Based on our cosmological observations and our theories do we have a first or a second order phase transition for a GUT mechanism?
The phase transitions that describe spontaneous breaking in gauge theories, of which Grand Unified Theories are an example, are second order, since the symmetric vacuum, where the vev of the appropriate scalar is zero, becomes unstable, when the vev is non-zero: the corresponding quadratic term changes sign. It's this fact that implies that the transition is second order. So the mechanism is the Brout-Englert-Higgs mechanism, that's known to be realized to describe the unification of electromagnetic and weak interactions in the Standard Model.
However there aren't any observations, or measurements, yet, in cosmology or elsewhere, that single out any particular Grand Unified Theory, that would describe how the strong interactions are unified with the electroweak interactions.
Yes, we need the answer to provide the specific mechanics to unify the strong with electroweak interactions, as Stam Nicolas has stated. Because the string model also lacks a definition for the pre-emergent strings, we cannot deduce this by property (yet) as a causality system with any familiar conditions. These , if determined, would give us a thermodynamic system where order could be determined. That is the effort made in theory right now, to determine by any systematic thermodynamic principle that can be determined. It is likely to sound simple, as how does an open string self assemble to closed loop? (an example). But we must also arrive at how string then defines space, and then we have a set of properties to explore. Right now I am examining different potential scenarios. That is the best I can offer you. Perhaps you can also explore this through thermodynamic models.
In a spontaneous symmetry breaking quadratic term of the scalar field φ(x) becomes negative. The minimum of Higgs potential shifts to a non-zero value of φ(x) which is or the VEV . But if we write down quantum fluctuations (say η(x)) around the newly developed minimum then the quadratic term in η(x) is again positive, that is there is a change in the sign of the quadratic term.
Furthermore in a second order phase transition a disordered phase of larger symmetry changes continuously to a ordered phase of reduced symmetry. This is the case here. A gut theory such as SU(5) or SO(10) has larger symmetry, or less order. After symmetry breaking, the gauge group becomes SU(3)xSU(2)xU(1), which has reduced symmetry and more order.
See:http://www.cmp.caltech.edu/~mcc/BNU/Notes2_2.pdf
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