arXiv:hep-th/0311127v5 17 May 2004 ∗ oal unu edtheory field quantum sonable that manner the proposed. in been breaking we has LI related paper required problems the some this obtaining address of to and section idea following this discuss the shall spontaneous In the LI. on of depends mechanism particles breaking of force-mediating sort old This composite an [10]. reviving generating flat- by for the theories) near idea tuning quantum or fine our flatness unnaturally of without constant parameters the Universe explain cosmological the of to the ness how addressing (i.e., for problem [9] proposal rddfrvoain fL n P [7]. be CPT may and theories LI suggested of gauge been violations chiral also for certain has traded in It anomalies 6]. to that [5, modifications itself or relativity [4] special LI from by departures measurements possi- small ray invoking the cosmic to puzzling explaining [3], community of bility string theo- the by supersymmetric and considered certain [2] ries in (see and space-time therein), physical pro- references of have descriptions LI some as of that non- posed breaking geometries Lorentz non-commutative explicit on the the violating in work from of ranged for have motivations has prospect invariance there recent the [1], More in theory LI. interest field spontaneously considerable the string of since bosonic been possibility in particularly the LI and on breaking years, work recent of boosted In publication been has rotated. experiment results an or physical after change that not states should relativity, special Einstein’s I orwn rmsm l hoeia ok[1 12] [11, work theoretical old some from Borrowing LI. 1 lcrncades [email protected] address: Electronic omttv emty opqatmgaiy t. ih be might [8]. LI etc.) of gravity, violations quantum theory, introducing low-energy (string loop the physics geometry, capture high-energy mode commutative to new standard whatever meant of the are effects which of proposed Extensions lite been scientific have rich non-invariance. the of Lorentz account thorough on a from far is This hslasu oivsiaeteqeto fhwarea- a how of question the investigate to us leads This recent a with began subject the in interest own Our oet naine(I,tefnaetlsmer of symmetry fundamental the (LI), invariance Lorentz nt aumepcainvalue po non- expectation chemical a vacuum a finite giving generating dynamically by therefore invariance (and Lorentz vacuum break spontaneously may ASnmes 13.p 11.x 14.70-e 11.15.Ex, 11.30.Cp, numbers: PACS prxmtl ieguebsn.W lomk eea remar general make also yie by We to provided order bosons. in gauge invariance like Lorentz approximately of breaking a such require edsrb o tbeeetv hoyi hc atce o particles which in theory effective stable a how describe We .INTRODUCTION I. h ψγ ¯ µ Dtd o.20;pbihdi hs e.D. Rev. Phys. in published 2003; Nov. (Dated: pnaeu raigo oet Invariance Lorentz of Breaking Spontaneous aionaIsiueo ehooy aaea A915 US 91125, CA Pasadena, Technology, of Institute California ψ might i ol eaet oko inl fLrnzvoaini elect in violation Lorentz of signals on work to relate could pnaeul break spontaneously 1 h ψγ ¯ µ ψ ljnr Jenkins Alejandro i hc ecnie ntecneto rpsdmdl that models proposed of context the in consider we which , rature non- l naim nta,Boknpooe okn iha with form working the of proposed theory Bjorken field fermion Instead, self-interacting axiom. an emo ubri nreial aoal,laigt a same (VEV) to value the h leading expectation of favorable, vacuum particles invariant energetically non-Lorentz of is condensate of number super- a existence fermion where color of the Lagrangians formation into for bare the invariant argue research Lorentz we recent with 15], theories the 14, from [13, conductivity as well as oet nain E.Ms ftewr nti area, this in investigation. work future for the left of non- is Most however, our VEV. to invariant limits Lorentz experimental general very on for some and phenomenology LI-violating make electrodynamics resulting to the us are on allows remarks processes This all which place. in rest background taking the charged be a would a of frame be frame preferred this also the would destroying and density density, without number charge cou- fermion theory gauge the the breaking, of of to LI kind frame added some rest were If the pling density. frame: number fermion preferred the a of introduction the E,wtotivkn local standard invoking of theory without a phenomenology formulate the QED, to reproduce was would idea which His electrody- [10]. quantum (QED) of namics” generation “dynamical the called 1] hr ol o pert e tti tg nour in compelling stage any this at physics, be, fundamental to of appear while not understanding a would renormalizable, that There renormalized demonstrated not be [16]. always has is can others theory invariant and (1) gauge Hooft locally ’t Eq. of work as the such theory a invariance. Lorentz con- broke a emerge de- of spontaneously could presence that that the “photons” densate as from resulting composite such bosons (1), theory Goldstone as a Eq. in by that scribed argued then Bjorken ψγ ¯ hsbekn fL a etogto ocpulyas conceptually of thought be can LI of breaking This n16,Boknpooe ehns o hthe what for mechanism a proposed Bjorken 1963, In jre’ damgtntse trcietdy since today, attractive seem not might idea Bjorken’s µ ψ 6 i ∗ dcmoiedgeso reo htact that freedom of degrees composite ld eta em.Ti ehns ilsa yields mechanism This term). tential I MRETGUEBOSONS GAUGE EMERGENT II. 0. = saothwtebcgon source background the how about ks eofrinnme est othe to density number fermion zero h aefrinnme attract number fermion same the f 69 L 150,2004) :105007, = ψ ¯ ( i∂/ − m rodynamics. A ) ψ − U 1 ag naineas invariance gauge (1) λ ( ψγ ¯ µ ψ ) 2 hep-th/0311127 . CALT-68-2470 (1) 2 reason to abandon local gauge invariance as an axiom relating each auxiliary field to its corresponding fermion for writing down interacting quantum field theories.2 bilinear make that Lagrangian classically equivalent to Furthermore, the arguments for the existence of a LI- Eq. (2). In this case the new Lagrangian would be of the breaking condensate in theories such as Eq. (1) have form never been solid. (For Bjorken’s most recent revisiting of → ← ′ µν µν i his proposal, in the light of the theoretical developments = (η + h )ψ¯ (γµ ∂ ν γµ ∂ ν )ψ ψ¯(A/ + m)ψ + ... since 1963, see [19]). L 2 − − V (A2) V (h2)+ ... (3) In 2002 Kraus and Tomboulis resurrected Bjorken’s − A − h idea for a different purpose of greater interest to contem- where A2 A Aµ and h2 h hµν . The ellipses in porary theoretical physics: solving the cosmological con- ≡ µ ≡ µν stant problem [9]. They proposed what Bjorken might Eq. (3) correspond to terms with other auxiliary fields call “dynamical generation of linearized gravity.” In this associated with more complicated fermion bilinears that scenario a composite graviton would emerge as a Gold- were also omitted in Eq. (2). stone boson from the spontaneous breaking of Lorentz We may then imagine that instead of having a single invariance in a theory of self-interacting fermions. fermion species we have one very heavy fermion ψ1 and Being a Goldstone boson, such a graviton would be one lighter one ψ2. Since Eq. (3) has terms that couple both fermion species to the auxiliary fields, integrating forbidden from developing a potential and the existence µ µν out ψ1 will then produce kinetic terms for A and h . of exact solutions with constant matter fields and a mass- µ less graviton would be assured. Then it would no longer In the case of A we can readily see that since it is be necessary to fine-tune the cosmological constant pa- minimally coupled to ψ1, the kinetic terms obtained from rameter in order to obtain a flat or nearly flat spacetime, integrating out the latter must be gauge invariant (pro- providing a possible solution to a problem that plagues vided a gauge invariant cutoff is used). To lowest order in 3 derivatives of Aµ, we must then get the standard photon all mainstream theories of quantum gravity. 1 2 Lagrangian 4 Fµν (where Fµν ∂µAν ∂ν Aµ). Since In [9], the authors consider fermions coupled to gauge µ − ≡ − bosons that have acquired masses beyond the energy A was also minimally coupled to ψ2, we then have, at low energies, something that has begun to look like QED. scale of interest. Then an effective low energy theory µ can be obtained by integrating out those gauge bosons. If A has a non-zero VEV, LI is spontaneously broken, We expect to obtain an effective Lagrangian of the form: producing three massless Goldstone bosons, two of which may be interpreted as photons (see [9] for a discussion of ∞ how the exotic physics of the other extraneous “photon” µ 2n = ψ¯(i∂/ m)ψ + λn(ψγ¯ ψ) can be suppressed). The integrating out of ψ and the L − 1 n=1 assumption that hµν has a VEV, by similar arguments, ∞ X i → ← 2n yield a low energy approximation to linearized gravity. + µ ψ¯ (γ ∂ γ ∂ )ψ + ... (2) n 2 µ ν − µ ν Fermion bilinears other than those we have written out n=1 X explicitly in Eq. (2) have their own auxiliary fields with their own potentials. If those potentials do not them- where we have explicitly written out only two of the selves produce VEV’s for the auxiliary fields, then there power series in fermion bilinears that we would in general would be no further Goldstone bosons, and one would expect to get from integrating out the gauge bosons. expect, on general grounds, that those extra auxiliary One may then introduce an auxiliary field for each fields would acquire masses of the order of the energy- of these fermion bilinears. In this example we shall as- momentum cutoff scale for our effective field theory, mak- sign the label Aµ to the auxiliary field corresponding to ing them irrelevant at low energies. ψγ¯ µψ, and the label hµν to the field corresponding to → ← The breaking of LI would be crucial for this kind ¯ i ψ (γµ ∂ ν γµ ∂ ν )ψ. It is possible to write a Lagrangian of mechanism, not only because we know experimen- 2 − that involves the auxiliary fields but not their derivatives, tally that photons and gravitons are massless or very so that the corresponding algebraic equations of motion nearly massless, but also because Weinberg and Wit- ten have shown that a Lorentz invariant theory with a Lorentz invariant vacuum and a Lorentz covariant energy-momentum tensor does not admit a composite 2 We do know that in the 1980’s Feynman regarded Bjorken’s pro- graviton [20]. posal as a serious alternative to postulating local gauge invari- Let us concentrate on the simpler case of the auxil- ance. For enlightening treatments of the principle of gauge in- µ variance and its historical role in the development of modern iary field A . For the theory described by Eq. (3), the µ physical theories, see [17, 18]. equation of motion for A is 3 In the bargain, this scheme would appear to offer an unorthodox ′ avenue to a renormalizable quantum theory of linearized grav- ∂ ′ L = ψγ¯ µψ V (A2) 2Aµ =0. (4) ity, because the fermion self-interactions could be interpreted as ∂A − − · coming from the integrating out, at low energies, of gauge bosons µ that have acquired large masses via the Higgs mechanism, so that ¯ µ linearized gravity would be the low energy behavior of a renor- Solving for ψγ ψ in Eq. (4) and substituting into both malizable theory. Eq. (2) and Eq. (3) we see that the condition for the 3
Lagrangians and ′ to be classically equivalent is a corresponding fermion bilinear, and therefore a trick such differential equationL forL V (A2) in terms of the coefficients as rewriting a theory in a form like Eq. (3) should not, λn: perhaps, be expected to uncover a physically significant ∞ phenomenon such as the spontaneous breaking of LI for ′ ′ V (A2)=2A2[V (A2)] λ 22nA2n[V (A2)]2n. (5) a theory where it was not otherwise apparent that the − n fermion bilinear in question had a VEV. Let us therefore n=1 X turn our attention to considering what would be required It is suggested in [9] that for some values of λn the so that one might reasonably expect a fermion field the- resulting potential V (A2) might have a minimum away ory to exhibit the kind of condensation that would give from A2 = 0, and that this would give the LI-breaking a VEV to a certain fermion bilinear. VEV needed. It seems to us, however, that a minimum If we allowed ourselves to be guided by purely classical of V (A2) away from the origin is not the correct thing to intuition, it would seem likely that a VEV for a bilinear → ← look for in order to obtain LI breaking. The Lagrangian with derivatives (such as ψ¯ i (γ ∂ γ ∂ )ψ) might re- in Eq. (3) contains Aµ’s not just in the potential but also 2 µ ν − µ ν quire nonstandard kinetic terms in the action.4 Whether in the “interaction” term A ψγ¯ µψ, which is not in any µ or not this intuition is correct, we abandon consideration sense a small perturbation as it might be, say, in QED. In of such bilinears here as too complicated. other words, the classical quantity V (A2) is not a useful The simplest fermion bilinear is, of course, ψψ¯ . Be- approximation to the quantum effective potential for the ing a Lorentz scalar, ψψ¯ = 0 will not break LI. This auxiliary field. kind of VEV was treatedh i back 6 in 1961 by Nambu and In fact, regardless of the values of the λ , Eq. (5) n Jona-Lasinio, who used it to spontaneously break chiral implies that V (A2 = 0) = 0, and also that at any point ′ symmetry in one of the early efforts to develop a the- where V (A2) = 0 the potential must be zero. Therefore, ory of the strong nuclear interactions, before the advent the existence of a classical extremum at A2 = C = 0 of quantum chromodynamics (QCD) [11]. It might be would imply that V (C)= V (0), and unless the potential6 ′ useful to review the original work of Nambu and Jona- is discontinuous somewhere, this would require that V Lasinio, as it may shed some light on the study of the (and therefore also V ) vanish somewhere between 0 and possibility of giving VEV’s to other fermion bilinears that C, and so on ad infinitum. Thus the potential V cannot are not Lorentz scalars. have a classical minimum away from A2 = 0, unless the potential has poles or some other discontinuity. In their original paper, Nambu and Jona-Lasinio start A similar observation applies to any fermion bilinear from a self-interacting massless fermion field theory and for which we might attempt this kind of procedure and propose that the strong interactions be mediated by pi- therefore the issue arises as well when dealing with the ons which appear as Goldstone bosons produced by the spontaneous breaking of the chiral symmetry associated proposal in [9] for generating the graviton. It is not possi- 5 ble to sidestep this difficulty by including other auxiliary with the transformation ψ exp(iαγ )ψ. This sym- metry breaking is produced7→ by a VEV for the fermion fields or other fermion bilinears, or even by imagining ¯ that we could start, instead of from Eq. (2), from a bilinear ψψ. In other words, Nambu and Jona-Lasinio theory with interactions given by an arbitrary, possibly originally proposed what, by close analogy to Bjorken’s idea, would be the “dynamical generation of the strong non-analytic function of the fermion bilinear F (bilinear). 5 The problem can be traced to the fact that the equation interactions.” of motion of any auxiliary field of this kind will always Nambu and Jona-Lasinio start from a non- be of the form renormalizable quantum field theory with a four-fermion interaction that respects chiral symmetry: ′ 0= (bilinear) V (field2) 2 field. (6) − − · g = iψ∂/ψ¯ [(ψγ¯ µψ)2 (ψγ¯ µγ5ψ)2]. (7) The point is that the vanishing of the first derivative of L − 2 − the potential or the vanishing of the auxiliary field itself will always, classically, imply that the fermion bilinear is In order to argue for the presence of a chiral symmetry- zero. Classically at least, it would seem that the extrema breaking condensate in the theory described by Eq. (7), of the potential would correspond to the same physical Nambu and Jona-Lasinio borrowed the technique of self- state as the zeroes of the auxiliary field. consistent-field theory from solid state physics (see, for instance, [12]). If one writes down a Lagrangian with a free and an interaction part, = 0 + , ordinarily one L L Li III. NAMBU AND JONA-LASINIO MODEL (REVIEW)
4 The complications we have discussed that emerge when Recent theoretical work in cosmology has shown interest in scalar field theories with such nonstandard kinetic terms. See, for in- one tries to implement LI breaking as proposed in [9] do stance, [21, 22, 23]. not, in retrospect, seem entirely surprising. A VEV for 5 Historically, though, Bjorken was motivated by the earlier work the auxiliary field would classically imply a VEV for the of Nambu and Jona-Lasinio. 4
2 2
¼ invariant energy-momentum cutoff p < Λ we find
´½µ = ·
½ÈÁ
¡ ¢ £ 2π2m m2 Λ2
2 = m 1 2 log 2 +1 . (10)
¼ ¼
¼
· = · · : : : ´¾µ ½ÈÁ ½ÈÁ gΛ Λ m
½ÈÁ −
¦ ¤ ¥ This equation will always have the trivial solution m = FIG. 1: Diagrammatic Schwinger-Dyson equation. The dou- 0, which corresponds to the vanishing of the proposed ble line represents the primed propagator, which incorporates self-interaction term . But if the self-energy term. The single line represents the unprimed Li propagator. 1PI′ stands for the sum of one-particle irreducible graphs with the primed propagator. 2π2 0 < < 1 (11) gΛ2 would then proceed to diagonalize 0 and treat i as a then there may also be a non-trivial solution to Eq. perturbation. In self-consistent fieldL theory oneL instead (10), i.e., a non-zero m for which the condition of self- rewrites the Lagrangian as = ( 0 + s) + ( i s)= consistency is met. For a rigorous treatment of the rela- ′ ′ L L L L − L 0+ i, where s is a self-interaction term, either bilinear tion between non-trivial solutions of this self-consistent L L L ′ or quadratic in the fields, such that 0 yields a linear equation and local extrema in the Wilsonian effective po- ′ L ′ equation of motion. Now 0 is diagonalized and i is tential for the corresponding fermion bilinears, see [25] treated as a perturbation. L L and the references therein. In order to determine what the form of s is, one re- In this model (which from now on we shall refer to quires that the perturbation ′ not produceL any addi- Li as NJL), we see that if the interaction between fermions tional self-energy effects. The name “self-consistent-field and antifermions is attractive (g > 0) and strong enough 2 theory” reflects the fact that in this technique i is found 2π ( 2 < 1) it might be energetically favorable to form by computing a self-energy via a perturbativeL expansion gΛ in fields that already are subject to that self-energy, and a fermion-antifermion condensate. This is reasonable to then requiring that such a perturbative expansion not expect in this case because the particles have no bare yield any additional self-energy effects. mass and thus the energy cost of producing them is small. Nambu and Jona-Lasinio proceed to make the ansatz The resulting condensate would have zero net charge, as well as zero total momentum and spin. Therefore it must that for Eq. (7) the self-interaction term will be of the 1 5 form = mψψ¯ . Then, to first order in the coupling pair a left-handed fermion ψL = 2 (1 γ )ψ with the Ls − − 1 5 constant g, they proceed to compute the fermion self- antiparticle of a right-handed fermion ψR = 2 (1 + γ )ψ, energy Σ′(p), using the propagator S′(p) = i(p/ m)−1, and vice versa. This is the mass-term self-interaction ′ ¯ − i = mψψ¯ = m(ψ¯LψR + ψ¯RψL) that NJL studies. which corresponds to the Lagrangian 0 = ψ(i∂/ m)ψ L − − that incorporates the proposed self-energyL term. − After QCD became the accepted theory of the strong The next step is to apply the self-consistency condition interactions, the ideas behind the NJL mechanism re- using the Schwinger-Dyson equation for the propagator: mained useful. The u and d quarks are not massless (nor is u-d flavor isospin an exact symmetry) but their ′ 4 ′ ′ bare masses are believed to be quite small compared to
S (x y)= S(x y)+ d½ zS(x z)Σ (0)S (z y) (8) − − − − Z their effective masses in baryons and mesons, so that the formation ofuu ¯ and dd¯ condensates represents the spon- which is represented diagrammatically in Fig. 1. The taneous breaking of an approximate chiral symmetry. In- primes indicate quantities that correspond to a free ′ terpreting the pions (which are fairly light) as the pseudo- Lagrangian that incorporates the self-energy term, 0 Goldstone bosons generated by the spontaneous break- whereas theL unprimed quantities correspond to the or- ′ ing of the approximate SU(2) SU(2) chiral isospin dinary free Lagrangian . For Σ we will use the ap- R L 0 symmetry down to just SU(2),× proved a fruitful line of proximation shown in Fig.L 2, valid to first order in the thought from the point of view of the phenomenology of coupling constant g. After Fourier transforming Eq. (8) and summing the left side as a geometric series, we find that the self-
consistency condition may be written, in our approxi-
¼ ¾
¼
i¦ = = · Ç ´g µ ´½µ
mation, as ½ÈÁ ¡ 4 ¢ ′ gmi d p m =Σ (0) = . (9) 2π4 p2 m2 + iǫ FIG. 2: Diagrammatic equation for the primed self-energy. Z − We will work to first order in the fermion self-coupling con- If we evaluate the momentum integral by Wick rotation stant g.
and regularize its divergence by introducing a Lorentz ½ 5 the strong interaction.6 j2 0. Considerations of causality make it natural to ≥ µ Condition Eq. (11) has a natural interpretation if we expect that something similar would be true of ψγ¯ ψ. think of the interaction in Eq. (7) as mediated by mas- For any time-like Lorentz vector nµ it is possible to sive gauge bosons with zero momentum and coupling e. find a Lorentz transformation that maps it to a vector ′ ′ For it to be reasonable to neglect boson momentum in n µ with only one non-vanishing component: n 0. For a the effective theory, the mass µ of the bosons should be constant current density jµ, this means that for jµ to be µ> Λ. If e2 < 2π2 then g = e2/µ2 < 2π2/Λ2, which vi- non-zero there must be a charge density j0, which has olates Eq. (11). Therefore for chiral symmetry breaking a rest frame. Therefore we only expect to see a VEV to happen, the coupling e should be quite large, mak- for ψγ¯ µψ if our theory somehow has a vacuum with a ing the renormalizable theory nonperturbative. This is non-zero fermion number density. The consequent spon- acceptable because the factor of 1/µ2 allows the pertur- taneous breaking of LI may be seen as the introduction of bative calculations we have carried out in the effective a preferred reference frame: the rest frame of the vacuum theory Eq. (7). This is why the NJL mechanism is mod- charge. ernly thought of as a model for a phenomenon of non- In the literature of finite density quantum field the- perturbative QCD. ory and of color superconductivity (see, for instance, [13] and [14]), the Lagrangians discussed are explicitly non- Lorentz invariant because they contain chemical poten- tial terms of the form f ψγ¯ 0ψ . This term appears IV. AN NJL-STYLE ARGUMENT FOR in theories whose ground· state has a non-zero fermion BREAKING LI number because, by the Pauli exclusion principle, new fermions must be added just above the Fermi surface, We have reviewed how NJL formulated a model that i.e., at energies higher than those already occupied by the exhibited a non-zero VEV for the fermion bilinear ψψ¯ . pre-existing fermions, while holes (which can be thought The next simplest fermion bilinear that we might con- of as antifermions) should be made by removing fermions sider is ψγ¯ µψ, which was the one that Bjorken, Kraus, at that Fermi surface. The result is an energy shift that and Tomboulis considered when they discussed the “dy- depends on the number of fermions already present and namical generation of QED.” This particular fermion bi- which has opposite signs for fermions and antifermions. linear is especially interesting because it corresponds to The physical picture that emerges is now, hopefully, the U(1) conserved current, and also because it is the clearer: a theory with a VEV for ψγ¯ µψ is one with a con- simplest bilinear with an odd number of Lorentz ten- densate that has non-zero fermion number. This means sor indices, so that a non-zero VEV for it would break that only theories with some form of attractive inter- not only LI but also charge (C), charge-parity (CP), and action between particles with the same sign in fermion charge-parity-time (CPT) reversal invariance. C and CP number may be expected to produce such a VEV. The may not be symmetries of the Lagrangian, as indeed they situation is closely analogous to BCS superconductiv- are not in the standard model, but by a celebrated re- ity [26], in which a phonon-mediated attractive inter- sult CPT must be an invariance of any reasonable theory action between electrons allows the presence of a con- (see [27] and references therein). This invariance, how- densate with non-zero electric charge. Note that in the ever, may well be spontaneously broken, as it would be NJL model, the condensate was composed of fermion- by any VEV with an odd number of Lorentz indices. antifermion pairs, and therefore clearly ψγ¯ 0ψ = 0, Before proceeding, however, it may be advisable to try which implies ψγ¯ µψ = 0. It should nowh be physicallyi to develop some physical intuition about what would be clear why a VEVh for iψγ¯ µψ would break not only LI but required for a fermion bilinear like ψγ¯ µψ to exhibit a also C, CP, and CPT. VEV. If we choose a representation of the gamma matrix There is an easy way to write a theory which will have algebra and use it to write out (ψγ¯ µψ)2 for an arbitrary a VEV for a U(1) conserved current: to couple a massive bispinor ψ, we may check that (ψγ¯ µψ)2 0 for the choice photon to such a current via a purely imaginary charge. ≥ of mostly negative metric gµν = diag(1, 1, 1, 1). To see this, let us write a Proca Lagrangian for a massive µ − − − That is, ψγ¯ ψ is timelike. This has an intuitive explana- photon field with an external source: ¯ µ tion, based on the observation that ψγ ψ is a conserved 2 fermion-number current density. Classically a charge 1 2 µ 2 µ = Fµν + A jµA . (12) density ρ moving with a velocity ~v will produce a current L −4 2 − µ j = (ρ, ρ~v) (in units of c = 1). Therefore the relativistic The equation of motion for the photon field is requirement that the charge density not move faster than µν ν 2 ν the speed of light in any frame of reference implies that ∂µF = j µ A . (13) − At energy scales well below the photon mass µ, the 2 kinetic term Fµν /4 may be neglected with respect to − 2 2 6 For a treatment of this subject, including a historical note on the mass term µ A /2. We may then integrate out the the influence of the NJL model in the development of QCD, see photon at zero momentum by solving the equation of mo- Chap. 19, Sec. 4 in [24]. tion Eq. (13) for the photon field Aµ with its conjugate 6 momenta F µν set to zero, and substituting the result
back into the Lagrangian in Eq. (12). The resulting low-
· ´½µ energy effective field theory has the Hamiltonian =
2
¢ £ j ¡ effective = 2 . (14) FIG. 3: The four-fermion vertex in the self-interacting theory H 2µ may be seen as the sum of two photon-mediated interactions Nothing interesting happens if the source is a timelike with a massive photon that carries zero momentum and is coupled to the fermion via a purely imaginary charge. current density, since in that case Eq. (14) has its min- imum at jµ = 0. But if we were to make the charge coupling to the photon imaginary (e.g., jµ = ieψγ¯ µψ for 2 e real), then j is actually always negative (recall that compute the self-energy we will rely on the identity rep- ¯ µ 2 (ψγ ψ) is always positive) and we get a “potential” resented in Fig. 3. (In QED the second diagram on the with the wrong sign, so that the energy can be made right-hand side of Fig. 3 would vanish by Furry’s theo- 2 µ arbitrarily low by decreasing j . If we make j dynam- rem, but in our case the propagator in the loop will have ical by adding to the Lagrangian terms corresponding a chemical potential term that breaks the C invariance to the field that sets up the current, we might expect, on which Furry’s theorem depends.) for certain parameters in the theory, that the energy be To leading order in g, the self-energy is minimized for a finite value of jµ. 4 0 i By making the charge purely imaginary, our effec- d k 3(k0 f)γ +3k γ 2m Σ(0) = 2ig − i − tive theory at energy scales much lower than the pho- (2π)4 2 ~ 2 2 2 Z k0 k m + f 2fk0 + iǫσ ton mass µ will look similar to Eq. (7), except that − − − (18) the four-fermion interaction in the effective Lagrangian where σ (a function of ~k , f, and m) takes values 1 will be e2(ψγ¯ µψ)2/2µ2 (with an overall positive, rather so as to enforce the standard| | Feynman prescription± for than a negative, sign). What this means is that fermions shifting the k0 poles: positive k0 poles are shifted down are attracting fermions and antifermions are attracting from the real line, while negative poles are shifted up. antifermions, rather than what we had in NJL (and At first sight it might appear as if the self-energy in in QED): attraction between a fermion and an an- Eq. (18) could not be used to argue for the breaking of tifermion. Condensation, if it occurs, will here produce a LI, because the shift in the integration variable k k′ = net fermion number, spontaneously breaking C, CP, and 0 7→ 7 (k f,~k) would wipe out f dependence. This, however, CPT. − 0 Let us analyze this situation again more rigorously us- is not the case, as we will see. We may carry out the dk ing self-consistent field theory methods, following Nambu integration, for which we must find the corresponding and Jona-Lasinio. For this we consider a fermion field poles. These are located at with the usual free Lagrangian 0 = ψ¯(i∂/ m0)ψ and L − 2 2 pose as our self-consistent ansatz: k0 = f ~k + m . (19) ± 0 q s = (m m0)ψψ¯ fψγ¯ ψ. (15) From now on, without loss of generality, we will take L − − − f to be positive. The contour integral which results from The corresponding momentum-space propagator for 0 ′ closing the d k integral of Eq. (18) in the complex plane = 0 + is, therefore, L0 L Ls will vanish unless f < ~k2 + m2, because otherwise both ′ 0 −1 poles in Eq. (19) will lie on the same side of the imag- S (k)= i(k/ fγ m) . (16) p − − inary axis. In light of the Feynman prescription used Now let us suppose that the interaction term looks like for the shifting of the poles away from the real axis, it would then be possible to close the contour at infinity so g µ 2 that there would be no poles in the interior. The pole i = (ψγ¯ ψ) . (17) L 2 shifting prescription, through its effect on the dk0 inte- gral, is what introduces an actual f dependence into the To obtain the Feynman rules corresponding to Eq. (17) expression for the self-energy.½ we note that this is what we would obtain in massive By the Cauchy integral formula, we have QED if we replaced the charge e by ie and the usual pho- ton propagator by igµν/µ2, with g = e2/µ2. Therefore to g 3 ~k2 + m2γ0 +2m Σ(0) = − d3k 4π3 ~ 2 2 Z " p 2 k + m
7 p 3 0 At one point, Dyson argued that such a theory with attraction θ( ~k2 + m2 f) γ (20) × − − 2 between particles of the same fermion number would be unstable q and used this to suggest that perturbative series in QED might diverge after renormalization of the charge and mass [28]. We where the second term in the right hand side subtracts will address the issue of stability at the end of this section. the contribution from closing the contour out at infinity 7 in the complex plane (note the branch cut in the loga- equations: rithm that results from computing that part of the con- g tour integral explicitly). We will introduce the cutoff f = (f 2 m2)3/2 (23) 2π2 ~k2 < Λ2 to make the integral in Eq. (20) finite.8 − Note that the Heaviside step function θ( ~k2 + m2 f) and in Eq. (20) is always unity if m>f, so that there− will p 2 2 be no f dependence at all in Eq. (20) unless m f. gm 2 f + f m m0 m = m log − Assuming that m f we have ≤ − 2π2 Λ+ √Λ2 + m2 ≤ " p ! 2 2 2 2 g 2 2 3/2 0 3 2 2 +Λ Λ + m f f m . (24) Σ(0) = − 2 (f m ) γ + m log (f + f m ) − − 2π − − − h p p m3 log (Λ + Λ2 + m2) p − It is important to bear in mind that Eqs. (23) and (24) were written under the assumption that f m. For + mΛ Λ2 +pm2 mf f 2 m2 . (21) ≥ − − f f = π 2/g. (26) 8 0 p Carrying out the dk integration separately from the spatial in- Plot (b) in Fig. 4 shows a 0 < m0 Λ for which ≪ tegral is legitimate and useful in light of the form of Eq. (18), the corresponding m will be significantly less than m0. which does not lend itself naturally to Wick rotation. But the Plot (c) in the same figure illustrates that a very large use of a non-Lorentz invariant regulator may cause concern that m is needed before m>m , but such solutions are not any breaking of LI we might arrive at could be an artifact of our 0 0 choice of regulator. An alternative is to dimensionally regulate physically meaningful because m0 itself is already well Eq. (20) by replacing d3k with dd−1k. The resulting equations beyond the energy scale for which our effective theory is are more complicated and the dependence on the range of en- supposed to hold. By comparing plot (b) to plot (e) we ergies where our non-renormalizable theory is valid is obscured, may see the effect of increasing g for a given m0 and Λ. but the overall argument does not change. It is also possible A comparison of plots (b) and (f) should illustrate the to multiply the integrand in Eq. (18) by a cutoff in Minkowski 2 2 2 2 ~ 2 ~ 2 2 effect of increasing Λ with the other parameters fixed. space θ(Λ +k )= θ(Λ +k0 −k ). For k < Λ we get the same result as in Eq. (20). For ~k2 > Λ2 we must impose the condition The plots in Fig. 5 illustrate the progression, as the 2 ~ 2 2 parameter Λ is increased for fixed α, from an unstable that k0 > k − Λ , yielding an additional, rather complicated term which does not affect the logic of our discussion in this sec- theory in which bare masses m0 on the order of Λ are tion. It should be pointed out that previous work on LI breaking mapped to m > Λ, to a theory that maps such bare has used 3-momentum cutoffs in computing self-energies [30], al- masses to m < Λ. Such an analysis of Eq. (25) reveals though in that case there seems to be a physical interpretation for such a cutoff which does not apply to the present discussion. that the condition for this mass stability is The original work of Nambu and Jona-Lasinio [11] considers cut- 2π2 offs in Euclidean 4-momentum and in 3-momentum, arriving in 0 < < 1 (27) both cases at similar conclusions. gΛ2 8 20 1500 1500 1000 1000 15 500 500 10 0 m 0 m -500 5 -500 -1000 -1000 0 m -2000 -1000 0 1000 2000 -2000 -1000 0 1000 2000 -2 -1 0 1 2 (a) (b) (c) 20 20 1500 1000 15 15 500 10 10 0 m -500 5 5 -1000 0 m 0 m -2000 -1000 0 1000 2000 -2 -1 0 1 2 -2 -1 0 1 2 (d) (e) (f) g FIG. 4: Plots of the left-hand side (in gray) and right-hand side (in black) of equation Eq. (25). Define α ≡ 2π2 . For each plot the parameters are: (a) Λ = 100, m0 = 0, α = 0.001. (b) Λ = 100, m0 = 15, α = 0.001. (c) Λ = 100, m0 = 1200, α = 0.001. (d) Λ = 100, m0 = 0, α = 0.002. (e) Λ = 100, m0 = 15, α = 0.002. (f) Λ = 200, m0 = 15, α = 0.001. which is reminiscent of the condition Eq. (11) for chiral V. CONSEQUENCES FOR EMERGENT symmetry breaking in the NJL model (except that now PHOTONS the interaction has the opposite sign). Combining Eq. (27) with Eq. (26) (which was exact for m0 but may serve The theory approximately for m0 small) we arrive at the requirement g µ 2 = ψ¯(i∂/ m0)ψ + (ψγ¯ ψ) (29) L − 2 is equivalent to 2 2 0 10 20 20 20 7.5 15 15 15 5 10 10 10 2.5 5 5 5 0 m 0 m 0 m 0 m -2.5 -5 -5 -5 -5 -10 -10 -10 -7.5 -15 -15 -15 -10 -5 0 5 10 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 (a) (b) (c) (d) g FIG. 5: Plots of the left-hand side (in gray) and right-hand side (in black) of equation Eq. (25). For all of them α ≡ 2π2 = 0.01. (a) Λ = m0 =2. (b) Λ= m0 =8. (c) Λ= m0 =12. (d) Λ= m0 = 16. a finite time-like VEV, which would imply a finite VEV free photon Lagrangian of the form for ψγ¯ µψ in the theory of Eq. (29). We argued in the previous section that g must be large for the theory de- photon 1 2 µ = F jµA (32) scribed by Eq. (29) to be stable. This too seems natural L0 −4 µν − in light of Eq. (30), because a large g makes the A2 term where jµ = e ψγ¯ µψ , thought of as an external source. small, so that the instability created by it may be easily h i controlled by the interaction with the fermions, yielding The corresponding propagator for the free photon is a VEV for Aµ that lies within the energy range of the T Aµ(x)Aν (y) = Dµν (x y)+ Aµ(x) Aν (y) effective theory. F j j h { }i − h i h (33)i Armed with Eq. (30) it would seem possible to carry where Dµν (x y) is the connected photon propagator and out the program proposed by Bjorken, and by Kraus µ − µ A (x) j is the expectation value of A in the presence and Tomboulis, in order to arrive at an approxima- hof the externali source. tion of QED in which the photons are composite Gold- If we take jµ constant and naively attempt to calculate stone bosons. It is conceivable that a complicated the- the classical expectation value of Aµ in the presence of ory of self-interacting fermions, perhaps one with non- a constant source by integrating the Green function for standard kinetic terms, might similarly yield a VEV for → ← electrodynamics, we will get a volume divergence. We ¯ i ψ (γµ ∂ ν γµ ∂ ν )ψ, allowing the project of dynam- may attempt to regulate this volume divergence by in- 2 − ically generating linearized gravity to go forward. We troducing a photon mass µ, which gives the result leave this for future investigation. jµ Aµ(x) = . (34) h ij µ2 VI. PHENOMENOLOGY OF LORENTZ (It is trivial to check that this is a solution to ∂2Aµ + VIOLATION BY A BACKGROUND SOURCE µ2Aµ = jµ, the wave equation for the massive photon field with a source.) This is not satisfactory because the A separate line of thought that might be pursued from disconnected term in Eq. (33) will be proportional to this work concerns a phenomenology of Lorentz viola- µ−4 and Feynman diagrams computed with our modified tion in electrodynamics with a background source. That photon propagator would produce results that depend is, we might imagine that the fermions of the universe strongly on what we took for a regulator. In fact the mass have some interaction that plays the role of Eq. (17) in is physical and analogous to the effective photon mass giving a VEV to ψγ¯ µψ, and that in addition they have a first described by the London brothers in their theory U(1) gauge coupling (at this stage we have abandoned the of the electromagnetic behavior of superconductors [31]. project of producing composite photons). Then the U(1) (Using the language of particle physics we may say that, gauge field may interact with a charged background and in the presence of a U(1) gauge field, the VEV ψγ¯ µψ we would be breaking LI in electrodynamics by introduc- spontaneously breaks the gauge invariance andh gives ai ing a preferred frame: the rest frame of the background mass to the boson, as in the Higgs mechanism.) source. Photons in a superconductor propagate through a con- The possibility of a vacuum that breaks LI and has stant electromagnetic source. In a simplified picture, we non-trivial optical properties has already been investi- may think of it as a current density set up by the motion gated in [29, 30]. This work, however, deals with signif- of charge carriers of mass m and charge e, moving with a icantly more complicated models, both in terms of the velocity ~u. The proper charge density is ρ0. The proper interactions that spontaneously break LI and of the opti- velocity of the charge carriers is ηµ = (1, ~u)/√1 u2. µ µ µ µ − cal properties of the resulting vacuum. To obtain a phe- The source is then j = ρ0η = ρ0p /m, where p is the nomenology for our own simpler proposal, we consider a classical energy momentum of the charge carriers. We 10 may think of m and ρ0 as deriving from the solutions to fermion number and spontaneously breaks CPT, a vio- the parameters in a self-consistent equation such as we lation which can ease the Sakharov condition of thermo- had in Eq. (25). dynamical non-equilibrium [39]. The canonical energy momentum P µ of the system is It has recently been suggested that the standard model µ µ µ µ µ P = mη + eA = mj /ρ0 + eA . As is discussed in might be formulated without a Higgs scalar field, by the superconductivity literature (see, for instance, Chap. introducing instead fermion self-interactions which do 8 in [32]), the superconducting state has zero canonical not destroy the renormalizability of the theory if there energy momentum, which leads to the London equation are nonzero UV fixed points under the renormalization group operation [40]. That work, published after the first eρ jµ = 0 Aµ. (35) manuscript of the present paper had appeared in the pre- − m print archive, might well relate to the mechanism we have With this jµ inserted into the right-hand side of ∂2Aµ = described, particularly in light of what was discussed in jµ (the wave equation for the photon field in the Lorenz the previous section of this paper. gauge), we find that we have a solution to the wave All these tentative ideas are left for possible consider- equation of a massive Aµ with no source and a mass ation in the future. 2 µ = eρ0/m: eρ ∂2Aµ + 0 Aµ =0. (36) m VIII. CONCLUSIONS If we solve for Aµ in Eq. (35) and substitute this back into Eq. (33), we get that We have presented a stable effective theory in which a chemical potential term is dynamically generated, thus 2 spontaneously breaking LI (as well as C, CP, and CPT). µν m T Aµ(x)Aν (y) = D (x y)+ jµjν . (37) The main reasons why this theory might be interesting h { }i F − e2j2 are the following: (a) that it might serve as the starting Notice that if jµ(x) is not constant, then Fourier trans- point for models with emergent gauge bosons, (b) that it could conceivably point to LI breaking in other more nat- formation of the second term in Eq. (37) will not yield, in Feynman diagram vertices, the usual energy-momentum ural theories that share its fundamental attribute: attrac- tion between particles of the same fermion number sign conserving delta function. Therefore, presumed small vi- olations of energy or momentum conservation in electro- (something that is seen in non-Abelian gauge theories such as QCD, which allows bound states with non-zero magnetic processes could conceivably be parametrized by the space-time variation of the background source.9 baryon number) and (c) that it produces something that could perhaps interest those who study the phenomenol- With Eq. (37) and a rule for external massive photon ogy of Lorentz violation in electrodynamics: the breaking legs, one may then go ahead and calculate the amplitude of LI by introducing a background source with its own for various electromagnetic processes with this modified rest frame. photon propagator, and parametrize supposed observed violations of LI (see [34, 35, 36]) by jµ. If wecan makean All of these remain somewhat problematic because (a) estimate of the size of the the mass m of the background our work applies directly not to the more interesting case charges, experimental limits on the photon mass (< 2 of generating emergent gravitons, but only to photons, 10−16 eV according to [37]) will provide a limit on the× (b) so far we have not been able to produce models that VEV of ψγ¯ µψ, in light of Eq. (35). spontaneously break LI which are significantly more nat- ural than Eq. (29), which is a non-renormalizable the- ory in which the fermion self-coupling has the opposite sign to what is obtained by integrating out a heavy U(1) VII. OTHER POSSIBLE CONSEQUENCES OF 10 THIS MECHANISM gauge boson , and (c) it remains to be seen whether a phenomenology of electrodynamics with a background source is of any interest to the effort of explaining the There are other consequences of a VEV ψγ¯ µψ = 0 h i 6 supposed indications of Lorentz violation in cosmic ray on which we may speculate. Such a background may data and other measurements. These are all areas that have cosmological effects, a line of thought which might would need to be explored in order to make more concrete connect, for instance, with [38]. Also, it is conceivable and useful the ideas presented here. that such a VEV might have some relation to the prob- lem of baryogenesis, since it gives the background finite 10 Recent work has shown interest in the possibility of introducing fermion self-couplings that respect non-perturbative renormaliz- 9 This line of thought could connect to work on LI violation from ability [40]. This is possible in the presence of non-zero UV fixed variable couplings as discussed in [33]. points of the renormalization group. 11 Acknowledgments and R. Lehnert for pointing out references to other work on Lorentz non invariance which were missing from the The author would like to thank his advisor, first draft. This work was financially supported in part M. B. Wise, for his guidance, and P. Kraus, J. D. Bjorken, by a R. A. Millikan graduate fellowship from the Califor- J. Jaeckel, H. Ooguri, K. Sigurdson, and D. O’Connell for nia Institute of Technology. useful exchanges. Thanks are also due to O. Bertolami [1] V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 [20] S. Weinberg and E. Witten, Phys. Lett. B96, 59 (1980). (1989); [21] C. Armendariz-Picon, T. Damour and V. Mukhanov, V. A. Kostelecky and R. Potting, Nucl. Phys. B359, 545 Phys. Lett. B458, 209 (1999) [hep-th/9904075]. (1991). [22] R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, [2] J. Madore, gr-qc/9906059. Phys. Rev. Lett. 91, 071301 (2003) [astro-ph/0302506]. [3] H. Ooguri and C. Vafa, Adv. Theor. Math. Phys. 7, 53 [23] N. Arkani-Hamed, H. C. Cheng, M. A. Luty and S. Muko- (2003) [hep-th/0302109]. hyama, hep-th/0312099; [4] S. R. Coleman and S. L. Glashow, Phys. Rev. D 59, N. Arkani-Hamed, P. Creminelli, S. Mukohyama 116008 (1999) [hep-ph/9812418]; Phys. Lett. B405, 249 and M. Zaldarriaga, JCAP 0404, 001 (2004) (1997) [hep-ph/9703240]. [hep-th/0312100]. [5] J. Magueijo and L. Smolin, Phys. Rev. Lett. 88, 190403 [24] S. Weinberg, The Quantum Theory of Fields, Vol. 2, (2002) [hep-th/0112090]; Phys. Rev. D 67, 044017 (2003) Cambridge (1996). [gr-qc/0207085]. [25] K. I. Aoki, K. Morikawa, J. I. Sumi, H. Terao [6] G. Amelino-Camelia, Nature (London) 418, 34 (2002) and M. Tomoyose, Phys. Rev. D 61, 045008 (2000) [gr-qc/0207049]. [hep-th/9908043]. [7] F. R. Klinkhamer, Nucl. Phys. B535, 233 (1998) [26] J. Bardeen, L. Cooper, and J. Schrieffer, Phys. Rev. 106, [hep-th/9805095]; 162 (1957); Phys. Rev. 108 1175 (1957). Nucl. Phys. B578, 277 (2000) [hep-th/9912169]; [27] R. F. Streater and A. S. Wightman, PCT, Spin and F. R. Klinkhamer and J. Nishimura, Phys. Rev. D 63, Statistics, and All That, Benjamin, (1964). 097701 (2001) [hep-th/0006154]; [28] F. J. Dyson, Phys. Rev. 85, 631 (1952). F. R. Klinkhamer and C. Mayer, Nucl. Phys. B616, 215 [29] A. A. Andrianov and R. Soldati, Phys. Rev. D 51, 5961 (2001) [hep-th/0105310]; (1995); F. R. Klinkhamer and J. Schimmel, Nucl. Phys. B639, Phys. Lett. B435, 449 (1998) [hep-ph/9804448]; 241 (2002) [hep-th/0205038]. A. A. Andrianov, R. Soldati and L. Sorbo, Phys. Rev. D [8] D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 59, 025002 (1999) [hep-th/9806220]. 116002 (1998) [hep-ph/9809521]; [30] A. A. Andrianov, P. Giacconi and R. Soldati, JHEP V. A. Kostelecky and R. Lehnert, Phys. Rev. D 63, 0202, 030 (2002) [hep-th/0110279]. 065008 (2001) [hep-th/0012060]. [31] F. London and H. London, Proc. Roy. Soc. A149 71 [9] P. Kraus and E. T. Tomboulis, Phys. Rev. D 66, 045015 (1935). (2002) [hep-th/0203221]. [32] C. Kittel, Quantum Theory of Solids, Wiley (1963). [10] J. D. Bjorken, Ann. Phys. (N. Y.) 24, 194 (1963). [33] V. A. Kostelecky, R. Lehnert and M. J. Perry, [11] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 astro-ph/0212003. (1961); 124, 246 (1961). [34] S. M. Carroll, G. B. Field and R. Jackiw, Phys. Rev. D [12] Y. Nambu, Phys. Rev. 117, 648 (1960). 41, 1231 (1990). [13] M. G. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys. [35] V. A. Kostelecky and M. Mewes, Phys. Rev. D 66, 056005 A638, 515C (1998) [hep-ph/9802284]. (2002) [hep-ph/0205211]. [14] M. G. Alford, J. A. Bowers, J. M. Cheyne and [36] O. Bertolami and C. S. Carvalho, Phys. Rev. D 61, G. A. Cowan, Phys. Rev. D 67, 054018 (2003) 103002 (2000) [gr-qc/9912117]; [hep-ph/0210106]. O. Bertolami, Gen. Rel. Grav. 34, 707 (2002) [15] C. Wetterich, Phys. Rev. D 64, 036003 (2001) [astro-ph/0012462]. [hep-ph/0008150]; [37] K. Hagiwara et al. [Particle Data Group Collaboration], J. Berges and C. Wetterich, Phys. Lett. B512, 85 (2001) Phys. Rev. D 66, 010001 (2002). [hep-ph/0012311]. [38] H. M. Fried, hep-th/0310095. [16] G. ’t Hooft, Nucl. Phys. B33, 173 (1971); 35, 167 (1971). [39] O. Bertolami, D. Colladay, V. A. Kostelecky and R. Pot- [17] L. O’Raifeartaigh and N. Straumann, Rev. Mod. Phys. ting, Phys. Lett. B395, 178 (1997) [hep-ph/9612437]. 72, 1 (2000). [40] H. Gies, J. Jaeckel and C. Wetterich, Phys. Rev. D (to [18] J. D. Jackson and L. B. Okun, Rev. Mod. Phys. 73, 663 be published), hep-ph/0312034. (2001) [hep-ph/0012061]. [19] J .D. Bjorken, hep-th/0111196.