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Equation 2 · Comparing the Main Approaches to Particle Physics Beyond the Standard Model

What does this equation mean?

LSMEFT=LSM+∑iciΛ2Oi(6)+O ⁣(1Λ4)\mathcal{L}_{\text{SMEFT}} = \mathcal{L}_{\text{SM}} + \sum_i \frac{c_i}{\Lambda^{2}} \mathcal{O}_i^{(6)} + \mathcal{O}\!\left(\frac{1}{\Lambda^{4}}\right)

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Start withc_i
Divide byLambda^2
This relates toL_SMEFT
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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LSMEFT\mathcal{L}_{\text{SMEFT}}

Symbol L_SMEFT

LSL_SMEFT is part of the quantity the equation computes from the expression on the right.

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LSM\mathcal{L}_{\text{SM}}

Symbol L_SM

LSL_SM is one of the signed contributions combined to compute the quantity on the left.

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ii

Symbol i

i occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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cic_i

Symbol c_i

a dimensionless Wilson coefficient set by the details of the underlying theory.

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Λ2\Lambda^{2}

Symbol Lambda^2

Lambda2a^2 occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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Oi(6)\mathcal{O}_i^{(6)}

Symbol O_i^(6)

a dimension-six operator built from Standard Model fields.

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O\mathcal{O}

Symbol O

O is one of the signed contributions combined to compute the quantity on the left.

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Λ4\Lambda^{4}

Symbol Lambda^4

Lambda4a^4 occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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fraction

fraction

Divide the expression above the line by the one below it.

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addition

addition

Add the term after the plus sign to the term or group before it.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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ii

Starting index or lower bound: i

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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11

Numerator: 1

The complete quantity above the fraction bar.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

SMEFT occupies a different position in this comparison because it is not competing to be the correct underlying theory — it is a bookkeeping framework for any underlying theory heavy enough to not yet have been produced directly. Concretely, SMEFT adds effective operators to the Standard Model Lagrangian, each suppressed by powers of an energy scale Λ\Lambda representing the mass of whatever new particles generate it: LSMEFT=LSM+∑iciΛ2Oi(6)+O ⁣(1Λ4)\mathcal{L}_{\text{SMEFT}} = \mathcal{L}_{\text{SM}} + \sum_i \frac{c_i}{\Lambda^{2}} \mathcal{O}_i^{(6)} + \mathcal{O}\!\left(\frac{1}{\Lambda^{4}}\right). where each Oi(6)\mathcal{O}_i^{(6)} is a dimension-six operator built from Standard Model fields and cic_i is a dimensionless Wilson coefficient set by the details of the underlying theory. This notation is worth showing because it makes the comparison’s real…
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SMEFT occupies a different position in this comparison because it is not competing to be the correct underlying theory — it is a bookkeeping framework for any underlying theory heavy enough to not yet have been produced directly. Concretely, SMEFT adds effective operators to the Standard Model Lagrangian, each suppressed by powers of an energy scale Λ\Lambda representing the mass of whatever new particles generate it: LSMEFT=LSM+∑iciΛ2Oi(6)+O ⁣(1Λ4)\mathcal{L}_{\text{SMEFT}} = \mathcal{L}_{\text{SM}} + \sum_i \frac{c_i}{\Lambda^{2}} \mathcal{O}_i^{(6)} + \mathcal{O}\!\left(\frac{1}{\Lambda^{4}}\right). where each Oi(6)\mathcal{O}_i^{(6)} is a dimension-six operator built from Standard Model fields and cic_i is a dimensionless Wilson coefficient set by the details of the underlying theory. This notation is worth showing because it makes the comparison’s real structure explicit: SUSY, extra dimensions, and dark-sector models are each, in this language, a specific choice of which Oi\mathcal{O}_i operators get generated and with what coefficients. A null search in a global SMEFT fit constrains cic_i/Λ2\Lambda^2 combinations directly from precision electroweak, Higgs, and flavor data, without committing to which particle content produced them [ 7 ] . The tradeoff is exactly the one implied by that generality: an EFT fit can tell you that new physics coupling to a particular operator must lie above some energy scale, but it cannot, by itself, tell you whether that new physics looks like a superpartner, a Kaluza-Klein mode, or a dark-sector mediator. It is the most model-independent of the four approaches and, for that reason, the hardest to describe as “confirmed” or “excluded” in the way a direct search can be.

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