Quantum scale invariance, cosmological constant and hierarchy problem
aa r X i v : . [ h e p - t h ] D ec Quantum scale invariance, cosmological constant and hierarchyproblem
Mikhail Shaposhnikov and Daniel Zenh¨ausern
Institut de Th´eorie des Ph´enom`enes Physiques, ´Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland
Abstract
We construct a class of theories which are scale-invariant on quantum level in all orders of perturbation theory. In a subclass ofthese models scale invariance is spontaneously broken, leading to the existence of a massless dilaton. The applications of theseresults to the problem of stability of the electroweak scale against quantum corrections, to the cosmological constant problemand to dark energy are discussed.
Key words: scale invariance, hierarchy problem, cosmological constant problem, unimodular gravity, dark energy, inflation
PACS:
1. Introduction
If in any theory all dimensionfull parameters (gener-ically denoted by M ), including masses of elementaryparticles, Newton’s gravitational constant, L QCD andalike are rescaled by the same amount M → M s , thiscannot be measured by any observation. Indeed, thischange, supplemented by a dilatation of space-time co-ordinates x m → s x m and an appropriate redefinition ofthe fields does not change the complete quantum effec-tive action of the theory. However, the symmetry trans-formations in quantum field theory only act on fields andnot on parameters of the Lagrangian. The realization ofscale invariance happens to be a non-trivial problem. Aclassical field theory which does not contain any dimen-sionfull parameters is invariant under the substitution F ( x ) → s n F ( s x ) , (1)where n is the canonical mass dimension of the field F .This dilatational symmetry turns out to be anomalous on Email addresses:
[email protected] (Mikhail Shaposhnikov),
[email protected] (Daniel Zenh¨ausern). quantum level for all realistic renormalizable quantumfield theories (for a review see [1]). The divergence ofthe dilatation current J m is non-zero and is proportionalto the b -functions of the couplings. For example, in puregluodynamics, scale-invariant on the classical level, onehas ¶ m J m (cid:181) b ( g ) G a ab G ab a , (2)where G a ab is the non-Abelian gauge field strength.At the same time, it is very tempting to have a theorywhich is scale-invariant (SI) on the quantum level, asthis would solve a number of puzzles in high energyphysics. Most notably, these problems include twotremendous fine-tunings, facing the Standard Model(SM). The first one is related to the stability of theHiggs mass against radiative corrections and the sec-ond one to the cosmological constant problem. If thefull quantum theory, including gravity, is indeed scale-invariant, and SI is broken spontaneously, the Higgsmass is protected from radiative corrections by an exactdilatational symmetry.Moreover, as we have shown in [2], the classical theory with SI broken spontaneously and given by theaction (we omit from the Lagrangian of [2] all degrees of Preprint submitted to Elsevier 14 November 2008 reedom which are irrelevant for the present discussionand keep only the gravity part, the Higgs field h and thedilaton c ): L tot = L G + L , (3)where L G = − (cid:0) x c c + x h h (cid:1) R , (4) L = (cid:2) ( ¶ m c ) + ( ¶ m h ) (cid:3) − l (cid:0) h − z c (cid:1) , (5)not only has zero cosmological constant but also gives asource for dynamical dark energy, provided that gravityis unimodular, i.e. the determinant of the metric is fixedto be −
1. (Here R is the scalar curvature and x c , x h , l and z are dimensionless coupling constants.) In this the-ory all mass parameters (on the tree level) come fromone and the same source – the vacuum expectation valueof the dilaton field h c i = c , which is exactly massless.In addition, the primordial inflation is a natural conse-quence of (3), with a Higgs field playing the role of theinflaton [3].It looks like all these findings are ruined by quantumcorrections. The aim of this Letter is to show that this isnot the case. We will construct a class of effective fieldtheories, which obey the following properties:(i) Scale invariance is preserved on quantum level in allorders of perturbation theory.(ii) Scale invariance is broken spontaneously, leading toa massless dilaton.(iii) The effective running of coupling constants is au-tomatically reproduced at low energies.In other words, the benefits of classical SI theories (nocorrections to the Higgs mass, zero cosmological con-stant, presence of dark energy and primordial inflation)can all be present on the quantum level. At the sametime, the standard results of quantum field theory, suchas the running of coupling constants, remain in place.Whether the theories we construct are renormalizable and unitary is not known to us (though we will for-mulate some conjectures on this point). However, therenormalizability is not essential for the validity of theresults.The Letter is organized as follows. In Section 2 weexplain our main idea with the use of a simple modelof two scalar fields. In Section 3 we describe its gener-alization to an arbitrary case. In Section 4 we discussthe inclusion of gravity and present our conclusions inSection 5. The precise sense of this word in the present context will bespecified later.
2. Scalar field example
We will explain our idea using the example of a sim-ple system containing two scalar fields described in clas-sical theory by the Lagrangian (5) without gravity. Theconstruction is essentially perturbative and based on thedimensional regularization of ’t Hooft and Veltman [4](for a discussion of the hierarchy problem within thisscheme see, e.g. [5]).At the classical level the theory (5) is scale-invariant.In fact, the requirement of dilatational symmetry doesnot forbid the presence of an additional term bc in(5). If b <
0, the theory does not have a stable groundstate, for b > h = c =
0. At the classical level one wouldconclude that the theory contains two scalar masslessexcitations for the ground state respecting scale invari-ance. For the case b = h = ± zc and the vacuum is degenerate.If c = c =
0, the dilatational invariance is sponta-neously broken. Then the theory contains a massiveHiggs boson, m H ( c ) = lz ( + z ) c and a mass-less dilaton. So, the only choice for b , interesting forphenomenology, is b =
0, otherwise the vacuum doesnot exist or the theory does not contain any massiveparticles . In what follows we will also assume that z ≪
1, which is true for phenomenological applica-tions: c ∼ M P = . × GeV is related to thePlanck scale, and h = zc ∼ M W ∼
100 GeV to theelectroweak scale. However, the smallness of z is notessential for the theoretical construction.It is well known what happens in this theory ifthe standard renormalization procedure is applied.In d -dimensional space-time (we use the convention d = − e ) the mass dimension of the scalar fields is1 − e , and that of the coupling constant l is 2 e . Intro-ducing a (finite) dimensionless coupling l R , one canwrite l = m e " l R + ¥ (cid:229) n = a n e n , (6)where m is a dimensionfull parameter and the Laurentseries in e corresponds to counter-terms. The parameters a n are to be fixed by the requirement that renormalizedGreen’s functions are finite in every order of perturba-tion theory. Similar replacements are to be done withother parameters of the theory, and the factors Z c , Z h ,related to the renormalization of fields must be intro-duced (they do not appear at one-loop level in our scalar More discussion of the b > MS subtraction scheme, the one-loop effective potential along the flat direction has theform V ( c ) = m H ( c ) p (cid:20) log m H ( c ) m − (cid:21) , (7)spoiling its degeneracy, and leading thus to explicitbreaking of the dilatational symmetry. The vacuum ex-pectation value of the field c can be fixed by renormal-ization conditions [6]. The dilaton acquires a nonzeromass. It is the mismatch in mass dimensions of bare ( l )and renormalized couplings ( l R ) which leads to the di-latational anomaly and thus to explicit breaking of scaleinvariance (see [7] for a recent discussion).Let us now use another prescription, which we willcall the ”SI prescription” . Replace m e in (6) and inall other similar relations by (different, in general) com-binations of fields c and h , which have the correct massdimension: m e → c e − e F e ( x ) , (8)where x = h / c and F e ( x ) is a function depending onthe parameter e with the property F ( x ) =
1. In princi-ple, one can use different functions F e ( x ) for the variouscouplings. The resulting field theory, by construction,is scale-invariant for any number of space-time dimen-sions d . This means, that if for instance the MS subtrac-tion scheme is used for calculations, the renormalizedtheory is also scale-invariant in any order of perturba-tion theory.The requirement of scale invariance itself does notfix the details of the prescription. However, the form ofthe couplings of the scalar fields c and h to gravity asin Eq. (4) indicates that the combination x c c + x h h ≡ w (9)plays a special role, being the effective Planck constant.Therefore, we arrive to a simple “GR-SI prescription”,in which m e → (cid:2) w (cid:3) e − e , (10)corresponding to the choice of the function F e ( x ) =( x c + x h x ) e − e . We will apply the GR-SI prescriptionto the one-loop analysis of our scalar theory below. Inthe appendix we will consider a modified variant of theprocedure.The SI construction is entirely perturbative and can infact be used only if SI is spontaneously broken. In otherwords, in order to use the GR-SI prescription the groundstate has to be ( h , c ) = ( , ) , because otherwise it A similar procedure was suggested in [8] in connection with theconformal anomaly. We thank Thomas Hertog who pointed out thisreference to us after our work has been submitted to hep-th. is impossible to perform an expansion of (10). Indeed,consider the exact effective potential V e f f ( h , c ) of ourtheory, constructed using the prescription (8) or (10) inthe limit e →
0. Because of exact SI, it can be written as V e f f ( h , c ) = c V c ( x ) = h V h ( x ) . (11)For the ground state to exist, we must have V c ( x ) ≥ V h ( x ) ≥
0) for all x . For the min-imum of V e f f ( h , c ) to lie in the region where c = h = V c ( x ) = V h ( x ) = x is a solution of V ′ c ( x ) = V ′ h ( x ) =
0) and prime de-notes the derivative with respect to x . If these conditionsare satisfied, the theory has an infinite set of groundstates corresponding to the spontaneous breakdown ofdilatational invariance. The dilaton is massless in all or-ders of perturbation theory. In this case one can developthe perturbation theory around the vacuum state corre-sponding to c = h = x c with arbitrary c (or h = c = h / x with arbitrary h ).To summarize: the use of prescriptions (8) or (10)supplemented by the requirement V c , h ( x ) = h / c ) is required. But, even if this isthe case, scale invariance is maintained in all orders ofperturbation theory and can be spontaneously broken.Another potential issue is unitarity. We do not knowwhether higher derivative terms in the effective action,dangerous from this point of view, would require theintroduction of corresponding counter-terms. However,the functional arbitrariness in the choice of F e ( x ) forpotential and kinetic terms may give enough freedomto remove the unwanted contributions.The theories we construct are quite different fromordinary renormalizable theories. Their physics is de-termined not only by the values of “classical” couplingconstants ( l and z in our case), but also by “hidden”parameters contained in the functions F e ( x ) . Still, aswe will see shortly, for the SI-GR prescription, in thelimit z ≪ E ≪ c , only “clas-sical” parameters matter. Moreover, they automaticallyacquire the necessary renormalization group running.To this end, we carry out a one-loop analysis of thetheory (5) with the GR-SI prescription. We write the3 -dimensional generalization of the classical potentialas U = l R (cid:2) w (cid:3) e − e (cid:2) h − z R c (cid:3) , (12)and introduce the counter-terms U cc = (cid:2) w (cid:3) e − e " Ah c (cid:18) e + a (cid:19) + (13) B c (cid:18) e + b (cid:19) + Ch (cid:18) e + c (cid:19) , where e = e − g + log ( p ) , g is the Euler constant and a , b , c , A , B , and C are arbitrary for the moment. Wedo not introduce any modification of the kinetic termssince no wave function renormalization is expected atthe one loop level.It is straightforward to find the one-loop effective po-tential for this theory. The counter-terms removing thedivergences coincide with those of the standard pre-scription and are given by: A → − l R z R z R − z R + p , B → l R z R z R + p , C → l R z R + p . (14)The potential itself has a generic form U = c W ( x ) and is given by a rather lengthy expression (we do notpresent it here, since it is not very illuminating), whichalso depends on the “hidden” parameters. For a genericchoice of a , b , and c the classical flat direction x = z R is lifted by quantum effects. However, the requirement W ( z R ) = W ′ ( z R ) = z R ≪ b = a + (cid:18) l R z R x c (cid:19) + O (cid:0) z R (cid:1) , c = (cid:20) a + − (cid:18) l R z R x c (cid:19)(cid:21) + O (cid:0) z R (cid:1) . (15)The function W ( x ) is positive near the flat direction,provided a + + (cid:16) l R z R x c (cid:17) > If we define the parameters a ≡ √ l and b ≡ √ lz , theclassical potential takes the form U = (cid:0) a h − bc (cid:1) . In thisnotation the GR-SI prescription corresponds to the substitutions a → (cid:2) w (cid:3) e ( − e ) a R and b → (cid:2) w (cid:3) e ( − e ) b R . The truncation only serves to shorten the expressions. There isno difficulty in finding the exact relations.
It is interesting to look at the one-loop effective po-tential as a function of h for c = c , h ∼ z R c ≡ v and z ≪
1, i.e. h ≪ c . One finds U = m ( h ) p (cid:20) log m ( h ) v + O (cid:0) z R (cid:1)(cid:21) (16) + l R p (cid:2) C v + C v h + C h (cid:3) + O (cid:18) h c (cid:19) , where m ( h ) = l R ( h − v ) and C = (cid:20) a − + (cid:18) z R x c (cid:19) +
43 log 2 l R + O ( z R ) (cid:21) , C = − (cid:20) a − + (cid:18) z R x c (cid:19) + O ( z R ) (cid:21) , (17) C = (cid:20) a − + (cid:18) z R x c (cid:19) − l R + O ( z R ) (cid:21) . The first term in (16) is exactly the standard effectivepotential for the theory (5) with the dynamical field c replaced by a constant c , while the rest is a quar-tic polynomial of h and comes from our GR-SI pre-scription, leading to redefinition of coupling constants,masses, and the vacuum energy.One can see from (16) that the quantum correctionsto the Higgs mass are proportional to v (cid:181) z R c . Thismeans that they are small compared to the classicalvalue. Moreover, the potentially dangerous correctionsof the type l n c to the Higgs mass cannot appear inhigher orders of perturbation theory. Indeed, for z = z =
0, the value of the (large) field c can appear only through log’s in the effective poten-tial, coming from the expansion of [ w ] e / − e in Eq. 12,or at most as z R c if z =
0. Hence, in this theory thereis no problem of instability of the Higgs mass againstquantum corrections, appearing in the Standard Model.Consider now the high energy ( √ s ≫ v but √ s ≪ c )behaviour of scattering amplitudes with the exampleof Higgs-Higgs scattering (assuming, as usual, that z R ≪ G = l R + l R p (cid:20) log (cid:18) s x c c (cid:19) + const (cid:21) + O (cid:0) z R (cid:1) . (18)This implies that at v ≪ √ s ≪ c the effective Higgsself-coupling runs in a way prescribed by the ordinaryrenormalization group. Not only the tree Higgs mass isdetermined by the vev of the dilaton, but also all L QCD -4ike parameters. We expect that these results remainvalid in higher orders of perturbation theory.Let us comment now on the case when the flat di-rection does not exist at the quantum level (classicallythis corresponds to b >
3. Scale-invariant quantum field theory: Generalformulation
It is straightforward to generalize the constructionpresented above to the case of theories containingfermions and gauge fields, such as the Standard Model.The mass dimension of a fermionic field is − e ,leading to the dimension of bare Yukawa couplings F B equal to e . The mass dimension of the gauge field canbe fixed to 1 for any number of space-time dimensions d , leading to the dimensionality of the bare gaugecoupling g B equal to e . So, in the standard procedureone chooses F B (cid:181) m e F R , g B (cid:181) m e g R , where the index R refers to renormalized couplings. For the SI or GR-SI prescription one replaces m e by a combination ofscalar fields of appropriate dimension, as in (8) or in(10). For the perturbation theory to make sense, onehas to choose counter-terms in such a way that the fulleffective potential has a flat direction corresponding tospontaneously broken dilatational invariance.
4. Inclusion of gravity
The inclusion of scale-invariant gravity is carried outprecisely along the same lines. The metric tensor g mn isdimensionless for any number of space-time dimensionsand R always has mass dimension 2. Therefore, the non-minimal couplings x c , x h (see Eq. (4)) are dimension-less and thus can only be multiplied by functions F e ( x ) of the type defined in (8). In addition to (4), the gravita-tional action may contain the operators R , R mn R mn , R and R mnrs R mnrs , multiplied by c − e − e F e ( x ) (here R mn and R mnrs are the Ricci and Riemann curvature ten-sors). These operators are actually needed for renormal- ization of field theory in curved space-time (for a reviewsee [9]).The presence of gravity is crucial for phenomenolog-ical applications. Since Newton’s constant is dynami-cally generated, the dilaton decouples from the particlesof the Standard model [2, 13, 14, 15, 16], and thus satis-fies all laboratory and astrophysical constraints. As wefound in [2], if gravity is unimodular, the absence of acosmological constant and the existence of dynamicaldark energy are automatic consequences of the theory.It is interesting to note that the action of unimodulargravity is polynomial with respect to the metric tensor.This leads us to the conjecture that the SI unimodulargravity with matter fields may happen to be a renormal-izable theory in the sense described in Section 2.
5. Conclusions
In this Letter we constructed a class of theories, whichare scale-invariant on the quantum level. If dilatationalsymmetry is spontaneously broken, all mass scales inthese models are generated simultaneously and origi-nate from one and the same source. In these theoriesthe effective cutoff scale depends on the backgrounddilaton field, as was already proposed in [16], which isessential for inflation [3] and dark energy [2]. The cos-mological constant is absent and the mass of the Higgsboson is protected from large radiative corrections bythe dilatational symmetry. Dynamical dark energy is aremnant of initial conditions in unimodular gravity.There are still many questions to be understood. Hereis a partial list of them. Our construction is essentiallyperturbative. How to make it work non-perturbatively ?Though the stability of the electroweak scale againstquantum corrections is achieved, it is absolutely unclear why the electroweak scale is so much smaller than thePlanck scale (or why z ≪ p > ∼ M P the perturbation theory diverges and thus is not appli-cable. What is the high energy limit of these theories? Acknowledgements.
This work was supported bythe Swiss National Science Foundation. We thank F.Bezrukov, K. Chetyrkin, S. Sibiryakov and I. Tkachevfor valuable comments. A proposal based on lattice regularization has been discussedrecently in [17]. c was the same as for the ordi-nary renormalizable scalar theory containing the Higgsfield h only. This is not necessarily the case if the SIprescription given by Eq. (8) is used. Indeed, considernow a distinct way of continuing the scalar potential to d -dimensional space-time: U = l R h h − e − e x a e − z R c − e − e x b e i , (19)and introduce counter-terms for all terms appearing inthe potential: U cc = h A (cid:18) e + a (cid:19) h − e − e c − e − e x ( a + b ) e + (20) B (cid:18) e + b (cid:19) c − e − e x b e + C (cid:18) e + c (cid:19) h − e − e x a e i . As before, we do not introduce any modification of thekinetic terms. Now we have more freedom in compari-son with the GR-SI prescription due to the existence ofnew arbitrary parameters a and b .The coefficients A , B , and C are fixed as in Eq. (14).The parameters a and b can be chosen in such a waythat the one-loop effective potential does not containterms c / h and h / c , which are singular at ( , ) .These conditions lead to a = , b =
0. Then the re-quirement that the classical flat direction x = z is notlifted by quantum effects gives (for z ≪ b = a − + ( l R ) + O (cid:0) z R (cid:1) c = [ a + − ( l R )] + O (cid:0) z R (cid:1) . (21)With all these conditions satisfied the one-loop ef-fective potential as a function of h for c = c fixed, h ∼ zc = v and z ≪ different from that inEq. (16): U = m ( h ) p (cid:20) log m ( h ) v + O (cid:0) z R (cid:1)(cid:21) + P log h v + P , (22)where P , P are quadratic polynomials of h and v .Though the first term is exactly the standard effectivepotential for the theory (5) with the dynamical field c replaced by a constant c , the rest is not simply aredefinition of the coupling constants of the theory due In the notation with a ≡ √ l and b ≡ √ lz , the prescriptionused here corresponds to the substitutions a → h e − e x a e a R and b → c e − e x b e b R . to the presence of log h v . In other words, even the lowenergy physics is modified in comparison with ordinaryrenormalizable theories.References[1] S. Coleman, “Dilatations”, in “Aspects of symme-try”, Cambridge University Press, 1985, p. 67.[2] M. Shaposhnikov and D. Zenh¨ausern,arXiv:0809.3395 [hep-th].[3] F. L. Bezrukov and M. Shaposhnikov, Phys. Lett.B (2008) 703.[4] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B (1972) 189.[5] M. Shaposhnikov, arXiv:0708.3550 [hep-th].[6] S. R. Coleman and E. Weinberg, Phys. Rev. D (1973) 1888.[7] K.A. Meissner and H. Nicolai, Phys. Lett. B (2007) 312; Phys. Lett. B (2008) 260.[8] F. Englert, C. Truffin and R. Gastmans, Nucl. Phys.B (1976) 407.[9] I. L. Buchbinder, S. D. Odintsov and I. L. Shapiro,“Effective action in quantum gravity,” Bristol, UK:IOP (1992) 413 p. [10] O. Aharony, S. S. Gubser, J. M. Maldacena,H. Ooguri and Y. Oz, Phys. Rept. (2000) 183.[11] O. W. Greenberg, Annals Phys. (1961) 158.[12] N.N. Bogolubov, A.A. Logunov and I.T. Todorov,”Introduction to Axiomatic Quantum Field The-ory,” W.A. Benjamin Inc. (1975) [13] J. J. van der Bij, Acta Phys. Polon. B (1994)827.[14] J. L. Cervantes-Cota and H. Dehnen, Nucl. Phys.B (1995) 391.[15] C. Wetterich, Nucl. Phys. B (1988) 645.[16] C. Wetterich, Nucl. Phys. B302