge condition can easily be exponentiated [5]. In order to exponentiate the constraint, we have chosen in ref. [1] to represent 8(L - L0) as the limit of a gaussian. This gives the effective interaction
,1 We may also use another ~ and introduce a jacobian. 395
Volume 148B, number 6
PHYSICS LETTERS
H e f t = lim [ H + ½ D - I ( L - L o ) D~0
2+½A-1521,
(3)
where A is an arbitrary gauge parameter [1,5]. The limit D ~ 0 has to be taken in the final expressions for the matrix elements o f physical operators, which must be independent o f the gauge parameter A. However, application o f the method to realistic cases proved to be difficult [6] since it requires exact cancellations between all the contributions in terms carrying negative powers of D. We adopt in this paper an alternative procedure which is free o f these difficulties. By expressing the angular m o m e n t u m constraint as a functional Fourier transform, we add a term f p ( t ) ( L - LO) dt to the action. The Lagrange multiplier p(t) becomes thus a new dynamical variable in the problem. In order to give this new degree o f freedom a kinetic energy term, we replace C(t) by C(t) + F[~ in the gauge condition, where F is an arbitrary constant. The new effective lagrangian and hamiltonian are "/~eff = i ~
ama.
- H + p(L - LO) - ½ A - I ( $ -
F[9) 2
m
=
6 December 1984
bation theory. The same will be true in our case: the Lagrange multiplier/9 - analogous to A 0 - will cancel the contribution o f the spurious state. Moreover, our new gauge condition is the analogous of the Lorentz gauge condition in the sense that now ~bis centered around FIS. In eq. (4) n e f f contains two coupled phonons without restoring forces. The first one corresponds to the new degree of freedom (P, p). The second one is included in H, since we are interested in systems such that the (small) fluctuations around the g.s. include a zero frequency mode. Quite generally [8], within RPA, H = [(L - L 0 ) / 2 ~ ] RPA + finite frequency modes. Therefore, we have a coupled boson hamiltonian * 3 H b = (L - L0)RPA/2 9 + ( A / 2 F 2 ) p 2 + P(~RPA/F -
(o
-
Oo)(L
-
L0)RPA •
(5)
It is convenient to introduce boson creation and annihilation operators through the equations (L - L0)RP A = - ( ½ QE')1/2(7+ + 7 ) , q~RPA = i( 2Q E ' ) - I/2(~t + - 7 ) ,
+
i ~ama
•
m
+
*
Pp
--
P = 2 - t / 2 i ( j 3 + - 13),
t
Heft,
m
(P - P0) = 2-1/2(/3+ +/3). H e ft = H -
( A / 2 F 2 ) p 2 + P(o/F - p(L - L O ) ,
where P = 0Z?eff/0tS.The quantization condition Lo, P] = i is assumed ,2 The two alternative approaches have their counterpart in the different gauge f'Lxing procedures used in the quantization of gauge theories [7]. The physical gauge where there are no spurious states is the Coulomb gauge d i v A = 0. In our case, it follows from eq. (3) that (~b) = 0 in the intrinsic frame and that the spurious state has a frequency co = ( A D ) - 1 / 2 which goes to infinity and disappears from the spectrum ofH'ef f as the constraint is enforced by l e t t i n g D ~ 0. In the Lorentz gauge, where div A + .zi0 = 0, the Lagrange multiplier A 0 that enforces the constraint (Gauss law) is introduced as a new dynamical variable which will ultimately cancel the longitudinal photons in pertur,2 The constant parameters A, F are arbitrary, and no physical result should depend on them (as we later verify). Thus the fact that the kinetic energy in (4) is only positive for A < 0 does not have any physical consequence. 396
(6)
(4) Here the parameter E ' represents an average p a r t i c l e hole energy. The transformation to uncoupled pho+ + + nons F v = Xv~3, + Xv~13 -/av~/7 - ~vt3~ (v = 1, 2) yields the (doubly degenerate) roots w = +l/x/ft. As usual in the RPA, we use the two positive roots. The fact that they are finite ensures the ehmination o f the infrared divergences associated with the zero frequency modes. In addition, one obtains the amplitudes k l . r = ½ [(colE') 1/2 + (E'/co)I/2] , •113 = l ( Q / ~ ) l / 2 ' lal,y = 21[(~IE')l/2 - (E'/6o)1/2] , //1~ =
-½(O/w) 1/2 ,
;k2.Y = ~(co/E') 112 ,
(7)
,3 A part -oo(L - Lo) of the last term in (4) will be incorporated to H. The constant Po is to be chosen such that the g.s. average (L) is fixed at the value Lo. The remaining contribution, -(O - Po)( L - Lo) is included in (5).
Volume 148B, number 6
PHYSICS LETTERS
where the subindex n = f denotes the last filled level. The angular momentum and the angle 0 are
~'2~ = ½[(9/6o) 1/2 - (6°/9)1/21, /.127 = I(6O/E')112 , U2~ = _ 1 [(~/6o)1/2 + (6o/9)1/21 ,
6 December 1984
(7 cont'd)
L = L 0 + l i B - 2(A + + A ) , 0 = (i/2EZ) [A + - A - (A +2 - A 2)II/ZE
where use has been made of the orthonormalization relations
[Pl, P~] = [r'+2, r21 = 1,
[ r 2, 71] = IF2,
vii = 0,
(8) There are no solutions if we impose usual normalization conditions for both phonons. This is reasonable, since spurious states (due to the presence of extra degrees of freedom) can only be eliminated through a non-canonical transformation. According to (8) the g.s. contractions have the value (PlP~ -) = - ( P 2 P ~ ) = 1. The zeroth order hamiltonian consists of an independent-particle, deformed hamiltonian, the finitefrequency RPA bosons and the two spurious bosons [eqs. (6)-(8)]. The residual interaction may be constructed according to the NFT formalism. We exemplify the procedure for the case of particles moving in a two-dimensional h.o. shell N and coupled through quadruj~ole forces H 0 = - 1 9 - 1 ( Q 2 + $2), Q = IMW X £ i ( x ~ - y 2 ) and S = M W ~i xi Yi" This case was already studied in ref. [3]. Exact results and the construction of a deformed basis are given, respectively, in sections 2 and 3.1 o f ref. [3]. The "Nilsson" singleparticle states are labelled by the quantum number n (n = 0, 1, ..., At) plus additional numbers] (spin, isospin, etc.) which distinguish between £2-degenerate states. The levels are equidistant, and their distance has the value E = 2 Z / 9 , where G is the largest possible value of Q. Use is made of the operators A +=~rlnAn,
+
,4 A mistake which is present in eq. (3.9) of ref. [3] has been corrected in (10).
Table 1 Vertices used in the calculation. In addition the pure fermion vertices ofeq. (3.8) in ref. [3] have to be used. Vertices denoted by A are particle-vibration (fig. 3, ref. [3]) or pure boson (fig. 2, ref. [3]) vertices. IfLo is increased by l the pure phonon vertex is replaced by Aqv + Aqv , Hll by Hll + h l l B and there appears a term h2o(A + + A). The Q+-vare the pbonon vertices corresponding to the quadrupole operator (c.f. fig. 6, ref. [3]). Ab 1 = (E + w)(E~- 2 - to)/2~21/2"~(wE) 1/2
Asl = _ [ I I ( t o E ) l l 2 / 2 Z
B = ~ onB n n>f
~ OnBn, n<~f
an,.ian, ] ,
on = N - 2 n ,
n <,f,
(9)
1/2 ] (+ On )
As2 = 0
A q l = _(IRo/2~l12-.2)(tolE)l/2
Aq2 --- - A q 1
Aql = l(toE)l/2/2Z1/2
Aq2 --" 0
n >f,
1
a n j a n , ],
1/2
Af2 = [ ( E - e0)12$21/2~1 ( t o / E ) 1/2
1
1
where 4) is given ,4 as an expansion in powers of Z -1. Here 11 = L O / E and li2 + 22 = 1. The average energy E ' in (6) and (7) results to be equal to E ~-2. The operator A ; creates particle-hole excitations. The particle-phonon and the phonon-creation vertices are constructed through the replacement of A ; by its phonon expression x/~3 '+ = (~.171"'~ +/./i3,Pl -- X2vP ~ +/a2vP2)X/-~ , as usual in the NFT. The resultant vertices are given in table 1. In addition, a boson contraction in the interaction term P~)/F between the P operator and the A ; operator in the third term of ~ (eq. (10)), gives rise to a singleparticle contribution - w B / 2 Y , which should be added
Afl = - ( E - to)(E~ 2 + t o ) / 2 1 2 1 / 2 ~ ( ~ E )
An+ = . an+l,yan,],
=
(10)
Ab2 = [(E + w)/2121/2,.~] (to/E) 1/2
r/n= [ ( N - n ) ( n + l ) ] l / 2 ,
n
Bn =
- (A+B - B A ) / Z + ...],
hll
= -IFIE/2z
Q+I
= (]~I/2/N)(FI
Q+2 = Q - 2
h2o = IE--/2E -T- E~2Iuo)(oo/E)
1/2
= - ( g 1 5 2 1 / 2 / ~ , ) ( c o l E ) 1/2
397
Volume 148B, number 6
PHYSICS LETTERS
6 December 1984 p
Table 2 Diagrammatic calculations of the particle-hole excitation energy, of the moment of inertia and of the quadrupole transition rate between two consecutive rotational states. The notation Zn for a given contribution indicates that the calculation corresponds to graph Z in fig. n of ref. [3]). A4 + C4 + D4 = -EH2/E B 4 + K 4 + L 4 = -~E--2/r. + co/~, E4 + F4 + G 4 + H 4
= 1 E ~ 2 / E - (to/~) (1 + H2) + (EH2/E)(E~-2/2w - II 2)
I4 + J4 + M4 + N4 = -(II2/Y_,)IE(E~-~/2to - n 2 - I)+ ton2/x 2 ] + (n21~)to/-2
0 6 + X6 = ( I H / ~ ) ( 1 E - co/2.-2) P6 = - 1 2 E [ 4~ + (IH/~)t°/2~'2 A7 = -(~./2to)(E~ 2 - 2w) + I'I2(E~ 2 - to)/E ~-
The hamiltonian corresponding to the case L 0 = L 0 + l differs from the one corresponding to L 0 through the value of some vertices as indicated in the caption to table 1. We take the difference between the g.s. energies corresponding to L~ and L 0. The sum of diagrams 0 6 , P6, X6 plus the constant A 0 equals l L o / 9 + 1 2 / 2 9 , which yields the exact moment of inertia 9. Finally the transition rate between two consecutive members of the rotational band differs from the unperturbed value --.E by the quantity _--/(1 + II), which is obtained adding together the third group of contributions (A 7 ..... D7) of table 2. In all cases the exclusion of all diagrams with spurious phonons yields a wrong result. Inclusion of only the zero frequency boson leads to infrared divergences. The present method for carrying out perturbation procedures in deformed bases appears to be conceptually equivalent to the one previously developed, but very much simpler from the practical point of view. Therefore it is hoped that applications to realistic cases will be much more feasible.
D7 = -E~_/2co B7 = _yI2(E,~2 _ to)/E ~_
Illuminating discussions with Professor H. Fanchiotti are greatly appreciated.
C7 = -(l/_--)(rl - E ~ 2 / w ) References
to the corresponding term from HQ - p0(Z - L 0) in eq. (3.8) of ref. [3]. In the following we calculate particle-hole excitation energies, the m o m e n t of inertia and quadrupole transition rates. The corresponding diagrams are to be found in figs. 4, 6 and 7 of ref. [3]. It is understood that a phonon line indicates summation over the phonons 1 and 2. The unperturbed energy E gives the exact p a r t i c l e hole excitation energy. The sum of contributions appearing in the first part of table 2 (A4, ..., N4) vanishes. This corresponds to a non-trivial cancellation of the E / E corrections.
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[1] V. Alessandrini, D.R. Bes and B. Machet, Nucl. Phys. B142 (1978) 489. [2] J.L. Gervais, A. Jevickiand B. Sakita, Phys. Rep. 23C (1976) 281. [3] D.R. Bes, G.G. Dussel and R.P.J. Perazzo, Nucl. Phys. A340 (1980) 157. [4] D.R. Bes, O. Civitarese and H.M. Sofia, Nucl. Phys. A370 (1981) 99. [5] G. 't Hooft, Nucl. Phys. B33 (1971) 173. [6] D.R. Bes, O. Civitarese, R.J. Liotta and M.T. Mehr, to be published. [7] E.S. Abers and B.W. Lee, Phys. Rep. 9C (1973) 1; J. Bernstein, Rev. Mod. Phys. 46 (1974) 1. [8] E.R. Marshalek and J. Weneser, Phys. Rev. C2 (1970) 1682.