Invariant Monomials Research Articles

Chiral effective Lagrangian for doubly charmed baryons up to O(q4)

The chiral effective meson-baryon Lagrangian for the description of interactions between the doubly charmed baryons and Goldstone bosons is constructed up to the order of $q^{4}$. The numbers of linearly independent invariant monomials of $\mathcal{O}(q^2)$, $\mathcal{O}(q^3)$ and $\mathcal{O}(q^4)$ are 8, 32 and 218, in order. The obtained Lagrangian can be used to study the chiral dynamics and relevant phenomenology of the doubly charmed baryons at complete one-loop level in future. For completeness, the non-relativistic reduction of the Lagrangian is also discussed.

Rephasing invariants of the Cabibbo-Kobayashi- Maskawa matrix

The paper is motivated by the importance of the rephasing invariance of the CKM (Cabibbo-Kobayashi-Maskawa) matrix observables. These observables appear in the discussion of the CP violation in the standard model (Jarlskog invariant) and also in the renormalization group equations for the quark Yukawa couplings. Our discussion is based on the general phase invariant monomials built out of the CKM matrix elements and their conjugates. We show that there exist 30 fundamental phase invariant monomials and 18 of them are a product of 4 CKM matrix elements and 12 are a product of 6 CKM matrix elements. In the main theorem we show that a general rephasing invariant monomial can be expressed as a product of at most five factors: four of them are fundamental phase invariant monomials and the fifth factor consists of powers of squares of absolute values of the CKM matrix elements. We also show that the imaginary part of any rephasing invariant monomial is proportional to the Jarlskog’s invariant J or is 0.

Cabibbo–Kobayashi–Maskawa matrix: rephasing invariants and parameterizations

In this work we study two topics: the first one considers the general phase invariant monomials built out of the CKM matrix elements and their conjugates. We show, that there exist 30 fundamental phase invariant monomials and 18 of them are a product of 4 CKM matrix elements and 12 are a product of 6 CKM matrix elements. In our main result we show that all rephasing invariant monomials can be expressed as a product of at most 5 factors with positive powers. Next, we propose a general method of a recursive construction of the CKM matrix for any number of generations. This allows to construct a parameterization with desired properties. As an application we generalize the Wolfenstein parameterization to the case of 4 generations and obtain restrictions on the CKM suppression of the fourth generation.

Partition function of beta-gamma system on orbifolds

Partition function of beta-gamma systems on the orbifolds C^2/Z_N and C^3/Z_M x Z_N are obtained as the invariant part of that on the respective affine spaces, by lifting the geometric action of the orbifold group to the fields. Interpreting the sum over roots of unity as an elementary contour integration, the partition function evaluates to an infinite series counting invariant monomials composed of basic operators of the theory at each mass level.

A COMPLETE AND MINIMAL CATALOGUE OF MSSM GAUGE INVARIANT MONOMIALS

We present a complete and minimal catalogue of MSSM gauge invariant monomials. That is, the catalogue of Gherghetta, Kolda and Martin is elaborated to include generational structure for all monomials. Any gauge invariant operator can be built as a linear combination of elements of the catalogue lifted to nonnegative integer powers. And the removal of any one of the monomials would deprive the catalogue of this feature. It contains 712 monomials, plus 3 generations of right-handed neutrinos if one extends the model to the νMSSM. We note that νMSSM flat directions can all be lifted by the sixth-order superpotential compared to the ninth-order needed in MSSM.

Lexsegment and Gotzmann ideals associated with the diagonal action of $${\mathbb Z/p}$$

We consider a diagonal action of a cyclic group of prime order on a polynomial ring F[x1, . . . , xn]. We give a description of the actions for which the corresponding Hilbert ideal is Gotzmann when n = 2. Nevertheless, we show that there is a separating set of invariant monomials that generates a proper lexsegment ideal in the polynomial ring for all n. As well, we provide an algorithm to compute this set.

Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups

The Harish-Chandra correlation functions, i.e. integrals over compact groups of invariant monomials $\prod \operatorname{tr}(X^{p_{1}}\varOmega Y^{q_{1}}\varOmega^{\dagger}X^{p_{2}}\cdots)$ with the weight exp tr (X Ω Y Ω † ) are computed for the orthogonal and symplectic groups. We proceed in two steps. First, the integral over the compact group is recast into a Gaussian integral over strictly upper triangular complex matrices (with some additional symmetries), supplemented by a summation over the Weyl group. This result follows from the study of loop equations in an associated two-matrix integral and may be viewed as the adequate version of Duistermaat–Heckman’s theorem for our correlation function integrals. Secondly, the Gaussian integration over triangular matrices is carried out and leads to compact determinantal expressions.

UNIVERSAL CALABI–YAU ALGEBRA: TOWARDS AN UNIFICATION OF COMPLEX GEOMETRY

We present a universal normal algebra suitable for constructing and classifying Calabi–Yau spaces in arbitrary dimensions. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions, related to Batyrev's reflexive polyhedra, and their n-ary combinations. It also includes a "dual" construction based on the Diophantine decomposition of invariant monomials, which provides explicit recurrence formulae for the numbers of Calabi–Yau spaces in arbitrary dimensions with Weierstrass, K3, etc., fibrations. Our approach also yields simple algebraic relations between chains of Calabi–Yau spaces in different dimensions, and concrete visualizations of their singularities related to Cartan–Lie algebras. This Universal Calabi–Yau algebra is a powerful tool for deciphering the Calabi–Yau genome in all dimensions.

UNIVERSAL CALABI–YAU ALGEBRA: CLASSIFICATION AND ENUMERATION OF FIBRATIONS

We apply a universal normal Calabi–Yau algebra to the construction and classification of compact complex n-dimensional spaces with SU (n) holonomy and their fibrations. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions and a "dual" construction based on the Diophantine decomposition of invariant monomials. The latter provides recurrence formulas for the numbers of fibrations of Calabi–Yau spaces in arbitrary dimensions, which we exhibit explicitly for some Weierstrass and K3 examples.

Hermite reciprocity for the regular representations of cyclic groups

A formula is obtained for the dimension a( n, m) of the space of degree m invariants of the regular representation of the n-th order cyclic group. This formula implies in particular that a( n, m) = a( m, n). Moreover, the automorphism group of the multiplicative semigroup generated by invariant monomials of the cyclic group in that representation is determined.