Rainbow Diagrams in Multiple Disciplines

• Rainbow diagrams are context-dependent constructs that define layered, color-structured representations in fields from optics to tensor models and graph theory.
• In geometrical optics, they quantify primary and secondary rainbows through precise ray refraction, reflection, and dispersion calculations matching experimental ring radii.
• In graph theory, tensor models, and EDM computations, rainbow diagrams reveal nested substructures, enhanced color symmetries, and specific cancellation mechanisms that drive key results.

“Rainbow diagrams” is a context-dependent term used in several technically distinct literatures. In geometrical optics it denotes ray diagrams for primary and secondary rainbows formed by refraction, dispersion, and internal reflection in water; in graph theory it denotes subgraphs whose edges have pairwise distinct colors; in tensor models it denotes nested melonic dressings of propagators enforced by enhanced color symmetry; in electric-dipole-moment calculations it denotes multi-loop topologies attached to a single fermion line; and in gravity’s rainbow it denotes the mass–central density and mass–radius structural curves of neutron stars computed from rainbow-deformed hydrostatic equilibrium equations (Sarkar et al., 23 May 2026, Benzing et al., 2017, Itoyama et al., 2017, Fujiwara et al., 2021, Hendi et al., 2015).

1. Terminological scope

The term is therefore not a single universal diagrammatic object. Its meaning is fixed by the host discipline and by the role played by “color,” “arc-like layering,” or “rainbow deformation.”

Area	Meaning of “rainbow diagrams”	Governing structure
Geometrical optics	Ray diagrams for rays entering, reflecting inside, and emerging from water droplets	Wavelength-dependent refraction and minimum deviation
Edge-coloured graphs	Paths, cycles, trees, or other subgraphs with pairwise distinct edge colors	Proper coloring, local/global boundedness
Tensor models	Nested, non-crossing melonic dressings of propagators	Enhanced color symmetry and large-$N$ melonic dominance
EDM calculations	Multi-loop topologies with sequential gauge-boson exchanges on one fermion line	Ward–Takahashi identity and chirality structure
Gravity’s rainbow	Structural $M$–$\rho_c$ and $M$–$R$ curves for neutron stars	Rainbow-deformed TOV equations

A plausible implication is that “rainbow” functions less as a single formal notion than as a family resemblance. In the optics and graph-theoretic usages, the term is tied directly to visible or abstract color ordering. In the tensor-model and EDM usages, it refers to the topology of nested or arced insertions. In gravity’s rainbow, it refers to deformation by rainbow functions and to the resulting structural diagrams.

2. Geometrical optics: primary and secondary rainbow construction

In optics, rainbow diagrams quantify how rays refract into, reflect inside, and emerge from water droplets. In the plane of incidence, $i$ is the angle between the incident ray and the local surface normal at the entry point, $r$ is the refracted angle inside water, and $k$ is the number of internal reflections. The total deviation from the original direction is

$D_k(i,\lambda)=2i-2r+k\big(180^\circ-2r\big),$

with wavelength dependence entering through Snell’s law,

$n(\lambda)=\frac{\sin i}{\sin r}.$

The rainbow occurs at the minimum of $M$0, which is obtained from

$M$1

For the observer, the ring radii are

$M$2

This minimum-deviation construction reproduces the standard primary and secondary bows. For the primary rainbow, $M$3, the minimum total deviation is near $M$4, so the observed cone radius is $M$5. The color order is outer-to-inner as red $M$6 violet because $M$7 is smaller for red than violet, giving larger $M$8 for red. For the secondary rainbow, $M$9, the minimum deviation is near $\rho_c$0, so $\rho_c$1; the color order reverses, the bow is fainter, and the gap between the bows is Alexander’s dark band.

A notable feature of the cited experiment is that a water-filled cylindrical glass reproduces the spherical-droplet rainbow geometry in two dimensions. For a horizontally propagating laser beam intersecting a water-filled right circular cylinder, the ray remains in a single vertical plane, and the intersection of that plane with the cylinder is a circle. The in-plane geometry is therefore identical to the great-circle cross-section of a spherical drop. Using graph paper, a clear cylindrical glass tumbler, and three low-power semiconductor laser pointers at 405 nm, 532 nm, and 635 nm, the experiment identifies the minimum deviation by the stationary cusp of the emergent ray as the glass is translated. With a 1 mm wall, the estimated lateral ray separation is $\rho_c$2 cm, comparable to the beam width, so it does not change the measured minimum deviation materially.

The measured ring radii closely match theory. For the primary rainbow the measured values are $\rho_c$3 at 405 nm, $\rho_c$4 at 532 nm, and $\rho_c$5 at 635 nm, compared with theoretical values $\rho_c$6, $\rho_c$7, and $\rho_c$8. For the secondary rainbow the measured values are $\rho_c$9, $M$0, and $M$1, compared with $M$2, $M$3, and $M$4. The reported agreement is within about $M$5, and the classroom method avoids the need for a Cauchy-type dispersion model when only three wavelengths are used (Sarkar et al., 23 May 2026).

3. Edge-coloured graphs and digraphs

In graph theory, a subgraph of an edge-coloured graph or digraph is called rainbow if all of its edges have pairwise distinct colors. Proper edge-coloring means that no two edges sharing a vertex have the same color in an undirected graph, and in the directed setting that all outgoing arcs from any vertex have distinct colors and all incoming arcs to any vertex have distinct colors. Two additional notions structure the theory: a coloring is locally $M$6-bounded if every color appears at most $M$7 times at each vertex, and globally $M$8-bounded if every color appears at most $M$9 times in total.

The cited work proves a directed analogue of long rainbow cycles in properly edge-colored complete digraphs. For the complete directed graph $R$0, every properly edge-coloured $R$1 contains a rainbow path forest of length $R$2 and a directed rainbow cycle of length $R$3. An explicit quantitative form yields, for sufficiently large $R$4, a directed rainbow cycle of length at least $R$5, that is, $R$6. The result is not proved for tournaments, and the tournament case is left as an open problem.

The proof proceeds in three stages. First, a pseudorandom directed expander $R$7 is built by independently selecting each color class with probability $R$8. With high probability, all vertices in $R$9 have in- and out-degree $i$0, and for sufficiently large disjoint sets $i$1, one has $i$2. Second, in the complement $i$3, a directed Andersen-type lemma yields a directed rainbow path forest $i$4 with at most $i$5 paths and at least $i$6 edges whenever the minimum in-degree is at least $i$7. In the application, $i$8 and $i$9, yielding $r$0 paths and at least $r$1 edges. Third, a rotation-augmentation lemma uses the expansion of $r$2 to glue path segments and obtain one very long rainbow path, then closes it to a cycle.

The same paper also proves a rainbow spanning-tree theorem. There exist constants $r$3 such that if $r$4 is a tree on $r$5 vertices with maximum degree $r$6, and $r$7 is properly edge-colored with each color appearing at most $r$8 times, then $r$9 has a rainbow embedding into $k$0. The proof has a random greedy embedding of the highest-degree quarter of the tree, followed by a Lopsided Local Lemma argument in the Lu–Székely framework for random injections. The global $k$1-boundedness assumption is essential: the paper gives counterexamples based on 1-factorizations and on Maamoun–Meyniel colorings.

These results sit in the broader landscape connecting Euler’s work on Latin squares and transversals, Hahn’s conjecture on rainbow Hamilton paths, canonical Ramsey theory in the Erdős–Rado sense, and recent work on rainbow Hamilton cycles and bounded-degree tree embeddings. The cited theorems sharpen the directed picture while leaving several natural extensions open, including better than $k$2 error terms and analogues for tournaments (Benzing et al., 2017).

4. Large-$k$3 tensor models and extreme melonic dominance

In the rainbow tensor model, rainbow diagrams are the nested, non-crossing melonic dressings of propagators that dominate the large-$k$4 expansion because index contractions are forced to be strictly color-consistent. For rank $k$5 tensors, the symmetry is promoted to a product of unitary groups, one $k$6 for each color. For $k$7, there are six colors—red, orange, yellow, green, blue, and violet—and four tensor fields $k$8, each transforming under a product of three $k$9’s. The interaction is a tetrahedron, or starfish, vertex, written compactly as $D_k(i,\lambda)=2i-2r+k\big(180^\circ-2r\big),$0 and its conjugate, with color lines routed so that every color appears exactly twice per interaction.

The central structural claim is that all planar diagrams are melonic. In ordinary tensor models, non-melonic planar graphs are suppressed by extra negative powers of $D_k(i,\lambda)=2i-2r+k\big(180^\circ-2r\big),$1. In the rainbow model, many non-melonic planar fat graphs are not merely suppressed; they cannot be colored consistently at all. The cited paper illustrates this with planar non-melonic configurations whose internal lines cannot be assigned a consistent color. Consequently, among planar diagrams, only melons survive. The leading large-$D_k(i,\lambda)=2i-2r+k\big(180^\circ-2r\big),$2 dynamics therefore dresses propagators only; vertices remain undressed.

This produces a closed system of Schwinger–Dyson equations. If $D_k(i,\lambda)=2i-2r+k\big(180^\circ-2r\big),$3 is the dressed two-point function of field $D_k(i,\lambda)=2i-2r+k\big(180^\circ-2r\big),$4 and $D_k(i,\lambda)=2i-2r+k\big(180^\circ-2r\big),$5 its self-energy, then

$D_k(i,\lambda)=2i-2r+k\big(180^\circ-2r\big),$6

In the melonic limit,

$D_k(i,\lambda)=2i-2r+k\big(180^\circ-2r\big),$7

For $D_k(i,\lambda)=2i-2r+k\big(180^\circ-2r\big),$8, a typical example is

$D_k(i,\lambda)=2i-2r+k\big(180^\circ-2r\big),$9

In a normalized two-flavor toy model, the equations become

$n(\lambda)=\frac{\sin i}{\sin r}.$0

The paper emphasizes that the large-$n(\lambda)=\frac{\sin i}{\sin r}.$1 limit depends non-analytically on ratios of the $n(\lambda)=\frac{\sin i}{\sin r}.$2’s through the $n(\lambda)=\frac{\sin i}{\sin r}.$3’s, with branches and multicritical lines when $n(\lambda)=\frac{\sin i}{\sin r}.$4.

Because vertices are undressed at leading order, higher-point correlators are assembled from propagators and melonic insertions only. The large-$n(\lambda)=\frac{\sin i}{\sin r}.$5 factorization pattern familiar from matrix models persists in tensor form, but the usual spectral-curve technology does not. The cited work therefore points instead to Ward identities, diffeomorphisms in coupling space, the Bogoliubov–Zimmermann forest formula, and Connes–Kreimer Hopf-algebraic control of melonic insertions. In this setting, recursion is “graft-a-melon” over propagators rather than topological recursion on a non-trivial spectral curve. The term “rainbow” thus refers both to the enhanced color structure and to the layered, non-crossing nesting of melons around a propagator core (Itoyama et al., 2017).

5. Rainbow-type diagrams in electric dipole moment calculations

In two-loop EDM calculations, rainbow-type diagrams are multi-loop topologies in which an external fermion line carries sequential gauge-boson exchanges forming an arc-like “rainbow” attached to that single line. Their defining structural property is that the inner sub-diagrams are the fermion self-energy and the electromagnetic vertex correction of the same internal fermion. This distinguishes them from Barr–Zee-type diagrams, where a heavy boson loop generates an effective CP-odd coupling to the photon through a separate loop.

For a Dirac fermion, the EDM operator is

$n(\lambda)=\frac{\sin i}{\sin r}.$6

and the static EDM is extracted from the electromagnetic vertex form factor through

$n(\lambda)=\frac{\sin i}{\sin r}.$7

The key organizing principle is the Ward–Takahashi identity,

$n(\lambda)=\frac{\sin i}{\sin r}.$8

which ties the longitudinal part of the vertex correction to the momentum derivative of the self-energy. The self-energy is parameterized by

$n(\lambda)=\frac{\sin i}{\sin r}.$9

with the chirality-flipping information residing in the scalar form factors $M$00. The transverse vertex is expanded on an eight-vector basis; for EDM purposes, the CP-odd structures are those multiplying the form factors $M$01 and $M$02.

The cited reduction formulas show that the EDM-relevant content of a rainbow diagram is exhausted by the combination

$M$03

where $M$04 comes from chirality-flipping self-energy terms and $M$05 are transverse vertex form factors. If this combination vanishes, then the EDM vanishes. This is precisely what happens for the Standard Model two-loop $M$06-exchange rainbow diagrams: the $M$07 couples purely left-chirally, so $M$08, and the transverse CP-odd vertex structures $M$09 and $M$10 are absent. The result is the exact cancellation associated with the classic Shabalin result. The paper also notes that photon attachment to the outer loop cancels by flavor antisymmetry involving the Jarlskog invariant.

Non-vanishing rainbow EDMs require chirality flipping in the internal loop, transverse CP-odd vertex structures, and complex phases in the couplings. The cited example is a singlet–triplet fermion model extended by a real $M$11-triplet scalar $M$12, where new Yukawa-type couplings induce non-zero $M$13 and $M$14. The resulting electron EDM can reach the $M$15 cm level for TeV-scale masses and $M$16 couplings. The same reduction logic also extends to chromo-EDMs, with the Abelian Ward–Takahashi identity replaced by the background-field version appropriate to gluons (Fujiwara et al., 2021).

6. Structural diagrams in gravity’s rainbow

In gravity’s rainbow, “rainbow diagrams” refers to the neutron-star structural curves $M$17–$M$18 and $M$19–$M$20, computed from a rainbow-deformed hydrostatic equilibrium equation. The deformation begins from the modified dispersion relation

$M$21

with $M$22. For a static, spherically symmetric star in $M$23 dimensions, the metric ansatz is

$M$24

The effective mass is

$M$25

and the modified TOV equation can be written as

$M$26

with

$M$27

$M$28

Three rainbow-function models are tested. In Case I,

$M$29

In Case II,

$M$30

and the TOV equation becomes independent of the rainbow functions, so there is no effect on stellar structure. In Case III,

$M$31

The matter model is neutron-star matter composed of neutrons, protons, electrons, and muons in charge neutrality and beta equilibrium, with an AV18-based microscopic equation of state fitted by

$M$32

Integration starts from $M$33 and $M$34 and stops at $M$35, generating the structural sequences.

The $M$36–$M$37 and $M$38–$M$39 diagrams shift systematically with $M$40. Increasing $M$41 shifts the curves upward and to larger radii; decreasing $M$42 reduces both $M$43 and $M$44. Representative values are $M$45 and $M$46 km for $M$47, $M$48 and $M$49 km for $M$50, and $M$51 and $M$52 km for $M$53. The paper identifies a consistency window $M$54, because for $M$55 the average density exceeds the central density. Within the adopted equation of state, the global upper limit is $M$56 at $M$57.

A positive cosmological constant shifts the $M$58–$M$59 and $M$60–$M$61 diagrams downward. For $M$62, increasing $M$63 from $M$64 to $M$65 changes $M$66 from $M$67 to $M$68 and reduces the radius from $M$69 km to $M$70 km; for $M$71 and $M$72, the corresponding values are $M$73 and $M$74 km. By contrast, the cosmological value $M$75 has no discernible effect. The paper also reports rainbow dependence in the Schwarzschild radius, average density, compactness, and surface redshift, with typical redshifts $M$76, and establishes dynamical stability through the Chandrasekhar criterion

$M$77

In this usage, rainbow diagrams are not Feynman graphs or ray pictures but deformation-sensitive structural plots encoding how $M$78 and $M$79 modify neutron-star equilibrium and stability (Hendi et al., 2015).