PHYSICAL REVIEW D 72, 104002 (2005) Quantum gravity, torsion, parity violation, and all that
Laurent Freidel,1,2,* Djordje Minic,3,† and Tatsu Takeuchi3,‡ 1Perimeter Institute for Theoretical Physics, 31 Caroline St. N. Waterloo, N2L 2Y5, Ontario, Canada 2Laboratoire de Physique, Ecole Normale Supe´rieure de Lyon, 46 Alle´e d’Italie, 69364 Lyon, Cedex 07, France 3Institute for Particle Physics and Astrophysics, Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA (Received 16 September 2005; published 1 November 2005) We discuss the issue of parity violation in quantum gravity. In particular, we study the coupling of fermionic degrees of freedom in the presence of torsion and the physical meaning of the Immirzi parameter from the viewpoint of effective field theory. We derive the low-energy effective Lagrangian which turns out to involve two parameters: one measuring the nonminimal coupling of fermions in the presence of torsion, the other being the Immirzi parameter. In the case of nonminimal coupling the effective Lagrangian contains an axial-vector interaction leading to parity violation. Alternatively, in the case of minimal coupling there is no parity violation and the effective Lagrangian contains only the usual axial-axial interaction. In this situation the real values of the Immirzi parameter are not at all constrained. On the other hand, purely imaginary values of the Immirzi parameter lead to violations of unitarity for the case of nonminimal coupling. Finally, the effective Lagrangian blows up for the positive and negative unit imaginary values of the Immirzi parameter.
DOI: 10.1103/PhysRevD.72.104002 PACS numbers: 04.60.Ds
I. QUANTUM GRAVITYAND EFFECTIVE FIELD They correspond to the Euler class THEORY Z IJ KL The Wilsonian point of view on effective field theory is a 1 R ! ^R ! IJKL; (1) very powerful organizing principle allowing us to drasti- cally restrict the number of theories relevant for the de- the Pontryagin class scription of low-energy physics. As far as the low-energy Z physics is concerned, the Wilsonian approach instructs us RIJ ! ^R ; (2) to consider the most general local actions invariant under a 2 IJ given symmetry, while forgetting about the underlying microscopic degrees of freedom. Of course this philosophy and the Nieh-Yan class might have to be radically changed in the presence of Z gravity, but in this paper we adopt the usual point of I IJ 3 d!e ^ d!eI R ^ eI ^ eJ; (3) view in order to understand better the issue of gravitation- ally induced parity violation. IJ IJ IK J Gravity is a theory which is classically invariant under where R ! d! ! ^ !K is the canonical cur- vature two form. Since we are interested in semiclassical the diffeomorphism group and since we want to be able to 2 incorporate standard model-like fermions, the effective physics we concentrate our attention on the bulk terms. field theory also should be invariant under a local Two of the bulk terms are well known; they correspond to Lorentz gauge symmetry. The dynamical field variables the Einstein-Hilbert action and the cosmological constant are given by the frame field (a one form valued in the term. The corresponding coupling constants are, respec- I tively, the Newton constant G and the cosmological con- vectorial representation) e dx and Lorentz connection (a one form valued in the Lorentz Lie algebra) !IJdx .Ifwe stant . This is, however, not the final answer, for there exists an additional term which is usually disregarded. apply the effective field theory point of view to gravity, and From an effective field theory point of view, this term is consider only actions which are analytic functionals of the as important as the two others, and thus has to be taken into frame field connection and its derivatives,1 we easily read account in the low-energy effective action. The corre- out the most relevant terms in the low-energy effective sponding coupling constant is called the Immirzi parame- action. To leading order this low-energy effective action ter, . contains exactly six possible terms. Three of these terms Therefore, the leading bulk terms in the most general are topological invariants analogous to the QCD theta term. four-dimensional low-energy gravitational action, in any microscopic quantum theory of gravity (be it string theory, *Electronic address: [email protected] loop gravity etc.) are given by †Electronic address: [email protected] ‡Electronic address: [email protected] 1This excludes functionals which depend on the inverse frame 2Topological ‘‘deformations’’ of Einstein’s classical theory in field. four dimensions have been investigated in detail in [1].
1550-7998=2005=72(10)=104002(6)$23.00 104002-1 © 2005 The American Physical Society LAURENT FREIDEL, DJORDJE MINIC, AND TATSU TAKEUCHI PHYSICAL REVIEW D 72, 104002 (2005) Z 1 II. TORSION AND FERMIONS Se RIJ ! ^eK ^ eL eI ^ eJ P G IJKL 32 6 In this section we follow the derivation of Rovelli and 2 Perez [7], obtaining similar but crucially different results. ^ eK ^ eL RIJ ! ^e ^ e : (4) IJKL I J We start with the formulation of general relativity in the IJ first order formalism in terms of a Lorentz connection ! I Note that unlike the Newton and cosmological constant, and a frame field e , where I; J ... 0; 1; 2; 3 denotes the the Immirzi parameter is dimensionless. This action and internal Lorentz indices and ; ... 0; 1; 2; 3 the respec- the meaning of the Immirzi parameter was first discussed tive space-time indices. The curvature is defined to be by Holst [2]. Usually is taken to be equal to zero; this IJ IJ IJ IJ 3 R ! @ ! @ ! ! ;! . When the frame forces the torsion to be the null and we recover the usual field is invertible the gravitational action (4) which in- second order formulation of gravity. It is also frequent but cludes the Immirzi parameter and a zero cosmological less common to encounter the choice 1. In this case constant can be written in the form we recover the Cartan-Weyl formulation of gravity [3]. Z From the effective field theory point of view there is a 1 S e; ! d4xee e PIJ RKL ! ; (5) priori no reason to prefer one choice or another, and thus, G 16 G I J KL one should take into account all possible values of and let where we have introduced the following tensor and its experiment decide its numerical value. This is the point of inverse view which is taken here. Effectively, at the quantum level, this amounts to treating as a superselection parameter. 1 IJ PIJ I J KL ; We should add a caveat here: as briefly mentioned above, KL K L 2 in this paper we assume the usual local Wilsonian effective (6) 2 1 IJ theory, in which decoupling between the UV and IR de- P 1 IJ I J KL : grees of freedom is usually taken for granted. This never- KL 2 1 K L 2 theless does not have to be the case in the presence of The coupling of gravity to fermions is given by the action gravity [4], but we refrain from any further discussion of this important issue in this paper [5]. Z ie S e; !; d4x Ie r r Ie ; (7) Classically when there is no matter coupled to gravity F 2 I I the equations of motion for the connection imply zero I 4 torsion irrespective of the value of the Immirzi parameter. where are the Dirac matrices, and At the quantum level this is not the case and one expects r @ !IJ I J ; r ; r RIJ I J : observable effects associated with a particular microscopic 4 4 formulation of quantum gravity (such as loop quantum (8) gravity) which contains this parameter [6,7]. In [6] it is IJ IJ eIJ IJ e shown that controls the rate of fluctuation of the torsion We can decompose ! as ! ! C , where ! is at the quantum level and when different from 0 or 1,it the torsion free spin connection satisfying leads to a compactification of the phase space of gravity. e I The Immirzi parameter plays an important role in loop r e 0; (9) quantum gravity, which predicts, for example, that the area IJ and C is the so called ‘‘contorsion’’ tensor related to the of a surface or the volume of a spatial region are quantized I IJ 2 torsion in the obvious way r e C e J. [8] (the unit quanta of area and volume being lP and 3=2 3 It is important to consider in the fermion action the lP). The spectra of the area and volume operators are discrete provided that the Immirzi parameter is not equal to above real combination (a point overlooked in [7]) in order zero. In contrast, there is no indication that area or volume to get the same equation of motion for and . The are quantized in string theory (which might be because of general fermion action written in terms of the covariant the subtle issue of background independence). Thus any derivative r, which gives back the usual fermion action in direct experimental constraint of the Immirzi parameter the absence of gravity, can be written in the form can be taken as concrete steps towards the falsification of a Z ie ~ 4 I specific microscopic approach. This is the main motivation SF e; !; d x 1 i e r 2 I for the analysis that follows. I 1 i r eI : (10) 3The action (4) is singular when 0. One can however Such an action does not give the same equation of motion introduce an auxiliary field BI (a two form valued in the vectorial representation) in order toR rewrite the Immirzi term, up to a I I I 4 I J IJ boundary contribution, as B ^ TI T ^ TI, where T We use the particle physics conventions f ; g 2 with I IJ 5 0 1 2 3 d!e is the torsion. When 0 the equation of motion for diag 1; 1; 1; 1 , 5 i . The real- y y the B field forces the torsion to be null. ity conditions are I 0 I 0, i 5 0 i 5 0.
104002-2 QUANTUM GRAVITY, TORSION, PARITY VIOLATION, ... PHYSICAL REVIEW D 72, 104002 (2005) for and unless 1 A V K is different from 0 unless 0 or 6 IJ 1. This shows, as previously stated, that one should eI C 0: (11) take real in order to have a consistent action principle. If is not real this imposes a constraint on the torsion We can now obtain an equivalent action by replacing ! e tensor which is incompatible with the gravitational equa- with ! C in (5) and (10). The terms of (5) linear in tion of motion that fixes the torsion. This is due to the fact the fermion current are total derivatives, leaving us with that the following symmetric combination is not a total the standard second order tetrad action of general relativity derivative in the presence of torsion [3] with fermions, plus a four-Fermi interaction term 1 2 e Sint e; Sint e; Sint e; . This interaction term Ie r r Ie comes first from the evaluation of 2 I I Z I I J 1 1 @ eeI V eI C J V ; (12) S e; d4xee e PIJ C ;C KL: (19) int 16 G I J KL where VI is the vector fermion current (AI denotes the axial fermion current) One can use the following identity 2UV VI I ;AI 5 I : (13) e e PIJ C ;C KL 6 U2 V2 (20) I J KL Both VI and AI are real currents. The value 0 corre- L sponds to the usual minimal coupling of fermions to grav- if eI C JK IJKLU IJVK IKVJ, in order to get ity. In general, an arbitrary real value for corresponds to a 3 2 2 nonminimal coupling. 1 2 2 2 Sint e; G 2 A A V V : We now consider the equation of motion coming from 2 1 the variation of Eq. (7) with respect to the connection (21) S S = ! 0. Since G F The other contribution is given by Z SG 1 e r ee e PKL ; 2 4 IJK L IJ K L IJ (14) Sint e; d x eI C IJKLA 2 I JVK ! 8 G 4 (22) and leading to the final expression for the effective action S iee F K f ; g ; (15) 3 2 2 ! IJ 8 K I J S e; G A2 A V 2V2 : int 2 2 1 this equation of motion reads5 (23) Note that this effective action is similar yet different in 1 KL KL detail from the one derived in [7]. One sees that the only C KL C K eL P IJ IJ eKAL: (16) 2 G interaction which is independent of is the axial-axial interaction. The parity-violating vector-axial interaction is The solution is given by absent if one considers the minimal coupling 0. Thus parity violation cannot be taken as a measure of the 2 1 1 L Immirzi parameter but only of the combination 2 de- eI C JK 4 G 2 IJKLA I JAK : (17) 1 1 2 pending on the nonminimal coupling parameter. Note also that when the Immirzi parameter is purely imaginary If one does the same computation starting with the action i, the interaction becomes infinite. This infinite factor ~ (10) SF one obtains instead 6 K 2 L Another theoretical possibility is to have the constraint A 1 VK satisfied. This is possible if we have 1. In this e C JK 4 G IJKL A V I 2 1 2 case this constraint is satisfied if all fermions are right handed 1 ( 1) or left handed ( 1). However this theoretical I J A V K : (18) possibility is clearly excluded by experiments, since we see both right and left handed fermions and we will therefore not discuss this option in what follows. Note that this argument relies on the IK 2 It is clear from this expression that e C / = fact that in our framework the nonminimal coupling is I supposed to be the same for all species. It is logically possible to relax this constraint and study the case of a ‘‘species- 5 y We use the identities 0 0, f K; I J g dependent’’ [9]. We will not study this possibility which 5 L 2i IJKL , K; I J 4 K I J . seems unnatural.
104002-3 LAURENT FREIDEL, DJORDJE MINIC, AND TATSU TAKEUCHI PHYSICAL REVIEW D 72, 104002 (2005) 2 multiplies the term A i V and effectively imposes the where Nq is the total number of quarks of flavor q con- constraint between vector and axial-vector currents dis- tained in the nucleus. Therefore, cussed in footnote 2. X 0 We now turn to the discussion of experimental signifi- C1qhZ; Njq x q x jZ; Ni cance of the gravitational parity violation for both non- q minimal and minimal couplings of fermions. 3 C1u 2Z N C1d Z 2N r 1 III. PARITY VIOLATION: THE WEAK CHARGE Q Z; N 3 r ; (31) 2 W In this section we review the necessary physics needed e e 1 to discuss possible gravitational parity violation. where gL gR 2 in C1q is factored out so that We start with the low-energy effective Lagrangian de- QW Z; N depends only on the nucleus. scribing interactions between electrons and quarks induced Next, using the relation between Dirac spinors u p (in by Z exchange the Dirac representation) and nonrelativistic Pauli spinors , 8G X p F gq q q gq q q "# Leff L L L R R R 2 q p u p E m ~ p~ ; (32) e e gLe L eL gRe R eR ; (24) E m where we find that the time component of the electron axial- vector current matrix element in the nonrelativistic limit f 2 f 2 gL If Qfsin W;gR Qfsin W: (25) becomes I and Q are, respectively, the isospin and charge of the f f h j j i fermion. The parity-violating part of Eq. (24) is e pf e x 0 5e x e pi X 1 1 i p p x GF p p u p u p e i f p C q q e e f 0 5 i LPV 1q 5 2EfV 2EiV 2 q 1 y i pi pf r C2q q 5q e e ; (26) f ~ p~ f ~ p~ i ie : (33) 2meV where Inserting Eqs. (31) and (33) into (28), we find q q e e C1q 2 gL gR gL gR ; (27) G Z q q e e F 4 y ^ ^ 3 C2q 2 gL gR gL gR : M p QW Z; N d x’f x ~ p ~ r^ 4 2me The first term of Eq. (26) leads to a parity-violating inter- 3 r^ ^~ p^~ ’ x ; (34) action between electrons and nuclei whose amplitude is i given by where G X Z F 4 M p C1q d xhZ; Njq x q x jZ; Ni 1 2 ’ x p e ipx : (35) q V he pf je x 5e x je pi i; (28) From Eq. (34), we conclude that the first term of Eq. (26) where jZ; Ni denotes a nucleus consisting of Z protons and induces a parity-violating potential of the form N neutrons. G In the nonrelativistic or static limit of the nucleus, we F ^ ^ 3 3 ^ ^ V^ PV p QW Z; N ~ p ~ r^ r^ ~ p~ ; can neglect the space components of the quark vector 4 2me current matrix element. Its time component (36) 0 y hZ; Njq x q x jZ; Ni hZ; Njq x q x jZ; Ni q r where m , =^~ 2, p^~, and r^~ are, respectively, the mass, spin, (29) e momentum, and position of the electron. The factor yields the density of quark flavor q inside the nucleus. QW Z; N is called the ‘‘weak charge’’ of the nucleus and Since the size of the nucleus can be assumed to be small can be large enough for heavy nuclei to make the effects of compared to the wavelength of the momentum transfer this potential observable. pf pi , we can make the approximation The second term of Eq. (26) induces a potential depen- dent on the nuclear spin [10] which is too weak to be 3 q r Nq r ; (30) observed.
104002-4 QUANTUM GRAVITY, TORSION, PARITY VIOLATION, ... PHYSICAL REVIEW D 72, 104002 (2005) p IV. QUANTUM GRAVITYAND PARITY GFQW Z; N $9 2 GN Z N ; (45) VIOLATION so that the ‘‘effective’’ weak charge for the Immirzi pa- A. Nonminimal coupling rameter is According to the above discussion (Sec. II) the effective p Lagrangian contains a parity-violating term provided the 2GN QI 9 Z N : (46) coupling of fermions in the presence of torsion is non- GF minimal, i.e. Þ 0, The values of GN and GF are [11] 3 2 A 39 @ 2 2 L PV GN e A 5 : (37) GN 6:707 10 10 c GeV=c ; 2 2 1 (47) 5 3 2 GF 1:16 639 1 10 @c GeV : If we allow to be purely imaginary the overall effective p 33 coupling in the low-energy Lagrangian is imaginary ( So in natural units, the ratio of 2GN to GF is about 10 . being real), and thus unitarity is violated. Thus the purely The factor 9 is about 3 101, and the factor Z N is imaginary values of the Immirzi parameter are excluded by about 102 for heavy nuclei, so the effective weak charge is appealing to unitarity. Furthermore, the contact interaction about 10 30 . blows up at i. Note that i corresponds to the The weak charges of heavy nuclei are typically of order self-dual Ashtekar canonical formalism [7]. 102, and the experimental errors on them are of order 1 Let be real and let [12]. So the experimental constraint on will typically be 2 2 ; 1 < <1: (38) < about 1030; (48) 2 1 2 1 Then the above effective interaction becomes which is practically no constraint at all. Obviously the physical reason for this is the weakness of gravity as 3 L PV GN 5 ; (39) compared to weak interactions, and the fact that parity is 2 already maximally violated in the weak sector. which has the same form as the first term of Eq. (26) if we The crucial point here is that the parity-violating inter- assign the vector current to the quarks and the axial-vector action in principle contains two undetermined parameters, current to the electrons: the strength of the nonminimal coupling of fermions to gravity and the Immirzi parameter. This seems to have 3 L G q q e e : (40) been overlooked in the literature. PV 2 N 5 Therefore, the parity-violating interaction amplitude is B. Minimal coupling Z 3 X In this situation ( 0) the effective Lagrangian con- M G d4xhZ; Njq x q x jZ; Ni PV 2 N tains only the axial-axial coupling and the experimental q bound on the effective coupling is known in the literature he pf je x 5e x je pi i: (41) both on four-dimensional [3] and large extra-dimensional physics [13,14]. The effective action reads Using 2 3 A 0 y L G e ; (49) hZ; Njq x q x jZ; Ni hZ; Njq x q x jZ; Ni AA 2 N 2 1 A 5 5 3 q r Nq r ; (42) 3 2 3 where we set GN 2 2 in order to compare with 2 1 T we have [13,14]. X For example, the axial-axial contact four-Fermi interac- hZ; Njq x 0q x jZ; Ni 3 Z N 3 r ; (43) tion can affect the electron-quark contact interactions. A typical bound is [13,14] and using Eq. (33) we find T 5:3 TeV: (50) Z 9 GN Z N 4 y ^ ^ 3 A stronger constraint is implied by astrophysical data M PV d x’f x ~ p ~ r^ 4me provided one assumes the existence of light sterile neutri- 3 ^ ^ nos. From supernova data one infers [14] r^ ~ p~ ’i x : (44) 210 TeV: (51) Comparison with the Z-exchange case shows that the T corresponding coefficients are If we assume that is real, then
104002-5 LAURENT FREIDEL, DJORDJE MINIC, AND TATSU TAKEUCHI PHYSICAL REVIEW D 72, 104002 (2005) 2 constraint of an effective parameter involving both and < 1: (52) 2 1 can be discussed. In the case of minimal coupling ( 0) there is no parity violation and the effective Lagrangian Thus there is no bound on in this case, given the usual contains only the usual axial-axial interaction. Here the value of the Planck scale. bounds on the effective coupling, a function of the Immirzi parameter, are well known in the literature. These do not V. CONCLUSIONS impose any constraint on the real value of . On the other In this paper we have discussed the issue of parity hand, purely imaginary values of the Immirzi parameter violation in quantum gravity. In particular, we have clari- lead to violations of unitarity for Þ 0. Finally, the effec- fied the role of the coupling of fermionic degrees of free- tive Lagrangian blows up for i. dom in the presence of torsion as well as the physical meaning of the Immirzi parameter in the low-energy ef- ACKNOWLEDGMENTS fective Lagrangian. The low-energy effective theory is It is our pleasure to thank A. Perez and C. Rovelli for found to contain two parameters: the nonminimal coupling discussions of their work and A. Randono for a correspon- parameter ( ) and the Immirzi parameter ( ). Only in the dence on the issue of a purely imaginary . We also thank case of nonminimal coupling ( Þ 0) the effective R. Jackiw for drawing our attention to his work with S.-Y. Lagrangian contains the axial-vector interaction leading Pi. The research of D. M. and T. T. is supported in part by to parity violation. The important point here is that the the U.S. Department of Energy under Contract No. DE- parity-violating vector-axial interaction contains two pa- FG05-92ER40677. rameters ( and ). In this situation, the experimental
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