Reply to Comment on "Covariant formulation of electrodynamics in isotropic media''
This note contains my response to the comment written by J. Franklin on my paper ``Covariant formulation of electrodynamics in isotropic media''.
physics.class-ph↗
arXiv subjects
Palash B. Pal
Publications and source records attributed to Palash B. Pal.
On Pythagorean triplets
We discuss properties of diophantine solutions of the Pythagoras equation, $a^2+b^2=c^2$, where the three numbers have no common factor. Some of the highlights are: (1) All triplets for which $c$ (called the `peak') is non-prime can be deduced from the triplets with prime peaks; (2) If a peak has $n+1$ prime factors, there are $2^n$ independent solutions of the Pythagoras equation; (3) All Pythagorean peaks have to be of the form $12k+1$ or $12k+5$ for integer $k$; (4) A Pythagorean peak cannot have 3, or any number of the form $12k+7$ or $12k+11$, as its prime factors.
math.GM↗
Democratic three Higgs-doublet models: the custodial limit and wrong-sign Yukawa
We study two novel aspects of democratic 3HDMs -- the custodial limit and the possibility of wrong-sign Yukawa couplings. In the custodial limit, the democratic 3HDMs can easily negotiate the constraints from the electroweak $T$-parameter. We also uncover the possibility of having wrong-sign Yukawa couplings in democratic 3HDMs, as in the case of 2HDMs. We show that a democratic 3HDM encompasses all the wrong-sign possibilities entertained by 2HDMs, and has considerably more leeway in the wrong-sign limit as compared to the 2HDM case. Our study underscores the importance of reporting analysis in the kappa-formalism without any implicit assumptions on the signs of the kappas.
hep-ph↗
Discrete symmetry transformations on non-abelian gauge fields
All gauge bosons of a non-abelian gauge theory do not transform the same way under the discrete transformations of time-reversal and charge-conjugation. Moreover, the transformations rules depend on how the generators are chosen. We show how well-defined rules pertain only to specific choices of generators, and then show how unified rules can be constructed, using matrix forms of the gauge bosons, which are completely independent of the choice of generators.
hep-th↗
Custodial symmetry, Georgi-Machacek model, and other scalar extensions
In an SU(2) gauge theory, if the gauge bosons turn out to be degenerate after spontaneous symmetry breaking, obviously these mass terms are invariant under a global SU(2) symmetry that is unbroken. The pure gauge terms are also invariant under this symmetry. This symmetry is called the {\em custodial symmetry} (CS). In $\rm SU(2)\times U(1)$ gauge theories, CS implies a mass relation between the $W$ and the $Z$ bosons. The Standard Model (SM), as well as various extensions of it in the scalar sector, possess such a symmetry. In this paper, we critically examine the notion of CS and show that there may be three different classes of CS, depending on the gauge couplings and self-couplings of the scalars. Among old models that preserve CS, we discuss the Two-Higgs Doublet Model and the one doublet plus two triplet model by Georgi and Machacek. We show that for two-triplet extensions, the Georgi-Machacek model is not the most general possibility with CS. Rather, we find, as the most general extension, a new model with more parameters and hence a richer phenomenology. Some of the consequences of this new model have also been discussed.
hep-ph↗
The incredibly strange story of Einstein's Nobel prize
It is well-known that Einstein got the 1921 Nobel prize not for his theory of relativity, but for his theory of photoelectricity. It is not that well-known that he did not get the prize in 1921. Why not, and when did he get it?
physics.pop-ph↗
Covariant formulation of electrodynamics in isotropic media
The equations of electromagnetic fields in a medium is usually written in the rest frame of the medium. We outline a method of generalizing the discussion to arbitrary inertial frames. In the discussion, we also include the possibility that the medium is optically active, a possibility that is often overlooked in discussions of electromagnetic fields in a medium.
physics.class-ph↗
Generalized 331 models
We examine different ways in which the standard model can be embedded into the ${\rm SU(3)}_c \times {\rm SU(3)}_L \times {\rm U(1)}_X$ gauge group. We show that there exist families of models characterized by a free parameter. Only a few of these models, corresponding to specific values of the free parameter, have been studied so far.
hep-ph↗
Photon propagation in non-trivial backgrounds
Propagation of photons (or of any spin-1 boson) is of interest in different kinds of non-trivial background, including a thermal bath, or a background magnetic field, or both. We give a unified treatment of all such cases, casting the problem as a matrix eigenvalue problem. The matrix in question is not a normal matrix, and therefore care should be given to distinguish the right eigenvectors from the left eigenvectors. The polarization vectors are shown to be right eigenvectors of this matrix, and the polarization sum formula is seen as the completeness relation of the eigenvectors. We show how this method is successfully applied to different non-trivial backgrounds.
hep-ph↗
Crossed two Higgs-doublet models: reduction of Yukawa parameters in the low-scale limit of left-right symmetry and other avatars
We present new variants of the Two Higgs-Doublet Model where all Yukawa couplings with physical Higgs bosons are controlled by the quark mixing matrices of both chiralities, as well as, in one case, the ratio between the two scalar doublets' vacuum expectation values. We obtain these by imposing approximate symmetries on the Lagrangian which, in one of the cases, clearly reveals the model to be the electroweak remnant of the Minimal Left-Right Symmetric Model. We also argue for the benefits of the bidoublet notation in the Two Higgs-Doublet Model context for uncovering new models.
hep-ph↗
SO(10) unification with horizontal symmetry
We extend the nonsupersymmetric SO(10) grand unification theories by adding a horizontal symmetry, which connects the three generations of fermions. Without committing to any specific symmetry group, we investigate the 1-loop renormalization group evolutions of the gauge couplings with one and two intermediate breaking scales. We find that depending on the SO(10) breaking chains, gauge coupling unification is compatible with only a handful of choices of representations of the Higgs bosons under the horizontal symmetry.
hep-ph↗
An $S_3$ flavored left-right symmetric model of quarks
We construct a model based on the electroweak gauge group $\rm SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ augmented by an $S_3$ symmetry. We assign nontrivial $S_3$ transformation properties to the quarks and consequently we need two scalar bidoublets. Despite the extra bidoublet we have only six Yukawa couplings thanks to the discrete symmetry. Diagonalization of the quark mass matrices shows that at the leading order only the first two generations mix, resulting in a block diagonal CKM matrix, and the first generation quarks are massless. Inclusion of subleading terms produce an acceptable CKM matrix up to corrections of order $λ^4$. As for the first generation quark masses, we obtain satisfactory value of $m_u/m_d$. The masses themselves, though being in the same ballpark, turn out to be somewhat smaller than the accepted range.
hep-ph↗
Coxeter groups and the PMNS matrix
We discuss the symmetries of the Lagrangian of the leptonic sector. We consider the case when this symmetry group is a Coxeter group. The number of elements of the PMNS matrix predicted by this group structure would depend on the number of generators of this group. We analyze finite Coxeter groups with 2 to 4 generators and even finite subgroups of infinite Coxeter groups with 4 generators and show which of them can give results that are consistent with experimental data.
hep-ph↗
Involution symmetries and the PMNS matrix
Lam suggested that the PMNS matrix (or at least some of its elements) can be predicted by embedding the residual symmetry of the leptonic mass terms into a bigger symmetry. We analyze the possibility that the residual symmetries consist of involution generators only and explore how Lam's idea can be implemented.
hep-ph↗
Quark mixing in an $S_3$ symmetric model with two Higgs doublets
We construct a model where the smallness of the masses of first quark generations implies the near block diagonal nature of the CKM matrix and vice-versa. For this set-up, we rely on a 2HDM structure with an $S_3$ symmetry. We show that an SM-like Higgs emerges naturally from such a construction. Moreover, the ratio of two VEVs, $\tanβ$ can be precisely determined from the requirement of the near masslessness of the up- and down-quarks. The FCNC structure that arises from our model is also very predictive.
hep-ph↗
$S_3$ symmetry and the quark mixing matrix
We impose an $S_3$ symmetry on the quark fields under which two of three quarks transform like a doublet and the remaining one as singlet, and use a scalar sector with the same structure of $SU(2)$ doublets. After gauge symmetry breaking, a $\mathbb{Z}_2$ subgroup of the $S_3$ remains unbroken. We show that this unbroken subgroup can explain the approximate block structure of the CKM matrix. By allowing soft breaking of the $S_3$ symmetry in the scalar sector, we show that one can generate the small elements, of quadratic or higher order in the Wolfenstein parametrization of the CKM matrix. We also predict the existence of exotic new scalars, with unconventional decay properties, which can be used to test our model experimentally.
hep-ph↗
Representation-independent manipulations with Dirac matrices and spinors
Dirac matrices, also known as gamma matrices, are defined only up to a similarity transformation. Usually, some explicit representation of these matrices is assumed in order to deal with them. In this article, we show how it is possible to proceed without any such assumption. Various important identities involving Dirac matrices and spinors have been derived without assuming any representation at any stage.
physics.ed-ph↗
Reduction formulas for symmetric products of spin matrices
We show that, for SU(2) generators of arbitrary dimension $D$, there exist identities that express the completely symmetric product of $D$ matrices in terms of completely symmetric products of fewer number of matrices. We also indicate why such identities are important in characterizing electromagnetic interactions of particles.
quant-ph↗