BBeyond Horndeski interactions induced by quantum effects
B. Latosh ∗ Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia Dubna State University, Universitetskaya str. 19, Dubna 141982, Russia
Abstract
Opportunity to generate beyond Horndeski interactions is addressed. An amplitudegenerating a certain beyond Horndeski coupling is explicitly found. The amplitude is freefrom ultraviolet divergences, so it is protected from ultraviolet contributions and can beconsidered as a universal prediction of effective field theory.
Horndeski models are scalar-tensor models that have second-order field equations and have nointeraction between the scalar field and matter [1, 2]. Because of the second-order field equationsthey are protected from the Ostrogradsky instability associated with higher derivatives [3]. TheHorndeski action reads (Jordan frame): A = (cid:90) d x √− g [ L + L + L + L + L matter [Ψ , g µν ]] , (1) L = G , L = G (cid:3) φ, L = G R + G ,X [( (cid:3) φ ) − ( ∇ µν φ ) ] L = G G µν ∇ µν φ − G ,X (cid:2) ( (cid:3) φ ) − ∇ µν φ ) (cid:3) φ + 2( ∇ µν φ ) (cid:3) (2)Here Ψ notes matter degrees of freedom; G i are arbitrary functions of the scalar field φ and itscanonical kinetic term X = 1 / ∂φ ) ; G i,X are derivatives with respect to X ; R is the Ricciscalar, and G µν is the Einstein tensor.Beyond Horndeski models generalize Horndeski gravity. These models introduce a non-minimal coupling between the scalar field and matter which preserves the second differentialorder of field equations. The most general beyond Horndeski non-minimal coupling is describedby the following Lagrangian [4]: A int = (cid:90) d x √− g [ C ( φ, X ) g µν + D ( φ, X ) ∂ µ φ ∂ ν φ ] T µν . (3)Here C and D are arbitrary functions of the scalar field φ and the canonical kinetic term X .It must be noted that both Horndeski and beyond Horndeski models are well motivated andwidely studied [5, 6, 7, 8, 9, 10, 11]. In particular, beyond Horndeski models originated fromdegenerate higher-order scalar-tensor (DHOST) theories [12, 13, 14, 15, 16, 17, 18, 19] and itwas shown that only models (3) have no ghost modes in the spectrum of tensor perturbationspropagating about a cosmological background [18, 19]. More detailed reviews of these modelscan be found elsewhere [4, 20].Within modified gravity the usage of Horndeski and beyond Horndeski models is well-motivated,as these models, in some sense, describe the most general class of physically acceptable mod-els. In this paper we argue that some beyond Horndeski interactions are generated dynamicallywithin quantum theory. ∗ [email protected] a r X i v : . [ h e p - t h ] F e b uantum behavior of a gravity theory can be consistently described within effective fieldtheory formalism [21, 22, 23]. Such an approach to gravity is well studied for general relativityand it allows one to obtain some verifiable predictions [24, 25, 26]. We also should note thatone should not neglect the whole manifold of alternative approaches to quantum features ofscalar-tensor theories [27, 28, 29, 30, 31, 32, 33, 34].Recently it was found that a certain non-minimal interaction between the scalar field andgravity is dynamically induced at the one-loop level [26]. The effect is a complete analogy withthe anomalous fermion dipole moment. In this paper we show that this non-minimal interactionis also responsible for a dynamical generation of a certain beyond Horndeski interaction. We alsoshow that some other beyond Horndeski interactions can be generated at the loop level, althoughthey will be strongly suppressed.This paper is organized as follows. In Section 2 we briefly discuss the effective field theorymethod, its implementation to gravity, and the mechanism responsible for the generation ofa new non-minimal coupling between the scalar field and gravity. In Section 3 we show thatthe same mechanism generates certain beyond Horndeski interactions at the one-loop level. Wediscuss what interactions can be generated by such a mechanism and point to the fact that someof them are suppressed more then the others. We highlight the interaction which provides theleading contribution in the low energy regime. We conclude in Section 4 where we discuss thephysical role of such interactions. Effective field theory method originated within particle physics [35] and recently was appliedto gravity [21, 22, 23]. Its main premise is the factorization of ultraviolet physics. In thecontext of gravity this means that low energy phenomena, for instance, Solar system physics,are not affected by the Planck scale fluctuations in a meaningful way. This allows one to set afactorization scale µ , which is naturally should lie below the Planck scale, and to construct alow energy theory as a momentum expansion. This construction factorizes out all phenomenalying above the factorization scale µ and provides a consistent quantum description of the lowenergy physics.On the practical ground the effective field theory paradigm is implemented to gravity asfollows. Firstly, one sets the factorization scale µ which lies below the Planck mass. Secondly,one defines the microscopic action A of a gravity theory. The action describes both gravitationand non-gravitational degrees of freedom. Finally, one performs a background field quantization[37, 38, 39, 40].Background field quantization, in turn, is perform as follows. Firstly, one finds a suitablebackground metric g µν which solves the classical field equations. Then, one introduces thefull metric g µν which accounts both for the background contribution and for the perturbationspropagating about it: g µν = g µν + κ h µν . (4)Here κ is a dimensional parameter ( κ = 32 πG in our case) and h µν is a field with the canonicalmass-dimension which describes small metric perturbations. Lastly, the action A is expanded ina series with respect to h µν , which is equivalent to an expansion with respect to (cid:3) /m P : A [ g ] = A [ g ] + δ A δg µν κh µν + 12 δ A δg µν δg αβ κ h µν h αβ + 13! δ A δg µν δg αβ δg ρσ κ h µν + O ( κ ) . (5)After that the field h µν is quantized. In this expansion δ A/ ( δg µν δg αβ ) terms describe prop-agation of a free graviton, while other terms describe gravitational field self-interaction andinteraction with matter.Effective field approach to gravity is discussed in great details in papers [22, 21], so we willnot discuss it further. We would only highlight two things. The first one is the gauge fixing. Weuse the Fock gauge ∂ µ h µν − ∂ ν h = 0 and fix it with the following gauge-fixing term: L gf = (cid:32) ∂ µ h µν − ∂ ν h (cid:33) . (6) In the previous papers [26, 36] is was called “the renormalization scale”. Such a terminology is misleadingand we will not use it hereafter. η µν as the background.An important milestone of effective gravity was found in paper [21] where it was shown thatone can use effective theory paradigm to evaluate classical corrections to the Newton potential.Such calculations can be performed due to the structure of loop corrections. The power-lawcorrections of the Newton potential are given by non-analytical functions in the momentumspace [21]: (cid:90) d (cid:126)k (2 π ) k e − i(cid:126)k · (cid:126)r = 12 π r , (cid:90) d (cid:126)k (2 π ) ln( (cid:126)k ) e − i(cid:126)k · (cid:126)r = − π r . (7)Such non-analytical terms are typically generated by loop corrections. In the low energy regime,where we perform the most part of terrestrial experiments, these terms dominate over localcorrections. But most importantly, such terms are multiplied by finite coefficients and do notdepend on the factorization scale . To put it otherwise, these terms are protected from the highenergy content of a given model and can be treated as a universal prediction.A similar effect was found in paper [26]. In an analogy with [21] the paper addresses an effectthat does not depend on the factorization scale. At the one-loop level the following non-minimalinteraction between a scalar field and gravity is generated: L = G µν ∇ µ φ ∇ ν φ. (8)This interaction is generated by the following diagram: l pqµν anomalous → iκ l C µναβ p α q β . (9)The amplitude contains a contribution free from ultraviolet divergences. Infrared divergences ofsuch an amplitude are regularized via soft graviton radiation. Therefore, in full analogy withthe previous case, one finds a contribution that is independent on the high energy content of amodel. Moreover, exactly the same phenomenon is well-known within particle physics where itis responsible for the generation of anomalous magnetic momenta.This provides ground to claim that the non-minimal kinetic interaction (8) is a universalprediction of quantum gravity which is independent on the high energy content of a model. Italso should be noted that, unlike the other case [21], such an interaction is local and stronglysuppressed. Because of this one can establish some empirical constraints on its coupling [26]. The same mechanism that generates the non-minimal kinetic coupling (8) is responsible for ageneration of a certain beyond Horndeski interaction. Let us consider a minimal scalar-tensormodel which contains no non-minimal couplings A = (cid:90) d x √− g (cid:34) − πG R + 12 g µν ∇ µ φ ∇ ν φ + L matter [Ψ , g µν ] (cid:35) . (10)At the one-loop level the model generates the desired non-minimal kinetic coupling [26]. To showthat beyond Horndeski interactions are generated in a way similar to the non-minimal couplingwe find the specific one-loop amplitude that generates a beyond Horndeski interaction.Beyond Horndeski interactions deal with a non-minimal interaction between the scalar fieldand matter (3). Therefore one should study gravitational scattering of the scalar field on matterdegrees of freedom. The simplest suitable amplitude corresponds to the following diagram (11)3ere we note matter degrees of freedom with a double plain line. We only consider tree-levelinteraction between the matter and gravity because of the two reasons. Firstly, such correctionslie beyond the scope of this paper and were extensively studied before [21, 24, 41, 42]. Secondly,we are interested in the coupling between regular matter and the new scalar field. Correctionsto a coupling between matter and gravity can only alter a coupling between the matter energy-tensor and gravity. Therefore they are irrelevant for our purpose. Finally, such corrections willintroduce an additional factors κ , so the resulted amplitude receives additional suppression.Because of this only corrections to the scalar-tensor sector are relevant. Such corrections werediscussed in paper [26] and, as we have highlighted, it was shown that an anomalous interaction(9) is generated. This exact anomalous contribution generates an anomalous interaction betweenmatter and the scalar field: anomalous → iκ l C µναβ p α q β i C µνρσ l iκ C ρσλτ T λτ = − i κ p α q β C αβµν T µν . (12)Here p and q are momenta of the ingoing scalar particles. The expression (12) is local, so itcorresponds to the following contact interaction:= − i κ p α q β C αβµν T µν ↔ − κ ∂ µ φ ∂ ν φ T µν + 2 κ ( ∂φ ) T. (13)It can be seen clearly that this is a beyond Horndeski interaction (3) with the following param-eters: C ( φ, X ) = 2 κ ( ∂φ ) , D ( φ, X ) = − κ . (14)It should be noted that the amplitude (11) is not the only amplitude that generates beyondHorndeski interactions. However, the other amplitudes experience stronger suppression. Namely,the following amplitude may also generate beyond Horndeski interaction: (15)Here double plain lines note matter degrees of freedom. This amplitude may very well generatean interaction between one graviton, two scalars, and two matter degrees of freedom that belongto the beyond Horndeski class: h µν ( ∂φ ) T µν ↔ C ( φ, X ) g µν T µν . (16)Firstly, this interaction is suppressed by the factor κ , so it does not provide the leading contribu-tion and can be safely neglected for the time being. Secondly, simple dimensional considerationsshow that this amplitude develops log-dependence of the factorization scale. Although such a de-pendence of the factorization scale is weak, it does not allow one to consider such a contributionindependent from the high energy content of a theory.The following amplitudes can be treated in a similar way: , , · · · (17)They are suppressed, at very least, by factor κ N , where N is the number of virtual gravitons.Therefore the amplitude (12), considered above, provides the leading contribution.This proves our original claim, so a certain beyond Horndeski interaction are generated atthe one-loop level. The corresponding amplitude (12) is free from ultraviolet divergences andtherefore are independent on the high energy content of a quantum gravity model. We discussthis result and its implications in the next Section.4 Discussion and conclusion
We have shown that a certain beyond Horndeski interaction is universally generated withineffective field theory.Both analytical [26] and non-analytical [21] terms that are independent from the factorizationscale µ which separating low and high energy physics are generated at the loop level. BeyondHorndeski interactions are generated in the same way. Namely, a specific one-loop amplitude (12)contains a part generating the non-minimal kinetic coupling (8) which is independent from thefactorization scale. Because of this the amplitude (12) generates a new non-minimal interactionbetween the scalar field and matter. This interaction belongs to beyond Horndeski class and itdoes not depend on the factorization scale.This shows that certain beyond Horndeski interactions are generated in a way similar toanomalous magnetic momentum and to a certain non-minimal kinetic coupling [26].Because such a beyond Horndeski interaction is generated at the loop level it experience anextremely strong suppression by factor κ ∼ G . Moreover, such interactions are local, so theycan hardly be found in the low energy (large scale) experiments. Such interactions appear atleast at 2PN (post-Newtonian) order or higher [43, 44]. Although such effects can be calculated[45, 46, 47, 48, 49, 11], it would be challenging to efficiently constraint them with the currentempirical data. Acknowledgment
The work was supported by the Foundation for the Advancement of Theoretical Physics andMathematics “BASIS”.
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