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Dilaton
Dilaton
Dilaton
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In particle physics, the hypothetical dilaton is a particle of a scalar field that appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theory's compactifications of extra dimensions. In Brans–Dicke theory of gravity, Newton's constant is not presumed to be constant but instead 1/G is replaced by a scalar field and the associated particle is the dilaton.

Exposition

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In Kaluza–Klein theories, after dimensional reduction, the effective Planck mass varies as some power of the volume of compactified space. This is why volume can turn out as a dilaton in the lower-dimensional effective theory.

Although string theory naturally incorporates Kaluza–Klein theory that first introduced the dilaton, perturbative string theories such as type I string theory, type II string theory, and heterotic string theory already contain the dilaton in the maximal number of 10 dimensions. However, M-theory in 11 dimensions does not include the dilaton in its spectrum unless compactified. The dilaton in type IIA string theory parallels the radion of M-theory compactified over a circle, and the dilaton in   E8 × E8   string theory parallels the radion for the Hořava–Witten model. (For more on the M-theory origin of the dilaton, see Berman & Perry (2006).[1])

In string theory, there is also a dilaton in the worldsheet CFT – two-dimensional conformal field theory. The exponential of its vacuum expectation value determines the coupling constant g and the Euler characteristic   χ = 2 − 2g   as for compact worldsheets by the Gauss–Bonnet theorem, where the genus g counts the number of handles and thus the number of loops or string interactions described by a specific worldsheet.

Therefore, the dynamic variable coupling constant in string theory contrasts the quantum field theory where it is constant. As long as supersymmetry is unbroken, such scalar fields can take arbitrary values moduli. However, supersymmetry breaking usually creates a potential energy for the scalar fields and the scalar fields localize near a minimum whose position should in principle calculate in string theory.

The dilaton acts like a Brans–Dicke scalar, with the effective Planck scale depending upon both the string scale and the dilaton field.

In supersymmetry the superpartner of the dilaton or here the dilatino, combines with the axion to form a complex scalar field.[citation needed]

The dilaton in quantum gravity

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The dilaton made its first appearance in Kaluza–Klein theory, a five-dimensional theory that combined gravitation and electromagnetism. It appears in string theory. However, it has become central to the lower-dimensional many-bodied gravity problem[2] based on the field theoretic approach of Roman Jackiw. The impetus arose from the fact that complete analytical solutions for the metric of a covariant N-body system have proven elusive in general relativity. To simplify the problem, the number of dimensions was lowered to 1 + 1 – one spatial dimension and one temporal dimension. This model problem, known as R = T theory,[3] as opposed to the general G = T theory, was amenable to exact solutions in terms of a generalization of the Lambert W function. Also, the field equation governing the dilaton, derived from differential geometry, as the Schrödinger equation could be amenable to quantization.[4]

This combines gravity, quantization, and even the electromagnetic interaction, promising ingredients of a fundamental physical theory. This outcome revealed a previously unknown and already existing natural link between general relativity and quantum mechanics. There lacks clarity in the generalization of this theory to 3 + 1 dimensions. However, a recent derivation in 3 + 1 dimensions under the right coordinate conditions yields a formulation similar to the earlier 1 + 1, a dilaton field governed by the logarithmic Schrödinger equation[5] that is seen in condensed matter physics and superfluids. The field equations are amenable to such a generalization, as shown with the inclusion of a one-graviton process,[6] and yield the correct Newtonian limit in d dimensions, but only with a dilaton. Furthermore, some speculate on the view of the apparent resemblance between the dilaton and the Higgs boson.[7] However, there needs more experimentation to resolve the relationship between these two particles. Finally, since this theory can combine gravitational, electromagnetic, and quantum effects, their coupling could potentially lead to a means of testing the theory through cosmology and experimentation.

Dilaton action

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The dilaton-gravity action is

This is more general than Brans–Dicke in vacuum in that we have a dilaton potential.

See also

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Citations

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  1. ^ Berman, David S.; Perry, Malcolm J. (6 April 2006). "M-theory and the string genus expansion". Physics Letters B. 635 (2–3): 131–135. arXiv:hep-th/0601141. Bibcode:2006PhLB..635..131B. doi:10.1016/j.physletb.2006.02.038.
  2. ^ Ohta, Tadayuki; Mann, Robert (1996). "Canonical reduction of two-dimensional gravity for particle dynamics". Classical and Quantum Gravity. 13 (9): 2585–2602. arXiv:gr-qc/9605004. Bibcode:1996CQGra..13.2585O. doi:10.1088/0264-9381/13/9/022. S2CID 5245516.
  3. ^ Sikkema, A E; Mann, R B (1991). "Gravitation and cosmology in (1 + 1) dimensions". Classical and Quantum Gravity. 8 (1): 219–235. Bibcode:1991CQGra...8..219S. doi:10.1088/0264-9381/8/1/022. S2CID 250910547.
  4. ^ Farrugia; Mann; Scott (2007). "N-body Gravity and the Schroedinger Equation". Classical and Quantum Gravity. 24 (18): 4647–4659. arXiv:gr-qc/0611144. Bibcode:2007CQGra..24.4647F. doi:10.1088/0264-9381/24/18/006. S2CID 119365501.
  5. ^ Scott, T.C.; Zhang, Xiangdong; Mann, Robert; Fee, G.J. (2016). "Canonical reduction for dilatonic gravity in 3 + 1 dimensions". Physical Review D. 93 (8) 084017. arXiv:1605.03431. Bibcode:2016PhRvD..93h4017S. doi:10.1103/PhysRevD.93.084017.
  6. ^ Mann, R B; Ohta, T (1997). "Exact solution for the metric and the motion of two bodies in (1 + 1)-dimensional gravity". Phys. Rev. D. 55 (8): 4723–4747. arXiv:gr-qc/9611008. Bibcode:1997PhRvD..55.4723M. doi:10.1103/PhysRevD.55.4723. S2CID 119083668.
  7. ^ Bellazzini, B.; Csaki, C.; Hubisz, J.; Serra, J.; Terning, J. (2013). "A higgs-like dilaton". Eur. Phys. J. C. 73 (2): 2333. arXiv:1209.3299. Bibcode:2013EPJC...73.2333B. doi:10.1140/epjc/s10052-013-2333-x. S2CID 118416422.

References

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Dilaton

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In , the dilaton is a massless that emerges in the low-energy effective description of , where its determines the fundamental string gs=e⟨ϕ⟩g_s = e^{\langle \phi \rangle}, governing the strength of interactions between strings. This field, denoted ϕ\phi, couples universally to the trace of the energy-momentum tensor, effectively rescaling the metric and influencing gravitational interactions across all and gauge fields in the . In the bosonic string's , the dilaton appears in the term e−2ϕ[R+4(∂ϕ)2−112H2]e^{-2\phi} \left[ R + 4 (\partial \phi)^2 - \frac{1}{12} H^2 \right], ensuring conformal invariance through the vanishing of its at one-loop order. Beyond its central role in string theory, the dilaton arises as a pseudo-Nambu-Goldstone boson associated with the spontaneous breaking of approximate scale invariance in near-conformal, confining gauge theories, such as those with many massless fermions approaching the conformal window. In this context, dilaton effective field theory (dEFT) provides a framework to describe the light sector, including the dilaton alongside pseudo-Nambu-Goldstone bosons from chiral symmetry breaking, with systematic power-counting expansions in terms of small parameters like fermion mass and deviation from conformality. For instance, lattice simulations of SU(3) gauge theory with eight Dirac fermions have been fitted to leading-order dEFT Lagrangians, revealing the dilaton's mass and couplings consistent with near-conformal dynamics. The dilaton also features prominently in lower-dimensional models, such as two-dimensional dilaton gravity, which captures aspects of and physics analogous to higher-dimensional spherically symmetric reductions. In cosmological applications, particularly "runaway dilaton" scenarios from , a time-varying dilaton can drive dynamical or quintessence while coupling to fundamental constants like the , offering testable predictions against violations. These properties underscore the dilaton's potential as a bridge between , gravity, and cosmology, though its direct observation remains elusive due to its weak couplings and theoretical status.

Introduction

Definition and Role

The dilaton is a hypothetical massless scalar field in theoretical physics that modulates the strength of fundamental interactions, serving as a key component in various unification frameworks. In string theory, it specifically governs the string coupling constant gs=e⟨Φ⟩g_s = e^{\langle \Phi \rangle}, where Φ\Phi denotes the dilaton field, thereby controlling the perturbative regime of string interactions. This field emerges as part of the massless spectrum of closed strings, alongside the graviton and Kalb-Ramond field, ensuring the consistency of the theory at the quantum level. Fundamental properties of the dilaton include its invariance under conformal transformations, reflecting its association with , and its universal to the trace of the energy-momentum tensor TμμT^\mu_\mu, which captures deviations from classical in quantum theories. This arises naturally in effective field theories where the dilaton acts as the Nambu-Goldstone associated with the spontaneous breaking of dilatation , with a ⟨χ⟩=fd\langle \chi \rangle = f_d setting the scale of . Additionally, the dilaton influences the effective Planck scale MPlM_{\rm Pl} by linking higher-dimensional gravitational dynamics to four-dimensional physics, particularly through its role in stabilizing compactification scales. Physically, the dilaton can be interpreted as a "dilator" of the metric, dynamically adjusting the overall scale of interactions or the size of in higher-dimensional models. In scenarios with compact , it is often identified with the radion field, which fixes the interbrane distance or radius RR of the extra dimension, thereby modulating the between the Planck scale and electroweak scale via warp factors like e−kπRe^{-k \pi R}. This property underscores its potential to address naturalness problems in by varying the effective strength of gravity. Examples of the dilaton's emergence include its appearance as a pseudo-Goldstone in quantum field theories with approximate , where flat directions in the lead to a massless mode χ\chi that parametrizes degenerate vacua. It also arises from the trace anomaly in conformal field theories, where quantum effects generate a non-zero TμμT^\mu_\mu that the dilaton couples to, restoring an effective scale symmetry at low energies.

Historical Context

The concept of the dilaton originated in the late 1960s and early 1970s within , as theorists explored the implications of broken . In 1968, Freund and Nambu introduced scalar fields coupled to the trace of the energy-momentum tensor, providing an early framework for a field that could mediate scale transformations. This was followed by Mack's proposal of a partially conserved dilatation current, linking the dilaton to approximate conformal symmetries in field theories. By 1970, Isham, Salam, and Strathdee developed effective Lagrangian formalisms for spontaneously broken conformal and chiral symmetries, where the dilaton emerged as the associated Nambu-Goldstone mode. These developments laid the intellectual foundation for viewing the dilaton as a light scalar tied to scale symmetry violations. In the context of quantum chromodynamics (QCD), the dilaton was interpreted as a pseudo-Nambu-Goldstone boson resulting from the explicit breaking of via the trace anomaly. Crewther's 1971 analysis of spontaneous breakdown of conformal and chiral invariance described the dilaton as a mixture of isoscalar scalars, connecting it to low-energy physics. The Coleman-Weinberg mechanism, detailed in their 1973 paper on radiative corrections inducing , further supported this view by showing how quantum effects could generate mass scales in classically scale-invariant theories, applicable to QCD-like dynamics. This era established the dilaton's role in explaining light scalar resonances, such as the f₀(500), as remnants of approximate scale symmetry in strong interactions. The 1980s marked the dilaton's integration into , particularly through low-energy effective actions derived from superstring formulations. and Schwarz's 1984 demonstration of anomaly cancellation in type I superstrings elevated the dilaton to a fundamental field governing the string coupling. This was expanded in Witten's contributions to , where the dilaton appears in the ten-dimensional limit, unifying , gauge fields, and scalars. Their collective work in the superstring revolution solidified the dilaton's centrality in consistent string vacua. From the onward, the dilaton's significance grew in understanding dualities and holographic correspondences. Polchinski's analysis of addressed dilaton tadpoles, resolving divergences in open amplitudes and ensuring in effective Lagrangians. In the AdS/CFT framework, initiated by Maldacena in 1997, the dilaton facilitates mapping strong-coupling gauge theories to weakly coupled gravity, with its profile encoding the running coupling. Key explorations by Callan, Giddings, and Harvey in the early , through the CGHS model of two-dimensional dilaton gravity, highlighted the dilaton's dynamics in processes, including evaporation and inflationary-like expansions.

Theoretical Foundations

In String Theory

In , the dilaton emerges as a massless in the of closed strings, distinct from the open string sector. In the , formulated in 26 dimensions, the closed string massless level includes the , the Kalb-Ramond field, and the dilaton, arising from the zero-mode excitations of the coordinates in the light-cone gauge quantization. This dilaton corresponds to the trace of the second-rank tensor in the transverse oscillator modes, ensuring a tachyon-free at higher levels while the persists at the . In superstring theories, such as type II and heterotic strings in 10 dimensions, the dilaton similarly appears in the closed string Neveu-Schwarz sector as the scalar partner to the and B-field, but the GSO projection eliminates the , yielding a consistent supersymmetric where the dilaton φ relates to the length scale α' through the overall normalization of the . The dilaton profoundly influences the dynamics of string perturbation theory via its vacuum expectation value, which sets the fundamental string coupling constant g_s = e^{\langle \phi \rangle}. This exponential relation implies that weak coupling (g_s \ll 1) corresponds to a large negative \langle \phi \rangle, enabling the perturbative expansion in powers of g_s, while strong coupling regimes require non-perturbative methods like dualities. The dilaton thus parametrizes the strength of string interactions, with loop corrections scaling as g_s^{2L} for L loops, and its vev is dynamically determined in the full theory to ensure consistency. Conformal invariance on the worldsheet, required for anomaly cancellation, imposes beta-function vanishing conditions on the background fields, yielding equations of motion for the dilaton in curved spacetime. In the conformal gauge, the leading-order dilaton beta function sets to zero, resulting in the equation ∇2ϕ−(∂ϕ)2+14R+⋯=0,\nabla^2 \phi - (\partial \phi)^2 + \frac{1}{4} R + \cdots = 0, where R is the Ricci scalar and the dots denote higher-order α' corrections involving curvatures and field strengths. This equation governs the dilaton profile in string backgrounds, ensuring the theory is free of anomalies. The dilaton plays a crucial role in dualities that relate different string theories and compactifications. Under T-duality, which exchanges momentum and winding modes in compact directions, the dilaton transforms in a way that preserves the overall coupling in the effective theory, often remaining invariant at leading order for toroidal compactifications. In type IIB string theory, S-duality under SL(2,\mathbb{Z}) acts on the complexified dilaton-axion field \tau = C_0 + i e^{-\phi}, interchanging weak and strong coupling while stabilizing the dilaton vev. In flux compactifications, such as those on Calabi-Yau manifolds with RR and H-fluxes, the dilaton profile is fixed by the competition between flux-induced potentials and curvature, preventing runaway behavior and yielding AdS or Minkowski vacua consistent with moduli stabilization.

In Effective Field Theories

In effective field theories (EFTs), the dilaton emerges as the leading scalar field associated with the spontaneous breaking of approximate conformal symmetry in strongly interacting systems, such as near-conformal gauge theories. This model-independent framework treats the dilaton as a pseudo-Nambu-Goldstone boson, arising from the infrared dynamics where the theory approaches a conformal fixed point before breaking scale invariance at a lower energy scale ff. The effective Lagrangian is constructed by integrating out higher-dimensional operators, capturing universal low-energy behaviors decoupled from ultraviolet details, with power-counting rules dictating the relevance of operators based on the breaking parameter Δ\Delta, the deviation from exact conformality. The dilaton's coupling structure in these EFTs is dictated by the underlying conformal invariance, featuring a universal interaction with gravity through the term e−2ϕ/dRe^{-2\phi / \sqrt{d}} R
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