<p>We thoroughly investigate nonanalytic terms in the finite-temperature effective potential in one-loop approximation on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi></math>-dimensional spacetime, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>D</mi><mo>−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>×</mo><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>p</mi></mrow></msubsup><mtext> </mtext><msubsup><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msubsup></mrow></math>, using a mode recombination formula. Such nonanalytic terms cannot be expressed as positive powers of field-dependent mass squared. The formula provides a clear separation of the effective potential into a part that contains the nonanalytic terms and a part that is purely analytic, and clarifies the origin of the nonanalytic terms. We obtain all the nonanalytic terms and show that only two types of nonanalytic terms arise from the modes with zero Matsubara frequency. For a real scalar field with periodic boundary conditions, if the number of noncompacted spatial dimensions is odd (even), there are odd powers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>log</mi><mi>M</mi></math> terms) but no <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>log</mi><mi>M</mi></math> terms (no odd powers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi></math>). For fermions with general boundary conditions, we find that neither of the two types appears. These results clarify the nonanalytic structure of the finite-temperature effective potential on the spacetime with compactified spatial dimensions.</p>

Japan

Makoto Sakamoto

Kazunori Takenaga

Nonanalytic structure of effective potential at finite temperature on compactified space

We develop a new formula called a mode recombination formula, and we can recast the effective potential at finite temperature in one-loop approximation for fermion and scalar fields on the <math><mi>D</mi></math>-dimensional spacetime, <math><msubsup><mi>S</mi><mi>τ</mi><mn>1</mn></msubsup><mo>×</mo><msup><mi>R</mi><mrow><mi>D</mi><mo>−</mo><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>×</mo><msubsup><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><msubsup><mi>S</mi><mi>i</mi><mn>1</mn></msubsup></math> into a convenient form for discussing nonanalytic terms, which cannot be written in the form of any positive integer power of the field-dependent mass squared, in the effective potential. The formula holds irrespective of whether the field is a fermion or a scalar and of boundary conditions for spatial <math><msubsup><mi>S</mi><mi>i</mi><mn>1</mn></msubsup></math> directions and clarifies the importance of zero modes in the Matsubara and Kaluza-Klein modes for the existence of the nonanalytic terms. The effective potential is drastically simplified further to obtain the nonanalytic terms in easier and more transparent way. In addition to reproducing previous results, we find that there exists no nonanalytic term for the fermion field with arbitrary boundary condition for the spatial <math><msubsup><mi>S</mi><mi>i</mi><mn>1</mn></msubsup></math> direction, which is also the case for the scalar field with the antiperiodic boundary condition for the spatial direction.

Mode recombination formula and nonanalytic term in an effective potential at finite temperature on a compactified space

We study a novel six-dimensional gauge theory compactified on the <math><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math> orbifold utilizing the diagonal embedding method. The bulk gauge group is <math><mi>G</mi><mo>×</mo><mi>G</mi><mo>×</mo><mi>G</mi></math>, and the diagonal part <math><mrow><msup><mrow><mi>G</mi></mrow><mrow><mi>diag</mi></mrow></msup></mrow></math> remains manifest in the effective four-dimensional theory. Further spontaneous breaking of the gauge symmetry occurs through the dynamics of the zero modes of the extradimensional components of the gauge field. We apply this setup to the <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>5</mn><mo>)</mo></math> grand unified theory and examine the vacuum structure determined by the dynamics of the zero modes. We find bulk matter contents that radiatively induce spontaneous breakings of the unified symmetry <math><mrow><msup><mrow><mi>G</mi></mrow><mrow><mi>diag</mi></mrow></msup><mo>≅</mo><mi>S</mi><mi>U</mi><mo>(</mo><mn>5</mn><mo>)</mo></mrow></math> down to <math><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo><mo>×</mo><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></math> at the global minima of the one-loop effective potential for the zero modes. This spontaneous breaking provides notable features such as a realization of the doublet-triplet splitting without a fine tuning and a prediction of light adjoint fields.

Kentaro Kojima

Toshifumi Yamashita

Grand gauge-Higgs unification on <math><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math> via the diagonal embedding method

We study nonanalytic terms, which cannot be written in the form of any positive integer power of field-dependent mass squared, in effective potential at finite temperature in one-loop approximation for a real scalar field on the <math><mi>D</mi></math>-dimensional spacetime, <math><msubsup><mi>S</mi><mi>τ</mi><mn>1</mn></msubsup><mo>×</mo><msup><mi>R</mi><mrow><mi>D</mi><mo>−</mo><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>×</mo><msubsup><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><msubsup><mi>S</mi><mi>i</mi><mn>1</mn></msubsup></math>. The effective potential can be recast into the integral form in the complex plane by using the integral representation for the modified Bessel function of the second kind and the analytical extension for multiple mode summations. The pole structure of the mode summations is clarified and all the nonanalytic terms are obtained by the residue theorem. We find that the effective potential has a nonanalytic term when the dimension of the flat Euclidean space, <math><mi>D</mi><mo>−</mo><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo></math> is odd. There appears only one nonanalytic term for the given values of <math><mi>D</mi></math> and <math><mi>p</mi></math>, for which the nonanalytic term exists.

Nonanalytic term in the effective potential at finite temperature for a scalar field on compactified space

Abstract We show that the observed quark/lepton mass hierarchy can be realized dynamically on an interval extra dimension with point interactions. In our model, the positions of the point interactions play a crucial role in controlling the quark/lepton mass hierarchy and are determined by the minimization of the Casimir energy. By use of the exact extra-dimensional coordinate-dependent vacuum expectation value of a gauge-singlet scalar, we find that there is a parameter set, where the positions of the point interactions are stabilized and fixed, which can reproduce the experimental values of the quark masses precisely enough, while the charged lepton part is less relevant. We also show that possible mixings among the charged leptons will improve the situation significantly.

Yukihiro Fujimoto

Kouhei Hasegawa

Croatia

Kenji Nishiwaki

Pedro Hugo Tanaka

Inori Ueba

Dynamical generation of quark/lepton mass hierarchy in an extra dimension

We study grand unified models in the five-dimensional space-time where the extra dimension is compactified on S 1 / ℤ 2 . The spontaneous breaking of unified gauge symmetries is achieved via vacuum expectation values of the extra-dimensional components of gauge fields. We derive one-loop effective potentials for the zero modes of the gauge fields in SU(7), SU(8), SO(10), and E 6 models. In each model, the rank of the residual gauge symmetry that respects the boundary condition imposed at the orbifold fixed points is higher than that of the standard model. We verify that the residual symmetry is broken to the standard model gauge symmetry at the global minima of the effective potential for certain sets of bulk fermion fields in each model.

The standard model gauge symmetry from higher-rank unified groups in grand gauge-Higgs unification models

We discuss a Polyakov loop in a non-covariant operator formalism that consists of only physical degrees of freedom at finite temperature. It is pointed out that although the Polyakov loop is expressed by a Euclidean time component of gauge fields in a covariant path integral formalism, there is no direct counterpart of the Polyakov loop operator in the operator formalism because the Euclidean time component of gauge fields is not a physical degree of freedom. We show that by starting with an operator that is constructed in terms of only physical operators in the non-covariant operator formalism, the vacuum expectation value of the operator calculated by the trace formula can be rewritten into the familiar form of an expectation value of the Polyakov loop in a covariant path integral formalism at finite temperature for the cases of the axial and Coulomb gauge.