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Equation 4 · How Particle Physics Beyond the Standard Model Actually Works

What does this equation mean?

Leff=LSM+∑iciΛ2 Oi\mathcal{L}_{\text{eff}} = \mathcal{L}_{\text{SM}} + \sum_i \frac{c_i}{\Lambda^{2}}\,\mathcal{O}_i

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_eff
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.

Read it piece by piece

Leff\mathcal{L}_{\text{eff}}

Symbol L_eff

LeL_eff 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

unknown coefficients.

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Λ2\Lambda^{2}

Symbol Lambda^2

the square of Lambda; the mass scale of whatever new physics has been integrated out.

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Oi\mathcal{O}_i

Symbol O_i

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

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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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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

If a new particle is too heavy to produce directly, its effects can still appear as small deviations in the interactions of known particles, encoded as additional terms in an effective Lagrangian: Leff=LSM+∑iciΛ2 Oi\mathcal{L}_{\text{eff}} = \mathcal{L}_{\text{SM}} + \sum_i \frac{c_i}{\Lambda^{2}}\,\mathcal{O}_i. Here Oi\mathcal{O}_i are Standard-Model-field operators of dimension six, cic_i are unknown coefficients, and Λ\Lambda is the mass scale of whatever new physics has been integrated out. This is the Standard Model Effective Field Theory (SMEFT) framework used across current LHC and dark-matter searches. Its value is precise and limited: it lets an experiment quote a model-independent bound on cic_i/Λ2\Lambda^2 from a measured deviation (or non-deviation) without committing to what…
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If a new particle is too heavy to produce directly, its effects can still appear as small deviations in the interactions of known particles, encoded as additional terms in an effective Lagrangian: Leff=LSM+∑iciΛ2 Oi\mathcal{L}_{\text{eff}} = \mathcal{L}_{\text{SM}} + \sum_i \frac{c_i}{\Lambda^{2}}\,\mathcal{O}_i. Here Oi\mathcal{O}_i are Standard-Model-field operators of dimension six, cic_i are unknown coefficients, and Λ\Lambda is the mass scale of whatever new physics has been integrated out. This is the Standard Model Effective Field Theory (SMEFT) framework used across current LHC and dark-matter searches. Its value is precise and limited: it lets an experiment quote a model-independent bound on cic_i/Λ2\Lambda^2 from a measured deviation (or non-deviation) without committing to what specific particle generates that coefficient, and separate models mapping onto the same operators can then be constrained together rather than one at a time. It cannot, by itself, tell you the new particle’s mass or spin; it tells you how tightly current data constrains the combination of scale and coupling.

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