A technical analog of the cosmological constant problem and a solution thereof
aa r X i v : . [ h e p - t h ] J a n A technical analog of the cosmologicalconstant problem and a solution thereof
Ioanna Kourkoulou, Alberto Nicolis, and Guanhao Sun
Center for Theoretical Physics and Department of Physics,Columbia University, New York, NY 10027, USA
Abstract
The near vanishing of the cosmological constant is one of the most puzzling openproblems in theoretical physics. We consider a system, the so-called framid, that fea-tures a technically similar problem. Its stress-energy tensor has a Lorentz-invariantexpectation value on the ground state, yet there are no standard, symmetry-based selec-tion rules enforcing this, since the ground state spontaneously breaks boosts. We verifythe Lorentz invariance of the expectation value in question with explicit one-loop com-putations. These, however, yield the expected result only thanks to highly nontrivialcancellations, which are quite mysterious from the low-energy effective theory viewpoint.
Contents b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Analysis for b and b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Introduction
The cosmological constant problem has been a topic of heated debate for decades. In fact, itis difficult to find two theoretical physicists who agree on what exactly the problem is, how tophrase it, how to quantify it, or how many problems there are. Some maintain that there isno problem at all. (We realize that some of our colleagues will take issue with this paragraphas well.)We do not intend to enter the debate ourselves, nor to review the extensive literatureon the subject . Our aim with this paper is, instead, to describe a system that exhibits atechnically similar problem, and to show in detail how that problem is “solved” there. Thequotes are in order: from a low-energy effective field theory viewpoint, the problem is solvedby apparently miraculous cancellations. However, there is a more fundamental viewpoint,according to which those cancellations have to happen. Needless to say, it is a symmetrythat ultimately enforces those cancellations, but a symmetry that is spontaneously broken.In particular, that symmetry does not enforce the cancellations through standard selectionrules—the ground state is not invariant under the symmetry—but only in a roundabout way,which is utterly obscure in the low-energy effective theory computation we will perform.The system we have in mind is the so-called framid [6]. It is a hypothetical state ofrelativistic matter that spontaneously breaks Lorentz boosts but no other symmetry. Thelow-energy effective field theory involves three gapless Goldstone bosons ~η ( x ) that, like thebroken boost generators they are associated with, transform as a vector under rotations. Theirlow-energy effective action can be written down systematically, in a derivative expansion, usingfor instance the coset construction for spacetime symmetries. From this, one can derive thestress energy tensor of the theory, which displays a very peculiar property: if evaluated on the ~η ( x ) = 0 background, it is Lorentz invariant [6]: T µν ( x ) = − Λ η µν + O ( ∂~η ) . (1)This is surprising: the ground state of the system spontaneously breaks boosts, so thereis no obvious reason why it should have a Lorentz-invariant stress tensor. Certainly, allfamiliar condensed matter systems in the lab, such as solids, fluids, superfluids, which alsospontaneously break Lorentz, have stress-energy tensors in their ground (or equilibrium) statesthat are not Lorentz invariant. In particular, in c = 1 units, they typically have mass densitiesmuch bigger than their pressures or internal stresses.So, what enforces the structure in (1) for the framid? Unclear. It can be argued that itis the underlying Lorentz invariance of the theory, but only in the sense that if one writesdown the most general framid effective action compatible with spontaneously broken Lorentzboosts, and then derives the corresponding stress-energy tensor, one ends up with eq. (1). Amore direct argument based on symmetry considerations directly for the expectation valueof T µν is not available at the moment. In fact, there exists a completely different systemthat features precisely the same symmetry breaking pattern as the framid, but saturates theassociated Goldstone theorem in a radically different way, and that most definitely does not It is impossible for us to do justice to all the attempts that have been made at tackling the cosmologicalconstant problem. For an overview, we refer the reader to [1–5] and references therein. In particular, ref. [5]reviews recent attempts based on relaxation mechanisms. h T µν ( x ) i 6 = − Λ η µν , (2)for any Λ. This, however, does not seem to be consistent with renormalization theory and,more in general, renormalization group ideas. The reason is that we can think of our framideffective theory as a theory for physics below a certain energy scale M , with all the physicsabove M having been integrated out. If we integrate out some more physics, say down to ascale M ′ < M , the coefficients in the framid low-energy effective action change, but the setof allowed terms in such an action remains the same. So, what was a “quantum correction”in the energy window between M ′ and M , now becomes a new contribution to a tree-levelcoupling in the low-energy theory. But at tree-level the effective theory yields h T µν ( x ) i = − Λ η µν . (3)This suggests that the same should hold at the quantum level, up to a renormalization of Λ.This is our technical analog of the cosmological constant problem. There, one has that theexpectation value of the real world’s stress-energy tensor that couples to gravity is zero (orfantastically smaller than the “natural” value it should have), without any manifest symmetryreasons for why it should be so. In our case, we have that the Lorentz-violating componentsof the expectation value of the framid’s stress-energy tensor are exactly zero, without anymanifest symmetry reasons for why it should be so.Our paper is devoted to explicitly verifying eq. (3) for a framid to one-loop order. As onemight expect, the computation will involve UV-divergent loop integrals, and we will have topay particular attention to how we regulate such integrals. The reason is that our questionhas to do with Lorentz invariance, and so we need to make sure that our regulator respects it.However, since Lorentz invariance is spontaneously broken, it is not manifest in the effectivetheory or, more importantly, in our loop integrals. And so, for instance, cutting off ourintegrals in (Euclidean) momentum space in a manifestly Lorentz invariant fashion makes nosense: the framids’ Goldstones already have propagation speeds different from unity, and fromone another, and so why should their loops be cutoff in a Lorentz-invariant fashion?To address this problem, we use two regulators that can be made straightforwardly com-patible with our spontaneously broken Lorentz invariance. The first is a generalization ofPauli-Villars. The second is dimensional regularization. Notice that, for the latter, since theGoldstone’s effective theory has no mass parameters, the relevant integrals for our one-loopcomputation will all be trivially zero, which would make our check moot. To circumvent this,we couple the framid to a massive scalar particle, in all ways allowed by symmetry, and runour check in that case.With both regulators, we find that, indeed, eq. (3) is obeyed at one-loop order. However,the cancellations involved in making the Lorentz-violating components vanish are absolutelynontrivial, and we were not able to find a clear structure in the actual computation that would3nsure, or even suggest, those cancellations. As a further check of our techniques and of thenon-triviality of the cancellations, we run the same computation in the case of a superfluid,where we don’t expect analogous cancellations—a superfluid certainly does not respect (3).Indeed, in that case we find deviations from (3), and our result matches precisely that of anindependent computation carried out elsewhere [8]. Notation and conventions:
We use natural units ( ~ = c = 1) and the mostly-plus metricsignature throughout the paper. Before attempting to compute the expectation value of T µν for our framid, it is instructive toreview how things work out for a free relativistic scalar field, even if we perform computationsin a way that is not manifestly Lorentz covariant.Consider thus the Lagrangian L = − ( ∂φ ) − m φ , (4)which yields the stress-energy tensor T µν = ∂ µ φ∂ ν φ + η µν L . (5)A quadratic Lagrangian can always be rewritten as a total derivative plus a term propor-tional to the equations of motion. So, since the vacuum is translationally invariant, the secondterm in T µν does not contribute to our expectation value, and we simply have h T µν ( x ) i = h ∂ µ φ ( x ) ∂ ν φ ( x ) i (6)= − lim y → x ∂ µx ∂ νx G W ( x − y ) (7)= Z d k (2 π ) k µ k ν ˜ G W ( k ) , (8)where G W is the Wightman two-point function of φ :˜ G W ( k ) = θ ( k ) (2 π ) δ ( k + m ) . (9)At this stage one could use Lorentz-invariance and conclude that h T µν ( x ) i = 14 η µν Z d k (2 π ) k ˜ G W ( k ) . (10)The integral is UV divergent and must be regulated. However, whatever its value, we see that h T µν ( x ) i is proportional to η µν , as expected.Let’s instead go back to eq. (8), give up manifest Lorentz invariance, and use the deltafunction in ˜ G W to perform the integral over k . Using spatial rotational invariance, we get ρ ≡ h T i = 12 Z d k (2 π ) ω k , h T i i = 0 , p ≡ h T ii i = 16 Z d k (2 π ) k ω k , (11)4here ω k = p k + m . h T µν i is proportional to η µν if and only if ρ + p vanishes, but nowthis seems impossible: the integrals entering ρ and p are manifestly positive definite, so, howcan there be any cancellations? Then again, such integrals are UV divergent, so one shouldregulate them properly before jumping to conclusions.The UV regulator used should preserve Lorentz invariance. In particular, a hard cutoff inmomentum space will not do, because we have already performed the integral in k , and soat this point there is no way to introduce a hard cutoff compatible with Lorentz invariance(usually this involves Wick-rotating the d k integral to Euclidean space, and then imposing a4D rotationally invariant cutoff there.)One possibility is to use dimensional regularization directly for our 3-dimensional integrals.This is compatible with Lorentz invariance because it corresponds to formulating the originaltheory (4) in d + 1-dimensions, going through the manipulations (6)-(8) in d + 1 dimensions,and then performing the k integral explicitly to be left with d -dimensional integrals. We get: ρ = 12 Z d d k (2 π ) d ω k = − m d +1 Γ (cid:0) − d +12 (cid:1) π ) d +12 (12) p = 12 d Z d d k (2 π ) d k ω k = m d +1 Γ (cid:0) − d +12 (cid:1) π ) d +12 (13)(notice the → d replacement in the definition of p ), in agreement with ρ + p = 0.Another possibility is to use a generalization of Pauli-Villars. Recall that, in the simplestcase, Pauli-Villars amounts to regulating a log-divergent loop integral by a modification of theFeynman propagator of the form˜ G F ( k ) = i − k − m + iǫ → ˜ G P VF ( k ) = i − k − m + iǫ − i − k − M + iǫ , (14)with M being a very large mass scale, in particular M ≫ m . This improves the UV behaviorof the integral without affecting the IR one:˜ G P VF ( k ≪ M ) ≃ ˜ G F ( k ) , ˜ G P VF ( k ≫ M ) ≃ − i ( M − m ) k . (15)This is manifestly a Lorentz invariant modification of the propagator. More importantly forour applying these ideas to the framid case, such a modification corresponds to adding certainLorentz-invariant higher derivative terms to the Lagrangian, as clear from the exact rewriting(up to iǫ ’s) ˜ G P VF ( k ) = i − M + m M − m · k − M − m · k − M M − m · m , (16)or, keeping only the leading order in m/M ,˜ G P VF ( k ) ≃ i − k − k /M − m . (17) Here and for the rest of the paper we avoid introducing the MS renormalization scale µ , which wouldbe needed to make our dim-reg formulae dimensionally correct. We do so for notational simplicity, to avoidclutter. If one wants to reinstate µ in our formulae, one can do so just by dimensional analysis, interpretingour formulae as being expressed in µ = 1 units. G F ( k ; M ) the Feynman propagatorfor generic mass M , we want to perform the replacement˜ G F ( k ; m ) → ˜ G P VF ( k ) = ˜ G F ( k ; m ) + X a =1 c a ˜ G F ( k ; α a M ) , (18)where the c ’s and α ’s are suitable (order-one) coefficients, and M is a common very large massscale, M ≫ m .Since we need to improve the high energy behavior of our loop integrals by four powers of k compared to the standard Pauli-Villars case of (15), we want our modified propagator tohave the high energy behavior ˜ G P VF ( k ≫ M ) ∼ k , (19)which requires the α a ’s to be all different, and the c a ’s to be c a = Y b = a α b − m M α b − α a . (20)By combining denominators as done above in (16), one can see that this still corresponds toadding suitable higher-derivative Lorentz-invariant terms to the Lagrangian.Our expressions for ρ and p in eq. (11) do not involve directly a Feynman propagator.However, we have to remember that they came from integrating over k an expression involvingthe Wightman two-point function of φ . In our case, we must replace this with˜ G W ( k ; m ) → ˜ G P VW ( k ) = ˜ G W ( k ; m ) + X a =1 c a ˜ G W ( k ; α a M ) . (21)Then, integrating again over k , for ρ and p we simply get ρ = 12 Z d k (2 π ) (cid:2) ω k ; m + X a =1 c a ω k ; α a M (cid:3) (22) p = 16 Z d k (2 π ) k (cid:20) ω k ; m + X a =1 c a ω k ; α a M (cid:21) , ω k ; M ≡ p k + M . (23)As expected, but still surprisingly enough, choosing the c a ’s as in eq. (20) makes both of theseintegrals finite, and, in fact, opposite to each other. Namely: ρ = − p = f ( α ) M + g ( α ) m M + 132 π m log( m/M ) + h ( α ) m , (24)where f , g , and h are somewhat complicated functions of the α coefficients, whose explicit formwe spare the reader. On the other hand, the coefficient of m log m is finite (for M → ∞ ) and α -independent, in agreement with standard renormalization theory: like all non-analyticities inexternal momenta and mass parameters, it should be finite and calculable, that is, independentof the regulator used. In fact, if we expand the dim-reg result (12) for d →
3, we get exactlythe same coefficient for m log m . 6 The framid stress-energy tensor
The low-energy effective theory for the framid, a relativistic system that spontaneously breaksLorentz boosts, was first developed in [6]. Here, we provide a brief description of the theoryto set the ground for our computation.Framids are most intuitively described in terms of a vector field A µ ( x ) with a constanttime-like expectation value: h A µ ( x ) i = δ µ . (25)Such an expectation value breaks Lorentz boosts, and the corresponding Goldstone fields ~η ( x )can be thought of as parametrizing the fluctuations of A µ ( x ) in the directions of the brokensymmetries: A µ ( x ) = (cid:0) e i~η ( x ) · ~K (cid:1) µα h A α ( x ) i , (26)where ~K are the boost generators. Notice that, even with Goldstone fields present, A µ A µ = − A µ , performing the replacement (26),and then expanding to any desired order in the ~η fields, or by using the coset construction forspontaneously broken spacetime symmetries. These two approaches have different advantagesand disadvantages, but they yield the same result [6]. We will follow the former.To the second order in the derivative expansion, the most general effective Lagrangiantakes the form [6] L = − M (cid:2) ( c L − c T )( ∂ µ A µ ) + c T ( ∂ µ A ν ) + ( c T − A ρ ∂ ρ A µ ) (cid:3) , (27)where M is an overall mass scale and c L , c T represent the propagation speeds of the longitu-dinal and transverse Goldstones. We want to eventually work with the Goldstone Lagrangianexpanded to quadratic order, hence, with two derivatives in each term, it suffices to expand A µ to first order in the Goldstones, A = Λ = cosh | ~η | ≃ , A i = Λ i = η i | ~η | sinh | ~η | ≃ η i . (28)It is convenient to separate the Goldstones into their longitudinal and transverse modes, ~η = ~η L + ~η T , where ~ ∇ × η L = 0 , ~ ∇ · ~η T = 0 , (29)and also rescale them, ~η → ~ηM , (30)to eventually obtain a neat form of the effective Goldstone Lagrangian, L = 12 h ˙ ~η − c L ( ~ ∇ · ~η L ) − c T ( ∂ i η jT ) i . (31)Our goal is to check whether the stress-energy tensor resulting from this Lorentz-violatingtheory remains Lorentz-invariant when including quantum corrections: h T µν ( x ) i = − Λ η µν .7e know the off-diagonal components of the stress-energy tensor respect this condition dueto rotational invariance of the ground state, hence our task reduces to proving that h T ( x ) i + 13 h T ii ( x ) i = 0 , (32)where the spatial indices are implicitly summed over (as for the rest of the paper).In order to compute the one-loop correction to h T µν ( x ) i , we start from the full covarianttheory (27) and compute the canonical (Noether) stress-energy tensor, T µν = − ∂ L ∂ ( ∂ µ A λ ) ∂ ν A λ + g µν L = M (cid:2) ( c L − c T ) ∂ ν A µ ∂ λ A λ + c T ∂ µ A λ ∂ ν A λ + ( c T − A µ ∂ ν A λ A ρ ∂ ρ A λ (cid:3) + g µν L . (33)We wish to introduce the framid Goldstones as in (26) and expand T µν up to quadraticorder. The diagonal components of (33) reduce to T = −L + ˙ ~η · ˙ ~η, (34) T ii = 3 L + c T ∂ i ~η · ∂ i ~η + ( c L − c T )( ~ ∇ · ~η ) . (35)Not surprisingly, this is the same result we would have gotten by applying Noether’s theoremdirectly to eq. (31): even though boosts are spontaneously broken, spacetime translations arenot, and so one can compute the associated Noether current using directly the ~η parametriza-tion of the action and having ~η transform in the usual way under translations, ~η → ~η − ǫ µ ∂ µ ~η .We now perform manipulations similar to those of sect. 2. Dropping the terms proportionalto the Lagrangian for the same reason as made explicit there, the ground-state expectationvalues of the expressions above can be written as ρ ≡ h T i = lim y → x ∂ t x ∂ t y h ~η ( x ) · ~η ( y ) i (36) p ≡ h T ii i = 13 lim y → x h c T ∂ ix ∂ iy h ~η ( x ) · ~η ( y ) i + ( c L − c T ) ∂ ix ∂ jy h η i ( x ) η j ( y ) i i (37)Decomposing ~η into longitudinal and transverse components, the energy density becomes ρ = − lim y → x ∂ t x h ~η L ( x ) · ~η L ( y ) + ~η T ( x ) · ~η T ( y ) i = − lim y → x ∂ t x ( G L ( x − y ) + 2 G T ( x − y ))= 12 Z d k (2 π ) ( ω L + 2 ω T ) , (38)where G L and G T are the (scalar) Wightman two-point functions for longitudinal and trans-verse modes, ˜ G L/T ( ω, k ) = θ ( ω )(2 π ) δ ( ω − ω L/T ) , (39)and ω L and ω T are the corresponding energies, ω L/T = c L/T | k | . The relative factor of two in(38) comes from the fact that ˆ k · ˆ k = 1 and P i δ ii − ˆ k i ˆ k i = 2.8imilarly, the pressure can be rewritten as p = 16 Z d k (2 π ) (cid:18) c L k ω L + 2 c T k ω T (cid:19) . (40)These expressions for ρ and p are very simple generalizations of the corresponding onesin sect. 2 for a generic massive scalar. However, as we did there, we first need to regularizethem before we can check whether ρ + p vanishes. As we emphasized in the Introduction, theregulators used should be consistent with the spontaneously broken Lorentz invariance. Let us first consider a suitable generalization of Pauli-Villars regularization. Given the tech-nical similarities between the derivation we just performed and that of sect. 2, it’s clear that if we are allowed to introduce independent Pauli-Villars modifications for the longitudinaland transverse phonons’ propagators, taking into account their different speeds, then we get ρ + p = 0 for the framid as well.More explicitly, consider the longitudinal phonons’ contributions to ρ and p , ρ L ≡ Z d k (2 π ) ω L , p L ≡ Z d k (2 π ) c L k ω L , (41)with ω L = c L | k | . Apart from the integration measure, k always appears here in the com-bination c L | k | . The same is true for the Wightman two-point function (39), where theseexpressions come from, and for the associated Feynman propagator,˜ G LF ( ω, k ) = iω − c L k + iǫ . (42)So, upon changing the integration variable, k ′ = c L k , we have ρ L = 1 c L ρ rel (0) , p L = 1 c L p rel (0) , (43)where ρ rel ( m ) and p rel ( m ) are the relativistic expressions for the energy density and pressureof a massive scalar of mass m , eq. (11). Then, the same Pauli-Villars regularization that wasapplied in sect. 2 can be applied here, yielding ρ L + p L = 0 . (44)Mutatis mutandis, the same considerations can be applied to the transverse phonons’ contri-butions ρ T and p T , yielding ρ T + p T = 0 . (45)And so, combining the longitudinal and transverse sectors, ρ + p = 0.Notice however that the longitudinal and transverse Goldstones have different speeds ingeneral, and so the Pauli-Villars regularization procedure that we are advocating has to bedifferent for the two sectors. Namely, referring to the explicit analysis of sect. 2, whenever9here is a k in a propagator, when we deal with the longitudinal sector we have to replacethat with − ω + c L k , and when we deal with the transverse sector we have to replace thatwith − ω + c T k .Recall that in standard relativistic cases, such as that studied in sect. 2, a Pauli-Villarsmodification of the propagator can be thought of as coming directly from a suitable localhigher-derivative modification of the action. So, in our case the question is whether there isa local and Lorentz-invariant higher-derivative modification of the framid’s action that corre-sponds to independent Pauli-Villars modifications of the longitudinal and transverse propaga-tors of the desired type. The requirements of locality and Lorentz-invariance are nontrivial—the former because the longitudinal/transverse splitting of ~η is non-local, the latter because ~η transforms nonlinearly under Lorentz boosts.Let’s start with locality. The modifications of the Feynman propagators we are after, uponcombining denominators as explained in sect. 2, take the form˜ G L,P VF ( ω, k ) = i − k L − k L − k L − k L , k L ≡ − ω + c L k , (46)and similarly for the transverse propagator. The Λ a ’s are suitable combinations of the Pauli-Villars pole masses. Importantly for what follows, they are the same for the longitudinal andtransverse propagators, as long as the pole masses are chosen to be the same for the twosectors. This can be understood easily by thinking about the structure of the propagatorsat k = 0, in which case the fact that the longitudinal and transverse propagation speeds aredifferent does not matter.Using the canonical normalization of (31), up to total derivatives these propagators cor-respond to the quadratic Lagrangian L P V = 12 ~η L · h (cid:3) L − (cid:3) L + 1Λ (cid:3) L − (cid:3) L i ~η L + ( L → T ) , (47)where (cid:3) L denotes the differential operator (cid:3) L ≡ − ∂ t + c L ∇ , (48)and ‘( L → T )’ stands for a similar structure involving the transverse field ~η T and its propaga-tion speed. Our question of locality is thus reduced to the question of whether this quadraticLagrangian can be written as a local quadratic Lagrangian for the full ~η field.Notice that as long as at least one spatial laplacian acts on η L or η T , one can easily performthe longitudinal/transverse decomposition in a local fashion: defining the local differentialoperator matrix, D ij = ∂ i ∂ j , (49)we simply have ∇ ~η L = D · ~η , ∇ ~η T = ( ∇ − D ) · ~η . (50)Notice also that, in the quadratic action (47), it is enough that the ~η ’s on the right—thoseacted upon by derivatives—be split into longitudinal and transverse. The undifferentiated ~η ’s on the left can be replaced with the full ~η field, because, as usual, at quadratic order alllongitudinal-transverse mixings automatically vanish.10o, the only remaining question concerning locality is whether the terms in (47) with time-derivatives only can be rewritten in a local fashion. However, the coefficients of such terms arethe same for the longitudinal and transverse sectors. This, upon using again the vanishing oflongitudinal-transverse mixings, allows us to combine these terms into purely time-derivativeterms for the full ~η field: L P V ⊃ ~η L · h − ∂ t − ∂ t − ∂ t − ∂ t i ~η L + ( L → T ) (51)= 12 ~η · h − ∂ t − ∂ t − ∂ t − ∂ t i ~η (52)So, in summary, a local rewriting of (47) involves the following building blocks: ~η · ∂ at ( ∇ ) b D c · ~η , (53)with different integer non-negative values for a , b , and c , up to a + b + c = 4, and suitablecoefficients. In particular, we can restrict to c = 0 ,
1, because D is proportional to a projectoroperator (its only role is to isolate the longitudinal component of ~η ): D · D = ∇ · D . (54)We can now ask whether these building blocks are compatible with the spontaneously brokenLorentz invariance. In particular, we can ask whether there exist manifestly Lorentz-invariantcombinations of A µ and ∂ µ that, when expanded to quadratic order in the ~η fields, reduceprecisely to these building blocks.It is quite easy to convince oneself that the answer is yes. To this end, it is convenient tointegrate by parts half of the derivatives in (53). Then, up to a possible sign, the buildingblocks are (cid:0) ( ∂ t ) a ∂ i . . . ∂ i b η k (cid:1) ( c = 0) (55)and (cid:0) ( ∂ t ) a ∂ i . . . ∂ i b ( ~ ∇ · ~η ) (cid:1) ( c = 1) (56)Recalling that, to first order in ~η , A µ is simply A ≃ , ~A ≃ ~η , (57)we see that two simple Lorentz-invariant generalizations of (55) and (56) that reduce to themto quadratic order in ~η are (cid:0) ( ∂ k ) a ∂ ⊥ µ . . . ∂ ⊥ µ b A ν (cid:1) ( c = 0) (58)and (cid:0) ( ∂ k ) a ∂ ⊥ µ . . . ∂ ⊥ µ b ( ∂ ν A ν ) (cid:1) , ( c = 1) (59)where ∂ k and ∂ ⊥ µ are defined as ∂ k ≡ − A α ∂ α , ∂ ⊥ µ ≡ ∂ µ + A µ A α ∂ α . (60)In conclusion: there exist local Lorentz-invariant higher-derivative corrections to the framidaction that modify the ~η propagator in a Pauli-Villars fashion, with suitable independentmodifications for the longitudinal and transverse modes, such that the Pauli-Villars analysisof sect. 2 can be separately applied to the two sectors, yielding ρ + p = 0.11 Dimensional regularization
Let’s now consider dimensional regularization. As for the relativistic case considered in sect. 2,dimensional regularization of our spatial momentum integrals (38) and (40) is consistent withLorentz invariance. This is because we can think of it as corresponding to formulating theoriginal manifestly Lorentz-invariant theory for A µ , eq. (27), in d + 1 spacetime dimensions,and then going through all the subsequent steps that led us to (38) and (40) keeping d generic.Were we to do so, we would end up with the same integrals (38) and (40), but in d ratherthan 3 dimensions, and with a 1 / d rather than 1 / ρ + p = 0, but in a trivial way. To run a nontrivial check, we need to deform our theory.We must do so consistently with Lorentz invariance, of course. And so, in particular, massparameters for the Goldstones are not allowed.A particularly physical way to deform the theory is to couple the framid to a massivefield, in all ways allowed by symmetry. For simplicity, let’s take this to be a scalar field, φ . Ifits mass is below the cutoff of the effective theory (the M of sect. 3, up to suitable powersof c L/T ), φ must be included in our computation. Assuming it has zero expectation value,for our one-loop computation we are interested in all Lorentz-invariant combinations of A µ and φ with up to two derivatives and which, once expanded about the framid’s backgroundconfiguration, yield quadratic terms in the ~η and φ fields.We find that there are only three interactions with these properties: L ⊃ A µ ∂ µ φ, A µ A ν ∂ µ ∂ ν φ, A µ A ν ∂ µ φ∂ ν φ . (61)All other possibilities are either related to these through integration by parts or, when ex-panded in the Goldstone fields to the desired order, yield terms that are total derivativesthemselves, and can thus be neglected. Adding the terms above to the framid action, and in-cluding also standard kinetic and mass terms for φ , which are Lorentz-invariant by themselves,our effective Lagrangian at quadratic order becomes L → h ˙ ~η − c L ( ~ ∇ · ~η L ) − c T ( ∂ i η jT ) + ˙ φ − ( ~ ∇ φ ) − m φ + 2 b φ ~ ∇ · ~η L + 2 b ˙ φ ~ ∇ · ~η L + b ˙ φ i , (62)where the b a ’s are generic coupling constants. Notice that the transverse components of theGoldstone fields don’t mix with the massive scalar, since ~ ∇ · ~η T = 0.The relevant components of the stress-energy tensor now read T = ˙ ~η + b ˙ φ ~ ∇ · ~η L + (1 + b ) ˙ φ , (63) T ii = c L ( ~ ∇ · ~η L ) + c T ∂ i ~η T · ∂ i ~η T − b φ ~ ∇ · ~η L − b ˙ φ ~ ∇ · ~η L + ( ~ ∇ φ ) , (64)where we have omitted terms proportional to the Lagrangian, since, as before, they do notcontribute to our expectation values.Notice that, at this order, the transverse Goldstones ~η T are completely decoupled from φ , both at the level of the Lagrangian and as far as their contributions to the stress-energy12ensor are concerned. The systems is thus divided into two sectors: a transverse one, for whichthe presence of φ is irrelevant, and which thus has vanishing energy density and pressure indimensional regularization, ρ T = Z d d k (2 π ) d c T | k | = 0 , p T = 1 d Z d d k (2 π ) d c T | k | = 0 , (65)and a longitudinal one, consisting of ~η L and φ , for which the computation of the energy andpressure is, as we will now see, quite involved.From now on, we will restrict to the longitudinal sector, which amounts to setting ~η T tozero in the formulae above. Also, for notational simplicity we will drop the subscript ‘ L ’ from ~η L (but we will remember that we are dealing with a longitudinal field.) b Consider first the case in which b and b are set to zero while b is nonzero. In that case, ~η and φ are completely decoupled (at this order), and their contributions to ρ + p can beanalyzed separately. For ~η , there isn’t much to say: the relevant integrals for ρ and p vanishin dimensional regularization, because they do not involve any mass scale—that is, they areintegrals of pure powers.The situation is more interesting for φ . Its action is that of a massive scalar with apropagation speed different from one, S = Z d x (cid:2) ˙ φ − c φ ( ~ ∇ φ ) − M φ φ (cid:3) , c φ ≡
11 + b , M φ ≡ c φ m , (66)where for convenience we redefined the normalization of φ by φ → c φ φ .Applying to the purely φ parts of eqs. (63), (64) the same manipulations as in the case ofa relativistic scalar (see sect. 2), we find that the integrals we should compute are ρ φ = 12 Z d d k (2 π ) d q c φ k + M φ , p φ = c φ × d Z d d k (2 π ) d k q c φ k + M φ . (67)Up to a redefinition of the integration variable, k = k ′ /c φ , these are clearly the same integralsas those of sect. 2. What’s perhaps surprising is that, after such a change of variables, theoverall powers of c φ we are left with are the same for ρ φ and for p φ : ρ φ = 1 c dφ ρ rel ( M φ ) , p φ = 1 c dφ p rel ( M φ ) , (68)where ρ rel ( m ) and p rel ( m ) are the energy density and pressure for a relativistic scalar of mass m , eqs. (12) and (13). So, once again, we get ρ φ + p φ = 0 . (69)This apparent accident is in fact a consequence of a formal (spurionic) invariance of theaction (66): if we rescale the spatial coordinates but not time, and compensate for this by arescaling of c φ and of φ , ~x → λ ~x , c φ → λ c φ , φ → λ − / φ , (70)13he action does not change. As usual for spurion analyses, this informs how physical quantitiescan depend on c φ . In particular, we are interested in ρ φ = h T i , p φ = d h T ii i , (71)for a state that is invariant under translations. These expectation values are thus constant in ~x , and so their transformation properties under rescalings of coordinates must be completelytaken care of by explicit powers of c φ . T is a spatial density of energy. Since energy ∼ time − does not change under a rescaling of spatial coordinates, we must have T → λ d T ⇒ h T i ∝ c dφ . (72) T ij is not the density of a conserved quantity. However, it is related to the momentum density T j by the conservation equation ∂ T j + ∂ i T ij = 0 . (73)The momentum density rescales as T j → λ d +1 T j , (74)because momentum itself is the inverse of a length, and so it must rescale as ~P → ~P /λ . Using(73) and (74), we thus have T ij → λ d T ij ⇒ h T ij i ∝ c dφ (75)We thus see that both ρ φ and p φ depend on c φ exactly in the same way, in agreement withour explicit result in (68).In more intuitive terms, all of the above stems from the statement that, even if we abandonnatural units and we give independent units to mass ( m ), length ( ℓ ), and time ( t ), energydensity and pressure still have the same units : h energyvolume i = mℓ · t , (cid:2) pressure (cid:3) = h forcesurface i = mℓ · t . (76)Their ratio is thus dimensionless, and must be the same in all unit systems. In particular ifit is − c φ = 1, then it must be − c φ = 1.The arguments above show that, as far as our check is concerned, we can consistently set b to zero, even when we turn b and b back on. The reason is that we can work in unitssuch that c φ = 1, which corresponds to b = 0, and run our check in those units. Going fromnatural units ( c = 1) to c φ = 1 units certainly affects the values of the other parameters ofthe theory: c L , c T , M , b , b , and m . However, since we are leaving these generic anyway,such a change has no repercussions for our check . For instance, in SI units, 1 J/m = 1 kg/m s = 1 Pa. For more general questions, say computing ρ and p separately rather that just checking ρ + p = 0, onecan still first work in c φ = 1 units and then reinstate the dependence on c φ after the fact, by taking intoaccount how the other parameters change when we change units. For instance the parameter c L in c φ = 1units becomes c L /c φ in any other unit system. .2 Analysis for b and b We may then switch off b and perform our computation for non-zero values of the mixingcoefficients b and b . Combining eqs. (63) and (64), switching to a generic number of spatialdimensions, 3 → d , and dropping again terms proportional to the quadratic Lagrangian, weobtain the following expression for ρ + p : ρ + p ≡ h T + 1 d T ii i = d + 1 d (cid:16) ˙ ~η + ˙ φ (cid:17) − d m φ + 1 d b φ ~ ∇ · ~η + d + 1 d b ˙ φ ~ ∇ · ~η . (77)We can now perform the same manipulations as in sect. 2. We find: ρ + p = 1 d lim x → y h − ( d + 1) ∂ t h ~η ( x ) · ~η ( y ) i − (( d + 1) ∂ t + m ) h φ ( x ) φ ( y ) i + ( b − ( d + 1) b ∂ t ) ~ ∇ · h ~η ( x ) φ ( y ) i i = 1 d Z d d k (2 π ) d dω π h ( d + 1) ω ˜ G ηηW ( ω, k ) + (( d + 1) ω − m ) ˜ G φφW ( ω, k )+ ( ib − ( d + 1) b ω ) | k | ˜ G ηφW ( ω, k ) i , (78)where ˜ G abW is the matrix of Wightman two-point functions for the fields ψ a = ( η, φ ) and,thanks to the longitudinality of ~η , we were able to switch to a purely scalar notation,˜ ~η ( ω, k ) ≡ ˆ k ˜ η ( ω, k ) . (79)As usual, from the quadratic Lagrangian we can easily compute the matrix of Feynmanpropagators, ˜ G abF ( ω, k ). However, going from these to the Wightman two-point functionsrequires some work. The general relationship, which we review in Appendix A, is˜ G abW ( ω, k ) = Z dω ′ π (cid:20) iω − ω ′ + iǫ ˜ G abF ( ω ′ , k ) − iω − ω ′ − iǫ ˜ G ab ∗ F ( − ω ′ , − k ) (cid:21) (80)= X n (cid:2) K − ( ω, k )( ω − ω n ) (cid:3) ab (2 π ) δ ( ω − ω n ) , (81)where K is the kinetic matrix appearing in the quadratic action, S = 12 Z d k (2 π ) dω π ˜ ψ ∗ a ( ω, k ) K ab ( ω, k ) ˜ ψ b ( ω, k ) , (82)and the ω n = ω n ( k ) are the positive-energy poles of the Feynman propagators.Using this in eq. (78) and performing the ω integral leaves us with ρ + p = 1 d X n Z d d k (2 π ) d h ( d + 1) ω n R ηηn ( k ) + (( d + 1) ω n − m ) R φφn ( k )+ ( ib − ( d + 1) b ω n ) | k | R ηφn ( k ) i , (83)15here the R ’s are the residues R abn ( k ) = lim ω → ω n (cid:2) K − ( ω, k )( ω − ω n ) (cid:3) ab . (84)The positions of the poles can be computed for generic b and b , and so can the associatedresidues. However, the resulting expressions involve somewhat complicated double squareroot structures, which makes it impossible for us to perform the final integral in k explicitly.To circumvent this problem, we can consider the small b , b limit, and expand to the firstnontrivial order in these couplings.The kinetic matrix for η and φ associated with the quadratic Lagrangian (62) is K = ω − c L k − ( b ω + ib ) | k |− ( b ω − ib ) | k | ω − k − m (85)The poles of the Feynman propagators are the zeros of det K , which to quadratic order in b , b read ω ≃ c L | k | − | k | (cid:0) b + b c L k (cid:1) c L (cid:0) (1 − c L ) k + m (cid:1) ,ω ≃ p k + m + k (cid:0) b + b ( k + m ) (cid:1) p k + m (cid:0) (1 − c L ) k + m (cid:1) . (86)Upon inverting K , expanding the residues (84) also to quadratic order, and plugging theseexpansions into our expression for ρ + p , eq. (83), we end up with ρ + p = 1 d X A C A Z d d k (2 π ) d | k | α A (cid:0) (1 − c L ) k + m (cid:1) γ A p k + m β A , (87)where the C A ’s are suitable coefficients, and α A , β A , γ A are powers that vary in combinations,yielding in total eleven structurally distinct terms, as outlined in Table 1.Notice that the first three terms in Table 1 are the contributions one gets from the freetheories of the Goldstones and the massive scalar. Namely, the first term is ρ + p ⊃ Z d d k (2 π ) d (cid:18) c L | k | + 1 d c L | k | (cid:19) ≡ ρ L + p L , (88)in agreement with equations (38) and (40) and, since it involves only pure powers of themomentum, it integrates to zero in dimensional regularization. The second and third termsare the pressure and energy densities of the free massive scalar, ρ + p ⊃ Z d d k (2 π ) d d k p k + m + p k + m ! ≡ p φ + ρ φ , (89)whose sum also equals zero, as was shown in section 2.16 β γ C c L ( d + 1)2 -1 0 d b (1 − c L )( d + 1)4 -1 2 b ( d −
1) + b m (3 + d (2 − c L ))2 -1 2 b m ( d + 2) + b m ( d + 2)5 0 2 − b c L (1 − c L )( d + 1)3 0 2 − b c L (1 − c L )( d − − b c L m ( d +3)1 0 2 − b c L m ( d + 1)6 -3 2 − b c L ( d − − b c L m d Table 1: Coefficients C A for all the values of the powers α A , β A , γ A that appear in (87).For the rest of the terms, we switch to polar coordinates and perform the integral in theradial k -direction; the closed-form result for this type of integrals, dropping the coefficients C A , is12 m d + α j + β j − γ j " (1 − c L ) − d − αj Γ (cid:0) d + α j (cid:1) Γ (cid:0) − d − α j +2 γ j (cid:1) F (cid:0) d + α j , − β j ; d + α j − γ j ; − c L (cid:1) Γ( γ j )+ (1 − c L ) − γ j Γ (cid:0) d + α j − γ j (cid:1) Γ (cid:0) − d − α j − β j +2 γ j (cid:1) F (cid:0) γ j , − d − α j − β j +2 γ j ; − d − α j +2 γ j ; − c L (cid:1) Γ( − β j ) , (90)where F is a hypergeometric function. For some structures in the integrand of equation (87),the integration result takes much simpler forms; for instance the term with α = 5 , β = 0 , γ = 2yields Z d d k (2 π ) d | k | (cid:0) (1 − c L ) k + m (cid:1) = − π m d (1 − c L ) − d − (3 + d ) sec (cid:16) dπ (cid:17) . (91)Other structures, particularly when β = 0, integrate to totally nontrivial combinations ofhypergeometric, Gamma, and trigonometric functions, as seen in the generalized form in (90),17nd so displaying them here wouldn’t provide much intuition. It is miraculous to see that allthese terms, when summed, cancel exactly with each other, despite the highly complicatedstructures and combinations of coefficients involved. What’s also interesting, is that we don’tneed to specify a number of spatial dimensions to obtain the final answer. That is, for any d ,we find: ρ + p = 0 . (92) As a check of our methods and computational tools, we now look at the case of a superfluid,for which we know that the stress-energy tensor does not have a Lorentz invariant expectationvalue. In particular, we want to make sure that the mysterious cancellations that yield ρ + p = 0in the framid case are not a result of potential nuances of the computation. For a superfluid, h T µν ( x ) i 6 = − Λ η µν already at tree level, and so an analogous computation should yield anonzero result.The simplest implementation of a superfluid EFT involves a single scalar field ψ ( x ) with ashift symmetry, ψ → ψ +const, and a time-dependent vev, h ψ ( x ) i = µt [6,9,10], where µ is thechemical potential. The superfluid phonon field, π ( x ), parametrizes fluctuations around thisbackground, ψ ( x ) ≡ µt + π ( x ). To lowest order in derivatives, the most general low-energyeffective action is S = Z d x P ( X ) , X ≡ − ∂ µ ψ∂ µ ψ, (93)where P ( X ) is a generic function, in one-to-one correspondence with the superfluid’s equationof state . Upon expanding to quadratic order in the phonon field and choosing canonicalnormalization, one gets S → Z d x h ˙ π − c s ( ~ ∇ π ) i , (94)with the sound speed given in terms of derivatives of the Lagrangian, c s = P ′ ( X ) P ′ ( X ) + 2 XP ′′ ( X ) . (95)Similarly to the framid case, in order to run a nontrivial check in dimensional regularization wecouple the superfluid to a massive scalar φ . We look for all possible shift-symmetric, Lorentz-invariant couplings that, when expanded in the superfluid phonons π ( x ), yield quadratic termsin π and φ , with at most two derivatives acting on them. Since X = µ + 2 µ ˙ π + ˙ π − ( ~ ∇ π ) ,the couplings that produce Lorentz-violating structures are: f ( X ) φ → ˙ πφf ( X ) ∂ µ ψ∂ µ φ → ˙ π ˙ φ, ~ ∇ π · ~ ∇ φf ( X ) ∂ µ ψ∂ ν ψ∂ µ φ∂ ν φ → ˙ φ . Namely, the function P relates the pressure to the chemical potential: p = P ( µ ). πφ term inthe quadratic Lagrangian. Moreover, we choose the simplest possible form for f , that is f ( X ) = X . Finally, we work in the c s = 1 limit (which implies P ′′ ( X ) = 0). Thesechoices give us the opportunity to compare our result directly to an independent path-integralcalculation, which will be summarized below and published elsewhere [8].So, in summary, we start from L = P ( X ) + Xφ − ∂ µ φ∂ µ φ − m φ , (96)whose stress-energy tensor is T µν = 2 P ′ ( X ) ∂ µ ψ∂ ν ψ + 2 ∂ µ ψ∂ ν ψφ + ∂ µ φ∂ ν φ, (97)omitting, as before, terms proportional to the Lagrangian. Expanding to quadratic order andcanonically normalizing π , we get L ≃ h ˙ π − ( ~ ∇ π ) + 4 b ˙ πφ + ∂ µ φ∂ µ φ − m φ i , (98)and T ≃ ˙ π + ˙ φ + 4 b ˙ πφ , T ii ≃ ( ~ ∇ π ) + ( ~ ∇ φ ) , (99)where the spatial indices are summed over and b ≡ µ/ p P ′ ( µ ) .Performing manipulations similar to those of the framid case, we rearrange some termsand obtain T + 1 d T ii = d + 1 d ( ˙ π + ˙ φ ) − d m φ + 4 d + 1 d b ˙ πφ − d L , (100)whose expectation value is given by ρ + p ≡ h T + 1 d T ii i = 1 d Z d d k (2 π ) d dω π (cid:20) ( d + 1) ω (cid:16) ˜ G ππW ( ω, k ) + ˜ G φφW ( ω, k ) (cid:17) − m ˜ G φφW ( ω, k ) − i ( d + 1) ωb ˜ G πφW ( ω, k ) (cid:21) . (101)Like before, we expand the Feynman propagators and their poles for small values of thecoupling constant b , and then construct the Wightman two-point functions. As expected, allexpressions take simpler forms in this case. Namely, the (positive-frequency) poles are ω ≃ p k + m + 2 b m p k + m ,ω ≃ | k | − b m | k | , (102) Notice that the expressions (99) can be obtained by applying Noether’s theorem directly to the quadraticLagrangian (98), but only if one takes into account that π transforms nonlinearly under time-translations,since these are spontaneously broken by the background ψ = µt . This is related to the statement that thesuperfluid’s ground state is an eigenstate of H − µQ , but not of H , with H = R d x T being the Hamiltonian,and Q = R d x J the charge. R ππ ( k ) ≃ b m p k + m , R ππ ( k ) ≃ (cid:20) | k | − b (2 k + m ) m | k | (cid:21) ,R φφ ( k ) ≃ " p k + m − b (2 k + m ) m p k + m , R φφ ( k ) ≃ b m | k | ,R πφ ( k ) ≃ − i bm , R πφ ( k ) ≃ i bm . (103)Putting everything together, the integral in question eventually becomes ρ + p = b d m Z d d k (2 π ) d k − d ( k + m ) p k + m (104)= − b m π ( d + 1) (cid:18) m π (cid:19) ( d − / Γ (cid:18) − d (cid:19) , (105)which is different from zero, as expected.We now come to the result of the independent computation alluded to above. As a possibleUV completion of a superfluid EFT, one can consider a massive complex scalar Φ with quarticinteractions. Putting the system at finite chemical potential, one ends up with a superfluid,with our field ψ being associated with the phase of Φ. On the other hand, the radial modeof Φ is massive, and can thus be thought of as our massive scalar φ . For this system, onecan explicitly compute first the associated P ( X ) at tree level, and then the one-loop quantumcorrections to it, in the form of the quantum effective action Γ[ ψ ] [8]. (To lowest order inderivatives, this can be done via functional methods akin to those normally used to computea Coleman-Weinberg effective potential [11, 12].) Applying Noether’s theorem to Γ[ ψ ], oneobtains directly the one-loop expectation value of T µν , which, for ρ + p , matches exactly theresult above [8]. We close with a few remarks:1. Our paper is about Lorentz-invariance in a system that spontaneously breaks it—thatis, in a system in which such a symmetry is not manifest. So, one must be particularlycareful in unveiling possible sources of Lorentz breaking coming from the way one doescomputations. We already discussed at length how to address UV divergences in a waythat is compatible with Lorentz invariance. Another subtlety one needs to address isthe Lorentz invariance of the path-integral measure. Even though we did not use pathintegrals for our computations, canonical quantization of the Goldstones’ effective theoryis equivalent to path-integral quantization with the measure
D~η ≡ Y x d η ( x ) . (106)20his is not Lorentz invariant, because the ~η fields transform non-linearly under boosts . As usual, this should not matter as long as one uses dimensional regularization, butwith other regulators it might matter. We investigate the issue in Appendix C. Theconclusion is that this subtlety does not matter: ( i ) for our quantity ( h T µν i ), to theorder we are doing computations (one loop), regardless of the regulator used; or ( ii ) forany quantity, to any order, if one uses dimensional regularization. We were thus justifiedto neglect it.2. We have been emphasizing, puzzling over, and checking the fact that our expectationvalue h T µν i ∝ η µν is Lorentz-invariant. However, as we hope the discussion in sect. 5.1has made clear, such an expectation value is in fact compatible with any Lorentz invari-ance, that is, it is invariant under generalized boosts with an arbitrary speed-of-lightparameter c . This is because h T i (energy density) and h T ij i (pressure, or stresses)always have the same units, and so a statement like h T µν i ∝ η µν = diag( − , , ,
1) isindependent of the units used. And so, in particular, it is independent of the value of c .We find this to be an interesting twist. It could have important conceptual implications,or it could just be a technical curiosity.3. Besides the framid, there are other cases in which the expectation value of T µν is moresymmetric than the ground state . For instance, for a superfluid time-translations arespontaneously broken, but the expectation value in question is invariant under them (seesect. 6). However, in that case such a symmetry property can be explained by standardselection rules: the ground state spontaneously breaks time translations ( H ) and a U (1)symmetry ( Q ) down to a linear combination thereof ( H − µQ ) [10]. Expectation valuesand more in general correlation functions should only be invariant under the unbrokencombination. However, for operators that are neutral under the U (1) symmetry, thisautomatically translates into invariance under time-translations. T µν is one such oper-ator, and so its expectation value is invariant under time translations, even though theground state is not. We do not see any mechanism like this at play in the framid case.What is the general lesson of our analysis? Are there implications for the cosmological constantproblem? In general terms, our analysis exhibits an explicit example of a quantum systemin which a certain expectation value is invariant under a symmetry even though there areno selection rules (including those of remark 3 above) enforcing this. In order to find apotential application of this phenomenon to the cosmological constant problem, we think oneshould start by making progress in two directions. The first is to understand how general thisphenomenon is: are there other examples, and what are their common features—for instance,do they all require a spontaneous breaking of Lorentz invariance? The second is to find astructure, a pattern in our one-loop check: the cancellations that lead to ρ + p = 0 for theframid, especially those of sect. 5.2, are absolutely nontrivial. It is hard to believe that theyare not enforced by a hidden structure in the computation. Perhaps there is a better way oforganizing the computation that would make such a structure manifest. We plan to explorethese questions in the near future. Equivalently, one can phrase the problem directly in the canonical formalism [13]. See for instance ref. [14]for an analysis of the same issue for the chiral Lagrangian. We thank Lam Hui for bringing this up. cknowledgements We thank Lam Hui, Riccardo Penco, and Raman Sundrum for useful discussions and com-ments. Our work is partially supported by the US DOE (award number DE-SC011941) andby the Simons Foundation (award number 658906).
A Wightman and Feynman
As we have seen, it is particularly helpful to rewrite the expectation value of the stress-energytensor in terms of derivatives acting on Wightman two-point functions. Formally, given a setof real fields ψ a governed by a quadratic Lagrangian, in general one has h T µν (0) i = lim x → X ab D µνab ( ∂ x ) G abW ( x ) , (107)where D µνab ( ∂ x ) is a function of derivatives, usually up to second order, and G W represents thematrix of Wightman two-point functions, G abW ( x ) ≡ h ψ a ( x ) ψ b (0) i . (108)Notice that, for real fields, one has G baW ( x ) = G ab ∗ W ( − x ) . (109)However, we are more familiar with the calculation of Feynman propagators given a certaintheory, not the Wightman version. How can we relate G W to G F , the Feynman propagator,in a way that is helpful to our calculations?To begin with, notice that, by definition, G abF = h ψ a ( x ) ψ b (0) i θ ( t ) + h ψ b (0) ψ a ( x ) i θ ( − t ) (110)= G abW ( x ) θ ( t ) + G baW ( − x ) θ ( − t ) (111)= G abW ( x ) θ ( t ) + G ab ∗ W ( x ) θ ( − t ) . (112)As usual, θ ( t ) is the step function. This relation can be inverted to give G abW ( x ) = G abF ( x ) θ ( t ) + G ab ∗ F ( x ) θ ( − t ) . (113)Computations are usually easier in Fourier transform. Using the Fourier representation ofthe step function, θ ( t ) = Z dω π iω + iǫ e − iωt , (114)we get ˜ G abW ( ω, k ) = Z dω ′ π (cid:20) iω − ω ′ + iǫ ˜ G abF ( ω ′ , k ) − iω − ω ′ − iǫ ˜ G ab ∗ F ( − ω ′ , − k ) (cid:21) (115)= Z dω ′ π (cid:20) iω − ω ′ + iǫ ˜ G abF ( ω ′ , k ) + h . c . (cid:21) , (116)22here the last equality follows from ˜ G abF ( ω, k ) = ˜ G baF ( − ω, − k )—a direct consequence of thedefinition (110). We thus see that ˜ G abW is a hermitian matrix, in agreement with (109).The matrix of Feynman propagators is easily computed starting from the quadratic actionwritten in Fourier space, S = 12 Z d k (2 π ) dω π ˜ ψ ∗ a ( ω, k ) K ab ( ω, k ) ˜ ψ b ( ω, k ) , (117)where the kinetic matrix K is hermitian. In matrix notation, one simply has˜ G F ( ω, k ) = i (cid:0) K ( ω, k ) + iǫ (cid:1) − . (118)Focusing on the first term in the integral (116), and assuming that, as usual, the Feynmanpropagators decay at infinity and have (simple) poles slightly away from the real axis, we canclose the ω ′ contour in the lower half plane. We thus only pick up the poles of ˜ G F that lieunder the real axis—the positive frequency ones, for a stable theory. We get˜ G abW ( ω, k ) = X n iω − ω n + iǫ (cid:2) K − ( ω ′ , k )( ω ′ − ω n ) (cid:3) abω ′ = ω n + h . c . , (119)where the sum is extended over the positive frequency poles, ω n = ω n ( k ).Using the distributional identity 1 x + iǫ = P x − iπδ ( x ) (120)and the hermiticity of K , we finally get˜ G abW ( ω, k ) = X n (cid:2) K − ( ω ′ , k )( ω ′ − ω n ) (cid:3) abω ′ = ω n (2 π ) δ ( ω − ω n ) . (121)As a check, for a single relativistic massive scalar the kinetic “matrix” is simply K = ω − k − m , (122)the positive frequency pole is ω k = √ k + m , (123)and the Wightman two-point function thus reduces to˜ G W ( ω, k ) = 12 ω k (2 π ) δ ( ω − ω k ) (124)= (2 π ) θ ( ω ) δ ( k + m ) , (125)which is the correct expression. Notice however that, in the general derivation above, we havenever used Lorentz invariance. 23 Symmetric stress-energy tensors
One may consider performing our calculations starting from the more trusted symmetricversions of the stress-energy tensor, i.e. the Hilbert and Belinfante tensors. When derivingthe Hilbert tensor for a framid, we have to keep in mind the unit-norm constraint on A µ , g µν A µ A ν = − , (126)which forbids varying the metric g µν independently of A µ . One can introduce a vierbein andvary the action with respect to it instead, yielding [6] T µνH = 1 √− g (cid:18) δSδg µν + δSδA µ A ν (cid:19) , (127)where now the functional derivatives are unconstrained.The tensor above is evidently not symmetric in general. In fact, it is symmetric onlyon-shell, i.e upon using the equations of motion, which, taking into account the unit-normconstraint once more, are simply ( η µν + A µ A ν ) δSδA ν = 0 . (128)The end result for the (symmetric) Hilbert tensor is T µνH = L g µν + 2 c h A ( µ ∂ α ∂ ν ) A α − A α ∂ α ∂ ( µ A ν ) − ∂ α A α ∂ ( µ A ν ) − ∂ ( µ A α ∂ ν ) A α + ∂ α A ( µ ∂ ν ) A α i + 2 c h A ( µ ∂ α ∂ ν ) A α − g µν A α ∂ β ∂ α A β − g µν ∂ α A α ∂ β A β i + 2 c h A ( µ ∂ α ∂ α A ν ) + ∂ α A ( µ ∂ α A ν ) − A α ∂ α ∂ ( µ A ν ) − ∂ α A α ∂ ( µ A ν ) − ∂ α A ( µ ∂ ν ) A α i + 2 c h A µ A ν ∂ α A β ∂ β A α + A α A µ A ν ∂ β ∂ α A β − A α A ( µ ∂ α A ν ) ∂ β A β − A α A β ∂ α A ( µ ∂ β A ν ) − A α A β A ( µ ∂ β ∂ α A ν ) + A α A ( µ ∂ α A β ∂ β A ν ) − A α A ( µ ∂ α A β ∂ ν ) A β i , (129)where the c a ’s are coefficients related to the M , c L , and c T coefficients of sect. 3 [6]. Ma-nipulating this tensor to compute our quantum corrections clearly requires considerably moreeffort compared to the Noether one, eq. (33). The same is true for the Belinfante tensor,which turns out to be exactly the same as the Hilbert one.Namely, the Belinfante stress-energy tensor in general is defined as [13] T µνB = T µνN − i ∂ κ (cid:20) ∂ L ∂ ( ∂ κ Φ a ) ( J µν ) ab Φ b − ∂ L ∂ ( ∂ µ Φ a ) ( J κν ) ab Φ b − ∂ L ∂ ( ∂ ν Φ a ) ( J κµ ) ab Φ b (cid:21) , (130)where T µνN is the Noether stress-energy tensor, and the J µν ’s are the Lorentz generators inthe representation appropriate for the fields Φ a . For 4-vector fields,( J ρσ ) κλ = i ( η σκ δ ρλ − η ρκ δ σλ ) . (131)24riting down all terms in (130), the result is equal to equation (129) plus terms that are pro-portional to the equations of motion . We checked that upon expanding the Belinfante andHilbert stress-energy tensors to quadratic order in our ~η fields, we get exactly the same ex-pressions for h T i and h T ij i as those derived from the Noether stress-energy tensor, eqs. (36),(37).Notice that adding terms proportional to the equations of motion is one of the ambiguitiesinherent in the definition of the stress-energy tensor, or, for that matter, of any Noethercurrent. Consider in fact the standard Noether procedure to derive a conserved current.It starts with inspecting how an action changes under a symmetry transformation if thetransformation parameter ǫ of a global symmetry of the formΦ a → Φ a + ǫ ∆ a [Φ] (132)(for some functional ∆ a ) is modulated weakly in space and time. But this prescription isambiguous. The standard way to implement it isΦ a → Φ a + ǫ ( x )∆ a [Φ] , (133)but an equally valid one is, for example,Φ a → Φ a + ǫ ( x )∆ a [Φ] + ∂ µ ǫ ( x ) F aµ [Φ] , (134)for an arbitrary functional F a . These two approaches both yield conserved currents, and thetwo currents differ by a term proportional to the equations of motion, δSδ Φ a F aµ [Φ]. C The path-integral measure
In order to construct a Lorentz-invariant measure, we can start from the obvious invariantmeasure for A µ , DA µ ≡ Y x d A ( x ) , (135)and impose an invariant constraint that removes its norm, e.g. δ ( A µ A µ + 1) . (136)We can then parametrize A µ in terms of our Goldstone fields ~η ( x ) and of a radial mode ρ ( x ), A = ρ cosh | ~η | , ~A = ρ ~η | ~η | sinh | ~η | . (137)The path integral then reads Z DA µ δ ( A µ A µ + 1) e iS · · · = Z Dρ D~η
Det
J δ ( ρ − e iS · · · , (138) Note that the Belinfante tensor is also non-symmetric unless one uses the equations of motion [13]. J is J ( x, x ′ ) ≡ δ (cid:0) A ( x ) , ~A ( x ) (cid:1) δ (cid:0) ρ ( x ′ ) , ~η ( x ′ ) (cid:1) = ∂ (cid:0) A , ~A (cid:1) ∂ (cid:0) ρ, ~η (cid:1) δ ( x − x ′ ) . (139)Using standard functional methods [12], its determinant can be written in exponential formas Det J = e i ∆ S , ∆ S ≡ − i (cid:16) Z d k (2 π ) (cid:17) Z d x log sinh | ~η || ~η | , (140)where we used that, thanks to the delta-function in (138), ρ = 1. The integral over ρ can nowbe performed explicitly, upon which we are left with the path integral Z D~η e i ( S +∆ S ) · · · . (141)We thus reach the conclusion that, to preserve Lorentz invariance in our computations, weshould supplement the ~η effective action with ∆ S .If we use dimensional regularization, ∆ S vanishes, because its overall coefficient does. Thisis one of the many reasons why dimensional regularization is convenient, and why we usuallydon’t track functional determinants coming from field redefinitions in the path integral.If, on the other hand, we use other UV regulators, we should keep ∆ S around. Notice that,like all effects coming from functional determinants, ∆ S is formally of one-loop order. Weshould then use it consistently in perturbation theory. For instance, for one-loop computations,we should use ∆ S at tree-level.∆ S is a (UV divergent) potential for our Goldstone fields. In particular, it includes a massterm for them. This is inconsistent with the Goldstone theorem for spontaneously brokenboosts [7]. This means that, at one-loop, there must be other contributions that cancel atleast the effects of such a mass term. Or, conversely, if we don’t keep ∆ S around, at one-loop we must find nontrivial contributions to the mass of the Goldstones, in violation of theGoldstone theorem.We can check this explicitly. For simplicity, let’s consider the c L = c T = 1 case, which isparticularly symmetric [6]. The two-derivative Goldstone action takes the form of a relativisticnon-linear sigma model, S = − M Z d xf ij ( ~η ) ∂ µ η i ∂ µ η j , (142)with f ij given by f ij ( ~η ) = P k ij ( ~η ) + sinh | ~η || ~η | P ⊥ ij ( ~η ) , (143)where P k and P ⊥ are the parallel and perpendicular projectors in ~η -space. We can compute atonce all one-loop contributions to the mass and to non-derivative interactions of the Goldstonefields by computing the one-loop Coleman-Weinberg potential [11]. Following again standardfunctional methods [12], we get∆Γ CW = i Z d x d k (2 π ) tr log( k f ij ( ~η )) (144)26here the trace is a simple finite-dimensional (3 ×
3) matrix trace. We can split the matrixinside the trace as log( k f ij ( ~η )) = log( k ) δ ij + log( f ij ( ~η )) . (145)The first term is field independent, and we can discard it. As to the second term, we canevaluate its trace in a basis in which it is diagonal, such as a basis in which ~η ∝ (1 , , CW = i (cid:16) Z d k (2 π ) (cid:17) Z d x log sinh | ~η || ~η | = − ∆ S . (146)Regardless of the UV regulator used, this cancels exactly all effects of ∆ S at this order, thusrecovering agreement with the boost Goldstone theorem.In conclusion, when using UV regulators other than dim-reg, the correction ∆ S comingfrom the path-integral measure should be kept, and used consistently in perturbation theory.In practice, for our purposes in this paper, this ends up not mattering. This is because wecomputed the one-loop expectation value of the stress-energy on the framid’s ground state.Since ∆ S is formally already of one-loop order, its contributions to such an expectation valueshould be considered only at tree level. That is, h T µν i − loop = h T µν i − loop + h ∆ T µν i tree , (147)where the l.h.s. stands for all one-loop contributions in the full theory (with action S + ∆ S ),the first term on the r.h.s. stands for the one-loop contributions in the theory without ∆ S ,and the second term on the r.h.s. stands for the correction to the stress-energy tensor operatorcoming from ∆ S , evaluated on the ground state at tree level only. But at tree-level the groundstate simply corresponds to ~η = 0, so ∆ S vanishes there, and so does its contribution to ourexpectation value. References [1] S. Weinberg, “The Cosmological Constant Problem,”
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