# Tree-level unitarity constraints in Two-Higgs Doublet Model

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• mr. bean
In summary: Best of luck! In summary, the conversation discusses the unitarity constraints for the Two-Higgs Doublet Model and the process of calculating the scattering matrices in eq. (7). To do so, one must use the Feynman rules and the relations given in eq. (3) and (6) to express the scattering amplitudes in terms of the states defined in eq. (4) and (5). This may require some practice and patience, but the article attached can serve as a helpful resource for understanding this topic.
mr. bean
Hi, I'm looking at the unitarity constraints for the Two-Higgs Doublet Model and I'm trying to follow what they do in the attached article, which can also be found here: https://arxiv.org/pdf/hep-ph/0312374v1.pdf.

However I do not know how to get the scattering matrices in eq. (7). They say that it is an easy calculation that can be done for each set of states given in eq. (4) and (5) using the potential given in eq. (1) and using the relations given in eq. (3) and (6).

I do not know how I should use eq. (3) and (6). It would be very nice if someone could give me hints for how to do this.

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• Tree-level unitarity constraints in the most.pdf
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Hi there,

I'm glad to see that you are interested in the Two-Higgs Doublet Model and are trying to understand the unitarity constraints. The article you have attached is a great resource for understanding this topic. Let me try to provide some guidance for calculating the scattering matrices in eq. (7).

Firstly, it is important to note that eq. (7) represents the scattering amplitudes between states in the Two-Higgs Doublet Model. These states are defined in eq. (4) and (5) as the physical Higgs bosons (H, h, A, and H+), the Goldstone bosons (G+, G-), and the would-be Goldstone bosons (G0, G'0). These states are related to the two Higgs doublets (Φ1 and Φ2) through the relations given in eq. (3) and (6).

To calculate the scattering matrices, you can use the Feynman rules for the Two-Higgs Doublet Model, which are derived from the potential given in eq. (1). These rules will allow you to calculate the Feynman diagrams for the scattering processes and then use the relations in eq. (3) and (6) to express the scattering amplitudes in terms of the states in eq. (4) and (5).

For example, to calculate the scattering amplitude for the process H+G- → H+G-, you would first use the Feynman rules to draw the relevant Feynman diagram. Then, using the relations in eq. (3) and (6), you can express the H+ and G- states in terms of the Higgs doublets Φ1 and Φ2. Finally, you can use the Feynman rules to calculate the scattering amplitude for the process in terms of the Higgs doublets, and then use the relations in eq. (3) and (6) again to express the amplitude in terms of the physical states in eq. (4) and (5).

I hope this helps you understand how to use eq. (3) and (6) to calculate the scattering matrices in eq. (7). It may take some practice and patience, but with the help of the Feynman rules and the relations in eq. (3) and (6), you should be able to calculate the scattering amplitudes for all the processes listed in eq. (7).

## 1. What is the Two-Higgs Doublet Model?

The Two-Higgs Doublet Model (2HDM) is an extension of the Standard Model of particle physics that introduces a second Higgs doublet to explain the origin of mass and provide a mechanism for electroweak symmetry breaking.

## 2. What are tree-level unitarity constraints?

Tree-level unitarity constraints refer to the conditions that must be satisfied in a particle theory to ensure that the scattering amplitudes do not violate unitarity at tree level. This is important because unitarity is a fundamental principle in quantum mechanics that ensures the probabilities of all possible outcomes of a physical process add up to 1.

## 3. How do tree-level unitarity constraints apply to the Two-Higgs Doublet Model?

In the Two-Higgs Doublet Model, the tree-level unitarity constraints require that the Higgs boson self-interactions do not violate unitarity at high energies. This puts restrictions on the values of the model's parameters, such as the quartic Higgs couplings.

## 4. What is the significance of tree-level unitarity constraints in the Two-Higgs Doublet Model?

The tree-level unitarity constraints are important because they ensure that the 2HDM is a consistent and mathematically well-defined theory. Violations of unitarity could lead to unphysical predictions and render the model invalid.

## 5. Are there any experimental tests for tree-level unitarity constraints in the Two-Higgs Doublet Model?

Yes, there are ongoing experiments at the Large Hadron Collider (LHC) and future colliders that aim to test the 2HDM and its predictions. These experiments can provide valuable insights into the model's parameters and potential violations of tree-level unitarity constraints.

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