Fortschritte der Physik / Progress of Physics: Volume 32, Number 12 [Reprint 2022 ed.] 9783112656280

Citation preview

FORTSCHRITTE DER PHYSIK PROGRESS OF PHYSICS

Volume 32 1984 Number 12

Board of Editors

F. Kaschluhn A. Lösche R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J . Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin J . Lopuszanski, Wroclaw A. Salam, Trieste D. V. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J . Zinn-Justin, Saclay

CONTENTS: P . GAIGG, M . SCHWEDA, 0 . PIGUET, a n d K . SIBOLD

The One-Loop Effective Action of the Supersymmetric CP (N-l) Model

623-638

E . W . MIBLKE

639—660

On Pseudoparticle Solutions in the Poincaré Gauge Theory of Gravity

AKADEMIE-VERLAG • BERLIN ISSN 0015-8208

Fortschr. Phys., Berlin 32 (1984) 12, 623-660

EVP 1 0 , - M

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 pages in length. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from "1" onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the authors name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred too in the text and on the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawings should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawings and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written to small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary upright typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vektors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g, .. .)> all elements and particles (H, He p, ...), elementary mathematical functions like Be, Im, sin, cos, exp,...): black underlined Greek letters: red underlined Boldface Greek letters: red interlined twice Upright Greek letters (symbols of elementary particles): red and black underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as oC,k K, o 0, p P, a S, u V, v V, w W, x X, y Y, z Z). I t will help the printer if position of subscripts and superscripts is marked with pencil in the following way: at, bi, Mil, if/), W^ Please differentiate between following symbols: a, a ; «, a , oo; a, d; e, C, c i ; e, I, g, e, k, K, x; x, x, X, %> X ; 1,1; o, 0, a, 0; p, Q; U, U; U, », v, V; (j> as additive part of the action and a third one in front of Ftfyy5 F has been ignored. This has been'done in presupposing the constraint which eliminates these functions at the expense to a shift in S. I t is clear, that in higher orders the definition of the model necessitates a constructive treatment of this arbitrariness, e.g. along the lines of ref. [8] for the quantum chiral model. As outlined in the introduction we wish to make clear in the present paper only a good quantum mechanical starting point, reached after a one-loop calculation; so the above problem is of no concern presently.

3.

(1/AT)-Expansion and the One-Loop Action

The aim is now to set up a suitable perturbation scheme, the (1/jV)-expansion [1, 3, 9], for Green's functions of the theory and to incorporate in this way the main quantum aspects of the theory: the potential appearance of bound states. To this end we first rescale the fields

fN

fN

fN

and then define the generating functional for Green's functions via the Feynmanpath integral

Z[J, J, Jz, v]= J

exp [-S - / dz(J +

+ JzZ + rjA)].

(3.1) Here J (J) is the source for 0 ( 0 ) , J z the source for S and rj the source for A „ .

(3.2)

Fortschr. Phys. 32 (1984) 12

627

Note that we have dropped an overall-factor N in 8 and replaced the number one in the constraint by N f y in order to permit renormalization. It is now clear why beeping S and Aa is so useful: these fields make the action at most bilinear in 0(0) and permit therefore explicit integration over (j> and cj>. The result is Z [ j ] =

exp [

J

- a ,

a

+

-

r j A -

(3.3)

J£Z],

where j stands collectively for all sources. We use a matrix notation, e.g. J L - V

== / dz1

dz

2

J

x

L-[}J

(3.4)

2

and e.g. 1 denotes the superspace point zx == (xlt 0t) r

¿18

_

~

-

+

Li*

=

1

J

(-Dy5D)

2

ô12

+

±(Ay>D) ( A y * D2 ) 6Ô X2

(K° +

2

M )

=

1 2

X2

j = D

±

++

KU»32

2

ô

l 2

( A f A )

s

ô12

2

( y * A

~

•

)

-

y =

S

2

d

n

J ,

(3.5)

(3.6)

This last line implicitly defines M once we prescribe the massless propagator Zl° to solve 3-

(3.7)

for the kernel n

=

(3-8)

It is explicitly given by ^

d*k

=

(2n)

2

«(«.-*,) ce««*>-*>>

1 ~

(3.9)

W

The effective action 8ett reads S

e t t

=

N T i l n i

-

J d z E

= N Tr In (1 - A°M) -

f dzE. " y

(3.10)

J

(Note that N Tr In K ° , i s zero (cf. [9]).) The logarithm is understood in its power series expansion and that yields just the powers of 1 iyN due to the explicit N appearance in M: S t n=ZW-ci one gets: £iW>cii=0,

(3.18)

where L is given by (3.5), and this is essentially the old equation of motion (2.19). For A„ N (r>s*Lii\t=3 - D3aL¡¡V,

-

+ 4>c\3D3z0i3 — Z>3«C13 • 4>013 — Y^ ^ActìQclZ = 0 holds.

(3.19)

Fortschr. Phys. 32 (1984) 12

629

Solving for A„ we obtain 1 y _ „ y "7=^3« = ^ (cl3-D3«^cl3 ~ D3ael9 • 0ci3) + — {D^Ltf |i=3 — D^L^

|1=3).

(3.20) Here the terms in the second bracket represent so to speak the "renormalization" of the first bracket which was present before «^-integration!

4.

The Effective Potential

The effective potential Feff is defined through = m

/

(4.i)

where r is the vertex functional (3.16) taken for constant classical fields - S) (4.4)

Mlt = J

2

(T - - i (P +

+ -±=S(fi1y»9t 4 y,N

s

(0y 0)2

2/ = - ^ r ( e 2 + P 2 ) .

(4.10)

This expression is manifestly gauge invariant (see App. B). In particular, it does not depend on the gauge dependent field S. The momentum integral in eq. (4.9) is logarithmically divergent. We make it convergent by subtracting its divergent part — which amounts to an infinite renormalization of the coupling-constant y — thus obtaining the renormalized effective potential

= zz(x + y)

N YR

N x -f- — |>(ln p* + 1) + ylny — (x + y) In (x + y)}. 471

(4.11)

Fortschr. Phys. 32 (1984) 12

631

Looking now for the minimum of the potential we have to solve the equations defining the stationary point: . 8x

yR

4jr

8y

4n

y

^ f

/¿2 x

(4.12)

= «(« + i0 = o.

The first two equations allow us to solve for x and y as functions of zz:

(4.13)

Substituting this in eq. (4.11) one gets for the potential

which clearly attains its minimum value zero at the point 2 = 2 = 0, a; = 0,

(4.15)

y = p' exp

Going back to the original field variables (4.2), (4.10), the effective potential is minimal for % = I = p = F = 0, (s = 0 , + pz = —Nfi2 exp

(4.16) 4srl VR.

Only the sum + P2 is fixed, and $ is arbitrary. Interpreting the values ty as vacuum expectation values and demanding the gauge invariance of the vacuum leads however to 6=0. For P = 0 supersymmetry is not broken, since only g — the first component of the superf ield E — aquires a non-vanishing vacuum expectation value. The effective potential is indeed zero in this case. But we saw, that it remains zero for any values of P and q subject to the constraint (4.16), which suggests that supersymmetry remains unbroken,

632

P.

M. SCHWEDA et al. : The One-Loop Effective Action

GAIGG,

too, at least for physical quantities. Note that P is the first component of the gauge invariant superfield D„Aa. Supersymmetry may however be broken in the gauge sector generated by the components of the ghost superfield Dy$A. For the rest of the paper we shall stay with the explicitly supersymmetric solution

(P) = P = 0, 2

(4.17)

2

(e) = g = —N/j? exp 5.

•

The Physical Interpretation

Having now found a stable ground state for the theory, we shall perform a shift

Z{z)Z{z)

- ifN m

ma%= u2%exp r

4?r

L

(5.1)

i

(5.2)

VR.

in such a way, that the new fields have all zero vacuum expectation values, m2 will be understood as the mass parameter of the theory, replacing the coupling constant y ("dimensional transmutation"). In order to find the particle content of the theory we must evaluate the quadratic part of the vertex functional (3.16). Equations (3.5, 3.10, 3.16) are modified by the replacement of the massless propagator A0 with the massive propagator A, due to the shift (5.1):

Kl2 = -j[(DfD)2

+ 2m]dl2,

d2k ¿ik^-Xi) (2n f

[gdi'fetffl —m

(5.3)

k2 + m? [

-^12^23 " —^13The vertex functional is gauge invariant, i.e. it fulfils the Ward identity

[ - 1 > a IT

+

*^ ~*

m

A

" S ) **

r{

'

A

"

E)=

(5.4)

°-

Expanding T in the powers of fields up to quadratic terms we find

m

IA.,Z)

= YN r^s, +

+ 1

+ I A f r t f ^ j +..., (5.5)

where the dots represent terms of higher powers, which are at least of the order 2V-1'2. The linear term in 27 turns out to vanish due to the condition (5.1), which was a condition for the minimum of the effective potential. Indeed, computing its kernel r £ , one finds i*M = - 2

7 yR

d2k

+ 4n

k2 + m2

L _ l2l

k2 +

p \\

(5.6)

Fortschr. Phys. 32 (1984) 12

633

where we defined the renormalized coupling constant yR with the same subtraction as we did for deriving eq. (4.11). For the kernels of the bilinear terms one finds r j * = Ku> r f i =—^12^211 (5-7)

4

n u

= vly

DfDMiM

+ J

DsiA^A^)

- j D1yA2lDisAli + j n ^ i ^ i a ] Vh> where A is the free propagator (5.3) and K its inverse. Taking the Fourier transformed with respect to the variable Xi — x2 we obtain the kernels in momentum space:

r f i ( p ) = Z7(p») rt&eip)

- me12y%2),

=

(5.8)

{[l - j y t p d ^ d ^ ( p -

where IJ(p2) is given by the integral _

m

^ '

1

f

d?k

16ji 2 J -[(p + k)2 + to2] [jfc2 + to2]

1 = ^ W

1/^2 4. +4m2)]_1/2 l n

¿7(0) = [1671m2]-1,

4- 4ma L

'

(5.9)

n ( p 2 = - 4 m 2 + e) — [l6m fe}'1.

The effective propagators are found by inverting the kernels (5.8). The propagators for 0 and ¿"are readily found. For 0, it is the free propagator (5.3), whereas for Z one gets (5.10) For the gauge spinor superfield Aa one has first to fix the gauge. We choose the gauge condition Dy5A = 0,

(5.11)

which, in components, reads: 8 = 0,

Vv =0,

K+

0.

(5.11')

The result is then

(5.12)

634

P.

Gaigg,

M.

Schweda

et al.: The One-Loop Effective Action

We see that the 0 superfield propagates with the mass m given by (5.2). The S and A„ superfields, which did not propagate in the tree approximation, have now propagators given by eqs. (5.10) and (5.12). The S propagator has a singularity at p2 = —4m2, which is however only a branching point, due to the behaviour of the function (5.9): ifilP.+4m>=e ^ ^

(5.13)

me12y%i).,

Ve

The Aa propagator has a singularity of the same type at p2 = —4m2, but also a pole at p2 = 0. This last singularity disappears in the propagator of the gauge invariant superfield D„Aa(z) = 2P(x) - 0y5(A - iys &%) (x) + 0 y 5 0 v 8MX)> (5.14) and one can check that it has no effect in processes involving the exchange of gauge field propagators. In order to see this in more detail, together with the effect of the branching point singularity at p2 = —4m2, we write first the Kallen-Lehmann (KL) representation for a typical propagator AM = 1 KV ' P*(P2 + 4m2) i n

+

+ ^2)f>2 / 1 2,2 + 4 m 2)/ 2 m ] '

'

Elementary theory of analytic functions gives the KL-representation =¿1 + /

d«2

4m*

q(x2)

= j j*

1!x2

e(x2) + x2

p2

(5.16) -

4m2

In2

(x +

]/*2

-

4m2)

+ yjj

1

We have used that A is analytic in the z =p2 plane, with a cut (—oo, —4m2) and a pole at z = 0. The strength of the interaction mediated by the exchange of the propagator (5.15) is represented in the static limit by the potential F(%) = const. J dpteiPlX'A(i>2) |Po=o = const, j l ^ l - f dx2e{x2) x - V ^ l j . (5.17) We see, that the singularity l/p2 leads to .a confining potential. That this is indeed not the case in our model can be seen as follows: The .¿„-propagator near p — 0 may be expressed by •jj-

= const. =

P

+ reg = const.

1

- Y^ y 5 ^ 1 2 y 5 0 1 2 j y5 + reg

+ reg,

where "reg" means terms less singular than gularity, obeys the longitudinality-condition AA\„> =

jjP

(5.18) l/p2-

The term

A%,

which contains the sinT (5-19)

As a consequence of this and of the gauge invariance (5.4) of the interaction, the singularity does not contribute in processes like , ^-scattering with the exchange of an Aa propagator.

Fortschr. Phys. 32.(1984) 12

635

Conclusions

6.

The results of the above sections may be summarized by the following facts: (i) For the gauge field vacuum expectation value P — 0, the effective potential we found coincides with the one given in ref. [9] (see eqs. (4.9), (4.11)). (ii) The effective potential is manifestly gauge invariant and is identical with what would be obtained by computing directly in the Wess-Zumino gauge. Its minimum value zero is degenerate. The choice P — 0 makes the theory explicitly supersymmetric, whereas another choice is expected to respect supersymmetry only for physical quantities. (iii) The calculation of the effective propagators (see eqs. (5.3), (5.10), (5.12)) shows that the superfields &i(z) become massive and the superfields E, A„ propagate, whereas was massless and E, A„ were non propagating auxiliary superfields in the classical theory. /

The E, Aa propagators have a branching point singularity at the physical threshold 4m2. The Aa propagator has in addition a singularity 1 /p2 which however has no effect on physical processes, at least in the leading order of the 1/N expansion. In particular the I R singularity of the ^„-propagator does not lead to confinement. So our supersymmetrical analysis confirms the results of [6, 11], obtained in the Wess-Zumino gauge.

Appendix À Notations, conventions and useful formulae We use the following representation for Euclidean y-matrices : (A.1) so that {yM, yr} = 26^. In addition, the commutator is given by

«7,

=

j

[y"> rU =

=

(A.2)

Furthermore one has -

(A.3)

-Vf

and T r

(75W) =

2ie^,

£0i = 1 •

For the two component, real Grassmann spinors 0„ and tp one finds ty 5 0 = 0y 5 t, 0 2 = 0y50,

(ty.d)* = ty5B.

(AA)

636

P.

GAIGG,

M.

SCHWEDA

et al.: The One-Loop Effective Action

For complex two component spinor-variables (ip* = ip) btx

=

(A.5)

i fyfYa

(4>Ytf)* =

= - W s f .

w .

(fy^ya)*

(A.6)

= -zwsv-

This leads for real ©-variables to 6yll6 = 0,

dy^O = ie^B2,

By^d = 0 ,

0yS)>,y50 = fir^e2.

Finally one has

(fly) («*) =

y

The supersymmetric covariant derivative

(A.7) Da =

-D,

satisfies:

DysDDyBD = 4 • .

(A.8)

The integration-measure is given by (A.9> so that J dWyV

= 4.

(A.10)

637

Fortschr. Phys. 32 (1984) 12

Appendix B Here we collect some results concerning the gauge properties of the model. The real spinor superfield A„ possesses the following decomposition + ivsS + p)i* + j

a. =x* + Wym

WW*;

(B.i)

X„ and Aa are real two component spinors. v^ is a vector field. 8 and P are pseudo-scalar and scalar fields. The real super-gauge function is given by A = w + Wq + -i- 6y56r, M

A = A.

(B.2)

Under an infinitesimal gauge-transformation one has for the superfield 0 (see eq. (2.9)) 60 = %A0,

d$ = —iA$.

(B.3)

In components this means 6z = irnz, (B.4)

d f , = i(oy>s + e„z, SF = icoF — Qysy> +

tz.

For the .¿„-spinor superfield one has (B.5)

6Aa=iD.A,

or in components: h* = -e«.

&P = 0,

dvlt = —dltm, 88 =

.,) + 7

- Xfa*

(4.3)

= ezrf

a result for which the first term vanishes identically due to the second Bianchi identity (2.9). The addition of a double self-dual term of constant curvature in (4.1) is vital inas much as after covariant differentiation it gives rise to a constribution being proportional to the modified torsion tensor T'J •=

+

= 7

D

MlA\) •

(4.4)

By appropiately choosing the free constant y this modified torsion is able to compensate for part of the torsion contained in the translational gauge field momenta. Without this device in vacuum torsion would have been forced to vanish by our Ansatz. With the aid of the identity =

Myj

-

(4.5)

j

valid for any third rank tensor (density) being antisymmetric with respect to the last two indices (4.3) may be solved for X'jK The result -2Yjk

- 1 " T l & ] ) = -2etii*

+

3 e t

%

(4-6)

relates the translational momenta and the densities of the torsion algebraically to the canonical spin density. As such it reminds us of the so-called Cartan equation =

(4.7)

in the Einstein-Cartan theory [34, 35]. In the following we will restrict ourselves to a theory with quasilinear translational momenta given by (2.14). Then the decomposition Ft = F t + F % -

o

e^J 1 «

(4.8)

of the torsion into components being irreducible with respect to the Lorentz group is instrumental for further analysis. These irreducible pieces consist of the tracefree, sym-

Fortschr. Phys. 82 (1984) 12

645

metrical tens&r torsion h.a: =

( F t - F-M) + 1 e^FK

(4.9)

the axial torsion and the vector torsion F'\ — Ffyy as specified above. Using this decomposition of the torsion (c.f. Ref. [33]) as well as an analogous decomposition of the spin tensor the constraint (4.6) splits into the independent relations

and

( - 4 y + ¿ , + 1 ^ + 1 - d 3 J F> = 2l*h>~

(4.10)

( y - d i + d2) F % =

(4.11)

(2y + dt + jd2J

Ft

(4.12)

=

So far the case of vanishing spin has been mainly studied. Four cases have to be distinguished: For (A) "spherical" traceless torsion and FlaM

2y = -(d1

= F. = 0

(4.13)

has to be required, whereas for (B) "spherical" trace torsion 0 1 3 4y = dl + j ¿2 + j ¿3 and FWM = Fmfr = 0

(4.14)

is a necessary condition. For (C) purely "axial" torsion y

=d1-d2

and Fafiy = Fa = 0

(4.15)

is mandatory in order to satisfy the second field equation. The trivial case of • (D) vanishing translational field momenta, i.e. y = 0

= 0 requires (4.16)

provided solutions with non-trivial torsion are desired. For this type of Lagrangians studied e.g. by BENN et al. [32] there arise no symmetry restrictions on the torsion from our reduction procedure. Note that the case of reducible spherical torsion, i.e. the combined case (A) and (B) is possible if only if di+jd2

+ j

d3=0

(4.17)

646

E. W. MIELKE, Pseudoparticle Solutions

holds. This condition is known to lead to the so-called viable set in teleparallelism theories of gravity [36]. This class of theories is known to be indistinguishable from general relativity up to the 4 th post-Newtonian order. As already remarked the remainder of the second field equation resembles the Cartan equation in EC-theory. There, for a coupling to a spinless source the torsion has to vanish on account of (4.7) thus leading back to Einstein's theory of general relativity founded on a pseudo-Riemannian space-time (F4). In PG theory there is a fundamental difference, however. Since torsion has been liberated to become a dynamical field, even in the duality-constrained case non-trivial torsion solutions can occur for gravitational interacting systems without material spin. 5.

Reduction of the First Field Equation

The remaining task consists in the reduction of the symmetric form g rot + 2*2 •*(«« —

=

C¿

K5-1)

W)

of the first field equation (2.5). For later convenience the total current (2.7) of the gravitational energy-momentum has been decomposed into the translational current 7*a r

[

i -

T

i

+

"1

K5-2)

+ 2"

rot and the remaining rotational energy momentum current In quasi-linear models in view of (2.13) the latter takes the form rot i _ _ ^ = 7 n ^ i ^ « ? - vjWytt

e. ¡ i +1?., j5 \TX

F

\ + A) •

(5 3)

-

The Maxwellian part of this tensor may be written in a more convenient form by employing the identity +

= j v . ^ x ^

(c. f. Ref. [37] footnote 10, p. 541 and Ref. [31], Eq. (4.14)). The result is rot i _ _ . el 1 = j rí1 +

(5.4)

+

\

!(5 5)

-

Into this equivalent expression the double duality Ansatz (4.1) will now be inserted, taking into account that the operation (4.2) of taking the double dual is involutive. After a reordering of terms we find '£„(**) = - 2 y ¿

+

+ ^

*F%{fF*lfíri. (5.6)

Thereby a "linearization" of the symmetric part of the rotational tensor has been achieved, inasmuch as the term being quadratic in the curvature then drops out.

Fortschr. Phys. 32 (1984) 12

647

Moreover, due to the duality Ansatz (4.1) the occurring scalar curvature is constant, i.e.

where

'

—

A

A = 1+ j f