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Tricritical Point in Quantum Phase Transitions of the Coleman-Weinberg Model at Higgs Mass

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Tricritical Point in Quantum Phase Transitions of the Coleman-Weinberg Model at Higgs Mass Tricritical Point in Quantum Phase Transitions of the Coleman-Weinberg Model at Higgs Mass Miguel C N Fiolhais∗ LIP, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal Hagen Kleinerty Freie Universit¨atBerlin, Institut f¨urTheoretische Physik Arnimallee 14, D-14195 Berlin, Germany (Dated: June 28, 2013) Abstract The tricritical point, which separates first and second order phase transitions in three-dimensional superconductors, is studied in the four-dimensional Coleman-Weinberg model, and the similarities as well as the differences with respect to the three-dimensional result are exhibited. The position of the tricritical point in the Coleman-Weinberg model is derived and found to be in agreement with the Thomas-Fermi approximation in the three-dimensional Ginzburg-Landau theory. From this we deduce a special role of the tricritical point for the Standard Model Higgs sector in the scope of the latest experimental results, which suggests the unexpected relevance of tricritical behavior in the electroweak interactions. The following paper is published in Phys. Lett. A: http://www.sciencedirect.com/science/article/pii/S0375960113005768 INTRODUCTION this way, the GL theory has become what may be called the Standard Model of superconductive phenomenology. Ever since the formulation in 1964 of the electroweak A similar Ginzburg-Landau-like scalar field theory spontaneous symmetry breaking mechanism [1{5] to ex- with quartic interaction is successful in unifying the weak plain how elementary particles acquire mass, supercon- and the electromagnetic interactions, so that it has be- ductivity and high-energy physics became intimately con- come the Standard Model of particle physics, also called nected. For instance, the Ginzburg-Landau (GL) the- the Higgs Model. ory [1], proposed in 1950 to provide a local macroscopic An important new aspect that arises at the transi- description of superconductivity, makes use of a quar- tion from the three-dimensional GL theory to the (3+1)- tic potential of the same type that reappeared in the dimensional scalar electrodynamical Higgs model is that Higgs model1. In fact, the GL theory is the three- the field possess canonical commutation rules. These call dimensional predecessor of what is now called (3+1)- for the existence of a particle associated with each field. dimensional scalar quantum electrodynamics, that was After all, this is the logic which led to the discovery of studied in detail by Coleman and Weinberg [6, 7]. Be- pions as the quantum of the forces of nuclear physics. In fore the work of Ginzburg and Landau, the London the- particle physics, it induced an intensive search for a Higgs ory explained the existence of a finite penetration depth particle for many years. The recent discovery of a new of magnetic fields into a superconductor, the Meissner- signal in the 124-126 GeV mass region by the ATLAS Ochsenfeld effect [8]. GL extended this theory by a local and CMS collaborations at the Large Hadron Collider complex scalar order field φ(x), whose gradient terms in [10, 11] is a hopeful candidate for such a particle. the energy density produces a finite length scale of fluc- In this Letter we want to put this mass value into con- tuations of the order field, the so-called coherence length text with a known fact in superconductivity, that the su- ξ. In their theory, the Meissner-Ochsenfeld effect was perconductive phase transition may occur in two different arXiv:1306.6568v1 [hep-ph] 27 Jun 2013 explained by a local mass term of the vector potential, orders: a second order if the GL parameter κ lies deep in whose size is proportional to jφj2. the type-II regime, and a first order in the type-I regime. The GL theory possesses two length scales, the London For a long time, this issue was a matter of theoretical con- penetration depth λ , and the coherence length ξ. The troversy after it had been argued by Halperin, Lubensky, L and Ma (HLM) [12] that superconductors should really competitionp between the two is ruled by the GL parame- ter κ ≡ λ = 2ξ. This serves to distinguish two types of arise in a first-order transition. The issue was finally set- L p tled by the calculation of a tricritical point near the divid- superconductors,p type-I with κ > 1= 2 and type-II with κ < 1= 2. The second type, possess bundles of vortices ing line between type-II to type-I superconductivity. The approximate value of κ where this happens was predicted which confine magnetic flux in tubes of radius λL [9]. In p to be κtr ≈ 0:81= 2 [13{15], a value later confirmedp by Monte Carlo simulations to lie at κ = 0:76= 2 ± 0:04 [16]. The important point in the theory was that the 1 With the short name \Higgs" we abbreviate collectively all au- mass term of the electromagnetic potential was reliable thors that contributed from [1{5]. only as long as it was big, which is the case in the type-I 1 regime. If it is small, the mass is destroyed by fluctuating τ drops below zero, the ground state of the potential, 1 2 g 4 vortex lines [17, 18]. The precise position of the tricriti- V ( ) = 2 τj j + 4 j j , is obtained for an infinite number cal point is unknown and should be determined by Monte of degenerate states satisfying: Carlo simulations as described in [19]. τ The calculation of HLM had an interesting parallel h i2 = ρ2 = − ; (2) 0 g in (3+1)-dimensional scalar QED, where Coleman and Weinberg2 calculated that a massless field would acquire and corresponds to a second-order phase transition. Af- a mass from the fluctuations of the electromagnetic field. ter the spontaneous symmetry breaking, i.e. fixing the In the language of superconductive, this implies that gauge to θ(x) = 0, the Hamiltonian becomes, scalar QED has a first-order phase transition3. After 2 2 the calculation of the tricritical point in superconductive 1 2 e ρ H( ; r ; A; rA) = (rρ) + V (ρ) + A2 it was proposed that a similar tricritical point should 2 2 come up in (3+1)-dimensional QED [20]. The Coleman- 1 2 + (r × A) : (3) Weinberg result was derived without considering the fluc- 2 tuating vortex sheets which are the (3+1)-dimensional The mass term of the vector field, m = eν, which ap- analogs of the vortex lines in superconductors. These A peared with the spontaneous symmetry breaking, and should modify the CW-result in the small e2 regime. One the scalar field mass term, can be associated with two should therefore expect a tricritical value of κ also in characteristic lengths of a superconductor, the London (3+1)-dimensional scalar QED, and the present Letter penetration length, λ = 1=m = 1/eρ , and the coher- gives further support for this expectation with experi- p L A 0 ence length, ξ = 1= −2τ, respectively. mental consequences. Moreover, the tricritical point is The first-order phase transition can be achieved in the predicted and interpreted in the Standard Model as the Ginzburg-Landau theory by considering quantum correc- absolute stability boundary of the Higgs potential, by tions, which in the Thomas-Fermi approximation [21], analogy with superconductivity. The latest theoretical neglecting fluctuations in ρ, leads to an additional cubic predictions on the meta-stability and instability bound- term in the potential, aries up to the Planck scale of the Standard Model Higgs potential are discussed in the context of the recent results 1 g c e3 V (ρ) = τρ2 + ρ4 − ρ3 ; c = : (4) on the observed signal at the LHC. 2 4 3 2π As shown in Fig. 1, the cubic term generates a second 2 QUARTIC INTERACTION AND TRICRITICAL minimum for τ < c =4g, at the minimum, POINT ! c r 4τg ρ~0 = 1 + 1 − : (5) The Ginzburg-Landau theory of superconductivity is 2g c2 characterized by the following energy density: 2 At the specific point τ1 = 2c =9g, the minimum lies at 1 the same level as the origin for ρ = 2c=3g, where the H( ; r ; A; rA) = (r + ieA) ∗ (r − ieA) 1 2 phase transition becomes of first-order (tricritical point). τ g + j j2 + j j4 Therefore, in this point, the coherence length of the ρ- 2 4 field fluctuations becomes, 1 + (r × A)2 ; (1) 2 1 3rg ξ1 = = ; (6) pτ + 3gρ2 − 2cρ c 2 with the order parameter (x) = ρ(x)eiθ(x), where ρ(x) 1 1 and θ(x) are real fields. The vector field is represented by which is the same as the fluctuations around ρ = 0. Fi- A, e is the electric charge of the Cooper pairs4, and the nally, the Ginzburg parameter at the tricritical point is, real constants τ and g give the strength of the quadratic and quartic terms, respectively. If the mass parameter 1r g κ = : (7) 2 e2 2 Their work was done almost simultaneously with [12] on the same floor at Harvard University. TRICRITICAL POINT IN THE COLEMAN- 3 On a hiking excursion into the mountains near Geneva with Sid WEINBERG MODEL Coleman, H.K. once asked him whether this was really what they proved, he said \yes, but we foolishly did not put it that way". 4 The Euler number is represented by e, and shall not be confused The intriguing question now is how this result changes with the electric charge e. in the four-dimensional version of the Ginzburg-Landau 2 Ginzburg parameter becomes, r 1 1=ehφ i 1 λ κ = p c = p : (12) 2 1=m(φ) 2α e2 The result has the same form as the previously obtained 3-dimensional result, and becomes the same with an appropriate choice of the renormalization scale.
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