Low-energy general relativity with torsion: A systematic derivative expansion

Low-energy general relativity with torsion: A systematic derivative expansion

Dmitri Diakonov,1,2 Alexander G. Tumanov,2 and Alexey A. Vladimirov3

1Petersburg Nuclear Physics Institute, Gatchina 188300, St. Petersburg, Russia2St. Petersburg Academic University, St. Petersburg 194021, Russia

3Ruhr-Universitat Bochum, Bochum D-44780, Germany(Received 17 April 2011; published 19 December 2011)

We attempt to build systematically the low-energy effective Lagrangian for the Einstein-Cartan

formulation of gravity theory that generally includes the torsion field. We list all invariant action terms

in certain given order; some of the invariants are new. We show that in the leading order, the fermion

action with torsion possesses additional U1L U1R gauge symmetry, with 4 4 components of thetorsion (out of the general 24) playing the role of Abelian gauge bosons. The bosonic action quadratic in

torsion gives masses to those gauge bosons. Integrating out torsion one obtains a pointlike 4-fermion

action of a general form containing vector-vector, axial-vector, and axial-axial interactions. We present a

quantum field-theoretic method to average the 4-fermion interaction over the fermion medium, and

perform the explicit averaging for free fermions with given chemical potential and temperature. The result

is different from that following from the spin fluid approach used previously. On the whole, we arrive to

rather pessimistic conclusions on the possibility to observe effects of the torsion-induced 4-fermion

interaction, although under certain circumstances it may have cosmological consequences.

DOI: 10.1103/PhysRevD.84.124042 PACS numbers: 04.50.Kd, 11.15.q, 98.80.k

I. INTRODUCTION

Recently, there has been some renewed interest in tor-sion appearing in the Einstein-Cartan formulation ofGeneral Relativity [16], and in physical effects it mayimply, see, e.g. Refs. [79]. A general drawback of theseinteresting studies was certain lack of systematics, in par-ticular, not all a priori possible invariants of a given ordercontaining torsion were considered. The first aim of thispaper is to fill in this gap and to introduce torsion includingits interaction with fermions in a systematic way, in thespirit of the low-energy derivative expansion [10]. In thelowest order of this expansion, torsion induces a local4-fermion interaction [1] which may affect cosmologicalevolution at very early times and high matter density[1114], or be detectable from high-precision data at latertimes. Therefore, the second aim of this paper is to evaluatethe contribution of the general torsion-induced 4-fermioninteraction derived in the first part, to the stress-energytensor for possible applications. We think that such evalu-ation in the past, based on spin fluid ideas, has beenunsound.General Relativity with fermions is a theory invariant

under (i) general coordinate transformations (diffeomor-phisms) and (ii) point-dependent (local) Lorentz transfor-mations. The standard way one introduces fermions is viathe Fock-Weyl action [15,16]

Sf iZd4x dete12 eAADDeAA;

D @ 18!BC BC; (1)where is the 4-component fermion field assumed to be aworld scalar, A are the four Dirac matrices, !

BC is the

gauge field of the local Lorentz group, called spin connec-tion, and eA is the contravariant (inverse) frame field. Infact, there are other fermion actions invariant under (i) and(ii), to be discussed below, but Eq. (1) is generic.To incorporate fermions, one needs, therefore, the gauge

field ! and the frame field e, which are a priori inde-

pendent. Therefore, the bosonic part of the GeneralRelativity action must be also constructed from thesefields. We are thus bound to the Einstein-Cartan formula-tion of General Relativity.In this formulation, the lowest-derivative terms in the

bosonic part of the action invariant under (i) and (ii) are

Sb M2P

16

Zd4x2dete 14ABCDF ABeCeD

2F ABe

Ae

B . . .; (2)

whereDAB @ !AB is the covariant derivative, andF AB DDAB is the curvature. The first term is thecosmological term, dete1=4!ABCDeAeBeCe

D gp , with 5:5 1084 Gev2, the second

term is the would-be Einstein-Hilbert action with MP 1=

G

p 1:22 1019 GeV being the Planck mass, and thethird term is the P- and T-odd action first suggested inRefs. [17,18] with iota being so far a free dimensionlessparameter. In the context of canonical gravity, the inverseof is sometimes called the Barbero-Immirzi parameter,and the third term in (2) called the Holst action [19]although in fact they were introduced much earlier [17,18].In a quantum theory, one writes the amplitude as expiS.Equation (2) is quadratic in !, therefore the saddle-

point integration in ! in the assumed path integral over

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!, e, is exact. We define the antisymmetric torsion

tensor as

TAd 12DeA DeA: (3)Under Lorentz transformations it behaves as a 4-vector,and under diffeomorphisms it behaves as a rank-2 tensor.In the absence of fermions, the saddle-point equation aris-ing from the first variation of Eq. (2) in 24 independentvariables !AB results in 24 dynamical equations T

A 0.

This set of equations is linear in !, and its solution is the

well known

!AB e 12eA@eB @eB 12eB@eA @eA 12eAeBeC@eC @eC : (4)

Quantities with a bar refer, here and below, to this zero-torsion case. [If two flat indices A; B;C; . . . appear both assubscripts or both as superscripts, we sum over them withMinkowski signature 1;1;1;1; 0123 1,0123 1.] It is well known that in the zero-torsioncase the second term in Eq. (2) reduces to the standardEinstein-Hilbert action gp R, where R is the standardscalar curvature built from the Christoffel symbol ; 12 @g @g @g, and the third term reduces to 12 R; 0.With fermions switched in, the torsion is nonzero even at

the saddle point since from varying Eq. (1) one obtains T J, where J is a fermion bilinear current. Neither is itgenerally speaking zero, if terms with higher derivativesare added to the bosonic action, see Sec. VI C.Apart from full derivatives, the deviation of the second

and third term in Eq. (2) from its zero-torsion limit isOT2, and the deviation of the fermionic action (1) fromits zero-torsion limit is OT, see Eqs. (60), (61), and (55)below. However, these terms are not the only ones that canbe constructed from the requirements (i) and (ii), and thereare no a priori reasons why other terms should be ignored.The minimal actions (1) and (2) are at best the low-

energy limit of an effective theory whose microscopicorigin is still under debate, since the action is nonrenorma-lizable and, worse, nonpositive definite. Therefore, the bestwe can do in the absence of a well-defined quantum theoryis to write down a derivative expansion for an effectivelow-energy action satisfying the requirements (i) and (ii)with arbitrary constants, to be in principle determined or atleast restricted from observation or experiment. A futurefundamental, microscopic theory of gravity will be able tofix those constants. Unless proved otherwise, the derivativeexpansion in the effective action is assumed to be a Taylorseries in @2=M2P in the bosonic sector; in the fermionicsector odd powers of @=MP are also allowed.For completeness, we include in our consideration invar-

iants that are odd under P, T inversion. Since these discretesymmetries are not preserved by weak interactions, and theeffective low-energy gravity may imply integrating out

high-momenta fermions, we do not see the principle whysuch invariants should be avoided.

II. ORDER OF MAGNITUDE ANALYSIS

In the Einstein-Cartan formulation of gravity, which aswe stress is unavoidable if we wish to incorporate fermi-ons, the frame field eA and the spin connection !

AB are

a priori independent variables. One expands the action inthe derivatives of e and !, preserving the diffeomor-

phism invariance and gauge invariance under local Lorentztransformations, the only two symmetries requested. Thespin connection ! is a gauge field that transforms inho-

mogeneously under local Lorentz transformations, hence itcan appear only inside covariant derivatives.At the saddle point (in other wording from the equation

of motion) the spin connection is expressed through theframe field, ! ! e1@e, see Eq. (4). Suppressingthe indices and omitting the frame fields assumed to be ofthe order of unity, one can present symbolically the general! as

! ! T; T @e; (5)where T is the torsion field (3); the precise relation is givenby Eq. (35) below. From the point of view of derivativecounting, T, ! and hence ! itself are all one-derivativequantities. From the point of view of the gauge group, !transforms inhomogeneously and hence must always comeinside covariant derivatives, whereas T transforms homo-geneously and hence terms of the type T2 and the like areallowed by gauge symmetry.Let us classify the possible action terms. There is only

one zero-derivative term, the invariant volume or the cos-mological term. One-derivative terms are absent in thebosonic sector. They appear only in the fermionic sectorfrom the Dirac-Fock-Weyl action which we fully analyze

in Sec. IV: there is a unique term of the type @ !and four terms of the type T.There are precisely seven two-derivative terms: two

terms linear in the curvature F , presented in Eq. (2), andfive terms quadratic in torsion, fully listed in Sec. VIA.Since by definition T re, all terms quadratic in torsionare quadratic in the derivatives. There are no other termsquadratic in the derivatives in the bosonic sector, and this isan exact statement of this paper.Turning to four-derivative terms, there are in general

terms of the type FF R2, TrF , rT2, T2rT,T2F , and T4.Omitting the cosmological term we write down sym-

bolically the effective Lagrangian as

Leff M2P R @ TM2PT2 R2 T r R rT2 T2 rT T2 R T4 O1=M2P (6)

where the Lagrangian for matter is represented by thefermionic source. The equations of motion are obtained

DIAKONOV, TUMANOV, AND VLADIMIROV PHYSICAL REVIEW D 84, 124042 (2011)

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by varying the action (6) with respect to the fields involved.Assuming c 1 and the dimensionless metric tensorg, the dimensions of the quantities in Eq. (6) are the

usual R 1=cm2, T 1=cm, 1=cm3. In the leadingorder, one gets from the first two terms the standard esti-mate for the curvature generated by matter,

R h@iM2P

p4

M2P; (7)

where p is the characteristic momentum of matter, be ittemperature, cubic root of density, or massin fact, thelargest of these. Being substituted back into the action,this estimate shows that the first two terms of the action (6)are of the order of p4, and that the R2 term is a tiny p8=M4Pcorrection.Excluding torsion T from the 3rdand 4th terms in (6)

gives

ThiM2P

p3

M2P; M2PT

2Th ih iM2P

p6

M2P:

(8)

This is the leading post-Einstein correction, and we ana-lyze its most general structure and its effect in this paper.From the estimate (8), we see that other terms in Eq. (6)give even smaller corrections:

F 2 TrF rT2 p8

M4P;

T2rT T2F p10

M6P; T4 p

12

M8P:

(9)

Nevertheless, we list for completeness all 10 possible termsof the type F 2 in Sec. VI B and all 4 possible terms of thetype TrF in Sec. VIC.

III. GENERAL FRAMEWORK

In this section, we introduce the basic variables andmake sign and normalization conventions. To simplifythe algebra, we temporarily deal with the Euclidean sig-nature where the Lorentz group SO4 acting on flat in-dices A; B; C; . . . is locally isomorphic to the direct productSU2L SU2R. We return to Minkowski signature inthe final results.

A. SU2L SU2R subgroups of the Lorentz groupThe 4-component Dirac bi-spinor field in the spinor

basis is

c

_

; y c y; y_: (10)

The Lorentz SU2L SU2R transformation rotates the2-component Weyl spinors:

c !ULc ; c y! c yUyL; UL2SU2L; _!UR __

_; y_!y_UyR__; UR2SU2R: (11)

In the spinor basis, the (Euclidean) Dirac matrices are

A 0 A

A 0

!; 5

12 0

0 12

!;

A 12;ii; A y A :(12)

where ii 1; 2; 3 are the three Pauli matrices. We in-troduce the commutators

AB di

2A B BA iABi; AB y AB ;

AB di

2A B BA iABi; AB y AB ;

(13)

where , are t Hooft symbols. They are projectors ofso4 to the two su2 subalgebras. The basic relations fort Hooft symbols are

iABiCD ACBD ADBC ABCD;

iAB iCD ACBD ADBC ABCD;

iABjAB 4ij; iAB jAB 4ij; iAB jAB 0:

(14)

With Euclidean signature, there is no distinction betweenupper and lower flat (capital Latin) indices, in particular,1234 1234 1.The following commutation relations are helpful and

will be used below:

BC A ABC 2iBAC CAB ;BC A ABC 2iBAC CAB :

(15)

B. Tetrad

We introduce the frame field in the matrix form:

e d eAA : (16)Under Lorentz transformations, the frame field transformsas

e ! UReUyL; e ! ULeUyR: (17)Under the general differentiable change of the coordinatesystem x ! x0x, the frame field transforms as a worldvector:

ex ! e x0x@x0

@x: (18)

C. Covariant derivatives and curvatures

The requirement that the theory is invariant under local(point-dependent) SO4 Lorentz transformation demandsthat a compensating gauge field !AB !BA , called

LOW-ENERGY GENERAL RELATIVITY WITH TORSION: A . . . PHYSICAL REVIEW D 84, 124042 (2011)

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spin connection, must be introduced. With its help, oneconstructs the covariant derivatives,

DAB @AB !AB ; (19)

r @12 i4!AB

AB @12 iLi

i

2@ iL;

(20)

r @12 i4!AB

AB @12 iRi

i

2@ iR;

(21)

where LR are left (right) connections,

!AB 12Li

iAB

1

2Ri

iAB: (22)

Inversely,

Li 12 iAB!

AB ; R

i 12

iAB!

AB : (23)

In matrix notations,

L Li i

2; R Ri

i

2: (24)

Under gauge transformation (17), the covariant derivativestransform as follows:

r ! ULrUyL; r ! URrUyR; (25)corresponding to the usual gauge transformation of theconnections,

L!ULL i@UyL; R!URR i@UyR: (26)The commutators of the covariant derivatives are curva-

tures:

DDABF AB@!AB @!AB !AC !CB !AC !CB ; (27)

ir r F @L @L iLL

FiL i

2; (28)

ir r F @R @R iRR

FiR i

2: (29)

The SO4 curvature is decomposed accordingly into twopieces transforming as the 3; 1 1; 3 representation ofthe SU2L SU2R group:

F AB 12FiL iAB 12FiRiAB (30)where

FiL @Li @Li ijkLjLk;FiR @Ri @Ri ijkRjRk

(31)

are the usual Yang-Mills field strengths of the SU2 Yang-Mills potentials Li and R

i. These field strengths are

projections of the full curvature:

F L FiL i

2 14AB F AB;

F R FiR i

2 14AB F AB:

(32)

D. Torsion

The antisymmetric combinations

r e e r r e er DAB eB DAB eBA 2TAA ;

r e e r r e er DAB eB DAB eBA 2TAA ;

(33)

define the torsion field TAd 12 DeA DeA DeA. It is a 4-vector with respect to Lorentz trans-formations, and an antisymmetric rank-2 tensor with re-spect to diffeomorphisms. Contracting it with eA, one getsa rank-3 tensor T

T TAeA.The torsion tensor T

has 24 independent compo-

nents. It is convenient to decompose T into the totally

antisymmetric part related to an axial vector (a, 4 com-ponents), the trace part related to a vector (v, 4 compo-nents), and the rest of the 16 components (t

) subject to

constraints [3,20,21]:

T 23 deteag 23v 23t; (34)

where, inversely,

a

4 deteT;;v T;

t T T gT;:

The reduced torsion tensor t satisfies 4 constraints

t; 0 and 4 constraints t 0, therefore, ithas 16 degrees of freedom, as it should.The general 24-component spin connection can be pre-

sented as a sum of the zero-torsion part (4) and the torsionpart:

!AB !AB eAeBT T T !AB 23ABCDeCeDa 43eB eAv 23eAeBt; t; t;: (35)


124042-4

E. Affine connection

In the Einstein-Cartan formulation, the tetrad eA and the

spin connection!AB are primary fields, whereas the metric

tensor g and the general affine connection are

secondary, defined through the first pair.The quantity DAB e

B DeA is a vector in the flat

space, therefore, it can be decomposed in the frame field eA

which forms a basis in the flat space. We denote theexpansion coefficients by ,

DeAd eA; (36)which serves as a definition of the affine connection . It

is equal to the sum of the standard Christoffel symbol (orLevi-Civita connection) to which the general affine con-nection reduces in the zero-torsion limit,

12g@g @g @g; (37)

and the torsion part,

T T T: (38)

F. Affine curvature and Riemann tensor

Equation (36) can be rewritten as

reA !AB eB; (39)where

rd @ (40)

is the standard affine covariant derivative. The commutatorof two covariant derivatives defines the affine curvaturetensor

rrd R;@@;(41)

which is related to the curvature (27) built from the spinconnection:

R; eAeBF AB: (42)

The all-indices-down curvature tensor and the generaliza-tion of the Ricci tensor are

R;eAeBF AB; Rd R;geAeBF AB : (43)

Let us denote with the bar the curvature defined by

Eq. (41) but built from the symmetric Christoffel symbol (37). R; is then the standard, zero-torsion Riemann

tensor satisfying the following relations:

R; R;;R; R; R; 0;

R; 0;R R:

(44)

These relations are not valid in the general case for theunbarred curvature R; (43) if torsion is nonzero.

IV. THE FERMIONIC ACTION

In this section, we construct all possible bilinear fermionactions with zero and one covariant derivatives. To makesure that we do not miss any terms, we prefer to use thetwo-component formalism, see Sec. III A. Any action is, inprinciple, allowed that is (i) diffeomorphism-invariant and(ii) invariant under local Lorentz transformations (11). Thefirst requirement means that, if only covariant (lower)indices are used for e , r , F , they must be allcontracted with the antisymmetric , in order to com-pensate for the change of coordinates in the volume ele-ment d4x. The second requirement means that, because ofthe gauge transformation laws (11), (17), and (25), one hasto alternate subscripts plus and minus in the chain.Fermion operators are at the ends of the chain, such that theplus is followed by the Weyl field c whereas theminus is followed by the Weyl field . On the contrary,c y is always followed by a minus whereas y isfollowed by a plus.

A. Zero-derivative terms

The zero-derivative fermion action can have only twostructures:

Sf0 1

4!

Zd4xif01c ye e ee c

f02ye e ee Zd4xi detef01c yc f02y: (45)

Both terms are Hermitian if f01 and f02 are real, since c ,c y and , y are Grassmann variables. The two termstransform into one another under parity transformation.Therefore, if parity is not broken f01 f02 m. Inthis case, the two terms combine into the mass term:

Sf0 iZd4x detemy: (46)

B. One-derivative terms

Since all covariant indices must be contracted with, the total number of covariant indices belonging toe , r must be four. In one-derivative terms, one cova-riant index belongs tor , therefore, the number of framefields must be three. It means, first of all, that the number of


124042-5

pluses and minuses is odd, therefore, all possibleone-derivative terms are necessarily off-diagonal in theWeyl fields c , . It is sufficient to consider operators ofthe type y . . . c since the opposite order c y . . . willbe obtained by Hermitian conjugation.

A priori one can construct many terms satisfying theseconstraints, however, all of them can be reduced, using thealgebra from Sec. III A, to the following four terms:

ye e er c ; ye er e c ;yer ee c ; yr e ee c :

(47)

The covariant derivative in these strings acts either onthe Weyl field or on the tetrad. One can commute thederivative to the utmost right-hand position, therefore,the only term with the derivative of the fermion field isthe usual Dirac term, ye e er c . All otherterms contain derivatives of the tetrad, which should beantisymmetrized in the covariant indices to make worldtensors under diffeomorphisms. This forms the torsiontensor (33). Therefore, apart from the Dirac term,Eq. (47) describes the following three structures:

yTe e c ;ye Te c ;ye e Tc

(48)

where T TAA . In addition to the four terms inEq. (47), one can consider

yec TrTe andyec TrTe :

These two terms are in fact identical and equal to12

ye e Tc yTe e c belonging tothe set (48). Further on, the three terms in Eq. (48) arenot independent as there is an algebraic identity

Te e 2e Te e e T 0;leaving us with only two terms with the torsion field, say,the first and the last term in (48).In principle, one can build diffeomorphism-invariant

actions using two Levi-Civita symbols contractedwith one r and seven e but divided by dete such thatthe expression is again invariant under the change ofcoordinates, together with the volume element. However,all such expressions are in fact identical to linear combi-nations of invariants listed in (47).The resulting three independent fermion actions can be

presented in a more simple form. We use the decomposi-tion of the torsion tensor (34) and notice that actually only8 out of the possible 24 components of the torsion fieldcouple to fermions in this order: the traceless symmetricpart t

decouples. Indeed, one has:

ye e er c 6deteyer c ;yTe e c 8dete

12va

yec ;

ye e Tc 8dete12va

yec :

Therefore, the most general one-derivative fermion actionhas the form

Sf1 Zd4x detef10yer c f11aec

f12vyec h:c:; (49)where f1i are arbitrary complex numbers.It should be noted that the operator r in the first term

contains the full spin connection ! which, according to

Eq. (35), can be written as a sum of the zero-torsion part! (4) and terms proportional to torsion. The difference of

two operators is from (35) and (34) identically

er r ev a; (50)where r is the covariant derivative computed in thezero-torsion limit. We have

Sf1 Zd4x deteg0ye r c g1ayec

g2vyec h:c:; (51)where the new couplings are g0 f10, g1 f11 f10,g2 f12 f10.

C. Hermitian action

We now add explicitly the Hermitian conjugate action inEq. (51). It is straightforward for the second and thirdterms; changing the order of the fermion operators bringsthe minus sign. The Hermitian conjugation of the first termis more involved. We first take its complex conjugate,interchange the order of fermion operators ; c , inte-grate by parts, and get

deteye r c h:c: c y r e dete:We next write e dete 1=6e ee anddrag r to the right through this expression. Since tor-sion in r is by construction zero, it commutes with thetetrads owing to Eq. (33), and we get

c y r e dete c ye dete r :We, thus, obtain an explicitly Hermitian fermion actionwritten in the 2-component spinor Weyl form


124042-6

Sf1 Zd4x detefg0ye r c g0c ye r

ag1yec g1c ye vg2yec g2c yeg: (52)

The constant g0 can be made real by redefining theoverall phases of c , c y and , y. Indeed, if the argumentof g0 is (g0 jg0jei), the phase rotation c !expi=2c , c y ! c y expi=2, ! expi=2,y ! y expi=2 obviously makes g0 real, and itcan be further on put to unity by rescaling of the c , fields. Therefore, we can put g0 1 to make the Dirackinetic energy term standard.Finally, recalling the definition of the bispinors (10) and

the Dirac matrices (12), we rewrite the action (53) in the4-component Dirac form:

Sf1 Zd4x detey D ag1 g1 5

vg2 g2 5 (53)where

g1;2 g1;2 g1;2

2;

eAA;D @1 18AB !AB :

(54)

The action (53) is by construction andmanifestlyHermitian.In the minimal model often discussed in the literature

[1,5,22], the only source of the fermion interaction withtorsion is the Dirac term (1) with the full spin connectionincluding its torsion part. In this case, g1 1 and allother constants are zero, therefore, only the axial part of thetorsion couples to fermions. The term proportional to g2was first considered in Ref. [7] although in another form,see also its discussion in Ref. [6]. These authors take thefollowing Lagrangian generalizing the minimal model:

Sf Zd4x dete

1 i

2yD

1 i

2

Dy:

Integrating the second term by parts, we bring thisLagrangian to our form (53) with the particular values ofthe constants: g1 1, g2 , the rest being zero.The complete list of four one-derivative fermion-torsion

actions (53) was presented in Ref. [23], where in addition,nine terms with one extra derivative were suggested.

D. From Euclidean to Minkowski signature

The standard dictionary translating Euclidean intoMinkowski variables (see, e.g. [24]) reads

0 4E; i iiE; E; iyE; 5 i5E; S iSE:

Therefore, we obtain from Eqs. (46) and (53) the fullMinkowski fermion action with zero and one derivative:

Sf Zd4x dete fi D ag1 ig1 5

vg2 ig2 5 mg: (55)Assuming a is an axial and v is a vector field, the terms

with g1 and g2 break P-parity; other terms are parity-even. We remind that for a quantum amplitude, one takesexpiS.

V. TORSION AS AN ABELIAN GAUGE FIELD

If parity is conserved, one has to put g1 g2 0 inEq. (55). We denote the two nonzero constants that are leftas g1 ga [it is real by construction (54)] and g2 igv(it is purely imaginary), and rewrite the fermionLagrangian with torsion as

D il 1

5

2 ir 1

5

2

;

l gvv gaa; r gvv gaa:(56)

This Lagrangian is clearly invariant under the AbelianU1L U1R gauge transformationL c ! eixc ; R ! eix;l ! l @; r ! r @: (57)

This invariance, in the U1V U1A form, has beenpreviously noticed in Ref. [25].Therefore, the inclusion of torsion is equivalent to pro-

moting the fermion part of the standard gravity invariantunder the Lorentz SU2L SU2R gauge group to beinginvariant under the larger U2L U2R gauge group!If this gauge symmetry is preserved by the bosonic part

of the action, it has to depend only on the curvatures fL @l @l, fR @r @r. Linear terms are zero,so the expansion starts with quadratic terms in f. The

vector part, v, is then identical to the photon. It should be

mentioned that, since it is an Abelian field, one is free toascribe arbitrary coupling constants or charges withwhich this field interacts with various fermion species.To be separated from the photon, this field has to haveother charges with respect to fermion species, and bemassive. The only thing we know is that its mass must belarger than the experimental restriction on neutral inter-mediate bosons, of the order of 1 TeV. The same applies totheU1 axial boson a or their linear combinations l, r.The appearance of mass terms for l, r fields means

breaking of the U1L U1R gauge symmetry, whichcan be either explicit or spontaneous by some kind of a


124042-7

Higgs effect. As we shall see in the next section, addingterms quadratic in torsion implies explicit breaking of theU1L U1R symmetry. In principle, there is nothingwrong about it as the gauge symmetry is Abelian.Probably, spontaneous breaking would be more aestheticbut in the absence of the microscopic theory we can onlyspeculate about it.From the viewpoint that torsion fields a, v are just

another set of gauge vector bosons interacting with fermi-ons, we do not see compelling reasons why their massesshould be of the order of the Planck mass, as suggested bythe gravitational approach to torsion: with our presentlack of deeper understanding, the masses can be anythingbeyond the phenomenologically established limits [26].If parity is not conserved (meaning g1 , g

2 are nonzero),

one can still consider l and r as compensating gauge

fields. However, then they have to be complex, and com-pensate point-dependent real dilatations of the chiral fieldsc , and not only their phases. A discussion of thisinteresting topic lies beyond the scope of the paper.

VI. THE BOSONIC ACTION

Taking a purely phenomenological stand, one may in-quire what terms in the bosonic action can be written thatpreserve (i) diffeomorphism-invariance and (ii) invarianceunder the gauge Lorentz group. In this section, we give thefull list of invariants quadratic in torsion, invariants qua-dratic in curvature, and invariants that are linear in torsion

but containing r R .

A. Invariants quadratic in torsion

A general way to construct quadratic invariants is toconsider the following invariant under the diffeomorphism,

KACD;BEF deteTAeCeDTBeEeF; (58)

and to contract the flat indices into a Lorentz-group scalar.Since torsion is antisymmetric in world indices, this ex-pression is antisymmetric in CD and EF, meaning thatthe pairs of the frame fields belong to the 6-dimensionalrepresentation of the SO4 Lorentz group or to the 3; 1 1; 3 representation of the SU2L SU2R group, whileTA belongs to the 2; 2 representation of that group.The direct product of TA and eCeD belongs to the

2; 2 3; 1 1; 3 2; 21 2; 22 2; 4 4; 2representations, which should be multiplied by the samecombination. There are 5 singlets arising from

2; 21 2; 21; 2; 21 2; 22; 2; 22 2; 22;2; 4 2; 4; 4; 2 4; 2:

Therefore, there are precisely five linear independent in-variants which we write as

K1 deteTATAgg

dete83aa 23vv 49tt

;

K2 deteTATBeAeBg detevv;K3 deteTATBeAeBg

dete83a

a 13vv 29tt;

K4 12TATA dete83av 29tt;;K5 TATBeAeBg

dete83av 19tt;: (59)In the last column, we used the decomposition of the

torsion tensor (34). The last two terms are P, T-odd; the firstthree are even. The first four terms have been known for along time as they emerge from the leading Einstein-Cartanterms (2), see below. To the best of our knowledge, the fifthinvariant appears for the first time in thevery recentRef. [27].We now recall that in the Einstein-Cartan formulation

there are two leading terms linear in the curvature, see thesecond and the third terms in Eq. (2). Following the generalstrategy, we split them into a piece that survives in the zero-torsion limit, plus corrections from torsion.One has [2,28,29]

14

ABCDF ABeCeD

gp R K1 4K2 2K3 4@ gp v; (60)12

F ABeAeB 2K4 4@gp a: (61)

We see thus that four out of possible five terms quadraticin torsion are induced by terms linear in curvature, withconcrete coefficients. We shall, however, consider the gen-eral case where the terms K1...5 are included in the bosonicpart of the action with arbitrary real coefficients k1...5:

Sb2Zd4x

1

2

X5m1

kmKmT

Zd4xdete1

2

M2aaa

a2M2avavM2vvvv

M2tttt1

2M2ttt

t;

dete: (62)

The new constants having the meaning of the massessquared of the torsion fields are linear combinations ofthe original constants:

M2aa 8k3 k13 ; M2av 4k4 k53 ;

M2vv 2k1 3k2 k33 ; M2tt 22k1 k39 ;

M2tt 22k4 k59 :


124042-8

The second and fifth terms are P, T-odd, the rest are even.The first three terms in Eq. (62) are mass terms for the a,

v bosons or for their linear combinations l, r, that

break explicitly the U1 U1 gauge symmetry dis-cussed in Sec. V.The system must be stable with respect to small low-

momenta fluctuations of torsion about the flat space. Itmeans that all eigenvalues of the mass matrix (62) mustbe positive. This condition requires that

M2vvM2aa>0; M2vvM2aa>M4av; M2tt >M2tt; (63)which is satisfied in a broad range of the constants k15.See also the discussion of the positivity of the mass matrixin Ref. [27].In the minimal model corresponding to extracting

torsion terms from the leading-order action (2) only, seeEqs. (60) and (61), one obtains

M2minaa 83M2P16

; M2minvv 83M2P16

;

M2mintt 89M2P16

; M2minav 83M2P16

;

M2mintt 89M2P16

;

(64)

where the iota parameter is the coefficient in front of theP, T-odd action (61). We coincide in this table of masseswith Ref. [30], after adjusting the normalization.The eigenvalues of mass-squared matrix for a, v are

1 2

p8=3M2P=16. A check of the above algebra is

that at purely imaginary values, i, the eigenvaluesare zero. Indeed, at these values the self-dual or anti-self-dual combination F AB i 12 ABCDF CD drops out of theaction (2).At real values of the iota parameter, one of the eigen-

values is always negative. It means that the path integralover a, v fields strictly speaking does not exist, there-

fore, the minimal model cannot be complete.

B. Invariants quadratic in curvature

Such terms arise from the diffeomorphism-invariantstructure

GABCDEFGH deteF ABF CD eEeFeGeH (65)belonging to the 6 6 6 6 representation of theLorentz group, out of which one can extract 10 indepen-dent Lorentz-group invariants. Here is their list, expressedthrough the full Riemann tensor (43):

G1 116 dete ABCDF ABeCeD2

gp R2;G2 14 deteF

ABF AB

gg gp R;R;;G3 116 deteF

ABF CD

ABEFCDGHeEeFe

G e

H gp R;R;;

G4 14 ABCDF ABF CD

gp R2 4RR R;R;;G5 14 deteF

ABF CD

eAeBe

Ce

D gp R2 4RR R;R;;

G6 1dete F ABeAeB2

1gp R;2;G7 14e

ABCDF ABeCeDF ABeAeB RR;;G8 F ABF AB R;R;

G9 14 deteFABF CD

ABCDgg R;R;

G10 1deteFABF CD

CDEFeAeBe

Ee

F R;R:

(66)

Invariants 710 are P, T-odd, the rest are even. G4 and G8 are full derivatives even if torsion is nonzero. The P, T-eveninvariants G16 have been first constructed by Neville [31].In the zero-torsion limit, one replaces R; ! R; which satisfies the relations (44). Therefore, in this limit one has

G2 G3, G4 G5, G6;7 0, G8 G9 G10. Thus, in the zero-torsion limit one is left, apart from two full derivatives,with only two well-known invariants, namely,


124042-9

gp R2 and gp R R: (67)C. Invariants linear in torsion

Using the covariant derivative of the curvature it ispossible to construct invariants that are linear in torsionand linear in rR. The general structure from which allinvariants of this kind can be derived is

LABCDEFGH

dete r R;TAeBeCeDeEeFeGeH (68)

where r is the covariant derivative with the no-torsionChristoffel symbol (37). It belongs to the 63 42 represen-tation of the Lorentz group, which contains 20 singlets:

L01 deteTAeA r R;;L02 deteTAeA r R;;L03 deteTAeA r R;L04 deteTAeA r R;L05 deteTAeA r R;L06 deteTAeA r R;;L07 TAeA r R;L08 TAeA r R;L09 TAeA r R;L010 TAeAg r R;;L011 TAeAg r R;;L012 deteTAeA r R;L013

1

dete TAe

Ar R;;

L014 1

dete TAe

Ar R;;

L015 1

dete TAe

Ar R;;

L016 TAeA r R;;L017 TAeA r R;L018 TAeA r R;L019 TAeAg r R;;L020 TAeA r R;:

However, many of these invariants are zero or reduce toone another when one takes into account the additionalsymmetries of the standard Riemann tensor R,

R; R;; (69)

R; 0 or R; R; R; 0;(70)

as well as the Bianchi identity,

" r R; 0 orr R; r R; r R; 0:

(71)

Contracting Eq. (71) with the metric tensor, one obtains theidentities for the Ricci tensor and the curvature:

r R; 2 r R; @ R 2 r R: (72)It is also helpful to keep in mind that the covariant deriva-tive of the metric tensor and of the combination1= dete are zero.We immediately find that L01 L02, L03 L04, L08 L09,

L010 L011, L017 L018, and L08 0 because of Eq. (69),L012 L014 L015 0 because of Eq. (70), and L09 L010 L011 L013 0 because of Eq. (71). The invariantsL01, L02, and L06 are proportional owing to Eq. (72). Theinvariants L010 L011 0 owing to Eqs. (70) and (71).Therefore, actually only four linear independent invariantsare left:

L001 deteTAeA@ R detev@ R;L002 TAeA@ R 4 detea@ R;L003 deteTAeA r R

13detev@ R 23 detet;

r R;L004 TAeA r R

23detea@ R 23

t

r R;

and no linear combination of these invariants is a fullderivative. They can be recombined in a simpler way:

L1detev@ R; L2detea@ R;L3detet; r R; L4t r R:

(73)

Since these invariants are linear in torsion, they arepotential sources of torsion even in the absence of fermi-ons, including the reduced torsion part, t

.

We end up the derivative expansion here. We do not

consider four derivative terms of the type rT2, rTT2,RT2 and T4 (there are many dozens of such terms) sincethey lead to even smaller corrections to the Einstein equa-tion than the T2 terms listed in Sec. VIA and considered inthe next section, see the estimate in Sec. II. However, allfour derivative terms are, by dimension (which is 4), onequal footing from the point of view of the ultravioletrenormalization of the theory about curved backgroundwith generally nonzero torsion. Therefore, they shouldall be included, for example, in the asymptotic safetyapproach [10,32].


124042-10

In the logic of the effective Lagrangians, which weassume in this paper, the gravitational action is an infiniteseries in @2=M2P such that it makes no sense in studyingthe stability against small runaway fluctuations from flatspace-time in the concrete p4 order since the inversepropagator of the fields is an infinite series in p2=M2P.What makes certain sense, is to study the stability of flatspace-time at vanishing momenta but that is decided bythe two-derivative terms. In addition to the usual condi-tion that the Newton constant (or M2P) is positive, thenew requirement is that the eigenvalues of the T2 matrixare positive: this condition is summarized in the inequal-ities (63).It should be added that in the Einstein-Cartan formula-

tion, none of the thinkable diffeomorphism- and localLorentz-invariant action terms is strictly speaking stableunder large nonperturbative fluctuations of the frame andspin connection fields [33]. This observation strengthensthe argument that the present-day gravitation theory is butan effective low-energy one, and therefore it makes senseto study systematically order by order the possible effectsof the higher derivative terms.

VII. INDUCED FOUR-FERMION INTERACTION

If we ignore the Tr R terms (73) that lead to high-orderterms r R2 if we exclude the torsion, we are left withterms quadratic in torsion (62) and terms linear in torsioncoupled to fermion currents (55). It is important that in theleading order only the a, v part of the torsion couples to

fermions. In the next order, however, when a derivative isadded, the reduced torsion part t

may also couple to

fermions [23].In the leading order, if one integrates out the torsion, the

reduced torsion t vanishes, whereas the Gaussian inte-

gral over a, v produces the 4-fermion interaction

Lagrangian

L4 dete

2M2aaM2vvM4avABABg21 M2vv2g1 g2 M2av

g22 M2aaVBVBg21 M2vv2g1 g2 M2avg22 M2aa2ABVBg1 g1 M2vvg1 g2 g1 g2 M2avg2 g2 M2aa

gp hAAABABhVVVBVB2hAVABVB; (74)where AB B5 is the axial and VB B is thevector current. The dimensionless constants g1;2 are de-fined in Eq. (55) and the masses Ma;v are defined inEq. (62). The A V interaction term is C, P-odd andT-even.Certain particular cases of this Lagrangian have been

considered before. For example, to compare it with thepaper by Freidel et al. [7] we take g1 1, g2 (see

Sec. VI C) and the minimal model values of the torsionmasses (64) with the identification 1=,M2P 2=G.In this case, our general Eq. (74) reduces to Eq. (23) ofRef. [7].

VIII. STRESS-ENERGY TENSOR FROMFOUR-FERMION INTERACTION

If the a, v masses are of the order of the Planck mass

as in Eq. (64), the 4-fermion Lagrangian (74) leads to acorrection to the cosmological equation of the order ofp2=M2P, where p is the characteristic momentum of thefermion matter, for example, temperature. Therefore, it is atiny correction unless p approachesMP but then one has totake into account higher terms in the derivative expansion,that are being neglected.As discussed in Sec. V, the addition of torsion to the

General Relativity in the fermion sector enlarges the gaugesymmetry of gravity from the Lorentz SU2L SU2Rgroup to theU2L U2R group which we know must bebroken by the spontaneous or explicit masses of the a, vvector bosons. However, these masses need not be of theorder of the Planck mass but could be much smaller, say, ofthe order of 10 TeV [26]. In this case, the 4-fermioninteraction (74) could be an important correction in theepoch preceding the electroweak phase transition.Anyway, there is an interesting problem of evaluating

the contribution of the 4-fermion interaction to the stress-energy tensor in the r.h.s. of the Einstein-Friedman cosmo-logical equation. This problem has been addressed, e.g. inRefs. [1114] using the ideas of a spin fluid [3436]. Wethink that this approach is unsatisfactory. Particles withspin one-half are always quantum, for example, there areexchange effects, and that cannot be mimicked by anysemiclassical model. At some point in the above referen-ces, one has to average the spin-squared operator hs2i. Thisquantity is replaced by 1=4, why not 3=4?Meanwhile, averaging 4-fermion operators over a fer-

mion medium is a common problem in Quantum FieldTheory. With other fermion interactions (temporally)switched off, the contribution of the 4-fermion average tothe Lagrangian is given by two termsthe direct(Hartree) term and the exchange (Fock) term, corre-

sponding to two possible contractions of the , opera-tors by the fermion propagator Gp, see Fig. 1:

FIG. 1. Two contributions to the average of 4-fermion inter-action: Hartree (left) and Fock (right).


124042-11

h 1 2i

12

Z d4p124iTrGp11

Z d4p224iTrGp22

12

Z d4p124i

Z d4p224iTrGp11Gp22; (75)

where 1;2 can be arbitrary Dirac and fermion flavormatrices. If other fermion interactions need to be taken intoaccount, one has to dress the propagators and the4-vertex.We emphasize that averaging over the medium should

be performed in the Lagrangian; the corresponding correc-tion to the stress-energy tensor is then obtained by varyingthe Lagrangian with respect to g. Averaging in theequation of motion makes no sense.In what follows, we illustrate the use of Eq. (75) by

taking a medium of noninteracting fermions.

A. Averaging 4-fermion interaction in anoninteracting medium

We consider one species of fermions with mass m attemperature T and chemical potential which corre-sponds to certain charge density , to be specified below.It should be stressed that in a relativistic theory there is nostrict way of separating particles from antiparticles: In aheat bath, both particles from the upper continuum andholes (antiparticles) from the lower continuum are excited;the chemical potential regulates the difference between thenumber of particles and antiparticles, which is the onlywell-defined quantity.Neglecting interactions, one writes the free fermion

propagator

Gp 1m 6p

1

m i!n0 pii

m i!n0 pii

m2 p2 i!n2; (76)

where !n 2Tn 12 are the (imaginary) Matsubarafrequencies. Integration over p0 becomes a summationover Matsubara frequencies:

Z dp02i

. . . T X1n1

. . .

The charge density of the fermion gas is given by

hj0i h 0i TXn

Z d3p23 TrGp

0

4TXn

Z d3p23

i!n"2 i!n2

2Z d3p23

1

e"=T 11

e"=T 1; (77)

where " p2 m2p . This is nothing but the differencebetween Fermi-Dirac distributions for particles (positive) and antiparticles (negative ). The integral can beeasily evaluated in certain limiting cases; in particular, inthe ultrarelativistic case m , T one has

T2

3

3

32m2

22Om4: (78)

The second term dominates at high densities when themassless fermion gas becomes degenerate. One can extractthe chemical potential as function of from this equation.The thermodynamic potential of the fermion gas is [37]

2VTZ d3p23 loge

"=T1e"=T1

72

180T41

6T22 1

1224VOm2: (79)

We now compute the Hartree (direct) part of the4-fermion interaction keeping in mind that in our case1;2 ; 5, see Eq. (74). We note immediatelythat the axial current does not contribute to the Hartreepart as TrGp5 0. It could have contributed werethe chemical potential different for left- and right-handedparticles but we do not consider this possibility here. Forthe vector current, we have two independent loop integrals(and traces), each of which is exactly of the same type asfor the calculation of the charge density. Therefore, weobtain

hVBVBiHart h B BiHart 122;

hABABiHart 0; hABVBiHart 0: (80)

In the Fock (exchange) part, the loop integrals and sumsover Matsubara frequencies again factorize, however, thetrace does not. We obtain

hVBVBiFock142m

2

2

2; hABABiFock 142m

2

2 2;

hABVBiFock0; (81)where we have denoted

4TXn

Z d3p23

1

"2 i!n2

2Z d3p"23

1 1

e"=T11

e"=T1: (82)

At m , T, one gets

T2

6

2

22:

The 2 terms combine with the Om2 corrections to thecharge density (78).


124042-12

The full, Hartree plus Fock, contributions are

hVBVBiHartFock 342 m

2

2

2;

hABABiHartFock 142 m

2

2

2;

hABVBiHartFock 0:

(83)

These expressions should be put in the 4-fermionLagrangian (74) and we get

L4-ferm gp hVV3

42m

2

2

2hAA

1

42m

2

2

2:

(84)

We stress again that here is the charge density, i.e. thedensity of particlesminus the density of antiparticles, there-fore it is expected to be small in the cosmological context.

B. Derivation of the stress-energy tensorfrom the Lagrangian

If the Lagrangian is known, the corresponding stress-energy tensor if found by the general rule

2 1gp @L@g derivative terms: (85)The problem therefore is to establish the dependence of(84) on the metric tensor. This can be easily done if onerealizes that T and are actually the zero components of4-vectors, as seen from their use in Eq. (76).Let us illustrate how this logic works by finding the

correct stress-energy tensor in the leading term in thefermion action. We take the first term in the thermody-namic potential (79), the simple case of the Stefan-Boltzmann law for massless fermions, 72T4=180. The partition function is

Z expT

exp

iZdtZd3x=V

;

where t is now the Minkowski time. It means that thecorresponding Minkowski LagrangianL =V gener-alized to the case of an arbitrary metric is

L 72

180

gp TTg2; T T; 0; 0; 0:Using the general Eq. (85) we find immediately that in thecomoving frame

72

180T4g 400

72

60

T4; 0;13T

4; 1; 2; 3: ;

which gives, of course, the correct energy density 00and pressure p112233 3 for the ultrarelativ-istic fermion gas. In fact, 3p holds true for any relation

between T and in that gas, since the dilatational current isconserved when there are no dimensional world constants,hence

0.

We now find the stress-energy tensor following from the4-fermion Lagrangian (84). It will not satisfy the relation 3p anymore since the couplings hVV and hAA are notdimensionless but are of the order of 1=M2 whereM is themass of a, v bosons. Neglecting the fermion mass we

have

L4-ferm 3

4hVV 14 hAA

gp 2jm0;2jm0

2T4

9 2

4T2

92

6

94:

(86)

At first glance, when we promote, T to be 4-vectors thereis an ambiguity: one can write 2T4 as T T2 oras T2T T or their combination. However, the resultis independent of the decoding as long as and T remainparallel, which is the case in the comoving frame. Whatcounts, is the total power of the polynomial in , T, whichis 6 in this case. Using the general rule (85) we obtain forultrarelativistic fermions:

4-ferm jm0 34hVV 14hAA2jm0g 600;(87)

implying 4-ferm 5p4-ferm. This equation of state for the4-fermion piece can be independently checked by using thegeneral thermodynamic relations, see, e.g. Eq. (1.4) of[38]. Indeed, if Z exp=T is the partition func-tion, one has

pressurep T @ lnZ@V

;

chargeQ T @ lnZ@

;

charge density QV;

entropyS @T lnZ@T

;

energyE pV TSQ;energy density E

V:

In our case, we have from Eq. (86)

lnZ4-ferm 34hVV 14hAA2jm0V

T:

From the above general relations, one immediately finds

4-ferm; T 534hVV 14hAA2jm0 5p4-ferm; T;confirming Eq. (87). The equation implies that and T areused as independent variables.However, one may wish to express the stress-energy

tensor in terms of the charge density . To that end, one


124042-13

has to solve the equation T=V@ lnZ=@ with re-spect to and to substitute the function into thestress-energy tensor. In general, lnZ is a sum of the mainpiece (79) and the 4-fermion piece (86), such that is acomplicated function. But if the 4-fermion piece is a smallperturbation, one can easily find the linear correction to thezero-order following from Eq. (78). In this case, oneobtains

p4-ferm; T 3

4hVV 14 hAA

@2V@V

Qconst:3

4hVV 14 hAA

2 4-ferm; T:

The relation 4-ferm p4-ferm has been used inRefs. [1214] as following from the spin fluid approach.We see, however, that it is valid only if the term (81)from the Fock exchange contribution is neglected, and ifthe 4-fermion term is a small perturbation to the main partof the stress-energy tensor. If the 4-fermion interaction isthe leading term, as assumed for the early cosmologicalevolution in the above references, the equation of statebecomes 5p.

C. Can the 4-fermion interaction be observable?

On the whole, we come to rather pessimistic conclusionswith regard to the observability of the possible 4-fermioninteraction induced by integrating out torsion. If the4-fermion constants hVV , hAA are of the order of 1=M

2P as

assumed in the gravitational approach like in the minimalmodel discussed (and criticized) above, it seems hopelesssince it becomes significant only at particle momenta pMP, but at these momenta, we do not know the theory at alland anyway the derivative expansion fails. In addition, iffor some reasons the 4-fermion interaction becomes large,it must be included into the equation of state and nottreated as a perturbation [1214] when it is overwhelming.We have mentioned that the massesM of a, v bosons

could be of nongravitational origin and therefore be, say, ofthe order of 10 TeV. That would increase the torsion-induced 4-fermion interaction by 30 orders of magnitudeas compared to the previous case. Nevertheless, it is stillhardly observable. In an ultrarelativistic medium, the cor-rection of the 4-fermion interaction to the stress-energytensor is of the order of 2=M2T4, as compared to themain contributionT4, see Eq. (86). A chemical potentialin the TeV range is hardly imaginable.In general, it makes sense to introduce the chemical

potential only for conserved quantum numbers. There aremany conserved quantities in the late epoch, such as quarkflavors and baryon (B) and lepton (L) numbers. However,in a late epoch the density is small and the torsion effectsare probably negligible. The earlier we go into the evolu-tion the fewer quantum numbers are conserved. It wasthought some time ago that electroweak interactions break

B L but preserve B L quantum numbers, but today itis believed that both are broken since the Majorana type ofneutrino is preferable. It looks like there are no conservednumbers at all in the epoch preceding the electroweakphase transition [39], save the electric charge for whichthe chemical potential is zero. Therefore, it may wellhappen that in that early epoch the only possible contribu-tion to the 4-fermion interaction is due not to chargedensity (which is zero) but to the Fock exchangepart having the m2 piece, where m is the fermion mass,see Eq. (81). It may become large if there are superheavyfermions but then their contribution is suppressed by theBoltzmann factor expm=T, unless the temperature is ofthe same order of magnitude.Finally, we should mention the possibility that there is

no thermal equilibrium in the epoch preceding the electro-weak transition, meaning that temperature is not an ade-quate quantity. The Matsubara propagator (76) is thenirrelevant and should be replaced by

Gp 1m i2 6p

;

where is the relaxation time. In this case, 1= replaces,qualitatively, T and in the above equations for theestimate of the average 4-fermion interaction which, inprinciple, can then become sizable. We also mention aninteresting possibility that an interplay of the evolution outof thermal equilibrium, and the potential C, P violation bytorsion may lead to the baryon asymmetry of the Universe.

IX. CONCLUSIONS

We have systematically listed all possible invariants thatmay arise in General Relativity when one includes torsion,following the guiding principle of the derivative expansion.These include all possible invariants quadratic in torsion(5 invariants), quadratic in curvature (10), and linear intorsion and linear in the covariant derivatives of curvature(4). In the fermion sector, we have derived four possibleinvariants with torsion coupled to the bilinear fermioncurrents. We do not limit ourselves to P, T-even invariants.Some of the invariants are new, although most of them havebeen considered by different people before.In the leading one-derivative order, only 8 components

of torsion (out of the general 24) couple to fermions, whichcan be cast into the Abelian axial (a) and vector (v)

fields. If parity is conserved, the interaction of a, v fields

with fermions possesses gauge U1L U1R symmetry,in addition to the Lorentz gauge symmetry SU2L SU2R. Linear combinations of a, v are the gaugebosons of this additional symmetry.However, the bosonic torsion-squared invariants break

explicitly this symmetry as they provide masses to theAbelian bosons a, v or their linear combinations. From

this point of view, such mass terms may look unnatural:


124042-14

spontaneous breaking could be more aesthetic. Certainjustification for writing terms quadratic in torsioncomes from the fact that they appear anyway from expand-ing the Einstein-Cartan action. This is called the minimalmodel for torsion. However, we have shown that theminimal model leads to a nonpositive mass matrix for thea, v bosons, therefore the minimal model cannot be

complete.Assuming on purely phenomenological grounds that

there is a positive mass matrix and neglecting higher-derivative invariants, we integrate out the torsion fieldand obtain the effective four-fermion action. It contains,generally speaking, axial-axial, axial-vector, and vector-vector interactions. The effect of the first one has beenstudied in the past, with regard to its application to theEinstein-Friedman cosmological equation, using theso-called spin fluid approach. We find this approachunsound since particles with spin one-half are alwaysquantum (for example, there are exchange effects) andthat cannot be mimicked by any semiclassical model. Wepresent a systematic quantum field-theoretic method to

average the 4-fermion interaction over the fermion me-dium, and perform the explicit averaging in the case of freefermions with given chemical potential and temperature.The result is essentially different from that of the spinfluid approach.We arrive to rather pessimistic conclusions concerning

the possibility to observe any effects of the torsion-induced4-fermion interaction. However, under certain circumstan-ces it may have cosmological consequences, seeSec. VIII C, but this has not been worked out.

ACKNOWLEDGMENTS

We thank Prof. Victor Petrov and Prof. Maxim Polyakovfor helpful discussions and Prof. Friedrich Hehl and Prof.Ilya Shapiro for correspondence. This work has been sup-ported in part by Russian Government Grants No. RFBR-06-02-16786 and RSGSS-3628.2008.2, and by DeutscheForschungsgemeinschaft (DFG) Grant No. 436 RUS 113/998/01. A. T. acknowledges support from the DynastyFoundation.

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Low-energy general relativity with torsion: A systematic derivative expansion

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