<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 present a new approach to stochastic quantization à la Parisi–Wu with a discrete fictitious time. The noise average is modified by weights, which results in the equivalence in the large time limit to the correlation function of the corresponding quantum field theory <italic>without taking any continuum limit of fictitious time</italic>. We test our method in a 0D toy model both perturbatively and numerically.

Daisuke Kadoh

Mitsuhiro Kato

Hiroto So

Stochastic Quantization with Discrete Fictitious Time

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 investigate blow-up manifolds of <math><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></math> orbifolds with magnetic flux. First, we construct the blow-up manifolds and zero-mode wave functions on them more precisely. In particular, through an appropriate singular gauge transformation, winding numbers of wave functions on <math><msup><mi>T</mi><mn>2</mn></msup><mo>/</mo><msub><mi>Z</mi><mi>N</mi></msub></math> can be replaced with localized curvature and localized flux at orbifold fixed points. In addition, since the blow-up manifolds have no singularities, we apply the Atiyah-Singer index theorem to them; the chiral zero-mode number is given by the total magnetic flux. It can be also applied for <math><msup><mi>T</mi><mn>2</mn></msup><mo>/</mo><msub><mi>Z</mi><mi>N</mi></msub></math> orbifolds through the blow-up process, and then we find that it is consistent with the zero-mode counting formula in M. Sakamoto et al. [Zero-mode counting formula and zeros in orbifold compactifications, Phys. Rev. D 102, 025008 (2020).]. Furthermore, the Atiyah-Singer index theorem shows that an additional degree of freedom of localized flux gives new chiral zero modes. We study their wave functions and then we find that they correspond to localized modes at the orbifold singular points. We also calculate their Yukawa couplings.

Tatsuo Kobayashi

Hajime Otsuka

Maki Takeuchi

Italy

Yoshiyuki Tatsuta

Hikaru Uchida

Index theorem on magnetized blow-up manifold of <math><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></math>

We analyze the number of independent chiral zero modes and the winding numbers at the fixed points on <math><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></math> (<math><mi>N</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn></math>) orbifolds with magnetic flux. In the case of <math><mi>N</mi><mo>=</mo><mn>2</mn></math>, we derive the index formula <math><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>=</mo><mi>M</mi><mo>/</mo><mn>2</mn><mo>+</mo><mo>(</mo><mo>−</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>+</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>)</mo><mo>/</mo><mn>4</mn><mo>=</mo><mi>M</mi><mo>/</mo><mn>2</mn><mo>−</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> by using the trace formula, where <math><msub><mrow><mi>n</mi></mrow><mrow><mo>±</mo></mrow></msub></math> are the numbers of the ± chiral zero modes and <math><msub><mrow><mi>V</mi></mrow><mrow><mo>±</mo></mrow></msub></math> are the sums of the winding numbers at the fixed points on <math><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. We also obtain the formula <math><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>=</mo><mi>M</mi><mo>/</mo><mi>N</mi><mo>+</mo><mo>(</mo><mo>−</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>+</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>)</mo><mo>/</mo><mo>(</mo><mn>2</mn><mi>N</mi><mo>)</mo><mo>=</mo><mi>M</mi><mo>/</mo><mi>N</mi><mo>−</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>/</mo><mi>N</mi><mo>+</mo><mn>1</mn></math> for <math><mi>N</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn></math> under an assumption.

Hiroki Imai

Index and winding numbers on Orbifolds With Magnetic Flux

In this paper, we study five-dimensional Dirac fermions of which the extra-dimension is compactified on quantum graphs. We find that there is a nontrivial correspondence between matrices specifying boundary conditions at the vertex of the quantum graphs and zero-dimensional Hamiltonians in gapped free-fermion systems. Based on the correspondence, we provide a complete topological classification of the boundary conditions in terms of noninteracting fermionic topological phases. The ten symmetry classes of topological phases are fully identified in the language of five-dimensional Dirac fermions, and topological numbers of the boundary conditions are given. In analogy with the bulk-boundary correspondence in noninteracting fermionic topological phases, the boundary condition topological numbers predict four-dimensional massless fermions localized at the vertex of the quantum graphs and thus govern the low energy physics in four dimensions.

Tomonori Inoue

Masatoshi Sato

Inori Ueba

Correspondence of topological classification between quantum graph extra dimension and topological matter

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

We investigate chiral zero modes and winding numbers at fixed points on <math><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></math> orbifolds. It is shown that the Atiyah-Singer index theorem for the chiral zero modes leads to a formula <math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>=</mo><mspace></mspace><mo>(</mo><mo>−</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>+</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>)</mo><mo>/</mo><mn>2</mn><mi>N</mi></mrow></math>, where <math><msub><mi>n</mi><mo>±</mo></msub></math> are the numbers of the <math><mo>±</mo></math> chiral zero modes and <math><msub><mi>V</mi><mo>±</mo></msub></math> are the sums of the winding numbers at the fixed points on <math><msup><mi>T</mi><mn>2</mn></msup><mo>/</mo><msub><mi>Z</mi><mi>N</mi></msub></math>. This formula is complementary to our zero-mode counting formula on the magnetized orbifolds with nonzero flux background <math><mi>M</mi><mo>≠</mo><mn>0</mn></math>, consistently with substituting <math><mi>M</mi><mo>=</mo><mn>0</mn></math> for the counting formula <math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>=</mo><mo>(</mo><mn>2</mn><mi>M</mi><mo>−</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>+</mo><msub><mrow><mi>V</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>)</mo><mo>/</mo><mn>2</mn><mi>N</mi></mrow></math>.

Germany

Index theorem on <math><mrow><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></math> orbifolds

We thoroughly analyze the number of independent zero modes and their zero points on the toroidal orbifold <math><msup><mi>T</mi><mn>2</mn></msup><mo>/</mo><msub><mi>Z</mi><mi>N</mi></msub><mo>(</mo><mi>N</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>)</mo></math> with magnetic flux background, inspired by the Atiyah-Singer index theorem. We first show a complete list for the number <math><msub><mi>n</mi><mi>η</mi></msub></math> of orbifold zero modes belonging to <math><msub><mi>Z</mi><mi>N</mi></msub></math> eigenvalue <math><mi>η</mi></math>. Since it turns out that <math><msub><mi>n</mi><mi>η</mi></msub></math> quite complicatedly depends on the flux quanta <math><mi>M</mi></math>, the Scherk-Schwarz twist phase <math><mo>(</mo><msub><mi>α</mi><mn>1</mn></msub><mo>,</mo><msub><mi>α</mi><mn>2</mn></msub><mo>)</mo></math>, and the <math><msub><mi>Z</mi><mi>N</mi></msub></math> eigenvalue <math><mi>η</mi></math>, it seems hard that <math><msub><mi>n</mi><mi>η</mi></msub></math> can be universally explained in a simple formula. We, however, succeed in finding a single zero-mode counting formula <math><msub><mi>n</mi><mi>η</mi></msub><mo>=</mo><mo>(</mo><mi>M</mi><mo>−</mo><msub><mi>V</mi><mi>η</mi></msub><mo>)</mo><mo>/</mo><mi>N</mi><mo>+</mo><mn>1</mn></math>, where <math><msub><mi>V</mi><mi>η</mi></msub></math> denotes the sum of winding numbers at the fixed points on the orbifold <math><msup><mi>T</mi><mn>2</mn></msup><mo>/</mo><msub><mi>Z</mi><mi>N</mi></msub></math>. The formula is shown to hold for any pattern.

Zero-mode counting formula and zeros in orbifold compactifications

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.