Wednesday, February 19, 2014

Hadron Nomenclature

Hadrons are composite particles made up of fundamental particles that are bound together by the strong force. The two kinds of hadrons that have been observed experimentally are mesons (made up of pairs of quarks and anti-quarks) and baryons (made up of three quarks each).

No observed hadrons include top quarks, but names for such particles exist. Hypothetical hadrons with four quarks and no constituent mesons are called tetraquarks (and would be bosons just like mesons, but with potentially higher spin). Hypothetical hadrons with five quarks and no constituent mesons or baryons are called pentaquarks (and would be fermions just like baryons but with potentially higher spin).

Hypothetical hadrons with no quarks are called glueballs aka gluonium.  All glueballs would be bosons. Glueballs ground states are distinguished from each other by their quantum numbers, and they are usually categorized by total angular momentum which is always an integer value since they are bosons made up of spin-1 bosons called gluons combined together into a composite particle (spin-0 is scalar or pseudo-scalar, spin-1 is vector, and spin-2+ is tensor).

Issues related to modeling these hypothetical hadron states and fitting them to data are discussed, for example, in this power point presentation.

A more authoritative, but less fully spelled out and contextualized summary of hadron nomenclature can be found in a review article by the Particle Data Group.

Meson Nomenclature

A meson is a boson with spin-0 (i.e. total angular momentum which is quantum number J) is called pseduo-scalar in the ground state (which implies that the quarks in the meson have oppositely aligned spins) or spin-1 called vector (which implies that the quarks in the meson have spins aligned in the same direction).

A meson made of a quark with a particular color charge, and an anti-quark of the anti-color charge of the same type, or of a linear combination of these color charge neutral quark and anti-quark pairs.

Meson Types By Spin and Quark Content.

In a nutshell, pions (pseudo-scalar) (Greek letter pi), rho mesons (vector) (cursive p) and omega mesons (vector) (cursive lowercase w) have only up and down quarks.

Kaons (pseudo-scalar or vector) (K) and eta mesons (pseudo-scalar) (cursive n) have strange quarks and light quarks (i.e. up and/or down quarks). Phi mesons (vector) (three pronged Y) are made of strange quark pairs. Charged kaons have strange and up quarks, neutral kaons have strange and down quarks.

D mesons (pseudo-scalar or vector) (D) are made of charm quarks and a lighter quark and are called "strange" if the other quark is a strange quark. D mesons with up quarks are electrically neutral, D mesons with down type quarks are charged. Charmed eta mesons (pseudo-scalar) and J/Psi mesons (vector) are made of charm quark pairs. Both of these states are also called "charmonium".

B mesons (pseudo-scalar or vector) (B) are made of bottom quarks and a lighter quark, and are called "charmed" if the other quark is a charm quark, and "strange" if the other quark is a strange quark. B mesons with up or charm quarks type quarks are charged; B mesons with down or strange quarks are electrically neutral. Bottom eta mesons (pseudo-scalar) and Upsilon mesons (vector) (Y) are made of bottom quark pairs. Both of these states are also called "bottomonium".

Other Meson Types

In addition to the familiar quantum number L even mesons discussed elsewhere in this section (the pion, eta and eta prime pseudo-scalar mesons; the rho, omega or φ, J/psi, upsilon and theta vector mesons; D mesons, B mesons and T mesons), there are corresponding quantum number odd mesons with the same total angular momentum J, and in the case of neutral mesons, the same quantum number C, as their more familiar counterparts.  In JPC notation, + means "even" and - means "odd".

Neither the true scalar mesons nor the pseudoscalar and/or axial vector mesons surrender easily to simple construction as a simple single quark and anti-quark pair.

Scalar Mesons

The symbols "a" (isospin 1) and "f" or "f'" (isospin 0) apply to mesons with ground state JPC quantum numbers 0++ which are also known as (true) scalar mesons.  The Greek letter capital Chi (which looks like a cursive capital X) is used together with subscripts to describe heavy quarks that the scalar meson contains.  Wikipedia notes in part with regard to scalar mesons that:
The light (unflavored) scalar mesons may be divided into three groups; those having a mass below 1 GeV/c2, those having a mass between 1 GeV/c2 and 2 GeV/c2, and other radially-excited unflavored scalar mesons above 2 GeV/c2. The heavier scalar mesons containing charm and/or bottom quarks all occur well over 2 GeV/c2. Many attempts have been made to determine the quark content of the lighter scalar mesons; however, no consensus has yet been reached. 
The scalar mesons in the mass range of 1 GeV/c2 to 2 GeV/c2 are generally believed to be conventional quark-antiquark states with orbital excitation L = 1 and spin excitation S = 1,[1] although they occur at a higher mass than one would expect in the framework of mass-splittings from spin-orbit coupling.[2] The scalar glueball [3] is also expected to fall in this mass region, appearing in similar fashion to the conventional mesons but having very distinctive decay characteristics. The scalar mesons in the mass range below 1 GeV/c2 are much more controversial, and may be interpreted in a number of different ways. 
Since the late 1950s, the lightest scalar mesons were often interpreted within the framework of the linear sigma model, and many theorists still choose this interpretation of the scalar mesons as the chiral partners of the pseudoscalar meson multiplet.[4] Ever since Jaffe first suggested the existence of tetraquark multiplets in 1977,[5] the lightest scalar mesons have been interpreted by some theorists to be possible tetraquark or meson-meson molecule states. The tetraquark interpretation works well with the MIT Bag Model of QCD,[6] where the scalar tetraquarks are actually predicted to have lower mass than the conventional scalar mesons. This picture of the scalar mesons seems to fit experimental results well in certain ways, but often receives harsh criticism for ignoring unsolved problems with chiral symmetry breaking and the possibility of a non-trivial vacuum state as suggested by Gribov.[7] 
In-depth studies of the unflavored scalar mesons began with the Crystal Ball and Crystal Barrel experiments of the mid 1990s, focusing on the mass range between 1 GeV/c2 and 2 GeV/c2. With the re-introduction of the sigma meson as an acceptable candidate for a light scalar meson in 1996 by Tornqvist and Roos,[8] in-depth studies into the lightest scalar mesons were conducted with renewed interest. The "Particle Data Group" provides current information on the experimental status of various particles, including the scalar mesons.
Pseudovector Mesons aka Axial Vector Mesons

The symbols "b" (isospin 1) and "h" or "h'"(isospin 0) apply to pseudovector mesons aka axial vector mesons with ground state JPC quantum numbers 1++.

A recent (October 2013) paper on the subject of these mesons suggests that their nature is dreadfully complicated and that they are the subject of multiple distinct theoretical approaches to explaining them.  An extended linear sigma model such as one proposed in 2012 seems to explain them (and scalar mesons).  A power point presentation such a model can be found here.  Another which is somewhat more clear is here.

In a nutshell, scalar mesons and pseudovector or axial vector mesons are interpreted as a rather elaborate mixes of quarkonium states of pairs of up quarks and anti-up quarks, down quarks and anti-down quarks, and strange quarks and anti-strange quarks.

Unclassified Mesons

The symbol X applies to mesons when its properties cannot be determined completely enough to assign a meson type to it.

Hypothetical Top Quark Mesons

Hypothetical mesons including top quarks would follow the pattern for D and B mesons and presumably would be called T mesons and top eta mesons, and a hypothetical vector meson made of a top quark and anti-top quark pair would be a theta meson. Both of these states would also be called "toponium."

Suppose that such mesons did exist (and no such mesons have been observed to date), even ever so briefly and ever so infrequently, because top quarks decay via the weak force on average about twenty times as fast as the speed with which hadronization occurs on average.  The hypothetical T mesons could be no lighter than about 173 GeV (the mass of a single top quark compared to which the mass of a up, down or strange quark, or the minimum binding energy of a pseudo-scalar meson might be negligible), and any form of toponimum would have a mass in excess of 346 GeV.

Also, even if a T meson or top eta meson or theta meson were created by physicists, it might be difficult to distinguish this decay signature from the decays of a pair of bare top quarks that immediately decayed hadronically.

Mesons That Are Linear Combinations

Certain neutral mesons are linear combinations of pairs of quarks and anti-quarks of the same type.

The combinations of spin-0 quark-anti-quark pairs are the neutral pion (uu-dd/sqrt(2)), the K-short neutral kaon (d-anti-s minus s anti-d/sqrt(2)), the K-long neutral kaon (d-anti-s plus s anti-d/sqrt(2)), the eta meson (uu+dd-2ss/sqrt(6)), the eta prime meson (uu+dd+ss/sqrt(3)). The two neutral kaon types mix with each other.

The combinations of spin-1 quark-anti-quark pairs are the neutral rho (uu-dd./sqrt(2)), and the omega meson (uu+dd/sqrt(2)).

Models of scalar mesons and axial vector mesons suggest that many resonances that currently cannot be easily fit into simple quark models are linear combinations of quarkonia.

Anti-particles of mesons

Mesons that are linear combinations or are made of a quark and an anti-quark of the same type are their own anti-paticles. Charged mesons and neutral mesons with mixed quark flavors that are not linear combinations have antiparticles.

The State of Experimental Meson Observation

Which mesons have we observed so far?

The masses of the ground states of the mesons predicted by the Standard Model range from the 0.135 GeV of the neutral pion to the upsilon meson which has a mass of 9.46 GeV is the heaviest meson ground state that is possible in the Standard Model.

Many excited meson states of mesons whose quark content and spin have been determined experimentally have also been observed, with the heaviest being one of the excited states of the upsilon meson with a mass of 11.02 GeV.  This is about 1.56 GeV heavier than the ground state of the upsilon meson which is more than twice the 1.1 GeV of mass in excess of the bottom quark and anti-bottom quark in the ground state that it is made of due to the mass-energy of the gluons that bind the ground state of the upsilon together.

This heaviest excited upislon mesons is the heaviest hadrons ever observed experimentally to date.  No meson ground state, excited meson, or baryon with a mass in excess of 6.3 GeV, other than bottomonium ground states and excited states, have ever been observed.   The Standard Model predicts, however, that it is possible to create some baryons with two bottom quarks, and certainly the triple bottom omega baryon (which the Standard Model predicts is the heaviest possible hadron that does not involve a top quark) that are heavier than the upsilon meson.

Consider the case of mesons composed of pairs of the two heaviest flavors of quarks that form bosons. There are two ground states of charmonium (the charmed eta meson and the J/Psi meson) and the two ground states of bottomonium (the bottom eta meson and the upsilon meson) that have been observed.  But, high energy physicists have also observed about twenty-three excited charmonium states and thirteen different excited bottomonium states that have observed precisely enough to measure their masses (not all of which have been fully characterized in terms of quantum numbers), and described in academic publications.  Not all of these observations are definitive or complete discoveries, however.  For example, physicists have not yet fully characterized the quantum numbers of eleven of the twenty-three excited charmonium states and three of the thirteen excited bottomonium states which have been experimentally observed.

What mesons remain to be observed and/or characterized?

The only kind of meson ground state predicted by the Standard Model that has not yet been observed is the vector charmed B meson (made up of a charm quark and a bottom anti-quark or visa versa) which should have a mass of about 6.29-6.35 GeV (which is slightly more than the mass of the pseudo-scalar charmed B meson which has a mass of 6.28 GeV and the same quark content based on the mass differences between other pseudo-scalar and vector mesons with the same quark conduct).

A discovery of this last Standard Model meson ground state at the LHC is probably right around the proverbial corner after the LHC powers up again in 2015, because there is only one other hadron, the pseudo-scalar charmed B meson, and there are no other fundamental particles or pairs of fundamental particles, in that mass vicinity, to provide background noise, the signal physicists are looking for is very well understood (which allows searches to use very tight data cuts), and another new hadron mass state was just discovered in 2012 at 5.945 GeV, so the LHC has sufficient power to make observations in this mass range.

More interestingly, there are many unclassified resonances which are possibly mesons that have not been fully characterized or have quantum numbers different from the meson ground states (e.g. about fifteen excited meson states that have at least one heavy quark have been observed to have tensor spins of J=2 to J=5).  There are many unclassified meson states made up only of light quarks with masses from 0.5 GeV to about 2.5 GeV which may reflect not yet described linear combinations of mesons, tetraquarks, "meson molecules", or particularly high order excited states of light mesons that experiments have not had enough energy to produce for mesons with heavier ground states.

This is some of the most interesting unexplored territory of meson physics that is within the realm of what can be investigated experimentally.  But, only a quite small share of theoretical and phenomenological physics scholarship is devoted to exploring exotic hadronic states like these, either within the framework of QCD or some beyond the Standard Model modification of Standard Model QCD.

Why is it so hard to observe and characterize relatively light meson resonances?

The masses at which high energy physicist can consistently and completely observe and characterize heavy hadrons lags an order of magnitude behind their ability to observe and characterize heavy fundamental particles such as the W boson, Z boson, Higgs boson and top quark which are seven to fifteen times as heavy as the heaviest experimentally observed hadron.

There is still one suspected meson resonance lighter than a proton, with a mass of 0.5 GeV, that have not been fully described and characterized.  So, the fact that there are light hadron resonances that have not yet been fully described isn't as surprising as it seems.

All but six of the masses (only one of which, f(500), is not well characterized yet) of the scores and scores of exotic hadrons and excited hadron states observed to date (all of them other than fifteen bottomonium states) lie in the crowded mass range between the proton mass of 0.938 GeV and 6.4 GeV.  This mass range with a peak mass less than 7 times as great as the low end mass where all the action is in hadronic physics, grows more crowded at the low end of that range.  So, teasing out the single of a particular suspected hadron state from the thicket of all of the other hadron resonances with similar masses that emerge from a particle collider operating at a given energy scale is not an easy task.  The limited precision of QCD calculations to date makes it even harder to separate signals from backgrounds, or to properly characterize a resonance that is observed.

In contrast, the fundamental particles have masses spread out over a much more vast range, from electron neutrinos with a mass of about 1 meV to top quarks with a mass of about 173 GeV.  The top of this range is about 100,000,000,000,000 times as great as the bottom of this range.

Simlarly, the only two fundamental particles that have masses that are within experimental error bars of each other are the strange quark and the muon.  In contrast, there are a dozen or so sets of hadrons with the same total angular momentum and same number of quarks that have distinct masses that differ by 1% or less (which is about the accuracy of current theoretical calculations of hadron masses from first principles in QCD).

On the other hand, measuring the mass of an observable composite hadron or fundamental charged lepton that can actually be produced in a laboratory is much easier than determining the mass of a light quark which can never be directly observed even in ideal conditions due to the fact that quarks are confined within hadrons.

Baryon Nomenclature

Baryons, which have three quarks each and can have total angular momentum (J) commonly called spin of 1/2 or 3/2 in the ground state. They can be made of:

* three identical flavor quarks (in which case they must have spin-3/2),
* two quarks of one flavor and one of another (in which case there is one combination each of spin-1/2 or spin-3/2), or
* three quarks of different flavors (in which case there are two possible spin-1/2 combinations and one possible spin-3/2 combination) (this is not possible for particles made only of the two light quark flavors).

Ordinary baryons have three quarks, one with each color charge, and no antiquarks. Every baryon has a corresponding antibaryon made of three antiquarks, one of each anticolor charge, with an opposite electric charge and parity.

While there are several meson resonances that are linear combinations of multiple quark pairs, there are no known baryons are linear combinations of different baryon combinations.  This may be because they are fermions rather than bosons, although I don't really know for sure why this is the case.

The name of a baryon is a function of its isospin, i.e. the total angular momentum J, after disregarding spin attributable to quarks other than the up and down quarks in the baryon. Up and down quarks have isospin 1/2 which can be positive or negative for each such quark. Strange, charm, bottom and top quarks have isospin 0. The existing categories of baryons are as follows:

1.  A nucleon (isospin 1/2) (always J=1/2) and delta baryons (isospin 3/2) (triangle) (always J=3/2) each have three light quarks.

All six of these ground states which are predicted by the Standard Model have been observed.

2.  A lambda baryon (isospin 0) (upside down V) (J=1/2) with an up and a down quark of opposite spins, and one heavy quark, and a sigma baryon (isospin 1) have two light quarks and one heavy quark (J=1/2 or 3/2).

All of these except the bottom sigma (which has not been observed in either J=1/2 and J=3/2 form) have been observed experimentally.  Thus, nineteen of the twenty-one ground states of these kinds of baryons which are predicted by the Standard Model have been observed.  These are expected to be two of the heaviest sigma baryons.

3.  A chi baryon (isospin 1/2) (three horizontal lines) (J=1/2 or 3/2) have one light quark and two heavy quarks.

All four chi baryons without charm and bottom quarks, and all four chi baryons with one strange quark and one charm quark have been observed experimentally.  Both of the J=1/2 bottom chi baryons and one of the two J=3/2 bottom chi baryons have been observed experimentally.  One of the two J=1/2 double charmed chi baryon has also been observed.

Scientists have not yet observed one of the J=3/2 bottom chi baryons, one of the J=1/2 double charmed chi baryons, both of the J=3/2 double charmed baryons, any of the four of the charmed bottom chi baryons, or any of the four of the double bottom chi baryons.

Thus, exactly half of the twenty-four kinds of chi baryon ground states have been observed so far.  In general, the baryons not yet observed are the heaviest ones.

4.  An omega baryon (isospin 0) (J=1/2 or 3/2) has three heavy quarks.

Two of the eight omega baryon ground states predicted by the Standard Model with J=1/2 have been observed (the charmed omega which has two strange quarks and a charm quark, and the bottom omega with two strange quarks and a bottom quark).   Two of the ten omega baryon ground states predicted by the Standard Model with J=3/2 (the plain omega baryon made of three strange quarks and the charmed omega with two strange quarks and one charm quark) have been observed.  Thus, four of the eighteen kinds of omega baryon ground states have been observed.

The State of Experimental Baryon Observations

Which baryons have been observed?

The project of experimentally observing all baryon ground states predicted by the Standard Model is significantly less far along than the parallel project for mesons.

Forty-one of the sixty-nine baryon ground states predicted by the Standard Model have been observed experimentally. The lighest observed baryon, the proton, has a mass of 0.938 GeV.  The heaviest observed baryon, the bottom chi meson observed at the LHC in 2012, has a mass of 5.945 GeV.

All twenty-eight of the unobserved ground states involve baryons with at least one bottom quark or two charm quarks.  Physicists have observed and characterized the ground state of every single hadron composed of two or three quarks that contain only up, down or strange quarks that is predicted to exist by the Standard Model, and every hadron with a single charm quark and also doesn't have a bottom quark.

Physicists have also observed other excited baryon states and have observed some resonances that are probably unobserved baryons of some type which have not yet been definitively classified.  But, I get the impression that there are considerably more unexplained and excited meson states than there are excited and unclassified baryon states that have been observed to date.

Which baryons have not yet been observed?

All of the undiscovered hadron ground states contain (1) at least one bottom quark (the vector charmed B meson, and twenty-two kinds of baryons containing bottom quarks), and/or (2) two or more charm quarks (the two kinds of double charmed Omega baryons and three of the four kinds of double charmed Xi baryons, and the triple charmed omega baryon).

We are making progress on these frontiers of hadronic physics, however.  We have observed the ground states of eight of the thirty baryon ground states predicted to exist in the Standard Model that contain at least one bottom quark (although no double bottom quark baryons have been observed yet), and nine out of the ten meson ground states that contain at least one bottom quark (including both of the meson ground states that contain two bottom quarks).  We have observed both of the meson ground states that contain two charm quarks and one of the seven baryons ground states predicted to exist in the Standard Model that contain two or more charm quarks but no bottom quarks.

Based upon the masses of the doubly charmed mesons and mesons containing bottom quarks observed to date and the mass of the bottom quark, the undiscovered doubly charmed baryons without bottom quarks have masses in a bit excess of 3.5 GeV, the undiscovered baryons that contain one bottom quark have masses in excess of 5.8 GeV, double bottom baryons (none of which have been observed yet) have masses in excess of 10 GeV, and the triple bottom omega baryon (which should be the heaviest possible baryon that does not include a top quark) should have a mass in excess of 14.2 GeV, but probably not much more than 24 GeV.

It is reasonable to expect that only the double and triple bottom baryons will remain unobserved experimentally by the time that the LHC completes its run, and with luck, even some of those may be detected at the LHC.

Why is it harder to create exotic baryons in the lab than it is to create exotic mesons?

I do not know, but suspect, that this may be a product of the fact that exotic hadrons of all kinds are extremely unstable and that it may be harder to form baryons (which must be made of either three quarks or of three antiquarks) than it is to form mesons (which must be made of one quark and one anti-quark) in a particle collider environment where particles and anti-particles are produced in almost identical quantities.

For example, for a double bottom omega baryon to form, the collider must have generated at least two bottom quark pairs and a strange or charm quark pair, and realistically lots of other stuff as well in multiple jets of decay products.  In contrast, production of a single bottom quark pair can generate the quark content necessary to create bottomonium.

Subscripts and superscripts and further specifications in hadron nomenclature


Particles with charmed in the name have the subscript lower case c added to their symbol. Mesons with strange in their name have the subscript lower case s added to their symbol. Short kaons have subscript upper case S added to the K symbol, while long kaons have subscript upper case L added to their symbol. Baryons with charm or bottom quarks have one subscript for each charm or bottom or (hypothetically) top quark that is present (lower case c, b or t respectively).

Particles with total angular momentum J that is a subscript for all mesons except pseudoscalar and vector mesons.


Charged mesons have a superscript with a + or - indicating their electric charge. Neutral pions, kaons, D mesons and B mesons have superscript zero. The superscript zero is omitted for omega mesons, eta mesons, phi mesons, J/Psi mesons, and upsilon mesons which always have neutral electrical charge.

All baryons but neutrons (which always have neutral electric charge) carry a superscript identifying their electric charge. Baryons with electric charge +2 have superscript ++, while baryons with electric charge -2 have superscript --.

Mesons that can be either pseudo-scalar or vector (i.e. kaons, D mesons and B mesons) have an asterisk superscript if they are vector rather than pseudo-scalar (i.e. if they have quarks with aligned spins and thus spin-1, rather than quarks with complementary spins and thus spin-0).

Baryons of a type that can have total angular momentum J=1/2 or 3/2 (i.e. sigma, chi or omega baryons) have an asterisk superscript is they have total angular momentum J=3/2.

Additional Specification After A Particle Symbol

Sometimes a symbol for a kind of hadron is followed by a number in parentheses. This number is its mass in units of MeV/c^2. When quantum numbers are known for a resonance whose classification has not been determined it is typically included after the mass number, if not otherwise described, in JPC format.

Every hadron, in principle, also has an infinite number of higher mass, excited, higher energy states in addition to a ground state which are sometimes described with additional notation that (to be perfectly frank) I don't fully understand as a mere amateur science enthusiast.

Footnote on Exotic Hadrons

In this post, I have used the term "exotic hadrons" in a broad sense to encompass all hadrons which are not found in nature at this point in the history of the universe.  In other words, hadrons which would be exotic to you and me.  This includes almost all hadrons except the proton, the neutron, the pion, and maybe a handful of other mesons with light quarks or strange quarks.  This includes all of the highly unstable hadrons that contain charm or bottom quarks.

There is also a narrower sense of the phrase "exotic hadrons", which means hadronic resonances whose properties cannot be explained as a composite particle made up of two or three quarks, depending upon its total angular momentum J.  There are stronger indications of exotic meson resonances in this sense than there are of exotic baryons in this sense.  With regard to "exotic meson" reasonances, the Particle Data Group discussion referenced at the start of this post states:
Gluonium states or other mesons that are not qq states are, if the quantum numbers are not exotic, to be named just as are the qq mesons. Such states will probably be difficult to distinguish from qq states and will likely mix with them, and we make no attempt to distinguish those “mostly gluonium” from those “mostly qq.”
An “exotic” meson with JPC quantum numbers that a qq system cannot have, namely JPC
0−−, 0+−, 1−+, 2+−, 3−+, · · · , would use the same symbol as does an ordinary meson with all the same quantum numbers as the exotic meson except for the C parity. But then the J subscript may still distinguish it; for example, an isospin-0 1−+ meson could be denoted ω1.
Wikipedia discusses exotic mesons here.

The Particle Data Group discussion referenced at the start of this post has the following discussion of ""exotic baryons" of that type:
8.5. Exotic baryons
In 2003, several experiments reported finding a strangeness S = +1, charge Q = +1 baryon, and one experiment reported finding an S = −2, Q = −2 baryon. Baryons with such quantum numbers cannot be made from three quarks, and thus they are exotic. The S = +1 baryon, which once would have been called a Z, was quickly dubbed the Omega(1540)+, and we proposed to name the S = −2 baryon the Omega (1860). However, these “discoveries” were then completely ruled out by many experiments with far larger statistics: See our 2008 Review [2].
As of early 2014, no such exotic hadrons have been observed and characterized.  The most exotic resonances discovered so far (using the term in an intermediate and relative sense) are suspected "meson molecules" which I have discussed in previous posts at this blog.

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